GSIS IB MathematicsGSIS IB MathematicsGSIS IB MathematicsGSIS IB Mathematics
Student CompanionStudent CompanionStudent CompanionStudent Companion
Mathematical Studies SMathematical Studies SMathematical Studies SMathematical Studies SLLLL
Contents:
1 Course Introduction
2 Maths Learner Profile
3 Syllabus Outline
4 Assessment Schedule
5 Grade Boundaries
6 Project Suggestions
7 Project Criteria
8 Extended Essay suggestions
9 Autograph Download Instructions
10 Books for Further Reading
11 Suggested Websites
12 Command Terms and Notation
13 Studies Formula Booklet for use in exams and tests
NAME_____________________________________________
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IB StandardIB StandardIB StandardIB Standard Level MathsLevel MathsLevel MathsLevel Maths StudiesStudiesStudiesStudies
Welcome to SL SL SL SL MathsMathsMathsMaths StudiesStudiesStudiesStudies, your Group 5 choice for the IB Diploma.
Standard level Mathematical Studies is a very good subject which contains all the Maths you
will need for students who do not see Mathematics as a major part of their career or university.
We ask only for success at IGCSE Mathematics – and an open mindopen mindopen mindopen mind. If you have studied
Mathematics beyond IGCSE or if you did really well at IGCSE you may find you have a head start
in this course and might even consider a change to SL Maths. 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 Maths Studies page. LEO Maths Studies page. LEO Maths Studies page. LEO Maths Studies page. We plan to have teststeststeststests fairly regularly, perhaps after every topic. For
these, and in fact for the whole course, 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 with you for testsbe sure to bring this guide with you for testsbe sure to bring this guide with you for testsbe sure to bring this guide with you for tests....
The course text book is Mathematical StudiesMathematical StudiesMathematical StudiesMathematical Studies StandardStandardStandardStandard Level (Oxford UniverLevel (Oxford UniverLevel (Oxford UniverLevel (Oxford University Press, ISBN sity Press, ISBN sity Press, ISBN sity Press, ISBN
978 019 839013 8978 019 839013 8978 019 839013 8978 019 839013 8)))) 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 SSSStudiestudiestudiestudies assessment scheduleassessment scheduleassessment scheduleassessment 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 a projectprojectprojectproject 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 a project will be introduced early in the course. It
is usual to do a Statistics project and some students will want to collect data and get started
during the Summer – 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.
How this happens Mathematically
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.
How this happens Mathematically
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.
How this happens Mathematically
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.
How this happens Mathematically
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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How this happens Mathematically
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.
How this happens Mathematically
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.
How this happens Mathematically
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.
How this happens Mathematically
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.
How this happens Mathematically
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
Number and algebra
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Topic 2
Descriptive statistics
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Topic 3
Logic, sets and probability
20
Topic 4
Statistical applications
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Topic 5
Geometry and trigonometry
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Topic 6
Mathematical models
20
Topic 7
Introduction to differential calculus
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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.
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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.
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Assessment
Assessment outline
First examinations 2014
Assessment component Weighting
External assessment (3 hours)
Paper 1 (1 hour 30 minutes)
15 compulsory short-response questions based on the whole syllabus. (90 marks)
80%
40%
Paper 2 (1 hour 30 minutes)
6 compulsory extended-response questions based on the whole syllabus. (90 marks)
40%
Internal assessment
This component is internally assessed by the teacher and externally moderated by the IB at
the end of the course.
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. (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 6 65% or above
Grade 5 55% or above
Grade 4 45% or above
Grade 3 35% or above
Grade 2 20% 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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ProjectProjectProjectProject PlanningPlanningPlanningPlanning
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 mathematical modelling, an investigation, an application of mathematics but
most people choose to do a Statistics project. Get data on whatever is most interesting to you.
Possible areas for a project include the following, but you are certainly not restricted to these.
sport - get data from your favourite sport music food games population health peoples measurements – test out hypotheses about difference in ages or male/female Supermarkets fast food 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
Criterion A: IntroductionIn this context, the word “task” is defined as “what the student is going to do”; the word “plan” is defined as
“how the student is going to do it”. A statement of the task should appear at the beginning of each project. It is
expected that each project has a clear title.
Achievement level Descriptor
0 The project does not contain a clear statement of the task.
There is no evidence in the project of any statement of what the student is going
to do or has done.
1 The project contains a clear statement of the task.
For this level to be achieved, the task should be stated explicitly.
2 The project contains a title, a clear statement of the task and a description of the
plan.
The plan need not be highly detailed, but must describe how the task will be
performed. If the project does not have a title, this achievement level cannot be
awarded.
3 The project contains a title, a clear statement of the task and a detailed plan that
is followed.
The plan should specify what techniques are to be used at each stage and the
purpose behind them, thus lending focus to the task.
Criterion B: Information/measurementIn this context, generated measurements include those that have been generated by computer, by observation,
by prediction from a mathematical model or by experiment. Mathematical information includes geometrical
figures and data that is collected empirically or assembled from outside sources. This list is not exhaustive
and mathematical information does not solely imply data for statistical analysis. If a questionnaire or survey is
used then a copy of this along with the raw data must be included.
Achievement level Descriptor
0 The project does not contain any relevant information collected or relevant
measurements generated.
No attempt has been made to collect any relevant information or to generate any
relevant measurements.
1 The project contains relevant information collected or relevant generated
measurements.
This achievement level can be awarded even if a fundamental flaw exists in the
instrument used to collect the information, for example, a faulty questionnaire or
an interview conducted in an invalid way.
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Internal assessment
Achievement level Descriptor
2 The relevant information collected, or set of measurements generated, is
organized in a form appropriate for analysis or is sufficient in both quality and
quantity.
A satisfactory attempt has been made to structure the information/measurements
ready for the process of analysis, or the information/measurement collection
process has been thoroughly described and the quantity of information justified.
The raw data must be included for this achievement level to be awarded.
3 The relevant information collected, or set of measurements generated, is organized in
a form appropriate for analysis and is sufficient in both quality and quantity.
The information/measurements have been properly structured ready for analysis
and the information/measurement collection process has been thoroughly
described and the quantity of information justified. If the information/
measurements are too sparse or too simple, this achievement level cannot be
awarded. If the information/measurements are from a secondary source, then
there must be evidence of sampling if appropriate. All sampling processes should
be completely described.
Criterion C: Mathematical processesWhen presenting diagrams, students are expected to use rulers where necessary and not merely sketch. A
freehand sketch would not be considered a correct mathematical process. When technology is used, the student
would be expected to show a clear understanding of the mathematical processes used. All graphs must contain
all relevant information. The teacher is responsible for determining the accuracy of the mathematics used and
must indicate any errors on the final project. If a project contains no simple mathematical processes, then the
first two further processes are assessed as simple.
Achievement level Descriptor
0 The project does not contain any mathematical processes.
For example, where the processes have been copied from a book, with no attempt
being made to use any collected/generated information.
Projects consisting of only historical accounts will achieve this level.
1 At least two simple mathematical processes have been carried out.
Simple processes are considered to be those that a mathematical studies SL student
could carry out easily, for example, percentages, areas of plane shapes, graphs,
trigonometry, bar charts, pie charts, mean and standard deviation, substitution into
formulae and any calculations and/or graphs using technology only.
2 At least two simple mathematical processes have been carried out correctly.
A small number of isolated mistakes should not disqualify a student from
achieving this level. If there is incorrect use of formulae, or consistent mistakes
in using data, this level cannot be awarded.
3 At least two simple mathematical processes have been carried out correctly. All
processes used are relevant.
The simple mathematical processes must be relevant to the stated aim of the project.16
Internal assessment
Achievement level Descriptor
4 The simple relevant mathematical processes have been carried out correctly. In
addition, at least one relevant further process has been carried out.
Examples of further processes are differential calculus, mathematical modelling,
optimization, analysis of exponential functions, statistical tests and distributions,
compound probability. For this level to be achieved, it is not required that the
calculations of the further process be without error. At least one further process
must be calculated, showing full working.
5 The simple relevant mathematical processes have been carried out correctly. In
addition, at least one relevant further process has been carried out.
All processes, both simple and further, that have been carried out are without error.
If the measurements, information or data are limited in scope, then this
achievement level cannot be awarded.
Criterion D: Interpretation of resultsUse of the terms “interpretation” and “conclusion” refer very specifically to statements about what the
mathematics used tells us after it has been used to process the original information or data. Discussion of
limitations and validity of the processes is assessed elsewhere.
Achievement level Descriptor
0 The project does not contain any interpretations or conclusions.
For the student to be awarded this level, there must be no evidence of
interpretation or conclusions anywhere in the project, or a completely false
interpretation is given without reference to any of the results obtained.
1 The project contains at least one interpretation or conclusion.
Only minimal evidence of interpretations or conclusions is required for this
level. This level can be achieved by recognizing the need to interpret the results
and attempting to do so, but reaching only false or contradictory conclusions.
2 The project contains interpretations and/or conclusions that are consistent with
the mathematical processes used.
A “ follow through” procedure should be used and, consequently, it is irrelevant
here whether the processes are either correct or appropriate; the only
requirement is consistency.
3 The project contains a meaningful discussion of interpretations and conclusions
that are consistent with the mathematical processes used.
To achieve this level, the student would be expected to produce a discussion
of the results obtained and the conclusions drawn based on the level of
understanding reasonably to be expected from a student of mathematical studies
SL. This may lead to a discussion of underlying reasons for results obtained.
If the project is a very simple one, with few opportunities for substantial
interpretation, this achievement level cannot be awarded.
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Internal assessment
Criterion E: ValidityValidity addresses whether appropriate techniques were used to collect information, whether appropriate
mathematics was used to deal with this information, and whether the mathematics used has any limitations
in its applicability within the project. Any limitations or qualifications of the conclusions and interpretations
should also be judged within this criterion. The considerations here are independent of whether the particular
interpretations and conclusions reached are correct or adequate.
Achievement level Descriptor
0 There is no awareness shown that validity plays a part in the project.
1 There is an indication, with reasons, if and where validity plays a part in the project.
There is discussion of the validity of the techniques used or recognition of any
limitations that might apply. A simple statement such as “I should have used
more information/measurements” is not sufficient to achieve this level. If the
student considers that validity is not an issue, this must be fully justified.
Criterion F: Structure and communicationThe term “structure” should be taken primarily as referring to the organization of the information, calculations
and interpretations in such a way as to present the project as a logical sequence of thought and activities
starting with the task and the plan, and finishing with the conclusions and limitations.
Communication is not enhanced by a large number of repetitive procedures. All graphs must be fully labelled
and have an appropriate scale.
It is not expected that spelling, grammar and syntax are perfect, and these features are not judged in assigning
a level for this criterion. Nevertheless, teachers are strongly encouraged to correct and assist students with the
linguistic aspects of their work. Projects that are very poor linguistically are less likely to excel in the areas
that are important in this criterion. Projects that do not reflect the significant time commitment required will
not score highly on this assessment criterion.
Achievement level Descriptor
0 No attempt has been made to structure the project.
It is not expected that many students will be awarded this level.
1 Some attempt has been made to structure the project.
Partially complete and very simple projects would only achieve this level.
2 The project has been structured in a logical manner so that it is easily followed.
There must be a logical development to the project. The project must reflect the
appropriate commitment for this achievement level to be awarded.
3 The project has been well structured in accordance with the stated plan and is
communicated in a coherent manner.
To achieve this level, the project would be expected to read well, and contain
footnotes and a bibliography, as appropriate. The project must be focused and
contain only relevant discussions.
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Internal assessment
Criterion G: Notation and terminologyThis criterion refers to the use of correct terminology and mathematical notation. The use of calculator or
spreadsheet notation is not acceptable.
Achievement level Descriptor
0 The project does not contain correct mathematical notation or terminology.
It is not expected that many students will be awarded this level.
1 The project contains some correct mathematical notation or terminology.
2 The project contains correct mathematical notation and terminology throughout.
Variables should be explicitly defined. An isolated slip in notation need not
preclude a student from achieving this level. If it is a simple project requiring
little or no notation and/or terminology, this achievement level cannot be
awarded.
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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.
Nation Master Nation Master Nation Master Nation Master http://www.nationmaster.com/http://www.nationmaster.com/http://www.nationmaster.com/http://www.nationmaster.com/
Could be useful for Statistics projects, lots of data on all sorts of variables for all countries.
If you find any other websites of a mathematical nature that you find useful and think should be
included here, please let us know.
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50 Mathematical studies 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 frequently 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.
Construct Display information in a diagrammatic or logical form.
Deduce Reach a conclusion from the information given.
Describe Give a detailed account.
Determine Obtain the only possible answer.
Differentiate Obtain the derivative of a function.
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.
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.
Interpret Use knowledge and understanding to recognize trends and draw conclusions
from given information.
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.
Show Give the steps in a calculation or derivation.
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Mathematical studies SL guide 51
Glossary of command terms
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.
Verify Provide evidence that validates the result.
Write down Obtain the answer(s), usually by extracting information. Little or no calculation
is required. Working does not need to be shown.
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52 Mathematical studies SL guide
Notation list
Appendices
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
is an element of
is not an element of
the empty (null) set
U the universal set
union
intersection
is a proper subset of
is a subset of
A the complement of the set A
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Mathematical studies SL guide 53
Notation list
p q conjunction: p and q
p q disjunction: p or q (or both)
p q exclusive disjunction: p or q (not both)
p negation: not p
p q implication: if p then q
p q implication: if q then p
p q equivalence: p is equivalent to q
1/ , n na a a to the power 1
n, n
th root of a (if 0a then 0n a )
1n
na
a a to the power n , reciprocal of na
1/ 2 , a a a to the power 1
2, square root of a (if 0a then 0a )
x the 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 thn term of a sequence
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
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Mathematical studies SL guide54
Notation list
1
n
i
i
u 1 2 ... nu u u
( )f x the image of x under the function f
d
d
y
x the derivative of y with respect to x
( )f x the derivative of ( )f x with respect to x
sin, cos, tan the circular functions
A( , )x y the point A in the plane with Cartesian coordinates x and y
 the angle at A
ˆCAB the angle between the lines CA and AB
ABC the triangle whose vertices are A, B and C
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
x mean of a set of data
population mean
population standard deviation
2N( , ) normal distribution with mean and variance 2
2N( , )X random variable X has a normal distribution with mean and variance 2
r Pearson’s product–moment correlation coefficient
2 chi-squared
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36
© International Baccalaureate Organization 2012 5042
Mathematical studies 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—Number and algebra 3
Topic 2—Descriptive statistics 3
Topic 3—Logic, sets and probability 4
Topic 5—Geometry and trigonometry 5
Topic 6—Mathematical models 6
Topic 7—Introduction to differential calculus 6
38
Prior learning
5.0 Area of a parallelogram A b h= × , where b is the base, h is the height
Area of a triangle 1 ( )2
A b h= × , where b is the base, h is the
height
Area of a trapezium 1 ( )2
= +A a b h , where a and b are the
parallel sides, h is the height
Area of a circle 2= πA r , where r is the radius
Circumference of a circle 2= πC r , where r is the radius
Distance between two points 1 1( , )x y and
2 2( , )x y 2 2
1 2 1 2( ) ( )= − + −d x x y y
Coordinates of the midpoint of a line segment with endpoints 1 1( , )x y and 2 2( , )x y
1 2 1 2, 2 2+ +
x x y y
39
Topics
Topic 1—Number and algebra 1.2 Percentage error
A E
E
100%ε =−
×v v
v, where Ev is the exact value and Av is the
approximate value of v
1.7 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
1.8 The nth term of a geometric sequence
11
−= nnu u r
The sum of n terms of a geometric sequence
1 1( 1) (1 )1 1
n n
nu r u rS
r r− −
= =− −
, 1≠r
1.9 Compound interest 1
100
k nrFV PVk
= × +
, where FV = future value, PV = present
value, n = number of years, k = number of compounding periods per year, r% = nominal annual rate of interest
Topic 2—Descriptive statistics 2.5 Mean of a set of data
1
k
i ii
f xx
n==∑
, where 1
=
=∑k
ii
n f
2.6 Interquartile range 3 1IQR Q Q= −
40
Topic 3—Logic, sets and probability 3.3 Truth tables p q p¬ p q∧ p q∨ p q∨ p q⇒ p q⇔
T T F T T F T T
T F F F T T F F
F T T F T T T F
F F T F F F T T
3.6 Probability of an event A P( ) =A
Complementary events P( ) 1 P( )′ = −A A
3.7 Combined events P( ) P( ) P( ) P( )∪ = + − ∩A B A B A B
Mutually exclusive events P( ) 0∩ =A B
Independent events P( ) P( ) P( )∩ =A B A B
Conditional probability P( )P( | )
P( )∩
=A BA B
B
number of outcomes in A total number of outcomes
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Topic 5—Geometry and trigonometry 5.1 Equation of a straight line ; 0= + + + =y mx c ax by d
Gradient formula 2 1
2 1
−=
−y ymx x
5.3 Sine rule
sin sin sin= =
a b cA B C
Cosine rule 2 2 2
2 2 2 2 cos ; cos2+ −
= + − =b c aa b c bc A A
bc
Area of a triangle 1
sin2
= ab CA , where a and b are adjacent sides, C is the
included angle
5.5 Area of the curved surface of a cylinder
2= πA rh , where r is the radius, h is the height
Surface area of a sphere 24π=A r , where r is the radius
Area of the curved surface of a cone
= πA rl , where r is the radius, l is the slant height
Volume of a pyramid 13
=V Ah , where A is the area of the base, h is the vertical height
Volume of a cuboid = × ×V l w h , where l is the length, w is the width, h is the height
Volume of a cylinder 2= πV r h , where r is the radius, h is the height
Volume of a sphere 343
= πV r , where r is the radius
Volume of a cone 213
= πV r h , where r is the radius, h is the vertical height
Volume of a prism =V Ah , where A is the area of cross-section, h is the height
42
Topic 6—Mathematical models 6.3 Equation of the axis of
symmetry for the graph of the quadratic function
2y ax bx c= + +
2= −
bxa
Topic 7—Introduction to differential calculus 7.2 Derivative of nax 1( ) ( )n nf x ax f x nax −′= ⇒ =
Derivative of a sum 1 1( ) , ( ) ( ) ( )n m n mf x ax g x bx f x g x nax mbx− −′ ′= = ⇒ + = +
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