IB Math HL Syllabus

Mathematics HL First examinations 2008DIPLOMA PROGRAMMEMATHEMATICS HLFirst examinations 2008International Baccalaureate OrganizationBuenos Aires Cardiff Geneva New York SingaporeDiploma ProgrammeMathematics HLFirst published in September 2006International Baccalaureate OrganizationPeterson House, Malthouse Avenue, Cardiff GateCardiff, Wales GB CF23 8GLUnited KingdomPhone: + 44 29 2054 7777Fax: + 44 29 2054 7778Web site: © International Baccalaureate Organization 2006The International Baccalaureate Organization (IBO) was established in 1968 and is anon-profit, international educational foundation registered in Switzerland.The IBO is grateful for permission to reproduce and/or translate any copyrightmaterial used in this publication. Acknowledgments are included, where appropriate,and, if notified, the IBO will be pleased to rectify any errors or omissions at theearliest opportunity.IBO merchandise and publications in its official and working languages can be purchased through the IB store at . General ordering queriesshould be directed to the sales and marketing department in Cardiff.Phone: +44 29 2054 7746Fax: +44 29 2054 7779E-mail: sales@Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire.5007CONTENTSINTRODUCTION 1 NATURE OF THE SUBJECT 3 AIMS 6 OBJECTIVES 7 SYLLABUS OUTLINE 8 SYLLABUS DETAILS 9 ASSESSMENT OUTLINE 53 ASSESSMENT DETAILS 55INTRODUCTIONThe International Baccalaureate Diploma Programme (DP) is a rigorous pre-university course of studies, leading to examinations, that meets the needs of highly motivated secondary school students between the ages of 16 and 19 years. Designed as a comprehensive two-year curriculum that allows its graduates to fulfill requirements of various national education systems, the DP model is based on the pattern of no single country but incorporates the best elements of many. The DP is available in English, French and Spanish.The programme model is displayed in the shape of a hexagon with six academic areas surrounding the core. Subjects are studied concurrently and students are exposed to the two great traditions of learning: the humanities and the sciences.INTRODUCTIONDP students are required to select one subject from each of the six subject groups. At least three and not more than four are taken at higher level (HL), the others at standard level (SL). HL courses represent 240 teaching hours; SL courses cover 150 hours. By arranging work in this fashion, students are able to explore some subjects in depth and some more broadly over the two-year period; this is a deliberate compromise between the early specialization preferred in some national systems and the breadth found in others.Distribution requirements ensure that the science-orientated student is challenged to learn a foreign language and that the natural linguist becomes familiar with science laboratory procedures. While overall balance is maintained, flexibility in choosing HL concentrations allows the student to pursue areas of personal interest and to meet special requirements for university entrance.Successful DP students meet three requirements in addition to the six subjects. The interdisciplinary theory of knowledge (TOK) course is designed to develop a coherent approach to learning that transcends and unifies the academic areas and encourages appreciation of other cultural perspectives. The extended essay of some 4,000 words offers the opportunity to investigate a topic of special interest and acquaints students with the independent research and writing skills expected at university. Participation in the creativity, action, service (CAS) requirement encourages students to be involved in creative pursuits, physical activities and service projects in the local, national and international contexts.First examinations 2008NATURE OF THE SUBJECTIntroductionThe nature of mathematics can be summarized in a number of ways: for example, it can be seen as a well-defined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probably a combination of these, but there is no doubt that mathematical knowledge provides an important key to understanding the world in which we live. Mathematics can enter our lives in a number of ways: we buy produce in the market, consult a timetable, read a newspaper, time a process or estimate a length. Mathematics, for most of us, also extends into our chosen profession: artists need to learn about perspective; musicians need to appreciate the mathematical relationships within and between different rhythms; economists need to recognize trends in financial dealings; and engineers need to take account of stress patterns in physical materials. Scientists view mathematics as a language that is central to our understanding of events that occur in the natural world. Some people enjoy the challenges offered by the logical methods of mathematics and the adventure in reason that mathematical proof has to offer. Others appreciate mathematics as an aesthetic experience or even as a cornerstone of philosophy. This prevalence of mathematics in our lives provides a clear and sufficient rationale for making the study of this subject compulsory within the DP.Summary of courses availableBecause individual students have different needs, interests and abilities, there are four different courses in mathematics. These courses are designed for different types of students: those who wish to study mathematics in depth, either as a subject in its own right or to pursue their interests in areas related to mathematics; those who wish to gain a degree of understanding and competence better to understand their approach to other subjects; and those who may not as yet be aware how mathematics may be relevant to their studies and in their daily lives. Each course is designed to meet the needs of a particular group of students. Therefore, great care should be taken to select the course that is most appropriate for an individual student.In making this selection, individual students should be advised to take account of the following types of factor.•• Their own abilities in mathematics and the type of mathematics in which they can be successful •• Their own interest in mathematics, and those particular areas of the subject that may hold the most interest for them•• Their other choices of subjects within the framework of the DP•• Their academic plans, in particular the subjects they wish to study in future•• Their choice of careerTeachers are expected to assist with the selection process and to offer advice to students about how to choose the most appropriate course from the four mathematics courses available.NATURE OF THE SUBJECTMathematical studies SLThis course is available at SL only. It caters for students with varied backgrounds and abilities. More specifically, it is designed to build confidence and encourage an appreciation of mathematics in students who do not anticipate a need for mathematics in their future studies. Students taking this course need to be already equipped with fundamental skills and a rudimentary knowledge of basic processes. Mathematics SLThis course caters for students who already possess knowledge of basic mathematical concepts, and who are equipped with the skills needed to apply simple mathematical techniques correctly. The majority of these students will expect to need a sound mathematical background as they prepare for future studies in subjects such as chemistry, economics, psychology and business administration. Mathematics HLThis course caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems. Further mathematics SLThis course is available at SL only. It caters for students with a good background in mathematics who have attained a high degree of competence in a range of analytical and technical skills, and who display considerable interest in the subject. Most of these students will intend to study mathematics at university, either as a subject in its own right or as a major component of a related subject. The course is designed specifically to allow students to learn about a variety of branches of mathematics in depth and also to appreciate practical applications.Mathematics HL——course detailsThis course caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems. The nature of the subject is such that it focuses on developing important mathematical concepts in a comprehensible, coherent and rigorous way. This is achieved by means of a carefully balanced approach. Students are encouraged to apply their mathematical knowledge to solving problems set in a variety of meaningful contexts. Development of each topic should feature justification and proof of results. Students embarking on this course should expect to develop insight into mathematical form and structure, and should be intellectually equipped to appreciate the links between concepts in different topic areas. They should also be encouraged to develop the skills needed to continue their mathematical growth in other learning environments.The internally assessed component, the portfolio, offers students a framework for developing independence in their mathematical learning through engaging in mathematical investigation and mathematical modelling. Students will be provided with opportunities to take a considered approach to these activities, and to explore different ways of approaching a problem. The portfolio also allows students to work without the time constraints of a written examination and to develop skills in communicating mathematical ideas.NATURE OF THE SUBJECTThis course is a demanding one, requiring students to study a broad range of mathematical topics through a number of different approaches and to varying degrees of depth. Students wishing to study mathematics in a less rigorous environment should therefore opt for one of the standard level courses, mathematics SL or mathematical studies SL.© International Baccalaureate Organization 20065AIMSThe aims of all courses in group 5 are to enable students to: •• appreciate the multicultural and historical perspectives of all group 5 courses •• enjoy the courses and develop an appreciation of the elegance, power and usefulness of the subjects •• develop logical, critical and creative thinking •• develop an understanding of the principles and nature of the subject •• employ and refine their powers of abstraction and generalization •• develop patience and persistence in problem solving •• appreciate the consequences arising from technological developments •• transfer skills to alternative situations and to future developments •• communicate clearly and confidently in a variety of contexts.InternationalismOne of the aims of this course is to enable students to appreciate the multiplicity of cultural and historical perspectives of mathematics. This includes the international dimension of mathematics. Teachers can exploit opportunities to achieve this aim by discussing relevant issues as they arise and making reference to appropriate background information. For example, it may be appropriate to encourage students to discuss: •• differences in notation •• the lives of mathematicians set in a historical and/or social context •• the cultural context of mathematical discoveries •• the ways in which specific mathematical discoveries were made and the techniques used to make them •• how the attitudes of different societies towards specific areas of mathematics are demonstrated •• the universality of mathematics as a means of communication.6© International Baccalaureate Organization 2006OBJECTIVESHaving followed any one of the mathematics courses in group 5, students are expected to know and use mathematical concepts and principles. In particular, students must be able to: •• read, interpret and solve a given problem using appropriate mathematical terms •• organize and present information and data in tabular, graphical and/or diagrammatic forms •• know and use appropriate notation and terminology •• formulate a mathematical argument and communicate it clearly •• select and use appropriate mathematical strategies and techniques •• demonstrate an understanding of both the significance and the reasonableness of results •• recognize patterns and structures in a variety of situations, and make generalizations •• recognize and demonstrate an understanding of the practical applications of mathematics •• use appropriate technological devices as mathematical tools •• demonstrate an understanding of and the appropriate use of mathematical modelling.© International Baccalaureate Organization 20067SYLLABUS OUTLINEMathematics HLThe course consists of the study of seven core topics and one option topic.Total 240 hrsCore syllabus contentRequirements190 hrsAll topics in the core are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as presumed knowledge (PK). Topic 1——Algebra Topic 2——Functions and equations Topic 3——Circular functions and trigonometry Topic 4——Matrices Topic 5——Vectors Topic 6——Statistics and probability Topic 7——Calculus 20 hrs 26 hrs 22 hrs 12 hrs 22 hrs 40 hrs 48 hrsOption syllabus contentRequirements40 hrsStudents must study all the sub-topics in one of the following options as listed in the syllabus details. Topic 8——Statistics and probability Topic 9——Sets, relations and groups Topic 10——Series and differential equations Topic 11——Discrete mathematics 40 hrs 40 hrs 40 hrs 40 hrsPortfolio•• mathematical investigation •• mathematical modelling.10 hrsTwo pieces of work, based on different areas of the syllabus, representing the following two types of tasks:8© International Baccalaureate Organization 2006SYLLABUS DETAILSFormat of the syllabusThe syllabus to be taught is presented as three columns. •• Content: the first column lists, under each topic, the sub-topics to be covered. •• Amplifications/inclusions: the second column contains more explicit information on specific sub-topics listed in the first column. This helps to define what is required in terms of preparing for the examination. •• Exclusions: the third column contains information about what is not required in terms of preparing for the examination. Although the mathematics HL course is similar in content to parts of the mathematics SL course, there are differences. In particular, students and teachers are expected to take a more sophisticated approach for mathematics HL, during the course and in the examinations. Where appropriate, guidelines are provided in the second and third columns of the syllabus details (as indicated by the phrase ““See SL guide””). Teaching notes and calculator suggestions linked to the syllabus content are contained in a separate publication.Course of studyTeachers are required to teach all the sub-topics listed for the seven topics in the core, together with all the sub-topics in the chosen option. The topics in the syllabus do not need to be taught in the order in which they appear in this guide. Similarly, it is not necessary to teach all the topics in the core before starting to teach an option. Teachers should therefore construct a course of study that is tailored to the needs of their students and that integrates the areas covered by the syllabus, and, where necessary, the presumed knowledge (PK).Integration of portfolio assignmentsThe two pieces of work for the portfolio, based on the two types of tasks (mathematical investigation and mathematical modelling), should be incorporated into the course of study, and should relate directly to topics in the syllabus. Full details of how to do this are given in the section on internal assessment.Time allocationThe recommended teaching time for higher level courses is 240 hours. For mathematics HL, it is expected that 10 hours will be spent on work for the portfolio. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 230 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.© International Baccalaureate Organization 20069SYLLABUS DETAILSUse of calculatorsStudents are expected to have access to a graphic display calculator (GDC) at all times during the course. The minimum requirements are reviewed as technology advances, and updated information will be provided to schools. It is expected that teachers and schools monitor calculator use with reference to the calculator policy. Regulations covering the types of calculator allowed are provided in the Vade Mecum. Further information and advice is provided in the teacher support material. There are specific requirements for calculators used by students studying the statistics and probability option.Mathematics HL information bookletBecause each student is required to have access to a clean copy of this booklet during the examination, it is recommended that teachers ensure students are familiar with the contents of this document from the beginning of the course. The booklet is provided by the IBO and is published separately.Teacher support materialsA variety of teacher support materials will accompany this guide. These materials will include suggestions to help teachers integrate the use of GDCs into their teaching, guidance for teachers on the marking of portfolios, and specimen examination papers and markschemes. These will be distributed to all schools.External assessment guidelinesIt is recommended that teachers familiarize themselves with the section on external assessment guidelines, as this contains important information about the examination papers. In particular, students need to be familiar with notation the IBO uses and the command terms, as these will be used without explanation in the examination papers.Presumed knowledgeGeneralStudents are not required to be familiar with all the topics listed as PK before they start this course. However, they should be familiar with these topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics designated as PK that are unknown to their students at the start of the course are included at an early stage. They should also take into account the existing mathematical knowledge of their students to design an appropriate course of study for mathematics HL. This list of topics is not designed to represent the outline of a course that might lead to the mathematics HL course. Instead, it lists the knowledge, together with the syllabus content, that is essential to successful completion of the mathematics HL course Students must be familiar with SI (Système International) units of length, mass and time, and their derived units.10© International Baccalaureate Organization 2006SYLLABUS DETAILSTopicsNumber and algebraRoutine use of addition, subtraction, multiplication and division using integers, decimals and fractions, including order of operations. Example: 2 ( 3 + 4 × 7 ) = 62 . Simple positive exponents. Examples: 23 = 8 ; (−3)3 = −27 ; (−2) 4 = 16 .Simplification of expressions involving roots (surds or radicals).Examples:27 + 75 = 8 3 ;3 × 5 = 15 .Prime numbers and factors, including greatest common factors and least common multiples. Simple applications of ratio, percentage and proportion, linked to similarity. Definition and elementary treatment of absolute value (modulus), a . Rounding, decimal approximations and significant figures, including appreciation of errors. Expression of numbers in standard form (scientific notation), that is, a × 10k , 1 ≤ a < 10 , k ∈ .Concept and notation of sets, elements, universal (reference) set, empty (null) set, complement, subset, equality of sets, disjoint sets. Operations on sets: union and intersection. Commutative, associative and distributive properties. Venn diagrams. Number systems: natural numbers; integers, ; rationals, , and irrationals; real numbers, .Intervals on the real number line using set notation and using inequalities. Expressing the solution set of a linear inequality on the number line and in set notation. The concept of a relation between the elements of one set and between the elements of one set and those of another set. Mappings of the elements of one set onto or into another, or the same, set. Illustration by means of tables, diagrams and graphs. Basic manipulation of simple algebraic expressions involving factorization and expansion. Examples: ab + ac = a(b + c) ; (a ± b) 2 = a 2 + b 2 ± 2ab ; a 2 − b 2 = (a − b)(a + b) ; 3x 2 + 5 x + 2 = (3 x + 2)( x + 1) ; xa − 2a + xb − 2b = ( x − 2)(a + b) . Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas, particularly the sciences, should be included. The linear function xax + b and its graph, gradient and y-intercept.Addition and subtraction of algebraic fractions with denominators of the form ax + b .Example:2x 3x + 1 + . 3x − 1 2 x + 4ac > bc ; a > b, c < 0 5 . 7 ac < bc .The properties of order relations: <, , >, . Examples: a > b, c > 0Solution of equations and inequalities in one variable, including cases with rational coefficients. Example: 3 2x 1 − = (1 − x ) 7 5 2 x=Solution of simultaneous equations in two variables.© International Baccalaureate Organization 200611SYLLABUS DETAILSGeometryElementary geometry of the plane including the concepts of dimension for point, line, plane and space. Parallel and perpendicular lines, including m1 = m2 , and m1m2 = −1 . Geometry of simple plane figures. The function x ax + b : its graph, gradient and y-intercept.Angle measurement in degrees. Compass directions and bearings. Right-angle trigonometry. Simple applications for solving triangles. Pythagoras’’ theorem and its converse. The Cartesian plane: ordered pairs ( x, y ) , origin, axes. Mid-point of a line segment and distance between two points in the Cartesian plane. Simple geometric transformations: translation, reflection, rotation, enlargement. Congruence and similarity, including the concept of scale factor of an enlargement. The circle, its centre and radius, area and circumference. The terms ““arc””, ““sector””, ““chord””, ““tangent”” and ““segment””. Perimeter and area of plane figures. Triangles and quadrilaterals, including parallelograms, rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound shapes.StatisticsDescriptive statistics: collection of raw data, display of data in pictorial and diagrammatic forms (for example, pie charts, pictograms, stem and leaf diagrams, bar graphs and line graphs). Calculation of simple statistics from discrete data, including mean, median and mode.12© International Baccalaureate Organization 2006Core syllabus contentTopic 1——Core: AlgebraAimsThe aim of this section is to introduce students to some basic algebraic concepts and applications.20 hrsDetailsContent 1.1 Amplifications/inclusions ExclusionsArithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation.Examples of applications: compound interest and population growth.1.2Exponents and logarithms. Laws of exponents; laws of logarithms. Change of base.Elementary treatment only is required.log b a =log c a . log c b Formula for n Pr . Permutations where some objects are identical.1.3Counting principles, including permutations and Simple applications only. combinations. n The formula for also denoted by n Cr . r The binomial theorem: expansion of ( a + b) n , n ∈ .See SL guide© International Baccalaureate Organization 200613Topic 1——Core: Algebra (continued)Content 1.4 Amplifications/inclusions ExclusionsProof by mathematical induction. Forming conjectures to be proved by mathematical induction.Proof of binomial theorem.1.5Complex numbers: the number i = −1 ; the terms real part, imaginary part, conjugate, modulus and argument. Cartesian form z = a + ib . Modulus––argument form z = r ( cosθ + isin θ ) . Awareness that z = r ( cosθ + isin θ ) can be written as z = r eiθ and z = r cisθ . The complex plane. The complex plane is also known as the Argand diagram. Loci in the complex plane.1.6 1.7Sums, products and quotients of complex numbers. De Moivre’’s theorem. Powers and roots of a complex number. Proof by mathematical induction for n ∈+.1.8Conjugate roots of polynomial equations with real coefficients.Equations with complex coefficients.© International Baccalaureate Organization 200614Topic 2——Core: Functions and equationsAims26 hrsThe aims of this section are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of a GDC in both the development and the application of this topic.DetailsContent 2.1 Amplifications/inclusions ExclusionsConcept of function f : x range; image (value).f ( x) : domain,On examination papers: if the domain is the set The term ““codomain””. of real numbers then the statement ““ x ∈ ”” will be omitted. The composite function ( f g )( x) is defined as f ( g ( x)) . Distinction between one-to-one and many-to-one functions. Domain restriction.See SL guideComposite functions f g ; identity function.Inverse function f −1 .2.2The graph of a function; its equation y = f ( x) .On examination papers: questions may be set that require the graphing of functions that do not explicitly appear on the syllabus.Function graphing skills: use of a GDC to graph a variety of functions investigation of key features of graphs solutions of equations graphically. Identification of asymptotes. May be referred to as roots of equations, or zeros of functions.© International Baccalaureate Organization 200615Topic 2——Core: Functions and equations (continued)Content 2.3 Amplifications/inclusions ExclusionsTransformations of graphs: translations; stretches; reflections in the axes.Translations: y = f ( x) + b; y = f ( x − a ) . Stretches: y = pf ( x); y = f ( x q ) . Reflections (in both axes): y = − f ( x); y = f (− x) . Examples: y = x 2 used to obtain y = 3 x 2 + 2 by a stretch of scale factor 3 in the y-direction 0 . followed by a translation of 2y = sin x used to obtain y = 3sin 2 x by a stretch of scale factor 3 in the y-direction and a 1 stretch of scale factor in the x-direction. 2 The graph of y = f −1 ( x) as the reflection in the line y = x of the graph of y = f ( x) . The graph of y = 1 from y = f ( x) . f ( x)The graphs of the absolute value functions, y = f ( x) and y = f ( x ) .2.4The reciprocal function x graph; its self-inverse nature.1 , x ≠ 0 : its x© International Baccalaureate Organization 200616。