1.

2. The surface area for a raft made of n yellow rods is 12n + 10. The volume for a raft made of n yellow rods is 5n.

3. The graphs are both straight lines: surface area, a = 12n + 10, and volume, v = 5n. Note that the surface area graph starts at (1,22) and has a slope of 12. The volume graph starts at (1,5) and has a slope of 5.

Before you analyze some student work related to this problem, think about the following questions:

1. What mathematical content could be developed in working on this problem?

2. Every time you add a yellow rod to the raft, the surface area increases by 12 units. This is a mathematical conjecture you can make from the data in the table, from the slope of the graph, and by looking at the exposed surface of the added rod. What other mathematical conjectures can you make from analyzing the data in this problem?

3. You can justify the conjecture in Question 2 by seeing that every time you add a rod, you are adding 3 new long sides (15 units) but covering one side of the previous rod (5 units). The ends of the rod add an additional 2 units. So you are adding 15, covering (subtracting) 5, and adding 2, which gives you 12 units for each added rod. How else might you justify your thinking about the conjecture made in Question 2?

4. In what ways do representations (for example, tables and graphs) support students in their efforts to notice and generalize any patterns they see?

5. What questions could you ask students to help them move from noticing patterns to being able to develop and explain or justify conjectures?

  Observe how a middle school student solved a similar problem