Tag Archives: complex fractions

Common numerator fraction division [#algorithmchat]

My future elementary teachers explore the common denominator fraction division algorithm at the end of the semester. Reading their work got me thinking about common numerator fraction division, and about what sense I could make of the symbols that result.

I tried to keep my work neat so others could follow it. If this sort of thing amuses you (as it obviously does me), then you’ll want to take a few minutes with the larger versions of these images. If it does not amuse (and I cannot begrudge anyone this), then you’ll just want to move along; there’s nothing here for you today.

Page 1, in which I interpret the complex fraction that results from dividing across the fractions.

Page 1

Page 2.

Page 2

 

 

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More on complex fractions

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:

to:

As I wrote in the comments of my original rant,

[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.

Sean concludes:

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:


And here’s 1/(2.5)

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.

Objections to my Common Core objections

I want to give a little breathing room to a thoughtful debate going on in the comments. I know it requires an extra click to get to those comments so maybe we’ll get the attention of a few more readers by dedicating a post to it. Additionally, if Common Core spurs thoughtful debate about relationships between math content and students’ minds, we’ll all be smarter for it.

Sean objects to my Common Core unit rate rant and offers the following classroom scenario:

A problem

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.

the lesson

  1. We discuss what’s happening. Hopefully a few students are curious about either a) how fast he’s going or b) a reasonable approximation for when he’ll finish.
  2. We decide on a way to measure how fast he’s going. We discuss measuring by minutes or by hours, and hopefully come to the conclusion that miles per hour serve our purposes best.
  3. The students work in pairs. They use 1/2 mile per 15 minutes and get to 2 miles per hour (the desired unit rate) in any number of ways. Formal proportional reasoning, informal proportional reasoning, number sense, a graph, a table- whatever is thrown out.
  4. We discuss how these strategies are related by placing them side by side.
  5. We debate which are the most efficient of these strategies.
  6. Now the teacher has a decision to make. Personally, I feel that while it may be unholy and it’s definitely arranged, this is not a terrible transition into a discussion about complex fractions.
  7. 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.
  8. A number of strategies, again side-by-side, are used to solve this expression. We break it apart visually with manipulatives. We convert both the numerator and the denominator into decimals. We show the algorithm. The students notice that “2″ as a solution is the same “2″ that they saw before.
  9. The teacher states that it may not be the most efficient way to find the unit rate in this particular problem, but that it may be in future ones. We speculate about when.
  10. In subsequent days, the teacher can veer towards abstraction with complex fractions. When misconceptions arise, there is the race analogy to give it footing.

Sean concludes:

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?

Really thoughtful stuff. Much appreciated. I have more to say but I’ll hang back and let Sean’s ideas simmer for a few days.