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:
  
    1. Start with 4 items   * * * *

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

  -3 times 4:

    1. Start with 4 items                * * * *

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

  3 times -4:

    1. Start with 4 holes    o o o o       
     
    2. Add copies            o o o o
                             o o o o
                             o o o o

  -3 times -4

    1. Start with 4 holes                      o o o o

    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) = ?
  
and the answer is that the following convention has been adopted:

   (-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   

I hope this helps. Write back if you'd like to talk about this some 
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.