Transformations in math: translation, reflection, rotation and dilation

A transformation moves or resizes a shape. Translations slide, reflections flip, rotations turn, and dilations make it bigger or smaller.

The idea: four ways to move a shape

A transformation takes a shape, called the pre-image, to a new position, called the image. The image’s corners are named with a dash: A becomes A′ (“A prime”).

Translations, reflections and rotations keep the size and shape the same. Only a dilation changes the size (8.G.A.1).

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

ABCA′B′C′
Translation: 6 left and 2 up.
ABCA′B′C′
Reflection in the y-axis.
ABCA′B′C′
Rotation of 90° anticlockwise about the origin.
ABCA′B′C′
Dilation by a scale factor of 2 from the origin.

Translation: a slide

Every point moves the same distance in the same direction. The shape doesn’t turn or flip.

On a grid, describe it with steps: “6 left and 2 up”. In coordinates, (x, y) becomes (x − 6, y + 2).

Reflection: a flip

The shape is flipped over a mirror line. Each point ends up the same distance from the line, on the other side.

Reflecting in the y-axis changes (x, y) to (−x, y). Reflecting in the x-axis changes it to (x, −y). The image faces the other way, like a mirror.

Coordinate grid

Plot a point, then find where it goes after a reflection or a translation.

Rotation: a turn

The shape turns about a fixed point, the centre of rotation, by an angle such as 90° or 180°.

A 90° turn anticlockwise about the origin takes (x, y) to (−y, x). A 180° turn takes it to (−x, −y). The shape is turned, but not flipped.

Tracing paper helps. Trace the shape, hold the centre with a pencil point and turn the paper a quarter turn.

Dilation: bigger or smaller

A dilation multiplies every distance from a centre by a scale factor. A scale factor of 2 doubles the size; ½ halves it.

From the origin, (x, y) becomes (kx, ky). The image is similar to the original: same shape and angles, different size (8.G.A.4).

Congruent or similar?

After a translation, reflection or rotation, the image is congruent: exactly the same size and shape (8.G.A.2).

After a dilation it is similar: the same shape, but a different size. Every angle stays the same.

Key words

A week of practice

Ten minutes a day is plenty. Short, regular practice beats one long session at the weekend.

If one day goes badly, repeat it rather than moving on. Each day builds on the one before.

Quiz: which transformation?

Is it the same way round? Flipped? Turned?

Where this fits: geometry step by step

  1. Lines, rays and line segments
  2. Types of angles
  3. Measuring angles with a protractor
  4. Angle relationships
  5. Polygons
  6. Types of triangles
  7. Angles in a triangle
  8. Quadrilaterals
  9. Symmetry
  10. Perimeter
  11. Area
  12. Area of composite shapes
  13. Area of triangles
  14. Area of parallelograms and trapezoids
  15. Parts of a circle
  16. Circumference and area of a circle
  17. Faces, edges and vertices
  18. Volume
  19. Surface area
  20. Volume of cylinders, cones and spheres
  21. The coordinate plane
  22. Transformations (this page)
  23. Pythagorean theorem

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

  1. “Is the new shape the same size?”
  2. “Is it facing the same way?” (Then it slid: a translation.)
  3. “Is it a mirror image?” (A reflection.)
  4. “Is it turned round but not flipped?” (A rotation.)
  5. “Is it bigger or smaller?” (A dilation.)
  6. “Where does the point (2, 3) go if we reflect it in the y-axis?” ((−2, 3).)

More worked examples

Example 1: Translate (3, 1) 4 right, 2 down

(7, −1).

Example 2: Reflect (5, 2) in the x-axis

(5, −2).

Example 3: Rotate (2, 4) by 180° about the origin

(−2, −4).

Example 4: Dilate (3, 1) by 2

(6, 2).

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

  1. Slide with no turning
  2. Mirror image
  3. Turn about a point
  4. Make twice as big
  5. Reflect (4, 3) in the y-axis
  6. Translate (0, 0) 3 up
  7. Rotate (1, 0) 90° anticlockwise
  8. Dilate (2, 5) by 3
  9. Does a reflection change size?
  10. Stretch: rotate (3, 2) by 90° anticlockwise about the origin
Show the answers
  1. translation
  2. reflection
  3. rotation
  4. dilation
  5. (−4, 3)
  6. (0, 3)
  7. (0, 1)
  8. (6, 15)
  9. No
  10. (−2, 3)

Common mistakes

Mixing up reflection and rotationCalls a flipped shape a rotation.Fix: a rotation never makes a mirror image.
Reflecting in the wrong axisChanges y when reflecting in the y-axis.Fix: reflecting in the y-axis changes the sign of x.
Counting steps from the wrong cornerMoves one corner a different amount.Fix: every corner moves the same way in a translation.

How to teach it at home

Paper moves

Cut out an L shape and slide, flip and turn it on squared paper.

Mirror drawings

Draw half a shape next to a line; your child draws the reflection.

Tracing paper turns

Trace a shape, pin the centre with a pencil and turn it 90°.

By grade

More for each grade: Grade 4 math and the CBSE class 4 maths guide; Grade 5 math and the CBSE class 5 maths guide; Grade 6 math and the CBSE class 6 maths guide; Grade 7 math and the CBSE class 7 maths guide; Grade 8 math and the CBSE class 8 maths guide.

Worksheets and practice

Coordinate plane worksheet, page 1

Coordinate plane worksheet: printable PDF with an answer key.

Geometry practice · area and perimeter calculator · geometry for kids · area and perimeter chart · symmetry worksheet · lines of symmetry

Last reviewed

Questions parents ask

What are the four transformations?

Translation (slide), reflection (flip), rotation (turn) and dilation (resize).

What is the difference between a reflection and a rotation?

A reflection flips the shape into a mirror image. A rotation turns it without flipping.

What happens to coordinates in a reflection in the y-axis?

The x-coordinate changes sign: (x, y) becomes (−x, y).

Which transformations keep the same size?

Translations, reflections and rotations. The image is congruent to the original.

What grade learns transformations?

Grade 8 (8.G.A.1–4), building on symmetry in grade 4.