Year 6 · Mathematics · Space
Transformations and symmetry
Describe and perform translations, reflections and rotations of shapes, and identify lines of symmetry and rotational symmetry in patterns and designs.
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These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
Learning goals
Learners will
- Describe and perform translations, reflections and rotations of shapes.
- Identify lines of symmetry in shapes, letters and designs.
- Identify rotational symmetry and find its order.
Success looks like
- I can slide, flip and turn a shape and name each movement correctly.
- I can describe a rotation using quarter, half or three-quarter turns, clockwise or anticlockwise.
- I can find all the lines of symmetry in a shape.
- I can say how many times a shape looks the same during one full turn.
The big idea
A transformation moves a shape to a new position. In the three transformations in this lesson, the shape keeps exactly the same size and shape; only its position or direction changes.
- A translation is a slide. Every point of the shape moves the same distance in the same direction, for example “3 squares right and 2 squares down”. The shape does not turn or flip.
- A reflection is a flip over a mirror line. The new shape is a mirror image, and each point is the same distance from the mirror line as before, but on the other side.
- A rotation is a turn around a point. We describe it by the amount of turn (a quarter, half or three-quarter turn) and the direction (clockwise, like the hands of a clock, or anticlockwise).
An arrow helps to see rotations. An arrow pointing up, turned a quarter turn clockwise, points right. Another quarter turn clockwise (a half turn in total) makes it point down.
A shape has a line of symmetry if you can fold it along the line so both halves match exactly. A square has 4 lines of symmetry: 2 through the middles of opposite sides and 2 through opposite corners. A rectangle that is not a square has only 2. Its diagonals are not lines of symmetry, which you can test by folding a paper rectangle corner to corner.
A shape has rotational symmetry if it looks exactly the same in more than one position as you turn it through a full turn. The order of rotational symmetry is how many times it looks the same in one full turn. A square looks the same 4 times, so it has order 4. An equilateral triangle has order 3. The letter S has no line of symmetry, but it looks the same after a half turn, so it has rotational symmetry of order 2.
To test for rotational symmetry, trace the shape onto tracing paper, put a pencil point at its centre and turn the paper slowly. Count how many times the tracing fits exactly over the shape before you get back to the start.
Worked examples
Describing a translation
On grid paper, a triangle has a corner at the point 2 squares across and 1 square up. After a slide, that corner is at 6 squares across and 4 squares up. Describe the translation.
- Across: from 2 to 6 is 4 squares to the right.
- Up: from 1 to 4 is 3 squares up.
- The shape did not turn or flip, so it is a translation.
Answer: The triangle has been translated 4 squares right and 3 squares up.
Rotating an arrow
An arrow points up. It is rotated three-quarters of a turn clockwise. Which way does it point now?
- A quarter turn clockwise: up becomes right.
- Another quarter turn clockwise (half a turn so far): right becomes down.
- A third quarter turn clockwise: down becomes left.
Answer: The arrow points left. (This is the same as a quarter turn anticlockwise.)
Lines and order of symmetry
Find the number of lines of symmetry and the order of rotational symmetry of a rectangle that is not a square.
- Fold a paper rectangle in half from top to bottom: the halves match. That is 1 line.
- Fold it in half from side to side: the halves match. That is a 2nd line.
- Fold it along a diagonal: the halves do not match, so diagonals are not lines of symmetry.
- Turn the rectangle around its centre. It looks the same after a half turn and again after a full turn, so it fits 2 times.
Answer: A rectangle has 2 lines of symmetry and rotational symmetry of order 2.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
What do we call sliding a shape to a new position without turning or flipping it?
Translation — A translation is a slide. The shape keeps facing the same way.
How many lines of symmetry does a square have?
4 — A square has 2 lines through the middles of opposite sides and 2 lines through opposite corners: 4 in total.
Which capital letter has rotational symmetry but no line of symmetry?
S — S looks the same after a half turn but cannot be folded into matching halves. A and T have a vertical line of symmetry, and E has a horizontal one.
An arrow points up. It is rotated a quarter turn clockwise. Which way does it point now?
Right — Clockwise turns the same way as the hands of a clock. A quarter turn clockwise from up is right.
How many lines of symmetry does a rectangle that is not a square have?
2 — One line goes across the middle and one goes down the middle. The diagonals are not lines of symmetry.
An arrow points right. It is rotated a half turn. Which way does it point now?
Left — A half turn makes an arrow point the opposite way, so right becomes left. The direction of a half turn does not matter.
What is the order of rotational symmetry of an equilateral triangle?
3 — An equilateral triangle looks exactly the same 3 times during one full turn.
A shape is flipped over a line to make its mirror image. What is this transformation called?
Reflection — Flipping a shape over a mirror line is called a reflection.
Watch out for
- Thinking the diagonals of a rectangle are lines of symmetry. Fold a paper rectangle along a diagonal to show that the halves do not match.
- Mixing up clockwise and anticlockwise. Keep an analogue clock or a drawing of one nearby and trace the direction the hands move.
- Believing a reflected shape is just “the same shape moved”. Point out that a reflection reverses the shape, like writing seen in a mirror, which is why a translation and a reflection give different results.
Tips for parents
- Use a small mirror to test lines of symmetry in letters, leaves, logos and pictures around the house.
- Give movement instructions in a game: “slide 3 steps forward”, “make a quarter turn clockwise” and so on.
- Tracing paper (or baking paper) makes rotations easy to check, so let the learner test their predictions.
Go further
- Design a tile pattern on grid paper by translating, reflecting and rotating one simple shape, and label each transformation you used.
- Make a symmetrical paper snowflake or paper doll chain and count its lines of symmetry.
- Find the order of rotational symmetry of some regular shapes (square, regular pentagon, regular hexagon). What pattern do you notice?
About 45 minutes
You'll need
- Grid paper and a pencil
- Cardboard shapes or pattern blocks (squares, rectangles, triangles)
- Tracing paper or baking paper
- A small mirror (optional)
- Scissors
Key words
- Transformation
- A movement of a shape, such as a slide, flip or turn.
- Translation
- Sliding a shape up, down, left or right without turning or flipping it.
- Reflection
- Flipping a shape over a line to make a mirror image.
- Rotation
- Turning a shape around a fixed point.
- Line of symmetry
- A line that divides a shape into two halves that are mirror images of each other.
- Rotational symmetry
- A shape has rotational symmetry if it looks exactly the same more than once during a full turn.
Still tricky?
For parents
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