Key Stage 3 maths · Geometry and measures

KS3 transformations, congruence and enlargement: a parent’s guide

Transformations are the ways a shape can be moved or resized on a grid. In Years 7 to 9 pupils investigate the properties of triangles, quadrilaterals and circles, describe and carry out translations, rotations and reflections, decide when shapes are congruent, and enlarge shapes to make similar ones, with and without a coordinate grid.

Transformations and enlargement, explained

Before moving shapes, pupils need to know them well. They work out and explain properties rather than just memorising them: an isosceles triangle has two equal sides and two equal angles, the diagonals of a rectangle are equal and bisect each other, and the diagonals of a rhombus cross at right angles. They learn that a square is a special rectangle and a special rhombus, so shapes can belong to more than one family. They name the parts of a circle, such as the radius, diameter, chord and arc. Many lessons use dynamic geometry software so pupils can drag a vertex and see which properties stay the same.

Three transformations keep a shape exactly the same size, so the image is congruent to the original. A translation slides every point the same distance in the same direction, described as, for example, 3 right and 2 up (some schools also introduce column vector notation at this stage). A reflection flips the shape in a mirror line, and each point ends up the same perpendicular distance from the mirror on the other side. A rotation turns the shape about a centre through a stated angle and direction, such as 90° clockwise. To describe a transformation fully, pupils must give every piece of information: a reflection needs the mirror line, a rotation needs the centre, angle and direction.

Congruent means exactly the same shape and size, although one copy may be turned over or round. Pupils identify congruent triangles by looking for matching sides and angles, and they construct triangles given three sides (SSS), two sides and the angle between them (SAS), or two angles and a side (ASA), then check that every triangle built from the same information is identical. This links directly to the ruler and compass skills taught alongside.

An enlargement changes the size but not the shape. Each length is multiplied by a scale factor, so a scale factor of 3 makes every side three times longer. The angles do not change. Some schools also introduce fractional scale factors in Key Stage 3, where a scale factor of ½ halves every length and makes the shape smaller, while others leave these for GCSE. Pupils enlarge shapes on a coordinate grid from a centre of enlargement by multiplying distances from that centre, and they also enlarge by drawing rays from the centre through each corner. The result is similar to the original: the same shape, with corresponding sides in the same ratio and corresponding angles equal. Some classes go on to notice that when every length is doubled the area is multiplied by four, an idea developed further at GCSE.

Words your child will hear

Image: The shape you get after a transformation. The original is sometimes called the object.
Translation: A slide: every point moves the same distance in the same direction, with no turning.
Reflection: A flip in a mirror line. The image is a mirror image of the object.
Rotation: A turn about a fixed centre, described by its centre, angle and direction (clockwise or anticlockwise).
Congruent: Exactly the same shape and size, even if one has been turned or flipped.
Scale factor: The number every length is multiplied by in an enlargement.
Similar: The same shape but a different size: equal angles and sides in the same ratio.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies
Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

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

A worked example

Triangle ABC has corners A(1, 1), B(3, 1) and C(1, 2). Enlarge it by scale factor 2 with the centre of enlargement at the origin (0, 0). Write down the new coordinates and check that the image is similar to the original.

With the centre at (0, 0), each point’s distance from the centre is doubled in the same direction. Because the centre is the origin, this is the same as multiplying both coordinates by 2.

Work out each new corner.

A (1, 1) → A′ (2, 2)
B (3, 1) → B′ (6, 2)
C (1, 2) → C′ (2, 4)

Check on the grid with a ruler: a line from the origin through A passes through A′, and the same is true for B and C. The three lines meet at the centre.
Compare the sides. AB is 2 units long and A′B′ is 4. AC is 1 unit and A′C′ is 2. Each is doubled, as expected.
```
AB = 2 → A′B′ = 4
AC = 1 → A′C′ = 2
```
The angle at A is a right angle and so is the angle at A′. Equal angles and sides in the same ratio mean the triangles are similar, but they are not congruent because their sizes differ.

The image is triangle A′B′C′ with A′(2, 2), B′(6, 2) and C′(2, 4). It is similar to ABC, not congruent.

Common mistakes and how to help

Enlarging by adding the scale factor instead of multiplying

Adding 2 to each side of a 2 by 1 rectangle gives 4 by 3, which feels like an enlargement but changes the shape.

What helps: Draw both on squared paper. The true enlargement of 2 by 1 by scale factor 2 is 4 by 2. Multiplying keeps the ratio of sides, and adding does not.

Counting squares sideways when reflecting

When the mirror line is a diagonal, counting horizontally gives the wrong place, because the distance that matters is the perpendicular distance to the mirror.

What helps: Use tracing paper. Draw the shape, trace the mirror line, flip the paper over and line up the mirror line. Or fold along the mirror line and check the image lands exactly on the original.

Giving only part of the description of a transformation

Pupils write “rotation” or “rotate 90°” and consider it finished, but a rotation about a different centre gives a different result.

What helps: Use a checklist: translation needs a direction and distance, reflection a mirror line, rotation a centre, an angle and a direction, enlargement a centre and a scale factor.

Believing congruent shapes must look the same way round

In a textbook, congruent shapes are often drawn in the same orientation, so a flipped or turned copy does not seem like the same shape.

What helps: Cut out a paper triangle and move it around the page: slide it, turn it, flip it over. It is still congruent in every position.

How to help at home

Use a mirror. Stand a small mirror upright on a drawn line next to a simple shape. Look at the reflection, then draw the image on paper and compare. Try letters of the alphabet and see which look the same when flipped.
Enlarge a picture with a grid. Draw a light grid of 1 cm squares over a simple cartoon, then copy it square by square onto a grid of 3 cm squares. Each length has been multiplied by 3, and the result is similar to the original.
Find congruent shapes. Look at floor tiles, wallpaper or a quilt. Which shapes are congruent? Which are rotations or reflections of each other? Patterns that repeat are built from transformations.
Spot the symmetry. Choose objects around the house, such as a spoon, a leaf or a logo. How many lines of symmetry does each have, and can it be turned to fit onto itself?

Builds on

Year 6: Units, area and volume
Year 6: Angles, circles and nets
Year 6: Four-quadrant coordinates, translation, reflection

Leads on to

Key Stage 4: Enlargement, similarity and vectors
Key Stage 4: Circles, solids and bearings
Key Stage 4: Pythagoras and trigonometry
Key Stage 4: Circle theorems and vector proof
Key Stage 4: Sine rule and cosine rule