### A Pretty Short Explanation of "Invert and Multiply"

I can't remember if I've seen this one before, but I thought it was worth sharing. I found it in a forthcoming book by Pamela Harris, at the end of an extended discussion about how to develop proportional reasoning.

We all have had students who know one rule for dividing fractions: invert and multiply. I learned it as "multiply by the reciprocal," while some of my students annoyed me with the vocabulary-loose "copy dot flip" (or flop). But why does it work?

There are many ways to explain it, including some that get into the gritty details of partitive and quotative division, but I think this explanation would satisfy most students and teachers. Let's start with a problem: $$\frac{3}{4} \div \frac{2}{3}$$. I find this explanation is easiest to see if we write this as a compound fraction:
$\frac{\frac{3}{4}}{\frac{2}{3}}$
The reasoning from here is simple: dividing by a fraction is hard, but dividing by one is really easy. But how do we turn $$\frac{2}{3}$$ into one? By multiplying by it's reciprocal, of course. But we can't just multiply part of our problem by $$\frac{3}{2}$$ without changing its value. The only thing we can multiply by without changing our value is one, and we can write one as something over itself, like this:
$\frac{\frac{3}{2}}{\frac{3}{2}}$
So that's just a fancy way of writing one, and when we multiply the denominators we make a one:
$\frac{\frac{3}{4}}{\frac{2}{3}} \cdot \frac{\frac{3}{2}}{\frac{3}{2}} \rightarrow \frac{\frac{3}{4} \cdot \frac{3}{2}}{1} \rightarrow \frac{3}{4} \cdot \frac{3}{2}$
Which explains "invert and multiply." It's just the result of wanting to divide by one instead of a fraction.

References

Harris, P. W. (in press). Building Powerful Numeracy for Middle and High School Students. Portsmith, NH: Heinemann.

1. Concise explanation. But now, the question is, is this a strategy that students can construct through context, or does it have to be presented by the teacher? I'd argue that if you started with the problem above, it would be unlikely that students would construct the strategy. However, if you started with context, like:

"A serving of cookie is 1/2 of a cookie. I have 3/8 of a cookie. How many servings do I have?"

We can imagine students thinking, "If only the serving size were 1, then this would be an easy problem...." Further exploration, or pointed questioning, might lead students to see that if we double the serving size and the number of cookies, we haven't changed the quotient, we just have friendlier dividend. This could then be further abstracted to the "constant ratios" strategy for fraction division (see document below), and then the "invert and multiply" algorithm as you posted above. The cookie problem also lends itself nicely to the "find common denominators and divide the numerators" algorithm for fraction division (see document below).

If you're interested, here's a document that I wrote last year about fraction division after our "number sense" class:

http://db.tt/MarhwWF

2. You know, Fred, I now wish I would have taken a little more time to expand on my description of "the end of an extended discussion about how to develop proportional reasoning." Because you're right -- this explanation (proof, really) probably won't do much for a student who doesn't already have a lot of understanding about fractions and division. I just couldn't remember seeing this particular demonstration or explanation and wanted to put it out there.

And thanks for sharing that document. When it comes to dividing fractions, I frequently refer people to Gary Davis's work at Republic of Math: http://www.blog.republicofmath.com/archives/11.

3. After seeing this, I recognize it as the explanation that I learned (& had forgotten) when I learned how to work with fractions over 40 years ago.

It made sense then, and it still makes sense now (unlike the explanation here): http://mathed.byu.edu/~peterson/117%20Curriculum/Section%208%20Explaining%20Invert%20and%20Mult.htm