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 parallel lines without regard to the statement to be proven.
 Uses criteria that have yet to be proven.
 States that the lines have equal slopes without any reasoning.
 Constructs two parallel lines and proves that the slopes of those two specific lines are equal.

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 equal. Did you prove only that your two parallel lines have equal slopes or that all parallel lines have equal slopes? 
Instructional Implications Describe an overall strategy for the proof (e.g., draw two parallel lines along with transversals to form “slope triangles,” show the triangles are similar, use similarity to write proportions that contain ratios that also describe the slopes of the lines, and conclude the slopes are equal). 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 nonvertical parallel lines 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 parallel lines along with appropriately drawn transversals. Ask the student to use similar triangles to write a complete and convincing mathematical argument that shows the slopes of the parallel lines are equal. 
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 sketches lines a and b and constructs two transversals. The student attempts to show that two triangles are similar and uses the similarity to conclude that that lengths of sides are proportional, but:
 States triangles are similar without proof and does not clearly establish that the slopes are equal.
 Shows the triangles are similar but then assumes the slopes are equal.
 Draws transversals that do not create “slope triangles” and incorrectly states the slopes are equal.

Questions Eliciting Thinking You said that the triangles in the diagram are similar. Why are they similar?
What is true of the sides of similar triangles? Can you write a true proportion?
What is the slope of line a? What is the slope of line b? How are the slopes related to the proportional 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.
Review how to show two triangles are similar using the AA Similarity Theorem. Remind the student that once two triangles are proven similar, then corresponding sides are proportional. Guide the student to observe that the slopes of lines a and b can be written as a ratio of corresponding sides of the triangles.
Ask the student to use the slope triangles drawn in the diagram to write expressions for the slope of each line. Guide the student to understand that the proportionality of the sides leads to the statement that the slopes are equal. Make sure the student understands that this needs to be explicitly stated in the proof.
Address any misuses of notation, for example, 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 parallel lines along with appropriately drawn transversals. Ask the student to use similar triangles to write a complete and convincing mathematical argument that shows the slopes of the parallel lines are equal. 
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 two transversals to form slope triangles. The student shows the two triangles are similar, uses the similarity to conclude that that lengths of sides are proportional, and relates the proportional lengths to the slopes of the lines, but:
 Misuses notation.
 Neglects to justify the congruence of VWX and XYZ.
 Describes the slopes as proportional instead of equal.

Questions Eliciting Thinking There is a small error in your proof. Can you find it?
What is the difference between proportional and equal? 
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  Two (GGPE.2.5), Proving Slope Criterion for Perpendicular Lines  One (GGPE.2.5), or Proving Slope Criterion for Perpendicular Lines  Two (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 parallel line a and b, draw a vertical and a horizontal transversal that intersect each other between the two lines as shown in the diagram. Since vertical angles are congruent, WXV YXZ. Also, VWX XYZ is a pair of alternate interior angles and are congruent by the Alternate Interior Angles Theorem. VXW ZXY by the AngleAngle Similarity Postulate. Since the triangles are similar, their corresponding sides are proportional, so that which is equivalent to . Since the slope of line a is and the slope of line b is , the slope of line a is equal to the slope of line b.
Or:
Given lines a and b, draw two vertical transversals intersecting parallel lines a and b to form parallelogram HIJK. Draw horizontal segments and to form two right triangles, HLI and JMK. HI = KJ by the definition of a parallelogram. HL= MJ because horizontal distances between two vertical lines are equal in length. Therefore HLI JMK by the Hypotenuse Leg Theorem. IL = MK since corresponding sides of congruent triangles are equal. The slope of line a can be written as and the slope of line b can be written as . Since IL = MK and HL = MJ, then = so that the slopes of line a and line b are equal. 
Questions Eliciting Thinking Is there another method you could have used to prove that parallel lines have equal slopes?
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 parallel lines are equal.
Challenge the student with the MFAS tasks Proving Slope Criterion for Parallel Lines  Two (GGPE.2.5), Proving Slope Criterion for Perpendicular Lines  One (GGPE.2.5), or Proving Slope Criterion for Perpendicular Lines  Two (GGPE.2.5). 