Finding a Place on a Grid

A cinema ticket says "row 4, seat 6". With just two numbers, you can walk straight to your seat, even in a dark room with hundreds of seats. Can two numbers name any place on a flat page in the same way?

Two numbers name a place

Picture the seats as a grid. Count the rows from the front and the seats from the side: row 4, seat 6 is exactly one seat. Each number alone is not enough: "seat 6" is in every row.

Rows of cinema seats facing a screen. One seat, in row 4 and seat 6, is coloured amber

Across, then up

Mathematicians do the same on squared paper. Draw two lines that are perpendicular: one going across, one going up. Each is called an axis (two of them are axes). They are marked 0, 1, 2, 3 ... like two number lines, and they meet at 0 on both, a point called the origin.

To name a point, start at the origin, count how far across you go, then how far up. Write the two numbers in brackets, across first: (3, 2) means 3 across and 2 up. These two numbers are the point's coordinates.

A grid with two axes marked 0 to 7 across and 0 to 5 up. From the origin, an amber arrow goes 3 squares across and then 2 squares up to point A, labelled (3, 2)

The order matters

(3, 2) and (2, 3) use the same two numbers, but they are different points. (2, 3) is 2 across and 3 up. That is why everyone agrees on one rule: across first, then up.

The same grid with two points: (3, 2) in blue, 3 across and 2 up, and (2, 3) in green, 2 across and 3 up

Shapes on a grid

With coordinates, you can describe a shape with numbers. Join the points (1, 1), (6, 1), (6, 4) and (1, 4) in order, and back to (1, 1). You get a rectangle 5 squares long and 3 squares tall. Anyone, anywhere, can draw exactly the same rectangle from those four pairs of numbers. Maps, computer screens and games all name their places this way.

Four points on a grid, (1, 1), (6, 1), (6, 4) and (1, 4), joined in order to make a rectangle 5 squares long and 3 squares tall

Try it at home

Draw axes on squared paper, from 0 to 8 across and 0 to 6 up.
Mark the points (2, 1), (6, 1) and (4, 5) and join them. What shape is it?
A friend secretly marks a point. Find it by asking for its coordinates.

Answers: 2. A triangle, with two equal sides.

Words to know

axis (axes) - one of the two number lines on a grid, across and up
origin - the point where the axes meet, (0, 0)
coordinates - two numbers that name a point: how far across, then how far up

Big idea

Two numbers, across then up, can name any point on a flat page. With them, a shape becomes a list of numbers. From points and lines to grids, the whole of geometry is about where things are and how big they are.