IB Mathematics: Analysis and Approaches suits students who enjoy algebra, functions, proof and calculus, and who may go on to study mathematics, physics, engineering or economics. This guide explains what the course contains, how the papers are organised, what each paper asks of you, and a weekly routine that works. It applies to first assessment 2021 onwards, at both Standard Level (SL) and Higher Level (HL).
What the course covers
- Number and algebra: sequences and series, exponents and logarithms, the binomial expansion, proof, and at HL partial fractions, complex numbers and systems of equations.
- Functions: domain and range, composite and inverse functions, graph transformations, quadratics, rational, exponential and logarithmic models.
- Geometry and trigonometry: triangle work, radians, arcs and sectors, the unit circle, identities, and at HL vectors, lines and planes in three dimensions.
- Statistics and probability: summarising data, regression, conditional probability, the binomial and normal distributions, and at HL Bayes' theorem and continuous random variables.
- Calculus: limits, differentiation, integration, kinematics, optimisation, and at HL implicit differentiation, related rates and differential equations.
- Mathematical exploration: an individual investigative write-up on a topic of your choice, worth 20% at both levels.
How you are assessed
| Level | Component | Duration | Marks | Weighting | Calculator |
|---|---|---|---|---|---|
| SL | Paper 1 | 1 h 30 min | 80 | 40% | Not allowed |
| SL | Paper 2 | 1 h 30 min | 80 | 40% | Allowed |
| SL | Exploration (internal assessment) | Done in class time and at home | - | 20% | Not applicable |
| HL | Paper 1 | 2 h | 110 | 30% | Not allowed |
| HL | Paper 2 | 2 h | 110 | 30% | Allowed |
| HL | Paper 3 (extended problem-solving) | 1 h | 55 | 20% | Allowed |
| HL | Exploration (internal assessment) | Done in class time and at home | - | 20% | Not applicable |
A formula booklet is provided in every paper. Learn what it contains well before the exam, so you can find a result in seconds and so you know which facts you must carry in your head.
What each paper asks of you
Paper 1 (no calculator). Expect exact answers: surds, fractions, multiples of \(\pi\), logarithms left in exact form. Show every algebraic step, because a slip is easier to forgive when the method is visible. Practise mental routines for exact trigonometric values, differentiating and integrating standard functions, and simplifying expressions without a screen.
Paper 2 (calculator). Questions are more modelling-based and often longer. The calculator is a tool for solving equations, finding areas under curves, and working with distributions, but you still need to write down what you are calculating. Write the setup (for instance the integral or the probability statement) before quoting a result, and give answers to three significant figures unless told otherwise.
Paper 3 (HL only). One long investigation that builds from a familiar idea to something unfamiliar over several linked parts. Early parts are accessible, and later parts reward you for using results you have just proved. If you cannot prove a statement, you may still use it in the next part, so never leave a gap blank.
Exploration. Choose a topic you genuinely find interesting, state a clear aim, use mathematics at the right level for you, and reflect honestly on what you found. Start early and keep your drafting notes.
Command words and how to answer them
- Write down or state: give the result with little or no working. Do not over-explain.
- Find: work out a value or expression, showing the method that leads to it.
- Show that: you are given the answer, so every step must be visible and logical, ending exactly on the stated result.
- Hence: you must use the previous result. Starting again by another route may lose the marks.
- Hence or otherwise: using the previous result is the intended route, but another correct method is acceptable.
- Prove: build a complete, logical argument from known facts or from a stated starting point.
- Sketch: draw the shape with key features labelled (intercepts, asymptotes, turning points), but not to scale.
- Determine: reach a definite answer, usually exact, with justification.
- Interpret or comment: say what a number or a result means in the context of the problem.
Calculators
A calculator is not permitted in the first paper at either level and is permitted in the others. For the papers that allow one, use a graphical display calculator you know thoroughly: plotting, finding intersections and zeros, numerical integration and statistical distributions should all be routine. Check the rules for your own exam session on what is permitted, and keep spare batteries available.
How to revise this course
- Monday: read one revision note and rewrite its key results from memory without looking.
- Tuesday: work through the matching flashcards until you can answer every card quickly and correctly.
- Wednesday: attempt 10 to 15 question-bank questions on that topic, untimed, with full written working. Check your solutions strictly and write down every error.
- Thursday: repeat the questions you got wrong, then attempt a few from an older topic to keep it fresh.
- Friday: complete a short timed test in the style of the paper you are targeting, calculator or no calculator as appropriate.
- Weekend: review the week's errors, update a personal list of mistakes, and spend an hour on the exploration if it is still in progress. Once a month, sit a full timed paper.
Common errors
- Forgetting the constant of integration, or the limits, when integrating or solving a differential equation.
- Rounding too early on a multi-step calculation and then losing accuracy in the final answer.
- Working in degrees when the calculator or the question needs radians (or the other way round).
- Dividing both sides of an equation by an expression that could be zero, and so losing a solution.
- In a show-that question, jumping to the given answer without the intermediate line that makes the argument complete.
Exam tip: when a question has several parts, read them all first. A later part often tells you which method an earlier part is leading towards, and it helps you decide whether an exact or a decimal answer is wanted.