# Coordinates and the Number Plane

> Read and plot points with x first and y second, name the four quadrants and find distances along grid lines.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/coordinates/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The battleship game
Rosa and her brother play Battleships on squared paper. Rosa calls out "3 across, 2 up!" Her brother checks his grid. "Hit!"

If Rosa had said "2, 3" without saying which comes first, he might have looked in the wrong square. So mathematicians agree on one rule: the **across** number always comes first, then the **up** number.

These pairs of numbers are called **coordinates**. They pin down any point on a flat grid, from a game board to a city map.

**In short:** A coordinate $(x, y)$ gives a point's position on a grid: go across by $x$ first, then up by $y$. The origin is $(0, 0)$, and points to the left of it or below it have negative coordinates.

## Before you start

- Negative numbers on a number line (see Negative and Positive Numbers).
- Horizontal means across, like the horizon. Vertical means up and down.

## The number plane

The **number plane** has two number lines that cross at right angles:

- the $x$ axis goes across (horizontal)
- the $y$ axis goes up and down (vertical)
- they cross at the **origin**, the point $(0, 0)$

*Diagram: Number plane from negative 5 to 5 on both axes, with points A at 3 comma 2, B at negative 2 comma 3, C at negative 4 comma negative 1 and D at 1 comma negative 3*
*The number plane. The four quadrants are numbered anticlockwise, starting top right.*

## Reading coordinates

### Key idea: Across, then up
A point is written $(x, y)$.

- $x$: how far **across** from the origin. Right is positive, left is negative.
- $y$: how far **up** from the origin. Up is positive, down is negative.

"Along the corridor, then up the stairs."

In the diagram: $A = (3, 2)$, $B = (-2, 3)$, $C = (-4, -1)$ and $D = (1, -3)$.

### Watch out: The order matters
$(3, 2)$ and $(2, 3)$ are different points. $x$ always comes first, just as x comes before y in the alphabet.

### Worked example: Points on an axis
A point on the $x$ axis has $y = 0$, such as $(5, 0)$.

A point on the $y$ axis has $x = 0$, such as $(0, -3)$.

## The four quadrants

The axes split the plane into four **quadrants**, numbered anticlockwise from the top right.

| Quadrant | $x$ | $y$ | Example |
|---|---|---|---|
| 1st (top right) | positive | positive | $(3, 2)$ |
| 2nd (top left) | negative | positive | $(-2, 3)$ |
| 3rd (bottom left) | negative | negative | $(-4, -1)$ |
| 4th (bottom right) | positive | negative | $(1, -3)$ |

A point that sits on an axis is not in any quadrant.

## Distances along grid lines

If two points share the same $y$ value, they are on the same horizontal line. Their distance apart is the difference of their $x$ values.

### Worked example: Same row
$P = (-2, 2)$ and $Q = (1, 2)$. Same $y$, so the distance is the difference between $-2$ and $1$, which is $3$ units.

### Worked example: Same column
$Q = (1, 2)$ and $R = (1, -2)$. Same $x$, so the distance is the difference between $2$ and $-2$, which is $4$ units.

### Exam tip: Crossing an axis
When the two numbers have different signs, add their sizes: from $-2$ to $1$ is $2 + 1 = 3$.

### Challenge: Selective level: shapes on the plane
A rectangle has corners at $(-3, 1)$, $(2, 1)$, $(2, -3)$ and $(-3, -3)$. Its width is $2 - (-3) = 5$ units and its height is $1 - (-3) = 4$ units. Area $= 20$ square units, perimeter $= 18$ units.

Three corners of a square are $(1, 1)$, $(4, 1)$ and $(4, 4)$. The fourth corner is $(1, 4)$.

The point halfway between $(-2, 5)$ and $(6, 5)$ is $(2, 5)$, because 2 is halfway between $-2$ and 6.

**Mind map**

```text
Coordinates
  Axes
    x across
    y up
    Origin (0, 0)
  Order
    (x, y)
    Across, then up
  Quadrants
    1st (+, +)
    2nd (-, +)
    3rd (-, -)
    4th (+, -)
  Distance
    Same row: compare x
    Same column: compare y
```

## Quick summary

- A point is written $(x, y)$: across first, then up.
- The origin is $(0, 0)$. Left and down are negative.
- The quadrants go anticlockwise from top right: 1st $(+, +)$, 2nd $(-, +)$, 3rd $(-, -)$, 4th $(+, -)$.
- Points on an axis are not in a quadrant.
- Distance along a grid line is the difference of the changing coordinate.

## Quiz

Try these questions from the chapter quiz.

1. In the diagram, what are the coordinates of $A$?
   - A. $(3, 2)$
   - B. $(2, 3)$
   - C. $(-3, 2)$
   - D. $(3, -2)$
   - Answer: A. $(3, 2)$. $A$ is 3 across to the right and 2 up.

2. What are the coordinates of the origin?
   - A. $(1, 1)$
   - B. $(0, 0)$
   - C. $(0, 1)$
   - D. $(1, 0)$
   - Answer: B. $(0, 0)$. The origin is where the axes cross.

3. Which point is on the $y$ axis?
   - A. $(4, 0)$
   - B. $(0, -3)$
   - C. $(2, 2)$
   - D. $(-1, 5)$
   - Answer: B. $(0, -3)$. Points on the $y$ axis have $x = 0$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/coordinates/#quiz