If students are unsure where to start or argue that one figure is a dilation of the other because it’s stretched out, remind them of the distorted video game controllers from the beginning of this unit. If these rectangles had photographs inside them, would it be a proper enlargement or distort the image?

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).

Begin the discussion by inviting students to explain what Tyler did well, and what makes a good proof that two figures are similar. (Tyler gave reasons why each of his transformations worked. Tyler used rigid transformations and dilations.) Then, discuss where Tyler went wrong. (When Tyler did the second dilation, it changed the lengths from the first dilation, so the first pair of corresponding sides aren’t congruent anymore.)

Ask students what Tyler could have done before he even wrote the proof to make sure that he didn’t waste time proving something that wasn’t true. Help students see that experimenting, looking for examples and counterexamples, and drawing pictures is part of the proof process. 

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).

If students are stuck writing their proof, suggest that they use the model from the previous activity (the structure is valid despite the error in reasoning).

The purpose of the discussion is for students to understand that a single counterexample is enough to show that a conjecture is incorrect, but a proof must be given to show that the conjecture is true.

Invite a group to share an example of a pair of circles that are similar.

Ask students if having an example of similar circles is enough to show that the conjecture is always true. (No, conjectures are only true if they are true for all examples.)

Invite several groups to share a step in their proof for why all circles are similar. The steps should include:

• 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).

Invite additional groups to share their conclusions about the triangle conjectures. Students should understand that a counterexample is enough to prove that the conjectures about isosceles and right triangles are false, but a proof is needed to show that the equilateral conjecture is true.

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

All circles are similar. (Theorem)