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Ministry of Education Malaysia

Integrated Curriculum for Secondary Schools CURRICULUM SPECIFICATIONS ADDITIONAL MATHEMATICS FORM 4

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 Kerajaan Persekutuan 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 A1. Functions 1 A2. Quadratic Equations 4 A3. Quadratic Functions 6 A4. Simultaneous Equations 8 A5. Indices and Logarithms 9 G1. Coordinate Geometry 11 S1. Statistics 15 T1. Circular Measures 18 C1. Differentiation 20 AST1. Solution of Triangles 24 ASS1. Index Number 26 PW1. Project Work 27 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 Director instruction for Science and Mathematics subjects. The policy to change the Curriculum Development Centre medium of instruction for the two subjects follows a phased implementation Ministry of Education schedule and is expected to be completed by 2008. The teaching and Malaysia learning of Additional Mathematics in English started in 2006.

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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

(ix) 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 9 use hardware and software to explore mathematics, and improving the teaching process, and hence enables the provision of effectively aimed lessons. 10 practise intrinsic mathematical values.

AIM ORGANISATION OF CONTENT The Additional Mathematics curriculum for secondary schools aims to The contents of the Form Four Additional Mathematics are arranged into develop pupils with in-depth mathematical knowledge and ability, so that two learning packages. They are the Core Package and the Elective Package. they are able to use mathematics responsibly and effectively in 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 • Algebra The he Additional Mathematics curriculum enables pupils to: • 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 3 develop the ability to think critically, creatively and to reason out mathematics. Topics in a particular teaching component are arranged logically, according to hierarchy whereby easier topics are learned earlier before 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,

(x) 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 EMPHASES IN TEACHING AND LEARNING examples of teaching and learning activities including methods, techniques, strategies and resources pertaining to the specific concepts or skills. These, The teaching and learning process in this curriculum emphasise concept however, are mere sample learning experiences and are not the only building and skills acquisition as well as the inculcation of good and positive activities to be used in the classrooms. Teachers are encouraged to look for values. Beside these, there are other elements that have to be taken into further examples, determine the teaching and learning strategies most account and carefully planned and infused into the teaching and learning of suitable for their pupils and provide appropriate teaching and learning the subject. The main elements focused in the teaching and learning of materials. Teachers should also make cross-references to other resources Additional Mathematics are as follows: such as the textbooks, and the Internet. Problem Solving The Learning Outcomes column defines clearly what pupils should be able to do after a learning experience. The intended outcomes state the In the Mathematics Curriculum, problem-solving skills and problem-solving mathematical abilities that should transpire from the activities conducted. strategies such as trial and improvement, drawing diagrams, tabulating data, Teachers are expected to look for indicators that pupils have acquired all of identifying polar, experiment/simulation, solving easier problems, finding the abilities stated. analogy and working backwards have already been learnt. Further strengthening of the above strategies must be carried out in the process of teaching and learning of Additional Mathematics. Besides routine questions,

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

(xii) 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. • Constructivism; 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

(xiii) COMPONENT SCHEME TITLE SCHEME

Algebraic Component A1. Functions A1.Functions Statistics Component A2.Quadratic Equations A2. Quadratic Equations A3.Quadratic Functions S1.Statistics C1. Differentiation A4.Simultaneous Equations A5.Indices and Logarithms A3. Quadratic Functions Trigonometric Component Project Work Geometric Component T1.Circular Measures A4. Simultaneous Equations G1.Coordinate Geometry

Calculus Component G1. Coordinate Geometry

C1.Differentiation T1. Circular Measures

A5. Indices and Logarithms Science and Technology Social Science Package Package ASS1. Index Number S1. Statistics AST1. Solution of Triangles

AST1. Solution of Triangles Project Work Project Work or ASS1. Index Number

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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 four pupils taking Additional Mathematics is required to carry • non-routine questions (which demand the application of problem- out a project work whereby the theme given is either based on the Science solving strategies). and Technology or Social Science package. Pupils however are allowed to 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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A1 LEARNING AREA: Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand the concept Use pictures, role-play and (i) Represent relations using: Discuss the idea of set and function of relations. computer software to introduce a) arrow diagrams, introduce set notation. relation the concept of relations. b) ordered pairs, object

c) graphs. image (ii) Identify domain, object, range image and range of a domain relation. codomain (iii) Classify a relation shown on map a mapped diagram as: ordered pair one-to-one, many-to-one, arrow diagram one-to-many or many-to-many relation. 2 Understand the concept (i) Recognise functions as a Represent functions using arrow of functions. special relation. diagrams, ordered pairs or graphs, e.g. (ii) Express functions using f : x → 2x notation

function notation. f ()x = 2x “ f : x → 2x ” is read as “function f maps x to 2x”. “ f (x)= 2x ” is read as “2x is the image of x under the function f ”. (iii) Determine domain, object, Include examples of functions image and range of a that are not mathematically function. based.

1

A1 LEARNING AREA: Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… Use graphing calculators or (iv) Determine the image of a Examples of functions include computer software to explore function given the object algebraic (linear and quadratic), the image of functions. and vice versa. trigonometric and absolute value. Define and sketch absolute value functions. 3 Understand the concept Use arrow diagrams or (i) Determine composition of Involve algebraic functions only. inverse of composite functions. algebraic method to determine two functions. mapping composite functions. (ii) Determine the image of Images of composite functions composite function composite functions given include a range of values. (Limit the object and vice versa. to linear composite functions). (iii) Determine one of the functions in a given composite function given

the other related function.

4 Understand the concept (i) Find the object by inverse Limit to algebraic functions. of inverse functions. mapping given its image Exclude inverse of composite and function. functions. Use sketches of graphs to show (ii) Determine inverse functions the relationship between a using algebra. function and its inverse. (iii) Determine and state the Emphasise that inverse of a condition for existence of an function is not necessarily a inverse function. function.

2 LEARNING AREA: A2 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand the concept Use graphing calculators or (i) Recognise a quadratic . quadratic equation of quadratic equations and computer software such as the equation and express it in general form Geometer’s Sketchpad and general form. their roots. root spreadsheet to explore the concept of quadratic equations. substitution (ii) Determine whether a given Questions for 1.2(b) are given in inspection value is the root of a the form of; a and b are trial and improvement quadratic equation by: numerical values. method a) substitution, b) inspection.

(iii) Determine roots of quadratic equations by trial and improvement method.

2 Understand the concept (i) Determine the roots of a Discuss when factorisation of quadratic equations. quadratic equation by: (x – p)(x – q) = 0, completing the square a) factorisation, hence x – p = 0 or x – q = 0. b) completing the square, Include cases when p = q. c) using the formula. Derivation of formula for 2.1c is not required.

3 LEARNING AREA: A2 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… (ii) Form a quadratic equation If x = p and x = q are the roots, from given roots. then the quadratic equation is (x – p)(x – q) = 0, that is x2 – (p + q)x + pq = 0. Involve the use of: –b c α + β = and αβ = a a where α and β are roots of the quadratic equation ax 2 + bx + c = 0 3 Understand and use the (i) Determine types of roots of b 2 − 4ac > 0 discriminant conditions for quadratic quadratic equations from real roots b 2 − 4ac = 0 equations to have the value of b 2 − 4ac . a) two different roots; b 2 − 4ac < 0 b) two equal roots; Explain that “no roots” means c) no roots. “no real roots”. (ii) Solve problems involving b 2 − 4ac in quadratic equations to: a) find an unknown value, b) derive a relation.

4 LEARNING AREA: A3 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand the concept Use graphing calculators or (i) Recognise quadratic quadratic function of quadratic functions and computer software such as functions. tabulated values Geometer’s Sketchpad to their graphs. axis of symmetry explore the graphs of quadratic functions. parabola Use examples of everyday (ii) Plot quadratic function maximum point situations to introduce graphs graphs: minimum point of quadratic functions. a) based on given completing the square tabulated values, axis of symmetry b) by tabulating values

based on given functions. (iii) Recognise shapes of graphs Discuss cases where of quadratic functions. a > 0 and a < 0 for f(x) = ax2 + bx + c

(iv) Relate the position of quadratic function graphs with types of roots for

f (x)= 0 .

2 Find the maximum and Use graphing calculators or (i) Determine the maximum or minimum values of dynamic geometry software minimum value of a such as Geometer’s Sketchpad quadratic function by quadratic functions. to explore the graphs of completing the square. quadratic functions.

5 LEARNING AREA: A3 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 3 Sketch graphs of Use graphing calculators or (i) Sketch quadratic function Emphasise the marking of sketch quadratic functions. dynamic geometry software graphs by determining the maximum or minimum point and intersection such as the Geometer’s maximum or minimum two other points on the graphs vertical line Sketchpad to reinforce the point and two other points. drawn or by finding the axis of understanding of graphs of symmetry and the intersection quadratic inequality quadratic functions. with the y-axis. range Determine other points by number line finding the intersection with the x-axis (if it exists). 4 Understand and use the Use graphing calculators or (i) Determine the ranges of Emphasise on sketching graphs concept of quadratic dynamic geometry software values of x that satisfies and use of number lines when inequalities. such as the Geometer’s quadratic inequalities. necessary. Sketchpad to explore the concept of quadratic inequalities.

6 LEARNING AREA: A4 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Solve simultaneous Use graphing calculators or (i) Solve simultaneous Limit non-linear equations up to simultaneous equations in two unknowns: dynamic geometry software equations using the second degree only. equations one linear equation and one such as the Geometer’s substitution method. intersection Sketchpad to explore the non-linear equation. substitution method concept of simultaneous equations.

Use examples in real-life (ii) Solve simultaneous situations such as area, equations involving real-life perimeter and others. situations.

7 LEARNING AREA: A5 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand and use the Use examples of real-life (i) Find the values of numbers Discuss zero index and negative base concept of indices and laws situations to introduce the given in the form of: indices. integer indices concept of indices. of indices to solve a) integer indices, fractional indices problems. b) fractional indices. index form Use computer software such as (ii) Use laws of indices to find raised to a power the spreadsheet to enhance the the values of numbers in law of indices understanding of indices. index form that are

multiplied, divided or raised to a power. (iii) Use laws of indices to simplify algebraic expressions.

2 Understand and use the Use scientific calculators to (i) Express equation in index Explain definition of logarithm. index form enhance the understanding of form to logarithm form and concept of logarithms and N = ax; loga N = x with a > 0, logarithm form the concept of logarithms. vice versa. laws of logarithms to solve a ≠ 1. logarithm problems. Emphasise that: undefined loga 1 = 0; loga a = 1. (ii) Find logarithm of a number. Emphasise that: a) logarithm of negative numbers is undefined; b) logarithm of zero is undefined. Discuss cases where the given number is in: a) index form, b) numerical form.

8 LEARNING AREA: A5 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… (iii) Find logarithm of numbers Discuss laws of logarithms. by using laws of logarithms.

(iv) Simplify logarithmic expressions to the simplest form.

3 Understand and use the (i) Find the logarithm of a Discuss: change of base of number by changing the 1 logarithms to solve base of the logarithm to a log a b = log a problems. suitable base. b

(ii) Solve problems involving the change of base and laws of logarithms.

4 Solve equations (i) Solve equations involving Equations that involve indices involving indices and indices. and logarithms are limited to logarithms. equations with single solution only. Solve equations involving indices by: a) comparison of indices and bases, b) using logarithms. (ii) Solve equations involving logarithms.

9 LEARNING AREA: G1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Find distance between Use examples of real-life (i) Find the distance between Use the Pythagoras’ Theorem to distance two points. situations to find the distance two points using formula. find the formula for distance midpoint between two points. between two points. 2 Understand the concept (i) Find the midpoint of two Limit to cases where m and n are coordinate of division of line segments. given points. positive. ratio Derivation of the formula ⎛ nx + mx ny + my ⎞ ⎜ 1 2 , 1 2 ⎟ ⎝ m + n m + n ⎠ is not required. (ii) Find the coordinates of a point that divides a line according to a given ratio

m : n. 3 Find areas of polygons. Use dynamic geometry (i) Find the area of a triangle Limit to numerical values. area software such as the based on the area of specific Emphasise the relationship polygon Geometer’s Sketchpad to geometrical shapes. between the sign of the value for geometrical shape explore the concept of areas of area obtained with the order of quadrilateral polygons. the vertices used. (ii) Find the area of a triangle Derivation of the formula: vertex by using formula. 1 x x x x ⎛ x y + x y + x y − x y ⎞ vertices Use 1 2 3 4 1 ⎜ 1 1 2 3 3 1 2 1 ⎟ 2 ⎜ ⎟ clockwise 2 y1 y2 y3 y4 ⎝− x3 y2 − x1 y3 ⎠ anticlockwise for substitution of coordinates is not required. modulus into the formula. Emphasise that when the area of polygon is zero, the given points collinear are collinear. (iii) Find the area of a quadrilateral by using formula.

10 LEARNING AREA: G1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 4 Understand and use the Use dynamic geometry (i) Determine the x-intercept x-intercept concept of equation of a software such as the and the y-intercept of a line. y-intercept Geometer’s Sketchpad to straight line. gradient explore the concept of equation of a straight line. (ii) Find the gradient of a straight line that passes through two points.

(iii) Find the gradient of a straight line using the x-intercept and y-intercept.

(iv) Find the equation of a Answers for learning outcomes straight line straight line given: 4.4(a) and 4.4(b) must be stated general form in the simplest form. a) gradient and one point, intersection Involve changing the equation b) points, gradient form into gradient and intercept form. c) x-intercept and intercept form y-intercept.

(v) Find the gradient and the intercepts of a straight line given the equation.

(vi) Change the equation of a straight line to the general form.

(vii) Find the point of intersection of two lines.

11 LEARNING AREA: G1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 5 Understand and use the Use examples of real-life (i) Determine whether two Emphasise that for parallel lines: parallel concept of parallel and situations to explore parallel straight lines are parallel m1 = m2 . perpendicular perpendicular lines. and perpendicular lines. when the gradients of both

lines are known and vice versa. Use graphic calculator and (ii) Find the equation of a Emphasise that for perpendicular dynamic geometry software straight line that passes lines such as Geometer’s Sketchpad through a fixed point and m m = −1. to explore the concept of parallel to a given line. 1 2 parallel and perpendicular Derivation of m1m2 = −1 is not lines. required.

(iii) Determine whether two straight lines are perpendicular when the gradients of both lines are known and vice versa.

(iv) Determine the equation of a straight line that passes through a fixed point and perpendicular to a given line.

(v) Solve problems involving equations of straight lines.

12 LEARNING AREA: G1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 6 Understand and use the Use examples of real-life (i) Find the equation of locus equation of locus concept of equation of locus situations to explore equation that satisfies the condition moving point of locus involving distance if: involving distance between loci two points. between two points. a) the distance of a Use graphing calculators and moving point from a dynamic geometry software fixed point is constant, such as the Geometer’s b) the ratio of the distances Sketchpad to explore the of a moving point from concept of parallel and two fixed points is perpendicular lines. constant. (ii) Solve problems involving loci.

13 LEARNING AREA: S1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand and use the Use scientific calculators, (i) Calculate the mean of Discuss grouped data and measure of central concept of measures of graphing calculators and ungrouped data. ungrouped data. tendency central tendency to solve spreadsheets to explore mean measures of central tendency. problems. mode Pupils collect data from real- (ii) Determine the mode of median life situations to investigate ungrouped data. ungrouped data measures of central tendency. frequency (iii) Determine the median of distribution table ungrouped data. modal class (iv) Determine the modal class Involve uniform class intervals uniform class interval of grouped data from only. histogram frequency distribution tables.

(v) Find the mode from histograms.

(vi) Calculate the mean of Derivation of the median midpoint grouped data. formula is not required. cumulative frequency

(vii) Calculate the median of distribution table grouped data from ogive cumulative frequency distribution tables.

(viii) Estimate the median of Ogive is also known as grouped data from an ogive. cumulative frequency curve.

14 LEARNING AREA: S1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… (ix) Determine the effects on range mode, median and mean for interquartile a set of data when: measures of a) each data is changed dispersion uniformly, extreme value b) extreme values exist, lower boundary c) certain data is added or removed. (x) Determine the most suitable Involve grouped and ungrouped measure of central tendency data for given data.

2 Understand and use the (i) Find the range of ungrouped concept of measures of data. dispersion to solve problems. (ii) Find the interquartile range of ungrouped data.

(iii) Find the range of grouped data.

(iv) Find the interquartile range Determine the upper and lower standard deviation of grouped data from the quartiles by using the first class interval cumulative frequency table. principle. upper quartile lower quartile

15 LEARNING AREA: S1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… (v) Determine the interquartile range of grouped data from an ogive. (vi) Determine the variance of: variance a) ungrouped data, b) grouped data. (vii) Determine the standard deviation of: a) ungrouped data, b) grouped data. (viii) Determine the effects on range, interquartile range, variance and standard deviation for a set of data when: a) each data is changed uniformly, b) extreme values exist, c) certain data is added or removed. (ix) Compare measures of Emphasise that comparison central tendency and between two sets of data using dispersion between two sets only measures of central of data. tendency is not sufficient.

16 LEARNING AREA: T1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand the concept Use dynamic geometry (i) Convert measurements in Discuss the definition of one radian of radian. software such as the radians to degrees and vice radian. degree Geometer’s Sketchpad to versa. “rad” is the abbreviation of explore the concept of circular radian. measure. Include measurements in radians expressed in terms of π.

2 Understand and use the Use examples of real-life (i) Determine: length of arc concept of length of arc of a situations to explore circular a) length of arc, angle subtended measure. circle to solve problems. b) radius, c) angle subtended at the centre of a circle based on given information. (ii) Find perimeter of segments circle of circles. perimeter segment (iii) Solve problems involving lengths of arcs.

17 LEARNING AREA: T1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 3 Understand and use the (i) Determine the: area concept of area of sector of a) area of sector, sector a circle to solve problems. b) radius, c) angle subtended at the centre of a circle based on given information.

(ii) Find the area of segments of circles.

(iii) Solve problems involving areas of sectors.

18 LEARNING AREA: C1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand and use the Use graphing calculators or (i) Determine the value of a Idea of limit to a function can be limit concept of gradients of dynamic geometry software function when its variable illustrated using graphs. tangent such as Geometer’s Sketchpad approaches a certain value. curve and differentiation. The concept of first derivative of first derivative to explore the concept of a function is explained as a gradient differentiation. tangent to a curve and can be illustrated using graphs. induction (ii) Find the gradient of a chord curve joining two points on a fixed point curve.

(iii) Find the first derivative of a Limit to y = ax n ; function y = f (x), as the a, n are constants, n = 1, 2, 3… gradient of tangent to its graph. (iv) Find the first derivative of Notation of f '(x) is equivalent polynomials using the first dy principle. to when y = f (x), dx

f '(x) read as “f prime of x”. (v) Deduce the formula for first derivative of the function y = f (x) by induction.

2 Understand and use the (i) Determine the first concept of first derivative of derivative of the function polynomial functions to y = ax n using formula. solve problems.

19 LEARNING AREA: C1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… (ii) Determine value of the first derivative of the function n y = ax for a given value of x. (iii) Determine first derivative of a function involving: a) addition, or b) subtraction of algebraic terms. (iv) Determine the first product derivative of a product of quotient two polynomials. composite (v) Determine the first function derivative of a quotient of chain rule two polynomials. normal (vi) Determine the first derivative of composite function using chain rule.

(vii) Determine the gradient of Limit cases in Learning tangent at a point on a Outcomes 2.7 through 2.9 to curve. rules introduced in 2.4 through 2.6. (viii) Determine the equation of tangent at a point on a curve.

20 LEARNING AREA: C1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… (ix) Determine the equation of normal at a point on a curve. 3 Understand and use the Use graphing calculators or (i) Determine coordinates of Emphasise the use of first turning point concept of maximum and dynamic geometry software to turning points of a curve. derivative to determine the minimum point explore the concept of turning points. minimum values to solve maximum point problems. maximum and minimum values. (ii) Determine whether a Exclude points of inflexion. turning point is a maximum or a minimum point.

(iii) Solve problems involving Limit problems to two variables maximum or minimum only. values.

4 Understand and use the Use graphing calculators with (i) Determine rates of change Limit problems to 3 variables rates of change concept of rates of change computer base ranger to for related quantities. only. to solve problems. explore the concept of rates of change.

5 Understand and use the (i) Determine small changes in Exclude cases involving approximation concept of small changes quantities. percentage change. and approximations to solve problems. (ii) Determine approximate values using differentiation.

21 LEARNING AREA: C1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… Understand and use the (i) Determine the second second derivative 6 2 concept of second derivative of y = f (x). d y d dy Introduce 2 as or derivative to solve dx dx dx problems. d f”(x) = (f’(x)) dx (ii) Determine whether a turning point is maximum or minimum point of a

curve using the second derivative.

22 LEARNING AREA: AST1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand and use the Use dynamic geometry (i) Verify sine rule. sine rule concept of sine rule to solve software such as the acute-angled triangle problems. Geometer’s Sketchpad to explore the sine rule. Use examples of real-life (ii) Use sine rule to find Include obtuse-angled triangles. obtuse-angled triangle situations to explore the sine unknown sides or angles of ambiguous rule. a triangle.

(iii) Find the unknown sides and angles of a triangle involving ambiguous case.

(iv) Solve problems involving the sine rule.

2 Understand and use the Use dynamic geometry (i) Verify cosine rule. Include obtuse-angled triangles cosine rule concept of cosine rule to software such as the solve problems. Geometer’s Sketchpad to explore the cosine rule. Use examples of real-life (ii) Use cosine rule to find situations to explore the cosine unknown sides or angles of rule. a triangle.

(iii) Solve problems involving cosine rule.

(iv) Solve problems involving sine and cosine rules.

23 LEARNING AREA: AST1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 3 Understand and use the Use dynamic geometry (i) Find the areas of triangles formula for areas of software such as the using the formula 1 triangles to solve problems. Geometer’s Sketchpad to ab sin C or its equivalent. explore the concept of areas of 2 triangles.

Use examples of real-life (ii) Solve problems involving three-dimensional situations to explore areas of three-dimensional objects. object triangles.

24 LEARNING AREA: ASS1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Understand and use the Use examples of real-life (i) Calculate index number. Explain index number. index number concept of index number to situations to explore index price index numbers. solve problems. quantity at base time

(ii) Calculate price index. quantity at specific time

(iii) Find Q0 or Q 1 given Q0 = Quantity at base time. relevant information. Q1 = Quantity at specific time.

2 Understand and use the Use examples of real-life (i) Calculate composite index. Explain weightage and concept of composite index situations to explore composite composite index. to solve problems. index.

(ii) Find index number or weightage given relevant information.

(iii) Solve problems involving index number and composite index.

25 LEARNING AREA: PW1 Form 4

LEARNING OBJECTIVES SUGGESTED TEACHING LEARNING OUTCOMES POINTS TO NOTE VOCABULARY Pupils will be taught to… AND LEARNING ACTIVITIES Pupils will be able to… 1 Carry out project work. Use scientific calculators, (i) Define the Emphasise the use of Polya’s conjecture graphing calculators or problem/situation to be four-step problem-solving systematic computer software to carry out studied. process. critical evaluation project work. Pupils are allowed to carry out (ii) State relevant conjectures. Use at least two problem-solving mathematical project work in groups but strategies. reasoning written reports must be done justification individually. conclusion

(iii) Use problem-solving generalisation strategies to solve problems. mathematical communication Pupils should be given (iv) Interpret and discuss results. rubric opportunity to give oral presentation of their project work. (v) Draw conclusions and/or make generalisations based on critical evaluation of results.

(vi) Present systematic and Emphasise reasoning and comprehensive written effective mathematical reports. communication.

26 LEARNING AREA: PW1 Form 4

CONTRIBUTORS

Advisor Mahzan bin Bakar AMP Director Curriculum Development Centre

Zulkifly bin Mohd Wazir Deputy Director Curriculum Development Centre

Editorial Cheah Eng Joo Principal Assistant Director Advisors (Head of Science and Mathematics Department) Curriculum Development Centre

Abdul Wahab bin Ibrahim Assistant Director (Head of Mathematics Unit) Curriculum Development Centre

Hj. Ali Ab. Ghani Principal Assistant Director (Head of Language Department) Curriculum Development Centre

Editor Rosita Mat Zain Assistant Director Curriculum Development Centre

27 LEARNING AREA: PW1 Form 4

WRITERS

Abdul Wahab Ibrahim Rosita Mat Zain Susilawati Ehsan Curriculum Development Centre Curriculum Development Centre Curriculum Development Centre

Zurina Zainal Abidin Wong Sui Yong Mazlan bin Awi Curriculum Development Centre Curriculum Development Centre Curriculum Development Centre

Lau Choi Fong Dr. Pumadevi a/p Sivasubramaniam Bibi Kismete Kabul Khan SMK Hulu Kelang Maktab Perguruan Raja Melewar SMK Dr. Megat Khas Hulu Kelang, Selangor Seremban, Negeri Sembilan Ipoh, Perak Mak Sai Mooi SMK Jenjarom Krishian a/l Gobal Yam Weng Hoong Jenjarom, Selangor SMK Kampong Pasir Putih SM Teknik Butterworth Ipoh, Perak Pulau Pinang

Ahmad Zamri Aziz Roslie Ahmad SM Sains Pasir Puteh SMK Sains Seremban Pasir Puteh, Kelantan Seremban, Negeri Sembilan

LAYOUT AND ILLUSTRATION

Rosita Mat Zain Mazlan bin Awi Mohd Razif Hashim Curriculum Development Centre Curriculum Development Centre Curriculum Development Centre

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