Topics
3. The Coordinate Plane
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United States of America
Grade 6

3.02 Integers in the coordinate plane

Lesson
Practice

Introduction

We have already learned how to graph and name points in the  first quadrant.  We are now ready to extend our knowlege on plotting integer coordinates on all quadrants of the coordinate plane

The four quadrants

We can build the coordinate plane using two copies of the number line to describe the location of shapes and points in a 2D space.

Now we can extend this coordinate system using intergers, which will allow us to describe the location of points in any direction from the origin.

Each axis now has positive and negative numbers, and this means we can talk about four distinct regions of the plane, called quadrants.

Exploration

Drag the point P to explore the other quadrants.

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Notice that the 1st quadrant in the top right is equivalent to the coordinate plane that we looked in the previous lesson. The x-coordinate and y-coordinate of a point in the 1st quadrant are both positive.

Moving around counterclockwise we cover the other three quadrants, which have the following features:

  • 2nd quadrant: x-coordinates are negative, y-coordinates are positive
  • 3rd quadrant: both coordinates are negative
  • 4th quadrant: x-coordinates are positive, y-coordinates are negative

Points that lie on an axis, like \left(-5,0\right) or \left(0,4\right), are not in any quadrant.

Examples

Example 1

In which quadrant does the point (-5, 3) lie?

Worked Solution
Create a strategy

Recall the characteristic of each quadrant:

  • Quadrant I: positive x and positive y.

  • Quadrant II: negative x and positive y.

  • Quadrant III: negative x and negative y.

  • Quadrant IV: positive x and negative y.

Apply the idea

Since both axes are positive, the point (-5, 3) lies in Quadrant II.

Idea summary

Each quadrant of the coordinate plane has distinct characteristics:

  • 1st quadrant: both coordinates are positive
  • 2nd quadrant: x-coordinates are negative, y-coordinates are positive
  • 3rd quadrant: both coordinates are negative
  • 4th quadrant: x-coordinates are positive, y-coordinates are negative

Integers in the coordinate plane

The advantage of using integers on the coordinate plane is that we no longer have boundaries for the coordinates. If an object begins at some point on the plane, we can move it any which way we like, as far as we like, and still be able to describe its location with respect to the origin.

We should remember that the order of the numbers in identifying the coordinates is important.

Coordinates should be written with parentheses in the form (x,y) where the first number is the x-coordinate and the second number is the y-coordinate regardless of the sign of the numbers.

Plotting an ordered pair, from the origin, the x-coordinate will tell you how many units to move left if negative or right if positive. The y-coordinate will tell you how many units to move up if positive or down if negative.

Examples

Example 2

What are the coordinates of the point shown in the coordinate plane?

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
Worked Solution
Create a strategy

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, folllow the grid line across the vertical axis to identify the y-coordinate.

Apply the idea

The number on the horizontal axis directly above the point is 4 and across the y-axis is -6. So, the coordinates are (4, -6).

Example 3

What are the coordinates of the point shown in the coordinate plane?

Give the coordinates in the form (x, y).

-5
5
x
-5
5
y
Worked Solution
Create a strategy

Count the number of horizontal and vertical units required to move away from the origin and determine if it is in the positive or negative direction.

Apply the idea

The point is located 2 spaces to the left, then 1 space down. So, the coordinates are (-2, -1).

Example 4

Plot the point (-9,3) on the coordinate plane.

Worked Solution
Create a strategy

The first coordinate tells us how far to the right (positive) or left (negative) the point is from the origin.

The second coordinate tells us how far above (positive) or below (negative) the point is from the origin.

Apply the idea

Starting from the origin, go 9 units in the left direction and then 3 units in the upward direction.

-5
5
x
-5
5
y
Idea summary

The order of the numbers in identifying the coordinates is important.

Coordinates should be written with parentheses in the form (x,y) where the first number is the x-coordinate and the second number is the y-coordinate regardless of the sign of the numbers.

Plotting an ordered pair, from the origin, the x-coordinate will tell you how many units to move left if negative or right if positive. The y-coordinate will tell you how many units to move up if positive or down if negative.

Outcomes

6.NS.C.6

Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

6.NS.C.6.B

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

6.NS.C.6.C

Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.C.8

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

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