# 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.

### Student Facing

• I can describe a translation by stating the directed line segment.
• I can draw translations.

Building On

Building Towards

### Glossary Entries

• directed line segment

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

• 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 $$t$$.