IBDP Mathematics Analysis and Approaches SL
IB Diploma Programme Mathematics: Analysis and Approaches Standard Level (MAA SL) is a rigorous, proof-oriented mathematics course designed for students who relish the elegance of pure mathematics and its power to model the real world. Spanning five interconnected topics — Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and Calculus — the course develops both procedural fluency and deep conceptual understanding.In Number and Algebra, students master sequences, series, logarithms, the binomial theorem, and introductory proof by induction. The Functions unit builds a comprehensive toolkit of function families — quadratic, exponential, logarithmic, rational, and sinusoidal — alongside transformations, inverses, and mathematical modelling. Geometry and Trigonometry extends from radian measure and triangle trigonometry through to vector equations of lines in three dimensions. Statistics and Probability covers descriptive statistics, regression, probability rules, binomial and normal distributions, and formal hypothesis testing using chi-squared and t-tests. Calculus, the heart of the Analysis and Approaches course, develops differentiation from first principles through the chain, product, and quotient rules, applications to optimization and kinematics, and integration culminating in the Fundamental Theorem of Calculus, area calculations, and separable differential equations.Assessment comprises two external examination papers (Paper 1 without GDC and Paper 2 with GDC, each 90 minutes and worth 40% of the final grade) and an Internal Assessment mathematical exploration (20%), in which students independently investigate a topic of their choice, demonstrating personal engagement, mathematical communication, and reflective thinking.This AccelaStudy domain provides adaptive practice aligned to all IB assessment objectives (AO1–AO4), with targeted question banks, worked examples, contrastive concept pairs, and exam-strategy guidance to help students achieve their best possible grade on the IB scale of 1–7.
240
Minutes
60
Questions
4/7
Passing Score
$173
Exam Cost
Who Should Take This
This course is ideal for IB Diploma students who have a genuine interest in mathematics and plan to pursue university programmes in mathematics, physics, chemistry, engineering, computer science, economics, or any field requiring strong quantitative reasoning. MAA SL suits students who are comfortable with algebraic manipulation and enjoy understanding why mathematical results are true, not just how to apply them. It is also appropriate for students who want a challenging Group 5 subject that demonstrates mathematical rigour to university admissions offices without committing to the full demands of the Higher Level course.
What's Covered
1Number systems, sequences and series (arithmetic and geometric), exponents and logarithms, binomial theorem, proof by induction, systems of linear equations
2Concept of a function, domain and range, inverse and composite functions, transformations, quadratic, exponential, logarithmic, rational, and sinusoidal functions, modelling
3Radian measure, arc length and sector area, trigonometric ratios and identities, sine and cosine rules, trigonometric functions and graphs, vectors in 2D and 3D
4Descriptive statistics, regression and correlation, probability rules, discrete and continuous distributions (binomial and normal), hypothesis testing (chi-squared and t-test)
5Limits, differentiation rules (power, chain, product, quotient), applications of derivatives (tangents, normals, optimization, kinematics), integration (anti-differentiation, definite integrals, area, kinematics, separable differential equations)
6A 12–20 page written exploration of a mathematical topic chosen by the student, assessed on presentation, mathematical communication, personal engagement, reflection, and use of mathematics
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
13 Activity Formats
Course Outline
1Topic 1: Number and Algebra 4 topics
Number Systems and Operations
- State the properties of the real number system including natural numbers, integers, rational and irrational numbers, and identify to which subsets a given number belongs, using correct set notation.
- Apply the laws of exponents and logarithms to simplify expressions, solve exponential and logarithmic equations, and convert between exponential and logarithmic forms including natural logarithm.
- Calculate the sum and product of roots of a quadratic equation using Vieta's formulas and apply these relationships to construct quadratics with given roots.
Sequences and Series
- Describe arithmetic sequences and series, identify the common difference, and apply the formulas for the nth term and sum of the first n terms to solve contextual problems.
- Describe geometric sequences and series, identify the common ratio, apply the nth term and sum formulas, and determine the sum to infinity when the series converges, justifying the convergence condition.
- Apply arithmetic and geometric sequence and series formulas to model real-world financial contexts such as compound interest, loan repayments, and population growth, interpreting results in context.
- Construct a proof by mathematical induction to verify the formula for the sum of an arithmetic or geometric series, clearly stating the base case, inductive hypothesis, and inductive step.
Binomial Theorem and Counting
- Apply the binomial theorem to expand (a + b)^n for positive integer n, using Pascal's triangle or the combination formula nCr to identify and calculate specific terms in the expansion.
Systems of Equations and Proof
- Solve systems of two linear equations in two unknowns using algebraic elimination or substitution, and interpret the geometric meaning of unique, no, or infinitely many solutions.
- Justify simple algebraic identities and divisibility results using direct proof, clearly distinguishing between a proof and a numerical verification.
2Topic 2: Functions 3 topics
Concept of a Function
- Define a function as a mapping between sets, identify domain and range, determine whether a relation is a function using the vertical line test, and use correct function notation including f(x) and f: x ↦ y.
- Determine the inverse of a one-to-one function algebraically and graphically, state the domain and range of the inverse, and verify that f(f⁻¹(x)) = x.
- Construct composite functions f∘g and g∘f, determine their domains, evaluate them at specific values, and explain why composition is generally not commutative.
Transformations of Functions
- Apply and describe the effect of translations, reflections, and stretches on the graph of y = f(x), writing the transformed equation and sketching the resulting graph with key features labelled.
Specific Function Families
- Analyse quadratic functions in vertex, factored, and standard form; identify axis of symmetry, vertex, intercepts, and discriminant; and sketch the parabola, interpreting the discriminant to determine the number of real roots.
- Analyse exponential functions of the form f(x) = ka^x + c, identify horizontal asymptotes, intercepts, and growth/decay behaviour, and model real-world exponential growth and decay scenarios.
- Analyse logarithmic functions as inverses of exponential functions, sketch their graphs, identify vertical asymptotes and intercepts, and solve equations involving logarithms in context.
- Analyse rational functions of the form f(x) = (ax + b)/(cx + d), identify vertical and horizontal asymptotes, sketch the graph, and determine the domain and range.
- Evaluate and select an appropriate function model (linear, quadratic, exponential, logarithmic, sinusoidal) to fit a given data set or context, justifying the choice based on the shape of the data and contextual meaning of parameters.
3Topic 3: Geometry and Trigonometry 5 topics
Circle Mensuration and Radian Measure
- Calculate arc length and sector area using radian measure, convert between degrees and radians, and apply these formulas to solve problems involving circles and sectors in real-world contexts.
Trigonometric Ratios and Identities
- Define sine, cosine, and tangent using the unit circle for all angles, identify exact values at standard angles (0°, 30°, 45°, 60°, 90°), and determine the sign of trigonometric ratios in each quadrant.
- Apply the Pythagorean identity sin²θ + cos²θ = 1 and the identity tan θ = sin θ/cos θ to simplify trigonometric expressions and solve equations, showing all working clearly.
- Solve trigonometric equations of the form sin(x) = k, cos(x) = k, and tan(x) = k over a specified interval, identifying all solutions using the unit circle or general solution formulas.
Triangle Trigonometry
- Apply the sine rule and cosine rule to find unknown sides and angles in non-right triangles, identify the ambiguous case of the sine rule, and calculate the area of a triangle using (1/2)ab sin C.
- Apply trigonometric methods to solve problems in three-dimensional geometry, including finding angles between lines and planes, lengths of diagonals in 3D solids, and heights of inaccessible objects.
Trigonometric Functions and Graphs
- Sketch the graphs of y = sin x, y = cos x, and y = tan x, identifying amplitude, period, phase shift, and vertical shift for transformations of the form y = a sin(b(x − c)) + d.
- Construct a sinusoidal model to represent periodic real-world phenomena such as tidal heights, temperature cycles, or Ferris wheel motion, determining parameters from contextual information and interpreting the model.
Vectors
- Describe vectors in 2D and 3D using component form and unit vector notation, perform vector addition, subtraction, and scalar multiplication, and calculate the magnitude of a vector.
- Calculate the scalar (dot) product of two vectors, determine the angle between two vectors using the dot product formula, and identify when two vectors are perpendicular or parallel.
- Determine the vector equation of a line in 2D and 3D using a point and direction vector, convert between vector, parametric, and Cartesian forms, and find the intersection of two lines or determine they are parallel or skew.
4Topic 4: Statistics and Probability 5 topics
Descriptive Statistics
- Calculate measures of central tendency (mean, median, mode) and measures of spread (range, interquartile range, standard deviation, variance) for ungrouped and grouped data, using GDC where appropriate.
- Construct and interpret frequency histograms, cumulative frequency graphs, and box-and-whisker plots, identifying outliers and describing the shape, centre, and spread of a distribution.
Correlation and Regression
- Calculate and interpret Pearson's correlation coefficient r, describe the strength and direction of a linear relationship, and explain why correlation does not imply causation with reference to a real-world example.
- Determine the equation of the least-squares regression line y on x using GDC, plot the line on a scatter diagram, and use the equation to make predictions, commenting on the reliability of interpolation versus extrapolation.
Probability
- Define probability using the classical, frequentist, and subjective approaches, apply the addition rule for mutually exclusive and non-mutually exclusive events, and use Venn diagrams and sample space diagrams to calculate probabilities.
- Calculate conditional probability using P(A|B) = P(A∩B)/P(B), apply the multiplication rule for independent and dependent events, and use tree diagrams to solve multi-stage probability problems.
- Determine whether two events are independent using the condition P(A∩B) = P(A)·P(B), and distinguish between mutually exclusive and independent events with examples showing they are not the same concept.
Discrete Probability Distributions
- Construct a probability distribution table for a discrete random variable X, calculate the expected value E(X) and variance Var(X), and interpret these measures in context.
- Apply the binomial distribution B(n, p) to calculate probabilities of exactly k successes, cumulative probabilities, and the mean and variance, verifying that the conditions for a binomial model are satisfied.
Normal Distribution and Hypothesis Testing
- Describe the properties of the normal distribution N(μ, σ²), calculate probabilities and percentiles using GDC, and standardize values using z-scores to compare observations from different normal distributions.
- Apply the normal distribution to find unknown means or standard deviations given probability information, using inverse normal calculations on the GDC and interpreting results in context.
- Conduct a chi-squared test for independence on a contingency table, stating hypotheses, calculating the test statistic and p-value using GDC, and interpreting the result at a given significance level.
- Conduct a t-test for the mean of a single sample or the difference between two means, state null and alternative hypotheses, determine the p-value using GDC, and justify a conclusion at a specified significance level.
5Topic 5: Calculus 5 topics
Limits and Continuity
- Describe the concept of a limit informally, evaluate limits of functions as x approaches a finite value or infinity using algebraic simplification and GDC, and identify where a function is continuous or discontinuous.
- Describe the derivative as the limit of a difference quotient, interpret it as the instantaneous rate of change and the gradient of the tangent to a curve, and connect this to the concept of local linearity.
Differentiation Rules and Techniques
- Apply the power rule, constant multiple rule, and sum/difference rule to differentiate polynomial functions, and extend to differentiation of e^x, ln x, sin x, cos x, and tan x.
- Apply the chain rule to differentiate composite functions, the product rule to differentiate products of two functions, and the quotient rule to differentiate quotients, selecting the appropriate rule for a given expression.
- Determine the equations of tangent and normal lines to a curve at a given point by calculating the derivative, finding the gradient, and applying the point-slope form of a line.
Applications of Differentiation
- Analyse the behaviour of a function using the first and second derivatives to identify increasing/decreasing intervals, local maxima and minima, and points of inflection, sketching the curve with all key features labelled.
- Solve optimization problems in geometric, physical, and economic contexts by setting up an objective function, finding critical points using calculus, and verifying the nature of the extremum using the second derivative test.
- Apply differentiation to kinematics problems, interpreting displacement, velocity, and acceleration as functions of time, and determining when a particle is at rest, changing direction, or at maximum speed.
Integration — Anti-differentiation and Definite Integrals
- Apply the reverse power rule and standard integration results to find indefinite integrals of polynomial, exponential, and trigonometric functions, including the constant of integration and using initial conditions to determine it.
- Apply integration by substitution to evaluate integrals of composite functions of the form ∫f(g(x))g'(x)dx, identifying the appropriate substitution and transforming the limits for definite integrals.
- State and apply the Fundamental Theorem of Calculus to evaluate definite integrals, interpret the definite integral as the signed area between a curve and the x-axis, and use GDC to verify results.
- Calculate the area enclosed between two curves or between a curve and the x-axis over a specified interval, setting up the correct integral expression and evaluating it using algebraic or GDC methods.
- Apply integration to kinematics problems, determining displacement from velocity and velocity from acceleration by integration, and interpreting the difference between total distance travelled and net displacement.
Differential Equations (SL Scope)
- Solve separable differential equations of the form dy/dx = f(x)g(y) by separating variables and integrating both sides, applying initial conditions to find the particular solution and interpreting it in context.
6Mathematical Toolkit and Internal Assessment 3 topics
GDC Skills and Mathematical Communication
- Apply GDC functionality to graph functions, find intersections and roots, compute statistical summaries, evaluate definite integrals, and perform regression analysis, demonstrating awareness of when GDC use is and is not permitted.
- Demonstrate correct mathematical notation and communication throughout written solutions, including proper use of equals signs, implication arrows, set notation, and clear logical structure in multi-step proofs and problem solutions.
Mathematical Exploration (Internal Assessment)
- Identify a focused mathematical question or conjecture for the exploration, justify its mathematical interest and personal engagement, and outline a clear plan connecting the topic to SL syllabus content.
- Construct a coherent mathematical argument in the exploration using appropriate notation, definitions, and mathematical processes, demonstrating understanding beyond routine procedures and making connections between areas of mathematics.
- Evaluate the results and conclusions of the mathematical exploration critically, discussing limitations, potential extensions, and the significance of findings, demonstrating mathematical reflection and independent thinking.
Exam Strategy and Command Term Mastery
- Distinguish between IB command terms (show that, find, hence, hence or otherwise, determine, write down) and apply the appropriate level of working and justification required by each term in examination responses.
- Evaluate the structure of Paper 1 (no GDC) and Paper 2 (GDC required) questions, applying appropriate algebraic versus technology-based strategies and managing time effectively across short-answer and extended-response sections.
Scope
Included Topics
- Five core syllabus topics: Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and Calculus, as specified in the IB Mathematics: Analysis and Approaches SL syllabus (first assessment 2021)
- Number and Algebra: number systems, sequences and series (arithmetic and geometric), exponents and logarithms, binomial theorem, proof by induction (SL scope), systems of linear equations
- Functions: concept of a function, domain/range/inverse/composite, transformations, linear/quadratic/rational/exponential/logarithmic/sinusoidal functions, modelling with functions
- Geometry and Trigonometry: arc length and sector area, right-triangle and unit-circle trigonometry, sine and cosine rules, trigonometric identities, 3D geometry, vectors in 2D and 3D (SL scope)
- Statistics and Probability: descriptive statistics, regression and correlation, probability rules, discrete and continuous distributions (binomial and normal), hypothesis testing (chi-squared and t-test at SL scope)
- Calculus: limits and continuity, differentiation rules (product, quotient, chain), applications of derivatives (tangents, normals, optimization, kinematics), integration (anti-differentiation, definite integrals, area under/between curves, kinematics)
- Mathematical toolkit and exploration: use of GDC (graphical display calculator), mathematical communication, conjecture, generalization, and the Internal Assessment (mathematical exploration, 20% of final grade)
- Two external assessment papers: Paper 1 (no GDC, 80 marks, 90 min) and Paper 2 (GDC required, 80 marks, 90 min), plus Internal Assessment (20%)
- IB command terms taxonomy (AO1–AO4) and mark-scheme conventions for show-that, hence, and hence or otherwise questions
- Mathematical modelling contexts and real-world applications integrated throughout all five topics
Not Covered
- HL-only content: complex numbers, proof by contradiction and counterexample beyond SL scope, further trigonometric identities (double angle beyond SL), matrices, further vectors (cross product, planes), further statistics (confidence intervals, Poisson distribution), further calculus (Maclaurin series, L'Hôpital's rule, integration by parts, differential equations beyond SL scope)
- IB Mathematics: Applications and Interpretation syllabus content (separate course)
- University-level real analysis, abstract algebra, or topology
- Detailed programming or CAS-specific syntax beyond GDC use described in the syllabus
- Further Mathematics HL content
Official Exam Page
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