Unit 1, Lesson 12
Defining Translations
Integrated Math 1
12.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
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
NoneWhat 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.
Lesson Synthesis
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.