Dan Meyer has posed some questions in a recent post about the merits and potential of the following problem:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Dan asks:

**What can you do with this? How can you improve the task?**

Here are the thoughts I had (comment-turned-post) and an attempt at reworking the task.

Experiences with problems often become more powerful when they’re not just one-offs—when they’re surrounded by some kind of context. One kind of context that I find fruitful is the notion of a problem space. Most of the interesting aspects of the circle-square problem that Dan highlighted—like “assigning a variable” and “constructing a model” and “using operations on it”—aren’t specific to the geometric figures of circle and square. They equally well apply to similar questions (some easier and some harder) involving many different kinds of shapes. The circle-square combo does have a nice “classic” feel to it, but if we care most about the mathematical practices and methods involved in solving it, then we might do well to also explore variations on it.

In addition, the circle-square task as it stands puts the bar at one specific level of difficulty and requires certain algebraic and geometric skills to be firmly in place for a successful solution—and really even to make any significant headway. Whenever possibly I prefer to lower the threshold for success in a task while also raising the ceiling—an easy way to let students self-differentiate a task.

Here’s a first shot at a warm-up worksheet to introduce the problem space:

I might have string and scissors on hand for students who were interested in using them, but my sense is that the problems as I’ve stated it—and the solution methods students are likely to take—is sufficiently concrete to make manipulatives perhaps useful and fun, but not necessary.

What would happen after student had a chance to work on this sheet on their own for a bit? Well, they’d have time to share what they came up with in pairs or small groups. Then we’d do some sharing as a whole class and discuss solutions and conjectures. That’s another thing—there are so many equally and perhaps even more interesting mathematical questions that can arise in this particular problem-space than the circle-square one that we’re (likely) headed to and that prompted this whole adventure. I’m sure the directions that students would take this in and the questions and problems they would pose would be fascinating and worth pursuing. They’d also likely provide richer and more powerful mathematical experiences than the circle-square problem on its own would, and solving this problem would carry a new significance in light of the other problems that had been posed and solved prior to it.

I can imagine using this warm-up worksheet with middle schoolers, but I also think it would be an appropriate onramp to the circle-square problem for sophisticated high school students. For both audiences it will clarify the circle-square problem and build context and experience for the students. If the circle-square problem is the main course, it deserves both to have appetizers to whet everyone’s appetites and to make sure everyone has a seat at the table before the meal is over. Because let’s face it: mathematical meals in school are all too often hurried affairs.

Those are my thoughts so far. What do you make of them?

Justin

I jumped on this problem yesterday when I read it and I shared it with my Calc BC kids. I did not think of any sort of manipulative for them because (a) I was confident that they’d cut through the abstraction quickly and willingly (which they did and I wrote about yesterday at my place) and (b) I don’t think often enough in terms of manipulative experiences. I’ll be teaching geometry next year and I’m putting our curriculum together with a variety of resources. I’m going to include this problem as one for the geometry kids to get their hands on for sure.

Justin – I love the way you’ve differentiated this – for anyone who is intimidated by the original problem. As you point out, even more sophisticated students (I’m thinking Algebra 2 or Geometry) might need an entry point to the problem.

This is really great.

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I’m not a big fan of the problem as originally stated because it is not enlightening, it is simply a problem-solving exercise. Your restatements allow the students to think about more general concepts with wide application. The students can actually use the knowledge they learn with your worksheet, whereas the original problem the general ideas are obscured by the nuts-and-bolts of problem solving.