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:
 Is unable to show that and
 Makes some observations about the triangles without regard to the statement to be proven.
 Uses the Pythagorean Theorem in its own proof.
 Attempts to show that and but goes no further.

Questions Eliciting Thinking What do you know about this figure? What are you asked to prove? What are you asked to show first?
Did you think of a plan for your proof before you started?
In what ways can you prove that triangles are similar? How many triangles do you see in this diagram? Can you draw the three triangles separately?
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 different ways to prove that two triangles are similar.
Assist the student in understanding what can be used in a proof of a theorem (i.e., the assumptions presented in the statement of the theorem as well as definitions, postulates, and other previously established theorems). Emphasize that a theorem cannot be used as a justification in its own proof. Assist the student in devising an overall strategy for the proof: (1) Show and using the AA Similarity Theorem. (2) Conclude that and . (3) Observe that AD + DB = AB. (4) Reason from the two proportions to the statement to be proved. Guide the student through the statements of the proof and prompt the student to supply the justifications.
Provide opportunities for the student to determine the flow of a proof. Encourage the student to begin the proof process by developing an overall strategy. Provide another statement to be proven and have the student compare strategies with another student and collaborate on completing the proof.
Consider using MFAS task Geometric Mean Proof (GSRT.2.4) if not previously used. 
Moving Forward 
Misconception/Error The student’s proof shows some evidence of an overall strategy but is incomplete and fails to establish major conditions leading to the prove statement. 
Examples of Student Work at this Level The student:
 Assumes (or incorrectly shows) the two pairs of triangles are similar and reasons from resulting proportions to the prove statement but omits steps and justifications that are necessary to complete the proof.
 Correctly shows that and , deduces that and and that and but is unable to go any further in the proof.

Questions Eliciting Thinking What is your general strategy for this proof?
What do you know about the lengths of the sides of the similar triangles? Did you state this in your proof?
Is there an algebraic property you could use to simplify your proportion?
What do you know about the relationship between e, f, and c? 
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 look for ways to prove similarity.
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.
Consider using MFAS task Geometric Mean Proof (GSRT.2.4) if not previously used. 
Almost There 
Misconception/Error The student’s proof shows evidence of an overall strategy, but the student fails to establish a minor condition that is necessary to prove the theorem. 
Examples of Student Work at this Level The student:
 Provides the statements of the proof but omits some necessary justifications.
 Does not formally establish the similarity of the two pairs of similar triangles.

Questions Eliciting Thinking How do you know that (refer to a statement that was not justified)?
Can you write some statements and justifications that convincingly show these two pairs of triangles are similar? 
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 Theorem to show that (since and ) and likewise (since and ).
The student then provides a complete proof with appropriate justification such as:
Since corresponding sides of similar triangles are proportional then and . Consequently, and . Using the Addition Property of Equality . This equation can be rewritten by factoring as . But, AD + DB = AB or e + f = c by the Segment Addition Postulate. Therefore using the Substitution Property. 
Questions Eliciting Thinking Is there another way to prove the Pythagorean Theorem? 
Instructional Implications Challenge the student to prove other statements about similar triangles and right triangles. Provide the student opportunities to write proofs using a variety of formats some of which include narrative paragraphs, flow diagrams, and twocolumn format. 