YEAR 8 · INTEGERS AND RATIONAL NUMBERS · 18 MIN

Order of operations

Apply operation order carefully, including brackets, powers and signs.

In this lesson

Follow brackets, powers, multiplication/division, then addition/subtraction.
Work equal-priority operations left to right.
Explain the difference between a negative base and an outside negative sign.

Before you start

Four operations with integers
Square numbers

01 / LEARN

Understand the idea.

A consistent calculation order

Evaluate bracketed groups first, then powers, then multiplication and division, and finally addition and subtraction. At either shared-priority stage, work left to right. For example, $24 \div 6 \times 2$ means divide first because that operation is farther left.

A fraction bar also groups its numerator and denominator. Keep each unfinished group intact as you simplify, and write a fresh line after a useful stage. This prevents a nearby operation from being applied too early.

Powers and negative signs

An exponent applies to its base. In $- 3^{2}$ , the base of the power is $3$ ; the outside minus remains, producing $- 9$ . In ${(- 3)}^{2}$ , the whole negative number is the base, producing $9$ .

Use brackets to show exactly what is being raised to a power. Before evaluating a long expression, predict the sign of each product or power; a final answer with an unexpected sign is a reason to inspect your working.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A power and a bracket

Calculate $- 2^{2} + 3 (5 - 8)$ .

Worked solution

1
Evaluate the bracket.
$- 2^{2} + 3 (5 - 8) = - 2^{2} + 3 (- 3)$
The subtraction inside the brackets must be completed before the product.
2
Evaluate the power.
$- 2^{2} = - (2 \times 2) = - 4$
The exponent applies to two, with the negative sign outside the power.
3
Evaluate the product.
$3 (- 3) = - 9$
A positive factor multiplied by a negative factor gives a negative product.
4
Combine the two results.
$- 4 + (- 9) = - 13$
Adding nine negative units to negative four gives negative thirteen.

Answer

-13

EXAMPLE 02

Group a denominator

Calculate $36 \div (3 \times 4) + 2^{2}$ .

Worked solution

1
Complete the grouped product.
$36 \div (3 \times 4) + 2^{2} = 36 \div 12 + 2^{2}$
The brackets make twelve the complete divisor.
2
Evaluate the power.
$2^{2} = 4$
Powers have priority over the remaining addition.
3
Complete the division.
$36 \div 12 = 3$
The grouped divisor must be used as one value.
4
Add the remaining values.
$3 + 4 = 7$
Only the addition remains after the bracket, power and division.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

Calculate $6 + 2 \times (- 5)$ .

Show answer & reasoning

-4

Complete the multiplication before adding: $2 \times (- 5) = - 10$ , then $6 + (- 10) = - 4$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Calculate $8 - 3 \times 5$ .
2 marks
Calculate $(8 - 3) \times 5$ .
2 marks
Calculate $- 3^{2} + 4$ .
2 marks
Calculate ${(- 3)}^{2} + 4$ .
2 marks
Calculate $24 \div 6 \times 2$ .
2 marks
Calculate $18 - 2 [3 + (- 1)]$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Calculate $8 - 3 \times 5$ .
2 marks
$3 \times 5 = 15$ , so $8 - 15 = - 7$ .
-7
Calculate $(8 - 3) \times 5$ .
2 marks
$8 - 3 = 5$ , then $5 \times 5 = 25$ .
25
Calculate $- 3^{2} + 4$ .
2 marks
$- 3^{2} = - 9$ , then $- 9 + 4 = - 5$ .
-5
Calculate ${(- 3)}^{2} + 4$ .
2 marks
${(- 3)}^{2} = 9$ , then $9 + 4 = 13$ .
13
Calculate $24 \div 6 \times 2$ .
2 marks
Work left to right: $24 \div 6 = 4$ , then $4 \times 2 = 8$ .
8
Calculate $18 - 2 [3 + (- 1)]$ .
3 marks
$3 + (- 1) = 2$ .
$2 \times 2 = 4$ , so $18 - 4 = 14$ .
14

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Four operations with integers; signed fraction/decimal calculations; order of operations; terminating and recurring decimals; rational and irrational numbers; estimating square roots and recognising pi; choosing efficient exact/decimal representations.

Official curriculum source ↗

In this lesson

Understand the idea.

A consistent calculation order

Powers and negative signs

See the method in action.

A power and a bracket

Evaluate the bracket.

Evaluate the power.

Evaluate the product.

Combine the two results.

Group a denominator

Complete the grouped product.

Evaluate the power.

Complete the division.

Add the remaining values.

Try it yourself.

Check your understanding

Keep practising