C1.6 The four operations

Learning objectives

C1.6.1Use the four operations for calculations with integers, fractions and• Use the four operations for calculations with integers, fractions and decimals, including correct ordering of operations and use of brackets. Includes: • negative numbers • improper fractions • mixed numbers • practical situations, e.g. temperature changes.

The four operations combine quantities: addition joins, subtraction finds a change or difference, multiplication scales, and division shares or asks how many groups fit. Choose the operation from the relationship, then apply the agreed calculation order.

Priority	What to calculate	Boundary
1	brackets	work from the innermost brackets outward
2	powers and roots	evaluate the complete powered or rooted value
3	multiplication and division	work left to right when both occur
4	addition and subtraction	work left to right when both occur

For $9+5\times7-4\div2$ , calculate multiplication and division first: $9+35-2=42$ . Brackets change the structure: $(9+5)\times7-4\div2=96$ .

Operation with signed numbers	Reliable rule	Example
add or subtract	treat subtraction as adding the opposite; track movement on the number line	$-7-5+8=-12+8=-4$
multiply or divide	same signs give positive; different signs give negative	$-18\div(-4)=4.5$
difference	subtract one value from the other and use the positive distance when the context asks how far apart	$5-(-7)=12$

Fraction operation	Method
add or subtract	use a common denominator, then combine numerators
multiply	convert mixed numbers to improper fractions, multiply, then simplify
divide	multiply by the reciprocal of the divisor, then simplify

$\left(2 rac13- rac78 ight) imes rac6{25}=\left( rac{56}{24}- rac{21}{24} ight) imes rac6{25}= rac{35}{24} imes rac6{25}= rac7{20}$

For decimal addition and subtraction, align place values. For multiplication or division, calculate with the correct place-value scale and estimate first: $1.6\times0.02$ is close to $2\times0.02=0.04$ , so the exact result $0.032$ has a sensible size.

Translate practical wording before calculating. ‘ $5^\circ$ C lower than $-7^\circ$ C’ means $-7-5=-12$ , while ‘ $8^\circ$ C higher’ then means $-12+8=-4$ . When a whole item is required, such as coaches or bags, interpret the remainder and round up rather than reporting a partial item.

Do not automatically work from left to right across unlike operations, and do not add denominators when adding fractions. A negative sign belongs to its number; subtracting a negative changes the operation to addition.

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