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ministry of education malaysia integrated curriculum for secondary schools curriculum specifications mathematics form 2 curriculum development centre ministry of education malaysia 2002 iii

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contents page rukunegara national philosophy of education preface introduction directed numbers squares square roots cubes and cube roots algebraic expressions ii linear equations ratios rates and proportions pythagoras theorem geometrical constructions coordinates loci in two dimensions circles transformations solid geometry ii statistics contributors v vii ix 1 10 14 22 26 30 34 36 38 42 45 51 56 58 61 iii

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rukunegara declaration our nation malaysia being dedicated to achieving a greater unity of all her peoples to maintaining a democratic way of life to creating a just society in which the wealth of the nation shall be equitably shared to ensuring a liberal approach to her rich and diverse cultural traditions to building a progressive society which shall be oriented to modern science and technology we her peoples pledge our united efforts to attain these ends guided by these principles belief in god loyalty to king and country upholding the constitution rule of law good behaviour and morality v

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national philosophy of education education in malaysia is an on-going effort towards developing the potential of individuals in a holistic and integrated manner so as to produce individuals who are intellectually spiritually emotionally and physically balanced and harmonious based on a firm belief in and devotion to god such an effort is designed to produce malaysian citizens who are knowledgeable and competent who possess high moral standards and who are responsible and capable of achieving a high level of personal well being as well as being able to contribute to the harmony and betterment of the family society and the nation at large vii

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preface science and technology plays a critical role in meeting malaysia s aspiration to achieve developed nation status since mathematics is instrumental in developing scientific and technological knowledge the provision of quality mathematics education from an early age in the education process is important the secondary school mathematics curriculum as outlined in the syllabus has been designed to provide opportunities for pupils to acquire mathematical knowledge and skills and develop the higher order problem solving and decision making skills that they can apply in their everyday lives but more importantly together with the other subjects in the secondary school curriculum the mathematics curriculum seeks to inculcate noble values and love for the nation towards the final aim of developing the wholistic person who is capable of contributing to the harmony and prosperity of the nation and its people beginning in 2003 science and mathematics will be taught in english following a phased implementation schedule which will be completed by 2008 mathematics education in english makes use of ict in its delivery studying mathematics in the medium of english assisted by ict will provide greater opportunities for pupils to enhance their knowledge and skills because they are able to source the various repositories of mathematical knowledge written in english whether in electronic or print forms pupils will be able to communicate mathematically in english not only in the immediate environment but also with pupils from other countries thus increasing their overall english proficiency and mathematical competence in the process the development of this curriculum specifications accompanying the syllabus is the work of many individuals expert in the field to those who have contributed in one way or another to this effort on behalf of the ministry of education i would like to express my deepest gratitude and appreciation dr sharifah maimunah syed zin director curriculum development centre ministry of education malaysia ix

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introduction the vision of the country to become an industrialised nation by the end of the second decade can be achieved through a society that is educated and competent in the application of mathematical knowledge to achieve this vision it is important that the society must be inclined towards mathematics therefore problem solving and communicational skills in mathematics have to be nurtured from an early age so that effective decisions can be made later in life mathematics is instrumental in the development of science and technology as such the acquisition of mathematical knowledge must be intensified so as to create a skilled workforce as a requirement for a country in achieving a developed nation status in order to create a k-based economy research and development skills in mathematics must be instilled at school level based on the national education policy and the 2020 vision the mathematics curriculum has been reviewed and revised the rationale behind this move is the need to provide mathematical knowledge and skills to students from various backgrounds and levels of ability therefore it is the country s hope that with the knowledge and skills acquired in mathematics 1 students will be able to explore discover adapt modify and be innovative in facing ongoing changes and future challenges acquisition of these skills will help propel them forward in their future careers thus benefiting the individual society and the nation the mathematics curriculum kbsm is a continuum from form 1 through form 5 the content is categorised into three interrelated areas these are numbers shapes and spaces and relationships this categorisation is based on the fact that in any situation it is imperative that a person has knowledge and skills related to counting be able to recognise shapes and measurements as well as recognise relationships between numbers and shapes aim the mathematics curriculum for secondary school aims to develop individuals who are able to think mathematically and who can apply mathematical knowledge effectively and responsibly in solving problems and making decision this will enable the individual to face challenges in everyday life that arise due to the advancement of science and technology.

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objectives the mathematics curriculum for the secondary school enables students to 1 understand definitions concepts laws principles and theorems related to numbers shape and space and relationships widen applications of basic fundamental skills such as addition subtraction multiplication and division related to numbers shape and space and relationships 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 problem and · making decision communicate mathematically apply knowledge and the skills of mathematics in solving problems and making decisions 2 6 relate mathematics with other areas of knowledge use suitable technologies in concept building acquiring skills solving problems and exploring the field of mathematics cultivate mathematical knowledge and skills effectively and responsibly inculcate positive attitudes towards mathematics and appreciate the importance and the beauty of mathematics 7 2 8 9 3 10 content organisation the mathematics curriculum at the secondary level encompasses three main areas namely numbers shape and space and relationship the topics for each area have been been arranged according to hierarchy the basics have to be taught before abstract concepts can be introduced to students 4 5.

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the learning area outlines the scope of knowledge and skills which have to be mastered in the learning duration of the subject they are developed according to the appropriate learning objectives and represented in five columns as follows column 1 learning objectives column 2 suggested teaching and learning activities column 3 learning outcomes column 4 points to note and column 5 vocabulary determine teaching and learning strategies most suitable for their students and provide appropriate teaching and learning materials teachers should also make cross-references to other resources such as the textbooks and the internet the learning outcomes define specifically what students should be able to do they prescribe the knowledge skills or mathematical processes and values that should be inculcated and developed at the appropriate level these behavioural objectives are measureable in all aspects in the points to note column attention is drawn to the more significant aspects of mathematical concepts and skills these emphases are to be taken into account so as to ensure that the concepts and skills are taught and learnt effectively as intended the vocabulary consists of standard mathematical terms instructional words or phrases which are relevant in structuring activities in asking questions or setting tasks it is important to pay careful attention to the use of correct terminology and these need to be systematically introduced to students in various contexts so as to enable them to understand their meaning and learn to use them appropriately the learning objectives define clearly what should be taught they cover all aspects of the mathematics curriculum programme and are presented in a developmental sequence designed to support students understanding of the concepts and skill of mathematics the suggested teaching and learning activities lists some examples of teaching and learning activities including methods techniques strategies and resources pertaining to the specific concepts or skills these are however not the only intended approaches to be used in the classrooms teachers are encouraged to look for other examples 3

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emphases in teaching and learning this mathematics curriculum is arranged in such a way so as to give flexibility to teachers to implement an enjoyable meaningful useful and challenging teaching and learning environment at the same time it is important to ensure that students show progression in acquring the mathematical concepts and skills in determining the change to another learning area or topic the following have to be taken into consideration · · · the skills or concepts to be acquired in the learning area or in certain topics ensuring the hierarchy or relationship between learning areas or topics has been followed accordingly and ensuring the basic learning areas have been acquired fully before progressing to more abstract areas 1 problem solving in mathematics problem solving is the main focus in the teaching and learning of mathematics therefore the teaching and learning process must include problem solving skills which are comprehensive and cover the whole curriculum the development of problem solving skills need to be emphasised so that students are able to solve various problems effectively the skills 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 are expanded so as to be applicable in other learning areas through these activities students can apply their conceptual understanding of mathematics and be confident when facing new or complex situations among the problem solving strategies that could be introduced are · · · · · · 4 trying a simple case trial and improvement drawing diagrams identifying patterns making a table chart or systematic list simulation the teaching and learning processes emphasise concept building and skill acqusition as well as the inculcation of good and positive values besides these there are other elements that have to be taken into account and infused in the teaching and learning processes in the classroom the main elements focused in the teaching and learning of mathematics are as follows:

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· · · · 2 using analogies working backwards logical reasoning and using algebra writing the skill should be expanded to include listening communication in mathematics through the listening process occurs when individuals respond to what they hear and this encourages individuals to think using their mathematical knowledge in making decisions communication in mathematics through the reading process takes place when an individual collects information and data and rearranges the relationship between ideas and concepts communication in mathematics through the visualisation process takes place when an individual makes an observation analyses interprets and synthesises data and presents them in the form of geometric board pictures and diagrams tables and graphs an effective communication environment can be created by taking into consideration the following methods · identifying relevant contexts associated with environment and everyday life experience of students · identifying students interests · identifying suitable teaching materials · ensuring active learning · stimulating meta-cognitive skills 5 communication in mathematics communication is an essential means of sharing ideas and clarifying the understanding of mathematics through communication mathematical ideas become the object of reflection discussion and modification the process of analytical and systematic reasoning helps students to reinforce and strengthen their knowledge and understanding of mathematics to a deeper level through effective communication students will become efficient in problem solving and be able to explain their conceptual understanding and mathematical skills to their peers and teachers students who have developed the skills to communicate mathematically will become more inquisitive and in the process gain confidence communicational skills in mathematics include reading and understanding problems interpreting diagrams and graphs using correct and concise mathematical terms during oral presentations and in

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· · 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 psychomotor activities like listening touching observing tasting and smelling it is a twoway interaction that takes place between teacher and student student and student and student and object some of the more effective and meaningful oral communication techniques in the learning of mathematics are as follows · story-telling question and answer sessions using one s own words · asking and answering questions · structured and unstructured interviews · discussions during forums seminars debates and brainstorming sessions and · presentation of findings of assignments written communication is the process whereby mathematical ideas and information are disseminated through writing the written work is usually the result of discussion input from people and brainstorming activities when working on assignments through writing students will be encouraged to think in depth about the mathematics content and observe the relationships between concepts examples of written communication activities that can be developed through assignments are · · · · · · · doing exercises keeping journal keeping scrap books keeping folio keeping portfolios undertaking projects and doing written tests representation is a process of analysing a mathematical problem and interpreting it from one mode to another mathematical representation enables students to find relationships between mathematical ideas that are informal intuitive and abstract using everyday language for example 6xy can be interpreted as a rectangular area with sides 2x and 3y this will make students realise that some methods of representation are more effective and useful if they know how to use the elements of mathematical representation 6

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3 reasoning in mathematics logical reasoning or thinking is the basis for understanding and solving mathematical problems the development of mathematical reasoning is closely related to the intellectual and communicative development of students emphasis on logical thinking during mathematical activities opens up students minds to accept mathematics as a powerful tool in the world today students are encouraged to estimate predict and make intelligent guesses in the process of seeking solutions students at all levels have to be trained to investigate their predictions or guesses by using concrete material calculators computers mathematical representation and others logical reasoning has to be absorbed in the teaching of mathematics so that students can recognise construct and evaluate predictions and mathematical arguments 4 mathematical connections areas in general the mathematics curriculum consists of several areas such as arithmetic geometry algebra measures and problem solving without connections between these areas students will have to learn and memorise too many concepts and skills separately by making connections students are able to see mathematics as an integrated whole rather than a jumble of unconnected ideas when mathematical ideas and the curriculum are connected to real life within or outside the classroom students will become more conscious of the importance and significance of mathematics they will also be able to use mathematics contextually in different learning areas and in real life situations 5 application of technology in the mathematics curriculum opportunities for making connections must be created so that students can link conceptual to procedural knowledge and relate topics within mathematics and other learning the teaching and learning of mathematics should employ the latest technology to help students understand mathematical concepts in depth meaningfully and precisely and enable them to explore mathematical ideas the use of calculators computers educational software websites in the internet and relevant learning packages can help to upgrade the pedagogical approach and thus promote the understanding of mathematical concepts 7

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