Category: 3acts

Total 58 Posts

My Winter Break in Recreational Mathematics

Chase Orton asked all of us, “What is your professional New Year’s resolution?

I said that I wanted to stay skilled as a math teacher. As much as I’d like to pretend I’ve still got it in spite of my years outside of the classroom, I know there aren’t any shortcuts here: I need to do more math and I need to do more teaching. I have plans for both halves of that goal.

In support of “doing more math,” I’ll periodically post about my recreational mathematics. Please a) critique my work, and b) shoot me any interesting mathematics you’re working on.

When Should You Bet Your Coffee?

Ken Templeton sent me an image from his local coffee shop.

Should you bet your free coffee or not? Under what circumstances?

This question offers such a ticklish application of the Intermediate Value Theorem:

If the bowl only has one other “Free Coffee” card in it, you’d want to bet your own card on the possibility of a year of free coffee. But if the bowl had one million cards in it, you’d want to hold onto your card. So somewhere in between one and one million, there is a number of cards where your decision switches. How do you figure out that number? (PS. I realize the IVT doesn’t hold for discrete functions like this one. Definitions offer us a lot of insight when we stretch them, though.)

I asked some of my fellow New Year’s Eve partygoers this question and one person offered a concise and intuitive explanation for her number, a number I personally had to calculate using algebraic manipulation. Someone else then did his best to translate some logistical and psychological considerations into mathematics. (eg. “Even if I win it all, I won’t likely go get a drink every day. Plus I’m risk averse.”) It was such an interesting conversation. Plus what great friends right?

Here’s my work and the 3-Act Task for download.

How Many Bottles of Coca-Cola Are in That Pool?

When I watched this video, I had to wonder, “How many bottles of Coca Cola did they have to buy to fill that pool?”

I tweeted the video’s creator and asked him for the dimensions of the pool.

12 feet across by 30 inches high,” he responded.

Even though the frame of the pool is a dodecagon, the pool lining itself seems roughly cylindrical. So I calculated the volume of the pool and performed some unit conversions to figure out an estimate of the number of 2-liter bottles of Coca Cola he and his collaborators would have to buy.

Here’s my work and the 3-Act Task for download.

How Do You Solve Zukei Puzzles?

Many thanks to Sarah Carter who collected all of these Japanese logic puzzles into one handout.

Carter describes the puzzles as useful for vocabulary practice, but I found myself doing a lot of other interesting work too. For instance, justification. The rhombus was a challenging puzzle for me, and this answer was tempting.

So it’s important for me to know the definition of a rhombus – every side congruent – but also to be able to argue from that definition.

And the challenge that tickled my brain most was pushing myself away from an unsystematic visual search towards a systematic process, and then to write that process down in a way a computer might understand.

For instance, with squares, I’d say:

Computer: pick one of the points. Then pick any other point. Take the distance between those two points and check if you find another point when you venture that distance out on a perpendicular line. If so, see if you can complete the square that matches those three points. If you can’t, move to the next pair.


  • What are your professional resolutions for 2017?
  • What recreational mathematics are you working on lately?
  • If any of you enterprising programmers want to make a Zukei puzzle solver, I’d love to see it.

2017 Jan 2. Ask and ye shall receive! I have Zukei solvers from Matthew Fahrenbacher, Jed, and Dan Anderson. They’re all rather different, each with its own set of strengths and weaknesses.

2017 Jan 2. Shaun Carter is another contender.

#BottleFlipping & the Lessons You Throw Back

2016 Oct 7. I was wrong about everything below. After admitting defeat to #bottleflipping, my commenters rescued the lesson.

I’m sorry. I went looking for a lesson and couldn’t find it.

Relevant background information:

Last spring, 18-year-old Mike Senatore, in a display of infinite swagger, flipped a bottle and landed it perfectly on its end. In front of his whole school. In one try.


That thirty-second video has six million views at the time of this writing. Bottle flipping now has the sort of cultural ubiquity that can drive even the most stoic teacher a little bit insane.

Some of my favorite math educators suggested that we turn those water bottles into a math lesson instead of confiscating them.

I was game. Coming up with a math task about bottle flipping should be easy, right? Watch:

Marta flipped x2 + 6x + 8 bottles in x + 4 minutes. At what rate is she flipping bottles?

Obviously unsatisfactory, right? But what would satisfy you. Try to define it. Denis Sheeran sees relevance in the bottle flipping but “relevance” is a term that’s really hard to define and even harder to design lessons around. If you turn your back on relevance for a second, it’ll turn into pseudocontext.

For me, at the end of this hypothetical lesson, I want students to feel more powerful, able to complete some task more efficiently or more accurately.

Ideally, that task would be bottle flipping. Ideally, students who had studied the math of bottle flipping would dazzle their friends who hadn’t. I don’t think that’s going to happen here.

But what if the task wasn’t bottle flipping (where math won’t help) rather predicting the outcome of bottle flipping (where math might). You can see this same approach in Will It Hit the Hoop?

The quadratic formula grants you no extra power when you’re in mid-air with the basketball. But when you’re trying to predict whether or not a ball will go in, that’s where math gives you power.


Act One

So in the same vein as that basketball task, here are four bottle flips from yours truly. At least one lands. At least one doesn’t. Each flip cuts off early and invites students to predict how will it land?

Act Two

Okay. Here’s a coordinate plane on top of each flip.

If you’ve been around this blog for even a day, you know what’s coming up: we’re going to show which flips landed and which flips didn’t. Ideally, the math students learn in the second act will enable them to make more confident and more accurate predictions than they made in the first act.

But what is that math?

I asked that question of Jason Merrill, one of the many smart people I work with at Desmos. I won’t quote his full response, but I’ll say that it included phrases like “cycloid type thing” and “contact angle parameter space,” none of which fit neatly in any K-12 scope and sequence that I know. He was nice enough to create this simulator, which has been well-received online, though even the simulator had to be simplified. It illustrates baton flipping, not bottle flipping

Act Three

Here is the result of those bottle flips. For good measure, here’s a bottle flip from the perspective of the bottle.


I’m obviously lost.

Here’s a link to the entire multimedia package. Have at it. If you have a great idea for how we can resurrect this, let me know. I’m game to do some video editing on your behalf.

But when it comes to bottle flipping, if “math” is the answer, I’m not sure what the question is. Please help me out. What is the lesson plan? How will students experience math as power, rather than punishment.

Sure, it’s probably a bad idea to destroy the bottles. But it’s possible we shouldn’t turn them into a math lesson either. Maybe bottle flipping is the kind of silly fun that should stay silly.

2016 Oct 7. Okay: I was wrong about #bottleflipping. A bunch of commenters came up with a great idea.

Featured Comments

Elizabeth Raskin:

I see a couple students playing the game during some down time and my immediate reaction is, “There’s gotta be some great math in there!” One of the boys who was playing sees my eyes light up. He looks at me in fear and says, “Mrs. Raskin. Please. I know what you’re thinking. Please don’t mathify our game. Let us just have this one thing we don’t have to math.”

Mr K:

I suspect I should put as much effort into making this teachable as I would for dabbing.

Meaghan found a nice angle in on bottle flipping, along with several other commenters:

It would be neat if you could spend a, for example, physics class period talking about experimental design (for fill ratio questions or probability questions) and collecting the data, and then troop right over to math class with your data to figure out how to interpret it.

Paul Jorgens has the data:

It started with an argument in class last week with the optimal amount of water in the bottle. Should it be 1/4 filled? 1/3? Just below 1/2? I told the group that we could use our extra period to try to answer the question. We met and designed an experiment. Thought about problems like skill of tosser, variation in bottles, etc. We started with 32 bottles filled to varying levels. During class over 20 minutes 32 students flipped bottles 4,220 times.


a/k/a Oh Come On, A Pokémon Go #3Act, Are You Kidding Me With This?

Karim Ani, the founder of Mathalicious, hassles me because I design problems about water tanks while Mathalicious tackles issues of greater sociological importance. Traditionalists like Barry Garelick see my 3-Act Math project as superficial multimedia whizbangery and wonder why we don’t just stick with thirty spiraled practice problems every night when that’s worked pretty well for the world so far. Basically everybody I follow on Twitter cast a disapproving eye at posts trying to turn Pokémon Go into the future of education, posts which no one will admit to having written in three months, once Pokémon Go has fallen farther out of the public eye than Angry Birds.

So this 3-Act math task is bound to disappoint everybody above. It’s a trivial question about a piece of pop culture ephemera wrapped up in multimedia whizbangery.

But I had to testify. That’s what this has always been – a testimonial – where by “this” I mean this blog, these tasks, and my career in math education to date.

I don’t care about Pokémon Go. I don’t care about multimedia. I don’t care about the sociological importance of a question.

I care about math’s power to puzzle a person and then help that person unpuzzle herself. I want my work always to testify to that power.

So when I read this article about how people were tricking their smartphones into thinking they were walking (for the sake of achievements in Pokémon Go), I was puzzled. I was curious about other objects that spin, and then about ceiling fans, and then I wondered how long a ceiling fan would have to spin before it had “walked” a necessary number of kilometers. I couldn’t resist the question.

That doesn’t mean you’ll find the question irresistible, or that I think you should. But I feel an enormous burden to testify to my curiosity. That isn’t simple.

“Math is fun,” argues mathematics professor Robert Craigen. “It takes effort to make it otherwise.” But nothing is actually like that – intrinsically interesting or uninteresting. Every last thing – pure math, applied math, your favorite movie, everything – requires humans like ourselves to testify on its behalf.

In one kind of testimonial, I’d stand in front of a class and read the article word-for-word. Then I’d work out all of this math in front of students on the board. I would circle the answer and step back.


But everything I’ve read and experienced has taught me that this would be a lousy testimonial. My curiosity wouldn’t become anybody else’s.

Meanwhile, multimedia allows me to develop a question with students as I experienced it, to postpone helpful tools, information, and resources until they’re necessary, and to show the resolution of that question as it exists in the world itself.

I don’t care about the multimedia. I care about the testimonial. Curiosity is my project. Multimedia lets me testify on its behalf.

So why are you here? What is your project? I care much less about the specifics of your project than I care how you testify on its behalf.

I care about Talking Points much less than Elizabeth Statmore. I care about math mistakes much less than Michael Pershan. I care about elementary math education much less than Tracy Zager and Joe Schwartz. I care about equity much less than Danny Brown and identity much less than Ilana Horn. I care about pure mathematics much less than Sam Shah and Gordi Hamilton. I care about sociological importance much less than Mathalicious. I care about applications of math to art and creativity much less than Anna Weltman.

But I love how each one of them testifies on behalf of their project. When any of them takes the stand to testify, I’m locked in. They make their project my own.


Why are you here? What is your project? How do you testify on its behalf?

Related: How Do You Turn Something Interesting Into Something Challenging?

[Download the goods.]

[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.