Getting Started 
Misconception/Error The student does not understand similarity in terms of transformations. 
Examples of Student Work at this Level The student:
 Writes that the circles are similar because all circles are similar.
 Identifies the scale factor but is unable to complete the proof.
 Identifies the scale factor but is unable to continue.
 Does not determine whether or not the figures are similar and describes only an incorrect sequence of transformations.
 Attempts to describe a sequence of similarity transformations that carries one circle onto the other but does not understand how this shows the two circles are similar.

Questions Eliciting Thinking What does it mean for two circles to be similar?
How is similarity determined or verified? How can transformations be used to define and justify similarity? Which transformations can you use when showing two figures are similar?
Can you describe a sequence of similarity transformations that maps circle A onto circle B?
What is scale factor, and how does it relate to similarity?
Can you model for me the sequence of similarity transformations that you described? 
Instructional Implications Review with the student the definition of similarity in terms of similarity transformations. Explain that two figures are similar if there is a dilation or a dilation and a congruence (e.g., a sequence of rigid motions) which carries one figure onto the other. Have the student develop his or her understanding of dilations by using dynamic geometry software (e.g., Geogebra) or interactive websites (e.g., http://www.mathsisfun.com/geometry/resizing.html, http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?open=activities, http://www.mathopenref.com/dilate.html) to obtain images of a given figure under dilations having specified centers and scale factors.
Discuss with the student how similarity transformations can be used to show that two circles are similar. Explain that given two circles, circle A of radius five units and circle B of radius two units, circle A can be translated along vector so that the center of circle A coincides with the center of circle B. Continue to explain to the student that because the two circles are now concentric (e.g., their centers coincide) a dilation with center B and scale factor maps every point on circle A to a point on circle B. Explain to the student that after the translation and dilation, every point of circle A will be b units (or two units) from center B so must now be on circle B. Assist the student, if needed, in understanding why the scale factor of the dilation, , results in relocating each point on circle A to a point units from center B. Remind the student that a dilation with a scale factor that is less than one is a reduction and a dilation with a scale factor greater than one is an enlargement.
Provide the student with examples of pairs of similar figures, including circles, and have the student identify the sequence of similarity transformations that map one figure onto the other. When translating figures in the coordinate plane or on grid paper, guide the student to decompose the vector that defines the translation into its horizontal and vertical components. Encourage the student to consider in the dilation that each point, P, on the ray extending from the center of dilation, O, will be mapped to another point, P’, on the same ray, so that .
Consider implementing the MFAS task Showing Similarity (GSRT.1.2). 
Making Progress 
Misconception/Error The student shows some understanding of the sequence of similarity transformations that maps one circle onto the other but omits important details or incorrectly describes the transformations in his or her explanation. 
Examples of Student Work at this Level The student does not provide enough detail when describing each transformation and:
 Writes only “translate and then dilate the circle.”
 Incorrectly or incompletely describes the translation (e.g., does not describe a translation along ).
 Does not clearly describe which circle is being mapped onto the other.
 Does not specify the center of the dilation and/or the scale factor.
 Describes a translation and dilation of one of the circles but does not explain how or why points on the transformed circle align with points on the other circle.

Questions Eliciting Thinking Which transformations produce congruent figures? Which transformations produce similar figures?
How can you determine, based on a sequence of similarity transformations, that circle A is similar to circle B?
How would you describe a translation? Can you describe the translation in more detail? What point on the image corresponds to point A on the preimage? How did you determine the vector for the translation?
How would you describe the dilation? Can you describe in more detail the dilation? How is the center of dilation determined? What point did you use for the center of dilation?
How did you determine the scale factor? Is the dilation of circle A onto circle B an enlargement or a reduction? 
Instructional Implications Review what is necessary to include in a description of a translation (e.g., the vector along which the translations occurs) and a dilation (e.g., the location of the center and the scale factor). Guide the student through the translation and dilation of circle A providing appropriate detail. Model for the student a clear and complete explanation of the sequence of similarity transformations that maps circle A to circle B. Then ask the student to show that every point on circle B can be mapped to a point on circle A.
Challenge the student to describe how any circle can be mapped to any other circle using a translation and a dilation. Ask the student to consider a general way to describe the translation that brings the centers of the two circles in alignment and the center and scale factor of the dilation that maps points from one circle to the other circle.
For additional practice with similarity, consider implementing one or more of the MFAS tasks Similar Circles (GC.1.1), To Be or Not To Be Similar (GSRT.1.2), and The Consequences of Similarity (GSRT.1.2). 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student describes the following sequence of transformations to show the two circles are similar:
 Translate circle A along so that point A coincides with point B.
 Choose an arbitrary point P on circle A and dilate it using point B (or point A) as the center and scale factor .
 Since P' will be a distance units from point B, it will lie on circle B.
 Since this sequence of a translation and dilation maps every point on circle A to a point on circle B, the circles are similar.

Questions Eliciting Thinking Could you have dilated the image about a different point?
Is it always necessary to include a dilation when describing the sequence of similarity transformations used to verify two figures are similar? Why or why not?
Does it matter which transformation is completed first?
Could proving similarity of two circles require more than two similarity transformations? Why or why not?
What property of circles ensures a dilation can be used to map one circle onto another circle whose centers coincide? 
Instructional Implications Ask the student to also show that every point on circle B can be mapped to a point on circle A.
Ask the student to show that all circles are similar. Consider implementing the MFAS task Similar Circles (GC.1.1).
Introduce the concept of an equivalence relation (e.g., a relationship that satisfies the reflexive, symmetric, and transitive properties) and ask the student to consider if similarity is an equivalence relation. 