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. That’s fertile territory for mathematics but different textbooks will travel that territory in different ways.
This week, I’m looking at three of my favorite secondary geometry textbooks. Who wore the pool table problem best?
Jonathan ClaydonJuly 6, 2016 - 10:43 pm -
My initial reaction is “put the kids in a pool hall, Sal.”
Jumps to the math notation too quickly. Could be better if the choice between A and B wasn’t just right there. I’d erase Step 1 and start at Step 2. I like Step 4 and 5 until the “cushion CP” kludge. More notation I don’t necessarily need or want yet.
Reflecting the pool table seems rough, because in a game a pool I’m not sure anyone lines up a shot by virtually flip flopping the table surface in their head, getting how a pool ball behaves is not hard. This strategy seems to be showing off without a compelling reason.
College Preparatory Mathematics:
Unhelpful diagrams and lots of scenarios to consider. There is a ton of language in this problems. But of the choices, I think it does the best job of showing how mathy an easy to understand concept like bouncing pool balls can be.
RuthJuly 7, 2016 - 5:22 am -
I agree with Jonathan – take the kids to the pool hall!
Have them make videos of the shots and then analyze them. Perhaps they’ll come up with the required conclusions on their own, rather than using these textbook versions. Much more fun for everyone!
Jonathan ClaydomJuly 7, 2016 - 11:34 am -
I should clarify that “unhelpful” diagrams is a good feature. They are unintimidating and give students a chance to develop their own way of talking about the path of the ball.
Dan MeyerJuly 8, 2016 - 11:26 am -
See, I remember my geometry teacher taking us to a pool hall. I remember the pool hall. I don’t remember the math I learned there. So I’d be obliged if you’d expand on what students will do at the pool hall?
Ruth expands a bit, inviting students to analyze video. But I’m concerned that the videos will be taken from an angle that obscures the property under investigation. If the camera shakes, it’s obscured. If the camera isn’t directly overhead, it’s obscured.
What’s the plan for the pool hall?
Lisa BejaranoJuly 8, 2016 - 11:34 am -
I must argue that Fawn Nguyen wore this one best. Her approach is simple and allows for student buy in, multiple approaches and discussion. There is room for students to debate various strategies (and critique the reasoning of others) and a clear need for them to justify their thinking.
pool. golf. same thing.
Brian R LawlerJuly 8, 2016 - 11:48 am -
There may be some brilliant geometry learned in textbooks that are not just “secondary geometry textbooks” but maybe integrated or interdisciplinary? :-)
BjÃ¶rn BelingJuly 8, 2016 - 10:51 pm -
I like the pool hall idea if the students get there with the right question in mind. Instead of using videos, I’d follow up on the reflection idea and give them mirrors to align with the cushions in order to find out the ideal point where to hit the cushion. Mark the spots in different scenarios, find out what they have in common, try to form hypotheses about how to find these points without a mirror. Then take overhead pictures for classroom discussion later.
RuthJuly 9, 2016 - 5:48 am -
Dan is right that the camera angle is a problem. In a task like this, I think I would give them the instruction to record a few bank shots from the point where the ball bounces. (this may require some sample video to talk about camera position & steadiness) The lights above a pool table make a bird’s eye view impossible, so videos would be at an angle. Will that matter in the later analysis? It’s worth discussing. Real-world math is messy, after all.
You could even scaffold the time at the pool hall, so that they record a few shots, then use mirrors as BjÃ¶rn suggested and keep building the concepts. I think that having them draw the scenarios afterward is key.
I love Fawn Nguyen’s task linked above – having students record their own pool shots is similar to having them draw their own mini-golf scenario. It becomes their own problem, which is always much more interesting than a textbook problem.
Jonathan ClaydonJuly 9, 2016 - 11:38 am -
Fawn, BjÃ¶rn, and Ruth all get at what I would do in a pool hall. Although in my mind none of this would involve going to a pool hall. I think you have students work through scenarios with a wood box and some marbles. Those would be easier to video overhead if you wanted that kind of thing.
One way to start would be plotting shots on a piece of paper and seeing if you could replicate them with your marbles/pool balls. How perfect are the bounces really? With the use of slo-mo you could find a way to introduce percent error.
I think the elliptical pool table is relevant to this discussion: https://www.youtube.com/watch?v=4KHCuXN2F3I
The math says the ball goes in the hole every time, but practically that’s not true.
Julie WrightJuly 9, 2016 - 10:02 pm -
I’m outside my comfort zone on pool (and a bit rustier than I would like on geometry and physics, this year) so I was thinking I’d sit this one out, but then I thought perhaps that helps me relate to the average student more, so…
First of all, I find it frustrating that the first two books just state that the angle of incidence and angle of reflection will be the same, and CPM seems to just expect students to know that. (In fairness, maybe it was covered in an earlier CPM lesson.) As a student, I’d find this believable, but I’d be completely derailed and distracted by wondering why that is true. I think you could explain that the ball will keep traveling in the x direction at the same rate and will travel at the same speed, but with the opposite sign in the y direction, and have students work out the similar triangles and therefore know that the angles are the same. Admittedly, this is still asking students to take some physics on faith, but it seems like less of a jump and gives them some interesting and relevant math to reach that conclusion.
Aside from that…
Discovering Geometry: the problems seem clear enough mathematically, but as someone who doesn’t know much about pool, I feel really confused about why we are solving them. Why should I care about hitting the center of the 8 ball or not hitting the CP cushion? It seems completely arbitrary, which makes me feel like I’m missing the point.
CME Project: Ugh!!! My least favorite by far. The double reflection thing is insane. After staring at it a while, I see how it works, but 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. And it doesn’t connect well to my understanding of what the path of the pool ball actually looks like or how the bouncing off the walls works. Very procedural/monkey see monkey do.
CPM: I really like that they’ve basically introduced a reasonable coordinate system (the diamonds) that I could refer to instead of slapping down letters to mark points. The single ball is less distracting, too. Aside from some worry that students who don’t know angle of incidence=angle of reflection will be stuck, and my usual eye-rolling at CPM for boldfacing “tools,” I like the first page. 6-42b is flat out ridiculous and distracts from the new ideas, but the rest of 6-42 is clear, interesting, and varied.
Otherwise, I had the same reaction as Lisa: Fawn wore it best! I even Googled for the same lesson before realizing Lisa had already pointed to it. It’s memorable for sure, since I recalled it even though reading it (as part of Jo Boaler’s online course How to Teach Math) was the first time I ever heard of Fawn.
Gordon SakaueJuly 10, 2016 - 8:45 am -
Does it muddy the waters (or obscure the images) to bring up Pong? Not advanced, with paddle speed affecting the bounce angle, but beginning level, simply meeting the ball with paddle?
Scott MillerJuly 12, 2016 - 10:15 pm -
I think that all three textbook offerings for the pool table problem are equally inadequate. Instead of following the process described by each text, have students try a digital pool table with a progression of questions: https://www.geogebra.org/m/CSFE7ZZW
Dan MeyerJuly 13, 2016 - 6:03 am -
@Scott, thanks for sending along the applet.
Can you expand on why you found the print textbooks “inadequate”?
Also, how do you imagine your applet being used by students? It seems possible to me that a student would just alter the point of reflection bit by bit until the shot was successful.