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Mathematics Curriculum for PCM (elective) REPUBLIC OF RWANDA MINISTRY OF EDUCATION RWANDA EDUCATION BOARD (REB) P.O.BOX, 3817 KIGALI www.reb.rw MATHEMATICS CURRICULUM FOR PHYSICS - CHEMISTRY BIOLOGY (PCB) COMBINATION (ELECTIVE) ADVANCED LEVEL Kigali, January 2014 i Mathematics Curriculum for PCM (elective) Foreword In the line with Rwandan Education system reform, where combinations system has been introduced in schools in 2009, the leaver’s profiles for secondary education have been revised. At this time after the curriculum of Mathematics as core subject have been revised, the Curriculum Department of Rwanda Education Board is at the stage of designing of the Curriculum of Mathematics as elective subject in accordance with the revised leaver’s profile, to the pedagogical approach progress and the new time table allocation. It is in this regard, this Curriculum has been developed for Mathematics as elective and non examinable subject in upper secondary Physics-Chemistry-Biology combination. Starting from the 2013 academic year this curriculum will replace the curriculum which had been proposed in 2009 by the instructions No 426/12.00/2009 of the Minister of State in charge of Primary and Secondary Education, dated 02/03/2009. Dr John RUTAYISIRE Director General of REB (Rwanda Education Board) ii Mathematics Curriculum for PCM (elective) List of participants in elaboration of this curriculum The following were involved in the development of this curriculum: Coordinator: Dr. Joyce MUSABE, Deputy Director General of CPMD (Curriculum and Pedagogical Material Department), Supervisor: RUTAKAMIZE Joseph, Director of Science and Art Unit Curriculum Specialists: 1.KAYINAMURA Aloys : Mathematics Specialist/ CPMD, team leader 2.NYIRANDAGIJIMANA Anathalie: Specialist in charge of Pedagogic Norms/ CPMD; iii Mathematics Curriculum for PCM (elective) Teachers : 1. HABIMANA Joseph : Mathematics Teacher, G.S.O. de Butare 2. HABINEZA NSHUTI Jean Clément :Mathematics teacher, Ecole Secondaire de Nyanza 3. KAMUHANDA James Kant: Mathematics teacher, Nyagatare Secondary School 4. MUSENGIMANA Théophile: Mathematics teacher, Kiziguro Secondary School Textbook Approval Committee (TAC) 1.Dr MUSABE Joyce, Member and Chairperson 2.Mr. NHIMIYIMANA Alexis, Member and Secretary 3.Mr. RWAMBONERA Francois, Member 4.Mr. NTAGANZWA Damian, Member 5.Mr. GASANA Janvier, Member 6.Mr. BUTERA Anastase, Non permanent member 7.Mr. RUTAKAMIZE Joseph, Non permanent member iv Mathematics Curriculum for PCM (elective) Table of contents List of participants.................................................................................................................................... ii Table of contents ..................................................................................................................................... iii 1. Introduction .......................................................................................................................................... 1 2. General objectives ............................................................................................................................... 3 3. METHODOLOGICAL NOTES .......................................................................................................... 7 4. EVALUATION APPROACH ............................................................................................................. 9 5. PROGRAMs ...................................................................................................................................... 12 5.1. PROGRAM FOR SENIOR 4.......................................................................................................... 12 Chapter 1.ELEMENTARY LOGIC....................................................................................................... 13 Chapter 2: COMBINATORIAL ............................................................................................................ 43 5.3 PROGRAM FOR Sinior six............................................................................................................. 45 Chapter 1: Complex numbers ................................................................................................................ 46 Chapter 2: Analysis ................................................................................................................................ 52 v Mathematics Curriculum for PCM (elective) Chapter.3: Differential equations .......................................................................................................... 60 References .............................................................................................................................................. 62 vi Mathematics Curriculum for PCM (elective) I.INTRODUCTION After completing the curriculum of Mathematics for ordinary level, the Curriculum of Mathematics for advanced level: Elective and non examinable subject comes for capacity building of secondary students, especially in PCB combination where Mathematics is elective non examinable subject. This curriculum is new and deals learning ally with Logic, Trigonometry, Analysis, Algebra, Statistics, Combinatorial Analysis, Probability and Differential Equations. The chapters are developed in a logical progressive sequence enabling the learner to have a good comprehension of the subject matter. This Mathematics curriculum is prepared in a format which helps teachers to teach a particular topic effectively. The structure of each chapter is presented in three columns as follow: • Learning objectives; • Contents 1 Mathematics Curriculum for PCM (elective) • Suggested Teaching and Learning Activities. Note that at the end of detailed program for each grade, there is a proposal of lesson distribution. To avoid the areas of Mathematics to be continually seen as separate and learners acquiring concepts and skills in isolation, Mathematics is linked to everyday life and experiences in and out of school. Learners will have the opportunity to apply Mathematics in different contexts, and see the relevance of Mathematics in natural sciences. This curriculum is prepared for PCB combination and has to be taught in two periods a week. This curriculum also helps learners to use ICT tools to support the mastery and achievement of the desired learning outcomes. Technology used in the teaching and learning of Mathematics, for example calculators, are to be regarded as tools to enhance the teaching and learning process and not to replace teachers. 2 Mathematics Curriculum for PCM (elective) II. GENERAL ORIENTATION The proposed Mathematics Curriculum has content and subject matter specific to PhysicsChemistry-Biology (PCB) Combination. As Math subject is an unavoidable support for learning sciences and many other subjects, this curriculum aims at preparing the learner to follow and understand studies offered at higher levels of learning in various fields such as, Human Medicine, Veterinary medicine, Sciences, Allied Health sciences, Nursing sciences, Agriculture and rural development. In order to create a link between this curriculum and the leaver’s profile the following skills have to be improved through different teaching and learning activities. 3 Mathematics Curriculum for PCM (elective) Skills Main learning activities 1. ICT skills as tools for learning Using computers and projectors in presenting individual or group activities 2. Communication skills Using calculators in operations Discussion in group, oral and writing presentations of findings (results), 3. Logical thinking skills Using formulae to solve problems related to natural sciences Interpretation of graphs (curves) Relating the solution of a problem to the real life 4. Critical and interpretation skills Collecting data, analyzing data, synthesizing data, interpreting data and presenting data by using tables, 4 Mathematics Curriculum for PCM (elective) charts, diagrams, graphs,… 5. Individual and group learning skills Organize and conduct individual and group activities in a given time 6. Social skills Working in groups through exchange 7. Problem solving skills Activities related to natural sciences ( Physics, Chemistry, Biology) 8. Motivation and self confidence Activities related to the use of Mathematics in real life skills 5 Mathematics Curriculum for PCM (elective) III. GENERAL OBJECTIVES After the completion of Mathematics curriculum for advanced level: Elective and non examinable subject, a learner should be able to: 1. Develop clear, logical, creative and coherent thinking; 2. Master basic mathematical concepts and use them correctly in solving problems related to natural sciences. 3. Understand the useful role played by Mathematics in the learning of natural sciences. 4. Use the acquired mathematical concepts and skills to follow easily higher studies (Colleges, Higher Institutions and Universities); 5. Use acquired mathematical skills to develop work spirit, team work, self-confidence and time management without supervision; 6. Use ICT tools to explore Mathematics (examples: calculators, computers, mathematical software, etc.). 6 Mathematics Curriculum for PCM (elective) IV. METHODOLOGICAL APPROACHES The use of teaching resources and teaching materials like: reference books, worked examples presented on charts, diagrams/ graphs/curves on wall charts, geometrical instruments, is crucial in guiding learners to develop mathematical ideas. Teachers should use real or concrete materials to help learners gain experience, construct abstract ideas, make inventions, build self confidence, encourage independence and inculcate the spirit of cooperation. In order to assist learners in having positive attitudes towards Mathematics, confidence and thinking systematically, learners have to be involved into the teaching and learning process. Learning in groups should be emphasized to help learners to develop social skills, encourage cooperation and build self confidence. 7 Mathematics Curriculum for PCM (elective) Various teaching strategies and approaches such as direct instruction, learner-centered learning (Problem solving, group work learning, individualization, etc.) must be incorporated. While teaching and learning, the following point should be taken into consideration: • Usage of relevant, suitable and effective teaching materials; • Formative evaluation to determine the effectiveness of teaching and learning process. The choice of a suitable approach will stimulate the teaching and learning environment inside or outside the classroom. In this curriculum, suggested various exercises in all chapters may be done in groups or individually. In implementation of this curriculum, some activities to be done should be related to the main courses (core subjects like Physics, Chemistry and Biology) in order to establish the relationship between Mathematics and the above mentioned subjects. 8 Mathematics Curriculum for PCM (elective) V. EVALUATION APPROACH Evaluation or assessment has to be planned and carried out as a part of the classroom activities. Different methods of assessment can be conducted. These may be in the form of formative assessments (oral questioning and answering, assignments: Individual and group work activities), summative assessments (General tests at the end of one or several topics and exams). The harmonization of group findings and group presentations is required and some of the group activities should be marked. The making guides of general test and exams must be given to the learners. 9 Mathematics Curriculum for PCM (elective) VI. IMPORTANT FACTORS The teachers of Mathematics should make sure that the material taught and the examples given are drawn from everyday life such that the learner sees clearly the application of Mathematics involving problems dictated by nature. In this case Mathematics will take on a form of practical applicability rather than being abstract. Bearing in mind the difficulties encountered by teacher of Mathematics in some areas, it is recommended that teachers have regular meetings and consultations among teachers of the same school or between neighboring schools. IT in general and particularly ICT should be used as a pedagogical tool to facilitate teaching and learning of Mathematics. Nowadays using ICT we would like to encourage teachers of Mathematics and students to share experiences and good practices with their colleagues in the country and around the world. For the better harmonization of the Mathematics program it is recommended that trainings be held frequently in which teachers can be exposed to any new methods of teaching and latest information relevant in their subject or discipline. The teacher should try to motivate the learner to like Mathematics. This may be done through showing him/her the usefulness of the subject its 10 Mathematics Curriculum for PCM (elective) applicability in everyday life and in other disciplines. The approach should be such that the learner doesn’t see the subject simply as an abstract difficult subject. 11 Mathematics Curriculum for PCM (elective) VII. CURRICULUM DEVELOPMENT VII.1. SENIOR 4 General objectives At the end of senior 4 the learner should be able to: 1. Develop clear, logical and coherent thinking. 2. Solve algebraically and graphically given problems in the set of real numbers. 3. Solve problems in physics using trigonometric concepts and trigonometric formulas. 4. Solve problems in physics using algebra and vectors 12 Mathematics Curriculum for PCM (elective) CHAPTER 1.ELEMENTARY LOGIC (6 PERIODS) General objective: At the end of this chapter, learners should be able to develop clear, logical and coherent thinking. Learning objectives Contents Suggested teaching-learning activities At the end of this topic, 1.Elementary Logic - After giving various examples which are learner should be able / or not propositions, learners should be to: asked to form their own sentences and - Define a statement/ 1.1.Definitions: lead them to determine whether the proposition - Propopositions (simple given - Formulate his/her and compound statement) statements or not. own statement - Construct a truth sentences are mathematical - The teacher should give learners various - truth tables and truth 13 exercises to construct the truth table of Mathematics Curriculum for PCM (elective) table of a values one or several propositions statement - Give the negation of a proposition - Through various exercises, teacher - Negation of a should emphasize on the use of the word proposition “not” or “no” and their symbols to change a true statement into a false statement and vice versa 1.2. logical connectors - Form a compound Logical connectives and statement using the their properties: - Through group works, teacher should helpp learners to discuss about the use of logical connectives “or”; “ “and” conjunction “and” Conjunction (∧) - Lead learners to differentiate the use of - Determine the truth Disjunction (∨) “If p, then q” and “p “ if and only if q” value of the implication ( ) through various exercises compound equivalence ( ) Statement 14 Mathematics Curriculum for PCM (elective) - Form a compound statement using -Truth tables of the disjunction “or” propositions including -Determine the truth connectives value of the compound Statement - Use correctly implication and equivalence signs in mathematical expressions 15 Mathematics Curriculum for PCM (elective) CHAPTER 2: ALGEBRA (28 PERIODS) General objective: At the end of this chapter the learner should be able to solve algebraically and graphically given problems in the set of real numbers. Learning objectives Contents Suggested teaching-learning activities At the end of this 2. Algebra topic, learner should be able to: - Carry out correctly 2.1. Operations in the - Define absolute value by means of the operations on Set of Real numbers : a concrete example (for example: sets of numbers. - Define absolute value considering the number line, the 2.1.1. Absolute Value: - Definition and 16 distance from zero to positive A is the same from zero to the Mathematics Curriculum for PCM (elective) -Solve problems involving absolute value by applying properties of the absolute value. properties negative A where A is a non zero Absolute value of a real number) product, of a The teacher should prepare various quotient, of a sum group works on absolute value and of a difference 2.2. Indices or Powers and radicals (Surds): - The teacher should make a review - Use indices and Indices or Powers on indices and powers by giving surds’ (radicals) and its properties several examples and exercise as properties to simplify Surds: group works. algebraic - Square roots - Facilitate learners to understand expressions. - Cube root surds and their properties through - nth root of a positive examples. 17 Mathematics Curriculum for PCM (elective) real number and its properties - Give various group activities to be presented by students Conversion from indices to Surds notation and viceversa 2.3. Quadratic - Identify the - Lead learners to use the expression equations quadratic equations Definition of a with one unknown quadratic equations among other Solution of a equations. quadratic equation - Solve the quadratic equations 18 2 b b 2 4ac ax 2 bx c a x 2a 4a 2 Mathematics Curriculum for PCM (elective) - Determine the Sum Determine the sum for determining and product of and the product of of quadratic equation roots of quadratic roots of quadratic equations equations - Solve the quadratic the solution - The teacher should give various inequalities exercises involving the use of sum and product of roots of quadratic equations - Emphasize the role of variation - Study the Sign of table in Solving quadratic quadratic function inequalities - Sketch the graph of - To plot a curve of a quadratic function , teacher should help learners to determine the concavity by considering the a quadratic function 19 Mathematics Curriculum for PCM (elective) 2.4. QUADRATIC FUNCTIONS sign of a vertex, axis of symmetry and intersection with axes. Graphical representation of a quadratic function - Understand the concept of Basic concepts of the logarithm by definition - Relate definition of logarithm 2.5. Logarithms y a x log a y x to the logarithms to: - The base “a” and - ( 8 23 log 2 8 3 ) its properties - examples of powers The base “10” and - Help students to look for the its properties - Natural / Napierian 20 properties of logarithms, by Mathematics Curriculum for PCM (elective) logarithms (to the - Solve problems involving base “e”) means of examples. - Give various exercises to solve Logarithmic equations systems of logarithmic equations logarithmic by using the acquired methods. equations. - While solving equations involving logarithms help students to determine the domain of a solution /existence conditions, before computing. - The teacher should give examples of application of logarithm in real life. with 21 Mathematics Curriculum for PCM (elective) Examples: 1. The Population growth of a country at a rate R 2. The compound interest in the Bank 3. Half life time in radioactivity - During activities, only simple logarithmic equations should be given. 22 Mathematics Curriculum for PCM (elective) CHAPTER 3: TRIGONOMETRY (12 PERIODS) General objective: At the end of this chapter the learner should be able to use the trigonometric concepts and formulas in solving Physics problem involving Trigonometry Learning objectives Contents Suggested teaching-learning activities At the end of this topic, 3. Trigonometry the learner will be able - The teacher should make a to: 3.0. Revision of angles and review on angles by giving - Represent the angle their measurements several examples and in a trigonometric circle. exercises as group works 3.1. Trigonometric circle: 23 (Examples: Exercises Mathematics Curriculum for PCM (elective) Definitions : leading to conversion of - Define: sine, cosine, Sine, cosine, tangent, angle measurements, tangent and cotangent secant and cosecant of an constructing angles,….) of any oriented angle. oriented angle. - Use trigonometric circle to determine the sine, cosine, tangent and cotangent of an -Verify trigonometric identities using fundamental formula Fundamental formula sin2x + cos2x =1 Trigonometric identities - Simplify the oriented angle - Teacher should use trigonometric circle to show the fundamental trigonometric trigonometric formula expressions - Teacher should lead learners to 3.2.Graphical representation: Graphical representation 24 use basic identities and fundamental trigonometric Mathematics Curriculum for PCM (elective) of trigonometric - Represent functions: ( sine and graphically the sine, cosine) by using cosine functions by coordinates plotting - Apply trigonometric formulae in various exercises trigonometric functions, the 3.3. Trigonometric ratios - In graphical representation of Trigonometric ratios method of plotting is recommended. ratios to solve in a right angled problems related to triangle and cotangent of principal Trigonometric ratios angles (0°,30°,45°,60°,90°) the right angled triangle. - Solve problems related to in any other triangle Solution of right trigonometric angled triangle and equations any other triangle 25 - The use of sine, cosine, tangent through group works should be emphasized Mathematics Curriculum for PCM (elective) - The teacher should lead 3.3.Transformation learners to use transformation formulae formulae in solving Addition formulae Double angle (duplication) trigonometric equations. - The teacher should provide group activities which use formulae Simpson’s formulae addition formulae, double angle (duplication). - The teacher should guide learners to obtain Simpson’s formulae using addition formulae (these activities should be done as group work). 26 Mathematics Curriculum for PCM (elective) CHAPTER 4: LINEAR ALGEBRA (14 PERIODS) General objective: At the end of this chapter the learner should be able to solve problems in physics using algebra and vectors Learning objectives Contents Suggested teaching-learning activities At the end of this topic, 4. Linear algebra the learner should be able 4.1. Vectors to: Add vectors and multiply a vector with Operations on - Vectors: The teacher should make a a real number Revision on vectors in review on vectors in (scalar). Cartesian Cartesian plane 27 Mathematics Curriculum for PCM (elective) plane : addition of vectors and multiplication of vectors by a real number Calculate the Scalar product of two vectors product of vectors Determine the magnitude of a vector Scalar products or dot modulus or magnitude -The teacher should prepare (length) of a vector various exercises on: Unit vector relations in a triangle Conditions for two lines distance between two Apply the scalar to be parallel or product to solve given perpendicular problems. points -The use of geometrical instruments by the teacher and 28 Mathematics Curriculum for PCM (elective) learners is recommended -To visualize some concepts, Application of scalar teacher should use drawings. product: - relations in a triangle - distance between two - Apply scalar product to points Find the area of a triangle, 4.2. Matrices of order 2 of two vectors, and 3 Definition of a matrix Operations on matrices: Addition, subtraction and multiplication 29 Show the orthogonality Calculate the angle between two vectors … Mathematics Curriculum for PCM (elective) - The teacher should define the Determinants of matrix using any rectangular order 2 and 3by matrices of order 2 and array of numbers to help means of examples order 3: learners to understand easily Define matrices of Add matrices and - with a scalar - Calculation / subtraction and multiplication computing of matrices should be given in Inverse of matrices of order 2 Calculate the group activities. - During solving system of 2 Solution of linear equations, the teacher matrices of order 2 simultaneous should emphasize on the use and order 3 equations using of inverse method Determine the inverse matrices determinants of the definition of matrix. - Various exercises on addition, Properties multiply matrices, multiply matrix Definition and - While solving system of 3 30 Mathematics Curriculum for PCM (elective) of a matrix of order 2 - System of 2 or 3 linear equations, the teacher equations should emphasize on the use Problems leading to the of Cramer’s Rule and/ or 2 or 3 linear system of 2 or 3 linear elimination method. equations by using equations Solve problems related to a system of - matrices 31 Mathematics Curriculum for PCM (elective) VII.2. PROGRAM FOR SENIOR FIVE General objectives At the end of senior 5 the learner should be able: 1. To study and to represent graphically a numerical function. 2. To apply the properties of real numbers, to solve problems in physics 3. To determine and interpret the dispersion parameters of statistical series in one variable 4. To represent statistical data by scatter diagram and determine the linear regression line 5. To solve problems on combinations and permutations 6. To apply combinatorial concepts to solve problems related to Elementary probability 32 Mathematics Curriculum for PCM (elective) Chapter 1: ANALYSIS (36 PERIODS) General objective: At the end of this chapter the learner should be able to study and to represent graphically a numerical function. Learning objectives Contents Suggested teaching-learning activities At the end of this topic, the learner 1. ANALYSIS 1.1.Numerical functions should be able: Determine if a Definition of a numerical - Through worked examples given function function: examples, parity involving different functions, the is odd function, of a numerical function: odd teacher helps learners to master the even function, function, even function and use of compound concepts in this periodic periodic functions, unit such as: domain of a function, 33 Mathematics Curriculum for PCM (elective) function increasing and decreasing odd function, even function, Find the period functions, domain of periodic function. of a given definition of a function. - The teacher should prepare various function exercises to help learners to identify Determine if a if a function decreases or increases given function on a given interval. decreases or increases on a 1.2. Limits : given interval Concept of a limit with examples Determine the domain of Right hand limits and left - Introduce the concept of limit hand limits definition of through examples in real life (for 34 Mathematics Curriculum for PCM (elective) different Operations on limits example, if you draw different functions Extension of the concept concentric circles, the limit will be of limit: when values of the centre, etc.) the variable or those of Evaluate the function tend towards correctly the infinity, given limits - Before introducing the notation lim f ( x ) b the teacher should give xa Indeterminate cases: examples showing that if x takes values very closed to a then 0 , , , 0 0 f (x) takes values which are very closed to b. - Worked examples and various exercises are needed in order to help learners to use correctly the concept of limit. - Each of the following indeterminate 35 Mathematics Curriculum for PCM (elective) 0 , , , 0 , will be 0 illustrated using one example and the rest of exercises will be reserved to group activities forms: Study the continuity of a given function 1.3.Continuity of a function at - and discontinuity, examples in real a point To introduce the concept of continuity life should be used (a broken bridge Continuity and at a given point discontinuity of a or on a given function at a point, should be used as an example of discontinuity, etc.) - interval The teacher should help learners to interpret graphical representation of curves showing the discontinuity and continuity of functions at points using wall charts. 36 Mathematics Curriculum for PCM (elective) - Various activities on the study of continuity of different functions should be given in groups. 1.4.Asymptotes: Define an asymptote Find different The use of limit concept is required to curve help learners to determine vertical, Determination of vertical, horizontal and oblique asymptotes asymptotes of a horizontal and oblique given function asymptote Define the derivative of a Find the first through examples and then exercises in groups should be done. 1.5.Differentiation : function at a point Definition: asymptotes on a - Derivative of a function - The teacher should lead learners to at a point, identify geometrical interpretation of Geometric interpretation derivative of a function at a point as 37 Mathematics Curriculum for PCM (elective) derivative of a of a derivative of a given function function at a point, using definition Kinematical meaning of To determine a derivative: velocity at a derivatives using time t formulae follow. Derivative of a function: derivative of a constant, of a sum, of a product, and a quotient of While denoting derivatives, the teacher should help learners to be familiar with the notations functions - Differentiation of some dx y f ( x ) and , so that the dy important functions: dx can be used to dy power function, second notation composite function, explain the velocity, acceleration at a 38 Mathematics Curriculum for PCM (elective) circular functions, given time parametric functions and (v implicit functions. - Successive - Application of derivative: given in groups. - Tangent and normal at a application of derivatives Hospital’s theorem (proof not required) Explain to the learners that the determine the maximum and - tangent and normal at a point - first derivative can be used to Determine the equation of Various activities concerning the use of Hospital’s rule should be given as point of a function - Various activities on differentiation of functions in different forms should be differentiation, d 2s ds , a 2 ) dt dt - Minimum and maximum minimum of a function at a point points of a function, and the interval where that Increasing and decreasing function is increasing and 39 Mathematics Curriculum for PCM (elective) on a curve. interval of a function. decreasing. Then, the second Concavity, inflection point / derivative is used to determine the Hospital’s rule for Turning point on a curve by concavity and inflection point. eliminating the use of the second derivative, Use correctly - indeterminate forms in calculating limits. Determine whether a turning point of a given function is a maximum or a minimum point. solve problems 40 Mathematics Curriculum for PCM (elective) involving maximum or minimum values Study the concavity of a given function and find its inflection points. 1.6.Study of a function and curve sketching Study different functions and Types of functions to - Ensure that each required form of function is sketched through one consider : 41 Mathematics Curriculum for PCM (elective) sketch their - Polynomial functions example and various exercises curves. - Rational functions should be given as group x ax b with a 0 and cx d c0 x ax bx c with dx c 2 a 0 and d 0 activities. - The teacher should help learners to sketch different functions related to: Sinusoidal motion, projectile motion , harmonic motion, etc. Circular functions (tangent and cotangent) 42 Mathematics Curriculum for PCM (elective) CHAPTER 2: COMBINATORIAL (16 PERIODS) General objective: At the end of this chapter the learner should be able to solve problems on combinations and permutations. Learning objectives Contents Suggested teaching-learning activities At the end of this topic, learner should be able 3. Combinatorial to: 3.1. Permutations and combinations Solve problems involving factorial notation Factorial notation Calculation of the number learners to be familiar with of permutations factorial (arrangements) of n calculations through different 43 - The teacher should notation help and Mathematics Curriculum for PCM (elective) Determine the number of elements taken r at a time. permutations of n - in a house, random selection of elements taken n at a time taken r at a time. a football team, election of (n!) By use of examples from real life, (arrangement of furniture of permutations of n different objects Determine the Calculation of the number examples and group activities. representative committee, etc.) Simple combinations - number of permutations - 44 The teacher should introduce the concepts of permutation and combination. It is also advisable to go out of classroom for visualizing permutation of n elements through different activities By the help of the teacher, various exercises and problems on permutations and combinations will be done in groups. Mathematics Curriculum for PCM (elective) VII.3 PROGRAM FOR SINIOR SIX General objectives At the end of senior 6 the learner should be able: 1. To utilize the algebraic, trigonometric and exponential forms of a non-zero complex number to solve problems in trigonometry and geometry. 2. To study and represent logarithmic and exponential functions and to apply them in other scientific domains 3. To calculate the integrals of functions and apply them in various domains 4. To solve simple differential equations of 1st and 2nd order with constant coefficients 45 Mathematics Curriculum for PCM (elective) CHAPTER 1: COMPLEX NUMBERS (14 PERIODS) General objective: At the end of this chapter the learner should be able to utilize the algebraic, trigonometric and exponential forms of a non-zero zero complex number to solve problems in trigonometry and geometry. Learning objectives Contents Suggested teaching-learning teaching activities 1. Complex numbers At the end of the topic the 1.1.The set of complex learner will be able to: Define a complex number numbers in algebraic form Calculate the sum / Definition of a complex number Notation of the set of 46 - Before introducing complex numbers, the teacher should give different equations in leading to the negative discriminant to help learners to discover the necessity neces of a new set, set of complex numbers. Mathematics Curriculum for PCM (elective) difference and the product of complex numbers properties Determine the real and imaginary parts of a Determine the conjugate of complex numbers Apply conjugate properties of complex numbers in solving simple equations Multiplication in Algebraic form of a complex number - real and imaginary parts - conjugate and modulus of complex numbers, - notation and properties 47 (Example: ) - Various exercises should be done in groups to help learners to , and its properties complex number complex numbers Addition in and its master different operations in Mathematics Curriculum for PCM (elective) 1.2. Geometric representation of a complex number: Represent a complex Location of points and - While teaching, establish a number in Argand vectors in a complex relationship between the diagram plane or Argand diagram. representation of a point in Cartesian plane and the representation of complex number in complex plane or Argand diagram 1.3.Trigonometric form of complex number - Define a complex number in trigonometric form Modulus and Argument of a complex number 48 - The teacher should help learners to identify that Mathematics Curriculum for PCM (elective) - Determine a modulus and an argument of power of complex number - Convert a complex number from a trigonometric form to an Trigonometric form of a considering the sign of real complex number part and imaginary part will Trigonometric form of facilitate to determine the product, and quotient of position of argument of a two complex numbers complex number in Argand Power of complex diagram algebraic form and vice number in trigonometric versa form - Define a complex number in Exponential form - Activities on trigonometric form of product, and quotient of two DeMoivre’s theorem. Changing a complex complex numbers will be done in groups , do not insist on them number from a trigonometric form to an algebraic form and vice versa 49 - While teaching exponential Mathematics Curriculum for PCM (elective) form of a complex number, Exponential form of a give only definition and one complex number worked example. Most activities will be done do as group works 1.4. Calculations in the field of complex numbers - Calculate a square root of - a complex number - Solve quadratic equations Square root of a complex While calculating the square root of number - - Quadratic equations in z a bi , it is advisable to solve the system in , Where 50 Mathematics Curriculum for PCM (elective) x iy is the square root of z and make sure that the choice of x and y depends on the sign of b 1.5.Application of complex -The teacher should prepare numbers - Apply De Moivre’s Calculation of sine, cosine, various exercises related to the formula in calculating sine, tangent, cotangent of cosine, tangent and multiple angles of a given cotangent of multiple angle and demonstrating the -While solving the simple angles of a given angle trigonometric identities by trigonometric equations, make - Factorize polynomials in C application of complex numbers. use of De Moivre’s theorem sure that the equations of the Linearization of trigonometric polynomials. 51 form : a cos x b sin x c are also solved Mathematics Curriculum for PCM (elective) CHAPTER 2: ANALYSIS (36 PERIODS) General objective: At the end of this chapter the learner should be able: - To study and represent logarithmic and exponential functions and to apply them in natural sciences To calculate the integrals of functions and apply them in natural sciences Learning objectives Contents Suggested teachinglearning activities At the end of the topic the learner should be able to: 2. Calculus (Analysis) 2.1.Logarithmic and -Define Napierian logarithm function exponential functions Napierian logarithmic functions : -Represent graphically the - Definition and properties, 52 - Through Various Mathematics Curriculum for PCM (elective) function y ln x - -Solve logarithmic equations -Calculate the derivative of - logarithmic functions in base a -Define an exponential function in base e Equation of a logarithmic exercises, teacher teache function should help learners Differentiation to apply logarithmic (derivation) of the properties to any base functions of the form (The group activities ln u , where u is should be done as differentiable over an revision). interval I. -Study and represent graphically - While teaching Differentiation (derivation) of the -Solve exponential equations function ln x , the - Study of logarithmic teacher should functions of the forms emphasize on log a x and y ln x polynomial functions only and the other 53 Mathematics Curriculum for PCM (elective) functions should be done by students as Exponential functions in homework, group exponential functions in base e base e ; definition, activities, etc. - Solve logarithmic equations notation, properties, study -Define an exponential function of functions y e x in ,exponential equations, base a differentiation of eu , -Calculate the derivative of - Study and represent where u graphically - Solve exponential equations y a x in base a - The teacher facilitates learners to establish the relationship between logarithm to base a and logarithm to base e through change of basis. log e x ln x log a x log e a ln a where a 0 and a -After After the definition of Napiierian logarithm, -derivative of exponential 54 Mathematics Curriculum for PCM (elective) functions in base a the graphical representation of loga x - Establish the relationship , between and exponential functions will be done by learners following procedures of the study of functions (see content of Senior five). five) y a x and y e x - Study and represent graphically logarithmic and exponential - The teacher should lead learners to solve equations involving logarithmic and exponential expressions through various exercises. functions in base a 55 Mathematics Curriculum for PCM (elective) 2.2.CALCULUS INTEGRALS 2.2.1. Anti-derivative integrals - Define a primitive function - Apply immediate primitives in Definition and Properties of : exercises - Primative of a function - Calculate integrals by - set of primitive functions decomposition - immediate primitives method Techniques of integration: - Calculate integrals by - Integration by substitution decomposition: integration -The teacher should lead method of a sum of functions, learners to calculate integration of a product of indefinite integrals by functions by a real number decomposition, by -Calculate integrals by parts 56 Mathematics Curriculum for PCM (elective) - Calculate integrals of rational - functions Integration by change of change of variable and variable, by parts through one worked example on each - Calculate integrals of - Integration by parts irrational functions - Integration of trigonometric method and various - Calculate integrals of functions: in trigonometric in and exercises will be done in group activities. , where Functions , and is a non-zero real number. 2.2.2. Definite - Define a definite integral - Calculate definite integrals by applying their properties integrals - The teacher should lead learners to calculate Definition and Properties definite integrals by of definite integrals decomposition, by 57 Mathematics Curriculum for PCM (elective) - Calculate the area of a plane Methods of integration, change of variable and surface by use of definite Application of definite by parts through one integrals on the: worked example on each integrals - Calculate the volume of a - solid of revolution by use of definite - Calculation of area method and various of a plane surface exercises will be done in Calculation of Work group activities. integrals -The teacher should lead - Calculate the length of a learners to determine the curved surface by the use of area of: square, definite integral rectangle, trapezium, - Calculate the work done by a triangle using definite given force integrals. 58 Mathematics Curriculum for PCM (elective) - Explain that the calculation of the work done depends on the application of definite integrals b ( W F .dr where r is a the distance travelled). 59 Mathematics Curriculum for PCM (elective) CHAPTER.3: DIFFERENTIAL EQUATIONS (10 PERIODS) General objective: At the end of this chapter the learner should be able to solve simple differential equations of 1stand 2ndorder with constant coefficients Learning objectives Contents Suggested teaching-learning activities At the end of the unit the learner will be able 3. Differential to : - Define a differential equation of the 1st order - Solve differential equations -Through exercises, the teacher should Definition and help learners to verify whether a given terminology function is a solution of a given Examples of differential equation. differential equations 60 Mathematics Curriculum for PCM (elective) of the 1st equations with separable variables 3.1.Differential equations homogeneous of the 1st order it is advisable to use of the 1st order - Solve simple -In solving linear differential equations Differential equations natural sciences examples (For differential equations with separable example: Growth and Decay Problems - Solve linear variables differential equations st of the 1 order leading to Simple homogeneous differential equations dN kN , where N(t) is the dt amount of substance at a given time, Electrical Circuits cuits leading to The basic equation governing the amount of current I (in amperes) in a simple RLRL circuit 61 Mathematics Curriculum for PCM (elective) VIII. REFERENCES 1. Arthur Adam, Freddy Goossens, Francis Lousberg, Mathématisons 65, DeBoeck, 3e edition 1991 2. Christopher Claphan and James Nicholson: Concise Dictionary of Mathematics, Oxford University Press, Fourth Edition, 2009 3. Curriculum Development Center, Ministry of Education Malaysia, Integrated Curriculum for Secondary School, Mathematics form 4 and 5, Ministry of Education Malaysia, 2006 4. DPES-RWANDA, Complexes 5eme, Livre de l’élève, IMPRISCO – Kigali, Février 1990 5. DPES-RWANDA, Coniques, Livre de l’élève, IMPRISCO – Kigali, 1988 6. J.K.Backhouse, SPT Houldsworth B.E.D. Cooper and P.J.F.Horril, Pure mathematics 1, Longman, Third edition 1985, fifteenth impression 1998 7. J.K.Backhouse, SPT Houldsworth B.E.D. Cooper and P.J.F.Horril, Pure mathematics 2, Longman, Third edition 1985, fifteenth impression 1998 8. JAMES STEWART, Mc. MASTER UNIVERSITY, Calculus Early Transcendental, sixth edition, United States of America, 2008. 62 Mathematics Curriculum for PCM (elective) 9. John A. Grahan Robert. Sorgenfrey. Trigonometry with application, Hounghton Mifflin Campany, Boston 1993 10. Manjeet Singh: Pioneer Mathematics, Dhanpat Rdi & Co. (Pvt) Ltd, Educational and Technical Publishers 2003 11. MINEDUC/NCDC: Advanced Level Mathematics Currriculum for science combinations, Kigali, April 2010 12. Richard G. Brown, David P. Robbins, Advanced Mathematics: A precalculus course, Houghton Miffin Company, Boston 1994 13. SCHAUM’S OUTLINE OF “Theory and Problems of COLLEGE MATHEMATICS” third edition, McGraw-Hill Companies, 1976. 14. Shampiyona Aimable, Mathématiques 6, Kigali, Juin 2005 15. Teach yourself Mathematics, UK, 2003, Trevor Johnson and Hugh Neil 63 Mathematics Curriculum for PCM (elective) IX. APPENDICES IX.1. Weekly time allocation for A’ level Physics- Chemistry-Biology combination (PCB) Number of periods (1 period = 50 min) S4 S5 S6 Core subjects (all compulsory and examinable) Physics 7 7 7 Chemistry 7 7 7 Biology 7 7 7 Entrepreneurship 5 5 5 General paper 2 2 2 SUB-TOTAL 28 28 28 Subject 64 Mathematics Curriculum for PCM (elective) Elective skills subjects (school may choose one or two non examinable subjects below but students must choose one not both subjects) Fine arts 2 2 2 Mathematics 2 2 2 Computer science 2 2 2 English language skills communication 2 2 2 French 2 2 2 Co-curricular activities Sport, culture activities, Clubs, spiritual activities, study, research in library… 2 2 2 TOTAL 36 36 36 65 Mathematics Curriculum for PCM (elective) IX.2. ADVANCED LEVEL/ PHYSICS-CHEMISTRY- BIOLOGY (PCB) COMBINATION LEAVER’S PROFILE Upon completion of advanced level secondary education in Physics-Chemistry-Biology (PCB), the student should have acquired basic knowledge, skills and attitudes which will enable him/her to: Apply experimental, prospective and axiomatic processes; Analyse, explain facts and practical applications of phenomena relating to daily life; Posses appropriate attitude in usual scientific and professional situations, by improving knowledge, being realistic and self motivated; Apply ordinary skills, mathematical techniques and operational methods in the resolution of problems related other subjects; Collect, interpret statistic data and present the results; Have access to higher studies in higher institutions of learning and universities mainly in the following faculties: Human Medicine Veterinary medicine Sciences Allied Health sciences Nursing sciences Agriculture and rural development 66