Getting Started 
Misconception/Error The student is unable to accurately describe a rigid motion that demonstrates congruence. 
Examples of Student Work at this Level The student:
 Writes that the figures are congruent because they look the same.
 Describes an incorrect rigid motion (e.g., rotation or reflection).
 Claims that “a transformation was made” with no reference to which one.

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? 
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 http://www.mathopenref.com/translate.html, or the CPALMS Virtual Manipulatives Transformations—Translation (ID 11260), Transformations—Rotation (ID 11262), and Transformations—Reflections (ID 11263). Consider implementing the CPALMS Lesson Plan Polygon Transformers (ID 48156), which 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 the rigid motion that demonstrates congruence. 
Examples of Student Work at this Level The student simply writes “translation” or “6 right, 5 down.”
The student does not make clear that the two figures coincide.

Questions Eliciting Thinking Can you describe the translation more specifically? How many units is the parallelogram being translated and in which direction?
How does translating parallelogram ABCD show that the parallelogram ABCD and parallelogram are congruent? What must happen to show that they are congruent? 
Instructional Implications Explain that a translation is a transformation of the plane. A transformation along a vector (or a vector decomposed into horizontal and vertical translations) assigns to each point, P, in the plane an image point, , so that the distance from to P corresponds to the magnitude (length) of the vector and the direction of from P corresponds to the direction of the vector. Use grid paper to illustrate translations of points and to demonstrate the relationship between a point, its image, and the vector that defines the translation. Then illustrate translations of more complex figures such as segments, angles, and polygons. Discuss the basic properties of translations (e.g., translations map lines to lines, rays to rays, and segments to segments; translations are both distance preserving and degree preserving) and how these properties ensure that the image of a figure under a translation is always congruent to the preimage.
Encourage the student to be precise when describing translations. Model a concise description of a translation using mathematical terminology. Then make clear that the figures are congruent because the translation carries one figure onto the other.
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, 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:
 Miscounts the number of units in which the figure is translated (e.g., says to translate seven units to the right and five units down).
 Represents the number of units in which the figure is translated as an ordered pair [e.g., as (6, 5)].

Questions Eliciting Thinking How did you determine that parallelogram ABCD can be translated seven units to the right and five units down? Can you check this?
You miscounted the number of units that parallelogram ABCD would be translated in order to coincide with parallelogram . Can you correct your error?
What does (6, 5) represent? What is an ordered pair? How does “moving to the right six units and then down five units” compare to being at (6, 5) on a coordinate plane? 
Instructional Implications Provide specific feedback to the student concerning any error made and allow the student to revise his or her work. Encourage the student to attend to precision (MP.6).
Confirm the student’s description of “six units to the right and five units down,” but correct any notation error. Make explicit the difference between an ordered pair and translation notation. Show the student the conventional ways to describe a translation (e.g., by using a verbal description, algebraic notation, and vector notation). Use the translation in this task to model each:
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. Pair the student with a Got It partner and have the students compare their descriptions and reconcile any differences. 
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:
 How parallelogram ABCD can be translated five units to the right and six units down so that it coincides with parallelogram ,
 How each vertex (A, B, C, and D) can be translated five units to the right and six units down to coincide with , , , and and explains that if the vertices coincide the parallelograms will coincide, or
 Describes or draws a vector along which parallelogram ABCD can be translated (e.g., ) so that it coincides with parallelogram .

Questions Eliciting Thinking How do you know the parallelograms will coincide if their vertices coincide?
Are there other rigid motions you could have used to show the parallelograms are congruent?
Can you think of a sequence of rigid motions (or transformations) that can be used to show that parallelogram ABCD is congruent to parallelogram ? 
Instructional Implications Consider implementing the MFAS tasks Rigid Motion  2 and Rigid Motion  3 (8.G.1.2), to assess the student’s ability to describe other onestep transformations to demonstrate congruence; or the MFAS task Multistep Congruence (8.G.1.2), to assess the student’s ability to describe multistep transformations to demonstrate congruence. 