Student Ideas

My students have done some very interesting work on this project over the last few years. Their explorations began with some guesses - most were very imaginative and creative, and some of their very first conjectures turned out to be correct! The following thoughts, sketches, and reflections are from a variety of students, in 9th and 10th grades.

Some assumed that the center has to be inside the triangle. This seems to be a reasonable assumption, but perhaps it is not always the case?

Many students felt that the center had to be equally far from either the sides of the triangle or the vertices of the triangle. Some wondered if it could be both at the same time. One group said " The center of a triangle is not a very easy thing to define!"

One group commented "The center of the triangle was at first elusive, because it seemed to change in different situations. This was a different kind of writing assignment in chinch fascinating questions were raised."

Different groups thought of different approaches, at the beginning of the project. "We guessed that the center might be on one of the midsegments of the triangle, or the point where a median and a midsegment met (a midsegment is a segment joining the midpoints a pairs sides).We then wondered what would happen if we circumscribed a circle about the triangle - was this the center of the triangle?"

Another group thought of constructing the midsegments of all the sides of the triangle, then constructing the midsegments of the sides of the new triangle, and repeating this process. This "fractal" image is actually a very good visualization of a center of the triangle - as the triangles get smaller and smaller, the "limit" appears to be a point, at the "center" of the triangle. Which center would this be?

 

One group, in searching for the point in the Rec Center problem, came upon this idea: a point equidistant from the 3 towns would be the point of intersection of 3 congruent circles, centered at each of the 3 towns. They constructed 3 circles, and then dragged each circle to make it smaller or larger until all the radii were equal AND the circles intersected at one point: this trial and error approach gave them some very good clues as to where the center of the triangle might be:

This seems to work fairly well. Is this the answer? Is this a point equally distant from each of the 3 towns? Does it match the answers we got before? And how would you construct the point where three congruent circles would intersect, other than by trial and error. What would have to be true about this "center"? What would have to be true about the distances to each of the three towns?

The students tried many different solutions to answer the question "What is the center of a triangle?" One student commented: "To me, this project was like a puzzle that you have to fit together. There are little pieces scattered everywhere and it is up to us to see where they fit."

In exploring the architect's dilemma, again the idea of a "trial and error" method yielded much information about the geometry of the situation. Many groups drew a triangle, then alternated between changing its position and then its size until it "fit" in the space, and was as large as they could make it:

Now the question was, where is the center of that circle? That is, if I wanted to construct a circle that "fit" like this, where would I place the center, and how would I know what length to use for the radius, besides trial and error?


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