Lesson 12

Defining Translations

12.1: Notice and Wonder: Two Triangles and an Arrow (5 minutes)

Warm-up

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).

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 the things they notice with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

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.

 

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. 

12.2: What’s the Point: Translations (15 minutes)

Activity

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, “I noticed _____, so I . . .,” and “_____could/could not be true because . . .”
Supports accessibility for: Language; Organization

Student Facing

  1. After a translation, the image of \(V\) is \(W\). Find at least 3 other points that are taken to a labeled point by that translation.
  2. Write at least 1 conjecture about translations.
  3. In a new translation, the image of \(V\) is \(Z\). Find at least 3 other points that are taken to a labeled point by the new translation.
  4. Are your conjectures still true for the new translation?
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.

Student Response

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Anticipated Misconceptions

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.

12.3: Translating Triangles (15 minutes)

Activity

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).

Launch

Note that GeoGebra refers to a directed line segment as a vector. 

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 a whole-class discussion.

Speaking: MLR8 Discussion Supports. To support students in producing statements about the properties of the translated figures, provide sentence frames for students to use such as: “\(BC\) and \(B’C’\) are _____ because _____.” or “Shape \(EE'D’D\) is a _____ because _____.”
Design Principle(s): Support sense-making; Optimize output (for comparison)
Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “This method works/doesn’t work because…,” “Another strategy would be _____ because…,” and “Is there another way to say/do...?”
Supports accessibility for: Social-emotional skills; Organization; Language

Student Facing

  1. Translate triangle \(ABC\) by the directed line segment from \(A\) to \(C\).
    1. What is the relationship between line \(BC\) and line \(B’C’\)? Explain your reasoning.
    2. How does the length of segment \(BC\) compare to the length of segment \(B’C’\)? Explain your reasoning.
  2. Translate segment \(DE\) by directed line segment \(w\). Label the new endpoints \(D’\) and \(E’\).
    1. Connect \(D\) to \(D’\) and \(E\) to \(E’\).
    2. What kind of shape did you draw? What properties does it have? Explain your reasoning.

Student Response

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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.

Speaking: MLR8 Discussion Supports. To support students in producing statements about the properties of the translated figures, provide sentence frames for students to use such as: “\(BC\) and \(B’C’\) are_____ because _____.” or “Shape \(EDD'E'\) is a _____ because _____.”
Design Principle(s): Support sense-making; Optimize output (for comparison)
Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “This method works/doesn’t work because…,” “Another strategy would be _____ because…,” and “Is there another way to say/do...?”
Supports accessibility for: Social-emotional skills; Organization; Language

Student Facing

Triangle A B C. Point A faces to the left, Point B faces upwards, Point C faces to the right.
 
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.
  1. Translate triangle \(ABC\) by the directed line segment from \(A\) to \(C\).
    1. What is the relationship between line \(BC\) and line \(B’C’\)? Explain your reasoning.
    2. How does the length of segment \(BC\) compare to the length of segment \(B’C’\)? Explain your reasoning.
  2. Translate segment \(DE\) by directed line segment \(w\). Label the new endpoints \(D’\) and \(E’\).
    1. Connect \(D\) to \(D’\) and \(E\) to \(E’\).
    2. What kind of shape did you draw? What properties does it have? Explain your reasoning.

Student Response

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Student Facing

Are you ready for more?

  1. On triangle \(ABC\) in the task, use a straightedge and compass to construct the line which passes through \(A\) and is perpendicular to \(AC\). Label it \(\ell\). Then, construct the perpendicular bisector of \(AC\) and label it \(m\). Draw the reflection of \(ABC\) across the line \(\ell\). Since the label \(A’B’C’\) is used already, label it \(DEF\) instead.
  2. What is the reflection of \(DEF\) across the line \(m\)?
  3. Explain why this is cool.

Student Response

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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. 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.)

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is to recap ideas from class and add them to the reference chart. Choose how much detail to include based on timing.

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 \(\ell\) and a point not on the line, \(B\). Ask students, “What are the possible lines through \(B\)? How many of them are parallel to \(\ell\)?” (Infinite lines go through \(B\), but only one is parallel to \(\ell\).) Tell students the idea that there is one unique line parallel to \(\ell\) that goes through \(B\) 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 \(A\) by a directed line segment \(t\) assumes there is only one line through \(A\) that is parallel to \(t\)

Remind students that in the previous activity, they came up with explanations for why translations take segments to parallel segments. If time allows, demonstrate a proof of a similar idea, specifically, that translations take lines either to themselves or to parallel lines:

  • Suppose that a line \(\ell\) is translated by a directed line segment \(v\).
  • The image of \(\ell\) is either parallel or there is at least one intersection point. Suppose that there is an intersection point and call it \(A\).
  • That means there is some other point on \(\ell\) that was taken to \(A\). Label the point taken to \(A\) as \(P\).
  • That means that the original directed line segment, \(v\), is the same direction and length as the directed line segment from \(P\) to \(A\).

So whenever the image of a line isn’t parallel to the original figure, it’s because the directed line segment just shifted all the points along the line. Another way of saying this is that directed line segments translate points on a given line to points off the given line, unless the directed line segment started off parallel to the given line in the first place. Then it shifts all the points along the given line in that direction. This is an example of translation symmetry. Students will explore other types of symmetry in a subsequent lesson.

Ask students to add these definitions, assertions, and theorems to their reference charts as you add them to the class reference chart:

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 $A$ by the directed line segment $v$.

Point A translated by directed line segment v.

Parallel Postulate: Given a line (\(m\)) and a point (\(A\)) that is not on \(m\), there is exactly one line that goes through \(A\) that is parallel to \(m\)

(Assertion)

Two parallel lines; one of them is labeled m. The unlabeled line goes through point A.

Translations take lines to parallel lines or to themselves.

(Theorem)

\(m \parallel m'\)

A translation. m. m prime.

12.4: Cool-down - What Went Wrong? Translation (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

A translation slides a figure in a given direction for a given distance with no rotation. The distance and direction is 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. 

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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.

More precisely, a translation of a point \(A\) along a directed line segment \(t\) is a transformation that takes \(A\) to \(A’\) so that the directed line segment \(AA’\) is parallel to \(t\), goes in the same direction as \(t\), and is the same length as \(t\)

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Directed line segments 

Here is a translation of 3 points. Notice that the directed line segments \(CC’\), \(DD’\), and \(EE’\) are each parallel to \(v\), going in the same direction as \(v\), and the same length as \(v\)

Two triangles and four direct line segments