Lesson 13

Incorporating Rotations

Lesson Narrative

This lesson continues to build on prior knowledge about congruence to reinforce the idea that the rigid motions, translations, reflections, and rotations preserve distances and angles. In this lesson, students return to studying transformations on a grid, as they encounter rotations for the first time in this course. In subsequent lessons in this unit, students learn a precise definition for rotation that applies off the grid.

In one activity, students complete a sequence of translation, reflection, and rotation where each rigid motion lines up one pair of points in a pair of congruent triangles. This sequence of point-by-point transformation will be the basis for triangle congruence proof in a subsequent unit. Students attend to precision when they determine the necessary pieces of information to describe a rotation as well as when they determine appropriate levels of confidence when measuring with a protractor (MP6).

Note that these materials use the convention that all named angles are assumed to measure less than 180 degrees, unless otherwise specified.

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 rigid transformations produce congruent figures by preserving distance and angles.
• Draw the result of a transformation (in written language) of a given figure.
• Explain (orally and in writing) a sequence of transformations to take a given figure onto another.

Student Facing

Let's draw some transformations.

Student Facing

• Given a figure and the description of a transformation, I can draw the figure's image after the transformation.
• I can describe the sequence of transformations necessary to take a figure onto another figure.
• I know that rigid transformations result in congruent figures.

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.

• 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 $$t$$.