Getting Started 
Misconception/Error The student does not appear to understand the AA Similarity Theorem. 
Examples of Student Work at this Level The student:
 Reasons that since , , and the sum of the angles of a triangle is 180 degrees, the triangles must be similar.
 Reasons about the angle measures rather than the similarity of the triangles.
 Uses circular reasoning (i.e., cites the AA Similarity Theorem as a justification of the similarity).

Questions Eliciting Thinking Can you restate the AA Similarity Theorem in your own words?
What are the assumptions of this theorem? What is its conclusion?
Can you use a theorem in its own proof? 
Instructional Implications Clearly explain the AA Similarity Theorem making explicit its assumptions and conclusion. Make clear to the student that the purpose of the exercise is to justify the AA Similarity Theorem. Therefore, the theorem cannot be used in its own justification since it has not yet been established as true.
Review 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 sequence of rigid motions that carries one figure onto the other. Demonstrate using the definition of similarity in terms of similarity transformations how to show that two triangles are similar. Provide the student with another pair of similar triangles that are related by a dilation and have the student determine the center of the dilation and the scale factor. Explain to the student that a convincing way to show the two figures are similar is to describe the dilation in detail (e.g., by specifying its center and scale factor) and explain why the two figures must be congruent after the dilation. Next, provide two similar figures that are related by a dilation and a rigid motion. Have the student describe both the dilation and the rigid motion that will carry one figure onto the other.
Review the Fundamental Theorem of Similarity. Then guide the student through the steps of an informal justification such as the one described below in the Got It Instructional Implications. Have the student use tracing paper to demonstrate the initial translation of triangle XYZ. Then ask the student to draw triangle as described, and consider the relationship between triangle and triangle ABC. Ask the student to explain how this exercise satisfies the definition of similarity in terms of similarity transformations.
Note: See this document for a discussion of the Fundamental Theorem of Similarity (p. 87) and a proof of the AA Similarity Theorem (pp. 95 99). 
Making Progress 
Misconception/Error The student omits important details from the justification. 
Examples of Student Work at this Level The student appears to understand the AA Similarity Theorem and attempts a justification using similarity transformations. However, the student omits important details. For example, the student says translate point A to point X and dilate triangle ABC until it aligns with triangle XYZ.

Questions Eliciting Thinking Once you have translated point A to point X, how do you know will align with ?
Can you describe the dilation in more detail?
What is the definition of similarity in terms of similarity transformations? What is the role of congruence in showing similarity? 
Instructional Implications Review the definition of similarity in terms of similarity transformations (i.e., two figures are similar if there is a dilation or a dilation and a sequence of rigid motions that carries one figure onto the other). Provide feedback to the student regarding his or her response and explain why more justification is required in order to make the explanation convincing. Then guide the student through the steps of an informal justification of the AA Similarity Theorem such as the one described below in the Got It Instructional Implications. Have the student use tracing paper to demonstrate the initial translation of triangle XYZ. Then ask the student to draw triangle as described, and consider the relationship between triangle and triangle ABC. Ask the student to explain how this exercise satisfies the definition of similarity in terms of similarity transformations. 
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 an informal justification of the AA Similarity Theorem. For example, the student reasons as follows:
 Translate point X to point A (and rotate, if necessary) so that one side of aligns with the corresponding side of . Since , the other sides of these angles will also coincide.
 Locate on so that = AB.
 Using X as the center of dilation and a scale factor of r = (or ), dilate triangle XYZ.
 Let be the point on such that = r(XZ).
 By the Fundamental Theorem of Similarity, so that .
 Since (by assumption), (by step 2), and (by step 5 and assumption), triangle ABC triangle (by ASA Congruence Theorem).
This shows that there is a translation followed by a dilation of triangle XYZ so that triangle XYZ is carried onto triangle ABC. Therefore the two triangles are similar.

Questions Eliciting Thinking Would it also be possible to translate and dilate triangle ABC so that it is carried onto triangle XYZ? 
Instructional Implications Provide opportunities for the student to use the AA Similarity Theorem to prove that pairs of triangles are similar.
Consider implementing other 8.G.1.5 MFAS tasks to further assess the student. 