I’ve been working on this series for the last two months. I asked four of the most active contributors on 101questions (Andrew Stadel, Chris Robinson, Timon Piccini, and Nathan Kraft — all dudes, sorry about that) to:

I love these problems, but they scare the hell out of me.

You have the #3act hallmarks: a short visual setup, minimal language demand, and a question that can be approached intuitively at first. Have your students write down a gut-level ranking of each contestant. Who drew the best square?

Now we ask the students what information matters and doesn’t matter and how they’ll use that information to make a rule.

We’ll eventually give them all the information they could want — area, perimeter, angles, side lengths, and coordinates (so they can get whatever we missed). The point is, we could very easily hint our way towards an answer by providing the area and the perimeter in advance, but now the student’s task is much harder and much more interesting.

Also, *your task* is much harder and much more interesting. You have to take whatever rule your groups of students come up with and parry back with cases — large, small, degenerate, etc. — that heat that rule to the melting point.

If the student says, “Let’s subtract each side from the mean side length. All the sides should be congruent,” you offer her a tilted rhombus, which scores *perfectly* against that rule but shouldn’t.

If the student says, “Let’s subtract each *angle* from the mean angle measure. All of them should be 90°,” you offer her a short, wide rectangle, which scores *perfectly* against that rule but shouldn’t.

These problems terrify me because even as I put an answer in the teacher’s guide [pdf], I’m not convinced it’s the *best* answer. (Should we give the bigger squares more credit because they’re tougher, for instance?) I only know the process is worth the terror.

**BTW**. This activity owes a debt to The Eyeballing Game and to Patrick Honner’s question, “Which triangle is more equilateral?” where you can find him parrying superbly in the comments.

**Featured Comment**

Bowen Kerins:

Why should it be “our task” (teachers?) to take the students’ rule and parry back bad cases? This is one of the most interesting roles a student can take in this process. I’d much rather have the students coming up with edge cases against their own, or ideally others’, rules. I’d keep a few in my back pocket just in case, but students can drive that conversation in great ways.

James Key:

“Draw two points and then the point exactly between them.”

Sorry to be nit-picky, but while the above task certainly meets the requirement for “minimal language demand,” I think ONE MORE WORD is required for the sake of precision: “halfway.” There is not a unique point that is “exactly between” two given points; there are many. But there is one point that is “exactly halfway between them.”

I know that “conciseness” and “precision” sometimes compete with one another, and I confess that I often strike the balance poorly.

**2012 Dec 15**. Fawn Nguyen gave this a go in her classroom.