Topics
11 Transformations and the number plane
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Level 7

11.02 Directed numbers on the number plane

Lesson
Worksheet
Practice
Lesson

Quadrants on the number plane

We built  the number plane  using two copies of the number line and we explored how it can be used to describe the location of shapes and points in a 2D space.

Now we can extend this coordinate system using directed numbers, which will allow us to describe the location of points in any direction from the origin.

Exploration

The applet below demonstrates the main features of this extended number plane.

Drag the point P to explore the other quadrants.

Loading interactive...

Both axes now have positive and negative coordinates.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Each axis now has positive and negative numbers, and this means we can talk about four distinct regions of the plane, called quadrants.

The 1st quadrant on the top right is equivalent to the number plane that we looked at  last lesson  . The x-coordinate and y-coordinate of a point in the 1st quadrant are both positive.

Moving around anticlockwise we cover the other three quadrants, which have the following features:

  • 2nd quadrant: x-coordinates are negative, y-coordinates are positive

  • 3rd quadrant: both coordinates are negative

  • 4th quadrant: x-coordinates are positive, y-coordinates are negative

Points that lie on an axis, like (-5,\,0) or (0,\,4), are not in any quadrant.

The advantage of using directed numbers on the number plane is that we no longer have boundaries for the coordinates. If an object begins at some point on the plane, we can move it any which way we like, as far as we like, and still be able to describe its location with respect to the origin.

Examples

Example 1

What are the coordinates of the point shown in the number plane?

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Worked Solution
Create a strategy

Count the number of horizontal and vertical units required to move away from the origin and determine if it is in the positive or negative direction.

Apply the idea

The point is located 1 space to the right, then 3 spaces up from origin. So, the coordinates are (1,\,3).

Example 2

In which quadrant does the point (3,\,-2) lie?

Worked Solution
Create a strategy

Recall the characteristic of each quadrants:

  • 1st quadrant: positive x and positive y.

  • 2nd quadrant: negative x and positive y.

  • 3rd quadrant: negative x and negative y.

  • 4th quadrant: positive x and negative y.

Apply the idea

Since the coordinates have positive x and negative y, the point (3,\,-2) lies in 4th quadrant.

Example 3

What is the distance between A\left(6,\,8\right) and B\left(-3,\,8\right)?

Worked Solution
Create a strategy

Since the y-coordinates are the same, find the difference of the x-coordinates.

Apply the idea
\displaystyle \text{Distance}\displaystyle =\displaystyle 6-(-3)Subtract -3 from 6
\displaystyle =\displaystyle 6+3Combine the adjacent signs
\displaystyle =\displaystyle 9 \text{ units}Evaluate
Idea summary

Each quadrant of the coordinate plane has distinct characteristics:

  • 1st quadrant: both coordinates are positive
  • 2nd quadrant: x-coordinates are negative, y-coordinates are positive
  • 3rd quadrant: both coordinates are negative
  • 4th quadrant: x-coordinates are positive, y-coordinates are negative

Outcomes

VCMNA255

Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point

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