ZIMBABWE MINISTRY OF PRIMARY AND SECONDARY EDUCATION PURE MATHEMATICS SYLLABUS FORMS 3 - 4 2015 - 2022 Curriculum Development and Technical Services P.O. Box MP 133 Mount Pleasant Harare © All Rights Reserved 2015 Pure Mathematics Syllabus Forms 3 - 4 ACKNOWLEDGEMENTS The Ministry of Primary and Secondary Education wishes to acknowledge the following for their valued contribution in the development of this syllabus: • • Panellists for Form 3-4 Pure Mathematics Representatives of the following organizations: - Zimbabwe School Examinations Council (ZIMSEC) - Uunited Nations Children’s Fund (UNICEF ) - United Nations Cultural, Scientific and Cultural Organisation (UNESCO) i Pure Mathematics Syllabus Forms 3 - 4 CONTENTS ACKNOWLEDGEMENTS.....................................................................................................i CONTENTS...........................................................................................................................ii 1.0 PREAMBLE....................................................................................................................1 2.0 PRESENTATION OF SYLLABUS..................................................................................1 3.0 AIMS................................................................................................................................1 4.0 SYLLABUS OBJECTIVES.............................................................................................2 5.0 METHODOLOGY............................................................................................................2 6.0 TOPICS...........................................................................................................................2 7.0 SCOPE AND SEQUENCE..............................................................................................3 8.0 COMPTETNCY MATRIX.................................................................................................7 8.2 FORM 4 COMPETENCY MATRIX..................................................................................13 9.0 ASSESSMENT................................................................................................................20 ii Pure Mathematics Syllabus Forms 3 - 4 1.0 PREAMBLE calculus. 1.1 Introduction 1.4 Assumption In developing the Form 3 - 4 Pure Mathematics syllabus attention was paid to the need to provide continuity of mathematical concepts from Form 1through to Form 4 and lay foundations for further studies, focusing on learners who have the ability and interest. It is assumed that learners who take this syllabus will take it concurrently with the Form 1 - 4 Mathematics syllabus. The syllabus is intended to produce a citizen who is a critical thinker and problem solver in life.The two year learning phase will provide learners with opportunities to apply Mathematical concepts in other subject areas and enhance Mathematical literacy and numeracy. It also desires to produce a learner with the ability to communicate effectively. In learning Pure Mathematics, learners should be helped to acquire a variety of skills, knowledge and processes, and develop positive attitude towards the subject and in life. This will enable them to investigate and interpret numerical and spatial relationships as well as patterns that exist in the world. The syllabus also caters for learners with diverse needs to experience Pure Mathematics as relevant and worthwhile. 1.2 Rationale Zimbabwe is undergoing a socio-economic transformation where Pure Mathematics is key to development, therefore it is imperative that learners acquire necessary mathematical knowledge and skills to enable as many learners as possible to proceed to Form 5-6 Mathematics and beyond. The knowledge of Pure Mathematics enables learners to develop mathematical skills such as dealing with the abstract, presenting mathematical arguments, interpreting mathematical information and solving problems essential in life and for sustainable development. The importance of Ppure Mmathematics can be underpinned in inclusivity, human dignity and enterprise as it plays a pivotal role in careers such as reserach, actuarial science, materology and engineering The syllabus assumes that the learner has: 1.4.1 mastered concepts and skills involving number, algebra and geometry at Form 1 and 2 level 1.4.2 shown interest in pursuing Pure Mathematics 1.4.3 the ability to operate some ICT tools 1.5 Cross Cutting Themes The following are some of the cross cutting themes in Pure Mathematics:1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 Financial literacy Disaster risk management Collaboration Environmentalissues Enterprise skills Sexuality, HIV & AIDS Education 2.0 PRESENTATION OF SYLLABUS The Pure Mathematics syllabus is presented as singledocument covering Form 3 - 4.It contains the preamble, aims, syllabus objectives, syllabus topics, scope and sequence and competency matrix. The syllabus also suggests a list of resources that could be used during learning and teaching process. 3.0 AIMS This syllabus is intended to provide a guideline for Form 3 - 4 learners which will enable them to: 3.1 3.2 1.3 Summary of Content (Knowledge, Skills and Attitudes) 3.3 The syllabus will cover the theoretical and practical aspects of Pure Mathematics. This two year learning area will cover: algebra, coordinate geometry and 3.4 1 acquire a firm foundation for further studies and future careers use ICT tools for learning and solving mathematical problems develop an ability to apply Pure Mathematics in life and other subjects, particularly Science and Technology develop a further understanding of mathematical concepts and processes in a way that Pure Mathematics Syllabus Forms 3 - 4 3.5 3.6 3.7 encourages confidence, enjoyment, interest and lifelong learning appreciate Pure Mathematics as a basis for applying the learning area in a variety of life situations develop the ability to solve problems, reason clearly and logically as well as communicate mathematical ideas successfully acquire enterprise skills in an indigenised and globalised economy through research and project-based learning contexts. The teaching and learning of Pure Mathematics must be learner centred and ICT driven. The following are suggested methods of the teaching and learning of Pure Mathematics • • • • • • • • 4.0 SYLLABUS OBJECTIVES 5.1 Time Allocation Six periods of 40 minutes each per week should be allocated for the adequate coverage of the syllabus By the end of the two year learning period, the learners should be able to: 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 Guided discovery Group work Interactive e-learning Games and puzzles Quiz Problem solving Simulation and modelling Experimentation 6.0 TOPICS use relevant mathematical symbols, terms and definitions in problem solving use formulae and generalisations to solve a variety of problems in Pure Mathematics and other related learning areas formulate problems into mathematical terms and apply appropriate techniques for solutions use ICT tools for learning through problem solving apply Pure Mathematics concepts and principles in life demonstrate an appreciation of mathematical concepts and processes demonstrate an ability to solve problems systematically, applying mathematical reasoning communicate mathematical concepts and principles clearly explore ways of solving routine and non-routine problems in Pure Mathematics using appropriate formulae, algorithms and strategies model mathematical information from one form to another e.g. verbal/words to symbolic form conduct research projects including those related to enterprise The following topics will be covered from Form 3 - 4 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 5.0 METHODOLOGY It is recommended that teachers use methods and techniques in which Pure Mathematics is seen as a process which arouses an interest and confidence in tackling problems both in familiar and unfamiliar 2 Indices and irrational numbers Polynomials Identities, equations and inequalities Graphs and coordinate geometry Vectors Functions Sequences Binomial expansions Trigonometry Logarithmic and exponential functions Differentiation Integration Numerical methods 3 Polynomials Surds Components of polynomials Addition Subtraction Partial fractions Definition of identity Unknown coefficients Equations Quadratic inequalities Cubic inequalities FORM 3 Pure Mathematics Syllabus Form 3 – 4 2016 Inequalities Identities and equations SUB TOPIC Multiplication Division Factor Theorem Solving equations FORM 4 5 Completing the square Simultaneous equations FORM 4 TOPIC 3: IDENTITIES, EQUATIONS AND INEQUALITIES FORM 3 TOPIC 3: IDENTITIES, EQUATIONS AND INEQUALITIES SUB TOPIC TOPIC 2: POLYNOMIALS Irrational numbers Laws of indices Equations involving indices TOPIC 2: POLYNOMIALS FORM 3 Indices SUB TOPIC FORM 4 TOPIC 1: INDICES AND IRRATIONAL NUMBERS SCOPE AND SEQUENCE TOPIC 1: INDICES AND IRRATIONAL NUMBERS 7.0 7.0 SCOPE AND SEQUENCE Pure Mathematics Syllabus Forms 3 - 4 Straight line graphs Gradient of a line segment Distance between two points Coordinates of the mid-point • Types of vectors • Vector operations Vecors in three dimensions 4 • Definition of a function • Domain and range • Composite function Functions Pure Mathematics Syllabus Form 3 – 4 2016 FORM 3 SUB TOPIC TOPIC 6: FUNCTIONS FORM 3 TOPIC 5: VECTORS FORM 3 SUB TOPIC Coordinate geometry Graphs SUB TOPIC Graphs of𝑦𝑦 = 𝑘𝑘𝑥𝑥 𝑛𝑛 Unit vectors Scalar product Vector properties of plane shapes Areas of triangles and parallelogram 8 • One- one funcion • Inverse of a function • Graphs of functions FORM 4 • • • • FORM 4 FORM 4 TOPIC 4: GRAPHS AND COORDINATE GEOMETRY TOPIC 4: GRAPHS AND COORDINATE GEOMETRY Pure Mathematics Syllabus Forms 3 - 4 Definition of a sequence Examples of sequences 5 • Sine and cosine rules • Area of a trriangle • Trigonometrical functions for angles of any size • Exact values of sine, cosine and tangent of special angles Plane Trigonometry Trigonometric functions Pure Mathematics Syllabus Form 3 – 4 2016 FORM 3 TOPIC 9: TRIGONOMETRY FORM 3 TOPIC 8: BINOMIAL EXPANSION FORM 3 SUB TOPIC Binomial expansion SUB TOPIC Sequences SUB TOPIC TOPIC 7: SEQUENCES TOPIC 7: SEQUENCES Arithmetic progression Geometric progression 11 • Equations • Radians • Length of an arc • Area of a sector • Area of a segment FORM 4 • Pasdcal’s Triangle • Exapnsion of (a+b) n where n is a positive integer FORM 4 FORM 4 Pure Mathematics Syllabus Forms 3 - 4 Laws of logarithms Logarithms and indices 6 SUB TOPIC Pure Mathematics Syllabus Form 3 – 4 2016 Numerical methods FORM 3 TOPIC 13: NUMERICAL METHODS • • • Integration 14 Indefinite integration as the reverse process of differentiation Integration of functions of the formaxn Integration of a polynomial FORM 3 TOPIC 12: INTEGRATION FORM FORM 33 Gradient of a curve at a point Derived function of the form axn Derivative of a sum TOPIC 11: DIFFERENTIATION FORM 3 SUB TOPIC SUB TOPIC TOPIC Differentiation TOPIC 11: DIFFERENTIATION Exponential functions Logarithms SUB TOPIC Exponential growth and decay Natural logarithms Equations of the form a x = b Area Volume • • • Simple iterative procedures Newton Reaphson method Trapezium Rule FORM 4 • • FORM 4 FORM FORM44 Application of differentiation to gradients, tangents and normals, stationary points, rates of change, velocity and acceleration FORM 4 TOPIC 10: LOGARITHMIC AND EXPONENTIAL FUNCTIONS TOPIC 10: LOGARITHMIC AND EXPONENTIAL FUNCTIONS Pure Mathematics Syllabus Forms 3 - 4 define irrational numbers reduce a surd to its simplest form carry out the four operations on surds rationalise denominators state the laws of indices use laws of indices to simplify algebraic expressions solve equations involving indices Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Irrational numbers Indices SUB TOPIC Surds Laws of indices Equations involving indices CONTENT: {Skills, Knowledge, Attitudes} TOPIC 1: INDICES AND IRRATIONAL NUMBERS COMPETENCY MATRIX: FORM 3 SYLLABUS FORM THREE (3) SYLLABUS FORM 3 COMPTETNCY MATRIX TOPIC 1: INDICES AND IRRATIONAL NUMBERS 8.0 8.1 8.0 COMPTETNCY MATRIX 7 18 Distinguishing between rational and irrational numbers Deriving and finding ways of simplifying surds Expressing surds in simple form Carrying out the four operations on surds Rationalising denominators Deriving the laws of indices Simplifying algebraic expressions Applying laws of indices to solve problems SUGGESTED NOTES AND ACTIVITIES ICT tools Relevant texts ICT tools Relevant texts SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 define polynomials carry out operations of identify proper and improper addition and subtraction of fractions carry out operations of addition and polynomials subtraction polynomials express aoffunction as a sum of express a function as a sum of simpler fractions simpler fractions fractions identify proper andto: improper Learners should be able OBJECTIVES define polynomials Subtraction Partial fractions CONTENT: {Skills, Knowledge, Attitudes} CONTENT: {Skills, of Components Knowledge, Attitudes} polynomials Addition Components of polynomials Subtraction Addition Partial fractions Distinguishing between proper and improper fractions Discussing polynomials Adding and subtracting Distinguishing between proper and polynomials improper fractions Adding and subtracting polynomials Decomposing functions into Decomposing functions into fractions fractions with linear with linear denominators denominators SUGGESTED NOTES AND ACTIVITIES Discussing polynomials SUGGESTED NOTES AND ACTIVITIES Learners should be able to: OBJECTIVES 8 Pure Mathematics Syllabus Form 3 – 4 2016 distinguish between an identity and an equation use identities to determine unknown coefficients in polynomials solve cubic equations using Factor Theorem Inequalities factorise quadratic expressions solve quadratic inequalities Pure Mathematics Syllabus Formcubic 3 – 4inequalities 2016 solve with the use of Factor Theorem Identities and equations SUB TOPIC Definition of identity Unknown coefficients Equations Quadratic inequalities Cubic inequalities CONTENT: (Skills, Knowledge, Attitudes) 19 Factorising quadratic expressions Finding solutions of quadratic 12 inequalities Exploring ways of solving cubic inequalities Using the Factor Theorem to solve cubic inequalities Discussing the difference between an and anequation equation identity andan Finding unknown coefficients using identities Using the Factor Theorem to solve cubic equations SUGGESTED NOTES AND ACTIVITIES TOPIC 3: IDENTITIES, EQUATIONS AND INEQUALITIES TOPIC 3:IDENTITIES, EQUATIONS AND INEQUALITIES TOPIC 3: IDENTITIES, EQUATIONS AND INEQUALITIES Polynomials Polynomials SUB TOPIC OBJECTIVES TOPIC 2: POLYNOMIALS Learners should be able to: TOPIC 2: POLYNOMIALS SUB TOPIC TOPIC 2: POLYNOMIALS ICT tools Relevant texts Relevant texts ICT tools ICT tools Relevant texts SUGGESTED RESOURCES SUGGESTED ICT tools RESOURCES Relevant texts SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 9 calculate the distance between two points given in coordinate form solve practical problems involving distance between two points find the coordinates of the mid-point of a straight line sketch straight lines find the gradient of a line segment find the equation of a straight line identify parallel and perpendicular lines solve problems involving straight line graphs Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Coordinate geometry Graphs SUB TOPIC TOPIC 4:GRAPHS AND COORDINATE GEOMETRY Distance between two points Coordinates of the midpoint Straight line graphs Gradient of a line segment CONTENT: {Skills, Knowledge, Attitudes} TOPIC 4: GRAPHS AND COORDINATE GEOMETRY 21 Sketching straight lines given sufficient information Exploring how to find equation of a straight line given two points or the gradient and a point Distinguishing between parallel and perpendicular lines Solving problems involving straight line graphs Identifying examples in life where linear relationships occur Carrying out experiments involving linear relationships Exploring ways to find distance between two points Calculating coordinates of the midpoint of a straight line Conducting field work to solve problems involving distance between two points SUGGESTED NOTES AND ACTIVITIES Geo-board ICT tools Environment Geo-board ICT tools Environment SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 10 TOPIC 6: FUNCTIONS determineposition and free vectors identify parallel vectors and co-linear points calculate the modulus of a vector add and subtract vectors multiply a vector by a scalar 𝐴𝐴𝐴𝐴 xi+yj+zk,→ , a define a function define domain and range of a function use functional notation simplify a composite function Learners should be able to: 𝑥𝑥 use standard notation of vectors (𝑦𝑦𝑧𝑧), OBJECTIVES Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Functions SUB TOPIC TOPIC 6:FUNCTIONS Vectors in three dimensions SUB TOPIC TOPIC 5: VECTORS TOPIC 5: VECTORS Types of vectors Vector operations Definition of a function Domain and range Composite function CONTENT: {Skills, Knowledge, Attitudes} CONTENT: {Skills, Knowledge, Attitudes} 𝐴𝐴𝐴𝐴 xi+yj+zk, → , a Distinguishing between positionand free vectors Interpreting operations in geometrical terms Computing the magnitude of vectors in three dimensions Adding and subtracting vectors Finding a product of a vector by a scalar Solving problems involving vector operations 𝑥𝑥 (𝑦𝑦𝑧𝑧), Using standard notation of vectors 22 Identifying and defining functions Discussing the domain and the range Using functional notations Simplifying composite functions SUGGESTED NOTES AND ACTIVITIES SUGGESTED NOTES AND ACTIVITIES ICT tools Mathematical models Environment Geo-board Relevant texts ICT tools Relevant texts SUGGESTED RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 define a sequence list the elements of a sequence Learners should be able to: OBJECTIVES Composite function Definition of a sequence Examples of sequences CONTENT: {Skills, Knowledge, Attitudes} OBJECTIVES 11 Trigonometrical functions Plane Trigonometry SUB TOPIC find the trigonometrical ratios of angles of any size use the exact values of the trigonometrical ratios of special angles in a variety of situations find the area of a triangle 2 use sine and cosine rules to solve problems 1 use the formula 𝐴𝐴 = 𝑎𝑎𝑎𝑎 sin 𝐶𝐶 to Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 TOPIC 9: TRIGONOMETRY Learners should be able to: Learners should be able to: OBJECTIVES TOPIC 9: TRIGONOMETRY expansion Binomial Binomial expansion SUB TOPIC SUB TOPIC TOPIC 8:BINOMIAL EXPANSION Trigonometrical functions for angles of any size Exact values of sine, cosine and tangent of special angles Sine and cosine rules Area of a triangle CONTENT: {Skills, Knowledge, Attitudes} Attitudes} Tobe becovered covered in form To in form 4 4 CONTENT: {Skills, CONTENT: {Skills, Knowledge, Knowledge, Attitudes} TOPIC 8: BINOMIAL EXPANSION TOPIC 8:BINOMIAL EXPANSION Sequences SUB TOPIC TOPIC 7: SEQUENCES use functional notation simplify a composite function TOPIC 7: SEQUENCES Deriving formulae for the sine and cosine rules Solving problems using the sine and cosine rules Deriving the formula for finding area of a triangle Solving problems involving area of a triangle Computing the trigonometrical ratios of angles of any size Calculating the exact values of the sine, cosine and tangent of 0°, 30°, 45° ,60°, 90° in a variety of situations 15 SUGGESTED NOTES AND ACTIVITIES SUGGESTED NOTES AND Discussing sequences Outlining elements of sequences SUGGESTED ACTIVITIESNOTES AND ACTIVITIES SUGGESTED NOTES AND ACTIVITIES range Using functional notations Simplifying composite functions ICT tools Relevant texts ICT tools Relevant texts SUGGESTED RESOURCES SUGGESTED ICT tools Relevant texts SUGGESTED RESOURCES RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 Exact values of sine, cosine and tangent of special angles Learners should be able to: logarithms and indices state the laws of logarithms use the laws of logarithms in solving problems identify the relationships between OBJECTIVES 12 TOPIC 12: INTEGRATION differentiate a polynomial recognise indefinite integration as the reverse process of differentiation integrate functions of the form axn where n is rational integrate polynomials Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Integration SUB TOPIC Calculating the exact values of the sine, cosine and tangent of 0°, 30°, 45° ,60°, 90° in a variety of situations Indefinite integration as the reverse process of differentiation Integration of functions of the form axn Integration of a polynomial CONTENT: {Skills, Knowledge, Attitudes} and vice versa Exploring the laws of logarithms Applying laws of logarithms in solving problems Expressing logarithms in index form SUGGESTED NOTES AND ACTIVITIES 17 Discussing indefinite integration as the reverse process of differentiation Performing integration of functions of the form axn where n is rational Computing indefinite integrals of polynomials SUGGESTED NOTES AND ACTIVITIES Calculating the gradient of a using tangents the gradient of a in curve Calculating Discussing situations life at where a given point using tangents gradients of curves Discussing situations in life where are important gradients of curves are important Differentiating Differentiating and linearlinear and quadratic quadratic expressions from expressions from first principles Differentiating functions of the form first principles n using the formula ax Differentiating functions of the form axn using the formula SUGGESTED NOTES AND curve at a given point ACTIVITIES 16 CONTENT: {Skills, SUGGESTED NOTES AND Knowledge, Attitudes} ACTIVITIES Laws of logarithms Logarithms and indices CONTENT: {Skills, Knowledge, Attitudes} find the gradient of a curve at a Gradient of a curve CONTENT: {Skills, point using the tangent at a point Knowledge, Attitudes} Learners should be able to: differentiate linear and quadratic Derived function of n find the gradient of a curve a point Gradient of aax curve at a expressions from firstatprinciples the form using the tangent point find the derivative of functions of Derivative of a sum differentiate linear and quadratic Derived function of the the form from axn where n is a rational expressions first principles form axn number find the derivative of functions of the Derivative of a sum n ax where n ais polynomial a rational number form differentiate OBJECTIVES TOPIC 12: INTEGRATION Differentiation SUB TOPIC Differentiation TOPIC 11: DIFFERENTIATION Pure Mathematics Syllabus Form 3 – 4 2016 SUB TOPIC OBJECTIVES TOPIC 11: DIFFERENTIATION Learners should be able to: TOPIC 11: DIFFERENTIATION Logarithms SUB TOPIC TOPIC 10: LOGARITHMIC AND EXPONENTIAL FUNCTIONS use the exact values of the trigonometrical ratios of special angles in a variety of situations TOPIC 10: LOGARITHMIC AND EXPONENTIAL FUNCTIONS ICT tools Relevant texts Environment texts Environment ICT tools ICT tools Relevant texts Environment SUGGESTED RESOURCES SUGGESTED RESOURCES ICT tools Relevant texts SUGGESTED Relevant RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 13 Learners should be able to: OBJECTIVES Covered in form 3 CONTENT: {Skills, Knowledge, Attitudes} TOPIC 1: INDICES AND IRRATIONAL NUMBERS Pure Mathematics Syllabus Form 3 – 4 2016 Irrational numbers Indices SUB TOPIC FORM FOUR (4) SYLLABUS To be covered in form 4 FORM 4COMPETENCY MATRIX COMPETENCY MATRIX: FORM 4 SYLLABUS 8.2 Learners should be able to: OBJECTIVES TOPIC 1: INDICES AND IRRATIONAL NUMBERS 8.0 Iterative methods SUB TOPIC SUGGESTED NOTES AND ACTIVITIES 30 SUGGESTED NOTES AND ACTIVITIES CONTENT: {Skills, Knowledge, Attitudes} TOPIC 13: NUMERICAL METHODS TOPIC 13: NUMERICAL METHODS SUGGESTED RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 Learners should be able to: TOPIC 2: POLYNOMIALS OBJECTIVES multiply polynomials polynomial by Learners should be able to: another use thepolynomials Factor Theorem to multiply factorise polynomials divide one polynomial by another use thecubic Factorequations Theorem to using factorise solve polynomials the Factor Theorem solve cubic equations using the Factor use Factor Theorem to Theorem evaluate use Factor unknown Theorem tocoefficients evaluate Division Factor Theorem Solving equations 14 express ax2 + bx + c in the form d(x+e)2+f derive the quadratic formula solve quadratic equations by completing the square solve simultaneous equations by substitution Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Pure Mathematics Syllabus Form 3 – 4 2016 Identities and equations SUB TOPIC Completing the square Simultaneous equations CONTENT: {Skills, Knowledge, Attitudes} 32 20 Exploring ways of expressing ax2 + bx + c in the form d(x+e)2+f Completing the square to derive the quadratic formula Solving quadratic equations by completing the square Solving simultaneous equations (one linear and one implicit) by substitution SUGGESTED NOTES AND ACTIVITIES using the Factor Theorem texts ICT tools Relevant texts ICT tools ICT tools Relevant texts SUGGESTED RESOURCES Usinglong division to find the quotient and Determining theremainder product of polynomials Deriving the Factor theorem Usinglong find the quotient Using thedivision Factorto Theorem to and remainder factorise polynomials Deriving the Factor theorem Solving cubic equations using Using the Factor Theorem to the Factor Theorem factorise polynomials Evaluating Solving cubic unknown equations using the Factor Theorem coefficients using the Factor Evaluating unknown coefficients Theorem Determining the product of SUGGESTED Relevant RESOURCES SUGGESTED RESOURCES SUGGESTED NOTES AND ACTIVITIES polynomials SUGGESTED NOTES AND ACTIVITIES TOPIC 3: IDENTITIES, EQUATIONS AND INEQUALITIES unknown coefficients OBJECTIVES divide one CONTENT: {Skills, Knowledge, Attitudes} Multiplication CONTENT: {Skills, Division Knowledge, Attitudes} Factor Theorem Solving equations Multiplication TOPIC 3: IDENTITIES, EQUATIONS AND INEQUALITIES Polynomials SUB TOPIC Polynomials TOPIC 2: POLYNOMIALS SUB TOPIC Pure Mathematics Syllabus Forms 3 - 4 15 TOPIC 5: VECTORS sketch graphs of the form 𝑦𝑦 = 𝑘𝑘𝑥𝑥 𝑛𝑛 explain the geometrical effect of the value of k on the shape of the graph of 𝑦𝑦 = 𝑘𝑘𝑥𝑥 𝑛𝑛 calculate unit vectors define the scalar product use the scalar product to determine the angle between vectors identify vector properties of quadrilaterals solve problems involving perpendicular vectors calculate areas of triangles and parallelograms Learners should be able to: OBJECTIVES Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Vectors in three dimensions SUB TOPIC TOPIC 5:VECTORS Graphs SUB TOPIC TOPIC 4:GRAPHS AND COORDINATE GEOMETRY Graphs of 𝑦𝑦 = 𝑘𝑘𝑥𝑥 𝑛𝑛 Unit vectors Scalar product Vector properties of plane shapes Areas of triangles and parallelograms CONTENT: {Skills, Knowledge, Attitudes} CONTENT: {Skills, Knowledge, Attitudes} TOPIC 4: GRAPHS AND COORDINATE GEOMETRY Sketching the standard curves where n and k are rational Discussing the geometrical effect of the value of k on the shape of the graph for a given value of n 34 Computing unit vectors Discussing the scalar product Determining the angle between vectors Exploring vector properties of quadrilaterals Solving problems involving perpendicular vectors Computing areas of triangles and parallelograms SUGGESTED NOTES AND ACTIVITIES SUGGESTED NOTES AND ACTIVITIES ICT tools Relevant texts Geo-board ICT tools Mathematical models Environment SUGGESTED RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 16 derive the formulas for the general terms of the AP and GP use the formula for the nth term of an AP and GP derive the formula for the sum of the first n terms of an AP and GP and use it to solve problems distinguish between APs and GPs Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Sequences SUB TOPIC TOPIC 7: SEQUENCES define a one-one function restrict the domain to get a one-one function sketch the graph of a function for a given domain define the inverse of a function find the inverse of a given function illustrate graphically the relationship between a function and its inverse Learners should be able to: OBJECTIVES TOPIC 7: SEQUENCES Functions SUB TOPIC TOPIC 6: FUNCTIONS TOPIC 6: FUNCTIONS One-one function Inverse of a function Graphs of functions Arithmetic progression Geometric progression CONTENT: {Skills, Knowledge, Attitudes} CONTENT: {Skills, Knowledge, Attitudes} Discussing one-one functions Sketching graphs of functions for a given domain Discussing the inverse of a function Determining the range of functions Determining the inverse of a given function excluding inverses of quadratic functions Illustrating graphically the relationship between a function and its inverse 36 Exploring ways of finding the formula for the general term of an AP and a GP Solving problems using the formula for the nth term of an AP and a GP Exploring ways of finding the formulae of the sum of the first n terms of an AP and a GP and solving problems Differentiating APs from GPs Discussing applications of APs and GPs in life SUGGESTED NOTES AND ACTIVITIES SUGGESTED NOTES AND ACTIVITIES ICT tools Relevant texts Geo-board ICT tools Relevant texts SUGGESTED RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 define a radian Learners should be able to: OBJECTIVES 17 convert degrees to radians and find area of a sector and a segment solve defineproblems a radianinvolving length of arcs,the areas of sectors andnotation segments use correct radian formulae for the area of a sector and segment 24 Equations Radians Length of an arc Area of a sector Area of a segment Pure Mathematics Syllabus Form 3 – 4 2016 Knowledge, Attitudes} Discussing radians and degrees and their relationship Using the correct radian notation SUGGESTED NOTES Converting degrees to AND radians and radians to degrees ACTIVITIES Deriving and using the formulae for length of an arc Discussing radians and degrees Deriving and using the formulae for and their the area of a sector and segment relationship Solving problems involving length of Using the correct radian arcs, areas of sectors and notation segments Discussing the notationto sin-1x, tan-1x Converting degrees -1x and cos radians and radians to Finding all the solutions, within a degrees specified interval of the equations Deriving usingand thetan(kx)=c sin(kx)=c,and cos(kx)=c for length of equations an arc formulae Applying trigonometrical in solving life problems Deriving and using the SUGGESTED NOTES AND ACTIVITIES 38 convert degrees to radians and radians to degrees find the length of an arc -1 use area the notation sin-1x, tan find of a sector andx and a cos-1x to solve problems segment Solve trigonometrical equations solve problems involving length of arcs, areas of sectors and segments find the length of an arc Learners should be able to: OBJECTIVES radians to degrees Radians Length of an arc Area of a sector CONTENT: Area of a {Skills, segment CONTENT: {Skills, Knowledge, Attitudes} ACTIVITIES Constructing Pascal’s Triangle Making use ofPascal’s the Pascal’s Triangle Constructing Triangle to expand (𝑎𝑎 + 𝑏𝑏)n where n is a Making use of the Pascal’s Triangle positive integer to expand (𝑎𝑎 + 𝑏𝑏)n where n is a positive Solving problems integer involving expansion of (𝑎𝑎 + 𝑏𝑏)n Solving problems involving expansion of (𝑎𝑎 + 𝑏𝑏)n SUGGESTED NOTES AND ACTIVITIES Pure Mathematics Syllabus Form 3 – 4 2016 Trigonometrical functions Plane Trigonometry SUB TOPIC TOPIC 9: TRIGONOMETRY use the correct radian notation Plane Trigonometry SUB TOPIC integer Knowledge, Attitudes} Pascal’s Triangle Expansion of (𝑎𝑎 + Pascal’s Triangle 𝑏𝑏)nwhere n is a positive Expansion of (𝑎𝑎 + 𝑏𝑏)n integern is a positive where TOPIC 9: TRIGONOMETRY Learners should be ableTriangle to: construct a Pascal’s n expand (𝑎𝑎a+Pascal’s 𝑏𝑏) using Pascal’s construct Triangle Triangle n expand (𝑎𝑎 + 𝑏𝑏) using Pascal’s Triangle solve problems involving expansion of (𝑎𝑎 + 𝑏𝑏)n solve problems involving expansion of (𝑎𝑎 + 𝑏𝑏)n Learners should be able to: OBJECTIVES TOPIC 9: TRIGONOMETRY Binomial expansion Binomial expansion SUB TOPIC CONTENT: {Skills, Knowledge, Attitudes} TOPIC 8: BINOMIAL EXPANSION TOPIC 8: BINOMIAL EXPANSION Relevant texts Environment ICT tools Relevant texts Environment Geo-board Geometrical instruments ICT tools SUGGESTED RESOURCES ICT tools ICT tools Relevant texts Environment SUGGESTED Geo-board RESOURCES Geometrical instruments RESOURCES Relevant texts ICT tools texts Relevant SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 18 find the equation of a tangent and a normal to quadratic and cubic curves determine stationary points and their nature find the rate of change of one variable with respect to another solve problems involving rates of change Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Differentiation SUB TOPIC Application of differentiation to gradients, tangents and normals, stationary points, rates of change, velocity and acceleration Determining equations of tangents and normals to quadratic and cubic curves Discussing the nature of, and solving problems involving, stationary points Calculating the rate of change of one variable with respect to another Solving problems involving rates 40 of change in life SUGGESTED NOTES AND ACTIVITIES Discussing the concept of exponential growth and decay Sketching the graph of the exponential function and compare with the logarithmic function Formulating equations involving exponential growth and decay Solving equations involving exponential growth and decay Discussing the natural logarithms in relation to logarithms in general Sketching the graphs of the form y=ln(ax+b) Solving equations of the form a x = b using logarithms SUGGESTED NOTES AND ACTIVITIES CONTENT: {Skills, Knowledge, Attitudes} Exponential growth and decay Natural logarithms Equations of the form ax= b CONTENT: {Skills, Knowledge, Attitudes} TOPIC 11: DIFFERENTIATION explain the concept of exponential growth and decay sketch the graph of the exponential function solve equations involving exponential growth and decay formulate equations involving exponential growth and decay define the natural logarithm sketch the graph of the form y=ln(ax+b) use logarithms to solve equations of the form a x = b Learners should be able to: OBJECTIVES TOPIC 11: DIFFERENTIATION Exponential functions Logarithms SUB TOPIC TOPIC 10: LOGARITHMIC AND EXPONENTIAL FUNCTIONS TOPIC 10: LOGARITHMIC AND EXPONENTIAL FUNCTIONS ICT tools Relevant texts Geo-board ICT tools Relevant texts Geo-board ICT tools Relevant texts Environment SUGGESTED RESOURCES SUGGESTED RESOURCES Pure Mathematics Syllabus Forms 3 - 4 19 Area Volume CONTENT: {Skills, Knowledge, Attitudes} • locate locatethe theposition positionofofroots rootsby bysign • Simple Simpleiterative iterativeprocedures procedures sign change Newton-Raphsonmethod method change • Newton-Raphson solve equations by the bisection TrapeziumRule Rule • solve equations by the bisection • Trapezium method method usesimple simpleiterative iterativeprocedures • use proceduresand the Newtonand the Newton-Raphson method Raphson method to solve toequations solve equations • estimate estimateareas areasunder undercurves curvesusing the Trapezium Rule Rule using the Trapezium Learners should be able to: OBJECTIVES Pure Mathematics Syllabus Form 3 – 4 2016 Numerical Numerical methods methods SUB TOPIC CONTENT: {Skills, Knowledge, Attitudes} TOPIC 13: NUMERICAL METHODS compute area bounded by acurveand a lineparallel to the coordinate axes calculatethe volume of revolution about one of the axes Learners should be able to: OBJECTIVES TOPIC 13: NUMERICAL METHODS Integration SUB TOPIC TOPIC 12: INTEGRATION TOPIC 12: INTEGRATION Exploring ways of finding area bounded by a curve and a line parallel to the coordinate axes Exploring ways of finding the volume of revolution about one of the axes Solving problems involving use of integration in determining areas and volumes ICT tools Relevant texts Environment SUGGESTED RESOURCES SUGGESTED RESOURCES 42 • Determining • ICT Determiningthe theexistence existenceofof ICTtools tools rootswithin withinan aninterval interval Relevanttexts texts roots • Relevant Using the bisection methodto Environment • Using the bisection method to • Environment solveequations equationstotoaaspecific specific solve degree of accuracy degree of accuracy Solving equations using simple • Solving equations using simple iterative procedures and the iterative proceduresmethod and the Newton-Raphson Newton-Raphson method Determiningestimates of area • Determining estimates of area under curves and related areas under related Rule areas in in lifecurves by the and Trapezium life by the Trapezium Rule SUGGESTED NOTES AND ACTIVITIES SUGGESTED NOTES AND ACTIVITIES Pure Mathematics Syllabus Forms 3 - 4 Pure Mathematics Syllabus Forms 3 - 4 9.0 ASSESSMENT (a) ASSESSMENT OBJECTIVES The assessment will test candidate’s ability to:• • • • • • • • • • use Mathematical symbols, terms and definitions correctly in problem solving sketch graphs accurately use appropriate formulae,algorithms and strategies, to solve routine and non-routine problems in Pure Mathematics demonstrate the appropriate and accurate use of ICT tools in problem solving translate Mathematical information from one form to another accurately demonstrate an appreciation of Mathematical concepts and processes demonstrate an ability to solve problems systematically apply Mathematical reasoning and communicate mathematical ideas clearly model information from other forms to Mathematical form and vice versa conduct research projects including those related to enterprise (b) SCHEME OF ASSESSMENT The Form 3 - 4Pure Mathematics assessment will be based on 30% Continuous Assessment and 70% Summative Assessment. The syllabus’ scheme of assessment caters for all learners and does not condone direct or indirect discrimination. Arrangements, accommodations and modifications must be visible in both continuous and summative assessments to enable candidates with special needs to access assessments and receive accurate performance measurement of their abilities. Access arrangements must neither give these candidates an undue advantage over others nor compromise the standards being assessed. Candidates who are unable to access the assessments of any component or part of component due to disability (transitory or permanent) may be eligible to receive an award based on the assessment they would have taken Continuous Assessment Continuous assessment will consists of topic tasks, written tests and end of term examinations: i) Topic Tasks These are activities that teachers use in their day to day teaching. These may include projects, assignments and team work activities. ii) Written Tests These are tests set by the teacher to assess the concepts covered during a given period of up to a month. The tests should consist of short structured questions as well as long structured questions. iii) End of term examinations These are comprehensive tests of the whole term’s or year’s work. These can be set at school/district/provincial level. Summary of Continuous Assessment Tasks From term one to five, candidates are expected to have done at least the following recorded tasks per term: • 1 Topic task • 1 Written test • 1 End of term test 20 Summary of Continuous Assessment Tasks From term one to five, candidates are expected to have done at least the following recorded tasks per term: 1 Topic task 1 End of term test 1 Written test Syllabus Forms 3 - 4 Pure Mathematics Detailed Continuous Assessment Tasks Table Detailed Continuous Assessment Tasks Table Term Number of Topic Tasks Number of Written Tests Number of End Of Term Tests 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1 5 1 1 1 Weighting 25% 25% 50% 100% Actual Weight 7.5% 7.5% 15% 30% Comment: Total Term 6 is for the National Examination c)(c)Specification Grid Specification Grid Pure Mathematics Syllabus Form 3 – 4 2016 25 Specification grid for continuous assessment Component Skills Topic Tasks Written Tests End of Term Skill 1 30% 30% 30% 50% 50% 50% 20% 20% 20% Total 100% 100% 100% Actual weighting 7.5% 7.5% 15% Knowledge Comprehension Skill 2 Application Analysis Skill 3 Synthesis Evaluation Summative Assessment The examination will consist of 2 papers: paper 1 and paper 2, each to be written in 2½ hours Pure Mathematics Syllabus Form 3 – 4 2016 26 21 Type of Paper Approximately 16questions 15 Short where Answer candidates Questions, answer any where 10, each Pure Mathematics Syllabus 3-4 candidatesForms question answer all carrying 10 questions marks Summative Assessment Marks 100 100 The examination will consist of 2 papers: paper 1 and paper 2, each to be written in 2½ hours P1 P2 Total Weighting 50% 50% 100% Actual weighting 35% 35% 70% Type of paper Approximately 15 Short Answer Questions, where candidates anwer all questions 16 Questions where candidates answer any 10, each question carrying 10 marks Marks 100 100 Specification Grid for Summative Assessment Specification Grid for Summative Assessment Skill 1 P1 P2 Total 50% 30% 80% Knowledge & Pure Mathematics Syllabus Form 3 – 4 2016 27 Comprehension Skill 2 40% 50% 90% 10% 20% 30% Total 100% 100% 200% Weighting 50% 50% 100% Application & Analysis Skill 3 Synthesis & Evaluation (d) Assessment Model Learners will be assessed using both continuous and summative assessments. Assessment of learner performance in Pure Mathematics 100% Pure Mathematics Syllabus Form 3 – 4 2016 28 22 Pure Mathematics Syllabus Forms 3 - 4 d) Assessment Model Assessment of learner performance in Pure Mathematics 100% Summative assessment 70% Continuous assessment 30% Profiling Topic Tasks 7.5% Written Tests 7.5% End of term Tests 15% Paper 1 35% Paper 2 35% Profile Continuous assessment 30% Examination Mark 100% Exit Profile FINAL MARK 100% 23 Pure Mathematics Syllabus Forms 3 - 4 25 Pure Mathematics Syllabus Forms 3 - 4 26