Problem Solving at it's Finest

November 24^th

After spending a few days talking about area and perimeter, I decided it was time to move on to getting students to understand the connections between length, width and area and how one influences the other. At the beginning of class I wrote two questions on the board, the first, a review of what was taught in the last few days, and the second, something a little different. Here are the two questions:

Mrs. John said that she had two different sized patios built onto her house, but they both had the same area. Can Mrs. John be correct? Prove it!

Mrs. John’s desk is the same width as Alex’s desk, but Alex’s desk is three times longer than Mrs. John’s. What can you say about the area of both desks? Use pictures, words and numbers to explain your thinking.

Question number one was just meant to be a means of assessment of the previously taught material and to get a sense of student reasoning and communication.

I would like to spend a few minutes talking about question two. Before today’s class, I did not do any work involving the influence of changing the dimensions on the area of the shape. We did, however, look at what happens to one of the dimensions when there is a fixed perimeter and the other dimension changes in the Changing Garden’s problem. I was hoping students would think back to this problem to help them solve the current one.

As usual, I spent a few moments discussing the context of the problem ensuring everyone knew what they had to do. I demonstrated, using my desk, that the widths of the two desks were the same, but the other desk, Alex’s desk was three times longer than mine. I entertained any questions and then let them go.

I was very pleased to see that all students were able to infer that three times the length required them to multiply the length of my desk by three to obtain the length of Alex’s desk. I seen all of the students using diagrams to sort out their thinking, which was a breakthrough for me as this is something, using illustrations to help solve the problem, I have been emphasizing all year. By using these diagrams students then decided to find the area of the two rectangles. Three students automatically seen the connection that the area was going to triple. They were so excited about this discovery that they went ahead and began questioning what would happen if they doubled both the length and the width. Of course, I let them go ahead and investigate that. The other students kept working on this and many, but not all, seen that the area tripled. The class ended with two students not being able to complete the question and it was left for next class.

I feel this illustrates learning math through problem solving at its finest. I did not tell students what was going to happen, nor did I lead them through conversing or probing or questioning. They discovered these connections on their own and when asked what would happen if the width, for example, tripled, they were quick to tell me that the area would also triple. I then knew they got it. But then I wanted to see how well they actually knew it, namely why the area was doubled when the length doubles. This will be the focus of my next lesson.