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## Forwards & Backwards

I’ve done a lot of opinionating, and not so much lesson sharing. I’ve wanted to, but the adding and subtracting of negatives didn’t go well, and I had to reteach them focusing on the mechanics, slowly tying those mechanics to the manipulative practice before. It went well, but was just old fashioned grunt work teaching, with lots of practice and repetition to reinforce the different cases.

So, that lesson I planned last week?

I finally got to do it.

It rocked.

I hate trying to use technology when ordinary paper and pencil will do the job.

Conversely, there really are times where it works like nothing else.

This is the final slide of about 30. Here’s how I got to it:

Last week I filmed kids walking back and forth – the one that ended up in the presentation wanted to do it in my rolling chair.

We start off with a simple still shot of him going forwards, holding a whiteboard showing the direction he was rolling in. We discuss and decide going forwards should be positive.

The next shot is a still of him going backwards, with a corresponding arrow. This is negative.

Then we get to the motion. Repeat the shots, but this time they’re moving1.

Both of them show the film rolling forwards – the students analyze the motions and convert the results from forwards & backwards to postive and negative.

Then, the whole process is repeated for the film rolling backwards.

Somewhere, well before the loop of him rolling backwards with the film going backwards, about 1/2 of the kids will start shouting out that when both he & the film are going backwards that it’ll look like he’s going forwards.

They just explained to me why a negative times a negative is a positive.

I still make them go through the written explanations for all the individual clips.

Then I put them all together, in that slide up above2. It comes up without the +/- signs in the header column/row: They have to tell me what each row or column has in common. But once they do, it’s very clear that the top left and bottom right clips both show my student rolling in the same direction from left to right (i.e. positive) and the top right & bottom left show, in different ways him rolling backwards from right to left (i.e. negative).

Not only do they now get why the rule is there, they also understand why it’s for multiplication and not addition.

I can’t say that they’re going to succeed at algebra next year, but they will at least be able to tell the difference between adding and multiplying negative numbers (in much the same way that they now know the difference between multiplying and adding fractions).

1 During testing with my wife, it wasn’t obvious which way the film was rolling, so I added the little forwards & backwards icons at the bottom

2 This is where the technology was cool: I had four videos running at the same time. It was stressing my poor little processor, so I had to recreate them as low res/bandwidth versions in order to get smooth motion. In a couple of years, that won’t be an issue, and I’ll be able to play those together with the x2 and x1/2 videos.