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Unit 5, Lesson 18

Volume and Graphing

Geometry

18.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Aspects of Mathematical Modeling

Materials

To Gather

Scientific calculators

Activity Narrative

The purpose of this Warm-up is to remind students that if a solid is scaled by a factor of , the solid’s volume increases by a factor of . Also, the situation described in this activity sets the stage for the next activity in the lesson. As students evaluate the accuracy of a cylindrical model, they’re modeling with mathematics (MP4).

Monitor for different unit conversion strategies. Some students may start by converting 13 inches to 1.08 feet. Others may calculate the volume in cubic inches, then convert the result to cubic feet.

Launch

None

Activity

None

A company makes giant balloons for parades. They’re designing a balloon that will be a dilated version of a drum similar to the one in the image. The real-life drum’s diameter is 2 feet and it’s 13 inches wide.

<p>Photograph of a man playing a drum in a marching band </p>

What’s the approximate volume of the real-life drum in cubic feet? Round to the nearest tenth.
Suppose the drum is dilated by a scale factor of . Write an equation that gives the volume, , of the dilated drum.
What are some reasons that the actual drum volume might be different from what you calculated?

Activity Synthesis

The goal of this discussion is to ensure that all students understand why the correct formula for the dilated drum volume is .

If possible, highlight different unit conversion strategies in the discussion. Ask students what the dimensions and volume would be if the drum were dilated by a scale factor of 2, making note that they can calculate the new volume either by scaling the dimensions and calculating the volume directly, or by using the formula .

18.2

Activity

Instructional Routines

Aspects of Mathematical Modeling Graph It MLR6: Three Reads

Materials

To Gather

Graphing technology

Activity Narrative

In this activity, students use a cube root graph to analyze the relationship between a solid’s volume and its dimensions, in a situational context. In this activity, students use their model from the Warm-up to determine the diameter or volume of the balloon based on different pieces of information (MP4).

Monitor for students who use the graph to solve each problem, and for students who use other methods for one or more of the problems.

Launch

Arrange students in groups of 3–4.

Remind students that in the Warm-up, they looked at a drum with a diameter of 2 feet, a width of 13 inches, and a volume of about 3.4 cubic feet.

Provide access to devices that can run Desmos or other graphing technology.

After students have worked on the task for several minutes, pause for a whole-class discussion to make sure that all groups have successfully graphed a correct equation. If many groups are struggling, solve and graph the equation together as a class. Verify that all groups know how to locate points on the graph using a trace feature or by tapping and dragging.

MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the questions. Say, “We are going to read this problem 3 times.”

After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the questions. Ask, “What are some ways we might get started on this?”

Advances: Reading, Representing

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, present one question at a time, and monitor students to ensure that they are making progress throughout the activity.
Supports accessibility for: Organization, Attention

18.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

Students compare two cube-root graphs to draw conclusions about scaled volumes.

In the digital version of the activity, students use an applet to analyze the graphs of two equations that relate volume and scale factor. The applet allows students to move a horizontal line to make conclusions about the cube-root functions.

This activity works best when each student has access to an individual device that can run the Desmos 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 Activity Synthesis.

Lesson Synthesis

In this lesson, students have used graphs to solve problems involving geometric contexts. Here are some questions for discussion:

“Consider the volume formula for a sphere of radius : . Suppose you want to know the radius of different spheres with volumes 100, 200, 300, 400, and so on. What are some ways to find the radii that produce these volumes?” (Substitute each volume into the formula and solve for repeatedly. Or, solve for in terms of and substitute each volume into that expression. Alternatively, solve for and represent the relationship with a graph, then use the graph to estimate each radius.)
“What are the advantages and disadvantages of these different techniques?” (Solving for radius repeatedly will yield more precise results than estimating with a graph, but it requires solving a new equation for each desired volume. Solving for the radius once and then inputting different volumes is just as precise, and requires solving only one equation. However, it requires multiple computations with a calculator if decimal approximations are required. Estimating with the graph requires solving only one equation and can find solution estimates for many inputs easily. This may come at the cost of precision.)

to access Cool-down.

Student Lesson Summary

Suppose a farm has a water tank shaped like a cone. Water is poured in from the top, and a valve can be opened to let the water flow out the bottom. There is currently a small volume of water in the tank. The section of the cone that is filled has radius 1 foot and height 1 foot.

<p>Cone with point facing down. Partially colored blue to represent water. Height of water is 1 foot. Surface of water is circular. Radius of circle is 1 foot. Radius and height meet at right angle.</p>

Using the expression , we find that the volume of water in the tank is about 1.05 cubic feet because. As more water is poured into the tank, the shape of the water will be a dilation of the original small cone. An equation that expresses the volume, , in terms of the scale factor of dilation, , is . This equation can be rearranged, resulting in . Here is a graph of the rearranged equation.

<p>Coordinate plane, horizontal, volume in cubic feet, vertical, scale factor. Curve from origin climbs steeply toward 10 comma 2, slopes toward 20 comma 2 point 5, climbs slowly to 100 comma 4 point 57.</p>

The point on the graph tells us that if the farmer puts 100 cubic feet of water in the tank, the scale factor of dilation will be 4.57. That means that the height of the water would be the original height of 1 foot times the scale factor 4.57, or 4.57 feet.

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Lesson 18

Volume and Graphing

Warm-up

Standards Alignment

Building On

HSG-GMD.A.3

Addressing

HSG-MG.A.1

Building Toward

HSF-IF.C.7.b

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Activity Synthesis

Standards Alignment

Building On

HSA-CED.A.4

Addressing

HSF-IF.C.7.b

HSG-MG.A.3

Building Toward

Standards Alignment

Building On

HSA-CED.A.4

HSA-REI.D.11

Addressing

HSF-IF.C.7.b

HSG-MG.A.3

Building Toward

Student Response

Building on Student Thinking