October 7, 2012 – 3:51 pm
This happens more often than I’d like: we start by having a normal class discussion and end up trying to resolve something that doesn’t engage the students’ interest, or even their understanding*. It’s the exact opposite of the way I’d like my classroom to feel.*

Last week in 9^{th} grade class we were discussing the following problem, which had been assigned for homework.

Write an equation for a rule a?b, so that the answer is odd only when both a and b are even.

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January 22, 2012 – 3:43 pm
The concept of proof-in-a-bag is simple. Write out a two-column proof and then cut it up so that each statement or reason is by itself on a scrap of paper. Then put all the scraps in a bag (a small sandwich bag works well, though an opaque paper bag might have more of a dramatic effect) and have kids work on rearranging the scraps so that they form a coherent proof. You can decide whether you want students to know ahead of time what it is they’re proving, or if you want them to figure it out by putting statements with “given:…”, “prove…” and a diagram in the bag as well.

Credit where credit is due: I got the idea for this from Laura Chihara while a student in her Algebraic coding class at the Carleton-St. Olaf Summer Math Program.

It’s nice to have any activity where kids are physically doing something in a math class, of course, but I really like what kids get out of this activity. It emphasizes the idea that you have to have enough evidence before you can conclude that triangles are congruent (otherwise, what are those “extra” statements doing in the bag?) And it is very good for helping students understand what can be a statement vs. what can be a reason. I often find that students want to use triangle congruence theorems like SAS when using properties of triangle congruence; the structure of this activity leads them to realize that they’ve already used SAS to justify the triangle congruence statement; they now need to use something else (CPCTC or the equivalent) to start using the congruence.

There are some times when I would definitely *not* use this activity. If the proof is a particularly exciting one for kids to work out on their own, I wouldn’t rob them of the opportunity. Proof-in-a-bag works best for simple, straightforward proofs, where the two-column proof format can be used without having to do a lot of extra explaining. I generally use it for one day only, at a time when the class has had some practice writing proofs but has not yet reached a level of comfort with them.

Does anybody else have activities or techniques that they use to teach writing proofs? I’d be especially interested in what people do who don’t insist on a strict two-column format all of the time.