Mathematics

Our A Level Mathematics curriculum is designed to develop confident, resilient and independent mathematicians who are equipped with the knowledge, reasoning skills and problem-solving ability needed for higher education, apprenticeships and a wide range of careers.

The curriculum builds on the foundations established at GCSE, deepening students’ understanding of mathematical concepts whilst encouraging them to make connections between different areas of mathematics. Students develop fluency in algebra, calculus, trigonometry, statistics and mechanics, alongside the ability to apply these skills in unfamiliar contexts. Throughout the course, we place a strong emphasis on mathematical reasoning, communication and the use of precise mathematical language.

Learning is carefully sequenced so that new ideas build logically on prior knowledge. Regular retrieval and interleaving ensure that students revisit key concepts throughout the course, helping them to retain knowledge and recognise links between topics. Students are encouraged to develop perseverance and confidence when tackling challenging problems, reflecting the demands of university study and many STEM professions.

Alongside procedural fluency, students explore real-world applications of mathematics through statistics and mechanics, developing an appreciation of how mathematics is used to model, analyse and solve problems across science, engineering, economics, computing and many other disciplines.

Our curriculum prepares students not only for success in their A Level examinations but also for the mathematical thinking required beyond school. By the end of the course, students will be able to think logically, communicate mathematical ideas effectively and approach unfamiliar problems with confidence and resilience—skills that will benefit them throughout their future education and careers.

Year 12

Term 1

Problem Solving & Proof

Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion

Use methods of proof, including proof by deduction, proof by exhaustion

Disproof by counter example

Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs)

Useful Links

AS level maths – Proof

Surds and Indices

Understand and use the laws of indices for all rational exponents.

Use and manipulate surds, including rationalising the denominator.

Useful Links

AS level maths – Indices

AS level maths – Surds

Quadratic Functions

Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots.

Completing the square

Solution of quadratic equations including solving quadratic equations in a function of the unknown.

Useful Links

AS level maths – Quadratic functions

AS level maths – Quadratic graphs

Equations and Inequalities

Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.

Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions.

Express solutions through correct use of ‘and’ and ‘or’, or through set notation.

Represent linear and quadratic inequalities such as y>x+1 and y>〖ax〗^2+bx+c graphically.

Useful Links

AS level maths – Simultaneous equations

AS level maths – Inequalities

Coordinate Geometry

Understand and use the equation of a straight line, including the forms

y-y_1=m(x-x_1 ) ■(&@&)and■(&@&)ax+by+c=0

Gradient conditions for two straight lines to be parallel or perpendicular.

Be able to use straight line models in a variety of contexts.

Understand and use the coordinate geometry of the circle including using the equation of a circle in the form

(xⓜ-a)^2+(yⓜ-b)^2=r^2

Completing the square to find the centre and radius of a circle.

Use of the following properties:

• the angle in a semicircle is a right angle

• the perpendicular from the centre to a chord bisects the chord

• the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.

Useful Links

AS level maths – Straight Lines

AS level maths – Circles

Trigonometry

Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form

1/2 ab sin⁡C

Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity.

Understand and use tan⁡θ≡sin⁡θ/cos⁡θ

Understand and use 〖sin〗^2⁡θ+〖cos〗^2⁡θ≡1

Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.

Useful Links

AS level maths – Trigonometry

AS level maths – Trigonometric graphs

AS level maths – Solving trigonometric equations

AS level maths – Trigonometric Identities

Polynomials

Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division.

Use of the factor theorem.

Useful Links

AS level maths – Factor theorem

AS level maths – Algebraic division

Assessments:

End of unit written assessment for each topic.

Term 2

Graphs and Transformations

Understand and use graphs of functions.

Sketch curves defined by simple equations including polynomials, y=a/x and y=a/x^2 (including their vertical and horizontal asymptotes).

Interpret algebraic solution of equations graphically.

Use intersection points of graphs to solve equations.

Understand and use proportional relationships and their graphs.

Understand the effect of simple transformations on the graph of y=f(x) including sketching associated graphs: y=af(x), y=f(x)+a, y=f(x+a), y=f(ax)

Useful Links

AS level maths – Graphs of functions

AS level maths – Proportional functions

AS level maths – Transformations of graphs

The Binomial Expansion

Understand and use the binomial expansion of

(aⓜ+bx)^n

for positive integer n.

The notations n!, nCr, and (■(n@r)).

Link to binomial probabilities.

Useful Links

AS level maths – Binomial Expansion

Differentiation

Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y=f(x) at a general point (x,y) including;

the gradient of a tangent as a limit;
interpretation as a rate of change;
sketching the gradient function for a given curve
differentiation from first principles for small positive integer powers of n

Second derivative including understanding and using it as the rate of change of gradient

Differentiate x^n , for rational values of n , and related constant multiples, sums and differences

Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points.

Identify where functions are increasing or decreasing.

Useful Links

AS level maths – Differentiation 1st Principles

AS level maths – Differentiation

AS level maths – Optimisation using differentiation

Integration

Know and use the Fundamental Theorem of Calculus

Integrate x^n (excluding x=-1), and related sums, differences and constant multiples.

Evaluate definite integrals

Use a definite integral to find the area under a curve

Useful Links

AS level maths – Integration

AS level maths – Definite Integrals

Vectors

Use vectors in two dimensions

Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form.

Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.

Understand and use position vectors

Calculate the distance between two points represented by position vectors.

Use vectors to solve problems in pure mathematics and in context, including forces.

Useful Links

AS level maths – Vectors

Exponentials and logarithms

Know and use the function a^x and its graph, where a is positive.

Know and use the function e^x and its graph.

Know that the gradient of e^kx is equal to 〖ke〗^kx and hence understand why the exponential model is suitable in many applications.

Know and use the definition of 〖log〗_a x as the inverse of a^x where a is positive and x≥0

Know and use the function ln x and its graph

Know and use ln x as the inverse function of e^x

Understand and use the laws of logarithms:

〖log〗_a x + 〖log〗_a y= 〖log〗_a (xy)
〖log〗_a x – 〖log〗_a y= 〖log〗_a (x/y)
〖klog〗_a x = 〖log〗_a x^k

(including k=-1 and k=-1/2 )

Solve equations of the form a^x=b

Understand and use exponential growth and decay

Use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth)

Consideration of limitations and refinements of exponential models

Useful Links

AS level maths – Exponential functions

AS level maths – Logarithms

AS level maths – Linear relationships

Assessments:

End of unit written assessment for each topic.

Term 3

Data Collection

Understand and use the terms ‘population’ and ‘sample’

Use samples to make informal inferences about the population

Understand and use sampling techniques, including simple random sampling and opportunity sampling

Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population.

Useful Links

AS level maths – Statistical sampling

Data processing, presentation and interpretation

Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency

Connect to probability distributions

Interpret measures of central tendency and variation, extending to standard deviation

Be able to calculate standard deviation, including from summary statistics

Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded)

Understand informal interpretation of correlation

Understand that correlation does not imply causation

Recognise and interpret possible outliers in data sets and statistical diagrams

Select or critique data presentation techniques in the context of a statistical problem

Be able to clean data, including dealing with missing data, errors and outliers

Useful Links

AS level maths – Data representation, averages and spread

AS level maths – Data representation scatter graphs

Probability

Understand and use mutually exclusive and independent events when calculating probabilities

Link to discrete and continuous distributions

The binomial distribution

Understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables is excluded), including the binomial distribution, as a model

Calculate probabilities using the binomial distribution

Statistical hypothesis testing using the binomial distribution

Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value

Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context

Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis

Kinematics

Understand and use fundamental quantities and units in the S.I. system: length, time, mass

Understand and use derived quantities and units: velocity, acceleration, force, weight

Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration

Understand, use and interpret graphs in kinematics for motion in a straight line:

displacement against time and interpretation of gradient
velocity against time and interpretation of gradient and area under the graph

Understand, use and derive the formulae for constant acceleration for motion in a straight line

Forces and Newton’s laws of motion

Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy

Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors); application to problems involving smooth pulleys and connected particles

Variable acceleration

Use calculus in kinematics for motion in a straight line:

v=dr/dt , a=dv/dt=(d^2 r)/(dt^2 ) , r=∫▒v dt , v=∫▒a dt

Assessments:

End of unit written assessment for each topic.

Year 13

Term 1

Algebra

Proof by contradiction including ‘Prove root 2 is a rational number’ and ‘Prove an infinite set of primes’

Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only)

The modulus of a linear function

Use of functions in modelling, including consideration of limitations and refinements of the models

Understand and use composite functions

Combinations of transformations (translations and stretches)

Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms

Use parametric equations in modelling in a variety of contexts

Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear)

Useful Links

A2 level maths – Proof by contradiction

A2 level maths – Domain, Range and Inverse functions

A2 level maths – Inverse and reciprocal trig functions

A2 level maths – Rational Expressions

A2 level maths – Modulus function

A2 level maths – Composite functions

A2 level maths – Combinations of transformations

A2 level maths – Parametric equations

A2 level maths – Partial fractions

Sequences

Extend the binomial theorem to any rational n , including its use for approximation

Be aware that the expansion is valid for |bx/a|<1

Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms

Understand and work with geometric sequences and series including the formulae for the nth term and the sum of a finite geometric series

The sum to infinity of a convergent geometric series, including the use of |r|<1

Modulus notation

Understand and use sigma notation for sums of series

Use sequences and series in modelling

Work with sequences including;

Those given by a formula for the nth term
Those generated by a simple relation of the form x_(n+1)= f(x_n )
Increasing and decreasing sequences

Periodic sequences

Useful Links

A2 level maths – Binomial Expansion

Trigonometric identities

Work with radian measure, including use for arc length and area of sector

Know and use exact values of sin and cos for
0 , π/6 , π/4 , π/3 , π/2 , π and multiples thereof

Know and use exact values of tan for 0 , π/6 , π/4 , π/3 , π and multiples thereof

Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan

Understand their relationships to sine, cosine and tangent

Understanding of their graphs; their ranges and domains

Inverse functions and their graphs

Understand and use;

sec^2 θ = 1 + tan^2 θ
cosec^2 θ = 1 + cot^2 θ

Understand and use double angle formulae

Use of formulae for sin⁡(A ± B) , cos⁡(A ± B) and tan⁡(A ± B)

Understand geometrical proofs of these formulae

Understand and use expressions for a cos θ + b sin θ in the equivalent forms of rcos(θ±α) or rsin(θ±α)

Understand and use the standard small angle approximations of sine, cosine and tangent

sin⁡θ≈θ
cos⁡〖θ≈1-θ^2/2〗
tan⁡〖θ≈θ〗

where θ is in radians

Construct proofs involving trigonometric functions and identities

Useful Links

A2 level maths – Exact angles

A2 level maths – Using reciprocal trig functions

A2 level maths – Compound and double angles

A2 level maths – Harmonic form

A2 level maths – Small angles

Assessments:

End of unit written assessment for each topic.

Term 2

Differentiation

Understand and use the derivative of sin⁡x and cos⁡x

The second derivative and its connection to convex and concave sections of curves and points of inflection.

Differentiate e^kx, a^kx , sin⁡kx , cos⁡kx , tan⁡kx and related sums, differences and constant multiples

Understand and use the derivative of ln⁡x

Apply differentiation to find points of inflection

Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions

Differentiate simple functions and relations defined parametrically, for first derivative only

Integrate using parametric equations

Differentiate simple functions and relations defined implicitly for first derivative only

Useful Links

A2 level maths – Differentiation of trig and exponential

A2 level maths – Points of inflection

A2 level maths – Chain rule

A2 level maths – x in terms of y

A2 level maths – Product and Quotient rules

A2 level maths – Implicit differentiation

Integration and differential equations

Integrate e^kx , 1/x , sin⁡kx , cos⁡kx and related sums, differences and constant multiples

Use a definite integral to find the area between two curves

Understand and use integration as the limit of a sum

Carry out simple cases of integration by substitution and integration by parts

Understand these methods as the inverse processes of the chain and product rules respectively

Integrate using partial fractions that are linear in the denominator

Construct simple differential equations in pure mathematics and in context,

(contexts may include kinematics, population growth and modelling the relationship between price and demand)

Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions

(Separation of variables may require factorisation involving a common factor)

Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics

Useful Links

A2 level maths – Integration

A2 level maths – Integration limit as a sum

A2 level maths – Integration by substitution and by parts

A2 level maths – Differential equations

Numerical methods

Locate roots of f(x)= 0 by considering changes of sign of f(x) in an interval of x on which f(x) is sufficiently well-behaved

Understand how change of sign methods can fail

Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams

Solve equations using the Newton-Raphson method and other recurrence relations of the form x_(n+1) = g(x_n )

Understand how such methods can fail

Use numerical methods to solve problems in context

Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between

Useful Links

A2 level maths – Locating roots

A2 level maths – Iterative method

A2 level maths – Newton-Raphson method

Assessments:

End of unit written assessment for each topic.

Term 3

Motion in two dimensions

Understand, use and derive the formulae for constant acceleration for motion in 2 dimensions using vectors

Use calculus in kinematics for motion in 2 dimensions using vectors

Model motion under gravity in a vertical plane using vectors

Projectiles

Understand and use Newton’s second law for motion in situations where forces need to be resolved (restricted to 2 dimensions)

Useful Links

A2 level maths – Kinematics in 2D

A2 level maths – Projectiles

Forces

Use vectors in three dimensions

Use vectors to solve problems in kinematics

Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces

Resolving forces in 2 dimensions

Equilibrium of a particle under coplanar forces

Understand and use addition of forces

Resultant forces

Dynamics for motion in a plane

Coefficient of friction

Understand and use the F ≤ μR model for friction

Motion of a body on a rough surface

Limiting friction and statics

Understand and use derived quantities and units: moment

Understand and use moments in simple static contexts

Useful Links

A2 level maths – Vectors in 3D

A2 level maths – Equilibrium and resolving

A2 level maths – Statics and dynamics

A2 level maths – Moments

Probability and continuous random variables

Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables

Understand and use the conditional probability formula P(A│B)=P(A∩B)/P(B)

Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions

Understand and use the Normal distribution as a model; find probabilities using the Normal distribution

Link to histograms, mean, standard deviation, points of inflection and the binomial distribution

Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate

Useful Links

A2 level maths – Conditional Probability

A2 level maths – Normal distribution

Hypothesis testing

Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context

Understand and apply correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p-value or critical value (calculation of correlation coefficients is excluded)

Useful Links

A2 level maths – Normal distribution hypothesis test

Assessments:

End of unit written assessment for each topic.