As-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 Content from the following sections:

• A: Proof
• B: Algebra and functions
• C: Coordinate geometry
• D: Sequences and series
• E: Trigonometry
• F: Exponentials and logarithms
• G: Differentiation
• H: Integration
• J: Vectors
• P: Quantities and units in mechanics
• Q: Kinematics
• R: Forces and Newton’s laws

How it’s assessed

• Written exam: 1 hour 30 minutes
• 80 marks
• 50% of AS

Questions

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

Paper 2

What’s assessed Content from the following sections:

• A: Proof
• B: Algebra and functions
• C: Coordinate geometry
• D: Sequences and series
• E: Trigonometry
• F: Exponentials and logarithms
• G: Differentiation
• H: Integration
• K: Statistical sampling
• L: Data presentation and interpretation
• M: Probability
• N: Statistical distributions
• O: Statistical hypothesis testing How it’s assessed
• Written exam: 1 hour 30 minutes
• 80 marks
• 50% of AS

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.
• 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.
• Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions.
• 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.

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 graphically
• Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem.
• Understand and use graphs of functions; sketch curves defined by simple equations including polynomials.
• 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.

Coordinate geometry in the ( x , y ) plane

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

Sequences and series

• Understand and use the binomial expansion.
• Link to binomial probabilities.

Trigonometry

• Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle.
• Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity.
• Understand and use the tan, sine and cosine relationship.
• Understand and use
• Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.

Exponentials and logarithms

• Know and use the function and its graph, where a 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  is positive and
• Know and use the function ln x and its graph.
• Know and use ln x 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 and
• Understand and use exponential growth and decay; use in modelling

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 ).
• Differentiate , 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.

Integration

• Know and use the Fundamental Theorem of Calculus.
• Integrate (excluding  ), and related sums, differences and constant multiples.
• Evaluate definite integrals; use a definite integral to find the area under a curve.

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.

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.

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.

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.
• 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.

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.

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.
• Use calculus in kinematics for motion in a straight line.

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).
• 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.

Assessment details

Assessment Objectives

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)