order positive and negative
integers, decimals and
fractions
Add, Subtract, Multiply and Divide integers, decimals and
simple fractions (proper and improper), and mixed numbers – all
both positive and negative; understand and use place value
Recognise and use relationships between operations, including
inverse operations, 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,
use positive integer powers and
associated real roots (square, cube ...)
apply systematic listing
strategies including the
product rule
calculate with roots, and with
integer and fractional indices
calculate exactly with fractions, surds and
multiples of π; simplify surd expressions
involving squares and rationalise denominators
calculate with and interpret standard
form A x 10n , where 1 ≤ A < 10 and n
is an integer
Fractions, decimals and percentages
work interchangeably with terminating decimals and their
corresponding fractions, change recurring decimals into their
corresponding fractions and vice versa
identify and work with fractions in
ratio problems
interpret fractions and
percentages as operators.
Measures and accuracy
use standard units of
mass, length, time, money
and other measures
estimate answers; check
calculations using approximation
round numbers and measures to an appropriate
degree of accuracy ; use inequality notation to
specify simple error intervals
apply and interpret limits of accuracy,
including upper and lower bounds
Algebra
Notation, vocabulary and manipulation
use and interpret
algebraic notation
substitute numerical values into formulae and
expressions, including scientific formulae
simplify and manipulate algebraic expressions (including those
involving surds and algebraic fractions) by:
taking out common factors
multiplying a single
term over a bracket
collecting like terms
expanding products of two or
more binomials
factorising quadratic expressions,
including the difference of two squares;
simplifying expressions involving
sums, products and powers,
including the laws of indices
understand and use the concepts and vocabulary of expressions,
equations, formulae, identities inequalities, terms and factors
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 and proofs
where appropriate, interpret simple expressions as
functions with inputs and outputs; interpret the
reverse process as the ‘inverse function’; interpret the
succession of two functions as a ‘composite function’.
Graphs
plot graphs of equations that correspond to
straight-line graphs in the coordinate plane; use the
form y = mx + c to identify parallel and
perpendicular 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 and turning points by completing the square
recognise, sketch and interpret graphs of linear functions, quadratic
functions, simple cubic functions, the reciprocal function y = 1 x with
x ≠ 0, exponential functions =xyk for positive values of k, and the
trigonometric functions (with arguments in degrees) y = sin x , y =
cos x and y = tan x for angles of any size
work with
coordinates in all four
quadrants
plot and interpret graphs (including reciprocal
graphs and exponential 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
recognise and use the equation
of a circle with centre at the
origin; find the equation of a
tangent to a circle at a given
point.
sketch translations and
reflections of a given function
calculate or estimate gradients of graphs and
areas under graphs (including quadratic and
other non-linear graphs), and interpret results in
cases such as distance-time graphs, velocity-time
graphs and graphs in financial contexts
Solving equations and inequalities
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, by completing the square and by
using the quadratic formula; find
approximate solutions using a graph
solve two simultaneous equations in two variables
(linear/linear or linear/quadratic) algebraically
find approximate solutions to
equations numerically using iteration
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 or two variable(s), and
quadratic inequalities in one variable; represent the
solution set on a number line, using set notation and on
a graph
Sequences
generate terms of a sequence from either a term-to-term or a position-to-term rule
recognise and use sequences of triangular, square and
cube numbers, simple arithmetic progressions, Fibonacci
type sequences, quadratic sequences, and simple
geometric progressions
deduce expressions to calculate the
nth term of linear and quadratic
sequences.
Ratio, proportion and rates
of change
use scale factors,
scale diagrams and
maps
express one quantity as a
fraction of another,
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
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
express a multiplicative
relationship between two
quantities as a ratio or a
fraction
use ratio notation, including
reduction to simplest form
define percentage ; interpret percentages
and percentage changes as a fraction or a
decimal, express one quantity as a
percentage of another; compare two
quantities using percentages; work with
percentages greater than 100%; solve
problems involving percentage change
understand and use
proportion as equality
of ratios
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
relate ratios
to fractions
and to linear
functions
interpret the gradient of a straight
line graph as a rate of change;
recognise and interpret graphs that
illustrate direct and inverse
proportion
compare lengths, areas
and volumes using ratio
notation; make links to
similarity (including
trigonometric ratios) and
scale factors
interpret the gradient at a point
on a curve as the instantaneous
rate of change; apply the concepts
of average and instantaneous
rate of change (gradients of
chords and tangents) in
numerical, algebraic and
graphical contexts
set up, solve and interpret
the answers in growth and
decay problems, including
compound interest and
work with general iterative
processes
understand that X is inversely proportional
to Y is equivalent to X is proportional to 1/Y;
construct and interpret equations that
describe direct and inverse proportion
Geometry and
measures
Properties and constructions
use conventional terms and notations; 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 ; use
these to construct given figures
and solve loci problems
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
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
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)
identify, describe and construct congruent and similar
shapes, including on coordinate axes, by considering
rotation, reflection, translation and enlargement
(including fractional and negative scale factors)
identify and apply circle definitions and properties,
including: centre, radius, chord, diameter,
circumference, tangent, arc, sector and segment
describe the
changes and
invariance
achieved by
combinations of
rotations,
reflections and
translations
apply and prove the standard circle theorems concerning angles,
radii, tangents and chords, and use them to prove related results
identify properties of the faces,
surfaces, edges and vertices of:
cubes, cuboids, prisms, cylinders,
pyramids, cones and spheres
solve geometrical
problems on
coordinate axes
construct and interpret plans
and elevations of 3D shapes.
Mensuration and calculation
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 = πr2; 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, areas and volumes in
similar figures
know the formulae for: Pythagoras’ theorem and
the trigonometric ratios, sinθ = opposite hypotenuse
, cosθ = adjacent hypotenuse and tanθ = opposite
adjacent ; apply them to find angles and lengths in
right-angled triangles and, where possible, general
triangles in two and three dimensional figures
know the exact
values of sinθ and
cosθ for given
values of θ (see
attached)
know and apply the sine rule
and cosine rule, to find
unknown lengths and angles
know and apply Area
= 1/2 ab SinC to
calculate the area,
sides or angles of any
triangle.
Vectors
describe translations as 2D vectors
apply addition and subtraction of vectors,
multiplication of vectors by a scalar, and
diagrammatic and column representations of
vectors; use vectors to construct geometric
arguments and proofs
Probability
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, 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
calculate and interpret conditional probabilities through
representation using expected frequencies with two-way
tables, tree diagrams and Venn diagrams.
Statistics
construct and interpret diagrams for grouped
discrete data and continuous data, i.e.
histograms with equal and unequal class
intervals and cumulative frequency graphs
infer properties of populations or
distributions from a sample,
know the limitations
use and interpret scatter graphs of bivariate
data; recognise correlation; draw estimated
lines of best fit; make predictions; interpolate
and extrapolate apparent trends
interpret, analyse and compare the
distributions of data sets from univariate
empirical distributions through:
appropriate
graphical
representation
involving discrete,
continuous and
grouped data,
including box plots
appropriate measures of
central tendency (median,
mean, mode and modal class)
and spread (range, including
outliers, quartiles and
inter-quartile range)
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