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 the given and one or two more statements that fail to establish the congruence of the base angles.
The student states the base angles must be congruent since the sides opposite them are congruent (restatement of the isosceles triangle theorem), but no other justification is given.
The student's proof may include one or more irrelevant or incorrect statements. 
Questions Eliciting Thinking Did you think through a plan for your proof before you started? Did you consider what you already know that might help you to prove these angles are congruent?
What if you drew a line segment from point B to the midpoint of ? If the two triangles formed are congruent, can you conclude that ?
How can you prove the two angles are congruent in a way other than stating the isosceles triangle theorem? 
Instructional Implications Provide the student with frequent opportunities to make deductions using a variety of previously encountered definitions and established theorems. For example, provide diagrams as appropriate and ask the student what can be concluded as a consequence of:
 Point M is the midpoint of .
 < A and < B are supplementary and m < A = d.
 .
 is the bisector of < E.
Provide the student with the statements of a proof and ask the student to supply the justifications. Then have the student analyze and describe the overall strategy used in the proof.
To prove that the base angles of an isosceles triangle are congruent, guide the student to first describe an overall strategy [e.g., (1) Add an auxiliary line so that two triangles are formed, (2) Show the two triangles are congruent, and (3) Conclude the base angles are congruent by definition of congruent triangles]. Then assist the student in filling in the details of the proof including justifications. Encourage the student to always begin the proof process by developing an overall strategy. Assist the student by providing feedback on the strategy.
Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the student to first question what is available to use in a proof of a particular statement.
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, and HL) and what must be established in order to conclude two triangles are congruent when using each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding parts can be concluded to be congruent. 
Moving Forward 
Misconception/Error The student’s proof shows evidence of an overall strategy but the student fails to establish major conditions leading to the prove statement. 
Examples of Student Work at this Level The student attempts to prove the two triangles formed by drawing an altitude, angle bisector, or median from vertex B are congruent (in order to conclude that ) but fails to do so.

Questions Eliciting Thinking How do you know you are justified in adding this segment to the diagram?
What do you know as a consequence of this segment being an altitude (or angle bisector or median)? What can you conclude from this? What properties does it have?
What do you need to show in order to use the SSS (or other relevant) congruence theorem? Have you done that in your proof?
Suppose you show the triangles are congruent. What will allow you to conclude that these angles are then congruent? 
Instructional Implications Review the triangle congruence theorems and provide more opportunities and experiences with proving triangles congruent.
Consider using the NCTM lesson Pieces of Proof (http://illuminations.nctm.org/Lesson.aspx?id=2561) in which the statements and reasons of a proof are given separately and the student must rearrange the steps in a logical order. Encourage the student to use multiple proof formats including flow diagrams, twocolumn, and paragraph proofs. Allow the student to work with a partner to complete these exercises.
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 to collaborate on completing the proof. 
Almost There 
Misconception/Error The student’s proof shows evidence of an overall strategy, but the student fails to establish a condition that is necessary for a later statement in the proof. 
Examples of Student Work at this Level The student fails to first establish the existence of the midpoint M of and the uniqueness of .
The student fails to first establish that the bisector of < B intersects at some point D.
The student fails to establish the congruence of one pair of angles or sides necessary to use the congruence theorem cited. 
Questions Eliciting Thinking How did you know you could add this segment to your diagram? How do you know this segment is unique?
How do you know that this angle bisector will intersect the opposite side of the triangle?
I see you stated these triangles are congruent. Can you show me all of the steps needed to use the theorem you used? Did you include all of them in your proof? 
Instructional Implications Review how to address and justify adding a point, such as a midpoint, or an auxiliary line to a diagram.
Using a colored pencil or highlighter, encourage the student to mark the statements which support the congruence theorem chosen. Remind the student that each letter of the theorem name represents a pair of parts that must be shown to be congruent [e.g. if using SSS to prove the triangles congruent, the proof must include showing three pairs of corresponding sides are congruent (and a reason or justification must be provided for each)]. 
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 devises a complete and correct proof in which he or she (1) Adds an auxiliary line so that two triangles are formed, (2) Shows the two triangles are congruent, and (3) Concludes the base angles are congruent by definition of congruent triangles. For example, the student establishes the existence of point M, the midpoint of , and the uniqueness of (“Through two distinct points passes a unique line”). The student states that (as given), (by the Reflexive Property), and (by definition of a midpoint). The student then states that by the SSS Congruence theorem. The student concludes that by the definition of congruent triangles. 
Questions Eliciting Thinking Can you think of another way to prove that the base angles of an isosceles triangle are congruent? How many different ways could you complete this proof? 
Instructional Implications Challenge the student with statements requiring more complex proofs (e.g. given a diagram that includes overlapping triangles, ask the student to prove a statement that requires first proving one pair of triangles congruent in order to name a pair of corresponding parts congruent needed to show a second pair of triangles congruent).
Encourage the student to assist other students in developing and writing proofs. 