Rotations, perpendicular bisectors, and constructions

Trying to catch up three days in this post.

Most of the class did not come up with any ideas about how to find reflection lines or the center of rotation. A couple of students had an idea about finding a circle that passed through the pre-image and image points and looking at where the center of the circle was located. It was an interesting idea. I pointed out that any sized circle could be used and demonstrated how this would work. I didn't pursue the relationship of centers of circles of the same size that passed through the associated points. Could be an interesting investigation. Perhaps during a circle-focused unit, but not now.

I had anticipated this and modeled problem-solving thinking. I basically walked through thinking that rotations and reflections were somehow connected. Maybe I could start with reflections that I could create and see where this led me. After reflecting all three points I saw that the reflection lines intersected. No big surprise there, except that all three intersected at the same point. This was interesting. I had the class check this out and they found that their reflecting lines also intersected at a single point.

In taking a step back, I was able to remind students that the segment connecting pre-image and image points and the reflecting line were perpendicular to each other and the reflecting line bisected the segment between pre-image and image point. The reflecting line was a perpendicular bisector.

With this, I transitioned to how do we construct perpendicular bisectors. I gave students a series of constructions: copy segment, copy angle, segment bisector, perpendicular line, perpendicular line, parallel line through point not on a line, angle bisector, equilateral triangle, 30^o angle, and 45^o angle.

I walked around and helped students struggling with these and let those who were on it to just go. Overall the class stayed engaged and worked through these constructions. I provided little direct instruction except for those who were struggling. Since each construction had two versions, I was then able to have struggling students practice on the second instance to be sure they go it.

From there we moved into geometric proofs. I provided a sheet of definitions, postulates, and theorems (angle addition, segment addition) that could be used. The class was given a series of proofs to work through. The first five had written plans, so students were to follow the plan and write the proof. I walked through the first proof with the class and they worked through the second proof.  Their homework was to work through the next three proofs that had plans written out.

We'll continue with proofs next class.