First Day Back – A Tale of Triumph and Sadness

My first period class (we call them Bells here, rather than Periods) is my Geometry class. I started by sharing with the the NYTimes story about ‘The Interview’ and was pleased that they quickly attacked this as a system of equation. I had a secret plot for starting with this problem. We are getting ready to explore triangle bisectors of various sorts. I started out with this question for my students: ‘What does it mean to call a point a center for an object?’ Luckily, this prompted a quick recollection of centers of circles along with a nice attempt at remembering a sound definition for a circle. I then asked them to consider what would be the center of a square. One of my students, a freshman named Matthew, quickly proposed that the intersection of the diagonals would be his point of choice. I opened GeoGebra, drew a rather random square and tested Matthew’s idea. We saw that this was in fact equidistant from the vertices. I then asked about distance to the sides. This required a quick conversation to remind them of what we mean when we talk about distance from a point to a line or to a line segment. We quickly came to an agreement that the perpendicular distance was what we wanted. GeoGebra confirmed that this ‘center’ was equidistant from the sides of the square, but I pretended to be troubled that this second equal distance was not equal to the first equal distance. My students quickly overruled me and were comfy with this point as the center. Next, I asked what the center of a triangle might be. I had three students each volunteer and ordered pair as a vertex of the triangle. It turned out that they formed a right triangle. We agreed the idea of perpendicular bisectors (which we had JUST looked at for the square!) was the way to go. Some quick GeoGebra showed that these lines coincided at the midpoint of the hypotenuse. I was pleased that this raised questions. No one jumped to the conclusion that this would always happen and a student named Tara quickly guessed that this was happening due to the original triangle being a right one. I then moved one of the vertices so that the triangle was acute and, happily, we noticed two things together. First, the perpendicular bisectors still coincided as I moved a vertex. Second, they coincided inside the triangle. Matthew then asked to see what happened with an obtuse triangle and we saw the point of coincidence migrate outside the boundaries of the triangle. It was great to notice that they still met at a point, but the idea of a ‘center’ being outside the triangle did not make anyone happy. Matthew observed that this point did not feel very ‘centery’ to him. Awesome stuff. Finally, since we had GeoGebra to confirm our work, it did not seem that intimidating to go ahead and find the coordinates of the point of intersection for these lines. My secret plot of having them think about systems of equations at the beginning of class paid off. Overall, a wonderful way to start the new year. Tomorrow, I’ll try my idea of HW review at the beginning of class and see how that feels.

Unfortunately, the feeling of triumph dissipated quickly. I have two Bells of AP Stats this year and I had asked these students to listen to a Radiolab Shorts episode called Are We Coins? and I gave them a handful of question prompts. I asked everyone to jot down some reaction notes and to bring these notes to class today. In my first class of 11 students I had three who showed clearly that they had listened to the episode. I had zero students with notes. I asked everyone to take out their notes and a number tried to fool me by having a notebook in front of them, but none of these had anything to do with my questions. In my next class of 16 students, three of them had notes and one or two others showed clear evidence of having paid attention to my request. Sigh…

I’ll dwell tonight on the Geometry kids instead and get ready to really dig into this idea of ‘center-ness’ for a triangle tomorrow. A couple already asked, on their way out of class, about using the vertices as anchors instead of midpoints. Should be fun tomorrow. Lots of noticing and wondering and a concession on my part to their need for HW reinforcement. Hoping for another great start to a day.