Lesson 10

In this warm-up, students encounter the concept of right and oblique cylinders. They begin thinking about volume relationships in terms of cross sections. This helps prepare them to give informal arguments for cylinder, cone, and pyramid volumes.

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner. Follow with a whole-class discussion.

The goal of the discussion is for students to recognize that the two stacks of coins have the same volume, and to learn vocabulary words to describe each type of solid.

Tell students that the coin stack A resembles a right cylinder, while coin stack B resembles an oblique cylinder. Challenge students to create definitions for these terms: a right cylinder has its curved face at right angles to its base, while the oblique cylinder has non-right angles.

Emphasize that the heights and volumes of the two stacks are the same.

In this activity, students conclude that an oblique prism and a right prism that have the same height and whose bases are 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.

This activity works best when each student has access to devices that can run the GeoGebra 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.

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.

The goal of the discussion is for students to understand that if two solids of the same height have equal-area cross sections at all heights, 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 this idea was developed by a mathematician named Bonaventura Cavalieri, so it’s called Cavalieri’s Principle. This principle can be proven, but the proof is beyond the scope of this course. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Cavalieri’s Principle: If two solids are cut into cross sections by parallel planes, and the corresponding cross sections on each plane always have equal areas, then the two solids have the same volume. (Theorem)

In this task, students develop their ability to analyze solids of different obliqueness and cross-sectional shape. They use Cavalieri’s Principle to identify solids of equal volume. This concept will be helpful in upcoming lessons when students develop a formula for the volume of a pyramid.

As students work, monitor for groups that calculate actual volumes, and for those who use the vocabulary of cross sections or other logical deduction methods.

Students have the opportunity to look for and make use of structure (MP7) as they identify fundamental characteristics of solids such as base area and height.

Arrange students in groups of 2.

The purpose of the 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.

As each pair of solids is discussed, invite students who have used different strategies to present their work. For example, students may have calculated volumes, or they may have used the language of cross sections and heights. If it doesn’t come up, ask students how Cavalieri’s Principle relates to these problems, as the concept of equal-area cross sections at all heights will be used in upcoming activities. It’s not critical that students memorize the name of the principle, but they should know how the concept relates to this task.