INTERNATIONAL BACCALAURATE: DIPLOMA PROGRAMME
MATHEMATICS HIGHER LEVEL
Contents:
| Core Units |
| Optional Units |
| Assessments |
The program of study has 8 Core Units and 5 Options Units:
CORE UNITS (Compulsory)
1 Core: Number and Algebra (20h)
The aims of this section are: to introduce important results and methods of proof in algebra, and to extend the concept of number to include complex numbers.
1.1.- Arithmetic Sequences and series; Sum of finite and infinite geometric series.
Applications.
1.2.- Exponents and logarithms: laws of exponents; laws of logarithms.
1.3.- The binomial Theorem. (a+b)n
1.4.- Proof by mathematical induction. Forming conjectures to be proved by mathematical induction. Include sums of squares and cubes of natural numbers.
1.5.- Complex numbers: Cartesian and Trigonometric forms. The complex plane.
1.6.- Arithmetic of complex numbers. Include multiplication by “i” as a rotation
of 900 in the complex plane.
1.7.-
1.8.- Conjugate roots of polynomial equations with real coefficients.
2 Core : Functions and Equations (25h)
The aims of this section are: to introduce methods of solution for different types of equations, to explore the notion of function as a unifying theme in mathematics, to study certain functions in more depth and to explore the transformations of the graphical representations of functions.
2.1.- Concept of function 
: domain, range, image (value)
Composite functions f ° g , identity function. Inverse function. Domain restrictions
“one-one” and “many to one” functions
2.2.- Develop of function graphing skills. Identification of vertical and horizontal asymptotes. Also develop ability with the graphic calculator. Solution of f (x) = 0
to a given accuracy.
2.3.- Transformation of graphs: translations; stretches; reflections in the axes.
The graph of the inverse as a reflection in the y = x line. Link with matrix transformations.
2.4.- The reciprocal function f(x) = 1/x its graph; its self- inverse nature.
2.5.- The quadratic function; its graph y-intercept, x-intercept and vertex: V(h, k)
The discriminant. Link with transformation of functions: Y
= a (X- h)2 +k as Y =
2.6.- Solution of: f(x) = g(x); f and g linear or quadratic.
2.7.- Inequalities in one variable. Graphical representation. Solution of f(x) > g(x) where
f and g are linear or quadratic. Include the case when cross multiplication is not appropriate
e.g.
The use of absolute value in inequalities.
2.8.- Polynomial functions. The factor and remainder theorem with application to the solution of polynomial equations and inequalities. The use of synthetic division to obtain zeroes, remainders and values. The significance of multiple roots.
2.9.- The exponential function. Domain and range; its inverse Y = loga X. Solutions of
the equation : ax = b
2.10.- Natural logarithms. Applications: growth, decay, compound interest.
3 Core: Circular Functions and Trigonometry (25h)
The aims of this sections are: to use trigonometry to solve general triangles, to explore the behaviour of circular functions both graphically and algebraically and to introduce some important identities in trigonometry.
3.1.- The circle. Radians as measure of angles, length of an arc, area of a sector.
3.2.- Definitions of the trigonometric functions in terms of the unit circle. The Pythagorean identities
3.3.- The domains and ranges of the trigonometric functions; their periodic nature; their graphs. The inverses of the trigonometric functions and their graphs. The graph of
Y = a sin [b(X + c)] may be presented as a transformation of Y = sin (X).
3.4.- Addition, double-angle and half-angle formulae. The compound formula:
a cos(X)+b sin(X) = R cos(X-
)
3.5.- Composite functions of the form: f(x) = a sin(x + c) Solutions of f(x) = 0
in a given interval. Solutions of equations leading to quadratic or linear equations in sin(x).
Graphical interpretation of the above.
3.6.- Solution of triangles. Include the ambiguous case. Applications to practical problems in two and three dimensions. Sine and cosine rules. The area of triangles.
4 Core: Vector Geometry (25h)
The aims of this section are: to introduce the use of vectors in two and three dimensions. To facilitate solution of problems involving points, lines and planes and to enable the associated angles, distances and areas be calculated.
4.1.- Vectors as displacement in a plane and in three dimensions. Components of a vector;
magnitude; position vector; unit vector.
4.2.- The scalar (dot) product of two vectors: u.v = u1v1+u2v2+u3v3. Properties of the scalar product. Parallel and perpendicular vectors.
4.3.- The angle between two vectors. The projection of vector v in the direction of w.
Application to angle between lines cx + dy = q and ax + by = p as angle between normal vectors. Link with the cosine rule.
4.4.- The vector product of two vectors, include the geometric representation of the magnitude of v x w as the area of a parallelogram. The determinant representation.
4.5.- Vector equation of a line 
. Vector equation of a plane: 
Use of normal vector to obtain r n = a n . Cartesian equations of lines and planes.
; Cartesian equation of a plane: ax + by + cz = d.
4.6.- Intersection of two lines, a line with a plane; two planes; three planes. Angles between two lines; a line and a plane; two planes. Include Gaussian elimination and inversion of a matrix to find the intersection of three planes.
4.7.- Distances in two and three dimensions between points, lines and planes.
5 Core : Matrices and Transformations (20h)
The aims of this section are: to introduce the algebra of small matrices, to extend knowledge of transformations, to considered linear transformations of the plane represented by square matrices, to explore compositions of transformations and to link matrices to solutions of sets of linear equations.
5.1.- Definition of matrix; elements, row column, dimension. Applications to equations.
5.2.- Algebra of matrices; the identity matrix. Use of calculators.
5.3.- Determinants. Singularity of a matrix. Included matrices of dimension 3 x 3.
5.4.- The inverse of a matrix. Inverse of a composite : (PQ)-1 = P-1 Q-1
cofactors and minors.
5.5.- Linear transformation of vectors in two dimensions and their matrix representation;
rotations, reflections and enlargements. The geometric significance of the determinant.
5.6.- Compositions of linear transformations P,Q. (PQ denotes Q followed by P
5.7.- Solution of linear equations (three unknowns). Conditions of the existence of a unique solution, no solution and infinity of solutions.
6 Core Statistics (10h)
The aims of this section are: to explore methods of describing and presenting data, and to introduce methods of measuring central tendency and dispersion of data.
6.1.- Concept of population and sample . Discrete data and continuous data, frequency tables.
6.2.- Presentation of data. Group data; mid-interval values; interval width; upper and lower interval boundaries. Frequency histograms.
6.3.- Measures of central tendency: sample mean, median and mode.
6.4.- Cumulative frequency; cumulative frequency graphs; quartiles and percentiles. Box-whisker plots.
6.5.- Measures of dispersion: range; inter-quartile range; sample standard deviation and the unbiased estimate of the population standard deviation Sn-1. Use of calculator.
7 Core Probability (20h)
The aims of this section are: to extend knowledge of the concepts, notation and laws of probability, and to introduce some important probability distributions and theirs parameters.
7.1.- Sample space, U; the vent A. The probability of an event A. The complementary event of A (A´ ). P(A) + P(A´) = 1. Emphasis on the concept of equally likely outcomes.
7.2.- Combined events (or / and ). Mutually exclusive events. The relation P(A and B ) = 0
7.3.- Conditional probability. Independent events. Bayes´theorem for two events.
7.4.- Use of Venn diagrams to solve problems. Applications.
7.5.- Counting principles. Combinations and permutations. Problems of applications.
Link with the binomial theorem.
7.6.- Discrete probability distributions. Expectation, mode,
median, variance and standard deviation. Knowledge of the formulae 
7.7.- The binomial distribution, its mean and variance(without proof)
7.8.- Continuous probability distributions. Expectation, mode, median, variance and standard deviation. The concept of a continuous random variable; definition and use of probability density functions.
7.9.- The normal distribution. Standardization of a normal distribution; the use of the standard normal distribution table.
8 Core Calculus (50h)
The aims of this section are: to introduce the basic concepts and techniques of differential and integral calculus, and some of their applications.
8.1.- Informal ideas about limit and convergence. Use of calculator to investigate limits
numerically.
8.2.- Differentiation from first principles as the limit of the difference quotient.
Use of different notations.
8.3.- Differentiation of sums of functions and real multiple of functions. The chain rule for composite functions. Derivatives of trigonometric functions.
8.4.- Product and quotient rules. The second derivative. Differentiation of exponential and logarithmic functions.
8.5.- Graphical behavior of functions: tangents, normals and singularities, behavior for large |x|; asymptotes. Maxima and minima; points of inflexion.
8.6.- Applications of the first and second derivative to maximum and minimum problems.
Kinematics problems: displacement, velocity, acceleration.
8.7.- Implicit differentiation. Derivatives of the inverse trigonometric functions. Applications to related rates of change.
8.8.- Indefinite integration as anti-differentiation. Applications.
8.9.- Anti-differentiation with a boundary condition to determine the constant term.
Definite integrals. Areas under curves.
8.10.- Further integration: Integration by substitution; integration by parts and integration using partial fractions.
8.11.- Solution of first order differential equations by separation of variables. Transformation of a homogeneous equation by the substitution y = vx
OPTIONAL UNITS (the candidates select one)
9 Option: Statistics (35h)
The aims of this section are: to enable candidates to apply core knowledge of probability distributions and basic statistical calculations, and to make and test hypotheses. A practical approach is envisaged including statistical modeling tasks suitable for portfolio inclusion.
10 Option: Set, Relations and Groups (35h)
The aims of this section are: to study two important mathematical concepts, set, and groups.
The first allows for the extension and development of the notion of a function , while the second provides the framework to discover the common underlying structure unifying many familiar systems.
11 Option: Discrete Mathematics (35h)
The aims of this section are: To introduce topics appropriate for the student of mathematics and computer science who will later confront data structures, theory of programming languages and analysis of algorithms, and to explore a variety of applications and techniques of discrete methods and reasoning.
12 Option: Analysis and approximation (35h)
The aims of this section are: to use calculus knowledge to solve differential equations analytically and numerically, to approximate definite integrals, to solve non-linear equations by iteration, and to approximate functions by expansions of power series. The expectations are that candidates will use a graphic display calculator to perform computations and also to develop a sound understanding of the underlying mathematics.
13 Option: Euclidian Geometry and Conic Sections (35h)
The aims of this section are: To expose candidates to formal proofs in Euclidian Geometry
Thereby providing a broader understanding of the scope of mathematical proof, and to study conic sections using their Cartesian Equations.
ASSESSMENTS: External and Internal
EXTERNAL ASSESSMENTS. 80%
Written Papers 5h:
Paper 1 2h 30%
Twenty compulsory short-response questions based on part one of the syllabus,
the compulsory core.
Paper 2 3h 50%
Section A 35%
Five compulsory extended-response questions based on part one of the syllabus,
the compulsory core.
Section B 15%
Five extended response questions, one on each of the optimal topics in part II
of the syllabus; one question to be answered on the chosen topic.
AWARDING OF MARKS FOR EXTERNAL ASSESSMENT.-
Marks will be awarded according to the following four categories.
Method: Evidence of knowledge, the ability to apply concepts and skills, and the ability to analyze the problem in a logical manner.
Accuracy: Computational skills and numerical accuracy.
Reasoning: Clear reasoning, explanation and /or logical arguments.
Correct Statements: Results or conclusions express in words.
INTERNAL ASSESSMENT 20%
Portfolio
A collection of three pieces of work assigned by the teacher and completed by the candidate during the course. The assignments must be based on the different areas of the syllabus and represent the three activities: mathematical investigation, extended closed-problem solving and mathematical modeling. The portfolio is internally assessed by the teacher and externally moderated by the IBO.
ASSESSMENT CRITERIA FOR THE PORTFOLIO.-
A: Use of Notation and Terminology (0-2 points)
B: Communication (0-3 points)
C: Mathematical Content (0-5 points)
D: Results and Conclusions (0-3 points)
E: Making Conjectures (0-4 points)
D: Use of technology (0-3 points)