Algebra · KS3

Coordinates

Two numbers pin down any point on a flat surface: how far across, then how far up. Add negative numbers and the grid extends in every direction, giving four quadrants and the plane that every graph for the rest of school is drawn on.

National curriculum KS3: work with coordinates in all four quadrants; this page is the foundation for plotting graphs of linear and other functions

1 · Across first, then up

Two number lines crossing at right angles. The horizontal one is the x-axis; the vertical one is the y-axis. Where they cross is the origin, (0, 0). A point is written (x, y): first the x-value, then the y-value, in brackets with a comma. Order matters: (3, 5) and (5, 3) are different places.

Click anywhere on the grid or drag the point. Watch the path: along the corridor, then up the stairs.

Across (x)
Up (y)
Quadrant

Remembering the order. x comes before y in the alphabet, and across comes before up in the brackets. "Along the corridor, then up the stairs." Spanish uses the same convention: abscisa (x) then ordenada (y), the par ordenado.

2 · Four quadrants

With negative numbers on both axes, the grid splits into four regions, numbered anticlockwise from top-right with Roman numerals. The signs of x and y tell you which quadrant you're in without looking.

Quadrant II(−, +)
Quadrant I(+, +)
Quadrant III(−, −)
Quadrant IV(+, −)

Points on an axis belong to no quadrant: (4, 0) sits on the x-axis, (0, −3) on the y-axis. Move the point in Section 1 onto an axis and the quadrant readout says so.

3 · Plot the point

Now the other way round: you're given the coordinates and must find the place. Click where you think the point is. Five points, and a negative or two in every round.

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4 · Read the point

And the reverse again, which exam questions ask more often: a point is marked, you write its coordinates. Type them, including minus signs.

5 · Shapes from coordinates

Coordinates let you describe shapes with numbers. Plot three corners of a rectangle and the fourth is forced. This is the kind of question exams ask: "A(−2, 1), B(3, 1), C(3, 4) are three vertices of a rectangle. Find D." Join the points and the answer is visible.

Midpoint = average of the ends. Halfway between (−2, 1) and (6, 5) is ((−2 + 6)/2, (1 + 5)/2) = (2, 3). Average the x's, average the y's. It's the first formula built on coordinates and it comes up every year.

6 · Where coordinates live

Every graph

A graph is a set of points whose coordinates satisfy a rule. y = 2x + 1 is just the collection of all (x, y) pairs where the y really is 2x + 1. The next page plots them.

Maps and GPS

Latitude and longitude are coordinates on a sphere. Jersey is roughly (49.2° N, 2.1° W): the W is a negative x. Grid references on an OS map are "eastings then northings": across then up, exactly.

Screens and games

Every pixel has an (x, y). Note the trap: on screens y grows downwards, so (0, 0) is the top-left. Scratch puts (0, 0) in the centre with y up, like maths. Game characters move by adding to their coordinates.

Spreadsheets

Cell C4 is a coordinate: column C across, row 4 down. Letters then numbers: across then up (well, down), same convention.

Chess and Battleships

e4, B7: column letter then row number. Battleships is a coordinate game with hidden points, which is why it's used to teach this.

Data science

A scatter graph is thousands of coordinate pairs (height, weight), (hours studied, score). Patterns in the cloud of points are correlation. Statistics starts here.

Worked examples

Plot (−3, 2)
Start at the origin.
x = −3: go 3 to the left (negative is left).
y = 2: go 2 up. Mark it. You're in quadrant II (−, +).
Write the coordinates of a point 4 below the origin and 1 to the right
Across first: 1 to the right → x = 1.
Then up/down: 4 below → y = −4.
Answer: (1, −4), quadrant IV.
A(−2, 1), B(3, 1), C(3, 4) are three vertices of a rectangle. Find D.
A and B share y = 1, so AB is horizontal, length 5.
B and C share x = 3, so BC is vertical, length 3.
D sits above A by 3 and level with C: D = (−2, 4). Sketch it to be sure.
Exam-style: M is the midpoint of PQ. P = (−5, 2), M = (1, −1). Find Q.
Midpoint is the average, so M is halfway. From P to M the x went −5 → 1, which is +6; the y went 2 → −1, which is −3.
Go the same again from M: x = 1 + 6 = 7, y = −1 − 3 = −4.
Q = (7, −4). Check: average of −5 and 7 is 1 ✓, average of 2 and −4 is −1 ✓

Quick check

What you've learned

A coordinate pair (x, y) means across then up, measured from the origin, with negatives meaning left and down. The four quadrants are numbered anticlockwise from the top right and each has its own sign pattern; points on an axis belong to none. Reading and plotting points in all four quadrants is the groundwork for every graph, and simple geometry (fourth vertex, midpoint, reflection) is done by looking at how the x's and y's change.