Flatland Encounters

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SWBAT describe 2-dimensional cross-sections of 3-dimensional figures. Students will understand the meaning of an intersection.

Big Idea

Dipping solids in water to simulate the intersection of a solid with a plane. What part of Spherious is "in" Flatland right now?

Lesson Open

5 minutes

This lesson is designed to give students hands-on experiences that will help them to visualize 2-dimensional cross-sections of 3-dimensional solids, while coming to see that geometric figures intersect in predictable ways.

Team Warm-up

Using the slideshow, I display the Warm Up prompt for the lesson as the bell rings.  The prompt asks students to recall how a 3-dimensional sphere looked to the inhabitants of Flatland, a 2-dimensional world, as it passed through.  The question refers to the film students watched in an earlier lesson.

After reviewing the team answers, I tell the class that the part of a 3-dimensional solid that lies "in" Flatland at any moment is called the intersection of the solid and the plane.  It is also called the "cross-section" of the solid.  I ask my students to move to where they can see the surface of a tank of water at the front of the classroom--I use a fish tank that I bought at a garage sale and keep in the classroom for this unit. Using the tank, I simulate a sphere passing through a plane by slowly submerging a clear plastic sphere in the water.  The outline of the sphere where it contacts the surface of the water shows the cross-section of the solid at different levels.

I point out that the circular cross-section of Spherious that Flatlanders see consists of points that are part of both Spherious and Flatland at the same time.  This is the meaning of an intersection.

The warm-up follows our Team Warm-up routine, with students writing their answers in their Learning Journals.


Displaying the Agenda and Learning Targets, I tell the class that today we will be investigating the intersections of different solids with a plane.  We will use the story of Flatland to add a little fun...and because thinking of the part of a solid that is "in" Flatland any given moment is a good way to think of an intersection.

Investigating Intersections

45 minutes

In this activity, students practice visualizing the intersections of different solids with a plane.  They check their predictions by submerging clear plastic models of geometric solids in a container of water.  The outline of the solid where it contacts the surface of the water simulates the cross-section of the solid.

I stress to students that they need to practice visualizing the intersection first, using clues in the structure of the solid (MP7).  For example, where the face of a polyhedron intersects the surface of the water, the intersection is a... line segment.  Seeing patterns like this will help them understand the rules that govern intersections.  Later, they will be doing this without a model to check their predictions.

This activity follows a Team Jigsaw format, with members of each team filling different roles.

To launch the activity, I distribute the handout and display the instructions using the slideshow.   I ask for a student in each cooperative learning team to read the handout out loud to the team.   While the students are reading the instructions, I give a plastic solid to each team, along with the remaining resources for the activity: PiB Surveillance Manual, Surveillance Record A, and Surveillance Record B.  

I use a set of 4 plastic geometric solids: cube, square pyramid, cone, and cylinder.  I also have a plastic half-cube for use in a class with more than 4 teams, and I have added a model of an octahedron, which I created by printing the net of an actahedron on transparency film.  To avoid bottlenecks, I need one solid per team.  I set a limit of 7 minutes with each solid and display a digital timer.  Teams pass their solid to the next team in number order.

For containers of water, I use inexpensive plastic serving savers, one per team.  For demonstrating to the class, I use a medium-sized fish tank (which I bought for $5 at a garage sale).  I invite students to use the tank when they want to experiment a little more with the objects, since the serving saver containers don't give a lot of room. 

For teams that finish early, I offer a challenge: describe how to position a cube in a plane, so that the intersection is a 6-sided polygon.  If students can successfully describe how it can be done, I give them a chance to demonstrate using the fish tank.

Lesson Close and Homework

4 minutes

Displaying the Lesson Close prompt, I ask students to explain how cross-sections are related to intersections.  Students often think of a cross-section as the exposed face formed by cutting a solid.  I want to see if students understand that a cross-section is simply the intersection of a 3-dimensional solid with a 2-dimensional plane.  This activity follows our Team Size-Up routine.



For homework I assign problems 29-31 of Homework Set 2.  Problem #29 is a review of the Flatland Encounters students completed in class.  Problem #30 revisits solids of revolution, this time using a few real-world objects.  Problem #31 provides additional practice in vocabulary and geometric notation.