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?

If your son is clear on the concept of money and knows what a credit card is, this might be a good explanation:
Imagine you tell your son that you will buy him $\color{green}{\mathrm{seven}}$ gift vouchers worth £$\color{green}{5}$ each and pay for them using your credit card. Explain to your son that you now owe money, and say that it is $7 \times -5=-\mathrm{£}35$. Not part of your question; but this covers the $ \text{positive} \times \text{negative} = \text{negative}$ case.
You are having dinner with your best friend when the bill arrives from the credit card company, when your friend sees the bill he generously insists on paying it for you. You now have £$\color{blue}{35}$ worth of gift vouchers without having paid anything. 
So you tell your son that your best friend $\color{red}{\fbox{took-away}}$ seven $\color{red}{\fbox{debts}}$ of £$5$ ($\color{red}{-7}\times\color{red}{-5}$) and this equals a gain of £$\color{blue}{35}$.
