E1.6 The four operations

Learning objectives

E1.6.1Use 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 describe different relationships: addition combines, subtraction finds change or difference, multiplication scales, and division shares or finds how many groups fit. Choose the relationship first, then preserve its structure through the agreed calculation order.

Priority	Calculate	Boundary
1	brackets	work from innermost brackets outward
2	powers and roots	evaluate each complete powered or rooted value
3	multiplication and division	work left to right when tied
4	addition and subtraction	work left to right when tied

For $9+5\times7-4\div2$ , multiplication and division come first: $9+35-2=42$ . Brackets change the structure: $(9+5)\times7-4\div2=14\times7-2=96$ .

Signed operation	Reliable rule	Example
add or subtract	rewrite subtraction as adding the opposite	$-7-5+8=-12+8=-4$
multiply or divide	same signs give positive; different signs give negative	$-18\div(-4)=4.5$
difference	when context asks how far apart, use the positive distance	$5-(-7)=12$

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

$1\frac56+\frac25=\frac{11}{6}+\frac25=\frac{55}{30}+\frac{12}{30}=\frac{67}{30}=2\frac7{30}$

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

Translate practical wording before calculating. ‘ $5^\circ$ C lower than $-7^\circ$ C’ means $-7-5=-12$ ; ‘ $8^\circ$ C higher’ then means $-12+8=-4$ . If the context requires whole coaches or bags, interpret the remainder and round up rather than reporting part of an item.

Do not work mechanically from left to right across different operation priorities, 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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