My boys have been doing well with integers but Multiplying Minus One threw them (and me!) for a loop. We get that the text is trying to explain why two negative integers equal a positive, and I think we understand that, since 1 times any number is that number, and since a negative times a positive is a negative, -1(a) would be -a and -1(-a) would be a, but we don't get where the -1 came from in the first place? Is it just being pulled out of thin air?

I told the boys they just need to know that like signs give positive answers and unlike give negative, but we'd still like to understand the proof. We tend to be gluttons for punishment that way.