Mathematics • Year 8 • Unit 3 • Lesson 18

Reflections and Rotations in the Real World

Use reflection and rotation rules where they actually show up: mirror reflections, clock hands, logo symmetry, snowflakes and Ferris wheels. Then explain your thinking in your own words.

Apply · Real-World Maths

1. Word problems

Each problem hides a reflection or rotation. Identify the transformation, state the rule, then apply it. Show working — a single final answer with no working earns only half marks.

1.1 — Mirror selfie. Imagine the y-axis is a vertical mirror. A person stands at the point (3, 4) (3 units to the right of the mirror, 4 units above the ground line).

(a) Where does their mirror image appear?
(b) Which transformation rule did you use?
(c) If the mirror were the x-axis instead (a horizontal mirror at ground level), where would the image appear? 3 marks

Stuck? Vertical mirror = reflect over the y-axis (x changes sign). Horizontal mirror = reflect over the x-axis (y changes sign).

1.2 — Clock hand at the centre. The tip of a clock's minute hand is at coordinate (0, 6) when the hand points straight up to 12.

(a) Where is the tip when the hand has rotated 90° anticlockwise (now pointing at 9)?
(b) Where is the tip after another 90° ACW (pointing at 6)?
(c) Where is the tip after a third 90° ACW (pointing at 3)? 3 marks

Stuck? 90° ACW: (x, y) → (−y, x). Apply repeatedly. After 4 steps you should return to (0, 6).

1.3 — Symmetric logo. A company logo has one point at (3, 2). To make the logo symmetric, the designer needs to add the mirror image across the y-axis AND the mirror image across the x-axis AND the 180° rotation of the original.

(a) Find the three new points to add (one from each transformation).
(b) The four points together form a familiar shape — what is it? 3 marks

Stuck? y-axis: (3, 2) → (−3, 2). x-axis: (3, 2) → (3, −2). 180°: (3, 2) → (−3, −2). The four points are corners of a rectangle.

1.4 — Snowflake design. A snowflake artist starts with a single "arm" with vertices at A(1, 0), B(3, 0), C(2, 1). To make the snowflake symmetric, they rotate the arm 90° ACW, 180°, and 270° ACW about the origin to create three more identical arms.

(a) Find the image of A under each rotation (three images).
(b) Find the image of B under each rotation (three images). 3 marks

Stuck? 90° ACW: (x, y) → (−y, x); 180°: (−x, −y); 270° ACW: (y, −x). Apply each to A and B.

1.5 — Ferris wheel cabin. A Ferris wheel has its centre at the origin. One cabin is at the point (8, 0) (the "3 o'clock" position) when the wheel is paused.

(a) The wheel rotates 90° anticlockwise. Where is the cabin now?
(b) The wheel continues another 90° ACW (180° total from the start). Where is the cabin now?
(c) Identify the transformation that maps the starting position straight to the position after (b). 3 marks

Stuck? Apply 90° ACW once, then twice. The result of 180° ACW is the rule (x, y) → (−x, −y), the same as a 180° rotation.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate is asked to rotate (2, 5) by 90° anticlockwise about the origin. They write "(2, 5) → (5, −2)" and explain "I swapped them and put a negative sign on the y". In your own words, explain (i) what mistake they have made, (ii) what the correct image is and how to get it, and (iii) one easy way they could check (for example, by quickly sketching where the point would land on a grid). Use the phrase "swap, then negate the new x" somewhere in your answer.

Stuck? Revisit lesson § Card 8 — for 90° ACW the rule is (x, y) → (−y, x). Swap first, then negate the NEW x-coordinate (which was the original y).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Mirror selfie

(a) Vertical mirror (y-axis): (x, y) → (−x, y). So (3, 4) → (−3, 4).
(b) Rule used: reflection over the y-axis.
(c) Horizontal mirror (x-axis): (x, y) → (x, −y). So (3, 4) → (3, −4).

1.2 — Clock hand

Rule 90° ACW: (x, y) → (−y, x).
(a) (0, 6) → (−6, 0) (pointing at 9).
(b) (−6, 0) → (0, −6) (pointing at 6).
(c) (0, −6) → (6, 0) (pointing at 3).

1.3 — Symmetric logo

(a) y-axis: (3, 2) → (−3, 2). x-axis: (3, 2) → (3, −2). 180°: (3, 2) → (−3, −2).
(b) The four points (3, 2), (−3, 2), (3, −2), (−3, −2) form the corners of a rectangle centred at the origin.

1.4 — Snowflake

(a) A(1, 0): 90° ACW → (0, 1); 180° → (−1, 0); 270° ACW → (0, −1).
(b) B(3, 0): 90° ACW → (0, 3); 180° → (−3, 0); 270° ACW → (0, −3).

1.5 — Ferris wheel

(a) (8, 0) under 90° ACW: (−0, 8) = (0, 8) (top of the wheel).
(b) Another 90° ACW: (0, 8) → (−8, 0) (9 o'clock position).
(c) From (8, 0) to (−8, 0) directly: both signs flipped, so the rule is (x, y) → (−x, −y) — a 180° rotation about the origin.

2.1 — Explain your thinking (sample response)

The classmate has applied the wrong rule — they negated the y instead of the new x. For 90° anticlockwise about the origin, the rule is (x, y) → (−y, x): swap, then negate the new x (which was the original y). The correct image of (2, 5) is (−5, 2). The classmate's answer (5, −2) actually matches the 90° clockwise (or 270° ACW) rule, which is (x, y) → (y, −x) — so they did a clockwise rotation by mistake. To check, a quick sketch helps: (2, 5) sits in the upper-right quadrant; a 90° anticlockwise turn about the origin should land it in the upper-left quadrant (a point like (−5, 2)). Landing in the lower-right (like (5, −2)) signals a clockwise rotation by mistake.

Marking: 1 mark for spotting that they used the clockwise rule by mistake; 1 mark for the correct image (−5, 2); 1 mark for the sanity check (quadrant sketch); 1 mark for clear full-sentence explanation using "swap, then negate the new x".