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# Cutting Planes

Lesson 13 of 15

## Objective: SWBAT describe precisely the intersection of a polyhedron with a plane. Students will understand how conjectures related to the intersections of points, lines, and planes can be used to visualize the intersection of complex figures.

## Big Idea: Breaking down the cross-section of a solid: What is the intersection of the cutting plane with this edge? With this face?

*54 minutes*

#### Lesson Open

*10 min*

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):

- That a cross-section is the intersection of a solid and a plane.
- That the intersection of a polyhedron with a cutting plane is a polygon (unless it is a point or segment).
- That the intersection of a
*face*of the polyhedron with the cutting plane is a*side*of the cross-section.

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:

- Could the cross-section be a circle or ellipse? Why or why not?
- Could the cross-section have 5 sides? 10? Why or why not?
- Imagine cutting through the tetrahedron with a knife. What does the exposed cut look like where the knife passes through the faces of the solid? Is the edge of the cut round or straight?
- In the intersection derby activity, what did the intersection of two planes look like? What if one of those objects was finite, like a disc?

I display the warm-up prompt using the Slideshow. The lesson open follows our Team Warm-up routine, with students writing their answers in their Learning Journals.

**Goal-Setting**

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.

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#### Describing Intersections

*40 min*

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.****

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The Lesson Close follows our Team Size-Up routine. The prompt asks whether it is possible to cut a tetrahedron in such a way that the cross-section is a pentagon. This is a check for understanding:

- Do students understand that the intersection of a polyhedron with a cutting plane is a polygon (unless it is a point or segment).
- Do students understand that the intersection of a
*face*of the polyhedron with the cutting plane is a*side*of the cross-section.

A tetrahedron has four faces, so its cross-section can have--at most--4 sides.

**Homework**

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.

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- LESSON 2: Unto the 4th Dimension
- LESSON 3: New Directions
- LESSON 4: Angle Management
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