In 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.
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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 a • c, b • c, b • d, and a • d. So that means that the total area of the four small rectangles—and therefore the one large rectangle—is a • c + b • c + b • d + a • d.
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).
If we set these two expressions for the total area equal to each other, we see that the previous drawing represents the equation:
This equation says that the total area of the rectangle is the same as the sum of the areas of the 4 smaller rectangles. This is one of those things that is totally obvious when you see the picture, but not so much when looking at the equation.
And if you’ve taken an algebra class, then I can guarantee that you’ve seen this equation before: this is the infamous FOIL formula that tells you how to distribute multiplication over two expressions that each contain two terms (aka, multiplying a pair of “binomials.”)
Why FOIL?
If you’re not sure where the acronym FOIL comes from, take another look at the order of the 4 multiplications performed on the left side of the equation. FOIL stands for
- The First terms from each expression are multiplied and added to the total.
- The Outer terms from each expression are multiplied and added to the total.
- The Inner terms from the two expressions are multiplied and added to the total.
- The Last terms from the two expressions are multiplied and added to the total.
There’s nothing special about the order in which these multiplications are performed. You’d get the exact same answer (after rearranging a few terms) by doing something like LIOF instead. But everybody seems to remember it as FOIL - most likely because it’s an actual word that's easy to pronounce.
In truth, you don’t need to memorize FOIL at all. Instead, take a few minutes to make sure you really understand the picture we’ve drawn. If you do that, you’ll actually understand where FOIL comes from in the first place, and you'll be able to conjure it up whenever the need may arise. Which is a lot more useful than memorizing that formula.
Use FOIL to Multiply Quickly
Little would you have guessed it, but FOIL is actually the secret to you performing lightning fast multiplication in your head. Intrigued? Let’s think about solving the problem 62 • 27. But instead of writing it out and doing it the old fashioned way, let’s use this:
This rectangle gives you a new way to think about multiplying numbers. The first thing you need to do is split each number into two easy-to-multiply parts (ideally, you want one part to be a power of 10.)
In this case, let’s split 62 into 60 + 2, and 27 into 20 + 7. Then the problem becomes 62 • 27 = (60 + 2) • (20 + 7.) Using FOIL, we get 62 • 27 = (60 • 20) + (60 • 7) + (2 • 20) + (2 • 7.)
These are all fairly easy to solve in your head, which means you can quickly calculate 62 • 27 = 1200 + 420 + 40 + 14 = 1674.
Now it’s your turn. Use FOIL and the following rectangle to quickly solve 74 • 45:
You can find the answer and an explanation below on the next page...
Solution: Challenge Problem
If we split 74 into 70 + 4 and 45 into 40 + 5, the problem 74 • 45 becomes 74 • 45 = (70 + 4) • (40 + 5).
Using FOIL, we get 74 • 45 = (70 • 40) + (70 • 5) + (4 • 40) + (4 • 5.) These are all fairly easy to solve in your head, which means you can quickly calculate
74 • 45 = 2800 + 350 + 160 + 20 = 3330. Done!
Wrap Up
Okay, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Aluminum foil image courtesy of Shutterstock.
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