10-Week Mathematics Catch-Up Programme (CAPS)

Week 3 – Day 1

Introduction to the Cartesian Plane and Plotting Points

Functions and graphs make up a significant part of every Grade 10 and Grade 11 CAPS Mathematics syllabus, and they are tested repeatedly in exams.

The aim this week is not simply to draw graphs, but to understand what a graph tells you about the relationship between two quantities. Duration: 1 Hour

Grade Level: Grade 9 Foundation (Preparing for Grade 10 & 11 Functions)

Learning Outcomes

By the end of today's lesson you should be able to:

• Identify the x-axis and y-axis.
• Locate and name the four quadrants.
• Plot points accurately.
• Read coordinates from a graph.
• Calculate the distance between points on horizontal and vertical lines.

Part A – Study Guide

1. The Cartesian Plane

The Cartesian Plane is a grid used to locate points using two numbers called coordinates.

It consists of:

• The x-axis (horizontal axis)
• The y-axis (vertical axis)

These axes meet at the origin, written as:

$(0,0)$

2. Coordinates

A point is written as:

$(x,y)$

The first number tells you how far to move left or right.

The second number tells you how far to move up or down.

Example

Plot

$(4,3)$

• Move 4 units right.
• Move 3 units up.
• Mark the point.

Another Example

Plot

$(-2,5)$

• Move 2 units left.
• Move 5 units up.

3. The Four Quadrants

The axes divide the plane into four quadrants.

A useful way to remember this is to go anti-clockwise, starting in the top-right corner.

4. Points on the Axes

If a point lies on the x-axis:

$(x,0)$

The y-coordinate is always 0.

Example:

$(6,0)$

If a point lies on the y-axis:

$(0,y)$

The x-coordinate is always 0.

Example:

$(0,-4)$

5. Reading Coordinates

Always read:

Across first (x)

Then up or down (y)

Example:

A point is 5 units left and 2 units down.

Coordinates:

$(-5,-2)$

6. Horizontal and Vertical Distance

Horizontal Distance

Subtract the x-values.

Example

$A(2,4)$

$B(8,4)$

Distance:

$8-2=6$

Vertical Distance

Subtract the y-values.

Example

$A(5,2)$

$B(5,10)$

Distance:

$10-2=8$

Note: We are only calculating horizontal and vertical distances today. The distance formula between any two points will be introduced later in the programme.

Common Mistakes

Mistake 1

Writing coordinates as

$(y,x)$

Remember:

Always write (x, y).

Mistake 2

Confusing left and right.

Negative x = left.

Positive x = right.

Mistake 3

Confusing up and down.

Positive y = up.

Negative y = down.

Part B – Worked Examples

Example 1

State the quadrant of

$(5,2)$

Both coordinates are positive.

Answer: Quadrant I

Example 2

State the quadrant of

$(-4,7)$

Negative x

Positive y

Answer: Quadrant II

Example 3

State the quadrant of

$(-3,-5)$

Both negative.

Answer: Quadrant III

Example 4

State the quadrant of

$(8,-2)$

Positive x

Negative y

Answer: Quadrant IV

Example 5

What are the coordinates of a point 6 units left and 4 units up?

Answer:

$(-6,4)$

Example 6

Find the horizontal distance between

A(1,3)

and

B(9,3)

Distance:

$9-1=8$

Example 7

Find the vertical distance between

C(5,-2)

and

D(5,6)

Distance:

$6-(-2)=8$

Part C – Practice Questions

Section A – Identify the Quadrant

State which quadrant each point lies in.

1. (3,4)

2. (-2,6)

3. (-7,-1)

4. (5,-8)

5. (10,9)

6. (-8,2)

7. (-4,-9)

8. (12,-3)

9. (-6,8)

10. (2,-5)

Section B – Write the Coordinates

Write the coordinates of the following descriptions.

11. Four units right and three units up.

12. Seven units left and two units down.

13. Five units right and six units down.

14. Eight units left and four units up.

15. On the x-axis, six units to the left.

16. On the y-axis, nine units above the origin.

17. At the origin.

18. On the x-axis, twelve units to the right.

19. On the y-axis, three units below the origin.

20. Nine units left and one unit up.

Section C – Horizontal and Vertical Distances

21. A(2,5) and B(9,5)

22. C(-4,3) and D(6,3)

23. E(5,-2) and F(5,8)

24. G(-3,-4) and H(-3,7)

25. J(1,0) and K(10,0)

Find the distance between each pair of points.

Section D – Mixed Questions

26. Which axis does the point (0,8) lie on?

27. Which axis does the point (-5,0) lie on?

28. In which quadrant is (-12,15)?

29. In which quadrant is (11,-6)?

30. Explain why the point (0,0) is not in any quadrant.

Challenge Questions

31. Write one point that lies in each quadrant.

32. A point has a positive x-coordinate and a negative y-coordinate. Which quadrant is it in?

33. A point is 4 units below the x-axis and 7 units to the left of the y-axis. Write its coordinates.

34. Point A is at (-6,2). Move it 5 units to the right and 3 units down. What are the new coordinates?

35. A rectangle has vertices:

A(2,1)

B(8,1)

C(8,6)

D(2,6)

Find:

a) The width of the rectangle.

b) The height of the rectangle.

c) The perimeter.

Answers

Section A

1. Quadrant I

2. Quadrant II

3. Quadrant III

4. Quadrant IV

5. Quadrant I

6. Quadrant II

7. Quadrant III

8. Quadrant IV

9. Quadrant II

10. Quadrant IV

Section B

11. (4,3)

12. (-7,-2)

13. (5,-6)

14. (-8,4)

15. (-6,0)

16. (0,9)

17. (0,0)

18. (12,0)

19. (0,-3)

20. (-9,1)

Section C

21. 7 units

22. 10 units

23. 10 units

24. 11 units

25. 9 units

Section D

26. y-axis

27. x-axis

28. Quadrant II

29. Quadrant IV

30. The origin is where the x-axis and y-axis intersect. It is not inside any of the four quadrants.

Challenge Answers

31. Example answers:

• Quadrant I: (2,3)
• Quadrant II: (-2,3)
• Quadrant III: (-2,-3)
• Quadrant IV: (2,-3)

(Any correct examples are acceptable.)

32. Quadrant IV

33. (-7,-4)

Start: (-6,2)

Move 5 right:

$x=-6+5=-1$

Move 3 down:

$y=2-3=-1$

New coordinates:

$(-1,-1)$

Width:

$8-2=6$

Height:

$6-1=5$

Perimeter:

$2(6+5)=22 \text{ units}$

Extension Activity (Optional)

On graph paper:

1. Draw the x-axis and y-axis.
2. Plot the following points:

• A(2,3)
• B(-4,5)
• C(-5,-2)
• D(4,-3)
• E(0,6)
• F(7,0)

Then answer:

• Which points lie on an axis?
• Which quadrants contain the remaining points?
• Which two points are the furthest apart horizontally?
• Which two points are the furthest apart vertically?

Parent's Notes

Today lays the foundation for all graph work in Grades 10 and 11. If your student is comfortable plotting points and identifying quadrants without hesitation, they'll find graphing functions much easier.

A practical tip: If possible, have them use graph paper rather than plain paper. Drawing axes and plotting points neatly helps develop accuracy and confidence.

Looking Ahead – Week 3, Day 2

Tomorrow they'll learn how to:

• Recognise patterns in tables of values.