Topics
1. Number Sense
topic badge
United States of America
Grade 6

1.09 Identify and represent rational numbers

Lesson
Practice

Introduction

We already know how to identify integers and represent them on a number line. We'll now look to identifying and representing rational numbers on a number line.

Identify rational numbers

What are rational numbers? Let's start by looking at the real number system.

The real number system includes rational numbers, irrational numbers, integers, whole numbers, and natural numbers.

Real numbers include rational and irrational numbers. Rational numbers include integers that include whole numbers.

The first numbers we put on the number line are the natural numbers.

Number line starting from 1 to 4.

The set of natural numbers are the counting numbers, starting from 1:

1,\,2,\,3,\,4,\,5,\,6,\,7,\,\ldots

Next, we will add 0 to our number line to show the whole numbers.

Number line starting from 0 to 4.

The set of whole numbers are the counting numbers, starting from 0:

0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,\ldots

The left side of this line looks pretty empty. If we add all the negatives we now have a set of numbers called the integers.

Number line starting from negative 4 to 4.

Whole numbers together with the negatives of the whole numbers make up the set of integers:

\ldots ,\,-7,-6,-5,-4,-3,-2,-1,\,0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,\ldots

But are there numbers between the ones we already have marked? The answer is yes - an infinite amount of numbers between every little mark.

What sort of numbers are these? Well, rational numbers are all numbers that indicate whole numbers as well as parts of whole numbers. So fractions, decimals, and percentages are added to our number line to create the set of rational numbers.

This image shows a number line starting from negative 4 to 4 with rational numbers. Ask your teacher for more information.

Rational numbers are numbers that can be written as the ratio of two integers with a non-zero denominator.

Integers together with all fractions, terminating and repeating decimals, and percents make up the set of rational numbers.

They cannot all be listed, but here are some examples:

\ldots ,-8,-7.4,-7,-6,-5.33387,-4,-2,\,0,\,\dfrac{1}{2},75\%,\,1,\,2,\,3,\,3.5656,\,\ldots

Examples

Example 1

Is -\dfrac{9}{5} an integer, a rational number, or both?

Worked Solution
Create a strategy

Whole numbers together with the negatives of the whole numbers make up the set of integers.

Integers together with all fractions, terminating and repeating decimals, and percents make up the set of rational numbers.

Apply the idea

The fraction -\dfrac{9}{5} is a signed number (negative) but written as a ratio of two integers with a nonzero denominator. It is not an integer.

The fraction -\dfrac{9}{5} is just a rational number.

Idea summary

Rational numbers are numbers that can be written as the ratio of two integers with a non-zero denominator.

Integers together with all fractions, terminating and repeating decimals, and percents make up the set of rational numbers.

Represent rational numbers

Like the representation of the integers on the number line, the number zero (0) is called the origin. All the positive rational numbers are represented on the right side of the origin, and the negative ones are on the left side.

Let's take a look at some worked questions to know how to represent a rational number and state the number represented by each point plotted on the number line:

Examples

Example 2

State the number in fraction form represented by the point plotted on the number line:

4567
Worked Solution
Create a strategy

Count how many equal parts each unit is divided into, to find the value of each small tick. Then count up by up to the tick that the point is plotted on.

Apply the idea

The point is located between integers 5 and 6. The unit is divided into 5 equal parts.

Each small tick is equivalent to \dfrac{1}{5}. The point is located on the second small tick which is equivalent to \dfrac{2}{5}.

The point is located \dfrac{2}{5} units to the right of positive 5 so, the point represents 5\dfrac{2}{5}.

Reflect and check

What if we want to state the rational number as a decimal?

We can just change the fraction into decimal by dividing the numerator by the denominator.

The point above represents 5\dfrac{2}{5} or 5.4, since \dfrac{2}{5}=2\div 5=0.4.

Example 3

Plot -\dfrac{15}{4} on the number line.

Worked Solution
Create a strategy

Change the improper fraction to a mixed number and determine the number of parts each unit should be equally divided into.

Consider the sign of the number to decide whether the point is located on the left or right side of the 0.

Apply the idea

The fraction -\dfrac{15}{4} is equivalent to -3\dfrac{3}{4} and must be on the left side of 0 on the number line.

Draw a number line and locate the point that represent 0. Mark equal spaces and label with consecutive positive integers on the right and consecutive negative integers on the left.

-4-3-2-101234

Since we have a negative fraction, the point must be on the left side of the denominator.

How many parts should we divide each unit into?

The fraction has 4 as its denominator so we need to divide each unit into 4 equal parts. Each part is equivalent to \dfrac{1}{4}.

The improper fraction -\dfrac{15}{4} is the mixed number -3\dfrac{3}{4} plotted as:

-4-3-2-101234

Example 4

Plot 8.85 on the number line.

Worked Solution
Create a strategy

Determine between what two integers the point should be located.

Divide each whole unit into ten equal parts.

Apply the idea

The point should be located at the right of zero between positive 8 and positive 9.

-1012345678910

Dividing each unit of the number line into ten equal parts makes each small tick equivalent to 0.10.

8910

The point should be between 8.8 and 8.9, the eighth and ninth small tick after 8.

The point 8.85 is plotted as:

8910
Idea summary

To plot a rational number on the number line:

  • Draw a line and locate where the 0, or origin is.

  • Positive rational numbers should be marked on the right side of 0, while negative rational numbers are marked on the left side of 0.

  • For fractions, divide each unit into the values equal to the fraction’s denominator.

  • For decimals, divide each unit into ten equal parts so that each part is equivalent to 0.10.

  • Plot the point.

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.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.

What is Mathspace

About Mathspace