User manual | Further mathematics HL guide

Diploma Programme
Further mathematics HL guide
First examinations 2014
Diploma Programme
Further mathematics HL guide
First examinations 2014
Further mathematics HL guide
Diploma Programme
Published June 2012
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IB mission statement
The International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help to
create a better and more peaceful world through intercultural understanding and respect.
To this end the organization works with schools, governments and international organizations to develop challenging
programmes of international education and rigorous assessment.
These programmes encourage students across the world to become active, compassionate and lifelong learners who
understand that other people, with their differences, can also be right.
IB learner profile
The aim of all IB programmes is to develop internationally minded people who, recognizing their common humanity
and shared guardianship of the planet, help to create a better and more peaceful world.
IB learners strive to be:
Inquirers
They develop their natural curiosity. They acquire the skills necessary to conduct inquiry
and research and show independence in learning. They actively enjoy learning and this love
of learning will be sustained throughout their lives.
Knowledgeable
They explore concepts, ideas and issues that have local and global significance. In so doing,
they acquire in-depth knowledge and develop understanding across a broad and balanced
range of disciplines.
Thinkers
They exercise initiative in applying thinking skills critically and creatively to recognize
and approach complex problems, and make reasoned, ethical decisions.
Communicators
They understand and express ideas and information confidently and creatively in more
than one language and in a variety of modes of communication. They work effectively and
willingly in collaboration with others.
Principled
They act with integrity and honesty, with a strong sense of fairness, justice and respect for
the dignity of the individual, groups and communities. They take responsibility for their
own actions and the consequences that accompany them.
Open-minded
They understand and appreciate their own cultures and personal histories, and are open
to the perspectives, values and traditions of other individuals and communities. They are
accustomed to seeking and evaluating a range of points of view, and are willing to grow
from the experience.
Caring
They show empathy, compassion and respect towards the needs and feelings of others.
They have a personal commitment to service, and act to make a positive difference to the
lives of others and to the environment.
Risk-takers
They approach unfamiliar situations and uncertainty with courage and forethought, and
have the independence of spirit to explore new roles, ideas and strategies. They are brave
and articulate in defending their beliefs.
Balanced
They understand the importance of intellectual, physical and emotional balance to achieve
personal well-being for themselves and others.
Reflective
They give thoughtful consideration to their own learning and experience. They are able to
assess and understand their strengths and limitations in order to support their learning and
personal development.
© International Baccalaureate Organization 2007
Contents
Introduction1
Purpose of this document
1
The Diploma Programme
2
Nature of the subject
4
Aims8
Assessment objectives
9
Syllabus10
Syllabus outline
10
Approaches to the teaching and learning of further mathematics HL
11
Syllabus content
15
Glossary of terminology: Discrete mathematics
44
Assessment46
Assessment in the Diploma Programme
46
Assessment outline
48
Assessment details
49
Appendices51
Glossary of command terms
51
Notation list
53
Further mathematics HL guide
Introduction
Purpose of this document
This publication is intended to guide the planning, teaching and assessment of the subject in schools. Subject
teachers are the primary audience, although it is expected that teachers will use the guide to inform students
and parents about the subject.
This guide can be found on the subject page of the online curriculum centre (OCC) at http://occ.ibo.org, a
password-protected IB website designed to support IB teachers. It can also be purchased from the IB store at
http://store.ibo.org.
Additional resources
Additional publications such as teacher support materials, subject reports 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
Further mathematics HL guide
1
Introduction
The Diploma Programme
The Diploma Programme is a rigorous pre-university course of study designed for students in the 16 to 19
age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and
inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to develop
intercultural understanding, open-mindedness, and the attitudes necessary for them to respect and evaluate a
range of points of view.
The Diploma Programme hexagon
The course is presented as six academic areas enclosing a central core (see figure 1). It encourages the concurrent
study of a broad range of academic areas. Students study: two modern languages (or a modern language and
a classical language); a humanities or social science subject; an experimental science; mathematics; one of
the creative arts. It is this comprehensive range of subjects that makes the Diploma Programme a demanding
course of study designed to prepare students effectively for university entrance. In each of the academic areas
students have flexibility in making their choices, which means they can choose subjects that particularly
interest them and that they may wish to study further at university.
Studies in language
and literature
Group 1
Group 2
Group 3
Individuals
and societies
essay
ed
nd
PR OFIL
ER
dge
ext
wle
e
o
n
E
L
A
R
B
I
N
E
theory
of
k
TH
Language
acquisition
E
Experimental
sciences
Group 4
cr
ea
ice
tivi
ty, action, serv
Group 5
Mathematics
Group 6
The arts
Figure 1
Diploma Programme model
2
Further mathematics HL guide
The Diploma Programme
Choosing the right combination
Students are required to choose one subject from each of the six academic areas, although they can choose a
second subject from groups 1 to 5 instead of a group 6 subject. Normally, three subjects (and not more than
four) are taken at higher level (HL), and the others are taken at standard level (SL). The IB recommends 240
teaching hours for HL subjects and 150 hours for SL. Subjects at HL are studied in greater depth and breadth
than at SL.
At both levels, many skills are developed, especially those of critical thinking and analysis. At the end of
the course, students’ abilities are measured by means of external assessment. Many subjects contain some
element of coursework assessed by teachers. The 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 hexagon
All Diploma Programme students participate in the three course requirements that make up the core of the
hexagon. Reflection on all these activities is a principle that lies at the heart of the thinking behind the Diploma
Programme.
The theory of knowledge course encourages students to think about the nature of knowledge, to reflect on
the process of learning in all the subjects they study as part of their Diploma Programme course, and to make
connections across the academic areas. The extended essay, a substantial piece of writing of up to 4,000 words,
enables students to investigate a topic of special interest that they have chosen themselves. It also encourages
them to develop the skills of independent research that will be expected at university. Creativity, action, service
involves students in experiential learning through a range of artistic, sporting, physical and service activities.
The IB mission statement and the IB learner profile
The Diploma Programme aims to develop in students the knowledge, skills and attitudes they will need to
fulfill the aims of the IB, as expressed in the organization’s mission statement and the learner profile. Teaching
and learning in the Diploma Programme represent the reality in daily practice of the organization’s educational
philosophy.
Further mathematics HL guide
3
Introduction
Nature of the subject
Introduction
The nature of mathematics can be summarized in a number of ways: for example, it can be seen as a welldefined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probably
a combination of these, but there is no doubt that mathematical knowledge provides an important key to
understanding the world in which we live. Mathematics can enter our lives in a number of ways: we buy
produce in the market, consult a timetable, read a newspaper, time a process or estimate a length. Mathematics,
for most of us, also extends into our chosen profession: visual artists need to learn about perspective; musicians
need to appreciate the mathematical relationships within and between different rhythms; economists need
to recognize trends in financial dealings; and engineers need to take account of stress patterns in physical
materials. Scientists view mathematics as a language that is central to our understanding of events that occur
in the natural world. Some people enjoy the challenges offered by the logical methods of mathematics and
the adventure in reason that mathematical proof has to offer. Others appreciate mathematics as an aesthetic
experience or even as a cornerstone of philosophy. This prevalence of mathematics in our lives, with all its
interdisciplinary connections, provides a clear and sufficient rationale for making the study of this subject
compulsory for students studying the full diploma.
Summary of courses available
Because individual students have different needs, interests and abilities, there are four different courses in
mathematics. These courses are designed for different types of students: those who wish to study mathematics
in depth, either as a subject in its own right or to pursue their interests in areas related to mathematics; those
who wish to gain a degree of understanding and competence to understand better their approach to other
subjects; and those who may not as yet be aware how mathematics may be relevant to their studies and in their
daily lives. Each course is designed to meet the needs of a particular group of students. Therefore, great care
should be taken to select the course that is most appropriate for an individual student.
In making this selection, individual students should be advised to take account of the following factors:
•
their own abilities in mathematics and the type of mathematics in which they can be successful
•
their own interest in mathematics and those particular areas of the subject that may hold the most interest
for them
•
their other choices of subjects within the framework of the Diploma Programme
•
their academic plans, in particular the subjects they wish to study in future
•
their choice of career.
Teachers are expected to assist with the selection process and to offer advice to students.
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Further mathematics HL guide
Nature of the subject
Mathematical studies SL
This course is available only at standard level, and is equivalent in status to mathematics SL, but addresses
different needs. It has an emphasis on applications of mathematics, and the largest section is on statistical
techniques. It is designed for students with varied mathematical backgrounds and abilities. It offers students
opportunities to learn important concepts and techniques and to gain an understanding of a wide variety
of mathematical topics. It prepares students to be able to solve problems in a variety of settings, to develop
more sophisticated mathematical reasoning and to enhance their critical thinking. The individual project is an
extended piece of work based on personal research involving the collection, analysis and evaluation of data.
Students taking this course are well prepared for a career in social sciences, humanities, languages or arts.
These students may need to utilize the statistics and logical reasoning that they have learned as part of the
mathematical studies SL course in their future studies.
Mathematics SL
This course caters for students who already possess knowledge of basic mathematical concepts, and who are
equipped with the skills needed to apply simple mathematical techniques correctly. The majority of these
students will expect to need a sound mathematical background as they prepare for future studies in subjects
such as chemistry, economics, psychology and business administration.
Mathematics HL
This course caters for students with a good background in mathematics who are competent in a range of
analytical and technical skills. The majority of these students will be expecting to include mathematics as
a major component of their university studies, either as a subject in its own right or within courses such as
physics, engineering and technology. Others may take this subject because they have a strong interest in
mathematics and enjoy meeting its challenges and engaging with its problems.
Further mathematics HL
This course is available only at higher level. It caters for students with a very strong background in mathematics
who have attained a high degree of competence in a range of analytical and technical skills, and who display
considerable interest in the subject. Most of these students will expect to study mathematics at university, either
as a subject in its own right or as a major component of a related subject. The course is designed specifically
to allow students to learn about a variety of branches of mathematics in depth and also to appreciate practical
applications. It is expected that students taking this course will also be taking mathematics HL.
Note: Mathematics HL is an ideal course for students expecting to include mathematics as a major component
of their university studies, either as a subject in its own right or within courses such as physics, engineering
or technology. It should not be regarded as necessary for such students to study further mathematics HL.
Rather, further mathematics HL is an optional course for students with a particular aptitude and interest in
mathematics, enabling them to study some wider and deeper aspects of mathematics, but is by no means a
necessary qualification to study for a degree in mathematics.
Further mathematics HL—course details
The 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 guide
5
Nature of the subject
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 learning
Mathematics is a linear subject, and it is expected that most students embarking on a Diploma Programme
(DP) mathematics course will have studied mathematics for at least 10 years. There will be a great variety of
topics studied, and differing approaches to teaching and learning. Thus students will have a wide variety of
skills and knowledge when they start the 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 Programme
The prior learning topics for the DP courses have been written in conjunction with the Middle Years
Programme (MYP) mathematics guide. The approaches to teaching and learning for DP mathematics build
on the approaches used in the MYP. These include investigations, exploration and a variety of different
assessment tools.
A continuum document called Mathematics: The MYP–DP continuum (November 2010) is available on the
DP mathematics home pages of the OCC. This extensive publication focuses on the alignment of mathematics
across the MYP and the DP. It was developed in response to feedback provided by IB World Schools, which
expressed the need to articulate the transition of mathematics from the MYP to the DP. The publication also
highlights the similarities and differences between MYP and DP mathematics, and is a valuable resource for
teachers.
Mathematics and theory of knowledge
The Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed that
these all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by data
from sense perception, mathematics is dominated by reason, and some mathematicians argue that their subject
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.
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Further mathematics HL guide
Nature of the subject
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 dimension
Mathematics is in a sense an international language, and, apart from slightly differing notation, mathematicians
from around the world can communicate within their field. Mathematics transcends politics, religion and
nationality, yet throughout history great civilizations owe their success in part to their mathematicians being
able to create and maintain complex social and architectural structures.
Despite recent advances in the development of information and communication technologies, the global
exchange of mathematical information and ideas is not a new phenomenon and has been essential to the
progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries
ago by Arabic, Greek, Indian and Chinese civilizations, among others. Teachers could use timeline websites
to show the contributions of different civilizations to mathematics, but not just for their mathematical content.
Illustrating the characters and personalities of the mathematicians concerned and the historical context in
which they worked brings home the human and cultural dimension of mathematics.
The importance of science and technology in the everyday world is clear, but the vital role of mathematics
is not so well recognized. It is the language of science, and underpins most developments in science and
technology. A good example of this is the digital revolution, which is transforming the world, as it is all based
on the binary number system in mathematics.
Many international bodies now exist to promote mathematics. Students are encouraged to access the extensive
websites of international mathematical organizations to enhance their appreciation of the international
dimension and to engage in the global issues surrounding the subject.
Examples of global issues relating to international-mindedness (Int) are given in the “Links” column of the
syllabus.
Further mathematics HL guide
7
Introduction
Aims
Group 5 aims
The aims of all mathematics courses in group 5 are to enable students to:
1.
enjoy mathematics, and develop an appreciation of the elegance and power of mathematics
2.
develop an understanding of the principles and nature of mathematics
3.
communicate clearly and confidently in a variety of contexts
4.
develop logical, critical and creative thinking, and patience and persistence in problem-solving
5.
employ and refine their powers of abstraction and generalization
6.
apply and transfer skills to alternative situations, to other areas of knowledge and to future developments
7.
appreciate how developments in technology and mathematics have influenced each other
8.
appreciate the moral, social and ethical implications arising from the work of mathematicians and the
applications of mathematics
9.
appreciate the international dimension in mathematics through an awareness of the universality of
mathematics and its multicultural and historical perspectives
10. appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge”
in the TOK course.
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Further mathematics HL guide
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.
Further mathematics HL guide
9
Syllabus
Syllabus outline
Teaching hours
Syllabus component
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
48
Linear algebra
Topic 2
48
Geometry
Topic 3
48
Statistics and probability
Topic 4
48
Sets, relations and groups
Topic 5
48
Calculus
Topic 6
48
Discrete mathematics
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
10
240
Further mathematics HL guide
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 inquiry
The 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
Test the conjecture
Reject
Accept
Prove
Extend
Figure 2
Further mathematics HL guide
11
Approaches to the teaching and learning of further mathematics HL
Mathematical modelling and applications
Students 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
Test the model
Reject
Accept
Reflect on and apply the model
Extend
Figure 3
Technology
Technology 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.
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Further mathematics HL guide
Approaches to the teaching and learning of further mathematics HL
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
Intinternational-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 study
The 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.
Further mathematics HL guide
13
Approaches to the teaching and learning of further mathematics HL
Time allocation
The 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 calculators
Students 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 booklet
Each student is required to have access to a clean copy of this booklet during the examination. It is
recommended that teachers ensure students are familiar with the contents of this document from the beginning
of the course. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there
are no printing errors, and ensure that there are sufficient copies available for all students.
Command terms and notation list
Teachers and students need to be familiar with the IB notation and the command terms, as these will be used
without explanation in the examination papers. The “Glossary of command terms” and “Notation list” appear
as appendices in this guide.
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Further mathematics HL guide
Further mathematics HL guide
15
48 hours
1.1
( AB )T = B T AT .
Transpose of a matrix including AT notation:
Identity and zero matrices.
Properties of matrix multiplication:
associativity, distributivity.
Multiplication of matrices.
Students should be familiar with the definition
of symmetric and skew-symmetric matrices.
Students should be familiar with the notation I
and 0.
TOK: Given the many applications of matrices
as seen in this course, consider the fact that
mathematicians marvel at some of the deep
connections between disparate parts of their
subject. Is this evidence for a simple
underlying mathematical reality?
Matrix operations to handle or process
information.
Algebra of matrices: equality; addition;
Including use of the GDC.
subtraction; multiplication by a scalar for m × n
matrices.
Links
Data storage and manipulation, eg stock
control.
Further guidance
Definition of a matrix: the terms element, row,
column and order for m × n matrices.
Content
The aim of this section is to introduce students to the principles of matrices, vector spaces and linear algebra, including eigenvalues and geometrical
interpretations.
Topic 1—Linear algebra
Syllabus content
Syllabus
16
1.4
1.3
1.2
Pivoting is the R1 + aR2 method.
The result det( AB ) = det A det B .
Solutions of m linear equations in n unknowns: Include the non-existent, unique and infinitely
both augmented matrix method, leading to
many cases.
reduced row echelon form method, and inverse
matrix method, when applicable.
Row rank and column rank and their equality.
Row space, column space and null space.
Row reduced echelon form.
Corresponding elementary matrices.
Scaling, swapping and pivoting.
Elementary row and column operations for
matrices.
Formulae for the inverse and determinant of a
2 × 2 matrix and the determinant of a 3 × 3
matrix.
Use of elementary row operations to find A−1 .
Calculation of A−1 .
( An ) −1 = ( A−1 ) n .
( AT ) −1 = ( A−1 )T ,
Using a GDC.
The terms singular and non-singular matrices.
Definition and properties of the inverse of a
square matrix:
( AB ) −1 = B −1 A−1 ,
Further guidance
Content
Aim 8: Linear optimization was developed by
George Dantzig in the 1940s as a way of
allocating scarce resources.
Int: Also known as Gaussian elimination after
the German mathematician CF Gauss (1777–
1855).
Links
Syllabus content
Further mathematics HL guide
Further mathematics HL guide
1.6
1.5
Rank-nullity theorem (proof not required).
Result and proof that the range is a subspace of
the codomain.
Result and proof that the kernel is a subspace
of the domain.
Domain, range, codomain and kernel.
Composition of linear transformations.
T(u + v) = T(u) + T(v), T(ku) = kT(u).
Linear transformations:
Subspaces.
Basis and dimension for a vector space.
Linear independence of vectors.
Spanning set.
Linear combinations of vectors.
The vector space  n .
Content
Dim(range) = rank and Dim(kernel) = nullity.
where
Dim(domain) = Dim(range) + Dim(kernel)
The kernel of a linear transformation is the null
space of its matrix representation.
Students should be familiar with the term
orthogonal.
Include linear dependence and use of
determinants.
Further guidance
Aim 8: Lorenz transformations and their use in
relativity and quantum mechanics (Physics
13.1).
Aim 8: Link to Fourier analysis for waves,
which has many uses in physics (Physics 4.5)
and engineering.
Links
Syllabus content
17
18
1.8
1.7
Geometric interpretation of determinant.
Compositions of the above transformations.
Geometric transformations represented by
2 × 2 matrices include general rotation, general
reflection in y = (tan α ) x , stretches parallel to
axes, shears parallel to axes, and projection
onto y = (tan α ) x .
Solution of Ax = b.
Application of linear transformations to
solutions of system of equations.
Result that the numbers of linearly independent
rows and columns are equal, and this is the
dimension of the range of the transformation
(proof not required).
Result that any linear transformation can be
represented by a matrix, and the converse of
this result.
Content
New=
area det A × old area .
(particular solution) + (any member of the null
space).
Using
Further guidance
Aim 8: Computer graphics in threedimensional modelling.
Links
Syllabus content
Further mathematics HL guide
1.9
Geometric interpretation.
Eigenvalues and eigenvectors of 2 × 2
matrices.
Applications to powers of 2 × 2 matrices.
Diagonalization of 2 × 2 matrices (restricted to
the case where there are distinct real
eigenvalues).
Characteristic polynomial of 2 × 2 matrices.
Further guidance
Content
Further mathematics HL guide
Markov chains, link with genetics (Biology 4).
Aim 8: Damping noise in car design, test for
cracks in solid objects, oil exploration, the
Google PageRank formula, and “the $25
billion dollar eigenvector”.
For example, the natural frequency of an object
can be characterized by the eigenvalue of
smallest magnitude (1940 Tacoma Narrows
Bridge disaster).
TOK: “We can use mathematics successfully
to model real-world processes. Is this because
we create mathematics to mirror the world or
because the world is intrinsically
mathematical?”
Representation of conics.
Invariant states.
Stock market values and trends.
Stochastic processes.
Links
Syllabus content
19
20
2.4
Circle geometry.
2.3
Further mathematics HL guide
Proofs of these theorems and converses.
Angle bisector theorem; Apollonius’ circle
theorem, Menelaus’ theorem; Ceva’s theorem;
Ptolemy’s theorem for cyclic quadrilaterals.
In a cyclic quadrilateral, opposite angles are
supplementary, and the converse.
Tangents; arcs, chords and secants.
Centres of a triangle: orthocentre, incentre,
circumcentre and centroid.
Euclid’s theorem for proportional segments in
a right-angled triangle.
Similar and congruent triangles.
2.2
2.1
Content
The use of these theorems to prove further
results.
The tangent–secant and secant–secant
theorems and intersecting chords theorems.
Angle at centre theorem and corollaries.
The terms inscribed and circumscribed.
Proof of concurrency theorems.
The terms altitude, angle bisector,
perpendicular bisector, median.
Further guidance
The aim of this section is to develop students’ geometric intuition, visualization and deductive reasoning.
Topic 2—Geometry
Appl: centre of mass, triangulation.
TOK: Crisis over non-Euclidean geometry
parallels with that of Cantor’s set theory.
TOK: Hippasus’ existence proof for irrational
numbers and impact on separate development
of number and geometry.
TOK, Int: The influence of Euclid’s axiomatic
approach on philosophy (Descartes) and
politics (Jefferson: American Declaration of
Independence).
Links
48 hours
Syllabus content
Further mathematics HL guide
2.8
2.7
2.6
2.5
Diagonalizing the matrix A with the rotation
matrix P and reducing the general conic to
standard form.
The general conic
ax 2 + 2bxy + cy 2 + dx + ey + f =
0,
and the quadratic form
x T Ax =ax 2 + 2bxy + cy 2 .
The standard parametric equations of the
circle, parabola, ellipse, rectangular hyperbola,
hyperbola.
Tangents and normals.
Parametric differentiation.
Parametric equations.
Tangents and normals.
Focus–directrix definitions.
The parabola, ellipse and hyperbola, including
rectangular hyperbola.
Conic sections.
21
matrix A in the quadratic form x T Ax .
where λ1 and λ 2 are the eigenvalues of the
0,
λ1 x 2 + λ 2 y 2 + dx + ey + f =
The general conic can be rotated to give the
form
x = at 2 , y = 2at ,
x = a cos θ , y = b sin θ ,
c
x = ct , y = ,
t
x = a secθ , y = b tan θ .
x = a cos θ , y = a sin θ ,
x2 y 2
1 and their translations.
−
=
a 2 b2
x2 y 2
1,
+
=
a 2 b2
x 2 + y 2 + dx + ey + f =
0.
Coordinate geometry of the circle.
The standard forms y 2 = 4ax ,
The equations ( x − h) 2 + ( y − k ) 2 =
r 2 and
Finding equations of loci.
Tangents to a circle.
Further guidance
Content
TOK: Kepler’s difficulties accepting that an
orbit was not a “perfect” circle.
Appl: Satellite dish, headlight, orbits,
projectiles (Physics 9.1).
TOK: Consequences of Descartes’ unification
of algebra and geometry.
Links
Syllabus content
Syllabus content
Geometry theorems—clarification of theorems used in topic 2
Teachers 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 triangle
The proportional segments p and q satisfy the following:
b
a
h
q
p
c
h2 = pq,
a2 = pc,
b2 = qc.
Angle at centre theorem
The angle subtended by an arc at the circumference is half that subtended by the same arc at the centre.
x
O
2x
Corollaries
Angles subtended at the circumference by the same arc are equal.
x
22
x
Further mathematics HL guide
Syllabus content
The angle in a semicircle is a right angle.
O
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.
x
x
The tangent is perpendicular to the radius at the point of tangency.
O
The intersecting chords theorem
ab = cd
a
d
Further mathematics HL guide
c
b
23
Syllabus content
The tangent–secant and secant–secant theorems
PT2 = PA × PB = PC × PD
TP
C
B
D
Apollonius’ circle theorem (circle of Apollonius)
PA
is a constant not equal to one, then the locus of P is a circle. This
If A and B are two fixed points such that
PB
is called the circle of Apollonius.
Included: the converse of this theorem.
Menelaus’ theorem
If a transversal meets the sides [BC], [CA] and [AB] of a triangle at D, E and F respectively, then
BD CE AF
×
×
= −1.
DC EA FB
A
E
F
A
E
B
C
D
B
F
C
D
Converse: if D, E and F are points on the sides [BC], [CA] and [AB], respectively, of a triangle such that
BD CE AF
×
×
= −1, then D, E and F are collinear.
DC EA FB
24
Further mathematics HL guide
Syllabus content
Ceva’s theorem
If 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, then
BD
DC
CE
EA
AF
FB
1.
A
A
F
D
E
C
B
F
O
B
E
O
C
D
BD CE AF
1, then
Converse: if D, E and F are points on [BC], [CA] and [AB], respectively, such that
DC
EA
FB
[AD], [BE] and [CF] are concurrent.
Note on Ceva’s theorem and Menelaus’ theorem
The 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
called like or unlike) according to whether the displacements A → B and P → Q are in the same or opposite
directions. 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 theorem
If 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
Further mathematics HL guide
B
C
25
Syllabus content
Angle bisector theorem
The 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
BD AB
for internal bisectors
DC AC
and
BD
DC
AB
AC
for external bisectors.
A
E
A
B
D
BÂD  CÂD
C
B
C
CÂD  EÂD
Included: the converse of this theorem.
26
Further mathematics HL guide
D
48 hours
Further mathematics HL guide
3.2
3.1
Expectation of the product of independent
random variables.
Variance of linear combinations of n
independent random variables.
Mean of linear combinations of n random
variables.
Linear transformation of a single random
variable.
Using probability generating functions to find
mean, variance and the distribution of the sum
of n independent random variables.
Probability generating functions for discrete
random variables.
Negative binomial distribution.
Geometric distribution.
Cumulative distribution functions for both
discrete and continuous distributions.
Content
x
P( X
∑=
x)t x .
E( XY ) = E( X )E(Y ) .
Var(aX + b) =
a 2 Var( X ) .
E(aX +=
b) aE( X ) + b ,
=
G (t ) E(
=
tX )
Further guidance
Aim 8: Statistical compression of data files.
Int: Also known as Pascal’s distribution.
Links
The aims of this topic are to allow students the opportunity to approach statistics in a practical way; to demonstrate a good level of statistical understanding;
and to understand which situations apply and to interpret the given results. It is expected that GDCs will be used throughout this option and that the
minimum requirement of a GDC will be to find the probability distribution function (pdf), cumulative distribution function (cdf), inverse cumulative
distribution function, p-values and test statistics, including calculations for the following distributions: binomial, Poisson, normal and t. Students are
expected to set up the problem mathematically and then read the answers from the GDC, indicating this within their written answers. Calculator-specific or
brand-specific language should not be used within these explanations.
Topic 3—Statistics and probability
Syllabus content
27
28
3.4
3.3
−X)
n −1
i
2
.
The central limit theorem.
 σ2 
X ~ N  µ,  .
n 

TOK: Nature of mathematics. The central
limit theorem can be proved mathematically
(formalism), but its truth can be confirmed by
its applications (empiricism).
i =1
(X
X ~ N( µ ,σ 2 ) ⇒
2
S =∑
n
i =1
Xi
.
n
Aim 8/TOK: Mathematics and the world.
“Without the central limit theorem, there could
be no statistics of any value within the human
sciences.”
2
n
X =∑
A linear combination of independent normal
random variables is normally distributed. In
particular,
2
S as an unbiased estimator for σ .
X as an unbiased estimator for µ .
Var(T1 ) < Var(T 2 ) .
T1 is a more efficient estimator than T 2 if
TOK: Mathematics and the world. In the
absence of knowing the value of a parameter,
will an unbiased estimator always be better
than a biased one?
T is an unbiased estimator for the parameter
θ if E(T ) = θ .
Unbiased estimators and estimates.
Comparison of unbiased estimators based on
variances.
Links
Further guidance
Content
Syllabus content
Further mathematics HL guide
Further mathematics HL guide
Testing hypotheses for the mean of a normal
population.
Type I and II errors, including calculations of
their probabilities.
Critical regions, critical values, p-values, onetailed and two-tailed tests.
Significance level.
Null and alternative hypotheses, H 0 and H1 .
3.6
Use of the normal distribution when σ is
known and use of the t-distribution when σ is
unknown, regardless of sample size. The case
of matched pairs is to be treated as an example
of a single sample technique.
TOK: Mathematics and the world. Claiming
brand A is “better” on average than brand B
can mean very little if there is a large overlap
between the confidence intervals of the two
means.
Use of the normal distribution when σ is
known and use of the t-distribution when σ is
unknown, regardless of sample size. The case
of matched pairs is to be treated as an example
of a single sample technique.
Confidence intervals for the mean of a normal
population.
3.5
Appl: When is it more important not to make a
Type I error and when is it more important not
to make a Type II error?
TOK: Mathematics and the world. Does the
ability to test only certain parameters in a
population affect the way knowledge claims in
the human sciences are valued?
TOK: Mathematics and the world. In practical
terms, is saying that a result is significant the
same as saying that it is true?
Appl: Geography.
Links
Further guidance
Content
Syllabus content
29
3.7
30
Cov( X , Y ) = E[( X − µ x )(Y − µ y )]
Covariance and (population) product moment
correlation coefficient ρ.
Definition of the (sample) product moment
correlation coefficient R in terms of n paired
observations on X and Y. Its application to the
estimation of ρ.
Proof that ρ = 0 in the case of independence
and ±1 in the case of a linear relationship
between X and Y.
Informal discussion of commonly occurring
situations, eg marks in pure mathematics and
statistics exams taken by a class of students,
salary and age of teachers in a certain school.
The need for a measure of association between
the variables and the possibility of predicting
the value of one of the variables given the
value of the other variable.
Introduction to bivariate distributions.
Cov( X , Y )
.
Var( X )Var(Y )
=
R=
i =1
i i
 n 2
2 
2
2
 ∑ X i − n X  ∑ Yi − nY 

 i =1

i =1
∑ X Y − nXY
n
∑ ( X i − X )2 ∑ (Yi − Y )2
i =1
− X )(Yi − Y )
n
i
n
i =1
∑(X
n
.
The use of ρ as a measure of association
between X and Y, with values near 0 indicating
a weak association and values near +1 or near
–1 indicating a strong association.
ρ=
where
=
µ x E(
=
X ), µ y E(Y ) .
= E( XY ) − µ x µ y ,
Further guidance
Content
Further mathematics HL guide
(continued)
Aim 8: The physicist Frank Oppenheimer
wrote: “Prediction is dependent only on the
assumption that observed patterns will be
repeated.” This is the danger of extrapolation.
There are many examples of its failure in the
past, eg share prices, the spread of disease,
climate change.
TOK: Mathematics and the world. Given that
a set of data may be approximately fitted by a
range of curves, where would we seek for
knowledge of which equation is the “true”
model?
Appl: Using technology to fit a range of curves
to a set of data.
Aim 8: The correlation between smoking and
lung cancer was “discovered” using
mathematics. Science had to justify the cause.
Appl: Geographic skills.
Links
Syllabus content
Further guidance
Further mathematics HL guide
n−2
has the Student’s t-distribution with
1 − R2
(n − 2) degrees of freedom.
 n

 ∑ ( xi − x )( yi − y ) 
 i =1 n
 ( y − y)
x−x =


2
 ∑ ( yi − y )

i =1


n


 ∑ xi yi − nx y 
 ( y − y ),
=  i =1n

2
2 
 ∑ yi − n y 
 i =1

Knowledge of the facts that the regression of X
on Y ( E( X ) | Y = y ) and Y on X ( E(Y ) | X = x )
are linear.
The use of these regression lines to predict the
value of one of the variables given the value of
the other.
 n

 ∑ ( xi − x )( yi − y ) 
 i =1 n
 (x − x )
y−y =


2
( xi − x )
∑


i =1


n


 ∑ xi yi − nx y 
 ( x − x ).
=  i =1n

2
2 
 ∑ xi − nx 
 i =1

R
Use of the t-statistic to test the null hypothesis
ρ = 0.
Least-squares estimates of these regression
lines (proof not required).
It is expected that the GDC will be used
wherever possible in the following work.
The following topics are based on the
assumption of bivariate normality.
Informal interpretation of r, the observed value Values of r near 0 indicate a weak association
of R. Scatter diagrams.
between X and Y, and values near ±1 indicate a
strong association.
Content
Links
(see notes above)
Syllabus content
31
32
48 hours
4.4
4.3
4.2
4.1
Further guidance
TOK: Cantor theory of transfinite numbers,
Russell’s paradox, Godel’s incompleteness
theorems.
Links
Further mathematics HL guide
Operation tables (Cayley tables).
Binary operations.
A binary operation ∗ on a non-empty set S is a
rule for combining any two elements a, b ∈ S
to give a unique element c. That is, in this
definition, a binary operation on a set is not
necessarily closed.
bijection from set A to set B then f −1 exists
and is a bijection from set B to set A.
Knowledge that the function composition is not
a commutative operation and that if f is a
The term codomain.
Functions: injections; surjections; bijections.
Composition of functions and inverse
functions.
An equivalence relation on a set forms a
partition of the set.
Relations: equivalence relations; equivalence
classes.
Ordered pairs: the Cartesian product of two
sets.
Appl, Int: Scottish clans.
De Morgan’s laws: distributive, associative and Illustration of these laws using Venn diagrams. Appl: Logic, Boolean algebra, computer
commutative laws (for union and intersection).
circuits.
Students may be asked to prove that two sets
are the same by establishing that A ⊆ B and
B ⊆ A.
Operations on sets: union; intersection;
complement; set difference; symmetric
difference.
Finite and infinite sets. Subsets.
Content
The aims of this topic are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract algebra.
Topic 4—Sets, relations and groups
Syllabus content
Further mathematics HL guide
4.7
The identity element e.
4.6
Abelian groups.
The operation table of a group is a Latin
square, but the converse is false.
The definition of a group {G , ∗} .
Proofs of the uniqueness of the identity and
inverse elements.
Proof that left-cancellation and rightcancellation by an element a hold, provided
that a has an inverse.
The inverse a −1 of an element a.
The arithmetic operations on  and .
Binary operations: associative, distributive and
commutative properties.
4.5
∗ is associative;
G contains an identity element;
each element in G has an inverse in G.
•
•
•
a ∗ b = b ∗ a , for all a, b ∈ G .
G is closed under ∗ ;
•
For the set G under a given operation ∗ :
Both a ∗ a −1 =
e and a −1 ∗ a =
e must hold.
Both the right-identity a ∗ e =
a and leftidentity e ∗ a =
a must hold if e is an identity
element.
Examples of distributivity could include the
fact that, on  , multiplication is distributive
over addition but addition is not distributive
over multiplication.
Further guidance
Content
Appl: Galois theory for the impossibility of
such formulae for polynomials of degree 5 or
higher.
Appl: Existence of formula for roots of
polynomials.
TOK: Which are more fundamental, the
general models or the familiar examples?
Links
Syllabus content
33
34
4.9
4.8
integers under addition modulo n;
non-zero integers under multiplication,
modulo p, where p is prime;
symmetries of plane figures, including
equilateral triangles and rectangles;
invertible functions under composition of
functions.
•
•
•
•
Proof that all cyclic groups are Abelian.
Generators.
Cyclic groups.
The order of a group element.
The order of a group.
, ,  and  under addition;
by T 2 .
The composition T 2  T1 denotes T1 followed
Cross-topic questions may be set in further
mathematics examinations, so there may be
questions on groups of matrices.
Examples of groups:
•
Further guidance
Content
Appl: Music circle of fifths, prime numbers.
Appl: Rubik’s cube, time measures, crystal
structure, symmetries of molecules, strut and
cable constructions, Physics H2.2 (special
relativity), the 8-fold way, supersymmetry.
Links
Syllabus content
Further mathematics HL guide
Further mathematics HL guide
4.11
4.10
Further guidance
Use and proof of the result that the order of a
finite group is divisible by the order of any
element. (Corollary to Lagrange’s theorem.)
Lagrange’s theorem.
Definition and examples of left and right cosets
of a subgroup of a group.
Suppose that {G , ∗} is a group and H is a
non-empty subset of G. Then {H , ∗} is a
Use and proof of subgroup tests.
Suppose that {G , ∗} is a finite group and H is a
non-empty subset of G. Then {H , ∗} is a
subgroup of {G , ∗} if H is closed under ∗ .
subgroup of {G , ∗} if a ∗ b −1 ∈ H whenever
a, b ∈ H .
A proper subgroup is neither the group itself nor
the subgroup containing only the identity element.
Subgroups, proper subgroups.
The order of a combination of cycles.
On examination papers: the form
1 2 3
p=
 or in cycle notation (132) will
Cycle notation for permutations.
3 1 2
Result that every permutation can be written as be used to represent the permutation 1 → 3 ,
2 → 1 , 3 → 2.
a composition of disjoint cycles.
Permutations under composition of
permutations.
Content
Appl: Prime factorization, symmetry breaking.
Appl: Cryptography, campanology.
Links
Syllabus content
35
4.12
36
Infinite groups as well as finite groups.
Definition of a group homomorphism.
The order of an element is unchanged by an
isomorphism.
Isomorphism of groups.
−1
for all a ∈ G .
The homomorphism f : G → H is an
isomorphism if f is bijective.
Infinite groups as well as finite groups.
Inverse: f (a −1 ) = ( f (a ) )
Identity: let eG and eH be the identity elements
of (G, ∗) and ( H , ) , respectively, then
f (eG ) = eH .
Proof of homomorphism properties for
identities and inverses.
Proof that the kernel and range of a
homomorphism are subgroups.
If f : G → H is a group homomorphism, then
Ker( f ) is the set of a ∈ G such that
f (a ) = eH .
Definition of the kernel of a homomorphism.
Let {G ,*} and {H , } be groups, then the
function f : G → H is a homomorphism if
f (a * b) = f (a )  f (b) for all a, b ∈ G .
Further guidance
Content
Links
Syllabus content
Further mathematics HL guide
Further mathematics HL guide
Convergence of infinite series.
5.2
p
1
∑n
.
Power series: radius of convergence and
interval of convergence. Determination of the
radius of convergence by the ratio test.
Alternating series.
Series that converge conditionally.
Series that converge absolutely.
The p-series,
Tests for convergence: comparison test; limit
comparison test; ratio test; integral test.
Informal treatment of limit of sum, difference,
product, quotient; squeeze theorem.
Infinite sequences of real numbers and their
convergence or divergence.
5.1
is convergent for p > 1 and divergent
The absolute value of the truncation error is
less than the next term in the series.
Conditions for convergence.
otherwise. When p = 1 , this is the harmonic
series.
p
1
∑n
x →∞
then the series is not necessarily convergent,
but if lim xn ≠ 0 , the series diverges.
x →∞
Students should be aware that if lim xn = 0
The sum of a series is the limit of the sequence
of its partial sums.
Divergent is taken to mean not convergent.
Further guidance
Content
48 hours
TOK: Euler’s idea that 1 − 1 + 1 − 1 +  =12 .
Was it a mistake or just an alternative view?
TOK: Zeno’s paradox, impact of infinite
sequences and limits on our understanding of
the physical world.
Links
The aims of this topic are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations.
Topic 5—Calculus
Syllabus content
37
38
5.4
5.3
Further guidance
Improper integrals of the type
a
∫ f ( x) dx .
∞
Fundamental theorem of calculus.
The integral as a limit of a sum; lower and
upper Riemann sums.
Continuous functions and differentiable
functions.
x → a+
h
f ( a + h) – f ( a )
h
f ( a + h) – f ( a )
exist and are equal.
and
x

d 
 ∫ f ( y ) dy  = f ( x ) .
dx  a

Students should be aware that a function may
be continuous but not differentiable at a point,
eg f ( x ) = x and simple piecewise functions.
h → 0+
lim
h →0 −
lim
f is continuous at a and
Test for differentiability:
x→a –
Continuity and differentiability of a function at Test for continuity:
a point.
lim f ( x) = f ( a ) = lim f ( x ) .
Content
1
, 1≤ x ≤ ∞ .
x
An infinite area sweeps out a finite volume.
Can this be reconciled with our intuition? What
does this tell us about mathematical
knowledge?
TOK: Consider f  x =
Aim 8: Leibniz versus Newton versus the
“giants” on whose shoulders they stood—who
deserves credit for mathematical progress?
Int: Contribution of Arab, Chinese and Indian
mathematicians to the development of calculus.
Int: How close was Archimedes to integral
calculus?
Links
Syllabus content
Further mathematics HL guide
5.5
Further mathematics HL guide
 y
f 
x
using the integrating factor.
y′ + P(x)y = Q(x),
Solution of
using the substitution y = vx.
dy
=
dx
Homogeneous differential equation
Variables separable.
using Euler’s method.
xn +=
xn + h ,
1
dy
= f ( x, y )
dx
where h is a constant.
yn +=
yn + hf ( xn , yn ) ,
1
Numerical solution of
carbon dating.
population growth,
Newton’s law of cooling,
Geometric interpretation using slope fields,
including identification of isoclines.
Links
Appl: Real-life differential equations, eg
Further guidance
First-order differential equations.
Content
Syllabus content
39
40
5.7
5.6
f ( x)
f ( x)
and lim
.
g
g ( x)
x →∞
( x)
Using l’Hôpital’s rule or the Taylor series.
x→a
lim
The evaluation of limits of the form
Taylor series developed from differential
equations.
0
∞
and .
0
∞
Repeated use of l’Hôpital’s rule.
The indeterminate forms
Students should be aware of the intervals of
convergence.
Maclaurin series for e x , sin x , cos x ,
ln(1 + x) , (1 + x) p , p ∈  .
Use of substitution, products, integration and
differentiation to obtain other series.
Applications to the approximation of functions;
formula for the error term, in terms of the value
of the (n + 1)th derivative at an intermediate
point.
Further guidance
Taylor polynomials; the Lagrange form of the
error term.
Mean value theorem.
Rolle’s theorem.
Content
Int: Compare with work of the Kerala school.
Int, TOK: Influence of Bourbaki on
understanding and teaching of mathematics.
Links
Syllabus content
Further mathematics HL guide
Further mathematics HL guide
6.3
6.2
6.1
For example, proofs of the fundamental
theorem of arithmetic and the fact that a tree
with n vertices has n – 1 edges.
Strong induction.
Linear Diophantine equations ax + by =
c.
Prime numbers; relatively prime numbers and
the fundamental theorem of arithmetic.
General solutions required and solutions
subject to constraints. For example, all
solutions must be positive.
The Euclidean algorithm for determining the
greatest common divisor of two integers.
Division and Euclidean algorithms.
The greatest common divisor, gcd(a, b) , and
the least common multiple, lcm(a, b) , of
integers a and b.
The division algorithm =
a bq + r , 0 ≤ r < b .
The theorem a | b and a | c ⇒ a | (bx ± cy )
where x, y ∈  .
a|b ⇒ b =
na for some n ∈  .
Pigeon-hole principle.
Further guidance
Content
48 hours
Int: Described in Diophantus’ Arithmetica
written in Alexandria in the 3rd century CE.
When studying Arithmetica, a French
mathematician, Pierre de Fermat (1601–1665)
wrote in the margin that he had discovered a
simple proof regarding higher-order
Diophantine equations—Fermat’s last theorem.
Aim 8: Use of prime numbers in cryptography.
The possible impact of the discovery of
powerful factorization techniques on internet
and bank security.
Int: Euclidean algorithm contained in Euclid’s
Elements, written in Alexandria about
300 BCE.
TOK: Proof by contradiction.
TOK: Mathematics and knowledge claims.
The difference between proof and conjecture,
eg Goldbach’s conjecture. Can a mathematical
statement be true before it is proven?
Links
The aim of this topic is to provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.
Topic 6—Discrete mathematics
Syllabus content
41
42
Two vertices are adjacent if they are joined by
an edge. Two edges are adjacent if they have a
common vertex.
Graphs, vertices, edges, faces. Adjacent
vertices, adjacent edges.
6.7
Euler’s relation: v − e + f =
2 ; theorems for
planar graphs including e ≤ 3v − 6 , e ≤ 2v − 4 ,
leading to the results that κ 5 and κ 3,3 are not
planar.
Subgraphs; complements of graphs.
Simple graphs; connected graphs; complete
graphs; bipartite graphs; planar graphs; trees;
weighted graphs, including tabular
representation.
Handshaking lemma.
Degree of a vertex, degree sequence.
a p = a (mod p ) , where p is prime.
Fermat’s little theorem.
6.6
Further mathematics HL guide
If the graph is simple, planar, has no cycles of
length 3 and v ≥ 3 , then e ≤ 2v − 4 .
If the graph is simple and planar and v ≥ 3 ,
then e ≤ 3v − 6 .
TOK: Mathematics and knowledge claims.
Applications of the Euler characteristic
(v − e + f ) to higher dimensions. Its use in
understanding properties of shapes that cannot
be visualized.
It should be stressed that a graph should not be Aim 8: Importance of planar graphs in
assumed to be simple unless specifically stated. constructing circuit boards.
The term adjacency table may be used.
TOK: Mathematics and knowledge claims.
Proof of the four-colour theorem. If a theorem
is proved by computer, how can we claim to
know that it is true?
Aim 8: Symbolic maps, eg Metro and
Underground maps, structural formulae in
chemistry, electrical circuits.
TOK: Nature of mathematics. An interest may
be pursued for centuries before becoming
“useful”.
Int: Babylonians developed a base 60 number
system and the Mayans a base 20 number system.
On examination papers, questions that go
beyond base 16 will not be set.
Representation of integers in different bases.
6.5
Links
Int: Discussed by Chinese mathematician Sun
Tzu in the 3rd century CE.
The solution of linear congruences.
Modular arithmetic.
Further guidance
Solution of simultaneous linear congruences
(Chinese remainder theorem).
6.4
Content
Syllabus content
Further mathematics HL guide
6.11
Modelling with recurrence relations.
The first-degree linear recurrence relation
=
un aun −1 + b .
Solution of first- and second-degree linear
homogeneous recurrence relations with
constant coefficients.
Recurrence relations. Initial conditions,
recursive definition of a sequence.
Deleted vertex algorithm for determining a
lower bound.
Nearest-neighbour algorithm for determining
an upper bound.
Travelling salesman problem.
Not required:
Graphs with more than four vertices of odd
degree.
Chinese postman problem.
6.10
Solving problems such as compound interest,
debt repayment and counting problems.
Includes the cases where auxiliary equation has
equal roots or complex roots.
To determine the Hamiltonian cycle of least
weight in a weighted complete graph.
To determine the shortest route around a
weighted graph going along each edge at least
once.
Simple treatment only.
Hamiltonian paths and cycles.
Graph algorithms: Kruskal’s; Dijkstra’s.
A connected graph contains an Eulerian circuit
if and only if every vertex of the graph is of
even degree.
Further guidance
Eulerian trails and circuits.
Walks, trails, paths, circuits, cycles.
6.9
6.8
Content
TOK: Mathematics and the world. The
connections of sequences such as the Fibonacci
sequence with art and biology.
TOK: Mathematics and knowledge claims.
How long would it take a computer to test all
Hamiltonian cycles in a complete, weighted
graph with just 30 vertices?
Int: Problem posed by the Chinese
mathematician Kwan Mei-Ko in 1962.
Int: The “Bridges of Königsberg” problem.
Links
Syllabus content
43
Syllabus
Glossary of terminology: Discrete mathematics
Introduction
Teachers 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.
44
Further mathematics HL guide
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.
Further mathematics HL guide
45
Assessment
Assessment in the Diploma Programme
General
Assessment 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.
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Further mathematics HL guide
Assessment in the Diploma Programme
Methods of assessment
The IB uses several methods to assess work produced by students.
Assessment criteria
Assessment 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.
Markbands
Markbands 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.
Markschemes
This 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.
Further mathematics HL guide
47
Assessment
Assessment outline
First examinations 2014
Assessment component
Weighting
External assessment (5 hours)
50%
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.
48
Further mathematics HL guide
Assessment
Assessment details
External assessment
Papers 1 and 2
These 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.
Calculators
Papers 1 and 2
Students 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 booklet
Each 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 marks
Marks 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 1
Duration: 2 hours 30 minutes
Weighting: 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.
Further mathematics HL guide
49
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 2
Duration: 2 hours 30 minutes
Weighting: 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 assessment
There is no internal assessment component in this course.
50
Further mathematics HL guide
Appendices
Glossary of command terms
Command terms with definitions
Students 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.
Further mathematics HL guide
51
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.
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Further mathematics HL guide
Appendices
Notation list
Of the various notations in use, the IB has chosen to adopt a system of notation based on the recommendations of the
International Organization for Standardization (ISO). This notation is used in the examination papers for this course
without explanation. If forms of notation other than those listed in this guide are used on a particular examination
paper, they are defined within the question in which they appear.
Because students are required to recognize, though not necessarily use, IB notation in examinations, it is recommended
that teachers introduce students to this notation at the earliest opportunity. Students are not allowed access to
information about this notation in the examinations.
Students must always use correct mathematical notation, not calculator notation.

the set of positive integers and zero, {0,1, 2, 3, ...}

the set of integers, {0, ± 1, ± 2, ± 3, ...}
+
the set of positive integers, {1, 2, 3, ...}

the set of rational numbers
+
the set of positive rational numbers, {x | x ∈ , x > 0}

the set of real numbers
+
the set of positive real numbers, {x | x ∈ , x > 0}

the set of complex numbers, {a + ib | 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
cos θ + i sin θ
{x1 , x2 , ...}
the set with elements x1 , x2 , ...
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
Further mathematics HL guide
53
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
a1/ n ,
n
a1/ 2 ,
a
a to the power of
a
a to the power
1 th
, n root of a (if a ≥ 0 then
n
1
, square root of a (if a ≥ 0 then
2
n
a ≥0)
a ≥0)
 x for x ≥ 0, x ∈ 
− x for x < 0, x ∈ 
x
the modulus or absolute value of x, that is 
≡
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
un
the n
d
the common difference of an arithmetic sequence
r
the common ratio of a geometric sequence
Sn
the sum of the first n terms of a sequence, u1 + u2 + ... + un
S∞
the sum to infinity of a sequence, u1 + u2 + ...
n
∑u
i =1
i
n
th
term of a sequence or series
u1 + u2 + ... + un
∏u
u1 × u2 × ... × un
 n
 
r 
n!
r !(n − r )!
i =1
54
i
Further mathematics HL guide
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
f −1
the inverse function of the function f
f g
the composite function of f and g
lim f ( x)
the limit of f ( x) as x tends to a
dy
dx
the derivative of y with respect to x
f ′( x)
the derivative of f ( x) with respect to x
d2 y
dx 2
the second derivative of y with respect to x
f ′′( x)
the second derivative of f ( x) with respect to x
dn y
dx n
the n
th
derivative of y with respect to x
f ( ) ( x)
the n
th
derivative of f ( x) with respect to x
∫ y dx
the indefinite integral of y with respect to x
x→a
n
∫
b
a
y dx
the definite integral of y with respect to x between the limits
ex
the exponential function of x
log a x
the logarithm to the base a of x
ln x
the natural logarithm of x, log e 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
Further mathematics HL guide
x = a and x = b
55
Notation list
ˆ
CAB
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
A−1
the inverse of the non-singular matrix A
AT
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
x1 , x2 , ...
observations
f1 , f 2 , ...
frequencies with which the observations x1 , x2 , ... 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
k
σ2
population variance,
σ =
2
∑ f (x
i =1
σ
population standard deviation
x
sample mean
56
i
i
n
− µ )2
, where n =
k
∑f
i =1
i
Further mathematics HL guide
Notation list
k
s
2
n
2
sample variance, sn =
sn
∑ f (x − x )
i =1
i
i
n
2
, where n =
k
∑f
i =1
i
standard deviation of the sample
k
n 2
unbiased estimate of the population variance,
=
sn2−1 =
sn
n −1
sn2−1
k
∑ f (x
i =1
i
i
− x )2
n −1
, where
n = ∑ fi
i =1
B ( n, p )
binomial distribution with parameters n and p
Po ( m )
Poisson distribution with mean m
N ( µ ,σ 2 )
normal distribution with mean
X ~ B( n , p)
the random variable X has a binomial distribution with parameters n and p
X ~ Po ( m )
the random variable X has a Poisson distribution with mean m
X ~ N(µ , σ 2 )
the random variable X has a normal distribution with mean
Φ
cumulative distribution function of the standardized normal variable with distribution
ν
number of degrees of freedom
A\ B
the difference of the sets A and B (that is, A \ B =
A ∩ B′ =
{x x ∈ A and x ∉ B} )
A∆B
the symmetric difference of the sets A and B (that is, A=
∆B ( A \ B ) ∪ ( B \ A) )
κn
a complete graph with n vertices
κ n, m
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,  , p − 1} 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
AG
the adjacency matrix of graph G
CG
the cost adjacency matrix of graph G
µ and variance σ 2
µ and variance σ 2
N ( 0,1)
Further mathematics HL guide
57

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