The resource I need doesn’t seem to exist yet, so I’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 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.

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.

It’s not just margins of error. The math actually isn’t there. I’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.

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

http://function-of-time.blogspot.com/2009/01/where-to-bounce-ball-to-get-it-to-your.html

http://function-of-time.blogspot.com/2009/01/how-to-bounce-ball-part-2-solution.html

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.

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?

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)

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).