Evaluate expressions containing rational numbers

Ontario Grade 8 Mathematics

Algebra

Use order of operations to find the value of an expression

An expression is a mathematical phrase made from numbers and operation signs. It does not include an equals sign. To evaluate an expression means to find its value. Rational numbers include integers, fractions, and decimals that end, such as

-3

,

\frac{2}{5}

, and

0.75

. You have already used addition, subtraction, multiplication, and division with these numbers. In this lesson, you will decide which operation to do first and take care with negative numbers. The goal is to get one correct value from the whole expression.

What you will learn

Recognize rational numbers in expressions, including integers, fractions, and terminating decimals.
Evaluate expressions by following the order of operations.
Keep track of negative signs and use brackets to show which operations belong together.
Check whether an answer is reasonable.

Start with the numbers and operation signs

Imagine a number line marked in equal steps. If you start at

2

and move

5

steps left, you land at

-3

. This gives a concrete way to think about

2-5

: subtracting a larger number moves you left past zero. Negative numbers are less than zero, and positive numbers are greater than zero.

Fractions and decimals can also describe parts of a step. For example,

\frac{1}{2}

and

0.5

name the same amount. When an expression contains different kinds of rational numbers, you can still follow the same operation rules. If it helps, rewrite a terminating decimal as a fraction or rewrite a fraction as a decimal. Choose a form that makes the calculation clear.

An expression may contain several operations. The order of operations is a set of rules for deciding what to calculate first. It prevents different readers from getting different answers from the same expression.

A rational number can be written as a fraction of integers, with a non-zero denominator.
A negative sign changes the number's position relative to zero.
Evaluation means finding the value of the entire expression.

Use the order of operations

First, calculate inside brackets. Brackets group parts of an expression and show that the grouped part should be treated together. Next, calculate powers, such as a number multiplied by itself. Then do multiplication and division in the order they appear, moving from left to right. Last, do addition and subtraction in the order they appear, also moving from left to right.

Multiplication does not always come before division. They share a level in the order, so go from left to right. Addition and subtraction also share a level. For example, in

18 \div 3 \times 2

, divide first because that operation appears first from the left. The result is then multiplied by

2

.

A negative sign can belong to a number, as in

-4

, or it can show subtraction, as in

7-4

. Brackets help make the meaning clear. In

-(-4)

, the expression inside the brackets is the negative number

-4

, and the outside subtraction sign indicates its opposite. Keep the signs attached to the numbers they belong to as you calculate.

Brackets, powers, multiplication and division, then addition and subtraction.
For operations at the same level, calculate from left to right.
Use brackets to keep grouped calculations clear.

Choose a clear way to calculate

Before calculating, look over the whole expression. Notice its brackets, signs, and operation types. Then work one part at a time. Writing each new line with the same value as the line before helps you spot a mistake.

For fraction calculations, use equivalent fractions when adding or subtracting. Equivalent fractions name the same amount but use different numerators and denominators. To add fractions with different denominators, rewrite them with a shared denominator. For multiplication, multiply the numerators together and the denominators together. For division by a fraction, multiply by its reciprocal. A reciprocal is the fraction turned upside down; for example, the reciprocal of

\frac{2}{3}

is

\frac{3}{2}

.

You do not have to turn every number into a fraction. For some expressions, decimals are easier. But do not round partway through a calculation unless a question asks you to. Rounding too early can change the final value.

Use a form that keeps the calculation exact and easy to follow.
Keep intermediate values exact when possible.
Estimate the size and sign of an answer to check whether it makes sense.

Try the method, then practise

In each example, notice how the first operation is selected. The examples move from whole numbers to fractions and decimals. After reading them, try a similar expression on your own. Follow the order of operations and show enough steps to make your choices clear.

For practice, evaluate

6-2\times 5

,

\frac{3}{4}+\frac{1}{2}\div 2

, and

1.2-(-0.8)\times 3

. Before calculating, say which operation comes first and why. Then compare your answer with a quick estimate. Do not use a calculator to replace the reasoning about order.

Identify the first operation before you begin.
Write one clear calculation step at a time.
Check both the sign and the size of the result.

Worked example

Brackets and multiplication

Evaluate

8-3(2+1)

.

Calculate inside the brackets
The brackets group the addition, so find that value before the multiplication or subtraction.
$8-3(3)$
Multiply
The number beside the brackets means multiplication. Do that before the subtraction.
$8-9$
Subtract
Subtracting $9$ from $8$ gives a value one less than zero.
$-1$

Answer:

-1

Check: The grouped product is

9

, which is greater than

8

, so the result should be negative.

Worked example

Fractions and left-to-right order

Evaluate

\frac{3}{4}+\frac{1}{2}\div 2

.

Divide first
Division comes before addition. Dividing by $2$ is the same as multiplying by $\frac{1}{2}$ .
$\frac{1}{2}\div 2=\frac{1}{4}$
Add the fractions
The denominators now match, so add the numerators and keep the denominator. The fractions represent one whole split into fourths.
$\frac{3}{4}+\frac{1}{4}=1$

Answer:

1

Check: Three fourths plus one fourth makes four fourths, which equals one.

Worked example

Negative numbers and decimals

Evaluate

1.2-(-0.8)\times 3

.

Multiply the signed number
Multiplication comes before subtraction. A negative number multiplied by a positive number gives a negative product.
$(-0.8)\times 3=-2.4$
Subtract the negative product
The expression becomes $1.2-(-2.4)$ . Subtracting a negative is the same as adding the positive amount, so the result is greater than $1.2$ .
$1.2+2.4=3.6$

Answer:

3.6

Check: The product is negative, and subtracting it increases

1.2

. A positive answer greater than

1.2

is reasonable.

Common mistakes and how to avoid them

Doing addition before multiplication because it appears first when reading left to right.

Correction: Multiplication and division come before addition and subtraction. Use left-to-right order only among operations at the same level.

Treating subtraction of a negative as subtraction of a positive.

Correction: Keep the negative sign with its number. For example, subtracting

-2

is the same as adding

2

.

Adding fractions by adding both numerators and denominators.

Correction: Use a shared denominator, then add the numerators. For example,

\frac{1}{4}+\frac{2}{4}=\frac{3}{4}

.

Changing the order of multiplication and division.

Correction: Do multiplication and division from left to right, in the order they appear.

Lesson summary

Evaluation means finding the value of an expression.
Follow the order: brackets, powers, multiplication and division from left to right, then addition and subtraction from left to right.
Keep negative signs attached to their numbers and use brackets when they help show the meaning.
Check whether your answer's sign and size make sense.

Check your understanding

Question 1

What is the value of

10-2\times 3

?

$24$
$4$
$8$
$-4$

Show answer and explanation

4

Multiply first:

2\times 3=6

. Then subtract:

10-6=4

.

Question 2

What is the value of

\frac{2}{3}+\frac{1}{3}\times 3

?

$1$
$\frac{5}{3}$
$3$
$\frac{7}{3}$

Show answer and explanation

\frac{7}{3}

Multiply first:

\frac{1}{3}\times 3=1

. Then add

\frac{2}{3}

to get

\frac{5}{3}

.

Question 3

What is the value of

-1.5-(-0.5)

?

$-2$
$-1$
$1$
$2$

Show answer and explanation

-1

Subtracting a negative is adding the positive amount:

-1.5+0.5=-1

.

Key terms

Expression: A mathematical phrase made from numbers and operation signs, without an equals sign.
Evaluate: Find the value of an expression.
Rational number: A number that can be written as a fraction of integers with a non-zero denominator. Integers and terminating decimals are examples.
Equivalent fractions: Fractions that name the same amount, even though they use different numerators or denominators.
Reciprocal: A fraction turned upside down. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C2.2. 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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C2.2 · Evaluate expressions containing rational numbers

Ontario Grade 8 Mathematics

Use order of operations to find the value of an expression

What you will learn

Start with the numbers and operation signs

Use the order of operations

Choose a clear way to calculate

Try the method, then practise

Brackets and multiplication

Fractions and left-to-right order

Negative numbers and decimals

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review