## [WCYDWT] Cheese Blocks

The Goods

• 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
• 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.

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.

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:

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:"

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:

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

### 12 Responses to “[WCYDWT] Cheese Blocks”

1. on 28 Feb 2011 at 5:33 pmKarim

Very, very cool. Whatever the standards alignment ends up being–volume vs. surface area for muddle schoolers, exp. growth for Alg II–perhaps that’s almost secondary here. Maybe a fine sentence would be: Students will be…curious.

And if nothing else, that seems like a lot. Galileo. Copernicus. Odysseus might have known where he was going when he started, but I bet Homer didn’t.

Which is to say: I wouldn’t hold off on teaching this simply because I didn’t know where it’d go. There’s a whole rest of the year for GPS. Maybe I’d teach it exactly because I *didn’t*, and that’s a pretty cool lesson in and if itself, no?

Thanks, Dan.

2. on 28 Feb 2011 at 6:05 pmPaul Hawking

Interesting stuff, Dan. I think a question that I would want to explore is

“How can we make it melt faster in the microwave?”

Does it melt faster on a plastic plate, a glass plate, or a paper plate? Will wrapping it in plastic insulate the cheese from outside heat, or trap in the heat from the waves that travel through the plastic?

Yes, I would DEFINITELY invite one of the science teachers to help in answering the students questions AS WELL AS to suggest questions that the student’s don’t have the requisite background to even think of. This is, in fact, one of my main criticisms of WCYDWT and I’ve gone into more depth explaining my concerns at irrationalcube here:

“WCYDWT In-n-Out: Then and Now”
http://irrationalcube.wordpress.com/2011/02/12/wcydwt-in-n-out-then-and-now/#comment-206

Paul Hawking
Blog:
The Challenge of Teaching Math
Latest post:
What are you doing for Pi Day?
http://challenge-of-teaching-math.blogspot.com/2011/02/what-are-you-doing-for-pi-day.html

3. on 01 Mar 2011 at 1:35 amRafael

If you are inviting the science teacher down the hall, you might want to change the label on your y-axis from “sec” to “s”. Maybe it is a pet peeve of mine, but that is the correct symbol for the SI unit of time…
Funny story, completely off-topic: when I told this to my physics class a couple of years back, a girl from the class shouted out “But I always do secs!”… at which point she turned bright red, and the class burst out laughing (teacher tried to suppress his grin).

4. on 01 Mar 2011 at 9:50 amKevin Stuart

You always have great stuff Dan!

5. on 02 Mar 2011 at 8:22 pmDan L

Great summary. I agree that a deeper understanding of the physics will eliminate the pseudocontext. This could be good fun for a physics class. This could also work as a college laboratory with appropriate models for physical behavior.
(There are a bunch of folks who are interested in microwave processing of engineering materials – so the relevance is already there, you’re just using cheese!)

I should also point out that folks who study viscoelasticity (time dependence of stress and strain in materials) use Velveeta.

What I’m saying is that you are in good company in cheese science – viscoelasticity strikes fear into the hearts of engineering students!

6. on 03 Mar 2011 at 6:43 amDvora

Quick tech question… I can’t seem to access the videos from school since they are being blocked marked as R rated. Any ideas?

7. on 04 Mar 2011 at 8:23 amMax

Could it be an inverse variation model rather than exponential? That melting time is inversely proportional to the ratio of volume to surface area? So that our curve is rational rather than exponential?

I don’t know how to tell…

8. on 08 Mar 2011 at 5:43 pmSam

Dan,
What program do you use to make your splitscreen effect? Do you have a link to any tutorials on this?
Thanks!
Sam

9. on 08 Mar 2011 at 10:42 pmDan Meyer

Adobe AfterEffects. If you ever pick it up, let me know and I’ll throw together a tutorial.

10. on 19 Apr 2011 at 7:36 pmBev

This would be a good lesson to use in the science classroom when we introduce why cells need to be small to survive.

11. on 26 Oct 2011 at 7:17 amBrian

Sorry to necro this thread, but did you notice you can measure the speed of light from the first video? The block of cheese starts melting from two points that I would eyeball at 6cm apart. These correspond to anti-nodes in the microwave, so the wavelength is 12cm. Standard microwaves are 2.45GHz (there’s usually a plate on the back you can check). Plugging into google “12cm * 2.45GHz” should give an answer that’s scary close to the speed of light.

12. on 22 May 2012 at 4:31 amJames Cleveland

The geogebra applet doesn’t seem to be in the archive, fyi.