Further mathematics HL guide - City University of …...Further mathematics HL guide 1 Purpose of...

Diploma Programme

Further mathematics HL guideFirst examinations 2014

Diploma Programme

Further mathematics HL guideFirst examinations 2014

Further mathematics HL guideDiploma Programme

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

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

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© International Baccalaureate Organization 2007

Further mathematics HL guide

Contents

Introduction 1Purpose of this document 1

The Diploma Programme 2

Nature of the subject 4

Aims 8

Assessment objectives 9

Syllabus 10Syllabus outline 10

Approaches to the teaching and learning of further mathematics HL 11

Syllabus content 15

Glossary of terminology: Discrete mathematics 44

Assessment 46Assessment in the Diploma Programme 46

Assessment outline 48

Assessment details 49

Appendices 51Glossary of command terms 51

Notation list 53

Further mathematics HL guide 1

Purpose of this document

Introduction

This publication is intended to guide the planning, teaching and assessment of the subject in schools. Subject teachers are the primary audience, although it is expected that teachers will use the guide to inform students and parents about the subject.

This guide can be found on the subject page of the online curriculum centre (OCC) at http://occ.ibo.org, a password-protected IB website designed to support IB teachers. It can also be purchased from the IB store at http://store.ibo.org.

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

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

First examinations 2014

2 Further mathematics HL guide

Introduction

The Diploma Programme

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

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

Studies in language and literature

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The Diploma Programme

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

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

The core of the hexagonAll Diploma Programme students participate in the three course requirements that make up the core of the hexagon. Reflection on all these activities is a principle that lies at the heart of the thinking behind the Diploma Programme.

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

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


Introduction

Nature of the subject

IntroductionThe nature of mathematics can be summarized in a number of ways: for example, it can be seen as a well-defined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probably a combination of these, but there is no doubt that mathematical knowledge provides an important key to understanding the world in which we live. Mathematics can enter our lives in a number of ways: we buy produce in the market, consult a timetable, read a newspaper, time a process or estimate a length. Mathematics, for most of us, also extends into our chosen profession: visual artists need to learn about perspective; musicians need to appreciate the mathematical relationships within and between different rhythms; economists need to recognize trends in financial dealings; and engineers need to take account of stress patterns in physical materials. Scientists view mathematics as a language that is central to our understanding of events that occur in the natural world. Some people enjoy the challenges offered by the logical methods of mathematics and the adventure in reason that mathematical proof has to offer. Others appreciate mathematics as an aesthetic experience or even as a cornerstone of philosophy. This prevalence of mathematics in our lives, with all its interdisciplinary connections, provides a clear and sufficient rationale for making the study of this subject compulsory for students studying the full diploma.

Summary of courses availableBecause individual students have different needs, interests and abilities, there are four different courses in mathematics. These courses are designed for different types of students: those who wish to study mathematics in depth, either as a subject in its own right or to pursue their interests in areas related to mathematics; those who wish to gain a degree of understanding and competence to understand better their approach to other subjects; and those who may not as yet be aware how mathematics may be relevant to their studies and in their daily lives. Each course is designed to meet the needs of a particular group of students. Therefore, great care should be taken to select the course that is most appropriate for an individual student.

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

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

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

• their other choices of subjects within the framework of the Diploma Programme

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

• their choice of career.

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



Mathematical studies SLThis course is available only at standard level, and is equivalent in status to mathematics SL, but addresses different needs. It has an emphasis on applications of mathematics, and the largest section is on statistical techniques. It is designed for students with varied mathematical backgrounds and abilities. It offers students opportunities to learn important concepts and techniques and to gain an understanding of a wide variety of mathematical topics. It prepares students to be able to solve problems in a variety of settings, to develop more sophisticated mathematical reasoning and to enhance their critical thinking. The individual project is an extended piece of work based on personal research involving the collection, analysis and evaluation of data. Students taking this course are well prepared for a career in social sciences, humanities, languages or arts. These students may need to utilize the statistics and logical reasoning that they have learned as part of the mathematical studies SL course in their future studies.

Mathematics SLThis course caters for students who already possess knowledge of basic mathematical concepts, and who are equipped with the skills needed to apply simple mathematical techniques correctly. The majority of these students will expect to need a sound mathematical background as they prepare for future studies in subjects such as chemistry, economics, psychology and business administration.

Mathematics HLThis course caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems.

Further mathematics HLThis course is available only at higher level. It caters for students with a very strong background in mathematics who have attained a high degree of competence in a range of analytical and technical skills, and who display considerable interest in the subject. Most of these students will expect to study mathematics at university, either as a subject in its own right or as a major component of a related subject. The course is designed specifically to allow students to learn about a variety of branches of mathematics in depth and also to appreciate practical applications. It is expected that students taking this course will also be taking mathematics HL.

Note: Mathematics HL is an ideal course for students expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering or technology. It should not be regarded as necessary for such students to study further mathematics HL. Rather, further mathematics HL is an optional course for students with a particular aptitude and interest in mathematics, enabling them to study some wider and deeper aspects of mathematics, but is by no means a necessary qualification to study for a degree in mathematics.

Further mathematics HL—course detailsThe nature of the subject is such that it focuses on different branches of mathematics to encourage students to appreciate the diversity of the subject. Students should be equipped at this stage in their mathematical progress to begin to form an overview of the characteristics that are common to all mathematical thinking, independent of topic or branch.

Further mathematics HL guide6


All categories of student can register for mathematics HL only or for further mathematics HL only or for both. However, students registering for further mathematics HL will be presumed to know the topics in the core syllabus of mathematics HL and to have studied one of the options, irrespective of whether they have also registered for mathematics HL.

Examination questions are intended to be comparable in difficulty with those set on the four options in the mathematics HL course. The challenge for students will be to reach an equivalent level of understanding across all topics. There is no internal assessment component in this course. Although not a requirement, it is expected that students studying further mathematics HL will also be studying mathematics HL and therefore will be required to undertake a mathematical exploration for the internal assessment component of that course.

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

As previously stated, students registering for further mathematics HL will be presumed to know the topics in the core syllabus of mathematics HL and to have studied one of the options.

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

A continuum document called Mathematics: The MYP–DP continuum (November 2010) is available on the DP mathematics home pages of the OCC. This extensive publication focuses on the alignment of mathematics across the MYP and the DP. It was developed in response to feedback provided by IB World Schools, which expressed the need to articulate the transition of mathematics from the MYP to the DP. The publication also highlights the similarities and differences between MYP and DP mathematics, and is a valuable resource for teachers.

Mathematics and theory of knowledgeThe Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed that these all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by data from sense perception, mathematics is dominated by reason, and some mathematicians argue that their subject is a language, that it is, in some sense, universal. However, there is also no doubt that mathematicians perceive beauty in mathematics, and that emotion can be a strong driver in the search for mathematical knowledge.

As an area of knowledge, mathematics seems to supply a certainty perhaps missing in other disciplines. This may be related to the “purity” of the subject that makes it sometimes seem divorced from reality. However, mathematics has also provided important knowledge about the world, and the use of mathematics in science and technology has been one of the driving forces for scientific advances.



Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there “waiting to be discovered” or is it a human creation?

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

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

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

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

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

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


Introduction

Aims

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

1. enjoy mathematics, and develop an appreciation of the elegance and power of mathematics

2. develop an understanding of the principles and nature of mathematics

3. communicate clearly and confidently in a variety of contexts

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

5. employ and refine their powers of abstraction and generalization

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

7. appreciate how developments in technology and mathematics have influenced each other

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

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

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


Introduction

Assessment objectives

Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics HL course, students will be expected to demonstrate the following.

1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.

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

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

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

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

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


Syllabus outline

Syllabus

Syllabus componentTeaching hours

HL

All topics are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with all of the core topics in mathematics HL.

Topic 1

Linear algebra

48

Topic 2

Geometry

48

Topic 3

Statistics and probability

48

Topic 4

Sets, relations and groups

48

Topic 5

Calculus

48

Topic 6

Discrete mathematics

48

Note: One of topics 3–6 will be assumed to have been taught as part of the mathematics HL course and therefore the total teaching hours will be 240 not 288.

Total teaching hours 240


Syllabus

Approaches to the teaching and learning of further mathematics HL

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

Mathematical inquiryThe IB learner profile encourages learning by experimentation, questioning and discovery. In the IB classroom, students should generally learn mathematics by being active participants in learning activities rather than recipients of instruction. Teachers should therefore provide students with opportunities to learn through mathematical inquiry. This approach is illustrated in figure 2.

Explore the context

Make a conjecture

Extend

Prove

Accept

RejectTest the conjecture

Figure 2



Mathematical modelling and applicationsStudents should be able to use mathematics to solve problems in the real world. Engaging students in the mathematical modelling process provides such opportunities. Students should develop, apply and critically analyse models. This approach is illustrated in figure 3.

Pose a real-world problem

Develop a model

Extend

Reflect on and apply the model

Accept

RejectTest the model

Figure 3

TechnologyTechnology is a powerful tool in the teaching and learning of mathematics. Technology can be used to enhance visualization and support student understanding of mathematical concepts. It can assist in the collection, recording, organization and analysis of data. Technology can increase the scope of the problem situations that are accessible to students. The use of technology increases the feasibility of students working with interesting problem contexts where students reflect, reason, solve problems and make decisions.

As teachers tie together the unifying themes of mathematical inquiry, mathematical modelling and applications and the use of technology, they should begin by providing substantial guidance, and then gradually encourage students to become more independent as inquirers and thinkers. IB students should learn to become strong communicators through the language of mathematics. Teachers should create a safe learning environment in which students are comfortable as risk-takers.

Teachers are encouraged to relate the mathematics being studied to other subjects and to the real world, especially topics that have particular relevance or are of interest to their students. Everyday problems and questions should be drawn into the lessons to motivate students and keep the material relevant; suggestions are provided in the “Links” column of the syllabus.



For further information on “Approaches to teaching a DP course”, please refer to the publication The Diploma Programme: From principles into practice (April 2009). To support teachers, a variety of resources can be found on the OCC and details of workshops for professional development are available on the public website.

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

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

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

Appl real-life examples and links to other DP subjects

Aim 8 moral, social and ethical implications of the sub-topic

Int international-mindedness

TOK suggestions for discussion

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

Notes on the syllabus• Formulae are only included in this document where there may be some ambiguity. All formulae required

for the course are in the mathematics HL and further mathematics HL formula booklet.

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

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

Course of studyThe content of topics 1 and 2 as well as three out of the four remaining topics in the syllabus must be taught, although not necessarily in the order in which they appear in this guide. Teachers are expected to construct a course of study that addresses the needs of their students and includes, where necessary, the topics noted in prior learning.



Time allocationThe recommended teaching time for higher level courses is 240 hours. The time allocations given in this guide are approximate, and are intended to suggest how the 240 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.

Use of calculatorsStudents are expected to have access to a graphic display calculator (GDC) at all times during the course. The minimum requirements are reviewed as technology advances, and updated information will be provided to schools. It is expected that teachers and schools monitor calculator use with reference to the calculator policy. Regulations covering the types of calculators allowed in examinations are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematics HL/SL: Graphic display calculators teacher support material (May 2005) and on the OCC.

Mathematics HL and further mathematics HL formula bookletEach student is required to have access to a clean copy of this booklet during the examination. It is recommended that teachers ensure students are familiar with the contents of this document from the beginning of the course. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there are no printing errors, and ensure that there are sufficient copies available for all students.

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


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null

spac

e.

Row

rank

and

col

umn

rank

and

thei

r equ

ality

.

Pivo

ting

is th

e 1

2R

aR+

met

hod.

1.4

Solu

tions

of m

line

ar e

quat

ions

in n

unk

now

ns:

both

aug

men

ted

mat

rix m

etho

d, le

adin

g to

re

duce

d ro

w e

chel

on fo

rm m

etho

d, a

nd in

vers

e m

atrix

met

hod,

whe

n ap

plic

able

.

Incl

ude

the

non-

exis

tent

, uni

que

and

infin

itely

m

any

case

s. In

t: A

lso

know

n as

Gau

ssia

n el

imin

atio

n af

ter

the

Ger

man

mat

hem

atic

ian

CF

Gau

ss (1

777–

1855

).

Aim

8: L

inea

r opt

imiz

atio

n w

as d

evel

oped

by

Geo

rge

Dan

tzig

in th

e 19

40s a

s a w

ay o

f al

loca

ting

scar

ce re

sour

ces.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.5

The

vect

or sp

ace

n

.

Line

ar c

ombi

natio

ns o

f vec

tors

.

Span

ning

set.

Line

ar in

depe

nden

ce o

f vec

tors

.

A

im 8

: Lin

k to

Fou

rier a

naly

sis f

or w

aves

, w

hich

has

man

y us

es in

phy

sics

(Phy

sics

4.5

) an

d en

gine

erin

g.

Bas

is a

nd d

imen

sion

for a

vec

tor s

pace

.

Subs

pace

s.

Incl

ude

linea

r dep

ende

nce

and

use

of

dete

rmin

ants

.

Stud

ents

shou

ld b

e fa

mili

ar w

ith th

e te

rm

orth

ogon

al.

1.6

Line

ar tr

ansf

orm

atio

ns:

T(u

+ v)

= T

(u) +

T(v

), T(

ku) =

kT(

u).

Com

posi

tion

of li

near

tran

sfor

mat

ions

.

A

im 8

: Lor

enz

trans

form

atio

ns a

nd th

eir u

se in

re

lativ

ity a

nd q

uant

um m

echa

nics

(Phy

sics

13

.1).

Dom

ain,

rang

e, c

odom

ain

and

kern

el.

The

kern

el o

f a li

near

tran

sfor

mat

ion

is th

e nu

ll sp

ace

of it

s mat

rix re

pres

enta

tion.

R

esul

t and

pro

of th

at th

e ke

rnel

is a

subs

pace

of

the

dom

ain.

Res

ult a

nd p

roof

that

the

rang

e is

a su

bspa

ce o

f th

e co

dom

ain.

Ran

k-nu

llity

theo

rem

(pro

of n

ot re

quire

d).

Dim

(dom

ain)

= D

im(r

ange

) + D

im(k

erne

l)

whe

re

Dim

(ran

ge) =

rank

and

Dim

(ker

nel)

= nu

llity

.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.7

Res

ult t

hat a

ny li

near

tran

sfor

mat

ion

can

be

repr

esen

ted

by a

mat

rix, a

nd th

e co

nver

se o

f th

is re

sult.

Res

ult t

hat t

he n

umbe

rs o

f lin

early

inde

pend

ent

row

s and

col

umns

are

equ

al, a

nd th

is is

the

dim

ensi

on o

f the

rang

e of

the

trans

form

atio

n (p

roof

not

requ

ired)

.

App

licat

ion

of li

near

tran

sfor

mat

ions

to

solu

tions

of s

yste

m o

f equ

atio

ns.

Solu

tion

of A

x =

b.

Usi

ng

(par

ticul

ar so

lutio

n) +

(any

mem

ber o

f the

nul

l sp

ace)

.

1.8

Geo

met

ric tr

ansf

orm

atio

ns re

pres

ente

d by

2

× 2

mat

rices

incl

ude

gene

ral r

otat

ion,

gen

eral

re

flect

ion

in

(tan

)y

xα

=, s

tretc

hes p

aral

lel t

o ax

es, s

hear

s par

alle

l to

axes

, and

pro

ject

ion

onto

(ta

n)

yx

α=

.

Com

posi

tions

of t

he a

bove

tran

sfor

mat

ions

.

A

im 8

: Com

pute

r gra

phic

s in

thre

e-di

men

sion

al m

odel

ling.

Geo

met

ric in

terp

reta

tion

of d

eter

min

ant.

New

are

ade

tol

d ar

ea.

A=

×


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.9

Eige

nval

ues a

nd e

igen

vect

ors o

f 2 ×

2

mat

rices

.

Cha

ract

eris

tic p

olyn

omia

l of 2

× 2

mat

rices

.

Dia

gona

lizat

ion

of 2

× 2

mat

rices

(res

trict

ed to

th

e ca

se w

here

ther

e ar

e di

stin

ct re

al

eige

nval

ues)

.

App

licat

ions

to p

ower

s of 2

× 2

mat

rices

.

Geo

met

ric in

terp

reta

tion.

St

ocha

stic

pro

cess

es.

Stoc

k m

arke

t val

ues a

nd tr

ends

.

Inva

riant

stat

es.

Rep

rese

ntat

ion

of c

onic

s.

TOK

: “W

e ca

n us

e m

athe

mat

ics s

ucce

ssfu

lly

to m

odel

real

-wor

ld p

roce

sses

. Is t

his b

ecau

se

we

crea

te m

athe

mat

ics t

o m

irror

the

wor

ld o

r be

caus

e th

e w

orld

is in

trins

ical

ly

mat

hem

atic

al?”

For e

xam

ple,

the

natu

ral f

requ

ency

of a

n ob

ject

ca

n be

cha

ract

eriz

ed b

y th

e ei

genv

alue

of

smal

lest

mag

nitu

de (1

940

Taco

ma

Nar

row

s B

ridge

dis

aste

r).

Aim

8: D

ampi

ng n

oise

in c

ar d

esig

n, te

st fo

r cr

acks

in so

lid o

bjec

ts, o

il ex

plor

atio

n, th

e G

oogl

e Pa

geR

ank

form

ula,

and

“th

e $2

5 bi

llion

dol

lar e

igen

vect

or”.

Mar

kov

chai

ns, l

ink

with

gen

etic

s (B

iolo

gy 4

).


Syllabus content

Top

ic 2

—G

eom

etry

48

hou

rs

The

aim

of t

his s

ectio

n is

to d

evel

op st

uden

ts’ g

eom

etric

intu

ition

, vis

ualiz

atio

n an

d de

duct

ive

reas

onin

g.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

2.1

Sim

ilar a

nd c

ongr

uent

tria

ngle

s.

Eucl

id’s

theo

rem

for p

ropo

rtion

al se

gmen

ts in

a

right

-ang

led

trian

gle.

T

OK

, Int

: The

influ

ence

of E

uclid

’s a

xiom

atic

ap

proa

ch o

n ph

iloso

phy

(Des

carte

s) a

nd

polit

ics (

Jeff

erso

n: A

mer

ican

Dec

lara

tion

of

Inde

pend

ence

).

TOK

: Hip

pasu

s’ e

xist

ence

pro

of fo

r irr

atio

nal

num

bers

and

impa

ct o

n se

para

te d

evel

opm

ent

of n

umbe

r and

geo

met

ry.

TOK

: Cris

is o

ver n

on-E

uclid

ean

geom

etry

pa

ralle

ls w

ith th

at o

f Can

tor’

s set

theo

ry.

2.2

Cen

tres o

f a tr

iang

le: o

rthoc

entre

, inc

entre

, ci

rcum

cent

re a

nd c

entro

id.

The

term

s alti

tude

, ang

le b

isec

tor,

perp

endi

cula

r bis

ecto

r, m

edia

n.

Proo

f of c

oncu

rren

cy th

eore

ms.

The

term

s ins

crib

ed a

nd c

ircum

scrib

ed.

App

l: ce

ntre

of m

ass,

trian

gula

tion.

2.3

Circ

le g

eom

etry

. A

ngle

at c

entre

theo

rem

and

cor

olla

ries.

Tang

ents

; arc

s, ch

ords

and

seca

nts.

In a

cyc

lic q

uadr

ilate

ral,

oppo

site

ang

les a

re

supp

lem

enta

ry, a

nd th

e co

nver

se.

The

tang

ent–

seca

nt a

nd se

cant

–sec

ant

theo

rem

s and

inte

rsec

ting

chor

ds th

eore

ms.

2.4

Ang

le b

isec

tor t

heor

em; A

pollo

nius

’ circ

le

theo

rem

, Men

elau

s’ th

eore

m; C

eva’

s the

orem

; Pt

olem

y’s t

heor

em fo

r cyc

lic q

uadr

ilate

rals

.

Proo

fs o

f the

se th

eore

ms a

nd c

onve

rses

.

The

use

of th

ese

theo

rem

s to

prov

e fu

rther

re

sults

.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

2.5

Find

ing

equa

tions

of l

oci.

Coo

rdin

ate

geom

etry

of t

he c

ircle

.

Tang

ents

to a

circ

le.

The

equa

tions

2

22

()

()

xh

yk

r−

+−

= a

nd

22

0x

ydx

eyf

++

++

=.

TOK

: Con

sequ

ence

s of D

esca

rtes’

uni

ficat

ion

of a

lgeb

ra a

nd g

eom

etry

.

2.6

Con

ic se

ctio

ns.

The

para

bola

, elli

pse

and

hype

rbol

a, in

clud

ing

rect

angu

lar h

yper

bola

.

Focu

s–di

rect

rix d

efin

ition

s.

Tang

ents

and

nor

mal

s.

The

stan

dard

form

s 2

4y

ax=

, 2

2

22

1x

ya

b+

=,

22

22

1x

ya

b−

= a

nd th

eir t

rans

latio

ns.

App

l: Sa

telli

te d

ish,

hea

dlig

ht, o

rbits

, pr

ojec

tiles

(Phy

sics

9.1

).

TOK

: Kep

ler’

s diff

icul

ties a

ccep

ting

that

an

orbi

t was

not

a “

perf

ect”

circ

le.

2.7

Para

met

ric e

quat

ions

.

Para

met

ric d

iffer

entia

tion.

Tang

ents

and

nor

mal

s.

The

stan

dard

par

amet

ric e

quat

ions

of t

he

circ

le, p

arab

ola,

elli

pse,

rect

angu

lar h

yper

bola

, hy

perb

ola.

cos

xa

θ=

, si

ny

aθ

=,

2x

at=

, 2

yat

=,

cos

xa

θ=

, si

ny

bθ

=,

xct

=,

cy

t=

,

sec

xa

θ=

, ta

ny

bθ

=.

2.8

The

gene

ral c

onic

2

22

0ax

bxy

cydx

eyf

++

++

+=

, an

d th

e qu

adra

tic fo

rm

T2

22

axbx

ycy

=+

+x

Ax

.

Dia

gona

lizin

g th

e m

atrix

A w

ith th

e ro

tatio

n m

atrix

P a

nd re

duci

ng th

e ge

nera

l con

ic to

st

anda

rd fo

rm.

The

gene

ral c

onic

can

be

rota

ted

to g

ive

the

form

22

12

0x

ydx

eyf

λλ

++

++

=,

whe

re

1λ a

nd

2λ a

re th

e ei

genv

alue

s of t

he

mat

rix A

in th

e qu

adra

tic fo

rm

T xA

x.


Syllabus content

Geometry theorems—clarification of theorems used in topic 2Teachers and students should be aware that some of the theorems mentioned in this section may be known by other names, or some names of theorems may be associated with different statements in some textbooks. To avoid confusion, on examination papers, theorems that may be misinterpreted are defined below.

Euclid’s theorem for proportional segments in a right-angled triangleThe proportional segments p and q satisfy the following:

h2 = pq,

a2 = pc,

b2 = qc.

Angle at centre theoremThe angle subtended by an arc at the circumference is half that subtended by the same arc at the centre.

CorollariesAngles subtended at the circumference by the same arc are equal.

ab

h

c pq

2x

O

x

x x


Syllabus content

The angle in a semicircle is a right angle.

The alternate segment theorem: The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.

The tangent is perpendicular to the radius at the point of tangency.

The intersecting chords theoremab = cd

O

x

x

O

a

d

c

b


Syllabus content

The tangent–secant and secant–secant theoremsPT2 = PA × PB = PC × PD

Apollonius’ circle theorem (circle of Apollonius)If A and B are two fixed points such that PA

PB is a constant not equal to one, then the locus of P is a circle. This

is called the circle of Apollonius.

Included: the converse of this theorem.

Menelaus’ theoremIf a transversal meets the sides [BC], [CA] and [AB] of a triangle at D, E and F respectively, then

BDDC

CEEA

AFFB

× × = −1.

Converse: if D, E and F are points on the sides [BC], [CA] and [AB], respectively, of a triangle such that BDDC

CEEA

AFFB

× × = −1, then D, E and F are collinear.

TP

C

D

B

A

F

B

E

C D

EF

A

B C D


Syllabus content

Ceva’s theoremIf three concurrent lines are drawn through the vertices A, B and C of a triangle ABC to meet the opposite sides at D, E and F respectively, thenBDDC

� � ��CEEA

AFFB

1.

Converse: if D, E and F are points on [BC], [CA] and [AB], respectively, such that BDDC

� � ��CEEA

AFFB

1, then [AD], [BE] and [CF] are concurrent.

Note on Ceva’s theorem and Menelaus’ theoremThe statements and proofs of these theorems presuppose the idea of sensed magnitudes. Two segments [AB] and [PQ] of the same or parallel lines are said to have the same sense or opposite senses (or are sometimes calledlikeorunlike)accordingtowhetherthedisplacementsA→BandP→Qareinthesameoroppositedirections. The idea of sensed magnitudes may be used to prove the following theorem:

If A, B and C are any three collinear points, then AB BC CA 0, where AB, BC and CA denote sensed magnitudes.

Ptolemy’s theoremIf a quadrilateral is cyclic, the sum of the products of the two pairs of opposite sides equals the products of the diagonals. That is, for a cyclic quadrilateral ABCD, AB CD BC DA AC BD .

A

D

OE

C

F

B

A

E

CDB

F

O

A

D

B

C


Syllabus content

Angle bisector theoremThe angle bisector of an angle of a triangle divides the side of the triangle opposite the angle into segments proportional to the sides adjacent to the angle.

If ABC is the given triangle with (AD) as the bisector of angle BAC intersecting (BC) at point D, then BDDC

ABAC

� for internal bisectors

and

BDDC

ABAC

�� for external bisectors.

BÂD CÂD CÂD EÂD

Included: the converse of this theorem.

A

B D C

E

DCB

A


Syllabus content

Top

ic 3

—St

atis

tics

and

prob

abili

ty

48 h

ours

Th

e ai

ms o

f thi

s top

ic a

re to

allo

w st

uden

ts th

e op

portu

nity

to a

ppro

ach

stat

istic

s in

a pr

actic

al w

ay; t

o de

mon

stra

te a

goo

d le

vel o

f sta

tistic

al u

nder

stan

ding

; an

d to

und

erst

and

whi

ch s

ituat

ions

app

ly a

nd t

o in

terp

ret

the

give

n re

sults

. It

is e

xpec

ted

that

GD

Cs

will

be

used

thr

ough

out

this

opt

ion

and

that

the

m

inim

um r

equi

rem

ent

of a

GD

C w

ill b

e to

fin

d th

e pr

obab

ility

dis

tribu

tion

func

tion

(pdf

), cu

mul

ativ

e di

strib

utio

n fu

nctio

n (c

df),

inve

rse

cum

ulat

ive

dist

ribut

ion

func

tion,

p-v

alue

s an

d te

st s

tatis

tics,

incl

udin

g ca

lcul

atio

ns f

or t

he f

ollo

win

g di

strib

utio

ns:

bino

mia

l, Po

isso

n, n

orm

al a

nd t

. St

uden

ts a

re

expe

cted

to se

t up

the

prob

lem

mat

hem

atic

ally

and

then

read

the

answ

ers

from

the

GD

C, i

ndic

atin

g th

is w

ithin

thei

r writ

ten

answ

ers.

Cal

cula

tor-s

peci

fic o

r br

and-

spec

ific

lang

uage

shou

ld n

ot b

e us

ed w

ithin

thes

e ex

plan

atio

ns.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

3.1

Cum

ulat

ive

dist

ribut

ion

func

tions

for b

oth

disc

rete

and

con

tinuo

us d

istri

butio

ns.

Geo

met

ric d

istri

butio

n.

Neg

ativ

e bi

nom

ial d

istri

butio

n.

Prob

abili

ty g

ener

atin

g fu

nctio

ns fo

r dis

cret

e ra

ndom

var

iabl

es.

()

E()

()

Xx

xG

tt

PX

xt

==

=∑

. In

t: A

lso

know

n as

Pas

cal’s

dis

tribu

tion.

Usi

ng p

roba

bilit

y ge

nera

ting

func

tions

to fi

nd

mea

n, v

aria

nce

and

the

distr

ibut

ion

of th

e su

m

of n

inde

pend

ent r

ando

m v

aria

bles

.

A

im 8

: Sta

tistic

al c

ompr

essi

on o

f dat

a fil

es.

3.2

Line

ar tr

ansf

orm

atio

n of

a si

ngle

rand

om

varia

ble.

Mea

n of

line

ar c

ombi

natio

ns o

f n ra

ndom

va

riabl

es.

Var

ianc

e of

line

ar c

ombi

natio

ns o

f n

inde

pend

ent r

ando

m v

aria

bles

.

E()

E()

aXb

aX

b+

=+

, 2

Var

()

Var

()

aXb

aX

+=

.

Expe

ctat

ion

of th

e pr

oduc

t of i

ndep

ende

nt

rand

om v

aria

bles

. E(

)E(

)E(

)XY

XY

=.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

3.3

Unb

iase

d es

timat

ors a

nd e

stim

ates

.

Com

paris

on o

f unb

iase

d es

timat

ors b

ased

on

varia

nces

.

T is

an

unbi

ased

est

imat

or fo

r the

par

amet

er

θ if

E(

)T

θ=

.

1T is

a m

ore

effic

ient

esti

mat

or th

an

2T if

12

Var

()

Var

()

TT

<.

TO

K: M

athe

mat

ics a

nd th

e w

orld

. In

the

abse

nce

of k

now

ing

the

valu

e of

a p

aram

eter

, w

ill a

n un

bias

ed e

stim

ator

alw

ays b

e be

tter

than

a b

iase

d on

e?

X a

s an

unbi

ased

est

imat

or fo

r µ

.

2 S a

s an

unbi

ased

est

imat

or fo

r 2

σ.

1ni

i

XX

n=

=∑

.

()2

2

11

ni

i

XX

Sn

=

−=

−∑

.

3.4

A li

near

com

bina

tion

of in

depe

nden

t nor

mal

ra

ndom

var

iabl

es is

nor

mal

ly d

istri

bute

d. In

pa

rticu

lar,

22

~N

(,

)~

N,

XX

nσµσ

µ

⇒

.

The

cent

ral l

imit

theo

rem

.

A

im 8

/TO

K: M

athe

mat

ics a

nd th

e w

orld

. “W

ithou

t the

cen

tral l

imit

theo

rem

, the

re c

ould

be

no

stat

istic

s of a

ny v

alue

with

in th

e hu

man

sc

ienc

es.”

TO

K: N

atur

e of

mat

hem

atic

s. Th

e ce

ntra

l lim

it th

eore

m c

an b

e pr

oved

mat

hem

atic

ally

(f

orm

alis

m),

but i

ts tr

uth

can

be c

onfir

med

by

its a

pplic

atio

ns (e

mpi

ricis

m).


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

3.5

Con

fiden

ce in

terv

als f

or th

e m

ean

of a

nor

mal

po

pula

tion.

U

se o

f the

nor

mal

dis

tribu

tion

whe

n σ

is

know

n an

d us

e of

the

t-dis

tribu

tion

whe

n σ

is

unkn

own,

rega

rdle

ss o

f sam

ple

size

. The

cas

e of

mat

ched

pai

rs is

to b

e tre

ated

as a

n ex

ampl

e of

a si

ngle

sam

ple

tech

niqu

e.

TO

K: M

athe

mat

ics a

nd th

e w

orld

. Cla

imin

g br

and

A is

“be

tter”

on

aver

age

than

bra

nd B

ca

n m

ean

very

littl

e if

ther

e is

a la

rge

over

lap

betw

een

the

conf

iden

ce in

terv

als o

f the

two

mea

ns.

App

l: G

eogr

aphy

.

3.6

Nul

l and

alte

rnat

ive

hypo

thes

es,

0H

and

1

H.

Sign

ifica

nce

leve

l.

Crit

ical

regi

ons,

criti

cal v

alue

s, p-

valu

es, o

ne-

taile

d an

d tw

o-ta

iled

test

s.

Type

I an

d II

erro

rs, i

nclu

ding

cal

cula

tions

of

thei

r pro

babi

litie

s.

Test

ing

hypo

thes

es fo

r the

mea

n of

a n

orm

al

popu

latio

n.

Use

of t

he n

orm

al d

istri

butio

n w

hen σ

is

know

n an

d us

e of

the

t-dis

tribu

tion

whe

n σ

is

unkn

own,

rega

rdle

ss o

f sam

ple

size

. The

cas

e of

mat

ched

pai

rs is

to b

e tre

ated

as a

n ex

ampl

e of

a si

ngle

sam

ple

tech

niqu

e.

TO

K: M

athe

mat

ics a

nd th

e w

orld

. In

prac

tical

te

rms,

is sa

ying

that

a re

sult

is si

gnifi

cant

the

sam

e as

sayi

ng th

at it

is tr

ue?

TO

K: M

athe

mat

ics a

nd th

e w

orld

. Doe

s the

ab

ility

to te

st o

nly

certa

in p

aram

eter

s in

a po

pula

tion

affe

ct th

e w

ay k

now

ledg

e cl

aim

s in

the

hum

an sc

ienc

es a

re v

alue

d?

App

l: W

hen

is it

mor

e im

porta

nt n

ot to

mak

e a

Type

I er

ror a

nd w

hen

is it

mor

e im

porta

nt n

ot

to m

ake

a Ty

pe II

err

or?


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

3.7

Intro

duct

ion

to b

ivar

iate

dis

tribu

tions

. In

form

al d

iscu

ssio

n of

com

mon

ly o

ccur

ring

situ

atio

ns, e

g m

arks

in p

ure

mat

hem

atic

s and

st

atis

tics e

xam

s tak

en b

y a

clas

s of s

tude

nts,

sala

ry a

nd a

ge o

f tea

cher

s in

a ce

rtain

scho

ol.

The

need

for a

mea

sure

of a

ssoc

iatio

n be

twee

n th

e va

riabl

es a

nd th

e po

ssib

ility

of p

redi

ctin

g th

e va

lue

of o

ne o

f the

var

iabl

es g

iven

the

valu

e of

the

othe

r var

iabl

e.

App

l: G

eogr

aphi

c sk

ills.

Aim

8: T

he c

orre

latio

n be

twee

n sm

okin

g an

d lu

ng c

ance

r was

“di

scov

ered

” us

ing

mat

hem

atic

s. Sc

ienc

e ha

d to

just

ify th

e ca

use.

Cov

aria

nce

and

(pop

ulat

ion)

pro

duct

mom

ent

corr

elat

ion

coef

ficie

nt ρ

. C

ov(

,)

E[(

)

,

()]

E()

xy

xy

XY

XY

XYµ

µµ

µ

=−

−

=−

whe

re

E(),

E()

xy

XY

µµ

==

. C

ov(

,)

Var

()V

ar(

)X

YX

Yρ=

.

App

l: U

sing

tech

nolo

gy to

fit a

rang

e of

cur

ves

to a

set o

f dat

a.

Proo

f tha

t ρ =

0 in

the

case

of i

ndep

ende

nce

and ±1

in th

e ca

se o

f a li

near

rela

tions

hip

betw

een

X an

d Y.

The

use

of ρ

as a

mea

sure

of a

ssoc

iatio

n be

twee

n X

and

Y, w

ith v

alue

s nea

r 0 in

dica

ting

a w

eak

asso

ciat

ion

and

valu

es n

ear +

1 or

nea

r –1

indi

catin

g a

stro

ng a

ssoc

iatio

n.

TOK

: Mat

hem

atic

s and

the

wor

ld. G

iven

that

a

set o

f dat

a m

ay b

e ap

prox

imat

ely

fitte

d by

a

rang

e of

cur

ves,

whe

re w

ould

we

seek

for

know

ledg

e of

whi

ch e

quat

ion

is th

e “t

rue”

m

odel

?

Aim

8: T

he p

hysi

cist

Fra

nk O

ppen

heim

er

wro

te: “

Pred

ictio

n is

dep

ende

nt o

nly

on th

e as

sum

ptio

n th

at o

bser

ved

patte

rns w

ill b

e re

peat

ed.”

Thi

s is t

he d

ange

r of e

xtra

pola

tion.

Th

ere

are

man

y ex

ampl

es o

f its

failu

re in

the

past

, eg

shar

e pr

ices

, the

spre

ad o

f dis

ease

, cl

imat

e ch

ange

.

(con

tinue

d)

Def

initi

on o

f the

(sam

ple)

pro

duct

mom

ent

corr

elat

ion

coef

ficie

nt R

in te

rms o

f n p

aire

d ob

serv

atio

ns o

n X

and

Y. It

s app

licat

ion

to th

e es

timat

ion

of ρ

.

1

22

11

1

22

22

1

()(

)

()

()

.

n

ii

in

n

ii

ii

n

ii

i

n

ii

i

XX

YY

RX

XY

Y

XY

nXY

XnX

YnY

=

==

=

=

−−

=

−−

−=

−

−

∑ ∑∑

∑

∑∑


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

In

form

al in

terp

reta

tion

of r,

the

obse

rved

val

ue

of R

. Sca

tter d

iagr

ams.

Val

ues o

f r n

ear 0

indi

cate

a w

eak

asso

ciat

ion

betw

een

X an

d Y,

and

val

ues n

ear ±

1 in

dica

te a

st

rong

ass

ocia

tion.

(see

not

es a

bove

)

The

follo

win

g to

pics

are

bas

ed o

n th

e as

sum

ptio

n of

biv

aria

te n

orm

ality

. It

is e

xpec

ted

that

the

GD

C w

ill b

e us

ed

whe

reve

r pos

sibl

e in

the

follo

win

g w

ork.

Use

of t

he t-

stat

istic

to te

st th

e nu

ll hy

poth

esis

ρ

= 0.

22

1nR

R− − h

as th

e St

uden

t’s t-

dist

ribut

ion

with

(2)

n−

deg

rees

of f

reed

om.

Kno

wle

dge

of th

e fa

cts t

hat t

he re

gres

sion

of X

on

Y (

)E(

)|X

Yy

= a

nd Y

on

X (

)E(

)|Y

Xx

=

are

linea

r.

Leas

t-squ

ares

est

imat

es o

f the

se re

gres

sion

lin

es (p

roof

not

requ

ired)

.

The

use

of th

ese

regr

essi

on li

nes t

o pr

edic

t the

va

lue

of o

ne o

f the

var

iabl

es g

iven

the

valu

e of

th

e ot

her.

1

2

1

1

22

1()(

) (

)(

)

(),

n

ii

in

ii

n

ii

in

ii

xx

yy

xx

yy

yy

xy

nxy

yy

yny

=

=

=

=

−−

−=

−

−

−

=

−

−

∑

∑

∑ ∑

1

2

1

1

22

1()(

) (

)(

)

().

n

ii

in

ii

n

ii

in

ii

xx

yy

yy

xx

xx

xy

nxy

xx

xnx

=

=

=

=

−−

−=

−

−

−

=

−

−

∑

∑

∑ ∑


Syllabus content

Top

ic 4

—Se

ts, r

elat

ions

and

gro

ups

48 h

ours

Th

e aim

s of t

his t

opic

are t

o pr

ovid

e the

opp

ortu

nity

to st

udy

som

e im

porta

nt m

athe

mat

ical

conc

epts,

and

intro

duce

the p

rinci

ples

of p

roof

thro

ugh

abstr

act a

lgeb

ra.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.1

Fini

te a

nd in

finite

sets

. Sub

sets

.

Ope

ratio

ns o

n se

ts: u

nion

; int

erse

ctio

n;

com

plem

ent;

set d

iffer

ence

; sym

met

ric

diff

eren

ce.

TO

K: C

anto

r the

ory

of tr

ansf

inite

num

bers

, R

usse

ll’s p

arad

ox, G

odel

’s in

com

plet

enes

s th

eore

ms.

De

Mor

gan’

s law

s: d

istri

butiv

e, a

ssoc

iativ

e an

d co

mm

utat

ive

law

s (fo

r uni

on a

nd in

ters

ectio

n).

Illus

tratio

n of

thes

e la

ws u

sing

Ven

n di

agra

ms.

Stud

ents

may

be

aske

d to

pro

ve th

at tw

o se

ts

are

the

sam

e by

est

ablis

hing

that

AB

⊆ a

nd

BA

⊆.

App

l: Lo

gic,

Boo

lean

alg

ebra

, com

pute

r ci

rcui

ts.

4.2

Ord

ered

pai

rs: t

he C

arte

sian

pro

duct

of t

wo

sets

.

Rel

atio

ns: e

quiv

alen

ce re

latio

ns; e

quiv

alen

ce

clas

ses.

An

equi

vale

nce

rela

tion

on a

set f

orm

s a

parti

tion

of th

e se

t. A

ppl,

Int:

Sco

ttish

cla

ns.

4.3

Func

tions

: inj

ectio

ns; s

urje

ctio

ns; b

iject

ions

. Th

e te

rm c

odom

ain.

Com

posi

tion

of fu

nctio

ns a

nd in

vers

e fu

nctio

ns.

Kno

wle

dge

that

the

func

tion

com

posi

tion

is n

ot

a co

mm

utat

ive

oper

atio

n an

d th

at if

f is

a

bije

ctio

n fr

om se

t A to

set B

then

1

f− e

xist

s an

d is

a b

iject

ion

from

set B

to se

t A.

4.4

Bin

ary

oper

atio

ns.

A b

inar

y op

erat

ion ∗

on a

non

-em

pty

set S

is a

ru

le fo

r com

bini

ng a

ny tw

o el

emen

ts

,ab

S∈

to

giv

e a

uniq

ue e

lem

ent c

. Tha

t is,

in th

is

defin

ition

, a b

inar

y op

erat

ion

on a

set i

s not

ne

cess

arily

clo

sed.

Ope

ratio

n ta

bles

(Cay

ley

tabl

es).


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.5

Bin

ary

oper

atio

ns: a

ssoc

iativ

e, d

istri

butiv

e an

d co

mm

utat

ive

prop

ertie

s. Th

e ar

ithm

etic

ope

ratio

ns o

n

and

.

Exam

ples

of d

istri

butiv

ity c

ould

incl

ude

the

fact

that

, on

, mul

tiplic

atio

n is

dis

tribu

tive

over

add

ition

but

add

ition

is n

ot d

istri

butiv

e ov

er m

ultip

licat

ion.

TO

K: W

hich

are

mor

e fu

ndam

enta

l, th

e ge

nera

l mod

els o

r the

fam

iliar

exa

mpl

es?

4.6

The

iden

tity

elem

ent e

.

The

inve

rse

1a−

of a

n el

emen

t a.

Proo

f tha

t lef

t-can

cella

tion

and

right

-ca

ncel

latio

n by

an

elem

ent a

hol

d, p

rovi

ded

that

a h

as a

n in

vers

e.

Proo

fs o

f the

uni

quen

ess o

f the

iden

tity

and

inve

rse

elem

ents

.

Bot

h th

e rig

ht-id

entit

y a

ea

∗=

and

left-

iden

tity

ea

a∗

= m

ust h

old

if e

is a

n id

entit

y el

emen

t.

Bot

h 1

aa

e−

∗=

and

1

aa

e−∗

= m

ust h

old.

4.7

The

defin

ition

of a

gro

up {

,}

G∗

.

The

oper

atio

n ta

ble

of a

gro

up is

a L

atin

sq

uare

, but

the

conv

erse

is fa

lse.

For t

he se

t G u

nder

a g

iven

ope

ratio

n ∗:

• G

is c

lose

d un

der ∗;

• ∗

is a

ssoc

iativ

e;

• G

con

tain

s an

iden

tity

elem

ent;

• ea

ch e

lem

ent i

n G

has

an

inve

rse

in G

.

App

l: Ex

iste

nce

of fo

rmul

a fo

r roo

ts o

f po

lyno

mia

ls.

App

l: G

aloi

s the

ory

for t

he im

poss

ibili

ty o

f su

ch fo

rmul

ae fo

r pol

ynom

ials

of d

egre

e 5

or

high

er.

Abe

lian

grou

ps.

ab

ba

∗=

∗, f

or a

ll ,ab

G∈

.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.8

Exam

ples

of g

roup

s:

•

, ,

and

u

nder

add

ition

;

• in

tege

rs u

nder

add

ition

mod

ulo

n;

• no

n-ze

ro in

tege

rs u

nder

mul

tiplic

atio

n,

mod

ulo

p, w

here

p is

prim

e;

Cro

ss-to

pic

ques

tions

may

be

set i

n fu

rther

m

athe

mat

ics e

xam

inat

ions

, so

ther

e m

ay b

e qu

estio

ns o

n gr

oups

of m

atric

es.

App

l: R

ubik

’s c

ube,

tim

e m

easu

res,

crys

tal

stru

ctur

e, sy

mm

etrie

s of m

olec

ules

, stru

t and

ca

ble

cons

truct

ions

, Phy

sics

H2.

2 (s

peci

al

rela

tivity

), th

e 8-

fold

way

, sup

ersy

mm

etry

.

• sy

mm

etrie

s of p

lane

figu

res,

incl

udin

g eq

uila

tera

l tria

ngle

s and

rect

angl

es;

• in

verti

ble

func

tions

und

er c

ompo

sitio

n of

fu

nctio

ns.

The

com

posi

tion

21

TT

den

otes

1T fo

llow

ed

by2T.

4.9

The

orde

r of a

gro

up.

The

orde

r of a

gro

up e

lem

ent.

Cyc

lic g

roup

s.

Gen

erat

ors.

Proo

f tha

t all

cycl

ic g

roup

s are

Abe

lian.

A

ppl:

Mus

ic c

ircle

of f

ifths

, prim

e nu

mbe

rs.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.10

Pe

rmut

atio

ns u

nder

com

posi

tion

of

perm

utat

ions

.

Cyc

le n

otat

ion

for p

erm

utat

ions

.

Res

ult t

hat e

very

per

mut

atio

n ca

n be

writ

ten

as

a co

mpo

sitio

n of

dis

join

t cyc

les.

The

orde

r of a

com

bina

tion

of c

ycle

s.

On

exam

inat

ion

pape

rs: t

he fo

rm

12

33

12

p

=

or i

n cy

cle

nota

tion

(132

) will

be u

sed

to re

pres

ent t

he p

erm

utat

ion

13

→,

21

→, 3

2.→

App

l: C

rypt

ogra

phy,

cam

pano

logy

.

4.11

Su

bgro

ups,

prop

er su

bgro

ups.

A p

rope

r sub

grou

p is

neith

er th

e gro

up it

self

nor

the s

ubgr

oup

cont

aini

ng o

nly

the i

dent

ity el

emen

t.

Use

and

pro

of o

f sub

grou

p te

sts.

Supp

ose

that

{,

}G∗

is a

gro

up a

nd H

is a

no

n-em

pty

subs

et o

f G. T

hen

{,

}H

∗ is

a

subg

roup

of {

,}

G∗

if

1a

bH

−∗

∈ w

hene

ver

,ab

H∈

.

Supp

ose

that

{,

}G∗

is a

fini

te g

roup

and

H is

a

non-

empt

y su

bset

of G

. The

n {

,}

H∗

is a

su

bgro

up o

f {,

}G∗

if H

is c

lose

d un

der ∗.

Def

initi

on a

nd e

xam

ples

of l

eft a

nd ri

ght c

oset

s of

a su

bgro

up o

f a g

roup

.

Lagr

ange

’s th

eore

m.

Use

and

pro

of o

f the

resu

lt th

at th

e or

der o

f a

finite

gro

up is

div

isib

le b

y th

e or

der o

f any

el

emen

t. (C

orol

lary

to L

agra

nge’

s the

orem

.)

A

ppl:

Prim

e fa

ctor

izat

ion,

sym

met

ry b

reak

ing.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.12

D

efin

ition

of a

gro

up h

omom

orph

ism

. In

finite

gro

ups a

s wel

l as f

inite

gro

ups.

Let {

,*}

G a

nd {

,}

H

be

grou

ps, t

hen

the

func

tion

:fG

H→

is a

hom

omor

phis

m if

(

*)

()

()

fa

bf

af

b=

fo

r all

,ab

G∈

.

Def

initi

on o

f the

ker

nel o

f a h

omom

orph

ism

.

Proo

f tha

t the

ker

nel a

nd ra

nge

of a

ho

mom

orph

ism

are

subg

roup

s.

If :fG

H→

is a

gro

up h

omom

orph

ism

, the

n K

er(

)f is

the

set o

f a

G∈

such

that

(

)H

fa

e=

.

Proo

f of h

omom

orph

ism

pro

perti

es fo

r id

entit

ies a

nd in

vers

es.

Iden

tity:

let

Ge a

nd

He b

e th

e id

entit

y el

emen

ts

of (

,)

G∗

and

(,

)H

, res

pect

ivel

y, th

en

()

GH

fe

e=

.

Inve

rse:

(

)11

()

()

fa

fa

−−

= fo

r all

aG

∈.

Isom

orph

ism

of g

roup

s. In

finite

gro

ups a

s wel

l as f

inite

gro

ups.

The

hom

omor

phis

m

:fG

H→

is a

n is

omor

phis

m if

f is

bije

ctiv

e.

The

orde

r of a

n el

emen

t is u

ncha

nged

by

an

isom

orph

ism

.


Syllabus content

Top

ic 5

—C

alcu

lus

48 h

ours

Th

e ai

ms o

f thi

s top

ic a

re to

intro

duce

lim

it th

eore

ms a

nd c

onve

rgen

ce o

f ser

ies,

and

to u

se c

alcu

lus r

esul

ts to

solv

e di

ffer

entia

l equ

atio

ns.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.1

Infin

ite se

quen

ces o

f rea

l num

bers

and

thei

r co

nver

genc

e or

div

erge

nce.

In

form

al tr

eatm

ent o

f lim

it of

sum

, diff

eren

ce,

prod

uct,

quot

ient

; squ

eeze

theo

rem

.

Div

erge

nt is

take

n to

mea

n no

t con

verg

ent.

TO

K: Z

eno’

s par

adox

, im

pact

of i

nfin

ite

sequ

ence

s and

lim

its o

n ou

r und

erst

andi

ng o

f th

e ph

ysic

al w

orld

.

5.2

Con

verg

ence

of i

nfin

ite se

ries.

Test

s for

con

verg

ence

: com

paris

on te

st; l

imit

com

paris

on te

st; r

atio

test

; int

egra

l tes

t.

The

sum

of a

serie

s is t

he li

mit

of th

e se

quen

ce

of it

s par

tial s

ums.

Stud

ents

shou

ld b

e aw

are

that

if l

im0

nx

x→∞

=

then

the

serie

s is n

ot n

eces

saril

y co

nver

gent

, bu

t if

lim0

nx

x→∞

≠, t

he se

ries d

iver

ges.

TO

K: E

uler

’s id

ea th

at

1 21

11

1−

+−

+=

.

Was

it a

mis

take

or j

ust a

n al

tern

ativ

e vi

ew?

The

p-se

ries,

1 p n∑

. 1 p n

∑ is

con

verg

ent f

or

1p>

and

div

erge

nt

othe

rwis

e. W

hen

1p=

, thi

s is t

he h

arm

onic

se

ries.

Serie

s tha

t con

verg

e ab

solu

tely

.

Serie

s tha

t con

verg

e co

nditi

onal

ly.

Con

ditio

ns fo

r con

verg

ence

.

Alte

rnat

ing

serie

s.

Pow

er se

ries:

radi

us o

f con

verg

ence

and

in

terv

al o

f con

verg

ence

. Det

erm

inat

ion

of th

e ra

dius

of c

onve

rgen

ce b

y th

e ra

tio te

st.

The

abso

lute

val

ue o

f the

trun

catio

n er

ror i

s le

ss th

an th

e ne

xt te

rm in

the

serie

s.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.3

Con

tinui

ty a

nd d

iffer

entia

bilit

y of

a fu

nctio

n at

a

poin

t. Te

st fo

r con

tinui

ty:

()

()

()

limlim

xa–

xa+

fx

=f

a=

fx

→→

.

Con

tinuo

us fu

nctio

ns a

nd d

iffer

entia

ble

func

tions

. Te

st fo

r diff

eren

tiabi

lity:

f is

con

tinuo

us a

t a a

nd

()

0

()

lim h

fa

h–

fa

h→

−

+ a

nd

()

0

()

lim h+

fa

h–

fa

h→

+ e

xist

and

are

equ

al.

Stud

ents

shou

ld b

e aw

are

that

a fu

nctio

n m

ay

be c

ontin

uous

but

not

diff

eren

tiabl

e at

a p

oint

, eg

()

fx

=x

and

sim

ple

piec

ewis

e fu

nctio

ns.

5.4

The

inte

gral

as a

lim

it of

a su

m; l

ower

and

up

per R

iem

ann

sum

s.

Int:

How

clo

se w

as A

rchi

med

es to

inte

gral

ca

lcul

us?

Int:

Con

tribu

tion

of A

rab,

Chi

nese

and

Indi

an

mat

hem

atic

ians

to th

e de

velo

pmen

t of c

alcu

lus.

Aim

8: L

eibn

iz v

ersu

s New

ton

vers

us th

e “g

iant

s” o

n w

hose

shou

lder

s the

y st

ood—

who

de

serv

es c

redi

t for

mat

hem

atic

al p

rogr

ess?

TOK

: Con

side

r

1f

x=

x, 1

x≤

≤∞

.

An

infin

ite a

rea

swee

ps o

ut a

fini

te v

olum

e.

Can

this

be

reco

ncile

d w

ith o

ur in

tuiti

on?

Wha

t do

es th

is te

ll us

abo

ut m

athe

mat

ical

kn

owle

dge?

Fund

amen

tal t

heor

em o

f cal

culu

s. d

()d

()

d

x a

fy

y=

fx

x

∫

.

Impr

oper

inte

gral

s of t

he ty

pe

()d

a

fx

x∞ ∫

.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.5

Firs

t-ord

er d

iffer

entia

l equ

atio

ns.

Geo

met

ric in

terp

reta

tion

usin

g sl

ope

field

s, in

clud

ing

iden

tific

atio

n of

isoc

lines

.

A

ppl:

Rea

l-life

diff

eren

tial e

quat

ions

, eg

New

ton’

s law

of c

oolin

g,

popu

latio

n gr

owth

,

carb

on d

atin

g.

Num

eric

al so

lutio

n of

d(

,)

dy=

fx

yx

usin

g Eu

ler’

s met

hod.

Var

iabl

es se

para

ble.

Hom

ogen

eous

diff

eren

tial e

quat

ion

d dyy

=f

xx

usin

g th

e su

bstit

utio

n y

= vx

.

Solu

tion

of

y′ +

P(x)

y =

Q(x

),

usin

g th

e in

tegr

atin

g fa

ctor

.

1(

,)

nn

nn

yy

hfx

y+=

+,

1n

nx

xh

+=

+,

whe

re h

is a

con

stan

t.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.6

Rol

le’s

theo

rem

.

Mea

n va

lue

theo

rem

.

In

t, TO

K: I

nflu

ence

of B

ourb

aki o

n un

ders

tand

ing

and

teac

hing

of m

athe

mat

ics.

Int:

Com

pare

with

wor

k of

the

Ker

ala

scho

ol.

Tayl

or p

olyn

omia

ls; t

he L

agra

nge

form

of t

he

erro

r ter

m.

App

licat

ions

to th

e ap

prox

imat

ion

of fu

nctio

ns;

form

ula

for t

he e

rror

term

, in

term

s of t

he v

alue

of

the

(n +

1)th

der

ivat

ive

at a

n in

term

edia

te

poin

t.

Mac

laur

in se

ries f

or e

x, s

inx

, cos

x,

ln(1

)x+

, (1

)px

+,

p∈

.

Use

of s

ubst

itutio

n, p

rodu

cts,

inte

grat

ion

and

diff

eren

tiatio

n to

obt

ain

othe

r ser

ies.

Tayl

or se

ries d

evel

oped

from

diff

eren

tial

equa

tions

.

Stud

ents

shou

ld b

e aw

are

of th

e in

terv

als o

f co

nver

genc

e.

5.7

The

eval

uatio

n of

lim

its o

f the

form

()

()

lim xa

fx

gx

→ a

nd

()

()

lim x

fx

gx

→∞

.

The

inde

term

inat

e fo

rms

0 0 a

nd ∞ ∞

.

Usi

ng l’

Hôp

ital’s

rule

or t

he T

aylo

r ser

ies.

Rep

eate

d us

e of

l’H

ôpita

l’s ru

le.


Syllabus content

Top

ic 6

—D

iscr

ete

mat

hem

atic

s 48

hou

rs

The

aim

of t

his t

opic

is to

pro

vide

the

oppo

rtuni

ty fo

r stu

dent

s to

enga

ge in

logi

cal r

easo

ning

, alg

orith

mic

thin

king

and

app

licat

ions

.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.1

Stro

ng in

duct

ion.

Pige

on-h

ole

prin

cipl

e.

For e

xam

ple,

pro

ofs o

f the

fund

amen

tal

theo

rem

of a

rithm

etic

and

the

fact

that

a tr

ee

with

n v

ertic

es h

as n

– 1

edg

es.

TOK

: Mat

hem

atic

s and

kno

wle

dge

clai

ms.

The

diff

eren

ce b

etw

een

proo

f and

con

ject

ure,

eg

Gol

dbac

h’s c

onje

ctur

e. C

an a

mat

hem

atic

al

stat

emen

t be

true

befo

re it

is p

rove

n?

TOK

: Pro

of b

y co

ntra

dict

ion.

6.2

|a

bb

na⇒

= fo

r som

e n∈

.

The

theo

rem

|

ab

and

|

|()

ac

abx

cy⇒

±

whe

re

,xy∈

. Th

e di

visi

on a

lgor

ithm

abq

r=

+, 0

rb

≤<

.

Div

isio

n an

d Eu

clid

ean

algo

rithm

s.

The

grea

test

com

mon

div

isor

, gcd

(,

)a

b, a

nd

the

leas

t com

mon

mul

tiple

, lcm

(,

)a

b, o

f in

tege

rs a

and

b.

Prim

e nu

mbe

rs; r

elat

ivel

y pr

ime

num

bers

and

th

e fu

ndam

enta

l the

orem

of a

rithm

etic

.

The

Eucl

idea

n al

gorit

hm fo

r det

erm

inin

g th

e gr

eate

st c

omm

on d

ivis

or o

f tw

o in

tege

rs.

Int:

Euc

lidea

n al

gorit

hm c

onta

ined

in E

uclid

’s

Elem

ents

, writ

ten

in A

lexa

ndria

abo

ut

300

BC

E.

Aim

8: U

se o

f prim

e nu

mbe

rs in

cry

ptog

raph

y.

The

poss

ible

impa

ct o

f the

dis

cove

ry o

f po

wer

ful f

acto

rizat

ion

tech

niqu

es o

n in

tern

et

and

bank

secu

rity.

6.3

Line

ar D

ioph

antin

e eq

uatio

ns a

xby

c+

=.

Gen

eral

solu

tions

requ

ired

and

solu

tions

su

bjec

t to

cons

train

ts. F

or e

xam

ple,

all

solu

tions

mus

t be

posi

tive.

Int:

Des

crib

ed in

Dio

phan

tus’

Ari

thm

etic

a w

ritte

n in

Ale

xand

ria in

the

3rd c

entu

ry C

E.

Whe

n st

udyi

ng A

rith

met

ica,

a F

renc

h m

athe

mat

icia

n, P

ierr

e de

Fer

mat

(160

1–16

65)

wro

te in

the

mar

gin

that

he

had

disc

over

ed a

si

mpl

e pr

oof r

egar

ding

hig

her-o

rder

D

ioph

antin

e eq

uatio

ns—

Ferm

at’s

last

theo

rem

.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.4

Mod

ular

arit

hmet

ic.

The

solu

tion

of li

near

con

grue

nces

.

Solu

tion

of si

mul

tane

ous l

inea

r con

grue

nces

(C

hine

se re

mai

nder

theo

rem

).

Int:

Dis

cuss

ed b

y C

hine

se m

athe

mat

icia

n Su

n Tz

u in

the

3rd c

entu

ry C

E.

6.5

Rep

rese

ntat

ion

of in

tege

rs in

diff

eren

t bas

es.

On

exam

inat

ion

pape

rs, q

uest

ions

that

go

beyo

nd b

ase

16 w

ill n

ot b

e se

t. In

t: Ba

bylo

nian

s dev

elop

ed a

base

60

num

ber

syste

m an

d th

e May

ans a

bas

e 20

num

ber s

yste

m.

6.6

Ferm

at’s

littl

e th

eore

m.

(mod

)p a

ap

=, w

here

p is

prim

e.

TOK

: Nat

ure

of m

athe

mat

ics.

An

inte

rest

may

be

pur

sued

for c

entu

ries b

efor

e be

com

ing

“use

ful”

.

6.7

Gra

phs,

verti

ces,

edge

s, fa

ces.

Adj

acen

t ve

rtice

s, ad

jace

nt e

dges

.

Deg

ree

of a

ver

tex,

deg

ree

sequ

ence

.

Han

dsha

king

lem

ma.

Two

verti

ces a

re a

djac

ent i

f the

y ar

e jo

ined

by

an e

dge.

Tw

o ed

ges a

re a

djac

ent i

f the

y ha

ve a

co

mm

on v

erte

x.

Aim

8: S

ymbo

lic m

aps,

eg M

etro

and

U

nder

grou

nd m

aps,

stru

ctur

al fo

rmul

ae in

ch

emis

try, e

lect

rical

circ

uits

.

TOK

: Mat

hem

atic

s and

kno

wle

dge

clai

ms.

Proo

f of t

he fo

ur-c

olou

r the

orem

. If a

theo

rem

is

pro

ved

by c

ompu

ter,

how

can

we

clai

m to

kn

ow th

at it

is tr

ue?

Sim

ple

grap

hs; c

onne

cted

gra

phs;

com

plet

e gr

aphs

; bip

artit

e gr

aphs

; pla

nar g

raph

s; tr

ees;

w

eigh

ted

grap

hs, i

nclu

ding

tabu

lar

repr

esen

tatio

n.

Subg

raph

s; c

ompl

emen

ts o

f gra

phs.

It sh

ould

be

stre

ssed

that

a g

raph

shou

ld n

ot b

e as

sum

ed to

be

sim

ple

unle

ss sp

ecifi

cally

stat

ed.

The

term

adj

acen

cy ta

ble

may

be

used

.

Aim

8: I

mpo

rtanc

e of

pla

nar g

raph

s in

cons

truct

ing

circ

uit b

oard

s.

Eule

r’s r

elat

ion:

2

ve

f−

+=

; the

orem

s for

pl

anar

gra

phs i

nclu

ding

3

6e

v≤

−,

24

ev

≤−

, le

adin

g to

the

resu

lts th

at

5κ a

nd

3,3

κ a

re n

ot

plan

ar.

If th

e gr

aph

is si

mpl

e an

d pl

anar

and

3

v≥

, th

en

36

ev

≤−

.

If th

e gr

aph

is si

mpl

e, p

lana

r, ha

s no

cycl

es o

f le

ngth

3 a

nd

3v≥

, the

n 2

4e

v≤

−.

TOK

: Mat

hem

atic

s and

kno

wle

dge

clai

ms.

App

licat

ions

of t

he E

uler

cha

ract

eris

tic

()

ve

f−

+ to

hig

her d

imen

sion

s. Its

use

in

unde

rsta

ndin

g pr

oper

ties o

f sha

pes t

hat c

anno

t be

vis

ualiz

ed.


Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.8

Wal

ks, t

rails

, pat

hs, c

ircui

ts, c

ycle

s.

Eule

rian

trails

and

circ

uits

. A

con

nect

ed g

raph

con

tain

s an

Eule

rian

circ

uit

if an

d on

ly if

eve

ry v

erte

x of

the

grap

h is

of

even

deg

ree.

Int:

The

“Br

idge

s of K

önig

sber

g” p

robl

em.

Ham

ilton

ian

path

s and

cyc

les.

Sim

ple

treat

men

t onl

y.

6.9

Gra

ph a

lgor

ithm

s: K

rusk

al’s

; Dijk

stra

’s.

6.10

C

hine

se p

ostm

an p

robl

em.

Not

req

uire

d:

Gra

phs w

ith m

ore

than

four

ver

tices

of o

dd

degr

ee.

To d

eter

min

e th

e sh

orte

st ro

ute

arou

nd a

w

eigh

ted

grap

h go

ing

alon

g ea

ch e

dge

at le

ast

once

.

Int:

Pro

blem

pos

ed b

y th

e C

hine

se

mat

hem

atic

ian

Kw

an M

ei-K

o in

196

2.

Trav

ellin

g sa

lesm

an p

robl

em.

Nea

rest

-nei

ghbo

ur a

lgor

ithm

for d

eter

min

ing

an u

pper

bou

nd.

Del

eted

ver

tex

algo

rithm

for d

eter

min

ing

a lo

wer

bou

nd.

To d

eter

min

e th

e H

amilt

onia

n cy

cle

of le

ast

wei

ght i

n a

wei

ghte

d co

mpl

ete

grap

h.

TOK

: Mat

hem

atic

s and

kno

wle

dge

clai

ms.

How

long

wou

ld it

take

a c

ompu

ter t

o te

st a

ll H

amilt

onia

n cy

cles

in a

com

plet

e, w

eigh

ted

grap

h w

ith ju

st 3

0 ve

rtice

s?

6.11

R

ecur

renc

e re

latio

ns. I

nitia

l con

ditio

ns,

recu

rsiv

e de

finiti

on o

f a se

quen

ce.

TO

K: M

athe

mat

ics a

nd th

e w

orld

. The

co

nnec

tions

of s

eque

nces

such

as t

he F

ibon

acci

se

quen

ce w

ith a

rt an

d bi

olog

y.

Solu

tion

of fi

rst-

and

seco

nd-d

egre

e lin

ear

hom

ogen

eous

recu

rren

ce re

latio

ns w

ith

cons

tant

coe

ffic

ient

s.

The

first

-deg

ree

linea

r rec

urre

nce

rela

tion

1n

nu

aub

−=

+.

Incl

udes

the

case

s whe

re a

uxili

ary

equa

tion

has

equa

l roo

ts o

r com

plex

root

s.

Mod

ellin

g w

ith re

curr

ence

rela

tions

. So

lvin

g pr

oble

ms s

uch

as c

ompo

und

inte

rest

, de

bt re

paym

ent a

nd c

ount

ing

prob

lem

s.


Syllabus

Glossary of terminology: Discrete mathematics

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

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

Terminology

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

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

Complement of a graph G

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

Complete bipartite graph

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

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

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

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

Degree of a vertex The number of edges joined to the vertex; a loop contributes two edges, one for each of its end points.

Disconnected graph A graph that has at least one pair of vertices not joined by a path.

Eulerian circuit A circuit that contains every edge of a graph.

Eulerian trail A trail that contains every edge of a graph.

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

Graph isomorphism between two simple graphs G and H

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

Hamiltonian cycle A cycle that contains all the vertices of the graph.

Hamiltonian path A path that contains all the vertices of the graph.


Glossary of terminology: Discrete mathematics

Loop An edge joining a vertex to itself.

Minimum spanning tree

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

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

Path A walk with no repeated vertices.

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

Simple graph A graph without loops or multiple edges.

Spanning tree of a graph

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

Subgraph A graph within a graph.

Trail A walk in which no edge appears more than once.

Tree A connected graph that contains no cycles.

Walk A sequence of linked edges.

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

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


Assessment in the Diploma Programme

Assessment

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

There are two types of assessment identified by the IB.

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

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

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

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

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


Assessment in the Diploma Programme

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

Assessment criteriaAssessment criteria are used when the assessment task is open-ended. Each criterion concentrates on a particular skill that students are expected to demonstrate. An assessment objective describes what students should be able to do, and assessment criteria describe how well they should be able to do it. Using assessment criteria allows discrimination between different answers and encourages a variety of responses. Each criterion comprises a set of hierarchically ordered level descriptors. Each level descriptor is worth one or more marks. Each criterion is applied independently using a best-fit model. The maximum marks for each criterion may differ according to the criterion’s importance. The marks awarded for each criterion are added together to give the total mark for the piece of work.

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

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


Assessment

Assessment outline

First examinations 2014

Assessment component Weighting

External assessment (5 hours)

Paper 1 (2 hours 30 minutes)Graphic display calculator required.

Compulsory short- to medium-response questions based on the whole syllabus.

50%

Paper 2 (2 hours 30 minutes)Graphic display calculator required.

Compulsory medium- to extended-response questions based on the whole syllabus.

50%


Assessment

Assessment details

External assessment

Papers 1 and 2These papers are externally set and externally marked. The papers are designed to allow students to demonstrate what they know and what they can do.

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

CalculatorsPapers 1 and 2Students must have access to a GDC at all times. However, not all questions will necessarily require the use of the GDC. Regulations covering the types of GDC allowed are provided in the Handbook of procedures for the Diploma Programme.

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

Awarding of marksMarks may be awarded for method, accuracy, answers and reasoning, including interpretation.

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

Paper 1Duration: 2 hours 30 minutesWeighting: 50%• This paper consists of short- to medium-response questions. A GDC is required for this paper, but not

every question will necessarily require its use.

Syllabus coverage• Knowledge of all topics in the syllabus is required for this paper. However, not all topics are necessarily

assessed in every examination session.


Assessment details

Mark allocation• This paper is worth 150 marks, representing 50% of the final mark.

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

• The intention of this paper is to test students’ knowledge across the breadth of the syllabus.

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

Paper 2Duration: 2 hours 30 minutesWeighting: 50%• This paper consists of medium- to extended-response questions. A GDC is required for this paper, but

not every question will necessarily require its use.

Syllabus coverage• Knowledge of all topics in the core of the syllabus is required for this paper. However, not all topics are

necessarily assessed in every examination session.

Mark allocation• This paper is worth 150 marks, representing 50% of the final mark.

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

• The intention of this paper is to test students’ knowledge and understanding across the breadth of the syllabus.

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

Internal assessmentThere is no internal assessment component in this course.


Appendices

Glossary of command terms

Command terms with definitionsStudents should be familiar with the following key terms and phrases used in examination questions, which are to be understood as described below. Although these terms will be used in examination questions, other terms may be used to direct students to present an argument in a specific way.

Calculate Obtain a numerical answer showing the relevant stages in the working.

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

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

Compare and contrast

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

Construct Display information in a diagrammatic or logical form.

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

Deduce Reach a conclusion from the information given.

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

Describe Give a detailed account.

Determine Obtain the only possible answer.

Differentiate Obtain the derivative of a function.

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

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

Estimate Obtain an approximate value.

Explain Give a detailed account, including reasons or causes.

Find Obtain an answer, showing relevant stages in the working.

Hence Use the preceding work to obtain the required result.

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


Glossary of command terms

Identify Provide an answer from a number of possibilities.

Integrate Obtain the integral of a function.

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

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

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

Label Add labels to a diagram.

List Give a sequence of brief answers with no explanation.

Plot Mark the position of points on a diagram.

Predict Give an expected result.

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

Show Give the steps in a calculation or derivation.

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

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

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

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

Suggest Propose a solution, hypothesis or other possible answer.

Verify Provide evidence that validates the result.

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


Appendices

Notation list

Of the various notations in use, the IB has chosen to adopt a system of notation based on the recommendations of the International Organization for Standardization (ISO). This notation is used in the examination papers for this course without explanation. If forms of notation other than those listed in this guide are used on a particular examination paper, they are defined within the question in which they appear.

Because students are required to recognize, though not necessarily use, IB notation in examinations, it is recommended that teachers introduce students to this notation at the earliest opportunity. Students are not allowed access to information about this notation in the examinations.

Students must always use correct mathematical notation, not calculator notation.

the set of positive integers and zero, {0,1, 2, 3, ...}

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

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

the set of rational numbers

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

the set of real numbers

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

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

i 1−

z a complex number

z∗ the complex conjugate of z

z the modulus of z

arg z the argument of z

Re z the real part of z

Im z the imaginary part of z

cisθ ic s ino sθ θ+

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

( )n A the number of elements in the finite set A

{ | }x the set of all x such that

∈ is an element of

∉ is not an element of

∅ the empty (null) set

U the universal set

∪ union


Notation list

∩ intersection

⊂ is a proper subset of

⊆ is a subset of

A′ the complement of the set A

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

|a b a divides b

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

, thn root of a (if 0a ≥ then 0n a ≥ )

1/ 2a , a a to the power 12

, square root of a (if 0a ≥ then 0a ≥ )

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

for 0, x x xx x x

≥ ∈− < ∈

≡ identity

≈ 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

[ ],a b the closed interval a x b≤ ≤

] [,a b the open interval a x b< <

nu the thn term of a sequence or series

d the common difference of an arithmetic sequence

r the common ratio of a geometric sequence

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

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

1

n

ii

u=∑ 1 2 ... nu u u+ + +

1

n

ii

u=∏ 1 2 ... nu u u× × ×

nr

!

!( )!n

r n r−


Notation list

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

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

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

1f − the inverse function of the function f

f g the composite function of f and g

lim ( )x a

f x→

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

ddyx

the derivative of y with respect to x

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

2

2

dd

yx

the second derivative of y with respect to x

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

dd

n

n

yx

the thn derivative of y with respect to x

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

dy x∫ the indefinite integral of y with respect to x

db

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

ex the exponential function of x

loga x the logarithm to the base a of x

ln x the natural logarithm of x, elog x

sin, cos, tan the circular functions

arcsin, arccos,arctan

the inverse circular functions

csc, sec, cot the reciprocal circular functions

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

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

AB the length of [ ]AB

( )AB the line containing points A and B

Â the angle at A


Notation list

ˆCAB the angle between [ ]CA and [ ]AB

ABC∆ the triangle whose vertices are A, B and C

v the vector v

AB→

the vector represented in magnitude and direction by the directed line segment from A to B

a the position vector OA→

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

a the magnitude of a

|AB|→

the magnitude of AB→

⋅v w the scalar product of v and w

×v w the vector product of v and w

1−A the inverse of the non-singular matrix A

TA the transpose of the matrix A

det A the determinant of the square matrix A

I the identity matrix

P( )A the probability of event A

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

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

1 2, , ...x x observations

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

Px the probability distribution function P( = )X x of the discrete random variable X

( )f x the probability density function of the continuous random variable X

( )F x the cumulative distribution function of the continuous random variable X

E ( )X the expected value of the random variable X

Var ( )X the variance of the random variable X

µ population mean

2σ population variance,

2

2 1( )

k

i ii

f x

n

µσ =

−=∑

, where 1

k

ii

n f=

=∑

σ population standard deviation

x sample mean


Notation list

2ns sample variance,

2

2 1( )

k

i ii

n

f x xs

n=

−=∑

, where 1

k

ii

n f=

=∑

ns standard deviation of the sample

21ns −

unbiased estimate of the population variance,

2

2 2 11

( )

1 1

k

i ii

n n

f x xns s

n n=

−

−= =

− −

∑, where

1

k

ii

n f=

=∑

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

( )Po m Poisson distribution with mean m

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

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

( )~ PoX m the random variable X has a Poisson distribution with mean m

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

Φ cumulative distribution function of the standardized normal variable with distribution ( )N 0,1

ν number of degrees of freedom

\A B the difference of the sets A and B (that is, \ { and }A B A B x x A x B∩ ′= = ∈ ∉ )

A B∆ the symmetric difference of the sets A and B (that is, ( \ ) ( \ )A B A B B A∆ = ∪ )

nκ a complete graph with n vertices

,n mκ a complete bipartite graph with one set of n vertices and another set of m vertices

p the set of equivalence classes {0,1, 2, , 1}p − of integers modulo p

gcd( , )a b the greatest common divisor of integers a and b

lcm( , )a b the least common multiple of integers a and b

GA the adjacency matrix of graph G

GC the cost adjacency matrix of graph G

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