Teacher2Teacher |
Q&A #19694 |
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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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