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 (MP1).

In the digital version of the activity, students use an applet to check their predictions for the cross-sections of solids. The applet allows students to move a plane through each solid to view the cross-sections. This activity works best when each student has individual 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 Activity Synthesis.

The purpose of this discussion is to compare the cross-sections of a sphere and a cube. Invite groups of students 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.

The goal is to make sure students understand that cross-sections parallel to the base of prisms are all congruent. 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). Display the stack of quarters again.

• How do parallel cross-sections of a cylinder compare to the parallel cross-sections of a sphere? (The parallel cross-sections of a cylinder are congruent to each other. The cross-sections of a sphere are different sizes.)
• 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, but not congruent, to each other.)

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