It's amazing to me how many things, one might want to say "easy" things, in mathematics I never really thought about. I was reading an ed article today, which reminded me of something I saw a while back about how people explain why a product of two negative numbers is a positive number, so I tried searching for it. Needless to say I wasn't able to find it, but I ran into something else which was even better! Here is the source, and an excerpt follows:
We take any Euclidean line, mark point 0 on it, choose a unit length and mark point 1, and then the non-negative "numbers" are associated (identified) with all the points on the ray from 0 through 1in the usual way. The negative numbers are then identified with the remaining points on the opposite ray in the usual way so that we have now a Cartesian coordinate number line L. Now choose a second line M through 0 and mark a 1 on it at the same unit distance as we used on L, and complete this so M is also a Cartesian coordinate line. Now for any number b (point) on L, and any nonzero number c on M, the product point b*c can be found: let J be the line from 1 on L to c on M, and K the line parallel to J through b on L, then b*c is the intersection of K and M.
Here is a picture that shows L as x-axis, M as y-axis, and we are looking for 4*3.
The only problem with this explanation is that the students need to know about similar triangles! Regardless, this was way cool. If I had thought about this on time, I would have given it to my students on their final that had them think about similar triangles. Next time!