Here's a fun game that the NYC Teachers Math Circle members enjoyed playing in the outdoor classroom at Union College this evening. Albany Area Math Circle members who will be mentoring middle school students this year may also want to use this game with them.

"Hangmath" requires no equipment except for pencil and paper, or a chalkboard and chalk. It works like Hangman only with numbers.

Here's an example: one player privately writes out a 2 digit by 3 digit multiplication problem and keeps it hidden from the other players.

The first player creates an empty grid corresponding to the digits in the problem, for example:

_ _ _

x _ _

__________

_ _ _

_ _ _ _

__________

_ _ _ _ _

(EDIT: thanks to Prof. Moorthy for help with the html to line that up correctly!)

The other players then have to make guesses of the form "Is there a four in the tens column?" If yes, then the first player fills in all the fours in the tens column. Otherwise draw a part of the hangman.

Obviously, you can always get a guaranteed fill-in if you ask "Is there a zero in the ones column?" (Why?)

Initial guesses after that tend to be random (unless there was more than one zero in the ones column), but once one of your random guesses turns out to be a hit, you can use that information to make more intelligent guesses about the remaining

numbers.

What I like about this is that there is a lot of logic and backwards reasoning and understanding that can go on in solving the problem. Playing this game really gives students a solid understanding of what they're doing when they use a multi-digit multiplication algorithm. They understand it inside out, upside down, and backwards, rather than as just a mindlessly memorized rote recipe.

The player who makes up the problem also can use some good logic and backwards reasoning in trying to design a problem that will be hard to crack.

Variation: You could also use this game with a long-division problem grid as well.

I like this game as an icebreaker with a group of students that may include some very shy and tentative students who may be unsure of themselves. Why? Because a certain amount of guessing/trial and error is necessary at the beginning, and that takes pressure off students.

It's very exhilarating for an inexperienced student to get a lucky guess early on, occasionally. It's also empowering for an inexperienced student to see veterans getting some unlucky guesses wrong initially.

Real mathematical research often involves a fair amount of trial and error experimentation, and Hangmath is a nice way to ease students into that idea. Sometimes you just have to experiment somewhat in the dark in order to get insights

that will lead you where you want to go.

That's a really profound and important idea to transmit to students early on. The whole notion that a certain amount of intelligent trial and error is a legitimate problem-solving strategy is pretty radical to reinforce. When you don't know what

to do, guess and see what you learn from your guess is a very powerful idea. It's one I never learned in school.

When I was in school (a very traditional and rigid and mechanical rote school, with 56 students per classroom!), the idea was that you solved problems by applying rigid and deterministic recipes. If you didn't know the recipe you couldn't solve the problem, and you were supposed to wait until someone taught you the recipe.

When I run this game, I usually start out by asking each of the individual students to take turns with the early guesses, but then later I have them work in teams figuring out what logical inferences they can make about the best numbers to

guess next. It was really good to hear the math teachers working in teams to figure out strategies for what numbers to guess next.

I'd never asked teachers to play this game before, only students. They did a great job of efficiently reasoning through the guesses, and coming up with strategies to make it easier to be systematic with their guesses.

It should be noted that, at one point, I made a mistake in running the game. I failed to notice that there were two "2's" in the hundreds column. I only wrote one of them after a team asked me "Are there any 2's in the hundreds column?" I wrote down one of the two's in the hundreds column and they then began reasoning about the remaining numbers on the assumption that there were no other 2's in the hundreds column. Eventually, they started to realize something was wrong--and I could tell there was a problem, so I rechecked my copy of the problem, and realized that I'd neglected to fill in the second 2. It's good for students to see that even the "authority figure" running the game makes mistakes. (However, it's also helpful to have a second person with access to the answer key checking for such oversights.)

Hangmath also provides a good opportunity to talk about the

value of reasoning from negative, as well as positive, information!

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