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Reasoning and ProofSession 04 OverviewTab atab btab ctab dtab eReference
Part A

Observing Student Reasoning and Proof
  Introduction | Problem: Building Rafts with Rods | Solution: Building Rafts with Rods | Student Work #1 | Questions and Answers #1 | Student Work Reflection #1 | Student Work #2 | Questions and Answers #2 | Student Work Reflection #2 | Observe Classroom | Classroom Practice | Your Journal


Following are some examples of questions you might ask Sara. For each question, think about an answer that Sara might provide. Then, select "Show Answer" to see one possible answer. Keep in mind that this exchange is just one approach; middle school students will come up with a variety of ways to tackle this problem.

Teacher: You have seen a pattern as you work down the surface area column in the chart. Using your pattern, what would be the surface area for five, six, and seven rods?

Show Answer
The pattern is to add 6 for each rod, so for five rods it would be 34 square centimeters, for six rods it would be 40 square centimeters, and for seven rods it would be 46 square centimeters.

Teacher: What if there were 100 rods?

Show Answer
It would take a lot of time to extend the table to 100!

Teacher: Do you think you could use the pattern you found to come up with a general rule that will work for any number of rods, without having to extend the table all the way to that number? Let's start small, with just one rod. How would you find its surface area?

Show Answer
To find the surface area, I would do 4 • 2 for the long sides and then add 2 for the ends.

Teacher: Look at how the two-rod raft is put together. You said that the surface area for two rods was 6 more than the surface area for one rod. Think about how you are finding the surface area. Can you see where the extra 6 comes from?

Show Answer
Part of the extra 6 –– 2 –– comes from two more ends. The other 4 must be from another top side and another bottom side.

Teacher: Look at the raft from different views. What do you notice?

Show Answer
The top and bottom are the same. The front and back are the same. The sides are the same.

Teacher: How will that help you figure out the surface area for any raft?

Show Answer
The front and back are always the same. The surface area of each is always 2 square centimeters, no matter how many rods are in the raft. That's 4 square centimeters. You will also have one square on each side for each rod, so you need to take 2 times the number of rods.

Teacher: How could you write this last step?

Show Answer
Since the number of rods is changing, I'll call that x. So for the square on each side, you have 2x. There are also 4 squares (2 on top and 2 on the bottom) for each rod, which would be 4 times the number of rods, or 4x. So you have 4 + 2x + 4x, or 4 + 6x.

next  Reflect on the student work

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