Getting Started 
Misconception/Error The student does not understand why the sums of the lengths are equal. 
Examples of Student Work at this Level The student does not recognize that the sums of the lengths are equal.
The student recognizes that the sums of the lengths are equal but provides an explanation that does not reference any relevant information. For example, the student justifies the relationship by describing:
 Relationships among angle measures and invoking the AA Postulate.
 Lines as perpendicular and forming right triangles.

Questions Eliciting Thinking What can you conclude as a consequence of knowing that point E is the midpoint of ?
Suppose you draw diagonal ? What is the relationship between and ?
Can you determine the relationship between ED and BE? 
Instructional Implications Guide the student to understand that in order to explain why AE + ED = BE + EC , it is necessary to show that AE = EC and ED = BE. Review the Addition Property of Equality, the Commutative Property of Addition, the definition of a midpoint, and the Diagonals of a Parallelogram Bisection Theorem. Then guide the student through the statements of an explanation and prompt the student to provide justifications.
Address any misuse of notation and allow the student to make corrections on his or her paper.
Provide the student with opportunities to make deductions using a variety of previously encountered definitions and established theorems related to parallelograms. For example, provide a diagram of a parallelogram and ask the student what can be concluded about pairs of opposite angles, pairs of consecutive angles, or pairs of opposite sides. Provide a diagram of a parallelogram with its diagonals drawn and ask the student what can be concluded about the point where the diagonals intersect and specific pairs of angles that may or may not be congruent.
Challenge the student to determine if given statements about parallelograms are always true, sometimes true, or never true and to provide an explanation or justification of each answer. Statements might include:
 The diagonals are congruent.
 Opposite angles are congruent.
 The diagonals bisect each other.
 Each diagonal bisects a pair of opposite angles.
 The diagonals are perpendicular.
 The sum of the measures of the interior angles is 180 degrees.
Provide additional opportunities to explain or prove statements about parallelograms. Consider implementing other MFAS tasks aligned to standard GCO.3.11.

Moving Forward 
Misconception/Error The student writes an incomplete explanation. 
Examples of Student Work at this Level The student appears to understand why the sums of the two lengths are equal but omits important components of the explanation. For example, the student indicates that AE = EC and ED = BE (in writing or by marking the diagram) so that AE + ED and BE + EC. However, the student omits needed statements, justifications, or both. For example, the student:
 Does not explain why AE = EC, why BE = ED, and how these equalities lead to the conclusion that AE + ED = BE + EC.
 Observes that point E is the midpoint of but does not explicitly conclude that ED = BE and how the two equalities lead to the conclusion that AE + ED = BE + EC.
 Refers to a length incorrectly (e.g., refers to AE as AC) and justifies all statements by simply saying that the diagonals bisect each other.

Questions Eliciting Thinking How do you know AE = EC and BE = ED?
Given that AE = EC and BE = ED, how do you know AE + ED = BE + EC? What property justifies this conclusion?
What can specifically be concluded about the diagram as a consequence of knowing that the diagonals bisect each other? 
Instructional Implications Provide feedback to the student concerning any missing statements or justifications. Review the Addition Property of Equality, the Commutative Property of Addition, the definition of a midpoint, and the Diagonals of a Parallelogram Bisection Theorem and ask the student to revise his or her explanation. Consider asking the student to write up a formal proof of the statement.
Address any misuse of notation and allow the student to make corrections on his or her paper.
Provide additional opportunities to explain or prove statements about parallelograms. Consider implementing other MFAS tasks aligned to standard GCO.3.11. 
Almost There 
Misconception/Error The student’s explanation includes a minor error. 
Examples of Student Work at this Level The student writes an essentially correct explanation. However, the student uses notation incorrectly or makes an error in a justification.

Questions Eliciting Thinking What does this symbol mean (pointing to the congruence symbol)? What can be congruent – segments or their lengths? How should this be written?
What is the definition of a midpoint? What is the midpoint formula? Which allows you to conclude that AE = EC? 
Instructional Implications Provide feedback to the student concerning any notational errors and allow the student to correct his or her paper. Assist the student in correcting any statements or justifications.
Provide additional opportunities to explain or prove statements about parallelograms. Consider implementing other MFAS tasks aligned to standard GCO.3.11. 
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 determines that AE + ED = BE + EC and provides a complete and correct explanation of this result such as the following:
Since E is the midpoint of , AE = EC (by definition of a midpoint). Since the diagonals of a parallelogram bisect each other (by the Diagonals of a Parallelogram Bisection Theorem), point E is the midpoint of so that ED = BE. Consequently, AE + ED = EC + BE (by the Addition Property of Equality) and AE + ED = BE + EC (by the Commutative Property of Addition). 
Questions Eliciting Thinking How do you know that diagonal intersects diagonal at point E?
How could you use congruent triangles to show that AE = EC and ED = BE? 
Instructional Implications Adapt the problem to the context of an isosceles trapezoid. For example, given isosceles trapezoid TRAP with  and diagonals that intersect at point E, show AE + EP = RE + ET. 