Mathematical Musings » All Posts Sat, 19 Aug 2017 21:00:30 +0000 en-US <![CDATA[Reply To: Content of Algebra 2]]> Wed, 19 Jul 2017 11:33:17 +0000 mathteacher I realize that it is summer, but it has been a month since we first started this thread and students will be returning to school shortly. Has there been any political movement on your part to help locals with the burdensome rules set in motion by the common core with respect to college and career readiness, graduation requirements, ESSA, the demise of PARCC, and so on. As I have maintained, these may have been unintended consequences of the core but nonetheless are the responsibility of those who set these into motion. Many of my colleagues and I have been active both locally and nationally but we have yet to see any leader of the movement speak.

]]> <![CDATA[Reply To: Resources for Progressions K-10]]> Wed, 28 Jun 2017 12:15:39 +0000 kgartland I have reviewed this map extensively and it is a start. Perhaps putting my students in the driver’s seat to see the connections is the right route to go.

]]> <![CDATA[Reply To: Resources for Progressions K-10]]> Wed, 28 Jun 2017 03:49:46 +0000 Bill McCallum I don’t think this is exactly what you are looking for, but the SAP Coherence Map might be useful in tracing these progressions.

]]> <![CDATA[Resources for Progressions K-10]]> Mon, 26 Jun 2017 21:04:01 +0000 kgartland I’m going to be teaching a mathematics education graduate course this summer that will include participants who teach grades K-12. I’m interested in knowing if there are any resources that have already been created which demonstrate the progression of an operation or of our number system from K-10.
For example, do you know of any documents that look at addition from kindergarten through algebra? or…the progression of number from whole numbers to percent, etc.
I have already looked at and will include the progressions documents and a look at the relationship amongst the standards but I’m looking for articles, activities, visuals, etc.
Any ideas?
Thanks in advance.
Karen Gartland

]]> <![CDATA[Reply To: 5.NF.B.4.B Tiling with Unit Squares]]> Sat, 24 Jun 2017 01:17:02 +0000 Bill McCallum So anyway, the basic idea here is this: I know that rectangle which is 1/n by 1/m has area 1/nm because I can fit nm of them in a unit square. So then I know that a rectangle with dimensions a/n and b/m has ab of those little rectangles, so its area is ab x 1/nm = ab/nm. In other words, the area of a rectangle with fractional side lengths is the product of the side lengths. Of course, curricula often treat this as completely obvious, which is a shame, because the reasoning is fun.

]]> <![CDATA[Reply To: 5.NF.B.4.B Tiling with Unit Squares]]> Sat, 24 Jun 2017 00:31:51 +0000 Bill McCallum Wu’s approach looks right to me. I’m not seeing the EngageNY approach, just a list of objectives. Am I missing something?

]]> <![CDATA[Reply To: 5.NF.B.4.B Tiling with Unit Squares]]> Fri, 23 Jun 2017 17:58:08 +0000 dlward6 Thanks for the quick response. I am still a bit confused by the variations in lesson that address this topic. Dr. Wu’s example on Page 52 of the document found at the website below offers a different approach than one found on Engage New York’s site at
Do either of the approaches target this standard?
Thank you!

]]> <![CDATA[Reply To: 5.NF.B.4.B Tiling with Unit Squares]]> Fri, 23 Jun 2017 14:19:49 +0000 Bill McCallum This is an error in the standards (I’ve noted it before in these pages, but, of course, that’s difficult to find!). It should say “squares with unit fraction side lengths.”

]]> <![CDATA[Reply To: 5.NF.B.4.B Tiling with Unit Squares]]> Wed, 21 Jun 2017 17:32:11 +0000 dlward6 My office has the same question. The language “tiling with unit squares” is the issue. Activities found online that address this standard seem to indicate that there are a variety of interpretations of this standard. Any insights would be appreciated.

]]> <![CDATA[Reply To: Content of Algebra 2]]> Fri, 16 Jun 2017 13:56:28 +0000 Bill McCallum Thanks Lane, I love your list, and agree about the need to prune topics carefully.

]]> <![CDATA[Reply To: Content of Algebra 2]]> Thu, 15 Jun 2017 22:52:12 +0000 lanewalker Hello Math Teacher, As a MO HS math teacher, I definitely feel your pain. MO “voted out” CCSS in 2015. When our rewriting committees got together, they found out what a huge project rewriting is. When they started comparing with old MA standards, old MO and others, I’m thinking they figured out the CCSS writers did a lot better job than they realized because they kept most of CCSS, mostly just tweaking wording…except for Algebra 2. MO Algebra 2 is really bad now. I have gotten involved by volunteering on committees, doing a lot of listening and reading to understand from multiple perspectives. It has eased my frustration to find no one who is intentionally messing up students here in MO.

I agree with you that with unlimited talent and funding, the implementation could have been a lot smoother. It was unfortunate how much time, money, and talent was diverted to politics. I might be wrong, but it seems like the more the CCSS writers and supporters try to help from the top, the more people who do not understand the whole picture feel like CCSS are “top down.” So what I’m trying to say is that I think it’s best to work for change on a state level. I’m seeing some cool progress and you seem to have the passion it takes to make a serious difference in MD as well.

You mentioned not being able to cover all the Alg 2 topics. One tool I’m leveraging is a list of topics commonly found in Algebra that should not be there. It’s linked in the third paragraph from the bottom this article: I use this list to help wherever I can to keep the Algebra 2 content manageable.

Here’s another piece where you can see my journey to figure out why Algebra 2 is such a mess:

]]> <![CDATA[Reply To: Content of Algebra 2]]> Thu, 15 Jun 2017 20:51:21 +0000 Bill McCallum If you want to talk at NCTM you will have to let me know who you are.

]]> <![CDATA[Reply To: Content of Algebra 2]]> Thu, 15 Jun 2017 19:12:25 +0000 mathteacher I do appreciate the response, but honestly, I haven’t seen evidence of advocacy for students from any of the authors of the standards. As evidence, you champion your completion of a middle school curriculum, which is not available today but will be available this summer, a full 7 years after the standards and a full 6 years after the adoption by my state. With all due respect, it is like the teacher who reviews before a test solely to tell parents that they reviewed without any thought about the students who will take the test.

This middle school math curriculum is at least 6 years too late and honestly, if it took the authors of the core standards 7 years to write a curriculum for grades 6-8, what chance did the locals have? I should point out that we received nothing in terms of curriculum from our state and created everything ourselves locally with little funding. I must admit that I am shocked it took the people who wrote the standards 6 years to figure out a scope, sequence, and curriculum. We were only given 6 weeks.

As I mentioned, I’m not sure about the level of advocacy on your end. But what I saw during the last presidential administration was that they shoved the standards at states by requiring them to adopt and take money (RTTT) without understanding that No Child Left behind was still in effect. That is, we had 2 sets of rules to follow, often contradicting sets of rules. Only last year did the previous presidential administration reauthorize NCLB as ESSA. This has left long lasting issues and major problems for locals, including the lack of primary resources for classroom teachers.

The larger question is what leaders such as yourself, can do now, especially as states try to comply with ESSA. I would encourage you to work with local states, such as Maryland. I would encourage you to look at specifics to see exactly what has happened. For example, look at the PowerPoint regarding ESSA presented to the Maryland state board of education (in the May 2017 agenda) and you will see how your message (“standards are not curriculum and standards are not assessment frameworks”) has not been received by states. Maryland’s entire plan is about “compliance” and is not about learning nor about students. As I have mentioned, this is an unintended consequence of the rollout and structure of the standards.

Perhaps I will see you at NCTM in the fall and we can talk more specifics.

]]> <![CDATA[Reply To: Content of Algebra 2]]> Thu, 15 Jun 2017 15:51:15 +0000 Bill McCallum Thanks for the reply. A few thoughts in response.
First, I haven’t kept track of all the PARCC changes and I think you are probably more up on them than I am, so I take your point there.

On testing I have mixed feelings. I get the frustration people feel about too much testing and I think it would be a healthy move to reduce the amount of testing (and that seems to be happening). I don’t think it is a healthy move to opt out altogether. So there has to be some balance in between (something this country is having trouble achieving in all sorts of areas). What I don’t know is where that balance point lies. Partly it will be determined by political forces, of course, but is their empirical evidence to help decide? You say 4.5 hours is too long for a high school test. As someone who grew up in a country where the high school test was 3 hours long, I’m inclined to agree. But how do we come up with these numbers? Do we just take the average of everybody’s gut feelings? I wish we had a more empirical approach.

Finally, on the question of responsibility: well, I have pretty much devoted my life since the standards were written to helping teachers understand and implement them and advising curriculum writers, assessment writers, and policy makers on what I think is their proper use (spoiler: standards are not curriculum and standard are not assessment frameworks). I can’t control the extent to which my voice is heard. Illustrative Mathematics, the non-profit I went on leave from my university position to found, has just completed writing a complete, freely available, grades 6–8 curriculum, and are hoping to continue on to high school. Stay tuned!

]]> <![CDATA[Reply To: Content of Algebra 2]]> Wed, 14 Jun 2017 18:08:25 +0000 mathteacher I appreciate the response and reposted over here. Those efforts to change what is occurring around the country could come from you and other leaders, especially since graduation, college and career readiness, and many other factors affecting kids are in play. The common core standards put this issue in motion, and, although some of the consequences may have unintended, they are still the responsibility of those who set this course in motion, especially that ESSA, PARCC, and high-stakes testing are in play.

As far as PARCC goes, I would disagree that they put limits on the tested standards. In fact, didn’t PARCC “invent” standards to be tested? These are some of the integrated, C, and D standards. One standard even asks high school students to “use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity.” PARCC has interpreted this to mean that students can use right triangle trig on non-right angles, for example. PARCC has revised and re-revised the PLDs several times. Algebra 1 students are also tested on “securely held knowledge.” The common core standards put this chain of issues in motion.

And, from my experience, asking elementary students to take a high stakes test that takes 4 hours is a bit much. The high school test is 4.5 hours long. Way too much testing. The common core standards put this chain of issues in motion.

Finally, I do appreciate an honest and candid dialogue. Many of my colleagues and I have been frustrated as we find the balance between content and mastery, especially in Algebra 2. We have seen high failure rates on PARCC, as defined by 3 or lower, and then told that high failure rates mean that PARCC and the standards are so rigorous. So rigorous as to be unattainable. We have even been told that unless a student is in the 60th percentile in math (on MAP tests, for example), they cannot get a 4 or higher on PARCC. We are frustrated because it is not possible for all students to be greater than the median. The common core standards put this in motion.

We are champions for our students and want them to succeed. We have high standards for ourselves and our students. We hope that those who set this in motion can see what has transpired and help us in the efforts to truly help every child succeed in math.

]]> <![CDATA[Content of Algebra 2]]> Wed, 07 Jun 2017 15:17:10 +0000 Bill McCallum This is a repost from a comment in a thread on my latest blog post. It started with the following comment from a user:

. . . let’s start with the debacle in the appendix that became Algebra 2 and is now part of college and career readiness in my state. The issue simply is: how can one expect to teach that to the typical student in one school year?

I replied:

I am always up for a serious and civil discussion about issues in math education K–12. To your question, I agree that forcing everybody to accomplish the standards in 3 years is a bad idea. There are four years in high school, and some students need all four. Many states, districts, and schools struggle to handle that problem in a humane and even-handed way. And because we live in a local-control system, where implementation of the standards is up to each individual state that adopts them (and in some states devolves to districts or even schools), there are many different solutions out there. The standards themselves do not specify an arrangement into courses and do not require that all standards be covered in three years. The near universal agreement that end-of-high-school testing should be required in grade 11, rather than grade 12, has always struck me as strange. That’s not the way it is in other countries, for example Australia, the country I grew up in.

All that aside, because they were worried that states might want guidance on arranging the standards into courses in high school, Achieve created Appendix A. It was not intended to be taken as a mandate but rather as a model, as is stated clearly on page 2 ( However, understandably I suppose, many policy makers took it as gospel, and that has resulted in the situation you decry here.

I would point out that the PARCC and Smarter Balanced frameworks did not follow Appendix A to the letter, and put some important limits on the complexity of items for certain standards. However, as I say, I think the real problem here is the assumption that testing happens in grade 11. That should be an option for students who are ready for it, of course, but not the norm. I’ve been saying this for years, but I don’t know of any efforts to change it.

By the way, if you want to continue this thread, it should probably go over in the forum on arranging the standards into courses. I’ll repost it over there.

  • This topic was modified 2 months, 1 week ago by  Bill McCallum.
]]> <![CDATA[Reply To: 2.G.1]]> Mon, 22 May 2017 20:16:50 +0000 Bill McCallum I could imagine a classroom activity where students try to draw three-dimensional shapes and compare their work, but I don’t think it would be reasonable to put that on a summative assessment. (I think assessment in grade 2 should be mostly formative anyway.) Note that in general the standards are not assessment frameworks; they are just statements of the things we want students to know and be able to do at each grade level. The PARCC and Smarter Balanced assessments make judgements about limits on assessment, although they don’t offer guidance here because they start in Grade 3.

]]> <![CDATA[2.G.1]]> Mon, 22 May 2017 16:49:29 +0000 Caitlin Duncan Does this standard mean that second grade students should be able to draw three-dimensional shapes? It is clear which shapes they should be able to identify, but not which shapes they should be able to recognize and draw based on attributes. I read the Geometric progressions and it doesn’t clarify there either. Our students are assessed on their ability to draw a shape with four rectangular faces and 2 square faces. Does this align with 2.G.1?

]]> <![CDATA[Strategies and models and tools…oh my!]]> Wed, 17 May 2017 17:39:41 +0000 kaustin Hi Bill,

I’ve had the opportunity to work with elementary educators around the country as they adopt/adapt curricula to meet the CCSS-M. Educators are using the terms “strategy,” “tool,” and “model” differently and sometimes interchangeably. My goal is to provide some clarification based on the intent of the standards and your own interpretation.

The progressions documents have provided clarification and created new questions.

1. What is the difference between a strategy and a method?

In response to an earlier question (, you point out that strategies are tied to written methods in many standards (e.g., 1.NBT.4).
Can “written methods” be used interchangeably with “notation?”

In the OA progression at the top of page 6 where Level 1, 2, and 3 , “method” seems to take on a slightly different meaning, perhaps tied to strategy, “Methods used for solving single-digit addition and subtraction problems.”
Can “method” and “strategy” be used interchangeably?

2. Typically, the word “model” is used in the elementary progressions along with “visual” (e.g., “visual model,” “visual fraction model”). Is it fair to say that a model could be concrete objects, drawings, diagrams, charts, simulations, equations? I’m beginning to define model (n) as a visual representation that helps to conceptualize, solve, or explain a mathematical situation or relationship.

3. Would you consider the number line as a model or a tool?

Thanks for your insight!


]]> <![CDATA[Reply To: 5.NF.A.1]]> Fri, 12 May 2017 14:24:50 +0000 Bill McCallum I was about to write a long reply to this and then I discovered that George Bergman has already done it for me. Short version: there is no universally accepted convention that would require the parentheses, but they would indeed remove ambiguity. In this case I think the context (well-known formula for adding fractions) removes the ambiguity anyway. See the end of George’s article for a discussion of PEMDAS and its interpretations.

]]> <![CDATA[5.NF.A.1]]> Fri, 12 May 2017 13:11:19 +0000 jhays This standard is missing a set of parentheses in the denominator of the fraction in the example.

“For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)”
Should instead be:
“For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/(bd).)”


]]> <![CDATA[Reply To: Solving inequalities in grade 7]]> Tue, 28 Mar 2017 14:11:02 +0000 Aaron Bieniek Since 7.EE.4b is specific about solving word problems and interpreting the graph using the context of the problem, I would say yes, problems here would include situations and context.

And even though we could manipulate the expressions so dealing with the negative can be avoided, I wonder if that’s in the spirit of the standard. The progression makes a point to say “It is useful to present contexts that allow students to make sense of [multiplying or dividing by a negative].” If we avoid it, I’m not sure we are making sense of it. Especially since the goal is reasoning about the quantities.

]]> <![CDATA[Reply To: Solving inequalities in grade 7]]> Mon, 27 Mar 2017 15:27:07 +0000 voverton I would like clarification on this, too. Word problems that ask students to write inequalities requiring multiplication or division by a negative value to solve could also be written in a way that does not require that.

]]> <![CDATA[Reply To: Points of Concurrency]]> Wed, 01 Mar 2017 00:39:33 +0000 Bill McCallum First let me say that having grown up with a fairly traditional education in Euclidean Geometry in Australia I have never heard of “points of concurrency” as a topic. So I agree with Kristie Donavan!

I’m assuming this refers to the various theorems about medians, altitudes, angle bisectors, and side bisectors of triangles all intersecting at a point. The only one of these that is explicitly called out in the standards the one about medians. Constructing inscribed and circumscribed circles suggests also studying the concurrency of angle and side bisectors, although I think there is latitude in curriculum about how far you go with that. I myself would not advocate remembering all the names of the points where various lines intersect, and that is certainly not required by the standards, although of course it is not forbidden either.

Generally speaking the high school standards were designed to allow states some latitude in curriculum.

]]> <![CDATA[Reply To: Points of Concurrency]]> Tue, 28 Feb 2017 19:25:41 +0000 Kristie Donavan Does anyone have a response to this post? Our teachers are also asking about the extent of studying points of concurrency, if at all.
In reading the standards, I don’t interpret “points of concurrency” as a topic in itself; I see the concepts instead used in solving problems like in standards G-C.3 (construct inscribed and circumscribed circles of a triangle), G-CO.10 (prove medians of a triangle meet at a point), and G-CO.9 (points on perpendicular bisector are equidistant from endpoints). Am I interpreting this correctly?

]]> <![CDATA[Reply To: Proving the slope criteria?]]> Sun, 30 Oct 2016 13:33:01 +0000 Bill McCallum To sketch the answer to (B) first: given two non-vertical parallel lines, draw a vertical transversal and a horizontal transversal so that they intersect at a point not on either of the lines. The transversals form a right triangle with each of the lines, and the slope of each line is the quotient of the lengths of the vertical and horizontal sides. Using the fact that alternate interior angles are congruent, you can use the AAA criterion to show that these two triangles are similar. That means the corresponding sides are related by the same scale factor, so the quotients of the lengths of the horizontal and vertical sides are the same.

As to (A), I agree there would be a danger of circularity of you defined the notion of parallel lines in terms of slope. So it would be a good idea not to do that! A standard definition is to say that two lines are parallel if they are either identical or do not intersect at all.

]]> <![CDATA[Proving the slope criteria?]]> Tue, 25 Oct 2016 17:17:13 +0000 bsmithwbms The standards expect students to prove the slope criteria:

G-GPE 5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Most of the proofs I’ve seen involve similarity; however, I was thinking that one could also prove the slope criteria using (just) rigid motions. But then I recalled that another standard expects students to

G-CO 4 Develop definitions of rotations, reflections, and translations in terms of …parallel lines…

Which seems to beg the question. If the criterion for the slope of parallel lines is proven using transformations, but transformations are themselves defined in terms of parallel lines, there seems to be some circularity going on.

I am hoping someone here could shed some light on: (A) How this isn’t circular and (B) What a good proof of the slope criterion for parallel lines would look like. Wu doesn’t address the topic, so far as I know.

  • This topic was modified 9 months, 4 weeks ago by  bsmithwbms.
  • This topic was modified 9 months, 4 weeks ago by  bsmithwbms.
]]> <![CDATA[Reply To: Specify Standards in Blueprints]]> Tue, 16 Aug 2016 23:37:53 +0000 Bill McCallum This will happen once we drill down to the next level of these blueprints.

]]> <![CDATA[Reply To: Quantities]]> Tue, 16 Aug 2016 23:36:43 +0000 Bill McCallum This progression will be out soon!

]]> <![CDATA[Reply To: 6.EE.1/6.EE.2c]]> Tue, 16 Aug 2016 23:32:02 +0000 Bill McCallum The standards set expectations for what kids should know at the end of the course; they are not markers for particular topics along the way. The sort of question you ask here comes up when people try to arrange the standards into a curriculum, but there are many cases where that doesn’t really make sense (this one, for instance). That said, assessment writers have to make decisions about what sorts of questions belong to which standards, and I think your suggestion here is reasonable that 6.EE.1 would be assessed with simpler expressions than 6.EE.2c.

As for your question about bases, I don’t see any reason to restrict the base to whole numbers.

]]> <![CDATA[Reply To: Ratio – fractional notation]]> Tue, 16 Aug 2016 23:23:40 +0000 Bill McCallum The usage of ratio in that PARCC question is incorrect, I agree. I can’t see the table, but why didn’t they just say the sales tax is a fixed percentage of the purchase (assuming that they give percentages in the table)?

As for the confusion about rate and unit rate, it’s not a problem for multiple choice items (as long as they get it right!). For student produced response items, assessment writers will have to make some decisions. I think it would be reasonable to accept an answer with the units even though only the unit rate was asked for. I don’t think it would be reasonable to accept an answer without units when the rate was asked for.

]]> <![CDATA[Reply To: 6.G.1 – special quadrilaterals]]> Tue, 16 Aug 2016 23:18:11 +0000 Bill McCallum Yes to abienek and jkerr!

]]> <![CDATA[Reply To: 8.G.1 and 8.G.2]]> Tue, 16 Aug 2016 23:08:53 +0000 Bill McCallum If you want students to give coordinates of reflected or rotated points then you have to restrict to reflections and rotations where that is possible given what they know, so yes, that limits what you can do. You could have rotations about points other than the origin in multiples of 90°, and you could probably dream up other situations where special placement of the points or symmetry would make it possible, but basically you are right.

]]> <![CDATA[Reply To: What is an Angle?]]> Tue, 16 Aug 2016 23:05:07 +0000 Bill McCallum I think you mean the Geometric Measurement Progression, right? The full quote is “An angle is the union of two rays, a and b, with the same initial point
P. The rays can be made to coincide by rotating one to the other about P; this rotation determines the size of the angle between a and b.” So you need to specify a direction from one ray to the other in addition to the rays. I think the meaning is clear enough, but a more formal definition would be something like “An angle is defined to be the union of two rays, a and b, with the same initial point P, along with a direction of rotation from one ray to the other.” Would that be better? I worry that it would sacrifice clarity for precision.

]]> <![CDATA[Reply To: Algebra absolute value]]> Tue, 16 Aug 2016 22:53:29 +0000 Bill McCallum I think absolute value equations should be treated pretty lightly, and that is reflected in the blueprints. That said, a more detailed elaboration of the blueprints will probably say something about them.

]]> <![CDATA[Reply To: Data]]> Tue, 16 Aug 2016 22:51:46 +0000 Bill McCallum You are correct that it does not mean simulation. I think that sentence might be clearer if it just said “choose a sampling method” although that’s probably not technically exactly the same thing from a statistician’s point of view. If you do a random sample, you are choosing a probability model where every unit in the population has the same probability (that is, a uniform probability model). If you do a stratified random sample, then you could think of that as assigning equal probability to each group and then uniform probability within each group. I think that in the great majority of cases it is going to be a simple random sample for a curriculum aligned to the standards. Of course, an AP Stats course will go more deeply into things.

]]> <![CDATA[Reply To: The term "improper" fractions]]> Tue, 16 Aug 2016 22:33:04 +0000 Bill McCallum A fraction does not have to be less on than one, that’s for sure! As for improper fractions, there is no prohibition on writing things like 2 1/2. Indeed, it would be hard to avoid. But the standards do not use the term “improper fraction” because it promotes the misconception that a “proper” fraction must be less than 1. The notation 2 1/2 is just a shorthand for the sum 2 + 1/2, and should be read that way. Then rewriting it in the form 5/2 is accomplished the same way as for any sum of fractions (with the understanding that 2 = 2/1).

The standards also avoid talking about converting between proper and improper fractions, because the word “convert” suggests you are actually changing the number. The number stays the same, there are just different ways of writing it, depending on your purpose. Students should be able to deal with fractions written in any form, but there is no need to insist they write them in one particular way.

I’m not sure you can avoid the term “improper fraction” entirely. I’d be interested to try though.

]]> <![CDATA[The term "improper" fractions]]> Tue, 26 Jul 2016 17:03:33 +0000 kgartland We had a professional development session yesterday and had a lively discussion about the use of the term “improper” fraction. I mentioned that this vocabulary was no longer used in the standards and that led to a discussion of what should be used as an alternative to this term – I thought that we were now calling them “fractions greater than one” but one teacher insisted that a fraction had to be less than one.
Would someone mind clarifying these definitions and what the writers of the standards intended for teachers to use to describe “13/8”

Thank you in advance.

]]> <![CDATA[Data]]> Fri, 22 Jul 2016 16:43:18 +0000 Corey Andreasen Bill, I’m hoping you can clear up some confusion for me and a colleague. On page 8 of this document, it says “The reasoning process is as follows: develop a statistical question in the form of a hypothesis (supposition) about a population parameter; choose a probability model for collecting data relevant to that parameter; collect data; compare the results seen in the data with what is expected under the hypothesis”

We’re a bit unclear on “choose a probability model for collecting data relevant to that parameter.” I think you are referring to choosing a random sampling method or randomly assigning treatments and the data you get from your survey or experiment are what you mean by “data.”

She thinks that “probability model” means you’re talking about a simulation, and the simulated statistics are what you mean by “data.” I’m pretty sure you’re not calling simulated statistics data, which is what started our debate.

Can you clarify this? Thanks.


]]> <![CDATA[Algebra absolute value]]> Wed, 25 May 2016 15:13:51 +0000 lanewalker Will graphing absolute value (AV) functions be included in the blueprints as recommended in Appendix A, bottom of page 10, for F.IF.7? The virtual absence of AV after 7th grade is puzzling. There is no mention of solving AV equations in CCSS.

]]> <![CDATA[What is an Angle?]]> Tue, 24 May 2016 21:46:34 +0000 Aaron The Measurement and Data progression defines it as the union of two rays at a common endpoint. If this is the case, then a pair of rays AB and AC that both originate at point A would constitute a unique angle that has multiple measurements (x, 360 – x, and others if not confined to a full circle). Verbiage from other sources (e.g., Wikipedia) refers to angles as being enclosed by the rays and therefore not the rays themselves. Under this notion, the pair of rays forms two angles: one measuring x and the other measuring 360 – x.

While this might seem to be a clarification of little consequence, being sure of this would be helpful when writing problems for Common Core. For example, suppose we want students to identify the measure of an angle as 60 degrees. For emphasis, we would have the 60-degree space associated with this angle marked with an arc or some sort of shading and direct students to the “marked angle” in the text. But if the angle is merely the union of two rays, then “300 degrees” would technically be an accurate measurement of the marked angle in which case we should not include it as an answer choice (because 300 is one of the measures of the angle regardless of any interior markings).

If, instead, the progression took a less definitive approach explaining that angles are “formed by these rays” and not the combination of rays themselves (similar to what I see in the wording of standard 4.MD.5), writers such as myself could more freely refer to the rotational space on one side of the pair of rays as the angle in question without any technical violation. Moreover, including the aforementioned “300 degrees” as an incorrect answer choice would not present the same issue, since it would not describe the angle, a.k.a. rotational space, that is marked. Under the present definition in the progression, 300 degrees DOES describe/measure the angle, a.k.a. pair of rays, making it less appropriate to entice students with it.

Thank you for your feedback and insight.

]]> <![CDATA[8.G.1 and 8.G.2]]> Fri, 13 May 2016 15:29:55 +0000 bbaggett Since 8th grade is meant to be an informal and intuitive exploration into the world of transformations, should we be reflecting over lines other than the axes when exploring reflections on the coordinate plane? And….should we be rotating around points other than the origin when rotating on the coordinate plane? I understand that we should be exploring rigid motions outside of the coordinate plane which allows for lines of reflection other than the axes and points of rotation other than the other than the origin, but I’m not sure about what the limitations are when on the coordinate plane.

]]> <![CDATA[Reply To: 6.G.1 – special quadrilaterals]]> Mon, 25 Apr 2016 19:54:27 +0000 jkerr I’ve taken this to mean that students should be able to find the area of anything by composing and decomposing. If we start giving a formula for everything, it’s going to turn into a memorization task and the conceptual understanding will be lost. Outside of formulas for rectangles and triangles/parallelograms (justified through decomposition/composition), I think other formulas would be more distracting than helpful at this level.

]]> <![CDATA[Reply To: Unit Rate Revisited]]> Mon, 25 Apr 2016 18:23:00 +0000 jkerr I think it is worth pointing out that in Grade 6, all of this terminology will be new to students. The confusion that teachers/parents may have with the change in terminology won’t be similarly experienced by students.

I think that the terms as defined in the progressions will make matters less confusing for students. In traditional teachings of ratios, rates are defined to have different units. Then you do all of the same math that you did with ratios. So why separate them and create more words to memorize? It creates a perception that it’s totally new thing, when it is not.

One new thing done with rates in the traditional approach is finding the unit rate. But why didn’t we do that when the units are the same? We can and should, but then we’d need a new term, maybe unit ratios? So we’d have ratios, unit ratios, rates, unit rates, numerical rates, and I suppose we’d also need numerical ratios to refer to the numerical part of a unit ratio. Yikes. Is that really better than not distinguishing like/different units with different terminology, allowing us to only need the three terms ratios, rates, and unit rates?

Also, if it is “common sense” for unit price to be an example of unit rate, then define unit price as the numerical part of the rate 5 dollars per pound. Then it is an example of a unit rate!

]]> <![CDATA[Reply To: Is dimensional analysis part of Math 6?]]> Mon, 25 Apr 2016 16:49:04 +0000 jkerr I posted a few days ago about discrepancies between language in progressions and language seen from PARCC assessments. Well, here is something similar.

The RP progression doc states that in high school and beyond, students will write rates using derived units, something like a/b units/units (using fractions). Essentially, they will move away from the wordier version of a/b units for every 1 unit and write them in a more concise manner. This makes sense in conjunction with your statement about dimensional analysis. So, how is the following Grade 6 Smarter Balanced question fair?
Item #25 on page 27)

I would expect students to be using ratio tables, double number lines, or other ratio reasoning to convert units, not working with derived units. This seems like an inappropriate assessment question.
Am I missing something here?

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Mon, 25 Apr 2016 01:26:58 +0000 ejackson1 My name is Erneice and I am a student in the Masters of Mathematics Education program. I am preparing for a project on Fluency through grades 6-8. I saw that Standards 6NSB2 and 6NSB3 contain the word “fluency” in the wording. Who would contact to get more information on why students should be fluent with these two skills? Why now? How does being fluent in these skills help with real life situations? Please contact me at

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Mon, 25 Apr 2016 01:17:39 +0000 ejackson1 My name is Erneice Jackson and I am a student in the Masters of Mathematics Education program. I am preparing for a project on Fluency through grades 6-8. I saw that Standards 6NSB2 and 6NSB3 contain the word “fluency” in the wording. Who would contact to get more information on why students should be fluent with these two skills? Why now? How does being fluent in these skills help with real life situations? Please contact me at

]]> <![CDATA[Reply To: Ratio – fractional notation]]> Fri, 22 Apr 2016 21:07:47 +0000 jkerr Ok, so a ratio is a comparison of two numbers. It is not a number itself. If I’m a student, I think I can handle that. So what happens when I read the following on a Grade 6 PARCC practice test item….
….The ratio of the sales tax to the amount of a purchase is a fixed number in Town Q. The table shows the sales tax for a purchase of $1,200….
I suppose what they are actually referring to is the value of that ratio. This will confuse students if they have been writing every ratio as a:b or a to b. I understand this is not a problem with Common Core Standards. Rather, whoever wrote this at Pearson did a poor job.
Something similar will happen with the words rate and unit rate that the progressions define, yet it is claimed that the concepts can be presented to students however one wants (I don’t see how this is true when 6.RP.2 specifically refers to the unit rate a/b associated with a ratio a:b). I agree that you can get at these concepts using various language, but students will get confused come testing time if the language used in a question from PARCC or Smarter Balanced differs from the language in their book. I’ve seen a few items that use unit rate in a way that is more along the lines of a/b units to 1 unit, rather than as the value a/b.
For now, this won’t cause an issue for most students as they won’t have textbooks using the language from the progression docs. However for students using Eureka and students that will eventually use the Illustrative Mathematics curriculum (I assume it will define rate and unit rate as in the progressions), how will they be able to handle the change in language on assessments?
It seems to me as though testing consortia need to avoid the words rate and unit rate. But is it really that simple? Any thoughts?

]]> <![CDATA[6.EE.1/6.EE.2c]]> Wed, 13 Apr 2016 20:24:32 +0000 chrystalsage Can you explain whether numerical expressions using order of operations with exponents belong in 6.EE.1 or 6.EE.2c since order of operations with exponents is discussed in detail within 6.EE.2c. For instance 5 – (3+1)^2 -2^3
The progressions seem to call out some numerical expressions and specifics about order of operations in 6.EE.2c. Should 6.EE.1 be simpler type of expressions not relying on the methods of order of operations?
Also, can the base in 6.EE.1 of a exponential number be a fraction or decimal?
Thank you!

]]> <![CDATA[Quantities]]> Thu, 31 Mar 2016 18:29:29 +0000 Sarah Stevens HI! We are preparing to do a training with science teachers on graphing. We want to emphasize N-Q 1-3, as it is highly relevant and frequently linked in the NGSS. I was curious about what the progressions said for this cluster of standards, what words of wisdom I could glean, and was surprised that those standards are skipped in the progression. Will those standards be written about in future updates? I don’t have any specific questions but I often don’t know I have questions until I read an idea in a progression. 🙂 Thanks!

]]> <![CDATA[Specify Standards in Blueprints]]> Tue, 15 Mar 2016 16:51:11 +0000 jeffmerithew Would it be possible to specify which standards are addressed in each unit?

For example, Grade 5 Unit 1 is volume. Does it address all of the 5.MD.C cluster or just some of the standards within the cluster?

]]> <![CDATA[Reply To: 1.NBT.4]]> Mon, 07 Mar 2016 20:52:37 +0000 Aaron Bieniek It seems like your example of 48 + 29 fits within the standard because the sum is within 100. The standard points out two specific things – adding a 2-digit and a 1-digit, and adding a 2-digit to a multiple of 10. I don’t read those as limiting, but rather as illustrating the underlying understanding of counting on by tens and ones.

]]> <![CDATA[Reply To: Proportion vs. Proportional Reasoning]]> Mon, 07 Mar 2016 20:30:45 +0000 Aaron Bieniek In the second half of your statement “If ratios are the comparison of two quantities and ratios represent those two quantities”, you say that ratios represent the quantities. But they don’t. A ratio represents the relationship between the quantities. The quantities are what they are. A ratio associates them.

Then your second question essentially asks why ratios aren’t quantities by asserting that numbers are quantities (?) and numbers in a relationship form a ratio. I guess I wouldn’t say that numbers form ratios. Ratios are associations that describe the relationship between two or more quantities – cups of apple juice to cups of grape juice, or meters walked to seconds elapsed.

Ratios can be equivalent if they have the same value. The value of a ratio is the quotient A/B. A and B are the measurements of the quantities described in the ratio. 2 cups of apple juice to 3 cups of grape juice is equivalent to 6 cups of apple juice to 9 cups of grape juice because 2/3 = 6/9.

If we collect a bunch of those pairs of numbers that are in equivalent ratios, we have a proportional relationship: (2,3), (6,9), (1,2/3), (10,30), … which we can describe with an equation y = kx.

]]> <![CDATA[Reply To: Extent of 5.NBT.6 and 5.NBT.7]]> Mon, 07 Mar 2016 18:56:26 +0000 Aaron Bieniek Bill’s reply in this thread (from 3/25/13) states that 5.NBT.7 is limited to the fraction divisions under 5.NF.7: unit fractions by whole numbers and whole numbers by unit fractions. So, it seems like the example of 0.16/0.4 would not fall under 5.NBT.7, but the other examples (4/0.25 and 0.25/4) would.

]]> <![CDATA[Reply To: Extent of 5.NBT.6 and 5.NBT.7]]> Tue, 01 Mar 2016 22:09:35 +0000 aescame Hi Bill,
I’m still confused about 5.NBT.7 based on the posts. Your last post says that “dividends, divisors, and quotients” can all be decimals limited to the hundredths place. Should it be dividend or divisor (e.g., 4 / 0.25 or 0.25/4) as opposed to dividend and divisor (0.16/0.4)?

Also, does 5.NF.7
“Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions” constrain 5.NBT.7 at all?

]]> <![CDATA[1.NBT.4]]> Tue, 02 Feb 2016 16:12:33 +0000 scabreda I’m an instructional math coach for an elementary school district. We’re having some debate about the language in the standard 1.NBT.4:
“Add within 100, including adding a 2-digit number and a 1-digit number, and adding a 2-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding 2-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.”

Our question is: should students be adding a 2-digit and a 2-digit number? We understand the special circumstances of 2-digit and 1-digit, and 2-digit and multiple of ten, but does this standard ALSO include 2-digit and 2-digit (for example, 48 + 29)?

Thank you in advance for your help!

]]> <![CDATA[Reply To: RP Progressions]]> Wed, 20 Jan 2016 20:09:28 +0000 Susan Forbes I have some comments about the RP Progressions that I would like to put out there. I welcome any and all input. Sue

When I looked at “Progression on Ratios and Proportional Relationships” I noticed no mention of the importance of ordering a ratio expression when it is placed into a table. My thought here is that the order in which two quantities are related makes a difference when these two quantities are displayed in a table and then later graphed. Further, the language surrounding the words that describe a ratio should be consistent so that students can more easily discern this relationship. When I look at standards 6.RP. 1 – 3, and standard 6.EE.9, I note a disconnect. I found this same disconnect between pages 5 and 6 of the RP Progressions article and also within a recent Texas Instruments webinar. On page 6 of the RP Progressions article and in the TI webinar, I noticed that the ratio expression was not properly treated when it was placed into a table where units were attached.

The first treatment of the ratio expression on page 5 of the RP Progressions article differs from the second treatment of the ratio expression on page 6. I believe that the first treatment of the ratio: ““for every 5 cups grape juice, mix in 2 cups peach juice” was correctly represented within the table with grape juice being shown within the first column and then later graphed as the independent variable on the x. However, I noticed that this was later reversed in the tables shown on page 6 when the ratios: “1 cup red paint for every 3 cups yellow paint and … 3 cups red for every 5 cups yellow” were arranged within the table with red paint as the independent variable.

In a like manner I found similar flip-flopped reversals of ratios displayed within a graph during a recently viewed Texas Instruments on-demand webinar entitled: “Deciphering Ratios with TI- Inspire Technology: Are They Fractions?” Fifty minutes into this webinar, when it came time to display the rate 3m for every 2 seconds in a graph, the points were labelled in reverse order with distance listed first. I have captured this in a screen shot attachment below. I am wondering if this was done to maintain the ratio as it was originally read. I am also wondering if this is an ideal representation.

I am also wondering if we shouldn’t explicitly teach rate as a special type of ratio in which units are attached and order matters. If we discuss this order, the language clues, semantics, and relationship contexts prior to placing a ratio into a table and graphing it, the potential for later student confusion might be avoided.

Any thoughts on this???

]]> <![CDATA[Reply To: Negative Constant of Proportionality?]]> Wed, 20 Jan 2016 19:50:42 +0000 Susan Forbes I would argue that the ratio and proportions standards when viewed in light of the overarching mathematical practices and underlying grade 5 standards 5.OA.3 and 5.G.2 would support an earlier exposure to “negative slope” relationships than grade 8. If we support the traditionally less fluent operations of subtraction and division through grade 5 exposure to descending patterns within tables, student facility with these operations can not only be remediated, but a foundation for the understanding of negative proportional relationships can be laid. In grade 6, this can then be further supported by providing situational contexts for negative slope that the students can relate to such as tracking a runner’s distance from Home Base in the problem “Running Home From Third Base”. Grade 6 student understanding of negative slope has been shown to be easily facilitated by combining this earlier introduction to descending table patterns with a subsequent physical modeling of decreasing distance over time. When this physical modeling was then combined with freeze-framed second-by-second representations of this motion on a number line students had little difficulty conceptualizing and interacting with the tabular and graphical representations of this negative slope scenario.

]]> <![CDATA[Reply To: Reading Inequality Symbols]]> Mon, 18 Jan 2016 23:55:03 +0000 Carole I realize that this thread is a few years old. As I kindergarten teacher, I am familiar with the “alligator” eating the larger number, and I’ve never thought that was a good way to teach the “less than” and “greater than” signs. The students don’t know what the sign really means. I remember how I learned the signs, way back in the 60’s (during the “New Math” era!), and that’s what I teach my students: Picture each sign as an arrow. Now, picture a number line or number path. The numbers to the right on a number line become bigger or “greater”, and that’s what “>” means. Back to the number line: the numbers to the left are smaller, or become less, and that’s what “<” means. The “Alligator” lesson only works when you are comparing two numbers. In fact, I just taught the arrow pointing down a number line to a substitute teacher the other day who said she can never remember which is which. She must have been taught the “alligator” way. Ha!

]]> <![CDATA[Proportion vs. Proportional Reasoning]]> Fri, 08 Jan 2016 18:48:45 +0000 N. Barone I have been wrestling my way through the RP domain and the creation of units and professional development. This has come up and I was wondering what other people thought…

I was recently told that “Two equivalent ratios are a proportion; however, they are not proportional. Only quantities can be proportional.”

My question is… “If ratios are the comparison of two quantities and ratios represent those two quantities then why can’t they be proportional? Also — If numbers are quantities then why aren’t two numbers that are in a relationship and form a ratio considered to be a quantity?”

Any clarification would be greatly appreciated.

]]> <![CDATA[Reply To: Angle bisector theorem]]> Thu, 03 Dec 2015 18:26:02 +0000 johnrmead The Angle Bisector Theorem always stood out to me as one of those tricks that can be useful in solving specific problems, but which had relatively few useful applications in authentic problems. I think of it as a good candidate for analysis under G-SRT.4 but it is more like a trick than a useful general tool. It’s another vestigial relic in our curriculum from Elements that, even in the original, served as a dead end rather than as a foundation to other work.

]]> <![CDATA[Reply To: A-REI.11]]> Tue, 17 Nov 2015 15:14:20 +0000 tomergal I think that students don’t have to actually graph rational functions (or any function) in order to practice problems related to A-REI.11. This standard is much more concerned with the fact that the x-coordinate of the intersection of the graphs is the solution of the corresponding equation. To practice that, the students can just be given the graph ready-made, and they can also graph the functions themselves using a graphing calculator.

]]> <![CDATA[Error in 6-8 SMP #7]]> Sat, 14 Nov 2015 17:20:05 +0000 Nordstromj I am working from a May 2014 draft so this may already have been fixed. I’m the next to last sentence it says, “…all sums of three consecutive whole numbers are divisible by six.” But that isn’t true. Either product, or divisible by 3.

]]> <![CDATA[RP Progressions]]> Thu, 22 Oct 2015 15:42:25 +0000 maddieblue Will there be an updated progressions document for RP soon? I have been reading lots of discussions about the terms rate and unit rate, and how they are defined in the progressions (and how they are NOT defined in the standards). I am hoping an updated version of the progressions will help to clarify the meanings of these terms.

]]> <![CDATA[Reply To: mile wide and foot thick?]]> Thu, 15 Oct 2015 02:00:04 +0000 allenfoster In fact I am seeing that the number of topics are increasing, but every topic need t be fundamental in terms of learning and development for other can cope for it and that every stages of each topic is needed to determine in which every takers earn their stages of mistakes and correct them from it.

Allen of

]]> <![CDATA[PDFs of the Blueprints]]> Mon, 05 Oct 2015 02:11:36 +0000 kjulianhall It would be great to have a PDF of the Blueprints available to download. It would help me in my planning.

]]> <![CDATA[7.SP.8c Designing and using a simiulation]]> Wed, 30 Sep 2015 18:16:24 +0000 kirkkimb When identifying models to use to conduct a simulation, I have found several resources that mention ignoring certain outcomes. For example, if there is a 60 percent chance of an event ocurring, a number cube is used. 1, 2, and 3 represent a success 4 and 5 represent a failure and 6 is ignored. Is this model accpetable or even necessary to be introduced in grade 7?

  • This topic was modified 1 year, 10 months ago by  kirkkimb.
]]> <![CDATA[Reply To: 6.G.1 – special quadrilaterals]]> Thu, 10 Sep 2015 03:06:53 +0000 Aaron Bieniek Here’s a quote from the K-6 Geometry Progression (page 19): “Also building on their knowledge of composition and decomposition, students decompose rectilinear polygons into rectangles, and decompose special quadrilaterals and other polygons into triangles and other shapes, using such decompositions to determine their areas, and justifying and finding relationships among the formulas for the areas of different polygons.”

The standard asks students to find areas through composition and decomposition. To me, this makes formulas less “things to know” and more “things to justify and connect”.

]]> <![CDATA[6.G.1 – special quadrilaterals]]> Wed, 09 Sep 2015 05:54:57 +0000 Duane This standard mentions “special quadrilaterals” but doesn’t spell out what they are (apart from parallelograms mentioned in the grade level introduction). I guess the fact that polygons are mentioned in a general way means that anything is fair game. Is the intention that students learn particular formulas for the special quads? Or is it more that students should be able to reason their way through any polygon if they understand area for rectangles and triangles?

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Tue, 08 Sep 2015 07:00:14 +0000 Duane I’m scratching my head about Bill’s response to kirkkimb’s query as it relates to 6.NS.2 and 3. Those standards require fluency with the standard algorithm for multi-digit decimals. Bill’s response is more of a “reasoning via common fractions” approach instead of an extension of existing skills with the standard algorithm developed in Grade 5.

My interpretation of “extending the zeros” with the standard algorithm is shown in this link for dividing 121 by 8:

(hopefully this link will persist for a while – I have no other way to show it)

In other words, continuously extending the decimal places of 121 to thousandths (121.000) to assist with recording a decimal remainder. Is this part of what is expected for fluency with the standard algorithm for multi-digit decimals?

]]> <![CDATA[Reply To: 6.EE.3]]> Wed, 02 Sep 2015 14:04:26 +0000 Aaron Bieniek I’m going to leave the question of whether this is an appropriate 6th grade task for someone else, but what about this for writing the equivalent expressions?

Use the commutative property of addition to rewrite the expression as 7y + 6x – 2x. Then proceed with the distributive property: 7y + x(6 – 2). Yeah?

]]> <![CDATA[Reply To: 8.NS.1]]> Wed, 02 Sep 2015 13:26:45 +0000 Aaron Bieniek I think the standard is focused on decimal expansions which repeat in digits other than zero, but those are certainly included here. The statement “understand informally that every number has a decimal expansion” seems to me to include rationals with a repeating digit of zero.

That being said, grade 4 is the place where we start to see terminating decimals rewritten as rational numbers. 4.NF.6 asks students to “use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100.” Then in grade 5, decimals and decimal fractions are extended to the thousandths. (5.NBT.3) Also in grade 5, in 5.NBT.1, the ground work is laid for generalizing terminating decimals to any place – “Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right, and 1/10 of what it represents in the place to its left.”

]]> <![CDATA[8.NS.1]]> Tue, 01 Sep 2015 04:59:30 +0000 ak2014 Hello,

A question regarding the standard:
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Please clarify if the bolded part of this standard is inclusive of terminating decimals, previously defined as: A decimal is called terminating if its repeating digit is 0.

Do the 7th grade standards (7.NS.2d or any other 6-8 standard) require converting terminating decimals into a rational number?

Thank you!

]]> <![CDATA[Reply To: 6.EE.3]]> Wed, 26 Aug 2015 21:10:59 +0000 maddieblue What about an expression like 6x + 7y – 2x? I am looking at a book that uses the commutative property as justification for rewriting this as 6x – 2x + 7y and then the distributive property to go from there to (6 – 2)x + 7y and then 4x + 7y. I don’t think the commutative property should be used in this case because the expression involves subtraction, and students do not yet know that subtracting 2x is the same as adding –2x. Do you agree? If so, how should students reason about simplifying this expression?

]]> <![CDATA[Reply To: Algorithms Grades 2-5]]> Fri, 14 Aug 2015 06:00:55 +0000 Duane Thanks Bill, I finally had a decent chance to read through the new draft. I think the explanations are much clearer than before, especially the discussion surrounding “efficient, accurate, and generalizable methods”. There are a few things I’m interested in clarifying though.

One is that on p.14 there is mention of students adding and subtracting through 1,000,000 using the standard algorithm in Grade 4. I recall reading somewhere else on this blog that the standard algorithms need only be extended as far as necessary to demonstrate that they are generalizable. In light of that comment, is this reach to 1,000,000 viewed as what is necessary to accomplish the light-bulb moment or is it simply a border to stop teachers going any further?

Another is there seems to be a mismatch between an explanation on p.9 and a method in the margin. At the start of the fourth paragraph a description is given of the first (presumably top) method shown in the margin. The statement is given that “The first method can be seen as related to oral counting-on… in which an addend is decomposed…[and] successively added to the other addend.” Further down in that paragraph this is shown as essentially: 278 + 100 = 378 –> 378 + 40 = 418 –> 418 + 7 = 425. However, this is not what the method in the margin shows. Instead the method shown is simply an expanded form of the standard algorithm and relies on splitting both addends into hundreds, tens, and ones. Am I interpreting the paragraph text and margin method correctly? (On a related note, something that may help generally in all final versions of the Progressions documents is labeling any figures “Figure 1”, “Figure 2”, and so on.)

A final query is about a term on p.7, 3rd paragraph. What is a “5-group”?

]]> <![CDATA[Reply To: Suggestion about 7.SP.3]]> Thu, 30 Jul 2015 13:05:48 +0000 ajcarloz Could someone help me understand the differences between 7.SP.B.3 and 7.SP.B.4? Thanks!

]]> <![CDATA[Reply To: Suggestion about 7.SP.3]]> Thu, 30 Jul 2015 12:31:26 +0000 ajcarloz Could someone help me understand the difference between 7.SP.B.3 and 7.SP.B.4? Thanks!

]]> <![CDATA[A-REI.11]]> Wed, 29 Jul 2015 16:54:18 +0000 csteadman Graphing rational functions appears as part of A-REI.11 which PARCC has in Algebra I and Algebra II. However F-IF.7d has graphing rational functions as a plus standard. I am a little unclear what grade level graphing rational functions is intended for. Any thoughts?

]]> <![CDATA[Reply To: rational root theorem]]> Tue, 28 Jul 2015 11:35:06 +0000 tomergal Okay, interesting. I knew graphing calculators were allowed, but was not aware they were required for the test. Without a graphing calculator though, is there any way to solve this question without RRT?

]]> <![CDATA[Reply To: rational root theorem]]> Tue, 28 Jul 2015 05:11:29 +0000 lhwalker In my thinking, if a student understands how to find the point of intersection with a calculator, the RRT would be inefficient for that student. The PARCC calculator policy is here:

]]> <![CDATA[Reply To: rational root theorem]]> Fri, 24 Jul 2015 12:15:26 +0000 tomergal Hello,

I just wanted to note that the rational root theorem seems necessary in order to solve problem number 19 in this PARCC Algebra 1 practice test.

]]> <![CDATA[Angle bisector theorem]]> Tue, 21 Jul 2015 14:22:28 +0000 tomergal I’m wondering what’s the CCSS stance about the angle bisector theorem. I guess that proving this kind of falls under G-SRT.4 But what about using the theorem when solving geometrical problems?

]]> <![CDATA[Base-ten Numeral]]> Tue, 21 Jul 2015 12:28:38 +0000 dhassett Hi Bill,

The Standards use “base-ten numerals” to refer to conventional number form (2.NBT.3, 4.NBT.2, 5.NBT.3). By what grade should students be held accountable for recognizing this language as indicating conventional number form? Prior to that grade, what language (standard form, decimal form, number form, etc.) would you recommend using as student-facing?

]]> <![CDATA[SMPs as related to content]]> Thu, 16 Jul 2015 07:51:58 +0000 anrey Hello,

I have attended numerous training on the SMPs (Math practices) and each trainer is telling me different things.

My question is, does each content standard relate to a particular group of math practices? Or are the math practices something the teachers for themselves?

If each content is related/focused/zeros in to a particular group of practices, how would we know which group to choose? And if the teachers/designers choose for themselves which SMP to use, what are the parameters?

{Note: for example, in HS-AREI4 one district declares SMP 2,5, 6 to apply while another district may say its SMP 5, and 8 that apply. How do we choose?}

]]> <![CDATA[6.EE.1]]> Thu, 09 Jul 2015 21:58:57 +0000 Aaron The language of this standard allows for evaluations such as 8^0 = 1, but is this expected in 6th grade? Or should it be reserved for 8.EE.1, where students are better equipped to understand why numbers to the power of zero are defined as 1?

]]> <![CDATA[Reply To: Simplifying Fractions]]> Wed, 01 Jul 2015 14:32:55 +0000 pbierre Just to be clear, if the thinking is that reducing fractions is considered unnecessary, then we have to be prepared to accept whole number-equvalent results expressed in the following form, without the student necessarily recognizing that they are looking at a whole number result:




As a computational math person, I’m all for this new freedom in numeric formats. I just want to make sure educators and assessment folks are on the same page.

]]> <![CDATA[Reply To: Rationalizing the denominator]]> Sun, 28 Jun 2015 11:59:22 +0000 tomergal Thanks for the response!

I agree that RtD (“rationalizing the denominator”) is simply an application of the “difference of squares” pattern, which is also applied when we divide complex numbers using conjugates. I think we should definitely stress this fact, that these seemingly different things are all applications of the same algebraic tool.

However, while RtD is nice to practice, it doesn’t serve any useful purpose like conjugates do in complex number division. Therefore, I’m unsure whether we should teach it to students as something they need to know and do. Maybe it suffices to let students extend expressions such as (√6+√5)(√6-√5)=6-5=1, or factor expressions as in x-1=(√x+1)(√x-1), to show how the pattern is applicable even in cases where the terms aren’t perfect squares.

Let’s wait to hear from Prof. McCallum on how he perceives the status of RtD in the curriculum.

]]> <![CDATA[Reply To: Triangle congruence criteria (G-CO.8)]]> Sun, 28 Jun 2015 11:12:06 +0000 tomergal Thank you Sarah!

This definitely helps. I’m pretty certain I can devise nice, flowing, proofs of SAS, ASA, and SSS based on Wu’s highly rigorous proofs.

]]> <![CDATA[Reply To: Triangle congruence criteria (G-CO.8)]]> Thu, 25 Jun 2015 22:18:53 +0000 Sarah Stevens It won’t be easy but you should find your answers in this progression written by Dr. Hung Hsi Wu. It is difficult reading but mathematically beautiful!

Also, Dr. Zal Usiskin has written Geometry texts using this approach which can be ordered off Amazon:

Finally, I had the pleasure of meeting Dr. Usiskin at the annual NCTM conference and he confirmed that this tiny little book is another great resource.

In regards to your question, it is possible to do rigorous proofs of triangle congruence using rigid transformations. From reading Wu, you will see that a reflection along the perpendicular bisector will guarantee to carry one point to another. Then a reflection across the angle bisector will guarantee to take one side to another. Then it’s simply am matter of proving that the vertex opposite that side must be at the same location. To wrap my head around these proofs, I got a box of patty paper and worked on the transformations until I understand why one triangle was guaranteed to be concurrent with another. It really is quite a powerful tool.

]]> <![CDATA[trigonometric equations]]> Tue, 23 Jun 2015 02:37:09 +0000 lhwalker I do not see that the standards require solving simple trigonometric equations like (sin x)^2 = 1/4 for 0 < x < 2pi Am I correct?

]]> <![CDATA[Reply To: Rationalizing the denominator]]> Thu, 18 Jun 2015 21:01:27 +0000 lhwalker I’ve been on a safari looking for that answer, too! I’m thinking rationalizing goes with N.CN.8(+) which one wouldn’t know is in the Algebra 2 standards unless they read Appendix A, pages 8 and 38. I suspect the writers felt it was best not to teach “rationalizing for the sake of rationalizing.” Some of us think radicals in the denominator look cute the way they are, so why…? On the other hand, what are students to do when their answers don’t look like the ones at the back of the book? I am hoping Dr. McCallum or someone else can answer that.

The importance of N.CN.8 was not obvious for me. I now think it is important because students need to connect polynomial identities (specifically difference of squares and perfect square trinomials) from real number quadratics to complex quadratics. Along with that connection comes the idea of “conjugates” that eliminate middle terms when they are multiplied. Of course, that means N.CN.8 connects with rationalizing. However I’m thinking that rationalizing is only done to effect division with two complex numbers. How else can we get an a+bi number from dividing two a+bi numbers? Quotients of complex numbers is only in N.CN.3 which is in the “fourth course” of high school math. So for N.CN.8 I think we only need to connect polynomial identities. We won’t need to rationalize with conjugates since we won’t be dividing complex numbers.

While I was out hunting, I realized that (I think) A.SSE.2 is where we get perfect square trinomials (along with difference of squares).

]]> <![CDATA[Rationalizing the denominator]]> Mon, 15 Jun 2015 10:12:54 +0000 tomergal I’m wondering what’s the Standards’ view on “rationalizing the denominator.” On the one hand, it kind of falls within N-RN (specifically N-RN.2) and is generally a nice manipulation to practice when working with radicals. On the other hand, the motivation behind this procedure is pretty vague for high school students, and they don’t really get to use it anywhere else.

]]> <![CDATA[Reply To: Absolute Value Inequalities on the ACT]]> Fri, 12 Jun 2015 17:40:27 +0000 lhwalker Please correct me if I’m wrong, but I think I just found the answer to my question on page 39 of Appendix A where the cluster heading “Represent and solve equations and inequalities” is directly linked to specific combinations including absolute value for a Traditional Algebra 2, Unit 1. My confusion stems from A.REI.11, “Explain why the x-coordinates of the points where the graphs of
the equations… I had interpreted that to mean equations only. Indeed, on page 19 of Appendix A, for the traditional Algebra I unit 2, the same cluster and standard seems to limit absolute value to equations.

]]> <![CDATA[Reply To: Rational expressions, equations, and functions]]> Fri, 12 Jun 2015 02:24:25 +0000 lhwalker I just found the explanation for “simple” rational and radical functions. It’s a footnote in Appendix A, page 36:

]]> <![CDATA[Reply To: Rational expressions, equations, and functions]]> Wed, 10 Jun 2015 17:53:23 +0000 lhwalker One thing have found helpful is to occasionally take placement tests at colleges (I took two in April). I consistently see traditional cases of arithmetic with rational expressions (including the need to factor), so I continue to teach that in Algebra 2 and above until I hear otherwise. The ACT practice problems include an addition problem where the student must get a common denominator for x and (x + 5). As you pointed out, rational expressions present a challenge for making them seem relevant instead of a “means to an end.” I just finished writing a lesson sequence that bounces off gas laws and capacitors in series. While my disaffected seniors may not have dreams of becoming engineers, it is certainly not a huge stretch for them to consider working in a technical environment, and having a clue how those formulas work has its advantages. I include brief videos of things blowing up when maximums are exceeded and am confident I will at least get their attention. In another lesson, they modify a 8-line calculator program that calculates D=rt to t = D/r to see one is a linear function and the other is rational. Please email me if you want the series: But, yes, I am anxious to see if Dr. McCallum has any words of wisdom on this.

]]> <![CDATA[Triangle congruence criteria (G-CO.8)]]> Wed, 10 Jun 2015 15:30:32 +0000 tomergal I wonder how students are expected to “explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions.”

Let’s take SSS for example. In the “traditional” approach (i.e. triangles are congruent iff they have congruent side lengths and angle measures), we could justify it by showing how when we know all the side lengths of a triangle, there’s only one triangle we can construct (…and therefore the angle measures must be congruent as well).

But how would we do that with the rigid motions definition of congruence? The general plan is clear: assume two triangles have the same side lengths, and come up with a sequence of rigid transformations that maps one onto the other. But when trying to work this line of reasoning for two non-specific triangles, I found I had to reason in a way similar to that described above (i.e. there’s only one way to construct a triangle from three given side lengths).

It might be that I’m aiming too high and all that is necessary is to have concrete examples with two concrete triangles and a sequence of concrete rigid transformations.

I will appreciate any help with this issue.

]]> <![CDATA[Rational expressions, equations, and functions]]> Tue, 09 Jun 2015 10:23:03 +0000 tomergal I find the standards unclear regarding the scope for working with rational expressions. In general, I identified only four standards that concern rational expressions:

– A-APR.7, regarding the analogy between rational expressions and rational numbers, implies that students should know how to add/subtract/multiply/divide rational expressions.
– A-REI.2 only calls for solving “simple” rational equations, although it’s unclear what “simple” implies. Also, the standard is mainly concerned with extraneous solutions, where rational equations are just “means to an end.”
– A-CED.1 again calls for using “simple” rational functions, this time for the purpose of modeling.
– F-IF.7d calls for graphing rational functions. The sentence “identifying zeros and asymptotes when suitable factorizations are available” seems to imply the functions are not necessarily very “simple.”

It’s very easy to get really messy and complicated when dealing with rational expressions, equations, and functions. I wonder what’s the standards’ intention for those. Should we limit ourselves to linear denominators? Quadratic denominators? Constant numerators? Any guidance is welcome here.

]]> <![CDATA[3.MD.3 – pictograph vs picture graph]]> Fri, 05 Jun 2015 14:40:13 +0000 jammermath In the third grade, the standard asks students to draw scaled picture graphs and bar graphs. Can anyone define a picture graph? How is it different or the same from a pictograph? Why was the language of picture graph used? Does a picture graph in third grade include pie charts, line graphs, and bar graphs?

]]> <![CDATA[Absolute Value Inequalities on the ACT]]> Wed, 03 Jun 2015 15:29:01 +0000 lhwalker Content covered on the ACT is listed here:

and includes “absolute value inequalities.” However, I do not see “absolute value” and “inequalities” together in any standard. Is this combination somehow implied or did the ACT folks include something not intended for “intermediate algebra?”

]]> <![CDATA[Solving inequalities in grade 7]]> Mon, 01 Jun 2015 22:39:19 +0000 Lisa j r 7.EE.4b discusses solving word problems leading to inequalities. The progressions clearly indicate understanding about multiplying or dividing by a negative value is introcuded here, but there seems to be no standard asking students to just solve inequalities such as -2b < 12 or 14 – 5x >= 22. In 7.EE.4a it says to solve these types or EQUATIONS fluently but that isn’t stated for the inequalities. For assessment purposes should all items for 7.EE.4b include situations or context?

  • This topic was modified 2 years, 2 months ago by  Lisa j r.
]]> <![CDATA[The structure is the standards? Clusters & Focus topics]]> Sun, 31 May 2015 22:33:46 +0000 Jkatsikas I am hoping to understand better the concept of the clusters (and Focus Topics) in the math standards. I have seen some school districts restructuring the standards to the sequences under the Focus topic, but I also read in the Appendix of the Publisher’s criteria that the “Structure is in the Standards”. This has led to some confusion on my part as the standards do not seem to be in cohesive groups when under the clusters. I understand that the concept clusters should be linked to the standards & will spiral throughout my teaching, but would like some clarification as to the sequencing of math skills.

One example I’d like to share is from Third Grade. I am including the focus topics for our first and second cycles as well as the standards that were tested.

· Focus Topic 1: Exploring Equal Groups as a Foundation for Multiplication and Division (3.OA.1 & 3.OA.2, 3.OA.3, 3.OA.7)
· Focus Topic 2: Develop Conceptual Understanding of Area (3.OA.5, 3.MD.5, 3.MD.6, 3.MD.7)
· Focus Topic 3: Relating Addition and Subtraction to Length (3.NBT.1, 3.NBT.2, 3.MD.8)

Items tested at the end of this cycle​ ​ ​
3.OA.2 3.OA.5 3.MD.6 3.MD.7 a 3.OA.3 3.OA.7 3.MD.5 b 3.OA.3

· Focus Topic 4: Understanding Unit Fractions (3.G.2, 3.NF.1, 3.NF.2)
· Focus Topic 5: Using fractions in measurement and data (3.NF.1, 3.NF.2, 3.MD.4)
· Focus Topic 6: Solving Addition and Subtraction problems involving measurement (3.MD.1, 3.NBT.1, 3.MD.2)
· Focus topic 7: Understanding the relationship between multiplication and division (3.OA.2, 3.OA.3, 3.OA.6, 3.OA.7)

Items tested at the end of this cycle ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
3.OA.2 3.MD.6 3.NF.1 3.G.2 3.NF.2 a 3.NF.2 b 3.MD.1 3.MD.2 3.NBT.1 3.OA.3 3.OA.6 3.OA.7

Focus topic #6 was singled out as one our students had difficulty with: “Solving addition and subtraction problems involving measurement” and the standards included were:
3.MD.1: Tell & w​rite time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition & subtraction of time intervals.
​3.NBT.1: Use place value understanding to round whole numbers to the nearest 10 or 100.
​3.MD.2: Measure & estimate liquid volumes and massess of objects using standard unit of grams, kilograms, and liters. Add subtract, multiply and divide to solve one step word problems involving masses or volumes that are given in the same units.
Each standard was assessed with only one question, so I’m not sure how we can draw reliable and valid conclusions based on this.

My understanding was that the Common Core was supposed to help us delve deeper into fewer topics in a way that relates concepts, so I am hoping for some clarification to verify that we are grouping these concepts in a way that can best support students’ learning.

Thank you for any guidance you might offer.​

]]> <![CDATA[5.NF.B.4.B Tiling with Unit Squares]]> Thu, 21 May 2015 18:13:34 +0000 ChristyG In 5.NF.B.4.B it states, “Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths…”

My question is about the phrase “unit squares.”
Is the intention that the area must be tiled by squares rather than simply rectangles?
In the Progressions document (p. 13) 3/4 x 5/3 is shown tiled with rectangles that are each 1/4 by 1/3 (not squares).
The area could be tiled with 1/12 by 1/12 squares (resulting in an area of 9/12 x 20/12 = 180/144) but this seems like it would unnecessarily complicate the problem.

]]> <![CDATA[Points of Concurrency]]> Wed, 20 May 2015 20:50:49 +0000 erfarmer Hello,

I am trying to figure out how (and if) points of concurrency are part of High School Geometry. I know that constructions are (G.CO.12 & G.CO.13), and there is a nod to the idea in G.CO.10 “Prove theorems about triangle. Theorems include: . . . the medians of a triangle meet at a point“. In Wu’s writing on Geometry – he explains how to prove each of the points are concurrent. So my question is to what extent should we focus on these points – their names? their properties? how to construct them? Any guidance or suggestions is appreciated.

]]> <![CDATA[Reply To: trapeziod definition]]> Mon, 18 May 2015 13:30:51 +0000 eamick On a related note… I am curious about a preferred definition of polygon. The progressions document does not address this definition at all. Are star polygons included? That is, polygons whose sides intersect at points other than their endpoints.


]]> <![CDATA[Grade 1 Blueprint]]> Fri, 15 May 2015 17:41:05 +0000 starksj I am delighted you’ve created these blueprints! I have mapped out the standards listed by cluster for each unit for the entire year, and I believe the cluster that covers data (1MD.C – Represent and interpret data) has been left out of this grade’s blueprint. If that is not the case, can someone please direct me to the placing of this cluster within the blueprint?

Also, do you eventually plan to link tasks to units? Even though many tasks can fit in more than one unit, linking them would be most helpful!

Thank you again for the hard work!

]]> <![CDATA[Progressions Use]]> Thu, 14 May 2015 22:33:13 +0000 vickyk We have been trying for several weeks to obtain permission to make copies of the progressions documents for school district professional learning purposes. We have not been able to get much help and one gentleman has been trying at the University but there is no one answering emails or phones at the contact info. Can we either get permission to print or a contact name to get this same in writing in order to move forward. Thanks vicky

]]> <![CDATA[Reply To: 5.NF.2 – Does this include mixed numbers?]]> Wed, 13 May 2015 19:55:08 +0000 Bill McCallum You can certiainly have problems about comparing mixed numbers. Mixed numbers are really sums of fractions, i.e. 3 2/5 is really a shorthand way of writing 3 + 2/5. So, depending on how the student does the comparison, it might call on their knowledge of 5.NF.3 as well as 5.NF.2.

Remember that the standards are goals to be achieved by the end of the year, so you don’t have to think of them as curricular units, nor does the order in which the standards are written necessarily relate to the order of topics in the curriculum.

]]> <![CDATA[Reply To: Grade 5 Blueprints]]> Wed, 13 May 2015 19:44:09 +0000 Bill McCallum We have added an alignment to the cluster 5.OA.A in Unit 5.3.

]]> <![CDATA[Reply To: Concept of discriminants]]> Wed, 13 May 2015 19:33:29 +0000 Bill McCallum abienek is right that the discriminant is not required explicitly, although A.REI.4 would be the place to put it. Of course, it’s there implicitly in that it’s the thing under the square root sign in the quadratic formula. No harm in pointing out that that thing has a name, I suppose, but I agree it doesn’t seem necessary and could need to excessive rulifying.

]]> <![CDATA[Reply To: 7.G.4 and G.GMD.1]]> Wed, 13 May 2015 18:33:02 +0000 Bill McCallum Sorry it is taken me so long to reply to this. First, there is no official definition of π in the standards, so no, you don’t have to take Wu’s approach. I quite like it myself but I have talked to others who disagree. Mathematically, you can do it either way; the miracle is that the two numbers (area of unit circle and constant of proportionality between circumference and diameter) turn out to be the same.

As to your other questions, I think Wu’s limit argument is a bit too much for Grade 7. I would use the argument that rearranges the triangles into an approximate rectangle with length equal to half the diameter and height equal to the radius if I were going to give any argument at all. From this you can get that the area is 1/2 the product of the circumference and the diameter, A = 1/2 Cr. As you point out, the standards do not technically require that you justify the individual formulas C = πd (or C = 2πr) and A = πr^2. But you are almost there at this point. If you have defined π as a constant of proportionality, you may have given an informal justification of why that constant of proportionality exists. Doing so would amount to a justification of the formula for the circumference. And once you have that you can get the area formula by substituting the circumference formula into A = 1/2 Cr.

]]> <![CDATA[Almost Final Draft(s)]]> Wed, 06 May 2015 13:51:58 +0000 Chad T. Lower So I was looking at the Almost Final Draft, and saw the statement:

For updates and more information about the Progressions, see

When I go to the website and download this “same” document, I got a revision date of 21 April 2012. From the website, the date was 6 March 2015.

Is there a location I can go to get the most up to date drafts for all of the documents? I tried scrolling through the other blog posts on this website, but didn’t see any other Progressions listed in the posts I went through.

Thank you for any help you can provide.
Chad T. Lower

]]> <![CDATA[Reply To: Geometry Progressions]]> Fri, 24 Apr 2015 01:57:54 +0000 eprebys I asked about this at NCTM and got the response below. Bill, please correct me if I paraphrase incorrectly:

The progressions are still under work. The article by Wu is essentially the first draft of the progression document. There is a geometry blueprint up on IllustrativeMathematics (

I responded, “So did you end up going with the Wu approach or did you couch it as Harel recommends?”

Bill said that essentially the blueprint goes with a transformation first approach as opposed to a Euclidean approach.

My opinions: I find Harel’s argument very convincing. That said, I am glad that Bill is making this decision. I much prefer a thoughtful decision I disagree with from him, that includes a solid explanation and a lot of coherence, over a decision from publishers with less coherence.

]]> <![CDATA[Reply To: 7.G.4 and G.GMD.1]]> Tue, 21 Apr 2015 14:50:36 +0000 Sarah Stevens Hi! I am just following up and wondering if anyone has any thoughts on my question. I am planning for some Geometry PD this summer and would like to know if these perspectives are above the scope of the standards.


]]> <![CDATA[5.NF.2 – Does this include mixed numbers?]]> Mon, 20 Apr 2015 14:13:50 +0000 SteveG Forgive me if this seems like a silly question, but the language of 5.NF.2 includes mixed numbers, right?
5.NF.2 = Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

I was discussing this with a colleague the other day and noticed that 5.NF.2 just says “fractions” but it follows 5.NF.1 (and seems to be the application of the skills in 5.NF.1), so it would make sense to think that 5.NF.2 includes word problems with mixed numbers. The example in the G3-5 Fraction progression doc (page 11) does not have mixed numbers, so I am not sure what to do.
Any insight would be much appreciated.

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Mon, 20 Apr 2015 14:11:49 +0000 SteveG I’m going to post my question about 5.NF.2 in a separate post, as I realized later it was kind of a separate question.

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Wed, 08 Apr 2015 14:55:48 +0000 SteveG Forgive me if this seems like a silly question, but the language of 5.NF.2 includes mixed numbers, right? I was discussing this with a colleague the other day and noticed that 5.NF.2 just says “fractions” but it follows 5.NF.1 (and is the application of the skills in 5.NF.1), so it would make sense to think that 5.NF.2 includes word problems with mixed numbers. The example in the G3-5 Fraction progression doc (page 11) does not have mixed numbers, so I am not sure what to do.
Any insight would be much appreciated.

]]> <![CDATA[Reply To: trapeziod definition]]> Tue, 07 Apr 2015 04:38:23 +0000 Bill McCallum I don’t see an SBAC definition, but my guess is they are likely to agree with PARCC. It is the more common sense definition from a mathematical point of view.

]]> <![CDATA[Reply To: trapeziod definition]]> Tue, 07 Apr 2015 04:37:34 +0000 Bill McCallum I don’t see an SBAC definition, but my guess is they are likely to agree with PARCC. It is the more common sense definition from a mathematical point of view.

]]> <![CDATA[Reply To: 5.NBT.2]]> Tue, 07 Apr 2015 04:33:33 +0000 Bill McCallum I don’t think the standards answer this question directly. Assessment of the standards is a different issue from statement of the standards. In this case, the main point is to understand what happens when you multiply or divide by a power of ten. If you can think of an assessment question that gets at this which involves asking the student to give the power of ten, then great! My instinct is that such questions would be more difficult than ones where the power is given, but I would have to see some examples to be sure.

]]> <![CDATA[Reply To: 5.OA.2]]> Tue, 07 Apr 2015 04:29:19 +0000 Bill McCallum Yep, you are both right!

]]> <![CDATA[Reply To: Interpretation of MAD and Interquartile Range]]> Tue, 07 Apr 2015 04:28:02 +0000 Bill McCallum Interesting question. Even when you are interpreting a single data set you probably have some reference data set in mind. If you look at a data set and say to yourself “that data set is very variable (as indicated by its MAD or IQR)” you are implicitly comparing it to a standard data set in your mind that has a smaller “normal” variability. So the point of this standard is to make such comparisons explicit. But that doesn’t mean you would never ask a student to comment on a single data set.

]]> <![CDATA[Reply To: 3.OA.B.5]]> Wed, 01 Apr 2015 03:49:13 +0000 Aaron Bieniek 3.OA.7 calls for third graders to fluently multiply and divide within 100. So double digit factors are all good as long as the product is between 0 and 100.

]]> <![CDATA[3.OA.B.5]]> Tue, 31 Mar 2015 14:38:16 +0000 dseabold My question is whether 13 x 6 = (10×6) + (3×6) goes beyond what this standard intended for third grade? Basically, I’m wondering if it’s ok to move into double digit factors or if 3rd graders should just be working with single digit factors. Thanks, in advance, for any clarification.

]]> <![CDATA[Grade 5 Blueprints]]> Thu, 26 Mar 2015 15:17:41 +0000 rburns I just looked over the Grade 5 Blueprints and I don’t see any OA standards. Did I miss something?

]]> <![CDATA[Reply To: explicit and recursive sequences]]> Fri, 20 Mar 2015 22:10:07 +0000 lhwalker Interestingly enough, the f(n) concern turned out to be a paper tiger. The real problem is that lots of people do not realize a sequence is a function. The terminology input-output is not universally understood, and those worried about confusion students might have appear to have been confused themselves. Can I thank you, once again Dr. McCallum, for the beautiful job you have done with these standards?

]]> <![CDATA[Reply To: Concept of discriminants]]> Tue, 17 Mar 2015 16:03:13 +0000 Aaron Bieniek I’m thinking about what I gain by knowing that the value of the discriminant gives me information about the number of real roots in a quadratic. My thought is “not enough” to go beyond having the word as part of my students’ vocabulary.

The standards call for being able to solve quadratic equations by a method appropriate to the initial form of the equation AND recognize and write complex solutions. So, how does the discriminant connection help students progress through this standard? I would argue it doesn’t, and may in fact detract from the focus of the standards. First, it replaces “reasoning” with a “rule”. If I have learned that the square root of a negative number is a non-real result, then I should be able to reason that a negative discriminant gives non-real results. I don’t need to learn a new thing in order to learn something I already know. (reminds me of things like the vertical line test.)

Secondly, a focus on zero, one, or two solutions is a little misleading since even if there are no real solutions, we still need to to recognize (Algebra 1?) and write (Algebra 2?) the complex solutions.

]]> <![CDATA[Reply To: Expressing a whole number as a fraction before 5th grade]]> Tue, 17 Mar 2015 00:49:28 +0000 Bill McCallum I guess I would just say “five plus four over nine.” Well, that could still be interpreted as a mixed number. People sometimes say something like “quantity five plus four,” with a pause afterwards, to indicate they are referring to a single quantity. So you could say “quantity five plus four [slight pause] over nine.” Probably I would just avoid having to say this verbally at all.

]]> <![CDATA[Reply To: Set Model for Fractions]]> Mon, 16 Mar 2015 21:52:47 +0000 Bill McCallum The standards don’t actually mention the set model for fractions, so exactly when to introduce it is partly up to the judgement of the curriuclum writer. The reason for not introducing it in Grade 3 is that it can cause confusion as to what the whole is. If eat 6 out of 12 bananas, that’s 1/2 of the bananas. But notice that in order to interpret this I need to regard the 12 bananas as a whole. That seems a little alien, as opposed to area of length models, where I can clearly see the rectangle as a whole, or the length from 0 to 1 on the number line.

The set model is really more related to multiplication of a whole number by a fraction: $\frac12 \times 12 = 6$. That doesn’t happen until Grade 5, so that’s why you will see some people say it should go there.

But I think you could also start working with in Grade 4, as a preparation for multiplication of whole numbers by fractions. It depends on exactly how you introduce it.

]]> <![CDATA[Reply To: Number Line in grades K and 1]]> Mon, 16 Mar 2015 19:54:34 +0000 Bill McCallum There are different types of number line. In Kindergarten students might place numbers on a line, but it is often more like a “number row” than a number line … that is, the precise placement is not attended to. In Grade 2 Common Core, the number line is really meant to have a measurement aspect to it. You have a 0 and 1, then you mark of the line in lengths equal to the length from 0 to 1 in order to get 2, 3, etc. At least, you start doing that in Grade 2 … and in Grade 3, the measurement aspect becomes salient when you start talking about fractions. I suspect that all the different sources you are looking at are really talking about different incarnations of the number line.

]]> <![CDATA[Reply To: How do you measure fluency?]]> Mon, 16 Mar 2015 19:32:09 +0000 Bill McCallum In my view “fluent” means “fast and accurate” and I don’t see a meaningful distinction between this and “automatic.” But, as you point out, people do make quite suble distinctions between all these words. I think a lot of these distinctions are more meaningful for assessment and curriculum then they are for standards themselves. The standards require fluency; implementers of the standards will be making more fine-grained decisions about how to teach and how to measure it.

]]> <![CDATA[Reply To: Length Measurement in Grade 1 Blueprint]]> Mon, 16 Mar 2015 19:11:41 +0000 Bill McCallum Thanks for pointing this out … I’m going to pass it on the blueprint authors for comment. And yes, I try to include the update date, but sometimes I forget!

]]> <![CDATA[Reply To: 1NBT4]]> Mon, 16 Mar 2015 18:47:18 +0000 Bill McCallum I’m a little confused by this comment, because 1.NBT.4 does explicitly call for “adding a two-digit number and a multiple of 10,” which seems to be what you are asking for? Can you clarify your concern?

]]> <![CDATA[Reply To: explicit and recursive sequences]]> Mon, 16 Mar 2015 18:44:09 +0000 Bill McCallum Hi Lane, could you explain a bit more? Maybe give an example of what people are concerned that students might do.

]]> <![CDATA[Reply To: Algorithms Grades 2-5]]> Mon, 16 Mar 2015 18:41:59 +0000 Bill McCallum Duane,
Sorry for the long delay in replying to this, but it made me realize I needed to get that revised version of NBT finished. It is now posted. Could you take a look and see if it helps with this confusion? Happy also to answer more questions, now that it is done.

]]> <![CDATA[Concept of discriminants]]> Mon, 16 Mar 2015 02:54:40 +0000 skeeter As I am preparing for an upcoming unit on Quadratic Functions, I am looking for a connection between the concept of discriminants and their relationship to the roots of quadratic equations. I have read through the standards and am trying to find a connection to a specific standard. Would this concept surface through A.REI.4? Is this a connection that students should build in order to progress through quadratic functions in Algebra I to Algebra II? Any feedback would be helpful. This is my first year teaching Algebra as well as teaching the Common Core Standards. So, I am trying to make sense of the content provided in my textbook along with the content described in the Standards and the progression documents. Thank you for your response.

]]> <![CDATA[7.G.4 and G.GMD.1]]> Wed, 04 Mar 2015 16:10:55 +0000 Sarah Stevens Hi! In the absence of a Geometry Progression, I have been reading Wu’s Geometry Progression ( and I have a question about an interpretation he takes which is not evident in the standards. I am curious if the highly anticipated Geometry Progression will take this position and if this position is the intended reading of 7.G.4.

On page 55, Wu defines pi as the area of the unit circle. He does this in order to lead into the relationship between the circumference and area of a circle and derive the formula for the area of the circle. Shifting the definition from pi as the ratio of the circumference to the diameter to the area of the unit circle will be a big task. Do you think this is a worthwhile battle, in the grand scheme of all things CCSS which must be shifted to new understandings? Is it a necessary shift for the high school standards but not the middle school standards? I guess, in general, I am curious about your thoughts about defining pi as the area of the unit of a unit circle.

From here, Wu takes a polygon and decomposes it into triangles from the center of the polygon (pg 56-59). He creates a general formula for the area of a polygon based on the area of each triangle. Then he does an informal limit, as the polygon increases the number of sides, it gets more circular. Therefore the formula created can be used to find the relationship between the area and circumference of a circle and then he continues this logic to find the standard formula for the area of a circle. My questions are:
1) Is this line of reasoning, decomposing polygons and informal limits, the intended line of reasoning for the part of the standard asking for the informal derivation of the relationship between the circumference and area of a circle?
2) Should students be deriving the formula for the area of a circle in 7th grade? The standard only says “know and use” so I was wondering if this interpretation is an extension of the intent of this standard.
3) Would this explanation also work for (or be more appropriate for) the high school G.GMD.1 standard? Is this the intent of that standard?

Finally, any word on when we will get an official Geometry Progression? 🙂


]]> <![CDATA[Reply To: trapeziod definition]]> Tue, 24 Feb 2015 14:07:08 +0000 David Woodward Does anyone know what definition the SBAC is choosing to use?

]]> <![CDATA[Reply To: trapeziod definition]]> Tue, 24 Feb 2015 05:38:41 +0000 David Woodward I sure wish that they had used the exclusive definition, since every text book that I have ever worked with does, but so be it. Does anyone know what SBAC has chosen for their definition?

]]> <![CDATA[Reply To: 5.OA.2]]> Tue, 24 Feb 2015 03:36:30 +0000 Aaron Bieniek Yep. You won’t find any disagreement here. Read the posts above yours in this thread and you will see your point confirmed at least twice. The original post in this thread suggested a “correct” order in which to use grouping symbols, and all of the replies to it suggested that such an order – even if it exists by convention – is neither important nor the point of 5.OA.2.

]]> <![CDATA[5.NBT.2]]> Mon, 23 Feb 2015 21:44:58 +0000 dhassett Should assessment of 5.NBT.2 include generation of products and quotients with powers of ten? Or, is the conceptual nature of the standard better aligned to questions in which a product or quotient is given so as to focus students’ attention exclusively on explanation of givens?

]]> <![CDATA[Reply To: 5.OA.2]]> Sun, 22 Feb 2015 23:12:02 +0000 rPantoja

Mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there cannot be braces in a problem that does not have both parentheses and brackets.

I found this website when I search the sentences above in GOOGLE.
I did so because I found those sentences in some instructional materials. I could also find the exact same quote in several educational websites. School districts and State Departments of Education equally repeat these guidelines, but they are simply incorrect.
While there is a widely accepted hierarchy for grouping symbols, nothing in MATHEMATICS forbids the use of brackets of braces in the absence of parenthesis. The expressions below may not be the most elegant, but they are perfectly “legal” mathematically speaking:

(3[2^4+1] +17)^2
2x + [x+1]^2 + (y-2)
(2^3 + [3^2 + √(2) ])

]]> <![CDATA[Interpretation of MAD and Interquartile Range]]> Sun, 22 Feb 2015 22:36:04 +0000 traceylboh1 Do you have any examples of what is expected of students for interpreting the MAD and Interquartile range of data? Is it only expected that students interpret data when comparing two data sets or should they be able to interpret a single data set?

]]> <![CDATA[Reply To: Expressing a whole number as a fraction before 5th grade]]> Tue, 17 Feb 2015 16:26:00 +0000 dmcdonald I have a question similar to the original one, in regards to how to say a particular fraction when it comes up with my students. The Progression documents have a number of examples of fractions in which the numerator is an addition or multiplication expressions, such as 5 + 4/9 (5 + 4 as a numerator and 9 as a denominator). How would this be read exactly? If I were to say 5 + 4 ninths, it would seem to be indicating a mixed number. Would I repeat the word “ninths”, reading it as five-ninths plus four-ninths, as it were two different fractions?

Thank you for your help.

]]> <![CDATA[Set Model for Fractions]]> Tue, 17 Feb 2015 00:08:34 +0000 noamszoke Hello,
We were watching the video on Illustrative Mathematics that seems to say that the set model for fractions is NOT introduced until Grade 5. Other sources suggest they are introduced in Grade 4. The progressions say nothing about it. Your thoughts?

thank you,

]]> <![CDATA[Number Line in grades K and 1]]> Mon, 16 Feb 2015 23:55:21 +0000 noamszoke Hello,
We are wondering about the use of the number line as an instructional model in grades K and 1. The number line first appears in the CCSS-M in grade 2, in 2.MD.6. It is not mentioned in the progressions before then. Various sites use it extensively, while at least one (Arizona – exhorts against it.
What are your thoughts on this? There is a looong tradition of number line use in Kinder (e.g. Math Their Way, etc.) but is there research giving us direction on this?

thank you,


]]> <![CDATA[Reply To: How do you measure fluency?]]> Sat, 14 Feb 2015 04:24:52 +0000 EANelson Do the standards require “automaticity?” How would that be different from fluency? I ask in part because in the 2008 NMAP Report, on page 4-5, cognitive experts Geary, Renya, Siegler, Embertson, and Boykin wrote,

“At all ages, there are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory.”

A key word there is “implicit:” Being able to do the right thing instinctively, without necessarily being able to explain why. In solving scientific calculations, because of limits on the duration of novel elements held in working memory during processing, automaticity is vital. How is fluency different? Is implicit recall of an appropriate procedure what students and instructors should aim for under the standards?

]]> <![CDATA[Length Measurement in Grade 1 Blueprint]]> Tue, 10 Feb 2015 02:35:17 +0000 teachtucson I am a curriculum designer who specializes in aiding teachers in designing curriculum maps and curriculum units. I have been using the progressions extensively. In the K–5 Geometric Measurement (page 9, paragraphs 3 & 4) states regarding manipulative standards units:

“Another important issue concerns the use of standard or nonstandard units of length. Many curricula or other instructional guides advise a sequence of instruction in which students compare lengths, measure with nonstandard units (e.g., paper clips), incorporate the use of manipulative standard units (e.g., inch cubes), and measure
with a ruler. This approach is probably intended to help students see the need for standardization. However, the use of a variety of different length units,
before students understand the concepts, procedures, and usefulness of measurement, may actually deter students’ development. Instead, students might learn to measure correctly with standard units, and even learn to use rulers, before they can
successfully use nonstandard units and understand relationships between different units of measurement. To realize that arbitrary (and especially mixed-size) units result in the same length being described by different numbers, a student must reconcile the varying lengths and numbers of arbitrary units. Emphasizing nonstandard
units too early may defeat the purpose it is intended to achieve. Early use of many nonstandard units may actually interfere with students’ development of basic measurement concepts required to understand the need for standard units. In contrast, using manipulative standard units, or even standard rulers, is less demanding and
appears to be a more interesting and meaningful real-world activity for young students.
Thus, an instructional progression based on this finding would start by ensuring that students can perform direct comparisons.Then, children should engage in experiences that allow them to connect number to length, using manipulative units that have a standard unit of length, such as centimeter cubes. These can be labeled “length-units” with the students. Students learn to lay such physical units end-to-end and count them to measure a length. They compare the results of measuring to direct and indirect comparisons.”

After much deliberation regarding the above progressions paragraphs, as well as related paragraphs for Grade 1, and Grade 2’s mention (page 12, paragraph 1) of: Measure and estimate lengths in standard units: Second graders learn to measure length with a variety of tools, such as rulers, meter sticks, and measuring tapes (2.MD.1 – Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.), teachers I have been working with purposefully have Grade 1 students exploring/learning length measurement concepts with MANIPULATIVE standard units only. We agree it makes sense maturationally/ cognitively since the manipulative units used in Grade 1 are in actuality standard (e.g., button that is 1 cm, tile that is 1 in., dowel that is 1 ft., string that is 1 m), so that when the students begin Grade 2 the transition to a standard unit length (2.MD.A.3 …inches, feet, centimeters, meters) using rulers, yardsticks, meter sticks, and measuring tapes will be a natural and smooth transition.

So, what is my specific question? Why does the newly published Grade 1 blueprint have a unit that encourages a transition to using a ruler in Grade 1 based on what the progressions encourage regarding manipulative units as well as Grade 2’s expectation for using a standard tool: ruler, yardstick, etc.?
Unit 1.1 “Length and the Number Line” states:
Students begin their work with standard units as well. A good transitional activity would be using a 12-inch ruler to measure the side-lengths of a train of 1-inch tiles; this makes the connection between iterating length units (the tiles) and the structure of the ruler clearer.
We have the “transitional activity” in early on in Grade 2, not Grade 1 based on the progression’s recommendations.

Thank you in advance for your response.

P.S. Could you please include an “updated date” when making any changes to the progressions? I have noticed nuance modifications and additions to some of the progressions over time, but the draft date does not get updated (e.g., Draft, 6/23/2012). Thank you!

]]> <![CDATA[Reply To: Algorithms Grades 2-5]]> Mon, 09 Feb 2015 22:24:44 +0000 Duane Just bumping this topic up.

]]> <![CDATA[Reply To: 1NBT4]]> Thu, 05 Feb 2015 21:58:27 +0000 Clint Vandiver I find this standard really unclear. I even tried to reason some clarification by reading kindergarten and second grade standards related to this topic. I honestly hoped that the standard was meant to allow more time with and understanding of what happens numerically when adding 2 digit numbers and multiples of ten. I think the ability to add 10 to a number is crucial in later math as students decompose and recompose numbers to make addition easier with numbers that make sense. The second grade standard did not mention multiples so I thought that meant in third grade that any addition problem was fair game as long as it met the number of digit requirements. Maybe some better wording would have been with special emphasis on being sure to include 2 digit numbers and multiples of 10.

]]> <![CDATA[explicit and recursive sequences]]> Wed, 04 Feb 2015 13:32:15 +0000 lhwalker Algebra1 CCSS uses f(n), f(n+1) notation for sequences, but there is a concern that students will “evaluate” f(n+1) as f(x+1). I would appreciate feedback.

]]> <![CDATA[Reply To: base e]]> Sun, 01 Feb 2015 17:29:27 +0000 Bill McCallum Well, remember that course placement is not included in the standards; I guess you are probably looking at Appendix A, correct? I could imagine introducing exponential functions with base e quite early as a standard function in science and finance, and delaying the explanation until later.

]]> <![CDATA[Reply To: Expressing a whole number as a fraction before 5th grade]]> Sun, 01 Feb 2015 17:26:53 +0000 Bill McCallum Agree with abieniek, and I would only add that this is an explicit requirement in Grade 3.

3.NF.3c: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

]]> <![CDATA[Reply To: 2.NBT.9]]> Sun, 01 Feb 2015 16:37:34 +0000 Bill McCallum It could be either, I think. It would be a natural part of a classroom where students are talking about their reasoning and asking questions about others’ reasoning. Certainly no need to write an essay!

]]> <![CDATA[Reply To: 2.NBT.2 – Skip Counting]]> Sun, 01 Feb 2015 16:32:34 +0000 Bill McCallum Yes, it seems we should document this in the progressions document, thanks. It’s interesting that both examples are online tools for skip counting. It’s probably easier to program if you don’t insist on multiples.

]]> <![CDATA[Reply To: Algorithms Grades 2-5]]> Wed, 28 Jan 2015 00:20:32 +0000 Duane On a related note to algorithms, reading through the NBT Progressions, page 3 notes a distinction between “general methods” and “special strategies”. General methods are defined as applicable to all numbers (in base-ten) but not necessarily efficient. They may be efficient but it is not always the case. Special strategies are defined as applicable only to certain cases or applicably to more cases only with “considerable modification”.

The example given on page 3 for a special strategy is 398 + 17, which is rewritten as (398 + 2) + 15. A general strategy example is given as combining like base-ten units, i.e. 300 + (90 + 10) + (8 + 7). Another example of a special strategy is given on page 7 (margin) where you start with one number then count on tens then ones individually, e.g. 46 + 37 –> 46, 56, 66, 76, 77, 78… and so on.

The special strategy given on page 7 doesn’t seem all that difficult to extend to three-digit numbers (i.e. count hundreds, tens, then ones) and beyond, or by adding instead of counting (as noted on page 7). Time-consuming, yes, but not requiring considerable modification. It’s not all that different from counting by ones which was defined on page 3 as a general method. Given its close similarity to counting by ones, and its applicability to all cases, what makes this strategy “special”?

Also, a distinction is made between algorithms and strategies (p.3), with strategies being broken into special and general as discussed above. The top example in the margin of page 7 shows the standard addition algorithm but it is labeled as a general method, i.e. a “strategy”. So I’m confused – is the diagram showing an algorithm or a strategy?

]]> <![CDATA[Reply To: Algebra 2 for all?]]> Wed, 21 Jan 2015 03:25:48 +0000 Bill McCallum Not sure what to say here. I agree with some of what Steve says, disagree with some other other things he says, and think he misses some important structural elements of the standards. Everybody could write standards that they personally think are better than the Common Core. (Including me.) But that’s not the point of having common standards. Different people with different opinions settled on an agreement in 2010 about expectations for what students should know at the end of each grade level or course. Until we prove we can implement an agreement (not at all clear yet) I’m not interested in relitigating old arguments.

]]> <![CDATA[Reply To: HSG.GMD.A.1, A.2]]> Wed, 21 Jan 2015 03:17:01 +0000 Bill McCallum Well, I would need to know exactly which methods you are talking about here. I don’t know which method CME uses. The argument I remember from Archimedes involves balancing the areas of different slices … this is arguably making use of Cavaleieri’s principle, whose essence is slicing. Happy to answer in more detail if you can be more explicit.

]]> <![CDATA[Reply To: A-SSE.B.3.c]]> Wed, 21 Jan 2015 03:09:18 +0000 Bill McCallum The diffence is certainly small, but small differences in effective interest rates can add up over time: see

]]> <![CDATA[base e]]> Sat, 17 Jan 2015 21:25:00 +0000 lhwalker It seems like base e comes along with F.LE.4 which is an Algebra 2 standard. I suspect base e is delayed from Algebra I to be introduced with logarithms, but I see in some software packages it is mixed in with exponential functions in general. Am I correct that base e was intended to be delayed?

]]> <![CDATA[Reply To: Algebra 2 for all?]]> Thu, 15 Jan 2015 16:46:28 +0000 Sarah Stevens Let met try again. It doesn’t link to the article but to his blog. When you click on the link in the blog, it will download a word document with his thougths.

If this link doesn’t work, you can go to and navigate to the blogs. It is the second blog back from the most recent.

]]> <![CDATA[Reply To: Expressing a whole number as a fraction before 5th grade]]> Thu, 15 Jan 2015 02:03:11 +0000 Aaron Bieniek Students will have experience in 3rd grade seeing whole numbers as fractions – more than likely as they experiment with number line diagrams and notice that many fractions can label the same point. On the number line, the whole is the unit interval from 0 to 1. So using the language of the 3rd grade standards I would think that 3/1 would be verbalized as “3 wholes” or “3 units” or “the quantity you get by putting 3 wholes together.” Just like 9/3 is the quantity you get by putting 9 copies of 1/3 together or 7/6 is 7 shares of size 1/6. I would steer clear of language like “3 over 1″ or “4 out of 5″ since it does not support the unit fraction idea that is the root of all the fraction standards.

]]> <![CDATA[Expressing a whole number as a fraction before 5th grade]]> Wed, 14 Jan 2015 18:17:22 +0000 zdeluna4 Hi!

How should fractions equivalent to whole numbers be expressed before 5 grade? For example, how should “3/1” be expressed? If I said “3 divided by one”, I feel it would confuse the students since they aren’t learning the connection between fractions and division until 5.NF.3.

Would “3 over 1” or “3 oneths” be more appropriate?


]]> <![CDATA[Reply To: A-SSE.B.3.c]]> Wed, 14 Jan 2015 13:29:03 +0000 csteadman I am still struggling with this standard. Mostly with the financial context. From the example in the standard raising 1.15^(1/12) reveals a very similar rate as using the compounding formula, (1+ .15/12). When is it more appropriate to use one over the other? Is one more right from a financial aspect? Is this in the APY vs. APR category? I may be over thinking it, but it seems like a tough difference to assess.

]]> <![CDATA[2.NBT.9]]> Tue, 13 Jan 2015 18:59:48 +0000 kelli We’re working on a second grade standards-based report card in my district, and trying to assure that our expectations are in line with the intentions of the math standards. Is it the intention of 2.NBT.9 for students to “explain why addition and subtraction strategies work” verbally? In writing? Or both?

  • This topic was modified 2 years, 7 months ago by  kelli.
]]> <![CDATA[Reply To: 2G1]]> Tue, 13 Jan 2015 03:23:19 +0000 Bill McCallum I think abieniek has answered this as well as I can without further clarification.

]]> <![CDATA[Reply To: Algebra 2 for all?]]> Sun, 11 Jan 2015 05:10:25 +0000 lhwalker I was really interested in the link you shared but it doesn’t open to the article. Can you re-post? I’m working on Algebra I curriculum for our district and have really had to look closely at what our students are doing in 8th grade to capitalize on (review) immediately and move on promptly, scaffolding weak spots along the way. I nearly croaked at the idea of teaching 14-year-olds about rational exponents when they struggle so badly with integer exponents, but then I realized there was a shift in emphasis to usefulness with exponential functions instead of 3xy(2xyz)^3…. I am confident, now, they can do it. There are other topics like piece-wise functions that can be integrated along with a review of linear functions, etc. So far I have 20 extra days each semester to work in more project-based activities.

]]> <![CDATA[Reply To: 2G1]]> Thu, 08 Jan 2015 03:35:41 +0000 Aaron Bieniek I’m a little unclear as to what you are asking here. 1.G.2 mentions cylinders but only to the extent that they might be used to help kids learn to perceive combinations of shapes as a single new shape, or to decompose a combination of shapes into its “original” shapes. I don’t see anything here which hints at attributes, much less attributes of a complicated shape like a cylinder. 1.G.1 is about attributes but only in the sense of defining versus non-defining attributes.

In 2.G.1 we dive a little deeper into those defining attributes from 1.G.2 but I’m thinking that the standard purposefully limits those attributes and shapes to mostly 2D figures and simple attributes like number or sides and number of angles. Including cubes makes sense as an introduction to faces since they are so familiar with the shape. It doesn’t seem to me that cylinders have any place in this standard.

]]> <![CDATA[Reply To: 5.NBT.7]]> Thu, 08 Jan 2015 03:07:38 +0000 Aaron Bieniek The expectations for decimals is limited to thousandths so the largest products would be tenths by hundredths. Then, 5.NBT.6 sets the limits for division: up to four digit dividends and two digit divisors. So, if both the dividend and divisor are decimals, the dividend could be to the hundredths place and four digits, like 34.29 and the divisor could also be to the hundredths place but 2 digits, like .27.

It seems to me that the expectation is for 34.29 / .27 to be handled just like 3429 / 27 with the added step of thinking about the placement of the decimal point. Thinking about the placement of the decimal point is a strategy based on place value. Maybe in this case, you have already established that dividing by .01 results in a dividend that is 100 times larger and since .27 is equivalent to (.01 x 27) we could first divide by .01 getting 3429, and then divide by 27.

Or another alternative might be thinking about how many hundredths make up 34.29. There are 100 one-hundredths in 1 so there are 3400 + 29 hundredths in 34.29. So now we are thinking about how many 27 hundredths are in 3429 hundredths.

]]> <![CDATA[Reply To: 2.NBT.2 – Skip Counting]]> Wed, 07 Jan 2015 22:19:27 +0000 Aaron I couldn’t get the links to show, nor was I able to edit the post to show the web addresses. I’ll try here, but if it still doesn’t take, I’d have to show them some other way.

]]> <![CDATA[Reply To: 2.NBT.2 – Skip Counting]]> Wed, 07 Jan 2015 22:01:23 +0000 Aaron The only “unpacking document” that I managed to come across was North Carolina’s resource material, which states that 2nd graders are expected to “count on from any number,” but only in the context of counting by 1s. Thereafter, skip-counting is explained in a way that favors starting with a multiple of the increment skipped.

However, I did come across a couple examples where skip-counting is not confined to multiples only. Both are skip-counting worksheet generators that allow for the listing of non-multiples of the skipped increment. The second site is particularly telling, because the example worksheet that it shows is a list of non-multiples.

Seeing that “skip-counting” is actually a broader term than the intended meaning in 2.NBT.2, would this warrant documentation that the standard does not require skip-counting from a non-multiple?

(I had trouble getting the links to show, so I apologize if they are missing. I could show them otherwise, if necessary.)

  • This reply was modified 2 years, 7 months ago by  Aaron. Reason: links didn't show
  • This reply was modified 2 years, 7 months ago by  Aaron.
]]> <![CDATA[Labeling Representations]]> Wed, 07 Jan 2015 16:16:28 +0000 maryinvt I am looking over the document “Standards for Mathematical Practice:
Commentary and Elaborations for K–5”. We are having a debate about the following statement: “In using representations, such as pictures, tables, graphs, or diagrams, they use appropriate labels to communicate the meaning of their representation.”

Here’s our question: For a label to be appropriate, would units be required in the representation? For example, if a student were solving a problem in which a rectangle is described as being 12 ft by 30 ft, would their representation of the rectangle need to include ‘feet’ in the label or would just 12 and 30 suffice? If it were simply labeled with the quantities 12 and 30, would this be an example of decontextualizing a situation (as long as it were contextualized later in the solution)?

]]> <![CDATA[Reply To: Algebra 2 for all?]]> Tue, 06 Jan 2015 15:23:57 +0000 Sarah Stevens Hi again! I recently came across this blog post by Steven Leinwand addressing many of the frustrations I have about the high school standards.

I definitely feel like the high school standards are too broad. It is frustrating when my K8 co-workers talk about how focused the standards are and I have to clarify that the High School standards do not share the same focus. For us, there is more. Are there plans to revise the High School Standards. Many states have 7 year adoption cycles for standards so they are beginning the process of re-adopting standards. I fear if the states undertake revisions in isolation, we will lose the combined force we gained when math classes were the same across the country.

]]> <![CDATA[Reply To: Geometry Progressions]]> Mon, 05 Jan 2015 22:38:32 +0000 GKelemanik Very eager to read the 7-12 Geometry Progressions Document. Any update on when a draft will become available??

]]> <![CDATA[HSG.GMD.A.1, A.2]]> Tue, 30 Dec 2014 04:50:03 +0000 BobT “CCSS.Math.Content.HSG.GMD.A.1
“Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

“(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.”

I am reviewing the CME Integrated II math text published by Pearson. The writers do implement Cavalieri’s principle to calculate the volume of a sphere. It seems quite awkward compared to Archimedes’ solution in his Method. Is there any background for the decision to use Cavalieri’s principle in the GMD Standard? Thanks for any perspectives.

]]> <![CDATA[Reply To: A-SSE.B.3.c]]> Tue, 23 Dec 2014 22:42:06 +0000 lhwalker The answer to my questions is no, but I skim too fast. I see that instruction in exponents in the Common Core took a huge shift toward relevance instead of (xy)^2z^3(x^2)y. That makes perfect sense!

]]> <![CDATA[A-SSE.B.3.c]]> Sat, 13 Dec 2014 20:22:45 +0000 lhwalker Besides the example in the Standard itself, this is the only example in Illustrative Mathematics:

I’m wondering if these aren’t a bit much for Algebra I students just beginning to understand rational exponents. Am I being a wimp?

]]> <![CDATA[5.NBT.7]]> Sat, 13 Dec 2014 03:49:05 +0000 learningwish What size of numbers does standard 5.NBT.7 refer to? Students are not called to divide a fraction by a fraction until 6th grade, but should they divide a decimal by a decimal? Up to how many digits should the divisor and the dividend be? What is reasonable using strategies based on place value?
7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

]]> <![CDATA[Reply To: 2G1]]> Fri, 12 Dec 2014 19:49:06 +0000 bergba Here is the confusion my primary teachers (and myself) are having with attributes of a cylinder.

Sometimes on the cylinder we see that it has 2 faces, 2 edges, and 0 vertices. Other times we see that it is 0 faces, 0 edges, and 0 vertices. We can’t find a clear definition in the standards or anywhere else.

There are mixed messages with how to define a “face” in geometry, therefore,….confusion lies within cylinders!

First question regarding this:
*In 1st grade, students compose two-dimensional shapes or 3-D shapes (including right circular cylinders) to create a composite shape.
*2nd Grade, When we are discussing various attributes in 2.G.1, one of the attributes is the number of faces. However, a cylinder is not mentioned in this specific standard.

Insights? How does CC define faces and attributes of cylinders?
Feeling like I need a better understanding of this progression and the expectations for this topic.

]]> <![CDATA[Math for kids – free]]> Fri, 05 Dec 2014 18:59:19 +0000 dejanmolnar On website [url=][/url] are prepared math exercises for 6-9 years old children. Website verifies the correctness of solution.

What can children find on this website?

Numbers up to 5: counting, patterns, compare numbers, addition, subtraction, …

Numbers up to 10: compare numbers, addition, subtraction, …

Numbers up to 20: patterns, compare numbers, addition, subtraction, …

Numbers up to 100: compare numbers, predecessor, successor, even and odd numbers, addition and subtraction (without regrouping and with regrouping), multiplication and division, mixed operations, …

Numbers up to 1000: comparing numbers, predecessor, successor, addition and subtraction (vertical form), multiplication and division, mixed operations, …

Geometry: geometric shapes, geometric bodies

Using website is free of charge.


]]> <![CDATA[Reply To: Functional relationships (F-IF.1)]]> Tue, 25 Nov 2014 17:19:54 +0000 Bill McCallum Kristin has pretty much said it all. Here are a couple more thoughts. The vertical line test and tables of ordered pairs are tools in service of the understanding expressed in your second quotation above. As long as they remain tools, that’s fine (although personally I think the vertical line test is excessive codification of a simple visual observation). The first quotation you give refers to “time normally spent on exercises” on the vertical line test or tables of coordinate pairs. The key words here are “time” and “exercises.” Once a tool becomes the subject of a set of exercises devoted specifically to it, it becomes a topic in its own right, disconnected from the understanding it originally served (as Kristin says in her last paragraph).

]]> <![CDATA[Reply To: Functional relationships (F-IF.1)]]> Tue, 25 Nov 2014 14:41:42 +0000 Kristin Umland A couple thoughts:

* First, remember that because something is not in the standards doesn’t mean teachers can’t address it. The main point about these topics not being in the standards is that the assessment folks should not be writing items that test whether students can apply the vertical line test or pick out sets of ordered pairs with a particular property. These are procedures that aren’t very interesting when extracted away from their reasoning purpose.

* However, students should be able to look at a graph of x=y^2 and note that, for example, the value x = 1 corresponds to y = 1 and also y = -1, so it is not the graph of a function. Note that this is not the same kind of argument as applying the vertical line test, because it connects back to the definition of a function. The argument, “The line x=1 intersects the graph in two places so the graph is not of a function” is a black-box explanation for most students–they are told that you do such-and-such, and you interpret the results in some way–it is like reading tea leaves or consulting the oracle, but does not constitute mathematical reasoning.

The problem with standard questions about functions that ask students to employ the vertical line test or to look at ordered pairs is that students almost never realize that these are fundamentally the same kind of investigation: if you took the list of ordered pairs and plotted them in the coordinate plane, applying the vertical line test amounts to the same thing as inspecting the ordered pairs and looking for x-values that correspond to different y-values. In other words, for most students, these are completely disconnected procedures rather than different manifestations of the same kind of mathematical argument, one that relies on the definition of a function to determine if a relation is a function. We want students to be able to reason from the definition of a function to determine if a relation is a function; we don’t care if they can enter the correct letter when prompted, “Apply the vertical line test and mark y or n for whether the graph shown is the graph of a function.”

]]> <![CDATA[Reply To: A-REI.1]]> Tue, 25 Nov 2014 14:29:06 +0000 Bill McCallum Yes, I agree it is an important standard. One way not to implement it would be to get too bogged down in formality and terminology (like insisting that students keep referring to the properties of equality by name, for example). I would have students get in the habit of talking through their solutions:

“If $x$ is a number such that $x^2 – 3x – 4 = 0$
then $(x-4)(x+1) = 0$ because $x^2 – 3x – 4 = (x-4)(x+1)$ no matter what $x$ is.
for all $x$ (by the distributive law). This means that either $x-4=0$ or $x+1=0$, so $x =4$ or $x=-1$. ”

At first I would want students simply to understand that solving an equation is a flow of if-the statements; then I would start asking why each step was correct (distributive property, zero-factor property). And then I would raise the question of the converse: you’ve shown me that if $x$ is a solution to the equation it has to be 4 or $-1$, but does that tell me that 4 and $-1$ have to be solutions? How do I know they are solutions?

Maybe one of these days I will write a blog post on this.

]]> <![CDATA[Reply To: CN.7 vs REI.4b]]> Tue, 25 Nov 2014 14:08:36 +0000 Bill McCallum This is a case where coherence trumped conciseness in writing the standards. The skill and the piece of knowledge named by these two standards are the same, but the context is different. In A-REI.4b the context is solving equations, and you want students to know that sometimes solving equations leads to complex numbers. In CN.7 the context is the complex numbers as a system, and you want students to know that they can sometimes provide solutions to equations that didn’t have solutions before. These are two different understandings, two different ways of viewing the same piece of knowledge, and approaching that piece of knowledge from both sides seemed worthwhile.

]]> <![CDATA[Reply To: REI.7 – A circle "being" a quadratic]]> Tue, 25 Nov 2014 13:58:20 +0000 Bill McCallum Sorry, I don’t know how I missed this one, thanks for bringing it to the front again. Yes, the equation for an ellipse or a hyperbola is a quadratic equation, so is included in this standard.

]]> <![CDATA[Reply To: How do you measure fluency?]]> Tue, 25 Nov 2014 13:55:46 +0000 Bill McCallum In the standards, “fluently” means “fast and accurate.” So assessing fluency is always going to involve some observation of how long it takes a student to do a calculation. As abienek points out, that observation could be made by the teacher listening to a student solve a problem; or, it could also be made by a timed test. In talking to teachers I’ve found that timed tests are a very controversial topic: some people love them, others hate them. I don’t have a strong opinion either way. I do think that if you use timed tests you should try to make the fun and competitive, not scary and stress-inducing. That’s possible in athletics, so it should be possible in math. I also think they are probably not necessary in a classroom where students are explaining their solutions a lot; you can probably tell pretty well from that who is fluent and who is not.

]]> <![CDATA[Functional relationships (F-IF.1)]]> Wed, 19 Nov 2014 08:17:18 +0000 tomergal I have a problem reconciling the following two passages from the progression document:

Notice that a common preoccupation of high school mathematics, distinguishing functions from relations, is not in the Standards. Time normally spent on exercises involving the vertical line test, or searching lists of ordered pairs to find two with the same x-coordinate and different y-coordinate, can be reallocated elsewhere.


The essential question when investigating functions is: “Does each element of the domain correspond to exactly one element in the range?”

I have no problem disposing with relations or with the vertical line test, but the first passage goes past that and suggests avoiding going over ordered pairs. The second passage does seem to suggest we want students to look for domain elements that are associated with more than one unique range element.

Does the restriction from the first passage apply to other representations of associations? Such representations may be a table of corresponding values, a graph, an equation, or a verbal description. The quantities in question can be abstract and they can be a part of a real world context. In every case we can ask whether any of the quantities is a function of the other. Are some of the cases I mentioned problematic in that manner? If so, could you please elaborate why?

]]> <![CDATA[A-REI.1]]> Mon, 17 Nov 2014 10:45:26 +0000 tomergal This is an extremely important standard in my opinion. However, I find it hard to implement without getting into quite meticulous logical considerations. I wonder if that’s expected, if you have any helpful tips for implementation, or whether you can refer me to good sources on this subject.

Furthermore, I wonder what you think about reasoning with *inequalities*. This is absent from the standard although it seems that, like more complicated equations (e.g. quadratic), inequalities can shed more light on the importance of reasoning with algebra.

]]> <![CDATA[Reply To: How do you measure fluency?]]> Sun, 16 Nov 2014 23:38:38 +0000 Duane I’m sure Bill will add his own opinion again if necessary, but there was some talk of fluency and “knowing from memory” back in this discussion:

]]> <![CDATA[CN.7 vs REI.4b]]> Thu, 13 Nov 2014 18:45:36 +0000 dhust Help me to see the difference.

REI.4b (last part)
Cluster: Solve quadratic equations in one variable.
Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

This seems to tell me that they need to understand the discriminant and then write the solution as a complex number.

Solve quadratic equations with real coefficients that have complex solutions.

This seems to tell me that they are solving quadratic equations and then writing the solutions as complex numbers.

It seems to me that REI.4b completely encompasses CN.7. What am I missing?


]]> <![CDATA[Reply To: REI.7 – A circle "being" a quadratic]]> Thu, 13 Nov 2014 18:37:17 +0000 dhust Anyone?

]]> <![CDATA[Reply To: How do you measure fluency?]]> Thu, 13 Nov 2014 05:58:04 +0000 Aaron Bieniek I would say that fluently refers to how you do a calculation and know from memory means being able to produce an answer without having to do a calculation. As a third grader if I can multiply 13 X 7 by reasoning that 10 x 7 is 70 and 3 x 7 is 21 so 13 x 7 = 91, and I can apply that strategy consistently and accurately (without hesitating and fumbling through it), then I would argue that I am fluent. I get that timed tests are easy to give and easy to score, but I don’t get how a timed test would give you much information on the strategies the students are using and whether they can use those strategies efficiently and accurately (fluently).

And as far as the knowing from memory, I guess you could use a timed test to measure that, but my concern is the overemphasis on timed tests that might send the unintended message that somehow math is about speed. I’d rather that kids get the message that math is a creative endeavor first, and then learn lessons along the way that efficiency matters also.

As an alternative to timed tests, I would submit that listening to a kid solve a problem involving the “know from memory” facts would tell you everything you need to know about how much they know from memory and how much they have to compute.

]]> <![CDATA[How do you measure fluency?]]> Thu, 13 Nov 2014 00:04:03 +0000 ginger11772 In the standards (3.OA.7) it states, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers.” How do you measure “fluency” without timing the students? Also, if you do time students on these facts, then what time constraints and/or amount of problems are appropriate. We as teachers still have to give grades. HELP!!!

  • This topic was modified 2 years, 9 months ago by  ginger11772.
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]]> <![CDATA[Reply To: Expressions with unknowns]]> Mon, 10 Nov 2014 18:33:38 +0000 Aaron Bieniek Perhaps what you are asking about is mostly contained in 6th grade, in 6.EE.3. Students apply the properties of operations to generate equivalent expressions. In the case you mention, students would use the distributive property to rewrite 5x – 3x as
(5 – 3)x, and then 2x.

In grade 7 students build on that in 7.EE.1 by including expressions with rational coefficients.

]]> <![CDATA[Reply To: 4.NBT.1 and 5.NBT.1 – Only 1 place value to the right/left?]]> Mon, 10 Nov 2014 18:02:54 +0000 Priya Thank you both! This is extremely clarifying!

]]> <![CDATA[Reply To: 4.NBT.1 and 5.NBT.1 – Only 1 place value to the right/left?]]> Mon, 10 Nov 2014 17:58:49 +0000 Priya This is really clarifying! Thank you both so much!

]]> <![CDATA[Expressions with unknowns]]> Sun, 09 Nov 2014 00:46:59 +0000 ronistol By the end of the 7th grade, do students need to simplify expressions with unknowns and negative numbers such as 5x-3x=2x ?
Or do they only need to evaluate numeric expressions without unknowns (5-3=2)?

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Tue, 04 Nov 2014 13:59:00 +0000 Aaron Bieniek …but they should not be taught that simplification is a mathematical imperative.

]]> <![CDATA[Reply To: Solutions in set notation]]> Tue, 04 Nov 2014 00:26:31 +0000 Bill McCallum That’s a curricular decision, not a requirement of the standards. Personally I’m with you … I think the set notation is too much baggage at this level.

]]> <![CDATA[Reply To: Writing rational functions]]> Tue, 04 Nov 2014 00:13:50 +0000 Bill McCallum Yup.

]]> <![CDATA[Reply To: Typo?]]> Tue, 04 Nov 2014 00:13:15 +0000 Bill McCallum Good eye! It has been fixed in my files, but apparently I didn’t upload the latest version. Will try to get to that.

]]> <![CDATA[Reply To: Is dimensional analysis part of Math 6?]]> Tue, 04 Nov 2014 00:09:59 +0000 Bill McCallum I think Grade 6 is early for dimensional analysis. That comes in in the high school standard:

N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

]]> <![CDATA[Reply To: 7.NS.3 – What is "difference" when comparing integers?]]> Tue, 04 Nov 2014 00:06:55 +0000 Bill McCallum I agree the problem is weak, and I agree with Alexei’s analysis of 7.NS.A.1 and 7.EE.B.3. I would only add that grouping standards into cluster headings, which carry important meaning on their own, has the effect that you sometimes see an overlap in the meaning between two standards. So, the cluster 7.NS.A is about operations with rational numbers, and that includes problem solving, whereas 7.EE.B is about problem solving, and that includes working with rational numbers.

]]> <![CDATA[Reply To: Integers]]> Mon, 03 Nov 2014 23:59:22 +0000 Bill McCallum Lane, I think this is a useful way of helping students remember the rules, and that’s especially needed for remedial students. There’s a bit of sweeping under the rug going on here, because if you spell it out, you are saying that

(opposite of 2) times (opposite of 3)

is the same as

opposite of (2 times (opposite of 3)).

We can actually prove this using the distributive property, because that property tells me that

(opposite of 2) times (opposite of 3) plus 2 times (opposite of 3)

is the same as

((opposite of 2) + 2) times (opposite of 3)

which is just zero times (opposite of 3), namely zero.

But if I add two numbers and get zero, they must be opposites!

Of course, I’m not suggesting that you have to go over all this with your remedial students!

]]> <![CDATA[Reply To: Include: All inclusive or examples?]]> Mon, 03 Nov 2014 23:50:14 +0000 Bill McCallum Take two intersecting lines. Rotate them through 180 degrees around the point of intersection. Each line is taken to itself, and the point of intersection stays put, so each one of the pair of vertical angles is taken onto the other one. Therefore they are congruent (because there is a rigid motion which takes the one to the other).

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Mon, 03 Nov 2014 23:45:55 +0000 Bill McCallum Oh, one more point: although simplification is de-emphasized, transforming between equivalent forms is not. That is, students should be able to recognize and produce equivalent fractions (4.NF.1); but they should be taught that simplification is a mathematical imperative.

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Mon, 03 Nov 2014 23:42:22 +0000 Bill McCallum Alexei is correct here. Mixed numbers, fractions, reduced fractions are all different ways of writing the same number. The focus in the standards is on the number itself, not how you write it. Of course, sometimes it might be convenient to write the number in a particular way. But that goes both ways: you might want to write 7/14 as 1/2, but you might equally want to write 2/5 as 4/10 (to see the connection with decimal notation). Choosing a convenient form for the purpose at hand is an important skill, as is the fundamental understanding of equivalence of forms.

And yes, I think both the PARCC and Smarter Balanced assessments will reflect this philosophy. Indeed, if they marked an equivalent form wrong they would themselves be wrong, since a number is a number is a number; if you get the answer right, you should get the points.

]]> <![CDATA[Reply To: Graphing Inequalities]]> Mon, 03 Nov 2014 23:36:42 +0000 Bill McCallum Well, if you introduce the ≤ and ≥ symbols then you should also introduce the open and closed dot notation for graphing. The standards don’t call for either of these things, however; it’s really a curricular decision whether to introduce them. Certainly the focus in middle school is on equations, not inequalities.

]]> <![CDATA[Reply To: A.SSE.2]]> Mon, 03 Nov 2014 23:32:43 +0000 Bill McCallum It certainly falls under the standard, although it is an advanced example. I wouldn’t call it an expectation.

]]> <![CDATA[Reply To: 4.NBT.6 – Division Remainders]]> Mon, 03 Nov 2014 23:31:28 +0000 Bill McCallum I understand the desire for a compact notation here, but I don’t really have a great idea, and I would worry that introducing one would lead us into the same issues as 2R1. I think it is best to stick to something like a verbal statement “3 goes into 7 two times with a reminder of 1” and then express this mathematically using the equation 7 = 2 x 3 + 1.

]]> <![CDATA[Reply To: 4.NBT.1 and 5.NBT.1 – Only 1 place value to the right/left?]]> Mon, 03 Nov 2014 23:01:14 +0000 Bill McCallum I think Abieniek has it right, as usual!

]]> <![CDATA[Reply To: special right triangles]]> Mon, 03 Nov 2014 22:58:10 +0000 Bill McCallum I think the point of studying these special triangles is that it is an exercise in reasoning with the Pythagorean theorem in the context of certain special triangles (equilateral, right isosceles). It’s a bit of a miracle that you can express the trigonometric ratios of these angles exactly using square roots, and I wouldn’t want that miracle to get lost in drill, with our without technology.

]]> <![CDATA[Reply To: 8.G.3 Coordinate Notation?]]> Mon, 03 Nov 2014 22:51:47 +0000 Bill McCallum Are you talking about using matrices, or equations in two variables, to describe the coordinates? I think that is beyond Grade 8. The intent of the standard is that they should be able to give the coordinates of the transformed figure.

]]> <![CDATA[Reply To: Problem Solving Question]]> Mon, 03 Nov 2014 22:49:21 +0000 Bill McCallum I imagine you’ve written this test already by now! But Smarter Balanced is coming out with some new test specs soon that should have some examples.

]]> <![CDATA[Reply To: Writing rational functions]]> Sat, 01 Nov 2014 17:02:35 +0000 lhwalker I think I just answered my own question: A.CED.a.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

]]> <![CDATA[Solutions in set notation]]> Sat, 01 Nov 2014 05:00:22 +0000 lhwalker I noticed that EngageNY uses set notation for solutions: {x real | x>0} Is it expected that Algebra I students recognize that or would x>0 suffice?

]]> <![CDATA[Writing rational functions]]> Fri, 31 Oct 2014 14:59:59 +0000 lhwalker I don’t see where creating a function to model inverse variation is in the Standards. Is it there somewhere with different wording?

]]> <![CDATA[Typo?]]> Thu, 30 Oct 2014 23:12:38 +0000 dyong Did anyone already mention or catch the typo on page 9? I think it should be


instead of


Super minor!

]]> <![CDATA[Is dimensional analysis part of Math 6?]]> Thu, 30 Oct 2014 05:23:09 +0000 jmasia I was looking into the ratio and proportions progression in math 6. I know that 6rp3 talks about unit convention as part of the standard. I also know that dimensional analysis uses ratio equivalences to perform conversions. Is it an appropriate tool at that grade level and if so can you advise on how to approach it. Thanks.

]]> <![CDATA[Reply To: 7.NS.3 – What is "difference" when comparing integers?]]> Wed, 29 Oct 2014 17:30:37 +0000 Aaron Bieniek So, I’m with you that this seems like a kind of weak item to assess 7.NS.3…

As far as your questions, I don’t think any standard stands alone, and I also don’t think that 7.NS.3 is a catch all standard either. 7.NS.1-2 ask for some very specific understandings. In 7.NS.1 we are describing situations that make 0, understanding p + q on a number line and interpreting sums by describing contexts, understanding subtraction as additive inverse and representing on a number line, and applying properties of operations to add and subtract. In 7.NS.2 we apply the distributive property to rational products and interpret products and quotients by describing contexts, and deal with the -(p/q) subtlety. Then we apply properties of operations to multiply and divide and finally convert rational numbers to decimals.

In all of that we still haven’t gotten to what 7.NS.3 asks for, which is solving real-world and mathematical problems using any of the operations and probably combinations of them and using all sorts of rational numbers. I’m not sure the focus of 7.NS.1-2 is problem solving like it is in 7.NS.3.

For your second question, I would say yes. All of these standards fall under the same cluster heading and I would imagine that as the students solve problems in 7.NS.3, they will draw on the understandings that came out of the rest of the standards in this cluster.

So, it looks like there are 2 correct answers since differences can be positive or negative. I might take (a) or (d) depending on how the student justifies their choice. Of course, I don’t think I would use this without revising the prompt…

]]> <![CDATA[7.NS.3 – What is "difference" when comparing integers?]]> Wed, 29 Oct 2014 00:59:38 +0000 rmarcelo I was reviewing an item that was “aligned” to MACC.7.NS.3 and it read something like this:
One liquid is at -8 degrees Celsius and a second liquid is at 14 degrees Celsius. What is their difference in degrees?
a) -22
b) 6
c) -6
d) 22
Before i establish my position, a couple of things. 1) Is 7.NS.3 a “catch all” standard in NS or is it a stand alone standard? 2) Should one consider 7.NS.1 when developing items for 7.NS.3?
What is the correct answer and why?

]]> <![CDATA[Reply To: Integers]]> Mon, 27 Oct 2014 03:37:29 +0000 lhwalker I was just watching your Khan video for multiplying and dividing negative numbers. My remedial Algebra students gain solid traction with multiplying negative numbers like this:

-3 lost 3 friends
2(-3) lost 3 friends twice
-2(-3) the opposite of losing 3 friends twice.

The added benefit is that “opposite” connects with -x.

The emotional connections help with retention.

]]> <![CDATA[Reply To: 6.EE.8 Inequalities]]> Sun, 26 Oct 2014 21:58:27 +0000 lhwalker Great question. I asked the same thing! Here’s what Dr. McCallum said:

Compound Inequalities

]]> <![CDATA[6.EE.8 Inequalities]]> Wed, 22 Oct 2014 19:57:36 +0000 kirkkimb We are introdcuing inequalities using real world problems however, if we present a problem such as “A student can carry up to 1/5 of their weight in their back pack.” Our concern is the inequality w is less than or equal to 1/5 reflects all values less than 1/5 including values less than zero. Is this where we should introduce compound inequalities within our discussion of what numbers satisfy the description? Thanks for your help.

]]> <![CDATA[Reply To: Include: All inclusive or examples?]]> Sat, 18 Oct 2014 22:29:06 +0000 jamiechaikin Will you please provide an example using rigid motions to show that vertical angles are congruent? Thanks.

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Mon, 13 Oct 2014 19:30:08 +0000 Alexei Kassymov My understanding is that simplifying is really deemphasized in the CC (for example, the quote below for the progressions document). PARCC seems to automatically detect equivalent forms of numbers in the answers. Also, one of their released items uses 5/15 as an outcome that students need to evaluate for value (not representation) reasonableness. Appears that they are not going to rely on simplification as a source of errors to catch.

“It is possible to over-emphasize the importance of simplifying fractions
in this way. There is no mathematical reason why fractions
must be written in simplified form, although it may be convenient to
do so in some cases.”

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Sat, 11 Oct 2014 22:40:56 +0000 mrsmstweet This is the question I was just looking for an answer to online! So I still need some clarification. If you have changed to a ‘purely fractional form’ does that mean in older terminology that you have changed it to an ‘improper fraction’? Also, let’s say students had to do that step and then they subtract to find the solution… what do they do next? Should they change back to a mixed number… should they simplify and if so first or last… I basically feel that kids needs to know how to make sense of fractions and interchange them as needed especially since with TESTING… many questions will have multiple choice answers. What if the PARCC assessment has a LCD answer or a simplified answer? I hope I’m making sense! I just found your blog and it’s super helpful!

]]> <![CDATA[Reply To: Zero Pairs]]> Sat, 11 Oct 2014 02:57:45 +0000 Aaron Bieniek I think 6.NS.5 is the standard you are looking for:

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

]]> <![CDATA[Reply To: 6.EE.7]]> Sat, 11 Oct 2014 02:56:25 +0000 Aaron Bieniek I think 6.NS.5 is the standard you are looking for:

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

]]> <![CDATA[Graphing Inequalities]]> Fri, 10 Oct 2014 17:42:26 +0000 alwong Hi Bill,

In 6th grade, when we graph inequalities, do we show the open dot and closed dot? Or do we just do greater than or less than?

]]> <![CDATA[Zero Pairs]]> Fri, 10 Oct 2014 17:31:07 +0000 alwong Hi Bill,

I understand that we will be teaching 6th grade students x-2=5. My questions is: how do we go about teaching it if students are not familiar with the concept of zero pairs? (-2+2)

]]> <![CDATA[Reply To: 6.EE.7]]> Fri, 10 Oct 2014 17:29:29 +0000 alwong Hi Bill,

I understand that we will be teaching 6th grade students x-2=5. My questions is: how do we go about teaching it if students are not familiar with the concept of zero pairs? (-2+2)

]]> <![CDATA[Reply To: 6.EE.7]]> Fri, 10 Oct 2014 17:24:22 +0000 alwong Hi Bill,

I understand that we need to teach x-2 = 5 to our 6th grade students. My question is, how would you go about teaching it if the students don’t have the concept of zero pairs (-2+2)?

]]> <![CDATA[A.SSE.2]]> Wed, 08 Oct 2014 18:09:31 +0000 lhwalker Are students expected to recognize that non-perfect squares such as x-9 can be written as a difference of squares?

]]> <![CDATA[Reply To: Vertex of a parabola]]> Tue, 07 Oct 2014 20:44:43 +0000 lhwalker I would write (x-h)^2 + k Using h and k is pretty much textbook standard for shifts on conics (although students need to know variables are variables). Since h is the horizontal shift from the origin, the line of symmetry runs through x=h

]]> <![CDATA[Reply To: 4.NBT.6 – Division Remainders]]> Tue, 07 Oct 2014 19:02:31 +0000 MathCoach5 I am wondering if there is a way we can help teachers understand how to record the answer to a division problem with a remainder so that it will make sense to students. Most often, a division problem is being solved as a result of being in context. Therefore, we could interpret the remainder in terms of the context and not cause confusion with how to record our quotient. However, if a student is asked to solve a division problem out of context, such as the example above, is there a way that would be mathematically correct to record the answer symbolically other than using 2 1/3? Would you want students to write 7 = 2 x 3 + 1 as an equation to represent the solution to 7 ÷ 3? I understand why 2 R1 is not acceptable, but am looking for a way to help students see a connection to their work and the correct mathematics. Any thoughts would be appreciated.

]]> <![CDATA[Reply To: 3rd Grade Multiplication]]> Sun, 05 Oct 2014 18:10:24 +0000 Aaron Bieniek If I understand your question, Bill answered it in his reply to dcoker above. Multiplication within 100 is defined in the glossary of the standards (p.86) to mean “Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0–100.”

So yes, your examples of 15 x 5 and 14 x 3 are definitely appropriate. Something like 7 x 19 would not be.

]]> <![CDATA[Reply To: 3rd Grade Multiplication]]> Sun, 05 Oct 2014 02:45:01 +0000 Nic Given that 3rd graders are expected to use the distributive property in 3.MD.7 to solve problems involving area wouldn’t multiplication within 100 also include 2-digit by 1-digit problems such as 15 x 5, 14 x 3 etc. where students use the distributive property to solve? e.g. 15 x 5 = (10 x 5) + (5 x 5)
= 50 + 25 = 75

]]> <![CDATA[Reply To: Details about 5.OA.3]]> Sat, 04 Oct 2014 03:55:58 +0000 Bill McCallum I agree with abieniek here. Basically this standard is a prelude to understanding relationships between two quantities. If you generate two different patterns, you can then compare them and see how they vary together. Without actually seeing the NC example, I can imagine on person catching 2 fish per day, the other catching 1 fish per day, and then you would notice that the numbers for the first person are twice the number for the second person. Later, in Grade 6, students will make tables of ratios expressing the same fact, and in Grade 7 they will record this using the equation y = 2x. At any rate, in Grade 5, they should be recording the ordered pairs made by pairing the number of fish caught by person 1 versus the number of fish caught by person 2; not the two lines graphing each person’s catch as a function of the number of days (which would come much later).

]]> <![CDATA[Reply To: 4.NBT.1 and 5.NBT.1 – Only 1 place value to the right/left?]]> Fri, 03 Oct 2014 20:03:20 +0000 Aaron Bieniek To me, it looks like 4.NBT.1 is where we start to generalize place value understanding, so it makes sense to start with adjacent digits. Since 5.NBT.1 is still dealing with adjacent digits (although, now both to the left and the right) it seems like your example of 7000 being 100 times the value 70 would not be a typical problem in grade 4.

5.NBT.2, however, seems like the place that we begin to explore multiplying by more than 10 “…multiplying a number by powers of 10,…” So, using your second example, students might explain why when 16.915 is multiplied by 100, the result is 1691.5 and maybe express 1691.5 as 16.915 x 10^2

]]> <![CDATA[Reply To: Decimals]]> Fri, 03 Oct 2014 19:36:00 +0000 Aaron Bieniek I just looked at the Pet Shop task and this was just added to the comments:

    Mathematics Specialist wrote this public comment about 8 hours ago

Is the decimal point in the money notation appropriate for 2nd grade? Decimals in place value are not introduced until 4th grade. In referring to the Progressions and to the comments in the Forum, it seems as if the intent of this standard is to use the dollar sign and cent sign appropriately, not to work with decimal numbers.

    Kristin Umland wrote this public reply about 3 hours ago

You are so right–thank you for catching that! The task and solution are fixed now.

It looks like Choices, Choices, Choices was fixed as well.

]]> <![CDATA[Reply To: Decimals]]> Fri, 03 Oct 2014 16:09:59 +0000 traceyd I would like a clarification of which grade level is the first grade level to introduce and work with decimals. It is my understanding that it is in 4th grade. As a person that provides professional development to teachers, the Progressions, Illustrative Mathematics, and this Forum are where I refer teachers to clarify their understanding of what the Standards are asking the students to learn and what it might look like. There are two Illustrative Tasks for 2nd grade that I am concerned send the message to teachers that they are to add $1.35 + $1.25, “Pet Shop” and “Choices, Choices, Choices”. I would like to teach the teachers the correct expectations for money and the use of decimals in 2nd Grade. Thank you!

]]> <![CDATA[Reply To: Vertex of a parabola]]> Thu, 02 Oct 2014 19:43:05 +0000 Alexei Kassymov Lane, were thinking of students doing something like this to find the line of symmetry?
(x – a)^2 + k is the quadratic with the square completed.
Then they solve
(x – p – a)^2 + k = (x + p – a)^2 + k for x?

]]> <![CDATA[4.NBT.1 and 5.NBT.1 – Only 1 place value to the right/left?]]> Thu, 02 Oct 2014 15:25:29 +0000 Priya My colleagues and I have long been wondering about whether it is outside the scope of the standards to move multiple place values in 4.NBT.1 and 5.NBT.1.

Thinking for assessment purposes, is asking a student to recognize that 7,000 is 100 times the value of 70 out of the realm of 4.NBT.1? The progressions seem to imply that the standard is really calling for students to only recognize that the place value to the immediate right of a given place value is 10 times that value.

For 5.NBT.1, I certainly feel less clear about whether moving more than one place value is acceptable. For example, if I were to give students a task in which they need to recognize that the 5 in 16.915 is 1/100 the value of the 5 in 182.54, is this going beyond the scope of the standard? 5.NBT.2 seems to imply that students should understand the effect of multiplying or dividing by more than one place value, but some of my colleagues feel that 5.NBT.1 is strictly one place value to the left or right.

Would love others’ thoughts on this.

]]> <![CDATA[Reply To: Geometry Progressions]]> Wed, 01 Oct 2014 13:15:52 +0000 Aaron Bieniek Post it! Can’t wait to read it!

]]> <![CDATA[Reply To: Geometry Progressions]]> Wed, 01 Oct 2014 04:09:03 +0000 Bill McCallum We have a draft!

]]> <![CDATA[special right triangles]]> Fri, 26 Sep 2014 19:24:09 +0000 idomath We can see that F.TF.3 is a natural place to have some prior experience with 30-60-90 and 45-45-90 triangles. Traditionally, a fair amount of time might be devoted to solving right triangles of this nature and developing fluency so that students are quick to do so without the aid of technology. Is this still appropriate? Do we also expect they might use this skill in finding volume and surface area of special pyramids, for example?

]]> <![CDATA[8.G.3 Coordinate Notation?]]> Fri, 26 Sep 2014 14:07:04 +0000 bbaggett 8.G.3 reads: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Does this standard go as far as students should be able to record these movements in coordinate notation? And…should we go so far as to showing the students the coordinates of the original figure and the coordinates of the image and having them record in coordinate notation what happened to the coordinates?

]]> <![CDATA[Problem Solving Question]]> Thu, 25 Sep 2014 21:27:26 +0000 alwong Hi Bill,

I am trying to create a test that has Problem Solving Problems. I am trying to write at the Claim 2 level at a DOK 2 or 3. I want to write the item based on any of the standards within the cluster 6.NS.C. Any ideas? Thanks!

]]> <![CDATA[Reply To: Details about 5.OA.3]]> Thu, 25 Sep 2014 16:17:06 +0000 Aaron Bieniek It seems to me that the first example in the NC document misses the mark when it asks students to make a line graph and interpret the graph. First off, given the fact that a line graph has Sam catching half a fish, halfway through the first day, I think we can say that a line graph isn’t even appropriate for this example. Second, the emphasis here isn’t interpreting graphs but rather looking for relationships between the corresponding terms of the two patterns. The point of the ordered pairs is to display the values in a way that helps students see the relationships between them, and the graphing supports 5.G.2.

I’m thinking this standard is much less complicated than the NC document makes it out to be.

]]> <![CDATA[Reply To: Verbs in CCGPS Math Standards]]> Thu, 25 Sep 2014 01:01:24 +0000 Bill McCallum Are you talking about the Common Core or some other standards? I suspect the latter because the Common Core doesn’t use the phrase “perform well.” I wasn’t sure what you meant by “pedagogical rationale” or “aesthetic reasons.” If you give me an example of what you are talking about I can try to help.

]]> <![CDATA[Reply To: Vertex of a parabola]]> Thu, 25 Sep 2014 00:57:16 +0000 Bill McCallum I completely agree, Lane.

]]> <![CDATA[Reply To: Standards Assessed in Grade 8]]> Thu, 25 Sep 2014 00:55:48 +0000 Bill McCallum Certainly these are easier to assess in the classroom than on a multiple-choice test … I would still call what the teacher does in the classroom assessment, however. This particularly applies to 8.G.1.

]]> <![CDATA[Details about 5.OA.3]]> Thu, 25 Sep 2014 00:42:24 +0000 noamszoke 5.OA.3 states:
5.OA.3 – Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

The North Carolina unpacked document gives two examples of this standard. In the first example, Sam and Terri catch fish at a different rate. Their information is captured in one table, then graphed in two separate lines, and those are compared. The second example is treated differently. Rather than graph each rule separately, as in the fish problem, they graph the two rules against each other.

Those examples are posted here.

My question: Are 5th graders supposed to see both ways? If so, where do we put the emphasis?

Thank you for your insight!


]]> <![CDATA[Reply To: Geometry Progressions]]> Fri, 12 Sep 2014 13:56:48 +0000 jspencer Also wondering if there is an update on the middle school geometry progressions. Less specific questions, more general guidance.

Thank you!

]]> <![CDATA[Verbs in CCGPS Math Standards]]> Fri, 05 Sep 2014 01:15:14 +0000 KAH I am currently in the process of analyzing our standards for mathematics and was wondering if the verbs used in our standards have a pedagogical rationale or whether the various verbs were used for aesthetic reasons, such as to avoid repetitive text. This is important to me as I look to help educators determine what it means for a student to “perform well” for each standard. Thanks in advance for your time and help.

]]> <![CDATA[Vertex of a parabola]]> Tue, 02 Sep 2014 15:41:18 +0000 lhwalker I am helping with curriculum at my school and noticed EngageNY Algebra I, Module 4, Lesson 8, Graph Vocabulary defines the axis of symmetry as “the vertical line given by the graph of the equation x = -b/(2a).” Memorizing and using that formula does not seem to be required for the unit and it is not in the Standards, so am I correct to assume that formula does not need to even be presented to the students because they can complete the square and derive the equation of the line of symmetry from that form? I’m thinking that formula is part of the mile-wide problem we are trying to address by adopting the CCSS.

]]> <![CDATA[Standards Assessed in Grade 8]]> Mon, 11 Aug 2014 14:28:50 +0000 dcornstubble In grade 8, we have found that 8.G.1 and 8.G.6 in particular are not suitable standards for assessment. We see them as teacher led classroom activities and NOT benchmark or big stakes assessments. Would you please provide an opinion on this statement? Thank you in advance.

]]> <![CDATA[Reply To: Ideas for 7.G.3]]> Tue, 05 Aug 2014 20:48:03 +0000 nathan118 I want to look into making a bunch of playdough, having kids form it into a “cube” (you can get it pretty close by hand), and then let them cut it with a plastic knife or floss or something. Could make them draw the different cross sections they create. Or have them snap a photo and email it to you to show to the class.

]]> <![CDATA[Reply To: Ratio – fractional notation]]> Tue, 29 Jul 2014 14:31:41 +0000 Bill McCallum A ratio is comparison of two numbers, whereas a fraction is a single number. All sorts of problems can arise by confusing the two. For example, suppose I have juice recipe which requires 2 cups of orange juice to 3 cups of peach juice. So the ratio of orange juice to peach juice is 2 to 3. Now I double the recipe, to get an equivalent ratio of 4 to 6. If I identify these ratios with the corresponding fractions, 2/3 and 4/6, then it sounds like I am saying that two times 2/3 is 4/6. A similar problem arises with addition of fractions. Suppose I have one class that has 9 girls and 10 boys, and another class that has 11 girls and 8 boys. So the ratio of girls to boys is 9:10 in the first class and 11:8 in the second class. Now suppose that I combine the two classes: what is the ratio of girls to boys in the combined class? I just add the 9 and the 11 to get 20 girls and the 10 and the 8 to get 18 boys, so the ratio is 20:18. But if I now confuse the ratios with the fractions, it seems like I am saying that 9/10 + 11/8 = 20/18. The misconception that you add fractions by adding the numerators and adding the denominators is a fairly common one, and I suspect it comes from the confusion of fractions with ratios. You are right that this confusion is rife in the field. I’ve even seen materials that promote the confusion and come up with this different way of adding fractions and say it is just as good as the “traditional” way. What a mess!

]]> <![CDATA[Reply To: Unit Rate Revisited]]> Tue, 29 Jul 2014 14:11:53 +0000 Bill McCallum Thanks for these comments. I agree that we don’t want to make too pedantic with children and parents; certainly it would be crazy to test them on this vocabulary point (but yes, I know, people do crazy things). And the progressions documents should not be read as dictating what we say or do with children and parents. Their main purpose is to clarify the underlying issues for curriculum writers and others who need to make a deep examination of the standards. From that point of view, there is an issue that needs to be clarified. Suppose I say that apples cost $1.15 per pound, and then write the equation y = 1.15x to represent this, where x is the number of apples and y is the cost in dollars. That number 1.15 appears two times in the sentence: once with units (dollars per pound) and once without (in the equation). The progressions document calls the second one the unit rate: you could also go the other way and call the first one the unit rate (your preference) and call the second one the constant of proportionality, or the numerical rate, as you suggest. Either way, it’s useful to have some conventions about this. Notice that the standards themselves sidestep this issue, by requiring the use of ratio and rate language, but not specifying what that language is. So this is something the field has to sort out.

]]> <![CDATA[Reply To: 6.RP.3(c) – Percents Question]]> Tue, 29 Jul 2014 13:58:15 +0000 Bill McCallum No need to be sorry! Good thinking there.

]]> <![CDATA[Reply To: Ideas for 7.G.3]]> Sun, 27 Jul 2014 05:03:42 +0000 lhwalker I really like your idea of slicing food like this. I would add a clear statement of the objective at the beginning, something like, “Seventh grade standard G.3 requires students to describe the two-dimensional figures that result from slicing three dimensional figures.” “Vertex” is in the high school standards but I don’t see it in the lower grades, so maybe it would be better to just show we can slice three different ways without using the word “vertex.” You might want to include quick examples like a stack of 3×5 cards. I love the way this standard can lead into calculus and your example of sliced objects does that well.

]]> <![CDATA[Ratio – fractional notation]]> Thu, 24 Jul 2014 22:51:35 +0000 hagitsela My understanding is that the CCSS express ratio with colon (3:2), or with words (3 to 2), but not as a fraction (3/2).

Can you explain the reasoning behind it?

The progressions document refers to the quotient 3/2 as the value of the ratio 3:2 (“3/2 is sometimes called the value of the ratio 3 : 2.”). And also “In everyday language the word “ratio” sometimes refers to the value of a ratio”.

Can you elaborate on the difference between ratio and the value of the ratio? There are numerous websites that express ratio as a fraction, and we found assessment items that require identification of ratio as a fraction. Why is this considered “everyday language”? Why is it wrong?

]]> <![CDATA[Ideas for 7.G.3]]> Sat, 19 Jul 2014 20:52:46 +0000 SSgtMath Hello Everyone,

I recently started making math videos that give teachers ideas on how they can teach math common core standards. This is my first one and I was wondering if I could get some of your feedback. I really appreciate your time! Thank you!

]]> <![CDATA[Unit Rate Revisited]]> Sat, 12 Jul 2014 14:48:49 +0000 The Connected Math Pearson Prentice Hall video that Cathy Kessel posted in June 2012 is a meaningful example of how to use a unit rate that has units. The discussion is one that students and their parents can understand. Why make an artificial distinction that unit rate is the numerical component of a rate? The verbal gymnastics that result lead to confusion and frustration for teachers, students, and parents. Allowing unit rate to have units opens the door to the real world of unit price for students. Unit price is a logical concept from everyday life that people can understand . It makes common sense that the definition of unit rate should include unit price as an example. Making an unnecessary distinction is not helpful for the Common Core Math Standards. I taught middle school during the “New Math” era. Teachers and parents didn’t like ideas that went against common sense. Making an issue of the difference between “number” and “numeral” was one of the many technically correct distinctions that killed New Math. I feel the Common Core movement will go the same way as New Math if those in charge (whoever that is) present unnecessary distinctions. I suggest that a unit rate such as 25 miles/hour has a “numerical rate” of 25.

]]> <![CDATA[Reply To: 6.RP.3(c) – Percents Question]]> Mon, 07 Jul 2014 14:57:36 +0000 Lisa j r I am replying to myself. Upon a better reading of the standard I see this as the first part of the standard. Find a percent of a quantity as a rate per 100. Sorry.

]]> <![CDATA[Reply To: 6.RP.3(c) – Percents Question]]> Mon, 07 Jul 2014 14:05:51 +0000 Lisa j r I know this is an older post and I have also looked through the progressions document. I don’t see a solution to this part of Nathan’s question.

“I see this leading to things like 8 out of 25, and having students change them to ratios out of 100, and then making a percent.

That’s where I get a little confused on where to go. You could do more complex examples, like what percent is 7 out of 12…but if the whole point of the standard is to build on proportionality, this seems too complex. Dividing 7 by 12 and making a decimal is a 7th grade standard. So do I only deal with denominators of 2, 5, 10, 20, 25, 50, 100? You could change 7/12 to 56/96 and conclude that it’s a little more than 56% and estimate, which might be the farthest you’d want to go in 6th grade?”
I don’t see that as part of 6.RP.3c because you are not finding a percent of a number. Nathan notes that an initial part is to determine the percent but the standard doesn’t say that.

]]> <![CDATA[Reply To: Connecting prior understanding to reduce fractions prior to operations]]> Sun, 06 Jul 2014 03:42:01 +0000 lhwalker My examples were ambiguous. I explain here:

]]> <![CDATA[Reply To: PARCC and SBAC high school content frameworks]]> Fri, 04 Jul 2014 04:46:09 +0000 eprebys I’ve gotten confirmation that the high school level PARCC tests will be given as end-of-course exams and not correlated to grade level. This is described here:

These are some additional useful documents relating to PARCC if you are, like me, trying to figure out precisely which content standards will be covered on the exam: – Version 3, updated framework – Math Performance Level Descriptors – Assessment Blueprints

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Fri, 04 Jul 2014 00:20:33 +0000 Bill McCallum Yes, using a tree diagram for this is entirely consistent with 7.SP.8. The main thing would be not to get hung up on teaching general rules for calculating the probability of compound events—that really is high school, as in S-CP.B—but rather to give students concrete experiences that prepare them for those abstract rules.

]]> <![CDATA[Reply To: Histograms]]> Fri, 04 Jul 2014 00:17:01 +0000 Bill McCallum I really don’t think this is an important mathematical or pedagogical issue. Clearly there is some confusion about the meanings of the terms “histogram” and “bar chart.” While it would be a good idea for the field to come to some common understanding of the meanings of these terms, I do not think it’s an important concern for students of mathematics in these grades. In the end, we want them to be able to correctly read and produce these graphical representations of data. That’s the most important thing. What words they use for them is less important. Textbooks and teachers should be consistent in whatever terminology they use, of course.

]]> <![CDATA[Reply To: 5.NBT.3 Representation for expanded form]]> Fri, 04 Jul 2014 00:12:37 +0000 Bill McCallum I wouldn’t read to much into this in the sense of it being a format required by the standards. It was, as you say, mostly a matter of making the meaning clear. Stacked fractions are clearer, but not always typographically possible.

]]> <![CDATA[Reply To: Connecting prior understanding to reduce fractions prior to operations]]> Fri, 04 Jul 2014 00:10:13 +0000 Bill McCallum I’m not sure I completely understand the question, but here are some reactions. On the one hand, I think the activity on page 2 might strike students as a little weird: “we already know that 2/3 of 15 is 10, why are we doing this the hard way?” On the other hand, it is illuminating in showing a connection between previous knowledge and general rules that have now been developed for operations on fractions. My general feeling about reducing fractions is that there will be situations where it is clearly beneficial to replace a fraction with an equivalent simpler one, and this is one of them. So I’m not opposed to reducing fractions, just to the idea that it is always a necessary thing to do. We want students to know how to find equivalent fractions and choose useful ones in cases where there is one that is clearly useful.

]]> <![CDATA[Reply To: "simplify" in general]]> Fri, 04 Jul 2014 00:01:36 +0000 Bill McCallum Well, really, we were not trying to be word police here; I don’t see any reason why you couldn’t just use the word “simplify” on occasion. But when there is a more precise term available, use it. For example, when you multiply two polynomials you might expand the product, or you might leave it in factored form. When you substitute numbers into the quadratic formula, then I think it is appropriate to talk about simplifying the resulting numerical expression.

]]> <![CDATA[Reply To: 6th Grade Geometry – Critical area?]]> Thu, 03 Jul 2014 23:58:34 +0000 Bill McCallum You have a sharp eye. Geometry in Grade 6 is, I think, a not-quite-critical area, and the grade level introduction reflects that ambiguity. I would also say that the statements of critical areas are really just attempts to summarize succinctly what is in that grade level. So they are not necessarily an indication of priorities about allocating time; there is no statement about which critical areas should receive more attention than others.

]]> <![CDATA[Reply To: Reciprocal Trig Functions]]> Thu, 03 Jul 2014 23:50:30 +0000 Bill McCallum I would say the emphasis in the standards is clearly on sine, cosine and tangent.

]]> <![CDATA[Reply To: S.ID.4]]> Thu, 03 Jul 2014 23:47:09 +0000 Bill McCallum I think you meant “will not need to be able to read a z-score table” in the first paragraph, right? Your approach sounds reasonable, although I could also imagine an approach that does not mention the term “z-scores”. There are lots of possible approaches here.

]]> <![CDATA[Reply To: CCSS Flip Books]]> Thu, 03 Jul 2014 23:42:50 +0000 Bill McCallum “Vetted and approved” is a bit strong. I helped review the Arizona ones early on, but I don’t think there was time for a thorough review. So they are probably basically pretty good, but you should not attribute oracular authority to them. I don’t know about the Kansas ones.

]]> <![CDATA[Reply To: 7.G.1]]> Thu, 03 Jul 2014 23:41:12 +0000 Bill McCallum I can imagine what excesses people might be going to with this standard. The intent of the standard is to prepare for the discussion of similarity and congruence in Grade 8, with some hands on problems that give students a concrete feel for similarity. All of the things you mention in your second paragraph are possibilities, but some judgments have to be made about how much time this takes up in the curriculum and how it supports other more central topics, such as ratios, proportional relationships, and work with rational numbers and their representation as decimals and percents.

I do not think it is necessary to introduce congruence and similarity as formal concepts here; as I said, the point is to prepare for those concepts, not try to do everything at once.

And, any geometric figures that kids have seen are fair game, but you could go a long way with figures made up out of triangles and rectangles.

]]> <![CDATA[Reply To: Grade 8 Functions Progression pg 6]]> Thu, 03 Jul 2014 23:33:24 +0000 Bill McCallum Slope is a geometric concept; it is a property of a line in a coordinate plane. So, a line can have slope. But a function is not a geometric object, so it can’t really have a slope. When people talk about the slope of a linear function, they really mean the slope of its graph, a geometric object which is attached to the function. And if the linear function relates quantities with units, for example distance versus time, then makes more sense to talk about the rate of change rather than the slope.

]]> <![CDATA[Reply To: number sentences]]> Thu, 03 Jul 2014 23:29:40 +0000 Bill McCallum In a document like the Progressions you want to use the same word throughout to emphasize the unity of mathematical concepts across grade levels. Ellenberg’s article served a different purpose; it explained why the term “number sentence” made sense mathematically, and might help students understand what an equation is. Personally I don’t have a strong opinion here; my main interest is in discussions about meanings. So I’m happy either way as long as the meaning is made clear.

]]> <![CDATA[Reply To: F-IF.8b Is there a typo in an example?]]> Thu, 03 Jul 2014 23:20:19 +0000 Bill McCallum No, that’s the example we intended.

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Wed, 02 Jul 2014 17:55:44 +0000 Lisa j r Not sure this is the correct place for this question but as a follow up to the above question and using the spinner described would it be appropriate in the middle grades to ask a student to determine the probability of spinning a 1 followed by a 2? or the probability of spinning and getting a 1 on both spins? Could this be presented using a tree diagram with the probabilities given and find the probability of each possible event given that the spinner is used 2 times? Or is this included in the high school standards for probability?

]]> <![CDATA[Reply To: Histograms]]> Tue, 01 Jul 2014 21:00:53 +0000 Lisa j r I would like to see further discussion on this topic. While I do understand what Alexei is referring to I also believe that the example as presented in the progressions is how students are taught histograms in the early years. Can we get more input? If the example in the progressions is incorrect, are there further examples or resources that teachers could be directed to.

]]> <![CDATA[5.NBT.3 Representation for expanded form]]> Fri, 27 Jun 2014 18:17:48 +0000 Lisa j r In the standards doc there are ( ) around the fractional values. Was that intentional as a prefered format? Was that done for convenience because the fractions were not stacked? Would it be appropriate to use ( ) for each partial product. … (1 x 1/10) + (9 x 1/100) … etc. Using stacked fractions and not ( ) around each fraction.

]]> <![CDATA[Connecting prior understanding to reduce fractions prior to operations]]> Wed, 25 Jun 2014 22:45:31 +0000 lhwalker On the linked explanation, I’m looking for thoughts on canceling for fraction reduction which is not in the standards for good reason (first page), yet wondering if related thought processes could connect with and reinforce other important prior topics (second page). My question is, “Would it be worth venturing into activities like page 2, that would evolve into mostly mental math, to develop a sense for fraction reduction.”

]]> <![CDATA[Reply To: S.ID.4]]> Mon, 23 Jun 2014 14:02:44 +0000 SteveG My short answer (at least how we’ve interpreted it in our district here in Florida) is no.
Our district is thinking that students will need to be able to read a z-score table. For right now, we’re going to try starting the unit with just working on the percentages for 1, 2, and 3 standard deviations from the mean, i.e. knowing the 68%/95%/99.7% rule. You can still do a lot with just those numbers.
Then, for the last part of the standard with other areas under the curve, we thought that made more sense to do once students have worked with the easier percentages first. Although we teach students to use a graphing calc to find the percentages, the syntax can sometimes be tricky. So we thought a z-score table was the way to go. At this point, students will calculate the standardized z-score and then look it up in a table.
That’s what we’re going to try. Hope that helps.

]]> <![CDATA["simplify" in general]]> Fri, 20 Jun 2014 19:40:32 +0000 csteadman I know “simplify” is not part of the CCSS for good reason, but I am wondering if their is an easy phrase people can fall back on since “simplify” has been a crutch for so long. For example, when multiplying polynomials or substituting into the quadratic formula, can we coin the phrase “clean it up?”

]]> <![CDATA[6th Grade Geometry – Critical area?]]> Thu, 19 Jun 2014 18:34:07 +0000 The geometry of grade 6 seems to be called out as a critical area for that grade level, as if it could be critical area #5, although it is not labeled as such. Could you clarify whether it should be considered a separate critical area? If so, why was it not labeled as such?

]]> <![CDATA[Reciprocal Trig Functions]]> Wed, 18 Jun 2014 18:48:42 +0000 idomath After reading the functions progression document, we are still unclear whether reciprocal trig functions would be part of F.IF.7e.

]]> <![CDATA[S.ID.4]]> Wed, 18 Jun 2014 16:43:12 +0000 JRoderique Will students be expected to calculate z-scores prior to using a calculator or table to find the probabilities?

]]> <![CDATA[CCSS Flip Books]]> Tue, 17 Jun 2014 19:32:38 +0000 hccrawford I have a question about the CCSS Flip books that are used by Kansas and ARizona. Can we assume these are vetted and approved by “the CC Powers?” I use them as a reference for training teachers, in Florida, and hope they are 🙂

]]> <![CDATA[7.G.1]]> Tue, 17 Jun 2014 01:48:17 +0000 mrsnolanroom412 I am trying to determine how in-depth to go on this standard, mainly because of all the example problems out there that seem to be going way beyond the language of the standard.

Also, should students be learning just “scale factor” or ratios as in scale of 1:8. What about percent increase or decrease?

Should conversations with students include terminology such as congruent and similar with a congruent figure having a scale of 1 and similar figures having scales making the figure larger or smaller?

Finally, the standard states “geometric figures.” How far should this extend beyond rectangles and triangles?

]]> <![CDATA[Grade 8 Functions Progression pg 6]]> Mon, 16 Jun 2014 23:15:38 +0000 dsekreta A group of Grade 8 teachers were discussing the meaning/interpretation of the narrative on page 6 of the Functions progression document that says, “a linear function does not have a slope but the graph of a non-vertical line has a slope.” The discussion between the teachers was around linear functions versus linear equations and slope in those two concepts. Could you please clarify what the authors of this statement might have meant by this statement so I can share the response(s) with the group of teachers?

]]> <![CDATA[number sentences]]> Mon, 16 Jun 2014 21:48:33 +0000 Alexei Kassymov Bill McCallum on twitter liked an article in defense of “number sentences” (among other things).
Stephen Colbert Thinks “Number Sentences” Are Silly. They’re Not.

Not sure if the twitter link at the bottom will work or not.

Here is what “Front Matter for Progressions for the Common Core State Standards in Mathematics” states:

Reconceptualized topics; changed notation and
This section mentions some topics, terms, and notation that have
been frequent in U.S. school mathematics, but do not occur in the
Standards or Progressions.
“Number sentence” in elementary grades “Equation” is used instead
of “number sentence,” allowing the same term to be used
throughout K–12.

The topic has already appeared on this forum.

What is the current thinking about “number sentences” in the context of CC?

<p>Great article by @JSEllenberg about #CommonCore (and Stephen Colbert) @IsupportCCSS via @slate</p>— Bill McCallum (@wgmccallum) June 12, 2014

<script async src=”//” charset=”utf-8″></script>

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Sun, 15 Jun 2014 02:07:17 +0000 lhwalker I would like to add a thought about “once they understood it.” Until recently, I would explain briefly and then show them examples. I didn’t realize I had trained them to zone out until I showed the examples. Sometimes I would question them to make sure someone understood it, but the main take-away was procedure. Then they would ask, “Do I multiply or add? Where do I put the zero?…” Now when they ask questions like that, I help them reconstruct the conceptual explanation. Now the main take-away is “what would be a sensible step?”

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Fri, 13 Jun 2014 18:08:10 +0000 Bill McCallum Yes, but I wouldn’t describe it that way. Students should understand that a decimal is a fraction, so dividing decimals is dividing fractions. For example, to calculate 2.16 ÷ 0.3, they would know that 2.16 = 216/100 and that 0.3 = 3/10, which is also 30/100. If they were asked to calculate 2.16 ÷ 3 they would use this understanding to rewrite this as (216/100) ÷ (30/100) = 216 ÷ 30. I think this is what you mean by extending the zeros … and of course I wouldn’t expect them to got through this reasoning every time, once they understood it.

]]> <![CDATA[F-IF.8b Is there a typo in an example?]]> Fri, 13 Jun 2014 17:15:22 +0000 SteveG F-IF.8b talks about interpreting expressions for exponential functions. The last example listed is y = (1.2)^(t/10).
Was that supposed to be y = (1/2)^(t/10), which is a typical half-life equation? Or was it only supposed to be 1.2 as shown?
I didnt find anything in the progressions on this and the version on the website also has this exampel.

]]> <![CDATA[Algebra 2 for all?]]> Fri, 13 Jun 2014 16:12:42 +0000 Sarah Stevens As we continue to carry our first cohort of students through a high school Common Core sequence, we are looking at what kind of courses will be acceptable for high school math graduation credit. The non-plus standards include many topics that have, previously, been included only in an Algebra 2 (and sometimes Pre-Calculus/Trigonometry course). In other words, the standards written for all students are topics from Algebra 2 or above.

This shift in math topics on the upper end of school has the potential to have a huge impact on the expectations high schools have for students before they graduate. Historically; high schools have offered alternatives to Algebra 2 which will meet graduation requirements but the scope and sequence is not as broad as Algebra 2. Students in these courses are, typically, not planning to attend a 4 year university immediately after high school. They might be planning on going straight to the work force or to attend a community/technical college.

My question is, why are some of these topics not (+) standards for STEM students? For example, do all students need arithmetic operations on polynomials, rewriting rational expressions, analyzing logarithmic and trigonometric functions, composing and finding the inverse of functions, radian measurement and the unit circle, equations for conic sections, etc. I’m sure there is a reason for including these topics in the “for all” category but I’m curious about how the writing team decided if a standard was (+) or not.

As a follow-up, do you think these topics necessitate Algebra 2 for all students? I have read about states, such as Texas, which required Algebra 2 for graduation and discovered that a more targeted approach to math credit- based on students needs- better served its students. I think we can create rigorous math courses for a students third year of math which include an in-depth study of some of these topics but I am struggling with how I can guarantee that all students meet the basic expectations of the standards without requiring Algebra 2 for all.


]]> <![CDATA[Reply To: simplifying radicals]]> Fri, 13 Jun 2014 15:30:08 +0000 Sarah Stevens There was a similar question in the 6-8 Number System strand, which Bill replied to. You can find it here.

]]> <![CDATA[Reply To: F-IF.7 and discussion of asymptotes]]> Fri, 13 Jun 2014 13:28:44 +0000 SteveG Hey thanks for your quick reply. Some teachers are working on units aligned to CCSS over the summer and your speedy reply helps alot.

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Thu, 12 Jun 2014 19:27:54 +0000 kirkkimb Thanks so much for responding to this question. I have been wondering about this as well. However, would you mind elaborating on missbaldin’s question “Do students in 6th grade need to learn about extending the zeroes in whole numbers and decimals in order to divide”? Thanks

]]> <![CDATA[Reply To: S.ID.5 Two Way Frequency Tables]]> Thu, 12 Jun 2014 19:01:32 +0000 Bill McCallum Well, it says “Summarize categorical data for two categories in two-way frequency tables.” So I would interpret that as including the possibility that they are given the data in some other form and have to put it in a two-way table. Of course, it doesn’t have to be that way every time.

]]> <![CDATA[Reply To: F-IF.7 and discussion of asymptotes]]> Thu, 12 Jun 2014 18:27:21 +0000 Bill McCallum It’s fine to mention asymptotes. The end behavior of an exponential function is that it approaches an asymptote on one side or the other of the $y$-axis. The two terms are not always synonymous, but they are in this case.

]]> <![CDATA[F-IF.7 and discussion of asymptotes]]> Thu, 12 Jun 2014 18:19:54 +0000 SteveG F-IF.7e asks for graphing and analysis of exponential and logarithmic functions. While it mentions “end behavior,” it does not use the word asymptote. However, it seems very natural when teaching exponential and logarithmic functions to talk about the equations of the asymptotes. This is especially true when doing transformations (F-BF.3).

However, the only place where “asymptote” appears is in F-IF.7d, which is a plus standard. Some people are interpreting this to mean that there should be no mention of asymptotes at all when teaching to F-IF.7e. That seems a bit extreme to me.
What are your thoughts? I agree that end behavior can be assessed and helpful in conceptual understanding of these functions, but exponential and logarithmic functions have asymptotes — why shouldn’t we use the correct mathematical word when talking about these functions?
Any insight you can offer is greatly appreciated.

]]> <![CDATA[Reply To: Zimba's comment: "The number line is not an appropriate model for place value"?]]> Thu, 12 Jun 2014 16:38:36 +0000 Bill McCallum I can’t speak for Jason, but I can give you my thoughts on this problem. There are two concepts at play in this problem: one is the understanding of subtraction as a missing addend problem. That is, understanding 427 – 316 as the number you add to 315 to get 427. The number line is a good model for visualizing this. The other concept is using the base 10 system in subtraction. That is, understanding 427 as 4 hundreds 2 tens and 7 ones, 316 as 3 hundreds, 1 ten, and 6 ones, and subtracting hundreds from hundreds, tens from tens, and ones from ones. I agree with Jason that the number line is not a good model for this understanding. The precise relationship between 100s, 10s and 1s is not so easy to see on the number line, because you can’t really accurately depict a one on a scale which also includes hundreds. The ones and the tens can get confused; indeed, that seems to be exactly the problem Jack was having (although it’s a little hard to figure out which marks are Jack’s and which marks are the student doing the problem). Also, the method presented might be misconstrued as suggesting you have to go in order: first subtract the hundreds, then the tens, then the ones, which really misses the point. I don’t think the problem is completely bad, but I do think it’s a little off key. I’d be inclined to use a number line for something like 23-8, where you can easily see the intermediate numbers 10 and 20. For a 3 digit subtraction I’d want to use base ten blocks, or just the verbal decomposition described above.

]]> <![CDATA[Reply To: 7.G.6 Pyramid Surface Area]]> Thu, 12 Jun 2014 15:46:38 +0000 Bill McCallum I’m a little confused by this question, because the discussion was about pyramids, not cones or spheres. The middle school surface area standards are in Grades 6 and 7, as listed at the top of this thread. You are correct that surface area of cones and spheres is not in the standards.

]]> <![CDATA[Reply To: Common Core Geometry 2.0]]> Thu, 12 Jun 2014 15:40:12 +0000 Bill McCallum Thanks, I’ll take a look when I get a chance!

]]> <![CDATA[Reply To: What is the difference between and A.REI.1 ?]]> Thu, 12 Jun 2014 15:37:49 +0000 Bill McCallum There are two differences that I can see: (a) the one you noticed, that A.REI.1 explicitly asks students to explain the reasoning and (b) A.REI.1 is not limited to linear equations.

As for (a), “justifying the steps” sometimes ends up being a recitation of rules, e.g. “I added a 5 to both sides.” A.REI.1 is asking for more than that. It is asking for students to see each step as an if-then statement: if x – 5 = 3 then (x-5) + 5 = 3 + 5, therefore x = 8.

]]> <![CDATA[Reply To: Acceleration]]> Thu, 12 Jun 2014 15:30:16 +0000 Bill McCallum This is the most extreme example of acceleration I have heard of. Calculus for all is crazy. As a university professor I see the damage done by this sort of thing all the time: kids coming in with a fragile grasp of algebra or calculus because they have been raced through it, and being placed into remedial courses. This is a recipe for failure in college for many kids. There are some students who are ready for acceleration and they should have that opportunity. The rest are being done a disservice by this sort of thing. It’s an abdication of educational responsibility.

There used to be an excuse for it: the middle school curriculum was often sparse and repetitive and acceleration was the only way out of it. But with the Common Core students have plenty to do in middle school, and if they take it at the right pace they will be truly ready for college.

]]> <![CDATA[S.ID.5 Two Way Frequency Tables]]> Thu, 12 Jun 2014 14:24:13 +0000 JRoderique Are students expected to create their own two-way table or are they just interpreting relative frequencies?

]]> <![CDATA[Zimba's comment: "The number line is not an appropriate model for place value"?]]> Tue, 10 Jun 2014 20:41:21 +0000 smithba.wbms In the Huffington Post, Jason Zimba is quoted as saying “The number line is not an appropriate model for place value”. His comment is part of an interview pertaining to that atrocious viral response to a “Common Core” math problem. I’m sure you’ve all seen it on Facebook by now.

But what I’m wondering is why Zimba disparages the actual problem rather than the widespread misunderstanding of its purpose (Common Core teaches models AS WELL AS methods, right?).

We’re supposed to use the number line to model lots of operations, like operations on fractions (Hung-Hsi Wu, anyone?)

Why is the number line “an inappropriate model for place value”? What exactly is wrong with the problem? Can you give an example of a better problem or task that addresses the same standards?

]]> <![CDATA[Reply To: 7.G.6 Pyramid Surface Area]]> Tue, 10 Jun 2014 16:46:42 +0000 cshore Your response makes sense, but we don’t find a standard for surface area in 8th grade. In fact, we don’t see surface area of cones and spheres anywhere throughout middle and high school. Are we interpreting correctly?

]]> <![CDATA[Common Core Geometry 2.0]]> Sun, 08 Jun 2014 21:11:37 +0000 Dr. M Last summer I completely reworked by geometry class to bring it in line with the CCSS.

As I taught this last school year, I made revisions. Many, many revisions. Take a look: Elementary Euclidean Geometry.

I judge from the responses I’ve had that it can be of help.

– Dr. M

]]> <![CDATA[What is the difference between and A.REI.1 ?]]> Tue, 03 Jun 2014 04:46:07 +0000 KatherineMarroquin 8.EE.7b ….. including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

I am confused as to what will be the difference in one variable linear equatIons solved at each level. In the eighth grade standard, distribution ans collecting like terms is already mentioned thus students will be solving multi-step one problems in 8th grade. What is the difference in the algebra standard? Other than justifying steps I do not see a difference. But I would never teach solving equation without the accompanying justification for each step anyway.


]]> <![CDATA[Reply To: Acceleration]]> Sat, 31 May 2014 21:08:37 +0000 ttotorica Bill,

I just attended an Illustrative Math Conference and I am so energized by all I learned. Thank you for helping to provide this professional development for teachers! I especially appreciated hearing some of your thoughts about acceleration and compacting. We in our district have adopted a model of AP Calculus access for all, and as a result are developing course pathways that involve compacting Appendix A’s Math 7 and Math 8 courses into a single, one-year course along with a 4:3 model of compacting that absorbs the Fourth Course (Precalculus) into the integrated Math I, II, and III courses. These compacted courses are being built to provide entry points for calculus at both the junior high and high school levels and to also provide a double-accelerated pathway for those wishing to spend both their junior and senior years studying calculus.

My concern is that these courses are being built as the default courses for our general, non-gifted students. In an effort to send as many students to calculus as possible, my district is proposing the compacted 4:3 HS path for everyone, and I fear that by doing this, we will compromise both the depth of student understanding and the rigor of our courses. Do you share my concern that by compacting too much for the typical student, we will defeat the entire purpose underlying the CCSS development and generate thin and fragile mathematics competency as a result?

Given the district goal of providing access to calculus for all, do you know of any successful designs for high school acceleration that we might offer instead of the 4:3 model? I know of the “doubling up” model, where students simultaneously enroll in both Algebra and Geometry to gain a year for calculus study. But since we have adopted an integrated pathway, I’m not sure how that would look. I would like to propose a compacted, intensive summer course for precalculus, where the topics necessary for calculus success are offered to students after they have completed a normally-paced Math III course in their junior year. Do you think this could be a viable alternative to pitch to my district?

Thank you for your time and expert opinion; I appreciate it!

]]> <![CDATA[Reply To: Algebra 1 in 8th grade?]]> Sat, 31 May 2014 14:30:13 +0000 Bill McCallum Yes, a lot of states have put thought into this. Here is a document from the Massachusetts Department of Education outlining various acceleration options:

]]> <![CDATA[Algebra 1 in 8th grade?]]> Sat, 31 May 2014 00:56:30 +0000 nvitale In New York our old standards had a great deal of overlap in what was taught and tested in 8th grade and 9th grade (very little focus). In this environment, many schools offered Algebra 1 in 8th grade as an advanced option. It was feasible to mix 8th grade and 9th grade content in one year.

Now, as New York transitions to a common core aligned 9th grade course and exam, I’m not sure how folks should approach this – the content of 8th grade seem very distinct from Algebra 1 (although there is a clear progression through them).

I guess my question is what guidance or advice would you give to schools or districts which are looking to continue offering Algebra 1 in 8th grade. How would you approach this? Has anyone had experience doing this in the CC era?

I’d love to hear thoughts,

]]> <![CDATA[Reply To: Solving systems of linear equations]]> Fri, 30 May 2014 23:55:31 +0000 Bill McCallum The Grade 8 standard explicitly limits to systems of two equations in two variables, whereas the high school standard only suggests a focus on such systems, but allows for bigger systems as well. There is also a general uptick in fluency and complexity expectations from Grade 8 to high school. In general, there is some overlap in the topics studied in Grade 8 and high school, with greater depth expected in high school.

]]> <![CDATA[Reply To: The Percent Proportion]]> Fri, 30 May 2014 23:23:25 +0000 Bill McCallum Yes, I think abieniek has it right. Although, I confess, I’m not completely clear on what is meant by “the percent proportion.” In the Common Core a percent is a rate per 100, so anything you would do with a rate you can do with percents. This includes, in Grade 6, understanding that 5% of 200 is 10 because $\frac{5}{100} \times 200 = 10$, and, in Grade 7, being able to express the statement “the sales tax is 5% of the price” using an equation such as $T = 0.05P$.

  • This reply was modified 3 years, 2 months ago by  Bill McCallum.
]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Fri, 30 May 2014 23:15:29 +0000 Bill McCallum Yes, this sounds basically right. Although I wouldn’t call the spinner a model, but rather a representation of the model. The model itself is just the sample space {1, 2} with the assigned probabilities P(1) = 2/5 and P(2) = 3/5.

]]> <![CDATA[Reply To: symbolic logic]]> Fri, 30 May 2014 23:12:23 +0000 Bill McCallum No, symbolic logic is not in the standards (and I don’t think it was in many state standards previously).

]]> <![CDATA[Reply To: HSF-LE.A.3]]> Fri, 30 May 2014 23:08:18 +0000 Bill McCallum Here is the standard for convenience of other readers:

HSF-LE.A.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.⋆

We certainly want students to know this is always true. However, the mathematical proof of this fact uses calculus, so it is beyond the scope of the standards. I don’t know what informal ways of seeing it you have in mind, but they may well fall under the rubric of “observe.”

]]> <![CDATA[Reply To: F.IF.4]]> Fri, 30 May 2014 23:03:31 +0000 Bill McCallum I agree with the calculus teachers! After all, for $f(x) = x^2$, it is true that $f(x) < f(y)$ for any $x ]]> <![CDATA[Reply To: Modeling in HSF-IF.C.7]]> Fri, 30 May 2014 22:57:16 +0000 Bill McCallum Yes, if graphing is involved. The idea of the modeling star is that it flags standards likely to be involved in modeling problems.

]]> <![CDATA[Reply To: Independent/Dependent Events]]> Fri, 30 May 2014 22:47:05 +0000 Bill McCallum No, these are both high school topics. See

S-CP.A.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.


S-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Note that the latter is a (+) standard, and so beyond the college and career ready threshold. So certainly not necessary in Grade 7!

(By the way, you can answer a lot of these questions using a word search on the pdf of the standards, available at

]]> <![CDATA[Reply To: Definition of Place Value]]> Fri, 30 May 2014 22:36:16 +0000 Bill McCallum I think all occurrences of the term “place value” in the standards could be replaced by the term “place value notation” without changing the meaning. Place value notation (as I’m sure you know!) is a the system of writing numbers where, for whole numbers,

  • Each number is represented as a sequence of digits 0–9.
  • Each digit is assigned a value equal to the digit times a power of 10, the power being 1 for the right most digit, then 10, 100, 1000, etc. as you move successively to the left.

  • The number is the sum of the values of the digits.

For decimals the system is extended by putting a decimal point at the end and adding digits to the right of the decimal point, whose values are the digit times 1/10, 1/100, etc.

The place value system is this system of notation. So, “the value represented by a digit” is not synonymous with “place value” … rather it is determined by place value (notation).

]]> <![CDATA[Reply To: Geometry Progressions]]> Fri, 30 May 2014 22:14:48 +0000 Bill McCallum Hoping to get it done this summer. (But I also hoped that last summer.)

]]> <![CDATA[Reply To: 6.NS.2, 6.NS.3]]> Fri, 30 May 2014 22:14:04 +0000 Bill McCallum Students learn about whole number remainders in Grade 4:

4.OA.A.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

In Grade 6 they study division of fractions, which would include 14.6/3. They are not required to know about infinite repeating decimals until Grade 8, so they might express the answer as a fraction rather than a decimal, as in $4 \frac{26}{30}$. In the Common Core finite decimals are treated as a way of writing certain sorts of fractions, namely those that can be written with denominator 10, 100, and so on. There is no explicit requirement that they express this problem as division with remainder, but it seems a natural extension of their Grade 4 work to be able to say that 14.6 = 3 x 4 + 2.6, and to interpret both the quotient and the remainder in a context.

It also seems to me that, although knowledge of infinite repeating decimals is not required until Grade 8, simple examples such as 1/3 = 0.333 … could appear earlier. But it is not necessary, since you can always just use fraction notation.

]]> <![CDATA[Reply To: The Percent Proportion]]> Fri, 30 May 2014 19:07:31 +0000 Aaron Bieniek I’d like to take a crack at this one before Bill does if that’s ok.

It seems to me that our treatment of proportion in the past has been setting 2 ratios equal to each other and “solving” (maybe relying heavily on the magic of cross-multiplying), but now the standards are asking us to analyze proportional relationships, i.e testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin (7.RP.2a).

I think “percent proportion” falls into the former way of thinking about proportion rather than what we are being asked to think about in the Common Core, so I worry about using it at all. If it is to be used, I think it would be limited to 7th grade since proportional relationships are not introduced in 6th grade, but more importantly, how does it support the ideas of proportional relationships as outlined in the standards? Those ideas being: testing for equivalent ratios in a table, graphing on a coordinate plane and observing whether the graph is a straight line through the origin, and using the unit rate to write equations of the form y=kx.

]]> <![CDATA[Reply To: Solving systems of linear equations]]> Thu, 29 May 2014 19:31:17 +0000 eamick A wondering… How is the high school standard A-REI.6 different from the 8th grade standard 8.EE.8b?

A-REI.6 reads, “Solve systems of linear equation exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.”

I can’t find any substantive difference in the language of the standards, but I can’t imagine that there was no intended difference. Thanks.

]]> <![CDATA[The Percent Proportion]]> Thu, 29 May 2014 18:32:01 +0000 alwong Hi Bill,

We are laying out our MS curriculum for next year and we have a question about the percent proportion. When is the percent proportion introduced? Should we use the percent proportion as a method to teach 6.RP.2c or wait to introduce it in 7.RP.3?

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Tue, 27 May 2014 19:31:01 +0000 sjones171 I am trying to qualify uniform probability models with my colleagues and I need to clarify some of your discussion last year with SteveG. The reason that selecting a student is a uniform probability model is because each of the 10 students (the outcomes) has an equal chance of being selected, correct?

In SteveG’s question, he moved from the experiment (selecting a student) to a specific event – select a 7th grader (a subset of the sample space). The selection of a student is equally likely, but the selection of a 7th grader is not equally likely (0.4 to 0.6), but each student is still equally likely to be chosen.

Let’s say we have a spinner divided into equal fifths. If the sections were labeled 1 to 5, then the spinner would be a uniform probability model because the sample space is {1, 2, 3, 4, 5} and each outcome is equally likely to happen. Spinning an odd number would be the sum of all the probabilities of the odd numbers = 1/5 +1/5 + 1/5 = 3/5. Just like above, the spinner is equally likely to land on any number on the spinner.

Taking this one step further, take the same spinner, but label two sections “1” and three sections “2.” This is no longer a uniform probability distribution because the sample space is now {1, 2}, but P(1) = 2/5 and P(2) = 3/5. The outcomes no longer are equally likely to occur. Would you agree with this?

]]> <![CDATA[symbolic logic]]> Mon, 26 May 2014 16:14:21 +0000 Are topics in symbolic logic (e.g., truth tables, material implication) included in the common core. If so, at what stage are they introduced?

]]> <![CDATA[HSF-LE.A.3]]> Sun, 25 May 2014 17:49:37 +0000 tomergal I wonder why the wording of the standard uses “observe.” Do we only want students to observe specific exponential functions exceeding specific polynomials? Don’t we want them to know that always happens (for increasing functions of course)? Shouldn’t they know, not only that it’s true, but why it’s true? Unfortunately, the progression doc doesn’t discuss this standard.

There are many informal ways to assure oneself of this property of exponentials over polynomials. However, I don’t want to go there without knowing it’s standard aligned, since this transgresses the apparent boundary outlined by the standard.

]]> <![CDATA[F.IF.4]]> Fri, 23 May 2014 13:05:19 +0000 csteadman When dealing with increasing and decreasing functions as in F.IF.4, there seems to be a debate in the field. For example, do you include the turning point of a quadratic in the interval where a function is increasing or decreasing?

Most teachers seem to exclude this point and justify the decision by saying a function is neither increasing nor decreasing at the turning point. But teachers of calc and above seem to disagree and use max and mins as part of the increasing and decreasing intervals [a,b].

Any thoughts?

]]> <![CDATA[Modeling in HSF-IF.C.7]]> Tue, 20 May 2014 16:57:21 +0000 tomergal Hi,

Standard HSF-IF.C.7 is marked with the modeling star, but other than that little star, there’s no reference to any expected modeling, neither in the sub-parts of the standard nor in the progression doc.

There are, however, many other standards that call for modeling with all of the function families mentioned in HSF-IF.C.7. Should we align these modeling problems to HSF-IF.C.7 as well?


]]> <![CDATA[Independent/Dependent Events]]> Sat, 17 May 2014 21:50:29 +0000 monkeymom33 I am trying to figure out the SP standards for grade 7 . Should I be teaching independent events and dependent events? What about permutations and combinations?

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Sat, 17 May 2014 16:32:26 +0000 lhwalker I must disagree that having to consult the internet means there is a problem. There is a huge problem with math education in this country, but it is because too often we math teachers have presented math as a series of compartmentalized, memorized steps. That’s what most adults remember, but I wonder if they are aware of how few consider themselves to be good at math. There are some instructional techniques that research shows develop solid number sense that goes with the math procedures, connecting them and making them meaningful, easier to remember. Those techniques need names. I have had to look up a few myself. Better yet, I have watched a few demonstrated on It’s an exciting time to be involved with math instruction because, for the first time, I believe we are making a huge step toward making mathematics within easy reach of all students.

]]> <![CDATA[Reply To: Definition of Place Value]]> Sat, 17 May 2014 16:09:48 +0000 lhwalker I don’t know enough to define “place value” definitively here, but I have learned to see the importance in the standards. Place value connects with “like terms” in all of math. In two-digit arithmetic, place value makes the difference between 20+5=25 OR 20+5=70, because we add like terms: tens to tens and ones to ones. Otherwise students memorize “line up on the left or line up on the right” and get those rules confused later. That’s why working with tens boxes or something similar is crucial in the lower grades. With decimal numbers, students often believe 0.25 > 0.8 because they don’t understand place value very well. With fractions, students confuse “multiply straight across” with “add straight across” if they do not understand we can only add like terms. In algebra, 2x + 3y = 5xy if a student doesn’t understand we can only add like terms. Pedagogically, I favor using interactive presentation software like Notebook. I type 536 on top of a white rectangle behind which I hide 500, 300 and 6. I drag the hidden numbers out from behind the white rectangle so students can all clearly see the value of each digit. I have to do this for some kids who make it all the way to high school, still fuzzy about place value.

]]> <![CDATA[Definition of Place Value]]> Fri, 16 May 2014 12:59:37 +0000 jgrove What is the definition of place value?

Some background:
The term “place value” is ubiquitous in the standards. I’m wondering about language usage around the idea of place value and what exactly is meant by each term. Here are some ideas I have/have found:

place: location of a digit within a number (hundreds, tens, ones, etc.)

value of a place: This is referred to in the progressions: “In the base-ten system, the value of each place is 10 times the value of the place to its immediate right.”

place value: I have seen “place value” used in three different ways:
– The value a digit has by virtue of its position of a number (e.g. the value of 6 is 6, but the place value of 6 in 642 is 600). If this is the case, is “value represented by a digit” (used in 5.NBT.1 and 4.NBT.1) interchangeable with “place value”?
– The value of a place or position within a number (e.g. the place value of the ones place is 1). If this is the case, then “value represented by a digit” is not interchangeable with place value.
– As an umbrella term referring to properties and consequences of a base 10 number system.

Hoping to get some clarity to these discussions!

]]> <![CDATA[Reply To: Geometry Progressions]]> Wed, 14 May 2014 23:11:22 +0000 ak2014 Is there an update as to when the 7–12 geometry progression will be out?


]]> <![CDATA[6.NS.2, 6.NS.3]]> Wed, 14 May 2014 17:22:23 +0000 missbaldwin Examples I’ve seen of division in 6th grade, both with and without decimals, have nicely terminating quotients. Do students in 6th grade need to learn about remainders, for whole numbers and decimals? Do students in 6th grade need to learn about extending the zeroes in whole numbers and decimals in order to divide? Should students in 6th grade be solving the problem 14.6/3? How should they interpret the quotient?

]]> <![CDATA[Reply To: 3.OA.A]]> Fri, 09 May 2014 19:19:59 +0000 Bill McCallum Well, I don’t know the whole story here, and getting caught in the middle of a battle between a grandparent and a teacher is the last thing I want to do, but maybe you could use MP6 in your response to this teacher. Attending to precision includes attending to precision in the asking of questions. From your description it does seem that your grandchild answered the question correctly as posed, since the question did not specify a particular order in which the product must be written (nor would that have been a good idea). I agree that it is unacceptable that a mathematically correct answer should be marked wrong.

Now, I think I know where the teacher is coming from (you probably do too). The teacher has in mind that the student should be thinking of 10 groups of 2, and 3.OA.5 does suggest this would be written as $10 \times 2$:

3.OA.5 Interpret products of whole numbers, e.g., interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each.

Still, your grandchild’s answer is not wrong, because $10 \times 2 = 2 \times 10$.

It seems to me all this could come out in classroom discussions, and this would be the appropriate place to discuss the answer. That is, without saying the answer is wrong, you could ask the student to explain their thinking, and see how they decided to write $2 \times 10$, and that could lead to some good discussion.

]]> <![CDATA[Reply To: 2D Shapes in Kindergarten]]> Fri, 09 May 2014 19:01:55 +0000 Bill McCallum No, the standards do not require this identification.

]]> <![CDATA[Reply To: Mean absolute deviation]]> Fri, 09 May 2014 18:58:24 +0000 Bill McCallum I certainly hope your 6th grader is not being exposed to statistics at a college level! Grade 6 statistics in the Common Core is meant to be an introduction to some basic ideas of data and variability. It is actually pretty important for people to have that ability these days, since we are presented with statistics all the time in newspapers, reports about polling, discussions of important issues like climate change and polling bias, and so on. Not to mention the pervasiveness of statistics in more specialized jobs in the scientific, technical, medical, and biological fields.

But really the purpose of this blog is not to have general discussions about the importance of learning mathematics and statistics, but rather to answer specific questions about the standards. So if you can point to specific standards that you have questions about, that you think your child’s curriculum might not be treating correctly, then I’d be happy to answer them.

It’s possible that the curriculum your child is experiencing is just spending too much time on this, so I’d be interested to know if you think that is the case.

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Fri, 09 May 2014 18:42:04 +0000 Bill McCallum I need a more specific question to be able to answer this. The purpose of this blog is to help people understand the standards; what does a particular standard mean, what’s an example that illustrates it, and so on. It looks like you are having a problem with a particular curriculum. Not all curricula that claim to be Common Core aligned really are, and even with curricula that are aligned I don’t think I can take on answering every question about every curriculum. (This is an entirely volunteer effort.) Still, I’d be happy to try to help if you can give me something more specific. I would encourage you to read the Grade 6 statistics standards themselves and see if you think the work your child is doing is related to them. You can read them at or

]]> <![CDATA[3.OA.A]]> Fri, 09 May 2014 18:04:37 +0000 dlward I am asking this as a grandparent of a 3rd grade student. Is there anything in Common Core 3rd grade mathematics that would make the following scenario acceptable? I have been a Secondary Mathematics specialist for many years and I believe there is a an error in the thinking of this teacher but I want to make sure that I am not missing anything.

The student is completing a problem with the following instructions.
“Pick a number to use for the number of people or animals.”
The student enters the number they select in an answer blank.

The student is then asked
“How many Human Legs?” and then asked to write a number sentence and produce a drawing.
Here is the issue, the teacher is saying that the student must write the number sentence with the number of humans written first and the number of legs that the human has written second. For example 10×2. The teacher marks the student wrong if they write 2×10. The teacher claims that requiring students to write the numbers in a particular order is Standard for Mathematical Practice #6: Attend to Precision. Is there something I am missing – I am concerned with the conceptual understanding that is being developed by this teacher’s practice. I would appreciate your feedback. Thanks

]]> <![CDATA[Reply To: 2D Shapes in Kindergarten]]> Wed, 07 May 2014 18:47:06 +0000 kristyswig Thank you for this clarification! So when assessing students in kindergarten on 2-D shape recognition, should we be expecting that all kindergarteners be able to identify a trapezoid? From your response I am interpreting that we just need to expose them to this term/shape throughout the year, but not expect identification mastery.

]]> <![CDATA[Reply To: 2D Shapes in Kindergarten]]> Wed, 07 May 2014 18:47:06 +0000 kristyswig Thank you for this clarification! So when assessing students in kindergarten on 2-D shape recognition, should we be expecting that all kindergarteners be able to identify a trapezoid? From your response I am interpreting that we just need to expose them to this term/shape throughout the year, but not expect identification mastery.

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Wed, 07 May 2014 00:51:28 +0000 TR Also, my daughter is coming out of DODDEA which has not picked up common core. So far, I will be honest I am not pleased. When a parent must consult the internet to answer questions for 6th grade math then there are issues. Please sell me on the new curriculum standards.


]]> <![CDATA[Reply To: Mean absolute deviation]]> Wed, 07 May 2014 00:39:04 +0000 TR Please comment on the practical application of MAD or IQR. I need to re-teach every thing my daughter is taught in school. By the way, I did not delve into statistics until college. Why the push for a 6th grader to learn statistics?

thanks for any help

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Wed, 07 May 2014 00:32:16 +0000 TR I am a parent of a 6th grader that is now in a school that teaches common core. My child has an IEP for learning delay. How can I help my child deal with the statistics in her math class. She can do the math but the application in a real world setting does not present itself easily. I do not even see the practical application of some of the course work. Please help. My daughter does have one period of ‘Transmath’ but it does not back up or re-enforce what is taught in her regular math class.

]]> <![CDATA[Reply To: 2D Shapes in Kindergarten]]> Tue, 06 May 2014 20:50:22 +0000 Bill McCallum I think there’s a difference between “Identify and describe” and “increase their knowledge of.” Notice that it is a particular sort of trapezoid being described, one with two non-parallel sides of equal length. Students might see such shapes around the classroom, or build them up out of triangular and rectangular tiles, or has some tiles of that shape available, as in the picture on the next page. So in that sense they will become familiar with them. And maybe they will have a name for them, but the standards don’t specify that.

]]> <![CDATA[2D Shapes in Kindergarten]]> Tue, 06 May 2014 19:01:21 +0000 kristyswig First of all, I would like to commend your team on a job well done writing the progressions document. It has proven to be an excellent resource for the teachers and staff in our district. Just last week, a group of math specialists I was working with at a conference had a lengthy discussion as to whether or not identifying and describing a trapezoid is part of the Common Core standards for kindergarten. The Common Core standard K.G.1 states, “Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and sphere.” This leads some to believe that kindergarten teachers are responsible for teaching their students to identify and describe the following 2-D shapes; squares, circles, triangles, rectangles, and hexagons. However, in the progression document for kindergarten geometry there is a line that states, “They increase their knowledge of a variety of shapes, including circles, triangles, squares, rectangles, and special cases of other shapes such as regular hexagons and trapezoids with unequal bases and non-parallel sides of equal length.” Any input that you can provide to help clear this up for our team would be greatly appreciated. We greatly value your expertise and hope to hear from you soon!

]]> <![CDATA[Remediation by Reinforcing Foundations]]> Sun, 04 May 2014 19:46:53 +0000 lhwalker If I could wish for something that would really help in implementing CCSS-M, it would be a handbook of “Concepts to Reference.” It would integrate the ideas from with foundational concepts in the Standards. It would guide us in how to answer students’ questions by quickly reinforcing number sense, instead of re-teaching entire lessons or resorting to memorized tricks. For example, given 2/7 + 3/7, “Do I keep the denominators the same or do I add them?” “Answer: What was added together to get 2/7? (1/7 + 1/7). What was added together to get 3/7 (1/7 + 1/7 + 1/7)…” Example 2, a student writes 1/k instead of k for y=kx. Teacher, “Make a table of your number pairs and see if we can figure it out from there.” Example 3, “Do I add the exponents or multiply them?” Do you know of anything similar that is in the works?

]]> <![CDATA[Reply To: 6.G.3]]> Sat, 03 May 2014 01:07:52 +0000 Bill McCallum Why do they need integer operations? Students can see that the distance between $(-1,3)$ and $(2,3)$ is 3 without knowing how to subtract $-1$ from $2$. For example, they could plot the points on the plane and count the units to get the distance. This is in fact good preparation for integer subtraction. I think maybe you are thinking about the formula for distance, which would indeed require integer subtraction. But you don’t need to use the formula.

]]> <![CDATA[Reply To: Simplifying radicals]]> Sat, 03 May 2014 01:02:24 +0000 Bill McCallum Simplifying radicals is one of those high school topics that has evolved into a cancerous growth on the curriculum, starving other more important topics for resources. What is important is for students to understand and use the laws of exponents. When they get to algebra they should be able to see that $\sqrt{x^2y}$ is the same as $x\sqrt{y}$ (if $x$ and $y$ are positive). Seeing that $\sqrt{45} = 3 \sqrt{5}$ is a sort of rehearsal for this, and as Alexei points out comes quite appropriately N-RN.2. But treating such simplification as an end in itself, accompanied by long lists of problems, is an example of misplaced priorities. That’s why the standards don’t make explicit mention of it.

]]> <![CDATA[Reply To: rational root theorem]]> Sat, 03 May 2014 00:55:12 +0000 Bill McCallum Lane, you have it exactly right here.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Sat, 03 May 2014 00:54:14 +0000 Bill McCallum Yes, the first proof is just wrong, and also misguided. The similarity of circles follows very directly from the definition of similarity in terms of dilations, as explained above. Trying to go via similar triangles strikes me as extremely irrational. And the review of triangle similarity is out of sync with the standards.

]]> <![CDATA[Reply To: 8.G.2 – Demonstrating Rotational Symmettry]]> Sat, 03 May 2014 00:44:38 +0000 Bill McCallum It depends what you mean by “all angles of rotation.” Students can make arguments where the angle of rotation is specified by another angle in the diagram. So, you might prove SAS congruence by first translating so that the vertices with the A coincide, and then rotating so that two sides coincide, and then reasoning that the other vertices must coincide as well. This is an arbitrary rotation, but the argument doesn’t require you to ever specify the angle measure of the rotation: you would simply say “rotate by angle AOB” or something like that. So that is well within the standards.

But, giving a coordinate formula for an arbitrary rotation through a given angle measure is beyond the standards.

]]> <![CDATA[Reply To: Blending 7th and 8th grade CCSS to create a Pre-Algebra course]]> Sat, 03 May 2014 00:38:46 +0000 Bill McCallum I don’t know how the district resolved the issue. I continue to think that for many students acceleration leads to shallow grasp of the mathematics, which ultimately leads to them hitting remediation when they get to college. Acceleration is like adderall: appropriate for some students but way over-used. Parents and schools are beginning to appreciate the problems with over-prescription of adderall; it’s time that started appreciating the problems with over-prescription of acceleration.

]]> <![CDATA[Reply To: 4.NF.5–7 (decimal fractions)]]> Sat, 03 May 2014 00:30:52 +0000 Bill McCallum The standards don’t regard decimals and fractions as different types of numbers between which conversions must be made, but rather as different notations for writing fractions. Note the language of the cluster heading above 4.NF.5: “Understand decimal notation for fractions, and compare decimal fractions.” Thus, students should see 0.57 as another way of writing the fraction 57/100. I think this means that the answer to all your questions is yes (subject to limits of common sense, of course … you don’t want to get carried away with all the ramifications of understanding decimal notation right away).

]]> <![CDATA[Reply To: Surface Area of a Cylinder]]> Sat, 03 May 2014 00:26:07 +0000 Bill McCallum I agree with abieniek’s ideas for where you might teach this. I would add that you don’t have to assess everything you teach, and since this is not explicitly in the standards it is not required that it be assessed.

]]> <![CDATA[Reply To: 3.NBT.3]]> Sat, 03 May 2014 00:09:18 +0000 Bill McCallum Lane basically has it right here: 3.NBT.3 is about 4 x 80, and 3.OA.5 allows for students reasoning from this to 80 x 4. But the emphasis in Grade 3 would be on 4 x 80.

]]> <![CDATA[Reply To: 2G1]]> Sat, 03 May 2014 00:03:48 +0000 Bill McCallum It’s worth quoting the entire text of 2.G.1 here.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

The second sentence specifies that the shapes to be recognized and drawn should include the ones listed, but does not limit to those shapes. The core of this standard is the first part of the first sentence: students should have experience recognizing and drawing shapes with specified attributes. The progression gives examples of this that go beyond the list in the second sentence, but should not be interpreted as a required interpretation of this standard. So, basically, this is really up to states to interpret. I would add that the consortium assessments don’t start until Grade 3, so there is really some flexibility here.

]]> <![CDATA[Reply To: A.APR.4]]> Fri, 02 May 2014 23:45:39 +0000 Bill McCallum Great, thanks for letting us use them, and thanks for the clarification.

]]> <![CDATA[Reply To: relative size of metric units in 4th]]> Fri, 02 May 2014 23:42:13 +0000 Bill McCallum The “know relative sizes” in the standard is really about students knowing, for example, that 1 kg is 1000 times as large as 1 g. That said, they can do the comparisons you ask by expressing 0.1 kg as 100 g, or 1/10 m as 10 cm. Expressing larger units in terms of smaller units is part of the standard as well.

]]> <![CDATA[Reply To: Geometry 6 G 2]]> Fri, 02 May 2014 23:40:45 +0000 Aaron Bieniek Please see this post:

]]> <![CDATA[Reply To: Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Fri, 02 May 2014 23:32:57 +0000 Bill McCallum Middle school acceleration made some sense when the middle school curriculum was impoverished, as it often was under previous state standards. It makes less sense when the middle school standards are as rich and demanding as they are under the Common Core. I certainly understand all the forces driving middle school acceleration and I also understand that those forces are not going to go away overnight. But parents and schools do not have to submit to those forces. I didn’t with my own children, and they are all happy and successful.

]]> <![CDATA[Reply To: Geometry 6 G 2]]> Fri, 02 May 2014 22:03:32 +0000 carlymorales A group of us working on writing curriculum noticed that the standards have the formula is written v=bh noting that the B is not capitalized. We wondered why it isn’t a B?

]]> <![CDATA[Reply To: trapeziod definition]]> Fri, 02 May 2014 18:27:51 +0000 Bill McCallum Yes, PARCC is using the inclusive definition, see here:

]]> <![CDATA[6.G.3]]> Thu, 01 May 2014 17:19:55 +0000 mitzell How are 6th graders to calculate distance in the coordinate plane without having operations of integers until 7th grade?

]]> <![CDATA[Reply To: simplifying radicals]]> Wed, 30 Apr 2014 22:41:57 +0000 Aaron Bieniek Seems to me this is addressed in N-RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.

]]> <![CDATA[Reply To: Simplifying radicals]]> Wed, 30 Apr 2014 21:05:07 +0000 Alexei Kassymov Expressions and Equations progression says: “Notice that students do not learn the properties of rational exponents until high school.” (a*b)^(1/2) = a^(1/2)*b^(1/2) in the form sqrt(a*b)=sqrt(a)*sqrt(b) appears to fit this description.

On the other hand N-RN.2 looks very appropriate. Part b in the task below uses the property:

Algebra I makes more sense.

]]> <![CDATA[simplifying radicals]]> Wed, 30 Apr 2014 18:23:39 +0000 bbaggett 8.NS.2 asks students to use rational approximations of irrational numbers to compare the size of irrationsal numbers, locat them on a number line diagram and estimate the value of expressions. 8.EE.2 says to use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p and to evaluate tje square roots of small perfect squares and cube roots of small perfect cubes.

So….this does not look to me like 8th grade students should be simplifying radicals since 8th grade is the first time students have worked with square roots and cube roots (and the standards say specifically perfect squares and perfect cubes). But, I don’t see anything in the high school standards that says where simplifying radicals should be taught. My Algebra I teachers believe it should be in 8th grade. My 8th grade teachers believe it should be in Algebra I. What was the intention?

]]> <![CDATA[Simplifying radicals]]> Wed, 30 Apr 2014 18:22:10 +0000 bbaggett 8.NS.2 asks students to use rational approximations of irrational numbers to compare the size of irrationsal numbers, locat them on a number line diagram and estimate the value of expressions. 8.EE.2 says to use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p and to evaluate tje square roots of small perfect squares and cube roots of small perfect cubes.

So….this does not look to me like 8th grade students should be simplifying radicals since 8th grade is the first time students have worked with square roots and cube roots (and the standards say specifically perfect squares and perfect cubes). But, I don’t see anything in the high school standards that says where simplifying radicals should be taught. My Algebra I teachers believe it should be in 8th grade. My 8th grade teachers believe it should be in Algebra I. What was the intention?

]]> <![CDATA[rational root theorem]]> Wed, 30 Apr 2014 14:45:40 +0000 lhwalker I do not see the rational root theorem in the CCSS-M and assume students are not expected to memorize it. However, most textbooks have a unit where students must use it to list all possible rational zeros, and I see it included in several curriculum posted online. That might fall under A-APR.3 but I’m thinking what the students really need to know are the connections between zeros, factors, and remainders…not memorize the theorem or the formula x = p/q. Surely being aware of the theorem would be good but it doesn’t appear to me that we will be spending a lot of time with it if it isn’t mentioned in the standards…?

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Sun, 27 Apr 2014 17:12:33 +0000 dan Consider the first “proof” in this link but instead, start with a circle radius 2 and a square with side 4.
Then please correct me if I’m wrong, but doesn’t the “similar triangles” proof written out in this link also prove that the 2 unit circle and 4 unit square are similar?

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Sat, 26 Apr 2014 22:58:43 +0000 Joshua Garien Take a look at this site.

]]> <![CDATA[Reply To: 8.G.2 – Demonstrating Rotational Symmettry]]> Thu, 24 Apr 2014 13:51:26 +0000 dseabold So, when do students need to work with any angle of rotation? In HS with GEO-CO standards? Or is it enough to stick with multiples of 90 degrees in high school as well?

]]> <![CDATA[Reply To: 8.G.2 – Demonstrating Rotational Symmettry]]> Thu, 24 Apr 2014 13:40:54 +0000 dseabold So, when should rotations include all angles of rotation, in Geo-CO in high school? Or is it enough to stick to rotations that are multiples of 90 degrees there as well?

]]> <![CDATA[Reply To: Blending 7th and 8th grade CCSS to create a Pre-Algebra course]]> Wed, 23 Apr 2014 14:29:02 +0000 pherman I realize this post is over a year old….but I am in a district working with the same issue/concern originally posted.

I am definitely curious as to how her district resolved this issue.

In my area there is a trend to have students take Algebra I as early in their educational experience as possible. My district typically offers it to “high achieving” students in Grade 8. (Many districts even offer it in 7th.)

Thanks for the links to some other sources, I plan to check them out next.
THEN… I may be back with some additional questions!

]]> <![CDATA[Reply To: Tape Diagrams]]> Mon, 21 Apr 2014 15:30:09 +0000 Alexei Kassymov NF progression mentions tape diagrams. There are several examples in the OA progression.

]]> <![CDATA[Reply To: 3.NBT.3]]> Sat, 19 Apr 2014 05:03:37 +0000 lhwalker I think you are correct that the standard is focused on the structure where a single digit is multiplied by a multiple of ten. As a high school teacher I use the K-8 standards to remediate, and many of my students are surprised when I point out (as they are reaching for a calculator) that 2 x 80 is simply doubling eight tens. Worse, most of them think they need to put a one under the 2 to multiply 2 x 1/8. So it makes sense to have a time of focus on the particular structure where a single digit comes first. However, that doesn’t negate 3.OA.B.5 where students apply the commutative property.

]]> <![CDATA[Tape Diagrams]]> Fri, 18 Apr 2014 18:14:01 +0000 alwong In k-5, the common core says to use models. In 6th grade, it specifically says tape diagrams. Why isn’t k-5 referring to models are bar models or tape diagrams. Are they the same? Different? Can you clarify for me? I am trying to make a smooth transition from elem to middle using similar vocabulary. Thanks!

]]> <![CDATA[4.NF.5–7 (decimal fractions)]]> Thu, 17 Apr 2014 17:23:18 +0000 johnrmead We hit a point of disagreement around the decimal standards in my school today. A couple of questions generated a bit of discussion around interpretations of standards:

Are grade 4 students expected to convert from decimal form into fractions? (only decimal to fraction is explicitly included)

Are grade 4 students expected to do arithmetic with decimal fractions (0.73-58/100)?

Are grade 4 students expected to interpret decimal expressions written in words (1 and 57 hundredths of a meter)?

The last question sparked what I’ll politely call a spirited conversation, particularly in light of the NBT progressions, where grade 4 is limited to whole numbers, and grade 5 expands to thousandths.

]]> <![CDATA[Reply To: Surface Area of a Cylinder]]> Tue, 15 Apr 2014 16:51:12 +0000 Aaron Bieniek To me it seems like anytime after students have a good grasp of surface area, and also a firm grip on area of a circle. So that might put it in 7th grade or maybe 8th grade (or maybe not at all…is that ok?). As Bill mentioned earlier in the discussion, it depends on how the curriculum is designed. I really like the idea of surface area of a cylinder as a study of nets and 3D figures – digger deeper into the concept of surface area. I guess it also matters how you would assess it. How would you assess it?

]]> <![CDATA[Reply To: Surface Area of a Cylinder]]> Tue, 15 Apr 2014 13:39:16 +0000 bumblebee Where, in your opinion, would it be fair to assess being able to find the surface area of a cylinder?

]]> <![CDATA[3.NBT.3]]> Mon, 14 Apr 2014 22:17:57 +0000 Alexei Kassymov In some cases of introducing operations on a limited scale, for example in 4.NF.4b, the order of numbers matters – 3 x (1/2) is OK, but (1/2) x 3 is a 5th grade expectation. Are students expected to perform, say, 4 x 80 only or 80 x 4 too in third grade?

3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

]]> <![CDATA[2G1]]> Mon, 14 Apr 2014 21:25:12 +0000 ccoward Our standard 2.G.1 says students should be able to identify triangles, quadrilaterals, pentagons, hexagons, and cubes. From reading this I though the students just needed to know that 3 sides = triangle, 4 sides = quadrilateral, etc. In reading the progression description for this standard though, it specifies that they should be able to identify specific categories of quadrilaterals: trapezoids and rhombus. Are they expected to know and be able to pick out those shapes and if so – how do we know that except by reading the progressions (why aren’t they specified in the standard itself?) Is there a list of expected shapes some where?

Also, my colleagues and I have been wondering how much work we should do with 3D shapes. The standard says students need to know about faces, edges, and vertices, but the only 3D shape they need to be able to identify is a cube. Should we go into prisms and other 3D shapes?

Thank you!

]]> <![CDATA[Reply To: A.APR.4]]> Sun, 13 Apr 2014 09:41:07 +0000 tomergal First of all, of course you can use my examples. I should also probably mention that I’m writing questions for Khan Academy.

Second, I think we’re in agreement that the intention in “use polynomial identities” is that students should be able to use identities as a tool in reasoning about numbers.

I was mainly confused by the example of the “Pythagorean” identity. I think that under my interpretation, the “use” of this identity is to explain why for any integers x and y, the three expressions (x^2+y^2)^2, (x^2-y^2)^2, and (2xy)^2 form a Pythagorean triple. This is different from the “use” suggested by the example, which is the act of finding triples by substituting specific integers for x and y. Finding Pythagorean triples is a very specific use, which I wasn’t able to generalize to a broader category of application.

Hope that was clearer.

  • This reply was modified 3 years, 4 months ago by  tomergal.
  • This reply was modified 3 years, 4 months ago by  tomergal.
  • This reply was modified 3 years, 4 months ago by  tomergal.
]]> <![CDATA[relative size of metric units in 4th]]> Fri, 11 Apr 2014 22:49:08 +0000 Alexei Kassymov Students work with relative sizes of units in 4.MD.1. Would it be expected that in 4th grade students know that 0.1 kg is more than 1 g, for example? Or 1/10 m is greater than 1 cm?

]]> <![CDATA[Reply To: Converting fractions and decimals]]> Wed, 09 Apr 2014 03:07:39 +0000 Bill McCallum In the Common Core decimals are treated simply as a different way of writing fractions with denominator 10, 100, and so on. So Grade 5 students can certainly see the equivalence of 3/5 and 0.6 because they can see the equivalence of 3/5 and 6/10. For 5/8 you are getting into 1000ths, so that would have to wait.

]]> <![CDATA[Reply To: A.APR.4]]> Wed, 09 Apr 2014 03:05:04 +0000 Bill McCallum I’m not clear on the distinction between your two interpretations, but the examples you came up with are really great, so I guess the second one is best! I’ll get the Illustrative Mathematics team working on more. Can we use yours?

]]> <![CDATA[Reply To: Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Mon, 07 Apr 2014 01:43:14 +0000 sunny Forgive me, but this mode of acceleration seems awfully excessive, especially with the new standards. I hope that this rush through standards isn’t at the expense of the students love or like for math, or conceptual understanding of math. I am also left to wonder of the preparation of the middle school and K-5 teachers. Are they prepared to teach high school courses? Do your K-5 and MS teachers have math degrees? Are they prepared to teach these new standards, which are already radically more rigorous? “Algebra 1″ and “Algebra 11″ are just the names of the courses. In our district, the standards that are in these courses are radically different and more rigorous than we had in the past.
I am curious about your data… do many students who complete this mode of acceleration enter an elite university in a STEM field? or complete a degree in a STEM field? Lots of questions…

]]> <![CDATA[Reply To: Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Mon, 07 Apr 2014 01:23:53 +0000 sunny Forgive me, but this mode of acceleration seems awfully excessive, especially with the new standards. I hope that this rush through standards isn’t at the expense of the students love or like for math, or conceptual understanding of math. I am also left to wonder of the preparation of the middle school and K-5 teachers. Are they prepared to teach high school courses? Do your K-5 and MS teachers have math degrees? Are they prepared to teach these new standards, which are already radically more rigorous? “Algebra 1” and “Algebra 11” are just the names of the courses. In our district, the standards that are in these courses are radically different and more rigorous than we had in the past.
I am curious about your data… do many students who complete this mode of acceleration enter an elite university in a STEM field? or complete a degree in STEM field? Lots of questions….

]]> <![CDATA[Reply To: Geometry definitions]]> Mon, 07 Apr 2014 00:24:59 +0000 Sarah Stevens Hi! Have you read the Geometry progression for K-6. I know your question is specific to 3rd grade but you might find your question answered by reading the entire document. It can be found here

Also, you might look in the K6 Geometry section of this forum. I often find I am not the first to have a question and, therefore, can find an answer there.

Hope this helps!

]]> <![CDATA[Reply To: 5th grade Mixed Number addition & subtraction–regrouping]]> Sun, 06 Apr 2014 03:17:47 +0000 Bill McCallum If I understand the question correctly, you are saying that sometimes students should add fractions expressed as mixed numbers by grouping the whole number parts together and adding them, and grouping the fractional parts together and adding them, and then putting the two results together. And, other times they might just want to expressed the fractions in purely fractional form and add the numbers together in that form. And, they should have some judgement about when to do which.

If that’s what you are saying, then I wholeheartedly agree!

]]> <![CDATA[Reply To: 3.G.1 vs. 4.G.2]]> Sun, 06 Apr 2014 03:13:26 +0000 Bill McCallum Good question, and I think different curricula might approach this differently. Some might choose to introduce some of these Grade 4 concepts in an informal way in Grade 3. But I would point out that the classification into squares, rhombuses, and rectangles really only requires an ability to recognize angles as square angles or not, and an ability to detect when pairs of sides, or all four sides, are equal in length. This could be the right way to approach this with 3rd graders.

]]> <![CDATA[Reply To: Power standards]]> Sun, 06 Apr 2014 03:08:55 +0000 Bill McCallum Yes, this is not a classification made by the standards themselves, although I do think PARCC did a pretty good job of interpreting the standards with this classification, and I would add that SBAC classifies the standards the same way (except that they merge the supporting and additional categories).

]]> <![CDATA[Reply To: Enhanced High School Sequence?]]> Sun, 06 Apr 2014 03:07:22 +0000 Bill McCallum Well, I think this is a great idea, but I don’t know of anybody who has done it yet.

]]> <![CDATA[Reply To: Enhanced High School Sequence?]]> Sun, 06 Apr 2014 03:07:20 +0000 Bill McCallum Well, I think this is a great idea, but I don’t know of anybody who has done it yet.

]]> <![CDATA[Reply To: Mathematics Practices in the classroom]]> Sun, 06 Apr 2014 03:06:40 +0000 Bill McCallum This sounds awful. No, there is no goal that students be able to name the practices. The standards are not intended to be read by children, but rather by their teachers and those who design curriculum for children.

]]> <![CDATA[Reply To: Geometry Progressions]]> Sun, 06 Apr 2014 03:05:06 +0000 Bill McCallum I think highly of both thinkers, and have read both their writings on this, but don’t have a detailed comparison of their different interpretations of the geometry standards to throw out right now.

]]> <![CDATA[Reply To: Geometry Progressions]]> Sun, 06 Apr 2014 03:05:04 +0000 Bill McCallum I think highly of both thinkers, and have read both their writings on this, but don’t have a detailed comparison of their different interpretations of the geometry standards to throw out right now.

]]> <![CDATA[Reply To: KGA]]> Sun, 06 Apr 2014 03:02:07 +0000 Bill McCallum I don’t think this standard gets down to this level of detail (distinguishing hexagons from octagons) and I think it is certainly beyond the standards to be assessing Kindergartners on this distinction. The list in parentheses is neither inclusive nor exclusive. Like most other lists in the standards it is there for guidance, and such guidance must always be wedded to common sense. There are many overly focused assessment schemes that are simply trying to read more fine grained resolution from the standards than is there; the standards were not designed to be friendly to such schemes, rather preferring schemes that focus on larger coherent units of mathematical knowledge.

]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Sun, 06 Apr 2014 02:55:33 +0000 Bill McCallum This is a good discussion, and I don’t have much to add, except to confirm that students are not required to calculate standard deviation by hand. Students should be seeing real data sets where this would be absurd.

]]> <![CDATA[Reply To: 5.OA.2]]> Sun, 06 Apr 2014 02:52:00 +0000 Bill McCallum Yes, you are right, this isn’t called out explicitly. And, in fact, the main point for students to understand eventually is nesting of grouping symbols, rather than the hierarchy itself. That is to say, if you can parse a complex expression with parentheses nested 2 or 3 deep, then that is the main point; it doesn’t particularly matter if you go parentheis, bracket, brace, as I was taught. It was really this idea of nesting that I wanted to get across, and mainly the idea that it doesn’t have to happen in Grade 5.

This whole subject is an area where the standards are pretty agnostic. Reading the conventions of mathematical notation is important, but conventions themselves are not mathematical concepts. So, the ability to read nested grouping symbols is implicit in A-SSE.1–3, for example, but not explicitly mentioned.

]]> <![CDATA[Reply To: trapeziod definition]]> Sat, 05 Apr 2014 21:47:41 +0000 Lois I am getting ready to begin teaching a grade 3 geometry unit and all the materials I have found use the exclusive definition. The definition needs to be clarified since third grade has only two geometry standards and one is about 2-D attributes. The inclusive definition will totally change the image that our students now recognize as a trapezoid. It will include all squares, rectangles, parallelograms, and rhombi.
Is PARCC aware of the discrepancy in terms?

]]> <![CDATA[Geometry definitions]]> Sat, 05 Apr 2014 21:37:32 +0000 Lois I am struggling to write a geometry unit for gr. 3. There are only 2 standards and the first is broad. Children need to be able to understand that “shapes may share attributes, and that the shared attributes can define a larger category.”
Is there a listing of attributes/vocabulary (with definitions) that third graders are expected to know? Do they need to know the terms vertex or angle, parallel, right angle, ray, line, etc.?

]]> <![CDATA[Reply To: Definition of a Trapezoid]]> Sat, 05 Apr 2014 19:32:56 +0000 Aaron Bieniek This comes directly from the Progression Document on Geometry K-6:
Note that in the U.S., that the term “trapezoid” may have two different meanings. In their study The Classification of Quadrilaterals (Information Age Publishing, 2008), Usiskin et al. call these the exclusive and inclusive definitions:
T(E): a trapezoid is a quadrilateral with exactly one pair of parallel sides
T(I): a trapezoid is a quadrilateral with at least one pair of parallel sides.
These different meanings result in different classifications at the analytic level. According to T(E), a parallelogram is not a trapezoid; according to T(I), a parallelogram is a trapezoid.
Both definitions are legitimate. However, Usiskin et al. conclude, “The preponderance of advantages to the inclusive definition of trapezoid has caused all the articles we could find on the subject, and most college-bound geometry books, to favor the inclusive definition.”

]]> <![CDATA[Definition of a Trapezoid]]> Sat, 05 Apr 2014 18:04:26 +0000 maddie Which definition of a trapezoid works best for the Common Core Standards? At least one or only one pair of parallel sides. This question is geared towards K-6.

]]> <![CDATA[Reply To: Trying to understand G-C.5]]> Fri, 04 Apr 2014 21:11:36 +0000 moberlin Thank you for your response. The one area that is still a little fuzzy for me has to do with the assumption that arc length responds to a dilation like any segment length. I understand that this is true but I am wondering what sense students will make of this. It seems that we need to first develop the concept of arc length. How do you suggest arc length be developed – by approximating it with segments of uniform length and then using an informal limits argument (as one might to develop the concept of the circumference of a circle)? Thanks again!

]]> <![CDATA[Converting fractions and decimals]]> Fri, 04 Apr 2014 18:16:23 +0000 alwong Hello! I am having a debate with my math colleagues about what grade level do students convert or understand fraction and decimal equivalencies? I know in 5th grade they learn about tenths and hundredths in both fraction and decimal forms. But what about 5/8, 3/5…as decimals…etc? What is your opinion?

Also, I have noticed in a lot of resources for suggested units, it begins in 6th grade with ratios and proportions. Why?


]]> <![CDATA[Reply To: A.APR.4]]> Thu, 03 Apr 2014 08:02:40 +0000 tomergal I’m having trouble with figuring out the meaning of this standard (“Prove polynomial identities and use them to describe numerical relationships.”) It seems there are two general aspects to the standard: proving identities, and using them. The standard, the progression doc, and the Illustrative Math example all put much more focus on the applied aspect of the standard. However, while there are infinite polynomial identities to prove, we have only two specific examples of polynomial identities we can use, and these uses are both pretty hard to replicate with other polynomial identities.

The example suggested by the standard itself (let’s call it the “Pythagorean” identity) implies that the use of polynomial identities is such that there are some special identities we can use by plugging in values and obtaining meaningful sets of numbers. I would really appreciate more examples of polynomial identities we can use this way.

Being unable to find even one more example of a such an identity, I came up with a different interpretation of the standard. According to this interpretation, the standard is aiming for students to arrive at different polynomial identities by themselves, in order to prove theorems that regard numerical relationships. The “Pythagorean” identity doesn’t fit this interpretation, since I don’t think we can expect students to derive it by themselves. It also isn’t used as a part of a grander proof. The case of (n+1)^2-n^2=2n+1 seems more to the point here. With some guidance, students should be able not only to prove this identity, but to actually derive it as a part of a modeling effort to explain why the difference between consecutive perfect squares is always odd.

Under this interpretation, I thought it could be possible to use polynomial identities to prove some divisibility issues. For instance, the identity n^2+n=n(n+1) can explain why for any value of n, the result of n^2+n is an even number. Similarly (but more elaborately), the identity n^3+3n^2+3n=n(n+1)(n+2) can explain why for any value of n, n^3+3n^2+3n is divisible by 6.

To sum it up, I would appreciate:
a. More examples of polynomial identities that can be useful in a similar manner to the “Pythagorean” identity.
b. Your opinion of the two different interpretations and the examples that follow.

]]> <![CDATA[5th grade Mixed Number addition & subtraction–regrouping]]> Wed, 02 Apr 2014 19:07:45 +0000 hcoffey As a math coach, I am trying to advise my teachers on the expectations of the standards. I understand the most important thing in adding and subtracting mixed numbers is understanding why you do what you do. My 5th grade teachers asked if they are supposed to teach regrouping when adding and subtracting fractions. My thoughts were to advise them to teach it as one strategy, and focus on students understanding of when to use that strategy (when it is efficient and when it is not), rather than teaching regrouping as the way to do it all the time. I would recommend that they also teach students that converting to improper fractions can also be efficient,depending on the problem. Am I on the right track? Any other insights would be appreciated. Thanks!

]]> <![CDATA[Reply To: Reading Inequality Symbols]]> Mon, 31 Mar 2014 23:29:05 +0000 Bill McCallum I guess in Kindergarten students might just be saying “bigger” and “smaller,” but I don’t think they need to wait until Grade 6 to see “greater than” and “less than.” In fact, you want bigger and smaller to go away sooner than that, because of the confusion this could cause with comparisons of negative numbers.

On “wide part,” I would say that’s a little different from alligators eating something, because it relates quantities visible in the symbol (the width of each end of the symbol) to the quantities in the comparison; there is no eating action here, which I agree is extraneous!

]]> <![CDATA[Reply To: Trying to understand G-C.5]]> Mon, 31 Mar 2014 23:18:45 +0000 Bill McCallum This follows from the fact that all circles are similar. It’s quite fun to figure out why using the definition of a circle and the definition of a similarity transformation. (I can supply the answer later if you want.)

Now, a similarity transformation with scale factor $k$ transforms any length by a scale factor of $k$, including the arc length. So, just as the radius gets multiplied by $k$, so does the arc length. This means that the ratio between the arc length and the radius stays equivalent no matter what the radius; in other words, the arc length is proportional to the radius.

So this tell us that
\mbox{arc length} = \mbox{constant} \times \mbox{radius}.
Setting the radius equal to 1 tells us what the constant is: it’s just the arc length for a circle of radius 1, which is exactly the radian measure of the angle.

]]> <![CDATA[3.G.1 vs. 4.G.2]]> Mon, 31 Mar 2014 20:17:15 +0000 dseabold It seems as though talking about parallel sides and the types of angles in figures would be very helpful to pull 3.G.1 off. In fact, in looking at the Geometry Progression, on page 13 (3rd grade), it even mentions using parallel sides as a category in the example in the right margin. My question is how can informatoin about parallel sides or specific angles be used to categorize shapes in 3rd grade with this is not called for until 4th GRADE with 4.G.A.2:
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

]]> <![CDATA[Reply To: Power standards]]> Mon, 31 Mar 2014 16:37:17 +0000 Aaron Bieniek The effort of sorting clusters into major, minor, and additional was done by the PARCC assessment consortium. They explain their reasoning, rationale, as well as Do’s and Don’ts in this document: Specifically starting on the bottom of page 2 – Content Emphasis by Cluster.

]]> <![CDATA[Reply To: Power standards]]> Mon, 31 Mar 2014 15:31:02 +0000 harrelli Could you please provide more clarification as to why the clusters are identified as “Major, Supporting, and Additional”? I provide professional development trainings for teachers in our school district and I would like to be able to help explain this to them with confidence as we develop curriculum maps and pacing guides.

Thank you,

]]> <![CDATA[Enhanced High School Sequence?]]> Wed, 26 Mar 2014 17:25:00 +0000 jspencer I am hoping that someone out there has explored doing an enhanced model for high school content. Generally, the idea is that additional standards are put into Algebra 1, Geometry, and Algebra 2 to eliminate Pre-Calc. Students would go directly from Algebra 2 to AP Calculus.

We are looking at this as a potential model to “fix” the 8th grade/algebra 1/pathway to calculus issue. Massachusetts Dept. of Ed. has not approved this model but has documentation about how they envision the standards being arranged.

We are also looking at the compacted 7th grade model, but this may be less attractive for a variety of reasons…

Thank you,

]]> <![CDATA[Mathematics Practices in the classroom]]> Wed, 26 Mar 2014 02:01:30 +0000 KarenS I recently saw that a publisher had created some “animations” of the Math Practices to teach the children in every grade level what the practices look like in their grade level. The children can then check off when are using that practice. It made me wonder if it was a goal that the students be able to name/identify when they are engaged in a practice, or if the goal was to develop habits of thinking and ways of engaging in the work with their peers?’

]]> <![CDATA[Reply To: Geometry Progressions]]> Sat, 22 Mar 2014 04:15:02 +0000 eprebys Any thoughts on Guershon Harel’s article that is essentially a response to Wu’s article?

]]> <![CDATA[REI.7 – A circle "being" a quadratic]]> Thu, 20 Mar 2014 17:59:03 +0000 dhust When I came across the explanation of this standard and then saw the example I was a little thrown off. The standard talks about a quadratic and linear equation but the example talks about a circle and a linear equation. When the word “quadratic” was used was it meant to pertain to any second degree equation? If so, could I use an ellipse or a hyperbola in place of the circle and have it still align to the standard?


]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Thu, 20 Mar 2014 12:48:15 +0000 csteadman I think that is a great interpretation of the standards considering they are at the Algebra I grade level in PARCC states and represent an additional cluster. There will be more work with standard deviation in Algebra II in S-ID.4, although that is in the additional cluster as well.

I think it is interesting how curricula will be written as writers across the country interpret the standards. There is a lot you could do with S-ID.1-3, but you have to make decisions timewise, considering the rest of the S-ID’s, while making sure the central purpose of the standards, as you mentioned above, are covered.

]]> <![CDATA[Reply To: KGA]]> Thu, 20 Mar 2014 03:13:28 +0000 Duane Sorry, the link didn’t paste correctly:

]]> <![CDATA[Reply To: KGA]]> Thu, 20 Mar 2014 01:55:19 +0000 Duane Aside from anything else you’ve asked, regarding hexagons it may be helpful to emphasize the number of sides/corners as being the defining feature. If students are getting confused between a hexagon and an octagon it may be that they are reasoning purely visually, rather than counting the sides. To check if this is the case try using a variety of hexagons for identification. This page has some examples:

]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Wed, 19 Mar 2014 17:54:05 +0000 Aaron Bieniek I guess when I read the standard, my thought was more along the lines of how we can get students to understand that standard deviation is basically just a value that we assign a set of data in order to rank its spread from the mean. So given 8 sets of data with the same mean and different spreads, students can intuitively rank those sets from least spread out to most spread out. The fact that we have a fancy way of calculating that rank and the fact that we call it standard deviation are nice facts, but we don’t need to know those things in order to talk about and explore the idea of spread. I would hope that after students have a handle on the idea of spread they would be able to see that any outliers present would make a set of data more spread out. I don’t see the need for any calculation to have that discussion.

I think if one approaches the standard with the mindset that knowing how to calculate standard deviation is important, then you will find ways to try to work it in. My approach is that the standard seems to be silent on the issue so I’ll look for instructional methods that don’t depend on the calculation itself.

]]> <![CDATA[KGA]]> Wed, 19 Mar 2014 14:21:52 +0000 starksj KGA
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres)

Question 1: Is this list intended to be exclusive? If a student knows more or less would we consider them below or above meeting the expectations?

Question 2: Should kindergartners teachers take time to teach the difference between hexagons and octagons? We are seeing many kindergartners in our school who are getting mixed up about the two shapes. They’ve been exposed to octagons in real life (stop signs) and have discussed hexagons in class, but when they see one or the other in isolation, and are not able to side-by-side make comparisons, they do not naturally notice the difference. Since octagons are not listed in the standard, I just wondered how we should address this. Of course, I am looking at this from an assessment stand point. How do we assess that the kids have met the expectation of the standard and how do we ensure that teachers are meeting the expectation of the standard in their instruction.

]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Wed, 19 Mar 2014 12:55:49 +0000 csteadman New York has a lesson in Algebra I where students walk through the formula. Then students calculate and interpret standard deviation. I wouldn’t restrict calculator use completely.

I think there is a fine line in how deep you go in certain standards at a ninth grade level. How far do you go when a student asks where the correlation coefficient comes from? Show them the formula and smile.

]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Wed, 19 Mar 2014 12:50:30 +0000 csteadman New York has one lesson in Algebra I where students walk through the formula. Then students calculate and interpret standard deviation. I wouldn’t restrict calculator use completely.

I think there is a fine line in how deep you go in certain areas at a ninth grade level. How far do you go when a student asks where the correlation coefficient comes from? Show them the formula and smile.

]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Tue, 18 Mar 2014 17:12:50 +0000 missbaldwin Agreed. There does not seem to be emphasis on calculating standard deviation. We’re wrestling with how students will understand what standard deviation is, and how outliers affect its value, without some introduction to its calculation.

]]> <![CDATA[Reply To: Standard Deviation in S-ID.A]]> Tue, 18 Mar 2014 02:16:29 +0000 Aaron Bieniek I don’t see anything in the standards that requires students to calculate standard deviation at all – by hand, by technology, or otherwise. Seems like the standards want students to be able to use and understand standard deviation, but not necessarily calculate it.

]]> <![CDATA[Standard Deviation in S-ID.A]]> Mon, 17 Mar 2014 21:01:19 +0000 missbaldwin We are working on curriculum maps for Integrated HS Math 1 which would include S-ID 1 – 3 and are wondering whether students must know how to calculate standard deviation by hand without the use of technology.

]]> <![CDATA[Reply To: Confidence Intervals]]> Sat, 15 Mar 2014 00:26:53 +0000 Cathy Kessel That’s discussed here:

]]> <![CDATA[Confidence Intervals]]> Thu, 13 Mar 2014 22:29:18 +0000 CCSDmathgirl Is Confidence Intervals part of S.IC.B.4?

]]> <![CDATA[Reply To: Reading Inequality Symbols]]> Thu, 13 Mar 2014 21:16:44 +0000 lhwalker Ha! I just graded a unit test and I saw a lot of: sq root 11 < 3 2/3 > 3.51. Because I never know exactly what vocabulary my students will encounter, I read 0 < x < 10 as “x is between” and sometimes say, “Zero is less than x which is less than 10, but isn’t that a mouthful?” Between notation is succinct for describing domains and ranges. Is there a way to connect the terminology “order symbols” to a solution set like {x | 0 < x < 10}?

]]> <![CDATA[Reply To: Reading Inequality Symbols]]> Thu, 13 Mar 2014 18:53:19 +0000 johnrmead It might be nitpicking, but another issue that pops up in some high school courses is the 0<x<1 notation. One could argue that it’s an abuse of notation, but it is common enough that at least some students will come across it sooner or later. In this setting, the preferred reading might be that x is between 0 and 1, a statement which completely omits any mention of “greater than” or “less than”. It has always struck me as imprecise when I’ve told students that “<” means “less than”. I would much prefer to call both signs order symbols and instruct that “4<6” could be read “four is less than six” or “six is greater than four”.

]]> <![CDATA[Reply To: Reading Inequality Symbols]]> Thu, 13 Mar 2014 17:21:26 +0000 lhwalker You raise a very important issue. We definitely need to get rid of mouths because x<6 is interpreted as x eats 6. Wide-part-as-large, in my opinion, seems mathematically sound and works very well with my Algebra students, particularly when analyzing word problems: “She needs to make at least $50, so what needs to be large?” After seeing <> for many years, many of my 14-year-olds still get them mixed up. I write <ess frequently, but still… So after my initial surprise to see wide-is-large in the progressions, I felt affirmed in my practice. It makes sense to me to have mental images like 8<10, 0.05 < 0.5, 1/2 > 1/3, driving the recall. We all have “hooks” that are necessary for retention. So if they can recall an example like 8<10, then they can recall “less than” if they pause a second to think 😉

]]> <![CDATA[Reply To: 5.OA.2]]> Thu, 13 Mar 2014 14:26:47 +0000 Mary I am trying to figure out at what point students would be expected to work with a a hierarchy of symbols. Unless I am missing it, that isn’t explicitly called out in the 6 – 8 standards.

]]> <![CDATA[Definition of shape]]> Thu, 13 Mar 2014 13:59:35 +0000 Sarah Caban I am working on a 3rd grade unit that encompasses area, perimeter, and Geometry.
We are hoping to get our students to be able to sort shapes according to defining and non-defining attributes using a graphic organizer that illustrates hierarchical inclusion.
I am struggling with finding a consistent definition of “shape”
Is a shape, by definition, a closed figure? Or is “closed” defining when a figure is 2D or 3D?
Am I correct that being closed is a defining attribute of plane figures?

]]> <![CDATA[Reply To: Reading Inequality Symbols]]> Thu, 13 Mar 2014 13:35:13 +0000 Aaron Bieniek I kind of cringed a little when I read your post and then found that statement in the progression document. The whole idea of putting the wide part of the symbol next to the larger number doesn’t seem very mathematical. It seems more reminiscent of making the inequality symbol an alligator that eats the bigger number because it’s hungry. I don’t see how that helps students attach meaning to the symbols – it just becomes an exercise in identifying the larger number – which is fine if that’s the goal.

But since the goal in kindergarten is to identify sets as greater than or less than, I’m thinking the first grade goal should be to attach meaning to the symbols themselves. I would want the kids to see that we don’t have to write the words “greater than” or “less than” all the time because we have some notation that makes the job easier.

Can we get there with the wide mouth idea? Probably, but maybe a more direct route is helping the kids know the symbols and what they mean from memory. If the task is comparing 4 and 7, one thought process is “which way do I aim the hungry mouth” and the other thought process is “< means less than, and 4 is less than 7, so 4 < 7” or “> means greater than, and 7 is greater than 4, so 7 > 4”.

]]> <![CDATA[Reading Inequality Symbols]]> Wed, 12 Mar 2014 23:19:04 +0000 lhwalker First graders learn to use inequality symbols, per the NBT progressions this way: “putting the wide part of the symbol next to the larger number.” I don’t see when, if ever, we tell students to read a symbol “greater than” or “less than.” Would it be in 6th grade 6.NS.7d when they have to …”recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.” Or is it expected that the students will always refer back to the idea of wide part next to larger number and think x > 6 is like 10 > 6 so read > as “greater.”

]]> <![CDATA[Trying to understand G-C.5]]> Wed, 12 Mar 2014 20:36:41 +0000 moberlin I am hoping that you can help me understand the kinds of derivations that are referenced in G-C.5 (Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.) How might one use similarity to show that arc length is proportional to radius length? I know I can show arc length is proportional to radius length given the arc length formula but I suspect that is not the intention. Also, to derive the formula for the area of a sector, what can I assume has already been established? Thanks!

]]> <![CDATA[Reply To: More than Two Fractions]]> Wed, 12 Mar 2014 20:11:05 +0000 Alexei Kassymov Based in this, in 5th.

]]> <![CDATA[Reply To: A-SSE.3c and F-IF.8b]]> Wed, 12 Mar 2014 17:44:08 +0000 Aaron Bieniek To me it looks like A.SSE.3c is after transforming an expression for the purpose of revealing something about a particular quantity (which is represented by the expression), whereas F.IF.8 is focused on revealing properties of a function – like whether the function represents exponential growth or decay. In short – a single quantity versus a function.

]]> <![CDATA[Reply To: More than Two Fractions]]> Wed, 12 Mar 2014 17:22:24 +0000 Aaron Bieniek I’m going to go with 5th grade when we start adding fractions with unlike denominators. The standards don’t specify the number of fractions being added. Sounds like a great place to talk about not only adding fractions but also properties of operations. That’s a great video, btw. Everyone should see it. Twice.

]]> <![CDATA[Reply To: Science Connections]]> Wed, 12 Mar 2014 02:24:37 +0000 sunny I suggest looking into Modeling Instruction curriculum from Arizona State University. Have math and science teachers attend the training together. The Physical Science training focuses on proportional relationships. Our district is doing this training for the third year in a row. We have found a HUGE disconnect between math and science classes, often due to the different “language” used and differing methods of instructing the same concept. CCSS Math standards will be listed on the Next Gen standards.

]]> <![CDATA[A-SSE.3c and F-IF.8b]]> Tue, 11 Mar 2014 15:04:59 +0000 csteadman What is the difference between A-SSE.3c and F-IF.8b?

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12 t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

It seems like all of the questions I have seen could fall under either standard.

]]> <![CDATA[Reply To: Integers]]> Sun, 09 Mar 2014 01:06:47 +0000 Bill McCallum You are very welcome! Sorry if I sounded a little bit crabby in my last answer. The term “research-based” sometimes does that to me.

]]> <![CDATA[More than Two Fractions]]> Sat, 08 Mar 2014 22:57:03 +0000 lhwalker When do students start adding more than two fractions with different denominators such as 2/7 + 1/2 + 2/3? Phil Daro’s video “Against Answer Getting” mentions that is a typical TIMSS question.

]]> <![CDATA[Reply To: Integers]]> Sat, 08 Mar 2014 00:01:42 +0000 alwong Thanks! This is the message we are telling our teachers! I am glad I am on the same page as you! Teachers just wanted a firm “research-based” answer. But I will continue to share your answer!

]]> <![CDATA[Reply To: Integers]]> Fri, 07 Mar 2014 23:06:32 +0000 Bill McCallum We looked at many sources, including national reports, well-regarded state standards, and standards of high achieving countries. There is no agreement on where to put operations with integers. Singapore puts them in Grade 7. Massachusetts started with addition and subtraction in Grade 6, excluding subtraction of negative numbers. Curriculum Focal Points puts them in Grade 7. And so on.

As for research, the way you ask the question makes it sound as if you think that the placement of operations with integers in Grade 6 is itself research-based. But is it? How would you design a research experiment to determine the correct grade placement of operations with integers, or any other topic? Doesn’t it depend on what else you are doing in Grade 6, and what you have been doing in Grades 1–5, and what you plan to do in Grades 7–12? The research would have to look at the entire sets of standards. There has been some such research, for example the research of William Schmidt and Richard Houang, but they don’t have conclusions about specific grade level placement of specific topics. Rather they address large scale properties of the standards, such as coherence. My guess is that you could have a coherent set of standards which places operations with integers in Grade 6 and one which doesn’t; and you could have an incoherent curricula which do the same two things. The important thing is not the exact grade level placement but the coherence.

]]> <![CDATA[Integers]]> Fri, 07 Mar 2014 22:31:01 +0000 alwong Hello! I was wondering if you could point me in the direction of any research on why operations with integers were taken out of Grade 6 standards? I am a math coach and several of my teachers would like some reason or some research about why operations with integers are not taught in grade 6.

Thanks so much,

]]> <![CDATA[Reply To: Geometry then Algebra I]]> Fri, 07 Mar 2014 15:11:22 +0000 Bill McCallum The standards do not specify how the high school standards should be arranged into courses. Maybe you are looking at Appendix A, which is one example of how it might be done. But it’s just that, an example. Notice that Appendix A also gives a sample integrated sequence. There may be all sorts of practical reasons to start with Algebra I (e.g., because that’s what everyone else is doing and it is useful to coordinate with them) but there is no specification of that in the standards.

]]> <![CDATA[Reply To: Function Progressions – F.IF]]> Fri, 07 Mar 2014 15:08:33 +0000 Bill McCallum I think I need a more precise question here. But here are some musings. If I were teaching functions I would certainly give examples where there is not a well-defined output, to emphasize the importance of that aspect of functions. For example, give a table of days and average temperatures and then ask if the temperature is a function of the day (yes) or if the day is a function of the temperature (no, because there is more than one day with a given temperature). That’s different from defining the concept of a relation and then giving students a whole bunch of relations and asking them to sort them into functions and non-functions. That can lead you into territory where the concept of “one output for each input” gets lost in a blizzard of vertical line tests (which I think most students never connect with inputs and outputs).

]]> <![CDATA[Reply To: 3.0A. 4 and 3.0A. 6]]> Fri, 07 Mar 2014 14:59:34 +0000 Bill McCallum The are two aspects of the same thing, but the first is about performing a procedure, and the second is about understanding. So, an assessment of the second one might expect a student to come up with an appropriate multiplication equation give a division problem, whereas an assessment of the first one might expect a student to solve a multiplication equation without necessarily relating it to division.

]]> <![CDATA[Reply To: ""]]> Fri, 07 Mar 2014 14:55:14 +0000 Bill McCallum Yikes, I see what you mean. Looks like overkill to me too.

]]> <![CDATA[Reply To: Grade 2 notation for fractions question]]> Fri, 07 Mar 2014 14:52:51 +0000 Bill McCallum I guess the word “dividing” could cause confusion with the operation of division. If you talk about dividing a circle into quarters, and then later talk about dividing by four, you might end up confusing division by 4 with division by 1/4. Maybe. I don’t think it’s a big deal.

]]> <![CDATA[Geometry then Algebra I]]> Fri, 07 Mar 2014 14:07:23 +0000 mary_morris Are there specific reasons why the HS math sequence starts with Algebra 1? We are discussing the pro’s and con’s of teaching Geometry in 9th grade vs. Algebra I.

]]> <![CDATA[Function Progressions – F.IF]]> Thu, 06 Mar 2014 20:07:28 +0000 abarkley On page 8 of the function progression document, the following statement was provided.
“Notice that a common preoccupation of high school mathematics, distinguishing functions from relations, is not in the Standards.” This leaves the impression that there should be a reduced focus on identifying relations as functions or non-functions, and yet in the same paragraph it states “The essential
question when investigating functions is: “Does each element of the
domain correspond to exactly one element in the range?” Can you elaborate on the instructional strategies used to address how functions should be identified based on these statements in the progressions document?

]]> <![CDATA[3.0A. 4 and 3.0A. 6]]> Thu, 06 Mar 2014 15:16:40 +0000 ahenry1 We are trying to create item banks for the third grade common core standards. Can you clarify the differences in the two standards below,

3.OA. 4 – Determine the unkown whole number in a multiplication or division equation relating three whole numbers. and 3.OA. 6 – Understand division as an unknown-factor problem.

Thank you.

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Wed, 05 Mar 2014 18:44:40 +0000 jspencer Hello wise math folks,

I am reviving this topic in the hopes that people can share back what they have tried regarding Algebra 1 and 8th grade standards (compaction, enhanced HS pathways, etc.) and what the challenges and successes of doing this have been.

Also hoping that folks have found/established other “go to” resources for this topic beyond Appendix A.

Thank you!
Jen Spencer
HS Algebra teacher

]]> <![CDATA[Reply To: ""]]> Wed, 05 Mar 2014 04:57:36 +0000 lhwalker Oh, of course. I was looking at pages 78-85 from EngageNY and the treatment of the commutative and associative properties. It looked like overkill to me but maybe I just don’t have the right thinking on that.

]]> <![CDATA[Reply To: Grade 2 notation for fractions question]]> Mon, 03 Mar 2014 17:14:18 +0000 lhwalker How important is the word partitioning over dividing in 1.G.3 “Partition circles and rectangles into two or four equal shares…” I see the difference but noticed that a 2nd grade textbook I am evaluating consistently uses “dividing.”

]]> <![CDATA[SMP elaborations — two questions]]> Mon, 03 Mar 2014 06:04:09 +0000 ekmath I have been using the elaborations of the SMP with elementary teachers and have found them enormously useful. Two questions have come up in this work that may be useful as the document gets further refined and expanded.
In MP6, the following sentence, “They calculate accurately and efficiently and use clear and concise notation to record their work” led some readers I was working with to connect precision with efficiency. I think the relevant part of this sentence for elaborating MP6 is the use of “clear and concise notation” rather than efficient calculation. It may be worth noting that.

A second issue I would like to raise is further elaboration about the meaning of the phrase “critique the reasoning of others” in MP3. Readers may associate critiquing the reasoning of others with “criticizing” the reasoning of others. The examples in the elaborations convey the way elementary students construct mathematical arguments. The elaborations could help us communicate that critique and criticize are not synonymous. Students are engaged in critiquing each other’s reasoning when they are coming to understand and evaluate a peer’s sound reasoning in a valid solution, not as the word may connote, finding the shortcomings in a faulty solution (although students may also engage in troubleshooting problematic arguments and revising them to make them better).

]]> <![CDATA[Reply To: ""]]> Sun, 02 Mar 2014 21:02:29 +0000 Bill McCallum I depends on the questions … if it’s about whether or not a certain standard is being interpreted correctly, for example, then that’s fine. If it’s about the intentions of EngageNY authors, then I don’t think I can answer.

]]> <![CDATA[Reply To: ""]]> Sat, 01 Mar 2014 18:21:45 +0000 lhwalker Thank you for directing me to EngageNY. I also realized MA has some good stuff
I have some questions after looking over EngageNY for Algebra I at:
but I’m not sure this the best place to open such a discussion. What do you think, Dr. McCallum?

]]> <![CDATA[Reply To: PARCC and SBAC high school content frameworks]]> Tue, 25 Feb 2014 19:18:43 +0000 eprebys Do we know how the annual tests will work with students who are not following the same schedule? The standard schedule at my school has 8th grade students taking Algebra 1, 9th grade students taking Geometry and 10th grade students taking Algebra 2. Will the 10th grade students be taking the Geometry version of the PARCC exam? Does it depend on the state?

]]> <![CDATA[Reply To: Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Mon, 24 Feb 2014 00:31:59 +0000 rnarasimhan Dear Dr. McCallum,
You had inquired the following:

The K–8 standards were designed to gives students a solid preparation for algebra. How do you handle acceleration in Grades K–6 for these students?

I am an NJ parent of a child who is slated to be radically accelerated in 6th grade so as to be ready for AP Calculus BC in 11th grade(!) In our NJ district, there is no acceleration at all in grades K-5 for math. Since I am also a math professor, I supplement whatever is needed for my kid at an appropriate developmental level. I see topics rushed through at a speedy pace in 6th grade and beyond, with lots of concept gaps. Even the “standard” track kids are rushed through so that almost all can take some version of calculus as seniors. Competitive school districts feel that this is the way to gain admission to top universities, so I am not sure that compactification is going to go away.
The proper way of acceleration – to teach on-grade topics in depth by using materials such as the ones written by the Art of Problem Solving group – requires a fairly sophisticated understanding of mathematics. Also, many universities still use the traditional high school math curricula as the benchmark for their placement criteria. Thus, as far as districts like mine are concerned, I do not, unfortunately, foresee much change in the approach to secondary school math.

]]> <![CDATA[Reply To: Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Sun, 23 Feb 2014 22:10:17 +0000 jlanphear We along with many other middle schools with rigorous course offerings offer acceleration to talented students and have for many years. This acceleration has occurred for many, many years and it requires careful development of placement criteria, and curriculum mapping and compression. Things indeed have changed with Common Core, yet the meticulous mapping is similar. In terms of our current procedure, students are identified for initial acceleration at the end of 5th grade using a multi-dimensional and multi-modal analysis of mathematical achievement, algebra readiness/potential and progress in developing the mathematical practices. Those students who are determined as eligible undertake a Pre-Algebra Accelerated course in 6th grade. The curriculum includes the 6th grade CCSS standards as well as 7th and 8th grade CCSS standards pertinent to Pre-Algebra. For these students, the curriculum connects the more concrete concepts in 6th grade with algebraic extensions from Pre-Algebra. We benchmark to follow progress and mediate any gaps. These sixth grade students take the state-mandated 6th grade level assessment, NJASK and soon to be PARCC, as well as local course exams. As seventh graders, these students take Algebra I Accelerated in a similar fashion, and have taken seventh grade level state assessments and course-specific local exams. During the Achieve Algebra I EOC years, they took also took that high-stakes assessments.

Until about 8 or 9 years ago, as eighth graders, these students would have taken Geometry as eighth graders, until our sending district changed their middle school accelerated sequence to Alg I- Alg II- Geometry. Currently, these students undertake Algebra II in eighth grade, with respective state mandated eighth grade level assessments and course-specific local exams; beginning next year, the state has advised us that once students reach Algebra I, they will no longer take assessments by grade level, but by course. We will still ensure that students achieve all the standards, but at least students won’t lose twice the instructional time in standardized testing.

We are not unique in offering middle school mathematics acceleration; it is common at least in our region. It allows competitive NJ students looking to apply to selective schools to reach Calculus I by their junior year. We are considering reverting to the more traditional Algebra I- Geometry- Algebra II sequence for these students– Algebra I in 7th, Geometry in 8th and Algebra II in 9th. It seems more developmentally appropriate to integrate the visual support for complex mathematical ideas earlier, and save the more abstract content of Algebra II for 9th grade. We have been unusual in our Alg I-Alg II- Geometry sequence. We are concerned as to whether being out of the typical sequence will disadvantage our students in achieving the standards or in demonstrating mastery in high-stakes assessment, as part of the larger discussion about the sequence, especially since our sending district will not be changing the sequence this year.

We are planning on examining the frameworks documents in detail, but I was hoping for availability of some additional expertise to take under consideration.

]]> <![CDATA[Reply To: Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Sun, 23 Feb 2014 16:21:22 +0000 Bill McCallum The high school standards do not specify any arrangement of the mathematics into courses; that is up to states and districts. I’m not an expert on PARCC, but I believe they will have end of course tests for Algebra I, Geometry, and Algebra II. These should be relatively independent of each other, but there could be places where they assume that an Algebra II student has had Geometry. For example, the Geometry domain Expressing Geometric Properties with Equations (G-GPE) is in Algebra II in the PARCC framework. You can check out the frameworks here.

The biggest question I have about your proposal is not the order of those courses, however, but the preparation of the students entering Algebra I in Grade 7. The K–8 standards were designed to gives students a solid preparation for algebra. How do you handle acceleration in Grades K–6 for these students?

]]> <![CDATA[Alg I Alg II then Geo or Alg I Geo then Alg II?]]> Sun, 23 Feb 2014 04:27:45 +0000 jlanphear We have each year a cohort of students who study high school math in the middle school grades. To align to a change in course sequence 9 years ago at our sending district, we changed our sequence for these students to Algebra I in 7th and Algebra II in 8th. These students take Geometry in 9th grade at the high school and then go on to Pre-Calculus, etc. We recognize that most schools instead follow the Algebra I- Geometry- Algebra II sequence.

We are looking for information regarding whether the standards are constructed to prefer one sequence over another, or if PARCC testing requires that instruction follow one of the two sequences in order for students to be properly prepared. Would you be able to provide some guidance based upon your work with Common Core and PARCC that could inform my process and stake-holder decision-making?

]]> <![CDATA[Reply To: 3.OA.9]]> Sat, 22 Feb 2014 18:10:42 +0000 khughes Thank you for the clarification. I believe my example is an arithmetic pattern but I agree with you that I do not how see a property of operations is used but rather it is generalizing a rule. I believe that is grade 4. So, I think my question was really whether the patterns in grade 3 are limited to those associated with properties of operations, and you answered that question. Thank you for the speedy reply.

]]> <![CDATA[Reply To: 3.OA.9]]> Sat, 22 Feb 2014 17:41:03 +0000 Bill McCallum Illustrative Mathematics has some illustrations of 3.OA.D.9.

As for your example, I guess the question is whether it counts as an arithmetic pattern, and how properties of operations are used make the conclusions. I’m not really seeing that right now, but maybe it works.

]]> <![CDATA[Reply To: Compound Inequalities]]> Sat, 22 Feb 2014 17:35:20 +0000 Bill McCallum I was a bit hasty last time. I shouldn’t have suggested that inequalities were not included in A-CED.1 itself … just that you shouldn’t infer inequalities from an explicit statement listing types of equations to be included.

Here’s the full text of A-CED.1

A-CED.1. Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

So yes, inequalities are included in this standard. However, for Algebra I, PARCC focuses only on equations, as you noted. For Algebra II, I don’t read their statement as excluding inequalities. It says tasks should have a real-world context, which is appropriate because this standard has a modeling star. And it lists things to be included, which happen all to be equations (consistent with the original standard). But that does not exclude the things not listed. So no, I don’t think PARCC states are disregarding that part of the standard. However, there is a pretty clear signal here that inequalities do not play as big a role as equations.

]]> <![CDATA[3.OA.9]]> Fri, 21 Feb 2014 19:24:17 +0000 khughes In the 3.OA.9 standard, students need to identify arithmetic patterns and explain them using properties of operations. I cannot find much elaboration on this standard other than the example provided in the standard itself.I was wondering if someone could provide a more complete description and/or more examples. I did not see it addressed in the Learning Progressions.
In the past, 3rd grade students in our state have done tasks such at the following, given a pattern such as red, blue, green, red, blue, green, students would need to find the color of the 20th cube for example. They would need to describe patterns such as all of the green cubes are a multiple of 3, the blue cubes are 1 less than a multiple of 3 and so on. Is this an example for this standard?
Any insights are greatly appreciated. Thank you!

]]> <![CDATA[Reply To: Compound Inequalities]]> Wed, 19 Feb 2014 18:49:23 +0000 csteadman Ok, thanks. Just to clarify in Algebra II as well, PARCC limits A-CED.1 with, “Tasks have a real-world context. In Algebra II, tasks include exponential equations with rational or real exponents, rational functions, and absolute value functions.”

So is the inequality aspect of A-CED.1 disregarded in PARCC states?

]]> <![CDATA[Reply To: ""]]> Wed, 19 Feb 2014 13:23:38 +0000 steveoc The site is run by Common Core, Inc. here is information about them:

They are the contractors hired by NYSED to develop the EngageNY math modules. The lessons are available on the EngageNY website for free. They have packaged the same lessons as Eureka Math. They are essentially identical—both based on the Common Core standards.

I have been using the fifth grade math modules and feel that they are the most coherent math lessons I have used. I have been converting the fifth grade lessons to multimedia and share them freely on

]]> <![CDATA[Reply To: N-RN.2]]> Sun, 16 Feb 2014 22:19:16 +0000 Bill McCallum I think the standard allows either approach. But yes, you are right, the idea is not to treat radicals as a separate thing, but to connect them with rational exponents and the rules of exponents.

]]> <![CDATA[Reply To: 6th grade coordinate plane language]]> Sun, 16 Feb 2014 22:16:26 +0000 Bill McCallum Your second interpretation is correct. You might want students to observe that the two points are the same distance from the axis but on opposite sides, or that you could get one from the other by flipping across the axis, or something like that. You might even use the word “reflection” when likening the relationship between the points to reflection in a mirror. But students are not expected to understand reflections as transformations, or use the word, in Grade 6.

]]> <![CDATA[Reply To: progressions/strand maps]]> Sun, 16 Feb 2014 22:13:21 +0000 Bill McCallum Well, this is not quite the same thing, but Illustrative Mathematics is developing course plans and unit blueprints which will be implemented on the website as a navigational tool somewhat similar to the NDSL diagrams.

]]> <![CDATA[Reply To: 1NBT4]]> Sun, 16 Feb 2014 22:02:01 +0000 Bill McCallum Any problem where the sum is less than 100 is included here. The point of the “including” was to make sure that these simpler types of problems were not neglected for their power to reveal the role of the base 10 system in two-digit addition.

]]> <![CDATA[Reply To: Extent of 5.NBT.6 and 5.NBT.7]]> Sun, 16 Feb 2014 21:34:39 +0000 Bill McCallum Yes, I would say that this limits the division problems to ones where divisors, dividends and quotients all be decimals that with no non-zero digits beyond the 100ths place.

]]> <![CDATA[Reply To: definition of unit]]> Sun, 16 Feb 2014 20:43:39 +0000 Bill McCallum I agree with Cathy here. Also, I would add to your definition that a unit is something replicable; you can measure out quantities using copies of the unit. For example, you can measure the length of a table in matchsticks.

]]> <![CDATA[Reply To: ""]]> Sun, 16 Feb 2014 20:39:35 +0000 Bill McCallum Not in the sense of being founded by any of the lead writers, no. But I believe they are employing some of the members of the original Work Team for the math standards to write their curriculum. I don’t know what the history is, but I assume they must have had that url before the standards.

]]> <![CDATA[Reply To: Progression Document?]]> Sun, 16 Feb 2014 20:37:52 +0000 Bill McCallum Thanks Julie!

]]> <![CDATA[Reply To: memorizing conversion tables]]> Sun, 16 Feb 2014 20:37:18 +0000 Bill McCallum Well, the standard 5.MD.1 simply asks students to convert … it doesn’t say whether they should memorize the relationships. That said, my own opinion is that there are some basic relationships that students should simply know, for example that there are 12 inches in a foot. It seems a waste of time to look this up in a chart when the human brain is already naturally adapted to storing such bits of information. Whether they acquire this by memorization or by repeated exposure is a matter of pedagogy, not specified in the standards. And the metric system is designed for ease of remembering.

Once you have a few basic facts, you can derive the others. For example, if you know there are 100 centimeters in a meter, you also know that a centimeter is 0.01 meters; it is not a separate fact, but a related fact coming from and understanding of the relationship between multiplication and division and an understanding of decimal notation.

]]> <![CDATA[Reply To: F-BF.1b]]> Sun, 16 Feb 2014 20:28:37 +0000 Bill McCallum We’ll try and work on some for Illustrative Mathematics!

]]> <![CDATA[Reply To: 7th grade geometry standards]]> Sun, 16 Feb 2014 20:26:45 +0000 Bill McCallum Oops, already answered this in your other post.

]]> <![CDATA[Reply To: 7th grade geometry standards]]> Sun, 16 Feb 2014 20:26:10 +0000 Bill McCallum 1) The standard says “Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.” You can notice experimentally that two angles determine the triangle without knowing the angle sum theorem. In fact, noticing this is good preparation for the angle sum theorem, since it makes you suspect that something like it might be true. There’s also no harm if it comes up; it’s just not required until Grade 8.

3) I think you’ve answered your own question here; certainly describing the cross-sections in 7.G.3 would benefit from drawing figures in 7.G.2. I think 7.G.2 is mostly about plane figures.

2) Have you considered connecting it more to the algebra part of the curriculum? It’s a good opportunity to work with equations.

]]> <![CDATA[Reply To: 7.G.2]]> Sun, 16 Feb 2014 20:21:19 +0000 Bill McCallum It says “focus on triangles” but it doesn’t say only triangles. Other shapes are possible.

]]> <![CDATA[Reply To: Further Clarification 4.MD.1]]> Sun, 16 Feb 2014 20:19:57 +0000 Bill McCallum A couple of thoughts on this. First, 4.MD.1 is about converting from larger units to smaller units, not the other way around. Converting both ways is in 5.MD.1, which in fact mentions as an example the exact conversion you give, 5 cm to 0.05 m. The reason for this is that multiplication of whole numbers by fractions doesn’t occur until Grade 5. However, you are right that this particular conversion could be discussed in Grade 4, using both 4.NF.6 and 4.NF.4, which enables students to see 5/100 as $5 \times 1/100$. Still, it is not required in Grade 4. And your second example is even less required!

]]> <![CDATA[Reply To: S-IC.1]]> Fri, 14 Feb 2014 20:34:35 +0000 Bill McCallum Partly it is just a matter of the level of sophistication with which the topic is treated. But notice also that the high school standard makes explicit reference to population parameters, whereas the Grade 7 standard simply talks about gaining information about the population. So in Grade 7 you might just look at a graphical representation of the sample and discuss what you can infer from it, whereas in high school you get more into the technical details of estimating parameters such as mean and standard deviation.

]]> <![CDATA[Reply To: Mixture Problems]]> Fri, 14 Feb 2014 20:27:04 +0000 Bill McCallum Mixture problems are certainly among the problems you might give students in Grade 8 working with simultaneous equations. But I think you have be clear in your own mind what the purpose of doing so is. If mixture problems occur among a range of problems where students have to model a situation by setting up a system of simultaneous equations, solve the system, and then interpret their solution, then that’s good. If the idea is to have a separate unit called “mixture problems” where you look at a lot of problems following a fixed template, and train students in a fixed procedure for dealing with that template, then that’s bad. It’s not the type of problem that is important, but learning flexible skills and understandings that can be applied to other types of problems with the same structure that are not about mixtures.

]]> <![CDATA[Reply To: PARCC and SBAC high school content frameworks]]> Fri, 14 Feb 2014 20:19:52 +0000 Bill McCallum Nice work, John, I hope people find these useful.

]]> <![CDATA[N-RN.2]]> Fri, 14 Feb 2014 01:23:53 +0000 Aaron Bieniek Hi all – if students were asked to rewrite sqrt(12) + sqrt(3), am I correct that the approach the standards are suggesting is rewriting the radicals as rational exponents and then using the properties of exponents? So, 12^(1/2) + 3^(1/2), then (4*3)^(1/2) + 3^(1/2) and so on? Now that I write it, it kind of seems obvious that this is the intent of the standard – but what I typically see in my high school is teachers treating radicals as a separate thing with their own set of rules and procedures that involve simplifying and collecting “like terms”.

]]> <![CDATA[Reply To: 6.EE.3]]> Fri, 14 Feb 2014 00:54:43 +0000 Aaron Bieniek Hi Bill – in response to your comment that it would be ok if students saw y + y + y informally as 3 y’s…do you see that happening after a good amount of formal work actually using the properties? The reason I ask is because I rarely see any work with the properties and work with “like terms” generally boils down to some analogy like “3 apples and 5 apples is 8 apples, so 3x + 5x = 8x”. I’m not convinced that type of argument actually counts as mathematics. I’d like to see my teachers treating 3x + 5x as x(3 + 5). Is that what the standard is expecting?

]]> <![CDATA[Reply To: SMP draft for later MS and HS?]]> Thu, 13 Feb 2014 17:42:03 +0000 Bill McCallum Yes, those are in the works. But I’ve given up on timelines, since I never meet them! I will say that they are basically ready in draft form and just need some formatting and checking.

]]> <![CDATA[SMP draft for later MS and HS?]]> Thu, 13 Feb 2014 16:19:44 +0000 Matt Friedman I haven’t gotten to read this over in detail, but it looks very helpful on first glance. Does anyone know if a similar draft of progressions of the practices is being worked on for the middle school and high school levels? And, if so, what the timeline is expected to be for their release?

]]> <![CDATA[Reply To: Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Wed, 12 Feb 2014 04:06:50 +0000 Bill McCallum Great story Steve. How did he see that the x-intercepts where 5 and -5?

]]> <![CDATA[Reply To: Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Wed, 12 Feb 2014 04:06:47 +0000 Bill McCallum Great story Steve. How did he see that the x-intercepts where 5 and -5?

]]> <![CDATA[Reply To: Compound Inequalities]]> Wed, 12 Feb 2014 04:04:10 +0000 Bill McCallum No, inequalities are not included in these standards.

]]> <![CDATA[Reply To: Prime factorization]]> Tue, 11 Feb 2014 23:00:15 +0000 Bill McCallum Well, that’s a bit of a stretch, I think, although I can image using prime factorization to generate examples for this standard.

]]> <![CDATA[Reply To: 3.OA.8 – two-step equations]]> Tue, 11 Feb 2014 22:53:21 +0000 Bill McCallum Very late to the party here, but I agree with Cathy and Lane. There is no explicit restriction to the number of equations, therefore there is no requirement to write a single 2-step numerical expression. On the other, some students will be ready for this and should be encouraged.

]]> <![CDATA[Reply To: definition of unit]]> Tue, 11 Feb 2014 22:44:55 +0000 Cathy Kessel Here’s a quick comment: I don’t think there is any grade in which students are required to know a general definition of unit. However, they see examples of different types of units over the grades as described in the section on units (pp. 10-11) in the draft front matter for the Progressions here: So, over the grades, their understanding of “unit” expands.

]]> <![CDATA[6th grade coordinate plane language]]> Tue, 11 Feb 2014 20:11:41 +0000 Alexei Kassymov 6.NS.6b says:
Understand signs of numbers in ordered pairs as indicating
locations in quadrants of the coordinate plane; recognize that
when two ordered pairs differ only by signs, the locations of the
points are related by reflections across one or both axes.

The next time we see reflections in 8th grade geometry, where, it is introduced. Is it expected that students talk about, say, (1, 0) and (-1, 0) in terms of a reflection across the y-axis or is the reference to reflections addresses the underlying mathematical structure but not how students will be talking about this situation (similar to them using commutativity in 1st grade but not using the term)? I guess, given that students should be familiar with the idea of symmetry from at least grade 4, there is a familiar language that students can use.

]]> <![CDATA[progressions/strand maps]]> Tue, 11 Feb 2014 20:11:38 +0000 nvitale In trying to understand the structure of the standards, I’ve looked at the progressions documents. I’ve tried to unpack Jason Zimba’s monster atlas, but it is pretty cumbersome….

I wonder if there are any plans to put together something like this:

here is one on number

It might be a helpful tool to explore the structure of the standards.

]]> <![CDATA[1NBT4]]> Mon, 10 Feb 2014 02:17:22 +0000 lprice Can you help answer a question that we have about adding two-digit numbers? We were discussing if this standard means that first graders should only add a two-digit and a one-digit number or a two-digit number and a multiple of ten or if they should also be introduced to adding two 2-digit numbers neither of which are multiples of ten. The end of the standard seemed to suggest that they maybe should be introduced to the later and the example in the progression shows a problem with two 2-digit numbers. We all understand that we are not expecting first graders to use the standard algorithm but rather math tools, pictures, etc to find the sum. We are just unsure about what type of numbers the standard calls for our students to add.

Thank you for your help

]]> <![CDATA[Reply To: Extent of 5.NBT.6 and 5.NBT.7]]> Sun, 09 Feb 2014 15:42:24 +0000 jegtitle Regarding 5.NBT.7, please clarify …and divide decimals to hundredths. Does this mean divisors, dividends and quotients to hundredths? I’ve used drawings and models, but using 5.NBT.1’s understanding of place value seems to help my students.

]]> <![CDATA[definition of unit]]> Fri, 07 Feb 2014 16:22:14 +0000 lsharlow We are trying to find a general, student-friendly definition for unit. We were considering using this: any group of things or persons regarded as an entity. At first we were considering that the definition needed to include something that said it had to be a consistent size (“tens”, “inches”, “halves”, etc.) that you can count, but then we thought about counting “pumpkins” that wouldn’t be the same size. We just want to be sure that we are consistent with the terminology used in the CCSS. Thank you.

]]> <![CDATA[Reply To: F-BF.1b]]> Thu, 06 Feb 2014 17:27:43 +0000 Elizabeth My questions is specifically about part b: Combine standard function types using arithmetic operations.

This is a modeling standard, and the only task directed at it is a non-modeling task, “A Sum of Functions”. There aren’t any tasks (yet 🙂 specifically aligned to part b, and the Functions Progression also cites the example of an exponential function combined with a constant function. I’d love some other examples. Thanks!

]]> <![CDATA[Reply To: F-BF.1b]]> Thu, 06 Feb 2014 00:07:22 +0000 lhwalker There are lots of great task examples here:

Explanations for F.BF standards begin on page 11 of the High School Functions progressions:

]]> <![CDATA[""]]> Wed, 05 Feb 2014 17:45:25 +0000 lhwalker I stumbled across “” Do they have any official status ore relationship like does?

]]> <![CDATA[Reply To: Progression Document?]]> Mon, 03 Feb 2014 21:05:31 +0000 juliejames178 Thank you for all your work on all the Progression Documents. I have found them very helpful in planning professional development. Just something I noticed that may want to be edited on the Number Systems Progression:
On Pg. 7 at the very bottom right, there is a model showing -(-a) on a number line. The heading of that image states “Showing -(-a) = 0 on the number line”. I believe that should state -(-a) = a

Thanks again for all your hard work!

]]> <![CDATA[memorizing conversion tables]]> Sat, 01 Feb 2014 16:28:27 +0000 hsurrette Currently, my fifth grade teacher is teaching the students about converting both metric and customary units of measurement. She asked me the other day if she should require the students to memorize the relationship between the different units of measurement or if she can provide them with a conversion chart. As I read through the progressions, I shared with her that through the examples provided it seems that the students must develop an understanding of the relative size of a unit of measurement and how it relates to others. It also seems that students are expected to memorize metric system units and the common customary units of measurement conversions.
I want to make sure that we are addressing the requirements of the intent of the progressions and the standards, please advise. Thank you.

  • This topic was modified 3 years, 6 months ago by  hsurrette.
]]> <![CDATA[F-BF.1b]]> Wed, 29 Jan 2014 19:27:08 +0000 Elizabeth I’d appreciate some help with applications of F-BF.1b (Combine standard function types using arithmetic operations) beyond the example of adding a constant function to an exponential function. Should students in Math 2 or Algebra 2 combine quadratic functions with other functions to model real-world situations?

]]> <![CDATA[Reply To: Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Wed, 29 Jan 2014 15:03:29 +0000 SteveG An update on the sums & differences story.
A few days after we talked about the graph of y = x^3 +c (which, if c is negative covers both factoring), I was working with a student on some factoring questions. The student came to x^3 – 125. In the air above his paper he moved his finger in the shape of the cubic parent function (as if tracing it in his mind). Excited by this, I asked him to explain.
He said, “Well, the graph of y = x^3 -125 would have a negative y-intercept down here and its x-intercept would be over here at positive 5. That means that the first factor has to be (x-5).” I asked him to explain if it worked for y=x^3+125 as well, and he explained that it did because the root is at -5 which makes the factor in x^3+125 be x+5.
I was so excited by this connection that the student made! I told him that he was really making some good connections and understanding and that he should share his explanation with the class. For this shy kid who usually is middle-of-pack gradewise, that was a real boost to his confidence. He explained it well in class. When someone asked if it worked for things like 8x^3 – 125, I suggested that everyone think about it and explore that on their own. We discussed that one briefly another day.
Had to share that story. I love those kinds of connections.

]]> <![CDATA[7th grade geometry standards]]> Tue, 28 Jan 2014 22:30:01 +0000 missbaldwin So many questions…
1) In 7.G.2, students work with constructing triangles given certain conditions, yet don’t learn the triangle sum theorem until 8th grade. Are they supposed to know it informally in 7th grade?

2) How does 7.G.3 connect to 7.G.2? Are the shapes students are drawing in 7.G.2 three-dimensional? Are they drawing shapes that would be plane sections of prisms and pyramids?

3) How does 7.G.5 connect to the rest of the standards in its cluster? All of the other domains are so well connected, we’re having a hard time making the geometry standards in 7th grade flow together.


]]> <![CDATA[7th grade geometry standards]]> Tue, 28 Jan 2014 22:29:46 +0000 missbaldwin So many questions…
1) In 7.G.2, students work with constructing triangles given certain conditions, yet don’t learn the triangle sum theorem until 8th grade. Are they supposed to know it informally in 7th grade?

3) How does 7.G.3 connect to 7.G.2? Are the shapes students are drawing in 7.G.2 three-dimensional? Are they drawing shapes that would be plane sections of prisms and pyramids?

2) How does 7.G.5 connect to the rest of the standards in its cluster? All of the other domains are so well connected, we’re having a hard time making the geometry standards in 7th grade flow together.


]]> <![CDATA[Reply To: 7.G.2]]> Tue, 28 Jan 2014 21:48:06 +0000 missbaldwin Which geometric shapes in particular should students be able to draw in 7.G.2? Is it only triangles?

]]> <![CDATA[Further Clarification 4.MD.1]]> Mon, 27 Jan 2014 13:20:41 +0000 brandeli Bill, I have read past forum posts to try to get some clarification for 4.MD.1. I understand that we should take into account the other grade 4 standards. I could see students having to convert 5 cm to m and looking at it as 5/100 or 0.05 because this is within the scope of 4.NF.6 to write decimals as fractions of 10 or 100. But would they convert say 5 mm to m? This would be 5/1000 and not within the scope of 4.NF.6. Would they be asked to convert say 5 m to 0.005 KM? I know this seems picky but teachers are asking and I don’t want to lead them down the wrong path.


]]> <![CDATA[S-IC.6]]> Thu, 23 Jan 2014 09:10:37 +0000 tomergal Could you please elaborate on this standard (S-IC.6)? The progression doc doesn’t focus on it. How exactly does it differ from “Use data from a sample survey to estimate a population mean or proportion” (S-IC.4) or “Use data from a randomized experiment to compare two treatments” (S-IC.5)?

]]> <![CDATA[S-IC.1]]> Thu, 23 Jan 2014 09:03:52 +0000 tomergal Could you please elaborate on the differences between this standard (S-IC.1) and the 7th grade standard 7.SP.1?

(7.SP.1: “Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.”)

]]> <![CDATA[Mixture Problems]]> Wed, 22 Jan 2014 19:32:00 +0000 shelahf In 8th grade, mixture problems were specifically notated in the ’97 standards yet in Common Core it isn’t mentioned. I have referenced the EE Progressions and Illustrative tasks as for a clue as to its appropriateness in 8th grade content and I’m not finding any answers. Do you feel that traditional mixture problems are appropriate for 8.EE.8.A and how much time should be spent on that concept?

]]> <![CDATA[Reply To: PARCC and SBAC high school content frameworks]]> Wed, 22 Jan 2014 17:40:23 +0000 jmeinzen Because I could not find a similar resource, I created several responsive webpages that combines the language-based PARCC Model Content Frameworks into the visual-based Scope & Sequence documents (High School Traditional) that Patrick Callahan, et al. were kind enough to share. I followed the color-coding scheme that the PARCC MFC uses. I’d appreciate any feedback on it’s usefulness.

The webpages are temporarily hosted:
1. Algebra 1 (

2. Geometry (

3. Algebra 2 exists but is still a work-in-progress…I’ll wait for feedback on the above two before spending inordinate amount of time completing the document.


]]> <![CDATA[Join in a Twitter chat about the standards?]]> Tue, 21 Jan 2014 19:57:18 +0000 davidwees Hi Bill,

I just found your blog. I love your openness about your project, and have personally been pretty impressed with many aspects of Common Core math, in particular the Standards for Mathematical Practice, which for me could act (finally) as a common definition as to what it means to do mathematics.

I notice you have Ashli on your team (@mythagon on Twitter). I wonder if you’d like to ever join us at #mathchat for a discussion about the Standards for Mathematical Practice, and how you developed them. Perhaps an informal Q&A? Ashli can help get you set up if you don’t have a Twitter presence yet.


]]> <![CDATA[Reply To: 3.OA.8 – two-step equations]]> Sat, 18 Jan 2014 19:09:20 +0000 lhwalker Oh, I should have also mentioned Illustrative Mathematics has a couple other great tasks you might like to see. Illustrative Mathematics is linked to this site, but here’s the particular tasks:

]]> <![CDATA[Reply To: 3.OA.8 – two-step equations]]> Sat, 18 Jan 2014 17:42:33 +0000 lhwalker I’ve been watching (and Phil Daro videos on these days and I envision this excellent number story playing out like this: Kids work in pairs or small groups to figure out the problem. Some kids would draw pictures of 7 bags, each with 3 apples and figure it out that way. More advanced students would, as you say, write two equations. Even more advanced students write one equation. At each level, the teacher serves as coach, drawing the best out of the students as they think and talk with each other.
Next, a whole class discussion begins with the picture-students who explain their thinking. The discussion then morphs to students who used two equations and capped with the single equation. Lastly, the students are given time to think about how subtracting before multiplying would mess up the answer.
3.0A.8 says, “Represent these problems using equations with a letter standing for the unknown quantity.” In your task, the variable is already isolated. At some point we need the students to be able to solve this: Julie has some bags of apples, each with three apples. If she has 21 apples total, how many bags does she have? 21 = 3b. The next level of complexity would be something like, “Julie empties the bags onto the table and Amanda takes one of them away. Julie sees there are 20 apples left. How many bags did she have?” 20 = 3b -1
This might seem “over the top” with a class of third graders, but working in groups to solve problems like this tends to pull their thinking skills impressively upward! What we don’t want is for students to see there are two numbers in the problem and automatically use their favorite operation to “get the answer.”

]]> <![CDATA[Reply To: 3.OA.8 – two-step equations]]> Tue, 14 Jan 2014 15:54:55 +0000 jjacobi Thanks for the feedback. I’m less concerned about equations being “constrained” to one, but more whether kids are expected to be able to read a number story and write one number sentence to represent it. At this point in third grade, it seems it would be much more natural for kids to use two or more number sentences. For example, in the sample problem I provided above, take this number story:
Julie had 7 bags of apples. Each bag contained 3 apples. She gave one apple to Amanda. How many apples does she have left?
Most third graders I know would write two equations: 7 x 3 = 21, and then 21 – 2 = S. It is a more advanced process to see this equation as a single number model: 7 x 3 – 2 = S. It gets even more complicated if we expect them to interpret a number story that involves application of the order of operations (which, of course, they are expected to apply correctly), or where the unknown amount is the start or the change.
Is the intent of the standard that kids must be able to represent a multi-step number story with one equation?

]]> <![CDATA[Reply To: 3.OA.8 – two-step equations]]> Mon, 13 Jan 2014 02:22:29 +0000 Cathy Kessel I don’t see anything that constrains the number of equations used to solve a problem in 3.OA.8 to one. Note that 2.OA.1 uses a very similar formulation but goes on to elaborate:

Represent and solve problems involving addition and subtraction.

1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

As you note, page 28 of the OA Progression says:

As with two-step problems at Grade 2,2.OA.1, 2.MD.5 which involve only addition and subtraction, the Grade 3 two-step word problems vary greatly in difficulty and ease of representation. More difficult problems may require two steps of representation and solution rather than one.

“Two steps of representation and solution” sounds to me as it includes solutions that involve two (or more) equations or tape diagrams, as in the example in the margin.

It might be that part of the concern is whether students should be able to interpret things like 3 × 10 + 5. That’s discussed here:

]]> <![CDATA[Reply To: 8.EE.5/6 cluster]]> Sun, 12 Jan 2014 23:22:00 +0000 Cathy Kessel Maybe you also want to look at p. 14 of the Ratio and Proportional Relationship Progression, which describes proportional relationships in terms of ratios and begins: A proportional relationship is a collection of pairs of numbers that are in equivalent ratios.

For example, if the relationship in question is given by y = 2x + 1, it has pairs, e.g., (0, 1), (1,3), (2,5), that are not in equivalent ratios. Because of that the relationship between x and y is not a proportional relationship.

The m in y = mx + b is not always a constant of proportionality for the relationship between x and y because y = mx + b does not always represent a proportional relationship between x and y. (On the other hand, one could consider the relationship between the quantities represented by yb and x.)

]]> <![CDATA[Reply To: 8.EE.5/6 cluster]]> Sat, 11 Jan 2014 03:48:12 +0000 lhwalker I think the answer you are looking for is on Page 8 of the progression, “Ratios and Proportional Relationships,”
Where it says, “Recognizing proportional relationships. Students examine situations carefully to determine if they describe a proportional relationship…Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships.”

]]> <![CDATA[8.EE.5/6 cluster]]> Fri, 10 Jan 2014 22:26:27 +0000 beth780 A heated debate came up at a common core in-service I was participating in the other day over the EE cluster on proportional relationships and y = mx+ b. A number of teachers were stating that y = mx + b is proportional because after the initial point it then has a constant of proportionality. The other group of teachers said that y = mx + b is a non-proportional linear relationship and that this is only proportional when b is equal to 0. As nothing was really resolved, I thought I would bring it to the forums and see if I could bring back some clarification to the group.

]]> <![CDATA[Reply To: Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Thu, 09 Jan 2014 20:34:41 +0000 SteveG Thanks for the replies! I had to move up to high school at the semester break, and I jumped straight into Algebra II.
When we got to this identity, we discussed it. We even explored the graph of y = x^3 + c. Since the kids already did some with roots in the fall, they were able to make some great connections with the structure and the graph.
y = x^3+c has only 1 real solution (aka x-intercept)
And we know it factors as (x+c)(x^2-x+c^2). Since we only see one x-intercept, it makes sense that the second factor is a quadratic with non-real roots. That’s part of the reason why we know the factoring pattern is as simplified as it can be.
It was a great conversation. Hooray for structure!

]]> <![CDATA[Reply To: 3.OA.8 – two-step equations]]> Thu, 09 Jan 2014 16:59:33 +0000 jjacobi I would really appreciate any feedback you could offer on this topic. I am unsure whether I should expect my third graders to be able to write a single number model to represent a two-step number story, or if two equations are acceptable. My colleagues and I interpret the expectation of this standard differently. Thank you so much.

]]> <![CDATA[Reply To: Compound Inequalities]]> Tue, 07 Jan 2014 16:46:46 +0000 csteadman Could you expand on inequalities in the Common Core? Following A-CED.1 is, “Include equations arising from linear and quadratic functions, and simple rational and exponential functions.” Also, in Algebra I PARCC adds “Tasks are limited to linear, quadratic, or exponential equations with integer exponents.”

Neither of these mention inequalities. Is it implied that equations also refers to inequalities?

]]> <![CDATA[Reply To: Prime factorization]]> Mon, 23 Dec 2013 22:30:34 +0000 ezka29 Could you consider prime number factorization as a part of 6.EE.1 (write and evaluate numerical expressions involving whole-number exponents)?

]]> <![CDATA[Reply To: Counseling Students about Common Core]]> Sat, 14 Dec 2013 02:11:07 +0000 Cathy Kessel Less complicated suggestion: I haven’t read or used them, but you might want to check PTA’s parent’s guides to student success:

Here’s what the page says:

National PTA® created the guides for grades K-8 and two for grades 9-12 (one for English language arts/literacy and one for mathematics).

The Guide includes:

• Key items that children should be learning in English language arts and mathematics in each grade, once the standards are fully implemented.

• Activities that parents can do at home to support their child’s learning.

• Methods for helping parents build stronger relationships with their child’s teacher.

• Tips for planning for college and career (high school only).

PTAs can play a pivotal role in how the standards are put in place at the state and district levels. PTA® leaders are encouraged to meet with their school, district, and/or state administrators to discuss their plans to implement the standards and how their PTA can support that work. The goal is that PTAs and education administrators will collaborate on how to share the guides with all of the parents and caregivers in their states or communities, once the standards are fully implemented.

]]> <![CDATA[Reply To: Counseling Students about Common Core]]> Fri, 13 Dec 2013 20:19:45 +0000 Cathy Kessel Persevering in trying to solve the problem . . . I’m putting short pieces.

Part of what got me into math education from mathematics was the disconnect between the unmathematical beliefs and practices that students often acquire in K–12 and what’s expected in college. (Somewhat related: A large proportion of undergraduates take remedial courses, i.e., courses that repeat topics of high school. See TABLE S.2 and Figure S.2.1 of Over half of the undergraduates in mathematics courses at four-year institutions are taking courses below calculus.)

]]> <![CDATA[Reply To: Counseling Students about Common Core]]> Fri, 13 Dec 2013 02:42:29 +0000 Cathy Kessel I’ve tried to post a reply several times. I’m just putting this comment as an experiment.

]]> <![CDATA[Reply To: Order of operations]]> Fri, 13 Dec 2013 00:21:35 +0000 Cathy Kessel Order of operations with respect to parentheses is discussed here:, and here: There’s a discussion of related issues, e.g., the “any order” property in the Expressions and Equations Progression:

]]> <![CDATA[Order of operations]]> Wed, 11 Dec 2013 22:58:22 +0000 Karena Clarke I am just wondering at what level in the Standards teachers begin teaching the order of operations?

  • This topic was modified 3 years, 8 months ago by  Karena Clarke.
]]> <![CDATA[Counseling Students about Common Core]]> Wed, 11 Dec 2013 01:26:08 +0000 druss I am a high school counselor and I am concerned about the first wave of students who are just being introduced to common core. I am especially worried about our high achievers. i think over the years students will get used to this new way of learning, thinking, participating, and being assessed, but in the meantime, i think our students deserve some explanation/insight into the “why” of common core (math inparticular). I have found loads of resources but none that are kid friendly. I want to explain why in math and pe there is going to be some writing and explaining expected of them and why (in a kid friendly way), it’s important. Some of these kids are great rule followers and can memorize anything and they have “learned” that that equates to being smart but now the script has flipped a bit. Any suggestions on to explain the “why” to kids would be greatly appreciated. We have seen high achievers get frustrated and give up on assessments.
I am a believer in common core and I want to explain it to parents and students in way that is not full of edutalk and not real to them.

Thanks so much.

]]> <![CDATA[3.OA.8 – two-step equations]]> Tue, 10 Dec 2013 22:18:52 +0000 jjacobi My question pertains to this excerpt from 3.OA.8: “Represent these [two-step word] problems using equations with a letter standing for the unknown quantity.” It is unclear whether third graders should be expected to represent two-step word problems with a single number model, or if multiple equations are acceptable. The progressions documents don’t clearly address this either, saying only, “More difficult problems may require two steps of representation and solution rather than one.” Is the spirit of the standard that third graders should be able to represent a multi-step problem with a single number model (e.g., 7 x 3 – 2 = S) as well as with multiple models (e.g., 7 x 3 = 21; 21 – 2 = S)? I appreciate your thoughts on this.

]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Tue, 10 Dec 2013 02:57:19 +0000 lhwalker I totally get this. One would have to be some sort of omniscient prophet to anticipate exactly how the standards will play out in all respects: vertically, horizontally with science classes, etc., in a society with evolving technology beyond “Google Glass.” One of the reasons I have grown to trust your reflections is your tenacious humility. In a Country where democracy has been quickly fading, you have refused to become a dictator, and I really appreciate that.

]]> <![CDATA[Algebra necessary to succeed in geometry]]> Sun, 08 Dec 2013 15:29:26 +0000 terehi Good morning,
As districts begin (yes, BEGIN)to gear up for high school CCSS pathways,this question keeps coming up. What are the essential learnings in algebra that are necessary for students to have in order to be successful in geometry?
One reason this keeps coming up is that through SBAC students will be tested in 11th grade. If they follow a traditional pathway, this test may end up being the gate keeper to college readiness. Districts are expressing concerns about struggling students that traditionally kept repeating algebra ad nauseum and never reached geometry. They want to craft a course that will give students access to geometry while still remaining true to algebra. Do I make sense? Anyway, wondering what your thoughts are on this notion.

]]> <![CDATA[Reply To: 3rd Grade Multiplication]]> Fri, 06 Dec 2013 13:31:42 +0000 Bill McCallum My guess is that these publishers have not read the glossary to the standards. The Grade 3 standards call for fluency with multiplication within 100 (3.OA.7), knowing single digit multiplication facts from memory (3.OA.7), and multiplying single digit numbers by 2-digit multiples of 10 (3.NBT.3). Multiplication within 100 is defined in the glossary to mean “Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0–100.”

So that excludes all two-digit by two-digit multiplications except 10 x 10. And it excludes the more difficult single-digit by two-digit multiplications.

]]> <![CDATA[3rd Grade Multiplication]]> Fri, 06 Dec 2013 13:08:42 +0000 dcoker I’ve noticed that so called “Common Core Aligned” resources have 3rd grade students multiplying single-digit by double-digit (non-decade) numbers and double-digit by double-digit numbers. When I’ve questioned it, the publishers explained that the only standards asking 3rd grade students to work with single digit numbers are 3.OA.1 and 3.OA.7. Would you be willing to clarify how far teachers need to go with third grade multiplication?

Thanks so much!

]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Fri, 06 Dec 2013 12:46:09 +0000 Bill McCallum Hang on, which of the things that Solon did are you referring to? Inscribe the laws on large wooden slabs, or leave the country? According to Herodotus, Solon “left his native country for ten years and sailed away saying that he desired to visit various lands, in order that he might not be compelled to repeal any of the laws which he had proposed” [emphasis added]. I’ve always thought 10 years was about the right length of time for a revision cycle.

As for community ownership, I think that’s beginning to happen, in the messy way that such things do happen in this country. Our discussion on this blog is part of that process.

]]> <![CDATA[Reply To: misdirected link?]]> Fri, 06 Dec 2013 12:19:08 +0000 Bill McCallum Ah, sorry, I see it now. I’ll get this fixed now.

]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Tue, 03 Dec 2013 20:04:07 +0000 AndyIsaacs While you may try to speak your “private opinion,” you cannot because of your privileged role as a CCSS writer. Maybe you should have done what Solon did after he gave his laws to the Athenians.

Probably my most serious disappointment with the Common Core is the failure to provide the means for some community, any community, to own it. As I recall, there were representations made three years ago about a process for revision, including technical corrections and updates. But to date, so far as I know, nothing has happened. So we are left with requirements such as to teach “the” standard algorithms when there is no agreement about what those are.


]]> <![CDATA[Reply To: Common Denominators]]> Tue, 03 Dec 2013 03:14:34 +0000 Cathy Kessel I think your question is answered here (

]]> <![CDATA[Common Denominators]]> Tue, 03 Dec 2013 01:00:51 +0000 shelahf After reading the Fractions Progression and the CCSS for 3rd through 5th grade, I am slightly unsure on the expectations for adding fractions. The Progression states, “In grade 4, students calculate sums of fractions with with different denominators where one denominator is a divisor of another, so that only one fraction has to be changed…” (seemed clear to me) yet the standards don’t specifically mention denominators in 4.NF.3.a. I decided to check the Illustrative Tasks for 4.NF.3.a and saw the commentary, “Notice that students are not asked to find the sum because in grade 4, students are limited to computing sums of fractions with like denominators (i.e. fractions where 1 denominator is a multiple of the other).” Will you please elaborate on the expectation?

]]> <![CDATA[Reply To: CC Criticism]]> Tue, 03 Dec 2013 00:19:19 +0000 Cathy Kessel I did a little searching and found a description from Ravitch of “field testing.” From:

“I have worked on state standards in various states. When the standards are written, no one knows how they will work until teachers take them and teach them. When you get feedback from teachers, you find out what works and what doesn’t work. You find out that some content or expectations are in the wrong grade level; some are too hard for that grade, and some are too easy. And some stuff just doesn’t work at all, and you take it out.”

The comment above doesn’t mention education research at all, but it does help me understand the quote about reaching consensus. My impression after reading three Ravitch books (from 1995, 2010, and 2013) is that she’s not a user of subject-specific education research, so probably doesn’t see the relevance of work on learning trajectories for early grades and their implications for later grades. She does cite studies that use test scores as measures, noting sometimes that reading scores rose and math scores didn’t for certain studies. That’s about as subject-specific as she gets when reporting research—at least in those three books.

This Ravitch comment about development may also be helpful. From

“I’m very supportive of the idea of developing new assessments, and I think it’s a very important thing. But it will take years.

Just as these common core standards were written in a little over a year — it took me three years working on the California history standards. I worked on history standards in other states, and it was never done in only a year. So I would like to think that it’s going to take a lot of time to do this well because anything that’s done hurriedly is not going to survive….

I’m very happy that there’s money out there to develop new tests, but don’t think that they’re going to be available next year or the year after. If they’re good tests, it could be three to five years. And then they have to be tried out….So this is not going to be in time for the next election.”

I can remember also being skeptical about the short turnaround for standards development. (I worked on PSSM in its third and last year as an additional writer.) I think that one difference is that people working on CCSS worked very intensely via email. A second difference is that PSSM had longer illustrative examples. PSSM is 402 pages. CCSS is 93 pages.

Ravitch worked on the CA history framework for 1997 and its 2005 update. The 2005 version is 249 pages long and has suggested courses and appendixes. I found no references to education research in it. That doesn’t rule out its use but does reinforce the impression that I gained that subject-specific education research isn’t one of Ravitch’s considerations.

]]> <![CDATA[Reply To: misdirected link?]]> Mon, 02 Dec 2013 12:21:57 +0000 nvitale It not working the way I expected:
On the “Tools” page/tab, under K-8 Standards by domain….

If I click on most of the Domain names, they link to a document that has all of the standards across grade levels for the given Domain (“standards by domain”, as expected)

If I click on Number and Operations – Fractions, it links to the full progressions document for the domain, not the list of standards in the domain (which is what I was looking for).

Thanks for looking into it.

]]> <![CDATA[Reply To: Compound Inequalities]]> Mon, 02 Dec 2013 05:05:06 +0000 Bill McCallum Lane, there isn’t anything specifically about these as you point out. They could come in as an advanced example of

A-CED.1. Create equations and inequalities in one variable and use them to solve problems.

But the standards treat inequalities with a light touch, leaving the heavy work with them to advanced courses.

]]> <![CDATA[Reply To: Logarithmic Properties]]> Mon, 02 Dec 2013 04:57:42 +0000 Bill McCallum I guess I would choose

F-BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Note that this is a (+) standard so might be beyond the coursework that some students take.

]]> <![CDATA[Reply To: 8.NS.A.1 and 2]]> Mon, 02 Dec 2013 04:54:53 +0000 Bill McCallum Joanna, to address your last question on the importance of converting repeating decimals to fractions, I would say that the important thing here is not so much actually doing the converting as understanding why it can always be done. Repeated reasoning with the conversion can lead to this understanding. So, to achieve the understanding, you need to do a few conversions, just to see how it works. But it is the seeing how it works that is important point.

]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Mon, 02 Dec 2013 04:45:24 +0000 Bill McCallum Hi Andy, sorry for the long delay in replying … this blog got away from me for a while. Your points (i)—(v) are basically right for me. As for the difference between speaking ex cathedra or as private opinion, I have tried as hard as possible on this blog to express my views of the standards as someone reading them along with everybody else. Of course, I have insights into the intentions of the authors, having been one of them, and I am happy to share those insights. But I think if the standards are going to work then we have to treat them as a document owned by the community.

On (ii), my guess is that different curricula will take different approaches. I see the partial products algorithm as a natural precursor to the standard algorithm, where you compress some of the partial products by noticing you can sum them as you go, for each digit in the multiplier.

]]> <![CDATA[Reply To: misdirected link?]]> Mon, 02 Dec 2013 04:36:50 +0000 Bill McCallum Hmm, this link works for me, let me know if you still have trouble.

]]> <![CDATA[Reply To: Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Fri, 29 Nov 2013 04:06:18 +0000 lhwalker I lead my students in a discussion of factoring sums and differences of cubes by comparing similarities with factored differences of squares. For example, it is easy to see why $2x-3y$ might be a factor of $8x^3 – 27y^3$. Rather than memorize the pattern for the other factor, my students divide out $2x-3y$, generating the other factor.

[2013-12-06: Typo corrected]

  • This reply was modified 3 years, 8 months ago by  Bill McCallum.
]]> <![CDATA[Reply To: S-IC's]]> Thu, 28 Nov 2013 06:28:09 +0000 roxypeck I think that confidence intervals are beyond what is intended in the standard S-IC4. Students should have an idea of what margin of error is and how it is interpreted, which can lead informally to the idea of an interval estimate, but the concept of confidence level and calcualting formal confidence intervals would not be a part of what is expected in Algebra 2. Margin of error can be developed via simulation, building on the concept of sampling variability which is first introduced in grade 7.

While not an IC standard, S-ID.4 indicates that students would use calculators, spreadsheets and tables to estimate areas under a normal curve. If tables of the standard normal distribution are used to do this, students would need to use z-scores to move from an area under a particular normal distribution to an equvalent area for the standard normal distribution, so z-scores would probbly be covered there.

]]> <![CDATA[Reply To: 8.EE.1]]> Thu, 28 Nov 2013 01:23:00 +0000 Bill McCallum No, notice that it mentions only numerical expressions, so that would not include expressions containing variables.

]]> <![CDATA[Reply To: Geometry Progressions]]> Thu, 28 Nov 2013 01:22:03 +0000 Bill McCallum I’ve made a start on it!

]]> <![CDATA[Reply To: 5.OA.3]]> Thu, 28 Nov 2013 01:21:25 +0000 Bill McCallum I agree this standard is a little opaque. But I’m also having trouble understanding what your question is. Here’s the standard:

Generate two numerical patterns using two given rules. Identify apparent relation- ships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

The standard then goes on to give an example:

For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

So, students would generate 0, 3, 6, 9, etc., and then generate 0, 6, 12, 18, etc., and then they would notice that all the numbers in the second pattern are twice those in the first pattern. Notice that this is not something explicitly given to them … it is a consequence of the fact that 6 is twice 3. Later, in studying the proportional relationship $y = 2x$, students might make tables of $x$ and $y$ values where they notice the same thing: adding 3 (or any other number) to a value in the $x$ column results in adding 6 (or twice that number) to the value in the $y$ column. The process of forming ordered pairs and graphing them is preparation for making tables and graphs of relationships between varying quantities.

]]> <![CDATA[Reply To: Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Thu, 28 Nov 2013 01:07:39 +0000 Bill McCallum I think your example of looking at $x^6-y^6$ in two different ways is an excellent example of seeing structure in expressions, and it is no more complicated than the identity mentioned in the “for example” part of A-APR.4. It’s certainly very reasonable for classroom discussion. As for assessments, I don’t know what limits the assessment consortia will set on types of identities. I hope we don’t end up with some long list of identities students have to memorize. In some strange way that can work against seeing structure, because the list becomes the object instead of the expression. But you are quite right that it does not make sense to limit to only the identities explicitly list. That would be a strange way to interpret A-APR.4, for example, which only lists one identity as an example. It would be odd if that one identity made the list but difference of cubes did not.

]]> <![CDATA[Reply To: Other trig functions]]> Thu, 28 Nov 2013 00:54:12 +0000 Bill McCallum There is no prohibition on these in the standards, but they are not required. So the answer to your question really depends on entire curricular design. High school courses can include material not in the standards, but the material in the standards should come first. As a practical matter, I suspect for most students there is enough in the standards to take up a full year without including extra material, but an advanced course or a course for STEM-intending students might want to go further.

]]> <![CDATA[Reply To: Congruence]]> Thu, 28 Nov 2013 00:49:29 +0000 Bill McCallum Students start to study congruence in earnest in Grade 8. I wouldn’t think they have to know the term before then (although of course it is not forbidden).

]]> <![CDATA[Reply To: 6.NS.A.1 Division of Fractions]]> Thu, 28 Nov 2013 00:47:58 +0000 Bill McCallum Becky, have looked at the Number System Progression?

]]> <![CDATA[Sequences]]> Sat, 23 Nov 2013 02:30:16 +0000 kelicker I have a few questions about sequences in the CCSS associated with Algebra 1 by the PARCC Frameworks.

1. What is the distinction between the following two standards? I find them to be very closely related.

F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

The PARCC “Assessment Limits for Standards Assessed on More Than One End-of-Course Test” states that F-IF.A.3 is “part of the Major work in Algebra I and will be assessed accordingly.” Yet according to the “Pathway Summary Table,” F-BF.A.2 will not be tested in Algebra 1.

2. What notation will be primarily used to represent sequences in Algebra 1? I understand that it is beneficial to expose students to multiple notations, however I have been trying to align my instruction to use the language (and notation) of the standards.

There is a specific notation used in F-IF.A.3, and the Progressions states “In courses that address material corresponding to the plus standards, students may use subscript notation for sequences.” This leads me to infer that the primary notation in Algebra 1 will be as in F-IF.A.3. On the contrary, the PARCC reference sheet provides the formulas using the subscript notation. Will subscript notation appear on the Algebra 1 PARCC exam, or will the PARCC reference sheet be altered for Algebra 1?

I thank you for creating this site and taking time to answer questions.

]]> <![CDATA[Compound Inequalities]]> Sun, 17 Nov 2013 20:26:33 +0000 lhwalker I do not see And/Or (conjunction/disjunction) compound inequalities specifically in the standards. Where do they fit?

]]> <![CDATA[Logarithmic Properties]]> Thu, 14 Nov 2013 16:25:34 +0000 srlogan What common core standard would you map a question about Log properites. For example, writing an expression as the sum or difference of logarithms or writing an expanded logrithmic expression as a single quantity.

]]> <![CDATA[Reply To: 8.NS.A.1 and 2]]> Wed, 13 Nov 2013 01:33:56 +0000 jburtkinderman Bill,

I guess the issue is that I am seeing such huge payoff within my district (I am a K-12 math coach for a rural district) in leading with the message that students should build on steps they can justify. We are avoiding cross-multiplying, FOILing, and all sorts of shortcuts that are previously accepted as if they were ideas unto themselves… I am a strong believer in keeping the arrows of this change all pointing in the same direction, as the change is just enormous and is so very important.

If we are to unpack the thinking of this particular repeating decimal to fraction process, and have students replicate it, as I see it, the following is what is going on:

x = 17.171717…

100x = 1717.1717…
(here student should be able to answer why we choose 100)
(student should also be able to answer why the second equation is true, assuming the first / why the equations say the same thing)

Now, to move forward, student needs a reason subtracting the first from the second results in an equally true equation. If we are to regularly say:
“well, then, why would it be really cool to know what 99x is? why would this be a better idea that 101 x?”

then I suppose we are teaching students to replicate a teacher seeing a cool structure and making use of it, but I don’t think we can reasonably call this the students looking for the structure. In the classrooms that I’m observing and coaching, this task as a standard does not build meaning, but rather a notion that math is magic…

If we teach this as a most simple application of systems, I don’t think you encounter the same disconnect.

As an underlying issue, I’m not even sure that I see the real relevance here… Perhaps you can illuminate the importance of the idea of changing repeating decimals (we would only see this with a calculator) to fractions? If this is truly just a nice mathematical problem, should it not be a resource for MP 7 or a specific suggested treatment of 7.EE.4? Can you give me another example of a nice mathematical problem that is important enough to be turned into a standard?

I so appreciate the feedback and debate, because the process is so clarifying and cleansing. Further, as someone doing my work without a peer group, This site is a gem. It is beyond wonderful to have this level of feedback. If I’m missing something here, I’d enjoy seeing it from a new angle.

So Many Thanks,


]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Sat, 09 Nov 2013 13:10:14 +0000 AndyIsaacs Bill,

I appreciate this response. Thanks. My apologies for taking your opinion
as a “requirement.” It’s sometimes hard to know when you all are speaking
as private citizens and when you are speaking ex cathedra.

So, the bottom line seems to be that (i) since “the” standard
multiplication algorithm is not defined either in the standards or in the
progressions, our operational definition for it should be what most people
meant by the term in 2009; so (ii) you think the partial products
algorithm is OK in G4 but is not sufficient for G5; but (iii) kids should
not be “beaten to death with this” in seeking to meet 5.NBT.5 (or,
presumably, anywhere); and (iv) we should not think of CCSS as a catechism
(though maybe as a hymnal?). Also, (v) questions about the advantages or
disadvantages of specific features of “the” standard multiplication
algorithm would involve lengthy discussion, for which probably nobody has
the stomach.

One more question. Is there an example of “the” standard multiplication
algorithm anywhere in the progressions document at

or in some other place you could point to? If not, then given the
confusion over what qualifies, maybe you all should consider providing an
example or two.


]]> <![CDATA[misdirected link?]]> Sat, 09 Nov 2013 04:09:58 +0000 nvitale I wanted to use the standards by domain (Number and Operations – Fractions) in a workshop I’m doing, but the link is taking me to the full progressions document. Is this a misdirected link or am i looking in the wrong place for the Fractions standards by domain?

]]> <![CDATA[S-IC's]]> Fri, 08 Nov 2013 17:59:05 +0000 csteadman What role will confidence intervals and z-scores have, if any, in the S-IC standards? Could you provide an example of what is relevant to Algebra 2, and an example of what is beyond the course?

]]> <![CDATA[8.EE.1]]> Thu, 07 Nov 2013 20:17:56 +0000 klpatel Does this standard include generating equivalent expressions with exponents that have a variable as a base?

]]> <![CDATA[Reply To: Geometry Progressions]]> Thu, 07 Nov 2013 18:17:45 +0000 fsullivan68 Any timeline on the High School Geometry progressions document?

]]> <![CDATA[5.OA.3]]> Tue, 05 Nov 2013 14:58:13 +0000 dward I am not sure I understand the intent of this standard. The Progression document that discusses this standard says:

Students extend their Grade 4 pattern work by working briefly
with two numerical patterns that can be related and examining these
relationships within sequences of ordered pairs and in the graphs
in the first quadrant of the coordinate plane.5.OA.3 This work pre-pares students for studying proportional relationships and functions
in middle school.

I am hoping that someone might provide another interpretation of this standard and how the skills will connect to the study of proportional relationships and functions.

  • This topic was modified 3 years, 8 months ago by  Bill McCallum.
]]> <![CDATA[Sum of cubes and difference of cubes – APR.4 and SSE.2]]> Fri, 01 Nov 2013 17:17:41 +0000 SteveG I am posting this question for a high school teacher friend (I teach middle school). A-APR.4 mentions proving polynomial identities and A-SSE.2 mentions seeing structure. Evidently it is quite a debate with the high school folks as to whether or not the patterns for factoring sum of cubes and difference of cubes are included in these. I think it is, and that having students look at expressions like x^6 – y^6 as either a difference of cubes or a difference of squares can lead to some really interesting mathematical conversations.

I try to err on the side of saying that the standards cannot list every possible polynomial identities and that we should think about the ones that are the most useful. Others want to limit the identities to only those explicitly listed in the CCSS (such as the difference of squares).

What do you think? Would sum/diff of cubes only be allowed for classroom discussion or would it be reasonable to expect students to know and use a sum/diff of cubes factoring on an assessment?

]]> <![CDATA[Other trig functions]]> Mon, 28 Oct 2013 15:58:18 +0000 pstinchcombe Should we avoid using trig functions other than sine, cosine, and tangent? Should we think of facts about them (e.g., the other Pythagorean identities) as non-curricular facts? How about their definitions?

The progressions doc says “Because the second three may be expressed as reciprocals of the first three, this progression discusses only the first three,” but it’s unclear whether that means these are being left out of the discussion to save space, or that they’re being left out of the expectations for students because they’re redundant.

]]> <![CDATA[Congruence]]> Mon, 28 Oct 2013 14:54:46 +0000 bumblebee I see many comments in the Progressions and on the Blog about students knowing congruent sides, shapes, angles, etc. When should students know the formal term “congruent”?

]]> <![CDATA[Reply To: Unrepresentative samples in stats problems]]> Sun, 27 Oct 2013 18:45:34 +0000 pstinchcombe OK, thanks — so basically the conclusion is that we can do descriptive statistics on a non-representative sample just fine, and it’s only when we want to make inferences about the whole population that we need to worry about representativeness.

It seems to me that questions about whether there *is* a linear trend (or, equivalently, a nonlinear trend, or a relationship between categorical variables) are irrevocably inferential — unless the correlation is exactly zero, which almost never happens. Do you agree with that, or is there a purely descriptive way to make sense of such a question?

The progressions doc does indicate that we should make such judgments. (“if the two proportions… are about the same…”) If one of two proportions is a little bit higher than the other, the only way I know how to make sense of differentiating between sample proportions being about the same vs. one being higher is to make an (informal) inferential distinction: if I flip a coin 100 times, and 51% of them are heads, that’s about the same as I’d expect for a fair coin, but if I flip a coin 100,000,000 times and 51% are heads, that’s not about the same as I’d expect for a fair coin.

Does this seem right to you? That questions about whether there is a relationship are inferential by their very nature? It seems like in this case we’ve got to be careful about using the data set in the example on speeds and lifespans to make any conclusions; it seems like saying “the linear trend is due to a couple of outliers” is okay, as is “the line that best fits the data is…” but asking if there is a linear trend is a bad idea since this is an inferential question (or rather, either it’s an inferential question or the answer is “yes” for almost every data set including this one) and the data are unrepresentative.

]]> <![CDATA[6.NS.A.1 Division of Fractions]]> Sat, 26 Oct 2013 15:31:38 +0000 brbhazy I am looking for information on the 6th grade division of fractions standard. Can you direct me to information already posted here, if any? I am specifically interested in unpacking the standard and identifying the concepts and skills involved in this standard. Is there any explication of “interpret and compute” available?
Thank you, Becky Hayes

]]> <![CDATA[Reply To: 8.G.6 Converse of Pythagorean Theorem]]> Wed, 23 Oct 2013 05:11:20 +0000 Bill McCallum I meant to say “we are not expecting very formal proofs” and have edited my response to reflect that.

]]> <![CDATA[Reply To: F-LE A.1.a Use of "Prove" in standards]]> Wed, 23 Oct 2013 05:09:06 +0000 Bill McCallum A good working definition is that “prove” means provide a logical argument appropriate to the grade level. Exactly what “appropriate” means is up for discussion and adjustment as achievement improves. But the fundamental requirement is that the argument be faithful to the mathematics. One might define a linear function to be a function of the form $f(x) = b + mx$ and then adopt a fairly straightforward algebraic proof: over an interval of length $d$, $f(x)$ grows by the constant amount $f(x+d) – f(x) = b + m(x+d) – (b +mx) = md$.

As for assessment, I would expect it to be difficult, but not impossible, to assess this standard with a summative, machine-graded assessment. The ideal would be in-class observation by knowledgeable teachers.

]]> <![CDATA[Reply To: Solving Absolute Value and Compound Inequalities]]> Wed, 23 Oct 2013 04:59:07 +0000 Bill McCallum Here’s the link that I think Lane is referring to:

But I noticed that this is mostly about absolute value equations, not inequalities. I would also point out

7.NS.1.c. … Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

It seems to me that this is the key understanding, and could be applied in many contexts, including inequalities. I’m not sure that breaking this down into discrete activities such as solving absolute value inequalities is helpful.

As for compound inequalities, I see that as a notational device which is within the scope of the standards but which does not rise to the level of an explicit mention (but maybe I am misunderstanding the question here).

]]> <![CDATA[Reply To: Generating Content]]> Wed, 23 Oct 2013 04:35:19 +0000 Bill McCallum Here’s the answer from Dick Scheaffer:

Most of the graphics were done on a software program called Fathom. Fathom was designed for statistics education and probably is the favorite among those who teach AP Statistics (whereas most statisticians on college faculty would find it too limiting).

]]> <![CDATA[Reply To: Unrepresentative samples in stats problems]]> Mon, 21 Oct 2013 20:43:11 +0000 matthew.schneidman I teach 6th grade math at a charter school in the Bay Area and have been working with my students on statistics these past few weeks. I think I’ve been overthinking my unit – there’s too much going on and too little time to teach it all. I wrote up my unit plan before being told about this website and decided to teach my students about representative samples. I’ve enjoyed teaching them this – although they are no doubt a little confused – but it seems like a logical place to go. I interpreted the standards as a way not just to analyze data but a way for them to create their own statistical questions, collect data, create visual representations of that data, and analyze it. Although a representative sample is a challenging concept to understand, I’m wondering why this is not part of the standard. I’m not going into too much depth with it but it should be intuitive that you cannot take a complete sample of something – you have to ask a smaller group to get information about the larger population. You can use the data from a smaller sample to make inferences about larger samples. They also learn about inferring in ELA class and this provides a great opportunity to extend that skill to another subject.

I would love some feedback on this. Thanks.

]]> <![CDATA[Generating Content]]> Mon, 21 Oct 2013 14:45:33 +0000 dburris Thank you for the forums!

I am interested in the software you used for generating the graphics within the HS stats progressions documents … What did you use? I am especially interested in the dot plot on p.11 that shows re-randomization and the plotting of the mean differentials. I would really like to get to making/using this in the classroom!

Thanks for the help,

]]> <![CDATA[Reply To: Factoring Quadratics REI 4b]]> Mon, 21 Oct 2013 03:37:45 +0000 lhwalker I am still not happy with my “factor by grouping progression” because it ends in a trick.

Our curriculum currently requires “factor cubics by grouping,” so my progression has started off with a cubic like this: x^3+ x^2 -3x -3 which we factor by grouping :
(x3+ x2 )+(-9x -9)
x2 (x + 1) -9(x+1)
(x2 -9)(x+1)

The next step in my factoring progression has been to break up the middle term of a quadratic (with ANY integer coefficients)…and then factor by grouping:
4x^2 -11x – 3
4x^2 +x – 12x -3
x(4x + 1) -3(4x+1)

This method eliminates student frustration from guessing and checking, but in order to break up the middle term, the students use the AC trick, “What two numbers multiplied together give you AC and add to get B?” This Khan Academy video shows more examples. At the end of the video, he explains why the trick works, but of course 99.9% of the students will zone out during the explanation.

After further study of the standards, I see no reason why we would have to teach factoring a cubic by grouping because we could use graphing calculators to find integer zeros for polynomials of degree greater than 2 and divide out factors from there. If that is true, I will encourage my district to remove the requirement for “factoring by grouping cubics” and focus on trial and error for simple quadratics and completing the square for more complicated cases. Am I correct that there is no need to teach factoring a cubic by grouping or did I overlook something?

]]> <![CDATA[Reply To: Solving Absolute Value and Compound Inequalities]]> Sun, 20 Oct 2013 02:45:14 +0000 lhwalker Cathy Kessel did a great job answering this question in HS Algebra > Absolute Value Equations

]]> <![CDATA[3.OA.3]]> Fri, 18 Oct 2013 19:26:24 +0000 martha Can you clarify the meaning of measurement quantities? Does this refer to units (i.e. liters, pounds, inches, etc) or is it the 3rd type of multi/div situation? (found in table 3 of the K-5 Operations and Algebraic Thinking Progression document)

]]> <![CDATA[Reply To: 8.G.6 Converse of Pythagorean Theorem]]> Fri, 18 Oct 2013 00:34:01 +0000 Dr. M Bill,

I expect that you mean ‘very informal proofs’. But your description of how to go about it is right on. Of course, the diagram you describe is just the one used to prove SSS – join along a pair of sides of equal measure, construct a diagonal, and then invoke the Isosceles Triangle Theorem. As you say, the best way to prove the ITT is to construct the angle bisector and then ask what happens when we fold over it. Students get that right away.

]]> <![CDATA[Solving Absolute Value and Compound Inequalities]]> Thu, 17 Oct 2013 22:33:01 +0000 Sarah Stevens We have looked at the standards, the Algebra progression, and the 6-8 Equations and Expressions for guidance on solving inequalities. 7.EE4b states that students should solve problems of the form px+q>r and px+q<r. The progression points out that this includes less than or equal to. Does it also include compound inequalities or absolute value inequalities? The HS standards have students graph absolute value equations and systems of inequalities but we can’t identify when they learn to solve compound or absolute value inequalities in one variable. Any guidance would be helpful. Thanks

]]> <![CDATA[Reply To: 8.G.6 Converse of Pythagorean Theorem]]> Wed, 16 Oct 2013 07:04:25 +0000 Bill McCallum Well, one way to do it would be to indeed prove the SSS criterion. The triangle congruence criteria are quite fun to do with transformations, and remember that in Grade 8 we are not expecting very formal proofs. But another way to go would be something like the following proof, which essentially pulls in the necessary piece of the SSS proof.

Given a triangle whose three side lengths $a$, $b$, and $c$ satisfy $a^2+b^2= c^2$, construct a right triangle with legs of length $a$ and $b$. Then, by Pythagoras’ theorem, its hypotenuse has length $c$. Now put the two triangles together along their sides of length $c$, flipping one of the triangles if necessary to get a kite shaped figure (because of the corresponding sides of lengths $a$ and $b$). Drawing the other diagonal you can see the kite as two isosceles triangles matched along their bases. The base angles of the isosceles triangles are equal, so the opposite angles of the kite that you just joined are also equal. But one of those is the right angle of your right triangle. Therefore the original triangle also has a right angle, and you have proved the converse.

(Probably should have tried to draw a figure for this.)

You need to know that the base angles of an isosceles triangle are equal. That has a very nice proof using reflection about the angle bisector of the vertex.

  • This reply was modified 3 years, 10 months ago by  Bill McCallum.
]]> <![CDATA[Reply To: CC Criticism]]> Wed, 16 Oct 2013 03:13:23 +0000 Bill McCallum Thanks Corey. Maybe I will get around to giving a history on my other blog one day. Basically this mixes up two documents, the end-of-high-school expectations produced in summer 2009, and the standards themselves, produced in 2009-10. If you google “NGA press releases”, then search the press releases on “Common Core”, then read those in chronological order, you will get the bare bones of the story, including the composition of the various teams and the various organizations involved. Maybe I will say that over at Diane Ravitch’s blog.

]]> <![CDATA[F-LE A.1.a Use of "Prove" in standards]]> Tue, 15 Oct 2013 18:12:58 +0000 jtravis08 RE: a. Prove that linear functions grow by equal differences over equal intervals…
My question is in two parts…
1) When a standard like this one says “prove,” what specifically do you mean?

2) How do you think this standard might be assessed from a student perspective?

]]> <![CDATA[Reply To: CC Criticism]]> Tue, 15 Oct 2013 14:34:29 +0000 Corey Andreasen Thanks, Bill. I appreciate your responses.

I don’t know if you have any interest in responding to the conspiracy theories about the Common Core. There’s some stuff in the comments here:

One comment starts: The Common Core is largely the work of – as Mercedes Schneider points out – three main groups, “Achieve, ACT, and College Board.” Toss in the Education Trust. All of these groups are tied tightly to corporate-style “reform.”

I don’t know anything about this. I suspect there’s truth involvement of these groups, but I don’t know the nature of it.

]]> <![CDATA[8.G.6 Converse of Pythagorean Theorem]]> Fri, 11 Oct 2013 01:23:06 +0000 maddieblue What proof of the converse of the Pythagorean Theorem will Grade 8 students be able to understand? The common one requires students to understand the SSS congruence criteria, and I don’t believe they have learned that yet.

]]> <![CDATA[Reply To: Unrepresentative samples in stats problems]]> Mon, 07 Oct 2013 13:52:14 +0000 Bill McCallum Here is the comment from the statistician I asked, Roxy Peck:

I think that at the grade 6 level, the idea is that kids are working with census data and that they are not interested in generalizing beyond the group that they have data on. The idea of sampling is introduced in Grade 7 and from there on it is reasonable to have students think about the sampling process and whether or not that process is likely to result in a representative sample whenever they are generalizing from a sample to a larger population.

However, in many situations, like the ones that look at relationships between variables in Grade 8, regression and other model fitting is viewed as descriptive rather than inferential. So I think that these kinds of examples are OK, as long as students are trying to describe the relationship in a given set of data and are not generalizing beyond the data set. It is related to the distinction between descriptive statistics and inferential statistics. My take on the common core standards is that all of the modeling relationships between two numerical variables falls in the descriptive statistics realm. But I think it is appropriate to ask students to think about the sampling issues if they are asked to make predictions based on regression models. The assumption that is being made is that the data are representative of the relationship between the variables–something that would follow if the data are from a random sample but which also might be reasonable even when the data are not from a random sample.

I would add that the example in the progression about animal speeds is linked to the Grade 6 standard about describing data sets (6.SP.5), so falls squarely in the domain of descriptive statistics.

]]> <![CDATA[Reply To: Unrepresentative samples in stats problems]]> Mon, 07 Oct 2013 00:58:39 +0000 pstinchcombe Just to clarify, I certainly don’t think this problem is a good way to introduce students to the idea of representative sampling. The question isn’t whether we should use such datasets to teach students about representative sampling: it’s whether we should use such datasets at all.

My impression has been that representativeness of a sample is still relevant — although harder to assess — when the population in question isn’t as easy to understand. If we don’t believe our sample is representative at least in the key regards, what justification could we have for drawing conclusions about the relationship between characteristics of animals from that sample?

Part of the problem, for me, is that the standards do address this, briefly, when they say “random sampling tends to produce representative samples and valid inferences.” The probability that any random selection of animals — whether you weighted by geographic distribution, or by population, or biomass, or any other reasonable criterion except familiarity to us — would produce a sample which is this weighted towards large animals and mammals is vanishingly small.

]]> <![CDATA[Reply To: Unrepresentative samples in stats problems]]> Mon, 07 Oct 2013 00:41:24 +0000 Bill McCallum I’m going to ask a statistician to take a look at this question, but here is my take on it. When you talk about a representative sample, you are talking about a situation where there is a more or less homogeneous population with a number of different subtype (e.g. ethnic groups in a population of a country). And you want to make sure your sample has the same proportions of subtypes.

But I don’t think you can say that animals form a homogenous group. For each species you might measure average height and weigh, and you have probably already chosen a representative sample within each species in order to get those averages. And then you want to see if there is any relationship between these variables across species. You could just try to get all the species, but I’m not sure what it would mean to make a representative selection of species. Would you try to make sure the percentage of mammalian species represented the true proportion in the world? Or would you go for some sort of geographic representation? I can see that it’s worth discussing these things, but it seems a too complicated example to introduce 6th graders to the idea of a representative sample (or even high schoolers, for that matter).

]]> <![CDATA[Reply To: Mean absolute deviation]]> Mon, 07 Oct 2013 00:32:50 +0000 Bill McCallum The Mean Absolute Deviation is intended as a more intuitive precursor concept to the Standard Deviation. It makes sense to take the mean of all the absolute deviations if you want to get a quantitative measurement for the spread; it turns out for rather subtle reasons that taking the square root of the sum of the squares of the deviations (that’s what the Standard Deviation is) is better, but you can’t really explain the reasons at this level, and the more complicated procedure might obscure the underlying idea.

]]> <![CDATA[Reply To: S-IC-5]]> Mon, 07 Oct 2013 00:29:24 +0000 Bill McCallum This captures most of the standard, namely comparing the means of two treatments and deciding whether the difference is significant. The standard talks about comparing parameters, and the mean is just one parameter. But it is probably the one you will most often look at when comparing to treatments. You might, however, see a difference in standard deviations, or one of the quartiles, and want to know if that difference is significant as well. Also, I’m not completely sure what you mean by a resampling technique, but I suppose it is the same as as a simulation, which is the word the standard uses.

]]> <![CDATA[Reply To: 5.OA.2]]> Sat, 05 Oct 2013 02:46:18 +0000 Bill McCallum Sorry for the delay in replying, somehow this one got lost! I really think this is just a miscommunication. We should be keeping things simple at Grade 5, and there was no intention here to suggest nesting of symbols. It was simply a matter of being agnostic about which grouping symbols to use. Later on students learn a hierarchy of symbols, but here they just learn the idea of bracketing things off (or bracing them, or putting them in parentheses). In practice it will probably always be parentheses, and we should probably just have said that.

]]> <![CDATA[Reply To: Mean absolute deviation]]> Thu, 03 Oct 2013 18:20:54 +0000 pstinchcombe On question two:

Mathematically, I think the mean absolute deviation is more sensitive to outliers — and it makes sense for a measure of spread to be sensitive to outliers. For example, the IQR of the data set {4,5,6,6,7,8,9} is the same as the IQR of {1,5,5,8,8,8,20} — the quartiles are 5 and 8 — but I think any reasonable person would say the second set is more dispersed.

From a practical standpoint, I think IQR is rarely used by professionals, so students are unlikely to see it reported. If part of the goal of S&P curriculum is to prepare students to understand stats they encounter, IQR does very little to serve that goal. MAD isn’t very often used either (I think it is used more commonly than IQR, but it’s still not very standard), but it’s closer to the measure of spread which actually is used most often, which is standard deviation.

]]> <![CDATA[Reply To: Mean absolute deviation]]> Thu, 03 Oct 2013 18:19:57 +0000 pstinchcombe Mathematically, I think the mean absolute deviation is more sensitive to outliers — and it makes sense for a measure of spread to be sensitive to outliers. For example, the IQR of the data set {4,5,6,6,7,8,9} is the same as the IQR of {1,5,5,8,8,8,20} — the quartiles are 5 and 8 — but I think any reasonable person would say the second set is more dispersed.

From a practical standpoint, I think IQR is rarely used by professionals, so students are unlikely to see it reported. If part of the goal of S&P curriculum is to prepare students to understand stats they encounter, IQR does very little to serve that goal. MAD isn’t very often used either (I think it is used more commonly than IQR, but it’s still not very standard), but it’s closer to the measure of spread which actually is used most often, which is standard deviation.

]]> <![CDATA[Unrepresentative samples in stats problems]]> Thu, 03 Oct 2013 18:03:49 +0000 pstinchcombe I’ve seen, in lots of stats curriculum fora including the Common Core progressions and other Common Core materials, problems that ask students to make inferences about animals based on data on some list of about 30 animals.

In every case that I’ve seen, that list of animals is wildly unrepresentative: for example, the one shown in the progressions is vast-majority (>80%) mammals, and vast-majority over 100 grams. In the progressions doc, this is targeted at a 6th-grade standard, so the students aren’t supposed to know about representative samples yet — but often these questions are targeted at higher-level standards (my guess is the 8th-grade question on life span vs. speed is based on a similarly size-biased sample, although I can’t tell for sure).

It feels like we ask students to learn about representative samples in 7th grade, but at every point before and after this, we expect them to be ignoring it and drawing inferences based on unrepresentative samples. The progressions document seems to be implicitly telling me I shouldn’t be worried about this — is that the intention? If so, I’d like to hear why this isn’t a concern.

]]> <![CDATA[Mean absolute deviation]]> Thu, 03 Oct 2013 17:18:43 +0000 lilaclorax Two questions in reference to 6.SP.5.c

If the standard says IQR and/or Mean Absolute Deviation- do students need to be able to do both or would they be able to answer any question using just one?

Two- What are the benefits of using Mean Absolute Deviation rather than IQR? In other words, what is the payoff in 6th graders learning Mean Absolute Deviation?

]]> <![CDATA[Reply To: S-IC-5]]> Wed, 02 Oct 2013 18:10:27 +0000 SB_EHS We are breaking down S-IC-5 and interpreted it to the following statement

– compare two treatment means with a re-sampling technique(with computer software) to test whether results are random or significant.

What are your thoughts on that interpretation?

]]> <![CDATA[Reply To: S-IC-5]]> Wed, 02 Oct 2013 18:05:27 +0000 SB_EHS We broke down Standard S.IC.5 in the following way:

-Compare two treatments means/proportions with a resampling technique (simulation software) to test whether results are random or significant.

What are your thoughts?

]]> <![CDATA[Reply To: Inverses of Functions]]> Tue, 01 Oct 2013 15:14:11 +0000 jkerr That makes sense.

I hope other teachers and administrators aren’t lost on this idea. I can envision a “why are you wasting time covering this standard, it won’t be assessed” scenario. I see a similar situation with rational functions, where graphing won’t be assessed. However, it would be logical to teach some graphing so that students would have that tool to check the rational expression operations and equation solving that will be assessed.

Thanks for your reply.

]]> <![CDATA[Reply To: Definitions]]> Sun, 29 Sep 2013 18:59:38 +0000 Julia Brenson Hello,
May I suggest a reason why knowing the difference between the terms skewed left (negative) and skewed right (positive) may be not only beneficial, but essential, in understanding the relationship between the shape and context of the data? If students are asked to interpret the differences in shape, center and spread in context (S.ID.3) between the distribution of the age at which people first get a driver’s license and the distribution of the age at which people retire, part of understanding this data in context is to understand that in the first instance, the distribution is likely to be skewed positive and in the second, the distribution is likely to be skewed negative.

]]> <![CDATA[Reply To: CC Criticism]]> Fri, 27 Sep 2013 04:44:15 +0000 Bill McCallum The first installment of my answer is now posted here.

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Wed, 25 Sep 2013 00:41:01 +0000 Cathy Kessel Here are a few more comments on “multistage” and “compound.”

I did a google search on “multistage event”. I don’t get a lot of hits related to probability and statistics for that but I do for “multistage experiment”, which gives the term a slightly different emphasis. An experiment is something like tossing a coin, tossing two coins, rolling a die, etc. It’s what you do to generate an outcome.

“Compound event” appears to have two definitions which are not equivalent. Sometimes the idea that there are two different definitions of a term comes as a shock, but it does happen. Trapezoid is one example.

The NCTM book Navigating Through Probability, 6–8 says on p. 11: “An event is the outcome of a trial. A simple event (usually called an event) is a single outcome. A compound event is an event that consists of more than one outcome.” For the experiment “roll a die,” it gives the example of “six on top face” for simple event, “prime number on top face” for compound event. Neither of these is the result of a multistage experiment (“roll a die” has only one stage).

That’s the definition of compound event that I grew up with. Using that definition and using the sample space described above (i.e., there are four 7th graders on the list), the event “picking a 7th grader” is a compound event and “picking an 8th grader” is a simple event.

In the CCSS, “compound event” is more akin to “an outcome of a multistage experiment,” e.g., an outcome of rolling two dice, as Bill has already discussed. Under that definition, “picking a 7th grader” as described above is not a compound event. The experiment is “picking a student” which has only one stage.

I can see that we need a note about this in the S&P Progression.

]]> <![CDATA[Reply To: CC Criticism]]> Tue, 24 Sep 2013 18:45:08 +0000 Bill McCallum I try to keep this blog focused on simply answer questions about what’s in the standards. These are all good questions, but I think I’ll answer them over at my other blog, (it will probably take me a couple of days … I’ll post a note here when I have done it).

]]> <![CDATA[CC Criticism]]> Tue, 24 Sep 2013 18:14:14 +0000 Corey Andreasen Bill,
I’ve been a fan of Diane Ravitch’s recent work opposing the privatization movement in the schools. The Common Core is often in her sights. And I agree with her when she talks about the problems with the way it’s being implemented, the high-stakes tests and the school teacher accountability aspect of it. But I’m also seeing people complaining on her blog (and Diane does it, too) about the way the standards were developed. No teachers involved, no opportunity for public review, etc. And I don’t think those concerns are based in fact.

She also talks about how the standards were never field-tested. I may be wrong, but it seems to me there’s nothing to test about the standards themselves. It’s simply a decision about what we want students to know and be able to do.

A recent post ( brings up this point, and in the comments on reader said, “I suddenly hit in an aha moment. How can anyone, or any group, especially sans teachers, reach a consensus on what is appropriate for say, fourth grade?
Teachers would have difficulty agreeing, and they’re the experts.” I see this a great deal.

I’d like to be able to respond to these comments knowledgeably, but I don’t know enough about the development of the standards. I do know that, as a member of the Wisconsin Mathematics Council board of directors I knew about the Common Core years before they were implemented, and I seem to recall opportunities to give feedback on them.

Can you speak to any of this? Or maybe even answer Diane’s concerns on her blog, if you’re into that sort of thing! 🙂


]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Tue, 24 Sep 2013 12:33:34 +0000 Bill McCallum Yes, a probability model and a probability distribution are the same thing, or at least two ways of looking at the same thing. A probability model assigns a probability to every event in the sample space. One way to conceptualize this (later in high school) is through the idea of a distribution, a function on the sample space for which the area under the graph above a certain event represents the probability of that event. (Sorry, compressed a large part of a course into that sentence!) And so, a uniform probability model and a uniform distribution are the same thing (note, however, that we also use the word distribution to refer to a data distribution, a different but related thing … but that’s another story).

The second definition of compound event (from your textbook) is the one used in CCSS. A compound event is an event in a sample space that has been constructed out of two other sample spaces. For example, you have the sample space {heads, tails} for tossing a coin and the sample space {1, 2, 3, 4, 5, 6} for rolling a die, and you construct the space

{(heads, 1), (heads, 2), (heads, 3), (heads, 4), (heads, 5), (heads, 6),
(tails, 1), (tails, 2), (tails, 3), (tails, 4), (tails, 5), (tails, 6)}

out of both spaces, and calculate probability of events like “flip heads and roll an even number”, which is the subset {(heads, 2), (heads, 4), (heads, 6)}.

And yes, you might call this particular compound event a multi-stage event too. Although I can imagine cases where the two parts of the compound event happen at the same time (e.g. lightning strikes the tree and I am standing under it).

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Mon, 23 Sep 2013 15:36:26 +0000 sunny Is P(7th grader) compound?
I am curious to the answer to Steve’s question.

I am a bit confused by the following terms, and I find disagreements in definitions in resources:
1. uniform probability model vs. uniform distribution – are these the same?
2. compound events vs. multi-stage events
According to the NCTM Navigating through Probability 6-8 book, a compound event is “an event that consists of more than one outcome”. Two examples given are P(one heads and one tails) is compound because there two outcomes from the sample space for one heads and one tails, and P(rolling a prime number).
However, in our textbook CCSS supplemental materials a compound event is defined as “an event that consists of two or more single events”. Is this describing a multi-stage event? i.e. flip a coin, spin a spinner.

I am wondering how the term “compound event” is to be interpreted in SP.8? As more than one outcome, or more than one event (multi-stage)?

Thank you very much for your help!

]]> <![CDATA[Reply To: Confusion about 8th grade Function Standards]]> Sun, 22 Sep 2013 19:34:25 +0000 Bill McCallum I’d have to see the text to be able to comment on it. But as a general comment I would say that there is a progression from proportional relationships to functions. At some point in that progression you introduce the concept of a function, and then you can look back and point out that a proportional relationship can be viewed as a function (in two different ways, depending on which quantity you choose as the input). I don’t know where the non-proportional relationships come in, however, that sounds a bit out of place to me.

]]> <![CDATA[Reply To: 6.G.2]]> Sun, 22 Sep 2013 19:28:17 +0000 Bill McCallum Andy, you are talking about my August 23 reply, right? I didn’t mean it to be taken any differently from my previous comments on this. To me the unit cubes or the rectangular prisms with unit fraction sides are mathematical objects, and when I talk about packing them this way or that I am talking about a mathematical activity. So, this could be represented by “manipulatives, drawings, computer animation, verbal descriptions …” as in my previous answer. Sorry for the miscommunication.

  • This reply was modified 3 years, 11 months ago by  Bill McCallum.
]]> <![CDATA[Reply To: Step Functions]]> Sun, 22 Sep 2013 19:21:14 +0000 Bill McCallum Lane, no, there is no standard notation for step functions that I know of in current use in school mathematics.

]]> <![CDATA[Reply To: 8.NS.A.1 and 2]]> Sun, 22 Sep 2013 19:19:06 +0000 Bill McCallum In answer to your first paragraph: Yes! The standard says “Use rational approximations …” not “Find rational approximations …” so it is a mystery to me how people could misinterpret it. Of course, as you say, this might sometimes involving finding them using simple methods of trial and error, as you suggest, but how anybody could interpret this as requiring a systematic method for finding approximations is beyond me.

In answer to your second paragraph, I’m not sure what you mean by “a clever application of solving by elimination.” 8.NS.1 does in fact say “convert a decimal expansion which repeats eventually into a rational number,” so you have to have some way of doing this. One way is to solve a linear equation: if $x = 0.171717 …$ then I can write the equation $100x-x = 17.171717 … – 0.171717 … = 17$ and solve the equation $100x – x = 17$ to write $x$ as a fraction. You are right that there is some cleverness involved in thinking of multiplying by 100 and subtracting the original number, but this in itself is a nice application of MP7, Look for and use of structure.

Students have been solving equations like this since Grade 7:

7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Converting repeating decimals to fractions strikes me as a nice “mathematical problem” that falls under this standard.

]]> <![CDATA[Reply To: Uniqueness of triangle constructions in 7.G.A.2]]> Sun, 22 Sep 2013 19:06:26 +0000 Bill McCallum Sorry to be so slow in answer this … but yes, you have answered your own question correctly!

]]> <![CDATA[Reply To: Inverses of Functions]]> Sun, 22 Sep 2013 18:59:20 +0000 Bill McCallum The dividing line been regular standards and (+) standards shouldn’t be viewed as a demarcation in the curriculum, but rather in assessment. There are a number of instances where for reasons of coherence one would want to include some (+) standards in the curriculum. Notice the statement on p. 57:

All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students. [Emphasis added.]

That said, I do think there’s a difference between F-BF.4a and A-CED.4. The procedure is the same, but conceptualizing the procedure as “finding an input to a function which yields a given output” is a step up. Seeing functions as objects in their own right, and algebraic procedures as ways of analyzing those objects, is a sophisticated viewpoint.

]]> <![CDATA[Reply To: Adding and subtracting mixed numbers – Grade 4]]> Thu, 19 Sep 2013 12:44:15 +0000 Bill McCallum This is still causing a bit of confusion. I just uploaded a version of the progression with the new language here. It makes clear that adding and subtracting fractions with one denominator dividing the other is not a requirement in Grade 4 except in the case of denominators 10 and 100. But it also makes clear that this is still a possibility in the case of simple additions such as $\frac13 + \frac16$ where the idea is not to present a general rule but to reason directly with simple equivalences and the definition of addition using visual representations such as fraction strips. So, you couldn’t put this on an assessment in Grade 4 but you could put it in the curriculum.

]]> <![CDATA[Reply To: G-GPE.4]]> Thu, 19 Sep 2013 11:13:10 +0000 Bill McCallum There is that word “simple” in there. The examples are not intended to be exhaustive, of course, but they do illustrate what is meant by that word. I would say that the examples you gave, while certainly falling within the meaning of the standard, are not required by it. Such theorems should certainly be proved, but proofs using congruence and similarity make more sense to me. The analytic proofs would be quite laborious, wouldn’t they?

]]> <![CDATA[Reply To: Question on expectations for S.ID.6 and S.ID.8]]> Thu, 19 Sep 2013 11:06:31 +0000 Bill McCallum I think students will be using some sort of technology (not limited to calculators) for all the operations with data: scatterplots, curve-fitting, plotting residuals. They should be looking at realistic data sets, which are too large for anything else. It makes sense to me to analyze residuals while looking at a different curves of fit, in order to see how the distribution varies. Then you eventually just start looking at the curve of best fit and the correlation coefficient.

]]> <![CDATA[Step Functions]]> Fri, 13 Sep 2013 02:05:40 +0000 lhwalker Is there a standard notation we will use for step functions?

]]> <![CDATA[8.NS.A.1 and 2]]> Thu, 12 Sep 2013 01:24:52 +0000 jburtkinderman I’m troubled by the interpretations I’m seeing of these standards and would love some clarification, correction and/or support. It seems that folks are interpreting the second of these standards as a call to lead students through memorizing a procedure for finding approximations of irrational numbers to the tenths and then hundredths place by essentially pretending that the growth between, say, sqrt(16) and sqrt(25) is linear. I see no use in 8th graders muddying their mathematical waters with such procedures as opposed to, say, reasoning that sqrt(22) must be closer to 5 than 4. I suppose that it makes sense to then choose 5.6 and then 5.7, … , square them and see which is closer, but it makes no sense at all to pretend that sqrt(22) is 6/9 from sqrt(16). This is corrupting the sense of what a square root IS. Yes?
Secondly, most people seem to be ‘treating’ this standard early in an 8th grade text / course, and so are teaching how to rewrite a repeating decimal as a fraction. This process is a clever application of solving by elimination, which we haven’t learned yet (I assume) until the end of our 8th grade year. Using mathematical techniques that have not been derived, explored and established seems epically contrary to everything I love and admire about CCSS-M.
As always, advice and feedback are most appreciated.
My Best,
Joanna Burt-Kinderman

]]> <![CDATA[Reply To: Uniqueness of triangle constructions in 7.G.A.2]]> Wed, 11 Sep 2013 20:28:28 +0000 ivan After looking at the high school geometry standards, I understand that the notion of uniqueness of a triangle is up to congruence relations. Thus the triangle with side lengths 3,4,5 and the triangle with side lengths 3,5,4 are the “same” since one can be obtained from the other by a reflection.

]]> <![CDATA[Reply To: Introducing supporting concepts, not in the standards]]> Tue, 10 Sep 2013 18:05:57 +0000 Bill McCallum I agree with Cathy. What you have sketched is an approach to proving the Pythagorean theorem which is quite beautiful.

]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Tue, 10 Sep 2013 18:03:48 +0000 Bill McCallum Requirement?

Here is a link to the comment: In it, I said

Some think it is the algorithm exactly as notated by our forebears, some think it includes the expanded algorithm, where you write down all the partial products of the base ten components and then add them up. Ultimately this is a question that has to be settled by discussion, not fiat.

I then went on to state my opinion:

My opinion is that the standard algorithm has two key features; like the expanded algorithm it relies on the distributive law applied to the decomposition of the number into base ten components, but in addition it relies on the fact that the order of computing the partial products allows you to keep track of the addition of the partial products while you are computing them, by storing the higher value digit of each product until the next product is calculated.

I don’t see how I could have made it clearer that I was not stating a requirement, just giving an opinion. And I didn’t say anything about benefit, either. That would take a much longer discussion, since benefit or harm would depend on the context: the students, the classroom culture, the curriculum, the time constraints, and so on.

I agree that students can be fluent with methods such as those described by Beckmann and Fuson. They would satisfy

4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. …

However, as I’ve said elsewhere, I don’t think that the partial products algorithm is included in what people meant when the used the term “standard algorithm” circa 2009, whether they were for it or against it. So I don’t think 5.NBT.5 includes the partial products algorithm. That said, I don’t think kids should be beaten to death with this; the standards form a pathway along which some students will be ahead, some behind. And some of those behind will need to take shortcuts to catch up. They are not a catechism, they are a shared agreement about what we want students to learn.

]]> <![CDATA[Reply To: identification and classification of shapes]]> Tue, 10 Sep 2013 17:32:01 +0000 Bill McCallum Lots of questions here. First, on the rhombuses, rectangles and squares, notice the phrase “and others” earlier in the standard. The intention was not to exclude parallelograms or any other particular type of of quadrilateral. Rather, the intention was to not require parallelograms. Listing every single type might have been taken as a requirement. The main point is to begin to see how different types can be included in larger more general category. The exact list of specific shapes is not important.

The cluster heading “Reason with shapes and their attributes” is consistent through Grades 1–3 … not sure what the question is here.

As for revising the standards, your guess is as good as mine. My preference would be a long revision cycle, say 10 years. Not because the standards don’t need revision, but we need time to work with them to do a thoughtful revision that isn’t just a cacophony of everybody’s favorite modifications.

]]> <![CDATA[Reply To: 7.NS.2d]]> Tue, 10 Sep 2013 17:21:29 +0000 Bill McCallum Sorry, I thought I had already replied to this one. I don’t think students have to see repeating decimals before Grade 7; that is, the standards do not require this.

]]> <![CDATA[Reply To: Laws of Sines/Cosines]]> Tue, 10 Sep 2013 17:16:56 +0000 Bill McCallum Well, it’s a bit like saying “Get ready for school, and don’t forget to brush your teeth.” Brushing your teeth is implied by getting read for school, but you want to make sure it isn’t forgotten. G-SRT.11 is a brush-your-teeth standard. Same goes for calling out proof in G-SRT.10 … you want to make sure people see that.

]]> <![CDATA[Reply To: Inverses of Functions]]> Tue, 10 Sep 2013 13:04:31 +0000 jkerr Yes, that’s where I first went for clarification. Overall, F.BF.4abcd is fine coverage of inverses. My question is about the thinking behind separating F.BF.4a apart from the rest. Maybe my problem stems more with the progression doc stating that formal notation and language are not important at this stage. My thinking is that if we don’t at least call this thing an inverse, then what are the students actually going to get out of this? Students wouldn’t be doing much more than was done in standard A.CED.4.

]]> <![CDATA[Reply To: Sums/Products are Rational/Irrational]]> Tue, 10 Sep 2013 01:22:08 +0000 Bill McCallum I think Appendix A is a little of base here. There’s a discussion of this from about a year ago here.

]]> <![CDATA[Reply To: Geometry CO.6-8 and 8th grade connections as well as "prove" standards]]> Tue, 10 Sep 2013 01:09:08 +0000 Bill McCallum First, I would say the isometry standards in Grade 8 are 8.G.1–3. 8.G.4 is about similarity transformations.

On the question about G-CO.6, I think the words “geometric descriptions of rigid motions” are important. In Grade 8 students might have a fairly intuitive notion of rigid motions; in high school they work with definitions of rigid motions in geometric terms. For example, in Grade 8 you might say that a reflection about a line takes every point to the point on the other side at the same distance from the line. In high school you might say that reflection about the line $\ell$ takes a point P to itself, if P is on $\ell$, and otherwise takes to a different point P’ such that the line through P and P’ is perpendicular to l, intersecting it at O, and PO is congruent to OP’. Predicting effects using this description is partly simply a matter of grasping which rigid motion is being described.

I also agree that predicting effects could include naming the coordinates. It could also include other observations: for example, reflecting about the side of a triangle produces a triangle that shares a side with the original.

]]> <![CDATA[Reply To: Introducing supporting concepts, not in the standards]]> Tue, 10 Sep 2013 00:31:17 +0000 Cathy Kessel This might be answered by the Publishers Criteria here:

Some comments though . . . one thing that I notice in discussions of curriculum and instruction is that a given topic can be taught in different ways. Just saying that a given topic is included isn’t necessarily evidence that something (e.g., curriculum materials) is standards-aligned or not. Obviously, you’re thinking of a particular approach rather than just a topic. The question might become “How does this approach fit with the standards?” I’d suggest thinking of “standards” (plural) rather than just the Pythagorean theorem or just standards that involve the Pythagorean theorem.

]]> <![CDATA[Reply To: 9-12.N.Q.3]]> Mon, 09 Sep 2013 23:45:40 +0000 Bill McCallum So, a quick example would be: don’t quote the population of the United States down to the ones digit. Do you remember that count down to when the population passed 300 million? It was meaningless. We can’t possibly know the population that accurately, given (a) the possibility of error in the census and (b) the fact that people are dying and being born all the time. A quick google search suggests that births and deaths are in the thousands per day. If you google “population of the United States” you get 313.9 million, accurate to the nearest 100,000.

]]> <![CDATA[Reply To: Inverses of Functions]]> Mon, 09 Sep 2013 23:27:11 +0000 Cathy Kessel Have you looked at p. 13 of the Functions Progression? It can be downloaded here:

]]> <![CDATA[Inverses of Functions]]> Mon, 09 Sep 2013 20:48:22 +0000 jkerr Hello,

I’m having a bit of trouble justifying the approach to inverses in high school. I’m not sure what lasting knowledge students will gain from F.BF.4a if that is the full extent of the coverage in Alg 1 & 2. It is a good place to start with inverses, but without extending the coverage in the same school year it seems like this standard is on it’s own little island. It appears as though we will be losing the opportunity to use inverses to make the connection between various types of functions.

Maybe F.BF.4a is saying more than I think it is. Could you explain the rationale behind the approach to inverses of functions in Algebra 1 and 2?


]]> <![CDATA[Reply To: Sums/Products are Rational/Irrational]]> Mon, 09 Sep 2013 01:33:52 +0000 Fred Hollingshead Thank you for pointing out the 6-8 NS Progression actually includes high school as well.

The document I am referring to is Appendix A. Whether looking at page 25, Algebra I/Unit 5 or page 63, Math II/Unit 1, the Clusters with Instructional Notes column to the of the standard states “Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2.”

We have since moved beyond this standard having needed a solution a couple of weeks ago. We handled my posted question by asking students to either state that a particular property was true or if false, by offering a counter example. Our in-class conversations went much deeper than this and included some discussion of formal proofs for a few of the properties.

This was our first go at this particular standard as we are phasing in Math 1/2/3 and replacing the A1/G/A2 track we use to have. We may handle this differently next year based as we learn more about this particular standard over the next year.

]]> <![CDATA[Reply To: Sums/Products are Rational/Irrational]]> Mon, 09 Sep 2013 01:14:29 +0000 Cathy Kessel Does it help to look at p. 17 of the Number System & Number Progression posted here: I see that the document label on that page is misleading, you need to click on the link that says “Draft 6–8 Progression on The Number System”.

I’m not sure how the standard points to physical situations, could you explain?

]]> <![CDATA[Reply To: Confusion about 8th grade Function Standards]]> Sat, 07 Sep 2013 18:19:13 +0000 terehi I also have a question that is a bit more pragmatic. I’m in the middle of reviewing an 8th grade text for my state adoption. In the earlier part of the text they do an extensive section on proportional and nonproportional “relationships”, but never mention the concept of function. Later in the text, they delve deeply into “functions” with. Definitions and such. My dilemma is that they include the earlier lessons on relationships as primary citations matched to the function standards. What are your thoughts on that? Do those citations qualify even though there is no mention of a “function” or is this an insurmountable flaw in alignment?

]]> <![CDATA[Reply To: 6.G.2]]> Fri, 06 Sep 2013 23:03:11 +0000 AndyIsaacs Bill,

Your reply above seems to suggest that 6.G.2 is calling for some sort of hands-on manipulative activity, which seems wildly impractical to me. Your reply above is also at odds with the answer you gave us in July 2010 when we asked essentially the same question. Your answer then was, “Certainly the intention was not to mandate any manipulative activity,
but rather to describe how to conceive of the volume of a prism with
fractional side lengths. This could be aided by manipulatives,
drawings, computer animation, verbal description … how to bring
about the conceptual understanding is not mandated.” Jason Zimba also responded to us at that time, saying, “I myself at least always thought of this as a pencil and paper diagrammatic argument, not a manipulative activity.”
So, which is it? Are you proposing that teachers try to obtain actual “unit cubes of the appropriate unit fraction edge lengths”? Or even, as you now seem to be proposing, “rectangular prisms with unit fraction side lengths”? Or is this a thought experiment, as we thought you and Jason were saying three years ago?


]]> <![CDATA[Uniqueness of triangle constructions in 7.G.A.2]]> Fri, 06 Sep 2013 18:30:06 +0000 ivan 7.G.A.2 says:

… Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

I am parsing this to mean that students should be able to identify whether there is a unique, multiple, or no solutions for the given constraints.

The uniqueness question can be tricky. Are we considering reflections of triangles in the plane to be different triangles?
Is the black half of this triangle the “same” as the white half in the following unicode character ◭ ?

Example: If I specify three side lengths say 3,4,5, then you can build a left-hand and a right-hand version of this triangle in the plane.
Should these be considered as multiple solutions or a unique solution?

Workaround: We could interpret the standard as saying “unique up to symmetries” and interpret the multiple solutions as infinitely-multiple solutions, e.g. when we do not specify the third side-length or the third angle, as in example above.

  • This topic was modified 3 years, 11 months ago by  ivan.
]]> <![CDATA[G-GPE.4]]> Fri, 06 Sep 2013 13:00:46 +0000 moberlin GPE-4 states that students should be able to, “Use coordinates to prove simple geometric theorems algebraically.” Upon first read, this standard seems pretty straightforward. I imagine students using the coordinate plane setting to prove statements such as, “The diagonals of a rectangle are congruent” or “The diagonals of a parallelogram bisect each other” or “The mid-segment of a triangle is parallel to a side of the triangle.” But what follows threw me for a loop:

For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

These examples do not describe what I would have considered a proof of a geometric theorem. The types of problems described in the example certainly have pedagogical merit; they simply do not describe what I would consider geometric theorems. Can you help me reconcile these examples with the statement of the standard?

]]> <![CDATA[Question on expectations for S.ID.6 and S.ID.8]]> Thu, 05 Sep 2013 16:57:32 +0000 dward When students fit exponential and quadratic functions to data, is the expectation that they will use a calculator to first examine a scatter plot and then use the calculator to determine a regression equation that best models the data? If so, should the analysis of the residuals plots and correlation coefficient come into play for such situations? Standard S.ID.8 indicates that students interpret the correlation coefficient of linear fit. Does this include the correlation coefficient that the TI gives with the exponential regression?

]]> <![CDATA[Introducing supporting concepts, not in the standards]]> Mon, 02 Sep 2013 18:21:08 +0000 StevenGubkin Often I like to make a subject more accessible by breaking it down into more manageable chunks. Sometimes these chunks have established mathematical names. For instance, before teaching the Pythagorean theorem, I would like to investigate right triangles with a perpendicular drawn from the right angle vertex to the hypotenuse. The geometric mean of two numbers appears again and again in such calculations. I think it is worthwhile to introduce the term “geometric mean” at this point. This prepares students beautifully for understanding the Pythagorean theorem – you can have them “solving” right triangles using similarity before they have seen the theorem, and the theorem follows by just doing the same stuff to a general triangle.

The concept of “geometric mean” is not in the standards, but I believe that it is of great utility in getting students to develop an appreciation for the Pythagorean theorem. Is it “Common core aligned” to introduce this operation? More generally, if a concept of great utility connects to a standard, but is not explicitly mentioned, may it be included in “Common core aligned” material?

]]> <![CDATA[Reply To: Division and Multiplication Algorithms in the Progressions]]> Sun, 01 Sep 2013 18:48:59 +0000 AndyIsaacs We’re wondering what the benefit is to the requirement that “the” standard algorithm for multiplication “relies on the fact that the order of computing the partial products allows you to keep track of the addition of the partial products while you are computing them, by storing the higher value digit of each product until the next product is calculated” (Bill McCallum, September 13, 2012 at 12:08 pm #939).

This requirement seems to rule out the “all-partials” algorithm as “the” standard algorithm. For example, as we read the requirement above, it seems that Beckman and Fuson’s Method D on page 24 (or Method A or Method B on page 23) of their NCSM Journal article ( would not qualify as “the” standard algorithm. Given that Fuson and Beckman contend that Method D is conceptually clear and fast enough for fluency we’re wondering what the benefit is to the requirement above.

]]> <![CDATA[Reply To: Typos in Canonical Forms Document]]> Fri, 30 Aug 2013 22:32:36 +0000 Chris Meador There also appears to be a typo for CCSS.Math.Content.HSN-VM.B.5a, where the 5 is missing from both the dot notation and the URI.

]]> <![CDATA[identification and classification of shapes]]> Fri, 30 Aug 2013 13:31:40 +0000 ldacey I appreciate the comments regarding when students are expected to learn formal definitions of polygons (a term missing from the k-5 geometry standards). My question is even more basic. Many of the k-5 geometry standards list specific shapes, for example, 3.G.1 includes the statement: Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. The specificity of this list, without a qualifying phrase such as for example, suggests that students do not need to recognize trapezoids, parallelograms, and kites as examples of quadrilaterals. In fact, parallelograms are never mentioned in the standards at these grade levels. Conversely, though the progressions document lists several shape names that should be introduced at the kindergarten level, the standard does not: K.G2: Correctly name shapes regardless of their orientation or size. This open-ended statement offers no guidance as to the specific shapes students should be able to identify though circles, rectangles, squares, and triangles are listed in the introduction. Also, what am I missing about the cluster name at this level? I see Identify and describe shapes, but other times, including in this blog I see a parenthetical phrase listing two-dimensional and three-dimensional shapes. I understand the argument that ways of knowing/understanding grow over the years, rather than the types of shapes explored, but teachers would be greatly served by further guidance in the standards themselves. This leads me to my overall concern for this domain, the challenge of revision. Have standards been created that would be politically impossible to revise?

]]> <![CDATA[Reply To: Geometry CO.6-8 and 8th grade connections as well as "prove" standards]]> Thu, 29 Aug 2013 16:06:36 +0000 Dr. M Let me add that I find Harold Jacobs text Geometry: Seeing, Doing, Understanding just about right for a high school classroom. The one real deficiency from the point of view of the CCSS is that it makes SAS and a postulate. That’s a problem, though, that’s relatively easily rectified. (He also gives Euclid’s proof of the AA triangle similarity principle. By my lights, that’s fine. But one might like to do it instead with some sort of scale transformation postulate.)

]]> <![CDATA[Reply To: Geometry CO.6-8 and 8th grade connections as well as "prove" standards]]> Thu, 29 Aug 2013 15:41:46 +0000 Dr. M I’ll respond to the question about parallels.

There’s no one right answer about what to call a postulate and what to call a theorem. Euclid offered a proof of the SAS triangle congruence principle in his Elements, but Hilbert, in his Foundations of Geometry, made that principle is postulate. Neither is right. Neither is wrong. They simply made different decisions.

But there are certain principles that should guide the choice of a set of postulates. One is that it should not include superfluous postulates, i.e. postulates that can be proven on the basis of other postulates already in place. I do think that a bit of that is fine in a high school geometry classroom, but it should be kept to a minimum. That’s why I so dislike, for instance, when a text makes both SAS and SSS triangle congruence postulates. SAS can be used in a relatively straightforward proof of SSS!

This principle implies that we should not make the statement below (or any other equivalent to it) a postulate:

When a transversal cuts a pair of lines so that alternate interior angles are congruent, then those lines are parallel.

Why not? It’s provable! See Book I, Proposition 27 of the Elements.

A second principle that should guide the choice of a postulate set is that the postulates chosen should be both simple and obviously true. (I mean this to hold only for the high school classroom. These requirements are dropped at higher levels.) That’s why, when I teach parallels, I choose the Playfair Postulate. (Through a point not on a line, there’s at most one line parallel to that given line.) It’s clear and (to students’ minds) obviously true. Together with the proposition above, it can be used to prove that if a pair of parallel lines are cut by a transversal, then alternate interior angles are congruent. (Here’s a quick sketch of the proof. Assume that point P does not lie on line m. Construct line n parallel to m through P. Construct a transversal to m and n through P. If alternate interior angles are not congruent, then we can construct a second line r through P for which they are. But then this line r will be parallel to m, and so we then have two lines through P parallel to m. This contradicts the Playfair Postulate. Hence alternate interior angles are in fact congruent.)

]]> <![CDATA[Reply To: 7.NS.2d]]> Wed, 28 Aug 2013 20:02:58 +0000 smithba.wbms Check out H.-S. Wu’s paper “Teaching Fractions According the Common Core State Standards”. It contains some answers to your question.

]]> <![CDATA[Laws of Sines/Cosines]]> Tue, 27 Aug 2013 20:48:50 +0000 Alexei Kassymov G-SRT-10: … use them <Laws of Sines and Cosines> to solve problems.

G-SRT.11: apply the Law of Sines and the Law of Cosines
to find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces).

It looks like the entirety of SRT.11 fits into SRT.10. What kind of problems are envisioned for SRT.10 that are different from finding unknown measurements in triangles? Even the first parts are hard to separate. Depending on the point of view, “understanding” (SRT.11) includes being able to prove (SRT.10).

Any clarification on these two will be greatly appreciated.

]]> <![CDATA[Sums/Products are Rational/Irrational]]> Tue, 27 Aug 2013 15:21:25 +0000 Fred Hollingshead N.RN.3 states “Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.” Without requiring a formal proof, what should we look for from Math II students in their explanations of these fairly sophisticated questions? The standard points to physical situations, which then logically seems to indicate for specific situations. Yet the standard itself seems to require a general explanation (proof?), but doesn’t actually use “any” or “all” as part of the prompts.

]]> <![CDATA[Geometry CO.6-8 and 8th grade connections as well as "prove" standards]]> Mon, 26 Aug 2013 19:45:14 +0000 tsebring In 8th grade, a foundation of isometry is laid with 8.G.3 and 8.G.4. The Geometry standards that relate are CO.6-8. I am trying to discern the intent of CO.6 and its relation to 8.G.3 and 8.G.4. For example the wording in 8.G.3 specifically mentions coordinates whereas the wording of CO.6 does not use “coordinates”. Would predicting “the effect of a given rigid motion on a given figure” include a student giving the coordinates of a point on a figure that has undergone a reflection, rotation or translation? As CO.7 refers to congruence I am assuming the predicted effect is not that the figures are congruent. As CO.5 includes naming the transformation I also did not that as as “the effect”.

Lastly, the Geometry standards have many “prove theorems”. I have two questions about these standards. The application of several of these theorems is not always seen in the standards, is there a reason for that? Or would that be “implied”? And lastly, I have seen some books that named “corresponding angles are congruent” as a postulate. What resource would you suggest using as an authority for what is a theorem and what is a postulate?

]]> <![CDATA[Reply To: 9-12.N.Q.3]]> Mon, 26 Aug 2013 17:54:30 +0000 Cathy Kessel It may help to look at the discussion under HS Number and Quantity here:

]]> <![CDATA[9-12.N.Q.3]]> Mon, 26 Aug 2013 17:16:48 +0000 Jane Bill, I have had many questions about what exactly we are looking for in this standard. “Choose a level of accuracy appropriate to limitations on measurement when reporting quantities”. Could you give me your interpretation and an example of expectation?

]]> <![CDATA[Reply To: 6.G.2]]> Sun, 25 Aug 2013 14:14:27 +0000 kgartland Thanks so much for responding so quickly. I agree with the proposed language suggestion of “unit fraction side lengths.” Your added bonus of describing how you could use unit cubes is intriguing, however, it would certainly be a very small cube 🙂
Perhaps an activity could be suggested where students make unit cubes that are 1.5 by 1.5 out of grid paper (or something like that) and then use it to measure the volume as you suggest in your post.
So important to keep it concrete while they are first learning the concept rather than just multiplying l x w x h without reason.

]]> <![CDATA[Reply To: 6.G.2]]> Fri, 23 Aug 2013 13:53:59 +0000 Bill McCallum Alexei is right, I think it makes sense to replace “unit cubes” with “rectangular prisms with unit fraction side lengths” here. One of these days I will publish my glitch file!

Although I would point out that you could use unit cubes. For example, you can pack a $\frac13$ by $\frac15$ by $\frac17$ rectangular prism with unit cubes with side length $\frac1{3\times5\times7}$, forming a $5 \times 7$ by $3 \times 7$ by $3 \times 5$ array. But in the end you still have to find the volume of the cube, and the natural way to do that is by seeing how many of them fit into a cube with side length 1. Since that’s also the way you would find the volume of a rectangular prism with unit fraction side lengths, I think it makes more sense to do the latter directly.

Probably way more answer than you wanted!

]]> <![CDATA[Reply To: 6.EE.6]]> Fri, 23 Aug 2013 13:29:49 +0000 Bill McCallum There are two related standards:

6.EE.6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.


6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form $x+p=q$ and $px=q$ for cases in which $p$, $q$ and $x$ are all nonnegative rational numbers.

Both of these occur the margin in the passage you mention in the Progression. The first is about variables and expressions, the second is about equations. Since expressions are used in writing equations, the reference to a “set” in 6.EE.6 could refer to a solution set in the context of 6.EE.7, but it does not have to. The Progression illustrates this with the example of the expression $0.44n$ to represent the price in dollars of $n$ stamps: here $n$ comes from the set of whole numbers. And note that 6.EE.6 does not refer to a “set of solutions” but rather simply to a “set.”

As to your last question, it is certainly true that in many instances where one uses a variable to stand for a single unknown, it will be in the context of writing an equation to find that unknown. That is, the use of variables described in 6.EE.6 to represent an unknown number will arise in the practice described in 6.EE.7 of writing equations to solve problems. There is indeed a close connection between the two standards. Still, it seems worth distinguishing the idea of choosing a letter to represent an unknown number as an important idea in its own right.

  • This reply was modified 3 years, 12 months ago by  Bill McCallum.
]]> <![CDATA[Reply To: Real-world vs. mathematical]]> Fri, 23 Aug 2013 05:15:39 +0000 Bill McCallum That’s very observant to notice the “or” versus “and.” I wouldn’t attach much significance to it, however. I think it’s just an editing inconsistency. The point of the phrase is simply to make clear that there is no demand to contextualize every problem, nor is there a demand that students must work on purely mathematical problems all the time. I suppose we could just have said “problems” rather than “real-world and mathematical problems,” but people tend to see what they want to see, so we wanted to make clear that any sort of problem was permissible.

]]> <![CDATA[Reply To: tan = sin/cos]]> Fri, 23 Aug 2013 05:12:05 +0000 Bill McCallum The standards don’t lay out an entire trigonometry curriculum, and this is a case of something in such a curriculum which is not explicitly called for. G-SRT.6 is a natural place to attach this piece of knowledge, but it is not required by that standard.

]]> <![CDATA[Reply To: Units in equations]]> Fri, 23 Aug 2013 05:09:58 +0000 Bill McCallum Physicists and mathematicians have different approaches to this. You are right that the tradition in physics is to include units in the equations themselves. In mathematics we tend to put the units in the definition of variables. The important thing is to put the units somewhere. So, if you write $d = 65 t -12$, you’d better have said beforehand that $d$ is distance in miles and $t$ is time in hours (and not “let $d$ be distance and $t$ be time”). Then you can deduce from the equation that the units of 65 are miles/hour. There are advantages to the physicists’ approach, but the mathematicians’ approach also had its virtues. For example, when the units are not present in the equation itself, it is easier to see the structure of the equation.

]]> <![CDATA[Reply To: Acceleration]]> Fri, 23 Aug 2013 05:02:28 +0000 Bill McCallum I don’t have a lot to add here, but one comment I would like to make is that CCSS necessitates a rethinking of acceleration policies. Acceleration in middle school was often a response to the repetitiveness of the middle school curriculum. But CCSS in middle school is not repetitive; it is a dense and rich diet of important mathematics. So students who previously hungered for acceleration might now be quite satisfied with a solid implementation of CCSS.

]]> <![CDATA[Reply To: 8.F.3: Are constant functions linear functions?]]> Fri, 23 Aug 2013 04:55:34 +0000 Bill McCallum I agree. Nice to have a question asked and answered in the same comment!

]]> <![CDATA[Reply To: Laws of Logarithms]]> Fri, 23 Aug 2013 04:54:25 +0000 Bill McCallum I’m wondering if you are looking at your state’s augmentation of the standards, rather than the standards themselves. The standard in question, on page 71, is

F.LE.4. For exponential models, express as a logarithm the solution to $ab^{ct} = d$ where $a$, $c$, and $d$ are numbers and the base $b$ is $2$, $10$, or $e$; evaluate the logarithm using technology.

There is nothing about laws of logarithms, and furthermore the bases are limited.

Still, I would say that in general there is point introducing laws if you don’t apply them.

]]> <![CDATA[Reply To: The Continuous/Discrete Distinction]]> Fri, 23 Aug 2013 04:48:36 +0000 Bill McCallum This implicit distinction also occurs earlier than Grade 7. As soon as students start working with the number line, they are beginning to move from a discrete counting model of the whole numbers to a continuous measurement model, which allows for the introduction of fractions. You make an interesting comment that this is not mentioned explicitly in the standards. One possible reason is that although the distinction is important background knowledge for teachers, it is not necessarily useful to introduce this terminology with students. For students, the related concepts are counting (discrete) and linear measurement (continuous). They should work with both of these, but it might not be worth while to introduce terminology, since it is difficult to see how the terminology would add to the intuitive conceptions, and the formalization of these conceptions is way beyond K–12.

]]> <![CDATA[Reply To: Not sure about MCC9-12.G.SRT.5 and others]]> Fri, 23 Aug 2013 04:34:30 +0000 Bill McCallum First, this standard is not necessarily about coordinate algebra at all. It is more about using the basic similarity and congruence criteria to solve problems with more complex figures. A simple example might be showing that the diagonals of a parallelogram bisect each other. Using this diagram, randomly chosen from the internet


one might first observe that $\triangle AOE$ is similar to $\triangle COB$, using the AAA criterion, and then use the congruence of opposite sides (previously proven) to conclude that the two triangles are congruent, and hence that $AO = OC$ and $BO = OD$. Of course, one can condense this argument with a direct appeal to the ASA criterion for congruence, but I quite like breaking it apart this way in this case, since the similarity is what first arises from the basic properties of transversals, and then the congruence depends on a previously established result.

Here is another very nice example, more complicated. It is a good example of looking for and making use of structure, because if you draw auxiliary lines parallel to the sides through E and F you see all sorts of similar triangles and can chain the ratios between them to solve the problem (I won’t spoil it for you by giving the solution).

I realize this doesn’t really give you the guidance you are asking for, but perhaps it will set some ideas in motion for the geometry course.

  • This reply was modified 3 years, 12 months ago by  Bill McCallum.
]]> <![CDATA[7.NS.2d]]> Fri, 23 Aug 2013 01:56:28 +0000 amsmith5 Based on their work with 5.NF.3, 4.NF.1, 4.NF.6, and 6.NS.2&3, should students already know some fractions have decimal representations that terminate in a finite number of digits while others are infinitely repeating? Are students expected to express remainders as decimals in grade 6 and the intent of 7.NS.2d is just to make sense of why some fractions result in terminating decimals and others do not?

]]> <![CDATA[Reply To: 6.G.2]]> Wed, 21 Aug 2013 23:05:23 +0000 Alexei Kassymov It’s likely the same glitch as here:

Drafty draft of Fractions Progression

]]> <![CDATA[6.G.2]]> Wed, 21 Aug 2013 16:38:23 +0000 kgartland Hi,
I have a question about the reference to “unit cubes” and fractional side lengths in the 6.G.2 standard:
It states “Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas v=lwh and V=Bh to find the volume of right rectangular prisms with fractional edge lengths in the context of real-world problems”.

As it says to use “unit cubes” and yet we are to have students work with fractional edge lengths, I’m wondering if you could elaborate on what kind of unit cubes you could suggest be used in which fractional side lengths could then be created?
Is there such a manipulative? Or would you suggest conducting this activity using technology?
Thanks for any help you can provide.

]]> <![CDATA[6.EE.6]]> Wed, 21 Aug 2013 14:24:37 +0000 BrunoBrunelli Dr. McCallum,

Standard 6.EE.6 refers specifically to writing expressions (without mention of equations), but also includes an understanding that a variable can represent a single unknown or a set of solutions. The progressions illustrate these perspectives on variables by differentiating the expression .44x from the equation .44x=11. Does this standard include writing equations as well as expressions? If not, how can an expression alone be used to illustrate the fact that a variable can represent a single unknown value?

]]> <![CDATA[Real-world vs. mathematical]]> Wed, 21 Aug 2013 14:23:09 +0000 BrunoBrunelli Dr. McCallum,

My understanding is that the frequent differentiation between Real-world and Mathematical problems refers to mathematical problems that are or are not contextualized. Is a problem considered to be Mathematical rather than Real-world when it is abstract and free of real-world context from beginning to end? Are problems in which a student must contextualize or decontextualize both considered to be Real-world problems?

Additionally, I’ve noted that the phrase “real-world and mathematical problems” is used throughout the standards, while there are several occurrences of “a real-world or mathematical problem” within the EE sections of 6th and 7th grade. Is any specific distinction intended by the two wording forms?

]]> <![CDATA[tan = sin/cos]]> Tue, 20 Aug 2013 16:09:58 +0000 Alexei Kassymov I don’t believe it is a relationship stated anywhere specifically. Would it be something expected as part of G-SRT.6?

]]> <![CDATA[Units in equations]]> Mon, 19 Aug 2013 00:27:05 +0000 Sarah Stevens The last two summers I have participated in a science modeling workshop to increase my own content knowledge for applying to analytic modeling and to experience pedagogy techniques to encourage conceptual understanding. Throughout these courses, the instructors (science teachers) were very clear about the necessity of labeling all numbers- even those in equations. For example, an equation to calculate the position as a function of time might look like
d = 65miles/hour * t -12miles
instead of our familiar math equations d=65t-12.

Through-out the experience, I learned how lax some math teachers (including myself) have become with units. I gained an appreciation for how labels on numbers kept the focus on the context and the meaning of the numbers in the context of the situation. So I left the training, determined to shift our math teachers to this “scientific” way of writing equations. We would be supporting science and scaffolding our students as we transition into a modeling focus.

However, at my first meeting with math teachers who hadn’t attended the trainings, I experienced quite a bit of questioning from my colleagues. The modeling progression came out the same week as my math training. The teachers quickly noted that the equations in the progression didn’t contain labels. I still argue that it does no harm, helps another content area, and keeps students anchored in the context of the problem.

I continued researching and noticed that the beginning of the High School “Number and Quantity” defines quantities as numbers with units. When I look at the Functions standards, I see the word quantity used in most of the standards- which I think further bolsters my thinking about units in equations. I would like to continue with our shift towards units on all measured numbers; but, with a lack of examples, my teachers are hesitant. We are very interested in your opinion on the matter, both personal and the requirements of the standards.

]]> <![CDATA[Reply To: Acceleration]]> Sun, 18 Aug 2013 19:20:15 +0000 williamsl I continue to be very concerned that ALL students have a right to learn and mentally sweat in their math classes on a daily basis. We all know that students don’t learn at the same rate. I truly believe we can accelerate the learning of many who traditionally have struggled in math by using the CCSS with fidelity, providing those many representations and strong differentiate core instruction for all. This would fit with the RtI research that shows that if we teach this way we can really help all but a few learn during core instruction and then provide a little bit extra or a little bit different yet for those still needing more experiences to be successful. However, there are also those on the extremely fast learning end who don’t need to see things in many different ways to make the connections and truly understand. They don’t need the repetition and they have a right to go at the pace that is appropriate for them and not be slowed down so that they can teach others.

We have found that about 15% of our students, when provided with really good CCSS instruction don’t need two days on a lesson, they are making connections extremely quickly because this type of instruction is helping them as well. We’re using the CPM Core Connections curriculum and our accelerated group was able to complete all of the grade 7 curriculum and move through a chunk of the grade 8 curriculum in one year. This fall they will continue where they left off and move into the algebra curriculum. Effectively compacting three years into two without skipping anything, just by taking out repetitions that they don’t need that many students do.

Much of the research on heterogeneous groupings and taking away the slow paced classes makes sense. We can’t close achievement gaps by slowing some groups down and doing algebra in two years instead of one. However, in some studies the growth for the gifted students isn’t compared to the growth they make when they have appropriately paced curriculum. Reports say things like the top 10% were excluded because they were in other classes or had already been accelerated. So, the progress for the top students in the study was good, but the study didn’t actually include the top 10% of the students. Other times the gifted are reported as “no worse off” than when they were in undifferentiated classrooms with everyone getting identical instruction, again not compared to when they get what they need and can handle.

For more info please read Karen Roger’s book, “Reforming Gifted Education.” We need to make sure there are many options for all the types of students we have. We may need to work on making sure there are equitable pathways to all populations to all the options, but we shouldn’t be taking all the options away (especially when we are replacing it with undifferentiated one-size fits all teaching). Why would we stop offering IB High Level Math or AP Calc at the high school because we can’t figure out a way to increase the pace at which some students are able to experience curriculum?

]]> <![CDATA[Reply To: Acceleration]]> Sun, 18 Aug 2013 02:13:54 +0000 swuttig We are working in our district to find ways to engage and challenge our gifted math students. Last year, we worked on a consistent curriculum across the district rather than a school by school decision. We compacted traditional 6th, 7th, and 8th grade curriculum since there is much repetition in 7th and 8th grade. Our district is in the process of implementing our new state standards which are very similar to the Common Core. We are trying to decide what we will be doing in the future about challenging our gifted students in addition to challenging problems. We don’t want them to skip grades/classes. We are a bit hesitant to compact the Common Core 8th grade math in the suggested accelerated pathway since it is an important foundation for later mathematics. We were wondering was there any discussion about other options of accelerating students by compacting, such as compacting 6th and 7th grade Common Core math.

]]> <![CDATA[8.F.3: Are constant functions linear functions?]]> Thu, 15 Aug 2013 21:08:29 +0000 Jason Zimba Today I was asked the following question:

Is a constant function a subset of linear functions or are the two mutually exclusive?

Here is how I answered:

I think the standards settle this question. They say,

“Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.”

If it had been desired to exclude the case m = 0, then I think we must believe that it would have said

“Interpret the equation y = mx + b (with m not equal to 0) as defining a linear function…”

Also, the phrase “straight line” is used twice (once in the standard, once in the italicized example). A horizontal line is straight.

*Error in original post has been fixed (originally said b=0 not m=0). Thanks Heather!

  • This topic was modified 4 years ago by  Jason Zimba. Reason: Forgot to tag it the first time
  • This topic was modified 4 years ago by  Jason Zimba.
  • This topic was modified 4 years ago by  Jason Zimba.
  • This topic was modified 4 years ago by  Jason Zimba.
]]> <![CDATA[Laws of Logarithms]]> Wed, 14 Aug 2013 14:04:03 +0000 sutterw For Algebra II, F.LE 4.1, the standard says that students should prove simple laws of logarithms. My question is, should they also be able to apply those simple laws? For example, should they be able to create equivalent forms of 5^(log base 5 of 12) – log base 2 of 4?

]]> <![CDATA[Reply To: The Continuous/Discrete Distinction]]> Tue, 13 Aug 2013 12:31:18 +0000 smithba.wbms Here is the post that got me wondering about this:

]]> <![CDATA[The Continuous/Discrete Distinction]]> Mon, 12 Aug 2013 16:08:15 +0000 smithba.wbms I have noticed that the CCSS do not include a single instance of the word “continuous”, and the standards do not highlight the continuous/discrete distinction. In a discussion on this blog, Jason Zimba here shows that the distinction is implicit in teaching number and measurement, and clearly there is opportunity to discuss the distinction while working with, e.g. slope in seventh grade (which is what I teach). But I would have expected to find these terms explicitly in the standards.

Will you comment on this omission?

]]> <![CDATA[Not sure about MCC9-12.G.SRT.5 and others]]> Fri, 09 Aug 2013 15:25:23 +0000 kpillow I’m a high school teacher in Georgia, and I teach a course called Coordinate Algebra. Many of the standards leave me wondering what it is I should do to convey a particular standard; they seem vague. For example,

MCC9-12.G.SRT.5 Use congruence and similarity criteria for triangles to
solve problems and to prove relationships in geometric figures.

While I do appreciate some freedom in which to work, my knowing what the
students will be responsible for knowing can be considered important. In other words, what relationships? One way I interpret this standard is that a student would be given a random relationship and asked to use his knowledge to prove that relationship.

Can you clear up my confusion over the “versatility” of the
standards? Am I looking at this too closely or not close enough? Thanks.

]]> <![CDATA[Reply To: 8.G.2 – Demonstrating Rotational Symmettry]]> Tue, 06 Aug 2013 19:18:02 +0000 Dr. M In my class I plan to keep the examples simple. This means rotations that are multiples of 90 degrees. Other rotations are more difficult to handle. How does one demonstrate, for instance, that one figure is the rotation of another 30 degrees about the origin? In general, that isn’t trivial, and it is of course beyond an introductory geometry class.

For my own part, I don’t think that analytic verification of a given transformation is of such great importance. A bit is good. But I do only so much as is necessary to motivate a set of results that follow: when polygons are congruent, sides which correspond and angles which correspond have equal measures; when sides and angles of polygons can be paired up in such a way that those which correspond have the same measure, then the polygons are congruent; SAS congruence and the rest of the triangle congruence principles.

]]> <![CDATA[8.G.2 – Demonstrating Rotational Symmettry]]> Mon, 05 Aug 2013 16:09:49 +0000 Silas Kulkarni I’m a math coach and I’ve been working with one of my teachers on 8.G.2 and we are hitting a bit of a wall.

Here’s the standard.
Understand congruence and similarity using physical models, transparencies, or geometry software.

CCSS.Math.Content.8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

So the standard asks the students to understand that if one figure is, for example, a rotation of another figure then the two figures are congruent, and then to show a sequence that exhibits how you get from one to the other.

This has been fine for reflection and translation. But for rotation it seems much harder. The problem we’ve been having is that it seems hard to demonstrate that one figure is a rotation of another in a mathematically rigorous way. Specifically, you can usually tell that a figure is a rotation by looking at it (literally drawing it on a piece of paper and turning the paper), but how can you spell out the conditions that prove that?

We thought about considering the distance of each point from the origin, since rotations will create images whose points are the same distance from the origin as the pre-image, but that is true of reflections also (or even of non-congruent figures, whose points are a rotation of the points in the pre-image, but the points of which have been rotated by differing amounts). We also thought about using the degree measures of each point relative to the x axis, i.e. something very similar to what you do in trigonometry with the unit circle, but that seems to require going far beyond the knowledge currently available to students in 8th grade.

So how do you demonstrate rigorously that one figure is a rotation of another?

If you have any insight on this, we could really use the help, as we are sort of stuck on this point.

Thanks for the help,

]]> <![CDATA[Reply To: A-REI.5 – what does it mean/look like?]]> Mon, 05 Aug 2013 14:33:35 +0000 Bill McCallum I don’t think I understand the question. You can solve a set of linear equations in many variables by the elimination method. Or, you can represent the system by a single matrix equation and solve by row operations on the matrix. The latter is merely a notational representation of the former; solving by elementary row operations and solving by the elimination method are the same thing at bottom. The use of matrix notation to represent systems of equations is not required in the standards.

]]> <![CDATA[Reply To: 6.G.1 Surface Area]]> Mon, 05 Aug 2013 01:47:58 +0000 lhwalker Another nice thing about the Standards is the way students are encouraged to use area models in lower grades. If the students have been using them in prior grades…examples at:

then surface area is easily connected to those.

]]> <![CDATA[Reply To: 6.G.1 Surface Area]]> Sun, 04 Aug 2013 00:23:13 +0000 Bill McCallum It doesn’t; Grade 6 is the first explicit mention of surface area. Notice this is in 6.G.4, not 6.G.1. Students use nets to find surface area.

]]> <![CDATA[6.G.1 Surface Area]]> Thu, 01 Aug 2013 03:34:28 +0000 mtstaller This standard asks students to solve problems involving area, surface area, and volume. Can you tell me where surface area comes up before 6th Grade? I can’t find a reference in the Progressions. Thank you for your help.

]]> <![CDATA[Reply To: A-REI.5 – what does it mean/look like?]]> Sun, 28 Jul 2013 14:44:28 +0000 kelicker Thank you for your response. This is helpful. May I ask, are we interpreting this standard correctly? It is not talking about proving why we can use the Elimination method to solve for a variable, right? Instead it is talking more about why elementary row operations preserve the solution set? Everyone that I have talked to was interpreting this standard as referring to the Elimination method (including, it seems, the Progressions).

]]> <![CDATA[Reply To: Math Practice Standards]]> Fri, 26 Jul 2013 23:08:17 +0000 Bill McCallum Sorry, nothing springs to mind. Maybe one of my other readers has some suggestions.

]]> <![CDATA[Reply To: A-REI.5 – what does it mean/look like?]]> Fri, 26 Jul 2013 23:06:45 +0000 Bill McCallum It’s important to remember that we are talking about a system of simultaneous equations at each step, not just about one equation. Your goal is to show that at each step the new system of equations has the same solutions as the old system. Specifically, in this standard, you start with two equations $A=B$ and $C=D$. You want to show that this system of equation is equivalent to the system $A+eC = B+eD$ and $C=D$ (this is the sum of the first equation and $e$ times the second equation). Well, if $A = B$ and $C= D$ then $A + eC = B + eD$ and $C=D$, so any solution of the first system is a solution of the second. And if $A+eC = B + eD$ and $C = D$, then $eC = eD$, so subtracting $eC$ from the left of the first equation and $eD$ from the right, we get $A = B$. And we still have $C =D$. So any solution of the second system is a solution of the first. So the two systems have the same solution and are equivalent.

]]> <![CDATA[Reply To: 8.F.1 Functions and the word "Rule"]]> Fri, 26 Jul 2013 22:56:40 +0000 Bill McCallum Your debate has nicely captured the way the meaning of rule evolves over the grades. In Grade 4 it means what your friend says: a prescribed sequence of arithmetic operations. The rule could be either recursive (start with 1 and keep adding 3s) or explicit (think of a number, double it and add 1). This leads to the Grade 8 notion of using a rule to define a function. At some point the concept of a rule becomes merged into the concept of a function. Your example illustrates this.

]]> <![CDATA[Reply To: Math Practice Standards]]> Thu, 25 Jul 2013 22:19:35 +0000 mishaquarles Good Afternoon Dr. McCallum,
Our school district is going to add open response questions to our formative assessments, (given once each quarter). The desired goal is to have students creating written justification for their thinking or work. I have been researching rubrics to use to assess these responses but I have not seen anything out there. Do you or your team have any suggestions?
Thank you.

]]> <![CDATA[Reply To: A-REI.5 – what does it mean/look like?]]> Thu, 25 Jul 2013 18:55:11 +0000 kelicker I am still having the same trouble as nvitale. The proof that has been described by the progressions is of simply adding (or subtracting) the two equations together. This produces one new equation that is true. This standard seems to be talking more about the row operations of matrices where you “replace one equation by the sum of that equation and a multiple of the other.” This results in a system of equations where one equation is the same as one of the original equations and the other equation is a new hybrid of the first two original equations. It is asking why this new system has the same solutions as the original.

I do not think this standard is addressing “The Elimination Method” as we know it where you multiply one or both equations by a constant and add them together. This seems like more of a precursor to performing row operations on a matrix.

I myself do not know how to prove this, and I cannot find any proofs online.

]]> <![CDATA[8.F.1 Functions and the word "Rule"]]> Thu, 25 Jul 2013 18:34:16 +0000 Alexander I’m seeking clarity on the definition of the word rule in the first sentence of standard 8.F.1.

When I first read it, without hesitation, I thought of the definition:
rule – a prescribed guide for conduct or action (merriam-webster)

While arguing with a friend =), he defined rule as in a pattern rule (Grade 4.OA.5). Leading the the following claims:

Ex.1 Suppose a function sends 1 to 3, 2 to 7, 3 to -1.
Me: “The ‘rule’ (the guidelines for conduct, the accepted procedure) is send 1 to 3, 2 to 7, and 3 to -1.”
Friend: “The function has a mapping but no rule because there is no pattern.”

I think it’s an interesting topic for educators that have spent years with function machines and finding patterns to determine the ‘rule.’ Hopefully our debate can provide clarity for others.

]]> <![CDATA[Reply To: Geometry Progressions]]> Thu, 25 Jul 2013 02:53:44 +0000 Bill McCallum The standards themselves do not specify which definition to use, but the progression does, as you point out. It probably makes sense for everybody to agree on this, and the progression is as close to an official interpretation as one can get, I suppose, since the progressions were written by members of the original Work Team.

I hope the 7–12 geometry progression will be out by the end of the summer. (That’s a hope, not a promise.) It will necessarily be shorter than Wu’s document; the progressions are not intended to spell out every point, but to provide some exegesis of the standards.

]]> <![CDATA[Reply To: Confusion about 8th grade Function Standards]]> Thu, 25 Jul 2013 02:36:51 +0000 Bill McCallum I don’t completely understand the question, but I’ll try to say a few things that might help.

First, the phrase “linear relationship” does not occur in the standards, but the phrase “proportional relationship” does. The concept of a proportional relationship is a precursor the concept of a function. One important difference is that when you define a function you designate one of the variables as the input variable and the other as the output variable. Also, as students start to study functions, they start to think of them as objects in their own right. Later in high school they use a letter to stand for a function, and they perform various operations on functions. In Grade 8 the focus is on simply understanding a function as something that takes inputs and yields outputs. Yes, the domain is important, but truth be told the same is true with proportional relationships. So when you say “a table that is a Linear Relationship” you have to be careful. In some cases the variables may only take on whole number of values (e.g. the number of baseball cards), and then it would no more be appropriate in this case to ignore that restriction than it would be in talking about the domain of a function.

Also, it’s important to be clear about the distinction between a function and an equation that defines the function. The equation $y = 2x + 3$ can be viewed as defining a linear function, with $x$ specified as the input variable, $y$ specified as the output variable, and the equation understood as giving the value of $y$ in terms of $x$. But it can also be viewed as an equation in two variables whose solutions form a straight line; an algebraic description of a geometric object. The it is not correct to say that the equation is a function. Rather one should say that the equation defines a function, but also has other uses.

]]> <![CDATA[Reply To: Geometry Progressions]]> Thu, 25 Jul 2013 02:11:38 +0000 Sarah Stevens I have been reading Wu’s article on Geometry. I am unsure how heavily to rely on this work for guidance. For example, the elementary grades use an inclusive definition for trapezoids- according to Wu. I was consulting with a colleague about this shift. She wanted to see what guidance the progressions had on the matter, rather than trusting Wu’s interpretation of the matter. I looked at the K-6 Geometry progression and saw that Wu was in line with standards. I haven’t yet completed the entire article but am anticipating more such “new” ideas and would like a second source to consult. Also, I find the progressions are easier to consume than Wu.

]]> <![CDATA[Reply To: Confusion about 8th grade Function Standards]]> Wed, 24 Jul 2013 16:40:27 +0000 Alexei Kassymov The situation in Baseball Cards could be a good place for students to discuss whether a continuos graph always gives a faithful picture.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Tue, 23 Jul 2013 18:11:34 +0000 lhwalker I am going to chicken out at this point because I am not sure exactly how the CCSS writers want us to word this with the best (consistent) precision, and Dr. McCallum may want to address it here. In high school, we sometimes say, “larger negative” meaning the number is farther from zero on a number line. Vector magnitude and absolute value fall into this discussion as far as what does it mean to be “larger.” I’m thinking Phil Daro’s video about misconceptions might be helpful because he addresses how a concept understood at one level can best be tweaked at the upper level:

]]> <![CDATA[Confusion about 8th grade Function Standards]]> Tue, 23 Jul 2013 17:08:58 +0000 linseykirby After what we feel to be close analysis of the Function Standards at the 8th grade level, we would like some clarification on the difference between these standards and the standards that address linear relationships.

One example we have: Given a table that is a Linear Relationship, I would expect my students to make a graph that is continuous and using an infinite domain. When given a Linear Function, would I expect my students to make a graph that is infinite/continuous, continuous with limits, or discrete with limits?

We have many more questions on the topic about how functions (at the 8th grade level) differ from linear relationships. We would appreciate any and all clarification you can provide on this subject.

]]> <![CDATA[Reply To: Math Practice Standards]]> Tue, 23 Jul 2013 16:47:24 +0000 Bill McCallum I took a quick look at these. Arizona and Kentucky both attach practice standards to individual content standards, whereas Colorado attaches them at a higher level, so it’s hard to compare Colorado with the other two. It’s not surprising to me that people come up with different ways of attaching practice standards to content standards, because the practice standards really live in curricular implementation of the standards. For any given standard, you can imagine it being touched in at numerous different moments in the curriculum, and in different ways, with different styles of teaching. All of these differences would bring different practice standards to mind. I would expect more agreement if people started tagging practice standards to particular moments in a particular curriculum.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Tue, 23 Jul 2013 16:18:51 +0000 Alexei Kassymov I can see how in 5th grade a bit longer explanation will be useful. In Algebra I, with some caution, it can be taken for granted, that they know which number is bigger. In 5th grade it’s something that they still have to learn properly (5.NBT.3b). Stating that, for example, 321 is greater, and 0.0321 is smaller, and then using the fact that dividing by 10 makes a number smaller could combine two things that students should work on.

Speaking of “making bigger”. Algebra I students know negative numbers. Do you think that the statement should be “Multiplying a positive number by 10 makes it bigger” so that misconceptions are not reinforced?

]]> <![CDATA[Power standards]]> Tue, 23 Jul 2013 15:15:27 +0000 Bill McCallum This question was asked over in the Tools section, I’m reposting it here:

Good Morning Everyone!

I am the math and gifted curriculum coordinator at a public PreK-12 school in Hartford, CT. We have been having conversations surrounding “power standards” in order to better plan for our students. I am of the impression that identifying power standards is an obsolete process because they seem to be built into the CCSS via the clusters. Can you shed some light on this idea please! Is there a way to identify the power standards in the CCSS? If so, where should we begin?

I guess finding power standards means finding some sort of prioritization of the standards. I agree that it is more appropriate to do this at the cluster level than at the standard level. Clusters are coherent groupings of standards that go together around the same idea, and breaking out one standard from a cluster is usually going to interfere with that coherence. Among the clusters in a given grade level, one can identify some that are more central to focus of that grade level. Both the PARCC and Smarter Balanced assessment consortia have published frameworks describing the major, additional, and supporting clusters in each grade level. You can find links to them here.

The cluster headings provide an important framing for the standards within them. For example, the cluster heading “Understand place value” in 2.NBT lends meaning to the standard within the cluster 2.NBT.2, “Count within 1000; skip-count by 5s, 10s, and 100s.” It makes it clear that the purpose of the skip counting is to build place value understanding (which explains why skip counting by 2 is not present).

You might also find it useful to read The Structure is the Standards and Jason Zimba’s examples of structures in the standards.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Tue, 23 Jul 2013 03:51:59 +0000 lhwalker I would not use that exact worksheet at 5th grade (right now I use it in Algebra I based on our old curriculum). Here’s a quick, partial modification I envisioned for what you were asking about:
In my opinion, it is unwise to imply decimals move as though they have wheels. I would talk about place value and relative size of the number.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Mon, 22 Jul 2013 18:06:08 +0000 Alexei Kassymov Do you use this exercise in 5th grade? It uses scientific notation, including negative exponents, which currently 8th grade. How do students process negative exponents? Do they memorize a rule like $10^2$ means multiply by 1/100?

]]> <![CDATA[Reply To: Math Practice Standards]]> Mon, 22 Jul 2013 02:28:48 +0000 tlangton Hi Dr. McCallum,

I am researching educator understanding of the CCSSM. Regarding the standards for mathematical practice (SMP), after reviewing the 45 states’ DOE sites I found three states (Arizona, Colorado, and Kentucky) that connected the standards for mathematical content to the SMP at each grade level (see links below). So far, I have not found a content standard at any grade level where all three states agreed on the related SMP’s. My question is:

Should there be agreement?

This website has been like a gold mine for my research. Thank you.

Terry Langton

Arizona –
Colorado –
Kentucky –

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Sun, 21 Jul 2013 03:42:47 +0000 lhwalker I see what you mean. What an interesting thought! It is important how we word things. We do not want them to think dividing 0.05 by 10 will result in fewer digits on the right! I have not observed any students making this wrong conclusion, probably because as we talk about it, we are focusing on relative value. Here’s the quick exercise they complete:

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Sun, 21 Jul 2013 03:35:40 +0000 lhwalker I see what you mean and it does matter how we word things. The exercise I give my students involves focusing on relative value. Certainly it is incorrect to state that there are fewer digits because, for example, there may be more digits on the right side of the decimal when dividing by 10. Here is the exercise and maybe this is why I have not been aware of any of my students equating size with number of digits:

]]> <![CDATA[Reply To: Algebra and Functions Progressions]]> Fri, 19 Jul 2013 21:56:13 +0000 Bill McCallum I get these concerns from users occasionally, but I usually can’t diagnose them because the files work for me. I may have forgotten to print to pdf before I uploaded this one … that usually cleans out a lot of the problems. I’ll try to get to that soon.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Fri, 19 Jul 2013 16:35:38 +0000 Alexei Kassymov Dividing by 10 makes a number one decimal place smaller while multiplying by 10 makes a number one decimal place larger.

Do students ever interpret this as dividing by ten results in one fewer digits after the decimal point (and the other way around)?

Sometimes I think how sad it is that the idea of placing the decimal point under ones lost to the practice we use now. Would make place value names symmetric, emphasize units in a clear way. Oh, well.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Fri, 19 Jul 2013 04:16:07 +0000 lhwalker In my observations, describing decimals “moving left and moving right” easily becomes something students memorize and confuse, especially with scientific notation. When converting from scientific to decimal notation, the decimal moves one way and when converting back it moves the other way. The students get it right 50% of the time. That’s not to say we should never say “the decimal moves to the left,” but I have gotten much better results by having students talk about numbers being “one decimal place larger or one decimal place smaller.” Dividing by 10 makes a number one decimal place smaller while multiplying by 10 makes a number one decimal place larger. This connects well with NF.5.b, “Explaining why multiplying a given number by a fraction greater
than 1 results in a product greater than the given number
…; explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b b.

]]> <![CDATA[Reply To: Algebra and Functions Progressions]]> Thu, 18 Jul 2013 19:53:57 +0000 Cathy Kessel I just downloaded the Algebra Progression. This seems to be fine on my mac, both with Adobe Reader and Preview. I just checked out Functions which also seems fine. I wonder if your OS needs updating.

I tend to wait a long time to update, so am familiar with the problems caused by not updating. With one of my old systems, I couldn’t open some progressions with Adobe but they were fine with Preview. You may want to download another copy and stay with Preview as your progressions reader.

]]> <![CDATA[Reply To: Geometry Progressions]]> Wed, 17 Jul 2013 17:56:14 +0000 Dr. M Wu’s article is superb. It’s been my primary source as I rework my geometry class.

]]> <![CDATA[Reply To: Algebra and Functions Progressions]]> Wed, 17 Jul 2013 17:35:18 +0000 xyzzy The July 2013 update to the Algebra progressions document seems to be a corrupted PDF file. Macintosh users opening this in Acrobat find huge swaths of text and images missing on pages 4 and 10-14. I found it could be viewed using Preview program and then re-saved. Can you get a better version of this document made and then make it available on the Arizona site?


]]> <![CDATA[Reply To: Putting transformational approach into practice – notation]]> Tue, 16 Jul 2013 19:20:11 +0000 Bill McCallum I think the notation should be delayed until the point that the students see the need for it. You can say “the translation that takes A to B” or “the rotation clockwise about O through 30 degrees” for quite a long time. So, for example, I would avoid notation in Grade 8. As for high school, I think it’s up to curriculum writers. I can imagine writing a curriculum which avoids it almost entirely (well, I can imagine trying, anyway). I can also imagine careful deployment of notation being very useful.

If you do introduce notation, I would be in favor of not having it tied to coordinates. For example, for translations, I would write something like $T_{A,B}$ for the translation that takes $A$ to $B$, rather than finding the coordinates of $A$ and $B$ and writing $T(2,5)$. For rotations, I would write something like $R_{\angle AOB}$ for the rotation about $O$ through the angle $\angle AOB$, rather than figuring out the coordinates of $O$ and the angle measure of $\angle AOB$ and writing $R_{(2,1),30^\circ}$. There are a couple of reasons for this. First, coordinates are unnecessary and might not be present. Second, knowing the position of $A$ and $B$ or the measure of $\angle AOB$ is unnecessary, and might be a distraction from the proof. I would want students to get used to the idea that the points, lines, and angles from which they construct the transformations should be points, lines, and angles already existing in the situation, not ones they have to come up with numbers for.

]]> <![CDATA[Reply To: The decimal point and powers of 10. How important is the language?]]> Tue, 16 Jul 2013 16:40:53 +0000 Alexei Kassymov It seems like 5.NBT.2. Having only one instance presented may not be a problem. If they’ve studied this topic, they should be able apply the pattern in one particular instance. After all, using the pattern in calculations of this sort is one reason to study it.

The language of moving the decimal point is in the standard: “explain patterns in the placement of the decimal point”. If the standard is interpreted in the spirit of mathematics that we should consider inverse problems as closely related, then question like “if the decimal point is moved 2 places to the right, what kind of operation could do that?” be appropriate.

]]> <![CDATA[Putting transformational approach into practice – notation]]> Sun, 14 Jul 2013 22:19:07 +0000 JimOlsen Hi Bill (& others),
(I’m doing workshops with teachers this summer on the CCSSM.)
We are digesting the transformation approach to geometry (which, BTW, I like).

I think we (big ‘we’–teachers across the entire country) have a ways to go in terms of putting this into use in the classroom.

In particular, my question is about notation for the transformations.
On the one hand, we want students to have a conceptual understanding of, for example, of translation–and notation “shouldn’t matter.”
However, from a practical handout, in mathematics, ultimately we want to be able to write things down and also notation is important for thinking about, and communicating, mathematics.
In the case of translation, I’ve seen a few different notations. For example, (x+2, y+3) or T(2,3).

Is there a particular notation, for the various transformations, we want to make sure we expose our student to?
I ask this question, because it would be a shame if our students had a very good conceptual understanding of transformations, but on the PARCC (or Smarter Balanced) test a notation unfamiliar to the students is given and they cannot answer the question.

Thank you.

]]> <![CDATA[Reply To: 3.G.2]]> Sun, 14 Jul 2013 16:20:31 +0000 Bill McCallum It’s about both, and serves to connect fractions with area. Fundamentally 3.G.2 is about partitioning, which is necessary both for an understanding of fractions and for an understanding of area. The passage in the progression doesn’t limit the ways in which you can use area to represent fractions to only arrays of unit squares. In fact, that passage should also refer to 3.MD.6, Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

]]> <![CDATA[Reply To: 8.G Glide Reflection]]> Sun, 14 Jul 2013 15:53:54 +0000 Bill McCallum In the Common Core rigid motions are used to define the idea of congruence, and to prove theorems about congruence, such as the criteria for congruence of triangles. This is different from the common use of transformations to study frieze patterns and tilings of the plane. Glide reflections figure prominently in the latter, but not so much in the former. Of course, this doesn’t forbid curriculum writers from naming and using glide reflections. But that is a a side trail from the main goal in the Common Core.

]]> <![CDATA[3.G.2]]> Thu, 11 Jul 2013 19:53:08 +0000 csprader Hello,

I am doing work with 3.G.2 this summer (Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.) and am a little confused as I refer to the progression to better understand what the standard means and what it is building up to. It is is similar to me to 2.G.3 which seems to be setting a foundation for fractions, but then I read the progression for Grade 3:

“Students also develop more competence in the composition and
decomposition of rectangular regions, that is, spatially structuring
rectangular arrays. They learn to partition a rectangle into identical
squares (3.G.2) by anticipating the final structure and thus forming the
array by drawing rows and columns (see the bottom right example
on p. 11; some students may still need work building or drawing
squares inside the rectangle first). They count by the number of
columns or rows, or use multiplication to determine the number of
squares in the array. They also learn to rotate these arrays physically and mentally to view them as composed of smaller arrays,
allowing illustrations of properties of multiplication (e.g., the commutative property and the distributive property).

So to me it sounds like 3.G.2 is really building up for the understanding area more than continuing to build on understanding of fractions? Because as I look at various “unpacked standards” documents they seem to focus on the fraction described by partitioning rectangles into equal shares and do not mention same size squares, rows, or columns.

Thank you for your knowledge and assistance.

]]> <![CDATA[8.G Glide Reflection]]> Thu, 11 Jul 2013 17:51:50 +0000 KarlSchaffer Can someone tell me what the thinking was behind not including glide reflection in the Common Core as one of the basic rigid symmetries of the plane, along with reflections, rotations, and translations? I know some people say glide reflections are more difficult to recognize, or that every glide reflection is a combinations of a translation and a reflection. However, every isometry may also be composed of a combination of reflections. Thanks!

]]> <![CDATA[Reply To: 6.EE.3]]> Wed, 10 Jul 2013 18:15:13 +0000 mrsnolanroom412 Thank you for clarifying.

Would we also review all other properties of operations with this standard or will those be picked up in other standards with Expressions and Equations? We were discussing this at a meeting with some disagreement. Should we include the inverse operation standards here? Zero property? Associative and commutative?

The concern for us is that PARCC may interpret the standard differently and we don’t want our students unprepared.

]]> <![CDATA[4 July 2013 version]]> Wed, 10 Jul 2013 15:52:11 +0000 lhwalker Fraction multiplication and division is tough to visualize and I think the writers did a fantastic job explaining multiple ways to do so. Page 6 seems to make a very subtle y=kx connection. It might be helpful to make the connection stand out more there. On page 7 students see -p as “the opposite of p.” Can’t that idea be used to explain why -5 x -2 = 10? Your explanation on page 10: 5(-2) =….-10 was very clear. It seems to me that “the opposite of 5(-2),” written as -5(-2) would then be a “positive 10.” If that line of thinking is okay, it seems to me that would be easier for the students to wrap their minds around: very quickly recalled and connected. Near the bottom of page 9 there is a repeated word, wrapped to the next line: proved proved. Again, great job and very, very helpful!

]]> <![CDATA[Reply To: 6.EE.3]]> Wed, 10 Jul 2013 07:14:54 +0000 Bill McCallum Formally, this is the existence of a multiplicative identity and the distributive law,
y + y + y = 1\cdot y + 1 \cdot y + 1 \cdot y = (1 + 1 + 1) y = 3y.
But in this example I think it would be fine if students also saw this informally as 3 $y$s. And it’s important to remember the footnote on page 23: students need not remember the formal names for the properties. The main point is that they should use them. Thinking of seeing $y$ as $1 \cdot y$ is useful in many manipulations, for example $xy + y = (x+1)y$.

  • This reply was modified 4 years, 1 month ago by  Bill McCallum.
]]> <![CDATA[Reply To: Fractions Progressions Revised Draft?]]> Wed, 10 Jul 2013 04:37:32 +0000 Bill McCallum We are focused mainly on getting the remaining progressions out in draft form (hopefully by the end of the summer). The editing for final production will take place in the fall. I can’t say exactly when it will be finished.

]]> <![CDATA[Reply To: 4.NF.1]]> Wed, 10 Jul 2013 02:44:32 +0000 greyhound2 Yes, that should have been “multiplying n (4 in this case) by the numerator (2)” since multiplying “4 by 2″ means 2 groups of 4 (2 x 4).

So, if I’m understanding correctly, if we want to talk about 2 rows of 4 in this case, instead of 4 columns of 2, we would talk about “multiplying the same number, n, by the numerator and denominator of a fraction . . .” instead of “multiplying the numerator and denominator of a fraction by the same number, n . . . .”

I see how this goes back to my original question of how to say “4 x 2.” If we say “4 by 2” (in an equal groups situation), it means 2 groups of 4 but, unless we specifically interpret it as 4 by 2, 4 x 2 would generally, at least in the U.S., be interpreted as 4 groups of 2 as explained on page 24 of the OA Progression.

Then there’s the fact pointed out in the OA Progression that, in many other countries, 4 x 2 would mean 2 fours. According to the Grade 3 section of the OA Progression, “it is useful to discuss the different interpretations and allow students to use whichever is used in their home. This is a kind of linguistic commutativity that precedes the reasoning . . . arising from rotating an array.”

I’ll remember that 4 x 2 can mean different things in an equal groups situation depending on whether it is interpreted as: a) 4 by 2, b) four twos as in the U.S., or c) two fours as in other countries.

Thank you.

]]> <![CDATA[6.EE.3]]> Wed, 10 Jul 2013 00:26:29 +0000 mrsnolanroom412 In this standard, apply the properties of operations to generate equivalent expressions, two examples are distributive, the third says apply properties of operations to y + y + y. For the third, I am thinking multiplicative identity and commutative?

Should we expect to review all mathematical properties that could ever be used to generate equivalent expressions, or just focus on these examples at the sixth grade level?

]]> <![CDATA[Reply To: Geometry Progressions]]> Wed, 10 Jul 2013 00:22:24 +0000 Sarah Stevens I am also eagerly awaiting the Geometry Progressions. We begin our work overhauling HS Geometry this year (we are a little behind). I am hoping we will see the progression sometime in first semester. Will we? 🙂

]]> <![CDATA[Fractions Progressions Revised Draft?]]> Mon, 08 Jul 2013 18:59:24 +0000 greyhound2 Just wondering if we might be seeing a new draft of the Fractions Progression anytime soon? (I understand that it must be a lot of work and that it’s only one of many progressions documents.) Just asking. Thank you.

]]> <![CDATA[Reply To: Transformations]]> Sun, 07 Jul 2013 18:27:07 +0000 Bill McCallum It might be worth looking at these materials. (I haven’t had a close look myself.)

]]> <![CDATA[Reply To: Exponent Properties]]> Sun, 07 Jul 2013 18:16:23 +0000 kevinh Thanks, I do agree with you, but I don’t have a choice in giving students lots of practice on this skill. It’s a major aspect of the Virginia Standards of Learning for Algebra 1. But now I know why I can’t find it explicitly in the CCSS.

]]> <![CDATA[Reply To: Transformations]]> Sun, 07 Jul 2013 16:57:11 +0000 Kathleen Thank you for the insight. I have never taken the time to help them grasp that the plane and the grid should be thought of as “independent.” How simple and clean. Most of these teachers have only taught transformations without ever thinking of coordinates on a plane, unless it was specifically on the Cartesian plane and in Algebra, not Geometry. They are having trouble with the big picture. The whole problem arose when we were looking at task 602 on IM, Dilating a Line. So it started with one of them saying, “You can’t dilate a line because it is already infinitely long.” I responded, “What about the distance between specific points on the line?” Then we started the task. In the commentary it states: “The points A’, B’, and C’ appear to be collinear. If we choose more points on line l and dilate those points about point P, we will see that the dilations of those points also appear to lie on the line through A’, B’, and C’. It appears that the dilations of the points on the line l form a new line l’ that is parallel to line l”. That’s when they all agreed, “You can’t dilate a point!” But, in fact, GeoGebra allows you to do just that. It was not a good day for me. I was not ready for the discussion. I agree with you that the expression “dilation of a point” is awkward, but it is out there and I don’t think we can take it back. Older teachers do not want to make dilations “more difficult” than just finding ratios, and young teachers who have just finished college courses still don’t remember doing dilations in high school geometry. I have found that some teachers do not even understand that a dilation needs a center. They think of “similar” and “dilation” as the same thing, and the pictures in the texts do not have “centers” of similarity. We have a lot of work to do on this specific idea. Can you suggest some good resources I might suggest to them?

Thank you again. I feel like a real pain about this, but I do want to help these teachers be less anxious. Kathleen

]]> <![CDATA[Reply To: Exponent Properties]]> Sun, 07 Jul 2013 16:34:22 +0000 Bill McCallum Once students start using letters to stand for numbers in a systematic way, anything they can do with numbers they can also do with letters standing for numbers. The exponent rules are important in all sorts of situations, for example working working with exponential functions (A-SSE.3c) and with polynomials and rational functions (A-APR). The sort of problem you mention here strikes me as more a sort of algebraic calisthenics—not directly required by the standards, but possibly useful in generating fluency with algebraic manipulations. I would use them sparingly, however; it is possible to go overboard with this sort of thing. And it’s not obvious to me that a student who has done plenty of these would be able to notice that, for example $e^{kt} = (e^k)^t$, which strikes me as much more important.

]]> <![CDATA[Reply To: 4.NF.1]]> Sun, 07 Jul 2013 16:14:52 +0000 Bill McCallum I was a bit confused by your last paragraph, because I think of the phrase “multiplying 2 by 4” as meaning $4 \times 2$, not $2 \times 4$. But I take your overall point.

]]> <![CDATA[Reply To: 4.NF.1]]> Sun, 07 Jul 2013 16:14:43 +0000 Bill McCallum I was a bit confused by your last paragraph, because I think of the phrase “multiplying 2 by 4” as meaning $4 \times 2$, not $2 \times 4$. But I take your overall point.

]]> <![CDATA[Reply To: 5.NF.4a]]> Sun, 07 Jul 2013 16:00:18 +0000 Bill McCallum Multiplying fractions by whole numbers occurs in the previous grade, Grade 4. See 4.NF.4. And yes, 4.NF.4c is about story problems, and has an example of one. Your story problem seems to be more about $(a \times q)/b$ than $a \times (q/b)$. That is, we seem to double Ron’s beans before sharing, not after. Note that neither 5.NF.4a nor 4.NF.4c requires students to construct story problems, although that’s a good idea.

]]> <![CDATA[Exponent Properties]]> Sat, 06 Jul 2013 21:46:27 +0000 kevinh It seems to me that exponent properties are taught in 8th grade with numerical bases, and I can’t tell which standard in Algebra 1 is the first to cover exponent properties with variables as the bases, like (2x^3)/(4xy^-2). Can you tell me where that change occurs?

I teach in a non-CCSS state (VA), but I’m trying to correlate my lessons to CCSS over the summer so I can share them on the web.

]]> <![CDATA[Reply To: 4.NF.1]]> Sat, 06 Jul 2013 02:46:22 +0000 greyhound2 Thank you very much for your reply.

I was thinking that it could be advantageous to focus on the rows (2×4 or 2 groups/rows of 4 smaller squares) rather than the columns (4×2 or 4 groups/columns of 2 smaller squares) because the rows correspond to the original thirds before they were partitioned into smaller pieces. I suppose one could still focus on the rows whether one says 2 groups/rows of 4 smaller pieces (2xn) or 4 groups/columns of 2 smaller pieces (4×2), but there seems to be a better connection between saying “2 groups/rows of 4” and seeing the two-thirds as two 1/3 unit fractions.

Focusing on the rows rather than the columns also seems (to me) to align better with this sentence from page 5: “They see that the numerical process of multiplying [each] the numerator and denominator of a fraction by the same number, n, corresponds physically to partitioning each unit fraction piece into n smaller equal pieces.”

I was also thinking that focusing on the rows vs. the columns might make it easier for students to understand why multiplying the numerator, in this case 2, by 4 (2 groups of 4 or 2×4) and the denominator, 3, by 4 (3 groups of 4 or 3×4) results in an equivalent fraction, although I understand that the students, using visual models, are to first develop their own methods/rules/algorithms for generating equivalent fractions, so a student may see 4 groups/columns of 2 (4×2) as readily or even more easily than they see 2 groups/rows of 4. As you said, “Either way is fine, and it’s probably useful to go through both ways.”

Thank you again.

]]> <![CDATA[Reply To: 4.NF.1]]> Fri, 05 Jul 2013 14:15:13 +0000 Bill McCallum This a very interesting question. To summarize, you are saying the to you the diagram says
\frac23 = \frac{2 \times 4}{3 \times 4}
rather than
\frac23 = \frac{4 \times 2}{4 \times 3}.
First, your way of seeing it is fine and ends up with the correct understanding. The difference between your way of seeing the diagram and the way expressed in the progression is the difference between the two ways of viewing a $3 \times 4$ array: as 3 rows of 4, or 4 columns of 3. You are looking at the rows: there are two green rows of 4 squares each in an array consisting of 3 rows of 4 squares each, so the fraction is $(2\times4)/(3\times4)$. The other way to look at this is that there 4 columns of 2 green squares each in an array consisting of 4 columns of 3 squares each, so the fraction is $(4 \times 2)/(4 \times 3)$.

Either way is fine, and it’s probably useful to go through both ways.

  • This reply was modified 4 years, 1 month ago by  Bill McCallum.
  • This reply was modified 4 years, 1 month ago by  Bill McCallum.
  • This reply was modified 4 years, 1 month ago by  Bill McCallum.
]]> <![CDATA[Reply To: Transformations]]> Fri, 05 Jul 2013 13:58:56 +0000 Bill McCallum It might help to make a distinction between the plane and the coordinate grid imposed on it. The plane in which geometric figures live does not come automatically equipped with a coordinate grid; indeed, there was no such thing as a coordinate grid in Euclid’s day. You could think of the plane as a featureless plane in which geometric figures exist, and the coordinate grid as an overlay which can be used to measure positions in the plane. So when I perform a transformation, I move points in the plane to other points, but the grid stays where it is so I can use it to measure the new position of a point. This might be where the in/of confusion comes from: I can think of this as a transformation of the plane, but I can also think of points moving in the coordinate grid.

By the way, you don’t really need the coordinate grid at all to talk about transformations. You can talk about a rotation about a certain point through a certain angle without necessary giving coordinates to the point. It could just be some point in a geometric figure (say, the vertex of a triangle). It seems to me that the coordinates are almost getting in the way of things with your teachers.

As for dilations, I think the phrase “dilation of a point” is awkward. A dilation with center $O$ takes all the points in the plane and moves them along rays from $O$, scaling the distance from $O$ by a certain scale factor. Again, there is no need for coordinates, and no need for vectors.

]]> <![CDATA[Reply To: Integers & the Number Line]]> Fri, 05 Jul 2013 13:31:12 +0000 Bill McCallum There are two big steps in students’ understanding of number in Grade 6. The first is the unification of whole numbers, fractions, decimals, and negative numbers into a single number system as represented by the number line. But the number line doesn’t do everything you need. The other big step is a systematic use of properties of operations to understand how operations can be extended to include negative numbers.

For example, the relation between addition and subtraction helps in understanding subtraction with negative numbers. In earlier grades, students understand $6-4$ as the number you add to $4$ in order to get $6$, that is, the missing addend in the equation $4 + ? = 6$. In Grade 6 they understand $(-6) – (-4)$ as the number you need to add to $-4$ in order to get $-6$, the missing addend in $(-4) + ? = -6$. Since $-6$ is two units to the left of $-4$ on the number line, the missing addend is $-2$. So $(-6) – (-4) = -2$.

By the same token, the relation between multiplication and division helps with division of negative numbers. So $8\div (-4)$ is the missing factor in the equation $? \times (-4) = 8$.

In general in Grade 6 there is a consolidation of operational understanding of rational numbers, and a move away from concrete models, although concrete models like the ones suggested by molleyk are still useful.

]]> <![CDATA[Reply To: 5.NF.4a]]> Fri, 05 Jul 2013 10:50:39 +0000 juniorprom I am just wondering if you are also looking for students to create story problems and draw pictures for the equivalent sequence a x q/b. Such as, Ron had 4 pounds of Every Flavour Beans and Harry had twice as much as Ron. If Harry divided his beans between 3 people, how much would each person get? Thank you.

]]> <![CDATA[Reply To: Integers & the Number Line]]> Thu, 04 Jul 2013 05:04:44 +0000 molleyk Great questions, I wonder if these would help?
For subtraction, I always tell my students it’s a loss of a debt. In terms of the numberline, I ask students to start with the first number, -6 and subtract -4. If they move toward the left I say, that looks like you’re subtracting a positive four. Negative 4 is the opposite of positive 4 so you need to move to the right.

For multiplication, I think in terms of football yards. If a team loses 3 yards 7 times, they’ve lost a total of 21 yards.

Division…I just tell them division doesn’t exist. It’s multiplying by the reciprocal. so 8 / -4 is 8 * -1/4. same rules apply as multiplication…

]]> <![CDATA[4.NF.1]]> Thu, 04 Jul 2013 04:15:32 +0000 greyhound2 On page 5 of the Fractions Progression, the caption for the top picture explains that “the whole on the right is divided into 4 x 3 small rectangles of equal area, and the shaded area comprises 4 x 2 of these, and so it represents 4 x 2/4 x 3.

two thirds shaded eight twelfths shaded

Is there a way of reading “4 x 3” and “4 x 2“ aloud (e.g., “4 groups of 2,” “4 times as many as 2,” etc.) that would be more in keeping with a conceptual understanding? (I understand that students are not the audience for the caption, but I’m trying to imagine how students might verbalize their thought processes and actions and asking myself if there are particular phrases or word orders that would be more consistent than others with a student’s emerging conceptual understanding.)

You previously wrote (here) that “the concept of fraction equivalence . . . is developed more fully in Grade 4, where students reason directly with visual fraction models to see that taking, say, 3 times as many copies of a unit fraction one-third the size gives you the same number.” In the case of the example on page 5, it would be “4 times as many copies of a unit fraction one-fourth the size gives you the same number.” I understand that, but I’m having a hard time matching that phrase up conceptually with 4 x 2/4 x 3. I see the picture on the top right as 2 groups of 4/3 groups of 4 (2 x 4/3 x 4).

As students are deepening their understanding of equivalent fractions, is there a difference between seeing the two thirds as partitioned into 4 groups of 2 vs. partitioned into 2 groups of 4?

Thank you, Mr. McCallum, for all the work you’re doing and for reading my questions.

]]> <![CDATA[Reply To: Transformations]]> Wed, 03 Jul 2013 17:24:09 +0000 Kathleen Thank you for this reply. Your description of moving in the plane, or moving the entire plane is exactly the problem the teachers with whom I am working find troubling. Moving the points in the plane (not the entire plane) is much easier for them to grasp and does not seem to concern them at all. Picking up the origin and moving it seems to cause them great distress. I think this goes back to how they view “families of functions” where say the vertex of the parabola is “translated” rather than the origin of the plane.

So, if I understand you, I can just tell them to think “in the plane” when they see something written as “of the plane” and no harm will be done? This appears in some of the tasks on IM.

Also, is there an easy way I can discuss “dilation of a point” with them? They are focused on dilation as “change in size” and contend a point has not “size”. If we discuss “point has location” and if we change it’s location from the origin, then we change it’s “distance”, they say we are talking about vectors which are not “points” and hold that vectors are not addressed in the CCSM before we begin dilations. I am having a very hard time with this. I need guidance. Thank you very much for answering my questions.

]]> <![CDATA[Integers & the Number Line]]> Wed, 03 Jul 2013 04:48:28 +0000 jcanzone We are trying to make sense of integer subtraction on the number line. In the case of –6 – -4 is –2 the only explanations that work start by stating that the direction between the second number (-4) to the first number (-6) is in a negative direction resulting in a negative answer whereas, 7-5, distance from second (5) to first (7) is in a positive direction, resulting in a positive answer. This seems to be challenging in the sense that students are not used to starting with the second number when subtracting. Is this the only way to explain these cases using a number line? Also, what about using the number line for multiplication when a negative factor comes first. How do you show -3(7) or negative 3 groups of positive 7 without switching the problem to be 7(-3) or 7 groups of negative 3. Lastly, negative integers on the number line with division: how do we show 8 divided by negative four? “How many -4’s are in 8?” doesn’t make sense on the number line either. Do we have to skip these cases when teaching these topics? Thank you!

]]> <![CDATA[Reply To: Progressions Draft]]> Wed, 03 Jul 2013 03:33:10 +0000 Bill McCallum Thanks Lane for these careful edits!

]]> <![CDATA[Progressions Draft]]> Tue, 02 Jul 2013 23:18:40 +0000 lhwalker In my opinion, this progression explains very clearly why we need to move to an integrated math sequence at the high school level. It is very well done. I would suggest a couple of edits.

The bottom of page 3 begins a very long sentence that might best be broken up for easier reading. Maybe, “The following advice is attributed to Einstein ….simpler.’ We can aptly argue the choice of the model: C(t) = p + at as being ‘as simple as possible.’ Trying to make the model ‘simpler’ by dropping the purchase price p or the term at would delete…cost differences”

Page 4: Things that affect the model but whose behavior not characteristics the model is not designed to study–inputs or independent variables.

Page 5: Especially in…derivation of a formula. I do not understand what this means.

Page 6: Graphing utilities and dynamic geometry software produce revealing models using technology. I’m wondering what graphing utilities and software do not use technology 😉

Page 15: Later, as students… I had to really think about what this is saying. Maybe something like:
“Later, as students are challenged to develop more complex models. Two or more simple equations may be combined in to one formula using substitution. For example, volume = … surface area = … cost = where 10 < V < 20, could be combined into one conjunctive inequality.

Pate 18
R, i. e. (spacing)

  • This topic was modified 4 years, 1 month ago by  lhwalker.
  • This topic was modified 4 years, 1 month ago by  lhwalker.
]]> <![CDATA[Reply To: 7.SP.8 regarding P(A and B) = P(A) + P(B)]]> Mon, 01 Jul 2013 17:24:56 +0000 zabeljonathon Thank you for the perspective! I like your suggestion of pointing it out in the context of an example problem while also addressing that this method doesn’t always apply. Teaching the rule “formally” at the middle school level could definitely lead to misconceptions if not tempered with appropriate explanations.

]]> <![CDATA[Reply To: S-IC-5]]> Sat, 29 Jun 2013 15:32:50 +0000 Bill McCallum No, I would say this does not included formalized significance tests, but is more a matter of understanding how sampling works through simulations of sampling distributions, for example. There are some examples in the high school statistics progression, for example the relation between caffeine and finger-tapping (an experiment I find particularly relevant as I type this).

]]> <![CDATA[S-IC-5]]> Sat, 29 Jun 2013 11:31:37 +0000 danlemaypi My burning question at this moment:

S-IC-5: Use data from a randomized experiment to compare two treatments

Are we talking about what could be a formal significance test? Would a task that asked students to conduct a significance test assess this standard? Should it be more open ended, just getting them to think about how sampling distributions behave?

]]> <![CDATA[Reply To: 7.SP.8 regarding P(A and B) = P(A) + P(B)]]> Fri, 28 Jun 2013 14:27:36 +0000 Bill McCallum The formal probability rules are in the high school standards, and in that case it is the general rule that does not assume the events are independent: P(A and B) = P(A)*P(B|A) = P(B)*P(A|B). The Grade 8 standards are really focused on understanding the underlying concepts of sample space, event, and probability model in concrete terms, without getting into the formal calculus of probability. So I would say that it goes beyond the standards to introduce the multiplication rule at this stage. Of course, if it arises naturally in an example there is no harm in pointing out that the probability of the compound event is the product of the individual probabilities, but that is not quite the same as introducing a formal rule. And it would be important to point out that this simplified multiplication rule does not always apply.

]]> <![CDATA[Reply To: Calculators]]> Thu, 27 Jun 2013 23:24:06 +0000 Bill McCallum Nothing new that I know of, but I think you can sign up for news updates from both PARCC and SBAC.

]]> <![CDATA[7.SP.8 regarding P(A and B) = P(A) + P(B)]]> Thu, 27 Jun 2013 23:08:16 +0000 zabeljonathon Hello! I am working with colleagues this summer in trying to design some performance tasks related to the standards. We started by examining the sample performance tasks that came with our textbook series, but are also trying to branch out to make sure what we put together matches CCSS as closely as possible.

I was hoping to get some guidance regarding 7.SP.8 (shown below):

7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events.

The standard seems to stop short of introducing a formalized algorithm for finding the probability of compound events via multiplication… here is my example: You roll a six sided die and flip a coin, what is the probability that you will role a 4 and land on tails? The standard seems to suggest that we would solve this problem by creating a sample space and deriving our probability from that. The textbook materials we use have the students create a sample space, but also take students to the next level of having them calculate P(A and B) = P(A) * P(B) in a later section…

I also took a look through the progressions document, and while it does reference teaching the counting principle, it doesn’t reference this probability rule at seventh grade.

My question is this – is teaching the use of multiplication to find the probability on compound events in this case part of the standard? If so… can I get a little help in dissecting the standard so I can understand the phrasing? If not… can I ask at what level it would be appropriate to teach this “rule” to the students? My colleagues and I were baffled by this one!

I apologize for my imprecise vocabulary! I have a spent the past two years as a technology curriculum integration specialist, and am looking forward to getting back into the classroom next year as a seventh grade math teacher.

]]> <![CDATA[Reply To: 6.NS.5 and the Number Line]]> Thu, 27 Jun 2013 22:46:49 +0000 Bill McCallum One point I would make is that the order of the standards does not necessarily correspond to the order of topics in the curriculum (see p. 5 of the standards). I could imagine teaching about rational numbers on the number line first, and then giving applications of negative numbers as in 6.NS.5. Or you could do it the other way around. In that case the temperature model lends itself particularly well to introducing the negative extension of the number line, since a thermometer is really just a vertical number line.

]]> <![CDATA[Reply To: A Complete Common-Core-Aligned Course in Geometry]]> Thu, 27 Jun 2013 22:42:28 +0000 Bill McCallum Thanks for sharing this!

]]> <![CDATA[Reply To: 4NBT2 – Trillions]]> Thu, 27 Jun 2013 22:37:32 +0000 Bill McCallum These are both curricular decisions, not dictated by the standards. There are limits on the number of digits in some of the Grade 4 operations standards, but not on the size of numbers that students see without operations. the term “multi-digit” does not come with an inherent limit, so it is up to the curriculum developer to decide the limit there. As for base-thousand units, I think it was an ad hoc term to refer to thousands, millions, etc., and not intended as a new piece of terminology to be taught to children.

]]> <![CDATA[Reply To: Transformations]]> Thu, 27 Jun 2013 22:27:27 +0000 Bill McCallum I’m not sure there is a mathematical difference here, but rather a difference in point of view. A transformation takes points in the plane as inputs, and outputs other points in the plane. We conceptualize this as motion: the transformation moves the input point to the output point. This conceptualization is useful but not strictly part of the mathematical definition. Whether you think of this as moving points in the plane (a transformation in the plane) or whether you think of it as moving the entire plane all together (a transformation of the plane) strikes me as a matter of point of view, or taste as Dr. M. says.

]]> <![CDATA[Reply To: Rectangular Arrays/Area Models with 5NBT 6 and Properties Question]]> Thu, 27 Jun 2013 22:19:08 +0000 Bill McCallum Lane and Cathy have pretty well covered things, but just a couple of extra points. First, notice the footnote on page 15 that says students need not use the formal terms for the properties of operations. They should understand that you can add numbers in any order and use this fact, but they don’t have to know the name for it. Second, when you read a standard like 5.NBT.6, it is important not to interpret it is requiring every method listed for every division problem (that’s why the “and/or” is there).

]]> <![CDATA[Reply To: 2.NBT.2 – Skip Counting]]> Thu, 27 Jun 2013 22:11:52 +0000 Bill McCallum Again, please tell me which unpacked document you are talking about. I looked around on the web a bit and didn’t find anything where “skip counting” meant starting from a number other than a multiple of 5, so I would say that starting from any number is certainly not required by this standard. That doesn’t mean it is forbidden, of course. And it does seem reasonable to start from any multiple of 5.

]]> <![CDATA[Reply To: 2.NBT.8]]> Wed, 26 Jun 2013 21:50:46 +0000 Bill McCallum I’m not sure which unpacked document you are talking about, do you mean the NBT Progressions document? Anyway, to answer your question, I think it is limited to what it says, just adding 10 or 100. Notice that it is about mental computation only. In 2.NBT.7 students are expected to add and subtract within 1000, so that certainly includes 243 + 30 and 472 + 400. But this standard is about mental work only. The point is not so much computation as encouraging a clear understanding of the base ten system. If a student can add 10 mentally to 243, then adding 30 is more cognitively demanding but doesn’t show a better grasp of the base ten system.

]]> <![CDATA[Reply To: Calculators]]> Wed, 19 Jun 2013 18:24:35 +0000 MLorimer Okay…so I’m wondering if there’s any further “news” on using calculators. I’m helping some teachers plan for next year, and they’re pretty fixated on this issue. My thought is that since CCSSM is aimed at college/career/workforce readiness, we should be asking ourselves if a calculator would be a likely tool if the problem being solved were a real world situation. For example, I do my financial figuring with a calculator because it makes a lot more sense than scribbling on scrap paper.

Thanks, y’all.

]]> <![CDATA[Reply To: Rectangular Arrays/Area Models with 5NBT 6 and Properties Question]]> Tue, 18 Jun 2013 18:50:45 +0000 Cathy Kessel Not surprisingly, some of the examples that lhwalker mentions bear a strong resemblance to examples in in the NBT Progression.

kimbergunn, it sounds as you might be thinking that an area model must be carved up into units. When students begin using area models, it seems that initially they should maintain the connection between understanding the connection between units of area and units of numbers, but that certainly becomes unwieldy to show explicitly by drawing unit squares when numbers get large (and we hope the connection has been built in the context of smaller numbers).

The area model on p. 15 of the NBT Progression does not show individual units of area. (It shows a 3-digit dividend and 1-digit divisor, I hope it’s obvious how an area model might be drawn for a 4-digit dividend and 2-digit divisor.) Also, the standard allows an equation as an illustration.

The discussion of introducing the commutative property for addition here ( might be helpful in thinking about how to introduce it for multiplication.

]]> <![CDATA[Reply To: Rectangular Arrays/Area Models with 5NBT 6 and Properties Question]]> Tue, 18 Jun 2013 04:04:08 +0000 lhwalker 1) There are some great examples of multiplying with larger numbers beginning on page 23 of this document:
In my opinion, we need to continue to give examples of area models for smaller numbers with each block shown to maintain that connection.

]]> <![CDATA[Reply To: Plotting Ratios in the Coordinate Plane]]> Mon, 17 Jun 2013 23:12:31 +0000 Cathy Kessel Note that there are various different definitions and notations for ratios and fractions. This got discussed a while back (November 2011) on the blog here:

CCSS treats a ratio of two numbers as a pair of numbers rather than a fraction. So, one answer is that a/b is one number (assuming that b isn’t zero) and doesn’t determine coordinates of a point in the plane. (I’m assuming that we’re not dealing with complex numbers.)

Maybe this helps to make an answer more obvious because the only choice is how to plot the pair of numbers a and b. That depends on what the coordinate axes are supposed to be representing. If you’ve got a ratio of 5 cups of grape juice to 2 cups of peach juice, and cups of grape juice corresponds to the horizontal axis and cups of peach by the vertical axis (as in RP Progression, p. 4), then the ratio corresponds to the point (5, 2). If cups of peach were represented by the horizontal axis, then the corresponding point would be (2, 5).

]]> <![CDATA[6.NS.5 and the Number Line]]> Mon, 17 Jun 2013 22:55:52 +0000 Silas Kulkarni Some other math coaches and I were having a recent discussion about whether it makes sense to include the number line in lessons introducing 6.NS.5. Here is the quote:
“We are wondering how much (if at all) to bring the number line into this lesson set [on 6.NS.5]. The standard feels focused on real world and less about the number line as a model. However, there is the reasoning that prior knowledge of a number line will help a student get a deeper understanding of 6.NS.5. But then 6.NS.6 explicitly mentions the number line.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.”

What do you guys think? Do you think the first quadrant all positive vertical and horizontal number line from 5.G.2 should be mentioned as prior knowledge in [a review] section? Do you think they should be presented with a numberline of positive and negative numbers in 6.NS.5 at all or should it all wait until 6.NS.6?”

Since several of us were wondering about this, I thought it would be useful to bring it to the forum.

Thanks for your help.

]]> <![CDATA[Reply To: Transformations]]> Sun, 16 Jun 2013 18:57:50 +0000 Dr. M I take it that the core definitions – of ‘rigid motion’, say – can be given either in terms of transformation of the plane or transformation in the plane. In the first, we map the whole of the plane to itself. In the second, we map one subset of the plane to another. I’m tempted to say that it’s just a matter of taste.

]]> <![CDATA[A Complete Common-Core-Aligned Course in Geometry]]> Sun, 16 Jun 2013 18:52:53 +0000 Dr. M I’ve been hard at work since summer began on a complete overhaul of my geometry class to bring it in line with the Common Core Standards. I’m now just about done. Below is a link to my class web site. The chapters links contain note sets and worksheets. The material you’ll find there is the culmination of six years of work.

My goal all along is to present interesting and challenging material in a mathematically authentic way. This means that results are never just stated. They are proved. My geometry class is a proof class.

I like the new standards quite a bit. They mesh well with how I’ve taught for years.

My site.

]]> <![CDATA[Plotting Ratios in the Coordinate Plane]]> Fri, 14 Jun 2013 16:23:58 +0000 jacobsj I have a question about plotting ratios in the coordinate plane. I have been looking into this for a while and cannot find an answer. If I am plotting the ratio of a/b in the coordinate plane, do I plot it as the point (a,b) or (b,a)? I have gone through the progressions document and the NCTM book Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning and I cannot find where it specifically says one way or the other. Any help you can give me regarding this question would be greatly appreciated.

]]> <![CDATA[4NBT2 – Trillions]]> Thu, 13 Jun 2013 20:13:31 +0000 kimbergunn Good afternoon,

In 4NBT2, the standard states:
“Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using > < and = symbols to record the results of comparisons.”

Question: Within the Progression document (p.12), it states “To read numerals between 1,000 and 1,000,000, students need to understand the role of commas. Each sequence of three digits made by commas is reads as hundreds, tens, and ones, followed by the name of the appropriate base-thousand unit (thousand, million, billion, trillion, etc.)

In instruction, would there need to be an emphasis about billions and trillions? (I understand you are showing them a pattern here, but the top end of 4th grade appears to be millions).

Also, the term “Base-Thousand” is new to students. Would this need to be directly taught as well?

Thank you much!



]]> <![CDATA[Rectangular Arrays/Area Models with 5NBT 6 and Properties Question]]> Thu, 13 Jun 2013 19:56:36 +0000 kimbergunn Good morning,

5NBT6 states, “Find whole number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division; illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.”

Two questions:

1) To have students illustrate and explain the calculation using area models/rectangular arrays for 4 digit dividends would be very tricky because they would have to divide up the spaces into very small units. Can you please provide an example how this would be shown using an area model/rectangular array?

2) You indicate in the standards using “properties of operations” in several places, yet the only two properties referenced in the Progressions documents are “Distributive,” “Commutative,” and “Associative.” Should these be taught in isolation prior to asking them to use them? I know that they are introduced in 3rd and 4th grade, meaning the students should have an understanding of them, but should a whole lesson be taught as a reminder, or should we just reference during a problem, “I used the _____ Property to complete this problem.”

]]> <![CDATA[Transformations]]> Thu, 13 Jun 2013 02:13:16 +0000 Kathleen I have seen both “transformations in the plane” and “transformations of the plane” used when talking about the standards. To me they are not interchangeable. Please clarify for me whether the standards address only transformations in the plane, both in and of the plane, or if indeed the two are interchangeable. Thank you.

]]> <![CDATA[2.NBT.2 – Skip Counting]]> Thu, 13 Jun 2013 00:26:15 +0000 ginnybaldwin When creating Learn Zillion lessons, we want to make sure that we are addressing skip counting by 5’s correctly. 2.NBT.2 states, “Count within 1000; skip-count by 5s, 10s, and 100s.” The unpacked document emphasizes skip counting from ANY number.

Does this mean that we are skip counting from ANY multiple of 5 such as 245, 250, … or does it mean that students skip count by ANY number such as 43, 48, 53, 58 to build fluency?

]]> <![CDATA[2.NBT.8]]> Thu, 13 Jun 2013 00:21:00 +0000 ginnybaldwin I am working on lesson creation for Learn Zillion. When looking at 2.NBT.8, it states, “CCGPS.2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.”

One of the unpacked document mentions that this also expands to multiples of 10 and crossing centuries. When creating lessons, should we only focus on adding and subtraction 10 and 100, or does it imply and expand into teaching adding and subtracting multiples of 10 such as 243 + 30 and 473 + 400?

Also, does this standard address crossing centuries in problems such as 274 + 40?

]]> <![CDATA[Reply To: Multiple Models]]> Tue, 11 Jun 2013 17:57:58 +0000 Bill McCallum No, I don’t think this is quite accurate. The full sentence is “Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.” The “can” indicates that arguments using concrete referents are acceptable, not that they are required. And the “such as” indicates that objects, drawings, diagrams and actions are examples of possible concrete referents; not a list of requirements.

By the way, over the last few years there have been articles discrediting the whole theory of learning styles, e.g., this 2010 one in the New York Times.

]]> <![CDATA[Multiple Models]]> Tue, 11 Jun 2013 09:16:09 +0000 paulm The use of multiple models always seemed to be a recognition of the fact that people learn differently – some visually, some verbally, some kinesthetically, etc., etc. The different ways in which whole number place value can be taught is a good example, with Base 10 Blocks, bundles of straws, drawings representing tens and hundreds, verbal naming (e.g., 4 Tens + 6 Ones), expanded notation, all used to help students understand place value. These different methods were employed primarily so we could be sure we gave all our students their best chance at grasping concepts.

It now seems that representing mathematical concepts in multiple ways is considered an end in itself, and that students are expected to have (and will be assessed on) the ability to create those multiple representations for themselves. I’m thinking in particular about Practice 3: “…construct arguments using concrete referents such as objects, drawings, diagrams, and actions…, ” though there are other examples.

Is this perception accurate? Has anyone else wondered about this?



]]> <![CDATA[Reply To: Smart Quotes in Geometry Overview]]> Tue, 11 Jun 2013 01:15:42 +0000 Bill McCallum Now I am really impressed by your proof-reading skills!

]]> <![CDATA[Reply To: Math Practice Standards]]> Tue, 11 Jun 2013 01:14:58 +0000 Bill McCallum Send me an email privately (you can find my email address by googling my name) and I will put you in touch with someone who probably knows.

]]> <![CDATA[Reply To: 7.G.2]]> Tue, 11 Jun 2013 01:12:26 +0000 Bill McCallum I don’t think “three measures of angles or sides” means the same thing as “three measures of angles or three measures of sides.” That is, I interpret the standard as referring to six possible measures (angles or sides), any three of which may be chosen. Of course, in some cases this leads to ambiguity, such as SSA, but that is covered by the later “more than one triangle.” So, basically, I don’t think this standard is ambiguous, and I think that it allows examples such as the Arizona one. I find the publishers’ interpretations you refer to quite puzzling.

]]> <![CDATA[5.OA.2]]> Mon, 10 Jun 2013 19:41:12 +0000 bbaggett This standard reads: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Everything I have read in trying to interpret this standard indicates that, mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there cannot be braces in a problem that does not have both parentheses and brackets. Using this information, I would believe that 5th graders must interpret nested expressions involving parentheses, brackets, and/or braces. As a matter of fact the PARCC assessment blueprints and specifications indicate that there will be test items involving nested items using parentheses and brackets. However, in the OA progresions (on page 32) it is noted that “Grade 5 work should be exploratory and expressions should not contain nested grouping symbols and should be no more complex than (8 + 27) + 2.” So, I am confused. So, should 5th graders experience evaluating expressions involving nested grouping symbols? If so, to what degree? And what about the mathematical correctness of using braces without brackets and parentheses, and brackets without parentheses?

]]> <![CDATA[Reply To: Smart Quotes in Geometry Overview]]> Mon, 10 Jun 2013 16:30:30 +0000 Jim Similarly, sometimes a degree symbol is used, and sometimes a superscript ‘o’ is used.

]]> <![CDATA[Reply To: Math Practice Standards]]> Fri, 07 Jun 2013 15:38:54 +0000 bumblebee Do you have the name of someone in Arizona I could contact to find out?

]]> <![CDATA[Reply To: Rectangle nonexample]]> Tue, 04 Jun 2013 18:20:27 +0000 Cathy Kessel Thanks for mentioning this. It was also mentioned in February (I think somewhere on the forum). Anyway, the revised figure is now up on my blog:

]]> <![CDATA[Reply To: PARCC and SBAC high school content frameworks]]> Sun, 02 Jun 2013 01:03:35 +0000 emilya1 PARCC is writing end of course tests for the 9th, 10th, and 11th grade courses (not sure if/how they are distinguishing between traditional and integrated). SBAC is writing one assessment for high school to be taken in 11th grade.

]]> <![CDATA[7.G.2]]> Thu, 30 May 2013 22:02:55 +0000 beckywong The standard reads, “Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.” One of the examples given on Arizona’s standards document is, “Is it possible to draw a triangle with a 90˚ angle and one leg that is 4 inches long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle?” Multiple textbooks supposedly aligned to Common Core only give examples where three angle measures or three side lengths are given. Due to the fact that there are inconsistencies, we want to verify whether the state’s example aligns to Common Core’s intent.

We also had a question about the use of the word “constructing” in this standard. Are students required to construct triangles using a compass?

]]> <![CDATA[Rectangle nonexample]]> Wed, 29 May 2013 20:40:32 +0000 Alexei Kassymov In the Geometry Progression document, on p. 6 the illustration for K.G.4 has as the top difficult distractor for rectangles a shape that looks like a rectangle to me (and other people). Was it supposed to be not closed or have some other deformity or am I not seeing something obvious?

]]> <![CDATA[Reply To: measurement conversion in 5th grade]]> Mon, 27 May 2013 21:09:07 +0000 team metric partner The PARCC prototype item mentioned by TEAM METRIC actually dealt with grade 6 ratios and proportional reasoning. I was mistaken to think that it was evidence of requiring conversions between customary and metric units. However, nearly all the prototype items containing measurement units included customary units rather than metric. I wondered if this was an indication of the focus being placed on the customary system.

]]> <![CDATA[Reply To: Progression of Monomials/Polynomials]]> Sun, 26 May 2013 14:38:33 +0000 Bill McCallum Some people like to run marathons, I guess, and some students might enjoy showing off their algebraic manipulation skills. Such problems could serve a purpose in some sort of math competition. But no, I can’t see a need for them at all in the regular curriculum.

]]> <![CDATA[Reply To: Progression of Monomials/Polynomials]]> Sun, 26 May 2013 02:41:18 +0000 lhwalker To clarify, I’m wondering if, even at the high school level, we need to be reducing expressions like that.

]]> <![CDATA[Reply To: 5.MD.A.1]]> Sun, 26 May 2013 00:13:01 +0000 Bill McCallum Children should grow up knowing that there are 12 months in a year and 10 decades in a century, but it is not obvious to me that it is the math teacher’s responsibility to teach them. Some (most?) kids will learn this at home, some in reading, writing, social studies, or science (it sort of comes up everywhere, doesn’t it?). So, there’s no harm in having a math problem that puts this knowledge to use. “How many months in 5 years?” becomes a question about understanding multiplication once you know that there are 12 months in one year.

But it seems strange to me to treat this as some sort of conversion rule to be memorized, and I wouldn’t be in favor of having a special unit about it in the curriculum. That way lies curricular bloat.

]]> <![CDATA[Reply To: Measurement units and “best bets” in education?]]> Sun, 26 May 2013 00:01:44 +0000 Bill McCallum My personal opinion is that a primary focus on metric units is a good bet, although obviously they still have to learn about customary units. Better still, students should leave school carrying some approximate conversion factors in their heads, and have the computational fluency to be able to use them mentally.

]]> <![CDATA[Reply To: measurement conversion in 5th grade]]> Sat, 25 May 2013 23:58:22 +0000 Bill McCallum I think the standard is pretty clear in limiting conversion to within systems, and I think your understanding of that is correct. Can you give an example from one of the sample PARCC test questions? If they are requiring conversion between different systems in Grade 5, that goes beyond the standards.

]]> <![CDATA[Reply To: Bar Graphs/Line Plots]]> Sat, 25 May 2013 23:53:42 +0000 Bill McCallum Yes, I think it is a reasonable interpretation to limit assessment focused on this standard to those fractions. And decimals forms are fine as well; the point of view taken in the Common Core is that decimals are fractions (with denominator 10, 100, etc.), and that 0.25 and 1/4 are just two ways of naming the same fraction. In Grade 5 students are working with decimals to the thousandths (5.NBT.3).

]]> <![CDATA[Reply To: A.APR.4]]> Sat, 25 May 2013 23:46:54 +0000 Bill McCallum Thanks abieniek, that is exactly right, and thanks Tracy for asking the question.

]]> <![CDATA[Reply To: Progression of Monomials/Polynomials]]> Sat, 25 May 2013 23:45:38 +0000 Bill McCallum Polynomials as a topic in their own right are not introduced until high school, and there the emphasis is on seeing them as a system (like the integers) of “numbers” that can be added, subtracted, and multiplied. Introducing monomials earlier and separately doesn’t fit well with this approach. In fact, I don’t see a good reason for introducing them at all, except possibly as a piece of terminology. And even there I’m not sure; you can talk about the monomials in a polynomial as terms in a sum. Certainly problems like the one Lane suggested are beyond the scope of Grade 8, and I would say that even the simpler multiplication of degree two monomials suggested by sbrockley is straying off track.

The focus of algebra in Grades 6–8 is linear expressions, equations and functions. The laws of exponents are limited to numerical expressions (8.EE.1).

]]> <![CDATA[Reply To: Use of technology: S.ID.6]]> Sat, 25 May 2013 23:34:44 +0000 Bill McCallum Yes, this sounds about right to me. MP5 (Use appropriate tools strategically) puts the ball in the court of curriculum developers to make these decisions, but I think Statistics and Probability is an obvious area where technology will play a role, simply because the subject doesn’t really come to life until you start looking a data sets of a decent size.

]]> <![CDATA[Reply To: Standards taught in order?]]> Sat, 25 May 2013 23:32:18 +0000 Bill McCallum Stay tuned … Illustrative Mathematics is working this, with some interesting partners.

]]> <![CDATA[Reply To: Progression of Monomials/Polynomials]]> Sat, 25 May 2013 02:36:17 +0000 lhwalker I’ve wondered how far we will go with this as well. It takes a lot of practice to be able to reduce fractions in the form of (((3xy^-4z^5)^-3(4y^-2z^-3))/((7x^-2)^3)(y^6)^-7) and it is a challenge to answer the question, “When will we ever…”

  • This reply was modified 4 years, 2 months ago by  lhwalker.
  • This reply was modified 4 years, 2 months ago by  lhwalker.
]]> <![CDATA[Reply To: 5.MD.A.1]]> Fri, 24 May 2013 19:34:02 +0000 Tracy The question we were really trying to find is where “time” fits in. For example, we have people who want to convert months to years, decades to centuries, etc.

]]> <![CDATA[Reply To: 5.MD.A.1]]> Fri, 24 May 2013 19:15:33 +0000 Jason Zimba @teammetric:

I could not find any text in the standards suggesting the need for students to convert … from customary … units to SI units

5.MD.1 is explicitly restricted to conversions within a given system. But 6.RP.3d is not restricted in this way:

“6.RP.3d. Use ratio reasoning to convert measurement units; ….”

I would interpret this language to include converting cm to in, or vice-versa.

(By the way, I don’t think the larger question in boldface belongs on a discussion thread about 5.MD.1. You might consider finding another thread in which to post that question. I will say that both assessment consortia have websites with a great deal of documentation about how they are interpreting specific standards and the Standards as a whole.)

]]> <![CDATA[Reply To: 5.MD.A.1]]> Fri, 24 May 2013 18:44:31 +0000 TeamMetric Perhaps, I do not understand the original question. Is Tracy asking if students need to convert from one measurement system (inch-pound) to the other (SI)? My understanding is that the new standards do not suggest the need to convert between only within the same system. Is this correct?

I briefly looked again, I could not find any text in the standards suggesting the need for students to convert problems from customary (inch-pound) units to SI units. This contradicts some of the sample PARCC prototypes and the approved text book we reviewed in Louisiana- both require a student to convert between systems not just within one.

This is what I found:
4.MD.1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

5.MD.1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Who is responsible for making sure that the standardized tests align with the common core standards?

]]> <![CDATA[Measurement units and “best bets” in education?]]> Fri, 24 May 2013 14:11:45 +0000 TeamMetric I am interested in the topic of measurement units and the traditional practice of treating customary units (inch-pound) units as the primary “system” of measurement and SI (metric units) as secondary. Traditionally, metric units are not how our kids intuitively think and metric units are those other units which we teach them to leap to from mathematical conversions.

With all the changes in science standards (which have changed/ is rapidly changing to a completely metric unit instruction model) and all the occupational pathways which now in the U.S include metric-mostly and in the case of healthcare (20% of our workforce) metric-only professions in addition to all the known STEM occupations, precision manufacturing jobs (all transport/ additive manufacturing) and the military opportunities for our kids which work in predominately in metric units, how does thinking intuitively in inch-pound units still benefit them? Are customary units really their best bet?

]]> <![CDATA[Reply To: measurement conversion in 5th grade]]> Fri, 24 May 2013 13:56:40 +0000 TeamMetric (Note: I understand the difference between the metric system and SI.)

I am so glad to have this topic discussed. My reading of the standards taken below is that students should convert within the same system not between customary (inch-pound) units and metric units. Is this not correct?

Convert like measurement units within a given measurement system.
CCSS.Math.Content.5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

In reviewing the sample PARC test questions and the approved textbooks/ workbooks, conversions between the two different “systems” are still required and I assume prevalent based on the fact that I am only looking at sample questions?

]]> <![CDATA[Reply To: Bar Graphs/Line Plots]]> Fri, 24 May 2013 05:06:50 +0000 Duane In 5.MD.2 the first sentence lists “fractions of a unit (1/2, 1/4, 1/8).”
Are these the only fractions that should be assessed, as there’s no qualifier as is used elsewhere (such as “e.g.” or “including”)?
Also, is the decimal form of those fractions included, or is there something particular about the common fraction format on line plots that we need to get across to students?

]]> <![CDATA[Reply To: A.APR.4]]> Thu, 23 May 2013 20:50:37 +0000 Tracy Thank you! That is what we were thinking, but we wanted to make sure.

]]> <![CDATA[Reply To: A.APR.4]]> Thu, 23 May 2013 20:41:26 +0000 Aaron Bieniek Hi Tracy – If x does not equal y then I think any values for x and y will produce an a^2 + b^2 = c^2 situation where (a,b,c) is a Pythagorean triple. So, if x = 2 and y = 5, then (4+25)^2 = (4-25)^2 + (2*10)^2. [Note (-21)^2 = 21^2]

Then 29^2 = 21^2 + 20^2 and (20,21,29) is the triple.

If x = y then you still get a Pythagorean triple like (0,2,2) but I guess it would not represent the sides of a right triangle since one side has zero length. Somebody much smarter than me will have to weigh in on the technicalities there 🙂

]]> <![CDATA[A.APR.4]]> Thu, 23 May 2013 17:36:13 +0000 Tracy How does polynomial identity, (x^2 + y^2)^2 = (x^2- y^2)^2 + (2xy)^2 generate Pythagorean triples?

]]> <![CDATA[Progression of Monomials/Polynomials]]> Thu, 23 May 2013 15:25:09 +0000 sbrockley We have been getting some questions concerning the Common Core progression of performing operations with algebraic expressions. It appears that the addition of monomials and the multiplication of monomials by an integer starts in 6th grade, with problems similar to 10a+3a=13a and 5(6x)=30x. The distributive property leads into factoring expressions (generating equivalent expressions) similar to 4x+12. In 7th grade, it looks like this continues with rational coefficients. There is also some subtraction of binomials, problems like (5x+8)-(2x-9). Whole number exponents are introduced in grade 6, and work with exponents (laws of exponents) is in grade 8. Should students be encountering problems that look like (8×2)(4×5) (8 times x squared)(4 times x to the fifth).
People are questioning the depth or how far to take these algebraic expressions in 8th grade. What about the multiplication of two binomials ? I was just curious as to your thoughts about how the transition from 6th to the beginning of 9th should go with these algebraic expressions. Thank you for your time and I look forward to your response.

]]> <![CDATA[Use of technology: S.ID.6]]> Wed, 22 May 2013 16:21:24 +0000 Aaron Bieniek I think it’s interesting how little talk there is on the Statistics and Probability strand. I hope that isn’t an indication of the level of importance being placed on it by teachers…

My comment is about fitting functions to data (linear, exponential, quadratic) and plotting residuals. From my reading of the Progression document, I am gathering that these concepts are developed with technology. We are not expecting students to do any sort of least square regression or (complex) residual plotting without the use of technology. I can see nailing down the idea of residual by calculating a plotting a few by hand – but not an entire plot. Is that right? – and are there areas I am missing where technology is appropriate (almost necessary) and others where it is not?

]]> <![CDATA[Reply To: Standards taught in order?]]> Tue, 21 May 2013 16:32:40 +0000 cthomson What recommendations do you have for teachers who are trying to formulate a sequence for teaching the standards at their respective grade levels? How might they tackle the process?

]]> <![CDATA[Reply To: 4.NF.3]]> Sun, 19 May 2013 02:52:53 +0000 Bill McCallum Yes, it’s certainly true that they need to be whole numbers. Not necessarily greater than one, but certainly greater than zero.

]]> <![CDATA[Reply To: Equations of the form p(x + q) + c = r]]> Sun, 19 May 2013 02:48:53 +0000 Bill McCallum Yes, 8.EE.7b,

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms

pretty much covers any form of a linear equation.

]]> <![CDATA[Reply To: Surface Area of a Cylinder]]> Sun, 19 May 2013 02:44:18 +0000 Bill McCallum Your interpretation is plausible in the following sense. First, you can think of a cylinder as composed of a quadrilateral in the sense that you can make a cylinder by forming a rectangle into a tube. Second, given the radius of a cylinder you can use the formula for the circumference to find the side length of the rectangle that made the cylinder (the other side being the height of the cylinder). That said, I’m not sure it is exactly what we had in mind with these two standards. My instinct is to see where this reasoning naturally arises in a fully developed curriculum based on the standards. That might well be in Grade 7 as you suggest, but it might be later. The main point I think is not to try to find a catalog of every formula for volume and surface area, but rather to take opportunities to calculate these things as they arise naturally, using geometric reasoning, as in the idea above of unfolding the cylinder into a rectangle. In the end, it’s the ability to think this way that will stand students in good stead, rather than “knowing the formula.”

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Sun, 19 May 2013 02:10:40 +0000 Bill McCallum Joanna, very interesting thoughts (and nice to know the connection with Jeff, who I still see regularly at our Harvard consortium meetings). It’s a complicated problem, and I don’t have a neat answer. Like you, I would like to see more students get further ahead in mathematics, and I wouldn’t want the Common Core to be interpreted as a barrier to acceleration. In fact, I believe that in the long run the Common Core is an engine for acceleration. Students who experience a faithful implementation of the Common Core in elementary school are more likely to be ready to take off. But we have to wait for that, and in the meantime we have to eliminate phony courses that give the illusion of acceleration without the reality of.

]]> <![CDATA[4.NF.3]]> Fri, 17 May 2013 19:49:21 +0000 leighsure The intro to the Standard states: “Understand a fraction a/b with a > 1 as a sum of  fractions 1/b.” I had some trouble understanding what the intent was here, until I put it to myself this way: “… as a sum of unit fractions equal to 1/b.” Still, I felt that this was not precise enough.
Finally, I realized that what was missing was the indication that a and b have to be whole numbers greater than one. Now, I realize that most 4th graders are going to think only of whole numbers, but with the CCSS trying to build understanding from the earliest age that can handle the concepts, I think that this needs to be a specific, clearly delineated way to introduce unit fractions as building blocks.

]]> <![CDATA[Equations of the form p(x + q) + c = r]]> Mon, 13 May 2013 21:25:22 +0000 grettaetta In 7th grade, students solve equations of the form p(x+q)=r. Where are students expected to solve equations p(x+q)+c=r? Should this be an extension that is built upon in 8th grade?

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Mon, 13 May 2013 16:24:11 +0000 jburtkinderman All,

I’m a district math coach and coordinator in WV and have great interest in this topic. I have been a passionate advocate in the last few years for allowing middle school acceleration as we transition to common core standards. It matters SO much in states like mine what the folks at the top are recommending and I’d love to have comment on my rationale…
In a state with the one of the least well-educated populations in the nation, with over half of our students living in poverty, where better than 1/5 of 9th graders failed 2 or more subjects, with less than 40% even proficient in math, I am concerned that it’s more important than ever to give future leaders every possible opportunity for brain stretching.
I totally buy the argument that, with common core standards, content is deepening and that students do not need to ‘skip’ so much as they need to deepen. However, it really only resonates in a theoretical context. What I can’t wrap my mind around is that as the theory morphs down into practice, learners will continue to need different amounts of time to own and personalize ideas. Next year, we will begin to implement these new standards with the students that we have with a great mix of ability, need and desire. Even while working at elite Southern private schools, where 100% of the students finished Algebra by the end of 8th grade, the hungry, naturally gifted learners finished Geometry by the end of 8th.
I worry that changing curriculum can get confused with changing audience. In my (granted, limited in comparison!) experience, the most naturally gifted learners easily learn at both a pace and depth two times greater than the average. In addition, I find that these learners struggle mightily at the beginning of a truly rigorous course surrounded by their peers. But after a short adjustment period, as the sore muscles in their brains transform into stronger cerebral muscles, their potential for engagement and depth grow exponentially.
In short, there’s a difference between the local community college where I have taught and Haverford, where I went to school myself (and where my thesis advisor and lasting friend, Jeff Tecosky-Feldman, still speaks highly of you, Bill!). The institutions don’t serve the same audience, and thus don’t use the same strategies. In our most educationally disadvantaged areas, it’s so important to do the very best by the higher-level students. The ratio of need for highly qualified leaders to availability thereof in all fields is much higher here than in those states lacking our dismal stats.
As one teacher put it… “are we not just moving from no child left behind to no child pushed ahead?” We should continue to delve deeper into these standards, to strive to make our curriculum fertile ground for learning. As our classrooms improve and deepen, the hope is that the learning potential of ALL students will rise, leaving the group of mathematically gifted still with needs beyond the grade level.
My idea in regards to middle school advancement is not to leave out a middle school course, nor to identify kids as 5th or 6th graders. Instead, 8th graders who are ready, willing and able to go an extra mile could take an elective math class in 8th grade in addition to their regular class.
I appreciate feedback, insight, and the opportunity to join the conversation.

My Best,

]]> <![CDATA[Surface Area of a Cylinder]]> Thu, 09 May 2013 16:08:48 +0000 jisimons Good afternoon,
We are debating at what grade does surface area of a cylinder first appear. Can you please clarify for us? We see in 7th grade that 7.G.4 introduces the students to the relationship between radius and diameter which allows students to develop formulas for circumference and area. In 7.G.6, students solve problems for surface area of 2-d and 3-d objects composed of triangles, quads, polygons, cubes and right prisms. We are debating between the fact that these 2 standards combined give access to the cylinder or that cylinders are excluded because they are not polygons. Can you please assist us with this?

]]> <![CDATA[Reply To: 5.MD.A.1]]> Thu, 09 May 2013 16:03:25 +0000 Jason Zimba I’ll try to offer some thoughts here and I hope they’re helpful – although I’m not sure what is meant by “the metric system.” The system of units that is used for scientific purposes is the SI system.

Time is one of the base quantities in the SI system. The SI unit of time is the second. Minutes, hours, and days are outside of SI, but these units are accepted for use within SI. (Information about the SI system is found here.)

If you look in the previous grade, you’ll see that standard 4.MD.1 is explicit about hours, minutes, and seconds. So it would certainly be natural for hours, minutes, and seconds to be part of the continuing thread in grade 5.

(It is often helpful to look at progressions across grades in order to shed light on a specific standard within a single grade.)

I’ll offer some additional comments in case helpful or at least interesting…

When a standard “includes” a lot of things, there is always a risk that it will translate into a laundry list of to-do’s for students. The granular approach to standards exacerbates this risk. (See Grant Wiggins on Granularity)

In cases where a standard “includes” a lot of things, maybe instead we can think of it as an opportunity to find unity in diversity – to write the kinds of problems and lessons that put ourselves in a position to say to the students, “See? It’s all the same!”

So, including time in this standard could be a virtue if it helps to give the subject of measurement a unified character. (I’ll note here that time plays a role in my post “Units, a Unifying Idea in Measurement, Fractions, and Base Ten”.)

A final note in favor of coherence…cluster 5.NBT.A is designated ‘Supporting’ by the two assessment consortia (see here), and that is a reminder that instead of treating this work as yet another disconnected set of tasks, unit conversions might be positioned in such a way as to support of the major work of grade 5. The parenthetical example in 5.MD.1 involves converting 5 cm to 0.05 m; this begins to gesture at how the work of 5.MD.1 relates to other grade 5 work (cf. 5.NBT.A, 5.NBT.7, and 5.NF.B). So in addition to teaching students the enumerated concepts and skills of measurement, the body of work relating to 5.MD.1 could also give students practice with, and insight into, place value, decimal computation, and fraction operations.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Wed, 08 May 2013 20:19:54 +0000 Cathy Kessel I started a forum thread with some curriculum ideas here

  • This reply was modified 4 years, 3 months ago by  Cathy Kessel.
  • This reply was modified 4 years, 3 months ago by  Cathy Kessel.
]]> <![CDATA[prove that all circles are similar]]> Wed, 08 May 2013 20:17:10 +0000 Cathy Kessel Continuing from

As a preliminary, somewhere in curriculum these ideas could occur in terms of transformations: Can you take any two circles and transform one into the other using rigid motions followed by dilation? And, you could ask it for a variety of things: triangles, equilateral triangles, right triangles, angles, lines, . . . (This is not meant to suggest that students don’t eventually need to know the meanings of “prove” and “similar.”)

There’s a nice animation for rigid motions at See the Gingerbread Transformer which talks about packing shapes into boxes. If the transformer in the animation got a dilation button, and the shapes and boxes got labels like “triangle” or “equilateral triangle,” one could ask what shapes could be packed into what boxes, e.g., could you take any shape labeled “triangle” and pack it into any box labeled “triangle.”

]]> <![CDATA[5.MD.A.1]]> Wed, 08 May 2013 19:04:31 +0000 Tracy Is the conversion between units only inclusive of metric and US Customary or does this standard also include time?

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Tue, 07 May 2013 18:55:42 +0000 Cathy Kessel Putting equations and expressions in particular forms comes up under a variety of headings in the Algebra forum:

A general principle of the Standards is described in the Algebra Progression, p. 4,

“The Standards emphasize purposeful transformation of expressions into equivalent forms that are suitable for the purpose at hand. . . . Each is useful in different ways. The traditional emphasis on simplification as an automatic procedure might lead students to automatically convert the second two forms to the first, before considering which form is most useful in a given context.”

In the quote above, for “simplification” one could substitute “putting the equation of a line into standard form” or “putting the equation of a line into slope–intercept form.”

Re systems of equations: there is the beginning of discussion here: I’ll try to contribute more to that thread.

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Sun, 05 May 2013 02:57:06 +0000 ccfree As we were working on curriculum planning with the new CCSS, a question came up about how standard form fits into the CCSS in both The functions and expressions and equations domain and what the expectations are for the students. For instance, in the current TN curriculum, it is an expectation for students to rewrite equations that are in standard form in slope-intercept form. Is this an expectation in CCSS? Also, what are the expectations when solving systems and the methods used to solve systems? Thanks for your time!

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Sat, 04 May 2013 11:57:19 +0000 Bill McCallum I don’t know of any such test, but I will ask around.

]]> <![CDATA[Reply To: Derive vs. Prove]]> Sat, 04 May 2013 11:55:58 +0000 Bill McCallum I don’t think there’s any significant difference. It might have been better to use the word “prove” in both places, but we tend to use the word “derive” for formulas.

]]> <![CDATA[Reply To: Standards taught in order?]]> Sat, 04 May 2013 04:52:40 +0000 Bill McCallum No, this is not true. In fact, the standards themselves explicitly contradict this. Page 5 of the standards says:

These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.

And in particular it would not make sense to teach OA and NBT separately, since there are many close connections between those two domains. It would make more sense to intertwine them in the curriculum.

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Sat, 04 May 2013 02:33:53 +0000 SteveG Thanks for taking the time to reply. I do appreciate it very much. My sincerest apologies for misquoting outcomes and events in my post.

]]> <![CDATA[Reply To: CCSS Algebra 1 in 8th grade]]> Fri, 03 May 2013 20:40:50 +0000 debbiejc In the Catholic Diocese of St. Augustine, FL, we are struggling with the issue of Advanced 7th grade and Algebra 1 8th grade, too. We don’t WANT to offer these courses because we believe enrichment is the way to go for the few truly gifted students we have in each of our 23 elementary schools and the Common Core is already advanced, both in its content and its critical thinking expectations. But, we compete with public schools offering IB programs, 8th grade Honors Alg. 1, etc. and our tuition paying parents insist that their students be placed in “accelerated” courses in 7th and 8th grades. For now, we will need to accommodate.

On that note, does anyone know of a “placement test” that might reliably tell us if an incoming 7th grader is capable of handling the Common Core 7th Grade Advanced course and also one for the 8th Grade Algebra 1? And, are there already tests available that we might give quarterly in each of these courses that could be used as formative assessments to help us figure out what to do with students who might be having difficulty? The PARCC assessments will likely tell us what we need to know, but we can’t wait until 2014-2015.

Also, ALL teachers in the Diocese, including those in the 4 high schools will be using the problems from in their daily lessons. Thank you for these excellent problems! They are awesome.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Fri, 03 May 2013 01:41:37 +0000 Bill McCallum By the way, please try to post queries in the right forum. There is a high school geometry forum.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Fri, 03 May 2013 01:41:04 +0000 Bill McCallum Jim basically had it right in his first post, but Dan and Sherry were correct that the coordinates were unnecessary, and that the key point went by very quickly. To my mind the key sentence in Jim’s proof is “then dilate it until the radii match.”

Why is it even possible to dilate until the radii match? Because by definition all the points on the circle are the same distance (the radius) from the center. So that means that if you dilate the smaller circle from the center, all its points will arrive at the larger circle at the same time.

In more detail: Given two circles, translate the first one so that its center coincides with the center of the second circle. If the first circle has radius r and the second circle has radius R, then perform a dilation on the first one from its center with scale factor k = R/r. Since every point on the first circle is a distance r from the center, every point on the dilated circle will be a distance kr = R from the center, so the dilated circle is identical to the second circle.

This would be easier to explain with visual aids, of course.

As for activities to support this, I can imagine having students play around with a dynamic geometry program, and asking them perform similarity transformations that map circles onto each other. At first maybe using the mouse, but then by giving the precise commands: “perform the translation that takes O to O'” and “dilate around O’ with scale factor 1.2”. To find the scale factor they would have to realize it is the ratio of the radii. Then you could ask what it is about a circle that makes this work (it has a constant radius).

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Fri, 03 May 2013 01:25:53 +0000 Bill McCallum You had me worried for a moment there that we had a major blunder in the standards, until I checked and realized you had misquoted the standard in an important way. It actually says “Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.” Can you see the difference? There’s a difference between outcomes and events; an outcome is an element of the probability space (in this case, a name on a list). In a uniform probably model, every outcome is assigned the same probability. So we assign every name on the list a probability of 1/10.

An event is a subset of the probability space; for example, the subset consisting of all students on the list who are in Grade 7. We can use the uniform probability model to calculate the probability of an event by counting the number of outcomes in the event (the number of students in Grade 7) and multiplying the probability of a single outcome.

As for your question about repeated names, I am imagining a list of full names, like a roll. Of course, two students could still have the same name, in which case presumably the school would have a way of distinguishing them. Still, we should make that clear.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Thu, 02 May 2013 16:28:32 +0000 Sherry Fraser Jim,
Your “proof” seems to be proof by definition. I am a curriculum developer and have been trying to come up with some activity or activities for students that address this standard.
Maybe the authors of the Common Core can tell us why this is a standard, how you prove it, and what you would do in the classroom to prepare students for an assessment item on this standard. Yes, PARCC plans on assessing this standard. Perhaps Bill McCallum, Jason Zimba, or Phil Daro can tell us why this is a standard and how to prove it. Can you help us out here?

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Thu, 02 May 2013 14:34:27 +0000 dan Jim-
“Proof is making obvious what was not obvious.” -Rene Descartes
I already know all that you wrote, and could have written it myself. But to a sophomore it’s all irrelevant BS. Why must such a simple idea be made so complicated?
I guess what I’m really asking is why must all students in US high schools be expected to “Prove all circles are similar” when it is obvious?
Re your simple version: “You can take two circles and move one so that they have the same center, then dilate it so they are the same size. QED”
If that’s all there is to do, then why in the world is this trivial idea a specific standard?

]]> <![CDATA[Standards taught in order?]]> Thu, 02 May 2013 03:33:44 +0000 aparks1 Hi Bill,

We’ve heard that the Common Core State Standards  have to be taught in order? Is that true? For instance, OAT and then the NBT standards all in order?

Thanks for any insights.

]]> <![CDATA[Reply To: K.G.B.6]]> Wed, 01 May 2013 17:42:53 +0000 Cathy Kessel Thanks for this comment. Yes, the composing shapes are adjacent not superimposed.

I don’t know if you’ve looked at the Progressions, but you might check out the K–6 Geometry Progression,

There’s a discussion of composing shapes to build pictures and designs on p. 7, but I see that the discussion might say more about other shapes before it gets into pictures and designs.

There’s an illustration of shapes composing another shape on p. 10.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Wed, 01 May 2013 05:18:49 +0000 Jim Dan,

Sherry asked “How do you prove all circles are similar?” (emphasis added).

I gave my answer.  In no way did I suggest it is the only proof.   One of the beautiful things about math is that there are sometimes many ways to prove the same thing.  So no, Euclidean geometry is not dependent on Cartesian coordinates, but coordinate proofs are one of the tools in the toolbox even of the high school geometry student.

Additionally, I was outlining, rather than detailing the proof, so it may have looked like handwaving because I was allowing the reader to fill in the details.

Since it seems more clarity is called for, feel free to see my extended explanation below:
To show: All circles are similar.

Similarity of two figures is defined as obtaining the second “from the first by a sequence of rotations, reflections, translations, and dilations.” I intend to show that any circle is similar to the unit circle, and that the unit circle is similar to any circle.  Since a combination of two sequences of the above transformations is still a sequence of the above transformations, this would succeed in showing that the two circles are similar.  Without Loss Of Generality place a circle in the plane centered at (h,k) with radius r.  It can be described by the equation (x-h)^2+(y-k)^2=r^2.  Apply the transformation (x,y)->(x-h,y-k) which we’ll call T_1.  Then apply the dilation (x,y)->(x/r,y/r) which we’ll call D_1.  The sequence T_1, D_1 transforms any circle to the unit circle.  Let T_2, D_2 be transformations that likewise take  (x-g)^2+(y-j)^2=s^2 to the origin.  Because translations and dilations are both invertible, the sequence (D_2)^(-1), (T_2)^(-1) transforms the unit circle into this second arbitrary circle (x-g)^2+(y-j)^2=s^2.  So  T_1, D_1,  (D_2)^(-1), (T_2)^(-1) is a sequence of translations and dilations which allows you to obtain any circle in the plane from any other circle in the plane. QED

This version has less handwaving.  It also has a lot more notation and makes the concept of the proof ugly and obscured.

I think the expectation for a high school student is more along the lines of my original argument:

You can take two circles and move one so that they have the same center, then dilate it so they are the same size. QED

  • This reply was modified 4 years, 3 months ago by  Jim.
]]> <![CDATA[Reply To: One-Step Inequalities]]> Wed, 01 May 2013 04:31:15 +0000 Bill McCallum There is no intention to exclude these symbols from the curriculum. There is a discussion of this point here. In general, some confusion has resulted from the avoidance of specifying vocabulary and specific symbol usage in the standards. A lot of those decisions are up to curriculum writers.

]]> <![CDATA[Reply To: Geometry Progressions]]> Wed, 01 May 2013 04:22:39 +0000 Bill McCallum Your intuition is correct that the middle school geometry standards are more experiential and the high school ones more formal. In middle school students get a feeling for transformations and their properties by playing around with transparencies or dynamic geometry software. In high school they should be able to make arguments using the precise descriptions of transformations. Sorry that the geometry progression hasn’t come out yet; working on that.

As for the question about volume formulas, I would be inclined to take the Grade 8 standard fairly literally; it’s really just about knowing the formulas. Understanding where they come from and being able to give a formal derivation is significantly more advanced than that, which is why it is left till high school. I can see why some might find this interpretation unpalatable, but some formulas are quite simply beautiful and classical, and it doesn’t do any harm to appreciate them for a while without deep analysis (we do the same with art all the time).

]]> <![CDATA[Reply To: Math Practice Standards]]> Wed, 01 May 2013 04:10:54 +0000 Bill McCallum I did participate in a review of the Arizona documents, but didn’t participate in writing them. Not sure I remember the specifics of these comments any more.

]]> <![CDATA[Reply To: Coherence and Connections]]> Wed, 01 May 2013 03:15:26 +0000 lhwalker Both books are on their way.  And, yes, you know I’ve read the progressions!

]]> <![CDATA[Reply To: What are Common Core standards for Algebra 2]]> Wed, 01 May 2013 02:18:06 +0000 Bill McCallum First, I would say that what you have heard from your administrator is a local interpretation of the standards rather than something that is in the standards. The standards do not dictate curriculum or pedagogy; they certainly say nothing about whether you should be tied to a text book or not.

Second, the high school standards in the Common Core are not divided into courses. So it will be up to states and districts to decide what is in Algebra I, Geometry, Algebra II, Math Analysis (not quite sure what that is), etc. But I would say that the biggest change in the approach to Algebra in the standards is embodied the domains A-SSE (Seeing Structure in Expressions), A-REI (Reasoning with Equations and Inequalities), F-BF (Building Functions) and F-IF (Interpreting Functions). Although the topics within these domains might seem familiar, the emphasis in seeing structure, reasoning, building, and interpreting is a big shift.

]]> <![CDATA[Reply To: Coherence and Connections]]> Wed, 01 May 2013 02:05:44 +0000 Bill McCallum Well, there are the progressions documents! When I was growing up (in the previous century) I remember reading books by W.W. Sawyer which really brought out a lot of connections for me. His “Vision in Elementary Mathematics” deals with middle and high school topics, and is available on Amazon. Another nice book at a higher level is Gelfand and Shen’s Algebra. Maybe others could add their suggestions.

]]> <![CDATA[Reply To: Error on page 10?]]> Wed, 01 May 2013 01:57:40 +0000 Bill McCallum Wow, that is a very fine catch indeed. Thanks for the close reading.

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Wed, 01 May 2013 01:52:57 +0000 Bill McCallum First, sorry for the long delay in replying. I got hijacked by my day job for a while.

I guess when there seems to be confusion we should try to go back to the text of the standards and see what we can get from it. The second cluster under 6.SP is called “Summarize and describe distributions.” It doesn’t use the word “construct”, although one could argue that in order to summarize a distribution you need to construct a summary. But, as I said in the earlier post, you could do this using technology, and it seems to me that this would be a strategic use of tools in Grade 6, falling under the meaning of MP5. So my inclination would be to stick with my original interpretation (surprise!) and say that Grade 6 students could be using technology to produce summary statistics.

As the question of box plots, 6.SP.B.5c says:

Summarize numerical data sets in relation to their context, such as by

c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

Taking the first choice in each parenthetical, we get median and interquartile range. Box plots are a possible way of representing these, so it would be natural to use them in this context. Although I would certainly not expect Grade 6 students to be skilled in producing them by hand, for the same reasons outlined above.

I haven’t actually checked if this contradicts the progressions document or not, and feel I should move on to answer other overdue questions!

]]> <![CDATA[Derive vs. Prove]]> Tue, 30 Apr 2013 22:31:45 +0000 sparvankin I was with a group of high school math teachers facilitating a discussion about the Geometry standards.  When looking at G.SRT.9 and G.SRT.10, there was much discussion about the difference between derive and prove.  Can you help us distinguish between the two?

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Tue, 30 Apr 2013 16:56:13 +0000 dan Jim-“Prove all circles are similar” is a specific common core standard that ALL high school students are supposed to be able to do.  But your reply you state “slap some Cartesian coordinates on them”  and then, in my opinion, just do a bit of handwaving.

Are you saying all Euclidean geometry is now dependent on coordinates?

Most high school students, teachers ,and textbooks do not think about similarity in this way .

Traditionally, Euclidean Geometry did not depend on a coordinate system for meaning. This is new to me. When, and who, made this decision?

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Tue, 30 Apr 2013 16:56:13 +0000 dan Jim-“Prove all circles are similar” is a specific common core standard that ALL high school students are supposed to be able to do.  But your reply you state “slap some Cartesian coordinates on them”  and then, in my opinion, just do a bit of handwaving.

Are you saying all Euclidean geometry is now dependent on coordinates?

Most high school students, teachers ,and textbooks do not think about similarity in this way .

Traditionally, Euclidean Geometry did not depend on a coordinate system for meaning. This is new to me. When, and who, made this decision?

]]> <![CDATA[K.G.B.6]]> Tue, 30 Apr 2013 15:46:44 +0000 Tracy I was wondering if  you could help me in regards to a quick clarification on CCSS.Math.Content.K.G.B.6 Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”

Does this mean that the two shapes that are being used need to form a new shape?  What I mean by that, is do the lines need to touch or can a circle be placed on top of a square, resulting in a new figure, but not a new shape?   I believe at this grade level, the intent was to really create a new shape like the example above, but we want to make sure we are understanding this correctly

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Tue, 30 Apr 2013 12:28:49 +0000 Jim You can’t prove they are similar until you have a definition of similar.  The definition is based on transformations and congruence.  The axioms are not being invoked directly, but everything done is undergirded by them.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Mon, 29 Apr 2013 23:35:28 +0000 Sherry Fraser That explanation sounds like circular reasoning to me. How is that a proof? You state what similar means and then say since they are the same shape, they are similar. Are there any axioms involved? Sorry if I’m being thick, but I really don’t get this.

]]> <![CDATA[Reply To: Prove that all circles are similar]]> Mon, 29 Apr 2013 12:57:50 +0000 Jim Let’s use the definition of Similarity given in the standards:

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

We’ll start by showing any two circles are the same.  Take any two circles, and slap some Cartesian Coordinates on them, such that the first is at the origin.  Translate the second circle to the origin, then dilate it until the radii match.  Thus the pair of circles is similar.

If any two circles are similar, then all circles are similar by transitivity of similarity. QED

  • This reply was modified 4 years, 3 months ago by  Jim.
]]> <![CDATA[Prove that all circles are similar]]> Fri, 26 Apr 2013 20:48:12 +0000 Sherry Fraser How do you prove all circles are similar?

]]> <![CDATA[Reply To: 7.SP.7a – trying to understand uniform probability model]]> Fri, 26 Apr 2013 19:34:23 +0000 SteveG I’m a teacher in Florida, by the way, sorry I didn’t say that earlier.

]]> <![CDATA[7.SP.7a – trying to understand uniform probability model]]> Fri, 26 Apr 2013 19:13:06 +0000 SteveG I am hoping that someone can help me understand the standard 7.SP.7a, “Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of outcomes.”  The comments in the draft progressions begin on the bottom of page 7 and continue to the top of page 8.

In the progressions document, it gives the example of selecting a given student’s name if there are 10 students as 1/10.  What if there were two students who were named John, for example? I can assume that the example was talking about selecting each student as 1/10.  If so, I understand that it is uniform because each student has a likely chance of being selected. The next sentence says, “If there are exactly four seventh graders on the list, the chance of selecting a seventh grader’s name is 0.40.”  For the sake of argument let’s say there are 4 seventh graders, 5 sixth graders, and 1 eighth grader.

Here are my questions about that:

If we select one student at random from the ten, is that a uniform probability model regardless of the characteristic we note about that student (i.e. his/her name or grade level or color of their shirt or shoe size)?  Or, does the characteristic we note about that student change the model from being uniform to not uniform.

That is, if all 10 students have different names and we choose 1 student out of 10 so that each name has a 1/10 probability, it is uniform. I think that’s easy to see.

But, if the students have the grade levels mentioned above and we choose 1 student out of 10 so that P(7th grader) is different that P(8th grader) is different, is the model now not uniform? Or is it still uniform because each student is equally likely to be chosen?

Also, is the P(7th grader event) considered a simple event or a compound event?

I would sincerely appreciate any clarification that you can offer. Thank you for your time.

]]> <![CDATA[Reply To: One-Step Inequalities]]> Wed, 24 Apr 2013 18:56:50 +0000 nancymclaughlin Hello All,


To the CCSS-M folks, Are there future plans to include a standard for learning the meaning of ≤ and ≥, in a subsequent grade level?

]]> <![CDATA[Geometry Progressions]]> Tue, 23 Apr 2013 18:34:45 +0000 Ksullivan Is there any word on when we can expect a geometry progressions for beyond grade 6? 

Right now I am struggling  with differences in grade 8 and the geometry course when it comes to transformations and volume formulas.  For example, Grade 8 has the standard, “8.G.C.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems” and high school has  “HS. GMD. A.1   Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments”

Iunderstand what they are asking students to do is different for each standard.  I am assuming that when you teach these volume formulas in grade 8, you want students to undersand the concept behind the formulas and know where they came from.  How is the teaching  going to be different for  “students being able to give an informal argument for the formulas” in high school and having students know where the formulas come from in middle school?    I realized that discussion may occur at higher level at high schoool since it lists some arguments and principles to use.  Do those arguments have to be used, or will the understanding from middle school be enough to informally explain the formulas?

I also wonder the same type thing when it comes to tranformations.  It seems 8th grade spends a lot of time working with tranformations and using transformations to show congruence.  They even work with coordinates (8.G.3)  The high school geometry standards include many standards that also have students experiment with transformations and use them to show congruence as well.  I’m  having a hard time picking up on how they are different, beyond the fact that things are “formalized”  in the high school course.  Can anyone help?



]]> <![CDATA[Math Practice Standards]]> Mon, 22 Apr 2013 14:33:33 +0000 bumblebee On the Arizona DOE website (  I noticed that, at the end of their Standards document for each grade level (the link above is for Gr. 3), they have included a section on the Standards for Mathemtaical Practice with explanations and examples for each standard at that grade level.  In researching multiple states’ websites, I am finding this same wording posted in some form.  Was this done by the writers of the CCSS or was it done by one of these states and then others are using it as well?

]]> <![CDATA[What are Common Core standards for Algebra 2]]> Mon, 22 Apr 2013 05:22:16 +0000 Tom Destefano I teach Algebra 2, Math Analysis/Trig, and AP Calculus (AB).

As I understand it, the Common Core is not only a new set of curriculum standards, but it’s also a new way of getting the students to learn the concepts. I’ve seen a set of standards from Arizona dated 2010, so I’m not sure if those are actually the Common Core. Those standards were not organized by class, but rather by concept, so it was difficult to ascertain the changes to any particular course.

I assume Calculus will remain the same, but will there be any significant changes to the curricula of Algebra 2 or Math Analysis?

My administrator has described the changes in pedagogy as follows:

Teachers will not be tied to a text book.

The teacher will lead the student to discover the concepts by asking leading questions, rather than merely disseminating information.

As I look through my curriculum though, it seems to me that very few students will be able to pick up information on their own without it being given to them. For example, if a student could derive the Law of Cosines on his own, he wouldn’t need a teacher at all.

]]> <![CDATA[Coherence and Connections]]> Sun, 21 Apr 2013 03:54:15 +0000 lhwalker Every time I read “coherence” and “connections,” I feel a twinge of guilt knowing that after ten years of teaching high school math, I still don’t have all those down.  I know I’m not alone because every year my Algebra 3 students are surprised by many of the connections I do make.  For example, when I explain adding complex numbers is consistent with adding “like terms” beginning with adding tens to tens and ones to ones, followed by adding fractions with common denominators, radical expressions, etc., it becomes apparent it is the first time anyone has made those connections for them.  Can you recommend some reading on “coherence” and “connections?”

]]> <![CDATA[Error on page 10?]]> Wed, 17 Apr 2013 20:26:26 +0000 mjacks I am not sure if this has been mentioned yet, but on the very  bottom of page 10 it states “They calculate 1/2 + 2/5 = 9/10, and see this as 1/10 less than 1 ….. They detect an incorrect result such as 2/5 + 2/5 = 3/7 by noticing that 3/7 < 1/2″ Shouldn’t it be 2/5 + 1/2 = 3/7 (just as it was written in the example with the standard 5.NF.2 in the CCSS)?

Thanks, Malissa Jacks

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Fri, 12 Apr 2013 17:34:40 +0000 kkirnie Cathy,

I have read the progression many times.  Hence the confusion between Bill’s blog and the progression document.  Do students need to be skilled at creating box plots and Mean Absolute Deviation OR should they be skilled at interpreting them when they are given a mean absolute deviation?

]]> <![CDATA[Reply To: 6th grade statistics – what are they constructing?]]> Thu, 11 Apr 2013 22:34:41 +0000 Cathy Kessel The grades 6–8 Statistics and Probability Progression discusses box plots in grade 6. It can be downloaded here:

]]> <![CDATA[6th grade statistics – what are they constructing?]]> Thu, 11 Apr 2013 19:18:08 +0000 kkirnie I am looking for clarification on box plots and mean absolute deviation.

According to the blog on mode and range, Bill has written.


“A curriculum could meet the Grade 6 standards on Statistics and Probability by working with large data sets arising from real contexts, using technology to plot them and compute their summary statistics. Students should be able to answer statistical questions, display data graphically, choose appropriate summary statistics and interpret them in terms of the context. That’s what the Grade 6 standards say.”
This leads me to believe that students are reading and interpreting data using measures of center and variability in 6th grade. Which would mean not finding the Mean Absolute Deviation but looking at the value of the MAD and applying it to understanding the data. Leaving the figuring out the mean absolute deviation in 7th grade.
Does the same reasoning apply to box plots?

]]> <![CDATA[Reply To: Smart Quotes in Geometry Overview]]> Tue, 09 Apr 2013 20:50:14 +0000 Jim Not directly related to Geometry, but another minor inconsistency throughout the standards:

‘multi-step’ is sometimes hyphenated and sometimes not.

(I noticed because I was searching for ‘multi-step’ and only found some of the references).

]]> <![CDATA[Reply To: Decimals]]> Tue, 09 Apr 2013 05:21:56 +0000 Bill McCallum It depends which fractions and decimals. First, let me reiterate that the standards do not regard fractions and decimals as different kinds of numbers, but rather different ways of writing the same number. Thus, rather than talking about converting fractions to decimals, I would talk about writing fractions in decimal notation (and vice versa). And this is indeed the language used in the standards:

4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

So, in Grade 4, students deal with decimals that have one or two digits after the decimal point.

]]> <![CDATA[Reply To: Decimals]]> Mon, 08 Apr 2013 20:21:37 +0000 bumblebee Would you say that students should be able to convert fractions to decimals by the end of Gr.4 or the end of Gr. 5?

]]> <![CDATA[Reply To: Extent of 5.NBT.6 and 5.NBT.7]]> Mon, 08 Apr 2013 05:29:36 +0000 Bill McCallum I was thinking C was o.k. because 0.2 is a unit fraction, but of course you are right that this requires a conversion. That conversion is fairly simple, and well within the purview of Grade 5; your example of 0.125 is somewhat trickier.

]]> <![CDATA[Reply To: Circle Graphs/ Pie Charts]]> Mon, 08 Apr 2013 05:26:08 +0000 Bill McCallum The standards don’t suggest circle graphs should be taught anywhere, although of course they also don’t forbid the teaching of circle graphs at some point, as long as it doesn’t interfere with the things they do suggest. I agree they are a part of daily life, but I worry a bit about the statement that “students must understand what they represent.” Yes, eventually, perhaps during their schooling, perhaps after. But this sort of rhetoric is what led us into the mile-wide-inch-deep curriculum. There are many wonderful and important things to learn; not all of them can be learned before the end of high school. Focus is important for depth.

Another point to consider is that not every important quantitative representation has to be taught in mathematics class. Circle graphs could show up in history, social studies, or science.

]]> <![CDATA[Reply To: Extent of 5.NBT.6 and 5.NBT.7]]> Mon, 08 Apr 2013 02:00:40 +0000 Duane Thanks Bill. Just to clarify – is example C okay? It’s similar to an example on p.18 of the NBT Progressions (7 / 0.2) because it has 0.2 as the divisor. But would 0.2 really be considered a unit fraction (i.e. 1/5) and, if so, how does 0.125 (i.e. 1/8) fare?

]]> <![CDATA[Reply To: Circle Graphs/ Pie Charts]]> Sat, 06 Apr 2013 02:50:34 +0000 NEB So where and when do the standards suggest circle graphs be taught?  They are certainly a part of daily life and students must understand what they represent.

I agree that bar graphs are wonderful and have a tight connection with the number line.  Circle graphs are really nice to quickly view relative size compared to the whole.  That is the reason they are used so often in newspapers and daily periodicals.  They display information in a format easy for people to digest.

So, where should they be taught?

I value your insight and look forward to your guidance.


]]> <![CDATA[Concept of a "ten"]]> Sat, 06 Apr 2013 01:26:04 +0000 Jason Zimba I was asked the following question and thought it might be productive to post it here, along with the answer I gave:
>During a lesson on the numbers 11-19 a kindergarten child looks at a filled ten frame drawn on the board and says that’s 10 because that’s a 10 frame and a 10 frame holds 10.
>Would you consider that exercising K level reasoning? Or making use of structure?
I replied to this with the following quick thoughts, which I’m simply pasting in here now: 
First, remember that the concept of a ten as a unit is a grade 1 expectation (1.NBT.2a), so the standards don’t require this kind of work with kindergarteners.
Not that they couldn’t work with ten-frames or get into grade 1 material early if desired, but it isn’t expected.

 As to the question itself, I think what the student’s statement might show is some measurement thinking. Effectively, she has learned the value of a conventional unit of measure (the “ten-frame”). This seems more like acculturation than mathematics. I think it isn’t possible to know, just from her words, whether she is unitizing the 10. (Which again is a grade 1 expectation.)

]]> <![CDATA[Reply To: 6.RP.3(c) – Percents Question]]> Fri, 05 Apr 2013 20:45:11 +0000 nathan118 Thanks for the reply Bill! I’ve always taught students to move the decimal twice to the left. I try and show them why that happens (divide a few numbers by 100 and show what happens), but most simply remember “move the decimal twice to the left” and they have no idea why.

The standard itself says to put the percent over 100 and multiply, and I’ve seen it taught that way, as if putting it over 100 is a more “meaningful” representation than changing it to a decimal.

But like you said, the 6th grade NS standards go quite deep into division with decimals and multiplication with decimals, so 0.30 x 45 would certainly be possible.

Thanks for the website and your time it takes to respond. It is much appreciated!

]]> <![CDATA[Reply To: Circle Graphs/ Pie Charts]]> Fri, 05 Apr 2013 15:24:53 +0000 Bill McCallum Circle graphs are not explicitly mentioned in the standards; categorical data is represented by bar charts, which are introduced in Grade 2. A reason is that the connection with the number line is tighter for bar graphs, since the value in each category is read from the vertical scale, rather than estimated from the area of a sector.

]]> <![CDATA[Reply To: 6.RP.3(c) – Percents Question]]> Fri, 05 Apr 2013 15:13:17 +0000 Bill McCallum It seems to me you are doing a good job of thinking this through. Yes, you are right, 7/12 doesn’t come until Grade 7. All your ideas for 30% of 45 sound good to me. But I don’t see why you are trying to avoid 0.30 x 45, that is also covered by the NS standards. You are right also that the NS standards have to be interwoven with the RP standards in the appropriate way.

]]> <![CDATA[Reply To: 4.MD.2]]> Thu, 04 Apr 2013 21:17:13 +0000 Bill McCallum Andy, thanks for the invitation, I’ll answer off line.

]]> <![CDATA[Reply To: pre-k?]]> Thu, 04 Apr 2013 21:14:25 +0000 Bill McCallum I’m not sure where you are seeing pre-K, but some states added pre-K standards (using their 15%). The standards themselves don’t have any.

]]> <![CDATA[Reply To: Typos in Canonical Forms Document]]> Thu, 04 Apr 2013 21:10:10 +0000 Bill McCallum Thanks Jim, I’ve passed it on to CCSSO.

]]> <![CDATA[Reply To: Grade 3 : Scaled Picture Graphs]]> Thu, 04 Apr 2013 21:07:04 +0000 Bill McCallum Good question, but the standards don’t answer it. This is really a question for hte curriculum writer. I’d be inclined to agree with your implementation here, but another interpretation would not violate the standard.

]]> <![CDATA[Reply To: Welcome!]]> Thu, 04 Apr 2013 21:03:53 +0000 Bill McCallum The sub-parts of a standard draw attention to particular aspects of the standard or emphasize that a particular case must be treated. In this case, S-ID.6c draws attention to linear functions particularly. It’s a bit like saying to your child, make sure you pick up everything, and don’t forget to look under the bed. The second is technically a consequence of the first, but you say it anyway.

I’m not sure about the second part of your question because I don’t know what you have in mind when you talk about “using” a standard. Certainly it’s possible to have a task which addresses some but not all aspects of a standard. And yes, you could just say it is related to S-ID.6, without necessarily naming which part or parts.

]]> <![CDATA[Reply To: A-REI.5 – what does it mean/look like?]]> Thu, 04 Apr 2013 20:53:04 +0000 Bill McCallum I think the discussion shows that you understand this standard very well! In particular, an equivalent system of equations is not necessarily made up equations that are individually equivalent the equations in the original system. A student proof would look something like the discussion here.

]]> <![CDATA[Circle Graphs/ Pie Charts]]> Thu, 04 Apr 2013 18:48:22 +0000 NEB Is this the intended domain for circle graphs?  Are we supposed to see that as part of percents?  Or would the circle graph/pie chart be part of statistics or probability model?

]]> <![CDATA[Reply To: Decimals]]> Wed, 03 Apr 2013 20:46:34 +0000 Cathy Kessel Two important building blocks for understanding relationships between fraction and decimal notation occur in Grades 4 and 5. In Grade 4, students’ understanding of decimal notation for fractions includes using decimal notation for fractions with denominators 10 and 100 (4.NF.5; 4.NF.6). In Grade 5, students’ understanding of fraction notation for decimals includes using fraction notation for decimals to thousandths (5.NBT.3a).

Students identify correspondences between different approaches to the same problem (MP.1). In Grade 4, when solving word problems that involve computations with simple fractions or decimals (e.g., 4.MD.2), one student might compute 1/5 + 12/10 as .2 + 1.2 = 1.4, another as 1/5 + 6/5 = 7/5; and yet another as 2/10 + 12/10 = 14/10. Explanations of correspondences between 1/5 + 12/10, .2 + 1.2, 1/5 + 6/5, and 2/10 + 12/10 draw on understanding of equivalent fractions (3.NF.3 is one building block) and conversion from fractions to decimals (4.NF.5; 4.NF.6). This is revisited and augmented in Grade 7 when students use numerical and algebraic expressions to solve problems posed with rational numbers expressed in different forms, converting between forms as appropriate (7.EE.3).

]]> <![CDATA[Reply To: A-REI.5 – what does it mean/look like?]]> Tue, 02 Apr 2013 21:01:21 +0000 nvitale Thank you Cathy, for your reply!
I do understand that line of reasoning, and it is basically how I began thinking about this. The concept of equality of sums of equal expressions which you and the Algebra Progressions expressed so well is fairly accessible, and is an extension of the reasoning used in solving one-variable equations. I also think the point made about “realizing that a solution to a system of equations must be a solution of all the equations in the system simultaneously” gets to an important point. There is something about the assumption of equality of the equations for the solution set (not equivalence) that makes this reasoning hold water.
What I’m grappling with here is that the resulting system has the same solution as the original system, but the equations are not equivalent (as you would find if you just scale one or both of the original equations, or otherwise manipulate each equation algebraically, but separately) – and the graph would look different (although still cross at the solution point). Another way of thinking about his is that the solution sets of the new equations in the system have the same intersection, but each individual equation’s solution set does not have to be identical to any of the solution sets of the original equations. I’m seeing that this is a result of our assumption of equality is for a special case (for the solution) – our resulting equations are equal (and therefore sum-able or substitute-able) at the solution values, but otherwise are NOT equivalent – interesting!
Still not sure how a students would “prove” this, what proof might look like. I still feel like I’m missing something about this standard – just reading it has made me re-think what I think I know about systems of equations…
Maybe I will post this in the general forum, I would love to hear your further thoughts and thoughts from others.

]]> <![CDATA[Decimals]]> Tue, 02 Apr 2013 15:22:06 +0000 bumblebee What standards include changing decimals to fractions (e.g. 0.75 = 3/4) and fractions to decimals?

]]> <![CDATA[Reply To: 6.RP.3(c) – Percents Question]]> Fri, 29 Mar 2013 21:22:24 +0000 nathan118 I found the draft progression on 6-7 RP, from Dec 2011….but it didn’t answer any of my questions. 🙂

]]> <![CDATA[6.RP.3(c) – Percents Question]]> Fri, 29 Mar 2013 20:31:17 +0000 nathan118 Hi everybody. Had a question about how best to approach this standard. Let me know where I’m wrong, because I’m sure I don’t have this completely right.

This is the first introduction of percents, so teaching a percent as a ratio out of 100 is a must. I’m thinking a graphical representation like 10×10 grids. Maybe even some estimating percents with some pie charts, horizontal bars partly shaded.

I see this leading to things like 8 out of 25, and having students change them to ratios out of 100, and then making a percent.

That’s where I get a little confused on where to go. You could do more complex examples, like what percent is 7 out of 12…but if the whole point of the standard is to build on proportionality, this seems too complex.  Dividing 7 by 12 and making a decimal is a 7th grade standard. So do I only deal with denominators of 2, 5, 10, 20, 25, 50, 100? You could change 7/12 to 56/96 and conclude that it’s a little more than 56% and estimate, which might be the farthest you’d want to go in 6th grade?


As far as finding percents of numbers, like 30% of 45, I envision (30/100) x 45.  Is the expectation that this should be solved as a fraction question? That would lead to 1350/100…is the expectation then that students should divide that to get 13.5? Does that mean this standard should come after some of the 6th grade NS standards that deal with division and division of decimals? I’m trying to stay away from simply converting to a decimal and multiplying (though even that would probably put this standard after some of the NS standards).

An alternate way I’ve seen is to say that if 45 was broken up into 100 pieces, each would be  worth .45, and you would then multiply this by 33. To get that you’d have to divide 45 by 100, so aren’t you essentially teaching students to move the decimal twice to the left (but perhaps with more meaning I guess). Is that really any different from dividing 30 by 100, and multiplying .30 x 45? That would be a fraction to a decimal, which is more of a 7th grade thing though.


And finally the standard says, given a percent and the part, find the whole. I’ve seen the example 30% of a class is 6 students, how many in the class? Those numbers work beautifully to do 30% – 6, 30% – 6, 30% -6, and then the final 10% – 2. Does that mean students aren’t expected to do something more complex like “35% of a whole is 17?” which would obviously result in a decimal. I know in 7th grade we could do .35x = 17, but is the standard in 6th grade supposed to be fairly simple?


I know the progression document isn’t out yet, so a lot of this might get answered in there, but I thought I’d ask! Amazing how something as simple as this standard can make me feel like I know nothing about percents, haha.  🙂


]]> <![CDATA[Reply To: pre-k?]]> Fri, 29 Mar 2013 19:44:33 +0000 Cathy Kessel I’ll check with the main author, but my guess would be that the progression was meant to cover the standards which (as I suspect that you know) begin at K, not PK. Or is your question meant to be about why are there no PK standards?

PK territory is covered in the National Research Council report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. This can be downloaded free of charge here: In addition to discussion of research on early childhood mathematics, it’s got recommendations about the early childhood workforce and its professional development, and is accompanied by a podcast.

]]> <![CDATA[Reply To: 4.MD.2]]> Fri, 29 Mar 2013 12:32:03 +0000 AndyIsaacs Dear Bill,

We have been working non-stop for almost three years to create faithful enactments of the CCSS — working to translate the standards into a viable classroom reality.  Through this work, we have discovered a number of inconsistencies or ambiguities, such as the one we discussed here.  I would be happy to share other examples with you if you would like.

Nobody would expect the standards to be perfect documents.  However, the bigger issue right now is that there is no formal mechanism to raise these kinds of concerns, short of your blog.  This strikes me as not healthy for the standards or for the field.  The standards could benefit from a clear understanding of how these kinds of issues can get raised and a process for considering issues raised by the field and for updating (or correcting) the standards as indicated.  I personally would like to see the ongoing development of the standards reside with an independent organization, such as NCTM or MAA, or perhaps with a designated coalition of professional organizations.  That could open the door to a formal mechanism where practitioners, assessment developers, curriculum developers, and others to raise questions about the standards and get the kinds of clarifications that we have been forwarding to you.

I’d be happy to continue this conversation with you, though I think it preferable to continue the discussion face-to-face, rather than via email or blog.  We invite you to visit us at CEMSE, where we can continue the discussion and also illustrate how we have been working on updating EM in response to the CCSS-M.  Please let me know if you are interested and available.

]]> <![CDATA[pre-k?]]> Thu, 28 Mar 2013 16:57:43 +0000 chilllyt Out of curiosity why does this progression begin with kindergarten as opposed to pre-k, considering the Counting and Cardinality domain begins at pre-k? Thank you!


]]> <![CDATA[Typos in Canonical Forms Document]]> Thu, 28 Mar 2013 15:57:03 +0000 Jim To whom it may concern:
This is probably the wrong spot for this, but I’m not sure exactly where to put it.

There are typos in the following standards (they are missing their final periods) in the Canonical Forms file.







]]> <![CDATA[Grade 3 : Scal