
...we learned a lot from each other that we wouldn't have learned otherwise. I had worked the constant area problem abstractly, but I had never pictured it as taking a sheet of paper and cutting it up and pasting it together again, as the middle school teachers saw it.
We chose this Experiment with Volume because it involves a number of mathematical concepts and problemsolving approaches that can lead to rich discourse in the classroom. The Problem: Form two cylinders from a rectangular piece of paper, one by joining the long sides, one by joining the short sides. Which of these cylinders will have greater volume, or will they hold the same amount? In our experience, students are engaged by the mathematical demonstration afforded by this investigation, the results of which seem counterintuitive to many of them. The investigation also generates "aha" moments that motivate students to focus their thinking. When exploring this problem, students describe patterns, develop algorithms, rules, and algebraic expressions, and explain their conclusions. Thus, the lesson involves multiple stages of investigation: prediction, testing, rejection or extension of hypotheses, discovering and exploring the underlying mathematics, and making generalizations and proving results. This kind of project offers many opportunities for fruitful conversation in the classroom, such as when students interpret a graph in Judith Koenig's class [view clip], or when they provide the reasoning for predictions in Susan Stein's class [view clip]. The students are actively engaged during these moments, negotiating and communicating ideas with the teacher and with other students. The lesson incorporates manipulatives, serving visual and tactile learners. In addition, because of the data that can be generated, the problem is suitable for mathematical modeling. To help with the creation of such models, we have the opportunity to use such technologies as graphing calculators and spreadsheets, allowing students to recognize patterns without getting bogged down in numbercrunching. The lesson also offers multiple entry points for a variety of levels, from elementary school through calculus. Elementary students can explore the family of cylinders that can be created from a single sheet of paper. Middlelevel students can use spreadsheets, calculate volumes, and relate them to the physical models. Students of algebra and geometry can approach mathematical modeling using linear, quadratic, cubic, and rational functions. Calculus students can use derivatives with their mathematical models.
Finally, the problem presented in this lesson also has real life applications,
as Jon Basden demonstrates.
Figure This! Math Challenges for Families, from the National Council of Teachers of Mathematics (NCTM), illustrates another example of a reallife application of the problem.

