Lesson 6
Scaling Solids
6.1: Math Talk: Cube Volumes (5 minutes)
Warmup
The purpose of this Math Talk is to elicit strategies and understandings students have for finding volumes of cubes. These understandings help students develop fluency and will be helpful later in this 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
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 task. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Organization
Student Facing
Find the volume of each cube mentally.
Student Response
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Anticipated Misconceptions
Students may be unsure how to cube \(\frac12\). Ask them to visualize a tape diagram. First, what is \(\frac12\) of \(\frac12\)? Next, what’s \(\frac12\) of the result? Alternatively, students may visualize a cube with edge length 1 inch, and compare its volume to the pictured cube with edge length \(\frac{1}{2}\) inch. How many of the small cubes fit in the large cube?
Activity Synthesis
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?”
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, \(2 \boldcdot 2 \boldcdot 2 = 2^3\). Ask students why we call it “cubing.”
Design Principle(s): Optimize output (for explanation)
6.2: How Do Surface Area and Volume Change with Scaling? (20 minutes)
Activity
The purpose of this activity is to explore the effect of dilation on the surface area and volume of a cube. Just like in two dimensions, the area (now surface area) is multiplied by the square of the scale factor. Volume, however, is multiplied by the cube of the scale factor.
Launch
Arrange students in groups of 2. Display the applet for all to see.
Show students a cube of side length 1. Explain that each side is 1 unit in length, so we call it a unit cube. Ask students what the area of 1 face of the cube is (1 square unit). Remind students that the surface area of a solid is the sum of the area of all its faces. Ask students to find the total surface area and volume of the cube (6 square units and 1 cubic unit).
Finally, ask students to imagine scaling the unit cube by a factor of 2. Ask students to describe the result (a cube made of 8 unit cubes; each side length measures 2 units). Point out that all 3 side lengths were multiplied by the scale factor—we are dilating all 3 dimensions. Build the cube in the applet as students give their descriptions and display the result for all to see.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
Supports accessibility for: Memory; Language
Student Facing
 Use the applet to build cubes that result from dilating a unit cube by each scale factor shown in the table. Then, complete the table with the surface area and volume of each dilated cube.

scale factor surface area in square units volume in cubic units 1 2 3 4  Suppose a unit cube is dilated by some scale factor \(k\).
 Write an expression for the surface area of the dilated cube.
 Write an expression for the volume of the dilated cube.
 Compare and contrast the expression for surface area and the expression for volume.
Student Response
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Launch
Arrange students in groups of 3–4. Provide each group with about 100 small cubes.
Show students a cube of side length 1. Explain that each side is 1 unit in length, so we call it a unit cube. Ask students what the area of 1 face of the cube is (1 square unit). Remind students that the surface area of a solid is the sum of the area of all its faces. Ask students to find the total surface area and volume of the cube (6 square units and 1 cubic unit).
Finally, ask students to imagine scaling the unit cube by a factor of 2. Ask students to describe the result (a cube made of 8 unit cubes; each side length measures 2 units). Point out that all 3 side lengths were multiplied by the scale factor—we are dilating all 3 dimensions. Build the cube as students give their descriptions and display the result for all to see.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
Supports accessibility for: Memory; Language
Student Facing
 Use unit cubes to build cubes that result from dilating a unit cube by each scale factor shown in the table. Then, complete the table with the surface area and volume of each dilated cube.
scale factor surface area in square units volume in cubic units 1 2 3 4  Suppose a unit cube is dilated by some scale factor \(k\).
 Write an expression for the surface area of the dilated cube.
 Write an expression for the volume of the dilated cube.
 Compare and contrast the expression for surface area and the expression for volume.
Student Response
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Anticipated Misconceptions
Students may struggle to write an expression for the surface area of a unit cube dilated by scale factor \(k\). Ask them how they found areas of dilated twodimensional figures and ask them if that relates to this situation.
Activity Synthesis
The goal of the discussion is to conclude that dilating a cube by a factor of \(k\) multiplies the surface area by \(k^2\) and the volume by \(k^3\).
 “What does the 6 in \(6k^2\) represent?” (It represents the surface area of a single unit cube before dilation.)
 “Why doesn‘t the \(k^3\) expression have a coefficient?” (The original volume is 1, so technically the expression is \(1\boldcdot k^3\).)
 “How does the work with area in 2 dimensions relate to this work with surface area in 3 dimensions?” (In both cases, dilating by \(k\) changes the result by \(k^2\). We can think of surface area as the area of a twodimensional net of the solid, so the concept is really the same.)
 “How do the exponents in our expressions relate to dimensions?” (When we work with area, which is a twodimensional measurement, we use squaring or an exponent of 2. When we work with volume, which is a threedimensional measurement, we use cubing or an exponent of 3.)
6.3: Scaling All Solids (10 minutes)
Activity
Students create an informal argument to explain why all solids have the property that if they’re dilated by a factor of \(k\), the volume is multiplied by \(k^3\). 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 similar to how an irregular twodimensional figure was filled with rectangles in an earlier activity.
As students create their explanations, they are constructing viable arguments (MP3).
Launch
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Supports accessibility for: Language; Conceptual processing
Student Facing
Clare says, “We know that if we dilate a cube by a factor of \(k\), the cube’s volume is multiplied by \(k^3\). It seems like that must apply to all solids, but I’m not sure how to prove it.”
Elena says, “Earlier in the unit, we showed that we can cover any twodimensional shape with rectangles, so the property that area changes by \(k^2\) when we dilate a figure by \(k\) applies to all shapes, not just rectangles. Can we do something similar here?”
 Use Elena’s line of reasoning to argue that for any solid, if it’s dilated by a factor of \(k\), the volume is multiplied by \(k^3\).
 Suppose a triangular prism has surface area 84 square centimeters and volume 36 cubic centimeters. The prism is dilated by scale factor \(k=4\). Calculate the surface area and volume of the dilated prism.
Student Response
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Student Facing
Are you ready for more?
The image shows a figure called Sierpinski’s triangle. It’s formed by starting with an equilateral triangle, then repeatedly removing equilateral triangles created by joining the midpoints of the existing triangle’s sides. The first few stages are shown.
If we continue this process of removal forever, we are left with some points that never get removed from the triangle. The remaining points are what we call Sierpinski’s triangle. At any given stage, the triangle at the top of the figure is a scaled copy of the triangle at the previous stage.
For the completed Sierpinski’s triangle, though, the top triangle is a scaled copy not of the previous stage, but of the full Sierpinski’s triangle.
 For the completed figure, what scale factor takes Sierpinski’s triangle to its scaled copy at the top?
 Based on the scale factor, what fraction of the original shaded region should be contained in the scaled copy at the top?
 The scaled copy at the top actually contains \(\frac13\) of the shading of the original. Provide reasoning that shows that this is true.
Student Response
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Anticipated Misconceptions
If students question why they’re asked to explain why volume changes by a factor of \(k^3\) but they’re not asked why surface area changes by a factor of \(k^2\), remind them that we can think of surface area as a twodimensional net. Therefore, the properties that applied to area also apply to surface area.
Activity Synthesis
The purpose of the discussion is to make sure students understand how to calculate the surface area and volume of a dilated solid. Here are some questions for discussion:
 “In the triangular prism problem, what is the value of \(k\)? How about \(k^2\)? \(k^3\)?” (4; 16; 64)
 “How do the values of \(k\), \(k^2\), and \(k^3\) relate to the dilated prism?” (All the side lengths will be multiplied by \(k\). The area of each face, and the total surface area, get multiplied by \(k^2\). The volume is multiplied by \(k^3\).)
Lesson Synthesis
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 5^{2}, and the volume is multiplied by 125, which is 5^{3}.)
 “What are some situations where a solid is filled with something different than 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.)
Ask students to record this theorem in their reference charts as you add it to the class reference chart:
When any solid is dilated using a scale factor of \(k\), all lengths are multiplied by \(k\), all areas are multiplied by \(k^2\), and all volumes are multiplied by \(k^3\). (Theorem)
6.4: Cooldown  One Cubic Foot (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
In earlier activities, we saw that if we dilate a twodimensional 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 threedimensional 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.
Since these are threedimensional shapes, we can look at both volume and surface area. The volume of the original prism is 60 cubic units because \(3 \boldcdot 4 \boldcdot 5=60\). The volume of the dilated prism is 1,620 cubic units because \(9 \boldcdot 12 \boldcdot 15=1,\!620\). The volume became 27 times larger! Why? Since the side lengths tripled, when we calculated the volume we were really finding \((3 \boldcdot 3) \boldcdot (4 \boldcdot 3) \boldcdot (5 \boldcdot 3) = (3 \boldcdot 4 \boldcdot 5)\boldcdot 3^3\). The volume was multiplied by the cube of the scale factor, or by \(3^3=27\).
Now let’s look at surface area. How do you think the surface area will 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 twodimensional shapes, the surface area of the prism changed by the square of the scale factor.
In general, when you dilate any threedimensional solid by scale factor \(k\), the surface area is multiplied by \(k^2\) and the volume is multiplied by \(k^3\).