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 information but is unable to go any further.
 Makes some observations about lines with slopes that are opposite and reciprocal without regard to the statement to be proven.
 States that the lines are perpendicular without any reasoning or using circular reasoning.

Questions Eliciting Thinking What information were you given?
What can you assume is true about lines a and b? What are you being asked to prove?
What does it mean for two lines to be perpendicular? If line a and line b are perpendicular, what kind of angle will their intersection form?
Did you think of a plan for your proof before you started? How might you show that line a and line b intersect in a right angle? 
Instructional Implications Describe an overall strategy for the proof (e.g., draw a horizontal line and a vertical line that are concurrent, draw slope triangles on line a and line b each with a vertex at the point of concurrency, show the slope triangles are similar using the assumption that the slopes are opposite and reciprocal, use the similarity of the triangles to show that the two angles that compose the right angle formed by the horizontal and vertical lines are congruent to the two angles that compose the angle of intersection of line a and line b, and then conclude that because this angle is a right angle, the lines are perpendicular). Provide the student with an appropriately drawn diagram and the statements of this proof and ask the student to supply the justifications.
Remind the student that the statement to be proven is general and applies to any pair of lines with slopes that are opposite and reciprocal and not just to one specific case.
Provide the student with a diagram that includes line a, line b, a horizontal line, and a vertical line drawn so that they are concurrent. Ask the student to assume that lines a and b have slopes that are both opposite and reciprocal and prove that line a and line b are perpendicular. 
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 constructs a slope triangle on each line. The student:
 Does not first establish that the triangles are similar but is able to use this result to complete the explanation.
 Establishes that the triangles are similar but does not clearly establish that the lines are perpendicular.

Questions Eliciting Thinking You said that the triangles in the diagram are similar. Why are they similar?
What is true of the corresponding angles of similar triangles?
What kind of angle is formed by the horizontal and vertical lines? How is this angle related to the angles of the slope triangles?
What do you need to show in order to say that line a and line b are perpendicular? 
Instructional Implications Review the overall strategy used in the student’s proof and provide feedback concerning any aspect of the proof that is incomplete or requires revision.
Address any misuses of notation (e.g., confusing measures of angles with their names, naming an angle with one letter when three letters are required, or writing similarity symbols incorrectly).
Provide the student with a diagram that includes line a, line b, a horizontal line, and a vertical line drawn so that they are concurrent. Ask the student to assume that lines a and b have slopes that are both opposite and reciprocal and prove that line a and line b are perpendicular. 
Almost There 
Misconception/Error The student provides a correct response but with insufficient reasoning or imprecise language. 
Examples of Student Work at this Level The student draws a horizontal line and a vertical line that are concurrent along with slope triangles on line a and line b each with a vertex at the point of concurrency. However, the student:
 Assumes that the lengths of the corresponding sides of the slope triangles are congruent instead of proportional.
 Uses the SideAngleSide Similarity Theorem but does not address the congruence of a pair of angles.
 States that mMJL + mNJM = 90 without first clearly establishing that mMJK = mMJL + mLJK = 90 and mLJK = mNJM.
 Misuses notation.

Questions Eliciting Thinking Did you show that these triangles are similar or congruent?
You used the SAS Similarity Theorem but did you state that the included angles were congruent?
How do you know mMJL + mNJM = 90? 
Instructional Implications Provide feedback to the student concerning any omissions or statements that need more justification and allow the student to revise his or her proof.
Address any misuses of notation (e.g., confusing measures of angles with their names, naming an angle with one letter when three letters are required, or writing similarity symbols incorrectly).
Challenge the student with the MFAS tasks Proving Slope Criterion for Parallel Lines  1 (GGPE.2.5), Proving Slope Criterion for Parallel Lines  2 (GGPE.2.5), or Proving Slope Criterion for Perpendicular Lines  1 (GGPE.2.5). 
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 provides a complete proof with justification such as:
Let J be the point of intersection of lines a and b. Draw vertical line and horizontal line . Since is vertical and is horizontal, MJK is a right angle. Next, draw slope triangles, and . The slope of line a can be written as while the slope of line b can be written as . Since the slopes are reciprocal then. Since both JKL and JMN are slope triangles, mM = = mK. Therefore, JKL ~ JMN by the SideAngleSide Similarity Theorem. Since corresponding angles of similar triangles are congruent, . Since mMJK = mMJL + mLJK = and (i.e., mLJK = mNJM) then + = . Therefore, line a is perpendicular to line b.

Questions Eliciting Thinking What exactly is a slope triangle?
Is there another method you could use to prove two lines are perpendicular if they have slopes that are opposite reciprocals?
Were there any statements in your proof that you did not really need? 
Instructional Implications Ask the student to use coordinate geometry to show that lines with slopes that are both opposite and reciprocal are perpendicular.
Challenge the student with MFAS tasks Proving Slope Criterion for Parallel Lines  1 (GGPE.2.5), Proving Slope Criterion for Parallel Lines  2 (GGPE.2.5), or Proving Slope Criterion for Perpendicular Lines  1 (GGPE.2.5). 