8.G.A.3 8th Grade Geometry

Rotations

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

How to explain it

The anchor students hold onto: Rotate about the origin: 90° CCW uses (−y, x); 180° uses (−x, −y); 270° CCW uses (y, −x). CCW is the CCSS default unless stated otherwise.

Students extend rotation thinking in Dilations, then compose multiple transformations in Congruence through Transformations.

Worked examples

Example 1 Rotate 90° CCW

Rotate (3, 2) by 90° CCW.

Step 1Rule: 90° CCW gives (-y, x).

Step 2Apply: (3, 2) gives (-2, 3).

Answer(-2, 3)

Example 2 Rotate 180°

Rotate point (4, 1) by 180°.

Step 1Rule: 180° gives (-x, -y).

Step 2Apply: (4, 1) gives (-4, -1).

Answer(-4, -1)

Example 3 Rotate 270° CCW

Rotate (2, 5) by 270° CCW.

Step 1Rule: 270° CCW gives (y, -x).

Step 2Apply: (2, 5) gives (5, -2).

Answer(5, -2)

Common mistakes

What students write Applying the CW rule (y, -x) when the problem says 90° CCW — confusing rotation direction.

The fix CCW is the CCSS default. The 90° CCW rule is (-y, x), not (y, -x).

Try this Alex rotates P(3, 2) by 90° CCW about the origin and writes P′(2, −3). Identify the error and state the correct image coordinates. Error: ___________________________ Correct image: P′( ________ , ________ )

What students write Mixing up 90° CCW and 270° CCW rules: getting (-y, x) and (y, -x) swapped.

The fix Remember: 90° CCW = (-y, x); 270° CCW = (y, -x). The signs flip between them.

Teacher tip

Head off the two predictable errors before they happen. First: CCW is the CCSS default. The 90° CCW rule is (-y, x), not (y, -x). Second: Remember: 90° CCW = (-y, x); 270° CCW = (y, -x). The signs flip between them.