Now watch an extended video excerpt (duration 3:44) at left of the class you have just seen as they work on the Building Rafts with Rods problem.

This problem encourages students to find a pattern or structure for the surface area of the raft as they add additional rods. The goal is for students to move from the more obvious recursive pattern -- a pattern in which succeeding results are based on the prior result -- to a general pattern for any given number of rods. Students use concrete materials as a representation to help them see how the pattern is growing. They communicate their ideas to one another to help test conjectures, refine ideas, and develop formulas. The teacher's role -- that is, phrasing questions to guide students, rather than telling them what to do -- is a critical part of developing reasoning skills.

After you have completed the activity and watched the video segment, reflect on the following questions:

• How does the structure of the lesson encourage reasoning among the students?
• What is the evidence of student reasoning?
• Do students justify their answers? How?
• What parts of the reasoning process are the students struggling with?
• Is there evidence of proof for the general formula?
• How does the teacher guide the students without telling them what to do?

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