MINISTRY OF EDUCATION

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1 Republic of Namibia MINISTRY OF EDUCATION NAMIBIA SENIOR SECONDARY CERTIFICATE (NSSC) MATHEMATICS SYLLABUS ORDINARY LEVEL SYLLABUS CODE: 4324 GRADES FOR IMPLEMENTATION IN 2006 FOR FIRST EXAMINATION IN 2007 DEVELOPED IN COLLABORATION WITH UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS

2 Republic of Namibia MINISTRY OF EDUCATION NAMIBIA SENIOR SECONDARY CERTIFICATE (NSSC) MATHEMATICS SYLLABUS ORDINARY LEVEL SYLLABUS CODE: 4324 GRADES 11-12

3 Ministry of Education National Institute for Educational Development (NIED) Private Bag 2034 Okahandja Namibia Copyright NIED, Ministry of Education, 2005 Mathematics Syllabus Ordinary Level Grades ISBN: Printed by NIED Publication date: 2005

4 TABLE OF CONTENTS 1. Introduction Rationale Aims Learning Content...3 Content Section Asssessment Objectives Scheme Of Assessment Specification Grid Grade Descriptions...17

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6 1. INTRODUCTION The Namibia Senior Secondary Certificate (NSSC) syllabus for Mathematics Ordinary Level is designed as a two-year course leading to examination after completion of the Junior Secondary Certificate. Mathematics Ordinary Level Syllabus makes provision for Core and Extended levels which will be assessed in separate question papers at the end of the course. 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* NSSCO Mathematics Syllabus, NIED

7 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 and general 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, 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 21 st 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 Ordinary Level for the NSSC examination. They are not listed in order of priority. The aims are to: 1. develop their mathematical knowledge and oral, written and practical skills in a way which encourages confidence and provides satisfaction and enjoyment; 2. read mathematics, and write and talk about the subject in a variety of ways; 3. develop a feel for number, carry out calculations and understand the significance of the results obtained; 4. apply mathematics in everyday situations and develop an understanding of the part which mathematics plays in the world around them; 5. solve problems, present the solutions clearly, check and interpret the results; 6. develop an understanding of mathematical principles; 7. 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; 8. use mathematics as a means of communication with emphasis on the use of clear expression; 9. develop the ability to apply mathematics in other subjects, particularly science and technology; 10. develop the abilities to reason logically, to classify, to generalise and to prove; 11. appreciate patterns and relationships in mathematics; 12. produce and appreciate imaginative and creative work arising from mathematical ideas; 13. appreciate the interdependence of different branches of mathematics; 14. acquire a foundation appropriate to their further study of mathematics and of other disciplines. NSSCO Mathematics Syllabus, NIED

8 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. Topics refer to those components of the subject which learners are required to study. The General Objectives are derived from the topic and is 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. Specific objectives have been numbered. NSSCO Mathematics Syllabus, NIED

9 CONTENT SECTION Specific objectives printed in italics are meant for learners doing the Extended syllabus only. 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 1 use the language and notation of simple vulgar and decimal fractions and percentages in appropriate context percentages 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 (f) Standard form - express very large and very small quantities in standard form (g) Four basic operations - use the four basic operations in calculations symbols =,, >, <,, n 1 use the standard form A 10 where n is a positive or negative integer, and 1 A 10 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 NSSCO Mathematics Syllabus, NIED

10 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 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 (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 (k) Use of an electronic calculator - know how and when to use an electronic calculator (l) Money and finance - do calculations using money and understand issues involving personal and household finances rectangle) given data to specified accuracy 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 1 use an electronic calculator efficiently 2 apply appropriate checks of accuracy 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 NSSCO Mathematics Syllabus, NIED

11 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 4. Geometry (a) Geometrical terms and relationships - know and use geometrical terms and the vocabulary of simple plane figures and simple solids (b) Geometrical constructions - measure lines and angles and construct simple geometrical figures using straight edges, compasses, protractors and set squares 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) 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 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 NSSCO Mathematics Syllabus, NIED

12 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 (d) Angle properties - calculate unknown angles using the geometrical properties of intersecting and parallel lines and of simple plane figure (e) Locus - determine the locus (path) of a point under certain conditions - tangents from an external point are equal in length 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 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 NSSCO Mathematics Syllabus, NIED

13 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 (b) Algebraic manipulation - manipulate algebraic expressions and extract factors 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 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 ax + ay; ax + bx + kay; + kby; a x b y ; a + 2ab + b ; ax + bx + c 5 manipulate algebraic fractions, e.g. x 4 + x 2x 2 3a 5ab 1 2,,, x x 2 x x 2x factorise and simplify expressions such as 2 x 5x write quadratic expressions in the form a ( x + p) + q (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 NSSCO Mathematics Syllabus, NIED

14 TOPIC GENERAL OBJECTIVES SPECIFIC OBJECTIVES Learners will: Learners will be able to: (d) Equations and inequalities - solve linear equations and quadratic 1 solve simple linear equations in one unknown equations and linear inequalities 2 solve simultaneous linear equations in two unknowns 3 solve quadratic equations given in factorized form, e.g. ( x 2)( x + 3) = 0 4 solve quadratic equations by factorisation, by completing the b ± b 4ac square and by use of the formula 2a 5 solve equations with fractions, e.g. x 2 x 1 = 1 x + 8 x 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 n th term of a linear sequence 3 distinguish between arithmetic and geometric sequences use the formulae for the n th 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 (g) Logarithms - understand the relationship between indices and logarithms (only to base 10) 3 use and interpret fractional indices, e.g. = 2 4 x y x 3 3 simplify expressions involving indices, e.g. 2x 3 5 x 1 solve simple exponential equations, e.g. 2 = 4 1 use the laws of logarithms to simplify expressions 2 solve equations of the form a x = b using logarithms y NSSCO Mathematics Syllabus, NIED

15 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 (b) Linear programming - represent and apply linear inequalities graphically (c) Functions - understand the function idea and use function notation (d) Graphs of functions - construct tables of values for functions, draw and interpret graphs and solve equations graphically NSSCO Mathematics Syllabus, NIED graph 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) 1 use function notation, e.g. f ( x) = 3x 5 ; f : x α 3x 5 to describe simple functions, and the notation f 1 (x) to describe their inverses 2 form composite functions as defined by gf ( x) = g( f ( x)) 1 construct tables of values for functions of the form 2 y = ax + b, y = ± x + ax + b, y = a x, ( x 0) where a and b are integral constants 2 draw and interpret such graphs 3 find the gradient of a straight line graph and determine the equation of a straight line in the form y = mx + c 4 solve linear and quadratic equations approximately by graphical methods 5 construct tables of values and draw graphs for functions of the form n y = ax 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 x the form y = a where a is a positive integer 6 estimate gradients of curves by drawing tangents 7 solve associated equations approximately by graphical methods

16 7. Coordinate Geometry (a) Graphs and Cartesian coordinates 8. Trigonometry (a) Bearings and the trigonometrical ratios TOPIC GENERAL OBJECTIVES SPECIFIC OBJECTIVES Learners will: Learners will be able to: - use rectangular Cartesian coordinates in two dimensions, and understand the relationship between a graph and an associated algebraic equation - interpret and use the sine, cosine and tangent ratios 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 1 interpret and use three-figure bearings measured clockwise from the north (i.e. 000 o to 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 rightangled triangle (angles will be 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 90 o and 360 o 5 solve problems using sine and cosine rules for any triangle and the 1 formula: area of triangle = ab sin C 2 6 solve simple trigonometrical problems in three dimensions including angle between a line and a plane NSSCO Mathematics Syllabus, NIED

17 TOPIC GENERAL OBJECTIVES SPECIFIC OBJECTVES Learners will: Learners will be able to: 9. Vectors in two dimensions and transformations (a) Vectors - understand and use vectors x 1 describe a translation using a vector represented by or AB y or a, add and subtract vectors and multiply a vector by a scalar x 2 calculate the magnitude of a vector y as x 2 + y 2 (Vectors will be printed as AB or a and their magnitudes as AB 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 NSSCO Mathematics Syllabus, NIED

18 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 range, 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 interquartile 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) NSSCO Mathematics Syllabus, NIED

19 5. ASSSESSMENT OBJECTIVES The assessment will include, wherever appropriate, personal, social, environmental, economic and technological applications of Mathematics in modern society. Learners are required to demonstrate the assessment objectives in the context of the content and skills prescribed. Within each of the Assessment Objectives the assessment must take account of the learners ability to communicate clearly and logically and apply conventions where appropriate. The abilities to be assessed in the NSSC Mathematics Ordinary Level examination cover a single assessment objective, technique with application. The examination will test the ability of candidates to: 1. organise, interpret and present information accurately in written, tabular, graphical and diagrammatic forms; 2. perform calculations by suitable methods; 3. use an electronic calculator; 4. understand systems of measurement in everyday use and make use of them in the solution of problems; 5. estimate, approximate and work to degrees of accuracy appropriate to the context; 6. use mathematical and other instruments to measure and to draw to an acceptable degree of accuracy; 7. interpret, transform and make appropriate use of mathematical statements expressed in words or symbols; 8. recognise and use spatial relationships in two and three dimensions, particularly in solving problems; 9. recall, apply and interpret mathematical knowledge in the context of everyday situations; 10. make logical deductions from given mathematical data; 11. recognise patterns and structures in a variety of situations, and form generalisations; 12. respond to a problem relating to a relatively unstructured situation by translating it into an appropriately structured form; 13. analyse a problem, select a suitable strategy and apply an appropriate technique to obtain its solution; 14. apply combinations of mathematical skills and techniques in problem solving; 15. set out mathematical work, including the solution of problems, in a logical and clear form using appropriate symbols and terminology. NSSCO Mathematics Syllabus, NIED

20 6. SCHEME OF ASSESSMENT Candidates who have followed the Core curriculum and take the relevant papers are eligible for the award of grades C to G only. Candidates who have followed the Extended curriculum are eligible for the award of grades A* to E only. DESCRIPTION OF PAPERS All candidates will take two written papers as follows (a) Paper 1 or Paper 2 Short answer questions (b) Paper 3 or Paper 4 Structured questions Core curriculum Grades available: C G Weighting Extended curriculum Grades available: A* E Weighting Paper 1 (1 hour) 60 marks short-answer questions 40% Paper 2 (1 1 2 hours) 80 marks short-answer questions 40% Paper 3 (1 3 4 hours) 90 marks structured questions 60% Paper 4 (2 1 2 hours) 120 marks structured questions 60% NOTES: 1. There will be no choice of questions. 2. The syllabus assumes that candidates will be in possession of an electronic calculator for all papers. 3. Three significant figures will be required if the degree of accuracy is not specified, and if the answer is not exact. Answers in degrees must be given correct to one decimal place. 4. Candidates are encouraged to use the value of π from their calculators if their calculator provides this. For other candidates, the value of will be given on the front page of the question paper only. NSSCO Mathematics Syllabus, NIED

21 7. SPECIFICATION GRID The approximate weightings allocated to each of the Assessment Objectives across the papers are summarised in the table below. Objectives Short-answer questions Structured/Extended Answer questions Core/Extended 1 to 8 Both 9 Greater emphasis at Core 10 Both 11 Both 12 and 13 Greater emphasis at Extended 14 Both 15 Greater emphasis at Extended Commentary on 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 components of the scheme will differ in the emphasis placed on the various objectives. A difference in emphasis will be apparent between the Core and the Extended papers; for example the assessment of candidates response to relatively unstructured situations (Objective 12) is particularly important on Paper 4 (Extended). The grid above is for general guidance only and illustrates where particular objectives might receive most emphasis in the various components. Ticks are placed in the grid only where there is likely to be emphasis although the objective will also be met in other components. The short-answer questions fulfil a particularly important function in ensuring syllabus coverage and allowing the testing of knowledge, understanding and manipulative skills, while greater emphasis is placed on applications to the processes of problem solving in the structured answer papers. NSSCO Mathematics Syllabus, NIED

22 8. GRADE DESCRIPTIONS The scheme of assessment is intended to encourage positive achievement by all learners. Grade descriptions are therefore provided for judgmental grades A, C, E and G to give a general indication of the standards of achievement likely to have been shown by 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 practice upon the extent to which the learner has met the assessment objective overall. Shortcomings in some aspects of the assessment may be balanced by better performance in others. It must be clearly understood that a learner who meets the demands of a particular grade, by implication also meets the demands of all lower grades, e.g. a candidate who achieves a Grade A, by implication also fulfils the criteria set for Grades C, E and G. At Grade A the learner is expected to: - Express any number to 1, 2 or 3 significant figures. Relate a percentage change to a multiplying factor and vice versa, e.g. multiplication by 1.03 results in a 3% increase. - 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 and quadratic. 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. Make quantitative and qualitative deductions from distance-time and speed-time graphs. - Make clear, concise and accurate mathematical statements, demonstrating ease and confidence in the use of symbolic forms and accuracy in algebraic or arithmetic manipulation. At Grade C the learner is expected to: - Perform calculations involving several operations. Calculate percentage change. Use a calculator fluently. Give a reasonable approximation to a calculation involving the four rules. Do compound interest calculations. Use and understand the standard form of a number. Apply ratio and proportion concepts. - Use area and volume units. Find the volume and surface area of a prism and a cylinder. Use a scale diagram to solve a two-dimensional problem. Calculate the length of the third side of a right-angled triangle. Apply trigonometry to solve triangles. Calculate angles in geometrical figures. Use constructions to solve a locus problem. - Recognise, and in simple cases formulate, rules for generating a pattern or sequence. Solve simple simultaneous linear equations in two unknowns. Solve quadratic equations. Transform simple formulae. Substitute numbers in complicated formulae and evaluate the remaining term. - Construct a pie-chart from simple data. Plot and interpret graphs, including graphs of linear and simple quadratic functions. NSSCO Mathematics Syllabus, NIED

23 At Grade E a learner is expected to: - Apply the four rules of number to positive and negative integers and decimal and vulgar fractions. Identify cube and prime numbers. Calculate a quantity as a percentage of another quantity. Convert between currencies. Use natural number indices. Understand and apply simple proportion. Find average speed. - Find the perimeter and area of rectilinear shapes. Find the volume and surface area of a cuboid. Calculate angles in triangles and quadrilaterals. Use a map and scales to find distances and bearings. Construct triangles and nets of cuboids and triangular prisms. Recognise reflections, rotations and enlargements. - Continue a sequence progressing into negative numbers. Write a simple expression using two variables. Simplify a linear expression using two variables. Substitute numbers into formulae and evaluate the remaining term. Use brackets and extract common factors from algebraic expressions. Solve simple linear equations. - Use and understand the concepts mean, median and mode. Interpret a simple pie chart. Draw and interpret straight line graphs, including conversion and travel graphs. At Grade G a learner is expected to: - Perform the four rules on positive integers and decimal fractions using a calculator where necessary. Convert simple fractions to decimals and percentages. Identify square numbers. Give factors of numbers. Order a list of decimal numbers. Calculate simple percentages. Perform simple money calculations. Work out the time of a simple journey. Do simple estimations. Identify the smallest integer in a list containing negative numbers. - Recognise and name simple plane figures and common solid shapes. Measure a line or an angle accurately. Find the perimeter of rectilinear shapes. Find the area of a rectangle and the volume of a cuboid. Draw the line of symmetry in a shape. Draw the reflection of a shape about a given vertical or horizontal mirror line. - Continue a sequence in numbers or diagrams. Substitute numbers into simple formulae. - Interpret tables. Draw and interpret a bar chart. Calculate the mean of a small set of discrete data. Identify the mode from a set of discrete data or a bar chart. Plot or give coordinates of points in the first quadrant. NSSCO Mathematics Syllabus, NIED

24 The National Institute for Educational Development P/Bag 2034 Okahandja NAMIBIA Telephone: Facsimile: info@nied.edu.na Website: NIED 2006