Entering the Problem

Instructional Strategies: Guided Practice, Use of Whiteboards, Cooperative Learning

How did Ms. Scott's students solve the actual CAHSEE problem? She had them work in pairs and make a scale sketch of the figure on their whiteboards--a 20 x 20 unit square with a large circle in the middle touching the square on all four sides. Students were permitted to change their answers (A, B, C, and D) as they came closer to a meaningful solution. While the scale sketch perhaps added little information, it seemed to give many students a way to "enter" the problem. As a result, they were willing to engage the problem, which was a HUGE step for this class, and perhaps for many other classes as well.

In working through the problem, few students seemed to use a systematic formula-driven approach, described in the solution above. Rather, they worked the problem one piece at a time, and then tried to put the pieces together.

While this is not a precise transcript, several conversations between student A and student B went something like this:

A: We have to subtract the circle. So how big is the circle?
B: It has a diameter of 10, so the area is pi-r-squared.
A: But the radius is not 10, it's 5. So we use the 5.
B: OK, so we take 3.14 times 5 times 5 (using calculator), which is 78.5.
A: But we have to subtract it from the original square, which was 10.
B: So it's 78.5 minus 10.
A: But the area of the square is 10 squared equals 100, like the ones we cut out.
B: Whatever. So take 100 minus 78.5 (using calculator) which is 21.5.
A: So the answer is A.
B: Cool.