YEAR 7 · WHOLE NUMBERS, FACTORS AND POWERS · 12 MIN

Efficient whole-number calculations and order of operations

Choose efficient calculation strategies and use the order of operations correctly.

In this lesson

efficient whole-number calculations and order of operations
Explain a method and check that the result is sensible.

Before you start

Reading and calculating with whole numbers

01 / LEARN

Understand the idea.

An expression gives an order

An expression is a set of calculation instructions. Evaluate brackets, then powers, then multiplication and division, then addition and subtraction. Operations sharing a level are evaluated from left to right, so division does not wait for every multiplication.

A number beside a bracket means multiplication: $5 (8 + 2)$ means $5 \times (8 + 2)$ . In your working, copy the parts you have not evaluated. Each new line must have the same value as the line before it.

Efficient does not mean changing the problem

Regroup addition or multiplication to create friendly numbers. For example, $25 \times 16$ can become $(25 \times 4) \times 4$ , producing a hundred first. The grouping changes, but the factors and their product do not.

Subtraction and division need more care: $12 - 5 - 2$ differs from $12 - (5 - 2)$ . Keep the original order unless you can justify a rewrite. Estimate the size of the result as a separate check on your exact calculation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Multiplication comes before addition

Evaluate $18 + 6 \times 4$ .

Worked solution

1
Identify the product.
$6 \times 4 = 24$
Multiplication has higher priority than the addition.
2
Replace only the product.
$18 + 6 \times 4 = 18 + 24$
The 18 is unchanged while the multiplication is evaluated.
3
Add the two amounts.
$18 + 24 = 42$
Only addition remains in the expression.

Answer

EXAMPLE 02

Brackets and two equal-priority operations

Evaluate $8 (15 - 9) + 12 \div 3$ .

Worked solution

1
Evaluate the bracket.
$8 (15 - 9) + 12 \div 3 = 8 \times 6 + 12 \div 3$
The bracket fixes the subtraction as the first calculation.
2
Evaluate the product and quotient.
$\begin{aligned} 8 \times 6 & = 48 \\ 12 \div 3 & = 4 \end{aligned}$
Both multiplication and division are completed before the remaining addition.
3
Combine the results.
$48 + 4 = 52$
The original expression adds the two calculated amounts.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

Evaluate $24 \div 6 + 2^{3}$ .

Show answer & reasoning

Evaluate the power $2^{3} = 8$ and the division $24 \div 6 = 4$ before adding. Therefore $24 \div 6 + 2^{3} = 4 + 8 = 12$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $72 \div (3 \times 4)$ .
2 marks
Evaluate $30 - 12 \div 3 + 2$ .
2 marks
Evaluate $15 + 4^{2} - 9$ .
2 marks
Evaluate $5 (8 + 6)$ .
2 marks
Evaluate $96 \div 8 \div 3$ .
2 marks
Six cinema tickets cost 9 dollars each. A single booking fee of 4 dollars is added. Find the total cost.
3 marks

Read the worked solutions

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

Evaluate $72 \div (3 \times 4)$ .
2 marks
Calculate the bracket: $3 \times 4 = 12$ .
Then $72 \div 12 = 6$ .
6
Evaluate $30 - 12 \div 3 + 2$ .
2 marks
Division first gives $12 \div 3 = 4$ .
Then work left to right: $30 - 4 + 2 = 26 + 2 = 28$ .
28
Evaluate $15 + 4^{2} - 9$ .
2 marks
The power is $4^{2} = 16$ .
Then $15 + 16 - 9 = 31 - 9 = 22$ .
22
Evaluate $5 (8 + 6)$ .
2 marks
The bracket is $8 + 6 = 14$ .
Multiply: $5 \times 14 = 70$ .
70
Evaluate $96 \div 8 \div 3$ .
2 marks
Both operations have equal priority.
Work left to right: $96 \div 8 = 12$ , then $12 \div 3 = 4$ .
4
Six cinema tickets cost 9 dollars each. A single booking fee of 4 dollars is added. Find the total cost.
3 marks
Ticket cost is $6 \times 9 = 54$ dollars.
Add the one booking fee: $54 + 4 = 58$ dollars.
58 dollars

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Place value and powers of ten; efficient whole-number calculations and order of operations; factors, multiples and primes; prime factorisation and exponent notation; square numbers and square roots; estimation and rounding.

Official curriculum source ↗

In this lesson

Understand the idea.

An expression gives an order

Efficient does not mean changing the problem

See the method in action.

Multiplication comes before addition

Identify the product.

Replace only the product.

Add the two amounts.

Brackets and two equal-priority operations

Evaluate the bracket.

Evaluate the product and quotient.

Combine the results.

Try it yourself.

Check your understanding

Keep practising