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.](https://cms-im.s3.amazonaws.com/f4E9LuPw6GdrXoepXaV2xLu1?response-content-disposition=inline%3B%20filename%3D%22Semaphore%20alphabet%20flag.png%22%3B%20filename%2A%3DUTF-8%27%27Semaphore%2520alphabet%2520flag.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T232845Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5acd3c75ae6ede18306b505d0656b4e28245fa5e59102966462f0de516fc3526)
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
J
Student Response
For access, consult 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.
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.
Supports accessibility for: Organization; Attention
Student Facing
- Rotate \(ABCD\) 90 degrees clockwise around \(Q\).
- Rotate \(ABCD\) 180 degrees around \(R\).
- Rotate \(HJKLMN\) 120 degrees clockwise around \(O\).
- Rotate \(HJKLMN\) 60 degrees counterclockwise around \(P\).
Student Response
For access, consult 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.
Supports accessibility for: Organization; Attention
Student Facing
- Rotate \(ABCD\) 90 degrees clockwise around \(Q\).
- Rotate \(ABCD\) 180 degrees around \(R\).
- Rotate \(HJKLMN\) 120 degrees clockwise around \(O\).
- Rotate \(HJKLMN\) 60 degrees counterclockwise around \(P\).
Student Response
For access, consult 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
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 clockwise 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
For access, consult one of our IM Certified Partners.
Launch
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 clockwise 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
For access, consult 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.
- Find the center and angle of this rotation.
- Describe a general procedure for finding a center of rotation.
Student Response
For access, consult 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.)
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
N
O
13.4: Cool-down - Find a Sequence (5 minutes)
Cool-Down
For access, consult 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 \(P\).
Rotate \(EFG\) 120 degrees counterclockwise around point \(C\).