Certain areas of algebra are like pebbles in your shoe: looked at closely, they are very small — tiny in the scheme of things — yet they are “oh-so-bothersome!”
As a tutor, I’ve long felt this way about negative signs before parentheses. It’s a small thing really, and it seems simple to grasp for those of us who get it. Yet students make so many mistakes when facing this situation; to them it’s quite irritating.
So there I was again, trying to help a girl see how to simplify the expression: – (– 5x + 3y – 7)
But this time I thought of something different, the word “opposite.”
I talked for a moment with my tutee about the idea of opposites, and then I started out like this:
Q: So, what’s the opposite of black?
She replied: White (with the teenage “that’s-totally-obvious-what are you-doing?-insulting-my-intelligence? accent)
I told her not to worry, this would lead back to the problem. Next I gave her two terms for which she were to find the opposite, as in:
Q: opp (tall, happy)
She wrote: (short, sad), still wondering …
And I continued:
Q: opp (heavy, up)
She wrote: (light, down), sighing.
Then I explained that in math we express the idea of “opposite” with nothing more than the negative sign.
Then I gave her some problems with the negative sign:
Q: – (cold, left)
A: (hot, right)
Q: – (under, near)
A: (over, far)
She was still giving me that “this-is-so-easy-I-could-die” kind of look. When I thought about that, I realized it was good!
Next I explained that in math, just as in real life, there are opposites. And we find mathematical opposites by examining signs. For example, the opposite of 5 is – 5; opposite of – 3/4 is 3/4; opposite of – 3x is 3x; opposite of y is – y, and so on.
Then I gave her these problems:
Q: – (+ 2x, – 5)
Still she was with me: – 2x, + 5
Q: – (– 4y, + 3x, – 6)
A: + 4y, – 3x, + 6
The sighing was slowing down, finally. Then I simply told her that we’re going to “lose” the comma (how’s that for modern slang!), both in the original expression and in their answer. Then I gave her a new problem:
Q: – (5a – 3a – 9)
This puzzled her a bit. So I explained that she needs to mentally group the term with the sign that lies to the left. And that if no sign is showing, as for leading positive terms, she needs to mentally insert the invisible positive sign: 5 becomes + 5; 2a becomes + 2a. Once she got that, she was able to proceed:
Q: – (5a – 3a – 9)
A: – 5a + 3a + 9
And so on … one success after another. The concept was sticking. And best of all, she had a conceptual framework — the concept of opposite — that she could “lean against” any time she got stuck.
The longer I tutor the more I realize that this kind of conceptual framework — a story or concept we know from everyday life, which relates to the algebra in a direct way — is a big key to helping students grasp algebra. I use these kinds of stories in my book, the Algebra Survival Guide, providing stories we know from everyday life, which serve as analogies that show how the math works. For example, in the Guide I use a “tug-of-war” analogy to show how you solve problems like: – 3 + 8.
I’ve had so much success with this “story”-approach to algebra that I am working on an eBook that provides a whole litany of stories that work for algebra. It is fun to work on, and kids like this approach because it gives them a new way — an everyday way — to relate to the math.
So in any case, my suggestion is that when you teach or review the concept of negative signs before parentheses, you might just try the “opposites” approach and see how it works with your students.