Getting Started 
Misconception/Error The student is unable to correctly identify the coordinates of vertex C in terms of the given coordinates. 
Examples of Student Work at this Level The student
 Identifies the coordinates of vertex C as (b, a).
 Introduces new variables to describe the coordinates of vertex C, e.g., C(c, d).

Questions Eliciting Thinking Where would the point with coordinates (a, b) be located?
Can you describe the coordinates of vertex C in terms of the coordinates of vertices A and B? 
Instructional Implications Explain how to represent the coordinates of vertex C in terms of the coordinates of vertices A and B. Provide other examples of figures graphed on the coordinate plane such as:
 A square with one of its vertices is at the origin and sides that coincide with the axes.
 A right triangle with the vertex of the right angle at the origin and sides that coincide with the axes.
 An isosceles triangle with a base that coincides with the xaxis and positioned so that the yaxis is an axis of symmetry.
And ask the student to use the least number of variables to describe the coordinates of the vertices. Then introduce more challenging figures such as parallelograms, rhombuses, isosceles trapezoids, and equilateral triangles.
Guide the student to develop an overall strategy for solving the problem presented in this task, i.e., (1) graph the vertices, (2) identify the coordinates of the fourth vertex, (3) calculate the lengths of the diagonals using the distance formula, and (4) conclude that the diagonals are congruent. Ask the student to implement the strategy and provide feedback.
If needed, review the distance formula and provide additional opportunities for the student to use the formula to calculate lengths in the coordinate plane.
If needed, provide feedback on the appropriate use of notation. 
Moving Forward 
Misconception/Error The student does not use the distance formula correctly or at all to find the lengths of the diagonals. 
Examples of Student Work at this Level The student correctly identifies the coordinates of vertex C as (a, b) but, to prove the diagonals are congruent, the student:
 Says to find the slope of each diagonal.
 References the distance formula but then uses the midpoint formula.
 Says to use the distance formula but provides no additional work.
 Applies the distance formula incorrectly.

Questions Eliciting Thinking What must be true of the diagonals in order for them to be congruent? How can you show this?
If you are asked to prove that the diagonals are congruent, is it sufficient to say, “just use the distance formula?”
Can you use the distance formula to calculate the lengths of the diagonals? 
Instructional Implications Review the distance formula and provide additional opportunities for the student to use the formula to calculate lengths in the coordinate plane.
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 makes an algebraic error. 
Examples of Student Work at this Level The student rewrites the expression as a + b.
The student may also not show work completely or label work appropriately. 
Questions Eliciting Thinking Is really equal to a + b? Can you prove that?
You showed that you were using the distance formula to calculate the length of a diagonal. Can you tell me which length this is? Can you label that on your paper? 
Instructional Implications Provide a counterexample to demonstrate that . For example, ask the student to evaluate each expression for a = 3 and b = 4 to show that the expressions are not equivalent.
If needed, guide the student to show work completely, label all work appropriately, and use correct notation.
Consider implementing MFAS tasks Describe the Quadrilateral (GGPE.2.4), Midpoints of Sides of a Quadrilateral (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 states the coordinates of vertex C are (a, b). The student uses the distance formula to correctly calculate the length of each diagonal. The student then concludes that since both diagonals have the same length, , they are congruent.

Questions Eliciting Thinking Would this proof be valid for a rectangle that is positioned differently from the one described on the worksheet?
How could you show that the diagonals bisect each other? 
Instructional Implications If needed, provide feedback to the student on any aspect of his or her work that might improve it (e.g., using the notation AB and CD rather than d to represent the lengths of and , respectively, or enclosing ordered pairs in parentheses).
Challenge the student to prove geometric theorems using coordinate geometry. Consider implementing MFAS task Triangle MidSegment Proof (GCO.3.10).
Give the student the coordinates of three of the vertices of a parallelogram ABCD, e.g., A(0, 0). B(b, 0), and D(a, c). Ask the student to identify the coordinates of vertex C and prove statements such as:
 The diagonals of a parallelogram bisect each other.
 Opposites sides of a parallelogram are congruent.
