Monthly Archives: February 2014

Dan’s Circle-Square Challenge

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:

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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?

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I Choose Fractions

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On Monday I was walking with one my colleagues (Paul Scutt) as we tromped around Cranbury, NJ, on an excursion to the local historical museum and the store fronts of the town’s main street. We’d just had a bit to eat at Teddy’s, a local diner. Paul was wondering about a possible discrepancy in the bill and was calculating what the price of one drink must be if the total we’d been charged for three drinks was $5.25.

Paul was working out the quotient aloud in the style of long division. “3 into 5 is one with 2 left over…” Going through the steps was tricky—there was some retracing steps and checking involved—but things carried through and the answer came out as $1.75. This squared with the prices on the menu. All was well.

Through some of this I was only half listening. My thoughts were distracted by something or other; perhaps I had already begun my alternate calculation. Soon enough I was sharing my thoughts with Paul: “$5.25 is 5 and 1/4 or 21/4. 21/4 divided by 3 is 7/4, and that’s $1.75.”

It was an occasion to recall that over the course of my years of teaching math—and middle school math, in particular—I’ve developed a preference for fractions over decimals. Decimals, I’ve been inclined to argue, are just a special case of fractions—fractions with powers of ten in the denominator that we can suppress with a bit of ingenious and pragmatic notation. And decimals are a very slick trick—calculating with them is practically verbatim from whole-number arithmetic.

But then again, seen in a certain light, so is calculating with fractions—whole numbers that we calculate with while carrying them around in separate buckets. And since calculations with fractions have their own shape to them—their own kind of mental juggling—their meaning isn’t sedimented and murked up with their relation to ordinary place-value arithmetic. What is gained in ease is often purchased with opacity. This is true throughout math, and I’ve certainly found it to occur with decimals in particular. From these experiences, I’ll claim that it’s easier to help kids to think about fractions than about decimals, and that a teacher with a particular bent of mind can safely leave decimals as an afterthought. (Note: not “should” be.)

The ability to think flexibly about both decimals and fractions and to move between them comfortably is certainly the ideal. But how do we help students get there? What are the paths that we carve?

This post was prompted by an exchange on Twitter:

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What thoughts do you have to add? How do you think about the relationship between fractions and decimals? How do you talk about this with your students, or how might it affect your classroom silently? What is your own relationship with computations involving fractions and decimals—any preferences, for any reason? And what are good examples of their relative strengths and weaknesses?

Screen Shot 2014-02-19 at 9.45.18 PMScreen Shot 2014-02-19 at 9.45.24 PMWhat lines do you draw?