Lesson 15

Symmetry

In this lesson and the next, students determine the cases where applying a certain rigid motion to a shape doesn’t change it. This is the idea of symmetry. A shape is said to have symmetry if there is a rigid transformation that takes the shape to itself. Students first study reflection symmetry using lines of symmetry, and then they study rotation symmetry in a subsequent lesson. Translation symmetry isn’t mentioned explicitly, but students were exposed to the idea that a line has translation symmetry in a previous lesson. Students apply their understanding of rigid transformations to identify shapes where there is a line of symmetry which reflects the shape onto itself, satisfying the definition of reflection symmetry. 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.

In one activity, each group is assigned a different shape to consider. Students make use of structure when they discuss which lines of symmetry apply to a type of shape generally, rather than limiting their thinking to a given example (MP7).

Describe (orally and in writing) the reflections that take a figure onto itself.

Let’s describe some symmetries of shapes.

Print and cut up slips from the blackline master. The blackline master for this lesson contains 8 different shapes. Each group of 2–4 students will be investigating a shape. Prepare enough copies of the blackline master so that each student in each group gets a copy of the shape their group will investigate. (Note: Students will repeat this process for rotation symmetry in the next lesson; it may be easier to prepare twice as many shapes once rather than repeat the process.)

I can describe the reflections that take a figure onto itself.

Building On

Addressing

line of symmetry

A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.
reflection symmetry

A figure has reflection symmetry if there is a reflection that takes the figure to itself.
symmetry

A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).