Go read Math is a social construct for the debate. I'm just filling a nodeshell with some (hopefully) amusing, interesting information about how I spend my days.
Another w/u from the junior high math teacher:
When we get to multiplying (and dividing)* integers, these are the notes that I put on the board**:
SAME signs :) +
I tell the kids, if I say I like chocolate, and Laura says she likes chocolate, then we are agreeing and getting along; everything is positive. But if Theo says he hates brussels sprouts, and Caldwell says he hates brussels sprouts too, even though they are both being negative about brussel sprouts, they are agreeing and getting along, so that is positive, too.
DIFFERENT signs :(     -
If, however, Khaled likes The Matrix and Dan thinks it was a stupid movie, then they’re not agreeing, and there are some negative vibes, so the outcome is negative.
Then we practice with some examples.
+3 x +4 = +12 -5 x +2 = -10 -6 x -10 = +60
Eventually, someone wants to know who made up these rules. They want to know why that person made it so complicated. It is up to me to explain to them that no-one made up the rules; people just made a note of patterns that they found. The rules describe the behavior of the numbers.
To illustrate, we take a little side trip (I am famous for tangents) and talk about even and odd numbers.
. . .
”What do you get when you add two even numbers?” I ask. “Let’s try and see:
2 + 2 = 4 4 + 6 = 10 28 + 62 = 90
”How about two odd numbers?”
1 + 3 = 4 5 + 7 = 12 89 + 15 = 104
”And an even plus an odd?”
2 + 3 = 5 33 + 88 = 121 17 + 6 = 23
(Sometimes I go so far as to draw dots representing the numbers on the board, and point out that if there are two sets of even dots, or two sets of odd dots, every dot has a buddy; it’s only when there’s one set of even dots and one set of odd, that someone is odd man out…)
”So, we can describe the patterns that we’ve found, right? We didn’t check every number, but we checked enough to find a pattern that seems to hold true:
even + even = even
odd + odd = even
even + odd = odd
”Did we just make up those rules? We just described a pattern that we found, right? Well, that’s how it works.”
. . .
Of course, I can and do show them the progression with integers:
+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
Where each time, the number positive three is being multiplied by goes down one; each time, the answer goes down three. By setting up the pattern, you can predict the answer to the next multiplication problem.
Then I change the pattern:
+3 x -3 = -9
+2 x -3 = -6
+1 x -3 = -3
0 x -3 = 0
-1 x -3 = +3
-2 x -3 = +6
-3 x -3 = +9
We already established that positive three times negative three is negative nine; now I change the first factor, decreasing by one each time. The product increases by three each time: in this pattern, the answer to negative one times negative three can be predicted as three more than zero, or positive three.
Of course, at this point, most kids are willing to just take my word for it, and start working on the assignment so that they'll have less homework.
*When I was in high school, no attempt was made to explain or simplify "the rules". I remember having to memorize
positive times positive equals positive
positive times negative equals negative
negative times positive equals negative
negative times negative equals positive
positive divided by positive equals positive
positive divided by negative equals negative
negative divided by positive equals negative
negative divided by negative equals positive
Auuggghhh! If I can save one child from that, I've done my good deed for the day.
**Yes, I really draw the smiley faces. But not sideways. I usually go off on another tangent about how the addition of eyebrows to a simple smiley face can completely change the expression.