Sean objected to my common core objections. He suggested the following context to motivate complex fractions as unit rates, as required by Common Core at grade 7 (and as explicitly proscribed by footnote at grade 6).
John is in a 10 mile walkathon for breast cancer. He looked at his watch when he started walking- it was 7:02. After a half mile, he saw that it was 7:17.
He suggests (and I concur) that students will likely use 1/2 mile per 15 minutes and get to 2 miles per hour (the desired unit rate), and then as teachers,
We write (1/2)/(1/4) on the board, and discuss its relationship to 1/2 per 1/4. This may be our major line of disagreement, as I don’t think this a terribly sophisticated jump. Assuming the students have some experience with slope and rate of change, this feels like fair game.
Indeed right here is the point of contention. It’s the transition from:
[I]f I’m mathematically sophisticated and possess a graduate degree, then moving between (a) 1/2 mile per 1/4 hour and (b) (1/2)/(1/4) miles per hour is a very simple formal move justified by the multiplication algorithm for fractions, I suppose.
But walk into your average seventh grade classroom, make some orange juice from concentrate using the standard recipe (3 cans water: 1 can concentrate), pour precisely 1 cup of juice into a glass and ask, How much of this cup of juice is concentrate and how much is the water I added? My experience is that this is a challenging question for 12 and 13-year olds to make sense of. It’s really important at that grade level. It’s challenging to teach precisely because the relationship is so formally simple. But it doesn’t come naturally to lots of kids.
Answering the orange juice question requires shifting from ratio (cups water: cups juice) to unit rate (water per cup of juice), or from ratio to fraction. That’s hard, and it’s not even complex fractions.
The Common Core writers seem to want to move this from seventh grade to sixth grade. I’m OK with that; I wouldn’t have written it that way myself, but I have no major problem with it.
But they seem to think that we need to ramp it up in seventh grade and that the only way to do this is to do it with complex fractions. From the perspective of a middle school curriculum guy, I question whether it makes sense to do so. I don’t think it’s worth making kids do because it’s an unnatural representation. Furthermore, it’s machinery we don’t need. I don’t see a middle school problem that complex fraction unit rates will solve but conceptually simpler techniques will not. From my perspective as a college teacher, I don’t see it either. College Algebra? Calculus? Neither of these relies on complex fraction unit rates. It is conceptually much simpler to deal with (1/2 mile)/(1/4 hour) by either (1) equivalent fractions (multiply numerator and denominator by 4 to obtain (2 miles)/(1 hour)) or (2) division (divide 1/2 by 1/4, get 2).
I worked for a while with the best route to complex fraction unit rates that Sean suggested:
Assuming the students have some experience with slope and rate of change, this feels like fair game.
This makes sense; think of (1/2)/(1/4) as a slope, rise/run. But this leads me to want to divide. I don’t usually think of slope as 6/3 miles/hour; I divide and say the slope is 2 and the rate is 2 mph. So it doesn’t feel any better when the rise and run are fractions.
Obviously this isn’t perfect. But if complex fractions are a necessary component to a middle-school curriculum, where else do they land outside of unit rates and proportions?
Part-whole fractions. Area models. Here’s (1/2)/3:
I’m all for complex fractions as they arise naturally in Sean’s walkathon problem-and they do arise naturally there. I am not for introducing a way of working with those complex fractions that is unnatural and has no special payoff.
I will gladly consider any and all suggestions for contexts and problems in which the complex-fraction-unit-rate gets me something that equivalence and division do not-and I’ll even accept examples from high school and Calculus. But I’m not optimistic that they’ll arise.
Update 6/7/11: I edited out a confusing statement about the ratios in the orange juice question. My question is about part-whole relationships, the explanation in the following paragraph alluded to part-part relationships. It’s fixed now.