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!
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.”)
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...