The goal of this activity is to help visualize cross-sections of a three-dimensional object. One way to do this is to cut a solid object and use one or both of the pieces to stamp the resulting cross-section onto paper. This helps students make sense of the two-dimensional shape that results from cutting a three-dimensional object (MP1). During the launch of this activity, students see a demonstration of cutting a fruit or vegetable and are asked to describe the shape of the cross-section. Students are then asked to describe the shape of a cross-section of a three-dimensional object given to them in the task statement.

As students work on the task, monitor for students who can describe the two-dimensional shape produced from each cross-section described.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their descriptions of the cross-sections. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. 

Select previously identified students to describe the shapes of cross-sections of the objects.  As students describe the cross-sections, consider displaying this applet for all to see:

The Geogebra applet "What's the Cross-Section?" is available here: https://www.geogebra.org/m/HkyGsPVW

If time allows, here are some additional questions for discussion:

• “How do the cross-sections in the different objects compare to one another?” (One is a scaled copy of the other.)
• “How do the cross-sections in each object compare to its own base?” (In the cube, the cross-section is the same as the base, in the pyramid, the cross-section is a scaled copy of the base.)

In this partner activity, students take turns visualizing cross-sections. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3).

The purpose of this discussion is to explore the cross-sections that are made when shapes are cut in different ways. Select 2–3 groups to share one of their sets of cards and how they grouped the cross-sections. Discuss as many different groups of cards as the time allows.

If not mentioned by students, explain that there are a few ways to sort the cards:

• Based on the solid object that is being cut. (rectangular prism, triangular prism, square-based pyramid, triangle-based pyramid)
• Based on the type of cross-section made by the cuts. (parallel to the base, perpendicular to the base, oblique to the base)
• Based on the shape of the cross-section.

Explain to students that it is possible to create other cross-section shapes by cutting these objects in other ways.

In this activity, students are given pictures and descriptions of planes cutting prisms and pyramids. Students are asked to draw cross-sections freehand, but this is not a skill that is required in order for students to be able to describe two-dimensional shapes created from cross-sections—which is why this is an optional activity. Some pictures are of a moving plane. Students describe how the cross-section changes as the plane moves.

In the digital version of the activity, students use an applet to draw cross-sections of shapes. The applet allows students to change the plane of the cross-section dynamically. This activity works best when students have access to the applet because they will benefit from seeing the cross-sections in a dynamic way. If students don't have access, displaying the applet for all to see would be helpful during the synthesis.

Adapted from applets created in GeoGebra by Anthony C.M. OR.

Select students to share their drawings and descriptions. Consider asking some of the following questions:

• “How did you figure out the shape of the cross-section?”
• “What helped you visualize the shape?”
• “Were any of the shapes you drew here similar to the shapes you described in the previous activity?”

The purpose of the discussion is to relate the shapes of the graphs to the shapes of the containers. The key takeaway is that cross-sections parallel to the base of each shape determine how fast the water level rises. For the hexagonal prism, there is a linear function that relates the volume and the height of the water. For the square pyramid, the function is not linear because its cross-sections are not congruent..

Consider asking:

• “Did the shapes of the graphs match your predictions?” (Yes, the graph for Container A was a straight line. No, the graph for Container B curved downwards instead of upwards.)
• “What is the same about the two graphs? What is different?” (Both graphs are increasing from left to right. The graph for Container A is linear, while the graph for Container B is not.)
• “Which function is a linear function?” (The height of the water in Container A is a linear function of the volume of the water in Container A.)
• “Why is the graph of the function for the prism linear?” (All the cross-sections of the prism, parallel to the base, are congruent. Therefore the height of the water increases at a constant rate.)
• “What is the rate of change for this linear function?” (The height of the water in the prism increases 0.6 cm for every 10 ml of volume, or 0.06 cm per ml.)
• “Why do you think the graph of the function for the pyramid is not linear?” (The cross-sections are small near the bottom of the pyramid and get larger as the water level increases. Adding the same volume of water increases the height by less when the cross-section is larger.)