## tutoring

# 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)