I had one class of students who never got a grasp on how addition and subtraction works with negative numbers. I can’t blame them much – that was what I was covering during the period that one of their classmates got shot. I obviously didn’t accommodate that as well as I could or should have.
It did give me an opportunity to try reteaching the concepts from scratch, though, and I hit gold.
I’ve already recounted how I made videos of them going backwards & forwards and then played the videos backwards and forwards to convey the idea of multiplication.
This ties into that conceptually wonderfully. The next time around, these lessons will combine to make for some powerful understanding.
Here’s how it works:
Print up a number line. Basically, one number per 8.5×11 page, bold, 380 pt font. From -12 to 12, more if you have room. Tape the pages together in order.
Pick a place in your room where you can lay this out. It’s okay if the students have to move your furniture around. Wherever it gets laid out, put up a sign at the positive end saying (in large bold) FACE THIS WAY TO ADD. At the other end, put a sign saying FACE THIS WAY TO SUBTRACT.
They may not know it, but this gives them all the information they need. The trick is to slowly build the problems they have to solve.
I wrote them out on 3×5 cards, handed them out in order, and had the kids step out the problems with coaching from everyone else. (Make sure the answers don’t run them off of the end of the number line).
Start off with simple addition using positive numbers. Then have them subtract, first with positive results, then with negative. Go back to adding, But start from a negative number, with negative results. Then starting from a negative, with positive results.
Now comes a big step. So far, the second number has always been positive. They always have walked forward, even when they’re subtracting. Give them 5 + -1. Let them struggle with which direction to face (they’re adding, right?) and what the answer should be. It took all of 5 seconds for someone to realize they had to walk backwards, and about 30 to convince everyone else. Throw a lot of adding with negatives, let them walk backwards to their hearts content.
Finally, when they subtract negatives, they get the big aha. They realize that they could just stay turned around facing the ADD sign, and walk forwards. It’s not long after that that they’ll get that they can get anywhere on the number line while facing the ADD sign, as long as they correctly adjust whether they walk forwards or backwards.
My kids went from being completely confused about the relationship between adding and subtracting to completely comfortable with converting between equivalent versions of the two. A little bit more work with all the different combinations of positive & negatives of the same two numbers revealed the other patterns that turn into rules governing the addition & subtraction with negative numbers.
And even though the number line is gone today, the signs are still up to serve as cognitive clues for how they learned these concepts.