199r. The Cartesian Plane

Learning Intentions

• Identify ordered pairs on the Cartesian plane.
• Plot points accurately Use and coordinates.
• Describe horizontal and vertical changes between two points.

Pre-requisite Summary

• Know that an ordered pair is written in the form .
• Know that the first coordinate gives the horizontal position and the second coordinate gives the vertical position.
• Know that the Cartesian plane has a horizontal -axis and a vertical -axis.
• Know that the origin is the point .
• Know that moving right increases the -coordinate and moving left decreases the -coordinate.
• Know that moving up increases the -coordinate and moving down decreases the -coordinate.

Worked Examples

Worked Example 1

State the -coordinate and -coordinate of each ordered pair.

a)

b)

c)

Worked Example 2

Identify the location of each point.

a)

b)

c)

Worked Example 3

Plot each point on a Cartesian plane.

a)

b)

c)

Worked Example 4

Plot each point on a Cartesian plane.

a)

b)

c)

Worked Example 5

Describe the horizontal and vertical change from the first point to the second point.

a) From to

b) From to

c) From to

Worked Example 6

Complete each sentence.

a) To move from to , move units horizontally and units vertically.

b) To move from to , move units horizontally and units vertically.

c) To move from to , move units horizontally and units vertically.

Problems

Problem 1

State the -coordinate and -coordinate of each ordered pair.

a)

b)

c)

Problem 2

Identify the location of each point.

a)

b)

c)

Problem 3

Plot each point on a Cartesian plane.

a)

b)

c)

Problem 4

Plot each point on a Cartesian plane.

a)

b)

c)

Problem 5

Describe the horizontal and vertical change from the first point to the second point.

a) From to

b) From to

c) From to

Problem 6

Complete each sentence.

a) To move from to , move units horizontally and units vertically.

b) To move from to , move units horizontally and units vertically.

c) To move from to , move units horizontally and units vertically.

Exercises

Understanding and Fluency

Exercise 1

State the -coordinate and -coordinate of each ordered pair.

a)

b)

c)

d)

Exercise 2

State whether each point lies on the -axis, the -axis, or neither.

a)

b)

c)

d)

Exercise 3

Identify the quadrant or axis where each point is located.

a)

b)

c)

d)

Exercise 4

Plot each point on a Cartesian plane.

a)

b)

c)

d)

Exercise 5

Plot each point on a Cartesian plane.

a)

b)

c)

d)

Exercise 6

Plot each point on a Cartesian plane.

a)

b)

c)

d)

Exercise 7

Describe the horizontal and vertical change from the first point to the second point.

a) From to

b) From to

c) From to

d) From to

Exercise 8

Complete each movement description.

a) From to : move units right and units up or down.

b) From to : move units left or right and units down.

c) From to : move units right and units up.

d) From to : move units left and units down.

Exercise 9

Write the coordinates of the point reached after each movement.

a) Start at and move units right and unit up.

b) Start at and move units left and units down.

c) Start at and move units right and units up.

d) Start at and move units left and units down.

Exercise 10

Copy and complete each statement.

a) The point is units along the -axis and units along the -axis.

b) The point is units left of the origin and units up.

c) The point is units right of the origin and units down.

d) The point is units left of the origin and unit down.

Reasoning

Exercise 11

Explain why the point is not in the same location as .

Exercise 12

A student says that is on the -axis because it has a in the ordered pair. Explain the mistake.

Exercise 13

A student plots by moving units right and units up. Explain the mistake and describe the correct movement.

Exercise 14

Explain how you can tell that the points and are vertically aligned.

Exercise 15

Explain how you can tell that the points and are horizontally aligned.

Exercise 16

A point moves from to . Explain why the horizontal change is units right and the vertical change is units up.

Problem-solving

Exercise 17

A point starts at .

a) Move units left and units up. Write the new coordinates.

b) Move units right and units down from the new point. Write the final coordinates.

c) Describe the total horizontal and vertical change from the starting point to the final point.

Exercise 18

A rectangle has vertices , , and .

a) Describe the horizontal change from to .

b) Describe the vertical change from to .

c) Explain why opposite sides of the rectangle are equal in length.

Exercise 19

A point is located at . A point is units to the right of and units below .

a) Write the coordinates of .

b) Describe the horizontal change from to .

c) Describe the vertical change from to .

Exercise 20

Create your own pair of points on the Cartesian plane.

Your response must include:

• the coordinates of both points
• a horizontal change
• a vertical change
• a sentence describing how to move from the first point to the second point

Potential Misunderstandings

• Students may reverse the order of coordinates and treat as .
• Students may think the first coordinate tells them to move vertically instead of horizontally.
• Students may ignore negative signs when reading ordered pairs.
• Students may plot points with negative -coordinates to the right of the origin.
• Students may plot points with negative -coordinates above the origin.
• Students may not recognise that points with lie on the -axis and points with lie on the -axis.
• Students may describe only the total distance moved and ignore direction.
• Students may confuse horizontal change with vertical change.
• Students may not notice that equal -coordinates indicate vertical alignment and equal -coordinates indicate horizontal alignment.

Next: 200. Distance Between Points