Topics
1. Number Sense
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United States of America
Grade 6

1.10 Compare and order rational numbers

Lesson
Practice

Introduction

We know that rational numbers include integers, all fractions, terminating and repeating decimals, and percents

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

Now let's look at how we can compare and order rational numbers that are in the same form.

Compare rational numbers

Just as we can compare and order integers on the number line, we can compare rational numbers on the number line.

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

As we move to the right on the number line the numbers get larger.

Likewise, as we move to the left on the number line the numbers get smaller.

For example, -3.12, is to the left of 1.3, so -3.12 is smaller than 1.3. We can write this statement as the following inequalities:

\displaystyle -3.12\displaystyle <\displaystyle 1.3because -3.12 is further to the left on the number line
\displaystyle 1.3\displaystyle >\displaystyle -3.12because 1.3 is further to the right on the number line

Examples

Example 1

Which is the largest rational number marked on the number line?

-5-4-3-2-1012345
Worked Solution
Create a strategy

Recall that the further a number is to the right on a number line, the larger the number is.

Apply the idea

The rational number farthest to the right on the number line is 2 \dfrac {1}{3}. So the largest number is 2 \dfrac {1}{3}

Example 2

Consider the values 1.25 and 0.75.

a

Plot the pair of numbers on a number line.

Worked Solution
Create a strategy

We can see that 1.25 and 0.75 are both positive and so will be to the right of 0.

Apply the idea

To plot the point 1.25, start at 0 and count right 1 place and then move to the right 0.25 one time past 1. To plot the point 0.75, we can start at 0 and jump to the right by 0.25 three times.

-5-4-3-2-1012345
b

State the inequality sign that makes the statement true.

1.25 \,⬚\, 0.75

Worked Solution
Create a strategy

Compare the plotted decimals given in the answer found from part (a).

Apply the idea

The rational number farther to the right on the number line is 1.25 so the larger number is 1.25

\displaystyle 1.25\displaystyle >\displaystyle 0.751.25 is greater than 0.75
\displaystyle 1.25\displaystyle >\displaystyle 0.75Complete the inquality with the greater than symbol
Idea summary

The symbol < represents the phrase "is less than".

The symbol > represents the phrase "is greater than".

Order rational numbers

We can also use a number line to help us arrange rational numbers in ascending order by plotting them on a number line and then listing them in order from the number that is the farthest to the left to the number that is farthest to the right. Here are the rational numbers 0,\, 3.7,\, -4.5,\, 1.3 plotted on a number line:

-5-4-3-2-1012345

Order the numbers from least to greatest:-4.5,\,0,\,1.3,\,3.7

We can use the same number line to arrange the rational numbers in descending order by finding the greatest number, which is the number farthest to the right, and writing them in order to the number that is farthest to the left.

Here are the rational numbers written in descending order:

3.7,\,1.3,\,0,\,-4.5

Examples

Example 3

Arrange the following numbers in ascending order:

2.25, \, 2.75, 1.5, \, -2.5, \,-2.75

Worked Solution
Create a strategy

Ascending means going from left to right on the number line.

Apply the idea

Plot the points on the number line:

-3-2-10123

Arrange the list from least to greatest -2.75,\, -2.5,\, 1.5, \,2.25, \,2.75

Idea summary

Descending rational numbers get smaller as we move to the left on the number line.

Ascending rational numbers get larger as we move to the right on the number line.

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.

6.NS.C.7

Understand ordering and absolute value of rational numbers.

6.NS.C.7.A

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

6.NS.C.7.B

Write, interpret, and explain statements of order for rational numbers in real-world contexts.

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