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?
This is pretty soft, but I saw an analogy online to explain this once. 

If you film a man running forwards ($+$) and then play the film forward ($+$) he is still running forward ($+$). If you play the film backward ($-$) he appears to be running backwards ($-$) so the result of multiplying a positive and a negative is negative. Same goes for if you film a man running backwards ($-$) and play it normally ($+$) he appears to be still running backwards ($-$). Now, if you film a man running backwards ($-$) and play it backwards ($-$) he appears to be running forward ($+$). The level to which you speed up the rewind doesn't matter ($-3x$ or $-4x$) these results hold true. 
$$\text{backward} \times \text{backward} = \text{forward}$$
$$ \text{negative} \times \text{negative} = \text{positive}$$
It's not perfect, but it introduces the notion of the number line having directions at least.
Why a negative times a negative can be reduced to the question of why
-1 x -1 = 1.
The reason for that is because it is forced upon you by the other rules of arithmetic.

1 + (-1) = 0 because of the definition of -1 as the additive inverse of 1

Now multiple both sides by -1 to get

-1(1+(-1)) = 0 because 0 times anything is 0

Use distributive law to get:

-1* 1 + (-1)x(-1) = 0

Now -1 * 1 = -1 because 1 is multiplicative identity.

So we have
-1 + (-1)x (-1) = 0

Put -1 on the other side by adding 1 to both sides to get

(-1) x (-1) = 1

So -1 x -1 = 1.

Now for any other negative numbers x, y we have

x = (-1) |x| and y= (-1) |y|

So x * y = (-1) |x| * (-1) |y| = (-1) *(-1) * |x| * |y| = |x * y| is positive.

Now that you know the reason it really doesn't make much difference in understanding. This question is not really that important. 
It's like asking why is 1 raised to the 0 power equal to 1?
Because that's forced upon you by other rules of exponents,etc.

A lot of time is wasted on this. This is not the kind of problem kids should be thinking about.