Getting Started 
Misconception/Error The student is unable to completely and correctly identify all of the rotations and reflections that carry the figure onto itself. 
Examples of Student Work at this Level The student:
 Describes only some of the rotations and reflections.
 Includes incorrect degrees of rotation.
 Includes incorrect lines of reflection, for example, a diagonal of the parallelogram.

Questions Eliciting Thinking About which point did you rotate this figure? Could you find more rotations if you used the point at which the diagonals intersect as the center of rotation?
Does a rotation of 360° satisfy the conditions of the problem?
Can you model the rotation (or reflection) that you described? Can you explain how the transformation you described will carry the figure onto itself? 
Instructional Implications Review the concepts of reflections and rotations. Have the student experiment with rotations and reflections on an interactive website such as http://www.cuttheknot.org/Curriculum/Geometry/Rotation.shtml or http://www.cuttheknot.org/Curriculum/Geometry/Reflection.shtml to help the student visualize the results of various transformations. Clearly explain what it means for a rotation or reflection to carry a figure onto itself.
Provide tracing paper so the student can model rotations of parallelograms and rhombi about the intersections of their diagonals. Assist the student in determining the degree of each rotation that carries the figure onto itself. Remind the student that rotations are described by specifying the center, the degree, and the direction (clockwise or counterclockwise) of rotation. Assist the student in precisely describing each rotation that was identified.
Provide parallelograms and rhombi that can be cut out. Have the student fold each figure to identify lines of reflection that carry the figure onto itself. Guide the student to precisely describe these lines. Then ask the student to identify lines of symmetry for a variety of figures by both drawing and describing each line for each figure.
Consider implementing MFAS task Transformations of Rectangles and Squares (GCO.1.3). 
Making Progress 
Misconception/Error The student does not clearly and precisely describe transformations. 
Examples of Student Work at this Level The student identifies each rotation and reflection that will carry each figure onto itself. However, some identifications are not clearly stated or precisely described. For example, the student:
 Neglects to describe (or does not precisely describe) the center or direction of the rotation until prompted.
 Uses imprecise terminology to describe lines of reflection.
 Draws but does not describe lines of reflection.

Questions Eliciting Thinking What is the center of the rotation you described? What is its direction?
How could you more precisely describe the line of reflection? 
Instructional Implications Remind the student that rotations are described by specifying the center, the degree, and the direction (clockwise or counterclockwise) of rotation. Assist the student in precisely describing each rotation that was identified.
Guide the student to draw and label lines of reflection and then refer to the lines by name. Encourage the student to use mathematical terms such as midpoint, vertex, parallel, perpendicular, and diagonal when describing lines of reflection.
Consider implementing MFAS task Transformations of Rectangles and Squares (GCO.1.3). 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level With regard to the parallelogram, the student describes the center of rotation as the point where the diagonals intersect. The student indicates that 180° and 360° clockwise and counterclockwise rotations about this point will carry the parallelogram onto itself. The student also (upon questioning) indicates that any multiple of 180 will provide a degree of rotation that carries a parallelogram onto itself. The student says that there are no lines of reflection that carry the parallelogram onto itself.
With regard to the rhombus, the student describes the center of rotation as the point where the diagonals intersect. The student indicates that 180° and 360° clockwise and counterclockwise rotations about this point will carry the rhombus onto itself. The student also (upon questioning) indicates that any multiple of 180 will provide a degree of rotation that carries a rhombus onto itself. The student says that there are two lines of reflection that carry the rhombus onto itself and precisely describes them as, for example, the lines that contain the diagonals of the rhombus.
The student may initially neglect to include the direction of rotation (clockwise and counterclockwise) but does so immediately upon questioning. 
Questions Eliciting Thinking Are there any rotations of degree larger than 360 that carry the figure onto itself? How can you describe these rotations? How can you describe, in general, the rotations that will carry a parallelogram (or rhombus) onto itself?
How would your answer change if the parallelogram had been a rectangle and the rhombus had been a square? What properties of each determine this difference? 
Instructional Implications Have the student identify the rotations and reflections that carry a variety of quadrilaterals, regular polygons, and nonregular polygons onto itself.
Introduce the student to compositions of transformations. Ask the student to identify a composition of transformations that will carry a given polygon to another given (congruent) polygon. 