Visualize negative multiplications
Why -1 x -1 = 1?
I recently had to explain to my daughter negative multiplication. I had a way to help her visualize multiplication. I thought it can be applied to all multiplications. But I stumbled upon -1 x -1 and had no way to explain why it turns into 1. I found a video on Khan Academy. It is more of a proof than an explanation.
A way to understand multiplication is to think of it making copies. For 1 x 1, you take 1 and copy it 1 time. For 1 x 2, you take 1 and copy it 2 times.

Now for a single negative multiple, we can still use this visualization. For -1 x 1, you take -1 and copy it 1 time. For -1 x 2, you take -1 and copy it 2 times.

But this visualization doesn’t work when we try to visualize -1 x -1. So I came up it a way to visualize 1 x -1 first. Multiplying -1 means reversing the direction. So 1 x -1 is to take 1, reverse its direction.

For 2 x -1, we take 2 and reverse its direction.

Finally, for -1 x -1, we take -1 and reverse its direction.

Another example, for -2 x -1, we take -2 and reverse its direction.

Hopefully, you can explain why -1 x -1 = 1 now.
