Transformations - 90 Degree Rotations

🔑 Key Concepts

  • A rotation turns a figure around a fixed point called the center of rotation.
  • In this lesson, the center of rotation is the origin, \((0,0)\).
  • A \(90^\circ\) clockwise rotation and a \(90^\circ\) counterclockwise rotation use different coordinate rules.
  • The rule for a \(90^\circ\) clockwise rotation around the origin is \((x,y)\rightarrow(y,-x)\).
  • The rule for a \(90^\circ\) counterclockwise rotation around the origin is \((x,y)\rightarrow(-y,x)\).
  • Rotations are rigid transformations, so the image keeps the same size and shape as the preimage.

✏️ Worked Examples

🧠 Math Vocabulary

  • Rotation: a transformation that turns a figure around a fixed point.
  • Center of Rotation: the fixed point that a figure rotates around.
  • Origin: the point \((0,0)\), where the \(x\)-axis and \(y\)-axis intersect.
  • Clockwise: turning in the same direction as the hands of a clock.
  • Counterclockwise: turning in the opposite direction of the hands of a clock.
  • Quadrant: one of the four regions of the coordinate plane, each with its own pattern of positive and negative coordinates.
  • Rigid Transformation: a transformation that preserves size and shape.
  • Image: the new figure after a transformation.
  • Preimage: the original figure before a transformation.
  • Coordinate Rule: an algebraic rule that shows how each ordered pair changes during a transformation.

💡 Main Idea

A \(90^\circ\) rotation around the origin changes both the order and signs of the coordinates. Clockwise and counterclockwise rotations use different rules, so it is important to pay attention to the direction of the turn.

📚 What You Should Already Know

Students should know how to plot ordered pairs, identify the origin, name the four quadrants, and determine whether coordinates are positive or negative in each quadrant.

🚀 What Comes Next

After practicing \(90^\circ\) rotations, students can compare \(90^\circ\), \(180^\circ\), and \(270^\circ\) rotations, then connect rotations with other rigid transformations such as reflections and translations.

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