For this coming year, I’m going to try using the term, “Overflow Fraction.”

]]>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 ]]>

https://www.engageny.org/resource/grade-5-mathematics-module-5-topic-c-overview

Do either of the approaches target this standard?

Thank you! ]]>

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: http://achievethecore.org/aligned/the-power-of-strong-assessments-the-danger-of-bad-ones/ 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: http://achievethecore.org/aligned/8-questions-about-high-school-math-and-stem/

]]>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.

]]>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!

]]>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.

]]>. . . 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 (http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf). 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. http://commoncoretools.me/forums/forum/public/arranging-the-high-school-standards-into-courses/. I’ll repost it over there.

- This topic was modified 3 months, 2 weeks ago by Bill McCallum.

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 (http://commoncoretools.me/forums/topic/algorithms-grades-2-5), 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!

Kate

]]>“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).)”

]]>

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.

]]>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.

]]>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? ]]>

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.

]]>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 10 months, 4 weeks ago by bsmithwbms.
- This topic was modified 10 months, 4 weeks ago by bsmithwbms.

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

]]>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.

]]>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. ]]>

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.

]]>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.

]]>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.

Corey

]]>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.

]]>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!

]]>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?

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? ]]>

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! ]]>

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?

]]>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.

]]>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?

“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!

]]>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???

]]>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.

]]>Regards.

Allen of http://www.digitekprinting.com/

- This topic was modified 1 year, 11 months ago by kirkkimb.

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”.

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

https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/8d3dfd8f-9610-413b-9952-8db75f1b71a4.gif

(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?

]]>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?

]]>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.”

]]>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!

]]>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”?

]]>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.

]]>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?

]]>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?}

]]>32/8

5/1

-27/3

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.

]]>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.

]]>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.

]]>Also, Dr. Zal Usiskin has written Geometry texts using this approach which can be ordered off Amazon: http://www.amazon.com/gp/product/067345956X?psc=1&redirect=true&ref_=oh_aui_detailpage_o02_s00

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. http://www.amazon.com/gp/product/0866514651?psc=1&redirect=true&ref_=oh_aui_detailpage_o05_s00

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.

]]>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).

]]>the

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.

]]>– 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.

]]>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?”

]]>Thanks

- This topic was modified 2 years, 3 months ago by Lisa j r.

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.1

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 & write 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.

]]>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.

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.

Thanks,

Emily

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!

]]>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.

]]>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.

]]>For updates and more information about the Progressions, see http://ime.math.arizona.edu/progressions.

When I go to the Arizona.edu website and download this “same” document, I got a revision date of 21 April 2012. From the commoncoretools.me 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

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 (https://www.illustrativemathematics.org/blueprints/G).

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.

]]>Thanks!

Sarah

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.

Any insight would be much appreciated. ]]>

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.

]]>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.

]]>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. ]]>

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? 🙂

Thanks!

]]>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:

[x+1]

(3[2^4+1] +17)^2

[2x+3]*[x+1]

2x + [x+1]^2 + (y-2)

([2x+3]*[x+1])^2

(2^3 + [3^2 + √(2) ])

Thank you for your help.

]]>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,

Noam

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 – https://www.azed.gov/azccrs/files/2013/11/kflipbookedited.pdf) 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,

Noam

]]>“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?

]]>“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!

]]>]]>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.

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?

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

]]>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?

Thanks!

]]>- This topic was modified 2 years, 8 months ago by kelli.

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.

]]>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.

]]>http://www.math-aids.com/Skip_Counting/Skip_Counting_Advanced.html

http://www.worksheetworks.com/math/numbers/skip-counting.html

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.)

]]>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)?

]]>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. *

“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.

“CCSS.Math.Content.HSG.GMD.A.2

“(+) 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.

]]>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?

]]>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. ]]>

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.

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

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]]>* 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.”

]]>“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.

]]>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.

And

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?

]]>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.

]]>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.

CN.7

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?

Thanks,

Dave

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.

]]>- This topic was modified 2 years, 10 months ago by ginger11772.

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- This topic was modified 2 years, 10 months ago by dejanmolnar.

(5 – 3)x, and then 2x.

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

]]>Or do they only need to evaluate numeric expressions without unknowns (5-3=2)? ]]>

]]>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.

(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!

]]>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.

]]>3s+37.5=63

instead of

3s+37.5=64

Super minor!

Darryl

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…

]]>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? ]]>

-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.

]]>]]>

“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.”

]]>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 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.

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?

]]>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)

]]>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)

]]>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)?

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

]]>= 50 + 25 = 75 ]]>

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

]]>- 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.

]]>(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? ]]>

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.

]]>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?

]]>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!

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

]]>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.

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!

Noam

]]>Thank you!

Jen

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?

]]>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!

]]>

“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.

http://www.parcconline.org/mcf/mathematics/high-school-standards-analysis

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:

http://parcconline.org/for-educators

http://parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL_0.pdf – Version 3, updated framework

http://www.parcconline.org/math-plds – Math Performance Level Descriptors

https://www.parcconline.org/assessment-blueprints-test-specs – Assessment Blueprints

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.

]]>Thanks ]]>

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. ]]>

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?

]]>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

terminology

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 http://t.co/zv6IJNFc89 via @slate</p>— Bill McCallum (@wgmccallum) June 12, 2014

<script async src=”//platform.twitter.com/widgets.js” charset=”utf-8″></script>

]]>Was that supposed to be y = (

I didnt find anything in the progressions on this and the version on the website also has this exampel. ]]>

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.

Thanks!

]]>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.

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.

]]>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.

]]>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?

]]>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

]]>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.

…confused.

]]>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!

]]>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,

Thanks!

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.”

]]>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.

and

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 corestandards.org.)

]]>- Each number is represented as a sequence of digits 0–9.
- The number is the sum of the values of the digits.

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.

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).

]]>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.

]]>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.

]]>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.

]]>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?

]]>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?

]]>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.

]]>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?

]]>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?

Thanks!

]]>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!

]]>Thanks!

]]>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.

]]>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.

]]>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

Thanks

]]>thanks for any help

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

]]>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.

]]>https://www.parcconline.org/sites/parcc/files/ES%20Table%20Geometry%20EOY%20for%20PARCC_Final.pdf

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

https://www.illustrativemathematics.org/illustrations/608

Algebra I makes more sense.

]]>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?

]]>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?

]]>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? ]]>

http://learnzillion.com/lessonsets/427-prove-that-all-circles-are-similar ]]>

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!

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.

]]>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.

]]>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!

]]>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.

]]>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… ]]>

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…. ]]>

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!

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

]]>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.

]]>Is PARCC aware of the discrepancy in terms? ]]>

MCC3G.1

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.? ]]>

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.” ]]>

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

Thanks.

]]>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.

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!

]]>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.

CCSS.MATH.CONTENT.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. ]]>

Thank you,

]]>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,

Jen

Thanks,

Dave

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.

]]>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.

]]>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.

]]>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.

]]>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.

]]>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? ]]>

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”.

]]>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.

]]>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.

]]>Thanks so much,

Adrienne

“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? ]]>

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.

]]>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

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).

]]>I have some questions after looking over EngageNY for Algebra I at:

http://www.engageny.org/sites/default/files/resource/attachments/algebra-i-m1-teacher-materials.pdf

but I’m not sure this the best place to open such a discussion. What do you think, 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.

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.

]]>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?

]]>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?

]]>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.

]]>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.

]]>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! ]]>

So is the inequality aspect of A-CED.1 disregarded in PARCC states?

]]>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 http://ccss5.com

]]>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.

]]>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.

]]>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

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.

]]>I wonder if there are any plans to put together something like this:

http://strandmaps.nsdl.org/

here is one on number

http://strandmaps.nsdl.org/?id=SMS-MAP-2264

It might be a helpful tool to explore the structure of the standards.

]]>Thank you for your help

]]>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!

]]>Explanations for F.BF standards begin on page 11 of the High School Functions progressions:

http://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_functions_2012_12_04.pdf

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!

]]>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, 7 months ago by hsurrette.

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. ]]>

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.

2)

]]>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.

2)

]]>Thanks!

Beth

(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.”)

]]>The webpages are temporarily hosted:

1. Algebra 1 (http://www.meinzeit.com/commonCoreSite/algebra1.html)

2. Geometry (http://www.meinzeit.com/commonCoreSite/geometry.html)

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.

john

]]>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.

David

(@davidwees)

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.” ]]>

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

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: http://commoncoretools.me/forums/topic/expanded-notation-and-order-of-operations/

]]>For example, if the relationship in question is given by *y* = 2*x* + 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 *y* – *b* and *x*.)

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.” ]]>

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! ]]>

Neither of these mention inequalities. Is it implied that equations also refers to inequalities?

]]>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.

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 http://www.ams.org/profession/data/cbms-survey/cbms2010. Over half of the undergraduates in mathematics courses at four-year institutions are taking courses below calculus.)

]]>- This topic was modified 3 years, 9 months ago by Karena Clarke.

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.

]]>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. ]]>

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.

]]>Thanks so much!

]]>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.

]]>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.

Andy

]]>“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 http://gothamschools.org/2010/10/29/city-official-and-biggest-critic-find-slivers-of-common-ground/#more-48933:

“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.

]]>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.

]]>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.

]]>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.

]]>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.

]]>[2013-12-06: Typo corrected]

]]>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.

]]>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.

]]>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.

]]>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,

Joanna

]]>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

http://commoncoretools.me/wp-content/uploads/2011/04/ccss_progression_nbt_2

011_04_073_corrected2.pdf

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.

Andy

]]>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, 9 months ago by Bill McCallum.

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?

]]>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.

]]>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.

]]>Thank you, Becky Hayes ]]>

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.

]]>http://commoncoretools.me/forums/topic/absolute-value-equations/#post-1799

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).

]]>]]>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).

I would love some feedback on this. Thanks.

]]>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,

Darren

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)

(x-3)(x+3)(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)

(x-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. https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-by-grouping/v/factor-by-grouping-and-factoring-completely

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?

]]>I expect that you mean ‘very *in*formal 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.

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.

]]>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?

]]>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: http://dianeravitch.net/2013/10/15/mercedes-schneider-who-created-the-common-core-how-did-it-happen/comment-page-1/#comment-332091

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.

]]>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.

]]>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.

]]>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).

]]>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.

]]>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.

]]>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.

]]>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?

]]>– 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?

]]>-Compare two treatments means/proportions with a resampling technique (simulation software) to test whether results are random or significant.

What are your thoughts?

]]>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.

]]>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. ]]>

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.

]]>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 (http://dianeravitch.net/2013/09/24/major-corporations-fighting-common-core-backlash/) 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! 🙂

Thanks,

]]>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).

]]>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!

]]>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.

]]>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.

]]>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 ]]>

Here is a link to the comment: http://commoncoretools.me/forums/topic/algorithms-grades-2-5/#post-939. 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.

]]>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.

]]>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.

]]>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.

]]>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?

Thanks

]]>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.

]]>I’m not sure how the standard points to physical situations, could you explain?

]]>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?

Andy

]]>… 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 4 years ago by ivan.

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?

]]>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?

]]>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 (http://www.mathedleadership.org/docs/resources/journals/NCSMJournal_ST_Algorithms_Fuson_Beckmann.pdf) 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.

]]>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.)

]]>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.

]]>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?

]]>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. ]]>

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!

]]>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.

and

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.

]]>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.

]]>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.

]]>]]>

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.

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?

]]>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?

]]>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.

]]>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?

]]>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, 1 month ago by Jason Zimba. Reason: Forgot to tag it the first time
- This topic was modified 4 years, 1 month ago by Jason Zimba.
- This topic was modified 4 years, 1 month ago by Jason Zimba.
- This topic was modified 4 years, 1 month ago by Jason Zimba.

Will you comment on this omission?

]]>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.

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.

]]>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,

Silas

then surface area is easily connected to those.

]]>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. ]]>

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.

]]>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.

]]>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.

]]>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.

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.

]]>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?

]]>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.

]]>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. ]]>

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

References

Arizona – http://www.azed.gov/standards-practices/mathematics-standards/

Colorado – http://www.cde.state.co.us/CoMath/StateStandards.asp

Kentucky – http://education.ky.gov/curriculum/math/pages/mathematics-deconstructed-standards.aspx

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.

]]>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. ]]>

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.

]]>Xyzzy

]]>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.

]]>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.

]]>(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.

]]>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.

]]>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.

]]>$$

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 $xy + y = (x+1)y$. ]]>

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.

]]>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?

]]>Thank you again. I feel like a real pain about this, but I do want to help these teachers be less anxious. Kathleen

]]>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.

]]>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.

]]>$$

\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.

]]>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.

]]>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.

]]>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…

]]>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.

]]>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.

]]>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)

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?

]]>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.

]]>Thanks, y’all.

]]>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 (http://lipingma.net/math/One-place-number-addition-and-subtraction-Ma-Draft-2011.pdf) might be helpful in thinking about how to introduce it for multiplication.

]]>http://www.mathedleadership.org/docs/resources/journals/NCSMJournal_ST_Algorithms_Fuson_Beckmann.pdf

In my opinion, we need to continue to give examples of area models for smaller numbers with each block shown to maintain that connection. ]]>

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).

]]>“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.

6.NS.5:

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.

6.NS.6:

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.

]]>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.

]]>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!

Sincerely,

Kimberly

]]>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.”

]]>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?

]]>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?

]]>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.

]]>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?

Thanks,

PaulM

]]>We also had a question about the use of the word “constructing” in this standard. Are students required to construct triangles using a compass?

]]>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.

]]>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).

]]>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.)

]]>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? **

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?

]]>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?

(http://www.corestandards.org/Math/Content/5/MD)

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?

]]>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? ]]>

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 🙂

]]>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. ]]>

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?

]]>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.

]]>Finally, I realized that what was missing was the indication that

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,

Joanna

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? ]]>

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.

]]>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 http://www.girlsangle.org/page/videos.html. 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.”

]]>A general principle of the Standards is described in the Algebra Progression, p. 4, http://ime.math.arizona.edu/progressions/:

“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: http://commoncoretools.me/forums/topic/a-rei-5-what-does-it-meanlook-like/. I’ll try to contribute more to that thread.

]]>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.

]]>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 illustrativemathematics.org in their daily lessons. Thank you for these excellent problems! They are awesome.

]]>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).

]]>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.

]]>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? ]]>

“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? ]]>

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.

]]>I don’t know if you’ve looked at the Progressions, but you might check out the K–6 Geometry Progression, http://ime.math.arizona.edu/progressions/

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.

]]>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

]]>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).

]]>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.

]]>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!

]]>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?

]]>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?

]]>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

]]>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

]]>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.

]]>

To the CCSS-M folks, Are there future plans to include a standard for learning the meaning of ≤ and ≥, in a subsequent grade level?

]]>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?

]]>

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.

]]>Thanks, Malissa Jacks

]]>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?

]]>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?

‘multi-step’ is sometimes hyphenated and sometimes not.

(I noticed because I was searching for ‘multi-step’ and only found some of the references).

]]>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.

]]>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.

]]>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.

]]>

>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.)

]]>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!

]]>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.

]]>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).

]]>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. ]]>

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. 🙂

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