Progression on Ratios and Proportional Relationships

Here is a draft of the Progression on Ratios and Proportional Relationships. This one took a long time because there is a lot of conflicting and confusing language about ratios and proportional reasoning out in the field, and we struggled with decisions about the extent to which we should try to standardize the language. So comments on this draft would be especially appreciated. [Corrected file uploaded 13 February 2012]

About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.
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58 Responses to Progression on Ratios and Proportional Relationships

  1. Lane says:

    This compilation of perspectives will provide great fodder for whole-class discussion and efficiently inspire depth. That said, I believe what is happening with our explanations of cross-products is the students tend to wave the explanations and latch on to the procedure. It makes more sense to me to have them figure out the pattern of cross products as they repeatedly eliminate denominators (using reciprocals), building on their foundations for ratios and proprotions in general. Those who are very familiar with multiplicative identity will “see” cross products eventually. Those who are not so familar would benefit from more application instead of learning a procedure they do not fully understand.

  2. nancy says:

    I am thinking deeply about the word “big” in the Appendix on p. 13. When I hear primary teachers using the word “big” to describe the magnitude of numbers, I wonder about the children’s perspective. Is “big” referring to size as a measurement of shape? I was conjecturing about the multiplicative comparison with “How many times the magnitude A is as B.”

  3. Tad Watanabe says:

    Bill,

    Thanks for keep posting these progression drafts. I have too many things on my plate to look at the whole document (even though it is relatively short) at once.

    I just noticed on p. 3 (at the top), you say, “Ratios have associated rates.” But since you say one paragraph above, “A ratio associates two or more quantities,” this sentence should be “Ratios of two quantities have associated rates,” shouldn’t it?

    I do like the way the document clearly distinguishes ratios and rates. These two terms are too often used without clear distinction.

  4. Charles says:

    Bill,
    Going a little of topic here. I have been in a struggle with Math Practice 7 and 8. I know they have been coupled together in your work. However, as the writers produced the document they felt that there were enough discretion that each of them got their own place in the document. I am looking for answers to what exactly distinguishes the two. My first take on these two practices was that 7 was a predacessor to 8. For example, students in elementary would work with quantaties using ten frames. A teacher may pose several problems based around 9 plus a number. Students see and make use of the ten frame structure by taking one from the other addend to fill up the ten frame and essentially making it a plus ten fact.
    I would then view the transition to Practice 8 as the ability for a child to make a connection that when posed with a task such as 19 + 6= that they could apply the same idea that we could take one from the six to make 20. Likewise students could use the same concept when solving +8 problems by moving two over. I felt this transfer of structure to new situations emboddied practice 8. However, I have been doubting that and am looking for insight.

    • I think the distinction between MP7 and MP8 is clearer at the middle and high school level than the elementary school level, where one might well want to merge the two. And none of the practices is likely to show up individually. They are attributes of a well-rounded student and often occur simultaneously. My colleague Phil Daro likes to compare them to desirable attributes such as wisdom and goodness, which we expect to see together even though they have different meanings.

      • Charles says:

        I understand the merging of the two and that they are closing linked. However, could you elaborate. For instance if students were working on linear equations, what would be the “wisdom” and what would be the “goodness” when students are engaged in learning how slope and intercept are related to the equation and how they effect the outcome. Is the process of inputting data to create multiple graphs with different slopes reflective of SMP 7 where they are looking for structure in how changing the slope effects the line. Once students make the connection that the line continues to get steeper as the slope increases and accept it as a fact due to trials and logic have they arrived at SMP8 now? In this case both would be prevelant in the lesson, but each would have their own “place” per say in the lesson.

  5. Anne says:

    Bill,
    You show a graphic representation of ratio under a table representation but do not illustrate the fact that if two ratios are graphed on the same set of axes the comparison between the two ratios can be determined by the steepness of the line. Use of the Cartesian coordinate plane to compare ratios lays a foundation for thinking about slope as ratio. This graphical representation also very nicely engenders discussion about equivalent ratios being on the same ray.

    • Anne, thanks for this suggestion, it certainly belongs in the progression.

      • Anne says:

        Bill,
        If you do include the graphical representation of two ratios on the same set of axes it also lends itself to identifying ratios that come between the two graphed…thus if you graph for instance 2:3 and 3:4 by inspection it is possible to identify many ratios greater than 2:3, less than 3:4, setting the stage for inequalities.
        I have done a lot of work on this representation and can report that for students who struggle with abstract notation, they find it makes a lot of sense when they can “see” the comparisons.
        Anne

  6. Dick says:

    Bill,

    One of the striking features of the CCSS document is that it does not even mention the idea of “setting up and solving a proportion”. Thus, what has been the staple of the “ratio and proportion” part of the curriculum for more than a century has been removed from its position of prominence (or dominance?). What replaces it is the idea of a “proportional relationship”.

    The current draft of the RP progression addresses this most directly on page 10: “Such problems can be framed in terms of proportional relationships and the constant of proportionality or unit rate, which is obscured by the traditional method of setting up proportions.”

    This is a succinct statement of the issue, and a careful reading of the accompanying example on page 10 will show the benefits of framing such a problem in terms of a proportional relationship.

    Still, given the fact that moving from solving a proportion to proportional relationships represents a huge shift, it would be helpful to have a bit more discussion of what this means and why it makes sense. My experience is that getting people to think of proportionality in any other terms than solving a proportion is not easy.

    • Anne says:

      I think that the idea of thinking about proportionality as being a multiplicative relationship for which a scale factor is applied is so very much more important than the way in which most students are exposed to solving proportions…cross-multiply, or even worse “cross your heart method.” If this document and the reality of CCSS-M moves a critical mass of teachers to rethink how they engage students in these important concepts then I will become a staunch supporter.
      Anne

  7. Dick says:

    Bill,

    I hope I am not overdoing things with multiple comments, but I agree with your statement that there is a lot of conflicting and confusing language about ratios and proportional reasoning out in the field, and am encouraged by your statement that comments would be especially appreciated.

    I feel that a nice service provided by this progression is to highlight the notion of the “value” of a ratio. Specifically, the value of a ratio A : B is defined as the quotient A÷B (pages 3 and 13). It is surprising how many treatments of ratio in school materials do not have an explicit term for the value of a ratio.

    Without this notion, the central role of division in forming ratios may not be appreciated. But with this notion, the progression is able to state succinctly the purpose of using a ratio: “The value of the ratio A : B tells how A and B compare multiplicatively; specifically, it tells how many times as big A is as B.” (page 13).

    But the progression also points to the fact that “Ratios are often described as comparisons by division.” (page 14). It may be useful to say explicitly why ratios are described BOTH as comparisons by division, and at the same time as multiplicative comparisons.

    Using an example, the value of the ratio (18:12) is 1.5, a number found by division. In turn, this number tells us that 18 is 1.5 times larger than 12, a fact expressed multiplicatively as the product 18 = 1.5 (12). In general the value of a ratio of quantities P:Q (for P>Q) tells us how many times larger P is than Q.

    When discussing the role of ratios in giving comparisons, it is also helpful to point out the parallel between the “relative” comparison by division provided by the value P÷Q of a ratio, with the “absolute” comparison by subtraction provided by the difference P-Q of P and Q.

  8. Dick says:

    Bill,

    You wondered about the extent to which we should try to standardize the conflicting and confusing language about ratios and such. One subject that could use some standardization is that of ratio and rate.

    The RP progression (page 2) specifically interprets the standards as applying ratios to situations in which units are the same as well as to situations in which units are different. Thus, situations traditionally treated with rates (situations in which units are different) are described in terms of ratios. On the other hand, ratios are interpreted in terms of rates (“Ratios have associated rates”).

    In fact, ratios and rates are thoroughly intermingled in the progression. For example, the situation (page 3) where an object is moving 3 meters every 2 seconds is treated both as a ratio and as a rate. Specifically, this situation is described there by the ratio 3:2 of distance to time (a pair of numbers). This ratio (as every ratio) is said to have a value, which is the quotient 3/2 (a single number). But this ratio (as every ratio?) has an associated rate, which is also the quotient 3/2.

    There may be a useful point of view underlying this treatment, but as it stands, it is a little confusing, and not likely to be helpful to those who have wondered about the difference (or lack of one) between ratios and rates. Also unclear is how the unit “meters per second” is to be associated with the numerical value 3/2 in either a ratio or a rate. One might expect that the “rate” is a number with a unit, “3/2 meters per second”. But the statement later in the paragraph (“The unit rate is the numerical part of the rate.”) would seem to indicate that the rate is just the number 3/2. The definitions in the Appendix do not clarify the situation very much.

    To be fair, the usage of ratio and rate terminology varies widely, and any careful definition is bound to conflict with some existing usage. Moreover, the RP progression is not intended to be full treatment of the mathematics of ratios and rates in middle school. Still, there are some regularities of usage that would be nice to acknowledge. it would be helpful if the progression could give a little more clarity on the very basic ideas of ratio and rate.

  9. [Comment removed, posted in wrong place.]

  10. Christopher says:

    I have to admit that I find sentences such as the following to be unhelpful:

    If 5/2 and 100/T are viewed as unit rates obtained from the equivalent ratios 5:2 and 100:T, then they must be equivalent fractions because equivalent ratios have the same rates.

    It seems that the distinctions that the writer(s) are trying to draw among fractions, rates and ratios really don’t hold up to this kind of scrutiny, and the result muddies the waters rather than clarifying them.

    We start with two equivalent ratios (not, for some reason, expressed as fractions), we extract from them their associated rates, deem these rates to be the same (not “equivalent” evidently) and finally use this sameness to justify the equivalence of the fractions in which these rates are expressed.

    I just don’t see how distinctions at this fine grain size are either (1) useful in middle school classrooms, or (2) really appropriate to the larger mission of creating common standards that have the force of law.

    I greatly appreciate the careful attention to meaning throughout this document. That we are asking students and teachers to pay careful attention to the meanings of both the numbers and operations that are under study has been too long neglected in a lot of American curriculum.

    But these fine-grained pseudo-definitions are puzzling to me.

  11. Christopher, thanks, this expresses a frustration that it is important for us to hear as we refine and clarify this document. Do you have a different proposal? You seem to be suggesting at one point that we should not try to make a distinction between ratios and fractions. It’s true that in everyday language the two are often conflated, but I’m not sure this works for all situations where you use ratios. For example, in mixture examples (3 parts red paint to 2 parts blue paint), it seems natural to express the ratio as a pair of numbers rather than as a fraction. Do you disagree? We are trying to find clear language that respects both the mathematical meaning and is useful to teachers, so if we haven’t done that I’d appreciate other suggestions.

    • Christopher says:

      A fraction is a notation. We commonly use the term in ways that suggest a part-whole relationship, but it needn’t be so. When we see 2/3, we cannot know whether it refers to a part-whole relationship, a part-part relationship or a comparison of otherwise unrelated quantities. It’s just a fraction: two numbers separated by a vinculum (which last term please let’s agree is only useful as a Jeopardy question).

      This is true of all abstract mathematical notation. There is no way to know without context whether 9 refers to a length, a discrete quantity, a scale factor or an area.

      Because a fraction is a notation with a part-whole flavor (but not a part-whole definition), it does indeed seem more natural to express most part-part ratios using colons or to. That notation has the flavor of part-part. And fractions with units attached tend to suggest rates.

      Just as not all fractions are part-whole, not all rates (in my view) are unit rates. It’s when we overcommit to these nuances-trying to give them something like the rigor of definition-that we end up with the kind of writing I critiqued above. Consider the CCSS standard 7.RP.1 about complex-fraction unit rates. I wrote about my objections to that last spring.

      I’m sounding a common theme here, which I think bears repeating. The only rigorous definition of a fraction comes from undergraduate Modern Algebra and is inappropriate for middle school. We need to come to terms with that and allow some nuance in the conceptual development of fractions and proportional reasoning. Efforts to purge nuance (e.g. a fraction sometimes refers to a part of a whole, sometimes a rate, sometimes a part-part ratio, sometimes to a quotient, etc.) are necessarily going to result in either (1) patent untruths (e.g. a rate is something quite different from a ratio) or (2) overwrought language that poorly captures how people who are fluent with proportional reasoning actually think.

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    • One can imagine continuing the discussion beyond height and width to include the length of the diagonal, and the length of the line from the vertex of the rectangle to the midpoint of the diagonal. This would go well with the standards on scale drawings in Grade 7 and the standards on similarity in Grade 8. It may well be that an earlier draft of the document had such a discussion. This document is itself still just a draft, and we can take the triangles out if they are a distraction, or perhaps add a comment about the extension. Which do you think would be better?

  13. To say that a fraction is a notation is true as far as it goes, but a natural question arises: a notation for what? One answer, possibly the one being advocated here (I can’t quite tell), is to say that the notation has multiple meanings and explore the relations between all of them. Another answer, the one adopted in CCSS, is to give the notation a primary meaning, and then think of those other meanings as ways in which fractions are used. In this approach, a fraction is a notation for a number on the number line: a/b is the number you get by dividing the interval from 0 to 1 into b equal parts and marking off a of them from 0. That number can be used to express a part of a whole, or a multiplicative comparison between two quantities, or a rate, or the division of two other numbers, and CCSS explicitly mentions all these interpretations in the progression from Grade 3 to 7.

    As for ratios, I agree that people who are fluent with proportional reasoning use the terms ratio and rate almost interchangeably. But do they come by that fluency by conflating the two notions from the very beginning? There are lots of situations where a mature understanding of an idea combines what were previously distinct understandings in the child’s learning. For example, is it obvious that 3 ÷ 5 = 3/5? Many mathematically knowledgeable adults might think this is true by definition. But I think that the two sides have different meanings for most elementary school children, whatever approach to fractions they have been taught. By the same token, I think the distinction between ratios and rates is real for children at a certain stage, but that distinction is a stepping stone to a more mature understanding in which the distinction becomes less important.

    • Christopher says:

      So you would rather have the number have multiple meanings, not the notation? Is this right?

      I agree that it is absolutely not the case that 3÷5=3/5 in the minds of most elementary and middle school children. And that’s why it needs to be taught. Viewing a fraction as a quotient is quite sophisticated, but worthy of middle school instruction.

  14. So you would rather have the number have multiple meanings, not the notation? Is this right?

    No, I would say (in fact did say) that it can have multiple uses in different contexts and different problems. I feel the same way about the equals sign: people often say it has multiple meanings, but I prefer to say it has one meaning, but many uses.

  15. Christopher says:

    Hmph. I guess at this point the discussion devolves into an unpleasant semantics debate, centered on the question, Is what something is “used to express” different from its “meaning”? I certainly think it isn’t different in natural language. If I use the word “train” to express my son’s favorite vehicle, I don’t understand how that can be detached from the meaning of “train”.

    I’ll just have to let it go, I suppose.

    But just to be clear, if I say that my house is on a 1/4-acre lot, you’re saying that fraction (1/4) refers to a number. That number means that I have marked off a point that sits at the far end of one out of four equal-sized intervals between 0 and 1 on the number line. And I am using this number to express an area.

    Do I have it right now?

  16. Bill McCallum says:

    There is no need for a debate, I was just trying to explain how I use words. I have no interest in convincing you to use them differently from the way you do now. The main thing is to communicate. In that spirit, the last paragraph is not something I would say. If I were given the assignment to write a paragraph on the term “1/4-acre lot” making reference to the meaning of the number 1/4, I would talk first about quantities, and what it means to multiply a quantity by a number, and relate the meaning of multiplying a quantity by 1/4 to the meaning of 1/4 as a number. Something like that. I could work on it some more.

  17. Cathy Kessel says:

    A general comment about definitions: The exchange above made me think of the beginning of The Classification of Quadrilaterals: A Study of Definition by Zalman Usiskin et al. This book notes that many do not realize there is a choice of definitions for mathematical terms, and that not all textbooks assign the same meaning to terms.

    Comment on fractions and ratios: One study (Clark et al., 2003, Journal of Mathematical Behavior) found that different teachers and textbooks had different meanings for the terms “fraction” and “ratio.” Clark et al. say: “At our meetings, the three of us started using Venn diagrams to communicate our ideas about the relationship between ratios and fractions. We found these diagrams so helpful that they became the basis for an activity with the math teachers at that workshop and in classes and workshops since that summer.” Using these diagrams, they found that prospective and practicing teachers had at least five different views.

    View 1: Ratios are a proper subset of fractions. Clark et al. point out that there are difficulties if one allows ratios of three quantities (as the RP Progression does).

    View 2: Fractions are a proper subset of ratios. In Clark et al.’s study, some justified this choice by saying that ratios are multiplicative comparisons, and fraction notation is one way for expressing that comparison. (This view is ruled out by the RP Progression because it distinguishes between a ratio a : b and its associated unit rate a/b.)

    View 3: Fractions and ratios are disjoint. In Clark et al.’s study, some justified this choice by saying that fractions are part–whole comparisons and ratios are part–part comparisons. (The RP Progression considers ratios to include part–whole and part–part comparisons.)

    View 4: Fractions and ratios are overlapping sets. Clark et al. comment: “Because of the three realms, Model 4 seems to be open to the widest variety of interpretation” and discuss various interpretations further.

    View 5: Fractions and ratios are the same. Some teachers noted that views 1 and 5 are ruled out if something like 12 : 0 is allowed to be a ratio. Clark et al. say: “Although we did not find any textbooks that introduce the terms as synonyms, we did find instances when the terms are used in a way that their definitions are indistinguishable.”

  18. Lane says:

    Whew, It’s been awhile since I visited. As I read, my concern is that we will frustrate students to tears if we don’t come to a succinct consensus on all the vocab but I honestly wouldn’t even know where to begin to argue for one or the other on any of it.

    As a high school teacher, I have not seen the progression from a/b = y/x to y = kx clearly spelled out in a high school textbook yet. I’ve always seen the two compartmentalized, but the “Progression on Ratios” has helped me to visualize why this needs to be unified.

    I would like to know where “equivalent fractions have equivalent reciprocals” fits into all of this. I use that in leiu of cross-products because my younger students tend to be less confused. They learn to eliminate one denominator by multiplicative identity and equality and have a long time to get confident with the idea before they need to work with two denominators. I agree with pg. 9 in the “Progression,” that students who eliminate denominators can “see” cross product but they seem to quickly forget what they saw at first if they are taught cross-product as a memorized procedure. I do not teach cross-products directly, and so I do have students crossing every time they see two fractions in close proximity. As mine work with denominators and the denominators of reciprocals, they eventually start taking cross-products because, by experience, they “see” the denominators eliminated by multiplicative identity on both sides. Another plus for eliminating denominators by multiplicative identity is that the process reaches across (connecting with) a broad spectrum of equation solving.

    I have another concern about how much emphasis is placed on whether 3 blue for 4 red must be written as 3:4 or if it is okay to allow the students to think it is the same to say 4 red for 3 blue as 4:3 as long as they make it clear the 3 goes with the blue and the 4 goes with the red. When my students formulate a proportional relationship with 4 red for 3 blue, increased to 9 blue for x red, they don’t worry where to “put the numbers.” They just make sure the quantities are correctly related, not criss-crossed. But now I’m thinking I must be overlooking something.

  19. Christopher says:

    Lanewrites:

    I have another concern about how much emphasis is placed on whether 3 blue for 4 red must be written as 3:4 or if it is okay to allow the students to think it is the same to say 4 red for 3 blue as 4:3 as long as they make it clear the 3 goes with the blue and the 4 goes with the red. When my students formulate a proportional relationship with 4 red for 3 blue, increased to 9 blue for x red, they don’t worry where to “put the numbers.” They just make sure the quantities are correctly related, not criss-crossed. But now I’m thinking I must be overlooking something.

    I’m with Lane here. I cannot imagine a world in which this matters. How say you, Common Core writers?

    • Describing ratios in words is fine. To quote the progression:

      A ratio associates two or more quantities. Ratios can be indicated in words as “3 to 2” and “3 for every 2” and “3 out of every 5” and “3 parts to 2 parts.” This use might include units, e.g., “3 cups of flour for every 2 eggs” or “3 meters in 2 seconds.” Notation for ratios can include the use of a colon, as in 3 : 2.

      Neither the Standards nor the Progression insist on the use of colon notation; they do, however, try to stabilize the meanings of the various terms and notations, in order to tame the jungle of confusion that has grown up around the subject of ratios.

  20. Lane says:

    I slept on the ratios definitions last night and this morning I recalled teaching an Integrated Algebra class for which I spent a considerable amount of time trying to sort it all out. Here’s a link to what I concluded and would appreciate someone evaluating it for correct understanding.
    http://dl.dropbox.com/u/7405693/MEGSL/Ratios%20Definitions.pdf
    Surely if we could come up with an accurate diagram on this order it would be helpful.

  21. Cathy Kessel says:

    Lane, thanks. As I noted above, there isn’t a single assignment of meanings for terms in school mathematics that’s the One Correct View (although that doesn’t allow one to change meanings in midstream). I think it would be a useful test of the RP Progression if you (or someone else) would compare meanings and notation in your diagram and the Progression. Whether or not that’s doable by readers would be very helpful information for me as editor.

  22. Lane says:

    Ratios Diagram: It’s really hard to find time for focused connected study of the standards while trying to keep all of my 150 students afloat; but I did go back over the RP Progression, especially the appendix. The diagram seems to fit. To be honest, I wasn’t even aware of the different kinds of ratios when I first started teaching; and putting them in a diagram was very helpful to me. Knowing I’m not alone, I’m thinking it would be well worthwhile to consider fine tuning the diagram by a group of readers or by those who have been more immersed in the trains of thought.

  23. Cathy Kessel says:

    Lane, I put a quick comparison of ratio and rate in the RP Progression and my interpretation of your diagram here.

    As I say there, my guess is that your diagram was helpful to you because it focuses on the different kinds of possible situations. You may not have noticed that in the RP Progression “ratio” includes situations in which two quantities have the same units and situations in which two quantities have different units. Rates are associated with each of these kinds of situations.

    That’s for the meaning of “ratio” and “rate” as used in the RP Progression. As I noted above, those terms can be used in different ways.

  24. Lane says:

    Oh, wow. I think I see it now. So rate, then, is any ratio for which nouns, meaning, units have been assigned? I’m still fishing for a succinct definition. As an example, if we cut an 8′ board in 2 pieces 2:6 feet , we would say that the board was cut at a rate of 2 units (feet) per 6 units (feet). Is this correct?

  25. Cathy Kessel says:

    That first “is” needs some fixing. The RP Progression is considering rate and ratio as two different kinds of things. There’s some discussion in the comments above about what people fluent with rate and ratio might do (including not distinguishing between them), but I don’t think anyone is assuming fluency in Grades 6 and 7, which is what the RP Progression discusses.

    In the RP Progression, a rate is associated with any ratio—with caveats about the ratio, e.g., the ratio needs to involve two quantities rather than three or more (as Tad points out. Thanks, Tad!).

    In high school, a rate is one quantity, expressed as a number or variable together with a derived unit (e.g., 2 miles per hour or .5 hours per mile, if the ratio of miles to hours is 4 : 2, or 2 : 1, and so on). In Grades 6 and 7, students are moving toward that understanding, just as in earlier grades they moved toward understanding a fraction as one number rather than some sort of compound of two numbers.

  26. Lane says:

    I took another look at the Progression this evening to try to figure out why my quick read-throughs were not yielding appropriate conclusions. It is my understanding that this should be a document that a K-12 department chair or PLC leader can read through in an evening to get a solid understanding of what must change in their curriculum and pedagogy. As you can see from my posts, I’m struggling! I’m an NBT high school math teacher with 30 hours of graduate math (post BS)….learning under professors who barely spoke English so it’s not like I can’t read (please laugh with me!). I think what’s happening is that it is so wordy that I mentally wandered to other things I wanted to be doing, thereby missing what’s really important. I think that stating some very obvious things lulls the reader into thinking “same ole, same ole.” I did tech writing for several different companies for about 8 years, so I’m wondering if I might be a candidate for helping to make this easier to understand. Here’s a link for my initial offering for the first couple pages, cutting the size down 1184:965 or 965:1184 which is about 18% decrease.
    Beyond cutting what I believe is distraction, I’ve added emphasis. http://dl.dropbox.com/u/7405693/MEGSL/Progressions.doc
    For the document to be really effective, though, I believe it should be read (and understood) by numerous PLC leaders and chairs who are transparent enough to say what is clear to them and what is not. If we don’t do that, I’m afraid the CCSS will end up in a pile of other standards so few ever read.

  27. Cathy Kessel says:

    Lane, thanks for taking the effort to comment and suggest revisions. This will be helpful when we revise the draft.

    I know that it says on the Progressions page that up-to-date versions of the Progressions “would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.” I do not get the impression that this means that a Progression is intended to be read in an evening. Many comment that the CCSS will require professional development that is not business as usual. Perhaps part of this is taking more time to read documents.

    I’d be interested to see your reaction to the style of other Progressions, e.g., Statistics, Expressions and Equations.

    Re unit rate: Note the rightmost column in the first table of my comparison here. For the rate 3/2 cups apple juice for every 1 cup grape juice, the unit rate is 3/2. For the rate 3/2 feet for every 1 second, the unit rate is 3/2.

  28. Lane says:

    This is helpful. I agree the new ways of seeing concepts will not be grasped in an evening, but I do think changes from the typical textbook methods could be bullet-pointed up front or otherwise emphasized so that, with a glance, teachers get the general idea of what they must change. Then they will look for clarification as they read the narratives and study the diagrams. Because our culture bombards us with documents, we skim. Even when we are trying to be thorough, we tend to read over what we perceive to be familar, looking for what is new. When I was tech writing, I would write a few pages and then take them to the shop and ask someone with a 7th grade reading level to ask me what they thought it said. Then I would revise and ask another person to read and explain. That took a lot of mystery out of how the final document would be understood by the rank-and-file. I think that is the kind of review that will get us documents that can generate the changes the CCSS intend. I’ll try to get around to reading more.

  29. Amelia Salazar says:

    RE: 6-7, Ratios and Proportional Relationship page 5, showing structure in tables and graphs, the multiplicative structure table for cups grapes and cup peach is incorrect. It is supposed to be 100 is to 40.

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  31. Nick Wasserman says:

    One thought after reading the progression and some of the comments is the need to articulate why such careful distinction between proportions and fractions is necessary. The visual on page 4 does a nice job clarifying the difference between equivalent ratios and equivalent fractions. The introduction states that, “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.” However, little else is said about why this distinction between proportions and fractions is so important. One such reason, at least in my mind, is that “combining” proportions and “combining” (e.g. adding) fractions are inherently different because of the changing whole (the visual on page 4). For example, that a ratio of 2 cups blue: 3 total cups, combined with a ratio of 3 cups blue: 5 total cups, results in a ratio that is 5 cups blue: 8 total cups is true; whereas combining the fractions 2/3 + 3/5 cannot be done in a similar manner. Another thought is that “comparison of ratios” is frequently accomplished by computing the decimal/fractional equivalent and comparing the decimals/fractions. In particular, this idea “works” for comparing ratios, only because the decimal comparisons are now comparing ratios that have the same “whole” (Example on page 6 for 3 ways to compare ratios – same red, same yellow, same total: 1red:3yellow = 0.333…red:1yellow, compared with 3red:5yellow = 0.6red:1yellow). Some of these thoughts, and others, might be useful for helping readers see a need to distinguish ratio and fractional notation.

    • Lane says:

      When I taught these different scenarios to 9th graders, most of them could see blue to red as a “part-to-part” 2:3 or 2/3 and blue to whole as “part to whole” 2:5 or 2/5. Again, this isn’t something I was ever taught but had to find a way to “distinguish” and maybe there was something I was missing. In any case, this connects well to “like terms” which connects with common denominators. I’m thinking this all fits well with seeing expressions as “objects.”

  32. Melanie says:

    I have always found the use of fraction notation for part-part ratios to be very confusing. If I am going to represent 2 red chips to 3 blue chips as 2/3, I want to know that the actual number 2/3 has to do with it. To my way of thinking, every ratio is a rate; 2 red chips to 3 blue chips is a “rate” of 2/3 red chips per blue chip. By broadening the idea of a “unit” to include a descriptor, like red chip instead of just chip, everything makes sense, both representation and various procedures, like cancelling units in science class. Allowing such “units” and considering every ratio a rate seems like a helpful way to approach some of this hair-splitting discussion of the fraction representation of ratio.

  33. Lynda says:

    So, inquiring minds want to know–what do the CCSS standards suggest as standard definitions for these terms: ratio, rate, unit rate? Can anyone nail those definitions down? What about these?
    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.

  34. Lynda, these terms are defined in the progressions document that started this thread, here, on pages 2–3. Your first and third definitions are basically the same as the ones given in the progression, but the second definition is different. First, the standards do not make a distinction in terminology based on whether the units are the same or not. Second, a rate is not a ratio, rather a rate is a quantity that is derived from the ratio (e.g. the ratio 80 feet for every 10 seconds has an associated rate of 8 feet per second).

  35. Lynda says:

    Are rate and unit rate interchangable? Or should a teacher define them for a middle school students as…
    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.

    • The standards do not give explicit definitions of ratio and rate, although it is possible to extract plausible definitions from the language of the standards. The distinction between rate and unit rate is a little trickier. The standards are not completely clear on whether they are really asking for such a distinction or not; the Progressions document has to do a certain amount of interpretation and clarification. As I say in the next comment, I wouldn’t belabor the distinction with 6th graders.

      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.

      • Lynda says:

        The Smarter Balanced Assessment Consortium has a glossary in its item/task specifications documents for math. Their definition of rate? A ratio that compares two quantities of different units (e.g., miles per hour).
        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.

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  37. I have no association with Common Core, Lynda. But I’ve read closely. The answer is “yes”.

    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.

    • Cathy Kessel says:

      Note that the term “unit rate” comes from math education research where “unit rate” and “unit rate strategy” are used to describe what students do in solving problems rather than as part of the mathematics to be taught. For example, some researchers distinguish between a “building up strategy” that uses additive structure as opposed to a “unit rate strategy” that uses multiplicative structure. (See p. 4 of the RP Progression for illustration of these structures–but not the strategies. These are illustrated in the upper right sidebar on p. 6.) My Singapore teachers manual for primary 5B gives a unit rate strategy that it calls “the unitary method” (perhaps a better name).

      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.)

  38. This is not quite right. The standards do not insist on this distinction, although both terms are used in the standards. The Progressions document is an effort to interpret and clarify the standards, and is currently in draft form. Discussions such as this could influence the final form of the document. I don’t think this is a distinction I would belabor with sixth graders. The purpose of the Progressions document is offer some clear languages for teachers to use and talk about the concepts. This area in particular is one where there are contradictory definitions flying around and, apparently, a lot of emotion, so achieving this clarity might be difficult!

  39. David Cox says:

    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?

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  41. Others have mentioned the same objection to this term, and I’m certainly sympathetic to it!

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