Additional Mathematics Syllabus

ZIMBABWE

MINISTRY OF PRIMARY AND SECONDARY EDUCATION

ADDITIONAL MATHEMATICS SYLLABUS

FORMS 5 - 6

2015 - 2022

Curriculum Development and Technical Services P. O. Box MP 133 Mount Pleasant Harare

Additional Mathematics Syllabus Forms 5 - 6

ACKNOWLEDGEMENTS

The Ministry of Primary and Secondary Education wishes to acknowledge the following for their valued contribution in the production of this syllabus:

• National Mathematics panel • Zimbabwe School Examinations Council (ZIMSEC) • United Nations International Children’s Emergency Fund (UNICEF) • United Nations Educational Scientific and Cultural Organisation (UNESCO) • Representatives from Higher and Tertiary institutions

i Additional Mathematics Syllabus Forms 5 - 6

CONTENTS

ACKNOWLEDGEMENTS...... i

CONTENTS...... ii

1.0 PREAMBLE...... 1

2.0 PRESENTATION OF SYLLABUS...... 1

3.0 AIMS...... 1

4.0 OBJECTIVES...... 1

5.0 METHODOLOGY AND TIME ALLOCATION...... 2

6.0 TOPICS...... 3

7.0 SCOPE AND SEQUENCE...... 4

8.0 COMPETENCY MATRIX...... 15

FORM FIVE (5)...... 15

FORM SIX (6)...... 29

9.0 ASSESSMENT...... 50

ii Additional Mathematics Syllabus Forms 5 - 6

1.0 PREAMBLE 1.4 ASSUMPTIONS

1.1 INTRODUCTION The syllabus assumes that the learner:

The Form 5 – 6 Additional Mathematics syllabus builds • has a talent in mathematics on the Form 3 - 4 Additional Mathematics syllabus • has a high pass in Mathematics or a pass in either and is designed to cater for the needs of post Form Pure Mathematics or Additional Mathematics at 4, mathematically outstanding learners. The teaching Form 4. and learning that is based on the syllabus provides • Is excelling in Pure Mathematics at Form 5 further opportunities for such learners to enjoy doing and using mathematics and, in the process developing 1.5 CROSS- CUTTING ISSUES a deeper appreciation and understanding of the nature of mathematics, its pivotal role in Science, Technology, The following are some of the cross cutting themes Engineering and Mathematics (STEM) and life in general catered for in the teaching and learning of Additional and hence promoting sustainable development. The Mathematics:- intention is to provide a firm basis on which to pursue further studies in mathematics and related areas which • Business and financial literacy are critical for national development and global compet- • Disaster and risk management itiveness in the 21st century. In addition to providing a • Communication firm base, the learning area is also expected to enhance • Team building learners’ confidence and sense of self fulfilment, • Problem solving collegiality and intellectual honesty, hence contributing to • Environmental issues their growth in the acquisition of Unhu/Ubuntu//Vumunhu • Enterprise skills values. • Information Communication and Technology (ICT) • HIV & AIDS 1.2 RATIONALE 2.0 PRESENTATION OF In its socio-economic transformation agenda, Zimbabwe has embarked on an industrialisation development SYLLABUS strategy, where high level mathematical knowledge and skills are a prerequisite. To this end, Additional Math- The Additional Mathematics syllabus is a single ematics learners are expected to emerge with a solid document covering Form 5–6. It contains the preamble, basis for further studies in STEM, and with enhanced aims, objectives, syllabus topics, scope and sequence, mathematical competencies. These include abstract competency matrix and assessment procedures. The and analytical thinking, logical reasoning, designing syllabus also suggests a list of resources to be used algorithms, communicating mathematical information during the teaching and learning process. effectively, conjecturing and proving conjectures, modelling life phenomena into mathematical language 3.0 AIMS in order to better understand those phenomena and to solve related problems and applying mathematics in a The syllabus will enable learners to: wide variety of contexts. It is from the pool of individuals with such competencies that some of Zimbabwe’s most 3.1 further develop mathematical knowledge and creative thinkers, innovators, inventors and mathemati- skills in a way that promotes fulfilment and en- cally literate entrepreneurs are expected to emerge. This joyment and lays a firm foundation for further is critical for sustainable development. studies and lifelong learning 3.2 acquire an additional range of mathematical 1.3 SUMMARY OF CONTENT skills, particularly those applicable to other learning areas and various life contexts, in- The Form 5- 6 Additional Mathematics syllabus will cover cluding enterprise the theoretical concepts and their applications. This 3.3 enhance confidence, critical thinking, innova- two year learning area consists of Pure Mathematics, tiveness, creativity and problem solving skills Mechanics and Statistics. for sustainable development

1 Additional Mathematics Syllabus Forms 5 - 6

3.4 develop a greater appreciation of the role of exchange of ideas and information; inclusivity and mathematics in personal, community and na- respect for each other’s views, regardless of personal tional development in line with Unhu/Ubuntu/ circumstances (in terms of, for example: gender, ap- Vumunhu pearance, disability and religious beliefs); collaboration 3.5 further develop an appreciation for and ability and cooperation; intellectual honesty; diligence and to use I.C.T tools in solving problems in math- persistence; and Unhu/ Ubuntu /Vumunhu. This is ematical and other contexts particularly important in a learning area like mathematics, 3.6 develop the ability to communicate mathemati- given the negative attitudes associated with its teaching cal ideas and to learn cooperatively and learning.

4.0 OBJECTIVES Providing appropriate stimuli has to do with posing relevant challenges that excite learners, and help to make learning Additional Mathematics an enjoyable, fulfilling The learners should be able to: experience. Such challenges could be posed in the form of problems that encourage learners to create new 4.1 use mathematical skills and techniques that mathematical ideas in line with the teacher expectations are necessary for further studies and even beyond. New knowledge acquired in such 4.2 use mathematical models to solve problems in a manner tends to be deep rooted and meaningful to life and for sustainable development learners, hence enhancing their ability to apply it within 4.3 use I.C.T tools in solving problems in mathe- the learning area and in life. Definitely spoon feeding is matical and other contexts not and cannot be an appropriate stimulus, as it does 4.4 demonstrate expertise, perseverance, coop- not help learners to develop critical thinking, creativity, eration and intellectual honesty for personal, and the ability to think outside the box, which are critical community and national development for self-reliance, national sustainable development 4.5 interpret mathematical results and their impli- and global competitiveness. Thus learners need to be cations in life active participants and decision makers in the Additional 4.6 communicate mathematical results and their Mathematics teaching and learning process, with the implications in life teacher playing the facilitator’s role. 4.7 draw inferences through correct manipulation Pre-requisite knowledge and skills refers to what the of data learners should already know and can do, which can 4.8 conduct research projects including those form a strong basis on which to construct the expected related to enterprise new knowledge. Thus the Additional Mathematics 4.9 apply relevant mathematical notations and teacher needs to carefully analyse the new concepts terms in problem solving and principles he/she intends to introduce, identify the 4.10 present data through appropriate presenta- relevant pre-requisite knowledge, assess to identify any tions gaps, and take appropriate steps to fill such gaps. 4.11 apply mathematical skills in other learning The following, is a list of teaching and learning approach- areas es that are consistent with, and supportive of the above 4.12 conduct mathematical proofs approach:

5.0 METHODOLOGY AND TIME 5.1.1 Guided discovery ALLOCATION 5.1.2 Group work 5.1.3 Interactive e-learning 5.1 Methodology 5.1.4 Problem solving 5.1.5 Discussion A constructivist based teaching and learning approach is 5.1.6 Modelling recommended for the Form 5 – 6 Additional Mathematics 5.1.7 Visual tactile Syllabus. The theoretical basis for this approach is that: in a conducive environment with appropriate The above suggested methods should be enhanced stimuli, learners’ capacity to build on their pre-requisite through the application of multi-sensory (inclusive) knowledge and create new mathematical knowledge approaches. is enhanced. A conducive environment in this context is one that encourages creativity and originality; a free

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5.2 Time Allocation 6.1.6 Calculus

10 periods of 35 minutes each per week should be 6.2 Mechanics allocated. Learners are expected to participate in the following 6.2.1 Particle dynamics activities:- 6.2.2 Elasticity - Mathematics Olympiads 6.2.3 Energy, work and power - Mathematics and Science exhibitions 6.2.4 Circular motion - Mathematics seminars 6.2.5 Simple harmonic motion - Mathematical tours to tertiary and other institutions 6.3 Statistics

6.0 TOPICS 6.3.1 Probability 6.3.2 Random Variables The following topics will be covered from Form 5 - 6 6.3.3 Sampling and estimation 6.3.4 Statistical inference 6.1 Pure Mathematics 6.3.5 Bivariate data

6.1.1 Algebra 6.1.2 Series 6.1.3 Trigonometry 6.1.4 Mathematical Induction 6.1.5 Geometry and Vectors

Additional Mathematics Syllabus Forms 5 - 6 7.0 SCOPE AND SEQUENCE

7.0 SCOPE AND SEQUENCE 7.1 PURE MATHEMATICS

7.1 PURE MATHEMATICS TOPIC 1: ALGERBRA TOPIC 1: ALGERBRA

TOPICS FORM 5 FORM 6

RATIONAL FUNCTIONS  Partial Fractions  Oblique Asymptotes  Graphs

MATRICES AND LINEAR SPACES  Linear equations  Spaces and subspaces

GROUPS  Properties  Order of elements  Simple subgroups  LaGrange’s theorem  Structure of finite groups  Isomorphism

Additional Mathematics Syllabus Forms 5 - 6 4

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 2: SERIES

SUBTOPIC FORM 5 FORM 6

SUMMATION OF SERIES  Standard results  ( , , ) 𝟐𝟐 𝟑𝟑  Method of differences ∑ 𝒓𝒓 ∑ 𝒓𝒓 ∑ 𝒓𝒓  Sum to infinity

TOPIC 3: TRIGONOMETRY

SUBTOPIC FORM 5 FORM 6

HYPERBOLIC FUNCTIONS  Six hyperbolic functions  Identities  Inverse notation

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 4: MATHEMATICAL INDUCTION TOPIC 4: MATHEMATICAL INDUCTION

SUBTOPIC FORM 5 FORM 6

MATHEMATICAL INDUCTION  Proof by induction  Conjecture

TOPIC 5: GEOMETRY AND VECTORS

SUBTOPIC FORM 5 FORM 6

POLAR COORDINATES  Cartesian and polar coordinates

 Polar coordinates curves

 Area of a sector

VECTOR GEOMETRY  Triple scalar product  Cross product

 Equations of lines and planes

Additional Mathematics Syllabus Forms 5 - 6 14

Additional Mathematics Syllabus Forms 5 - 6 TOPIC 6: CALCULUS TOPIC 5: GEOMETRY AND VECTORS

SUBTOPIC FORM 5 FORM 6

POLAR COORDINATES  Cartesian and polar coordinates  Polar coordinates curves  Area of a sector

VECTOR GEOMETRY  Triple scalar product  Cross product  Equations of lines and planes

Additional Mathematics Syllabus Forms 5 - 6

7.2 MECHANICS

7.2 MECHANICS TOPIC 1: PARTICLE DYNAMICS TOPIC 1: PARTICLE DYNAMICS

SUBTOPIC FORM 5 FORM 6

KINEMATICS OF MOTION  Motion in a straight line  Velocity

 Acceleration  Displacement - time velocity-time and acceleration- time graphs  Equation of motion for constant linear acceleration  Vertical motion under gravity  Motion and constant velocity

NEWTON’S LAWS OF MOTION  Newton’s laws of motion - Motion caused by a set of forces - Concept of mass and weight  Motion of connected objects

MOTION OF A PROJECTILE  Projectile  Motion of a projectile

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Additional Mathematics Syllabus Forms 5 - 6

 Velocity and displacements  Range on horizontal plane

 Greatest height  Maximum range

 Cartesian equation of a trajectory of a projectile

TOPIC 2: ELASTICITY TOPIC 2: ELASTICITY

SUBTOPIC FORM 5 FORM 6

ELASTICITY  Properties of elastic strings and springs  Work done in stretching a string

 Elastic potential energy  Mechanical energy

 Conservation of mechanical energy

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Additional Mathematics Syllabus Forms 5 - 6

TOPIC 3: ENERGY, WORK AND POWER TOPIC 3: ENERGY, WORK AND POWER

SUBTOPIC FORM 5 FORM 6

ENERGY, WORK AND POWER  Energy - Gravitational potential - Elastic potential - Kinetic  Work  Power

 Principle of energy conservation

TOPIC 4: CIRCULAR MOTION

TOPIC4: CIRCULAR MOTION

SUBTOPIC FORM 5 FORM 6

CIRCULAR MOTION  Angular speed and velocity

 Horizontal and vertical circular motion (Vertical and Horizontal)

 Acceleration of a particle moving on a circle  Motion in a circle with constant speed 19  Centripetal force

 Relation between angular and linear speed  Conical pendulum  Banked tracks

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 5: SIMPLE HARMONIC MOTION

SUBTOPIC FORM 5 FORM 6

SIMPLE HARMONIC MOTION  Basic equation of simple harmonic motion  Properties of simple harmonic motion

7.3 STATISTICS

TOPIC7.3 1:STATISTICS PROBABILITY

TOPIC 1: PROBABILITY

SUBTOPIC FORM 5 FORM 6

PROBABILITY  Events

- Independent - Mutually exclusive

- Exhaustive - Combined  Conditional probability 21  Tree diagrams  Outcome tables  Venn diagrams  Permutations and combinations

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 2: RANDOM VARIABLES

SUBTOPIC FORM 5 FORM 6

RANDOM VARIABLES  Probability distributions (discrete and continuous)  Expectation  Variance  Probability density functions (pdf) and cumulative distribution functions (cdf)  Mean, median, mode, standard deviation and percentiles

DISTRIBUTIONS  Binomial distribution  Poisson distribution  Normal distribution  Standard normal tables  Continuity correction  Linear combinations of normal and Poisson distributions

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 3: SAMPLING AND ESTIMA- TOPIC 3: SAMPLING AND ESTIMATION

SUBTOPIC FORM 5 FORM 6

SAMPLING TECHNIQUES AND  Probability sampling techniques ESTIMATION  Non-probability sampling techniques

 Estimation of population parameters  Central limit theorem  Confidence intervals

TOPIC4: STATISTICAL INFERENCE TOPIC4: STATISTICAL INFERENCE

SUBTOPIC FORM 5 FORM 6

HYPOTHESIS TESTING  Null hypothesis  Alternative hypothesis  Test statistics

 Significance level  Hypothesis test (1-tail and 2-tail)  Type 1 and type 2 errors  z- tests 24  t – tests

 chi-squared tests

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 5: BIVARIATE DATA TOPIC4: STATISTICAL INFERENCE

SUBTOPIC FORM 5 FORM 6

HYPOTHESIS TESTING  Null hypothesis  Alternative hypothesis  Test statistics  Significance level  Hypothesis test (1-tail and 2-tail)  Type 1 and type 2 errors  z- tests  t – tests  chi-squared tests

Additional Mathematics Syllabus Forms 5 - 6

8.08.0 COMPETENCYCOMPETENCY MATRIX MATRIX

FORM FIVE (5) FORM FIVE (5)

PUREPURE MATHEMATICS

TOPICTOPIC 1:1: ALGEBRAALGEBRA

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

RATIONAL FUNCTIONS  express rational functions in  Partial Fractions  Expressing rational functions in  ICT tools partial fractions  Oblique Asymptotes partial fractions  Relevant texts  determine key features of  Graphs  Exploring and determining key  Geo-board rational function graphs, features of rational function  Braille including oblique asymptotes graphs, including oblique materials in cases where degree of asymptotes in cases where  Talking books numerator and denominator degree of numerator and are at most two denominator are at most two

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SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 sketch graphs of rational  Sketching graphs of rational functions functions, including the  solve problems involving determination of oblique rational functions asymptotes  Solving problems involving rational functions

MATRICES AND LINEAR  define the systems of linear  Linear equations  Discussing the systems of linear  ICT tools SPACES equations  Spaces and subspaces equations  Relevant texts  determine consistency or  Discussing and determining the  Environment inconsistency of systems of consistency or inconsistency of  Braille linear equations systems of linear equations materials  interpret geometrically the  Interpreting geometrically the  Talking books consistency or inconsistency consistency or inconsistency of of systems of linear equations systems of linear equations  relate to singularity the  Relating to singularity the corresponding square matrix corresponding square matrix of a of a system of linear system of linear equations equations  Solving linear equations of  solve linear equations of consistent systems consistent systems  Solving problems involving  solve problems involving systems of linear equations

Additional Mathematics Syllabus Forms 5 - 6 16

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

systems of linear equations  Discussing properties of linear  define a linear space and a spaces and sub-spaces sub-space  Solving problems involving linear  solve problems involving spaces and sub-spaces linear spaces and sub-spaces

GROUPS  define a group  Properties  Discussing groups and  ICT tools  determine whether or not a  Order of elements determining whether or not a  Relevant texts given structure is a group  Simple subgroups given structure is a group  Braille  define the order of the group  LaGrange’s theorem  Determining the order of a group materials and elements of a group  Structure of finite groups and the order of elements of a  Talking books  define a subgroup  Isomorphism group  determine whether or not a  Discussing subgroups and given structure is a sub group determining whether or not a in simple cases given structure is a subgroup in  state the LaGrange’s theorem simple cases  apply the LaGrange’s  Discussing and applying the theorem concerning the order LaGrange’s theorem concerning

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SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

of a subgroup of a finite the order of a subgroup of a finite group group  define a cyclic group  Discussing and finding examples  demonstrate familiarity with of cyclic groups the structure of a finite group  Exploring properties of structures up to order 7 of finite groups up to order 7  define isomorphism between  Discussing and determining groups whether or not given finite groups  determine whether or not are isomorphic given finite groups are  Solving problems involving isomorphic groups  solve problems involving groups

Additional Mathematics Syllabus Forms 5 - 6 18

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 2: SERIES

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

SUMMATION OF SERIES  use the standard results to  Standard results  Finding related sums using the  ICT tools find related sums ( , , ) standard results  Relevant 𝟐𝟐 𝟑𝟑  derive the method of  Method∑ 𝒓𝒓 ∑ 𝒓𝒓 of∑ differences𝒓𝒓  Exploring and deriving the method texts differences  Sum to infinity of differences  Braille  find the sum of finite series  Using method of differences to materials using method of differences find the sum of finite series  Talking books  determine the sum to infinity  Computing the sum to infinity of a of a convergent series by convergent series by considering considering the sum to n the sum to n terms terms  Solving problems involving series  solve problems involving  Representing life phenomena series using mathematical models involving series and exploring their applications in life

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 3: TRIGONOMETRY

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

HYPERBOLIC  define the six hyperbolic  Six hyperbolic functions  Discussing the six hyperbolic  ICT tools FUNCTIONS functions in terms of  Identities functions in terms of exponentials  Relevant exponentials  Inverse notation  Sketching the graphs of texts  sketch the graphs of hyperbolic functions  Environment hyperbolic functions  Deriving and using identities  Resource  derive identities involving involving hyperbolic functions to persons hyperbolic functions solve problems Braille  use hyperbolic identities in  Using the inverse notations to materials solving problems denote the principal values of the  Talking books  denote the principal values of inverse hyperbolic relations the inverse hyperbolic  Deriving and using hyperbolic relations using the inverse expressions in terms of logarithms notations to solve problems  derive hyperbolic expressions  Solving problems involving in terms of logarithms hyperbolic functions  use hyperbolic expressions in  Representing life phenomena terms of logarithms to solve using mathematical models problems involving series and exploring

Additional Mathematics Syllabus Forms 5 - 6 20

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Knowledge) Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES  solve problems involving Knowledge) their applications in life hyperbolic functions  solve problems involving their applications in life

hyperbolic functions TOPICTOPIC 5:5: MATHEMATICALMATHEMATICAL INDUCTION INDUCTION

TOPICSUBTOPIC 5: MATHEMATICALLEARNING INDUCTION OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Knowledge) Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES MATHEMATICAL  describe the process of Knowledge) Proof by induction  Discussing the processes of  ICT Tools INDUCTION mathematical induction  Conjecture mathematical induction  Relevant MATHEMATICAL  describe the process of  Proof by induction  Discussing the processes of  ICT Tools  prove by mathematical  Proving by mathematical Texts INDUCTION mathematical induction  Conjecture mathematical induction  Relevant induction to establish a given induction to establish given  Braille  prove by mathematical  Proving by mathematical Texts result results materials induction to establish a given induction to establish given  Braille  use the strategy of conducting  Conducting limited trials to  Talking books result results materials limited trials to formulate a formulate conjecture and proving  use the strategy of conducting  Conducting limited trials to  conjecture and proving it by the it by the method of induction Talking books limited trials to formulate a formulate conjecture and proving method of induction conjecture and proving it by the it by the method of induction method of induction

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 6: GEOMETRY AND VECTORS

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

POLAR COORDINATES  explain the relationship  Cartesian and polar  Discussing the appropriateness of  ICT Tools between Cartesian and polar coordinates using Cartesian or polar  Relevant coordinates for r 0  Polar coordinates curves coordinates Texts

 convert equations of curves≥  Area of a sector  Converting equations of curves  Braille from Cartesian to polar and from Cartesian to polar and vice materials vice versa versa  Talking books  sketch simple polar curves for  Sketching simple polar curves for 0 < 2 or - < , 0 < 2 or - < ,

showing≤ 𝜃𝜃 significant𝜋𝜋 𝜋𝜋 features𝜃𝜃 ≤ 𝜋𝜋 showing≤ 𝜃𝜃 significant𝜋𝜋 features𝜋𝜋 𝜃𝜃 ≤such𝜋𝜋 such as symmetry, the form of as symmetry, the form of the the curve at the pole and curve at the pole and least/greatest values of r least/greatest values of r

Additional Mathematics Syllabus Forms 5 - 6 22

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 derive the formula  Deriving and using the formula 𝟏𝟏 𝜷𝜷 𝟐𝟐 d for the area of a sector𝟐𝟐 ∫𝜶𝜶 𝒓𝒓 in d for the area of a sector 𝟏𝟏 𝜷𝜷 𝟐𝟐 simple𝜽𝜽 cases 𝟐𝟐in∫ 𝜶𝜶simple𝒓𝒓 𝜽𝜽 cases

 use the formula  Solving problems involving polar 𝟏𝟏 𝜷𝜷 𝟐𝟐 coordinates 𝟐𝟐 𝜶𝜶 d for the area of a sector∫ 𝒓𝒓 in  Representing life phenomena simple𝜽𝜽 cases using mathematical models  solve problems involving polar involving polar coordinates and coordinates exploring their applications in life

VECTOR GEOMETRY  determine the vector product  Triple scalar product  Calculating the vector product of  ICT Tools of two given vectors  Cross product two given vectors  Relevant  calculate the triple scalar  Equations of lines and planes  Computing the triple scalar Texts product of given vectors product of given vectors  Environment  interpret the triple scalar  Discussing the triple scalar and  Braille product and cross product in cross product in geometrical materials geometrical terms terms  Talking books  solve problems concerning  Solving problems concerning distances, angles and distances, angles and intersections using equations intersections using equations of of lines and planes. lines and planes.

Additional SUBTOPIC Mathematics SyllabusLEARNING Forms OBJECTIVES 5 - 6 CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES

SUBTOPIC LEARNING OBJECTIVES CONTENTKnowledge) SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES  Representing life phenomena Knowledge) using mathematical models

 Representinginvolving vector life geometry phenomena and usingexploring mathematical their applications models in life involving vector geometry and

exploring their applications in life

TOPIC 7: CALCULUS

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED TOPIC 7: CALCULUS Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES

SUBTOPIC LEARNING OBJECTIVES CONTENTKnowledge) SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES DIFFERENTIATION  derive an expression for  Higher order differentiation  Deriving an expression for in  ICT Tools 2 2 𝑑𝑑 𝑦𝑦 Knowledge) 𝑑𝑑 𝑦𝑦 2  Concavity 2  Relevant AND INTEGRATION in cases where the relation𝑑𝑑𝑥𝑥 cases where the relation between𝑑𝑑𝑥𝑥  Reduction formulae Texts DIFFERENTIATION  derivebetween an expression and is defined for  Higher order differentiation  Deriving an expression for and is defined implicitly or in  ICT Tools 2 2 𝑑𝑑 𝑦𝑦  Arc lengths 𝑑𝑑 𝑦𝑦  Braille implicitly or parametrically 2  Concavity parametrically 2  Relevant AND INTEGRATION in cases where𝑥𝑥 𝑦𝑦the relation𝑑𝑑𝑥𝑥 cases𝑥𝑥 𝑦𝑦where the relation between𝑑𝑑𝑥𝑥  ReductionSurface areas formulae of revolution materialsTexts  deducebetween the and relationship is defined  Exploring and is defineand deducingd implicitly the or   Differentiation and Integration  TalkingBraille books betweenimplicitly theor parametrically sign of and Arc lengths parametricallyrelationship between the sign of 𝑥𝑥 𝑦𝑦 2 𝑥𝑥 𝑦𝑦 𝑑𝑑 𝑦𝑦 of inverse trigonometric 2  Surface areas of revolution materials  deduce the relationship𝑑𝑑𝑥𝑥  Exploring and deducing the  Differentiation and Integration  Talking books between the sign of and relationship between the sign of 2 𝑑𝑑 𝑦𝑦 of inverse trigonometric 2 36 𝑑𝑑𝑥𝑥

Additional Mathematics Syllabus Forms 5 - 6 24

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

concavity functions and hyperbolic and concavity 2 𝑑𝑑 𝑦𝑦  determine the points of functions 2  Finding𝑑𝑑𝑥𝑥 points of inflexion using inflexion using 2 𝑑𝑑 𝑦𝑦 2 2 𝑑𝑑 𝑦𝑦  derive the derivatives𝑑𝑑𝑥𝑥 of 2  Exploring𝑑𝑑𝑥𝑥 and deriving the inverse trigonometric and derivatives of inverse hyperbolic functions trigonometric and hyperbolic  apply the derivatives of functions inverse trigonometric and  Applying the derivatives of inverse hyperbolic functions in trigonometric and hyperbolic problem solving functions in problem solving  find definite or indefinite  Finding definite or indefinite integrals using appropriate integrals using appropriate trigonometric or hyperbolic trigonometric or hyperbolic substitution substitution  integrate rational functions by  Integrating rational functions by means of decomposition into means of decomposition into partial fractions partial fractions  derive reduction formulae for  Exploring and deriving reduction the evaluation of definite formulae for the evaluation of integrals in simple cases definite integrals in simple cases  find definite integrals using  Finding definite integrals using

25 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

reduction formula reduction formula  find arc lengths for curves  Finding arc lengths for curves with with equations in Cartesian equations in Cartesian coordinates including use of a coordinates including use of a parameter or in polar form parameter or in polar form  find the area of surface of  Determining the area of surface of revolution when an arc is revolution when an arc is rotated rotated about one of the about one of the coordinate axes, coordinate axes, given that given that the equation of the the equation of the curve curve containing the arc is in containing the arc is in Cartesian or parametric form Cartesian or parametric form  Solving problems involving  solve problems involving differentiation and integration differentiation and integration  Representing life phenomena using mathematical models

involving differentiation and integration and exploring their applications in life

DIFFERENTIAL  find the general solution of a  First order differential  Determining general solutions of  ICT Tools EQUATIONS first order linear differential equations first order linear differential  Relevant equation by means of an  Second order differential equations by means of integrating Texts

Additional Mathematics Syllabus Forms 5 - 6 26

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

integrating factor equations factors  Braille  define a ‘complementary  Complementary function  Discussing complementary materials function’ and a ‘particular  General and Particular functions and particular integrals  Talking books integral’ in the context of integrals in the context of linear differential linear differential equations  Substitution equations  use a given substitution to  Particular solution  Applying given substitutions to reduce a first order differential reduce first order differential equation to linear form or to a equations to linear form or to form in which variables are forms in which variables are separable separable  find the complimentary  Finding complimentary functions function of a first or second of first or second order linear order linear differential differential equations with equation with constant constant coefficients coefficients  Determining particular integrals  determine a particular integral for first or second order linear for a first or second order differential equations in the case linear differential equation in where + or or + 𝑏𝑏𝑏𝑏 the case where or ( 𝑎𝑎𝑎𝑎) + 𝑏𝑏 𝑎𝑎 (𝑒𝑒 ) is a ( ) + ( ) or 𝑎𝑎𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎suitable𝑝𝑝𝑝𝑝 form.𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏 𝑝𝑝𝑝𝑝 𝑏𝑏𝑏𝑏 𝑎𝑎is𝑒𝑒 a suitable𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 form.𝑝𝑝𝑝𝑝 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑝𝑝𝑝𝑝  Finding appropriate coefficients of

27 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 find the appropriate coefficient first or second order differential (s) of a first or second order equations given suitable forms of differential equation given a particular integrals suitable form of a particular  Determining particular solutions to integral differential equations using initial  find a particular solution to a conditions differential equation using  Interpreting particular solutions in initial conditions the context of problems modelled  interpret particular solutions in by differential equations the context of problems  Solving problems involving modelled by differential differential equations equations  Representing life phenomena  solve problems involving using mathematical models differential equations involving differential equations and exploring their applications in life

Additional Mathematics Syllabus Forms 5 - 6

FORM SIX (6)

MECHANICS FORM SIX (6)

MECHANICS

TOPIC 1: PARTICLE DYNAMICS

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

KINEMATICS OF  use differentiation and  Motion in a straight line  Discussing distance(x)  ICT tools MOTION integration with respect to  Velocity displacement(s), speed,  Geo-board time to solve problems  Acceleration velocity(v) and acceleration(a)  Environment

concerning displacement,  Displacement - time velocity-  Solving problems concerning  Relevant texts velocity and acceleration time and acceleration-time displacement, velocity and  Braille materials  sketch the graphs of: (x-t), (s- graphs acceleration using differentiation  Talking books

29 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

t), (v-t) and (a-t)  Equation of motion for and integration  interpret the (x-t), (s-t), (v-t) constant linear acceleration  Sketching (x-t), (s-t), (v-t) and (a- and (a-t) graphs  Vertical motion under gravity t) graphs  derive the equations of  Motion and constant velocity  Interpreting the (x-t), motion of a particle with (s-t), (v-t) and (a-t) graphs constant acceleration in a  Deriving the equations of motion straight line of a particle with constant  use the equations of motion acceleration in a straight line of a particle with constant  Solving kinematics problems acceleration in a straight line  Representing life phenomena to solve kinematics problems using mathematical models  solve problems involving involving kinematics of motion in kinematics of motion in a a straight line and exploring their straight line including vertical applications in life motion under gravity

Additional Mathematics Syllabus Forms 5 - 6 30

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

NEWTON’S LAWS OF  state Newton’s laws of  Newton’s laws of motion  Discussing the Newton’s laws  ICT tools MOTION motion  Motion caused by a set of of motion  Geo-board  apply Newton’s laws of forces  Applying Newton’s laws of  Environment motion to the linear motion of  Concept of mass and motion to the linear motion of a  Relevant texts a body of constant mass weight body of constant mass moving  Braille materials moving under the action of  Motion of connected under the action of constant  Talking books constant forces objects forces  solve problems using the  Solving problems using the relationship between mass relationship between mass and and weight weight  solve problems involving the  Solving problems involving the motion of two particles, motion of two particles, connected by a light connected by a light inextensible string which may inextensible string which may pass over a fixed, smooth, pass over a fixed, smooth, light light pulley or peg pulley or peg  model the motion of the body  Solving problems involving moving vertically or on an Newton’s laws of motion inclined plane as motion with  Modelling the motion of a body constant acceleration moving vertically or on an  solve problems involving inclined plane as motion with

31 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

Newton’s laws of motion constant acceleration  Representing life phenomena using mathematical models involving Newton’s laws of motion and exploring their applications in life

MOTION OF A  model the motion of a  Projectile  Modelling the motion of a  ICT tools PROJECTILE projectile as a particle - Motion of a projectile projectile as a particle moving  Geo-board moving with constant - Velocity and with constant acceleration  Environment acceleration displacements  Applying horizontal and vertical  Relevant texts  solve problems on the motion  Range on horizontal plane equations of motion in solving  Braille materials of projectiles using horizontal  Greatest height problems on the motion of  Talking books and vertical equations of  Maximum range projectiles motion  Cartesian equation of a  Calculating the magnitude and  find the magnitude and the trajectory of a projectile the direction of the velocity at a

Additional Mathematics Syllabus Forms 5 - 6 32

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

direction of the velocity at a given time or position given time or position  find the range on the  Finding the range on the horizontal plane and height horizontal plane and height reached reached  derive formulae for greatest  Deriving formulae for greatest height and maximum range height and maximum range  derive the Cartesian equation  Deriving the Cartesian equation of a trajectory of a projectile of a trajectory of a projectile  solve problems using the  Solving problems using Cartesian Cartesian equation of a equation of a trajectory of a trajectory of a projectile projectile

33 Additional Mathematics Syllabus Forms 5 - 6

TOPIC 2: ELASTICITY

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED (Attitudes, Skills and ACTIVITIES RESOURCES Learners should be able to: Knowledge)

ELASTICITY  define elasticity in strings and  Properties of elastic strings  Discussing elasticity in strings and  ICT tools springs and springs springs  Relevant  explain Hooke’s law  Work done in stretching a  Explaining Hooke’s law texts  calculate modulus of elasticity string  Conducting experiments to verify  Environment  solve problems involving  Elastic potential energy Hooke’s law  Braille forces due to elastic strings or  Mechanical energy  Calculating modulus of elasticity materials springs including those where  Conservation of mechanical  Solving problems involving forces  Talking books consideration of work and energy due to elastic strings or springs energy are needed including those where consideration of work and energy are needed  Representing life phenomena using mathematical models involving elasticity and exploring their applications in life

Additional Mathematics Syllabus Forms 5 - 6 34

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 3: ENERGY WORK

TOPIC 4: CIRCULAR MOTION

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

CIRCULAR MOTION  derive the formula for speed,  Angular speed and velocity  Exploring and deriving the  ICT tools velocity and acceleration of a  Horizontal and vertical formula for speed, velocity and  Relevant texts (Vertical and Horizontal) particle moving in a circle circular motion acceleration of a particle moving  Environment  explain the concept of  Acceleration of a particle in a circle  Simple angular speed for a particle moving on a circle  Discussing the concept of pendulum moving in a circle with  Motion in a circle with angular speed for a particle  Braille materials constant speed constant speed moving in a circle with constant  Talking books  distinguish between  Centripetal force speed horizontal and vertical motion  Relation between angular  Distinguishing between the  calculate angular speed for a and linear speed concepts of horizontal and

particle moving in a circle  Conical pendulum vertical motion in a circle with constant speed  Banked tracks  Computing angular speed for a  calculate acceleration of a particle moving in a circle with

particle moving in a circle constant speed with constant speed  Calculating acceleration of a  solve problems which can be particle moving in a circle with

modelled as the motion of a constant speed particle moving in a  Solving problems which can be horizontal circle with constant modelled as the motion of a speed particle moving in a horizontal

35 Additional Mathematics Syllabus Forms 5 - 6

TOPIC 4: CIRCULAR MOTION

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

particle moving in a circle constant speed with constant speed  Calculating acceleration of a  solve problems which can be particle moving in a circle with

modelled as the motion of a constant speed particle moving in a  Solving problems which can be horizontal circle with constant modelled as the motion of a speed particle moving in a horizontal

Additional Mathematics Syllabus Forms 5 - 6 36

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 solve problems which can be and vertical circle with constant modelled as the motion of a speed particle moving in a vertical  Discussing the relationship circle with constant speed between angular and linear  find the relationship between speed angular and linear speed  Computing tension in a string  calculate tension in a string and angular speed in a conical and angular speed in a pendulum conical pendulum  Representing life phenomena  solve problems involving using mathematical models banked tracks. involving circular motion and  solve problems which can be exploring their applications in life modelled as the motion of a particle moving in a horizontal circle with constant speed using Newton’s second law

37 Additional Mathematics Syllabus Forms 5 - 6

TOPIC 5: SIMPLE HARMINIC MOTION

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

SIMPLE HARMONIC  define simple harmonic  Basic equation of simple  Discussing simple harmonic  ICT tools MOTION motion harmonic motion motion  Relevant  Properties of simple harmonic  Conducting experiments to texts

motion demonstrate simple harmonic  Environment motion using a pendulum  Pendulum  solve problems using the  Solving problems using the  Braille standard simple harmonic standard simple harmonic motion materials motion formula formula  Talking books  formulate differential  Formulating differential equations equations of motion in of motion in problems leading to problems leading to simple simple harmonic motion harmonic motion  solve differential equations  Solving differential equations involving simple harmonic involving simple harmonic motion motion to obtain the period to obtain the period and amplitude and amplitude of the motion of the motion  Representing life phenomena using mathematical models involving simple harmonic motion and exploring their applications in

Additional Mathematics Syllabus Forms 5 - 6 38

Additional Mathematics Syllabus Forms 5 - 6

STATISTICS STATISTICS TOPIC 1: PROBABILITY TOPIC 1: PROBABILITY

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

PROBABILITY  define probability key terms  Events  Discussing the importance of  ICT tools  calculate probabilities of - Independent probability in life  Relevant events - Mutually exclusive  Computing probabilities of a texts  solve problems involving - Exhaustive variety of events  Braille conditional probability - Combined  Applying conditional probability materials  use tree diagrams, Venn  Conditional concepts in solving problems  Talking books diagrams and outcome probability  Solving problems using tree tables to solve problems  Tree diagrams diagrams, Venn diagrams and  use the notations n!, nP and  Outcome tables outcome tables   𝑟𝑟 Venn diagrams Carrying out experiments 𝑛𝑛 involving probability (𝑟𝑟)  Using the notations n!, nP and 𝑟𝑟 𝑛𝑛  Permutations and  define permutations and  Explaining(𝑟𝑟) the meanings of combinations combinations permutations and combinations  solve problems involving  Solving problems involving permutations and permutations and combinations combinations

39 Additional Mathematics Syllabus Forms 5 - 6

TOPIC 2: RANDOM VARIABLES

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

RANDOM VARIABLES  define a random variable  Probability distributions  Discussing examples of random  ICT tools (discrete and  construct a probability  Expectation variables  Relevant texts continuous) distribution table  Variance  Constructing probability  Braille materials  define expectation  Probability density functions distribution tables  Talking books

 use the probability density (pdf) and cumulative  Calculating mean, mode, median, functions and cumulative distribution functions (cdf) standard deviation, variance and distribution functions to percentiles calculate probabilities  calculate mean, mode,  Mean, median, mode,  Solving problems involving mean, median, standard deviation, standard deviation and variance and standard deviation variance and percentiles percentiles  Discussing the difference  use integration to calculate between a discrete random cumulative distribution variable and a continuous function from probability random variable density function  Discussing the significance of  use differentiation to probability density function and calculate probability density cumulative distribution function of function from cumulative a continuous random variable distribution function  Computing probabilities using both pdf and cdf

Additional Mathematics Syllabus Forms 5 - 6 40

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 solve problems involving  Solving problems involving pdf probability density function and cdf

DISTRIBUTIONS  Outline the characteristics of  Binomial distribution  Discussing the characteristics of  ICT tools Binomial, Poisson and  Poisson distribution Binomial, Poisson and Normal  Relevant texts

Normal distributions  Normal distribution distributions  Environment  calculate the mean, variance  Standard normal tables  Computing the mean, variance  Braille materials and standard deviation of  Continuity correction and standard deviation of each  Talking books each distribution  Linear combinations of distribution  calculate probabilities for the normal and Poisson  Calculating probabilities using the distributions distributions probability density functions of  explain the characteristics of the distributions a normal distribution curve  Discussing the characteristics of  standardize a random a normal distribution curve, giving variable life examples  use the standard normal  Standardizing random variables tables to obtain probabilities  Obtaining probabilities using standard normal tables  approximate the binomial  Using the normal distribution using the normal distribution model to approximate the

41 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

where n is large enough to binomial distribution ensure that > 5 and >  Using the normal distribution as a

5 and apply 𝑛𝑛𝑛𝑛continuity 𝑛𝑛𝑛𝑛 model to solve problems correction  Discussing examples of linear  use the normal distribution as combinations a model to solve problems  Calculating probabilities, mean  use the following facts to and variance of a sum of two or solve problems: more independent variables for - E(aX±b) = aE(X) ± b and Poisson or normal distribution Var(aX±b) = a2Var(X)  Solving problems involving linear - E(aX±bY) = aE(X)±bE(Y) combinations and their - Var(aX±bY) = a2Var(X)+ applications in life b2Var(Y) for independent  Solving problems involving X and Y distributions  use the results that:  Representing life phenomena - If X has a normal using mathematical models distribution, then so does involving distributions and aX + b exploring their applications in life - If X and Y have independent normal distributions, then aX + bY has a normal

Additional Mathematics Syllabus Forms 5 - 6 42

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

distribution - If X and Y have independent Poisson distributions, then X + Y has a Poisson distribution  solve problems involving the distributions

43 Additional Mathematics Syllabus Forms 5 - 6

TOPIC 3: SAMPLING AND ESTIMATIONS

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

SAMPLING TECHNIQUES  distinguish between a sample  Probability sampling  Discussing the difference between  ICT tools AND ESTIMATION and a population techniques a sample and a population,  Relevant texts  distinguish between  Non-probability sampling probability sampling and non-  Environment probability sampling techniques probability sampling techniques  Braille techniques and non-  Estimation of population  Identifying and discussing materials probability sampling parameters situations in which probability and  Talking books techniques  Central limit theorem non-probability sampling methods  apply the sampling methods  Confidence intervals are used to identify representative  Carrying out sampling in a samples practical situation  calculate sample mean,  Computing sample mean, variance and standard variance and standard deviation deviation  Determining the unbiased  find the unbiased estimates of estimates of population population parameters parameters  state the Central Limit  Deriving the Central Limit Theorem Theorem  recognize that the sample  Explaining how the sample mean mean can be regarded as a can be regarded as a random random variable variable

Additional Mathematics Syllabus Forms 5 - 6 44

Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 use the Central Limit Theorem  Solving problems using the in solving problems Central Limit Theorem  identify the implications of the  Discussing the implications of the Central Limit Theorem on Central Limit Theorem small and large samples  Calculating confidence intervals  determine a confidence for population mean in cases interval for a population mean where the population is normally in cases where the population distributed with known variance or is normally distributed with where a large sample with known variance or where a unknown variance is used large sample with unknown  Computing confidence intervals variance is used for population mean in cases  determine a confidence where the population is normally interval for a population mean distributed with unknown variance in cases where the population where a small sample is used is normally distributed with  Determining from a large sample unknown variance where a an approximate confidence small sample is used interval for a population proportion  determine from a large  Solving problems involving sample an approximate sampling and estimation confidence interval for a  Conducting fieldwork population proportion

45 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 solve problems involving investigations involving sampling sampling and estimation and estimation

Additional Mathematics Syllabus Forms 5 - 6 46

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 4: STATISTICAL INFERENCE

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

HYPOTHESIS TESTING  formulate hypotheses  Null hypothesis  Discussing hypothesis testing in  ICT tools  distinguish between a type 1  Alternative hypothesis research  Relevant texts and a type 2 error  Test statistics  Calculating probabilities of  Environment  compute probabilities of  Significance level making type 1 and type 2 errors  Braille making type 1 and type 2  Hypothesis test (1-tail and 2-  Applying hypothesis tests in the materials errors tail) context of a single observation  Talking books  apply a hypothesis test in the  Type 1 and type 2 errors from a population which has context of a single  z- tests binomial distribution using either observation from a population  t – tests the binomial distribution or the which has binomial  chi-squared tests normal approximation to the distribution using either the binomial distribution binomial distribution or the  Discussing the characteristics of normal approximation to the a t and chi-squared distribution binomial distribution  apply a hypothesis test  Formulating and applying concerning population mean hypothesis tests concerning using a sample drawn from a population mean using a small normal distribution of known sample drawn from a normal variance using the normal distribution of unknown variance distribution using a t – test

47 Additional Mathematics Syllabus Forms 5 - 6

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

 describe the characteristics of  using chi-squared tests to test for a t and chi-squared independence in a contingency distribution table  apply a hypothesis test  applying chi-squared tests to concerning population mean carry out the goodness of fit using a small sample drawn analysis from a normal distribution of  Solving problems involving unknown variance using a t – hypothesis tests test  Conducting research projects  use a chi-squared test to test involving hypothesis tests for independence in a contingency table  use a chi-squared test to carry out the goodness of fit analysis  solve problems using an appropriate test

Additional Mathematics Syllabus Forms 5 - 6 48

Additional Mathematics Syllabus Forms 5 - 6

TOPIC 5: BIVARIATE DATA

SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES Knowledge)

BIVARIATE DATA  plot scatter diagrams  Scatter diagrams  Plotting and interpreting scatter  ICT tools  draw lines of best fit  Regression lines diagrams  Relevant texts  find the equations of  Least squares  Drawing lines of best fit  Environment regression lines  Pearson`s Product moment  Finding the equations of  Geo-board  calculate Pearson`s product correlation (r) regression lines  Braille materials moment correlation  Coefficient of determination  Computing Pearson`s product  Talking books coefficient (r) (r2) moment correlation coefficient  compute the coefficient of (r) determination (r2)  Interpreting the value of  solve problems involving Pearson`s product moment regression and correlation correlation coefficient  Discussing the significance of coefficient of determination  Solving problems involving regression and correlation

 Conducting investigations involving linear relationships SUBTOPIC LEARNING OBJECTIVES CONTENT SUGGESTED NOTES AND SUGGESTED  Representing life phenomena Learners should be able to: (Attitudes, Skills and ACTIVITIES RESOURCES using mathematical models Knowledge) involving bivariate data and exploring their applications in life 63

Additional Mathematics Syllabus Forms 5 - 6

9.0 ASSESSMENT

9.1 Assessment Objectives

The assessment will test candidate’s ability to:-

9.1.1 use mathematical symbols, terms and definitions appropriately 9.1.2 sketch graphs accurately 9.1.3 use appropriate formulae, algorithms and strategies to solve problems in familiar and less familiar contexts 9.1.4 solve problems in Pure Mathematics, Mechanics and Statistics systematically 9.1.5 apply mathematical reasoning and communicate mathematical ideas clearly 9.1.6 conduct mathematical proofs rigorously 9.1.7 make effective use of a variety of ICT tools in solving problems 9.1.8 construct and use appropriate mathematical models for a given life situation 9.1.9 conduct research projects including those related to enterprise 9.1.10 draw inferences through correct manipulation of data. 9.1.11 use data correctly to predict trends for planning and decision making purposes 9.1.12 construct mathematical arguments through appropriate use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions 9.1.13 evaluate mathematical models including an appreciation of the assumptions made and interpret, justify and present the result from a mathematical analysis in a form relevant to the original problem

9.2 Scheme of Assessment

Forms 5 to 6 Additional Mathematics assessment will be based on 30% Continuous Assessment and 70% Summative Assessment.The syllabus’ scheme of assessment is grounded in the principle of equalisation of opportunities hence, does not condone direct or indirect discrimination of learners. 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. NB For further details on arrangements, accommodations and modifications refer to the assessment procedure booklet. a) Continuous Assessment

Continuous Assessment for Form 5 – 6 will consists of topic tasks, written tests, end of term examinations and projects to measure soft skills. i) Topic Tasks

These are activities that teachers use in their day to day teaching. These should include practical activities, assign- ments and group work activities. ii) Written Tests

These are tests set by the teacher to assess the concepts and skills covered during a given period of up to a month. The tests should consist of short structured questions as well as long structured questions.

50 Additional Mathematics Syllabus Forms 5 - 6 iii) End of term examinations

These are comprehensive tests of the whole term’s or year’s work and can be set at school or district or provincial level. iv) Project

This should be done from term two to term five.

Summary of Continuous Assessment Tasks

From term two to five, candidates are expected to have done the following recorded tasks:

• 1 topic task per term • 2 written tests per term • 1 end of term examination per term

Term Number of Topic Number of Number of End Project Total Tasks Written Tests Of Term Exam- ination 2 1 2 1 1 3 1 2 1 4 1 2 1 5 1 2 1 Actual Weight 3% 8% 9% 10% 30%

Specification Grid for Continuous Assessment

Component Skills Topic Tasks Written Tests End of Term Project Skill 1 50% 50% 50% 20% Knowledge& Comprehension

Skill 2 40% 40% 40% 40% Application& Analysis

Skill 3 10% 10% 10% 40% Synthesis& Evaluation

Total 100% 100% 100% 100% Actual weighting 3% 8% 9% 10%

Actual 3% 8% 9% 10% Additionalweighting Mathematics Syllabus Forms 5 - 6

b. b.Summative Summative Assessment

The examinationThe examination will consist will consist of 2 papers: of 2 papers: Paper Paper1 and Paper1 and Paper2. 2.

Paper Type of Paper and Topics Marks Duration

1 Pure Mathematics 1 - 10 120 3 hours

2 Mechanics 1 – 5 120 3 hours

Statistics 1 – 6

Paper 1(120 marks) Paper 1(120 marks) Pure Mathematics– this is a paper containing about 14 compulsory questions chosen from topics 1 – 6. Pure Mathematics– this is a paper containing about 14 compulsory questions chosen from topics 1 – 6. Paper 2 (120 marks)

MechanicsPaper 2 (120and Statisticsmarks) – this is a paper containing about 14 compulsory questions chosen from topics 1 – 5 of the mechanics section and topics 1 – 5 of the Statistics section. Mechanics and Statistics – this is a paper containing about 14 compulsory questions chosen from topics 1 – 5 of the mechanics section and topics 1 – 5 of the DetailedStatistics Summative section. Assessment Table

TheDetailedThe table table below Summative below shows shows the Assessment the information information Table on on types types of ofpapers papers to tobe be offered offered and and their their weighting. weighting.

P1 P2 Total 69

Weighting 50% 50% 100%

Actual weighting 35% 35% 70%

Type of Paper PURE MATHEMATICS MECHANICS AND STATISTICS

Section A (60 Marks) Section A

About 10 compulsory short questions MECHANICS-(60 Marks)

Section B (60 Marks) About 7 compulsory questions

5 compulsory long questions Section B

STATISTICS- (60 Marks) about 7 compulsory questions

Marks 120 120

52 Specification Grid for Summative Assessment

Additional Mathematics Syllabus Forms 5 - 6

Specification Grid for Summative Assessment

P1 P2 Total

Skill 1 50% 36% 86%

Knowledge &

Comprehension

Skill 2 40% 36% 76%

Application &

Analysis

Skill 3 10% 28% 38%

Synthesis &

Evaluation

Total 100% 100% 200%

Weighting 35% 35% 70%

9.3 Assessment Model Learners will be assessed using both Continuous and Summative Assessments.

53 Additional Mathematics Syllabus Forms 5 - 6

9.3 Assessment Model

Learners will be assessed using both Continuous and Summative Assessments.

Assessment of learner performance in Additional Mathematics 100%

Continuous Assessment 30% Summative Assessment 70%

Profiling Topic Written End of Project Paper 1 Paper 2 Tasks3% Tests term 10% 35% 35% 8% Tests 9%

Profile

Continuous Assessment Mark = 30% Summative Assessment Mark = 70%

Exit Final Mark 100% Profile

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