(iii)CONTENTSPageRukunegara ivNational Philosophy of Education vPreface viiIntroduction ixNUMBER BASES 1GRAPHS OF FUNCTIONS II 4TRANFORMATIONS III 8MATRICES 10VARIATIONS 18GRADIENT AND AREA UNDER A GRAPH 22PROBABILITY II 25BEARING 28EARTH AS A SPHERE 30PLANS AND ELEVATIONS 35
(iv)RUKUNEGARADECLARATIONOUR 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 nationshall be equitably shared;• to ensuring a liberal approach to her rich and diversecultural traditions;• to building a progressive society which shall be orientedto 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
(v)Education in Malaysia is an ongoing efforttowards further developing the potential ofindividuals in a holistic and integratedmanner so as to produce individuals who areintellectually, spiritually, emotionally andphysically balanced and harmonious, basedon a firm belief in God. Such an effort isdesigned to produce Malaysian citizens whoare knowledgeable and competent, whopossess high moral standards, and who areresponsible and capable of achieving a highlevel of personal well-being as well as beingable to contribute to the betterment of thefamily, the society and the nation at large.
(vii)PREFACEScience and technology plays a critical role in realising Malaysia’saspiration to become a developed nation. Since mathematics is instrumentalin the development of scientific and technological knowledge, the provisionof quality mathematics education from an early age in the education processis thus important. The Malaysian school curriculum offers threemathematics education programs, namely Mathematics for primary schools,Mathematics and Additional Mathematics for secondary schools.The Malaysian school mathematics curriculum aims to developmathematical knowledge, competency and inculcate positive attitudestowards mathematics among pupils. Mathematics for secondary schoolsprovides opportunities for pupils to acquire mathematical knowledge andskills, and develop higher order problem solving and decision making skillsto enable pupils to cope with daily life challenges. As with other subjects inthe secondary school curriculum, Mathematics aims to inculcate noblevalues and love for the nation in the development of a holistic person, whoin turn will be able to contribute to the harmony and prosperity of the nationand its people.Beginning 2003, English is used as the medium of instruction for Scienceand Mathematics subjects. The policy to change the medium of instructionfor Science and Mathematics subjects follows a phased implementationschedule and is expected to be completed by 2008.In the teaching and learning of Mathematics, the use of technologyespecially ICT is greatly emphasised. Mathematics taught in English,coupled with the use of ICT, provide greater opportunities for pupils toimprove their knowledge and skills in mathematics because of the richnessof resources and repositories of knowledge in English. Pupils will be betterable to interact with pupils from other countries, improve their proficiencyin English and thus make the learning of mathematics more interesting andexciting.The development of this Mathematics syllabus is the work of manyindividuals and experts in the field. On behalf of the CurriculumDevelopment Centre, I would like to express much gratitude andappreciation to those who have contributed in one way or another towardsthis initiative(MAHZAN BIN BAKAR SMP, AMP)DirectorCurriculum Development CentreMinistry of EducationMalaysia
(ix)INTRODUCTIONA well-informed and knowledgeable society well versed in the use ofmathematics to cope with daily life challenges is integral to realising thenation’s aspiration to become an industrialised nation. Thus, efforts are takento ensure a society that assimilates mathematics into their daily lives. Pupilsare nurtured from an early age with the skills to solve problems andcommunicate mathematically, to enable them to make effective decisions.Mathematics is essential in preparing a workforce capable of meeting thedemands of a progressive nation. As such, this field assumes its role as thedriving force behind various developments in science and technology. In linewith the nation’s objective to create a knowledge-based economy, the skillsof Research & Development in mathematics is nurtured and developed atschool level.As a field of study, Mathematics trains the mind to think logically andsystematically in solving problems and making decisions. This disciplineencourages meaningful learning and challenges the mind, and hencecontributes to the holistic development of the individual. To this end,strategies to solve problems are widely used in the teaching and learning ofmathematics. The development of mathematical reasoning is believed to beclosely linked to the intellectual development and communication ability ofpupils. Hence, mathematics reasoning skills are also incorporated in themathematics activities to enable pupils to recognize, build and evaluatemathematics conjectures and statements.In keeping with the National Education Philosophy, the Mathematicscurriculum provides opportunities to pupils from various backgrounds andlevels of abilities to acquire mathematical skills and knowledge. Pupils arethen able to seek relevant information, and be creative in formulatingalternatives and solutions when faced with challenges.The general Mathematics curriculum has often been seen to comprise ofdiscrete areas related to counting, measurement, geometry, algebra andsolving problems. To avoid the areas to be continually seen as separate andpupils acquiring concepts and skills in isolation, mathematics is linked toeveryday life and experiences in and out of school. Pupils will have theopportunity to apply mathematics in different contexts, and see the relevanceof mathematics in daily life.In giving opinions and solving problems either orally or in writing, pupilsare guided in the correct usage of language and mathematics registers. Pupilsare trained to select information presented in mathematical and non-mathematical language; interpret and represent information in tables, graphs,diagrams, equations or inequalities; and subsequently present informationclearly and precisely, without any deviation from the original meaning.Technology in education supports the mastery and achievement of thedesired learning outcomes. Technology used in the teaching and learning ofMathematics, for example calculators, are to be regarded as tools to enhancethe teaching and learning process and not to replace teachers.Importance is also placed on the appreciation of the inherent beauty ofmathematics. Acquainting pupils with the life-history of well-knownmathematicians or events, the information of which is easily available fromthe Internet for example, will go a long way in motivating pupils toappreciate mathematics.The intrinsic values of mathematics namely thinking systematically,accurately, thoroughly, diligently and with confidence, infused throughoutthe teaching and learning process; contribute to the moulding of characterand the inculcation of positive attitudes towards mathematics. Together withthese, moral values are also introduced in context throughout the teachingand learning of mathematics.
(x)Assessment, in the form of tests and examinations helps to gauge pupils’achievements. The use of good assessment data from a variety of sourcesalso provides valuable information on the development and progress ofpupils. On-going assessment built into the daily lessons allows theidentification of pupils’ strengths and weaknesses, and effectiveness of theinstructional activities. Information gained from responses to questions,group work results, and homework helps in improving the teaching process,and hence enables the provision of effectively aimed lessons.AIMThe mathematics curriculum for secondary schools aims to developindividuals who are able to think mathematically, and apply mathematicalknowledge effectively and responsibly in solving problems and makingdecisions; and face the challenges in everyday life brought about by theadvancement of science and technology.OBJECTIVESThe mathematics curriculum for the secondary school enables pupils to:1 understand definitions, concepts, laws, principles, and theorems related toNumber, Shape and Space, and Relationship;2 widen the use of basic operations of addition, subtraction, multiplicationand division related to Number, Shape and Space, and Relationship;3 acquire basic mathematical skills such as:• making estimation and rounding;• measuring and constructing;• collecting and handling data;• representing and interpreting data;• recognising and representing relationship mathematically;• using algorithm and relationship;• solving problems; and• making decisions.4 communicate mathematically;5 apply knowledge and skills of mathematics in solving problems andmaking decisions;6 relate mathematics with other areas of knowledge;7 use suitable technologies in concept building, acquiring skills, solvingproblems and exploring the field of mathematics;8 acquire mathematical knowledge and develop skills effectively and usethem responsibly;9 inculcate a positive attitude towards mathematics; and10 appreciate the importance and beauty of mathematics.CONTENT ORGANISATIONThe Mathematics Curriculum content at the secondary school level isorganised into three main areas, namely: Number; Shape and Space; andRelationship. Mathematical concepts related to the respective area, arefurther organised into topics. These topics are arranged in a hierarchicalmanner such that the more basic and tangible concepts appear first and themore complex and abstract concepts appear subsequently.
(xi)The Learning Area outlines the scope of mathematical knowledge, abilitiesand attitudes to be developed in pupils when learning the subject. They aredeveloped according to the appropriate learning objectives and representedin five columns, as follows:Column 1: Learning ObjectivesColumn 2: Suggested Teaching and Learning ActivitiesColumn 3: Learning OutcomesColumn 4: Points To Note; andColumn 5: Vocabulary.The Learning Objectives define clearly what should be taught. They coverall aspects of the Mathematics curriculum programme and are presented in adevelopmental sequence designed to support pupils understanding of theconcepts and skill of mathematics.The Suggested Teaching and Learning Activities lists some examples ofteaching and learning activities including methods, techniques, strategies andresources pertaining to the specific concepts or skills. These are, however,not the only intended approaches to be used in the classrooms. Teachers areencouraged to look for other examples, determine teaching and learningstrategies most suitable for their students and provide appropriate teachingand learning materials. Teachers should also make cross-references to otherresources such as the textbooks and the Internet.The Learning Outcomes define specifically what pupils should be able todo. They prescribe the knowledge, skills or mathematical processes andvalues that should be inculcated and developed at the appropriate level.These behavioural objectives are measurable in all aspects.In the Points To Note column, attention is drawn to the more significantaspects of mathematical concepts and skills. These emphases are to be takeninto account so as to ensure that the concepts and skills are taught and learnteffectively as intended.The Vocabulary consists of standard mathematical terms, instructionalwords or phrases which are relevant in structuring activities, in askingquestions or setting tasks. It is important to pay careful attention to the use ofcorrect terminology and these need to be systematically introduced to pupilsin various contexts so as to enable them to understand their meaning andlearn to use them appropriately.EMPHASES IN TEACHING AND LEARNINGThis Mathematics Curriculum is arranged in such a way so as to giveflexibility to teachers to implement an enjoyable, meaningful, useful andchallenging teaching and learning environment. At the same time, it isimportant to ensure that pupils show progression in acquiring themathematical concepts and skills.In determining the change to another learning area or topic, the followinghave to be taken into consideration:• The skills or concepts to be acquired in the learning area or incertain topics;• Ensuring the hierarchy or relationship between learning areas ortopics has been followed accordingly; and• Ensuring the basic learning areas have been acquired fully beforeprogressing to more abstract areas.The teaching and learning processes emphasise concept building and skillacquisition as well as the inculcation of good and positive values. Besidesthese, there are other elements that have to be taken into account and infusedin the teaching and learning processes in the classroom. The main elementsfocused in the teaching and learning of mathematics are as follows:
(xii)1. Problem Solving in MathematicsProblem solving is the main focus in the teaching and learning ofmathematics. Therefore the teaching and learning process must includeproblem solving skills which are comprehensive and cover the wholecurriculum. The development of problem solving skills need to beemphasised so that pupils are able to solve various problems effectively. Theskills involved are:• Understanding the problem;• Devising a plan;• Carrying out the plan; and• Looking back at the solutions.Various strategies and steps are used to solve problems and these areexpanded so as to be applicable in other learning areas. Through theseactivities, students can apply their conceptual understanding of mathematicsand be confident when facing new or complex situations. Among theproblem solving strategies that could be introduced are:• Trying a simple case;• Trial and improvement;• Drawing diagrams;• Identifying patterns;• Making a table, chart or systematic list;• Simulation;• Using analogies;• Working backwards;• Logical reasoning; and• Using algebra.2. Communication in MathematicsCommunication is an essential means of sharing ideas and clarifying theunderstanding of Mathematics. Through communication, mathematical ideasbecome the object of reflection, discussion and modification. The process ofanalytical and systematic reasoning helps pupils to reinforce and strengthentheir knowledge and understanding of mathematics to a deeper level.Through effective communication, pupils will become efficient in problemsolving and be able to explain their conceptual understanding andmathematical skills to their peers and teachers.Pupils who have developed the skills to communicate mathematically willbecome more inquisitive and, in the process, gain confidence.Communication skills in mathematics include reading and understandingproblems, interpreting diagrams and graphs, using correct and concisemathematical terms during oral presentations and in writing. The skillsshould be expanded to include listening.Communication in mathematics through the listening process occurs whenindividuals respond to what they hear and this encourages individuals tothink using their mathematical knowledge in making decisions.Communication in mathematics through the reading process takes placewhen an individual collects information and data and rearranges therelationship between ideas and concepts.Communication in mathematics through the visualisation process takes placewhen an individual makes an observation, analyses, interprets andsynthesises data and presents them in the form of geometric board, picturesand diagrams, tables and graphs. An effective communication environmentcan be created by taking into consideration the following methods:• Identifying relevant contexts associated with environment andeveryday life experience of pupils;• Identifying pupils’ interests;
(xiii)• Identifying suitable teaching materials;• Ensuring active learning;• Stimulating meta-cognitive skills;• Inculcating positive attitudes; and• Setting up conducive learning environment.Effective communication can be developed through the following methods:Oral communication is an interactive process that involves psychomotoractivities like listening, touching, observing, tasting and smelling. It is a two-way interaction that takes place between teacher and pupils, pupils andpupils, and pupils and object.Some of the more effective and meaningful oral communication techniquesin the learning of mathematics are as follows:• Story-telling and question and answer sessions using one’s ownwords;• Asking and answering questions;• Structured and unstructured interviews;• Discussions during forums, seminars, debates and brainstormingsessions; and• Presentation of findings of assignments.Written communication is the process whereby mathematical ideas andinformation are disseminated through writing. The written work is usuallythe result of discussion, input from people and brainstorming activities whenworking on assignments. Through writing, pupils will be encouraged tothink in depth about the mathematics content and observe the relationshipsbetween concepts. Examples of written communication activities that can bedeveloped through assignments are:• Doing exercises;• Keeping journals;• Keeping scrap books;• Keeping folios;• Keeping portfolios;• Undertaking projects; and• Doing written tests.Representation is a process of analysing a mathematical problem andinterpreting it from one mode to another. Mathematical representationenables pupils to find relationships between mathematical ideas that areinformal, intuitive and abstract using everyday language. For example 6xycan be interpreted as a rectangular area with sides 2x and 3y. This will makepupils realise that some methods of representation are more effective anduseful if they know how to use the elements of mathematical representation.3. Reasoning in MathematicsLogical Reasoning or thinking is the basis for understanding and solvingmathematical problems. The development of mathematical reasoning isclosely related to the intellectual and communicative development of pupils.Emphasis on logical thinking, during mathematical activities opens uppupils’ minds to accept mathematics as a powerful tool in the world today.Pupils are encouraged to estimate, predict and make intelligent guesses inthe process of seeking solutions. Pupils at all levels have to be trained toinvestigate their predictions or guesses by using concrete material,calculators, computers, mathematical representation and others. Logicalreasoning has to be absorbed in the teaching of mathematics so that pupilscan recognise, construct and evaluate predictions and mathematicalarguments.
(xiv)4. Mathematical ConnectionsIn the mathematics curriculum, opportunities for making connections mustbe created so that pupils can link conceptual to procedural knowledge andrelate topics within mathematics and other learning areas in general.The mathematics curriculum consists of several areas such as arithmetic,geometry, algebra, measures and problem solving. Without connectionsbetween these areas, pupils will have to learn and memorise too manyconcepts and skills separately. By making connections, pupils are able to seemathematics as an integrated whole rather than a jumble of unconnectedideas. When mathematical ideas and the curriculum are connected to real lifewithin or outside the classroom, pupils will become more conscious of theimportance and significance of mathematics. They will also be able to usemathematics contextually in different learning areas and in real lifesituations.5. Application of TechnologyThe teaching and learning of mathematics should employ the latesttechnology to help pupils understand mathematical concepts in depth,meaningfully and precisely and enable them to explore mathematical ideas.The use of calculators, computers, educational software, websites in theInternet and relevant learning packages can help to upgrade the pedagogicalapproach and thus promote the understanding of mathematical concepts.The use of these teaching resources will also help pupils absorb abstractideas, be creative, feel confident and be able to work independently or ingroups. Most of these resources are designed for self-access learning.Through self-access learning pupils will be able to access knowledge orskills and informations independently according to their own pace. This willserve to stimulate pupils interests and develop a sense of responsibilitytowards their learning and understanding of mathematics.Technology however does not replace the need for all pupils to learn andmaster the basic mathematical skills. Pupils must be able to efficiently add,subtract, multiply and divide without the use of calculators or otherelectronic tools. The use of technology must therefore emphasise theacquisition of mathematical concepts and knowledge rather than merelydoing calculation.APPROACHES IN TEACHING AND LEARNINGThe belief on how mathematics is learnt influence how mathematicalconcepts are to be taught. Whatever belief the teachers subscribe to, the factremains that mathematical concepts are abstract. The use of teachingresources is therefore crucial in guiding pupils to develop matematical ideas.Teachers should use real or concrete materials to help pupils gainexperience, construct abstract ideas, make inventions, build self confidence,encourage independence and inculcate the spirit of cooperation.The teaching and learning materials used should contain self diagnosticelements so that pupils know how far they have understood the concepts andacquire the skills.In order to assist pupils in having positive attitudes and personalities, theintrinsic mathematical values of accuracy, confidence and thinkingsystematically have to be infused into the teaching and learning process.Good moral values can be cultivated through suitable contexts. Learning ingroups for example can help pupils to develop social skills, encouragecooperation and build self confidence. The element of patriotism should alsobe inculcated through the teaching and learning process in the classroomusing certain topics.
(xv)Brief historical anecdotes related to aspects of mathematics and famousmathematicians associated with the learning areas are also incorporated intothe curriculum. It should be presented at appropriate points where it providesstudents with a better understanding and appreciation of mathematics.Various teaching strategies and approaches such as direct instruction,discovery learning, investigation, guided discovery or other methods must beincorporated. Amongst the approaches that can be given considerationinclude the following:• Pupils-centered learning that is interesting;• Different learning abilities and styles of pupils;• Usage of relevant, suitable and effective teaching materials; and• Formative evaluation to determine the effectiveness of teaching andlearning.The choice of an approach that is suitable will stimulate the teaching andlearning environment inside or outside the classroom. Approaches that areconsidered suitable include the following:• Cooperative learning;• Contextual learning;• Mastery learning;• Constructivism;• Enquiry-discovery; and• Future studies.EVALUATIONEvaluation or assessment is part of the teaching and learning process toascertain the strengths and weaknesses of pupils. It has to be planned andcarried out as part of the classroom activities. Different methods ofassessment can be conducted. These maybe in the form of assignments, oralquestioning and answering, observations and interviews. Based on theresponse, teachers can rectify pupils misconceptions and weaknesses andalso improve their own teaching skills. Teachers can then take subsequenteffective measures in conducting remedial and enrichment activities inupgrading pupils’ performances.
Form 511LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of number in basetwo, eight and five.Use models such as a clock face or acounter which uses a particular numberbase.Number base blocks of twos, eights andfives can be used to demonstrate thevalue of a number in the respectivenumber bases.For example:2435 is2 4 3Discuss:• digits used,• place valuesin the number system with a particularnumber base.(i) State zero, one, two, three,…, asa number in base:a) two,b) eight,c) Five.(ii) State the value of a digit of anumber in base:a) two,b) eight,c) Five.(iii) Write a number in base:a) two,b) eight,c) fivein expanded notation.Emphasise the ways toread numbers in variousbases.Examples:• 1012 is read as “onezero one base two”• 72058 is read as“seven two zero fivebase eight”• 4325 is read as “fourthree two base five”Numbers in base two arealso known as binarynumbers.Examples of numbers inexpanded notation:• 101102 = 1 × 24+ 0 × 23+ 1 × 22+ 1 × 21+ 0 × 20• 3258 = 3 × 82+ 2 × 81+ 5 × 80• 30415 = 3 × 53+ 0 × 52+ 4 × 51+ 1 × 50expandednotation
Form 521LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYNumber base blocks of twos, eights andfives can also be used here. For example,to convert 1010 to a number in base two,use the least number of blocks (23), tiles(22), rectangles (21) and squares (20).Here, the least number of objects here areone block, no tiles, one rectangle and nosquares. So, 1010 = 10102.(iv) Convert a number in base:a) two,b) eight,c) fiveto a number in base ten and viceversa.Perform repeateddivision to convert anumber in base ten to anumber in other bases.For example, convert71410 to a number inbase five:5)7145)142 --- 45) 28 --- 25) 5 --- 35) 1 --- 00 --- 1∴ 71410 = 10324 5convert
Form 531LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYDiscuss the special case of converting anumber in base two directly to a number inbase eight and vice-versa.For example, convert a number in base twodirectly to a number in base eight throughgrouping of three consecutive digits.Perform direct addition and subtraction inthe conventional manner.For example:10102+ 1102(v) Convert a number in a certainbase to a number in anotherbase.(vi) Perform computationsinvolving:a) addition,b) subtraction,of two numbers in base two.Limit conversion ofnumbers to base two,eight and five only.
Form 542LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of graph offunctions.Explore graphs of functions by usinggraphing calculator or Geometer’sSketchpad.Compare the characteristics of graphs offunctions with different values ofconstants.For example:A BThe curve in graph B is wider than thecurve in graph A and intersects the verticalaxis above the horizontal axis.(i) Draw the graph of a:a) linear function: y = ax + b,where a and b are constants,b) quadratic function:y = ax 2+ bx + c,where a, b and c areconstants , a ≠ 0,c) cubic function:y = ax 3+ bx 2+ cx + d,where a, b, c and d areconstants, a 0≠ ,d) reciprocal function: y =xa,where a is a constant,a 0≠ .(ii) Find from a graph:a) the value of y, given a valueof x,b) the value(s) of x, given avalue of y.Limit cubic functions tothe following forms:• y = ax 3• y = ax 3+ b• y = x 3+ bx + c• y = −x 3+ bx + cFor certain functions andsome values of y, therecould be nocorresponding values ofx.linear functionquadraticfunctioncubic functionreciprocalfunction
Form 552LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYAs reinforcement, let pupils play a game;for example, matching cards of graphswith their respective functions. When thepupils have their matching partners, askthem to group themselves into fourgroups of types of functions. Finally, askeach group to name the type of functionthat is depicted on the cards.(iii) Identify:a) the shape of graph given atype of function,b) the type of function given agraph,c) the graph given a functionand vice versa.(iv) Sketch the graph of a givenlinear, quadratic, cubic orreciprocal function.For graphs of cubicfunctions, limit toy = ax3and y = ax3+ b.For graphs of quadraticfunctions, limit toy = ax2+ b and quadraticfunctions which can befactorised to(mx + n)(px + q) wherem, n, p, q are integers.For graphs of cubicfunctions, limit toy = ax 3and y = ax 3+ b.
Form 562LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY2 Understand and use theconcept of the solution ofan equation by graphicalmethod.Explore on the graphing calculator or theGeometer’s Sketchpad to relate thex-coordinate of a point of intersection oftwo appropriate graphs to the solution ofa given equation. Make generalisationabout the point(s) of intersection of twographs.(i) Find the point(s) of intersectionof two graphs.(ii) Obtain the solution of anequation by finding the point(s)of intersection of two graphs.(iii) Solve problems involvingsolution of an equation bygraphical method.Use the traditionalgraph plotting exerciseif the graphingcalculator or theSketchpad isunavailable.Involve everydayproblems.point ofintersection3 Understand and use theconcept of the regionrepresenting inequalities intwo variables.Discuss that if one point in a regionsatisfies y > ax + b or y < ax + b, then allpoints in the region satisfy the sameinequality.(i) Determine whether a given pointsatisfies:y = ax + b, ory > ax + b, ory < ax + b.(ii) Determine the position of agiven point relative to the graphy = ax + b.(iii) Identify the region satisfyingy > ax + b or y < ax + b.For Learning Objective3, include situationsinvolving x = a, x ≥ a,x > a, x ≤ a or x < a.
Form 572LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYUse the Sketchpad or the graphingcalculator to explore points relative to agraph to make generalisation aboutregions satisfying the given inequalities.Carry out activities using overheadprojector and transparencies, graphingcalculator or Geometer’s Sketchpad.(iv) Shade the regions representingthe inequalities:a) y > ax + b or y < ax + b,b) y ≥ ax + b or y ≤ ax + b.(v) Determine the region whichsatisfies two or moresimultaneous linear inequalities.Emphasise that:• for the regionrepresentingy > ax + b ory < ax + b, the liney = ax + b is drawnas a dashed line toindicate that allpoints on the liney = ax + b are not inthe region.• for the regionrepresentingy ≥ ax + b ory ≤ ax + b, the liney = ax + b is drawnas a solid line toindicate that allpoints on the liney = ax + b are in theregion.regiondashed linesolid line
Form 583LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of combination oftwo transformations.Relate to transformations in real lifesituations such as tessellation patterns onwalls, ceilings or floors.Explore combined transformation usingGeometer’s Sketchpad, graphingcalculator or overhead projector andtransparencies.Investigate the characteristics of anobject and its image under combinedtransformation.(i) Determine the image of anobject under combination of twoisometric transformations.(ii) Determine the image of anobject under combination of:a) two enlargements,b) an enlargement and anisometric transformation.(iii) Draw the image of an objectunder combination of twotransformations.(iv) State the coordinates of theimage of a point undercombined transformation.(v) Determine whether combinedtransformation AB is equivalentto combined transformation BA.Begin with a point,followed by a line andan object.Limit isometrictransformations totranslations, reflectionsand rotations.combinedtransformationequivalent
Form 593LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYCarry out projects to design patternsusing combined transformations that canbe used as decorative purposes. Theseprojects can then be presented in classwith the students describing or specifyingthe transformations involved.Use the Sketchpad to prove the singletransformation which is equivalent to thecombination of two isometrictransformations.(vi) Specify two successivetransformations in a combinedtransformation given the objectand the image.(vii) Specify a transformation which isequivalent to the combination oftwo isometric transformations.(viii) Solve problems involvingtransformation.Limit the equivalenttransformation totranslation, reflectionand rotation.specify
Form 5104LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of matrix.Represent data in real life situations, forexample, the price of food on a menu, intable form and then in matrix form.Use pupils seating positions in class byrows and columns to identify a pupils whois sitting in a particular row and in aparticular column as a concrete example.(i) Form a matrix from giveninformation.(ii) Determine:a) the number of rows,b) the number of columns,c) the orderof a matrix.(iii) Identify a specific element in amatrix.Emphasise that matricesare written in brackets.Introduce row matrix,column matrix andsquare matrix.Emphasise that a matrixof order m × n is read as“an m by n matrix”.Use row number andcolumn number tospecify the position ofan element.matrixrowcolumnorderrow matrixcolumn matrixsquare matrix2 Understand and use theconcept of equal matrices.Discuss equal matrices in terms of:• the order,• the corresponding elements.(i) Determine whether two matricesare equal;(ii) Solve problems involving equalmatrices.Include finding values ofunknown elements.equal matricesunknownelements
Form 5114LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY3 Perform addition andsubtraction on matrices.Relate to real life situations such askeeping score of medal tally or points insports.(i) Determine whether addition orsubtraction can be performed ontwo given matrices.(ii) Find the sum or the difference oftwo matrices.(iii) Perform addition and subtractionon a few matrices.(iv) Solve matrix equationsinvolving addition andsubtraction.Limit to matrices withno more than three rowsand three columns.Include finding values ofunknown elements.matrix equation4 Perform multiplicationof a matrix by a number.Relate to real life situations such as inindustrial productions.(i) Multiply a matrix by a number.(ii) Express a given matrix asmultiplication of a matrix by anumber.(iii) Perform calculation on matricesinvolving addition, subtractionand scalar multiplication.Multiplying a matrix bya number is known asscalar multiplication.scalarmultiplication
Form 5124LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY(iv) Solve matrix equationsinvolving addition, subtractionand scalar multiplication.Include finding valuesof unknown elements.5 Perform multiplicationof two matrices.Relate to real life situations such as findingthe cost of a meal in a restaurant.For matrices A and B, discuss therelationship between AB and BA..(i) Determine whether two matricescan be multiplied and state theorder of the product when thetwo matrices can be multiplied.(ii) Find the product of twomatrices.(iii) Solve matrix equationsinvolving multiplication of twomatrices.Limit to matrices withno more than 3 rows and3 columns.Limit to two unknownelements.product6 Understand and use theconcept of identity matrix.Begin with discussing the property of thenumber 1 as an identity for multiplicationof numbers.Discuss:• an identity matrix is a square matrix,• there is only one identity matrix foreach order.(i) Determine whether a givenmatrix is an identity matrix bymultiplying it to another matrix.(ii) Write identity matrix of anyorder.Identity matrix isusually denoted by I andis also known as unitmatrix.identity matrixunit matrix
Form 5134LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYDiscuss the properties:• AI = A,• IA = A..(iii) Perform calculation involvingidentity matrices.Limit to matrices withno more than 3 rows and3 columns.7 Understand and use theconcept of inverse matrix.Relate to property of multiplicative inverseof numbers.For example:2 × 2−1= 2−1× 2 = 1In this example, 2−1is the multiplicativeinverse of 2 and vice versa.(i) Determine whether a 2 × 2matrix is the inverse matrix ofanother 2 × 2 matrix.The inverse of matrix Ais denoted by A−1.Emphasise that:• if matrix B is theinverse of matrix A,then matrix A is alsothe inverse of matrixB, AB = BA = I;• inverse matrices canonly exist for squarematrices, but not allsquare matrices haveinverse matrices.inverse matrix
Form 5144LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYUse the method of solving simultaneouslinear equations to show that not all squarematrices have inverse matrices. Forexample, ask pupils to try to find theinverse matrix of ⎟⎟⎠⎞⎜⎜⎝⎛4623.(ii) Find the inverse matrix of a 2 ×2 matrix by using:a) method of solvingsimultaneous linearequations,b) formula.Steps to find the inversematrix:• solving simultaneouslinear equations⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛10014321srqpp + 2r = 1, q + 2s = 03p + 44 = 0, 3q + 4s = 1where ⎟⎟⎠⎞⎜⎜⎝⎛srqpis theinverse matrix;
Form 5154LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYCarry out operations leading to theformula.Use matrices and their respective inversematrices in the previous method to relate tothe formula. Express each inverse matrix asa multiplication of a matrix by a number.Compare the scalar multiplication to theoriginal matrix and discuss how thedeterminant is obtained.Discuss the condition for the existence ofinverse matrix.• using formulaFor A = ⎟⎟⎠⎞⎜⎜⎝⎛dcba,A−1=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−−−−−−bcadabcadcbcadbbcaddorA−1=⎟⎟⎠⎞⎜⎜⎝⎛−−− acbdbcad1when ad − bc ≠ 0.ad − bc is known as thedeterminant of thematrix A.A−1does not exist if thedeterminant is zero.
Form 5164LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY8 Solve simultaneouslinear equations by usingmatrices.Relate to equal matrices by witing downthe simultaneous equations as equalmatrices first. For example:Write 2x + 3y = 134x − y = 5as equal matrices:⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−+513432yxyxwhich is then expressed as:⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛− 5131432yxDiscuss why:• the use of inverse matrix is necessary.Relate to solving linear equations oftype ax = b,• it is important to place the inversematrix at the right place on both sides ofthe equation.(i) Write down simultaneous linearequations in matrix form.(ii) Find the matrix ⎟⎟⎠⎞⎜⎜⎝⎛qpin⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛khqpdcbaby using inverse matrix.Limit to two unknowns.Simultaneous linearequationsap + bq = hcp + dq = kin matrix form is⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛khqpdcbawhere a, b, c, d, h and kare constants, p and qare unknowns.A−1⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛qpdcba= A−1⎟⎟⎠⎞⎜⎜⎝⎛khwhere A = ⎟⎟⎠⎞⎜⎜⎝⎛dcba ..
Form 5174LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYRelate the use of matrices in other areassuch as in business or economy, scienceand so on.Carry out projects involving matrices usingspreadsheet softwares.(iii) Solve simultaneous linearequations by using the matrixmethod.(iv) Solve problems involvingmatrices.The matrix method usesinverse matrix to solvesimultaneous linearequations.matrix method
Form 5185LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of direct variation.Discuss the characteristics of the graphof y against x when y ∝ x.Relate to other areas like science andtechnology. For example, the Law ofCharles and Gay-Lussac (or Charles’Law), Hooke’s Law and the simplependulum.For the cases y ∝ x n, n = 2, 3,21 ,discuss the characteristics of the graphsof y against x n.(i) State the changes in a quantitywith respect to changes inanother quantity, in everydaylife situations involving directvariation.(ii) Determine from giveninformation whether a quantityvaries directly as anotherquantity.(iii) Express a direct variation in theform of equation involving twovariables.(iv) Find the value of a variable in adirect variation when sufficientinformation is given.(v) Solve problems involving directvariation for the following cases:y ∝ x; y ∝ x 2; y ∝ x 3; y ∝ x 21.y varies directly as x ifand only ifxyis aconstant.If y varies directly as x,the relation is written asy ∝ x .For the cases y ∝ xn,limit n to 2, 3 and21 .If y ∝ x, then y = kxwhere k is the constantof variation.Using:• y = kx; or•2211xyxy=to get the solution.direct variationquantityconstant ofvariationvariable
Form 5195LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY2 Understand and use theconcept of inversevariation.Discuss the form of the graph of yagainstx1when y ∝x1.Relate to other areas like science andtechnology. For example, Boyle’s Law.(i) State the changes in a quantitywith respect to changes inanother quantity, in everydaylife situations involving inversevariation.(ii) Determine from giveninformation whether a quantityvaries inversely as anotherquantity.(iii) Express an inverse variation inthe form of equation involvingtwo variables.y varies inversely as x ifand only if xy is aconstant.If y varies inversely as x,the relation is written asy ∝x1.For the cases y ∝ nx1,limit n to 2, 3 and21 .If y ∝x1, then y =xkwhere k is the constantof variation.inverse variation
Form 5205LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYFor the cases y ∝ nx1, n = 2, 3,21 ,discuss the characteristics of the graph ofy against nx1.(iv) Find the value of a variable in aninverse variation when sufficientinformation is given.(v) Solve problems involvinginverse variation for thefollowing cases:21321;1;1;1xyxyxyxy ∝∝∝∝Using:• y =xk; or• 2211 yxyx =to get the solution.
Form 5215LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY3 Understand and use theconcept of joint variation.Discuss joint variation for the three casesin everyday life situations.Relate to other areas like science andtechnology.For example:RVI ∝ means the current I variesdirectly as the voltage V and variesinversely as the resistance R.(i) Represent a joint variation byusing the symbol ∝ for thefollowing cases:a) two direct variations,b) two inverse variations,c) a direct variation and aninverse variation.(ii) Express a joint variation in theform of equation.(iii) Find the value of a variable in ajoint variation when sufficientinformation is given.(iv) Solve problems involving jointvariation.For the cases y ∝ xnzn,y ∝ nnzx1and y ∝ nnzx,limit n to 2, 3 and21 .joint variation
Form 5226LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of quantityrepresented by the gradientof a graph.Use examples in various areas such astechnology and social science.Compare and differentiate betweendistance-time graph and speed-timegraph..(i) State the quantity represented bythe gradient of a graph.(ii) Draw the distance-time graph,given:a) a table of distance-timevalues,b) a relationship betweendistance and time.(iii) Find and interpret the gradientof a distance-time graph.Limit to graph of astraight line.The gradient of a graphrepresents the rate ofchange of a quantity onthe vertical axis withrespect to the change ofanother quantity on thehorizontal axis. The rateof change may have aspecific name forexample “speed” for adistance-time graph.Emphasise that:gradient== speeddistance-timegraphspeed-timegraphchange of distancechange of time
Form 5236LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYUse real life situations such as travellingfrom one place to another by train or bybus.Use examples in social science andeconomy.(iv) Find the speed for a period oftime from a distance-time graph.(v) Draw a graph to show therelationship between twovariables representing certainmeasurements and state themeaning of its gradient.Include graph whichconsists of a few straightlines.For example:Distance, sTime, t2 Understand the conceptof quantity represented bythe area under a graph.Discuss that in certain cases, the areaunder a graph may not represent anymeaningful quantity.For example:The area under the distance-time graph.(i) State the quantity represented bythe area under a graph.Include speed-time andacceleration-timegraphs.area under agraphacceleration-time graph
Form 5246LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYDiscuss the formula for finding the areaunder a graph involving:• a straight line which is parallel to thex-axis,• a straight line in the form ofy = kx + h,• a combination of the above.(ii) Find the area under a graph.(iii) Determine the distance byfinding the area under thefollowing types of speed-timegraphs:a) v = k (uniform speed),b) v = kt,c) v = kt + h,d) a combination of the above.(iv) Solve problems involvinggradient and area under a graph.Limit to graph of astraight line or acombination of a fewstraight lines.v represents speed;t represents time;h and k are constants.For example:Speed, vTime, tuniform speed
Form 5257LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of probability of anevent.Discuss equiprobable sample spacethrough concrete activities and beginwith simple cases such as tossing a faircoin.In cases that can be seen as built up ofsimple cases, for example tossing a faircoin and a fair die, tree diagrams can beused to obtain the sample space.Graphing calculator can be used for theseactivities.Discuss events that produce P(A) = 1and P(A) = 0 .(i) Determine the sample space ofan experiment with equallylikely outcomes.(ii) Determine the probability of anevent with equiprobable samplespace.(iii) Solve problems involvingprobability of an event.Limit to sample spacewith equally likelyoutcomes.A sample space inwhich each event isequally likely is calledequiprobable samplespace.The probability of anevent A, withequiprobable samplespace S, is P(A) =)()(SnAn.Use tree diagram whereappropriate.Include everydayproblems and makingpredictions.equally likelyequiprobablesample spacetree diagram
Form 5267LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY2 Understand and use theconcept of probability ofthe complement of anevent.Include events in real life situations suchas winning or losing a game and passingor failing an exam.(i) State the complement of anevent in:a) words,b) set notation.(ii) Find the probability of thecomplement of an event.The complement of anevent A is the set of alloutcomes in the samplespace that are notincluded in the outcomesof event A.complement ofan event3 Understand and use theconcept of probability ofcombined event.Use real life situations to show therelationship between• A or B and A ∪ B,• A and B and A ∩ B.For example, being chosen to be amember of an exclusive club withrestricted conditions.Tree diagrams and coordinate planes areuseful tools that can be used to find allthe outcomes of combined events.(i) List the outcomes for events:a) A or B as elements of the setA ∪ B,b) A and B as elements of theset A ∩ B.(ii) Find the probability by listingthe outcomes of the combinedevent:a) A or B,b) A and B.combined event
Form 5277LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYUse two-way classification tables ofevents from newspaper articles orstatistical data to find the probability ofcombined events. Ask students to createtree diagrams from these tables.Example of a two-way classificationtable:Means of going to workOfficers Car Bus OthersMen 56 25 83Women 50 42 37Disscuss:• situations where decisions have to bemade based on probability, forexample in business, such asdetermining the value for specificinsurance policy and time slot foradvertisements on television,• the statement “probability is theunderlying language of statistics”.(iii) Solve problems involvingprobability of combined event.Emphasise that:• knowledge aboutprobability is usefulin making decisions;• prediction based onprobability is notdefinite or absolute.
Form 5288LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of bearing.Carry out activities or games involvingfinding directions using a compass, suchas treasure hunt or scavenger hunt. It canalso be about locating several points on amap.(i) Draw and label the eight maincompass directions:a) north, south, east, west,b) north-east, north-west,south-east, south west.(ii) State the compass angle of anycompass direction.(iii) Draw a diagram which showsthe direction of B relative to Agiven the bearing of B from A.Compass angle andbearing are written inthree-digit form, from000 οto 360 ο. They aremeasured in a clockwisedirection from north.Due north is consideredas bearing 000 ο.For cases involvingdegrees and minutes,state in degrees up toone decimal place.north-eastsouth-eastnorth-westsouth-westcompass anglebearing
Form 5298LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYDiscuss the use of bearing in real lifesituations. For example, in map readingand navigation.(iv) State the bearing of point A frompoint B based on giveninformation.(v) Solve problems involvingbearing.Begin with the casewhere bearing of point Bfrom point A is given.
Form 5309LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of longitude.Models such as globes should be used.Introduce the meridian throughGreenwich in England as the GreenwichMeridian with longitude 0°.Discuss that:• all points on a meridian have the samelongitude,• there are two meridians on a greatcircle through both poles,• meridian with longitudes x°E (or W)and (180° − x°)W (or E) form a greatcircle through both poles.(i) Sketch a great circle through theNorth and South Poles.(ii) State the longitude of a givenpoint.(iii) Sketch and label a meridian withthe longitude given.(iv) Find the difference between twolongitudes.Emphasise thatlongitude 180°E andlongitude 180°W refer tothe same meridian.Express the difference asan angle less than orequal to 180°.great circlemeridianlongitude2 Understand and use theconcept of latitude.(i) Sketch a circle parallel to theequator.(ii) State the latitude of a givenpoint.Emphasise that• the latitude of theequator is 0°;• latitude ranges from0° to 90°N (or S).equatorlatitude
Form 5319LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYDiscuss that all points on a parallel oflatitude have the same latitude.(iii) Sketch and label a parallel oflatitude.(iv) Find the difference between twolatitudes.Involve actual places onearth.Express the difference asan angle less than orequal to 180°.parallel oflatitude3 Understand the conceptof location of a place.Use a globe or a map to find locations ofcities around the world.Use a globe or a map to name a placegiven its location.(i) State the latitude and longitudeof a given place.(ii) Mark the location of a place.(iii) Sketch and label the latitude andlongitude of a given place.A place on the surface ofthe earth is representedby a point.The location of a place Aat latitude x°N andlongitude y°E is writtenas A(x°N, y°E).
Form 5329LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY4 Understand and use theconcept of distance on thesurface of the earth to solveproblems.Find the distance between two cities ortowns on the same meridian, as a groupproject.Sketch the angle at the centre of the earththat is subtended by the arc between twogiven points along the equator. Discusshow to find the value of this angle.(i) Find the length of an arc of agreat circle in nautical mile,given the subtended angle at thecentre of the earth and viceversa.(ii) Find the distance between twopoints measured along ameridian, given the latitudes ofboth points;.(iii) Find the latitude of a point giventhe latitude of another point andthe distance between the twopoints along the same meridian.(iv) Find the distance between twopoints measured along theequator, given the longitudes ofboth points.(v) Find the longitude of a pointgiven the longitude of anotherpoint and the distance betweenthe two points along the equator.Limit to nautical mile asthe unit for distance.Explain one nauticalmile as the length of thearc of a great circlesubtending a one minuteangle at the centre of theearth.nautical mile
Form 5339LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYUse models.Find the distance between two cities ortowns on the same parallel of latitude asa group project.Use the globe and a few pieces of stringto show how to determine the shortestdistance between two points on thesurface of the earth.(vi) State the relation between theradius of the earth and the radiusof a parallel of latitude.(vii) State the relation between thelength of an arc on the equatorbetween two meridians and thelength of the corresponding arcon a parallel of latitude.(viii) Find the distance between twopoints measured along a parallelof latitude.(ix) Find the longitude of a pointgiven the longitude of anotherpoint and the distance betweenthe two points along a parallel oflatitude.(x) Find the shortest distancebetween two points on thesurface of the earth.(xi) Solve problems involving:a) distance between two points,b) travelling on the surface ofthe earth.Limit to two points onthe equator or a greatcircle through the poles.Use knot as the unit forspeed in navigation andaviation.
Form 53410LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARY1 Understand and use theconcept of orthogonalprojection.Use models, blocks or plan and elevationkit.(i) Identify orthogonal projection.(ii) Draw orthogonal projection,given an object and a plane.(iii) Determine the differencebetween an object and itsorthogonal projection withrespect to edges and angles.Emphasise the differentuse of dashed lines andsolid lines.Begin with simple solidobjects such as cube,cuboid, cylinder, cone,prism and right pyramid.orthogonalprojection2 Understand and use theconcept of plan andelevation.Carry out activities in groups where pupilscombine two or more different shapes ofsimple solid objects into interesting modelsand draw plans and elevations for thesemodels.Use examples to show that it is importantto have a plan and at least two sideelevations to construct an object.(i) Draw the plan of a solid object.(ii) Draw:a) the front elevation,b) side elevationof a solid object.Include drawing planand elevation in onediagram showingprojection lines.Limit to full-scaledrawings only.planfront elevationside elevation
Form 53510LEARNING AREA:LEARNING OBJECTIVESPupils will be taught to…SUGGESTED TEACHING ANDLEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to…POINTS TO NOTE VOCABULARYCarry out group project:Draw plan and elevations of buildings orstructures, for example pupils orteacher’s dream homes and construct ascale model based on the drawings.Involve real life situations such as inbuilding prototypes and using actualhome plans.(iii) Draw:a) the plan,b) the front elevation,c) the side elevationof a solid object to scale.(iv) Solve problems involving planand elevation.