Sequencing Rotations

Videos and solutions to help Grade 8 students learn that sequences of rotations preserve lengths of segments as well as degrees of measures of angles.

New York State Common Core Math Module 2, Grade 8, Lesson 9

Lesson 9 Student Outcomes

• Students learn that sequences of rotations preserve lengths of segments as well as degrees of measures of angles.
• Students describe a sequence of rigid motions that would map a triangle back to its original position after being rotated around two different centers.

Lesson 9 Summary
• Sequences of rotations have the same properties as a single rotation:
• A sequence of rotations preserves degrees of measures of angles.
• A sequence of rotations preserves lengths of segments.
• The order in which a sequence of rotations around different centers is performed matters with respect to the final location of the image of the figure that is rotated.
• The order in which a sequence of rotations around the same center is performed does not matter. The image of the figure will be in the same location.

Classwork
Exploratory Challenge
1.
a. Rotate △ ABC d degrees around center D. Label the rotated image as △ A'B'C'.
b. Rotate △ A'B'C' d degrees around center E. Label the rotated image as △ A''B''C''.
c. Measure and label the angles and side lengths of △ ABC. How do they compare with the images △ A'B'C' and △ A''B''C''.
d. How can you explain what you observed in part (c)? What statement can you make about properties of sequences of rotations as they relate to a single rotation?

2.
a. Rotate △ ABC d degrees around center D and then rotate again d degrees around center E. Label the image as △ A'B'C' after you have completed both rotations.
b. Can a single rotation around center D map △ A'B'C' onto △ ABC?
c. Can a single rotation around center E map △ A'B'C' onto △ ABC?
d. Can you find a center that would map △ A'B'C' onto △ ABC in one rotation? If so, label the center F.

3.
a. Rotate △ ABC 90°(counterclockwise) around center D then rotate the image another 90°(counterclockwise around center E. Label the image △ A'B'C'.
a. Rotate △ ABC 90°(counterclockwise) around center E then rotate the image another 90°(counterclockwise around center D. Label the image △ A''B''C''.
c. What do you notice about the locations of △ A'B'C' and △ A''B''C''. Does the order in which you rotate a figure around different centers have an impact on the final location of the figures image?

4.
a. Rotate △ ABC 90°(counterclockwise) around center D then rotate the image another 45°(counterclockwise) around center D. Label the image △ A'B'C'.
b. Rotate △ ABC 45°(counterclockwise) around center D then rotate the image another 90°(counterclockwise) around center D. Label the image △ A''B''C''.
c. What do you notice about the locations of △ A'B'C' and △ A''B''C''. Does the order in which you rotate a figure around the same center have an impact on the final location of the figures image?

5. △ ABC has been rotated around two different centers and its image is △ A'B'C'. Describe a sequence of rigid motions that would map △ ABC onto △ A'B'C'.