Beginning New Habits AND a Fun Activity

This morning in Geometry I started by not talking for the first five minutes while my students shared their HW with each other. They talked about their answers, they puzzled over why/where they differed and they talked about using GeoGebra on their own to explore the intersection of perpendicular bisectors of triangles. I was SO delighted I almost wanted to call the rest of the day off.  I did not, though and I’m glad I stuck it out.

We looked together at GeoGebra, reviewed (again) how to find the intersection of lines, we let GeoGebra confirm that we were right. We remembered from yesterday that these lines coincide on the hypotenuse for a right triangle, in the interior for an acute triangle, and outside of an obtuse triangle. After playing with another GeoGebra sketch we all agreed that this behavior made this point of coincidence a pretty poor candidate for the center of a triangle. I pointed out that one of our students had suggested – on his way out of class yesterday – that we should concentrate on vertices rather than midpoints of sides. Again, we let GeoGebra take over and looked at a compromise by constructing a line through a vertex AND through the midpoint of a side. I named this for them as the median. I also displayed that these medians seem to ALWAYS intersect in the interior of a triangle and I named this point for them as the centroid. We all agreed that this name was ‘center-y’ enough. As time ran out, at the suggestion of another student, we asked GeoGebra to construct angle bisectors. It does so, but draws an exterior line as well. They did not complain when I erased them, but I want to examine what is really happening there. It felt a little too much like I was waving away a distraction. We saw that these angle bisectors intersect in the interior as well – setting up a great debate for tomorrow about which center is the center-est. Just thrilled with how they hung together during the intro time and during the quick GeoGebra exploring. Need to commit to both HW review time tomorrow and to revisiting the blur of activity on GeoGebra. I am planning on a lab day for Thursday so that they can manipulate these ideas themselves.

In my AP Stats I tried out the German Tank problem using resources found here at the Stats Monkey site. My two classes dealt with this in pretty different ways. My smaller class (12 students today) worked in 3 groups of four. I made a mistake in responding to one of the first ideas I heard. One group decided to invoke the empirical rule and guessed that the # of tanks was their sample mean plus three standard deviations. I responded positively to them but this simply steered the other two groups into following this lead. In my other class I was smarter and quieter. Here I had four groups of four. One group invoked the empirical rule but they also pooled their three samples together. One group used the inverse normal function on their calculator seeking a point where the area was 0.999. One group added their sample minimum to their sample maximum guessing that they should be (roughly) equidistant from the extremes. The final group doubled their median guessing it should be halfway to the max. I was thrilled with the level of discussion and the variety of responses. A great step forward from yesterday’s disappointment where they largely ignored my Radiolab assignment.

I’ll count this day in the victory column for sure.


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