Geometry and measures - Properties and constructions: identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional scale factors) (Foundation tier)

The idea

Transformations change a shape's position or size, often on a coordinate grid. Translation, reflection and rotation produce a congruent image: identical in size and shape. Enlargement produces a similar image: same angles, sides in one ratio. To 'describe fully', give all required data. Translation: a column vector, e.g. $\begin{pmatrix} -4 \\ 1 \end{pmatrix}$ means 4 left, 1 up. Reflection: the mirror line equation, e.g. y = 2 or y = x. Rotation: centre (coordinates), angle and direction. Enlargement: centre and scale factor; a fractional scale factor such as ½ makes the image smaller. For example, enlarging a triangle by scale factor ½ about (0, 0) halves each vertex's coordinates. Construct images by applying the rule to each vertex, then joining them.

What earns the marks

• Drawing the image in the correct position and orientation on a coordinate grid, with vertices joined in the correct order.
• Giving every required part of a full description: for rotation, centre, angle and direction; for enlargement, centre and scale factor.
• Stating the equation of the mirror line correctly, e.g. y = x rather than y = -x.
• Writing a translation as a column vector in the form $\begin{pmatrix} x \\ y \end{pmatrix}$, with the top number for horizontal movement.
• Recognising that translation, reflection and rotation give congruent images, while enlargement gives a similar image.

Common mistakes and how to put them right

• Writing the column vector the wrong way round, e.g. $\begin{pmatrix} y \\ x \end{pmatrix}$ instead of $\begin{pmatrix} x \\ y \end{pmatrix}$.

  Put it right: The top number is always the horizontal movement.

• Confusing the lines y = x and y = -x, or x = c and y = c, when reflecting.

  Put it right: Sketch the line first to check.

• Forgetting one part of a description, most commonly the centre of rotation or enlargement.

  Put it right: A rotation needs centre, angle and direction, an enlargement needs centre and scale factor.

• Treating a fractional scale factor as making the shape larger.

  Put it right: A scale factor between 0 and 1 reduces the shape.

Exam tips

• Use tracing paper for rotations: trace the shape and the centre, then turn the paper through the given angle and direction.
• For enlargements, draw rays from the centre through each vertex; the image vertices lie on these rays at the scaled distance.
• Check congruence: after translation, reflection or rotation the image should be identical in size and shape to the original.