Integrated Curriculum for Secondary Schools Curriculum Specifications MATHEMATICS FORM 4 Curriculum Development Centre Ministry of Education Malaysia 2006 Copyright 2006 Curriculum Development Centre Ministry of Education Malaysia Aras 4 - 8, Blok E9 Kompleks Kerajaan Parcel E Pusat Pentadbiran Kerajaan Persekutuan 62604 Putrajaya First published 2006 Copyright reserved. Except for use in a review, the reproduction or utilisation of this work in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, and recording is forbidden without the prior written permission from the Director of the Curriculum Development Centre, Ministry of Education Malaysia. CONTENTS Page RUKUNEGARA iv National Philosophy of Education v Preface vi Introduction vii Standard Form 1 Quadratic Expressions and Equations 2 Sets 4 Mathematical Reasoning 8 The Straight Line 16 Statistics 20 Probability I 24 Circles III 26 Trigonometry II 29 Angles of Elevation and Depression 33 Lines and Planes in 3-Dimensions 34 (iii) RUKUNEGARA DECLARATION OUR NATION, MALAYSIA, being dedicated • • • to achieving a greater unity of all her peoples; • 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; to maintaining a democratic way of life; to creating a just society in which the wealth of the nation shall be equitably shared; 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 (iv) Education in Malaysia is an ongoing effort towards further developing the potential of individuals in a holistic and integrated manner so as to produce individuals who are intellectually, spiritually, emotionally and physically balanced and harmonious, based on a firm belief in God. Such an effort is designed to produce Malaysian citizens who are knowledgeable and competent, who possess high moral standards, and who are responsible and capable of achieving a high level of personal well-being as well as being able to contribute to the betterment of the family, the society and the nation at large. (v) (vi) PREFACE Science and technology plays a critical role in realising Malaysia’s aspiration to become a developed nation. Since mathematics is instrumental in the development of scientific and technological knowledge, the provision of quality mathematics education from an early age in the education process is thus important. The Malaysian school curriculum offers three mathematics education programs, namely Mathematics for primary schools, Mathematics and Additional Mathematics for secondary schools. in English and thus make the learning of mathematics more interesting and exciting. The development of this Mathematics syllabus is the work of many individuals and experts in the field. On behalf of the Curriculum Development Centre, I would like to express much gratitude and appreciation to those who have contributed in one way or another towards this initiative. The Malaysian school mathematics curriculum aims to develop mathematical knowledge, competency and inculcate positive attitudes towards mathematics among pupils. Mathematics for secondary schools provides opportunities for pupils to acquire mathematical knowledge and skills, and develop higher order problem solving and decision making skills to enable pupils to cope with daily life challenges. As with other subjects in the secondary school curriculum, Mathematics aims to inculcate noble values and love for the nation in the development of a holistic person, who in turn will be able to contribute to the harmony and prosperity of the nation and its people. Beginning 2003, English is used as the medium of instruction for Science and Mathematics subjects. The policy to change the medium of instruction for Science and Mathematics subjects follows a phased implementation schedule and is expected to be completed by 2008. In the teaching and learning of Mathematics, the use of technology especially ICT is greatly emphasised. Mathematics taught in English, coupled with the use of ICT, provide greater opportunities for pupils to improve their knowledge and skills in mathematics because of the richness of resources and repositories of knowledge in English. Pupils will be better able to interact with pupils from other countries, improve their proficiency (MAHZAN BIN BAKAR SMP, AMP) Director Curriculum Development Centre Ministry of Education Malaysia (vii) (viii) INTRODUCTION The general Mathematics curriculum has often been seen to comprise of discrete areas related to counting, measurement, geometry, algebra and solving problems. To avoid the areas to be continually seen as separate and pupils acquiring concepts and skills in isolation, mathematics is linked to everyday life and experiences in and out of school. Pupils will have the opportunity to apply mathematics in different contexts, and see the relevance of mathematics in daily life. A well-informed and knowledgeable society well versed in the use of mathematics to cope with daily life challenges is integral to realising the nation’s aspiration to become an industrialised nation. Thus, efforts are taken to ensure a society that assimilates mathematics into their daily lives. Pupils are nurtured from an early age with the skills to solve problems and communicate mathematically, to enable them to make effective decisions. In giving opinions and solving problems either orally or in writing, pupils are guided in the correct usage of language and mathematics registers. Pupils are trained to select information presented in mathematical and nonmathematical language; interpret and represent information in tables, graphs, diagrams, equations or inequalities; and subsequently present information clearly and precisely, without any deviation from the original meaning. Mathematics is essential in preparing a workforce capable of meeting the demands of a progressive nation. As such, this field assumes its role as the driving force behind various developments in science and technology. In line with the nation’s objective to create a knowledge-based economy, the skills of Research & Development in mathematics is nurtured and developed at school level. Technology in education supports the mastery and achievement of the desired learning outcomes. Technology used in the teaching and learning of Mathematics, for example calculators, are to be regarded as tools to enhance the teaching and learning process and not to replace teachers. As a field of study, Mathematics trains the mind to think logically and systematically in solving problems and making decisions. This discipline encourages meaningful learning and challenges the mind, and hence contributes to the holistic development of the individual. To this end, strategies to solve problems are widely used in the teaching and learning of mathematics. The development of mathematical reasoning is believed to be closely linked to the intellectual development and communication ability of pupils. Hence, mathematics reasoning skills are also incorporated in the mathematics activities to enable pupils to recognize, build and evaluate mathematics conjectures and statements. Importance is also placed on the appreciation of the inherent beauty of mathematics. Acquainting pupils with the life-history of well-known mathematicians or events, the information of which is easily available from the Internet for example, will go a long way in motivating pupils to appreciate mathematics. The intrinsic values of mathematics namely thinking systematically, accurately, thoroughly, diligently and with confidence, infused throughout the teaching and learning process; contribute to the moulding of character and the inculcation of positive attitudes towards mathematics. Together with these, moral values are also introduced in context throughout the teaching and learning of mathematics. In keeping with the National Education Philosophy, the Mathematics curriculum provides opportunities to pupils from various backgrounds and levels of abilities to acquire mathematical skills and knowledge. Pupils are then able to seek relevant information, and be creative in formulating alternatives and solutions when faced with challenges. (ix) • representing and interpreting data; Assessment, in the form of tests and examinations helps to gauge pupils’ achievements. The use of good assessment data from a variety of sources also provides valuable information on the development and progress of pupils. On-going assessment built into the daily lessons allows the identification of pupils’ strengths and weaknesses, and effectiveness of the instructional 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. • recognising and representing relationship mathematically; • using algorithm and relationship; • solving problems; and • making decisions. AIM The mathematics curriculum for secondary schools aims to develop individuals who are able to think mathematically, and apply mathematical knowledge effectively and responsibly in solving problems and making decisions; and face the challenges in everyday life brought about by the advancement of science and technology. OBJECTIVES The mathematics curriculum for the secondary school enables pupils to: Number, Shape and Space, and Relationship; and division related to Number, Shape and Space, and Relationship; measuring and constructing; • collecting and handling data; apply knowledge and skills of mathematics in solving problems and making decisions; 6 relate mathematics with other areas of knowledge; 7 use suitable technologies in concept building, acquiring skills, solving problems and exploring the field of mathematics; 8 acquire mathematical knowledge and develop skills effectively and use them responsibly; 9 inculcate a positive attitude towards mathematics; and The Mathematics Curriculum content at the secondary school level is organised into three main areas, namely: Number; Shape and Space; and Relationship. Mathematical concepts related to the respective area, are further organised into topics. These topics are arranged in a hierarchical manner such that the more basic and tangible concepts appear first and the more complex and abstract concepts appear subsequently. 3 acquire basic mathematical skills such as: • 5 CONTENT ORGANISATION 2 widen the use of basic operations of addition, subtraction, multiplication making estimation and rounding; communicate mathematically; 10 appreciate the importance and beauty of mathematics. 1 understand definitions, concepts, laws, principles, and theorems related to • 4 (x) 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 pupils in various contexts so as to enable them to understand their meaning and learn to use them appropriately. The Learning Area outlines the scope of mathematical knowledge, abilities and attitudes to be developed in pupils when learning the subject. They are developed according to the appropriate learning objectives and represented in five columns, as follows: Column 1 : Column 2 : Column 3 : Column 4 : Column 5 : Learning Objectives Suggested Teaching and Learning Activities Learning Outcomes Points To Note; and Vocabulary. EMPHASES IN TEACHING AND LEARNING 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 pupils understanding of the concepts and skill of mathematics. 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 pupils show progression in acquiring the mathematical concepts and skills. 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, 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. In determining the change to another learning area or topic, the following have to be taken into consideration: • • The Learning Outcomes define specifically what pupils 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 measurable in all aspects. • 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. The teaching and learning processes emphasise concept building and skill acquisition 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: 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. (xi) 1. Problem Solving in Mathematics 2. Communication 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 pupils are able to solve various problems effectively. The skills involved are: 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 pupils to reinforce and strengthen their knowledge and understanding of mathematics to a deeper level. Through effective communication, pupils will become efficient in problem solving and be able to explain their conceptual understanding and mathematical skills to their peers and teachers. • • • • 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, pupils 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: • • • • • • • • • • 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. Pupils who have developed the skills to communicate mathematically will become more inquisitive and, in the process, gain confidence. Communication skills in mathematics include reading and understanding problems, interpreting diagrams and graphs, using correct and concise mathematical terms during oral presentations and in writing. The skills 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 pupils; • Identifying pupils’ interests; (xii) • • • • • • • • • • 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 psychomotor activities like listening, touching, observing, tasting and smelling. It is a twoway interaction that takes place between teacher and pupils, pupils and pupils, and pupils and object. Some of the more effective and meaningful oral communication techniques in the learning of mathematics are as follows: • Story-telling and 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, pupils 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 journals; Keeping scrap books; Keeping folios; 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 pupils 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 pupils realise that some methods of representation are more effective and useful if they know how to use the elements of mathematical representation. 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 pupils’. Emphasis on logical thinking, during mathematical activities opens up pupils’ minds to accept mathematics as a powerful tool in the world today. Pupils are encouraged to estimate, predict and make intelligent guesses in the process of seeking solutions. Pupils 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 pupils can recognise, construct and evaluate predictions and mathematical arguments. (xiii) 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 be created so that pupils can link conceptual to procedural knowledge and relate topics within mathematics and other learning areas in general. Technology however does not replace the need for all pupils to learn and master the basic mathematical skills. Pupils must be able to efficiently add, subtract, multiply and divide without the use of calculators or other electronic tools. The use of technology must therefore emphasise the acquisition of mathematical concepts and knowledge rather than merely doing calculation. The mathematics curriculum consists of several areas such as arithmetic, geometry, algebra, measures and problem solving. Without connections between these areas, pupils will have to learn and memorise too many concepts and skills separately. By making connections, pupils 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, pupils 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 The teaching and learning of mathematics should employ the latest technology 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 the Internet and relevant learning packages can help to upgrade the pedagogical approach and thus promote the understanding of mathematical concepts. The use of these teaching resources will also help pupils absorb abstract ideas, be creative, feel confident and be able to work independently or in groups. Most of these resources are designed for self-access learning. Through self-access learning pupils will be able to access knowledge or skills and informations independently according to their own pace. This will APPROACHES IN TEACHING AND LEARNING The belief on how mathematics is learnt influence how mathematical concepts are to be taught. Whatever belief the teachers subscribe to, the fact 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 gain experience, 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 diagnostic elements so that pupils know how far they have understood the concepts and acquire the skills. In order to assist pupils in having positive attitudes and personalities, the intrinsic mathematical values of accuracy, confidence and thinking systematically have to be infused into the teaching and learning process. Good moral values can be cultivated through suitable contexts. Learning in groups for example can help pupils to develop social skills, encourage cooperation and build self confidence. The element of patriotism should also be inculcated through the teaching and learning process in the classroom using certain topics. (xiv) Brief historical anecdotes related to aspects of mathematics and famous mathematicians associated with the learning areas are also incorporated into the curriculum. It should be presented at appropriate points where it provides students 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 be incorporated. Amongst the approaches that can be given consideration include 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. questioning and answering, observations and interviews. Based on the response, teachers can rectify pupils misconceptions and weaknesses and also improve their own teaching skills. Teachers can then take subsequent effective measures in conducting remedial and enrichment activities in upgrading pupils’ performances. The choice of an approach that is suitable will stimulate the teaching and learning environment inside or outside the classroom. Approaches that are considered suitable include the following: • Cooperative learning; • Contextual learning; • Mastery learning; • Constructivism; • Enquiry-discovery; and • Future studies. EVALUATION Evaluation or assessment is part of the teaching and learning process to ascertain the strengths and weaknesses of pupils. It has to be planned and carried out as part of the classroom activities. Different methods of assessment can be conducted. These maybe in the form of assignments, oral (xv) 1 Form 4 LEARNING AREA: LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be taught to… 1 Understand and use the concept of significant figure. Discuss the significance of zero in a number. (i) Discuss the use of significant figures in everyday life and other areas. (ii) 2 Understand and use the concept of standard form to solve problems. Use everyday life situations such as in health, technology, industry, construction and business involving numbers in standard form. Use scientific calculator to explore numbers in standard form. Pupils will be able to… Round off positive numbers to a given number of significant figures when the numbers are: a) greater than 1, b) less than 1. Perform operations of addition, subtraction, multiplication and division, involving a few numbers and state the answer in specific significant figures. (iii) Solve problems involving significant figures. (i) State positive numbers in standard form when the numbers are: a) greater than or equal to 10, b) less than 1. Convert numbers in standard form to single numbers. (iii) Perform operations of addition, subtraction, multiplication and division, involving any two numbers and state the answers in standard form. (iv) Solve problems involving numbers in standard form. POINTS TO NOTE VOCABULARY Rounded numbers are only approximates. Limit to positive numbers only. significance significant figure relevant round off accuracy Generally, rounding is done on the final answer. Another term for standard form is scientific notation. (ii) 1 Include two numbers in standard form. standard form single number scientific notation 2 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand the concept of quadratic expression. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Discuss the characteristics of quadratic expressions of the form ax 2 + bx + c = 0 , where a, b and c are constants, a ≠ 0 and x is an unknown. (i) (ii) POINTS TO NOTE VOCABULARY Identify quadratic expressions. Include the case when b = 0 and/or c = 0. quadratic expression constant constant factor Form quadratic expressions by multiplying any two linear expressions. Emphasise that for the terms x2 and x, the coefficients are understood to be 1. Include everyday life situations. unknown highest power expand 1 is also a perfect square. perfect square Factorisation methods that can be used are: • cross method; • inspection. cross method inspection common factor complete factorisation Pupils will be able to… (iii) Form quadratic expressions based on specific situations. 2 Factorise quadratic expression. Discuss the various methods to obtain the desired product. Begin with the case a = 1. Explore the use of graphing calculator to factorise quadratic expressions. (i) Factorise quadratic expressions of the form ax 2 + bx + c , b = 0 or c = 0. (ii) Factorise quadratic expressions of the form px2 − q, p and q are perfect squares. (iii) Factorise quadratic expressions of the form ax2 + bx + c, a, b and c not equal to zero. (iv) Factorise quadratic expressions containing coefficients with common factors. 2 coefficient term factorise common factor 2 Form 4 LEARNING AREA: LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be taught to… 3 Understand the concept of quadratic equation. Discuss the characteristics of quadratic equations. (i) Identify quadratic equations with one unknown. (ii) Write quadratic equations in general form i.e. 4 Understand and use the concept of roots of quadratic equations to solve problems. Discuss the number of roots of a quadratic equation. Use everyday life situations. Pupils will be able to… ax 2 + bx + c = 0 . (iii) Form quadratic equations based on specific situations. (i) Determine whether a given value is a root of a specific quadratic equation. (ii) Determine the solutions for quadratic equations by: a) trial and error method, b) factorisation. (iii) Solve problems involving quadratic equations. 3 POINTS TO NOTE VOCABULARY quadratic equation general form Include everyday life situations. substitute root There are quadratic equations that cannot be solved by factorisation. trial and error method Check the rationality of the solution. solution 3 Form 4 LEARNING AREA: LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be taught to… 1 Understand the concept of set. Use everyday life examples to introduce the concept of set. (i) (ii) POINTS TO NOTE VOCABULARY Sort given objects into groups. The word set refers to any collection or group of objects. set element Define sets by: a) descriptions, b) using set notation. The notation used for sets is braces, { }. The same elements in a set need not be repeated. Sets are usually denoted by capital letters. The definition of sets has to be clear and precise so that the elements can be identified. description label set notation Pupils will be able to… (iii) Identify whether a given object is an element of a set and use the symbol ∈ or ∉. Discuss the difference between the representation of elements and the number of elements in Venn diagrams. (iv) Represent sets by using Venn diagrams. 4 denote The symbol ∈ (epsilon) is read “is an element of” or “is a member of”. The symbol ∉ is read “is not an element of” or “is not a member of”. Venn diagram empty set 3 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 2 Understand and use the concept of subset, universal set and the complement of a set. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Discuss why { 0 } and { ∅ } are not empty sets. (v) Begin with everyday life situations. POINTS TO NOTE VOCABULARY List the elements and state the number of elements of a set. The notation n(A) denotes the number of elements in set A. equal sets (vi) Determine whether a set is an empty set. The symbol ∅ (phi) or { } denotes an empty set. An empty set is also called a null set. An empty set is a subset of any set. Every set is a subset of itself. Pupils will be able to… (vii) Determine whether two sets are equal. (i) Determine whether a given set is a subset of a specific set and use the symbol ⊂ or ⊄ . (ii) Discuss the relationship between sets and universal sets. Represent subset using Venn diagram. (iii) List the subsets for a specific set. (iv) Illustrate the relationship between set and universal set using Venn diagram. subset (v) Determine the complement of a given set. (vi) Determine the relationship between set, subset, universal set and the complement of a set. 5 The symbol ξ denotes a universal set. The symbol A′ denotes the complement of set A. Include everyday life situations. universal set complement of a set 3 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES 3 Uerform operations on sets: • the intersection of sets, • the union of sets. LEARNING OUTCOMES Pupils will be able to… (i) Discuss cases when: • A ∩ B = ∅, (ii) Determine the intersection of: a) two sets, b) three sets, POINTS TO NOTE VOCABULARY Include everyday life situations. intersection common elements and use the symbol ∩. Represent the intersection of sets using Venn diagram. • A ⊂ B. (iii) State the relationship between a) A ∩ B and A, b) A ∩ B and B. (iv) Determine the complement of the intersection of sets; (v) Solve problems involving the intersection of sets. (vi) Determine the union of: a) two sets, b) three sets, and use the symbol ∪. (vii) Represent the union of sets using Venn diagram. (viii) State the relationship between a) A ∪ B and A, b) A ∪ B and B. (ix) Determine the complement of the union of sets. 6 Include everyday life situations. 3 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be able to… (x) Solve problems involving the union of sets. (xi) Determine the outcome of combined operations on sets. (xii) Solve problems involving combined operations on sets. 7 POINTS TO NOTE Include everyday life situations. Include everyday life situations. VOCABULARY 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be taught to… 1 Understand the concept of statement. Introduce this topic using everyday life situations. (i) Focus on mathematical sentences. (ii) Discuss sentences consisting of: • words only, • numbers and words, • numbers and mathematical symbols. 2 Understand the concept of quantifiers “all” and “some”. Start with everyday life situations. POINTS TO NOTE VOCABULARY Determine whether a given sentence is a statement. Statements consisting of: statement Determine whether a given statement is true or false. • words only, e.g. “Five is greater than two”; • numbers and words, e.g. “5 is greater than 2”; • numbers and symbols, e.g. 5 > 2 true false mathematical sentence mathematical statement mathematical symbol (iii) Construct true or false statements using given numbers and mathematical symbols. The following are not statements: • “Is the place value of digit 9 in 1928 hundreds?” Pupils will be able to… (i) Construct statements using the quantifier: a) all, b) some. 8 • 4n − 5m + 2s • “Add the two numbers.” • x+2=8 Quantifiers such as “every” and “any” can be introduced based on context. quantifier all every any 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be able to… (ii) Determine whether a statement that contains the quantifier “all” is true or false. (iii) Determine whether a statement can be generalised to cover all cases by using the quantifier “all”. (iv) Construct a true statement using the quantifier “all” or “some”, given an object and a property. 9 POINTS TO NOTE VOCABULARY Examples: • All squares are four sided figures. • Every square is a four sided figure. • Any square is a four sided figure. Other quantifiers such as “several”, “one of” and “part of” can be used based on context. Example: Object: Trapezium. Property: Two sides are parallel to each other. Statement: All trapeziums have two parallel sides. Object: Even numbers. Property: Divisible by 4. Statement: Some even numbers are divisible by 4. some several one of part of negate contrary object 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 3 Perform operations involving the words “not” or “no”, “and” and “or” on statements. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Begin with everyday life situations. (i) (ii) POINTS TO NOTE VOCABULARY Change the truth value of a given statement by placing the word “not” into the original statement. The negation “no” can be used where appropriate. The symbol “~” (tilde) denotes negation. “~p” denotes negation of p which means “not p” or “no p”. The truth table for p and ~p are as follows: p ~p True False False True negation not p no p truth table truth value Identify two statements from a compound statement that contains the word “and”. The truth values for “p and q” are as follows: p q p and q True True True True False False False True False False False False and compound statement Pupils will be able to… 10 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be able to… (iii) Form a compound statement by combining two given statements using the word “and”. (iv) Identify two statement from a compound statement that contains the word “or”. (v) Form a compound statement by combining two given statements using the word “or”. (vi) Determine the truth value of a compound statement which is the combination of two statements with the word “and”. (vii) Determine the truth value of a compound statement which is the combination of two statements with the word “or”. 11 POINTS TO NOTE VOCABULARY The truth values for “p or q” are as follows: or p True True False False q True False True False p or q True True True False 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 4 Understand the concept of implication. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Start with everyday life situations. (i) Pupils will be able to… Identify the antecedent and consequent of an implication “if p, then q”. Write two implications from a compound statement containing “if and only if”. (iii) Construct mathematical statements in the form of implication: a) If p, then q, b) p if and only if q. (iv) Determine the converse of a given implication. (v) Determine whether the converse of an implication is true or false. POINTS TO NOTE VOCABULARY Implication “if p, then q” can be written as p ⇒ q, and “p if and only if q” can be written as p ⇔ q, which means p ⇒ q and q ⇒ p. implication antecedent consequent The converse of an implication is not necessarily true. Example 1: If x < 3, then x < 5 (true) Conversely: If x < 5, then x < 3 (false) converse (ii) 12 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 5 Understand the concept of argument. SUGGESTED TEACHING AND LEARNING ACTIVITIES Start with everyday life situations. LEARNING OUTCOMES Pupils will be able to… (i) Identify the premise and conclusion of a given simple argument. (ii) Make a conclusion based on two given premises for: a) Argument Form I, b) Argument Form II, c) Argument Form III. 13 POINTS TO NOTE Example 2: If PQR is a triangle, then the sum of the interior angles of PQR is 180°. (true) Conversely: If the sum of the interior angles of PQR is 180°, then PQR is a triangle. (true) Limit to arguments with true premises. Names for argument forms, i.e. syllogism (Form I), modus ponens (Form II) and modus tollens (Form III), need not be introduced. VOCABULARY argument premise conclusion 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 6 Understand and use the concept of deduction and induction to solve problems. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Encourage students to produce arguments based on previous knowledge. (iii) Complete an argument given a premise and the conclusion. Use specific examples/activities to introduce the concept. (i) Pupils will be able to… Determine whether a conclusion is made through: a) reasoning by deduction, b) reasoning by induction. 14 POINTS TO NOTE VOCABULARY Specify that these three forms of arguments are deductions based on two premises only. Argument Form I Premise 1: All A are B. Premise 2: C is A. Conclusion: C is B. Argument Form II: Premise 1: If p, then q. Premise 2: p is true. Conclusion: q is true. Argument Form III: Premise 1: If p, then q. Premise 2: Not q is true. Conclusion: Not p is true. reasoning deduction induction pattern 4 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be able to… (ii) POINTS TO NOTE Make a conclusion for a specific case based on a given general statement, by deduction. (iii) Make a generalization based on Limit to cases where formulae can be the pattern of a numerical sequence, by induction. induced. (iv) Use deduction and induction in problem solving. 15 Specify that: making conclusion by deduction is definite; making conclusion by induction is not necessarily definite. VOCABULARY special conclusion general statement general conclusion specific case numerical sequence 5 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand the concept of gradient of a straight line. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Use technology such as the Geometer’s Sketchpad, graphing calculators, graph boards, magnetic boards or topo maps as teaching aids where appropriate. (i) Determine the vertical and horizontal distances between two given points on a straight line. Begin with concrete examples/daily situations to introduce the concept of gradient. (ii) Determine the ratio of vertical distance to horizontal distance. θ Pupils will be able to… Vertical distance Horizontal distance Discuss: • the relationship between gradient and tan θ, • the steepness of the straight line with different values of gradient. Carry out activities to find the ratio of vertical distance to horizontal distance for several pairs of points on a straight line to conclude that the ratio is constant. 16 POINTS TO NOTE VOCABULARY straight line steepness horizontal distance vertical distance gradient ratio 5 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 2 Understand the concept of gradient of a straight line in Cartesian coordinates. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Discuss the value of gradient if: • P is chosen as (x1, y1) and Q is (x2, y2), • P is chosen as (x2, y2) and Q is (x1, y1). (i) Pupils will be able to… Derive the formula for the gradient of a straight line. VOCABULARY The gradient of a straight line passing through P(x1, y1) and Q(x2, y2) is: y − y1 m= 2 x 2 − x1 acute angle obtuse angle inclined upwards to the right inclined downwards to the right undefined Emphasise that x-intercept and y-intercept are not written in the form of coordinates. x-intercept y-intercept (ii) 3 Understand the concept of intercept. Calculate the gradient of a straight line passing through two points. (iii) Determine the relationship between the value of the gradient and the: a) steepness, b) direction of inclination of a straight line. (i) Determine the x-intercept and the y-intercept of a straight line. POINTS TO NOTE (ii) Derive the formula for the gradient of a straight line in terms of the x-intercept and the y-intercept. (iii) Perform calculations involving gradient, x-intercept and y-intercept. 17 5 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 4 Understand and use equation of a straight line. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Discuss the change in the form of the straight line if the values of m and c are changed. (i) Carry out activities using the graphing calculator, Geometer’s Sketchpad or other teaching aids. (ii) Verify that m is the gradient and c is the y-intercept of a straight line with equation y = mx + c . (iii) Write the equation of the straight line given the gradient and y-intercept. (iv) Determine the gradient and y-intercept of the straight line which equation is of the form: a) y = mx + c, b) ax + by = c. POINTS TO NOTE VOCABULARY Draw the graph given an equation of the form y = mx + c. Emphasise that the graph obtained is a straight line. Determine whether a given point lies on a specific straight line. If a point lies on a straight line, then the coordinates of the point satisfy the equation of the straight line. linear equation graph table of values coefficient constant satisfy Pupils will be able to… (v) Find the equation of the straight line which: a) is parallel to the x-axis, b) is parallel to the y-axis, c) passes through a given point and has a specific gradient, d) passes through two given points. 18 The equation ax + by = c can be written in the form y = mx + c. parallel point of intersection simultaneous equations 5 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 5 Understand and use the concept of parallel lines. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Discuss and conclude that the point of intersection is the only point that satisfies both equations. Use the graphing calculator and Geometer’s Sketchpad or other teaching aids to find the point of intersection. (vi) Find the point of intersection of two straight lines by: a) drawing the two straight lines, b) solving simultaneous equations. Explore properties of parallel lines using the graphing calculator and Geometer’s Sketchpad or other teaching aids. (i) Pupils will be able to… Verify that two parallel lines have the same gradient and vice versa. (ii) Determine from the given equations whether two straight lines are parallel. (iii) Find the equation of the straight line which passes through a given point and is parallel to another straight line. (iv) Solve problems involving equations of straight lines. 19 POINTS TO NOTE VOCABULARY parallel lines 6 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand the concept of class interval. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Use data obtained from activities and other sources such as research studies to introduce the concept of class interval. (i) Complete the class interval for a set of data given one of the class intervals. (ii) Determine: a) the upper limit and lower limit, b) the upper boundary and lower boundary of a class in a grouped data. Pupils will be able to… (iii) Calculate the size of a class interval. Discuss criteria for suitable class intervals. 2 Understand and use the concept of mode and mean of grouped data. POINTS TO NOTE Size of class interval = [upper boundary – lower boundary] VOCABULARY statistics class interval data grouped data upper limit lower limit upper boundary lower boundary size of class interval frequency table (iv) Determine the class interval, given a set of data and the number of classes. (v) Determine a suitable class interval for a given set of data. (vi) Construct a frequency table for a given set of data. (i) (ii) Determine the modal class from the frequency table of grouped data. Calculate the midpoint of a class. 20 mode modal class Midpoint of class = 1 2 (lower limit + upper limit) mean midpoint of a class 6 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 3 Represent and interpret data in histograms with class intervals of the same size to solve problems. SUGGESTED TEACHING AND LEARNING ACTIVITIES Discuss the difference between histogram and bar chart. Use graphing calculator to explore the effect of different class interval on histogram. 4 Represent and interpret data in frequency polygons to solve problems. LEARNING OUTCOMES Pupils will be able to… POINTS TO NOTE (iii) Verify the formula for the mean of grouped data. (iv) Calculate the mean from the frequency table of grouped data. (v) Discuss the effect of the size of class interval on the accuracy of the mean for a specific set of grouped data. (i) Draw a histogram based on the frequency table of a grouped data. (ii) uniform class interval histogram Interpret information from a given histogram. (iii) Solve problems involving histograms. (i) Draw the frequency polygon based on: a) a histogram, b) a frequency table. Interpret information from a given frequency polygon. (iii) Solve problems involving frequency polygon. vertical axis horizontal axis Include everyday life situations. When drawing a frequency polygon add a class with 0 frequency before the first class and after the last class. (ii) 21 VOCABULARY Include everyday life situations. frequency polygon 6 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES 5 Understand the concept of cumulative frequency. LEARNING OUTCOMES Pupils will be able to… (i) (ii) 6 Understand and use the concept of measures of dispersion to solve problems. Discuss the meaning of dispersion by comparing a few sets of data. Graphing calculator can be used for this purpose. (i) Construct the cumulative frequency table for: a) ungrouped data, b) grouped data. Draw the ogive for: a) ungrouped data, b) grouped data. Determine the range of a set of data. (ii) Determine: a) the median, b) the first quartile, c) the third quartile, d) the interquartile range, from the ogive. (iii) Interpret information from an ogive. 22 POINTS TO NOTE VOCABULARY cumulative frequency ungrouped data ogive When drawing ogive: • use the upper boundaries; • add a class with zero frequency before the first class. For grouped data: Range = [midpoint of the last class – midpoint of the first class] range measures of dispersion median first quartile third quartile interquartile range 6 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Carry out a project/research and analyse as well as interpret the data. Present the findings of the project/research. Emphasise the importance of honesty and accuracy in managing statistical research. (iv) Solve problems involving data representations and measures of dispersion. Pupils will be able to… 23 POINTS TO NOTE VOCABULARY 7 Form 4 LEARNING AREA: LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be taught to… 1 Understand the concept of sample space. Use concrete examples such as throwing a die and tossing a coin. (i) 2 Understand the concept of events. Discuss that an event is a subset of the sample space. Discuss also impossible events for a sample space. Pupils will be able to… Determine whether an outcome is a possible outcome of an experiment. (ii) List all the possible outcomes of an experiment: a) from activities, b) by reasoning. (iii) Determine the sample space of an experiment. (iv) Write the sample space by using set notations. (i) Identify the elements of a sample space which satisfy given conditions. 3 Understand and use the concept of probability of an event to solve problems. (ii) Find the probability of an event from a big enough number of trials. 24 VOCABULARY sample space outcome experiment possible outcome An impossible event is an empty set. event element subset empty set impossible event Probability is obtained from activities and appropriate data. probability (ii) Discuss that the sample space itself is an event. Carry out activities to introduce the concept of probability. The graphing calculator can be used to simulate such activities. List all the elements of a sample space which satisfy certain conditions using set notations. (iii) Determine whether an event is possible for a sample space. (i) Find the ratio of the number of times an event occurs to the number of trials. POINTS TO NOTE 7 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Discuss situation which results in: • probability of event = 1. • probability of event = 0. (iii) Calculate the expected number of times an event will occur, given the probability of the event and number of trials. (iv) Solve problems involving probability. (v) Predict the occurrence of an outcome and make a decision based on known information. Emphasise that the value of probability is between 0 and 1. Predict possible events which might occur in daily situations. Pupils will be able to… 25 POINTS TO NOTE VOCABULARY 8 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand and use the concept of tangents to a circle. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Develop concepts and abilities through activities using technology such as the Geometer’s Sketchpad and graphing calculator. (i) Pupils will be able to… POINTS TO NOTE VOCABULARY Identify tangents to a circle. Make inference that the tangent to a circle is a straight line perpendicular to the radius that passes through the contact point. (iii) Construct the tangent to a circle passing through a point: a) on the circumference of the circle, b) outside the circle. (iv) Determine the properties related to two tangents to a circle from a given point outside the circle. tangent to a circle circle (ii) perpendicular radius circumference semicircle O B AC = BC ∠ACO = ∠BCO ∠AOC = ∠BOC ΔAOC and ΔBOC are congruent. 26 congruent Properties of angle in semicircles can be used. Examples of properties of two tangents to a circle: A C 8 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Pupils will be able to… (v) 2 Understand and use the properties of angle between tangent and chord to solve problems. Explore the property of angle in alternate segment using Geometer’s Sketchpad or other teaching aids. (i) POINTS TO NOTE Solve problems involving Relate to Pythagoras’ tangents to a circle. Theorem Identify the angle in the alternate segment which is subtended by the chord through E the contact point of the tangent. VOCABULARY chords alternate segment major sector subtended D A (ii) 3 Understand and use the properties of common tangents to solve problems. Discuss the maximum number of common tangents for the three cases. Verify the relationship between the angle formed by the tangent and the chord with the angle in the alternate segment which is subtended by the chord. (iii) Perform calculations involving the angle in alternate segment. (iv) Solve problems involving tangent to a circle and angle in alternate segment. (i) Determine the number of common tangents which can be drawn to two circles which: a) intersect at two points, b) intersect only at one point, c) do not intersect. 27 B C ∠ABE = ∠BDE ∠CBD = ∠BED Emphasise that the lengths of common tangents are equal. common tangents 8 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Include daily situations. (ii) Pupils will be able to… Determine the properties related to the common tangent to two circles which: a) intersect at two points, b) intersect only at one point, c) do not intersect. (iii) Solve problems involving common tangents to two circles. (iv) Solve problems involving tangents and common tangents. 28 POINTS TO NOTE Include problems involving Pythagoras’ Theorem. VOCABULARY 9 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand and use the concept of the values of sin θ, cos θ and tan θ (0° ≤ θ ≤ 360°) to solve problems. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Explain the meaning of unit circle. (i) Pupils will be able to… Identify the quadrants and angles in the unit circle. y P(x,y) 0 y x Q x Begin with definitions of sine, cosine and tangent of an acute angle. PQ y sin θ = = = y OP 1 OQ = OP PQ = tan θ = OQ cos θ = VOCABULARY The unit circle is the circle of radius 1 with its centre at the origin. quadrant (ii) 1 x =x 1 y x Determine: a) the value of y-coordinate, b) the value of x-coordinate, c) the ratio of y-coordinate to x-coordinate of several points on the circumference of the unit circle. (iii) Verify that, for an angle in quadrant I of the unit circle: POINTS TO NOTE a) sin θ = y-coordinate, b) cosθ = x-coordinate, c) tan θ = y-coordinate . x-coordinate (iv) Determine the values of: a) sine, b) cosine, c) tangent of an angle in quadrant I of the unit circle. 29 sine θ cosine θ tangent θ 9 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Explain that the concept sin θ = y-coordinate , cosθ = x-coordinate, y-coordinate tan θ = x-coordinate can be extended to angles in quadrant II, III and IV. (v) √2 1 45o 1 2 30o Pupils will be able to… POINTS TO NOTE Determine the values of: a) sin θ, b) cos θ , c) tan θ , for 90° ≤ θ ≤ 360°. √3 60o 1 Use the above triangles to find the values of sine, cosine and tangent for 30°, 45°, 60°. Teaching can be expanded through activities such as reflection. (vi) Determine whether the values of: a) sine, b) cosine, c) tangent, of an angle in a specific quadrant is positive or negative. (vii) Determine the values of sine, cosine and tangent for special angles. (viii) Determine the values of the angles in quadrant I which correspond to the values of the angles in other quadrants. 30 Consider special angles such as 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°. VOCABULARY 9 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Use the Geometer’s Sketchpad to explore the change in the values of sine, cosine and tangent relative to the change in angles. (ix) State the relationships between the values of: a) sine, b) cosine, and c) tangent of angles in quadrant II, III and IV with their respective values of the corresponding angle in quadrant I. (x) Find the values of sine, cosine and tangent of the angles between 90° and 360°. (xi) Find the angles between 0° and 360°, given the values of sine, cosine or tangent. (xii) Solve problems involving sine, cosine and tangent. Relate to daily situations. Pupils will be able to… 31 POINTS TO NOTE VOCABULARY 9 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 2 Draw and use the graphs of sine, cosine and tangent. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Use the graphing calculator and Geometer’s Sketchpad to explore the feature of the graphs of y = sin θ, y = cos θ, y = tan θ. Discuss the feature of the graphs of (i) Draw the graphs of sine, cosine and tangent for angles between 0° and 360°. (ii) Compare the graphs of sine, cosine and tangent for angles between 0° and 360°. y = sin θ, y = cos θ, y = tan θ. Discuss the examples of these graphs in other areas. Pupils will be able to… (iii) Solve problems involving graphs of sine, cosine and tangent. 32 POINTS TO NOTE VOCABULARY 10 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand and use the concept of angle of elevation and angle of depression to solve problems. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Use daily situations to introduce the concept. (i) Pupils will be able to… Identify: a) the horizontal line, b) the angle of elevation, c) the angle of depression for a particular situation. (ii) Represent a particular situation involving: a) the angle of elevation, b) the angle of depression using diagrams. (iii) Solve problems involving the angle of elevation and the angle of depression. 33 POINTS TO NOTE VOCABULARY angle of elevation angle of depression horizontal line Include two observations on the same horizontal plane. Involve activities outside the classroom. 11 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… 1 Understand and use the concept of angle between lines and planes to solve problems. SUGGESTED TEACHING AND LEARNING ACTIVITIES LEARNING OUTCOMES Carry out activities using daily situations and 3-dimensional models. (i) Identify planes. Differentiate between 2-dimensional and 3-dimensional shapes. Involve planes found in natural surroundings. (ii) Identify horizontal planes, vertical planes and inclined planes. Begin with 3-dimensional models. Pupils will be able to… (viii) Determine the angle between a line and a plane. (ix) Solve problems involving the angle between a line and a plane. 34 VOCABULARY horizontal plane vertical plane 3-dimensional normal to a plane orthogonal projection space diagonal (iii) Sketch a three dimensional shape and identify the specific planes. (iv) Identify: a) lines that lie on a plane, b) lines that intersect with a plane. (v) Identify normals to a given plane. (vi) Determine the orthogonal projection of a line on a plane. (vii) Draw and name the orthogonal projection of a line on a plane. Use 3-dimensional models to give clearer pictures. POINTS TO NOTE Include lines in 3-dimensional shapes. 11 Form 4 LEARNING AREA: LEARNING OBJECTIVES Pupils will be taught to… SUGGESTED TEACHING AND LEARNING ACTIVITIES 2 Understand and use the concept of angle between two planes to solve problems. LEARNING OUTCOMES Pupils will be able to… (i) Use 3-dimensional models to give clearer pictures. Identify the line of intersection between two planes. (ii) Draw a line on each plane which is perpendicular to the line of intersection of the two planes at a point on the line of intersection. (iii) Determine the angle between two planes on a model and a given diagram. (iv) Solve problems involving lines and planes in 3-dimensional shapes. 35 POINTS TO NOTE VOCABULARY angle between two planes

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