# Lesson 6

Similarity

## 6.1: Equivalent Expressions (5 minutes)

### Warm-up

This warm-up prompts students to use what they know about operations and properties of operations to create related expressions. While many warm-ups encourage students to work mentally and verbally, students will write their responses to this prompt. Since many different responses are possible, the task is accessible to all students and provides an opportunity to hear how each student reasons about the operations. Some different ideas that may emerge are:

• commutative property
• distributive property
• inverse operations
• adjusting factors (for example, doubling and halving)

Examples of each are given in the student response section. Students are not expected to use these terms, but highlight the terms if students do use them.

### Launch

Arrange students in groups of 2. Tell students they are writing a list of several expressions equivalent to $$10 (2 + 3) - 8 \boldcdot 3$$. Give students 2 minutes of quiet think time followed by 1 minute to discuss their responses with a partner.

### Student Facing

Use what you know about operations and their properties to write three expressions equivalent to the expression shown.

$$\displaystyle 10 (2 + 3) - 8 \boldcdot 3$$

### Activity Synthesis

Much of the discussion takes place between partners. Ask students to share any expressions that they aren't sure about, but try to resolve these and move on quickly.

## 6.2: Similarity Transformations (Part 1) (20 minutes)

### Activity

In this activity, students learn that two figures are similar when there is a sequence of translations, reflections, rotations and dilations that takes one figure to the other. Students practice discovering these sequences for two pairs of figures.

When two shapes are similar but not congruent, the sequence of steps showing the similarity usually has a single dilation and then the rest of the steps are rigid transformations. The dilation can come at any time. It does not matter which figure you start with. An important thing for students to notice in this activity is that there is more than one sequence of transformations that show two figures are similar. Monitor for students who insert a dilation at different places in the sequence. Also monitor for how students find the scale factor for the hexagons.

### Launch

Before beginning this task, define what it means for two figures to be similar.

Tell students: We have talked about how one figure can be a scaled copy of another. This relationship goes in both directions. For example, if triangle $$DEF$$ is a scaled copy of triangle $$ABC$$ with scale factor of 2 then triangle $$ABC$$ is a scaled copy of triangle $$DEF$$ with scale factor $$\frac12$$. We have learned that the transformation that creates scaled copies is called a dilation.

We say that triangles $$ABC$$ and $$DEF$$ are similar. The previous unit explored how translations, rotations, and reflections define congruent figures. The inclusion of dilations can change the size of the figure as well as its location and orientation.

We will start our investigation of similar figures by identifying sequences of translations, rotations, reflections, and dilations that show two figures are similar. Demonstrate using this example.

There are many methods to make this work. Explain at least two. First, identify the corresponding parts. Then come up with a plan to take one figure to the other. Ensure students understand, through demonstration with this example, that the work of showing two figures are similar requires communicating the details of each transformation in the sequence with enough precision. Some sample methods:

1. Method 1 ($$ABCD$$ to $$GFEH$$: Dilate, Translate, Rotate, Reflect)
1. Dilate using $$D$$ as the center with scale factor 2.
2. Translate $$D$$ to $$H$$
3. Rotate using $$H$$ as the center clockwise by 90 degrees
4. Reflect using the line that contains $$H$$ and $$F$$.
2. Method 2 ($$ABCD$$ to $$GFEH$$: Reflect, Translate, Rotate, Dilate)
1. Reflect using the line that contains $$D$$ and $$B$$.
2. Translate $$D$$ to $$H$$.
3. Rotate using $$H$$ as the center clockwise by 90 degrees
4. Dilate using $$H$$ as the center with a scale factor of 2.
3. Method 3 ($$ABCD$$ to $$GFEH$$: Translate, Rotate, Reflect, Dilate)
1. Translate $$B$$ to $$F$$.
2. Rotate using $$F$$ as the center clockwise by 90 degrees.
3. Reflect using the line that contains $$F$$ and $$H$$.
4. Dilate using $$F$$ as the center with scale factor 2.

These arguments can also be applied to figures that are not on a grid: the grid helps to identify directions and distances of  translation, 90 degree angles of rotation, and horizontal and vertical lines of reflection.

If using the print version, provide access to geometry toolkits. If using the digital version, remind students of the meaning and functionality of each transformation tool, as necessary.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the unit: similar. On this display, include the step-by-step instructions of at least 2 of the 3 given methods for creating similar polygons using transformations.
Supports accessibility for: Memory; Language

### Student Facing

1. Triangle $$EGH$$ and triangle $$LME$$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

2. Hexagon $$ABCDEF$$ and hexagon $$HGLKJI$$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

### Launch

Before beginning this task, define what it means for two figures to be similar.

Tell students: We have talked about how one figure can be a scaled copy of another. This relationship goes in both directions. For example, if triangle $$DEF$$ is a scaled copy of triangle $$ABC$$ with scale factor of 2 then triangle $$ABC$$ is a scaled copy of triangle $$DEF$$ with scale factor $$\frac12$$. We have learned that the transformation that creates scaled copies is called a dilation.

We say that triangles $$ABC$$ and $$DEF$$ are similar. The previous unit explored how translations, rotations, and reflections define congruent figures. The inclusion of dilations can change the size of the figure as well as its location and orientation.

We will start our investigation of similar figures by identifying sequences of translations, rotations, reflections, and dilations that show two figures are similar. Demonstrate using this example.

There are many methods to make this work. Explain at least two. First, identify the corresponding parts. Then come up with a plan to take one figure to the other. Ensure students understand, through demonstration with this example, that the work of showing two figures are similar requires communicating the details of each transformation in the sequence with enough precision. Some sample methods:

1. Method 1 ($$ABCD$$ to $$GFEH$$: Dilate, Translate, Rotate, Reflect)
1. Dilate using $$D$$ as the center with scale factor 2.
2. Translate $$D$$ to $$H$$
3. Rotate using $$H$$ as the center clockwise by 90 degrees
4. Reflect using the line that contains $$H$$ and $$F$$.
2. Method 2 ($$ABCD$$ to $$GFEH$$: Reflect, Translate, Rotate, Dilate)
1. Reflect using the line that contains $$D$$ and $$B$$.
2. Translate $$D$$ to $$H$$.
3. Rotate using $$H$$ as the center clockwise by 90 degrees
4. Dilate using $$H$$ as the center with a scale factor of 2.
3. Method 3 ($$ABCD$$ to $$GFEH$$: Translate, Rotate, Reflect, Dilate)
1. Translate $$B$$ to $$F$$.
2. Rotate using $$F$$ as the center clockwise by 90 degrees.
3. Reflect using the line that contains $$F$$ and $$H$$.
4. Dilate using $$F$$ as the center with scale factor 2.

These arguments can also be applied to figures that are not on a grid: the grid helps to identify directions and distances of  translation, 90 degree angles of rotation, and horizontal and vertical lines of reflection.

If using the print version, provide access to geometry toolkits. If using the digital version, remind students of the meaning and functionality of each transformation tool, as necessary.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the unit: similar. On this display, include the step-by-step instructions of at least 2 of the 3 given methods for creating similar polygons using transformations.
Supports accessibility for: Memory; Language

### Student Facing

1. Triangle $$EGH$$ and triangle $$LME$$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

2. Hexagon $$ABCDEF$$ and hexagon $$HGLKJI$$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

### Student Facing

#### Are you ready for more?

The same sequence of transformations takes Triangle A to Triangle B, takes Triangle B to Triangle C, and so on. Describe a sequence of transformations with this property.

### Anticipated Misconceptions

If students do not recall the three types of rigid transformations, refer them to the classroom display that provides an example of a rotation, a reflection, and a translation.

For the second problem, students may need encouragement to experiment moving the shapes (using tracing paper for example). If they get stuck finding the scale factor, tell them that they can approximate by measuring sides of the two figures.

### Activity Synthesis

Select students to give a variety of solutions for the different problems. Point out that there are multiple ways to do each pair and any valid sequence is allowed. Ensure that students communicate each transformation in the sequence in sufficient detail.

Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have determined the sequence of transformations that shows the polygons are similar, ask students to write a detailed sequence of the transformations on their paper. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “How did you know to translate point L to point E?”, and “How did you know to dilate the polygon by a scale factor of 3?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own explanation and learn about other ways to show polygons are similar using a sequence of transformations.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness

## 6.3: Similarity Transformations (Part 2) (10 minutes)

### Activity

This activity helps students visualize what happens to figures under different kinds of transformations. Students practice identifying which transformations might be used in the sequence of translations, rotations, reflections, and dilations in order to show figures are similar. By recognizing patterns in the image results after using certain transformations (MP8), students may be able to apply this to finding transformations for other problems. Students should pay special attention to see the connection between the orientations of the original figure and the resulting images after transformation.

Encourage students to draw rough sketches though it is also ok to use patty paper, for example, to execute the rigid motions. For the dilations, however, it could become time consuming to choose an explicit scale factor and measure carefully. Make sure, after students have worked on the first problem, to show some examples of sketches that are not exact but capture the main features of the figure.

### Launch

The figure in this task is intended to resemble a hand with all of the fingers together and the thumb sticking out. Encourage students to “sketch” the resulting images for this task (or make tracing paper available, indicating that the images do not need to be exact). They do not need to make the side lengths, angles, etc. perfect, but it should be clear where the corresponding parts of the image are and whether it is larger or smaller than the original. Other activities in this unit will ask students to be precise in their use of transformations, but the goal of this activity is to get an idea of how the different transformations affect a figure’s image.

Select a few good examples of student work (including some that have been sketched free hand) to share with the class after problem 1, in order to clearly communicate the expectation for level of precision.

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 are not using exact or precise measurements, as this activity only requires a sketch. After the first problem, invite a few students to think aloud and share their sketches to guide the rest of the individual work time.
Supports accessibility for: Memory; Organization
Speaking, Listening: MLR7 Compare and Connect. Ask students to prepare a visual display of their figures that are similar to Figure A. As students investigate each other’s work, ask students to share what transformations are especially clear in the display of similar figures. Listen for and amplify any comments about what might make the transformations clearer in the display. Then encourage students to make connections between the words “translation,” “rotation,” “reflection,” and “dilation” and how they affect the figure. Listen for and amplify language students use to describe what happens to figures under different kinds of transformations. This will foster students’ meta-awareness and support constructive conversations as they compare images of the same figure and make connections between transformations and their effects on figures.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

### Student Facing

Sketch figures similar to Figure A that use only the transformations listed to show similarity.

1. A translation and a reflection. Label your sketch Figure B.
2. A reflection and a dilation with scale factor greater than 1. Label your sketch Figure C.
3. A rotation and a reflection. Label your sketch Figure D.
4. A dilation with scale factor less than 1 and a translation. Label your sketch Figure E.

### Anticipated Misconceptions

If students choose an exact scale factor, or measure the exact angle sizes, explain that precise measurements are not needed in this task. At this point, they are just sketching similar figures.

### Activity Synthesis

Select students to display their answers for each of the questions. Ask students to notice things the answers have in common so that they can make connections to the types of transformations that might be useful in showing that two figures are similar.

Assuming that the rotations are not through an angle that is a multiple of $$360^\circ$$ and that the translations have a non-zero horizontal or vertical part, point out that:

• Dilations will create larger or smaller copies depending on the scale factor as seen in previous lessons.
• Translations will slide the figure in some direction.
• Rotations will “tilt” or “turn” the figure.
• Reflections will change the handedness so that the resulting image will look like the back of a right hand instead of the back of a left hand as in the original image.

## 6.4: Methods for Translations and Dilations (10 minutes)

### Optional activity

The purpose of this task is for students to practice showing that two shapes are similar using only a few pre-determined rigid motions and dilations.  Some students will start with triangle $$ABC$$ and take this to triangle $$DEF$$ while other start with $$DEF$$ and take this to $$ABC$$. While there is flexibility in either direction, one way of getting from $$DEF$$ to $$ABC$$ is to “undo” the moves that take $$ABC$$ to $$DEF$$.

Monitor for students who use different centers of dilation in their sequence, particularly as the first step in the sequence. Invite these students to share, highlighting the fact that two dilations with the same scale factor but different centers differ by a translation. Also monitor for students whose sequences of rigid motions and dilations are the same but in the opposite order, one set taking $$ABC$$ to $$DEF$$ and the opposite taking $$DEF$$ back to $$ABC$$. Select these students to share this important observation during the discussion.

### Launch

Again, remind students that two figures are similar if there is a sequence of translations, rotations, reflections, and dilations that takes one figure to another. Tell them that they need to find at least one way to show that triangle $$ABC$$ and triangle $$DEF$$ are similar using only the transformations they are given on their cards.

Arrange students in groups of 2. Give each group one complete set of cards.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “One thing that is the same is….”; “One thing that is different is….”; and “Another strategy to get the same result is….”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Your teacher will give you a set of five cards and your partner a different set of five cards. Using only the cards you were given, find at least one way to show that triangle $$ABC$$ and triangle $$DEF$$ are similar. Compare your method with your partner’s method. What is the same about your methods? What is different?

### Anticipated Misconceptions

Students might think that is it necessary to perform transformations in the same order or that one particular point needs to be the center. If partners choose the same methods, prompt them to try it another way that will have the same end result.

### Activity Synthesis

Invite selected students to share, highlighting methods of moving $$ABC$$ to $$DEF$$ and $$DEF$$ to $$ABC$$ which are “opposite” of one another, for example dilations with center $$P$$ and reciprocal scale factors.

As students share their responses, highlight these points:

• The scale factors for the dilations are reciprocals regardless of when the dilations are done in the sequence.
• If used, the translations are inverses of each other (eg “Translate $$A$$ to $$D$$” instead of “Translate $$D$$ to $$A$$”).
• Dilations with different centers but the same scale factor produce congruent figures that differ by a translation.
• The order in which transformations are applied can influence the result.

One important conclusion (the third bullet point) is that when you are showing that two figures are similar, you can pick any point as the center of dilation if you know the scale factor, because you can always adjust the position using a translation.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their sequence of transformations to show that triangle $$ABC$$ and triangle $$DEF$$ are similar, present an incorrect sequence of transformations. For example, “Translate from $$A$$ to $$D$$. Then dilate using center $$P$$ and scale factor 3.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss in partners, listen for students who clarify the meaning of the center of dilation. Prompt students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens when $$P$$ is the center of dilation and explain why $$D$$ should be the center of dilation. This routine will engage students in meta-awareness as they clarify how the center of dilation affects the dilated figure.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

Review the definition of similar figures and any important insights that arose during the lesson. Insights to highlight:

Two figures are similar if there is a sequence of translations, rotations, reflections, and dilations that maps one to the other. Scaled copies of figures, studied in grade 7, are all examples of similar figures. In this lesson, we found transformations that showed that two figures were similar.

We saw that there is more than one sequence of transformations that shows two figures are similar. One way to think about it is that you need to use a dilation to make corresponding side lengths the same size. The figures will be congruent after this step. Once you do that, you just need to find a sequence of rigid transformations that align the congruent figures. You can also do it the other way, by bringing the figures into alignment and then dilating one to match up with the other.

Add the term similar along with a definition and example to your classroom display such as a word wall or anchor chart.

## Student Lesson Summary

### Student Facing

Let’s show that triangle $$ABC$$ is similar to triangle $$DEF$$:

Two figures are similar if one figure can be transformed into the other by a sequence of translations, rotations, reflections, and dilations. There are many correct sequences of transformations, but we only need to describe one to show that two figures are similar.

One way to get from $$ABC$$ to $$DEF$$ follows these steps:

• step 1: reflect across line $$f$$
• step 2: rotate $$90^\circ$$ counterclockwise around $$D$$
• step 3: dilate with center $$D$$ and scale factor 2

Another way would be to dilate triangle $$ABC$$ by a scale factor of 2 with center of dilation $$A$$, then translate $$A$$ to $$D$$, then reflect over a vertical line through $$D$$, and finally rotate it so it matches up with triangle $$DEF$$. What steps would you choose to show the two triangles are similar?