by Josh Rappaport
Certain areas of algebra are like pebbles in your shoe: looked at closely, they appear tiny. And yet they can be “oh-so-annoying.”
One example of such a tricky "little" area: negative signs standing before a set of parentheses, as in the expression: – (– 3x + 7) In this article, I'll show how you can use an everyday concept to elucidate this seemingly "knotty" algebraic concept.
When this came up recently, I was trying to teach a girl how to simplify the expression: – (– 5x + 3y – 7)
This time, though, I thought of something different: the word “opposite.” I talked for a moment with my teenage tutee about the concept of an opposite, and we played around with the notion.
Josh: So, what’s the opposite of black?
Student: White! (with that teenage “why-are-you-insulting-my-intelligence?" tone)
I told her not to worry, this would help her solve the problem. Next I gave her two terms and asked her to write 'opp' (meaning 'opposite') for both terms, and to separate them with a comma.
Josh: Write the simplified version for: opp (tall, happy)
She wrote: (short, sad), wondering how in the world this would be helpful.
I continued ...
Josh: How would you show the opposites here? opp (heavy, up)
She wrote: (light, down), sighing with boredom.
Then I explained that in math we express the idea of “opposite” with the negative sign.
Then I gave her some problems that relate the concept of opposite to everyday life. But I used the negative sign to keep the focus on mathematical expressions, like this:
Josh: – (cold, left)
Student: (hot, right)
and
Josh: – (under, near)
Girl: (over, far)
She was still giving me that “this-is-so-easy-I-could-die” kind of look. But I realized that this was a good thing.
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; the opposite of – 3/4 is 3/4; opposite of – 3x is 3x; opposite of y is – y, and so on. In other words, to find the opposite of a number or term, we just change the sign of the term, either from positive to negative, or from negative to positive.
Then I gave her these problems:
Josh: – (+ 2x, – 5)
Still she was with me: – 2x, + 5
and
Josh: – (– 4y, + 3x, – 6)
A: + 4y, – 3x, + 6
The sighing was easing, finally. At this point I told my tutee that we would now “lose” the comma (how’s that for modern slang), both in the original expression and in the answer. Then I gave her a new problem:
Josh: – (5a – 3a – 9)
This puzzled her a bit because the 5a does not show its sign. So I explained that if no sign is showing, for a leading positive term, she needs to mentally insert the invisible positive sign. That is to say, 5 would become + 5; 2a becomes + 2a. Once she grasped these ideas, we continued.
Josh: – (5a – 3a – 9)
Student: – 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 an opposite — that she could mentally “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 we relate to the algebra— is a big key to helping students grasp algebra. I suggest that you try this idea for the concept of negative signs before parentheses. And I also suggest that you keep your mind open to analogies from everyday life that you can use to explain the concepts of math. I find that the process is both fun and useful.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Algebra Survival Guide, and the companion Algebra Survival Guide Workbook, both of which will soon be available for homeschoolers as a computer-based Learning Management System, developed and run by Sleek Corp., of Austin, TX.
At his blog, Josh writes about math education, offering tips and tricks. Josh also authors Turtle Talk, a free monthly newsletter with an engaging "Problem of the Month." You can see a sample issue here or subscribe at this site. Josh also is co-author of the "learn-by-playing" Card Game Roundup books, and author of PreAlgebra Blastoff!, a "Sci-Fi" cartoon math book featuring a playful, hands-on approach to positive and negative numbers.
