Getting Started 
Misconception/Error The student does not address the congruence of the two trapezoids using the definition in terms of rigid motion. 
Examples of Student Work at this Level The student explains that the trapezoids are congruent because:
 They remain congruent when they are transformed.
 They are the same shape and size.
 The “unit measures” are not changed.

Questions Eliciting Thinking What is the definition of congruence in terms of rigid motion?
How can one figure be shown to be congruent to another using this definition? 
Instructional Implications Review the definition of congruence in terms of rigid motions (e.g., two polygons are congruent if a composition of a finite number of basic rigid motions maps one to the other). Explain that in order to show the two trapezoids are congruent a specific and detailed sequence of rigid motions that maps one trapezoid to the other must be described. If needed, review the definitions of the basic rigid motions (translation, reflection, and rotation) and how to perform and describe each.
Guide the student to identify a specific sequence of rigid motions that will map one of the trapezoids to the other. Emphasize that the student should clearly identify when corresponding vertices coincide. Explain that if the vertices coincide, then the sides will coincide as well (since two points define a unique line). Assist the student in describing each rigid motion in sufficient detail. Guide the student to conclude the description with an explicit statement that explains that since one trapezoid can be mapped to the other, the trapezoids are congruent.
Ask the student to identify another sequence of rigid motions that will map one of the trapezoids to the other. Ask the student to provide a detailed description of each rigid motion, indicate when vertices coincide, and explicitly conclude the trapezoids are congruent and explain why.
Provide additional opportunities to show that two polygons are congruent using the definition of congruence in terms of rigid motion. 
Moving Forward 
Misconception/Error The student attempts to apply the definition of congruence in terms of rigid motion but makes an error in one of the transformations. 
Examples of Student Work at this Level The student:
 Attempts to describe a single rotation that will map one trapezoid to the other but does not correctly identify the center of the rotation.
 Describes a reflection followed by a rotation but the rotations will not result in the trapezoids coinciding.

Questions Eliciting Thinking What is the center of the rotation? Can you perform the rotation and draw the result?
Where is the reflection line? Can you perform the reflection and draw the result?
I think there is an error in one of your rigid motions. Can you review your work and try to find it? 
Instructional Implications Ask the student to perform the rigid motions he or she describes. Provide feedback to the student and assist the student in revising his or her work. Emphasize that the student should clearly identify when corresponding vertices coincide. Explain that if the vertices coincide, then the sides will coincide as well (since two points define a unique line). Assist the student in describing each rigid motion in sufficient detail. Guide the student to conclude the description with an explicit statement that explains that since one trapezoid can be mapped to the other, the trapezoids are congruent.
Ask the student to identify another sequence of rigid motions that will map one of the trapezoids to the other. Ask the student to provide a detailed description of each rigid motion, indicate when vertices coincide, and explicitly conclude the trapezoids are congruent and explain why.
Provide additional opportunities to show that two polygons are congruent using the definition of congruence in terms of rigid motion. 
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 describes a sequence of rigid motions that will successfully map one trapezoid to the other. However, the student’s description lacks an important detail. For example, the student does not:
 Provide a complete description of a rigid motion although the student draws the rigid motion correctly.
 Indicate which vertices coincide as a result of each rigid motion.
 Justify that the trapezoids are congruent because the described sequence shows that one can be mapped to the other.

Questions Eliciting Thinking It appears that you drew the transformed trapezoid correctly after each rigid motion but did not provide all of the details in your description(s). Can you indicate the center of the rotation (or the direction or degree measure)? Can you indicate the reflection line?
How do you know that the sequence you described maps one trapezoid to the other? Which vertices will coincide after each transformation?
Are the trapezoids congruent? Can you state that the trapezoids are congruent and explain why? 
Instructional Implications Provide feedback to the student concerning ways to improve his or her response. Emphasize that the student should clearly identify when corresponding vertices coincide. Explain that if the vertices coincide, then the sides will coincide as well (since two points define a unique line). Assist the student in describing each rigid motion in sufficient detail. Guide the student to conclude the description with an explicit statement that explains that since one trapezoid can be mapped to the other, the trapezoids are congruent.
Provide additional opportunities to show that two polygons are congruent using the definition of congruence in terms of rigid motion. 
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 describes a sequence of rigid motions that maps one trapezoid to the other. For example, the student says:
 Rotate trapezoid EFGH clockwise about point F so that vertex G coincides with vertex D and vertex H coincides with vertex C.
 Reflect the rotated image of trapezoid EFGH across so that vertex F coincides with vertex A and vertex E coincides with vertex B.
 Since the four vertices of trapezoid EFGH now coincide with the four vertices of trapezoid BADC, trapezoid EFGH has been mapped to trapezoid BADC which shows the trapezoids are congruent.

Questions Eliciting Thinking Is the alignment of the vertices of the trapezoids sufficient to infer congruence? Is it possible that the vertices of two trapezoids coincide but one has not been mapped to the other? Do two points determine a unique segment? 
Instructional Implications Ask the student to position a pair of axes in the diagram and then describe each transformation algebraically [e.g., in terms of the effect on the coordinates of P(x, y)]. Suggest the student think carefully about the location of the origin. 