Tenth Grade · Math · week 11

Rotations and compositions of rigid motions

Apply coordinate rules for rotations about the origin, and carry out a composition of two rigid motions in the correct order.

A rotation turns every point of a figure through a given angle around a fixed center of rotation, and like translations and reflections, it preserves distance and angle measure. Around the origin, three rules cover what we need: rotating 90 degrees counterclockwise sends (x, y) to (-y, x); rotating 180 degrees sends (x, y) to (-x, -y); and rotating 270 degrees counterclockwise (the same as 90 degrees clockwise) sends (x, y) to (y, -x). Notice 180 degrees is its own reverse -- rotating twice by 180 degrees returns a point to where it started.

A composition of transformations performs two or more rigid motions one after another, using the image of the first as the preimage for the second. Order can matter a great deal: reflecting over the x-axis and then translating right 3 does not, in general, land in the same place as translating right 3 first and then reflecting -- always perform the transformations in the order listed, finishing one completely before starting the next. A figure has line symmetry if some line of reflection maps it exactly onto itself, and rotational symmetry if some rotation less than 360 degrees maps it exactly onto itself.

📋 For the teacher
Teach it (10–15 min):
  • Physically rotate a cutout triangle or index card through 90-degree turns to build intuition before touching the coordinate rules.
  • For the three rotation rules, say the pattern out loud: 90 degrees CCW swaps the coordinates AND negates the new first one -- easier to remember than three unrelated rules.
  • For compositions, require them to finish and WRITE DOWN the intermediate image before starting the second transformation -- skipping that write-down is the biggest source of errors.
  • Hunt for line and rotational symmetry in logos, snowflakes, or stop signs together -- symmetry is everywhere once you know to look for it.
Talk about it:
  • Why does rotating a point 180 degrees twice return it to its starting position?
  • Give an example of a composition where the order genuinely changes the final image.
  • What is the difference between line symmetry and rotational symmetry?

Mastery looks like: They apply the three rotation rules correctly and carry out a two-step composition of rigid motions by completing each transformation in order.

Common stumbles: Mixing up the 90-degree-counterclockwise and 90-degree-clockwise rules; performing both steps of a composition on the original figure instead of feeding the first image into the second transformation.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

Coordinate PlanePlot points and explore the four quadrants.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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