Key Stage 5 (A-level) Math Topics
We teach the complete Key Stage 5 Mathematics Syllabus as set out by the UK Department of Education and as detailed by all the applicable exam boards. Our maths tutors will cover the following KS5 topics as required and will tailor each lesson plan to each student depending on the areas they need to improve on. We will continuously test our students’ learning progress by assigning them numerous practice questions to check their understanding and problem solving improvement on topics that are covered. We ensure our students are thoroughly prepared for to progress onto KS5 with the knowledge and confidence they need to do well.
Pure Mathematics 1 (AS- level)
Algebraic Expressions:
Index laws
Expanding brackets
Factorising
Negative and fractional indices
Surds
Rationalising denominators
2. Quadratics
Solving quadratic equations
Completing the square
Functions
Quadratic graphs
The discriminant
Modelling with quadratics
3. Equations and Inequalities
Linear simultaneous equations
Quadratic simultaneous equations
Simultaneous equations on graphs
Linear inequalities
Quadratic inequalities
Inequalities on graphs
Regions
4. Graphs and transformations
Cubic graphs
Quartic graphs
Reciprocal graphs
Points of intersection
Translating graphs
Stretching graphs
Transforming functions
5. Straight line graphs
y = mx + c
Equations of straight lines
Parallel and Perpendicular lines
Length and area
Modelling with straight lines
6. Circles
Midpoints and perpendicular bisectors
Equation of a circle
Intersections of straight lines and circles
Use tangent and chord properties
Circles and triangles
7. Algebraic methods
Algebraic fractions
Dividing polynomials
The factor theorem
Mathematical proof
Methods of proof
8. The binomial expansion
Pascal’s triangle
Factorial notation
The binomial expansion
Solving binomial problems
Binomial estimation
9. Trigonometric ratios
The cosine rule
The sine rule
Areas of triangles
Solving triangle problems
Graphs of sine, cosine and tangent
Transforming trigonometric graphs
10. Trigonometric identities and equations
Angles in all four quadrants
Exact values of trigonometric ratios
Trigonometric identities
Simple trigonometric equations
Harder trigonometric equations
Equations and identities
11. Vectors
Vectors
Representing vectors
Magnitude and direction
Position vectors
Solving geometrical problems
Modelling vectors
Partial fractions
12. Differentiation
Gradients of curves
Finding the derivative
Differentiating x ^ n
Differentiating quadratics
Differentiating functions with two or more terms
Gradients, tangents and normal
Increasing and decreasing functions
Second order derivatives
Stationary points
Sketching gradient functions
Modelling with differentiation
13. Integration
Integrating x ^ n
Indefinite integrals
Finding functions
Definite integrals
Areas under curves
Areas under the x-axis
Areas between curves and lines
14. Exponentials and Logarithms
Exponential functions
y = e ^ x
Exponential modelling
Logarithms
Laws of logarithms
Solving equations with logarithms
Working with natural logarithms
Logarithms and non-linear data
Pure Mathematics 2
Algebraic Methods:
Proof by contradiction
Algebraic fractions
Partial fractions
Repeated factors
Algebraic division
2. Functions and Graphs
The modulus function
Functions and mappings
Composite functions
Inverse functions
Combining transformations
Solving modulus problems
3. Sequences and Series
Arithmetic sequences
Arithmetic series
Geometric sequences
Geometric series
Sum to infinity
Sigma notation
Recurrence relations
Modelling with series
4. Binomial Expansion
Expanding (1 + x) ^ n
Expanding (a + bx) ^ n
Using partial fractions
5. Radians
Radian measure
Arc length
Areas of sectors and segments
Solving trigonometric equations
Small angle approximations
6. Trigonometric functions
Secant, cosecant and cotangent
Graphs of sec x, cosec x and cot x
Using sec x, cosec x and cot x
Trigonometric identities
Inverse trigonometric functions
7. Trigonometry and modelling
Addition formulae
Using the angle addition formulae
Double-angle formulae
Solving trigonometric equations
Simplifying a cos x +/- b sin x
Proving trigonometric identities
Modelling with trigonometric functions
8. Parametric Equations
Parametric equations
Using trigonometric identities
Curve sketching
Points of intersection
Modelling with parametric equations
9. Differentiation
Differentiating sin x and cos x
Differentiating exponentials and logarithms
The chain rule
The product rule
The quotient rule
Differentiating trigonometric functions
Parametric differentiation
Implicit differentiation
Using second derivatives
Rates of change
10. Numerical methods
Locating roots
Iteration
The Newton-Raphson method
Applications to modelling
11. Integration
Integrating standard functions
Integrating f(ax + b)
Using trigonometric identities
Reverse chain rule
Integration by substitution
Integration by parts
Partial fractions
Finding areas
The trapezium rule
Solving differential equations
Modelling with differential equations
12. Vectors
3D coordinates
Vectors in 3D
Solving geometric problems
Application to mechanics