There are three steps:

- Invite students to try a task that is intuitive, but inefficient or inaccurate.
- Help them understand some math.
- Invite them to
*re-try*the task and see that with math it’s more efficient and accurate.

That’s an instructional design pattern meant to help students see that the math they learn is power rather than punishment. Most instructional resources do a great job at #2, which they decorate with images of *other people* using that math in their lives. *Some* resources invite students to use the math themselves in #3. But without experiencing #1 the advantage of math may be unclear. “Why do I need to learn this stuff?” they may ask. “I could have done this by guesswork just as easily.”

We should show them the limits of guesswork.

Last week’s installment of Who Wore It Best looked at three textbooks each trying to exploit billiards as a context in geometry. None of the textbooks applied all three steps. I needed a resource that didn’t exist and I spent two days building it. Here is how it works.

**Inefficient & Inaccurate**

Play this video. Maybe twice.

Ask students to write down their estimates for all eight shots on this handout.

For instance:

@ddmeyer CBCAABAB Great discourse between hubby and I. I want a paper version so I can sketch and protractor. Probably all wrong.

— Adrianne Burns (@a_schindy) July 19, 2016

**Some Math**

Several of the textbooks simply *assert* the principle that the incoming angle of the pool ball is congruent to the outgoing angle. Based on Schwartz & Martin’s work on contrasting cases, I’ll offer students this page as preparation for future instruction.

What do you notice about the reals that isn’t true about the fakes?

**2016 Jul 31**. Edited to add this literature review, which elaborates the positive effect of contrasting cases (and building explanations on student solutions) in more detail.

**2016 Jul 31**. Also, in the spirit of “you can always add, you can’t subtract,” I’m sure that before I showed all four contrasting cases and the labels “real” and “fake,” I’d show the individual cases *without* those labels. Students can make predictions without the labels.

**Efficient & Accurate**

Now that they have an introduction to the principle that the incoming angle and the outgoing angle are congruent, ask them to apply it, now with analysis instead of intuition. Have them record those calculations next to their estimates.

Then show them the answer video.

Have the students tally up the difference between their correct calculations and their correct estimates. If that isn’t a positive number, we’re in trouble, and essentially forced to admit that the math we asked them to learn isn’t actually powerful.

I’ll wager your class average is positive, though, and on the last three shots, which bank off of multiple cushions, *very* positive.

Because math is power, not punishment.

[Download the goods.]

**2016 Jul 26**. I have changed a pretty significant aspect of the problem setup after receiving feedback from Scott Farrar and Riley Eynon-Lynch. Thanks, team.

**2016 Jul 26**. I’ll be changing the name of this activity shortly, on request from a Chicago educator who thinks his students will read violence into the title. That makes sense to me.

**2016 Jul 28**. Changed to “Pool Bounce.” I am amazing at titles.

**Featured Comment**

I love this partly because the fake ones

lookfake, and students have to think about why and are given materials to test their hypothesis. You’re making students refine their intuition to include mathematical precision, which they can then use to solve the rest. I feel like this honors and builds on the knowledge they already have in a way that’s far more motivational than throwing out some big-words statement about angles of incidence and reflection.