Pick’s Theorem

Happy 2018!  Here I am posting on this infrequently-updated blog, and who knows, maybe I’ll keep it up.

In the past few years, I’ve seen Pick’s Theorem alluded to in various places. This theorem gives a surprisingly simple way to calculate the area of a polygon drawn on a lattice (most people imagine a Geoboard) based on the number of points on the polygon’s boundary and the number of points on the polygon’s interior. I won’t post the formula here in case you’d like to discover it for yourself, but here is one of many websites illustrating the theorem.

For example, the polygons below each have 5 boundary points and 3 interior points. Despite their different shapes, Pick’s Theorem predicts that each will have an area of 4.5 units.

Picks Theorem blog
I wanted to explore Pick’s Theorem with our Math Circle, a group of about 8-14 middle schoolers (mostly 6th graders). After examining lots of other math-circle Pick’s Theorem explorations, I handed the students the following much simpler version:

Math Circle – Picks Theorem

The handout ends right about where you would want to start making conjectures, with students being asked to find polygons with a given number of boundary points, number of interior points, and area. The last question is deliberately impossible.

The ideal math circle as described by its founders, the Kaplans, is a relaxed session in which students are presented with an interesting mathematical situation, then ask questions and make conjectures and discoveries together. I find that my group is a little young for that kind of Socratic seminar style: they really want to be doing things. So my aim last time was to given them lots to do, but eventually put them in a position where they would begin to generate some questions and ideas of their own.

Last time, most students got to various places on the last page. For a few, their interest waned before the end of the session. Next time will be tougher to manage! I’d rather not make a handout for the actual discovery of Pick’s theorem because I’d like the investigation to begin with student questions/observations. This means more “talking together” time, which will tax their attention spans.

My plan is to ask students to continue working on the last page, and then whenever enough people grumble about the problems being hard or impossible pool all of our data and see if we can find a pattern that might help us determine which polygons are possible. Depending on what they come up with, I might ask everyone to figure out (B,I,A) for rectangles with dimensions of their choosing, and try to generalize. Or, I might ask them to generate a bunch of polygons all with the same number of boundary points, but to vary the number of interior points, and see what happens.

I’ll bet we can come up with the formula, and at that point some students might lose interest (and I can give them a 1to9puzzle), those who are still with me can think about why the theorem might be plausible. We can come up with a good explanation for rectangles. I’m not sure if we can get beyond that, but the students may surprise me!

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