Getting Started 
Misconception/Error The student’s proof shows no evidence of an overall strategy or logical flow. 
Examples of Student Work at this Level The student:
 States that but is unable to explain why.
 Makes some observations about the triangles without regard to the statement to be proven.

Questions Eliciting Thinking What do you know about this figure? What are you being asked to prove?
Did you think of a plan for your proof before you started?
How can you show two triangles are similar? Are there any theorems that might help? 
Instructional Implications Review ways to prove two triangles are similar (AA, SAS, SSS) and what must be established in order to conclude two triangles are similar using each method. Remind the student that once two triangles are proven similar, all corresponding angles are congruent and all corresponding sides are proportional. Encourage the student to draw and label all three triangles to identify corresponding congruent angles.
Consider using NCTM lesson Are They Similar (http://www.illustrativemathematics.org/illustrations/603) to help the student identify the different ways to prove that two triangles are similar.
Assist the student in devising an overall strategy for the proof: (1) Show and using the AA Similarity Theorem and then deduce that (2) Conclude that . (3) Conclude that h is the geometric mean of e and f. Guide the student through the statements of the proof and prompt the student to supply the justifications.
If necessary, review the definition of geometric mean and provide examples. 
Moving Forward 
Misconception/Error The student’s proof shows evidence of an overall strategy but fails to establish major conditions leading to the prove statement. 
Examples of Student Work at this Level The student fails to establish that .

Questions Eliciting Thinking How can you show that two triangles are similar? Can you immediately say that ?
What do you need to show in order to use the AA Similarity Theorem? Have you done this? 
Instructional Implications Review ways to prove two triangles are similar (AA, SAS, SSS) and what must be established in order to conclude two triangles are similar using each method. Remind the student that once two triangles are proven similar, all corresponding angles are congruent and all corresponding sides are proportional. Encourage the student to draw and label all three triangles to identify corresponding congruent angles.
Consider using NCTM lesson Are They Similar (http://www.illustrativemathematics.org/illustrations/603) to help the student identify the different ways to prove that two triangles are similar.
Review an overall strategy for the proof and guide the student through the steps of any part of his or her proof that was incomplete. Prompt the student to provide justifications for each step.
Encourage the student to begin the process of writing a proof by developing an overall strategy and have the student compare his or her strategies with the strategies of another student at the same level. For additional practice, provide other theorems to be proven in which the statements and reasons are given separately and the student must rearrange the steps into a logical order. 
Almost There 
Misconception/Error The student fails to establish a condition that is necessary for a later statement. 
Examples of Student Work at this Level The student fails to establish:
 That or when using the AA Similarity Postulate.
 That the corresponding parts of similar triangles are proportional

Questions Eliciting Thinking What do you need to show in order to use the AA Similarity Postulate? Have you done this? Other that the right angles what angles are congruent?
What allows you to say that ? Did you state this? 
Instructional Implications Provide the student with feedback on his or her proof. If the student omitted a statement, have the student go through each step of the proof to see if he or she can find the gap in the logical flow of the proof. Prompt the student to supply justifications or statements that are missing. If necessary, review notation for naming sides, lengths of sides, angles, and angle measures. Also, review that when naming similar triangles, vertices are named in corresponding order. 
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 uses the AA Similarity Postulate to show that and and then reasons by transitivity that . The student then states that since the triangles are similar, their corresponding sides are proportional so that . Therefore, using the Cross Product Property, . Since h is the length of the altitude, then it is the geometric mean of the two segments, e and f, of the hypotenuse.

Questions Eliciting Thinking Do you know the meaning of the term geometric mean? How is it related to this proof? 
Instructional Implications Challenge the student to use the same diagram to prove that when the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean of the hypotenuse and the segment of that hypotenuse that is adjacent to that leg.
Consider using MFAS task Pythagorean Theorem Proof (GSRT.2.4) if not previously used. 