Associated Topics || Dr. Math Home || Search Dr. Math

### Algebra Tiles and Negatives

```
Date: 11/05/2001 at 11:31:46
From: David Herman
Subject: Integers

How can we use a model (algebra tiles) to demonstrate that a negative
times a negative = a positive?  I tried to show repeated addition but
that doesn't work.

For example, -3(-5). How do you represent -3 sets of -5 at a 7th grade
level?
```

```
Date: 11/05/2001 at 12:38:30
From: Doctor Ian
Subject: Re: Integers

Hi David,

The truth is, models only go so far, and it's not clear that there is
a way to model the fact that -1 * -1 = +1 that doesn't create more
confusion than it clears up.

One thing you might try is modeling 'this times that' in the following
way:

When 'this' is positive, add copies.
When 'this' is negative, take away copies.

When 'that' is positive, use items.
When 'that' is negative, use holes.

So,

3 times 4:

2. Add copies           * * * *
* * * *
* * * *

-3 times 4:

2. Take away copies, leaving holes   o o o o
o o o o
o o o o

3 times -4:

2. Add copies            o o o o
o o o o
o o o o

-3 times -4

3. Take away holes... by adding copies!    * * * *
* * * *
* * * *

But even this isn't very satisfying, because it's not any easier to
think of removing holes by adding copies than it is to remember that
-1 * -1 = +1.  (You could use electrical charges as a model --
removing a negative charge has the same effect as adding a positive
one -- but many students find electricity even more confusing than
math.)

And even if it were satisfying, by trying to reduce this to something
that can be modeled with physical items, you would be missing the
chance to make a very important point about one of the ways in which
mathematics grows.

If we can agree that a negative number is just a positive number
multiplied by -1, then we can always write the product of two negative
numbers this way:

(-a)(-b) = (-1)(a)(-1)(b)

= (-1)(-1)ab

For example,

-2 * -3 = (-1)(2)(-1)(3)

= (-1)(-1)(2)(3)

= (-1)(-1) * 6

So the real question is,

(-1)(-1) = ?

(-1)(-1) = +1

This convention has been adopted for the simple reason that any other
convention would cause something to break.

For example, if we adopted the convention that

(-1)(-1) = -1,

the distributive property of multiplication wouldn't work for negative
numbers:

(-1)(1 + -1) = (-1)(1) + (-1)(-1)

(-1)(0) = -1 + -1

0 = -2

As Sherlock Holmes observed, "When you have excluded the impossible,
whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows
that, however improbable it seems, (-1)(-1) = +1.

If that seems too abstract, try to think of it in terms of a murder
mystery. We've considered all the other possible suspects, and we know
that none of them could have committed the murder; and we have no
reason for believing that '1' didn't commit the murder; and _someone_
did it; so '1' must be guilty.

Also, if you just want to teach your students a good way to remember
that a negative times a negative must be positive, they can use this
trick:

-a(b + -b) = (-a)(b) + (-a)(-b)
\________/   \_____/   \______/
0         neg       must
be pos

For more on the topic, see the Dr. Math FAQ:

Negative x Negative = Positive
http://mathforum.org/dr.math/faq/faq.negxneg.html

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 11/15/2001 at 11:53:09
From: David Herman
Subject: Re: Integers

Thank you for your response to my question i thought that I was going
to go insane.  I also thought that I was the only person having
problems explaining why a negative times a negative was a positive.
It helps me to rest a little easier knowing that there are a lot of
people that have wrestled with this question.
```
Associated Topics:
High School Negative Numbers
High School Number Theory
Middle School Negative Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search