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 lines of reflection, for example, a diagonal of the rectangle.
 Includes incorrect degrees of rotation.

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.
Provide tracing paper so the student can model rotations of rectangles and squares 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 rectangles and squares 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 Parallelograms and Rhombi (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, for example, describes them as vertical, horizontal, or diagonal.
 Draws but does not describe lines of reflection.

Questions Eliciting Thinking What is the direction of the rotations you described – clockwise or counterclockwise?
How could you precisely describe the center of this rotation?
How could you precisely describe the lines 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 for each figure 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 Parallelograms and Rhombi (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 rectangle, 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 rectangle onto itself. The student also (upon questioning) indicates that any multiple of 180 will provide a degree of rotation that carries a rectangle onto itself. The student says that there are two lines of reflection that carry the rectangle onto itself and precisely describes them (e.g., as the lines that contain the midpoints of opposite sides).
With regard to the square, the student describes the center of rotation as the point where the diagonals intersect. The student indicates that 90°, 180°, 270° and 360° clockwise and counterclockwise rotations about this point will carry the square onto itself. The student also (upon questioning) indicates that any multiple of 90 will provide a degree of rotation that carries a square onto itself. The student says that there are four lines of reflection that carry the square onto itself and precisely describes them (e.g., as the lines that contain the midpoints of opposite sides and the lines that contain the two diagonals of the square).
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, in general, the rotations that will carry a rectangle (or square) onto itself?
How would your answer change if the rectangle had been a parallelogram? If the square had been a rhombus?
Can you describe any rotations that will carry the figure onto itself using one of the vertices (or an exterior point) as the center? 
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 composition of reflections. On a coordinate plane, have the student graph a regular polygon with all its vertices in the first quadrant. Ask the student to reflect the regular polygon over the yaxis and then again over the xaxis. Challenge the student to describe a rotation that will produce the same result. 