I have great difficulty with the explanation you provide below. First, is it really an explanation? Labeling one thing "pluses" and another "minuses" is not an explanation, but rather an artificial and complicated way to avoid explaining. For instance, why is one a "plus" and the other a "minus" and why is it that the combination "+ -" is a zero? I still maintain that understanding "negatives" is really just understanding the idea of directed numbers on a number line.
>First, you'll need to visit a store that sells ceramic tiles for >countertops. Ask for grouting spacers (they come in various sizes), and >you'll discover that you will get a bag of pluses (+). These are easy to >trim to produce minuses (-). > >Let a "+" represent +1 and a "-" a -1. A zero is represented by the >combination, "+ -". Let's consider the examples (2)(3), (2)(-3), (-2)(3), >and (-2)(-3). I place six zeros on the overhead projector using six >combinations of "+ -", with two rows of three zeros. I define the first >factor as indicating how many groups are either added or taken away. For >example, (2)(3) means that I add two groups of three. On the overhead >projector, I simply remove the six minuses, and there remain two rows of >three pluses each. Clearly the result is +6. With (2)(-3), we add two >groups of -3 (remove all the pluses from the six zeros). For (-2)(3), we >remove two groups of 3, leaving behind two rows of minuses (-6). Finally, >(-2)(-3) indicates a removal of two groups of -3 with the net result of two >rows of 3 pluses remaining on the projector.