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Hsp maths f5

  1. 1. Ministry of Education MalaysiaIntegrated Curriculum for Secondary Schools CURRICULUM SPECIFICATIONS MATHEMATICS FORM 5 Curriculum Development Centre Ministry of Education Malaysia 2006 (i)
  2. 2. Copyright © 2006 Curriculum Development CentreMinistry of Education MalaysiaAras 4-8, Blok E9Kompleks Kerajaan Parcel EPusat Pentadbiran Putrajaya62604 PutrajayaFirst published 2006Copyright reserved. Except for use in a review, the reproduction orutilisation of this work in any form or by any electronic, mechanical, orother means, now known or hereafter invented, including photocopying,and recording is forbidden without the prior written permission from theDirector of the Curriculum Development Centre, Ministry of EducationMalaysia. (2)
  4. 4. 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 (iv)
  5. 5. 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. (v)
  6. 6. (vi)
  7. 7. PREFACEScience and technology plays a critical role in realising Malaysia’s in English and thus make the learning of mathematics more interesting andaspiration to become a developed nation. Since mathematics is instrumental exciting.in the development of scientific and technological knowledge, the provisionof quality mathematics education from an early age in the education process The development of this Mathematics syllabus is the work of manyis thus important. The Malaysian school curriculum offers three individuals and experts in the field. On behalf of the Curriculummathematics education programs, namely Mathematics for primary schools, Development Centre, I would like to express much gratitude andMathematics and Additional Mathematics for secondary schools. appreciation to those who have contributed in one way or another towards this initiativeThe 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. (MAHZAN BIN BAKAR SMP, AMP)In the teaching and learning of Mathematics, the use of technologyespecially ICT is greatly emphasised. Mathematics taught in English, Directorcoupled with the use of ICT, provide greater opportunities for pupils to Curriculum Development Centreimprove their knowledge and skills in mathematics because of the richness Ministry of Educationof resources and repositories of knowledge in English. Pupils will be better Malaysiaable to interact with pupils from other countries, improve their proficiency (vii)
  8. 8. (viii)
  9. 9. INTRODUCTION The general Mathematics curriculum has often been seen to comprise of discrete areas related to counting, measurement, geometry, algebra andA well-informed and knowledgeable society well versed in the use of solving problems. To avoid the areas to be continually seen as separate andmathematics to cope with daily life challenges is integral to realising the pupils acquiring concepts and skills in isolation, mathematics is linked tonation’s aspiration to become an industrialised nation. Thus, efforts are taken everyday life and experiences in and out of school. Pupils will have theto ensure a society that assimilates mathematics into their daily lives. Pupils opportunity to apply mathematics in different contexts, and see the relevanceare nurtured from an early age with the skills to solve problems and of mathematics in daily life.communicate mathematically, to enable them to make effective decisions. In giving opinions and solving problems either orally or in writing, pupilsMathematics is essential in preparing a workforce capable of meeting the are guided in the correct usage of language and mathematics registers. Pupilsdemands of a progressive nation. As such, this field assumes its role as the are trained to select information presented in mathematical and non-driving force behind various developments in science and technology. In line mathematical language; interpret and represent information in tables, graphs,with the nation’s objective to create a knowledge-based economy, the skills diagrams, equations or inequalities; and subsequently present informationof Research & Development in mathematics is nurtured and developed at clearly and precisely, without any deviation from the original meaning.school level. Technology in education supports the mastery and achievement of theAs a field of study, Mathematics trains the mind to think logically and desired learning outcomes. Technology used in the teaching and learning ofsystematically in solving problems and making decisions. This discipline Mathematics, for example calculators, are to be regarded as tools to enhanceencourages meaningful learning and challenges the mind, and hence the teaching and learning process and not to replace teachers.contributes to the holistic development of the individual. To this end,strategies to solve problems are widely used in the teaching and learning of Importance is also placed on the appreciation of the inherent beauty ofmathematics. The development of mathematical reasoning is believed to be mathematics. Acquainting pupils with the life-history of well-knownclosely linked to the intellectual development and communication ability of mathematicians or events, the information of which is easily available frompupils. Hence, mathematics reasoning skills are also incorporated in the the Internet for example, will go a long way in motivating pupils tomathematics activities to enable pupils to recognize, build and evaluate appreciate mathematics.mathematics conjectures and statements. The intrinsic values of mathematics namely thinking systematically,In keeping with the National Education Philosophy, the Mathematics accurately, thoroughly, diligently and with confidence, infused throughoutcurriculum provides opportunities to pupils from various backgrounds and the teaching and learning process; contribute to the moulding of characterlevels of abilities to acquire mathematical skills and knowledge. Pupils are and the inculcation of positive attitudes towards mathematics. Together withthen able to seek relevant information, and be creative in formulating these, moral values are also introduced in context throughout the teachingalternatives and solutions when faced with challenges. and learning of mathematics. (ix)
  10. 10. Assessment, in the form of tests and examinations helps to gauge pupils’ • representing and interpreting data;achievements. The use of good assessment data from a variety of sourcesalso provides valuable information on the development and progress of • recognising and representing relationship mathematically;pupils. On-going assessment built into the daily lessons allows the • using algorithm and relationship;identification of pupils’ strengths and weaknesses, and effectiveness of theinstructional activities. Information gained from responses to questions, • solving problems; andgroup work results, and homework helps in improving the teaching process, • making decisions.and hence enables the provision of effectively aimed lessons. 4 communicate mathematically;AIM 5 apply knowledge and skills of mathematics in solving problems andThe mathematics curriculum for secondary schools aims to develop making decisions;individuals who are able to think mathematically, and apply mathematical 6 relate mathematics with other areas of knowledge;knowledge effectively and responsibly in solving problems and makingdecisions; and face the challenges in everyday life brought about by the 7 use suitable technologies in concept building, acquiring skills, solvingadvancement of science and technology. problems and exploring the field of mathematics; 8 acquire mathematical knowledge and develop skills effectively and use them responsibly;OBJECTIVES 9 inculcate a positive attitude towards mathematics; andThe mathematics curriculum for the secondary school enables pupils to: 10 appreciate the importance and beauty of mathematics.1 understand definitions, concepts, laws, principles, and theorems related to Number, Shape and Space, and Relationship;2 widen the use of basic operations of addition, subtraction, multiplication CONTENT ORGANISATION and division related to Number, Shape and Space, and Relationship; The Mathematics Curriculum content at the secondary school level is3 acquire basic mathematical skills such as: organised into three main areas, namely: Number; Shape and Space; and • making estimation and rounding; Relationship. Mathematical concepts related to the respective area, are further organised into topics. These topics are arranged in a hierarchical • measuring and constructing; manner such that the more basic and tangible concepts appear first and the • collecting and handling data; more complex and abstract concepts appear subsequently. (x)
  11. 11. The Learning Area outlines the scope of mathematical knowledge, abilities The Vocabulary consists of standard mathematical terms, instructionaland attitudes to be developed in pupils when learning the subject. They are words or phrases which are relevant in structuring activities, in askingdeveloped according to the appropriate learning objectives and represented questions or setting tasks. It is important to pay careful attention to the use ofin five columns, as follows: correct terminology and these need to be systematically introduced to pupils in various contexts so as to enable them to understand their meaning and Column 1 : Learning Objectives learn to use them appropriately. Column 2 : Suggested Teaching and Learning Activities Column 3 : Learning Outcomes Column 4 : Points To Note; and Column 5 : Vocabulary.The Learning Objectives define clearly what should be taught. They cover EMPHASES IN TEACHING AND LEARNINGall aspects of the Mathematics curriculum programme and are presented in a This Mathematics Curriculum is arranged in such a way so as to givedevelopmental sequence designed to support pupils understanding of the flexibility to teachers to implement an enjoyable, meaningful, useful andconcepts and skill of mathematics. challenging teaching and learning environment. At the same time, it isThe Suggested Teaching and Learning Activities lists some examples of important to ensure that pupils show progression in acquiring theteaching and learning activities including methods, techniques, strategies and mathematical concepts and skills.resources pertaining to the specific concepts or skills. These are, however, In determining the change to another learning area or topic, the followingnot the only intended approaches to be used in the classrooms. Teachers are have to be taken into consideration:encouraged to look for other examples, determine teaching and learningstrategies most suitable for their students and provide appropriate teaching • The skills or concepts to be acquired in the learning area or inand learning materials. Teachers should also make cross-references to other certain topics;resources such as the textbooks and the Internet. • Ensuring the hierarchy or relationship between learning areas orThe Learning Outcomes define specifically what pupils should be able to topics has been followed accordingly; anddo. They prescribe the knowledge, skills or mathematical processes and • Ensuring the basic learning areas have been acquired fully beforevalues that should be inculcated and developed at the appropriate level. progressing to more abstract areas.These behavioural objectives are measurable in all aspects. The teaching and learning processes emphasise concept building and skillIn the Points To Note column, attention is drawn to the more significant acquisition as well as the inculcation of good and positive values. Besidesaspects of mathematical concepts and skills. These emphases are to be taken these, there are other elements that have to be taken into account and infusedinto account so as to ensure that the concepts and skills are taught and learnt in the teaching and learning processes in the classroom. The main elementseffectively as intended. focused in the teaching and learning of mathematics are as follows: (xi)
  12. 12. 1. Problem Solving in Mathematics 2. Communication in MathematicsProblem solving is the main focus in the teaching and learning of Communication is an essential means of sharing ideas and clarifying themathematics. Therefore the teaching and learning process must include understanding of Mathematics. Through communication, mathematical ideasproblem solving skills which are comprehensive and cover the whole become the object of reflection, discussion and modification. The process ofcurriculum. The development of problem solving skills need to be analytical and systematic reasoning helps pupils to reinforce and strengthenemphasised so that pupils are able to solve various problems effectively. The their knowledge and understanding of mathematics to a deeper level.skills involved are: Through effective communication, pupils will become efficient in problem solving and be able to explain their conceptual understanding and • Understanding the problem; mathematical skills to their peers and teachers. • Devising a plan; • Carrying out the plan; and Pupils who have developed the skills to communicate mathematically will • Looking back at the solutions. become more inquisitive and, in the process, gain confidence. Communication skills in mathematics include reading and understandingVarious strategies and steps are used to solve problems and these are problems, interpreting diagrams and graphs, using correct and conciseexpanded so as to be applicable in other learning areas. Through these mathematical terms during oral presentations and in writing. The skillsactivities, students can apply their conceptual understanding of mathematics should be expanded to include listening.and be confident when facing new or complex situations. Among theproblem solving strategies that could be introduced are: Communication in mathematics through the listening process occurs when individuals respond to what they hear and this encourages individuals to • Trying a simple case; think using their mathematical knowledge in making decisions. • Trial and improvement; • Drawing diagrams; Communication in mathematics through the reading process takes place when an individual collects information and data and rearranges the • Identifying patterns; relationship between ideas and concepts. • Making a table, chart or systematic list; • Simulation; Communication in mathematics through the visualisation process takes place • Using analogies; when an individual makes an observation, analyses, interprets and • Working backwards; synthesises data and presents them in the form of geometric board, pictures • Logical reasoning; and and diagrams, tables and graphs. An effective communication environment • Using algebra. can be created by taking into consideration the following methods: • Identifying relevant contexts associated with environment and everyday life experience of pupils; • Identifying pupils’ interests; (xii)
  13. 13. • Identifying suitable teaching materials; • Keeping scrap books; • Ensuring active learning; • Keeping folios; • Stimulating meta-cognitive skills; • Keeping portfolios; • Inculcating positive attitudes; and • Undertaking projects; and • Setting up conducive learning environment. • Doing written tests.Effective communication can be developed through the following methods: Representation is a process of analysing a mathematical problem and interpreting it from one mode to another. Mathematical representationOral communication is an interactive process that involves psychomotor enables pupils to find relationships between mathematical ideas that areactivities like listening, touching, observing, tasting and smelling. It is a two- informal, intuitive and abstract using everyday language. For example 6xyway interaction that takes place between teacher and pupils, pupils and can be interpreted as a rectangular area with sides 2x and 3y. This will makepupils, and pupils and object. pupils realise that some methods of representation are more effective andSome of the more effective and meaningful oral communication techniques useful if they know how to use the elements of mathematical representation.in the learning of mathematics are as follows: • Story-telling and question and answer sessions using one’s own words; 3. Reasoning in Mathematics • Asking and answering questions; • Structured and unstructured interviews; Logical Reasoning or thinking is the basis for understanding and solving • Discussions during forums, seminars, debates and brainstorming mathematical problems. The development of mathematical reasoning is sessions; and closely related to the intellectual and communicative development of pupils. Emphasis on logical thinking, during mathematical activities opens up • Presentation of findings of assignments. pupils’ minds to accept mathematics as a powerful tool in the world today.Written communication is the process whereby mathematical ideas and Pupils are encouraged to estimate, predict and make intelligent guesses ininformation are disseminated through writing. The written work is usually the process of seeking solutions. Pupils at all levels have to be trained tothe result of discussion, input from people and brainstorming activities when investigate their predictions or guesses by using concrete material,working on assignments. Through writing, pupils will be encouraged to calculators, computers, mathematical representation and others. Logicalthink in depth about the mathematics content and observe the relationships reasoning has to be absorbed in the teaching of mathematics so that pupilsbetween concepts. Examples of written communication activities that can be can recognise, construct and evaluate predictions and mathematicaldeveloped through assignments are: arguments. • Doing exercises; • Keeping journals; (xiii)
  14. 14. 4. Mathematical Connections serve to stimulate pupils interests and develop a sense of responsibility towards their learning and understanding of mathematics.In the mathematics curriculum, opportunities for making connections must Technology however does not replace the need for all pupils to learn andbe created so that pupils can link conceptual to procedural knowledge and master the basic mathematical skills. Pupils must be able to efficiently add,relate topics within mathematics and other learning areas in general. subtract, multiply and divide without the use of calculators or otherThe mathematics curriculum consists of several areas such as arithmetic, electronic tools. The use of technology must therefore emphasise thegeometry, algebra, measures and problem solving. Without connections acquisition of mathematical concepts and knowledge rather than merelybetween these areas, pupils will have to learn and memorise too many doing calculation.concepts 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 life APPROACHES IN TEACHING AND LEARNINGwithin or outside the classroom, pupils will become more conscious of theimportance and significance of mathematics. They will also be able to use The belief on how mathematics is learnt influence how mathematicalmathematics contextually in different learning areas and in real life concepts are to be taught. Whatever belief the teachers subscribe to, the factsituations. remains that mathematical concepts are abstract. The use of teaching resources is therefore crucial in guiding pupils to develop matematical ideas. Teachers should use real or concrete materials to help pupils gain5. Application of Technology experience, construct abstract ideas, make inventions, build self confidence, encourage independence and inculcate the spirit of cooperation.The teaching and learning of mathematics should employ the latest The teaching and learning materials used should contain self diagnostictechnology to help pupils understand mathematical concepts in depth, elements so that pupils know how far they have understood the concepts andmeaningfully and precisely and enable them to explore mathematical ideas. acquire the skills.The use of calculators, computers, educational software, websites in theInternet and relevant learning packages can help to upgrade the pedagogical In order to assist pupils in having positive attitudes and personalities, theapproach and thus promote the understanding of mathematical concepts. intrinsic mathematical values of accuracy, confidence and thinking systematically have to be infused into the teaching and learning process.The use of these teaching resources will also help pupils absorb abstract Good moral values can be cultivated through suitable contexts. Learning inideas, be creative, feel confident and be able to work independently or in groups for example can help pupils to develop social skills, encouragegroups. Most of these resources are designed for self-access learning. cooperation and build self confidence. The element of patriotism should alsoThrough self-access learning pupils will be able to access knowledge or be inculcated through the teaching and learning process in the classroomskills and informations independently according to their own pace. This will using certain topics. (xiv)
  15. 15. Brief historical anecdotes related to aspects of mathematics and famous assessment can be conducted. These maybe in the form of assignments, oralmathematicians associated with the learning areas are also incorporated into questioning and answering, observations and interviews. Based on thethe curriculum. It should be presented at appropriate points where it provides response, teachers can rectify pupils misconceptions and weaknesses andstudents with a better understanding and appreciation of mathematics. also improve their own teaching skills. Teachers can then take subsequent effective measures in conducting remedial and enrichment activities inVarious teaching strategies and approaches such as direct instruction, upgrading pupils’ performances.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 and learning.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 of (xv)
  16. 16. Form 5 LEARNING AREA: 1LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Use models such as a clock face or a (i) State zero, one, two, three,…, as Emphasise the ways toconcept of number in base counter which uses a particular number a number in base: read numbers in varioustwo, eight and five. base. bases. a) two, Examples: Number base blocks of twos, eights and b) eight, • 1012 is read as “one fives can be used to demonstrate the c) Five. zero one base two” value of a number in the respective • 72058 is read as number bases. (ii) State the value of a digit of a “seven two zero five number in base: base eight” For example: • 4325 is read as “four a) two, three two base five” 2435 is b) eight, Numbers in base two are c) Five. also known as binary numbers. Examples of numbers in (iii) Write a number in base: expanded notation: expanded notation 2 4 3 a) two, • 101102 = 1 × 24 + 0 × 23 + 1 × 22 Discuss: b) eight, + 1 × 21 • digits used, c) five + 0 × 20 • place values in expanded notation. • 3258 = 3 × 82 + 2 × 81 in the number system with a particular + 5 × 80 number base. • 30415 = 3 × 53 + 0 × 52 + 4 × 51 + 1 × 50 1
  17. 17. Form 5 LEARNING AREA: 1LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… Number base blocks of twos, eights and (iv) Convert a number in base: Perform repeated convert fives can also be used here. For example, division to convert a a) two, number in base ten to a to convert 1010 to a number in base two, use the least number of blocks (23), tiles b) eight, number in other bases. (22), rectangles (21) and squares (20). c) five For example, convert Here, the least number of objects here are 714 10 to a number in one block, no tiles, one rectangle and no to a number in base ten and vice squares. So, 1010 = 10102. versa. base five: 5)714 5)142 --- 4 5) 28 --- 2 5) 5 --- 3 5) 1 --- 0 0 --- 1 ∴ 714 10 = 10324 5 2
  18. 18. Form 5 LEARNING AREA: 1LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… Discuss the special case of converting a (v) Convert a number in a certain Limit conversion of number in base two directly to a number in base to a number in another numbers to base two, base eight and vice-versa. base. eight and five only. For example, convert a number in base two directly to a number in base eight through grouping of three consecutive digits. Perform direct addition and subtraction in (vi) Perform computations the conventional manner. involving: For example: a) addition, 10102 b) subtraction, + 1102 of two numbers in base two. 3
  19. 19. Form 5 LEARNING AREA: 2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Explore graphs of functions by using (i) Draw the graph of a: linear functionconcept of graph of graphing calculator or Geometer’s a) linear function: y = ax + b,functions. Sketchpad. where a and b are constants, quadratic Compare the characteristics of graphs of function b) quadratic function: functions with different values of constants. y = ax 2 + bx + c, For example: where a, b and c are constants , a ≠ 0, c) cubic function: Limit cubic functions to cubic function y = ax 3 + bx 2 + cx + d, the following forms: where a, b, c and d are • y = ax 3 constants, a ≠ 0 , • y = ax 3 + b a • y = x 3 + bx + c reciprocal A B d) reciprocal function: y = , x • y = −x 3 + bx + c function The curve in graph B is wider than the where a is a constant, curve in graph A and intersects the vertical a≠ 0. axis above the horizontal axis. (ii) Find from a graph: For certain functions and a) the value of y, given a value some values of y, there of x, could be no corresponding values of b) the value(s) of x, given a x. value of y. 4
  20. 20. Form 5 LEARNING AREA: 2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (iii) Identify: a) the shape of graph given a type of function, b) the type of function given a graph, As reinforcement, let pupils play a game; c) the graph given a function For graphs of cubic for example, matching cards of graphs and vice versa. functions, limit to with their respective functions. When the y = ax3 and y = ax3 + b. pupils have their matching partners, ask For graphs of quadratic them to group themselves into four functions, limit to groups of types of functions. Finally, ask y = ax2 + b and quadratic each group to name the type of function functions which can be that is depicted on the cards. factorised to (mx + n)(px + q) where m, n, p, q are integers. (iv) Sketch the graph of a given For graphs of cubic linear, quadratic, cubic or functions, limit to reciprocal function. y = ax 3 and y = ax 3 + b. 5
  21. 21. Form 5 LEARNING AREA: 2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the Explore on the graphing calculator or the (i) Find the point(s) of intersection Use the traditional point ofconcept of the solution of Geometer’s Sketchpad to relate the of two graphs. graph plotting exercise intersectionan equation by graphical x-coordinate of a point of intersection of if the graphingmethod. two appropriate graphs to the solution of calculator or the (ii) Obtain the solution of an a given equation. Make generalisation Sketchpad is equation by finding the point(s) about the point(s) of intersection of two unavailable. of intersection of two graphs. graphs. (iii) Solve problems involving Involve everyday solution of an equation by problems. graphical method.3 Understand and use the (i) Determine whether a given point For Learning Objectiveconcept of the region satisfies: 3, include situationsrepresenting inequalities in y = ax + b, or involving x = a, x ≥ a,two variables. x > a, x ≤ a or x < a. y > ax + b, or y < ax + b. (ii) Determine the position of a given point relative to the graph y = ax + b. Discuss that if one point in a region (iii) Identify the region satisfying satisfies y > ax + b or y < ax + b, then all y > ax + b or y < ax + b. points in the region satisfy the same inequality. 6
  22. 22. Form 5 LEARNING AREA: 2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… Use the Sketchpad or the graphing (iv) Shade the regions representing Emphasise that: region calculator to explore points relative to a the inequalities: • for the region graph to make generalisation about a) y > ax + b or y < ax + b, representing regions satisfying the given inequalities. b) y ≥ ax + b or y ≤ ax + b. y > ax + b or y < ax + b, the line y = ax + b is drawn as a dashed line to dashed line indicate that all points on the line y = ax + b are not in the region. • for the region representing y ≥ ax + b or y ≤ ax + b, the line y = ax + b is drawn as a solid line to solid line indicate that all points on the line y = ax + b are in the region. Carry out activities using overhead (v) Determine the region which projector and transparencies, graphing satisfies two or more calculator or Geometer’s Sketchpad. simultaneous linear inequalities. 7
  23. 23. Form 5 LEARNING AREA: 3LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Relate to transformations in real life (i) Determine the image of an Begin with a point,concept of combination of situations such as tessellation patterns on object under combination of two followed by a line andtwo transformations. walls, ceilings or floors. isometric transformations. an object. Explore combined transformation using (ii) Determine the image of an Limit isometric Geometer’s Sketchpad, graphing object under combination of: transformations to calculator or overhead projector and translations, reflections a) two enlargements, transparencies. and rotations. b) an enlargement and an Investigate the characteristics of an isometric transformation. object and its image under combined transformation. (iii) Draw the image of an object under combination of two transformations. (iv) State the coordinates of the combined image of a point under transformation combined transformation. (v) Determine whether combined equivalent transformation AB is equivalent to combined transformation BA. 8
  24. 24. Form 5 LEARNING AREA: 3LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… Carry out projects to design patterns (vi) Specify two successive specify using combined transformations that can transformations in a combined be used as decorative purposes. These transformation given the object projects can then be presented in class and the image. with the students describing or specifying the transformations involved. Use the Sketchpad to prove the single (vii) Specify a transformation which is Limit the equivalent transformation which is equivalent to the equivalent to the combination of transformation to combination of two isometric two isometric transformations. translation, reflection transformations. and rotation. (viii) Solve problems involving transformation. 9
  25. 25. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the Represent data in real life situations, for (i) Form a matrix from given Emphasise that matrices matrixconcept of matrix. example, the price of food on a menu, in information. are written in brackets. row table form and then in matrix form. (ii) Determine: Introduce row matrix, column column matrix and order a) the number of rows, square matrix. b) the number of columns, row matrix c) the order Emphasise that a matrix column matrix of order m × n is read as of a matrix. “an m by n matrix”. square matrix Use pupils seating positions in class by (iii) Identify a specific element in a Use row number and rows and columns to identify a pupils who matrix. column number to is sitting in a particular row and in a specify the position of particular column as a concrete example. an element.2 Understand and use the Discuss equal matrices in terms of: (i) Determine whether two matrices equal matricesconcept of equal matrices. are equal; • the order, • the corresponding elements. (ii) Solve problems involving equal Include finding values of unknown matrices. unknown elements. elements 10
  26. 26. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…3 Perform addition and (i) Determine whether addition orsubtraction on matrices. subtraction can be performed on two given matrices. Relate to real life situations such as (ii) Find the sum or the difference of Limit to matrices with keeping score of medal tally or points in two matrices. no more than three rows sports. and three columns. (iii) Perform addition and subtraction on a few matrices. (iv) Solve matrix equations Include finding values of matrix equation involving addition and unknown elements. subtraction.4 Perform multiplication Relate to real life situations such as in (i) Multiply a matrix by a number. Multiplying a matrix by scalarof a matrix by a number. industrial productions. a number is known as multiplication (ii) Express a given matrix as scalar multiplication. multiplication of a matrix by a number. (iii) Perform calculation on matrices involving addition, subtraction and scalar multiplication. 11
  27. 27. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (iv) Solve matrix equations Include finding values involving addition, subtraction of unknown elements. and scalar multiplication.5 Perform multiplication Relate to real life situations such as finding (i) Determine whether two matrices productof two matrices. the cost of a meal in a restaurant. can be multiplied and state the order of the product when the two matrices can be multiplied. For matrices A and B, discuss the (ii) Find the product of two Limit to matrices with relationship between AB and BA.. matrices. no more than 3 rows and 3 columns. (iii) Solve matrix equations Limit to two unknown involving multiplication of two elements. matrices.6 Understand and use the Begin with discussing the property of the (i) Determine whether a given Identity matrix is identity matrixconcept of identity matrix. number 1 as an identity for multiplication matrix is an identity matrix by usually denoted by I and unit matrix of numbers. multiplying it to another matrix. is also known as unit matrix. (ii) Write identity matrix of any Discuss: order. • an identity matrix is a square matrix, • there is only one identity matrix for each order. 12
  28. 28. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… Discuss the properties: (iii) Perform calculation involving Limit to matrices with identity matrices. no more than 3 rows and • AI = A, 3 columns. • IA = A..7 Understand and use the Relate to property of multiplicative inverse (i) Determine whether a 2 × 2 The inverse of matrix A inverse matrixconcept of inverse matrix. of numbers. matrix is the inverse matrix of is denoted by A−1. For example: another 2 × 2 matrix. Emphasise that: 2 × 2−1 = 2−1 × 2 = 1 • if matrix B is the −1 inverse of matrix A, In this example, 2 is the multiplicative then matrix A is also inverse of 2 and vice versa. the inverse of matrix B, AB = BA = I; • inverse matrices can only exist for square matrices, but not all square matrices have inverse matrices. 13
  29. 29. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (ii) Find the inverse matrix of a 2 × Steps to find the inverse 2 matrix by using: matrix: a) method of solving • solving simultaneous Use the method of solving simultaneous simultaneous linear linear equations linear equations to show that not all square equations, matrices have inverse matrices. For ⎛1 2⎞ ⎛ p q ⎞ ⎛ 1 0⎞ example, ask pupils to try to find the b) formula. ⎜ ⎜3 4⎟⎜ r s ⎟ = ⎜0 1⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎛ 3 2⎞ inverse matrix of ⎜⎜ 6 4⎟ . ⎟ p + 2r = 1, q + 2s = 0 ⎝ ⎠ 3p + 44 = 0, 3q + 4s = 1 ⎛ p q⎞ where ⎜ ⎜ r s⎟ ⎟ is the ⎝ ⎠ inverse matrix; 14
  30. 30. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… Carry out operations leading to the • using formula formula. ⎛a b ⎞ Use matrices and their respective inverse For A = ⎜ ⎜c d ⎟ ⎟ , matrices in the previous method to relate to ⎝ ⎠ the formula. Express each inverse matrix as ⎛ d −b ⎞ a multiplication of a matrix by a number. −1⎜ ⎟ A =⎜ ad − bc ad − bc ⎟ Compare the scalar multiplication to the ⎜ −c a ⎟ original matrix and discuss how the ⎜ ⎟ ⎝ ad − bc ad − bc ⎠ determinant is obtained. or A−1 = 1 ⎛ d − b⎞ ⎜ ⎟ ad − bc ⎜ − c a ⎟ ⎝ ⎠ when ad − bc ≠ 0. ad − bc is known as the determinant of the matrix A. Discuss the condition for the existence of A−1 does not exist if the inverse matrix. determinant is zero. 15
  31. 31. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…8 Solve simultaneous Relate to equal matrices by witing down (i) Write down simultaneous linear Limit to two unknowns.linear equations by using the simultaneous equations as equal equations in matrix form. Simultaneous linearmatrices. matrices first. For example: equations Write 2x + 3y = 13 ap + bq = h 4x − y = 5 cp + dq = k as equal matrices: in matrix form is ⎛ 2 x + 3 y ⎞ ⎛13 ⎞ ⎜ ⎜ 4x − y ⎟ = ⎜ 5 ⎟ ⎟ ⎜ ⎟ ⎛a b ⎞ ⎛ p⎞ ⎛h⎞ ⎝ ⎠ ⎝ ⎠ ⎜ ⎜c d ⎟ ⎜ q ⎟ = ⎜k ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ which is then expressed as: where a, b, c, d, h and k ⎛ 2 3 ⎞ ⎛ x ⎞ ⎛13 ⎞ are constants, p and q ⎜ ⎜ 4 − 1⎟ ⎜ y ⎟ = ⎜ 5 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ are unknowns. Discuss why: ⎛ p⎞ A−1 ⎛ a b ⎞ ⎛ p ⎞ = A−1 ⎛ h ⎞ (ii) Find the matrix ⎜ ⎟ in ⎜q⎟ ⎜c d⎟⎜ q⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜k ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ • the use of inverse matrix is necessary. Relate to solving linear equations of ⎛a b ⎞ ⎛ p⎞ ⎛h⎞ ⎛a b ⎞ . where A = ⎜ ⎜c d⎟ . type ax = b, ⎜ ⎜c d ⎟ ⎜ q ⎟ = ⎜k ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ • it is important to place the inverse matrix at the right place on both sides of by using inverse matrix. the equation. 16
  32. 32. Form 5 LEARNING AREA: 4LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to… (iii) Solve simultaneous linear The matrix method uses matrix method equations by using the matrix inverse matrix to solve method. simultaneous linear equations. Relate the use of matrices in other areas (iv) Solve problems involving such as in business or economy, science matrices. and so on. Carry out projects involving matrices using spreadsheet softwares. 17
  33. 33. Form 5 LEARNING AREA: 5LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…1 Understand and use the (i) State the changes in a quantity y varies directly as x if direct variationconcept of direct variation. with respect to changes in y and only if is a quantity another quantity, in everyday x life situations involving direct constant. variation. Discuss the characteristics of the graph (ii) Determine from given If y varies directly as x, of y against x when y ∝ x. information whether a quantity the relation is written as varies directly as another y ∝x . Relate to other areas like science and quantity. For the cases y ∝ xn, technology. For example, the Law of limit n to 2, 3 and 1 . Charles and Gay-Lussac (or Charles’ 2 Law), Hooke’s Law and the simple pendulum. (iii) Express a direct variation in the If y ∝ x, then y = kx constant of form of equation involving two where k is the constant variation variables. of variation. variable (iv) Find the value of a variable in a Using: direct variation when sufficient information is given. • y = kx; or y1 y2 For the cases y ∝ x n , n = 2, 3, 1 , (v) Solve problems involving direct • = 2 x1 x2 variation for the following cases: discuss the characteristics of the graphs 1 to get the solution. of y against x n . y ∝ x; y ∝ x ; y ∝ x ; y ∝ x . 2 3 2 18
  34. 34. Form 5 LEARNING AREA: 5LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES POINTS TO NOTE VOCABULARYPupils will be taught to… LEARNING ACTIVITIES Pupils will be able to…2 Understand and use the (i) State the changes in a quantity y varies inversely as x if inverse variationconcept of inverse with respect to changes in and only if xy is avariation. another quantity, in everyday constant. life situations involving inverse variation. Discuss the form of the graph of y (ii) Determine from given If y varies inversely as x, 1 1 information whether a quantity the relation is written as against when y ∝ . varies inversely as another 1 x x quantity. y∝ . x Relate to other areas like science and 1 technology. For example, Boyle’s Law. For the cases y ∝ , xn limit n to 2, 3 and 1 . 2 1 k (iii) Express an inverse variation in If y ∝ , then y = the form of equation involving x x two variables. where k is the constant of variation. 19