Lesson 13

Incorporating Rotations

13.1: Left to Right (10 minutes)

Warm-up

In this activity, students describe rotations. During discussion, students work together to determine that a complete description of a rotation includes a center, a direction, and an angle. Students also practice measuring angles with protractors to prepare for a subsequent lesson in which they will need to use protractors to draw rotations with a given angle measure.

Launch

Display this image and explain that people use flags to signal messages with the semaphore alphabet.

Flag semaphore figure.

Provide access to protractors.

Student Facing

The semaphore alphabet is a way to use flags to signal messages. Here's how to signal the letters Z and J. For each, precisely describe a rotation that would take the left hand flag to the right hand flag.

Z

Stick-figure person with L flag in left hand and R flag in right hand. Both arms are on same side of body. Left arm extends horizontally. Right arm extends down and at an angle.

J

Stick-figure person with L flag in left hand and R flag in right hand. Right arm extends straight up. Left arm extends out horizontally from the body.

 

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 identify the characteristics of a rotation and practice using protractors. Invite students to share their descriptions. Record their responses for all to see. Annotate the images as students share.

Demonstrate how to use a protractor to measure the angles. Invite the students to practice using their protractors. Students may get angle measures that vary slightly. Discuss the limitations of measuring with a protractor. When the diagram is labeled, we can know the exact measurement; otherwise, we have to estimate by using whatever tools are available.

Display the image with construction marks.

Flag Semaphore R. L.

Invite students to define rotation. (Every point of a figure moves in a circle around the center. There needs to be a direction, clockwise or counterclockwise, and an angle.) If not mentioned by students, point out that the construction mark is a circle, which means the distance from the center remains constant.

13.2: Turning on a Grid (15 minutes)

Activity

Students rotate images on both a rectangular grid and an isometric grid in this activity to practice rotating by a variety of angles. They will practice considering all the aspects of a rotation: center, angle, and direction of rotation.

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

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 rotate.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each rotation correctly before moving on to the next rotation.
Supports accessibility for: Organization; Attention

Student Facing

  1. Rotate \(ABCD\) 90 degrees clockwise around \(Q\).
  2. Rotate \(ABCD\) 180 degrees around \(R\).
  3. Rotate \(HJKLMN\) 120 degrees clockwise around \(O\).
  4. Rotate \(HJKLMN\) 60 degrees counterclockwise around \(P\).

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 rotate. Recommend that students use their pencil to hold the tracing paper at the center of rotation and then turn the tracing paper around that point.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each rotation correctly before moving on to the next rotation.
Supports accessibility for: Organization; Attention

Student Facing

Quadrilateral A B C D on a square grid.
  1. Rotate \(ABCD\) 90 degrees clockwise around \(Q\).
  2. Rotate \(ABCD\) 180 degrees around \(R\).
  3. Rotate \(HJKLMN\) 120 degrees clockwise around \(O\).
  4. Rotate \(HJKLMN\) 60 degrees counterclockwise around \(P\).
Isometric grid, Quadrilateral and two points.

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 solidify the aspects of a rotation.

“What information do you need to do a rotation?” (Center, angle, and direction of rotation.)

“Why don't you need to know the direction of rotation when the angle of rotation is 180 degrees?” (Both clockwise and counterclockwise land in the same place since 180 degrees is half a circle.)

13.3: Translate, Rotate, Reflect (10 minutes)

Activity

Students practice drawing transformations by following a character's instructions. This is the first time students see the point-by-point transformations they will use in a subsequent unit to prove that triangles are congruent. 

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, “First, I _____ because _____. Then, I…,” “How do you know…,” and “This method works/doesn’t work because.…”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Mai suspects triangle \(ABC\) is congruent to triangle \(DEF\). She thinks these steps will work to show there is a rigid transformation from \(ABC\) to \(DEF\).

  • Translate by directed line segment \(v\).
  • Rotate the image ____ degrees counterclockwise around point \(D\).
  • Reflect that image over line \(DE\).

Draw each image and determine the angle of rotation needed for these steps to take \(ABC\) to \(DEF\).

Student Response

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

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, “First, I _____ because _____. Then, I…,” “How do you know…,” and “This method works/doesn’t work because.…”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Mai suspects triangle \(ABC\) is congruent to triangle \(DEF\). She thinks these steps will work to show there is a rigid transformation from \(ABC\) to \(DEF\).

  • Translate by directed line segment \(v\).
  • Rotate the image ____ degrees counterclockwise around point \(D\).
  • Reflect that image over line \(DE\).

Draw each image and determine the angle of rotation needed for these steps to take \(ABC\) to \(DEF\).

Triangles A B C and D E F on isometric grid. Directed line segment, v, starts at A and points to D.

Student Response

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

Student Facing

Are you ready for more?

Mai’s first 2 steps could be combined into a single rotation.

  1. Find the center and angle of this rotation.
  2. Describe a general procedure for finding a center of rotation.

Student Response

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

Anticipated Misconceptions

If students are struggling to identify the angle of rotation, ask student to find and trace the angle connecting a point, its image, and the center of rotation. Then provide two options. They can use a protractor to measure an angle, or they can use the properties of the grid to calculate the measure of an angle.

Activity Synthesis

The purpose of this discussion is to emphasize that figures are congruent if there is a rigid transformation that takes one to the other. Here are some questions for discussion:

  • “What is the definition of congruent?“ (There is a rigid transformation that takes one figure to the other.)
  • “We have a transformation that takes triangle \(ABC\) to triangle \(DEF\). What does that tell us?“ (The sequence includes a translation, a reflection, and a rotation. Each of those is a rigid motion, so the measurements of triangle \(ABC\) must match the measurements of triangle \(DEF\). They're the same, just in different locations.)
Speaking, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to describe rotations. After a student speaks, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as: center, angle, direction of rotation, clockwise, counterclockwise, and "around point X." Consider inviting remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.
Design Principle(s): Support sense-making; Optimize output (for explanation)

Lesson Synthesis

Lesson Synthesis

Invite students to draw or act out (using their arms and paper as flags) a few transformations. To act out the letters, tell students the starting position of their left hand flag.

  • Rotate 135 degrees clockwise. (Semaphore letter S) 
  • Reflect over a vertical line. (Semaphore letter N)
  • Translate to the left, then rotate 45 degrees clockwise. (Semaphore letter O)

S

A stick figure. R. L.

N

A flag Semaphore. R. L.

O

A stick figure. R. L.

13.4: Cool-down - Find a Sequence (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

The 3 rigid motions are reflect, translate, and rotate. Each of these rigid motions can be applied to any figure to create an image that is congruent. To do a rotation, we need to know 3 things: the center, the direction, and the angle. 

Rotate \(ABCD\) 90 degrees clockwise around point \(P\).

Polygons A B C D and A prime B prime C prime D prime on square grid. Point P located 1 unit up and 1 unit to the right of A. 90 degree angle A P A prime is drawn in.

Rotate \(EFG\) 120 degrees counterclockwise around point \(C\).

Triangles E F G and E prime F prime G prime on isometric grid. Point C located 2 units down and to the right of F prime and 2 units down and to the left of F. Angle F prime C F marked 120 degrees.