Lesson | Loading...

Unit 5, Lesson 6

Scaling Solids

Geometry

6.1

Warm-up

10 mins

Math Talk: Cube Volumes

Standards Alignment

Building On

Instructional Routines

Math Talk

MLR8: Discussion Supports

Materials

None

Activity Narrative

This Math Talk focuses on finding volume. It encourages students to think about efficient strategies to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students analyze the effect of scaling on cube volumes.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they relate the word “cube” to both exponents and geometric solids.

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Activity

None

Student Task Statement

Find the volume of each cube mentally.

$<p>Drawing of a cube, width, length, and height each measure fraction 1 over 2 centimeters.</p>$

Student Response

to access Student Response.

Activity Synthesis

To involve more students in the conversation, consider asking:

“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

Remind students that when we multiply a number by itself 3 times, that’s called "cubing" the number. We can use an exponent to indicate cubing. For example, . Ask students why we call it “cubing.”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:

First, I _____ because _____.
I noticed _____, so I _____.

Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

6.2

Activity

15 mins

How Do Surface Area and Volume Change with Scaling?

Instructional Routines

MLR2: Collect and Display

Materials

6.3

Activity

10 mins

Scaling All Solids

Instructional Routines

None

Materials

None

Activity Narrative

Students create an informal argument to explain why all solids have the property that if they’re dilated by a factor of , the volume is multiplied by . Then students practice calculating the surface area and volume of a dilated solid.

Students aren’t expected to use formal language or symbolic representations in their arguments. The important part is to imagine a solid getting filled by cubes of different sizes, in a manner similar to how an irregular two-dimensional figure was filled with rectangles, in an earlier activity.

As students create their explanations, they are constructing viable arguments (MP3).

Lesson Synthesis

In this lesson, students studied the effect of dilation on the surface area and volume of solids. Here are some questions for discussion:

“If we dilate a rectangular pyramid by a factor of 5, how do the side lengths change? How does the surface area change? How about the volume?” (The side lengths are all multiplied by 5. The surface area is multiplied by 25, which is 5², and the volume is multiplied by 125, which is 5³.)
“What are some situations in which a solid is filled with something different from its surface material? What makes up the interior and the surface in these situations?” (Sample responses: A can of soup has metal on its surface and soup filling its interior. A tire has rubber on its surface and air inside. Phone screens are related to surface area, while batteries, processors, and other components are related to volume.)

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

When any solid is dilated using a scale factor of , all lengths are multiplied by , all areas are multiplied by , and all volumes are multiplied by . (Theorem)

<p>Two rectangular prisms after dilation.</p>

to access Cool-down.

Student Lesson Summary

In earlier activities, we saw that if we dilate a two-dimensional shape, the area of the dilated shape is the area of the original shape multiplied by the square of the scale factor. What happens when we dilate three-dimensional solids?

Here is a rectangular prism with side lengths 3, 4, and 5 units. When we dilate the prism using a scale factor of 3, the lengths become 9, 12, and 15 units.

<p>A rectangular prism with height 4, length 5, and depth 3. Another rectangular prism with height 12, length 15, and depth 9.</p>

Because these are three-dimensional shapes, we can look at both volume and surface area. The volume of the original prism is 60 cubic units because . The volume of the dilated prism is 1,620 cubic units because . The volume became 27 times larger! Why? Because the side lengths tripled, when we calculated the volume we were really finding . The volume was multiplied by the cube of the scale factor, or by .

Now let’s look at surface area. How will the surface area change when the prism is dilated?

In the original prism, 2 faces have area 12 square units, 2 have area 20 square units, and 2 have area 15 square units for a total surface area of 94 square units. The corresponding faces of the dilated prism have areas 108, 180, and 135 square units. The surface area of the new prism totals 846 square units, 9 times that of the original. Just like the area of two-dimensional shapes, the surface area of the prism changed by the square of the scale factor.

In general, when we dilate any three-dimensional solid by a scale factor of , the surface area is multiplied by , and the volume is multiplied by .

Have feedback on the curriculum?

Help us improve by sharing suggestions or reporting issues.

Submit Feedback

Student Task Statement

Complete the table with the surface area and volume of each dilated cube. Then write expressions that give the surface area and volume when the scale factor is .

Use the unit cubes to help, if you choose.

scale factor	surface area in square units	volume in cubic units
1
2
3
4

Lesson | Loading...

Navigation

Gradeband

Course

Unit

Lesson

Settings

Audience

Assessment Access

Lesson 6

Scaling Solids

Warm-up

Math Talk: Cube Volumes

Standards Alignment

6.G.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Activity Synthesis

Activity

How Do Surface Area and Volume Change with Scaling?

Instructional Routines

Materials

Activity

Scaling All Solids

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Building on Student Thinking

Are You Ready for More?

Activity Narrative

Launch

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

HSF-IF.C.7.b

HSG-GMD.A.1

Standards Alignment

HSA-SSE.A.1.a

HSF-IF.C.7.b

HSG-GMD.A.1

Standards Alignment

HSN-Q.A.1

HSF-IF.C.7.b

HSG-GMD.A.1

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?