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# Mathematics

The Mathematics department aims to promote the appreciation of the power and beauty of Mathematics within each student and to enable each student to develop their mathematical skills and understanding to the limit of their capability. As mathematics is an essential prerequisite for many other subjects within the school curriculum, the department aims to provide each student with the mathematical background that they require to be able to apply their mathematics successfully in other areas. Also it aims to ensure that all students are aware of the power of a mathematical argument and the need for clear, logical arguments. In an academic school, the need for each student to attain the highest exam result possible for their ability is very important. With this in mind the department tries to ensure that all students are fully aware of the demands of the subject and have experienced a variety of teaching styles to facilitate their learning and can apply their mathematics in examination situations.

## Staff

• Dr David Crawford, M.A., D.Phil (Oxon: Jesus), M.Ed. (Bristol), M.A. Ed. (O.U.) M.Sc. (O.U.) Head of Mathematics
• Miss Lorraine Howd
• Mrs Kate Hyde
• Mr Graham Inchley
• Mr Carl  James
• Mrs Roxanne King
• Miss Hannah McCleery
• Mr Neil Murray
• Mrs Kerry Pollard
• Mrs Zoe Village

## Year 6

Pupils in Year 6 are taught in a mixed ability form group (or groups). They follow the KS2 National Curriculum in general with an emphasis on formal written methods. The aim of the year is to bridge the gap from the mathematics they have studied in their many different primary schools to the rigour required in the senior school. They are prepared for a SATs style exam at the end of the year. The following material will be covered:

• Number - Using and applying number through problem solving and communicating using precise mathematical language and reasoning. Understanding of numbers and the number system including place value, factors, multiples, prime numbers, fractions, ratios, percentages, integers and negative numbers. Calculations using the four rules of arithmetic with whole numbers and decimals.
• Algebra - Using formulae in words and symbols. Introduction to simple linear equations.
• Shape and Space - Names of 2D and 3D shapes. Correct use of geometrical notation and vocabulary. Properties of shapes including parallel and perpendicular lines and knowledge of basic angle facts. Use of a protractor to draw and measure angles accurately. Reflective and rotational symmetry and simple transformations. Finding the perimeter and area of simple 2D shapes. Understand the concept of volume. Units of measurement including units of length, mass, capacity and time.
• Handling Data - Interpreting data from charts and graphs. Collection and display of data using frequency tables, bar charts and pie charts. Calculation of mean, mode, median and range for a set of numbers. Introduction to probability.

## Year 7

Pupils in Year 7 are taught in mixed ability form groups. The aim of the year is to ensure that all pupils have covered the basic building blocks of mathematics in a rigorous way. The following material will be covered:

• Number - Four rules of arithmetic with positive whole numbers, negative numbers, fractions and decimals. Converting between fractions and decimals. Finding the Highest Common Factor and Lowest Common Multiple of sets of numbers.
• Algebra - Solution of linear equations (including those with the variable on both sides). Simplification of additive expressions. Substitution of whole numbers into simple formulae. Introduction to set notation.
• Shape and Space - Plane geometry (including angle problems involving triangles, quadrilaterals and parallel lines). Construction of triangles using ruler, compasses and protractor. Co-ordinates in 4 quadrants. Area of rectangles, triangles, parallelograms and compound shapes.
• Handling Data - Creating and interpreting bar charts and pie charts. Finding the mean and range of a set of numbers.

## Year 8

Pupils in Year 8 (and upward) are taught in 6 or 7 ability sets. The initial setting arrangement is based on performance in Year 7 but is reviewed regularly to ensure that all pupils are being taught at a level and pace appropriate to their individual needs with extension/revision as required. The following material will be covered:

• Number - Evaluation of terms in index form involving positive and negative integral indices. Multiplication and division of terms in index form. Conversion of numbers to and from standard form. Percentages (including simple percentage increase and decrease). Simple calculations involving ratio and proportion.
• Algebra - Multiplication of a bracket by a single term. Linear equations involving brackets. Multiplication of two brackets. Substitution into formulae. Drawing straight line graphs from a formulae. Finding the gradient of a line between two points.
• Shape and Space - Angle properties of polygons. Pythagoras’ Theorem. Construction of angle and line bisectors and special angles using compasses and ruler. Area and circumference of a circle and of simple sectors. Volumes of cuboids and compound cuboids. Volumes of prisms (including cylinders). Similar triangles.
• Handling Data - Probability (including sample space diagrams and simple tree diagrams). Mean, median and mode of a set of values. Problems involving means.

## Year 9

The following material will be covered:

• Number - Evaluation of terms in index form involving fractional and negative indices. Manipulation of terms in index form. Standard form and calculations involving standard form. Percentage increase and decrease and inverse percentages.
• Algebra - Solving linear equations involving brackets. Creating and rearranging formulae. Solving a pair of linear simultaneous equations. Factorisation of quadratic expressions. Solution of quadratic equations by factorisation. Solving linear inequalities.
• Shape and Space - Trigonometry – finding lengths and angles in right-angled triangles. Circle theorems.
• Handling Data - Calculating means from frequency tables and grouped frequency tables. Drawing histograms.

## GCSE

In Years 10 and 11, pupils study for the Edexcel IGCSE in Mathematics. All pupils take Higher Level and achieve great success with an average of 58.6% of all pupils obtaining A* (or 8/9) grades and 85.6% obtaining at least an A (or 7) grade over the last 5 years.

As well as extending the ideas covered during Years 7 to 9, the following new material will be covered over the course of the two years:

• Number - Set Theory. Simplification and manipulation of surds. Upper and lower bounds.
• Algebra - Solution of quadratic equations using different methods. Simplification and manipulation of algebraic fractions. Drawing and using graphs to solve equations. Sequences. Introduction to calculus. Function notation and finding composite and inverse functions. Vectors. Solving quadratic inequalities.
• Shape and Space - Calculation of Volumes and Surface areas of common solid shapes. Tangent properties of circles. Areas and Volumes of similar shapes. Trigonometry in non right-angled triangles. Transformations.
• Handling Data - Cumulative frequency curves and calculations.

For the last eight years, the pupils in the top two sets have also had the chance to sit the AQA Level 2 Certificate in Further Mathematics which provides greater challenge for able mathematicians without accelerating pupils into A level work. The results they have achieved have been very pleasing with 110/315 obtaining A** grades and 147/315 obtaining A* grades.

Those taking this qualification also cover the following material:

• Algebra - Quadratic identities. Completing the square. The factor theorem. Drawing piecewise continuous graphs and using such graphs to solve equations. Algebraic proof. Limiting values of sequences. Further calculus. Introduction to matrices and matrix transformations.
• Shape and Space - Equations of circles. Solving geometric problems involving circles and straight lines. Geometric proof. Three dimensional trigonometry. Graphs of trigonometric functions. Solving trigonometric equations. Simplification and application of trigonometric identities.

## A Level

Mathematics is the most popular subject in the Sixth Form with between 60 and 90 students opting to study it every year. It is also one of the most successful with an average of 29.7% of students obtaining A* grades and 82.4% A* to B grades over the last five years.

An A Level in Mathematics is a very valuable qualification and is highly rated by both university tutors and employers.  Any student who has enjoyed IGCSE (or GCSE) Mathematics and who feels they would benefit from being challenged by the more in-depth material of A Level Mathematics should consider taking the subject in the Sixth Form.  As with any other A Level subject, it should be noted that A Level Mathematics requires hard work and dedication and students should not opt for the subject unless they are prepared to apply themselves fully.

An A grade/ grade 7 at IGCSE/GCSE is a minimum requirement, along with a very good level of fluency with algebra.

All students taking Mathematics at A-level will have 8 or 9 lessons in the Lower Sixth and 9 lessons in the Upper Sixth split between Pure Maths (6 lessons) and Applied Maths (2/3 lessons).

The course followed at Leicester Grammar School is the Edexcel A-level.  From September 2017, A-level Mathematics is a linear course with three equally weighted exams taken at the end of the Upper Sixth determining the grade a student obtains.  Unlike previous A-level specifications, there is now no longer any choice in the Applied Mathematics that a student will study.

The tables below show some details of the topics that are in each of the three A-level exam papers.

Pure Papers 1 & 2
2 hours

Proof – Proof by deduction and exhaustion, disproof by counter example.  Further proof by deduction, proof by contradiction.

Algebra and Functions – Indices, surds, quadratic equations and graphs, discriminants, simultaneous equations and inequalities, graphs and graph transformations, factor theorem.  Simplifying rational functions, modulus function, composite and inverse functions, partial fractions, modelling using functions.

Co-ordinate Geometry – Equations of lines and circles, parallel and perpendicular lines.  Parametric and Cartesian equations of curves, modelling using parametric equations.

Sequences and Series – Binomial theorem for integer n.  Binomial theorem for rational n, arithmetic and geometric sequences, sigma notation, iterative formulae.

Trigonometry – Sine and cosine rules, graphs of the 3 basic trigonometric functions, simple equations and identities.  Radian measure, small angle approximations, exact values of trigonometric functions, reciprocal and inverse trigonometric functions, compound and double angle formulae, further equations and identities, trigonometric proofs.

Exponentials and Logarithms – Graphs of exponential functions and their transformations, logarithms and the laws of logarithms, solving exponential equations, using logarithmic graphs to estimate parameters, modelling using exponential growth/decay.

Differentiation – Differentiation from first principles, differentiation of powers of x and constant multiples, sums and differences, application of differentiation to tangents and normals, maxima and minima, increasing and decreasing functions.  Differentiation from first principles for trigonometric functions, differentiation of exponential functions, the product and quotient rules, related rates of change, parametric and implicit differentiation, forming simple differential equations.

Integration – Indefinite integration of xn (n ≠ - 1) and constant multiples, sums and differences, definite integration and link to areas under curves.  Integration of exponential and trigonometric functions, finding the area between two curves, integration by substitution and by parts, integration using partial fractions, solution of first order differential equations and interpretation.

Numerical Methods – Root location using change of sign, root location using iteration, Newton-Raphson method, numerical integration.

Vectors – 2-dimensional vectors, magnitude and direction, vector addition/subtraction and scalar multiplication, use of vectors to solve problems (including in Mechanics).  3 dimensional vectors.

Applied Paper
2 hours

### Statistics:

Statistical Sampling – Different sampling techniques and their limitations.

Data Presentation and Interpretation – Histograms, skew, scatter diagrams and correlation.  Measures of central tendency and spread, outliers.

Probability – Mutually exclusive events, independent events, conditional probability, use of tree diagrams and Venn diagrams, modeling with probability.

Statistical Distributions – Discrete probability distributions, the binomial distribution, the Normal distribution,  approximation of a binomial by a Normal.

Statistical Hypothesis Testing – Binomial hypothesis testing, Normal hypothesis testing.

### Mechanics:

Quantities and Units in Mechanics – Use of S.I. units and conversions.

Kinematics – Travel graphs, constant acceleration formulae, calculus in kinematics, motion in a vertical plane including projectiles.

Forces and Newton’s laws – Use of Newton’s second law in a straight line, resolving forces, motion under gravity, equilibrium and Newton’s third law, resultant forces, friction.

Moments – Equilibrium of rigid bodies.

Calculators:

Calculators may be used for all three papers.  The exam board has specified that calculators used for A-level must include an iterative function and the ability to compute summary statistics and access probabilities from standard statistical distributions.

Setting:

Mathematics is fortunate in that all classes occur in the same option block.  Hence it is possible to set by ability.  It is likely that between 4 and 6 ability sets will be used in each year, usually with between 8 and 14 students in each.

## Further Mathematics

To study Further Mathematics in the Sixth Form, a student needs to have real enthusiasm for the subject as twice as much lesson time (and twice as much self-study time) will be spent on Mathematics as compared to other subjects.  The most able mathematicians who have found IGCSE/GCSE Mathematics straightforward and who are considering taking university courses in Mathematics, Engineering, Computer Programming or the Physical Sciences should consider this option.  As a large part of the Sixth Form timetable will be spent on Mathematics, the coverage of material is very rapid and so a firm grasp of the algebraic skills learnt at IGCSE/GCSE is needed.  An A* grade/grade 8 or 9 at IGCSE/GCSE should be a requirement for anyone wishing to follow a course in Further Mathematics.

Students opting for Further Mathematics at Leicester Grammar School with have 16 lessons in the Lower Sixth split between Pure Maths (11 lessons) and Applied Maths (5 lessons).  The intention is that Further Mathematics students will take A-level mathematics at the end of the Lower Sixth.  In the Upper Sixth, students will have 18 lessons split between Pure Maths (13 lessons) and Applied Maths (5 lessons), although this may depend on any options chosen.

Unlike A-level Mathematics, while half of the course content is compulsory, there is an element of choice in A-level Further Mathematics.  It may be the case that students within the class (or classes) have some firm ideas about the areas they wish to pursue and, if this is possible from a timetabling standpoint, there will be the possibility of different students studying different topics.  However, this will always be at the discretion of the Head of Mathematics, the Curriculum Deputy and the Headmaster in consultation with the subject teachers.

The table below shows some details of the topics that will be in the two compulsory exam papers together with a brief outline of the most likely optional papers to be taken.

 Further Mathematics Core Pure 1 1½ hours Proof – Proof by induction, including series sums and divisibility. Complex Numbers – Arithmetic with complex numbers, conjugate pairs, Argand diagrams, modulus argument form, loci in Argand diagrams, solving quadratic, cubic and quartic equations. Matrices – Addition, subtraction and multiplication by a scalar, matrices and transformations, invariant points and lines, determinants, inverse matrices,  using matrices to solve simultaneous equations. Further algebra and functions – Relationships between roots and coefficients of polynomial equations, linear transformations of polynomial equations, summation of series using standard formulae. Further calculus – Volumes of revolution using Cartesian or parametric equations. Further vectors – Vector and Cartesian equations of lines and planes in 3D, scalar product, intersection of lines and planes. Further  Mathematics Core Pure 2 1½ hours Complex numbers – De Moivre’s theorem and applications, complex roots of unity and application to geometric problems. Further algebra and functions – Method of differences for summation of series,  Maclaurin series. Further calculus – Mean value of a function, integration of partial fractions with irreducible quadratic denominators, differentiation of inverse trigonometric functions, integration using trigonometric substitutions. Polar co-ordinates – Conversion between polar and Cartesian co-ordinates, sketching curves expressed in polar for, finding the area enclosed by a polar curve. Hyperbolic functions – Graphs of hyperbolic functions, differentiation and integration of hyperbolic functions, inverse hyperbolic functions. Differential equations – Integrating factors, general and particular solutions, modeling using differential equations, second order differential equations and auxiliary equations, simple harmonic motion, damped oscillations, coupled first order differential equations. Optional  Further  Mathematics Further Pure 1 1½ hours Further calculus – Taylor series, series expansion and limits, Leibnitz’ theorem, Weierstrass substitution for integration. Further differential equations – Taylor series and differential equations, reducible differential equations. Co-ordinate systems – Parametric and Cartesian equations of the parabola, ellipse and hyperbola, focus and directrix, eccentricity, tangents and normals, loci. Further vectors – Vector product, triple scalar product, applications of vectors to 3-D geometry. Further numerical methods – Numerical solution of first and second order differential equations, Simpson’s rule. Inequalities – Solution of inequalities involving fractions and modulus. Optional  Further Mechanics Paper 1 1½ hours Momentum and Impulse – Conservation of momentum in direct contacts, impulse. Collisions – Direct impact of elastic particles, Newton’s law of restitution, energy loss in impact, repeated impacts. Centres of mass – Centre of mass of a discrete mass distribution, centre of mass of plane figures, centre of mass of a framework, equilibrium of a lamina or a framework. Work and energy – Kinetic and potential energy, work and power, the work-energy principle, conservation of mechanical energy. Elastic strings and springs – Hooke’s law, energy stored in an elastic string or spring. Optional  Further  Statistics Paper 1 1½ hours Linear regression – Least squares regression lines, residuals. Statistical distributions (discrete) – Mean and variance of a discrete probability distribution, Mean and variance of functions of a variable, the Poisson distribution and its additive properties, mean and variance of the binomial and Poisson distribution, use of the Poisson distribution as an approximation to the binomial. Statistical distributions (continuous) – Random variables, probability distribution functions and continuous distribution functions, mean and variance of continuous variables and functions of continuous variables, the continuous uniform distribution. Correlation – Calculation of correlation coefficients, coding, Spearman’s rank correlation coefficient. Hypothesis testing – Hypothesis test for the mean of a Poisson distribution, hypothesis test for zero correlation. Chi squared tests – Goodness of fit tests and contingency tables, degrees of freedom. Optional Decision Mathematics Paper 1 (1½ hours) Algorithms and Graph Theory – Application of algorithms defined by flow chart or text, Bin packing and sorting algorithms, the planarity algorithm. Algorithms on Graphs – Prim’s and Kruskal’s algorithms, Dijkstra’s algorithm, the Route Inspection Algorithm, the Travelling Salesman algorithm, the Nearest Neighbour algorithm. Critical Path Analysis – Precedence tables and activity networks, latest and earliest start and finish times for activities, float, construction of Gantt charts, construction of resource histograms, scheduling. Linear Programming – Formulation of problems as linear programs, slack, surplus and artificial variables, Graphical solution of two variable problems using vertex method or objective line method, the Simplex algorithm for maximising problems with ≤ constraints, the two stage Simplex and big M methods for maximising and minimising problems with both ≤ and ≥ constraints.

## Co-Curricular

The Mathematics Department believes that all pupils should be able to think about how to apply their knowledge of mathematical topics to solve problems. All pupils have the chance to take part in the UKMT Mathematics Challenges (All Year 8 and selected Year 7 in the Junior Challenge, all Year 10 and top set Year 9 in the Intermediate Challenge and all Lower Sixth Maths students and any Upper Sixth Maths students who opt to take it in the Senior Challenge). The department also sends teams to the Team Maths Challenge competitions and in 2012/13 our Junior Team qualified for the National Final and our Year 10 team qualified for the Midlands Final, in 2013/14 our Senior Team qualified for the National Final and our Year 10 team won the Midlands Final and in 2015/16, our Senior Team again qualified for the National Final. In 2018/19 our Junior Team qualified for the National Final.

## Other

The department organises a Year 6 Team Maths Challenge for local primary schools each November.

The department is an Institutional member of the Mathematical Association and is represented on both the UKMT Council and the UKMT Challenges Committee.