Search Results for "3acts"

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[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.

[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]

[3ACTS] Money Duck


Many, many thanks to everyone who helped me sort out some thoughts on this lesson in our previous confab post, including but not limited to Fawn Nguyen, Robert Kaplinsky, Bowen Kerins, Dan Anderson, and Max Ray, and the many participants I pestered at OAME2014 this last week.

Here’s the download link at 101questions.

Act One

Show this video.

Ask: “What would be a fair price for the Money Duck?”

You guys were right. In the end it makes more sense to pose the student as the seller. It’s more productive and more interesting even though its easier to empathize with the buyer initially.

Act Two

Ask: “What information would you need to decide on a fair price?”

Now we’re going to introduce the probability distribution model.


It’s unusual so we’re going to do several things in order:

  1. We’re going to ask for speculation about what it means. Then we’re going to tell them what it means. (A: It shows every possible event in the space of events along with their likelihoods.)
  2. We’re going to show more contrasting cases. (See: Schwartz, 2011.) Impossible cases and possible cases. We’ll ask them which are impossible and why. Then we’ll tell them which are impossible and why. (A: The probabilities have to add up to 1. Each rectangle here has to stack up and reach exactly the 1 line.)
  3. Now we’ll ask the students, “If you’re selling the Money Ducks for $5, why is each of these distributions bad for business.” (A: The first distribution means word gets out that you’re cheaping your customers and eventually no one will buy your ducks. The second distribution means you’re losing loads of money.)
  4. Show the students four distributions and ask them to make up a price for the distributions that would be fair to both buyer and seller, that wouldn’t result in money lost or gained.

After laying all this informal groundwork, we’re ready to transition from qualitative descriptions to numerical and define expected value.

  1. We ask them to calculate the expected value of the distributions from #4 and compare those values to their prices. If their intuitions were sound and their calculations correct, their intuition will support the validity of the expected value model.

Act Three

There’s no act three here. We don’t know the probability distribution of the Money Duck (I asked) so we can’t validate. That’s okay.


Let’s show the students the actual price of the Money Duck and ask them to determine a probability distribution that would give them a $3 profit per Money Duck as a seller. Answers, happily, can vary.

BTW. The bummer-world version of this problem reads like this:

A carnival game is played as follows: You pay $2 to draw a card from an ordinary deck of 52 playing cards. If you draw an ace, you win $5. You win $3 if you draw a face card (Jack, Queen, King) and $10 if you draw the seven of spades. If you pick anything else, you lose your $2. On average, how much money can the operater expect to make per customer?

2014 May 12. You should definitely read Dan Anderson’s experience running this lesson with students.

2014 May 19. Also, Megan Schmidt had some interesting results with students particularly w/r/t the question, “Which distributions are impossible?”

[3ACTS] Nana’s Paint Mixup

Nana’s back.

I messed up her chocolate milk order a few years back. This is a new ratio task I first heard from Colin Foster at the Shell Centre last winter.

Here’s the download link, which includes the third act.

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Featured Comments


I dunno, this one made me wish the follow-up showed him just throwing the paint away and starting over. There’s just not enough investment in materials & time to make me think past. Plus if 6 tablespoons was enough paint to do the job then 30 is just a waste of paint far in excess of throwing away 6.

Kevin Hall:

@Don, what if he had used up all the white paint after putting in the 5 scoops of white? Then he’d have to figure out how to do it by just adding more red.

2014 Mar 11. Great extension for Algebra students from Paul in the comments:

I used the task to set up this question in an Algebra class.

The students were very puzzled when their intuition about the solution did not match arithmetic or a demonstration with cubes.

“I have two cans of pink paint variations in the following ratios. Neither is perfect.

Nana’s Pink 5 Reds : 1 White
50/50 Pink 1 Red : 1 White

I think that a perfect pink will be 3 Reds : 1 White

Can I make it by mixing Nana Pink and 50/50 Pink?”

Students expected the solution to be one cup of each. How wonderful the sound of a perplexed group of students when their arithmetic did not match their intuition.

Sense making ensued for many minutes with pictures, cups with cubes and more arithmetic.

[3ACTS] Dueling Discounts


Ask your students to write down which one they’d use. Some students will assume you should always use $20 off. Others will assume you should always use 20% off. Still others will (rightly) understand that it depends on the cost of the item you’re buying.

Our goal here is to get all of those responses on paper, emptied out of the students’ head. If one student in the class blurts out “It depends!” we’ll lose a lot of the interesting and productive preconceptions lurking about.

Take a show of hands. Ideally you’ll find some disagreement. At this point, students should try to convince each other of their position.

Offer the material from act two here: a bunch of items that will test out their hypotheses.

Once we reach the understanding that it’s better to take a percentage off the large expensive items and better to use the fixed value with the small cheap items, it might seem natural to ask:

Where’s the break-even point? Where do cheap items become expensive items? For what dollar cost should you use one coupon versus the other?

Then generalize some more:

If the coupons read “x% off” and “$x off”, where is the break-even point? Does your answer work for every x?

BTW. There’s a perplexing little pile of coupons assembling at 101questions right now. Great work, everybody.

Featured Comment

Kate Nowak:

“If you are allowed to apply one coupon, and then the other on a purchase, does it matter in which order you apply them?” is also a really nice question.

Mary Hillman

You need to be careful in your use of “small” and “large.” An iPod is small (yet expensive) compared to a large bouncy ball (inexpensive).

2014 Dec 9. Shaun Errichiello has created a series of printable cards for students to sort:

We asked students to physically sort the cards into groups. One group contains all the cards where the 20% coupon is the better choice, the other group contains all the cards where the $20 coupon is the better choice. We changed one of the prices (the desk) to be exactly $100.