We'll now look at how two middle school students, Sara and Evan, solved two similar Building Rafts problems.

Just to be clear: At this level, we are not concerned with formal proof, but we are concerned with student reasoning. Students must be able to provide convincing mathematical arguments to justify their work. One series of steps in thinking about such arguments is based on the work of John Mason, to wit: First, can you convince yourself? Next, can you convince a friend? Finally, can you convince a skeptic? If students can make a convincing argument, we can then show them how to take the next step: making a formal proof. As you observe Sara and Evan's work, ask yourself: Have these students reasoned well? Have they provided convincing arguments?

Note: If you have Cuisenaire rods available, you may want to use them as you work through this section.

First, let's see how Sara approached the problem.

Sara had an unlimited supply of red Cuisenaire rods. The dimensions of one red rod are 2 cm by 1 cm by 1 cm:

Sara made the table below. After the fourth raft, she noticed that every time she added another red rod to the raft, she increased its surface area by 6 square centimeters: Sara couldn't figure out how to write a rule to help her find the surface area for any number of rods without first finding the surface area of the previous raft. For example, is there a way to determine the surface area for 100 rods without having to find the surface area for each raft up to 100?

  Consider some questions you might ask Sara