Getting Started 
Misconception/Error The student does not understand congruence in terms of rigid motions. 
Examples of Student Work at this Level The student:
 Describes a sequence of rigid motions that does not map onto .
 Selects points that no longer form a triangle congruent to .
 Makes sketches on the graph but is unable to describe the rigid motions performed.
 Is too vague when describing whether or not the sequence of rigid motions maps onto .

Questions Eliciting Thinking Can you explain in more detail how this sequence of rigid motions maps onto ?
How will performing a rigid motion affect the size and shape of ? Do the coordinates you listed form a triangle congruent to ?
How could you use rigid motion to move one triangle directly on top of the other? Can you identify a rigid motion or sequence of rigid motions that will map onto ?
Can you identify one transformation that would be needed to map onto ? 
Instructional Implications Review the definition of each of the rigid motions: translations, reflections, and rotations. To develop an intuitive understanding of rigid transformations, allow the student to experiment with a variety of transformations using transparent paper or interactive websites. Be sure the student understands not only how to perform a rigid motion but how to describe it using correct terminology and notation.
Have the student trace on patty paper or a transparency. Ask the student to use the transparency to model a sequence of rigid motions that maps onto providing assistance as needed. Allow the student to experiment with a variety of rigid motions. Once the student has successfully identified a sequence of rigid motions, assist the student in describing the sequence using correct terminology and notation. Remind the student how to describe the effect of rigid motions on coordinates algebraically, for example, (1) translation (x,y)(x+a,y+b) where a is the horizontal distance, left or right, and b is the vertical distance, up or down and (2) rotation (x,y)(y,x). Ask the student to describe the coordinates of the vertices of the image after each rigid motion. Then ask the student to verify that the coordinates of the final image of are the same as the coordinates of the corresponding vertices of , so it can be concluded that .
Provide the student with several other examples of congruent triangles shown on the coordinate plane. Have the student first model the rigid motions that map one triangle onto the other and then describe each rigid motion algebraically. Then ask the student to calculate the coordinates of the vertices of the image of the transformed triangle to ensure they are the same as the coordinates of the vertices of the other triangle. 
Moving Forward 
Misconception/Error The student does not completely describe the sequence of rigid motions that maps onto . 
Examples of Student Work at this Level The student:
 Does not include detailed descriptions of the translation and rotation.
 Does not include the coordinates or description of how he or she obtained the coordinates of the vertices of the images of .

Questions Eliciting Thinking Can you describe in more detail how this sequence of rigid motions will map one triangle onto the other?
What vector describes how you translated the figure? What happened after the translation? Did any of the vertices coincide?
Is it possible to identify both the center and the degree of rotation? What is the direction of rotation? What happened after the rotation? Did any of the sides coincide?
What are the coordinates of the vertices of D'E'F'? D''E''F''?
How do you know point C coincides with point F? What are the coordinates of the vertices of ? 
Instructional Implications Model for the student a clear and complete explanation of the student’s sequence of rigid motions. Provide the student with several other examples of congruent triangles and have the student identify the sequence of rigid motions that maps one triangle onto the other. Remind the student to be as clear and concise as possible in the description, identifying specifically the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. If possible, have the student ask another student to read his or her description to see if it can be followed without further explanation.
Explain to the student that two triangles can be shown congruent by describing a sequence of rigid motions that results in corresponding vertices coinciding. Consequently, it is important to explicitly state when vertices are mapped onto corresponding vertices by a transformation and to provide a justification for these occurrences. Review with the student the properties of rigid motions [e.g., (1) map lines to lines, rays to rays, and segments to segments, (2) are distance preserving, and (3) are degree preserving]. If needed, remind the student how to describe the effect of rigid motions on the coordinates algebraically, for example, (1) translation (x,y)(x+a,y+b) where a is the horizontal distance, left or right, and b is the vertical distance, up or down. (2) 90° clockwise rotation (x,y)(y,x). 
Almost There 
Misconception/Error The student does not adequately justify some statements in the proof. 
Examples of Student Work at this Level The student correctly describes a sequence of rigid motions that maps onto but:
 Does not describe completely the mapping of one vertex onto another.
 Makes a minor error in calculating the coordinates of a vertex of the triangle.
 Does not describe completely how he or she obtained the coordinates of the vertices.

Questions Eliciting Thinking How do you know that vertices C and F will coincide?
Can you find the calculation error you made when finding the coordinates of the vertices?
How did you obtain the coordinates for the vertices? 
Instructional Implications Show the student any statements that require justification and ask the student to provide them. Explain to the student that two triangles can be shown congruent by describing a sequence of rigid motions that results in corresponding vertices coinciding. Consequently, it is important to explicitly state when vertices are mapped onto corresponding vertices by a transformation and to provide a justification for these occurrences. Assist the student in finding the calculation error he or she made when calculating coordinates of vertices of the translated or rotated triangle. Model for the student how to justify the mapping of one vertex onto another by applying the algebraic description of the 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 reasons as follows:
 Translate along so that point E coincides with point B. This translation T(x,y)(x6,y+3) maps D (3,1), E (6,3), and F (3,5) to D' (3,4), E' (0,0) which now coincides with B (0, 0), and F' (3,2).
 Rotate 90° counterclockwise about point E' (or B). This rotation R(x,y)(y,x) maps D' (3,4), E'' (0,0), and F' (3,2) to D'' (4,3), E''(0,0) , and F''(2,3).
 Observe that the vertices of now coincide with the vertices of : A (4,3), B(0,0) , and C(2,3).
 Since the vertices coincide with the vertices of , .

Questions Eliciting Thinking Now that you have shown the triangles are congruent, what can you conclude about their corresponding sides and vertices? 
Instructional Implications Discuss the SSS, SAS, and ASA congruence theorems with the student. Ask the student which of these is implied in his or her explanation.
Consider implementing one of the following MFAS tasks: Justifying Side Angle Side Congruence (GCO.2.8), Justifying Angle Side Angle Congruence (GCO.2.8), Justifying Side Side Side Congruence (GCO.2.8). 