The award-winning lesson plan – the Billiard Ball Problem

Essay Submission for Rosenthal Prize, 2022The Billiard Ball Problem
When I first began teaching I felt an urgency to involve my students in mathematical problem solving. I tried to invite their participation, but the right invitation was elusive. It wasn’t until I left the standard framework of arithmetic/algebra/geometry in favor of billiard ball ricochets that students began to respond in a completely new way.
My students became captivated by the question of predicting where a ball will bounce on a rectangular table of different dimensions. At first it seemed straightforward, and reducing the question to smaller tables led them to believe they had the answer. But some student would always remain skeptical and find counterexamples that shot down their peers’ conjectures. The work of extending theories, demonstrating counterexamples, and refining conjectures was suddenly the central work of my students. They behaved, in short, like mathematicians.
As a result, the Billiard Ball problem has become one of my favorite lessons. I’ve seen it create a kind of intellectual aliveness in students. I’ve also seen it create a sense of ownership, both about the experience and arguments from the activity, and about mathematics in general.
I can share two examples of this sense of ownership from the last time I shared the lesson with a class. One student asked about extending the billiard ball problem to tables with non-integer sides. I replied that it was an interesting question, but we weren’t ready to tackle it as a class. That didn’t dissuade the student. He ended up attacking the problem on his own, and by the end of class shared tables with decimal side lengths containing similar path-shapes as the ones we were already looking for. (E.g., a 6 by 7.5 table created the same path-shape as a 12 by 15 table.) Following his own interest, he increased the challenge of the problem and solved it to his satisfaction.
For another student, the problem felt daunting at first, and she avoided it. But she still participated by creating her own billiard ball designs. When I stopped by to see her work, she showed me a 4 by 5 table that gave a “prettier fish” (in her opinion) than the 2 by 3 table. Because the exploration is flexible enough to adapt to exactly these kinds of alternate approaches, I quickly posed the challenge of creating different table dimensions that would also create the “pretty fish” to the rest of the class. By the end of the period, she was totally engrossed, and had invented her own method of creating new tables that would generate more “pretty fish.” She also found patterns in the dimensions that were original in the classroom.
Problems that self-differentiate to involve students of higher and lower confidence and higher and lower technical ability are rare finds. The Billiard Ball problem is, to me, a notably outstanding way to invite students into mathematical problem-solving.