Common Core Standards: Math
2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."
There's more than one way to skin a cat, which is probably the most horrifying idiom in the English language. But it's also true, at least mathematically speaking. In this standard, your students will be rearranging linear expressions to see how everything fits together from a new perspective—sort of like the different camera modes in a video game.
The name of the game here is combining like terms. (It doesn't roll off the tongue like Call of Duty or Super Smash Brothers, but it'll do the trick.) Students should understand that there are about a billion different ways to rewrite a single expression—and that each of them can tell us something new about that expression.
For example, we can write 5x + 1 as any of the following:
- 5(x + 1) – 4
- x + 1 + 2x + 2x
- 3(x – 1) + 2(x + 2)
- x + x + x + x + x + 1
When they combine those x's and those constants together, students should see that all four of these bad boys simplify to 5x + 1. They might not look it, but they're secretly quadruplets separated at birth.
This standard is especially useful with percent increases or decreases. If we're out at a fancy dinner and we want to tip 18%, your students should know we can write the expression for the total bill in two different ways. First we've got x + 0.18x, where we're adding the tip (0.18x) to the total bill (x), as you do. But we can also combine like terms to set this up as 1.18x instead, since x + 0.18x = x(1 + 0.18) = 1.18x.
Rewriting expressions like this can save us a few steps in the long run. If our bill was $35.50, then our total is 35.50 + 0.18(35.50), which is totally the same thing as 1.18(35.50). Instead of adding the price to the 18% tip like some kind of Neanderthal, we can just take 118% of the total and call it a day. That's how James Bond would do it.
Make sure your students also realize this trick works for percent decreases, too, though it's a little less intuitive. If their favorite store is having a huge 35% off sale, their total can be modeled either by x – 0.35x, or the simpler version, 0.65x. In other words, 35% off is the same thing as 65% on—that's 65% of the total price before discount. It's just another tool to help students power through word problems a little more quickly, and hopefully get a better grasp on how variables and constants are related.
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