YEAR 8 · INTEGERS AND RATIONAL NUMBERS · 18 MIN

Four operations with integers

Use signs, direction and operation rules to calculate with positive and negative whole numbers.

In this lesson

Add and subtract signed integers.
Multiply and divide integers using sign rules.
Distinguish a negative number from a subtraction instruction.

Before you start

Ordering integers
Times-table facts

01 / LEARN

Understand the idea.

Direction before rules

An integer is a whole-number value: positive, negative or zero. On a number line, adding a positive value moves right; adding a negative value moves left. The sign tells you the direction, and the size tells you how far.

Subtraction reverses the change being subtracted. In $3 - (- 4)$ , reversing a move of four left means moving four right, so the result is $7$ . Keep the subtraction sign separate from the negative sign inside the brackets.

Multiplication, division and zero

For multiplication and division, first find the size of the answer. Two numbers with the same sign give a positive result; different signs give a negative result. These sign rules describe products and quotients, not every addition.

Multiplying by zero gives zero. Also, $0 \div a = 0$ when $a \neq 0$ . Division by zero is undefined: a nonzero dividend cannot be obtained by multiplying zero, while $0 \div 0$ would not select a unique answer.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Add across zero

Calculate $- 7 + 12$ .

Worked solution

1
Split the move at zero.
$- 7 + 7 = 0$
Seven of the twelve positive units cancel the seven negative units.
2
Find the remaining move.
$12 - 7 = 5$
After reaching zero, five units of the original move remain.
3
Complete the addition.
$- 7 + 12 = 5$
Moving the remaining five units right from zero gives a positive result.

Answer

EXAMPLE 02

Use two operation rules

Calculate $(- 6) (- 4) - 18 \div (- 3)$ .

Worked solution

1
Multiply the two negatives.
$(- 6) (- 4) = 24$
Two negative factors have a positive product.
2
Divide with its sign.
$18 \div (- 3) = - 6$
A positive value divided by a negative value gives a negative quotient.
3
Subtract the negative quotient.
$\begin{aligned} 24 - (- 6) & = 24 + 6 \\ = 30 \end{aligned}$
Subtracting a negative adds its opposite.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

Calculate $- 4 - (- 9)$ .

Show answer & reasoning

Subtracting $- 9$ means adding $9$ , so $- 4 - (- 9) = - 4 + 9 = 5$ . On a number line, move nine places right from $- 4$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Calculate $- 9 + 4$ .
2 marks
Calculate $3 - (- 8)$ .
2 marks
Calculate $(- 7) \times 6$ .
2 marks
Calculate $(- 48) \div (- 8)$ .
2 marks
Calculate $- 5 + 3 (- 4)$ .
2 marks
A submarine is at -18 m relative to sea level. It rises 7 m, then descends 12 m. Find its final position.
3 marks

Read the worked solutions

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

Calculate $- 9 + 4$ .
2 marks
Move four places right from -9.
The result is -5.
-5
Calculate $3 - (- 8)$ .
2 marks
$3 - (- 8) = 3 + 8 = 11$ .
11
Calculate $(- 7) \times 6$ .
2 marks
Different signs give a negative result.
$7 \times 6 = 42$ , so the product is $- 42$ .
-42
Calculate $(- 48) \div (- 8)$ .
2 marks
Same signs give a positive result.
$48 \div 8 = 6$ .
6
Calculate $- 5 + 3 (- 4)$ .
2 marks
$3 (- 4) = - 12$ .
$- 5 + (- 12) = - 17$ .
-17
A submarine is at -18 m relative to sea level. It rises 7 m, then descends 12 m. Find its final position.
3 marks
$- 18 + 7 - 12 = - 11 - 12 = - 23$ .
It finishes 23 m below sea level.
-23 m

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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.

Direction before rules

Multiplication, division and zero

See the method in action.

Add across zero

Split the move at zero.

Find the remaining move.

Complete the addition.

Use two operation rules

Multiply the two negatives.

Divide with its sign.

Subtract the negative quotient.

Try it yourself.

Check your understanding

Keep practising