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Unit 1, Lesson 12

Cross-Sections and Volume

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Integrated Math 3

12.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

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.

Launch

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.

Activity

None

The images show the same number of coins arranged in different ways.

<p>A photo of a stack of coins, quarters.</p> — A

<p>A photo of a stack of coins, quarters, each one slightly off center from the one below it so that the stack appears to be leaning.</p> — B

How are the two coin stacks different from each other?
Does either stack of coins resemble a geometric solid? If so, which stack and what solid?
How do the heights of the two stacks compare?
How do the volumes of the two stacks compare? Explain your reasoning.

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

The goal of this 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.

12.2

Activity

Instructional Routines

MLR5: Co-Craft Questions

Materials

To Gather

Index cards

Activity Narrative

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.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

12.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

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.

Launch

Arrange students in groups of 2.

Engagement: Internalize Self-Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to 4 out of 6 questions.
Supports accessibility for: Organization, Attention

Activity

None

Lesson Synthesis

In this lesson, students explored right and oblique solids, and related Cavalieri’s Principle to the volumes of such solids. Display this image for all to see:

Ask students:

“Can you identify a right cylinder that would have the same volume as this one?” (One option would be to have the same radius and a height of 25.)
“Is there any measurement on this diagram that you would not use when calculating the volume?” (We wouldn’t use the 30.)
“Calculate the volume of this cylinder mentally.” (The area of the base is . Multiply that by the height to get cubic units.)

In prior grades, students established that for right prisms and cylinders. Cavalieri’s Principle now allows this to be extended.

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

A prism or cylinder whose base has an area of square units and whose height is units has a volume of cubic units, regardless of the shape of the base or whether the solid is oblique. (Theorem)

to access Cool-down.

Student Lesson Summary

Suppose we have a stack of paper in the shape of a rectangular prism. Then we shift the paper so the prism slants to the side. The first is called a right prism because its sides are at right angles to its base. The second one, without right angles between the sides and the base, is called an oblique prism. The volume of the prism doesn’t change when we shift it—the amount of paper stays the same.

<p>A ream of orange paper, slanted to the right. </p>

In fact, Cavalieri’s Principle says that if any two solids are cut into cross-sections by parallel planes, and the corresponding cross-sections at all heights all have equal areas, then the solids have the same volume.

The oblique cylinder and the right prism in this image have equal volumes because they have the same height and their cross-sections at all heights have the same area of 6 square units.

<p>A cylinder with height 4. The top base is shaded. The equation B equals 6 units squared is above it. A rectangular prism with height 4 and width 3.</p>

These two pyramids also have equal volumes. Their bases are congruent, and they have the same height. Shifting a solid from right to oblique doesn’t change its volume.

Launch

Show students a tall stack of index cards. Ask:

“What geometric solid does the stack of cards resemble?” (a right rectangular prism)
“What is the shape of the base?” (a rectangle)
“What is the shape of any cross-section taken parallel to the base?” (All the cross-sections are rectangles congruent to the base.)

Now shift the cards to create an oblique rectangular prism. Ask students if the shape of the base or the shape of the cross-sections has changed. Emphasize that each cross-section is still congruent to the rectangular base. Tell students that in this task, they will be working with a prism that gets shifted in the same manner as the index cards.

Arrange students in groups of 2. Use Co-Craft Questions to give students an opportunity to familiarize themselves with the context, and to practice producing the language of mathematical questions.

Display only the problem stem and related image, without revealing the question.
Ask students, “What mathematical questions could you ask about this situation?”
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
As partners discuss, support students in using conversation and collaboration skills to generate and refine their questions, for instance, by revoicing a question, seeking clarity, or referring to their written notes. Listen for how students use language about cross-sections, equal area, and equal volume.
Invite several groups to share one question with the class and record for all to see. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify questions that focus on cross-sections, area, and volume.
Reveal the questions and give students 1–2 minutes to compare them to their own questions and those of their classmates. Invite students to identify similarities and differences by asking:
- “Which of your questions is most similar to or different from the ones provided? Why?”
- “Is there a main mathematical concept that is present in both your questions and those provided? If so, describe it.”

Activity

None

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Lesson 12

Cross-Sections and Volume

Warm-up

Standards Alignment

Building On

Addressing

HSG-MG.A.1

Building Toward

HSG-GMD.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSG-GMD.B.4

Building Toward

HSG-GMD.A.1

Standards Alignment

Building On

HSG-GMD.B.4

Addressing

HSG-GMD.A.1

Building Toward

Student Response

Building on Student Thinking