The Warm-Up for this lesson asks students to make conjectures:
Is it possible to cut a tetrahedron in such a way that the cross-section is a triangle? A quadrilateral?
I want my students to discuss this puzzle as a team and explain their answers (MP3), so I give 5 minutes.
As I review the team answers, I am looking to see whether students realize (or guess):
Highlighting student work (I will have 3-6 team answers to choose from), I make sure that the class is confident in the first point, and I look for evidence that students at least suspect that the second and third points are true. If so, I congratulate them on their powers of reasoning (or observation, as the case may be). If not, I lead them with questions like these:
Displaying the Agenda and Learning Targets, I remind the class that we have been learning to visualize and describe the intersections of geometric figures. Up to now, we have focused on the intersections of simple objects (MP7). Our hard work has led us to the point where we are ready to describe the intersections of complex figures, like a polyhedron and a plane.
In this activity, students work in pairs to practice visualizing and describing the intersections of polyhedra with a plane. I display the instructions and distribute the half-sheets for the Activity...then watch.
This activity follows the Rally Coach format.
At this point, I do not expect most students to be able to visualize the cross-sections accurately, because they have not trained their "inner eye". When all students--whether acting as the primary problem-solver or as a coach--have had a chance to attempt the first problem, I call the class's attention to the front of the classroom. At this point, if a student who has described the intersection correctly is prepared to explain his or her thinking, I display the student's work using an overhead projector and build off of it. Regardless, I am planning to demonstrate how students can use their knowledge of the intersections of points, lines, and planes to visualize the intersections of complex shapes more accurately (MP3, MP7). As shown in this video, I perform the demonstration as a think-aloud.
Following the demonstration, I ask students to work on the remaining problems.
A tetrahedron has four faces, so its cross-section can have--at most--4 sides.
For homework, I assign problems #41-42 of Homework Set 2 as well as Portfolio Problem 2. Problem #41 and the portfolio problem give students additional practice in visualizing and describing the 2-dimensional cross-section of a solid. Problem #42 asks students to analyze the structure of a polyhedron in a modeling context.