# Lesson 3

Measuring Dilations

## 3.1: Dilating Out (5 minutes)

### Warm-up

Students drew dilations in previous lessons but this example is slightly different since the center of dilation is in the interior of the figure. This warm-up gives students an opportunity to practice dilating and do some error analysis.

### Student Facing

Dilate triangle \(FGH\) using center \(C\) and a scale factor of 3.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

Display the image for all to see and ask students why it is incorrect. (The image of \(H\) should be on the same side of \(C\) as \(H\). You can tell it's wrong because the triangles don't look like they're the same shape.)

Make sure that all students understand how to draw rays from the center point through the points to be dilated, and find the image of the points by measuring along the rays. Students will have more opportunities to dilate figures in other activities in this lesson so there is no need to have them correct any errors they made.

## 3.2: All the Scale Factors (15 minutes)

### Activity

In this activity, students get another opportunity to practice drawing dilations precisely. They also get to see the effects of different scale factors on an image. By assigning students scale factors less than and greater than one, students get a chance to see how the image is taken closer to or farther away from the center of dilation. By assigning equivalent scale factors, such as \(\frac32\) and 1.5, students get a chance to explore how images dilated with the same scale factor are congruent. They also get a chance to remind themselves of decimal representations of fractions, which will be useful when they calculate scale factors from ratios of side lengths in later lessons.

### Launch

Arrange students in groups of 2. Assign each student a scale factor from the list: \(\frac14\), \(\frac12\), \(\frac34\), \(\frac76\), \(\frac32\), \(\frac52\), 0.25, 0.5, 1.5, 2.5

*Representation: Internalize Comprehension.*Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Assign \(\frac12\) and \(\frac32\) to a pair of students who would benefit from additional support.

*Supports accessibility for: Conceptual processing; Organization; Attention*

### Student Facing

Here is a center of dilation and a triangle.

- Measure the sides of triangle \(EFG\) (to the nearest mm).
- Your teacher will assign you a scale factor. Predict the relative lengths of the original figure and the image after you dilate by your scale factor.
- Dilate triangle \(EFG\) using center \(C\) and your scale factor.
- How does your prediction compare to the image you drew?
- Use tracing paper to copy point \(C\), triangle \(EFG\), and your dilation. Label your tracing paper with your scale factor.
- Align your tracing paper with your partner’s. What do you notice?

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Student Facing

#### Are you ready for more?

- Dilate triangle \(FEG\) using center \(C\) and scale factors:
- \(\frac12\)
- 2

- What scale factors would cause some part of triangle \(E'F'G'\)’ to intersect some part of triangle \(EFG\)?

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

### Anticipated Misconceptions

If students are unclear about relative lengths, tell them to make a prediction and write a description such as “a little smaller” or “a lot bigger” before they calculate.

### Activity Synthesis

The goal of this discussion is to identify the effects of various scale factors. Collect and align several tracing papers on the original image. Invite students to make generalizations about different scenarios:

- compare scale factors which are greater and less than one
- compare equivalent scale factors with different representations (such as 0.5 and \(\frac12\))

*Speaking: MLR8 Discussion Supports.*Use this routine to help students produce statements about the scale factor of a dilation. Provide sentence frames for students to use when they generalize the effects of various scale factors, such as: “If _____ , then _____ because….” and “_____ will always _____ because….”

*Design Principle(s): Support sense-making; Optimize output (for generalization)*

## 3.3: What Stays the Same? (15 minutes)

### Activity

In this activity, students dilate quadrilateral \(ABCD\) using center \(P\) by different scale factors \(k\). They notice that not only are \(\frac{PA’}{PA}, \frac{PB’}{PB}, \frac{PC’}{PC}\), and \(\frac{PD’}{PD}\) equal to \(k\) because that is how dilations are defined, but for example, \(\frac{B’A’}{BA}\) and \(\frac{C’B’}{CB}\) are also equal to \(k\). This assertion is not proved in this course. \(\frac{PA’}{PA} = k\) is a consequence of the definition of dilation, but the fact that dilation results in figures which are scaled copies of one another by scale factor \(k\) is taken as an assertion after students make a conjecture based on their measurements. Students should be familiar with this assertion from middle school. In this activity students extend that understanding by making a distinction between the distances from points in the original figure and image to the center of dilation (determined by the definition of dilation), and lengths in the image and original figure (an unproven property of dilations).

### Launch

Arrange students in groups of 2–4. Assign each group a scale factor: 2, 3, \(\frac13\), \(\frac12\). Encourage students to split up the measuring tasks to save time.

Explain to students that although they may be used to a *ratio* referring to an association between two or more quantities, people often also use the word to refer to a quotient of two quantities in a ratio relationship. For example, we might say the ratio of juice to sparkling water in a punch is 3 to 2. We could also say that the ratio of juice to sparkling water is \(\frac32\) to 1, since this is equivalent. As a shorthand, people sometimes say the ratio of juice to sparkling water is three halves. In this activity, *ratio* is used to refer to the quotient of two side lengths, and in this unit we frequently use *ratio* as a shorthand for *quotient*.

*Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their written responses for the proof that the values of all ratios in the first table are equal to \(k\). 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, “What is the definition of a dilation?” and “How do you know that the ratio \(\frac{PA’}{PA}\) will always be equal to the scale factor \(k\)?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why all the ratios in the first table are equal to the scale factor.

*Design Principle(s): Optimize output (for justification); Cultivate conversation*

*Representation: Internalize Comprehension.*Use color coding and annotations to highlight connections between representations in a problem. For example highlight the segments in the shapes that correspond to the ratios in the calculations.

*Supports accessibility for: Visual-spatial processing*

### Student Facing

- Dilate quadrilateral \(ABCD\) using center \(P\) and your scale factor.
- Complete the table.
Ratio \(\frac{PA’}{PA}\) \(\frac{PB’}{PB}\) \(\frac{PC’}{PC}\) \(\frac{PD’}{PD}\) Value - What do you notice? Can you prove your conjecture?
- Complete the table.
Ratio \(\frac{B’A’}{BA}\) \(\frac{C’B’}{CB}\) \(\frac{D’C’}{DC}\) \(\frac{A’D’}{AD}\) Value - What do you notice? Does the same reasoning you just used also prove this conjecture?

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

The purpose of this discussion is to identify characteristics of a dilation. Invite students to explain their observations about the first table. Students might conjecture that the values of all the ratios are equal to \(k\), and can prove this using the definition of dilation. If no student attempts to justify their conjecture using the definition of dilation, remind students of the definition and ask how this explains the conjecture they made.

Then invite students to explain their observations about the second table. Students might conjecture that the values of all the ratios are still equal to \(k\). Students need to attend to precision when reading the definition of dilation (MP6) to realize that the definition of dilation doesn’t guarantee that corresponding sides from the image and original figure must have a scale factor of \(k\).

Explain to students that while it seems obvious that corresponding sides from the image and original figure must have a scale factor of \(k\), it’s actually tricky to prove. Remind them that just measuring a bunch of examples does not constitute a proof. Instead, explain that when we’ve verified something with examples and believe it seems obvious but aren’t going to prove it in this class, we can call it an assertion.

Ask students to add this assertion to their reference charts as you add it to the class reference chart:

The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. (Assertion)

## Lesson Synthesis

### Lesson Synthesis

Display the image for all to see:

Ask students where they would place points along ray \(OE\) in order to make a scaled copy of the triangle with different scale factors. Record their thinking for all to see.

- “If you wanted to make a scaled copy with side lengths twice the size of the original, where should you place \(E’\)? Explain how you know.” (16 units from \(O\) along ray \(OE\), or 8 units past \(E\) along ray \(OE\). In order for the scaled copy to be twice as big, it has to be twice as far from the center. So since \(E\) is 8 units from \(O\), make \(E’\) 16 units from \(O\).)
- “If you wanted to make a scaled copy with side lengths half the size of the original, where should you place \(E’\)? Explain how you know.” (4 units from \(O\) along ray \(OE\). In order for the scaled copy to be half as big, it has to be half as far from the center. So since \(E\) is 8 units from \(O\), make \(E’\) 4 units from \(O\).)
- “If you wanted to make a scaled copy with side lengths \(\frac14\) the size of the original, where should you place \(E’\)?” (2 units from \(O\) along ray \(OE\).)
- “If you wanted to make a scaled copy with side lengths \(\frac34\) the size of the original, where should you place \(E’\)?” (6 units from \(O\) along ray \(OE\).)

Invite students to explain what they learned about dilations and specifically what can be determined from knowing the distance \(E'\) to \(O\). (The ratio of the distances from the original to the center and the scaled copy to the center is the same as the scale factor from the original to the scaled copy.)

## 3.4: Cool-down - Describing Stretching (5 minutes)

### Cool-Down

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs.

## Student Lesson Summary

### Student Facing

We know a *dilation*** **with center \(P\) and positive *scale factor* \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.

The triangle \(A’B’C’\) is a dilation of the triangle \(ABC\) with center \(P\) and with scale factor 2. So \(A’\) is 2 times farther away from \(P\) than \(A\) is, \(B’\) is 2 times farther away from \(P\) than \(B\) is, and \(C’\) is 2 times farther away from \(P\) than \(C\) is.

Because of the way dilations are defined, all of these quotients give the scale factor: \(\frac{PA’}{PA} = \frac{PB’}{PB} = \frac{PC’}{PC} = 2\).

If triangle \(ABC\) is dilated from point \(P\) with scale factor \(\frac{1}{3}\), then each vertex in \(A’’B’’C’’\) is on the ray from P through the corresponding vertex of \(ABC\), and the distance from \(P\) to each vertex in \(A’’B’’C’’\) is one-third as far as the distance from \(P\) to the corresponding vertex in \(ABC\).

\(\frac{PA’’}{PA} = \frac{PB’’}{PB} = \frac{PC’’}{PC} = \frac{1}{3}\)

The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. In other words, If segment \(AB\) is dilated from point \(P\) with scale factor \(k\), then the length of segment \(AB\) is multiplied by \(k\) to get the corresponding length of \(A’B’\).

\(\frac{A’’B’’}{AB} = \frac{B’’C’’}{BC} = \frac{A’’C’’}{AC} = k\).

Corresponding side lengths of the original figure and dilated image are all in the same proportion, and related by the same scale factor \(k\).