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G.5(C)·Geometry·Patterns & TransformationsSupporting

Transformations in Real-World Designs

Apply properties of transformations, including rotations, to real-world layouts.

Students rotate asymmetric real-world layouts about a specified center, especially 90° counterclockwise, and check the result using landmarks, distances, and orientation.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice45 minutes

Learn the concept

Specify the complete rotation

A rotation requires a center, an angle, and a direction. Every point turns through the same angle about the center and remains the same distance from it. A 90° counterclockwise rotation is a quarter-turn opposite the direction of a clock's hands.

Track relative positions

About the origin, a 90° counterclockwise rotation maps (x,y)(x,y) to (−y,x)(-y,x). Right becomes up, up becomes left, left becomes down, and down becomes right. For another center, apply the rule to offsets from that center.

Check the whole design

A rotation preserves lengths, angle measures, and the figure's orientation around its boundary. It turns every feature together; it does not mirror the plan. Check two landmarks and the fixed center rather than relying on the outside shape alone.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 51 · 1 point

A rectangular yard, taller than it is wide, containing an irregular hexagonal pool that is wider than it is tall, with its short vertical edge on the right. Drain D is a point inside the pool, at the center of the yard.
Original pool layoutOriginal visual from the released STAAR test.

Worked example

A courtyard plan has center C(2, 3). A fountain is at F(5, 4), and a gate is at G(2, 1). Rotate both features 90° counterclockwise about C.

  1. Relative to C, the fountain has offset (3, 1). Rotate this to (−1, 3).
  2. Add the center back: F′ = (2 − 1, 3 + 3) = (1, 6).
  3. The gate has offset (0, −2), which rotates to (2, 0).
  4. Add the center back: G′ = (4, 3). The center C remains (2, 3).

F′(1, 6); G′(4, 3)

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Geometry - Describing Rotations Using Coordinate Notation

Five Minute Math · Aaron Daffern, Dallas ISD

Connect quarter-turn layouts to coordinate rules and a specified center.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. A triangular tile has vertices (1, 1), (4, 1), and (1, 2). Rotate it 90° counterclockwise about the origin.

    Show answer and reasoning

    (−1, 1), (−1, 4), and (−2, 1)

    Apply (x, y) → (−y, x) to each vertex. Every point makes the same quarter-turn.

  2. 2. A bench directly above a rotation center appears directly to its right in a proposed quarter-turn layout. Is this a 90° counterclockwise rotation?

    Show answer and reasoning

    No; it indicates a 90° clockwise rotation.

    A point above the center must move to the left of the center under a 90° counterclockwise rotation.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 51 · 1 point

A rectangular yard, taller than it is wide, containing an irregular hexagonal pool that is wider than it is tall, with its short vertical edge on the right. Drain D is a point inside the pool, at the center of the yard.
Original pool layoutOriginal visual from the released STAAR test.

The diagram represents one layout of a hexagonal swimming pool that contains a drain at point DD in the center of a rectangular yard. Other layouts are also being considered. Which layout is the result of a 90° counterclockwise rotation of the original layout using DD as the center of rotation?

  1. Layout AThe rectangular yard, taller than wide, with the hexagonal pool standing taller than wide and its short edge at the bottom. Drain D is inside the pool.
  2. Layout BThe rectangular yard, taller than wide, with the hexagonal pool wider than tall and its short edge on the left. Drain D is inside the pool.
  3. Layout CThe rectangular yard, taller than wide, with the hexagonal pool wider than tall and its short edge on the right, its slanted sides arranged as a top-to-bottom mirror image of the original. Drain D is inside the pool.
  4. Layout DThe rectangular yard, taller than wide, with the hexagonal pool standing taller than wide and its short edge at the top. Drain D is inside the pool.
Show STAAR answer and explanation

Layout D

Layout D carries the pool’s left-of-drain portion below the drain and its upper-left vertex to the lower-left, as a 90° counterclockwise turn requires. The drain stays fixed; the yard outline is retained as a reference.

Layout A: Layout A places the pool’s upper-left vertex at the upper-right, matching a clockwise rather than counterclockwise quarter-turn.

Layout B: Layout B keeps the pool’s long direction horizontal and reflects its upper and lower features rather than making a quarter-turn.

Layout C: Layout C keeps the pool’s long direction horizontal and reverses its left and right features rather than making a quarter-turn.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Identify the center, direction, and angle of a rotation.
  • Rotate a layout 90° counterclockwise using tracing or coordinates relative to the center.
  • Distinguish a rotated design from a reflected or translated design.

Instructional sequence

  1. 1

    Engage

    5 min

    Display a garden plan with an off-center bench. Ask where the bench moves when the entire plan turns a quarter-turn left.

  2. 2

    Physical model

    10 min

    Pin tracing paper at the rotation center and turn the entire drawing 90° counterclockwise. Track two different landmarks.

  3. 3

    Direct instruction

    10 min

    Connect a quarter-turn to right becoming up and up becoming left. State the origin-centered coordinate rule.

  4. 4

    Design comparison

    14 min

    Compare candidate layouts by checking the center, two landmarks, and the direction of the turn. Include reflected distractors.

  5. 5

    Closure

    6 min

    Complete the exit ticket and explain one feature preserved by the rotation.

Misconceptions to address

““Counterclockwise means every point moves left.””

Teacher move: Movement depends on a point's position relative to the center. A point to the right moves above the center in a 90° counterclockwise turn.

““A matching outline proves the layout is correct.””

Teacher move: Symmetric outlines can hide a wrong transformation. Track an asymmetric feature such as a gate, bench, or drain.