Why is a negative times a negative a positive?Minus times minus results in a plus,The reason for this, we needn't discuss.  Ogden Nash The swimming pool Here is an anology of filling or draining a swimming pool: If you are filling the pool at a rate of 3 litres per second (+3) in 4 seconds (+4) the amount of water will have increased by 12 litres (+3 * +4 = +12). If you are draining the pool at a rate of 3 litres per second (3) in 4 seconds (+4) the amount of water will have decreased by 12 litres (3 * +4 = 12). If you are filling the pool at a rate of 3 litres per second (+3) 4 seconds ago (4) the amount of water was 12 litres less (+3 * 4 = 12). If you are draining the pool at a rate of 3 litres per second (3) 4 seconds ago (4) the amount of water was 12 litres more than it is now (3 * 4 = +12). This anology may provide students with an intuitive basis for the multiplication rules of positive and negative numbers. This example may not work for you, and you might want to read others below.
Number Line Let's see how a mathematician might understand what's going on when a negative number is multiplied by a negative number. Here's a plausibility argument drawn from multiplication patterns: 3 x 3 = 9 2 x 3 = 6 1 x 3 = 3 0 x 3 = 0 1 x 3 = ?A Mathematical Explanation If we can agree that a negative number is just a positive number multiplied by 1, then we can always write the product of two negative numbers this way: (a)(b) = (1)(a)(1)(b) = (1)(1)ab For example, 2 * 3 = (1)(2)(1)(3) = (1)(1)(2)(3) = (1)(1) * 6 So the real question is, (1)(1) = ? and the answer is that the following convention has been adopted: (1)(1) = +1 This convention has been adopted for the simple reason that any other convention would cause something to break. For example, if we adopted the convention that
(1)(1 + 1) = (1)(1) + (1)(1) (1)(0) = 1 + 1 0 = 2 As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth." Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (1)(1) = +1.
More Mathematical Explanation
We can write x = ab + (a)[ (b) + (b) ] (factor out a) = ab + (a)(0) = ab + 0 = abAlso, x = [ a + (a) ]b + (a)(b) (factor out b) = 0 * b + (a)(b) = 0 + (a)(b) = (a)(b)So we have
x = ab ab = (a)(b)

Extracted from Math Forum: Ask Dr. Math