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 - and -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
• Recall that the first coordinate gives horizontal movement and the second gives vertical movement
• Understand that positive numbers move right on the -axis and up on the -axis
• Understand that negative numbers move left on the -axis and down on the -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 means.

Worked Example 2

Interpret the location of each point:

a)

b)

c)

Worked Example 3

Describe how to plot each point:

a)

b)

c)

Worked Example 4

A point is located by moving units left and 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 .

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 .

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 means.

Problem 2

Interpret the location of each point:

a)

b)

c)

Problem 3

Describe how to plot each point:

a)

b)

c)

Problem 4

A point is located by moving units left and 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 .

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 .

a) Describe its location.

b) State whether it lies on an axis.

c) Name the axis if it does.

Exercises

Understanding and Fluency

Exercise 1.

State the coordinates of each point described:

a) units left and units up

b) units right and units down

c) units left and units down

Exercise 2.

Describe the location of each point:

a)

b)

c)

Exercise 3.

Describe how to plot each point:

a)

b)

c)

Exercise 4.

State whether each point lies on an axis:

a)

b)

c)

Exercise 5.

Name the axis if the point lies on an axis:

a)

b)

c)

Exercise 6.

Write the coordinates of each point described:

a) units left and units up

b) units across and units down

c) units right and units down

Exercise 7.

Match each description to a coordinate:

a) on the vertical axis, units below the origin

b) on the horizontal axis, units left of the origin

c) units left and units down

Exercise 8.

Mixed practice:

a) What does the first number in tell you?

b) What does the second number in tell you?

c) What are the coordinates of the origin?

Reasoning

Exercise 9.

Explain why the point lies on the horizontal axis.

Exercise 10.

A student says the point means move units left and units up. Explain the mistake.

Exercise 11.

Explain why lies on the vertical axis.

Exercise 12.

Compare the points and . Explain how their locations are different.

Problem-solving

Exercise 13.

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

Exercise 14.

A robot starts at the origin and moves to . Describe its movement.

Exercise 15.

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

Exercise 16.

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

Exercise 17.

Plot the points , and . Describe one thing you notice about their positions.

Exercise 18.

A point is described by . Explain how to Solve and plot it on the Cartesian plane.

Potential Misunderstandings

• Students may reverse the order of coordinates and treat as vertical first, then horizontal
• Students may forget that negative -values mean move left and negative -values mean move down
• Students may confuse the origin with any point containing a zero
• Students may not Recognise that the origin is
• 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 lies on the vertical axis
• Students may not understand that a point with second coordinate lies on the horizontal axis
• Students may think a negative sign changes the order of the coordinates rather than only the direction of movement