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Year 7 Maths Number Plane

The Number Plane with Negative Numbers

20 practice questions 1 video lesson Theory + worked examples

The Number Plane with Negative Numbers extends both axes past the origin, so coordinates can be negative, in NSW Year 7 maths (NESA outcome MA4‑LIN‑C‑01). The axes split the plane into four quadrants, numbered anticlockwise from the top right, and the point \((-3, 2)\) sits 3 units left and 2 units up.

You will plot and read points in all four quadrants, including half values, name the quadrant from the signs of the coordinates, move points left, right, up and down, find lengths that cross an axis, work out areas of shapes, and decide when three points are collinear. Map grids use the same idea. It is part of Year 7 Number Plane.

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Theory

The \(x\)-axis and \(y\)-axis carry on past the origin, so coordinates can be negative. The axes split the number plane into four quadrants. A negative \(x\) is left; a negative \(y\) is down. This Year 7 maths page shows how to read and plot points in all four quadrants, move points, find lengths across an axis and work with shapes.

The number plane uses positive and negative numbers on both axes. The centre point \((0, 0)\) is the origin.

Reading and plotting. Start at the origin. Go across first: a positive \(x\) is right and a negative \(x\) is left. Then go up or down: a positive \(y\) is up and a negative \(y\) is down. So \((-3, 1)\) is \(3\) left, then \(1\) up.

Quadrants. The axes divide the plane into four regions called quadrants, numbered anticlockwise from the top right. The signs of the coordinates tell you the quadrant. A point on an axis, such as \((0, -4)\), is not in any quadrant.

Moving a point. Left and right change \(x\). Up and down change \(y\). From \((4, -1)\), move \(6\) left: \(x = 4 - 6 = -2\), so the point is \((-2, -1)\).

Lengths across an axis. Use the larger minus the smaller. From \(x = -3\) to \(x = 4\) is \(4 - (-3) = 7\) units.

Collinear points lie on one straight line. Along the line, equal steps in \(x\) give equal steps in \(y\).

The four quadrants A number plane from negative 10 to 10 on both axes, labelled every 5. The top right is the 1st quadrant, top left the 2nd, bottom left the 3rd and bottom right the 4th. −10−5510−10−55100xy 1st quadrant 2nd quadrant 3rd quadrant 4th quadrant origin (0, 0)
The quadrants are numbered anticlockwise from the top right: 1st \((+, +)\), 2nd \((-, +)\), 3rd \((-, -)\), 4th \((+, -)\).
Plotting A(4, 7), B(3, -5), C(-4, 2) and D(-9, -6) Number plane from negative 10 to 10 with dashed guide lines to the points A(4, 7), B(3, -5), C(-4, 2) and D(-9, -6). −10−5510−10−55100xy A(4, 7) B(3, −5) C(−4, 2) D(−9, −6)
Each dashed path goes across first, then up or down: \(B(3, -5)\) is \(3\) right and \(5\) down; \(D(-9, -6)\) is \(9\) left and \(6\) down.

The signs give the quadrant:

Quadrant\(x\)\(y\)Example
1st (top right)\(+\)\(+\)\((2, 5)\)
2nd (top left)\(-\)\(+\)\((-2, 5)\)
3rd (bottom left)\(-\)\(-\)\((-2, -5)\)
4th (bottom right)\(+\)\(-\)\((2, -5)\)
on an axisone is \(0\)\((0, -4)\)
Length across an axis. From \(y = -2\) to \(y = 3\) is \(3 - (-2) = 3 + 2 = 5\) units.
Missing corner of a parallelogram. Opposite sides make the same move. If \(B\) to \(C\) is \(1\) right and \(3\) up, then \(A\) to \(D\) is also \(1\) right and \(3\) up.

How to plot a point with negative coordinates

  1. Start at the origin.
  2. Across: right if \(x\) is positive, left if \(x\) is negative.
  3. Up or down: up if \(y\) is positive, down if \(y\) is negative.
  4. Dot and label the point.
Example 1 — Reading points
(i) Write down the coordinates of \(A\), \(B\), \(C\) and \(D\).
(ii) Which quadrant is \(C\) in?
Points A, B, C and D A number plane from negative 4 to 4 with A at (-2, 3), B at (3, -1), C at (-3.5, -2) and D at (0, -3). −4−3−2−11234−4−3−2−112340xy A B C D
Solution

Across first (left is negative), then up or down (down is negative). \(D\) is on the \(y\)-axis, so \(x = 0\).

(i) \(A(-2, 3)\), \(B(3, -1)\), \(C(-3.5, -2)\), \(D(0, -3)\)
(ii) \(C\) has \(x < 0\) and \(y < 0\): the 3rd quadrant.

Example 2 — Quadrants and axes
In which quadrant is each point?
(i) \((2, -4)\)
(ii) \((-1, -3)\)
(iii) \((-3, 1)\)
(iv) \(W\) is \(1.5\) units left of the origin. Where is \(W\)?
Solution

Use the signs: \((+, -)\) is bottom right, \((-, -)\) bottom left, \((-, +)\) top left. Left of the origin is on the \(x\)-axis, so \(y = 0\).

(i) 4th quadrant
(ii) 3rd quadrant
(iii) 2nd quadrant
(iv) \(W(-1.5, 0)\), on the \(x\)-axis (no quadrant)

Example 3 — Moving across the axes
Point \(M(-4, 3)\) moves \(6\) units right and then \(5\) units down. Where does it finish? Which quadrant is that?
Solution
M moves 6 right and 5 down Point M at (-4, 3) moves 6 right to (2, 3) then 5 down to M dash at (2, -2). −5−4−3−2−11234−4−3−2−112340xy +6 −5 M M′
\(x\)\(=\)\(-4 + 6 = 2\)
\(y\)\(=\)\(3 - 5 = -2\)
x =−4+6=2, y =3−5=−2

\(M\) finishes at \((2, -2)\): \(x > 0\) and \(y < 0\), the 4th quadrant.

Example 4 — A rectangle across the axes
\(ABCD\) is a rectangle with \(A(-3, -2)\), \(B(2, -2)\) and \(C(2, 4)\).
(i) Find \(D\).
(ii) Find the perimeter.
(iii) Find the area.
Solution

\(D\) has the \(x\) of \(A\) and the \(y\) of \(C\). \(AB = 2 - (-3) = 5\) and \(BC = 4 - (-2) = 6\).

Rectangle ABCD across the axes Rectangle with A(-3, -2), B(2, -2), C(2, 4) and D(-3, 4). AB is 5 units and BC is 6 units. −4−3−2−11234−3−2−1123450xy A B C D
\(P\)\(=\)\(5 + 6 + 5 + 6 = 22\)
\(A\)\(=\)\(5 \times 6 = 30\)
P =5+6+5+6=22, A =5×6=30

(i) \(D(-3, 4)\)
(ii) \(22\) units
(iii) \(30\) u²

Example 5 — Area across the axes
Triangle \(PQR\) has vertices \(P(-2, -3)\), \(Q(4, -3)\) and \(R(1, 2)\). Find its area.
Solution

\(PQ\) is horizontal: \(b = 4 - (-2) = 6\). The height goes straight down from \(R\): \(h = 2 - (-3) = 5\).

Triangle PQR Triangle with P(-2, -3), Q(4, -3) and R(1, 2). The base PQ is 6 units and a dashed height of 5 units goes down from R. −3−2−112345−4−3−2−11230xy h = 5 P Q R
\(A\)\(=\)\(\dfrac{1}{2} \times 6 \times 5\)
\(\)\(=\)\(15\)
A =1/2×6×5=15

The area is \(15\) u².

Example 6 — Missing corner and collinear points
(i) \(A(-2, -1)\), \(B(3, -1)\) and \(C(4, 2)\) are three corners of parallelogram \(ABCD\). Find \(D\).
(ii) The points \((0, -3)\), \((1, -1)\) and \((2, b)\) are collinear. Find \(b\).
Solution

(i) \(B\) to \(C\) is \(1\) right and \(3\) up, so \(A\) to \(D\) is the same move: \(D = (-2 + 1, -1 + 3)\).
(ii) Each time \(x\) goes up \(1\), \(y\) goes up \(2\): \(b = -1 + 2\).

(i) \(D(-1, 2)\)
(ii) \(b = 1\)

Common pitfalls

Dropping the sign. \((-3, 1)\) is \(3\) left, not \(3\) right. \((3, 1)\) is a different point.
Ignoring signs across an axis. From \(y = -2\) to \(y = 3\) is \(3 - (-2) = 5\) units, not \(3 - 2 = 1\).
Giving a quadrant to a point on an axis. \((0, -4)\) is on the \(y\)-axis and \((-4, 0)\) is on the \(x\)-axis. Neither is in a quadrant.
Numbering quadrants clockwise. The quadrants go anticlockwise: 1st top right, 2nd top left, 3rd bottom left, 4th bottom right.

Frequently asked questions

What are the four quadrants?

The four regions made by the axes, numbered anticlockwise from the top right.

Which quadrant has a negative x and a positive y?

The 2nd quadrant (top left), for example \((-2, 5)\).

How do you plot a point with a negative coordinate?

A negative \(x\) means go left; a negative \(y\) means go down.

How do you find the length between y = -2 and y = 3?

Larger minus smaller: \(3 - (-2) = 5\) units.

What does collinear mean?

The points lie on one straight line.

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