Transformations (Years 7–11) — Maths

Slide a chess piece across the board, hold a photo up to a mirror, spin the minute hand of a clock, or zoom in on a map — every one of those moves is a transformation. A transformation changes a shape's position or size on a grid. There are four to know: three that only move, turn or flip a shape (it stays exactly the same size) and one that resizes it.

Why this matters: transformations are how computer games move sprites, how designers rotate and scale logos, and how maps enlarge a street. They also train you to work carefully with coordinates and signs — a skill that runs right through GCSE algebra and geometry.

A translation slides a shape to a new position — same size and shape

The four transformations

Think of the first three as rigid moves — the shape is picked up and put down unchanged, like moving a jigsaw piece. Only the last one changes size.

Translation — slide the shape using a column vector (top number = right, bottom number = up). Size and orientation unchanged.
Reflection — flip it over a mirror line (e.g. the x-axis, y-axis, or the line y = x). Size unchanged; it comes out back-to-front.
Rotation — turn it about a fixed centre by an angle in a stated direction (clockwise or anticlockwise). Size unchanged.
Enlargement — resize it by a scale factor from a centre of enlargement. Scale factor 2 doubles every length; a fractional scale factor makes it smaller.

Transformation	What changes	Size?	You must state
Translation	Position	Same	The column vector
Reflection	Position + flipped	Same	The mirror line
Rotation	Position + turned	Same	Centre, angle, direction
Enlargement	Size	Different	Scale factor + centre

Translating a point

A translation just adds the vector to each coordinate: add the top number to x, add the bottom number to y.

Translate (2, 3) by the vector (4, −1): add to each coordinate → (2+4, 3−1) = (6, 2). The shape has slid 4 right and 1 down.

Quick reflection rules

A reflection flips the shape over a mirror line — same size, but back-to-front.

Reflecting in the main lines follows three sign patterns worth memorising:

In the x-axis: the y-coordinate changes sign — (3, 4) → (3, −4).
In the y-axis: the x-coordinate changes sign — (3, 4) → (−3, 4).
In the line y = x: swap the coordinates — (2, 5) → (5, 2).

Enlargement and area

Here is the idea students most often trip over. Imagine enlarging a 1 cm by 1 cm square by scale factor 3. Each side becomes 3 cm — but the new square is a 3-by-3 grid of nine small squares. The lengths tripled, yet the area is nine times bigger.

A shape of area 5 cm² is enlarged by SF 3: lengths scale by 3, so area scales by 3² = 9. New area = 5 × 9 = 45 cm².

Ray lines from the centre double every distance — that is enlargement by scale factor 2.

Common mistake: lengths grow by the scale factor, but area grows by the scale factor squared. SF 3 makes lengths 3× but area 9× bigger — never just 3× the area.

Triple the sides and the square holds 9 copies — area scales by 3² = 9.

In a nutshell: translate (slide), reflect (flip) and rotate (turn) keep the size; enlarge resizes — and area always scales by the factor squared.