Key Stage 4 maths · Geometry and measures

GCSE enlargement, similarity and vectors: a guide for parents

In Years 10 and 11, pupils extend work on transformations in two directions. They use fractional scale factors for enlargements and apply congruence and similarity to find missing lengths, and they meet vectors, which describe movement with a size and a direction. Vectors are written as column vectors, drawn as arrows, added, subtracted and multiplied by a number.

Enlargement, similarity and vectors, explained

An enlargement with a scale factor between 0 and 1, such as ½ or ⅔, makes the shape smaller, though in maths it is still called an enlargement. Every distance from the centre of enlargement is multiplied by the scale factor. With centre (0, 0) and scale factor ½, the point (4, 6) moves to (2, 3). Lengths are multiplied by the scale factor, but angles don’t change, so the image is similar to the original. To enlarge by ⅔ on a grid, pupils work out how far each corner is from the centre and take two thirds of that distance.

Congruent shapes are identical in shape and size, even if one has been rotated, reflected or moved. Two triangles are congruent if they match in three sides (SSS), two sides and the angle between them (SAS), two angles and a corresponding side (ASA or AAS), or a right angle, hypotenuse and one other side (RHS). Similar shapes have equal corresponding angles and corresponding lengths in the same ratio. A line drawn parallel to one side of a triangle makes a smaller triangle that is similar to the whole, a diagram pupils meet often.

To find a missing length in similar shapes, pupils first work out the scale factor between two matching sides, then apply it to the side they want. If one triangle has sides 4, 6 and 8 and a similar triangle has a shortest side of 10, the scale factor is 10 ÷ 4 = 2.5, so the other sides are 15 and 20. Care is needed to match the right sides, which is why labelling and checking angles come first.

A vector has size and direction. Pupils describe a translation as a column vector: the top number is the movement to the right (negative for left) and the bottom number is the movement up (negative for down). So a column with 3 on top and −2 underneath means 3 right and 2 down. It looks a little like a coordinate pair, but it describes a movement, not a position, and there is no fraction line between the numbers. In the text below, (3, −2) is shorthand for that column. Vectors are named with a lower-case letter, printed in bold or underlined when written by hand, or by the two points they join, such as AB with an arrow above.

To add vectors, add the top numbers and add the bottom numbers; in a diagram this means following the arrows nose to tail. Multiplying a vector by a number, such as 3a, multiplies both numbers. Subtracting is the same as adding the opposite: a − b = a + (−b), and −b is b reversed. Vectors that are multiples of each other are parallel. A common skill is finding a vector by choosing a route through known points. If OA = a and OB = b, then AB = AO + OB = −a + b = b − a. Using this to prove geometric facts is higher-tier work.

Words your child will hear

Scale factor: The number all lengths are multiplied by in an enlargement. A scale factor of ½ makes everything half the size.
Centre of enlargement: The fixed point that distances are measured from when a shape is enlarged.
Congruent: Exactly the same shape and size, though possibly turned over or around.
Similar: The same shape but not necessarily the same size: equal angles and lengths in a constant ratio.
Vector: A quantity with both size and direction, often written as a column of two numbers.
Column vector: Two numbers stacked, the top for movement across and the bottom for movement up or down.
Scalar: An ordinary number, as opposed to a vector, used to multiply a vector such as 2a.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Interpret and use fractional scale factors for enlargements
Apply the concepts of congruence and similarity, including the relationships between lengths in similar figures
Describe translations as 2D vectors
Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

a = (3, 2) and b = (−1, 5) as column vectors. Work out 2a − b. Then find where the point (1, 1) is after a translation by 2a − b.

Multiply a by 2: multiply both numbers by 2.

       | 3 |   | 6 |
2a = 2 | 2 | = | 4 |

Subtract b by subtracting the top numbers and the bottom numbers. Take care with the negative: 6 − (−1) = 6 + 1.

2a - b = | 6 | - | -1 |
         | 4 |   |  5 |

       = | 6 + 1 |
         | 4 - 5 |

       = |  7 |
         | -1 |

So 2a − b is the vector (7, −1): 7 to the right and 1 down.
A translation by this vector adds 7 to the x-coordinate and −1 to the y-coordinate. For (1, 1): 1 + 7 = 8 and 1 + (−1) = 0.

2a − b = (7, −1), and the point (1, 1) moves to (8, 0).

Common mistakes and how to help

Believing a fractional scale factor is not an enlargement, or makes the shape bigger

Everyday language uses “enlarge” to mean “make bigger”, so a factor of ½ seems to be a mistake.

What helps: In maths the scale factor describes the multiplier. Factors greater than 1 make the shape larger, and factors between 0 and 1 make it smaller. Check a corner: if it was 6 from the centre and the factor is ⅓, the image is 2 from the centre.

Matching the wrong sides in similar shapes

Pupils divide the side they can see most easily rather than the matching one, especially when a shape is turned round or drawn inside another.

What helps: Match the angles first, then the sides opposite them. Write corresponding pairs down in a list: shortest to shortest, longest to longest.

Getting vector directions and signs wrong

AB and BA are mixed up, the top and bottom numbers are swapped, or a negative is dropped when subtracting.

What helps: BA is the same journey as AB in the opposite direction, so BA = −AB. Remember across first (top), then up or down. Subtracting a negative adds, so check each line.

Thinking congruent shapes must face the same way

The shapes in textbooks are often shown as copies slid across the page, so a mirror image looks like a different shape.

What helps: Trace one shape onto tracing paper, then turn it over and slide it on top. If it fits, the two shapes are congruent. Only the sizes and angles matter.

How to help at home

Enlarge and shrink a drawing. On squared paper, draw a simple shape with straight sides, then enlarge it by scale factor 2 and shrink it by scale factor ½. Measure the angles of each to confirm they stay the same.
Measure a tree with a shadow. On a sunny day in a park or garden, measure your height and your shadow, then the shadow of a tree or lamp post at the same time. Because the sun’s rays hit both at the same angle, the triangles are similar. If you are 1.6 m tall with a 2 m shadow and the tree’s shadow is 10 m, the tree is 10 × 0.8 = 8 m tall. Take both measurements within a few minutes of each other, and never look directly at the sun.
Give vector directions for a hunt. Hide an object and give directions in vectors on a grid of floor tiles or squared paper: “start at the door, go (3, 2), then (−1, 4)”. Ask where it is, and what single vector would take you straight there.
Test congruence with tracing paper. Draw two triangles with different orientations, trace one, and see whether it fits the other exactly. Talk about which three measurements would be enough to guarantee a match.

Builds on

Key Stage 3: Perimeter, area and volume
Key Stage 3: Constructions and scale drawings
Key Stage 3: Transformations and enlargement
Key Stage 3: Angles, parallel lines and polygons
Key Stage 3: Pythagoras and trigonometry