# Mathematics

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 MATHEMATICS TERMLY OVERVIEWS

## Overview - Year 12 Autumn Term

 Subject: Pure Mathematics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams using correct symbols and language, including set notation. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Pure 01 -Algebraic Expressions: Index laws, manipulation of surds, expanding brackets, collecting like terms, factorising Pure 02 – Quadratics: Solve quadratic equations by factorising, using the formula, and by completing the square; quadratic graphs and their properties; the discriminant Pure 03 – Equations and Inequalities: Simultaneous equations including where one is quadratic; solve linear and quadratic inequalities and represent graphically; and set notation. Pure 05 – Straight Line Graphs: Equation of a straight line; gradients; parallel and perpendicular lines; midpoints Pure 09 – Trigonometric Ratios: Sine, cosine and tangent for all arguments, use sine rule, cosine rules and area of a triangle; graphs of sine, cosine and tangent and their properties. Rationale Algebra is the basis of all higher mathematics. It allows for mathematics to be done with variables in place of numerical values, and so allows for solving and the expression of relationships with regard to these variables. Algebra also develops modelling, logic, and rationalisation skills which can be widely applied to other areas that do not have a direct application of algebra, and are necessary throughout the A-Level Mathematics course. The course begins with learners embedding and developing their algebraic knowledge and skills from the GCSE syllabus in areas including: indices; surds; simultaneous and quadratic equations; and straightline graphs. These key skills are applied throughout the pure, statistics and mechanics elements of the course, so it is essential that learners are confident in these areas.

 Subject: Pure and Statistics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams, graphs and logical deductions using correct symbols and language. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Applied 01 - Data Collection: Populations, samples, census, sampling techniques and their advantages/disadvantages Applied 02 - Measures of Location and Spread: Mean, median, mode, variation, standard deviation, range, interquartile range, inter-percentile range Pure 04 - Graphs and Transformations: Cubic, quartic and reciprocal graphs, intersection points, transformations of graphs Applied 03 - Representations of Data: Outliers, box plots, cumulative frequency diagrams, histograms, comparing data sets Applied 04 - Correlation: Scatter graphs, bivariate data, causation, regression lines Rationale 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. They are taught how and why various sampling techniques are used, how to analyse and interpret data using their calculators as well as written methods, and how to present data in an effective and meaningful way. Learners are introduced to the pure topic of Graphs and Transformations early in the autumn term in order to prepare them with the skills they need to access concepts in other topics such as: gradient functions in Differentiation; transformations of trigonometric functions in Trigonometry; and points of intersection in Circles.

 Subject: Pure and Mechanics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams using correct symbols and language, including set notation. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Applied 08 - Modelling in Mechanics: Mathematical models, SI units, scalar and vector quantities, modelling assumptions Pure 11 - Vectors: Two dimensional vectors, vector arithmetic, magnitude, direction, position vectors, velocity, displacement Applied 09 - Constant Acceleration: Kinematics graphs, constant acceleration formulae, vertical motion due to gravity Applied 10 - Forces and Motion: Resultant forces, Newton’s laws of motion, connected particles Rationale The Mechanics part of the AS level aims to introduce learners to key ideas about modelling and motion. The autumn term begins by considering mathematical modelling, and the compromise between accuracy and complexity of different types of models. Having a firm knowledge of the quantities and units used in mechanics is a necessary part of mechanics as a whole. It allows us to judge if the result gained from a calculation is expected, reasonable or even possible. The interactions between the base units that create more complicated units and variables deepen our understanding of the physical processes in place and how they are interconnected. Learners then move to the pure mathematics topic of Vectors which enables them to consolidate their learning at GCSE and further this learning to new concepts including displacement and velocity. Vectors is taught at this point in the term as vectors are used throughout the remaining mechanics topics. It is therefore essential that students are able to manipulate and perform calculations with vectors. Constant Acceleration gives learners the opportunity to develop mathematical and critical thinking skills about the appropriateness of a given model and giving sensible interpretation to results. Kinematics is the field of mechanics that deals with moving objects. Kinematics is therefore a core part of mechanics as a whole, and to the entire field of physics. Kinematics is also a logical and consistent framework while helps to develop problem-solving and modelling skills. Learners develop the knowledge they have gained in Vectors and Constant Acceleration and apply it to further scenarios in Forces and Motion. Having studied mechanical modelling at the start of the autumn term, learners are better equipped with the skills to set up force diagrams. The learning of mechanics is interweaved with the pure elements throughout the course as we want learners to see mechanics as an application of Mathematics, rather than a stand-alone, additional topic.

## Overview - Year 12 Spring Term

 Subject: Pure Mathematics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams using correct symbols and language, including set notation. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Pure 10 – Trigonometric Identities and Equations: Angles in all four quadrants, exact values, trigonometric identities, trigonometric equations Pure 11 – Exponentials and Logarithms: Exponential functions, exponential modelling, logarithms, laws of logarithms, natural logarithms, solving equations using logs, non-linear data Rationale With their knowledge of trigonometric ratios, learners now have the knowledge to be able to solve trigonometric equations and understand how trigonometric identities are used. Learners are taught how the angles in all four quadrants are related using a CAST diagrams which is used to deepen their understanding of where the trigonometric identities come from and why they are multiple solutions to trigonometric equations. This method can also support learners when resolving forces in the later mechanics topics. Using their knowledge of indices from the autumn term learners can be introduced to exponentials and logarithms. Exponentials and logarithms are naturally occurring functions which are used to model and understand real-world patterns and problems. With the knowledge and applications learners gain from this topic, they are able to access a number of Year 13 topics.

 Subject: Pure and Statistics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams, graphs and logical deductions using correct symbols and language. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Pure 06 - Circles: Midpoints, perpendicular bisectors, equation of a circle, intersections of straight lines and circles, tangents, chords Applied 05 - Probability: Calculating probabilities, Venn diagrams, mutually exclusive events, independent events Applied 06 - Statistical Distributions: Probability distributions, binomial distribution, cumulative probabilities Applied 07 - Hypothesis Testing: Binomial hypothesis testing, critical values, one-tailed tests, two-tailed tests Pure 08 - Binomial Expansion: Pascal’s triangle, factorial notation, binomial expansion, binomial estimation Rationale Learners begin the spring term with Circles which builds on ideas developed during earlier topics such as straight-line graphs, and equations and inequalities. It is taught at the start of the spring term in order to provide essential knowledge which learners will need for Differentiation which is taught slightly later in the term. The statistics element of the course resumes with 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 build on the probability skills they have learned previously and are 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. Binomial Expansion is taught at the end of the syllabus as a stand-alone topic as it is not a prerequisite to any applied or other pure topics. The concepts introduced in these topics are developed early on in Year 13.

 Subject: Pure and Mechanics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams using correct symbols and language, including set notation. Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. Understand, interpret and extract information from diagrams to solve problems. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Pure 07 - Algebraic Methods: Algebraic fractions, dividing polynomials, factor theorem, mathematical proof, methods of proof Pure 12 - Differentiation: Gradients of curves, the derivative, differentiating polynomials, tangents and normal, increasing and decreasing functions, second order derivatives, stationary points, gradient functions Pure 13 - Integration: Integrating polynomials, indefinite integrals, definite integrals, areas under curves, areas between curves and lines Applied 11 - Variable Acceleration: Functions of time, using differentiation and integration, maxima and minima problems, using integration, constant acceleration formulae Rationale Learners revisit algebra where they develop and apply skills they acquired earlier in the course in Algebraic Expressions and Quadratics such as factorisation and solving quadratics. The proof element of this topic also relies heavily on learners’ previously acquired algebra skills. Learners are then introduced to calculus which forms an essential part of the A-Level course, particularly in the remaining mechanics topics and a number of pure Year 13 topics. Differentiation is taught first where learners link their new learning to their knowledge of tangents and normal from Straight-Line Graphs and Circles. Integration is taught after this, so that it can be introduced and understood as the reverse process of differentiation. Having studied differentiation and integration, learners now have the pure skills to move on to Variable Acceleration. Learners apply their calculus skills to a number of mechanics problems which also link with their learning on constant acceleration in the Autumn term.

## Overview - Year 12 Summer Term

 Subject: Pure Mathematics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams using correct symbols and language, including set notation. Understand and define functions, as well as their domain and range. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Pure 01 – Algebraic Methods: Proof by contradiction, algebraic fractions, partial fractions Pure 02 – Functions and Graphs: Modulus functions, mappings, domain and range, inverse functions, composite functions, combinations of transformations Rationale 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. With that in mind, the pure aspect of the course begins with two topics which have clear links to their learning from Year 12: Algebraic Methods; and Functions and Graphs. Both topics allow learners to develop key skills which will be used in later topics.

 Subject: Pure and Statistics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams, graphs and logical deductions using correct symbols and language. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Applied 01 - Regression and Correlation: Exponential models in bivariate data; coefficients in an exponential model; product moment correlation coefficient; hypothesis test for zero correlation Rationale Learners begin the A2 Statistics content by being introduced to applying the language of statistical hypothesis testing and extending 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). 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.

 Subject: Pure and Mechanics Skills OT1 Mathematical Argument, Language and Proof Construct and present mathematical arguments through appropriate use of diagrams using correct symbols and language, including set notation. OT2 Mathematical Problem-Solving Understand the concept of a mathematical problem-solving cycle and be able to solve problems presented in an unstructured form, clearly communicating solutions in the context of the original problem. Understand, interpret and extract information from diagrams to solve problems. OT3 Mathematical Modelling Translate a situation in context into a mathematical model whilst using appropriate modelling assumptions. Knowledge Applied 04 - Moments: Resultant moments, equilibrium, centres of mass, tilting Rationale Learners begin the Mechanics aspect of the A2 course with Moments. Up until now, learners have only studied situations involving particles. Moments is an opportunity for learners to extend their knowledge to rigid bodies where rotational effects must be considered. Moments provides a clear bridge between the AS and A2 course as learners apply their AS knowledge to new A2 situations.

## Knowledge organisers

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 A-Level 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.