Lesson 17

Working with Rigid Transformations

17.1: Math Talk: From Here to There (5 minutes)

Warm-up

The purpose of this warm-up is to elicit strategies and understandings students have for using rigid motions on a point-by-point basis without a grid. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to put together rigid transformations that take one polygon to another again without making reference to a grid. While participating in this activity, students need to be precise in their word choice and use of language (MP6) because the rigid motions can only be defined in reference to labeled points.

Launch

Display the diagram and one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Segment \(CD\) is the perpendicular bisector of segment \(AB\). Find each transformation mentally.

A transformation that takes \(A\) to \(B\).

A transformation that takes \(B\) to \(A\).

A transformation that takes \(C\) to \(D\).

A transformation that takes \(D\) to \(C\).

Segments AB and CD intersect at point M. CD is the perpendicular bisector of AB.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may think that \(C\) can be reflected over line \(AB\) onto \(D\). Ask them what we would need to know for that to work.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”

  • “Did anyone have the same strategy but would explain it differently?”

  • “Did anyone solve the problem in a different way?”

  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”

  • “Do you agree or disagree? Why?”

If no student brings it up, ask students how a rotation may be used to take \(A\) to \(B\).

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

17.2: Card Sort: How Did This Get There? (20 minutes)

Activity

The purpose of this activity is to activate students’ prior knowledge about rigid and non-rigid transformations. Asking students to choose their own categories invites them to determine what characteristics might be important to notice. Students also begin to practice the skill of defining a rigid transformation that takes one figure onto the other without using a grid to estimate or define centers, angles, lines of reflection, or directed line segments.

Students may struggle to describe the location of the line of reflection. There is no need to address this during this activity as it will be covered in a subsequent activity.

Monitor for students who first translate to line up one pair of corresponding vertices, and then rotate for the card with triangle \(ABC\).

Launch

Arrange students in groups of 2. Distribute pre-cut cards. Give students 3 minutes to sort and then pause for discussion. Ask a few groups to explain how they categorized their cards.

Invite students to write the transformations and follow with a whole-class discussion.

Student Facing

  1. Your teacher will give you a set of cards that show transformations of figures.

    1. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.

    2. Then sort the cards into categories in a different way. Be prepared to explain the meaning of your new categories.

  2. For each card with a rigid transformation: write a sequence of rotations, translations, and reflections to get from the original figure to the image. Be precise.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Diego observes that although it was often easier to use a sequence of reflections, rotations, and translations to describe the rigid transformations in the cards, each of them could be done with just a single reflection, rotation, or translation. However, Priya draws her own card, shown, which she claims can not be done as a single reflection, rotation, or translation.

Triangles A B C and A prime B Prime C Prime.
  1. For each rigid transformation from the card sort, write the transformation as a single reflection, rotation, or translation.
  2. Justify why Priya’s transformation cannot be written as a single reflection, rotation, or translation.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

One purpose of discussion is to re-emphasize that translations, rotations, and reflections are rigid transformations, which maintain the size and shape of polygons while other transformations do not. Remind students that figures are called congruent when there is a sequence of rigid transformations that takes one figure onto another.

The second purpose of discussion is to help students identify what is hard about defining rigid transformations off the grid. If students struggle to define rigid motions precisely, there is an optional lesson on point-by-point transformations to use in addition to this discussion.

Consider the card with triangles \(ABC\) and \(A'B'C'\). If any student first translated to line up one pair of corresponding vertices, and then rotated, call on that student to share their method. If not, discuss the difficulty of estimating where to place the center of rotation so that a single rotation will definitely line up all three points. Demonstrate for students how you can translate by the directed line segment from \(C\) to \(C'\), and then rotate by the angle formed at vertex \(C\).

If no students used a point-by-point method, encourage them to take another look at the card with \(VWXYZ\) and try to use a point-by-point method to define rigid motions that take the original figure onto the image without estimating the locations of any new lines or points. 

Wait on a detailed discussion of the transformation of the card with \(RSTU\), as it will be the focus of the next activity.

Speaking: MLR 7 Compare and Connect. Use this routine when students present their categories and their sequence of rotations, translations, and reflections to get from the original figure to the image. Ask students to consider what is the same and what is different about the sequences of transformations. Draw students' attention to the way that transformations are described precisely. These exchanges strengthen students' mathematical language use and reasoning about transformations.
Design Principle(s): Maximize meta-awareness

17.3: Reflecting on Reflection (10 minutes)

Activity

In this activity, students determine whether a reflection can take one figure to another and how to find the precise line of reflection for disjoint figures.

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

Launch

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “I know _____ because…,” “First, I _____ because  _____. Then, I…,” and “This method works/doesn’t work because….”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Diego says, “I see why a reflection could take \(RSTU\) to \(R'S'T'U'\), but I’m not sure where the line of reflection is. I’ll just guess.”

  1. How could Diego see that a reflection could work without knowing where the line of reflection is?
  2. How could Diego find an exact line of reflection that would work?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Tell students to refer to the card from the previous activity with figure \(RSTU\).

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “I know _____ because…,” “First, I _____ because _____. Then, I…,” and “This method works/doesn’t work because….”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Diego says, “I see why a reflection could take \(RSTU\) to \(R'S'T'U'\), but I’m not sure where the line of reflection is. I’ll just guess.”

  1. How could Diego see that a reflection could work without knowing where the line of reflection is?
  2. How could Diego find an exact line of reflection that would work?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of this discussion is to reinforce using defintions to justify a response. Invite a few students to share their responses with the class. Highlight any improvements over previous justifications such as good use of a well-labeled diagram.

Lesson Synthesis

Lesson Synthesis

Display two congruent triangles for all to see:

Rigid transformations.

Ask students:

  • Which triangle is the original and which is the image? (The triangle on the right is the original and the triangle on the left is the image. The prime notation on the labels of the vertices gives it away.)
  • What would be a sequence of transformations that would take triangle \(ABC\) onto triangle \(A’B’C’\)? Explain how you know the image of \(A\) coincides with \(A’\). (Translate \(B\) onto \(B’\), and then rotate until the image of \(A\) lands on \(A’\). The transformations keep the distances the same, so the image of \(A\) is on ray \(B’A’\) at a distance of \(B’A’\). That means the image of \(A\) has to land on \(A’\).)
  • Would that sequence work just for this one pair of congruent triangles, or would it work for other pairs of congruent triangles, too? (This sequence will work for any pair of congruent triangles, so long as there isn’t a reflection involved.)

Display another image for all to see:

Reflected triangle.

Ask students:

  • What is different about this from the previous case? (The previous case does not involve a reflection.)
  • What happens when we try the same sequence of transformations from last time? (If we translate \(B\) onto \(B’\) and then rotate until the image of \(A\) lands on \(A’\), then the triangles will share side \(A’B’\), but the image of \(C\) will be on the opposite side of line \(A’B’\) as \(C’\).)
  • What do we need to add to the sequence to take triangle \(ABC\) onto triangle \(A’B’C’\)? (Performing a reflection across the line \(A’B’\) will take the image of \(C\) onto \(C’\).)

Tell students that defining these sequences and looking at the results is enough to conjecture that the sequences take one figure onto another. To be convincing, it is necessary to be able to explain why each vertex lands exactly where we think it should land. Explaining why each point lands exactly where it should is something they will work on over time.

17.4: Cool-down - Get This There (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

If 2 figures are congruent, we can always find a rigid transformation that takes one onto the other.

Look at congruent figures \(ABC\) and \(DEF\). It looks like ​​​​​\(DEF\) might be a reflection and translation of \(ABC\). But is there a way to describe a sequence of transformations without guessing where the line of reflection might be?

Two congruent triangles labeled ABC and DEF.

Our goal is to take the image of \(E\) onto \(B\). Then we want to take the image of \(D\) onto \(A\) without moving \(E\) and \(B\). Finally, we need to take the image of \(F\) onto \(C\) without moving any of the matching points.

We can start with translation: Translate triangle \(DEF\) by the directed line segment from \(E\) to \(B\).

Three congruent triangles, DEF, D’E’F’ and ABC.

 

Now, a pair of corresponding points coincides. Is there a transformation we could use to take \(D'\) onto \(A\) that leaves \(B\) and \(E'\) in place? Rotations have a fixed point, so rotate triangle \(D'E'F'\) by angle \(D'BA\) using point \(B\) as the center.

Four congruent triangles, DEF, D’E’F’, D”E”F” and ABC.

Now, 2 pairs of corresponding points coincide. Reflecting across line \(AB\) will take \(D''E''F''\) onto \(ABC\), which is what we were trying to do. We know \(D''\) and \(E''\) won’t move, since points on the line of reflection don't move. How do we know \(F''\) will end up on \(C\)? Since the triangles are congruent, \(F''\)and \(C\) are the same distance from the line of reflection.

It is always possible to describe transformations using existing points, angles, and segments. It could take an extra step, but we can be confident transformations work if we don't guess where the line of reflection or center of rotation might be.