Foundation Tier

Number

Algebra

Ratio, proportion and rates of change

Geometry and measures

Probability

Statistics

Structure and calculation

Fractions, decimals and percentages

Measures and accuracy

Notation, vocabulary and manipulation

Graphs

Solving equations and inequalities

Sequences

Properties and constructions

Mensuration and calculation

Vectors

order positive and negative integers, decimals and fractions; use the symbols < , >, = >=, <=

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 (e.g. when working with very large or very small numbers, and when calculating with decimals)

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

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 

apply systematic listing strategies 

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

calculate with roots, and with integer indices

calculate exactly with fractions and multiples of π 

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

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

identify and work with fractions in ratio problems

interpret fractions and percentages as operators

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

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

round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding

apply and interpret limits of accuracy

New algebra notation

substitute numerical values into formulae and expressions, including scientific formulae

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

simplify and manipulate algebraic expressions (including those involving surds)

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

know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments

where appropriate, interpret simple expressions as functions with inputs and outputs.

work with coordinates in all four quadrants

plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel lines; find the equation of the line through two given points or through one point with a given gradient

identify and interpret gradients and intercepts of linear functions graphically and algebraically

identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically

recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function 1 y x = with x ≠ 0 A14 

plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph

solve quadratic equations algebraically by factorising; find approximate solutions using a graph

solve two simultaneous equations in two variables (linear/linear algebraically; find approximate solutions using a graph

translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution

solve linear inequalities in one variable; represent the solution set on a number line

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

recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (r n where n is an integer, and r is a rational number > 0)

deduce expressions to calculate the nth term of linear sequences

change freely between related standard units (e.g. time, length, area, volume/capacity, mass) and compound units (e.g. speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

use scale factors, scale diagrams and maps

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

use ratio notation, including reduction to simplest form

divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio; apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)

express a multiplicative relationship between two quantities as a ratio or a fraction

understand and use proportion as equality of ratios

relate ratios to fractions and to linear functions

define percentage as ‘number of parts per hundred’; interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics

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

use compound units such as speed, rates of pay, unit pricing, density and pressure

compare lengths, areas and volumes using ratio notation; make links to similarity (including trigonometric ratios) and scale factors

understand that X is inversely proportional to Y is equivalent to X is proportional to 1 Y ; interpret equations that describe direct and inverse proportion

interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion

set up, solve and interpret the answers in growth and decay problems, including compound interest

use conventional terms and notation: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries; use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description

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); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line

apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle (e.g. to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)

derive and apply the properties and definitions of special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus; and triangles and other plane figures using appropriate language

use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)

apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional scale factors)

identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

solve geometrical problems on coordinate axes

identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

construct and interpret plans and elevations of 3D shapes

use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money, etc.)

measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings

know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)

know the formulae: circumference of a circle = 2πr = πd , area of a circle = πr 2 ; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids

calculate arc lengths, angles and areas of sectors of circles

apply the concepts of congruence and similarity, including the relationships between lengths, in similar figures

know the formulae for: Pythagoras’ theorem a 2 + b2 = c 2 , and the trigonometric ratios, sin θ = opposite hypotenuse , cos θ = adjacent hypotenuse and tan θ = opposite adjacent ; apply them to find angles and lengths in right-angled triangles in two-dimensional figures

know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°

describe translations as 2D vectors

apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors

record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees

apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments

relate relative expected frequencies to theoretical probability, using appropriate language and the 0-1 probability scale

apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one

understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams

construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities

calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions

infer properties of populations or distributions from a sample, while knowing the limitations of sampling

interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, tables and line graphs for time series data and know their appropriate use

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:

apply statistics to describe a population

use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends while knowing the dangers of so doing

 appropriate graphical representation involving discrete, continuous and grouped data

appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers)