Getting Started 
Misconception/Error The student is unable to accurately describe a sequence of rigid motions that demonstrates congruence. 
Examples of Student Work at this Level The student:
 States the figures are congruent because they look the same.
 States transformations can be used to show congruence but does not describe a specific sequence.
 Describes an incorrect sequence of rigid motions.
 Provides an incomplete description.

Questions Eliciting Thinking What are rigid motions? Can you think of any examples of rigid motion?
Can you define the word congruence in terms of rigid motion?
How might you tell if two figures are congruent? Can you explain this in terms of rigid motion?
What is meant by sequence? Can you describe a sequence of rigid motions? 
Instructional Implications Review the definition of each of the rigid motions: translations, reflections, and rotations. To develop an intuitive understanding of rigid motion, allow the student to experiment with a variety of transformations using transparent paper, interactive websites such as , or the CPALMS Virtual Manipulatives Transformationsâ€”Translation (ID 11260), Transformationsâ€”Rotation (ID 11262), and Transformationsâ€”Reflections (ID 11263). Consider implementing the CPALMS Lesson Plan Polygon Transformers (ID 48156), a lesson that teaches that congruent polygons can be formed using a series of transformations (translations, rotations, reflections).
Review the definition of congruence in terms of rigid motion. Explain that two figures are congruent if there is a sequence of rigid motions that carries one figure onto the other. Assist the student in applying the definition of congruence in terms of rigid motion to show that two figures are congruent. Provide the student with two congruent figures (e.g., a pair of triangles or a pair of quadrilaterals) that are related by a single rigid motion, and ask the student to identify and describe the specific rigid motion that carries one figure onto the other. Explain to the student that describing the rigid motion in detail (e.g., by specifying the center and degree of rotation, the line of reflection, or the vector along which a figure is translated) and then performing the rigid motion is a convincing way to show that the two figures are congruent. Next provide two congruent figures that are related by more than one rigid motion. Have the student identify and describe the sequence of rigid motions that carries one figure onto the other. Ask the student to perform the sequence of rigid motions to ensure the figures are congruent. Provide assistance as needed. 
Moving Forward 
Misconception/Error The student provides only a general description of a sequence of rigid motions. 
Examples of Student Work at this Level The student states that rotating and translating triangle ABC will make it congruent to triangle .
The student does not make clear that the two figures coincide.

Questions Eliciting Thinking Can you describe the rotation more specifically? What needs to be included in the description of a rotation?
How many degrees is the triangle being rotated and in which direction? What is the center of rotation?
Can you describe the translation more specifically? How many units is the triangle being translated and in which direction?
How does the sequence of rigid motions show that triangle ABC and triangle are congruent? What must happen to show that they are congruent? 
Instructional Implications Assist the student in applying the definition of congruence in terms of rigid motion to show that two figures are congruent. Explain to the student that describing the rigid motion in detail (e.g., by specifying the center and degree of rotation, the line of reflection, or the vector along which a figure is translated) and then performing the rigid motion is a convincing way to show that the two figures are congruent. Encourage the student to be precise when describing rigid motions. Model a concise description of each motion using mathematical terminology. Then make clear that the figures are congruent because the sequence of rigid motions carries one figure onto the other.
Provide additional opportunities to show that two figures are congruent by describing a sequence of rigid motions that carry one figure onto the other. Remind the student to include all necessary components in each description, the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. 
Almost There 
Misconception/Error The student provides a detailed description of the rigid motion that demonstrates congruence, but the description contains a minor error. 
Examples of Student Work at this Level The student indicates that a rotation and a translation are necessary to demonstrate congruence but makes a minor error or omission when describing the rotation and translation. The student:
 Miscounts the number of units by which the figure is translated.
 Confuses clockwise and counterclockwise.
 Omits or refers to an incorrect center of rotation.
 Makes an error in notation (e.g., describes a vector as ).
 Uses unconventional terminology (e.g., translate â€śthree squaresâ€ť and says the triangles â€śoverlap.â€ť)

Questions Eliciting Thinking How did you determine the distance and direction of translation? Can you check your work?
How did you rotate triangle ABC? Did you provide all of the details? What is the center of rotation? 
Instructional Implications Provide specific feedback to the student concerning any error made and allow the student to revise his or her work. Confirm the studentâ€™s description of the rotation and correct any notation error. If needed, model for the student the conventional way to describe rotations and translations. Encourage the student to attend to precision.
Provide additional opportunities to show that two figures are congruent by describing a specific rigid motion (or sequence of rigid motions) that carry one figure onto the other. Remind the student to include all necessary components in each description (e.g., the center and degree of rotation, the line of reflection, or the vector along which a figure is translated) and to use notation correctly. 
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 carry triangle ABC onto triangle A'B'C'. For example:
 Rotate triangle ABC 90Â° counterclockwise about point C.
 Translate the image of triangle ABC under the rotation three units to the left or by vector .Â

Questions Eliciting Thinking Can you think of a different sequence of rigid motions to demonstrate that these two triangles are congruent? 
Instructional Implications Ask the student to use the definition of congruence in terms of rigid motion to show that two triangles in the coordinate plane are congruent. Have the student describe each rigid motion in terms of its effect on the coordinates of the vertices of the preimage [e.g., Â orÂ Â ]. 