I’ve always found triangle centers to be highly intriguing. The initial related mathematical questions are low-entry. But the ideas can go really deep, and there are fun, and valuable visual and logic skills to exercise.
With my 8th graders, I began by giving out an Entrance Ticket. We had practiced with the Pythagorean Theorem, finding missing side lengths, missing angles, and area, but we hadn’t explicitly referred to triangle centers before this.
The ticket asks them to plot the center of each triangle, but doesn’t give them any more specific instruction. Most students wanted to know how to do it. I was totally non-committal. I pretended to be busy setting up the projector, and told them to hurry up so we could move on to the lesson. I gave them a few minutes, and then collected the tickets.
Next, we visited Dan and Dave’s Interactive Ice Cream Shop. This app asks you to decide which ice cream shop you should patronize depending on your position in the park, to describe regions of the park where you would decide on one shop or another, and then shows how your intuitive work compares to others’. After getting over the idea of why anyone would possibly shop at a place called “Frozen Flavored Milk,” students definitely enjoyed this process. (Thanks Dave and Dan for putting it together and sharing it so publicly!). This is a great, low-pressure way to begin the exploration. Click on the picture to visit their site.
Many of them now thought that this was the “right” way to find a triangle center, and wanted the entrance tickets back to revise them. Their requests went totally unrequited. We reviewed our tools for geometric constructions, including bisecting angles, and drawing perpendicular lines, and then headed for Geogebra. Students created their own circumcenters (although we didn’t use the word yet – thus far, this is “Elsie’s method”). Oh, and they had to pass the “drag test.”
Then, we watched Vi Hart’s Infinity Elephants. Kids love the Apollonian Gaskets especially, and to make these, we just needed to find the incenter. “Wait a minute. This is different. Which one is the center? This one or the Ice Cream one?” I cruelly asked them the same question that they were asking me: “Which method is right? Which one is the real center?” I act busy and move on to someone else. Kids go to Geogebra, and create an incenter.
Now, I passed around some sheet metal triangles that I had lying around (detritus from my night job). “Try to balance these on the eraser of your pencil.” I let kids experiment for a few minutes, and then asked them to pass their triangle to the next group. Before attempting to balance this one, they had to mark a point on the steel. How did they come to that point? Back to Paper, pencil, and Geogebra for the centroid.
I’m still looking for an orthocenter hook. The useful things I could think of are only useful if you’re already into the geometry (leave an idea in the comments if you have one handy), but my kids were into it by then anyway. I just gave them another rule and asked them to construct bisectors. As students constructed these, I wondered out loud if there was any difference between this center and the first one.
I asked them to complete the exit ticket. Several still asked which one was the right one. “Which one is actually the center? No, really this time.” I desperately tried to hand the power back to them by asking them the same question. “YOU get to decide.” That’s the whole point. For me, it’s all about giving them the power to decide on their own definitions – based on their own criteria.
What exactly is the center of a triangle anyway?
Here are some extensions to have handy. I used some of these over the next few days.
- Given 3 cities, what city is at the circumcenter, centroid, etc? Middle school kids love to find things like this on maps. You can ask the question referencing ice-cream shops or balancing the entire state on a pencil eraser of that seems like fun.
- How could you find the centroid of square? A rectangle? A pentagon? Any irregular polygon?
- In which of the centers is it possible that the center is outside the triangle?
- Euler Lines – So cool!
- This problem is from Cynthia Lanius, and was only peripherally related, but it was there and relevant at the right moment. The Three Evil Dictators: To keep them from fighting, you must separate three evil dictators as far away from each other as possible on the surface of the earth. If you put one in Los Angeles, California, name two other places where the other two could be placed so that all are equidistant, and at maximum distances from each other. (Yes, you can put them in the ocean if you have to.)