# Lesson 6

Connecting Similarity and Transformations

## 6.1: Dilation Miscalculation (5 minutes)

### Warm-up

In a subsequent lesson, students will justify why not all rectangles are similar to each other, and 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 the figures are not the same shape
- notice the sides are not scaled by the same scale factor
- notice that \(E\) and \(F\) are not on rays \(PB\) and \(PC\)

### Student Facing

What’s wrong with this dilation? Why is \(GHFE\) not a dilation of \(ADCB\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The goal of this 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 can’t be a dilation of one another.

Invite students to share who:

- noticed the figures are not the same shape
- noticed the sides are not scaled by the same scale factor
- noticed that \(E\) and \(F\) are not on rays \(PB\) and \(PC\)

If scale factor or angles are not mentioned by students, ask, “All the angles are the same, doesn’t that mean they’re similar?” (Angles need to be congruent *and *sides need to be in the same ratio.) If no student noticed \(E\) and \(F\) are not on rays \(PB\) and \(PC\), 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: Card Sort: Not-So-Rigid Transformations (15 minutes)

### Activity

The goal of this card sort is to surface the connections among dilations and rigid transformations as well as similarity and congruence. 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.

Monitor for different ways groups choose to categorize the pairs of figures:

- figures that are similar
- figures that are congruent
- figures that are neither congruent nor similar

Listen for reasons that students provide. Do they use language of similarity, dilation, or same shape?

### Launch

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the pairs of figures, they should work with their partner to come up with categories.

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

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for and collect the language students use to sort the cards into categories of their choosing. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “Figure \(F\) and \(G\) are similar because Figure \(F\) is a bigger version of Figure \(G\),” can be improved by restating it as, “Figure \(F\) and \(G\) are similar because Figure \(F\) is a dilation of Figure \(G\).” This will help students use the mathematical language necessary to justify their reasoning for placing a card in a category.

*Design Principle(s): Optimize output (for justification); Maximize meta-awareness*

*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: “I think this is a dilation because . . .”, “The scale factor for this dilation is greater than/less than 1 because . . .”

*Supports accessibility for: Language; Social-emotional skills*

### Student Facing

- 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.
- For all the cards that could include a dilation, what scale factor is used to go from Figure \(F\) to Figure \(G\)? What scale factor is used to go from Figure \(G\) to Figure \(F\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Find a sequence of transformations that takes Figure \(G\) to Figure \(F\). How does this sequence compare to the sequence that took Figure \(F\) to Figure \(G\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

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

### Activity Synthesis

Select groups to share their categories. Ensure that at least one set of categories distinguishes between:

- figures that are similar
- figures that are congruent
- figures that are neither congruent nor similar

Record the scale factors for dilating from Figure \(F\) to Figure \(G\), and Figure \(G\) to Figure \(F\) for each pair of figures except the rhombi. Include the right triangles and the trapezoids, which have a scale factor of 1. Ask students what they notice. (There’s a scale factor either way. The scale factors are multiplicative inverses. The congruent figures have a scale factor of 1.)

## 6.3: Alphabet Soup (15 minutes)

### Activity

The goal of this activity is for students to use the definition of similarity, and to practice applying some of the skills they learned when studying congruent figures to similar 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

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

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)

Tell students \(\triangle ABC \sim \triangle FDE\) is a similarity statement, and we read \(\sim\) 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 decide if they are both correct, even if they are different. Follow with a whole-class discussion.

*Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their written responses for the sequence of transformations that takes one triangle to the other. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the directed line segment of the translation?” and “What is the center and scale factor of the dilation?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use precise language to describe the sequence of transformations that takes one triangle to the other.

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

*Representation: Internalize Comprehension.*Activate or supply background knowledge about symbolic notation. Provide students with access to explanations of symbols used in the caption for parallel and perpendicular, and connect to the right angle symbol in the image which shows that two segments are perpendicular.

*Supports accessibility for: Memory; Conceptual processing*

### Student Facing

Are the triangles **similar**?

- Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.
- Write a similarity statement about the 2 figures, and explain how you know they are similar.
- Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students struggle to define sequences of rigid motions and dilations, review an example from the card sort.

### Activity Synthesis

Ask students to determine if their sequence of rigid motions and dilations will always work for any pair of triangles that have all pairs of corresponding angles congruent and all pairs of corresponding sides lengths in the same proportion. (Probably not, some figures would require rotations and dilations. But the general method of dilating and using rigid motions will always work.)

For students who dilated first, ask:

- How do you choose the dilation to use? (Dilate from any vertex, by the scale factor given by the ratio of a pair of corresponding sides.)
- Once you dilate, how do you know you’ll be able to use rigid motions to take one triangle to the other? (The triangles will be congruent, because their corresponding angles will remain congruent, and all their side lengths will be multiplied by the value \(k\) that you need to make them the same length as the corresponding side. Since all the corresponding angles and all the corresponding sides will match, you’ll be able to take one triangle to the other.)

For students who used rigid motions first, ask:

- How do you know rigid motion will line up parts of the figures? (Because corresponding angles are congruent, we’ll be able to line up both of the rays that meet at the vertex by translating and rotating, because you can always take an angle onto another congruent angle.)
- How do you know that dilation will then line up the remaining two points? (The sides lengths will be in the same proportion, because we’re given that information, and the angles all line up, so the triangles are dilations of each other.)

## Lesson Synthesis

### Lesson Synthesis

The goal of this discussion is to make sure 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, 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.

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.

## 6.4: Cool-down - Forward and Backwards? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

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 \(DHF\) is similar to triangle \(EHG\). What is a rotation and a dilation that will take \(DHF\) onto \(EHG\)?

The triangles are similar because a \(180^{\circ}\) rotation of \(DHF\) using center \(H\) will take segment \(HF\) onto segment \(HG\), since \(180^{\circ}\) rotations take lines through the center of the rotation to themselves. It will also take \(HD\) onto \(HE\) for the same reason. Then \(G\) will be on a ray from \(H\) through \(F’\), and \(E\) will be on a ray from \(H\) through \(D’\). Since \(\frac{H’F’}{HG} = \frac{H’D’}{HE} = \frac12\), a dilation by a scale factor of 2 will take \(D’H’F’\) onto \(EHG\), which means there is a sequence of rigid motions and dilations that takes \(DHF\) onto \(EHG\).

Since similar figures are the result of rigid motions and dilations, in similar figures, all pairs of corresponding angles are congruent, and the lengths of all pairs of corresponding sides are in the same proportion. Angle \(D\) is congruent to angle \(E\). Angle \(F\) is congruent to angle \(G\). Angle \(DHF\) is congruent to angle \(EHG\). And \(\frac{HD}{HE}\)=\(\frac{HF}{HG}\)=\(\frac{DF}{EG}\).

We use \(\sim\) as a symbol for *is similar to*, so we read \(\triangle DHF \sim \triangle EHG\) as “triangle \(DHF\) is similar to triangle \(EHG\).”