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Unit 1, Lesson 12

Defining Translations

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Integrated Math 1

12.1

Warm-up

Notice and Wonder: Two Triangles and an Arrow

What do you notice? What do you wonder?

<p>Triangle C D E, blue, on the right, and unmarked triangle, yellow, on the left. Directed line segment V, facing upwards and to the right, with an arrow at the right end.</p>

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12.2

Activity

What’s the Point: Translations

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?

<p>26 points, labeled A through Z, scattered throughout an area. Point A is at the top right, and point Z is towards the bottom left.</p>

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12.3

Activity

Translating Triangles

<p>Triangle A B C. Point A faces to the left, Point B faces upwards, Point C faces to the right.<br>
</p>

<p>Line segment E D facing upwards and to the right, with D on the top right corner. Directed line segment w, pointing to the right.</p>

Translate triangle by the directed line segment from to .
1. What is the relationship between line and line ? Explain your reasoning.
2. How does the length of segment compare to the length of segment ? Explain your reasoning.
Translate segment by directed line segment . Label the new endpoints and .
1. Connect to and to .
2. What kind of shape did you draw? What properties does it have? Explain your reasoning.

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.

<p>Triangle C D E, Triangle C prime D prime E prime, and directed line segment v. V slants downward and to the left, begins below point D and ends below point D prime.</p>

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 .

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 .

<p>Two triangles and four direct line segments</p>

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.

Glossary

directed line segment

A directed line segment is a line segment that has distance (length) and direction.

The arrow on this directed line segment shows that it starts at point and ends at point .

theorem

A theorem is a statement that has been proved mathematically.

translation

A translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:

The directed line segment from the original point to its image is parallel to the given directed line segment.
Both directed line segments have the same length and direction.

In the figure shown here, is the image of under the translation given by the directed line segment .

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Gradeband

Course

Unit

Lesson

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Language

Audience

Assessment Access

Defining Translations

Warm-up

Notice and Wonder: Two Triangles and an Arrow

Are You Ready for More?

Activity

What’s the Point: Translations

Are You Ready for More?

Activity

Translating Triangles

Student Lesson Summary

Glossary

directed line segment

theorem

translation

Have feedback on the curriculum?

Are You Ready for More?