## tutoring

# A-level Maths

There has been some big changes made to AS and A Level Maths in recent years. Maths remains one of the most essential subjects for many higher education courses and in future careers.

**Paper 1**

What’s assessed Any content from:

- Proof
- Algebra and functions
- Coordinate geometry
- Sequences and series
- Trigonometry
- Exponentials and logarithms
- Differentiation
- Integration
- Numerical methods How it’s assessed
- Written exam: 2 hours
- 100 marks
- 33⅓ % of A-level

Questions

A mix of question styles, from short, single-mark questions to multi-step problems.

**Paper 2**

What’s assessed Any content from Paper 1 and content from:

- Vectors
- Quantities and units in mechanics
- Kinematics
- Forces and Newton’s laws
- Moments How it’s assessed
- Written exam: 2 hours • 100 marks
- 33⅓ % of A-level

Questions

A mix of question styles, from short, single-mark questions to multi-step problems.

**Paper 3**

What’s assessed Any content from Paper 1 and content from:

- Statistical sampling
- Data presentation and interpretation
- Probability
- Statistical distributions
- Statistical hypothesis testing

How it’s assessed

- Written exam: 2 hours
- 100 marks
- 33⅓ % of A-level

Questions

A mix of question styles, from short, single-mark questions to multi-step problems.

**Subject Content**

**Mathematical argument, language and proof**

- Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable.
- Understand and use mathematical language and syntax as set out in the content.
- Understand and use language and symbols associated with set theory, as set out in the appendices.
- Apply to solutions of inequalities and probability.
- Understand and use the definition of a function; domain and range of functions.
- Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics.

**Mathematical problem solving **

- Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved.
- Construct extended arguments to solve problems presented in an unstructured form, including problems in context.
- Interpret and communicate solutions in the context of the original problem.
- Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy.
- Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods.
- Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle.
- Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics.

**Mathematical modelling**

- Translate a situation in context into a mathematical model, making simplifying assumptions.
- Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the student).
- Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the student).
- Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate.
- Understand and use modelling assumptions.

**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).

**Algebra and functions **

- Understand and use the laws of indices for all rational exponents.
- Use and manipulate surds, including rationalising the denominator.
- 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.
- 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.
- Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem.
- Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only).
- Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function
- Understand and use proportional relationships and their graphs.
- Understand and use composite functions; inverse functions and their graphs.
- Understand the effect of simple transformations on the graph of y = f x including sketching associated graphs.
- 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).
- Use of functions in modelling, including consideration of limitations and refinements of the models.

**Coordinate geometry in the ( x , y ) plane**

- Understand and use the equation of a straight line.
- 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.
- Completing the square to find the centre and radius of a circle.
- 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.

**Sequences and series**

- Understand and use the binomial expansion
- Work with sequences including those given by a formula for the n th term and those generated by a simple relation, increasing sequences; decreasing sequences; periodic sequences.
- Understand and use sigma notation for sums of series.
- Understand and work with arithmetic sequences and series, including the formulae for n th term and the sum to n terms.
- Understand and work with geometric sequences and series including the formulae for the n th term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series.
- Use sequences and series in modelling.
- Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle.
- Work with radian measure, including use for arc length and area of sector.
- Understand and use the standard small angle approximations of sine, cosine and tangent.
- Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity.
- Know and use exact values of sin and cos for 0, and multiples thereof, and exact values of tan for 0, and multiples thereof.
- Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains.
- Understand and use the trig identities connecting tan, sin and cos. Including the Pythagorean identity and the identity with secant, cosecant, and cotangent.
- Understand and use double angle formulae, understand geometrical proofs of these formulae.
- Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.
- Construct proofs involving trigonometric functions and identities.
- Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

**Trigonometry**

Exponentials and logarithms

- Know and use the function and its graph, where is positive.
- Know and use the function and its graph.
- Know that the gradient of is equal to and hence understand why the exponential model is suitable in many applications.
- Know and use the definition of as the inverse of , where a is positive and x ≥ 0
- Know and use the function ln x and its graph.
- Know and use as the inverse function of
- Understand and use the laws of logarithms.
- Solve equations of the form
- Use logarithmic graphs to estimate parameters in relationships of the form and , given data for x and y .
- 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.

**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 ); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x and for sin x and cos x.
- Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection.
- Differentiate , for rational values of n , and related constant multiples, sums and differences.
- Differentiate and related sums, differences and constant multiples.
- Understand and use the derivative of .
- Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, 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 implicitly or parametrically, for first derivative only.
- Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

**Integration **

- Know and use the Fundamental Theorem of Calculus.
- Integrate (excluding ), and related sums, differences and constant multiples.
- Integrate , , , and related sums, differences and constant multiples.
- Evaluate definite integrals; use a definite integral to find the area under a curve and 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.
- (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae).
- Integrate using partial fractions that are linear in the denominator
- 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.

**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.
- Understand how such methods can fail.
- 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.
- Use numerical methods to solve problems in context.

**Vectors**

- Use vectors in two dimensions and in three 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 and kinematics.

**Statistical sampling**

- 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.

**Data presentation and interpretation **

- Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency.
- Connect to probability distributions.
- 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.
- Interpret measures of central tendency and variation, extending to standard deviation.
- Be able to calculate standard deviation, including from summary statistics.
- 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.

**Probability**

- Understand and use mutually exclusive and independent events when calculating probabilities.
- Link to discrete and continuous distributions.
- Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables.
- Understand and use the conditional probability formula.
- Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.

**N: Statistical distributions**

- 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.
- 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.

**Statistical hypothesis testing**

- 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; extend to 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).
- 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.
- Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.

**Quantities and units in mechanics**

- Understand and use fundamental quantities and units in the SI system: length, time, mass.
- Understand and use derived quantities and units: velocity, acceleration, force, weight, moment.

**Kinematics**

- 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; extend to 2 dimensions using vectors.
- Use calculus in kinematics for motion in a straight line.
- Model motion under gravity in a vertical plane using vectors; projectiles.

**Forces and Newton’s laws**

- Understand the concept of a force; understand and use Newton’s first law.
- Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2D vectors); extend to situations where forces need to be resolved (restricted to 2 dimensions).
- Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g , and its value in SI units to varying degrees of accuracy.
- (The inverse square law for gravitation is not required and g may be assumed to be constant, but students should be aware that g is not a universal constant but depends on location).
- 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 2D vectors); application to problems involving smooth pulleys and connected particles; 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.
- Understand and use the model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics.

**Moments**

- Understand and use moments in simple static contexts.

## Assessment details

**Assessment Objectives**

Assessment objectives (AOs) are set by Ofqual and are the same across all A-level Mathematics specifications and all exam boards. The exams will measure how students have achieved the following assessment objectives.

**AO1:** Use and apply standard techniques.

Students should be able to:

- select and correctly carry out routine procedures;

accurately recall facts, terminology and definitions.**AO2:** Reason, interpret and communicate mathematically.

Students should be able to:

- construct rigorous mathematical arguments (including proofs);
- make deductions and inferences;
- assess the validity of mathematical arguments;
- explain their reasoning;
- use mathematical language and notation correctly.

Where questions/tasks targeting this assessment objective will also credit students for the ability to ‘use and apply standard techniques’ (AO1) and/or to ‘solve problems within mathematics and in other contexts’ (AO3) an appropriate proportion of the marks for the question/task must be attributed to the corresponding assessment objective(s).

**AO3:** Solve problems within mathematics and in other contexts.

Students should be able to:

- translate problems in mathematical and non-mathematical contexts into mathematical processes;
- interpret solutions to problems in their original context, and, where appropriate, evaluate their accuracy and limitations;
- translate situations in context into mathematical models;
- use mathematical models;
- evaluate the outcomes of modelling in context, recognise the limitations of models and, where appropriate, explain how to refine them.

Where questions/tasks targeting this assessment objective will also credit students for the ability to ‘use and apply standard techniques’ (AO1) and/or to ‘reason, interpret and communicate mathematically’ (AO2) an appropriate proportion of the marks for the question/task must be attributed to the corresponding assessment objective(s).