## Who Wore It Best: Pool Table Math

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, Iâ€™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 mmathematically 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.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Mr. K

July 19, 2016 - 2:38 pm -

I missed the last round of this, for some reason.

I remember some math teacher doing a pool lesson that used reflection instead of angles of incidence.

My inclination is to use that lesson, together with the congruency of vertical angles, to derive this one. I’m not entirely sure how to set it up at the moment, but it feels like there are a couple of aha moments hidden away in there.

2. #### Bryan Penfound

July 19, 2016 - 2:46 pm -

What I am noticing from this game is that the velocity of the white ball seems to matter as well. It may not be as simple as measuring angles. The texts often fail to give the needed space for this conversation.

3. #### Alan Levine

July 19, 2016 - 4:10 pm -

Takes me back to 4th grade, Mr Fike, who showed us the film strip of Donald Duck explaining angles through pool. What I remember was to do a bank shot we were to aim for the reflection of the ball we were trying to hit to get the right bank spot. The problem of course is all the math depends on dead center cue hitting the ball, and everything in pool depends on understanding spins. The whole language of the word problems take away anything real.

4. #### Jason Dyer

July 19, 2016 - 4:47 pm -

I don’t have this sheet anymore, but I did an activity once where students used marbles and pencils for mini-billiards and were challenged (using a set paper as the pool table) to figure out how to get from point A to point B with X bounces (being from 1 to 3).

It was fun and got the incidental angle idea across (unlike some enrichment activities that are just, er, fun), but the concept got removed from the main part of the curriculum and I didn’t have the time to justify the investment in other years (takes about 2 hours).

5. #### Tim M.

July 19, 2016 - 6:21 pm -

I do something similar with miniature golf. Students have a diagram of a ball and a hole and they want to get a hole-in-one but there is a water hazard and they must bank the ball off of a wall. I give students rulers, protractors, compasses, etc. and students try to find a reasonable path on their own.

Students inevitably make their incoming and outgoing angles different measures and in groups or as a class we come to the conclusion that theoretically that wouldn’t be possible and the angle the ball hits the wall should be the same at the angle in which it comes off the wall.

Next, students think of a way to systematically find that point that we must aim at, and many members of the class will usually use reflections. I have a version on Geogebra for students to use as well (https://www.geogebra.org/m/W6fgwFbG)

6. #### Xavier

July 20, 2016 - 3:33 am -

It think it’s not real world. It’s an artificial thing for putting in textbook. No one pro calculates nothing for hitting the ball like no basketball player needs to calculate nothing to shot (only army needs it to accuracy at maximum for missiles).

But if you want to use this billiar issue, I think you have just one option: programming. How do you program a robot for, given two balls, hit one with another?

7. #### Scott Farrar

July 21, 2016 - 10:52 pm -

Kate Nowak had a blog post long ago (2010?) that described her kids inventing a game kind of out of the blue. Two students would face a wall, they’d take turns to bounce a ball against the wall at a diagonal so that the other student would be able to grab it.

I think Kate got them going on formalizing some of the things they were doing informally, like being able to predict where to stand.

It wasn’t that the formalization was “in order to do it better” it was just a way to dig deeper to understand something the students were already interested in.

Are students already interested in pool? Then these lesson intros can be shorthand for reloading that context into the present. But if not, then I don’t think its super effective.

I think the game stuff can be better at gathering the informal interest upon which you can build interest in formality. The game can be a shared experience, a contained experience, one that students can play “from scratch” instead of a book trying to jog memory about billiards which maybe the reader knows or cares nothing about.

8. #### Sharon Hessney

July 23, 2016 - 12:44 pm -

How about using a knock hockey table — rather than going to a pool hall. Let the students experiment with their own shots.

9. #### Riley Eynon-Lynch

July 23, 2016 - 1:05 pm -

I get a knot of anxiety in my stomach when I see pool examples. The balls don’t follow the same laws of light, and the diagrams are fundamentally misrepresentative. A ball following a line to a cushion doesn’t even touch the cushion at a point on the line! Because of the width of the ball, it strikes the cushion at a point to the side of the line. If you do that thing where you lay out points on a pool table, the ball only follows the path it’s supposed to in a few setups, and only if you assume the error you see is evenly distributed around the points you’re predicting.

It’s not just margins of error. The math actually isn’t there. I’d stick with lasers and mirrors!

10. #### Dan Meyer

July 25, 2016 - 10:49 am -

Me, at the end of this post:

The resource I need doesnâ€™t seem to exist yet, so Iâ€™ll try to build it.

Here is the second or third draft of that resource.

Mr. K:

I remember some math teacher doing a pool lesson that used reflection instead of angles of incidence.

Yeah, that math teacher had a lot more faith in the power of a good explanation than I do. There are some good visuals in those slides, but the students didn’t do much except watch them.

Alan Levine:

Takes me back to 4th grade, Mr Fike, who showed us the film strip of Donald Duck explaining angles through pool.

That movie is timeless.

The problem of course is all the math depends on dead center cue hitting the ball, and everything in pool depends on understanding spins.

I think this is a really strong argument for not taking kids to the pool hall. There really is a sound, interesting mathematical model at work here. Seeing the power of that model depends on the teacher’s ability to silence a lot of noise. That’s where the video game enters, I think. It’s close enough to the actual pool to resonate with students. It’s far enough way that the math will work.

Riley:

A ball following a line to a cushion doesnâ€™t even touch the cushion at a point on the line! Because of the width of the ball, it strikes the cushion at a point to the side of the line.

I realized this while making my three-act version. Don’t draw the ray directly to the cushion. Draw it until it’s a cue ball’s width away.

Itâ€™s not just margins of error. The math actually isnâ€™t there. Iâ€™d stick with lasers and mirrors!

Hm. There is the kind of error where I assume the path of a basketball follows an exponential curve. Then there is the kind of error where I apply a quadratic fit to the basketball but neglect some wind resistance. Those are different kinds of error. You’re of the mind that with pool table math we’re working with the first kind of error?

Tim M., I like those ideas. At any point in the lesson do students have a chance to see whether or not their mathematical predictions are correct? Unless I’m missing a feature, the student draws rays, segments, etc, keeping the angles congruent. Her faith in the correctness of that geometry will vary directly with her faith in the power of math. If she has low faith, then she’s pushing math around because her teacher told her this is how math works.