ministry of education · republic of namibia ministry of education namibia senior secondary...

Republic of Namibia

MINISTRY OF EDUCATION

NAMIBIA SENIOR SECONDARY CERTIFICATE (NSSC)

FOR IMPLEMENTATION IN 2006 FOR FIRST EXAMINATION IN 2007

DEVELOPED IN COLLABORATION WITH UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS

MATHEMATICS SYLLABUS

HIGHER LEVEL

SYLLABUS CODE: 8323

GRADES 11 - 12

Republic of Namibia

MINISTRY OF EDUCATION

NAMIBIA SENIOR SECONDARY CERTIFICATE (NSSC)

MATHEMATICS SYLLABUS

HIGHER LEVEL

SYLLABUS CODE: 8323

GRADES 11 - 12

Ministry of Education National Institute for Educational Development (NIED) Private Bag 2034 Okahandja Namibia © Copyright NIED, Ministry of Education, 2005 Mathematics Syllabus Higher Level Grades 11 - 12 ISBN: 99916-69-17-5 Printed by NIED Publication date: December 2005

TABLE OF CONTENTS 1. Introduction ................................................................................................................................. 1

2. Rationale...................................................................................................................................... 2

3. Aims ............................................................................................................................................ 2

4. Learning Content ......................................................................................................................... 3

Content Section ........................................................................................................................... 4

5. Assessment Objectives ................................................................................................................ 18

6. Scheme Of Assessment ............................................................................................................... 19

7. Specification Grid........................................................................................................................ 20

8. Grade Descriptions ...................................................................................................................... 21

NSSCH Mathematics Syllabus, NIED 2005 1

1. INTRODUCTION

The Namibia Senior Secondary Certificate (NSSC) syllabus for Mathematics Higher Level is designed as a two-year course leading to examination after completion of the Junior Secondary Certificate. The syllabus is designed to meet the requirements of the Curriculum Guide for Formal Senior Secondary Education for Namibia and has been approved by the National Examination, Assessment and Certification Board. (NEACB) The National Curriculum Guidelines, applicable at the stage of senior secondary education (Grades 11 and 12) and at equivalent stages of non-formal education, as a part of life-long learning, recognise the uniqueness of the learner and adhere to the philosophy of learner-centred education. The Namibia National Curriculum Guidelines:

- recognise that learning involves developing values and attitudes as well as knowledge and skills;

- promote self-awareness and an understanding of the attitudes, values and beliefs of others in a multilingual and multicultural society;

- encourage respect for human rights and freedom of speech; - provide insight and understanding of crucial “global” issues in a rapidly changing world

which affects quality of life: the AIDS pandemic, global warming, environmental degradation, distribution of wealth, expanding and increasing conflicts, the technological explosion and increased connectivity;

- recognise that as information in its various forms becomes more accessible, learners need to develop higher cognitive skills of analysis, interpretation and evaluation to use information effectively;

- seek to challenge and to motivate learners to reach their full potential and to contribute positively to the environment, economy and society.

Thus the Namibia National Curriculum Guidelines should provide opportunities for developing essential skills across the various fields of study. Such skills cannot be developed in isolation and they may differ from context to context according to a field of study. The skills marked with an * are relevant to this syllabus. The skills are: Communication skills * Numeracy skills * Information skills * Problem-Solving skills * Self-Management and Competitive skills * Social and Cooperative skills * Physical skills Work and Study skills * Critical and Creative Thinking*


2. RATIONALE

Mathematics is a dynamic, living and cultural product. It is more than an accumulation of facts, skills and knowledge. The learning of mathematics involves conceptual structures, strategies of problem solving and attitudes towards and appreciation of mathematics. Mathematical knowledge and mathematical methods of inquiry constitute an essential part of and contribute to all modern science and engineering. Today’s learners will live and work in an era dominated by computers, by worldwide communication and by a global economy. Mathematics is the key to success in this world where the economy requires workers who are prepared to absorb new ideas, to perceive patterns and to solve unconventional problems. The senior secondary curriculum strives to prepare learners to function effectively in the 21st century by providing a basis to use mathematics in their personal and professional lives.

3. AIMS

The aims of the syllabus are the same for all learners. These are set out below and describe the educational purposes of a course in Mathematics Higher Level for the NSSC examination. They are not listed in order of priority. The aims are to enable students to: - develop their mathematical knowledge and skills in a way which encourages confidence

and provides satisfaction and enjoyment; - develop a feel for number and measurement, carry out calculations and understand the

significance of the results obtained; - develop an understanding of spatial concepts and relationships; - develop their ability to apply mathematics, in the contexts of everyday situations and of

other subjects that they may be studying; - develop an understanding of mathematical principles; - develop their ability to analyse problems logically, recognise when and how a situation may

be represented mathematically, identify and interpret relevant factors and, where necessary, select an appropriate mathematical method to solve the problem;

- use mathematics as a means of communication with emphasis on the use of clear expression;

- develop the abilities to reason logically, to classify, to generalise and to prove; - experience a sufficiently wide range of mathematical topics and methods so that they can

develop their appreciation of the power, elegance and structure of the subject; - acquire the mathematical background necessary for further study in this or related subjects.


4. LEARNING CONTENT

NOTE:

1. The learning content outlined below is designed to provide guidance to teachers as to what will be assessed in the overall evaluation of learners. They are not meant to limit, in any way, the teaching program of any particular school.

2. The learning content is set out in three columns.

(a) Topics (b) General Objectives (c) Specific Objectives

3. Topic refers to those components of the subject which learners are required to study.

The General Objectives are derived from the topics and are the general knowledge, understanding and demonstration of skills on which learners will be assessed. The Specific Objectives are the detailed and specified content of the syllabus, which will be assessed. The Specific Objectives are numbered.


CONTENT SECTION

PART 1

TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES

Learners will: Learners will be able to: 1. Numbers and operations (a) Numbers - recognise and use different types of

numbers 1 identify and use natural numbers, integers (positive, negative and

zero), prime numbers, common factors and common multiples, rational and irrational numbers and real numbers (e.g. π, 2 )

(b) Squares, square roots, cubes and cube roots

- know the notation for and find squares, square roots, cubes and cube roots

1 calculate squares, square roots, cubes and cube roots of real numbers

(c) Directed numbers - understand and use directed numbers 1 use directed numbers in practical situations (e.g. temperature change, flood levels)

2 apply the four basic operations on directed numbers (d) Vulgar and decimal fractions

and percentages - know and use the relationships between

vulgar and decimal fractions and percentages

1 use the language and notation of simple vulgar and decimal fractions and percentages in appropriate context

2 recognise equivalence and convert between these forms 3 calculate a given percentage of a quantity 4 express one quantity as a percentage of another 5 calculate percentage increase and decrease 6 carry out calculations involving reverse percentages, e.g. finding

the cost price, given the selling price and the percentage profit (e) Ordering - order quantities by magnitude 1 order quantities by magnitude and demonstrate familiarity with the

symbols = , ≠ , > , < , ≤ , ≥ (f) Standard form - express very large and very small

quantities in standard form 1 use the standard form nA 10× where n is a positive or negative

integer, and 101 ≤≤ A (g) Four basic operations - use the four basic operations to

calculations 1 apply the four basic operations for calculations with integers,

decimal fractions and vulgar (and mixed) fractions, including correct order of operations and use of brackets

(h) Estimation - estimate numbers and quantities 1 make estimates of numbers and quantities 2 give approximations to specified numbers of significant figures and

decimal places 3 round off to reasonable accuracy in the context of the given

problem


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: (i) Limits of accuracy - know that approximated and rounded off

data have specified tolerances 1 give appropriate upper and lower bounds for data given to a

specified accuracy (e.g. measured lengths) 2 obtain appropriate upper and lower bounds to solutions of simple

problems (e.g. the calculation of the perimeter or the area of a rectangle) given data to a specified accuracy

(j) Ratio, proportion and rate - demonstrate an understanding of the elementary ideas and notation of ratio, direct and inverse proportion and common measures of rate

1 apply the ideas and notation of ratio, proportion and rate to practical situations

2 divide a quantity in a given ratio 3 use scales in practical situations 4 calculate average speed 5 increase and decrease a quantity by a given ratio 6 express direct and inverse variation in algebraic terms and use this

form of expression to find unknown quantities (k) Use of an electronic calculator - know how and when to use an electronic

calculator 1 use an electronic calculator efficiently 2 apply appropriate checks of accuracy

(l) Money and finance - do calculations using money and understand issues involving personal and household finances

1 solve problems involving money and convert from one currency to another

2 use given data to solve problems on personal and household finance involving earnings, simple interest, compound interest (knowledge of the formula is not required, can be provided in examinations), discount, profit and loss, tax and budgeting

3 extract data from tables and charts 2. Measures (a) SI units of measures - use units of measure in practical situations 1 use SI units of mass, length, area, volume and capacity in practical

situations 2 express quantities in larger or smaller units 3 calculate time in terms of the 24-hour and 12-hour clock (notation

for time e.g. 14:00 or 2.00 pm) 4 read clocks, dials and timetables


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 3. Mensuration (a) Perimeters, areas and volumes - do calculations involving the perimeter

and area of simple shapes, the surface area and volume of simple solid bodies

1 carry out calculations involving - the perimeter and area of a rectangle and triangle, - the circumference and area of a circle - the perimeter and area of a parallelogram and a trapezium - the volume of a cuboid, prism and cylinder - the surface area of a cuboid and a cylinder

2 solve problems involving the arc length and sector area as fractions of the circumference and area of a circle; the surface area and volume of a sphere, pyramid and cone (given formulae for the sphere, pyramid and cone)

4. Geometry (a) Geometrical terms and

relationships - know and use geometrical terms and the

vocabulary of simple plane figures and simple solids

1 use and interpret the geometrical terms: point, line, parallel, intersecting, bearing, right angle, acute, obtuse and reflex angles, perpendicular, similarity, congruence

2 use and interpret vocabulary of triangles, quadrilaterals, circles, polygons and simple solid figures, including nets

3 use the relationships between areas of similar triangles, with corresponding results for similar figures and extension to volumes and surface areas of similar solids

(b) Geometrical constructions - measure lines and angles and construct simple geometrical figures using straight edges, compasses, protractors and set squares

1 measure lines and angles 2 construct a triangle given the three sides using a straight edge and

compasses only 3 construct other simple geometrical figures from given data using

protractors and set squares as necessary 4 construct angle bisectors and perpendicular bisectors using straight

edges and compasses only 5 read and make scale drawings


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: (c) Symmetry - recognise properties of simple plane

figures directly related to their symmetries 1 recognise line and rotational symmetry (including order of

rotational symmetry) in two dimensions 2 recognise properties of triangles, quadrilaterals and circles directly

related to their symmetries 3 use the following symmetry properties of circles:

- equal chords are equidistant from the centre - the perpendicular bisector of a chord passes through the centre

of a circle - tangents from an external point are equal in length

(d) Angle properties - calculate unknown angles using the geometrical properties of intersecting and parallel lines and of simple plane figures

calculate unknown angles using the following geometrical properties (reasons may be required but no formal proofs): 1 angles at a point 2 angles on a straight line and intersecting straight lines 3 angles formed within parallel lines 4 angle properties of triangles and quadrilaterals 5 angle properties of regular polygons 6 angle in a semi-circle 7 angle between tangent and radius 8 angle properties of irregular polygons 9 angle at the centre of a circle is twice the angle at the circumference 10 angles in the same segment are equal 11 angles in opposite segments are supplementary

(e) Locus - determine the locus (path) of a point under certain conditions

use the following loci and the method of intersecting loci for sets of points in two dimensions: 1 which are at a given distance from a given point 2 which are at a given distance from a given straight line 3 which are equidistant from two given points 4 which are equidistant from two given intersecting straight lines


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 5. Algebra (a) Algebraic representation and

formulae - know how to express basic arithmetic

processes algebraically and how to substitute letters in formulae with numbers and how to transform simple formulae

1 use letters to express generalised numbers 2 express basic processes algebraically 3 substitute numbers for words and letters in formulae 4 transform simple formulae 5 construct linear equations from given situations 6 transform formulae involving roots, powers, fractions and

factorization (b) Algebraic manipulation - manipulate algebraic expressions and

extract factors 1 multiply a monomial by a polynomial 2 use brackets and extract common factors 3 expand products of algebraic expressions 4 factorise where possible expressions of the form

;ayax + ;; kbykaybxax +++ ;2222 ybxa − ;2 22 baba ++ cbxax ++2

5 manipulate algebraic fractions, e.g. .

,2

43

−+

xx ,3

23

2−

−x

x ,

35

43 aba

× 3

22

1−

−− xx

6 factorise and simplify expressions such as 65

22

2

+−−

xxxx

7 write quadratic expressions in the form qpxa ++ 2)( (c) Polynomials - manipulate polynomials 1 carry out operations of addition, subtraction and multiplication of

polynomials 2 carry out division of a polynomial by a binomial expression, and

identify the quotient and the remainder


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: (d) Equations and inequalities - solve linear and quadratic equations and

linear inequalities 1 solve simple linear equations in one unknown 2 solve simultaneous linear equations in two unknowns 3 solve quadratic equations given in factorised form, e.g.

(x – 2)(x + 3) = 0 4 solve quadratic equations by factorisation, by completing the square

and by use of the formula

5 solve equations with fractions, e.g. 2 811 14

x xx x− +

= −− +

6 solve simultaneous equations, one linear and one quadratic 7 solve simple linear inequalities

(e) Sequences - find patterns in sequences 1 recognise patterns in sequences 2 find the nth term of a linear sequence 3 distinguish between arithmetic and geometric sequences 4 use the formulae for the nth term and the sum of the first n terms to

solve problems involving arithmetic and geometric progressions (f) Indices - use indices 1 use and interpret positive, negative and zero indices

2 apply the laws of indices correctly

3 use and interpret fractional indices, e.g. 22 21

=

4 simplify expressions involving indices, e.g. x

yxyx

2333 +− ×

5 solve simple exponential equations, e.g. 412 =x

(g) Logarithms - understand the relationship between indices and logarithms (only to base 10)

1 use the laws of logarithms to simplify expressions 2 solve equations of the form bax = using logarithms


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 6. Graphs and Functions (a) Graphs in practical situations - understand the use of graphs in practical

situations 1 demonstrate familiarity with the Cartesian coordinates in two

dimensions 2 interpret and use graphs in practical situations, including travel

graphs and conversion graphs 3 draw graphs from given data 4 apply the idea of rate of change to easy kinematics involving

distance-time and speed-time graphs, acceleration and deceleration 5 calculate distance travelled as area under a linear speed-time graph

(b) Linear programming - represent and apply linear inequalities graphically

1 represent inequalities graphically and use this representation in the solution of simple linear programming problems (the convention of using broken lines for strict inequalities and shading unwanted regions will be expected

(c) Functions - understand the function idea and use function notation

1 use function notation, e.g. 53)( −= xxf ; 53: −xxf a to describe simple functions, and the notation f –1(x) to describe their inverses

2 form composite functions as defined by ))(()( xfgxgf = (d) Graphs of functions - construct tables of values for functions,

draw and interpret graphs and solve equations graphically

construct tables of values for functions of the form ,baxy += baxxy ++±= 2 , xay = , )0( ≠x where a and b

are integral constants 1 draw and interpret such graphs 2 find the gradient of a straight line graph and determine the equation

of a straight line in the form cmxy += 3 solve linear and quadratic equations approximately by graphical

methods 4 construct tables of values and draw graphs for functions of the form

naxy = where a is a rational constant and n = –2, –1, 0, 1, 2,3 and simple sums of not more than three of these and for functions of the form xay = where a is a positive integer

5 estimate gradients of curves by drawing tangents 6 solve associated equations approximately by graphical methods


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 7. Coordinate Geometry (a) Graphs and Cartesian

coordinates - use rectangular Cartesian coordinates in

two dimensions, and understand the relationship between a graph and an associated algebraic equation

1 calculate the distance between two points given in coordinate form, the gradient of the line-segment joining them, and the coordinates of their midpoint

2 find the equation of a straight line given sufficient information (e.g. the coordinates of two points on it or one point on it and its gradient)

3 interpret and use equations of the form ax + by + c = 0, including knowledge of the relationships involving gradients of parallel and perpendicular lines

4 apply coordinate geometry to quadrilaterals 8. Trigonometry (a) Bearings and the

trigonometrical ratios - interpret and use the sine, cosine and

tangent ratios 1 interpret and use three-figure bearings measured clockwise from the

north (i.e. 000 o – 360 o) 2 apply Pythagoras’ theorem and the sine, cosine and tangent ratios

for acute angles to the calculation of a side or of an angle of a right-angled triangle (angles are quoted in, and answers required in, degrees)

3 solve trigonometrical problems in two dimensions involving angles of elevation and depression

4 extend trigonometrical ratios to angles between 90o and 360o 5 solve problems using sine and cosine rules for any triangle and the

formula: area of triangle Cab sin21

=

6 solve simple trigonometrical problems in three dimensions including angle between a line and a plane


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 9. Vectors in two dimensions and

transformations

(a) Vectors - understand and use vectors 1 describe a translation using a vector represented by

xy

⎛ ⎞⎜ ⎟⎝ ⎠

, AB or a,

add and subtract vectors and multiply a vector by a scalar

2 calculate the magnitude of a vector xy

⎛ ⎞⎜ ⎟⎝ ⎠

as 2 2x y+ (Vectors will

be printed as ABuuur

or a and their magnitudes as ABuuur

or |a|. In their

answers to questions candidates are expected to indicate a in some

definite way, e.g. by an arrow or by underlining, thus AB or a 3. represent vectors by directed line segments 4. use the sum and difference of two vectors to express given vectors

in terms of two coplanar vectors 5 use position vectors

(b) Transformations - transform simple plane figures and describe these transformations

1 reflect simple plane figures in horizontal and vertical lines 2 rotate simple plane figures about the origin, vertices and midpoints

of edges of the figures, through multiples of 90 o 3 construct given translations and enlargements of simple plane

figures 4 recognise and describe reflections, rotations, translations and

enlargements 5 use the following transformations of the plane: reflection, rotation,

translation and enlargement and their combinations 6 identify and give precise descriptions of transformations connecting

given figures


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 10. Statistics and probability (a) Statistics - collect and manipulate statistical data and

understand the purpose of measures of central tendency

1 collect, classify and tabulate statistical data 2 read, interpret and draw simple inferences from tables and

statistical diagrams 3 construct and use bar charts, pie charts, pictograms, simple

frequency distributions and histograms with equal intervals 4 calculate the mean, median and mode for individual and discrete

data and distinguish between the purposes for which they are used 5 construct and read histograms with equal and unequal intervals

(areas proportional to frequencies and vertical axis labelled ‘frequency density’)

6 construct and use cumulative frequency diagrams 7 estimate and interpret the median, percentiles, quartiles and inter-

quartile range 8 calculate an estimate of the mean for grouped and continuous data 9 identify the modal class from a grouped frequency distribution

(b) Probability - understand and use probability 1 calculate the probability of a single event as either a fraction or a decimal (not ratio)

2 understand and use the probability scale from 0 to 1 3 understand that (the probability of an event occurring) = 1 – (the

probability of the event not occurring) 4 understand probability in practice, e.g. relative frequency 5 calculate the probability of simple combined events, using

possibility diagrams and tree diagrams where appropriate (in possibility diagrams outcomes will be represented by points on a grid and in tree diagrams outcomes will be written at the end of branches and probabilities by the side of the branches)


PART 2

TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES

Learners will: Learners will be able to: 11. Polynomials - extend their knowledge of factorisation of

polynomials 1 apply the remainder theorem and the factor theorem.

12. Identities, equations and inequalities

- recognise the differences between identities, equations and inequalities and solve for unknown coefficients and values

1 understand the difference between identities and equations, and use identities to determine unknown coefficients in polynomials

2 solve cubic equations in cases where at least one rational root may be found, by means of the factor theorem

3 recognise and solve equations in x which are quadratic in some function of x, e.g. 045 24 =+− xx

4 use the discriminant of a quadratic polynomial )(xf to determine the number of real roots of the equation 0)( =xf

5 solve quadratic inequalities in one unknown 6 find the equation of a graph (quadratic and cubic) given sufficient

information 7 use completing the square to determine the maximum or minimum

value of a quadratic expression 13. Vectors - use vectors in two and three dimensions 1 use rectangular Cartesian coordinates to locate points in three

dimensions, and use standard notations for vectors, i.e. xyz

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

, xi + yj + zk, ABuuur

, a

2 carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms

3 use unit vectors, position vectors and displacement vectors 4 calculate magnitude of a vector and the scalar product of two

vectors (in two and three dimensions) 5 use the scalar product to determine the angle between two

directions and to solve problems concerning perpendicularity of vectors


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 14. Functions - know the function terminology and

understand the concepts of inverse and composite functions

1 apply the terms function, domain, range (image set), one-one function, many-to-one function and inverse function

2 use the notations exemplified by 3)( xxf = and 3: xxf a , with 1−f for the inverse of f

3 find 1−f or given f in simple cases 4 draw sketch graphs of functions and their inverses for a specified

domain 5 form composite functions, and recognise when a given function can

be expressed as a composite 15. Logarithmic and exponential

functions

- understand the relationship between logarithms and indices and their functions

- use logarithms

1 convert exponential equations to logarithmic equations 2 use the laws of logarithms (including change of base) to simplify

expressions and solve logarithmic equations 3 sketch graphs of simple logarithmic and exponential functions,

including xln and xe 4 solve simple logarithmic inequalities 5 use logarithms to transform a given relationship to linear form, and

hence to determine unknown constants by considering the gradient and/or intercept

16. Absolute value (modulus) - understand the idea of absolute value

- know the notation for and the use of absolute values

- represent absolute values graphically

1 use the notation bax +

2 solve equations of the form dcxbax +=+ and inequalities of

the form dcxbax +≤+ and dcxbax +≥+ where a, b, c and d are integers draw and interpret graphs of the functions axy = , baxy +=

and cxy += where a, b and c are integers


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 17. Trigonometry - understand the meaning of a radian

measure - know trigonometrical identities and use

them to solve trigonometrical equations - understand amplitude and period of a

trigonometrical function and apply to graphs

1 convert between radians and degrees and apply to equations and graphs

2 use the six trigonometrical functions for angles of any magnitude

3 recall the identities AAA tan

cossin

= , AAA cot

sincos

= ,

AA

cossec

1= , A

Asin

cosec1

= , 1cossin 22 =+ AA ,

AA 22 tan1sec −= , AA 22 cot1cosec += , and select and use one or more of these as appropriate to the context

4 find the amplitude and period and sketch and interpret graphs of the form ,)sin( cbxay += ,)cos( cbxay += and

cbxay += )tan( 5 find solutions, in degrees or radians and within a specified interval,

of the equations ,)sin( knx = ,)cos( knx = ,)tan( knx = and of equations easily reducible to these forms, where n is a small integer or a simple fraction

18. Sequences - understand the use of Sigma notation

- understand the concept of convergence of sequences

1 interpret and apply the Sigma notation 2 use the formula for the sum to infinity of a geometric progression


TOPIC

GENERAL OBJECTIVES

SPECIFIC OBJECTIVES Learners will: Learners will be able to: 19. Differentiation - understand the concept of a limit of a

function and a derived function - understand the concept of differentiating

by first principles - know the notations for derivatives - use derivatives in a variety of applications

1 demonstrate understanding of the concept of limit of a function

2 use the notations ),(xf ′ dxdy

, ),(xf ′′ f ′′ (x) , 2

2

dxyd

3 recall and use the derivatives of nax (for any rational n) , ,ln xa ,xae together with sums, differences and composites of these

4 apply differentiation to gradients and tangents 5 locate stationary points, and distinguish (by any method) between

maximum and minimum points and points of inflexion 6 express rates of change in terms of derivatives, and use

differentiation to solve problems concerning rates of change, especially involving displacement, velocity and acceleration (rates of change of connected variables are not included)

20. Integration - know the relationship between

differentiation and indefinite integration - use integration to determine areas and

volumes and solve problems involving easy kinematics

1 understand indefinite integration as the reverse process of differentiation and solve simple differential equations of the type

)(xfdxdy

= , excluding separation of variables

2 and integrate nbax )( + , including the case where 1−=n , ,baxe + and sums and differences of these

3 evaluate definite integrals, and apply integration to the evaluation of plane areas, volumes of revolution about one of the coordinate axes, displacement, velocity and acceleration.


5. ASSESSMENT OBJECTIVES

The abilities to be assessed in the Mathematics Higher Level examination cover a single area, technique with application. The examination tests the ability of candidates to:

1. organise, interpret and present information accurately in written, tabular, graphical and

diagrammatic forms; 2. understand relevant mathematical concepts, terminology and notation; 3. perform calculations by suitable methods, including the appropriate use of an electronic

calculator; 4. recall, apply and interpret mathematical knowledge in the context of everyday situations,

and work to degrees of accuracy appropriate to the context; 5. recall accurately and use successfully appropriate manipulative techniques; 6. interpret, transform and make appropriate use of mathematical statements expressed in

words or symbols; 7. make logical deductions from given mathematical data; 8. respond to a problem relating to a relatively unstructured situation by translating it into an

appropriately structured form; 9. analyse a problem, select a suitable strategy and apply an appropriate technique to obtain its

solution; 10. apply combinations of relevant mathematical skills and techniques in problem solving; 11. present mathematical work, and communicate conclusions, in a clear, concise and logical

way.


6. SCHEME OF ASSESSMENT

Assessment consists of two written papers. All candidates must take both papers, and there is no choice of questions in either paper. Paper 1 (duration 2 hours) consists of questions on topics listed in Part 1 of the syllabus, which corresponds principally to the NSSC Mathematics Ordinary Level Syllabus (Extended). Paper 2 (duration 3 hours) consists of approximately 12 structured questions, some short and some more extended, set on topics listed in Part 2 of the syllabus. Where a topic appears in both Part 1 and Part 2 of the syllabus, e.g. logarithms and sequences, questions in Paper 2 may include some content prescribed in Part 1.

Papers Weighting of papers Marks Time

Paper 1 Paper 2

40%

60%

80

120

2 hours

3 hours

NOTES

1. The syllabus assumes that candidates will be in possession of a non-programmable scientific calculator for both papers. Three significant figures will be required in answers except where otherwise stated.

2. If the degree of accuracy is not specified in the question and if the answer is not exact, the answer must be given to three significant figures. Answers in degrees must be given to one decimal place. Forπ either the calculator value or 3.142 must be used.

3. Candidates will be expected to be familiar with scientific notation for the expression of compound units, e.g. 5 ms 1− for 5 metres per second.


7. SPECIFICATION GRID

A rigid association between particular Assessment Objectives and individual examination components is not appropriate since any of the objectives can be assessed in any question. Nevertheless, the two components of the scheme will differ in the emphasis placed on the various objectives. For example, the assessment of manipulative techniques (Objective 5) and of candidates' response to relatively unstructured situations (Objective 8) is particularly important in Paper 2. The grid below is for general guidance only and illustrates where particular objectives might receive most emphasis. In general, Paper 1 of the NSSC Mathematics Higher Level examination will assess the NSSC Mathematics Ordinary Level curriculum content and assessment objectives whilst Paper 2 will concentrate on the additional content in Part 2 of the NSSC Mathematics Higher Level syllabus. The assessment objectives will be interpreted as appropriate to the different topics in the two parts of the curriculum content.

Objectives Paper 1 Paper 2

1 to 4, 6 √ √ 5, 8, 9, 11 √

7, 10 √ √


8. GRADE DESCRIPTIONS

The scheme of assessment is intended to encourage positive achievement by all learners. Grade descriptions are therefore provided for judgmental grades 1, 3 and 4 to give a general indication of the standards of achievement expected of learners awarded particular grades. The description must be interpreted in relation to the content specified by the Mathematics syllabus but are not designed to define that content. The grade awarded will depend in practise upon the extent to which the learner has met the assessment objective overall. At Grade 1 the learner is expected to:

• Express any number in standard form and/or to a specified degree of accuracy. Relate a percentage change or a ratio to a multiplying factor and vice versa, e.g. multiplication by 1.03 results in a 3% increase. Perform arithmetic calculations efficiently and with due regard to the accuracy of the data, where appropriate.

• Relate scale factors to situations in both two and three dimensions. Calculate actual lengths, areas and volumes from scale models. Carry out calculations involving the use of right-angled triangles as part of work in three dimensions.

• Add, subtract, multiply and divide algebraic fractions. Manipulate algebraic equations - linear, simultaneous, quadratic and modulus. Use positive, negative and fractional indices in both numerical and algebraic work. Write down algebraic formulae and equations from a description of a situation.

• Process data, discriminating between necessary and redundant information. Select appropriate ways of illustrating and analysing statistical data. Make quantitative and qualitative deductions from distance-time and speed-time graphs.

• Carry out basic algebraic operations involving polynomials fluently. Solve problems using the remainder theorem. Factorise polynomials by using the factor theorem efficiently, and solve equations and inequalities by finding the factor, and solve equations and inequalities by finding factors.

• Solve problems, not necessarily purely numerical, involving arithmetic and geometric progressions.

• Apply methods of elementary coordinate geometry (using rectangular Cartesian coordinates in two dimensions only) to solve problems involving distances, gradients and equations of straight lines.

• Accurately represent geometrical relationships in vector terms, and interpret vector results geometrically.

• Use laws of logarithms, for a given base, to simplify and solve equations, and relate laws of logarithms (including the change of base) to the corresponding laws of indices.

• Solve trigonometrical equations (e.g. 1.0)2sin( =x for oo 3600 << x ), including cases where preliminary factorisation and/or simplification by means of simple identities is required. Express solution to equations in degrees or radians.

• Find the inverse of functions and composite functions with the appropriate domain and range. Sketch graphs of functions: linear, quadratic, reciprocal, exponential, logarithmic, modulus and trigonometrical.

• Differentiate powers of x, xln , xe and composites of these. Apply differentiation to solve coordinate geometry problems involving tangents to curves and stationery points, including the point of inflexion.

• Integrate powers of )( bax + and beax + . Apply definite integration to the calculation of areas and volumes.

• Express information about rates of change in terms of derivatives, and use differentiation and/or integration as appropriate to solve problems involving rates of change.

• Decide on, and work consistently to, degrees of accuracy appropriate to the context. • Make clear, concise and accurate mathematical statements, demonstrating ease and

confidence in the use of notation and terminology and accuracy in algebraic or arithmetic manipulation.

• Carry out mathematical reasoning clearly and concisely.


At Grade 3 the learner is expected to: • Carry out arithmetic calculations confidently, with appropriate use of a calculator, including

those involving several operations. Express results to a specified degree of accuracy. Give a reasonable approximation to a calculation involving the four rules.

• Calculate volumes and surface areas of prisms (including cylinders), pyramids (including cones), spheres and combinations of these shapes. Perform trigonometrical calculations involving right-angled triangles, in the context of problems in two dimensions. Calculate angles in geometrical figures.

• Manipulate algebraic fractions with monomial denominators. Solve simultaneous linear equations in two unknowns, and easily factorisable quadratic equations in one unknown. Transform simple formulae. Use positive and negative integer indices in algebraic work.

• Classify and tabulate statistical data, and construct histograms. Draw and interpret speed-time graphs. Plot graphs of simple algebraic functions, and recognise the algebraic form of a linear graph.

• Carry out basic algebraic operations involving addition and multiplication of polynomials, and division of a polynomial by a binomial expression. Use the remainder theorem to test whether a given binomial expression is a factor of a given polynomial.

• Use the factor theorem to find factors of the form )( ax − of a given polynomial, and use the correspondence between roots and factors.

• Select and carry out a strategy for eliminating one unknown from a pair of simultaneous equations, one of which is linear.

• Use formulae for arithmetic or geometric progressions to solve simple problems (e.g. to find the first term and the difference of an A.P. given the values of two of its terms).

• Use rectangular Cartesian coordinates, in two dimensions only, to calculate distance between points, mid-points of line segments, gradients and equations of straight lines. Solve coordinate geometry problems where only one or two undirected intermediate steps are required (e.g. to find the equation of a straight line through a given point perpendicular to a given line).

• Use the scalar product of two vectors to calculate the angle between the vectors. • Find composites and inverses of given functions. Sketch graphs of linear, quadratic and

reciprocal functions. • Use logarithms to make x the subject of the formula or an equation such as bax = . • Solve simple trigonometrical equations (e.g. 1.0)2sin( =x for oo 3600 << x ), and use a

given identity to carry out simplification of an equation prior to solving. • Differentiate powers of x confidently, for any rational power. Apply differentiation to find

the equation of the tangent at a given point on a curve, and to locate maximum and minimum points on a curve.

• Integrate powers of x confidently, for any rational power. Apply definite integration to calculate the area under a curve or a volume of revolution about the x-axis.

• Use differentiation and/or integration to solve simple problems concerning displacement, velocity and acceleration in linear motion.

• Work consistently to a directed degree of accuracy. • Use mathematical notations which are generally correct and appropriate.


At Grade 4 the learner is expected to: • Apply the four rules of number to positive and negative integers, and vulgar and decimal

fractions. Calculate percentage change. Make generally appropriate use of a calculator. • Use area and volume units. Find the volume and surface area of a prism (including a

cylinder). Calculate the length of the third side of a right-angled triangle. Find the angle in a right-angled triangle, given two sides.

• Solve simple simultaneous linear equations in two unknowns. Substitute numbers into a simple formula and evaluate the remaining term.

• Use bracket and factorise algebraic expressions. • Construct a pie chart or a bar chart from simple data. Draw and interpret a travel graph or a

conversion graph. Plot graphs of linear functions. • Add and multiply simple polynomials, and determine whether a binomial expression of the

form )( ax − is a factor of a given polynomial in x. • Deduce roots of polynomial equations given the factors of the polynomial. • Understand the process of eliminating one unknown from a pair of equations, and carry out

explicit substitutions not involving algebraic or arithmetic fractions. • Recognise arithmetic and geometric progressions, and calculate specific terms in a sequence

of either type. • Calculate the length, gradient and coordinates of the mid-point of the line segment joining

two given points, using rectangular Cartesian coordinates in two dimensions only. • Find the equation of the straight line through a given point with given gradient. • Find the gradient of a line parallel or perpendicular to a given direction. • Calculate the scalar product of two vectors. • Recall the meaning of the six trigonometrical ratios, with reference to angles between 0° and

360°, and use a calculator to find at least one solution of equations such as 3.0sin =x or 4sec =x .

• Differentiate nkx for non-negative integer vales of n, and the derivative to find the gradient of a curve at a given point.

The National Institute for Educational Development P/Bag 2034 Okahandja NAMIBIA Telephone: +64 62 502446 Facsimile: + 64 62 502613 E-mail: [email protected] Website: http://www.nied.edu.na © NIED 2005

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