Mathematics
SUBJECT overview
The overarching themes in A level mathematics are applied along with associated mathematical thinking and understanding across the specification. These overarching themes are inherent throughout the content, and learners must develop skills in working scientifically throughout the qualification. The skills show teachers which skills need to be included as part of the learning and assessment of the learners.
Overarching theme 1: Mathematical argument, language and proof
Overarching theme 2: Mathematical problem solving
Overarching theme 3: Mathematical modelling
Mathematics can be applied in practical tasks, real life problems and within mathematics itself. The aim of the course is to develop mathematical vocabulary, improve mental calculation and use a range of methods of computation and apply these to a variety of problems.
The course of study should help you whether working individually or collaboratively to reason logically, plan strategies and improve your confidence in solving complex problems.
During Maths lessons you will learn how to:
 Use and apply maths in practical tasks, real life problems and within mathematics itself.
 Develop and use a range of methods of computation and apply these to a variety of problems.
 Develop mathematical vocabulary and improve mental calculation.
 Consider how algebra can be used to model real life situations and solve problems.
 Explore shape and space through drawing and practical work using a range of materials and a variety of different representations.
 Use statistical methods to formulate questions about data, represent data and draw conclusions.
The overarching themes in A level mathematics are applied along with associated mathematical thinking and understanding across the specification. These overarching themes are inherent throughout the content, and learners must develop skills in working scientifically throughout the qualification. The skills show teachers which skills need to be included as part of the learning and assessment of the learners.
Overarching theme 1: Mathematical argument, language and proof
Overarching theme 2: Mathematical problem solving
Overarching theme 3: Mathematical modelling
Mathematics can be applied in practical tasks, real life problems and within mathematics itself. The aim of the course is to develop mathematical vocabulary, improve mental calculation and use a range of methods of computation and apply these to a variety of problems.
The course of study should help you whether working individually or collaboratively to reason logically, plan strategies and improve your confidence in solving complex problems.
During Maths lessons you will learn how to:
 Use and apply maths in practical tasks, real life problems and within mathematics itself.
 Develop and use a range of methods of computation and apply these to a variety of problems.
 Develop mathematical vocabulary and improve mental calculation.
 Consider how algebra can be used to model real life situations and solve problems.
 Explore shape and space through drawing and practical work using a range of materials and a variety of different representations.
 Use statistical methods to formulate questions about data, represent data and draw conclusions.
Engage in practical and experimental activities in order to appreciate principles of probability. There is no coursework
Year 12  Mechanics
The mechanics part of the AS level aims to introduce learners to key ideas about modelling, motion and forces. In the autumn term, learners are given a solid foundation in the knowledge and skills needed to succeed in mechanics at AS level. The Spring term begins by studying calculus before using it in the context of mechanics to solve kinematics problems involving variable acceleration. By the end of the spring, learners should be able to solve kinematics and dynamics problems in 1 dimension. They will understand the concept of a model and be confident in interpreting the results of their calculations to make sensible predictions about motion in a given time period.
Any time mathematics is applied to solve a problem in real life, a model must be created first. The term begins by considering mathematical modelling, and the compromise between accuracy and complexity of different types of models.
The next three units focus in depth at three different but mutually supporting parts of mathematics; vectors, the constant acceleration formulae, and forces. These form the foundations of kinematics and dynamics and are built upon later in the year and in Year 13. In addition to allowing learners to practice constructing and using models to solve real life kinematic problems, it gives the opportunity to develop mathematical and critical thinking skills about the appropriateness of a given model, and giving sensible interpretation to results.
Calculus is of enormous importance to mathematics and the real world. Rates of change are used in fields as diverse as finance, the military, aerospace, engineering and geography to model and make predictions. Learners will familiarise themselves with the concepts of derivatives and integrals and then use them to solve problems involving a range of mathematical models.
Later in the Spring, learners will be brought back to the context of mechanics and use this knowledge to understand, model and predict the motion of objects that move under variable acceleration. This provides an opportunity to revisit the fundamentals studied in the autumn term, as well as the chance to derive the constant acceleration formulae using calculus.
Year 12  Statistics
Statistics aims to introduce learners to the study of the collection, organisation, analysis, interpretation, and presentation of data. It deals with all aspects of data, including planning its collection in terms of the design of surveys and experiments.
In the autumn term, learners are given a solid foundation in the knowledge and skills needed to succeed in Statistics at AS level. For example, they are taught how sampling can be a valuable tool to collect and evaluate information about a large population or universe when it would otherwise be impractical (or impossible) to collect that information from the entire population.
Learners finish the term by learning how data presentation and interpretation has many uses. For example, it is used for all kinds of questions where statistical inference is appropriate, including hypothesis testing and experimentation of all types, machine learning algorithms.
The Spring term begins by studying the use of simple, discrete probability distributions (calculation of mean and variance of discrete random variables is excluded), including the binomial distribution, as a model leading to calculating probabilities using the binomial distribution.
Later in the Spring, learners will be introduced to the process of conducting a statistical hypothesis test for proportions in the binomial distribution and interpreting the results in context.
This process will lead them to understand that a sample is being used to infer the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.
Year 12  Mathematics Pure
The AS Mathematics Pure content is sequenced to allow learners the opportunity to develop knowledge and skills both within the Pure syllabus itself but also to apply these in the Mechanics and Statistics parts of the course. Mathematical modelling plays a key role in all parts of the pure course. Our main aim is to ensure the learners have the appropriate pure skills before being asked to apply them. For example, learners will have been taught calculus before facing variable acceleration problems in Mechanics. We regularly reaffirm with the learners that the most of the topics are not standalone but very much linked particularly with the applied parts of the course. Learners are assessed and reassessed if necessary after some intervention at the end of each topic.
The course begins with learners being given the opportunity to embed algebraic knowledge and skills from the GCSE syllabus including indices, surds, simultaneous & quadratic equations, transformations of graphs as well as straight line graphs. They are also introduced to calculus, specifically differentiation, which also gives them the opportunity to apply some of these basic algebraic skills. Examples of this would be finding the equation of a tangent or normal to a curve and manipulating equations when solving constant acceleration problems. Integration is then covered in the latter part of the Autumn Term in preparation for variable acceleration taught in the Spring Term.
Trigonometry is then introduced in the Spring term focussing on the ratios, graphs, formulae, identities and solving equations using the CAST diagram. This supports the learners when resolving forces later on in the Spring term. Learners are also taught the equation of a circle and vectors concurrently during this term. The former builds on the idea of transformations of graphs while the latter prepares learners for the vectors sections in the Mechanics syllabus.
The AS Pure syllabus concludes with those topics that are not necessarily essential for the applied parts of the course. The exponential and logarithmic graphs including solving equations is related to the indices work covered earlier in the year. The Binomial expansion, the factor theorem and algebraic proof are introduced towards the end of the course as these are then developed early on in Year 13.
After our learners have completed their examinations in June, we begin teaching the A2 content. This begins with developing algebraic proofs and extending learners’ skills with transformations of graphs by introducing the modulus function. Learners are provided with online work over the summer based on individual areas of weakness.
Year 13  Statistics
Teaching for year 13 starts in the summer term. The focus will be on understanding the use of Normal distribution as a model and finding probabilities using the Normal distribution link to histograms, mean, standard deviation, points of inflection, and the binomial distribution.
Bell curve grading assigns relative grades based on a normal distribution of scores.
As year 13 reconvene in the autumn term, they will learn how to conduct a statistical hypothesis test for the mean of the Normal distribution with known, given or assumed variance and interpret the results in context. Learners will learn that many scores are derived from the normal distribution, including percentile ranks ("percentiles" or "quantiles"), normal distribution curve equivalents, stanines, zscores, and Tscores. Additionally, some behavioural statistical procedures assume that scores are normally distributed, for example, ttests and ANOVAs.
The Spring term begins by studying the use of mutually exclusive and independent events when calculating probabilities. Learners will be able to link this to discrete and continuous distributions and understand and use conditional probability, including the use of tree diagrams, Venn diagrams, twoway tables.
Later in the Spring term, learners will be introduced to applying the language of statistical hypothesis testing and 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 pvalue or critical value (calculation of correlation coefficients is excluded). In context, the capital asset pricing model uses linear regression and the concept of beta for analysing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets.
Year 13  Mathematics Pure
We make it very clear to learners in Year 13 that the A2 course is very much a development of the skills and knowledge taught in Year 12. It is important that they don’t somehow disassociate the 2 courses thinking that Year 13 is a set of ‘new’ topics. Just like in Year 12, the Pure course is sequenced to allow learners the opportunity to develop knowledge and skills both within the Pure syllabus itself but also to apply these in the Mechanics and Statistics parts of the course. Mathematical modelling again plays a key role in all parts of the pure course in Year 13. Just as in Year 12, learners are assessed and reassessed if necessary after some intervention at the end of each topic.
Parametric Equations and Trigonometry are taught in the Autumn Term concurrently with Differentiation which is then followed by Integration. The learners need to know the purpose of the new Trig functions (SEC, COSEC and COT) before having to differentiate them. An emphasis is placed upon the fact that the same skills are applied here for these new functions as that in Year 12 when solving equations. This helps build the learners’ confidence as they are often daunted by what might seem like completely new topics. Similarly, they need to be able to manipulate parametric equations and link them to Cartesian equations before applying them in problems involving Calculus. Vectors are taught in the latter part of the term, again, with a focus on linking them to the applied part of the course in Mechanics.
The remainder of the Pure syllabus is covered in the Spring term to allow time for revision and further intervention before the examination season begins. The work covered on the Binomial Expansion in Year 12 is developed further by looking at negative and fractional powers. Partial fractions are also covered at this point to enable the learners to integrate certain functions using logs. The term is concluded with the Sequences & Series and Numerical Methods. Learners use algebraic skills covered earlier in the course when proving the sum of arithmetic and geometric series before applying these to some reallife problems. Numerical Methods gets the learners to think of alternative methods to solving equations. This builds on simple ‘trial and improvement’ and iteration covered in GCSE to location of roots using the Newton Raphson Method.
UNIT OVERVIEW  AUTUMN 1  MECHANICS
Unit 5: Forces and Friction 

Skills 

Knowledge 
By the end of the unit, students should:

Rationale 
Newton’s laws summarise and govern the basic interactions between objects and forces. At their most fundamental level, forces tell us how different objects interact. This can be viewed on a macro scale (such as the forces between a car and the road), or on a micro scale (such as looking how the forces between atoms cause the atoms to form an object). As such, knowledge of forces and how they work has application to chemistry, physics, engineering, sport, architecture and building, and any other field the deals with one object interacting with another. For example, structural materials need to be tested to decide if they are appropriate for certain situations. As such they need to be subjected various shear and compressive/stretching forces to determine the grade or material. 
UNIT OVERVIEW  AUTUMN 1  PURE
Units 5, 6, and 7: Radians and Trigonometry 

Skills 
2 i.e. sin θ ≈ θ, cos θ ≈ 1 – ^{8} , tan θ ≈ θ where θ is in radians. 2
6 4 3 2 and exact values of tan for 0 , ^{n}, ^{n}, ^{n}, ^{n}, π and multiples thereof. 6 4 3 2
B) and tan (A ± B); understand geometrical proofs of these formulae.
R cos (θ ± α) or R sin (θ ± α) .

Knowledge 
By the end of the unit, students should:

UNIT OVERVIEW  AUTUMN 2  MECHANICS
Unit 7: Applications of forces 

Skills 
extend to situations where forces need to be resolved (restricted to 2 dimensions).
in a straight line; application to problems involving smooth pulleys and connected particles; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces.

Knowledge 
By the end of the unit, students should:

Rationale 
Newton’s laws summarise and govern the basic interactions between objects and forces. At their most fundamental level, forces tell us how different objects interact. This can be viewed on a macro scale (such as the forces between a car and the road), or on a micro scale (such as looking how the forces between atoms cause the atoms to form an object). As such, knowledge of forces and how they work has application to chemistry, physics, engineering, sport, architecture and building, and any other field the deals with one object interacting with another. For example, structural materials need to be tested to decide if they are appropriate for certain situations. As such they need to be subjected various shear and compressive/stretching forces to determine the grade of material. 
unit overview  autumn 2  pure
Subject: Units 5, 6, and 7: Radians and Trigonometry 

Skills 
2 i.e. sin θ ≈ θ, cos θ ≈ 1 – ^{8} , tan θ ≈ θ where θ is in radians. 2
6 4 3 2 and exact values of tan for 0 , ^{n}, ^{n}, ^{n}, ^{n}, π and multiples thereof. 6 4 3 2
B) and tan (A ± B); understand geometrical proofs of these formulae.
R cos (θ ± α) or R sin (θ ± α) .

Knowledge 
By the end of the unit, students should:

Subject: Unit 11: Integration 

Skills 
s differences and constant multiples. To include integration of standard functions such as sin 3x, sec2 2x, tan x, e5x, ^{1} . 2s Students are expected to be able to use trigonometric identities to integrate, for example, sin2 x, tan2 x, cos2 3x.

Knowledge 
By the end of the subunit, students should:
s ln x;
.
õ f( x)


and the general solution. 
Rationale 
Integration is the second half of calculus, with the first half being differentiation. Integration is essentially the inverse operation of differentiation, and as such is applicable and useful wherever differentiation is. At the most fundamental level, integration finds the area between the graph of the function and the variable axis. Depending on the variables involved in the function, this will tell us different information. For example, for a function of velocity over time, the integral will give us the distance travelled over a given time period. Integration is heavily used in the field of Finance, Physics, Engineering, Chemistry, and almost all other forms of science and economic fields. 
Subject: Unit 12: Vectors (3D) 

Skills 
Knowledge of column vectors and i, j and k unit vectors in three dimensions. 
Knowledge 
By the end of the unit, students should:

Rationale 
A vector is a quantity that has an intrinsic magnitude and direction. Quantities like displacement, velocity, acceleration, force, and momentum are all vectors. When vectors of the same type are applied to the same object they combine in specific ways.
Vectors are utilised in many different fields, such as navigation, computer graphics, and engineering. In Navigation, when bearings are combined with distance they create a vector. Total displacement from a series of directions can then be calculated using vector addition, as well as average speed if the time frame is available.
Vectors are used to resolve forces, and as such are a core part of structural architecture, as buildings need to be in equilibrium. Through vector addition of forces and dispersion of these forces stable structures can be designed. 
unit overview  spring 1  mechanics
Unit 2: Conditional Probability 

Skills 
P(B)

Knowledge 
By the end of the unit, students should:
P(B)

Rationale 
Probability theory is applied in everyday life in risk assessment and modelling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis (Reliability theory of aging and longevity), and financial regulation. A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioural finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. Reference https://en.wikipedia.org/wiki/Probability#Applications 
unit overview  spring 1  pure
Unit 4: The binomial theorem 

Skills 
a 
Knowledge 
By the end of the unit, students should:
b
a 1;

Rationale 
The Binomial theorem, and therefore the binomial expansion is a naturally occurring distribution based off of a system with a fixed probability of success and only two states (success or failure). Even if the system technically has more than two states, it is a helpful tool that can simplify that system down to a success/failure cutoff point. A simple example of this is a dice, which can roll a 1, 2, 3, 4, 5, or 6. If you need to roll a minimum of a 5 for a game, then the success state is 5 or 6, and failure is 1, 2, 3, or 4. At this point the exact number does not matter – only if you “win” the roll or not. Binomial distributions have immediate applications to games of chance, but also to natural occurrences. For example, farmers or animal breeders could use a binomial expansion to calculate the probability of breeding a certain number of offspring with a desired genetic trait. 
unit overview  spring 1  statistics
Unit 2: Conditional Probability 

Skills 
P(B)

Knowledge 
By the end of the unit, students should:
P(B)

Rationale 
Probability theory is applied in everyday life in risk assessment and modelling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis (Reliability theory of aging and longevity), and financial regulation. A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioural finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. Reference https://en.wikipedia.org/wiki/Probability#Applications 
unit overview  spring 2  pure
Unit 10: Numerical methods 

Skills 

Knowledge 
By the end of the unit, students should:

Rationale 
Numerical methods are processed based methods of find approximations of a solution to complex problems. These can be used to get an approximate answer in order to fact check longer solutions, or when automating a problem solving process. For example, in computing and computer science, numerical methods are relatively straightforward to implement as they usually focus on iterative processes. In fact, calculators and mathematical software will often use numerical methods to solve more complicated mathematical operations that would otherwise require abstract or algebraic solutions. 
unit overview  spring 2  statistics
Unit 1: Regression and correlation 

Skills 

Knowledge 
By the end of the unit, students should:

Rationale 
Linear regression is widely used in biological, behavioural and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines. Linear regression is the predominant empirical tool in economics. For example, it is used to predict consumption spending, fixed investment spending, inventory investment, and purchases of a country's exports, spending on imports, the demand to hold liquid assets, labour demand, and labour supply. The capital asset pricing model uses linear regression as well as the concept of beta for analysing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets. Reference https://en.wikipedia.org/wiki/Linear_regression#Applications 
unit overview  summer 2  mechanics
Unit 4: Moments 

Skills 

Knowledge 
By the end of the unit, students should:

Rationale 
Moments are created when a force causes an object to rotate. The size of this moment depends on the size of the force, the angle of application and the distance that the fore was applied from the axis of rotation. Moments are a core concept to the idea of equilibrium, which is when an object is in a completely balanced state. This idea is applied to construction and architecture, where if a building gin not in equilibrium it may become unstable and collapse. Moments are also used in engineering, when a rotating wheel (on a car, for example) will have a moment applied to the axle which will cause the wheel to spin, hence accelerating the car. The size of the moment will determine how fast the car can accelerate. 
unit overview  summer 2  pure
Unit 1 (Part 1): Algebraic and partial fractions 

Skills 

Knowledge 
By the end of the unit, students should:
is a factor of a polynomial;

Rationale 
Algebraic manipulation is a key skill in any further mathematical studies, as well as fields that make frequent use of mathematical models. This includes finance, economics, engineering, and most forms of science. Algebraic and partial fractions is an important part of this algebraic manipulation toolkit. Being able to split and simplify fractions can in turn reduce complicated fractional functions into much easier to solve in implement mathematical models, which can be introduced into Full models linearly in sections rather than all at once. In the field of computer programming (or applied programming to finance or science) it is essential to be able to individually test and isolate different parts of the function. This can be achieved through partial fractions. 
unit overview  summer 2  statistics
Subject: Unit 3: The Normal distribution 

Skills 
probabilities using the Normal distribution
and the binomial distribution.
context, with appropriate reasoning, including recognising when the binomial or the Normal model may not be appropriate.
distribution with known, given or assumed variance and interpret the results in context. 
Knowledge 
By the end of the unit, students should:
distribution.

Rationale 
Measurement errors in physical experiments are often modelled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
Many scores are derived from the normal distribution, including percentile ranks ("percentiles" or "quantiles"), normal curve equivalents, stanines, zscores, and Tscores. Additionally, some behavioural statistical procedures assume that scores are normally distributed; for example, ttests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores. Reference:
https://en.wikipedia.org/wiki/Normal_distribution#Occurrence_and_applications 
knowledge Organiser
A knowledge organiser is an important document that lists the important facts that learners should know by the end of a unit of work. It is important that learners can recall these facts easily, so that when they are answering challenging questions in their assessments and GCSE and ALevel exams, they are not wasting precious time in exams focusing on remembering simple facts, but making complex arguments, and calculations.
We encourage all pupils to use them by doing the following:
 Quiz themselves at home, using the read, write, cover, check method.
 Practise spelling key vocabulary
 Further researching people, events and processes most relevant to the unit.