Lesson WHY is the negative multiplication rule what it is?

The rule for multiplying signed numbers is in my lesson how to multiply negatives.

For those curious enough, here's why that rule was set to be what it is.

The rule was invented so that accociative property (which is discussed in pre-algebra) would hold.
I'll construct a proof using -1 as an example of a negative number, so as not to confuse you with variables and so on. If you used any number besides -1, you would get to the same result.

Cartoon (animated) proof

Expanded proof

First, note that 1 + (-1) = 0. And any number multiplied by a zero is zero.

Knowing that, consider this simple equality:

2*(1 + (-1)) = 0

It is true because 1 + (-1) is zero. (see the section of adding signed numbers for details on this)

Since we want negative numbers to obey associative property (see it discussed in the pre-algebra section), the above equality can be rewritten as

2*1 + 2*(-1) = 0
If I subtract 2*1 from both sides of the equation, I would get
2*1 - 2*1 + 2*(-1) = 0 - 2*(-1)
I'll highlight

Since the highlighted parts cancel out, that simplifies to
2*(-1) = - 2*(-1)

Written in words, "the product of 2 by minus one is the opposite of the product of 2 and 1".

That's why the rule of multiplication of signed numbers was invented: so that associative property of multiplication and addition would hold true.