What is a good non-rules-based approach to teaching why a negative times a negative is a positive? I've seen a proof-through-absurdity showing that if -1 * -1 = -1, then you can make 0 = 2, and a proof using all three negativity permutations, and setting two different distributions equal, showing that ab = (-a)(-b), but the math for both of those approaches seems to be more advanced than the rule.

 Say that multiplication by -1 is a 180 degree rotation of the number line. Then -1 * -1 is a 360 degree rotation -- which is the same as a 0 degree rotation.Note that saying it like this is non-intuitive, but the point is that it's actually a geometric interpretation, so you draw a picture. This also helps students with the idea that there's actually some connection between algebra and geometry --- even though this is really fundamental, it's often omitted from mathematics education.
 My favourite motivation is to think of bank accounts. (A less monetary approach would be to think of a rising or falling temperature.) I think that anyone can predict the results of the following four operations:* make a deposit in your bank account every day, and see your balance in a few days (positive times positive);* make a deposit in your bank account every day, and see your balance a few days ago (positive times negative);* make a withdrawal from your bank account every day, and see your balance in a few days (negative times positive);* make a withdrawal from your bank account every day, and see your balance a few days ago (negative times negative).
 I've never had to teach this, except remedially, where honestly, time is a major issue. A teacher in my building has students record someone walking backwards down the hallway, then plays the video back in reverse. The student appears to walk forwards down the hallway, more convincingly than you'd expect.[1]From there, you can show that -(-1) = 1. If the students have any command of factoring, you can use that to handwavingly show, for example, that -2 ✕ -3 = -1 ✕ 2 ✕ -1 ✕ 3 = -(-6)) = 6.[1]She actually has them walk forwards, and hold signs - it's more involved, and covers all four cases for multiplying negatives and positives, but we'll gloss over that for the moment.edit: oh yeah, can't use * for multiplication.
 You make your own banking system or something. Say you give a class of 10, \$2 each. They all owe it back to you which makes the balance -2 each. So ask them how much you're owed back in entirety. I bet most of the class will crack it themselves without you and your rules :)

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