Coordinates

The coordinate grid

A coordinate grid has two number lines at right angles. The horizontal one is the $x$-axis and the vertical one is the $y$-axis. They cross at the origin, the point $(0,0)$.

Every point on the grid is given by a pair of coordinates $(x,y)$:

• the first number, $x$, says how far across the point is from the origin: positive to the right, negative to the left;
• the second number, $y$, says how far up it is: positive upwards, negative downwards.

The order matters: $(3,2)$ and $(2,3)$ are different points. A way to remember it is "along the corridor, then up the stairs": across first, then up or down. These are called Cartesian coordinates, after the mathematician René Descartes.

The axes split the grid into four quadrants. Moving anticlockwise from the top right, they are the first quadrant (both coordinates positive), the second ($x$ negative, $y$ positive), the third (both negative) and the fourth ($x$ positive, $y$ negative).

A point on an axis has a zero coordinate. Every point on the $y$-axis has $x=0$, like $(0,3)$. Every point on the $x$-axis has $y=0$, like $(-5,0)$.

$A(3,2)$ is in the first quadrant, $B(-4,1)$ in the second, $C(-2,-3)$ in the third and $D(1,-4)$ in the fourth. $E(0,3)$ is on the $y$-axis and $F(-5,0)$ is on the $x$-axis.

Reading and plotting points

To read a point: go straight down (or up) from the point to the $x$-axis and read the value: that is $x$. Then go straight across to the $y$-axis and read that value: that is $y$. Write them in brackets, $x$ first, separated by a comma.

To plot a point such as $(-3,4)$: start at the origin, move 3 to the left along the $x$-axis, then 4 up. Mark the point with a small neat cross or dot and label it if the question names it. A large blob covers several values and can cost the mark.

Check every point you plot by reading it back off the grid. The most common error by far is plotting $(4,-3)$ when you meant $(-3,4)$.

Scales and decimal coordinates

Not every grid goes up in ones. Before you read or plot anything, work out what one square is worth on each axis. The two axes can have different scales.

Find two labelled values on an axis, take the difference, and divide by the number of squares between them. If the $y$-axis is labelled 0, 4, 8 with two small squares between labels, each small square is worth $4÷2=2$.

Coordinates do not have to be whole numbers. On a grid with two small squares to each unit, a point halfway between 1 and 2 on the $x$-axis has $x=1.5$. Read decimals with care, and use the small squares, not a guess by eye.

Using coordinates to solve problems

Coordinates let you answer questions about shapes without measuring.

Horizontal and vertical lengths. Two points with the same $y$-coordinate lie on a horizontal line. The distance between them is the difference of their $x$-coordinates. From $(-3,4)$ to $(5,4)$ is $5-(-3)=8$ units: 3 units to reach the $y$-axis and 5 more beyond it. Points with the same $x$-coordinate lie on a vertical line, and their distance apart is the difference of the $y$-coordinates.

Missing corners. In a rectangle with sides parallel to the axes, each corner shares its $x$-coordinate with one neighbour and its $y$-coordinate with the other. In a parallelogram, opposite sides are equal and parallel, so the step from one corner to the next is the same on both sides: if you go 2 right and 4 up from Q to R, you also go 2 right and 4 up from P to S.

A quick sketch, even on rough paper, shows you which corner is missing and which way to go.