020. Coordinates on the Number Plane

Learning Intentions

• To know the meaning of the terms number plane (or Cartesian plane), origin and coordinates
• interpret the location of a point described by its coordinates, e.g. $(2,4)$
• plot points on the cartesian plane

Pre-requisite Summary

• Understand horizontal and vertical directions
• Be able to count whole number steps accurately
• Know that ordered pairs are written in brackets with a comma, e.g. $(x,y)$
• Understand that the first number tells how far to move across
• Understand that the second number tells how far to move up
• Recognise that the origin is the point where the two axes meet
• Be able to read values from a simple scale on a grid

Worked Examples

Worked Example 1

a) Define the terms number plane, origin and coordinates.
b) State the coordinates of the origin.
c) Explain what the ordered pair $(3,2)$ means.

Worked Example 2

Interpret the location of each point:
a) $(2,4)$
b) $(5,1)$
c) $(0,3)$

Worked Example 3

Describe how to plot each point on the cartesian plane:
a) $(4,3)$
b) $(1,5)$
c) $(6,0)$

Worked Example 4

A point is plotted by moving $3$ units across and $4$ units up.
a) Write its coordinates.
b) Explain how the coordinates show the location.

Worked Example 5

A point has coordinates $(0,5)$.
a) Describe its location.
b) State whether it lies on an axis.
c) Name the axis if it does.

Worked Example 6

A point has coordinates $(7,0)$.
a) Describe its location.
b) State whether it lies on an axis.
c) Name the axis if it does.

Problems

Problem 1

a) Define the terms number plane, origin and coordinates.
b) State the coordinates of the origin.
c) Explain what the ordered pair $(4,1)$ means.

Problem 2

Interpret the location of each point:
a) $(3,5)$
b) $(6,2)$
c) $(0,4)$

Problem 3

Describe how to plot each point on the cartesian plane:
a) $(2,6)$
b) $(5,3)$
c) $(4,0)$

Problem 4

A point is plotted by moving $5$ units across and $2$ units up.
a) Write its coordinates.
b) Explain how the coordinates show the location.

Problem 5

A point has coordinates $(0,7)$.
a) Describe its location.
b) State whether it lies on an axis.
c) Name the axis if it does.

Problem 6

A point has coordinates $(8,0)$.
a) Describe its location.
b) State whether it lies on an axis.
c) Name the axis if it does.

Exercises

Understanding and Fluency

1. State the coordinates of each point described:
   a) $2$ units across and $3$ units up
   b) $5$ units across and $1$ unit up
   c) $0$ units across and $4$ units up

2. Describe the location of each point:
   a) $(1,2)$
   b) $(4,5)$
   c) $(6,3)$

3. Describe how to plot each point:
   a) $(3,4)$
   b) $(2,1)$
   c) $(5,6)$

4. State whether each point lies on an axis:
   a) $(0,2)$
   b) $(4,0)$
   c) $(3,5)$

5. Name the axis if the point lies on an axis:
   a) $(0,7)$
   b) $(8,0)$
   c) $(0,1)$

6. Write the coordinates of each point described:
   a) $7$ units across and $0$ units up
   b) $0$ units across and $6$ units up
   c) $4$ units across and $2$ units up

7. Match each description to a coordinate:
   a) on the vertical axis, $3$ units up
   b) on the horizontal axis, $5$ units across
   c) $2$ units across and $4$ units up

8. Mixed practice:
   a) What does the first number in $(5,2)$ tell you?
   b) What does the second number in $(5,2)$ tell you?
   c) What are the coordinates of the origin?

Reasoning

9. Explain why the point $(0,4)$ lies on the vertical axis.

10. A student says the point $(2,5)$ means move $5$ across and $2$ up. Explain the mistake.

11. Explain why $(3,0)$ lies on the horizontal axis.

12. Compare the points $(4,2)$ and $(2,4)$. Explain how their locations are different.

Problem-solving

13. A treasure map uses coordinates on a number plane. The treasure is $6$ units across and $4$ units up from the origin. What are its coordinates?

14. A robot starts at the origin and moves to $(3,5)$. Describe its movement.

15. A point is on the vertical axis and is $8$ units above the origin. What are its coordinates?

16. A point is on the horizontal axis and is $9$ units to the right of the origin. What are its coordinates?

17. Plot the points $(1,1)$, $(1,4)$ and $(4,1)$. Describe one thing you notice about their positions.

18. A point is described by $(5,3)$. Explain how to find and plot it on the number plane.

Potential Misunderstandings

• Students may reverse the order of coordinates and treat $(x,y)$ as up first, then across
• Students may confuse the origin with any point containing a zero
• Students may not recognise that the origin is $(0,0)$
• Students may think the first coordinate tells vertical movement and the second tells horizontal movement
• Students may count grid squares inaccurately when plotting points
• Students may not understand that a point with first coordinate $0$ lies on the vertical axis
• Students may not understand that a point with second coordinate $0$ lies on the horizontal axis
• Students may confuse the number plane with only the axes rather than the full grid