by Josh Rappaport
Now that we’re a little ways into the “school year,” and our children are dealing with the “order of operations” in their PreAlgebra and Algebra texts, I’d like to demystify one area replete with difficulties: negative signs before parentheses.
Certain areas of math are annoying just like a pebble in your shoe: examined close-up, they’re very small — tiny in the scheme of things. Yet despite being small, they’re quite annoying. A negative sign in front of parentheses is just such an issue.
So there I was again, one week ago, trying to help a girl simplify the expression: – (– 5x + 3y – 7)
This time, though, I thought of something different: the word "opposite."
I talked for a moment with my tutee about the idea of opposites, then questioned her like this:
Q: So what's the opposite of black?
She replied: White (with that teenage "why-are-you- insulting-my-intelligence?” tone).
I told her not to worry, this would lead back to the problem. Next I gave her two terms for which she needs to find opposites, as in:
Q: opposite of (tall and happy)
She wrote: (short and sad), still wondering ...
And I continued:
Q: opposite of (heavy, up)
Sighing, she wrote: (light, down)
Then I explained that in math we express the idea of "opposite" with the negative sign: —
And I gave her some problems using the negative sign instead of the word “opposite.”
Q: – (cold, left)
A: (hot, right)
and
Q: – (under, near)
A: (over, far)
She was still giving me that "this-is-so-easy-I-could-die" kind of look, so it was time to relate this to her algebra problems.
I explained that in math, just as in real life, there are opposites. And we find mathematical opposites by examining signs. Examples: 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
and
Q: – (– 4y, + 3x, – 6)
A: + 4y, – 3x, + 6
The sighing was stopping, finally, which told me that a connection was being made. Then I simply told my student that we're going to "lose" the comma inside parentheses (how's that for modern slang!), both in the original expression and in their answer. And 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. Best of all, my student now 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 key to helping students grasp algebra.
I've had so much success with this "story"-approach to algebra that I’m working on an eBook that offers a whole range of stories that work for algebra. Children like this approach because it gives them a new way — and an everyday way, at that — to relate to math.
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 kids.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
Now that we’re a little ways into the “school year,” and our children are dealing with the “order of operations” in their PreAlgebra and Algebra texts, I’d like to demystify one area replete with difficulties: negative signs before parentheses.
Certain areas of math are annoying just like a pebble in your shoe: examined close-up, they’re very small — tiny in the scheme of things. Yet despite being small, they’re quite annoying. A negative sign in front of parentheses is just such an issue.
So there I was again, one week ago, trying to help a girl simplify the expression: – (– 5x + 3y – 7)
This time, though, I thought of something different: the word "opposite."
I talked for a moment with my tutee about the idea of opposites, then questioned her like this:
Q: So what's the opposite of black?
She replied: White (with that teenage "why-are-you- insulting-my-intelligence?” tone).
I told her not to worry, this would lead back to the problem. Next I gave her two terms for which she needs to find opposites, as in:
Q: opposite of (tall and happy)
She wrote: (short and sad), still wondering ...
And I continued:
Q: opposite of (heavy, up)
Sighing, she wrote: (light, down)
Then I explained that in math we express the idea of "opposite" with the negative sign: —
And I gave her some problems using the negative sign instead of the word “opposite.”
Q: – (cold, left)
A: (hot, right)
and
Q: – (under, near)
A: (over, far)
She was still giving me that "this-is-so-easy-I-could-die" kind of look, so it was time to relate this to her algebra problems.
I explained that in math, just as in real life, there are opposites. And we find mathematical opposites by examining signs. Examples: 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
and
Q: – (– 4y, + 3x, – 6)
A: + 4y, – 3x, + 6
The sighing was stopping, finally, which told me that a connection was being made. Then I simply told my student that we're going to "lose" the comma inside parentheses (how's that for modern slang!), both in the original expression and in their answer. And 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. Best of all, my student now 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 key to helping students grasp algebra.
I've had so much success with this "story"-approach to algebra that I’m working on an eBook that offers a whole range of stories that work for algebra. Children like this approach because it gives them a new way — and an everyday way, at that — to relate to math.
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 kids.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
