Transformations and Symmetry
Translations slide a shape
A translation slides a shape from one place to another without turning or flipping it. Every point of the shape moves the same distance in the same direction, so the shape arrives looking exactly as it started, just in a new position. Sliding a tile three squares right and two squares up is a translation. Because nothing about the shape itself changes, its size, its side lengths, its angles and the way it faces all stay the same. A translation only changes where the shape is, never what it looks like.
Reflections flip a shape
A reflection flips a shape over a line, called the mirror line, to produce a mirror image. Each point moves to the opposite side of the line, the same distance away, so the reflected shape is the same size and shape but reversed, like a reflection in a mirror. A shape and its reflection match when the page is folded along the mirror line. Reflecting changes the way the shape faces, turning a left-pointing flag into a right-pointing one, while keeping every length and angle the same.
Rotations turn a shape
A rotation turns a shape about a fixed point, called the centre of rotation, through an angle such as a quarter turn or a half turn. The shape spins around the centre, so it ends up facing a new direction while staying the same size and shape. Turning a shape a quarter turn clockwise about one of its corners is a rotation. Like the other transformations, a rotation keeps the shape itself unchanged; it alters only the direction the shape faces and, unless the centre is inside it, where the shape sits.
Lines of symmetry
A shape has line symmetry, also called reflective symmetry, when a line can be drawn so that one half is the mirror image of the other. Folding along that line of symmetry makes the two halves match exactly. A square has four lines of symmetry, a rectangle has two, and an equilateral triangle has three, while many shapes have just one or none at all. Finding the lines of symmetry of a shape means looking for every fold that leaves the two sides matching, which is the same idea as a reflection that maps the shape onto itself.
Rotational symmetry
A shape has rotational symmetry when it can be turned about its centre by less than a full turn and still look exactly the same. The number of times it matches itself in one complete turn is the order of its rotational symmetry. A square looks the same four times as it turns once around, so it has rotational symmetry of order four; an equilateral triangle has order three, and a rectangle order two. Every shape looks the same after a full turn, so rotational symmetry is only interesting when a shape matches itself before the turn is complete.
What changes and what stays the same
Across all three transformations, the key question is what changes and what stays the same. Translations, reflections and rotations all keep a shape the same size and shape, with the same side lengths and angles, because the shape is only moved, not redrawn. What can change is where the shape sits and the direction it faces: a translation changes position, a reflection reverses facing, and a rotation turns it. Recognising these constants and changes, and spotting the line and rotational symmetries a shape has, lets a child describe and perform any of these movements with confidence, ready for the coordinate transformations of later years.