So that lesson on multiplying negative and positive numbers still worked great for my algebra kids, understanding wise, even if they looked a little nonplussed at the end of it1.
As a recap, it goes through various combinations of video of a student walking forwards, or backwards, and then running the video forwards and backwards. forwards, of course, equates with positive numbers (and is carefully constructed on the video to go from left to right, to match our internalization of the numberline), while backwards equates with negative numbers.
The final slide has all four variations of the video looping at the same time, arranged in the usual +/- lattice. I have the kids draw the images, and write an analysis of what they saw.
This year, though, I added the coup de grace question:
What would happen if I took some video of that screen from here at the back of the room, and played that backwards for you? What would the math representation of that be?
And all of a sudden, they get that you could do this over and over again, and that every time you multiply by a negative number, you’re just flipping the sign back and forth…
1 Apparently I teach weird. They don’t seem to be used to having someone show them some stuff, and ask them to find patterns without much explanation on my part. I wasn’t aware that this approach was unusual. I don’t want to change it, but I should probably find a way to manage their expectations of how a math class works.