If a student has no idea where to start, you can prompt her to list a price that sounds fair to her for the three sodas or, failing even that, a price that seems unfair to her. (You’re basically asking her to give a wrong answer. It’s a lot easier to give wrong answers than right answers because there are a lot more of them.)

You’ll find students who divide all the way down to the unit rate (ie. each egg costs 19 cents) and then multiply back up. You may also find students who set up a proportion, which will disguise the unit rate in an interesting way.

You’ll find students who set up different but equivalent unit rates. (ie. 19 cents per egg and .05 eggs per cent.) You’ll find students who set up different but equivalent proportions.

One of your many challenges during this activity will be to select students to show work that highlights a) the different ways to find the unit rate, b) the different ways to set up the proportion, c) the equivalence within those methods, and d) the equivalence between those methods (ie. ask your students to help you find the unit rate within the proportions).

The Goods

Partial Product

Featured Comment

Larry Copes:

I’m with Christopher and, I think, Dan here: Toss it out with the understanding that students can use any method that makes sense to them. Then not only share those methods but compare them to see why they yield the same result. Love the word “catharsis” in this context.

John Scammell – Original from Dan Meyer on Vimeo.

Some things I’d like to accomplish in the redesign:

1. Get the camera lens parallel to the pencil, an angle that makes it easier to see the length changing.
2. Convey to the student visually what John wrote in his tweet: that this pencil is about to get ground down to nothing.
3. Postpone the pencil measurements until the second act. The moment where John measures the pencil is useful and necessary but the first act (the #anyqs) should focus exclusively on curiosity and context. The math introduces itself later in act two to help resolve that curiosity.

Act One

Pencil Sharpener – Act One from Dan Meyer on Vimeo.

Act Two

Pencil Sharpener – Act Two from Dan Meyer on Vimeo.

Act Three

Pencil Sharpener – Act Three from Dan Meyer on Vimeo.

The Goods

Download the full archive. [10.8 MB]

Large Candle – Stop Motion Teaser from Dan Anderson on Vimeo.

Frameworks are inherently limiting. The more guidelines you specify, the more material you exclude, some of which can be very good. Frameworks are great, though, because they make implementation easy. I know what happens in the first, second, and third acts of a mathematical story, so it’d be a simple matter to use Dan Anderson’s lesson in the classroom – no lesson plan or handout required.

Let’s see how well the storytelling framework holds up.

The Goods

Download the full archive [5.5 MB].

Act One

Play the question video.

[anyqs] Stacking Dolls – Question from Dan Meyer on Vimeo.

Ask your students what question interests them about it. Take some time here. This is the moment where we develop a shared understanding of the context. If a student has some miscellaneous question to ask or information to share about the dolls, encourage it. That isn’t off-task behavior. This task requires that behavior.

Then ask them to write down a guess at how many Russian dolls they think there are. Ask them to write down a number they think is too high and too low.

Act Two

Offer your students these resources:

1. The first two dolls side-by-side.
2. The second two dolls side-by-side.

After you show them the first set of two dolls, ask them how big they predict the third will be. As one of the commenters mentioned, they need to discover the fact that these guys aren’t decreasing by a fixed amount every time, that a new model is necessary.

Once they have this new model in mind, they’ll keep applying it until they reach a doll height they think is impossibly small.

Act Three

That task isn’t going to win anybody a Fields medal. As students finish, ratchet up the demand of the task with this sequel. Say:

I need you to design me a doll that’s as tall as the Empire State Building and is made up of 100 dolls total. Tell me everything you know about that doll.

Ask them to generalize. Ask them to graph.

Host a summary discussion of the activity. At this point you’ve identified different solution strategies around the room. Have those students explain and justify their work to their peers. Everyone is accountable for understanding everyone else’s strategy.

Then show them the answer video:

[anyqs] Stacking Dolls – Answer from Dan Meyer on Vimeo.

Find out whose guess was closest.

[h/t @baevmilena who gave me the idea when I met her in Doha.]

Featured Email

Dawn Crane:

I recently took your nesting dolls activity and here’s what I did:

At the beginning of the unit on exponential functions, I followed your process fairly closely, except I used pictures of the dolls. I asked kids to predict the patterns, etc. Most kids went with exponential, though a few were strongly in favor of linear. At the end of the unit was where I believe the magic appeared and is what I will use in the future. By this point, kids had done work with linear and exponential functions and some kids had studied quadratics. I had 7 different sets of nesting dolls in the room. Kids were told they could pick any of the sets, but had to identify them. Their job was to determine an equation to model the growth/decay pattern of the dolls and use math “tools” to convince me that their equation did an adequate job at modeling the dolls. They had to do all of the measuring…some kids chose height, some volume, some girth.

I got a huge array of problem solving. Some kids used graphs to visually show more of a regression to see whether linear or exponential had a better fit. Some kids developed both linear and exponential equations and then used tables and graphs to see where each went off track. Some recognized a constant second difference in growth and used systems of equations to develop an amazing quadratic equation that appeared to fit their data perfectly.

This project really allowed students to take the problem as far as they wanted with an entry point for everyone. And the kids loved the nesting dolls so they were really engaged. I strongly recommend actually using the dolls rather than video-taping them as well. It adds a tactile dimension which is really valuable to many students.

2013 May 14. Here’s a brief series on how to teach with three-act math tasks. It includes video.

2013 Apr 12. I’ve been working this blog post into curriculum ideas for a couple years now. They’re all available here.

Storytelling gives us a framework for certain mathematical tasks that is both prescriptive enough to be useful and flexible enough to be usable. Many stories divide into three acts, each of which maps neatly onto these mathematical tasks.

Act One

Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.

With Jaws your first act looks something like this:

The visual is clear. The camera is in focus. It isn’t bobbing around so much that you can’t get your bearings on the scene. There aren’t any words. And it’s visceral. It strikes you right in the terror bone.

With math, your first act looks something like this:

The visual is clear. The camera is locked to a tripod and focused. No words are necessary. I’m not saying anyone is going to shell out ten dollars on date night to do this math problem but you have a visceral reaction to the image. It strikes you right in the curiosity bone.

Leave no one out of your first act. Your first act should impose as few demands on the students as possible – either of language or of math. It should ask for little and offer a lot. This, incidentally, is as far as the #anyqs challenge takes us.

Act Two

The protagonist/student overcomes obstacles, looks for resources, and develops new tools.

Before he resolves his largest conflict, Luke Skywalker resolves a lot of smaller ones – find a pilot, find a ship, find the princess, get the Death Star plans back to the Rebellion, etc. He builds a team. He develops new skills.

So it is with your second act. What resources will your students need before they can resolve their conflict? The height of the basketball hoop? The distance to the three-point line? The diameter of a basketball?

What tools do they have already? What tools can you help them develop? They’ll need quadratics, for instance. Help them with that.

Act Three

Resolve the conflict and set up a sequel/extension.

The third act pays off on the hard work of act two and the motivation of act one. Here’s act three of Star Wars.

That’s a resolution right there. Imagine, though, that Luke fired his last shot and instead of watching the Death Star explode, we cut to a scene inside the Rebellion control room. No explosion. Just one of the commanders explaining that “the mission was a success.”

That what it’s like for students to encounter the resolution of their conflict in the back of the teacher’s edition of the textbook.

If we’ve successfully motivated our students in the first act, the payoff in the third act needs to meet their expectations. Something like this:

Now, remember Vader spinning off into the distance, hurtling off to set the stage for The Empire Strikes Back. You need to be Vader. Make sure you have extension problems (sequels, right?) ready for students as they finish.

Conclusion

Many math teachers take act two as their job description. Hit the board, offer students three worked examples and twenty practice problems. As the ALEKS algorithm gets better and Bill Gates throws more gold bricks at Sal Khan and more people flip their classrooms, though, it’s clear to me that the second act isn’t our job anymore. Not the biggest part of it, anyway. You are only one of many people your students can access as they look for resources and tools. Going forward, the value you bring to your math classroom increasingly will be tied up in the first and third acts of mathematical storytelling, your ability to motivate the second act and then pay off on that hard work.

Related

1. I gave this post a try a year ago.
2. Also, Breedeen Murray has a lot of useful things to say about storytelling, though I can’t endorse her enthusiasm for “confusion.”

2011 Dec 26: The Three Acts of a (Lousy) Mathematical Story is also on the syllabus.

In the first case, you have this exchange:

Romney: Across the nation, over twenty millions Americans still can’t find a job or have given up looking. In 1985 I helped found a company. At first we had 10 employees. Today there are hundreds.

Stewart: You created hundreds of jobs in just twenty-six years? At that rate you’ll have the whole country employed in – hold on – 4,000,000 years.

In the second case you have Stewart pretty well embarrassing himself with this wacky scaling:

Both clips would have stalled out at “interesting” but I censored out their interest – bleeping out Stewart’s (obv. imprecise) calculation of “4,000,000 years” in the first; blurring out the 2.9% in the second – making them perplexing instead.

The Goods [Romney]

Download the full archive [10.6 MB], including:

• Question Video
• Answer Video

The Goods [Health Care]

Download the full archive [15.3 MB], including:

• Question Video
• Question Photo
• Answer Video

Many thanks to my flinty-eyed reader Joshua Schmidt for making sure I didn’t miss last night’s episode.

2011 April 16: As of this writing, Bain Capital has 375 employees, which is completely germane to the problem. Huge ups to Emily’s resourcefulness.

Download the full archive [11.3 MB], including:

• Dan + Chris – Question
• Dan + Chris – Answer
• Dan + Chris + Annie – Question
• Dan + Chris + Annie – Answer

See, I had to do something about this problem:

This is one of those problems that wakes mathophobes up at night in a panicked, cold sweat. It’s so universally hated it’s a pop cultural cliché, a symbol of everything the layperson dislikes about math but can’t quite verbalize.

As math teachers, we step into the ring with this problem every year. In California you’re guaranteed at least one such problem on your summative student exam. But it’s a boutique problem. How much time can you really offer it in class? How will you treat it? There’s a fairly straightforward explicit formula for its solution:

Do you teach your students the formula and hope their memory serves? What kind of conceptual underpinning can you offer them without spending two weeks on it? In what ways can we improve this problem?

First, Fix The Visual

Since this problem represents itself as a real, no-fooling application of math to the world outside the math classroom, we owe it to our students to ask, “is this a good visual representation of that world?”

A: No. It’s clip art. We could upgrade the clip art to stock photography but both representations are decorative where we’d prefer something descriptive, a visual that is, itself, a useful site for analysis rather than mere drapery.

Here is that visual:

Bean Counting – Problem #1 from Dan Meyer on Vimeo.

This makeover was challenging. The task (whether painting a house or filling a cup of beans) is its own unit of measure. Think hours per house or minutes per cup. If the students manage to determine seconds per bean, you still have a math problem, but the task has changed significantly. So I sped up the tape to preserve that task. Also, the house-painting problem assumes identical houses and a constant rate of house-painting, which is the kind of unreality that exists only to serve a math problem. Here, though, we have used multimedia to inoculate that pseudocontext. Here, the glasses are identical every time and the actors are listening to a song in their earbuds that does keep them working at a constant rate, even when they’re working together.

Second, Fix The Motivation

Again, I hear you: who cares about two clowns filling a cup with beans. Try it, though. Play the video. Ask your students what questions they have. Ask them how many minutes they think it’ll take Dan and Chris together. Ask them to put down a number they know is too large and a number they know is too small. Write five of their guesses on the board.

Then move on to whatever else you had planned for the hour. Let us know how that goes.

Third, Teach

Lecture your way through the problem. Or, better, give your students a moment to reveal to you the tools they bring to the table. I’m only insisting here that when you make the brave (and, to the eyes of many students, ludicrous) claim that math has any application to the world outside the math classroom that you represent that world well and you get the motivation right. Your decisions past that will have a lot to do with your students, their developmental readiness, and your relationship to them.

But even if you lecture, don’t offer the formula. Build, instead, off their existing understanding of speed. This problem is nothing more than a strange unit for speed: percent per second. From there, what seemed complicated and dizzying becomes a straightforward application of rates: find Chris and Dan’s combined rate; divide into 100.

Fourth, Play The Answer

This is dramatic catharsis. I have no idea how to design a study to test for the effect of watching an answer rather than hearing it from the teacher’s mouth or reading it in an answer key but, anecdotally, it’s enormous.

Bean Counting – Answer #1 from Dan Meyer on Vimeo.

Fifth, Offer The Extension

Bean Counting – Problem #2 from Dan Meyer on Vimeo.

Your students’ reaction to this extensions is a meter stick for the effectiveness of your approach in step three. Will they take it in stride? Will they fall apart? If they have an inflexible understanding of the problem they may take this approach:

Which is understandable, but incorrect. (This is the explicit formula for three people.)

In both problems, though, the formula obscures the fact that nothing more complicated than speed runs beneath this problem. I’d rather my students developed that understanding than a full set of what reader Bowen Kerins calls “single-use tools”:

High school math is filled with specific tools for one purpose only. Use this box to solve a word problem about people painting houses, but this other box for this other problem. Use FOIL to expand a binomial multiplied by another binomial, but don’t try it on a trinomial! It makes no sense, and contributes to students’ feelings that mathematics is a giant toolbox you either know or don’t know how to use. [via e-mail, with permission]

Jump back into that video I linked earlier. It’s a useful depiction of a locker room-full of students who understand math to be a giant toolbox you either know or don’t know how to use. Confronted with two numbers, they multiply, they add, they average. They’re just striking the two numbers against each other, looking for sparks, looking for a number they can live with. It’s impatient problem solving.

Then the mathemagician enters the scene, reveals the “simple formula,” and computes it.

Check out the look on Junior’s face. It’s like he’s seen a magic trick. He asks, “Are you sure?” and then takes the mathemagician’s word for it. Meanwhile, we’ve upgraded the representation of the problem and not one of our students has had to take our word on the answer. They just watch it.

Featured Comment

Matt Vaudrey:

I passed out calculators for the bean counting problem, but made them give guesses (and back them up) first. Some couldn’t wait, and started crunching numbers.

The catharsis was definitely more potent in the bean counting video than the Little Big League video (“Wait, what’d he say?”). Once the time stopped at 4.5 minutes, students started with bragging.

“Ahhh! I TOLD yoouuuuu!”

Which glass contains more of its original soda?

[WCYDWT] Coke v. Sprite from Dan Meyer on Vimeo.

Justify your answer.

The Goods

Download the video.

2011 Mar 04: Updated to add the goods.

2011 Mar 13: 71 comments as of today means we’ve struck a nerve. Many commenters have put their mark down with an algebraic proof. More interesting to me are those who have included devices for illustrating the proof to their students. That’s harder stuff. See MPG’s comment:

Consider a similar problem using discrete objects (e.g., playing cards. Take 10 red cards and 10 black cards face down in separate piles. Take four at random from red pile; mix into black pile. Shuffle. Return four random cards face down to red pile. Ask: more black in the red pile or red in the black pile. Try this several times. If you’re not convinced, do it with the faces showing. Apply principle to soda problem.

Download the full archive [104 MB], including:

• video – the question
• video – all five cubes
• video – a pair of blocks with the same surface area
• video – a pair of blocks with the same volume
• video – miscellaneous block #1
• video – miscellaneous block #2
• video – the answer
• image – dimensions of all blocks
• applet – interactive exponential model

Caveat #1

I wouldn’t use this lesson. I can’t explain the best-fit model adequately. I can’t adequately explain a microwave. This link was extremely helpful (thanks, Jean-Marc!) as was this explanation (thanks, Carmen!) but in my hands this problem verges on pseudocontext because I’m asking the students to use an operation (exponential modeling) that may or may not follow from the context – I don’t have a strong sense of it.

But maybe you can explain the operation to your students and how it results from the context. In that case, here are all the resources and this is how I see other aspects of the lesson playing out.

Caveat #2

I need to reshoot everything, after controlling for variables mentioned by Matt and Christopher. It’ll take some time, though. Mostly because I’m sick of cheese.

Caveat #3

The original post wasn’t a lesson. I wanted to share something I found interesting and tap into our braintrust here to help me explain it. I only raise this particular caveat because there seems to be some misunderstanding that every blog post constitutes a lesson or a complete curriculum or something.

Perhaps this confusion is genuine. Perhaps it’s disingenuous. Certainly it’s easier to criticize something if you measure it against a higher bar than it’s trying to clear.

In any case, Belinda asks a useful question:

I’m interested in how everyone would complete this sentence: As a result of this lesson, students should understand that [blank].

My objectives. Students will:

• graph data, transferring them from a context to a table and then to a graph,
• calculate surface area and volume of rectangular solids,
• understand the effect and meaning of the parameters of an exponential function,
• enjoy a guest lecture from the science teacher down the hall. [optional]

1. Play the question video.

[WCYDWT] Cheese Block – Question from Dan Meyer on Vimeo.

2. Ask the students to write down a question in their research journals that interests them.

Then share out.

I presume a majority will want to know how long it will take the block to fully melt. If that isn’t a pressing question for your students, then abort. (And let me know.)

3. Estimate.

Ask students to write down a guess. Then ask them to write down a time they know is too long and a time they know is too short. Put some of those guesses on the board and attach them to names.

4. Give more data.

This is where you can introduce the idea of extrapolation, using what few data we know to draw conclusions about what we don’t. Open the video of the first five cubes. Ask them to write down what they think is going to happen when the microwave starts.

[WCYDWT] Cheese Blocks – Cheese Cubes from Dan Meyer on Vimeo.

Afterwards, ask them to write down why they think the cubes melted in the order they did. Really push hard on their idea that “bigger” blocks take longer to melt. Make sure they define bigger. More surface area? More volume?

In either case, you’re covered. If someone thinks surface area matters, load up the blocks with the same surface area:

[WCYDWT] Cheese Block – Controlling for Surface Area from Dan Meyer on Vimeo.

If someone thinks volume matters, load up the blocks with the same volume. “So you’re saying these should fully melt at basically the same time:”

[WCYDWT] Cheese Block – Controlling for Volume from Dan Meyer on Vimeo.

So we threw a sharp rock at both of those theories. Is there a better option? Lecture about the ratio of volume to exposed surface area or let the students discover it. Your method here matters less to me than the fact that we’ve given students some reason to care about the ratio, what it models, and what they can do with it.

5. Calculate.

Using the measurement images, have the students create a table including the dimensions, the total surface area, the exposed surface area, the volume, the ratios between them, and the melting time for the block. Include the big block whose melting time we don’t know.

Have them graph time against one set of data. Show student work. Discuss which model looks best.

6. Model the exponential.

Use the Geogebra file, which graphs the melting time of the block against its ratio of volume to exposed surface area. Have them adjust the parameters until they have a good fit. Discuss the meaning of the parameters.

7. Resolve the hook.

How do we use our model to find out how long it’ll take the enormous block to fully melt? Then show the answer:

[WCYDWT] Cheese Block – Answer from Dan Meyer on Vimeo.

Compare to the original guesses. Show some love to whomever was closest.

The disparity (150/100) is based on the fact that growth depends on ionic content of the water–the purer the water the larger they grow. The very same Orbeez wll grow to a different size depending on the water purity. The number we chose ended up being a marketing decision (100 is a powerful figure) but we should have been consistent. It’s impossible to choose one accurate number.

BTW: Sharon Cohen sent along Orbeez’ internal measurements of expansion given different water sources.