order positive and negative integers, decimals and fractions

use the symbols =, ≠, <, >, ≤, ≥

Notes: including use of a number line. See also A22

apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative

understand and use place value (eg when working with very large or very small numbers, and when calculating with decimals)

Notes: including questions set in context.

Knowledge and understanding of terms used in household finance, for example profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT and interest rate. See also R9

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

use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem

Notes: prime factor decomposition including product of prime factors written in index form.

apply systematic listing strategies

including use of the product rule for counting

Notes: including using lists, tables and diagrams.

use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5

estimate powers and roots of any given positive number

Notes: including square numbers up to 15 × 15

Students should know that 1000 = 10³ and 1 million = 10⁶

calculate with roots, and with integer indices

calculate with fractional indices

calculate exactly with fractions

calculate exactly with multiples of \(\pi\)

calculate exactly with surds

simplify surd expressions involving squares (eg \(\sqrt{12} = \sqrt{4×3} = \sqrt{4}×\sqrt{3} = 2\sqrt{3} \)) and rationalise denominators

Notes: see also G17 and G18

calculate with and interpret standard form \(A × 10^n\), where \(1 ≤ A < 10\) and \(n\) is an integer

Notes: with and without a calculator.

Interpret calculator displays.

work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and \(\frac{7}{2}\) or 0.375 and \(\frac{3}{8}\))

change recurring decimals into their corresponding fractions and vice versa

Notes: including ordering.

identify and work with fractions in ratio problems

Notes: See also R8

interpret fractions and percentages as operators

Notes: including interpreting percentage problems using a multiplier. See also R9

use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate

Notes: know and use metric conversion factors for length, area, volume and capacity.

Imperial/metric conversions will be given in the question.

estimate answers

check calculations using approximation and estimation, including answers obtained using technology

Notes: including evaluation of results obtained. See also N15

round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures)

use inequality notation to specify simple error intervals due to truncation or rounding

Notes: including appropriate rounding for questions set in context.

Students should know not to round values during intermediate steps of a calculation. See also N14

apply and interpret limits of accuracy

including upper and lower bounds