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Practice may not always be fun, but it can be purposeful. Some of my favorite tasks lately contain purposeful practice.

For instance, Dandy Candies tells students they’re going to package up 24 cubical candy boxes. It asks them, “Which of four packages uses the least amount of packaging? Which uses the least amount of ribbon?”


This is the usual house style. Concrete imagery. No abstraction. Contrasting cases. Predictions. Students make their guesses. Then they get the dimensions from the video. They calculate surface area and ribbon length. (Ribbon length is a little bit more interesting than perimeter but not by a lot.) They validate their predictions with their calculations.

But then we ask them to find out if another package dimension will use even less material.

So now the students have to think systematically, tabling out their work so they don’t waste effort finding the surface area of a lot of different prisms.

Contrast that against a worksheet like this, which is practice also, though rather less purposeful:


Where else have you seen purposeful practice?

I’d look to:

BTW. Given any number of cubical candies, what is the best way to minimize packaging? Can you prove it? I can handle real number side lengths but when you restrict the sides to integers, my mind explodes a little.

A workshop participant gave this algorithm. I have no reason to believe it works. I also have no reason to believe it doesn’t work.

  • Take the cube root of the volume.
  • Floor that to the nearest integer factor.
  • Square root the remainder factor.
  • Floor that to the nearest integer factor.
  • With the remainder factor, you have three factors now.
  • The smallest of all three factors is your height.
  • The other two are your length and width. Doesn’t matter which.

Note to self: test this against a bunch of cases. Find a counterexample where it falls apart.

Two anecdotes about curiosity, followed by a challenge:

1. Nana’s Lemon Water

I facilitated a workshop in Atlanta a few weeks ago and a participant had one of these enormous Thirstbuster mugs. I asked, somewhat nervously, “Whatcha got in there?” She replied “water with lemon.”

I wondered, as I’m sure others might, “Well how much lemon would you need in that enormous thing to even taste it.”

It’s natural for humans to have questions and seek to answer them. Once I heard her answer, though, an unnatural, teacherly act followed. I tried to recapture the question, something like mounting a butterfly in a shadow box or preserving a specimen in a jar, so that a student could experience it also.

That’s this video and the attached lesson.

2. Rotonda West

Another example. It takes very little curiosity to appreciate the gorgeous, curated satellite images from, such as this image of a Florida housing development:


What’s trickier for me is to format that appreciation, that awe, into a question, to capture that question so I can share it with students.


Making that image (and the answer) required a certain technological know-how, sure, but the really challenging part is training myself to probe interesting items for the curious questions they contain. It’s one of teaching’s unnatural acts and it requires practice and feedback.

3. Challenge

Curiosity is cultivated. Curious people grow more curious. These are examples of how I cultivate my own curiosity.

With that said, what curious questions can you find in this interesting story and video about the tallest water slide in the world? How can we capture that curiosity and make it accessible and productive for our students?

Previously: How Do You Turn Something Interesting Into Something Challenging

[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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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.

The Task


This is from Discovering Geometry.

What I Did

Basically, I three-acted the heck out of it. Which means:

  • Reduce the literacy demand. Let’s encode as much of the text as we can in a visual.
  • Add perplexity. That visual will attempt to leave students hanging with the question, “What’s going to happen next?”
  • Lower the floor on the task. The problem as written jumps straight to the task of calculation. We can scaffold our way to the calculation with some interesting concrete tasks.
  • Add intuition. Guessing is one of those lower-floor tasks and this problem is ready for it.
  • Add modeling. We’ll ask students “what information would be useful here?” before we give them that information. That’s because the first job of modeling (as it’s defined by the CCSS) is “identifying variables in the situation and selecting those that represent essential features.” The task as written does that job for students.
  • Create a better answer key. Once we’ve committed to a visual representation of the task, it’ll satisfy nobody to read the answer in the back of the book. They’ll want to watch the answer.

Here’s the three-act page. Leave a response to see the entire lesson.

Show this video to students.

Ask them to write down a guess: will the sauce overflow? Ask them to guess how many meatballs it’ll take. Guess guess guess. It’s the cheapest, easiest thing I can do to get students interested in an answer and also bring them into the world of the task.

Ask them what information would be useful to know and how they would get it. Have them chat in groups about what’s important.

If they come back at you telling you they want the radius of the pot and the radius of the meatballs, push on that. Ask them how they’d get the radius. That’s tough. Is there an easier dimension to get?

Someone here may ask if the lip of the pot matters. It isn’t a perfect cylinder. Give that kid a lot of status for checking those kinds of assumptions. Tell her, “It may matter. It isn’t a perfect cylinder but modeling means asking, ‘Is it good enough?'”

Give them the information you have.





Let them struggle with it enough to realize what kind of help they’ll need. Then help them with the formula for cylinder and sphere volume. Do some worked examples.

Once they have their mathematical answer, have them recontextualize it. What are the units? If that lip matters, how many meatballs will it matter? Should you adjust your answer up or down?

Then show them the answer.

Surprisingly close. The student who decided to add a couple of meatballs to her total on account of that lip is now looking really sharp.

Let’s not assume students are now fluent with these volume operations. Give them a pile of practice tasks next. Your textbook probably has a large set of them already written.

Help I Need

  • Raise the ceiling on the task. My usual strategy of swapping the knowns and unknowns to create an extension task is failing me here. Watch what that looks like: “The chef adds 50 meatballs to a different pot and it overflows. Tell me about that pot and its sauce level.” I’m not proud of myself. Can you find me a better extension? I’ll give highest marks to extensions that build on the context we’ve already worked to set up (ie. don’t go running off to bowling balls and swimming pools) and that further develop the concept of volume of spheres and cylinders (ie. don’t go running off to cubes or frustums).

What You Did

Over on the blogs:

Over on the Twitter:

  • Max Ray, Michelle Parker, and Terry Johanson are all inside my head.
  • Ignacio Mancera poses a similar situation but suggests doing it live in the classroom. I don’t accept the premise that “real” always beats “digital” – there are costs and benefits to consider – but I think Ignacio and Beth have the right plan here. If you have the materials, do it their way.
  • Scott McDaniel suggests changing the context from meatballs in sauce to ice cubes in an iced mocha because kids drink iced coffee but don’t make spaghetti. This introduces a pile of complications (like the non-spherical shape of the ice cubes and the non-cylindrical shape of the cup and the fact that the ice will float at the top of the cup) for unclear benefits. Time and again in this series I’ve tried to convince you that changing the context of a task does very little compared to the changes we can make to the task’s DNA. Does someone (Scott?) want to make the case that the following task is a significant improvement over the original?


Call for Submissions

You should play along. I post Monday’s task on Twitter the previous Thursday and collect your thoughts. (Follow me on Twitter.)

If you have a textbook task you’d like us to consider, you can feel free to e-mail it. Include the name of the textbook it came from. Or, if you have a blog, post your own makeover and send me a link. I’ll feature it in my own weekly installment. I’m at

2013 Jul 16. A makeover from Chris Hunter in the comments. (I had forgotten how weird Orbeez look in water.)

Featured Comment

James Cleveland:

We just talked about this problem and your makeover at Math for America. One idea was to up the stakes: I’m putting this jar of water on top of a student’s phone. How many balls can I put it before it spills over? If you are sure you are right, put your phone under the jar…

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