Getting Started 
Misconception/Error The student does not understand the need to show that the vertices of one triangle coincide with corresponding vertices of the other triangle in order to show the triangles are congruent. 
Examples of Student Work at this Level The student:
 Attempts to describe a sequence of rigid motions that maps one triangle to the other without regard to the assumptions or the need to establish that corresponding vertices coincide.
 Provides a vague or incomplete argument.
 Says the triangles are congruent because of the SAS congruence Theorem.

Questions Eliciting Thinking Based on the given information, what can you assume is true about and ?
What does the congruence statement indicate about the corresponding sides and angles? Does the sequence of rigid motions you have listed align the vertices of each triangle as indicated in the congruence statement?
In general, how can you show two triangles are congruent using rigid motion? What do you need to show about their vertices?
Can you use a theorem in its own proof? 
Instructional Implications Allow the student to use transparent paper to initially model a sequence of rigid motions that maps to . Then assist the student in describing each rigid motion in adequate detail. Provide the student with several other pairs of congruent triangles and have the student identify the sequence of rigid motions that maps one triangle to the other. Allow the student to use transparent paper as an aid, if needed.
Explain to the student that a convincing explanation of the congruence of two triangles involving rigid motion includes showing how each of the vertices of one triangle must coincide with corresponding vertices of the other triangle. Consequently, a good first step is to map a vertex of one triangle to a vertex of the other triangle (guided by any assumption about congruent angles or sides) since this ensures that a first pair of vertices will coincide. The next step is to use any additional assumptions, the properties of rigid motion, and other useful postulates and theorems to show the remaining two pairs of vertices coincide. Model this process by explaining why two triangles must be congruent when two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle (SAS). Make clear how each of the assumptions is used in the explanation and that a statement cannot be used in its own proof.
Explore other congruence postulates (e.g., ASA, SSS, HL, and AAS) and guide the student to explain, using rigid motion, why each of these patterns of congruence ensures the congruence of triangles. 
Moving Forward 
Misconception/Error The student provides an incomplete explanation. 
Examples of Student Work at this Level The student understands the need to prove that corresponding vertices align and shows how to map one or two vertices of one triangle onto corresponding vertices of the other but fails to justify that the remaining vertices coincide. For example, the student describes a translation that results in point B coinciding with point E followed by a rotation that results in aligning with . However, the student then restates the given and concludes the triangles are congruent rather than showing how the remaining vertices coincide.
The student may also omit some necessary detail(s) in describing rigid motions.

Questions Eliciting Thinking Can you describe in more detail how this sequence of rigid motions will map one triangle onto the other?
In general, how can you show two triangles are congruent using rigid motion? What do you need to show about their vertices?
You showed how vertex A coincides with vertex D and how vertex B coincides with vertex E. What about the remaining pair of vertices? How do you know they will coincide? 
Instructional Implications Explain to the student that a convincing explanation of the congruence of two triangles involving rigid motion includes showing how each of the vertices of one triangle must coincide with corresponding vertices of the other triangle after a sequence of rigid motions. Consequently, a good first step is to map a vertex of one triangle to a vertex of the other triangle (guided by any assumption that a pair of angles is congruent) since this ensures that a first pair of vertices will coincide. The next step is to use any additional assumptions, the properties of rigid motion, and other useful postulates and theorems to show the remaining two pairs of vertices coincide. Model this process by explaining why two triangles must be congruent when two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle (SAS). Make clear how each of the assumptions is used in the explanation and that a statement cannot be used in its own proof.
Explore other congruence postulates (e.g., ASA, SSS, HL, and AAS) and guide the student to explain, using rigid motion, why each of these patterns of congruence ensures the congruence of triangles. 
Almost There 
Misconception/Error The student uses rigid motion to develop a convincing argument but leaves out some detail. 
Examples of Student Work at this Level The student describes a sequence of rigid motions that maps the vertices of one triangle to corresponding vertices of the other triangle. However, the student omits some important detail. For example, the student:
 Suggests rotating but does not indicate a center or direction of rotation.
 States that points A and D will coincide without using the assumption that .
 Does not appeal to any properties of rigid motion such as the preservation of angle measure.

Questions Eliciting Thinking About what point should be rotated? How far should it be rotated?
I understand from your explanation why points A and D will lie on the same line, but how do you know they will coincide?
How do you know Â must align with ? Can you explain this in more detail? 
Instructional Implications Provide feedback to the student and allow the student to revise his or her explanation. Remind the student to be as clear and concise as possible when describing rigid motions, identifying the centers of rotation, the lines of reflection, and vectors that describe translations. If possible, have the student ask a classmate to read his or her explanation to see if it can be followed as written. Remind the student to always use the given assumptions and be sure to include how each vertex of one triangle coincides with the vertices of the other.
Explain to the student the need for a specific description for the location of the line of reflection and the point of rotation. Demonstrate using graph paper or interactive software how moving the line of reflection or the center of rotation alters the location of the image.
Explore other congruence postulates (e.g., ASA, SSS, HL, and AAS) and guide the student to explain, using rigid motion, why each of these patterns of congruence ensures the congruence of triangles. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level A sequence of rigid motions is described that maps vertex B to vertex E (or vice versa). Then using the other two assumptions, the student explains how the remaining two vertices of the two triangles coincide providing adequate explanation and justification. For example, the student might explain:
 Translate point B to point E according to so that vertexÂ B coincides with vertexÂ E (B = E).
 Rotate Â clockwise about point E until aligns with .
 Since rigid motion preserves length andÂ , vertexÂ A now coincides with point D (A = D).
 Since rigid motion preserves angle measure and , must align with .
 Since then BC = EF, so that point C now coincides with point F (C = F).
 Since vertices A, B, and C coincide with vertices D, E, and F, respectively, must be congruent to .

Questions Eliciting Thinking Where in your explanation did you specifically use the assumptionsÂ , and ?
How might you describe the degree of the rotation of about point B?
How do you know that if the vertices coincide, the triangles must be congruent? 
Instructional Implications Alter the orientation of the two triangles, so a translation followed by both a rotation and a reflection is needed to map to . Ask the student to adapt his or her explanation to this case.
Challenge the student to rework his or her explanation into a more formal proof in which details are provided and notation is used appropriately.
Consider implementing one of the following MFAS tasks Justifying SSS Congruence (GCO.2.8) or Justifying ASA Congruence (GCO.2.8). 