Why is a negative times a negative equal to a positive?

Adding and subtracting integers is usually fairly simple to teach. I prefer to introduce multiple models such as the number line, integer chips, an elevator in a high rise building, and, my favorite, the game of Sorry. Draw 2 cards, one card says "Go Forward 5 spaces" and the second card says "Go Back 7 spaces." Most students can easily understand they have moved back 2 spaces.

Once multiplication and division are introduced, however, what seemed simple suddenly seems abstract and foggy. I suspect that this is, in small part, due to the fact that a negative times a negative equals a positive. After all, students just finished learning that backward 3 and backward 5 is backward 8 (a negative AND a negative is negative). I've even had students who had long since mastered arithmetic get confused about 5-3. They might ask, do I change this to 5-(-3) and make it 8? Too many abstract rules without understanding or an intuitive "feel" for the property make learning multiple properties difficult. Students may know specific skills in isolation but when multi-step problems combine a myriad of skills all of those "abstract rules" get jumbled and interchanged.

While I've found that a presentation of multiple models works best for classrooms (since, after all, each student in each class connects more to one model than the other), one of my favorite demonstrations of negative integer multiplication remains the reverse video.

I first recorded students running on the sidewalk forward.

Then, we watched the video forward and, to no one's surprise, they appeared to run forward. (+ of + = +)
Next, we watched the video in reverse and, only a little more surprising, they appeared to run backward - although their gait/form is usually comical. (− of + = −)

Second, I recorded students running backward on the sidewalk.

When we watched the video forward, to no one's surprise, they appeared to run backward. (+ of − = −)
Finally, we watched the backward video in reverse and the students appeared to run forward - although their gait/form is usually pretty funny here also. (− of − = +)

1. Forward, Forward: https://vimeo.com/670844707?share=copy
2. Forward, Backward: https://vimeo.com/670844656?share=copy
3. Backward, Forward: https://vimeo.com/670844590?share=copy
4. Backward, Backward: https://vimeo.com/670844551?share=copy

In various years, students have used this fact to create some rather entertaining videos.

https://vimeo.com/308920925?share=copy
https://vimeo.com/308922313?share=copy