Lesson 2

The purpose of this activity is for students to visualize what a cross section might look like and then test the prediction by observing the result of slicing through a solid. Cylindrical food items, such as cheese or carrots, are convenient examples.

Arrange students in groups of 2. Tell students that a cross section is the intersection between a solid and a plane, or a two-dimensional figure that extends forever in all directions. Using a cylindrical food item such as cheese or carrots, or another cylindrical object, demonstrate that slicing a cylinder parallel to its base produces a circular cross section.

Then, give students quiet work time and then time to share their work with a partner.

Students may not consider non-horizontal or non-vertical cross sections at first. Remind them that a cross section is the intersection of any plane with a solid—the plane doesn't have to be vertical or horizontal.

Ask students to share their predictions of what the cross sections will look like. Demonstrate slicing each cylindrical food item according to student instructions to see several examples.

In this activity, students continue to develop familiarity with three-dimensional solids and their cross sections. Students use spatial visualization to predict what cross sections might look like and then test their predictions.

This activity works best when each student has access to devices that can run the applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.

If using the paper and pencil version of this activity and students are stuck, suggest they slice their solids at different angles and locations to see if different cross sections are generated.

Invite groups of students with different solids to share their list of cross sections with the class. Ask students:

• “Were there any cross sections that caught you by surprise?” (It was surprising that a cube can have cross sections that are triangles, quadrilaterals, pentagons, and hexagons.)
• “Compare and contrast the different cross sections of a sphere.” (All the cross sections were circles, but they were different sizes.)
• “How are a cube’s cross sections different from a sphere’s?” (The cube has many differently-shaped cross sections, while the sphere’s cross sections are all circles.)

In the last activity, students started with solids and identified various cross sections. In this activity, students view three-dimensional slabs of a solid between parallel cross sections and try to determine what the original solid was. Being able to visualize the relationship between a solid and its cross sections is important to later work on Cavalieri’s Principle.

Ask students, “What solid would a stack of all the same coins create?” Display a stack of quarters and note that it creates the shape of a cylinder. Then display, in order, a quarter, a nickel, a penny, and a dime. Ask, “What solid would a stack of coins decreasing in size create?” Make a stack with a few of each type of coin to make a solid that resembles a cone.

Ask students to share their predictions for what solids are formed. Then display these images for all to see.

Now focus students’ attention on cross sections that are taken parallel to a solid’s base (for those solids that have bases). Ask students how cross sections can be used to differentiate between prisms and pyramids. (The cross sections of prisms taken parallel to the base are congruent to each other. The cross sections of pyramids taken parallel to the base are similar to each other.)