Symmetry and Transformations

The method, step by step

1. 1

   Reflection: state the mirror line

   reflection in the line x = 2

   Every point moves to the same perpendicular distance on the other side of the line. Describing it fully means naming the mirror line by equation — 'reflection in a vertical line' does not earn the mark.

2. 2

   Rotation: state angle, direction and centre

   rotation 90° clockwise about (0, 0)

   Three pieces of information are required and all three are marked. Omitting the centre is the most common omission, and a rotation with no centre stated is not a description of a unique transformation.

3. 3

   Translation: state the vector

   translation by the vector (3, −2)

   The top number is the horizontal movement and the bottom the vertical, with negatives meaning left and down. Writing it as words rather than a column vector is usually accepted but the vector is what the mark scheme expects.

4. 4

   Enlargement: state scale factor and centre

   enlargement scale factor 2, centre (1, 1)

   The scale factor multiplies every distance from the centre. A factor between 0 and 1 makes the shape smaller while still being called an enlargement, and a negative factor puts the image on the opposite side of the centre, inverted.

5. 5

   Describe with one transformation, not two

   'reflect then translate' scores nothing if one rotation would do

   Questions asking to describe a single transformation expect exactly one. Two combined transformations, however correct the end result, do not answer the question that was asked.

The marks are in the description

Drawing the image correctly is usually one mark; describing a transformation fully is usually two or three. Each transformation has a required set of details, and leaving one out costs a mark regardless of how accurate everything else is.

The required details are worth memorising as a checklist: reflection needs the line; rotation needs angle, direction and centre; translation needs the vector; enlargement needs the scale factor and centre. Students who run the checklist stop losing these marks.

• Reflection → the mirror line's equation
• Rotation → angle, direction, centre
• Translation → the column vector
• Enlargement → scale factor and centre
• One transformation only, when asked to describe

Congruent and similar

Reflection, rotation and translation produce a congruent image — identical in size and shape, only moved. Enlargement produces a similar image: the same shape with every length multiplied by the scale factor and every angle unchanged.

That distinction is examined directly. Asking which transformations preserve congruence is a routine question, and the answer is all of them except enlargement, unless the scale factor is 1 or −1.

Enlargements that shrink or invert

A scale factor of ½ produces a smaller image, and it is still called an enlargement — the word describes the transformation, not the direction of the size change. Students who expect enlargements to enlarge often assume they have made an error.

A negative scale factor places the image on the opposite side of the centre and turns it upside down. Factor −2 doubles the distances and inverts, which is worth practising because the position is genuinely hard to predict without drawing the rays.

How we teach transformations

We drill the description checklist separately from the drawing. Students are generally competent at producing the image and lose marks on the wording, so practising descriptions of transformations they have not drawn targets the actual weakness.

For enlargements we always draw the rays from the centre through each vertex, even when the scale factor is simple. It is the only method that stays reliable for fractional and negative factors, and switching methods later is harder than starting with the general one.