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
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.
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.)
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.
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.