Through the mathematics content, pupils should be taught to:

Develop fluency

consolidate their numerical and mathematical capability from key stage 2 and extend their understanding of the number system and place value to include decimals, fractions, powers and roots

select and use appropriate calculation strategies to solve increasingly complex problems

use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships

substitute values in expressions, rearrange and simplify expressions, and solve equations

move freely between different numerical, algebraic, graphical and diagrammatic representations

develop algebraic and graphical fluency, including understanding linear and simple quadratic functions

use language and properties precisely to analyse numbers, algebraic expressions, 2-D and 3-D shapes, probability and statistics

Reason mathematically

extend their understanding of the number system; make connections between number relationships, and their algebraic and graphical representations

extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically

identify variables and express relations between variables algebraically and graphically

make and test conjectures about patterns and relationships; look for proofs or counter-examples

begin to reason deductively in geometry, number and algebra, including using geometrical constructions

interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning

explore what can and cannot be inferred in statistical and probabilistic settings, and begin to express their arguments formally

Solve problems

develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems

develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics

begin to model situations mathematically and express the results using a range of formal mathematical representations

select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems

Number

Pupils should be taught to:

understand and use place value for decimals, measures and integers of any size

order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥

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

use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals

recognise and use relationships between operations including inverse operations

use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations

interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or 0

work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and or 0.375 and )

define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100%

interpret fractions and percentages as operators

use standard units of mass, length, time, money and other measures, including with decimal quantities

round numbers and measures to an appropriate degree of accuracy

use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b

use a calculator and other technologies to calculate results accurately and then interpret them appropriately

appreciate the infinite nature of the sets of integers, real and rational numbers

Algebra

Pupils should be taught to:

use and interpret algebraic notation, including:

ab in place of a × b

3y in place of y + y + y and 3 × y

a² in place of a × a, a³ in place of a × a × a; a²b in place of a × a × b

in place of a ÷ b

coefficients written as fractions rather than as decimals

brackets

substitute numerical values into formulae and expressions, including scientific formulae

understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors

simplify and manipulate algebraic expressions to maintain equivalence by:

collecting like terms

multiplying a single term over a bracket

taking out common factors

expanding products of 2 or more binomials

understand and use standard mathematical formulae; rearrange formulae to change the subject

model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)

work with coordinates in all 4 quadrants

recognise, sketch and produce graphs of linear and quadratic functions of 1 variable with appropriate scaling, using equations in x and y and the Cartesian plane

interpret mathematical relationships both algebraically and graphically

reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically

use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations

find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

generate terms of a sequence from either a term-to-term or a position-to-term rule

recognise arithmetic sequences and find the nth term

recognise geometric sequences and appreciate other sequences that arise

Ratio, proportion and rates of change

Pupils should be taught to:

change freely between related standard units

use scale factors, scale diagrams and maps

express 1 quantity as a fraction of another, where the fraction is less than 1 and greater than 1

use ratio notation, including reduction to simplest form

divide a given quantity into 2 parts in a given part:part or part:whole ratio; express the division of a quantity into 2 parts as a ratio

understand that a multiplicative relationship between 2 quantities can be expressed as a ratio or a fraction

relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions

solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics

solve problems involving direct and inverse proportion, including graphical and algebraic representations

use compound units such as speed, unit pricing and density to solve problems

Geometry and measures

Pupils should be taught to:

derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)

calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes

draw and measure line segments and angles in geometric figures, including interpreting scale drawings

derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line

describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric

use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles

derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures using appropriate language and technologies

identify properties of, and describe the results of, translations, rotations and reflections applied to given figures

identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

understand and use the relationship between parallel lines and alternate and corresponding angles

derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs

use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D

interpret mathematical relationships both algebraically and geometrically

Probability

Pupils should be taught to:

record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale

understand that the probabilities of all possible outcomes sum to 1

enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams

generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities

Statistics

Pupils should be taught to:

describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)

construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data

describe simple mathematical relationships between 2 variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs