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:
 Describes an incorrect rigid motion (e.g., rotation or translation).
 Claims â€śa transformation was madeâ€ť with no reference to which one.
 Writes that the figures are congruent because they look the same.Â

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 , 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), 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 â€śreflectionâ€ť or â€śflip.â€ť
The student does not make clear that the two figures coincide.

Questions Eliciting Thinking Can you describe the translation more specifically?
In which direction is the image reflected? Is there a line of symmetry?
How does reflecting pentagon ABCDE show that the pentagon ABCDE and pentagon are congruent? What must happen to show they are congruent? 
Instructional Implications Explain that a reflection is a transformation of the plane across a line of symmetry (also called a line of reflection). A reflection across line m (the line of reflection) assigns to each point not on line m, a point that is symmetric to itself with respect to line m. Use grid paper to illustrate reflections of points and to demonstrate the relationship between a point, its image, and the line of reflection. Then illustrate reflections of more complex figures such as segments, angles, and polygons. Discuss the basic properties of reflections (e.g., reflections map lines to lines, rays to rays, and segments to segments; reflections are both distance preserving and degree preserving) and how these properties ensure the image of a figure under a reflection is always congruent to the preimage.
Encourage the student to be precise when describing reflections. Model a concise description of a reflection using mathematical terminology using terms such as preimage, image, reflection line, and symmetry. Then make clear that the figures are congruent because the reflection 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:
 Refers to the line of reflection as the â€śxaxisâ€ť instead of â€śline m.â€ť
 Refers to the reflection as a horizontal flip over line m.

Questions Eliciting Thinking Are you sure the line of reflection is the xaxis? Does the line of reflection always have to be on one of the axes?
Is the line of reflection (or line of symmetry) a horizontal or vertical line? Is the movement of pentagon ABCDE over that line a horizontal or vertical movement? 
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 reflection and correct any notation error. If needed, model for the student the conventional way to describe a reflection. 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, 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 pentagon ABCDE can be reflected vertically over line m, so that it coincides with pentagon .
 How each vertex (A, B, C, D, and E) can be reflected over the line of symmetry (line m) to coincide with , , , and and explains that if the vertices coincide, the pentagons will coincide.Â

Questions Eliciting Thinking How do you know the pentagons will coincide if their vertices coincide? Are there other rigid motions you could have used to show the pentagons are congruent? Can you think of a sequence of rigid motions (or transformations) that can be used to show that pentagon ABCDE is congruent to pentagon ? 
Instructional Implications Consider implementing the MFAS tasks Rigid Motion  1Â (if not done previously) and Rigid Motion  3Â to assess the studentâ€™s ability to describe other onestep transformations that demonstrate congruence, or the MFAS task Multistep CongruenceÂ to assess the studentâ€™s ability to describe multistep transformations that demonstrate congruence. 