syllabus - nie.lk maths.pdf · f o r c o m m e n t s i 1.0 introduction the aim of education is to...

For Comments

GENERAL CERTIFICATE OF EDUCATION

ADVANCED LEVEL

(Grade 12 and 13)

MATHEMATICSSYLLABUS

(Effective from 2017)

Department of MathematicsFaculty of Science and Technology

National Institute of EducationMaharagamaSRI LANKA

For Comments

Mathematics

Grade 12 and 13 - syllabus

National Institute of Education

First print 2017

ISBN :

Department of Mathematics

Faculty of Science and Technology

National Institute of Education

www.nie.lk

Printed by :

For Comments

CONTENTS

1.0 Introduction ......................................... iv

2.0 Common National Goals ......................................... v

3.0 Basic Competencies ..................................... vi

4.0 Aims of the syllabus5.0 Relationship between the common national

goals and the objectives of the syllabus. ......................................... ix

6.0 Proposed plan in the allocation of competencylevels in the syllabus into the school terms ......................................... x

7.0 Proposed term wise brekdown oif the syllabus ......................................... 1

8.0 Syllabus ......................................... 5

9.0 Teaching learning starteregies ......................................... 65

10.0 Assesment and evaluatin ......................................... 66

Mathematical symbols and notations ......................................... 67

For Comments

i

1.0 Introduction

The aim of education is to turn out creative children who would suit the modern world. To achieve this, the school curriculum should be revised according tothe needs of the time.

Thus, it had been decided to introduce a competency based syllabus in 2009. The earlier revision of the G.C.E. (Advanced Level) Combined Mathematicssyllabus was conducted in 1998. One of the main reason for the need to revise the earlier syllabus had been that in the Learning - Teaching- Assessmentprocess, competencies and competency levels had not been introduced adequately. It has been planned to change the existing syllabus that had beendesigned on a content based approach to a competency based curriculum in 2009. In 2007, the new curriculum revision which started at Grades 6 and 10 hadintroduced a competency based syllabi to Mathematics. This was continued at Grades 7 and 11 in 2008 and it will be continued to Grades 8 and 12 in 2009.Thus, a need has arisen to provide a competency based syllabus for Combined Mathematics at G.C.E.(Advanced Level) to those pupils who had followeda competency Based syllabus at Grades 10 and 11.

The student who has learnt Mathematics at Grades 6-11 under the new curriculum reforms through a competency based approach, enters grade 12 to learnCombined Mathematics at Grades 12 and 13 should be provided with abilities, skills and practical experiences for his future needs. and these have beenidentified and the new syllabus has been formulated accordingly. It is expected that all these competencies would be achieved by pupils who completelearning this subject at the end of Grade 13.

Pupils should achieve the competencies through competency levels and these are mentioned under each competency.

It also specifies the content that is needed for the pupils to achieve these competency levels. The number of periods that are needed to implement theprocess of Learning-Teaching and Assessment also mentioned in the syllabus.

Other than the facts mentioned regarding the introduction of the new curriculum, what had already been presented regarding the introduction of CombinedMathematics Syllabus earlier which are mentioned below too are valid.

• To decrease the gap between G.C.E. (Ordinary Level) Mathematics and G.C.E. (Advanced Level) Combined Mathematics.• To provide Mathematics knowledge to follow Engineering and Physical Science courses.• To provide a knowledge in Mathematics to follow Technological and other course at Tertiary level.• To provide Mathematics knowledge for commercial and other middle level employment.• To provide guidance to achieve various competencies on par with their mental activities and to show how they could be developed throughout

life.

For Comments

2.0 Common National Goals

The national system of education should assist individuals and groups to achieve major national goals that are relevant to the individual and society. Over

the years major education reports and documents in Sri Lanka have set goals that sought to meet individual and national needs. In the light of the

weaknesses manifest in contemporary educational structures and processes, the National Education Commission has identified the following set of goals

to be achieved through education within the conceptual framework of sustainable human development.

I Nation building and the establishment of a Sri Lankan identity through the promotion of national cohesion, national integrity, national unity,

harmony and peace, and recognizing cultural diversity in Sri Lanka’s plural society within a concept of respect for human dignity.

II Recognizing and conserving the best elements of the nation’s heritage while responding to the challenges of a changing world.

III Creating and supporting an environment imbued with the norms of social justice and a democratic way of life that promotes respect for human

rights, awareness of duties and obligations, and a deep and abiding concern for one another.

IV Promoting the mental and physical well-being of individuals and a sustainable life style based on respect for human values.

V Developing creativity, initiative, critical thinking, responsibility, accountability and other positive elements of a well-integrated and balance personality.

VI Human resource development by educating for productive work that enhances the quality of life of the individual and the nation and contributes

to the economic development of Sri Lanka.

VII Preparing individuals to adapt to and manage change, and to develop capacity to cope with complex and unforeseen situations in a rapidly

changing world.

VIII Fostering attitudes and skills that will contribute to securing an honourable place in the international community, based on justice, equality and

mutual respect.

For Comments

3.0 Basic Competencies

The following Basic Competencies developed through education will contribute to achieving the above National Goals.

(i) Competencies in Communication

Competencies in Communication are based on four subjects: Literacy, Numeracy, Graphics and IT proficiency.

Literacy : Listen attentively, speck clearly, read for meaning, write accurately and lucidly and communicate ideas effectively.

Numeracy : Use numbers for things, space and time, count, calculate and measure systematically.

Graphics : Make sense of line and form, express and record details, instructions and ideas with line form and color.

IT proficiency : Computeracy and the use of information and communication technologies (ICT) in learning, in the work environment and in

personal life.

(ii) Competencies relating to Personality Development

- General skills such as creativity, divergent thinking, initiative, decision making, problem solving, critical and analytical thinking, team

work, inter-personal relations, discovering and exploring;

- Values such as integrity, tolerance and respect for human dignity;

- Emotional intelligence.

(iii) Competencies relating to the Environment

These competencies relate to the environment : social, biological and physical.

Social Environment : Awareness of the national heritage, sensitivity and skills linked to being members of a plural society, concern for

distributive justice, social relationships, personal conduct, general and legal conventions, rights, responsibilities, duties and obligations.

Biological Environment : Awareness, sensitivity and skills linked to the living world, people and the ecosystem, the trees, forests, seas,

water, air and life-plant, animal and human life.

For Comments

Physical Environment : Awareness, sensitivity and skills linked to space, energy, fuels, matter, materials and their links with human living, food,

clothing, shelter, health, comfort, respiration, sleep, relaxation, rest, wastes and excretion.

Included here are skills in using tools and technologies for learning, working and living.

(iv) Competencies relating to Preparation for the World of Work.

Employment related skills to maximize their potential and to enhance their capacity

to contribute to economic development,

to discover their vocational interests ad aptitudes,

to choose a job that suits their abilities, and

to engage in a rewarding and sustainable livelihood.

(v) Competencies relating to Religion and Ethics

Assimilating and internalizing values, so that individuals may function in a manner consistent with the ethical, moral and religious modes of

conduct in everyday living, selecting that which is most appropriate.

(vi) Competencies in Play and the Use of Leisure

Pleasure, joy, emotions and such human experiences as expressed through aesthetics, literature, play, sports and athletics, leisure pursuits

and other creative modes of living.

(vii) Competencies relating to ‘ learning to learn’

Empowering individuals to learn independently and to be sensitive and successful in responding to and managing change through a transformative

process, in a rapidly changing, complex and interdependent world.

For Comments

4.0 Aims of the syllabus

(i) To provide basic skills of mathematics to continue higher studies in mathematics.

(ii) To provide the students experience on strategies of solving mathematical problems.

(iii) To improve the students knowledge of logical thinking in mathematics.

(iv) To motivate the students to learn mathematics.

This syllabus was prepared to achieve the above objectives through learning mathematics. It is expected not only to improve the knowledge of mathematicsbut also to improve the skill of applying the knowledge of mathematics in their day to day life and character development through this new syllabus.

When we implement this competency Based Syllabus in the learning - teaching process.

• Meaningful Discovery situations provided would lead to learning that would be more student centred.

• It will provide competencies according to the level of the students.

• Teacher's targets will be more specific.

• Teacher can provide necessary feed back as he/she is able to identify the student's levels of achieving each competency level.

• Teacher can play a transformation role by being away from other traditional teaching methods.

When this syllabus is implemented in the classroom the teacher should be able to create new teaching techniques by relating to various situations undergiven topics according to the current needs.

For the teachers it would be easy to assess and evaluate the achievement levels of students as it will facilitate to do activities on each competency levelin the learning- teaching process.

In this syllabus, the sections given below are helpful in the teaching - learning process of Combined Mathematics.

1

5.0 Proposed Term Breakdown of the SyllabusGrade - 12

Competency Level Content Number of Periods

Mathematics -I1.1, 1.2,1.3 Real number system 142.1,2.2 Algebra of sets 123.1 Mathematical logic 105.1, 5.2 One variable functions 206.1, 6.2, 6.3, 6.4 Polynomials 38

Mathematics -II1.1, 1.2 Basics of Statistics 062.1, 2.2,2.3,2.4 Data,data representation 10

Mathematics -I12.1, 12.2, 12.3, 12.4,12.5 Straight lines 307.1, 7.2 Rational functions and logarithm 304.1 Methods of proofs 12

Mathematics -II3.1,3.2 Measures of central tendency 24

Mathematics -I8.1, 8.2, 8.3 Inequalities 2411.1 Limits 08

Mathematics -II3.3, 3.4,3.5, 3.6, 3.7 Frequency distribution 234.1, 4.2 Random experiments and probability 186.1, 6.2 Permutation and combination 24

Term II

Term I

Term III

For Comments

Competency Level Content No. of Periods

Mathematics -I13.3, 13.4,13.5,13.6, 13.7 Derivatives 40

Mathematics -II4.3, 4.4,4.5, 4.6, 4.7,4.8.4.9 Probability 66

Mathematics -I

14.1, 14.2, 14.3, 14.4, 14.5,14.6.14.7 ` Integration 40

Mathematics -II4.10 Discreate probality distribution 145.1, 5.2 Linear programing 258.1,8.2 Deterninants 109.1, 9.2 Matrices 20

Mathematics -I9.1, 9.2 Binomial expansion 1610.1, 10.2, 10.3 Series 26

Mathematics -II4.11 Continuous probablity distribution 157.1,7.2 Networks 25

Term I

Term III

Proposed Termwise breakdown of the syllabus - Grade - 13

2

Term II

For Comments

Mathematics I 94

Mathematics II 16 110

M athematics I 72

M athematics I I 24 96

M athematics I 32


M athematics I I 40

M athematics I 66 106

M athematics I 40


M athematics I 42


Total 600

Subject Number of Periods Total

Second Term

First Term

Third Term

Fourth Term

Fifth Term

Sixth Term

For Comments

4

Competency Competency Level Contents Learning outcomesNo. of

Periods

1. Analysesthe RealNumbersystem

1.1 Classifies the Real numbersystem

1.2 Uses surds and decimals torepresens real numbers

1.3 Uses Exponents (indices) andradicals to communicatenumbers

Review of the of the real numbersystem

Notations for sets of numbers

Geometrical representation of a realnumber

Decimal representation of a realnumberÚ Finite decimalsÚ Infinite decimalsÚ Recurring decimalsÚ Surds

Positive integral Exponents (indices)

Negative integral and zero exponents

Fractional exponents

Rationalizing denominantors

04

04

06

6.0 SyllabusMathematics - I

Writes correct notations for sets ofnumbers

Represents real numbersgeometrically

Classifies decimal numbers

Classifies real numbers

Writes, conjugate of a surd

Rationalises the denominator ofexpressions with surds

Uses arithmetical operation on surds

Defines indicies.

Classifies positive integral exponents,negative integral exponets, zeroexponents and fractinal exponents

Applies laws of indicies to variousproblems

For Comments

5


Periods

2.1 Applies basics of sets tosolve problems

2.2 Uses Venn diagrams andalgebra of sets to solveproblems

3.1 Identify statements

Language of sets and elements of a setÚ Universal set, Null set, Finite set

and Infinite set, Cardinality of a setÚ Subsets, proper subsets,equality of

two sets, and Power set

Set operationsÚ Intersection and unionÚ Complement, relative complementÚ Set identities

The formula

Statements Logical connectives and compound

statements Truth tables Conditional statements Quantifiers Compound statements

Ú Construction of the truth table Logical equivalents

PredicatesÚ Symbolization of Predicates

2. Manipu-lates Alge-bra of Sets

3. Manipu-lates Math-ematicalLogic

Explains set notations Explains and writes the notations of

universal set and null set Explains finite sets and infinite sets Explains cardinality of a set and writes

its notation Defines, Cardinality of set & writes its

notation Defines subsets, proper sub sets

equality of two sets and power set

States the set operations also withvenn diagrams

Writes the formula of set identitiesexplains

Identifies the different types ofstatements

Writes the definition of all types ofstatements

Writes the constructions of truth table

States the definitions of logicalequivalents and predicates of a event

Writes the Symbolization of Predicates

06

06

10

For Comments

6


Periods

4. Manipulatesthe methodsof proofs toprove themathemati-cal results

5. Analysesfunctions ofa realvariable

4.1 Proves Mathematicalresults by using directproof, proof bycontradiction and proofby mathematical induction

5.1 Investigates functions

Methods of proofÚ Direct proofÚ Proof by contradictionÚ Proof by Mathematical induction

Intiutive idea of a function as one-one or many-one relationÚ Domain

Graph of a function, vertical line testfor a function

Elementry functions ,f x ax b

2, f x x f x x ,

1

, ( 0)f x xx

, 0f x x x ,

1

, ( 0)f x xx

Graphing functions using translations(shifting)

States type of proof

Describes direct proof, proof bycontradiction and proof bymathematical induction

Explains the Intiutive idea of afunction

Explains the domain of a function

Recognizes special functions

Sketches functions

Sketches functions using translations(shifting)

12

10

For Comments

7


Periods

6. Analysespolynomials

5.2 Investigates operations offunctions

6.1 Investigates polynomials

6.2 Uses mathematicaloperation involvingpolynomials

Basic operation +, -, , and Composite functions Inverse functions

Ú Definition of inverse function Ú Finding the inverse function

Polynomials in one variableÚ Degree, leading term and leading

coefficientÚ Equality of two polynomials

Mathematical operations involvingpolynomialsÚ Addition, subtractionÚ MultiplicationÚ Division, long divisionÚ Synthetic division by a linear

expressionÚ Remainder theoremÚ Factor theorem

performs basic operations Defines composite functions Writes the notations for composite

functions States the definition of inverse

function and finds the inverse function

Defines a polynomial in a singlevariable

States the condition for twopolynomials to be equal

Explain the basic mathmaticaloperations on polynomilas

Divides a polynomial by anotherpolynomial

States the remainder theorem Proves the remainder theorem States the factor theorem Proves the factor theorem Expresses converse of the factor

theorem Solves the problems using remainder

theorem and factor theoram. Solves the polynomials equations (up

to 4th order) Defines zeros of a polynomials

10

15

10

For Comments

8


Periods

6.3 Investigates quadraticfunctions and their properties

6.4 Investigates the quadraticequations

Quadratic functionsÚ Completing the squareÚ DiscriminantÚ Least and greatest valueÚ Drawing the graph

Applications of quadraric functions

Quadratic equationÚ Solution by completion of

squareÚ Graphical solution

Use of quadratic formula

Ú Discriminant Analysis of roots

Ú Real and distinctÚ Real and coincidentÚ Not real

Solution of simultaneouss equations,of one equation is linear in twovariables and the other equation quadratic in one or both variables

Introduces linear functions

Explains quadratic functions

Explains the properties of aquadratic function

Sketches the graph of a quadraticfunction

Describes different types of graphof the quardratic functions

Explains the roots and of a

quadratic equations 2 0ax bx c

Finds the roots of a quadraticequations

Describes the nature of the roots ofa quadratic equations

Expreses the sum and products ofthe roots of quadratic equationsinform of its coefficients

contructs quadratic equations whereroots are symmetric functions of and

Solves problems involving quadraticfunctions and quadratic equations

10

03

For Comments

9


Periods

7.1 Resolution of a rationalfunction into partialfractions

7.2 Analyses the Exponentialand Logarithmic functions

Rational functionsÚ Proper rational functionsÚ Improper rational functions

Partial fractionsÚ Partial fractions of proper

rational functions- with distinct factors in the

denomiator- with recurring fators in the

denominatorÚ Partial fractions of improper

rational functions

Exponential function and itspropertiesÚ Graphs of growth and decay of

propulationÚ definition of eÚ Properties of xeÚ Graph of xe

Logarithmic function and itspropertiesÚ Properties of lnxÚ Change of baseÚ Graph of lnx

Examples on compound Interest,Population growth etc.

7.Investigatesrationalfunctions

Defines rational functions

Defines proper rational expressionsand inpropper rational expressions

Defines rational functions

Finds patial fractions of rationalexpressions(not more than 4 unknowns areexpected)

states the properties of exponentialfunctions

sketches graph of exponentialfunctions

States the properties of xe anddraws its graph

States the properties of lnx

Writes change of base of alogarithmic function

Draws the graph of lnx

Compares the relations betweenlnx and

xe

Finds the compound Interest,population growth, using properequation.

15

15

For Comments

10


Periods

8.1 Solves problems involvinglinear and quadratic in-equalities

8.2 Solves quadratic inqualitesusing graphical methed

8.3 Solves inequalities involvingrational functions.

9.1 Describes the basic prop-erties of the binomialexpansion.

9.2 Applies binomial theorem

inequalities

Linear inequalities

Quadratic inequalities

Simultaneous linear inequalities byusing graphs

quadratic inequalities using graphs

• Inequalities of the form (x)

(x)

f

g

where f(x),g(x) are polynomials of x

(degree 3 ) and g(x) 0

Expansion of 1n

x

expressing the coefficients in the

form nrC

Application of 1n

x

Expansion of

0

nn n n r r

rr

a x C a x

Application of Binomial theorem

8.Manupulatesinequalities.

9. Exploresthe binomialexpansionfor positiveintegralindices.

Solves linear and quadraticinequalities

Solves Simultaneous inequalities

Solves quadratic and simultaneousinequalities using graphs

Solves in inequalities of the form

(x)

(x)

f

g

where f(x),g(x) are polynomials of x

(degree 3 ) and g(x) 0

Defines

Expands (1 )nx using binomialtheorem• Writes the general term in the

expansion of (1 )nx

Expands ( )na x using binomialtheorem• Writes the general term in the

expansion of ( )na x

n

a x

10

06

08

08

08

For Comments

11


Periods

Series

Summation,General term

Sigma notation

1 1

n n

r rr r

kU k U

1 1 1

n n n

r r r rr r r

U V U V

and their applications

10. Finds thesum of thefinite series

10.1 Describes finite series andtheir properties.

10.2 Solves problems involvingarithmetic and geometricseries.

10.3 finds sums of elementaryseries

Finds the sum of arithmetic andgeometric series by using sigmanotation

Writes the series using notation

and find the sum

Applications of of arithmetic andgeometric series by using sigmanotation

Proves and uses the formulae by principal of Mathematical Induction for values

2 3

1 1 1

, ,n n n

r r r

r r r

Applies the above ormulae to find the sumation of series

2 3

1 1 1

, ,n n n

r r r

r r r

08

08

10For Comments

12


Periods

11.1 Interpretes the limit of afunction.Solves problems using thetheorems on limits.

Uses the limit;

1limn n

n

x a

x ana

x a

to

solve problems

11.Diterminesthe limit ofthe function • Intuitive idea of a limit

• Theorems on limits

• Proof of 1lim

n nn

x a

x ana

x a

,

• States the intutive idea of a limit andtheorem on limits

• Proves 1lim

n nn

x a

x ana

x a

where n is a rational number

• applies the above theorem

08

For Comments

13


Periods

12. Investi-gates astraight linein terms ofcartesianco-ordinates

Plots points on a Cartesian plane

Writes the formulae for the distancein between two points in aCartesian plane

Writes the co-ordinaties of a pointwhich divides a joining two givenpoints at a given ratio

Finds the area of a triangle when thevertices are given

Finds the gradiant of a straight line

Finds x-intercept and y-intercept ofa straight line

01

06

06

Rectangular Cartesian co-ordinatesÚ Co-ordinate axes, origin of

co-ordinates, Quadrants,abcissa, ordinate

Ú Distance between two pointsÚ Co-ordinates of the point

dividing the straight line segmentjoining two points in a givenratio

Ú Area of a triangle with givenvertices

Straight lineÚ Inclination and gradient of a

straight line (for straight lines notparallel to the x axis)

Ú x-intercept, y-intercept of astraight line

12.1 Describes the rectanqularCartesion coordinatesystems

12.2 Finds the distancebetween two points andthe area of a triangle

12.3 Describes the equation ofa straight line

For Comments

14


Periods

Obtains equation of straight linesaccording to the data given

(i) Point - Gradient formy-y,=m (x-x)

(ii) Gradient Intercept formy=mx+c

(iii) Two point form 1 1

1 2 1 2

y y x x

y y x x

(iv) Intercept form

1 1

1 2 1 2

y y x x

y y x x

(v) General formax+by+c=0

Finds the coordinate of the point ofintersection of two non parallel lines

Interprets and usees the equation

0U V

12.4 Interprets the equation of astraight line

12.5 Derives the equation of astraight line passingthrough the point ofintersection of two givenstraight lines

Different forms of the equation of astraight lineÚ Point - gradient form

1 1y y m x x

Ú Gradient-Intercept form

y mx c

Ú Two point form

2 11 1

2 1

y yy y x x

x x

Ú Intercept form 1x y

a b

Ú General form 0ax by c

Ú Interpretation of the general

form when 0i a , 0ii b ,

0iii c

Point of intersection of two straightlines

Interpretation of the equation

0U V , where U =0 and V =0

are the equations of two intersectingstraight lines

12

05

For Comments

15


Periods

13.Uses thederivatives ofa function tosolveproblems.

13.1 Interprets the derivative ofa function

13.2 Finds the derivatives ofpoloynomials, exponentialand logarithmic functions.

13.3 Uses the formulae for thederivative of the sum,product and the quotientof two function

13.4 Uses the chain Rule tofind the derivative

Definition of the derivative of )(xf

at a point 0x as

0 0

0

( ) ( )limx

f x x f x

x

Limiting position of the line joiningtwo points on a curve as the tangentto the curveÚ Slope of the tangent line

Rate of change as a derivative

Derivatives of the functions

xex xn ln,,

Rules for finding the derivatives of asum, product and quotient of twofunctions and their applications

Chain rule for composition offunctions

Defines the derivative at a point

Obtains the slope of a tangent line ofa point on a curve

Describes rate of change as aderivative

Applies rate of change

Obtains the formulae

1 1( ) , (ln )n nd dx nx x

dx dn x

Derives formulae for sum, productand quotient of two functions andapplies to differentiable functions

Applies chain rule to find thederivative of a composite function

04

06

05

06

For Comments

16


Periods

13.5 Determines the behaviourof a function usingderivatives

13.6 Sketches simple curvesusing derivatives

13.7 Uses derivatives to solveproblems involving relatedrates

Increasing functions, Decreasingfunctions .

Stationary points of functions

Local maximum and local minimum

Sketching simple curves usingderivatives (horizontal and verticalasymptotes)

Related rates

Describes increasing and decreasingfunction by using differentiation

Finds stationarypoints

Finds local maximum and localminimum

Draws simple curves usingderivative

States vertical and horizontalasymptotes

Solves problems involving related rates

04

07

08

For Comments

17


Periods

14.Find indefi-nite anddefiniteintegers offunctions

Finds intergrals using results ofderivative

Uses the theorems on integration

Solves integral problems usingstandard results

Observes the function and variousmethods according to the data

States the funddermental theorem ofcalculus

Finds the values of definite integrals

Uses the properties of definiteintegrals

14.1 Identifies Intergtation asthe reverse process ofdifferentiation (anti-derivative of function)

14.2 Identifies integrals ofstandard functions

and results of intergtation

14.3 Determines definiteintegrals using thefundermental theorem ofcalculus

Antiderivatives and indefiniteIntegrails

Properties of intergrals

Ú dxxgdxxfdxxgxf )()()()(

Ú dxxfdxxf )()(

Integrals of standard functions

,n xx e

Standard results in intergtation

1

/

1

nn f x

f x f x dx cn

'( )ln ( )

( )

f xdx f x c

f x

Definite integrals

Ú b

a

dxxf )( notation

04

06

06

For Comments

18


Periods

Solves problems using partialfractions

Uses integration by parts to integratesuitable problems

Uses difinite integral to find areaunder curve

Uses difinite integral to find areabetween two curves

Solves problems by usingÚ Trapezium ruleÚ Simpson’s rule

14.4 Uses different methods forintegration

14.5 Integration using themethod of integration byparts

14.6 Determines the area of aregion bounded bycurves using integration

14.7 Uses method of approxi-mation to solve problems

Integration of rational functionsusing partial fractions

vduuvudv

Uses of integrationsÚ Area under a curveÚ Area between two curves

Numerical integration usingtrapezium rule and Simpson’s rule

04

04

08

08For Comments

19


Periods

1.1 Investigates the nature ofstatistics

1.2 Manipulates data toobtain information

Introduction to of statistics

Nature of statisticsÚ Descriptive statisticsÚ Inferential statisticsÚ Probability and Distribution

TheoryÚ Connection between

diescriptive, inferential andprobability theory

Ú Application of Statistics

Data and Information

Experiments and Data

Controlled Experiments and surveys

Types of DataÚ Discrete dataÚ Continuous data

Information

Distinction between data andinformation

03

03

1. Interpretsthe basics ofstatistics

Mathematics - II

Explains what is statistics and nature

Describes probability an distributiontheory

Explains the types of data

Describes the difference betweendata and information

For Comments

20


Periods

2. Presentsdata andinformationsystemati-cally

2.1 Classifies data andinformation.

2.2 Tabulates data andinformation

2.3 Represents data andinformation using charts

Techniques of presenting data

Classification of dataÚ Classification of data as a

process of arranging objectsÚ Aims of classificationÚ Basis of classification.

Techniques of tabulationÚ Construction of a frequency

tableÚ Ungrouped frequency

distributionÚ Grouped frequency distributionÚ Construction of two way tablesÚ Importance of tabulation

TechniquesÚ Importance of chartsÚ Limits and rulesÚ Geometrical formsÚ Bar chartsÚ Types of bar chartsÚ Pie chartsÚ Maps and graphs

02

02

03

Classifies data

States aims and basic ofclassification of data

Tabulates the data

States the presentation techniquesof data

Uses charts to represent dataFor Comments

21


Periods

2.4 Represents data andinformation graphically

3.1 Analyses mean as ameasure of central ten-dency

3.2 Interprets the frequencydistribution in terms ofvalues of relative positions

Graphical techniqes (lines and curveforms)Ú Line graphsÚ Line graphs for more than one

variableÚ HistogramÚ Frequency polygonÚ Smooth frequency curvesÚ Ogives or cumulative frequency

curvesÚ Cumulative percentile curves.

Mean for classified & unclassifieddataÚ weighted meanÚ geometric mean

Measures of relative positions of afrequency distribution.Ú MedianÚ quartilesÚ deciles and percentiles

3. Interpretsthebehaviourof a fre-quencydistribution

03

10

14

Uses methods of graphical data

Finds the central tendencymeasurement

Finds the relative position offrequency distribution

For Comments

22


Periods

3.3 Analyses mode as ameasure of cental ten-dency

3.4 Uses suitable measures ofcentral tendency to reachdecisions on frequencydistributions.

3.5 Interprets the dispersionof a distribution usingmeasures of deviation

3.6 Interprets coefficient ofvariation as a measures ofdispersion.

Mode of a frequency distribution

Relative importance of measures ofcentral tendency

Measures of dispersion

Importance of measures ofdispersionTypes of dispersionÚ RangeÚ Inter quartile rangeÚ semi interquartile range

(Quartile deviation)Ú Mean deviationÚ VarianceÚ Standard deviation

Coefficient of variation

04

04

10

03

Finds the mode as a measure ofcentral tendancy

States the relative importance ofmeasure of central tendency

Uses suitable measure of dispersionto make decision on frequencydistribution

States the measures of dispersionand their importancy

Explains coefficient of variation andsolves problems

For Comments

23


Periods

4. Analysesrandomphenomenamathemati-cally

3.7 Decides on the shape of adistribution using measuresof skewness

4.1 Determines the events of arandom experiment

Measures of skewess Ú Kal Pearson’s measure of

skewness

Ú 1

mean-modesk

SD

Ú

2

3 mean mediansk

SD

Experiments and eventsÚ Types of experimentsÚ Deterministic experimentsÚ Non-deterministic or random

experimentsÚ Possible outcomes of an

experimentÚ Sample space of an experiment

EventsÚ DefinitionÚ Simple event, composite events,

null event. complementary events,Ú Union of two events, intersec-

tion of two eventsÚ Mutually exclusive eventsÚ Exhaustive eventsÚ Equally probable eventsÚ Event space

02

08

Defines the measure of skewness

Staes relationship between mean,median,and mode

Explains random experiments

Defines sample space and samplepoints

Defines an events

Explains types of events

Classifies the events and finds unionand untersections

For Comments

24


Periods

4.2 Interprets probability

4.3 Decides the possibility ofan event in terms ofconditional Probability

4.4 Interprets the indepen-dence of two randomevents

4.5 Uses Baye’s Theorem asa derivative of the TotalProbability Theorem

Classical definition of probability

Statistical definition of probability

Axiomatic definition of probability

Laws of probability

Laws of ProbabilityÚ P(A)=P(A B)+P(A B’Ú P(A B)=P(A)+P(B)-PAA B)

Conditional probabilityÚ DefinitionÚ Conditional Probability

outcomesÚ Chain ruleÚ Chain rule for two eventsÚ Extension of the chain rule for

more than two events

Independent events

Independence of several events

Partition of the sample space

Total probability

Baye’s Theorem

10

08

04

08

States classical definition and itslimitation

States the axiomatic definition

Proves the theorems in probabilityusing axiomatic definition and solvesproblems using the above theorem

Defines conditional probability

States and proves the theorems onconditional probability

States chain rule and its extension

Uses independence of two or threeevents

Defines partition of a sample spaceProves and uses the theorem ontotal probability

States bay’s theorem and applies itin problems

For Comments

25


Periods

4.6 Interprets random vari-ables

4.7 Analyses the propertiesof a probability distribu-tion of a continuous and adicrete random variable

4.8 Interprets the mathemati-cal expectation of arandom variable

4.9 Determines the cumula-tive distribution functionof a random variable

4.10 Constructs models forspecial discreteprobability distributions,calculates probability andinterpret it

4.11 Claculates probabilityusing theoritical modelsand interprets the densityfunctions of specialcontinuous distribution

Possible values of a random variable Discrete and continuous random

variables

Probability distribution of a discreterandom variable

Probability density function of acontinuous random variable

Mathematical expectationÚ MeanÚ Variance

Cumulative distribution function of arandom variable

Discrete probability distributionsÚ Bernoullis distributionÚ Discrete uniform distributionÚ Binomial distributionÚ Poisson distribution

Continuous distributionÚ Uniform distributionÚ Exponential distributionÚ Normal and standard normal

distributions

Defines random variable Defines discrete and continuous

random variables

States possible values of randomvariable

Defines mathematical expectation(Mean & Varience)

Defines Cumulative distribution of afunction

States Bernoullis distributionDiscrete uniform distribution,Binomial distribution

Uses athe above in problems

States uniform distributions,exponential distributions, normal andstandard normal distributions

Uses the above in problems

02

12

12

20

14

15

For Comments

26


Periods

5. Determinesthe optimumsolution of alinearprogram-ming prob-lem

5.1 Constructs a linear Pro-gramming model

5.2 Determines the solution of alinear programming problemgraphically

Linear Programming Construction of a linear

programming modelÚ Decision variableÚ Objective functionÚ Notation in standard formÚ ConstraintsÚ Non-negative conditions

Linear programming graphicalsolutionÚ Feasible regionÚ Feasible regionÚ Solution of a maximising modelÚ Solution of a minimising model

Problems withÚ unfeasible solutionÚ Single solutionÚ Multiple solutions problems

Networks and their application Networks and definition of terms Applications of networks

Ú Critical pathÚ miniming spanning problemsÚ High flow problems Planning of projects and ritical

path analysis

Inteprets the linear model into aninequality and draw garphs.Investigates characteristic

Identifies the feasible andunfeasible region.

Finds the solutions of maximisingmodel and minimising model.Obtains unfeasible solution, singlesolutions, and multiple solutions inproblems.

Defines networks and theirapplication

10

15

For Comments

27


Periods

6.1 Uses Permutations as atechnique of solving math-ematical problems

6.2 Uses combinations as atechnique of solving Math-ematical problems

• Fundamental principle of counting• Clarification by examples• Factorial notation• The number of permutations of n

distinct objects taken ( n)r at a time

• Symbol nrP

• The number of permutations of

n objectsnot all distinct

Concept of combination Thenumber of combination of n

distinct object taken r at a time. • symbol and formula, problems with

particular nrC values for n and r

are considered, where

!

! !n

r

nC

r n r

Properties of nrC

Ú 1n no nC C and

Ún n

r n rC C

Ú1

1n n n

r r rC C C

10

14

6. Uses Per-mutationand combi-nation tosolveMathemati-cal prob-lems

• Explains the fundamental principle’sof counting defines the factorial andstates the recursive relation forfactorial.

• Defines nrP and obtains the formulae

for nrP

• Finds the permutation of n objectsdifferent

Defines nCr and finds a formulae for

nCr

Defines combination.

Explains the distinction betweenpermutations and combinations.

Applies the formulae to relatedproblems.

Writes the properties of nCr

For Comments

28


Periods

7. Analysesprojects byusingnetworks

7.1 Describe networks

7.2 Solves problems by usingnetwork

Networks and its applications

Definition of Networks and itsterminology

Critical path

Reducing spanning problem

Maximum flow

Planning projects and critical path

Defines network and applies it inproblems Identifies matrices

Uses critical path and maximum flow ina project

10

15

For Comments

29


Periods

8. Manipu-lates Deter-minants as amathemati-cal model ofsolvingproblems

8.1 Interprets the propertiesof determinants of ordertwo and three

8.1 Solve equations by usingtwo or three variable

Introduction : By solution of twosimultaneous equations with twounknowns.

PropertiesÚ Interchange of two rows or two

columns changes the sign of thedeterminant

Ú If two rows or two columns ofa determinant are identical thedeterminant becomes zero

Ú If k is common factor of all theelements in a row or a column,then k is a common factor ofthe determinant

a b p q

p q a b Identifies matrices

ax by c

px qy r

Expresses the above equation itsfollowing

a b x c

p q y r

x a b e

y p q r

04

06

For Comments

30


Periods

9. Manipu-lates MatrixAlgebra

9.1 Describes Matrix Algebra

9.2 Investigates the propertiesof square matrices

Definition and notation of ofmatrices

Elements rows, cloumns size ofMatrix row matrix column matrix.

Matrix additionÚ Conformable for additionÚ Commutative and associative

laws for addition

Scalar multiplication

Distributive law for addition overscalar multiplication.

Matrix multiplicationÚ Compatibility of matrices for

multiplicationÚ Illustrating that matrix

multiplication is notcommutative

Square MatricesÚ Unit matrixÚ Diagonal matrix

08

12

Identifies matrices

Writes the order of a matrix

Describes the matrix which arecompatible for addition andmultiplication

Applies disribution laws for additionand multiplication

Verifies square matrices by using thedefinition

Verifies the AB BA for any twomatrices

Defines the unit anddiagonal matrices

For Comments

31


Periods

Algebra of square matricesÚ Associativity of maxtrix

multiplication (AB)C=A(BC)Ú Distributivity of matrix

multiplication over additionA ( B + C ) = AB + AC( B + C ) A = BA + CA

Ú IA = A = AI, where I is the unitmatrix of order same as A

Ú When f(x) is a polynomial in xcomputation of f(A)

Transpose.

T T TA+B =A +B

TTA =A

T TA = Ak k , Where k is a scalar..

T T TAB =B A

If A, B, C are matrix of same orderverifies that the A(B+C) = AB +ACand (B+C) A=BA+CA (Associativity) (Distribution)

Describes the identity matrix andverifies AI=IA=A for all matrix hereA and I are with same order

States what is tranpose of a matrixand verifies all properties regardingtransposeFor Comments

7.0 TEACHING LEARNING STRATEGIES

To facilitate the students to achieve the anticipated outcome of this course, a variety of teaching stategies must be employed. If students are to improve

their mathematical communication, for example, they must have the opportunity to discuss interpretations, solution, explanations etc. with other students

as well as their teacher. They should be encouraged to communicate not only in writing but orally, and to use diagrams as well as numerial, symbolic and

word statements in their explanations.

Students learn in a multitude of ways. Students can be mainly visual, auditory or kinesthetic learners, or employ a variety of senses when learning. The

range of learning styles in influenced by many factors, each of which needs to be considered in determining the most appropriate teaching strategies.

Research suggests that the cltural and social background has a significant impact on the way students learn mathematics. These differences need to be

recognised and a variety of teaching strategies to be employed so that all students have equal access to the development of mathematical knowledge and

skills.

Learning can occur within a large group where the class is taught as a whole and also within a small group where students interact with other members

of the group, or at an individual level where a student interacts with the teacher or another student, or works independently. All arrangements have theirFor Comments

8.0 SCHOOL POLICY AND PROGRAMMES

To make learning of Mathematics meaningful and relevant to the students classroom work ought not to be based purely on the development ofknowledge and skills but also should encompass areas like communication, connection, reasoning and problem solving. The latter four aims, ensure the

enhancement of the thinking and behavioural process of childern.

For this purpose apart from normal classroom teaching the following co-curricular activities will provide the opportunity for participation of every child

in the learning process.

• Student’s study circles• Mathematical Societies• Mathematical camps• Contests (national and international)• Use of the library• The classroom wall Bulletin• Mathematical laboratory• Activity room• Collectin historical data regarding mathematics• Use of multimedia• Projects

It is the responsibility of the mathematics teacher to organise the above activities according to the facilities available. When organising these activities theteacher and the students can obtain the assistance of relevant outside persons and institution.

In order to organise such activites on a regular basis it is essential that each school develops a policy of its own in respect of Mathematics. This wouldform a part of the overall school policy to be developed by each school. In developin the policy, in respect of Mathematics, the school should takecognisance of the physical environment of the school and neighbourhood, the needs and concerns of the students and the community associated with theschool and the services of resource personnel and institutions to which the school has access.

The school should develop ists annual programmes, consisting of a variety of activities for achieving policy goals. In determining the activities to beundertaken during a particular year, the school will need to identify priorities and consider feasibility in relation to time and resource constraints.However, the school could organise a range of activities adequate to cater to the development of the variety of interests and aptitudes of differentstudents.

For Comments

9.0 ASSESSMENT AND EVALUATION

It is intended to implement this syllabus in schools with the School Based Assessment (SBA) process. Teachers will prepare creative teaching - learninginstruments on the basis of school terms.

The First Examination under this syllabus will be held in 2019.

For Comments

1. Set Notations

an element

not an element

1 2{ , ...}x x the set with elements 1 2, ...x x

/ ...x or : ...x the set of all x such that...

(A)n the number of elements in set A

empty set

universal set

A / the complement of the set A

the set of natural numbers, {1, 2, 3,...}

the set of integers 0, ±1, ±2, ±3, ... the set of positive integers {1, 2, 3, . . . }

the set of rational numbers

the set of real numbers

the set of complex numbers

a subset

a proper subset

M not subset

not a proper subset

union

intersection

],[ ba the colsed interval { : }x R a x b

],( ba the interval }:{ bxaRx

),[ ba the interval }:{ bxaRx

),( ba the open interval }:{ bxaRx

MATHEMATICAL SYMBOLS AND NOTATIONS

The following Mathematical notation will be used.

2. Miscellaneous Symbols

equal

not equal

identical or congruent

approximately equal

proportional

less than

Ÿ less than or equal

greater than

¦ greater than

infinity

if

if and only if ( iff )

For Comments

3. Operations

ba a plus b

ba a minus b

,a b a b a multipllied by b

,a

a bb

a divided by b

ba : the ratio between a and b

1

n

ii

a 1 2 . . . na a a

a the positive square root of the real number a

a the modulus of the real number a

!n n factorial for 0n

!

( )!n

r

nP

n r

0 r n n , {0}r

!,0

!( )!n

r

nC r n

r n r

n , {0}r

4. Functions

)(xf the functin f of x

f : AB f is a function under which each element of

set A has an Image in set B

yxf : the function f maps the element x to the elementy

1f the inverse the function f

g f

z

the composite function of f and g which is

defined by g f x

lim ( )x a

f x

the limit of f x as x tends to a

x an increment of x

dy

dxthe derivative of y with respect to x

n

n

d y

dxthen

thn derivative of y with respect to x

(1) (2), , ..., nf x f x f x

the first, second ,..., nth derivatives of f x

with respect to x

ydx indefinite integral of y with respect to x

b

aydx definite integral of y w.r.t x in the interval a x b

, , ...x x the first, second,... derivative of x with respect to time

For Comments

5. Exponential and Logarithmic Functions

xe exponential function of x

alog x logarithm to the base a of x

ln x natural logarithm of x

lg x logarithm of x to base 10

6. Matrices

M a matrix M

TM the transpose of the matrix M

-1M the inverse of the matrix M

det M the determinant of the matrix M

7. Probability and Statistics

A, B, C ect.. events

A B union of the events A and B

A B intersection of the events A and B

P(A) probability of the event A

A / complement of the event A

(A B)P x probability of the event A given the event B

X, Y, R, ... random variables

, , , ...x y r ect. values of the random variables X, Y, R etc.

1 2, , ...x x observations

f1, f

2, ... frequencies with which the observations

1 2, , ...x x occur

1 21

U A A A ... An

i ni

1 21A A A ... A

n

i ni

x Mean

2 Variance

/ S / SD Standard deviation

For Comments

8. Probability and Statistics

A, B, C etc .. events

A B union of the events A and B

A B intersection of the events A and B

P(A) probability of the event A

A / complement of the event A

(A B)P x probability of the event A given that event B is

happened

X, Y, R, ... random variables

, , , ...x y r ect. values of the random variables X, Y, R etc.

1 2, , ...x x observations

f1, f

2, ... frequencies with which the observations

1 2, , ...x x occur

1 21

U A A A ... An

i ni

1 21A A A ... A

n

i ni

x arithmetric mean2 variance

/ S / SD standard deviation

For Comments

syllabus - nie.lk maths.pdf · f o r c o m m e n t s i 1.0 introduction the aim of education is to...

Documents