# Lesson 16

More Symmetry

### Lesson Narrative

In this lesson, students continue to examine cases in which applying a certain rigid motion to a shape doesn’t change it, and this time, students will be looking at rotation symmetry. For a shape to have rotation symmetry, there must be an angle for which the rotation takes the shape to itself. Students have opportunities to use precise language in the warm-up as they identify different types of symmetry (MP6). Students continue using precise language in their justifications of symmetry throughout the activities.

### Learning Goals

Teacher Facing

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

### Student Facing

Let’s describe more symmetries of shapes.

### Required Preparation

If there are not enough leftover shapes from the previous lesson, prepare more copies of the blackline master from Self Reflection so that each student in each group gets copies of the shape their group will investigate in Self Rotation.

### Student Facing

• I can describe the rotations that take a figure onto itself.

Building On

### Glossary Entries

• assertion

A statement that you think is true but have not yet proved.

• congruent

One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

• directed line segment

A line segment with an arrow at one end specifying a direction.

• image

If a transformation takes $$A$$ to $$A'$$, then $$A$$ is the original and $$A'$$ is the image.

• 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

A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line.

In the figure, $$A'$$ is the image of $$A$$ under the reflection across the line $$m$$.

• reflection symmetry

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

• rigid transformation

A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

• rotation

A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

$$P'$$ is the image of $$P$$ after a counterclockwise rotation of  $$t^\circ$$ using the point $$O$$ as the center.

​​​​​Quadrilateral $$ABCD$$ is rotated 120 degrees counterclockwise using the point $$D$$ as the center.

• rotation symmetry

A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as $$0^\circ$$ and $$360^\circ$$ that leave every point on the figure where it is.)

• 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).

• theorem

A statement that has been proved mathematically.

• translation

A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

In the figure, $$A'$$ is the image of $$A$$ under the translation given by the directed line segment $$t$$.