Is a Banquet Table a Polygon?

Today, our one-day math exploration asked students to use one-inch square tiles to simulate tables. Using 12 tiles, students were asked to make as many rectangular tables as possible, then draw them and tell how many people could sit at each one. From there, there were various extensions. As I walked around, I interacted with a couple students who asked, "Is this all of them?" We discussed factors, arrays, and concluded, yes, that was all of them: 1x12, 2x6, 3x4. Those students went on to the extensions. As I continued on, I noticed that A.C. had created one that I hadn't thought of: he made a table that was three tiles long and two tiles wide, which left an open space in the middle. When I went back to report to those first students that there was at least one that we hadn't thought of, an interesting debate ensued. Referring back to a lesson from an earlier day, they reminded me that rectangles are polygons, and polygons can't have open spaces inside of them, so A.C.'s solution wasn't acceptable. A.C. stuck to his guns; the assignment said to make "rectangular" tables, and it said nothing about polygons. It was exciting to watch the students apply their new understandings about geometric shapes and terms in discussing whether or not this shape was a polygon, and whether or not all rectangles are polygons, and whether or not the direction to make "rectangular" tables also obligated them to adhere to all rules regarding polygons.

In the extensions to this first problem, students did all sorts of creative things. B.A. told everyone to remember this lesson; it would come in handy when they were running their own restaurant someday. With a restaurant in mind, students created towers of tables, which they imagined would require an elevator to ascend. J.H. arranged his 100 tables in a checkerboard pattern with the corners touching, claiming that his design would allow 400 diners. When it was pointed out that his diners might not be inclined to climb over or under the tables to get to their seats, he said that by the time his restaurant opens, there will be hover craft. Problem solved.

I was struck today at how this lesson illustrated so beautifully the way our students have been encouraged for years to approach problems in creative ways. They create rich contexts for their thinking. They debate the meaning of the directions, and they try to find various ways to innovate while still staying true to the spirit of the assignment. They share their thinking with each other, and they respectfully agree to disagree, as they did about the validity of A.C.'s table design. It was a wonderful 40 minutes.

CTO

Comments

[Michelle Martin]

I laughed out loud. Hovercraft - of course! What would the MCA readers have made of that?

[Cathy]

I have to admit--I was thinking of MCAs during this lesson and feeling sad that such inspired thinking would actually be penalized on such tests. When people say "test-taking skills", do they actually mean "suppression of divergent ways of thinking"?