055. Cartesian Plane

Learning Intentions

• To understand that the Cartesian plane can be extended to include negative numbers on both axes
• To understand what a coordinate pair means if one or both numbers is negative
• plot a point at a location expressed as $x$- and $y$-coordinates

Pre-requisite Summary

• Understand that the Cartesian plane is formed by a horizontal axis and a vertical axis
• Know that the origin is the point $(0,0)$
• Recall that the first coordinate gives horizontal movement and the second gives vertical movement
• Understand that positive numbers move right on the $x$-axis and up on the $y$-axis
• Understand that negative numbers move left on the $x$-axis and down on the $y$-axis
• Be able to plot and read points with positive whole-number coordinates
• Be able to compare positive and negative integers on a number line
• Know that ordered pairs must be read in the given order

Worked Examples

Worked Example 1

a) Explain how the Cartesian plane is extended to include negative numbers on both axes.
b) State the coordinates of the origin.
c) Describe what the point $(-3,2)$ means.

Worked Example 2

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

Worked Example 3

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

Worked Example 4

A point is located by moving $2$ units left and $4$ units up from the origin.
a) Write its coordinates.
b) Explain how the signs in the coordinates show the direction of movement.

Worked Example 5

A point has coordinates $(0,-6)$.
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) Explain how the Cartesian plane is extended to include negative numbers on both axes.
b) State the coordinates of the origin.
c) Describe what the point $(-2,4)$ means.

Problem 2

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

Problem 3

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

Problem 4

A point is located by moving $4$ units left and $3$ units down from the origin.
a) Write its coordinates.
b) Explain how the signs in the coordinates show the direction of movement.

Problem 5

A point has coordinates $(0,-8)$.
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 $(-9,0)$.
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) $3$ units left and $2$ units up
   b) $5$ units right and $4$ units down
   c) $2$ units left and $6$ units down

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

3. Describe how to plot each point:
   a) $(-3,4)$
   b) $(2,-6)$
   c) $(-7,-1)$

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

5. Name the axis if the point lies on an axis:
   a) $(0,-7)$
   b) $(-6,0)$
   c) $(0,4)$

6. Write the coordinates of each point described:
   a) $8$ units left and $0$ units up
   b) $0$ units across and $5$ units down
   c) $3$ units right and $2$ units down

7. Match each description to a coordinate:
   a) on the vertical axis, $4$ units below the origin
   b) on the horizontal axis, $6$ units left of the origin
   c) $2$ units left and $3$ units down

8. Mixed practice:
   a) What does the first number in $(-5,2)$ tell you?
   b) What does the second number in $(3,-4)$ tell you?
   c) What are the coordinates of the origin?

Reasoning

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

10. A student says the point $(-3,5)$ means move $5$ units left and $3$ units up. Explain the mistake.

11. Explain why $(0,-6)$ lies on the vertical axis.

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

Problem-solving

13. A treasure map uses coordinates on a Cartesian plane. The treasure is $4$ units left and $3$ units up from the origin. What are its coordinates?

14. A robot starts at the origin and moves to $(-5,2)$. Describe its movement.

15. A point is on the vertical axis and is $7$ units below the origin. What are its coordinates?

16. A point is on the horizontal axis and is $8$ units to the left of the origin. What are its coordinates?

17. Plot the points $(-1,1)$, $(-1,-3)$ and $(2,-3)$. Describe one thing you notice about their positions.

18. A point is described by $(-4,-2)$. Explain how to find and plot it on the Cartesian plane.

Potential Misunderstandings

• Students may reverse the order of coordinates and treat $(x,y)$ as vertical first, then horizontal
• Students may forget that negative $x$-values mean move left and negative $y$-values mean move down
• 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 with negative coordinates
• 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 think a negative sign changes the order of the coordinates rather than only the direction of movement