I’m getting a chance to put all my lofty talk about negative numbers into practice. A teacher I work with is doing an introduction to negative numbers with her sixth graders in a week and a half. I’m helping her plan the unit and she’s asked me to teach the intro lesson. (She is extremely awesome, by the way.)

A fair number of kids in the class can already produce, for example, -7 as the answer if 1 is tripled and then 10 is subtracted. Some kids can’t though. I am looking to make the intro lesson rich enough for the kids who can already do this to remain interested, but my intention is to make no assumptions about what kids already know. I want to give them a problem that will make them interact with the idea of negativeness even if they’ve never heard of it.

One idea is what I described before: consider a problem like 20+10 and ask what is the effect on the answer of adding 3 to or subtracting 3 from 10; do the same with 20-10; highlight the effect of the “added 3″ or the “subtracted 3.” I love this idea in principle but actually I think it doesn’t really develop fast enough for a single late-in-the-year lesson with 6th graders who already have some exposure to negatives. When I originally had it I was imagining 3rd and 4th graders, and developing the idea slowly, with little short exercises over the course of weeks, to slowly draw kids’ attention toward the “subtracted 3″ as a worthwhile object.

So, another idea. Base the lesson around a problem that involves numbers going in two directions. Make the problem a little mathematically interesting. Don’t try to force the negative idea but make the problem impossible to solve without considering the directions of the numbers. I have an idea for the mathematical content of such a problem but I want to put it in a realistic context to allow kids to reason about it without knowledge of negatives, and I’m having trouble thinking of a context that doesn’t feel contrived. So this is a request for suggestions.

Here’s the mathematical content idea: Three target numbers (positive and negative) and a list of 8-10 numbers, in no particular order, that sort into piles adding up to each target. The puzzle is to do the sorting. For instance:

Targets: -12, -19, and 7

List: -6, -4, -9, -15, 4, 2, -5, 9

I am avoiding posting an answer (there is at least one; I haven’t checked whether there is definitely only one) so you can try out the problem if you want.

I’ve been trying to compose a story so that (a) the positive and negative numbers have a concrete meaning accessible to someone who’s never heard of positives and negatives, and (b) there is some sort of reason why you’d know the target sums and the list but not know how the list sorts into piles, and yet you’d want to know. I have so far been stuck. I’ve been thinking about something like, 3 friends go gambling; A loses $12, B loses $19, C wins $7; for some reason (??) they know the individual transactions that took place (house collects $6, house collects $4, … , house collects $5, house pays out $9) but not who did which transaction, and for some even more far-fetched reason (????) they want to know. As you can see this idea isn’t really working.

So: can you help me out with this? What I like about the problem is that a) it’s got that sudoku-jigsaw-like challenge even for someone for whom the actual arithmetic is elementary; and b) if I can find the right context to allow kids to reason about the problem concretely even if they’ve never heard of negatives, then the problem elicits as an answer what amounts to a set of equations about negatives, without the kids needing to learn anything new beforehand.

Or, is the problem impossible to render in an un-contrived way? In which case, can you help me think of a less-contrived problem that accomplishes some of the same goals?