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 that or that .
 Offers an explanation that is unclear or does not make mathematical sense.
 Restates the statement to be proved.

Questions Eliciting Thinking What is it that you are given and what are you trying to prove?
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 that point P is equidistant from points A and B?
You referred to an angle as AP. What angle did you mean? How are angles typically named?
What do you know as a consequence of the fact that is the perpendicular bisector of ? 
Instructional Implications Provide the student with the statements of a proof of this theorem and ask the student to supply the justifications. Then have the student analyze and describe the overall strategy used in the proof. If needed, review the ways to prove two triangles are 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.
Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the student to first identify the assumptions given in the statement to be proved along with other definitions, postulates, and theorems already established.
Provide the student with 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 .
 and are supplementary and = d.
 and PQ = m units.
 is the bisector of .

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:
 Adds auxiliary segments ( and ) and then marks congruent segments and right angles in the diagram, and writes only, “By SAS.” No proof is provided.
 Adds auxiliary segments ( and ), states with no supporting statements and concludes that .

Questions Eliciting Thinking What do you know as a consequence of the fact that is the perpendicular bisector of ?
How can you show ? 
Instructional Implications Ask the student to indicate what it means for to be the perpendicular bisector of . Prompt the student to consider each word, perpendicular and bisector, separately and to mark the diagram accordingly. Guide the student to write a statement that corresponds to each of the marks made on the diagram and to justify them. Provide additional feedback to the student regarding any other missing statements or justifications. Review the triangle congruence theorems and provide more opportunities and experiences with proving triangles congruent.
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:
 Justify adding auxiliary segments (and ) to the diagram.
 Establish the congruence of and .
 Explicitly state that but provides all supporting statements.
 State that after showing that .

Questions Eliciting Thinking How did you know you could add these segments to your diagram? How do you know these segments are unique?
How do you know ?
Were you trying to prove ? Should this be stated in your proof?
Why does the congruence of and imply congruence of and ? 
Instructional Implications Review how to address and justify adding an auxiliary line or segment 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. For example, if using SAS to prove the triangles congruent, the proof must include showing two pairs of corresponding sides and their included angle are congruent (and a reason or justification must be provided for each).
When proving statements and theorems, ask the student to review his or her proof to be sure that the assumptions implicit in the statement were used as justifications and that the proof concludes with the statement to be proven. 
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 auxiliary segments ( and ) so that two triangles are formed, (2) Uses the fact that is the perpendicular bisector of to show by SAS, and (3) Concludes that by definition of congruent triangles. 
Questions Eliciting Thinking What if was not perpendicular to ? What can you say about the distance from point P to points A and B? 
Instructional Implications Pose the following problem variation for the student: Suppose point P is closer to A than it is to B. Determine and justify which is greater, AP or BP. 