In this lesson students refine their definition of translation. While they are practicing performing translations they are also formulating conjectures and developing their justification skills. These activities lead to the necessity of establishing the Parallel Postulate. In this lesson, students also encounter their first theorem of the course. Theorems are reserved for conjectures that students prove.

The concept of a directed line segment, a line segment with an arrow at one end specifying a direction, is introduced to give students language for efficiently describing the direction and length of a translation. A translation takes a point to another point so that the directed line segment from the original point to the image is parallel to the given directed line segment and has the same length and direction. Students know the term “line segment,” 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). 

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.