I use the area method to generate the problem to hide “cross-multiplying.” Slow the problem down only a little bit or simply teach “flip and multiply.”

]]>All of us have an intuitive idea of what it means to represent a situation;

we do it all the time when we teach or do mathematics. We represent numbers by points on a line or by rows of blocks. We use equations and geometric figures to represent each other. We talk about numerical, visual, tabular, and algebraic representations. And we think about things using “private”

representations and mental images that are often difficult to describe.

But what do we mean, precisely, by “representation,” and what does it

mean to represent something? These turn out to be hard philosophical questions

that get at the very nature of mathematical thinking.

I believe that as mathematics itself evolves, new methods and results shed

light on such questions—that mathematics is its own mirror on the very

thinking that creates it. And sure enough, there is a mathematical discipline

called representation theory. In representation theory, one attempts to

understand a mathematical structure by setting up a structure-preserving

map (or correspondence) between it and a better-understood structure.

There are two features of this mathematical use of the word representation

that mirror uses of “representation’’ in this book:

• The representation is the map. It is neither the source of the representation

(the thing being represented) nor its target (the better-understood

object). When a child sets up a correspondence between numbers and

points on a line, the points are not the representation; the representation

lives in the setting up of the correspondence.

• Representations don’t just match things; they preserve structure. Entering

on a calculator an algebraic expression that stands for a physical interaction

is not, all by itself, a representation. If algebraic operations on the

expression correspond to transformations of the physical situation, then

we have a genuine representation. Representations are “packages’’ that

assign objects and their transformations to other objects and their transformations.

If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?

]]>Related: We’re working on a computer science curriculum that uses a programming language that is unambiguous about what gets done when and doesn’t use parentheses (well, you may say that it uses them in disguise).

The course is called \emph{Beauty and Joy of Computing}:

The language (like many programming languages) \emph{requires} you to say what you mean. As an example, here are the two versions of the calculation posted by Bob Knittle.

And you can run the calculations to see if they give you what you want.

Al

]]>So no, there is no need to make a rule about whether you distribute first then add, or add what’s in the parentheses first and then multiply, and in fact such a rule would be harmful because there would be some occasions when you wanted to do one and some when you wanted to do the other. Which you choose depends on your purpose.

I wouldn’t talk about distribution as an operation at all, but would rather talk about the distributive *property*, which tells me that $a(b+c) = ab + ac$. What I do with that information is up to me.

6divided by 2 times quantity 2 plus 1 close quantity.

6/2*(2+1)

Once you perform the inner operation, does distribution precede the division – which is my question, does distribution require its own order of operations?

]]>I have used “the opposite of”.

If 3=three

Then -3 is the opposite of 3 which is of course negative 3

And -(-3) is the opposite of negative 3 which is positive 3

The pattern continues when applied to operations.

]]>Once there’s a better understanding of transformations, I like to think of it in terms of reflections of the number line. Multiplying by a negative corresponds to reflecting and dilating the number line. A negative times a negative is two reflections, which is the same effect as multiplying by a positive.

]]>My thoughts on this topic are here:

https://iheartgeo.wordpress.com/2014/03/08/multiplying-negative-numbers/

James

]]>I side with the posts that have “because we said so” under the hood. I’ve always tried to convey in my (HS) classes that the fact that $(-3)(5) = -15$ is not an act of congress or a rule of nature—it’s imposed on us by the desire to preserve the “rules of arithmetic” when we extend the systems in which we calculate (it’s the same motivation when extending exponentiation beyond whole number exponents). So a motivating context that stays close to the underlying mathematics should lead to student questions about why the rules need to be what they are.

Here’s one context (albeit mathematical) that emphasizes the extension theme and that has worked well for us and that finds use in our other HS courses (algebra 2, for example).

Start with the ordinary addition and multiplication tables, oriented in an unusual way:

https://go.edc.org/arithmetic-tables

Look at “addtable” and “multable”, for example. We first ask kids to fill in the blanks, just to get a rhythm going. Then turn them loose on finding as many patterns as they can. Most look at diagonals first. Sooner or later, many kids start noticing the corners of rectangles, up and over patterns, and so on. These can (now or later) be proved using the basic rules of algebra and things like $(a-1)(a+1)$. Many people (including adults) get engrossed in what they find in the multiplication table. Roger Howe has created for his students a nice collection of algebraic identities that arises here. I put a set of notes in the dropbox that we used for an all-day seminar for teachers, that includes some of Roger’s stuff, and that shows just how far you can push all this.

Next, extend the tables in all directions as in the other two figs in the dropbox. Here’s where the patterning comes in, but by now, the tables are contexts in their own right. You hear all kinds of ways to extend the patterns, most of them producing the usual extensions. But someone always has a different one like going across a row decreasing to 0 and then going back up. Here’s the chance to ask if the usual rules of arithmetic extend via this new extension. They don’t, and this is a good chance to introduce a derivation similar to what Bill posted.

BTW, the arrangment of the tables leads to other things. Circle all the 12s in the mult table. It looks like it can be fit with a nice curve. What’s its equation? Why? Stuff like that.

Al Cuoco

]]>“For all of human history, we have relied on patterns that we observe in order to make initial conjectures that may or may not be true about numbers and relationships, but in my class, we will deduce everything from first principles.”

This just seems like a ridiculous position to assert, given our history as humans doing mathematics.

FYI, there is a textbook based on this approach that derives all of elementary school mathematics from the Peano axioms. It’s 4000 pages long.

]]>I like your original argument however, I agree with you that from the perspective of a kid, sounds very similar to “it’s just true because it has to be true.”

I have a related post on subtracting negative numbers which is the result of a conversation with my wife. The idea there is that the goal is the learner understanding the math, not the context itself. As it turns out the “mathematical consistency as derived from a pattern” argument worked the best with my wife.

It’s also a good instructional strategy to have kids practice something they know, observe a pattern as a result of that practice, and then test applying that pattern to a slightly novel idea. I kind of describe this strategy in this post.

So that you don’t have to click on the post I’ll summarize it here:

Have kids calculate the following:

3 x 4 =

3 x 3 =

3 x 2 =

3 x 1 =

3 x 0 =

Ask them what they notice then “What would 3 x -1 be so that the pattern continues?”

Now the do the reverse (I don’t think we can assume kids naturally see/use commutativity):

4 x 5 =

3 x 5 =

2 x 5 =

1 x 5 =

0 x 5 =

Same questions basically, except now it’s -1 x 5.

Now ask students to apply what they’ve learned and try out a sequence of problems like this:

5 x -3 = -15

4 x -3 = -12

3 x -3 = -9

2 x -3 = -6

1 x -3 = -3

0 x -3 = 0

Ask them what they notice. “What would -1 x -3 be so that this pattern continues?”

I actually tried this strategy with a small group of kids about a year ago and it worked great. Afterward, I asked students to write a reflection and to generalize what they learned to see how it might work with all numbers.

One student basically, without special prompting from me, the “rules” that are often taught as the starting place for operations with signed numbers.

Once I think this argument is understood, *then* I would build toward an argument like yours which as you noted isn’t limited to operations on integers. I really like that aspect.

A more complete definition for multiplication makes this much less mysterious, as I am sure you know. A number is an arrow from zero. Multiply by “2” means make the arrow twice as long, multiply by -2 means make the arrow twice as long but in the opposite direction.

Tempting to use your idea with students that are already proficient with negative number arithmetic to create some thought about “d = rt”.

Here are some other interesting ideas. One similar to d= rt involves a movie playing backwards.

]]>Time works the same way. You choose a time to start your stop watch, and then positive numbers refer to times after that and negative numbers refer to times before that. It’s a good question which of those times make sense in the context, and it’s not only negative times you have to worry about there, but also, for example, times after the projectile hits the ground. But one thing at a time! First establish an understanding of time as a signed quantity, then start worrying about the domain for a given context.

]]>I’m also wondering how negative time works with interpreting roots of quadratic equations for a projectile launched from non-zero height b at time = 0, where the x-variable is time and the y-variable is height, and claiming that if one root is -3 that we know that the projectile was at ground level 3 seconds before launch: we don’t. It might have been in lots of different places at that time, including at b. We don’t know what was going on at t = -3 or any other negative time, do we?

]]>I find this better motivated for students than the version using zero, and it reinforces using properties for estimation, and promotes deeper understanding of area models.

]]>In our high school program, we have a motto “different forms for different purposes.” This certainly applies to algebraic expressions, but it also applies to rational numbers. Decimal notation is useful when you are concerned with \emph{value}—it helps you compare the size of a number or to determine how close two numbers are on a number line. Fractions are less good for this. But fraction representation is very useful when you care about the \emph{form} of a number, for example, when you want to investigate patterns in certain calculations. In a way, the distinction is between the analytic and algebraic faces of rational numbers. The representations come together, of course—the key to the length of the period for the decimal expansion of a rational number lies in its representation as a fraction.

Al (I have no idea why this lists me as “Budapest Education”)

]]> 1/8 = 0.1 + ?

1/7 = 0.14 + ?

1/3 = 0.3 + ?

At her stage of learning, these problems are thought-provoking; they are not mechanical exercises. The problems led to some good conversations. In particular, on the topic of your post today, it was clear from talking to my daughter that she has a sturdy conception of an “amount,” independent of the form in which that amount is expressed. (Her conception of ‘amount’ seems to be something like what Pat Thompson calls magnitude.)

The presence of addition/subtraction in the above problems also reminds me of something that I once read in a paper by (I think) Guershon Harel. The idea as I recall was along the following lines. One way in which new kinds of numbers get integrated into a student’s evolving conception of number is when the student calculates with the new kinds of numbers. In the case of fractions, for example, calculating with fractions tends to ‘make fractions numbers’ simply because, to some extent, ‘numbers’ just means ‘the things one does mathematics with.’ (This was the idea behind the task “Are Fractions Numbers?” http://achievethecore.org/page/929/are-fractions-numbers.)

]]>The heuristic you are looking for really comes into play as students develop fluency with the operation. Fluency is students ability to solve a problem flexibly, accurately, efficeintly, and choose a strategy appropriately. The context (e.g., situation or numbers) drive appropriateness. Do I need to need to calculate or estimate? How accurately? Does it make sense to use an algorithm or a counting strategy? For example, what is efficient for 204 – 199? or Do I have enough money to by xxxxx?

]]>As for your general point about what standards should be doing, I agree that there is a line to draw between standards and curriculum, and reasonable people can disagree about where that line should be. Wherever it is, discussions about how to cross that line are a large part of the purpose of this blog.

]]>To my reading, this standard does not define multiplication at all, but gives a student a way to make sense of multiplication given the scope that they will need to apply it in Grade 3, but perhaps under-sells the fact that this is indeed an interpretation and not a definition. That there are other interpretations (see, 4.OA.1 for starters), and none of the interpretations is truly what multiplication is in and of itself. This subtlety is not something that 8- and 9-year-olds should necessarily recognize, but it is something that the skillful teacher should be mindful of in order to keep students from getting too locked in to any particular interpretation of a concept that demands flexibility.

]]>Multiple representations of mathematical concepts exist for pretty much every concept taught within the CCSS. This is part of the beauty of mathematical thinking. To be able to see, for another instance, the Pythagorean Relationship can be viewed within a Number Theoretical, 2-D Geometric, Volumetric, or Trigonometric etc. perspectives each of these sheds light on and improves the depth of understanding of the basic principle. Fostering these multiple perspectives and representations of ideas or concepts is what good teaching should do. ]]>

3.OA.1 Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. (Note: These standards are written with the convention that a x b means a groups of b objects each; however, because of the commutative property, students may also interpret 5 x 7 as the total number of objects in 7 groups of 5 objects each).

Commutativity needs to be revisited regarding fraction multiplication. For example,

5 x (1/3), interpreted as “5 copies of 1/3” [4.NF.4a]

(1/3) x 5, interpreted as “1/3 of a copy of 5” . [5.NF.4a]

The second is harder than the first, and the important work is showing through pictures that they are equivalent. A robust understanding of fraction multiplication involves both of these ways of thinking.

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

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

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

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

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

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

By the way, if you want to continue this thread, it should probably go over in the forum on arranging the standards into courses. http://commoncoretools.me/forums/forum/public/arranging-the-high-school-standards-into-courses/. I’ll repost it over there.

]]>For example, in upper grades, there are choices to make when you see something you know is true but want to prove—this one feels like induction will work, this looks like a “suppose not” indirect method, or maybe a generic calculation will work. Or in algebra, the CCSSM advice about “different forms for different purposes” encourages students to transform an expression according to what they want to see. Similar heuristics live in geometry

Is there anything like that in these multiplication strategies, or are they all focused on getting to understand what’s behind the efficient methods?

Al

]]>What do you do when the students perceive the multiple methods as isolated things to be memorized? ]]>

I’d like to split hairs a bit, as your phrase “with the ultimate goal being the standard algorithm” is likely to be misinterpreted. While the standard algorithm through its universality is an ultimate goal in a narrow sense, I’d say the ultimate goal is to have access to both for purposes of wider efficiency as well as the number sense which goes with it. For example, one shouldn’t have to “borrow” (ack) a couple times to subtract 6 from 204. Though when I talk to parents about CCSSM (which led to the notes you posted here: http://commoncoretools.me/2014/11/21/common-core-math-parent-handouts-by-tricia-bevans-and-dev-sinha/ ), they often comment that they’d have to use the standard algorithm. That’s fine, but I think having both is better, and highly numerate people such as professional mathematicians, scientists and engineers have often taught themselves these alternate strategies. I’m reminded here of a post by Jason where he outlines how one can implement a curricular sequence in which the standard algorithm, with understanding, comes early on and then the strategies are brought in in part of more efficiently deal with special cases.

Early after CCSSM came out, I heard some suggest that it was too much to aspire for all students to have access to both, but now I’ve seen a variety of implementations with a wide range of student populations which have achieved exactly that.

]]>That perception might not have been the intent of those drafting the standards, but it would be naive and/or irresponsible not to anticipate how things inevitably would play out in practice given the past track record of other major standards efforts in mathematics and literacy in this country. Of course, there is also inevitable skepticism, criticism, and resistance to standards, but for the majority of teachers, administrators, and parents, the perception is that the Common Core was written to be followed like many districts expect scope and sequence calendars to be followed (I have seen this in Detroit, to name one large district where high school teachers fear diverging from the district’s calendar lest they be “caught” and punished). This sort of thing kills creativity on the part of teachers and students, and it absolutely undermines the ability of instructors and students to take advantage of serendipitous opportunities to explore and play in the rich fields of mathematics.

No matter how coherent a curriculum might be based on standards that themselves are, ostensibly, coherent, they are only coherent from one point of view: that of their author(s). It is folly to believe there is some privileged objectively universal stance from which one can glean the perfect ordering of topics that will serve all learners well. All a good curriculum can hope to do is to provide some framework, preferably one with a variety of pathways and branchings, never with tight strictures against finding sound alternatives to the choices provided, and indeed offering encouragement and suggestions as to how and when to leave the “official” pathways.

That is not to suggest that all viewpoints are equally good, all choices of paths equally likely to be effective, all possible journeys equally worth pursuing without limit. We always have to make sensible evaluations as we begin to pursue alternative routes. But this is precisely where teachers’ pedagogical content knowledge comes into play. And I fear that the combination of federal strictures and monetary threats intended to ensure that educators stay within a very narrow distance from some intended pathway, coupled with publisher guidelines codified in official teacher manuals, and the tendency of administrators and many teachers themselves to be lazy, fearful, or poorly-prepared to make the necessary countless decisions about teacher moves that arise in the course of lessons means that in spite of the best intentions of everyone involved in mandating, crafting, disseminating, and implementing any set of standards or turning them into a “coherent curriculum,” I believe that in this country, at least, they will have a conservative or reactionary effect upon what mathematics is taught and how lessons are presented. And that is in spite of the one feature of the entire Common Core State Standards that I find admirable, the Standards for Mathematical Practice. Some lip-service is paid to these by publishers and administrators at various levels of the federal, state, and local educational institutions, but the focus from the beginning has been and inevitably will continue to be the content standards because of high-stakes testing and its consequences. And until the culture of testing and assessment changes dramatically, the coherence of standards and/or curriculum matter far less than we might believe or wish.

]]>(*) Bill’s story of encapsulation is mirrored in the story of encapsulation of functions in HS. Ask a second grader “what’s 1/2?” and you’ll usually get “half of what?” Ask a HS sophomore “what’s cosine?” and you often hear “cosine of what?”. These operators need to be encapsulated into \emph{things} in order to use them as inputs to other higher order operations, like addition of fractions or derivatives of functions. Common Core makes explicit this journey from process to object, a journey that takes time and attention in school mathematics.

(*) There’s a substantial intersection of the use of encapsulation between mathematics and computer science. One of the most elegant treatment of this intersection is the classic CS book “Structure and Interpretation of Computer Programs” by Ableson and Sussman:

https://mitpress.mit.edu/sicp/full-text/book/book.html

SCIP is a treatment of computational thinking, but it’s one of the best sources for mathematical thinking that I’ve ever read.

—Al

]]>http://ime.math.arizona.edu/progressions/

to get to these progressions. I noticed that the document for NBT is not this latest version. Is there a better link to find the most updated versions of the progressions? ]]>

However my next question is about the geometry HS progressions document. Will it be out soon? I have reviewed Wu’s articles Geometry 4 – 12 and his 8 -HS geometry document. Both are excellent. ]]>

c Illustrative Mathematics

12 February 2014

Could you please give me some clarification? Thank you!

Katy

While a longtime Common Core enthusiast, I’ve always thought that Appendix A missed the mark. I’ve talked with the authors of App A about this, and I understand that it was a rapid prototype, just to show that one could create three courses that meet all the (non +) standards.

But some of the ways things are split between courses don’t make mathematical sense. For example, adding and multiplying two polynomials is the same (and requires the same skills) no matter what the degrees of the polynomials are. Yet these skills are divided between algebra~1 and algebra~2; algebra~1 is restricted to linear and quadratic expressions (so, multiplying a linear expression by a quadratic is off limits, it seems, in A1).

It makes much more sense to split the subject of polynomial algebra by operations rather than degree, so that addition and multiplication are part of A1 and division with remainder, the factor and remainder theorems, and composition are reserved for A2.

There are other examples: That the graph of an equation in two variables often forms a curve (which could be a line) is a statement about *any* equation, regardless of its type or special characteristics. An A1 student who understands this idea should, given the graph of an equation in x and y, be able to sketch the graph of the equation resulting from replacing x by x-3 and y by y+2.

General principles need to be illustrated in special cases, but the generality of the principles should not be hidden or tied to the specific examples.

The PARCC content frameworks takes a more sensible approach to how the standards split between elementary and advanced algebra.

]]>Most of my colleagues are intimidated by mathematics and went into elementary education because they didn’t think they would have to teach “hard” math.

The message that teaching only standard algorithm meets the standards, robs children of mathematics and enables teachers to not build their content knowledge in the discipline.

I do not disagree with the progression. I am an ardent supporter of the standards. However, students who work in procedures only are ill-equipped to mentally compute numbers because they lack place value understanding.

While I see the need to placate the masses, we should also recognize that our students and parents deserve better than to be happy. They deserve to be mathematically literate.

]]>My progression sketch shows one of the possible worlds. To see it, click the link in the post that says “accompanying table.” One will see that concepts are prominent in the progression, that there are questions about the procedure and why it works, that it is to be taught in such a way that it makes sense to students, and that there is also attention to mental computations like 6012 – 13 or 400 – 388, for which the standard algorithm is probably both slower and less reliable than a readily apparent mental strategy. These aspects of the progression show that it is not about “adhering to procedures without understanding.”

]]>As a math educator, I am disappointed that Mr. Zimba is encouraging the sentiment that teachers who teach only standard algorithm out of a textbook are still doing their jobs. So much for getting at those mathematical practices by adhering to procedures without understanding.

]]>First, if you read the table accompanying Jason’s post, you will see that there is more than “just teach[ing] the standard algorithms.” There is also sense making and mental computation.

Second, the Progressions documents do a lot of what you ask for in making connections among the standards and explaining how they fit together across grade levels and how they might work with children.

Third, I agree a gradual rollout would have been better. But that’s not the way anybody ever does anything in this country! And you talk as if there was someone who could have decided to do things differently, but a distinctive feature of the US education system is that there is no such someone, not even any such organization. In fact, it is barely a system at all. The rollout was up to the individual states who adopted the standards (with the federal government mixing in).

Finally, on assessment, I agree that assessments are currently tethered to too many high stakes. I think the standards have caused people to face up to this problem. I do think good assessments administered wisely and humanely are necessary.

]]>…

All that said, my question becomes this: if you really believe in the idea of changing the game for math education in this country, how can you allow that it *suffices* to just teach the standard algorithms. That is tantamount to saying that as far as doing anything really new in US math education, the authors of the Common Core Math Content Standards were “just kidding. Please go on with business as usual.”

…

Some of the biggest failures thus far I’ve seen in the Common Core have revolved around a failure to adequately prepare teachers or the public for what changes are coming and why … . I mean actually explaining why doing a particular thing in mathematics classes at a certain point is important mathematically; how it adds to and deepens students’ understanding and facility; how it breathes life back into mathematics education for as many children as possible.

Part and parcel with that issue is the decision to make wholesale changes in all grades at once, rather than to roll things out one or at most two grades per year. That would have allowed us to gather data necessary to revise what’s done next year, and then feeding more data into the loop for training new cadres of teachers for the new grade(s) being added. I think trying to do it all at once undermines the entire project.

Finally, the strong tie between high stakes testing and the standards continues to be a huge error, one that is clearly undermining public confidence in the standards and raising objections to them from parents and educators. As long as these tests and the standards are intertwined, they are doomed to fail. …

]]>Is it possible to purchase the elementary series now just to receive the recordings for all of them? ]]>

Ellen ]]>

Thanks for sharing your work. I noticed that the lesson plan, Plane Figure Court, assumes the exclusive definition of trapezoid. Worth noting for those who have decided to use the inclusive definition.

Thanks again for your work.

Best,

Turtle ]]>

I am the math and gifted curriculum coordinator at a public PreK-12 school in Hartford, CT. We have been having conversations surrounding “power standards” in order to better plan for our students. I am of the impression that identifying power standards is an obsolete process because they seem to be built into the CCSS via the clusters. Can you shed some light on this idea please! Is there a way to identify the power standards in the CCSS? If so, where should we begin?

]]>– Most statisticians do not consider statistics as a subset of mathematics. It uses mathematical tools, but also methods from science.

– Thanks for pointing that out

– The progressions documents are meant to be a translation of the standards into narrative form. There was too much personal opinion here that is not reflected in the standards.

– Good!

– No, that isn’t possible with the resources we have. ]]>

Thank you. ]]>

– I notice that in both drafts you separate statistical and mathematical models. Is there a reason that statistics is not considered a subset of mathematics?

– It seems on the first page you have lost the quote by Wigner, but not the citation.

– What was the thinking behind getting rid of the section on word problems being the bane of school mathematics? I thought this was a powerful statement addressing the needs of those students that don’t understand why they would ever want 84 grapefruit.

– I love the statement “These diagrams of modeling processes are intended as guides for teachers and curriculum developers rather than as illustrators of steps to be memorized by students.” (I wonder if this in in reaction to how some utilized the modeling cycle originally listed in the CCSSM).

– As you created a revision for this and some of the other progression documents, will you release any sort of “track changes” or “summary of changes” for those that are familiar with the previous version?

Thanks!

]]>More examples of this standard in action are in Chapter 2 of the NCTM monograph “Reasoning and Sense Making in Algebra”

http://www.nctm.org/catalog/product.aspx?id=13524

Al Cuoco

]]>But no matter what happens at the state level, I plan to teach the CCS. I’ve put lots of time into the transition, and by the time classes start in August, I’ll be ready to go. (Indeed I’m just about ready to go now. Check out my site.)

]]>If this was not the intent, is there a concern about spacing out the content a bit more? For example, my district has 180 contact days with students. If we were to spend more time on certain units then an entire unit, such as probability and statistics, could fall after the PARCC assessment window. Is the expectation that all content will be tested or is it expected that certain content at the end of the year would not be tested on the end of course tests.

I appreciate any feedback. We are working hard to develop more in depth documents for our teachers and would like to know our time frame. Thanks!

]]>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}.”)

]]>What are your thoughts? ]]>

(Maybe I should have waited and put everything in one message. There are a couple of other small typos, let me know if you want them before the next draft.) ]]>

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!

]]>The index to the illustrations was very useful; however, I am currently creating develop a scope and sequence document for grade 6 and find something missing which would be of great help. When I go to look for an illustration for a particular domain and cluster (for example 6.NS.5) I can use the new index to find the illustrations for the grade and domain I want but have to open every illustration to find the cluster I am looking for. It would be of enormous help add the cluster to the domain for each illustration. If the cluster were added a helpful but not essential additional step would be to order the illustrations by cluster under each domain.

Robert Springer ]]>

The new index to the illustrations provides a very valuable tool. I am helping to write a scope and sequence for grade 6 and can now easily look up illustrative problems which I can include as links in each section of the standard. ]]>

* 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!

]]>A colleague shared information about some fraction progressions videos from a project you are involved in. They are part of a course which also includes links to tasks and quizzes. How can I find out additional information about these? 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

]]>…thoughts? ]]>

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.

]]>Also, is there any coorelation/cooperation between these and the Dana Center’s scope and sequence? ]]>

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.

]]>I asked a question earlier about the sequence of standards, the way they are sequenced in the CCSS document compared to the sequence that a classroom teacher might create. I do understand from page 5 that the sequence in the CCSS was not intended to be a rigid format for us to hold ourselves to. Our academic leaders have always given us given the liberty to decide what is best for the students within our classroom environment. I’d like to narrow my initial comment down to a more specific concern about the sequence of 5.NBT. Scenario: I teach 5.NBT.2 the first week, followed by 5.NBT.5-.6. Seven weeks later I teach 5.NBT.1 and 3,4,6,and 7 later on. In this case, it seems to me that NBT.1 would be a foundation that leads precedes NBT.2. The remaining standards within NBT would be at my discretion. As a classroom teacher, could you see the rationale for the sequence of NBT.2 before NBT.1? There is a lot of ground to cover with the CCSS and I want to make sure I do it justice on behalf of my students. Thank you in advance for your reply. ]]>

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.

]]>Some fundamental formulas for area are in the standards, but in many cases students are expected to find areas by decomposing figures, e.g.

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 = (1/3) 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’m not sure if you’re looking for suggestions on how to address money as a life skill (as opposed to a math skill) or not, but I’ll offer one up anyway.

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

]]>My question is: If part of the 8.G.05 standard is supposed to “establish facts about the angles created when parallel lines are cut by a transversal”, is it ok that the official names of the angles created (alternate interior, alternate exterior, etc) be used when assessing students knowledge of the facts established under this task? ]]>

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.

]]>I am in need of clarification about the extent of teaching 6th graders about inequalities. II have seen in some task and common core workbooks where the student is being asked to solve 1-step inequalities,but it is not explicitly stated in the standards (see below).

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

]]>How does a commercial (for profit) organization get permission to reprint and distribute your tools (such as the CCSS progression) to participants in a face to face course? I could not find a specific email address to contact for permissions. Thanks. ]]>

Re. improvised units –

An example of an improvised unit I’ve used in third grade is post-it notes. Students used them to cover their tables and other surfaces in the classroom. The goal of the lessons was not to get a standard measure of the area of the table, but to ensure students understood that area was a measure of coverage of the plane. It was also an opportunity to see if students had internalized the structure of arrays. Since the post-its were of a uniform size, we could also compare results.

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/

]]>Thank you for the help. I found your answer to Susie H and modeling is stating to get clarified for me. I will study what I’ve found. ]]>

I looked for the thread on modeling, but it took me to Task #4 Contest. I am concerned the definition of modeling because it seems that most people I speak to have their own perception.

I can see what modeling looks like at the high school level (there are many examples out there), but I’m still confused as to how it would look at grades 3-5. Is a number line modeling? I can make an argument for it’s use as a model, say for explaining or proving addition of fractions. I guess I need some examples of what modeling would look like at these grades. ]]>

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://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

– How much detail would they need?

– What kinds of questions would they be expected to answer?

Can anyone provide pointers in the right direction? Thank you! ]]>

“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.

]]>My favorite quote for the day. Perfect. ]]>

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.

]]>On your suggested Scope and Sequence for the high school Common Core State Standards, you refer to 6 Projects. Do you have 6 specific projects in mind? Have they been created or are they just place holders? ]]>

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!

]]>———————————————————–

Patrick says:

April 19, 2012 at 9:16 am

One of ther question that I would have is… has anyone taken a poll of the CC states to see which states have adopted the Traditional Model at the HS level vs. the Integrated approach? This could be useful as we begin to share information/resources, etc. Thank you.

————————————————————-

I’m a teacher at a boarding school in Utah, and many of our students come from California. Utah is going with the integrated model for its high school math. In California, at what level is the “integrated vs traditional model” decision made? I mean, will I have students next year from California who were taking “Integrated Math 9” and others of the same grade taking traditional “Algebra One?”

I think the integrated model is less-adopted across the nation. I surmise this because of how many Utah entities (the State, a few districts, and a few schools) are currently actually writing their own online textbooks just for “Secondary Math I (i.e., Integrated Math 9).” ]]>

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

Projects are where students USE and DO mathematics to create, build, develop something that hopefully necessitates the content. For example, building catapults to analyze projectile motion and quadratic functions. ]]>

The documents are open for public comment now.

http://www.p12.nysed.gov/apda/common-core-sample-questions/

This discussion (paraphrased) has recently come up on our K-5 mathematics wiki-

“Concerning 5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.The Progressions document states, “In Grade 5, this work should be viewed as exploratory rather than for attaining mastery; for example, expressions should not contain nested grouping symbols and they should be no more complex than the expressions one finds in a application of the associative or distributive property, e.g., (8 + 27)+2 or (6 x 30) + (6 x 7)”. Would you clarify how this standard should be taught? How do you teach brackets, parentheses, and braces without nesting them within an equation or expression? Aren’t they hierarchical in nature?” I went out hunting, and here’s what the North Carolina explanation says: “Mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there cannot be braces in a problem that does not have both parentheses and brackets.”

I understand the confusion being generated here, and would love to have your thoughts on the matter. If you want to respond directly on the wiki (If not, I’ll cut and paste!)

– here’s the link- http://ccgpsmathematicsk-5.wikispaces.com/

The question is posted in the discussion forum on the home page.

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

Best,

Turtle

I have a question on 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.

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.

]]>Thanks for the clarification. You’re correct that I was looking at an old PDF of the standards, but please note: Although the downloadable version of the standards at corestandards.org has revised wording for F.TF.3, the online version does not. If you go to http://corestandards.org/the-standards/mathematics/high-school-functions/trigonometric-functions/, you’ll see that the original version of F.TF.3 is still there. ]]>

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?

]]>I am excited about the focus on numbers and operations in the K-5 standards. ]]>

I asked Dr. McCallum this same question via email before he provided this stream on his blog. This was his reply,

“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.)

]]>G.CO.2 discusses using functional notation to describe transformations and G.CO.6 discusses using geometric descriptions of rigid motions to transform figures and predict effects of a rigid motion on a given figure. Specifically what is the difference between the two different modes of description? Is it as simple as using the terms “reflect”, “rotate”, “translate”, or “dialate”? ]]>

And then is the thinking behind G-SRT.3 along the same lines?

Thanks very much for any help or thoughts on this! ]]>

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.

There’s a long discussion of this topic, with some responses already from Dr. McCallum, at the “General Questions about the Standards” site. Find “denominators” if you don’t want to scroll through the whole thing.

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.

]]>1) Will students need to rotate around a point other than the origin?

2) Will students need to reflect across any line other than the x-axis, y-axis, y=x, and y=-x?

3) Will students need to dilate using a center point other than (0,0)?

We are in the process of writing units and wanted to make sure we were covering the rigor required by the standards.

Thank you for assistance.

Elizabeth ]]>

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.

]]>http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/05/TaskItemSpecifications/Mathematics/MathematicsGeneralItemandTaskSpecificationsGrades6-8.pdf

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.

]]>(2) Same answer here; the standard certainly does not insist on the secant theorems, and a curriculum could satisfy the standards without them. But these are beautiful theorems, worth including if there is time.

(3) I think you meant G-GMD.4. Certainly, any shape is fair came, but your formulation makes me a bit nervous … we don’t want to get carried away with this. Again, the idea is to get kids thinking about shapes of cross sections. The idea is not to have some long exhaustive list of shapes and expect them to be able to deal with every single one on an assessment.

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

]]>

I’ve been thinking about how to respond to all this. I was just chatting with my colleague Cody Patterson and he said it in a way that I found very useful: finding least common denominators is a strategy, not an essential component of fraction addition (either numerical or algebraic). It’s important to separate out the task of understanding fraction addition as an operation from the task of finding efficient strategies one might use to find answers in special cases. We don’t want students to think the strategies are the same thing as the operations themselves. Job one is understanding the operation.
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.

Thank you for your time,

Julie Brandolino ]]>

Registered participants should have received an email last week detailing the location and shuttle schedules to the venue this weekend. The location is:

12727 Highway 90

Luling, LA 70070

And shuttle buses (through an outside company, not the hotel) will be leaving hotels this afternoon at either 2:15 or 2:30 depending on the hotel.

Looking forward to seeing you this weekend! ]]>

Thanks! ]]>

I’m a math coach in Massachusetts and we use Everyday Math in our district. We’ve been having some good discussions around the fraction standards in K-3 and if or when fractions of collections should be introduced. I can’t find any direct mention of fractions of sets. As I read the standards and the progressions document I’m interpreting the emphasis to be on area and number line models – so where does that leave the set model? Could you clarify this? Thank you. ]]>

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

In Grades 3 & 4, students learned that a/b is a pieces of 1/b units. So, they find it relatively easy to calculate 3/5 + 4/5 because 3 1/5 units and 4 1/5 units together will be 3+4 1/5 units, or 7/5. Then, in Grade 5, when they encounter 2/3 + 3/4, they would say, we can’t because 2 and 3 refer to different units. But, they learned in Grades 3 and 4, some fractions may look different but stand for the same numbers – and in Grade 4 they learned how to create equivalent fractions. So, they could say, well, 2/3 doesn’t always have to look that way (and 3/4 can look different, too). So, they find a common unit that can be used to express both 2/3 and 3/4. They realize 1/12 is an easy option since the way to create equivalent fractions is to multiply both the numerator and the denominator by the same number. So, if you have two unlike denominators, then one easy common unit is to use the product of the denominator as the common units.

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.

]]>“5.NF.1, the general formula for fraction addition is given as: a/b + c/d = (ad+bc)/bd”

is another formula to mess up; why not use multiplicative identity on primes til they see? ]]>

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

Brad, bburkman@lsmsa.edu.

]]>rate: a quantity derived from the ratio of two quantities that describes how many units of the first quantity corresponds to one unit of the second quantity

unit rate: the numerical part of a rate (e.g. For the rate 8 feet per second, the unit rate is 8.)

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.

]]>In 5.NF.1, the general formula for fraction addition is given as:

a/b + c/d = (ad+bc)/bd

I would write it as:

(ad+bc)/(bd)

I agree that we all know the order of operations that the author intended, but is it clear from what is written, and a good example of how to write fractions? ]]>

Regarding K.CC.1 “Count to 100 by ones and by tens,” I am wondering if the idea is for the student to be able to speak the two sequences rather than actually count objects. Sometimes, students live through 90 hours of calendar time per year for a number of years and then when asked to count a collection by fives, point to each object and speak a multiple of five. They don’t use the sequence to actually organize the objects in the collection and find out how many there are. So instead of 20 objects, they will respond with 100 objects.

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.

]]>I’m confused! Can you help explain?!

Sarah Renninger

Math Coach

New Jersey ]]>

ratio: describes the multiplicative relationship between two quantities

rate: a ratio of two quantitites with different units

unit rate: a rate that describes how many units of the first quantity corresponds to one unit of the second quantity. ]]>

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).**

- 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.
- Correctly name shapes regardless of their orientations or overall size.
- 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.

]]>The text file of all of the courses is missing pages 36-38. Can you repost with the complete file? ]]>

I had a question today from one of our districts about 3.G.2. In the NF domain we restrict denominators to 2, 3, 4, 6, and 8. Does the same restriction apply to partitioning shapes in 3.G.2 or should students be allowed to make as many partitions as they like? ]]>

My county is currently developing a Scope & Sequence for both grades. During this process, we were tasked with unwrapping the standards and developing unit plans of study. In comparing some of the geometry standards at K and 1, some questions were brought up, mainly involving K.G.1, K.G.2, K.G.4 and 1.G.1. The first grade team assumed that first graders would begin using defining attributes to determine proper names and classification of shapes, and that kindergarteners would have been recognizing shapes mainly by sight. The kindergarten team felt that kindergarteners would begin identifying, classifying and naming shapes using attributes, mainly number of sides and vertices. The team also felt that many misconceptions would be developed if students were only expected to use visual recognition to identify shapes. We sort of came to an agreement that in K, students might take a single shape and be able to define it based on sides or vertices. “I know this is a triangle because it has 3 sides” Whereas in 1st, students might take the attribute of 3 sides and identify all triangles from a set of shapes.

Without the geometry progression documents, is there a better explanation of the difference between these two grade level geometry standards and expectations? ]]>

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.

]]>One thing that does seem contradictory to the statement of multiplying withing products of 100 is 3.NBT.3, multiply one-digit numbers by multiples of 10 in the range 10-90. Even one example listed in the progression document (K-5 Number and Operations in Base Ten) shows 3 groups of 50, which would of course result in a product greater than 100.

My interpretation then would be that for students to explore equal groups of two-digit numbers is expected in third grade. Which makes sense when students are learning conceptually of what the meaning of multiplication is, as well as using place value throughout operations. (Modeling 3 groups of 25 would not be that different conceptually than modeling 6 groups of 4.) But for fluency, third graders would be expected to learn related multiplication and division facts of one-digit by one-digit factors. ]]>

I think your comment was disconnected from the original source. Can you clarify which document you are referring to?

Thanks!

Ashli ]]>

Multiply and divide within 100.

7. Fluently multiply and divide within 100, using strategies such as the

relationship between multiplication and division (e.g., knowing that 8 ×

5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end

of Grade 3, know from memory all products of two one-digit numbers.

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/.

]]>Our state (Florida) defined trapezoids in our testing glossary. So though we had great discussions, we let the students know that ultimately on our state test, we will define trapezoids as having only one pair of parallel lines. I imagine more clarity will come with test item specification from either PARCC or SBAC. ]]>

I don’t think that is the intention of the standard. I believe they just used the “within 100” example to cover all of the possible products of one digit factors. In fact, looking in 3.OA.7 it does say products of two one digit numbers. ]]>

– A question about the progressions in general:

I’m wondering if there is a timeframe for when the remainder of the progressons drafts and ultimately, the final version of the progressions documents, will be released? Do you anticipate that significant changes will be made to the draft documents?

– 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

]]>This post might be what you are looking for: http://commoncoretools.me/2012/03/16/arranging-the-high-school-standards-into-courses/

Cheers,

Ashli ]]>

I just wanted to follow up and let you know that the ability to directly link to a task at illustrativemathematics.org is now available. We are also working on providing a pdf link for all tasks, but that is a work in progress for older tasks.

Cheers,

Ashli ]]>

I work with teachers in multiplie school districts and have had the same reocurring question. What does “mastery” mean regarding the CCSS for math? Teachers are not sure if “mastery” in each grade level is referred to a certain percent or something else. Any explanation would be greatly appreciated! Thanks for this blog – it has helped answer a lot of questions!!!! ]]>

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.

]]>~Nancy ]]>

“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.

]]>First, thank you for answering all of our questions posted here. My questions are regarding Geometry in Grade 5.

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

I have a question about mode in the CC. I see median and mean mentioned in grade 6, but no mention of mode. Where should mode come into play, if at all?

Thanks so much for all of your insights. Excellent blog! ]]>

includes the expectation of using a linear equation to solve an exponential equation in Algebra I/Math I?

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 .

]]>You mention state testing (CSTs). In high school, the Common Core State Standards will not be tested with CST End of Course Exams. The summative assessment in high school (CA is part of SMARTER Balanced Assessment Consortium) will be at the end of junior year, and all standards except those marked with a (+) can be tested. Our current testing model will be different, with schools and districts having some flexibility over the testing window in the 11th grade. ]]>

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?

]]>Thank you for this site and this forum. ]]>

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!

]]>—

Fluency, as used in Principles and Standards, includes three ideas: efficiency, accuracy, and flexibility.

• Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of subproblems and making use of inter- mediate results to solve the problem.

• Accuracy depends on several aspects of the problem-solving process, among them, careful recording, the knowledge of basic number com- binations and other important number relation- ships, and concern for double-checking results.

• Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to be able to choose an appropriate strategy for the prob- lem at hand and also to use one method to solve a problem and another method to double-check the results.

—

I don’t know if this view is consistent with the CCSS writers’ view, but I like (yes, my personal preference) this view on fluency. It should also be noted that “fluency” seen from this perspective does not necessarily mean “quick.”

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 “