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A-level Mathematics builds on topics in the GCSE curriculum, such as algebra and trigonometry, and also extends it into new areas such as calculus. Modules in statistics and mechanics help students begin to see practical applications of mathematics that are extremely relevant in society today.

Sparx Maths Summer Transition Booklet

Year 12	Term 1	Term 1.5	Term 2
Development of topic:	Pure Year 1 Ch1: Algebraic Expressions Ch2: Quadratics Ch3: Equations and Inequalities Ch4: Graphs & Transformations Ch5: Straight Line Graphs Ch6: Circles	Pure Year 1 Ch7: Algebraic Methods Ch8: Binomial Expansion Ch9: Trig Ratios Ch10: Trig Identities & Ch11: Vectors Ch12: Differentiation	Pure Year 1 Ch13: Integration Ch14: Exponentials Statistic Ch1: Data Collection Ch2: Location & Spread Mechanics Ch8: Modelling Ch9: Const Acceleration
Previous Knowledge Required:	Year 10 and 11 GCSE Higher Content	Year 10 and 11 GCSE Higher Content + Half Term 1	Year 10 and 11 GCSE Higher Content + Half Term 1,2
Skills Taught:	Pure Year 1 Ch1: Algebraic Expressions Expanding Brackets Factorising Indices Surds Ch2: Quadratics Solving Quadratics Completing the square Functions Quadratic Graphs Discriminant Ch3: Equations and Inequalities Simultaneous Eqn’s Quadratic Sim Eqn’s Linear & Quadratic ineqs Inequalities on graphs Regions Ch4: Graphs & Transformations Cubic graphs Quartic graphs Reciprocal graphs Points of intersection Translations Ch5: Straight Line Graphs y = mx + c Eqns of straight lines Parallel & perpendicular lines Lengths & areas Problem solving Ch6: Circles Midpoint & perp bisectors Eqn of a circle Straight lines & circles Tangents & chords Circles & triangles	Pure Year 1 Ch7: Algebraic Methods Algebraic Fractions Polynomial Division The Factor Theorem Mathematical proof Methods of proof Ch8: Binomial Expansion Pascal’s Triangle Factorial Notation The Binomial Expansion Ch9: Trig Ratios Cosine Rule, Sine Rule Area = ½abSinC Solving problems Trig Graphs Transformations Ch10: Trig Identities & Equations Angles in 4 quadrants Trig identities & equations Equations & Identities Ch11: Vectors Magnitude & direction Position Vectors Geometrical Problems Ch12: Differentiation Gradient of curves Diff’ating xⁿ & quadratics Diff’ating two+ terms Tangents & Normals Increasing/ decreasing fns 2nd derivatives Stationary points	Pure Year 1 Ch13: Integration Integrating x^n Indefinite integrals Finding functions Definite integrals Area under a curve Area under x-axis Area between curve & a line Ch14: Exponentials & Logarithms Exponential fns Y=e^x & Ln(x) Exponential modelling Laws of logs Equations with logs Logs/non-linear data Statistic Ch1: Data Collection Populations & Samples Sampling Non-random sampling Types of data The Large Dataset Ch2: Location & Spread Central tendency Measures of location Measures of spread Variance & Standard deviation Coding Mechanics Ch8: Modelling Constructing a model Assumptions Quantities & units Vectors Ch9: Const Acceleration D-time graphs Velocity-time graphs Const accel 1 Const accel 2 Motion under gravity
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Year 12	Term 2.5	Term 3	Term 3.5
Development of topic:	Statistics Ch3: Representing Data Ch4: Correlation Ch5: Probability Mechanics Ch10: Forces & Motion Ch11: Variable Accel	Statistics Ch6: Binomial Distr. Ch7: Hypothesis Testing Pure year 2 Ch1: Algebraic Methods Ch2: Functions & Graphs	Pure year 2 Ch3: Sequences & Series Ch4: Binomial Expansion Ch5: Radians Ch6: Trig Functions
Previous Knowledge Required:	Year 10 and 11 GCSE Higher Content + Half Term 1,2,3	Year 10 and 11 GCSE Higher Content + Half Term 1,2,3	Year 10 and 11 GCSE Higher Content + Half Term 1,2,3
Skills Taught:	Statistics Ch3: Representing Data Outliers Box plots Cumulative frequency Histograms Comparing data Ch4: Correlation Correlation Linear Regression Ch5: Probability Calculating probability Venn diagrams Mutually exclusive & independent events Tree Diagrams Mechanics Ch10: Forces & Motion Force diagrams Forces as vectors Forces & acceleration Motion in 2D Connected particles Ch11: Variable Accel Functions of time Using differentiation Maxima/minima Using integration Const acceleration	Statistics Ch6: Binomial Distr. Probability distributions Binomial distributions Cumulative probabilities Ch7: Hypothesis Testing Hypothesis testing Critical values One-tailed tests Two-tailed tests *Finish Mech/Stats topics or revise using past papers Pure year 2 Ch1: Algebraic Methods Proof by Contradiction Algebraic Fractions Partial Fractions Repeated Factors Algebraic Division Ch2: Functions & Graphs Modulus Function Functions & Mappings Composite Functions Inverse Functions \|f(x)\| & f(\|x\|) Combining transformations Solving modulus functions	Pure Year 1 Ch3: Sequences & Series Arithmetic Sequences Arithmetic Series Geometric sequences Geometric Series Sum to infinity Sigma notation Recurrence relations Modelling with series Ch4: Binomial Expansion Expanding (1+x)ⁿ Expanding (a+bx)ⁿ Using Partial Fractions Ch5: Radians Radian Measure Arc length Sectors & segments Trig equations Small angle approximations Ch6: Trig Functions sec, cosec, cot graphs sec, cosec, cot using sec, cosec, cot Trig identities Inverse trig functions
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Year 13	Term 1	Term 1.5	Term 2
Key Topic:	Trigonometry Differentiation	Integration	Statistics and Mechanics
Development of topic:	Ch. 7 Trigonometry and Modelling Ch. 8 Parametric Equations Ch. 9 Differentiation	Ch. 10 Numerical Methods Ch. 12 Vectors Ch. 11 Integration	Stats Ch.1 Regression Mechanics Ch. 4 Moments
Previous Knowledge Required:	Y1 A-level Trigonometry Y12 A-level Differentiation	Y1 A-level Integration	Y1 Stats and Mechanics
Skills Taught:	Addition formulae Angle addition formula Double angle formulae Solving trig equations Simplifying acosx ± bsinx Proving trig identities Modelling with trig identities Parametric equations Using trig identities Curve sketching Points of intersection Modelling with parametric equations. Diff’ating sinx & cosx Diff’ating ex, ax, lnx Chain rule Product rule Quotient rule Diff’ating trig functions Diff’ating parametrics Implicit diff’ation 2nd derivatives Rates of change	Locating roots Iteration Newton-Raphson Application 3D coordinates Vectors in 3D Vector problems Applications to mechanics Integrating Standard functions Integrating f(ax + b) Using trig identities Reverse chain rule Substitution By parts Partial fractions Finding areas Trapezium rule Differential equations Modelling with Diff Eqns	Exponential models Correlation Hypothesis testing Moments Resultant moments Equilibrium Centres of mass Tilting
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Year 13	Term 2.5	Term 3
Key Topic:	Statistics and Mechanics	Statistics and Mechanics Revision
Development of topic:	Stats Ch. 2 Conditional Probability Ch.3 Normal Distribution Mechanics Ch.5 Forces and Friction Ch.6 Projectiles Ch.7 Applications of Forces	Mechanics Ch.8 Further Kinematics Revision and Exams
Previous Knowledge Required:	Y1 Stats and Mechanics	Y1 Mechanics
Skills Taught:	Set notation Conditional probability Venn diagrams Formulae Tree diagrams The normal distribution Finding probabilities Inverse normal dist. Standard normal dist. Finding µ Ѳ Approx. binomial dist. Hypothesis test for the normal distribution Resolving forces Inclined planes Friction Horizontal projection Horiz/vert projection Projection at angle Projectile motion Formulae Static particles Modelling with statics Friction and static particles Static rigid bodies Dynamics & inclined planes Connected articles	Vectors in kinematics Vector methods with projectiles Variable acceleration in 1 dimension Differentiating vectors Integrating vectors
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Further Maths

Further Maths Transition Booklet

Year 12 - Further Maths	Term 1	Term 1.5	Term 2
Key Topic:	Complex Numbers and Matrices	Series Polynomials Proof	Trigonometry Graph Theory
New Knowledge:	Ch. 1 Complex Numbers Ch. 2 Argand Diagrams Ch. 6 Matrices Ch.7 Linear Transformations	Ch. 3 Series Ch. 4 Roots of Polynomials Ch. 8 Proof by induction Ch. 9 Vectors Ch. 5 Volumes of Revolution	Pure Book Y2: Ch. 6 Trigonometric Functions Ch. 7 Trigonometry and modelling Decision Maths: Ch. 3 Algorithms on Graphs Ch. 2 Graphs and Networks Ch.4 Route Inspection Ch. 1 Algorithms
Previous Knowledge Required:	GCSE level Trigonometry, Pythagoras, Surds	Solving quadratic equations	Y1 A-level trigonometry
Skills Taught:	Imaginary & complex numbers Multiplying complex numbers Complex conjugation Roots of quadratics Solving cubics & quartics Argand diagrams Modulus argument Modulus argument form Loci Regions Multiplication Determinants Inverting 2x2’s Inverting 3x3’s Solving equations Transformations in 2D Reflections & rotations Enlargements & stretches Successive transformations Transformations in 3D	Sum of natural numbers Sum of squares & cubes Roots of quadratics Roots of cubics Roots of quartics Roots of a polynomial Transformations Introduction Divisibility Matrices Eqn of a straight line in 3D Eqn of a plane in 3D Dot product Angles between lines & planes Points of intersection Perpendiculars Spin around the x-axis Spin around the y-axis Adding & subtracting volumes Modelling	sec, cosec, cot graphs sec, cosec, cot using sec, cosec, cot Trig identities Inverse trig functions Addition formulae Double angle formulae Solving trig equations Simplifying acosx ± bsinx Kruskal’s Prims Prims & Matrix Dijkstra’s Modelling with graphs Graph Theory Types of graphs Graphs & networks & Matrices Eulerian Graphs Route inspection algorithm Algorithms Flow charts Sorting algorithms Bin-packing Order of an algorithm
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Year 12 - Further Maths	Term 2.5	Term 3	Term 3.5
Key Topic:	Differentiation	Integration	Complex Numbers Method of Differences
Development of topic:	Pure Book Y2: Ch. 9 Differentiation Decision Maths: Ch. 6 Linear Programming Ch. 8 Critical Path Analysis	Pure Book Y2: Ch. 11 Integration Decision Maths: Ch. 5 Travelling Salesman Ch 9. Simplex Algorithm	Pure Y2 Book: Ensure all prerequisites for Y2 are covered Y2 Core FM Book: Ch. 1 Complex Numbers Ch. 2 Method of Differences
Previous Knowledge Required:	GCSE algebraic manipulation including indices	Y12 A-level integration	Y1 A-level FM Complex Numbers
Skills Taught:	Differentiating sinx & cosx Differentiating ex and lnx The chain rule The product rule The quotient rule Differentiating trig functions Implicit Differentiation Linear programming problems Graphical methods Locating the optimal point Solutions with integer values Modelling a project Dummy activities Early and late event times Critical activities The float of an activity Gantt charts	Integrating Standard functions Integrating f(ax + b) Using trig identities Reverse chain rule Substitution By parts Partial fractions Differential equations Classical & practical TSM problems Min spanning tree for upper bound Min spanning tree for lower bound Nearest Neighbour for upper bound Formulating problems Simplex method Integer solutions 2-stage simplex method *The Big-M	Exponential form Multiplying and dividing complex numbers De Moivres Theorem Trigonometric Identities Sums of series Nth roots of a complex number Solving geometric problems The method of differences Higher derivatives Maclaurin series Series expansion of compound functions
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Year 13 - Further Maths	Term 1	Term 1.5	Term 2
Key Topic:	Polar Coordinates	Differential Equations	Conic Sections
New Knowledge:	Ch3. Methods in Calculus Ch.4 Volumes of Revolution Ch. 5 Polar Coordinates Ch. 6 Hyperbolics	Ch.7 Differential Equations Ch. 8 Modelling with DEs FP1 Ch. 1 Vectors	Ch. 2 Conic Sections 1 Ch. 3 Conic Sections 2 Ch. 4 Inequalities Ch. 5 t-formulae Ch 6. Taylor Series
Previous Knowledge Required:	Y1 FM Calculus Y1 A-level Trigonometry	Y1 and 2 A-level calculus Y1 A-level vectors	A-level Y1 and 2 Trigonometry Modulus function
New Skills:	Improper integrals The Mean Value Theorem Diff’ating inverse trig functions Integrating inverse trig functions Integrating with partial fractions Rotating around the x-axis Rotating around the y-axis Parametric curves Modelling Polar coordinates and equations Sketching curves Area enclosed by a polar curve Tangents to polar curves Introduction Inverse hyperbolic functions Identities & equations Diff’ating hyperbolic functions Integrating hyperbolic functions	1st order ODE’s 2nd order homogeneous ODE’s 2nd order non-homogeneous ODE’s Using boundary conditions Modelling with 1st order ODE’s Simple harmonic motion Damped & forced harmonic motion Coupled first order simultaneous eqns] Cross product Areas Scalar triple product Straight lines Geometrical problems	Parametric equations Parabolas Rectangular hyperbolas Tangents & normals Loci Ellipses Hyperbolas Eccentricity Tangents & normal (ellipses) Tangents & normal (hyperbola) Loci Algebraic Methods Using Graphs to solve inequalities Modulus inequalities The t-formulae Applying t-formulae Solving trig equations Modelling Taylor series Limits Diff’ntl equations
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Year 13 - Further Maths	Term 2.5	Term 3
Key Topic:	Calculus	Revision of all topics
Development of topic:	Ch. 7 Methods in Calculus Ch.8 Numerical Methods Ch.9 Reducible Differential Equations
Previous Knowledge Required:	Previous work on Differential Equations
Skills Taught:	Leibnitz Thm L’Hospital’s Rule Weierstrass substitution 1st order ODE’s 2nd order ODE’s Simpson’s rule 1st order RDE’s 2nd order RDE’s Modelling
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