Lesson 15: Symmetry
About this lesson
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).
Lesson overview
 15.1 Warmup: Back to the Start (5 minutes)

15.2 Activity: Self Reflection (20 minutes)
 Includes "Are you Ready for More?" extension problem
 15.3 Activity: Diabolic Diagonals (10 minutes)
 Lesson Synthesis
 15.4 Cooldown: Criss Cross (5 minutes)
Learning goals:
 Describe (orally and in writing) the reflections that take a figure onto itself.
Learning goals (student facing):
 Let’s describe some symmetries of shapes.
Learning targets (student facing):
 I can describe the reflections that take a figure onto itself.
Required materials:
 Copies of blackline master
 Geometry toolkits
 Sticky notes
 Tools for creating a visual display
Required preparation:
 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.)
Glossary:
 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).
 Access the complete Geometry Course glossary.
Standards:
 This lesson builds on the standards:CCSS.8.G.A.1MS.8.G.1MO.8.GM.A.1aMO.8.GM.A.1bCCSS.HSGCO.A.4MS.GCO.4MO.G.CO.A.4
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