Mathematics - Rwanda Education Board
REPUBLIC OF RWANDA MINISTRY OF EDUCATION MATHEMATICS SYLLABUS FOR ORDINARY LEVEL S1-S3 Kigali, 2015 RWANDA EDUCATION BOARD P.O Box 3817 KIGALI Telephone : (+250) 255121482 E-mail: [email protected] Website: www.reb.rw MATHEMATICS SYLLABUS FOR ORDINARY LEVEL S1-S3 Kigali, 2015 Page 1 of 60 © 2015 Rwanda Education Board All rights reserved This syllabus is the property of Rwanda Education Board, Credit must be provided to the author and source of the document when the content is quoted. Page 2 of 60 FOREWORD The Rwanda Education Board is honored to avail Syllabuses which serve as official documents and guide tocompetencebased teaching and learning in order to ensure consistency and coherence in the delivery of quality education across all levels of general education in Rwandan schools. The Rwandan education philosophy is to ensure that young people at every level of education achieve their full potential in terms of relevant knowledge, skills and appropriate attitudes that prepare them to be well integrated in society and exploit employment opportunities. In line with efforts to improve the quality of education, the government of Rwanda emphasizes the importance of aligning the syllabus, teaching and learning and assessment approaches in order to ensure that the system is producing the kind of citizens the country needs. Many factors influence what children are taught, how well they learn and the competencies they acquire, among them the relevance of the syllabus, the quality of teachers’ pedagogical approaches, the assessment strategies and the instructional materials available. The ambition to develop a knowledge-based society and the growth of regional and global competition in the jobs market has necessitated the shift to acompetence-based syllabus. With the help of the teachers, whose role is central to the success of the syllabus, learners will gain appropriate skills and be able to apply what they have learned in real life situations. Hence they will make a difference not only to their own lives but also to the success of the nation. I wish to sincerely extend my appreciation to the people who contributed towards the development of this document, particularly REB and its staff who organized the whole process from its inception. Special appreciation goes to the development partners who supported the exercise throughout. Any comment of contribution would be welcome for the improvement of this syllabus. Mr. GASANA Janvier, Director General REB. Page 3 of 60 ACKNOWLEDGEMENT I wish to sincerely extend my special appreciation to the people who played a major role in development of this syllabus. It would not have been successful without the participation of different education stakeholders and financial support from different donors that I would like to express my deep gratitude. My thanks first goes to the Rwanda Education leadership who supervised the curriculum review process and Rwanda Education Board staff who were involved in the conception and syllabus writing. I wish to extend my appreciation to teachers from pre-primary to university level whose efforts during conception were much valuable. I owe gratitude to different education partners such as UNICEF, UNFPA, DFID and Access to Finance Rwanda for their financial and technical support. We also value the contribution of other education partner organisations such as CNLG, AEGIS trust, Itorero ry’Igihugu, Gender Monitoring Office, National Unit and Reconciliation Commission, RBS, REMA, Handicap International, Wellspring Foundation, Right To Play, MEDISAR, EDC/L3, EDC/Akazi Kanoze, Save the Children, Faith Based Organisations, WDA, MINECOFIN and Local and International consultants. Their respective initiative, co- operation and support were basically responsible for the successful production of this syllabus by Curriculum and Pedagogical Material Production Department (CPMD). Dr. Joyce Musabe, Head of Curriculum and Pedagogical Material Production Department, Page 4 of 60 LIST OF PARTICIPANTS WHO WERE INVOLVED IN THE ELABORATION OF THE SYLLABUS Rwanda Education Board Staff 1. Dr. MUSABE Joyce, Head of Department ,Curriculum and Pedagogical Material Department (CPMD) 2. Mr. RUTAKAMIZE Joseph, Director of Science and Art Unit, 3. Mr. KAYINAMURA Aloys , Mathematics Curriculum Specialist : Team leader, 4. Madame NYIRANDAGIJIMANA Anathalie: Specialist in charge of Pedagogic Norms. Teachers and Lecturers 1. HABINEZA NSHUTI Jean Clément, Mathematics teachers at Ecole Secondaire de Nyanza 2. NKUNDINEZA Felix, Mathematics teachers at G.S. KIMIRONKO I 3. NSHIMIRYAYO Anastase, Mathematics teachers at NYAGATARE Secondary School 4. NYIRABAGABE Agnès, Mathematics teachers at Lycée Notre Dame de Citeaux 5. UNENCAN MUNGUMIYO Dieudonné, Mathematics teachers at Lycée de Kigali Other resource persons Mr Murekeraho Joseph, National consultant Quality assurer /editors Dr Alphonse Uworwabayeho (PhD), University of Rwanda (UR), College of Education. Page 5 of 60 Table of Contents FOREWORD ................................................................................................................................................................................................................ 3 ACKNOWLEDGEMENT .............................................................................................................................................................................................. 4 LIST OF PARTICIPANTS WHO WERE INVOLVED IN THE ELABORATION OF THE SYLLABUS....................................................................................... 5 1. 2. 3. INTRODUCTION ................................................................................................................................................................................................. 8 1.1. BACKGROUND TO CURRICULUM REVIEW ................................................................................................................................................. 8 1.2. RATIONALE OF TEACHING AND LEARNING MATHEMATICS ............................................................................................................... 8 1.2.1. MATHEMATICS AND SOCIETY.......................................................................................................................................................... 8 1.2.2. MATHEMATICS AND LEARNERS ...................................................................................................................................................... 9 1.2.3. COMPETENCES .................................................................................................................................................................................. 9 PEDAGOGICAL APPROACH ............................................................................................................................................................................. 12 2.1. ROLE OF THE LEARNER .......................................................................................................................................................................... 12 2.2. ROLE OF THE TEACHER .......................................................................................................................................................................... 13 2.3. SPECIAL NEEDS EDUCATION AND INCLUSIVE APPROACH ................................................................................................................ 14 ASSESSMENT APPROACH ............................................................................................................................................................................... 14 3.1. TYPES OF ASSESSMENTS ........................................................................................................................................................................ 15 3.1.1. FORMATIVE ASSESSMENT: ............................................................................................................................................................ 15 3.1.2. SUMMATIVE ASSESSMENTS: .......................................................................................................................................................... 15 3.2. RECORD KEEPING ................................................................................................................................................................................... 16 Page 6 of 60 4. 5. 3.3. ITEM WRITING IN SUMMATIVE ASSESSMENT ..................................................................................................................................... 16 3.4. STRUCTURE AND FORMAT OF THE EXAMINATION ............................................................................................................................ 17 3.5. REPORTING TO PARENTS....................................................................................................................................................................... 18 RESOURCES ...................................................................................................................................................................................................... 18 4.1. MATERIALS NEEDED FOR IMPLEMENTATION .................................................................................................................................... 18 4.2. HUMAN RESOURCE ................................................................................................................................................................................. 18 SYLLABUS UNITS ............................................................................................................................................................................................. 19 5.1. PRESENTATION OF THE STRUCTURE OF THE SYLLABUS UNITS ...................................................................................................... 19 5.2. MATHEMATICS PROGRAM FOR SECONDARY ONE .............................................................................................................................. 20 5.2.1. KEY COMPETENCIES AT THE END OF SECONDARY ONE ............................................................................................................ 21 5.2.2. MATHEMATICS UNITS FOR SECONDARY ONE ............................................................................................................................ 22 5.3. MATHEMATICS PROGRAM FOR SECONDARY TWO ............................................................................................................................. 31 5.3.1. KEY COMPETENCIES AT THE END OF SECONDARY TWO ........................................................................................................... 31 5.3.2. MATHEMATICS UNITS FOR SECONDARY TWO ........................................................................................................................... 32 5.4. MATHEMATICS PROGRAM FOR SECONDARY THREE.......................................................................................................................... 43 5.4.1. KEY COMPETENCIES AT THE END OF SECONDARY THREE....................................................................................................... 43 5.4.2. MATHEMATICS UNITS FOR SECONDARY THREE ......................................................................................................................... 44 6. REFERENCES .................................................................................................................................................................................................... 57 7. APPENDIX: SUBJECTS AND WEEKLY TIME ALLOCATION FOR O’ LEVEL (S1-S3)..................................................................................... 59 Page 7 of 60 1. INTRODUCTION 1.1. BACKGROUND TO CURRICULUM REVIEW The motive of reviewing the Mathematics syllabus of ordinary level was to ensure that the syllabus is responsive to the needs of the learner and to shift from objective and knowledge-based learning tocompetence-based learning. Emphasis in the review is put more on skills and competencies and the coherence within the existing content by benchmarking with syllabi elsewhere with best practices. The new Mathematics syllabus guides the interaction between the teacher and the learners in the learning processes and highlights the competencies a learner should acquire during and at the end of each learning unit. Learners will have the opportunity to apply Mathematics in different contexts, and discover its important in daily life. Teachers help the learners appreciate the relevance and benefits for studying this subject. The new Mathematics syllabus is prepared for all learners in ordinary level and it has to be taught in six periods per week. 1.2. RATIONALE OF TEACHING AND LEARNING MATHEMATICS 1.2.1. MATHEMATICS AND SOCIETY Mathematics plays an important role in society through abstraction and logic, counting, calculation, measurement, systematic study of shapes and motion. It is also used in natural sciences, engineering, medicine, finance and social sciences. The applied mathematics like statistics and probability play an important role in game theory, in the national census process, in scientific research,etc. In addition, some cross-cutting issues such as financial awareness are incorporated into some of the Mathematics units to improve social and economic welfare of Rwandan society. Page 8 of 60 Mathematics is key to the Rwandan education ambition of developing a knowledge-based and technology-led economy since it provide to learners all required knowledge and skills to be used in different learning areas. Therefore, Mathematics is an important subject as it supports other subjects.This new curriculum will address gaps in the current Rwanda Education system which lacks of appropriate skills and attitudes provided by the current education system. 1.2.2. MATHEMATICS AND LEARNERS Learners need enough basic mathematical competencies to be effective members of the Rwandan society, including the ability to estimate, measure, calculate, interpret statistics, assess probabilities, and read the commonly used mathematical representations and graphs. For example, reading or listening to the news requires some of these competencies, and citizenship requires being able to interpret critically the information one receives. Therefore, Mathematics equips learners with knowledge, skills and attitudes necessary to enable them to succeed in an era of rapid technological growth and socio-economic development. Mastery of basic Mathematical ideas and operations makes learners being confident in problem-solving. It enables the learners to be systematic, creative and self confident in using mathematical language and techniques to reason; think critically; develop imagination, initiative and flexibility of mind. Mathematics has a high profile at all levels of study where learning needs to include practical problem-solving activities with opportunities for learners to plan their own investigations and develop their confidence towards Mathematics. 1.2.3. COMPETENCES Competence is defined as the ability ability to perform a particular task successfully, resulting from having gained an appropriate combination of knowledge, skills and attitudes The Mathematics syllabus gives the opportunity to learners to develop different competencies, including the generic competencies . Page 9 of 60 Basic competencies are addressed in the stated broad subject competences and in objectives highlighted year on year basis and in each of units of learning. The generic competencies, basic competences that must be emphasized and reflected in the learning process are briefly described below and teachers will ensure that learners are exposed to tasks that help the learners acquire the skills. GENERIC COMPETENCIES AND VALUES Critical and problem solving skills: Learners use different techniques to solve mathematical problems related to real life situations.They are engaged in mathematical thinking, they construct, symbolize, apply and generalize mathematical ideas. The acquisition of such skills will help learners to think imaginatively and broadly to evaluate and find solutions to problems encountered in all situations. Creativity and innovation :The acquisition of such skills will help learners to take initiatives and use imagination beyond knowledge provided to generate new ideas and construct new concepts. Learners improve these skills through Mathematics contest, Mathematics competitions,etc. Research: This will help learners to find answers to questions basing on existing information and concepts and to explain phenomena basing on findings from gathered information. Communication in official languages: Learners communicate effectively their findings through explanations, construction of arguments and drawing relevant conclusions. Teachers, irrespective of not being teachers of language, will ensure the proper use of the language of instruction by learners which will help them to communicate clearly and confidently and convey ideas effectively through speaking and writing and using the correct language structure and relevant vocabulary. Cooperation, inter personal management and life skills: Learners are engaged in cooperative learning groups to promote higher achievement than do competitive and individual work. This will help them to cooperate with others as a team in whatever task assigned and to practice positive ethical moral values and respect for the rights, feelings and views of others. Perform practical activities related to environmental conservation Page 10 of 60 and protection. Advocating for personal, family and community health, hygiene and nutrition and Responding creatively to the variety of challenges encountered in life. Lifelong learning:The acquisition of such skills will help learners to update knowledge and skills with minimum external support and to cope with evolution of knowledge advances for personal fulfillment in areas that need improvement and development BROAD MATHEMATICS COMPETENCIES During and at the end of learning process, the learner can: 1. 2. 3. 4. 5. Use correctly specific symbolism of the fundamental concepts in Mathematics; Develop clear, logical, creative and coherent thinking; Apply acquired knowledge in Mathematics in solving problems encountered in everyday life; Use the acquired concepts for easy adaptation in the study of other subjects ; Deduce correctly a given situation from a picture and/or a well drawn out basic mathematical concepts and use them correctly in daily life situations; 6. Read and interpret a graph; 7. Use acquired mathematical skills to develop work spirit, team work, self-confidence and time management without supervision; 8. Use ICT tools to explore Mathematics (examples: calculators, computers, mathematical software,…). MATHEMATICS AND DEVELOPING COMPETENCES The national policy documents based on national aspirations identify some ‘basic Competencies’ alongside the ‘Generic Competencies’’ that will develop higher order thinking skills and help student learn subject content and promote application of acquired knowledge and skills. Through observations, constructions, hand-on, using symbols, applying and generalizing mathematical ideas, and presentation of information during the learning process, the learner will not only develop deductive and inductive skills but also acquire cooperation and communication, critical thinking and problem solving skills. This will be realized when learners make presentations Page 11 of 60 leading to inferences and conclusions at the end of learning unit. This will be achieved through learner group work and cooperative learning which in turn will promote interpersonal relations and teamwork. The acquired knowledge in learning Mathematics should develop a responsible citizen who adapts to scientific reasoning and attitudes and develops confidence in reasoning independently. The learner should show concern of individual attitudes, environmental protection and comply with the scientific method of reasoning. The scientific method should be applied with the necessary rigor, intellectual honesty to promote critical thinking while systematically pursuing the line of thought. 2. PEDAGOGICAL APPROACH The change to acompetence-based curriculum is about transforming learning, ensuring that learning is deep, enjoyable and habit-forming. 2.1. ROLE OF THE LEARNER In thecompetence-based syllabus, the learner is the principal actor of his / her education. He/she is not an empty bottle to fill. Taking into account the initial capacities and abilities of the learner, the syllabus lists under each unit, the activities which are engaging learners to participate in the learning process . The teaching- learning processes will be tailored towards creating a learner friendly environment basing on the capabilities, needs, experience and interests. Therefore, the following are some of the roles or the expectations from the learners: - Learners construct the knowledge either individually or in groups in an active way. From the learning theory, learners move in their understanding from concrete through pictorial to abstract. Therefore, the opportunities should be given to learners to manipulate concrete objects and to use models. Learners are encouraged to use hand-held calculator. This stimulates mathematics as it is really used, both on job and in scientific applications. Frequent use of calculators can enhance learners’ understanding and mastering of arithmetic. Learners work on one competency at a time in form of concrete units with specific learning objectives broken down into knowledge, skills and attitude. Learners will be encouraged to do research and present their findings through group work activities. A learner is cooperative: learners work in heterogeneous groups to increase tolerance and understanding. Learners are responsible for their own participation and ensure the effectivness of their work. Page 12 of 60 - Help is sought from within the group and the teacher is asked for help only when the whole group agrees to ask a question The learners who learn at a faster pace do not do the task alone and then the others merely sign off on it. Teacher ensure the effective contribution of each learner, through clear explanation and argumentation to improve the English literacy and to develop sense of responsibility and to increase the self-confidence, the public speech ability, etc. 2.2. ROLE OF THE TEACHER In thecompetence-based syllabus, the teacher is a facilitator, organiser, advisor, a conflict solver, ... The specific duties of the teacher in acompetence-based approach are the following: - - - - He / she is a facilitator, his/her role is to provide opportunities for learners to meet problems that interest and challenge them and that, with appropriate effort, they can solve. This requires an elaborated preparation to plan the activities, the place they will be carried, the required assistance. He/she is an organizer: his / her role is to organize the learners in the classroom or outside and engage them through participatory and interactive methods through the learning processes as individuals, in pairs or in groups. To ensure that the learning is personalized, active and participative , co-operative the teacher must identify the needs of the learners, the nature of the learning to be done, and the means to shape learning experiences accordingly He/she is an advisor: he/she provides counseling and guidance for learners in need. He/she comforts and encourages learners by valuing their contributions in the class activities. He/she is a conflict-solver: most of the activitiescompetence-based are performed in groups. The members of a group may have problems such as attribution of tasks; they should find useful and constructive the intervention of the teacher as a unifying element. He/she is ethical and preaches by examples, by being impartial, by being a role-model, by caring for individual needs, especially for slow learners and learners with physical impairments, through a special assistance by providing remedial activities or reinforncement activities. One should notice that this list is not exhaustive. Page 13 of 60 2.3. SPECIAL NEEDS EDUCATION AND INCLUSIVE APPROACH All Rwandans have the right to access education regardless of their different needs. The underpinnings of this provision would naturally hold that all citizens benefit from the same menu of educational programs. The possibility of this assumption is the focus of special needs education. The critical issue is that we have persons/ learners who are totally different in their ways of living and learning as opposed to the majority. The difference can either be emotional, physical, sensory and intellectual learning challenged traditionally known as mental retardation. These learners equally have the right to benefit from the free and compulsory basic education in the nearby ordinary/mainstream schools. Therefore, the schools’ role is to enrol them and also set strategies to provide relevant education to them. The teacher therefore is requested to consider each learner’s needs during teaching and learning process. Assessment strategies and conditions should also be standardised to the needs of these learners. Detailed guidance for each category of learners with special education needs is provided for in the guidance for teachers. 3. ASSESSMENT APPROACH Assessment is the process of evaluating the teaching and learning processes through collecting and interpreting evidence of individual learner’s progress in learning and to make a judgment about a learner’s achievements measured against defined standards. Assessment is an integral part of the teaching learning processes. In the new competence-based curriculum assessment must also be competence-based, whereby a learner is given a complex situation related to his/her everyday life and asked to try to overcome the situation by applying what he/she learned. Assessment will be organized at the following levels: School-based assessment, District examinations, National assessment (LARS) and National examinations. Page 14 of 60 3.1. TYPES OF ASSESSMENTS 3.1.1. FORMATIVE ASSESSMENT: Formative assessment helps to check the efficiency of the process of learning. It is done within the teaching/learning process. Continuous assessment involves formal and informal methods used by schools to check whether learning is taking place. When a teacher is planning his/her lesson, he/she should establish criteria for performance and behavior changes at the beginning of a unit. Then at the end of every unit, the teacher should ensure that all the learners have mastered the stated key unit competencies basing on the criteria stated, before going to the next unit. The teacher will assess how well each learner masters both the subject and the generic competencies described in the syllabus and from this, the teacher will gain a picture of the all-round progress of the learner. The teacher will use one or a combination of the following: (a) observation (b) pen and paper (c) oral questioning. 3.1.2. SUMMATIVE ASSESSMENTS: When assessment is used to record a judgment of a competence or performance of the learner, it serves a summative purpose. Summative assessment gives a picture of a learner’s competence or progress at any specific moment. The main purpose of summative assessment is to evaluate whether learning objectives have been achieved and to use the results for the ranking or grading of learners, for deciding on progression, for selection into the next level of education and for certification. This assessment should have an integrative aspect whereby a student must be able to show mastery of all competencies. It can be internal school based assessment or external assessment in the form of national examinations. School based summative assessment should take place once at the end of each term and once at the end of the year. School summative assessment average scores for each subject will be weighted and included in the final national examinations grade. School based assessment average grade will contribute a certain percentage as teachers gain more experience and confidence in Page 15 of 60 assessment techniques and in the third year of the implementation of the new curriculum it will initialy contribute 10% of the final grade, but will be progressively increased. Districts will be supported to continue their initiative to organize a common test per class for all the schools to evaluate the performance and the achievement level of learners in individual schools. External summative assessment will be done at the end of P6, S3 and S6. 3.2. RECORD KEEPING This is gathering facts and evidence from assessment instruments and using them to judge the student’s performance by assigning an indicator against the set criteria or standard. Whatever assessment procedures used shall generate data in the form of scores which will be carefully be recorded and stored in a portfolio because they will contribute for remedial actions, for alternative instructional strategy and feed back to the learner and to the parents to check the learning progress and to advice accordingly or to the final assessment of the students. This portfolio is a folder (or binder or even a digital collection) containing the student’s work as well as the student’s evaluation of the strengths and weaknesses of the work. Portfolios reflect not only work produced (such as papers and assignments), but also it is a record of the activities undertaken over time as part of student learning. Besides, it will serve as a verification tool for each learner that he/she attended the whole learning before he/she undergoes the summative assessment for the subject. 3.3. ITEM WRITING IN SUMMATIVE ASSESSMENT Before developing a question paper, a plan or specification of what is to be tested or examined must be elaborated to show the units or topics to be tested on, the number of questions in each level of Bloom’s taxonomy and the marks allocation for each question. In a competency based curriculum, questions from higher levels of Bloom’s taxonomy should be given more weight than those from knowledge and comprehension level. Page 16 of 60 Before developing a question paper, the item writer must ensure that the test or examination questions are tailored towards competency based assessment by doing the following: Identify topic areas to be tested on from the subject syllabus. Outline subject-matter content to be considered as the basis for the test. Identify learning outcomes to be measured by the test. Prepare a table of specifications. Ensure that the verbs used in the formulation of questions do not require memorization or recall answers only but testing broad competencies as stated in the syllabus. 3.4. STRUCTURE AND FORMAT OF THE EXAMINATION There will be one paper in Mathematics at the end of Primary 6. The paper will be composed by two sections, where the first section will be composed with short answer items or items with short calculations which include the questions testing for knowledge and understanding, investigation of patterns, quick calculations and applications of Mathematics in real life situations. The second section will be composed with long answer items or answers with simple demonstrations, constructions , calculations, simple analysis , interpretation and explanations. The items for the second section will emphasize on the mastering of Mathematics facts, the understanding of Mathematics concepts and its applications in real life situations. In this section, the assessment will find out not only what skills and facts have been mastered, but also how well learners understand the process of solving a mathematical problem and whether they can link the application of what they have learned to the context or to the real life situation. The Time required for the paper is three hours (3hrs). The following topic areas have to be assessed: algebra; metric measurements (money & its application); proportional reasoning; geometry; statistics and probability. Topic areas with more weight will have more emphasis in the second section where learners should have the right to choose to answer 3 items out of 5. Page 17 of 60 3.5. REPORTING TO PARENTS The wider range of learning in the new curriculum means that it is necessary to think again about how to share learners’ progress with parents. A single mark is not sufficient to convey the different expectations of learning which are in the learning objectives. The most helpful reporting is to share what students are doing well and where they need to improve. 4. RESOURCES 4.1. MATERIALS NEEDED FOR IMPLEMENTATION The following list shows the main materials/equipments needed in the learning and teaching process: Materials to encourage group work activities and presentations: Computers (Desk tops&lab tops) and projectors; Manila papers and markers Materials for drawing & measuring geometrical figures/shapes and graphs: Geometric instruments, ICT tools such as geogebra, Microsoft student ENCARTA, ... Materials for enhancing research skills: Textbooks and internet (the list of the textbooks to consult is given in the reference at the end of the syllabus and those books can be found in printed or digital copies). Materials to encourage the development of Mathematical models: scientific calculators, Math type, Matlab, etc The technology used in teaching and learning of Mathematics has to be regarded as tools to enhance the teaching and learning process and not to replace teachers. 4.2. HUMAN RESOURCE The effective implementation of this curriculum needs a joint collaboration of educators at all levels. Given the material requirements, teachers are expected to accomplish their noble role as stated above. On the other hand school head teachers Page 18 of 60 and directors of studies are required to make a follow-up and assess the teaching and learning of this subject due to their profiles in the schools. These combined efforts will ensure bright future careers and lives for learners as well as the contemporary development of the country. In a special way, the teacher of Mathematics at ordinary level should have a firm understanding of mathematical concepts at the leavel he / she teaches. He/she should be qualified in Mathematics and have a firm ethical conduct. The teacher should possess the qualities of a good facilitator, organizer, problem solver, listener and adviser. He/she is required to have basic skills and competency of guidance and counseling because students may come to him or her for advice. Skills required for the Teacher of Religious Education The teacher of Mathematics should have the following skills, values and qualities: - Engage learners in variety of learning activities Use multiple teaching and assessment methods Adjust instruction to the level of the learners Have creativity and innovation the teaching and learning process Be a good communicator and organizer Be a guide/ facilitator and a counsellor Manifest passion and impartial love for children in the teaching and learning process Make useful link of Mathematics with other Subjects and real life situations Have a good master of the Mathematics Content Have good classroom management skills 5. SYLLABUS UNITS 5.1. PRESENTATION OF THE STRUCTURE OF THE SYLLABUS UNITS Page 19 of 60 Mathematics subject is taught and learnt in lower secondary education as a core subject, i.e. in S1, S2 and S3 respectively. At every grade, the syllabus is structured in Topic Areas, sub-topic Areas where applicable and then further broken down into Units to promote the uniformity, effectivness and efficiency of teaching and learning Mathematics. The units have the following elements: 1. Unit is aligned with the Number of Lessons. 2. Each Unit has a Key Unit Competency whose achievement is pursued by all teaching and learning activities undertaken by both the teacher and the learners. 3. Each Unit Key Competency is broken into three types of Learning Objectives as follows: a. Type I: Learning Objectives relating to Knowledge and Understanding (Type I Learning Objectives are also known as Lower Order Thinking Skills or LOTS) b. –Type II and Type III: These Learning Objectives relate to acquisition of skills, Attitudes and Values (Type II and Type III Learning Objectives are also known as Higher Order Thinking Skills or HOTS) – These Learning Objectives are actually considered to be the ones targeted by the present reviewed curriculum. 4. Each Unit has a Content which indicates the scope of coverage of what to be tought and learnt in line with stated learning objectives 5. Each Unit suggests a non exhaustive list of Learning Activities that are expected to engage learners in an interactive learning process as much as possible (learner-centered and participatory approach). 6. Finally, each Unit is linked to Other Subjects, its Assessment Criteria and the Materials (or Resources) that are expected to be used in teaching and learning process. The Mathematics syllabus for ordinary level has got 6 Topic Areas: Algebra, Measures, Proportional reasoning, Geometry, Statistics and Probability. As for units, they are 9 in S1, 11in S2 and 13 in S3 5.2. MATHEMATICS PROGRAM FOR SECONDARY ONE Page 20 of 60 5.2.1. KEY COMPETENCIES AT THE END OF SECONDARY ONE After completion of secondary one, the mathematics syllabus will help learner to: 1. Use correctly simple language structure, vocabulary and suitable symbolism for Ordinary Level Mathematics; 2. Carry out correctly numerical calculations; 3. Solve simple equations of an unknown in , , ID and ; 4. Use methodical and coherent reasoning in solving mathematical problems; 5. Solve problems related to percentage, unitary method, movement, interest, division, surface area and volume of figures; 6. Draw correctly figures by using geometrical instruments and describe them using appropriate terms; 7. Locate area position from numerical data; 8. Make simple chart, graph or diagram from series of a statistical data. 9. Interpret simple diagrams and statistics, recognising ways in which representations can be misleading. 10. Determine the probability of an event happening under equally likely assumption. Page 21 of 60 5.2.2. MATHEMATICS UNITS FOR SECONDARY ONE Topic Area:ALGEBRA S1 Mathematics Unit 1: SETS No. of lessons:30 Key Unit Competency: To be able to use sets, Venn diagrams and relations to represent situations and solve problems. Learning Objectives Knowledge and understanding Skills - Define and give examples of sets - Indicate what a specified region in a Venn diagram represents, using connecting words (and, or, not) or set notation - Show how sets are used in representation of a given information - Observe a contextual problem that involves sets, record the solution, using set notation and give explanations - Demonstrate algebraic and graphical reasoning through the study of relations - Identify different types of relations between sets - Use sets to group and classify according to given conditions - Use Venn diagrams to represent information. - Find intersection, union, complement, difference and symmetrical difference on sets - Represent relations between sets as mappings and graphs - Use sets and relations to solve problems Content Learning Activities Attitudes and values Appreciate how sets, Venn diagrams and relations can be used to represent situations mathematica lly Set Concept: definition of set, notation,examples (subsets of natural numbers like even numbers,odd numbers,prime numbers, etc), cardinal number, Venn diagrams, complement, intersection, union, set difference, symmetric difference Relations:mappings, ordered pair,Cartesian product, domain and range,graphof a relation,equivalence relation(reflexive,symmetric, and transitive),particular relations(function, mapping, injection/ one to one, surjection/ onto, bijections/ one to one and onto) - Inverse relation,composite relations - The class act out various Venn diagrams with rules for sets (e.g. students are numbered and sort themselves according to different rules like even numbers,odd numbers,prime numbers,etc) Represent practical experiences in Venn diagrams and using the notation and symbols of sets, including, union ( ), intersection ( ), subset ( ), complement, difference, symmetrical difference ( ) In pairs, create sets of ordered pairs using the Cartesian product In pairs explore relations between sets (objects, shapes, and numbers) define domain and range, create mappings. Individually, for given relation between sets of numbers, illistrate it using a cartesian plane and show its elements in terms of couples / ordered pairs. In group, investigate when inverse relations are possible and identify the criteria. In pairs, verify if a given relation is an equivalence relation or a composite one. Links to other subjects: Any subject where classification is important e.g. biology, geography, physics, financial education,... Assessment criteria: Use sets, Venn diagrams and relations to represent situations and solve problems. Materials: cards for acting out scenarios … Page 22 of 60 Topic Area: ALGEBRA S1 Mathematics Unit 2: SETS OF NUMBERS No. of lessons: 36 Key Unit Competency: Use operations to explore properties of sets of numbers and their relationships Learning Objectives Knowledge and understanding - Identify sets of numbers (natural, integer, rational and real) and know the relationships between them - Illustrate different set of numbers on a number line - Show that irrational numbers cannot be expressed exactly as a decimal Skills - Curry out mathematical operations on sets of numbers - Work systematically to determine the operation properties of sets of numbers - Determine the hierarchy of sets of numbers and explain its relationship with operations - Convert between decimal and fraction representations of rational numbers Content Learning Activities Attitudes and values Appreciate that rational numbers can be represented exactly as a fraction or a decimal which may terminate or recur. Appreciate that the number line is incomplete without the irrationals which cannot be written exactly as a decimal Vocabularies and notation In groups, add, subtract, multiply and for different sets of numbers divide pairs of natural numbers – for Set of numbers and its which operation(s) is the answer subsets : natural numbers, always/sometimes/never a natural integers, rational numbers, number? Irrational numbers and real - Repeat for integers numbers - Repeat for rational numbers Four operations and - Repeat for real numbers Properties on sets of Individually, construct a Venn diagram numbers to illustrate the relationship between The relationship between two or more sets of numbers. sets of numbers In pairs, investigate the decimal representation of rational numbers and determine why the decimal is terminating or recurring Links to other subjects: It is linked with biology, English,computerscience,geography,chemistry,physics.economics,finance,accounting, construction, etc. Assessment criteria: Use operationsto explore properties of sets of numbers and their relationships. Materials: Text books, manila paper, calculators,. Page 23 of 60 Topic Area:ALGEBRA S1 Mathematics Unit 3: Linear Functions, Equations and Inequalities No. of lessons:36 Key Unit Competency:Represent and interpret graphs of linear functions and apply them in real life situations. Solve linear equations and inequalities, appreciate the importance of checking their solution, and represent the solution Learning Objectives Knowledge and understanding - Define a linear function and recognize its graph - Illustrate that the linear function iswritten in the form , c is the yintercept, m is a measure of steepness and the solution of the equation is the x-intercept - Explain what is meant by the solution of a linear equation and inequality Skills - Plot linear functions on the Cartesian plane - Interpret the graph of a linear function linking the parameters of the function with the features of the graph, including intercepts and steepness - Solve linear equations representing the solution graphically - Solve linear inequalities in one unknown representing the solution on a number line - Check solutions to equations and inequalities by substituting into one side of the original equation - Use linear functions, equations and inequalities to model situations and solve problems Content Learning Activities Attitudes and values Appreciate the importance of checking the solution when solving an equation or inequality and represent on a graph (equation only) and number line - In groups,systematically Linear functions: - Definition, notation and examples: investigate different values of m and c in (best done - Cartesian plane and coordinates using graph plotting software) - Graph of linear function and its to develop intuitive features (intercepts, steepness) understanding. Generalize how to find intercepts and determine Equations and inequalities with one steepness. Plot some examples unknown: by hand to illustrate findings - Solve linear equations with one - In pairs, solve linear equations unknown and represent the and relate the solution to a solution graphically graph - Solve linear inequalities in one - In pairs, solve linear inequalities unknown and represent the and record solutions on a solution on a number line number line - Model and solveproblems using - In groups,research contexts linear functions, equations and where linear functions, inequalities equations and inequalities are relevant – present to class Links to other subjects: Linear functions, equations and inequalities arise in science and economics Assessment criteria: Can represent and interpret graphs of linear functionsand apply them in real life situations. Solve linear equations and inequalities, Page 24 of 60 appreciate the importance of checking their solution, and represent the solution Materials:Digital technology including graph plotting software Topic Area: METRIC MEASUREMENTS (MONEY) S1 Mathematics Unit 4: PERCENTAGE, DISCOUNT, PROFIT AND LOSS No. of lessons:12 Key unit competency: To solve problems that involve calculating percentage, discount, profit and loss and other financial calculations. Learning Objectives Knowledge and understanding Explain how to calculate discount, commission, profit and loss, simple interest, tax Skills - Use percentages to calculate discount, commission, profit, loss, interest, taxes - Solve problems involving - Discount - Commission - Profit and loss - Loans and savings - Tax and insurance Content Learning Activities Attitudes and values - Appreciate the role money plays in our life - Be honest in managing and using money - Appreciate that saving and investing money can increase its value - Appreciate the importance of paying taxes - Percentages Discount Commission Profit and loss Loans and savings (simple interest only) Tax and insurance - In groups research, discuss the use of percentages in business, household and personal finance – prepare a poster - In groups,determine the best value for money with different discount arrangements - In pairs, solve problems involving simple interest, discount, profit and loss Links to other subjects: Personal finance calculations inEconomics, Entrepreneurship, Finance, Accounting, Business Administration and other related fields. Assessment criteria:Able to solve problems that involve calculating percentage, discount, profit and loss and other financial calculations Materials: coins, bills, receipt papers, Electronic materials, ATM cards. Page 25 of 60 Topic Area: PROPORTIONAL REASONING S1 Mathematics Unit 5: RATIO AND PROPORTIONS No. of lessons:12 Key Unit Competency:To be able to solve problems involving ratio and proportion Learning Objectives Knowledge and understanding Skills - Express ratios in their - Compare quantities using simplest form proportions - Share quantities in a given - Identify a direct and proportion or ratio indirect proportion. - Apply ratio and unequal - Differentiate direct sharing to solve given from indirect problems proportion - Solve real life problems involving direct and indirect proportion using tables and graphs - Interpret ratio and proportions in practical contexts. Content Learning Activities Attitudes and values Appreciate the - Ratio, proportion and - In groups, solve problems involving direct sharing and inverse proportion, ratios and sharing importance of adjust recipe amounts for different numbers multiplication when - Applying ratio and proportion in of people working with ratio practical and - In pairs, match different representations of and proportion everyday contexts ratios and proportions including simplest - Direct and indirect form proportional - In groups, interprete and explain the ratio and relationships in proportion in maps and scale practical contexts drawings/models - In pairs, solve problems in practical contexts involving direct and indirect proportion using tables of values and graphs Links to other subjects: Any subject where proportional reasoning is required e.g. biology, physics, computer science, chemistry, economics, personal finance etc. Assessment criteria: can solve problems involving ratio and proportion in a variety of contexts Materials: calculators and electronic materials Page 26 of 60 Topic Area:GEOMETRY S1 MATHEMATICS Unit 6: POINTS, LINES AND ANGLES No. of lessons:36 Key Unit Competency:To be able to construct mathematical arguments using the angle properties of parallel lines Learning Objectives Knowledge and understanding - Recognize the position of an angle at a point sum to 360o; angles at a point on straight line sum to 1800 - Distinguish and recognize vertically opposite, corresponding, alternate and supplementary angles Skills Content Learning Activities Attitudes and values - Use knowledge of - Appreciate the need - Segments, rays, lines - Practical – fold a paper triangle to bring all angles angle properties of to give reasons when and acute, right, together at a point - in groups discuss why this works parallel lines and developing solutions obtuse and reflex shapes to to missing angle angles construct problems - Parallel and arguments when - Value a variety of transversal lines and finding missing different approaches their properties. angles in to reach the same - Constructing - Inpairs,draw two parallel lines and a transversal, identify geometric conclusion mathematical all angles that are equal (measure to check) – identify diagrams arguments using vertically opposite, corresponding, alternate and - Construct and angle properties of supplementary angles and write their own glossary calculate angles parallel lines and In groups, solve missing angle problems, giving reasons shapes for each step in the process Links to other subjects: Physics, construction, engineering, geography, fine arts, scientific drawing. Assessment criteria:construct mathematical arguments using the angle properties of parallel lines Materials:Manila papers,geometrical instruments, Electronic materials. Page 27 of 60 Topic Area: GEOMETRY Mathematics S1 Unit 7: SOLIDS No. of lessons: 24 Key unitcompetency:To be able to select and use formulae to find the surface area and volume of solids. Learning Objectives Knowledge and understanding Skills Content Learning Activities Attitudes and values - Explain the - Derive the surface - Appreciate the - Components of - In small groups, count the number of faces (f), number of vertices(v) surface area of a area for prism and difference solids: faces, and number of edges(e) for a variety of solid figureswith polygonal solid as the area cylinder between vertices and faces and look for relationships (e.g. Euler’s rule(f+v=e+2) of the net - Calculate the surface surface area edges - In groups, investigate the relationship between the surface area of - Illustrate the area and volume of and volume and - Surface area and cuboids, prisms, pyramids, cylinders and their nets. Generalize. volume as the common geometrical recognize volume of a prism, - Practically in pairs(or teacher demonstration) measure the diameter space occupied by solids, using formulas solids in the pyramid, cylinder, of an orange then peel carefully and arrange the peel into circles with a solid where necessary environment cone and sphere the same diameter as the orange. How many circles does the peel fill? - Distinguish - Distinguish between - Formulae for (Roughlyfour). Relate to formula between surface surface area and surface area and - In groups, select appropriate methods and units when solving area and volume volume and select volume problems concerning the volume and surface area of solids e.g. design and know the appropriate formulae solids with a volume of 1000cm3, minimizing their surface area; what is correct units and units to use in the greatest volume cylinder that can be made from a sheet of A4 paper various contexts Links to other subjects: Where volume and area calculations may be needed e.g. physics, construction, engineering, geography, fine arts, scientific drawing etc. Assessment criteria:Able to select and use formulae to find the surface area and volume of solids Materials: solid figures for practical work, paper, scissors, glue, calculators, oranges Page 28 of 60 Topic Area: STATISTICS AND PROBABILITY SENIOR1 MATHEMATICS Unit8: STATISTICS (ungrouped data ) No. of lessons:24 Key Unit Competency: To be able to Collect, to represent, and to interpret quantitative discrete data appropriate to a question or problem. Learning Objectives Knowledge and understanding - Define quantitative data and qualitative data - Differentiate discrete and continuous data - Present data on a frequency distribution - Define mode and median of given statistical data - Recognize formulae used to calculate the mean and median - Read diagram of statistical data. Skills - Apply data collection to carry out a certain research. - Represent statistical information using frequency distribution table, bar chart, Histogram, Polygon, Pie chart or pictogram. - Determine the mode, mean and median of statistical data - Interpret correctly the graphs involving statistical data Content Learning Activities Attitudes and values - Help in decision - Definition of data. making and - Types of data (qualitative, draw conclusion quantitative, discret and - Self confidence continuous data and - Collecting data determination - Frequency distribution - Develop - Measures of central competitiveness tendency: mode, Mean, - Appreciate the median, quartiles (1st , 2nd, importance of 3rd quartiles, interorder in daily quartile range) activities. - Data display: Bar chart, - Develop Histogram, Frequency research and Polygon, Pie chart, creativity. Pictogram - Respect each - Reading statistical graphs other. - Converting statistical graphs into frequency tables In groups, collect data for a given situation such as heights, weight, colors, blood group, ages, marks etc Discuss whether it is quantitative or qualitative data, continuous or discrete data. Hence make a frequency distribution table for each case In groups, observe and collect data for a given situation such as height, weight, ages, marks etc. then determine, mode, mean and median. In groups, draw a bar chart, a histogram, frequency polygons and a pie chart corresponding to the data collected and compare results In pairs, calculate the quartiles, the inter-quartile range and represent them graphically Individually, in the given bar chart, histogram, polygon, pie chart identify mode, draw frequency table and hence find mean and median. work in group Given a graph, indicate/estmate where the mode, Mean, median can be found. Links to other subjects: Any subject where data collection, data representation, and data interpretation are important e.g. biology,geography, physics, computer science, finance, economics,engineering, etc. Assessment criteria:Consistently make appropriate data collection and data representation to solve a problem, and thendraw conclusions consistent with findings. Materials:Text books, papers, geometrical instruments, Electronic materials Page 29 of 60 Topic Area: STATISTICS AND PROBABILITY Mathematics S1 Unit 9: PROBABLITY No. of lessons:6 Key Unit Competency:To be able to determine the probability of an event happening using equally likely events or experiment. Learning Objectives Knowledge and understanding - Define event and explain why the probabilities can only be between 0 (impossible) and 1 (certain). - Explain that probabilities can be calculated using equally likely outcomes (e.g. tossing a coin or dice, drawing a card from a pack) or estimated using experimental data (e.g. weather, sport, arriving late to school) - Demonstrate that the more data is collected; the better is the estimate of the probability. Skills Content Learning Activities Attitudes and values - Calculate the - Appreciate that the probability of an chance of an event event where happening is given there are equally by its probability likely outcomes which is number e.g. heads or tails between 0 on a coin, a score (impossible) and 1 on a dice (certain) - Estimate - Distinguish when an probabilities experiment is using data necessary to find a probability and that more data improves the estimate - Definition of event In groups:Think and debate chance situations and outcome such as playing cards, tossing a coin, rolling dice – - Examples of random what are the chances of getting a particular events outcome? Introduce probability scale. - Probability of equally - Consider playing football, basketball ball, volley likely outcomes ball, hand ball or any other game.Discuss the through experiments chance ofwin, lose or draw.Use results to estimate like tossing a coin or probabilities. dice, etc - Investigate the relationship between - Estimation of experimental and calculated probability by probabilities where tossing a dice or coin many times and estimating experimental data is the probability of a particular outcome – plot a required graph to show the experimental probability and note how that tends to the calculated probability Links to other subjects: Any subject where probability is important e.g.economics, finance, physics chemistry,biology. Assessment criteria:Use appropriate mathematical concepts and skills to solve problems in both familiar and unfamiliar situations. Materials:Dice, coins, playing cards,graph paper Page 30 of 60 5.3. MATHEMATICS PROGRAM FOR SECONDARY TWO 5.3.1. KEY COMPETENCIES AT THE END OF SECONDARY TWO After completion of secondary two, the mathematics syllabus will help learner to: 1. Use correctly the simple language structures, vocabulary and the symbols found in the second year mathematics program; 2. Carry out efficiently numerical and literal calculations; 3. Solve the equations and inequalities of the first degree in ℝ 4. Recognize and justify congruent shapes. 5. Calculate the component of a vector. 6. Identify the image of a figure under a transformation and use the properties of transformations to solve related problems. 7. Use methodical and coherent reasoning in solving mathematical problems; 8. Collect quantintative data appropriate to the problem or investigation, taking into account possible bias and extend the knowledge to grouped data. Page 31 of 60 5.3.2. MATHEMATICS UNITS FOR SECONDARY TWO Topic Area: ALGEBRA S2 MATHEMATICS Unit 1: INDICES AND SURDS No. Of lessons: 18 Key unit competency: To be able to Calculate with indices and surds, use place value to represent very small and very large numbers. Learning Objectives Knowledge and understanding Skills -Recognize laws of indices -Represent very small number or large number in standard form -Define and give examples of surds -Identify properties of surds -Recognize the conjugates of surds. - Perform operations on indices and surds. - Solve simple equations involving indices and surds. - Use standard form to represent a number. - Apply properties of indices to simplify mathematical expressions - Apply Properties of surds to simplify radicals. - Compute rationalisation of denominator on surds. Attitudes and values Learning Activities Content Appreciate the Indices/powers or exponents. importance of o Definition indices and o Properties indices. surds in solving o Applications of indices: mathematical - Simple equations involving indices. problems. - Standard form Show concern of Surds/ radicals self-confidence, o Definition and examples determination, o Properties of surds and group work o Simplification of surds spirit. o Operations on surds o Rationalisation of denominator. Square roots calculation methods: o By factorization o By general method. o Estimation method - In pairs, learners think themselves two numbers or more having different powers but the same base, multiply and divide them.Then draw conclusion. - In groups, express the given larger numbers or smaller numbers in standard form. - Solve given equations involving indices - Individually, simplify surds by rationalizing the denominators - In groups, express each of the given surd as the square root of a single number - In groups, discuss and reduce surds to the simplest possible surd form - find the square roots of given numbers by using Square roots methodsand calculators Links to other subjects: Physics, Chemistry, Biology, Computer science, Economics, Finance, etc. Assessment criteria: Use rules of indices and surds to simplify mathematical situation involving indices and surds Materials: Calculator. Page 32 of 60 Topic Area: ALGEBRA S2 MATHEMATICS Unit 2: POLYNOMIALS No. of lessons: 30 Key Unit Competency: To be able to perform operations, factorise polynomials and solve related problems Learning Objectives Knowledge and understanding - Define polynomial - Classify polynomials by degree and number of terms. - Recognize operation properties on polynomials - Give common factor of algebraic expression Skills - Perform operation of polynomials - Expand algebraic expression by removing bracket and collecting like terms - Apply operation properties to carry out given operation of polynomials. - Factorize a given algebraic expression using appropriate methods. - Expand algebraic identities Content Attitudes and values Appreciate the role of numerical values of polynomials. and algebraic identities in simplifying mathematical expressions. Develop critical thinking and reasoning Ability to classify and able to follow order to perform a given task. Learning Activities - Definition and classification of polynomials including homogenious polynomials (monomials,binomials, and trinomials, polynomial of four terms) - Operations on polynomials. - Numerical values of polynomials. - Algebraic identities - Factorization of polynomials by : o Common factors o Grouping terms o Algebraic identities o Zeros (roots) of polynomials o Factorization of quadratic expressions(Sum and Product) In groups: - Classify polynomials according : to their degree or to the number of terms. - Discuss and perform operations on polynomials - Expand and factorize given mathematical expressions Links to other subjects: Any subject where polynomials are important like in Physics, Chemistry, etc. Assessment criteria: Perform operations, factorise polynomials and solve related problems. Materials: Text book, Papers, calculators Page 33 of 60 Topic Area: ALGEBRA S2 MATHEMATICS Unit 3: SIMULTANEOUS LINEAR EQUATIONS , INEQUALITIES No. of lessons: 30 Key unit Competency: To be able to solve problems related to simultaneous linear equations, inequalities and represent thesolution graphically. Learning Objectives Knowledge and understanding Skills Attitudes and values Content Learning Activities - Define simultaneous - Solve - Appreciate the - Definition and examples of simultaneous linear equations - In pairs, show whether a linear equations and simultaneous importance of solving in two variables and inequalities in one variable. given system of 2 linear give examples. linear equations problems related to - Types of simultaneous linear equations (independent equations is independent, - Show whether a in two variables simultaneous linear simultaneous linear equations, dependent simultaneous dependent, or inconsistent given simultaneous - Model and solve equations, inequalities. linear equations, and inconsistent/incompatible - In group, discuss different linear equtions is mathematical - Be accurate in solving simultaneous linear equations) methods for solving independent,depen word problems system of linear - Solving simultaneous linear equations in two unknowns simultaneous linear dent or inconsistent using equations, inequalities. using algebraic methods:Substitution method, equations and use one of - Recognize the forms simultaneous - Developing self Comparison method, Elimination method, and them(on your choice) to of compound equations confidence in solving Cramer’s rule. solve a given simultaneous inequalities with - Solve compound system of linear linear equations. - Inequalities of the types: , , one unknown and inequalities in equations/ inequalities , Individually, solve problems , give examples one variable. in one variable. involving simultaneous , equations. In pairs, solve given - Compound inequalities or systerm of two inequalities in simultaneous inequalities in one unknown. two unknowns and given compound inequalities. Links to other subjects: Any subject where simultaneous linear equations and inequalities are needed. Assessment criteria: Solve problems related simultaneous linear equations, inequalities and represent the solution graphically. Materials: Calculators, text book, papers. Page 34 of 60 Topic Area: ALGEBRA (PROPORTIONS REASONING) S2 MATHEMATICS Unit 4: MULTIPLIER FOR PROPORTIONAL CHANGE No. of lessons: 12 Key Unit Competency: To be able to use a multiplier for proportional change Learning Objectives Knowledge and understanding - Recognize the properties of proportions - Express ratio in their simplest form - Share quantities in a given proportion or ratio. Skills Content Learning Activities Attitudes and values - Solve problems in real life - Be honest in sharing involving multiplier proportion with other. change - Develop critical - Apply multipliers for thinking in terms proportional change to solve proportion multiplier given problem for proportional change - Use multiplier for proportional change to find the new quantities - Use “Decreased by n%” and “Increased n%” Increasing quantities by - In group, solve problems involving multiplier n% for proportional change Decreasing quantities by - Individually , solve problems involving n% Decreased by n% and Calculation of Increased by n% proportional change using multiplier Links to other subjects: Economics, Entrepreneurship, Finance, Accounting, Business Administration and other related fields. Assessment criteria: Explain the importance of money in connection to real life. Materials: Text books, coins, bills, geometrical instruments, receipt papers, Electronic materials, ATM cards. Page 35 of 60 Topic Area: GEOMETRY S2 MATHEMATICS Unit 5: THALES THEOREM No. of lessons:12 Key Unit Competency: Use Thales’ theorem to solve problems related to similar shapes, and determines their lengths and areas. Learning Objectives Knowledge and understanding - Identify and name triangles or trapezium from parallel and transversals intersecting lines - State Thales’ theorem and its corollaries Skills Content Learning Activities Attitudes and values - Associate extended - Develop participation, self- Midpoint theorem proportions in the confidence, determination, and - Thales’ theorem and its triangles team spirit. converse - Apply Thales’ theorem - Appreciate the importance of - Application of Thales‘ and its corollaries to solving daily activities involving theorem in calculating solve problems on midpoint theorem, lengths of proportions proportions of triangles, Thales’ theorem and its converse segments (in triangles, trapezium and application of Thales‘theorem. trapezium) - Discuss the converse of Thales’ theorem - In groups, solve problems involving midpoint theorem for a given situation. - In group, discuss and solve mathematical problems involving the applications of Thares’theorem. Links to other subjects: Technical drawing, Scientific drawing, Light Physics etc. Assessment criteria: Use Thales’ theorem to Solve problems related to similar shapes, and determines their lengths and areas. Materials: Geometrical instruments. Page 36 of 60 Topic Area: GEOMETRY S2 MATHEMATICS Unit 6: PYTHAGORAS’ THEOREM No. of lessons: 12 Key units Competency: To be able tosolve problems of lengths in right angled triangles by using Pythagoras’ theorem. Learning Objectives Knowledge and understanding Skills - State Pythagoras’ - Use Pythagoras’ theorem theorem to find - Identify the lengths of sides of hypotenuse in three right angled triangle sides of a right angled - Apply Pythagoras’ triangle theorem to solve - List properties of a problems in range of right angled triangle contexts - Demonstrate Pythagoras’ theorem practically Content Learning Activities Attitudes and values - Appreciate the role of - Pythagoras’ theorem - In groups, find the squares of given sides of a triangle, Pythagoras’ theorem - Demonstration of Pythagoras’ verify relationship between the sum of the squares of in solving daily life theorem shorter sides and the square of the longer side. activities. - Applications of Pythagoras’ Hence, discuss whether a triangle is right-angled given - Develop confidence theorem in the calculation of the length of sides and give the properties. and accuracy in any side of right angled constructing shapes. triangle , word problems - Individually, using Pythagoras’ theorem, find the - Develop team work length of the hypotenuse, if the other sides of the right spirit and respect angle are given. analytically the views - In groups, learners can practically, demonstrate of others. Pythagoras’ theorem by : Measuring the areas of squares on sides of the right angled triangles Exploring phythagorean dissections by cutting and reassembling parts Links to other subjects: Technical drawing, Scientific drawing, Optics,etc. Assessment criteria: Solve problems of lengths in right angled triangles by using Pythagoras’ theorem. Materials: Geometrical instruments, Calculators Page 37 of 60 Topic Area: GEOMETRY S2 MATHEMATICS Unit 7: VECTORS No. of lessons: 18 Key unit competency: To be able to solve problems using operation on vectors. Learning Objectives Knowledge and understanding - Define a vector - Represent a vector in a Cartesian plane - Differentiate between vector quantities and scalar quantities. - Show whether vectors are equal. Skills Attitudes and values - Use vector notations correctly and perform operations on vectors - Appreciate the importance of vectors in motion. - Show self-confidence; and, determination while solving - Find the components of problems on vectors. a vector in the Cartesian plane - Find the magnitude of a vector Content - Concept of a vector : definition and properties of a vector, notation - Vectors in a Cartesian plane - Components of a vector in the Cartesian plane - Equality of vectors - Operations on vectors : o Addition o Subtraction o Multiplication by a scalar - Magnitude of a vector as its length. Learning Activities In groups, graphically add and subtract given consecutive or any vectors using parallelogram rule. Graphically multiply a given vector by a scalar individually or in groups. In groups, perform multiplication of vectors by a scalar, addition or subtraction of vectors given their components. Individually, calculate the magnitude of vectors given their components Links to other subjects: Physics (forces) ,... Assessment criteria: Solve problems using operation on vectors. Materials: Geometrical instruments and calculators Page 38 of 60 Topic Area: GEOMETRY S2 MATHEMATICS Unit 8: PARALLEL AND ORTHOGONAL PROJECTIONS No. of lessons: 12 Key unit Competency: To be able to transform shapes under orthogonal or parallel projections Learning Objectives Knowledge and understanding - Identify an image of a figure under Parallel projection - Identify an image of a figure under orthogonal projection Skills Content Learning Activities Attitudes and values - Construct an image of - Show the importance of parallel - Definition of : an object or geometric and orthogonal projection in o Parallel projection shape under : various situations. o Orthogonal projection o Parallel projection - Develop critical thinking and - Properties of: o Orthogonal reasoning while transforming o Orthogonal projection projection. shapes under parallel or o parallel projections orthogonal projection. - Image of geometric shape - Be accurate in construction of under: figures and their images under o Parallel projection parallel or orthogonal o Orthogonal projection projection - Develop confidence in solving problems related to transformation of shapes under parallel or orthogonal projection. o In groups, observe drawings of different objects and their images involving parallel or orthogonal projection. Discuss and deduce properties and type of projection used. Links to other subjects: Technical drawing, scientific drawing,... Assessment criteria: make image of figures using parallel projections Materials: Geometrical instruments, calculators. Page 39 of 60 Topic Area: GEOMETRY S2 MATHEMATICS Unit 9: ISOMETRIES No. of lessons: 30 Key Unit Competency: To be able to transform shapes using congruence (central symmetry, reflection, translation and rotation). Learning Objectives Knowledge and understanding Content Skills - Identify an image of - Construct the image of a a figure under: point, a segment, a o Central geometric shape, under: symmetry o Central symmetry o Reflection o Reflection o Translation o Translation o Rotation o Rotation - Find the coordinates of image of an object under: o Central symmetry o Reflection o Translation o Rotation Learning Activities Attitudes and values - Appreciate that - Definition of: - In groups, construct the image of a given object translation, rotation and o Central symmetry under central symmetry , then compare the reflection play important o Reflection image to the initial, and then discuss and role in various situations. o Translation deduce the applied properties. - Develop team work spirit o Rotation - Repeat the above activity for reflection , - Develop confidence in - Construction of an image of an translation or rotation construction of the image object / geometric shape under : - In pairs, given an object and its image find: the of a point, a segment, a o Central symmetry center of symmetry, line of symmetry or the geometric shape under o Reflection translation vector, the center of rotation and any isometry o Translation angle of rotation. - Develop accuracy in o Rotation - Individually, construct the images of a given constructing shapes - Properties and effects of:Central object under successive transformations . under isometries symmetry Reflection o Translation o Rotation - Composite transformations up to three isometries Links to other subjects: Physics, ICT, Engineering, technical drawing, scientific drawing,… Assessment criteria: Transform shapes using congruence (central symmetry, reflection, translation and rotation). Materials: Geometrical instruments, calculators. Page 40 of 60 Topic Area: STATISTICS AND PROBABILITY S2 MATHEMATICS Unit10: STATISTICS (grouped data ) No. of lessons: 30 Key Unit Competency: To be able to collect, represent and interpret grouped data. Learning Objectives Knowledge and understanding - Define grouped data and represent grouped data on a frequency distribution - Identify mode, middle class, modal class and median of given grouped statistical data - Read diagram of grouped statistical data Skills Content Learning Activities Attitudes and values - Apply data collection - Appreciate how to carry out a certain data collection, data research. representation and - Represent grouped data interpretation statistical information can be used for using: histogram, solving real life polygon, frequency situations. distribution table and - Appreciate the pie chart. importance of data - Calculate the mode, in culture of mean and median of investigation statistical data and decision making. - Interpret correctly - Team work spirit the graph of grouped and respect the statistical data views of others. - Develop accuracy in reading graphs - Definition and examples of grouped data - Grouping data into classes - Frequency distribution table for grouped data. - Cummulative frequency distribution table. - Measures of Central tendency for grouped data: o Mean, Median , Mode, and range for grouped data - Graphical representation of grouped data: o polygon o Histogram o Superposed polygon - In groups: - Collect data for a given situation such as height, weight, ages, or marks in any subject, group them in a given interval, and then represent them in a frequency distribution table. - Collect data for a given situation such as their height, weight, ages, marks in any subject then group them into classes in given interval, determine the middle class, modal class, class mean and median class. Then draw histogram, frequency polygons and a superimposed frequency polygon and interpret the result then infer conclusion. - Converting statisticalgraphs into frequency tables and finding measures of central tendency using graphs Links to other subjects: History, Biology, Geography, Physics, Computer Science, Finance, Etc Assessment criteria: Collect, represent and Interpret grouped data. Materials: Calculators, graph papers. Page 41 of 60 pic Area: STATISTICS AND PROBABILITY S2 MATHEMATICS Unit11: TREE AND VENN DIAGRAMS AND SAMPLE SPACE No. of lessons: 12 Key unit Competency: To be able to determine probabilities and assess likelihood by using tree diagrams Learning Objectives Knowledge and understanding Skills Content Learning Activities Attitudes and values - Define mutually - Construct and - Appreciate the exclusive and Interpret correctly importance of independent events the tree diagram probability to find - Count the number of - Use tree and Venn chance for an event to branches and total diagrams to happen. number of outcomes determine - Show curiosity to on a tree diagram probability. predict what will happen in future. - Promote team work spirit and selfconfidence. - Tree diagram. In groups: - Total number of - For a given task, construct a tree diagram corresponding to that outcomes situation and determine the number of branches. Hence - Determining calculate required probability probability using: - Analyze a given situation, present it using Venn diagram and o Venn diagram then find probability. e.g. in the venn diagram E= {pupils in class o Tree diagram of 15}, G={girls}, S={Swimmers} , F={Pupils who are christians}. - Mutually exclusive A pupil is chosen at random . find the and independent probability that the pupil : events. a) Can swim b) Is a girl swimmer c) Is a boy swimmer who is christian Two pupils are chosen at random . find the probability that : d) both are boys e) neither can swim f) both are girls swimmers who are chritians - For given tasks on events , suggest whether events aremutually exclusive or independent or neither. Links to other subjects: Financial education, Physics, Chemistry, Biology, Physical Education and Sport, Etc. Assessment criteria: Determine probabilities and assess likelihood by using tree diagrams Materials: Dice, Coins, Playing Cards, Calculators, Balls, Page 42 of 60 5.4. MATHEMATICS PROGRAM FOR SECONDARY THREE 5.4.1. KEY COMPETENCIES AT THE END OF SECONDARY THREE 1. Carry out efficiently numerical and literal calculations; 2. Solve problems that involve sets of numbers using Venn diagram; 3. Represent graphically a function of the first degree, a function of the second degree point by point; 4. Solve equations, inequalities and the systems of the first degree in two unknowns; 5. Apply compound interest in daily life situations; 6. Calculate the side lenghts, angles in a right triangles and areas of geometric shapes. 7. Represent and interpret graphs to linear and quadratic functions 8. Construct mathematical arguments using circle theorems 9. Construct the image of a geometric figure under composite transformations. 10. Collect bivariate data to investigate possible relationships through observations. Page 43 of 60 5.4.2. MATHEMATICS UNITS FOR SECONDARY THREE Topic Area: ALGEBRA S3 MATHEMATICS. Unit 1: PROBLEMS ON SETS No. of lessons: 6 Key Unit Competency: To be able to solve problems involving sets. Learning Objectives Knowledge and understanding - Express a mathematical problem on set using a venn diagram - Represent a mathematical problem on venn diagram Skills Attitudes and values - Use Venn diagram to - Develop clear, logical and represent a coherent thinking in solving real mathematical life problems involving sets. problem on sets - Appreciate the importance of - Interpret , model , and representing , and solving a solve a mathematical mathematical problem on sets problem on sets using venn diagram Content Learning Activities - Matheatical Problems involving In groups sets: - Observe information o Analysis and given in the Venn interpretation of a diagram and solve problem using set related questions language (intersection, - Discuss on a given union,...) situation involving set o Representation of a theory, represent it problem using venn using Venn diagram, diagram , form an equation and o Modeling and Solving a solve related questions problem Links to other subjects: Any subject where classification is important e.g. biology, geography, physics, financial education, .... Assessment criteria: Solve problems involving sets. Materials: Calculators. Page 44 of 60 Topic Area: ALGEBRA S3 MATHEMATICS Unit 2: NUMBER BASES No. of lessons: 12 Key Unit Competency: To be able to represent numbers in different number bases and solve related problems. Learning Objectives - Knowledge and understanding Skills List the digits used in a given base Convert number from base ten to any other base and vice versa. - Carry out addition and subtraction multiplication and division on numbers bases - Solve equations involving bases. Content Learning Activities Attitudes and values - Develop clear, logical and - Definition and - In group: Convert a given number from coherent thinking while examples of different base ten to any other base and vice versa. solving problems on sets number bases. Convert numbers from any base diffrent - Appreciate the importance - Converting a number from ten to another. of bases in various from base ten to any - Discuss and carry out operations on contexts. other base like base 2, number bases 3, or 5 and vice versa - In groups, solve equations involving bases - Converting a number from one base to another (e.g. base 2 to base 3, etc). - Addition and substraction on number bases - Multiplication and division on number bases - Solving equations involving number bases Links to other subjects: ICT, etc Assessment criteria: Represent numbers in different number bases and solve related problems. Materials: Calculators. Page 45 of 60 Topic Area: ALGEBRA S3 MATHEMATICS Unit 3: Algebraic fractions No. of lessons:24 Key Unit Competency: To be able to perform operations on rational expressions and use them in different situations. Learning Objectives Knowledge and understanding - Define an algebraic fraction - State the restriction on the variable in algebraic fraction - Recognize the rules applied for addition, subtraction, multiplication or division and simplification of algebraic fractions. Skills Content Learning Activities Attitudes and values - Perform operations - Develop clear, logical and on algebraic fractions coherent thinking while - Solve rational working on algebraic fractions equations with linear - Show concern of patience, denominators mutual respect, tolerance, team - Simplify algebraic sprit and curiosity in group fractions activities while solving and discussing about mathematical situations involving algebraic fractions. - Definition and examples of In groups, an algebraic fraction - State the restrictions on - Restrictions on the variable the variable given or conditions of existence of algebraic fractions an algebraic fraction. - Carry out different - Simplification of algebraic operations for given fractions algebraic fractions and - Addition or substraction of simplify. Then present algebraic fractions with and explain the findings linear denominators - Multiplication or division Individually, Solve given and simplification of two rational equations. algebraic fractions - Solution of rational equations with linear denominators . Links to other subjects: Physics (Distance problems, Motion problems…), work problems… Assessment criteria: Perform operations on rational expressions and use them in different situations Materials: Calculators. Page 46 of 60 Topic Area: ALGEBRA S3 MATHEMATICS Unit 4: Simultaneous Linear Equations and inequalities No. of lessons : 18 Key Unit Competency: To be able to solve word problems involving simultaneous linear equations. Learning Objectives Knowledge and understanding - Define simultaneous linear inequalities in two unknowns - Give examples of simultaneous linear inequalities in two unknowns - Show solution set of simultaneous linear equations /inequalities in two unknowns given their graphs Skills - Solve graphically simultaneous linear equations / inequalities in two unknowns - Interpret graphical solution of simultaneous linear equations /inequalities in two unknowns - Solve word problems leading to simultaneous linear equations Content Learning Activities Attitudes and values Appreciate how simultaneous linear - Graphical solution of equations in two unknowns is important simultaneous linear equations in to represent and solve mathematical two unknowns word problems - Solving word problems Develop clear, logical and coherent involving simultaneous linear thinking while solving simultaneous equations in two unknowns ( linear equations /inequalities in two graphically or algebraically) unknowns - Definition and examples of Show concern of patience, mutual simultaneous linear inequalities respect, tolerance, team sprit and in two unknowns. curiosity in group activities while solving - Solving simultaneous linear and discussing about mathematical inequalities in two unknowns situations involving simultaneous linear equations / inequalities in two unknowns In group: Solve given simultaneous linear equations / inequalities in two unknowns graphically. Solve word problems involving simultaneous linear equations. Obseve a given graphical representation of simultaneous linear equations/ inequalities in two unknowns and deduce or show the solution set Links to other subjects: Physics, Financial education … Assessment criteria: Solve word problems involving simultaneous linear equations. Materials: Geometrical instruments, Calculators. Page 47 of 60 Topic Area: ALGEBRA S3 MATHEMATICS Unit 5: QUADRATIC EQUATIONS No. of lessons: 24 Key unit competency: To be able to solve quadratic equations. Learning Objectives Knowledge and understanding - Define quadratic equations - State methods used to solve a quadratic equation. Skills Content Learning Activities Attitudes and values - Solve quadratic - Develop clear, logical and coherent equations thinking in solving quadratic - Model and solve equations problems involving - Appreciate the importance of quadratic equations. quadratic equations in solving word - Solve equations problems involving quadratic reducible to equations. quadratic . - Promote team work spirit when working in group while solving quadratic equations - Show concern of patience, mutual respect, tolerance and curiosity when discussing and solving problems involving quadratic equations in groups - Definition and example of In groups: quadratic equation - Solving quadratic equations - Discuss and use factorization or any othor method to solve given by: quadratic equations o Factorization o graph - Model given mathematic problems o Completing squares. using quadratic equations and solve o Quadratic formula them. o Synthetic division - Problems involving quadratic - Solve given equations reducible to quadratic by using: equations. o Factorization - Solution of equations o Synthetic division (Horner’s reducible to quadratic by: method) o Factorization o Horner’s method Links to other subjects: Physics, Financial education … Assessment criteria: Solve word problems involving quadratic equations. Materials: Geometrical instruments, Calculators. Page 48 of 60 Topic Area: ALGEBRA S3 MATHEMATICS Unit 6: LINEAR AND QUADRATIC FUNCTIONS No. of lessons: 24 Key Unit Competency: To be able to solve problems involving linear or quadratics functions and interpret the graphs of quadratic functions. Learning Objectives Knowledge and understanding - Define a Cartesian equation of a straight line. - Define quadratic function - List the characteristics of linear or quadratic function. - Diffentiate linear from quadratic function Skills - Determine: o Cartesian equation of straight line o Coordinates of vertex o The equation of axis of symmetry - Determine the intercepts of a quadratic function. - Sketch and draw graphs from a given function - Use linear or quadratic function to solve problems in various situations and interpret the results. Content Learning Activities Attitudes and values Develop clear, logical and coherent thinking in working out on linear and quadratic functions Appreciate the importance of linear and quadratic functions in learning other subjects. Show concern on patience, mutual respect, tolerance, team work spirit, and curiosity in solving and discussions about problems involving linear and quadratic functions Linear functions: - In group, determine the equation of o Slopes straight line passes through: o Cartesian equation o a point and given its slope o Conditions for o two points lines to be o a point and parallel to a given line parallel or o a point and perpendicular to a perpendicular given line line - Individually, given a quadratic Quadratic functions function, say whether its graph is o Table of values concave up or down and determine o Vertex of the intercepts, the vertex, then make a parabola o Axis of symmetry table of values hence sketch the parabola. o Intercepts o Graph in Cartesian plane Links to other subjects: Physics (linear motion), … Assessment criteria: Solve problems involving linear or quadratics functions and interpret the graphs of quadratic functions. Materials: Geometrical instruments, Calculators. Page 49 of 60 Topic Area: ALGEBRA (MONEY ) S3 MATHEMATICS Unit 7: Compound interest , reverse percentage and compound proportional change No. of lessons: 20 Key unit Competency: To be able to solve problems involving compound interest, reverse percentage and proportional change using multipliers Learning Objectives Knowledge and understanding Skills Attitudes and values Content Learning Activities - Define compound - Solve problems involving - Appreciate the role of compound - Reverse percentages In group: interest, reverse reverse percentages and interest in banking, in financial - Compound interest - Solve given problems percentage , compound compound interest activities and its applications in involving reverse proportional change and - Apply compound interest in - Appreciate that in case of contextual situations ( percentages and compound continued proportional solving mathematical compound interest , saving and e.g. in banking, in interest - Find a reverse problems involving savings investing money can increase its financial activities...) - Compare the overall values percentages in a given or calculations in any other value. of given different goods, mathematical problem financial activity. - Show concern of paying taxes and - Compound hence draw conclusion. - Determine a compound - Apply reverse percentage being honest in daily activities proportional change or - In group, solve problems interest in a given and compound proportional involving money. Continued proportions involving compound mathematical problem change in solving real life - Develop logical and critical thinking proportional change or - Simplify ratio in their mathematical problems while solving problems involving continued proportions simplest form compound interest, reverse change, and continued proportional change Links to other subjects: The unit is linked with Economics, Entrepreneurship, Financial education and other related fields. Assessment criteria: Use mathematical concepts and skills to solve problems involving compound interest, reverse percentage and proportional change Materials: Text books, coins, bills, geometrical instruments, receipt papers, Electronic materials, ATM cards, Calculators. Page 50 of 60 Topic Area: GEOMETRY S3 MATHEMATICS Unit 8: RIGHT ANGLED TRIANGLES No. of lessons: 18 Key Unit Competency: to be able to find side lengths and angles in right angled triangles using trigonometric ratios Learning Objectives Knowledge and understanding - - Give and define the elements of a right angled triangle Show relationship between the elements of a right angled triangle. Skills Attitudes and values - Use Pythagoras’ - Appreciate the theorem to find importance of right relationship angled triangles in between the various situations. elements of a right - Promote team work angled triangle. spirit. - Solve problems about - Show concern on right angled triangle patience, mutual using the properties respect, tolerance and of elements of a right curiosity in the solving angled triangle and and discussion about Pythagoras theorem. problems involving right angled triangles Content - Median through the vertex of the right angle - Height through the vertex of a right-angled triangle and the sides of the right angle - Height through the vertex of a right-angled triangle and the lengths of the segments on the hypotenuse - Determination of the sides of right angled triangle given their orthogonal projection on the hypotenuse. - Trigonometric ratios in a right angled triangle: Sine, cosine and tangent Learning Activities - - - - In group, find the missing sides of a triangle in each case draw the clear diagram before starting to calculate In pairs, given two sides of right angled triangle but not hypotenuse. Find the length of hypotenuse and then calculate the length of height and the median corresponding to that hypotenuse In groups, given lengths of segments defined by the height on hypotenuse, find length of sides of that triangle, the height and the median Individual, given acute angle and length of one side; use sine, cosine and tangent to find the length of other sides. Links to other subjects: Technical drawing, scientific drawing… Assessment criteria: Construct mathematical arguments about right angled triangle to solve related problems. Materials: Calculators, geometrical instruments. Page 51 of 60 Topic Area: GEOMETRY S3 MATHEMATICS Sub-topic Area: SHAPE AND ANGLES Unit 9: CIRCLE THEOREMS No. of lessons: 18 Key Unit Competency: To be able to construct mathematical arguments about circle and use circle theorems and disk to solve related problems. Learning Objectives Knowledge and understanding - Recognise and identify the elements of a circle - Identify angle properties in a circle. Skills - Find the length of elements of a circle - Calculate the area of disk and its sector. - Use the angle properties of lines in circles to solve problems - Use tangent properties to solve circle problems Content Learning Activities Attitudes and values - - Develop clear, logical and coherent thinking Appreciate the importance of circle theorems in dividing into sectors Promote team work spirit when working in group. Show concern on patience, mutual respect, tolerance and curiosity in the solving and discussion about problems involving circle theorems and disk - Elements of a circle and disk: center, radius, diameter, circumference, area, chord, tangent, secant, sector. Circle theorems: First circle theorem: angles at the centre and at the circumference. Second circle theorem: angle in a semicircle. Third circle theorem: angles in the same segment. Fourth circle theorem: angles in a cyclic quadrilateral. Fifth circle theorem: length of tangents. Sixth circle theorem: angle between circle tangent and radius. Seventh circle theorem: alternate segment theorem. Eighth circle theorem: perpendicular from the centre bisects the chord In group, discuss about and solve problems involving two concentric circles, such as areas, lengths, ratios ... In pairs, for given circles involving arcs, find minor arc length, major arc length, minor sector area and major sector area. In group, discuss about the properties of points in a cyclic quadrilateral. In group, discuss about the properties of chord through given situation involving circle theorem. Links to other subjects: Technical drawing, scientific drawing… Assessment criteria: Construct mathematical arguments about circle and use circle theorems and disk to solve related problems. Materials: Calculators, geometrical instruments. Page 52 of 60 Topic Area: GEOMETRY S3 MATHEMATICS Unit 10: COLLINEAR POINTS AND ORTHOGONAL VECTORS No. of lessons: 6 Key Unit Competency: to be able to apply properties of collinearity and orthogonality to solve problems involving vectors. Learning Objectives Knowledge and understanding State the conditions and properties of collinearity and orthogonality Skills - Use definition and properties to show whether: o Three given points are collinear or not. o Two vectors are orthogonal or not. Attitudes and values Content Learning Activities - Appreciate the use of properties of - Conditions for: In group, collinearity and orthogonality to solve o Points to be collinear problems about vectors in two o Vectors to be orthogonal -Discuss whether three points are collinear in the dimensions. - Problems about points and given situations - Show concern on patience, mutual vectors in two dimensions Discuss whether vectors respect, tolerance and curiosity in the are parallel ororthogonal. solving and discussion about problems involving vectors in two dimensions Links to other subjects: Technical drawing, scientific drawing, Physics (forces, motion, …), Chemistry, … Assessment criteria: Solve problems involving points and vectors in two dimensions Materials: Calculators, geometrical instruments. Page 53 of 60 Topic Area: GEOMETRY S3 MATHEMATICS Unit 11: ENLARGEMENT AND SIMILARITY IN 2D No. of lessons: 22 Key Unit Competency: To be able to solve shape problems about enlargement and similarities in 2D Learning Objectives Knowledge and understanding - Define enlargement - Define similarity - Identify similar shapes - List properties of enlargement and similarities. Skills - Determine the linear scale factor of an enlargement - Find centre of an enlargement - Construct an image of an object under unlargement - Use properties of enlargement and similarities to transform a given shape. - Find lengths of sides, area, and volume of similar shapes. - Construct an image of an object under composite and inverse enlargement. Attitudes and values Content Learning Activities - Appreciate the - Definition of enlargement. - In group, construct the images of given importance of - Definition similarity. shapes under given instructions related enlargement and - Examples of similar shapes ( similar to enlargement and compare the images similarities to triangles, similar cylinder,etc) to the initials. Discuss about the transform shapes - Properties of enlargement and properties of enlargement and - Develop patience, similarities. similarities used to transform those mutual respect, - Determining linear scale factor of shapes to their images. tolerance and enlargiment - In pairs, construct an image of a given team work spirit - Determining centre of enlargement. object under composite and inverse in solving and - Finding lengths of sides of similar enlargement. discussing shapes using Thales theorem - Individually, show similar shapes in problems - Areas of similar shapes given different varieties of shapes and involving - Volumes of similar objects. find the linear scale factor and center of enlargement and - Composite and inverse enlargement for each case. similarities enlargements - In groups, find the area and volume of given similar shapes and solids. Links to other subjects: Physics, engineering, construction, technical drawing, scientific drawing, etc Assessment criteria: Solve shape problems about enlargement and similarities in two dimension Materials: Geometrical instruments. Page 54 of 60 Topic Area: GEOMETRY S3 MATHEMATICS Unit 12: Inverse And Composite Transformations in 2D No. of lessons: 12 Key Unit Competency: to be able to solve shape problems involving inverse and composite transformations. Learning Objectives Knowledge and understanding Skills - State and explain - Construct an image of an properties of object under composite composite and and inverse inverse transformation in 2D transformations in - Solve problems involving 2D inverse and composite - Identify type of transformations in 2D transformation used in given drawings in 2D - Show an image of an object from different transformed shapes in 2D Attitudes and values Content Learning Activities - Appreciate the importance - Composite - Individually, construct an image of a given of inverse and composite transformations: object under inverse and composite transformations to o Composite transformations in 2D transform shapes translations in 2D - In groups; observe, discuss and show - Show concern on patience, o Composite images of objects from given different mutual respect, tolerance reflections in 2D transformed shapes in 2D and give the and curiosity in the solving o Composite properties of inverse and composite and discussion about rotations in 2D transformations used to transform those problems involving o Mixed shapes. inverse and composite transformations - In pairs, construct image of objects under transformations.to in 2D mixed transformations transform - Inverse transformations in 2D Links to other subjects: Physics, engineering, construction, technical drawing, scientific drawing. Assessment criteria: Solve problems involving inverse and composite transformations of shapes in 2D Materials: Geometrical instruments, etc Page 55 of 60 Topic Area: STATISTICS AND PROBABILITY S3 MATHEMATICS Unit 13: STATISTICS (BIVARIATE DATA) No. of lessons:12 Key unit Competency: To be able to Collect, represent and interpret bivariate data. Learning Objectives Knowledge and understanding - Define bivariate data - Make a frequency distribution table of collected bivariate data - Interpret scatter diagram - Identify type of correlation on a scatter diagram Skills - Draw scatter diagram for bivarite data and indicate the type of correlation. - Analyze a scatter diagram and infer conclusion. Content Learning Activities Attitudes and values - Develop clear, logical and coherent thinking while drawing conclusion related to bivariate data or scatter diagram . - Appreciate the use of scatter diagram to represent information. - Show concern of patience, mutual respect, tolerance, and curiosity in the bivariate data collection, representation and interpretation. - Definition and examples of - In groups, Collect bivariate data bivariate data and organize them in frequency - Frequency distribution table distribution table and plot them of bivariate data on a scatter diagram. - Scatter diagram. - Types of correlation: - In pairs, observe given o Positive correlation information on the graphs o Negative correlation (scatter diagrams), mention the type of correlation, analyze, interpret them and infer conclusion. Links to other subjects: All subjects Assessment criteria: Solve problems involving Collection, representation and interpretation of bivariate data Materials: Calculators, geometrical instruments. Page 56 of 60 6. REFERENCES 1. 2. 3. 4. 5. Alundria, I. & al (2009). Secondary Mathematics (student’s book2). MK publishersLtd. Atkinson, C. & Mungumiyo, U. (2010). New general Mathematics: teacher’s guide. Pearson education ltd. Ayres ,F. &al (1992). Mathématiques de base. McGraw-Hill Inc, Paris. Backhouse, J.K. & Al (1985). Pure Mathematics 1(Fourth edition). Longman. Ball, B. & Ball, D. (2011). Rich task Maths2 engaging mathematics for all learners. Association of Teachers of Mathematics: England 6. Ball, D. (2003). Forty harder problems for classroom. Association of Teachers of Mathematics: England 7. Crawshaw, J. & Chambers, J. (2002). Advanced Level Statistics.Nelson Thornes Ltd. 8. Curriculum Planning and Development Division (2006). Secondary Mathematics Syllabuses. Ministry of Education: Singapore 9. Hatch, G. (2004). What kind of Game is Algebra. Association of Teachers of Mathematics: England 10. Kamba, G. (2010). Secondary mathematics (student’s book3). MK publishersLtd. 11. Kasirye, S. & al (2009). Secondary Mathematics (student’s book1). MK publishersLtd. 12. Laufer, B. H. (1984). Discrete Mathematics and Applied Modern Algebra.PWS publishers. 13. Les Frères de l’Instruction Chrétienne (1961). Géométrie plane. La Prairie, P.Q, Ottawa. 14. Lyonga, E K. (2004). Mathematics for Rwanda: student book. Macmillan. 15. Macrae, M. & al (2010). New General Mathematics: student book1. Pearson education ltd. 16. Macrae, M. & al (2010). New General Mathematics: student book2. Pearson education ltd. 17. Macrae, M. & al (2010). New General Mathematics: student book3. Pearson education ltd. 18. Ministry of Education (2007). The New Zealand Curriculum. New Zealand 19. National curriculum Development Center (2006). Mathematics curriculum for ordinary Level. Ministry of Education : Rwanda 20. National Curriculum Development Centre (2008). Mathematics Syllabus: Uganda Certificate of Education. Ministry of Education and Sports: Uganda. 21. National Curriculum Development Centre (2013). Mathematics Learning Area Syllabus. Cambridge Education. Uganda 22. Nichols & al (1992). Holt pre-algebra. Holt, Rinehart and Winston Inc. 23. Okot-Uma, R. & al (1997). Secondary school Mathematics: student book1. Macmillan: Uganda. 24. Okot-Uma, R. & al (1997). Secondary school Mathematics: student book2. Macmillan: Uganda. Page 57 of 60 25. Okot-Uma, R. & al (1997). Secondary school Mathematics: student book3. Macmillan: Uganda. 26. Rayner, D. (2011). Extended Mathematics for Cambridge IGCSE. Oxford University Press. 27. Singh, M. (2002). Pioneer Mathematics. Dhanpat RAI&CO.(PVT). 28. University of Cambridge (2013). Cambrige IGCSE, International Mathematics. UCLES:UK Page 58 of 60 7. APPENDIX: SUBJECTS AND WEEKLY TIME ALLOCATION FOR O’ LEVEL (S1-S3) When learners go to secondary school, they study twelve ‘core’ subjects and an ‘elective’ subject, selected by the school. In addition, there are three compulsory ‘co-curricular’ activities. Core subjects Weight (%) Number of Periods (1 period = 40 min.) S1 S2 S3 1. English 11 5 5 5 2. Kinyarwanda 7 3 3 3 3. Mathematics 13 6 6 6 4. Physics 9 4 4 4 5. Chemistry 9 4 4 4 6. Biology and Health Sciences 9 4 4 4 7. ICT 4 2 2 2 8. History and Citizenship 7 3 3 3 9. Geography and Environment 7 3 3 3 10. Entrepreneurship 4 2 2 2 11. French 4 2 2 2 12. Kiswahili 4 2 2 2 13. Literature in English 2 1 1 1 Page 59 of 60 Sub Total 41 periods 41 periods 41 periods II. Elective subjects: Schools can choose 1 subject Religion and Ethics 4 2 2 2 Music, Dance and Drama 4 2 2 2 Fine arts and Crafts 4 2 2 2 Home Sciences 4 2 2 2 Farming (Agriculture and Animal husbandry) 4 2 2 2 Physical Education and Sports 2 1 1 1 Library and Clubs 2 1 1 1 100 45 45 45 30 30 30 1170 1170 1170 III. Co-curricular activities (Compulsory) Total number of periods per week Total number of contact hours per week Total number of hours per year (39 weeks) Page 60 of 60
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