Getting Started 
Misconception/Error The student does not have an effective strategy for completing the proof. 
Examples of Student Work at this Level The student calculates (correctly or incorrectly) the coordinates of the midpoints of the sides (E, F, G and H) but shows no additional work.

Questions Eliciting Thinking What are you asked to show in this problem?
How can you show a quadrilateral is a parallelogram? 
Instructional Implications Review the conditions that are necessary and sufficient for a quadrilateral to be a parallelogram, i.e., opposites sides are parallel, opposite sides are congruent, opposite angles are congruent, the diagonals bisect each other, and one pair of opposite sides is both parallel and congruent. Discuss with the student how to use the slope formula to show that segments are parallel, the distance formula to show that segments are congruent, and the midpoint formulas to show that segments bisect each other.
Guide the student to develop an overall strategy for solving the problem presented in this task, i.e., (1) find the midpoints of the sides of the quadrilateral, (2) select and implement a strategy for showing the quadrilateral is a parallelogram (3) explicitly draw the appropriate conclusion. Ask the student to implement the strategy and provide feedback.
Give the student additional opportunities to use the slope, distance, and midpoint formulas in a variety of problem contexts. For example, ask the student to conclude what is true as a consequence of:
 Two segments having the same slope.
 Two segments having opposite reciprocal slopes.
 Two segments having the same length.
 The two diagonals of a quadrilateral having the same midpoint.
If needed, provide feedback on the appropriate use of notation.

Moving Forward 
Misconception/Error The student does not explicitly draw an appropriate conclusion to complete the proof. 
Examples of Student Work at this Level The student correctly calculates the coordinates of the midpoints of the sides (E, F, G and H) and provides work to support the conclusion that the figure is a parallelogram but does not explicitly draw this conclusion.

Questions Eliciting Thinking Can you explain how you showed the figure is a parallelogram?
What is your conclusion? Is the quadrilateral a parallelogram? 
Instructional Implications Discuss with the student how to write a clear and complete proof. Show the student a model coordinate geometry proof of another statement and point out all of the features that make it clear and convincing (e.g., steps are presented in a logical order, all work is labeled, computations are clearly presented, conclusions are explicitly stated, and no extraneous work is left on the paper).
If needed, provide feedback on the appropriate use of notation. 
Almost There 
Misconception/Error The student does not use mathematical terminology or notation correctly. 
Examples of Student Work at this Level The student correctly calculates the coordinates of the midpoints of the sides (E, F, G and H) ,shows appropriate work to prove the figure is a parallelogram, and provides an explicitly stated conclusion, but:
Uses notation incorrectly, e.g., refers to as EH.
Uses the term congruent to describe slopes that are equal.

Questions Eliciting Thinking What is the difference between and EH?
What does congruent mean? Can slopes be congruent? 
Instructional Implications Provide direct feedback to the student regarding his or her use of notation. Review the use of notation with regard to naming lines, segments, and rays and referring to the length of a segment. Provide the student with examples of the use of notation some of which contain errors. Ask the student to identify statements in which notation has been used incorrectly and to rewrite the statements so that they are written correctly.
Consider implementing MFAS tasks Describe the Quadrilateral (GGPE.2.4), Diagonals of a Rectangle (GGPE.2.4), and Triangle MidSegment Proof (GCO.3.10). 
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 correctly calculates the coordinates of the midpoints of the sides (E, F, G and H) ,shows appropriate work to prove the figure is a parallelogram, and provides an explicitly stated conclusion. In addition, the student uses notation correctly. 
Questions Eliciting Thinking Do you think joining the midpoints of any quadrilateral will create a figure that is a parallelogram? How could you test this idea?
What if the quadrilateral is concave? Will the figure formed by connecting the midpoints of the sides be a parallelogram? How could you explore this? 
Instructional Implications Challenge the student to prove geometric theorems using coordinate geometry. Consider implementing MFAS task Triangle MidSegment Proof (GCO.3.10).
Consider implementing other MFAS quadrilateral tasks Diagonals of a Rectangle (GGPE.2.4) and Describe the Quadrilateral (GGPE.2.4). 