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Year 6 Maths 2D Shapes, Position & Transformations

Plot & locate points on the coordinate plane (four quadrants)

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Theory

The number plane has a horizontal \(x\)-axis and a vertical \(y\)-axis that cross at the origin \((0, 0)\). A point is given by an ordered pair \((x, y)\): go across first (left for negative), then up or down (down for negative). The axes split the plane into four quadrants. This Year 6 maths page shows how to read and plot points anywhere on the plane, name the quadrant, and use coordinates to find distances and complete shapes.

The \(x\)-axis is horizontal and the \(y\)-axis is vertical. They meet at the origin, \((0, 0)\).

An ordered pair \((x, y)\) gives the \(x\)-coordinate first (across: right is positive, left is negative), then the \(y\)-coordinate (up is positive, down is negative).

The quadrants are numbered anticlockwise from the top right: first \((+, +)\), second \((-, +)\), third \((-, -)\), fourth \((+, -)\).

A point on the \(x\)-axis has \(y = 0\), like \((3, 0)\). A point on the \(y\)-axis has \(x = 0\), like \((0, -2)\). Points on the axes are not in any quadrant.

Along a grid line, the distance between two points is the difference of the coordinates that change: from \((-3, 2)\) to \((4, 2)\) is \(4 - (-3) = 7\) units.

Number plane with three points A number plane from negative 5 to 5 on both axes with the quadrants named. Point A is at (4, negative 3) in the fourth quadrant, B at (negative 2, 5) in the second quadrant and C at (negative 5, negative 1) in the third quadrant. First Second Third Fourth −5−4−3−2−112345 −5−4−3−2−112345 xy A B C
A \((4, -3)\), B \((-2, 5)\), C \((-5, -1)\).
Rectangle on the number plane A number plane with a rectangle whose corners are P (negative 3, 2), Q (4, 2), R (4, negative 3) and S (negative 3, negative 3). It is 7 units wide and 5 units tall. −5−4−3−2−112345 −5−4−3−2−112345 xy P Q R S
PQ is \(4 - (-3) = 7\) units; QR is \(2 - (-3) = 5\) units.

Signs of the coordinates in each quadrant:

QuadrantxyExample
First\(+\)\(+\)\((2, 3)\)
Second\(-\)\(+\)\((-2, 5)\)
Third\(-\)\(-\)\((-5, -1)\)
Fourth\(+\)\(-\)\((4, -3)\)
\[ (x, y) = (\text{across}, \text{up or down}) \]
( x , y )=( across , up or down )
Across, then up or down. \((5, -2)\) and \((-2, 5)\) are different points.
Zero means on an axis. \((0, -2)\) is on the \(y\)-axis, \((3, 0)\) on the \(x\)-axis.

Plotting a point (x, y)

  1. Start at the origin.
  2. Move across \(x\) units: right if positive, left if negative.
  3. Move up or down \(y\) units: up if positive, down if negative. Mark the point.

Reading a point

  1. Read straight down or up to the \(x\)-axis for \(x\), then across to the \(y\)-axis for \(y\).
  2. Write \((x, y)\) with the signs.
Example 1 — Read coordinates
Use the first diagram. Write the coordinates of A, B and C.
Solution

A \((4, -3)\), B \((-2, 5)\), C \((-5, -1)\).

Example 2 — Name the quadrant
Which quadrant is each point in?
(i) \((-1, 4)\)
(ii) \((3, -5)\)
(iii) \((-2, -2)\)
Solution

(i) Second
(ii) Fourth
(iii) Third

Example 3 — On the axes
Where are the points \((0, -2)\) and \((3, 0)\)?
Solution

\((0, -2)\) is on the \(y\)-axis, \(2\) below the origin. \((3, 0)\) is on the \(x\)-axis, \(3\) to the right. Neither is in a quadrant.

Example 4 — Signs
A point has a negative \(x\)-coordinate and a positive \(y\)-coordinate. Which quadrant is it in?
Solution

The second quadrant \((-, +)\): left of the \(y\)-axis and above the \(x\)-axis.

Example 5 — Distances
Use the second diagram. How long are sides PQ and QR?
Solution
\(4 - (-3)\)\(=\)\(7\)
\(2 - (-3)\)\(=\)\(5\)
PQ =7, QR =5

PQ is \(7\) units and QR is \(5\) units.

Example 6 — Complete a rectangle
Three corners of a rectangle are \((-3, 2)\), \((4, 2)\) and \((4, -3)\). Find the fourth corner.
Solution

It is below \((-3, 2)\) and level with \((4, -3)\): \((-3, -3)\).

Example 7 — Move a point
The point \((1, -2)\) moves \(4\) units left and \(3\) units up. Where is it now?
Solution
\(1 - 4\)\(=\)\(-3\)
\(-2 + 3\)\(=\)\(1\)
(−3,1)

\((-3, 1)\), in the second quadrant.

Example 8 — Find the mistake
Asked to plot \((-2, 5)\), Kim went \(5\) across and \(2\) down. Which point did she plot, and what is right?
Solution

She plotted \((5, -2)\). For \((-2, 5)\), go \(2\) left first, then \(5\) up (point B in the first diagram).

Common pitfalls

Swapping x and y. The first number is always across.
Ignoring the sign. \((-4, 1)\) is left of the \(y\)-axis; \((4, 1)\) is right of it.
Numbering quadrants clockwise. They go anticlockwise from the top right.
Counting squares instead of grid lines. Count the units along the axis, from line to line.

Frequently asked questions

What is the origin?

The point \((0, 0)\) where the axes cross.

Which number comes first in a coordinate pair?

The \(x\)-coordinate (across).

Which quadrant has both coordinates negative?

The third quadrant.

Where is the point (0, −2)?

On the \(y\)-axis, \(2\) units below the origin.

How far is it from (−3, 2) to (4, 2)?

\(7\) units.

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