Hsp add maths_f5

Ministry of Education
Malaysia

Integrated Curriculum for Secondary Schools
CURRICULUM SPECIFICATIONS

ADDITIONAL MATHEMATICS
FORM 5

Curriculum Development Centre
Ministry of Education Malaysia
2006

Copyright © 2006 Curriculum Development Centre
Ministry of Education Malaysia
Aras 4-8, Blok E9
Kompleks Kerajaan Parcel E
Pusat Pentadbiran Putrajaya
62604 Putrajaya

First published 2006

Copyright reserved. Except for use in a review, the reproduction or
utilisation of this work in any form or by any electronic, mechanical, or
other means, now known or hereafter invented, including photocopying,
and recording is forbidden without the prior written permission from the
Director of the Curriculum Development Centre, Ministry of Education
Malaysia.

CONTENTS
Page
RUKUNEGARA (iv)
National Philosophy of Education (v)
Preface (vii)
Introduction (ix)
A6. Progressions 1
A7. Linear Law 4
C2. Integration 5
G2. Vectors 7
T2. Trigonometric Functions 11
S2. Permutations and Combinations 14
S3. Probability 16
S4. Probability Distributions 18
AST2. Motion Along a Straight Line 20
ASS2. Linear Programming 23
PW2. Project Work 25

RUKUNEGARA
DECLARATION
OUR NATION, MALAYSIA, being dedicated
• to achieving a greater unity of all her peoples;
• to maintaining a democratic way of life;
• to creating a just society in which the wealth of the
nation shall be equitably shared;
• to ensuring a liberal approach to her rich and diverse
cultural traditions;
• to building a progressive society which shall be oriented
to modern science and technology;
WE, her peoples, pledge our united efforts to attain these
ends guided by these principles:
• BELIEF IN GOD
• LOYALTY TO KING AND COUNTRY
• UPHOLDING THE CONSTITUTION
• RULE OF LAW
• GOOD BEHAVIOUR AND MORALITY

Education in Malaysia is an ongoing effort
towards further developing the potential of
individuals in a holistic and integrated
manner so as to produce individuals who are
intellectually, spiritually, emotionally and
physically balanced and harmonious, based
on a firm belief in God. Such an effort is
designed to produce Malaysian citizens who
are knowledgeable and competent, who
possess high moral standards, and who are
responsible and capable of achieving a high
level of personal well-being as well as being
able to contribute to the betterment of the
family, the society and the nation at large.

PREFACE The use of technology in the teaching and learning of Additional
Mathematics is greatly emphasised. Additional Mathematics taught in
Science and technology plays a critical role in realising Malaysia’s English, coupled with the use of ICT, provide greater opportunities for
aspiration to become a developed nation. Since mathematics is instrumental pupils to improve their knowledge and skills in mathematics because of the
in the development of scientific and technological knowledge, the provision richness of resources and repositories of knowledge in English. Our pupils
of quality mathematics education from an early age in the education process will be able to interact with pupils from other countries, improve their
is thus important. The Malaysian school curriculum offers three proficiency in English; and thus make the learning of mathematics more
mathematics education programs, namely Mathematics for primary schools, interesting and exciting.
Mathematics and Additional Mathematics for secondary schools.
The development of this Additional Mathematics Curriculum Specifications
The Malaysian school mathematics curriculum aims to develop is the work of many individuals and experts in the field. On behalf of the
mathematical knowledge, competency and inculcate positive attitudes Curriculum Development Centre, I would like to express much gratitude and
towards mathematics among pupils. While the Mathematics curriculum appreciation to those who have contributed in one way or another towards
prepares pupils to cope with daily life challenges, the Additional this initiative.
Mathematics curriculum provides an exposure to the level of mathematics
appropriate for science and technology related careers. As with other
subjects in the secondary school curriculum, Additional Mathematics aims
to inculcate noble values and love for the nation in the development of a
holistic person, who in turn will be able to contribute to the harmony and
prosperity of the nation and its people.

Additional Mathematics is an elective subject offered to the upper (MAHZAN BIN BAKAR SMP, AMP)
secondary school pupils. Beginning 2003, English is used as the medium of
instruction for Science and Mathematics subjects. The policy to change the Director
medium of instruction for the two subjects follows a phased implementation Curriculum Development Centre
schedule and is expected to be completed by 2008. The teaching and Ministry of Education
learning of Additional Mathematics in English started in 2006. Malaysia

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INTRODUCTION
are also stressed in the process of learning Additional Mathematics. When
A well-informed and knowledgeable society, well versed in the use of pupils explain concepts and their work, they are guided in the use of correct
Mathematics to cope with daily life challenges is integral to realising the and precise mathematical terms and sentences. Emphasis on Mathematical
nation’s aspiration to become an industrialised nation. Thus, efforts are taken communications develops pupils’ ability in interpreting matters into
to ensure a society that assimilates mathematics into their daily lives. Pupils mathematical modellings or vice versa.
are nurtured from an early age with the skills to solve problems and
communicate mathematically, to enable them to make effective decisions. The use of technology especially, Information and Communication
Technology (ICT) is much encouraged in the teaching and learning process.
Mathematics is essential in preparing a workforce capable of meeting the Pupils’ understanding of concepts can be enhanced as visual stimuli are
demands of a progressive nation. As such, this field assumes its role as the provided and complex calculations are made easier with the use of
driving force behind various developments in science and technology. In line calculators.
with the nation’s objective to create a knowledge-based economy, the skills
of Research & Development in mathematics is nurtured and developed at Project work, compalsory in Additional Mathematics provides opportunities
school level. for pupils to apply the knowledge and skills learned in the classroom into
real-life situations. Project work carried out by pupils includes exploration of
Additional Mathematics is an elective subject in secondary schools, which mathematical problems, which activates their minds, makes the learning of
caters to the needs of pupils who are inclined towards Science and mathematics more meaningful, and enables pupils to apply mathematical
Technology. Thus, the content of the curriculum has been organised to concepts and skills, and further develops their communication skills.
achieve this objective.
The intrinsic values of mathematics namely thinking systematically,
The design of the Additional Mathematics syllabus takes into account the accurately, thoroughly, diligently and with confidence, infused throughout
contents of the Mathematics curriculum. New areas of mathematics the teaching and learning process; contribute to the moulding of character
introduced in the Additional Mathematics curriculum are in keeping with and the inculcation of positive attitudes towards mathematics. Together with
new developments in Mathematics. Emphasis is placed on the heuristics of these, moral values are also introduced in context throughout the teaching
problem solving in the process of teaching and learning to enable pupils to and learning of mathematics.
gain the ability and confidence to use mathematics in new and different
situations. Assessment, in the form of tests and examinations helps to gauge pupils’
achievement. Assessments in Additional Mathematics include aspects such
The Additional Mathematics syllabus emphasises understanding of concepts as understanding of concepts, mastery of skills and non-routine questions
and mastery of related skills with problem solving as the main focus in the that demand the application of problem-solving strategies. The use of good
teaching and learning process. Skills of communication through mathematics assessment data from a variety of sources provides valuable information on

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the development and progress of pupils. On-going assessment built into the 7 debate solutions using precise mathematical language,
daily lessons allows the identification of pupils’ strengths and weaknesses,
and effectiveness of the instructional activities. Information gained from 8 relate mathematical ideas to the needs and activities of human beings,
responses to questions, group work results, and homework helps in
improving the teaching process, and hence enables the provision of 9 use hardware and software to explore mathematics, and
effectively aimed lessons. 10 practise intrinsic mathematical values.

AIM
ORGANISATION OF CONTENT
The Additional Mathematics curriculum for secondary schools aims to
develop pupils with in-depth mathematical knowledge and ability, so that The contents of the Form Five Additional Mathematics are arranged into two
they are able to use mathematics responsibly and effectively in learning packages. They are the Core Package and the Elective Package.
communications and problem solving, and are prepared to pursue further The Core Package, compulsory for all pupils, consists of nine topics
studies and embark on science and technology related careers. arranged under five components, that is:
OBJECTIVES • Geometry
The Additional Mathematics curriculum enables pupils to: • Algebra
• Calculus
1 widen their ability in the fields of number, shape and relationship as
well as to gain knowledge in calculus, vector and linear • Trigonometry
programming, • Statistics
2 enhance problem-solving skills, Each teaching component includes topics related to one branch of
mathematics. Topics in a particular teaching component are arranged
3 develop the ability to think critically, creatively and to reason out
according to hierarchy whereby easier topics are learned earlier before
logically,
proceeding to the more complex topics.
4 make inference and reasonable generalisation from given information,
The Elective Package consists of two packages, namely the Science and
Technology Application Package and the Social Science Application
5 relate the learning of Mathematics to daily activities and careers, Package. Pupils need to choose one Elective Package according to their
6 use the knowledge and skills of Mathematics to interpret and solve inclination in their future field.
real-life problems,

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The Additional Mathematics Curriculum Specifications is prepared in a In the Points To Notes column, attention is drawn to the more significant
format which helps teachers to teach a particular topic effectively. The aspects of mathematical concepts and skills to be taught. This column
contents of each topic are divided into five columns: consists of:
• limitations to the scope of a particular topic;
• Learning Objectives;
• certain emphases;
• Suggested Teaching and Learning Activities;
• notations; and
• Learning Outcomes;
• formulae.
• Points to Note; and
The Vocabulary column consists of standard mathematical terminologies,
• Vocabulary. instructional words or phrases that are relevant in structuring activities,
All concepts and skills taught for a particular topic are arranged into a few asking questions or setting task. It is important to pay careful attention to the
learning units that are stated in the Learning Objectives column. These use of correct terminologies and these needs to be systematically introduced
Learning Objectives are arranged according to hierarchy from easy to the to pupils in various contexts so as to enable pupils to understand the
more abstract concepts. meanings of the terms and learn to use them appropriately.
The Suggested Teaching and Learning Activities column lists some
examples of teaching and learning activities including methods, techniques, EMPHASES IN TEACHING AND LEARNING
strategies and resources pertaining to the specific concepts or skills. These,
however, are mere sample learning experiences and are not the only The teaching and learning process in this curriculum emphasise concept
activities to be used in the classrooms. Teachers are encouraged to look for building and skills acquisition as well as the inculcation of good and positive
further examples, determine the teaching and learning strategies most values. Beside these, there are other elements that have to be taken into
suitable for their pupils and provide appropriate teaching and learning account and carefully planned and infused into the teaching and learning of
materials. Teachers should also make cross-references to other resources the subject. The main elements focused in the teaching and learning of
such as the textbooks, and the Internet. Additional Mathematics are as follows:
The Learning Outcomes column defines clearly what pupils should be able Problem Solving
to do after a learning experience. The intended outcomes state the
mathematical abilities that should transpire from the activities conducted. In the Mathematics Curriculum, problem-solving skills and problem-solving
Teachers are expected to look for indicators that pupils have acquired all of strategies such as trial and improvement, drawing diagrams, tabulating data,
the abilities stated. identifying polar, experiment/simulation, solving easier problems, finding
analogy and working backwards have already been learnt. Further
strengthening of the above strategies must be carried out in the process of

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teaching and learning of Additional Mathematics. Besides routine questions, up pupils’ minds to accept mathematics as a powerful tool in the world
pupils must be able to solve non-routine problems using problem-solving today.
strategies. Teachers are also encouraged to demonstrate problems with
Pupils are encouraged to estimate, predict and make intelligent guesses in
multiple problem-solving strategies.
the process of seeking solutions. Pupils at all levels have to be trained to
Communication in Mathematics investigate their predictions or guesses by using concrete materials,
calculators, computers, mathematical representations and others. Logical
The skills of communication in mathematics are also stressed in the teaching reasoning has to be absorbed in the teaching of mathematics so that pupils
and learning of Additional Mathematics. Communication is an essential can recognise, construct and evaluate predictions and mathematical
means of sharing ideas and clarifying the understanding of mathematics. arguments.
Through communication, mathematical ideas become the object of
reflection, discussion and modification. Communicational skills in Making Connections
mathematics include reading, writing, listening and speaking. Through
In the teaching and learning of Additional Mathematics, opportunities for
effective mathematical communication, pupils will become efficient in
making connections must be created so that pupils can link conceptual
problem-solving and be able to explain their conceptual understanding and
knowledge to procedural knowledge and relate topics within mathematics
mathematical skills to their peers and teachers. Therefore, through the
and other learning areas in general.
teaching and learning process, teachers should frequently create
opportunities for pupils to read, write and discuss ideas in which the The Additional Mathematics curriculum covers several areas of mathematics
language of mathematics becomes natural and this can only be done through such as Geometry, Algebra, Trigonometry, Statistics and Calculus. Without
suitable mathematical tasks that are worthwhile topics for discussion. connections between these areas, pupils will have to learn and memorise too
many concepts and skills separately. By making connections, pupils are able
Pupils who have developed the skills to communicate mathematically will
to see mathematics as an integrated whole rather than a string of
become more inquisitive and, in the process, gain confidence. Emphasis on
unconnected ideas. When mathematical ideas and the curriculum are
mathematical communications will develop pupils ability in interpreting
connected to real-life within or outside the classroom, pupils will become
certain matters into mathematical models or vice versa. The process of
more conscious of the importance and significance of mathematics. They
analytical and systematic reasoning helps pupils to reinforce and strengthen
will also be able to use mathematics contextually in different learning areas
their knowledge and understanding of mathematics to a deeper level.
and in real-life situations.
Reasoning The Use of Technology
Logical Reasoning or thinking is the basis for understanding and solving
The use of ICT and other technologies is encouraged in the teaching and
mathematical problems. The development of mathematical reasoning is
learning of Additional Mathematics. Technologies help pupils by increasing
closely related to the intellectual and communicative development of pupils.
their understanding of abstract concepts, providing visual input and making
Emphasis on logical thinking, during teaching and learning activities opens

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complex calculation easier. Calculators, computers, software related to where it provides pupils with a better understanding and appreciation of
education, web sites and learning packages can further improve the mathematics.
pedagogy of teaching and learning of Additional Mathematics. Schools are
Suitable choice of teaching and learning approaches will provide stimulating
therefore encouraged to equip teachers with appropriate and effective
learning environment that enhance effectiveness of learning mathematics.
software. The use of software such as Geometer’s Sketchpad not only helps
Approaches that are considered suitable include the following:
pupils to model problems and enables them to understand certain topics
• Cooperative learning;
better but also enables pupils to explore mathematical concepts more
effectively. However, technology can’t replace a teacher. Instead it should • Contextual learning;
be use as an effective tool to enhance the effectiveness of teaching and • Mastery learning;
learning mathematics.
• investigation;
APPROACHES IN TEACHING AND LEARNING • Enquiry; and
• Exploratory.
Advancement in mathematics and pedagogy of teaching mathematics
demand changes to the way mathematics is taught in the classroom. TEACHING SCHEMES
Effective use of teaching resources is vital in forming the understanding of
mathematical concepts. Teachers should use real or concrete materials to To facilitate the teaching and learning process, two types of annual schemes
help pupils gain experience, construct abstract ideas, make inventions, build are suggested. They are the Component Scheme and the Title Scheme.
self-confidence, encourage independence and inculcate the spirit of
In the Component Scheme, all topics related to Algebra are taught first
cooperation. The teaching and learning materials used should contain self-
before proceeding to other components. This scheme presents the Additional
diagnostic elements so that pupils know how far they have understood
Mathematics content that has been learnt before moving to new ones.
certain concepts and have acquired the skills.
The Title Scheme on the other hands allows more flexibility for the teachers
In order to assist pupils develop positive attitudes and personalities, the
to introduce the Algebraic and Geometrical topics before introducing the
mathematical values of accuracy, confidence and thinking systematically
new branches of Mathematics such as the Calculus.
have to be infused into the teaching and learning process. Good moral
values can be cultivated through suitable contexts. Learning in groups for Between these two teaching schemes, teachers are free to choose a more
example can help pupils develop social skills, encourage cooperation and suitable scheme based on their pupils’ previous knowledge, learning style
build self-confidence. The element of patriotism should also be inculcated and their own teaching style.
through the teaching and learning process in the classroom using certain
topics. Brief historical anecdotes related to aspects of mathematics and
famous mathematicians associated with particular learning areas are also
incorporated into the curriculum. It should be presented at appropriate points

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COMPONENT SCHEME TITLE SCHEME

Algebraic Component A6. Progressions

A6.Progressions
A7.Linear Law Trigonometric Component
C2. Integration
T2. Trigonometric Functions PW2. Project Work

Calculus Component A7. Linear Law
C2.Integration Statistics Component

S2. Permutations and G2. Vectors
Combinations
Geometric Component
S3 Probability
G2.Vectors T2. Trigonometric Functions
S4 Probability Distributions

S2. Permutations and Combinations

S3. Probability
Science and Technology Social Science Package
Package
ASS2. Linear Programming
AST2. Motion Along a Straight S4. Probability Distributions
Line

AST2. Motion Along a Straight Line
PW2. Project Work PW2. Project Work or
ASS2. Linear Programming

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PROJECT WORK EVALUATION
Project Work is a new element in the Additional Mathematics curriculum. It Continual and varied forms of evaluation is an important part of the teaching
is a mean of giving pupils the opportunity to transfer the understanding of and learning process. It not only provides feedback to pupils on their
mathematical concepts and skills learnt into situations outside the classroom. progress but also enable teachers to correct their pupils’ misconceptions and
Through Project Work, pupils are to pursue solutions to given tasks through weaknesses. Based on evaluation outcomes, teachers can take corrective
activities such as questioning, discussing, debating ideas, collecting and measures such as conducting remedial or enrichment activities in order to
analyzing data, investigating and also producing written report. With improve pupils’ performances and also strive to improve their own teaching
regards to this, suitable tasks containing non-routine problems must skills. Schools should also design effective internal programs to assist pupils
therefore be administered to pupils. However, in the process of seeking in improving their performances. The Additional Mathematics Curriculum
solutions to the tasks given, a demonstration of good reasoning and effective emphasis evaluation, which among other things must include the following
mathematical communication should be rewarded even more than the pupils aspects:
abilities to find correct answers.
• concept understandings and mastery of skills; and
Every form five pupils taking Additional Mathematics is required to carry
out a project work whereby the theme given is either based on the Science • non-routine questions (which demand the application of problem-
and Technology or Social Science package. Pupils however are allowed to solving strategies).
choose any topic from the list of tasks provided. Project work can only be
carried out in the second semester after pupils have mastered the first few
chapters. The tasks given must therefore be based on chapters that have
already been learnt and pupils are expected to complete it within the duration
of three weeks. Project work can be done in groups or individually but each
pupil is expected to submit an individually written report which include the
following:
• title/topic;
• background or introduction;
• method/strategy/procedure;
• finding;
• discussion/solution; and
• conclusion/generalisation.

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LEARNING AREA:
A6
LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES
Form 5
POINTS TO NOTE VOCABULARY
Pupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…

1 Understand and use Use examples from real-life (i) Identify characteristics of Begin with sequences to sequence
the concept of arithmetic situations, scientific or graphing arithmetic progressions. introduce arithmetic and series
progression. calculators and computer geometric progressions.
(ii) Determine whether a given characteristic
software to explore arithmetic
sequence is an arithmetic Include examples in
progressions. arithmetic
progression. algebraic form. progression
(iii) Determine by using formula: common
a) specific terms in arithmetic difference
progressions,
specific term
b) the number of terms in
arithmetic progressions. first term

(iv) Find: nth term
a) the sum of the first n terms of consecutive
arithmetic progressions,
b) the sum of a specific number Include the use of the
of consecutive terms of formula Tn = S n − S n −1
c) the value of n, given the sum
of the first n terms of
(v) Solve problems involving Include problems
arithmetic progressions. involving real-life
situations.

1

LEARNING AREA:
A6
Form 5

2 Understand and use the Use examples from real-life (i) Identify characteristics of Include examples in geometric
concept of geometric situations, scientific or graphing geometric progressions. algebraic form. progression
progression. calculators; and computer common ratio
software to explore geometric (ii) Determine whether a given
progressions. sequence is a geometric
progression.

(iii) Determine by using formula:
a) specific terms in
geometric progressions,
b) the number of terms in
geometric progressions.

(iv) Find:
a) the sum of the first n terms
of geometric progressions,
b) the sum of a specific number
of consecutive terms of
c) the value of n, given the sum
of the first n terms of

2

LEARNING AREA:
A6
Form 5
Discuss:
(v) Find:
a) the sum to infinity of As n → ∞ , rn 0 sum to infinity
a
geometric progressions, then S∞ = 1 – r
recurring
b) the first term or common
S∞ read as “sum to decimal
ratio, given the sum to infinity
infinity”.
of geometric progressions.

Include recurring decimals.
Limit to 2.recurring digits
..
such as 0.3, 0.15, …

(vi) Solve problems involving Exclude:
geometric progressions. a) combination of
arithmetic progressions
and geometric
progressions,
b) cumulative
sequences such as,
(1), (2, 3), (4, 5, 6),
(7, 8, 9, 10), …

3

LEARNING AREA:
A7
Form 5

1 Understand and use the Use examples from real-life (i) Draw lines of best fit by Limit data to linear line of best fit
concept of lines of best fit. situations to introduce the inspection of given data. relations between two inspection
concept of linear law. variables.
variable
(ii) Write equations for lines of best
fit. non-linear
Use graphing calculators or relation
computer software such as
Geometer’s Sketchpad to (iii) Determine values of variables linear form
explore lines of best fit. from:
reduce
a) lines of best fit,
b) equations of lines of best fit.

2 Apply linear law to non- (i) Reduce non-linear relations to
linear relations. linear form.
(ii) Determine values of constants
of non-linear relations given:
b) data.

(iii) Obtain information from:
b) equations of lines of best fit.

4

LEARNING AREA:
C2
Form 5

1 Understand and use the Use computer software such as (i) Determine integrals by reversing Emphasise constant of integration
concept of indefinite Geometer’s Sketchpad to differentiation. integration. integral
integral. explore the concept of
integration. indefinite
(ii) Determine integrals of axn, where
a is a constant and n is an integer,
∫ ydx read as “integration integral
of y with respect to x” reverse
n ≠ –1.
constant of
(iii) Determine integrals of algebraic integration
expressions. substitution

(iv) Find constants of integration, c, in
indefinite integrals.

(v) Determine equations of curves
from functions of gradients.

(vi) Determine by substitution the Limit integration of
integrals of expressions of the
∫u dx,
n

form (ax + b)n, where a and b are
where u = ax + b.
constants, n is an integer and
n ≠ –1.

5

LEARNING AREA:
C2
Form 5

2 Understand and use the Use scientific or graphing (i) Find definite integrals of Include definite integral
concept of definite integral. calculators to explore the algebraic expressions.
∫ kf ( x)dx = k ∫ f ( x)dx
b b
limit
concept of definite integrals. a a
b a
volume
∫ f ( x)dx = − ∫ f ( x)dx
a b
region
Use computer software and (ii) Find areas under curves as the Derivation of formulae not rotated
graphing calculator to explore limit of a sum of areas. required. revolution
areas under curves and the
significance of positive and (iii) Determine areas under curves Limit to one curve. solid of
negative values of areas. using formula. revolution

(iv) Find volumes of revolutions when
Use dynamic computer Derivation of formulae not
software to explore volumes of region bounded by a curve is required.
revolutions. rotated completely about the
a) x-axis,
b) y-axis
as the limit of a sum of volumes.
(v) Determine volumes of revolutions
using formula.
Limit volumes of
revolution about the
x-axis or y-axis.

6

LEARNING AREA:
G2
Form 5

1 Understand and use the Use examples from real-life (i) Differentiate between vector and Use notations: differentiate
→
concept of vector. situations and dynamic computer scalar quantities. Vector: ~ AB, a, AB.
a, scalar
software such as Geometer’s Magnitude:
→ vector
Sketchpad to explore vectors. (ii) Draw and label directed line
|a|, |AB|, |a|, |AB|.
segments to represent vectors. ~ directed line
segment
(iii) Determine the magnitude and Zero vector: ~
0 magnitude
direction of vectors represented
Emphasise that a zero direction
by directed line segments.
vector has a magnitude of zero vector
zero. negative vector
Emphasise negative
vector:
→ →
−AB = BA
(iv) Determine whether two vectors
are equal.
(v) Multiply vectors by scalars. Include negative scalar.

7

LEARNING AREA:
G2
Form 5

(vi) Determine whether two vectors Include: parallel
are parallel. a) collinear points non-parallel
b) non-parallel
collinear points
non-zero vectors.
non-zero
Emphasise: triangle law
If ~ and ~ are not parallel
a b
parallelogram
and ha = kb, then
~ ~ law
h = k = 0.
resultant vector

2 Understand and use the Use real-life situations and (i) Determine the resultant vector of polygon law
concept of addition and manipulative materials to two parallel vectors.
subtraction of vectors. explore addition and
subtraction of vectors. (ii) Determine the resultant vector of
two non-parallel vectors using:
a) triangle law,
b) parallelogram law.

(iii) Determine the resultant vector of
three or more vectors using the
polygon law.

8

LEARNING AREA:
G2
Form 5

(iv) Subtract two vectors which are: Emphasise:
a) Parallel, a – ~ = ~ + (−b)
~
b a ~
b) non-parallel.
(v) Represent a vector as a
combination of other vectors.
(vi) Solve problems involving
addition and subtraction of
vectors.

3 Understand and use Use computer software to (i) Express vectors in the form: Relate unit vector ~ and j
i Cartesian plane
explore vectors in the Cartesian ~
vectors in the Cartesian a) xi + yj
~ to Cartesian coordinates. unit vector
~
plane. plane.
⎛ x⎞ Emphasise:
b) ⎜ ⎟ .
⎜ y⎟
⎝ ⎠ ⎛1⎞
Vector i = ⎜ ⎟ and
~ ⎜ 0⎟
⎝ ⎠
⎛ 0⎞
Vector j = ⎜ ⎟
~ ⎜1⎟
⎝ ⎠

9

LEARNING AREA:
G2
Form 5

(ii) Determine magnitudes of vectors. For learning outcomes 3.2
(iii) Determine unit vectors in given to 3.7, all vectors are given
directions. in the form
(iv) Add two or more vectors. ⎛ x⎞
xi + yj or ⎜ ⎟ .
~ ~ ⎜ y⎟
(v) Subtract two vectors. ⎝ ⎠
(vi) Multiply vectors by scalars.
(vii) Perform combined operations on
vectors.
Limit combined operations
to addition, subtraction
and multiplication of
vectors by scalars.

(viii) Solve problems involving vectors.

10

LEARNING AREA:
T2
Form 5

1 Understand the concept Use dynamic computer (i) Represent in a Cartesian plane, Cartesian plane
of positive and negative software such as Geometer’s angles greater than 360o or 2π rotating ray
angles measured in degrees Sketchpad to explore angles in radians for:
Cartesian plane. a) positive angles, positive angle
and radians.
b) negative angles. negative angle
clockwise
2 Understand and use the Use dynamic computer (i) Define sine, cosine and tangent of Use unit circle to
software to explore any angle in a Cartesian plane. determine the sign of anticlockwise
six trigonometric functions
of any angle. trigonometric functions in trigonometric ratios. unit circle
(ii) Define cotangent, secant and
degrees and radians.
cosecant of any angle in a quadrant
Cartesian plane. Emphasise:
reference angle
Use scientific or graphing sin θ = cos (90o – θ)
calculators to explore cos θ = sin (90o – θ) trigonometric
(iii) Find values of the six tan θ = cot (90o – θ) function/ratio
trigonometric functions of any trigonometric functions of any
angle. cosec θ = sec (90o – θ) sine
angle.
sec θ = cosec (90o – θ)
cosine
cot θ = tan (90o – θ)
tangent
Emphasise the use of
triangles to find cosecant

(iv) Solve trigonometric equations. trigonometric ratios for secant
special angles 30o, 45o and
cotangent
60o.
special angle

11

LEARNING AREA:
T2
Form 5

3 Understand and use Use examples from real-life (i) Draw and sketch graphs of Use angles in modulus
graphs of sine, cosine and situations to introduce graphs trigonometric functions: a) degrees domain
tangent functions. of trigonometric functions. a) y = c + a sin bx, b) radians, in terms of π.
range
b) y = c + a cos bx,
c) y = c + a tan bx Emphasise the sketch
Use graphing calculators and
dynamic computer software where a, b and c are constants characteristics of sine, draw
such as Geometer’s Sketchpad and b > 0. cosine and tangent graphs.
period
to explore graphs of Include trigonometric
trigonometric functions. functions involving cycle
modulus. maximum

(ii) Determine the number of minimum
Exclude combinations of
solutions to a trigonometric trigonometric functions. asymptote
equation using sketched graphs.

(iii) Solve trigonometric equations
using drawn graphs.

12

LEARNING AREA:
T2
Form 5

4 Understand and use Use scientific or graphing (i) Prove basic identities: Basic identities are also basic identity
basic identities. calculators and dynamic a) sin2A + cos2A = 1, known as Pythagorean
computer software such as Pythagorean
b) 1 + tan2A = sec2A, identities. identity
Geometer’s Sketchpad to c) 1 + cot2A = cosec2A.
explore basic identities.
(ii) Prove trigonometric identities Include learning outcomes
using basic identities. 2.1 and 2.2.
(iii) Solve trigonometric equations
using basic identities.

5 Understand and use Use dynamic computer
(i) Prove trigonometric identities Derivation of addition addition formula
addition formulae and software such as Geometer’s
using addition formulae for formulae not required.
double-angle formulae. Sketchpad to explore addition double-angle
formulae and double-angle sin (A ± B), cos (A ± B) and Discuss half-angle formula
formulae. tan (A ± B). formulae. half-angle
(ii) Derive double-angle formulae for formula
sin 2A, cos 2A and tan 2A.
(iii) Prove trigonometric identities Exclude
using addition formulae and/or a cos x + b sin x = c,
double-angle formulae. where c ≠ 0.
(iv) Solve trigonometric equations.

13

LEARNING AREA:
S2
Form 5

1 Understand and use the For this topic:
concept of permutation. a) Introduce the concept
by using numerical
examples.
b) Calculators should only
be used after students
have understood the
concept.
Use manipulative materials to (i) Determine the total number of Limit to 3 events. multiplication
explore multiplication rule. ways to perform successive rule
events using multiplication rule. successive
(ii) Determine the number of Exclude cases involving events
Use real-life situations and
computer software such as permutations of n different identical objects. permutation
spreadsheet to explore objects. Explain the concept of factorial
permutations. permutations by listing all
arrangement
possible arrangements.
order
Include notations:
a) n! = n(n – 1)(n – 2)
…(3)(2)(1)
b) 0! = 1
n! read as “n factorial”.

14

LEARNING AREA:
S2
Form 5

(iii) Determine the number of Exclude cases involving
permutations of n different objects arrangement of objects in
taken r at a time. a circle.
(iv) Determine the number of
permutations of n different objects
for given conditions.
(v) Determine the number of
permutations of n different objects
taken r at a time for given
conditions.

2 Understand and use the Explore combinations using (i) Determine the number of Explain the concept of combination
concept of combination. real-life situations and combinations of r objects chosen combinations by listing all selection
computer software. from n different objects. possible selections.

(ii) Determine the number of Use examples to illustrate
n
combinations r objects chosen n
Cr = r
P
from n different objects for given r!
conditions.

15

LEARNING AREA:
S3
Form 5

1 Understand and use the Use real-life situations to (i) Describe the sample space of an Use set notations. experiment
concept of probability. introduce probability.
experiment. sample space
Use manipulative materials,
(ii) Determine the number of event
computer software, and
scientific or graphing outcomes of an event. outcome
calculators to explore the equally likely
concept of probability. (iii) Determine the probability of an Discuss:
event. a) classical probability probability
(theoretical occur
probability)
b) subjective probability classical
c) relative frequency
probability
(experimental theoretical
probability). probability
Emphasise: subjective
Only classical probability relative
is used to solve problems. frequency
Emphasise: experimental
P(A ∪ B) = P(A) + P(B) –
(iv) Determine the probability of two P(A ∩ B)
events: using Venn diagrams.
a) A or B occurring,
b) A and B occurring.

16

LEARNING AREA:
S3
Form 5

2 Understand and use the Use manipulative materials and (i) Determine whether two events are Include events that are mutually
concept of probability of graphing calculators to explore mutually exclusive. mutually exclusive and exclusive
mutually exclusive events. the concept of probability of exhaustive. event
mutually exclusive events.
exhaustive
Use computer software to (ii) Determine the probability of two Limit to three mutually
simulate experiments involving independent
or more events that are mutually exclusive events.
probability of mutually exclusive. tree diagrams
exclusive events.

3 Understand and use the Use manipulative materials and (i) Determine whether two events are Include tree diagrams.
concept of probability of graphing calculators to explore independent.
independent events. the concept of probability of
independent events. (ii) Determine the probability of two
Use computer software to independent events.
simulate experiments involving
probability of independent (iii) Determine the probability of three
events. independent events.

17

LEARNING AREA:
S4
Form 5

1 Understand and use the Use real-life situations to (i) List all possible values of a discrete random
concept of binomial introduce the concept of discrete random variable. variable
distribution. binomial distribution.
independent trial
Use graphing calculators and (ii) Determine the probability of an Include the characteristics
Bernoulli trials
computer software to explore event in a binomial distribution. of Bernoulli trials.
binomial distribution. binomial
For learning outcomes 1.2 distribution
and 1.4, derivation of
formulae not required. mean
variance

(iii) Plot binomial distribution graphs. standard
deviation
(iv) Determine mean, variance and
standard deviation of a binomial
distribution.

(v) Solve problems involving
binomial distributions.

18

LEARNING AREA:
S4
Form 5

2 Understand and use the Use real-life situations and (i) Describe continuous random continuous
concept of normal computer software such as variables using set notations. random variable
distribution. statistical packages to explore
normal
the concept of normal (ii) Find probability of z-values for Discuss characteristics distribution
distributions.
standard normal distribution. of: standard normal
a) normal distribution distribution
graphs
b) standard normal z-value
distribution graphs. standardised
Z is called standardised variable
variable.

(iii) Convert random variable of Integration of normal
normal distributions, X, to distribution function to
standardised variable, Z. determine probability is
not required.
(iv) Represent probability of an
event using set notation.
(v) Determine probability of an event.
(vi) Solve problems involving
normal distributions.

19

LEARNING AREA:
AST2
Form 5

1 Understand and use Emphasise the use of the particle
the concept of following symbols: fixed point
displacement. s = displacement
displacement
v = velocity
a = acceleration distance
t = time velocity
where s, v and a are
acceleration
functions of time.
time interval
Use real-life examples, graphing (i) Identify direction of displacement Emphasise the difference
calculators and computer of a particle from a fixed point. between displacement and
software such as Geometer’s distance.
Sketchpad to explore
displacement. (ii) Determine displacement of a Discuss positive, negative
particle from a fixed point. and zero displacements.

Include the use of number
line.
(iii) Determine the total distance
travelled by a particle over a time
interval using graphical method.

20

Hsp add maths_f5

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