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 perpendicular lines without regard to the statement to be proven.
 States that the lines have slopes that are opposite reciprocals without any reasoning.
 Draws two perpendicular lines and proves that the slopes of those two specific lines are opposite reciprocals.

Questions Eliciting Thinking What information were you given? Did you sketch lines a and b?
What can you assume is true about lines a and b? What are you being asked to prove?
Did you think of a plan for your proof before you started?
I see that you found the slope of each of yours lines and said they were opposite reciprocals. Did you prove only that these two perpendicular lines have slopes that are opposite and reciprocal or that all perpendicular lines have slopes that are opposite and reciprocal? 
Instructional Implications Describe an overall strategy for the proof (e.g., draw two perpendicular lines along with a “slope triangle” on one of the lines, rotate the slope triangle , show that it is now a slope triangle of the perpendicular line, and then write out expressions for the slopes of the lines showing that they are opposite and reciprocal). 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 perpendicular lines (except for a horizontal line and a vertical line) and not just to one specific case. Guide the student to describe lengths using variables instead of specific values.
Provide the student with a diagram that includes a pair of perpendicular lines along with an appropriately drawn slope triangle. Ask the student to use a rotation to write a complete and convincing mathematical argument that shows the slopes of the perpendicular lines are both opposite and reciprocal. 
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 draws lines a and b and constructs a slope triangle on one line. The student rotates the slope triangle onto the other line and attempts to show that slopes are opposite and reciprocal, but:
 Does not make clear the relationship between line b and the image of the triangle and attempts to use the congruence of the slope triangle and its rotated image.
 Does not establish that the original slope triangle is also a slope triangle of the perpendicular line.
 Does not clearly establish that the slopes are opposite and reciprocal.

Questions Eliciting Thinking You correctly stated that the triangles in the diagram are congruent, but how is the image triangle related to line b?
How do you know that the image of the slope triangle is also a slope triangle on line b?
What is the slope of line a? What is the slope of line b? How are the slopes related to the lengths of the sides of the triangles? 
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.
Assist the student in understanding that the relationship between line b and the image of the slope triangle must be clearly established (otherwise, statements about the image of the slope triangle have no implications for line b). Guide the student to observe that the slopes of line a and line b can be written as ratios of corresponding sides of the triangles. Remind the student that the definition of rotation preserves distance, which allows for the conclusion that the components of the slope of line a are equal to the components of the slope of line b.
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 a pair of perpendicular lines along with an appropriately drawn slope triangle. Ask the student to use a rotation to write a complete and convincing mathematical argument that shows the slopes of the perpendicular lines are opposite reciprocals. 
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 sketches lines a and b and draws a slope triangle on one line and then rotates that slope triangle onto the other line. The student uses the fact that one line must have a positive slope and the other line must have a negative slope and that rotations preserve distance so that the lines have slopes that are opposite reciprocals, but:
 Misuses notation.
 Does not specify the direction of the rotation.
 Describes the slopes only as reciprocals instead of both opposite and reciprocal.

Questions Eliciting Thinking There is a small error in your proof. Can you find it?
Can you be more specific in your description of the rotation?
What do you know about the slopes of your two lines? Are they both positive? Are they both negative? 
Instructional Implications Provide feedback to the student concerning any errors made 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  2 (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:
Given perpendicular lines a and b, draw and to form a slope triangle on line a, as shown. Then rotate JKL counterclockwise about point J to create . Since line b is perpendicular to line a, L maps to a point, L', on line b so that of , is contained in line b. Since rotations preserve angle measure, is a right angle (as is ) and is a slope triangle of line b. The slope of line a can be written as and the slope of line b can be written as . Since rotations preserve distance, and so that . Therefore, the slopes of line a and line b are both opposite and reciprocal. 
Questions Eliciting Thinking What exactly is a slope triangle?
Is there another method you could use to prove that perpendicular lines 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 devise a coordinate geometry proof that the slopes of perpendicular lines are opposite reciprocals.
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  2 (GGPE.2.5). 