How FOIL Can Help Your Math Skills

Have you heard of FOIL - not the aluminum kind, but the one that can help you multiply faster? Keep on reading The Math Dude to learn more!

Jason Marshall, PhD,
August 13, 2016
Episode #229

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Aluminum FoilIn true Math Dude tradition (or not), let’s kick things off today with a microwave oven pro tip: always remove aluminum foil wrapping from leftover pizza before reheating it in the microwave. If you fail to heed this advice, your pizza won’t be the only thing in your kitchen that gets nice and toasty!

While this warning might save you from a microwave pizza-reheating disaster, it might not keep you from making the equally dangerous mistake of attempting to reheat your leftover spaghetti in its not-aluminum-foil-yet-definitely-still-aluminum container.

And that’s because my original warning failed to tell you why you shouldn’t put aluminum foil in the microwave (because it’s crinkly metal that conducts electricity induced by the microwaves, and gets hot enough to burn.) I only told you not to do it. If I had told you why you shouldn’t do it, you would have connected the dots and realized that all aluminum is off limits. And, as a result, you wouldn’t have melted your microwave.

Teaching people to metaphorically not melt their microwaves is essentially my goal in life. By which I mean that instead of just giving you the “rules” of math, I want to help you see the deeper reasoning behind things, so that you can make connections and truly understand what’s going on.

Which is exactly what we’re going to do today, as we take a look at a rather famous piece of math known as FOIL.


A Picture is Worth a Single Expression

I’d like to get started today with a challenge. I’m going to show you a picture, and I’d like you to figure out what equation this picture represents. Here it is:


As you might expect given this picture, the equation you come up with should contain the variables a, b, c, and d. And it should also use the distributive property of multiplication. Take a few minutes, give it some thought, and then read on for the answer.

FOIL: First, Outer, Inner, Last

The first thing to notice about the drawing is the obvious: it’s a big rectangle that contains 4 smaller rectangles. Let’s start by coming up with expressions for the areas of the smaller rectangles.

As you can see in the drawing, starting from the bottom left and working clockwise, the 4 smaller rectangles have areas of ac, bc, bd, and ad. So that means that the total area of the four small rectangles—and therefore the one large rectangle—is ac + bc + bd + ad.

Or, using the shorter notation for multiplication that leaves out the dots, it is ac + bc + bd + ad.

Now, let’s think of another way to write the total area of the large rectangle. Given that the area is just the height of the rectangle, (a + b), times its width, (c + d), it’s pretty easy to see that its total area must be (a + b) • (c + d).