Structure and calculation: recognise and use relationships between operations, including inverse operations (eg cancellation to simplify calculations and expressions) use conventional notation for priority of operations, including brackets, powers, roots and reciprocals

The idea

This topic links operations by their opposites and by order. Inverse operations undo each other: addition and subtraction, multiplication and division, squaring and square-rooting, and reciprocals. Recognising these lets you cancel common factors in a fraction such as 18/24 to 3/4, or rearrange an expression. Priority of operations is the agreed order: work inside brackets first, then powers and roots, then multiplication and division left to right, then addition and subtraction left to right. For example, 3 + 4 × 2² = 3 + 4 × 4 = 3 + 16 = 19, not 49. A reciprocal of a non-zero number n is 1/n, so the reciprocal of 5 is 1/5 and of 2/3 is 3/2. Brackets, powers, roots and reciprocals all change which operation happens first.

What earns the marks

• Identifies inverse pairs correctly: addition with subtraction, multiplication with division, squaring with square-rooting, and a number with its reciprocal.
• Applies cancellation to simplify a calculation or expression by dividing numerator and denominator by a common factor.
• Evaluates expressions using the conventional order: brackets, then powers and roots, then multiplication and division, then addition and subtraction.
• Recognises that a reciprocal of a non-zero number n is 1/n and uses it to rewrite division as multiplication.
• Uses brackets to change the order of operations and explains how the value changes.
• Distinguishes between expressions such as (−3)² = 9 and −3² = −9 by applying priority of operations.

Common mistakes and how to put them right

• Working strictly left to right and adding before multiplying.

  Put it right: Multiplication and division are completed before addition and subtraction, so 2 + 3 × 4 = 14, not 20.

• Treating a reciprocal as the negative of a number.

  Put it right: The reciprocal of 4 is 1/4, while the negative is −4; they are different ideas.

• Squaring a negative number without brackets and keeping the negative sign.

  Put it right: −3² means −(3²) = −9, whereas (−3)² = 9.

• Cancelling terms that are added rather than factors that are multiplied.

  Put it right: In (x + 2)/2 you cannot cancel the 2s because the numerator is a sum, not a product.

Exam tips

• Underline the operation that must be done first before starting a calculation, then rewrite the expression step by step.
• When a question asks for a reciprocal, write it as a fraction such as 1/7 or invert a fraction such as 5/8 to 8/5, and check the value is not zero.
• Use brackets in your working to show the order you intend, especially when substituting negative numbers into expressions with powers.