The Daily Show gave us a twofer last night. Neither one of these is any kind of lengthy inquiry. They’re interesting examples of math used to make sense of political debate (in one case) and distort it (in the other).

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.

The Goods

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!”

This problem nearly ripped the Meyer household apart tonight:

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.

The Goods

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.

Sharon Cohen, the brand manager at Orbeez, checks in on the last post:

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.