Teaching fractions: the generalization problem

Question

I've been thinking about how you would go about teaching fractions, and there seems to be a problem in that every basic fact needs to be proven/explained twice, using two different layers of definition. Let me illustrate.

The rule for multipliying fractions is:

Where $a, b, c, d$ are integers, $\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$

To prove this takes an argument that relies entirely on the fact that $a, b, c$ and $d$ are integers, but then you define what it means to divide fractions and realize that a more general fact is true...

Where $a, b, c, d$ are rational, $\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$

The same thing applies for every basic fact about fractions: cancelling, adding, dividing, equivalence of $\frac{a}{b}$ and $a\frac{1}{b}$... There are two problems, here.

1. The formal side. It's a bit annoying to have to prove fact (1) only to have to prove its generalization. The only way I can think of is to explicitly write out $a, b, c$ and $d$ in numerator/denominator form and then apply rule (1), so you need the more specific case to prove the general case. What is the most efficient way to prove all of these facts?

2. The conceptual side. If this isn't explained properly, it's going seem like some sort of fantastic coincidence! The multiplication rule (1) is hardly obvious, and its proof takes a bit of cleverness. And then when this strange fact turns out to be true in a more general context, and the proof is just a formal manipulation of symbols, it's going to seem very mysterious. Actually, it seems mysterious even to me. How can these generalizations be presented in a way that doesn't make them seem like magical coincidences?

Answer 1

I think it would be best to focus on explanations for specific examples rather than generalities, which can come later. For example, let's consider multiplying $\frac{4}{7}$ and $\frac{3}{5}.$ (However, I suggest first doing simpler examples, such as $\frac{1}{2} \times \frac{1}{3}$ and $\frac{1}{3} \times \frac{2}{3}.)$

We can view $\frac{4}{7}$ of something as dividing the "something" into $7$ equal pieces and then keeping $4$ of those pieces. We want to do this for $\frac{4}{7}$ of $\frac{3}{5}.$ Viewing $\frac{3}{5}$ as $\frac{1}{5} + \frac{1}{5} + \frac{1}{5},$ we want to divide $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces and then keep $4$ of those pieces.

How do we divide $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces? One way is to simply divide each the three $\frac{1}{5}$'s into $7$ equal pieces, and keep one of these pieces for each of the three $\frac{1}{5}$'s. O-K, so what do we get if we divide $\frac{1}{5}$ into $7$ equal pieces? [Insert appropriate discussion about why this makes each piece be $\frac{1}{35}.]$ Therefore, dividing $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces gives us three pieces each of size $\frac{1}{35},$ or $\frac{1}{35} + \frac{1}{35} + \frac{1}{35}.$

Remembering that we were to divide $\frac{1}{5} + \frac{1}{5} + \frac{1}{5}$ into $7$ equal pieces and then keep 4 of those pieces, we are led to:

$$\left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) \; + \; \left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) \; + \;\left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) \; + \;\left(\frac{1}{35} + \frac{1}{35} + \frac{1}{35}\right) $$

But this is a total of $12$ many $\frac{1}{35}$'s, or $\frac{12}{35}.$

At this point we can think about what changes are needed to obtain a similar argument for other situations, such as $\frac{4}{17} \times \frac{3}{7}$ or $\frac{7}{11} \times \frac{6}{12}.$

Answer 2

Perhaps an even deeper problem is that, in school, students will frequently make use of the same identity when $a, b, c, d \in \mathbb{R}$. H.H. Wu discusses this in his Pre-Algebra text (MESE 1, MESE 2) as follows:

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The main difference between the situation Wu describes and what you describe is that students could actually prove the identity you've put forward. The reason for what is called FASM above is that students have not yet seen limits (in particular, have not yet seen real numbers as limits of convergent rational sequences modulo the appropriate equivalence relation).

As a teacher, you now choose how to proceed with your situation. One choice is to let students work out concrete examples, as suggested by DL Renfro; this is probably one of the best ways to start demystifying the generalization you have mentioned.

Another choice would be to write a proof. If you have students who you think are ready to grasp such a concept (i.e., for whom such a task falls in their ZPD), then you could scaffold it as appropriate, but (as JP Burke points out in a comment) you have now moved into the realm of proving. Moreover, even with a general proof of the fact you've written out, it may not be so clear how directly it can be applied to an example such as:

$$\frac{\frac{2}{3} - \frac{1}{2}}{\frac{6}{5} - \frac{1}{4}} \cdot \frac{7 - \frac{8}{11}}{\frac{12}{5} - 2}$$

Here, no numerator or denominator is (written as) a fraction. Does this law still hold? If so, why?

This question and other subtleties (what happens when one or more of $a,b,c,d$ is negative?) suggest you at least ensure students have sufficient procedural fluency with regard to working out examples before delving more deeply into generalizations.