## Sunday, 21 June 2015

### Ivan the jumping flea and negatives

So yesterday at the excellent National Mathematics Teachers Conference (#mathsconf4) I was discussing with some other teachers approaches to teaching addition and subtraction of negative numbers. I outlined two approaches I have used in the past, the mood cards I outlined in this blog and the other approach which I think underlines more of the conceptual understanding - movements along a number line. To make it a bit more memorable for pupils I invent a nice jumpy character to assist me in the demonstrations - Ivan the jumping flea.

The way Ivan moves is governed by the following rules:

1) Ivan starts at 0 facing the 'positive direction' (i.e. the direction of increasing numbers).
2) Positive numbers cause forward jumps.
3) Negative numbers cause backwards jumps.
4) The operation of addition makes Ivan face the 'positive direction' (i.e. the direction of increasing numbers).
5) The operation of subtraction makes Ivan face the 'negative direction' (i.e. the direction of decreasing numbers).

So lets say Ivan is completing the calculation 3 + (-5). Ivan would start at 0 facing up the number line and the first thing he would do is jump forwards 3 places, as the first part of the calculation is the positive number 3. The next thing Ivan would do is face the 'positive direction' (which would mean he did nothing as he is already facing the positive direction), as the next part of the calculation is the operation addition. The next thing Ivan would do is jump backwards 5, as the last part of the calculation is -5. Once these steps are completed Ivan would be at the value -2, showing 3+(-5) = -2.

Compare this to the calculation 3 - (-5). The first step would be the same, as Ivan still starts at 0 and still jumps to 3 as before. This time the operation is subtraction, so Ivan turns to face the 'negative direction'. Ivan then jumps backwards 5 as before, but because he is facing the 'negative direction' he is actually jumping up the number line, and so ends up at the number 8; showing that 3 - (-5) = 8.

I have a PowerPoint here which shows Ivan solving both of these questions (which I made some changes to in order to have a completely animated sequence, once you click to get going on each slide) which people are free to adapt for other questions as you see fit (provided you can alter the motion paths etc).