IB Group 5 · Mathematics
Mathematics: Analysis and Approaches
2021 syllabus · First assessment 2021
Five topics built on the guide’s content statements, with a formula booklet in every exam. Each note teaches the methods and worked examples the syllabus expects, with the SL core and AHL (HL-only) content clearly split.
1. Topic 1: Number and algebra
Sequences and series, exponents and logarithms, the binomial theorem, and (HL) proof, complex numbers, systems of equations.
SL1.1 — Standard form (scientific notation)
Writing numbers in standard form — Expressing very large and very small numbers as $a\times10^{k}$ with $1\le a<10$.
Operations with numbers in standard form — Multiplying, dividing, adding and subtracting numbers in the form $a\times10^{k}$.
SL1.2 — Arithmetic sequences and series
Arithmetic sequences: the nth term — Common difference $d$ and the formula $u_n=u_1+(n-1)d$.
Arithmetic series and sigma notation — Summing the first $n$ terms with $S_n=\tfrac{n}{2}(2u_1+(n-1)d)$.
Applications — Simple interest and real-life models that are approximately arithmetic.
SL1.3 — Geometric sequences and series
Geometric sequences: the nth term — Common ratio $r$ and the formula $u_n=u_1 r^{n-1}$.
Geometric series and sigma notation — Summing $n$ terms with $S_n=\dfrac{u_1(r^n-1)}{r-1}$.
Applications — Population growth, decay and other repeated-multiplier situations.
SL1.4 — Financial applications: compound interest and depreciation
Compound interest — The $FV = PV(1+\tfrac{r}{100k})^{kn}$ formula and different compounding frequencies.
Depreciation and real value — Annual depreciation, and adjusting an investment for inflation.
SL1.5 — Exponents (integer) and introduction to logarithms
Laws of exponents with integer exponents — Product, quotient, power, zero and negative-index rules.
Introduction to logarithms (base 10 and e) — $a^x=b \Leftrightarrow \log_a b = x$, and $\log_e x = \ln x$.
SL1.6 — Simple deductive proof
Equality, identity and the LHS-to-RHS method — The $=$ and $\equiv$ symbols and how to lay out a proof.
Worked proofs: numerical and algebraic — Transforming one side into the other, with a check of the result.
SL1.7 — Rational exponents and the laws of logarithms
Laws of exponents with rational exponents — Interpreting $a^{m/n}$ as roots and powers.
Laws of logarithms and change of base — Product, quotient, power laws and $\log_a x=\dfrac{\log_b x}{\log_b a}$.
Solving exponential equations — Using logarithms to solve equations where the unknown is an exponent.
SL1.8 — Sum of an infinite convergent geometric series
The sum to infinity — Convergence when $|r|<1$ and the formula $S_\infty=\dfrac{u_1}{1-r}$.
SL1.9 — The binomial theorem
Binomial coefficients: Pascal's triangle and nCr — Finding $\binom{n}{r}$ from Pascal's triangle and the factorial formula.
Expanding $(a+b)^n$ and finding a term — Applying the binomial theorem for $n\in\mathbb{N}$.
AHL1.10 — Counting principles and the extended binomial theorem
Counting principles — The multiplication (product) rule and the addition rule for counting arrangements.
Permutations and combinations — Ordered selections $^nP_r$ and unordered selections $^nC_r=\binom{n}{r}$.
Extended binomial theorem — Extending $(a+b)^n$ to fractional and negative indices, $n\in\mathbb{Q}$.
AHL1.11 — Partial fractions
Partial fractions with distinct linear factors — Splitting a rational function into $\frac{A}{x-p}+\frac{B}{x-q}$.
AHL1.12 — Complex numbers: Cartesian form
The imaginary unit and Cartesian form — $i^2=-1$, $z=a+bi$, real and imaginary parts, conjugate and arithmetic.
The complex plane, modulus and argument — Plotting $z$ on the Argand diagram; $|z|$ and $\arg z$.
AHL1.13 — Polar and Euler form
Modulus-argument (polar) and Euler form — $z=r(\cos\theta+i\sin\theta)=r\,\mathrm{cis}\,\theta=re^{i\theta}$ and conversions.
Products and quotients — Multiply moduli and add arguments; the geometric meaning.
AHL1.14 — De Moivre's theorem, powers and roots
Conjugate roots of polynomials — Complex roots of real-coefficient equations occur in conjugate pairs.
De Moivre's theorem — $[r\,\mathrm{cis}\,\theta]^n=r^n\,\mathrm{cis}\,n\theta$, proved by induction for $n\in\mathbb{Z}^+$.
Powers and roots of complex numbers — The $n$ distinct $n$th roots and their spacing on a circle.
AHL1.15 — Proof: induction, contradiction and counterexample
Proof by mathematical induction — Base case, assume $P(k)$, then prove $P(k)\Rightarrow P(k+1)$.
Proof by contradiction — Assume the statement is false and derive a contradiction.
Disproof by counterexample — One explained example that breaks a universal claim.
AHL1.16 — Systems of linear equations
Solving systems by elimination — Row reduction of up to three equations in three unknowns to a unique solution.
No solution and infinitely many solutions — Inconsistent systems and general solutions with a parameter.
2. Topic 2: Functions
Lines, functions and their graphs, transformations, quadratics, rational and exponential/log functions, and (HL) further functions and modelling.
SL2.1 — Straight lines
Forms of the equation of a line — Gradient-intercept, general and point-gradient forms; finding the gradient and the intercepts.
Parallel and perpendicular lines — The gradient conditions $m_1=m_2$ and $m_1\times m_2=-1$, with gradients of inclines.
SL2.2 — Functions, domain and range
Functions, domain, range and notation — What a function is, function notation such as $f(x)$, and how to read domain and range off a graph.
The inverse of a function — The informal idea that $f^{-1}$ undoes $f$, and the inverse as a reflection in $y=x$.
SL2.3 — The graph of a function
Graphing and sketching functions — Drawing versus sketching, labelling key features, and graphing sums and differences with technology.
SL2.4 — Key features of graphs
Key features of a graph — Maxima, minima, intercepts, symmetry, vertex, zeros and asymptotes, read using graphing technology.
Points of intersection — Finding where two curves or lines meet using graphing technology.
SL2.5 — Composite and inverse functions
Composite functions — The composition $(f\circ g)(x)=f(g(x))$ and the identity function.
Finding the inverse function — Reversing a function algebraically, using $(f\circ f^{-1})(x)=x$ and one-to-one existence.
SL2.6 — The quadratic function
Three forms of a quadratic — Standard, factorised and vertex forms of $f(x)=ax^2+bx+c$ and what each reveals.
Axis of symmetry and changing form — The axis of symmetry $x=-\tfrac{b}{2a}$ and converting between the three forms.
SL2.7 — Quadratic equations and the discriminant
Solving quadratic equations and inequalities — Factorisation, completing the square and the quadratic formula, plus quadratic inequalities.
The discriminant and nature of the roots — How $\Delta=b^2-4ac$ decides whether there are two, one or no real roots.
SL2.8 — Reciprocal and rational functions
The reciprocal function — The graph of $f(x)=\tfrac{1}{x}$, its two asymptotes and its self-inverse nature.
Rational functions of the form $\tfrac{ax+b}{cx+d}$ — Their graphs and the equations of the vertical and horizontal asymptotes.
SL2.9 — Exponentials and logarithms
Exponential functions — The graphs of $f(x)=a^x$ and $f(x)=e^x$, with their horizontal asymptote.
Logarithmic functions — The graphs of $f(x)=\log_a x$ and $f(x)=\ln x$, with their vertical asymptote.
Exponentials and logarithms as inverses — The reflection relationship and the identities linking $a^x$, $e^x$ and $\log$.
SL2.10 — Solving equations
Solving equations analytically and graphically — Exact algebraic methods, hidden quadratics, and using technology when no analytic method fits.
Applications of graphing and solving — Using these skills for real-life situations such as growth, decay and motion.
SL2.11 — Transformations of graphs
Translations and reflections — Shifting a graph with $f(x)+b$ and $f(x-a)$, and reflecting with $-f(x)$ and $f(-x)$.
Stretches — Vertical stretch $p\,f(x)$ and horizontal stretch $f(qx)$.
Composite transformations — Combining transformations and why the order in which they are applied matters.
AHL2.12 — Polynomial functions, roots and factors
Polynomials, zeros, roots and factors — Polynomial functions $P(x)=a_nx^n+\dots+a_0$, their graphs, and the link between zeros, roots and factors.
The factor and remainder theorems — $(x-a)$ is a factor iff $P(a)=0$; the remainder on dividing by $(x-a)$ is $P(a)$.
Sum and product of the roots — For $\sum a_rx^r=0$: sum of roots $=-\dfrac{a_{n-1}}{a_n}$, product $=\dfrac{(-1)^na_0}{a_n}$.
AHL2.13 — Rational functions and their asymptotes
Quadratic over linear: $\dfrac{ax^2+bx+c}{dx+e}$ — Graphs with a vertical asymptote and an oblique (slant) asymptote, found by division.
Linear over quadratic: $\dfrac{ax+b}{cx^2+dx+e}$ — Graphs with up to two vertical asymptotes and horizontal asymptote $y=0$.
AHL2.14 — Odd and even functions; inverses
Odd and even functions — Even $f(-x)=f(x)$ (line symmetry), odd $f(-x)=-f(x)$ (rotational symmetry); includes periodic functions.
Inverses with domain restriction; self-inverse — Restricting a domain so $f$ is one-to-one and $f^{-1}$ exists; self-inverse functions with $f^{-1}=f$.
AHL2.15 — Solving $g(x)\ge f(x)$
Solving $g(x)\ge f(x)$ graphically and analytically — Reading off where one graph lies above another, and solving algebraically for polynomials up to degree 3.
AHL2.16 — Further transformations and modulus equations
$y=|f(x)|$ and $y=f(|x|)$ — Reflecting negative parts upward, and mirroring the right side onto the left.
$y=\dfrac{1}{f(x)}$, $y=[f(x)]^2$ and $y=f(ax+b)$ — The reciprocal, square, and combined horizontal transformations of a graph.
Modulus equations and inequalities — Solving equations and inequalities involving $|\;|$, graphically and by cases.
3. Topic 3: Geometry and trigonometry
3D geometry, trigonometry, the unit circle, identities and equations, and (HL) reciprocal/inverse trig, compound angles and vectors.
SL3.1 — 3D geometry: distance, volume and angles
Distance and midpoint in 3D — Extend Pythagoras to three dimensions to find the distance between two points and their midpoint.
Volume and surface area of solids — Volume and surface area of pyramids, cones, spheres, hemispheres and combined solids.
Angles in three dimensions — Find the angle between two intersecting lines, or between a line and a plane, using right-angled triangles.
SL3.2 — Trigonometry: right triangles, sine and cosine rules
Right-angled trigonometry — Use sine, cosine and tangent ratios to find sides and angles in right-angled triangles.
The sine rule — Relate sides to the sines of their opposite angles in any triangle.
The cosine rule — Link three sides and one angle in any triangle.
Area of a triangle — Find the area from two sides and the included angle.
SL3.3 — Applications of trigonometry
Angles of elevation and depression — Apply trigonometry to sightlines measured above or below the horizontal.
Bearings — Describe direction as a three-figure angle measured clockwise from north.
Constructing and solving diagrams — Turn written statements into labelled diagrams and combine Pythagoras with right and non-right trigonometry.
SL3.4 — Radian measure, arcs and sectors
Radian measure — Measure angles in radians and convert to and from degrees.
Length of an arc — Find the length of a circular arc from the radius and the angle.
Area of a sector — Find the area of a pie-slice region of a circle.
SL3.5 — The unit circle and exact values
The unit circle definition — Define sine, cosine and tangent from a point moving round the unit circle.
Exact values — Know the exact ratios of the special angles and their multiples.
Symmetry across the quadrants — Relate angles in different quadrants to first-quadrant values.
A line through the origin — Connect the gradient of a line to the angle it makes with the x-axis.
SL3.6 — Trigonometric identities
The Pythagorean identity — Relate sine and cosine of the same angle.
Double angle identities — Express sin and cos of 2θ in terms of θ.
Relating the ratios — Find one ratio from another without finding the angle.
SL3.7 — Circular functions and their graphs
Graphs of sin, cos and tan — Recognise the shape, period and amplitude of the three circular functions.
Transformations of sinusoids — Build and read graphs of the form a sin(b(x + c)) + d.
Modelling real-life contexts — Use sinusoidal functions to model periodic phenomena.
SL3.8 — Solving trigonometric equations
Equations in a finite interval — Find every solution of a trig equation within a given interval, graphically and analytically.
Equations leading to quadratics — Reduce to a single ratio and solve as a quadratic.
AHL3.9 — Reciprocal and inverse trigonometric functions
Reciprocal trigonometric ratios — Define and use secant, cosecant and cotangent as reciprocals of cosine, sine and tangent.
Pythagorean identities — Derive and apply the two identities that follow from dividing the fundamental identity.
Inverse trigonometric functions — The functions arcsin, arccos and arctan: their restricted domains, ranges and graphs.
AHL3.10 — Compound and double angle identities
Compound angle identities — Expand the sine, cosine and tangent of a sum or difference of two angles.
Double angle identities — Derive the double angle identities from the compound angle identities and apply them.
AHL3.11 — Relationships between trigonometric functions
Symmetry relationships — Use the symmetry of the trig graphs and the unit circle to relate ratios of related angles.
AHL3.12 — Introduction to vectors
Vectors, components and base vectors — Represent vectors as directed line segments and in component/base-vector form.
Vector algebra — Add and subtract vectors, multiply by a scalar, and recognise parallel vectors.
Magnitude, unit vectors and distance — Compute the magnitude of a vector, form unit vectors, and find distance via displacement.
AHL3.13 — The scalar (dot) product
Definition and properties — Define the scalar product and use its algebraic properties.
Angle between vectors; perpendicular and parallel — Find the angle between two vectors and test for perpendicularity and parallelism.
AHL3.14 — Vector equation of a line
Forms of the equation of a line — Write a line in vector, parametric and Cartesian form in two or three dimensions.
Angle between two lines — Use the direction vectors and the scalar product to find the angle between two lines.
Kinematics applications — Interpret the line equation as motion, with the parameter as time and direction as velocity.
AHL3.15 — Pairs of lines and their intersections
Classifying pairs of lines — Distinguish coincident, parallel, intersecting and skew lines.
Points of intersection — Find where two lines meet by solving their parametric equations.
AHL3.16 — The vector (cross) product
Definition and properties — Define the vector product and use its algebraic properties.
Area of a parallelogram and triangle — Interpret the magnitude of the cross product geometrically to find areas.
AHL3.17 — Vector equations of a plane
Vector (parametric) form of a plane — Describe a plane using a point and two non-parallel direction vectors within it.
Normal form and Cartesian equation — Use a normal vector to write the scalar-product form and the Cartesian equation of a plane.
AHL3.18 — Intersections and angles with planes
Intersections of lines and planes — Find and interpret the intersection of a line with a plane, two planes, and three planes.
Angle between a line and a plane; two planes — Compute angles using normal vectors and direction vectors.
4. Topic 4: Statistics and probability
Data, correlation and regression, probability, discrete and normal distributions, and (HL) Bayes, and further distributions.
SL4.1 — Sampling and types of data
Populations, samples and types of data — Population vs sample, random samples, and discrete vs continuous data.
Reliability, bias and outliers — Judging data sources, handling missing/erroneous data, and the 1.5 x IQR outlier rule.
Sampling techniques — Simple random, convenience, systematic, quota and stratified sampling, and their effectiveness.
SL4.2 — Presentation of data
Frequency tables and histograms — Grouping discrete and continuous data into class intervals and displaying it with a frequency histogram.
Cumulative frequency graphs — Building a cumulative frequency curve and reading off median, quartiles, percentiles, range and IQR.
Box and whisker diagrams — The five-number summary, comparing distributions, marking outliers, and judging symmetry.
SL4.3 — Measures of central tendency and dispersion
Measures of central tendency — Mean, median and mode, including estimating the mean from grouped data and the modal class.
Measures of dispersion — Interquartile range, standard deviation and variance, found using technology.
Effect of transformations on data — How adding a constant or multiplying every value changes the mean and the standard deviation.
SL4.4 — Linear correlation and regression (y on x)
Scatter diagrams and correlation — Plotting bivariate data, describing correlation, and drawing a line of best fit by eye.
Pearson's correlation coefficient r — What r measures, its scale from -1 to 1, and finding it with technology.
The regression line of y on x — Finding y = ax + b with technology and interpreting the parameters a and b.
Prediction, correlation and causation — Using the line to predict, the dangers of extrapolation, and why correlation is not causation.
SL4.5 — Introduction to probability
Probability concepts and terminology — Trials, outcomes, equally likely outcomes, relative frequency, sample space and events.
Theoretical probability and the complement — The formula P(A) = n(A)/n(U) and complementary events A and A'.
Expected number of occurrences — Predicting how often an event happens over many trials.
SL4.6 — Combined events and conditional probability
Diagrams for probability — Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes.
Combined events and mutually exclusive events — The addition rule P(A u B) = P(A) + P(B) - P(A n B) and when P(A n B) = 0.
Conditional probability — Probability of A given B, and the multiplication rule.
Independent events and replacement — Independence P(A n B) = P(A)P(B), and problems with and without replacement.
SL4.7 — Discrete random variables
Discrete random variables and their distributions — Probability distributions of a discrete random variable and the total-probability condition.
Expected value E(X) — Calculating the mean of a discrete distribution and interpreting a fair game.
SL4.8 — The binomial distribution
When the binomial distribution applies — The conditions for a binomial model and the notation X ~ B(n, p).
Binomial probabilities using technology — Finding P(X = x) and cumulative probabilities with a GDC.
Mean and variance of the binomial — The results E(X) = np and Var(X) = np(1 - p).
SL4.9 — The normal distribution
The normal distribution and its properties — The bell curve, notation X ~ N(mu, sigma^2), and the 68-95-99.7 rule.
Normal probability calculations — Finding probabilities and proportions using technology.
Inverse normal calculations — Finding a data value from a given probability, with mean and standard deviation given.
SL4.10 — The regression line of x on y
The regression line of x on y — When to use x = cy + d, prediction, and why the choice of regression line matters.
SL4.11 — Conditional probability and independence
Formal conditional probability — The definition P(A|B) = P(A n B)/P(B) and its rearranged multiplication form.
Testing for independence — Using P(A|B) = P(A) = P(A|B') to decide whether events are independent.
SL4.12 — Standardization of normal variables
Standardized z-values — Converting to z = (x - mu)/sigma and what a z-value means.
Finding an unknown mean or standard deviation — Using z-values to solve for mu or sigma from a given probability.
AHL4.13 — Bayes' theorem
Bayes' theorem for two events — Reversing conditional probabilities: finding P(A|B) from P(B|A).
Bayes' theorem for three events — Using a partition of the sample space into three cases, with a medical-testing example.
AHL4.14 — Variance and continuous random variables
Variance of a discrete random variable — Var(X) = E(X^2) - [E(X)]^2 and the standard deviation.
Continuous random variables and pdfs — Probability density functions, area = probability, and the total-area condition.
Mode, median, mean and variance of a CRV — Locating the mode and median, and integrating for E(X) and Var(X).
Linear transformations of X — E(aX + b) = aE(X) + b and Var(aX + b) = a^2 Var(X).
5. Topic 5: Calculus
Limits, differentiation and integration and their applications, and (HL) further techniques, differential equations and Maclaurin series.
SL5.1 — Limits and the derivative
The concept of a limit — What a limit means and how to estimate one from a table or graph.
The derivative as gradient and rate of change — The gradient of a curve as a limit, the derivative as a rate of change, and notation.
SL5.2 — Increasing and decreasing functions
Increasing and decreasing functions — Using the sign of the first derivative to find where a function rises or falls.
SL5.3 — Differentiating polynomials
The power rule — Differentiating a single power term with the power rule.
Differentiating polynomials — Differentiating a sum of power terms, term by term.
SL5.4 — Tangents and normals
Equation of a tangent — Finding the equation of the tangent to a curve at a given point.
Equation of a normal — Finding the equation of the normal to a curve at a given point.
SL5.5 — Introduction to integration
Anti-differentiation — Integration as the reverse of differentiation, and the constant of integration.
Boundary conditions — Using a known point to determine the constant of integration.
Definite integrals and area — Definite integrals with technology, and the area under a positive curve.
SL5.6 — Differentiation: standard functions and rules
Derivatives of standard functions — Derivatives of powers, trig, exponential and log functions, plus sums and multiples.
The chain rule — Differentiating composite functions with the chain rule.
The product and quotient rules — Differentiating a product or a quotient of two functions.
SL5.7 — The second derivative
The second derivative — The second derivative and the relationship between the graphs of f, f-prime and f-double-prime.
SL5.8 — Maxima, minima and optimization
Local maximum and minimum points — Locating turning points and classifying them with the first or second derivative test.
Points of inflexion and concavity — Points of inflexion, and the concave-up / concave-down terminology.
Optimization — Using calculus to maximize or minimize a real-world quantity.
SL5.9 — Kinematics
Displacement, velocity and acceleration — Differentiating to move from displacement to velocity to acceleration.
Distance and displacement from velocity — Integrating velocity to recover displacement and total distance travelled.
SL5.10 — Indefinite integrals and further integration
Indefinite integrals of standard functions — Integrating standard functions and their composites with a linear function.
Reverse chain rule and substitution — Integrating expressions of the form k g-prime f(g(x)) by inspection or substitution.
SL5.11 — Definite integrals and areas
Definite integrals analytically — Evaluating definite integrals using the fundamental theorem of calculus.
Area between a curve and the x-axis — Finding area when the curve lies partly below the x-axis.
Areas between curves — Finding the area enclosed between two curves.
AHL5.12 — Limits and differentiation from first principles
Continuity and differentiability — Informal understanding of continuity and differentiability of a function at a point.
Limits: convergence and divergence — Understanding of limits, and convergence versus divergence (polynomials only).
Derivative from first principles — The formal definition of the derivative as a limit.
Higher derivatives and notation — Second and higher derivatives, and the standard notations.
AHL5.13 — L'Hopital's rule and limits
Indeterminate forms and l'Hopital's rule — Evaluating limits of the indeterminate forms 0/0 and infinity/infinity.
Repeated use and the Maclaurin method — Applying l'Hopital's rule more than once, or using a Maclaurin series.
AHL5.14 — Implicit differentiation, related rates and optimisation
Implicit differentiation — Differentiating relations that are not solved for y, using the chain rule.
Related rates of change — Linking the rates of two changing quantities through a shared equation.
Optimisation — Finding maxima and minima, including endpoint solutions.
AHL5.15 — Derivatives and integrals of further functions
Derivatives of further functions — Derivatives of tan, sec, cosec, cot, exponentials, logs and inverse trig.
Indefinite integrals of these functions — Reversing the standard derivatives, including composites with a linear function.
Partial fractions to rearrange the integrand — Splitting a rational integrand into partial fractions before integrating.
AHL5.16 — Integration by substitution and by parts
Integration by substitution — Reversing the chain rule; substitutions are given if not of the standard form.
Integration by parts — The by-parts formula, including repeated application.
AHL5.17 — Areas and volumes of revolution
Area between a curve and the y-axis — Integrating with respect to y to find the area enclosed with the y-axis.
Volumes of revolution — Rotating a region about the x-axis or the y-axis to generate a solid.
AHL5.18 — First order differential equations
Euler's method — Numerical solution of dy/dx = f(x,y) using Euler's method.
Separable variables — Solving dy/dx = g(x)h(y) by separating the variables.
Homogeneous equations — Solving dy/dx = f(y/x) with the substitution y = vx.
Integrating factor — Solving the linear equation y' + P(x)y = Q(x).
AHL5.19 — Maclaurin series
Standard Maclaurin expansions — The Maclaurin series and the standard expansions to know.
Building further series — Substitution, products, integration and differentiation of known series.
Maclaurin series from differential equations — Generating a series by repeatedly differentiating a differential equation.
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