DIPLOMA PROGRAMME Mathematical studies standard level · •Their own abilities in mathematics and...

INTERNATIONAL BACCALAUREATE ORGANIZATION

DIPLOMA PROGRAMME

Mathematical studies standard level

For first examinations in 2001

Mathematical Studies Standard LevelFirst published September 1997Second edition, September 2001

© International Baccalaureate Organization 1998, 2001

International Baccalaureate OrganizationRoute des Morillons 151218 Grand-Saconnex

Geneva, SWITZERLAND

CONTENTS

INTRODUCTION 1

NATURE OF THE SUBJECT 3

AIMS 5

OBJECTIVES 6

SYLLABUS OUTLINE 7

SYLLABUS DETAILS 8

SYLLABUS GUIDELINES 28

ASSESSMENT OUTLINE 31

ASSESSMENT DETAILS 32

ASSESSMENT CRITERIA 39

ASSESSMENT GUIDELINES 47

INTRODUCTION

The International Baccalaureate Diploma Programme is a rigorous pre-university course of studies,leading to examinations, that meets the needs of highly motivated secondary school students betweenthe ages of 16 and 19 years. Designed as a comprehensive two-year curriculum that allows itsgraduates to fulfil requirements of various national education systems, the Diploma Programmemodel is based on the pattern of no single country but incorporates the best elements of many. Theprogramme is available in English, French and Spanish.

The curriculum is displayed in the shape of a hexagon with six academic areas surrounding the core.Subjects are studied concurrently and students are exposed to the two great traditions of learning: thehumanities and the sciences.

IB Diploma Programme guide: Mathematical Studies SL, September 2001 1

Diploma Programme candidates are required to select one subject from each of the six subjectgroups. At least three and not more than four are taken at higher level (HL), the others at standardlevel (SL). Higher level courses represent 240 teaching hours; SL courses cover 150 hours. Byarranging work in this fashion, students are able to explore some subjects in depth and some morebroadly over the two-year period; this is a deliberate compromise between the early specializationpreferred in some national systems and the breadth found in others.

Distribution requirements ensure that the science-orientated student is challenged to learn a foreignlanguage and that the natural linguist becomes familiar with science laboratory procedures. Whileoverall balance is maintained, flexibility in choosing higher level concentrations allows the student topursue areas of personal interest and to meet special requirements for university entrance.

Successful Diploma Programme candidates meet three requirements in addition to the six subjects.The interdisciplinary Theory of Knowledge (TOK) course is designed to develop a coherent approachto learning which transcends and unifies the academic areas and encourages appreciation of othercultural perspectives. The extended essay of some 4000 words offers the opportunity to investigate atopic of special interest and acquaints students with the independent research and writing skillsexpected at university. Participation in the Creativity, Action, Service (CAS) requirement encouragesstudents to be involved in sports, artistic pursuits and community service work.

INTRODUCTION

2 IB Diploma Programme guide: Mathematical Studies SL, September 2001


NATURE OF THE SUBJECT

Introduction

The nature of mathematics can be summarized in a number of ways; for example, as awell-defined body of knowledge, as an abstract system of ideas or as a useful tool. For manypeople it is probably a combination of these, but there is no doubt that mathematicalknowledge provides an important key to understanding the world in which we live.Mathematics can enter our lives in a number of ways: buying produce in the market,consulting a timetable, reading a newspaper, timing a process or estimating a length. Formost people mathematics also extends into their chosen profession: artists need to learn aboutperspective; musicians need to appreciate the mathematical relationships within and betweendifferent rhythms; economists need to recognize trends in financial dealings; and engineersneed to take account of stress patterns. Scientists view mathematics as a language that is vitalto our understanding of events that occur in the natural world. Other people are challenged bythe logical methods of mathematics and the adventure in reason that mathematical proof hasto offer. Still others appreciate mathematics as an aesthetic experience or even as acornerstone of philosophy. The prevalence of mathematics in people’s lives thus provides aclear and sufficient rationale for making the study of this subject compulsory within the IBdiploma.

Since individual students have different needs, interests and abilities, the InternationalBaccalaureate Organization (IBO) offers a number of different courses in mathematics. Theseare targeted at students who wish to study mathematics in depth, either as a subject in its ownright or in order to pursue their interests in areas related to mathematics, those who wish togain a degree of understanding and competence in order to understand better their approachto other subjects, and those who may not be aware that mathematics has relevance in theirstudies and in their future lives. Each course is designed to meet the needs of a particulargroup of students and therefore great care should be exercised in selecting the one which ismost appropriate for an individual student.

In making the selection, individual students should be advised to take account of thefollowing considerations.

• Their own abilities in mathematics and the type of mathematics in which they can besuccessful.

• Their own interest in mathematics with respect to the areas which hold an appeal.

• Their other choices of subjects within the framework of the Diploma Programme.


• Their future academic plans in terms of the subjects they wish to study.

• Their choice of career.

Teachers are expected to assist with the selection process and to offer advice to students onchoosing the most appropriate subject from group 5.

Mathematical studies standard levelMathematical studies, available as a standard level (SL) subject only, caters for students withvaried backgrounds and abilities. More specifically it is designed to build confidence andencourage an appreciation of mathematics in students who do not anticipate a need formathematics in their future studies. Students embarking on this course need to be equippedwith fundamental skills and a rudimentary knowledge of basic processes.

The nature of mathematical studies is such that it concentrates on mathematics which can beapplied to contexts related as far as possible to other curriculum subjects, to common generalworld occurrences and to topics that relate to home, work and leisure situations. Theprogramme includes a feature unique within group 5, the project: a piece of written workbased on personal research, guided and supervised by the teacher. It provides an opportunityfor the student to undertake an investigation of a mathematical nature in the context ofanother subject in the curriculum, a hobby or interest of his/her choice using skills learnedbefore and during the mathematical studies course. This process allows students to ask theirown questions about mathematics and to acquire ownership of a part of the programme.

The population of students most likely to select this subject are those whose main interests lieoutside the field of mathematics, and for many mathematical studies students this will betheir last formal mathematics course. All parts of the syllabus have been carefully selected toensure that an approach from first principles can be used. As a consequence, students are ableto use their own inherent, logical thinking skills and do not have to rely on standardalgorithms and remembered formulae. Students likely to need mathematics for the pursuit offurther qualifications would be advised to consider an alternative subject from group 5.

Because of the nature of mathematical studies, teachers may find that traditional methods ofteaching are inappropriate for this course and that less formal, shared learning techniques canbe more stimulating and rewarding for students. Lessons which use an enquiry approachstarting with practical investigations, where possible, followed by analysis of results leadingto the understanding of a mathematical principle and its formulation into mathematicallanguage are often most successful and can engage the interest of the students. Furthermore,this approach is likely to assist students in their understanding of mathematics by providing ameaningful context and by leading students to understand better how to structure work fortheir own individual project.

NATURE OF THE SUBJECT


AIMS

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

• appreciate the international dimensions of mathematics and the multiplicity of itscultural and historical perspectives

• foster enjoyment from engaging in mathematical pursuits, and to develop anappreciation of the beauty, power and usefulness of mathematics

• develop logical, critical and creative thinking in mathematics

• develop mathematical knowledge, concepts and principles

• employ and refine the powers of abstraction and generalization

• develop patience and persistence in problem-solving

• have an enhanced awareness of, and utilize the potential of, technologicaldevelopments in a variety of mathematical contexts

• communicate mathematically, both clearly and confidently, in a variety of contexts.


OBJECTIVES

Having followed any one of the courses in group 5, candidates will be expected to:

• know and use mathematical concepts and principles

• read and interpret a given problem in appropriate mathematical terms

• organize and present information/data in tabular, graphical and/or diagrammaticforms

• know and use appropriate notation and terminology

• formulate a mathematical argument and communicate it clearly

• select and use appropriate mathematical techniques

• understand the significance and reasonableness of results

• recognize patterns and structures in a variety of situations and draw inductivegeneralizations

• demonstrate an understanding of, and competence in, the practical applications of mathematics

• use appropriate technological devices as mathematical tools.


SYLLABUS OUTLINE

The mathematical studies standard level (SL) syllabus consists of the study of six core topics and oneoption.

Part I: Core 100 hours

All topics in the core are compulsory. Candidates are required to study all the sub-topics in each ofthe six topics in this part of the syllabus as listed in the Syllabus Details.

1 Number and algebra 15 hours

2 Sets and logic 12 hours

3 Geometry and trigonometry 15 hours

4 Statistics and probability 25 hours

5 Functions 20 hours

6 Financial mathematics 13 hours

Part II: Options 25 hours

Candidates are required to study all the sub-topics in one of the following options as listed in theSyllabus Details.

7 Matrices and graph theory 25 hours

8 Further statistics and probability 25 hours

9 Introductory differential calculus 25 hours

Project 25 hours

An individual piece of work involving the collection and/or generation of data, and the analysis andevaluation of that data.


SYLLABUS DETAILS

Format of the syllabus

The syllabus is formatted into three columns labelled Content, Amplifications/Exclusionsand Teaching Notes.

• Content: the first column lists, under each topic, the sub-topics to be covered.

• Amplifications/Exclusions: the second column contains more explicit information onspecific sub-topics listed in the first column. This helps to define what is required andwhat is not required in terms of preparing for the examination.

• Teaching Notes: the third column provides useful suggestions for teachers. It is notmandatory that these suggestions be followed.

Course of studyTeachers are required to teach all the sub-topics listed under the six topics in the coretogether with all the sub-topics in the chosen option.

It is not necessary, nor desirable, to teach the topics in the core in the order in which theyappear in the Syllabus Outline and Syllabus Details. Neither is it necessary to teach all thetopics in the core before starting to teach an option. Teachers are therefore strongly advisedto draw up a course of study, tailored to the needs of their students, which integrates the areascovered by both the core and the chosen option.

Integration of project workWork relating to the project should be fully integrated into the course of study. Full detailsare given in Assessment Details, Project.


Time allocationThe recommended teaching time for a standard level subject is 150 hours. For mathematicalstudies SL, it is expected that 25 hours will be spent on work for the project. The timeallocations given in the Syllabus Outline and Syllabus Details are approximate, and areintended to suggest how the remaining 125 hours allowed for teaching the syllabus might beallocated. However, the exact time spent on each topic will depend on a number of factors,including the background knowledge and level of preparedness of each student. Teachersshould therefore adjust these timings to correspond with the needs of their students.

Use of calculatorsCandidates are expected to have access to a calculator, which may or may not have a graphicdisplay facility, at all times during the course. Regulations concerning the types of calculatorsallowed are provided in the Vade Mecum.

Formulae booklet and statistical tables (third edition,February 2001)

As each candidate is required to have access to clean copies of the IBO formulae booklet andstatistical tables during the examination, it is recommended that teachers ensure candidatesare familiar with the contents of these documents from the beginning of the course. Thebooklet and tables are provided by IBCA and are published separately.

Resource listA resource list is available for mathematical studies SL on the online curriculum centre. Thislist provides details of, for example, texts, software packages and videos which areconsidered by teachers to be appropriate for use with this course. It will be updated on aregular basis.

Teachers can at any time add any materials to this list which they consider to be appropriatefor candidate use or as reference material for teachers.

SYLLABUS DETAILS


1 Core: number and algebra Teaching time: 15 hours

The aim of this section is to introduce candidates to some basic elements and concepts of mathematics. A clear understanding of these is essential for furtherwork in the programme.

The formulae can be verified using numericalexamples.

Included: simple interest as an application.1.7 Arithmetic sequences and series and theirapplications.Use of the formulae for the nth term and thesum of the first n terms.

Link with the form of the notation in §1.4, eg .5 5 106 km mm= ×

1.6 Système International (SI) and other basicunits of measurement: gramme (g), metre(m), second (s), litre (l), metre per second (m s-1), etc.

Included: the ability to recognize whether theresults of calculations are reasonable.

1.5 Estimation.

Work should include examples on very large andvery small numbers in scientific, economic and otherapplications.

1.4 Expressing numbers in the form a k×10where and .1 10≤ <a k ∈ Z

Operations with numbers expressed in theform where and .a k×10 1 10≤ <a k ∈ Z

Link with §5.7.1.3 The exponential expression .a bb , ∈ Q

Included: an awareness of the errors which canresult from premature rounding.

1.2 Approximation: decimal places; significantfigures.

Not required: proof of irrationality, eg of .21.1 The sets of natural numbers, N, integers, Z,rational numbers, Q, and real numbers, R.

TEACHING NOTESAMPLIFICATIONS/EXCLUSIONSCONTENT

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1 Core: number and algebra (continued)

Link with quadratic functions in §5.5.Not required: knowledge of the quadraticformula and its application to solving quadraticequations.

1.10 Solutions of quadratic equations: by factorizing; by graphing.

Link with linear inequalities in §5.4.Note: on the number line an unshaded circle, ,!should be used to represent < and >; a shadedcircle, , should be used to represent and • ≤ ≥ .

1.9 Inequalities: solution of inequalities in one variable; representation on the real number line.

The formulae can be verified using numericalexamples.

Included: compound interest as an application.Not required: use of logarithms to find n, giventhe sum of a series; and sums to infinity.

1.8 Geometric sequences and series, and their applications.

Use of formulae for the nth term and the sum of n terms.


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2 Core: sets and logic Teaching time: 12 hours

The aims of this section are to enable candidates to understand the concept of a set, to use appropriate notation, to translate between verbal and symbolicstatements and to use the principles of logic to analyse these statements.

2.7 Testing the validity of simple argumentsthrough the use of truth tables; concepts ofcontradiction and tautology.

2.6 Definition of implication: converse; inverse;contrapositive.Logical equivalence.

Truth tables can be used to illustrate associative anddistributive properties of connectives and forvariations of implication statements, eg .¬ ⇒ ¬q p

In examinations: truth tables will be limited tothree propositions at most, eg p, q, r.

2.5 Truth tables: use of truth tables to provideproofs for the properties of connectives.

The introduction of can bep q p q∧ ∨ and enhanced by the application and use of electriccircuits, although circuits themselves will not beexamined.

Included: an emphasis on analogies betweensets and logic, eg and .( )A B∩ ( )a b∧Not required: knowledge and use of the‘exclusive or’ and the distinction between it andthe ‘inclusive or’.

2.4 Compound statements: implication, ;⇒negation, ; ‘and’, ; ‘or’, . ¬ ∧ ∨Translation between verbal statements,symbolic form and Venn diagrams.

2.3 Basic concepts of symbolic logic: definitionof a proposition; symbolic notation ofpropositions.

Included: diagrams with up to three subsets ofthe universal set.Not required: knowledge of De Morgan’s laws.

2.2 Venn diagrams and simple applications.

N, Z, Q, R and sets of prime numbers, multiplesand factors can be used as examples.

2.1 Basic concepts of set theory: subsets;intersection; union; complement.


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3 Core: geometry and trigonometry Teaching time: 15 hours

The aims of this section are to develop spatial awareness, to develop the ability to draw clear diagrams and to represent information given in two and threedimensions, and to develop the ability to apply trigonometrical techniques to problem solving.

Included: For lines with gradients , m m1 2 and knowledge of for parallel lines, m m1 2=

for perpendicular lines.mm1

2

1= −

3.3 Equation of a line in two dimensions: theforms and .y mx c= + ax by d+ + =0Gradient; intercepts.Points of intersection of lines; parallel lines;perpendicular lines.

3.2 Coordinates in two and three dimensions:points; lines; mid-points; angles.

Distances between points; sizes of angles;identification of right angles.

In all areas of this section candidates should beencouraged to draw sufficient, well-labelleddiagrams to support their solutions.

Not required: radian measure.In examinations: candidates will not be asked toderive the sine and cosine rules.

3.1 The sine rule (including the ambiguous

case): aA

bB

cCsin sin sin

.= =

The cosine rule: c a b ab C2 2 2 2= + − cos .

Area of a triangle as .12

ab Csin

Construction of labelled diagrams fromverbal statements.


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3 Core: geometry and trigonometry (continued)

Displacement, velocity and force may be used todemonstrate the concept of vectors.Treatment of vectors in 3-dimensional space canenhance understanding, but will not be examined.Vector sums and differences can be represented bythe diagonals of a parallelogram. Multiplication by a scalar can be illustrated byenlarging the vector parallelogram. Applications to simple geometric figures, eg ABCD

is a quadrilateral and is aAB CD ABCD→ →

= − ⇒parallelogram.

Not required: vectors in three dimensions.Included: vectors written in column format.Note that: components are with respect to thestandard basis i, j; the first component of v is v1,etc.Included: the difference of v and w as

.v w v w− = + −( )

Included: the vector expressed as AB→

.OB OA→ →

− = −b a

3.5 Vectors as displacements in the plane,

. 1

2

vv

=

v

Components of a vector; columnrepresentation.

The sum of two vectors; the difference oftwo vectors; the zero vector. Multiplication by a scalar, .kv

Magnitude of a vector, .v

Position vectors .OA→

= a

Model building with sticks and connectors can assistunderstanding.

Included: only right prisms and square-basedright pyramids.

3.4 Geometry of three dimensional shapes:cuboid; prism; pyramid.Lengths of lines joining vertices withvertices, vertices with midpoints andmidpoints with midpoints; sizes of anglesbetween two lines and between lines andplanes.


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4 Core: statistics and probability Teaching time: 25 hours

The aims of this section are to enable candidates to use mathematics to analyse random events, to introduce concepts that will prove useful in further studiesof probability and inferential statistics, and to develop techniques to describe and analyse sets of data.

Use mid-interval values to estimate the mean ofgroup data. Link median to the 50th percentile andcumulative diagrams.

4.6 Measures of central tendency.For simple discrete data: mean; median;mode.For grouped discrete and continuous data:approximate mean; modal group; 50thpercentile.

Candidates can be introduced to the concept of box-and-whisker plots.

4.5 Cumulative frequency tables for groupeddiscrete and grouped continuous data;cumulative frequency curves. Percentiles; quartiles.

Note: a frequency histogram uses equal classintervals; a frequency density histogram mayuse class intervals of unequal width.

4.4 Grouped discrete or continuous data: frequency tables; mid-interval values,interval width, upper and lower boundaries.Frequency histograms; frequency densityhistograms.

4.3 Simple discrete data: frequency tables;frequency polygons; histograms.

Included: qualitative understanding ofcorrelation.Not required: calculation of coefficient ofcorrelation.

4.2 Scatter diagrams; line of best fit by eye.

Student, school and/or community data can be used.4.1 Classification of data as discrete orcontinuous.


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4 Core: statistics and probability (continued)

In general, probability should be introduced andtaught in a practical way using coins, dice, playingcards and other examples to demonstrate randombehaviour.

4.10 Equally likely events.

Probability of an event .A, ( ) ( )( )

Pn A

An U

=

Probability of a complementary event, .P P( ) ( )A A' = −1

4.9 Sample space: event, A; complementaryevent, .′A

Candidates should be aware that the populationmean, , and the population standardµdeviation, , are generally unknown, and thatσthe sample mean, , and the sample standardxdeviation, , serve only as estimates of thesesn

quantities.

4.8 Concept of population and sample.

Initially the calculation of standard deviation fromfirst principles should be demonstrated.

Included: an awareness of the concept ofdispersion and an understanding of thesignificance of the numerical value of thestandard deviation, .sn

In examinations: candidates will be expected touse a statistical function on a calculator to findthe standard deviation.

4.7 Measures of dispersion: range; interquartilerange; standard deviation.


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4 Core: statistics and probability (continued)

Examples: cards, dice and other simple cases ofrandom selection.

4.13 Venn diagrams; tree diagrams; tables of outcomes.Solution of problems using “withreplacement” and “without replacement”.

It should be emphasized that problems might best besolved with the aid of a Venn diagram or treediagram, and without the explicit use of the formulaelisted in §4.11 and §4.12.

4.12 Conditional probability

.P PP

( | ) ( )( )

A B A BB

= ∩

Experiments using coins, dice, packs of cards, etc.can enhance understanding of experimental relativefrequency versus theoretical probability.

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

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

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


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5 Core: functions Teaching time: 20 hours

The aim of this section is to develop understanding of some of the functions which can be applied to practical situations.

The form of the equation of the axis of symmetrymay initially be found by investigation.Link with §1.10.

5.5 The graph of the quadratic function .y ax bx c= + +2

Properties of symmetry; axis of symmetry,

; vertex; intercepts.x ba

= −2

Link with linear programming in §6.5. Included: graphing of systems of linearinequalities.Note: if points on the boundary line areincluded then they should be represented by acontinuous line; if points on the boundary arenot included then they should be represented bya broken line.

5.4 Linear inequalities and their graphs.

Illustrate with examples such as the cost of mailingparcels of different weights.

5.3 Piecewise linear functions: continuousfunctions; step functions.

Illustrate with examples from real world problemssuch as temperature conversion graphs and car hirecharges.

5.2 Linear functions and their graphs f x mx c: ." +

In the notation, the letters f and x can each bereplaced by any other letter, for example g(x), h(y),k(t).

Note that: examples should include functionsdefined on the sets N, Z, Q and R as domains.In examinations: if the domain is R then thestatement will be omitted.x ∈ R

5.1 Concept of a function as a mapping. Domain and range.


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5 Core: functions (continued)

Real world examples, such as population growth,radioactive decay, and cooling of a liquid can beused.

In examinations: candidates will be expected touse graphical methods to solve problems.

5.7 Graphs and properties of exponentialfunctions:

f x a

f x ka

f x ka c k a c

x

x

x

( ) ;

( ) ;

( ) ; , , .

=

=

= + ∈ QGrowth and decay; basic concepts ofasymptotic behaviour.

Examples of periodic phenomena may include tides,length of day, rotating wheels, etc.

In examinations: candidates will be expected touse graphical methods to solve problems.

5.6 Graphs and properties of the sine and cosinefunctions:

( ) sin ;( ) cos ;( ) sin ;( ) cos ; , , .

f x a x cf x a x cf x bx cf x a bx c a b c

= += += += + ∈ Q

Amplitude and period.


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6 Core: financial mathematics Teaching time: 13 hours

The aim of this section is to build a firm understanding of the concepts underlying certain financial transactions.

Note: any correct method (eg iterative processes, finding successive approximations) is valid for obtaining a solution to a problem in this section.

Link with linear inequalities and graphs in §5.4.Spreadsheet programmes can enhance teaching in this area.

Included: interpretation of practical problemsconcerning two variables.

6.5 Linear programming.

6.4 Construction and use of tables: loan andrepayment schemes; investment and savingschemes; inflation.

Link with geometric sequences §1.8 and exponentialfunctions §5.7.

In examinations: candidates will not be asked toderive the formula.Included: the use of iterative methods orsuccessive approximation methods to find n (thenumber of time periods).Not required: use of logarithms.

6.3 Compound interest: use of the formula

.1100

nrI C C = + −

Link with arithmetic sequences in §1.7.In examinations: candidates will not be asked toderive the formula.

6.2 Simple interest: use of the formula

where capital, % rate, I Crn=100

, C = r =

number of time periods, interest.n = I =

Link with linear functions in §5.2.Included: currency transactions involvingcommission.

6.1 Currency conversions.TEACHING NOTESAMPLIFICATIONS/EXCLUSIONSCONTENT

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7 Option: matrices and graph theory Teaching time: 25 hours

The aim of this section is to acquaint students with a number of concepts and methods associated with discrete mathematics.

Examples of graphs are to be found on most maps,showing arrangements of complex systems of roads,railways, canals, rivers, etc.

Note: different terminologies are used bydifferent authors in dealing with finite graphsand networks.In examinations: candidates will be expected toknow and use the terms contained in thesyllabus (refer to Assessment Guidelines,Terminology for their definitions).

7.5 Graph theory: vocabulary and definitions;vertex, edge, loop, multiple edge, directededge, walk, trail, path, degree of a vertex.

Illustrate using examples from such diverse areas assales and purchase, sporting contests, etc.

7.4 The storage and handling of data in matrixform.

Included: properties of matrix operations and, inparticular, that matrix multiplication is notcommutative.Not required: proofs of properties of matrixoperations.

7.3 Arithmetic of matrices: addition;subtraction; multiplication; multiplication bya scalar.

7.2 Transpose of a matrix.Determinant of a matrix.

The idea of a matrix should be developed frompractical examples to the formal representation as

rectangular arrays of elements.( )m n×

Included: the study of matrices with ( )m n×.m n≤ ≤4 4,

7.1 The concept of a matrix.Definitions: order of a matrix; types ofmatrices (row, column, diagonal, identity,zero, singular, symmetric and squarematrices).


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7 Option: matrices and graph theory (continued)

Included: the study of two-person, zero-sumgames only.

7.10 Simple games.

Included: the study of problems concerned withtwo states only (giving a matrix).2 2×

7.9 Simple Markov chains.

For example, a complex network of one-way andtwo-way streets can be represented by a directedgraph which, in turn, can be interpreted algebraicallyas a matrix.

Included: use of directed graphs and matrices tosolve applications.

7.8 Directed graphs.

Required: properties of adjacency matrices andincidence matrices.

7.7 Matrix representation of a graph: adjacencymatrix, A; incidence matrix.Interpretation of .A2

Classic counting arguments such as the “handshakeproblem” and the condition for a graph to beEulerian can be used.

Included: construction of a graph from a givenmatrix.

7.6 Types of graphs: simple; connected;disconnected; complete; trees; subgraphs.


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8 Option: further statistics and probability Teaching time: 25 hours

The aims of this section are to enhance awareness of the statistical method and to provide possible tools for further use.

Note: for this section, candidates require access to a calculator with in-built statistical functions including linear regression.

Wherever possible students should be given theopportunity to work on sets of meaningful data.

8.3 Applications of the normal distribution.

Included: an awareness that 68% of thepopulation lies between .µ σ±Not required: linear interpolation.

8.2 The standard normal distribution: propertiesof the curve.Transformation of any normal variable to the

standardised normal variable, : usez x= − µσ

of normal tables or a calculator;diagrammatic representation.

Not required: the formula for the probabilitydensity function.

8.1 The normal distribution: concept of a randomvariable; parameters .µ σ and Properties of the curve of the probabilitydensity function: bell shape; symmetry about

; change of curvature at ; areax = µ ±σunder the curve = 1.


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8 Option: further statistics and probability (continued)

Included: one-tailed and two-tailed tests. Included: contingency tables where h k×

.h k, ≤ 3

8.6 The test for independence: formulationχ 2

of null and alternative hypotheses; test levels(5%, 1%); contingency tables; calculation ofexpected frequencies; use of the formula

; degrees of freedom;( )2

2calc

e o

e

f ff

χ−

=∑use of tables.

8.5 The regression line for y on x: use of the

formula .y yss

x xxy

x

− = −2 ( )

Use of the regression line for predictionpurposes.

In examinations: the value of will be given.sxy

Note: represents the standard deviation of thesx

variable ; represents the covariance of X sxy

the variables and .X Y

8.4 Bivariate data: the concept of correlation.Product-moment correlation coefficient: use

of the formula , where xy

x y

ss s

( )( )

1 .

n

i ii

xy

x x y ys

n=

− −=∑

Interpretation of positive, zero and negativecorrelations.


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8 Option: further statistics and probability (continued)

Included: combination of classes where theexpected frequency is less than 5.Not required: Yates’ continuity correction.

8.7 The test for goodness of fit of theχ 2

normal distribution: formulation of null andalternative hypotheses; test levels (5%, 1%);calculation of expected frequencies; use of

the formula ; degrees( )2

2calc

e o

e

f ff

χ−

=∑of freedom; use of tables.Consideration of the cases when µ σ and are known and when are used asx sn and estimates of .µ σ and


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9 Option: introductory differential calculus Teaching time: 25 hours

The aim of this section is to introduce the concept of the gradient of the graph of a function which is fundamental to the study of differential calculus, so thatstudents can apply the concept of derivative of a function to solv ing practical problems.

In examinations: problems of this nature will belimited to polynomials of degree 3 or less.

9.3 Gradients of curves for given values of x.Values of x where is given.f x' ( )

9.2 The principle that . f x ax f x anxn n( ) ( )= ⇒ ′ = −1

The derivative of polynomials of the form . f x ax bx nn n( ) . . . ,= + + ∈−1 Z+

The concept of a limit should be introduced usingnumerical and graphical investigations.

Not required: formal treatment of limits.Included: solving problems involving aparticular function for given values of h and x.

9.1 Gradient of the line through two points Pand Q which lie on the graph of a function.Behaviour of the gradient of the line throughtwo points P and Q on the graph of afunction as Q approaches P.The derivative as the gradient function;

( )

0

( )( ) lim ;

h

f x h f xf x

h→

+ − ′ =

.′ =f x yx

( ) dd


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9 Option: introductory differential calculus (continued)

9.7 Anti-differentiation as the inverse ofdifferentiation.Finding an original function given thederivative function and constraints.The relationship between instantaneousacceleration, instantaneous velocity anddisplacement.

Exposure to solving simple kinematics problems isrecommended.

Included: knowledge of the general shapes ofpolynomials of order . n ≤ 3

9.6 Application of the derivative concept:properties of curves; maxima and minimaproblems; rates of change; instantaneousvelocity, v(t), and instantaneous

acceleration, a(t), where v t s t st

( ) ( )= =' dd

and .a t v t vt

( ) ( )= =' dd

Included: the concept of a function changingfrom increasing to decreasing and vice versa asa test for local maxima and minima.

9.5 Values of x where the gradient of a curve is0 (zero): solution of .f x' ( ) = 0Local maximum and minimum points; pointsof inflexion with zero gradient.

9.4 Increasing and decreasing functions.Graphical interpretation of

.′ > ′ = ′ <f x f x f x( ) , ( ) , ( )0 0 0


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SYLLABUS GUIDELINES

Presumed knowledge

1 General

Candidates are not required to be familiar with all the topics listed as presumed knowledge(PK) before they start the mathematical studies SL course of study. However, they should befamiliar with these topics before they take the written papers, as questions will assumeknowledge of them. It is therefore recommended that teachers ensure that any topics includedin presumed knowledge which are unknown to their candidates at the start of the course areincluded in the programme of study at an early stage.

Candidates should be familiar with the Système International (SI) units of length, mass andtime, and their derived units.

2 Topics

PK1 Number and algebra

1.01 Basic use of the four operations of arithmetic, using integers, decimals andsimple fractions, including order of operations.Examples: ; ( )2 3 4 7 62+ × = 2 3 4 7 34× + × =

1.02 Prime numbers, factors and multiples.

1.03 Simple applications of ratio, percentage and proportion.

1.04 Absolute value .| |aExample: | |4 7 3− =

1.05 Basic manipulation of simple algebraic expressions including factorizationand expansion.Examples: ab ac a b c a ab b a b a b a b a b

x x x x xa a xb b x a b

+ = + ± + = ± − = + −

+ + = + + − + − = − +

( ); ( ) ; ( )( )

( )( ); ( )( )

2 2 2 2 2

2

2

3 5 2 3 2 1 2 2 2

1.06 Solving linear equations in one variable.Example: 3 6 4 1 0( ) ( )x x+ − − =


1.07 Solving a system of linear equations in two variables.

Examples: 3 4 13 13

2 1x y x y+ = − = −;

1.08 Evaluating exponential expressions.

Examples: ; a bb , ∈ Z 2 116

4− =

1.09 Order relations and their properties.< ≤ > ≥, , ,Examples: ( , ) ; ( , )a b c ac bc a b c ac bc> > ⇒ > > < ⇒ <0 0

1.10 Rearranging formulae.

Example: A bh h Ab

= ⇒ =12

2

1.11 Evaluating formulae by substitution. Example: If then x x x= − − + =3 2 3 182,

1.12 Intervals on the real number line.Example: 2 5< ≤ ∈x x, R

PK2 Geometry and trigonometry

2.01 Basic geometric concepts: point, line, plane.

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

2.03 The (x, y) coordinate plane.

2.04 Sine, cosine and tangent of acute angles.

2.05 Pythagoras’ theorem.

PK3 Probability and statistics

3.01 The collection of data and its representation in bar charts, pie charts andpictograms.

PK4 Financial mathematics

4.01 Basic use of commonly accepted world currencies.Examples: Swiss franc (CHF); United States dollar (USD, $); British pound

sterling (GBP, £); Japanese yen (JPY)

SYLLABUS GUIDELINES


Internationalism

One of the aims of this programme is to enable candidates to appreciate the internationaldimensions of mathematics and the multiplicity of its cultural and historical perspectives.While this aim is not explicitly written into the syllabus, it is hoped that teachers will takeevery opportunity to fulfil this aim by discussing relevant issues as they arise and makingreference to appropriate background information. For example, it may be appropriate todiscuss:

• differences in notation

• the lives of mathematicians set in a historical and/or social context

• the cultural context of mathematical discoveries

• the ways in which certain mathematical discoveries were made in terms of thetechniques used

• the attitudinal divergence of different societies towards certain areas ofmathematics

• the universality of mathematics as a language.

It should be noted that this aim has not been translated into a corresponding objective andtherefore that this aspect of the programme will not be tested in examinations.

SYLLABUS GUIDELINES


ASSESSMENT OUTLINE


An individual piece of work completed during the courseinvolving the collection and/or generation of data, and theanalysis and evaluation of that data. Projects may take the formof mathematical modelling, investigations, applications,statistical surveys, etc.

The project is internally assessed by the teacher and externallymoderated by the IBO. Procedures are provided in the VadeMecum.

Project

20%Internal assessment

15%Section B: Three extended-response questions, one on each of theoptional topics in part II of the syllabus; one question tobe answered on the chosen option.

35%Section A: Five compulsory extended-response questions based onpart I of the syllabus, the compulsory core.

50%Paper 2 2 hours

Fifteen compulsory short-response questions based on part I ofthe syllabus, the compulsory core.

30%Paper 1 1 hour

Written papers 3 hours

80%External assessment


ASSESSMENT DETAILS

External assessment: written papers

1 General

1 Paper 1 and paper 2

The external assessment consists of two written examination papers, paper 1 andpaper 2, which are externally set and externally marked. Together they contribute80% to the final mark. These papers are designed to allow candidates to demonstratewhat they know and can do.

2 Calculators

Candidates are expected to have access to a calculator, which may or may not have agraphic display facility, at all times during the course. Regulations concerning thetypes of calculators allowed are provided in the Vade Mecum.

3 Formulae booklet and statistical tables (second edition, April 1999)

As each candidate is required to have access to clean copies of the IBO formulaebooklet and statistical tables during the examination, it is recommended that teachersensure candidates are familiar with the contents of these documents from thebeginning of the course. The booklet and tables are provided by IBCA and arepublished separately.


2 Paper 1 (1 hour) 30%

This paper consists of fifteen compulsory short-response questions based on part I of thesyllabus, the core.

1 Syllabus coverage

• Knowledge of all topics from the core is required for this paper.

• The intention of this paper is to test candidates’ knowledge across the breadth ofthe core. However, it should not be assumed that the separate topics from thecore will be given equal weight or emphasis.

2 Question type

• A small number of steps will be needed to solve each question.

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

3 Mark allocation

• Each question is worth four marks. The maximum number of marks availablefor this paper is 60.

• Questions of varying levels of difficulty will be set. Each will be worth the samenumber of marks.

• Full marks are awarded for each correct answer irrespective of the presence ofworking.

Where a wrong answer is given, partial credit may be awarded for a correctmethod provided this is shown by written working; if no working is present thenno partial credit can be given and candidates cannot be awarded any marks.Candidates should therefore be encouraged to show their working at all times.

3 Paper 2 (2 hours) 50%

This paper is divided into two sections: section A, based on part I of the syllabus, and sectionB, based on part II. It is estimated that, during the total time of two hours, candidates will beable to spend up to 20 minutes in thought and reflection.

1 Question type

• Questions in both sections will require extended responses involving sustainedreasoning.

ASSESSMENT DETAILS


• Individual questions may develop a single theme or be divided into unconnectedparts.

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

• Normally, each question will reflect an incline of difficulty from relatively easytasks at the start of a question to relatively difficult tasks at the end of a question.The emphasis will be on problem-solving.

2 Awarding of marks

• Marks will be awarded according to the following categories.

Method: evidence of knowledge, the ability to apply concepts and skills, and theability to analyse a problem in a logical manner.

Accuracy: computational skill and numerical accuracy.

Reasoning: clear reasoning, explanation and/or logical argument.

Correct statements: results or conclusions expressed in words.

Follow through: if an incorrect answer found in an earlier part of a question isused later in the same question then marks may be awarded in the later part eventhough the original answer used is incorrect. In this way, candidates are notpenalized for the same mistake more than once.

• A correct answer with no indication of the method used (for example, in the formof diagrams, graphs, explanations, calculations, etc.) will normally be awarded nomarks. All candidates should therefore be advised to show their working.

4 Paper 2: section A

This section consists of five compulsory extended-response questions based on part I of thesyllabus, the core. Candidates will be expected to answer all the questions in this section.

1 Syllabus coverage

• Knowledge of all topics from part I of the syllabus is required for this section ofpaper 2.

• Individual questions may require knowledge of more than one topic from thecore.

• The intention of this section is to test candidates’ knowledge of the core in depth.A narrower range of topics from the syllabus will be tested in this paper than istested in paper 1.

ASSESSMENT DETAILS


2 Mark allocation

• This section is worth 70 marks, representing 35% of the final mark.

• Questions in this section may be unequal in terms of length and level ofdifficulty. Hence individual questions will not necessarily be worth the samenumber of marks. The exact number of marks allocated to each question will beindicated at the start of each question.

5 Paper 2: section B

This section consists of three extended-response questions based on part II of the syllabus,the options. One question will be set on each option.

1 Syllabus coverage

• Candidates will be expected to answer the question based on the option they havestudied.

• Knowledge of the entire contents of the option studied is required for this sectionof paper 2.

• In order to provide appropriate syllabus coverage of each option, questions in thissection are likely to contain two or more unconnected parts.

2 Mark allocation

• This section is worth 30 marks, representing 15% of the final mark.

• Questions in this section will be equal in terms of length and level of difficulty.Each question will be worth 30 marks.

ASSESSMENT DETAILS


Internal assessment: the project

1 The purpose of the project

The specific purposes of the project are to:

• develop candidates' personal insight into the nature of mathematics and to develop theirability to ask their own questions about mathematics

• encourage candidates to initiate and sustain a piece of work in mathematics

• enable candidates to acquire confidence in developing strategies for dealing with newsituations and problems

• provide opportunities for candidates to develop individual skills and techniques and toallow candidates of varying abilities, interests and experiences to achieve a sense ofpersonal satisfaction in doing mathematics

• enable candidates to experience mathematics as an integrated organic discipline and notas a set containing fragmented, compartmentalized skills and knowledge

• enable candidates to see connections and applications of mathematics to other areas ofinterest

• provide opportunities for candidates to show, with confidence, what they know and cando.

2 Requirements

The mathematical studies SL project is a piece of written work based on personal researchinvolving the collection, analysis and evaluation of data.

Each project should contain:

• a statement of the task • measurements, information or data which has been collected and/or generated• an analysis of the measurements, information or data• an evaluation of the analysis.

Candidates may choose from a wide variety of project types, for example, modelling,investigations, applications and statistical surveys. Historical projects which reiterate factsbut have little mathematical content are not appropriate and should be actively discouraged.

It is expected that teachers will give guidance to candidates in choosing appropriate areas ofstudy for their projects. As much as possible, these areas of study should relate to thecandidates’ own interests and, while mathematical in nature, may be based in contexts suchas sport, art, music, the environment, health, travel, trade and commerce.

ASSESSMENT DETAILS


In developing their projects, candidates should make use of mathematics learned as part ofthe course. The level of sophistication of the mathematics should be about the same as thatcontained in the syllabus. It is not intended that additional topics be taught to candidates toenable them to complete their projects.

3 Integration into the course of study

It is intended that project work be incorporated into the programme of study so thatcandidates are given the opportunity to learn the skills associated with a successful project.

Time in class can therefore be used for:

• general discussion of areas of study for project work—how data can be collected; wheredata can be collected; how much data should be collected; different ways of displayingdata; what steps should be taken to analyse the data; how data should be evaluated, etc.

• discussion between the teacher and the candidate(s) and/or discussion betweencandidates on particular aspects of projects belonging to individual candidates.

• oral presentations by individual candidates on particular aspects of their projects or ontheir projects as a whole.

4 Management of the project

1 Time allocation

The Vade Mecum states that a standard level programme requires at least 150teaching hours. In mathematical studies, 25 of these hours should be allocated towork connected with the project. This will allow time for teachers to explain tocandidates the requirements of the project, to discuss fully the assessment criteriaand allow class time for the development of project work.

2 Planning

An early start to planning project work is vital. Deadlines, preferably reached byagreement between candidates and teachers, need to be firmly established. Thereneeds to be a date for submission of the project title and a brief outline description, adate for the completion of data collection/generation, a date for the submission of thefirst draft and, of course, a date for project completion.

3 Length

The length of the project should not normally exceed 2000 words, excludingdiagrams, graphs, appendices and bibliography.

ASSESSMENT DETAILS


4 Guidance

• All candidates should be familiar with the requirements of the project and themeans by which it is assessed. In particular, teachers should discuss with theircandidates the levels of achievement expected for Criterion F, Commitment.

• It should be made clear to candidates that all work connected with the project,including the writing of the project, should be their own. It is therefore helpful ifteachers try to encourage in candidates a sense of responsibility for their ownlearning so that they accept a degree of ownership and take pride in their ownwork.

• Group work, whilst educationally desirable in certain situations, is notappropriate for work on projects.

• It must be emphasized that candidates are not expected to work entirely on theirown. The teacher is expected to give appropriate guidance at all stages of theproject by, for example, directing candidates into more productive routes ofenquiry, making suggestions for suitable sources of information, and providingadvice on the content and clarity of a project in the writing-up stage.

5 Authenticity

Teachers are required to ensure that each project is the candidate’s own work. If theteacher views the project at each stage of its development, this serves as a veryeffective way of ensuring that the project is the intellectual property of the candidateand also acts as a safeguard against plagiarism. In this way, by monitoring all stagesof the development, a teacher can ensure that the project is the authentic, personalwork of the candidate. If in doubt, authenticity may be checked by one or more of thefollowing methods:

! discussion with the candidate! asking the candidate to explain the methods used and to summarize the

results/conclusions! asking the candidate to replicate part of the analysis using different data.

It is also appropriate for teachers to request that each project be signed on completionto indicate that it is the candidate’s own work.

ASSESSMENT DETAILS


ASSESSMENT CRITERIA

The project

1 Introduction

The project is internally assessed by the teacher and externally moderated by the IBO usingassessment criteria which relate to the objectives for group 5.

2 Form of the assessment criteria

Each project should be assessed against the following six criteria:

A Statement of the taskB Data collectionC AnalysisD EvaluationE Structure and communicationF Commitment

3 Applying the assessment criteria

The method of assessment used is criterion referenced, not norm referenced. That is, themethod of assessing each project judges candidates by their performance in relation toidentified assessment criteria and not in relation to the work of other candidates.

• Each project submitted for mathematical studies SL is assessed against the six criteria Ato F.

• For each assessment criterion, different levels of achievement are described whichconcentrate on positive achievement. The description of each achievement levelrepresents the minimum requirement for that level to be achieved.


• The aim is to find, for each criterion, the level of achievement gained by the candidatefor that piece of work. Consequently, the process involves:

• reading the description of each achievement level, starting with level 0, until one isreached which describes a level of achievement that has not been reached

• the level of achievement gained by the candidate is therefore the preceding one and itis this which should be recorded.

For example, if, when considering successive achievement levels for a particularcriterion, the description for level 3 does not apply then level 2 should be recorded.

• If a piece of work appears to fall between two achievement levels then the lowerachievement level should be recorded since the minimum requirements for the higherachievement level have not been met.

• For each criterion, whole numbers only may be recorded; fractions and decimals are notacceptable.

• The whole range of achievement levels should be awarded as appropriate. For aparticular piece of work, a candidate who attains a high achievement level in relation toone criterion may not necessarily attain high achievement levels in relation to othercriteria.

It is recommended that the assessment criteria be available to candidates at all times.

4 The final mark

The final mark for each project is the sum of the scores for each criterion. The maximumpossible final mark is 25. Teachers are not required to scale the final mark. This will be doneat IBCA.

ASSESSMENT CRITERIA


5 Achievement Levels

Criterion A: statement of the task

In this context, the word “task” is defined as “what the candidate is going to do”; the word “plan” isdefined as “how the candidate is going to do it”. A statement of the task should appear at thebeginning of each project.

Achievementlevel

0 The candidate does not produce a statement of the task. · There is no evidence in the project of any statement of what the candidate is going to do or

has done.

1 The candidate produces a statement of the task.· For this level to be achieved the task must be stated explicitly.

2 The candidate produces a clear statement of the task and describes the plan bywhich it will be carried out.· For this level to be achieved the plan need not be detailed but some attempt must be made to

describe the process. The candidate cannot go beyond this level if the project has too fewvariables or is lacking in content; the variables may be developed at any point in theproject.

3 The candidate produces a clear statement of the task and describes the plan by whichit will be carried out, this plan being well focused.· To achieve this level the plan must have sufficient scope to enable the candidate to produce

relevant measurements, information or data.

ASSESSMENT CRITERIA


Criterion B: data collection

In this context, generated data includes data which has been generated by computer, by observation,by investigation, or by experiment. Mathematical information includes geometrical figures,synthetically generated numbers and data which is collected empirically or assembled from outsidesources. This list is not exclusive and mathematical information does not solely imply data forstatistical analysis.

Achievementlevel

0 The candidate does not collect/generate measurements, information or data.· No attempt has been made to collect any information.

1 The candidate collects/generates measurements, information or data.· Some attempt has been made to collect information, without sufficient care being taken to

ensure that it is relevant to the stated task .

2 The candidate collects/generates measurements, information or data relevant to thedefined task.· The candidate has clearly stated the task and the information collected is in accordance

with the described plan. This achievement level can be awarded even if a fundamental flawexists in the instrument used to collect the data, for example, a faulty questionnaire or aninterview conducted invalidly.

3 The candidate collects/generates measurements, information or data relevant to thedefined task and organizes the data in a form appropriate for analysis.· A satisfactory attempt has been made to structure the data ready for the process of analysis.

4 The candidate collects/generates measurements, information or data relevant to thedefined task and organizes the data in a form appropriate for analysis, the data beingsufficient in both quality and quantity.· This level cannot be achieved if the measurements, information or data are too sparse (ie

insufficient in quantity) or too simple (for example, one-dimensional) as clearly such datadoes not lend itself to being structured. It should therefore be recognized that within thisdescriptor there are assumptions concerning the quantity and, more importantly, the quality(in terms of depth and breadth) of data collected/generated.

ASSESSMENT CRITERIA


Criterion C: analysis

When presenting diagrams, candidates are expected to use rulers where necessary and not merelysketch. A freehand sketch would not be considered a correct mathematical process.

Achievement level

0 The candidate does not attempt to carry out any mathematical processes.· This would include candidates who have copied processes from a book with no attempt being

made to use their own collected/generated information. Projects consisting of, for example,only historical accounts will achieve this level.

1 The candidate carries out simple mathematical processes. · Simple processes are considered to be those which the average mathematical studies student

could carry out easily, for example, percentages, areas of plane shapes, linear functions(graphing and analysing), bar charts, pie charts, mean of discrete data. This level does notrequire the representation to be comprehensive, nor does it demand that the calculations areto be without error.

2 The candidate carries out simple mathematical processes correctly.· For this level to be achieved the mathematics must be without any significant error.

3 The candidate carries out simple mathematical processes correctly which arerelevant to the stated task.· For this level to be achieved the mathematical processes must be appropriate and used in a

meaningful way.

4 The candidate carries out simple mathematical processes correctly and makes use ofmore sophisticated techniques which are relevant to the stated task.· Sophisticated techniques are considered to be higher-order processes, for example, volumes

of pyramids and cones, analysis of trigonometrical and exponential functions, means andstandard deviations of continuous and grouped data. For this level to be achieved it is notrequired that all the calculations be without error; the candidate will have carried outsimple processes correctly and have included more difficult mathematical techniques whichmay be inaccurate.

5 The candidate carries out simple mathematical processes correctly and makesaccurate use of more sophisticated techniques which are relevant to the stated task.· The key word in this descriptor is accurate. It is accepted that not all calculations need to

be checked before awarding this achievement level; random checking of some calculationsis deemed to be sufficient. A small number of isolated mistakes should not disqualify acandidate from achieving this level. However, incorrect use of formulae, or consistentmistakes in using data, would disqualify the candidate from achieving this level.

6 The candidate carries out simple mathematical processes correctly and makesaccurate use of a wide range of more sophisticated techniques which are relevant tothe stated task. · To achieve this level the candidate would be expected to have carried out as wide a range of

meaningful mathematical processes as possible, although they may all relate to a singlearea of mathematics, for example, geometry. Measurements, information or data which arelimited in scope would not allow the candidate to achieve this level.

ASSESSMENT CRITERIA


Criterion D: evaluation

Achievement level

0 The candidate does not produce any interpretations or conclusions. · For the candidate to be awarded this level there must be no evidence of interpretations or

conclusions anywhere in the project .

1 The candidate produces at least one interpretation or conclusion.· Only minimal evidence of interpretations or conclusions is required for this level .

2 The candidate produces some interpretations and/or conclusions which areconsistent with the analysis.· For this level more than one interpretation and/or conclusion is required and each must be

consistent with the analysis. A ‘follow through’ procedure should be used and,consequently, it is irrelevant here whether the analysis is either correct or appropriate; theonly requirement is consistency.

3 The candidate produces some interpretations and/or conclusions which are consistentwith the analysis, and comments on the validity of the mathematical processes usedand the results obtained. · To achieve this level the candidate must have discussed, in a coherent manner, the

validity of the processes used in the project. Discussion of validity is not expected to beexhaustive at this level but candidates must demonstrate an understanding of themeaning of the word valid in relation to their projects.

4 The candidate produces thorough interpretations and/or conclusions which areconsistent with the analysis, and comments on the validity of the mathematicalprocesses used and the results obtained.· Here the key word is thorough and the candidate is expected to have produced nearly all

the possible interpretations and/or conclusions based on the project. A coherent and fulldiscussion of the results obtained and conclusions drawn should be present to obtain thislevel.

5 The candidate produces thorough interpretations and conclusions which areconsistent with the analysis, and comments critically on the validity of themathematical processes used and the results obtained. · This level requires a level of sophistication and understanding of validity of the kind that an

excellent candidate might produce. One or two sentences discussing the validity of theproject would not be sufficient to achieve this level .

ASSESSMENT CRITERIA


Criterion E: structure and communication

Achievement level

0 The candidate has made no attempt to structure the project.· It is not expected that many candidates will be awarded this level.

1 The candidate has made some attempt to structure the project by recording actionsat each stage.· A project may not necessarily contain many mathematical calculations for this level to be

achieved.

2 The candidate has structured the project by recording actions at each stage usingmathematical language and representation. · To achieve this level the candidate must record actions at each stage of the development of

the project by using some, but not necessarily all, of the following: mathematical language,symbols, diagrams, tables, graphs .

3 The candidate has structured the project by systematically recording actions at eachstage using appropriate mathematical language and representation. · For this level to be achieved the mathematics must be used in a meaningful manner. A

project in which the candidate has produced a large number of calculations, some of whichare inappropriate to the data, would not achieve this level .

4 The candidate has structured the project by systematically recording actions at eachstage using appropriate mathematical language and representation in a clear andcoherent manner. · To achieve this level the project would be expected to read well. The implication here is that

a higher degree of sophistication is required.

ASSESSMENT CRITERIA


Criterion F: commitment

The project should be an ongoing process involving consultation between candidate and teacher. Thecandidate should be aware of the expectations of the teacher from the beginning of the process andeach achievement level awarded should be justified by a written comment from the teacher at the timeof marking. The examples given below for each criterion level are teacher orientated and each teachershould use discretion when judging the levels .

Achievement level

0 The candidate showed no commitment.· For example, the candidate did not participate in class discussions on project work, did not

submit the required work in progress, and/or missed many deadlines .

1 The candidate showed average commitment. · For example, the candidate participated minimally in class discussions on project work, kept

to most deadlines, had some discussion initiated by the teacher and/or did not exploit theavailable opportunities for the development or improvement of the project.

2 The candidate showed good commitment.· For example, the candidate participated in class discussions on project work, initiated

discussions with the teacher and/or the rest of the class and/or became fully involved in thedevelopment of the project.

3 The candidate showed outstanding commitment. · For example, the candidate participated fully in class discussions on project work, took

initiatives both in discussion with the teacher and/or the rest of the class and in subsequentwork of a more independent nature and/or demonstrated a full understanding of all the stepsin the development of his/her project.

In order to obtain the highest achievement level for this criterion the candidate should haveexcelled in areas such as those listed below. This list is not exhaustive and teachers areencouraged to add their own expectations .

The candidate:• actively participated at all stages of the development of the project • demonstrated a full understanding of the concepts associated with his/her project• participated in class activities on project work • demonstrated initiative • demonstrated perseverance • showed insight • prepared well to meet deadlines set by the teacher.

ASSESSMENT CRITERIA


ASSESSMENT GUIDELINES

External assessment: written papers

1 Notation

Of the various notations in use, the IBO has chosen to adopt the notation listed below basedon the recommendations of the International Organisation for Standardization. These will beused on written examination papers in mathematical studies SL without explanation. Ifforms of notation other than those listed here are used on a particular examination paper thenthey will be defined within the question in which they appear.

Because candidates are required to recognize, though not necessarily use, the IBO notation inexaminations, it is recommended that teachers introduce students to IBO notation at theearliest opportunity. Candidates will not be permitted information relating to notation in theexaminations.

In a small number of cases, candidates will need to use alternative forms of notation in theirwritten answers as not all forms of IBO notation can be directly transferred into hand-writtenform. This is true particularly in the case of vectors where the IBO notation uses a bold,italic typeface which cannot be adequately transferred into hand-written form. In thisparticular case, teachers should advise candidates to use alternative forms of notation in theirwritten work (such as ).

!x x x, or

the set of positive integers and zero,N { , , , , ...}0 1 2 3the set of integers, Z { , , , , ...}0 1 2 3± ± ±the set of positive integers, Z+ { , , , ...}1 2 3the set of rational numbersQthe set of positive rational numbers, Q+ { | , }x x x∈ >Q 0the set of real numbersRthe set of positive real numbers,R+ { | , }x x x∈ >R 0the set with elements { , , ...}x x1 2 x x1 2, , ...the number of elements in the finite set An A( )the set of all x such that{ | }xis 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⊆the complement of the set A′Aconjunction: p and qp q∧disjunction: p or q (or both)p q∨negation: not p¬pimplication: if p then qp q⇒implication: if q then pp q⇐equivalence: p is equivalent to qp q⇔

a to the power of , nth root of a a an n1/ , 1n

( )if then a an≥ ≥0 0

a to the power , square root of aa a1 2/ , 12

( )if then a a≥ ≥0 0

the modulus or absolute value of x, ie | |x for 0, for 0,

x x xx x x

≥ ∈− < ∈

R

Ris approximately equal to≈

> is greater thanis greater than or equal to≥

< is less thanis less than or equal to≤

( is not greater than ' is not less than

the nth term of a sequence or seriesun

d the common difference of an arithmetic sequence r the common ratio of a geometric sequence

the sum of the first n terms of a sequence, Sn u u un1 2+ + +...

uii

n

=∑

1

u u un1 2+ + +...

f is a function under which each element of set A has an image in set Bf A B: →f is a function under which x is mapped to yf x y: !the image of x under the function ff x( )the limit of f (x) as x tends to alim ( )

x af x

→

the derivative of y with respect to xdd

yx

the derivative of f (x) with respect to x′f x( )sin, cos, tan the circular functionsA(x, y) the point A in the plane with cartesian coordinates x and y[AB] the line segment with end points A and BAB the length of [AB](AB) the line containing points A and B

the angle at A"Athe angle between [CA] and [AB]CAB"the triangle whose vertices are A, B and C∆ABC

v the vector v



the vector represented in magnitude and direction by the directed lineAB→

segment from A to B

a the position vector OA→

the magnitude of a| |a

the magnitude of |AB|→

AB→

the transpose of the matrix AAT

det A the determinant of the square matrix AI the identity matrixP(A) probability of event A

probability of the event 'not A'P( )′Aprobability of the event A given the event BP( | )A Bobservationsx x1 2, , ...frequencies with which the observations occurf f1 2, , ... x x1 2, , ...

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

population meanµ

population varianceσ 2

2

2 1

1

( ), where

k

i i ki

ii

f xn f

n

ms =

=

-= =Â

Âpopulation standard deviationσsample meanx

sample variance2ns

2

2 1

1

( ), where

k

i i ki

n ii

f x xs n f

n=

=

-= =Â

Âstandard deviation of sample sn

cumulative distribution function of the standardised normal variable withΦdistribution N(0, 1)

r product-moment correlation coefficient



2 Terminology (option 7: matrices and graph theory)

Teachers and students should be aware that many different terminologies exist in graphtheory and that different textbooks may employ different combinations of these. Examples ofthese are: vertex / node / junction / point; edge / route / arc; degree of a vertex / order;multiple edges / parallel edges; loop / self-loop.

In IBO examination questions, the terminology used will be as it appears in the syllabus. Forclarity these terms are defined below.

• A graph consists of a set of vertices and a set of edges. The endpoints of eachedge are connected to either the same vertex or two different vertices.

• An edge whose endpoints are connected to the same vertex is called a loop.

• If more than one edge connects the same pair of vertices then these edges arecalled multiple edges.

• A directed edge is one in which it is only possible to travel in one direction.

• A directed graph is a graph where every edge is directed.

• A walk is a sequence of linked edges.

• A trail is a walk in which no edge appears more than once.

• A path is a walk with no repeated vertices.

• The degree of a vertex is the number of edges connected to that vertex (a loopcontributes two, one for each of its endpoints).

• A simple graph has no loops or multiple edges.

• A graph is connected if there is a path connecting every pair of vertices.

• A graph is disconnected if there is at least one pair of vertices which is notconnected by a path.

• A complete graph is a simple graph, that is, one which has no loops or multipleedges, where every vertex is connected to every other vertex.

• A graph is a tree if it is connected and contains no paths which begin and end atthe same vertex.

• A subgraph is a graph within a graph.



• The elements of the nth row of an adjacency matrix are the number of edgesconnecting the nth vertex with every other vertex, taken in order. Hence, for anundirected graph, the adjacency matrix will be symmetric about the diagonal.

• The elements of the nth row of an incidence matrix are either 1 or 0 depending onwhether each edge, taken in order, is connected to the nth vertex or not.



Internal assessment: the project

1 Teaching and learning strategies

The following teaching and learning strategies are implied by the aims of the project.

• As an integral part of the programme, candidates need to be provided with opportunitiesto experiment, to explore, to generate hypotheses and to ask questions.

• Candidates should be given experiences in taking initiatives for developing their ownmathematics in the classroom.

• Candidates should be encouraged to become active learners of mathematics, inside andoutside the classroom.

• Topics in the syllabus should be organized to allow an active enquiry approach.

• Candidates should be encouraged to identify and reflect on the variety of mathematicalprocesses that may be used to solve problems.

Whilst teachers can advise on topics and titles they should realise that the most valuableprojects are those that reflect the candidate’s interests and enthusiasms.

2 Assessment strategies

The assessment strategies used in the classroom should:

• be in harmony with the teaching and learning strategies listed above

• enable candidates to gain new insights into the nature of mathematics

• enable candidates to assess the quality of their own work.

Assessment at each stage of the project can be monitored by the teacher, by class discussionof the project by other candidates, and by self-assessment by the owner of the project.Assessment should be criterion referenced and formative, thereby enabling the candidate tomodify, extend and improve a project throughout the course of its development. The clearintent is to encourage dialogue between teacher and candidate without compromising theintegrity of the project as the candidate's own work.



DIPLOMA PROGRAMME Mathematical studies standard level · •Their own abilities in mathematics and...

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