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, a directed line segment’s relation to triangles is the important discussion point. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that each point on one triangle is the same distance and direction from the corresponding point on the other triangle.
Launch
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 with their partner the things they notice and wonder, and follow with a whole-class discussion.
Activity
None
Student Task Statement
What do you notice? What do you wonder?
Student Response
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Activity Synthesis
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.
Explain to students that there are two facts related to translations and parallel lines that will come up several times in future lessons and units:
Given a line and a point off the line, there is a unique parallel line that goes through the point.
Translations take lines to parallel lines or to themselves.
Display a line and a point not on the line, . Ask students, “What are the possible lines through ? How many of them are parallel to ?” (Infinite lines go through , but only one is parallel to .) Tell students the idea that there is one unique line parallel to that goes through is called the Parallel Postulate. It’s an observation that seems to be true, but there is no way to prove or disprove it. We will take it as an assertion.
Explain to students that translations don’t make sense without the Parallel Postulate because the definition of translating a point by a directed line segment assumes there is only one line through that is parallel to .
Add the following definition, assertion, and theorem to the class reference chart, and ask students to add them to their reference charts.
Translation is a rigid transformation that 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.
"Translate _(object)_ by the directed line segment _(name or from [point] to [point])_."
(Definition)
Translate by the directed line segment .
Parallel Postulate: Given a line and a point that is not on , there is exactly one line that goes through that is parallel to .
(Assertion)
Translations take lines to parallel lines or to themselves.
(Theorem)
Student Lesson Summary
A translation slides a figure a given distance in a given direction with no rotation. The distance and direction are given by a directed line segment. The arrow of the directed line segment specifies the direction of the translation, and the length of the directed line segment specifies how far the figure gets translated.
More precisely, a translation of a point by a directed line segment is a transformation that takes to so that the directed line segment is parallel to , goes in the same direction as , and is the same length as .
Directed line segment T, slants upward and to the right, arrow at top end. Directed line segment A A prime, parallel and congruent to T, slants upward and to the right. Endpoint on bottom end, A, arrow at top end touching A prime.
Here is a translation of 3 points. Notice that the directed line segments , , and are each parallel to , go in the same direction as , and are the same length as .
Triangle C D E and a translation of three points. Directed line segments C C prime, D D prime, E E prime and v are all the same length and direction, slanting slightly downward and to the right, with the endpoint on the top end and the arrow at the bottom end. Triangle C prime D prime E prime at the end of three directed line segments.
Also notice that segment is parallel to segment . We proved that this would always be true, so we can write a theorem that says translations take lines to parallel lines or to themselves. A theorem is a statement that has been proved mathematically.
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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.
Launch
Suggest that students either use tracing paper or two different colors to clearly differentiate the two transformations.
Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example:
After a translation, the image of is . Find at least 3 other points that are taken to a labeled point by this same translation.
Write at least 1 conjecture about translations.
In a new translation, the image of is . Find at least 3 other points that are taken to a labeled point by the new translation.
Are your conjectures still true for the new translation?
Student Response
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Building on Student Thinking
Students may need to be reminded of the tools in their geometry toolkits, such as tracing paper, straightedges, and compasses.
Activity Synthesis
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.
Materials
None
Activity Narrative
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.
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. It is important to encourage students to justify their ideas to begin the transition to more formal arguments.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with whole-class discussion.
MLR8 Discussion Supports. Display sentence frames to support students in producing statements about the properties of the translated figures: “ and are _____ because _____” or “Shape is a _____ because _____.” Advances: Writing, Conversing
Activity Synthesis
The purpose of this discussion is to highlight the fact that translations take lines to parallel lines and segments to segments of equal length. Here are some questions for discussion:
“How do you know that lines and are parallel?” (Lines and are parallel because the translation took each point on segment 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 that segments and are the same length?” (Translations are rigid transformations that take segments to segments of equal length.)
HSG-CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
