Lesson 15

The purpose of this Math Talk is to elicit strategies and understandings students have for calculating volumes of solids. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to do more complex volume calculations.

In this activity, students have an opportunity to notice and make use of structure (MP7). To successfully evaluate volumes mentally, students must carefully consider the order in which to perform the required arithmetical operations. The structure of the number system, in particular, prime factorization, helps direct efficient choices.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

When calculating the cone’s volume, students may multiply the radius measurement by \(\frac 13\) before squaring. Remind them of the order of operations convention that says to evaluate exponents prior to performing multiplication.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

Students practice finding volumes of pyramids and prisms in problems that require the Pythagorean Theorem or trigonometry. When students articulate their strategies in advance of their calculations, they are making sense of a problem (MP1).

Invite students to share how they knew which measurements needed to be calculated and how they chose calculation strategies. When was the Pythagorean Theorem helpful, and when did they need to use trigonometry? Ask students to describe the easiest and most difficult aspects of this task.

This task combines concepts of decomposition, cylinder and cone volume formulas, and solids of rotation.

If students choose to use 3D graphing technology, reviewing the Axis of Rotation lesson may be helpful. Making this technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students may struggle to visualize and draw the solid of rotation. Suggest that they divide the two-dimensional figure into two pieces horizontally and think about what each would look like rotated using the vertical axis. If necessary, they can draw the two resulting solids (cone and cylinder) separately.

Ask students to compare and contrast the cylinder and the cone. They each have the same radius measurement, but their heights are different.

Invite students to summarize the kinds of solids that can be traced out through rotating a two-dimensional figure: spheres, cones, and cylinders can be created through spinning half-circles, triangles, and rectangles, but pryamids and prisms can’t be created through rotation. In general, a solid created through rotation will have circular cross sections.