Lesson 15

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.

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.

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.

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.

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.

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.

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. 

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.

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.)