## Math HL

### Course Description

This course is meant for students who have a strong background and interest in mathematics and related disciplines. Students should also have a high level of competency in analytical and technical skills. The course stresses problem-solving flexibility in a variety of contexts that will show the links between different areas of mathematics. Students will develop skills that will enable a lifetime of self-directed learning of mathematics.

By introducing students to the omnipresence and importance of mathematics across the natural and human sciences and by linking it to the Theory of Knowledge, students will develop an appreciation of the beauty and a better understanding of the global significance of mathematics.

This course is demanding and requires students to comprehend many topics of mathematics that will be approached in a variety of ways and studied in great depth. Therefore, students who do not wish to study the subject in such a rigorous manner should choose this course at the standard level.

The major goals of the course are that students will be able to:

• use appropriate mathematical notation and terminology to think, learn, and communicate effectively
• formulate and deliver coherent mathematical arguments with clarity and understanding
• explain the nature and significance of results
• solve problems with precision
• develop logical, critical, and analytical thinking
• use technology such as graphing calculators, and graphing software, in a constructive manner and to interpret the outcomes
• use mathematical modeling correctly
• conduct mathematical investigations and deliver proofs rigorously
• apply mathematical knowledge to solve complex problems that combine concepts from different areas of mathematics

### Text:

Haese: Mathematics (HL)

### Course Syllabus (240 hrs)

Core syllabus (190 hrs):

Topics: The course consists of the study of seven core topics and one option topic.

Topic 1—Algebra (30 hrs)

• Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series
• Exponents and logarithms. Laws of exponents; laws of logarithms. Change of base.
• Counting principles, including permutations and combinations.
• The binomial theorem
• Proof by mathematical induction.
• Complex numbers: Cartesian form, Modulus–argument form, the complex plane.
• Sums, products and quotients of complex numbers.
• Polar form of complex numbers.
• De Moivre’s theorem, powers and roots of a complex number, n-th roots of complex number.
• Conjugate roots of polynomial equations with real coefficients.
• Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.

Topic 2—Functions and equations (22 hrs)

• Concept of function: domain, range; image (value).
• Odd and even functions.
• Composite functions
• Identity function.
• One-to-one and one-to-many functions.
• Inverse function: $y&space;=&space;f^{-1}(x)$ including domain restriction. Self-inverse functions.
• Graph of function and its equation $y&space;=&space;f(x)$.
• Investigation of key features of graphs such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
• Function transformations: translations; stretches; reflections in the axes.
• The graphs of functions $y&space;=&space;\left&space;|&space;f(x)&space;\right&space;|$ and $y&space;=&space;f(\left&space;|&space;x&space;\right&space;|)$
• A graph of the inverse function as a reflection in $y&space;=&space;x$
• The reciprocal function: $f(x)&space;=&space;\frac{1}{f(x)}$ and its graph, as $y&space;=&space;f(x)$
• The rational function $f(x)&space;=&space;\frac{ax&space;+&space;b}{cx&space;+&space;d}$ and its graph.
• Exponential functions and their graphs.
• Logarithmic functions and their graphs.
• Quadratic functions; $(y&space;=&space;ax^{2}&space;+&space;bx&space;+&space;c)$ in general, standard, and transformational form and their graphs (axis of symmetry, vertex, x-intercepts, y-intercept)
• Solving polynomial equations both graphically and algebraically.
• Sum and product of the roots of polynomial equations
• Solving $a^{x}&space;=&space;b$, using logarithms
• Use of technology to solve a variety equations including those in which there is no appropriate analytic approach.
• Solutions of $g(x)&space;\geq&space;f(x)$. Graphical or algebraic methods for simple polynomials up to degree 3. Use of technology for these and other functions.

Topic 3—Circular functions and trigonometry (22 hrs)

• The circle: radian measure of angles; length of an arc; area of a sector
• Definitions of trig functions in terms of unit circle
• Definitions of reciprocal trig functions.
• Trig Identities – Pythagorean identities
• Compound angle identities.
• Double angle identities.
• Composite trig functions and their applications.
• Inverse trig functions: periods, domain, range, graphing
• Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization.
• The cosine rule. The sine rule including ambiguous case. Area of a triangle as $\frac{1}{2}ab\sin&space;C$.
• Trig applications

Topic 4—Vectors (24 hrs)

• Concept of a vector. Representation of vectors using direct line segments. Unit vectors, base vectors i, j, k. Components of a vector; column representation.
• Algebraic and geometric approaches to the following topics: the sum and difference of two vectors;
the zero vector, the vector $\mathbf{-v}$; The difference of two vectors; multiplication by a scalar; magnitude of a vector; unit vectors; base vectors; position vectors
• The scalar product of two vectors and algebraic properties of the scalar product
• Vector equation of a line $\boldsymbol{r}&space;=&space;\boldsymbol{a}&space;+&space;\lambda&space;\boldsymbol{b}$ in two and three dimensions. Simple applications to kinematics. The angle between two lines.
• Coincident, parallel, intersecting and skew lines; distinguishing between these cases. Points of
intersection.
• The vector product of two vectors, the determinant representation. Properties of the vector product.
Geometric interpretation of $\left&space;|&space;\boldsymbol{v}\times&space;\boldsymbol{w}\right&space;|$
• Vector equation of a plane $\boldsymbol{r}&space;=&space;\boldsymbol{a}&space;+&space;\lambda&space;\boldsymbol{b}&space;+&space;\mu&space;\boldsymbol{c}$. Use of normal vector to obtain the form $\boldsymbol{r}\cdot&space;\boldsymbol{n}&space;=&space;\boldsymbol{a}\cdot&space;\boldsymbol{n}$ Cartesian equation of a plane.
• Intersections of: a line with a plane; two planes; three planes. Angle between: a line and a plane; two planes.

Topic 5-Statistics and Probability (36 hrs)

• Concepts of population, sample, discrete and continuous data
• Presentation and measurements of data such as frequency tables, diagrams, box and whisker plots, mean, quartiles, standard deviation
• Grouped data: mid-interval values, interval values, upper and lower interval boundaries.
• Mean, variance, standard deviation.
• Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
• The probability of an event, the complementary event.
• Combined and mutually, combined events the formula exclusive events.
• Conditional probability; the definition
• Independent events; the definition
• Use of Bayes’ theorem for two events
• Concept of discrete and continuous random variables and their probability distributions.
• Definition and use of probability density functions.
• Expected value (mean), mode, median, variance and standard deviation; Knowledge and use of the formulae for E( X) and Var(X)
• Binomial distribution its mean and variance.
• Poisson distribution, its mean and variance.
• Normal distribution and its properties
• Standardization of normal variables

Topic 6-Calculus (48 hrs)

• Informal ideas of limit and convergence
• Definition of derivative as a limit from first principles.
• The derivative interpreted as a gradient function and as a rate of change.
• Finding equations of tangents and normals.
• Identifying increasing and decreasing functions.
• The second derivative.
• Higher derivatives.
• Derivative of power, trig, exponential, and logarithmic functions
• Interpretation of derivatives as gradient functions and rate of change
• Derivatives of reciprocal circular functions, their reciprocals and inverses
• The product and quotient rules
• The chain rule for composite functions
• Related rates of change.
• Implicit differentiation.
• Differentiation of a sum and a real multiple of the power, trig, exponential, and logarithmic functions.
• Local maximum and minimum points.
• Points of inflexion with zero and non-zero gradients.
• Graphical behaviour of functions: tangents and normals, behaviour for large x, horizontal and vertical asymptotes.
• Indefinite integration as anti-differentiation.
• Indefinite integration as anti-differentiation of power, exponential, reciprocal, sine and cosine functions
• The composites of any of these with the linear function
• Anti-differentiation with a boundary condition to determine the constant term
• Definite integrals.
• Areas under curves and areas between curves
• Volumes of revolution
• Kinematic problems involving displacement, s, velocity, v, and acceleration. Total distance travelled.
• Integration by substitution.
• Integration by parts.

Option Syllabus (48 hours)

Topic: Series and Differential Equations (48 hrs)

• Infinite sequences of real numbers and their convergence or divergence.
• Convergence of infinite series. Tests for convergence: comparison test; limit comparison test; ratio test; integral test.
• The p-series $\sum&space;\frac{1}{n^{p}}$
• Series that converge absolutely
• Series that converge conditionally
• Alternating series
• Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test
• Continuity and differentiability of a function at a point.
• Continuous functions and differentiable functions.
• The integral as a limit of a sum; lower sum and upper Riemann sums.
• Fundamental Theorem of Calculus.
• Improper integrals.
• First order differential equations: geometric interpretation using slope fields, including identification of isoclines.
• Numerical solutions of $\frac{dy}{dx}=f(x,y)$, using Euler’s method.
• Variables separable.
• Homogeneous differential equation
• Rolle’s theorem.
• Mean Value Theorem.
• Taylor polynomials and series, including Lagrange form of the error term.
• Maclaurin series
• The evaluation of limits of the form $\lim_{x&space;\to0&space;}\frac{f(x)}{g(x)}$ using l’Hôpital’s Rule and/or the Taylor series

Portfolio (10 hrs)

Two pieces of work based upon different areas of the syllabus, representing the following activities

• Type I: Mathematical investigation
• Type II: Mathematical modeling

### Assessment and Evaluation:

Year 1

• Tests – 50%
• Quizzes – 30%
• Homework – 20%

Year 2

External assessment (5 hrs) 80%

Test papers:

Paper 1 (2 hrs) No calculator allowed 30%

• Section A 15% – Compulsory short-response questions based on the compulsory core of the syllabus
• Section B 15% – Compulsory extended-response questions based on the compulsory core of the syllabus

Paper 2 (2 hrs) Graphic display calculator (GDC) required 30%

• Section A 15% – Compulsory short-response questions based on the compulsory core of the syllabus
• Section B 15% – Compulsory extended-response questions based on the compulsory core of the syllabus

Paper 3 (1 hr) Graphic display calculator (GDC) required 20%

• Extended-response questions based mainly on the syllabus options

Internal assessment 20%

Portfolio

A collection of two pieces of work assigned by the teacher and completed by the student during the
Course:

• Type 1: mathematical investigation
• Type 2: mathematical modeling.

### Markscheme:

Test questions will be graded according to the markscheme set forth by the IBO. Students will be awarded points for correct reasoning and usage of methods as well as correct answers. Abbreviations will be used to indicate the points awarded.

 Abbreviation M Marks awarded for attempting to use a correct Method; working must be seen. (M) Marks awarded for Method; may be implied correct by subsequent working. A Marks awarded for Answer or for Accuracy; often dependent on preceding M marks. (A) Marks awarded for Answer or for Accuracy; may be implied correct by subsequent working. R Marks awarded for clear Reasoning. N Marks awarded for correct answers if no working shown. AG Answer given in the question and so no marks awarded.