Below is a guest post by Dmitry Bryazgin (originally written in Russian). Dmitry runs a small math circle near Princeton for students in grades 3-5. On the particular day described in the post, there were 4 students present.
The main task for the day was to find a general formula for the area of a trapezoid. The class had previously derived formulas for the areas of a rectangle, right triangle, acute triangle, and parallelogram.
For some additional motivation, I promised to give a prize to anyone who independently solves the problem. I then drew all the previous area formulas that we had encountered and reminded the students of how we had derived them.
Each student received a piece of graph paper and a pencil, and the search began.
The children puffed, cutting corners off and drawing all sorts of add-ons to the trapezoid. I allowed them to use the board if they thought that would help them. Some of the kids rushed to the board, but this did not lead to any progress on the problem.
Then I gave them the first hint: try to complete the trapezoid to a parallelogram because we already know how it find the area of the latter. This did not help.
Soon I started to notice that the students were getting tired, so I decided to give them a second hint: try to add a second, upside down, trapezoid to the original one.
The students once again took to their drawings and soon started to yell that they solved the problem. But a quick check of their work revealed that the solution was not yet in sight. Nonetheless, I was pleased to see that the children were not giving up and were showing a real interest in the problem’s solution.
At one point I realized that one of the students was close to the first step of the problem – the construction of the parallelogram – but the drawing on his paper was very small.
So I drew a grid on the board with a trapezoid on top of it and asked the student to show his construction. He came up and accurately transferred his drawing. This was the first successful step towards solving the problem. And then came the most interesting part.
I said that now we need to make that next step and find that “general formula”. The formula for the area of a parallelogram was written slightly higher on the board. Applying it required a substantial leap inside a child’s mind: realizing that the formula could be used for a DIFFERENT picture and understanding where in this new picture are the sides “a” and “b”. This, by the way, is already an abstract reasoning skill.
I pointed to the lower base of the trapezoid and asked the students for its length – all the answers were incorrect. I gave the students some time to think and then asked the question again. And suddenly, one girl exclaimed, “it’s a+b since we have the upside down trapezoid!”
If I could have have leaped to the heavens, I probably would have done so. I was really waiting for this answer, but had started to lose hope. After this, two other girls almost simultaneously exclaimed that we need to multiply by this by “h” since we have a parallelogram. And the student who had made the first step now added the finishing touch by pointing out that we need to divide by 2, since the parallelogram contains two identical trapezoids.
Until the very last moment, I did not know whether we would be able to solve this problem with the students. This unpredictability of the lesson is what made it so interesting and what I wanted to share that with you.
By the way, no one ended up getting the prize, but it seemed that this was no longer important.
Looking back at the lesson, I thought of two mistakes/improvements that would have simplified the task for the students.
Mistake #1: I should have had the students draw a specific trapezoid on their papers with given dimensions. Without this, everyone ended up drawing whatever they wanted, the trapezoids were crooked and this got in the way.
Mistake #2: I initially did not draw the parallelogram on the board, whereas this was the key to solving the problem. I should have drawn it on a grid, along with the trapezoid, and had the students transfer both to their papers. Only after this should the search for the area have begun.