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. Youâ€™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 thatâ€™s far more motivational than throwing out some big-words statement about angles of incidence and reflection.

## 25 Comments

## Ali

July 25, 2016 - 10:01 am -Always thinking from a special ed perspective…

My one comment is that the ABC in your diagrams are SO close together that you can easily miss the target with just a small error, which will get very frustrating for any student with lower than average fine motor skills (poor handwriting). I know there’s a goal of making it not too easy to guess, but at the same time not too frustrating to actually calculate.

That said, the overall format is a winner.

## Dan Meyer

July 25, 2016 - 10:08 am -Ali:Yeah, this is the sweet spot I’m trying to hit. I hope that if anyone uses this with students they’ll let me know how it’s working.

## T. Devil

July 25, 2016 - 10:45 am -Math should be taught hand-in-hand with C++ or Java. Truth be told, the true value of math in a modern world is in writing applications that facilitate precisely this sort of activity.

Everyone is now carrying the equivelant of a pile of Cray supercomputers in their purse or pocket. Why subject every student to the rigors if mathdoctrination when you could instead empower the motivated and intelligent students to profit from the ignorance of the 99% of the populace that willfully suck at mathematics?

Call me opportunistic, callous or short-sighted, but the reality is that most of our students would do better to learn to be consumers of math, rather than practitioners. #mathisnotforeveryone

## John Golden

July 25, 2016 - 11:03 am -Really nice. The need for precision here is super. I can see Ali’s concern, but I like that closeness here. Right answer is not obvious and there could be some genuine math arguing as a result.

## Mark Greenaway

July 25, 2016 - 12:26 pm -When the students do the calculations what are they given to work with. For example are they given a diagram with the start point and initial direction indicated? Otherwise what calculations are they doing?

might have missed the point :-/

## Corey Andreasen

July 25, 2016 - 12:34 pm -I want video of a real bank shot, from directly above, so I can let kids figure that out.

## Dan Meyer

July 25, 2016 - 1:27 pm -Mark:Click on the handout and you’ll see that students are given a static picture of all eight pool shots where they’ll construct the rest of the path of the pool ball using their protractors. (Or, alternately, their compasses and straightedges.)

## Scott

July 25, 2016 - 4:05 pm -I think I like the removal of the assertion [that reflected angle = incoming angle] more than anything else. (I do actually like the rest though!)

So, bumping this behind the informal conjecturing feels really good… because really… what is this lesson about? That very theorem!

I do see some of Riley’s concerns on the handout. (#6 for instance) [the point of reflection will be a ball-radius away from the edge, complicating predictions] Or perhaps the others are even harder?

But perhaps this is an opportunity for that conversation. Hey, our model is “pretty good” and we think we know how reflected angles work. Now we have two possible avenues to take this: 1) alter our model to adapt to reality or 2) alter the real example (to lasers) to adapt to our idealized model.

Finally, perhaps here’s a nice video to show at some point in act 3? https://youtu.be/9Z6-i9pZrSU?t=43s (Augmented Reality pool table) Also note the margin of error: discussion on if/how the ARpool people adapted to the deviations from the ideal?

## Howard Phillips

July 25, 2016 - 6:25 pm -The theorem can be extracted from a pair of reflections using geometry.

Incident angle A must be equal to or greater than reflected angle B

The second part of the shot is from the at-right-angles cushion, with angles C and D

A>=B

90 – A >= C

90 >= A + C

but C >= D

So A + D <= 90

But A + D equals 90 (hence theorem proved)

## Chester Draws

July 25, 2016 - 9:10 pm -to help students see that the math they learn is power rather than punishmentIs this an actual problem?

Few of my students think that Maths lacks power. They might think that it doesn’t matter to them. Or that that they can’t be bothered doing it. Or that they are no good at it. Or that they have other powerful things to be getting on with.

(I doubt anyone these days thinks that writing good computer code isn’t powerful. Yet how many of us here on this site, worrying that students don’t think Maths is “powerful”, have themselves bothered to learn to code properly? We will be giving the exact same reasons — coding is great, but not important for me — I don’t want to study that hard –I’m no good at computers — I’m too busy.)

I don’t see that making Maths seem powerful is helpful unless they also see that

theyneed its power.What I am missing that makes this exercise more powerful than solving more traditional geometry exercises?

## Grant

July 26, 2016 - 12:03 am -@ Chester Draws: I think you bring up a valid point that being “powerful” isn’t always a sufficient reason for leaning something. There are simply too many powerful skills and bodies of knowledge for anyone to learn them all. At some point, we do have to choose what we learn and devote our attention to.

However, I don’t quite see coding and mathematics as being quite parallels. While coding may be a particular application for mathematics, there are plenty of other applications and uses for mathematics that don’t involve coding. Coding is a branch whereas mathematics is the trunk. Mathematics empowers people to think logically and creatively so that they may choose to pursue coding, if they wish, or choose something else like physics, environmental sciences, etc.

The pool table problem and others like them, if presented effectively, are the tools by which students can learn to think logically; make, test, and refine conjectures; critique others’ reasoning (and their own), and become more efficient and confident problem solvers. Those skills can take students a long way on many different paths. The trick now becomes to sell all of this to the students. I think the intention here is to shine a new light on a subject that many students think of as a difficult chore or burden.

## Dan Meyer

July 26, 2016 - 6:51 am -Scott Farrar:Agreed. This is another endorsement for

showingthe answer as it exists in the world: it keeps curriculum authors honest. The third act won’t validate the first and second acts since I got squirrely with the cue ball’s behavior. So I changed the handout and one of the images in this post to reflect the fact that the ball’s center doesn’t hit the cushion.(Pedantic decision: Do you have the students draw the invisible “radius-width line” or do you draw it for them? How helpful is too helpful? I made my decision. Reasonable people can disagree.)

Chester Draws:Only if you have students asking “when will I ever use this?” every other period.

## Chester Draws

July 26, 2016 - 12:13 pm -Dan,

I do indeed have those questions asked, especially at the start of something like Algebra or Geometry. It dies away as we get into the topics though, because once they can do it they don’t generally care about the fact they can’t see its point.

I just don’t think that showing them it is powerful is the way to combat that motivation issue. I think that they accept that it is powerful, but the ones asking those questions don’t care, because they don’t want that power.

I confess that I have started answering “When will I use this?” with “Never, unless you get good at it. Then people might pay you for your skills.”

Grant

The thing is that some things are a burden. We cannot both teach Maths to any high level and make it painless.

The trick is to make the burden as bearable as possible. In my view that means we get through it with as little distress as we can (hence I don’t do “discovery” teaching) and as directly as possible.

After teaching about pool tables, I’m still going to have to cover the material more conventionally anyway, except I now have less time to do it in. I don’t see that helps.

## l hodge

July 26, 2016 - 1:06 pm -What math have I been helped to understand? How does this help me understand why the incoming and outgoing angle are the same?

Which of the questions, if any, require reasoning? Why not a question or two providing the destination and requesting the initial path?

## Howard Phillips

July 26, 2016 - 1:34 pm -I Hodge : Try my comment.

## Ethan P Smith

July 26, 2016 - 1:46 pm -I agree with some other folks here that the “contrasting cases” inclusion is wonderful. It demonstrates that, even when we are building new understanding with students, we do not need to feed them classic direct instruction. They can still reason with patterns before we give them specific notation or language for this new math knowledge. Really great illustration there!

## Julie Wright

July 26, 2016 - 2:10 pm -“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 & Martinâ€™s work on contrasting cases, Iâ€™ll offer students this page as preparation for future instruction.”

OH YISSSS. Such a huge improvement on the originals!

I love this partly because the fake ones LOOK fake, 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.

Actually, that last statement applies to the whole lesson.

## Dan Meyer

July 26, 2016 - 2:29 pm -@

Julie, thanks for your analysis. I added it to the post.l hodge:The contrasting cases require an attention to structure and the ability to generate and defend a hypothesis. The initial exercise requires spatial intuition. The final exercise requires attention to precision. Your question is nice, though.

Chester:I don’t mind you repeating yourself in every post,

Chester. But clearly we come at the project of math education from very different angles. I tend to chalk your frustration with individual activities, lessons, and posts tothosedifferences, not to the activities, lessons, and posts themselves.That said, I

amcurious which material you’ll have to cover “more conventionally” anyway. The “material” I’ve focused on here is a particular understanding about angles and their application to pool. That material is “covered” in this activity already. It’s just sandwiched in between a couple of other productive and interesting activities.One could make the argument that those extra activities are an unconscionable expense of time and I will shrug in response and move along. How a teacher spends her finite classroom time isn’t really my business.

One can’t argue that the material isn’t covered, though.

Just because I don’t subscribe to a program of straight explicit instruction, no chaser,

doesn’tmean I subscribe to discovery learning, the usual boogeyman in your comments.My question isn’t “should or shouldn’t we explain?” Rather, “What can we do

beforewe explain, both to interest students in that explanation and prepare them to learn from it?”## Kate Nowak

July 26, 2016 - 3:07 pm -Just in case anyone finds it helpful, this PBS video is really nice, maybe for follow-up viewing.

http://www.pbslearningmedia.org/resource/lsps07.sci.phys.energy.lightreflect/light-and-the-law-of-reflection/

It explains what a photon is, and then it shows ball bearings and light bouncing off a flat surface.

## Grant

July 26, 2016 - 4:33 pm -One activity I’ve done in the past (I think someone mentioned it already) is a laser and target activity. I give students the dimensions of the classroom, the location of the target and a laser and the number of walls/mirrors they must reflect the laser off of to hit the target. It is then their job to use geometry to figure out where on the walls to place the mirrors so that they can hit the target with the laser. In the past, I’ve done this activity after we learned about the angles of incidence/reflection. In the future, I would probably introduce the game before and then ask the students what questions they have/what information would be helpful to achieve this? The laser activity is nice in the sense that the execution only depends on the geometry and the students’ ability to measure correctly whereas the pool also requires specialized skills to execute correctly.

## Steve Walker

July 31, 2016 - 12:11 pm -I’m so grateful for the improvement this makes to the traditional billiards problems.

I may add an extension that asks students to prove whether we’ll, eventually, sink 2-3 of the shots. I’ll look to choose one that drops after 3-4 bounces, another that gets stuck in an endless loop, and another that may fall after far too many bounces. In debriefing the last one, we’ll have to speculate on the impact of striking a corner of the pocket.

## Joel Patterson

September 15, 2016 - 5:57 pm -I tried the pool bounce lesson this week, Dan, and it was good. Lots of engagement, lots of motivation to get their angles right, plus some fun talk about spin (English). Looking forward to doing the “Make a minimal path from point A to touch a line then go to point B” lesson with my students already understanding this.

## Dan Meyer

September 15, 2016 - 8:48 pm -Awesome. Happy to hear it about it,

Joel.## Nate Garnett

March 9, 2017 - 8:52 am -I plan to use this in a lesson on how to construct a angle congruent to a given angle using compass and straightedge. I think I’ll zoom in and print off half of the pool shots on a full page to make it more accurate for students who have not had much experience using compass and straightedge.

## Dan Meyer

March 9, 2017 - 11:18 am -Nice! Let us know how it goes if you get a chance.