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Year 7 Maths Number Plane

The Number Plane

20 practice questions 1 video lesson Theory + worked examples

The Number Plane introduces the grid used to locate points, starting in the first quadrant, in NSW Year 7 maths (NESA outcome MA4‑LIN‑C‑01). The horizontal \(x\)-axis and vertical \(y\)-axis meet at the origin, and every point has coordinates \((x, y)\): go across first, then up.

You will read and plot points, including ones halfway between grid lines such as \((2.5, 4)\), find points that sit on an axis, join points to make shapes, and work out the perimeter and area of rectangles and triangles drawn on the grid. Following directions on a park map is an everyday example. It is part of Year 7 Number Plane.

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Theory

A number plane (or Cartesian plane) is a grid made from a horizontal \(x\)-axis and a vertical \(y\)-axis that meet at the origin \(O(0, 0)\). Every point has an address called its coordinates \((x, y)\): across first, then up. This Year 7 maths page shows how to read, plot and move points and how to find lengths, perimeters and areas on the grid.

A number plane is a grid made from a horizontal number line called the \(x\)-axis and a vertical number line called the \(y\)-axis. It is also called the Cartesian plane. The axes meet at the origin \(O(0, 0)\).

A point is given by an ordered pair \((x, y)\). The first number is the \(x\)-coordinate (how far across). The second is the \(y\)-coordinate (how far up). Order matters: \((0, 4)\) is not the same point as \((4, 0)\).

Plotting a point. To plot \((5, 3)\), start at the origin, go \(5\) across, then \(3\) up. Draw a dot and label it.

Points on an axis. Every point on the \(x\)-axis has \(y = 0\), e.g. \((4, 0)\). Every point on the \(y\)-axis has \(x = 0\), e.g. \((0, 2)\). A point directly below \((3, 5)\) on the \(x\)-axis is \((3, 0)\).

Halves. A point halfway between two grid lines has a coordinate like \(2.5\).

Lengths. Two points with the same \(y\) are on a horizontal line: the length is the larger \(x\) minus the smaller \(x\). Two points with the same \(x\) are on a vertical line: use the \(y\)-values. Perimeter is the sum of the sides, in units (u). Area is in square units (u²).

Plotting P(5, 3) A first-quadrant number plane from 0 to 6. Dashed red lines go 5 across and 3 up to the point P at (5, 3). A grey point at (3, 5) uses the same numbers the other way round. 1234561234560xy P(5, 3) (3, 5)
To reach \(P(5, 3)\), go \(5\) across, then \(3\) up. The grey point \((3, 5)\) uses the same numbers the other way round, so it is a different point.
Points with halves and points on the axes A number plane with A at (2.5, 4), halfway between 2 and 3 across; B at (4, 0) on the x-axis; C at (0, 1.5) on the y-axis. 123456123450xy A(2.5, 4) B(4, 0) C(0, 1.5)
\(A\) is halfway between \(2\) and \(3\) across. \(B\) is on the \(x\)-axis, so \(y = 0\). \(C\) is on the \(y\)-axis, so \(x = 0\).

Coordinates and lengths:

\[ (x,\ y) = (\text{across},\ \text{up}) \]
( x , y )=( across , up )
\[ \text{length} = \text{larger } x - \text{smaller } x \]
length = larger x − smaller x
\[ \text{Rectangle: } A = l \times w \]
Rectangle A = l × w
\[ \text{Triangle: } A = \dfrac{1}{2} \times b \times h \]
Triangle A =1/2× b × h
\[ \text{Parallelogram: } A = b \times h \]
Parallelogram A = b × h
Halfway along a horizontal side. Add half the length to the smaller \(x\). From \((1, 4)\) to \((6, 4)\) the length is \(5\), so halfway is \(x = 1 + 2.5 = 3.5\): the point \((3.5, 4)\).
Naming a right angle. An angle is named by three points with the vertex in the middle. If the right angle is at \(B\) in triangle \(ABC\), write \(\angle ABC\).

How to plot a point

  1. Start at the origin \(O(0, 0)\).
  2. Go across by the \(x\)-coordinate.
  3. Go up by the \(y\)-coordinate.
  4. Draw a dot and label it with its letter.

How to find a length, perimeter or area

  1. Find each horizontal or vertical side by subtracting coordinates (count gaps, not dots).
  2. Perimeter: add all the sides.
  3. Area: use the formula, or split an L-shape into rectangles and add.
Example 1 — Reading points
Write down the coordinates of \(A\), \(B\), \(C\) and \(D\).
Points A, B, C and D A number plane with points A at (2, 4), B at (5, 1), C at (3.5, 3) and D at (4, 0). 123456123450xy A B C D
Solution

Count from the origin: across first, then up. \(C\) is halfway between \(3\) and \(4\). \(D\) is on the \(x\)-axis, so \(y = 0\).

\(A(2, 4)\), \(B(5, 1)\), \(C(3.5, 3)\), \(D(4, 0)\)

Example 2 — Moving back
In a treasure-hunt game, Jack moves \(2\) units left, then \(4\) units up, and lands on the treasure \(T(3, 5)\).
(i) Where did Jack start?
(ii) A clue \(C\) is on the \(y\)-axis, level with \(T\). Where is \(C\)?
Solution

(i) Undo the moves: \(4\) down, then \(2\) right. \(x = 3 + 2 = 5\) and \(y = 5 - 4 = 1\).
(ii) On the \(y\)-axis \(x = 0\); level with \(T\) means the same \(y\).

(i) Jack started at \((5, 1)\).
(ii) \(C(0, 5)\)

Example 3 — A rectangle
\(ABCD\) is a rectangle with \(A(2, 1)\), \(B(6, 1)\) and \(C(6, 6)\).
(i) Find \(D\).
(ii) Find the perimeter and the area.
(iii) Find the point halfway between \(A\) and \(B\).
Solution

(i) \(D\) is above \(A\) (same \(x\)) and level with \(C\) (same \(y\)).
(ii) \(AB = 6 - 2 = 4\) and \(BC = 6 - 1 = 5\).

Rectangle ABCD Rectangle with A(2, 1), B(6, 1), C(6, 6) and D(2, 6). Side AB is 4 units and side BC is 5 units. 123456712345670xy A B C D M
\(P\)\(=\)\(4 + 5 + 4 + 5 = 18\)
\(A\)\(=\)\(4 \times 5 = 20\)
P =4+5+4+5=18, A =4×5=20

(i) \(D(2, 6)\)
(ii) \(18\) units and \(20\) u²
(iii) Half of \(4\) is \(2\): \(x = 2 + 2 = 4\), so \(M(4, 1)\).

Example 4 — A parallelogram
\(ABCD\) is a parallelogram with \(A(1, 1)\), \(B(5, 1)\) and \(D(2, 4)\).
(i) Find \(C\).
(ii) Find the area.
(iii) \(AD\) is about \(3.2\) units long. Find the perimeter.
Solution

(i) \(DC\) is level with \(AB\) and just as long: \(AB = 5 - 1 = 4\), so \(C = (2 + 4, 4)\).
(ii) Base \(4\); the height goes straight down from \(D\): \(h = 4 - 1 = 3\).

Parallelogram ABCD Parallelogram with A(1, 1), B(5, 1), C(6, 4) and D(2, 4). A dashed height of 3 units goes straight down from D to AB. 1234567123450xy h = 3 A B C D
\(A\)\(=\)\(4 \times 3 = 12\)
\(P\)\(=\)\(4 + 3.2 + 4 + 3.2 = 14.4\)
A =4×3=12, P =4+3.2+4+3.2=14.4

(i) \(C(6, 4)\)
(ii) \(12\) u²
(iii) \(14.4\) units (opposite sides are equal)

Example 5 — Area of a triangle
Plot and join \(A(1, 1)\), \(B(6, 1)\) and \(C(4, 5)\). Find the area of triangle \(ABC\).
Solution

\(AB\) is a horizontal base (same \(y\)): \(b = 6 - 1 = 5\). The height goes straight down from \(C\): \(h = 5 - 1 = 4\).

Triangle ABC Triangle with A(1, 1), B(6, 1) and C(4, 5). A dashed height of 4 units goes straight down from C to the base AB of 5 units. 12345671234560xy h = 4 A B C
\(A\)\(=\)\(\dfrac{1}{2} \times 5 \times 4\)
\(\)\(=\)\(10\)
A =1/2×5×4=10

The area is \(10\) u².

Example 6 — An L-shape
Plot and join \(A(1, 1)\), \(B(1, 4)\), \(C(3, 4)\), \(D(3, 2)\), \(E(6, 2)\) and \(F(6, 1)\). Find the area and the perimeter.
Solution

Split it into two rectangles: \(2 \times 3\) on the left and \(3 \times 1\) on the right.

L-shape ABCDEF An L-shape with corners A(1, 1), B(1, 4), C(3, 4), D(3, 2), E(6, 2) and F(6, 1), split by a dashed line into a 2 by 3 rectangle and a 3 by 1 rectangle. 1234567123450xy A B C D E F
\(A\)\(=\)\(2 \times 3 + 3 \times 1\)
\(\)\(=\)\(9\)
\(P\)\(=\)\(3 + 2 + 2 + 3 + 1 + 5 = 16\)
A =2×3+3×1=9, P =3+2+2+3+1+5=16

Area \(9\) u², perimeter \(16\) units.

Common pitfalls

Swapping the coordinates. \((5, 3)\) and \((3, 5)\) are different points. Across comes first: “along the corridor, then up the stairs”.
Leaving out a zero. A point \(4\) along the \(x\)-axis is \((4, 0)\), not \((0, 4)\) or just \(4\).
Counting dots instead of gaps. From \((1, 2)\) to \((6, 2)\) is \(6 - 1 = 5\) units, even though there are \(6\) grid lines on the way.
Using a sloping side as the height. The height of a triangle or parallelogram goes straight up from the base, at right angles to it.

Frequently asked questions

What is the origin?

The point \((0, 0)\) where the \(x\)-axis and the \(y\)-axis meet.

Which coordinate comes first?

The \(x\)-coordinate (across) comes first, then the \(y\)-coordinate (up).

What are the coordinates of a point on the x-axis?

Its \(y\)-coordinate is \(0\), for example \((4, 0)\).

How do you find the distance between two points on a horizontal line?

Subtract the smaller \(x\)-coordinate from the larger one. \((2, 3)\) to \((8, 3)\) is \(6\) units.

How do you find the area of a shape on the number plane?

Find the side lengths from the coordinates, then use the area formula, or split the shape into rectangles.

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