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?

Very nice idea. Some place in this understanding there has to develop a notion of direction and also the significance of zero.

I’ve had some luck talking about moving so many steps left and then so many steps to the right. For an especially athletic lesson one could probably arrange the students in a number line, each one assigned a number and then have one student enact a movement through them.

I would suggest delaying subtraction for some time.

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

Three friends have to do their taxes together, because that’s the only way they can face this onerous chore. They’ve been up late, and are too tired to think. (I started out thinking they’d been drinking, but this is 6th grade. Can’t have that.)

Nope. Start over. Three housemates are figuring out house accounts. Before they started, they thought they knew what their standing was, but now the receipts and … somethings … are mixed together.

Hmm, my son wants me to read, and I’m not sure where I’m going with this. Let me know if it helps any. ;^)

How about a somewhat silly fantasy approach?

A pet store sells Green Dragon Fish and Red Devil Fish. Both kinds of fish are perfectly peaceful when living in a tank with their own kind, but whenever a Green Dragon Fish and a Red Devil Fish meet, they will fight until they’ve torn each other apart [perhaps too gruesome?]. Furthermore, the fish will attack nets, human hands, or anything else that’s put in their tank, so the only way to transfer them from one tank to another is to dump the entire tankful into a new tank at once.

Three customers have come into the store. The first wants 12 green fish, the second wants 19 green fish, and the third wants 7 red fish. The owner has tanks of 6 green, 4 green, 9 green, etc. How can the owner combine these tanks so that each customer gets what he/she desires?

I realize this has some flaws, in particular that the negative numbers are now regarded as positive numbers of a different kind of object. It might be useful as an illustration of some different properties of negative numbers though, such as the symmetry between negative and positive (if you negate all the numbers you are adding, the answer is negated).

Maybe this is dumb, I don’t know, but how about make it like a card game. Actually this reminds me a teensy bit of Baccarat. Not really. Just a little bit. I would probably tell them it IS Baccarat, like smooth people play in movies, like James Bond. They don’t know.

“You’re dealt three piles in the field and given 8 sets of cards in your hand. If you can sort your cards so they match the three piles, you win! Money! If you can’t, the house wins, and takes your money. Every casino game has a fun, weird wrinkle, and in this one – the red and black pips cancel each other out.”

Here is your problem : http://img.skitch.com/20100522-mb7pyim94qjr5ch3bf4cgmprmf.jpg

Here is the solution: http://img.skitch.com/20100522-drtdn9una8xirxhwfb8wnyde66.jpg

The fact that you need more than one card for some of your numbers makes it look more messy and annoying. Maybe the problem could be altered so it’s just combining 8 cards to make 3 cards. Maybe you could invent a different sort of deck.

Also your problem might be a little difficult to start with. Maybe build it up with some simpler example games. Also I would try to put cards in their hands that they can sort and move around.

Would this be a feasible game for real? How likely is it that a random 8 cards could be sorted to match a random 3 cards? It would be a pretty awesome game if the odds were close to 50-50 with a slight house advantage, but it feels instinctively like it’s rather less likely than that.

Yo I forgot to ask you, how did you do that with the cards?

The story is going to be invented here no matter what, in which case the important thing is for the actions of your objects/people/whatever to be as intuitive as possible to 6th graders.

I’ve seen this done before with weights and balloons. To make this a puzzle, you could have a set of given weights and a set of given balloons that lift/lower a set of platforms. The story could involve some mad scientist who needs the platforms in that exact position for his death ray / other diabolical device to work.

I like the moving some number of steps left or right idea. Maybe it could be forwards and backwards alternately? Maybe it could be part of some sort of dance and the students could make up their own dances? Maybe it could be part of a choreographed dance for performance on stage. Then you would want the dancers to be in different positions on stage after doing different sequences of moves. Some moves might move you forward and backwards different amounts. And there are perfectly good artistic reasons that you might want to require that certain moves be performed only a certain number of times each. You could have a ton of fun with something like this when teaching coordinate geometry, but you could restrict yourself to something like a line dance right now to keep everything in 1D. I would think talking about real dance moves would also be cool and help a lot.

I like Susan’s approach.

I fall back, on occasion, on basketball and points (those are already numbers, sort of, right?)

So if we score 4 and they score 3, that’s seven points, right? Well, not exactly. There’s a difference between the two kinds of points.

(I’ve used artificial notation for this)

What I like here, is a few things.

1. Portable model, doesn’t require equipment.

2. Intuitive model. Doesn’t demand explanation of an (artificial) set-up

3. Works with natural numbers just fine.

3. Sense of “opposite” is strong, including a + b = 0

4. Transitions nicely to addition, if there was such a desire or need.

Jonathan

I like how many ways this can play out. I’m definitely going to point my beginning algebra (at community college) students here. I’ll ask them to pick their favorite way of thinking about it.

Hmm, I wonder if I can figure out odds for Kate’s game…

>How likely is it that a random 8 cards could be sorted to match a random 3 cards?

If there’s some simple way to make sure the total sums agree(sum of 3 targets = sum of 8 cards), that sure ups the odds for success. Maybe too high, though. I’ll play with that.

I’m thinking this sounds like something that could be turned into a rummy-ish game. Each start with 8 cards. 3 target cards are laid down. You keep picking until you can put your whole hand down in 3 sets that match the targets. I wish I had someone here to test play it with me!

So, back to your 8 numbers…

If you are in Williamsburg… or anywhere else where the streets are numbered with a functional 0 in the middle…

There’s a cab company on Grand St, and they were getting ready to pay the drivers… who they knew all started on Grand… but they dropped the lists of the trips.

They know one cab ended up on N7, one on S12, …

They know one trip was 4 blocks north, one was 6 blocks south…

Which cab made which trips?

I don’t know. I don’t love it. But I like it.

Jonathan

I like the weights-vs-balloons idea. You could also have them balancing scales… or playing tug-of-war… or how about balancing mobiles? If you have time it could be a fun hands-on project:

* Get 3 coathangers, and mark a blue end and a red end on each of them

* Find or make wooden (or styrofoam?) blocks of the right proportional sizes. Paint the “negative” blocks blue and the “positives” red, and mark the weight on each block

* Glue the 12-unit-weighted block onto the first hanger’s blue end, a 19 onto the second hanger’s blue end, and a 7 onto the last hanger’s red end

* Screw little hooks or glue strong magnets to all the blocks (and the hangers’ loose ends) so they can hang from each other in a chain

* Get the kids to balance each hanger, respecting the red-blue color-coding

Alternately, if you have programming skillz, you could code up a variation of this Flash game:

http://www.vectorpark.com/levers/

(That game itself is probably not right for your class now, but might be inspiration for some other math class ideas?)

Anyhow, it’s a fun problem to think about! Good luck!

Thank you everyone for the awesome brainstorm! Today is my birthday and I feel like this was my present.

I used smart notebook. The cards come with the gallery. But you could also use any image editor and find card clipart online.

I just blogged about Kate’s game.

What if the three target numbers are coordinates, and the other numbers are adjustments.

We need to hit (altitude 2, East/West 5, North/South -7). Targeting solution

(up 4 down 2, right 12 left 3 left 4, lower 6 lower 1)

Love it. Spatial direction captures exactly what I was looking for in terms of the negatives having a natural meaning that could be accessed by people who don’t know anything about negatives.

2 things are bugging me though: what circumstance would lead the allowable adjustments to be so constrained? And, how do you frame the story so that a kid who doesn’t know negatives could make sense of the idea that the 6 could be either down, west or south, but not up, east or north?