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Unit 3, Lesson 6

Connecting Similarity and Transformations

Geometry

6.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Rulers

Activity Narrative

In a subsequent lesson, students will justify why not all rectangles are similar to each other and will explore a false proof that all rectangles are similar. This activity previews that thinking, as students explain what went wrong with this attempt to dilate a square. Monitor for students who:

Notice that the figures are not the same shape.
Notice that the sides are not scaled by the same scale factor.
Notice that and are not on rays and .

Launch

None

Activity

None

<p><strong>Two diagonal line segments intersect at point P. Between two line segments, Quadrilaterals G H F E and A D C B, both with 4 right angles. G H F E looks like a rectangle, A D C B looks like a square.</strong></p>

How do you know that is not a dilation of ?

Activity Synthesis

The goal of this Activity Synthesis is for students to begin to justify why accurate dilations lead to figures with all pairs of corresponding angles congruent and all pairs of corresponding sides having the same ratio as the scale factor. They approach this from the contrapositive: If a figure does not have all pairs of corresponding angles congruent and all pairs of corresponding sides having the same ratio as the scale factor, then the figures cannot be a dilation of one another.

Invite students to share who:

Noticed that the figures are not the same shape.
Noticed that the sides are not scaled by the same scale factor.
Noticed that and are not on rays and .

If scale factor or angles are not mentioned by students, ask, “All the angles are the same, doesn’t that mean they’re similar?” (No. Angles need to be congruent and sides need to be in the same ratio.) If no student noticed that and are not on rays and , draw in the auxiliary lines. The goal is for students to specify the properties that must be true if a dilation is to exist between two figures.

6.2

Activity

Instructional Routines

Card Sort MLR2: Collect and Display

Materials

To Copy (from Blackline Masters)

Not-So-Rigid Transformations Cards

To Gather

Geometry toolkits (HS)Rulers

Activity Narrative

Students sort different pairs of figures during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the pairs of figures, but especially for categories that distinguish between congruent figures, similar figures, and neither congruent nor similar.

After doing and discussing a card sort, students are assigned one pair of figures to figure out a sequence of rigid motions and dilations that takes one figure onto the other. Students use their prior knowledge of rigid transformations and their experience carrying out dilations to help them be successful in this task.

6.3

Activity

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

None

Activity Narrative

The goal of this activity is for students to use the definition of similarity, and to practice applying to similar figures some of the skills they learned when studying congruent figures. Students practice writing a sequence of rigid motions and dilations to take one figure onto another, write accurate similarity statements about the figures, and reason about what must be true about corresponding side lengths.

Launch

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

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. (Definition)

<p>Triangles A B C and D E F with directed line segment.<br>
</p> — Translation and dilation takes onto , so

Tell students that is a similarity statement, and we read as “is similar to.”

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and to decide if they are both correct, even if they are different. Follow with a whole-class discussion.

MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to finding a sequence of transformations to show similarity. Invite listeners to ask questions and give feedback that will help clarify and strengthen their partner's ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive.
Advances: Writing, Speaking, Listening

Representation: Develop Language and Symbols. Activate or supply background knowledge. To help students recall the symbols used in the image caption for parallel and perpendicular, ask "How else does the image show that the two segments are perpendicular?" (with a right angle)
Supports accessibility for: Memory, Language

Lesson Synthesis

The goal of this discussion is to make sure that students are comfortable asserting that if all pairs of corresponding angles are congruent and all pairs of corresponding sides have lengths in the same proportion, that they could definitely find a sequence of rigid motions and a dilation that takes one triangle onto the other. Students will write more formal proofs of this idea in a subsequent lesson.

Invite students to choose a method of transforming one similar figure onto the other that they prefer, by either dilating first or using rigid motions first. Arrange students with a partner who prefers the same method.

Display a pair of triangles where one has sides twice as long as the other, and all the pairs of corresponding angles are congruent. Give students 1 minute to talk with their partner about their strategy for lining up the triangles.

<p>Similar triangles J K L and P Q R.</p>

Ask students to find a group that preferred a different strategy than theirs. Invite each group to describe to the other how they will line up the triangles and how they know their strategy will work. (Dilate the smaller triangle by a scale factor of 2. The triangles will be congruent. Then you can translate, rotate, and reflect so that the triangles line up. Alternatively, use rigid motions to line up one pair of corresponding vertices and two pairs of corresponding sides. Then dilate the smaller triangle by a scale factor of 2.)

Invite students to explain why they are convinced that if two triangles have all pairs of corresponding angles congruent and all pairs of corresponding sides similar, the triangles must be similar.

to access Cool-down.

Student Lesson Summary

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. For example, triangle is similar to triangle ().

What is a rotation and a dilation that will take onto ?

<p><strong>Two parallel horizontal line segments. Top, F D, bottom, E G, point H between two segments. Triangles D H F and E H G drawn. D H labeled 2, H F labeled 2 point 8. H G labeled 5 point 6, H E labeled 4.</strong></p>

The triangles are similar because a rotation of using center will take segment onto segment , because rotations take lines through the center of the rotation to themselves. It will also take segment onto segment for the same reason. Then will be on a ray from through , and will be on a ray from through . Because , a dilation by a scale factor of 2 and center will take to fit exactly onto . Therefore, there is a sequence of rigid motions and dilations that takes onto , and .

<p><strong>Adding on from previous image. Triangles D H F, blue, and E H G drawn. Point D prime on H E. Point F prime on H G. Triangle D prime H prime F prime drawn, green.</strong></p>

Because both rigid motions and dilations leave corresponding angles congruent, all pairs of corresponding angles in similar figures are congruent. In this example, angle is congruent to angle , angle is congruent to angle , and angle is congruent to angle .
Because rigid motions keep lengths congruent and dilations scale them at the same proportion, all pairs of corresponding sides in similar figures are in the same proportion. In this example, .

Launch

Arrange students in groups of 2.

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2, and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:

Give students 1 minute to place all the cards face up and to start thinking about possible ways to sort the cards into categories.
Pause the class, and select 1–3 students to share the categories they identified.
Discuss as many different categories as time allows.

Attend to the language that students use to describe their categories and pairs of figures, giving them opportunities to describe their pairs of figures more precisely. Highlight the use of terms like “quadrilaterals,” “congruent,” and “similar.” After a brief discussion, invite students to open their books or devices and continue with the activity.

After students have sorted their cards, assign each group a card to write transformations. Note: Do not assign the rhombi to any groups.

MLR2 Collect and Display. Collect the language that students use to sort the cards. Display words and phrases, such as "similar," "dilation," "copy," and "bigger version." During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display, as needed.
Advances: Conversing, Reading

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation, such as

“First, I _____ because _____.”
“I noticed _____, so I _____.”
“Why did you _____?”

Supports accessibility for: Language, Social-Emotional Functioning

Activity

None

Your teacher will give you a set of cards. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.
Your teacher will assign you one card. Write the sequence of transformations (translation, rotation, reflection, dilation) to take one figure to the other.
What scale factor is used to go from Figure to Figure ? What scale factor is used to go from Figure to Figure ?

Student Response

to access Student Response.

Building on Student Thinking

If students struggle to describe their transformations, remind them that they can use tracing paper to redraw the figures and add labels to the points.

If students struggle to name the scale factor for congruent figures, ask them to describe what they know about scale factors that make the image larger (the scale factors are greater than 1), then describe scale factors that make the image smaller (the scale factors are less than 1). Then, ask what scale factor might work if the image is the same size.

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Lesson 6

Connecting Similarity and Transformations

Warm-up

Standards Alignment

Building On

Addressing

HSG-SRT.A.1.b

Building Toward

HSG-SRT.A.2

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSG-SRT.A.2

Building Toward

Standards Alignment

Building On

HSG-CO.B.6

Addressing

HSG-SRT.A.2

Building Toward