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Following is another teacher-student dialogue for you to consider. As you read, focus on how the teacher's questions help the student formulate and test his assertions. As noted previously, this exchange is just one possible approach; middle school students will come up with a variety of ways to tackle this problem.
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Teacher: Where are the 10 and 12 in the one-rod raft?
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Student:
To find the surface area, I did 4 • 5 for the long sides and then added 2 for the ends. So the 12 must be two sides of the yellow rod and the two ends. The 10 must be the other two sides.
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Teacher: Look at how the two-rod raft is put together. You said that the surface area for two rods was 12 more than the surface area for one rod. Think about how you are finding the surface area. Can you see where the extra 12 comes from?
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Student:
The 2 comes from two more ends. The other 10 must be from another top side and another bottom side.
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Teacher: Look at the two-rod raft from the top, from the left, and from
the front. What do you see from each view?
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Student:
The bottom and top will look the same. The left and right will look the same. The front and back will look the same.
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Teacher: Will this be true for any size raft? How can you be sure?
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Student:
If I build more rafts, I can see if it always works. (The student continues to build rafts.) Yes, it works all the time!
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Teacher: What do you notice about the surface of each of these
views?
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Student:
Here's what I notice: The top view shows the top faces of each of the rods in the raft. The surface area of the top of one yellow rod is 5 square centimeters, so I will have 5 square centimeters of surface area for the top of each rod. The bottom is the same, so I double my answer. That gives me 10 square centimeters for each rod in the raft.
The left view shows the ends of the rods. The surface area of one end is 1 square centimeter for each rod. The right view is the same, so I double my answer. That gives me 2 square centimeters for each rod in the raft.
The front view shows one side of one rod. The surface area is 5 square centimeters. The back view is the same, so I double my answer. That means that no matter how many rods are in the raft, the front and back surface areas remain the same: 10 square centimeters.
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Teacher: How will that help you figure out the surface area?
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Student:
If I put all that together, this is what I get: 10 times the number of rods (x) would take care of the top and bottom (10x); 2 times the number of rods (x) would take care of the left and right (2x); and 5 times 2, or 10, would take care of the front and back. So that can be written as 10x + 2x + 10, or 12x +10.
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 Reflect on the student work
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