Lesson 12

The purpose of this warm-up is to elicit the idea that a translation takes each point in the same direction by the same distance, which will be useful when students investigate translations throughout this lesson. While students may notice and wonder many things about these images, the “arrow” and its relation to the triangles is the important discussion point. By engaging in discussion and thinking abound the mathematics involved with the “arrow,” students are making sense of the problem (MP1).

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If connecting points of one triangle to their corresponding points on the other does not come up during the conversation, ask students to discuss this idea. 

In this activity, students explore translations without a coordinate grid by identifying and describing transformations. Monitor for students who notice parallel lines formed by directed line segments or formed by points and their images.

Suggest that students either use tracing paper or two different colors to clearly differentiate the two transformations.

Students may need to be reminded of the tools in their geometry toolkits, such as tracing paper, straightedges, and compasses.

Invite students to share what they conjectured.

Highlight for students that connecting each original point to each image results in arrows that are all the same length and going in the same direction. Tell students that we call these arrows directed line segments. In other words, a directed line segment is a line segment with a direction to it. A directed line segment conveys the direction and distance that each point is translated.

If no student conjectures about translation taking lines to parallel lines, display the images of student solutions with lines drawn in. There will be time in subsequent activities to explore this idea further; students are only conjecturing at this point.

This activity highlights that translations take lines to parallel lines and segments to segments of the same length. Both of these properties will be used in future lessons to prove theorems. The activity also previews a proof of the Triangle Angle Sum Theorem later in this unit.

Monitor for different ways students justify their claims about parallel lines and equal distances. It is not expected that students come up with rigorous, formal arguments at this point, but it is important to encourage students to justify their ideas as a way to transition to more formal arguments.

 Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

The purpose of this discussion is to highlight the fact that translations take lines to parallel lines and segments to segments of equal length. This synthesis is intentionally short to allow more time for the lesson synthesis. Here are some questions for discussion:

• “How do you know lines \(BC\) and \(B’C’\) are parallel?” (Lines \(BC\) and \(B’C’\) are parallel because the translation took each point on segment \(BC\) the same distance in the same direction. One way to think of what makes lines parallel is that all pairs of corresponding points are the same distance apart.)
• “How do you know segments \(BC\) and \(B’C’\) are the same length?” (As an assertion, translations are rigid transformations that take segments to segments of equal length.)