Lesson 13

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.

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

Provide access to protractors.

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.

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.

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).

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.)

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).

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.

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.)