Mirror, Slide, Turn: Mastering Transformations

Look at these four words: slide, flip, turn, grow. Three of them describe ways to move a shape without changing its size. One does not belong. Which word is the odd one out, and why? Think for 30 seconds before reading on. The odd one out is 'grow', because sliding, flipping, and turning a shape keep its size and shape exactly the same. Mathematicians call these moves transformations. Today you will learn the precise names and rules for three types: translation (slide), reflection (flip), and rotation (turn).

A transformation moves or flips a shape on a grid. The original shape is called the object. The result after the move is called the image. Importantly, all three transformations you will learn today are isometric — meaning the object and image are always the same size and shape. **1. Translation (Slide)** A translation moves every point of a shape the same distance in the same direction. We describe it using a column vector, written as two numbers stacked vertically in brackets: the top number shows movement left (−) or right (+) along the x-axis, and the bottom number shows movement down (−) or up (+) along the y-axis. Example: the vector (3, −2) means 'move 3 right and 2 down'. **2. Reflection (Flip)** A reflection flips a shape over a mirror line. Every point in the image is the same perpendicular distance from the mirror line as the matching point in the object, but on the opposite side. Common mirror lines include the x-axis (y = 0), the y-axis (x = 0), and the diagonal line y = x. Key rule for the x-axis: point (a, b) maps to (a, −b). For the y-axis: (a, b) maps to (−a, b). For y = x: (a, b) maps to (b, a). **3. Rotation (Turn)** A rotation turns a shape around a fixed point called the centre of rotation. You must state three things: the centre, the angle (90°, 180°, 270°), and the direction (clockwise or anticlockwise). Key rule for 90° anticlockwise about the origin: point (a, b) maps to (−b, a). For 180° about the origin: (a, b) maps to (−a, −b). For 90° clockwise: (a, b) maps to (b, −a).