Unformatted text preview: INTERNATIONAL BACCALAUREATE ORGANIZATION DIPLOMA PROGRAMME
Mathematics higher level
For first examinations in 2001 Mathematics Higher Level
February 1998 Copyright © 1998 International Baccalaureate
International Baccalaureate Organisation
Route des Morillons 15
1218 GrandSaconnex
Geneva, SWITZERLAND CONTENTS
INTRODUCTION 1 NATURE OF THE SUBJECT 3 AIMS 5 OBJECTIVES 6 SYLLABUS OUTLINE 7 SYLLABUS DETAILS 9 SYLLABUS GUIDELINES 38 ASSESSMENT OUTLINE 42 ASSESSMENT DETAILS 43 ASSESSMENT CRITERIA 52 ASSESSMENT GUIDELINES 58 INTRODUCTION
The International Baccalaureate Diploma Programme is a rigorous preuniversity course of studies,
leading to examinations, that meets the needs of highly motivated secondary school students between
the ages of 16 and 19 years. Designed as a comprehensive twoyear curriculum that allows its
graduates to fulfil requirements of various national education systems, the diploma model is based on
the pattern of no single country but incorporates the best elements of many. The programme is
available in English, French and Spanish.
The curriculum is displayed in the shape of a hexagon with six academic areas surrounding the core.
Subjects are studied concurrently and students are exposed to the two great traditions of learning: the
humanities and the sciences. IB Diploma guide: Mathematics HL, September 2001 1 INTRODUCTION Diploma Programme candidates are required to select one subject from each of the six subject
groups. At least three and not more than four are taken at higher level (HL), the others at standard
level (SL). Higher level courses represent 240 teaching hours; SL courses cover 150 hours. By
arranging work in this fashion, students are able to explore some subjects in depth and some more
broadly over the twoyear period; this is a deliberate compromise between the early specialisation
preferred in some national systems and the breadth found in others.
Distribution requirements ensure that the scienceorientated student is challenged to learn a foreign
language and that the natural linguist becomes familiar with science laboratory procedures. While
overall balance is maintained, flexibility in choosing higher level concentrations allows the student to
pursue areas of personal interest and to meet special requirements for university entrance.
Successful Diploma Programme candidates meet three requirements in addition to the six subjects.
The interdisciplinary Theory of Knowledge (TOK) course is designed to develop a coherent approach
to learning which transcends and unifies the academic areas and encourages appreciation of other
cultural perspectives. The extended essay of some 4000 words offers the opportunity to investigate a
topic of special interest and acquaints students with the independent research and writing skills
expected at university. Participation in the Creativity, Action, Service (CAS) requirement encourages
students to be involved in sports, artistic pursuits and community service work. For first examinations in 2001 2 IB Diploma Programme guide: Mathematics HL, September 2001 NATURE OF THE SUBJECT
Introduction
The nature of mathematics can be summarized in a number of ways; for example, 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: buying produce in the market,
consulting a timetable, reading a newspaper, timing a process or estimating a length. For
most people mathematics also extends into their chosen profession: 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. Scientists view mathematics as a language that is vital
to our understanding of events that occur in the natural world. Other people are challenged by
the logical methods of mathematics and the adventure in reason that mathematical proof has
to offer. Still others appreciate mathematics as an aesthetic experience or even as a
cornerstone of philosophy. The prevalence of mathematics in people’s lives thus provides a
clear and sufficient rationale for making the study of this subject compulsory within the IB
diploma.
Since individual students have different needs, interests and abilities, the International
Baccalaureate Organization (IBO) offers a number of different courses in mathematics. These
are targeted at students who wish to study mathematics in depth, either as a subject in its own
right or in order to pursue their interests in areas related to mathematics, those who wish to
gain a degree of understanding and competence in order to understand better their approach
to other subjects, and those who may not be aware that mathematics has relevance in their
studies and in their future lives. Each course is designed to meet the needs of a particular
group of students and therefore great care should be exercised in selecting the one which is
most appropriate for an individual student.
In making the selection, individual students should be advised to take account of the
following considerations. • Their own abilities in mathematics and the type of mathematics in which they can be
successful. • Their own interest in mathematics with respect to the areas which hold an appeal. • Their other choices of subjects within the framework of the Diploma Programme. IB Diploma Programme guide: Mathematics HL, September 2001 3 NATURE OF THE SUBJECT • Their future academic plans in terms of the subjects they wish to study. • Their choice of career. Teachers are expected to assist with the selection process and to offer advice to students on
choosing the most appropriate subject from group 5. Mathematics higher level
Mathematics, available as a higher level (HL) subject only, 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.
The nature of the subject is such that it focuses on developing important mathematical
concepts in a comprehensible and coherent way. This is achieved by means of a carefully
balanced approach: students are encouraged to apply their mathematical knowledge to
solving problems set in a variety of meaningful contexts while, at the same time, being
introduced to important concepts of rigour and proof.
Students embarking on this course should expect to develop insight into mathematical form
and structure in their studies, and should be intellectually equipped to appreciate the links
between parallel structures in different topic areas. They should also be encouraged to
develop the skills needed to continue their mathematical growth in other learning
environments.
The internally assessed component, the portfolio, offers students a framework for developing
independence in their mathematical development through engaging in the following
activities: mathematical investigation, extended closedproblem solving and mathematical
modelling. Students will thus be provided with the means to ask their own questions about
mathematics and be given the chance to explore different ways of arriving at a solution,
either through experimenting with the techniques at their disposal or by researching new
methods. This process also allows students to work without the time constraints of a written
examination and to acquire ownership of a part of the course.
This course is clearly a demanding one, requiring students to study a broad range of
mathematical topics through a number of different approaches and to varying degrees of
depth. Students wishing to study mathematics in a less rigorous environment should therefore
opt for one of the standard level courses: mathematical methods or mathematical studies. 4 IB Diploma Programme guide: Mathematics HL, September 2001 AIMS
The aims of all courses in group 5 are to enable candidates to: • appreciate the international dimensions of mathematics and the multiplicity of its
cultural and historical perspectives • foster enjoyment from engaging in mathematical pursuits, and to develop an
appreciation of the beauty, power and usefulness of mathematics • develop logical, critical and creative thinking in mathematics • develop mathematical knowledge, concepts and principles • employ and refine the powers of abstraction and generalization • develop patience and persistence in problemsolving • have an enhanced awareness of, and utilize the potential of, technological
developments in a variety of mathematical contexts • communicate mathematically, both clearly and confidently, in a variety of contexts. IB Diploma Programme guide: Mathematics HL, September 2001 5 OBJECTIVES
Having followed any one of the courses in group 5, candidates will be expected to: •
• read and interpret a given problem in appropriate mathematical terms • organize and present information/data in tabular, graphical and/or diagrammatic
forms • know and use appropriate notation and terminology • formulate a mathematical argument and communicate it clearly • select and use appropriate mathematical techniques • understand the significance and reasonableness of results • recognize patterns and structures in a variety of situations and draw inductive
generalizations • demonstrate an understanding of, and competence in, the practical applications of
mathematics • 6 know and use mathematical concepts and principles use appropriate technological devices as mathematical tools. IB Diploma Programme guide: Mathematics HL, September 2001 SYLLABUS OUTLINE
The mathematics higher level (HL) syllabus consists of the study of eight core topics and one option. Part I: Core 195 hours All topics in the core are compulsory. Candidates are required to study all the subtopics in each of
the eight topics in this part of the syllabus as listed in the Syllabus Details.
1 Number and algebra 20 hours 2 Functions and equations 25 hours 3 Circular functions and trigonometry 25 hours 4 Vector geometry 25 hours 5 Matrices and transformations 20 hours 6 Statistics 10 hours 7 Probability 20 hours 8 Calculus 50 hours Part II: Options 35 hours Candidates are required to study all the subtopics in one of the following options as listed in the
Syllabus Details.
9 Statistics 35 hours 10 Sets, relations and groups 35 hours 11 Discrete mathematics 35 hours 12 Analysis and approximation 35 hours 13 Euclidean geometry and conic sections 35 hours IB Diploma Programme guide: Mathematics HL, September 2001 7 SYLLABUS OUTLINE Portfolio 10 hours Three assignments, based on different areas of the syllabus, representing each of the following
activities: !
!
! 8 mathematical investigation
extended closedproblem solving
mathematical modelling IB Diploma Programme guide: Mathematics HL, September 2001 SYLLABUS DETAILS
Format of the syllabus
The syllabus is formatted into three columns labelled Content, Amplifications/Exclusions and
Teaching Notes. ! Content: the first column lists, under each topic, the subtopics to be covered. ! Amplifications/Exclusions: the second column contains more explicit information on
specific subtopics listed in the first column. This helps to define what is required and
what is not required in terms of preparing for the examination. ! Teaching Notes: the third column provides useful suggestions for teachers. It is not
mandatory that these suggestions be followed. Course of study
Teachers are required to teach all the subtopics listed under the eight topics in the core,
together with all the subtopics in the chosen option.
It is not necessary, nor desirable, to teach the topics in the core in the order in which they
appear in the Syllabus Outline and Syllabus Details. Neither is it necessary to teach all the
topics in the core before starting to teach an option. Teachers are therefore strongly advised
to draw up a course of study, tailored to the needs of their students, which integrates the areas
covered by both the core and the chosen option. Integration of portfolio assignments
The three assignments for the portfolio, based on the three activities (mathematical
investigation, extended closedproblem solving and mathematical modelling), should be
incorporated into the course of study, and should relate directly to topics in the syllabus. Full
details are given in Assessment Details, Portfolio. IB Diploma Programme guide: Mathematics HL, September 2001 9 SYLLABUS DETAILS Time allocation
The recommended teaching time for a higher level subject is 240 hours. For mathematics HL,
it is expected that 10 hours will be spent on work for the portfolio. The time allocations given
in the Syllabus Outline and Syllabus Details are approximate, and are intended to suggest
how the remaining 230 hours allowed for teaching the syllabus might be allocated. However,
the exact time spent on each topic will depend on a number of factors, including the
background knowledge and level of preparedness of each student. Teachers should therefore
adjust these timings to correspond with the needs of their students. Use of calculators
Candidates are required to have access to a graphic display calculator at all times during the
course, both inside and out of the classroom. Regulations concerning the types of calculators
allowed are provided in the Vade Mecum. Formulae booklet and statistical tables (third edition,
February 2001)
As each candidate is required to have access to clean copies of the IBO formulae booklet and
statistical tables during the examination, it is recommended that teachers ensure candidates
are familiar with the contents of these documents from the beginning of the course. The
booklet and tables are provided by IBCA and are published separately. Resource list
A resource list is available for mathematics HL on the online curriculum centre. This list
provides details of, for example, texts, software packages and videos which are considered by
teachers to be appropriate for use with this course. It will be updated on a regular basis.
Teachers can at any time add any materials to this list which they consider to be appropriate
for candidate use or as reference material for teachers. 10 IB Diploma Programme guide: Mathematics HL, September 2001 IB Diploma Programme guide: Mathematics HL, September 2001 1 Core: number and algebra Teaching time: 20 hours The aims of this section are to introduce important results and methods of proof in algebra, and to extend the concept of number to include complex
numbers.
CONTENT
1.1 Arithmetic sequences and series; sum of
finite arithmetic series; geometric sequences
and series; sum of finite and infinite
geometric series.
Applications of the above.
1.2 Exponents and logarithms: laws of
exponents; laws of logarithms. ( a + b) , n ∈N . ∑ ai .
i =1 Included: applications of sequences and series
to compound interest and population growth.
Included: change of base, ie log b a = log c a
.
log c b This topic is developed further in §2.9. Forming conjectures to be proved by
mathematical induction.
−1 ; 11 the terms real part, imaginary part,
conjugate, modulus and argument; the forms
z = a + ib and z = r ( cosθ + isin θ ) .
The complex plane. Although only the notation will be used in Included: proofs of standard results for sums of
squares and cubes of natural numbers. examination papers, candidates will need to be
aware of alternative notation used in textbooks and
n
on calculators, eg nCr , nCr, C r .
Link with mathematical induction in §1.4.
Link with De Moivre’s theorem in §1.7.
Link with counting principles in §7.5.
Link with binomial distribution in §7.7.
Link with limits and convergence in §8.1.
Link with binomial theorem in §1.3 and De Moivre’s
theorem in §1.7. Included: cartesian and polar forms of a
complex number. n
r SYLLABUS DETAILS: CORE 1.4 Proof by mathematical induction. 1.5 Complex numbers: the number i = n Included: sigma notation, ie TEACHING NOTES
Generation of terms and partial sums by iterating on
a calculator can be useful.
Link with limits and convergence in §8.1. n
n!
Included: the formulae =
. r r !( n − r ) ! 1.3 The binomial theorem: expansion of
n AMPLIFICATIONS /EXCLUSIONS Core: number and algebra (continued) CONTENT
1.6 Sums, products and quotients of complex
numbers. AMPLIFICATIONS /EXCLUSIONS
Included: multiplication by i as a rotation of
90 ! in the complex plane. Link with binomial theorem in §1.3.
Link with proof by induction in §1.4. 1.7 De Moivre’s theorem (proof by mathematical induction).
Powers and roots of a complex number.
1.8 Conjugate roots of polynomial equations
with real coefficients. TEACHING NOTES
Link with transformation of vectors in §5.5. Not required: equations with complex
coefficients. SYLLABUS DETAILS: CORE 12 1 IB Diploma Programme guide: Mathematics HL, September 2001 IB Diploma Programme guide: Mathematics HL, September 2001 2 Core: functions and equations Teaching time: 25 hours The aims of this section are to introduce methods of solution for different types of equations, to explore the notion of function as a unifying theme in
mathematics, to study certain functions in more depth and to explore the transformations of the graphical representations of functions.
CONTENT
2.1 Concept of function f : x " f ( x ) : domain,
range; image (value).
Composite functions f ! g ; identity
function; inverse function f −1 .
Domain restriction.
The graph of a function; its equation
y = f (x) . 2.2 Function graphing skills: use of a graphic TEACHING NOTES
General examples: for x " 2 − x , domain is x ≤ 2 ,
range is y ≥ 0 ; for x " “distance from nearest
integer”, domain is R, range is 0 ≤ y ≤ 0.5.
Example of domain restriction: x " x − 3 is the
2
inverse of x " x + 3, x ≥ 0 , but x " − x − 3 is the
inverse of x " x 2 + 3, x < 0 .
Note that the composite function ( f ! g )( x ) is
defined as f ( g ( x )).
Link with the chain rule for composite functions in
§8.3. Included: identification of horizontal and
vertical asymptotes; use of the calculator to find
maximum and minimum points.
On examination papers: questions may be set
which require the graphing of functions which
do not explicitly appear on the syllabus. Calculator settings should be chosen appropriately to
avoid, for example, interpolation across a vertical
asymptote.
These graphing skills should be utilized throughout
the syllabus as appropriate.
Link with maximum and minimum problems in §8.6. 13 SYLLABUS DETAILS: CORE display calculator to graph a variety of
functions.
Appropriate choice of “window”, use of
“zoom” and “trace” (or equivalent ) to locate
points to a given accuracy; use of
“connected” and “dot” (or equivalent)
modes as appropriate.
Solution of f ( x ) = 0 to a given accuracy. AMPLIFICATIONS /EXCLUSIONS
In examinations: if the domain is the set of real
numbers then the statement “x ∈ R ” will be
omitted.
Included: formal definition of a function; the
terms “oneone”, and “manyone”.
Not required: the term “codomain”. Core: functions and equations (continued) CONTENT
2.3 Transformations of graphs: translations;
stretches; reflections in the axes.
−1
The graph of f as the reflection in the
line y = x of the graph of f.
Absolute value function f .
1
The graph of
from f ( x ) .
f (x) 2.4 The reciprocal function x ! AMPLIFICATIONS /EXCLUSIONS
Translations: y = f ( x ) + b ; y = f ( x − a ) .
Stretches: y = pf ( x); y = f ( x / q ) . Reflections (in the xaxis and yaxis):
y = f ( − x ); y = − f ( x ) .
Included: y = f ( x ) , y = f ( x ) . TEACHING NOTES
Examples: y = x may be used to obtain
2 3 y = ( x − 3) 2 + 5 by a translation of , or y = sin x 5
may be used to obtain y = 3sin ( x / 2 ) by a twoway
stretch.
Link with quadratic functions in §2.5.
Link with exponential functions in §2.9.
Link with circular functions in §3.3.
Link with matrix transformations in §5.5. 1
, x ≠ 0 : its
x graph; its selfinverse nature.
2
2.5 The quadratic function x ! ax + bx + c : its IB Diploma Programme guide: Mathematics HL, September 2001 graph.
The form x ! a ( x − h) 2 + k : vertex (h, k)
and yintercept ( 0, c ) .
The form x ! a ( x − p)( x − q ) : xintercepts
(p, 0) and (q, 0).
2.6 Solution of f ( x ) = g ( x ) , f , g linear or
quadratic. Included: rational coefficients only. In examinations: questions demanding
elaborate factorization techniques will not be
set.
Included: knowledge of the significance of the
discriminant ∆ = b 2 − 4ac for the solution set
in the three cases ∆ > 0, ∆ = 0, ∆ < 0 . Link the second form, “completing the square”, with
transformations of functions in §2.3, ie
2
y = a ( x − h) 2 + k as y = x transformed. SYLLABUS DETAILS: CORE 14 2 IB Diploma Programme guide: Mathematics HL, September 2001 2 Core: functions and equations (continued) CONTENT
2.7 Inequalities in one variable, including their
graphical representation. Solution of
f ( x ) ≥ g ( x ) , f, g linear or quadratic. AMPLIFICATIONS /EXCLUSIONS
Included: cases where cross multiplication is
2
1
not appropriate, eg
; the use of the
>
x −2 x −3
absolute value sign in inequalities. 2.8 Polynomial functions. Included: the significance of multiple roots. The factor and remainder theorems, with
application to the solution of polynomial
equations and inequations.
x
2.9 The exponential function x " a , a > 0 : its domain and range.
The inverse function x " log a x . Included: for the domain of a x only rational x
need be considered.
Included: knowledge that log a a x = x = a log a x . Graphs of y = a x and y = log a x . Included: knowledge that a x = b ⇔ x = log a b. Solution of a x = b .
2.10 The functions x " e x , x " ln x . Application to the solution of equations
based on problems of growth and decay. Included: a x expressed as e x ln a .
Included: applications to population growth and
compound interest (eg doublingtimes), and
radioactive decay (eg halflife). TEACHING NOTES The use of synthetic division may be encouraged in
finding zeros, remainders and values.
The use of graphic display calculators should be
encouraged in finding zeros through trace or
calculation.
Link with the laws of exponents and logarithms in
§1.2.
Note that the graph of y = a x reflected in the line
y = x gives the graph of y = log a x ; link with
transformations of graphs in §2.3.
This topic may be linked with the applications of
geometric sequences in §1.1.
Link with differential equations in §8.11. SYLLABUS DETAILS: CORE 15 Core: circular functions and trigonometry Teaching time: 25 hours The aims of this section are to use trigonometry to solve general triangles, to explore the behaviour of circular functions both graphically and algebraically
and to introduce some important identities in trigonometry.
CONTENT
3.1 The circle: radian measure of angles; length
of an arc; area of a sector. AMPLIFICATIONS /EXCLUSIONS
Included: radian measure expressed as multiples
of π. 3.2 Definition of ( cos θ , sin θ ) in terms of the Included: given sin θ , finding possible values
of cosθ . unit circle.
The Pythagorean identities:
cos2 θ + sin 2 θ = 1; TEACHING NOTES
Note that 2πr generalizes to θr , πr 2 generalizes to
12
θr .
2 1 + tan 2 θ = sec2 θ ;
1 + cot 2 θ = csc 2 θ .
IB Diploma Programme guide: Mathematics HL, September 2001 3.3 The six circular functions: x " sin x , x " cos x , x " tan x , x " csc x , x " sec x ,
x " cot x; their domains and ranges; their
periodic nature, and their graphs.
The inverse functions x " arcsin x ,
x " arccos x , x " arctan x; their domains
and ranges, and their graphs.
3.4 Addition, doubleangle and halfangle
formulae:
1
sin ( A + B ) , etc; sin 2 A, etc; sin A, etc.
2
The compound formula a cos x ± b sin x = R cos ( x ∓ α ) . In examinations: radian measure should be
assumed unless otherwise indicated (eg
x " sin x° ). Included: proof of addition and doubleangle
formulae.
Not required: formal proof of the compound
formula. Although only the notations arcsin x , etc will be
used on examination papers, candidates will need to
be aware of alternative notations used on calculators.
The graph of y = a sin b ( x + c ) may be presented as a
transformation of y = sin x .
Link with inverse functions in §2.3. SYLLABUS DETAILS: CORE 16 3 IB Diploma Programme guide: Mathematics HL, September 2001 3 Core: circular functions and trigonometry (continued) CONTENT
3.5 Composite functions of the form
f ( x ) = a sin b ( x + c ); solutions of f ( x ) = k
in a given finite interval.
Solution of equations leading to quadratic or
linear equations in sin x , etc.
Graphical interpretation of the above.
3.6 Solution of triangles.
2
2
2
The cosine rule: c = a + b − 2ab cos C.
a
b
c
=
=
.
The sine rule:
sin A sin B sin C
1
Area of a triangle as ab sin C, etc.
2 AMPLIFICATIONS /EXCLUSIONS Included: the derivation of the sine rule from
the formula for the area of the triangle; the
ambiguous case of the sine rule; applications to
practical problems in two dimensions and three
dimensions. TEACHING NOTES Appreciation of Pythagoras’ theorem as a special
case of the cosine rule.
Link with the cosine rule in scalar product form in
§4.3. SYLLABUS DETAILS: CORE 17 Core: vector geometry Teaching time: 25 hours The aims of this section are to introduce the use of vectors in two and three dimensions, to facilitate solution of problems involving points, lines and planes,
and to enable the associated angles, distances and areas to be calculated.
CONTENT
4.1 Vectors as displacements in the plane and in v1 three dimensions, v = v2 .
v 3
Components of a vector; column
representation.
The sum of two vectors; the zero vector; the
inverse vector, − v .
Multiplication by a scalar, kv . AMPLIFICATIONS /EXCLUSIONS
Note: components are with respect to the
standard basis i, j and k : v = v1i + v 2 j + v3 k .
Included: the difference of v and w as
v − w = v + (− w) .
→ Included: the vector AB expressed as TEACHING NOTES
Vector sums and differences can be represented by
the diagonals of a parallelogram.
Multiplication by a scalar can be illustrated by
enlarging the vector parallelogram.
Applications to simple geometric figures, eg ABCD
→ → OB − OA = b − a . is a quadrilateral and AB = − CD ⇒ ABCD is a
parallelogram. Included: for nonzero perpendicular vectors
v ⋅ w = 0 ; for nonzero parallel vectors
v⋅w = ± v w . The scalar product is also known as the dot product
and the inner product.
Link with condition for perpendicularity in §4.3. → → Magnitude of a vector, v .
→ IB Diploma Programme guide: Mathematics HL, September 2001 Position vectors OA = a .
Unit vectors including i, j and k.
4.2 The scalar product of two vectors
u ⋅ v = u1 v1 + u2 v 2 + u3 v3 .
Properties of the scalar product
v ⋅w = w ⋅v ;
u ⋅ (v + w ) = u ⋅ v + u ⋅ w ;
( kv ) ⋅ w = k (v ⋅ w ) ;
2 v ⋅v = v .
Perpendicular vectors; parallel vectors. SYLLABUS DETAILS: CORE 18 4 IB Diploma Programme guide: Mathematics HL, September 2001 4 Core: vector geometry (continued)
CONTENT 4.3 The expression v ⋅ w = v w cosθ ; the angle between two vectors.
The projection of a vector v in the direction
of w; simple applications, eg finding the
distance of a point from a line.
4.4 The vector product of two vectors v × w = v w sinθ .
The formula for the area of a triangle in the
1
form
v×w .
2 4.5 Vector equation of a line r = a + λb. Vector equation of a plane r = a + λ b + µ c .
Use of normal vector to obtain r ⋅ n = a ⋅ n.
Cartesian equations of a line and plane. AMPLIFICATIONS /EXCLUSIONS
Included: the following formulae
v w +v w cosθ = 1 1 2 2 , v cos θ = v ⋅ w .
w
vw Included: an understanding of “ m1m2 = −1” ⇒
lines are perpendicular.
Included: geometric interpretation of the
magnitude of v × w as the area of a
parallelogram.
Included: the determinant representation
i
j
k
v × w = v1 v 2 v3 .
w1 w2 w3 The vector product is also known as the cross
product. Included: cartesian equation of a line in three
x − x0 y − y0 z − z0
=
=
dimensions
; cartesian
l
m
n
equation of a plane ax + by + cz = d .
Link with solution of linear equations in §5.7. 19 SYLLABUS DETAILS: CORE Included: inverse matrix method and Gaussian
elimination for finding the intersection of three
plane; two planes; three planes.
Angle between: two lines; a line and a plane; planes.
two planes.
4.7 Distances in two and three dimensions
between points, lines and planes.
4.6 Intersections of: two lines; a line with a TEACHING NOTES
Link with generalization of perpendicular and
parallel cases in §4.2.
Application to angle between lines ax + by = p and
cx + dy = q as angle between normal vectors.
Link with the cosine rule in §3.6. Core: matrices and transformations Teaching time: 20 hours The aims of this section are to introduce matrices, particularly the algebra of small square matrices, to extend knowledge of transformations, to consider
linear transformations of the plane represented by square matrices, to explore composition of transformations and to link matrices to the solution of sets of
linear equations.
CONTENT
5.1 Definition of a matrix: the terms element,
row, column and dimension.
5.2 Algebra of matrices: equality; addition;
subtraction; multiplication by a scalar;
multiplication of two matrices.
The identity matrix.
5.3 Determinants of matrices; the condition
for singularity of a matrix.
5.4 The inverse of a square matrix.
−1
Inverse of a composite , ( PQ ) = Q −1 P −1 .
IB Diploma Programme guide: Mathematics HL, September 2001 5.5 Linear transformations of vectors in two dimensions and their matrix representation:
rotations; reflections and enlargements.
The geometric significance of the
determinant. 5.6 Composition of linear transformations P, Q.
5.7 Solution of linear equations (a maximum of three equations in three unknowns).
Conditions for the existence of a unique
solution, no solution and an infinity of
solutions. AMPLIFICATIONS /EXCLUSIONS TEACHING NOTES
Examples: systems of equations; data storage. The matrix facility on a graphic display calculator
may be introduced. Included: matrices of dimension 3 × 3 at most.
Included: matrices of dimension 3 × 3 at most .
Not required: cofactors and minors.
In examinations: the convention will be that the
same symbol will represent both a
transformation and its matrix, eg R is a rotation
!
of 9 0 a b o u t ( 0 , 0 ) , and 0 −1 R=
.
1 0 Linear transformations are origin invariant.
Link with complex numbers in §1.6.
Link with transformations of graphs in §2.3. Note that PQ denotes “Q followed by P”.
Unique solutions can be found using inverse
matrices; other cases using Gaussian elimination.
Link with intersections of two lines or three planes
in §4.6. SYLLABUS DETAILS: CORE 20 5 IB Diploma Programme guide: Mathematics HL, September 2001 6 Core: statistics Teaching time: 10 hours The aims of this section are to explore methods of describing and presenting data, and to introduce methods of measuring central tendency and dispersion
of data.
CONTENT
6.1 Concept of population and sample.
Discrete data and continuous data.
Frequency tables. AMPLIFICATIONS /EXCLUSIONS
Included: elementary treatment only. TEACHING NOTES
Data for analysis should be drawn from a wide range
of areas. 6.2 Presentation of data. Included: treatment of both continuous and
discrete data.
Note: a frequency histogram uses equal class
intervals. Use of computer spreadsheet software may enhance
treatment of this topic. Grouped data; midinterval values; interval
width; upper and lower interval boundaries.
Frequency histograms.
6.3 Measures of central tendency: sample mean,
x ; median. Included: an awareness that the population
mean, µ , is generally unknown, and that the
sample mean, x , serves as an unbiased estimate
of this quantity. 6.4 Cumulative frequency; cumulative frequency graphs; quartiles and percentiles.
6.5 Measures of dispersion: range; interquartile
range; standard deviation of the sample, sn .
2
The unbiased estimate, sn − 1 , of the 21 SYLLABUS DETAILS: CORE population variance σ 2 . Included: an awareness that the population
standard deviation, σ , is generally unknown,
n2
2
and knowledge that sn −1 =
sn serves as an
n −1
unbiased estimate of σ 2 .
In examinations: candidates are expected to use
a statistical function on a calculator to find
standard deviations. Use of boxandwhisker plots on a graphic calculator
may enhance understanding.
Teachers should be aware of calculator, text and
regional variations in notation for sample variance. Core: probability Teaching time: 20 hours The aims of this section are to extend knowledge of the concepts, notation and laws of probability, and to introduce some important probability distributions
and their parameters.
CONTENT
7.1 Sample space, U; the event A. The probability of an event A as
n( A) .
P ( A) =
n(U )
The complementary events A and A′ (not A);
the relation P ( A) + P ( A′ ) = 1 .
7.2 Combined events, A ∩ B and A ∪ B . IB Diploma Programme guide: Mathematics HL, September 2001 The relation
P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B ) .
Mutually exclusive events; the relation
P ( A ∩ B) = 0 .
7.3 Conditional probability; the relation P ( A ∩ B)
.
P ( A  B) =
P ( B)
Independent events; the relations
P ( A  B ) = P ( A ) = P ( A  B′ ) . Use of Bayes’ Theorem for two events.
7.4 Use of Venn diagrams, tree diagrams and
tables of outcomes to solve problems.
Applications.
7.5 Counting principles, including permutations
and combinations. AMPLIFICATIONS /EXCLUSIONS
Included: an emphasis on the concept of equally
likely outcomes. TEACHING NOTES
Experiments using coins, dice, packs of cards, etc,
can enhance understanding of the distinction
between (experimental) relative frequency and
(theoretical) probability. Simulation using random
numbers can also be useful. Included: an appreciation of the nonexclusivity
of “or”. It should be emphasized that problems might best be
solved with the aid of a Venn diagram or tree
diagram, without the explicit use of these formulae. Included: selection without replacement; proof
of independence using P( A ∩ B ) = P( A) P( B ) . The term “independent” is equivalent to “statistically
independent”. Examples: cards, dice and other simple cases of
random selection.
Included: the number of ways of selecting and
arranging r objects from n; simple applications. Link with the binomial theorem in § 1.3. SYLLABUS DETAILS: CORE 22 7 IB Diploma Programme guide: Mathematics HL, September 2001 7 Core: probability (continued) CONTENT
7.6 Discrete probability distributions.
Expectation, mode, median, variance and
standard deviation. AMPLIFICATIONS /EXCLUSIONS
Included: knowledge and use of the formulae 7.7 The binomial distribution, its mean and Included: situations and conditions for using a
binomial model.
Included: the concept of a continuous random
variable; definition and use of probability
density functions.
Not included: normal approximation to
binomial distribution. variance (without proof).
7.8 Continuous probability distributions.
Expectation, mode, median, variance and
standard deviation.
7.9 The normal distribution.
Standardisation of a normal distribution; the
use of the standard normal distribution table. E( X ) = ∑ ( x P( X = x) ) and Var( X ) = E( X − µ ) 2 = E( X 2 ) − E( X ) .
2 TEACHING NOTES
It is useful to discuss the fact that E( X ) = 0 indicates
a fair game, where X represents the gain of one of
the players. Link with the binomial theorem in §1.3. Although candidates will not be expected to use the
2
2
1
e − ( x − µ ) / 2σ , they could be
formula f ( x ) =
σ 2π
made aware of the fact that it is the probability
density function of the normal distribution.
Use of calculators as well as tables to find areas and
values of z for given probabilities is advised. SYLLABUS DETAILS: CORE 23 Core: calculus Teaching time: 50 hours The aim of this section is to introduce the basic concepts and techniques of differential and integral calculus, and some of their applications .
CONTENT
8.1 Informal ideas of a limit and convergence.
sin θ
The result lim
= 1 justified by
θ →0 θ geometric demonstration.
8.2 Differentiation from first principles as the limit of the difference quotient f ( x + h) − f ( x ) f ′( x) = lim h →0
h n
Differentiation of: x ! x , n ∈Q ;
x ! sin x; x ! cos x; x ! tan x; x ! e x ;
x ! ln x .
IB Diploma Programme guide: Mathematics HL, September 2001 8.3 Differentiation of sums of functions and real multiples of functions.
The chain rule for composite functions.
8.4 Further differentiation: the product and quotient rules; the second derivative;
x
differentiation of a and log a x . AMPLIFICATIONS /EXCLUSIONS
Included: only a very informal treatment of
limit and convergence, eg 0.3, 0.33, 0.333, ...
1
converges to .
3
Included: a formal treatment for positive integer
powers; informal extension to rational powers;
a formal treatment for x ! sin x .
Included: familiarity with the notation
dy
y = f ( x) ⇒
= f ′( x) .
dx TEACHING NOTES
Link with infinite geometric series in §1.1.
Link with the binomial theorem in §1.3.
Calculators can be used to investigate limits
numerically.
Other derivatives can be predicted or verified by
graphical considerations using graphic display
calculator.
Investigation of the derivative of x n from
n
n
consideration of the function ( x + h ) − x and its
h
graph, for small h, n ∈ Z , can enhance
understanding of limits.
Link with composite functions in §2.1.
Link with implicit differentiation in §8.7.
Link with integration by parts in §8.10.
+ Included: derivatives of reciprocal
trigonometric functions x ! sec x , x ! csc x ,
x ! cot x.
Included: applications to rates of change.
Included: understanding that 2 x = e x ln 2 , etc. SYLLABUS DETAILS: CORE 24 8 IB Diploma Programme guide: Mathematics HL, September 2001 8 Core: calculus (continued) CONTENT
8.5 Graphical behaviour of functions: tangents,
normals and singularities, behaviour for
large x ; asymptotes.
The significance of the second derivative;
distinction between maximum and minimum
points and points of inflexion.
8.6 Applications of the first and second
derivative to maximum and minimum
problems.
Kinematic problems involving
ds
= v , and
displacement, s, velocity,
dt
dv
= a.
acceleration,
dt
8.7 Implicit differentiation.
Derivatives of the inverse trigonometric
functions.
8.8 Indefinite integration as antidifferentiation. Indefinite integrals of: x ; n ∈Q , sin x ;
n TEACHING NOTES
Effective use of graphic display calculator envisaged
here, combined with sketching by hand.
Link with function graphing skills in §2.2.
The terms “concaveup” and “concavedown”
conveniently distinguish between f '' ( x ) > 0 and
f '' ( x ) < 0 respectively. Included: testing for maximum or minimum(eg
volume, area and profit) using the sign of the
first derivative or using the second derivative. Link with graphing functions in §2.2. Included: applications to related rates of
change.
Not required: second derivatives of parametric
functions.
1
Included: ∫ dx = ln x + C.
x
f ′( x) = cos ( 2 x + 3)
Example:
1
⇒ f ( x ) = sin(2 x + 3) + C .
2 Link with chain rule in §8.3. Candidates could be made aware of the fundamental
theorem of calculus,
x F ( x) = ∫ f (t )dt ⇒ F ′( x) = f ( x) , and discuss its
a graphical interpretation. 25 SYLLABUS DETAILS: CORE cos x ; e x .
Composites of these with x ! ax + b .
Application to acceleration and velocity. AMPLIFICATIONS /EXCLUSIONS
Included: both “global” and “local” behaviour;
choice of appropriate window; (a , b) point of
inflexion ⇒ f '' (a ) = 0 , but the converse is not
necessarily true; points of inflexion with zero or
nonzero gradient. Core: calculus (continued) CONTENT
8.9 Antidifferentiation with a boundary
condition to determine the constant term.
Definite integrals.
Areas under curves.
8.10 Further integration: integration by substitution; integration by parts; definite
integrals. 8.11 Solution of first order differential equations by separation of variables. AMPLIFICATIONS /EXCLUSIONS
Example of a boundary condition: if
ds
= 3t 2 + t , and s = 10 when t = 0, then
dt
1
s = t 3 + t 2 + 10.
2
Included: limit changes in definite integrals;
questions requiring repeated integration by
parts; integrals requiring further manipulation,
x
eg, ∫ e sin x dx ; integration using partial fraction decomposition.
Included: transformation of a homogeneous
equation by the substitution y = vx . TEACHING NOTES
Area under velocitytime graph representing distance
is a useful illustration. Link with transformations of graphs in §2.3.
Link with the chain rule in §8.3. Link with exponential and logarithmic functions in
§2.10. SYLLABUS DETAILS: CORE 26 8 IB Diploma Programme guide: Mathematics HL, September 2001 IB Diploma Programme guide: Mathematics HL, September 2001 9 Option: statistics Teaching time: 35 hours The aims of this section are to enable candidates to apply core knowledge of probability distributions and basic statistical calculations, and to make and test
hypotheses. A practical approach is envisaged including statistical modelling tasks suitable for inclusion in the portfolio.
CONTENT
9.1 Poisson distribution: mean and variance
(without proof). AMPLIFICATIONS /EXCLUSIONS
Included: conditions under which a random
variable has a Poisson distribution. 9.2 Mean and variance of linear combinations of Included: E ( a1 X 1 ± a2 X 2 ) = a1E ( X 1 ) ± a2 E ( X 2 ) ; two independent random variables. Var ( a1 X 1 ± a2 X 2 ) = a1 Var ( X 1 ) + a2 Var ( X 2 ) . 9.3 Sampling distribution of the mean. σ2 Included: X ~ N ( µ , σ 2 ) ⇒ X ~ N µ ,
;
n σ2 X is approximately N µ , for large
n samples whatever the distribution of X; for two
samples of size n and m, the pooled unbiased
estimators of the population parameters are TEACHING NOTES
Real applications should be introduced, eg the
number of telephone calls on a randomly chosen day
or the number of cars passing a particular point in an
interval. Standard error of the mean.
Central limit theorem (without proof).
Pooled estimators of population parameters
for two samples. 2 nX n + mX m
n+m nS n + mS m
2 , n+m−2 2 In some texts and on some calculators the unbiased
estimate of the population variance uses alternative
2
2
ˆ2 2
notation, eg σ , σ n −1 , σ x , sx . 2 .
SYLLABUS DETAILS: OPTIONS 27 Option: statistics (continued) CONTENT
9.4 Finding confidence intervals for the mean of
a normal population from a sample. 9.5 Significance testing: the mean of a sample; IB Diploma Programme guide: Mathematics HL, September 2001 the difference between two means.
Null and alternative hypotheses H0 and H1.
Significance levels; critical region and
critical values; onetailed and twotailed
tests.
Drawing conclusions.
9.6 The χ distribution; degrees of freedom, ν .
2 2
The χ statistic ∑ ( f e − f 0 )2
.
fe The χ goodness of fit test.
9.7 Contingency tables. AMPLIFICATIONS /EXCLUSIONS
Note: if the population variance is known, the
normal distribution should be used; if the
population variance is unknown, the
tdistribution should be used (regardless of
sample size).
On examination papers: the relevant values of
the tdistribution will be given either in the IBO
statistical tables or within the question;
alternatively, candidates may use their
calculators.
Use of the normal distribution when σ is known
and the tdistribution when σ is unknown. Included: test for goodness of fit for
distributions that could be uniform, binomial,
Poisson or normal; the requirement to combine
classes with expected frequencies less than five. 2 The χ test for the independence of two
factors.
2 Included: Yates’ continuity correction for v = 1 . TEACHING NOTES
With the advent of statistical software packages and
advanced calculator functions, the restriction on the
use of tdistribution to small samples is no longer
necessary.
Link with significance testing in §9.5. Link with confidence intervals in §9.4.
2
Link with χ distribution in §9.6. Link with significance testing in §9.5. SYLLABUS DETAILS: OPTIONS 28 9 IB Diploma Programme guide: Mathematics HL, September 2001 10 Option: sets, relations and groups Teaching time: 35 hours The aims of this section are to study two important mathematical concepts, sets and groups. The first allows for the extension and development of the notion
of a function, while the second provides the framework to discover the common underlying struct ure unifying many familiar systems.
CONTENT
10.1 Finite and infinite sets.
Operations on sets: union; intersection;
complement.
De Morgan’s laws; subsets.
10.2 Ordered pairs; the cartesian product of two
sets.
Relations; equivalence relations.
10.3 Functions: injections; surjections;
bijections.
Composition of functions and inverse
functions.
10.4 Binary operations: definition; closure; operation tables. commutative properties of binary
operations. TEACHING NOTES
Examples of set operations on finite and infinite sets
will assist understanding. Included: the fact that an equivalence relation
on a set induces a partition of the set. Include examples and visual representations of
relations.
Link with graphs in §2.2.
Link with trigonometric functions in §3.3. Included: knowledge that function composition
is not a commutative operation and that if f is a
bijection from set A onto set B then f −1 exists
and is a bijection from set B onto set A.
Note: a binary operation ∗ on a nonempty set S
is a rule for combining any two elements
a , b ∈ S to give an element c ∈ S where
c = a ∗b .
In examinations: candidates may be required to
test whether a given operation satisfies the
closure condition.
Included: the arithmetic operations in R and C;
matrix operations. Examples of binary operations and their closure
properties will assist understanding. Examples of noncommutative operations could be
given. 29 SYLLABUS DETAILS: OPTIONS 10.5 The associative, distributive and AMPLIFICATIONS /EXCLUSIONS
Included: illustration of the proof of De
Morgan’s laws using Venn diagrams. Options: sets, relations and groups (continued) CONTENT
10.6 The identity element e.
The inverse a −1 of an element a.
Proof that the leftcancellation and
rightcancellation laws hold, provided the
element has an inverse.
Proofs of the uniqueness of the identity and
inverse elements in particular cases.
10.7 The axioms of a group {S , ∗} . Abelian groups. IB Diploma Programme guide: Mathematics HL, September 2001 10.8 Examples of groups: R, Q, Z, and C under addition; symmetries of an equilateral
triangle and square; matrices of the same
order under addition; 2 × 2 invertible
matrices under multiplication; integers
under addition modulo n; invertible
functions under composition of functions;
permutations under composition of
permutations. AMPLIFICATIONS /EXCLUSIONS
Included: knowledge that both the rightidentity
a ∗ e = a and leftidentity e ∗ a = a must hold if e
is an identity element. Included: familiarity with a hierarchy of
algebraic structures, eg for the set S under a
given operation
the given operation is a binary operation, ie
closed,
the given operation is associative,
an identity element exists under this operation,
each element in S has an inverse.
Note: where the given operation is defined as a
a “binary operation”, closure may be assumed.
In examinations: for permutations, the form
1 2 3
p= will be used to represent the 3 1 2
mapping 1 → 3, 2 → 1, 3 → 2. TEACHING NOTES
The leftcancellation law is that
a∗ b = a∗ c ⇒ b = c ; a , b , c ∈ S .
The rightcancellation law is that
b∗ a = c∗ a ⇒ b = c ; a , b , c ∈ S . Candidates should be made aware that other forms of
notation for permutations will be found in various
texts. SYLLABUS DETAILS: OPTIONS 30 10 IB Diploma Programme guide: Mathematics HL, September 2001 10 Option: sets, relations and groups (continued) CONTENT
10.9 Finite and infinite groups.
The order of a group element and the order
of a group.
10.10 Cyclic groups and generators of a group.
Proof that all cyclic groups are Abelian.
10.11 Definition of a subgroup.
Lagrange’s theorem, without proof, and its
corollary.
10.12 Isomorphism and isomorphic groups: formal definition in terms of a bijection;
the property that an isomorphism maps the
identity of one group onto the identity of
the other group; a similar property for
inverses. AMPLIFICATIONS /EXCLUSIONS
Included: an awareness that, in a finite group
table, every element appears once only in each
row and each column.
Included: proof that a group of order n is cyclic
if and only if it contains an element of order n.
Included: the test for a subgroup.
Note: the corollary of Lagrange’s theorem is
that the order of the group is divisible by the
order of any element.
Included: isomorphism between two infinite
groups.
Note: an isomorphism between two groups
( G , ∗ ) and ( H , ∆ ) is a bijection φ : G → H such
that φ ( x ∗ y ) = φ ( x ) ∆ φ ( y ) ; two groups
( G , ∗ ) and ( H , ∆ ) are isomorphic if there exists
an isomorphism for G and H. TEACHING NOTES Isomorphism can be demonstrated using the group
tables for the following groups: permutations of a set
of three elements; symmetries of an equilateral
triangle.
It may be possible to set up an isomorphism between
two groups in more than one way. In any
isomorphism between two groups, the
corresponding elements must be of the same order. SYLLABUS DETAILS: OPTIONS 31 Option: discrete mathematics Teaching time: 35 hours The aims of this option are to introduce topics appropriate for the student of mathematics and computer science who will later confront data structures,
theory of programming languages and analysis of algorithms, and to explore a variety of applications and techniques of discrete methods and reasoning.
CONTENT
11.1 Natural numbers and the wellordering principle. 11.2 Division and Euclidean algorithms. The greatest common divisor of integers a
and b, (a, b).
Relatively prime numbers; prime numbers
and the fundamental theorem of arithmetic.
IB Diploma Programme guide: Mathematics HL, September 2001 11.3 Congruence modulo p as an equivalence class.
Residue classes.
11.4 Recurrence relations. Difference equations: basic definitions and
solutions of a difference equation. AMPLIFICATIONS /EXCLUSIONS
TEACHING NOTES
Included: knowledge that any nonempty subset of Recursive definitions and their proofs using
mathematical induction could be discussed.
Z + contains a smallest element.
Included: knowledge that the wellordering
principle implies mathematical induction (without
proof). Included: the theorem a  b and a  c ⇒ a  ( b ± c )
and other related theorems; the division algorithm
a = bq + r and the Euclidean algorithm for
determining the greatest common divisor of two
(or more) integers.
Included: relations; equivalence relations;
equivalence classes and partitions. Included: the equation Yk +1 = AYk + B ; solutions
as sequences; approximating a differential
equation by a difference equation; first order
difference equations; second order homogeneous
difference equations. + Relate to different number systems. If a , b, c ∈Z ,
ax + by = c has an integer solution x = x0 , y = y0 if
and only if (a, b) divides c. Relate to linear
congruence.
Proof that the number of primes is infinite is an
easy application.
Note that the term residue class is equivalent to
congruence class.
Link with the division and Euclidean algorithms in
§11.2.
Using a difference equation to approximate a
differential equation can serve as a good portfolio
activity. SYLLABUS DETAILS: OPTIONS 32 11 IB Diploma Programme guide: Mathematics HL, September 2001 11 Option: discrete mathematics (continued) CONTENT
11.5 Simple graphs; connected graphs; complete
graphs; multigraphs; directed graphs;
bipartite graphs; planar graphs.
Subgraphs; complements of graphs.
Graph isomorphism. AMPLIFICATIONS /EXCLUSIONS
Included: Euler’s relation: v − e + f = 2 ;
theorems for planar graphs including
e ≤ 3v − 6 ,
e ≤ 2v − 4 ,
κ 5 and κ 3,3 are not planar. TEACHING NOTES
Isomorphism between graphs can be emphasized
using a bijection between the vertex sets which
preserves adjacency of edges, and using the
adjacency matrices of the graphs. 11.6 Walks; Hamiltonian paths and cycles; Included: the following theorems (without
proof)
a graph is bipartite if and only if χ (G ) is at
most 2,
if κ n is a subgraph of G, then χ ( G ) ≥ n ,
if G is planar, then χ ( G ) ≤ 4 (the 4colour
problem).
Included: definitions and examples of
depthfirst search and breadthfirst search
algorithms. χ (G ) is the chromatic number of G.
Graph colouring is a worthwhile classroom activity Eulerian trails and circuits.
Graph colouring and chromatic number of a
graph. 11.7 Networks and trees: definitions and properties.
The travelling salesman problem.
Rooted trees; binary search trees; weighted
trees; sorting; spanning trees; minimal
spanning trees.
Prim’s, Kruskal’s and Dijkstra’s algorithms. Students interested in computing may engage in
writing programs for scheduling on a small database.
These may include designing transportation
networks for a small business, production plans for a
product involving several processes.
Note that Prim’s algorithm is an example of a greedy
algorithm since “at each iteration we do the thing
that seems best at that step”.
SYLLABUS DETAILS: OPTIONS 33 Option: analysis and approximation Teaching time: 35 hours The aims of this section are to use calculus results to solve differential equations (numerically and analytically), to approximate definite integrals, to solve
nonlinear equations by iteration, and to approximate functions by expansions of power series. The expectation is that candidates will use a graphic display
calculator to perform computations and also to develop a sound understanding of the underlying mathematics.
CONTENT
12.1 Convergence of infinite series.
Tests for convergence: ratio test; limit
comparison test; integral test. AMPLIFICATIONS /EXCLUSIONS
Included: conditions for the application of these
tests; the divergence theorem, if
un is a ∑ convergent series then lim un = 0 .
n→∞ 12.2 Alternating series. Conditional convergence. Included: knowledge that the absolute value of
the truncation error is less than the next term in
the series; absolute convergence of an infinite
series. TEACHING NOTES
Convergence of an infinite series should be
introduced through the convergence of the sequence
of partial sums; the limit comparison test and
comparison test may then be used.
∞ It is useful to explain that ∑ n is divergent, but that
1 1 ∞ ∑ ( −1)
1 n 1
is convergent.
n IB Diploma Programme guide: Mathematics HL, September 2001 12.3 Power series: radius of convergence. Included: power series in ( x − k ), k ≠ 0 . Determination of the radius of convergence
by the ratio test.
12.4 Rolle’s theorem; the mean value theorem.
Applications of these theorems. Included: graphical representation of these
theorems. Applications of the mean value theorem can include
proving inequalities such as sin x − sin y < x − y . Not required: proof of Taylor’s theorem.
Included: applications to the approximation of
functions; bounds on the error term.
On examination papers: the form of the error
term will be given.
Included: finding the Taylor approximations for
2
functions such as e x arctan x by multiplying the
2
Taylor approximations for e x and arctan x . Series expansions for the trigonometric functions
and their inverses, and the exponential and
logarithmic functions are good examples. 12.5 Use of Taylor series expansions, including the error term.
Maclaurin series as a special case.
Taylor polynomials.
Taylor series by multiplication. SYLLABUS DETAILS: OPTIONS 34 12 IB Diploma Programme guide: Mathematics HL, September 2001 12 Option: analysis and approximation (continued) CONTENT
12.6 Numerical integration.
Derivation and application of the trapezium
rule and Simpson’s rule.
The forms of the error terms; their use.
12.7 The solution of nonlinear equations by iterative methods, including the
NewtonRaphson method; graphical
interpretations.
Fixed point iteration; conditions for
convergence.
The concept of order of convergence
(without proof). AMPLIFICATIONS /EXCLUSIONS
Included: the definition of an integral as the
limit of a sum.
On examination papers: the forms of the error
terms will be given; geometric interpretations
will be given.
Included: choice of initial approximation by the
bisection method to solve f ( x ) = 0 using the
NewtonRaphson method. TEACHING NOTES
Comparison of the error estimates for the trapezium
rule and Simpson’s rule is worthwhile to emphasize
the accuracy of the latter. The use of a graphic display calculator to help
choose a suitable initial approximation to x is
valuable, as is the discussion of the calculator
algorithms for approximate solution of equations. SYLLABUS DETAILS: OPTIONS 35 Option: Euclidean geometry and conic sections Teaching time: 35 hours The aims of this section are to expose candidates to formal proofs in Euclidean geometry thereby providing a broader understanding of the scope of
mathematical proof, and to study conic sections using their cartesian equations.
CONTENT
13.1 Principles of geometric proof: postulates, IB Diploma Programme guide: Mathematics HL, September 2001 theorems and their proof; deductive
reasoning; ifthen statements and their
converses; inductive reasoning; geometric
patterns.
13.2 Triangles: medians; altitudes; angle
bisectors; perpendicular bisectors of sides.
Concurrency: orthocentre; incentre;
circumcentre; centroid.
Principles of construction of triangles from
secondary elements using a straight edge
and compass.
Euler’s circle (the nine point circle).
13.3 Proportional length and proportional
division of a line segment (internal and
external); the harmonic ratio; proportional
segments in right angled triangles.
Euclid’s theorem for proportional segments
in a right angled triangle. AMPLIFICATIONS /EXCLUSIONS
Included: use of properties of equivalent (equal
area), similar and congruent figures to provide
geometric proofs of proportions. TEACHING NOTES
It is helpful to draw comparisons between the
reasoning used in proofs in geometry and in other
topic areas.
Congruence is an equivalence relation. Note: the primary elements of a triangle are the
angles and the lengths of the sides; the
secondary elements include the altitudes,
medians and angle bisectors. Included: knowledge that the proportional
segments p, q satisfy
h 2 = pq
b a h
q p
c a 2 = pc
b2 = qc . This topic can be linked with vector geometry, which
provides a useful opportunity to compare different
approaches to geometrical proof. SYLLABUS DETAILS: OPTIONS 36 13 IB Diploma Programme guide: Mathematics HL, September 2001 13 Option: Euclidean geometry and conic sections (continued) CONTENT
13.4 Circle geometry: tangents; arcs, chords and
secants; the tangentsecant and
secantsecant theorems; the
intersectingchords theorem; loci and
constructions; inscribed and circumscribed
polygons; properties of cyclic
quadrilaterals. AMPLIFICATIONS /EXCLUSIONS
Included: the tangentsecant theorem TEACHING NOTES
The tangency of two circles and its implications
could be discussed. P T
A PT2 = PA × PB = PC × PD .
C B
D Included: the intersectingchords theorem
a c d b ab = cd . On examination papers: questions will not be
set which require constructions with ruler and
compasses.
13.5 Apollonius’ theorem (circle of Apollonius); Included: equations of tangents and normals to
these curves; proofs of properties associated
with intersections between tangents, normals
and curves. Some exploration of applications in science and
industry can enhance understanding.
Link with solution of equations of the form
f ( x ) = g( x ) in § 2.6. SYLLABUS DETAILS: OPTIONS 37 Apollonius’ theorem; Menelaus’ theorem;
Ceva’s theorem; Ptolemy’s theorem;
bisector theorem.
Proof of these theorems.
The use of the theorems to prove further
results.
13.6 Conic sections: focus and directrix;
eccentricity.
Circle; parabola; hyperbola; ellipse.
Parametric equations; the general equation
of second degree; rotation of axes. This subtopic provides an opportunity to introduce
historical connections and the development of the
concept of proof.
Applications in art and design can be explored. SYLLABUS GUIDELINES
Presumed knowledge
1 General
Candidates are not required to be familiar with all the topics listed below as presumed
knowledge (PK) before they start the mathematics HL course of study. However, they should
be familiar with these topics before they take the written papers, as questions will assume
knowledge of them. It is therefore recommended that teachers ensure that any topics from
presumed knowledge which are unknown to their candidates at the start of the course are
included in the programme of study at an early stage.
Candidates should be familiar with the Système International (SI) units of length, mass and
time, and their derived units. 2 Topics
PK1 Number and algebra
1.01 1.02 Simple positive exponents.
Examples: 2 3 = 8; ( −3)3 = −27; ( −2) 4 = 16 1.03 Simplification of expressions involving roots (surds or radicals).
Examples: 27 + 75 = 8 3; 3 × 5 = 15 1.04 Prime numbers and factors, including greatest common factors and least
common multiples. 1.05 Simple applications of ratio, percentage and proportion, linked to similarity. 1.06 Definition and elementary treatment of absolute value (modulus),  a  . 1.07 38 Routine use of addition, subtraction, multiplication and division using
integers, decimals and fractions, including order of operations.
Example: 2 ( 3 + 4 × 7 ) = 62 Rounding, decimal approximations and significant figures. IB Diploma Programme guide: Mathematics HL, September 2001 SYLLABUS GUIDELINES 1.08 Expression of numbers in standard form (scientific notation), ie, a × 10 k ,
1 ≤ a < 10, k ∈Z . 1.09 Concept and notation of sets, elements, universal (reference) set, empty
(null) set, complement, subset, equality of sets, disjoint sets. Operations on
sets: union and intersection. Commutative, associative and distributive
properties. Venn diagrams. 1.10 Number systems: natural numbers, N; integers, Z; rationals, Q, and
irrationals; real numbers, R. 1.11 Intervals on the real number line using set notation and using inequalities.
Expressing the solution set of a linear inequality on the number line and in
set notation. 1.12 The concept of a relation between the elements of one set and between the
elements of one set and those of another set. Mappings of the elements of
one set onto or into another, or the same, set. Illustration by means of tables,
diagrams and graphs. 1.13 Basic manipulation of simple algebraic expressions involving factorization
and expansion.
Examples: ab + ac = a ( b + c ) ; ( a ± b) 2 = a 2 + b 2 ± 2ab ;
a 2 − b 2 = (a − b)(a + b) ; 3x 2 + 5x + 2 = (3x + 2)( x + 1) ;
xa − 2a + xb − 2b = ( x − 2)(a + b) 1.14 Rearrangement, evaluation and combination of simple formulae. Examples
from other subject areas, particularly the sciences, should be included. 1.15 The linear function x ! ax + b and its graph, gradient and yintercept. 1.16 Addition and subtraction of algebraic fractions with denominators of the
form ax + b .
Example: 2 x + 3x + 1
3x − 1 2 x + 4 1.17 The properties of order relations: < , ≤ , > , ≥ .
Examples: ( a > b, c > 0) ⇒ ac > bc ; ( a > b, c < 0 ) ⇒ ac < bc 1.18 Solution of equations and inequalities in one variable including cases with
rational coefficients.
Example: 3 − 2 x = 1 (1 − x ) ⇒ x = 5
7 1.19 5 2 7 Solution of ax 2 + bx + c = 0 , a ≠ 0 . The quadratic formula. IB Diploma Programme guide: Mathematics HL, September 2001
39 SYLLABUS GUIDELINES PK2 Geometry
2.01 2.02 Angle trigonometry. Simple
sin θ
applications for solving triangles. Definition of tan θ as
. Graph of
cosθ
y = x tan θ with gradient (slope) tan θ . 2.03 Pythagoras’ theorem and its converse. 2.04 The cartesian plane: ordered pairs (x, y), origin, axes. Midpoint of a line
segment and distance between two points in the cartesian plane. 2.05 Simple geometric transformations: translation, reflection, rotation,
enlargement. Congruence and similarity, including the concept of
scalefactor of an enlargement. 2.06 The circle, including arc, chord and tangent properties. Area and
circumference. 2.07 PK3 Elementary geometry of the plane including the concepts of dimension for
point, line, plane and space. Parallel and perpendicular lines. Geometry of
simple plane figures. Perimeter and area of rectangles, triangles, parallelograms and trapezia
(trapezoids), including compound shapes. measurement in degrees. Rightangle Statistics
3.01 Descriptive statistics: collection of raw data, display of data in pictorial and
diagrammatic forms (eg pie charts, pictograms, stemandleaf diagrams, bar
graphs and line graphs). 3.02 Calculation of simple statistics from discrete data, including mean, median
and mode. Presumed skills
In addition to presumed knowledge, candidates should have the skills to carry out particular
mathematical tasks with confidence before starting the course. The course assumes that a
candidate will be competent in performing the following basic operations. •
• Solving linear equations and inequalities in one variable, and simultaneous equations in
two variables. • 40 Manipulating indices (exponents) and surds (radicals). Solving quadratic equations. IB Diploma Programme guide: Mathematics HL, September 2001 SYLLABUS GUIDELINES • Plotting accurate graphs from a table of values. • Applying the geometrical properties of the triangle and the circle using the concepts of
symmetry, reflection, rotation, similarity and congruence. • Recognizing, and analysing, the equations of straight lines and circles in the xy plane.
For example, finding points of intersection with axes and determining centres and radii. • Recognizing quadratic and cubic curves. • Dealing with errors in numerical calculation due to rounding. • Applying a sensible degree of accuracy in numerical work. Internationalism
One of the aims of this course is to enable candidates to appreciate the international
dimensions of mathematics and the multiplicity of its cultural and historical perspectives.
While this aim is not explicitly written into the syllabus, it is hoped that teachers will take
every opportunity to fulfil this aim by discussing relevant issues as they arise and making
reference to appropriate background information. For example, it may be appropriate to
discuss: • differences in notation • the lives of mathematicians set in a historical and/or social context • the cultural context of mathematical discoveries • the ways in which certain mathematical discoveries were made in terms of the techniques
used • the attitudinal divergence of different societies towards certain areas of mathematics • the universality of mathematics as a language. It should be noted that this aim has not been translated into a corresponding objective.
Therefore this aspect of the course will not be tested in examinations. IB Diploma Programme guide: Mathematics HL, September 2001 41 ASSESSMENT OUTLINE
For first examinations in 2001 External assessment 80% Written papers 5 hours
Paper 1 2 hours 30% Twenty compulsory shortresponse questions based on part I of
the syllabus, the compulsory core. Paper 2 3 hours 50%
35% Section A:
Five compulsory extendedresponse questions based on
part I of the syllabus, the compulsory core. 15% Section B:
Five extendedresponse questions, one on each of the
optional topics in part II of the syllabus; one question to be
answered on the chosen topic. Internal assessment 20% Portfolio
A collection of three pieces of work assigned by the teacher and
completed by the candidate during the course. The assignments
must be based on different areas of the syllabus and represent all
three activities: mathematical investigation; extended
closedproblem solving and mathematical modelling.
The portfolio is internally assessed by the teacher and externally
moderated by the IBO. Procedures are provided in the Vade
Mecum. 42 IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT DETAILS
External assessment: written papers
1 General
1 Paper 1 and paper 2
The external assessment consists of two written examination papers, paper 1 and
paper 2, which are externally set and externally marked. Together they contribute
80% to the final mark. These papers are designed to allow candidates to demonstrate
what they know and can do. 2 Calculators
Candidates are required to have access to a graphic display calculator at all times
during the course, both inside and out of the classroom. Regulations concerning the
types of calculators allowed are provided in the Vade Mecum. 3 Formulae booklet and statistical tables (third edition, February 2001)
As each candidate is required to have access to clean copies of the IBO formulae
booklet and statistical tables during the examination, it is recommended that teachers
ensure candidates are familiar with the contents of these documents from the
beginning of the course. The booklet and tables are provided by IBCA and are
published separately. IB Diploma Programme guide: Mathematics HL, September 2001 43 ASSESSMENT DETAILS 2 Paper 1: (2 hours) 30% This paper consists of twenty compulsory shortresponse questions based on part I of the
syllabus, the core. 1 Syllabus coverage
•
• 2 Knowledge of all topics from the core is required for this paper.
The intention of this paper is to test candidates’ knowledge across the breadth of
the core. However, it should not be assumed that the separate topics from the
core will be given equal weight or emphasis. Question type
•
• 3 A small number of steps will be needed to solve each question.
Questions may be presented in the form of words, symbols, tables or diagrams,
or combinations of these. Mark allocation
• Each question is worth three marks. The maximum number of marks available
for this paper is 60, representing 30% of the final assessment. • Questions of varying levels of difficulty will be set. Each will be worth the same
number of marks. • Full marks are awarded for each correct answer irrespective of the presence of
working.
Where a wrong answer is given, partial credit may be awarded for a correct
method provided this is shown by written working; if no working is present then
no partial credit can be given and candidates cannot be awarded any marks.
Candidates should therefore be encouraged to show their working at all times. 3 Paper 2: (3 hours) 50% This paper is divided into two sections: section A, based on part I of the syllabus, and section
B, based on part II. It is estimated that, during the total time of three hours, candidates will be
able to spend up to 30 minutes in thought and reflection.
1 Question type •
• 44 Questions in both sections will require extended responses involving sustained
reasoning.
Individual questions may develop a single theme or be divided into unconnected
parts. IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT DETAILS •
• 2 Questions may be presented in the form of words, symbols, diagrams or tables,
or combinations of these.
Normally, each question will reflect an incline of difficulty from relatively easy
tasks at the start of a question to relatively difficult tasks at the end of a question.
The emphasis will be on problemsolving. Awarding of marks • Marks will be awarded according to the following categories.
Method: evidence of knowledge, the ability to apply concepts and skills, and the
ability to analyse a problem in a logical manner.
Accuracy: computational skill and numerical accuracy.
Reasoning: clear reasoning, explanation and/or logical argument.
Correct statements: results or conclusions expressed in words.
Follow through: if an incorrect answer found in an earlier part of a question is
used later in the same question then marks may be awarded in the later part even
though the original answer used is incorrect. In this way, candidates are not
penalized for the same mistake more than once. • 4 A correct answer with no indication of the method used (for example, in the form
of diagrams, graphs, explanations, calculations) will normally be awarded no
marks. All candidates should therefore be advised to show their working. Paper 2: section A
This section consists of five compulsory extendedresponse questions based on part I of the
syllabus, the core. Candidates will be expected to answer all the questions in this section. 1 Syllabus coverage
• Knowledge of all topics from part I of the syllabus is required for this section of
paper 2. • Individual questions may require knowledge of more than one topic from the
core. • The intention of this section is to test candidates’ knowledge of the core in
depth. A narrower range of topics from the syllabus will be tested in this paper
than is tested in paper 1. IB Diploma Programme guide: Mathematics HL, September 2001 45 ASSESSMENT DETAILS 2 Mark allocation
•
• 5 This section is worth 70 marks, representing 35% of the final mark.
Questions in this section may be unequal in terms of length and level of
difficulty. Hence individual questions will not necessarily be worth the same
number of marks. The exact number of marks allocated to each question will be
indicated at the start of each question. Paper 2: section B
This section consists of five extendedresponse questions based on part II of the syllabus, the
options. One question will be set on each option. 1 Syllabus coverage
•
• Knowledge of the entire contents of the option studied is required for this section
of paper 2. •
2 Candidates will be expected to answer the question based on the option they
have studied. In order to provide appropriate syllabus coverage of each option, questions in
this section are likely to contain two or more unconnected parts. Mark allocation
•
• 46 This section is worth 30 marks, representing 15% of the final mark.
Questions in this section will be equal in terms of length and level of difficulty.
Each question will be worth 30 marks. IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT DETAILS Internal assessment: the portfolio
1 The purpose of the portfolio
The purpose of the portfolio is to provide candidates with opportunities to be rewarded for
mathematics carried out under ordinary conditions, that is, without the time limitations and
stress associated with written examinations. Consequently the emphasis should be on good
mathematical writing and thoughtful reflection.
The portfolio is also intended to provide candidates with opportunities to increase their
understanding of mathematical concepts and processes. It is hoped that, in this way,
candidates will benefit from these activities and find them both stimulating and rewarding.
The specific purposes of portfolio work are to: •
• provide opportunities for candidates to complete extended pieces of work in mathematics
without the time constraints of an examination • enable candidates to develop individual skills and techniques, and to allow them to
experience the satisfaction of applying mathematical processes on their own • provide candidates with the opportunity to experience for themselves the beauty, power
and usefulness of mathematics • provide candidates with the opportunity to discover, use and appreciate the power of a
calculator/computer as a tool for doing mathematics • enable candidates to develop qualities of patience and persistence, and to reflect on the
significance of the results they obtain • 2 develop candidates’ personal insight into the nature of mathematics and to develop their
ability to ask their own questions about mathematics provide opportunities for candidates to show, with confidence, what they know and can
do. Requirements
For mathematics HL, the portfolio must consist of a collection of three pieces of work
assigned by the teacher and completed by the candidate during the course.
Each assignment contained in the portfolio must be based on • an area of the mathematics HL syllabus • each of the three activities, type I, type II and type III. IB Diploma Programme guide: Mathematics HL, September 2001 47 ASSESSMENT DETAILS The level of sophistication of the mathematics should be about the same as that contained in
the syllabus. It is not intended that additional topics be taught to candidates to enable them to
complete a particular assignment.
Each portfolio must contain one assignment representing each type of activity. Therefore,
the portfolio must contain one type I, one type II and one type III assignment. • Type I: mathematical investigation
A mathematical investigation is defined as an enquiry into a particular area of
mathematics leading to a general result which was previously unknown to the
candidate. The use of a calculator and/or computer is encouraged in this type of
activity.
Example: • An investigation into the behaviour of the partial sums of a particular
sequence. Type II: extended closedproblem solving
An extended closedproblem is defined as a multipart problem where the candidate is
guided by developmental questions designed to lead the candidate to a particular
result or set of results.
Example: • A group of two or three related questions taken from past
examination papers, or a single question with a number of
extensions. Type III: mathematical modelling
In this context, mathematical modelling is taken to mean the solution of a practical
problem set in a realworld context in which the method of solution requires some
relatively elementary mathematical modelling skills.
Example: 3 Analysing the growth of a particular population using a proposed
model; reflecting on the nature and usefulness of this model. Integration into the course of study
It is intended that these assignments be completed at intervals throughout the course and not
left until towards the end. Indeed, teachers are encouraged to integrate portfolio assignments
into their teaching and to allow candidates the opportunity to explore various aspects of as
many different topics as possible from both the core and chosen option.
Teachers should not attempt to isolate assignments for the portfolio from what is going on in
the classroom, otherwise candidates may regard portfolio assignments as extra work which
has to be completed for the sole purpose of the assessment process rather than as a deliberate
move to provide them with opportunities for increasing their understanding of mathematical
concepts and processes. 48 IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT DETAILS Because of the relationship of the portfolio assignments to the syllabus, it is important that
each assignment be presented to candidates at the appropriate time. This may be immediately
before a topic is introduced, during the study of a topic or immediately after a topic is
studied.
Examples: •
• Mathematical modelling may be used to reinforce mathematical meaning and provide an
opportunity for candidates to gain a deeper understanding of the relevant concepts. • 4 A mathematical investigation may be used for the purpose of introducing a topic. A set of extended closedproblems may be used as a revision exercise. Management of the portfolio
1 Time allocation
The Vade Mecum states that a higher level course requires at least 240 teaching
hours. In mathematics HL, 10 of these hours should be allocated to work connected
to the portfolio. This will allow time for teachers to explain to candidates the
requirements of the portfolio and allow class time for candidates to work on
individual assignments.
Each assignment should take approximately three hours to complete: one hour of
class time and two hours of homework time. Consequently, it is expected that during
the course candidates will have the time to complete more than three assignments,
and will thus be provided with the opportunity to select the best three for inclusion in
their portfolios.
For each assignment, class time should be used for candidates to begin (or possibly
finish) a piece of work under the guidance of the teacher. It is not intended that this
class time be used to introduce material which is not on the syllabus, since each
assignment should be based on topics which are within the scope of the syllabus. 2 Setting of assignments
It is the teacher’s responsibility to set suitable assignments which comply with the
regulations.
There is no requirement to provide identical assignments for all candidates. Neither
is there a requirement to provide each candidate with a different assignment.
Teachers may decide which is the best course of action under different
circumstances.
Candidates may suggest areas of the syllabus in which they would like to attempt an
assignment or may make detailed suggestions as to the form a particular assignment
should take. Any such suggestions should be approved by the teacher before work is
started. IB Diploma Programme guide: Mathematics HL, September 2001 49 ASSESSMENT DETAILS 3 Submission of assignments
The finished piece of work should be submitted to the teacher for assessment soon
after it is set, that is, between three and ten days. Candidates should not be given the
opportunity to resubmit a piece of work after it has been assessed.
As a guide to length, each piece of work should be approximately equivalent to three
or four wordprocessed pages. However, it should be noted that there is no
requirement for work to be word processed. 4 Followup and feedback
• • 5 Teachers should ensure that candidates are aware of the significance of the
results/conclusions which are intended as the outcome of a particular assignment.
This is particularly important in the case of investigative work which is used for
the purpose of introducing a topic. Some class time devoted to followup work
should therefore be included when developing a course of study.
It is also important that candidates receive feedback on their own work so that
they are aware of alternative strategies for developing their mathematical
thinking and are provided with guidance for improving their skills in writing
mathematics. Guidance and authenticity
All candidates should be familiar with the requirements of the portfolio and the
means by which it is assessed.
It should be made clear to candidates that writing up a portfolio assignment should be
entirely their own work. It is therefore helpful if teachers try to encourage in
candidates a sense of responsibility for their own learning so that they accept a
degree of ownership and take pride in their own work. • Time in the classroom can be used for discussion of a particular assignment.
This discussion can be between teacher and candidates (or a single candidate)
and between two or more candidates. In responding to specific questions from
candidates teachers should, where appropriate, guide candidates into more
productive routes of enquiry rather than respond with a direct answer. • When completing a portfolio assignment outside the classroom, candidates
should work independently. Group work, whilst educationally desirable in certain situations, is not appropriate in
relation to work being prepared for the portfolio.
Teachers are required to ensure that work submitted for the portfolio is the candidate’s own.
If in doubt, authenticity may be checked by one or more of the following methods: !
!
! 50 discussion with the candidate
asking the candidate to explain the methods used and summarise the results
asking the candidate to repeat the assignment with a different set of data. IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT DETAILS It is also appropriate for teachers to request candidates to sign each assignment before
submitting it to indicate that it is their own work. 5 Record keeping
Careful records should be kept to ensure that all candidates are able to put together a
portfolio which complies with the regulations.
For each assignment, the following should be recorded: • exact details of the assignment given to the candidate(s) • areas of the syllabus on which the assignment is based • the date the assignment was given to the candidate and the date of submission • the type of activity: type I, type II or type III • the background to the assignment, in relation to the skills/concepts from the syllabus
which had, or had not, been taught to the candidate at the time the assignment was set. IB Diploma Programme guide: Mathematics HL, September 2001 51 ASSESSMENT CRITERIA
The portfolio
1 Introduction
The portfolio is internally assessed by the teacher and externally moderated by the IBO.
Assessment criteria have been developed to address collectively all the group 5 objectives.
In developing these criteria, particular attention has been given to the five objectives
described below, since these cannot be easily addressed by means of timed written
examinations.
Where appropriate in the portfolio, candidates will be expected to: •
• know and use appropriate notation and terminology • recognize patterns and structures in a variety of situations and draw inductive
generalizations • demonstrate an understanding of, and competence in, the practical applications of
mathematics • 2 organize and present information/data in tabular, graphical and/or diagrammatic forms use appropriate technological devices as mathematical tools. Form of the assessment criteria
Each piece of work in the portfolio should be assessed against the following four criteria:
A
B
C
D Use of notation and terminology
Communication
Mathematical content
Results and conclusions In addition, at least one assignment in each portfolio should include work which is
appropriate to be assessed against the criterion:
E Making conjectures And at least one assignment in each portfolio should include work which is appropriate to be
assessed against the criterion:
F 52 Use of technology IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT CRITERIA 3 Applying the assessment criteria
The method of assessment used is criterion referenced, not norm referenced. That is, the
method of assessing each assignment in a portfolio judges candidates by their performance in
relation to identified assessment criteria and not in relation to the work of other candidates. • Each assignment in the portfolio submitted for mathematics HL is assessed against the
four criteria A to D; at least one is assessed against criterion E, and at least one against
criterion F. • For each assessment criterion, different levels of achievement are described which
concentrate on positive achievement. The description of each achievement level
represents the minimum requirement for that level to be achieved. • The aim is to find, for each criterion, the level of achievement gained by the candidate
for that piece of work. Consequently, the process involves reading the description of each
achievement level, starting with level 0, until one is reached which describes a level of
achievement that has not been reached. The level of achievement gained by the candidate
is therefore the preceding one and it is this which should be recorded.
For example, if, when considering successive achievement levels for a particular
criterion, the description for level 3 does not apply, then level 2 should be recorded. • If a piece of work appears to fall between two achievement levels then the lower
achievement level should be recorded since the minimum requirements for the higher
achievement level have not been met. • For each criterion, only whole numbers may be recorded; fractions and decimals are not
acceptable. • The whole range of achievement levels should be awarded as appropriate. For a
particular piece of work, a candidate who attains a high achievement level in relation to
one criterion may not necessarily attain high achievement levels in relation to other
criteria. It is recommended that the assessment criteria be made available to candidates at all times. IB Diploma Programme guide: Mathematics HL, September 2001 53 ASSESSMENT CRITERIA 4 The final mark
Each portfolio should contain three pieces of work. If more than three pieces of work have
been completed (this is recommended) then the best three should be included in the portfolio.
To arrive at the final mark for the portfolio: ! Criteria A–D: ! Criterion E–F: calculate the average of all three achievement levels for each
criterion
take the highest achievement level Add these six marks to obtain the final mark, rounding to the nearest integer if necessary. The
maximum final mark is 20.
Example The achievement levels for each criterion might be as follows: Assignment type Criterion A Criterion B Criterion C Criterion D Criterion E Criterion F
I 1 0 2 1 – – II 1 1 3 1 3 2 III 2 3 3 1 – 3 Final mark 11/3 11/3 22/3 1 3 3 In this case, the final mark would be 12, that is, 12a rounded to the nearest integer since only
whole numbers are allowed. 5 Incomplete portfolios
Teachers should ensure that, during the course, all candidates are given the opportunity to
complete at least three assignments which comply with the requirements. However, if a
candidate’s portfolio contains fewer than three assignments, the final mark should be
calculated in exactly the same way as for a complete portfolio, with the missing marks
considered to be zeros.
Example In a portfolio which contains only two assignments, the achievement levels
for each criterion might be as follows: Assignment type Criterion A Criterion B Criterion C Criterion D Criterion E Criterion F
I 1 0 2 0 – – II 1 1 3 1 3 – III – – – – – – 3 0 Final mark 2 /3 1 /3 12/3 1 /3 In this case, the final mark would be 6.
Rounding should only take place at the end of the process in order to obtain the final mark. 54 IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT CRITERIA 6 Achievement Levels
Note that “appropriate” used here means “appropriate to the level of the mathematics HL course”. Criterion A: use of notation and terminology
All three pieces of work in each portfolio should be assessed against this criterion.
Achievement
level 0 The candidate does not use appropriate notation and terminology. 1 The candidate uses some appropriate notation and/or terminology. 2 The candidate uses appropriate notation and terminology in a consistent manner
and does so throughout the activity. Criterion B: communication
All three pieces of work in each portfolio should be assessed against this criterion.
Achievement
level 0 The candidate neither provides explanations nor uses appropriate forms of
representation (eg symbols, tables, graphs, diagrams). 1 The candidate attempts to provide explanations and uses some appropriate forms of
representation (eg symbols, tables, graphs, diagrams). 2 The candidate provides adequate explanations/arguments, and communicates
them using appropriate forms of representation (eg symbols, tables, graphs,
diagrams). 3 The candidate provides complete, coherent explanations/arguments, and
communicates them clearly using appropriate forms of representation (eg symbols,
tables, graphs, diagrams). IB Diploma Programme guide: Mathematics HL, September 2001 55 ASSESSMENT CRITERIA Criterion C: mathematical content
All three pieces of work in each portfolio should be assessed against this criterion .
Achievement
level 0 The candidate recognizes no mathematical concepts which are relevant to the
activity. 1 The candidate recognizes a mathematical concept or selects a mathematical
strategy which is relevant to the activity . 2 The candidate recognizes a mathematical concept and attempts to use a
mathematical strategy which is relevant to the activity and consistent with the level
of the programme . 3 The candidate recognizes a mathematical concept and uses a mathematical strategy
which is relevant to the activity and consistent with the level of the programme, and
makes few errors in applying mathematical techniques . 4 The candidate recognizes a mathematical concept, successfully uses a mathematical
strategy which is relevant to the activity and consistent with the level of the
programme, and applies mathematical techniques correctly throughout the activity . 5 The candidate displays work distinguished by precision, insight and a
sophisticated level of mathematical understanding . Criterion D: results or conclusions
All three pieces of work in each portfolio should be assessed against this criterion. Note that
candidates are rewarded for the quality of their conclusions or results. This is because most
assignments lend themselves to being assessed more appropriately in one or other of these two
categories.
Achievement
level 0
1 The candidate draws partial conclusions or demonstrates some consideration of
the significance or the reasonableness of results. 2 The candidate draws adequate conclusions or demonstrates some understanding of
the significance and reasonableness of results. 3 56 The candidate draws no conclusions or gives unreasonable or irrelevant results. The candidate draws full and relevant conclusions or demonstrates complete
understanding of the significance, reasonableness or possible limitations of results. IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT CRITERIA Criterion E: making conjectures
A minimum of one piece of work in each portfolio should be assessed against this criterion.
Achievement
level 0 The candidate demonstrates no awareness of patterns or structures. 1 The candidate recognizes patterns and/or structures. 2 The candidate recognizes patterns and/or structures and attempts to draw inductive
generalizations. 3 The candidate recognizes patterns and/or structures, successfully draws inductive
generalizations, and attempts to provide formal justifications . 4 The candidate recognizes patterns and/or structures, successfully draws inductive
generalizations and justifies (or disproves) the generalizations by means of
formal arguments . Criterion F: use of technology
A minimum of one piece of work in each portfolio should be assessed against this criterion.
Achievement
level 0 The candidate does not use a calculator or computer beyond routine calculations. 1 The candidate attempts to use a calculator or computer in a manner which could
enhance the development of the activity . 2 The candidate makes limited use of a calculator or computer in a manner which does
enhance the development of the activity. 3 The candidate makes full and resourceful use of a calculator or a computer in a
manner which significantly enhances the development of the activity. IB Diploma Programme guide: Mathematics HL, September 2001 57 ASSESSMENT GUIDELINES
External assessment: written papers
1 Notation
Of the various notations in use, the IBO has chosen to adopt the notation listed below based
on the recommendations of the International Organization for Standardization. These will be
used on written examination papers in mathematics HL without explanation. If forms of
notation other than those listed here are used on a particular examination paper then they will
be defined within the question in which they appear.
Because candidates are required to recognize, though not necessarily use, the IBO notation in
examinations, it is recommended that teachers introduce students to IBO notation at the
earliest opportunity. Candidates will not be permitted information relating to notation in the
examinations.
In a small number of cases, candidates will need to use alternative forms of notation in their
written answers as not all forms of IBO notation can be directly transferred into handwritten
form. This is true particularly in the case of vectors where the IBO notation uses a bold, italic
typeface which cannot be adequately transferred into handwritten form. In this particular
case, teachers should advise candidates to use alternative forms of notation in their written
"
work (eg x , x or x ). N
Z the set of positive integers and zero, {0, 1, 2, 3, ...}
the set of integers, {0, ± 1, ± 2, ± 3, ...} Z+
Q
Q+
R
R+
C the set of positive integers, {1, 2, 3, ...}
the set of rational numbers
the set of positive rational numbers, {x  x ∈Q, x > 0}
the set of real numbers
the set of positive real numbers, {x  x ∈R, x > 0}
the set of complex numbers, {a + ib  a , b ∈R} i
z −1
the complex number a + ib = r (cos θ + i sin θ ) z∗ the complex conjugate of z (ie z ∗ = a − ib = r (cos θ − i sin θ )) z the modulus of z arg z
Re z the argument of z
the real part of z 58 IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT GUIDELINES Im z
{x1 , x2 , ...}
n( A)
{x  }
∈
∉
∅
U
∪
∩
⊂
⊆
A′
A× B
a 1/ n , n a 1/ 2 , a
a x
≡
≈
>
≥
<
≤
(
'
[a , b] a, b [ the imaginary part of z
the set with elements x1 , x2 , ...
the number of elements in the finite set A
the set of all x such that
is an element of
is not an element of
the empty (null) set
the universal set
union
intersection
is a proper subset of
is a subset of
the complement of the set A
the cartesian product of sets A and B (ie A × B = {( a , b) a ∈ A , b ∈ B} )
a to the power of 1 , nth root of a (if a ≥ 0 then n a ≥ 0)
n
1
a to the power , square root of a ( if a ≥ 0 then a ≥ 0 )
2 x for x ≥ 0, x ∈ R
the modulus or absolute value of x, ie − x for x < 0, x ∈ R
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
the closed interval a ≤ x ≤ b un
d
r
Sn the open interval a < x < b
the nth term of a sequence or series
the common difference of an arithmetic sequence
the common ratio of a geometric sequence
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 u1 + u2 + ... + un n
r n!
r !( n − r )!
f is a function under which each element of set A has an image in set B
f is a function under which x is mapped to y i i =1 f:A→ B
f :x ! y
f ( x)
−1 f
f #g
lim f ( x )
x→a the image of x under the function f
the inverse function of the function f
the composite function of f and g
the limit of f (x) as x tends to a IB Diploma Programme guide: Mathematics HL, September 2001 59 ASSESSMENT GUIDELINES dy
dx
f ′( x ) ∫ y dx
∫ y dx
b a ex
log a x
ln x
sin,cos, tan
arcsin, arccos arctan csc, sec, cot
A(x, y)
[AB]
AB
(AB) $
A $
CAB ∆ABC v
→ AB the derivative of y with respect to x
the derivative of f (x) with respect to x
the indefinite integral of y with respect to x
the definite integral of y with respect to x between the limits x = a and x = b
exponential function of x
logarithm to the base a of x
the natural logarithm of x, log e x
the circular functions
the inverse circular functions
the reciprocal circular functions
the point A in the plane with cartesian coordinates x and y
the line segment with end points A and B
the length of [AB]
the line containing points A and B
the angle at A
the angle between [CA] and [AB]
the triangle whose vertices are A, B and C
the vector v
the vector represented in magnitude and direction by the directed line
segment from A to B
→ a
i, j, k
a 
→ AB
v⋅w
v×w
A −1
AT
det A
I
P( A)
P( A ′)
P( A  B )
x1 , x2 , ...
f 1 , f 2 , ...
Px
f ( x)
E( X )
Var ( X ) N ( µ, σ 2 ) X ~ N (µ , σ 2 ) 60 the position vector OA
unit vectors in the directions of the cartesian coordinate axes
the magnitude of a
→ the magnitude of AB
the scalar product of v and w
the vector product of v and w
the inverse of the nonsingular matrix A
the transpose of the matrix A
the determinant of the square matrix A
the identity matrix
probability of event A
probability of the event 'not A'
probability of the event A given B
observations
frequencies with which the observations x1 , x2 , ... occur
probability distribution function P( X = x ) of the discrete random variable X
probability density function of the continuous random variable X
the expected value of the random variable X
the variance of the random variable X
normal distribution with mean µ and variance σ 2 the random variable X distributed normally with mean µ and variance σ 2 IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT GUIDELINES χ2 µ the chisquared distribution
population mean
k σ2 σ
x population variance, σ 2 = ∑ f (x − µ)
i i =1 2
k i n , where n = ∑f i i =1 population standard deviation
sample mean
k 2 sn
sn 2 s n−1 ∑ f (x − x ) 2
sample variance, s n = i =1 i 2 i n k , where n = ∑f i i =1 standard deviation of the sample
unbiased estimate of the population variance
k n2
2
sn−1 =
s=
n −1 n
Φ κn
κ n, m
χ (G )
Zp ∑ f (x − x )
i =1 i i n −1 2
k , where n = ∑ fi
i =1 cumulative distribution function of the standardised normal variable with
distribution N(0, 1)
a complete graph with n vertices
a complete bipartite graph with one set of n vertices and another set of m
vertices.
the chromatic number of the graph G
the set of equivalence classes {0 , 1, 2 , …, p − 1} of integers modulo p IB Diploma Programme guide: Mathematics HL, September 2001 61 ASSESSMENT GUIDELINES 2 Terminology (syllabus topic 11, option on discrete mathematics)
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 of a vertex/order; multiple
edges/parallel edges; loop/selfloop.
In IBO examination questions, the terminology used will be as it appears in the syllabus. For
clarity these terms are defined below. •
• An edge whose endpoints are connected to the same vertex is called a loop. • If more than one edge connects the same pair of vertices then these edges are
called multiple edges. • A directed edge is one in which it is only possible to travel in one direction. • A directed graph is a graph where every edge is directed. • A walk is a sequence of linked edges. • A trail is a walk in which no edge appears more than once. • A path is a walk with no repeated vertices. • A circuit is a walk which begins and ends at the same vertex, and has no repeated
edges. • A cycle is a walk which begins and ends at the same vertex, and has no repeated
vertices otherwise. • A Hamiltonian path is a path in which all the vertices of a graph appear once. • A Hamiltonian cycle is a path in which all the vertices appear once before it
returns to the first vertex. • A Eulerian trail is a trail containing every edge of a graph once. • A Eulerian circuit is a Eulerian trail which begins and ends at the same vertex. • Graph colouring is the assignment of a colour to each vertex in such a way that
no two adjacent vertices are assigned the same colour. • 62 A graph consists of a set of vertices and a set of edges. The endpoints of each
edge are connected to either the same vertex or two different vertices. The chromatic number of a graph is the minimum number of colours needed to
colour the graph. IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT GUIDELINES • The degree of a vertex is the number of edges connected to that vertex (a loop
contributes two, one for each of its endpoints). • A simple graph has no loops or multiple edges. • A graph is connected if there is a path connecting every pair of vertices. • A graph is disconnected if there is at least one pair of vertices which is not
connected by a path. • A complete graph is a simple graph, that is, one which has no loops or multiple
edges, where every vertex is connected to every other vertex. • A graph is a tree if it is connected and contains no paths which begin and end at
the same vertex. • A rooted tree is a directed tree containing a vertex from which there is a path to
every other vertex. • If a rooted tree contains an edge from vertex u to vertex v then u is the parent of
v and v is the child of u. • A binary tree is a rooted tree in which no vertex has more than two children. • A binary search tree is a binary tree in which all children are designated left or
right and no vertex has more than one left child or right child. • A weighted tree is a tree in which each edge is allocated a number or weight. • Sorting is reordering a set into a list of elements in increasing order. • A spanning tree of a graph is a subgraph containing every vertex of the graph,
which is also a tree. • A minimal spanning tree is the spanning tree of a weighted graph that has the
minimum total weight. • The complement of a graph G is a graph with the same vertices as G but which
has an edge between any two vertices if and only if G does not. • A graph isomorphism between two graphs G and H is a onetoone
correspondence between pairs of vertices such that a pair of vertices in G is
adjacent if and only if the equivalent pair in H is adjacent. • A planar graph is a graph that can be drawn in the plane without any edge
crossing another. • A bipartite graph is a graph whose vertices are divided into two sets and in
which edges always connect a vertex from one set to a vertex from the other set. IB Diploma Programme guide: Mathematics HL, September 2001 63 ASSESSMENT GUIDELINES •
• A subgraph is a graph within a graph. • The elements of the nth row of an adjacency matrix are the number of edges
connecting the nth vertex with every other vertex, taken in order. Hence, for an
undirected graph, the adjacency matrix will be symmetric about the diagonal. • 64 A complete bipartite graph is a bipartite graph in which there is an edge from
every vertex in one set to every vertex in the other set. The elements of the nth row of an incidence matrix are either 1 or 0 depending
on whether each edge, taken in order, is connected to the nth vertex or not. IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT GUIDELINES 3 Clarification (syllabus topic 13, option on Euclidean geometry
and conic sections)
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. In order to avoid confusion, in IBO examinations, theorems
which may be misinterpreted are defined below. Apollonius’ theorem (Circle of Apollonius)
PA
is a constant not equal to one then the locus of P is
PB
a circle. This circle is called the circle of Apollonius. If A, B are two fixed points such that Remark: the converse of this theorem is included. Apollonius’ theorem
If D is the midpoint of the base [BC] of a triangle ABC, then AB2 + AC 2 = 2(AD 2 + BD 2 ) .
A BD = DC B IB Diploma Programme guide: Mathematics HL, September 2001 D C 65 ASSESSMENT GUIDELINES 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
=
.
DC AC A E
A B C D C B D $
$
CAD = EAD $
$
BAD = CAD Remark: The converse of this result is included. Ninepoint circle theorem
Given any triangle ABC, let H be the intersection of the three altitudes. There is a circle that
passes through these nine special points:
the midpoints L, M, N of the three sides
the points R, S, T, where the three altitudes of the triangle meet the sides
the midpoints, X, Y, Z, of [HA], [HB], [HC].
C
T
ZS
H
N M
X Y B A
L 66 R IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT GUIDELINES 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, ie for a cyclic quadrilateral ABCD,
AB × CD + BC × DA = AC × BD . A
B D
C Ceva’s theorem
If three concurrent lines are drawn through the vertices A, B, C of a triangle ABC to meet the
BD CE AF
opposite sides at D, E, F, respectively, then
×
×
= +1 .
DC EA FB
A
A
F
E
D
F C
B O
B D C E O Converse: If D, E, F are points on [BC], [CA], [AB], respectively such that BD CE AF
×
×
= +1 , then [AD], [BE] and [CF] are concurrent.
DC EA FB IB Diploma Programme guide: Mathematics HL, September 2001 67 ASSESSMENT GUIDELINES Menelaus’ theorem
If a transversal meets the sides [BC], [CA], [AB] of a triangle at D, E, F, respectively, then
BD CE AF
×
×
= −1 .
DC EA FB A
E
F F A
E B C D B C D Converse: If D, E, F are points on the sides [BC], [CA], [AB], respectively, of a triangle
such that BD CE AF
×
×
= −1 , then D, E, F are collinear.
DC EA FB Note on Ceva’s theorem and Menelaus’ theorem
The statements and proofs of these theorems presuppose the idea of sensed magnitudes. Two
segments [AB], [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 as the displacements
A → B and P → Q are in the same or opposite directions. This may be used to prove the
following theorem:
Theorem: If A, B, C are any three collinear points then AB + BC + CA = 0 where AB, BC
and CA denote sensed magnitudes. 68 IB Diploma Programme guide: Mathematics HL, September 2001 ASSESSMENT GUIDELINES Internal assessment: the portfolio
1 Teaching and learning strategies
! As an integral part of the course, candidates need to be provided with the opportunities to
experiment, explore, make conjectures and ask questions. Ideally the atmosphere in the
classroom should be one of enquiry. ! It will still be necessary for candidates to learn the skills associated with portfolio
activities. One way of approaching this might be for the whole class or smaller groups to
work through a small number of relatively simple assignments in order for candidates to
be made aware of what might be required, although the work done on these assignments
would not be eligible for inclusion in their portfolios.
For example, candidates may be unaware of certain strategies associated with
experimentation, or “playing”, which are an important part of investigative work,
particularly if they have only experienced more formal modes of working. ! ! 2 In reporting their results, candidates should realize that there is an emphasis on
thoughtful reflection and good mathematical writing. These are skills that are rarely
learned through timed tests/examinations, and therefore candidates may need some
guidance and encouragement in these areas.
It is also important that candidates are given the opportunity to learn mathematical
concepts new to them and to gain a deeper understanding of concepts already learned
through portfolio activities. It will therefore be necessary to allow time for classroom
discussion of the results/conclusions that can be drawn from a particular activity. This
time should not be regarded as additional to time allocated to teaching the syllabus since
it will, of course, involve discussion of topics which are already part of the syllabus. The nature of portfolio assignments
! Portfolio assignments should provide candidates with opportunities to engage in
mathematics in an environment which will capture their interest and provide them with
rich opportunities to exercise their mental powers. ! Each assignment should be accessible in terms of the candidates’ background in
mathematics and should allow them to achieve results at different levels. This will allow
even the weaker candidates to gain a sense of satisfaction in relation to what they have
accomplished. ! Assignments should be constructed in a manner which will offer candidates the
possibility of gaining the maximum achievement level for each criterion. At the same
time, it is accepted that for some activities maximum achievement levels will be more
difficult to obtain than for others. It is therefore important that candidates are presented with a range of activities which will
allow them to show what they can do in relation to each of the criteria. IB Diploma Programme guide: Mathematics HL, September 2001 69 ASSESSMENT GUIDELINES !
! 70 Activities should be presented to candidates in written form. Ideally, all candidates
should receive their own copies so that they may make reference to them at any time.
Within each assignment, the degree to which candidates are guided into choosing
particular strategies will depend on the skills the candidate has acquired. Assignments
presented to candidates in the earlier stages of the course are therefore likely to be more
structured than those presented to candidates towards the end of the course. IB Diploma Programme guide: Mathematics HL, September 2001 ...
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 Normal Distribution, Inverse function, IB Diploma Programme, Mathematics HL, Diploma Programme guide

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