This Warm-up gives students an opportunity to use the definition of similarity in an argument. Monitor for students who produce arguments such as:

• They aren’t similar because one is skinnier than the other.
• They aren’t similar because pairs of corresponding side lengths aren’t all in the same proportion.
• They aren’t similar because if you dilate based on the scale factor that makes these sides match, the other sides won’t match, and vice versa.

Display 2–3 reasonings from previously selected students for all to see. Invite students to briefly describe their reasoning, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the reasonings have in common?”
• “How can we restate these reasonings in a more precise way?”

In this activity, students analyze a faulty proof for why all rectangles are similar. Although it is incorrect, Tyler’s proof gives students a chance to see the structure of a similarity proof. In addition, it gives students a chance to correct the reasoning of others and provides another way to explain why not all rectangles are similar (MP3). Students should recognize many of the moves and justifications from proofs about congruence in a previous unit, and proofs about dilations earlier in this unit.

This activity also gives the class an opportunity to discuss the proof process. Sometimes students who struggle in geometry assume that they should just know if a statement is true, and try to jump to proof before they have explored. Mathematicians spend much of their time experimenting, wondering, guessing, sketching, and being unsure. Tyler’s mistake probably came from not experimenting and sketching before he began writing a proof. In subsequent activities, students will benefit from exploring and thinking about whether each statement is true before trying to prove it.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

After conjecturing, viewing, and critiquing examples of a similarity proof using rigid transformations and dilations, students are ready to write proofs of the two true statements in this activity. Monitor for students who include any of these points in their proof that all circles are similar:

• There will always be a dilation that makes the circles have the same radii.
• There will always be a rigid motion that takes one circle’s center onto the other.
• Two circles with the same center and the same radius are the same circle (coincide).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).