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

3.03 Rational numbers in the coordinate plane

Lesson
Practice

Introduction

We have previously learned how to plot points in the four quadrants of the coordinate plane. We'll now look at how to graph points with rational number coordinates and real-world applications involving this concept.

Rational numbers in the coordinate plane

Rational numbers are numbers that can be written as a fraction in the form of \dfrac{p}{q} where p and q are integers and q is not zero.

Similar to integers, rational numbers can be plotted or graphed in a coordinate plane. These points are represented in an ordered pair, or coordinates, of the form (x,y).

There are a few tips that can help us when plotting coordinates that contain non-integer values. For rational numbers that are not integers, we can determine which two integers the number falls in between in order to plot the points more easily. For an improper fraction, we can change it to a mixed numbers to determine which two integers we will plot between. It is also helpful if we identify the quadrant where the point will be plotted by looking at the signs of the x and y coordinates.

Examples

Example 1

Plot point D with coordinates \left(-\dfrac{5}{2}, \dfrac{2}{3}\right) on the coordinate plane.

Worked Solution
Create a strategy

Identify the quadrant where the point must be plotted.

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.

Identify the two integers on the axes the fraction coordinates fall in between.

Apply the idea
-3
-2
-1
1
2
3
x
-3
-2
-1
1
2
3
y

  • Since point D, \left(-\dfrac{5}{2}, \dfrac{2}{3}\right), has a negative x-coordinate and positive y-coordinate, it will be in Quadrant II.

  • The x-coordinate, -\dfrac{5}{2} is equal to -2\dfrac{1}{2}, meaning it will be plotted between the integers -2 and -3 on the x-axis.

  • The y-coordinate, \dfrac{2}{3} should be between 0 and 1, though closer to 1 on the y-axis.

Example 2

The graph shows the location of Gina's home and the nearby library. The scale of each axis is in kilometers.

What are the coordinates of the library?

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
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
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y

  • We can look closely at the location of the library and the integers that the point is between.

  • We can draw imaginary lines from the axes to the point in order to help us determine the coordinates.

  • From the origin, we move 3\dfrac{1}{2} units to the right.

  • We then move 5 units downward.

Therefore, the coordinates of the library are \left(3\dfrac{1}{2}, -5\right). In a real world context, this means that the library is 3\dfrac{1}{2} kilometers to the east and 5 kilometers to the south of Gina's home.

Reflect and check

We can also write the mixed number coordinates as improper fraction or decimal numbers.

Thus, we can write the coordinates of the library as: \left(\dfrac{7}{2}, -5\right) or \left(3.5, -5\right) .

Idea summary

Rational numbers can be plotted or graphed in a coordinate plane.

It is important to know what two integers on the axes the fraction or decimal falls in between.

We can change an improper fraction coordinate to a mixed number to easily determine where between the two integers it should be placed.

Coordinates are always written with parentheses in the form (x,y) where the first number, x, is the x-coordinate and the second number, y is the y-coordinate.

The coordinates \left(x, y\right) refer to the point x units to the left or right of the origin, and y units above or below the origin.

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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