# Lesson 14

Defining Rotations

### Lesson Narrative

The purpose of this lesson is to develop a rigorous definition of rotation, building on what students know from previous courses. Students first focus on what information is important for defining a rotation and then determine properties of rotations by rotating segments to make isosceles triangles. Students attend to precision when they clarify what information they need to uniquely determine a given rotation (MP6).

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

### Learning Goals

Teacher Facing

• Comprehend that the term "rotation" (in written and spoken language) requires several descriptors including angle, center, and direction.
• Determine whether a figure is a rotation of another.
• Draw rotations of figures.

### Student Facing

• Let’s rotate shapes precisely.

### Student Facing

• I can describe a rotation by stating the center and angle of rotation.
• I can draw rotations.

Building On

Building Towards

### 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.

• 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 $$m$$.

• 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.

• 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 $$t$$.