Cutting Planes

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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?

Lesson Open

10 minutes

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. 

Describing Intersections

40 minutes

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.

Lesson Close and Homework

4 minutes

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:

  1. Do students understand that the intersection of a polyhedron with a cutting plane is a polygon (unless it is a point or segment).
  2. 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.