I taught how to multiply negative numbers to my 7th graders today, so I asked Twitter for ideas over the weekend. With the help of @Xer0Dynamite, @polynumera, @StefD_84, and @Quornix, via a Martin Bauer tweet (thank you!) I came up with six ways to teach it. We didn’t have time to cover them all today, so I expect I’ll return to it on Wednesday
Second, when I try to think about what I think of multiplying negatives, I think of negatives as "anti." Like your rotation explanation and the hillbilly w/o shoes.
If going 'forward' is to the right on the number line, 2*-3 means to go backward three times twice. Adding another negative (-2*-3) means to do the opposite -- basically go anti-backwards three times twice. And of course, anti-backwards is forwards. I'm not sure that adds anything to your post. But thanks for helping me think.
Since -1 * -1 = taking a step backward and the reversing what you just did...that brings you back to your starting place = 1 (multiplicative identity).
First, thank you for teaching my seventh grader!
Second, when I try to think about what I think of multiplying negatives, I think of negatives as "anti." Like your rotation explanation and the hillbilly w/o shoes.
If going 'forward' is to the right on the number line, 2*-3 means to go backward three times twice. Adding another negative (-2*-3) means to do the opposite -- basically go anti-backwards three times twice. And of course, anti-backwards is forwards. I'm not sure that adds anything to your post. But thanks for helping me think.
So, I didn't notice that you did it by showing that (-1)(-1) = +1. Once you've got that plus associativity and commutativity, it's a walk in the park:
(-2)(-3) = (-1)(2)(-1)(3) = (-1)(-1)(2)(3) = 1*2*3 = +6.
Since -1 * -1 = taking a step backward and the reversing what you just did...that brings you back to your starting place = 1 (multiplicative identity).
"Maybe I should Substack on Pi separately someday."
How about a week from tomorrow? It's Pi Day.