In making these distinctions, I think it isn’t so much the parts of speech used in the standards we should attend to as it is the underlying mathematics and student actions. In your 3.NF.3 example, there is a noun there implicitly, namely a preferred proof that fractions are equivalent. (Thinking of proofs as nouns is part of the trouble with this dichotomy; the “verby” language of the standard is perfectly natural.) The verb “explain”, by the way, is essentially being borrowed from MP3.

[By the way, I happen to be going over this material at the moment with some pre-service teachers, and they came to agree that the choice of argument based on reasoning about sizes (in particular sizes on the number line) in the CCSS is preferable to more formal or algorithmic approaches. Just thought that you'd appreciate the vote of confidence in the choices you made from a room full of undergrads ; ) ]

Sometimes it is also important to attend to their other parts of speech. For example, in MP.5, “Use appropriate tools strategically,” the adjective “appropriate” and the adverb “strategically” are both crucial.

I do think it is true that the nouns in the practice standards differ in kind from the nouns in the content standards. The nouns in the practice standards are these: problems, arguments, reasoning, mathematics, tools, precision, structure, regularity, and repeated reasoning. These are not what we think of as content areas or “topics.” Whereas, the nouns in the content standards generally *are* what we think of as “topics.”

However, although it is true that “content standards provide nouns,” it is important to observe that the content standards also provide verbs (such as “understand”) and adverbs (such as “fluently”) that are essential to the expectation in question.

It may be true that people “usually pay attention to the nouns in content standards,” but I wouldn’t want to give a pass to this predilection. Consider 3.NF.3 for example, which says, “Explain equivalence of fractions, and compare fractions by reasoning about their size.” Virtually nothing about this expectation is captured by the noun “fractions,” or by the noun phrase “fraction equivalence.” The verb “explain” is clearly essential. Likewise essential is the adverbial phrase “by reasoning about their size” – without this, one might imagine that students were expected only to use algorithms based on numerators and/or denominators.

In case helpful or interesting, some related examples can be found here: http://www.achievethecore.org/downloads/New%20Twists%20on%20an%20Old%20Standard.docx.

But, perhaps we’ve exhausted this discussion. Go forth and $B$ whatever you want to $b$!

“The volume A in cubic inches is given by A=Fd, where F is the area of the base in square inches and d is the height in inches.”?

My name is Matt Friedman and I am an editor in Scholastic’s Classroom Magazines’ division. I am always curious to hear what resources teachers find helpful, and I wondered if you gotten to look at the book Bill suggested below. Could you you know if you found the book to be useful or if you found other resources that were out all in planning the course you discussed above?

Also, what university do you work for? The course you mentioned sounds very useful— I’m interested to learn more about it. Any information you can provide would be appreciated.

Thanks so much.

Best,
Matt Friedman
mfriedman at scholastic dot com

High School Mathematics Scope and Sequence (Draft) 2012.
Made possible by grants from the Pearson Foundation and the Bill and Melinda Gates Foundation.

But, there are also other kinds of meaning that we give to mathematical expressions. You might call them conventions. I think conventions can make mathematics easier to understand. If you consistently use B to represent the area of the base, then you can take shortcuts when you are communicating. It can make communication more efficient and also more effective. Of course, the person you’re communicating with needs to know the conventions, and you need to know that they know the conventions. Still, conventions should be broken sometimes. We learn something by breaking a new trail. And we need to make sure students know that they can break new trails, too.

Your point that “Naked formulas…mean nothing by themselves without surrounding words” applies strongly to science, and to the learning of it. Each symbol in a formula like “F = ma” has a detailed meaning that must be understood in order to apply the formula correctly. For example, “F” in the formula refers not simply to “force,” but specifically to the net, external force on a system. (Often when I’ve taught the Second Law, I’ve taken the trouble to carry around a lot of subscripts, as in “F_{net, ext} = m_{tot}a_{cm}.”)

We are looking for a suitable book for the course. The problem I am having is the currency of these type books. We would like a CCSS perspective hence a Copyright after 2011. Any suggestions?

Thank you!

All the progressions are on a page of the Institute for Mathematics and Education (IM&E) website.

Happy Reading!

* conceptual understanding
* challenge problem
* fluency
* formative assessment
* math game
* literature based
* machine scorable
* meaningful application
* number lines
* problem solving
* procedural knowledge
* professional development
* transformations
* summative assessment
* video
* MP 1
* MP 2
* MP 3
* MP 4
* MP 5
* MP 6
* MP 7
* MP 8

I’m an Aussie like you! I’m writing Maths assessments linked to the CCSS for grades 9-12. Can you recommend a scope and sequence that I can use to write my assessments? I have found some for grades 9-11, but none for 12. Can you assist?

Many thanks!

Additionally, we are charged with developing a pre-test for the course. I am using information from the Pathway documents as a rationale for which standards will be included in the pre-test.

In addition to a text on modeling it would be a good thing if publishers paid attention and began meaningful work on the development of a textbook aligned to any of the pathways detailed in Appendix A.

I was recommended this website, which may be helpful to you. Here is the link. http://ccsstoolbox.agilemind.com/pdf/Grade%205%20Dana%20Center%20Scope%20and%20Sequence.pdf

It is through the Dana Center. I hope this helps and tell me what you think. JT

http://www.edweek.org/ew/articles/2012/09/12/03newell.h32.html?tkn=PMLFC5GNUDFOdOz0n4CeqgjQd4q1Rr6nGZk8&cmp=clp-edweek

http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/feed/

will give you a feed of all the old comments on the General Questions post.

I also am wondering how teachers are supposed to implement the standards with curriculum that, as you say, “which supports learning the knowledge described by the standards” when no such curriculum seems to exist. Right now the options are to search through our current curriculum to figure out what matches the standards or to create it completely on our own. Both are a lot of work and imperfect as they rely on our correct interpretation of the standards.

Sorry that can’t frame all this into a specific question, but any suggestions or resources you can give me would be greatly appreciated.
Thank you, Alice

I’ve been poking around to find a plugin that will manage this better, but haven’t found one yet.

I don’t think I know what you mean by functional assessments; are they the same as formative assessments? I certainly think the latter are important.

As for your last question, which parts of the Common Core did you have in mind when you talk about “so much rigor”? I have talked to elementary teachers who believe it’s possible to push students to learn without pushing them away, but I agree it’s a skill we need to make sure teachers have.

I haven’t read the progressions in a great deal of depth, but I did read looking for a couple particular things in 7th grade as they came up in a workshop I was doing.

Standard 7.SP.3 “Informally assess the degree of visual overlap of two numerical
data distributions with similar variabilities, measuring the difference
between the centers by expressing it as a multiple of a measure of
variability.” will be difficult for teachers to understand. And once it’s explained I think they’ll have trouble understanding the point. I suggest that this should be expressed directly in the progression doc. Specifically an example showing the same different in means with different spreads. Maybe the larger spread example could be two samples from the same population, for example. Then keep the means the same but show a small spread, and clarify how using the measure of variability as your gauge is a way of assessing overlap. The idea is somewhat discussed, but I don’t think this important point is made clear.

You are obviously putting quite a bit of time and effort into this forum which is greatly appreciated. As a parent of three elementary age children, I am pleased to see what the new common core standards may have to offer. (Hopefully, the chance for kids to be kids for one) One concern I do have with the 2012-2013 school year is in regards to our county’s transition plan. For grades 3-5 there will be “full implementation of the common core with elements of the state curriculum infused”. Do you have any thoughts on this approach.

I loved your statement regarding teaching things such as skip counting for skip counting sake. With second grade twins last year, skip counting was not only taught as an indicator but was also continuously “shoved” into the curriculum throughout the year and regularly appeared on summative assessments. As parents, it was clear that teaching this for teaching sake only confused both children causing them at times difficulty in counting by ones. An example of too much information for their seven year old brains (a mile wide but only an inch thick). I was thrilled to hear Mr Daro discuss “less is more”.

I also wonder if the new common core standards will help teachers understand the true meaning of functional assessments? Teachers in our area use the term but continue to grade everything in a summative fashion. They put quizzes and tests into the summative category and even though everything else is still graded with a summative 0-100 grading scale, they call it functional. Will the common core standards address this at all?

Finally, although I understand the need for a focus on the STEM program at some point, is it necessary to introduce so much rigor at such an early age when children are developing and even still transitioning in to a full day of school? I have no issue with rigor and challenge but I also believe that the way it is being introduced to such young children will only push them away from the love they should have for school in their early years. I believe that intelligence is innate but not fixed. Whether we introduce certain challenges to our children at six years of age or at twelve, will not change their outcome. It will however, let the six year old be a six year old for the time being.

I realize this is a great deal of info but would appreciate any thoughts you might have to offer.

Basically, the standards are not units of instruction; you don’t always “teach a standard” in one chunk, whatever the order. For example, the OA and NBT standards in any given great level are very closely related, and a curriculum might be touching on these two domains simultaneously at times, not to mention supporting standards in MD and other domains. The standards describe achievements we want students to have. As my colleague Jason Zimba likes to say, you don’t teach standards, you teach math.

Similarly, page 9 provides an example of converting a mixed number to a decimal fraction, and converting 2.70 to 2.7. Is this beyond what students should be doing? Should conversions be kept to amounts less than 1?

I was confused by your MCC6.EE.8. It seems to say that students should solve inequalities of the form px + q < r, which is not in the Grade 6 standards. Maybe it is a cut and paste error or maybe that is a place where your state added something to the standards (but did they really add these sorts of inequalities without also increasing the demand on equations?).

The “Grade 5″ at the top of page 6 is correct. The idea is to point out that this common justification of the rule about equivalent fractions doesn’t really make sense until Grade 5, when students start to multiply fractions by fractions.

6.G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

A trapezoid counts as a special quadrilateral. Students who have calculated a few areas this way might begin to be able to see why the formula is true.

There’s no reason why pyramids couldn’t be included in Grades 7 or 8, although as you point out they aren’t explicitly required. Note that the “formula” is the same for both pyramids and cones, if you take it to be Volume = (1/2) x Area of Base x Height.

In high school, G-GMD.1, students are expected to understand at least informally why this is true. This line of reasoning actually starts with pyramids.

I know teachers in our area are concerned about the de-emphasis on time and money in 2nd grade as compared to our state standards. I used to use money as part of my classroom management plan and it worked really well. It has the bonus of being able to focus on money year-round as well as using it in an authentic context.

What I describe was used with third graders, but it could easily be adapted to 2nd.

I gave each student a bank account and a checkbook. I initially used fake money, but there ended up being a rash of thefts. I liked the checks because they had to use the written form of the number.

Each student was paid for being a student, we’ll say $\$100$/wk. There were performance perks – turning in all homework on time all week might have gotten the child a $\$10$ bonus. There were fines for rule infractions; if they forgot supplies they purchased them from me. Occasionally I’d set up a store for them to shop for treats. We also had days in which they could set up their own store and bring in toys to sell to each other. Class rewards were based on their balance since that was a good indicator of behavior and responsibility e.g. Everyone who had a balance of $54 or greater could have lunch in the classroom.

Logistics – I asked my bank to donate some blank counter checks. You only need a few to copy. I kept the checks in a common pile because having each student keep up with their personal checkbook proved too complicated. They did have to keep up with their own register, though.

The system worked really well and provided lots of opportunity for math throughout the day. It also allows them to “discover” the concept of negative numbers as their accounts went into the red.

Karen

I would point out that there is no standard requiring students to be able to state the properties of operations. However, they should be able to explain their strategies, and this will inevitably involve talking about the way operations work.

In the standards, I do not see where the 6th grade student is asked to specifically solve an inequality as they are asked to solve an equation.
The standard is specific about the type of equation the students should solve but does not indicate that the student has to solve an inequality.
My understanding of the standards is that students are being asked to:

* Identify from a given set whether a value is a solution of an
inequality. They are using substitution to find solutions of
inequality.
* Solve one-step equations
* Write inequalities given a specific situation
* Recognize that inequalities can have an infinitely number of solutions.
* Graph solution sets of inequalities on a number line.

6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

MCC6.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.

MCC6.EE.8. Write an inequality of the form x > c or x c or x r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $\$50$ per week plus $\$3$ per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Is it the case that the example in the Progressions document is *not* one to use with students but is just meant to illustrate to teachers what is required? Or is it expected that students will need to tackle denominators outside the stipulated range for Grade 4?

Also, at the top of p.6 of the Progressions document on fractions is the third line down meant to begin “Grade 5 students…” or “Grade 4 students…”? Thanks!

P.S. Thanks Karen G (July 23) for the response about improvised units!

I am from Washington state, and we are beginning to roll out the CCSS standards this year for K-2. I currently teach second grade and am on a team that is working on scope and sequence along with benchmarking in needed areas. We are a fairly low income area, and knowledge of money is not something that a vast majority of our students have knowledge of or get practice with outside of our classroom. With money being taught in second grade, we take it very seriously since it is such a life skill. My question may be a stupid question, but when looking at 2.MD.8, working with time and money, it states that students need to work with dollars and cents.

I am assuming that when the standard speaks of “dollars” that we are including $1.00, $5.00, $10.00, $20.00, $50.00, and $100.00 bills? I saw the post earlier about teaching the dollars in whole amounts in conjunction with whole number addition and subtraction. And that makes total sense to me.

Our former state standard only had students making amounts with coins up to $1.00, so this is going to be a big transition for our teachers (and students). I will be going in to a meeting to train teachers who are new to standards based grading in August, and I would like to be able to answer questions about the standards as well as formative and summative assessments.

I really appreciate the forum and the outlet in which to get some clarity. I am excited about working with the new standards, but want to make sure that I am presenting things accurately with my colleagues, and most importantly, my students.

Best Regards,
Amanda

5.MD.1 appears to have students convert larger to smaller but also smaller to larger.

Is that a correct interpretation?

Also, volume of most 3-d figures is developed as well but pyramids are missing. Do they belong with prisms in grade 7 or with cylinders, cones, and spheres in grade 8?

Thanks

Not an official answer, of course, but it might give you some ideas of avenues to take.
Karen

In this example, should “and/or” be interpreted as meaning that any of “place value”, “properties of operations”, and “the relationship between addition and subtraction” should only be used at any one time but all should be covered by the end of Grade 3? Or that you could use any combination of the three at one time? Or that it is okay if only one of them is covered by the end of Grade 3? And what would guide these decisions for teachers? Any of the approaches in 3.NBT.2 potentially could be used – I’m not sure if the expectation is to just use one or all of them.

On a different topic, 3.MD.6 asks students to measure area by counting unit squares and improvised units. What would improvised units include and why should students use them?

Thanks for your help with the first question – I had misinterpreted that due to the second sentence of 4.G.2, so I am glad the Progression and this blog exist to clarify!

To be clear, classifying triangles by side lengths would be a “natural extension,” but not required by the standard (with exception to “equiangular,” for the reason you state)?

Brian

As for density, I guess I would say the concept and the formula are almost identical; for example, knowing the concept of (average) population density entails knowing that you would calculate it by dividing the population in an area by the area. In some sense it doesn’t matter whether the formula is given or not, because the student who knows how to make use of it won’t need it anyway.

Question on 4.G.2:

The sentence “Classify two-dimensional figures based on the presence of absence of… angles of a specified size” leads me to believe that students need to be able to identify acute, right, and obtuse triangles. However, the sentence immediately following it, “Recognize right triangles as a category, and identify right triangles” leads me to believe that students do not need to be able to recognize/identify acute or obtuse triangles, only right triangles. The first paragraph of the grade 4 section of the Progression (p. 14) supports the first interpretation. Is that the intent? If so, what is the intent of the second sentence, which wouldn’t seem to say anything that the first sentence doesn’t?

The same paragraph of the progression interprets the same standard to also state that “[Students] can use side length to classify triangles as equilateral, equiangular, isosceles, or scalene…” I cannot find a way to interpret this standard that would include that statement. Is this actually “required” by the standard, or just a natural “extension” that is “not forbidden”?

Thanks,
Brian

http://commoncore.greenwich.wikispaces.net/math+resources

Their work states that is was adapted from the Arizona Dept of Education

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

6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

You might use examples of both concave and convex polygons to illustrate different ways of calculating the area, by either decomposing it or viewing it as obtained by subtracting an area from a larger figure. But you might not spend too much time on the terms themselves, and the standards do not require that students know them.

6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

I could imagine a curriculum pursuing this as far as solving single step inequalities as you suggest, but that’s not required by the standards.

http://www.dpi.state.nc.us/docs/acre/standards/common-core-tools/unpacking/math/statistics-probability.pdf

http://commoncoretools.me/wp-content/uploads/2012/04/ccss_progression_sp_hs_2012_04_21.pdf

In these discussions, I have suggested that teachers refer to other standards within their grade level to determine what is appropriate for their students. For example, describing a polygon as “regular” would come after understanding what it means for sides to be congruent and for angles to be congruent.

At what level do you see these ideas being appropriate based on the standards?

Julie

In Grade 6 of the EE progression document, under “Reason about and solve one-variable equations and inequalities”, very little is said about inequalities. The sentence “In Grade 6 they start the systematic study of equations and inequalities and methods of solving them” suggests that students might begin solving simple inequalities in Grade 6, such as inequalities of the form x + p > q or px > q for cases in which p, q, and x are all non-negative rational numbers. (This would parallel with the work done with equations (6.EE.7).) However, the only other mention of inequalities in Grade 6 of the EE progression document is in reference to inequalities of the form n > 0 used to represent a domain for a real-world situation. Is the expectation that solving inequalities will not begin until Grade 7 with two-step inequalities?

Thanks,
Peggy

“Attributes” and “features” are used interchangeably to indicate any characteristic of a shape, including properties, and other defining characteristics (e.g., straight sides) and nondefining characteristics (e.g., “right-side up”).

Page 8 talks about:
differentiate between geometrically defining attributes (e.g., “hexagons have six straight sides”) and nondefining attributes (e.g., color, overall size, or orientation).

Do these quotes help to clear things up?

Thank you for your explanation. This helps me recover a bit from the shock I suffered when reading the geometry progression – which quickly became filled with highlighting and question marks on statements that I though went beyond the grade-level standards.

In a related response you left to a question in the Progression for SP, you mentioned what has sort of become your signature catchphrase – “that which is not mentioned in the standards is not thereby forbidden,” and added a new twist – “that which is mentioned in the progressions is not thereby required.” Both are great and make perfect sense. My concern is that, as a reader of the standards and of the Progressions, I can’t always discern for myself what is “required” and what is simply “not forbidden” without asking… which doesn’t lend itself to “common” understandings of the standards.

I know the people working on these Progressions are some of the busiest among us, but I would like to offer two possible ideas for embedding this clarity in future iterations of the Progressions. Either:
• organize each grade level’s narrative into two sections: 1) required by the standards at this grade level, and 2) related to the standards at this grade level; or
• add those two categories as footnote tags and simply superscript a 1 or a 2 after each paragraph or sentence that needs to be clarified such that the field understands the intent and the scope of the particular standard.

I realize this is easier said than done, but it is certainly a much more doable way of achieving the clarity sought by some of your frequent posters (ex., Lane, Turtle, Jessica, and me) to increase the likelihood that curriculum writers, test writers, and teachers share a “common” understanding of what needs to be taught at each grade.

Thanks for all of your time and support,
Brian

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.

The progressions documents offer more guidance about sequencing, but even they don’t get at how domains might be intertwined. For example, in elementary school it would make sense to treat the Operations and Algebra Thinking and the Number and Operations in Base Ten domains in parallel, catching the many connections between them, rather than in sequence.

So, I don’t have a simple answer to your question. Ultimately sequencing is a matter of curriculum design, which take time if done well. All I can say is that the progressions documents are designed to help in this endeavor.

The standards describe the achievements we want for students. These are not limiting, but describe where time should be focused if it is short.

Also, you have to be careful about counting roots, since a polynomial can have a double root. One way to state the theorem is that every non-constant polynomial has a root in the complex numbers. This turns out to be equivalent to saying that a polynomial of degree n has n complex roots (where doubles roots are counted as 2, triples roots as 3 and so on).

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.

So there is a distinction between what might appear in courses for all students and what might appear on the assessment. My guess is that (+) standards will not show up on the assessments, but we will have to wait and see.

The progressions describes a lot of things that teachers might do in the classroom; the standards describe the key mathematical achievements that should result from this. For the core progressions in K–5 there’s not much difference between the two, but for the progressions in geometry and in measurement and data the difference is bound to be greater, since the standards in those progressions were intentionally restricted to a thin stream in order to make room for the core focus. In practical terms, this means that curriculum materials for geometry might well have students engaging in some activities which are not mentioned in the standards, but not thereby forbidden. Talking about 7-sided figures might be such an activity. In some classes there will be time for this, in others there won’t. Classes that don’t have the time will not be penalized in assessments for focusing on the core progressions in number and operations.

For your question about attributes, I’m going to ask Doug Clements to see if he can find the time to clarify (but he’s busy, so don’t hold your breath!).

Also, while I was working on the Kindergarten standards yesterday, I have the same questions Brian has above. Maybe we need to place these questions in the “General Questions about the Mathematics Standards” section?

Thanks,
Jessica

1) At the time, I did actually make a high school section of the diagram. It was based on an old guess that I’d made about how high school courses might look based on CCSSM (n.b., only for the traditional sequence). That guess went beyond an analysis of “which standards go into which course”; it took the form of some coherent chunks of material derived from the expectations in the standards. As a result, the entities in the high school portion of the diagram had novel codes, like “A1.2.4.1.” That made interpreting the high school portion of the diagram pretty hard, so for that and other reasons I only gave K-8. If you’re interested in the high school portion, you might want to contact Markus Iseli at UCLA, who has a copy of the high school portion of the diagram as well as a correspondence table between the codes in the diagram and the standards in CCSSM. I agree it would be interesting to see what the diagram shows as the “roots” of the high school standards in K-8, if somebody wants to chase that down and put it in a digestible form.

2) I guess don’t have much that I can add as to why some standards have those notes and others don’t. It’s just that in some cases, I perceived important “way-stations” that students might typically land on between the beginning of the grade and meeting the standard as written. Perhaps if these cases were extracted and put in their own list, generalities might emerge. For example, in standards that set expectations for fluency, it seemed prudent and natural to create way-stations along the way to fluency. Likewise for word problems, I made a way-station in Group A for easier types. So I suppose it is partly a progression of difficulty, and partly a signal that these are not the kinds of standards that are taught all at once, or met all at once. They are about sustained work. In other cases, it is just a matter of splitting out the key parts of “composite standards” (e.g., 3.MD.2) and putting first the parts first that seem logically or conceptually prior.

3) No significant reason for the difference. As you can imagine, making this diagram took some time…. As I moved across the grades from left to right, my visual and design conventions tended to evolve somewhat, and there wasn’t always time to refresh the entire diagram. So there are some inconsistencies of representation here and there.

Another re-post of a lingering question:

Some of the box plots (bottom of p. 5 and middle of p. 6) include outliers dealt with by disconnecting those points from the whisker. Do sixth grade students need to learn the very arbitrary “1.5 times the IQR above the upper median” rule for determining whether a data point is far enough to be considered an outlier?

Teaching this convention to sixth graders who are just being introduced to this sort of graph seems far too detailed and isolated from the larger focus of the data standards (and the focus of 6th grade, in general) to justify the time and confusion!

The standard itself does not seem to require nor forbid this, but including it in this Progression seems to say that this convention would be fair-game on grade 6 assessments. Please let me know if that was the intent or not! If it wasn’t, can a note be included in the Progression stating “the standards at grade 6 do not require nor forbid instruction focused on the ’1.5 times the IQR above the upper median’ rule for determining outliers. This should not be a target on assessments.” Without a note like that, teachers will have to spend time, which should be focused on the RP and EE standards instead, teaching this convention to mastery just in case it is ever tested.

Thanks,
Brian

I’m resubmitting an old question that went unanswered, but came back up during curriculum work:

On p. 6 of this progression, the formula for the surface area of cubes is used to illustrate 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers). Am I wrong to interpret that grade 6 work with surface area is limited to using nets (6.G.4) and grade 7 may take it to the level of algebraic generalization (7.G.6)?

Thanks,
Brian

First – THANK YOU to you and the whole team who worked to put this together for us!

Here’s the first major question this progression has raised for me and my understanding of the standards:

“Students learn to name and describe the defining attributes of categories of two-dimensional shapes, including circles, triangles, squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral. They describe pentagons, hexagons, septagons, octagons, and other polygons by the number of sides, for example, describing a septagon as either a “seven-gon” or simply “seven-sided shape” (MP2).2.G.1″ (page 10, first paragraph).

This description really expands the shapes listed in the standard itself (triangles, quadrilaterals, pentagons, hexagons, and cubes). Those shapes are all families named specifically by their numer of sides (of faces for a cube)… and the list does not include “septagons, octagons, and other polygons by the number of sides,” though those at least seem to follow what I thought the intent of the standard to be.

More problematic than this slight expansion of the standard is the preceding sentence that now asks students to “describe the defining attributes of categories of two-dimensional shapes, including circles, triangles, squares, rectangles, rhombuses, trapezoids, and the general category of quadrilateral.” This MUST be an accident? That has to be reserved for 4.G.2 and 5.G.3 & 4, right? Second grade students haven’t yet learned anything about about angles or parallel in order to be able to “describe the defining attributes”! And the defining attributes of circles?! Possible with sixth graders… but the CCSS doesn’t even deal with circles until 7th grade.

Does this paragraph in the Progression accurately describe the expectations around 2.G.1, or was this paragraph misplaced?

Thanks,
Brian

This week I am teaching a Grade 3 – 5 CCSSM course, and Geometry and Measurement will be the focus on Friday, so your timing is impeccable.

Thank you!

Could you share your thoughts on the expectation of this standard? To “know” the theorem, is that to simply be able to state it? I have seen it written slightly differently – but is it too simplistic to know that a polynomial has the same number of roots as its degree? And what would constitute “showing” that it is true for quadratic polynomials? Thanks!

I have read the interpretation of the (+) symbol in the actual standards document, but of course, assessments are not addressed there. My real question centers on the inclusion of some (+) standards in the Pathways (six of them by my count). This is a bit ambiguous, as they seem to be standards for both all students and for students preparing for advanced topics (doesn’t “all students” imply that already?). Thus, if they are for all students, wouldn’t it stand to reason that they could potentially be included on a high stakes assessment and the app’s note is misleading?

I know you have often directed questions about resources to the creators of said resources. So, perhaps I should ask instead for you to comment on your own beliefs/insights about the (+) which are included in the Pathways and the possibility that all students could possibly be assessed on these “advanced” topics. Thanks!

Most versions of your diagram I have seen end at 8th grade. But I am recalling seeing one version that “ended” at 8th grade, but had the beginning of the “next page” and I thought I could see what appeared to be a high school section of the chart. I am very interested in your take on the 9-12 standards and how you see them connecting to the K-8 standards. Is this something you would be willing to share as well?

Additionally, a few standards have a note above them. Could you share your rationale as to why it was important enough to include some small detail at times? Why those standards in particular?

And finally, the K and 8th grade standards have a stand-alone standard not connected to anything, while the 3rd grade standards have a “Not shown: 3.MD:1″ footnote rather than including it similar to K and 8th. Is there any significant reason for the difference?

Thanks for your time.
Fred

Would you elaborate more on A-REI.4a, specifically, on what would be expected to derive the quadratic formula from this form (x – p)^2 = q. In Appendix A, this standard is included in the Integrated Pathway, Math II, Unit 3 (probably 1st semester). Perhaps you could offer an example assessment question/activity that would get to the heart of this standard? Are p and q to be considered variables or constants? Any help you can give is greatly appreciated!

Help me to understand how much latitude instructors should take with the sequence of the CCSS standards. I’m in NC and we are “rolling out” these standards with the so-called Essential Standards in other subject areas. I am not a mathematician or even a teacher of math. However, as an instructional leader (principal), I am endeavoring to understand as much as I can about the standards. (And math was my favorite subject in school)

For years I’ve have believed that we attempted to cover too many topics within our math curricula. When I read William Schmidt’s paper, A Coherent Curriculum I was relieved to find out that people a lot more learned than I saw what I considered to be part of the problem.

Some of the progressions are obvious, but some are not.

Fortunately, NC has developed some “unpacking” documents which are helpful in preparation for implementation. Help me with understanding what should and should not be done with these standards where the importance of sequence may not be so obvious.

For example, one can see that the standards belonging to major clusters often exhibit long or complex chains of arrows in the diagram, and this would tend to push the beginnings of those chains all the way back into Group A. And the endpoints of those chains are also going to belong to major clusters, so major work will often stretch into Group C as well. Meanwhile, additional work often lacks those long or complex chains of arrows, so it might not end up in Group A very often, or maybe even Group B. So these circumstances might lead to some patterns, but as I say, there wasn’t any mechanistic rule determining these things in the diagram.

After reading earlier posts about “simplifying” and the general form for addition of fractions, I’m now wondering if “factoring” would also be driven by context. The correct multiple choice response to that question has kids factor using the GCF of 2/3, but 1/3(4x + 14) or 2((2/3)x + 2 1/3) would be factors of the expression too. Then I began to think about the language being used in the question versus the language being used in the rationale. Which expression is equivalent doesn’t really ask students to factor. They can find the correct answer by applying the distributive property to each of the answers looking for the one that matches. I guess if I’m doing my job right and helping kids to see things in context, then my students would understand equivalent and be able to use an appropriate strategy. I think I was trying to factor it because I saw that all the answers were in the form a(bx + c) and I was programmed to find the GCF.

New question… Is 7.EE.1 a call for students to be flexible in their thinking? If the item writers weren’t so focused on “factoring” could that question have been changed to:
Which expression is not equivalent to (4/3)x + 4 2/3?
A (4/3)x + 4 + 2/3
B (4/3)x + 2(2 1/3)
C (4/3)(x + 2)
D (2/3)(2x + 7)
which would assess student’s ability to add, subtract, multiply, factor and/or expand a linear expression?

I’m very concerned though about the way that teachers across the country are going to be interpreting each standard and/or reacting to other’s (State Education Departments…) interpretations. The progression documents are helpful, but when State Ed contradicts the progression document, what do you do? I’ve been thinking and reacting to this one question for a while and I stumbled across this thread which helped me to process it further. I’m overwhelmed to the point of shutdown by the prospect of writing a years worth of curriculum this summer and feel as though I don’t have enough coherent resources to help me.

Can’t you get a clear interpretation of relative frequency from (nearly) any measure of spread and any known distribution? For example, with a uniform distribution, you can tell the fraction of the cases that are within 1 standard deviation of the mean. With a normal distribution, you can tell the fraction of cases that are within 1 MAD of the mean.
qsareweirdos = Ken

The documents are open for public comment now.
http://www.p12.nysed.gov/apda/common-core-sample-questions/

Thanks so much. I truly appreciate the difference you are making.

Best,
Turtle

When it says “represent these problems using equations,” does this mean that the students need to formally solve an equation to solve the problem?

As to this specific question, I’m not sure if those groupings correspond to fixed periods of time such as trimesters. For example, it seems to me that group A is generally, in some sense, “less time” than group C. But I would leave such judgments to others…and stress that different curriculum authors might choose to sequence things differently.

Your interpretation is correct. SMP 4 is as you describe, and is also as described on pages 72–73 of the standards, at least in high school. The phrase “mathematical modeling” refers to this as well.

However, in the elementary standards, there are standards like:

1.OA.1 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-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.

I think this is the source of the confusion, because it seems to be going the other way, using a physical situation to model a mathematical idea. Personally, I think the underlying process is quite similar; you are taking some problem you don’t know how to solve, modeling it with things you can manipulate (in this case, base ten blocks, later on, algebraic symbols), and using them to solve the problem. True, a two-digit addition is not a contextual problem, although in elementary school it will often arise out of one.

Sorry I don’t have time right now to respond at greater length; thanks Lane for joining in!

As for proportions, the emphasis in the standards is on understanding proportional relationships and using them to solve problems. Students know that proportional relationships are sets of equivalent ratios, and that equivalent ratios have the same unit rate. The cross-multiplying method is a consequence of this, but “setting-up-and-solving-proportions-by-cross-multiplying” is not a topic in itself, but rather a method that arises out of understanding proportional relationships. The discussion on the second half of page 9 is an attempt to explain this, but maybe it needs to be fleshed out.

G.CO.6 is more precise, and asks students to work with the precise definition of different transformations. Here one might or might not introduce some notation for different types of transformations; that’s really a curricular decision. My preference would be to avoid it for as long as possible and ask student to work directly with the descriptions. I’ve seen horrible exercises in textbooks about this.

Rigid motion. A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures.

So it is taken as an axiom that translations, reflections, and rotations preserve distance and angle, although students are asked in Grade 8 to verify this experimentally (8.G.1).

With these definitions and assumptions, it is possible to prove that two triangles have congruent corresponding sides and angles then there is a rigid motion taking one to the other. First you translate one of the triangles so a pair of corresponding vertices coincides, then you you rotate so that a pair of corresponding sides from that vertex coincides, and possibly reflect so that the other pair of sides lies on the same side of the first one. From this point on, using that angles and distances have been preserved, you can reason that the triangles coincide.

Yes, I know, we should have the geometry progression out where all this is explained with diagrams. Sigh.

However, there is one standard that refers to coordinates, and I assume this is the one you are talking about:

8.G.4. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Here I think it is reasonable to suppose that the problem is limited to ones students can do with the algebraic tools at their disposal. That would include your (2) and (3), but it wouldn’t even include all rotations about the origin, since you need trigonometry for that. It could include rotations through 90 degrees, and I can imagine a few more transformations that could be used in instruction as challenge problems of an exploratory nature. For example, you could have students figure out that a dilation from a center other than the origin can be achieved by first translating the center to the origin, dilating there, and then translating back again. But I would hate to see this turned into some formulas to be memorized on a test.

Several of the California Mathematics Project (CMP) sites are holding institutes on mathematical modeling. At one of the sites I’m visiting, the leaders asked me about the difference between mathematical modeling and the Standard for Mathematical Practice (SMP) “Model with Mathematics.” Some teachers think these are different concepts, and I’ve been told different things.

This is my perspective, and I want to know if this is the intent of the CCSS. Mathematical modeling is similar to the types of problems that CoMap has produced. The SMP model with mathematics is also connected to that definition. If one has a contextual problem, then model with mathematics is using mathematics to solve the problem, using mathematics as the model to solve the problem, then going back to the context. What it is not is using physical models to represent the problem.

I’ve heard in conferences the latter for model with mathematics–using physical models to represent the problem. Could you help me understand the difference between mathematical modeling and model with mathematics? Thanks.

Susie

Any dividend less than 100 is acceptable for 3rd grade division????

A pilot asked me to explain a formula he learned in flight school. His teachers there could not explain it. Here is the formula, with revised terminology:

A circular arc, radius r miles, angle A degrees, has length of arc approximately rA/60 miles.

I explained that the exact formula is rA* Pi/180 . (The fraction Pi/180 converts degree measure to radian measure.) Since Pi/180 is approximately 3/180 = 1/60, the flight school formula works. I asked him why not use the exact formula with a calculator. He laughed and said that the pilot has both hands on the controls; the calculation has to be mental. Here is an example of such a calculation. Suppose the arc has radius 80 miles, angle 70 degrees. The length of arc is approximately 80 * 70/60 = 80 * 7/6 = 40 \times 7/3 = 280/3 = 93 1/3 miles.
Calculating without simplification, the result is 5600/60 miles, a result that might not help the pilot.

Here is another example: I need a half-recipe, and the original called for 1/3 cup of flour. Now ½ * 1/3 = 1/6, so I need 1/6 cup of flour, but this is not a standard measure. I convert cups to tablespoons; there are 16 tablespoons in a cup. So 1/3 * 16 = 16/6. It would be tedious and inaccurate to measure 1/6 tablespoon 16 times. Simplifying, I get 8/3 = 2 2/3 tablespoons. I measure 2 full tablespoons and estimate 2/3 of another tablespoon. For more accuracy, use the fact that there are 3 teaspoons in a tablespoon: to get 2/3 tablespoon, measure 2 teaspoons. (Most American cooks have a set of measuring spoons: 1 tablespoon, and teaspoons: 1, ½, ¼, and 1/8.)
In many cases, fractions are most convenient and useful when simplified. Simplification is easier when a least common denominator is found. If fractions are multiplied, the simplification I easier when done before rather than after the multiplication.

Simplification of radicals is not in the Standards, and I agree with this omission. For example, I question the custom of expressing square roots with no perfect squares under the radical sign, such as

Sqrt(200) = 10 * Sqrt(2).

Young people may not know the old-fashioned reasons for this calculation. Before the era of calculators, people often used a table of square roots. Many tables showed square roots of the integers between 2 and 100. So you could not look up the square root of 200, but you could evaluate it as 10 Sqrt(2) approx. 10(1.414) = 14.14.

Furthermore, evaluation was often time-consuming and not very accurate. In addition to tables of square roots, we used slide rules and tables of logarithms. Say you wanted to evaluate:
(Equation 1) Sqrt (72) + Sqrt (2) approx. 8.585 + 1.414 = 9.899.
If you simplified Sqrt (72) as 6 Sqrt (2), then the result is 7 Sqrt (2), and you need to look up only one square root instead of two. Also, this calculation could be done on a slide rule, but the addition in Equation 1 could not. Today Equation 1 is done more accurately on a calculator; I like to do it with an exponent instead of a radical, like this:
72^.5 + 2^.5
Similarly, younger folks may not know a reason to rationalize denominators. For hand calculation, multiplication of decimals is usually easier than division. For example:
1/Sqrt(3) approx. 1/1.732
Rationalizing the denominator, you get Sqrt3)/3 approx 1.732/3.
By hand, the first calculation takes more time than second. The calculator is just as happy to divide as to multiply, so rationalizing of denominators is not as important as formerly.

Of course, another reason instructors demand simplified answers to is to get answers in a standard form, easier to grade. One way to get standard answers is to request a numerical answer, correct, say, to two decimal places. I like numerical answers, because they prepare the students for applied problems, and because I can make sure students know how to round decimals.

Rational expressions are sometimes simplified by finding a least common denominator and adding. For example, say f(x, y) is
1/(x + y) + 2/(x – y) + 3/(x^2 – y^2) = (3x + y + 3)/(x^2 – y^2)
The expression on the right is easier to evaluate than the one on the left. To motivate such simplifications, sometimes I ask the student to use each expression and a calculator to evaluate, say, f(1.96, 2.17).

If the student gets a common denominator by multiplying all three denominators, the result is
((x – y)( x^2 – y^2) + 2(x + y)( x^2 – y^2) +3(x + y)(x – y))/((x + y)(x – y)( x^2 – y^2))
This expression is awkward to write, let alone to evaluate. In the bad old days, before calculators, people had powerful reasons to simplify calculation. Even now, the effort can be worthwhile. As others have observed, if students are skilled with numerical fractions, they are more ready to learn algebraic fractions.
I don’t like any formula for adding fractions. The formulas are intimidating, and they don’t work well for adding more than two fractions. Instead, I like Lane’s comments on May 24, using primes. The hard part is finding the least common multiple; there is a nice discussion at http://mathforum.org/library/drmath/view/58140.html. One you have that number, it is not rocket science to rename each fraction with this denominator. Don’t state a formula or even a rule; just show some examples and let students practice until they catch on.
I prefer to say rename a fraction, rather than replace it with an equivalent fraction. In abstract algebra, we define fractions with an equivalence relation for ordered pairs of integers, so one pair can be equivalent to another. The pair (1, 2) is equivalent to the pair (2, 4). But the fractions ½ and 2/4 are not merely equivalent; they are equal; these are two names for the same number.
In middle and high school, students still need to practice fractions, including mixed numbers. In most textbooks I see, answers are small whole numbers, or decimals calculated with a calculator. Here is a use for fractions in algebra: graph by hand a linear equation in standard form, like 3x + 7y = 10. The fastest way to do so is to calculate the two intercepts, which are (3 1/3, 0), and (0, 1 3/7). Plot the two points on grid paper and connect with a straight edge. Note that these mixed numbers are easier to plot than the improper fractions, 10/3 and 10/7.
We compete economically, and we may compete in war, with nations whose curriculums include simplification of fractions. Members of the Armed Services are expected to know this topic. Community colleges have remedial courses, which include it. Contractors, carpenters, plumbers, and electricians need it. They should know 4/64 inch = 1/16 inch. In short, simplification of fractions is still essential.

“Because ratios and rates are different and rates will often be written using fraction notation in high school, ratio notation should be distinct from fraction notation.” (page 4)

This makes sense for many reasons. However, this is a large departure from the norm. Keeping that in mind, critical area (1) in the Grade 6 standards introduction states:

“Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions.” (page 39)

The phrase “connect ratios and fractions” really sounds like it is referring to fraction notation. Many people will interpret it that way due to it being the current norm. However, it seems more likely that this phrase is referring to the unit rate a/b of the ratio a : b, and possibly how you can generate fractional values in equivalent ratios. Is this correct? If so, the language in the Grade 6 standards introduction is very confusing and could mislead a large number of people. If this statement really does imply fraction notation, then I’m confused.

Part 2: Assuming students do not see fraction notation with ratios in Grade 6, how do the standard writers envision the flow from Grade 6 ratios to Grade 7 with proportions? Is it merely a distinction that they can write the values of these ratios as fractions, equate them, and then solve an equation? This seems to be brushed over in the progression document. Sure, proportions are not referred to in a standard, but the progression document does refer to them. Without this type of distinction, the progression document seems to contradict itself between grades. Am I missing something?

“Dear Leandra,
Throughout K-5 the standards carve out room for a focus on number and operations by limiting the spread of other topics, and that’s why capacity is not mentioned. Of course there are many contexts in which children might learn about gallons, pints, quarts, etc., including the home, and that which is not mentioned in the standards is not thereby forbidden, so teachers might well choose to use them as examples if they think the class is familiar with them. But the standards do not require class time to be spent on them.

Regards,

Bill McCallum”

Hope this helps.

Do you know if the standards in a given math cluster are in a progression?
For example is 1, foundational to 2. etc..
We want to know the intent of the authors here.

Here are examples of how “unit rate” has been used various kinds of documents–not always with quite the same meaning. I’ve organized them in two groups.

Group 1: unit rate has no units.

1A. Grade 7 from NCTM Focal Points (2006): http://www.nctmmedia.org/cfp/focal_points_by_grade.pdf

Number and Operations and Algebra and Geometry: . . . . Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).

1B. Susan Lamon’s book Teaching Fractions and Ratios For Understanding (2nd edition, 2008), p. 195: “Note that all of the rates in this class are equivalent and that they reduce to 1.5/1, the unit rate, the cost per pound.” http://books.google.com/books?id=nx41Iqe1PSwC&lpg=PA196&ots=aWGoT0S8Q8&dq=%22unit%20rate%22%20lamon&pg=PA195#v=onepage&q&f=false

Group 2: unit rate has units.

2A. But on p. 192, Lamon writes: “the unit rate is 6 mph.”

2B. Connected Math: Vocabulary: Comparing and Scaling: http://connectedmath.msu.edu/parents/help/7/comparing_concept.pdf

Unit rates: are ratio statements of one quantity per one unit of the other quantity. Any given ratio can be rewritten as 2 different unit rate
statements, though one of these may make more sense in the given context.

2C. Connected Math Pearson Prentice Hall video: http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-890s.html

A unit rate is the rate for one unit of a given quantity. [example: . . . . The unit rate would be 25 mi/gal.]

Example 2C raises the question: “What’s the difference between a rate and a unit rate?” (Note that CCSS does not ask that students use derived units such as mi/gal until high school.)

Since it says apply the formulas does that mean that a student might be given the volume and two sides and be asked to find the other side? Also if it is a cube would they then be expected to find the cube root to determine the side given the volume? I would appreeciate hearing your thoughts on this matter as it has been brought to my attention by a sixth grade teacher. The Model Content Frameworks and the Illustrative Mathematics websites do help but we need more examples to clarify the depth of knowledge needed at each level.

First, on simplification. The last sentence of the following standard is relevant here:

4.NF.1. Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

In both your examples it is reasonable to expect students to try to make sense of their answers by expressing them as 2/3 and 2 1/2, for the reasons you give. Thus, there is support in the standards for the answer you want.

On least common denominators, I think the other thread has most of what I want to say. I would just point out that the same principle as above applies to the answer 360/72; students are expected to make sense of their answers, in this case by finding an equivalent fraction that expresses better how many pies there are. Also, it’s not obvious to me that the extra efficiency of finding the least common denominator is worth the time taken in the curriculum by teaching it as a general method. And, as I said elsewhere, it is certainly not forbidden that students see and use that shortcut here.

On mixed numbers: the method you suggest for adding 4 5/16, 2 1/16, and 3 7 /16 is exactly what is intended by the phrase “using the properties of operations”. Namely, students should see 4 5/16 as , etc., and then your method of adding the whole numbers first and then the fractions is just the principle that you can add numbers in any order and any grouping (commutative and associate laws of addition, although it is not necessary to use those terms). I completely agree that method is preferable.

For your last example, I agree that students should see that 24/8 = 3. Another relevant standard here, and also for the pie problem, is

5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b=a÷b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Students should see that 24/8 is 24 divided by 8, and therefore 3 (from their knowledge of multiplication facts).

Here is what I did: I went through and combined some of the more advanced 7th grade domains, cluster headings, and content standards with those of 8th. I left behind about 6 cluster headings that seemed more basic and 6th grade-based. I inserted those into the 8th grade year-at-a-glance and then grouped them according to Major, Supporting, and Additional topics, under the following assumptions: 1) Students will be assigned a one-year Pre-Algebra class so it needs to contain curriculum that can be covered in one year. 2) Pre-Algebra may eventually be phased-out, re-named, or usurped, however, until then it seems that the “basic” framework for 7th grade and that of 8th most closely resembles what is traditionally taught in such classes.

Here is what resulted:

Unit 1: Real Numbers
– Analyze proportional relationships and use them to solve real-world and mathematical problems. (Reinforcement from 7th grade)
– Know that there are numbers that are not rational, and approximate them by rational numbers. (Reinforcement from 8th grade)
– Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (Reinforcement from 7th grade)
– Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (Reinforcement from 7th grade)
– Work with radicals and integer exponents. (Reinforcement from 8th grade)

Unit 2: Pythagorean Theorem
– Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (Reinforcement from 7th grade)
– Understand and apply the Pythagorean Theorem. (Reinforcement from 8th grade)

Unit 3: Congruence and Similarity
– Understand congruence and similarity using physical models, transparencies, or geometry software. (Reinforcement from 8th grade)
Unit 4: Linear Relationships
– Understand the connections between proportional relationships, lines, and linear equations. (Reinforcement from 8th grade)
– Analyze and solve linear equations and pairs of simultaneous linear equations. (Reinforcement from 8th grade)
– Define, evaluate, and compare functions. (Reinforcement from 8th grade)
– Use functions to model relationships between quantities. (Reinforcement from 8th grade)

Unit 5: Systems of Linear Relationships
– Define, evaluate, and compare functions. (Reinforcement from 8th grade)
– Use functions to model relationships between quantities. (Reinforcement from 8th grade)
Unit 6: Volume
– Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Unit 7: Patterns in Data
– Investigate chance processes and develop, use, and evaluate probability models. (7th grade required)

Now, there could be an argument made for dropping some 8th grade cluster headings, and adding some 6th grade cluster headings, or vice versa, however, we chose to keep it this large and then on the Year-At-A-Glance file we developed to group the ideas according to importance as stated above, ie., Major, Supporting, Additional. Mr. McCallum, what do you think?

Thanks,
Jonathan Haack

A-SSE.3 is an example of your first interpretation. In that case, the stem “Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression” can apply to any expression students might encounter; the lettered statements make specific ties to quadratic and exponential expressions, but do not limit the standard to those.

8.G.1 is an example of your second interpretation. The stem says ” Verify experimentally the properties of rotations, reflections, and translations”, and then the lettered statements list the properties.

K.CC.4, the example you have given, fits somewhere between the two. I would not say that the stem of the standard is additional to the lettered statements, but neither would I say it summarizes them. Rather, it is a higher level statement that cannot be reduced to them; it holds them together in a certain way. An activity designed to assess this standard might operate at that higher level; see, for example, http://illustrativemathematics.org/illustrations/447, which operates at the level of the cluster heading. The lettered statements give the teacher things to look for during the course of this activity.

These varying interpretations of the lettered statements pose a challenge to assessment, of course, but it is the challenge inherent in trying to preserve in the standards themselves some of the complexity of the knowledge structures they describe.

On subitizing: it’s hard to imagine what a standard would look like. Subitizing is something that kids do naturally, as the progression describes, but I don’t think it’s a required performance at any particular stage, although presumably kids who can’t do it will eventually run into trouble with one or another of the performances that are required.

http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comments

The students will end up with a deep understanding of what decimals are and what percent means and will then be able to see relationships between the three forms of number.

For adding and subtracting fractions, Common Core says
• 5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

This algorithm is not efficient when adding three or more fractions. Here is a sample problem from a test given to people joining the military: “A baker made 20 pies. A Boy Scout troop buys one-fourth of this pies, a preschool teacher buys one-third of his pies, and a caterer buys one-sixth of his pies. How many pies does the baker have left? (reference below)”
With Common Core, you multiply all the denominators, getting 72, and add like this:
1/4 + 1/3 + 1/6 = (18 + 24 + 12)/72 = 54/72.
So 18/72 of the pies remain, and 20 * 18/72 = 360/72, not an acceptable answer on this test.

The Standards omit least common denominator, which is 12 in this example. Using it, the calculation becomes:

(3 + 4 + 2)/12 = 9/12 = ¾. So ¼ of the pies remain, and ¼ * 20 = 5, the correct answer.
Without the least common denominator, the calculation is slower and more likely to have mistakes; the result is unsimplified. The Standards include least common multiple, but why teach this idea and leave out its main use, least common denominator?

For mixed numbers, the standards say:

4.NF.3
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

I infer “and/or. . . .” means calculation without using improper fractions. Omit the word or. Both methods are important. The drafty-draft does not mention calculation of mixed numbers as such. Say you join 3 rods, of lengths

4 5/16, 2 1/16, and 3 7 /16 inches.

The efficient way to calculate the resulting length is to add the whole numbers, then the fractions, getting 9 13/16″. Common core method, with improper fractions:

69/16 + 33/16 + 55/16 = 157/16″.

Comparing the two methods, the calculation with improper fractions is slower and more likely to have mistakes; the sum is an improper fraction, which may not be as useful as a mixed number.

Skill with fractions is important in life. To enter the military, people take the Armed Forces Qualification Test, which includes fractions. Calculators are not allowed on the test. (See sample tests at http://www.military.com/join-armed-forces/asvab/). To become a contractor, people take a licensing examination, which includes fractions. Foreign countries, such as Singapore, teach simplification of fractions and least common denominator. See http://www.singaporemath.com/v/vspfiles/assets/images/sp_pmstdtg5a1.pdf.

Fractions are also basic to higher mathematics:
In trigonometry, the student may need to know that 5π/10 = π/2.

A formula for calculating exponential growth, with doubling time k, initial amount A, and time t, is A*2^(t/k). if an investment doubles every 8 years, how much does it grow in 24 years? The exponent here is 24/8. If the student simplifies this fraction as 3, they get 2^3 = 8; so if you start with $1000, you have $8000 in 24 years. If the student cannot simplify 24/8, the calculation requires a calculator.

Understanding numerical fractions, the student is ready for algebraic fractions

I admire the Common Core for promoting games and hands-on activities. It improves the standards of many states. Refine the standards. Students should learn to calculate fractions with speed and accuracy in grades through 5. Then they should also use fractions in middle and high school to maintain skill.
.
I admire Common Core for more emphasis on fractions than many schools now require. Also, I like the encouragement of activities and interesting applications. I hope you will include these three topics: least common denominators, calculation with mixed numbers as such, and simplification.
.
[Typo corrected 6/7/2012]

On a related note, where would subitizing lie in K? The progressions mention the role of subitizing but there does not seem to be a standard that either perceptual or conceptual subitizing clearly relates to.

a. Use the information to plot the measurements on a line plot on your Student Answer Sheet.

b. Record the title and label the axis.

c. Make a statement comparing the number of beakers filled with cup to the number of beakers filled with cup.

HERE is our question. Since the line on the plot is an axis, do students need to include 3/8 on that axis to have equally scaled intervals?

Any clarification or example(s) would be aprreciated.

Also noteworty, they do not define unit rate.

An aside:
Texas, a state not adopting the CCSS, has these standards for Grade 6:
6.4(C) give examples of ratios as multiplicative comparisons of two quantities describing the same attribute;
6.4(D) give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients;

The only mention of unit rate in Grade 6 is in this standard:
6.4(H) convert units within a measurement system, including the use of proportions and unit rates.

While I agree that it is not necessary to belabor the distinction between rate and unit rate, it is still necessary when teaching the concepts to at least initially define them.

Either way, I am happy to see a reasonable amount of time going to such an essential pair of topics.

Phil refers to videos of grade 8 classrooms in Japan and other countries. Some are available here: http://timssvideo.com/videos/Mathematics. The web site has a place to log in that is prominently displayed, but according to the web site, logging in is not necessary for viewing the videos. (Some of the transcripts are non-optimal. JP4 SOLVING INEQUALITIES has the phrase “inequality equation” which might work better as “inequality.” Tad Watanabe has a discussion of this in the book Mathematics Curriculum in Pacific Rim Countries (Information Age Publishers, 2008).

To whom might I address this typo (http://i.imgur.com/jJKtM.gif) I found in Appendix A?

Thanks,
Tom James

Usiskin et al. say: “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.”

The unit rate is the numerical part of the rate; the “unit” in “unit rate” is often used to highlight the 1 in “for each 1” or “for every 1.”

I realize that the progression documents are in draft, but what troubles me is the fact that things like “unit rate” can conceivably take on meanings that I’ve never seen before. Can you please take us through the thought process that led to a “unit rate” neither containing units nor being a rate? If a rate is always to be considered as something for every 1, then is the definition even necessary?

Dear Dr. McCallum,
Several of my Kg colleagues have trouble letting go of the unit about patterns (AB, ABC…) even though it is not part of the Kg Math CCSS. I am concerned that in teaching this unit, they will waste valuable time and will not be able to give enough instructional time to the CCSS. Would you please explain to them why patterns are not part of the KG Math CCSS and why it is so important to teach the critical areas as they are described in the Kg Math CCSS.

Here is an answer I gave to a similar question about patterning and skip counting elsewhere on this blog (in response to this thread.

Patterning and skip counting can support the work of learning to count and add whole numbers, but they can also be used in ways that don’t support that. For example, given a repeating patter red, blue, blue, red, blue, blue, … a teacher could ask what the next color is, or could ask questions that get more at the underlying operations of addition and multiplication (not in Kindergarten, obviously). For example, you could ask about the size of groupings, how many groups it takes to get to 12, what the 23rd color would be, and so on. The same goes for skip counting: if it is connected to addition and multiplication, it can be useful, but if it just a matter of memorizing a sequence, then it could get in the way of understanding counting as cardinality, and understanding going to the next number as adding 1. Skip counting by 5 or 10 can reinforce base 10 understanding, because you notice the pattern in how the digits go up (and this includes starting from a number other than 0). Skip counting is not a goal in its own right, however. In short, both skip counting and patterning are viewed as supporting learning of operations and their properties, rather than as being learning objectives in their own right.

As for your question about the lawns, I wouldn’t say the rate is “7/4 hours per lawn” (that sounds weird), but rather “7/4 hours for each lawn”. The standards ask for “rate language”, the Progression suggests that “per”, “for each”, and “for every” are all examples of rate language.

On your fluency question, I’m not sure I understand the first part (process vs. speed), but I think “ultra efficient mental strategies” certainly have a role in fluency. To this day see 8 + 7 = 15 as 8 + 7 = 8 + (2 + 5) = (8 +2) + 5 = 10 + 5 = 15. I sort of see the 2 flash over.

The measurement progression is almost ready, the geometry progression is taking a bit longer.

I’m going to disagree, on several points.

First, there IS a general formula for adding fractions using the least common denominator. In a/b + c/d, let g=gcd(b,d). Then there are relatively prime integers (or polynomials) e and f such that b=eg and d=fg.

a/b + c/d = a/(eg) + c/(fg) = (af+ce)/(efg).

The formula works for ratios of both integers and polynomials, and covers the case when g=1.

There is no “the” general formula; there are two. The one in the Standards is the general brute-force formula, like killing a fly with a bazooka. The formula I’ve given is the general elegant formula, which uses the basic number theory the Standards say students should learn in sixth grade.

I’m also going to disagree with Bill about the product of the denominators being the “natural common unit…the only common unit that exists in general.” The “efg” above exists in general, because g can be 1. The “bd” does require less forethought, but I don’t know why that’s more “natural.”

Let’s look at the three methods mentioned for solving Lane’s sum of ratios of polynomials.

Here’s the result of multiplying both sides by the product of the denominators and expanding.

y^5 – 7y^3 + 22y^2 + 32y – 48 = 0

which is a nice exercise in the Rational Roots Theorem, especially since the sum of the roots is zero.

When we apply Tad’s strategy, here’s what we get.

3y(y^2 + y – 2)(y^2 + 2y – 3) + 2(y^2 + 5y + 6)(y^2 + 2y – 3) = (2y-1)(y^2 + 5y + 6)(y^2 + y – 2)

3y(y+2)(y-1)(y+3)(y-1) + 2(y+2)(y+3)(y+3)(y-1) = (2y-1)(y+2)(y+3)(y+2)(y-1)

Now divide both sides by (y+2)(y+3)(y-1).

3y(y-1) + 2(y+3) = (2y-1)(y+2)

y^2 – 4y + 8 = 0

Easier at the end, and a great occasion to talk about canceling only one of a the (y+2)’s, one of the (y+3)’s, and one of the (y-1)’s.

If we’d used the least common denominator, we would have skipped straight to 3y(y-1) + 2(y+3) = (2y-1)(y+2).

I understand wanting students to see the big picture and not get lost in the details, but here is where high-school students get lost in the details because their elementary-school teachers were trying to keep them from getting lost in the details.

I would like the Standards to say that, every year, teachers need to review, remediate, practice, connect, and deepen students’ understanding of topics from previous years. Then, in high school, students will be ready to move from ratios of integers to ratios of polynomials.

Also, I wouldn’t say that Phil was talking against traditional story problems, or in favor of messy problems, or indeed expressing a preference for any particular type of problem. Rather, he was suggesting that we focus on how a problem is used, whatever type it is. Story problems can be great learning problems, if students are not encouraged to solve them by key word search. And “messy” problems can be very formulaic.

4.NF.5: Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.* For example, express 3/10 as 30/100, and add 3/10+4/100=34/100.

The footnote says:

* Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.

So the 1/3 + 1/6 example is consistent with the standard and the footnote, but is not required.

I’ll answer the next three questions in a later post.

Tad summarizes that understanding well: in adding fractions you express each fraction in terms of a common unit. The natural common unit is the unit fraction whose denominator is the product of the denominators of the addends. And this is only common unit that exists in general, so it’s the only one that leads to a general formula. All this is difficult enough conceptually; it muddies the waters to worry about special cases where you might be able to find a smaller common unit. Worse, it might make kids think that somehow finding least common denominators is an essential part of fraction addition (I’ve certainly met students who seem to think that). And, since there is no formula for “adding fractions by finding a least common denominator”, insisting on it gets in the way arriving at a general formula; the general formula becomes something extra to memorize later, rather than an essential understanding of fraction addition as a general operation. Fraction addition becomes an arcane art. The calculation

is 100% mathematically correct. There is not anything even a little bit wrong with it. Once kids have a solid understanding of fraction addition, and if there is time in the curriculum, it is worthwhile pointing out that the answer is equivalent to and exploring what strategies you might have used to get that answer.

The same comments applies to Lane’s original example. One strategy for solving this problem is to multiply both sides by the least common denominator. But as Tad pointed out, another reasonable strategy is to multiply both sides by the product of all the denominators and then cancel common factors from both sides of the equation before expanding. This takes slightly more ink, but provides an opportunity to discuss an important strategy in algebraic manipulation, namely the advantage sometimes of keeping expressions in factored form rather than blindly expanding them out. There are advantages to both strategies, but neither is sacred.

I agree with you both, that “when students see the need, that’s the best time to teach a relevant mathematical idea.” If I read Lane correctly, the concern of high school teachers (including myself) is that fraction addition in CCSS-M seems to stop in fifth grade. In sixth grade students learn to find the lcm, but they don’t apply it to fractions to find the lcd. Then in high school, Tad throws them a sum of rational expressions whose elementary-school equivalent is 1/6 + 1/8 + 1/9. Our students want to use 6*8*9 = 432 as the common denominator, when they could use 72. Between fifth and tenth grade, students do not expand their understanding of fraction addition. If in sixth or seventh grade we had them use their new skills in finding least common multiples to subtract
7/30 – 5/42 = 7(5*6) – 5/(7*6) = (49 – 25)/(5*6*7) = 24/(5*6*7) = 4/35
then they’d be ready for Lane’s question in algebra.

I don’t fully understand why CCSS has been give the power to make up these sorts of definitions from whole cloth, but they have been given it nonetheless. And you have correctly interpreted their meaning here. I disagree that the distinction is a useful one for student learning, but since it will be tested, it will need to be taught.

If you compare 2/3 + 3/4 and 2/3 + 5/6, perhaps 2/3 + 5/6 is easier procedurally (to find the common unit) but they are equal conceptually (let’s make the unit the same). So, comparing fractions with unlike denominator after learning how to create equivalent fractions (and learning how to add fractions like 2/3 + 5/6) but not add/subtract fractions with unlike denominators (where neither is a multiple of the other) is like having to stop an interesting story right before the end and told to wait till next year to finish it — to me.

5.NF.1 (“Add and subtract fractions with unlike denominators…”) is the first standard that explicitly addresses the addition and subtraction of fractions with unlike denominators.

Based on this, it seems to me that the quote from page 10 the Progression for NF that Eric raised (“In Grade 4, students calculate sums of fractions with different denominators…”) would refer to a natural extension of 4th grade standards, but is neither required nor forbidden by the standards themselves.

If this is not the correct interpretation, please advise soon, as I know a LOT of districts that will need to seriously alter plans!

Thanks,
Brian

I tend to think when students see the need, that’s the best time to teach a relevant mathematical idea. If calculation involving rational expressions is where the usefulness of LCD comes in, then maybe that’s when the idea should be discussed.

I’m glad someone else is fighting the good fight. Let’s compare strategies for remediation.

Brad, bburkman@lsmsa.edu.

If these are correct, I would then ask for clarity on the phrase “at that rate” in this example from 6.RP.3b.
“For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?”

Does “at that rate” here really mean “at the rate implied by the ratio of 7 hours to 4 lawns”? You aren’t suggesting that “7 hours to mow 4 lawns” is a rate? The rate, which you ask for in the last question, is “7/4 hours per lawn”? Correct?
I really am not trying to be difficult. Just trying to get clear definitions that teachers can use with their students and cirriculum developers and textbook publishers can use in the materials they produce for teachers and students. Your patience is appreciated.

Hi! Thanks for providing the opportunity to ask questions about the Common Core Standards. I just wanted to clarify a few things in the high school standards:

(1) G-GPE.4 Students are asked to “Use coordinates to prove simple geometric theorems algebraically” and the given examples are about proving that four points form a rectangle or that a point lies on a circle. Are there other types of “simple geometric theorems” that students should be familiar with, or guidance I can use to interpret the word “simple”?

(2) G-GC.2 Similarly, students are asked to “identify and describe relationships…” in circles. Should this include secant theorems, or just the more obvious angle theorems and tangent theorems?

(3) G-GMD.1 reads “Identify the shapes of two-dimensional cross-sections of three-dimensional
objects.” Since 7.G.3 reads “Describe the two-dimensional figures that result from slicing three-dimensional figures” and specifies right rectangular prisms and right rectangular pyramids, is it safe to assume that the high school standard includes cross-sections of any and all 3D objects beyond right rectangular prisms and right rectangular pyramids?

(4) I’m not seeing solving absolute value equations (e.g. |x – 3| = 5) in the standards, although I do see absolute value of real numbers in middle school and then graphs of absolute value functions in high school. Is it implied but not explicitly required that students should be able to solve absolute value equations?

(5) A-APR.3 and F-IF.7 use the language of “when suitable factorizations are available” to describe when students should be finding roots of polynomial and rational functions. Does this mean that students should be comfortable finding factors using GCF, grouping, sums/differences of squares, but not long division? Or not long division with remainders? Where do we draw the line with what is “suitable”?

(6) I see references to the properties of operations (commutative, associative, etc.) in lower elementary standards about addition and multiplication, and in high school standards relating to complex numbers and matrices, but not as they relate to algebraic expressions. Can I interpret this to mean that while students should be familiar with and able to flexibly use the properties, they will not be assessed on being able to name specific properties?

Thank you so much for your time and help!

4.OA.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.

and

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-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.

It doesn’t seem right to expect a kindergarten student to manipulate 100 objects into groups of 10 to count them by tens. Can you give some guidance?

I think lots of people are having similar trouble interpreting this standard, and the others in this domain. I think this particular task on Illustrative Mathematics does a good job, but you should check out the other tasks in this task set as well. This is a good place to direct people that have this question.

http://illustrativemathematics.org/illustrations/436

We hit a snag with this standard:
A-SSE.1 – Interpret expressions that represent a quantity in terms of its context.
Interpret parts of an expression, such as terms, factors, and coefficients.
Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

Our question is….What does the word “interpret parts of an expression such as factors” here mean? How does one interpret a factor? Does this standard merely mean that we identify parts of an expression? If that is the case, why is the wording used “interpret” instead of “describe” or “identify?”

Thank you

I think your question splits into three parts:

1. the term “progression.” The terms “learning progression” and (related but not identical) “learning trajectory” seem to be inventions of US mathematics education researchers.

2. the concept of “progression.” in my opinion, the idea of “learning progression” is implicit in textbooks and other curriculum documents from outside the US. For example, check out the discussion of the central sequence of the knowledge package for subtraction with regrouping in Liping Ma’s book Knowing and Teaching Elementary Mathematics that starts on p. 15 or for the knowledge package on multidigit multiplication that starts on p. 45 (you can see these via Google Books).

3. the content and order of the CCSS progressions as compared with those of other countries. The CCSS states two types of sources in the list of works consulted. These include: documents of other countries (and analyses of those documents), articles on US research on learning trajectories. So, I think the short answer is yes, the CCSS progressions are a US invention, combining US research and progressions from other countries. However, as the work of Schmidt and others points out, progressions in those other countries aren’t all that different from each other (this of course depends on the level of detail).

Many other countries do not have “standards” but have documents which are more or less comparable with names like “course of study” or “syllabus.” Links to those for several countries (including Japan and Hong Kong) are here: http://hrd.apec.org/index.php/Mathematics_Standards_in_APEC_Economies

You don’t need to pay money for textbooks to get evidence about progressions and the CCSS. There are two types of sources: US research on learning trajectories and documents of other countries (and analyses of those documents). Just to be brief (or at least not incredibly long), I’ll only comment on the latter, but note that both types are listed in the “works consulted” in the CCSS.

Many other countries do not have “standards” but have documents which are more or less comparable with names like “course of study” or “syllabus.” Links to those for several countries (including Japan and Hong Kong) are here: http://hrd.apec.org/index.php/Mathematics_Standards_in_APEC_Economies

I’ve discussed comparisons of CCSS and documents from other countries on my blog: http://mathedck.wordpress.com/2011/09/06/strange-accounts-of-the-common-core-state-standards/

Here are the textbook sources that I know. Note that they aren’t necessarily going show things that are identical to what’s in the Progressions (for example, there’s no guarantee that terminology will be identical to the US or even among other countries), but there’s a lot of resemblance).

Singapore Math (www.singaporemath.com) has Singapore textbooks (which were originally written for English-speaking Singapore students) adapted for the US. I think this mainly means that the names of things and British spelling and terms (e.g., “ring it” for “circle it”) are adapted to a US audience. I don’t think they have the teachers manuals for the books. (I do, and I find them useful.)

The University of Chicago School Mathematics Project (http://ucsmp.uchicago.edu/resources/translations/) has translations of Japanese textbooks for grades 7-9 and Russian grades 1-3. (It says that the American Mathematical Society has translations of Japanese textbooks for later grades, but I didn’t see them on the AMS web site and suspect that they may have sold out recently when they were on sale.)

Global Education Resources (GER, http://www.globaledresources.com/) has translations of Japanese textbooks for grades 1-6. It’s also got translations of the teaching guides for grades 1-6 and for grades 7-9, and lots of other things, some of which are free of charge. You can download (free of charge) some translations of the teachers manuals for one textbook series that GER translated from http://lessonresearch.net/nsf_toolkit.html. These are really nice (I worked on editing the translations) as are the textbooks. You can get a sense of what might be called a “progressions way of thinking” in Learning Across Boundaries: U.S.–Japan Collaboration in Mathematics, Science and Technology Education (free and downloadable at http://www.lessonresearch.net/LOB1.pdf). For example, check out the piece that begins on p. 261. In discussing “research lessons” (special lessons created by groups of Japanese teachers as a result of “lesson study”), a Japanese professor of mathematics education says:

A research lesson is only one lesson. However, in doing research lessons we are not thinking
about only one lesson. We need to think about the entire unit and how it’s related to other
grade levels. That is very important.

He continues (pp. 262-266) by illustrating how that might be done for a lesson on estimating the area of a circle, using excerpts from the GER translations of Japanese textbooks.

Sorry not to put live links, but it is time consuming and I seem not be doing well with this recently (maybe the blog interface has changed).

In fact, North Carolina’s official commentary on the CCSS-M, the “Unpacked” series, uses this exact example, 1/3 + 1/6, to interpret the Grade 5 CCSS-M to say that the student should use 18 as the common denominator.

I hope you can clarify.

K.G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
2. Correctly name shapes regardless of their orientations or overall size.
3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).

Compare this with

1.G.1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes.

It seems to me that the progression is pretty clear here. Of course, as always, that which is not stated is not thereby forbidden; if the definition of a triangle in terms of its attributes comes up naturally in a Kindergarten class, and students seem to be learning from it, then that’s good. But the standards do not require it.

7x + 33 should be 7x + 3.

I appreciate your having included this side-by-side example.

A follow-up to Patricia’s question about “In Grade 4, students calculate sums of fractions with different denominators where one denominator is a divisor of the other,…”

Adding fractions with different denominators does appear in CCSS-M Grade 4, but only as decimal fractions, with the example 3/10 + 4/100 = 34/100. My reading is the same as Patricia’s, that 1/3 + 1/6 is not there in the fourth grade.

North Carolina’s official commentary on the CCSS-M, the “Unpacked” series, uses the exact example here, 1/3 + 1/6, to interpret the fifth grade CCSS-M to instruct us to use 18 as the common denominator.

Please clarify.

So Bill, in looking at the glossary, the standard and your post, it sounds as if this is an issue where the standard is not limiting, but stating the minimum we should expose our students to and hold them accountable for. We could go beyond. For instance, they should leave 3rd grade with a working knowledge of all products from two one-digit numbers, but they may also be exposed to working with products of a one-digit and two-digit number within 100. The glossary doesn’t seem to define for us what IS included, nor does it tell us what is NOT included. This is another issue where the standard is very open to interpretation. Also in looking at 4th grade standard 4.NBT.5, that is a standard that does specifically mention multiplication with a one-digit by a multi-digit number, which the 3rd grade standard does not do.

Multiplication and division within 100. Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0-100. Example: 72 ÷ 8 = 9.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

I don’t like to play statistics educator when my background and experience is in math, so I’ll refer you to the following two things from the American Statistical Association.

Description of ASA education resources: http://www.amstat.org/education/pdfs/EducationResources.pdf.

The Meeting Within a Meeting (MWM) Statistics Workshop for Mathematics and Science Teachers will help middle and high school teachers teach the increased statistics content in the Common Core State Standards. The MWM statistics workshop will be held in conjunction with the Joint Statistical Meetings on Tuesday, July 31 and Wednesday, August 1 at the Hilton San Diego Bayfront with separate middle and high school strands. The registration fee is $50, which includes materials and refreshments. Optional graduate credit and limited lodging reimbursement is also available. More information and registration for the MWM workshop is available at http://www.amstat.org/education/mwm/.

Description of ASA education resources: http://www.amstat.org/education/pdfs/EducationResources.pdf.

The Meeting Within a Meeting (MWM) Statistics Workshop for Mathematics and Science Teachers will help middle and high school teachers teach the increased statistics content in the Common Core State Standards. The MWM statistics workshop will be held in conjunction with the Joint Statistical Meetings on Tuesday, July 31 and Wednesday, August 1 at the Hilton San Diego Bayfront with separate middle and high school strands. The registration fee is $50, which includes materials and refreshments. Optional graduate credit and limited lodging reimbursement is also available. More information and registration for the MWM workshop is available at http://www.amstat.org/education/mwm/.

- A question and a comment specific to the fractions document:
a) I recently had the opportunity to read through the fractions progressions with a group of 3rd – 5th grade teachers. We all were questioning the opening statement on p. 10 (grade 5): “In grade 4, students calculate sums of fractions with different denominators where one denominator …” I see that this was previously questioned back in Aug. 2011, however I am concerned with your response. As much as I appreciate the progressions, I believe the standards will be the primary resource used by educators; the grade 4 standard very clearly states that adding and subtracting of fractions will be done using like denominators. I am wondering if you might address this with a bit more detail? I have concerns that if the progressions documents do not clearly align with the standards, as they are written, the validity and value of the progressions will likely come into question. I am asking that you please take this into consideration as revisions are made.
b) I would like to share some of the responses from these teachers and also some of my MS colleagues who read the ratio/proportionality and the statistics progressions. Almost every teacher found value in these documents. They feel the progressions very clearly outline how to build understanding; the teachers stated that although some of the content may be similar to what they already teach, HOW they teach it will change. There was much concern voiced over how challenging it was to read through these documents. The MS teachers were able to work their way through and have rich converstations; they are content sprecialists. The elementary teachers, in some cases, needed some guidance working their way through the mathematics – which is truly written for someone who is comfortable with algebraic representation. I am a content specialist and I actually found the fractions document harder to read than the MS documents. Again, my concern is that if these are meant to be helpful to teachers as they plan their Common Core units, the elementary documents need to be written in a language that does not intimidate teachers who, themselves, do not have a comfort level with mathematics.

Thank you.

Patricia Posluszny

First of all, thank you, for all this work. I am a big fan of CCSSM and one of my personal favorite is this Domain, CC/OA. In looking at its progression I was hoping you could clarify a couple of things for me.
K.CC.6 asks that students make the assessment of qualifying the group as greater than, less than or equal to, yet, in many cases, it may be assessed the other way around, where the student selects the group that is greater or lesser. Or one could ask a student to look a set of 4 or 5 groups and select the 2 that are equal. Are these things still within the boundaries or expectations of the standard?
I guess my general question is how far are we to interpret a standard? Should it be taken strictly at face value or can we make assumption that if written one way, the opposite or the inverse should also be true? Some examples: if students are to represent and equation with objects should they be able to write an equation from a representation; if students are asked to decompose should they be able to compose; if they are asked to sequence numbers in a certain order they should be able to describe in what order numbers are sequenced, etc. And what about combining standards into question that may be more complex? For example by combining 1.OA.7 & 1.OA.8 you could ask students to find the unknown to make 4 + 8 = ? + 6 true.
My other question is regarding the definition of fluency as “fast and accurate”. How does process vs. speed play a part of it and how would you distinguish ultra efficient mental strategies from just memorization?
Lastly, do you know how soon may we expect the geometric measurement progression and the geometry progression to come out?

Thanks again,
Ivan

I have been curious about the decision to stick with only measurement data when using line plots. There are many classic activities used to introduce line plots like “how many pockets” or “how many siblings” does each kid have, but these seem to be counts more than measures since you can’t have 3 ½ siblings. Where would these activities fall under? Do you recommend continuing to do some line plots using counts or should it always be measurement data?

Thank you,

Ivan

I can appreciate your point. Anecdotally, I have noticed that my students who understand why we are learning a particular topic and how it fits into a larger picture, seem to do better at general problem solving and using the said topic knowledge to solve a variety of problems. However, inorder to communicate to students the larger picture and the “why are we learning this stuff” questions, teachers must have clear, understandable, standards. So far, I’m not finding this with these “new” Common Core Standards.

“Unpacking,” while maybe a little vague in terms of the process, makes sense to me. But “designing an itinerary through closely observed details, with the design of the itinerary supported by research about what itineraries work best for kids” as put by Mr McCallum, is confusing to me.

I hope that whatever new standards emerge that they are clear, concise, and complete. I hope that the result will not be that we end up teaching less.

John

I got this article sent to me by our department chair via our district math person. I can’t believe how poorly written it is. It is as if you didn’t even proof your paper. The run-on sentences never ended. There were even missing words. “Standards are a policy document, after all, not a work art.” While your message might be apt, it is lost in the lack of communication ability.

When I hear that the new standards replace “a mile wide and an inch deep” with less wide but more depth, I only hear less. The standards we currently use were an attempt to make our classes more rigorous. When they emerged some 20 years ago, I remember hearing the complaints from primary grade teachers about how much core subjects they would have to cover. They seemed upset because they wouldn’t be doing those fun and creative projects they had spent a lot of time developing and loved to do.

It feels like the pendulum is now swinging back in the other direction, less rigor, more art.

– John

Webinar Link

The usefulness of the mode depends on the nature of the data. If the data are discrete with a limited number of values (e.g. the number of pets each of our students own), then the mode may tell us something interesting. If the data are more continuous with many different valued (e.g. the heights of our students measured in cm), then the mode may be of little use.

6.SP.5d speaks of relating choice of measure of center to shape and context. That is the heart of it. Students should understand conceptually when the mode is and is not useful as a summary measure.

The usefulness of mode is dependent on the nature of the data. If a data set is discrete with a small number of different values, e.g. how many pets each of our students own, then the mode may have significance. If it tends to be more continuous and/or have a lot of different values, e.g. the heights of our students measured in cm, then the mode may not tell us anything useful.

6.SP.5d asks students to relate the choice of measure of center to shape and context. That gets to the root of it. If mode is part of a curriculum, there should be conceptual understanding of what it does and doesn’t) tell us about our data, beyond the skill of computing it.

I see that 6.NS.8 could be interpreted much more broadly than 6.G.3, so I guess I am wondering what are appropriate 6.NS.8-type tasks that are NOT already 6.G.3-type tasks? I would not include graphing proportional relationships or relationships between an independent and a dependent variable as there are other standards about those situations.

I checked illustratedmathematics.org and didn’t find any illustrations for these. Looking forward to having some soon! All of this has been so helpful!

In addition to knowing from memory the basic multiplication facts are 3rd graders becoming fluent with 2-digit by 1 -digit multiplication so long as the product is 100 or less? For example they should be expected to find 27 x 3 or 15 x 4, but not 47 x 5 or 85 x 2.

“By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).” (p. 20)

So, it appears that the NMP is saying that the definition of trapezoids is “at least” one pair of parallel sides.

5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

- How detailed should the hierarchy of 2-dimensional shapes be? Should kite be included?
- Also, what is the definition of a trapezoid? Is it “only one” pair of parallel sides or “at least” one pair of parallel sides? This would affect the hierarchy diagram.
Note: According to Van de Walle (2010) in Elementary and Middle School Mathematics, 7th Ed., “Some definitions of trapezoids specify only one pair of parallel sides, in which case the parallelogram would not be a trapezoid. The University of Chicago School Mathematics Project (UCSMP) uses the “at least one pair” definition, meaning that parallelograms and rectangles are trapezoids” (p. 411).

Thanks,
Trish

In Appendix A it states: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16 .

On the other hand, Russell’s definition of fluency may be a bit problematic. For example, if a 2nd grader is adding 9 + 8 by counting on 8 times from 9, without hesitation, is he fluent? I would say no because I would want 2nd graders to be moving away from inefficient counting strategy to obtain the correct answer.

As for assessing Kindergarteners, my inclination is not to worry about “know from memory” since the CCSS does not say it explicitly. I may still use flash cards to pose questions, but I would be assessing not how quickly students give me the correct answers but how they seem to be obtaining the answers.

With this, it seems that using mass and weight interchangeably would not be ‘Attending to Precision,’ as they are not synonymous. I would not have a problem using pounds as a unit for measuring mass, as long as we are measuring on a balance.

2) Well, the progression suggests it as a possibility for 6.RP.3d, not a requirement. It’s a natural thing to do, but the standard does not give an explicit list of which unit conversions are expected, so there is room for curriculum writers to use their judgment here.

Or would you just expect them to be able to do them “unhaltingly” but not necessarily from memory?

For our second graders we are using flash cards to assess during individual student interviews and we expect them to know the fact within 3 seconds. Thoughts?

Modeling (drawing and building) with different materials allow Kindergarten students to start paying attention to the parts that make up shapes. So, students should engage in modeling activities using tiles (pre-made shapes), sticks (focusing on sides of polygons), drawing by connecting dots on dot grid (seeing the vertices of polygons), etc.. Kindergarteners don’t have to specify those components by formal terms, but those experiences help them move from just seeing the whole shape to being able to (eventually) analyze the components of shapes.

CCSS seems to put a lot more emphasis on 3-D shapes, so I imagine students should have some experiences with building 3-D shapes (with blocks, empty boxes, sticks and clay balls, etc.). But, drawing, i.e., representing 3-D shapes on a 2-D medium is probably too much for Kindergarteners.

The second part of this standard confuses me. What exactly does that entail?

I believe I understand the first portion as combining two right triangles to make a square, which would be your “composite” shape. So would you take the new square (the composite shape) and another square to make a rectangle? (Obviously this is just one example.)

p.s. Love this blog. Really looking forward to the progressions in Geometry!

I also think it is interesting that the CCSS distinguish “fluency within 20″ and “knows from memory ” up to 9+9. This seems to suggest that students should be fluent with calculations like 13+5 and 18-3 using their understanding of the meaning of operations and number sense. But, for 1+1 … 9+9, the CCSS seems to expect students to “just know” the facts. I also like the fact that the CCSS puts “memorization” AFTER fluency. I think if students become fluent (as explained by Russell), they will remember basic facts, too.

K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

Is this referring to both 2-D and 3-D shapes, and which shapes should we expect them to model as opposed to draw and vice versa?

I need help with clarifying the fluency with addition and subtraction facts in K-2.

K- Fluent with addition & subtraction w/in 5
1- Fluent with addition & subtraction w/in 10
2- Fluent with addition & subtraction w/in 20 AND knows from memory single-digit to 9+9 (add only)

We are working on standards based report cards for our 1-2 grade levels. We have standards based in Kinder already. This year in kinder, we listed the standard as shown and then, for assessment purposes only, we used flash cards to assess fluency. (We were sure to use number talks, images, and manipulatives to ensure understanding).

When we started working on standards based for 1 and 2 we ran into some confusion, because we had at first thought in K and 1 we should be “flash card fluent” under 5 and under 10 respectively, but then in 2nd it says know from memory for the ones they should be “flash card fluent” with and the word fluent has a slightly different meaning.

What wording could be used on a report card to differentiate these skills for parents? And if the K and 1 should not be “know from memory” how should teachers assess the facts in kinder and first?

Is there any way to link to this material without going to home page and using the cascading menus?

A related question is whether there would be any problem if I copied the text and illustrations of interest and put it in the syllabus.

Robert Springer

Maybe what you are looking for is included in 8.G.5 (Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.)?

Brian

Any insights you can provide will be greatly appreciated.

There seems to be a bit of support out there for the idea. And I don’t mind working with my groups on producing first drafts. I have only two questions (and one request):

1. Is there a way to check with PARCC / SBAC that this work isn’t already in progress by their folks? (I’d be happy to send an email or make a phone call if you can send me the contact info. for someone who will actually give me an answer!)

2. If I coordinate an effort at drafts, would you (or any of your groups) be willing to look them over? If this sort of vetting is more appropriately done by some other group (ex., PARCC / SBAC or others), can you tell me who?

It would be hugely beneficial for the 45 states who have adopted if there was one “common” public space (ie., website) that was an “official” place to disseminate (

It doesn’t seem that there is any good mechanism in place to produce (or disseminate) this sort of inter-state infrastructure. Your blog and the Illustrative Math Project are the closest we have. As a result, 45 states are duplicating efforts and producing different interpretations of the “common” standards. Is there any way to get one “official” website that would function as a *common* public space to house *common* resources for the 45 states and have *common* inter-state discussions? Please?!

I know that a lot this comes down to funding… but it seems like individual states have been given a LOT of RTT money that could go a lot farther (and we could reduce the risk of splintering the common standards) if work could be done once by a group of inter-state folks and then shared on ONE inter-state space!

Thank you for sharing your thoughts, answers, and providing this blog as a common space,
Brian

Anyone else thinking about this as well?

For example, teachers have a hard time understanding the division of a fraction by a fraction. The 6.NS Traffic Jam illustration provides an excellent means of giving teachers some intuition about the division of fractions.

The problem is that I see no way to link to this illustration directly, or any text with the illustrations. Are there any plans to provide links. I don’t think I can get elementary school teachers to go to the home page and navigate down. In case the answer is no, am I free to copy and paste illustrations such as the one referenced in my presentation?

Robert Springer

The progressions document on Ratios and Proportional Reasoning 6-7 has been incredibly helpful as I attempt to dig into 6.RP.3a-d. However, I have two questions about 6.RP.3d:

1) I’m not sure I understand the expectation for students where it states, “manipulate and transform units”. The standard is clear up to that point, but this phrase seems to suggest actions other than converting. Am I trying to read too much into it? Or is this suggesting some application of the standards for mathematical practice that I am not seeing?

2) From the last paragraph on p. 7 of the progressions document, it seems that converting units between measurement systems (customary to metric and vice versa) using ratio reasoning is an expectation for sixth graders. Is this correct?

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“The standard N-RN.2, Rewrite expressions involving radicals and rational exponents using the properties of exponents, could support some work along these lines. But the standards overall try to get away from demanding that students “simplify” things. For example, they don’t expect students to find the least common denominator when adding fractions, or to reduce fractions to lowest terms. When thinking about radicals, it’s not at all obvious that 3 \sqrt{3} is simpler than \sqrt{27}, and the latter form is more useful for some purposes. For example, you can see that the number is slightly bigger than 5 much more easily from this form.”

Thanks, sorry for the oversight.

Indeed, there are situations where simplifying fractions gets in the way of understanding. For example, insisting that the answer to 1/10 + 3/10 be written as 2/5 gets in the way of the most important understanding that we want students to come away from this problem with, namely that this addition works the same way as whole number addition, with the unit 1 being replaced by the unit 1/10.

Is cups, pints, quarts, and gallons included in 4MD1 and 5MD1?

Shannon

I am used to there being preliminary work with similar figures in Grade 7 after working with proportions. However, it seems that the intent of the standards is to introduce congruence and similarity with transformations in Grade 8. If this is the case, could you explain the benefit of this approach being taken in the standards?

Thank you for taking the time to answer our questions.

Thanks – Lisa

Common vocabulary and definitions have been frequently-reoccurring requests. Many teachers are aware that there are different definitions for some mathematical terms (ex., trapezoid, isosceles, face). Similarly, there are questions about the appropriate language to introduce/use with students (either for developmental reasons, to better foster conceptual understanding, or to assure alignment with the coming tests). For example, should 5th grade students know and use the term “ordered pair,” or “coordinate pair,” both, or something else entirely?

This sort of grade-level specific “Suggested List of Mathematical Language” and “Math Glossary” were provided by our State Dept. of Ed. in the past. However, it doesn’t make sense for 45 states to produce different lists and different definitions for these “common” standards… especially if we will share common assessments. Is there any plan to provide this sort of supporting documents from groups you work with so that they are “common” for all 45 states? If not from your level, it seems very necessary that it be “common” at least within the PARCC states and within the SBAC states. Do they plan to produce such guidance documents?

Thanks,
Brian

There’s a danger that assessment will drive all this away, of course, attempting to reduce this standard to some mindless exercises; we have to resist that.

Thank you for your response.

N-RN.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

First, review of definition: a real number x is rational if and only if bx=a for some integer a and some non-zero integer b; in other words, it means x is the ratio a/b of two integers, the denominator being non-zero.

(a) why the sum of two rational numbers is rational: suppose x,y are rational numbers. That means we can write x=a/b and y=c/d with integers a,b,c,d where b,d are non-zero. Then x+y = (ad+bc)/(bd). Since ad+bc and bd are integers and bd is not zero, we’ve shown that x+y is rational, which is what we wanted.

(b) why the sum of a rational number and an irrational number is irrational. Suppose y is rational, z is irrational and their sum is x=y+z. Then z=x-y=x + (-y). Since y is rational, so is -y, so we have expressed the irrational z as the sum of x and a rational number. By (a), if x were rational, then z would have to be rational too, which it isn’t, so x must be irrational.

(c) why the product of a rational number and an irrational number is irrational. As in (b), suppose y is a non-zero rational and z is irrational. Let x=yz be their product. Since y is not zero, we can write y=a/b with a and b both non-zero. Then z=(b/a)x is the product of x with a rational number. Since the product of two rational numbers is rational (easy proof), the hypothesis that x is rational would imply that z is irrational, so it must be rejected. Thus, x must be irrational.

Should the unknown be in all positions when dealing with addition of three whole numbers or only in the total?

2. How do RI.4 standards (grades 6-8) and the Language .5 standards differ? Both ask students to determine word meaning by connotation/denotation and of figurative language.
RI.8.4. Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze the impact of specific word choices on meaning and tone, including analogies or allusions to other texts.
L.8.5. Demonstrate understanding of figurative language, word relationships, and nuances in word meanings.
Interpret figures of speech (e.g. verbal irony, puns) in context.
Use the relationship between particular words to better understand each of the words.
Distinguish among the connotations (associations) of words with similar denotations (definitions) (e.g., bullheaded, willful, firm, persistent, resolute).

In 4th grade I see the connection to decimal fractions in NF.5-7. Would you present these measurement problems using the decimal fractions rather than the algorithms for decimal operations? Would the operations performed on these decimal fractions be limited to what the standards have addressed up to this point (addition and subtraction with like denominators and multiplication of a whole number and a fraction)?

The inclusion of language such as “composing and decomposing a ten” (hundred, etc.) and the exclusion of language such as “carry a 1″ and “borrow” is intentional? Are the standards intending for teachers to avoid explanations that involve the terms carry and borrow? I see how composing and decomposing are more conceptually-sound terms; I just want to make sure I’m interpreting the intention of the standards correctly

2.MD.8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

4.MD.2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

As you know, I am preparing an example curriculum for the lower elementary grades (K – 3) which I intend to make public on my website (amphimath.com). After stating each concept that an instructor should teach, I plan to provide a brief illustration or a link to one. I plan to provide a link to nearly every illustration you have posted for these grades.

Is cups, pints, quarts, and gallons included in 4MD1 and 5MD1?

Shannon

I am wondering why the link to this particular progressions document has not been posted on the progressons webpage? I have downloaded this file and shared it with colleagues – but – I have it only because I was fortuante enough to have wandered onto this page.

Thank you!
Patricia

Thanks for the clarification. I was thinking more in terms of what have been called “baby exponentials”. For example a doubling sequence like 2^n (for whole number n) as opposed to analyzing exponentials as continuous functions like 2^x (for x real). These seem like a good and natural source of examples to contrast with linear and polynomial relationships.

An important point to make about moving exponentials to Algebra 2 is to recall that student have seen them in middle school (in particular Grade 8 Expressions and Equations). So there is every expectation that 9th grade students will compare and analyze situations that involve linear, polynomial, and exponential behavior (e.g. 2x versus x^2 versus 2^x) but the more rigorous work with exponential equations and logarithms is done in Algebra 2.

Also note that many standards are repeated throughout the scope and sequence. So when we cross out part of a standard the intent is to provide focus early on, but that standard will occur later in full (nothing is crossed out in the end).

Many decisions were needed to organize the units into Traditional and Integrated sequences. Those can be compared on pages ii and iii of the “High School Units” document. Since this post is getting lengthy I will respond to some of the rational behind specific choices in other posts.

Although I agree with some of the changes (and have concerns about others), I am more concerned about the work that we are doing with districts who are working on developing their scope and sequences for all grades. I am simply concerned about where to move next. I want to guide our teachers in our district in the right direction. If anyone can be of service, I would greatly appreciate it.

As mentioned above, is there any further discussion of completing a curricular priorities document that breaks high school courses down seperately? New York State has released their version, but in Algebra I, for example, all of the standards are listed as either 70% or 20%. There are no 10% standards. The URL to this document is http://engageny.org/wp-content/uploads/2012/03/nys-math-emphases-k-hs.pdf. I have a hard time believing that what is the emphasis in one course should automatically be considered an emphasis in the next.

Dr. McCallum and team… thank you again for all of your work around the shift to the Common Core. Your materials are invaluable and we have been basing most of our work around what you are producing. We can’t thank you enough. If you have any ideas/suggestions to help lead this work in the right direction, we would appreciate the feedback!

Will there be a way to search for highly rated tasks and/or will they be order that way?

Is there a way to receive an email notification if someone else comments on the same task I commented on?

There are already some tasks designed for modeling and I think some tasks could be reworked into projects with a little fine tuning. Maybe it’s a start?

Hopefully, someone will write a CCSS “modeling” handbook for teachers, as well as a book/website with suggested “projects” aimed directly at meeting the standards.

This is good. In the future, perhaps a logical progression by mathematical standard will be written. I am patiently waiting.

Nicely done.

I’m thinking about this (and all fractions instruction these days, frankly) in the context of the excellent book Extending Children’s Mathematics by Empson & Levi (2011). If you’ve not read it yet, one of the things they talk about is introducing fraction concepts through contexts that give meaning to the fractional quantities. So I’m wondering, if kids are using an area model that is rooted in an accessible context (4 children are given 11 brownies to share, how much does each child get?), can our concerns about their formal understanding of “area” can be allayed somewhat? Thoughts?

Are students supposed to memorize the conversions for the stated units, or just apply understanding about the relative sizes and conversions to solve problems?

While the latter provides an awesome opportunity to connect this standard to much of the OA domain at this grade (especially 4.OA.5) and the NF domain, the former would require a bit of time to develop and it doesn’t connect to any of the Critical Areas in grade 4.

Thank you for your support,
Brian

I re-discovered this question while doing work with my fourth grade teachers last week, then, yesterday, I received an email from a Math Coach from another state asking me if I had any answers on the same topic. Can you offer any clarification regarding the intention of 4.MD.4 and 5.MD.2? Are students supposed to measure to the 1/8 inch in grade 4, or is 1/4 inch (in grade 3) the most precise measuring the standards require?

Thanks,
Brian

To use the area model of fractions students must:
Know what area is,
Identify the area of the part,
Identify the area of the whole, and
Compare the two areas by direct or indirect measurement.
This is not a problem when students know enough about area to be able to do this.

That is, the concern is not about the introduction of one half or one quarter but the use of the area model to teach fractions when students have yet to develop a solid understanding of area measurement.

i have actuallywatched them all before. i am working on a using them as an organizing thread for some work we need to do next week.

Talk to you tomorrow.

Kathy

In the last one, the idea of rigor is, in my interpretation, defined as a cross section of focus and coherence. Rigor is accomplished through giving students conceptual understanding, procedural skill and fluency, and problem solving skills. Rigor is also accomplished by implementing the content standards that have a high cognitive demand for reasoning, sense making, and justification.The practice standards provide students with a habits of mind that help them to make sense of the content standards. Focus on fewer things gives kids more time to master the important understandings that are stepping stones in the stream of coherence. Mastery of one topic (given enough focus for students to learn) informs the next topic in the stream or progressions (helping students to make connections rather than memorize isolated facts.)

I look forward to working with you in our small state of Vermont to help teachers understand how to implement the CCSS-M!

Kathy here from Vermont. I was just asked to define rigor, focus and coherence as they are used in the Common Core mathematics. I have some thoughts but I would appreciate your ideas before I get too deep into this work.

It’s a complex process to design a curriculum to support the standards, and I don’t think kids should be “presented with the standards.” Rather, they should be presented with a curriculum which supports learning the knowledge described by the standards. On page of 8 of the documented at corestandards.org, it says

“These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.”

Thank you so much the article, “The Structure is the Standards”. It says so eloquently what I have been trying to explain to teachers for the past year. There are so many layers to, and connections between these new math standards.

The question we are teaching our students to ask themselves this year is: “What do I already know that can help me figure out this new problem?” Those connections that can be made to previous learning are what make the new standards so powerful. We are discovering that students are arriving at deeper levels of mathematical understanding, even in the early grades, as our teachers become more and more intentional about setting up learning experiences that allow students to make understanding for themselves. It has become clearer than ever the importance of teaching for understanding, rather than just teaching isolated skills.

I do have a question for you. Is fourth grade the first time that students specifically have to make conversions within the same measurement system? I have read everything that I can find and do not see anything at the third grade level that asks students to make conversions. A question came up about this recently and I just want to make sure that I have the correct answer.

Thank you!

The researchers and practitioners I referenced assert that unless it is crystal clear to a student where they need to “end up” the benefits of formatively assessing students and including students in the process of instructional decision-making cannot be realized.

If a student is presented with any kind of standard in any subject, but the student doesn’t understand what the standard means, then teachers and students need to “unpack” the standard so that it does make sense to students. Once students and teachers clearly understand the way(s) a standard describes mastery of knowledge, reasoning, or performances, then both may engage in a diagnosis of the strengths and weaknesses a student’s work reflects in relation to the standard(s) attempting to be mastered. This diagnosis informs what gaps may exist between current understanding and mastery, and informs both teachers and students how they might close those gaps.

Your essay seems to treat all “unpacking” processes as a hammer smashing an urn; however, the word “unpacking” also refers to a process of analyzing standards and expressing them in a way that makes sense to students. This may, for certain students in certain grades, mean that a single standard may need to be expressed in its component parts; however, and this is what I believe you and yours are asserting, if expressing a complicated standard to a student leads either student or teacher to permanently shift perspective to a micro-level, and forget the macro-level from which the standard came, then we have problems.

In essence, just as mathematicians and students can “step back and shift perspective” and see “complicated objects as composed of several objects” in making sense of mathematics, so too can educators do the same with standards. Educators and students can unpack standards for the purposes of formatively assessing and student self-assessment and, as long as they “maintain an oversight of the process” and are aware of how “intermediate results” relate to the overall task at hand “unpacking” does not necessarily lead to a loss in the overall structure of the standards.

Can you please cite the specific education research consulted in making your argument? If no specific education research was consulted, can you please explain why?

On the other hand, the OA progression is referring to multiplying a number by a fraction, which is conceptually more difficult. Thus in Grade 5 we have 5.NF.4, “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”

I read your progression on fractions with great interest. I have a question. It seems as though, in other posts, that I’ve seen you refer to K-8 progressions. However, the fractions progression ends with grade 5. Can you tell me if, in the revision of the draft, you intend to include anything beyond that grade level? There is some in 6th.

Thanks

Best,
~Nancy

We are sorry to have missed your expertise this weekend as we created professional development units together in Tucson. Many great tools and PD modules came out of the workshop. We are now in the period of revision by the authors and a bit of formatting before the modules are ready to share. Once they are ready we will be holding a much larger workshop to disseminate the information to even more regions, states, and districts nationwide. We will have information on the blog when the registration for the larger workshop at the end of April becomes available. Registration should be open soon!

And wonderful to meet you Elaine. I hope some collaboration can occur in VT!

Ellen