Socializing in Geometry – Similar Triangles

Another successful experiment getting my participation-challenged geometry class to interact with each other yesterday.

Each student received a cut-out triangle from the image at left. The challenge:

One (or possibly two) people in this room have triangles similar to yours. Your task is to find the person and do the following:

• Find the similarity ratio between your triangle and your match in the order big:small.
• Determine the ratio of the perimeters of each of your triangles.
• Determine the ratio of the areas of each of your triangles.

I then cut them loose. Almost immediately they started scrambling around the classroom holding up triangles and calculating as quickly as possible. (I didn’t totally get why they were in a hurry, actually.) They clustered on tables and rapidly shifted partners until everyone found they were in the right place. The calculating began for perimeter – that was the easy part. Then the area question took center stage.

Some asked me how to find the heights of the triangles, and I shrugged my shoulders with the smirk of someone with ideas that isn’t sharing them. (I call this my ‘jerk’ mode that I love taking on during class for the sole reason that it gets them finding and figuring on their own.) Some recreated the triangle in Geogebra. Some superimposed it over graph paper and counted to get an estimate. One student cleverly found Heron’s formula. It was really entertaining watching them excitedly explain the formula without writing it down (something else I didn’t understand) and share how it quickly and easily allows the area to be calculated. The energy in the room was apparent as they ran from person to person trying to get everyone to complete the task. Eventually they found out themselves that the similarity ratio was a square relationship. I didn’t have to do a thing.

Part of my justification in doing this was to get them thinking about the important ideas necessary in solving another problem I threw their way during the previous class comparing the old iPad to the new one. The two different groups that had worked on it were generally on the right track, but there were some serious errors in their reasoning that I hinted at but didn’t explicitly point out to them. I think this activity closed the gap. There should be some interesting answers to discuss in class when we next meet.