lower secondary mathematics syllabusmath.nie.edu.sg/kcang/pgde/docs/ls.pdf · 1 lower secondary...

1

LOWER SECONDARYMATHEMATICS SYLLABUS

SPECIAL/EXPRESSNORMAL ACADEMICNORMAL TECHNICAL

CURRICULUM PLANNING AND DEVELOPMENT DIVISIONMINISTRY OF EDUCATIONSINGAPORE

© MINISTRY OF EDUCATIONALL RIGHTS RESERVEDYEAR OF IMPLEMENTATION FROM 2001

2

CONTENTS

1 INTRODUCTION ……………………………………………………..………….. 3

2 AIMS OF MATHEMATICS EDUCATION .……………………………………... 4

3 FRAMEWORK OF THE MATHEMATICS PROGRAMME ..….……………... 5

4 GENERAL ASSESSMENT GUIDELINES ...…………………………………... 8

5 OBJECTIVES OF THE LOWER SECONDARY MATHEMATICSPROGRAMME ..………………………………………………………………….. 10

6 CONTENT CHART

• Secondary One & Two (Special/Express Course) …………..……………. 11

7 SYLLABUS

• Secondary One (Special/Express Course) ..…………………..…………..• Secondary Two (Special/Express Course) ………………………………...

1321

8 CONTENT CHART

• Secondary One & Two (Normal Academic Course) ..……..…………..…. 27

9 SYLLABUS

• Secondary One (Normal Academic Course)... .……………..…………….• Secondary Two (Normal Academic Course) ...…………………………….

2936

10 CONTENT CHART

• Secondary One & Two (Normal Technical Course) ..……..…………..…. 41

11 SYLLABUS

• Secondary One (Normal Technical Course)... .……………..…………….• Secondary Two (Normal Technical Course) ...…………………………….

4350

12 APPENDIX: DEFINITION OF SUGGESTED THINKING SKILLS …………. 55

3

INTRODUCTION

The Ministry of Education’s vision of “Thinking School, Learning Nation” gives impetus forthe infusion of three initiatives: Thinking Skills, Information Technology (IT) and NationalEducation into the curriculum. As we move towards a knowledge-based society that ispowered by IT, the need to prepare our people for the challenges and opportunities of thefuture becomes obvious. Besides being proficient in the use of IT, pupils will need to be ableto think creatively, learn independently and work successfully in teams. Above all, asSingapore’s economy moves towards globalisation, they need to have a strong feeling forhome and remain Singaporean in heart, mind and being. Against this background and withthe Desired Outcomes of Education as the overarching aim, the mathematics syllabus wasrevised.

This revised mathematics syllabus better reflects the recent developments in mathematicseducation. The focus of the syllabus is mathematical problem solving. The emphasis is onthe development of concepts, skills and its underlying processes. This, together with theexplication of thinking skills and the integration of IT in mathematics teaching and learning,will give leverage to the development of mathematical problem solving.

This syllabus consists of two parts.

Part A explains the philosophy of the syllabus and the spirit in which it should beimplemented. It also spells out the objectives of the mathematics programme.

The framework of the mathematics programme summarises the essence of mathematicsteaching and learning in schools. The learning of mathematics at all levels involves morethan the basic acquisition of concepts and skills. It also involves an understanding of themathematical thinking and general problem-solving strategies, having positive attitudes toand an appreciation of mathematics as an important and powerful tool in everyday life. Thisframework forms the basis for mathematics teaching and learning in schools.

Part B gives the syllabus content for each level. Care has been taken to ensure that there iscontinuity from the primary to the secondary level. In the syllabus, the spiral approach isadopted to ensure that each topic is covered at appropriate levels in increasing depth toenable pupils to consolidate the concepts and skills learnt and to further develop them. Alltopics come with ‘Learning Outcomes’ to enable teachers to monitor pupils' progress. The‘Remarks’ column provides teachers with guidance in interpreting the syllabus.

This syllabus is a guide for teachers to plan their mathematics programmes. Teachers neednot be bound by the sequence of topics presented here but should ensure that the hierarchyand linkages are maintained. Teachers should exercise flexibility and creativity when usingthe syllabus.

PART A

4

AIMS OF MATHEMATICS EDUCATION IN SCHOOLS

Mathematics education aims to enable pupils to:

• acquire and apply the skills and knowledge relating to number, measure and space inmathematical situations that they will meet in life.

• acquire mathematical concepts and skills necessary for a further study in mathematicsor other disciplines.

• develop the ability to make logical deduction and induction as well as to explicate theirthinking and reasoning skills through the solving of mathematical problems.

• use mathematical language to communicate mathematical ideas and argumentsprecisely, concisely and logically.

• develop positive attitudes towards mathematics including confidence, enjoyment andperseverance.

• appreciate the power and structure of mathematics, including patterns and relationships,and to enhance their intellectual curiosity.

PART A

5

FRAMEWORK OF THE MATHEMATICS PROGRAMME

The conceptualisation of the mathematics syllabus is based on the following framework:

AppreciationInterest Monitoring one’sConfidence own thinkingPerseverance

Estimation andApproximationMental calculation Thinking skillsCommunication HeuristicsUse of mathematical toolsArithmetic manipulationAlgebraic manipulationHandling data Numerical

Geometrical Algebraic Statistical

The primary aim of the mathematics programme is to enable pupils to develop their ability inmathematical problem solving. Mathematical problem solving includes using and applyingmathematics in practical tasks, in real life problems and within mathematics itself. In thiscontext, a problem covers a wide range of situations from routine mathematical problems toproblems in unfamiliar context and open-ended investigations that make use of the relevantmathematics and thinking processes.

The attainment of this mathematical problem solving ability is dependent on five inter-related

components - Concepts, Skills, Processes, Attitudes and Metacognition

1 Concepts

Concepts refer to the basic mathematical knowledge needed for solving mathematical

problems. They cover the following:

•••• Numerical concepts

• Geometrical concepts

• Algebraic concepts

• Statistical concepts

PART A

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2 Skills

Skills refer to the topic-related manipulative skills that pupils are expected to use when

solving problems. They include:

• Estimation and approximation

• Mental calculation

• Communication

• Use of mathematical tools

• Arithmetic manipulation

• Algebraic manipulation

• Handling data

3 Processes

Processes refer to the thinking and heuristics involved in mathematical problem solving.

Some thinking skills and heuristics, which are applicable to problem solving at the

secondary level, are listed below.

Thinking skills:

• Classifying

• Comparing

• Identifying Attributes & Components

• Sequencing

• Induction

• Deduction

• Generalising

• Justifying

• Verifying

• Spatial visualisation

Heuristics for problem solving:• Act it out

• Use a diagram/model

• Use guess-and-check

• Make a systematic list

PART A

7

• Look for pattern(s)

• Work backwards

• Use before-after concept

• Make suppositions

• Restate the problem in another way

• Simplify the problem

• Solve part of the problem

• Think of a related problem

• Use equations

(Refer to Appendix A for the definitions of the suggested thinking skills.)

4 Attitudes

Attitudes refer to the affective aspects of mathematics learning such as:

• enjoying mathematics

• appreciating the beauty and power of mathematics

• showing confidence in using mathematics

• persevering in solving a problem

5 Metacognition

Metacognition refers to the ability to monitor one's own thinking processes in problem

solving. This includes:

•••• constant and conscious monitoring of the strategies and thinking processes used

in carrying out a task

•••• seeking alternative ways of performing a task

•••• checking the appropriateness and reasonableness of answers

This framework encompasses the whole mathematics programme for primary and

secondary schools. The secondary curriculum is a continuation of the primary

curriculum.

PART A

8

GENERAL ASSESSMENT GUIDELINES

Assessment is an integral part of the teaching-learning process. The main purpose ofmathematical assessment should be to improve the teaching and learning of mathematics.Therefore we need assessment that:

• is continual and provides accurate and useful information about pupils’ learning• supports the programme in its aim to enable pupils to develop their problem solving

ability and in its emphasis on developing mathematical concepts, skills and thinking, aswell as positive attitudes towards mathematics

Teachers should assess different aspects of thinking, learning and behaviour. They shouldtry to incorporate, where appropriate, the following aspects in the assessment tasks set asthese are the key features of the syllabus:

• mental calculation• mathematical communication• practical uses of mathematics• investigative work• problem solving• critical thinking• creative work• use of information technology

Continual Assessment

It is important to carry out assessment on a continual basis. Teachers should use continualassessment to:

• obtain information about pupils’ learning, suitability of teaching methods and materialsand effectiveness of the teaching programme

• identify any learning difficulties that pupils may encounter or misconceptions that theymay have so as to plan effective remedial help

• provide prompt feedback to pupils on their progress and attainment• promote pupils’ confidence in doing mathematics by focusing on what they can achieve

rather than what they cannot

In continual assessment, teachers should use a variety of assessment modes as differentmodes provide different kinds of feedback about pupils. The modes used should yield thefeedback required, and collectively, they give teachers a more comprehensive profile of thepupils.

Assessment can be carried out through:

• classroom observations;• oral communication• written assignments and tests• practical and investigative tasks

PART A

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From observations of pupils engaged in a task and through their responses in oralcommunication, immediate feedback can be obtained on their understanding of concepts,thinking processes, mastery of skills and even their attitudes towards mathematics. Writtenassignments and tests help teachers to track the progress of pupils and are useful in theassessment of content and procedures, mathematical thinking and problem solving.Through practical and investigative tasks, teachers can assess pupils’ problem solving skills,creative and critical thinking as well as their attitudes. If the tasks are done in groups,teamwork can also be assessed.

Semestral Assessment

Semestral assessment comprises the mid-year and end-of-year examinations, which arecommon school-based examinations administered under formal conditions. The purpose ofsemestral assessment is to determine how far pupils have achieved the overall programmeobjectives for the semester or year. As such, it should be broad in coverage, test arepresentative sample of the syllabus taught and reflect its emphasis.

PART A

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OBJECTIVES OF THE LOWER SECONDARY MATHEMATICS PROGRAMME

The objectives of the lower secondary mathematics programme are to enable pupils to:

• develop an understanding of mathematical concepts - numerical, geometrical, algebraic,

statistical;

• perform calculations by using suitable methods, including mental calculation, with and

without a calculating aid;

• develop the ability to estimate and to make approximations and also to be alert to the

reasonableness of results and measurements;

• apply systems of measurement in everyday use and in the solutions of problems;

• use geometrical instruments;

• collect and analyse data;

• interpret, use and present information in written, graphical, diagrammatic and

tabular forms;

• understand and use mathematical language, symbols and diagrams to represent and

communicate mathematical ideas effectively;

• recognise and apply spatial relationships in two and three dimensions;

• recognise the appropriate mathematical procedures for a given situation;

• recognise patterns and structures in a variety of situations and form, and/or justify

generalisations;

• apply and interpret mathematical concepts learnt in familiar and unfamiliar contexts,

including daily life;

• think logically and derive conclusions deductively and apply these processes in

mathematical situations;

• analyse problems; formulate them into mathematical terms and use the appropriate

strategies to solve them; verify and interpret the solutions; and present their

mathematical arguments and solutions in a logical and clear fashion;

• recognise the relationships between topics in the syllabus;

• become aware of the application of mathematics in other subjects;

• develop an inquiring mind through investigative activities;

• enjoy learning mathematics through a variety of activities.

SPECIAL/EXPRESS PART B

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Content Chart - Secondary One & Two (Special/Express Course)

Secondary One Secondary Two

1 Whole numbers• the four operations• ordering• factors and multiples

2 Fractions and decimals• concept and notation• ordering• the four operations

3 Approximation and estimation• rounding off• estimation

4 Use of a scientific calculator

5 Squares, square roots, cubes andcube roots

6 Number sequences

7 Measures and money

8 Ratio, proportion and rate

9 Percentage

10 Simple financial transactions

11 Real numbers• integers• rational and irrational numbers

1 Arithmetic problems

2 Standard form

3 Number sequences

1 Perimeter and area• square, rectangle, triangle,

parallelogram, trapezium, circle

2 Volume and surface area• cube, cuboid, prism, cylinder

1 Volume and surface area• sphere, pyramid and cone

2 Arc length and sector area

1 Algebraic expressions andformulae

2 Simple algebraic manipulation

3 Simple linear equations

1 Algebraic manipulation andformulae

2 Solutions of equations• simultaneous linear equations• simple fractional equations• quadratic equations

SPECIAL/EXPRESS PART B

12

Content Chart - Secondary One & Two (Special/Express Course)


1 Graphs of linear and quadraticfunctions

2 Graphs in practical situations

1 Simple plane figures

2 Simple solid figures

3 AngIe properties• angles formed with a common

vertex• angles formed with parallel

lines• angle properties of triangles• angle properties of squares,

rectangles, parallelograms andrhombuses

4 Geometrical constructionof simple geometricalfigures

1 Motion geometry• reflection• rotation• translation• enlargement

2 Similar and congruent figures

3 Angle properties of polygons

4 Scale drawing

5 Symmetry• line symmetry• rotational symmetry

1 Handling data• table and chart• pictogram• dot diagram• bar graph• line graph• pie chart• stem and leaf diagram• histogram

1 Averages• mean• mode• median

1 Pythagoras’ theorem

2 Trigonometrical ratios:• sine• cosine• tangent

1 Problem solving heuristics

2 Practical uses of Mathematics



LEVEL : SEC 1 (SPECIAL/EXPRESS) PART B

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TOPICS/OUTCOMES REMARKS

ARITHMETIC

Pupils should be able to

1 Whole numbers

The four operations

• use the four operations for calculationswith whole numbers

• Include combined operations, i.e. correctordering of operations and the use ofbrackets

• Include mental calculation and estimation• Include awareness of the following

(i) commutative law(ii) associative law(iii) distributive law

• Exclude tedious calculations when the useof a calculator is not allowed

Ordering

• order numbers• use the following symbols correctly:

=, ≠, >, <, ≥, ≤

• Include the use of the number line

Factors and multiples

• use prime numbers, common factorsand common multiples

• Include the use of index notation and theterms “prime factorization”, “ HCF” and“LCM”

• Include odd and even numbers

2 Fractions and decimals

Concept and notation

• use fractions and decimals

• convert fractions to decimals, and viceversa

• Include the use of facts such as

0.25 = 14

, 0.75 = 34

, 0.125 = 18

0.333 ≈ 13

, 0.667 ≈ 23

• Exclude conversion of recurring decimalsto fractions

Ordering

• compare and order fractions anddecimals

• Include comparison of fractions with

decimals


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The four operations

• use the four operations for calculationswith fractions and decimals

• Include

i) combined operationsii) rounding off decimals to a specific

degree of accuracyiii) mental calculation and estimation


3 Approximation and estimation

Rounding off

• round off numbers and measures to aspecified degree of accuracy

• Includei) decimal places and significant figuresii) the use of the approximation sign “≈”

Estimation

• make estimates of numbers andmeasures

• Include use of estimation to check thereasonableness of answers

• Include mental estimation


• use the relevant keys of a scientificcalculator

• Include appropriate checks of accuracy by

estimation, e.g. in evaluating 47 600 ÷ 85as 560, pupils should recognize that47 600 ÷ 85 ≈ 600 (not 6 or 6 000)

• Pupils should be able to round off theanswer in the context of a given problem,e.g. pupils should realize the absurdity ofgiving the speed of a car to 5 decimalplaces

5 Squares, square roots, cubes and cuberoots

• find squares, square roots, cubes andcube roots of numbers

• Include the use of the square root sign" " and cube root sign " 3 "

• Include estimation• Include the use of the following keys of a

calculator:2x , x , yx , yx /1


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6 Number sequences

• continue a given number sequence • Recognize simple number patterns andstate the rules for the patterns


Mass, length, time and money

• solve problems involving the use of unitsof mass, length, time, money

• Includei) use of the term: “capacity”ii) practical activities to reinforce concept

of units of measuresiii) conversion of measures and currencyiv) calculation of time in terms of the 24-

hour and 12-hour clocks• reading of clocks, dials and timetables


Ratio and proportion

• find the ratio of two or more quantities • Include expressing a ratio in its lowestterms

• state the relationship between ratio andfraction

• Include rewriting x : y = a : b

as x = ( ab

) y

• use direct and inverse proportion • Include the concept of reciprocals

• solve problems involving ratio and

proportion• Include scales

Rate

• recognize and use common measures ofrate

• Include conversion such as km/h to m/sand vice versa

• solve problems involving rate • Include calculation of average speed

9 Percentage

• convert between− percentage and fraction− percentage and decimal


i) 25% = 0.25 = 14

ii) 50% = 0.5 = 12

iii) 75% = 0.75 = 34

iv) 20% = 0.2 = 15

v) 80% = 0.8 = 45


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• calculate a given percentage of aquantity

• Include reverse problemsExample:40% of a class are boys. If there are 16boys, find the number of pupils in the class.

• express one quantity as a percentage ofanother

• calculate percentage increase/decrease • Include reverse problems such as findingthe original salary given the new salary andthe percentage increase

• solve problems involving percentage


• solve problems on personal andhousehold finance, and simple financialtransactions

• Includei) earnings, simple interest, compound

interest (without use of formulae), hire-purchase, discount, commission, profitand loss, money exchange andtaxation

ii) reverse problems such as finding thecost price given the selling price andthe percentage profit

• extract data from tables and charts tosolve problems

11 Real numbers

Integers

• manipulate integers (positive, negativeand zero)

• Include ordering on the number line

Rational and irrational numbers

• recognize rational and irrational numbers

• Include understanding of the followingterms and their relationships:

Real numbers

irrational numbers rational numbers

fractions integers (+ve, -ve) (+ve, 0, -ve)


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• manipulate rational numbers • find an approximate value of an irrational

number with a calculator

• Exclude tedious computations

MENSURATION


1 Perimeter and area

• find the perimeters and areas of squares,rectangles, triangles, parallelograms,trapeziums and circles

• Include figures related to these shapes

• solve problems involving the perimeters andareas of squares, rectangles, triangles,parallelograms, trapeziums and circles

2 Volume and surface area

• find the volumes and surface areas ofcubes, cuboids, prisms and cylinders

• Include nets of these solids• Include finding the volumes of

composite solids• Exclude oblique prisms and oblique

cylinders

• solve problems involving the volumes andsurface areas of cubes, cuboids, prismsand cylinders

• Include problems involving density

ALGEBRA


1 Algebraic expressions and formulae

• use letters to represent numbers

• express basic arithmetic processesalgebraically

• substitute numbers for words and letters informulae and expressions


• manipulate simple algebraic expressions • Include collecting like terms andremoving brackets


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3 Simple linear equations

• solve simple linear equations

• Include simple cases involving fractional

and decimal coefficients Examples:

i) 12

x + 5 = x - 13

ii) 2 + 0.6x = 2x• solve problems involving linear

equations

GEOMETRY



• identify the following plane figures:− triangles: isosceles triangles,

equilateral triangles, right-angledtriangles, acute-angled triangles,obtuse-angled triangles andscalene triangles

− special quadrilaterals: squares,rectangles, parallelograms,rhombuses, trapeziums and kites

− polygons: pentagons, hexagons,octagons and decagons

• Include the use of the following geometricalterms:

point, line, plane, parallel,perpendicular, right angles, acute,obtuse and reflex angles,complementary andsupplementary angles, baseangles, interior and exteriorangles, regular and irregularpolygons, diagonal and vertex

• Include tiles and tessellations

2 Simple solid figures • identify the following simple solid figures:

cubes, cuboids, prisms, cylinders,pyramids, cones and spheres

• Include sketching of these solids

3 Angle properties

Angles formed with a common vertex

• calculate unknown angles involving:− adjacent angles on a straight line− vertically opposite angles− angles at a point

Angles formed with parallel lines

• calculate unknown angles involving:− corresponding angles− alternate angles− interior angles between parallel lines


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Angle properties of a triangle

• calculate unknown angles involving:− angle sum of a triangle− base angles of an isosceles triangle− angles of an equilateral triangle− exterior angle of a triangle

Angle properties of squares, rectangles,parallelograms and rhombuses

• calculate unknown angles using theangle properties of− squares− rectangles− parallelograms− rhombuses

• Include angle properties related to their

diagonals

4 Geometrical construction of simplegeometrical figures

• measure line segments and angles • Use rulers, set-squares, protractors and

compasses

• draw line segments, angles, parallellines and perpendicular lines

• Include cases where the following aregiven:i) the distance from a point to a lineii) the distance between two parallel

lines

• construct angle bisectors and

perpendicular bisectors

• Use protractors, set squares, compasses

and straight edges/rulers

• construct simple geometrical figuresfrom given data

• Include drawing the nets of cubes, cuboids,prisms and pyramids

STATISTICS


1 Handling data

• collect, classify and tabulate data

• Include the use of tally marks

• read and interpret tables and statisticaldiagrams

• construct a bar graph, pie chart,pictogram, dot diagram, line graph, stemand leaf diagram, and histogram withequal intervals


20


PROBLEM SOLVING



• use appropriate heuristics to solveproblems

2 Practical uses of mathematics

• solve mathematical problems ineveryday life


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ARITHMETIC



• solve problems involving measures, money,ratio, proportion, rate and speed,percentage, personal and household financeand simple financial transactions

2 Standard form

• read and write numbers in standard form A x10n where n is a positive or negative integer,and 1 ≤ A < 10

• Include significant figures of numberswritten in a standard form

• Include estimation and approximation

3 Number sequences

• formulate the algebraic expression for thegeneral term of a number sequence

• Exclude direct application of arithmetic progression and geometric progression formulae

MENSURATION



• find the volumes and surface areas ofspheres, pyramids and cones

• Include the use of the term“hemisphere”

• Include nets where applicable• Exclude oblique pyramids and cones

• solve problems involving the volumes andsurface areas of spheres, pyramids andcones


• express arc length as a fraction ofcircumference and sector area as a fractionof the area of a circle

• find arc length and sector area

• Include finding the area of a segment of

a circle

• solve problems involving arc length andsector area


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ALGEBRA


1 Algebraic manipulation and formulae

• expand products of simple algebraicexpressions

• Examples:i) (ax + b) (cx + d)ii) (ax + by) (cx + dy)

(a, b, c and d are integers)

• factorize algebraic expressions of theform: ax + ay ax + bx + kay + kby a2x2 − b2y2

a2 ± 2ab + b2

ax2 + bx + c

• manipulate simple algebraic fractions

• Include only simple expressions such asthe following:

i) x3

+ x − 42

ii) 23x - 3 5

2( )x −

iii) (34a

) ( 53ab

)

iv) 34a ÷ 9

10a

v) 12x −

+ 3

2−x

• Expressions such as 4

12 −x

+ 2

5−x

and

444

2

2

++−xx

x will be introduced at the upper

secondary level

• transform simple formulae • Exclude formulae involving square roots

2 Solutions of equations

Simultaneous linear equations

• solve simultaneous linear equations intwo unknowns

• Include both the elimination andsubstitution methods

• solve problems involving simultaneous

linear equations


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Simple fractional equations

• solve fractional equations involvingnumerical and linear algebraicdenominators

• Examples:

i) x3

+ x − 24

= 3

ii) 3x

= 6

iii) 32x −

= 6

• Exclude fractional equations such as

12x −

+ 23x −

= 212

• solve problems involving simplefractional equations

Quadratic equations

• solve quadratic equations byfactorization

• solve problems involving quadraticequations

GRAPHS


1 Graphs of linear and quadratic functions

• use cartesian coordinates in twodimensions

• draw graphs of linear and quadratic

functions

• Include finding the value of y from the

graph given the value of x, and vice versa

• use graphical methods to solvesimultaneous linear equations


• interpret and use graphs in practicalsituations

• Include travel graphs (distance-timegraphs) and conversion graphs

• draw graphs using data from practical

situations

• Include the choice of appropriate scales


24


GEOMETRY


1 Motion Geometry

Reflection, rotation, translation,enlargement

• draw the reflection of a simple planefigure in horizontal or vertical lines

• Include “line of reflection” (mirror line)

• draw the rotation of a simple planefigure, about the origin of the x-y plane,or a vertex or a mid-point of an edge ofthe figure, through multiples of 90°

• Include ”centre of rotation” and “angle ofrotation”

• draw the translation of a simple planefigure in the x-y plane

• Include only movements parallel to the xand y axes

• draw an enlargement of a given planefigure

• Include “centre of enlargement” and “scalefactor” (scaling up or down)

• Exclude negative scale factors

• identify reflection, rotation, translation

and enlargement of a given plane figure

• Exclude problems involving finding “line of

reflection”, “centre of rotation”, “angle ofrotation”, "centre of enlargement” and“scale factor”

• draw the image of a figure involvingcombined movements

• Exclude more than 2 movements


• recognise similar and congruent figures

• find unknown sides/angles ofsimilar/congruent figures

• Exclude tests for similarity and congruencybetween two triangles

3 Angle properties of polygon

• calculate− the sum of interior angles of a

polygon− the sum of exterior angles of a

polygon

• Include regular polygons

• calculate unknown angles of a polygon • Include finding the number of sides of apolygon


25


4 Scale drawing

• read and make scale drawings

5 Symmetry

Line symmetry and rotational symmetry

• identify line and rotational symmetry ofplane figures

• Include ‘line of symmetry’ and ‘centre andorder of rotational symmetry’

• Exclude the use of the term ‘pointsymmetry’

• use symmetrical properties of triangles,quadrilaterals and regular polygons

• Include properties of these figures directlyrelated to their symmetries

• use symmetrical properties of prisms,

cylinders, pyramids and cones

• Include ‘plane of symmetry’ and ‘axis of

rotational symmetry’

STATISTICS


1 Averages

• find mean, median and mode • Distinguish between the purposes for whichmean, median and mode are used

• Exclude grouped data

TRIGONOMETRY


1 Pythagoras' theorem

• state Pythagoras’ theorem • Include the converse of the theorem• Proving of the theorem is not required

• apply Pythagoras’ theorem to find a sideof a right-angled triangle

• solve problems applying Pythagoras’theorem

2 Trigonometrical ratios: sine, cosine andtangent

• state the sine, cosine and tangent ratiosfor an acute angle of a right-angledtriangle


26


• find the values of the trigonometrical

ratios with a calculator

• Include the use of inverse trigonometrical

function keys of the calculator

• apply the sine, cosine or tangent ratio foran acute angle to find

− a side of a right-angled triangle − an angle of a right-angled triangle

• Exclude expressing angles in degrees andminutes

• solve trigonometrical problems in twodimensions

• Include angles of elevation and depression

PROBLEM SOLVING






NORMAL ACADEMIC PART B

27

Content Chart - Secondary One & Two (Normal [Academic] Course)Secondary One Secondary Two






6 Number sequences



9 Percentage

10 Real numbers• integers• rational and irrational numbers



3 Standard form

4 Number sequences

1 Perimeter and area• square, rectangle, triangle,

parallelogram, trapezium, circle

2 Volume and surface area• cube, cuboid, prism, cylinder

1 Volume and surface area• sphere, pyramid and cone


1 Algebraic expressions andformulae


1 Algebraic manipulation andformulae

2 Solutions of equations• simple linear equations• simultaneous linear equations• simple fractional equations

NORMAL ACADEMIC PART B

28

Content Chart - Secondary One & Two (Normal [Academic] Course)


1 Graphs of linear and quadraticfunctions





vertex• angles formed with parallel lines• angle properties of triangles• angle properties of squares,


4 Geometrical construction ofsimple geometrical figures



3 Scale drawing


1 Handling data• table and chart• pictogram• dot diagram• bar graph• line graph• pie chart• stem and leaf diagram• histogram






LEVEL : SEC 1 (NORMAL ACADEMIC) PART B

29


ARITHMETIC


1 Whole numbers

The four operations

• use the four operations for calculationswith whole numbers

• Include combined operations, i.e. thecorrect ordering of operations and the useof brackets




Ordering

• order numbers• use the following symbols correctly:

=, ≠, >, <, ≥, ≤








• use fractions and decimals



0.25 = 14

, 0.75 = 34

, 0.125 = 18

0.333 ≈ 13

, 0.667 ≈ 23


Ordering



decimals


30


The four operations


• Include

i) combined operationsii) rounding-off decimals to a specific




Rounding off



Estimation



• Include mental estimation


• Use the relevant keys of a scientificcalculator


estimation, e.g. in evaluating 47 600 ÷ 85as 560, pupils should recognize that47 600 ÷ 85 ≈ 600 (not 6 or 6 000)

• Pupils should be able to round off theanswer in the context of a given problem,e.g. pupils should realize the absurdity ofgiving the speed of a car to 5 decimalplaces

5 Squares, square roots, cubes and cube roots


• Include the use of the square root sign ‘ ‘, and cube root sign ‘ 3 ‘• Include estimation• Include the use of the following keys of a


6 Number sequences

• continue a given number sequence • Recognise simple number patterns andstate the rules for the patterns


31



Mass, length, time and money

• solve problems involving the use of unitsof mass, length, time, money

• Includei) use of the term “capacity”ii) practical activities to reinforce concept

of units of measuresiii) conversion of measures and currencyiv) calculation of time in terms of the 24-

hour and 12-hour clocksv) reading of clocks, dials and timetables






as x = ( ab

) y

• use direct and inverse proportion • Include the concept of reciprocals

• solve problems involving ratio and

proportion

• Include scales

Rate

• recognize and use common measures ofrate

• Include conversion such as km/h to m/sand vice versa

• solve problems involving rate • Include calculation of average speed

9 Percentage



i) 25% = 0.25 = 14

ii) 50% = 0.5 = 12

iii) 75% = 0.75 = 34

iv) 20% = 0.2 = 15

v) 80% = 0.8 = 45


32



• Include reverse problemsExample:40% of a class are boys. If there are 16boys, find the number of pupils in the class



• solve problems involving percentage

10 Real numbers

Integers

• manipulate integers (positive, negativeand zero)

• Include ordering the number line

Rational and irrational numbers

• recognize rational and irrational numbers

• Include understanding the following terms

and their relationships:

Real numbers

irrational numbers rational numbers

fractions integers (+ve, -ve) (+ve, 0, -ve)

• manipulate rational numbers • find an approximate value of an irrational

number with a calculator

• Exclude tedious computations

MENSURATION



• find the perimeters and areas ofsquares, rectangles, triangles,parallelograms, trapeziums and circles


• solve problems involving the perimetersand areas of squares, rectangles,triangles, parallelograms, trapeziumsand circles


33



• find the volumes and surface areas of cubes, cuboids, prisms and cylinders

• Include nets of these solids• Include finding the volumes of composite

solids• Exclude oblique prisms and oblique

cylinders

• solve problems involving the volumesand surface areas of cubes, cuboids,prisms and cylinders


ALGEBRA


1 Algebra expression and formulae

• use letters to represent numbers

• express basic arithmetic processesalgebraically

• substitute numbers for letters informulae and expressions

2. Simple Algebraic Manipulation

• manipulate simple algebraic expressions • Include collecting like terms and removingbrackets

GEOMETRY


1 Simple Plane Figures






point, line, plane, parallel,perpendicular, right angles, acute,obtuse and reflex angles,complementary andsupplementary angles, baseangles, interior and exteriorangles, regular and irregularpolygons, diagonal and vertex

• include tiles and tessellations


34




• Include sketching these solid figures

3 Angle properties




• calculate unknown angles involving:

− corresponding angles− alternate angles− interior angles between parallel lines


• calculate unknown angles involving:− angle sum of a triangle− base angles of an isosceles triangle− angles of an equilateral triangle− exterior angle of a triangle


• calculate unknown angles using theangle properties of− squares− rectangles− parallelograms− rhombuses


diagonals

4. Geometrical construction of simplegeometrical figures

• measure line segments and angles • Use rulers, set-squares, protractors andcompasses


35



• Include cases where the following aregiven:i) the distance from a point to a lineii) the distance between two parallel lines

• construct angle bisectors andperpendicular bisectors

• Use protractors, set squares, compassesand straight edges/rulers

• construct simple geometrical figuresfrom given data

• Include the drawing of nets of cubes,cuboids, prisms and pyramids

STATISTICS


1 Handling data• collect, classify and tabulate data



• construct a bar graph, pie chart,pictogram, dot diagram, line graph, stemand leaf diagram, and histogram withequal intervals

PROBLEM SOLVING







36


ARITHMETIC




• Includei) earnings, simple interest, compound

interest (without use of formulae), hire-purchase, discount, commission, profitand loss, money exchange andtaxation

ii) reverse problems such as finding thecost price given the selling price andthe percentage profit



• solve problems involving measures,money, ratio, proportion, scale, rate,speed and percentage

3 Standard form

• read and write numbers in standard formA x 10n where n is a positive or negativeinteger, and 1 ≤ A < 10

• Include significant figures of numberswritten in standard form

• Include estimation and approximation

4 Number sequences

• formulate the algebraic expression forthe general term of a number sequence

• Exclude direct application of arithmetic progression and geometric progression formulae

MENSURATION



• find the volumes and surface areas ofspheres, pyramids and cones

• Include the use of the term “hemisphere”• Include nets where applicable• Exclude oblique pyramids and cones

• solve problems involving the volumesand surface areas of spheres, pyramidsand cones


37



• express arc length as a fraction ofcircumference and sector area as afraction of the area of a circle

• find arc length and sector area • Include finding the area of a segment of acircle

• solve problems involving arc length andsector area

ALGEBRA

1 Algebraic manipulation and formulae


• Examples:i) (ax + b) (cx + d)ii) (ax + by) (cx + dy)


• factorize algebraic expressions of theform:

ax + ay ax + bx + kay + kby a2x2 − b2y2

a2 ± 2ab + b2

ax2 + bx + c

• manipulate simple algebraic fractions

• Include only simple expressions such as

the following:

i) x3

+ x − 42

ii) 23x - 3 5

2( )x −

iii) (34a

) ( 53ab

)

iv) 34a ÷ 9

10a

v) 12x −

+ 23x −



38



Simple linear equations


• Include simple cases involving fractional

and decimal coefficients Examples:

i) 12

x + 5 = x - 13

ii) 2 + 0.6x = 2x

• solve problems involving linearequations

Simultaneous linear equations

• solve simultaneous linear equations intwo unknowns

• Include both the elimination andsubstitution methods

• solve problems involving simultaneouslinear equations

Simple fractional equations

• solve fractional equations involvingnumerical and linear algebraicdenominators

• Examples:

i) x3

+ x − 24

= 3

ii) 3x

= 6

iii) 32x −

= 6

• Exclude fractional equations such as

12x −

+ 23x −

= 212

• solve problems involving simplefractional equations

GRAPHS


1 Graphs of linear and quadratic functions


• draw graphs of linear and quadraticfunctions

• Include finding the value of y from thegraph given the value of x, and vice versa


39





• Include travel graphs (distance-timegraphs) and conversion graphs

• draw graphs using data from practicalsituations

• Include choice of appropriate scales

GEOMETRY



• recognise similar and congruent figures

• find unknown sides/angles ofsimilar/congruent figures

• Exclude tests for similarity/congruencybetween two triangles

2 Angle properties of a polygon

• calculate- the sum of interior angles of a

polygon- the sum of exterior angles of a

polygon


• calculate unknown angles of a polygon • Include finding the number of sides of apolygon

3 Scale Drawing


4 Symmetry





• use symmetrical properties of triangles,

quadrilaterals and regular polygons

• Include properties of these figures directly

related to their symmetries

• use symmetrical properties of prisms,cylinders, pyramids and cones

• Include ‘plane of symmetry’ and ‘axis ofrotational symmetry’


40


STATISTICS


1 Averages

• find mean, median and mode • Distinguish between the purposes for whichmean, median and mode are used


PROBLEM SOLVING






NORMAL TECHNICAL PART B

41

Content Chart - Secondary One & Two (Normal [Technical] Course)








7 Ratio, proportion, scale, rate andspeed

8 Percentage



3 Directed numbers• concept and notation• ordering• the four operations

4 Number sequences

1 Perimeter and area• square• rectangle• triangle• parallelogram• trapezium• circle

1 Volume and surface area• cube• cuboid• prism• cylinder

1 Algebraic expressions andformulae• concept and notation• substitution• simplification

1 Algebraic expressions andformulae• substitution• manipulation

2 Solutions of simple linearequations

NORMAL TECHNICAL PART B

42

Content Chart - Secondary One & Two (Normal [Technical] Course)


1 Linear graphs





vertex• angles formed with parallel

lines• angle properties of triangles• angle properties of squares,


4 Construction of triangles



3 Geometrical construction• simple four-sided figures• scale drawing

1 Handling data• table and chart• pictogram• dot diagram• bar graph• line graph• pie chart• histogram




3 Mathematics for leisure andrecreation*



3 Mathematics for leisure andrecreation*

*Non-examination topic

LEVEL : SEC 1 (NORMAL TECHNICAL) PART B

43


ARITHMETIC


1 Whole numbers

The four operations• use the four operations for calculations

with whole numbers• Include combined operations, i.e. correct

ordering of operations and the use ofbrackets




Ordering• order numbers• use the following symbols correctly:

=, ≠, >, <, ≥, ≤







Concept and notation• use fractions and decimals



0.25 = 14

, 0.75 = 34

, 0.125 = 18

0.333 ≈ 13

, 0.667 ≈ 23


Ordering



decimals


44


The four operations


• Includei) combined operationsii) rounding off decimals to a specific




Rounding off



Estimation



• Include mental estimation4 Use of a scientific calculator

• use the relevant keys of a scientificcalculator


estimation, e.g. in evaluating 47 600 ÷ 85as 560, pupils should recognize that

47 600 ÷ 85 ≈ 600 (not 6 or 6 000)• Pupils should be able to round off the

answer in the context of a given problem,e.g. pupils should realize the absurdity ofgiving the speed of a car to 5 decimalplaces

5 Squares, square roots, cubes and cuberoots


• Include the use of the square root sign ‘ ‘

and the cube root sign ‘ 3 '• Include estimation• Include the use of the following keys of a


6 Measures and money Mass, length, time and money

• use common instruments to measurequantity

• Include measuring capacity/volume of a

liquid• Common instruments refer to rulers,

measuring cylinders, weighing scales andstop-watches


45


• solve problems involving the use of unitsof mass, length, time and money

• Include(i) conversion of measures and currency(ii) calculation of time in terms of 24-hour

and 12-hour clocks(iii) reading of clocks, dials, timetables and

charts






as x = ( ab

) y

• use ratio, proportion and scale • Include direct proportion and inverseproportion

• Include map reading and estimatingdistance from the map

• Include the concept of reciprocals • solve problems involving ratio, proportion

and scale

Rate and speed

• use common measures of rate andspeed

• Include conversion such as km/h to m/s andvice versa

• solve problems involving rate and speed • Include calculation of average speed

8 PERCENTAGE



i) 25% = 0.25 = 14

ii) 50% = 0.5 = 12

iii) 75% = 0.75 = 34

iv) 20% = 0.2 = 15

v) 80% = 0.8 = 45


• Include reverse problemsExample:40% of a class are boys. If there are 16boys, find the number of pupils in the class.


46




MENSURATION



• find the perimeters and areas ofsquares, rectangles, triangles,parallelograms, trapeziums and circles


• solve problems involving the perimeters

and areas of squares, rectangles,triangles, parallelograms, trapeziumsand circles

ALGEBRA




• use letters to represent numbers• express basic arithmetic processes

algebraically

Substitution

• substitute numbers for letters inexpressions and formulae

• Exclude expressions with brackets• Exclude expressions involving squares and

higher powers

Simplification

• simplify simple algebraic expressions

• Include collecting like terms• Exclude removing of brackets at this level• Exclude expressions involving squares and

higher powers


47


GEOMETRY








point, line, plane, parallel,perpendicular, right angle, acute,obtuse and reflex angles,complementary andsupplementary angles, baseangle, interior and exterior angles,regular and irregular polygons,diagonal and vertex

• Include tiles and tessellations



• Include sketching these solid figures

3 Angle properties




• calculate unknown angles involving:− corresponding angles− alternate angles− interior angles between parallel lines


• calculate unknown angles involving:− the angle sum of a triangle− the base angles of an isosceles

triangle− the angles of an equilateral triangle− the exterior angle of a triangle


48



• calculate unknown angles using theangle properties of a− square− rectangle− parallelogram− rhombus


diagonals

4 Construction of triangles

• measure line segments and angles • Use rulers, set-squares, protractors andcompasses


• Include cases where the following aregiven:(i) the distance from a point to a line(ii) the distance between two parallel lines

• construct angle bisectors and

perpendicular bisectors

• Use protractors, set squares, compasses

and straight edges/rulers

• construct triangles from given data

STATISTICS


1 Handling data

• collect, classify and tabulate data



• construct a bar graph, pie chart,pictogram, dot diagram, line graph, andhistogram with equal intervals


49


PROBLEM SOLVING






3 Mathematics for leisure and recreation*

• appreciate solving non-routine problems

• Non-routine problems in this topic refer tosimple puzzles and simple problems forleisure and recreation

* Non-examination topic


50


ARITHMETIC




• Include

i) earnings, simple interest, compoundinterest (without the use of formulae),hire-purchase, discount, commission,profit and loss, money exchange andtaxation

II) reverse problems such as finding thecost price given the selling price andthe percentage profit



• solve problems involving measures,money, ratio, proportion, scale, rate,speed and percentage

3 Directed numbers


• use directed numbers • Include the use of the term "integers"(positive, negative, zero)

Ordering

• compare and order directed numbers • Include ordering on the number line

The four operations

• use the four operations for calculationwith directed numbers

• Include combined operations and the use ofbrackets

4 Number sequences

• continue a given number sequence • Recognise simple number patterns andstate the rules for the patterns


51


MENSURATION



• find the volumes and surface areas ofcubes, cuboids, prisms and cylinders

• Include nets of these solids• Include finding the volumes of composite

solids• Exclude oblique prisms and oblique

cylinders

• solve problems involving the volumes

and surface areas of cubes, cuboids,prisms and cylinders


ALGEBRA



Substitution

• substitute numbers for letters inexpressions and formulae

Manipulation

• simplify simple algebraic expressions • Include collecting like terms and removingbrackets


• Examples:(i) (ax + b) (cx + d)(ii) (ax + by) (cx + dy)


• factorize algebraic expressions of theform:ax + ayax + bx + kay + kbya2x2 - b2y2

x2 ± 2xy + y2

ax2 + bx + c



52



Solution of simple linear equations


• Include cases involving fractional anddecimal coefficients

Examples:

(i) 12

x + 5 = x - 13

(ii) 2 + 0.6x = 2x

• solve problems involving linearequations

GRAPHS


1 Graphs

Linear graphs


• draw linear graphs • Include finding the value of y from the graphgiven the value of x, and vice versa




• Include travel graphs (distance-time graphs)

and conversion graphs

• draw graphs using data from practicalsituations

• Include choice of appropriate scales

GEOMETRY


1 Symmetry






53


• use symmetrical properties of triangles,quadrilaterals and regular polygons

• Include properties of these figures directlyrelated to their symmetries

• use symmetrical properties of prisms,cylinders, pyramids and cones

• Include ‘plane of symmetry’ and ‘axis ofrotational symmetry'

2 Construction

Simple four-sided figures

• Construct simple four-sided figures

Scale drawing


3 Angle properties of polygon

• calculate- the sum of interior angles of a polygon- the sum of exterior angles of a polygon


• calculate the unknown angles of apolygon

• Include finding the number of sides of apolygon

STATISTICS


1 Averages

• find the mean, median and mode • Distinguish between the purposes for whichmean, median and mode are used



54


PROBLEM SOLVING






3 Mathematics for leisure and recreation*

• appreciate solving non-routine problems • Non-routine problems in this topic refer to

simple puzzles and simple problems forleisure and recreation

* Non-examination topic

APPENDIX

55

DEFINITION OF SUGGESTED THINKING SKILLS

• Classifying - using relevant attributes to sort, organise and group information • Comparing - using common attributes to identify commonalities and

discrepancies across numerous sets of information • Identifying Attributes & Components - recognising and articulating the parts

that together constitute a whole • Sequencing - placing items in a hierarchical order according to a quantifiable

value • Induction - drawing a general conclusion from clues gathered (from specific to

general) • Deduction - inferring various specific situations or examples from given

generalisations (from general to specific) • Generalising - using repeated, controlled and accurate observations to develop

a rule, principle or formula that explains a number of related situations • Verifying - checking or confirming the truth of an idea, using specific standards

or criteria of evaluation • Spatial Visualisation - visualising a situation or an object and mentally

manipulating various alternatives for solving a problem related to a situation orobject without the benefit of concrete manipulatives

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