# Mathematics – Detailed Course Comparison

There are two mathematics courses in the Diploma Programme (DP), both taught at Higher level (HL) and Standard level (SL). The first is Mathematics: analysis and approaches and the second is Mathematics: applications and interpretation. Each course approaches topics at varying levels of teaching hours. This guide will provide some insights into which course will be the best fit for you.

The courses are separated by how they approach mathematics, described generally by the table below:

#### Mathematics: applications and interpretation

• Emphasis on modelling and statistics
• Develops strong skills in applying mathematics to the real-world
• Real mathematical problem solving using technology
• For students interested in social sciences, natural sciences, medicine, statistics, business, engineering, some economics, psychology and design

#### Mathematics: analysis and approaches

• Emphasis on algebraic methods
• Develop strong skills in mathematical thinking
• Real and abstract mathematical problem solving
• For students interested in mathematics, engineering physical sciences, and some economics

A detailed breakdown of the subject content is as follows:

### Standard Level Comparison

 Applications and Interpretation SL Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer Arithmetic sequences and series Use of the formulae for the nth term and the sum of the first n terms of the sequence Use of sigma notation for sums of arithmetic sequences, applications Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life Geometric sequences and series Use of the formulae for the nth term and the sum of the first n terms of the sequence Use of sigma notation for the sums of geometric sequences, applications Financial applications of geometric sequences and series: compound interest, annual depreciation Laws of exponents with integer exponents Introduction to logarithms with base 10 and e Numerical evaluation of logarithms using technology Approximation: decimal places, significant figures Upper and lower bounds of rounded numbers Percentage errors Estimation Amortization and annuities using technology Use technology to solve: systems of linear equations in up to 3 variables, polynomial equations Analysis and Approaches SL Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer Arithmetic sequences and series Use of the formulae for the nth term and the sum of the first n terms of the sequence Use of sigma notation for sums of arithmetic sequences Geometric sequences and series Use of sigma notation for the sums of geometric sequences Financial applications of geometric sequences and series: compound interest, annual depreciation. Laws of exponents with integer exponents Introduction to logarithms with base 10 and e Numerical evaluation of logarithms using technology Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof The symbols and notation for equality and identity Laws of exponents with rational exponents Laws of logarithms Change of base of a logarithm Solving exponential equations, including using logarithms Sum of infinite convergent geometric sequences The binomial theorem Use of Pascal’s triangle and nCr
 Applications and Interpretation SL Different forms of the equation of a straight line Gradient; intercepts Lines with gradients m1 and m2 Parallel lines m1 = m2. Perpendicular lines m1 × m2 = – 1 Concept of a function, domain, range and graph Function notation, for example f(x), v(t), C(n) The concept of a function as a mathematical mode Informal concept that an inverse function reverses or undoes the effect of a function Inverse function as a reflection in the line y = x, and the notation f-1(x) The graph of a function; its equation y = f(x) Creating a sketch from information given or a context, including transferring a graph from screen to paper Using technology to graph functions including their sums and differences Determine key features of graphs Finding the point of intersection of two curves or lines using technology Modelling with the following functions: linear, quadratic, exponential, cubic, sinusoidal Linear models f(x) = mx + c Quadratic models Exponential growth and decay models Equation of a horizontal asymptote Direct/inverse variation Cubic models Sinusoidal models Modelling skills: develop and fit the model, determine a reasonable domain, find the parameters, test and reflect upon the model, use the model Develop and fit the model: given a context recognize and choose an appropriate model and possible parameters Determine a reasonable domain for a model Find the parameters of a model Test and reflect upon the model: comment on the appropriateness and reasonableness of a model, justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation Use the model: reading, interpreting and making predictions based on the model Analysis and Approaches SL Different forms of the equation of a straight line Gradient; intercepts Lines with gradients m1 and m2 Parallel lines m1 = m2 Perpendicular lines m1 × m2 = -1 Concept of a function, domain, range and graph Function notation The concept of a function as a mathematical model Informal concept that an inverse function reverses or undoes the effect of a function Inverse function as a reflection in the line y = x, and the notation f-1(x) The graph of a function; its equation y = f(x) Creating a sketch from information given or a context, including transferring a graph from screen to paper Using technology to graph functions including their sums and differences Determine key features of graphs Finding the point of intersection of two curves or lines using technology Composite functions Identity function Finding the inverse function f–1(x) The quadratic function f(x) = ax2 + bx + c: its graph, y-intercept (0, c). Axis of symmetry The form f(x) = a (x − p)(x − q), x-intercepts (p, 0) and (q, 0) The form f(x) = a (x − h)2 + k, vertex (h, k) Solution of quadratic equations and inequalities The quadratic formula The discriminant Δ = b2 − 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots The reciprocal function f(x) = 1/x , x ≠ 0: its graph and self-inverse nature Rational functions Equations of vertical and horizontal asymptotes Exponential functions and their graphs Logarithmic functions and their graphs Solving equations, both graphically and analytically Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach Applications of graphing skills and solving equations that relate to real-life situations Transformations of graphs, translations, reflections (in both axes), vertical stretch with scale factor, horizontal stretch with scale factor Composite transformations
 Applications and Interpretation SL The distance between two points in three-dimensional space, and their midpoint Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids The size of an angle between two intersecting lines or between a line and a plane Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles The sine rule: a / sinA = b / sinB = c / sinC The cosine rule: c2 = a2 + b2 – 2abcosC cosC = (a2 + b2 – c2) / 2ab Area of a triangle as ½absinC Applications of right and non-right angled trigonometry, including Pythagoras’ theorem Angles of elevation and depression Construction of labelled diagrams from written statements The circle: length of an arc; area of a sector Equations of perpendicular bisectors Voronoi diagrams: sites, vertices, edges, cells Addition of a site to an existing Voronoi diagram Nearest neighbour interpolation Applications Analysis and Approaches SL The distance between two points in three-dimensional space, and their midpoint Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solid The size of an angle between two intersecting lines or between a line and a plane Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles The sine rule The cosine rule Area of a triangle using ½absinC Applications of right and non-right angled trigonometry, including Pythagoras’s theorem Angles of elevation and depression Construction of labelled diagrams from written statements The circle: radian measure of angles; length of an arc; area of a sector. Definition of cosθ, sinθ in terms of the unit circle Definition of tanθ as sinθ / cosθ Exact values of trigonometric ratios of 0, π/6 , π/4 , π/3 , π/2 and their multiples Extension of the sine rule to the ambiguous case The Pythagorean identity cos2θ + sin2θ = 1 Double angle identities for sine and cosine The relationship between trigonometric ratios The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs Composite functions of the form f(x) = asin(b(x + c)) + d Transformations Real-life contexts Solving trigonometric equations in a finite interval, both graphically and analytically Equations leading to quadratic equations in sinx, cosx or tanx
 Applications and Interpretation SL Concepts of population, sample, random sample, discrete and continuous data Reliability of data sources and bias in sampling Interpretation of outliers Sampling techniques and their effectiveness Presentation of data (discrete and continuous): frequency distributions (tables) Histograms Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR) Production and understanding of box and whisker diagrams Measures of central tendency (mean, median and mode) Estimation of mean from grouped data Model class Measures of dispersion (interquartile range, standard deviation and variance) Effect of constant changes on the original data Quartiles of discrete data Linear correlation of bivariate data Pearson’s product-moment correlation coefficient, r Scatter diagrams; lines of best fit, by eye, passing through the mean point Equation of the regression line of y on x Use of the equation of the regression line for prediction purposes Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event The probability of an event A is P(A) = n(A) / n(U) The complementary events A and A′ (not A) Expected number of occurrences Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities Combined events and mutually exclusive events Conditional probability Independent events Concept of discrete random variables and their probability distributions Expected value (mean), E(X) for discrete data Binomial distribution Mean and variance of the binomial distribution The normal distribution and curve Properties of the normal distribution Diagrammatic representation Normal probability calculations Inverse normal calculations Spearman’s rank correlation coefficient, rs Awareness of the appropriateness and limitations of Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient, and the effect of outliers on each Formulation of null and alternative hypotheses, H0 and H1 Significance levels p -values Expected and observed frequencies The χ2 test for independence: contingency tables, degrees of freedom, critical value The χ2 goodness of fit test The t -test Use of the p -value to compare the means of two populations Using one-tailed and two-tailed tests Analysis and Approaches SL Concepts of population, sample, random sample, discrete and continuous data Reliability of data sources and bias in sampling Interpretation of outliers Sampling techniques and their effectiveness Presentation of data (discrete and continuous): frequency distributions (tables) Histograms Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR) Production and understanding of box and whisker diagrams Measures of central tendency (mean, median and mode) Estimation of mean from grouped data Model class Measures of dispersion (interquartile range, standard deviation and variance) Effect of constant changes on the original data Quartiles of discrete data Linear correlation of bivariate data Pearson’s product-moment correlation coefficient, r Scatter diagrams; lines of best fit, by eye, passing through the mean point Equation of the regression line of y on x Use of the equation of the regression line for prediction purposes Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event The probability of an event A is P(A) = n(A) / n(U) The complementary events A and A′ (not A) Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities Combined events and mutually exclusive events Conditional probability Independent events Concept of discrete random variables and their probability distributions Expected value (mean), for discrete data Applications Binomial distribution Mean and variance of the binomial distribution The normal distribution and curve Properties of the normal distribution Diagrammatic representation Normal probability calculations Inverse normal calculations Equation of the regression line of x on y Use of the equation for prediction purposes Use of the probability formulae for conditional and independent events Standardization of normal variables (z– values) Inverse normal calculations where mean and standard deviation are unknown
 Applications and Interpretation SL Introduction to the concept of a limit Derivative interpreted as gradient function and as rate of change Increasing and decreasing functions Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0 Derivative of f(x) = axn is f ′(x) = anxn– 1, n ∈ ℤ The derivative of functions of the form f(x) = axn + bxn – 1 + – where all exponents are integers Tangents and normals at a given point, and their equations Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn – 1 + …, where n ∈ ℤ, n ≠ -1 Anti-differentiation with a boundary condition to determine the constant term Definite integrals using technology Area of a region enclosed by a curve y = f(x) and the x-axis, where f(x) > 0 Values of x where the gradient of a curve is zero Solution of f ′(x) = 0 Local maximum and minimum points Optimisation problems in context Approximating areas using the trapezoidal rule Analysis and Approaches SL Introduction to the concept of a limit Derivative interpreted as gradient function and as rate of change Increasing and decreasing functions Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0 Derivative of f(x) = axn is f ′(x) = anxn-1 , n ∈ ℤ The derivative of functions of the form f(x) = axn + bxn-1. . . where all exponents are integers Tangents and normals at a given point, and their equations Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn − 1 + …., where n ∈ ℤ, n ≠ − 1 Anti-differentiation with a boundary condition to determine the constant term Definite integrals using technology Area of a region enclosed by a curve y = f(x) and the x-axis, where f(x) > 0 Derivative of xn (n ∈ ℚ), sinx, cosx, ex and lnx Differentiation of a sum and a multiple of these functions The chain rule for composite functions The product and quotient rules The second derivative Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ″ Local maximum and minimum points Testing for maximum and minimum Optimization Points of inflexion with zero and non-zero gradients Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled Indefinite integral of xn (n ∈ ℚ), sinx, cosx, 1/x and ex The composites of any of these with the linear function ax + b Integration by inspection (reverse chain rule) or by substitution for expressions of the form: ∫ kg′(x)f(g(x))dx Definite integrals, including analytical approach Areas of a region enclosed by a curve y = f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology Areas between curves

The content of the Higher Level courses is everything in the Standard Level course plus this additional content:

### Higher Level Comparison

 Applications and Interpretation HL Laws of logarithms Simplifying expressions, both numerically and algebraically, involving rational exponents The sum of infinite geometric sequences Complex numbers: the number i such that i2 = -1 Cartesian form: z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument Calculate sums, differences, products, quotients, by hand and with technology Calculating powers of complex numbers, in Cartesian form, with technology The complex plane Complex numbers as solutions to quadratic equations of the form ax2 + bx + c = 0, a ≠ 0, with real coefficients where b2 – 4ac < 0 Modulus–argument (polar) form: z = r (cosθ + isinθ) = rcisθ Exponential form: z = reiθ Conversion between Cartesian, polar and exponential forms, by hand and with technology Calculate products, quotients and integer powers in polar or exponential forms Adding sinusoidal functions with the same frequencies but different phase shift angles Geometric interpretation of complex numbers Definition of a matrix: the terms element, row, column and order for m × n matrices Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for m × n matrices Multiplication of matrices Properties of matrix multiplication: associativity, distributivity and non-commutativity Identity and zero matrices Determinants and inverses of n × n matrices with technology, and by hand for 2 × 2 matrices Awareness that a system of linear equations can be written in the form Ax = b Solution of the systems of equations using inverse matrix Eigenvalues and eigenvectors Characteristic polynomial of 2 × 2 matrices Diagonalization of 2 × 2 matrices (restricted to the case where there are distinct real eigenvalues) Applications to powers of 2 × 2 matrices Analysis and Approaches HL Counting principles, including permutations and combinations Extension of the binomial theorem to fractional and negative indices Partial fractions Complex numbers: the number i, where i2 = -1 Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument The complex plane Modulus–argument (polar) form Euler form Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation Complex conjugate roots of quadratic and polynomial equations with real coefficients De Moivre’s theorem and its extension to rational exponents Powers and roots of complex numbers Proof by mathematical induction Proof by contradiction Use of a counterexample to show that a statement is not always true Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution
 Applications and Interpretation HL Composite functions in context The notation (f ∘ g)(x) = f(g(x)) Inverse function f-1, including domain restriction Finding an inverse function Transformations of graphs Translations: y = f(x) + b ;y = f(x – a) Reflections: in the x axis y = –f(x), and in the y axis y = f( –x) Vertical stretch with scale factor p: y = p f(x) Horizontal stretch with scale factor 1 / q : y = f(qx) Composite transformations Exponential models to calculate half-life Natural logarithmic models Sinusoidal models Logistic models Piecewise models Scaling very large or small numbers using logarithms Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters Interpretation of log-log and semi-log graphs Analysis and Approaches HL Polynomial functions, their graphs and equations; zeros, roots and factors The factor and remainder theorems Sum and product of the roots of polynomial equations Rational functions Odd and even functions Finding the inverse function, f-1(x), including domain restriction Self-inverse functions Solutions of g(x) ≥ f(x), both graphically and analytically The graphs of the functions, y = |f(x)| and y = f(|x|), y = 1 / f(x) , y = f (ax + b), y = [f(x)]2 Solution of modulus equations and inequalities
 Applications and Interpretation HL The definition of a radian and conversion between degrees and radians Using radians to calculate area of sector, length of arc The definitions of cosθ and sinθ in terms of the unit circle The Pythagorean identity: cos2θ + sin2θ = 1 Definition of tanθ as (sinθ / cosθ) Extension of the sine rule to the ambiguous case Graphical methods of solving trigonometric equations in a finite interval Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations Compositions of the above transformations Geometric interpretation of the determinant of a transformation matrix Concept of a vector and a scalar Representation of vectors using directed line segments Unit vectors; base vectors i, j, k Components of a vector; column representation The zero vector 0, the vector -v Position vectors OA= a Rescaling and normalizing vectors Vector equation of a line in two and three dimensions: r = a + λb, where b is a direction vector of the line Vector applications to kinematics Modelling linear motion with constant velocity in two and three dimensions Motion with variable velocity in two dimensions Definition and calculation of the scalar product of two vectors The angle between two vectors; the acute angle between two lines Definition and calculation of the vector product of two vectors Geometric interpretation of | v × w | Components of vectors Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges Degree of a vertex Simple graphs; complete graphs; weighted graphs Directed graphs; in degree and out degree of a directed graph Subgraphs; trees Adjacency matrices Walks Number of k-length walks (or less than k-length walks) between two vertices Weighted adjacency tables Construction of the transition matrix for a strongly-connected, undirected or directed graph Tree and cycle algorithms with undirected graphs Walks, trails, paths, circuits, cycles Eulerian trails and circuits Hamiltonian paths and cycles Minimum spanning tree (MST) graph algorithms: Kruskal’s and Prim’s algorithms for finding minimum spanning trees The Chinese Postman and Trevelling Salesman problems Analysis and Approaches HL Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ Pythagorean identities: 1 + tan2θ = sec2θ, 1+cot2θ = cosec2θ The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs Compound angle identities Double angle identity for tan Relationships between trigonometric functions and the symmetry properties of their graphs Concept of a vector; position vectors; displacement vectors Representation of vectors using directed line segments Base vectors i, j, k Components of a vector Algebraic and geometric approaches to the following: the sum and difference of two vectors, the zero vector 0, the vector –v, multiplication by scalar, kv, parallel vectors, magnitude of a vector, position vectors, displacement vector Proofs of geometrical properties using vectors The definition of the scalar product of two vectors The angle between two vectors Perpendicular vectors; parallel vectors Vector equation of a line in two and three dimensions: r = a + λb The angle between two lines Simple applications to kinematics Coincident, parallel, intersecting and skew lines, distinguishing between these cases Points of intersection The definition of the vector product of two vectors Properties of the vector product Geometric interpretation of | v × w | Vector equations of a plane: r = a + λb + μc, where b and c are non-parallel vectors within the plane r ∙ n = a · n, where n is a normal to the plane and a is the position vector of a point on the plane Cartesian equation of a plane ax + by + cz = d Intersections of: a line with a plane; two planes; three planes Angle between: a line and a plane; two planes
 Applications and Interpretation HL Design of valid data collection methods, such as surveys and questionnaires Selecting relevant variables from many variables Choosing relevant and appropriate data to analyse Categorizing numerical data in a χ2 table and justifying the choice of categorization Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the χ2 goodness of fit test Definition of reliability and validity Reliability tests Validity tests Non-linear regression Evaluation of least squares regression curves using technology Sum of square residuals (SSres) as a measure of fit for a model The coefficient of determination (R2) Evaluation of R2 using technology Linear transformation of a single random variable Expected value of linear combinations of n random variables Variance of linear combinations of n independent random variables Unbiased estimates for means and variance A linear combination of n independent normal random variables is normally distributed Central limit theorem Confidence intervals for the mean of a normal population Poisson distribution, its mean and variance Sum of two independent Poisson distributions has a Poisson distribution Critical values and critical regions Test for population mean for normal distribution Test for proportion using binomial distribution Test for population mean using Poisson distribution Use of technology to test the hypothesis that the population product moment correlation coefficient (ρ) is 0 for bivariate normal distributions. Type I and II errors including calculations of their probabilities Transition matrices Powers of transition matrices Regular Markov chains Initial state probability matrices Calculation of steady state and long-term probabilities by repeated multiplication of the transition matrix or by solving a system of linear equations Analysis and Approaches HL Use of Bayes’ theorem for a maximum of three events Variance of a discrete random variable Continuous random variables and their probability density functions Mode and median of continuous random variables Mean, variance and standard deviation of both discrete and continuous random variables The effect of linear transformations of X
 Applications and Interpretation HL The derivatives of sin x, cos x, tan x, ex , ln x, xn where n ∈ ℚ The chain rule, product rule and quotient rules Related rates of change The second derivative Use of second derivative test to distinguish between a maximum and a minimum point Definite and indefinite integration of xn where n ∈ ℚ, including n = – 1 , sin x, cos x, 1 / cos2x and ex Integration by inspection, or substitution of the form ∫ f(g(x))g′(x)dx Area of the region enclosed by a curve and the x or y-axes in a given interval Volumes of revolution about the x– axis or y– axis Kinematic problems involving displacement s, velocity v and acceleration a Setting up a model/differential equation from a context Solving by separation of variables Slope fields and their diagrams Euler’s method for finding the approximate solution to first order differential equations Numerical solution of dy / dx = f(x, y) Numerical solution of the coupled system dx / dt = f1 (x, y, t) and dy / dt = f2 (x, y, t) Phase portrait for the solutions of coupled differential equations of the form: dx / dt = ax + by and dy / dt = cx + dy Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations and saddle points Solutions of (d2x/dt2) = f[x, (dx/dt), t] by Euler’s method Analysis and Approaches HL Informal understanding of continuity and differentiability of a function at a point Understanding of limits (convergence and divergence) Definition of derivative from first principles Higher derivatives Evaluation of limits using L’Hôpital’s rule or the Maclaurin series Repeated use of l’Hôpital’s rule Implicit differentiation Related rates of change Optimisation problems Derivatives of tanx, secx, cosecx, cotx, ax, logax, arcsinx, arccosx, arctanx Indefinite integrals of the derivatives of any of the above functions The composites of any of these with a linear function Use of partial fractions to rearrange the integrand Integration by substitution Integration by parts Repeated integration by parts Area of the region enclosed by a curve and the y-axis in a given interval Volumes of revolution about the x-axis or y-axis First order differential equations Numerical solution of dy / dx = f(x, y) using Euler’s method Variables separable Homogeneous differential equation dy / dx = f( y / x ) using the substitution y = vx Solution of y′ + P(x)y = Q(x), using the integrating factor Maclaurin series to obtain expansions for ex, sinx, cosx, ln(1 + x), (1 + x)p , p ∈ ℚ Use of simple substitution, products, integration and differentiation to obtain other series Maclaurin series developed from differential equations
Mathematics: Compare the courses