Lesson 16

More Symmetry

16.1: Which One Doesn't Belong: Symmetry (5 minutes)

Warm-up

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. 

Launch

Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn't belong.

Student Facing

Which one doesn’t belong?

A

Circle with center A.

B

Triangle A B C. Single tick marks on sides A C and C B.

C

Quadrilateral A B C D. Single arrows on opposite sides D A and C B. Double arrows on opposite sides D C and A B.

D

6-sided irreguar shape B C D E F G that has all right angles.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as line of symmetry or diagonal. Also, press students on unsubstantiated claims.

16.2: Self Rotation (20 minutes)

Activity

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.

Launch

Arrange students in the same groups as the previous class. Provide each group with tools for creating a visual display. Assign each group a different shape from Self Reflection and provide enough copies of the shape for everyone in the group.

Give students 5 minutes of work time followed by 5 minutes to put together their visual display. Explain that they will not have enough time to make the visual display perfect, so the purpose is just to get their rough ideas down in an organized way.

Student Facing

Determine all the angles of rotation that create symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:

  • the name of your shape
  • the definition of your shape
  • drawings of each rotation that creates symmetry
  • a description in words of each rotation that creates symmetry, including the center, angle, and direction of rotation
  • one non-example (a description and drawing of a rotation that does not result in symmetry)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Finite figures, like the shapes we have looked at in class, cannot have translation symmetry. But with a pattern that continues on forever, it is possible. Patterns like this one that have translation symmetry in only one direction are called frieze patterns.

Frieze pattern, octagon, centered with half octagons on both sides.

  1. What are the lines of symmetry for this pattern?
  2. What angles of rotation produce symmetry for this pattern?
  3. What translations produce symmetry for this pattern if we imagine it extending horizontally forever?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Invite students to do a “gallery walk” in which they leave written feedback on sticky notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:

  • Was there anything about the organization of the visual display that made the ideas especially clear? Was there anything about the organization that could be improved?
  • Was there anything about the way the ideas are explained that made the ideas especially clear? Was there anything about the explanations that could be improved?
  • Are there any angles of rotation creating symmetry that the group missed?
  • Are there any angles of rotation that don’t create symmetry for all shapes of the given type?

After the gallery walk, ask students to share their observations about rotation symmetry. Consider asking:

  • “Which shapes had the fewest angles of rotation that created symmetry? Which had 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.)

If questions about 0 or 360 degree rotations come up, explain that all shapes, no matter how strange, 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.

Speaking: MLR 7 Compare and Connect. Use this routine when students present their categories and their sequence of rotations, translations, and reflections to get from the original figure to the image. Ask students to consider what is the same and what is different about the sequences of transformations. Draw students' attention to the way that transformations are described precisely (i.e., order, numbers, words). These exchanges strengthen students' mathematical language use and reasoning about transformations.
Design Principle(s): Maximize meta-awareness
Engagement: Develop Effort and Persistence. Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class. 
Supports accessibility for: Language; Social-emotional skills; Attention

16.3: Parallelogram Symmetry (10 minutes)

Activity

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

Launch

Representation: Access for Perception. Read the situation aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language

Student Facing

Clare says, "Last class I thought the parallelogram would have reflection symmetry. I tried using a diagonal as the line of symmetry but it didn’t work. So now I’m doubting that it has rotation symmetry."

Lin says, "I thought that too at first, but now I think that a parallelogram does have rotation symmetry. Here, look at this."

How could Lin describe to Clare the symmetry she sees?

 

A parallelogram inside two concentric circles.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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

Activity Synthesis

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.

Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this with successive pair shares to give students a structured opportunity to revise and refine their response to “How could Lin describe to Clare the rotation symmetry she sees?” Ask each student to meet with 2–3 other partners in a row for feedback. Display prompts for feedback that will help teams strengthen their ideas and clarify their language. For example, "Can you explain how…?" "You should expand on...," etc. Students can borrow ideas and language from each partner to strengthen the final product. Design Principle(s): Optimize output (for generalization)

Lesson Synthesis

Lesson Synthesis

The main idea to draw out of this lesson is the relationship between the definition of rotation and rotation symmetry. Ask students when a rotation shows a figure has rotation symmetry. (This occurs when the rotation takes the shape to itself.)

Invite students to sketch shapes for which the following angles of rotation create symmetry and write a transformation statement for each one. Suggest students look at the sentence frame in the definition of rotation in their reference chart if they get stuck.

  • 180 degrees
  • 90 degrees
  • 45 degrees

16.4: Cool-down - Mystery Quad (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

A shape has rotation symmetry if there is a rotation between 0 and 360 degrees that takes the shape to itself. A regular hexagon has many angles that work to create rotation symmetry. Here is one of them. What other angles would create a rotation where the image is the same as the original figure?

Regular hexagon. At its center, a 60 degree angle is marked, and the angle sides extend to two vertices of the hexagon, forming a triangle. An arc of a circle also connects the two vertices.

Can you think of a shape that has translation symmetry?

There aren’t any polygons with translation symmetry, but an infinite shape like a line can be translated such that the translation takes the line to itself.