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Teaching Math: A Video Library, 5-8

Building Rafts With Rods

Video Overview
Michelle Mullin, demonstrating on the overhead, explains that each group will calculate the surface area and volume of a "raft" of n rods, where n goes from 1 to 10.
After that, they have three tasks they can do in any order: graph their data, make formulas for surface area and volume, and write a question that will introduce the task to another class.
The groups work for some time on their tasks. Ms. Mullin circulates, helping groups with problems and checking in with them.
At the end of the class, she has a few groups make brief presentations showing how far they've gotten.

An Exploration for Teacher Workshops
Materials: grid paper, cubes
Work in small groups to make the structures below. Then discuss: what does the fourth structure look like? The seventh?

(Structure #1)

(Structure #2)

(Structure #3)

Work in small groups to calculate the surface area and volume of each structure. Look for a pattern. Check your conjecture on the fourth and seventh structures in the series. Generalize the pattern into a formula so that you can predict the surface area and volume for any structure in the series, no matter how large.

Some Questions

What was the hardest part of figuring out the volume or surface area? Why?
How did you record your information? How did you figure out the formula? How did you record your formula?
Why are you sure your formula works for all structures in the series?
Suppose you were to graph surface area against volume. What shape graph would you get (straight, curved, wiggly)?
Does a task like this legitimately belong in pre–algebra? Why or why not? (When you watch the video, look to see how the students' task differs from this one.)

Extensions

Think about this: Volume is supposed to be cubic. Why is volume quadratic here?
If you plot surface area against volume (instead of against n), the graph is almost but not quite straight. Why is the surface–area–vs.–volume graph straight in the video? Why isn't straight here?

Additional Discussion Topics
Here are some additional ideas for discussion that arise in the video:

In some activities—and you can see some constructions left over from a surface–and–volume unit on the walls—volume and surface area do not increase linearly. Explain why they do in this situation.
One group graphs surface area against volume. Explain why this turns out to be a linear function.
Assessment

Here is an excerpt from a student conversation in the video—the group is building rafts with fives. The volume sequence is {5, 10, 15, 20...}. The group is discussing the surface–area sequence, {22, 34, 46, ...}.
"...just put 'preceding answer plus 12'."
"You can't put 'preceding answer.' It has to be a number or a letter."
"Let's just go on to volume. We can figure that out easier."
Comment on evidence for content understanding and problem–solving strategies in this exchange.

Referring to the above conversation, using "preceding answer plus twelve" is a recursive definition. You can write the surface area formula as:
????????
What's wrong with that?

Discussion Questions
These questions appear at the end of the video. Here are some follow–up ideas and prompts to help get a discussion going.
From what you've just seen, what do you know about these students' mathematical power?
Think about the student discourse you heard in the video. What evidence did you see or hear of understanding? What evidence of confusion? What can you tell from their graphs? What more might you ask students in order to learn how they are approaching this task?
How can you help students learn to generalize patterns into formulas?
What did Ms. Mullin do to help students figure out the formulas? What else could she have done? What do you do when students are struggling to figure out a formula?
What are the components of a good task?
In what ways is the task in the video better for this class than the one in the Exploration?
Consider changing this task and imagine how it might go differently. For example, what if the task were more structured? What if it were more open–ended? What if the situation generated nonlinear functions instead of linear ones? What if (as in the video The Location) students had used plain paper instead of grid paper for their posters?
In those cases, what else might students learn? What might they be confused about?