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Unit 5, Lesson 8

Speaking of Scaling

Geometry

8.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Scientific calculators

Activity Narrative

Students calculate a scale factor given the areas of the circular base of a cone and the base of its dilation. This connects the concept of surface area dilation to cross-sections, and gives practice with non-integer roots.

Monitor for pairs of students who initially consider an answer of 20.25 but come to a consensus that the scale factor is 4.5.

Launch

Arrange students in groups of 2. Provide access to scientific calculators.

Activity

None

The image shows a cone that has a base with an area of square centimeters. The cone has been dilated using the top vertex as a center. The area of the dilated cone’s base is square centimeters.

<p>A cone. Base is a circle with area 729 pi centimeters squared. A slice near the top, horizontal to the base, is a circle with area 36 pi centimeters squared.</p>

What was the scale factor of the dilation?

Student Response

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Building on Student Thinking

Some students may struggle to find the square root of 20.25. Remind students that their calculators can find square roots, and prompt them to use an estimate to check the reasonableness of the calculator output.

Activity Synthesis

The goal is to make sure students understand the difference between the scale factor and the value that the area was multiplied by. Select a pair of students to explain their reasoning. If the pair considered 20.25 but moved to an answer of 4.5, ask how they knew 20.25 wasn’t correct.

Discuss with students how they can decide if 4.5 is a reasonable value for the square root of 20.25, considering the fact that 4² = 16 and 5² = 25. If time allows, ask students to calculate the scale factor for the solid’s volume (4.5³ = 91.125).

8.2

Activity

Instructional Routines

MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

Originals and Dilations Cards

To Gather

Scientific calculators

Activity Narrative

This activity gives students an opportunity to determine and request the information needed to infer characteristics of original and dilated solids based on one-, two-, and three-dimensional scale factors.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

Monitor for pairs that complete Problem Card 2 by using the radius and the height of the dilated cylinder in the volume formula, and for other pairs who instead apply the cube of the scale factor to the original cylinder's volume.

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

8.3

Activity

Optional

Instructional Routines

Aspects of Mathematical Modeling

Materials

None

Activity Narrative

In this activity, students are building skills that will help them in mathematical modeling (MP4). They recognize that a geometric solid can be a mathematical model of a real-life object, and have an opportunity to consider the accuracy of that model. They’re prompted to connect surface area and volume to the real-life context of metal to create the container and liquid to fill it.

Launch

Ask students what types of beverages they and their family drink from cans (sparkling water, energy drinks, coconut water, juice, tea). Tell students they’ll be playing the part of a beverage company that’s considering introducing a new product. Consider showing students several different styles of beverage cans, including mini-sizes, tall and narrow cans, and standard cans.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students understand that the materials cost of the can is related to the can’s surface area and the fill cost is related to the can’s volume.
Supports accessibility for: Memory, Organization

Lesson Synthesis

The main idea of the lesson is that if we know the factor by which the volumes, surface areas, or lengths change when a solid is dilated, it’s possible to find the factor by which the remaining values change. Here are some questions for discussion:

“Suppose you know the volumes of an original solid and its dilation. How can you find the factor by which the surface area changed?” (Divide the dilated volume by the original volume to find the factor by which the volume changed. Then take the cube root of that to find the scale factor of dilation. Finally, square that value to get the surface area scale factor.)
“Suppose you know the surface areas of an original solid and its dilation. How can you find the factor by which the volume changed?” (Divide the dilated surface area by the original surface area to find the factor by which the surface area changed. Then take the square root of that to find the scale factor of dilation. Finally, cube that value to get the volume scale factor.)
“What are some real-world applications for these concepts?” (One example is designing any kind of packaging including shampoo, cereal, and coffee—we often need to understand how a change in volume for these products affects the packaging materials and product dimensions. Other examples include designing spaces that need a certain volume like car trunks, cargo containers, and tanker trucks, and engineering objects that are expensive to paint, such as airplanes.)

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Student Lesson Summary

Suppose a solid is dilated. If we know the factor by which the surface area or volume scale changed, we can work backward to find the scale factor of dilation. Then we can use that information to solve problems.

A company sells 10 inch by 10 inch by 14 inch 5-gallon aquariums, but a museum wants to buy a 135-gallon aquarium with the same shape. The company needs to know the dimensions of the new tank and by what factor the surface area will change.

<p>illustration of a rectangular fish tank</p>

Gallons are a measure of volume. So, the volume of the tank increases by a factor of . To find the scale factor for the dimensions of the tank, calculate the cube root of 27, or 3. This tells us that the height, length, and width of the tank will each be multiplied by 3. Next, we can square the scale factor of 3 to find that the tank’s surface area will increase by a factor of 3² = 9.

	original aquarium	dilated aquarium
height (inches)	10
length (inches)	14
width (inches)	10
surface area (square inches)	760
volume (gallons)	5

Launch

Tell students that they will continue to work with scale factors for dilated solids. Display the Information Gap graphic that illustrates a framework for the routine.

Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me _____?”
Explain to your partner how you are using the information to solve the problem. “I need to know _____ because _____.”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know _____?”
Listen to your partner’s reasoning, and ask clarifying questions. Give only information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner has enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

Student Response

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Audience

Assessment Access

Lesson 8

Speaking of Scaling

Warm-up

Standards Alignment

Building On

Addressing

HSG-GMD.B.4

Building Toward

HSG-GMD.A.1

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSN-Q.A.1

Building Toward

HSG-GMD.A.1

Standards Alignment

Building On

Addressing

HSG-MG.A.1

HSG-MG.A.3

HSN-Q.A.1

Building Toward