Q&A #19694

Proof for multiplication rule

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From: Gail (for Teacher2Teacher Service)
Date: Jul 24, 2008 at 17:59:35
Subject: Re: Proof for multiplication rule

I don't have a formal proof to share with you, but I usually introduce this
to my middle school students as a pattern.  

With multiplication and division, I like to use patterns to help students
make generalizations about what is happening. For example, look at the
products in this list:

3 x 4  = 12
3 x 3  =  9
3 x 2  =  6
3 x 1  =  3
3 x 0  =  0
3 x -1 = -3
3 x -2 = -6
3 x -3 = -9

There are lots of things to notice in this list, like what is staying the
same, what is changing, patterns exist, what can be predicted. And how about
just noticing, p x p = p, and p x n = n

Use that list to move to this one...

-3 x 3  = -9  ( a great time to bring up properties)
-3 x 2  = -6
-3 x 1  = -3
-3 x 0  =  0
-3 x -1 =  3
-3 x -2 =  6
-3 x -3 =  9

Wow, look what happened now. We already knew that n x p = n, but now we have
some proof that n x n = p.

I have students make a new list using numbers of their own, to confirm that
it works for more than the threes.

I think the most important thing about integer problems is helping students
understand that the numbers don't both mean the same sort of thing. For 
example, if you are trying to solve (-3)(-2), it is like trying
to find "the opposite of" 3 groups of -2. That first number names the number
of groups you have, and the second number tells how many are in each group,
just like if you were using two positive numbers. The catch is, with a
negative, you need to think about opposites.

You can use the same sort of reasoning with division. Suppose you are trying
to find how many -2 groups there are in -12.  6 would be a sensible answer,
right?  But if the problem were "how many -2 groups are there in +12, you
wouldn't expect 6 to be there answer again. Instead, it would be the
"opposite of" 6.

Hope this helps.  :-)

 -Gail, for the T2T service

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