In this activity, students conclude that an oblique prism and a right prism that have the same height and bases of equal area have the same volume, because their cross-sections at all heights have equal area. This is Cavalieri’s Principle. It will be used in developing the pyramid volume formula.

In the digital version of the activity, students use an applet to compare the cross-sections of oblique and right prisms. The applet allows students to use sliders to change a right prism into an oblique prism and analyze both cross-sections side by side. 

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.

This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?" The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.

The goal of this discussion is for students to understand that if two solids of the same height have equal-area cross-sections at all heights, then the solids have equal volumes. Remind students about the stack of index cards from the activity Launch. Ask them:

• “When we shifted the stack of index cards so that it was oblique, did the volume change? Why or why not?” (No. The stack has the same number of cards no matter how it’s arranged.)
• “How can you phrase your explanation in terms of cross-sections?” (The individual index cards are like cross-sections. Shifting them sideways doesn’t change their area, so the total volume doesn’t change.)
• “What if we had a stack of circular cards, and each card had the same area as the rectangular cards in this stack? How would its volume compare to this rectangular card stack’s volume?” (The volumes would be the same. Each circular card has the same area as the rectangular card, so when we stack up equal numbers of them, we get equal volumes.)

Tell students that the basis of this idea was first proven by Liu Hui (LEH-oh HWEE), a Chinese mathematician, in the 3rd century. Later, in 1647, Bonaventura Cavalieri, an Italian mathematician, published a proof of the version we use today, so it’s called Cavalieri’s Principle. The proof is beyond the scope of this course.

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

Students may believe that the oblique cylinder with a slant height of 10 and the right cylinder with a height of 10 is just like the stack of quarters. Provide students with a stack of quarters and a millimeter ruler. Ask them to measure the heights of different cylinders. The oblique cylinder maintains its original height, and the slant height increases as the quarters slide apart from each other.

The purpose of this discussion is to solidify the notion that objects with equal-area cross-sections and the same height have the same volume, regardless of how “oblique” the solids are or if the cross-sections have different shapes.

Invite a student who used logical deduction to explain why the solids in the first question have the same volume. Ask students who calculated the volume to verify the claim numerically. If it doesn’t come up, ask students how Cavalieri’s Principle relates to these problems. It’s not critical that students memorize the name of the principle, but they should know how the concept relates to this task.

Some students may not be convinced that the cross-sections don’t change shape when the figure is made oblique. Remind them of the index cards and the stack of coins, and ask them how those examples relate to this situation.

Students may believe that the cross-sections of the oblique prism are parallelograms. Ask them to think back to the two stacks of coins and the shifted stack of index cards, and to consider whether the shape of the coins or the cards changed when the stack was shifted.