Lesson 12
Defining Translations
Lesson Narrative
This lesson continues the theme of revisiting rigid transformations students explored on coordinate grids in previous grades and refining their understanding to define those transformations more generally without coordinate grids. The concept of a directed line segment is introduced to give students language for efficiently describing the direction and length of a translation. Students know the term line segment, and so the phrase directed line segment builds on a concept they already know and connects it to the concept of translations. The word vector is purposely avoided because the geometric interpretation of a vector should arise as a consequence of future work with vectors, not as a definition. Students make arguments and critique the reasoning of others when they explain why translating a line segment results in a parallelogram (MP3). This argument leads to the Parallel Postulate.
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 "translation" (in written and spoken language) requires specifying a directed line segment.
 Determine whether a figure is a translation of another.
 Draw translations of figures.
Student Facing
 Let’s translate some figures.
Learning Targets
Student Facing
 I can describe a translation by stating the directed line segment.
 I can draw translations.
CCSS Standards
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 \(t\).