Some students might be concerned that rotating the parallelogram will reverse the direction of the arrows. Emphasize that the arrows are markings to indicate the segments are parallel, but not actually part of the parallelogram itself.

In this activity, students work together to investigate rotation symmetry and communicate their thinking in a visual display. As you monitor, ask groups if they have found all possible angles of rotation that produce symmetry and to explain how they know these angles apply to all shapes of that type.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as “I notice . . . . I wonder . . . .”). A typical discussion prompt is “What is the same, and what is different?” comparing their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

The purpose of this discussion is to identify what types of shapes have rotation symmetry.

Ask groups to display their visual displays in the classroom for all to see in order of the number of angles of rotation that create symmetry for their shape. They will have to communicate with other groups to accomplish this. If possible, display the reflection symmetry visual displays from the previous class for comparison.

After all displays have been ordered, use Compare and Connect to help students compare, contrast, and connect the different representations. Ask:

• “Which shapes have the fewest angles of rotation that created symmetry? Which have the most?” (The isosceles trapezoid does not have rotation symmetry because the only rotations that take it to itself are 0 degrees or 360 degrees. For the circle, any angle of rotation creates symmetry.)
• “What do you notice about the order of the shapes in terms of reflection symmetry versus the order of the shapes in terms of rotation symmetry?” (The circle has the most lines of symmetry and the most angles of rotation that create symmetry. It seems to be the most symmetric shape overall. The parallelogram that is not a rhombus has 180-degree rotation symmetry, but no lines of symmetry; but the rhombus has both 180-degree rotation symmetry and 2 lines of symmetry.)
• “How are rotation and reflection symmetry the same? How are they different?” (Both are rigid transformations that take a shape back onto itself. Rotation symmetry is more connected to circles, while reflection symmetry is more connected to lines.)

If questions about 0- or 360-degree rotations come up, explain that all shapes, no matter how irregular, look the same when rotated by 0 degrees or 360 degrees. If these rotations are the only ones that take the shape to itself, then it is not considered to have rotation symmetry.

In this activity, students write out a description of a parallelogram’s rotation symmetry. This invites students to apply the definition of rotation. 

Direct students to their reference charts. What information is useful? (The definition of rotation.)

The purpose of this discussion is to reinforce using definitions to justify a response. Invite a few students to share their responses with the class. Highlight any improvements over previous justifications, such as good use of a well-labeled diagram.