A-Level Pure Mathematics Revision Guides
A-Level Maths
A-Level Pure Mathematics Revision Guides
An Explanation Guide and a Practice Guide for A-Level Pure Mathematics, used across exam boards.
- A-Level
- All boards
- 15 areas
The Explanation Guide sets out how each method works, step by step. The Practice Guide holds the questions with their answers and worked solutions, which is the part worked in three passes.
The two guides
Digital PDF, sent to the email address on your order.
The subject
Pure Mathematics is the compulsory core of A-Level Maths and carries two thirds of the qualification on the main specifications.
It runs from algebra, functions and proof through coordinate geometry, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, numerical methods and vectors.
The 3-step active recall method
Re-reading notes is a futile process when your aim is to remember the information and successfully apply it in exam settings. Active recall is the method you use to ensure the information sticks for a long period of time.
But even active recall can be futile unless you have a structured way of implementing it. This is where the 3-step active recall method comes in, a structured method that is signature to the Levo Learning guides.
-
Step 1
Answer up to 10 questions cold, closed book, even if you are unsure on the topic.
-
Step 2
Answer the same questions again, but open book. Write the correct answer down even if it seems pointless.
-
Step 3
Repeat as step 1 and answer the questions closed book again.
Worked through the Practice Guide, one set of questions at a time.
Memorisation, then application
Marks are given for applying content to the question that has been set, which cannot happen while the content is still being looked up. Step 1 shows what is actually held.
Step 2 puts the correct wording in front of you while the gap is still fresh. Step 3 repeats the cold test, and the difference between the first and third attempt is the part that has moved.
In maths the same applies to method. The steps of a technique have to be produced without the worked solution in front of you before an unfamiliar question can be attempted.
What is covered
Open an area to see the methods inside it.
Algebra and Functions18 methods
- Completing the square
- The factor theorem
- Algebraic division
- Rationalising surds
- Partial fractions
- Composite functions
- Inverse functions
- Modulus equations
- The discriminant
- Quadratics in disguise
- Simultaneous equations (one linear, one quadratic)
- Quadratic inequalities
- Solving quadratics in exact form
- Modulus inequalities
- Simultaneous inequalities in set notation
- Domain, range and the inverse function
- Simplifying algebraic fractions
- Transformations of graphs
Proof6 methods
- Proof by direct algebra
- Proving a multiple (proof by exhaustion)
- Proof by completing the square
- Disproof by counterexample
- Proof by contradiction
- Proof by exhaustion (odd and even cases)
Coordinate Geometry: Straight Lines8 methods
- Equation of a line through two points
- Parallel lines
- Perpendicular bisectors
- Distance between two points
- Collinear points
- Area of a triangle
- Perpendicular lines and their intersection
- Parametric to cartesian form
Coordinate Geometry: Circles6 methods
- Circles: centre and radius
- Tangent to a circle at a point
- Intersection of a line and a circle
- Chord length
- Showing a line is a tangent
- Circle through three points
Sequences and Series8 methods
- The nth term of a sequence
- Sum of the first n terms
- Sum to infinity of a geometric series
- Finding n given a sum condition
- Recurrence relations and limits
- Sigma notation
- Compound interest
- Increasing, decreasing and periodic sequences
Binomial Expansion8 methods
- Binomial expansion (first terms)
- Finding a coefficient
- Finding two coefficients
- Finding an unknown constant
- Matching terms to find p and n
- Equal coefficients
- Using an expansion to estimate
- Binomial expansion (rational / negative index)
Trigonometry: Equations6 methods
- Solving trigonometric equations
- Quadratic trigonometric equations
- Equations with multiple angles
- Equations using double-angle identities
- Equations with reciprocal functions
- Solving a sin x + b cos x = c
Trigonometry: Sine and Cosine Rules4 methods
- The sine rule
- The cosine rule
- Area of a triangle
- The ambiguous case of the sine rule
Trigonometry: Radian Measure4 methods
- Converting degrees to radians
- Arc length and sector area
- Area of a segment
- Finding an unknown in a sector
Trigonometry: Identities and Exact Values7 methods
- Small-angle approximations
- Exact values using compound angles
- Exact values of reciprocal functions
- Exact values of inverse functions
- Expressing in the form R sin(x + alpha)
- Trigonometric modelling
- Proving a trigonometric identity
Exponentials and Logarithms5 methods
- Solving exponential equations
- Solving equations using laws of logarithms
- Exponential equations reducible to quadratics
- Exponential models
- Linearising a model (log-log plots)
Differentiation13 methods
- Tangents to a curve
- Differentiation from first principles (polynomial)
- Differentiation from first principles (trigonometric)
- Stationary points
- Increasing and decreasing functions
- Concavity and the second derivative
- Optimisation using calculus
- Implicit differentiation
- Parametric differentiation (at a parameter value)
- Parametric differentiation (from a given point)
- Connected rates of change
- Logarithmic differentiation
- Points of inflection
Integration13 methods
- Indefinite integration
- Definite integrals and area under a curve
- Area between a curve and a line
- Intersections of a curve and a line
- Integration by substitution
- Integration by parts
- Integration using partial fractions
- Differential equations (separation of variables)
- Differential equations (proportional rate)
- Area under a parametric curve
- Integration using trigonometric identities
- The trapezium rule
- Integration as the limit of a sum
Numerical Methods4 methods
- Locating roots (change of sign)
- Iterative methods
- The Newton-Raphson method
- When the Newton-Raphson method fails
Vectors10 methods
- Magnitude and unit vectors
- Direction of a vector
- Resultant vectors
- Displacement and distance
- Dividing a line in a given ratio
- Finding an endpoint from a given ratio
- Section formula and distance from the origin
- Parallel and collinear vectors
- Finding an unknown for parallel vectors
- Finding a parallelogram vertex
Questions
What is in the A-Level Pure Mathematics guides?
Two guides. The Explanation Guide works through each method step by step.
The Practice Guide holds questions with answers and full worked solutions. Between them they cover 15 areas of pure mathematics.
Which exam board are the Pure Mathematics guides for?
A-Level Maths content is common across the main boards, so the guides are used with AQA, Edexcel, OCR and MEI alike.
How do I use active recall with maths?
Work the Practice Guide in three passes. Attempt up to 10 questions closed book, work the same questions again with the solutions open and write the correct working down, then attempt them closed book again.