Lesson 15

Symmetry

15.1: Back to the Start (5 minutes)

Warm-up

This warm-up invites students to visualize different sequences of transformations that will take a segment onto itself. This idea will be extended in subsequent activities in which students determine lines of symmetry and angles of rotation that create symmetry for various shapes. The fact that reflecting a segment across its perpendicular bisector exchanges its endpoints will be useful in the next unit when students study triangle congruence.

Student Facing

Here is a segment \(AB\):
 

Line segment AB

If you translate the segment up 5 units then down 5 units, it looks the same as it did originally.

  1. What other rigid transformations create an image that fits exactly over the original segment?
  2. Are there any single rigid motions that do the same thing?

Student Response

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

Activity Synthesis

Select a student to share a single reflection that will take the segment onto itself, and another student to share a single rotation that takes the segment onto itself. Tell students that if a rigid transformation exists that takes a figure onto itself, the figure is said to have symmetry. Explain that when a single reflection takes a figure onto itself, the figure has reflection symmetry and the line of reflection is called a line of symmetry of the figure. Also tell students that whenever a single rotation strictly between 0 and 360 degrees takes a figure onto itself, the figure has rotation symmetry.

15.2: Self Reflection (20 minutes)

Activity

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

Launch

Arrange students in groups of 2–4. Provide each group with tools for creating a visual display.

Assign a different shape from the blackline master to each group and provide enough copies of that shape for each student in each group. Note that the cool-down uses a rectangle, so you may choose to assign the rectangle to a group of students who need more processing time.

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 to get their ideas down in an organized way.

Student Facing

Determine all the lines of 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 line of symmetry
  • a description in words of each line of symmetry
  • one non-example in a different color (a description and drawing of a reflection not over a line of symmetry)

Student Response

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

Student Facing

Are you ready for more?

Look at all of the shapes the class explored and focus on those which had more than one line of symmetry.

  1. What is true for all the lines of symmetry in these shapes?
  2. Give an example of a shape that has two or more lines of symmetry that do not intersect at the same point.
  3. What would happen if you did a sequence of two different reflections across lines of symmetry for the shapes you explored in class?

Student Response

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

Anticipated Misconceptions

Students might think that any line that divides the area of a figure in half, like a diagonal of a non-square rectangle, is a line of symmetry. Ask students to reflect across these lines to see that the shapes are not taken onto themselves.

Activity Synthesis

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

Ask groups to display their visual displays in the classroom for all to see in order of the number of lines of symmetry their shape has. They will have to communicate with other groups to accomplish this. 

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 lines of symmetry that the group missed?
  • Are there any lines of symmetry that don’t work for all shapes of the given type?

After the gallery walk, ask students to share their observations about reflection symmetries for the different shapes. Consider asking:

  • “Which shapes had the fewest lines of symmetry? Which had the most?” (The parallelogram that is not a rhombus didn’t have any lines of symmetry. The circle has the most since it has infinite.)
  • “Were there any lines that you thought would be lines of symmetry, but when you tried to reflect, they turned out not to be?” (The diagonal of a parallelogram that is not a rhombus seemed like it would be a line of symmetry, but it’s not.)
  • “Describe a shape that would have no lines of symmetry.” (A scalene triangle has no lines of symmetry.)

Save the visual displays if possible for comparison with displays of angles of rotation that create symmetry in a subsequent lesson.

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

15.3: Diabolic Diagonals (10 minutes)

Activity

This activity invites students to apply the definition of reflection when they write out a justification for why a line is or is not a line of symmetry in a kite. 

Launch

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch, take time to review the following terms from previous lessons that students will need to access for this activity: kite, diagonal, line of symmetry, reflection, congruent. 
Supports accessibility for: Memory; Language

Student Facing

Kiran thinks both diagonals of a kite are lines of symmetry. Tyler thinks only 1 diagonal is a line of symmetry. Who is correct? Explain how you know.

Student Response

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

Anticipated Misconceptions

If students struggle to get started, direct them to their reference charts: “What information is useful?”. (The definition of reflection.)

If students do not know the definition of a kite, provide one: A quadrilateral which has two sides next to each other that are congruent, and where the other two sides are also congruent.

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. If a student used a well-labeled diagram in their explanation, highlight that strategy. If not mentioned by students, ask if a diagram would be useful. (Yes, it's easier to talk about labeled points than to describe a generic side.)

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 “Who is correct? Explain how you know.” 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 reflection and reflection symmetry. Ask students when a line of reflection is also a line of symmetry. (This occurs when the reflection takes the shape to itself.)

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

  • exactly 1 line of symmetry
  • at least 2 lines of symmetry
  • exactly 5 lines of symmetry

15.4: Cool-down - Criss Cross (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 symmetry if there is a rigid transformation which creates an image that fits exactly over the original shape. A shape has reflection symmetry if there is a reflection that takes the shape to itself, and the line of reflection in this case is called a line of symmetry. A regular hexagon has many lines of symmetry. Here are 2 of them. What other lines create a reflection where the image is the same as the original figure?

Two congruent hexagons, first with a line of symmetry through the midpoints of the top and bottom sides, second with line of symmetry through top right and bottom left vertices.