What is a rate? Common Core revisited

A commenter (not me) asks over on the CCSS Progressions blog:

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”?

The answer to this last question is going to be “yes”.

Whether it matches the meaning of these terms in real life or not, the answer will be “yes”.

Whether it matches the grammatical structure of the English language, in which unit would be seen to modify rate, the answer will be “yes”.

A unit rate, in the Looking Glass world of Common Core is not a kind of rate; it’s a different thing altogether. A rate is a numerical/linguistic construction. A unit rate is a number. Each is associated with a ratio.

But why?

The best sense I can make of this is that CCSS wants these terms to be precisely enough defined to admit a sort of mathematical clarity. No such definitions previously existed. So CCSS made them up.

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19 responses to “What is a rate? Common Core revisited

  1. Maybe I’m missing something, but I don’t see a distinction being made between the two. A rate could be 7 for every 4 where the unit rate would be 7/4 for every 1. Isn’t a unit rate just a specific rate?

  2. OK. This is the hardest part of being a thoughtful critic. I have to carefully explain the viewpoint of the object of my critique. It’s painful, but I’m up to it. Here goes.

    Seven for every four is not a rate in the language of CCSS. It is a ratio. Let’s say it’s seven cups of flour for every four cups of water. The rate is 7/4 cups of flour per cup of water. The unit rate is 7/4.

    The ratio is a comparison of two quantities (7:4, or 7 for every 4, etc., but not 7/4). The rate is a comparison where the second quantity is 1 (7/4 cups of flour per 1 cup of water). The unit rate is the number 7/4. I cannot emphasize this strongly enough. In the language of Common Core, a unit rate is a number. It is not a kind of rate. It is not a comparison of quantities. It is not a description of changing. It may be associated with all of these things; it may be derived from them. But it is a number.

    • Nora Jaegerman

      Would then “unit rate” be the “value of the ratio”? From the progressions: “The quotient 3/2 is sometimes called the value of the ratio 3 : 2″

  3. You know, I missed that when I read the progression document. I agree, it doesn’t make sense. Unit rate, to me, has always meant amount per 1. If the focus needs to be on the number, then maybe we need to drop the “rate.”

  4. So a rate has units but a unit rate does not have units? How much more confusing could they make it? I don’t see any need for the concept of a “unit rate”. Just keep the units with the rate (7/4 hours/lawn or 4/7 lawns/hour) and learn to manipulate numbers that have units attached.

  5. David

    Your comment reminds me of a conversation I had in August with my Calculus kids. I told them that they needed to switch gears on thinking about the slope of a line and that they needed to think of the number associated with slope as the amount of change in the y coordinate when the x coordinate changes by 1. I guess that in the language of CCSS I was telling them that slope is a unit rate.

  6. Take a look at the definition of rate in the glossary of the Smarter Balanced Mathematics Item Specifications document for Grades 6-8.

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

    Doesn’t look like they have seen the progressions documents.

  7. Oh, and also interesting….unit rate is not defined in that glossary.

  8. The made up definitions of CCSS are not a Canadian Concern, Strictly Speaking. Some thoughts…

    I attended a Marian Small presentation last week. She talked about how much she disliked unit rates. Give a HS teacher a ‘which is the better deal?’ problem and the automatic reaction is to calculate the unit price. Consider 4 cans for $7 vs. 6 cans for $10. Another strategy – common when you give this problem to younger children – is to use multiples and find the cost of 12 cans.

    In the first option, 12 cans cost $21. So we call $21 for 12 cans a rate, but not a unit rate because there’s no 1. But we do get to call $21 for 1 case a unit rate. It’s all about the 1. Kind of reminds me of the apples in your TED-Ed video.

    I guess CCSS tells us, er, I mean you guys, that unit rate is just a number so it’s 21/12 = 1.75. Or is it 21?

    One more thing… if 6 cans cost $10, then 1 can costs $1.67. Also, I can buy 0.6 cans (suspend reality here) for $1. One case would cost $20. A six-pack might be considered a 1 as well. Unlike pop tarts, it’s a fairly standard 1 at that. So, what is the unit rate? 1.67? 0.6? 20? 10? Please tell me the test will be multiple choice. And that only one of these options will show up.

    • Really helpful stuff, Chris. You nail the difference I am trying to outline here and help me to sharpen my argument. What you and I call a “rate” Common Core calls a “ratio”. What we call a “unit rate” Common Core calls a “rate”. And what we call a “fraction” Common Core calls a “unit rate”.

      There is a reason for this. At least one Common Core author, Hung-Hsi Wu, has written that,

      [T]he publishers [of US textbooks] mistakenly believe that intuitive arguments and analogies suffice. Thus, fractions are simultaneously (and incomprehensibly) parts of a whole, a division, and a ratio. (p. 4)

      There is longer discussion, together with links to the original for that quotation back at the post linked above.

      The point is that you and I would like to call 21/12 a rate, a unit rate AND a fraction. No can do, though. Common Core wants precise definitions of these terms. And the problematic one in the set is really “fraction”.

      We have a precise mathematical definition for “fraction”. A fraction is an ordered pair (a,b) of numbers, b=/= 0, together with the equivalence relation a/b=c/d precisely when ad=bc. That’s the definition of fraction which stands up to mathematical scrutiny, and it was a long time coming in mathematical history. It required the development of Abstract Algebra. And one thing we can all agree on is that this is inappropriate for middle schoolers.

      I claim that, by necessity, anything less isn’t really going to be a mathematical definition. And therefore we ought to embrace the ambiguity inherent in fractions. We need to comfortable with the fact that numbers frequently don’t stand alone in a meaningful way. They need to be interpreted in the contexts in which they are used.

      At least in part this is because-as I have said elsewhere-we don’t always know what 1 is.

  9. Wow, Lynda! Thank you so much for the link to Smarter Balanced assessment. These people seem to have some sense. But how is this going to work? The CCSS assessment is going to have different definitions for math terms than the standards? Ugh!

  10. If this madness is because of Hung-Hsi Wu, then I finally do feel animosity. Until now, I’ve read his stuff and thought he worried about the wrong things. My brother had sent me an article, because it looked good to him. I pointed out some strangeness to my brother, and said how surprising it must be to someone who doesn’t teach math, that there could be passionate disagreements about these things.

    Didn’t he learn from the first ‘new math’, that you can’t have the adult mathematician’s concerns driving the way you create materials for young kids? Mathematicians who understand kids need to be involved, yes, but not someone who obsesses over mathematical purity.

    I guess this is the same thing I feel about the (silly to me) argument over whether multiplication is repeated addition. I feel like we need to help elementary teachers see that it’s much more than that, but if you take out that meaning entirely, you’d certainly puzzle kids.

    I’m glad I’m listening to this conversation, even though it made no sense to me at first.

  11. Christopher has imputed an answer to the question Lynda asked me, although I’m not sure to whom he imputed it. He seems to disagree with the answer himself, and I agree with him that it is a bad answer. So it is not my answer. If you would like to see my answer, you can go my blog where Lynda originally posted the question, http://commoncoretools.me/2011/09/12/progression-on-ratios-and-proportional-reasoning/.

  12. Just be a little careful here, the definitions that Christopher is attributing to CCSS are in fact not made in the Standards themselves, but in the Progression on Ratios and Proportional Relationships. The two are related, of course: the authors of the Progression were on the work team for the Standards. I was a lead writer for the Standards and I am also managing the production of the progressions (although I did not write the draft of this one).

    However, there is a big difference between the Standards and the Progression: the Standards in final form until the next revision, whereas the Progressions are in draft form and subject to revision now. The terms “rate”, “ratio” and “unit rate” are not defined in the Standards (for reasons that the tone of this discussion makes obvious). The Progression provides an effort to define them. I don’t think that effort deserves quite the ridicule it is receiving here, but never mind, the criticism will be taken into consideration nonetheless and inform the final draft. I’ll only say that if I had a dollar for every time someone told me the answers to all these questions were obvious, I’d be a rich man. Of course, the “obvious” answers are mutually self-contradictory. This seems to be an area where it is very difficult indeed to find common language, and where emotions run high.

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  15. William Borgos, MD

    I am a parent and a physician, and from my perspective the above discussion is confusing the term rate with ratio. The rate at which the lawns are being mowed is 4/7 lawns per hour, or 0.57 lawns per hour. If I am asked how fast an IV should run, I give the order in ml/hour, not hours per ml. In other words, a rate should typically be expressed as a unit per unit of time. i.e. miles per hour, meters per sec, cents per minute. While it is critical to teach students the concepts of ratios and proportions, the solution offered to the above problem does not make sense in the real world. I would like to hear any thoughts.

  16. William Borgos, MD

    To further clarify my thoughts. Distance=rate x time. In this question the distance is the lawn. The question asks for the rate at which the lawns are being mowed, not how much time it takes to mow one lawn. D=RT, or R=D/T

  17. Thanks for joining the conversation, Dr. Borgos. It is really interesting for me to see the perspective of an informed outsider-someone who uses these things every day, but who is not wrapped up in trying to teach them.

    I have a few responses to your thoughts. The first is that you remind me that these things are seriously context-dependent. Because of the work you do, you think of rates as unit per unit of time, with the specific example being “milliliters per hour”. This is a professional convention, not a mathematical necessity. Expressing an equivalent rate in hours per milliliter would be awkward, of course, but it could just as easily be hours per liter. It’s not, though.

    But I painted houses (poorly) one summer in college. My foreman did estimates and determined our piecework pay based on time/unit. So he had a chart of hours per window, hours per square foot of siding, etc. Those are rates, too. I am told that marathon runners express their running rates in minutes per mile, rather than miles per hour.

    Similarly, many rates do not involve time at all. And in those cases, things are even more ambiguous. In the US, we express fuel efficiency as “miles per gallon”. In Europe, it’s “liters per 100 kilometers”. Both are rates. Daniel Kahneman, in his recent Thinking, Fast and Slow argues (citing Larrick and Soll) that the US rate leads to “misleading intuitions…[that] are likely to mislead policy makers as well as car buyers.”

    In particular, he argues that miles per gallon obscures the fact that improving one car’s mileage from 12 mpg to 14 mpg is more important than improving another car’s mileage from 30 mpg to 40 mpg. But expressed as “gallons per hundred miles (gphm)”, we can see that the former case goes from (approx) 8.33 to 7.14 gphm-a savings of over a gallon each hundred miles, while the latter case goes from (approx) 3.33 to 2.5 gphm-a savings of less than a gallon each hundred miles.

    Which is the right rate? Neither one. They’re both right. But as always in mathematics, the representation we choose emphasizes different things.

    Your second comment is really interesting to me. You interpret time literally and distance metaphorically. Time is time, but distance can be a lawn. That’s a really smart mode of mathematical thinking. It’s something I struggle every day to foster in my students. That sort of flexibility and creativity in thinking is what many of us hope to encourage and support through our teaching.

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