# Lesson 5

Scaling and Unscaling

## 5.1: Transamerica Building (10 minutes)

### Warm-up

Students apply the concepts of scaling and area to the cross sections of a pyramid-shaped building.

Monitor for students who find the dimensions of the top floor then multiply to find the area, and for those who multiply the original area by the squared scale factor.

### Launch

Arrange students in groups of 2. Provide access to calculators. Give students quiet work time and then time to share their work with a partner.

### Student Facing

The image shows the Transamerica Building in San Francisco. It’s shaped like a pyramid.

The bottom floor of the building is a rectangle measuring approximately 53 meters by 44 meters. The top floor of the building is a dilation of the base by scale factor $$k=0.32$$.

Ignoring the triangular “wings” on the sides, what is the area of the top floor? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may multiply the area of the bottom floor by the scale factor, $$k$$, arriving at an area of 746 square meters. Ask these students to check their answer by finding the dimensions of the top floor and multiplying them to find the area.

### Activity Synthesis

Invite previously selected students to share their reasoning in this order: first, a student who found the dimensions of the top floor; second, a student who squared the scale factor.

Ask students how the scale factor is used in each approach. In the first approach, the scale factor is multiplied by each dimension, then the resulting dimensions are multiplied by each other:

$$(0.32 \boldcdot 53)(0.32 \boldcdot 44)$$

$$16.96 \boldcdot 14.08$$

$$238.7968$$

The second approach uses the same values, but in a different order. First, the scale factor is multiplied by itself, then by the area of the original rectangle:

$$(0.32 \boldcdot 0.32)(53 \boldcdot 44)$$

$$(0.32^2)(53 \boldcdot 44)$$

$$(0.1024)(2332)$$

$$238.7968$$

Each approach arrives at the same answer.

## 5.2: Two Viewpoints (10 minutes)

### Activity

This activity addresses the common misconception that if a figure is dilated by a factor of $$k$$, the image’s area also changes by a factor of $$k$$. As students decide which response (if either) they agree with, they are critiquing the reasoning of others (MP3).

### Launch

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How is the area of the triangle affected when it is dilated by a factor of $$k=0.25$$?” or “How do you know that the area of the dilated triangle is 6.25 square inches?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for why the area of the triangle is multiplied by $$k^2$$ when dilated by scale factor $$k$$.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Access for Perception. Read the task statement aloud. Students who both listen to and read the information will benefit from extra processing time. Check for understanding by inviting students to rephrase the problem in their own words.
Supports accessibility for: Language

### Student Facing

A triangle has area 100 square inches. It’s dilated by a factor of $$k=0.25$$.

Mai says, “The dilated triangle’s area is 25 square inches.”

Lin says, “The dilated triangle’s area is 6.25 square inches.”

1. For each student, decide whether you agree with their statement. If you agree, explain why. If you disagree, explain what the student may have done to arrive at their answer.
2. Calculate the area of the image if the original triangle is dilated by each of these scale factors:
1. $$k=9$$
2. $$k=\frac34$$

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may believe they can’t calculate the area of the dilated triangle if they don’t have its dimensions. Remind them of the two approaches highlighted in the warm-up and ask if either of those applies here.

### Activity Synthesis

Here are some questions for discussion:

• “Why didn’t we need the dimensions of the original triangle to calculate the area of the dilated triangle?” (We could multiply $$k^2$$ by the original triangle’s area.)
• “What are some different ways to deal with the scale factor $$\frac34$$?” (We can square the fraction to get $$\left(\frac34 \right)^2 = \frac{9}{16}$$, then multiply that fraction by the area. Or, we can divide 3 by 4 to get 0.75, square that to get 0.5625, then multiply that by 100.)

## 5.3: Graphing Areas and Scale Factors (15 minutes)

### Activity

In this activity, students create a graph representing the relationship between dilated area and scale factor, and use it to answer questions. They analyze the average rate of change at different parts of the graph, noting that the rate of change is not constant. As students use the graph to solve problems, they are reasoning abstractly and quantitatively (MP2).

### Launch

Draw a square with side lengths labeled 1 unit and display it for all to see. Ask students to find the area of this square (1 square unit). Tell them that you want to dilate the square to get an image with area 25 square feet. Ask students how they could calculate the scale factor needed to achieve that area (the scale factor is 5 because $$\sqrt{25}=5$$). Remind students that $$\sqrt{x}$$ is defined as the positive number that squares to result in $$x$$.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

### Student Facing

An artist painted a 1 foot square painting. Now she wants to create more paintings of different sizes that are all scaled copies of her original painting. The paint she uses is expensive, so she wants to know the sizes she can create using different amounts of paint.

1. Suppose the artist has enough paint to cover 9 square feet. If she uses all her paint, by what scale factor can she dilate her original painting?
2. Complete the table that shows the relationship between the dilated area ($$x$$) and the scale factor ($$y$$). Round values to the nearest tenth if needed. Use the applet at the end of the lesson to help, if you choose.
dilated area in square feet scale factor
0
1
4
9
16
3. On graph paper, plot the points from the table and connect them with a smooth curve.
4. Use your graph to estimate the scale factor the artist could use if she had enough paint to cover 12 square feet.
5. Suppose the painter has enough paint to cover 1 square foot, and she buys enough paint to cover an additional 2 square feet. How does this change the scale factor she can use?
6. Suppose the painter has enough paint to cover 14 square feet, and she buys enough paint to cover an additional 2 square feet. How does this change the scale factor she can use?

Use the applet to help, if you choose.

### Student Response

For access, consult one of our IM Certified Partners.

### Launch

Draw a square with side lengths labeled 1 unit and display it for all to see. Ask students to find the area of this square (1 square unit). Tell them that you want to dilate the square to get an image with area 25 square feet. Ask students how they could calculate the scale factor needed to achieve that area (the scale factor is 5 because $$\sqrt{25}=5$$). Remind students that $$\sqrt{x}$$ is defined as the positive number that squares to result in $$x$$.

Distribute graph paper to each student.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

### Student Facing

An artist painted a 1 foot square painting. Now she wants to create more paintings of different sizes that are all scaled copies of her original painting. The paint she uses is expensive, so she wants to know the sizes she can create using different amounts of paint.

1. Suppose the artist has enough paint to cover 9 square feet. If she uses all her paint, by what scale factor can she dilate her original painting?
2. Complete the table that shows the relationship between the dilated area ($$x$$) and the scale factor ($$y$$). Round values to the nearest tenth if needed.

dilated area in square feet scale factor
0
1
4
9
16
3. On graph paper, plot the points from the table and connect them with a smooth curve.
4. Use your graph to estimate the scale factor the artist could use if she had enough paint to cover 12 square feet.
5. Suppose the painter has enough paint to cover 1 square foot, and she buys enough paint to cover an additional 2 square feet. How does this change the scale factor she can use?
6. Suppose the painter has enough paint to cover 14 square feet, and she buys enough paint to cover an additional 2 square feet. How does this change the scale factor she can use?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

The image shows triangle $$ABC$$.

1. Sketch the result of dilating triangle $$ABC$$ using a scale factor of 2 and a center of $$A$$. Label it $$AB'C'$$.
2. Sketch the result of dilating triangle $$ABC$$ using a scale factor of -2 and a center of $$A$$. Label it $$AB''C''$$.
3. Find a transformation that would take triangle $$AB'C'$$ to $$AB''C''$$.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The goal is to draw conclusions about the shape of the graph representing $$y=\sqrt{x}$$. Here are some questions for discussion:

• “What equation represents the relationship between the square feet covered by the paint ($$x$$) and the scale factor ($$y$$)?” ($$y=\sqrt{x}$$)
• “You calculated the rate of change for an additional 2 square feet of paint at different parts of the graph. Were those rates the same?” (No. The rate of change was larger between 1 and 3 than it was between 14 and 16.)
• “How do the different rates of change relate to the shape of the graph?” (The graph is steeper between 1 and 3 than it is between 14 and 16.)
Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to describe the rate of change at different parts of the graph. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as area, scale factor, and rate of change. For example, ask, "Can you say that again, using the term 'rate of change'?" Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.
Design Principle(s): Support sense-making

## Lesson Synthesis

### Lesson Synthesis

The goal is to to extend the square root discussion to a situation in which the area of the original figure isn’t 1. Ask students, “Suppose the painter’s original painting is 2 square feet in area, and she has enough paint to cover 72 square feet. How can we find the scale factor she should use?” Give students 1–2 minutes of quiet work time and then time to share with a partner. (Divide 72 by 2 to find that the area gets multiplied by 36. The scale factor is then the square root of 36, or 6.)

Invite students to find possible dimensions of a painting with area 2 square feet (1 foot by 2 foot) and the dimensions of the dilated image with area 72 square feet (6 feet by 12 feet).

## 5.4: Cool-down - Miniature (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

If we know the area of an original figure and its dilation, we can work backwards to find the scale factor. For example, suppose we have a circle with area 1 square unit, and a dilation of the circle with area 64 square units. We know the circle must have been dilated by a factor of 8, because 82 = 64. Another way to say this is $$\sqrt{64}=8$$.

A graph can help us understand the relationship between dilated areas and scale factors. We can make a table of values for the dilated circle, plot the points on a graph, and connect them with a smooth curve. In this table, the dilated area is the input or $$x$$ value, and the scale factor is the output or $$y$$ value. Remember that the area of the original circle is 1 square unit, so the square root of the dilated area is the same as the scale factor.

dilated area in square units scale factor
0 0
1 1
4 2
9 3
16 4

This graph represents the equation that describes the relationship between area and scale factor: $$y=\sqrt{x}$$. Note that the rate of change isn’t constant. On the left side, the graph is fairly steep. As the area increases, the scale factor increases quickly. But on the right side, the graph flattens out. As the area continues to increase, the scale factor still increases, but not as quickly.