SYLLABUSES FOR SECONDARY SCHOOLS -...

SYLLABUSES FOR SECONDARY SCHOOLS MATHEMATICS

SECONDARY 1 - 5 (1999)

CONTENTS

CONTENTS

MEMBERSHIP OF THE CDC MATHEMATICS SUBJECT COMMITTEE (SECONDARY)

PREAMBLE

CHAPTER 1 INTRODUCTION

Principles of Curriculum Design

CHAPTER 2 AIMS AND OBJECTIVES

2.1 Aims of the Secondary School Mathematics Education

2.2 Objectives of the Secondary School Mathematics Education

2.2.1 Knowledge Domain

2.2.2 Skill Domain

2.2.3 Attitude Domain

CHAPTER 3 ORGANIZING THE MATHEMATICS CURRICULUM

3.1 Dimensions

3.2 Structure

3.3 Time Allocation

3.4 School-based Mathematics Curriculum

CHAPTER 4 LEARNING TARGETS AND OBJECTIVES

4.1 An Overview of Learning Targets

4.1.1 Number and Algebra Dimension

4.1.2 Measures, Shape and Space Dimension

4.1.3 Data Handling Dimension

4.2 An Overview of Learning Modules and Units




4.2.4 Further Applications Module

4.3 Learning Objectives for Key Stage 3 (S1-S3)




4.4 Learning Objectives for Key Stage 4 (S4 -S5)





CHAPTER 5 TEACHING SUGGESTIONS

5.1 Curriculum Strategies

5.1.1 Process of Learning

5.1.2 Catering for Learner Differences

5.1.3 Appropriate Use of Information Technology (IT)

5.1.4 Appropriate Use of Multifarious Teaching Resources

5.2 Teaching Strategies in Individual Dimensions





CHAPTER 6 ASSESSMENT

6.1 Purposes of Assessment

6.2 Assessment Strategies

6.3 Feedback from Assessment

ANNEXES

ANNEX I Learning Targets and Learning Objectives for

Key Stages 1and2

ANNEX II An Overview of Learning Dimensions and Modules for

Key Stages3 and 4

ANNEX III The Flowchart of Learning Units for Secondary School

Mathematics Curriculum

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CONTENTS

Page

CONTENTS iii

MEMBERSHIP OF THE CDC MATHEMATICS SUBJECTCOMMITTEE (SECONDARY)

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PREAMBLE vii

CHAPTER 1 INTRODUCTION Principles of Curriculum Design 1



2.2 Objectives of the Secondary School Mathematics Education

2.2.1 Knowledge Domain

2.2.2 Skill Domain

2.2.3 Attitude Domain

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3.1 Dimensions

3.2 Structure

3.3 Time Allocation

3.4 School-based Mathematics Curriculum

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4.3 Learning Objectives for Key Stage 3 (S1-S3)




4.4 Learning Objectives for Key Stage 4 (S4 -S5)





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5.1 Curriculum Strategies




5.1.4 Appropriate Use of Multifarious Teaching Resources






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6.1 Purposes of Assessment

6.2 Assessment Strategies

6.3 Feedback from Assessment

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ANNEXES

ANNEX I Learning Targets and Learning Objectives for Key Stages 1 and 2 48

ANNEX II An Overview of Learning Dimensions and Modules for KeyStages 3 and 4

56

ANNEX III The Flowchart of Learning Units for Secondary SchoolMathematics Curriculum

57

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CURRICULUM DEVELOPMENT COUNCILMATHEMATICS SUBJECT COMMITTEE (SECONDARY)

The membership since 1 September 1995 has been as follows:

Chairman Mr. FUNG Tak-wah (until 31 August 1997)

Mrs. LEE HO Mei-fun(from 1 September 1997)

Vice-Chairman Senior Curriculum Officer (Mathematics),Curriculum Development Institute, Education Department(Mr. IP Chiu-kwan, until 30 September 1997)(Ms. TANG Mei-yue from 1 October 1997 to 16 December 1998)

Curriculum Officer (Mathematics),Curriculum Development Institute, Education Department(Ms. TANG Mei-yue from 17 December 1998)

Ex-officio Member Principal Inspector (Mathematics),Advisory Inspectorate Division, Education Department(Mr. TSANG Kin-wah)

Members Mr. CHEUNG Pak-hong(from 1 September 1997)

Dr CHIANG Yik-man

Mr. CHOW Wing- yiu, Colin(from 1 September 1997)

Mr. CHU Siu-wing(from 1 September 1997)

Mrs. LEE HO Mei-fun(until 31 August 1997)

Mr. LAI Chu-kuen

Mr. LAM Yat-fung, James(from 1 September 1996 to 31 August 1998)

Mr. LEUNG Kwok-ying(from 1 September 1997)

Mr. LUK Hok-wing

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Mr. Man Ping- fai(until 31 August 1996)

Miss SHUM So- lan(until 31 August 1997)

Ms. TO Lai-ching

Mr. TSANG Tze-fai(from 1 September 1998)

Mr. WONG Chi-sun(until 31 August 1997)

Ms. YAN Pui-lung(until 31 August 1997 )

Co-opted Member Mr. CHEUNG Pak-hong(until 31 August 1997)

Mr. HUI Wai-tin(until 31 August 1997)

Mr. LAI Chi-keung(until 31 August 1996)

Dr WONG Ngai-ying(until 31 August 1996)

Ms. YAN Pui-lung(from 1 September 1997)

Secretary Curriculum Officer (Mathematics),Curriculum Development Institute, Education Department(Ms. TANG Mei-yue from 1 November 1995 to 30 September 1997)(Ms. LAO Kam-ling from 18 May 1999)

Assistant Curriculum Officer (Mathematics)Curriculum Development Institute, Education Department(Mr. FUNG Wang- tak until 31 October 1995)(Mr. CHAN Sau-tang from 1 October 1997 to 19 February 1999)

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PREAMBLE

This syllabus is one of a series prepared for use in secondary schools by the Curriculum

Development Council, Hong Kong. The Curriculum Development Council, together with its co-

ordinating committees and subject committees, is widely representative of the local educational

community, membership including heads of schools and practising teachers from government and

non-government schools, lecturers from tertiary institutions, officers of the Hong Kong

Examinations Authority and those of the Curriculum Development Institute, the Advisory

Inspectorate Division and other divisions of the Education Department. The membership of the

Council also includes parents and employers.

All syllabuses prepared by the Curriculum Development Council for secondary 1-5 normally

lead to appropriate examinations at the Certificate of Education level provided by the Hong Kong

Examinations Authority.

This syllabus is recommended by the Education Department for use in Secondary 1 to 5.

Once the syllabus has been implemented, progress will be monitored by the Advisory Inspectorate

Division and the Curriculum Development Institute of the Education Department. This will enable

the Mathematics Subject Committee (Secondary) of the Curriculum Development Council to

review the syllabus from time to time in the light of teaching and learning experiences.

All comments and suggestions on the syllabus may be sent to:

Principal Curriculum Development Officer (Mathematics)

Curriculum Development Institute,

Education Department,

13/F, Wu Chung House,

213 Queen’s Road East,

Wanchai,

Hong Kong.

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1

CHAPTER 1 INTRODUCTION

Mathematics pervades all aspects of life, whether at home, in civic life or in the workplace. It has

been central to nearly all major scientific and technological advances. Also, many of the

developments and decisions made in industry and commerce, the provision of social and community

services as well as government policy and planning, rely to an extent on the use of mathematics.

It is important for our students to gain experience and build up the foundation skills and knowledge in

mathematics that can facilitate their future development in various aspects. It is also important that

our students are able to value mathematics and appreciate the beauty of mathematics after

mathematics education in school. In the information explosion era, there are drastic changes both in

our society and in the background of our students. It is vital that the curriculum should undergo

continuous review and renewal in order to meet the needs of our students and the community.

In reviewing the mathematics curriculum, there are several principles to be followed. The needs of

our students and the community are important considerations in developing the aims of the secondary

school mathematics education. The worldwide trends of mathematics curriculum have also been

taken into consideration in deciding the principles. This syllabus presented is a revised edition of

the version published in 1985. It has been scheduled for implementation in schools with effect from

September 2001 at Secondary 1.

Principles of Curriculum Design

The following principles are used to guide the evolution of the Aims and Objectives, the structure of

the curriculum, and the identification of objectives in each module and unit in the syllabus.

l Target-oriented

To ensure that learners will spend their time and effort meaningfully and for maximum benefits, there

must be a plan for them to work according to specific Learning Targets and Objectives which are

geared towards the Aims and Objectives of the school mathematics curriculum. They are

organized progressively across four Key Stages in primary and secondary schooling: Key Stage 1

(Primary 1 to 3), Key Stage 2 (Primary 4 to 6), Key Stage 3 (Secondary 1 to 3) and Key Stage 4

(Secondary 4 to 5).

The learning targets and objectives for Key Stages 1 and 2 can be found in Annex I. Continuing the

learning in primary schooling, the overall Aims and Objectives for Key Stages 3 and 4 are stated in

Chapter 2. The Learning Targets and Objectives for each dimension in each learning stage are

further elaborated in Chapter 4 to spell out the specific learning objectives for each learning area.

All learning and assessment activities fulfil the learning objectives of that particular unit and are geared

towards the maximum learning effectiveness for achieving the Aims and Objectives.

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l Catering for learner differences

Upon the implementation of universal education in Hong Kong, a wider range of students gain access

to secondary mathematics than have been in the past. The school mathematics curriculum should

cater for the diversity of students’ needs and for the wide spectrum of ability among them.

Besides the various ways of organizing students’ activities in the class to cater for learner differences,

the Foundation Part of the curriculum is identified. The Foundation Part is the essential part of the

Syllabus which ALL students should strive to learn. Apart from the Foundation Part, teachers can

judge for themselves the suitability and relevance of other topics in the Whole Syllabus for their own

students. For more able students, teachers can adopt some enrichment topics at their discretion

to extend these students’ horizon and exposure in mathematics.

In order to provide further flexibility for teachers to organize the teaching sequences to meet

individual teaching situations, the learning units and modules for each dimension are subdivided into

key stages (KS), i.e. KS3 for S1-S3, KS4 for S4-S5. Teachers are free to design their school

based mathematics curriculum for each year level with all learning areas suggested for each key stage

in mind.

l Relevance of study to students

In order that mathematics learning efforts are effective, the knowledge and skills to be learnt should

be determined by the activities deemed suitable for the age-group concerned. Great care is taken

to ensure that the curriculum is organized with a cognitive developmental perspective. For instance,

exposure to concrete objects and personal experience is planned to support abstract discussion as

far as possible.

Students who find the study relevant to their experience will be motivated to learn the subject. Daily

life applications are emphasized in the curriculum. Stories of historical development of mathematics

knowledge are included to enable students to understand mathematics knowledge evolved from real-

life problems and refined after years. A new module “Further Applications” which includes the

application of mathematics in more complex real-life situations requires students to integrate their

knowledge and skills from various disciplines to solve problems.

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l Impact of information technology

The tools for solving mathematical problems change from time to time. The introduction of

electronic calculators in 1980’s has influenced the teaching of secondary school mathematics.

There are different roles electronic calculators can play. A general worry among teachers and

parents is that the unwise use of calculators by students would hamper their development of

computational skills. With years of experience in the classrooms of various countries, the positive

role that calculators could play in the mathematics learning is generally aware of.

Today we are confronted with a similar situation. The popularity of graphing calculators, the

availability of computers and other information technology aids in the classrooms will have impact on

the mathematics curriculum in terms of contents and strategies for teaching and learning of

mathematics. There are ranges of ways in which information technology may be used in

mathematics classes, including data analysis, simulation device, graphical presentation, symbolic

manipulation and observing patterns. The appropriate use of information technology in the teaching

and learning of mathematics becomes one of the emphases in the mathematics curriculum.

l Fostering general abilities and skills

Knowledge is expanding at an ever faster pace and new challenges are continually being posed by

the rapid changes in technology and in the way society evolves. It is important that students need to

develop their capabilities to learn how to learn, to think logically and creatively, to develop and use

knowledge, to analyze and solve problems, to access information and process it effectively, and to

communicate with others so that they can meet the challenges that confront them now and in the

future. Acquiring mathematics knowledge has always been emphasized, but fostering these general

abilities and skills are strongly advocated for all students in the revised curriculum.

In the curriculum, fundamental and intertwining ways of learning and using knowledge such as

inquiring, communicating, reasoning, conceptualizing and problem-solving are considered important

in mathematics education. On the one hand, students are expected to learn mathematics to enhance

the development of these skills. On the other hand, students are expected to use these learning

strategies to construct their mathematics knowledge. A variety of learning activities should be

planned and geared towards the development of these general abilities and skills.

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The secondary school mathematics curriculum continues the development of the learning

of mathematics in the primary school. To enable students to cope confidently with the

mathematics needed in their future studies, workplaces or daily life in a technological and

information-rich society, the curriculum aims at developing students:

u the ability to conceptualize, inquire, reason and communicate mathematically, and

to use mathematics to formulate and solve problems in daily life as well as in

mathematical contexts;

u the ability to manipulate numbers, symbols and other mathematical objects;

u the number sense, symbol sense, spatial sense and a sense of measurement as

well as the capability in appreciating structures and patterns;

u a positive attitude towards mathematics and the capability in appreciating the

aesthetic nature and cultural aspect of mathematics.

2.2. Objectives of the Secondary School Mathematics Education

2. 2. 1. Knowledge Domain

To induce children to understand and grasp the knowledge of the following:

² the directed numbers and the real number system;

² the algebraic symbols to describe relations among quantities and number patterns;

² the equations, inequalities, identities, formulas and functions;

² the measures for simple 2-D and 3-D figures;

² the intuitive, deductive and analytic approach to study geometric figures;

² the trigonometric ratios and functions;

² the statistical methods and statistical measures;

² the simple ideas of probability and laws of probability.

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2. 2. 2. Skill Domain

To develop the following skills and capabilities in:

² basic computations in real numbers and symbols and an ability to judge reasonableness

of results;

² using the mathematical language to communicate ideas;

² reasoning mathematically, i.e. they should conjecture, test and build arguments about the

validity of a proposition;

² applying mathematical knowledge to solve a variety of problems;

² handling data and generating information;

² number sense and spatial sense;

² using modern technology appropriately to learn and do mathematics;

² learning mathematics independently and collaboratively for the whole life.

2. 2. 3. Attitude Domain

To foster the attitudes to:

² be interested in learning mathematics;

² be confident in their abilities to do mathematics;

² willingly apply mathematical knowledge;

² appreciate that mathematics is a dynamic field with its roots in many cultures;

² appreciate the precise and aesthetic aspect of mathematics;

² appreciate the role of mathematics in human affairs;

² be willing to persist in solving problems;

² be willing to work cooperatively with people and to value the contribution of others.

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3. 1 DimensionsIn the primary school level, the mathematics syllabus is organized in five dimensions, namely Number,

Measures, Shape and Space, Algebra, and Data Handling. Less emphasis will be put on the

Number and Measures Dimensions in the secondary school level. It is not easy to identify certain

learning areas into one single dimension in key stages 3 and 4. For instance, the Trigonometry can be

considered as a measure for triangles in junior forms but will be emphasized more in a functional

perspective in senior forms. Hence, it is more reasonable to merge and integrate 5 dimensions into

3, namely Number and Algebra, Measures, Shape and Space and Data Handling.

The learning contents in each dimension are further subdivided into modules and units. The

subdivision, on the one hand, reflects, to a certain extent, the relations of learning areas of the similar

nature. On the other hand, it is hoped that the arrangement can facilitate students to relate learning

areas of the similar nature in different years’ of study. An overview of learning dimensions and

learning modules can be referred to Annex II. Further details of units for each module and the

corresponding time ratio are listed in Section 4.2.

It should be noted that the classification of dimensions, modules and units does not mean that they

are discrete. In fact, concepts of mathematics are connected within a framework that is ‘multi-

dimensional’. Concepts in one dimension are very often linked to concepts in other dimensions.

For example, the Number and Algebra Dimension provides tools that may be needed in all other

areas of mathematics. It is important for students to appreciate the interrelations of various

disciplines of mathematical knowledge.

3. 2 StructureAs the curriculum is designed for the whole population of secondary school students, which covers a

wide range of abilities, interests and needs. To assist teachers to tailor the mathematics curriculum

to meet the needs of their individual groups of students, it is useful to identify the ‘Foundation Part’

of the Whole Syllabus . The ‘Foundation Part’ of the Syllabus represents the topics that ALL

students should strive to master. The Foundation Part is identified under the principles that:

1. it is the essential part of the Syllabus stressing the basic concepts, knowledge, properties

and simple applications in real life situations;

2. it contains different components that constitute a coherent curriculum.

Teachers can judge for themselves the suitability and relevance of topics outside the Foundation Part

of the Whole Syllabus for their own students. Teachers may also include some Enrichment

Topics at their discretion. These optional enrichment topics, targeted at the most able students,

could extend students’ exposure in mathematics. The objectives for the Non-Foundation Part of the

Syllabus and some enrichment topics are respectively underlined and denoted with ** in Sections

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4.3 and 4.4 for teachers’ reference.

3. 3 Time AllocationThe suggested time allocated for teaching the subject for years one to three is 5 periods per week

and that for years four and five is 6 periods per week. It is assumed that there are 40 minutes in

each period and 40 periods in each week. In Secondary One to Three and Secondary Four to Five,

a total of 480 and 280 periods should be available for the total three and two years respectively.

To aid teachers in judging how far to take a given topic, a time ratio is given.

It can be seen that there are a number of periods left in each key stage. These spare periods can

be used for consolidation activities or enrichment activities etc. to suit the teaching approaches and

the standard of students in the individual schools.

Allocation of time ratio in respective KS3 and KS4 is listed as follows:

Key Stage

Number and

Algebra

Dimension Time

ratio ( %*)

Measures, Shape

and Space

Dimension

Time ratio(%*)

Data Handling

Dimension

Time ratio

(%*)

Further

Applications

Module

Time ratio

( %*)

Sub-total

KS3 (S1-S3) 162 (39) 192 (46) 60 (15) 414

KS4 (S4-S5) 113 (42) 88 (33) 35 (13) 30 (12) 266

Total 275 (40) 280 (41) 95 (14) 30 (5) 680

* The percentage is calculated with the number of periods divided by the sub-total of periods in each

key stage.

3.4 School-based Mathematics CurriculumThe purpose of subdividing the learning units and modules only up to each key stage is to provide

flexibility for schools to design their school-based mathematics curriculum to suit individual school’s

need. In designing school-based mathematics curriculum, schools are encouraged to

l decide the aims and targets for the whole school mathematics curriculum and each key stage;

l select the depth of treatment of the learning units which lie outside the Foundation Part of the

Syllabus;

l arrange the learning units in a logical sequence for each year level;

l choose an appropriate textbook;

l decide the learning activities such as statistics project work, analyzing information collected

from the Internet, group work, etc. to be carried out in the school year;

l select extra-curricular activities such as mathematics competition, mathematics bulletin,

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mathematics reading scheme etc, for students in each individual year level or for the whole

school; and

l decide the methods of assessment and recording that provide feedback for teaching and

learning.

In selecting learning units for each year level, schools should follow the following principles:

i) the cognitive development of students;

ii) the mathematical abilities of students;

iii) the learning objectives for each learning unit;

iv) the inter-relation of learning units; (refer Annex III)

v) the inter-relation of mathematical learning in different year levels; and

vi) the total number of periods allocated for mathematics in each school year.

It should be noted that there are many combinations of learning units for each year level. Schools

can arrange the learning sequence with the orientation to focus on one dimension first and then the

other dimensions in later forms. Schools can also arrange the learning sequence for students to

learn each dimension spirally in each year. Schools may even reserve more periods in junior forms to

revise and consolidate students’ learning in primary schools and focus only on the Foundation Part

of the Syllabus over the five years to cater for low achievers.

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Key Stage 3 (S1 - S3) Key Stage 4 (S4 - S5)

To develop students an ever-improving

capability to

� extend the concepts of numbers to rational

and irrational numbers;

� develop various strategies in using numbers

to formulate and solve problems, and to

examine results;

� develop and refine strategies for estimating;

� extend the use of algebraic symbols in

communicating mathematical ideas;

� explore and describe patterns of

sequences of numbers using algebraic

symbols;

� interpret simple algebraic relations from

numerical, symbolic and graphical

perspectives;

� manipulate algebraic expressions and

relations; and apply these knowledge and

skills to formulate and solve simple practical

problems and to examine results; and

� apply the knowledge and skills of the

Number and Algebra Dimension to formulate

and solve a variety of practical problems in

various Learning Dimensions.


capability to

� understand the real number system;

� investigate and describe relationships between

quantities using algebraic symbols and

relations;

� generalize and describe patterns of sequences

of numbers using algebraic symbols; and

apply the results to solve problems;

� interpret more complex algebraic relations

from numerical, symbolic and graphical

perspectives;

� manipulate more complex algebraic

expressions and relations, and apply these

knowledge and skills to formulate and solve a

variety of practical problems and justify the

validity of results; and

� apply the knowledge and skills in the Number

and Algebra Dimension to generalize,

describe and communicate mathematical ideas

and solve further problems in various Learning

Dimensions.

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capability to

� understand the nature of measurement and

be aware of the issues about precision and

accuracy;

� apply a variety of techniques, tools and

formulas for measurements and solving

mensuration problems;

� explore and visualize geometric properties of

2-dimensional and 3-dimensional objects

intuitively;

� use inductive reasoning, deductive reasoning

and analytic approach to study the properties

of 2-dimensional rectilinear shapes;

� formulate and write simple geometric proofs

involving 2-dimensional rectilinear shapes

with appropriate symbols, terminology and

reasons;

� inquire, describe and represent geometric

knowledge in 2-dimensional figures using

numeric and algebraic relations;

� inquire geometric knowledge in 2-

dimensional space using trigonometric

relations; and

� interconnect the knowledge and skills of the

Measures, Shape and Space Dimension and

other Learning Dimensions, and apply them

to formulate and solve 2-dimensional

problems.


capability to

� use and select inductive reasoning, deductive

reasoning or analytic approach to study the

properties of 2-dimensional shapes;

� formulate and write geometric proofs

involving 2-dimensional shapes with

appropriate symbols, terminology and

reasons;


knowledge in 2-dimensional space using

algebraic relations;


knowledge in 2-dimensional and 3-

dimensional space using trigonometric

functions; and

� interconnect the knowledge and skills of the

Measures, Shape and Space Dimension and

other Learning Dimensions, and apply them to

formulate and solve 2-dimensional and 3-

dimensional problems with various strategies.

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capability to

� understand the criteria for organizing discrete

and continuous statistical data;

� choose and construct appropriate statistical

diagrams and graphs to represent given data

and interpret them;

� find, interpret and select the measure to

describe the central tendency of a set of

data;

� judge the appropriateness of the methods

used in handling statistical data;

� understand the notion of probability and

handle simple probability problems by listing

and drawing diagrams; and

� inquire and solve statistical and probability

problems with appropriate strategies.


capability to

� understand and compute the measures of

dispersion;

� select and use the measures of central

tendency and dispersion to compare data

sets;

� investigate and judge the validity of arguments

derived from the data set;

� formulate and solve further probability

problems by applying simple laws; and

� integrate the knowledge in statistics and

probability to solve real life problems.

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4. 2. 1 Number and Algebra Dimension


Number and Number Systems

l Directed Numbers and the Number Line

(12)

l Numerical Estimation (5)

l Approximation and Errors (7)

l Rational and Irrational Numbers (6)

Comparing Quantities

l Using Percentages (17)

l More about Percentages (7)

l Rate and Ratio (8)

Observing Patterns and Expressing Generality

l Formulating Problems with Algebraic

Language (14)

l More about Polynomials (9)

l Manipulations of Simple Polynomials

(10)

l Arithmetic and Geometric Sequences

and Their Summation (10)

l Laws of Integral Indices (10)

l Factorization of Simple Polynomials

(15)

Algebraic Relations and Functions

l Linear Equations in One Unknown (7) l Quadratic Equations in One Unknown

(17)

l Linear Equations in Two Unknowns (15) l More about Equations (15)

l Identities (8) l Variations (13)

l Formulas (14) l Linear Inequalities in Two Unknowns

(15)

l Linear Inequalities in One Unknown (7) l Exponential and Logarithmic Functions

(18)

l Functions and Graphs (16)

Note: The number in the bracket denotes the estimated time ratio for the unit.

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4. 2. 2 Measures, Shape and Space Dimension


Measures in 2-Dimensional (2D) and 3-Dimensional (3D) figures

l Estimation in Measurement (6)

l Simple Idea of Areas and Volumes (15)

l More about Areas and Volumes (18)

Learning Geometry through an Intuitive Approach

l Introduction to Geometry (10) l Qualitative Treatment of Locus (6)

l Transformation and Symmetry (6)

l Congruence and Similarity (14)

l Angles Related with Lines and

Rectilinear Figures (18)

l More about 3-D Figures (8)

Learning Geometry through a Deductive Approach

l Simple Introduction to Deductive

Geometry (27)

l Pythagoras’ Theorem (8)

l Quadrilaterals (15)

l Basic Properties of Circles (39)

Learning Geometry through an Analytic Approach

l Introduction to Coordinates (9) l Coordinate Treatment of Simple Locus

Problems (14)

l Coordinates Geometry of Straight Lines

(12)

Trigonometry

l Trigonometric Ratios and Using

Trigonometry (26)

l More about Trigonometry (29)


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4. 2. 3 Data Handling Dimension


Organization and Presentation of Data

l Introduction to Various Stages of

Statistics (5)

l Construction and Interpretation of

Simple Diagrams and Graphs (24)

Analysis and Interpretation of Data

l Measures of Central Tendency (19) l Measures of Dispersion (13)

Simple Statistical Surveys

l Uses and Abuses of Statistics (11)

Probability

l Simple Idea of Probability (12) l More about Probability (11)



l Further Applications (30)


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4.3 Learning Objectives for Key Stage 3 (S1 - S3)

4. 3. 1 Number and Algebra Dimension (Key Stage 3)

Unit Learning objectives Suggested

time ratio

Number and Number Systems

Directed Numbers and

the Number Line

l understand and accept intuitively the concept and uses of

negative numbers

l have simple ideas of ordering on the number line

l explore and discuss the manipulation of directed numbers

l manipulate directed numbers

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Numerical Estimation l be aware of the need to use estimation strategies in real-life

situations and appreciate the past attempts to approximate

values such as πl determine whether to estimate values or to compute the exact

values

l select and use estimation strategies to estimate values and to

judge the reasonableness of results

l choose appropriate means for calculation such as mental

computation, calculators or paper and pencil etc.

5

Approximation and Errors l acquire further concepts and skills of rounding off numbers to

a required number of significant figures

l understand the meaning of scientific notation

l use scientific notation in practical problems

l be aware of the size of errors during estimation and

approximation;

l understand and calculate different types of errors such as

absolute errors, relative errors and percentage errors.

7

Note: The objectives with asterisk (**) are exemplars of enrichment topics.

The objectives underlined are considered as non-foundation part of the syllabus.

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time ratio

Rational and Irrational

Numbers

l be aware of the existence of irrational numbers and surds

l explore the representations of irrational numbers in the number

line

l manipulate commonly encountered surds including the

rationalization of the denominator in the form of √a

l appreciate the expressions of surds could be expressed in a

more concise form

Note: The formal hierarchy of the real-number system need not be

mentioned in this unit.

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Using Percentages l understand the meaning of percentages and percentage

changes

l apply percentage changes to solve simple selling problems

l apply percentages to solve problems involving simple and

compound interests, growth and depreciation.

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More about Percentages l apply percentages to solve further practical problems involving

successive and component changes

l apply percentages to solve simple real-life problems involving

taxation and rates

7

Rate and Ratio l understand the meaning of rate and ratio

l recognize the notation of a : b, a : b : c

l apply the ability in using rate, ratio to solve real-life problems

including mensuration problems

8



17


time ratio


Formulating Problems

with Algebraic Language

l appreciate the use of letters to represent numbers

l understand the language of algebra including translating word

phrases into algebraic expressions or write descriptive

statement for algebraic expressions

l note the differences between the language of arithmetic and the

language of algebra

l recognize some common and simple formulas which can be

expressed as algebraic forms and be able to substitute values

l formulate simple algebraic equations/ inequalities to solve

problems

l investigate, appreciate and observe the patterns of various

number sequences such as polygonal numbers, arithmetic and

geometric sequences, Fibonacci sequence etc.

l use algebraic symbols to represent the number patterns

l obtain a preliminary idea of function such as input-processing-

output concept

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Manipulations of Simple

Polynomials

l recognize polynomial as a special example of algebraic

expressions

l recognize the meaning of the terminology involved

l add, subtract, multiply polynomials involving more than one

variable

10

Laws of Integral Indices l extend and explore the meaning of the index notation of

numbers with negative exponents

l explore, understand and use the laws of integral indices to

simplify simple algebraic expressions (up to 2 variables only)

l understand and compare numbers expressed in various bases

in real-life situations

l foster a sense of place values in different numeral systems

l inter-convert between simple binary/hexadecimal numbers to

decimal numbers

10



18


time ratio

Factorization of Simple

Polynomials

l understand factorization as a reverse process of expansion

l factorize polynomials by using common factors and grouping

of terms

l factorize polynomials by using identities including difference of

two squares; perfect square expressions; difference and sum

of two cubes

l factorize polynomials by cross-method

15


Linear Equations in One

Unknown

l formulate and solve linear equations in one unknown

l **solve literal equations

7

Linear Equations in Two

Unknowns

l plot and explore the graphs of linear equations in 2 unknowns

l formulate and solve simultaneous equations by algebraic and

graphical methods

l be aware of the approximate nature of the graphical method

l **explore simultaneous equations that are inconsistent or that

have no unique solution

15

Identities l explore the meaning of identities and distinguish between

equations and identities

l discover and use the identities : difference of two squares; the

perfect square expression; difference and sum of two cubes

8

Formulas l manipulate algebraic fractions with linear factors as

denominators

l develop an intuitive idea of factorization of polynomials

l explore familiar formulas and substitute values of formulas

l perform change of subject in simple formulas but not including

radical sign

14



19

Unit Learning objectives Suggested time

ratio

Linear Inequalities in One

Unknown

l understand the meaning of inequality signs ≥ , > , ≤ and <

l explore the fundamental properties and some laws of

inequalities

l solve simple linear inequalities in one unknown and represent

the solution on the number line

7


The objectives being underlined are considered as non-foundation part of the syllabus.

20

4. 3. 2 Measures, Shape and Space Dimension (Key Stage 3)


time ratio

Measures in 2-D and 3-D Figures

Estimation in

Measurement

l recognize the approximate nature of measurement and choose

an appropriate measuring tool and technique for a particular

purpose

l choose an appropriate unit and the degree of accuracy for a

particular purpose

l develop estimation strategies in measurement

l handle and reduce errors in measurement

l estimate, measure and calculate lengths, areas, capacities,

volumes, weights, rates, etc.

6

Simple Idea of Areas and

Volumes

l find areas of simple polygons

l explore the formula for the area of a circle

l calculate circumferences and areas of circles

l understand and use the formulas for surface areas and volumes

of cubes, cuboids, prisms and cylinders

l appreciate the application of formulas, besides measurement, in

finding measures and be aware of the accumulated errors arisen

l **explore the maximum area of figures for a given perimeter

l **design a container by cutting squares from the 4 corners of a

sheet of A4 paper to maximize the capacity of the container

15

More about Areas and

Volumes

l calculate arc lengths and areas of sectors

l understand and use the formulas for volumes of pyramids,

circular cones and spheres

l understand and use the formulas for surface areas of right

circular cones and spheres

l understand and use the relationships between sides, surface

areas and volumes of similar figures

l distinguish between formulas for length, area, volume by

considering dimensions

18




time ratio

21


Introduction to Geometry l recognize the common terms and notations in geometry such as

line segments, angles, regular polygons, cubes and regular

polyhedra (Platonic solids) etc.

l identify types of angles and polygons

l construct 3-D solids and explore their properties, such as

Euler’s formula

l sketch the 2-D representation of simple solids

l sketch the cross-sections of the solids

l overview tools of geometry and explore ways of using them to

construct polygons, circles, parallel and perpendicular lines

l **recognize some semi-regular polyhedra (Archimedean

Solids)

10

Transformation and

Symmetry

l recognize reflectional and rotational symmetries in 2-

dimensional (2-D) shapes

l recognize the effect on 2-D shapes after the transformation

including reflection, rotation, translation, dilation/contraction etc.

l appreciate the symmetrical shapes around and transformations

on shapes used in daily-life

l **construct and design tile patterns

6

Congruence and Similarityl recognize the properties for congruent and similar triangles

l extend the ideas of transformation and symmetry to explore the

conditions for congruent and similar triangles

l recognize the minimal conditions in fixing a triangle

l identify whether 2 triangles are congruent/similar with simple

reasons

l explore and justify the methods to construct angle bisectors,

perpendicular bisectors and special angles by compasses and

straight edges

l appreciate the construction of lines and angles with minimal

tools at hand

l ** discuss the possibility of trisecting an angle by compasses

and straight edges

l **explore some shapes in fractal geometry

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Angles related with Lines

and Rectilinear Figures

l recognize different types of angles

l explore and use the angle properties associated with intersecting

lines and parallel lines

l explore and use the properties of lines and angles of triangles

l explore and use the formulas for the angle sum of the interior

angles and exterior angles of polygons

l explore regular polygons that tessellate

l appreciate the past attempts in constructing some special

regular polygons with minimal tools at hand

l construct some special regular polygons using straight edges

and compasses

l **discuss past attempts in constructing some special regular

polygons such as 17-sided regular polygons

18

More about 3-D Figures l extend the idea of symmetry in 2-D figures to recognize and

appreciate the reflectional and rotational symmetries in cubes

and tetrahedron

l explore and identify the net of a given solid

l imagine and sketch the 3-D objects from given 2-D

representations from various views

l recognize the limitation of 2-D representations in identifying the

solid

l explore the properties of simple 3-D object, such as identifying

u the projection of an edge on one plane

u the angle between a line and a plane

u the angle between 2 planes

l **investigate the reflectional and rotational symmetries in other

regular polyhedra

l **assemble a set of Soma Cube into a larger cube

l **explore the number of regular polyhedra

8



23


time ratio


Simple Introduction to

Deductive Geometry

l develop a deductive approach to study geometric properties

through studying the story of Euclid and his book - Elements

l develop an intuitive idea of deductive reasoning by presenting

proofs of geometric problems relating with angles and lines

l understand and use the conditions for congruent and similar

triangles to perform simple proofs

l identify lines in a triangle such as medians, perpendicular

bisectors etc.

l explore and recognize the relations between the lines of triangles

such as the triangle inequality, concurrence of intersecting points

of medians etc.

l explore and justify the methods of constructing centres of a

triangle such as in-centre, circumcentre, orthocentre, centroids

etc.

l **prove some properties of the centres of the triangle

27

Pythagoras’ Theorem l recognize and appreciate different proofs of Pythagoras’

Theorem including those in Ancient China

l recognize the existence of irrational numbers and surds

l use Pythagoras’ Theorem and its converse to solve problems

l appreciate the dynamic element of mathematics knowledge

through studying the story of the first crisis of mathematics

l **investigate and compare the approaches behind in proving

Pythagoras’ Theorem in different cultures

l **explore various methods in finding square root

8

Quadrilaterals l extend the idea of deductive reasoning in handling geometric

problems involving quadrilaterals

l deduce the properties of various types of quadrilaterals but with

focus on parallelograms and special quadrilaterals

l perform simple proofs related with parallelograms

l understand and use the mid-point and intercept theorems to find

unknowns

15

Note: The objectives with asterisk (**) are considered as exemplars of enrichment topics.


24


ratio


Introduction to

Coordinates

l understand and use the rectangular and polar coordinate

systems to describe positions of points in a plane

l able to locate a point in a plane by means of an ordered pair

in the rectangular coordinate system

l describe intuitively the effects of transformation such as

translation, reflection with respect to lines parallel to x-axis,

y-axis and rotation about the origin through multiples of 90°on points in coordinate planes

l calculate areas of figures that can be cut into or formed by

common 2-D rectilinear figures

9

Coordinate Geometry of

Straight Lines

l understand and use formulas of distance and slope

l use ratio to find the coordinates of the internal point of

division and mid-point

l understand the conditions for parallel lines and perpendicular

lines

l appreciate the analytic approach to prove results relating to

rectilinear figures besides deductive approach

l choose and use appropriate methods to prove results relating

to rectilinear figures

l **explore the formula for external point of division

12

Trigonometry

Trigonometric Ratios and

Using Trigonometry

l understand the sine, cosine and tangent ratios for angles

between 0° to 90°l explore the properties and relations of trigonometric ratios

l explore the exact value of trigonometric ratios on special

angles 30°, 45°, 60°l rationalize the denominators such as √2

l apply trigonometric ratios to find measures of 2-D figures

l introduce the ideas of bearing, gradient, angle of elevation,

angle of depression and solve related 2-dimensional

problems

26

Note: The objectives with asterisk (**) are considered as exemplars of enrichment topics.


25

4. 3. 3. Data Handling Dimension (Key Stage 3)


time ratio

Organization and Representation of data

Introduction to Various

Stages of Statistics

l recognize various stages involved in statistics

l use simple methods to collect data so as to analyze posed

problems

l be aware of the existence of different types of data (discrete

and continuous)

l understand the criteria of organizing data and discuss different

ways of organizing the same set of data

5

Construction and

Interpretation of Simple

Diagrams and Graphs

l construct and interpret simple diagrams including stem-and-

leaf diagrams, pie charts, histograms, scatter diagrams, broken

line graphs

l construct and interpret simple frequency polygons and curves,

cumulative frequency polygons and curves

l be able to differentiate between histograms and bar charts

l explore the construction of diagrams and graphs with various

tools besides paper and pencil

l compare the presentations of the same set of data by using

various graphs or the same type of graphs but with different

scales

l choose appropriate diagrams/graphs to present a given set of

data

l read data from given frequencies in graphs (including

percentiles, quartiles, median)

l read frequencies from given data in diagrams and graphs

l use some common wordings such as ‘most popular’, ‘most

likely’, ‘equally likely’ to describe trends from line graphs

24

l discuss the impressions from graphs presented in various

sources

l identify sources of deception in misleading graphs and their

accompanying statements

l recognize the dangers of misinterpreting statistical data


The underlined objectives are considered as non-foundation part of the syllabus.

26


time ratio

Analysis and Interpretation of data

Measures of Central

Tendency

l find mean, median and mode from a given set of ungrouped

data

l find mean, median and modal class from a given set of

grouped data

l be aware that the mean found for grouped data is an

estimation

l compare 2 data sets with given mean, median and mode

l construct data sets with a given mean, median and mode

l discuss the relative merits of different measures of central

tendency for a given situation

l explore and make conjectures on the effect of the central

tendency of the data such as

(i) removal of a certain item from the data;

(ii) adding a common constant to the whole set of data;

(iii) multiplying the whole set of data by a common constant;

(iv) insertion of zero in the data set

l understand weighted mean and be aware of its use in various

real-life situations such as Hang Seng Index, calculation of

marks in a report etc.

19

l discuss the misuse of averages in various daily life situations

l recognize the dangers of misusing averages



27


time ratio

Probability

Simple Idea of Probability l explore the meaning of probability through various activities

l have an intuitive idea about the relation between probability

and the relative frequency as found in statistics or simulation

activities

l investigate probability in real-life activities, including geometric

probability

l compare the empirical and theoretical probabilities

l calculate the theoretical probability by listing the sample space

and counting

l recognize the meaning of expectation

12



28

4. 4 Learning Objectives for Key Stage 4 (S4 - S5)

4. 4. 1 Number and Algebra Dimension (Key Stage 4)


time ratio


More about Polynomials l manipulate polynomials further including long division up to

simple quadratic divisor

l recognize the concept of division algorithm

l understand and use remainder and factor theorems to

factorize polynomials up to degree 3

l appreciate the power of factor theorem and also be aware of

the limitation of the theorem

9

Arithmetic and Geometric

Sequences and their

Summation

l explore further the properties of arithmetic and geometric

sequences

l develop and use the general terms of the sequences

l investigate and use the general formulas of the sum to n

terms of arithmetic and geometric sequences

l develop an intuitive idea on limit and deduce the formula for

sum to infinity for certain geometric series

l solve real-life problems such as interest, growth and

depreciation, geometric problems etc.

l **explore recurrence in some sequences

10


Quadratic Equations in

One Unknown

l formulate and solve quadratic equations by factor method

and formula

l solve the equation ax2 + bx + c = 0 by plotting the graph y =

ax2 + bx + c and reading the x-intercepts

l be aware of the approximate nature of the graphical method

l choose the most appropriate strategy to solve quadratic

equations

l recognize the conditions for the nature of roots

l understand the hierarchy of real-number system and be

aware of the characteristics of rational numbers when

expressed in decimals

Note: Further exploration on properties of quadratic graphs

would be in the Unit “Function and Graphs”.

17



29


time ratio

More about Equations l formulate and solve equations which can be transformed into

quadratic equations

l formulate and solve one linear and one quadratic

simultaneous equations by algebraic method

l solve equations by reading intersecting points of given graphs

l appreciate the power and understand the limitation of

graphical method in solving equations

l choose the most appropriate strategy to solve equations

l **explore the algebraic method to solve cubic or higher

degree equations

15

Variations l discuss the relations between 2 changing quantities

l sketch the graphs of direct and inverse variations and

recognize the algebraic representations between the

quantities

l recognize and appreciate the algebraic representations of

various variations such as those in the forms of V=πr2h or

y=k1+k2x, etc.

l apply the relations to solve real-life problems

13

Linear Inequalities in Two

Unknowns

l represent the linear inequalities in 2 unknowns on a plane

l discuss the solution of compound linear inequalities

connected by ‘and’

l solve systems of linear inequalities in two unknowns

l solve linear programming problems

15

Exponential and

Logarithmic Functions

l understand and use the laws of rational indices

l understand the definition of logarithmic functions and

recognize the common logarithm is not the only type of the

function

l examine the properties of the graphs of exponential and

logarithmic functions

l explore and study the relations between the properties of

logarithmic function and that of exponential function

l appreciate the application of logarithm in various real-life

problems

18



30


time ratio

Functions and Graphs l relate the idea of input-processing-output to the meaning of

dependent and independent variables

l understand the basic idea of a function from the tabular,

symbolic and graphical representations of a function and the

dummy nature of x

l use the notation for a function

l explore various properties of quadratic functions such as

vertex, axis of symmetry, the optimum value(s) from their

graphs

l appreciate the contribution of Arabians on the method of

completing the square and use it to find the properties of

quadratic functions

l appreciate the power of the method in generating a perfect

square expression

l sketch and compare graphs of various types of functions

l solve f(x) > k, f(x) < k, f(x) ≥ k, f(x) ≤ k by reading graphs

of f(x)

l explore the effects of transformation on the functions from

tabular, symbolic and graphical perspectives

l visualize the effect of transformation on the graphs of

functions when giving symbolic relations

16



31

4. 4. 2 Measures, Shape and Space Dimension (Key Stage 4)


time ratio


Qualitative Treatment of

Locus

l describe verbally or sketch the locus of points moving under a

condition or conditions

l appreciate different conditions which can give rise to the same

type of locus

6


Basic Properties of

Circles

l understand and use the basic properties of chords and arcs of

a circle

l understand and use the angle properties of a circle

l understand and use the basic properties of cyclic quadrilateral

and tangent to a circle

l appreciate the intuitive and inductive ways of recognizing the

properties of circles and see the importance of deductive

approach

l perform geometric proofs related with circles

l appreciate the structure of Euclidean Geometry such as

definitions, axioms and postulates etc. and its deductive

approach in handling geometric problems

39


Coordinate Treatment of

Simple Locus Problems

l explore and visualize straight line as loci of moving points

and describe the loci with equations

l recognize the characteristics of equation form that represents a

straight line

l understand and apply the point-slope form to find the

equations of straight lines from various given conditions

l describe the properties of the line from a given linear equation

l explore and visualize circles as loci of moving points

l find the equation of circles from given conditions

l **explore other forms of equations for straight lines

14



32


ratio

Trigonometry

More about Trigonometry l understand the sine, cosine and tangent functions, their

graphs

l use graphs to explore properties of trigonometric functions

including periodicity etc.

l use graphs of the functions to find roots of an equation such

as sin θ = constant, where 0° ≤ θ ≤ 360°l recognize the limitation of Pythagoras’ Theorem in solving

triangles

l understand and use sine and cosine formulas to solve

triangles

l understand and use the formula ½absinC and Heron’s

formula for areas of triangles

l investigate and find the angle between 2 intersecting lines,

between a line and a plane, between 2 intersecting planes

l apply trigonometric knowledge in solving 2-dimensional and

3-dimensional problems

29



33

4. 4. 3 Data Handling Dimension (Key Stage 4 )


ratio

Analysis and Interpretation of Data

Measures of Dispersion l recognize range, inter-quartile range and standard deviation

as measures of dispersion for a set of data

l find range from a given set of data

l find inter-quartile range from the cumulative frequency

polygon

l construct box-and-whisker diagrams and use them to

compare the distributions of different sets of data

l interpret the basic formula of standard deviation and be able

to find the standard deviation for both grouped and

ungrouped data set

l compare the dispersions of different sets of data using

appropriate measures

l explore and make conjecture on the effect of the dispersion

of the data such as

i. removal of a certain item from the data;

ii. adding a common constant to the whole set of data;

iii. multiplying the whole set of data by a constant;

iv. insertion of zero in the data set.

13


Uses and Abuses of

Statistics

l recognize different techniques in choosing samples and the

criteria in choosing data collection method

l investigate methods in which statistical surveys are used and

misused in various daily-life activities

l discuss the strengths and weaknesses of statistical

investigations presented in different sources such as news

media, advertisements, etc including methods of collecting,

presenting and analysing data etc.

l recognize the complexity in conducting surveys

11



34


ratio


Conducting Surveys** l **conduct statistical investigations including

i. formulating key questions to investigate problems

relating to their experience;

ii. deciding appropriate data collection method which may

involve designing simple questionnaire;

iii. applying sampling techniques in collecting data;

iv. conducting the investigations;

v. making interpretation on the data collected and

analyzing their findings;

vi. presenting the investigations to other.

Probability

More about Probability l recognize the basic laws in probability

l apply the addition or multiplication laws in a wide variety of

activities including real-life problems

l recognize the notion of conditional probability and the

notation of P(AB)

Note: The Bayes’ Theorem need not be introduced.

11



35

4. 4. 4 Further Applications module ( Key Stage 4 )

Learning objectives Suggested

time ratio

Further Applications 30

l further apply mathematics in various dimensions to more sophisticated real-life or mathematical situations

including:

² exploring and interpreting graphs which illustrate real-life situations such as travel-graph, time-series

graphs, etc.

² further applying percentages to real-life situations such as taxation and loans etc.

² further applying rate and ratio to real-life situations such as taxi-rate etc.

² analyzing and interpreting data collected in surveys

² determining the relation between 2 variables which themselves, or after transformation, obeys a

linear law



36


5.1 Curriculum StrategiesThe curriculum strategies include methods of curriculum planning, material writing, teaching and

learning to help students work towards the targets. In addressing the needs of our students to face

the challenges of the 21st Century, the strategies used in this curriculum put emphases on

l the process of learning;

l catering for learner differences;

l the appropriate use of information technology (IT) in teaching and learning;

l the appropriate use of multifarious teaching resources.

However, it should be noted that no matter what emphasis of strategies is put in this curriculum, the

teacher is the key person in the classroom teaching. Past studies review that liveliness and clear

explanation of the teachers are students’ main concerns. Students perceive that teachers have the

responsibility of delivering clear explanation, designing and conducting activities in lessons, creating a

good environment and showing concern for students’ progress. The strategies mentioned below

are just to remind teachers to be aware of the emphasis when designing and preparing teaching and

learning activities to facilitate students’ learning.


Providing experiences and knowledge constructed in the learning process is considered as important

as the end product. Sufficient time should be allocated for students to inquire, communicate, reason

and conceptualize mathematical concepts so as to enable them to understand the knowledge

thoroughly, to master the skills confidently and to foster a positive attitude towards learning.

Students should also engage in the activities that enable them to practise problem-solving skills and to

integrate and apply mathematical concepts.

Inquiring involves discovery or constructing knowledge through questioning or testing hypothesis.

Posing questions to stimulate students to discover similarities or differences on different rules or

asking students to test mathematical conjectures enables students to participate in a more active role

in the learning process.

Communicating involves receiving and sharing meanings by using language, symbols, graphs and

aesthetic forms. Listening, speaking, reading and writing are the important elements of

communication which help students to interpret others’ statements, state their ideas, clarify their

meanings, refine their strategies to solve problems, hypothesize and construct simple arguments.

Activities such as teachers posing questions for students to answer, small-group work, large-group

discussions, presentation of individual and group projects (both written and oral form) provide

platforms for students to communicate mathematically. Mathematics in itself can also be considered

as other form of language. Teachers can guide students to see the difference of the mathematical

37

language with those languages used in daily life and appreciate the precise nature of the mathematical

language.

Reasoning involves developing plausible or logical arguments to deduce or infer conclusions. It is

fundamental to the knowing and doing of mathematics. A mathematician or a student makes a

conjecture by generalizing from a pattern of observations made in some particular cases (inductive

reasoning) and then tests the conjecture by constructing either a logical verification or a counter-

example (deductive reasoning).

Conceptualizing involves organizing and reorganizing knowledge through perceiving and thinking

about particular experiences in order to abstract patterns and ideas and to generalize from particular

experiences. In teaching, teachers should pay due emphasis on helping students master the basic

concepts of mathematics and create link between concepts.

The importance of problem solving in mathematics education has well been recognized. It involves

l understanding the problems;

l considering possible strategies and choosing an appropriate one to solve the problem;

l carrying out the plan; and

l justifying or evaluating the solution.

Concepts of mathematics are connected within a framework, which is ‘multi-dimensional’.

Concepts in one dimension very often are linked to concepts in other dimensions. For example,

nearly all concepts in the dimensions, Measures, Shape & Space and Data Handling are inevitably

connected with concepts in the Number and Algebra Dimension. Teachers should guide students to

see the inter-relationship of mathematical knowledge among different dimensions.

Past studies point out that some people could not apply their mathematical knowledge learnt in

schools to solve their real-life problems. Activities to foster students’ application of mathematical

knowledge to their real-life situations and to be aware of the link between school knowledge and

their real-life applications should also be provided. Teachers could ask students to extract some

daily-life problems appeared in the newspaper, advertisements, internet and so on for discussion.

38


The curriculum is structured with the Foundation Part identified to facilitate teachers to tailor the

curriculum for their students’ learning needs. Teachers could focus on teaching the Foundation

Part of the whole syllabus so as to provide appropriate quantities and a variety of activities for

students to conceptualize, construct knowledge and communicate mathematically. For more able

students, activities on enrichment topics could also be provided to broaden students’ horizon of

mathematical knowledge and enhance their interest in mathematics.

Teachers are advised to give due considerations to various aspects such as grouping students of

similar ability together, teaching/learning activities, resources and assessment. Teachers find

teaching in mixed ability classes harder than teaching in classes where students are relatively close in

ability. However, there can be a negative impact on the self-image of those students placed in

lower streams. No matter how the students are organized, it is inevitable that students in a class will

differ in abilities, needs and interests. Teachers need to use selectively whole class teaching, group

work and individual teaching as appropriate to the task in hand.

In daily classroom teaching, teachers could cater for learner differences by providing students with

different tasks or activities graded according to the levels of difficulty, so that students work on tasks

or exercises that match their stages of progress in learning. For less able students, tasks should be

relatively simple and fundamental in nature. For abler students, tasks assigned should be

challenging enough to cultivate as well as to sustain their interest in learning. Alternatively, teachers

could also provide students with the same task or exercise, but vary the amount and style of support

they give, i.e. giving more clues, breaking the more complicated problems into several parts for

weaker students.

The use of IT could also provide another solution for teachers to cater for learner differences.

Different levels of exercises or activities are always included in the educational software packages.

Teachers could make use of these software packages for students with different abilities to work

through at their own pace and at their levels of ability. The facilities to record students’

performance in these software packages could also provide information for teachers to diagnose

students’ misconceptions or general weaknesses so as to re-adjust the teaching pace or re-consider

the teaching strategies.


Traditional teaching is always conducted with chalk and talk. Audio-visual devices such as

television in the past 2 decades provide another alternative activity for mathematics teaching and

learning. The wide spread use of computers and calculators in this decade provides further

alternative for the teaching and learning in mathematics. The advantages of using IT over other

tools include:

i. interactive learning which enables learners to obtain “immediate” feedback for testing

39

hypothesis, readjust the problem-solving strategies, see connections between formulae

and their corresponding graphs by changing the values of relevant parameters;

ii. colourful, attractive and dynamic graphics which provide graphical images of various

functions, 2-D and 3-D models, animation activities, studying geometry dynamically;

iii. large memories that enable students to compute complicated expressions, work with

real data, study real-life statistical problems;

iv. fast speed which enables students to produce many examples in a short period of time for

the observation of patterns so that they are more willing to try different strategies to solve

problems.

IT in school mathematics education could be considered as:

i. a tool - Teachers could use presentation software as a ‘blackboard’ to present

notes, geometry software to demonstrate graphs and models, zoom-in

and zoom-out facilities in some graphing calculators or graph plotter

software to approximate the solution of equations from their graphs.

Students could use symbolic manipulation software to manipulate

complicated expressions, present statistical graphs with graphical facilities

in spreadsheet, submit homework through e-mail etc.

ii. a tutor - Many mathematical software packages, in the form of CD-ROMs, act as

a tutor to teach students mathematical concepts. These software

packages illustrate mathematical concepts with texts, graphics, and sound

and with graded exercises or tests. Students could use these software

packages to revise the contents learnt in the classroom, remedy the weak

areas or even learn new topics prior to teachers’ teaching. They could

further consolidate their learning with appropriate exercises chosen for

their levels of difficulty at their own pace.

iii. a tutee - Teachers could develop their own educational programs using spreadsheets or

other programming languages to suit their own teaching strategies.

Students could write programs in the language such as LOGO to explore

properties of geometric figures.

Both teachers and students of mathematics at all secondary levels are expected to use IT intelligently

and critically. They must be able to decide when to use the available technology. For example,

students have to decide whether to use calculator, or work mentally to solve the equation x2-3x-4 =

0, and teachers have to decide whether to use computers or the real objects to demonstrate the

projections of 3-D models, and which software is more appropriate for the task.

Besides, varieties of group work to facilitate collaborative learning or investigative approach in

learning with IT should also be considered. Class-work or home assignment should emphasize

upon concept development and understanding instead of manipulating complicated expressions or

40

symbols or just rote memorization of formulae.

5.1.4. Appropriate Use of Multifarious Teaching Resources

Besides IT, there are other teaching resources that teachers could make use of in planning and

conducting the teaching and learning activities:

u textbooks or teaching packages

u reference books

u audio-visual tapes such as ETV programs

u instruments and other equipment for drawing shapes and making models

u materials found around such as newspapers, advertising leaflets, maps, etc.

u resources found in libraries / resource centres etc.

Textbooks are one of the key resources for teaching and learning. They should be used to guide

students to acquire knowledge, skills and develop attitudes as well as to assimilate concepts and

process information in the texts and graphics therein. Textbooks should not be treated as a mean

of imparting factual knowledge or just providing exercises to drill students on the manipulative skills.

Some textbooks tend to provide exercises more than those required in the syllabus so far as the level

of difficulty and the amount are concerned. Teachers should therefore exercise discretion in selecting

suitable parts to teach and avoid over-teaching or over-drilling.

Besides textbooks, teachers could make use of teaching packages or references distributed to

schools. Some of these teaching packages or references provide ready-made worksheets, notes or

information that could be used in the classroom with slight modifications. ETV programs could

provide information that cannot be presented vividly by just chalk-and-talk. For instance, it is

interesting to watch a video on the historical development in approximating the value of π or the

applications of trigonometric ratios in surveying.

Mathematical language is progressively abstract. Different learning theories point out the

importance of providing students with rich experiences in manipulating concrete objects as a

foundation for the symbolic development. Teachers could make use of teaching aids such as 3-D

models, blocks, graph boards, protractors, pairs of compasses, rulers, measuring equipment, etc. to

demonstrate the mathematical concepts and allow students to “play” around before asking students

to “structure and apply” the concepts.

Materials around such as advertisement leaflets, statistical reports presented in the media, graphs

printed in the newspaper could supply up-to-date information that cannot be found in other sources

and they could easily arouse students’ interest in learning. In addition, a large quantity of related

materials for teachers’ reference can be obtained from libraries or various resource centres operated

by the Education Department, such as School Based Resource Centre, TOC Resource Centre(s).

In 1990s, Internet becomes another popular source for sharing and retrieving information.

41

Gathering and selecting information from these sources would be another major learning activities

in the 21st century.



In primary school levels, students have learnt different representations of numbers and the inter-

conversion of numbers among different representations (refer Annex I for further details). In

secondary schools, students are expected to extend the concepts of numbers from positive numbers

to directed numbers and then to real numbers. However, teachers should assess and provide

consolidation activities, whenever necessary, to ensure that students have firm foundations on the

concepts of numbers before proceeding to the study in KS3 and KS4. Students will encounter

rational and irrational numbers in KS3 as a natural consequence of introducing “Pythagoras’

Theorem” or “Trigonometric Ratios and Using Trigonometry”. Teachers could arrange the unit

“Rational and Irrational Numbers” together with any one of the above units. Manipulations of

surds are confined to techniques sufficient for handling problems related to the above 2 units.

Furthermore, it is important to foster students’ number sense and to build up habits of checking the

reasonableness of results. Teachers should encourage and remind students to apply the concept of

estimation throughout their learning process of mathematics.

Regarding algebra, students in primary schools have an intuitive idea of solving linear equations with

at most 2 steps in the solution. In KS3, it is important for students to build up a firm transition

from number to algebra and to recognize the strength of using algebraic language in solving problems.

However, teachers should not ask students to go into tedious manipulations of algebraic symbols.

As a base line, teachers should make sure that students possess the technical fluency necessary for

tackling equations and inequalities. Teachers could use a variety of instructional formats, such as

small cooperative groups, individual explorations, probing ‘what-if’ questions, etc to allow students

to search for patterns, make conjecture in formulating and solving algebraic problems. Various

concrete tools such as blocks could be used to demonstrate the equivalence of polynomials or

identities. In studying the concept of input-processing-output of polynomials, teachers could make

use of tables or spreadsheet software packages to demonstrate the effect on the output by changing

input, and allow students to explore various kinds of polynomials before proceeding to a formal

concept of function in KS4.

In KS4, when introducing the shapes and properties of quadratic functions, teachers could provide

students with graphs and tables to grasp the idea of symmetries, vertices, and extremum before

introducing algebraic methods to find vertices, etc. Graphical tools, such as graph-plot software

packages or graphing calculators, provide a ready visualization of relationships. In introducing the

concepts of functions, teachers could make use of them for students to explore and visualize the

effects on the dependent variable by changing values of the independent variables. In solving

equations, students are expected to learn basic algebraic skills to solve linear and quadratic equations

and to solve equations of higher-degree by factorization. Through using graphical tools and their

42

zoom-in and zoom-out functions, teachers could guide students to appreciate the power of graphical

method in solving equations and recognize their limitations.


Spatial sense is one of the capabilities emphasized in this curriculum. It includes the recognition of

plane figures and the manipulation of three-dimensional spatial objects. Students are expected to

have rich experiences in manipulating objects with activities like paper-folding, construction of

models, transformations of figures, geo-boards, visual arts by hand or by computer in both primary

and junior secondary school levels. Teachers could also provide students with experiences in

exploring and visualizing the geometric properties of figures with the aid of dynamic geometry

software packages. With these experiences, students could understand the properties of figures

and then gradually proceed to derive the proof with deductive reasoning.

The unit “Transformation & Symmetry” could be introduced with figures found in real-life situations.

Teachers could then ask students to explore the effects of transformations on the shapes by moving

objects with the help of software packages in computers or moving real-objects. Teachers should

link the idea of transformations with proofs in plane geometry, and with the units “Introduction to

Coordinates” and “Functions and Graphs”. For 3-D objects, teachers could provide students with

experiences in visualizing the net and the cross-sections of the solids and different views of the solids

with real models, or moving the solids around in the computer. The relations between lines and

relations between planes in space should only be treated qualitatively. These intuitive ideas could

be elaborated further by the study of trigonometry in solving 3-D problems in KS4.

In KS4, students are expected to compare different approaches in studying geometric problems.

Teachers should guide students to appreciate the importance of inductive reasoning and deductive

reasoning in studying the properties of geometric figures and also observe their limitations. The

significance of using analytic geometry in linking algebra and geometry should also be highlighted.

Graphical software packages or graphing calculators could also be used for students to explore the

locus of moving objects under a given condition before proceeding to the coordinate treatment of

some specific locus.


In the primary school level, students have learnt various statistical diagrams to present discrete data.

In KS3, students are expected to extend the idea of discrete data to continuous data. For

construction of statistical diagrams, it is appropriate to have students to draw graphs manually for

some small data sets. For other situations, the emphasis should be put on the use of calculators or

computers so as to minimize the drudgery involved. With the aid of software packages, students

should be guided further to explore and choose the appropriate method or diagram to organize and

present a given set of data. For example, a pie chart is more appropriate to present the idea of

portion and whole than using a bar chart. It should further be noted that emphasis should be laid on

43

interpretation rather than construction of graphs. Teachers could also ask students to interpret

different statistical diagrams or graphs gathered from newspapers first and to construct statistical

diagrams afterwards. Lastly, teachers should arrange students to undertake projects which involve

all of the stages of data handling, namely collecting, organizing, presenting and interpreting, whenever

appropriate.

In KS4, students are expected to discuss the statistical reports presented in various media. An

intuitive idea on sampling techniques and different data collection methods should be introduced in

order to provide background knowledge for students to study the reports. Students should not go

into details of sampling techniques nor sophisticated method in designing questionnaires. Teachers

could also arrange students to conduct surveys as enrichment and cross-curricular activities so that

students could integrate knowledge learnt in various subjects to study problems they are interested in.

Students could use software packages like spreadsheet to explore the effect on the measures of

central tendency and dispersion when changing values of the data set, or analyze and present reports

of surveys they have conducted.

In investigating probability in KS3, simple games and real-life activities could be used. Students

should experiment with, discuss and compare the results of different experiments and note that

separate experiments will usually produce somewhat different results. Besides hands-on activities,

computers or calculators could be used to facilitate the simulation of large number of trials so as to

enable them to develop an understanding of probability as the long-run relative frequency.

Teachers should not introduce the addition nor multiplication laws to students in KS3. In KS4,

teachers should guide students to see the benefits of using addition and multiplication laws in finding

probability of wider variety of activities. The notion of conditional probability will be introduced as

a direct consequence of finding probability of dependent events.

44

5.2.4. Further Applications Module

As students are growing mature in KS4, the module “Further Applications” is included so as to

encourage students to further apply mathematical knowledge to solve problems in a more complex

real-life and/or mathematical context. In this module, teachers should use real-life problems, as

far as possible, to encourage students to discuss and explore the ways of applying mathematics in

various real-life situations. Articles in the newspaper, statistical reports presented, advertisement

brochures, etc. could be used for discussion. Teachers could guide students to use different

approaches in tackling the real-world situations.

Although this module is categorized as non-foundation, teachers should select some topics that are

relevant to their students’ interest and ability. Furthermore, cross-dimensional problems should be

introduced so as to encourage students to integrate mathematical knowledge in solving problems.

45


6.1 Purposes of AssessmentAssessment can be used for a variety of purposes such as evaluating the teaching effectiveness,

diagnosing the learning difficulties of students, screening and placement etc. In this syllabus,

assessment is mainly considered as an integral part of the teaching and learning cycle. It is a process

of gathering information to find out students’ achievements related to the set objectives so as to

enhance the teaching and learning processes.

Information collected in assessment help

1. teachers to

l understand how students are progressing;

l recognize students’ strengths and areas for improvement in learning;

l work out ways of helping students;

l plan their lessons.

2. students to

l understand their own progress;

l recognize areas and ways for improvements in learning.

The Learning Targets and Learning Objectives describe the desired breadth and depth of contents in

which students need to learn and thus constituting the basis for assessment. The essence of

assessment is to judge students’ performance with respect to the Learning Targets and Learning

Objectives.

6.2 Assessment StrategiesThe complexity of learner performance which cannot be described by a single set of scores or single

type of assessment activity is well recognized. Through various modes of assessment activities,

evidence of learning could be collected to reflect the students’ achievement in mathematics.

However, assessment may cause students’ anxiety and undue pressure, loss of confidence and

interest and, in extreme cases, refusal to learn. Over-assessment may also increase teachers’

workload unnecessarily. To avoid such adverse effects, school needs to formulate its own

appropriate assessment and reporting policy according to the school’s culture, teachers’

experiences, learners’ needs and interests. Based on this school based assessment policy, teachers

may also include a range of well-planned assessment activities and recording formats in their

teaching/learning plans such as their schemes of work.

In planning assessment,

l a variety of activities such as tasks or exercises covering the comprehensive range of learning

objectives should be included; and

l the opportunity for all students to demonstrate the full range of their individual capabilities,

46

including evidence for students to demonstrate higher cognitive skills, should be provided.

Assessment activities may include:

l class discussions or oral presentations;

l observations of students’ performances during lessons;

l classwork and homework;

l project work such as making models, statistical surveys, etc;

l short quizzes;

l tests and examinations; and

l extra-curricular activities such as Mathematics Club, Mathematics Week, etc.

Activities may involve individuals or groups, can be formal or informal. They may be teacher-

directed or may involve students in making judgment. It is noted that some of the Learning

Objectives especially those related with the affective domain may be difficult to be assessed in a

formal way. Teachers should try various informal ways such as verbal response to provide

feedback to students.

Assessment activities can be conducted in a formative or summative way. Formative assessment is

an ongoing evaluation process of students’ progress so as to help teachers to diagnose students’

strengths and weaknesses in learning. Summative assessment is an assessment of students’ overall

progress at certain intervals, such as the end of a school term, a school year or a Key Stage. It is

designed to provide a comprehensive, summary description of performance and progress in

students’ learning.

6.3 Feedback from AssessmentThe evidence collected from the assessment activities should be used as important feedback for

students to improve their learning and for teachers to adjust their teaching strategies and pace.

Immediate feedback from formative assessment activities can be provided to students during class

time or in delivering their assignments. They can be in verbal or in written form. Students with this

immediate feedback can clarify their mistaken concepts before further knowledge to be built on.

Extra efforts can be paid to improve areas of weaknesses. Feedback from summative assessment

activities can provide information for students to plan their subsequent phase of study.

Teachers can use the information collected in the formative assessment activities to adjust teaching

strategies, decide whether to include further consolidation activities or introduce enrichment topics in

the subsequent day-to-day teaching. Information gained in the summative assessment activities can

be used as a basis for the planning of the teaching sequence, the breadth and depth of the learning

units in the subsequent term or year. This information can be very useful for schools to adjust their

aims and strategies of the school-based mathematics curriculum.

47

To enable learning in both school and home to be synchronized, it is essential that there is effective

and efficient communication between teachers and parents. Some studies reveal that parents

consider “practice makes perfect” and rote memorization important in their children’s learning.

This belief may lead to both positive and negative impacts on children, including over-emphasis of

drill-and-practice in improving children’s learning. An informal involvement in different home-school

activities or a formal written report on students’ progress could be used as a channel of

communication. Based on the evidence collected from the assessment activities, more information

on how to improve children’s learning could be provided to parents through these channels.

48

Overview of Learning Targets and Objectives - Key Stages 1 and 2

NUMBER DIMENSION

Key Stage 1 (P1 - P3) Key Stage 2 (P4 - P6)

Dimension Learning Targets for Key Stage

To develop an ever-improving capability to

• understand the concepts of whole numbersand simple fractions

• manipulate whole numbers and simplefractions

• understand and use simple properties ofoperations on numbers

• apply the knowledge and concepts ofmanipulating numbers to formulate andsolve simple problems

To develop an ever improving capability to

• understand the concepts of different formsof numbers

• interconvert numbers in different forms• manipulate numbers and check the

reasonableness of results• understand and use properties of

operations on numbers• apply the knowledge and concepts of

manipulating numbers to formulate andsolve problems

Learning Objectives for Key Stage

Learners

(1) read, write and order numbers up to 5digits, and understand the meaning ofplace value.

(2) (a) understand the concepts of additionand subtraction and their relationship.

(b) compute addition and subtractionwithin 4 places.

(c) understand and use the commutativeand associative properties of addition.

(d) solve relevant practical problems ofaddition and subtraction within 4places.

Learners

(1) (a) read, write, round off large numbers ineveryday life and estimate largequantities.

(b) recognize prime numbers andcomposite numbers and find primenumbers within 150.

(c) use index notation to representcomposite numbers.

(d) recognize the Chinese and Romannumeration systems.

(e) recognize the development of somecalculating devices.

Extracted from the Appendix A of Target Oriented Curriculum: Programme of Study for Mathematics KeyStage 2 (Primary 4-6) (1995) Prepared by the Curriculum Development Council

ANNEX I

49

NUMBER DIMENSION (CONT.)



(3) (a) understand the concepts ofmultiplication and division and theirrelationship.

(b) compute multiplication with 1-digitmultiplier and multiplicand up to 3digits, and division with 1-digit divisorand dividend up to 3 digits.

(c) understand and use the commutativeproperty of multiplication.

(d) solve relevant practical problems ofmultiplication and division.

(e) recognize multiples, factors andrelationship between them.

(4) solve problems on mixed operation of

(a) addition and subtraction,(b) multiplication and addition,(c) multiplication and subtraction,(d) division and addition,(e) division and subtraction,(f) multiplication and division,with number of operations not exceedingtwo.

(5) solve simple problems on the use ofbrackets in mixed operations of additionand subtraction with at most threeoperations.

(6) (a) recognize coins and notes up to$1000, and use them in simplecontexts.

(b) read and write price tags, change unitsand apply the four rules to solveproblems involving money.

(7) (a) understand and use fractions ineveryday life.

compute addition and subtraction offractions with the same denominator.

(2) (a) compute multiplication: a number up to 3digits with a number up to 2 digits, andcompute division: divisor up to 3 digitsand dividend up to 4 digits, and checkby approximation whether the answersare reasonable.

(b) understand and use the commutative,associative and distributive properties ofmultiplication.

(c) solve practical problems involvingmultiplication or division of wholenumbers.

(d) solve simple mixed operations withintegers and problems involving the fourrules, including the use of simplebrackets.

(3) recognize and compute common multiplies,common factors, lowest common multiple(L.C.M.) and highest common factor(H.C.F.) of not more than 3 numbers.

(4) (a) recognize the meaning of properfractions, improper fractions, mixednumbers and equivalent fractions.

(b) perform addition, subtraction,multiplication, division and simple mixedoperations of simple fractions andreduce the answers to the simplest form,and check the results by estimation orapproximation.

(c) solve simple problems on fractions.

50

(Answers should not be greater than 1 andno simplification is needed.)

NUMBER DIMENSION (CONT.)

Key Stage 1 (P1 – P3) Key Stage 2 (P4 - P6)


(5) (a) read, write and order decimals, andunderstand place values of decimals.

(b) perform addition, subtraction,multiplication, division and simple mixedoperations of decimals up to twodecimal places, apply the rounding offconcept, and check the results byestimation or approximation.

(c) understand percentages and performconversions between decimals, fractionsand percentages.

(d) solve simple practical problems ondecimals and percentages.

(6) recognize squares and square roots of wholenumbers.

(7) (a) understand the idea of direct proportionand solve simple practical problems ondirect proportion by unitary method.

(b) understand the idea of inverseproportion and solve simple practicalproblems on inverse proportion byunitary method.

(8) calculate the average of a small group ofdata, taking examples from everyday life.

51

MEASURES DIMENSION




• choose and use a variety of non-standardunits of basic measures in measuringobjects

• understand the need of using standardunits of measurement

• select appropriate standard units ofmeasurement for different situations

• integrate the knowledge of number,measures and shape & space to solveintuitively simple measurement problems


• choose and use a variety of non-standardand standard units of various measures inmeasuring objects

• select and justify appropriate standard unitsof measurement for different situations

• recognize the degree of accuracy and theapproximate nature of measurement

• inquire and use simple measurementformulas

• integrate the knowledge of number,measures and shape & space to formulateand solve simple measurement problems


Learners

(1) (a) compare and order objects of more orless the same length, and recognize theneed to use standard units of measuresof length.

(b) recognize and use millimetre,centimetre, metre and kilometre.

(c) understand perimeter and calculateperimeters of rectangles and squares.

(2) (a) recognize and use second, minute, halfhour, hour, a.m. and p.m..

(b) recognize the months of a year, thedays of a week, the number of days ineach month, the number of days in ayear and a leap year.

(c) read time from a digital clock anddates from a calendar.

Learners

(1) understand the relationship between units ofmeasurement in metric system.

(2) (a) understand the concept of area.(b) use non-standard units of area to

compare objects and recognize theneed to use standard units.

(c) recognize and use square centimetreand square metre, and recognize thatmeasurement is approximate in natureand choose the degree of accuracyappropriate for a particular purpose.

(d) understand and use the formulas forareas of rectangles and squares.

(e) find areas of parallelograms, trianglesand trapeziums by counting sqaure andby formulas, and find areas of

52

polygons.

MEASURES DIMENSION (CONT.)



(3) (a) compare and order objects of more orless the same weight and recognize theneed to use standard units of measuresof weight.

(b) recognize and use gram and kilogram.

(4) recognize the degree Celsius and readthermometers.

(5) (a) compare and order vessels of more orless the same capacity and recognizethe need to use standard units ofmeasures of capacity.

(b) recognize and use litre.

(3) (a) recognize the relationship betweendiameters and circumferences ofcircles.

(b) recognize � and solve simple problemsinvolving diameter, radius andcircumference of a circle using =3.14 or 22/7.

(4) (a) recognize and use millilitre.(b) understand the concept of volume, and

recognize and use cubic centimetre.(c) recognize the relationship between

litre/millilitre and cubic centimetre.(d) recognize cubic metre and its use.(e) understand and use the formulas for

volumes of cubic and cuboids.(f) find volumes of irregular solids.

(5) recognize the 24-hour clock andunderstand the relationship between theunits of time and solve simple practicalproblems on time.

(6) (a) understand the concept of speed, andrecognize and use the units ‘metre persecond (m/s)’ and ‘kilometre per hour(km/h)’.

(b) solve simple problems on speed.

(7) understand scale and use appropriate scaleto draw plans.

(8) (a) compare and measure angles, and drawangles up to 360° using a protractor.

(b) make special angles by folding papers.

53

ALGEBRA DIMENSION



The ALGEBRA Dimension is not included atthis key stage.


• recognize the use of symbols to representunknown numbers

• communicate simple mathematicalfacts/relations using symbols

• manipulate simple relations involvingsymbols, and apply these knowledge andskills to formulate and solve simpleproblems and check the validity of results

• explore simple number patterns


The ALGEBRA Dimension is not included atthis key stage.

Learners

(1) recognize the use of symbols to stand forunknown numbers.

(2) record facts using algebraic symbols.

(3) solve simple equations, limiting to thoserequiring at most two steps in the solution,and check answers.

(4) solve simple practical problems byequations.

(5) recognize and appreciate simple numberpatterns such as square numbers, triangularnumbers and rectangular numbers.

54

SHAPE & SPACE DIMENSION




• identify, describe and classify 2-dimensional & 3-dimensional shapes

• recognize intuitively the elementaryproperties of 2-dimensional and 3-dimensional shapes

• make 2-dimensional and 3-dimensionalshapes from given information

• recognize, describe, appreciate and usepatterns of shapes

• develop a basic sense of position anddirection


• understand intuitively the properties of 2-dimensional and 3-dimensional shapes, andmake use of this understanding to classifyand make 2-dimensional and 3-dimensionalshapes

• develop a more elaborate sense of positionand direction

• specify location by means of appropriatemeasurements

• integrate the knowledge of measures andshape & space to formulate and solve 2-dimensional problems


Learners

(1) (a) recognize, describe and make simple2-dimensional shapes and classifythem.

(b) recognize, describe and make simple3-dimensional shapes and classifythem.

(c) recognize angles..(2) recognize and appreciate tessellation/ tile

patterns.

(3) recognize intuitively (reflective) symmetryin a variety of shapes in 2-dimension and3-dimension, and make symmetrical

Learners

(1) (a) understand intuitively the properties of quadrilaterals and classify quadrilaterals.

(b) recognize the properties of circles.

(2) (a) understand intuitively the properties of some pyramids & prisms and appreciate the relations between the number of edges, vertices and sides of bases of pyramids and prisms.

(b) design nets and make simple solids.

(3) (a) recognize the directions N, E, S, W,

55

shapes.

(4) recognize intuitively straight lines, curvesand parallel lines.

(5) (a) use common words to describeposition.

(b) recognize the four basic directions.

NE, SE, NW and SW, and use them to describe directions.

(b) recognize and use bearings.(c) perform simple surveying activities to

find their positions of objects from themeasurement of directions and

distances.

DATA HANDLING DIMENSION




• collect, compare and classify discretestatistical data according to given criteria

• construct simple statistical graphs showingrelationships among data and interpretthem

• formulate and solve simple problemsarising from collected data andconstructed graphs


• select criteria for grouping and organizingdiscrete statistical data

• apply simple arithmetic and appropriatescales in constructing and interpreting morecomplex statistical/line graphs

• show relationships among data using avariety of statistical and graphicalrepresentations

• recognize and use relationships and patternsfrom graphs

• formulate and solve problems arising fromcollected data and constructed graphs


56

Learners

(1) collect and classify simple data on topicsfrom the environment.

.

(2) draw graphs (pictograms, block graphsand bar charts) using one-to-onerepresentation and make simpleinterpretation.

(3) present and interpret simple bar charts ofgreater frequency counts, using one-to-two or one-to-ten representation.

Learners

(1) collect and group data of greater frequencycounts

(2) (a) present and interpret pictograms usingone-to-ten or one-to-hundredrepresentation.

(b) present and interpret bar charts of largefrequency counts using one-to-thousand or even smaller scales andcompound bar charts.

(c) present and interpret line graphs.(d) interpret pie charts.

(3) design and use suitable graphs to presentdata.

(4) understand average and make estimatesfrom bar charts.

(5) solve problems on direct proportion byusing straight line graphs.

(6) look for relationships, patterns or trendsfrom graphs.

An Overview of Learning Dimensions and Modulesfor Key Stages 3 and 4

AN

NE

X II

Measures, Shapeand Space Dimension

Data Handling Dimension

Number and Algebra Dimension

56

ObservingPatterns

and Expressing Generality


AlgebraicRelations

andFunctions

Numberand

NumberSystems

LearningGeometrythrough aDeductiveApproach

LearningGeometrythrough anIntuitive

Approach

LearningGeometrythrough anAnalytic

Approach

Trigonometry

Measuresin 2D

and 3DFigures

Analysisand

Interpretationof Data

Organizationand

Presentationof Data

ProbabilitySimple

StatisticalSurveys

FurtherApplications

Learning Dimensions

A Flowchart of Learning Units for Secondary School Mathematics CurriculumANNEX III

Note: Mathematical knowledge is interrelated both within and across dimensions. It is impossible to illustrate all links in a flowchart. Strong links between learning units are shown in dotted lines. These lines are just for illustrations and do not mean to be exhaustive. Teachers should exercise their professional judgement in arranging the sequence of learning units with special attention to the prerequisite knowledge required. For example, students are required to have the prerequisite knowle dge in "Introduction to Coordinates" to solve "Linear Equations in Two Unknowns" by graphical methods. Another example is that students need to acquire

Construction and

Interpretationof SimpleDiagrams

and Graphs

Simple Ideaof

Probability

Uses andAbuses

ofStatistics

Introductionto VariousStages ofStatistics

Measures ofCentral

Tendency

Linear Inequalities

in One Unknown

Linear Equations

In TwoUnknowns

Identities

More about

3-D Figures

More about

Percentages

SimpleIntroductionto Deductive

Geometry

TrigonometricRatios and

UsingTrigonometry

Pythagoras'Theorem

Quadrilaterals

CoordinatesGeometryof Straight

Lines

More aboutAreas andVolumes

Rate and

Ratio

Rational and Irrational

Numbers

Factorizationof Simple

Polynomials

More about

Probability

QuadraticEquations

in OneUnknown

LinearInequalities

in TwoUnknowns

Basic Properties

ofCircles

More aboutTrigonometry

More about

Polynomials

More about

Equations

Introduction toGeometry

Transformationand Symmetry

Congruence and Similarity

Coordinate Treatment ofSimple Locus Problems

Angles Related with Lines and

Rectilinear Figures

Introduction toCoordinates

Variations

Estimation inMeasurement

QualitativeTreatment of Locus

Directed Numbers and the Number Line

FormulatingProblems with

AlgebraicLanguage

Functions andGraphs

Exponential and Logarithmic Functions

Manipulationsof Simple

Polynomials

Arithmetic and Geometric Sequences and Their Summation

Linear EquationsIn One Unknown

Formulas

Using Percentages

KS2 MATHEMATICS EDUCATION

Measuresof

Dispersion

Simple Ideaof Areas and

Volumes

FURTHER STUDY / WORK

Further Applications

Numerical Estimation

Approximation and Errors

Laws of IntegralIndices

KS3

KS4

KEY : NON-FOUNDATIONS PARTS

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