Translate, reflect & rotate shapes on a grid
Theory
A slide, a flip and a turn move a shape to a new place on a grid. Their maths names are translation, reflection and rotation, and the moved shape is called the image. This Year 5 maths page shows how to do each move, how to describe it in squares, and what changes and what stays the same.
Points on a grid are written as (across, up). \((4, 2)\) means go \(4\) across from \(0\), then \(2\) up.
A translation (slide) moves every point the same number of squares in the same direction. Describe it as squares right or left and squares up or down. The shape does not turn or flip, so it still faces the same way.
A reflection (flip) makes a mirror image in a mirror line. Each point of the image is the same distance from the line as the point it came from, on the other side. The image faces the opposite way. A mirror line can go straight up and down or straight across.
A rotation (turn) turns a shape about a fixed point, the centre of the turn. A quarter turn is \(90^\circ\), a half turn is \(180^\circ\) and a three-quarter turn is \(270^\circ\). Clockwise is the way the hands of a clock turn; anticlockwise is the other way.
After a slide, flip or turn, the image is the same size and shape: the side lengths and the angles do not change.
What changes and what stays the same:
| Move | Size, side lengths, angles | Position | Way it faces |
|---|---|---|---|
| Translation (slide) | Stay the same | Changes | Stays the same |
| Reflection (flip) | Stay the same | Changes | Opposite way |
| Rotation (turn) | Stay the same | Changes | Turned |
| Turn | Degrees |
|---|---|
| Quarter turn | \(90^\circ\) |
| Half turn | \(180^\circ\) |
| Three-quarter turn | \(270^\circ\) |
| Whole turn | \(360^\circ\) |
A point on the mirror line stays where it is. The centre of a turn stays where it is.
How to slide a point
- Right or left: add to or take away from the across number.
- Up or down: add to or take away from the up number.
- Slide every corner the same way, then join the corners.
How to describe a slide
- Pick one corner and find the same corner on the image.
- Count the squares right or left, then up or down.
How to reflect in a mirror line
- Count how many squares each corner is from the mirror line.
- Count the same number of squares on the other side, straight across the line.
- Join the new corners. A corner on the line stays where it is.
How to turn about a point
- Find how far each corner is from the centre, and in which direction.
- Turn each direction: for a quarter turn clockwise, right becomes down and up becomes right.
- Count the same number of squares in the new direction. The centre does not move.
Two moves in a row
- Do the first move and write down where the point is.
- Do the second move from that new point.
Compare the across numbers, then the up numbers.
| \(\text{Across: } 4 - 1\) | \(=\) | \(3 \text{ right}\) |
| \(\text{Up: } 3 - 1\) | \(=\) | \(2 \text{ up}\) |
The triangle moves \(3\) squares right and \(2\) squares up.
| \(\text{Across: } 7 - 2\) | \(=\) | \(5 \text{ right}\) |
| \(\text{Up: } 6 - 4\) | \(=\) | \(2 \text{ down}\) |
The up number got smaller, so the move is down: \(5\) squares right and \(2\) squares down.
| \(\text{Across: } 5 - 2\) | \(=\) | \(3\) |
| \(\text{Up: } 2 + 4\) | \(=\) | \(6\) |
It lands on \((3, 6)\). Left takes away from the across number; up adds to the up number.
The corner is \(3\) squares left of the line, so its image is \(3\) squares right of it, at the same height.
| \(\text{Across: } 4 + 3\) | \(=\) | \(7\) |
The image is at \((7, 2)\).
(i) Where is the image of \((6, 8)\)?
(ii) Where is the image of \((2, 5)\)?
(i) \((6, 8)\) is \(8 - 5 = 3\) squares above the line, so its image is \(3\) squares below it. The across number stays \(6\).
| \(\text{Up: } 5 - 3\) | \(=\) | \(2\) |
The image is \((6, 2)\).
(ii) \((2, 5)\) is on the mirror line, so it does not move: its image is \((2, 5)\).
(ii) A quarter turn anticlockwise ends facing the same way as which turn clockwise?
(i) A quarter turn is \(90^\circ\), so a half turn is \(2 \times 90^\circ = 180^\circ\).
| \(360^\circ - 90^\circ\) | \(=\) | \(270^\circ\) |
(ii) A whole turn is \(360^\circ\). Turning \(270^\circ\) clockwise, a three-quarter turn, ends facing the same way.
Clockwise, right turns to down and up turns to right. O does not move.
| \(\text{A: } (3,\ 4 - 3)\) | \(=\) | \((3, 1)\) |
| \(\text{B: } (3 + 2,\ 4)\) | \(=\) | \((5, 4)\) |
A is \(3\) squares right of O, so it goes \(3\) squares below O: \((3, 1)\). B is \(2\) squares above O, so it goes \(2\) squares right of O: \((5, 4)\).
Slide: \((1, 4 + 2) = (1, 6)\).
| \(\text{Flip: } 3 + 2\) | \(=\) | \(5\) |
\((1, 6)\) is \(3 - 1 = 2\) squares left of the line, so the image is \(2\) squares right of it: \((5, 6)\). Always do the moves in the order given.
First flip: \(1\) square left of across \(2\) goes to \(1\) square right: \((3, 3)\).
Second flip: \(5 - 3 = 2\) squares left of across \(5\) goes to \(2\) squares right:
| \(5 + 2\) | \(=\) | \(7\) |
It ends at \((7, 3)\), facing the same way as at the start. One slide of \(7 - 1 = 6\) squares right does the same job, twice the gap of \(3\) squares between the lines.
| \(\text{Right: } 2 + 1\) | \(=\) | \(3\) |
Up is \(3\) squares. The single slide is \(3\) squares right and \(3\) squares up. The order of slides does not change where the counter ends.
Common pitfalls
Frequently asked questions
What is a translation?
A slide. Every point moves the same number of squares in the same direction. The shape does not turn or flip.
What is a reflection?
A flip in a mirror line. Each point of the image is the same distance from the line as the point it came from, on the other side, so the image faces the opposite way.
What is a rotation?
A turn about a fixed point, the centre of the turn. It can be a quarter turn (90°), a half turn (180°) or a three-quarter turn (270°), clockwise or anticlockwise.
What stays the same after a slide, flip or turn?
The size and shape: every side length and every angle. Only the position changes, and after a flip or a turn the way the shape faces can change too.
Which points do not move?
A point on the mirror line stays where it is in a reflection, and the centre of a turn stays where it is in a rotation.