Mathematical studies SL guideFirst examinations 2014

Diploma Programme

Mathematical studies SL guide

First examinations 2014

Mathematical studies SL guide

First examinations 2014 Diploma Programme

Diploma Programme Mathematical studies SL guide

International Baccalaureate, Baccalauréat International and Bachillerato Internacional are registered trademarks of the International Baccalaureate Organization.

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© International Baccalaureate Organization 2012

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IB mission statement

The International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help to create a better and more peaceful world through intercultural understanding and respect.

To this end the organization works with schools, governments and international organizations to develop challenging programmes of international education and rigorous assessment.

These programmes encourage students across the world to become active, compassionate and lifelong learners who understand that other people, with their differences, can also be right.

IB learner profile

The aim of all IB programmes is to develop internationally minded people who, recognizing their common humanity and shared guardianship of the planet, help to create a better and more peaceful world.

IB learners strive to be:

Inquirers They develop their natural curiosity. They acquire the skills necessary to conduct inquiry and research and show independence in learning. They actively enjoy learning and this love of learning will be sustained throughout their lives.

Knowledgeable They explore concepts, ideas and issues that have local and global significance. In so doing, they acquire in-depth knowledge and develop understanding across a broad and balanced range of disciplines.

Thinkers They exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems, and make reasoned, ethical decisions.

Communicators They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication. They work effectively and willingly in collaboration with others.

Principled They act with integrity and honesty, with a strong sense of fairness, justice and respect for the dignity of the individual, groups and communities. They take responsibility for their own actions and the consequences that accompany them.

Open-minded They understand and appreciate their own cultures and personal histories, and are open to the perspectives, values and traditions of other individuals and communities. They are accustomed to seeking and evaluating a range of points of view, and are willing to grow from the experience.

Caring They show empathy, compassion and respect towards the needs and feelings of others.

They have a personal commitment to service, and act to make a positive difference to the lives of others and to the environment.

Risk-takers They approach unfamiliar situations and uncertainty with courage and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are brave and articulate in defending their beliefs.

Balanced They understand the importance of intellectual, physical and emotional balance to achieve personal well-being for themselves and others.

Contents

Introduction 1

Purpose of this document 1

The Diploma Programme 2

Nature of the subject 4

Aims 8

Assessment objectives 9

Syllabus 10

Syllabus outline 10

Approaches to the teaching and learning of mathematical studies SL 11

Prior learning topics 14

Syllabus content 16

Assessment 35

Assessment in the Diploma Programme 35

Assessment outline 37

External assessment 38

Internal assessment 40

Appendices 50

Glossary of command terms 50

Notation list 52

Purpose of this document

Introduction

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.

Additional resources

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.

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.

Acknowledgment

The IB wishes to thank the educators and associated schools for generously contributing time and resources to the production of this guide.

First examinations 2014

2 Mathematical studies SL guide

Introduction

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

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.

Studies in language and literature

Individuals and societies

Mathematics

The arts Experimental

sciences Language acquisition

Group 2

Group 4

Group 6

Group 5 Group 1

Group 3

theory of knowledge extended

essay

creativity, action, service THE IB LEARNE

R PRO E FIL

Figure 1

Diploma Programme model

The Diploma Programme

Choosing the right combination

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.

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.

The IB mission statement and the IB learner profile

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.

4 Mathematical studies SL guide

Introduction

Nature of the subject

Introduction

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 the study of this subject compulsory for students studying the full diploma.

Summary of courses available

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.

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

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

• their own interest in mathematics and those particular areas of the subject that may hold the most interest for them

• their other choices of subjects within the framework of the 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

Nature of the subject

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 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

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.

Further mathematics HL

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.

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.

Mathematical studies SL—course details

The course syllabus focuses on important mathematical topics that are interconnected. The syllabus is organized and structured with the following tenets in mind: placing more emphasis on student understanding of fundamental concepts than on symbolic manipulation and complex manipulative skills; giving greater

Mathematical studies SL guide 6

Nature of the subject

The students most likely to select this course are those whose main interests lie outside the field of mathematics, and for many students this course will be their final experience of being taught formal mathematics. All parts of the syllabus have therefore been carefully selected to ensure that an approach starting from first principles can be used. As a consequence, students can use their own inherent, logical thinking skills and do not need to rely on standard algorithms and remembered formulae. Students likely to need mathematics for the achievement of further qualifications should be advised to consider an alternative mathematics course.

Owing to the nature of mathematical studies SL, teachers may find that traditional methods of teaching are inappropriate and that less formal, shared learning techniques can be more stimulating and rewarding for students. Lessons that use an inquiry-based approach, starting with practical investigations where possible, followed by analysis of results, leading to the understanding of a mathematical principle and its formulation into mathematical language, are often most successful in engaging the interest of students. Furthermore, this type of approach is likely to assist students in their understanding of mathematics by providing a meaningful context and by leading them to understand more fully how to structure their work for the project.

Prior learning

Mathematics is a linear subject, and it is expected that most students embarking on a Diploma Programme (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 mathematical studies SL 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.

At the beginning of the syllabus section there is a list of topics that are considered to be prior learning for the mathematical studies SL 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.

Links to the Middle Years Programme

The prior learning topics for the 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.

A continuum document called Mathematics: The MYP–DP continuum (November 2010) 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.

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

Nature of the 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?

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 Theory of knowledge 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.

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.

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.

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.

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

8 Mathematical studies SL guide

Aims

Introduction

Group 5 aims

The aims of all mathematics courses in group 5 are to enable students to:

1. 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.

Assessment objectives

Introduction

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 mathematical studies SL course, students will be expected to demonstrate the following.

1. 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. Investigative approaches: investigate unfamiliar situations involving organizing and analysing information or measurements, drawing conclusions, testing their validity, and considering their scope and limitations.

10 Mathematical studies SL guide

Syllabus outline

Syllabus

Syllabus component

Teaching hours

SL 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.

Topic 1

Number and algebra

20

Topic 2

Descriptive statistics

12

Topic 3

Logic, sets and probability

20

Topic 4

Statistical applications

17

Topic 5

Geometry and trigonometry

18

Topic 6

Mathematical models

20

Topic 7

Introduction to differential calculus

18

Project

The project is an individual piece of work involving the collection of information or the generation of measurements, and the analysis and evaluation of the information or measurements.

25

Total teaching hours 150

It is essential that teachers are allowed the prescribed minimum number of teaching hours necessary to meet the requirements of the mathematical studies SL course. At SL the minimum prescribed number of hours is 150 hours.

Syllabus

Approaches to the teaching and learning of mathematical studies SL

In this course the students will have the opportunity to understand and appreciate both the practical use of mathematics and its aesthetic aspects. They will be encouraged to build on knowledge from prior learning in mathematics and other subjects, as well as their own experience. It is important that students develop mathematical intuition and understand how they can apply mathematics in life.

Teaching needs to be flexible and to allow for different styles of learning. There is a diverse range of students in a mathematical studies SL classroom, and visual, auditory and kinaesthetic approaches to teaching may give new insights. The use of technology, particularly the graphic display calculator (GDC) and computer packages, can be very useful in allowing students to explore ideas in a rich context. It is left to the individual teacher to decide the order in which the separate topics are presented, but teaching and learning activities should weave the parts of the syllabus together and focus on their interrelationships. For example, the connection between geometric sequences and exponential functions can be illustrated by the consideration of compound interest.

Teachers may wish to introduce some topics using hand calculations to give an initial insight into the principles.

However, once understanding has been gained, it is envisaged that the use of the GDC will support further work and simplify calculation (for example, the χ² statistic).

Teachers may take advantage of students’ mathematical intuition by approaching the teaching of probability in a way that does not solely rely on formulae.

The mathematical studies SL project is meant to be not only an assessment tool, but also a sophisticated learning opportunity. It is an independent but well-guided piece of research, using mathematical methods to draw conclusions and answer questions from the individual student’s interests. Project work should be incorporated into the course so that students are given the opportunity to learn the skills needed for the completion of a successful project. It is envisaged that the project will not be undertaken before students have experienced a range of techniques to make it meaningful. The scheme of work should be designed with this in mind.

Teachers should encourage students to find links and applications to their other IB subjects and the core of the hexagon. 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.

For further information on “Approaches to teaching a DP course” please refer to the publication The Diploma Programme: From principles into practice (April 2009). To support teachers, a variety of 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.

Mathematical studies SL guide 12

Approaches to the teaching and learning of mathematical studies SL

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

Appl real-life examples and links to other DP subjects Aim 8 moral, social and ethical implications of the sub-topic Int international-mindedness

TOK suggestions for discussion

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.

Course of study

The content of all seven 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 project work

Work leading to the completion of the project must 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 standard level courses is 150 hours. For mathematical studies SL, it is expected that 25 hours will be spent on work for the project. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 125 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.

Time has been allocated in each section of the syllabus to allow for the teaching of topics requiring the use of a GDC.

Use of calculators

Students are expected to have access to a 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 in examinations are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematical studies SL: Graphic display calculators teacher support material (May 2005) and on the OCC.

Approaches to the teaching and learning of mathematical studies SL

Mathematical studies SL formula booklet

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.

Teacher support materials

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

Command terms and notation list

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.

14 Mathematical studies SL guide

Syllabus

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 mathematical studies SL 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 mathematical studies SL.

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

The reference given in the left-hand column is to the topic in the syllabus content; for example, 1.0 refers to the prior learning for Topic 1—Number and algebra.

Learning how to use the graphic display calculator (GDC) effectively will be an integral part of the course, not a separate topic. Time has been allowed in each topic of the syllabus to do this.

Content Further guidance

1.0 Basic use of the four operations of arithmetic, using integers, decimals and fractions, including order of operations.

Prime numbers, factors and multiples.

Simple applications of ratio, percentage and proportion.

Examples: 2(3 4 7) 62+ × = ; 2 3 4 7 34× + × = .

Basic manipulation of simple algebraic expressions, including factorization and expansion.

Examples: ab ac a b c+ = ( + );(x+1)(x+2)=x²+3x+ . 2

Rearranging formulae. Example: ^{1 2}

2

A bh h A

= ⇒ = b .

Evaluating expressions by

substitution. Example: If x = −3 then

2 2 3 ( 3)2 2( 3) 3 18

x − x+ = − − − + = .

Solving linear equations in one

variable. Examples: 3(x+ −6) 4(x− =1) 0; ⁶ 4 7 5x + = . Solving systems of linear equations

in two variables. Example: 3x+4y=13, ¹ 2 1 3x− y= − . Evaluating exponential expressions

with integer values. Examples: ,a b∈ ; ^b 2^{4 1} 16

− = ; ( 2) 16− ⁴= . Use of inequalities < ≤ > ≥, , , .

Intervals on the real number line.

Example: 2< ≤x 5, x∈  .

Solving linear inequalities. Example: 2x+ < −5 7 x. Familiarity with commonly

accepted world currencies. Examples: Swiss franc (CHF); United States dollar (USD);

British pound sterling (GBP); euro (EUR); Japanese yen (JPY); Australian dollar (AUD).

Prior learning topics

Content Further guidance

2.0 The collection of data and its representation in bar charts, pie charts and pictograms.

5.0 Basic geometric concepts: point, line, plane, angle.

Simple two-dimensional shapes and their properties, including perimeters and areas of circles, triangles, quadrilaterals and compound shapes.

SI units for length and area.

Pythagoras’ theorem.

Coordinates in two dimensions.

Midpoints, distance between points.

16 Mathematical studies SL guide

Sy lla bu s c ont ent

Syllabus

To pi c 1 — N um ber a nd a lg eb ra 20 h ou rs Th e ai m s o f t hi s t op ic ar e t o i nt ro du ce s om e b asi c el em en ts an d co ncep ts o f m at hem at ics, a nd to li nk th es e t o f in an ci al an d o the r a ppl ic at ions .

1.1 N at ur al n um be rs ,  ; i nt eg er s,  ; r at io na l num be rs ,

; a nd r ea l num be rs ,  . N ot req ui red : pr oof of ir ra tion al ity , f or e xa m pl e, of

.

Li nk w ith dom ai n a nd r ang e 6.1. Int : H is tor ic al de ve lopm ent of n um be r s ys te m . A w ar en es s t hat o ur m od er n n um er al s a re de ve lope d f rom the A ra bi c not at io n. TO K: D o m at hem at ic al sy m bo ls h av e sen se in th e s am e w ay th at w or ds h av e sen se? I s z er o di ff er en t? A re th ese n um ber s c rea ted o r di sco ver ed ? D o the se num be rs e xi st ? 1.2 A ppr oxi m at ion: de ci m al pl ac es , s ig ni fic ant fig ure s. Per ce nt ag e e rr or s.

St ud en ts sh ou ld b e aw ar e o f t he er ro rs t hat can re su lt fr om pr em at ur e r oun di ng . A ppl : C ur ren cy ap pr ox im at io ns t o n ea res t w hol e num be r, eg pe so, y en . C ur re nc y appr ox im at ion s t o n ear est ce nt /pe nny , e g e ur o, dol la r, p ound. A ppl : P hy si cs 1.1 ( ra ng e of m ag ni tude s) . A ppl : M et eo rol ogy , a lte rna tiv e r ou ndi ng m et hods . A ppl : B io log y 2.1.5 (m icr osco pi c m easu rem en t). TO K: A pp rec iat io n o f t he di ff er en ces o f sc al e in num be r, an d of the w ay num be rs a re us ed th at ar e w el l b ey on d o ur ev er yd ay ex per ien ce.

Es tim ati on . St ude nt s s hou ld be a bl e t o r ec og ni ze w he th er th e resu lts o f ca lcu la tio ns a re r easo nab le, inc ludi ng re as ona bl e v al ue s of , f or e xa m pl e, len gt hs, an gl es an d ar ea s. Fo r ex am pl e, len gt hs can no t b e n eg at iv e.

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be rs in the for m 10

a × , w her e k is a n i nt eg er . St ude nt s s hou ld be a bl e t o us e s ci ent ifi c m ode on t he G D C . A ppl : V er y l ar ge an d v er y sm al l num be rs , e g as tronom ic al di st anc es , s ub -a to m ic p ar tic le s; Phy si cs 1.1 ; g loba l f ina nc ia l f ig ur es . A ppl : C he m is try 1.1 ( A vog adr o’ s num be r) . A ppl : P hy si cs 1.2 ( sc ie nt ific n ota tio n) . A ppl : C he m is try a nd bi ol og y ( sci en tif ic not at io n) . A ppl : E ar th sci en ce (e ar th qu ak e m easu rem en t scal e) .

be rs in thi s f or m . C alc ula to r n ota tio n i s n ot a ccep tab le. For e xa m pl e, 5.2E 3 is not a ccep tab le. nat ional ) a nd ot he r ba sic uni ts t: f or ex am pl e, k ilo gr am (k g) , nd (s) , l itr e ( l), m et re p er seco nd al e.

St ude nt s s hou ld be a bl e to c onv er t b et w ee n di ff er ent uni ts . Li nk w ith the for m of th e n ot at ion in 1. 3, f or exa m pl e,

mm

.

A ppl : S pe ed, ac ce le ra tion, f or ce ; P hy si cs 2. 1, Ph ysi cs 2.2 ; c onc en tra tion of a sol ut ion ; C he m is try 1.5 . Int : S I not at ion . TO K: D oe s t he us e of S I not at ion he lp us to th in k o f m at hem at ics a s a “u ni ver sa l lan gu ag e”? TO K : Wh at is m easu rab le? H ow can o ne m easu re m at hem at ica l ab ili ty ? er si on s. St ude nt s s hou ld be a bl e to pe rf or m c ur re nc y tra ns ac tions inv ol vi ng c om m is si on. A ppl : E co nom ic s 3.2 ( ex ch an ge r at es) . A im 8: T he e th ic al im plic at io ns o f tr ad in g in cu rr en cy an d its ef fe ct o n d iff er en t n at io nal com m uni tie s. Int : T he ef fe ct o f f lu ct uat io ns in cu rr en cy rat es on in te rna tion al tr ade .

Mathematical studies SL guide 18

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1.6 U se o f a G D C to so lv e • pai rs o f l in ea r eq uat io ns i n tw o v ar ia bl es In e xa m ina tio ns , no spe ci fic m et hod of sol ut io n w ill b e r equi re d. TO K : E qua tions w ith n o s ol ut ion s. A w ar ene ss th at w hen m at hem at ici an s t al k a bout “i m ag in ar y” o r “ rea l” s ol ut io ns t hey ar e u si ng pr ec ise te ch ni ca l t er m s t hat do not ha ve the sam e m ean in g as t he ev er yd ay ter m s. • qua dr at ic e qua tion s. St anda rd t er m inol og y, suc h as z er os or roo ts , shoul d be ta ug ht . Li nk w ith qu adr at ic m ode ls in 6.3. 1.7 A rit hm et ic s eq uen ces an d s er ies, a nd th ei r appl ic at ions . TO K: Inf or m al a nd f or m al re as oni ng in m at hem at ics. H ow d oe s m at hem at ic al p ro of di ff er fr om g ood r ea soni ng in e ve ry da y l ife ? I s m at hem at ical rea so ni ng d iff er en t f ro m sci en tif ic r ea so ni ng ? TO K: B eau ty an d e leg an ce in m at hem at ic s. Fi bona cc i num be rs a nd c on ne ct ions w ith t he G ol de n r at io.

U se o f t he fo rm ul ae fo r t he nth ter m an d t he sum of the fi rs t n ter m s o f t he seq uen ce. St ud en ts m ay u se a G D C fo r ca lcu la tio ns, b ut the y w ill be e xpe ct ed t o ide nti fy th e f irs t te rm and t he c om m on di ff er enc e. 1.8 G eo m et ric seq ue nce s an d s er ies . U se o f t he fo rm ul ae fo r t he nth ter m an d t he sum of the fi rs t n ter m s o f t he seq uen ce. N ot req ui red : for m al pr oof s of for m ul ae .

St ud en ts m ay u se a G D C fo r ca lcu la tio ns , bu t the y w ill be e xpe ct ed t o ide nt ify the fi rs t t er m and t he c om m on r at io. N ot req ui red : us e o f l og ar ithm s t o fin d n, gi ve n the sum of th e f irst n te rm s; s um s to in fin ity .

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so lv e ea r eq uat io ns i n tw o v ar ia bl es In e xa m ina tio ns , no spe ci fic m et hod of sol ut io n w ill b e r equi re d. TO K : E qua tions w ith n o s ol ut ion s. A w ar ene ss th at w hen m at hem at ici an s t al k a bout “i m ag in ar y” o r “ rea l” s ol ut io ns t hey ar e u si ng pr ec ise te ch ni ca l t er m s t hat do not ha ve the sam e m ean in g as t he ev er yd ay ter m s. qua tion s. St anda rd t er m inol og y, suc h as z er os or roo ts , shoul d be ta ug ht . Li nk w ith qu adr at ic m ode ls in 6.3. uen ces an d s er ies, a nd th ei r TO K: Inf or m al a nd f or m al re as oni ng in m at hem at ics. H ow d oe s m at hem at ic al p ro of di ff er fr om g ood r ea soni ng in e ve ry da y l ife ? I s m at hem at ical rea so ni ng d iff er en t f ro m sci en tif ic r ea so ni ng ? TO K: B eau ty an d e leg an ce in m at hem at ic s. Fi bona cc i num be rs a nd c on ne ct ions w ith t he G ol de n r at io.

ul ae fo r t he nth ter m an d t he ter m s o f t he seq uen ce. St ud en ts m ay u se a G D C fo r ca lcu la tio ns, b ut the y w ill be e xpe ct ed t o ide nti fy th e f irs t te rm and t he c om m on di ff er enc e. nce s an d s er ies . ul ae fo r t he nth ter m an d t he ter m s o f t he seq uen ce. for m ul ae .

St ud en ts m ay u se a G D C fo r ca lcu la tio ns , bu t the y w ill be e xpe ct ed t o ide nt ify the fi rs t t er m and t he c om m on r at io. s t o fin d n, gi ve n the sum of s; s um s to in fin ity .

atio ns o f g eo m et ric s eq ue nces nt er est pr ec ia tion . s.

U se of the G D C is e xpe ct ed, i nc lu di ng bu ilt -in f ina nc ia l pa ck age s. Th e co ncep t o f si m pl e i nt er est m ay b e u sed a s an i nt ro duc tion to com pound i nt er es t but w ill not be e xa m ine d. In e xa m ina tio ns , q ues tio ns t ha t ask st ud en ts t o de riv e t he for m ul a w ill not be se t. C om pound i nt er es t c an be c al cu la te d y ea rly , ha lf- ye ar ly , qua rte rly or m ont hl y. Li nk w ith e xpone nt ia l m ode ls 6. 4.

A ppl : E co nom ic s 3.2 ( ex ch an ge r at es) . A im 8 : E th ic al p er ce pt ion s of bo rr ow in g a nd le nd ing m one y. Int : D o al l so ci et ies v iew in vest m en t an d in ter est in th e sam e w ay ?

Mathematical studies SL guide 20

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To pi c 2 — De sc rip tiv e st at ist ic s 12 h ou rs The a im of thi s t op ic is to d ev el op t ec hn ique s t o de sc ribe a nd int er pr et se ts of d at a, i n p re pa ra tio n f or fu rthe r s ta tis tic al ap pl icat io ns.

2.1 C la ss ifi ca tion o f da ta a s di sc re te o r c on tinuous . St ude nt s s hou ld und er st an d t he c onc ept o f popul at ion a nd of re pr es ent at iv e a nd r andom sa m pl ing . S am pl ing w ill no t be e xa m ine d bu t ca n be us ed i n in ter nal ass essm en t.

A ppl : P syc hol ogy 3 (r ese ar ch m et hodol og y). A ppl : B iol ogy 1 (s ta tist ic al an al ysi s) . TO K: V al id ity of d at a a nd int roduc tion o f bi as . 2.2 Si m pl e di scr et e d at a: fr eq uen cy tab les . 2.3 G roupe d d is cr et e or c ont in uous da ta : f re qu enc y ta ble s; m id -in te rv al v al ue s; uppe r a nd l ow er bounda rie s. Fr eque nc y hi st og ra m s.

In e xa m ina tio ns , f req uen cy h is to gr am s w ill hav e eq ua l cl ass in ter val s. A ppl : G eog ra phy (g eo gr ap hi ca l an al yses ). 2.4 C um ul at iv e f re que nc y t abl es fo r g roupe d di sc re te d at a and for g roup ed c ont inuo us da ta ; cu m ul at iv e f req uen cy cu rv es, m ed ian an d qu ar til es. Bo x- and -w hi sk er d iag ram . N ot req ui red : tre atm en t o f o utl ie rs .

U se of G D C to pr od uc e hi st og ra m s a nd box - and -w hi sk er d iag ram s. 2.5 Meas ur es o f cen tral ten den cy . Fo r s im pl e d is cr et e dat a: m ean ; m edi an; m ode . For g roupe d di sc re te a nd c ont in uous d at a: est im at e o f a m ean ; m od al cl as s.

St ude nt s s hou ld us e m id -in te rv al v alu es to est im at e t he m ean o f g ro up ed d at a. In e xa m ina tio ns , qu es tion s us ing ∑ not at ion w ill n ot b e se t.

Ai m 8: T he e th ic al im plic at io ns o f u sin g st at ist ics t o m isl ead .

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er si on : r an ge, in te rq ua rti le ev ia tion. St ude nt s s hou ld us e m id -in te rv al v alu es to est im at e t he s tan dar d d ev iat io n o f g ro up ed dat a. In e xa m ina tio ns : • st ude nt s ar e ex pec ted to u se a G D C to ca lc ul at e s ta nda rd de vi at ion s • th e d at a se t w ill b e t rea ted a s t he popul at ion. St ude nt s s hou ld be a w ar e t ha t t he IB not at ion m ay di ff er fr om the no ta tio n on t he ir G D C . U se o f co m pu ter sp read sh ee t so ftw ar e is enc our ag ed i n t he tr ea tm en t o f th is to pic .

Int : T he be ne fit s of sha ring a nd a na ly si ng da ta fr om di ff er ent c oun tri es . TO K : I s st an da rd d ev iat io n a m at hem at ica l di sco ver y o r a c rea tio n of the hum an m ind ?

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To pi c 3 — Lo gic , s ets a nd pr oba bi lit y 20 h ou rs The a im s of thi s t opi c ar e to i nt rodu ce the p rinc ipl es o f l og ic , t o u se se t t he or y t o int rod uc e pr ob abi lit y, and to de te rm ine th e l ik el ih ood o f r andom e ve nt s us ing a var iet y of tec hn iq ues.

3.1 B as ic c onc ept s of sy m bol ic log ic : de fin iti on of a pr op os iti on; sy m bol ic not at ion of pr opos iti on s. 3.2 C om pound st at em ent s: im pl ic at ion, ⇒ ; equi va le nc e, ⇔ ; ne ga tio n,

; c on junc tion, ∧ ; di sj un ct ion, ∨ ; e xc lu si ve d isj unc tion,

. Tr an sl at io n b et w een v er bal st at em en ts a nd sym bol ic for m . 3.3 Tr uth ta ble s: c on ce pts o f lo gic al co nt ra dic tio n and t aut ol og y. A m axi m um of thr ee pr opo si tion s w ill b e us ed in t rut h t abl es. Tr ut h t ab le s can b e used to il lu st ra te t he as soc ia tiv e a nd di st ribu tiv e pr ope rti es of conne ct iv es , a nd f or v ar ia tions of im pl ic at ion an d eq ui val en ce s ta tem en ts , f or ex am pl e, qp ¬ ⇒¬ . 3.4 C on ver se, in ver se, co nt rap osi tiv e. Lo gi cal e qui va le nc e. A ppl : U se of a rg um ent s i n de ve lopi ng a lo gi ca l es say st ru ct ur e. A ppl : C om put er pr og ra m m ing ; d ig ita l c irc uits ; Ph ysi cs HL 14.1 ; Ph ys ic s S L C1 . TO K: In du ct iv e an d d ed uc tiv e lo gi c, fa llaci es. Te st ing the v al idi ty of si m pl e a rg um ent s thr oug h t he us e of tr ut h t abl es . The top ic m ay be e xt ende d to i nc lu de sy llog is m s. In e xa m ina tion s t he se w ill not be tes ted .

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f se t t heo ry : el em en ts xA ∈ , ; in te rs ec tio n AB ∩ ; u ni on em ent A ′.

d si m pl e ap pl ica tio ns. M or ga n’ s l aw s.

In ex am in at io ns, th e un iv er sal set U w ill inc lude n o m or e t ha n t hr ee subs et s. Th e em pt y set is d en ot ed by ∅ . en t A ; co m pl em en tar y ev en t, n e ve nt . m pl em en tar y ev en t. .

Pr oba bi lit y m ay be int rod uc ed a nd ta ug ht in a pr ac tical w ay u si ng co in s, d ice, p lay in g car ds and ot he r e xa m pl es to de m ons tra te ra ndom be ha vi our . In e xa m ina tio ns , qu es tion s i nv ol vi ng pl ay ing car ds w ill n ot b e se t.

A ppl : A ctu ar ia l s tu die s, pr oba bi lit y of li fe spa ns a nd t he ir ef fe ct on i ns ur anc e. A ppl : G ov er nm ent pl ann in g ba se d on pr oj ect ed fi gu res . TO K: T heo re tical an d ex per im en ta l pr oba bi lit y. om bi ne d e ve nt s, m ut ua lly in dep en den t ev en ts. St ude nt s s hou ld be e nc our ag ed t o us e the m os t appr op ria te m et hod i n s olv in g in div id ua l que st ion s.

A ppl : B io log y 4.3 ( th eo re tical g en et ics) ; B iol og y 4.3 .2 (P un net t sq uar es ). A ppl : P hys ic s H L13.1 ( de te rm ini ng the pos iti on of a n e le ct ron ); P hys ic s SL B1 . A im 8: T he e th ic s of g am bl ing . TO K: T he pe rc ep tion of ri sk , i n bus in es s, i n m ed ici ne a nd sa fe ty in tr av el .

ram s, V en n d iag ram s, sam pl e d t ab les o f o ut co m es. “w ith rep lacem en t” an d em en t”. bi lit y.

Pr oba bi lit y que st ions w ill b e pl ac ed in c ont ext an d w ill m ak e u se o f d iag ra m m at ic re pr es ent at ions . In e xa m ina tio ns , qu es tion s r equ iri ng the exc lu si ve us e of the fo rm ul a i n s ec tion 3.7 o f the for m ul a book le t w ill no t be se t.

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To pi c 4 — St at is tic al ap pli cat io ns 17 h ou rs The a im s of thi s t opi c a re to de vel op tech ni qu es i n in fer en tial st at ist ics in o rd er to an al yse set s o f d at a, d raw co ncl usi on s an d in te rp ret th ese.

4.1 The nor m al d is tri but ion. The c on ce pt of a ra ndom va ria bl e; o f t he par am et er s µ a nd σ ; of the b el l s ha pe ; t he sym m et ry a bout x µ = .

St ude nt s s hou ld be a w ar e t ha t a ppr oxi m at el y 68% o f t he da ta li es be tw ee n µσ

, 95% li es be tw ee n 2 µσ ± a nd 99% li es b et w een 3 µσ ± .

A ppl : E xam pl es of m easu rem en ts, ran gi ng fr om ps yc hol og ic al to phy si ca l phe nom ena , th at c an be a ppr ox im at ed, to v ar yi ng de gr ee s, by the nor m al d is tri but ion. A ppl : B iol ogy 1 (s ta tist ic al an al ysi s) . A ppl : P hy si cs 3.2 (ki ne tic m ol ecu lar th eo ry ). D iag ram m at ic r ep re sen ta tio n. U se o f sk et ch es o f n or m al c ur ves an d sh ad in g w he n us ing the G D C is e xp ec te d. N or m al pr oba bi lit y c al cu la tions . St ude nt s w ill be e xpe ct ed t o us e the G D C w he n c al cu la ting pr ob abi lit ie s an d i nv er se nor m al . Ex pect ed v al ue . In ver se n or m al ca lcu la tio ns . In e xa m ina tio ns , i nv er se no rm al que st ion s w ill not inv ol ve fi nd ing the m ea n or st and ar d de vi at ion. N ot req ui red : Tr ans for m at ion of a ny n or m al v ar iab le t o t he st an dar di ze d no rm al .

Tr ans for m at ion of a ny nor m al v ar ia bl e t o t he st an dar di zed n or m al v ar iab le, z, m ay be ap pr op riat e i n in ter nal ass essm en t. In e xa m ina tio ns , qu es tion s r equ iri ng the u se o f z sco res w ill n ot b e se t.

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e c on ce pt of c or re la tio n. St ude nt s s hou ld be a bl e to m ak e the di st inc tio n be tw ee n c or re la tio n a nd ca us at ion. A ppl : B iol ogy ; P hy si cs ; C hem ist ry ; S oci al sci en ce s. TO K : D oe s c or re la tion im pl y c aus at ion? s; li ne o f b est fi t, b y ey e, he m ea n p oi nt . t– m om en t co rrel at ion H and c al cu la tion s of r m ay en han ce unde rs ta ndi ng . In exa m ina tio ns , s tud ent s w ill be e xpe ct ed t o use a G D C to ca lcu la te r. f po si tiv e, z er o a nd ne ga tiv e, rr el at io ns. ine fo r y o n x. H an d ca lcu la tio ns o f t he re gr essi on li ne m ay enha nc e und er st an di ng . In exa m ina tio ns , s tud ent s w ill be e xpe ct ed t o use a G D C to fi nd th e reg re ssi on li ne .

A ppl : C he m is try 11.3 ( gr aphi ca l t ec hni qu es ). TO K: C an w e re lia bl y us e the e qua tion of the reg ress ion line to m ak e pr edi ct ion s? si on line for pr edi ct ion St ud en ts sh ou ld b e aw ar e o f t he d an ger s o f ext ra po la tion.

Mathematical studies SL guide 26

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4.4 Th e

χ te st fo r i nd epe nde nc e: for m ul at ion of nul l a nd a lte rna tiv e hy pot he se s; si gni fican ce lev el s; co nt in gen cy tab les ; ex pec ted fr eq uen ci es; d eg rees o f f re ed om ; p -v al ues.

In e xa m ina tio ns : • the m axi m um nu m be r of ro w s or c ol um ns in a c ont ing enc y t ab le w ill be 4 • th e d eg rees o f f re ed om w ill al w ay s b e gr ea te r t ha n o ne • the

χ cr iti ca l v al ue w ill a lw ay s b e g iv en • onl y que sti ons on up pe r ta il te sts w ith com m onl y used si gn ifi can ce l ev el s ( 1% , 5 % , 10% ) w ill be se t. C al cu la tion of e xpe ct ed fr eque nc ie s by ha nd is re qu ire d. H and c al cu la tion s of

χ ma y e nha nc e unde rs ta ndi ng . In e xa m ina tio ns st ude nt s w ill be e xpe ct ed to use t he G D C to cal cu lat e t he

χ st at is tic. If us ing

χ te st s i n in ter nal as sessm en t, st ude nt s s hou ld b e a w ar e o f t he li m ita tions o f th e te st fo r sm al l ex pe ct ed fr eq ue nci es ; ex pect ed freq uen ci es m ust b e g reat er th an 5. If t he d eg ree o f f reed om is 1 , t hen Y at es ’s cont inu ity c or re ct ion s ho ul d be a pp lie d.

A ppl : B io lo gy (i nt er nal a ss essm en t); Ps yc hol og y; G eog ra phy . TO K : S cie nti fic m eth od .

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m et ry a nd tri gonom et ry 18 h ou rs re to de ve lop t he a bi lit y t o dr aw c le ar di ag ra m s i n t w o di m ens ions , a nd t o a pp ly a ppr opr ia te g eom et ric an d t rig on om et ric t ech ni qu es t o nd thr ee d im ens ions .

ne in t w o d im ens ions : t he for m s 0 ax by d + += . Li nk w ith l in ea r f un ct ions in 6.2. A ppl : G ra di ent s of m ount ai n r oa ds , e g C ana di an H ig hw ay . G rad ie nt s o f ac ces s r am ps . A ppl : Ec onom ic s 1.2 ( ela sti cit y) . TO K: D es car tes sh ow ed th at g eo m et ric pr ob lem s can b e so lv ed a lg eb ra ica lly an d v ic e ver sa. Wh at d oes th is tel l u s ab ou t m at hem at ical rep res en tat io n an d m at hem at ic al know le dg e?

ep ts . ec tion o f l in es. Li nk w ith s ol ut io ns o f pa irs of li ne ar e qua tio ns in 1.6. nts ,

a nd

.

. ne s,

. si ne an d tan gen t r at io s t o fin d ng le s of ri ght -an gl ed tr ian gl es. tion a nd de pr es si on .

Pr ob lem s m ay in co rp or at e P yt hag or as’ th eo rem . In e xa m ina tio ns , qu es tion s w ill onl y be se t i n deg rees.

A ppl : T ria ng ul at ion, m ap -m aki ng, fi ndi ng pr ac tical m easu rem en ts u si ng tr ig on om et ry . Int : D iag ram s o f P yt hag or as’ the or em oc cur in ea rly C hi ne se a nd I ndi an m anus cr ip ts . Th e ear lie st re fer en ces t o t rig on om et ry ar e i n I nd ian m at hem at ics.

Mathematical studies SL guide 28

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5.3 U se of th e si ne ru le :

. In al l ar eas o f t hi s t op ic, st ud en ts sh ou ld b e en co ur ag ed to sk et ch w el l-l ab el led d iag ram s t o suppor t t he ir sol ut ions . The a m bi guous c as e c ou ld be ta ug ht , but w ill not be e xa m ine d. In e xa m ina tio ns , qu es tion s w ill onl y be se t i n deg rees.

A ppl : V ect or s; Ph ysi cs 1 .3 ; be ar ing s. U se of the c os ine rul e

;

. U se o f ar ea o f a t rian gl e =

. C ons truc tion of la be lle d di ag ra m s f rom v er ba l st at em en ts.

TO K : U se t he f ac t t ha t t he co si ne ru le i s one pos si bl e g ene ra liz at ion of P yt ha gor as ’ t heo rem to ex pl or e t he c on cep t o f “g en er al ity ”.

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ee -di m ens iona l s ol ids : c uboi d; ht p yr am id ; r ig ht c on e; cy lin de r; re ; a nd co m bi na tion s of the se w een tw o p oi nt s; eg b et w een er tic es w ith m idpoi nt s or idpo int s. le be tw ee n t w o l ine s or d a p lan e. pl ane s.

In e xa m ina tio ns , on ly ri ght -a ng le d tri gonom et ry que st ions w ill be se t i n r ef er enc e to th re e- di m ens iona l s ha pe s.

TO K: Wh at is an ax io m at ic sy st em ? D o t he a ng le s i n a tr ia ng le a lw ay s a dd to 180° ? No n- Eu cl id ean g eo m et ry , su ch as R iem an n’ s. Fl ig ht m aps of a irl ine s. A ppl : A rc hi te ct ur e a nd de si gn. rf ace a rea s o f t he t hr ee - ids d ef ine d in 5.4.

Mathematical studies SL guide 30

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To pi c 6 — M at he m at ic al m od els 20 h ou rs The a im of thi s t op ic is to d ev el op und er st and ing of so m e m at he m at ic al func tions tha t c an be u se d to m ode l p ra ct ic al si tua tio ns. E xt en si ve u se o f a G D C is t o b e enc ou ra ge d in t hi s t opi c.

6.1 C onc ept of a fun ct ion, dom ai n, r ang e and gr aph. Func tion no ta tion , e g ( ), ( ), ( ) f x vt C n . C on cep t o f a fu nct io n as a m at hem at ical m ode l.

In e xa m ina tio ns : • the d om ai n i s t he set o f a ll real n um ber s un les s o th er w ise st at ed • m appi ng not at ion :fx y  w ill n ot b e used .

TO K : Wh y can w e u se m at hem at ic s t o desc rib e t he w or ld an d m ak e p re di ct io ns? I s i t becau se w e d isc ov er th e m at hem at ic al b as is o f th e w or ld o r b ecau se w e im po se o ur o w n m at hem at ical st ru ct ur es o nt o t he w or ld ? Th e r el at io ns hi p b et w ee n rea l-w or ld pr ob le m s an d m at hem at ica l m od el s. 6.2 Li ne ar m ode ls . Li ne ar func tion s a nd the ir g ra phs , ()fx mx c = + .

Li nk w ith e qua tion of a li ne in 5.1. A ppl : C onv er si on g ra phs , eg tem per at ur e o r cur re nc y c onv er si on ; P hy si cs 3.1 ; E co no m ics 3.2 . 6.3 Q ua dr at ic m ode ls . Q ua dr at ic fun ct ions a nd the ir g ra ph s (p ar ab ol as ):

()f x ax bx c = ++ ; 0 ≠ a Li nk w ith the qu adr at ic e qu at ions in 1.6 . Func tions w ith z er o, on e or tw o r ea l r oo ts a re inc lude d.

A ppl : C os t f unc tion s; p ro je cti le m otio n; Phy si cs 9.1 ; a re a f un ct ions . Pr ope rti es of a pa ra bo la : sy m m et ry ; v er tex ; int er ce pt s on t he x -a xi s a nd y- ax is. Equa tion o f t he a xi s o f s ym m et ry,

.

The fo rm of the e qua tion of the a xi s of sy mme try m ay ini tia lly be found by in ve st ig atio n. Pr ope rti es shou ld be il lu st ra te d w ith a G D C or gr ap hi ca l s of tw ar e.

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ls . tio ns a nd t he ir g ra phs : , 1, 0

∈ ≠≠  a a k . ; , 1, 0 ∈ ≠≠  a a k . tio n of a hor iz ont al In exa m ina tio ns , s tud ent s w ill be e xpe ct ed t o us e g ra ph ic al m et hods , i nc ludi ng G D C s, t o so lv e p ro bl em s.

A ppl : B io log y 5.3 ( popu la tions ). A ppl : B iol ogy 5.3.2 (p opu la tion g row th ); Phy si cs 13.2 (r ad io ac tiv e d ecay ); Ph ysi cs I 2 (X -ra y a tte nu atio n); c oo lin g of a li qui d; sp rea d of a vi rus ; de pr ec ia tion. ct ions of the fo rm ...; , ∈  bx m n . In e xa m ina tio ns , s tud ent s w ill be e xpe ct ed to us e g ra ph ic al m et hods , i nc ludi ng G D C s, t o so lv e p ro bl em s. s t ype a nd t he ir g ra phs . er tica l a sy m pt ot e. Ex am pl es :

() 3 5 3 fx x x = −+ ,

. te g rap hs . ch fr om inf or m at ion g iv en . ra ph f rom G D C to pa pe r. et ing a nd m ak ing pr edi ct ion s

St ud en ts sh ou ld b e aw ar e o f t he d iff er en ce bet w ee n th e co m m an d t er m s “d raw ” an d “sk et ch ”. A ll g rap hs sh ou ld b e l ab el led an d h av e so m e indi ca tion of sc al e.

TO K: D oe s a g ra ph w itho ut la be ls or indi ca tion of sc al e ha ve m ea ni ng ? func tion s a bov e a nd a dd iti ons s. Ex am pl es :

, () 3

gx x

= + .

Mathematical studies SL guide 32

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6.7 U se o f a G D C to so lv e eq ua tions inv ol vi ng com bi na tions o f t he func tio ns abov e. Ex am pl es :

22 3 1 x xx += + − , 53

x = . O the r f unc tions c an b e us ed for m ode lli ng in in ter na l a sse ssm en t bu t w ill not be se t on exa m ina tion pa pe rs .

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so lv e eq ua tions inv ol vi ng f t he func tio ns abov e. Ex am pl es :

22 3 1 x xx += + − , 53

x = . O the r f unc tions c an b e us ed for m ode lli ng in in ter na l a sse ssm en t bu t w ill not be se t on exa m ina tion pa pe rs . ro du ct ion t o di ffe re nt ial c al cu lu s 18 h ou rs nt rod uc e t he c onc ept of the de riv at iv e of a fun ct ion a nd to app ly it to opt im iz at ion a nd ot he r p ro bl em s.

der iv at iv e a s a r at e o f ch an ge . rv e. t o f lim its .

Te ach er s ar e en co ur ag ed to in tro du ce di ff er ent ia tio n t hr oug h a g ra phi ca l a pp roa ch , rat her th an a f or m al tr ea tm en t. Em pha si s i s pl ac ed on i nt er pr et at ion o f t he conc ep t i n di ff er ent c ont ext s. In e xa m ina tio ns , qu es tion s on di ff er en tia tion fr om fir st p rin ci ple s w ill no t b e s et.

A ppl : R at es o f c ha ng e i n ec onom ic s, ki nem at ics an d m ed ici ne. A im 8: P lag iar ism an d ack no w led gm en t o f sour ce s, eg the c onf lic t be tw ee n N ew ton a nd Le ib nitz , w ho a ppr oa che d t he de ve lopm en t of cal cu lu s f ro m d iff er en t d ire ct io ns TO K: Is int ui tion a v al id w ay of k now ing in m at hs? H ow is i t p ossi bl e t o rea ch th e s am e co ncl usi on fr om d iff er en t r es ea rch p at hs? t

() f x anx

′ . fu nc tion s of the for m

...,

w her e a ll ex po nen ts a re

St ud en ts sh ou ld b e f am ili ar w ith th e al ter nat iv e no ta tio n f or d er iv at iv es

or

. In e xa m ina tio ns , k now le dg e of the se con d der iv at iv e w ill no t be a ss um ed.