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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.


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!)

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]

Falcon Radar

The speedometer in this video is broken.

Can you (or your students!?) fix it? Be careful: there are a couple of interesting ways to get this one wrong.

Also: what would the graph of speed v. time and position v. time look like here?

Let us know how you’re thinking about it in the comments.

2015 Oct 17. Updated to include the answer video and answer graph. You can also download these files at 101questions.

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