One way to explain why multiplying two negatives gives a positive

In this video I provide one explanation for why -a x -b = a x b

You may remember learning that multiplying two negative numbers gives a positive number, but have you ever paused to think about why that is the case?

#tickboom #highschoolmath #mathcoaching #math #maths #mathematics

[classydays]

The way I sort of envisioned something like -1*3, as an example, is through something I would have done every day. Say you have three rheams of paper that are meant to sit on a desk Monday morning because your boss is a prude about this sort of thing, and these rheams of paper are missing for some reason. Maybe you had to leave early Friday because the kids were sent home. That means there is one missing group (-1) of three rheams (*3), which gives us a total of -3 because it's counting for something that is not there but should be. This of course works the same as -3*1 as there are three missing (-3) rheams from this one (1) desk.

Now, if there are three bosses with a desk each (3) that all have to have three rheams each to start their week and all of them are missing (*-3), there are -9 rheams in the office for the bosses to start the week with and they are all miffed. So, how would I conceptualise -3*-3=9, because in my head we had to move eighteen spaces across the number line to get to a positive 9 and that feels like cheating. Sorry.

Let's assume you were called in to your Boss's office and were told of this issue and it was at that point your job to stack each of these desks with paper. At the start of the day, you haven't done anything, so you're at 3*-3=-9. For each desk you complete, you mark off a list that starts at 0. For each desk you stack, you mark -1 desks to do until you get to -3 desks to think about. At the end of the task, you have -3 desks to rectify the -3 paper, so in total there are nine rheams ready to start the week.

In Short, the start of the morning, there were 3 desks with -3 paper. At the end of your task, there are -3 desks with -3 paper, totalling 9 paper.