Getting Started 
Misconception/Error The student is unable to use the definition of similarity in terms of similarity transformations to prove the triangles are similar. 
Examples of Student Work at this Level The student:
 Reasons that since and , the triangles are similar by the AA Similarity Theorem.
 Writes down the given information and one or two other unrelated statements without completing the proof.

Questions Eliciting Thinking What does the AA Similarity Theorem say? Can a theorem be used in its own proof?
Do you know what is meant by a similarity transformation?
What is the definition of similarity in terms of similarity transformations? How can two triangles be shown to be similar? 
Instructional Implications Review the definition of similarity in terms of similarity transformations and explain how the definition can be used to show two triangles are similar. Provide opportunities to the student to show two triangles are similar using the definition. Then clearly state the AA Similarity Theorem and ask the student to identify the assumption and the conclusion. Be sure the student understands that a theorem cannot be used as a justification in its own proof.
Review each of the following:
 The Fundamental Theorem of Similarity,
 The Corresponding Angles Theorem, and
 The ASA Congruence Theorem.
Next review the overall strategy of the proof and guide the student through its steps prompting the student for justifications of key statements.
If needed, implement the MFAS tasks Describe the AA Similarity Theorem (GSRT.1.3) to assess the student’s understanding of the theorem and Justifying a Proof of the AA Similarity Theorem (GSRT.1.3) to assess the student’s understanding of a proof of this theorem. Eventually, ask the student to repeat this task. 
Moving Forward 
Misconception/Error The student is unable to complete the proof. 
Examples of Student Work at this Level The student attempts to show that the triangles are similar using the definition of similarity in terms of similarity transformations. However, the student is unable to produce a complete and correct proof. The student provides:
 A labeled sketch of two triangles and at least one step that could lead to a proof but then is unable to continue.
 A few steps that suggest a strategy but omits key steps and details.

Questions Eliciting Thinking What is your overall strategy for this proof? What do you need to prove now?
What is the definition of similarity in terms of similarity transformations?
If one figure is the result of a dilation of another figure, are the figures similar?
If one figure is the result of a dilation and a congruence of another figure, are the figures similar?
How is a segment and its image related under a dilation? Do you remember the Fundamental Theorem of Similarity? 
Instructional Implications Review each of the following:
 The Fundamental Theorem of Similarity,
 the Corresponding Angles Theorem, and
 the ASA Congruence Theorem.
Next review the overall strategy of the proof and guide the student through its steps prompting the student for justifications of key statements.
If needed, implement the MFAS task Justifying a Proof of the AA Similarity Theorem (GSRT.1.3) to assess the student’s understanding of a proof of this theorem. Eventually, ask the student to repeat this task. 
Almost There 
Misconception/Error The student provides a correct response but with insufficient reasoning or imprecise language. 
Examples of Student Work at this Level The student shows the triangles are similar using the definition of similarity in terms of similarity transformations. However, the student’s proof contains some errors or omissions. For example, the student omits or provides incorrect justification for one or two statements or omits a key step of the proof.

Questions Eliciting Thinking What is the relationship between these two points and their images?
How is a segment and its image related under a dilation?
What is the definition of similarity in terms of similarity transformations?
If one figure is the result of a dilation of another figure, are the figures similar?
If one figure is the result of a dilation and a congruence of another figure, are the figures similar? 
Instructional Implications Provide feedback to the student concerning any errors or omissions and allow the student to revise his or her proof. Consider implementing the MFAS task Justifying a Proof of the AA Similarity Theorem (GSRT.1.3) for further experience with the theorem and its proof. Eventually, ask the student to repeat this task. 
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 shows the triangles are similar using the definition of similarity in terms of similarity transformations. The student may provide a proof such as the following:
Let be the point on so that . Denote the dilation with center A and scale factor r = (which is also equal to ) by D, and let be the point on such that D(C) = . Then:
 is parallel to by the Fundamental Theorem of Similarity.
 by the Corresponding Angles Theorem.
 by the ASA Congruence Theorem. Call this congruence G.
 by the definition of similarity ( is the result of a dilation of ).
 Since dilation D maps to to and congruence G maps , then by the definition of similarity.

Questions Eliciting Thinking Are there other ways to prove the theorem?
Does the AA Similarity Theorem apply to rectangles? To rhombuses? 
Instructional Implications Ask the student to adapt his or her proof to prove the related theorems on either the Similarity With Given Sides or Similarity With Angle and Side worksheets which are included in the attachments. 