Represent, compare, and order signed fractions and decimals to thousandths

Ontario Grade 7 Mathematics

Number

Grade 7 Mathematics — B1.3

A number can be positive, negative, or zero. Positive numbers are greater than zero. Negative numbers are less than zero. For example, a temperature of

-1.5

degrees is lower than zero, while

1.5

degrees is higher than zero. A number line helps us show where these numbers belong and compare their positions. In this lesson, you will compare and order signed fractions and decimals, including decimals with three places after the decimal point. A thousandth is one of one thousand equal parts of a whole.

What you will learn

Explain what positive and negative numbers show.
Represent signed fractions and decimals on a number line.
Compare signed fractions and decimals to thousandths.
Order signed fractions and decimals from least to greatest.

1. Read signed numbers and use a number line

The sign tells which side of zero a number belongs on. A plus sign means positive, and a minus sign means negative. When no sign is written, a number is positive. Zero is neither positive nor negative.

A number line shows numbers in order. Numbers increase as you move to the right. Numbers decrease as you move to the left. So a number farther right is greater, and a number farther left is less.

Imagine a floor marked in equal steps on either side of a starting point called zero. Moving three steps left gives

-3

; moving three steps right gives

3

. The distance from zero is the same, but the signs show opposite sides. This idea also works for fractions and decimals between whole numbers.

A decimal to thousandths has up to three digits after the decimal point. In

0.375

, the 3 shows tenths, the 7 shows hundredths, and the 5 shows thousandths. A zero can hold an empty place:

0.4=0.400

. Writing the extra zeros does not change the value.

-1<0<1

Right on a number line means greater; left means less.
Negative numbers are less than zero.
A thousandth is one of 1,000 equal parts.

2. Represent fractions and decimals

A fraction names equal parts of a whole. In

\frac{3}{4}

, the whole is divided into four equal parts and three are counted. A fraction can be positive or negative. The negative sign means the fraction is below zero on a number line.

Decimals and fractions can name the same amount. For example,

\frac{1}{2}

and

0.5

are two ways to name one half. On a number line, equivalent names belong at the same point.

To represent a value, first find the interval it belongs in. For example,

-0.25

lies between

-1

and

0

. Then divide that interval into equal parts if needed. If a number is written in thousandths, an interval of one whole can be divided into 1,000 equal steps. A point’s location, not the length of its written name, determines its value.

When comparing decimals, write the same number of decimal places by adding zeros if helpful. Compare the digits from left to right, starting with the greatest place value. For example, compare tenths first; if they match, compare hundredths; if those match, compare thousandths. For negative values, remember that the number closer to zero is greater.

0.4=0.400

Equivalent fractions and decimals have the same value and the same number-line position.
Adding zeros at the end of a decimal does not change its value.
For negative values, closer to zero means greater.

3. Compare and order signed values

To compare two values, use the number-line rule or compare their sizes carefully. The symbol

<

means “is less than,” and

>

means “is greater than.” For instance,

-2<-1

because

-2

is farther left.

If one value is negative and the other is positive, the negative value is smaller. If both are positive, compare their ordinary sizes. If both are negative, compare their distances from zero in reverse order: the value with the smaller distance from zero is greater.

To order a list, arrange every value from least to greatest. Picture the values from left to right on a number line. Check that each value is greater than the one before it. Mixed forms are allowed: a fraction and a decimal can be compared by recognizing equal values or by placing both on the same number line.

Keep the signs attached while comparing. For example,

-0.6

is not the same as

0.6

. They are on opposite sides of zero. Also, do not decide which negative number is greater by looking only at the digits after the minus sign.

-0.6<0<0.6

A negative value is less than any positive value.
Among negative values, the one nearer zero is greater.
Ordering from least to greatest means moving left to right.

4. Check your order

Before you finish, check that every value appears once and that the signs are correct. If the list mixes fractions and decimals, think about where each belongs between nearby whole numbers. A number line is especially useful when values are close together.

For numbers written to thousandths, compare the thousandths place only after tenths and hundredths match. For a negative pair, use the number-line direction rather than treating the minus sign as decoration.

Practice by ordering

-0.08

,

\frac{1}{10}

, and

-0.1

from least to greatest. Rewrite

0.1

as

0.100

and

0.08

as

0.080

if that helps you compare. Then place the negative values to the left of zero and the positive value to the right.

Use equal decimal places to make a comparison easier.
Check number-line direction, especially for negative values.

Worked example

Compare two negative decimals

Which is greater,

-0.7

or

-0.65

?

Two negative decimals

The values are placed in increasing order from left to right.

Write matching decimal places
Write both values in hundredths. This does not change either value because a zero at the end of a decimal does not change its size.
$-0.70,\ -0.65$
Use their positions
Both are negative. On the number line, $-0.70$ is farther left than $-0.65$ . The value closer to zero is greater.
$-0.70<-0.65$

Answer:

-0.65

is greater.

Check:

-0.65

is closer to zero, so it lies to the right of

-0.70

.

Worked example

Place a fraction and a decimal

Order

-\frac{3}{4}

,

-0.7

, and

-0.8

from least to greatest.

Negative fraction and decimals

The fraction is shown at its equivalent decimal position.

Find the fraction’s decimal value
Three quarters is $0.75$ , so negative three quarters is $-0.75$ . This puts it between $-0.7$ and $-0.8$ .
$-\frac{3}{4}=-0.75$
Order from left to right
All three values are negative. The value farther from zero is farther left. Therefore, $-0.8$ comes first, then $-0.75$ , then $-0.7$ .
$-0.8<-0.75<-0.7$

Answer:

-0.8<-\frac{3}{4}<-0.7

Check: The order moves toward zero from left to right, so each next value is greater.

Worked example

Order values to thousandths

Order

-0.125

,

0.08

,

-\frac{1}{8}

, and

0.105

from least to greatest.

Values to thousandths

Equivalent negative values share one point; the positive values lie to the right of zero.

Match the negative fraction
One eighth is $0.125$ . So negative one eighth and negative $0.125$ name the same value.
$-\frac{1}{8}=-0.125$
Compare the positive decimals
Write $0.08$ as $0.080$ . Since 80 thousandths is less than 105 thousandths, $0.080$ is less than $0.105$ .
$0.080<0.105$
Put all values in order
The negative values come before the positive values. The two negative values are equal, so they share a position. Then place the positive values from smaller to larger.
$-0.125=-\frac{1}{8}<0.080<0.105$

Answer:

-0.125=-\frac{1}{8}<0.08<0.105

Check: The first two names are equivalent. Both are negative; the remaining values are positive and increase from left to right.

Common mistakes and how to avoid them

Thinking

-0.7

is less than

-0.65

because 70 is greater than 65.

Correction: For negative values, the number closer to zero is greater. Compare their positions on a number line.

Thinking that

0.4

and

0.400

have different values.

Correction: Zeros at the end of a decimal do not change its value. They can help show matching place values.

Leaving out a negative sign while ordering.

Correction: Keep each sign attached to its number. A positive and negative value with the same digits are on opposite sides of zero.

Lesson summary

Numbers increase from left to right on a number line.
Negative numbers are less than zero, and positive numbers are greater than zero.
Decimals can be written with matching places by adding zeros at the end.
For negative numbers, closer to zero means greater.
Equivalent fractions and decimals have the same value and position.

Check your understanding

Question 1

Which statement is true?

$-0.308<-0.38$
$-0.38<-0.308$
$-0.308=-0.38$
correctIndex_bogus

Show answer and explanation

-0.38<-0.308

Write

-0.38

as

-0.380

. It is farther left than

-0.308

, so

-0.38<-0.308

.

Question 2

Which value is least?

$-\frac{2}{5}$
$-0.35$
$0.2$
correctIndex_bogus

Show answer and explanation

-\frac{2}{5}

-\frac{2}{5}=-0.4

, which is less than

-0.35

and

0.2

.

Question 3

Which is the correct order from least to greatest?

$-0.006<-0.06<0.006$
$-0.06<-0.006<0.006$
$0.006<-0.006<-0.06$
correctIndex_bogus

Show answer and explanation

-0.06<-0.006<0.006

The negative number farther from zero,

-0.06

, is least. Then comes

-0.006

, followed by the positive value.

Key terms

Signed number: A number shown as positive, negative, or zero.
Thousandth: One of 1,000 equal parts of a whole; it is the third decimal place.
Equivalent values: Different names that have the same value, such as $\frac{1}{2}$ and $0.5$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 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.

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B1.3 · Represent, compare, and order signed fractions and decimals to thousandths

Ontario Grade 7 Mathematics

Grade 7 Mathematics — B1.3

What you will learn

1. Read signed numbers and use a number line

2. Represent fractions and decimals

3. Compare and order signed values

4. Check your order

Compare two negative decimals

Place a fraction and a decimal

Order values to thousandths

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review