## Who Wore It Best: Pool Table Math

July 19th, 2016 by Dan Meyer

Pool table math is a common feature of a lot of geometry textbooks. Billiards hit a cushion and leave it at about the same angle. We have a real-world application! But as we’ll see in this week’s WWIB installment, not all treatments of that application are equal. In fact, commenters found them *all* wanting in various ways. I invite you to click through to this week’s three contestants:

**What You Said**

In the preview post, commenters called out the following turn-offs in different versions.

- “It jumps to the math notation too quickly.”
- “There is a ton of language in these problems.”
- “Two of the books just state that the angle of incidence and angle of reflection are the same and the other just expects students to
*know*that.” - “I feel like if I sat down and solved the problem that follows their explanation, I’d be copying their steps rather than really thinking it out for myself in a way that would make sense of it.”

On Twitter, Rose Roberts urges us to *be careful here* as, “Problems involving pool and mini-golf were *the reason* I decided I hated geometry in 8th grade. The sole reason.”

I’ll try to summarize the critiques using language that’s common to this blog without putting *too* many words in my commenters mouths. These textbook treatments rush to a formal level of abstraction too quickly. They don’t do a sufficient job developing the question for which “angle of incidence = angle of reflection” is the answer, or helping students develop an *intuition* about that answer.

In *Discovering Geometry*, for example, the formal equivalence statement is given and then the text asks students to apply it with their protractor.

A number of my commenters offer variations on, “Just take ’em to the pool hall!” This idea *sounds* great and will scan to many as suitably progressive, inquiry-based, student-centered, etc. But I’m unsatisfied. Mr. Bishop took us to the pool hall when I was a high school student and let us watch a local pro knock down a rack. I think he let us shoot a bit ourselves. I remember enjoying myself. I don’t remember learning more math than I did in his classroom lesson.

Pro pool players don’t use protractors.

For one reason, they’ve internalized that mathematics through practice. For another, the player can’t measure the angle of the ball in real time. The ball moves too quickly and the pool player’s eye-level view of the pool table is unlike the bird’s-eye view that would allow her to measure that angle.

This is a problem.

**What I Need**

Here is the resource I need. I’d like students to experience mathematical analysis as *power*, rather than *punishment*.

So let’s start with a tool that comes easily to students: their intuition. Let’s invite them to use their intuition in the context of a pool table. And let’s establish the context so that their intuition *fails* them, or at most earns a C-.

Then, let’s help students learn how to analyze the path of the pool ball *mathematically*. We’ll repeat the previous exercise and point at the end to the superior results that accrue when students analyze the pool table m*mathematically* instead of *intuitively*. (If superior results *don’t* accrue, we should either re-design the context to better highlight math’s power on a pool table or admit to ourselves we were wrong about math’s power.)

John Golden gets us *close* to that resource, inviting teachers to pull out still frames from this video of billiard shots for student analysis. But that analysis is much more complex than the level of the textbooks we’re critiquing today. Billiards ricochet off of other *billiards* in that video.

The resource I need doesn’t seem to exist yet, so I’ll try to build it. I’ll start with this game. Stay tuned.