The following activity was developed for a middle school class which was covering the concept of the area of a triangle. These students had worked with this concept at least a few times prior to this presentation, but they had not mastered it.
A triangle was created in Sketchpad with one vertex resting on a segment parallel to the base line of the triangle. The lengths of the base and height were measured and posted as was the triangle area formula. As the students entered the class, they saw the triangle being animated along with the information described.
They "oooohhhed" and "aaaawwed" for the first few minutes. When the bell rang, the teacher said nothing. Soon the students started discussing the animated triangle. For some, it was amazing that the area was not changing even though the triangle kept looking different. For others, the area had to stay the same because the base and height were not changing.
Then, a most interesting discussion occurred. A boy went to the screen and explained that the base was not changing and so the area had to remain the same. He pointed to the base as he talked.
Not many were convinced by the boy. A girl went to the screen and held her hands at the endpoints of the height, moving them together as the triangle animated across the screen. She explained that since the measurement was not changing (she used the constant measurement being shown to verify this) the line and the line segment had to be parallel, and the distance between them constant. Then she sat.
This was the first time the teacher entered the discussion. The opening comment was: "Are you telling me that if the height changes the area will change?" As the question was asked, the height was changed and the class could see the area change in real time. The base was then changed in a similar manner.
After some discussion, the class was given seat work. It consisted of 10 problems and each was a variation of the same triangle taken from a figure established as described at the beginning of this discussion. In only a few minutes most of the students appeared finished. The teacher encouraged them to do their assignment but they announced they were finished, explaining that all the problems were the same. The teacher asked how that could be since the triangles all looked different. The students replied that looks did not matter; since the base and height were unchanged, the area would be the same.
The discussion that followed is best summarized by the students. They admitted to having worked with the formula for the area of a triangle before. However, they said, they had never SEEN it before.
Follow up work with the class several weeks later revealed that they considered the area of a triangle as trivially simple and were not influenced by the appearance of the figure. They focused on the base and height of the triangle to get the area.
Doug Brumbaugh, firstname.lastname@example.org
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