Why is negative times negative = positive? Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like so:
Assume $(-x) \cdot (-y) = -xy$
Then divide both sides by $(-x)$ and you get $(-y) = y$
Since we have a contradiction, then our first assumption must be incorrect.
I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.
Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one?

I would explain it by number patterns.
First, to establish that a positive times a negative is negative: $3 \times 2 = 6, 3 \times 1 = 3, 3 \times 0 = 0$. Notice in each case, as we reduce the second factor by 1, the product is being reduced by 3. So for consistency the next product in the pattern must be $0 - 3 = -3$. Therefore we have $3 \times (-1) = -3, 3 \times (-2) = -6$, and likewise a negative for any other other positive times a negative.
Second, to establish that a negative times a negative is positive: we now know that $3 \times (-2) = -6, 2 \times (-2) = -4, 1 \times (-2) = -2, 0 \times (-2) = 0$. Notice in each case, as we reduce the first multiplier by 1, the product is being increased by 2. So for consistency the next product in the pattern must be $0 + 2 = 2$. Therefore we have $(-1) \times (-2) = 2, (-2) \times (-2) = 4$, and likewise a positive for any other other negative times a negative.
