Diploma Programme

Mathematics HL guide

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First examinations 2014

This pre-publication has been released for information only, to help with planning. Although the content has been substantially finalized, the IB reserves the right to make further changes to the text before publication as it sees fit. This version must be replaced as soon as the final version is published, in order to avoid any confusion.

© International Baccalaureate Organization 2011

Contents Introduction Purpose of this document The Diploma Programme Nature of the subject Aims Assessment objectives Syllabus outline

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Approaches to the teaching and learning of mathematics HL Prior learning topics Syllabus content

External assessment Internal assessment

Appendices

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Notation list

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Glossary of command terms

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Assessment outline

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Assessment in the Diploma Programme

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Glossary of terminology: Discrete mathematics

Assessment

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Syllabus

Introduction

Purpose of this document This publication is intended to guide the planning, teaching and assessment of the subject in schools. Subject teachers are the primary audience, although it is expected that teachers will use the guide to inform students and parents about the subject. This guide can be found on the subject page of the online curriculum centre (OCC) at http://occ.ibo.org, a password-protected IB website designed to support IB teachers. It can also be purchased from the IB store at http://store.ibo.org.

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Additional resources

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Additional publications such as teacher support materials, subject reports, internal assessment guidance and grade descriptors can also be found on the OCC. Specimen and past examination papers as well as markschemes can be purchased from the IB store.

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Teachers are encouraged to check the OCC for additional resources created or used by other teachers. Teachers can provide details of useful resources, for example: websites, books, videos, journals or teaching ideas.

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First examinations 2014

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The Diploma Programme The Diploma Programme is a rigorous pre-university course of study designed for students in the 16 to 19 age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to develop intercultural understanding, open-mindedness, and the attitudes necessary for them to respect and evaluate a range of points of view.

The Diploma Programme hexagon

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The course is presented as six academic areas enclosing a central core (see figure 1). It encourages the concurrent study of a broad range of academic areas. Students study: two modern languages (or a modern language and a classical language); a humanities or social science subject; an experimental science; mathematics; one of the creative arts. It is this comprehensive range of subjects that makes the Diploma Programme a demanding course of study designed to prepare students effectively for university entrance. In each of the academic areas students have flexibility in making their choices, which means they can choose subjects that particularly interest them and that they may wish to study further at university.

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[Production insert DP hexagon diagram here.] Figure 1

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Diploma Programme model

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Choosing the right combination

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Students are required to choose one subject from each of the six academic areas, although they can choose a second subject from groups 1 to 5 instead of a group 6 subject. Normally, three subjects (and not more than four) are taken at higher level (HL), and the others are taken at standard level (SL). The IB recommends 240 teaching hours for HL subjects and 150 hours for SL. Subjects at HL are studied in greater depth and breadth than at SL.

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At both levels, many skills are developed, especially those of critical thinking and analysis. At the end of the course, students’ abilities are measured by means of external assessment. Many subjects contain some element of coursework assessed by teachers. The course is available for examinations in English, French and Spanish, with the exception of groups 1 and 2 courses where examinations are in the language of study.

The core of the hexagon All Diploma Programme students participate in the three course requirements that make up the core of the hexagon. Reflection on all these activities is a principle that lies at the heart of the thinking behind the Diploma Programme. The theory of knowledge course encourages students to think about the nature of knowledge, to reflect on the process of learning in all the subjects they study as part of their Diploma Programme course, and to make connections across the academic areas. The extended essay, a substantial piece of writing of up to 4,000 words, enables students to investigate a topic of special interest that they have chosen themselves. It also encourages them to develop the skills of independent research that will be expected at university. Creativity, action, service involves students in experiential learning through a range of artistic, sporting, physical and service activities.

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The IB mission statement and the IB learner profile

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The Diploma Programme aims to develop in students the knowledge, skills and attitudes they will need to fulfill the aims of the IB, as expressed in the organization’s mission statement and the learner profile. Teaching and learning in the Diploma Programme represent the reality in daily practice of the organization’s educational philosophy.

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Nature of the subject

Introduction

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The 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: visual 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, with all its interdisciplinary connections, provides a clear and sufficient rationale for making this subject compulsory for students studying for the full diploma.

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Summary of courses available

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Because 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 to understand better 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.

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their own abilities in mathematics and the type of mathematics in which they can be successful

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In making this selection, individual students should be advised to take account of the following factors:

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 Diploma Programme

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their academic plans, in particular the subjects they wish to study in future

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their choice of career.

Teachers are expected to assist with the selection process and to offer advice to students.

Mathematical studies SL This course is available only at standard level, and is equivalent in status to mathematics SL, but addresses different needs. It has an emphasis on applications of mathematics, and the largest section is on statistical techniques. It is designed for students with varied mathematical backgrounds and abilities. It offers students opportunities to learn important concepts and techniques and to gain an understanding of a wide variety of mathematical topics. It prepares students to be able to solve problems in a variety of settings, to develop more sophisticated mathematical reasoning and to PRE-PUBLICATION Mathematics HL guide

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enhance their critical thinking. The individual project is an extended piece of work based on personal research involving the collection, analysis and evaluation of data. Students taking this course are well prepared for a career in social sciences, humanities, languages or arts. These students may need to utilize the statistics and logical reasoning that they have learned as part of the mathematical studies SL course in their future studies.

Mathematics SL This 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 HL

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Further mathematics HL

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This 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.

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This course is available only at higher level. It caters for students with a very strong 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 expect 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. It is expected that students taking this course will also be taking mathematics HL.

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Note: Mathematics HL is an ideal course for students 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 or technology. It should not be regarded as necessary for such students to study further mathematics HL. Rather, further mathematics HL is an optional course for students with a particular aptitude and interest in mathematics, enabling them to study some wider and deeper aspects of mathematics, but is by no means a necessary qualification to study for a degree in mathematics.

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Mathematics HL—course details The course 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 solve 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 exploration, offers students the opportunity for developing independence in their mathematical learning. Students are encouraged to take a considered approach to various mathematical activities and to explore different mathematical ideas. The exploration also allows students to work without the time constraints of a written examination and to develop the skills they need for communicating mathematical ideas.

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This 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. Students who wish to study an even more rigorous and demanding course should consider taking further mathematics HL in addition to mathematics HL.

Prior learning

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Mathematics is a linear subject, and it is expected that most students embarking on a DP mathematics course will have studied mathematics for at least 10 years. There will be a great variety of topics studied, and differing approaches to teaching and learning. Thus students will have a wide variety of skills and knowledge when they start the mathematics HL course. Most will have some background in arithmetic, algebra, geometry, trigonometry, probability and statistics. Some will be familiar with an inquiry approach, and may have had an opportunity to complete an extended piece of work in mathematics.

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At the beginning of the syllabus section, there is a list of topics that are considered to be prior learning for the mathematics HL course. It is recognized that this may contain topics that are unfamiliar to some students, but it is anticipated that there may be other topics in the syllabus itself that these students have already encountered. Teachers should plan their teaching to incorporate topics mentioned that are unfamiliar to their students.

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Links to the Middle Years Programme

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The prior learning topics for the Diploma Programme (DP) courses have been written in conjunction with the Middle Years Programme (MYP) mathematics guide. The approaches to teaching and learning for DP mathematics build on the approaches used in the MYP. These include investigations, exploration and a variety of different assessment tools.

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A continuum document called Mathematics: The MYP–DP continuum is available on the DP mathematics home pages of the online curriculum centre (OCC). This extensive publication focuses on the alignment of mathematics across the MYP and the DP. It was developed in response to feedback provided by IB World Schools, which expressed the need to articulate the transition of mathematics from the MYP to the DP. The publication also highlights the similarities and differences between MYP and DP mathematics, and is a valuable resource for teachers.

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Mathematics and theory of knowledge The Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed that these all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by data from sense perception, mathematics is dominated by reason, and some mathematicians argue that their subject is a language, that it is, in some sense, universal. However, there is also no doubt that mathematicians perceive beauty in mathematics, and that emotion can be a strong driver in the search for mathematical knowledge. As an area of knowledge, mathematics seems to supply a certainty perhaps missing in other disciplines. This may be related to the “purity” of the subject that makes it sometimes seem divorced from reality. However, mathematics has also provided important knowledge about the world, and the use of mathematics in science and technology has been one of the driving forces for scientific advances. Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there “waiting to be discovered” or is it a human creation? PRE-PUBLICATION Mathematics HL guide

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Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should be encouraged to raise such questions themselves, in mathematics and TOK classes. This includes questioning all the claims made above. Examples of issues relating to TOK are given in the “Links” column of the syllabus. Teachers could also discuss questions such as those raised in the “Areas of knowledge” section of the TOK guide.

Mathematics and the international dimension Mathematics is in a sense an international language, and, apart from slightly differing notation, mathematicians from around the world can communicate within their field. Mathematics transcends politics, religion and nationality, yet throughout history great civilizations owe their success in part to their mathematicians being able to create and maintain complex social and architectural structures.

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Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by Arabic, Greek, Indian and Chinese civilizations, among others. Teachers could use timeline websites to show the contributions of different civilizations to mathematics, but not just for their mathematical content. Illustrating the characters and personalities of the mathematicians concerned and the historical context in which they worked brings home the human and cultural dimension of mathematics.

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The importance of science and technology in the everyday world is clear, but the vital role of mathematics is not so well recognized. It is the language of science, and underpins most developments in science and technology. A good example of this is the digital revolution, which is transforming the world, as it is all based on the binary number system in mathematics.

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Many international bodies now exist to promote mathematics. Students are encouraged to access the extensive websites of international mathematical organizations to enhance their appreciation of the international dimension and to engage in the global issues surrounding the subject.

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Examples of global issues relating to international-mindedness (Int) are given in the “Links” column of the syllabus.

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Aims

Group 5 aims The aims of all mathematics courses in group 5 are to enable students to: enjoy mathematics, and develop an appreciation of the elegance and power of mathematics

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develop an understanding of the principles and nature of mathematics

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communicate clearly and confidently in a variety of contexts

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develop logical, critical and creative thinking, and patience and persistence in problem-solving

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employ and refine their powers of abstraction and generalization

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apply and transfer skills to alternative situations, to other areas of knowledge and to future developments

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appreciate how developments in technology and mathematics have influenced each other

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appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics

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appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives

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appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course.

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Assessment objectives Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics HL course, students will be expected to demonstrate the following. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.

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Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems.

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Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.

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Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems.

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Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions.

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Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analysing information, making conjectures, drawing conclusions, and testing their validity.

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Syllabus

Syllabus outline Teaching hours

Syllabus component

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All topics 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 prior learning. 30

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Topic 1 Algebra

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Topic 2 Functions and equations Circular functions and trigonometry

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Topic 4

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Topic 3

Topic 5 Statistics and probability Topic 6

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Calculus

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Vectors

Option syllabus content

22 22 24 36 48 48

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Students must study all the sub-topics in one of the following options as listed in the syllabus details. Statistics and probability Topic 8

Sets, relations and groups Topic 9 Calculus Topic 10 Discrete mathematics Exploration

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Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. Total teaching hours

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Approaches to the teaching and learning of mathematics HL Throughout the DP mathematics HL course, students should be encouraged to develop their understanding of the methodology and practice of the discipline of mathematics. The processes of mathematical inquiry, mathematical modelling and applications and the use of technology should be introduced appropriately. These processes should be used throughout the course, and not treated in isolation.

Mathematical inquiry

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The IB learner profile encourages learning by experimentation, questioning and discovery. In the IB classroom, students should generally learn mathematics by being active participants in learning activities rather than recipients of instruction. Teachers should therefore provide students with opportunities to learn through mathematical inquiry. This approach is illustrated in the diagram below. [Production insert flowchart 1 here.]

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Mathematical modelling and applications

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Students should be able to use mathematics to solve problems in the real world. Engaging students in the mathematical modelling process provides such opportunities. Students should develop, apply and critically analyse models. This approach is illustrated in the diagram below.

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Technology

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Technology is a powerful tool in the teaching and learning of mathematics. Technology can be used to enhance visualization and support student understanding of mathematical concepts. It can assist in the collection, recording, organization and analysis of data. Technology can increase the scope of the problem situations that are accessible to students. The use of technology increases the feasibility of students working with interesting problem contexts where students reflect, reason, solve problems and make decisions. As teachers tie together the unifying themes of mathematical inquiry, mathematical modelling and applications and the use of technology, they should begin by providing substantial guidance, and then gradually encourage students to become more independent as inquirers and thinkers. IB students should learn to become strong communicators through the language of mathematics. Teachers should create a safe learning environment in which students are comfortable as risk-takers. Teachers are encouraged to relate the mathematics being studied to other subjects and to the real world, especially topics that have particular relevance or are of interest to their students. Everyday problems and questions should be drawn into the lessons to motivate students and keep the material relevant; suggestions are provided in the “Links” column of the syllabus. The mathematical exploration offers an opportunity to investigate the usefulness, relevance and occurrence of mathematics in the real world and will add an extra dimension to the course. The emphasis is on communication by means of mathematical forms (for example, formulae, diagrams, graphs and so on) with accompanying commentary. Modelling, investigation, reflection, personal engagement and mathematical communication should therefore feature prominently in the DP mathematics classroom. For further information on “Approaches to teaching a DP course”, please refer to the publication The Diploma Programme: From principles into practice (2009). To support teachers, a variety of PRE-PUBLICATION Mathematics HL guide

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resources can be found on the OCC, and details of workshops for professional development are available on the public website.

Format of the syllabus Content: this column lists, under each topic, the sub-topics to be covered.

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Further guidance: this column contains more detailed information on specific sub-topics listed in the content column. This clarifies the content for examinations.

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Links: this column provides useful links to the aims of the mathematics HL course, with suggestions for discussion, real-life examples and ideas for further investigation. These suggestions are only a guide for introducing and illustrating the sub-topic and are not exhaustive. Links are labelled as follows.

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real-life examples and links to other Diploma Programme subjects

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moral, social and ethical implications of the sub-topic

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international-mindedness

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suggestions for discussion

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Note that any syllabus references to other subject guides given in the “Links” column are correct for the current (2012) published versions of the guides.

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Notes on the syllabus

Formulae are only included in this document where there may be some ambiguity. All formulae required for the course are in the mathematics HL and further mathematics HL formula booklet.

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The term “technology” is used for any form of calculator or computer that may be available. However, there will be restrictions on which technology may be used in examinations, which will be noted in relevant documents.

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The terms “analysis” and “analytic approach” are generally used when referring to an approach that does not use technology.

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Course of study

The content of all six topics in the syllabus must be taught, although not necessarily in the order in which they appear in this guide. Teachers are expected to construct a course of study that addresses the needs of their students and includes, where necessary, the topics noted in prior learning.

Integration of the mathematical exploration Work leading to the completion of the exploration should be integrated into the course of study. Details of how to do this are given in the section on internal assessment and in the teacher support material.

Time allocation The recommended teaching time for higher level courses is 240 hours. For mathematics HL, it is expected that 10 hours of this will be spent on work for the exploration. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 230 hours allowed for the PRE-PUBLICATION Mathematics HL guide

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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.

Use of calculators Students 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 calculators allowed are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematics HL/SL: Graphic display calculators teacher support material (May 2005) and on the OCC.

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Mathematics HL and further mathematics HL formula booklet

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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. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there are no printing errors, and ensure that there are sufficient copies available for all students.

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Teacher support materials

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A variety of teacher support materials will accompany this guide. These materials will include guidance for teachers on the introduction, planning and marking of the exploration, and specimen examination papers and markschemes.

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Command terms and notation list

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Teachers and students need to be familiar with the IB notation, and the command terms, as these will be used without explanation in the examination papers. The “Glossary of command terms” and “Notation list” appear as appendices in this guide.

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Prior learning topics As noted in the previous section on prior learning, it is expected that all students have extensive previous mathematical experiences, but these will vary. It is expected that mathematics HL students will be familiar with the following topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics listed here 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 table lists the knowledge, together with the syllabus content, that is essential to successful completion of the mathematics HL course.

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Students must be familiar with SI (Système International) units of length, mass and time, and their derived units. Content

Number

Routine use of addition, subtraction, multiplication and division, using integers, decimals and fractions, including order of operations.

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Topic

Rational exponents.

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Simplification of expressions involving roots (surds or radicals), including rationalizing the denominator.

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Prime numbers and factors (divisors), including greatest common divisors and least common multiples.

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Simple applications of ratio, percentage and proportion, linked to similarity. Definition and elementary treatment of absolute value (modulus), a .

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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 ∈ .

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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.

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Sets and numbers

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. Mappings of the elements of one set to another; sets of ordered pairs.

Algebra

Manipulation of linear and quadratic expressions, including factorization, expansion, completing the square and use of the formula. Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas, particularly the sciences, should be included. Linear functions, their graphs, gradients and y-intercepts. Addition and subtraction of simple algebraic fractions. The properties of order relations: , ≥ .

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Solution of linear equations and inequalities in one variable, including cases with rational coefficients. Solution of quadratic equations and inequalities, using factorization and completing the square. Solution of simultaneous linear equations in two variables. Trigonometry

Angle measurement in degrees. Compass directions. Right-angle trigonometry. Simple applications for solving triangles. Pythagoras’ theorem and its converse. Simple geometric transformations: translation, reflection, rotation, enlargement. Congruence and similarity, including the concept of scale factor of an enlargement.

Geometry

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The circle, its centre and radius, area and circumference. The terms arc, sector, chord, tangent and segment.

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Perimeter and area of plane figures. Properties of triangles and quadrilaterals, including parallelograms, rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound shapes. Volumes of cuboids, pyramids, spheres, cylinders and cones. Classification of prisms and pyramids, including tetrahedra. Elementary geometry of the plane, including the concepts of dimension for point, line, plane and space. The equation of a line in the form = y mx + c . Parallel and perpendicular lines, including m1 = m2 and m1m2 = −1 .

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Coordinate geometry

Descriptive statistics: collection of raw data, display of data in pictorial and diagrammatic forms, including frequency histograms, cumulative frequency graphs.

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Statistics and probability

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The Cartesian plane: ordered pairs ( x, y ) , origin, axes. Mid-point of a line segment and distance between two points in the Cartesian plane.

Obtaining simple statistics from discrete and continuous data, including mean, median, mode, quartiles, range, interquartile range and percentiles.

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Calculating probabilities of simple events.

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Syllabus content

Further guidance

Links

Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.

Sequences can be generated and displayed in several ways, including recursive functions.

Sigma notation.

Link infinite geometric series with limits of convergence in 6.1.

Applications.

Examples include compound interest and population growth.

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Content

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1.1

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30 hours The aim of this topic is to introduce students to some basic algebraic concepts and applications.

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Topic 1—Core: Algebra

Int: The chess legend (Sissa ibn Dahir). Int: Aryabhatta is sometimes considered the “father of algebra”. Compare with al-Khawarizmi. Int: The use of several alphabets in mathematical notation (eg first term and common difference of an arithmetic sequence). TOK: Mathematics and the knower. To what extent should mathematical knowledge be consistent with our intuition?

TOK: Mathematics and the world. Some mathematical constants ( π , e, φ , Fibonacci numbers) appear consistently in nature. What does this tell us about mathematical knowledge? TOK: Mathematics and the knower. How is mathematical intuition used as a basis for formal proof? (Gauss’ method for adding up integers from 1 to 100.) Aim 8: Short-term loans at high interest rates. How can knowledge of mathematics result in individuals being exploited or protected from extortion? Appl: Physics 7.2, 13.2 (radioactive decay and nuclear physics).

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1.2

Exponents and logarithms. Laws of exponents; laws of logarithms.

Exponents and logarithms are further developed in 2.4.

Appl: Chemistry 18.1, 18.2 (calculation of pH and buffer solutions). TOK: The nature of mathematics and science. Were logarithms an invention or discovery? (This topic is an opportunity for teachers and students to reflect on “the nature of mathematics”.)

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Counting principles, including permutations and n The ability to find and n Pr using combinations. r both the formula and technology is expected. Link to 5.4. The binomial theorem:

Link to 5.6, binomial distribution.

expansion of (a + b) n , n ∈ . Not required: Permutations where some objects are identical. Proof of binomial theorem. Proof by mathematical induction.

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Circular arrangements.

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1.3

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Change of base.

Links to a wide variety of topics, for example, complex numbers, differentiation, sums of series and divisibility.

TOK: The nature of mathematics. The unforeseen links between Pascal’s triangle, counting methods and the coefficients of polynomials. Is there an underlying truth that can be found linking these?

Int: The properties of Pascal’s triangle were known in a number of different cultures long before Pascal (eg the Chinese mathematician Yang Hui). Aim 8: How many different tickets are possible in a lottery? What does this tell us about the ethics of selling lottery tickets to those who do not understand the implications of these large numbers?

TOK: Nature of mathematics and science. What are the different meanings of induction in mathematics and science? TOK: Knowledge claims in mathematics. Do proofs provide us with completely certain knowledge? TOK: Knowledge communities. Who judges the validity of a proof?

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1.5

Complex numbers: the number i= −1 ; the terms real part, imaginary part, conjugate, modulus and argument.

When solving problems, students may need to use technology.

Cartesian form z= a + ib .

Appl: Concepts in electrical engineering. Impedance as a combination of resistance and reactance; also apparent power as a combination of real and reactive powers. These combinations take the form z= a + ib .

N

TOK: Mathematics and the knower. Do the words imaginary and complex make the concepts more difficult than if they had different names?

TI O

Sums, products and quotients of complex numbers.

Modulus–argument (polar) form z =r (cos θ + i sin θ ) =r cisθ =r eiθ .

r eiθ is also known as Euler’s form.

BL IC

1.6

A

TOK: The nature of mathematics. Has “i” been invented or was it discovered?

The ability to convert between forms is expected.

-P U

Powers of complex numbers: de Moivre’s theorem.

PR E

nth roots of a complex number. 1.8

Appl: Concepts in electrical engineering. Phase angle/shift, power factor and apparent power as a complex quantity in polar form.

TOK: The nature of mathematics. Was the complex plane already there before it was used to represent The complex plane is also known as the complex numbers geometrically? Argand diagram. TOK: Mathematics and the knower. Why might it be said that ei π + 1 = 0 is beautiful?

The complex plane.

1.7

TOK: Mathematics and the world. Why does “i” appear in so many fundamental laws of physics?

Conjugate roots of polynomial equations with real coefficients.

PRE-PUBLICATION Mathematics HL guide

Proof by mathematical induction for n ∈ + .

TOK: Reason and mathematics. What is mathematical reasoning and what role does proof play in this form of reasoning? Are there examples of proof that are not mathematical?

Link to 2.5 and 2.7.

18

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.

These systems should be solved using both algebraic and technological methods, eg row reduction. Systems that have solution(s) may be referred to as consistent.

PR E

-P U

BL IC

A

Link to vectors in 4.7.

TI O

When a system has an infinity of solutions, a general solution may be required.

TOK: Mathematics, sense, perception and reason. If we can find solutions in higher dimensions, can we reason that these spaces exist beyond our sense perception?

N

1.9

PRE-PUBLICATION Mathematics HL guide

19

Topic 2—Core: Functions and equations 22 hours

Content 2.1

Further guidance

Concept of function f : x f ( x) : domain, range; image (value).

Links

A

Odd and even functions.

TI O

N

The aims of this topic 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 technology in both the development and the application of this topic.

( f g )( x) = f ( g ( x)) . Link with 6.2.

Identity function.

BL IC

Composite functions f g .

Link with 3.4.

Inverse function f −1 , including domain restriction. Self-inverse functions.

Link with 6.2.

TOK: The nature of mathematics. Is mathematics simply the manipulation of symbols under a set of formal rules?

PR E

-P U

One-to-one and many-to-one functions.

Int: The notation for functions was developed by a number of different mathematicians in the 17th and 18th centuries. How did the notation we use today become internationally accepted?

PRE-PUBLICATION Mathematics HL guide

20

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.

Use of technology to graph a variety of functions.

Appl: Sketching and interpreting graphs; Geography SL/HL (geographic skills); Chemistry 11.3.1.

The graphs of the functions y = f ( x) and

Int: Bourbaki group analytical approach versus Mandlebrot visual approach.

y = f ( x) .

2.3

Transformations of graphs: translations; stretches; reflections in the axes.

2.4

The rational function x

ax + b , and its cx + d

PR E

graph.

The function x a x , a > 0 , and its graph.

The function x log a x , x > 0 , and its graph.

PRE-PUBLICATION Mathematics HL guide

Link to 3.4. Students are expected to be aware Appl: Economics SL/HL 1.1 (shift in demand of the effect of transformations on both the and supply curves). algebraic expression and the graph of a function.

-P U

The graph of the inverse function as a reflection in y = x .

BL IC

1 given the graph of f ( x)

A

y = f(x).

The graph of y =

TOK: Mathematics and knowledge claims. Does studying the graph of a function contain the same level of mathematical rigour as studying the function algebraically (analytically)?

N

The graph of a function; its equation y = f ( x) .

TI O

2.2

The reciprocal function is a particular case. Graphs should include both asymptotes and any intercepts with axes. Exponential and logarithmic functions as inverses of each other. Link to 6.2 and the significance of e. Application of concepts in 2.1, 2.2 and 2.3.

Appl: Geography SL/HL (geographic skills); Physics SL/HL 7.2 (radioactive decay); Chemistry SL/HL 16.3 (activation energy); Economics SL/HL 3.2 (exchange rates).

21

The graphical significance of repeated factors.

The factor and remainder theorems.

The relationship between the degree of a polynomial function and the possible numbers of x-intercepts.

The fundamental theorem of algebra. 2.6

Solving quadratic equations using the quadratic formula.

N

Polynomial functions and their graphs.

May be referred to as roots of equations or zeros Appl: Chemistry 17.2 (equilibrium law). of functions. Appl: Physics 2.1 (kinematics).

TI O

2.5

Use of the discriminant ∆= b 2 − 4ac to determine the nature of the roots.

Appl: Physics 4.2 (energy changes in simple harmonic motion).

Link the solution of polynomial equations to conjugate roots in 1.8.

Sum and product of the roots of polynomial equations.

For the polynomial equation

A

Solving polynomial equations both graphically and algebraically.

BL IC

∑ ar x r = 0 , r =0

the sum is

-P U

Aim 8: The phrase “exponential growth” is used popularly to describe a number of phenomena. Is this a misleading use of a mathematical term?

− an −1 , an

the product is

Solution of a x = b using logarithms.

n

Appl: Physics (HL only) 9.1 (projectile motion).

(−1) n a0 . an

2.7

PR E

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach. Solutions of g ( x) ≥ f ( x) .

Graphical or algebraic methods, for simple polynomials up to degree 3.

Use of technology for these and other functions.

PRE-PUBLICATION Mathematics HL guide

22

Topic 3—Core: Circular functions and trigonometry 22 hours

Further guidance

The circle: radian measure of angles.

Radian measure may be expressed as multiples of π, or decimals. Link with 6.2.

Length of an arc; area of a sector. 3.2

Definition of cos θ , sin θ and tan θ in terms of the unit circle. Exact values of sin, cos and tan of 0,

π π π π

, , , and their multiples. 6 4 3 2

-P U

Definition of the reciprocal trigonometric ratios secθ , cscθ and cotθ .

Links

A

Content

BL IC

3.1

TI O

N

The aims of this topic are to explore the circular functions, to introduce some important trigonometric identities and to solve triangles using trigonometry. On . examination papers, radian measure should be assumed unless otherwise indicated, for example, by x sin x°

Pythagorean identities: cos 2 θ + sin 2 θ = 1; 2 2 2 2 1 + tan θ = sec θ ; 1 + cot θ = csc θ . Compound angle identities.

PR E

3.3

Double angle identities. Not required:

Proof of compound angle identities.

PRE-PUBLICATION Mathematics HL guide

Derivation of double angle identities from compound angle identities. Finding possible values of trigonometric ratios without finding θ, for example, finding sin 2θ given sin θ .

Int: The origin of degrees in the mathematics of Mesopotamia and why we use minutes and seconds for time.

TOK: Mathematics and the knower. Why do we use radians? (The arbitrary nature of degree measure versus radians as real numbers and the implications of using these two measures on the shape of sinusoidal graphs.) TOK: Mathematics and knowledge claims. If trigonometry is based on right triangles, how can we sensibly consider trigonometric ratios of angles greater than a right angle? Int: The origin of the word “sine”. Appl: Physics SL/HL 2.2 (forces and dynamics). Appl: Triangulation used in the Global Positioning System (GPS). Int: Why did Pythagoras link the study of music and mathematics? (continued)

23

3.4

(see notes above) Appl: Concepts in electrical engineering. Generation of sinusoidal voltage.

Composite functions of the form f= ( x) a sin(b( x + c)) + d . Applications.

N

The inverse functions x arcsin x , x arccos x , x arctan x ; their domains and ranges; their graphs.

TI O

3.5

TOK: Mathematics and the world. Music can be expressed using mathematics. Does this mean that music is mathematical, that mathematics is musical or that both are reflections of a common “truth”.

Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization. Not required: The general solution of trigonometric equations. The cosine rule

-P U

3.7

BL IC

3.6

A

Appl: Physics SL/HL 4.1 (kinematics of simple harmonic motion).

The sine rule including the ambiguous case.

Applications.

1 ab sin C . 2

PR E

Area of a triangle as

PRE-PUBLICATION Mathematics HL guide

Examples include navigation, problems in two and three dimensions, including angles of elevation and depression.

TOK: Mathematics and knowledge claims. How can there be an infinite number of discrete solutions to an equation?

TOK: Nature of mathematics. If the angles of a triangle can add up to less than 180°, 180° or more than 180°, what does this tell us about the “fact” of the angle sum of a triangle and about the nature of mathematical knowledge? Appl: Physics SL/HL 1.3 (vectors and scalars); Physics SL/HL 2.2 (forces and dynamics). Int: The use of triangulation to find the curvature of the Earth in order to settle a dispute between England and France over Newton’s gravity.

24

Topic 4—Core: Vectors 24 hours

4.1

Further guidance

Concept of a vector.

A

Representation of vectors using directed line segments. Components of a vector: .

Aim 8: Vectors are used to solve many problems in position location. This can be used to save a lost sailor or destroy a building with a laser-guided bomb. Appl: Physics SL/HL 1.3 (vectors and scalars); Physics SL/HL 2.2 (forces and dynamics).

TOK: Mathematics and knowledge claims. You can perform some proofs using different mathematical concepts. What does this tell us about mathematical knowledge?

Proofs of geometrical properties using vectors.

-P U

Algebraic and geometric approaches to the following:

BL IC

Unit vectors; base vectors i, j, k.

v1 v = v2 =v1i + v2 j + v3 k v 3

Links

TI O

Content

N

The aim of this topic is to introduce the use of vectors in two and three dimensions, and to facilitate solving problems involving points, lines and planes.

the sum and difference of two vectors;

PR E

the zero vector 0 , the vector −v ; multiplication by a scalar, kv ; magnitude of a vector, v ; →

position vectors OA = a . →

AB= b − a

PRE-PUBLICATION Mathematics HL guide

Distance between points A and B is the →

magnitude of AB . 25

The definition of the scalar product of two vectors. Properties of the scalar product:

Link to 3.6.

v ⋅w = w ⋅v ;

For non-zero vectors, v ⋅ w = 0 is equivalent to the vectors being perpendicular.

(kv ) ⋅ w = k (v ⋅ w ) ;

For parallel vectors, v ⋅ w = v w.

TOK: The nature of mathematics. Why this definition of scalar product?

N

u ⋅ (v + w ) = u ⋅ v + u ⋅ w ;

2

v ⋅ v =v .

The angle between two vectors.

A

Perpendicular vectors; parallel vectors. Vector equation of a line in two and three dimensions: r = a + λ b .

Knowledge of the following forms for equations Appl: Modelling linear motion in three of lines. dimensions.

Simple applications to kinematics.

Parametric form:

Appl: Navigational devices, eg GPS.

The angle between two lines.

x x0 + λ l , = z z0 + λ n . = y y0 + λ m , =

TOK: The nature of mathematics. Why might it be argued that vector representation of lines is superior to Cartesian?

BL IC

4.3

Appl: Physics SL/HL 2.2 (forces and dynamics).

v⋅w = v w cos θ , where θ is the angle between v and w .

TI O

4.2

-P U

Cartesian form:

x − x0 y − y0 z − z0 . = = l m n

Coincident, parallel, intersecting and skew lines; distinguishing between these cases.

PR E

4.4

Points of intersection.

PRE-PUBLICATION Mathematics HL guide

26

4.5

The definition of the vector product of two vectors. Properties of the vector product: u × (v + w ) = u × v + u × w ;

TI O

(kv ) × w = k (v × w ) ; v×v = 0.

Use of normal vector to obtain the form r ⋅n =a ⋅n. Cartesian equation of a plane ax + by + cz = d. Intersections of: a line with a plane; two planes; three planes.

Link to 1.9.

-P U

4.7

BL IC

Vector equation of a plane r =a + λ b + µ c .

A

Areas of triangles and parallelograms.

Geometric interpretation of v × w . 4.6

Appl: Physics SL/HL 6.3 (magnetic force and field).

N

v × w =− w × v ;

v×w = v w sin θ n , where θ is the angle between v and w and n is the unit normal vector whose direction is given by the righthand screw rule.

Geometrical interpretation of solutions.

PR E

Angle between: a line and a plane; two planes.

TOK: Mathematics and the knower. Why are symbolic representations of three-dimensional objects easier to deal with than visual representations? What does this tell us about our knowledge of mathematics in other dimensions?

PRE-PUBLICATION Mathematics HL guide

27

Topic 5—Core: Statistics and probability 36 hours

Further guidance

Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.

For examination purposes data will be treated as TOK: The nature of mathematics. Why have the population. mathematics and statistics sometimes been treated as separate subjects? In examinations the following formulae should be used: TOK: The nature of knowing. Is there a difference between information and data. k f x ∑ ii Aim 8: Does the use of statistics lead to an , µ = i =1 overemphasis on attributes that can easily be n measured over those that cannot?

Grouped data: mid-interval values, interval width, upper and lower interval boundaries. Mean, variance, standard deviation. Not required:

Links

A

Content

BL IC

5.1

TI O

N

The aim of this topic is to introduce basic concepts. It may be considered as three parts: manipulation and presentation of statistical data (5.1), the laws of probability (5.2–5.4.), and random variables and their probability distributions (5.5–5.7). It is expected that most of the calculations required will be done on a GDC. The emphasis is on understanding and interpreting the results obtained. Statistical tables will no longer be allowed in examinations.

PR E

-P U

k k Estimation of mean and variance of a population 2 − µ f x f i xi 2 ( ) ∑ ∑ i i from a sample. 2 i 1 =i 1 = = σ = − µ2 . n n

PRE-PUBLICATION Mathematics HL guide

Appl: Psychology SL/HL (descriptive statistics); Geography SL/HL (geographic skills); Biology SL/HL 1.1.2 (statistical analysis). Appl: Methods of collecting data in real life (census versus sampling). Appl: Misleading statistics in media reports.

28

Aim 8: Why has it been argued that theories based on the calculable probabilities found in casinos are pernicious when applied to everyday life (eg economics)?

Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A as P( A) =

n( A) . n(U )

Int: The development of the mathematical theory of probability in 17th century France.

N

5.2

The complementary events A and A′ (not A).

Combined events; the formula for P( A ∪ B ) .

A

5.3

TI O

Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.

5.4

Conditional probability; the definition P( A ∩ B ) P( A | B) = P( B )

.

P ( A= | B ) P= ( A ) P ( A | B′ ) .

Use of P( A ∩ B ) = P( A)P( B ) to show independence.

Appl: Use of probability methods in medical studies to assess risk factors for certain diseases. TOK: Mathematics and knowledge claims. Is independence as defined in probabilistic terms the same as that found in normal experience?

-P U

Independent events; the definition

BL IC

Mutually exclusive events.

5.5

PR E

Use of Bayes’ theorem for a maximum of three events.

TOK: Mathematics and the knower. To what extent can we trust samples of data?

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.

For a continuous random variable, a value at which the probability density function has a maximum value is called a mode.

Applications.

Examples include games of chance.

PRE-PUBLICATION Mathematics HL guide

Appl: Expected gain to insurance companies. 29

5.6

Binomial distribution, its mean and variance.

Link to binomial theorem in 1.3.

Poisson distribution, its mean and variance.

Conditions under which random variables have these distributions.

N

Not required:

TI O

Formal proof of means and variances. 5.7

TOK: Mathematics and the real world. Is the binomial distribution ever a useful model for an actual real-world situation?

Normal distribution.

Probabilities and values of the variable must be found using technology.

Link to 2.3.

BL IC

Properties of the normal distribution.

A

The standardized value (z) gives the number of standard deviations from the mean.

Aim 8: Why might the misuse of the normal distribution lead to dangerous inferences and conclusions? TOK: Mathematics and knowledge claims. To what extent can we trust mathematical models such as the normal distribution? Int: De Moivre’s derivation of the normal distribution and Querelet’s use of it to describe l’homme moyen.

PR E

-P U

Standardization of normal variables.

Appl: Chemistry SL/HL 6.2 (collision theory); Psychology HL (descriptive statistics); Biology SL/HL 1.1.3 (statistical analysis).

PRE-PUBLICATION Mathematics HL guide

30

Topic 6—Core: Calculus 48 hours

Informal ideas of limit, continuity and convergence.

Include result lim

Definition of derivative from first principles

Link to 1.1.

h →0

f ( x + h) − f ( x ) . h

Finding equations of tangents and normals.

PR E PRE-PUBLICATION Mathematics HL guide

Link to binomial theorem in 1.3. Both forms of notation, first derivative.

dy and f ′ ( x ) , for the dx

Use of both algebra and technology.

-P U

Identifying increasing and decreasing functions.

Higher derivatives.

θ

=1.

Use of this definition for polynomials only.

The derivative interpreted as a gradient function and as a rate of change.

The second derivative.

sin θ

A

f ′( x) = lim

θ →0

Links

TI O

Further guidance

BL IC

6.1

Content

N

The aim of this topic is to introduce students to the basic concepts and techniques of differential and integral calculus and their application.

TOK: The nature of mathematics. Does the fact that Leibniz and Newton came across the calculus at similar times support the argument that mathematics exists prior to its discovery? Int: How the Greeks’ distrust of zero meant that Archimedes’ work did not lead to the calculus. Int: Investigate attempts by Indian mathematicians (500–1000 CE) to explain division by zero.

TOK: Mathematics and the knower. What does the dispute between Newton and Leibniz tell us about human emotion and mathematical discovery?

Appl: Economics HL 1.5 (theory of the firm); Chemistry SL/HL 11.3.4 (graphical techniques); d2 y Both forms of notation, 2 and f ′′( x) , for the Physics SL/HL 2.1 (kinematics). dx second derivative. Familiarity with the notation

Link with induction in 1.4.

dn y and f ( n ) ( x) . dx n

31

6.2

Appl: Physics HL 2.4 (uniform circular motion); Physics 12.1 (induced electromotive force (emf)).

Derivatives of x n , sin x , cos x , tan x , e x and ln x . Differentiation of sums and multiples of functions.

N

TOK: Mathematics and knowledge claims. Euler was able to make important advances in mathematical analysis before calculus had been put on a solid theoretical foundation by Cauchy and others. However, some work was not possible until after Cauchy’s work. What does this tell us about the importance of proof and the nature of mathematics?

The product and quotient rules.

TI O

The chain rule for composite functions. Related rates of change. Implicit differentiation.

Optimization problems.

Points of inflexion with zero and non-zero gradients.

PR E

Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ′′ . Not required:

BL IC

Local maximum and minimum values.

TOK: Mathematics and the real world. The seemingly abstract concept of the calculus allows us to create mathematical models that permit human feats, such as getting a man on the Moon. What does this tell us about the links between mathematical models and physical reality?

Testing for the maximum or minimum using the change of sign of the first derivative and using the sign of the second derivative.

-P U

6.3

A

Derivatives of sec x , csc x , cot x , a x , log a x , arcsin x , arccos x and arctan x .

Use of the terms “concave up” for f ′′( x) > 0 , “concave down” for f ′′( x) < 0 . A point of inflexion exists where f ′′( x) = 0 and there is a sign change in f ′′ (i.e. concavity change).

Points of inflexion, where f ′′( x) is not defined, for example, y = x1 3 at (0,0) .

PRE-PUBLICATION Mathematics HL guide

32

Indefinite integral of x n , sin x , cos x and e x .

Indefinite integral interpreted as a family of curves. 1

Other indefinite integrals using the results from 6.2.

dx ∫ x=

The composites of any of these with a linear function.

Examples include and

1 dx . + 2x + 5

Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves.

Kinematic problems involving displacement s, velocity v and acceleration a.

PR E

Total distance travelled.

PRE-PUBLICATION Mathematics HL guide

dx ,

1

∫ 3x + 4 dx

The value of some definite integrals can only be found using technology.

-P U

Volumes of revolution about the x-axis or y-axis.

5

A

Anti-differentiation with a boundary condition to determine the constant of integration. Definite integrals.

6.6

2

∫ ( 2 x − 1)

BL IC

6.5

∫x

ln x + c .

N

Indefinite integration as anti-differentiation.

TI O

6.4

v=

ds dv d 2 s dv ,= a = = v . dt dt dt 2 ds t2

Total distance travelled = ∫ v dt .

Appl: Industrial design. Appl: Physics HL 2.1 (kinematics). Int: Does the inclusion of kinematics as core mathematics reflect a particular cultural heritage? Who decides what is mathematics?

t1

33

On examination papers, non-standard substitutions will be provided.

Integration by parts.

Link to 6.2. Examples:

∫ x sin x dx and ∫ ln x dx .

N

Integration by substitution

TI O

6.7

Repeated integration by parts.

∫x e

2 x

dx and ∫ e x sin x dx .

PR E

-P U

BL IC

A

Examples:

PRE-PUBLICATION Mathematics HL guide

34

Topic 7—Option: Statistics and probability

TI O

48 hours

N

Option syllabus content

7.1

Further guidance

Cumulative distribution functions for both discrete and continuous distributions. Geometric distribution. Negative binomial distribution.

BL IC

Content

-P U

= G (t ) E( = tX )

Probability generating functions for discrete random variables.

P( X ∑=

x)t x .

Links

Int: Also known as Pascal’s distribution.

x

Aim 8: Statistical compression of data files.

PR E

Using probability generating functions to find mean, variance and the distribution of the sum of n independent random variables.

A

The aims of this option are to allow students the opportunity to approach statistics in a practical way; to demonstrate a good level of statistical understanding; and to understand which situations apply and to interpret the given results. It is expected that GDCs will be used throughout this option, and that the minimum requirement of a GDC will be to find pdf, cdf, inverse cdf, p-values and test statistics, including calculations for the following distributions: binomial, Poisson, normal and t. Students are expected to set up the problem mathematically and then read the answers from the GDC, indicating this within their written answers. Calculator-specific or brand-specific language should not be used within these explanations.

PRE-PUBLICATION Mathematics HL guide

35

7.2

Linear transformation of a single random variable. Mean of linear combinations of n random variables.

E(aX += b) aE( X ) + b ,

Var(aX + b) = a 2 Var( X ) .

Unbiased estimators and estimates.

T is an unbiased estimator for the parameter θ if E(T ) = θ .

Comparison of unbiased estimators based on variances.

T1 is a more efficient estimator than T2 if Var(T1 ) < Var(T2 ) . n

X as an unbiased estimator for µ .

X =∑ i =1

S 2 as an unbiased estimator for σ 2 .

n

i =1

PR E

A linear combination of independent normal random variables is normally distributed. In particular, σ2 X ~ N( µ ,σ 2 ) ⇒ X ~ N µ , . n

The central limit theorem.

PRE-PUBLICATION Mathematics HL guide

TOK: Mathematics and the world. In the absence of knowing the value of a parameter, will an unbiased estimator always be better than a biased one?

Xi . n

(X

-P U

S =∑ 2

7.4

TI O

E( XY ) = E( X )E(Y ) .

A

Expectation of the product of independent random variables.

BL IC

7.3

N

Variance of linear combinations of n independent random variables.

i

−X)

n −1

2

. Aim 8/TOK: Mathematics and the world. “Without the central limit theorem, there could be no statistics of any value within the human sciences.” TOK: Nature of mathematics. The central limit theorem can be proved mathematically (formalism), but its truth can be confirmed by its applications (empiricism).

36

Confidence intervals for the mean of a normal population.

Use of the normal distribution when σ is known and use of the t-distribution when σ is unknown, regardless of sample size. The case of matched pairs is to be treated as an example of a single sample technique.

TOK: Mathematics and the world. Claiming brand A is “better” on average than brand B can mean very little if there is a large overlap between the confidence intervals of the two means.

N

7.5

Significance level. Critical regions, critical values, p-values, onetailed and two-tailed tests. Type I and II errors, including calculations of their probabilities.

TOK: Mathematics and the world. In practical terms, is saying that a result is significant the same as saying that it is true?

TOK: Mathematics and the world. Does the ability to test only certain parameters in a population affect the way knowledge claims in the human sciences are valued? Appl: When is it more important not to make a Type I error and when is it more important not to make a Type II error?

PR E

-P U

Testing hypotheses for the mean of a normal population.

Use of the normal distribution when σ is known and use of the t-distribution when σ is unknown, regardless of sample size. The case of matched pairs is to be treated as an example of a single sample technique.

A

Null and alternative hypotheses, H 0 and H1 .

BL IC

7.6

TI O

Appl: Geography.

PRE-PUBLICATION Mathematics HL guide

37

Introduction to bivariate distributions.

Informal discussion of commonly occurring situations, eg marks in pure mathematics and statistics exams taken by a class of students, salary and age of teachers in a certain school. The need for a measure of association between the variables and the possibility of predicting the value of one of the variables given the value of the other variable.

Appl: Geographic skills.

Cov( X , Y ) = E[( X − µ x )(Y − µ y )]

Appl: Using technology to fit a range of curves to a set of data.

Aim 8: The correlation between smoking and lung cancer was “discovered” using mathematics. Science had to justify the cause.

TI O

N

7.7

Covariance and (population) product moment correlation coefficient ρ.

= E( XY ) − µ x µ y ,

Cov( X , Y ) . Var( X )Var(Y )

BL IC

ρ=

A

where = µ x E( = X ), µ y E(Y ) .

-P U

Proof that ρ = 0 in the case of independence and The use of ρ as a measure of association ±1 in the case of a linear relationship between X between X and Y, with values near 0 indicating a weak association and values near ±1 indicating a and Y. strong association. n

PRE-PUBLICATION Mathematics HL guide

R=

∑(X i =1

n

i

− X )(Yi − Y )

n 2 i =i 1 =i 1

PR E

Definition of the (sample) product moment correlation coefficient R in terms of n paired observations on X and Y. Its application to the estimation of ρ.

∑(X

− X)

∑ (Y − Y ) i

2

n

=

∑ X Y − nXY i =1

i i

n 2 2 2 2 ∑ X i − n X ∑ Yi − nY i =1

.

TOK: Mathematics and the world. Given that a set of data may be approximately fitted by a range of curves, where would we seek for knowledge of which equation is the “true” model? Aim 8: The physicist Frank Oppenheimer wrote: “Prediction is dependent only on the assumption that observed patterns will be repeated.” This is the danger of extrapolation. There are many examples of its failure in the past, eg share prices, the spread of disease, climate change. (continued)

38

Values of r near 0 indicate a weak association between X and Y, and values near ±1 indicate a strong association.

The following topics are based on the assumption of bivariate normality.

It is expected that the GDC will be used wherever possible in the following work.

Use of the t-statistic to test the null hypothesis ρ = 0.

n−2 has the Student’s t-distribution with 1 − R2 (n − 2) degrees of freedom.

TI O

Knowledge of the facts that the regression of X on Y ( E( X ) | Y = y ) and Y on X ( E(Y ) | X = x ) x−x = are linear.

A

R

(see notes above)

N

Informal interpretation of r, the observed value of R. Scatter diagrams.

-P U

BL IC

n ∑ ( xi − x )( yi − y ) i =1 n ( y − y) 2 ∑ ( yi − y ) Least-squares estimates of these regression lines i =1 n (proof not required). xi yi − nx y ∑ The use of these regression lines to predict the i =1n ( y − y ), value of one of the variables given the= value of 2 2 the other. ∑ yi − n y i =1

PR E

n ∑ ( xi − x )( yi − y ) (x − x ) y − y i =1 n = 2 ( xi − x ) ∑ i =1 n ∑ xi yi − nx y ( x − x ). = i =1n 2 2 ∑ xi − nx i =1

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Topic 8—Option: Sets, relations and groups 48 hours

Content 8.1

Further guidance

BL IC

Illustration of these laws using Venn diagrams.

-P U

Students may be asked to prove that two sets are the same by establishing that A ⊆ B and B ⊆ A.

Relations: equivalence relations; equivalence classes.

An equivalence relation on a set forms a partition of the set.

Functions: injections; surjections; bijections.

The term codomain.

Composition of functions and inverse functions.

Knowledge that the function composition is not a commutative operation and that if f is a

PR E

8.3

Ordered pairs: the Cartesian product of two sets.

A

Operations on sets: union; intersection; complement; set difference; symmetric difference.

8.2

Links TOK: Cantor theory of transfinite numbers, Russell’s paradox, Godel’s incompleteness theorems.

Finite and infinite sets. Subsets.

De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).

TI O

N

The aims of this option are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract algebra.

PRE-PUBLICATION Mathematics HL guide

Appl: Logic, Boolean algebra, computer circuits.

Appl, Int: Scottish clans.

bijection from set A to set B then f −1 exists and is a bijection from set B to set A.

40

A binary operation ∗ on a non-empty set S is a rule for combining any two elements a, b ∈ S to give a unique element c. That is, in this definition, a binary operation on a set is not necessarily closed.

Binary operations.

N

8.4

Binary operations: associative, distributive and commutative properties.

The arithmetic operations on and .

The identity element e.

Both the right-identity a ∗ e = a and left-identity e∗a = a must hold if e is an identity element.

Examples of distributivity could include the fact that, on , multiplication is distributive over addition but addition is not distributive over multiplication.

TOK: Which are more fundamental, the general models or the familiar examples?

8.6

The inverse a −1 of an element a.

Both a ∗ a −1 = e and a −1 ∗ a = e must hold.

-P U

Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.

BL IC

A

8.5

TI O

Operation tables (Cayley tables).

Proofs of the uniqueness of the identity and inverse elements. The definition of a group {G , ∗} .

PR E

8.7

The operation table of a group is a Latin square, but the converse is false.

Abelian groups.

PRE-PUBLICATION Mathematics HL guide

For the set G under a given operation ∗ : G is closed under ∗ ; ∗ is associative;

G contains an identity element;

Appl: Existence of formula for roots of polynomials. Appl: Galois theory for the impossibility of such formulae for polynomials of degree 5 or higher.

each element in G has an inverse in G. a ∗ b = b ∗ a , for all a, b ∈ G .

41

8.8

Appl: Rubik’s cube, time measures, crystal structure, symmetries of molecules, strut and cable constructions, Physics H2.2 (special relativity), the 8–fold way, supersymmetry.

Examples of groups: , , and under addition; integers under addition modulo n;

8.9

The order of a group. The order of a group element. Cyclic groups. Generators.

8.10

Permutations under composition of permutations. Cycle notation for permutations.

TI O

-P U

Proof that all cyclic groups are Abelian.

A

invertible functions under composition of functions.

The composition T2 T1 denotes T1 followed by T2 .

BL IC

symmetries of plane figures, including equilateral triangles and rectangles;

N

non-zero integers under multiplication, modulo p, where p is prime;

PR E

Result that every permutation can be written as a composition of disjoint cycles.

On examination papers: the form 1 2 3 p= or in cycle notation (132) will 3 1 2 be used to represent the permutation 1 → 3 , 2 → 1 , 3 → 2.

Appl: Music circle of fifths, prime numbers.

Appl: Cryptography, campanology.

The order of a combination of cycles.

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A proper subgroup is neither the group itself nor the subgroup containing only the identity element.

Use and proof of subgroup tests.

Suppose that {G , ∗} is a group and H is a non-empty subset of G. Then {H , ∗} is a

N

Subgroups, proper subgroups.

TI O

8.11

subgroup of {G , ∗} if a ∗ b −1 ∈ H whenever a, b ∈ H .

Definition and examples of left and right cosets of a subgroup of a group. Lagrange’s theorem.

Appl: Prime factorization, symmetry breaking.

PR E

-P U

Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)

BL IC

A

Suppose that {G , ∗} is a finite group and H is a non-empty subset of G. Then {H , ∗} is a subgroup of {G , ∗} if H is closed under ∗ .

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8.12

Definition of a group homomorphism.

Infinite groups as well as finite groups.

Definition of the kernel of a homomorphism. Proof that the kernel and range of a homomorphism are subgroups.

TI O

N

Let {G ,*} and {H , } be groups, then the function f : G → H is a homomorphism if f (a * b) = f (a ) f (b) for all a, b ∈ G .

If f : G → H is a group homomorphism, then Ker( f ) is the set of a ∈ G such that f (a ) = eH .

BL IC

A

Proof of homomorphism properties for identities Identity: let eG and eH be the identity elements and inverses. of (G, ∗) and ( H , ) , respectively, then f (eG ) = eH . Inverse: f (a −1 ) = ( f (a ) ) Isomorphism of groups.

−1

for all a ∈ G .

Infinite groups as well as finite groups.

-P U

The homomorphism f : G → H is an isomorphism if f is bijective.

PR E

The order of an element is unchanged by an isomorphism.

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Topic 9—Option: Calculus 48 hours

Content

Further guidance

Links

Infinite sequences of real numbers and their convergence or divergence.

Informal treatment of limit of sum, difference, product, quotient; squeeze theorem.

TI O

9.1

N

The aims of this option are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations.

Convergence of infinite series. Tests for convergence: comparison test; limit comparison test; ratio test; integral test.

The sum of a series is the limit of the sequence of its partial sums.

BL IC

9.2

A

Divergent is taken to mean not convergent.

TOK: Zeno’s paradox, impact of infinite sequences and limits on our understanding of the physical world. TOK: Euler’s idea that 1 − 1 + 1 − 1 + =12 . Was it a mistake or just an alternative view?

Students should be aware that if lim xn = 0 then x →∞

the series is not necessarily convergent, but if lim xn ≠ 0 , the series diverges. x →∞

1

∑n

p

1

∑n

.

p

otherwise. When p = 1 , this is the harmonic series.

PR E

Series that converge absolutely.

Series that converge conditionally. Alternating series.

is convergent for p > 1 and divergent

-P U

The p-series,

Conditions for convergence. The absolute value of the truncation error is less than the next term in the series.

Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

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9.3

Continuity and differentiability of a function at a Test for continuity: point.

lim f ( x) = f ( a ) = lim f ( x ) .

Test for differentiability: f is continuous at a and

TI O

Continuous functions and differentiable functions.

x → a+

N

x→a –

f ( a + h) – f ( a ) and h h →0 − f ( a + h) – f ( a ) exist and are equal. lim h h → 0+

A

lim

The integral as a limit of a sum; lower and upper Riemann sums. Fundamental theorem of calculus.

-P U

9.4

PRE-PUBLICATION Mathematics HL guide

x d ∫ f ( y ) dy = f ( x ) . dx a

Int: How close was Archimedes to integral calculus? Int: Contribution of Islamic, Chinese and Indian mathematicians to the development of calculus.

∞

Aim 8: Leibniz versus Newton versus the “giants” on whose shoulders they stood—who deserves credit for mathematical progress?

a

TOK: Consider f x =

PR E

Improper integrals of the type

BL IC

Students should be aware that a function may be continuous but not differentiable at a point, eg f ( x ) = x and simple piecewise functions.

∫ f ( x ) dx .

1 , 1≤ x ≤ ∞. x

An infinite area sweeps out a finite volume. Can this be reconciled with our intuition? What does this tell us about mathematical knowledge?

46

First-order differential equations.

Appl: Real-life differential equations, eg

Geometric interpretation using slope fields, including identification of isoclines.

Newton’s law of cooling,

Euler’s method.

dy = f ( x, y ) using dx

N

Numerical solution of

population growth,

xn + h , where h is carbon dating. yn += yn + hf ( xn , yn ) , xn += 1 1 a constant.

TI O

9.5

Variables separable. y f x

Solution of y′ + P(x)y = Q(x), using the integrating factor. 9.6

Rolle’s theorem.

-P U

Mean value theorem.

A

using the substitution y = vx.

dy = dx

BL IC

Homogeneous differential equation

PR E

Taylor polynomials; the Lagrange form of the error term.

Maclaurin series for e x , sin x , cos x , ln(1 + x) , p

(1 + x) , p ∈ .

Int, TOK: Influence of Bourbaki on understanding and teaching of mathematics. Int: Compare with work of the Kerala school.

Applications to the approximation of functions; formula for the error term, in terms of the value of the (n + 1)th derivative at an intermediate point. Students should be aware of the intervals of convergence.

Use of substitution, products, integration and differentiation to obtain other series. Taylor series developed from differential equations.

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The evaluation of limits of the form lim

x→a

The indeterminate forms

f ( x) f ( x) and lim . g ( x) x →∞ g ( x )

Repeated use of l’Hôpital’s rule.

PR E

-P U

BL IC

A

TI O

Using l’Hôpital’s rule or the Taylor series.

0 ∞ and . 0 ∞

N

9.7

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Topic 10—Option: Discrete mathematics 48 hours

Content

Further guidance

Strong induction.

For example, proofs of the fundamental theorem TOK: Mathematics and knowledge claims. The of arithmetic and the fact that a tree with n difference between proof and conjecture, eg vertices has n – 1 edges. Goldbach’s conjecture. Can a mathematical statement be true before it is proven?

10.2

a|b ⇒ b = na for some n ∈ .

BL IC

A

Pigeon-hole principle.

Links

TI O

10.1

N

The aim of this option is to provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.

The theorem a | b and a | c ⇒ a | (bx ± cy ) where x, y ∈ .

The division algorithm = a bq + r , 0 ≤ r < b .

Division and Euclidean algorithms.

The Euclidean algorithm for determining the greatest common divisor of two integers.

-P U

The greatest common divisor, gcd(a, b) , and the least common multiple, lcm(a, b) , of integers a and b.

10.3

PR E

Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic. Linear Diophantine equations ax + by = c.

PRE-PUBLICATION Mathematics HL guide

General solutions required and solutions subject to constraints. For example, all solutions must be positive.

TOK: Proof by contradiction.

Int: Euclidean algorithm contained in Euclid’s Elements, written in Alexandria about 300 BCE. Aim 8: Use of prime numbers in cryptography. The possible impact of the discovery of powerful factorization techniques on internet and bank security. Int: Described in Diophantus’ Arithmetica written in Alexandria in the 3rd century CE. When studying Arithmetica, a French mathematician, Pierre de Fermat (1601–1665) wrote in the margin that he had discovered a simple proof regarding higher-order Diophantine equations—Fermat’s last theorem. 49

10.4

Modular arithmetic. The solution of linear congruences.

N

Int: Discussed by Chinese mathematician Sun Tzu in the 3rd century CE.

Solution of simultaneous linear congruences (Chinese remainder theorem). Representation of integers in different bases.

On examination papers, questions that go beyond base 16 will not be set.

10.6

Fermat’s little theorem.

a p = a (mod p ) , where p is prime.

10.7

Graphs, vertices, edges, faces. Adjacent vertices, adjacent edges.

Two vertices are adjacent if they are joined by an edge. Two edges are adjacent if they have a common vertex.

Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.

PR E

Subgraphs; complements of graphs.

Euler’s relation: v − e + f = 2 ; theorems for planar graphs including e ≤ 3v − 6 , e ≤ 2v − 4 , leading to the results that κ 5 and κ 3,3 are not planar.

PRE-PUBLICATION Mathematics HL guide

A

-P U

Handshaking lemma.

BL IC

Degree of a vertex, degree sequence.

Int: Babylonians developed a base 60 number system and the Mayans a base 20 number system.

TI O

10.5

TOK: Nature of mathematics. An interest may be pursued for centuries before becoming “useful”. Aim 8: Symbolic maps, eg Metro and Underground maps, structural formulae in chemistry, electrical circuits.

TOK: Mathematics and knowledge claims. Proof of the four-colour theorem. If a theorem is proved by computer, how can we claim to know that it is true?

It should be stressed that a graph should not be assumed to be simple unless specifically stated. The term adjacency table may be used.

Aim 8: Importance of planar graphs in constructing circuit boards.

e ≤ 3v − 6 only if the graph is connected and simple and v ≥ 3 .

TOK: “Only if”, link to Venn diagrams and logic.

e ≤ 2v − 4 only if the graph is connected and simple and all faces have degree ≥ 4 .

TOK: Mathematics and knowledge claims. Applications of the Euler characteristic (v − e + f ) to higher dimensions. Its use in understanding properties of shapes that cannot be visualized.

50

Walks, trails, paths, circuits, cycles. Eulerian trails and circuits.

A connected graph contains an Eulerian circuit if and only if every vertex of the graph is of even degree.

Hamiltonian paths and cycles.

Simple treatment only.

Int: The “Bridges of Königsberg” problem.

Graph algorithms: Kruskal’s; Dijkstra’s.

To determine the shortest route around a weighted graph going along each edge at least once.

Int: Problem posed by the Chinese mathematician Kwan Mei-Ko in 1962.

To determine the Hamiltonian cycle of least weight in a weighted complete graph.

TOK: Mathematics and knowledge claims. How long would it take a computer to test all Hamiltonian cycles in a complete, weighted graph with just 30 vertices?

A

10.10 Chinese postman problem. Not required: Graphs with more than four vertices of odd degree. Travelling salesman problem. Nearest-neighbour algorithm for determining an upper bound.

-P U

Deleted vertex algorithm for determining a lower bound.

BL IC

10.9

TI O

N

10.8

10.11 Recurrence relations. Initial conditions, recursive definition of a sequence.

PR E

Solution of first- and second-degree linear Includes the cases where auxiliary equation has homogeneous recurrence relations with constant equal roots or complex roots coefficients.

TOK: Mathematics and the world. The connections of sequences such as the Fibonacci sequence with art and biology.

The first-degree linear recurrence relation un aun −1 + b . = Modelling with recurrence relations.

PRE-PUBLICATION Mathematics HL guide

Solving problems such as compound interest, debt repayment and counting problems.

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Glossary of terminology: Discrete mathematics

Introduction Teachers and students should be aware that many different terminologies exist in graph theory, and that different textbooks may employ different combinations of these. Examples of these are: vertex / node / junction / point; edge / route / arc; degree / order of a vertex; multiple edges / parallel edges; loop / self-loop. In IB examination questions, the terminology used will be as it appears in the syllabus. For clarity, these terms are defined below.

N

Terminology

A graph whose vertices can be divided into two sets such that no two vertices in the same set are adjacent.

Circuit

A walk that begins and ends at the same vertex, and has no repeated edges.

Complement of a graph G

A graph with the same vertices as G but which has an edge between any two vertices if and only if G does not.

Complete bipartite graph

A bipartite graph in which every vertex in one set is joined to every vertex in the other set.

Complete graph

A simple graph in which each pair of vertices is joined by an edge.

Connected graph

A graph in which each pair of vertices is joined by a path.

Cycle

A walk that begins and ends at the same vertex, and has no other repeated vertices.

A

IC

BL

The number of edges joined to the vertex; a loop contributes two edges, one for each of its end points. A graph that has at least one pair of vertices not joined by a path.

E-

Disconnected graph

PU

Degree of a vertex

TI O

Bipartite graph

A circuit that contains every edge of a graph.

Eulerian trail

A trail that contains every edge of a graph.

PR

Eulerian circuit Graph

Consists of a set of vertices and a set of edges.

Graph isomorphism between two simple graphs G and H

A one-to-one correspondence between vertices of G and H such that a pair of vertices in G is adjacent if and only if the corresponding pair in H is adjacent.

Hamiltonian cycle

A cycle that contains all the vertices of the graph.

Hamiltonian path

A path that contains all the vertices of the graph.

Loop

An edge joining a vertex to itself.

Minimum spanning tree

A spanning tree of a weighted graph that has the minimum total weight.

Multiple edges

Occur if more than one edge joins the same pair of vertices.

Path

A walk with no repeated vertices.

Planar graph

A graph that can be drawn in the plane without any edge crossing another.

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A graph without loops or multiple edges.

Spanning tree of a graph

A subgraph that is a tree, containing every vertex of the graph.

Subgraph

A graph within a graph.

Trail

A walk in which no edge appears more than once.

Tree

A connected graph that contains no cycles.

Walk

A sequence of linked edges.

Weighted graph

A graph in which each edge is allocated a number or weight.

Weighted tree

A tree in which each edge is allocated a number or weight.

PR

E-

PU

BL

IC

A

TI O

N

Simple graph

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Assessment

Assessment in the Diploma Programme

General Assessment is an integral part of teaching and learning. The most important aims of assessment in the Diploma Programme are that it should support curricular goals and encourage appropriate student learning. Both external and internal assessment are used in the Diploma Programme. IB examiners mark work produced for external assessment, while work produced for internal assessment is marked by teachers and externally moderated by the IB.

N

There are two types of assessment identified by the IB. Formative assessment informs both teaching and learning. It is concerned with providing accurate and helpful feedback to students and teachers on the kind of learning taking place and the nature of students’ strengths and weaknesses in order to help develop students’ understanding and capabilities. Formative assessment can also help to improve teaching quality, as it can provide information to monitor progress towards meeting the course aims and objectives.

•

Summative assessment gives an overview of previous learning and is concerned with measuring student achievement.

A

TI O

•

BL

IC

The Diploma Programme primarily focuses on summative assessment designed to record student achievement at or towards the end of the course of study. However, many of the assessment instruments can also be used formatively during the course of teaching and learning, and teachers are encouraged to do this. A comprehensive assessment plan is viewed as being integral with teaching, learning and course organization. For further information, see the IB Programme standards and practices document.

E-

PU

The approach to assessment used by the IB is criterion-related, not norm-referenced. This approach to assessment judges students’ work by their performance in relation to identified levels of attainment, and not in relation to the work of other students. For further information on assessment within the Diploma Programme, please refer to the publication Diploma Programme assessment: Principles and practice.

PR

To support teachers in the planning, delivery and assessment of the Diploma Programme courses, a variety of resources can be found on the OCC or purchased from the IB store (http://store.ibo.org). Teacher support materials, subject reports, internal assessment guidance, grade descriptors, as well as resources from other teachers, can be found on the OCC. Specimen and past examination papers as well as markschemes can be purchased from the IB store.

Methods of assessment The IB uses several methods to assess work produced by students.

Assessment criteria Assessment criteria are used when the assessment task is open-ended. Each criterion concentrates on a particular skill that students are expected to demonstrate. An assessment objective describes what students should be able to do, and assessment criteria describe how well they should be able to do it. Using assessment criteria allows discrimination between different answers and encourages a variety of responses. Each criterion comprises a set of hierarchically ordered level descriptors. Each level descriptor is worth one or more marks. Each criterion is applied independently using a best-fit model. PRE-PUBLICATION Mathematics HL guide

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The maximum marks for each criterion may differ according to the criterion’s importance. The marks awarded for each criterion are added together to give the total mark for the piece of work.

Markbands Markbands are a comprehensive statement of expected performance against which responses are judged. They represent a single holistic criterion divided into level descriptors. Each level descriptor corresponds to a range of marks to differentiate student performance. A best-fit approach is used to ascertain which particular mark to use from the possible range for each level descriptor.

Markschemes

PR

E-

PU

BL

IC

A

TI O

N

This generic term is used to describe analytic markschemes that are prepared for specific examination papers. Analytic markschemes are prepared for those examination questions that expect a particular kind of response and/or a given final answer from the students. They give detailed instructions to examiners on how to break down the total mark for each question for different parts of the response. A markscheme may include the content expected in the responses to questions or may be a series of marking notes giving guidance on how to apply criteria.

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Assessment outline First examinations 2014 Assessment component

Weighting

External assessment (5 hours)

80%

Paper 1 (2 hours)

30%

No calculator allowed. (120 marks) Compulsory short-response questions based on the core syllabus. Section B

Paper 2 (2 hours)

Graphic display calculator (GDC) required. (120 marks)

30%

A

Section A

TI O

Compulsory extended-response questions based on the core syllabus.

N

Section A

Compulsory short-response questions based on the core syllabus.

IC

Section B

Compulsory extended-response questions based on the core syllabus.

BL

Paper 3 (1 hour)

20%

Graphic display calculator (GDC) required. (60 marks)

PU

Compulsory extended-response questions based mainly on the syllabus options. Internal assessment

20%

E-

This component is internally assessed by the teacher and externally moderated by the IB at the end of the course. Mathematical exploration

PR

Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)

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External assessment

General Markschemes are used to assess students in all papers. The markschemes are specific to each examination.

External assessment details Papers 1, 2 and 3

TI O

N

These papers are externally set and externally marked. Together, they contribute 80% of the final mark for the course. These papers are designed to allow students to demonstrate what they know and what they can do.

Calculators Paper 1

IC

A

Students are not permitted access to any calculator. Questions will mainly involve analytic approaches to solutions, rather than requiring the use of a GDC. The paper is not intended to require complicated calculations, with the potential for careless errors. However, questions will include some arithmetical manipulations when they are essential to the development of the question.

BL

Papers 2 and 3

PU

Students must have access to a GDC at all times. However, not all questions will necessarily require the use of the GDC. Regulations covering the types of GDC allowed are provided in the Handbook of procedures for the Diploma Programme.

Mathematics HL and Further mathematics HL formula booklet

E-

Each student must have access to a clean copy of the formula booklet during the examination. It is the responsibility of the school to download a copy from IBIS or the OCC and to ensure that there are sufficient copies available for all students.

PR

Awarding of marks

Marks may be awarded for method, accuracy, answers and reasoning, including interpretation. In paper 1 and paper 2, full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations (in the form of, for example, diagrams, graphs or calculations). Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. All students should therefore be advised to show their working.

Paper 1 Duration: 2 hours Weighting: 30% •

This paper consists of section A, short-response questions, and section B, extended-response questions.

•

Students are not permitted access to any calculator on this paper.

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Syllabus coverage •

Knowledge of all core topics is required for this paper. However, not all topics are necessarily assessed in every examination session.

Mark allocation •

This paper is worth 120 marks, representing 30% of the final mark.

•

Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.

Section A This section consists of compulsory short-response questions based on the core syllabus. It is worth 60 marks.

•

The intention of this section is to test students’ knowledge and understanding across the breadth of the syllabus. However, it should not be assumed that the separate topics are given equal emphasis.

TI O

N

•

Question type

A small number of steps is needed to solve each question.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

IC

A

•

BL

Section B

This section consists of a small number of compulsory extended-response questions based on the core syllabus. It is worth 60 marks.

•

Individual questions may require knowledge of more than one topic.

•

The intention of this section is to test students’ knowledge and understanding of the core in depth. The range of syllabus topics tested in this section may be narrower than that tested in section A.

E-

PU

•

Question type

•

Questions require extended responses involving sustained reasoning.

PR

•

Individual questions will develop a single theme.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

•

Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problemsolving.

Paper 2 Duration: 2 hours Weighting: 30% •

This paper consists of section A, short-response questions, and section B, extended-response questions.

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•

A GDC is required for this paper, but not every question will necessarily require its use.

Syllabus coverage •

Knowledge of all core topics is required for this paper. However, not all topics are necessarily assessed in every examination session.

Mark allocation •

This paper is worth 120 marks, representing 30% of the final mark.

•

Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.

N

Section A

This section consists of compulsory short-response questions based on the core syllabus. It is worth 60 marks.

•

The intention of this section is to test students’ knowledge and understanding across the breadth of the syllabus. However, it should not be assumed that the separate topics are given equal emphasis.

TI O

•

A

Question type

A small number of steps is needed to solve each question.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

BL

IC

•

Section B

This section consists of a small number of compulsory extended-response questions based on the core syllabus. It is worth 60 marks.

•

Individual questions may require knowledge of more than one topic.

•

The intention of this section is to test students’ knowledge and understanding of the core in depth. The range of syllabus topics tested in this section may be narrower than that tested in section A.

PR

E-

PU

•

Question type •

Questions require extended responses involving sustained reasoning.

•

Individual questions will develop a single theme.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

•

Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problemsolving.

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•

This paper consists of a small number of compulsory extended-response questions based on the option chosen.

•

Where possible, the first part of each question will be on core material leading to the option topic. When this is not readily achievable, as, for example, with the discrete mathematics option, the level of difficulty of the earlier part of a question will be comparable to that of the core questions.

Syllabus coverage Students must answer all questions.

•

Knowledge of the entire content of the option studied, as well as the core material, is required for this paper.

Mark allocation

N

•

This paper is worth 60 marks, representing 20% of the final mark.

•

Questions may be unequal in terms of length and level of difficulty. Therefore, individual questions may not be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of each question.

TI O

•

A

Question type

Questions require extended responses involving sustained reasoning.

•

Individual questions will develop a single theme or be divided into unconnected parts. Where the latter occur, the unconnected parts will be clearly labelled as such.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

•

Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problemsolving.

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E-

PU

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IC

•

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Internal assessment

Purpose of internal assessment Internal assessment is an integral part of the course and is compulsory for all students. It enables students to demonstrate the application of their skills and knowledge, and to pursue their personal interests, without the time limitations and other constraints that are associated with written examinations. The internal assessment should, as far as possible, be woven into normal classroom teaching and not be a separate activity conducted after a course has been taught.

TI O

Guidance and authenticity

N

Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. It is marked according to five assessment criteria.

A

The exploration submitted for internal assessment must be the student’s own work. However, it is not the intention that students should decide upon a title or topic and be left to work on the exploration without any further support from the teacher. The teacher should play an important role during both the planning stage and the period when the student is working on the exploration. It is the responsibility of the teacher to ensure that students are familiar with: the requirements of the type of work to be internally assessed

•

the IB academic honesty policy available on the OCC

•

the assessment criteria—students must understand that the work submitted for assessment must address these criteria effectively.

BL

IC

•

E-

PU

Teachers and students must discuss the exploration. Students should be encouraged to initiate discussions with the teacher to obtain advice and information, and students must not be penalized for seeking guidance. However, if a student could not have completed the exploration without substantial support from the teacher, this should be recorded on the appropriate form from the Handbook of procedures for the Diploma Programme.

PR

It is the responsibility of teachers to ensure that all students understand the basic meaning and significance of concepts that relate to academic honesty, especially authenticity and intellectual property. Teachers must ensure that all student work for assessment is prepared according to the requirements and must explain clearly to students that the exploration must be entirely their own. As part of the learning process, teachers can give advice to students on a first draft of the exploration. This advice should be in terms of the way the work could be improved, but this first draft must not be heavily annotated or edited by the teacher. The next version handed to the teacher after the first draft must be the final one. All work submitted to the IB for moderation or assessment must be authenticated by a teacher, and must not include any known instances of suspected or confirmed malpractice. Each student must sign the coversheet for internal assessment to confirm that the work is his or her authentic work and constitutes the final version of that work. Once a student has officially submitted the final version of the work to a teacher (or the coordinator) for internal assessment, together with the signed coversheet, it cannot be retracted.

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Authenticity may be checked by discussion with the student on the content of the work, and scrutiny of one or more of the following: •

the student’s initial proposal

•

the first draft of the written work

•

the references cited

•

the style of writing compared with work known to be that of the student.

TI O

N

The requirement for teachers and students to sign the coversheet for internal assessment applies to the work of all students, not just the sample work that will be submitted to an examiner for the purpose of moderation. If the teacher and student sign a coversheet, but there is a comment to the effect that the work may not be authentic, the student will not be eligible for a mark in that component and no grade will be awarded. For further details refer to the IB publication Academic honesty and the relevant articles in the General regulations: Diploma Programme. The same piece of work cannot be submitted to meet the requirements of both the internal assessment and the extended essay.

A

Group work

IC

Group work should not be used for explorations. Each exploration is an individual piece of work based on different data collected or measurements generated.

BL

It should be made clear to students that all work connected with the exploration, including the writing of the exploration, should be their own. It is therefore helpful if teachers try to encourage in students a sense of responsibility for their own learning so that they accept a degree of ownership and take pride in their own work.

PU

Time allocation

E-

Internal assessment is an integral part of the mathematics HL course, contributing 20% to the final assessment in the course. This weighting should be reflected in the time that is allocated to teaching the knowledge, skills and understanding required to undertake the work as well as the total time allocated to carry out the work.

•

PR

It is expected that a total of approximately 10 teaching hours should be allocated to the work. This should include: time for the teacher to explain to students the requirements of the exploration

•

class time for students to work on the exploration

•

time for consultation between the teacher and each student

•

time to review and monitor progress, and to check authenticity.

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Using assessment criteria for internal assessment For internal assessment, a number of assessment criteria have been identified. Each assessment criterion has level descriptors describing specific levels of achievement together with an appropriate range of marks. The level descriptors concentrate on positive achievement, although for the lower levels failure to achieve may be included in the description. Teachers must judge the internally assessed work against the criteria using the level descriptors. The aim is to find, for each criterion, the descriptor that conveys most accurately the level attained by the student.

•

When assessing a student’s work, teachers should read the level descriptors for each criterion, starting with level 0, until they reach a descriptor that describes a level of achievement that has not been reached. The level of achievement gained by the student is therefore the preceding one, and it is this that should be recorded.

•

Only whole numbers should be recorded; partial marks, that is fractions and decimals, are not acceptable.

•

Teachers should not think in terms of a pass or fail boundary, but should concentrate on identifying the appropriate descriptor for each assessment criterion.

•

The highest level descriptors do not imply faultless performance but should be achievable by a student. Teachers should not hesitate to use the extremes if they are appropriate descriptions of the work being assessed.

•

A student who attains a high level of achievement in relation to one criterion will not necessarily attain high levels of achievement in relation to the other criteria. Similarly, a student who attains a low level of achievement for one criterion will not necessarily attain low achievement levels for the other criteria. Teachers should not assume that the overall assessment of the students will produce any particular distribution of marks.

•

It is expected that the assessment criteria be made available to students.

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Internal assessment details Mathematical exploration Duration: 10 teaching hours Weighting: 20% Introduction

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N

The internally assessed component in this course is a mathematical exploration. This is a short report written by the student based on a topic chosen by him or her, and it should focus on the mathematics of that particular area. The emphasis is on mathematical communication (including formulae, diagrams, graphs and so on), with accompanying commentary, good mathematical writing and thoughtful reflection. A student should develop his or her own focus, with the teacher providing feedback via, for example, discussion and interview. This will allow the students to develop areas of interest to them without a time constraint as in an examination, and allow all students to experience a feeling of success.

The purpose of the exploration

IC

A

The final report should be approximately 6 to 12 pages long. It can be either word processed or hand written. Students should be able to explain all stages of their work in such a way that demonstrates clear understanding. While there is no requirement that students present their work in class, it should be written in such a way that their peers would be able to follow it fairly easily. The report should include a detailed bibliography, and sources need to be referenced in line with the IB academic honesty policy. Direct quotes must be acknowledged.

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PU

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The aims of the mathematics HL course are carried through into the objectives that are formally assessed as part of the course, through either written examination papers, or the exploration, or both. In addition to testing the objectives of the course, the exploration is intended to provide students with opportunities to increase their understanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics. These are noted in the aims of the course, in particular aims 6– 9 (applications, technology, moral, social and ethical implications, and the international dimension). It is intended that, by doing the exploration, students benefit from the mathematical activities undertaken and find them both stimulating and rewarding. It will enable students to acquire the attributes of the IB learner profile. The specific purposes of the exploration are to:

•

develop students’ personal insight into the nature of mathematics and develop their ability to ask their own questions about mathematics

PR

•

provide opportunities for students to complete a piece of mathematical work over an extended period of time

•

enable students to experience the satisfaction of applying mathematical processes independently

•

provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics

•

encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool

•

enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work

•

provide opportunities for students to show, with confidence, how they have developed mathematically.

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Management of the exploration Work for the exploration should be incorporated into the course so that students are given the opportunity to learn the skills needed. Time in class can therefore be used for general discussion of areas of study, as well as familiarizing students with the criteria. Further details on the development of the exploration are included in the teacher support material.

Requirements and recommendations Students can choose from a wide variety of activities, for example, modelling, investigations and applications of mathematics. To assist teachers and students in the choice of a topic, a list of stimuli is available in the teacher support material. However, students are not restricted to this list.

N

The exploration should not normally exceed 12 pages, including diagrams and graphs, but excluding the bibliography. However, it is the quality of the mathematical writing that is important, not the length.

TI O

The teacher is expected to give appropriate guidance at all stages of the exploration by, for example, directing students into more productive routes of inquiry, making suggestions for suitable sources of information, and providing advice on the content and clarity of the exploration in the writing-up stage.

A

Teachers are responsible for indicating to students the existence of errors, but should not explicitly correct these errors. It must be emphasized that students are expected to consult the teacher throughout the process.

BL

IC

All students should be familiar with the requirements of the exploration and the criteria by which it is assessed. Students need to start planning their explorations as early as possible in the course. Deadlines should be firmly established. There should be a date for submission of the exploration topic and a brief outline description, a date for the submission of the first draft and, of course, a date for completion.

PU

In developing their explorations, students should aim to make use of mathematics learned as part of the course. The mathematics used should be commensurate with the level of the course, that is, it should be similar to that suggested by the syllabus. It is not expected that students produce work that is outside the mathematics HL syllabus—however, this is not penalized.

E-

Internal assessment criteria

PR

The exploration is internally assessed by the teacher and externally moderated by the IB using assessment criteria that relate to the objectives for group 5 mathematics. Each exploration is assessed against the following five criteria. The final mark for each exploration is the sum of the scores for each criterion. The maximum possible final mark is 20. Students will not receive a grade for mathematics HL if they have not submitted an exploration. Criterion A

Communication

Criterion B

Mathematical presentation

Criterion C

Personal engagement

Criterion D

Reflection

Criterion E

Use of mathematics

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Criterion A: Communication This criterion assesses the organization and coherence of the exploration. A well-organized exploration includes an introduction, has a rationale (which includes explaining why this topic was chosen), describes the aim of the exploration and has a conclusion. A coherent exploration is logically developed and easy to follow. Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document. Achievement level

Descriptor

The exploration does not reach the standard described by the descriptors below.

1

The exploration has some coherence.

2

The exploration has some coherence and shows some organization.

3

The exploration is coherent and well organized.

4

The exploration is coherent, well organized, concise and complete.

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A

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0

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Criterion B: Mathematical presentation This criterion assesses to what extent the student is able to: •

use appropriate mathematical language (notation, symbols, terminology)

•

define key terms, where required

•

use multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models, where appropriate.

Students are expected to use mathematical language when communicating mathematical ideas, reasoning and findings.

Achievement level

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N

Students are encouraged to choose and use appropriate ICT tools such as graphic display calculators, screenshots, graphing, spreadsheets, databases, drawing and word-processing software, as appropriate, to enhance mathematical communication. Descriptor

The exploration does not reach the standard described by the descriptors below.

1

There is some appropriate mathematical presentation.

2

The mathematical presentation is mostly appropriate.

3

The mathematical presentation is appropriate throughout.

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0

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Criterion C: Personal engagement This criterion assesses the extent to which the student engages with the exploration and makes it his or her own. Personal engagement may be recognized in different attributes and skills. These include thinking independently and/or creatively, addressing personal interest and presenting mathematical ideas in his or her own way. Achievement level

Descriptor

The exploration does not reach the standard described by the descriptors below.

1

There is evidence of limited or superficial personal engagement.

2

There is evidence of some personal engagement.

3

There is evidence of significant personal engagement.

4

There is abundant evidence of outstanding personal engagement.

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A

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N

0

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Criterion D: Reflection This criterion assesses how the student reviews, analyses and evaluates the exploration. Although reflection may be seen in the conclusion to the exploration, it may also be found throughout the exploration. Achievement level

Descriptor

The exploration does not reach the standard described by the descriptors below.

1

There is evidence of limited or superficial reflection.

2

There is evidence of meaningful reflection.

3

There is substantial evidence of critical reflection.

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A

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0

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Criterion E: Use of mathematics This criterion assesses to what extent and how well students use mathematics in the exploration. Students are expected to produce work that is commensurate with the level of the course. The mathematics explored should either be part of the syllabus, or at a similar level or beyond. It should not be completely based on mathematics listed in the prior learning. If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion. The mathematics can be regarded as correct even if there are occasional minor errors as long as they do not detract from the flow of the mathematics or lead to an unreasonable outcome.

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N

Sophistication in mathematics may include understanding and use of challenging mathematical concepts, looking at a problem from different perspectives and seeing underlying structures to link different areas of mathematics. Rigour involves clarity of logic and language when making mathematical arguments and calculations. Precise mathematics is error-free and uses an appropriate level of accuracy at all times. Descriptor

A

Achievement level

The exploration does not reach the standard described by the descriptors below.

1

Some relevant mathematics is used. Limited understanding is demonstrated.

2

Some relevant mathematics is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated.

3

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Good knowledge and understanding are demonstrated.

BL

PU E-

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication expected. Good knowledge and understanding are demonstrated.

PR

4

IC

0

5

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication and rigor expected. Thorough knowledge and understanding are demonstrated.

6

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and reflects the sophistication and rigor expected. Thorough knowledge and understanding are demonstrated.

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Appendices

Glossary of command terms

Command terms with definitions Students should be familiar with the following key terms and phrases used in examination questions, which are to be understood as described below. Although these terms will be used in examination questions, other terms may be used to direct students to present an argument in a specific way. Obtain a numerical answer showing the relevant stages in the working.

Comment

Give a judgment based on a given statement or result of a calculation.

Compare

Give an account of the similarities between two (or more) items or situations, referring to both (all) of them throughout.

Compare and contrast

Give an account of the similarities and differences between two (or more) items or situations, referring to both (all) of them throughout.

Construct

Display information in a diagrammatic or logical form.

Contrast

Give an account of the differences between two (or more) items or situations, referring to both (all) of them throughout.

Deduce

Reach a conclusion from the information given.

Demonstrate

Make clear by reasoning or evidence, illustrating with examples or practical application.

Describe

Give a detailed account.

Determine

Obtain the only possible answer.

Differentiate

Obtain the derivative of a function.

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IC

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Distinguish

Make clear the differences between two or more concepts or items. Represent by means of a labelled, accurate diagram or graph, using a pencil. A ruler (straight edge) should be used for straight lines. Diagrams should be drawn to scale. Graphs should have points correctly plotted (if appropriate) and joined in a straight line or smooth curve.

PR

E-

Draw

Estimate

N

Calculate

Obtain an approximate value.

Explain

Give a detailed account, including reasons or causes.

Find

Obtain an answer, showing relevant stages in the working.

Hence

Use the preceding work to obtain the required result.

Hence or otherwise

It is suggested that the preceding work is used, but other methods could also receive credit.

Identify

Provide an answer from a number of possibilities.

Integrate

Obtain the integral of a function.

Interpret

Use knowledge and understanding to recognize trends and draw conclusions from given information.

Investigate

Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions.

Justify

Give valid reasons or evidence to support an answer or conclusion.

Label

Add labels to a diagram.

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Give a sequence of brief answers with no explanation.

Plot

Mark the position of points on a diagram.

Predict

Give an expected result.

Prove

Use a sequence of logical steps to obtain the required result in a formal way.

Show

Give the steps in a calculation or derivation.

Show that

Obtain the required result (possibly using information given) without the formality of proof. “Show that” questions do not generally require the use of a calculator.

Sketch

Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.

Solve

Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.

State

Give a specific name, value or other brief answer without explanation or calculation.

Suggest

Propose a solution, hypothesis or other possible answer.

Verify

Provide evidence that validates the result.

Write down

Obtain the answer(s), usually by extracting information. Little or no calculation is required. Working does not need to be shown.

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A

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N

List

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Notation list Of the various notations in use, the IB has chosen to adopt a system of notation based on the recommendations of the International Organization for Standardization (ISO). This notation is used in the examination papers for this course without explanation. If forms of notation other than those listed in this guide are used on a particular examination paper, they are defined within the question in which they appear. Because students are required to recognize, though not necessarily use, IB notation in examinations, it is recommended that teachers introduce students to this notation at the earliest opportunity. Students are not allowed access to information about this notation in the examinations.

N

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3, ...}

the set of integers, {0, ± 1, ± 2, ± 3, ...}

+

the set of positive integers, {1, 2, 3, ...}

the set of rational numbers

+

the set of positive rational numbers, {x | x ∈ , x > 0}

the set of real numbers

+

the set of positive real numbers, {x | x ∈ , x > 0}

the set of complex numbers, {a + ib | a , b ∈ }

i

PU

−1

BL

IC

A

TI O

a complex number

z

the complex conjugate of z

z∗

z

E-

the modulus of z

arg z

PR

the argument of z

Re z

the real part of z

Im z

the imaginary part of z

cisθ

cos θ + i sin θ

{x1 , x2 , ...}

the set with elements x1 , x2 , ...

n( A)

the number of elements in the finite set A

{x |

}

the set of all x such that

∈

is an element of

∉

is not an element of

∅

the empty (null) set

U

the universal set

∪

union

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∩

intersection

⊂

is a proper subset of

⊆

is a subset of

A′

the complement of the set A

A× B

the Cartesian product of sets A and B (that is, = A × B {(a , b) a ∈ A , b ∈ B} )

a|b

a divides b

a1/ n ,

n

a

a to the power of

1 th , n root of a (if a ≥ 0 then n

n

a ≥0)

x for x ≥ 0, x ∈ the modulus or absolute value of x, that is − x for x < 0, x ∈

≡

identity

≈

is approximately equal to

>

is greater than

≥

is greater than or equal to

/

is not greater than

Mathematics HL guide

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First examinations 2014

This pre-publication has been released for information only, to help with planning. Although the content has been substantially finalized, the IB reserves the right to make further changes to the text before publication as it sees fit. This version must be replaced as soon as the final version is published, in order to avoid any confusion.

© International Baccalaureate Organization 2011

Contents Introduction Purpose of this document The Diploma Programme Nature of the subject Aims Assessment objectives Syllabus outline

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Approaches to the teaching and learning of mathematics HL Prior learning topics Syllabus content

External assessment Internal assessment

Appendices

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Notation list

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Glossary of command terms

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Assessment outline

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Assessment in the Diploma Programme

A

Glossary of terminology: Discrete mathematics

Assessment

N

Syllabus

Introduction

Purpose of this document This publication is intended to guide the planning, teaching and assessment of the subject in schools. Subject teachers are the primary audience, although it is expected that teachers will use the guide to inform students and parents about the subject. This guide can be found on the subject page of the online curriculum centre (OCC) at http://occ.ibo.org, a password-protected IB website designed to support IB teachers. It can also be purchased from the IB store at http://store.ibo.org.

N

Additional resources

TI O

Additional publications such as teacher support materials, subject reports, internal assessment guidance and grade descriptors can also be found on the OCC. Specimen and past examination papers as well as markschemes can be purchased from the IB store.

A

Teachers are encouraged to check the OCC for additional resources created or used by other teachers. Teachers can provide details of useful resources, for example: websites, books, videos, journals or teaching ideas.

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First examinations 2014

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The Diploma Programme The Diploma Programme is a rigorous pre-university course of study designed for students in the 16 to 19 age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to develop intercultural understanding, open-mindedness, and the attitudes necessary for them to respect and evaluate a range of points of view.

The Diploma Programme hexagon

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N

The course is presented as six academic areas enclosing a central core (see figure 1). It encourages the concurrent study of a broad range of academic areas. Students study: two modern languages (or a modern language and a classical language); a humanities or social science subject; an experimental science; mathematics; one of the creative arts. It is this comprehensive range of subjects that makes the Diploma Programme a demanding course of study designed to prepare students effectively for university entrance. In each of the academic areas students have flexibility in making their choices, which means they can choose subjects that particularly interest them and that they may wish to study further at university.

A

[Production insert DP hexagon diagram here.] Figure 1

IC

Diploma Programme model

BL

Choosing the right combination

PU

Students are required to choose one subject from each of the six academic areas, although they can choose a second subject from groups 1 to 5 instead of a group 6 subject. Normally, three subjects (and not more than four) are taken at higher level (HL), and the others are taken at standard level (SL). The IB recommends 240 teaching hours for HL subjects and 150 hours for SL. Subjects at HL are studied in greater depth and breadth than at SL.

PR

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At both levels, many skills are developed, especially those of critical thinking and analysis. At the end of the course, students’ abilities are measured by means of external assessment. Many subjects contain some element of coursework assessed by teachers. The course is available for examinations in English, French and Spanish, with the exception of groups 1 and 2 courses where examinations are in the language of study.

The core of the hexagon All Diploma Programme students participate in the three course requirements that make up the core of the hexagon. Reflection on all these activities is a principle that lies at the heart of the thinking behind the Diploma Programme. The theory of knowledge course encourages students to think about the nature of knowledge, to reflect on the process of learning in all the subjects they study as part of their Diploma Programme course, and to make connections across the academic areas. The extended essay, a substantial piece of writing of up to 4,000 words, enables students to investigate a topic of special interest that they have chosen themselves. It also encourages them to develop the skills of independent research that will be expected at university. Creativity, action, service involves students in experiential learning through a range of artistic, sporting, physical and service activities.

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The IB mission statement and the IB learner profile

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The Diploma Programme aims to develop in students the knowledge, skills and attitudes they will need to fulfill the aims of the IB, as expressed in the organization’s mission statement and the learner profile. Teaching and learning in the Diploma Programme represent the reality in daily practice of the organization’s educational philosophy.

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Nature of the subject

Introduction

TI O

N

The 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: visual 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, with all its interdisciplinary connections, provides a clear and sufficient rationale for making this subject compulsory for students studying for the full diploma.

IC

A

Summary of courses available

PU

BL

Because 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 to understand better 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.

•

•

their own abilities in mathematics and the type of mathematics in which they can be successful

PR

•

E-

In making this selection, individual students should be advised to take account of the following factors:

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 Diploma Programme

•

their academic plans, in particular the subjects they wish to study in future

•

their choice of career.

Teachers are expected to assist with the selection process and to offer advice to students.

Mathematical studies SL This course is available only at standard level, and is equivalent in status to mathematics SL, but addresses different needs. It has an emphasis on applications of mathematics, and the largest section is on statistical techniques. It is designed for students with varied mathematical backgrounds and abilities. It offers students opportunities to learn important concepts and techniques and to gain an understanding of a wide variety of mathematical topics. It prepares students to be able to solve problems in a variety of settings, to develop more sophisticated mathematical reasoning and to PRE-PUBLICATION Mathematics HL guide

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enhance their critical thinking. The individual project is an extended piece of work based on personal research involving the collection, analysis and evaluation of data. Students taking this course are well prepared for a career in social sciences, humanities, languages or arts. These students may need to utilize the statistics and logical reasoning that they have learned as part of the mathematical studies SL course in their future studies.

Mathematics SL This 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 HL

A

Further mathematics HL

TI O

N

This 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.

BL

IC

This course is available only at higher level. It caters for students with a very strong 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 expect 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. It is expected that students taking this course will also be taking mathematics HL.

E-

PU

Note: Mathematics HL is an ideal course for students 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 or technology. It should not be regarded as necessary for such students to study further mathematics HL. Rather, further mathematics HL is an optional course for students with a particular aptitude and interest in mathematics, enabling them to study some wider and deeper aspects of mathematics, but is by no means a necessary qualification to study for a degree in mathematics.

PR

Mathematics HL—course details The course 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 solve 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 exploration, offers students the opportunity for developing independence in their mathematical learning. Students are encouraged to take a considered approach to various mathematical activities and to explore different mathematical ideas. The exploration also allows students to work without the time constraints of a written examination and to develop the skills they need for communicating mathematical ideas.

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This 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. Students who wish to study an even more rigorous and demanding course should consider taking further mathematics HL in addition to mathematics HL.

Prior learning

N

Mathematics is a linear subject, and it is expected that most students embarking on a DP mathematics course will have studied mathematics for at least 10 years. There will be a great variety of topics studied, and differing approaches to teaching and learning. Thus students will have a wide variety of skills and knowledge when they start the mathematics HL course. Most will have some background in arithmetic, algebra, geometry, trigonometry, probability and statistics. Some will be familiar with an inquiry approach, and may have had an opportunity to complete an extended piece of work in mathematics.

TI O

At the beginning of the syllabus section, there is a list of topics that are considered to be prior learning for the mathematics HL course. It is recognized that this may contain topics that are unfamiliar to some students, but it is anticipated that there may be other topics in the syllabus itself that these students have already encountered. Teachers should plan their teaching to incorporate topics mentioned that are unfamiliar to their students.

A

Links to the Middle Years Programme

BL

IC

The prior learning topics for the Diploma Programme (DP) courses have been written in conjunction with the Middle Years Programme (MYP) mathematics guide. The approaches to teaching and learning for DP mathematics build on the approaches used in the MYP. These include investigations, exploration and a variety of different assessment tools.

E-

PU

A continuum document called Mathematics: The MYP–DP continuum is available on the DP mathematics home pages of the online curriculum centre (OCC). This extensive publication focuses on the alignment of mathematics across the MYP and the DP. It was developed in response to feedback provided by IB World Schools, which expressed the need to articulate the transition of mathematics from the MYP to the DP. The publication also highlights the similarities and differences between MYP and DP mathematics, and is a valuable resource for teachers.

PR

Mathematics and theory of knowledge The Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed that these all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by data from sense perception, mathematics is dominated by reason, and some mathematicians argue that their subject is a language, that it is, in some sense, universal. However, there is also no doubt that mathematicians perceive beauty in mathematics, and that emotion can be a strong driver in the search for mathematical knowledge. As an area of knowledge, mathematics seems to supply a certainty perhaps missing in other disciplines. This may be related to the “purity” of the subject that makes it sometimes seem divorced from reality. However, mathematics has also provided important knowledge about the world, and the use of mathematics in science and technology has been one of the driving forces for scientific advances. Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there “waiting to be discovered” or is it a human creation? PRE-PUBLICATION Mathematics HL guide

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Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should be encouraged to raise such questions themselves, in mathematics and TOK classes. This includes questioning all the claims made above. Examples of issues relating to TOK are given in the “Links” column of the syllabus. Teachers could also discuss questions such as those raised in the “Areas of knowledge” section of the TOK guide.

Mathematics and the international dimension Mathematics is in a sense an international language, and, apart from slightly differing notation, mathematicians from around the world can communicate within their field. Mathematics transcends politics, religion and nationality, yet throughout history great civilizations owe their success in part to their mathematicians being able to create and maintain complex social and architectural structures.

TI O

N

Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by Arabic, Greek, Indian and Chinese civilizations, among others. Teachers could use timeline websites to show the contributions of different civilizations to mathematics, but not just for their mathematical content. Illustrating the characters and personalities of the mathematicians concerned and the historical context in which they worked brings home the human and cultural dimension of mathematics.

IC

A

The importance of science and technology in the everyday world is clear, but the vital role of mathematics is not so well recognized. It is the language of science, and underpins most developments in science and technology. A good example of this is the digital revolution, which is transforming the world, as it is all based on the binary number system in mathematics.

BL

Many international bodies now exist to promote mathematics. Students are encouraged to access the extensive websites of international mathematical organizations to enhance their appreciation of the international dimension and to engage in the global issues surrounding the subject.

PR

E-

PU

Examples of global issues relating to international-mindedness (Int) are given in the “Links” column of the syllabus.

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Aims

Group 5 aims The aims of all mathematics courses in group 5 are to enable students to: enjoy mathematics, and develop an appreciation of the elegance and power of mathematics

2.

develop an understanding of the principles and nature of mathematics

3.

communicate clearly and confidently in a variety of contexts

4.

develop logical, critical and creative thinking, and patience and persistence in problem-solving

5.

employ and refine their powers of abstraction and generalization

6.

apply and transfer skills to alternative situations, to other areas of knowledge and to future developments

7.

appreciate how developments in technology and mathematics have influenced each other

8.

appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics

9.

appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives

10.

appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course.

PR

E-

PU

BL

IC

A

TI O

N

1.

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Assessment objectives Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics HL course, students will be expected to demonstrate the following. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.

2.

Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems.

3.

Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.

4.

Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems.

5.

Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions.

6.

Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analysing information, making conjectures, drawing conclusions, and testing their validity.

PR

E-

PU

BL

IC

A

TI O

N

1.

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Syllabus

Syllabus outline Teaching hours

Syllabus component

HL

All topics 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 prior learning. 30

N

Topic 1 Algebra

TI O

Topic 2 Functions and equations Circular functions and trigonometry

IC

Topic 4

A

Topic 3

Topic 5 Statistics and probability Topic 6

PU

Calculus

BL

Vectors

Option syllabus content

22 22 24 36 48 48

PR

Topic 7

E-

Students must study all the sub-topics in one of the following options as listed in the syllabus details. Statistics and probability Topic 8

Sets, relations and groups Topic 9 Calculus Topic 10 Discrete mathematics Exploration

10

Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. Total teaching hours

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10

Approaches to the teaching and learning of mathematics HL Throughout the DP mathematics HL course, students should be encouraged to develop their understanding of the methodology and practice of the discipline of mathematics. The processes of mathematical inquiry, mathematical modelling and applications and the use of technology should be introduced appropriately. These processes should be used throughout the course, and not treated in isolation.

Mathematical inquiry

TI O

N

The IB learner profile encourages learning by experimentation, questioning and discovery. In the IB classroom, students should generally learn mathematics by being active participants in learning activities rather than recipients of instruction. Teachers should therefore provide students with opportunities to learn through mathematical inquiry. This approach is illustrated in the diagram below. [Production insert flowchart 1 here.]

A

Mathematical modelling and applications

IC

Students should be able to use mathematics to solve problems in the real world. Engaging students in the mathematical modelling process provides such opportunities. Students should develop, apply and critically analyse models. This approach is illustrated in the diagram below.

PU

Technology

BL

[Production insert flowchart 2 here.]

PR

E-

Technology is a powerful tool in the teaching and learning of mathematics. Technology can be used to enhance visualization and support student understanding of mathematical concepts. It can assist in the collection, recording, organization and analysis of data. Technology can increase the scope of the problem situations that are accessible to students. The use of technology increases the feasibility of students working with interesting problem contexts where students reflect, reason, solve problems and make decisions. As teachers tie together the unifying themes of mathematical inquiry, mathematical modelling and applications and the use of technology, they should begin by providing substantial guidance, and then gradually encourage students to become more independent as inquirers and thinkers. IB students should learn to become strong communicators through the language of mathematics. Teachers should create a safe learning environment in which students are comfortable as risk-takers. Teachers are encouraged to relate the mathematics being studied to other subjects and to the real world, especially topics that have particular relevance or are of interest to their students. Everyday problems and questions should be drawn into the lessons to motivate students and keep the material relevant; suggestions are provided in the “Links” column of the syllabus. The mathematical exploration offers an opportunity to investigate the usefulness, relevance and occurrence of mathematics in the real world and will add an extra dimension to the course. The emphasis is on communication by means of mathematical forms (for example, formulae, diagrams, graphs and so on) with accompanying commentary. Modelling, investigation, reflection, personal engagement and mathematical communication should therefore feature prominently in the DP mathematics classroom. For further information on “Approaches to teaching a DP course”, please refer to the publication The Diploma Programme: From principles into practice (2009). To support teachers, a variety of PRE-PUBLICATION Mathematics HL guide

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resources can be found on the OCC, and details of workshops for professional development are available on the public website.

Format of the syllabus Content: this column lists, under each topic, the sub-topics to be covered.

•

Further guidance: this column contains more detailed information on specific sub-topics listed in the content column. This clarifies the content for examinations.

•

Links: this column provides useful links to the aims of the mathematics HL course, with suggestions for discussion, real-life examples and ideas for further investigation. These suggestions are only a guide for introducing and illustrating the sub-topic and are not exhaustive. Links are labelled as follows.

N

•

real-life examples and links to other Diploma Programme subjects

Aim 8

moral, social and ethical implications of the sub-topic

Int

international-mindedness

TOK

suggestions for discussion

A

TI O

Appl

IC

Note that any syllabus references to other subject guides given in the “Links” column are correct for the current (2012) published versions of the guides.

BL

Notes on the syllabus

Formulae are only included in this document where there may be some ambiguity. All formulae required for the course are in the mathematics HL and further mathematics HL formula booklet.

•

The term “technology” is used for any form of calculator or computer that may be available. However, there will be restrictions on which technology may be used in examinations, which will be noted in relevant documents.

•

The terms “analysis” and “analytic approach” are generally used when referring to an approach that does not use technology.

E-

PU

•

PR

Course of study

The content of all six topics in the syllabus must be taught, although not necessarily in the order in which they appear in this guide. Teachers are expected to construct a course of study that addresses the needs of their students and includes, where necessary, the topics noted in prior learning.

Integration of the mathematical exploration Work leading to the completion of the exploration should be integrated into the course of study. Details of how to do this are given in the section on internal assessment and in the teacher support material.

Time allocation The recommended teaching time for higher level courses is 240 hours. For mathematics HL, it is expected that 10 hours of this will be spent on work for the exploration. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 230 hours allowed for the PRE-PUBLICATION Mathematics HL guide

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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.

Use of calculators Students 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 calculators allowed are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematics HL/SL: Graphic display calculators teacher support material (May 2005) and on the OCC.

TI O

N

Mathematics HL and further mathematics HL formula booklet

A

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. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there are no printing errors, and ensure that there are sufficient copies available for all students.

IC

Teacher support materials

BL

A variety of teacher support materials will accompany this guide. These materials will include guidance for teachers on the introduction, planning and marking of the exploration, and specimen examination papers and markschemes.

PU

Command terms and notation list

PR

E-

Teachers and students need to be familiar with the IB notation, and the command terms, as these will be used without explanation in the examination papers. The “Glossary of command terms” and “Notation list” appear as appendices in this guide.

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Prior learning topics As noted in the previous section on prior learning, it is expected that all students have extensive previous mathematical experiences, but these will vary. It is expected that mathematics HL students will be familiar with the following topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics listed here 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 table lists the knowledge, together with the syllabus content, that is essential to successful completion of the mathematics HL course.

N

Students must be familiar with SI (Système International) units of length, mass and time, and their derived units. Content

Number

Routine use of addition, subtraction, multiplication and division, using integers, decimals and fractions, including order of operations.

TI O

Topic

Rational exponents.

A

Simplification of expressions involving roots (surds or radicals), including rationalizing the denominator.

IC

Prime numbers and factors (divisors), including greatest common divisors and least common multiples.

BL

Simple applications of ratio, percentage and proportion, linked to similarity. Definition and elementary treatment of absolute value (modulus), a .

PU

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 ∈ .

E-

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.

PR

Sets and numbers

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. Mappings of the elements of one set to another; sets of ordered pairs.

Algebra

Manipulation of linear and quadratic expressions, including factorization, expansion, completing the square and use of the formula. Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas, particularly the sciences, should be included. Linear functions, their graphs, gradients and y-intercepts. Addition and subtraction of simple algebraic fractions. The properties of order relations: , ≥ .

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Solution of linear equations and inequalities in one variable, including cases with rational coefficients. Solution of quadratic equations and inequalities, using factorization and completing the square. Solution of simultaneous linear equations in two variables. Trigonometry

Angle measurement in degrees. Compass directions. Right-angle trigonometry. Simple applications for solving triangles. Pythagoras’ theorem and its converse. Simple geometric transformations: translation, reflection, rotation, enlargement. Congruence and similarity, including the concept of scale factor of an enlargement.

Geometry

N

The circle, its centre and radius, area and circumference. The terms arc, sector, chord, tangent and segment.

TI O

Perimeter and area of plane figures. Properties of triangles and quadrilaterals, including parallelograms, rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound shapes. Volumes of cuboids, pyramids, spheres, cylinders and cones. Classification of prisms and pyramids, including tetrahedra. Elementary geometry of the plane, including the concepts of dimension for point, line, plane and space. The equation of a line in the form = y mx + c . Parallel and perpendicular lines, including m1 = m2 and m1m2 = −1 .

IC

A

Coordinate geometry

Descriptive statistics: collection of raw data, display of data in pictorial and diagrammatic forms, including frequency histograms, cumulative frequency graphs.

PU

Statistics and probability

BL

The Cartesian plane: ordered pairs ( x, y ) , origin, axes. Mid-point of a line segment and distance between two points in the Cartesian plane.

Obtaining simple statistics from discrete and continuous data, including mean, median, mode, quartiles, range, interquartile range and percentiles.

PR

E-

Calculating probabilities of simple events.

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Syllabus content

Further guidance

Links

Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.

Sequences can be generated and displayed in several ways, including recursive functions.

Sigma notation.

Link infinite geometric series with limits of convergence in 6.1.

Applications.

Examples include compound interest and population growth.

BL IC

A

Content

PR E

-P U

1.1

TI O

30 hours The aim of this topic is to introduce students to some basic algebraic concepts and applications.

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Topic 1—Core: Algebra

Int: The chess legend (Sissa ibn Dahir). Int: Aryabhatta is sometimes considered the “father of algebra”. Compare with al-Khawarizmi. Int: The use of several alphabets in mathematical notation (eg first term and common difference of an arithmetic sequence). TOK: Mathematics and the knower. To what extent should mathematical knowledge be consistent with our intuition?

TOK: Mathematics and the world. Some mathematical constants ( π , e, φ , Fibonacci numbers) appear consistently in nature. What does this tell us about mathematical knowledge? TOK: Mathematics and the knower. How is mathematical intuition used as a basis for formal proof? (Gauss’ method for adding up integers from 1 to 100.) Aim 8: Short-term loans at high interest rates. How can knowledge of mathematics result in individuals being exploited or protected from extortion? Appl: Physics 7.2, 13.2 (radioactive decay and nuclear physics).

16

1.2

Exponents and logarithms. Laws of exponents; laws of logarithms.

Exponents and logarithms are further developed in 2.4.

Appl: Chemistry 18.1, 18.2 (calculation of pH and buffer solutions). TOK: The nature of mathematics and science. Were logarithms an invention or discovery? (This topic is an opportunity for teachers and students to reflect on “the nature of mathematics”.)

A

Counting principles, including permutations and n The ability to find and n Pr using combinations. r both the formula and technology is expected. Link to 5.4. The binomial theorem:

Link to 5.6, binomial distribution.

expansion of (a + b) n , n ∈ . Not required: Permutations where some objects are identical. Proof of binomial theorem. Proof by mathematical induction.

PR E

1.4

-P U

Circular arrangements.

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1.3

TI O

N

Change of base.

Links to a wide variety of topics, for example, complex numbers, differentiation, sums of series and divisibility.

TOK: The nature of mathematics. The unforeseen links between Pascal’s triangle, counting methods and the coefficients of polynomials. Is there an underlying truth that can be found linking these?

Int: The properties of Pascal’s triangle were known in a number of different cultures long before Pascal (eg the Chinese mathematician Yang Hui). Aim 8: How many different tickets are possible in a lottery? What does this tell us about the ethics of selling lottery tickets to those who do not understand the implications of these large numbers?

TOK: Nature of mathematics and science. What are the different meanings of induction in mathematics and science? TOK: Knowledge claims in mathematics. Do proofs provide us with completely certain knowledge? TOK: Knowledge communities. Who judges the validity of a proof?

17

1.5

Complex numbers: the number i= −1 ; the terms real part, imaginary part, conjugate, modulus and argument.

When solving problems, students may need to use technology.

Cartesian form z= a + ib .

Appl: Concepts in electrical engineering. Impedance as a combination of resistance and reactance; also apparent power as a combination of real and reactive powers. These combinations take the form z= a + ib .

N

TOK: Mathematics and the knower. Do the words imaginary and complex make the concepts more difficult than if they had different names?

TI O

Sums, products and quotients of complex numbers.

Modulus–argument (polar) form z =r (cos θ + i sin θ ) =r cisθ =r eiθ .

r eiθ is also known as Euler’s form.

BL IC

1.6

A

TOK: The nature of mathematics. Has “i” been invented or was it discovered?

The ability to convert between forms is expected.

-P U

Powers of complex numbers: de Moivre’s theorem.

PR E

nth roots of a complex number. 1.8

Appl: Concepts in electrical engineering. Phase angle/shift, power factor and apparent power as a complex quantity in polar form.

TOK: The nature of mathematics. Was the complex plane already there before it was used to represent The complex plane is also known as the complex numbers geometrically? Argand diagram. TOK: Mathematics and the knower. Why might it be said that ei π + 1 = 0 is beautiful?

The complex plane.

1.7

TOK: Mathematics and the world. Why does “i” appear in so many fundamental laws of physics?

Conjugate roots of polynomial equations with real coefficients.

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Proof by mathematical induction for n ∈ + .

TOK: Reason and mathematics. What is mathematical reasoning and what role does proof play in this form of reasoning? Are there examples of proof that are not mathematical?

Link to 2.5 and 2.7.

18

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.

These systems should be solved using both algebraic and technological methods, eg row reduction. Systems that have solution(s) may be referred to as consistent.

PR E

-P U

BL IC

A

Link to vectors in 4.7.

TI O

When a system has an infinity of solutions, a general solution may be required.

TOK: Mathematics, sense, perception and reason. If we can find solutions in higher dimensions, can we reason that these spaces exist beyond our sense perception?

N

1.9

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Topic 2—Core: Functions and equations 22 hours

Content 2.1

Further guidance

Concept of function f : x f ( x) : domain, range; image (value).

Links

A

Odd and even functions.

TI O

N

The aims of this topic 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 technology in both the development and the application of this topic.

( f g )( x) = f ( g ( x)) . Link with 6.2.

Identity function.

BL IC

Composite functions f g .

Link with 3.4.

Inverse function f −1 , including domain restriction. Self-inverse functions.

Link with 6.2.

TOK: The nature of mathematics. Is mathematics simply the manipulation of symbols under a set of formal rules?

PR E

-P U

One-to-one and many-to-one functions.

Int: The notation for functions was developed by a number of different mathematicians in the 17th and 18th centuries. How did the notation we use today become internationally accepted?

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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.

Use of technology to graph a variety of functions.

Appl: Sketching and interpreting graphs; Geography SL/HL (geographic skills); Chemistry 11.3.1.

The graphs of the functions y = f ( x) and

Int: Bourbaki group analytical approach versus Mandlebrot visual approach.

y = f ( x) .

2.3

Transformations of graphs: translations; stretches; reflections in the axes.

2.4

The rational function x

ax + b , and its cx + d

PR E

graph.

The function x a x , a > 0 , and its graph.

The function x log a x , x > 0 , and its graph.

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Link to 3.4. Students are expected to be aware Appl: Economics SL/HL 1.1 (shift in demand of the effect of transformations on both the and supply curves). algebraic expression and the graph of a function.

-P U

The graph of the inverse function as a reflection in y = x .

BL IC

1 given the graph of f ( x)

A

y = f(x).

The graph of y =

TOK: Mathematics and knowledge claims. Does studying the graph of a function contain the same level of mathematical rigour as studying the function algebraically (analytically)?

N

The graph of a function; its equation y = f ( x) .

TI O

2.2

The reciprocal function is a particular case. Graphs should include both asymptotes and any intercepts with axes. Exponential and logarithmic functions as inverses of each other. Link to 6.2 and the significance of e. Application of concepts in 2.1, 2.2 and 2.3.

Appl: Geography SL/HL (geographic skills); Physics SL/HL 7.2 (radioactive decay); Chemistry SL/HL 16.3 (activation energy); Economics SL/HL 3.2 (exchange rates).

21

The graphical significance of repeated factors.

The factor and remainder theorems.

The relationship between the degree of a polynomial function and the possible numbers of x-intercepts.

The fundamental theorem of algebra. 2.6

Solving quadratic equations using the quadratic formula.

N

Polynomial functions and their graphs.

May be referred to as roots of equations or zeros Appl: Chemistry 17.2 (equilibrium law). of functions. Appl: Physics 2.1 (kinematics).

TI O

2.5

Use of the discriminant ∆= b 2 − 4ac to determine the nature of the roots.

Appl: Physics 4.2 (energy changes in simple harmonic motion).

Link the solution of polynomial equations to conjugate roots in 1.8.

Sum and product of the roots of polynomial equations.

For the polynomial equation

A

Solving polynomial equations both graphically and algebraically.

BL IC

∑ ar x r = 0 , r =0

the sum is

-P U

Aim 8: The phrase “exponential growth” is used popularly to describe a number of phenomena. Is this a misleading use of a mathematical term?

− an −1 , an

the product is

Solution of a x = b using logarithms.

n

Appl: Physics (HL only) 9.1 (projectile motion).

(−1) n a0 . an

2.7

PR E

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach. Solutions of g ( x) ≥ f ( x) .

Graphical or algebraic methods, for simple polynomials up to degree 3.

Use of technology for these and other functions.

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Topic 3—Core: Circular functions and trigonometry 22 hours

Further guidance

The circle: radian measure of angles.

Radian measure may be expressed as multiples of π, or decimals. Link with 6.2.

Length of an arc; area of a sector. 3.2

Definition of cos θ , sin θ and tan θ in terms of the unit circle. Exact values of sin, cos and tan of 0,

π π π π

, , , and their multiples. 6 4 3 2

-P U

Definition of the reciprocal trigonometric ratios secθ , cscθ and cotθ .

Links

A

Content

BL IC

3.1

TI O

N

The aims of this topic are to explore the circular functions, to introduce some important trigonometric identities and to solve triangles using trigonometry. On . examination papers, radian measure should be assumed unless otherwise indicated, for example, by x sin x°

Pythagorean identities: cos 2 θ + sin 2 θ = 1; 2 2 2 2 1 + tan θ = sec θ ; 1 + cot θ = csc θ . Compound angle identities.

PR E

3.3

Double angle identities. Not required:

Proof of compound angle identities.

PRE-PUBLICATION Mathematics HL guide

Derivation of double angle identities from compound angle identities. Finding possible values of trigonometric ratios without finding θ, for example, finding sin 2θ given sin θ .

Int: The origin of degrees in the mathematics of Mesopotamia and why we use minutes and seconds for time.

TOK: Mathematics and the knower. Why do we use radians? (The arbitrary nature of degree measure versus radians as real numbers and the implications of using these two measures on the shape of sinusoidal graphs.) TOK: Mathematics and knowledge claims. If trigonometry is based on right triangles, how can we sensibly consider trigonometric ratios of angles greater than a right angle? Int: The origin of the word “sine”. Appl: Physics SL/HL 2.2 (forces and dynamics). Appl: Triangulation used in the Global Positioning System (GPS). Int: Why did Pythagoras link the study of music and mathematics? (continued)

23

3.4

(see notes above) Appl: Concepts in electrical engineering. Generation of sinusoidal voltage.

Composite functions of the form f= ( x) a sin(b( x + c)) + d . Applications.

N

The inverse functions x arcsin x , x arccos x , x arctan x ; their domains and ranges; their graphs.

TI O

3.5

TOK: Mathematics and the world. Music can be expressed using mathematics. Does this mean that music is mathematical, that mathematics is musical or that both are reflections of a common “truth”.

Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization. Not required: The general solution of trigonometric equations. The cosine rule

-P U

3.7

BL IC

3.6

A

Appl: Physics SL/HL 4.1 (kinematics of simple harmonic motion).

The sine rule including the ambiguous case.

Applications.

1 ab sin C . 2

PR E

Area of a triangle as

PRE-PUBLICATION Mathematics HL guide

Examples include navigation, problems in two and three dimensions, including angles of elevation and depression.

TOK: Mathematics and knowledge claims. How can there be an infinite number of discrete solutions to an equation?

TOK: Nature of mathematics. If the angles of a triangle can add up to less than 180°, 180° or more than 180°, what does this tell us about the “fact” of the angle sum of a triangle and about the nature of mathematical knowledge? Appl: Physics SL/HL 1.3 (vectors and scalars); Physics SL/HL 2.2 (forces and dynamics). Int: The use of triangulation to find the curvature of the Earth in order to settle a dispute between England and France over Newton’s gravity.

24

Topic 4—Core: Vectors 24 hours

4.1

Further guidance

Concept of a vector.

A

Representation of vectors using directed line segments. Components of a vector: .

Aim 8: Vectors are used to solve many problems in position location. This can be used to save a lost sailor or destroy a building with a laser-guided bomb. Appl: Physics SL/HL 1.3 (vectors and scalars); Physics SL/HL 2.2 (forces and dynamics).

TOK: Mathematics and knowledge claims. You can perform some proofs using different mathematical concepts. What does this tell us about mathematical knowledge?

Proofs of geometrical properties using vectors.

-P U

Algebraic and geometric approaches to the following:

BL IC

Unit vectors; base vectors i, j, k.

v1 v = v2 =v1i + v2 j + v3 k v 3

Links

TI O

Content

N

The aim of this topic is to introduce the use of vectors in two and three dimensions, and to facilitate solving problems involving points, lines and planes.

the sum and difference of two vectors;

PR E

the zero vector 0 , the vector −v ; multiplication by a scalar, kv ; magnitude of a vector, v ; →

position vectors OA = a . →

AB= b − a

PRE-PUBLICATION Mathematics HL guide

Distance between points A and B is the →

magnitude of AB . 25

The definition of the scalar product of two vectors. Properties of the scalar product:

Link to 3.6.

v ⋅w = w ⋅v ;

For non-zero vectors, v ⋅ w = 0 is equivalent to the vectors being perpendicular.

(kv ) ⋅ w = k (v ⋅ w ) ;

For parallel vectors, v ⋅ w = v w.

TOK: The nature of mathematics. Why this definition of scalar product?

N

u ⋅ (v + w ) = u ⋅ v + u ⋅ w ;

2

v ⋅ v =v .

The angle between two vectors.

A

Perpendicular vectors; parallel vectors. Vector equation of a line in two and three dimensions: r = a + λ b .

Knowledge of the following forms for equations Appl: Modelling linear motion in three of lines. dimensions.

Simple applications to kinematics.

Parametric form:

Appl: Navigational devices, eg GPS.

The angle between two lines.

x x0 + λ l , = z z0 + λ n . = y y0 + λ m , =

TOK: The nature of mathematics. Why might it be argued that vector representation of lines is superior to Cartesian?

BL IC

4.3

Appl: Physics SL/HL 2.2 (forces and dynamics).

v⋅w = v w cos θ , where θ is the angle between v and w .

TI O

4.2

-P U

Cartesian form:

x − x0 y − y0 z − z0 . = = l m n

Coincident, parallel, intersecting and skew lines; distinguishing between these cases.

PR E

4.4

Points of intersection.

PRE-PUBLICATION Mathematics HL guide

26

4.5

The definition of the vector product of two vectors. Properties of the vector product: u × (v + w ) = u × v + u × w ;

TI O

(kv ) × w = k (v × w ) ; v×v = 0.

Use of normal vector to obtain the form r ⋅n =a ⋅n. Cartesian equation of a plane ax + by + cz = d. Intersections of: a line with a plane; two planes; three planes.

Link to 1.9.

-P U

4.7

BL IC

Vector equation of a plane r =a + λ b + µ c .

A

Areas of triangles and parallelograms.

Geometric interpretation of v × w . 4.6

Appl: Physics SL/HL 6.3 (magnetic force and field).

N

v × w =− w × v ;

v×w = v w sin θ n , where θ is the angle between v and w and n is the unit normal vector whose direction is given by the righthand screw rule.

Geometrical interpretation of solutions.

PR E

Angle between: a line and a plane; two planes.

TOK: Mathematics and the knower. Why are symbolic representations of three-dimensional objects easier to deal with than visual representations? What does this tell us about our knowledge of mathematics in other dimensions?

PRE-PUBLICATION Mathematics HL guide

27

Topic 5—Core: Statistics and probability 36 hours

Further guidance

Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.

For examination purposes data will be treated as TOK: The nature of mathematics. Why have the population. mathematics and statistics sometimes been treated as separate subjects? In examinations the following formulae should be used: TOK: The nature of knowing. Is there a difference between information and data. k f x ∑ ii Aim 8: Does the use of statistics lead to an , µ = i =1 overemphasis on attributes that can easily be n measured over those that cannot?

Grouped data: mid-interval values, interval width, upper and lower interval boundaries. Mean, variance, standard deviation. Not required:

Links

A

Content

BL IC

5.1

TI O

N

The aim of this topic is to introduce basic concepts. It may be considered as three parts: manipulation and presentation of statistical data (5.1), the laws of probability (5.2–5.4.), and random variables and their probability distributions (5.5–5.7). It is expected that most of the calculations required will be done on a GDC. The emphasis is on understanding and interpreting the results obtained. Statistical tables will no longer be allowed in examinations.

PR E

-P U

k k Estimation of mean and variance of a population 2 − µ f x f i xi 2 ( ) ∑ ∑ i i from a sample. 2 i 1 =i 1 = = σ = − µ2 . n n

PRE-PUBLICATION Mathematics HL guide

Appl: Psychology SL/HL (descriptive statistics); Geography SL/HL (geographic skills); Biology SL/HL 1.1.2 (statistical analysis). Appl: Methods of collecting data in real life (census versus sampling). Appl: Misleading statistics in media reports.

28

Aim 8: Why has it been argued that theories based on the calculable probabilities found in casinos are pernicious when applied to everyday life (eg economics)?

Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A as P( A) =

n( A) . n(U )

Int: The development of the mathematical theory of probability in 17th century France.

N

5.2

The complementary events A and A′ (not A).

Combined events; the formula for P( A ∪ B ) .

A

5.3

TI O

Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.

5.4

Conditional probability; the definition P( A ∩ B ) P( A | B) = P( B )

.

P ( A= | B ) P= ( A ) P ( A | B′ ) .

Use of P( A ∩ B ) = P( A)P( B ) to show independence.

Appl: Use of probability methods in medical studies to assess risk factors for certain diseases. TOK: Mathematics and knowledge claims. Is independence as defined in probabilistic terms the same as that found in normal experience?

-P U

Independent events; the definition

BL IC

Mutually exclusive events.

5.5

PR E

Use of Bayes’ theorem for a maximum of three events.

TOK: Mathematics and the knower. To what extent can we trust samples of data?

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.

For a continuous random variable, a value at which the probability density function has a maximum value is called a mode.

Applications.

Examples include games of chance.

PRE-PUBLICATION Mathematics HL guide

Appl: Expected gain to insurance companies. 29

5.6

Binomial distribution, its mean and variance.

Link to binomial theorem in 1.3.

Poisson distribution, its mean and variance.

Conditions under which random variables have these distributions.

N

Not required:

TI O

Formal proof of means and variances. 5.7

TOK: Mathematics and the real world. Is the binomial distribution ever a useful model for an actual real-world situation?

Normal distribution.

Probabilities and values of the variable must be found using technology.

Link to 2.3.

BL IC

Properties of the normal distribution.

A

The standardized value (z) gives the number of standard deviations from the mean.

Aim 8: Why might the misuse of the normal distribution lead to dangerous inferences and conclusions? TOK: Mathematics and knowledge claims. To what extent can we trust mathematical models such as the normal distribution? Int: De Moivre’s derivation of the normal distribution and Querelet’s use of it to describe l’homme moyen.

PR E

-P U

Standardization of normal variables.

Appl: Chemistry SL/HL 6.2 (collision theory); Psychology HL (descriptive statistics); Biology SL/HL 1.1.3 (statistical analysis).

PRE-PUBLICATION Mathematics HL guide

30

Topic 6—Core: Calculus 48 hours

Informal ideas of limit, continuity and convergence.

Include result lim

Definition of derivative from first principles

Link to 1.1.

h →0

f ( x + h) − f ( x ) . h

Finding equations of tangents and normals.

PR E PRE-PUBLICATION Mathematics HL guide

Link to binomial theorem in 1.3. Both forms of notation, first derivative.

dy and f ′ ( x ) , for the dx

Use of both algebra and technology.

-P U

Identifying increasing and decreasing functions.

Higher derivatives.

θ

=1.

Use of this definition for polynomials only.

The derivative interpreted as a gradient function and as a rate of change.

The second derivative.

sin θ

A

f ′( x) = lim

θ →0

Links

TI O

Further guidance

BL IC

6.1

Content

N

The aim of this topic is to introduce students to the basic concepts and techniques of differential and integral calculus and their application.

TOK: The nature of mathematics. Does the fact that Leibniz and Newton came across the calculus at similar times support the argument that mathematics exists prior to its discovery? Int: How the Greeks’ distrust of zero meant that Archimedes’ work did not lead to the calculus. Int: Investigate attempts by Indian mathematicians (500–1000 CE) to explain division by zero.

TOK: Mathematics and the knower. What does the dispute between Newton and Leibniz tell us about human emotion and mathematical discovery?

Appl: Economics HL 1.5 (theory of the firm); Chemistry SL/HL 11.3.4 (graphical techniques); d2 y Both forms of notation, 2 and f ′′( x) , for the Physics SL/HL 2.1 (kinematics). dx second derivative. Familiarity with the notation

Link with induction in 1.4.

dn y and f ( n ) ( x) . dx n

31

6.2

Appl: Physics HL 2.4 (uniform circular motion); Physics 12.1 (induced electromotive force (emf)).

Derivatives of x n , sin x , cos x , tan x , e x and ln x . Differentiation of sums and multiples of functions.

N

TOK: Mathematics and knowledge claims. Euler was able to make important advances in mathematical analysis before calculus had been put on a solid theoretical foundation by Cauchy and others. However, some work was not possible until after Cauchy’s work. What does this tell us about the importance of proof and the nature of mathematics?

The product and quotient rules.

TI O

The chain rule for composite functions. Related rates of change. Implicit differentiation.

Optimization problems.

Points of inflexion with zero and non-zero gradients.

PR E

Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ′′ . Not required:

BL IC

Local maximum and minimum values.

TOK: Mathematics and the real world. The seemingly abstract concept of the calculus allows us to create mathematical models that permit human feats, such as getting a man on the Moon. What does this tell us about the links between mathematical models and physical reality?

Testing for the maximum or minimum using the change of sign of the first derivative and using the sign of the second derivative.

-P U

6.3

A

Derivatives of sec x , csc x , cot x , a x , log a x , arcsin x , arccos x and arctan x .

Use of the terms “concave up” for f ′′( x) > 0 , “concave down” for f ′′( x) < 0 . A point of inflexion exists where f ′′( x) = 0 and there is a sign change in f ′′ (i.e. concavity change).

Points of inflexion, where f ′′( x) is not defined, for example, y = x1 3 at (0,0) .

PRE-PUBLICATION Mathematics HL guide

32

Indefinite integral of x n , sin x , cos x and e x .

Indefinite integral interpreted as a family of curves. 1

Other indefinite integrals using the results from 6.2.

dx ∫ x=

The composites of any of these with a linear function.

Examples include and

1 dx . + 2x + 5

Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves.

Kinematic problems involving displacement s, velocity v and acceleration a.

PR E

Total distance travelled.

PRE-PUBLICATION Mathematics HL guide

dx ,

1

∫ 3x + 4 dx

The value of some definite integrals can only be found using technology.

-P U

Volumes of revolution about the x-axis or y-axis.

5

A

Anti-differentiation with a boundary condition to determine the constant of integration. Definite integrals.

6.6

2

∫ ( 2 x − 1)

BL IC

6.5

∫x

ln x + c .

N

Indefinite integration as anti-differentiation.

TI O

6.4

v=

ds dv d 2 s dv ,= a = = v . dt dt dt 2 ds t2

Total distance travelled = ∫ v dt .

Appl: Industrial design. Appl: Physics HL 2.1 (kinematics). Int: Does the inclusion of kinematics as core mathematics reflect a particular cultural heritage? Who decides what is mathematics?

t1

33

On examination papers, non-standard substitutions will be provided.

Integration by parts.

Link to 6.2. Examples:

∫ x sin x dx and ∫ ln x dx .

N

Integration by substitution

TI O

6.7

Repeated integration by parts.

∫x e

2 x

dx and ∫ e x sin x dx .

PR E

-P U

BL IC

A

Examples:

PRE-PUBLICATION Mathematics HL guide

34

Topic 7—Option: Statistics and probability

TI O

48 hours

N

Option syllabus content

7.1

Further guidance

Cumulative distribution functions for both discrete and continuous distributions. Geometric distribution. Negative binomial distribution.

BL IC

Content

-P U

= G (t ) E( = tX )

Probability generating functions for discrete random variables.

P( X ∑=

x)t x .

Links

Int: Also known as Pascal’s distribution.

x

Aim 8: Statistical compression of data files.

PR E

Using probability generating functions to find mean, variance and the distribution of the sum of n independent random variables.

A

The aims of this option are to allow students the opportunity to approach statistics in a practical way; to demonstrate a good level of statistical understanding; and to understand which situations apply and to interpret the given results. It is expected that GDCs will be used throughout this option, and that the minimum requirement of a GDC will be to find pdf, cdf, inverse cdf, p-values and test statistics, including calculations for the following distributions: binomial, Poisson, normal and t. Students are expected to set up the problem mathematically and then read the answers from the GDC, indicating this within their written answers. Calculator-specific or brand-specific language should not be used within these explanations.

PRE-PUBLICATION Mathematics HL guide

35

7.2

Linear transformation of a single random variable. Mean of linear combinations of n random variables.

E(aX += b) aE( X ) + b ,

Var(aX + b) = a 2 Var( X ) .

Unbiased estimators and estimates.

T is an unbiased estimator for the parameter θ if E(T ) = θ .

Comparison of unbiased estimators based on variances.

T1 is a more efficient estimator than T2 if Var(T1 ) < Var(T2 ) . n

X as an unbiased estimator for µ .

X =∑ i =1

S 2 as an unbiased estimator for σ 2 .

n

i =1

PR E

A linear combination of independent normal random variables is normally distributed. In particular, σ2 X ~ N( µ ,σ 2 ) ⇒ X ~ N µ , . n

The central limit theorem.

PRE-PUBLICATION Mathematics HL guide

TOK: Mathematics and the world. In the absence of knowing the value of a parameter, will an unbiased estimator always be better than a biased one?

Xi . n

(X

-P U

S =∑ 2

7.4

TI O

E( XY ) = E( X )E(Y ) .

A

Expectation of the product of independent random variables.

BL IC

7.3

N

Variance of linear combinations of n independent random variables.

i

−X)

n −1

2

. Aim 8/TOK: Mathematics and the world. “Without the central limit theorem, there could be no statistics of any value within the human sciences.” TOK: Nature of mathematics. The central limit theorem can be proved mathematically (formalism), but its truth can be confirmed by its applications (empiricism).

36

Confidence intervals for the mean of a normal population.

Use of the normal distribution when σ is known and use of the t-distribution when σ is unknown, regardless of sample size. The case of matched pairs is to be treated as an example of a single sample technique.

TOK: Mathematics and the world. Claiming brand A is “better” on average than brand B can mean very little if there is a large overlap between the confidence intervals of the two means.

N

7.5

Significance level. Critical regions, critical values, p-values, onetailed and two-tailed tests. Type I and II errors, including calculations of their probabilities.

TOK: Mathematics and the world. In practical terms, is saying that a result is significant the same as saying that it is true?

TOK: Mathematics and the world. Does the ability to test only certain parameters in a population affect the way knowledge claims in the human sciences are valued? Appl: When is it more important not to make a Type I error and when is it more important not to make a Type II error?

PR E

-P U

Testing hypotheses for the mean of a normal population.

Use of the normal distribution when σ is known and use of the t-distribution when σ is unknown, regardless of sample size. The case of matched pairs is to be treated as an example of a single sample technique.

A

Null and alternative hypotheses, H 0 and H1 .

BL IC

7.6

TI O

Appl: Geography.

PRE-PUBLICATION Mathematics HL guide

37

Introduction to bivariate distributions.

Informal discussion of commonly occurring situations, eg marks in pure mathematics and statistics exams taken by a class of students, salary and age of teachers in a certain school. The need for a measure of association between the variables and the possibility of predicting the value of one of the variables given the value of the other variable.

Appl: Geographic skills.

Cov( X , Y ) = E[( X − µ x )(Y − µ y )]

Appl: Using technology to fit a range of curves to a set of data.

Aim 8: The correlation between smoking and lung cancer was “discovered” using mathematics. Science had to justify the cause.

TI O

N

7.7

Covariance and (population) product moment correlation coefficient ρ.

= E( XY ) − µ x µ y ,

Cov( X , Y ) . Var( X )Var(Y )

BL IC

ρ=

A

where = µ x E( = X ), µ y E(Y ) .

-P U

Proof that ρ = 0 in the case of independence and The use of ρ as a measure of association ±1 in the case of a linear relationship between X between X and Y, with values near 0 indicating a weak association and values near ±1 indicating a and Y. strong association. n

PRE-PUBLICATION Mathematics HL guide

R=

∑(X i =1

n

i

− X )(Yi − Y )

n 2 i =i 1 =i 1

PR E

Definition of the (sample) product moment correlation coefficient R in terms of n paired observations on X and Y. Its application to the estimation of ρ.

∑(X

− X)

∑ (Y − Y ) i

2

n

=

∑ X Y − nXY i =1

i i

n 2 2 2 2 ∑ X i − n X ∑ Yi − nY i =1

.

TOK: Mathematics and the world. Given that a set of data may be approximately fitted by a range of curves, where would we seek for knowledge of which equation is the “true” model? Aim 8: The physicist Frank Oppenheimer wrote: “Prediction is dependent only on the assumption that observed patterns will be repeated.” This is the danger of extrapolation. There are many examples of its failure in the past, eg share prices, the spread of disease, climate change. (continued)

38

Values of r near 0 indicate a weak association between X and Y, and values near ±1 indicate a strong association.

The following topics are based on the assumption of bivariate normality.

It is expected that the GDC will be used wherever possible in the following work.

Use of the t-statistic to test the null hypothesis ρ = 0.

n−2 has the Student’s t-distribution with 1 − R2 (n − 2) degrees of freedom.

TI O

Knowledge of the facts that the regression of X on Y ( E( X ) | Y = y ) and Y on X ( E(Y ) | X = x ) x−x = are linear.

A

R

(see notes above)

N

Informal interpretation of r, the observed value of R. Scatter diagrams.

-P U

BL IC

n ∑ ( xi − x )( yi − y ) i =1 n ( y − y) 2 ∑ ( yi − y ) Least-squares estimates of these regression lines i =1 n (proof not required). xi yi − nx y ∑ The use of these regression lines to predict the i =1n ( y − y ), value of one of the variables given the= value of 2 2 the other. ∑ yi − n y i =1

PR E

n ∑ ( xi − x )( yi − y ) (x − x ) y − y i =1 n = 2 ( xi − x ) ∑ i =1 n ∑ xi yi − nx y ( x − x ). = i =1n 2 2 ∑ xi − nx i =1

PRE-PUBLICATION Mathematics HL guide

39

Topic 8—Option: Sets, relations and groups 48 hours

Content 8.1

Further guidance

BL IC

Illustration of these laws using Venn diagrams.

-P U

Students may be asked to prove that two sets are the same by establishing that A ⊆ B and B ⊆ A.

Relations: equivalence relations; equivalence classes.

An equivalence relation on a set forms a partition of the set.

Functions: injections; surjections; bijections.

The term codomain.

Composition of functions and inverse functions.

Knowledge that the function composition is not a commutative operation and that if f is a

PR E

8.3

Ordered pairs: the Cartesian product of two sets.

A

Operations on sets: union; intersection; complement; set difference; symmetric difference.

8.2

Links TOK: Cantor theory of transfinite numbers, Russell’s paradox, Godel’s incompleteness theorems.

Finite and infinite sets. Subsets.

De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).

TI O

N

The aims of this option are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract algebra.

PRE-PUBLICATION Mathematics HL guide

Appl: Logic, Boolean algebra, computer circuits.

Appl, Int: Scottish clans.

bijection from set A to set B then f −1 exists and is a bijection from set B to set A.

40

A binary operation ∗ on a non-empty set S is a rule for combining any two elements a, b ∈ S to give a unique element c. That is, in this definition, a binary operation on a set is not necessarily closed.

Binary operations.

N

8.4

Binary operations: associative, distributive and commutative properties.

The arithmetic operations on and .

The identity element e.

Both the right-identity a ∗ e = a and left-identity e∗a = a must hold if e is an identity element.

Examples of distributivity could include the fact that, on , multiplication is distributive over addition but addition is not distributive over multiplication.

TOK: Which are more fundamental, the general models or the familiar examples?

8.6

The inverse a −1 of an element a.

Both a ∗ a −1 = e and a −1 ∗ a = e must hold.

-P U

Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.

BL IC

A

8.5

TI O

Operation tables (Cayley tables).

Proofs of the uniqueness of the identity and inverse elements. The definition of a group {G , ∗} .

PR E

8.7

The operation table of a group is a Latin square, but the converse is false.

Abelian groups.

PRE-PUBLICATION Mathematics HL guide

For the set G under a given operation ∗ : G is closed under ∗ ; ∗ is associative;

G contains an identity element;

Appl: Existence of formula for roots of polynomials. Appl: Galois theory for the impossibility of such formulae for polynomials of degree 5 or higher.

each element in G has an inverse in G. a ∗ b = b ∗ a , for all a, b ∈ G .

41

8.8

Appl: Rubik’s cube, time measures, crystal structure, symmetries of molecules, strut and cable constructions, Physics H2.2 (special relativity), the 8–fold way, supersymmetry.

Examples of groups: , , and under addition; integers under addition modulo n;

8.9

The order of a group. The order of a group element. Cyclic groups. Generators.

8.10

Permutations under composition of permutations. Cycle notation for permutations.

TI O

-P U

Proof that all cyclic groups are Abelian.

A

invertible functions under composition of functions.

The composition T2 T1 denotes T1 followed by T2 .

BL IC

symmetries of plane figures, including equilateral triangles and rectangles;

N

non-zero integers under multiplication, modulo p, where p is prime;

PR E

Result that every permutation can be written as a composition of disjoint cycles.

On examination papers: the form 1 2 3 p= or in cycle notation (132) will 3 1 2 be used to represent the permutation 1 → 3 , 2 → 1 , 3 → 2.

Appl: Music circle of fifths, prime numbers.

Appl: Cryptography, campanology.

The order of a combination of cycles.

PRE-PUBLICATION Mathematics HL guide

42

A proper subgroup is neither the group itself nor the subgroup containing only the identity element.

Use and proof of subgroup tests.

Suppose that {G , ∗} is a group and H is a non-empty subset of G. Then {H , ∗} is a

N

Subgroups, proper subgroups.

TI O

8.11

subgroup of {G , ∗} if a ∗ b −1 ∈ H whenever a, b ∈ H .

Definition and examples of left and right cosets of a subgroup of a group. Lagrange’s theorem.

Appl: Prime factorization, symmetry breaking.

PR E

-P U

Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)

BL IC

A

Suppose that {G , ∗} is a finite group and H is a non-empty subset of G. Then {H , ∗} is a subgroup of {G , ∗} if H is closed under ∗ .

PRE-PUBLICATION Mathematics HL guide

43

8.12

Definition of a group homomorphism.

Infinite groups as well as finite groups.

Definition of the kernel of a homomorphism. Proof that the kernel and range of a homomorphism are subgroups.

TI O

N

Let {G ,*} and {H , } be groups, then the function f : G → H is a homomorphism if f (a * b) = f (a ) f (b) for all a, b ∈ G .

If f : G → H is a group homomorphism, then Ker( f ) is the set of a ∈ G such that f (a ) = eH .

BL IC

A

Proof of homomorphism properties for identities Identity: let eG and eH be the identity elements and inverses. of (G, ∗) and ( H , ) , respectively, then f (eG ) = eH . Inverse: f (a −1 ) = ( f (a ) ) Isomorphism of groups.

−1

for all a ∈ G .

Infinite groups as well as finite groups.

-P U

The homomorphism f : G → H is an isomorphism if f is bijective.

PR E

The order of an element is unchanged by an isomorphism.

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Topic 9—Option: Calculus 48 hours

Content

Further guidance

Links

Infinite sequences of real numbers and their convergence or divergence.

Informal treatment of limit of sum, difference, product, quotient; squeeze theorem.

TI O

9.1

N

The aims of this option are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations.

Convergence of infinite series. Tests for convergence: comparison test; limit comparison test; ratio test; integral test.

The sum of a series is the limit of the sequence of its partial sums.

BL IC

9.2

A

Divergent is taken to mean not convergent.

TOK: Zeno’s paradox, impact of infinite sequences and limits on our understanding of the physical world. TOK: Euler’s idea that 1 − 1 + 1 − 1 + =12 . Was it a mistake or just an alternative view?

Students should be aware that if lim xn = 0 then x →∞

the series is not necessarily convergent, but if lim xn ≠ 0 , the series diverges. x →∞

1

∑n

p

1

∑n

.

p

otherwise. When p = 1 , this is the harmonic series.

PR E

Series that converge absolutely.

Series that converge conditionally. Alternating series.

is convergent for p > 1 and divergent

-P U

The p-series,

Conditions for convergence. The absolute value of the truncation error is less than the next term in the series.

Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

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9.3

Continuity and differentiability of a function at a Test for continuity: point.

lim f ( x) = f ( a ) = lim f ( x ) .

Test for differentiability: f is continuous at a and

TI O

Continuous functions and differentiable functions.

x → a+

N

x→a –

f ( a + h) – f ( a ) and h h →0 − f ( a + h) – f ( a ) exist and are equal. lim h h → 0+

A

lim

The integral as a limit of a sum; lower and upper Riemann sums. Fundamental theorem of calculus.

-P U

9.4

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x d ∫ f ( y ) dy = f ( x ) . dx a

Int: How close was Archimedes to integral calculus? Int: Contribution of Islamic, Chinese and Indian mathematicians to the development of calculus.

∞

Aim 8: Leibniz versus Newton versus the “giants” on whose shoulders they stood—who deserves credit for mathematical progress?

a

TOK: Consider f x =

PR E

Improper integrals of the type

BL IC

Students should be aware that a function may be continuous but not differentiable at a point, eg f ( x ) = x and simple piecewise functions.

∫ f ( x ) dx .

1 , 1≤ x ≤ ∞. x

An infinite area sweeps out a finite volume. Can this be reconciled with our intuition? What does this tell us about mathematical knowledge?

46

First-order differential equations.

Appl: Real-life differential equations, eg

Geometric interpretation using slope fields, including identification of isoclines.

Newton’s law of cooling,

Euler’s method.

dy = f ( x, y ) using dx

N

Numerical solution of

population growth,

xn + h , where h is carbon dating. yn += yn + hf ( xn , yn ) , xn += 1 1 a constant.

TI O

9.5

Variables separable. y f x

Solution of y′ + P(x)y = Q(x), using the integrating factor. 9.6

Rolle’s theorem.

-P U

Mean value theorem.

A

using the substitution y = vx.

dy = dx

BL IC

Homogeneous differential equation

PR E

Taylor polynomials; the Lagrange form of the error term.

Maclaurin series for e x , sin x , cos x , ln(1 + x) , p

(1 + x) , p ∈ .

Int, TOK: Influence of Bourbaki on understanding and teaching of mathematics. Int: Compare with work of the Kerala school.

Applications to the approximation of functions; formula for the error term, in terms of the value of the (n + 1)th derivative at an intermediate point. Students should be aware of the intervals of convergence.

Use of substitution, products, integration and differentiation to obtain other series. Taylor series developed from differential equations.

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The evaluation of limits of the form lim

x→a

The indeterminate forms

f ( x) f ( x) and lim . g ( x) x →∞ g ( x )

Repeated use of l’Hôpital’s rule.

PR E

-P U

BL IC

A

TI O

Using l’Hôpital’s rule or the Taylor series.

0 ∞ and . 0 ∞

N

9.7

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Topic 10—Option: Discrete mathematics 48 hours

Content

Further guidance

Strong induction.

For example, proofs of the fundamental theorem TOK: Mathematics and knowledge claims. The of arithmetic and the fact that a tree with n difference between proof and conjecture, eg vertices has n – 1 edges. Goldbach’s conjecture. Can a mathematical statement be true before it is proven?

10.2

a|b ⇒ b = na for some n ∈ .

BL IC

A

Pigeon-hole principle.

Links

TI O

10.1

N

The aim of this option is to provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.

The theorem a | b and a | c ⇒ a | (bx ± cy ) where x, y ∈ .

The division algorithm = a bq + r , 0 ≤ r < b .

Division and Euclidean algorithms.

The Euclidean algorithm for determining the greatest common divisor of two integers.

-P U

The greatest common divisor, gcd(a, b) , and the least common multiple, lcm(a, b) , of integers a and b.

10.3

PR E

Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic. Linear Diophantine equations ax + by = c.

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General solutions required and solutions subject to constraints. For example, all solutions must be positive.

TOK: Proof by contradiction.

Int: Euclidean algorithm contained in Euclid’s Elements, written in Alexandria about 300 BCE. Aim 8: Use of prime numbers in cryptography. The possible impact of the discovery of powerful factorization techniques on internet and bank security. Int: Described in Diophantus’ Arithmetica written in Alexandria in the 3rd century CE. When studying Arithmetica, a French mathematician, Pierre de Fermat (1601–1665) wrote in the margin that he had discovered a simple proof regarding higher-order Diophantine equations—Fermat’s last theorem. 49

10.4

Modular arithmetic. The solution of linear congruences.

N

Int: Discussed by Chinese mathematician Sun Tzu in the 3rd century CE.

Solution of simultaneous linear congruences (Chinese remainder theorem). Representation of integers in different bases.

On examination papers, questions that go beyond base 16 will not be set.

10.6

Fermat’s little theorem.

a p = a (mod p ) , where p is prime.

10.7

Graphs, vertices, edges, faces. Adjacent vertices, adjacent edges.

Two vertices are adjacent if they are joined by an edge. Two edges are adjacent if they have a common vertex.

Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.

PR E

Subgraphs; complements of graphs.

Euler’s relation: v − e + f = 2 ; theorems for planar graphs including e ≤ 3v − 6 , e ≤ 2v − 4 , leading to the results that κ 5 and κ 3,3 are not planar.

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-P U

Handshaking lemma.

BL IC

Degree of a vertex, degree sequence.

Int: Babylonians developed a base 60 number system and the Mayans a base 20 number system.

TI O

10.5

TOK: Nature of mathematics. An interest may be pursued for centuries before becoming “useful”. Aim 8: Symbolic maps, eg Metro and Underground maps, structural formulae in chemistry, electrical circuits.

TOK: Mathematics and knowledge claims. Proof of the four-colour theorem. If a theorem is proved by computer, how can we claim to know that it is true?

It should be stressed that a graph should not be assumed to be simple unless specifically stated. The term adjacency table may be used.

Aim 8: Importance of planar graphs in constructing circuit boards.

e ≤ 3v − 6 only if the graph is connected and simple and v ≥ 3 .

TOK: “Only if”, link to Venn diagrams and logic.

e ≤ 2v − 4 only if the graph is connected and simple and all faces have degree ≥ 4 .

TOK: Mathematics and knowledge claims. Applications of the Euler characteristic (v − e + f ) to higher dimensions. Its use in understanding properties of shapes that cannot be visualized.

50

Walks, trails, paths, circuits, cycles. Eulerian trails and circuits.

A connected graph contains an Eulerian circuit if and only if every vertex of the graph is of even degree.

Hamiltonian paths and cycles.

Simple treatment only.

Int: The “Bridges of Königsberg” problem.

Graph algorithms: Kruskal’s; Dijkstra’s.

To determine the shortest route around a weighted graph going along each edge at least once.

Int: Problem posed by the Chinese mathematician Kwan Mei-Ko in 1962.

To determine the Hamiltonian cycle of least weight in a weighted complete graph.

TOK: Mathematics and knowledge claims. How long would it take a computer to test all Hamiltonian cycles in a complete, weighted graph with just 30 vertices?

A

10.10 Chinese postman problem. Not required: Graphs with more than four vertices of odd degree. Travelling salesman problem. Nearest-neighbour algorithm for determining an upper bound.

-P U

Deleted vertex algorithm for determining a lower bound.

BL IC

10.9

TI O

N

10.8

10.11 Recurrence relations. Initial conditions, recursive definition of a sequence.

PR E

Solution of first- and second-degree linear Includes the cases where auxiliary equation has homogeneous recurrence relations with constant equal roots or complex roots coefficients.

TOK: Mathematics and the world. The connections of sequences such as the Fibonacci sequence with art and biology.

The first-degree linear recurrence relation un aun −1 + b . = Modelling with recurrence relations.

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Solving problems such as compound interest, debt repayment and counting problems.

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Glossary of terminology: Discrete mathematics

Introduction Teachers and students should be aware that many different terminologies exist in graph theory, and that different textbooks may employ different combinations of these. Examples of these are: vertex / node / junction / point; edge / route / arc; degree / order of a vertex; multiple edges / parallel edges; loop / self-loop. In IB examination questions, the terminology used will be as it appears in the syllabus. For clarity, these terms are defined below.

N

Terminology

A graph whose vertices can be divided into two sets such that no two vertices in the same set are adjacent.

Circuit

A walk that begins and ends at the same vertex, and has no repeated edges.

Complement of a graph G

A graph with the same vertices as G but which has an edge between any two vertices if and only if G does not.

Complete bipartite graph

A bipartite graph in which every vertex in one set is joined to every vertex in the other set.

Complete graph

A simple graph in which each pair of vertices is joined by an edge.

Connected graph

A graph in which each pair of vertices is joined by a path.

Cycle

A walk that begins and ends at the same vertex, and has no other repeated vertices.

A

IC

BL

The number of edges joined to the vertex; a loop contributes two edges, one for each of its end points. A graph that has at least one pair of vertices not joined by a path.

E-

Disconnected graph

PU

Degree of a vertex

TI O

Bipartite graph

A circuit that contains every edge of a graph.

Eulerian trail

A trail that contains every edge of a graph.

PR

Eulerian circuit Graph

Consists of a set of vertices and a set of edges.

Graph isomorphism between two simple graphs G and H

A one-to-one correspondence between vertices of G and H such that a pair of vertices in G is adjacent if and only if the corresponding pair in H is adjacent.

Hamiltonian cycle

A cycle that contains all the vertices of the graph.

Hamiltonian path

A path that contains all the vertices of the graph.

Loop

An edge joining a vertex to itself.

Minimum spanning tree

A spanning tree of a weighted graph that has the minimum total weight.

Multiple edges

Occur if more than one edge joins the same pair of vertices.

Path

A walk with no repeated vertices.

Planar graph

A graph that can be drawn in the plane without any edge crossing another.

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A graph without loops or multiple edges.

Spanning tree of a graph

A subgraph that is a tree, containing every vertex of the graph.

Subgraph

A graph within a graph.

Trail

A walk in which no edge appears more than once.

Tree

A connected graph that contains no cycles.

Walk

A sequence of linked edges.

Weighted graph

A graph in which each edge is allocated a number or weight.

Weighted tree

A tree in which each edge is allocated a number or weight.

PR

E-

PU

BL

IC

A

TI O

N

Simple graph

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Assessment

Assessment in the Diploma Programme

General Assessment is an integral part of teaching and learning. The most important aims of assessment in the Diploma Programme are that it should support curricular goals and encourage appropriate student learning. Both external and internal assessment are used in the Diploma Programme. IB examiners mark work produced for external assessment, while work produced for internal assessment is marked by teachers and externally moderated by the IB.

N

There are two types of assessment identified by the IB. Formative assessment informs both teaching and learning. It is concerned with providing accurate and helpful feedback to students and teachers on the kind of learning taking place and the nature of students’ strengths and weaknesses in order to help develop students’ understanding and capabilities. Formative assessment can also help to improve teaching quality, as it can provide information to monitor progress towards meeting the course aims and objectives.

•

Summative assessment gives an overview of previous learning and is concerned with measuring student achievement.

A

TI O

•

BL

IC

The Diploma Programme primarily focuses on summative assessment designed to record student achievement at or towards the end of the course of study. However, many of the assessment instruments can also be used formatively during the course of teaching and learning, and teachers are encouraged to do this. A comprehensive assessment plan is viewed as being integral with teaching, learning and course organization. For further information, see the IB Programme standards and practices document.

E-

PU

The approach to assessment used by the IB is criterion-related, not norm-referenced. This approach to assessment judges students’ work by their performance in relation to identified levels of attainment, and not in relation to the work of other students. For further information on assessment within the Diploma Programme, please refer to the publication Diploma Programme assessment: Principles and practice.

PR

To support teachers in the planning, delivery and assessment of the Diploma Programme courses, a variety of resources can be found on the OCC or purchased from the IB store (http://store.ibo.org). Teacher support materials, subject reports, internal assessment guidance, grade descriptors, as well as resources from other teachers, can be found on the OCC. Specimen and past examination papers as well as markschemes can be purchased from the IB store.

Methods of assessment The IB uses several methods to assess work produced by students.

Assessment criteria Assessment criteria are used when the assessment task is open-ended. Each criterion concentrates on a particular skill that students are expected to demonstrate. An assessment objective describes what students should be able to do, and assessment criteria describe how well they should be able to do it. Using assessment criteria allows discrimination between different answers and encourages a variety of responses. Each criterion comprises a set of hierarchically ordered level descriptors. Each level descriptor is worth one or more marks. Each criterion is applied independently using a best-fit model. PRE-PUBLICATION Mathematics HL guide

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The maximum marks for each criterion may differ according to the criterion’s importance. The marks awarded for each criterion are added together to give the total mark for the piece of work.

Markbands Markbands are a comprehensive statement of expected performance against which responses are judged. They represent a single holistic criterion divided into level descriptors. Each level descriptor corresponds to a range of marks to differentiate student performance. A best-fit approach is used to ascertain which particular mark to use from the possible range for each level descriptor.

Markschemes

PR

E-

PU

BL

IC

A

TI O

N

This generic term is used to describe analytic markschemes that are prepared for specific examination papers. Analytic markschemes are prepared for those examination questions that expect a particular kind of response and/or a given final answer from the students. They give detailed instructions to examiners on how to break down the total mark for each question for different parts of the response. A markscheme may include the content expected in the responses to questions or may be a series of marking notes giving guidance on how to apply criteria.

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Assessment outline First examinations 2014 Assessment component

Weighting

External assessment (5 hours)

80%

Paper 1 (2 hours)

30%

No calculator allowed. (120 marks) Compulsory short-response questions based on the core syllabus. Section B

Paper 2 (2 hours)

Graphic display calculator (GDC) required. (120 marks)

30%

A

Section A

TI O

Compulsory extended-response questions based on the core syllabus.

N

Section A

Compulsory short-response questions based on the core syllabus.

IC

Section B

Compulsory extended-response questions based on the core syllabus.

BL

Paper 3 (1 hour)

20%

Graphic display calculator (GDC) required. (60 marks)

PU

Compulsory extended-response questions based mainly on the syllabus options. Internal assessment

20%

E-

This component is internally assessed by the teacher and externally moderated by the IB at the end of the course. Mathematical exploration

PR

Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)

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External assessment

General Markschemes are used to assess students in all papers. The markschemes are specific to each examination.

External assessment details Papers 1, 2 and 3

TI O

N

These papers are externally set and externally marked. Together, they contribute 80% of the final mark for the course. These papers are designed to allow students to demonstrate what they know and what they can do.

Calculators Paper 1

IC

A

Students are not permitted access to any calculator. Questions will mainly involve analytic approaches to solutions, rather than requiring the use of a GDC. The paper is not intended to require complicated calculations, with the potential for careless errors. However, questions will include some arithmetical manipulations when they are essential to the development of the question.

BL

Papers 2 and 3

PU

Students must have access to a GDC at all times. However, not all questions will necessarily require the use of the GDC. Regulations covering the types of GDC allowed are provided in the Handbook of procedures for the Diploma Programme.

Mathematics HL and Further mathematics HL formula booklet

E-

Each student must have access to a clean copy of the formula booklet during the examination. It is the responsibility of the school to download a copy from IBIS or the OCC and to ensure that there are sufficient copies available for all students.

PR

Awarding of marks

Marks may be awarded for method, accuracy, answers and reasoning, including interpretation. In paper 1 and paper 2, full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations (in the form of, for example, diagrams, graphs or calculations). Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. All students should therefore be advised to show their working.

Paper 1 Duration: 2 hours Weighting: 30% •

This paper consists of section A, short-response questions, and section B, extended-response questions.

•

Students are not permitted access to any calculator on this paper.

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Syllabus coverage •

Knowledge of all core topics is required for this paper. However, not all topics are necessarily assessed in every examination session.

Mark allocation •

This paper is worth 120 marks, representing 30% of the final mark.

•

Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.

Section A This section consists of compulsory short-response questions based on the core syllabus. It is worth 60 marks.

•

The intention of this section is to test students’ knowledge and understanding across the breadth of the syllabus. However, it should not be assumed that the separate topics are given equal emphasis.

TI O

N

•

Question type

A small number of steps is needed to solve each question.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

IC

A

•

BL

Section B

This section consists of a small number of compulsory extended-response questions based on the core syllabus. It is worth 60 marks.

•

Individual questions may require knowledge of more than one topic.

•

The intention of this section is to test students’ knowledge and understanding of the core in depth. The range of syllabus topics tested in this section may be narrower than that tested in section A.

E-

PU

•

Question type

•

Questions require extended responses involving sustained reasoning.

PR

•

Individual questions will develop a single theme.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

•

Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problemsolving.

Paper 2 Duration: 2 hours Weighting: 30% •

This paper consists of section A, short-response questions, and section B, extended-response questions.

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•

A GDC is required for this paper, but not every question will necessarily require its use.

Syllabus coverage •

Knowledge of all core topics is required for this paper. However, not all topics are necessarily assessed in every examination session.

Mark allocation •

This paper is worth 120 marks, representing 30% of the final mark.

•

Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.

N

Section A

This section consists of compulsory short-response questions based on the core syllabus. It is worth 60 marks.

•

The intention of this section is to test students’ knowledge and understanding across the breadth of the syllabus. However, it should not be assumed that the separate topics are given equal emphasis.

TI O

•

A

Question type

A small number of steps is needed to solve each question.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

BL

IC

•

Section B

This section consists of a small number of compulsory extended-response questions based on the core syllabus. It is worth 60 marks.

•

Individual questions may require knowledge of more than one topic.

•

The intention of this section is to test students’ knowledge and understanding of the core in depth. The range of syllabus topics tested in this section may be narrower than that tested in section A.

PR

E-

PU

•

Question type •

Questions require extended responses involving sustained reasoning.

•

Individual questions will develop a single theme.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

•

Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problemsolving.

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•

This paper consists of a small number of compulsory extended-response questions based on the option chosen.

•

Where possible, the first part of each question will be on core material leading to the option topic. When this is not readily achievable, as, for example, with the discrete mathematics option, the level of difficulty of the earlier part of a question will be comparable to that of the core questions.

Syllabus coverage Students must answer all questions.

•

Knowledge of the entire content of the option studied, as well as the core material, is required for this paper.

Mark allocation

N

•

This paper is worth 60 marks, representing 20% of the final mark.

•

Questions may be unequal in terms of length and level of difficulty. Therefore, individual questions may not be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of each question.

TI O

•

A

Question type

Questions require extended responses involving sustained reasoning.

•

Individual questions will develop a single theme or be divided into unconnected parts. Where the latter occur, the unconnected parts will be clearly labelled as such.

•

Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

•

Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problemsolving.

PR

E-

PU

BL

IC

•

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Internal assessment

Purpose of internal assessment Internal assessment is an integral part of the course and is compulsory for all students. It enables students to demonstrate the application of their skills and knowledge, and to pursue their personal interests, without the time limitations and other constraints that are associated with written examinations. The internal assessment should, as far as possible, be woven into normal classroom teaching and not be a separate activity conducted after a course has been taught.

TI O

Guidance and authenticity

N

Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. It is marked according to five assessment criteria.

A

The exploration submitted for internal assessment must be the student’s own work. However, it is not the intention that students should decide upon a title or topic and be left to work on the exploration without any further support from the teacher. The teacher should play an important role during both the planning stage and the period when the student is working on the exploration. It is the responsibility of the teacher to ensure that students are familiar with: the requirements of the type of work to be internally assessed

•

the IB academic honesty policy available on the OCC

•

the assessment criteria—students must understand that the work submitted for assessment must address these criteria effectively.

BL

IC

•

E-

PU

Teachers and students must discuss the exploration. Students should be encouraged to initiate discussions with the teacher to obtain advice and information, and students must not be penalized for seeking guidance. However, if a student could not have completed the exploration without substantial support from the teacher, this should be recorded on the appropriate form from the Handbook of procedures for the Diploma Programme.

PR

It is the responsibility of teachers to ensure that all students understand the basic meaning and significance of concepts that relate to academic honesty, especially authenticity and intellectual property. Teachers must ensure that all student work for assessment is prepared according to the requirements and must explain clearly to students that the exploration must be entirely their own. As part of the learning process, teachers can give advice to students on a first draft of the exploration. This advice should be in terms of the way the work could be improved, but this first draft must not be heavily annotated or edited by the teacher. The next version handed to the teacher after the first draft must be the final one. All work submitted to the IB for moderation or assessment must be authenticated by a teacher, and must not include any known instances of suspected or confirmed malpractice. Each student must sign the coversheet for internal assessment to confirm that the work is his or her authentic work and constitutes the final version of that work. Once a student has officially submitted the final version of the work to a teacher (or the coordinator) for internal assessment, together with the signed coversheet, it cannot be retracted.

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Authenticity may be checked by discussion with the student on the content of the work, and scrutiny of one or more of the following: •

the student’s initial proposal

•

the first draft of the written work

•

the references cited

•

the style of writing compared with work known to be that of the student.

TI O

N

The requirement for teachers and students to sign the coversheet for internal assessment applies to the work of all students, not just the sample work that will be submitted to an examiner for the purpose of moderation. If the teacher and student sign a coversheet, but there is a comment to the effect that the work may not be authentic, the student will not be eligible for a mark in that component and no grade will be awarded. For further details refer to the IB publication Academic honesty and the relevant articles in the General regulations: Diploma Programme. The same piece of work cannot be submitted to meet the requirements of both the internal assessment and the extended essay.

A

Group work

IC

Group work should not be used for explorations. Each exploration is an individual piece of work based on different data collected or measurements generated.

BL

It should be made clear to students that all work connected with the exploration, including the writing of the exploration, should be their own. It is therefore helpful if teachers try to encourage in students a sense of responsibility for their own learning so that they accept a degree of ownership and take pride in their own work.

PU

Time allocation

E-

Internal assessment is an integral part of the mathematics HL course, contributing 20% to the final assessment in the course. This weighting should be reflected in the time that is allocated to teaching the knowledge, skills and understanding required to undertake the work as well as the total time allocated to carry out the work.

•

PR

It is expected that a total of approximately 10 teaching hours should be allocated to the work. This should include: time for the teacher to explain to students the requirements of the exploration

•

class time for students to work on the exploration

•

time for consultation between the teacher and each student

•

time to review and monitor progress, and to check authenticity.

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Using assessment criteria for internal assessment For internal assessment, a number of assessment criteria have been identified. Each assessment criterion has level descriptors describing specific levels of achievement together with an appropriate range of marks. The level descriptors concentrate on positive achievement, although for the lower levels failure to achieve may be included in the description. Teachers must judge the internally assessed work against the criteria using the level descriptors. The aim is to find, for each criterion, the descriptor that conveys most accurately the level attained by the student.

•

When assessing a student’s work, teachers should read the level descriptors for each criterion, starting with level 0, until they reach a descriptor that describes a level of achievement that has not been reached. The level of achievement gained by the student is therefore the preceding one, and it is this that should be recorded.

•

Only whole numbers should be recorded; partial marks, that is fractions and decimals, are not acceptable.

•

Teachers should not think in terms of a pass or fail boundary, but should concentrate on identifying the appropriate descriptor for each assessment criterion.

•

The highest level descriptors do not imply faultless performance but should be achievable by a student. Teachers should not hesitate to use the extremes if they are appropriate descriptions of the work being assessed.

•

A student who attains a high level of achievement in relation to one criterion will not necessarily attain high levels of achievement in relation to the other criteria. Similarly, a student who attains a low level of achievement for one criterion will not necessarily attain low achievement levels for the other criteria. Teachers should not assume that the overall assessment of the students will produce any particular distribution of marks.

•

It is expected that the assessment criteria be made available to students.

PR

E-

PU

BL

IC

A

TI O

N

•

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Internal assessment details Mathematical exploration Duration: 10 teaching hours Weighting: 20% Introduction

TI O

N

The internally assessed component in this course is a mathematical exploration. This is a short report written by the student based on a topic chosen by him or her, and it should focus on the mathematics of that particular area. The emphasis is on mathematical communication (including formulae, diagrams, graphs and so on), with accompanying commentary, good mathematical writing and thoughtful reflection. A student should develop his or her own focus, with the teacher providing feedback via, for example, discussion and interview. This will allow the students to develop areas of interest to them without a time constraint as in an examination, and allow all students to experience a feeling of success.

The purpose of the exploration

IC

A

The final report should be approximately 6 to 12 pages long. It can be either word processed or hand written. Students should be able to explain all stages of their work in such a way that demonstrates clear understanding. While there is no requirement that students present their work in class, it should be written in such a way that their peers would be able to follow it fairly easily. The report should include a detailed bibliography, and sources need to be referenced in line with the IB academic honesty policy. Direct quotes must be acknowledged.

E-

PU

BL

The aims of the mathematics HL course are carried through into the objectives that are formally assessed as part of the course, through either written examination papers, or the exploration, or both. In addition to testing the objectives of the course, the exploration is intended to provide students with opportunities to increase their understanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics. These are noted in the aims of the course, in particular aims 6– 9 (applications, technology, moral, social and ethical implications, and the international dimension). It is intended that, by doing the exploration, students benefit from the mathematical activities undertaken and find them both stimulating and rewarding. It will enable students to acquire the attributes of the IB learner profile. The specific purposes of the exploration are to:

•

develop students’ personal insight into the nature of mathematics and develop their ability to ask their own questions about mathematics

PR

•

provide opportunities for students to complete a piece of mathematical work over an extended period of time

•

enable students to experience the satisfaction of applying mathematical processes independently

•

provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics

•

encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool

•

enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work

•

provide opportunities for students to show, with confidence, how they have developed mathematically.

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Management of the exploration Work for the exploration should be incorporated into the course so that students are given the opportunity to learn the skills needed. Time in class can therefore be used for general discussion of areas of study, as well as familiarizing students with the criteria. Further details on the development of the exploration are included in the teacher support material.

Requirements and recommendations Students can choose from a wide variety of activities, for example, modelling, investigations and applications of mathematics. To assist teachers and students in the choice of a topic, a list of stimuli is available in the teacher support material. However, students are not restricted to this list.

N

The exploration should not normally exceed 12 pages, including diagrams and graphs, but excluding the bibliography. However, it is the quality of the mathematical writing that is important, not the length.

TI O

The teacher is expected to give appropriate guidance at all stages of the exploration by, for example, directing students into more productive routes of inquiry, making suggestions for suitable sources of information, and providing advice on the content and clarity of the exploration in the writing-up stage.

A

Teachers are responsible for indicating to students the existence of errors, but should not explicitly correct these errors. It must be emphasized that students are expected to consult the teacher throughout the process.

BL

IC

All students should be familiar with the requirements of the exploration and the criteria by which it is assessed. Students need to start planning their explorations as early as possible in the course. Deadlines should be firmly established. There should be a date for submission of the exploration topic and a brief outline description, a date for the submission of the first draft and, of course, a date for completion.

PU

In developing their explorations, students should aim to make use of mathematics learned as part of the course. The mathematics used should be commensurate with the level of the course, that is, it should be similar to that suggested by the syllabus. It is not expected that students produce work that is outside the mathematics HL syllabus—however, this is not penalized.

E-

Internal assessment criteria

PR

The exploration is internally assessed by the teacher and externally moderated by the IB using assessment criteria that relate to the objectives for group 5 mathematics. Each exploration is assessed against the following five criteria. The final mark for each exploration is the sum of the scores for each criterion. The maximum possible final mark is 20. Students will not receive a grade for mathematics HL if they have not submitted an exploration. Criterion A

Communication

Criterion B

Mathematical presentation

Criterion C

Personal engagement

Criterion D

Reflection

Criterion E

Use of mathematics

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Criterion A: Communication This criterion assesses the organization and coherence of the exploration. A well-organized exploration includes an introduction, has a rationale (which includes explaining why this topic was chosen), describes the aim of the exploration and has a conclusion. A coherent exploration is logically developed and easy to follow. Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document. Achievement level

Descriptor

The exploration does not reach the standard described by the descriptors below.

1

The exploration has some coherence.

2

The exploration has some coherence and shows some organization.

3

The exploration is coherent and well organized.

4

The exploration is coherent, well organized, concise and complete.

PR

E-

PU

BL

IC

A

TI O

N

0

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Criterion B: Mathematical presentation This criterion assesses to what extent the student is able to: •

use appropriate mathematical language (notation, symbols, terminology)

•

define key terms, where required

•

use multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models, where appropriate.

Students are expected to use mathematical language when communicating mathematical ideas, reasoning and findings.

Achievement level

TI O

N

Students are encouraged to choose and use appropriate ICT tools such as graphic display calculators, screenshots, graphing, spreadsheets, databases, drawing and word-processing software, as appropriate, to enhance mathematical communication. Descriptor

The exploration does not reach the standard described by the descriptors below.

1

There is some appropriate mathematical presentation.

2

The mathematical presentation is mostly appropriate.

3

The mathematical presentation is appropriate throughout.

PR

E-

PU

BL

IC

A

0

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Criterion C: Personal engagement This criterion assesses the extent to which the student engages with the exploration and makes it his or her own. Personal engagement may be recognized in different attributes and skills. These include thinking independently and/or creatively, addressing personal interest and presenting mathematical ideas in his or her own way. Achievement level

Descriptor

The exploration does not reach the standard described by the descriptors below.

1

There is evidence of limited or superficial personal engagement.

2

There is evidence of some personal engagement.

3

There is evidence of significant personal engagement.

4

There is abundant evidence of outstanding personal engagement.

PR

E-

PU

BL

IC

A

TI O

N

0

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Criterion D: Reflection This criterion assesses how the student reviews, analyses and evaluates the exploration. Although reflection may be seen in the conclusion to the exploration, it may also be found throughout the exploration. Achievement level

Descriptor

The exploration does not reach the standard described by the descriptors below.

1

There is evidence of limited or superficial reflection.

2

There is evidence of meaningful reflection.

3

There is substantial evidence of critical reflection.

PR

E-

PU

BL

IC

A

TI O

N

0

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Criterion E: Use of mathematics This criterion assesses to what extent and how well students use mathematics in the exploration. Students are expected to produce work that is commensurate with the level of the course. The mathematics explored should either be part of the syllabus, or at a similar level or beyond. It should not be completely based on mathematics listed in the prior learning. If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion. The mathematics can be regarded as correct even if there are occasional minor errors as long as they do not detract from the flow of the mathematics or lead to an unreasonable outcome.

TI O

N

Sophistication in mathematics may include understanding and use of challenging mathematical concepts, looking at a problem from different perspectives and seeing underlying structures to link different areas of mathematics. Rigour involves clarity of logic and language when making mathematical arguments and calculations. Precise mathematics is error-free and uses an appropriate level of accuracy at all times. Descriptor

A

Achievement level

The exploration does not reach the standard described by the descriptors below.

1

Some relevant mathematics is used. Limited understanding is demonstrated.

2

Some relevant mathematics is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated.

3

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Good knowledge and understanding are demonstrated.

BL

PU E-

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication expected. Good knowledge and understanding are demonstrated.

PR

4

IC

0

5

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication and rigor expected. Thorough knowledge and understanding are demonstrated.

6

Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and reflects the sophistication and rigor expected. Thorough knowledge and understanding are demonstrated.

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Appendices

Glossary of command terms

Command terms with definitions Students should be familiar with the following key terms and phrases used in examination questions, which are to be understood as described below. Although these terms will be used in examination questions, other terms may be used to direct students to present an argument in a specific way. Obtain a numerical answer showing the relevant stages in the working.

Comment

Give a judgment based on a given statement or result of a calculation.

Compare

Give an account of the similarities between two (or more) items or situations, referring to both (all) of them throughout.

Compare and contrast

Give an account of the similarities and differences between two (or more) items or situations, referring to both (all) of them throughout.

Construct

Display information in a diagrammatic or logical form.

Contrast

Give an account of the differences between two (or more) items or situations, referring to both (all) of them throughout.

Deduce

Reach a conclusion from the information given.

Demonstrate

Make clear by reasoning or evidence, illustrating with examples or practical application.

Describe

Give a detailed account.

Determine

Obtain the only possible answer.

Differentiate

Obtain the derivative of a function.

TI O

A

IC

BL

PU

Distinguish

Make clear the differences between two or more concepts or items. Represent by means of a labelled, accurate diagram or graph, using a pencil. A ruler (straight edge) should be used for straight lines. Diagrams should be drawn to scale. Graphs should have points correctly plotted (if appropriate) and joined in a straight line or smooth curve.

PR

E-

Draw

Estimate

N

Calculate

Obtain an approximate value.

Explain

Give a detailed account, including reasons or causes.

Find

Obtain an answer, showing relevant stages in the working.

Hence

Use the preceding work to obtain the required result.

Hence or otherwise

It is suggested that the preceding work is used, but other methods could also receive credit.

Identify

Provide an answer from a number of possibilities.

Integrate

Obtain the integral of a function.

Interpret

Use knowledge and understanding to recognize trends and draw conclusions from given information.

Investigate

Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions.

Justify

Give valid reasons or evidence to support an answer or conclusion.

Label

Add labels to a diagram.

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Give a sequence of brief answers with no explanation.

Plot

Mark the position of points on a diagram.

Predict

Give an expected result.

Prove

Use a sequence of logical steps to obtain the required result in a formal way.

Show

Give the steps in a calculation or derivation.

Show that

Obtain the required result (possibly using information given) without the formality of proof. “Show that” questions do not generally require the use of a calculator.

Sketch

Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.

Solve

Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.

State

Give a specific name, value or other brief answer without explanation or calculation.

Suggest

Propose a solution, hypothesis or other possible answer.

Verify

Provide evidence that validates the result.

Write down

Obtain the answer(s), usually by extracting information. Little or no calculation is required. Working does not need to be shown.

PR

E-

PU

BL

IC

A

TI O

N

List

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Notation list Of the various notations in use, the IB has chosen to adopt a system of notation based on the recommendations of the International Organization for Standardization (ISO). This notation is used in the examination papers for this course without explanation. If forms of notation other than those listed in this guide are used on a particular examination paper, they are defined within the question in which they appear. Because students are required to recognize, though not necessarily use, IB notation in examinations, it is recommended that teachers introduce students to this notation at the earliest opportunity. Students are not allowed access to information about this notation in the examinations.

N

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3, ...}

the set of integers, {0, ± 1, ± 2, ± 3, ...}

+

the set of positive integers, {1, 2, 3, ...}

the set of rational numbers

+

the set of positive rational numbers, {x | x ∈ , x > 0}

the set of real numbers

+

the set of positive real numbers, {x | x ∈ , x > 0}

the set of complex numbers, {a + ib | a , b ∈ }

i

PU

−1

BL

IC

A

TI O

a complex number

z

the complex conjugate of z

z∗

z

E-

the modulus of z

arg z

PR

the argument of z

Re z

the real part of z

Im z

the imaginary part of z

cisθ

cos θ + i sin θ

{x1 , x2 , ...}

the set with elements x1 , x2 , ...

n( A)

the number of elements in the finite set A

{x |

}

the set of all x such that

∈

is an element of

∉

is not an element of

∅

the empty (null) set

U

the universal set

∪

union

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∩

intersection

⊂

is a proper subset of

⊆

is a subset of

A′

the complement of the set A

A× B

the Cartesian product of sets A and B (that is, = A × B {(a , b) a ∈ A , b ∈ B} )

a|b

a divides b

a1/ n ,

n

a

a to the power of

1 th , n root of a (if a ≥ 0 then n

n

a ≥0)

x for x ≥ 0, x ∈ the modulus or absolute value of x, that is − x for x < 0, x ∈

≡

identity

≈

is approximately equal to

>

is greater than

≥

is greater than or equal to

/

is not greater than