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

The idea

Every operation has an inverse that undoes it: addition and subtraction are inverses, as are multiplication and division. This is used to check answers and solve equations. Calculations follow a strict order of priority (BIDMAS/BODMAS): Brackets, then Indices (powers, roots), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right). For example, 3 + 4 × 5² is calculated as 3 + 4 × 25 = 3 + 100 = 103. A reciprocal is 1 divided by the number; the reciprocal of 4 is 1/4 and of 2/3 is 3/2. Cancellation simplifies calculations and expressions by dividing by a common factor. For example, 36/48 simplifies to 3/4 by cancelling the common factor of 12. In algebra, an expression like (4x²)/(2x) simplifies to 2x.

What earns the marks

• Applying the priority of operations in the correct order (BIDMAS/BODMAS).
• Correctly identifying and cancelling a common numerical or algebraic factor.
• Using inverse operations to reverse a chain of steps, such as in solving an equation.
• Finding the correct reciprocal of an integer, fraction or decimal.

Common mistakes and how to put them right

• Working strictly from left to right, so calculating 2 + 3 × 4 as 5 × 4 = 20 rather than 2 + 12 = 14.

• Calculating −3² as (−3) × (−3) = 9, when the index applies only to the 3, so the value is −(3 × 3) = −9.

• Incorrectly cancelling terms instead of factors in an expression, e.g. simplifying (x+2)/2 to x+1 instead of x/2 + 1.

• Confusing the reciprocal with the negative of a number.

Exam tips

• Write each stage of a priority-of-operations calculation on a new line to reduce the chance of errors spreading.
• Check a numerical answer by applying the inverse operations.
• Cancel common factors before you multiply fractions; the numbers stay smaller and the arithmetic is easier.