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. 

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 and it does not matter which figure is the original. 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.

In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). As students revise their writing, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). 

In the digital version of the activity, students use an applet to perform transformations of a figure on a grid. The applet allows students to test transformations dynamically and precisely. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the first question. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “ makes sense, but what do you mean when you say . . . ?”

• “Can you say more about . . . ?”

• What do you mean when you say . . . ?

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, invite select students to share their sequence for the first problem. Emphasize that there are multiple ways to show the two triangles are similar and any valid sequence is allowed. Ensure that students communicate each transformation in the sequence using precise mathematical language. Here are some questions for discussion:

• “How have the methods shared by your classmates been alike or different?” (All methods required a dilation. Some methods went from triangle to triangle and some went the other way.)

• “What pieces of information are necessary to completely describe a dilation?” (a scale factor, a center of dilation, and an object being dilated)

This activity focuses on common misconceptions about similar figures. Two polygons with proportional side lengths but different angles are not similar, and two polygons with the same angles but side lengths that are not proportional are also not similar. Reasoning through these problems will help further student thinking about when two polygons are not similar and give students an opportunity to critique the reasoning of others (MP3).

Some students may think the side lengths must be different in order for 2 figures to be similar, but a dilation does not need to be used in the sequence of transformations to show similarity. Prompt students to recall the Warm-up where students saw that congruent figures are always similar.

The goal of this discussion is to make sure students understand that for similar figures, corresponding angles are congruent and corresponding side lengths are proportional. 

Discuss with students:

• “In the question about the square and the rhombus, both polygons had all the same side lengths but they still were not similar. What does this tell us about similar polygons?” (If corresponding angles are not congruent, then the figures cannot be similar.) 

• “In the question about the rectangles, both polygons had all the same angles but they still were not similar. What does this tell us about similar polygons?” (If the side lengths are not all scaled by the same value, then the figures cannot be similar.)

In this activity, students practice identifying similar figures. Each student has a card with a figure on it and they identify someone with a similar (but not congruent) figure. Students work together to construct an argument for why their two figures are similar (MP3).  

Monitor for students using these methods to identify a partner:

• Process of elimination (figures with different angle measures cannot be similar to each other)

• Looking for the same kind of figure (rectangle, square, rhombus)

• Looking for a the same kind of figure with congruent corresponding angles

• Looking for a the same kind of figure with corresponding sides scaled by the same scale factor 

The purpose of this discussion is to highlight different strategies for finding similar figures. For example, students can use the process of elimination (one shape is a rectangle and the other is not) or actively look for certain features (a rhombus whose angle measures are 60 degrees and 120 degrees).

Direct students' attention to the reference created using Collect and Display. Ask students to share how they determined that two figures were similar. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. 

Discuss with students:

• “Were the side lengths important?” (For some figures, such as the rectangles, the side lengths were important.)

• “Were the angles important?” (Yes. Several of the figures had different angles and that meant they were not similar.)

• “How did you know when two figures were not similar? (The shapes were not the same. Corresponding angles didn’t match. Corresponding sides were multiplied by different scale factors.) 

Some students may notice that all squares are similar since they have 4 right angles and proportional side lengths. If not mentioned by students, show how the first problem gives an example of rhombuses that are not similar to one another.