Getting Started 
Misconception/Error The student can only describe the steps of a proof in general terms. 
Examples of Student Work at this Level The student:
 Writes that all circles are similar.
 Does not understand how to determine the scale factor when numerical radii are not given.
 Does not understand how to describe similarity transformations without a diagram of the circles and/or more information about the size and position of the circles.

Questions Eliciting Thinking What does it mean for two circles to be similar?
Exactly how would you translate circle A to circle B? What point or points will align? How can you describe the direction and distance of the translation?
What is the center and scale factor of the dilation? Can you describe the scale factor in terms of the radii of the circles? 
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 any two circles, circle A and circle B, 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 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.
Consider implementing the MFAS task All Circles Are Similar (GC.1.1). 
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:
 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.
 Does not clearly demonstrate how a point on one circle is mapped to a point on the other circle after the transformation and dilation.

Questions Eliciting Thinking 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 in general terms? What details would still need to be included? How did you determine and describe the vector for the translation?
How would you describe the dilation in general terms? What details would still need to be included? How could you determine and describe the center of dilation? What point could you use for the center of dilation? 
Instructional Implications Review what is necessary to include in a description of a translation (e.g., the vector along which the translation occurs) and a dilation (e.g., the location of the center and the scale factor). Challenge the student to consider a general way to describe the translation that brings the centers of the two circles in alignment as well as the center and scale factor of the dilation that maps points from one circle to the other circle. 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.
For additional practice, consider implementing the MFAS task All Circles Are Similar (GC.1.1) or the MFAS tasks To Be or Not To Be Similar (GSRT.1.2) and/or The Consequences of Similarity (GSRT.1.2) to review similarity. 
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 draws two circles, circle A of radius a and circle B of radius b, with . The student then 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 the image of P, P', will be a distance units from center B, it will lie on circle B.
 Since this sequence of a translation and dilation maps any 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 with the same center point? 
Instructional Implications Ask the student to draw circle A and circle B and a line externally tangent to both circles. Challenge the student to use the tangent line and to determine the center of dilation that directly maps circle B onto circle A and to describe the scale factor. 