### Mathematics Education Key Learning Area

**Mathematics **

## Curriculum and Assessment Guide (Secondary 4 6)

Jointly prepared by the Curriculum Development Council and The Hong Kong Examinations and Assessment Authority

Recommended for use in schools by the Education Bureau HKSARG

2007 (with updates in December 2017)

(Blank page)

**Contents **

Page

**Preamble ** i

**Acronyms ** iii

**Chapter 1 ** **Introduction ** 1

1.1 Background 1

1.2 Rationale 1

1.3 Curriculum Aims 2

1.4 Interface with the Junior Secondary Curriculum and Post- secondary Pathways

3

**Chapter 2 ** **Curriculum Framework ** 5

2.1 Design Principles 5

2.2 The Mathematics Education Key Learning Area Curriculum Framework

7 2.3 Aims of Senior Secondary Mathematics Curriculum 10 2.4 Framework of Senior Secondary Mathematics Curriculum 11

2.5 Compulsory Part 13

2.6 Extended Part 40

**Chapter 3 ** **Curriculum Planning ** 79

3.1 Guiding Principles 79

3.2 Curriculum Planning Strategies 81

3.3 Progression 84

3.4 Curriculum Management 89

**Chapter 4 ** **Learning and Teaching ** 93

4.1 Knowledge and Learning 93

4.2 Guiding Principles 94

4.3 Choosing Learning and Teaching Approaches and Strategies 96

4.4 Classroom Interaction 104

4.5 Learning Community 107

Page

4.6 Catering for Learner Diversity 108

4.7 Use of Information Technology (IT) in Learning and Teaching 109

**Chapter 5 ** **Assessment ** 111

5.1 The Roles of Assessment 111

5.2 Formative and Summative Assessment 112

5.3 Assessment Objectives 113

5.4 Internal Assessment 114

5.5 Public Assessment 118

**Chapter 6 ** **Learning and Teaching Resources ** 123

6.1 Purpose and Function of Learning and Teaching Resources 123

6.2 Guiding Principles 123

6.3 Types of Resources 124

6.4 Use of Learning and Teaching Resources 128

6.5 Resource Management 129

**Appendices ** 131

1 Reference Books for Learning and Teaching 131

2 Useful Websites 141

**Glossary ** 149

**References ** 155

**Membership of the CDC-HKEAA Committee on Mathematics **
**Education and its Working Groups **

**Membership of the Ad Hoc Committee on Secondary Mathematics **
**Curriculum **

i

### Preamble

The Education and Manpower Bureau (EMB, now renamed Education Bureau (EDB)) stated
in its report^{1} in 2005 that the implementation of a three-year senior secondary academic
structure would commence at Secondary 4 in September 2009. The senior secondary
academic structure is supported by a flexible, coherent and diversified senior secondary
curriculum aimed at catering for students' varied interests, needs and abilities. This
Curriculum and Assessment (C&A) Guide is one of the series of documents prepared for the
senior secondary curriculum. It is based on the goals of senior secondary education and on
other official documents related to the curriculum and assessment reform since 2000, including
*the Secondary Education Curriculum Guide (2017) and the Mathematics Education Key *
*Learning Area Curriculum Guide (Primary 1 – Secondary 6) (2017). To gain a full *
understanding of the connection between education at the senior secondary level and other key
stages, and how effective learning, teaching and assessment can be achieved, it is strongly
recommended that reference should be made to all related documents.

This C&A Guide is designed to provide the rationale and aims of the subject curriculum, followed by chapters on the curriculum framework, curriculum planning, pedagogy, assessment and use of learning and teaching resources. One key concept underlying the senior secondary curriculum is that curriculum, pedagogy and assessment should be well aligned. While learning and teaching strategies form an integral part of the curriculum and are conducive to promoting learning to learn and whole-person development, assessment should also be recognised not only as a means to gauge performance but also to improve learning. To understand the interplay between these three key components, all chapters in the C&A Guide should be read in a holistic manner.

The C&A Guide was jointly prepared by the Curriculum Development Council (CDC) and the Hong Kong Examinations and Assessment Authority (HKEAA) in 2007. The first updating was made in January 2014 to align with the short-term recommendations made on the senior secondary curriculum and assessment resulting from the New Academic Structure (NAS) review so that students and teachers could benefit at the earliest possible instance. The second updating is made to align with the medium-term recommendations of the NAS review made on curriculum and assessment. In response to the need to keep abreast of the ongoing renewal of the school curriculum and the feedback collected from the New Academic Structure Medium-term Review and Beyond conducted from November 2014 to April 2015, and to

1* The report is The New Academic Structure for Senior Secondary Education and Higher Education – Action *
*Plan for Investing in the Future of Hong Kong, and will be referred to as the 334 Report hereafter. *

ii

strengthen vertical continuity and lateral coherence, the Curriculum Development Council
Committee on Mathematics Education set up three Ad Hoc Committees in December 2015 to
review and revise the Mathematics curriculum from Primary 1 to Secondary 6. The
development of the revised Mathematics curriculum is based on the curriculum aims of
Mathematics education, guiding principles of curriculum design, and assessment stipulated in
*Mathematics Education Key Learning Area Curriculum Guide (Primary 1 - Secondary 6) *
(2017).

The CDC is an advisory body that gives recommendations to the HKSAR Government on all matters relating to curriculum development for the school system from kindergarten to senior secondary level. Its membership includes heads of schools, practising teachers, parents, employers, academics from tertiary institutions, professionals from related fields/bodies, representatives from the HKEAA and the Vocational Training Council (VTC), as well as officers from the EDB. The HKEAA is an independent statutory body responsible for the conduct of public assessment, including the assessment for the Hong Kong Diploma of Secondary Education (HKDSE). Its governing council includes members drawn from the school sector, tertiary institutions and government bodies, as well as professionals and members of the business community.

The C&A Guide is recommended by the EDB for use in secondary schools. The subject curriculum forms the basis of the assessment designed and administered by the HKEAA. In this connection, the HKEAA will issue a handbook to provide information on the rules and regulations of the HKDSE Examination as well as the structure and format of public assessment for each subject.

The CDC and HKEAA will keep the subject curriculum under constant review and evaluation in the light of classroom experiences, students’ performance in the public assessment, and the changing needs of students and society. All comments and suggestions on this C&A Guide may be sent to:

Chief Curriculum Development Officer (Mathematics) Curriculum Development Institute

Education Bureau

4/F Kowloon Government Offices 405 Nathan Road, Kowloon Fax: 3426 9265

E-mail: ccdoma@edb.gov.hk

iii

### Acronyms

AL Advanced Level

ApL Applied Learning

ASL Advanced Supplementary Level C&A Curriculum and Assessment CDC Curriculum Development Council CE Certificate of Education

COC Career-Oriented Curriculum (pilot of the Career-oriented Studies)

EC Education Commission

EDB Education Bureau

EMB Education and Manpower Bureau

HKALE Hong Kong Advanced Level Examination HKCAA Hong Kong Council for Academic Accreditation HKCEE Hong Kong Certificate of Education Examination HKDSE Hong Kong Diploma of Secondary Education

HKEAA Hong Kong Examinations and Assessment Authority HKSAR Hong Kong Special Administrative Region

IT Information Technology

KLA Key Learning Area

KS1/2/3/4 Key Stage 1/2/3/4

OLE Other Learning Experiences One Committee CDC-HKEAA Committee P1/2/3/4/5/6 Primary 1/2/3/4/5/6

PDP Professional Development Programmes

RASIH Review of the Academic Structure for Senior Secondary Education and Interface with Higher Education

S1/2/3/4/5/6/7 Secondary 1/2/3/4/5/6/7

iv

SBA School-based Assessment SEN Special Educational Needs SLP Student Learning Profile

SRR Standards-referenced Reporting SSCG Senior Secondary Curriculum Guide TPPG Teacher Professional Preparation Grant

1 1

### Chapter 1 Introduction

This chapter provides the background, rationale and aims of Mathematics as a core subject in the three-year senior secondary curriculum, and highlights how it articulates with the junior secondary curriculum, post-secondary education and future career pathways.

### 1.1 Background

* This Guide has been prepared by the Curriculum and Development Council (CDC) – Hong *
Kong Examinations and Assessment Authority (HKEAA) Committee on Mathematics
Education (Senior Secondary) in support of the new three-year senior secondary curriculum

*recommended in the 334 report on the New Academic Structure published in May 2005.*

Mathematics is a core subject for students from the primary level to the junior secondary level.

In the senior secondary curriculum, Mathematics is also one of the core subjects.

The Mathematics curriculum (S4 – 6) is a continuation of the Mathematics curriculum at the
*junior secondary level. Its development is built on the direction set out in the Mathematics *
*Education Key Learning Area Curriculum Guide (Primary 1 – Secondary 6) (2017). Students’ *

knowledge, skills, positive values and attitudes are further extended.

This document presents an outline of the overall aims, learning targets and objectives of the subject for the senior secondary level. It also provides suggestions regarding curriculum planning, learning and teaching strategies, assessment practices and resources. Schools are encouraged to adopt the recommendations in this Guide, taking into account their context, needs and strengths.

### 1.2 Rationale

The rationale for studying Mathematics as a core subject at the senior secondary level is presented below:

Mathematics is a powerful means in a technology-oriented and information-rich society to help students acquire the ability to communicate, explore, conjecture, reason logically and solve problems using a variety of methods.

2 2

Mathematics provides a means to acquire, organise and apply information, and plays an important role in communicating ideas through pictorial, graphical, symbolic, descriptive and analytical representations. Hence, mathematics at the senior secondary level helps to lay a strong foundation for students’ lifelong learning, and provides a platform for the acquisition of new knowledge in this rapidly changing world.

Many of the developments, plans and decisions made in modern society rely, to some extent, on the use of measures, structures, patterns, shapes and the analysis of quantitative information. Therefore, mathematical experiences acquired at the senior secondary level enable students to become mathematically literate citizens who are more able to cope with the demands of the workplace.

Mathematics is a tool to help students enhance their understanding of the world. It provides a foundation for the study of other disciplines in the senior secondary and post- secondary education system.

Mathematics is an intellectual endeavour through which students can develop their imagination, initiative, creativity and flexibility of mind, as well as their ability to appreciate the beauty of nature. Mathematics is a discipline which plays a central role in human culture.

### 1.3 Curriculum Aims

*Overall Aims *

The overall curriculum aims of the Mathematics Education Key Learning Area are to develop in students:

(a) the ability to think critically and creatively, to conceptualise, inquire and reason mathematically, and to use mathematics to formulate and solve problems in daily life as well as in mathematical contexts and other disciplines;

(b) the ability to communicate with others and express their views clearly and logically in mathematical language;

(c) the ability to manipulate numbers, symbols and other mathematical objects;

3 3

(d) number sense, symbol sense, spatial sense, measurement sense and the capacity to appreciate structures and patterns;

(e) a positive attitude towards the learning of mathematics and an appreciation of the aesthetic nature and cultural aspects of mathematics.

### 1.4 Interface with the Junior Secondary Curriculum and Post-secondary Pathways

1.4.1 Interface with the Junior Secondary Mathematics Curriculum

The Mathematics curriculum (S4 – 6), as part of the secondary curriculum, is built on the
*direction for development set out in the Mathematics Education Key Learning Area Curriculum *
*Guide (Primary 1 – Secondary 6) (2017). It aims at helping students to consolidate what they *
have learned through basic education, broadening and deepening their learning experiences, as
well as further enhancing their positive values and attitudes towards the learning of
mathematics. To ensure a seamless transition between the junior and senior secondary levels,
a coherent curriculum framework is designed for mathematics education at both levels.

As at the junior secondary level, the Mathematics curriculum at the senior secondary level aims to meet the challenges of the 21st century by developing students’ ability to think critically and creatively, to inquire and reason mathematically, and to use mathematics to formulate and solve problems in daily life as well as in mathematical contexts.

A particular learning unit “Inquiry and Investigation” has been included to provide students with opportunities to improve their ability to inquire, communicate, reason and conceptualise mathematical concepts; and there is also a “Further Applications” learning unit in which they have to integrate various parts of Mathematics which they have learned, and thus recognise the inter-relationships between their experiences of concrete objects in junior forms and abstract notions in senior forms.

1.4.2 Interface with Post-secondary Pathways

The curriculum also aims to prepare students for a range of post-secondary pathways, including tertiary education, vocational training and employment. It consists of a Compulsory Part and an Extended Part. In order to broaden students’ choices for further study and work, two

4 4

modules in the Extended Part are provided to further develop their knowledge of mathematics.

These two modules are designed to cater for students who intend to:

pursue further studies which require more mathematics; or

follow a career in fields such as natural sciences, computer sciences, technology or engineering.

**Module 1 (Calculus and Statistics) focuses on statistics and the application of **
mathematics, and is designed for students who will be involved in study and work which
demand a wider knowledge and deeper understanding of the application of mathematics, in
particular, statistics.

**Module 2 (Algebra and Calculus) focuses on mathematics in depth and aims to cater for **
students who will be involved in mathematics-related disciplines or careers.

The students’ performances in the public examination in the Compulsory Part, Module 1 and Module 2 will be separately reported for the reference of different users.

The following illustration gives an indication of the migration of the former Mathematics curricula towards the Mathematics curriculum (S4 – 6).

Former Mathematics Curricula Mathematics Curriculum (S4 – 6)

The Mathematics curriculum (S4 – 6) supports students’ needs in numerous vocational areas and careers, by providing them with various learning pathways. Further details will be provided in Chapter 2.

Secondary Mathematics Curriculum

Compulsory Part Additional Mathematics

Curriculum

ASL/AL Mathematics Curricula

Extended Part (Module1 or

Module 2)

5 5

### Chapter 2 Curriculum Framework

The curriculum framework for Mathematics embodies the key knowledge, skills, values and attitudes that students are to develop at the senior secondary level. It forms the basis on which schools and teachers can plan their school-based curricula, and design appropriate learning, teaching and assessment activities.

### 2.1 Design Principles

The following principles are used in designing the curriculum:

(a) Building on knowledge developed at the basic education level

To ensure that the curricula at different levels of schooling are coherent, the development of the Mathematics curriculum (S4 – 6) is built on the knowledge, skills, values and attitude acquired through the Mathematics curriculum for basic education from Primary 1 to Secondary 3.

(b) Providing a balanced, flexible and diversified curriculum

With the implementation of the New Academic Structure in Hong Kong, a wider range of students will gain access to Mathematics at the senior secondary level than in the past. The Mathematics curriculum (S4 – 6) offers a Compulsory Part and an Extended Part. The Compulsory Part is a foundation for all students and provides mathematical concepts, skills and knowledge which are necessary for students’ different career pathways. The Extended Part embraces two optional modules to provide add-on mathematical knowledge to suit the individual needs of students who would like to learn more mathematics and in a greater depth. The curriculum thus provides flexibility for teachers to:

offer a choice of courses within the curriculum to meet students’ individual needs, e.g. Compulsory Part, Compulsory Part with Module 1 (Calculus and Statistics) or Compulsory Part with Module 2 (Algebra and Calculus);

organise the teaching sequence to meet individual situations; and

make adaptations to the content.

(c) Catering for learner diversity

6 6

The curriculum provides opportunities for organising a variety of student activities to cater
for learner diversity. The learning unit “Inquiry and Investigation” in the curriculum
allows teachers to plan different learning activities for individual students. To further
assist teachers to adapt the curriculum to suit the needs of individual groups of students,
the content in the Compulsory Part is categorised into Foundation Topics and Non-
foundation Topics. The Foundation Topics constitute a set of essential concepts and
**knowledge which all students should strive to learn. Teachers can judge for themselves **
the suitability and relevance of the content from the Non-foundation Topics for their own
students. The Extended Part comprises two modules with different orientations.

Students who are more able in mathematics, more mathematically oriented or need more mathematical knowledge and skills to prepare for their future studies and careers may choose to study a module from the Extended Part. Module 1 (Calculus and Statistics) focuses more on mathematical applications, whereas Module 2 (Algebra and Calculus) places more emphasis on mathematical concepts and knowledge. Students who would like to learn more mathematics may choose the module which best suits their interests and needs.

(d) Achieving a balance between breadth and depth

The curriculum covers the important and relevant content for senior secondary students, based on the views of mathematicians, professionals in Mathematics Education and overseas Mathematics curricula at the same level. The breadth and depth of treatment in the Extended Part are intended to provide more opportunities for intellectually rigorous study in the subject.

(e) Achieving a balance between theoretical and applied learning

An equal emphasis is given on theories and applications in both real-life and mathematical contexts to help students construct their knowledge and skills in Mathematics. The historical development of selected mathematical topics is also included to promote students’

understanding of how mathematical knowledge has evolved and been refined in the past.

(f) Fostering lifelong learning skills

Knowledge is expanding at an ever faster pace and new challenges are continually posed by rapid developments in technology. It is important for our students to learn how to learn, think critically, analyse and solve problems, and communicate with others effectively so

7 7

that they can confront current and future challenges. The curriculum provides a suitable context for developing such abilities.

(g) Promoting positive values and attitudes to learning

Positive values and attitudes to learning, which are important in learning mathematics, permeate the Mathematics curriculum (S4 – 6). In particular, the unit “Inquiry and Investigation” helps to develop in students an interest in learning mathematics, keenness to participate in mathematical activities, and sensitivity and confidence in applying mathematics in daily life. It also helps to foster open-mindedness and independent thinking.

### 2.2 The Mathematics Education Key Learning Area Curriculum Framework

The curriculum framework for Mathematics Education is the overall structure for organising learning and teaching activities for the subject of Mathematics. The framework comprises a set of interlocking components, including:

subject knowledge and skills, which are expressed in the form of learning targets and learning objectives within strands;

generic skills; and

positive values and attitudes.

The framework sets out what students should know, value and be able to do at various stages of schooling from Primary 1 to Secondary 6. It provides schools and teachers with the flexibility to adapt the Mathematics curriculum to meet their varied needs.

A diagrammatic representation highlighting the major components of the Mathematics curriculum framework is provided on the following page.

8 8

**Diagrammatic Representation of the Framework ** **of the Mathematics Curriculum **

9 Generic Skills

Number Algebra Measures

Shape and Space

Data Handling

Values and Attitudes

Number and Algebra

Measures, Shape and Space

Data Handling

(Extended Part) (Compulsory Part) (Extended Part)

Module 1 (Calculus

and Statistics)

Number and Algebra

Measures, Shape and Space

Data Handling

Module 2 (Algebra

and Calculus)

Further Learning Unit

**Mathematics Curriculum **

provides content knowledge which can serve as a means to develop students’ thinking abilities

and foster students’ generic skills and positive attitudes

towards the learning of mathematics

**Strands **

provide a structured framework of learning objectives in different areas of the Mathematics

curriculum

Effective linkage of learning, teaching and assessment

Overall Aims and Learning Targets of Mathematics

S4-6 S4-6

P1-6 P1-6

S1-3 S1-3

9 9

2.2.1 Strands

Strands are categories of mathematical knowledge and concepts for organising the curriculum.

Their main function is to organise mathematical content for the purpose of developing knowledge, skills, values and attitudes as a holistic process. The content of the Mathematics curriculum is organised into five strands at the primary level and three strands at the secondary level. In particular, the Compulsory Part of the Mathematics curriculum (S4 – 6) comprises three strands, namely “Number and Algebra”, “Measures, Shape and Space” and “Data Handling”. As the content of the Extended Part is interwoven, it is not categorised into strands.

2.2.2 Generic Skills

Generic skills can be seen as both process skills and learning outcomes in the Mathematics Education Key Learning Area. They are essential for enabling learners to learn how to learn.

Nine generic skills have been identified: collaboration skills, communication skills, creativity, critical thinking skills, information technology skills, mathematical skills, problem solving skills, self-management skills and self-learning skills.

It should be noted that generic skills are not something to be added on to the learning and teaching of mathematical concepts, but should be embedded within them. They serve as a means to develop the acquisition and mastery of mathematical knowledge and concepts. An emphasis on communication skills, creativity and critical thinking skills in the context of mathematical activities will help to strengthen students’ ability to achieve the overall learning targets of the curriculum. Daily-life applications, further applications of mathematics, inquiry and investigation are emphasised.

2.2.3 Values and Attitudes

Besides knowledge and skills, the development of positive values and attitudes is also important in Mathematics Education. Values and attitudes such as responsibility, commitment and open-mindedness are necessary for developing goals in life and learning. The inculcation of such positive values/attitudes through appropriate learning and teaching strategies can enhance learning, and this in turn will reinforce their development in students as part of character formation. Positive values and attitudes permeate the Mathematics curriculum (S4 – 6) and have been incorporated into its learning objectives, so that students can:

10 10

** develop interest in learning mathematics; **

** show keenness to participate in mathematical activities; **

** develop sensitivity towards the importance of mathematics in daily life; **

** show confidence in applying mathematical knowledge in daily life, by clarifying one’s **
argument and challenging others’ statements;

** share ideas and experience and work cooperatively with others in accomplishing **
mathematical tasks/activities and solving mathematical problems;

** understand and take up responsibilities; **

** be open-minded, willing to listen to others in the discussion of mathematical problems, **
respect others’ opinions, and value and appreciate others’ contributions;

** think independently in solving mathematical problems; **

** be persistent in solving mathematical problems; and **

** appreciate the precise, aesthetic and cultural aspects of mathematics and the role of **
mathematics in human affairs.

These values and attitudes can be fostered through the learning of mathematical content.

Teachers can help students cultivate them through planned learning activities.

### 2.3 Aims of Senior Secondary Mathematics Curriculum

The Mathematics curriculum (S4 – 6) is a continuation of the Mathematics curriculum (S1 – 3). It aims to:

(a) further develop students’ mathematical knowledge, skills and concepts;

(b) provide students with mathematical tools for their personal development and future career pathways;

(c) provide a foundation for students who may further their studies in mathematics or related areas;

(d) develop in students the generic skills, and in particular, the capability to use mathematics to solve problems, reason and communicate;

(e) develop in students interest in and positive attitudes towards the learning of mathematics;

11 11

(f) develop students’ competence and confidence in dealing with mathematics needed in life;

and

(g) help students to fulfil their potential in mathematics.

### 2.4 Framework of Senior Secondary Mathematics Curriculum

The structure of the Mathematics curriculum (S4 – 6) can be represented diagrammatically as follows:

**Mathematics Curriculum ** **(S4 – 6) **

**Compulsory Part ** **Extended Part **

Module 1

(Calculus and Statistics)

Module 2 ( Algebra and Calculus)

[*Note: Students may take the Compulsory Part only, the Compulsory Part with Module 1 (Calculus and *
*Statistics) or the Compulsory Part with Module 2 (Algebra and Calculus). Students are only allowed *
*to take at most one module from the Extended Part.*]

To cater for students who have different needs, interests and orientations, the curriculum
**comprises a Compulsory Part and an Extended Part. All students must study the Compulsory **
Part.

The Extended Part has two optional modules, namely Module 1 (Calculus and Statistics) and Module 2 (Algebra and Calculus). The inclusion of the Extended Part is designed to provide more flexibility and diversity in the curriculum. The two modules in the Extended Part provide additional mathematical knowledge to the Compulsory Part. Students, based on their individual needs and interests, are encouraged to take at most one of the two modules.

The following diagrams show the different ways in which students can progress:

12 12

(1) Students who study only the Foundation Topics in the Compulsory Part

Foundation Topics

Non-
foundation
Topics
**Compulsory Part **

(2) Students who study the Foundation Topics and some Non-foundation Topics in the Compulsory Part

Foundation Topics

Non-
foundation
Topics
**Compulsory Part **

(3) Students who study all topics in the Compulsory Part

Foundation Topics

Non-
foundation
Topics
**Compulsory Part **

13 13

(4) Students who study the Compulsory Part with Module 1 (Calculus and Statistics)

**Compulsory Part **

**Module 1 **
**(Calculus and **

**Statistics) **

(5) Students who study the Compulsory Part with Module 2 (Algebra and Calculus)

**Compulsory Part **

**Module 2 **
**(Algebra and **

**Calculus) **

As a core subject, the Mathematics curriculum (S4 – 6) accounts for up to 15% (approximately
375 hours)^{1} of the total lesson time available in the senior secondary curriculum. The
suggested time allocations for the Compulsory Part and the Extended Part are as follows:

Lesson time

(Approximate number of hours) Compulsory Part 10% – 12.5% (250 hours – 313 hours) Compulsory Part with a module 15% (375 hours)

### 2.5 Compulsory Part

The principles of curriculum design of the Compulsory Part comply with those of the Mathematics curriculum (S4 – 6) as a whole, but have two distinguishing features.

First, the Compulsory Part serves as a foundation for all students and at the same time provides flexibility to cater for the diverse needs of individual students. Its content is categorised into Foundation Topics and Non-foundation Topics. The Foundation Topics constitute a coherent

1 The NSS curriculum is designed on the basis of 2,500 lesson hours. A flexible range of total lesson time at 2,400±200 hours over three years is recommended for school-based planning purposes to cater for school diversity and varying learning needs while maintaining international benchmarking standards.

As always, the amount of time spent in learning and teaching is governed by a variety of factors, including whole- school curriculum planning, learners’ abilities and needs, students’ prior knowledge, teaching and assessment strategies, teaching styles and the number of subjects offered. Schools should exercise professional judgement and flexibility over time allocation to achieve specific curriculum aims and objectives as well as to suit students' specific needs and the school context.

14 14

set of essential concepts and knowledge while the Non-foundation Topics cover a wider range of content.

Second, the topics in the Compulsory Part emphasise the relevance of mathematics to various human activities. Students are expected to engage in different activities to foster their awareness of the worldwide use of mathematical terminology, notation and strategies to solve problems. Also, to enable students to recognise and appreciate the interconnection between the different parts of mathematics they have learned at both the junior and senior secondary levels, a “Further Applications” learning unit is incorporated into the Compulsory Part.

The learning objectives of the Compulsory Part foster students’ understanding of the development of mathematical knowledge and skills and their applications in the solution of various problems, including real-life ones. In addition, learning units such as “Uses and Abuses of Statistics”, “Permutation and Combination” and “Further Applications” are included for students to use the mathematics learned at junior and senior secondary levels to understand and assess more sophisticated scenarios critically.

2.5.1 Organisation of the Compulsory Part

The most significant aspects of learning and teaching in each strand of the Compulsory Part are organised into a hierarchy from Learning Targets to specific Learning Objectives.

Learning Targets set out the aims and direction for learning and teaching and, under these, Learning Objectives are identified to spell out specifically what students need to learn. In the curriculum, Learning Objectives are presented and grouped under different Learning Units.

The three strands in the Compulsory Part are “Number and Algebra”, “Measures, Shape and Space” and “Data Handling”. In addition, the “Further Learning Unit” is designed to integrate and apply knowledge and skills learned in the strands to solve problems in real-life as well as in mathematical contexts.

2.5.2 Learning Targets of the Compulsory Part

An overview of the learning targets of the three strands in the Compulsory Part is provided on the following page.

15 15

**Learning Targets of the Compulsory Part **

**Number and Algebra **
**Strand **

**Measures, Shape and Space **
**Strand **

**Data Handling **
**Strand **
Students are expected to:

extend the concepts of numbers to complex numbers;

further investigate and describe relationships between quantities using algebraic symbols;

generalise and describe patterns in 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 the knowledge and skills to formulate and solve more complex real-life problems and justify the validity of the results obtained; and

apply the knowledge and skills in the Number and Algebra strand to generalise, describe and communicate

mathematical ideas and further solve problems in other strands.

use inductive and deductive approaches to study the properties of 2-dimensional figures;

perform geometric proofs involving 2-dimensional figures with appropriate symbols, terminology and reasons;

further inquire and describe geometric knowledge in 2-dimensional space using algebraic relations and apply the knowledge to solve problems;

inquire and describe geometric knowledge in 2-dimensional space and 3-dimensional space using trigonometric functions and apply the knowledge to solve problems; and

apply the knowledge and skills in the Measures, Shape and Space strand to generalise, describe and communicate mathematical ideas and further solve problems in other strands.

understand the measures of dispersion;

select and use the measures of central

tendency and dispersion to describe and compare data sets;

further investigate and judge the validity of arguments derived from data sets;

acquire basic techniques in counting;

formulate and solve more complex probability problems by applying simple laws; and

integrate the knowledge in statistics and probability to solve more complex real-life problems.

16 16

2.5.3 Foundation Topics and Non-foundation Topics in the Compulsory Part

To cater for the needs of individual students, the content of the Compulsory Part is categorised into Foundation Topics and Non-foundation Topics. The Foundation Topics of the Compulsory Part and the Foundation Topics of the Mathematics curriculum (S1 – 3) constitute a coherent set of essential concepts and knowledge. The Foundation Topics, which all students should strive to learn, are selected in accordance with the following principles in mind:

to include basic concepts and knowledge necessary for the learning content in the Compulsory Part and for simple applications in real-life situations; and

to cover topics from different areas to enable students to develop a coherent body of knowledge and to experience mathematics from an all-around perspective.

There are also topics beyond those in the Foundation Topics in terms of depth and breadth.

They are identified as Non-foundation Topics and cover a wider range of content, to provide students who study only the Compulsory Part with a foundation for their future studies and career development. Teachers can judge for themselves the suitability and relevance of the Non-foundation Topics for their own students.

The content of Module 1 and Module 2 is built upon the study of the Foundation and Non- foundation Topics in the Compulsory Part. It is advisable for students to study both the Foundation Topics and Non-foundation Topics in the Compulsory Part if they study either one of the modules from the Extended Part.

2.5.4 Learning Content of the Compulsory Part

The time allocated to the Compulsory Part ranges from 10% to 12.5% of the total lesson time (approximately 250 hours to 313 hours), subject to the different pathways, orientations and learning speeds of students.

To aid teachers in their planning and adaptation, a suggested lesson time in hours is given against each learning unit in the following table. The learning objectives of the Non- foundation Topics are underlined for teachers’ reference.

1717

**Learning Content of the Compulsory Part ** **Notes: **

1. *Learning units are grouped under three strands (“Number and Algebra”, “Measures, Shape and Space” and “Data Handling”) and a Further *
*Learning Unit. *

2. *Related learning objectives are grouped under the same learning unit. *

3. *The learning objectives underlined are the Non-foundation Topics. *

4. *The notes in the “Remarks” column of the table may be considered as supplementary information about the learning objectives. *

5. *To aid teachers in judging how far to take a given topic, a suggested lesson time in hours is given against each learning unit. However, the *
*lesson time assigned is for their reference only. Teachers may adjust the lesson time to meet their individual needs. *

6. *Schools may allocate up to 313 hours (i.e. 12.5% of the total lesson time) to those students who need more time for learning. *

**Learning Unit ** **Learning Objective ** **Time Remarks **

**Number and Algebra Strand **
1. Quadratic equations in

one unknown

1.1 solve quadratic equations by the factor method 19

1.2 form quadratic equations from given roots The given roots are confined to real numbers.

*1.3 solve the equation ax*^{2 }*+ bx + c = 0 by plotting *
*the graph of the parabola y = ax*^{2 }*+ bx + c and *
*reading the x-intercepts *

1818

**Learning Unit ** **Learning Objective ** **Time Remarks **

1.4 solve quadratic equations by the quadratic formula **The following are not required for students **
taking only the Foundation Topics:

expressing nonreal roots in the form a ± bi

simplifying expressions involving surds such as 2 48

1.5 understand the relations between the discriminant of a quadratic equation and the nature of its roots

When < 0, students have to point out that

**“the equation has no real roots” or “the **
**equation has two nonreal roots” as they are **
expected to recognise the existence of
complex numbers in Learning Objective 1.8.

1.6 solve problems involving quadratic equations Teachers should select the problems related to students’ experiences.

Problems involving complicated equations

such as 5

1 6

6

*x*

*x* are required only in
the Non-foundation Topics and dealt with in
Learning Objective 5.4.

1919

**Learning Unit ** **Learning Objective ** **Time Remarks **

1.7 understand the relations between the roots and coefficients and form quadratic equations using these relations

The relations between the roots and coefficients include:

*a*

*b*

and

*a*

*c*

,

where and are the roots of the equation
*ax*^{2}* + bx + c = 0 and a 0. *

1.8 appreciate the development of the number systems including the system of complex numbers

The topics such as the hierarchy of the number systems and the conversion between recurring decimals and fractions may be discussed.

1.9 perform addition, subtraction, multiplication and division of complex numbers

*Complex numbers are confined to the form a *

bi .

Note: The coefficients of quadratic equations are confined to real numbers.

2. Functions and graphs 2.1 recognise the intuitive concepts of functions, domains and co-domains, independent and dependent variables

10

2020

**Learning Unit ** **Learning Objective ** **Time Remarks **

2.2 recognise the notation of functions and use tabular, algebraic and graphical methods to represent functions

Representations like

can also be accepted.

2.3 understand the features of the graphs of quadratic functions

The features of the graphs of quadratic functions include:

the vertex

the axis of symmetry

the direction of opening

relations with the axes

Students are required to find the maximum and minimum values of quadratic functions by the graphical method.

2.4 find the maximum and minimum values of quadratic functions by the algebraic method

The method of completing the square is required.

Students are required to solve problems related to maximum and minimum values of quadratic functions.

1 2

2121

**Learning Unit ** **Learning Objective ** **Time Remarks **

3. Exponential and logarithmic functions

3.1 understand the definitions of rational indices 16 The definitions include and . 3.2 understand the laws of rational indices The laws of rational indices include:

a^{ p }*a*^{ q}* = a*^{ p + q}

_{q}^{p}*a*

*a* * = a*^{ p q}

(a* ^{ p}*)

^{q}*= a*

^{ pq} a^{ p}* b*^{ p}* = (ab)*^{ p}

*p*

*p*
*p*

*b*
*a*
*b*

*a*

3.3 understand the definition and properties of logarithms (including the change of base)

The properties of logarithms include:

log* a *1 = 0

log* a **a = 1 *

log* a **MN = log** a **M + log** a **N *

log* a*

*N*

*M* = log* a **M log** a **N *

log* a** M*^{ k}* = k log** a **M *

log *b **N =*

*b*
*N*

*a*
*a*

log log

2222

**Learning Unit ** **Learning Objective ** **Time Remarks **

3.4 understand the properties of exponential functions and logarithmic functions and recognise the features of their graphs

The properties and features include:

the domains of the functions

the function f (x) = a^{ x}* and f (x) = log **a** x *
*increases (decreases) as x increases for *
*a > 1 (0 < a < 1) *

y = a^{ x}* is symmetric to y = log** a **x *
*about y = x *

the intercepts with the axes

the rate of increasing/the rate of decreasing of the functions (by direct inspection) 3.5 solve exponential equations and logarithmic

equations

Equations which can be transformed into
quadratic equations such as 4* ^{x}* 3 2

* 4 =*

^{x}*0 or log (x* 22) + log (x + 26) = 2 are dealt with in Learning Objective 5.3.

3.6 appreciate the applications of logarithms in real-life situations

The applications such as measuring earthquake intensity in the Richter Scale and sound intensity level in decibels may be discussed.

2323

**Learning Unit ** **Learning Objective ** **Time Remarks **

3.7 appreciate the development of the concepts of logarithms

The topics such as the historical development of the concepts of logarithms and its applications to the design of some past calculation tools such as slide rules and the logarithmic table may be discussed.

4. More about polynomials

4.1 perform division of polynomials 14 Methods other than long division are also accepted.

4.2 understand the remainder theorem

4.3 understand the factor theorem Students are required to use factor theorem to
*factorise polynomials such as x*^{3 } a^{3}.

4.4 understand the concepts of the greatest common divisor and the least common multiple of polynomials

The terms “H.C.F.” , “gcd”, etc. can be used.

4.5 perform addition, subtraction, multiplication and division of rational functions

Computation of rational functions with more
**than two variables is not required. **

Rational functions refer to algebraic fractions at Key Stage 3.

5. More about equations 5.1 use the graphical method to solve simultaneous
equations in two unknowns, one linear and one
*quadratic in the form y = ax*^{2}* + bx + c *

10

2424

**Learning Unit ** **Learning Objective ** **Time Remarks **

5.2 use the algebraic method to solve simultaneous equations in two unknowns, one linear and one quadratic

5.3 solve equations (including fractional equations, exponential equations, logarithmic equations and trigonometric equations) which can be transformed into quadratic equations

Solutions for trigonometric equations are confined to the interval from 0 to 360 .

5.4 solve problems involving equations which can be transformed into quadratic equations

Teachers should select the problems related to students’ experience.

6. Variations 6.1 understand direct variations and inverse variations, and their applications to solving real-life problems

7

6.2 understand the graphs of direct and inverse variations

6.3 understand joint and partial variations, and their applications to solving real-life problems

7. Arithmetic and geometric sequences and their summations

7.1 understand the concept and the properties of arithmetic sequences

17 The properties of arithmetic sequences include:

T*n** = ½ ( T**n–1** + T**n+1* )

if T1* , T*2* , T*3 , … is an arithmetic
*sequence, then k T*1* + a , k T*2* + a , k T*3 +
*a , … is also an arithmetic sequence *

2525

**Learning Unit ** **Learning Objective ** **Time Remarks **

7.2 understand the general term of an arithmetic sequence

7.3 understand the concept and the properties of geometric sequences

The properties of geometric sequences include:

T*n*2* = T**n1* T*n+1*

if T1* , T*2* , T*3 , … is a geometric
*sequence, then k T*1* , k T*2* , k T*3 , … is
also a geometric sequence

7.4 understand the general term of a geometric sequence

7.5 understand the general formulae of the sum to a finite number of terms of an arithmetic sequence and a geometric sequence and use the formulae to solve related problems

Example: geometrical problems involving the sum of arithmetic or geometric sequences.

7.6 explore the general formulae of the sum to infinity for certain geometric sequences and use the formulae to solve related problems

Example: geometrical problems involving infinite sum of the geometric sequences.

7.7 solve related real-life problems Examples: problems about interest, growth or depreciation.

2626

**Learning Unit ** **Learning Objective ** **Time Remarks **

8. Inequalities and linear programming

8.1 solve compound linear inequalities in one unknown 16 Compound inequalities involving logical connectives “and” or “or” are required.

Solving the problems on triangle inequalities is required.

8.2 solve quadratic inequalities in one unknown by the graphical method

8.3 solve quadratic inequalities in one unknown by the algebraic method

8.4 represent the graphs of linear inequalities in two unknowns in the rectangular coordinate plane 8.5 solve systems of linear inequalities in two

unknowns

8.6 solve linear programming problems 9. More about graphs of

functions

9.1 sketch and compare graphs of various types of functions including constant, linear, quadratic, trigonometric, exponential and logarithmic functions

11 Comparison includes domains, existence of maximum or minimum values, symmetry and periodicity.

*9.2 solve the equation f (x) = k using the graph of *
*y = f (x) *

2727

**Learning Unit ** **Learning Objective ** **Time Remarks **

*9.3 solve the inequalities f (x) > k , f (x) < k , *
*f (x) k and f (x) k using the graph of *
*y = f (x) *

9.4 understand the transformations of the function
*f (x) including f (x) + k , f (x + k) , k f (x) *
*and f (kx) from tabular, symbolic and graphical *
perspectives

**Measures, Shape and Space Strand **
10. Equations of straight

lines

10.1 understand the equation of a straight line 7 Students are required to find the equation of a straight line from given conditions such as:

the coordinates of any two points on the straight line

the slope of the straight line and the coordinates of a point on it

the slope and the y-intercept of the straight line

Students are required to describe the features of a straight line from its equation. The features include:

the slope

2828

**Learning Unit ** **Learning Objective ** **Time Remarks **

the intercepts with the axes

whether it passes through a given point
**The normal form is not required. **

Students are required to recognise the relation between slope and inclination.

10.2 understand the possible intersection of two straight lines

Students are required to determine the number of intersection points of two straight lines from their equations.

At Key Stage 3, students are required to solve simultaneous linear equations in two unknowns.

Note: Teachers are suggested to arrange the teaching of this Learning Unit in the first term of S4.

11. Basic properties of circles

11.1 understand the properties of chords and arcs of a circle

23 The properties of chords and arcs of a circle include:

the chords of equal arcs are equal

equal chords cut off equal arcs

the perpendicular from the centre to a chord bisects the chord

2929

**Learning Unit ** **Learning Objective ** **Time Remarks **

the straight line joining the centre and the mid-point of a chord which is not a diameter is perpendicular to the chord

the perpendicular bisector of a chord passes through the centre

equal chords are equidistant from the centre

chords equidistant from the centre are equal Students are required to understand that there is one and only one circle passing through given three non-collinear points.

The property that the arcs are proportional to their corresponding angles at the centre should be discussed at Key Stage 3 when the formula for calculating arc lengths is being explicated.

11.2 understand the angle properties of a circle The angle properties of a circle include:

the angle subtended by an arc of a circle at the centre is double the angle subtended by the arc at any point on the remaining part of the circumference

angles in the same segment are equal

the arcs are proportional to their corresponding angles at the circumference

3030

**Learning Unit ** **Learning Objective ** **Time Remarks **

the angle in a semi-circle is a right angle

if the angle at the circumference is a right angle, then the chord that subtends the angle is a diameter

11.3 understand the properties of a cyclic quadrilateral The properties of a cyclic quadrilateral include:

the opposite angles of a cyclic quadrilateral are supplementary

an exterior angle of a cyclic quadrilateral equals its interior opposite angle

11.4 understand the tests for concyclic points and cyclic quadrilaterals

The tests for concyclic points and cyclic quadrilaterals include:

if A and D are two points on the same side
*of the line BC and BAC = BDC , then *
*A , B , C and D are concyclic *

if a pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic

if the exterior angle of a quadrilateral equals its interior opposite angle, then the quadrilateral is cyclic

3131

**Learning Unit ** **Learning Objective ** **Time Remarks **

11.5 understand the properties of tangents to a circle and angles in the alternate segments

The properties include:

a tangent to a circle is perpendicular to the radius through the point of contact

the straight line perpendicular to a radius of a circle at its external extremity is a tangent to the circle

the perpendicular to a tangent at its point of contact passes through the centre of the circle

if two tangents are drawn to a circle from an external point, then:

- the distances from the external point to the points of contact are equal - the tangents subtend equal angles at

the centre

- the straight line joining the centre to the external point bisects the angle between the tangents

if a straight line is tangent to a circle, then the tangent-chord angle is equal to the angle in the alternate segment

3232

**Learning Unit ** **Learning Objective ** **Time Remarks **

if a straight line passes through an end point of a chord of a circle so that the angle it makes with the chord is equal to the angle in the alternate segment, then the straight line touches the circle

11.6 use the basic properties of circles to perform simple geometric proofs

Knowledge on geometry learnt at Key Stage 3 can be involved in the geometric proofs.

12. Loci 12.1 understand the concept of loci 6

12.2 describe and sketch the locus of points satisfying given conditions

The conditions include:

maintaining a fixed distance from a fixed point

maintaining an equal distance from two given points

maintaining a fixed distance from a line

maintaining an equal distance from two parallel lines

maintaining an equal distance from two intersecting lines

3333

**Learning Unit ** **Learning Objective ** **Time Remarks **

12.3 describe the locus of points with algebraic equations

Students are required to find the equations of
simple loci, which include equations of
straight lines, circles and parabolas (in the
*form of y = ax*^{2}* + bx + c ). *

13. Equations of circles 13.1 understand the equation of a circle 7 Students are required to find the equation of a circle from given conditions such as:

the coordinates of the centre and the radius of the circle

the coordinates of any three points on the circle

Students are required to describe the features of a circle from its equation. The features include:

the centre

the radius

whether a given point lies inside, outside or on the circle

13.2 find the coordinates of the intersections of a straight line and a circle and understand the possible intersection of a straight line and a circle

Students are required to find the equations of tangents to a circle.

3434

**Learning Unit ** **Learning Objective ** **Time Remarks **

14. More about trigonometry

14.1 understand the functions sine, cosine and tangent, and their graphs and properties, including maximum and minimum values and periodicity

25 Simplification of expressions involving sine, cosine and tangent of , 90 , 180 , … , etc. is required.

*14.2 solve the trigonometric equations a sin = b , *
*a cos = b , a tan = b (solutions in the interval *
from 0 to 360 ) and other trigonometric
equations (solutions in the interval from 0 to
360 )

Equations that can be transformed into quadratic equations are required only in the Non-foundation Topics and dealt with in Learning Objective 5.3.

*14.3 understand the formula ½ ab sin C for areas of *
triangles

14.4 understand the sine and cosine formulae 14.5 understand Heron’s formula

14.6 understand the concept of projection

14.7 understand the angle between a line and a plane, and the angle between 2 planes

The concept of inclination is required.

14.8 understand the theorem of three perpendiculars

3535

**Learning Unit ** **Learning Objective ** **Time Remarks **

14.9 solve related 2-dimensional and 3-dimensional problems

3-dimensional problems include finding the angle between two lines, the angle between a line and a plane, the angle between two planes, the distance between points, the distance between a point and a line, and the distance between a point and a plane.

**Data Handling Strand **
15. Permutations and

combinations

15.1 understand the addition rule and multiplication rule in the counting principle

11

15.2 understand the concept and notation of permutation Notations such as “*P*_{r}* ^{n}*” , “

*n*

*P*

*r*” , “

^{n}*P*

*r*” , etc.

can be used.

15.3 solve problems on the permutation of distinct objects without repetition

Problems such as “permutation of objects in which three particular objects are put next to each other” are required.

**Circular permutation is not required. **

15.4 understand the concept and notation of

combination Notations such as “*C*_{r}* ^{n}*” , “

*n*

*C*

*r*” , “

^{ n}*C*

*r*” ,

“

*r*

*n* ” , etc. can be used.

3636

**Learning Unit ** **Learning Objective ** **Time Remarks **

15.5 solve problems on the combination of distinct objects without repetition

16. More about probability

16.1 recognise the notation of set language including union, intersection and complement

10 The concept of Venn Diagram is required.

16.2 understand the addition law of probability and the concepts of mutually exclusive events and complementary events

The addition law of probability refers to

*“P(A B) = P(A) + P(B) P(A B) ”. *

16.3 understand the multiplication law of probability and the concept of independent events

The multiplication law of probability refers to

*“P(A * B) = P(A) P(B) , where A and
*B are independent events”. *

16.4 recognise the concept and notation of conditional probability

*The rule “P(A * B) = P(A) P(B | A)” is
required.

**Bayes’ Theorem is not required. **

16.5 use permutation and combination to solve problems related to probability

17. Measures of dispersion

17.1 understand the concept of dispersion 13 17.2 understand the concepts of range and inter-quartile

range

3737

**Learning Unit ** **Learning Objective ** **Time Remarks **

17.3 construct and interpret the box-and-whisker diagram and use it to compare the distributions of different sets of data

A box-and-whisker diagram can also be called a “boxplot”.

17.4 understand the concept of standard deviation for both grouped and ungrouped data sets

Students are required to recognise the term

“variance” and that variance equals to the square of standard deviation.

Students are required to understand the following formula for standard deviation:

= *N*

*x*

*x*_{1} )^{2} ( * _{N}* )

^{2}(

.

17.5 compare the dispersions of different sets of data using appropriate measures

17.6 understand the applications of standard deviation to real-life problems involving standard scores and the normal distribution

17.7 understand the effect of the following operations on the dispersion of the data:

(i) adding a common constant to each item of the set of data

(ii) multiplying each item of the set of data by a common constant

3838

**Learning Unit ** **Learning Objective ** **Time Remarks **

18. Uses and abuses of statistics

18.1 recognise different techniques in survey sampling and the basic principles of questionnaire design

4 Students are required to recognise the concepts of “populations” and “samples”.

Students are required to recognise probability sampling and non-probability sampling.

Students are required to recognise that, in constructing questionnaires, factors such as the types, wording and ordering of questions and response options influence their validity and reliability.

18.2 discuss and recognise the uses and abuses of statistical methods in various daily-life activities or investigations

18.3 assess statistical investigations presented in different sources such as news media, research reports, etc.

**Further Learning Unit **

19. Further applications Solve more sophisticated real-life and mathematical problems that may require students to search the information for clues, to explore different strategies, or to integrate various parts of mathematics which they have learned in different areas

14 Examples:

solve simple financial problems in areas such as taxation and instalment payment

analyse and interpret data collected in surveys