Dividing by Fractions

In his recent post "Math is a dangerous subject to teach," Joe Bower discusses the ability to learn the procedures of math without understanding the conceptual foundations. As an example, Joe humbly admits that he has "absolutely no idea why" (emphasis his) dividing by a fraction is replaced by multiplying by the fraction's reciprocal. He can get the answer right without proper understanding, and therein lies the danger.

One of the things I've enjoyed most about teaching is finding new and deeper ways of understanding so-called "simple" math that I thought I had already mastered. Most of my mathematical upbringing focused more on procedure than understanding, so I occasionally find myself in the same position as Joe. Using Joe's post as inspiration, I've given more thought to dividing by fractions and finally have a model and a description that I hope explains what's really happening when you divide by a fraction.

First, let's look at a simple fraction:

The top number, the numerator, simply counts "how many." The bottom number, the denominator, tells us "how big." We read this fraction properly as "two-thirds," and usually think of that as two objects, each one-third the size of the whole (however big that is).

Now let's look at division. Some students are led astray with early beliefs that "multiplying makes bigger" and "division makes smaller." That kind of misguided number sense can be frighteningly persistent. Multiplication is better thought of as a "scaling" operation, and division can be thought of as a "grouping" operation. To see what I mean, let me explain using whole numbers:

The model I imagine for this problem looks like this:

Using the "grouping" concept of division, I've made four groups. Because each group contains two, the answer is two. No surprise. Let's try another:

The model:

I've made two groups. Because each group contains four, the answer is four. Still no surprises. Let's try one more with whole numbers:

The model:

I've made one group, which is as trivial as it gets. Because the group contains eight, the answer is eight. Now that we've established the pattern with a "grouping" definition, it should be easy to see why you can't divide by zero. I can't possibly make zero groups and still have the eight squares.

Okay, now let's try an easy division problem with fractions:

Remember, the numerator tells us "how many" and the denominator tells us "how big." The model:

It's still one group (how many), but a group of halves (how big). If you count objects, you get the answer: sixteen. But we haven't added or taken away anything -- the eight is still there. Got it? Let's try another:

The model:

Eight divided by two thirds, translated into "grouping-speak," is "eight grouped into two groups of thirds." Because each group contains twelve, the answer is twelve, even though you can still imagine the original eight.

So is this the same as multiplying by the reciprocal? Breaking the wholes into thirds gave us three times as many pieces (24, same as 8 times 3), and grouping into two groups gave us half of the pieces in each group (12, same as 24 divided by 2). More concisely, we multiplied 8 by 3 and divided by two. So doesn't that mean dividing by two thirds is the same as multiplying by three halves? Not exactly. We get the same answer, but for a different reason. To me, the model for eight times three halves would mean scaling eight to be three times bigger (24), then scaled back down to half of that (12). It's a different picture, even if we still get the same answer.

You can choose to accept this as a complication or a convenience; either way, I hope you have a better understanding of dividing by fractions. As always, feel free to offer criticisms in the comments below. (There has to be many a sixth grade teacher who could teach me a thing or two about this topic!)

Update 4/18/2010: Gary Davis co-authored a great guide on the division of fractions. It provides more strategies, more examples, and more detail than my post did. Thanks for sharing, Gary!