In Proof Practice 4, students again focus on the given information from which they can draw logical conclusions.
I ask a student volunteer to present his/her ideas for the proof. After the student presents, I give groups 1-2 minutes to react to the presentation, asking them to keep track of the ideas they agreed with, raise questions they wonder about, and make note of any feedback they want to give regarding the proof itself or how it was presented. After students in the audience give feedback and ask questions, I revisit the proof. I use different colored whiteboard markers to surface the presenter's thinking, making connections between the ideas. I ask students to revisit their own work and to use colored pens and highlighters to make their thinking visible.
In this set of investigations, I ask groups of students to investigate points of concurrency. For each investigation, each person in the group constructs one of the following triangles--right, acute, obtuse, and isosceles--changing the type of triangle they construct for each investigation.
The group goes through four investigations (construct all the angle bisectors, perpendicular bisectors, altitudes, and medians for their assigned triangle) and compare the results. Students will ultimately see that for all types of triangles, the angle bisectors, perpendicular bisectors, altitudes, and medians are concurrent.
When groups are finished with all four investigations, I check students' work, offering an extension to groups who finish earlier. I ask students to use the incenter to inscribe a circle in each of their triangles and the circumcenter to circumscribe each of their circles.
We debrief the results of these investigations by taking notes, which includes formally naming all of the centers.
I ask the groups who finished early to project the circles they have constructed using the incenter and circumcenter for all of their triangles. I give students in the audience time to try these constructions on their own, having the presenting group assist me with circulating the room and checking their work.
When we finish discussing the incenter, circumcenter, and orthocenter, I show students acute, obtuse, right, and isosceles triangles for which I have constructed all the medians. I tell them that the centroid is the center of gravity, and show them how they can balance the triangle at this point. I then pass out these cardboard examples so students can try them out.
I pass back the Constructions Group Quiz students took during a previous lesson and make sure to talk students through how these problems were graded. For each problem, I show students the types of construction marks I expect to see by projecting my answer key, as well as the geometry symbols I expect them to use to make their ideas clear. I make sure to take lots of student questions during this time, particularly because students see multiple ways to perform the construction and want to make sure their construction shows the geometry ideas correctly.