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.
 Uses criteria that have yet to be proven.
 Makes some observations about lines with equal slopes without regard to the statement to be proven.
 States that the lines are parallel without any reasoning.
 Draws two lines with equal slopes and proves that those two specific lines are parallel.

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 your 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 a diagram that includes “slope triangles,” use the fact that the slopes of the lines are equal to show that the slope triangles are similar, and use the consequences of similarity to show that a pair of angles are congruent that will lead to the conclusion that the lines are parallel). Provide the student with an appropriately drawn diagram and the statements of the 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 lines with equal slopes along with appropriately drawn transversals. Ask the student to use similar triangles to write a complete and convincing mathematical argument that shows the two lines are parallel. 
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 lengths of sides are proportional and concludes that the two triangles are similar, but:
 States the lengths of the sides are proportional without proof.
 States the triangles are similar without proof.
 Proves the triangles are similar but then assumes the lines are parallel.

Questions Eliciting Thinking What do you know about the slopes of these lines? Can you write that in a proportion?
What do you know about similar triangles? How can you prove two triangles are similar?
Can you mark which angles are congruent? What type of angles are these?
What must be true if the alternate interior angles are congruent? 
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.
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. Review how to show two triangles are similar by proving that corresponding sides are proportional. Remind the student that once two triangles are proven similar, then the corresponding angles are congruent.
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 since the slopes are equal that leads to the proportionality of the sides. Make sure the student understands that this needs to be explicitly stated in the 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).
Provide the student with a diagram that includes a pair of lines with equal slopes along with appropriately drawn transversals. Ask the student to use similar triangles to write a complete and convincing mathematical argument that shows the two lines are parallel. 
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 writes a proportion to show that the two triangles are similar, and uses the similarity to conclude that that the corresponding angles are congruent, but:
 Misuses notation.
 Neglects to state that VWX and XYZ are corresponding angles of similar triangles and are, therefore, VWX XYZ.
 Provides an incorrect justification (e.g., justifies the congruence of two vertical angles by citing the definition of vertical angles).

Questions Eliciting Thinking There is a small error in your proof. Can you find it?
Why are the lines parallel? Did you state that in your proof? 
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  One (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 lines a and b, draw a vertical and a horizontal transversal that intersect each other between the lines as shown in the diagram. The slope of line a can be given as and the slope of line b can be given as . Since the slopes are equal (by assumption), then = . Also VXW YXZ by the Vertical Angle Theorem so that VXW ZXY by the SideAngleSide Similarity Theorem. Since VWX and XYZ are corresponding angles of similar triangles, VWX XYZ. Consequently, line a is parallel to line b (by the converse of the Alternate Interior Angles Theorem).

Questions Eliciting Thinking Is there another method you could have used to prove that lines with the same slope are parallel?
Were there any statements in your proof that you did not really need? 
Instructional Implications Ask the student to devise a coordinate geometry proof to show that lines with the same slope are parallel.
Challenge the student with MFAS tasks Proving Slope Criterion for Parallel Line  One (GGPE.2.5), Proving Slope Criterion for Perpendicular Lines  One (GGPE.2.5), or Proving Slope Criterion for Perpendicular Lines  Two (GGPE.2.5). 