# Key Stage 5

## Maths A' Level

##### Intent

Our A-level course takes students on a journey from the topics they have met at GCSE into more complex ideas, some familiar, some brand new, such as exponentials and logarithms, integration and parametric equations. We also study the mathematics of movement of objects in the mechanics section and look more in depth at probability and perform hypothesis tests in the statistics section. Throughout the course proof and problem solving will be interwoven.

##### Year 12 Learning Journey

**Quadratic Functions, Equations and Inequalities**

- Solve and sketch any quadratic including hidden quadratics
- Completed the square form
- Solve quadratic inequalities
- Discriminant
- Simultaneous equations

**Problem Solving and Proof**

- Solving problems
- Use mathematical symbols and notation including implication
- Formal methods of proof

**Polynomials**

- Add subtract and multiply any two polynomials
- Factor theorem and algebraic division
- Know shape and features of graphs of polynomials

**Graphs**

- Transformations of graphs
- Reciprocal graphs
- Direct proportion graphs
- Inequalities on graphs

**Differentiation**

- Differentiation from first principles
- Differentiate polynomials
- Equation of tangents and normal to curves
- Increasing and decreasing functions
- Finding and classifying stationary points

**Surds and Indices**

- Structure of the number system
- Add, subtract and multiply surds
- Rationalise the denominator
- Use all laws of indices
- Write numbers as a power of another
- Solve problems using indices

**Trigonometry**

- Sine, cosine and sine rule for area
- Graphs of trigonometric functions
- Trig identities
- Solve equations involving trig functions

**Coordinate Geometry **

- Distance between two points and midpoint
- Equation of a line in a given format
- Equation of parallel and perpendicular lines
- Equation of a circle
- Equation of the tangent to the circle
- Simultaneous equations with circles

**Binomial Expansion**

- Binomial expansion to find terms
- Use binomial coefficient to find unknowns
- Use binomial expansion when two binomials are being multiplied
- Use expansions to approximate

**Differentiation**

- Differentiation from first principles
- Differentiate polynomials
- Equation of tangents and normal to curves
- Increasing and decreasing functions
- Finding and classifying stationary points

**Exponentials and Logarithms**

- Sketch exponential functions
- Derivative of e^kx
- Use laws of logarithms
- Use logarithms to solve exponential equations
- Sketch logarithmic graphs
- Modelling with exponentials
- Logarithms to transform graphs

**Data Collection**

- Data handling cycle
- Different types of data
- Types of sampling

**Data Processing and Interpretation**

- Histograms
- Cumulative frequency
- Variance and standard deviation
- Scatter diagrams
- Outliers

**Binomial Expansion**

**Integration **

- Integrate functions with terms of the form ax^n
- Find constant of integration
- Definite integration for area under a curve
- Area between a curve and a line

**Vectors**

- Change between column vectors and magnitude and direction
- Add and subtract vectors
- Geometric problems with vectors
- Vector proof

**Kinematics and Variable Acceleration**

**Probability**

- Calculate probabilities
- Show two events are independent
- Use tree diagrams and Venn diagrams
- Discrete probability distributions

**Binomial Distribution**

- Formula for number of combinations
- Properties of Binomial distribution
- Calculate probabilities with Binomial distribution

**Hypothesis Testing**

- One and two tailed hypothesis tests with Binomial distribution

**Large Data Set**

**Kinematics and Variable Acceleration **

- Distance-time, displacement-time and velocity-time graphs
- Derive constant acceleration formulae
- Use constant acceleration formulae
- Simultaneous equations and the constant acceleration formulae
- Effect of assumptions
- Link between acceleration, velocity and displacement
- Solve problems with variable acceleration

**Forces and Motion**

- Force diagrams
- Newton’s laws
- Types of forces
- Solve problems with Newton’s laws
- Modelling assumptions

##### Year 13 Learning Journey

Functions

- Types of functions
- Graphs of functions including discontinuous
- Domain and range
- Composite and inverse functions
- Modulus function
- Solve equations/ inequalities with the modulus function
- Graph transformations

Differentiation

- Differentiate lnx, sinx, cosx, tanx
- Chain rule
- Product and Quotient rule
- Implicit differentiation
- Differentiate reciprocal trig functions
- Convex and concave curves
- Connected rates of change

Trig Identities

- Compound angle formulae
- Double angle formulae
- New identities and proof
- Form Rcos(x+c) and Rsin(x+d)

Integration

- Integrate Exponential and Trigonometric functions
- Area between two curves
- Reverse chain rule
- Integration by substitution
- Integrate with partial fractions
- Integration by parts
- Integration and trig identities

Proof

- Proof by exhaustion
- Direct proof
- Proof by contradiction

Trigonometry

- Radians
- Graphs of Trig functions
- Area and arc length of a sector
- Are of a segment
- Small angle approximations

Trig Functions

- Reciprocal trig functions and their graphs
- Derive new identities
- Inverse trig functions
- Solve trig equations with reciprocal trig functions

Further Algebra

- Factor theorem with (ax+b)
- Remainders with algebraic division
- Partial fractions
- Expand (ax+b)^n

**Sequences and Series**

- Different types of sequences and series
- Convergence or divergence
- Sigma notation
- Arithmetic series
- Geometric series

**Parametric Equations**

- Graphs of parametric equations
- Convert to Cartesian form
- Differentiate and find gradients
- Integrate and find areas

**Differential Equations**

- Separation of variables
- General solution
- Particular solution
- Forming differential equations

**Vectors and Kinematics**

- 3D vectors, magnitude and direction
- Use vectors in kinematics (suvat and F=ma)

**Forces. Motion and Friction**

- Resolve forces on inclined plane
- Friction
- Solve problems involving friction and inclined planes

**Moments**

- Moment on uniform rods and lamina
- Equilibrium and moments

**Projectiles**

- Resolve vertically and horizontally
- Solve projectile problems
- Equation of trajectory

**Numerical Methods**

- Change of sign to find a root
- Newton Raphson
- Staircase and cobweb diagrams
- Using rectangles to estimate area
- Trapezium rule

**Probability **

- Set notation to describe probability
- Conditional probability formula
- Solve problems with conditional probability

**Stats Distributions**

- Normal distribution finding probabilities
- Reverse normal distribution
- Standardise to find unknowns
- Approximate binomial distribution

**Hypothesis Testing**

- Normal distribution when using a sample
- Hypothesis testing with normal distribution
- Product moment correlation coefficient

**Revision for Exams and Gap Fill **

**Examinations**

## Further Maths A' Level

##### Intent

Further Maths A-level allows students to go deeper into the mathematical world. We will build on knowledge from GCSE and A-level maths throughout. There is the opportunity to cover more deeply the mechanics of how things move but also to look at some of the more recent mathematical developments in discrete mathematics.

##### Year 12 Learning Journey

**Complex Numbers**

- Extend number system
- Solve quadratics with complex roots
- Complex conjugate
- Argand diagram
- Modulus argument form
- Loci of complex numbers

**Sequences and Series**

- Standard results for sum of integers, squares and cubes
- Method of differences
- Maclaurin series standard results

**Proof by Induction**

- Proof by induction with series
- Proof by induction with matrices
- Proof by induction with divisibility

**Polar Coordinates**

- Convert between polar and cartesian coordinates
- Sketch polar curves
- Tangents at the poles

**Vectors**

- Vector and Cartesian form of a 3D straight line
- Intersection point between two lines
- Angle between two lines
- Parallel and perpendicular lines
- Shortest distance

**Hyperbolic Functions**

- Exponential form of hyperbolics
- Graphs, domains and ranges of hyperbolics
- Inverse hyperbolics
- Solve equations with hyperbolics

**Conics**

- Equations and graphs of conic sections
- Equation of tangents
- Intersection of lines with conic curves
- Transformations

**Matrices and Transformations**

- Multiplication of matrices
- Associativity and commutativity
- Determinant of a matrix
- Inverse of 2x2 matrices
- Matrix transformations
- Geometric interpretations
- Invariant points and lines

**Rational Functions**

- Sketch graphs of form (ax+b)/(cx+d) or (ax^2+bx+c)/(dx^2+ex+f)
- Inequalities using graph
- Inequalities creating cubics/quartics
- Possible value of a function and stationary point

**Roots of Polynomials**

- Factorise and solve polynomials with complex roots
- Relationship between roots and coefficients
- Linear transformation of roots
- Complex roots with real coefficients

**Graphs**

- Language and terminology of graphs
- Euler’s formula for connected planar graphs
- Adjacency matrices

**Networks**

- Terminology of networks
- Kruskal’s algorithm
- Primm’s algorithm
- Route inspection problems
- Chinese postman problem
- Nearest neighbour algorithm

**Linear Programming**

- Create linear programming problem
- Graph inequalities
- Solve optimisation problems

**Critical Path Analysis**

- Construct precedence networks
- Latest start time and earliest finish times
- Refine models and effect

**Network Flows**

- Interpret flow problems
- Terminology
- Find cuts
- Max flow-minimum cut theorem
- Supersource and supersink

**Game Theory **

- Construct payoff matrices
- Play safe strategies
- Value of a game
- Stable solution
- Dominated strategies
- Optimal mixed strategies

**Forces, Motion and Friction **

- Force diagrams and Newton’s laws of motion
- Equilibrium
- Friction
- Vectors in two dimensions

**Kinematics**

- Derive constant acceleration formulae
- Use constant acceleration formulae
- Effect of assumptions
- Solve problems with variable acceleration

**Work, Energy, Power and Hooke's Law**

- Work done
- Kinetic and gravitational potential energy
- Conservation of energy
- Work done by a variable force
- Hooke’s law

**Momentum and Collisions**

- Momentum and impulse
- Impulse of a variable force
- Conservation of momentum
- Newton’s experimental law

**Binary Operations**

- Construct payoff matrices
- Play safe strategies
- Value of a game
- Stable solution
- Dominated strategies
- Optimal mixed strategies

**Further Calculus**

- Proof of volume of a revolution
- Find volumes about x and y axes
- Mean value of a function

**Vectors **

- Vector product and application
- Equation of a plane in vector and Cartesian format
- Ways planes can intersect
- Angles between lines and planes

**Polar Coordinates**

- Area enclosed by polar curves

**Momentum and collisions**

- Momentum and impulse
- Impulse of a variable force
- Conservation of momentum
- Newton’s experimental law

**Dimensional Analysis**

- Dimensional analysis notation
- Validity of formulae
- Predict formulae

**Circular Motion**

- Forces in circular motion
- Linear speed and angular speed
- Acceleration
- Solve problems

**Further Graphs and Inequalities**

- Modulus reciprocal graphs and inequalities
- Draw reciprocal graphs with obliques asymptotes
- Sketch and solve equations and inequalities with y=|f(x)|

**Conics**

- Equation of conics after combinations of transformations
- Find combinations of transformations applied
- Parametric form of conics

##### Year 13 Learning Journey

**Matrices**

- Determinant and inverse of a 3x3 matrix
- Row/column operation
- 3 variable simultaneous equations
- Eigenvalues and eigenvectors

**Complex Numbers**

- De Moivre’s theorem
- Exponential form of a complex number
- Roots of unity
- Factorise z^n-a
- De Moivre’s theorem to derive trig identities

**Hyperbolic Functions**

- Inverse hyperbolic functions
- Reciprocal hyperbolic functions
- Hyperbolic identities
- Differentiation of hyperbolics
- Integration of hyperbolics and inverse hyperbolics

**Further Calculus**

- Differentiate inverse trig functions
- Inverse trig and hyperbolic functions with substitution
- Partial fractions
- Reduction formulae
- Length of an arc
- Surface area of revolution

**Series and Limits**

- Maclaurin’s series
- L’Hopital’s rule
- Improper integrals and limits

**Differential Equations**

- Integrating factor
- 2nd order differential equations

**Application of Differential Equations**

- Simple harmonic motion
- Damping
- Coupled first order differential equations

**Numerical Methods**

- Mid-ordinate rule
- Simpson’s rule
- Euler’s method
- Euler’s improved method

**Work, Energy and Power**

**Momentum and Collisions**

**Dimensional Analysis**

**Cicular Motion**

- Problems with horizontal circular motion
- Variable speed
- Vertical circle
- Solve problems

**Moments and Couples**

- Solve problems with moments
- Vector method

**Centre of Mass**

- Centre of mass of particles
- Centre of mass of standard shapes and composite bodies
- Integration to find centre of mass
- Equilibrium, sliding and toppling

**Graphs**

**Linear Programming**

- Simplex algorithm
- Interpret a simplex tableau

**Critical Path Analysis**

- Gantt Chart
- Resource histogram
- Resource levelling

**Network Flows**

- Augment flows
- Upper and lower arc capacity problems
- Refine network flows

**Game Theory**

- Optimal mixed strategies with simplex algorithm
- Solve higher order games

**Groups**

- Language of groups
- Group axioms
- Finite and infinite groups
- Legrange’s theorem
- Isomorphisms

**Revision and Gap Filling**

**Examinations**

## Mathematics News

- Reading in Maths

Do you want to learn about mathematical mistakes? (22/02/2021) - UKMT Senior Challenges

UK Mathematical Trust Girl’s Mathematical Olympiad and Senior Mathematics Challenges (18/11/2019) - NFS and UKMT

Planning for further success in 2019-20 (24/09/2019)