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 helps students visualize what happens to figures under different kinds of transformations. By recognizing patterns in the image results after using certain transformations, students can make connections between the orientations of the original figure and the resulting images after transformation. They can also apply this to find transformations for other problems (MP8).

Some students may choose an exact scale factor or measure the exact angle sizes. Explain that precise measurements are not needed in this task—they are just sketching similar figures.

The goal of this discussion is for students to notice things each answer has in common so that they can make connections to the types of transformations that might be useful in showing that 2 figures are similar.

Select students to display their answers for each of the remaining problems. Here are some questions for discussion:

• “What do you notice about all of the dilations with a scale factor greater than 1?” (They all create an image larger than the original.)

• “What do you notice about all of the reflections?” (Reflections will make the resulting image look like the back of a right hand instead of the back of a left hand as in the original image.)

• “What do you notice about all of the rotations?” (Rotations will appear to “tilt” or “turn” the figure, unless the rotation is a multiple of 360 degrees, where it will look the same.)

• “What do you notice about all of the dilations with a scale factor smaller than 1?” (They all create an image smaller than the original.)

• “What do you notice about all of the translations?” (Translations will slide the figure in some direction, but the size and orientation remain the same.)

The purpose of this optional activity is for students to practice showing that 2 shapes are similar.  It is an opportunity for extra practice that not all classes or students may need. Some students will start with triangle and take this to triangle while others start with and take this to . 

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 2 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 to and the opposite taking back to .

The goal of the discussion is to make sure students understand how similarity can be shown in different ways, and how the transformations in the differing methods are related.

Invite previously selected students to share. As students share their responses, discuss:

• “What do you notice about the scale factors you and your partner used?” (The scale factors for the dilations are reciprocals regardless of when the dilations are done in the sequence.)

• “What do you notice about the translations you and your partner used? (The translations are inverses of each other: “Translate to ” instead of “Translate to ”).

• What do you notice about the centers of dilation you and your partner used?” (Any point can be chosen as the center of dilation if the scale factor is known, because the position can be adjusted using a translation.)

• What do you notice about the order of the transformations you and your partner used? (The order of the transformation will affect the result, but many different orders can be used to get the same result.)