Lesson 10

Rigid Transformations

10.1: Notice and Wonder: Transformed (10 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that some shapes can be described as transformations of other shapes, which will be useful when students specify sequences of rigid transformations that take one figure onto another in the next activities. While students may notice and wonder many things about these images, the important discussion point is that rigid transformations take sides to sides of the same length and angles to angles of the same measure.

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.

Action and Expression: Internalize Executive Functions. Provide students with a table to record what they notice and wonder prior to being expected to share these ideas with others. 
Supports accessibility for: Language; Organization

Student Facing

What do you notice? What do you wonder?

3 parallelograms on isometric grid. Parallelogram S has vertices A, B, C and D. Parallelogram M has vertices A’, B’, C’ and D’. Parallelogram D has vertices E, F, G and H.

Student Response

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

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 that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, and point out contradicting information.

If sequences of rigid transformations or corresponding measurements do not come up during the conversation, ask students to discuss this idea. Reinforce that because the sizes, shapes, and angles did not change from Figure \(S\) to Figure \(M\), that transformation is called a rigid transformation. But the transformation from Figure \(S\) to Figure \(D\) is not a rigid transformation because the size changed.

If the difference between \(A\) and \(A'\) does not come up during the conversation, ask students to discuss this idea and tell them that \(A'\) is pronounced “\(A\) prime.” Explain that \(ABCD\) is called the original figure and \(A'B'C'D'\) is called the image of the transformation.

10.2: What’s the Same? (15 minutes)

Activity

The purpose of this activity is to activate students’ prior knowledge of rigid transformations. Students also see that the image of a polygon is determined by the images of each of its vertices. Students build toward the concept that transformations are functions that take points as inputs and produce points as outputs so that distances and angles are preserved.

Launch

Give students 3 minutes of quiet time to work, then pause for a brief whole-class discussion.

Invite a student to demonstrate how to use dynamic geometry software to translate. Invite students to define translation. (A translation has a distance and a direction. It moves every point in a figure the given distance in the given direction.)

Display the image from the warm-up with construction marks.

Reflection.

Invite students to define reflection. (Every point of the figure ends up on the other side of the line of reflection and the same distance from the line.) If not mentioned by students, point out that the construction marks are lines perpendicular to the line of reflection.

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch, take time to review the following terms from previous lessons that students will need to access for this activity: rigid transformation, image, translation, reflection, and congruent.
Supports accessibility for: Memory; Language

Student Facing

Draw each rigid transformation. Use the Style Bar to choose a different color for each one.

  1. Translate figure \(S\) along the line segment \(v\) in the direction shown by the arrow. Color: _____________
  2. Reflect figure \(S\) across line \(y\). Color: _____________
  3. Reflect figure \(S\) across line \(m\). Color: _____________
  4. Translate figure \(S\) along the line segment \(w\) in the direction shown by the arrow. Reflect this image across line \(y\). Color: _____________
  5. How are the images the same? How are they different?

Student Response

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

Launch

Give students 3 minutes of quiet time to work, then pause for a brief whole-class discussion.

Invite a student to demonstrate how to use tracing paper to translate. Recommend that students start with the edges of the tracing paper parallel to the sides of the paper so they can see if they accidentally tilt the tracing paper as they translate. Invite students to define translation. (A translation has a distance and a direction. It moves every point in a figure the given distance in the given direction.)

Display the image from the warm-up with construction marks.

Reflection.

Invite students to define reflection. (Every point of the figure ends up on the other side of the line of reflection and the same distance from the line.) If not mentioned by students, point out that the construction marks are lines perpendicular to the line of reflection.

Action and Expression: Develop Expression and Communication. Maintain a display of important terms and vocabulary. During the launch, take time to review the following terms from previous lessons that students will need to access for this activity: rigid transformation, image, translation, reflection, and congruent.
Supports accessibility for: Memory; Language

Student Facing

Figure S on isometric grid. Vertical line y, line segment v intersects figure S and line m and is parallel to line segment w.

Draw each rigid transformation in a different color.

  1. Translate figure \(S\) along the line segment \(v\) in the direction shown by the arrow. Color: _____________
  2. Reflect figure \(S\) across line \(y\). Color: _____________
  3. Reflect figure \(S\) across line \(m\). Color: _____________
  4. Translate figure \(S\) along the line segment \(w\) in the direction shown by the arrow. Reflect this image across line \(y\). Color: _____________
  5. How are the images the same? How are they different?

Student Response

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

Anticipated Misconceptions

Some students may have trouble reflecting on the isometric grid. Ask these students to use tracing paper to fold across the line of reflection to find the image.

If students are stuck as to how to translate when \(w\) isn't connected to a vertex of the figure, remind students that \(w\) is telling them the direction and the distance, but the location doesn't matter.

Activity Synthesis

The important idea for discussion is that rigid transformations preserve distances and angles. Display a student’s work for all to see and ask:

  • For the reflection of \(S\) across line \(y\), how do the side lengths of \(S\) compare to the corresponding side lengths in its image? (The lengths are equal.)
  • What is the measures of the angle in the upper left corner of \(S\)? How does this compare to the corresponding angle measure in any of the images of \(S\)? (This angle and all of its images measure 120 degrees.)

10.3: Does Order Matter? (10 minutes)

Activity

The purpose of this activity is to observe that the order of the transformations in a sequence of transformations can have an effect on the image.

Monitor for students who define a sequence that works even when the order is reversed to compare to students whose sequences don’t work when the order is reversed during the discussion.

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

Launch

Arrange students in groups of 2. Remind students that if there is a sequence of rigid transformations that takes one figure onto another, the figures are called congruent. Tell students there are many possible answers for the questions. 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.

If students are struggling after several minutes, invite students to share what rigid motions they will need. (Reflection because Figure \(B\) is an L shape and Figure \(C\) can be rotated to look like an L, but Figure \(A\) cannot be.) Suggest that they start each sequence with a reflection, then use a translation.

Reading, Writing: MLR 3 Clarify, Critique, Correct. Present a first draft of a description of a sequence of transformations that will take Figure \(A\) onto Figure \(B\): “I flipped it over and slid it around.” Prompt discussion by asking, “What were the steps the author took?” Ask students to clarify and correct the statement. Improved statements should include some of the following: a directed distance for a translation, specifying which point is taken to which point as a result of a translation, the words “translation” and “reflection,” the line over which Figure A is being reflected. This will help students develop descriptions of transformations. Design Principle(s): Maximize meta-awareness; Optimize output (for explanation)
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

Here is an applet with 3 congruent L shapes on a grid. 

  1. Describe a sequence of transformations that will take Figure \(A\) onto Figure \(B\).
  2. If you reverse the order of your sequence, will the reverse sequence still take \(A\) onto \(B\)?
  3. Describe a sequence of transformations that will take Figure \(A\) onto Figure \(C\).
  4. If you reverse the order of your sequence, will the reverse sequence still take \(A\) onto \(C\)?

Student Response

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

Launch

Arrange students in groups of 2. Remind students that if there is a sequence of rigid transformations that takes one figure onto another, the figures are called congruent. Tell students there are many possible answers for the questions. 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.

If students are struggling after several minutes, invite students to share what rigid motions they will need. (Reflection because Figure \(B\) is an L shape and Figure \(C\) can be rotated to look like an L, but Figure \(A\) cannot be.) Suggest that they start each sequence with a reflection, then use a translation.

Reading, Writing: MLR 3 Clarify, Critique, Correct. Present a first draft of a description of a sequence of transformations that will take Figure \(A\) onto Figure \(B\): “I flipped it over and slid it around.” Prompt discussion by asking, “What were the steps the author took?” Ask students to clarify and correct the statement. Improved statements should include some of the following: a directed distance for a translation, specifying which point is taken to which point as a result of a translation, the words “translation” and “reflection,” the line over which Figure A is being reflected. This will help students develop descriptions of transformations. Design Principle(s): Maximize meta-awareness; Optimize output (for explanation)
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

Here are 3 congruent L shapes on a grid. 

L Shaped Polygons A, B and C on a square grid. A is an upside down L, facing to the right. B is a regular L, facing to the right. C is an L on its side, facing up.
  1. Describe a sequence of transformations that will take Figure \(A\) onto Figure \(B\).
  2. If you reverse the order of your sequence, will the reverse sequence still take \(A\) onto \(B\)?
  3. Describe a sequence of transformations that will take Figure \(A\) onto Figure \(C\).
  4. If you reverse the order of your sequence, will the reverse sequence still take \(A\) onto \(C\)?

Student Response

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

Student Facing

Are you ready for more?

  1. Construct some examples of sequences of two rigid transformations that take Figure \(A\) to a new Figure \(D\) where reversing the order of the sequence also takes Figure \(A\) to Figure \(D\).

  2. Make some conjectures about when reversing the order of a sequence of two rigid transformations still takes a figure to the same place.

Student Response

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

Anticipated Misconceptions

Reversing the sequence means using the same steps, but step 2 becomes step 1. For example, 1) reflect across line \(\ell\) then 2) translate left 2 units would become 1) translate left 2 units then 2) reflect across line \(\ell\).

Activity Synthesis

Select previously identified students to share their responses.

Highlight that reversing the steps in a sequence of transformations sometimes results in the same transformation, and sometimes it results in a different transformation.

Lesson Synthesis

Lesson Synthesis

Ask students, “What important ideas did you learn about rigid transformations?” (The result is called an image. The image is congruent to the original figure.)

Display the blank reference chart for all to see and give 1 copy of the blank reference chart blackline master to each student. Explain that in order to write convincing arguments, they need to support their statements with facts. The reference chart is a way to keep track of those facts for future reference when they are trying to prove new facts. Ask students to add an assertion and a definition to their reference charts as you add them to the class reference chart:

A rigid transformation is a translation, reflection, rotation, or any sequence of the three.
Rigid transformations take lines to lines, angles to angles of the same measure, and segments to segments of the same length.
(Assertion)

A rigid transformation.

One figure is congruent to another if there is a sequence of translations, rotations, and reflections that takes the first figure exactly onto the second figure. 
The second figure is called the image of the rigid transformation.
(Definition)

\(\triangle EDC \cong \triangle E'D'C'\)

Image of a congruent figure. CDE. C prime, D prime, E prime.

Each entry in the reference chart includes a statement, a diagram and a type. The types will be assertions, definitions, and theorems. Explain that an assertion is an observation that seems to be true, but is not proven. The fact that rigid transformations always take lines to lines, angles to angles of the same measure, and segments to segments of the same length seems to be true, but there is no way to prove or disprove this. So, moving forward, they can assert that rigid transformations have these properties and find out what follows from that starting point. Remind students that earlier, they conjectured that the perpendicular bisector of a segment is the set of points that are the same distance away from each endpoint. They could write this as an assertion based on their experiments, but they will see in the next unit that it’s actually possible to prove that it’s true based on the assertion they just made about rigid transformations. Geometry is generally more interesting when you try to connect as many ideas as possible to just a few starting assertions.

10.4: Cool-down - How Will That Get There? (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

A figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes one of the figures onto the other. This is because translations, rotations, and reflections are rigid motions. Any sequence of rigid motions is called a rigid transformation. A rigid transformation is a transformation that doesn’t change measurements on any figure. With a rigid transformation, figures like polygons have corresponding sides of the same length and corresponding angles of the same measure.

The result of any transformation is called the image. The points in the original figure are the inputs for the transformation sequence and are named with capital letters. The points in the image are the outputs and are named with capital letters and an apostrophe, which is referred to as “prime.”

There are many ways to show that 2 figures are congruent since many sequences of transformations take a figure to the same image. However, order matters in a set of instructions. Sometimes we can switch 2 steps in a sequence and get the same output, but other times, switching 2 steps results in a different image. These 2 sequences of transformations both have the points \(A\), \(B\), and \(C\) as inputs and points \(A’’\), \(B’’\), and \(C’’\) as outputs. Each step in the sequences of rigid transformations creates a triangle that is congruent to triangle \(ABC\).

Two images of triangle A B C  in a transformation sequence.