Getting Started 
Misconception/Error The student is unable to explain and justify the steps of the proof. 
Examples of Student Work at this Level The student may be able to explain some of the steps or name some of the properties used as justifications, but the student does not demonstrate an overall understanding of the proof. The student:
 Makes assumptions that have no basis (e.g., immediately states ABC is a right triangle).
 Says substitutions can be made because the variables name the same parts of the respective triangles (e.g., both are hypotenuses).
 Provides answer(s) that do not address the question asked.
 Is unable to complete the proof.

Questions Eliciting Thinking What information is given? What is being proven?
What does the converse of the Pythagorean Theorem state?
Can the substitution in the second question be justified by saying all the variables represent legs of a triangle? Why can a and b substitute for r and s? Look closely at the given information.
Can you sayÂ just because c and t both name a hypotenuse?
When is it appropriate to apply the Substitution Property? 
Instructional Implications Provide the student with basic instruction on the Pythagorean Theorem and its converse. Review the parts of a right triangle (e.g., vertices, right angle, acute angles, hypotenuse, and legs) and be sure the student understands the distinction between the legs and the hypotenuse. When initially introducing the two theorems, explicitly state the assumptions and conclusion:
 Pythagorean Theorem: Assume a triangle is a right triangle with legs of lengths a and b and hypotenuse of length c. Then prove .
 Converse: Assume a triangle has sides whose lengths are related by the equation . Then prove the triangle is a right triangle.
Be sure the student understands the distinction between the Pythagorean Theorem and its converse.
Provide instruction on writing mathematical explanations, justifications, and proofs. Encourage the student to first consider the statement to be proven. Next, ask the student to examine the assumptions and then formulate an overall strategy. Make clear that every step must be justified with mathematical properties or theorems. Guide the student through the proof of the converse of the Pythagorean Theorem modeling the use of definitions, properties, or theorems to justify each step of the proof.
If necessary, review notation for naming angles (e.g., ) and describing measures of angles (e.g., ) and guide the student to use the notation appropriately.
Provide additional opportunities for the student to write informal proofs. Consider implementing MFAS tasks which ask the student to justify various angle relationships. 
Making Progress 
Misconception/Error The student does not clearly justify each step of the proof. 
Examples of Student Work at this Level The student demonstrates overall understanding of the proof but does not fully justify or explain one or more steps of the proof. The student:
 Makes a minor mistake in an explanation.
 Provides an incomplete explanation.

Questions Eliciting Thinking What is the Substitution Property of Equality? How does the Substitution Property justify that ?
Can you explain your answer (Pythagorean Theorem) to the first question? How can you justify the use of the Pythagorean Theorem?
You saidÂ is congruent to . Why is that significant? What does it mean about triangle ABC? 
Instructional Implications Provide the student with feedback on the answers and prompt the student to supply justifications that are missing. Emphasize that every step must be justified using the given information and mathematical properties or theorems. If necessary, review notation for naming sides, lengths of sides, angles, and angle measures.
Consider implementing MFAS task Explaining a Proof of the Pythagorean TheoremÂ if not used previously. 
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 justifies each step of the proof with a complete explanation and specific evidence. For example, the student writes:
 Mr. Lopez can use the Pythagorean Theorem because it is given thatÂ is a right triangle.
 It is given that aÂ =Â r and bÂ =Â s, so the Substitution Property can be applied.
 BecauseÂ and , the Substitution Property can be used to justify thatÂ is equal to .
 BecauseÂ is congruent to , the sides and angles of the triangles are congruent. Therefore, you can concludeÂ is congruent toÂ which measures 90Â°. SinceÂ measures 90Â°, triangle ABC is a right triangle.

Questions Eliciting Thinking Can you explain the proof in your own words?
Why is it important to justify each step of a proof?
How can you use the converse of the Pythagorean Theorem? 
Instructional Implications Provide opportunities to use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.
Consider implementing MFAS task Explaining a Proof of the Pythagorean TheoremÂ if not used previously. 