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IB Diploma · session May 2027

Mathematics: analysis and approaches

Guide first assessed in May 2021, which is the guide May 2027 and May 2028 candidates sit.

83 subtopics502 diagrams340,608 words

Mocks: in the future, hold tight!

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The higher-level subtopics are hidden, and so is the higher-level material inside the rest.

Topic 1 Number and algebraTest setThe theoryWorked examplesPractice testMark scheme

  1. 1.1 Scientific notation how do you write, compare and calculate with numbers far too large or too small to write out in full? 5 diagrams
  2. 1.2 Arithmetic sequences and series when something goes up or down by the same amount every time, how do you find any term, and the total, without listing them all? 6 diagrams
  3. 1.3 Geometric sequences and series when something is multiplied by the same number every time, how do you find any term, and the total, and how quickly does it run away from you? 6 diagrams
  4. 1.4 Financial applications of geometric sequences if money grows or shrinks by a fixed percentage each period, what will it be worth later, how long will that take, and what is it really worth once prices have risen? 6 diagrams
  5. 1.5 Laws of exponents and introduction to logarithms how do you simplify expressions built from powers, and how do you find the power itself when it is the thing you do not know? 6 diagrams
  6. 1.6 Simple deductive proof a pattern works every time you try it, so how do you show it works every time you do not? 5 diagrams
  7. 1.7 Laws of exponents and logarithms when the unknown is up in the exponent, how do you bring it down? 7 diagrams
  8. 1.8 Sum of infinite geometric sequences how can you add up infinitely many numbers and get a finite answer, and when can you not? 6 diagrams
  9. 1.9 The binomial theorem how do you multiply out (a + b)ⁿ, or find just one of its terms, without writing out n brackets? 6 diagrams
  10. 1.10 Counting principles and the extended binomial theorem how many ways can something happen, without listing them all, and what does (1 + x)ⁿ look like when n is negative or a fraction? HL only 7 diagrams
  11. 1.11 Partial fractions how do you split a fraction such as (7x − 1)/((x + 1)(x − 3)) back into the two simple fractions it came from, and why would you want to? HL only 5 diagrams
  12. 1.12 Complex numbers in Cartesian form what happens if you allow a number whose square is −1, and how do you calculate with, and draw, the numbers that result? HL only 6 diagrams
  13. 1.13 Polar and Euler forms of complex numbers why does multiplying complex numbers turn and stretch the plane, and which way of writing a complex number makes that obvious? HL only 6 diagrams
  14. 1.14 Complex roots, De Moivre's theorem, powers and roots why do the complex roots of a real polynomial come in pairs, and how do you raise a complex number to a power, or find all of its nth roots? HL only 5 diagrams
  15. 1.15 Proof by induction, contradiction and counterexample how do you prove that a statement holds for infinitely many cases, prove that something is impossible, or show that a claim is false? HL only 6 diagrams
  16. 1.16 Systems of linear equations when three equations share three unknowns, how do you find the solution, and how do you tell when there is exactly one, none, or infinitely many? HL only 5 diagrams

Topic 2 Functions

  1. 2.1 Straight lines a straight line is the simplest relationship there is, so how do you pin one down, write it in whichever form the question wants, and tell when two lines are parallel or meet at a right angle? 7 diagrams
  2. 2.2 Functions, domain, range and inverse what exactly makes a rule a function, which inputs is it allowed to take and which outputs can it give, and when can it be run backwards? 7 diagrams
  3. 2.3 Graphs of functions how do you turn a function, a screen or a description into a graph on paper that an examiner can read and mark? 6 diagrams
  4. 2.4 Key features of graphs and intersections once a function is on the screen, which points and lines describe it, and how do you find each of them, including where two graphs meet? 6 diagrams
  5. 2.5 Composite and inverse functions what happens when you feed the output of one function into another, and how do you find, by algebra, the function that undoes a function? 5 diagrams
  6. 2.6 The quadratic function the same quadratic can be written three ways, so which way shows which feature of its graph, and how do you move between them? 6 diagrams
  7. 2.7 Quadratic equations and inequalities, and the discriminant how do you solve any quadratic equation or inequality, and how can you tell how many solutions there are before you find them? 6 diagrams
  8. 2.8 The reciprocal function and rational functions what does a graph look like when x is in the denominator, and how do you find its asymptotes and intercepts without drawing it first? 5 diagrams
  9. 2.9 Exponential and logarithmic functions what do the graphs of aˣ, eˣ, logₐ x and ln x look like, and why is each exponential the mirror image of a logarithm? 6 diagrams
  10. 2.10 Solving equations graphically and analytically when can an equation be solved exactly by algebra, and how do you solve it, and all its solutions, when it cannot? 6 diagrams
  11. 2.11 Transformations of graphs if you know the graph of y = f(x), what does the graph of y = p·f(x − a) + b look like, and how do you get from one to the other without plotting a single new point? 7 diagrams
  12. 2.12 Polynomial functions, the factor and remainder theorems, and sums and products of roots how do the roots of a polynomial, its factors, its coefficients and the shape of its graph determine one another? HL only 5 diagrams
  13. 2.13 Rational functions of the form (ax + b)/(cx² + dx + e) and (ax² + bx + c)/(dx + e) when a fraction has a linear expression over a quadratic, or a quadratic over a linear one, what do its asymptotes look like, and how do you sketch the whole graph and find its range without plotting? HL only 5 diagrams
  14. 2.14 Odd and even functions, inverse functions with restricted domains, and self-inverse functions what does the symmetry of a graph say about its formula, how do you make a function reversible when it is not, and when is a function its own inverse? HL only 5 diagrams
  15. 2.15 Solutions of g(x) ≥ f(x), graphically and analytically for which values of x is one function at least as big as another, and how do you find every such x, with algebra when you can and with technology when you cannot? HL only 6 diagrams
  16. 2.16 The graphs of |f(x)|, f(|x|), 1/f(x), f(ax + b) and [f(x)]², and modulus equations and inequalities given the graph of y = f(x), how do you draw |f(x)|, f(|x|), 1/f(x), f(ax + b) and [f(x)]² without plotting, and how do you solve equations and inequalities that contain a modulus? HL only 7 diagrams

Topic 3 Geometry and trigonometry

  1. 3.1 Three-dimensional geometry once a shape leaves the flat page, how do you measure it: the distance between two points, how much it holds, how much surface it has, and the angle a line makes with a plane? 7 diagrams
  2. 3.2 Trigonometry in triangles right-angled triangles give up their secrets through SOH CAH TOA, so what do you do when the triangle has no right angle, and how do you know which rule to reach for? 7 diagrams
  3. 3.3 Applications of trigonometry how do you get from a paragraph about a lighthouse, a ship or a tower to the right triangle, with the right angle in the right place, so that Pythagoras, SOH CAH TOA, the sine rule or the cosine rule can finish the job? 6 diagrams
  4. 3.4 The circle: radians, arcs and sectors why does mathematics measure angles in radians rather than degrees, and how does that choice turn the length of an arc and the area of a sector into one-line formulas? 6 diagrams
  5. 3.5 The unit circle, exact values and the ambiguous case sine and cosine were born in right-angled triangles, where no angle reaches 90°, so what can sin 150°, cos(−π/3) or tan 5π mean, how do you find their exact values without a calculator, and why does the sine rule sometimes hand you two different triangles? 8 diagrams
  6. 3.6 The Pythagorean and double angle identities if you know one trigonometric ratio of an angle, how much else do you know about it, and about double the angle, without ever finding the angle itself? 5 diagrams
  7. 3.7 The circular functions and their graphs what do the graphs of sin x, cos x and tan x look like, how do the numbers in f(x) = a sin(b(x + c)) + d stretch and move them, and how do you use one to model something that repeats? 8 diagrams
  8. 3.8 Solving trigonometric equations a calculator gives one angle for sin x = k, but the wave meets that value again and again, so how do you find every solution in the interval you are given, by hand and on a GDC, including when the equation hides a quadratic? 6 diagrams
  9. 3.9 Reciprocal and inverse trigonometric functions what are sec, cosec and cot, how do they give two more Pythagorean identities, and how do you undo sin, cos and tan when none of them is one-to-one? HL only 6 diagrams
  10. 3.10 Compound angle identities how do you find sin, cos or tan of a sum or difference of two angles from the ratios of the angles themselves, and where do the double angle identities come from? HL only 5 diagrams
  11. 3.11 Symmetry properties of trigonometric graphs why do the graphs of sine, cosine and tangent have the symmetries they do, and how do those symmetries turn into identities, simplifications and every solution of an equation? HL only 6 diagrams
  12. 3.12 Introduction to vectors what is a vector, how do you write, add, subtract and scale vectors both as arrows and as components, and how do vectors prove facts about shapes? HL only 7 diagrams
  13. 3.13 The scalar product how do you multiply two vectors to get a number, and how does that number find the angle between them and decide at once whether they are perpendicular or parallel? HL only 6 diagrams
  14. 3.14 Vector equation of a line in three dimensions a line has no gradient and no y-intercept, so what do you write down instead, and how does that same equation track something that is moving? HL only 6 diagrams
  15. 3.15 Coincident, parallel, intersecting and skew lines given the equations of two lines, how do you decide for certain whether they are the same line, parallel, meeting at a point or passing each other by, and where exactly do they meet if they do? HL only 5 diagrams
  16. 3.16 The vector product given two vectors in space, how do you produce a vector at right angles to both of them, and why does its length turn out to be an area? HL only 6 diagrams
  17. 3.17 Equations of a plane what is the least information that fixes a flat plane in space, and how do you turn that information into an equation you can test points against and use with lines and other planes? HL only 5 diagrams
  18. 3.18 Intersections and angles with planes when a line meets a plane, or two or three planes meet each other, what shape is the common part (a point, a line, a whole plane or nothing), how do you find it, and at what angle do they meet? HL only 7 diagrams

Topic 4 Statistics and probability

  1. 4.1 Populations, samples and sampling before you calculate anything from a set of data, how do you decide whether the data deserve to be trusted? 6 diagrams
  2. 4.2 Presenting data a list of eighty numbers tells you almost nothing at a glance, so which pictures of it show the centre, the spread and the shape, and how do you read values back off them? 8 diagrams
  3. 4.3 Measures of centre and spread if you could keep only two numbers about a data set, one for where it sits and one for how spread out it is, which should they be, and how do you find them? 6 diagrams
  4. 4.4 Correlation and regression when two quantities seem to move together, how do you measure how strongly, draw the straight line that best describes them, and know when a prediction from it can be trusted? 7 diagrams
  5. 4.5 Sample spaces and probability how do you put a number on how likely something is, either by counting what could happen or by watching what did happen, and what does that number let you predict? 5 diagrams
  6. 4.6 Combined, conditional and independent events when two events are in play at once, how do you find the probability of one or the other, of both, or of one given that you already know the other has happened? 7 diagrams
  7. 4.7 Discrete random variables when the result of a trial is a number, how do you describe every value it could take and how likely each one is, and what should you expect it to be on average? 5 diagrams
  8. 4.8 The binomial distribution when the same chance event is tried a fixed number of times, how likely is each possible number of successes, and how many should you expect? 6 diagrams
  9. 4.9 The normal distribution so many measured quantities pile up in the same bell shape, so how do you describe that shape, and how do you use it to find probabilities and cut-off values? 8 diagrams
  10. 4.10 Regression line of x on y the regression line of y on x predicts y, so what do you use when the value you know is y and the value you want is x, and when can you trust the answer? 5 diagrams
  11. 4.11 Conditional probability and independence once you know that one thing has happened, how should the probability of another change, and how do you show that it does not change at all? 5 diagrams
  12. 4.12 Standardization of normal variables how far from average is a value, measured in a way that lets you compare different distributions, and if you know what percentage lies beyond a point, how do you recover a mean or standard deviation you were never told? 6 diagrams
  13. 4.13 Bayes' theorem when you know how likely the evidence is for each possible cause, how do you work out how likely each cause is, now that you have seen the evidence? HL only 5 diagrams
  14. 4.14 Discrete and continuous random variables when a random variable can take a handful of values, or any value in an interval, how do you describe where it is centred and how widely it spreads, and what happens to those numbers when you convert its units? HL only 5 diagrams

Topic 5 Calculus

  1. 5.1 Limits and the derivative a straight line has one gradient, but a curve is steeper in some places than others, so what does "the gradient at a point" even mean, and how do we find it? 6 diagrams
  2. 5.2 Increasing and decreasing functions if the derivative is the gradient at every point, what does its sign tell you about where a function is going up and where it is going down? 6 diagrams
  3. 5.3 Differentiating powers and polynomials finding a gradient with chords and limits works but is slow, so is there a rule that gives the derivative of a function like 4x⁵ − 3x² + 7x − 9 straight away? 5 diagrams
  4. 5.4 Tangents and normals once you know the gradient of a curve at a point, how do you write down the straight line that touches it there, and the line that meets it at right angles? 6 diagrams
  5. 5.5 Anti-differentiation and area if you know how fast something is changing, how do you get back to the thing itself, and why does that same process give the area under a curve? 6 diagrams
  6. 5.6 Standard derivatives, and the chain, product and quotient rules the power rule handles polynomials, but how do you differentiate functions such as e^{x² − 3x}, x² sin x or ln x ÷ x? 6 diagrams
  7. 5.7 The second derivative and concavity the first derivative tells you whether a graph is going up or down, so what does the derivative of the derivative tell you, and how do the graphs of f, f′ and f″ tell one story three ways? 8 diagrams
  8. 5.8 Maxima, minima, inflexion and optimisation where does a function reach its highest and lowest values, how do you prove which is which, and how do you turn that into the best possible answer to a real problem? 7 diagrams
  9. 5.9 Kinematics if you know where a moving object is at every moment, how do you find how fast it is going and how that speed is changing, and how do you get back from its velocity to its position and to the total distance it has covered? 6 diagrams
  10. 5.10 Indefinite integrals and substitution if differentiating is a set of rules, how do you run each rule backwards, and how do you recognise, inside a complicated expression, the chain rule waiting to be undone? 5 diagrams
  11. 5.11 Definite integrals and areas how do you work out a definite integral exactly, why does it sometimes give a negative or zero answer when the region plainly has an area, and how do you find the area enclosed by a curve and the x-axis or between two curves? 6 diagrams
  12. 5.12 Continuity, differentiability and first principles where does the derivative actually come from, when can a curve be said to have one, and what happens when you differentiate again and again? HL only 7 diagrams
  13. 5.13 Limits and l'Hôpital's rule when a limit comes out as 0/0 or ∞/∞, which tells you nothing, how do you find the value it is really heading for? HL only 6 diagrams
  14. 5.14 Implicit differentiation, related rates and optimisation how do you differentiate a relationship you cannot solve for y, how fast does one quantity change when another linked to it is changing, and where exactly is the best value of a model, including when it sits at the edge of what is allowed? HL only 7 diagrams
  15. 5.15 Further derivatives and integrals how do you differentiate tan x, sec x, cosec x, cot x, aˣ, logₐ x and the inverse trigonometric functions, and how does reading those results backwards let you integrate expressions such as 1/(x² + 2x + 5) that SL calculus could not touch? HL only 5 diagrams
  16. 5.16 Integration by substitution and by parts when an integral is not on the standard list, how do you change it into one that is, either by renaming the variable or by trading one integral for an easier one? HL only 6 diagrams
  17. 5.17 Areas against the y-axis and volumes of revolution how do you find the area of a region that sits against the y-axis, and the volume of the solid made by spinning a region about either axis? HL only 7 diagrams
  18. 5.18 First-order differential equations if you know only the rule for how fast something changes, how do you recover the thing itself, as an exact formula or as a numerical estimate? HL only 7 diagrams
  19. 5.19 Maclaurin series how can a polynomial stand in for eˣ, sin x or ln(1 + x), and how do you build the series for a new function from the ones you already know? HL only 6 diagrams

The course in one page

The course in one page. Standard and higher level, for the guide first assessed in May 2021, which is the guide May 2027 and May 2028 candidates sit. The next AA course is first taught from August 2027 and first assessed in May 2029.

What the course is

Analysis and approaches is the more abstract of the two IB mathematics courses. It rewards algebraic fluency and proof. Where applications and interpretation reaches for a model and a calculator, analysis and approaches asks you to work exactly: to leave an answer as ln 3 rather than 1.0986, to prove a statement holds for every integer rather than check it for five of them.

Choose it if you like the machinery. It is the course university mathematics, physics and engineering departments look for.

The five topics

TopicWhat it is really about
1Number and algebraSequences, series, exponents, logarithms, proof. HL adds complex numbers, induction, systems.
2FunctionsWhat a function does to its input, and what happens to the graph when you interfere with it.
3Geometry and trigonometryTriangles, the unit circle, trigonometric functions and identities. HL adds vectors.
4Statistics and probabilityDescribing data, then reasoning under uncertainty with named distributions.
5CalculusRate of change and accumulation, and the fact that they are inverse to one another.

Topics are taught in an order the school chooses, so Educerie plans around where your class actually is rather than around this numbering.

The papers

Standard levelHigher level
Paper 190 min, 80 marks, no technology120 min, 110 marks, no technology
Paper 290 min, 80 marks, technology required120 min, 110 marks, technology required
Paper 3—75 min, 55 marks, two extended-response problems

Every paper draws on every topic. There is no "the calculus paper".

Paper 1 is the one that catches people. No calculator means the numbers are chosen to come out clean, and it means exact form is not optional: π/6, 2√3, ln 5. A decimal where an exact value was asked for loses the accuracy mark even when it is correct to nine places.

Paper 3 is higher level only and is not more content — it is the same syllabus asked at length. Two problems, each building from an accessible opening part to something genuinely hard, where later parts depend on earlier ones. The skill is reading forward: part (a) is usually there because part (d) needs it.

How it is marked

Marks are awarded against a scheme, point by point, in three kinds:

  • M — method. Awarded for a correct approach, even if the arithmetic that follows is wrong.
  • A — accuracy. Awarded for a correct value, and dependent on the method mark above it.
  • R — reasoning. Awarded for saying why, in words. Proof questions live and die here.

Two consequences worth internalising now. First, write the method down even when you can do it in your head — an unsupported correct answer often scores less than a supported wrong one. Second, follow through works in your favour: a wrong value carried correctly into a later part usually still earns that part's method marks, so a mistake in (a) does not have to cost you (b).

Assessment objectives

Roughly
AO1Knowledge and understandingrecall and routine technique
AO2Problem solvingchoosing an approach when none is named
AO3Communication and reasoningproof, justification, exact form, notation

Educerie reports your marks split three ways — by topic, by objective and by command word — because "you lost 12 marks" is not actionable and "you lose AO3 marks whenever a question says show that" is.

Command words, and what each one obliges you to write

WordWhat must appear on the page
Write downThe answer alone. No working expected; no method marks available.
Find / CalculateAnswer with enough working to show where it came from.
DetermineAs find, but the question expects a decision or a justification with it.
Show thatA route from the given start to the given end. The answer is printed, so the marks are entirely in the steps — and you may not use the result to prove itself.
HenceYou must use the previous part. A fresh independent method scores zero, even if correct.
Hence or otherwiseYou may use the previous part or start again. Both are open.
ProveA general argument covering every case, closed properly. Examples are not a proof.
SketchShape, intercepts, asymptotes and turning points, labelled. Not a plotted table.
StateA short factual answer, no justification needed.

The gap between hence and hence or otherwise is worth more marks across a paper than most students realise.

Educerie · written from the published IB Diploma Programme structure for Mathematics: analysis and approaches, first assessment 2021, the guide in force for the May 2027 and May 2028 sessions. Original text. Paper durations and mark totals checked against the May 2027 examination schedule. Sources consulted: none beyond the published course structure. Last reviewed 9 September 2026.