Lesson 8

In a previous lesson, students were reminded of the definition of similarity in terms of dilations and rigid transformations (which they first encountered in middle school). This warm-up gives students an opportunity to use that definition in an argument. Monitor for students who use an alternative definition of similarity as “same shape, different size,” producing arguments such as:

• they are similar because they’re both rectangles
• 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

Identify students using dilation-based arguments:

• they are similar because one is a stretched-out version of the other
• 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

Save plenty of time for the synthesis of this activity. Students should not need more than 3 minutes to gather their thoughts, and then the remaining 7 minutes can be used to synthesize and activate students’ prior knowledge for use in later activities.

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 hippos from the beginning of this unit. If these rectangles had photographs inside them, would it be a proper enlargement or distort the image?

Invite students with both general and dilation-based arguments to share. Ask the class to translate the general arguments into dilation-based arguments.

For example, if a student said that the rectangles were not similar because one was much skinnier than the other, the class could revise the statement to say that there couldn’t be a dilation from one rectangle to the other because if you made the scale factor big enough to make the horizontal sides match, the dilated rectangle would be way too tall.

Tyler’s proof gives students a chance to see the structure of a similarity proof. In addition, it gives students a chance to do error analysis and provides another way to explain why not all rectangles are similar. 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).

Emphasize to students that they should draw exactly what the proof says for every step. Encourage students to draw different rectangles for \(ABCD\) and \(PQRS\) that will help them prove that Tyler’s statement really is false.

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 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).

Remind students of the importance of drawing pictures and experimenting before trying to prove. 

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

Begin with the statements that students were able to prove false. Invite students to share examples and counterexamples. Ask students if having an example of a statement makes it true. (No, statements are only true if they are always true for all examples. One counterexample is enough to prove a statement false.)

Invite several students to share their reasoning about circles. Their explanations 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)

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

All circles are similar. (Theorem)