In this activity, students investigate polygons on a grid to determine whether or not they are scaled copies. The sides of the polygons are diagonal, so the side lengths cannot be easily determined using the grid.

First, students measure the angles using a protractor to see if corresponding angles have the same measure. Next, they find other corresponding distances that are not side lengths and see that these distances are related by a consistent scale factor. Students learn that in scaled copies, any corresponding distances—not limited to lengths of sides or segments—are related by the same scale factor.

Students must take care when they identify corresponding parts. As students work, urge them to attend to the order in which points or segments are listed (MP6).

If students are not sure what to make of the values in the table, encourage them to consider the corresponding distances of two figures at a time. For example, ask: “What do you notice about the corresponding vertical distances in and ? What about the corresponding horizontal distances in those two figures?”  

Display the completed tables for all to see. To highlight how all distances in a scaled copy (not just the side lengths of the figure) are related by the same scale factor, ask questions such as:

• “How do the distances in compare to the distances in ? Do the corresponding distances share the same scale factor?” (The distances in are each times the corresponding distance in .)
• “How do distances in compare to the distances in ? Do the corresponding distances share the same scale factor?” (The distances in are each times the corresponding distance in .)

The purpose of this activity is for students to see that figures may not be scaled copies, even though they have either corresponding angles with equal measures or corresponding distances multiplied by the same scale factor. This shows that to determine whether one figure is a scaled copy of another, we have to check both the corresponding angles and the corresponding distances.

As students work, monitor for convincing arguments about why one polygon is or is not a scaled copy of the other. Ask them to present their cases during the whole-class discussion. As they present and analyze different claims about the figures in the discussion, students construct arguments and critique reasoning (MP3).

This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured partner meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . . ?” and “Can you say that another way?” Finally, students write a second draft of their response, reflecting ideas from partners, and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.

The goal of this discussion is to make clear that angle measurements and distances are both important when deciding whether two polygons are scaled copies. To highlight the different arguments about whether one polygon is a scaled copy of another, consider debriefing with a role play. Ask two students to take on the roles of the characters Mai and Noah and make a brief argument about whether they believe one figure is a scaled copy of the other in the first question. After the cases are presented, ask students whether they agree with Mai or Noah.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last question, about whether polygon is a scaled copy of . 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 did you mean by . . . ?”
• “Can you say that another way?”
• “How do you know . . . ? What else do you know is true?”

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. If 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.

Here is an example of a second draft:

I agree with Lin. Even though the corresponding angles are equal, is not a scaled copy of . To get from to we multiply by , but to get from to we multiply by , so the corresponding side lengths don’t all have the same scale factor.

If time allows, have students compare their first and second drafts.

After Stronger and Clearer Each Time, poll the class to find out whether they agree with Kiran or Lin.

In this activity, students use what they know about corresponding lengths and angles to show that one picture of a bird is not a scaled copy of the other. Unlike in previous tasks, minimal scaffolding is given here, so students need to decide what evidence is necessary to explain or show an absence of scaling. As students work, notice the different ways students use corresponding lengths and angles to think about scaling.

When students use corresponding measurements to show that one image is not a copy of the other, they practice looking for and making use of structure (MP7).

Monitor for students who use different approaches to show that one image is not a scaled copy of the other:

• Describe the copy as being stretched vertically or squished horizontally.
• Identify two pairs of corresponding distances in the images that have different scale factors.
• Identify corresponding angles in the two images that have different measures.

Plan to have students present in this order to support them in developing more precise ways of justifying that images are not scaled copies.

In the digital version of the activity, students use an applet to compare corresponding lengths and angles in two images. The applet allows students to add points and line segments and to measure distances and angles. This activity works best when each student has access to the applet because the level of precision is important. The digital version may help students measure quickly and accurately so they can focus more on the mathematical analysis.

The purpose of this discussion is to show various ways in which students can select evidence that shows the image is not a scaled copy. Invite previously selected groups to share their reasoning. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see. As students share their approaches, consider asking how they knew that certain pairs of points in the images were corresponding points. Students could identify a unique feature such as the eyes or the tip of the wing.

Connect the different responses to the learning goals by asking questions such as:

• “How were corresponding points used in each approach?”
• “Which methods examined corresponding distances and which examined corresponding angles?”
• “How many pieces of evidence did it take for each approach to show that the image was not a scaled copy?”

Make sure students understand that:

• Corresponding angles on an image and a scaled copy have the same measure.
• Corresponding distances on an image and a scaled copy are related by the same scale factor.
• To conclude that an image is not a scaled copy of another, it is sufficient to find one pair of corresponding angles with different measures or two pairs of corresponding distances that are related by different scale factors.