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Ministry of Education Malaysia Integrated Curriculum for Secondary Schools CURRICULUM SPECIFICATIONSADDITIONAL MATHEMATICS FORM 5 Curriculum Development Centre Ministry of Education Malaysia 2006
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CONTENTS PageRUKUNEGARA (iv)National Philosophy of Education (v)Preface (vii)Introduction (ix)A6. Progressions 1A7. Linear Law 4C2. Integration 5G2. Vectors 7T2. Trigonometric Functions 11S2. Permutations and Combinations 14S3. Probability 16S4. Probability Distributions 18AST2. Motion Along a Straight Line 20ASS2. Linear Programming 23PW2. Project Work 25
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RUKUNEGARA DECLARATIONOUR 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 theseends guided by these principles:• BELIEF IN GOD• LOYALTY TO KING AND COUNTRY• UPHOLDING THE CONSTITUTION• RULE OF LAW• GOOD BEHAVIOUR AND MORALITY
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Education in Malaysia is an ongoing effort towards further developing the potential of individuals in a holistic and integratedmanner so as to produce individuals who are intellectually, spiritually, emotionally andphysically balanced and harmonious, based on a firm belief in God. Such an effort isdesigned to produce Malaysian citizens who are knowledgeable and competent, whopossess high moral standards, and who areresponsible and capable of achieving a highlevel of personal well-being as well as being able to contribute to the betterment of the family, the society and the nation at large.
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PREFACE The use of technology in the teaching and learning of Additional Mathematics is greatly emphasised. Additional Mathematics taught inScience and technology plays a critical role in realising Malaysia’s English, coupled with the use of ICT, provide greater opportunities foraspiration to become a developed nation. Since mathematics is instrumental pupils to improve their knowledge and skills in mathematics because of thein the development of scientific and technological knowledge, the provision richness of resources and repositories of knowledge in English. Our pupilsof quality mathematics education from an early age in the education process will be able to interact with pupils from other countries, improve theiris thus important. The Malaysian school curriculum offers three proficiency in English; and thus make the learning of mathematics moremathematics education programs, namely Mathematics for primary schools, interesting and exciting.Mathematics and Additional Mathematics for secondary schools. The development of this Additional Mathematics Curriculum SpecificationsThe Malaysian school mathematics curriculum aims to develop is the work of many individuals and experts in the field. On behalf of themathematical knowledge, competency and inculcate positive attitudes Curriculum Development Centre, I would like to express much gratitude andtowards mathematics among pupils. While the Mathematics curriculum appreciation to those who have contributed in one way or another towardsprepares pupils to cope with daily life challenges, the Additional this initiative.Mathematics curriculum provides an exposure to the level of mathematicsappropriate for science and technology related careers. As with othersubjects in the secondary school curriculum, Additional Mathematics aimsto inculcate noble values and love for the nation in the development of aholistic person, who in turn will be able to contribute to the harmony andprosperity 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 ofinstruction for Science and Mathematics subjects. The policy to change the Directormedium of instruction for the two subjects follows a phased implementation Curriculum Development Centreschedule and is expected to be completed by 2008. The teaching and Ministry of Educationlearning of Additional Mathematics in English started in 2006. Malaysia (vii)
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INTRODUCTION are also stressed in the process of learning Additional Mathematics. WhenA well-informed and knowledgeable society, well versed in the use of pupils explain concepts and their work, they are guided in the use of correctMathematics to cope with daily life challenges is integral to realising the and precise mathematical terms and sentences. Emphasis on Mathematicalnation’s aspiration to become an industrialised nation. Thus, efforts are taken communications develops pupils’ ability in interpreting matters intoto 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 andcommunicate 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 aredemands of a progressive nation. As such, this field assumes its role as the provided and complex calculations are made easier with the use ofdriving force behind various developments in science and technology. In line calculators.with the nation’s objective to create a knowledge-based economy, the skillsof Research & Development in mathematics is nurtured and developed at Project work, compalsory in Additional Mathematics provides opportunitiesschool 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 ofAdditional Mathematics is an elective subject in secondary schools, which mathematical problems, which activates their minds, makes the learning ofcaters to the needs of pupils who are inclined towards Science and mathematics more meaningful, and enables pupils to apply mathematicalTechnology. 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 throughoutcontents of the Mathematics curriculum. New areas of mathematics the teaching and learning process; contribute to the moulding of characterintroduced in the Additional Mathematics curriculum are in keeping with and the inculcation of positive attitudes towards mathematics. Together withnew developments in Mathematics. Emphasis is placed on the heuristics of these, moral values are also introduced in context throughout the teachingproblem 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 differentsituations. Assessment, in the form of tests and examinations helps to gauge pupils’ achievement. Assessments in Additional Mathematics include aspects suchThe Additional Mathematics syllabus emphasises understanding of concepts as understanding of concepts, mastery of skills and non-routine questionsand mastery of related skills with problem solving as the main focus in the that demand the application of problem-solving strategies. The use of goodteaching and learning process. Skills of communication through mathematics assessment data from a variety of sources provides valuable information on (ix)
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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 inimproving the teaching process, and hence enables the provision of 9 use hardware and software to explore mathematics, andeffectively aimed lessons. 10 practise intrinsic mathematical values.AIM ORGANISATION OF CONTENTThe Additional Mathematics curriculum for secondary schools aims todevelop pupils with in-depth mathematical knowledge and ability, so that The contents of the Form Five Additional Mathematics are arranged into twothey 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 topicsstudies and embark on science and technology related careers. arranged under five components, that is:OBJECTIVES • GeometryThe Additional Mathematics curriculum enables pupils to: • Algebra • Calculus1 widen their ability in the fields of number, shape and relationship as well as to gain knowledge in calculus, vector and linear • Trigonometry programming, • Statistics2 enhance problem-solving skills, Each teaching component includes topics related to one branch of mathematics. Topics in a particular teaching component are arranged3 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 Application5 relate the learning of Mathematics to daily activities and careers, Package. Pupils need to choose one Elective Package according to their6 use the knowledge and skills of Mathematics to interpret and solve inclination in their future field. real-life problems, (x)
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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 significantformat which helps teachers to teach a particular topic effectively. The aspects of mathematical concepts and skills to be taught. This columncontents 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 thelearning units that are stated in the Learning Objectives column. These use of correct terminologies and these needs to be systematically introducedLearning Objectives are arranged according to hierarchy from easy to the to pupils in various contexts so as to enable pupils to understand themore abstract concepts. meanings of the terms and learn to use them appropriately.The Suggested Teaching and Learning Activities column lists someexamples of teaching and learning activities including methods, techniques, EMPHASES IN TEACHING AND LEARNINGstrategies 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 conceptactivities to be used in the classrooms. Teachers are encouraged to look for building and skills acquisition as well as the inculcation of good and positivefurther examples, determine the teaching and learning strategies most values. Beside these, there are other elements that have to be taken intosuitable for their pupils and provide appropriate teaching and learning account and carefully planned and infused into the teaching and learning ofmaterials. Teachers should also make cross-references to other resources the subject. The main elements focused in the teaching and learning ofsuch as the textbooks, and the Internet. Additional Mathematics are as follows:The Learning Outcomes column defines clearly what pupils should be able Problem Solvingto do after a learning experience. The intended outcomes state themathematical abilities that should transpire from the activities conducted. In the Mathematics Curriculum, problem-solving skills and problem-solvingTeachers 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 (xi)
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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 worldpupils 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 inmultiple problem-solving strategies. the process of seeking solutions. Pupils at all levels have to be trained toCommunication in Mathematics investigate their predictions or guesses by using concrete materials, calculators, computers, mathematical representations and others. LogicalThe skills of communication in mathematics are also stressed in the teaching reasoning has to be absorbed in the teaching of mathematics so that pupilsand learning of Additional Mathematics. Communication is an essential can recognise, construct and evaluate predictions and mathematicalmeans of sharing ideas and clarifying the understanding of mathematics. arguments.Through communication, mathematical ideas become the object ofreflection, discussion and modification. Communicational skills in Making Connectionsmathematics include reading, writing, listening and speaking. Through In the teaching and learning of Additional Mathematics, opportunities foreffective mathematical communication, pupils will become efficient in making connections must be created so that pupils can link conceptualproblem-solving and be able to explain their conceptual understanding and knowledge to procedural knowledge and relate topics within mathematicsmathematical skills to their peers and teachers. Therefore, through the and other learning areas in general.teaching and learning process, teachers should frequently createopportunities for pupils to read, write and discuss ideas in which the The Additional Mathematics curriculum covers several areas of mathematicslanguage of mathematics becomes natural and this can only be done through such as Geometry, Algebra, Trigonometry, Statistics and Calculus. Withoutsuitable 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 ablePupils who have developed the skills to communicate mathematically will to see mathematics as an integrated whole rather than a string ofbecome more inquisitive and, in the process, gain confidence. Emphasis on unconnected ideas. When mathematical ideas and the curriculum aremathematical communications will develop pupils ability in interpreting connected to real-life within or outside the classroom, pupils will becomecertain matters into mathematical models or vice versa. The process of more conscious of the importance and significance of mathematics. Theyanalytical and systematic reasoning helps pupils to reinforce and strengthen will also be able to use mathematics contextually in different learning areastheir knowledge and understanding of mathematics to a deeper level. and in real-life situations.Reasoning The Use of TechnologyLogical Reasoning or thinking is the basis for understanding and solving The use of ICT and other technologies is encouraged in the teaching andmathematical problems. The development of mathematical reasoning is learning of Additional Mathematics. Technologies help pupils by increasingclosely related to the intellectual and communicative development of pupils. their understanding of abstract concepts, providing visual input and makingEmphasis on logical thinking, during teaching and learning activities opens (xii)
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complex calculation easier. Calculators, computers, software related to where it provides pupils with a better understanding and appreciation ofeducation, 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 stimulatingtherefore 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 moreeffectively. 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 mathematicsdemand changes to the way mathematics is taught in the classroom. TEACHING SCHEMESEffective use of teaching resources is vital in forming the understanding ofmathematical concepts. Teachers should use real or concrete materials to To facilitate the teaching and learning process, two types of annual schemeshelp 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 firstcooperation. The teaching and learning materials used should contain self- before proceeding to other components. This scheme presents the Additionaldiagnostic 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 teachersIn order to assist pupils develop positive attitudes and personalities, the to introduce the Algebraic and Geometrical topics before introducing themathematical 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 moralvalues can be cultivated through suitable contexts. Learning in groups for Between these two teaching schemes, teachers are free to choose a moreexample can help pupils develop social skills, encourage cooperation and suitable scheme based on their pupils’ previous knowledge, learning stylebuild 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 certaintopics. Brief historical anecdotes related to aspects of mathematics andfamous mathematicians associated with particular learning areas are alsoincorporated into the curriculum. It should be presented at appropriate points (xiii)
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COMPONENT SCHEME TITLE SCHEMEAlgebraic Component A6. ProgressionsA6.ProgressionsA7.Linear Law Trigonometric Component C2. Integration T2. Trigonometric Functions PW2. Project WorkCalculus Component A7. Linear LawC2.Integration Statistics Component S2. Permutations and G2. Vectors CombinationsGeometric Component S3 ProbabilityG2.Vectors T2. Trigonometric Functions S4 Probability Distributions S2. Permutations and Combinations S3. ProbabilityScience and Technology Social Science PackagePackage ASS2. Linear ProgrammingAST2. Motion Along a Straight S4. Probability Distributions Line AST2. Motion Along a Straight LinePW2. Project Work PW2. Project Work or ASS2. Linear Programming (xiv)
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PROJECT WORK EVALUATIONProject Work is a new element in the Additional Mathematics curriculum. It Continual and varied forms of evaluation is an important part of the teachingis a mean of giving pupils the opportunity to transfer the understanding of and learning process. It not only provides feedback to pupils on theirmathematical concepts and skills learnt into situations outside the classroom. progress but also enable teachers to correct their pupils’ misconceptions andThrough Project Work, pupils are to pursue solutions to given tasks through weaknesses. Based on evaluation outcomes, teachers can take correctiveactivities such as questioning, discussing, debating ideas, collecting and measures such as conducting remedial or enrichment activities in order toanalyzing data, investigating and also producing written report. With improve pupils’ performances and also strive to improve their own teachingregards to this, suitable tasks containing non-routine problems must skills. Schools should also design effective internal programs to assist pupilstherefore be administered to pupils. However, in the process of seeking in improving their performances. The Additional Mathematics Curriculumsolutions to the tasks given, a demonstration of good reasoning and effective emphasis evaluation, which among other things must include the followingmathematical communication should be rewarded even more than the pupils aspects:abilities to find correct answers. • concept understandings and mastery of skills; andEvery form five pupils taking Additional Mathematics is required to carryout 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 becarried out in the second semester after pupils have mastered the first fewchapters. The tasks given must therefore be based on chapters that havealready been learnt and pupils are expected to complete it within the durationof three weeks. Project work can be done in groups or individually but eachpupil is expected to submit an individually written report which include thefollowing: • title/topic; • background or introduction; • method/strategy/procedure; • finding; • discussion/solution; and • conclusion/generalisation. (xv)
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LEARNING AREA: A6LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils 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 sequencethe concept of arithmetic situations, scientific or graphing arithmetic progressions. introduce arithmetic and seriesprogression. 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 arithmetic progressions, c) the value of n, given the sum of the first n terms of arithmetic progressions, (v) Solve problems involving Include problems arithmetic progressions. involving real-life situations. 1
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LEARNING AREA: A6LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the Use examples from real-life (i) Identify characteristics of Include examples in geometricconcept of geometric situations, scientific or graphing geometric progressions. algebraic form. progressionprogression. 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 geometric progressions, c) the value of n, given the sum of the first n terms of geometric progressions, 2
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LEARNING AREA: A6LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… 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
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LEARNING AREA: A7LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Use examples from real-life (i) Draw lines of best fit by Limit data to linear line of best fitconcept 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 tolinear relations. linear form. (ii) Determine values of constants of non-linear relations given: a) lines of best fit, b) data. (iii) Obtain information from: a) lines of best fit, b) equations of lines of best fit. 4
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LEARNING AREA: C2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Use computer software such as (i) Determine integrals by reversing Emphasise constant of integrationconcept of indefinite Geometer’s Sketchpad to differentiation. integration. integralintegral. 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
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LEARNING AREA: C2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the Use scientific or graphing (i) Find definite integrals of Include definite integralconcept 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
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LEARNING AREA: G2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…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
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LEARNING AREA: G2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (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 vector2 Understand and use the Use real-life situations and (i) Determine the resultant vector of polygon lawconcept 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
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LEARNING AREA: G2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (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
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LEARNING AREA: G2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (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
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LEARNING AREA: T2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand the concept Use dynamic computer (i) Represent in a Cartesian plane, Cartesian planeof positive and negative software such as Geometer’s angles greater than 360o or 2π rotating rayangles measured in degrees Sketchpad to explore angles in radians for: Cartesian plane. a) positive angles, positive angleand radians. b) negative angles. negative angle clockwise2 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 anticlockwisesix trigonometric functionsof 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
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LEARNING AREA: T2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…3 Understand and use Use examples from real-life (i) Draw and sketch graphs of Use angles in modulusgraphs of sine, cosine and situations to introduce graphs trigonometric functions: a) degrees domaintangent 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
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LEARNING AREA: T2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…4 Understand and use Use scientific or graphing (i) Prove basic identities: Basic identities are also basic identitybasic 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 formulaaddition 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
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LEARNING AREA: S2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…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
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LEARNING AREA: S2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (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 combinationconcept 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
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LEARNING AREA: S3LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Use real-life situations to (i) Describe the sample space of an Use set notations. experimentconcept 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
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LEARNING AREA: S3LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the Use manipulative materials and (i) Determine whether two events are Include events that are mutuallyconcept of probability of graphing calculators to explore mutually exclusive. mutually exclusive and exclusivemutually 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
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LEARNING AREA: S4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Use real-life situations to (i) List all possible values of a discrete randomconcept of binomial introduce the concept of discrete random variable. variabledistribution. 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
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LEARNING AREA: S4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the Use real-life situations and (i) Describe continuous random continuousconcept of normal computer software such as variables using set notations. random variabledistribution. 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
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LEARNING AREA: AST2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use Emphasise the use of the particlethe concept of following symbols: fixed pointdisplacement. 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
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LEARNING AREA: AST2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the Use real-life examples, graphing (i) Determine velocity function of a Emphasise velocity as the instantaneousconcept of velocity. calculators and dynamic particle by differentiation. rate of change of velocity computer software such as displacement. velocity Geometer’s Sketchpad to explore function Include graphs of velocity the concept of velocity. uniform velocity functions. rate of change (ii) Determine instantaneous velocity Discuss: of a particle. a) uniform velocity maximum displacement b) zero instantaneous velocity stationary c) positive velocity d) negative velocity. (iii) Determine displacement of a particle from velocity function by integration. 21
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LEARNING AREA: AST2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…3 Understand and use the Use real-life examples and (i) Determine acceleration function Emphasise acceleration as maximumconcept of acceleration. computer software such as of a particle by differentiation. the rate of change of velocity Geometer’s Sketchpad to explore velocity. minimum the concept of acceleration. velocity (ii) Determine instantaneous Discuss: uniform acceleration of a particle. a) uniform acceleration acceleration (iii) Determine instantaneous velocity b) zero acceleration of a particle from acceleration c) positive function by integration. acceleration d) negative (iv) Determine displacement of a acceleration. particle from acceleration function by integration. (v) Solve problems involving motion along a straight line. 22
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LEARNING AREA: ASS2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… linear1 Understand and use the Use real-life examples, (i) Identify and shade the region on Emphasise the use of solid programmingconcept of graphs of linear graphing calculators and the graph that satisfies a linear lines and dashed lines.inequalities. dynamic computer software inequality. linear inequality such as Geometer’s Sketchpad dashed line to explore linear programming. solid line (ii) Find the linear inequality that defines a shaded region. region define satisfy (iii) Shade region on the graph that Limit to regions defined satisfies several linear by a maximum of 3 linear inequalities. inequalities (not including the x-axis and y-axis). (iv) Find linear inequalities that define a shaded region. 23
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LEARNING AREA: ASS2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the (i) Solve problems related to linear feasible solutionconcept of linear programming by: objectiveprogramming. a) writing linear inequalities and function equations describing a parallel lines situation, vertex b) shading the region of feasible vertices solutions, c) determining and drawing the optimum value objective function ax + by = k maximum value where a, b and k are minimum value constants, d) determining graphically the Optimum values refer to optimum value of the maximum or minimum objective function. values. Include the use of vertices to find the optimum value. 24
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LEARNING AREA: ASS2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Carry out project work. Use scientific calculators, (i) Define the problem/situation to be Emphasise the use of conjecture graphing calculators or studied. Polya’s four-step problem- systematic computer software to carry out solving process. project work. (ii) State relevant conjectures. critical evaluation Pupils are allowed to carry out (iii) Use problem-solving strategies to Use at least two problem- mathematical project work in groups but solve problems. solving strategies. reasoning written reports must be done individually. (iv) Interpret and discuss results. justification conclusion (v) Draw conclusions and/or Pupils should be given the generalisation generalisations based on critical opportunities to give oral evaluation of results. mathematical presentations of their project work. communication (vi) Present systematic and Emphasise reasoning and comprehensive written reports. effective mathematical rubric communication. 25
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LEARNING AREA: PW2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… 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 26
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LEARNING AREA: PW2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… 27
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LEARNING AREA: PW2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES Form 5 POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… 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 Krishian a/l Gobal Yam Weng Hoong SMK Jenjarom SMK Kampong Pasir Putih SM Teknik Butterworth Jenjarom, Selangor 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 28
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