L5 · Rotations and describing transformations

Learning goal

“I can rotate points and polygons 90° and 180° about the origin in all four quadrants, and - given a shape and its image - I can describe exactly which transformation (translation, reflection, or rotation) connects them.”

Rotation — a turn of a shape about a fixed centre through a stated angle and direction.

Centre of rotation — the fixed point the shape turns about. In this lesson: the origin, unless stated.

Clockwise / counterclockwise — clockwise follows the clock hands (top -> right); counterclockwise is the opposite (top -> left). 180° lands the same either way.

Describe a transformation — to name the type AND its details: for a translation, the rule; for a reflection, the mirror line; for a rotation, the centre, angle and direction.

Worked example

Rotate the point (4, 2) by 90° clockwise about the origin. Step 1: The centre is the origin, so the point's offset from the centre is just (4, 2) - exactly the Grade 6 offset setup with centre (0, 0). Step 2: A 90° clockwise turn sends offset (a, b) to (b, -a): (4, 2) -> (2, -4). Step 3: Check with the picture in words: (4, 2) sits right-and-slightly-up; a quarter turn clockwise should put it down-and-slightly-right - (2, -4) fits. A 180° turn sends (a, b) to (-a, -b): (4, 2) -> (-4, -2), the opposite corner. A 90° COUNTERCLOCKWISE turn sends (a, b) to (-b, a): (4, 2) -> (-2, 4). Say it plainly: swap, then negate the right one - 90° CW negates the NEW y (the old x lands untouched as the new x, the old y lands negated as the new y). For a polygon, turn every vertex by the same rule.

Check yourself — explain it

A 180° rotation sends (a, b) to (-a, -b) no matter the direction because...

Paired problem — example still visible

Rotate (1, 5) 90° clockwise about the origin. -> (5, -1)

Faded completion — last step is yours

"Rotate (-3, 1) by 180°: (-(-3), -(1)) = ( ___, ___ )" -> (3, -1)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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