gsis ib mathematics student companion student ... -...

GSIS IB MathematicsGSIS IB MathematicsGSIS IB MathematicsGSIS IB Mathematics

Student CompanionStudent CompanionStudent CompanionStudent Companion

Mathematics SMathematics SMathematics SMathematics SLLLL

Contents:

1 Course Introduction

2 Maths Learner Profile

3 Syllabus Outline

4 Assessment Schedule

5 Grade Boundaries

6 Exploration Suggestions

7 Exploration Criteria

8 Extended Essay suggestions

9 Autograph Download Instructions

10 Books for Further Reading

11 Suggested Websites

12 Command Terms and Notation

13 SL Formula Booklet for use in exams and tests

NAME_____________________________________________

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IB StandardIB StandardIB StandardIB Standard Level MathsLevel MathsLevel MathsLevel Maths

Welcome to StandardStandardStandardStandard level Mathslevel Mathslevel Mathslevel Maths, your Group 5 choice for the IB Diploma.

Standard level Maths is a challenging subject which contains all the Maths you might need in

other courses such as Physics, Chemistry or Economics. We ask only for success at IGCSE

Mathematics and an open mindopen mindopen mindopen mind. If you have studied Mathematics beyond IGCSE you may find

you have a head start in this course and might even consider a change to HL Maths. If you

found IGCSE Mathematics difficult, you might be more suited to Mathematical Studies SL. If

you think you may need to change, you need to discuss this with your Maths teacher and the

school IB Coordinator as soon as possible – it may well be possible to change but there could

be a problem, particularly if not done early in the course.

We do hope you will succeed with this excellent course. The IB Learner ProfileIB Learner ProfileIB Learner ProfileIB Learner Profile requires you to

be an independent learner; you need to take responsibility for your own learning, but this can

be difficult at first and your teacher is here to help.

The coursecoursecoursecourse will start by very quickly going over some of the basics you should have learnt at

IGCSE before soon moving on to new material (see syllabus outlinesyllabus outlinesyllabus outlinesyllabus outline). More details are on the

LEO SL Maths page. LEO SL Maths page. LEO SL Maths page. LEO SL Maths page. We plan to have teststeststeststests fairly regularly, perhaps after every two topics. For

these you will be expected to have a GDCGDCGDCGDC (Graphic Display Calculator) the school recommended

model is the Casio fx CG20Casio fx CG20Casio fx CG20Casio fx CG20 available from the school shop, but other GDCs are acceptable

provided they cannot do algebra. If in doubt consult your teacher. For the tests, you will also

need the IB Formula Booklet IB Formula Booklet IB Formula Booklet IB Formula Booklet which is included in this guide – so be sure to bring this guide be sure to bring this guide be sure to bring this guide be sure to bring this guide

with you for testswith you for testswith you for testswith you for tests....

The course text book is Mathematics StandardMathematics StandardMathematics StandardMathematics Standard Level (Oxford UnivLevel (Oxford UnivLevel (Oxford UnivLevel (Oxford Univerererersity Press, ISBN 978 019 sity Press, ISBN 978 019 sity Press, ISBN 978 019 sity Press, ISBN 978 019

839011 4839011 4839011 4839011 4)))) which will be needed throughout the course.

You will also sit school examsschool examsschool examsschool exams during the course, probably in May of year 12 and a full mock

exam probably in February of year 13. See the SSSSL assessment scheduleL assessment scheduleL assessment scheduleL assessment schedule for the exam format

which will be reflected in the school exams to some degree.

Notice that the assessment scheduleassessment scheduleassessment scheduleassessment schedule includes an explorationexplorationexplorationexploration which is your Maths IA (Internal

Assessment = coursework) worth 20% of your final grade. This will probably be done during

the first half term of year 13, but the idea of an exploration will be introduced early in the

course. The hope is that some area of Mathematics or some application, not necessarily

introduced in the course, will inspire you to want to explore further – use the pages in this

guide to jot down and consider any ideas that come to you during the course.

You are, of course, free to choose to do an EEEEEEEE (Extended Essay(Extended Essay(Extended Essay(Extended Essay)))) in any of your 6 subjects, but we

hope you might consider doing one in Maths; you will find some suggested titles in here as well

as some suggestions for Maths further readingMaths further readingMaths further readingMaths further reading....

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GERMAN SWISS INTERNATIONAL SCHOOLGERMAN SWISS INTERNATIONAL SCHOOLGERMAN SWISS INTERNATIONAL SCHOOLGERMAN SWISS INTERNATIONAL SCHOOL

MATHEMATICS DEPARTMENTMATHEMATICS DEPARTMENTMATHEMATICS DEPARTMENTMATHEMATICS DEPARTMENT

The Learner ProfileThe Learner ProfileThe Learner ProfileThe Learner Profile

fffforororor

The International Baccalaureate The International Baccalaureate The International Baccalaureate The International Baccalaureate DiplomaDiplomaDiplomaDiploma

PREAMBLEPREAMBLEPREAMBLEPREAMBLE

The International Baccalaureate Organization Mission Statement aims to develop inquiring,

knowledgeable and caring young people who help to create a better and more peaceful world

through intercultural understanding and respect.

It encourages students across the world to become active, compassionate and lifelong learners

who understand that other people, with their differences, can also be right.

The Learner Profile is not a list, it is not another poster to make rooms look interesting and it is not

a set of rules.

It is a whole school vision and our value system. As such it is something every member of the

school community should aspire to, it should be found at every turn in the school and is embedded

in all our teaching and learning.

FEATURES of the LEARNER PROFILEFEATURES of the LEARNER PROFILEFEATURES of the LEARNER PROFILEFEATURES of the LEARNER PROFILE

IBIBIBIB learners strive to be:learners strive to be:learners strive to be:learners strive to be:

InquirersInquirersInquirersInquirers At GSIS their natural curiosity is nurtured. They acquire the skills necessary

to conduct purposeful, constructive research and become independent active

learners. They actively enjoy learning and this love of learning will be

sustained throughout their lives.

How this happens Mathematically

Inquirers look for patterns which some often say define the discipline of Mathematics. They write proofs to illuminate these patterns. Inquirers discover relationships to deepen their understanding and ownership of the ideas and concepts being studied.

Critical thinkersCritical thinkersCritical thinkersCritical thinkers They exercise initiative in applying thinking skills critically and creatively to

make sound decisions and approach complex problems.


Higher Mathematics involves solving complex, multi-step problems. Students are required to think critically in order to evaluate their solutions and problem solving approaches.

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CommunicatorsCommunicatorsCommunicatorsCommunicators

They understand and express ideas and information confidently in more than

one language and in a variety of literacies.


Our students are encouraged to use appropriate mathematical language because Mathematics has its own language. Mathematics is one area where the “globalness” that the IB Diploma seeks in its students can be found. Communicating in this language requires an understanding of its set of rules, symbols, notation, syntax etc. Mathematics has multiple modes of communication (graphical, algebraic and symbolic among others) that need to be mutually reinforcing and consistent. They communicate by sharing mathematical ideas, methods and conclusions; they do this orally, at the whiteboard and through presentations.

RiskRiskRiskRisk----takerstakerstakerstakers

They approach unfamiliar situations confidently and have the independence

of spirit to explore new roles, ideas and strategies. They are courageous and

articulate in defending the things in which they believe or believe to be true.


Risk takers are encouraged to contribute to class discussion despite the possibility of being incorrect. Risk takers attack unfamiliar problems because they have instant feedback when they can solve them and hence reinforce their self values.

KnowledgeableKnowledgeableKnowledgeableKnowledgeable

They explore concepts, ideas and issues which have global relevance and

importance. In doing so, they acquire, and are able to make use of, a

significant body of knowledge across a range of disciplines.


Mathematics is a global and multi-disciplinary language. Science and other disciplines express themselves through Mathematics. Our understanding of Mathematics continues to evolve and deepen as our ability to explore ideas of greater complexity continues to develop.

PrincipledPrincipledPrincipledPrincipled

They have a sound grasp of the principles of moral reasoning. They have

integrity, honesty, a sense of fairness and justice and respect for the dignity

of the individual.

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Students are expected to take responsibility for their own work and problem solving. Mathematics can be very unforgiving – if a student tries to pretend to work at or understand the subject, their lack of knowledge will be found out by independent assessment. We expect students to have self-respect for the work they do and to care about what they do and what it means to others.

CaringCaringCaringCaring They show empathy and compassion towards the needs and feelings of

others. They have a personal commitment to action and service to enhance

the human condition, and respect for the environment.


Better students learn better by teaching peers and owning their peers’ progress. Attaching real world emotions and morals to Mathematics problems by relating the mathematical concept to problems that have real human impact increases a student’s appreciation for the role that Mathematics can play in improving the world in which they live.

OpenOpenOpenOpen----mindedmindedmindedminded Through an understanding and appreciation of their own culture, they are

open to the perspectives, values and traditions of other individuals and

cultures and are accustomed to seeking and considering a range of points of

view.


Students explore and discover multiple methods of solving problems. Students understand that there are different perspectives that can be equally effective in visualizing, setting up, or solving problems. We promote open-ended exploration that encourages and rewards individual responses.

WellWellWellWell----balancedbalancedbalancedbalanced They understand the importance of physical and mental balance and

personal well-being for themselves and others.


Balanced students manage their time in and out of the class. One way of maintaining balance is through finding the quick, simple ways to solve problems. A good understanding of Mathematics and its elegance can streamline problem solving and make students more effective and efficient.

ReflectiveReflectiveReflectiveReflective They give thoughtful consideration to their own learning and personal

development. They are able to analyse their strengths and weaknesses in a

constructive manner, and act on them.


Reflecting involves considering where assumptions are made that can lead to truth or error. Being able to reflect on your own work and how you are approaching a problem and how to correct an inferior method can lead to penetrating insights.

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The Mathematics Department believes that if we expect our students to be lifelong learners then

the teaching staff should exemplify this. In this regard, the teaching staff of the Mathematics

Department:

• continues to update the skills and knowledge of all staff members through professional

development, collaborative training and the sharing of best practice

• places the learner firmly at the heart of our programmes and focuses attention on the

processes and the outcomes of learning

• sets and reflects personal teaching goals based on the attributes of the Learner Profile

• provides time for students to reflect on an assessment task and what they have learnt from it

• recognises the Learner Profile as a map of a lifelong journey in pursuit of international-

mindedness

• is committed to shift the focus in our department from content and skills to ideals and learning

• models “intellectual character”; that set of dispositions which helps to inspire students to

develop intellectual behaviour

• accepts that Mathematics is a discipline with nuances, subtlety and the capacity to

acknowledge that “other people …can also be right”.

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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

Algebra

9

Topic 2

Functions and equations

24

Topic 3

Circular functions and trigonometry

16

Topic 4

Vectors

16

Topic 5

Statistics and probability

35

Topic 6

Calculus

40

Mathematical exploration

Internal assessment in mathematics SL is an individual exploration. This is a piece of

written work that involves investigating an area of mathematics.

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Total teaching hours 150

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Assessment

Assessment outline

First examinations 2014

Assessment component Weighting

External assessment (3 hours)

Paper 1 (1 hour 30 minutes)

No calculator allowed. (90 marks)

Section A

Compulsory short-response questions based on the whole syllabus.

Section B

Compulsory extended-response questions based on the whole syllabus.

80%

40%

Paper 2 (1 hour 30 minutes)

Graphic display calculator required. (90 marks)

Section A

Compulsory short-response questions based on the whole syllabus.

Section B

Compulsory extended-response questions based on the whole syllabus.

40%

Internal assessmentThis component is internally assessed by the teacher and externally moderated by the IB at

the end of the course.

Mathematical exploration

Internal assessment in mathematics SL is an individual exploration. This is a piece of

written work that involves investigating an area of mathematics. (20 marks)

20%

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GSIS IB Mathematics – Grade Boundaries

For all school tests, exams and coursework in Mathematics – HL, SL or Studies we intend

using the following Grade Boundaries:

Grade 7 80% or above






Grade 1 0% or above

The actual IB grade boundaries vary a little each year, depending on the difficulty of the papers, but

they are roughly in line with these. We might also sometimes need to change our boundaries for the

same reasons, but we will try to stick to these as much as possible.

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Exploration PlanningExploration PlanningExploration PlanningExploration Planning

During the course, consider what you might like to do for your exploration. Likely length is

around 1000 words (6 to 8 sides). A good idea is to draw up a mind map of a topic that interests

you – it could be pure Maths or an application, and to focus your exploration on what seems the

most mathematically interesting to you. The target audience is your fellow students – so don’t

aim to do advanced mathematics that is too far beyond your syllabus.

Possible areas for exploration include the following, but you are certainly not restricted to

these.

sport archaeology computers algorithms mobile phones music

sine musical harmony motion e electricity water

space orbits food volcanoes diet Euler games

symmetry architecture codes the internet communication

tiling population agriculture viruses health dance play π geography biology business economics physics chemistry

information technology in a global society psychology your own ideas

Use the space below to jot down any ideas you have and to plan out any suggestions:

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Internal assessment

Internal assessment criteriaThe exploration is internally assessed by the teacher and externally moderated by the IB using assessment

criteria that relate to the objectives for mathematics SL.

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 SL 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

Criterion A: CommunicationThis 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

0 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.

Criterion B: Mathematical presentationThis 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.

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Internal assessment

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.



below.

1 There is some appropriate mathematical presentation.

2 The mathematical presentation is mostly appropriate.

3 The mathematical presentation is appropriate throughout.

Criterion C: Personal engagementThis criterion assesses the extent to which the student engages with the exploration and makes it their 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 their own way.



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.

Criterion D: ReflectionThis 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.



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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Internal assessment

Criterion E: Use of mathematicsThis criterion assesses to what extent 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.



below.

1 Some relevant mathematics is used.

2 Some relevant mathematics is used. Limited understanding is demonstrated.

3 Relevant mathematics commensurate with the level of the course is used.

Limited understanding is demonstrated.

4 Relevant mathematics commensurate with the level of the course is used.

The mathematics explored is partially correct. Some knowledge and

understanding are demonstrated.

5 Relevant mathematics commensurate with the level of the course is used. The

mathematics explored is mostly correct. Good knowledge and understanding

are demonstrated.

6 Relevant mathematics commensurate with the level of the course is used. The

mathematics explored is correct. Thorough knowledge and understanding are

demonstrated.

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IB Mathematics Extended Essay TitlesIB Mathematics Extended Essay TitlesIB Mathematics Extended Essay TitlesIB Mathematics Extended Essay Titles Your extended essay will be marked out of 36. 24 marks are for general essay style and content; 12 marks are specific to the subject in which you are doing your essay. Thus it is possible to do a maths extended essay if you are only doing Maths Standard level or Studies. You may not score so highly on the 12 Maths marks, but can still write a good essay and score over 20 marks. Likewise, if your essay is not purely Mathematical, perhaps it is really Maths with some Music or Biology or Geography - it will be marked as a Mathematics essay and may not score so highly on the maths 12 marks, but can still score well overall if it is a well written essay. So it is best to find something you are really interested in and do your essay on that, rather than choose a topic that doesn’t appeal because you think you can score highly – you may not! Your essay needs a clearly stated well focussed research questionresearch questionresearch questionresearch question. You need to write an abstractabstractabstractabstract of your essay which states your approach to answering the question and summarises your conclusions. This should be written last, but placed at the front of the essay. Pages should be numberedPages should be numberedPages should be numberedPages should be numbered, there should be a contents pagecontents pagecontents pagecontents page, referencesreferencesreferencesreferences should be cited and a bibliographybibliographybibliographybibliography given. You should have clearly stated conclusionsconclusionsconclusionsconclusions at the end. Essay titles that have proved successful include those below. These may give you an idea for your essay title, but might also serve to show the huge variety in what you can choose, so choose something you are really interested in.

1. What is the percentage return of a particular 3 reel slot machine? (Any casino-type situation gives opportunities for data collection, comparing expected results with observed data and comparing payout with probability)

2. What are alternatives to Euclidean Geometry and what practical applications do

they have? 3. A comparative study of population growth models for Country X over the last n

years, with future predictions. (Which model best fits the data?) 4. The sound of mathematics - investigation of geometric series in musical

instruments - the position of the frets on a guitar, for example. 5. How many convex polygons can be made from the seven tangram pieces? 6. An exploration of the distortion of the truth content of a message over the course of

transmission between individuals. 7. Is there a link between the golden ratio and how we perceive beauty in nature, with

special emphasis on the human face and form. 8. Does athleticism affect pulse/heart rate? The role of statistics in medical research.

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9. Is there a correlation between SAT results and school test scores/GPA? 10. The proof of Sophie Germain's Theorem - Sophie Germain is one of the most

famous female mathematicians and she made a valuable contribution in the search for a proof of Fermat's Last Theorem.

11. Leibniz and Newton discovered calculus at about the same time, independently of

each other. Compare their methods and discuss which notation is more used now. 12. What is e? What practical implications has its discovery and use had? 13. What is the Binomial Theorem and how has it contributed to the history of humanity? 14. What is the best way to calculate π ? 15. Complex number problem solving strategies - what sorts of real life problems do

complex numbers help solve? 16. Solving cubic equations. 17. Predicting the number of triangles formed when subdividing the sides of an

equilateral triangle n times by applying Newton's Forward difference Formula. 18. The use of modular arithmetic and large prime numbers to achieve privacy with

RSA Public Key Cryptography. 19. An investigation into the relationship between Pascal's Triangle and the Fibonacci

sequence. 20. Will 'Impact' (a hypothetical comet) collide with Earth?

21. Investigating the patterns and structures in 11n. 22. Balls and their purpose - bouncing balls, markings on balls, e.g. comparing the

dimple packing on golfballs, or how well do basketballs, soccer balls etc bounce? 23. Fractional number bases. An investigation. 24. To what extent can mathematical modelling using differential equations be used in

determining population growth patterns for a predator and its prey? 25. How can cell population be determined over time? Which mathematical model

gives a more accurate approximation to a real experiment? 26. An investigation into Riemann Sums (i.e. standard integration to get areas) and

Numerical Integration.

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27. An investigation into population growth models. 28. Vedic Mathematics: investigating its efficiency and exploring its applications. 29. Euler's method for solving differential equations numerically. 30. Laplace transformations - how are they used in solving second order differential

equations? 31. A statistical investigation on the effect of background music on short term memory

capacity of students. 32. Analytical and geometrical formulations for the parabolic and cubic Bezier curves

(used in computer graphics software). 33. Origami: solving cubic equations 34. Stress prevalence and coping mechanisms among pre-university students. 35. A mathematical study of the effectiveness of two herbs in the treatment of Impetigo skin disease. 36. Theory of probability in casinos. 37. An investigation into the nature of beats and the relative consonance of pure tone dyads. 38. Chaos and the heart 39. Exploring Vedic methods of multiplication. 40. Investigating human body proportions of 5 year olds. 41. The relationship between logical-mathematical intelligence and academic performance. 42. An investigation of the relationship between the coupon rate, yield to maturity and the clean price of a bond. 43. Applying the addition of sine curves to an analysis of the harmony in Chinese and Western music scales. 44. The relationship between students' attitude towards mathematics and their performance in mathematics. 45. How close is the Taylor Series approximation to the original function? 46. What factors affect whether the movement of workers in a construction site reaches 'equilibrium'.

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47. Investigating a model for an optimum lighting system. 48. The effectiveness of an English Tuition Programme towards improving participants' English. 49. A statistical analysis of factors affecting fatal road accidents during the festive season. 50. Comparing age, standard of living and weights of students with their fast food consumption. 51. Does learning a third language have any effect on lower secondary students' short term memory retention? 52. The general health of years 10, 11, 12 students. 53. Misconceptions of beauty: examining myths of the Golden ratio 54. Perfect Numbers 55. What is the volume of different ketupat shapes (rice cooked in coconut leaf strips) 56. Pursuit Curves – Zeno’s Mice Problem 57. The Monty Hall Problem 58. Mandelbrot’s Problem: What is the exact length of the coastline of Great Britain (or substitute your country/island) 59. How does GPS find my present location? 60. The area and perimeter of an ellipse.

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AutographGraphingSoftware

GSIS has an extended site licence for Autograph, which means all students can download and

use the software on their own machines.

To download:

1. Go to Autograph Installer

2. If you have a Mac: Click on ‘Download Single User Mac’ and follow the instructions. It may

take a few minutes for the installer to begin. When prompted, insert the licence ID below:

Student Computer at Home ID for MAC (use MAC installer):

59f0-9189-868c-44f9-9e33-369f-7cf2-fae1

Or if you have a PC: Click on ‘Download Single User PC’ and follow the instructions. It may

take a few minutes for the installer to begin. When prompted, insert the licence ID below:

Student Computer at Home ID for Windows (use single installer):

9974-1b43-6bf1-418e-9014-8d11-282a-b2ea

If the hyperlink in (1) fails, go to:

http://www.autograph-math.com/download/index.shtml

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SuggeSuggeSuggeSuggestions for Mathematical Readingstions for Mathematical Readingstions for Mathematical Readingstions for Mathematical Reading

You might enjoy reading some of these books – some may give you ideas for your Exploration (HL,

SL) or your Project (Studies). You might also find ideas for your Extended Essay.

Chaos: Making a New Chaos: Making a New Chaos: Making a New Chaos: Making a New Science byScience byScience byScience by James James James James GleickGleickGleickGleick

An introduction to chaos theory and fractals.

God Created the Integers, edited by Stephen HawkingGod Created the Integers, edited by Stephen HawkingGod Created the Integers, edited by Stephen HawkingGod Created the Integers, edited by Stephen Hawking

A collection of important works in the history of Mathematics.

The Man who LThe Man who LThe Man who LThe Man who Lovovovoved only Numbers byed only Numbers byed only Numbers byed only Numbers by Paul HoffmanPaul HoffmanPaul HoffmanPaul Hoffman

A biography of the eccentric mathematician Paul Erdös.

Fermat’s Last Theorem byFermat’s Last Theorem byFermat’s Last Theorem byFermat’s Last Theorem by Amir D. AczelAmir D. AczelAmir D. AczelAmir D. Aczel

The story of how Fermat’s last theorem was finally proved by Andrew Wiles.

Uncle PeUncle PeUncle PeUncle Petros and Goldbach’s Conjecture by tros and Goldbach’s Conjecture by tros and Goldbach’s Conjecture by tros and Goldbach’s Conjecture by Apostolos DoxiadisApostolos DoxiadisApostolos DoxiadisApostolos Doxiadis

A novel describing a Greek mathematician’s efforts to prove Goldbach’s Conjecture.

Seventeen Equations that Changed the Seventeen Equations that Changed the Seventeen Equations that Changed the Seventeen Equations that Changed the WorldWorldWorldWorld bybybyby Ian StewartIan StewartIan StewartIan Stewart

From Pythagoras to Einstein and beyond

Gödel, Escher, Bach by Douglas HofstadterGödel, Escher, Bach by Douglas HofstadterGödel, Escher, Bach by Douglas HofstadterGödel, Escher, Bach by Douglas Hofstadter

This combines the mathematical logic of Kurt Gödel, who proved that some questions in arithmetic can never be answered, with the etchings of Maurits Escher and the music of Bach.

The Colossal Book of The Colossal Book of The Colossal Book of The Colossal Book of Mathematics by Martin GardnerMathematics by Martin GardnerMathematics by Martin GardnerMathematics by Martin Gardner

The book Consists of numerous selections from the Scientific American articles by Martin Gardener, classified according to the mathematical area involved.

Euclid in the Rainforest by Joseph MazurEuclid in the Rainforest by Joseph MazurEuclid in the Rainforest by Joseph MazurEuclid in the Rainforest by Joseph Mazur

A readable account of the meaning of truth in mathematics, presented through a series of quirky adventures in the Greek Islands, the jungles around the Orinoco River, and elsewhere.

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Magical Mathematics by Persi Diaconis and Ron GrahamMagical Mathematics by Persi Diaconis and Ron GrahamMagical Mathematics by Persi Diaconis and Ron GrahamMagical Mathematics by Persi Diaconis and Ron Graham

Both authors are top-rank mathematicians with years of stage performances behind them, and

their speciality is mathematical magic. They show how mathematics relates to juggling and reveal the secrets behind some amazing card tricks.

Games of Life by Karl SigmundGames of Life by Karl SigmundGames of Life by Karl SigmundGames of Life by Karl Sigmund

Biologists' understanding of many vital features of the living world, such as sex and survival,

depends on the theory of evolution. One of the basic theoretical tools here is the mathematics of game theory.

The Mathematical Principles of Natural Philosophy by Isaac NewtonThe Mathematical Principles of Natural Philosophy by Isaac NewtonThe Mathematical Principles of Natural Philosophy by Isaac NewtonThe Mathematical Principles of Natural Philosophy by Isaac Newton

Newton’s great work: Nature has laws, and they can be expressed in the language of mathematics.

Using nothing more complicated than Euclid's geometry, Newton developed his laws of motion and gravity.

Why do Buses Come in Threes? By Rob EastawayWhy do Buses Come in Threes? By Rob EastawayWhy do Buses Come in Threes? By Rob EastawayWhy do Buses Come in Threes? By Rob Eastaway

The hidden mathematics of everyday life

Game, Set Game, Set Game, Set Game, Set and Math by Ian Stewartand Math by Ian Stewartand Math by Ian Stewartand Math by Ian Stewart

In this book, the mathematics (including that of tennis) is hidden in a conversation between two

men.

The Code Book by Simon SinghThe Code Book by Simon SinghThe Code Book by Simon SinghThe Code Book by Simon Singh

The story of codes and encryption

The Music of the Primes by Marcus du SautoyThe Music of the Primes by Marcus du SautoyThe Music of the Primes by Marcus du SautoyThe Music of the Primes by Marcus du Sautoy

Prime numbers – their fascination and distribution

If you come across any other books of a mathematical nature that you think should be included

here, please let us know.

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Useful WebsitesUseful WebsitesUseful WebsitesUseful Websites

Go to the Leo page for this course to find hyperlinks.

mmmmyimathsyimathsyimathsyimaths http://www.myimaths.com/http://www.myimaths.com/http://www.myimaths.com/http://www.myimaths.com/

Originally for up to IGCSE, myimaths is extending to include IB HL, SL, Studies. At time of

writing (May 2013) you’d have to find IB resources via the A-level tab, but this will soon change.

Log in is: gsis and password from your Maths teacher who will also have a second level

password for you.

UK Maths Trust UK Maths Trust UK Maths Trust UK Maths Trust http://www.ukmt.org.uk/http://www.ukmt.org.uk/http://www.ukmt.org.uk/http://www.ukmt.org.uk/

You’ll find past papers here for the UK Senior Maths contest, held in November each year.

Nrich Nrich Nrich Nrich http://nrich.maths.orghttp://nrich.maths.orghttp://nrich.maths.orghttp://nrich.maths.org

Have a look at the Upper Secondary level for interesting Maths problems.

Maths Net IB Maths Net IB Maths Net IB Maths Net IB http://www.mathsnetib.com/http://www.mathsnetib.com/http://www.mathsnetib.com/http://www.mathsnetib.com/

Hundreds of pages, resources for all IB topics (all courses), students tend to use this site a lot

for revision. Password from your teacher.

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27

50 Mathematics SL guide

Glossary of command terms

Appendices

Command terms with definitionsStudents 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.

Calculate 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.

Distinguish Make clear the differences between two or more concepts or items.

Draw 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.

Estimate 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.

29

Mathematics SL guide 51

Glossary of command terms

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.

List Give a sequence of brief answers with no explanation.

Plot Mark the position of points on a diagram.

Predict Give an expected result.

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.

30

52 Mathematics SL guide

Appendices

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.

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, { | , 0}x x x

the set of real numbers

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

1 2{ , , ...}x x the set with elements 1 2, , ...x x

( )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

31


Notation list

Intersection

is a proper subset of

is a subset of

A the complement of the set A

|a b a divides b

1/ na , n a a to the power of 1

n, n

th root of a (if 0a then 0n a )

x modulus or absolute value of x , that is for 0,

for 0,

x x x

x x x

is approximately equal to

is greater than

is greater than or equal to

is less than

is less than or equal to

is not greater than

is not less than

nu the nth term of a sequence or series

d the common difference of an arithmetic sequence

r the common ratio of a geometric sequence

nS the sum of the first n terms of a sequence, 1 2 ... nu u u

S the sum to infinity of a sequence, 1 2 ...u u

1

n

i

i

u 1 2 ... nu u u

n

r the r

th binomial coefficient, r = 0, 1, 2, …, in the expansion of ( )na b

:f A B f is a function under which each element of set A has an image in set B

:f x y f is a function under which x is mapped to y

32

Mathematics SL guide54

Notation list

( )f x the image of x under the function f

1f the inverse function of the function f

f g the composite function of f and g

lim ( )x a

f x the limit of ( )f x as x tends to a

d

d

y

x the derivative of y with respect to x

( )f x the derivative of ( )f x with respect to x

2

2

d

d

y

x the second derivative of y with respect to x

( )f x the second derivative of ( )f x with respect to x

d

d

n

n

y

x the nth

derivative of y with respect to x

( ) ( )nf x the nth derivative of ( )f x with respect to x

dy x the indefinite integral of y with respect to x

db

ay x the definite integral of y with respect to x between the limits x a and x b

ex exponential function (base e) of x

loga x logarithm to the base a of x

ln x the natural logarithm of x , elog x

sin, cos, tan the circular functions

A( , )x y the point A in the plane with Cartesian coordinates x and y

[AB] the line segment with end points A and B

AB the length of [AB]

(AB) the line containing points A and B

Â the angle at A

ˆCAB the angle between [CA] and [AB]

33


Notation list

ABC the triangle whose vertices are A , B and C

v the vector v

AB the vector represented in magnitude and direction by the directed line segment

from A to B

a the position vector OA

i, j, k unit vectors in the directions of the Cartesian coordinate axes

a the magnitude of a

| AB| the magnitude of AB

v w the scalar product of v and w

P( )A probability of event A

P( )A probability of the event “not A ”

P( | )A B probability of the event A given the event B

1 2, , ...x x Observations

1 2, , ...f f frequencies with which the observations 1 2, , ...x x occur

n

r number of ways of selecting r items from n items

B( , )n p binomial distribution with parameters n and p

2N( , ) normal distribution with mean and variance 2

~ B( , )X n p the random variable X has a binomial distribution with parameters n and p

2~ N( , )X the random variable X has a normal distribution with mean and

variance 2

population mean

2 population variance

population standard deviation

x mean of a set of data, 1 2 3, , ,...x x x

34

Mathematics SL guide56

Notation list

z standardized normal random variable, x

z

cumulative distribution function of the standardized normal variable with

distribution N(0, 1)

r Pearson’s product–moment correlation coefficient

35

© International Baccalaureate Organization 2012 5045

Mathematics SL formula booklet

For use during the course and in the examinations

First examinations 2014

Edited in 2015 (version 2)

Diploma Programme

37

Contents

Prior learning 2 Topics 3

Topic 1—Algebra 3

Topic 2—Functions and equations 4

Topic 3—Circular functions and trigonometry 4

Topic 4—Vectors 5

Topic 5—Statistics and probability 5

Topic 6—Calculus 6

38

Formulae

Prior learning

Area of a parallelogram A b h= ×

Area of a triangle 1 ( )2

= ×A b h

Area of a trapezium 1 ( )2

= +A a b h

Area of a circle 2= πA r

Circumference of a circle 2= πC r

Volume of a pyramid 13

area of base vertical height( )= ×V

Volume of a cuboid (rectangular prism) = × ×V l w h

Volume of a cylinder 2= πV r h

Area of the curved surface of a cylinder 2= πA rh

Volume of a sphere 343

= πV r

Volume of a cone 213

= πV r h

Distance between two points 1 1 1( , , )x y z and

2 2 2( , , )x y z

2 2 21 2 1 2 1 2( ) ( ) ( )= − + − + −d x x y y z z

Coordinates of the midpoint of a line segment with endpoints 1 1 1( , , )x y z and 2 2 2( , , )x y z

1 2 1 2 1 2, , 2 2 2+ + +

x x y y z z

39

Topics

Topic 1—Algebra 1.1 The nth term of an

arithmetic sequence 1 ( 1)= + −nu u n d

The sum of n terms of an arithmetic sequence ( )1 12 ( 1) ( )

2 2= + − = +n n

n nS u n d u u

The nth term of a geometric sequence

11

−= nnu u r

The sum of n terms of a finite geometric sequence

1 1( 1) (1 )1 1− −

= =− −

n n

nu r u rS

r r, 1≠r

The sum of an infinite geometric sequence

1

1uS

r∞ = −, 1<r

1.2 Exponents and logarithms logxaa b x b= ⇔ =

Laws of logarithms log log logc c ca b ab+ =

log log logc c caa bb

− =

log logrc ca r a=

Change of base logloglog

cb

c

aab

=

1.3 Binomial coefficient !

!( )!

= −

n nr r n r

Binomial theorem 1( )1

− − + = + + + + +

n n n n r r nn na b a a b a b b

r

40

Topic 2—Functions and equations 2.4 Axis of symmetry of graph

of a quadratic function 2( )

2axis of symmetry = + + ⇒ = −

bf x ax bx c xa

2.6 Relationships between logarithmic and exponential functions

lnex x aa = loglog a xx

a a x a= =

2.7 Solutions of a quadratic equation

22 40 , 0

2b b acax bx c x a

a− ± −

+ + = ⇒ = ≠

Discriminant 2 4b ac∆ = −

Topic 3—Circular functions and trigonometry 3.1 Length of an arc l rθ=

Area of a sector 212

A rθ=

3.2 Trigonometric identity sintan

cosθθθ

=

3.3 Pythagorean identity 2 2sin 1cos θ θ+ =

Double angle formulae 2sinsin 2 cosθ θ θ= 2 2 2 2cos sin 2cos 1 1 2c s io s2 nθ θ θ θ θ= − = − = −

3.6 Cosine rule 2 2 2 2 cosc a b ab C= + − ;

2 2 2

cos2

a b cCab

+ −=

Sine rule sin sin sin

a b cA B C= =

Area of a triangle 1 sin2

A ab C=

41

Topic 4—Vectors 4.1 Magnitude of a vector 2 2 2

1 2 3v v v= + +v

4.2 Scalar product cosθ⋅ =v w v w

1 1 2 2 3 3⋅ = + +v w v w v wv w

Angle between two vectors

cosθ ⋅=

v wv w

4.3 Vector equation of a line = + tr a b

Topic 5—Statistics and probability 5.2 Mean of a set of data

1

1

n

i ii

n

ii

f xx

f

=

=

=∑

∑

5.5 Probability of an event A

( )P( )( )

=n AAn U

Complementary events P( ) P( ) 1′+ =A A

5.6 Combined events P( ) P( ) P( ) P( )∪ = + − ∩A B A B A B

Mutually exclusive events P( ) P( ) P( )∪ = +A B A B

Conditional probability P( ) P( ) P( | )A B A B A∩ =

Independent events P( ) P( ) P( )∩ =A B A B

5.7 Expected value of a discrete random variable X

E( ) P( )µ= = =∑x

X x X x

5.8 Binomial distribution ~ B( , ) P( ) (1 ) , 0,1, ,r n rn

X n p X r p p r nr

− ⇒ = = − =

Mean E( ) =X np

Variance Var ( ) (1 )= −X np p

5.9 Standardized normal variable

µσ−

=xz

42

Topic 6—Calculus 6.1

Derivative of ( )f x 0

d ( ) ( )( ) ( ) limd h

y f x h f xy f x f xx h→

+ − ′= ⇒ = =

6.2 Derivative of nx 1( ) ( )n nf x x f x nx −′= ⇒ =

Derivative of sin x ( ) sin ( ) cosf x x f x x′= ⇒ =

Derivative of cos x ( ) cos ( ) sinf x x f x x′= ⇒ = −

Derivative of tan x 2

1( ) tan ( )cos

f x x f xx

′= ⇒ =

Derivative of ex ( ) e ( ) ex xf x f x′= ⇒ =

Derivative of ln x 1( ) ln ( )f x x f xx

′= ⇒ =

Chain rule ( )=y g u , d d d( )d d dy y uu f xx u x

= ⇒ = ×

Product rule d d dd d dy v uy uv u vx x x

= ⇒ = +

Quotient rule 2

d dd d dd

u vv uu y x xyv x v

−= ⇒ =

6.4 Standard integrals 1

d , 11

nn xx x C n

n

+

= + ≠ −+∫

1 d ln , 0x x C xx

= + >∫

sin d cosx x x C= − +∫

cos d sinx x x C= +∫

e d e= +∫ x xx C

6.5 Area under a curve between x = a and x = b d

b

aA y x= ∫

Volume of revolution about the x-axis from x = a to x = b

2π db

aV y x= ∫

6.6 Total distance travelled from 1t to 2t

distance 2

1

( ) dt

tv t t= ∫

43

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