Certain areas of algebra are like pebbles in your shoe: looked at closely they’re tiny. And yet they are “oh-so-bothersome!”

As a tutor, I’ve long felt this way about a negative sign before parentheses. It’s a small thing, and it seems simple to grasp to those who get it. Yet students make so many mistakes when facing this situation, so to them it’s extremely irritating!

And there I was again, trying to help a girl understand how to simplify this expression:

– (– 5x + 3y – 7)

However this time I came up with 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)

and

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

and

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.

Comments on:"Common Algebra Mistake: How to Understand a Negative Sign in Front of Parentheses" (12)Shana Donohuesaid:This is an EXCELLENT idea! Wow, what a great way to teach it. I’m always running into students who mistake (x + 2) – (x – 5) for (x + 2)(x – 5). That’s a tough one, but even tougher is watching kids get the subtraction wrong because they don’t know (or forget) to distibute that negative sign in front of the (x – 5). Ugh!

If you get time, I’d loove for you to add a comment to a blog post I created a while back about common mistakes made in algebra. Yours is a classic (http://zerosumruler.wordpress.com/2010/06/16/maths-freshmans-dream/)

I’m planning to use your example today! Better yet, create a poster! Thank you!

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Shana Donohuesaid:p.s. Will you please email out the link to your ebook when it is complete? I would LOVE to read it (or even collaborate if you were up for that).

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Denisesaid:I like this. I’ve always used the idea of “opposites” when working with negative numbers, but I never thought of scaffolding it with a function of words. Thanks!

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Valeriesaid:Your guidance has been invaluable for our family! Two CPA parents have three kids…#1 and #2 have math minds like their parents. However, #3 looks and thinks like her aunt, a marketing/art person — gasp! When she looks at us with the “I don’t get it” expression, we have been incredulous, exasperated, frustrated and eventually out of ideas. Our left brains vs. her right brain. Her science teacher is the one who recommended your book. We, too, would be interested in the ebook. We will be following your posts also. Thanks for helping open a window in our left-brained heads — a window open to new ideas and concepts to teach and explain math.

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Beverlysaid:I have used that also, but the students seem to forget when applied to a new situation. 5 – (x – 4) would give as the answer 5 -x – 4 = -x +1 totally ignoring the parenthesis. I know that one student is somewhat dyslexic, maybe she doesn’t see the parenthesis.

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Josh Rappaportsaid:Hi Beverly,

For problems like the one you referenced, 5 – (x – 4), you need to use a separation device so that students do not blend the 5 with any of the terms to the right of the 5. I have developed a term that I call the “double slash.” You can read about this approach at the blog post titled: How to Decrease Mistakes in Algebra – Part 2, which is here:

https://mathchat.me/2011/05/04/how-to-decrease-mistakes-in-algebra-part-2/

i hope this helps.

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Rebecca Hillsaid:Great explanation for a student that loves English and Language Arts.

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Kishore Shreedharansaid:smart

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Lulusaid:What happens when there is a positive sign outside the parentheses?

Do you just get rid of the parentheses then?

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Josh Rappaportsaid:Hi Lulu. Thanks for that excellent question. Yes, whenever there’s a positive sign in front of a set of parentheses, you can “unpack” what’s inside parentheses (unpack meaning, remove from the parentheses), leaving those terms’ signs as they were.

Example: + (4x – 8)

= 4x – 8

Notice that the terms inside parentheses came out just as they were.

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Ayana Boruahsaid:I wanna know , in this mathematical problem 👇

-n^2 +13n-36

What if we take common of “-” sign from “n”.

What will left then?? I wanna know that…!! (Sorry for my English)

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j rapsaid:Hi Ayana,

Good question. I think what you’re asking is what would happen if we factor out a – 1 from all terms in the expression. Well, here’s the answer to that.

You would get this:

– 1[n^2 – 13n + 36]

and then you would see that that’s pretty nice because the trinomial inside the brackets factors, as follows:

– 1[n – 4)(n – 9)] , which can be written slightly more simply as:

– (n – 4)(n – 9)

So in this situation it makes a lot of sense to do the initial step of factoring out the – 1. Doing that leads to a much simpler expression. And in general a factored expression is easier to work with than an un-factored expression because a factored expression, if it’s one side of an equation that equals 0, allows us to solve the equation.

Example with this expression; if we had this:

– (n – 4)(n – 9) = 0

The solutions for n would be: n = 4, 9

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