Category: 3acts

Total 60 Posts

[3ACTS] Pool Bounce

There are three steps:

  1. Invite students to try a task that is intuitive, but inefficient or inaccurate.
  2. Help them understand some math.
  3. 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:

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

Julie Wright:

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.

Great #3Act Action

Please enjoy two posts from two teachers who are playing around with the Three-Act task structure.

First, John Rowe shows his students this image as a lead-in to counting problems, asking them, “Which state has more registered vehicles and how do you know?”


(Here’s an alternate version of that image that allows students to wonder “Where do you think these license plates are from and how do you know?” Because you can always add, but you can’t subtract.)

Second, Jenn Vadnais creates a stop-motion animation that does exactly what it wants to do, which is compose a cylinder into a sphere and then decompose the sphere into doughy little cubes. How close will math take you to the actual answer?


Go check out their posts, and keep up the great work, everybody.

[Updated] Will It Hit The Hoop?


Six years ago, I released a lesson called Will It Hit The Hoop? that broke the math education Internet. (Not a big brag. It was a much smaller Internet back then.)

I think the core concept still works. First, students predict whether or not a shot goes in the hoop based on an image and intuition alone. Then they analyze the shot using quadratic modeling and update their prediction. Then they see the answer. For most students, quadratic modeling beats their intuition.


The technology was a chore, though. Teachers had to juggle two dozen different files and distribute some of them to students. I remember loading seven Geogebra files onto student laptops using a thumb drive. That was 2010, a more innocent time.


So here’s a version I made for the Desmos Activity Builder which you’re welcome to use. It preserves the core concept and streamlines the technology. All students need is a browser and a class code.

Six year older and maybe a couple of years wiser, I decided to add a new element. I wanted students to understand that linears are a powerful model but that power has limits. I wanted students to understand that the context dictates the model.

So I now ask students to model this data with a linear equation.


Then I show students where the data came from and ask them to describe the implications of their linear model. (A: Their linear ball goes onwards and upwards forever.)


And then we introduce parabolas.

Gas Station Ripoff

Here are three gas station pumps. Which ones are trying to rip you off? Can you tell just by looking?

After your students have that debate and share their reasons (expected: “the third is a ripoff because it’s moving faster”) invite your students to collect data for each pump and enter it at Desmos. Here we’re establishing a need for a graphical representation. It may reveal patterns that our eyes can’t detect.


The third act helps clarify the underlying trends. The third pump is spinning faster, but the price and the gas still exist in a proportional relationship. The first pump, meanwhile, pumps less gas per dollar the longer it runs.

I am indebted to William G. McGowan and Sean Berg, whose NCTM 2016 session description included the words “gas pumps have been hacked,” and there went my weekend.

Their description reminded me how important it is to expose students to counter-examples of the relationships they’re studying, protecting against over-generalization. (ie. “Everything is proportional. That’s the chapter we’re in!”) I’m becoming fascinated, in general, by problems that ask students to prove that a mathematical model is broken rather than just apply a model that works.

[Download the goods.]

Featured Comment

Scott Farrar:

I’ve written before about expanding teaching to the “neighborhood” of the special case. If we always show the highlight reel, students never get appreciation for how special and how powerful the ideas are. So I like that this lesson is about finding the non-proportional “ripoff” as it stands out in contrast to the “normal/expected” proportional relationships. (Ironically I would have said that proportionality is the special case, and nonproportionality is the ‘normal’— before thinking about what we expect as consumers. ‘normal’ is all subjective!)

[3ACTS] Nissan Girl Scout Cookies

Treatment #1

A small rectangular prism measures 7 inches x 2.3 inches x 4.6 inches. How many times could it fit in a larger rectangular prism with a volume of 39.3 cubic feet?

Treatment #2

Nissan is going to stuff the trunk of a Nissan Rogue full of boxes of Girl Scout cookies. Nissan lists the Rogue’s trunk space as 39.3 cubic feet. A box of cookies measures 7 inches x 2.3 inches x 4.6 inches. How many boxes will they fit in the trunk?

Treatment #3

Show this video.

  1. Ask for questions.
  2. Ask for wrong answers.
  3. Ask for estimates.
  4. Ask for important information.
  5. Ask for estimates of the capacity of the trunk and the dimensions of the box of cookies.
  6. Show the answer.
  7. Ask for reasons why our mathematical answer differs from the actual answer.


Treatment #1 and Treatment #2 are as different from each other as Treatment #2 is from Treatment #3.

A layperson might claim that Treatment #2 has made Treatment #1 real world and relevant to student interests. But the real prize is Treatment #3, which doesn’t just add the world, but changes the work students do in that world, emphasizing formal and informal mathematisation.

“Real world” guarantees us very little if the work isn’t real also.

Design Notes

You can check out the original Act One and Act Three from Nissan.

I deleted this screen from Act One because I wanted students to think about the information that might be useful and to estimate that information. I can always add this information, but I can’t subtract it.


I added a ticker to the end of the video because that’s my house style.


I deleted a bunch of marketing copy because it was kind of corny and because it broke the flow of their awesome stop motion video.

I left the fine-print advisory that you should “never block your view while driving” because the youth are impressionable.

The Goods

Download the goods.

[via whoever runs the Bismarck Schools’ Twitter account]