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Year 6 maths · Key Stage 2 · Geometry – position and direction

Year 6 coordinates, translation and reflection: a guide

In Year 6 children extend coordinates beyond the first quadrant they have used before, so that points can have negative x and y values. They describe positions anywhere on the grid, draw shapes from their coordinates, work out a missing corner, translate shapes by a given movement and reflect them in the x-axis and y-axis.

Four-quadrant coordinates, translation, reflection, explained

A coordinate grid is made from two number lines that cross at right angles. The horizontal one is the x-axis, the vertical one is the y-axis, and they meet at the origin, (0, 0). The axes split the grid into four quadrants. In earlier years children worked only in the top right quadrant, where every number is positive. In Year 6 the axes are extended below and to the left of the origin, so children meet negative numbers on both lines. Pupils draw and label their own axes with equal scaling, which means the same distance on the page always stands for the same amount, and the scale may go up in ones, twos or fives.

A position is written as a pair of coordinates in brackets, with the x value first and the y value second. Many teachers use the phrase “along the corridor and up the stairs”: move along the x-axis first, then up or down. So (−3, 2) is three squares to the left of the origin and two up, while (2, −3) is two to the right and three down. Children plot points like this, join them to draw a shape, and name the quadrant a point is in. They are also asked to predict missing coordinates. If three corners of a rectangle are (−3, 1), (2, 1) and (2, −2), the fourth must share the x value of the first and the y value of the last, so it is (−3, −2). With other shapes, such as a square or a parallelogram, they use what they know about the sides, for example that opposite sides of a parallelogram are equal in length and parallel.

A translation slides a shape without turning or flipping it. Every point moves the same distance in the same direction, so the shape stays the same size and faces the same way. Children describe a translation in words, such as “3 right and 2 down”, and carry it out by moving every vertex. Some schools also ask Year 6 children to write a translation algebraically, for example (a, b) → (a − 2, b + 3), which means “subtract 2 from the x value and add 3 to the y value”. The point (4, −1) would move to (2, 2).

A reflection produces a mirror image. In Year 6 the mirror lines are the axes themselves. Each point lands the same distance from the mirror as it started, but on the opposite side. Reflecting in the y-axis, which is the vertical line, changes the sign of the x value and keeps the y value, so (−3, 1) becomes (3, 1). Reflecting in the x-axis, the horizontal line, changes the sign of the y value and keeps the x value. A reflected shape is the same size as the original but turned back to front, so the order of its corners runs the opposite way round. Reflecting in sloping lines, and turning shapes about a point, come later in secondary school.

Words your child will hear

Coordinates
A pair of numbers that gives a position on a grid, written (x, y) with the x value first.
Origin
The point (0, 0) where the x-axis and y-axis cross.
Axes
The two number lines of a grid: the horizontal x-axis and the vertical y-axis. Just one of them is called an axis.
Quadrant
One of the four regions the axes divide the grid into. The first quadrant is the top right.
Vertex
A corner of a shape. The plural is vertices.
Translation
Sliding a shape in a straight line without turning or flipping it.
Reflection
Flipping a shape over a mirror line to make a mirror image.

What the National Curriculum says

In Year 6, pupils should be taught to:

  • Describe positions on the full coordinate grid (all 4 quadrants)
  • Draw and translate simple shapes on the coordinate plane, and reflect them in the axes

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(−3, 1), B(−3, 4) and C(−1, 1). (a) Translate it 5 right and 3 down. (b) Reflect the original triangle in the y-axis. Write down the new coordinates.

  1. Draw axes from −5 to 5 in both directions with the same scale. Plot A, B and C. A is 3 left of the origin and 1 up, and the triangle has a right angle at A.
  2. For the translation, 5 right adds 5 to every x value and 3 down takes 3 off every y value. Do the same to all three corners.
    A (−3, 1) → (2, −2)
B (−3, 4) → (2, 1)
C (−1, 1) → (4, −2)
  3. Check the new triangle. A′B′ is still straight up and down and still 3 squares long, and A′C′ is still 2 squares long. If the sides have changed length, one corner has been moved wrongly.
  4. For the reflection, the y-axis is a vertical mirror. Each point goes the same distance to the other side, so the x value changes sign and the y value stays the same.
    A (−3, 1) → (3, 1)
B (−3, 4) → (3, 4)
C (−1, 1) → (1, 1)
  5. Check it is a mirror image. In the original, C is to the right of A. In the reflection, C″ is to the left of A″, so the triangle has been flipped.

Translation: A′(2, −2), B′(2, 1), C′(4, −2). Reflection in the y-axis: A″(3, 1), B″(3, 4), C″(1, 1).

Common mistakes and how to help

Swapping the coordinates, so (2, 5) is plotted at (5, 2)

Children read the pair as two separate numbers and go for whichever feels natural first, often the vertical one.

What helps: Keep to “along the corridor, then up the stairs” and have your child trace the route with a finger. Ask them to plot (2, 5) and (5, 2) and compare the two points.

Plotting negative numbers one square out

When counting left or down from the origin, some children count the origin itself as the first step, so −3 ends up at −2. Others label the spaces between lines rather than the lines.

What helps: Number the lines, not the gaps, and start counting from the step after the origin. A thermometer or a lift with basement floors is a good picture of negative numbers.

Thinking a reflection in the x-axis changes the x value

The name “x-axis” makes it sound as though x should be the thing that moves. In fact the x-axis is a horizontal mirror, so the shape flips up to down and the y value changes sign.

What helps: Draw the mirror line in a different colour and count how far each corner is from it. A small mirror stood on the line shows the result straight away.

Moving only some corners in a translation, or by different amounts

Counting squares for every vertex is fiddly, and one slip stretches or squashes the shape.

What helps: Apply the same instruction to every corner and write the old and new coordinates side by side. Then check that the new shape has the same side lengths as the original.

How to help at home

  • Hide and find. Draw a large grid with axes from −5 to 5 on a sheet of paper, or outdoors with chalk. Hide a small object on a point and give your child clues in coordinates to find it, then swap roles.
  • Negative numbers in real life. Talk about temperatures below zero, lift buttons for basement floors or a bank balance in the red. Comfort with negative numbers on a vertical line makes the lower half of the grid much easier.
  • Mirror and slide. Draw a simple shape on squared paper next to an axis. Stand a small mirror on the axis line to see the reflection, then ask your child to draw it and write the coordinates. Next, ask for the shape to be slid by a movement such as 4 left and 1 up.
  • Find the missing corner. Give three corners of a rectangle or parallelogram and ask for the fourth, then plot all four to check. Making up puzzles for you to solve is even better practice.

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