Unit Rate Revisited

[sandramccune@hotmail.com]

The Connected Math Pearson Prentice Hall video that Cathy Kessel posted in June 2012 is a meaningful example of how to use a unit rate that has units. The discussion is one that students and their parents can understand. Why make an artificial distinction that unit rate is the numerical component of a rate? The verbal gymnastics that result lead to confusion and frustration for teachers, students, and parents. Allowing unit rate to have units opens the door to the real world of unit price for students. Unit price is a logical concept from everyday life that people can understand . It makes common sense that the definition of unit rate should include unit price as an example. Making an unnecessary distinction is not helpful for the Common Core Math Standards. I taught middle school during the “New Math” era. Teachers and parents didn’t like ideas that went against common sense. Making an issue of the difference between “number” and “numeral” was one of the many technically correct distinctions that killed New Math. I feel the Common Core movement will go the same way as New Math if those in charge (whoever that is) present unnecessary distinctions. I suggest that a unit rate such as 25 miles/hour has a “numerical rate” of 25.

[Bill McCallum]

Thanks for these comments. I agree that we don’t want to make too pedantic with children and parents; certainly it would be crazy to test them on this vocabulary point (but yes, I know, people do crazy things). And the progressions documents should not be read as dictating what we say or do with children and parents. Their main purpose is to clarify the underlying issues for curriculum writers and others who need to make a deep examination of the standards. From that point of view, there is an issue that needs to be clarified. Suppose I say that apples cost $1.15 per pound, and then write the equation y = 1.15x to represent this, where x is the number of apples and y is the cost in dollars. That number 1.15 appears two times in the sentence: once with units (dollars per pound) and once without (in the equation). The progressions document calls the second one the unit rate: you could also go the other way and call the first one the unit rate (your preference) and call the second one the constant of proportionality, or the numerical rate, as you suggest. Either way, it’s useful to have some conventions about this. Notice that the standards themselves sidestep this issue, by requiring the use of ratio and rate language, but not specifying what that language is. So this is something the field has to sort out.

[jkerr]

I think it is worth pointing out that in Grade 6, all of this terminology will be new to students. The confusion that teachers/parents may have with the change in terminology won’t be similarly experienced by students.

I think that the terms as defined in the progressions will make matters less confusing for students. In traditional teachings of ratios, rates are defined to have different units. Then you do all of the same math that you did with ratios. So why separate them and create more words to memorize? It creates a perception that it’s totally new thing, when it is not.

One new thing done with rates in the traditional approach is finding the unit rate. But why didn’t we do that when the units are the same? We can and should, but then we’d need a new term, maybe unit ratios? So we’d have ratios, unit ratios, rates, unit rates, numerical rates, and I suppose we’d also need numerical ratios to refer to the numerical part of a unit ratio. Yikes. Is that really better than not distinguishing like/different units with different terminology, allowing us to only need the three terms ratios, rates, and unit rates?

Also, if it is “common sense” for unit price to be an example of unit rate, then define unit price as the numerical part of the rate 5 dollars per pound. Then it is an example of a unit rate!

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