Bill McCallum writes in the comments here at OMT:
I don’t think that effort [to define terms such “ratio” and “rate” that CCSS leaves undefined] 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.
Fair enough. I’m happy to tone things down a bit.
I do need to observe that no one here at OMT (least of all me) has suggested that the answers to the questions at hand are “obvious“. I agree that they are not obvious at all, and I agree that it is very difficult to find common language with respect to these ideas. (Although this last bit is tricky; if we all use the same words but mean different things by them, are we speaking a common language?)
No, my critique is not at all that Common Core has failed to state the obvious definitions.
My critique is that I see no evidence that Common Core-either the Standards or the Progression on rational number-take into account research on how children learn this content, nor do they seem to coincide with everyday uses of these terms. In the era of No Child Left Behind and “evidence-based practice”, I find it troublesome that results of important research work such as that in the Rational Number Project or Cognitively Guided Instruction don’t seem to form a basis for either document.
I find it surprising that there are no research references in the Progression.
In my work with Connected Mathematics, I have many times had teachers ask for definitions of rate, ratio, fraction and rational number. As writers, we have hashed out these ideas many times as well. Answers are not obvious and reasonable people can disagree.
But we sort of agree that answers to these questions ought to be consistent with, and explain relationships to, uses of these terms in mathematics and the world. I don’t understand why this isn’t the starting place for the Progression.
So how about this for a first attempt at the relationships involved here?
A ratio is a multiplicative comparison of two quantities (usually both are non-zero). Conventionally, we use the term “ratio” to apply to part-part comparisons, but this need not be the case.
We can express ratios in several forms. If there are 5 girls for every 3 boys in a certain class, we say that (1) the ratio of girls to boys is 5 to 3, (2) the ratio of girls to boys is 5:3, (3) the ratio of girls to boys is , (4) there are 5 girls for every 3 boys.
The fraction notation is problematic in early ratio instruction because children may confuse it to mean that of the students are girls. Children are accustomed to fraction notation being reserved for part-whole relationships; for this reason the notation should be saved for later instruction.
The term rate suggests change. We tend to talk about a “ratio” in static situations where the values remain constant but a “rate” in a situation where the quantities are changing. In the girls and boys situation above, it would be correct to say that there is a rate of 5 girls for every 3 boys, but this feels awkward. If students were enrolling in a school and there were 5 girls enrolling for every 3 boys, the term “rate” is a more natural fit.
A unit rate is a rate where one of the quantities being compared is 1 unit. If we enroll five girls for every three boys, this is not a unit rate. We could say that there are girls per boy enrolling at the school, or girls for every boy. For every (non-zero) unit rate, there is a reciprocal unit rate. So we can also say that there are boys per girl.
What counts as a unit varies. When computing a “unit rate” for buying pop, we could compute the cost per ounce, the cost per can, the cost per six-pack or the cost per case of 24. Which of these is considered a unit rate depends on our choice of unit.
To summarize the discussion, ratios and rates are different mainly in connotation. Each expresses a multiplicative relationship between two numbers (as opposed to an additive relationship, for which we use the term “difference”). Unit rates are important forms of rates because of their intimate connections to algebraic and calculus ideas such as slope and rate of change.
It’s just a first stab at discussing these terms in this context, but is consistent with common usage and focuses the discussion on the main idea that is important at this level-rates and ratios are about multiplication relationships, which are the heart of proportional reasoning.
Going back to the lawn example that has been the focus of discussion here as well as over at the Common Core Tools website, this would suggest that “7 lawns in 4 hours” is a rate (there is change involved, and it’s not a part-to-part relationship), and that there are two unit rates: lawns per hour and hours per lawn.
Again, I am not claiming that these relationships are obvious. But if a couple of important goals for the Progressions work are (1) clarity and (2) usefulness for teachers, professional development and curriculum development, I think my proposal above is an improvement over the present document.