Compare and order integers, decimals, and fractions

Ontario Grade 6 Mathematics

Number

Use place value, fraction parts, and number lines to decide which values are greater or smaller.

Imagine placing several number cards on a line that has smaller numbers on the left and larger numbers on the right. A number's place on the line tells you how it compares with the others. We can compare whole numbers, integers, decimals, and fractions by using place value, equal-sized parts, and familiar benchmarks such as zero and one. In this lesson, a benchmark is a number we know well and can use to help compare another number.

What you will learn

Compare integers, decimals, and fractions using their values.
Order numbers from least to greatest or greatest to least.
Use a number line to explain why a number is greater or smaller.

Start with what the numbers mean

A number line is a straight line that shows numbers in order. Numbers increase as you move to the right. Numbers decrease as you move to the left. The point for zero separates positive numbers from negative numbers.

An integer is a whole number, its opposite, or zero. Examples include

-4

,

0

, and

6

. A negative integer is less than zero. The farther a negative integer is from zero, the smaller it is. For example,

-8

is less than

-3

because

-8

is farther left.

A decimal uses a decimal point to show parts of a whole. In

2.35

, the digit

3

is in the tenths place and the digit

5

is in the hundredths place. When comparing decimals, compare the whole-number parts first. If they match, compare digits from left to right. You may add zeroes at the end of a decimal without changing its value:

0.6=0.60

.

A fraction names equal parts of a whole. In

\frac{3}{4}

, the bottom number tells how many equal parts make one whole. The top number tells how many of those parts are counted. For fractions with the same bottom number, the fraction with the larger top number is greater. For example,

\frac{5}{8}>\frac{3}{8}

.

Farther right on a number line means greater.
For negative integers, farther from zero means smaller.
Compare decimal digits by place value.
For fractions with the same denominator, compare the numerators.

Use a number line and useful benchmarks

A number line makes the order visible. To place a value, find the interval it belongs in and decide how far along that interval it should go. Between zero and one, for example,

\frac{1}{2}

is halfway. The decimal

0.4

is four tenths of the way from zero to one.

Fractions can be compared by thinking about their size. A benchmark fraction is a familiar fraction, such as

\frac{1}{2}

or

1

, used to help judge another fraction. For example,

\frac{3}{8}

is less than

\frac{1}{2}

because half of eight equal parts is four parts, and three parts are fewer than four.

You can also rename fractions so they have the same denominator. Equivalent fractions name the same amount, even though their numerators and denominators differ. For example,

\frac{1}{2}

is equal to

\frac{5}{10}

. This can help compare a fraction with a decimal written in tenths or hundredths.

To compare numbers from different forms, look for a shared way to understand them. For instance,

0.7

is seven tenths, or

\frac{7}{10}

. Comparing

\frac{3}{5}

with

0.7

becomes comparing

\frac{6}{10}

with

\frac{7}{10}

. This shows that

\frac{3}{5}

is smaller.

0.7=\frac{7}{10}

A number line shows order from left to right.
Use zero, one, and one-half as helpful benchmarks.
Equivalent fractions have the same value.
Tenths and hundredths can connect fractions and decimals.

Order a set carefully

To order numbers, first check what the question asks: least to greatest or greatest to least. Least to greatest means begin with the smallest value. Greatest to least means begin with the largest.

When a set includes negative and positive integers, place the negative values first, then zero, then positive values. Among negative numbers, remember that the value closer to zero is greater. For example,

-2>-5

.

For decimals, compare each number by place value. Add ending zeroes if they make the digits easier to line up. For fractions, use the denominator or a benchmark when possible. For a mixed set, explain how a value compares with a familiar point such as zero, one-half, or one.

After arranging the values, check adjacent pairs. Each number should be greater than the one before it for least-to-greatest order. If two values are equal, they may share a place in the order.

Read the requested direction before ordering.
Check negative values with care.
Line up decimal places by place value.
Check each neighbouring pair at the end.

A reliable comparison routine

Use this routine when several forms appear together. First, identify each number and what it represents. Next, choose a helpful model: place value for decimals, equal parts or benchmarks for fractions, and zero for integers. Then compare the values, not just how the symbols look. Finally, write the requested order and check it on a number line or by checking neighbouring pairs.

A useful comparison symbol is

<

. It means 'is less than.' The symbol

>

means 'is greater than,' and

=

means 'is equal to.' Read the symbol from the smaller value toward the larger value. For example,

-3<1

says that negative three is less than one.

Before moving on, try ordering

0.25

,

\frac{1}{2}

, and

0.8

. Think of

\frac{1}{2}

as five tenths. The values are two and a half tenths, five tenths, and eight tenths. So the order from least to greatest is

0.25

,

\frac{1}{2}

,

0.8

.

-3<1

Choose a model that makes the values easy to see.
Compare amounts, not the appearance or length of the notation.
Read comparison symbols as statements about value.

Worked example

Compare negative integers

Put

-6

,

-1

, and

-4

in order from least to greatest.

Negative integers

Values increase as you move right. The integers are placed in order.

Locate zero
All three numbers are negative, so they are left of zero. A number farther left is smaller.
$-6<-4<-1$
Read the order
Starting with the value farthest left gives the least-to-greatest order.
$-6,\ -4,\ -1$

Answer: The order from least to greatest is

-6

,

-4

,

-1

.

Check:

-6<-4

and

-4<-1

, so each step moves to a greater value.

Worked example

Order decimals by place value

Put

0.8

,

0.35

, and

0.6

in order from least to greatest.

Decimals between zero and one

The points show the stated decimals in increasing order.

Line up the decimal places
Write each value with two decimal places. Adding a zero at the end does not change a decimal's value.
$CAD 0.80,\ 0.35,\ 0.60$
Compare the tenths
The tenths digits are $8$ , $3$ , and $6$ . The smallest is $3$ , then $6$ , then $8$ .
$0.35<0.60<0.80$

Answer: The order from least to greatest is

0.35

,

0.6

,

0.8

.

Check: The decimals all have the same whole-number part, zero. Their tenths digits show the order.

Worked example

Compare fractions and a decimal

Put

\frac{3}{5}

,

0.7

, and

\frac{1}{2}

in order from least to greatest.

Fractions and decimals

The values are located between zero and one in increasing order.

Rename the values
Write each value in tenths. This gives a shared way to compare the fractions and the decimal.
$\frac{3}{5}=\frac{6}{10},\quad 0.7=\frac{7}{10},\quad \frac{1}{2}=\frac{5}{10}$
Compare equal-sized parts
Each fraction now counts tenths. Five tenths is less than six tenths, which is less than seven tenths.
$\frac{5}{10}<\frac{6}{10}<\frac{7}{10}$

Answer: The order from least to greatest is

\frac{1}{2}

,

\frac{3}{5}

,

0.7

.

Check: The three values are five, six, and seven tenths. That confirms their order.

Common mistakes and how to avoid them

Thinking

-6

is greater than

-1

because six is greater than one.

Correction: Negative values lie left of zero. Since

-6

is farther left, it is smaller than

-1

.

Thinking

0.35

is greater than

0.8

because 35 is greater than 8.

Correction: Compare place values. Write

0.8

as

0.80

. Then compare hundredths:

35

hundredths is less than

80

hundredths.

Comparing fractions only by their top numbers when the bottom numbers differ.

Correction: The parts may not be the same size. Use a benchmark or rename the fractions with equal denominators before comparing.

Lesson summary

Numbers farther right on a number line are greater.
Negative integers farther from zero are smaller.
Compare decimals from left to right by place value.
Use equal denominators or familiar benchmarks to compare fractions.
For a mixed set, use a shared form such as tenths when it makes the values easy to compare.

Check your understanding

Question 1

Which statement is true?

$-5<-2$
$-5>-2$
$-5= -2$
correctIndex":0,"explanation":"Both values are negative. $-5$ is farther left on the number line, so it is less than $-2$ ."}

Show answer and explanation

-5<-2

Both values are negative.

-5

is farther left on the number line, so it is less than

-2

.

Question 2

Which list is ordered from least to greatest?

CAD 0.4,\ 0.45,\ 0.5
CAD 0.5,\ 0.45,\ 0.4
CAD 0.45,\ 0.4,\ 0.5
correctIndex":0,"explanation":"Write the decimals in hundredths: $0.40$ , $0.45$ , and $0.50$ . They increase in that order."}

Show answer and explanation

CAD 0.4,\ 0.45,\ 0.5

Write the decimals in hundredths:

0.40

,

0.45

, and

0.50

. They increase in that order.

Question 3

Which value is greatest?

$\frac{3}{4}$
$0.8$
$\frac{7}{10}$
correctIndex":1,"explanation":"The values are $0.75$ , $0.8$ , and $0.7$ . Eight tenths is the greatest."}

Show answer and explanation

0.8

The values are

0.75

,

0.8

, and

0.7

. Eight tenths is the greatest.

Key terms

Integer: A whole number, its opposite, or zero.
Benchmark: A familiar number used to help compare another number.
Equivalent fractions: Fractions that name the same amount.
Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.
Numerator: The top number in a fraction. It tells how many parts are counted.

Continue through Grade 6 Math

View the complete Grade 6 Math Ontario Grade 6 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B1.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

B1.3 · Compare and order integers, decimals, and fractions

Ontario Grade 6 Mathematics

Use place value, fraction parts, and number lines to decide which values are greater or smaller.

What you will learn

Start with what the numbers mean

Use a number line and useful benchmarks

Order a set carefully

A reliable comparison routine

Compare negative integers

Order decimals by place value

Compare fractions and a decimal

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review