## Mathematics

“The essence of maths is … to make complicated things simple.” (Stanley Gudder)

###### KS3, International GCSE, Additional Mathematics, A-Level, A-Level Further Mathematics, Core Maths

Maths counts, solves problems, and provides a bedrock of logic for analytical thinking. Mathematics uses a universal language of immutable abstract truths to help us master the world around us – its mechanisms, sciences, economies, engineering, technology and design; but also understand its beauty, creativity, people and societies. Using numbers, geometry, algebra, calculus and statistics, maths is about seeking out and recognising patterns and structure. Practically, maths helps us put a price on things and interpret their value, balance budgets, build bridges, plan projects, set up systems, communicate, code computers, analyse data, develop websites, stay secure, make medicines, create graphics, travel the world, expand research, harness energy, evaluate efficiency and measure success. With maths, we can understand the way things change and what stays the same, we can make sense of claims we are told by the media and statistics presented to us at work, we can learn to use algorithms to help us but not control us, and we can even debate more rationally and effectively.

Often, the process of learning maths doesn’t feel simple or intuitive. However, we believe that with thorough explanation, exploration, encouragement and patience, all students can enjoy becoming fluent at the maths they need and want to know. Students are supported to excel at every level, so that they are confident in applying maths, whether in everyday life, or pursuing mathematics, engineering, technology, science or finance at university and beyond.

The learning of mathematics is sometimes described as “hierarchical”, meaning that there are always some concepts that must be mastered, before attempting to understand any more detailed or advanced topics. It is also “spiral shaped”, so as the curriculum goes round, each time a general topic is revisited, the new skills, knowledge and understanding build on what has previously been learnt. (For example, algebra could grow from the concept of finding an unknown number, patterns and sequences in Years 6/7, through to solving equations and then quadratic graphs in key stage 4, and then in key stage 5 up to calculus and beyond!)

**Our curriculum:**

- aims for
**fluency**for every student as they learn each topic, to a level appropriate for them at the time; - remains
**flexible**, accounting for students’ prior knowledge, experience, needs, and progress week by week; and - is
**forward-looking**, preparing students with excellent foundational knowledge and sound understanding for their next stage of learning.

Developing strong mathematical reasoning is integral to all maths teaching and learning at FIC, with students encouraged to recognise connections between different areas of maths. As students’ confidence grows in each topic, they seamlessly progress to solving increasingly complex problems.

**Assessment Progress Map for KS3 and KS4**

**KS3 Maths**

Our KS3 Maths curriculum is largely based on the White Rose scheme of learning, also incorporating some of the Complete Maths curriculum, to inspire students’ confidence and interest in learning, while also meeting all aspects of the National Curriculum.

By the end of KS3, all students have a thorough understanding of at least the following topics:

- place value, multiplying and dividing by multiples of 10;
- calculations and number sense (including the relationship between addition, multiplication and exponents, and when to multiply and divide in problem solving);
- the values of fractions, and how they relate to decimals and percentages;
- basic ratios and scaling;
- simple algebra, patterns and sequences;
- shapes and measurement (including perimeter and area, measuring angles); and
- statistical averages and basic charts.

The following topics are also covered (* denotes Year 9):

- numbers, powers and roots, standard form, number sense and problem solving, finances*, proof*;
- using fractions, decimals and percentages;
- ratio, proportion, scaling, scale diagrams, converting units, compound units and proportionality*;
- algebra, patterns, sequences, factorising, equations, indices, coordinate grids, linear graphs*;
- geometry, shape, space, angles, parallel lines, polygons, circles, symmetry, measure, drawing, constructions*, congruence and similarity*, transformations*, 3D shapes and volume*, Pythagoras*; and
- probability, sets, statistics, averages and range, charts, data handling.

**KS3 Units of Work**

**International GCSE Maths** (Specification A)

**Exam Board**: Pearson Edexcel

At the end of Year 11 (or Year 10 in exceptional circumstances), students take the Higher Level papers, which have an extended syllabus, more difficult questions, and can award up to grade 9. If it is more appropriate, a student may take the Foundation level papers, which award a maximum grade of 5. Students take 2 externally assessed 2-hour exams at the end of the course. Calculators may be used in both.

The IGCSE is very similar to the “English” GCSE syllabus for maths, but the questions are generally written in a style which is slightly more accessible, especially for overseas students and for any students who find processing language more difficult. Also, at the Higher level, the syllabus includes a few extra small topics which form a good foundation for studying A level (for example differentiation and sums of series). Both in the UK and worldwide, the International GCSE is respected on an equally high basis to GCSE.

The IGCSE syllabus requires students to demonstrate knowledge, understanding and skills in the following topics (* denotes Higher level only):

- numbers, number sense, rounding, estimation, error intervals, fractions, decimals, percentages, indices, sets, standard form, finances, problem solving, surds*;
- ratio, proportion, scaling, scale diagrams, converting units, compound units, compound interest and depreciation (exponential growth and decay), proportionality*;
- algebra, notation, index laws, equations, formulae and identities, rearranging and solving, factorising, quadratic equations, inequalities, algebraic fractions*;
- sequences, patterns, coordinates, graphs (linear, quadratic), advanced graphs*, transforming graphs*, gradients*, tangents*, differentiation*, functions*;
- geometry, shape, space, angles, parallel lines, polygons, symmetry, measure, bearings, speed, drawing and construction, 2D and 3D shape properties, congruence and similarity, area and volume scale factors*, transformations (translation, rotation, reflection, enlargement), circle facts, circle theorems*, trigonometry and Pythagoras, sine and cosine rules*, vectors*;
- probability, tables, tree diagrams, Venn diagrams, statistics, data handling, averages and range, charts, histograms* and cumulative frequency*.

**GCSE Units of Work **

**Additional Maths (KS4, Year 11 option)**

From 2022, we will be offering the chance to study for the prestigious Additional Maths qualification at the end of Year 11: Level 3 Free Standing Mathematics Qualification FSMQ

**Exam Board**: OCR

This is a linear course, with 1 externally assessed exam at the end of the course, which is 2 hours long. Calculators are used.

**Syllabus Overview**

Additional Mathematics builds on the higher level GCSE topics, encouraging the development of greater confidence in using algebraic techniques, and extending understanding to include more advanced topics. The course specification states that the qualification “provides a strong mathematical foundation for learners who go on to study mathematics at a higher level” as well as “encouraging learners to recognise the importance of mathematics in their own lives and to society”.

The aims of the course include:

- developing fluent knowledge, skills and understanding of mathematical methods and concepts
- acquiring, selecting and applying mathematical techniques to solve problems
- reasoning mathematically, making deductions and inferences and drawing conclusions
- comprehending, interpreting and communicating mathematical information in a variety of forms appropriate to the information and context
- developing confidence in using mathematical techniques in a variety of ways.

The syllabus covers the following topics:

- Enumeration – binomial expansion, combinatorics, etc
- Coordinate Geometry – straight lines, circles, graphs, linear programming
- Pythagoras and Trigonometry – ratios, identities, equations, 3D
- Calculus – differentiation, integration, kinematics
- Numerical Methods – solving equations, tangents, area under curve, applications
- Exponentials and logarithms

**A Level Mathematics**

A level Mathematics is one of the most highly respected Further Education (level 3) qualifications around the world. It demonstrates strong abilities for learning complex concepts, logical thinking, and advanced problem solving. It is considered virtually essential for studying Science, Engineering, Technology or Computing at university, and is highly valued in all higher education and future work environments, opening up opportunities in almost every career imaginable.

**Exam Board**: Pearson Edexcel

This is a 2-year linear course, with 3 externally assessed 2-hour exams (with calculators) taken during the Summer Term at the end of Year 13. (There are no longer any modules taken in Year 12.) The first two papers (two-thirds of the total marks) test knowledge of all of the Pure Maths topics, and the third paper tests Statistics and Mechanics (one-sixth of the total marks for each).

As an alternative option, AS Level Maths is a one year course, essentially the first half of the A Level syllabus. There are 2 externally assessed exam papers: Pure Maths (2 hours, 5/8 ths of the total marks), and Statistics and Mechanics (1 hour 15 minutes, 3/8 ths of the total marks).

To study A Level Maths, students are expected to have passed GCSE (or equivalent) Maths with at least a Grade 6 (a high B), but a student who is especially committed to working hard may be permitted to start the course with a Grade 5. Throughout the course, students are expected to study for a further 5 hours per week beyond lessons (as well as time during the holidays), to complete homework, practise exam questions, and do their own independent review and consolidation of previous topics.

To prepare for starting A Level Maths, students are encouraged to master their general algebra skills, including factorising, indices, solving equations and rearranging formulae, which underpin every part of Mathematics at an advanced level. A confident working knowledge of quadratic functions, graphs, surds and trigonometry also makes for a good start to the course.

**Assessment Objectives and Overarching Themes**

Around half of the marks in the exams are awarded for using and applying standard mathematical techniques and procedures, and recalling facts and definitions. However, there is now a much greater emphasis on: “reasoning, interpreting and communicating mathematically”, which includes constructing arguments or assessing their validity, and “making deductions and inferences”; as well as problem solving, modelling, turning stories into equations using sensible assumptions to simplify them, and evaluating results and the limitations of models.

A wide range of mathematical skills need to be applied, including the appropriate use of diagrams and graphs, correct use of mathematical language, understanding mathematical arguments and proofs, and understanding the concept of a mathematical problem-solving cycle.

**The Syllabus**

In each of the two years, about half a term is spent studying Statistics, about the same studying Mechanics, and up to two terms on the various topics within Pure Maths.

In Year 12, some of the Pure Maths starts as a gentle step building on the now more difficult GCSE Maths Syllabus:

- Algebra, including indices, surds, quadratics, simultaneous equations and inequalities;
- Sketching graphs, gradients, and solving functions;
- Trigonometry, including the Sine and Cosine Rules; and
- Vectors.

This is then extended to being able to use:

- Binomial expansion, the Factor Theorem, and division of algebra;
- Transforming graphs;
- Circles and Coordinate Geometry;
- Trigonometric Identities and solving Trigonometry equations;
- Magnitude and direction of Vectors, including in 3 dimensions;
- Various types of Proof;
- An introduction to Calculus – Differentiation, gradients, rate of change, stationary points, Integration, and areas under graphs;
- Exponential functions, Logarithms, growth and decay.

Pure Maths in Year 13 then includes:

- Partial Fractions;
- Inverse Functions, Modulus, and Parametric Equations;
- Series and Sequences, including Arithmetic and Geometric Progressions;
- Further Trigonometry, measuring angles in units of Radians, use of the Small Angle Approximation formulae, more identities and multiple angle formulae, Reciprocal and Inverse Trigonometric Functions and their graphs;
- Numerical Methods to find roots of “unsolvable” equations, Change of Sign, Iteration, Newton-Raphson; and
- A much greater knowledge of Calculus – Differentiation, Chain, Product and Quotient Rules, Integration, Substitution, By Parts, Trapezium Rule, and Differential Equations.

Statistics across Years 12 and 13 covers:

- Sampling;
- Presenting and Interpreting Data, Averages and Measures of Spread;
- Probability Distributions, Binomial Distribution, Venn Diagrams and Conditional Probability;
- Regression, Correlation Coefficients;
- The Normal Distribution; and
- Hypothesis Testing.

Mechanics across the two years covers:

- Statics, Newton’s Laws, Resolving Forces including using Vectors, Moments, Friction;
- Kinematics and Dynamics, Movement, Velocity, Constant Acceleration (suvat) equations, Gravity;
- Projectiles; and
- Variable Acceleration using Calculus.

**Units of Work**

**A Level Further Mathematics**

For students who really enjoy maths and want to pursue it to the highest level, Further Maths is the perfect course to study alongside A level Mathematics.

**Exam Board**: Pearson Edexcel

This is a 2-year linear course, with emphasis both on taught lessons in small groups, and students’ self-studying with support. The A level course books published by Pearson Edexcel are followed, supplemented with other resources and practice exam questions, to encourage students’ best possible progress.

During the Summer Term at the end of Year 13, students take 4 externally assessed exams, which are each 1 hour 30 minutes long.

An alternative option, AS Level Further Maths, is essentially one half of the A Level syllabus. There are 2 externally assessed exam papers, which are each 1 hour 40 minutes long.

To study A Level (or AS) Further Maths, students are expected to have passed GCSE (or equivalent) Maths with a Grade 8 or 9 (A*), and be very keen to extend their passion for maths further.

**The Syllabus **(* denotes not in AS level)

All students must complete Core Pure Mathematics, as well as** two **elective modules. For the topics chosen, students must demonstrate that they can: use and apply standard techniques; reason, interpret and communicate mathematically; and solve problems within mathematics and in other contexts.

**Core Pure Mathematics:**

- Complex numbers and Argand diagrams, De Moivre’s theorem*, geometrical problems*
- Polar coordinates*
- Roots of polynomials
- Series, method of differences*, Maclaurin series*
- Proof by induction
- Vectors – lines, planes, scalar product
- Matrices and linear transformations – inverse of 3×3 matrix, simultaneous equations, reflection, rotation and stretch
- Further Integration – volumes of revolution, using parametric equations*, improper integrals*, mean of a function*, inverse trigonometric functions*
- Differential equations*
- Hyperbolic functions*

**Mechanics 1:**

- Momentum and impulse, vector form*
- Work, energy and power
- Elastic collisions in 1 dimension, in 2 dimensions*
- Elastic energy*, strings* and springs*

**Mechanics 2 ***(only if Mechanics 1 has also been selected)***:**

- Motion in a circle, vertical circles*
- Centres of mass, further centres of mass*, toppling and sliding*
- Further kinematics (as a function of displacement, velocity or time)
- Further dynamics*, simple harmonic motion*, energy*

**Further Pure 1:**

- Trigonometry – t-formulae
- Conic sections – parabola, rectangular hyperbola, ellipse*, hyperbola*
- Vectors – cross product, 3D geometry*
- Inequalities, including modulus*
- Numerical methods, Simpson’s Rule*
- Calculus – series expansions, Taylor, Leibnitz, L’Hospital, Weierstrass
- Further differential equations*

**Further Pure 2 ***(only if Further Pure 1 has also been selected)***:**

- Groups, isomorphism*
- Number theory, Fermat’s Little Theorem*
- Matrix algebra – eigenvectors, 3×3 matrices*
- Further complex numbers loci, transformations from z-plane to w-plane*
- Further series, second order recurrence*
- Further integration*

**Statistics 1:**

- Discrete probability distributions
- Poisson and binomial distributions, geometric and negative binomial*
- Chi squared tests
- Hypothesis testing for Poisson and geometric*
- Central limit theorem*
- Probability generating functions*
- Quality of tests*

**Statistics 2 ***(only if Statistics 1 has also been selected)***:**

- Continuous probability distributions
- Linear regression
- Correlation
- Combinations of random variables*
- Estimations, confidence intervals and tests – normal*
- Other hypothesis tests and confidence intervals*
- Confidence intervals using t-distribution*

**Decision Maths (Discrete Maths) 1:**

- Algorithms and graph theory
- Algorithms on graphs
- Critical path analysis
- Linear programming

**Decision Maths (Discrete Maths) 2 ***(only if Decision Maths 1 has also been selected)***:**

- Allocation (assignment) problems, Hungarian algorithm*
- Transportation problems*
- Flows in networks, optimal flow rate*
- Game theory
- Recurrence relations, second order recurrence*
- Dynamic programming*
- Decision analysis*

**Level 3 Certificate in Mathematical Studies (Core Maths)**

**Exam Board**: AQA

This is a 2 year linear course. During the Summer Term at the end of Year 13, students take 2 externally assessed exams, which are each 1 hour 30 minutes long.

We offer AQA Level 3 Mathematical Studies (known colloquially as Core Maths) as an alternative for students who do not choose A level Maths. It is a much more accessible syllabus for students who are not so keen on advanced algebra or trigonometry, but is still a rigorous and demanding course, promoting excellent critical thinking, problem solving and data analysis skills. It has the same UCAS points value as an AS level (approximately half an A level) and is studied in about half the hours of an A level, usually 2 hours per week across 2 years, alongside personal study time. Core Maths helps with understanding applied maths, everyday calculations, estimations, evaluating claims, statistics and finances. It gives students wishing to study psychology or social sciences a great headstart for their future education, as well as being a very useful and highly respected qualification for any future career.

**The Syllabus**

For the following curriculum, students should be able to: use and apply standard techniques; select appropriate techniques to solve problems in a mathematical or non-mathematical context and analyse data and represent situations mathematically; and devise strategies to solve problems where the method is not obvious and communicate processes and results.

- Prior knowledge (revised) – prime factors, indices, standard form, equations, functions, straight line graphs, percentages, use of the multiplier, geometry, perimeter and area, circles, similar shapes, scale factors, converting units, 3D shapes, Pythagoras, bearings;
- Using spreadsheets – basic formulae;
- Analysis of Data – data types, collecting and sampling data, representing data numerically and diagrammatically, correlation;
- Finances – profit and loss, income tax, VAT, interest rate;
- Estimation, Fermi estimation;
- The Modelling Cycle; and
- Critical Analysis of given data and models, presenting logical and reasoned arguments in context, and communicating mathematical approaches and solutions.

And **one** of the following three elective modules:

Statistical Techniques – normal distribution, probabilities and estimation, population and sample, mean of sample size n, confidence intervals;

Critical Path And Risk Analysis – compound projects, activity networks, Gantt charts, cost benefit analysis, expectation, uncertainty, control measures, probabilities; or

Graphical Techniques and Methods – graphs of functions (linear, quadratic, exponential), intersection points and solutions, rates of change, gradient, speed, and acceleration.