**Mathematics Education Section**
**Curriculum Development Institute**
**Education Bureau**

**2020 **

**Explanatory Notes to**

**Junior Secondary Mathematics Curriculum**

**Contents**

Page

Foreword i

Learning Unit 1 Basic computation 1

Learning Unit 2 Directed numbers 4

Learning Unit 3 Approximate values and numerical estimation 6

Learning Unit 4 Rational and irrational numbers 8

Learning Unit 5 Using percentages 11

Learning Unit 6 Rates, ratios and proportions 13

Learning Unit 7 Algebraic expressions 16

Learning Unit 8 Linear equations in one unknown 19

Learning Unit 9 Linear equations in two unknowns 20

Learning Unit 10 Laws of integral indices 23

Learning Unit 11 Polynomials 26

Learning Unit 12 Identities 29

Learning Unit 13 Formulae 30

Learning Unit 14 Linear inequalities in one unknown 32

Learning Unit 15 Errors in measurement 34

Learning Unit 16 Arc lengths and areas of sectors 36

Learning Unit 17 3-D figures 38

Learning Unit 18 Mensuration 41

Learning Unit 19 Angles and parallel lines 44

Learning Unit 20 Polygons 47

Learning Unit 21 Congruent triangles 50

Learning Unit 22 Similar triangles 53

Learning Unit 23 Quadrilaterals 55

Learning Unit 24 Centres of triangles 58

Learning Unit 25 Pythagoras’ theorem 60

Learning Unit 26 Rectangular coordinate system 62

Learning Unit 27 Trigonometry 66

Learning Unit 28 Organisation of data 68

Learning Unit 29 Presentation of data 69

Learning Unit 30 Measures of central tendency 72

Learning Unit 31 Probability 75

Learning Unit 32 Inquiry and investigation 78

Acknowledgements 79

**Foreword **

To keep abreast of the ongoing renewal of school curriculum at primary and secondary
*levels, the revised Mathematics Education Key Learning Area Curriculum Guide *
*(Primary 1 - Secondary 6) (2017) and its supplements setting out the learning content *
at each key stage have been prepared by the Curriculum Development Council and
*released in late 2017. Among these documents, the Supplement to Mathematics *
*Education Key Learning Area Curriculum Guide: Learning Content of Junior *
*Secondary Mathematics (2017) (hereafter referred to as “Supplement”) aims at *
elucidating in detail the learning targets and content of the revised junior secondary
Mathematics curriculum.

*In the Supplement, the Learning Objectives of the junior secondary Mathematics *
curriculum are grouped under different Learning Units in the form of a table. The notes
in the “Remarks” column of the table provide supplementary information about the
Learning Objectives.

The explanatory notes in this booklet aim at further explicating:

1. the requirements of the Learning Objectives of the junior secondary Mathematics curriculum;

2. the strategies suggested for the teaching of the junior secondary Mathematics curriculum;

3. the connections and structures among different Learning Units of the junior secondary Mathematics curriculum; and

4. the context of development between the junior secondary Mathematics curriculum and other key stages, such as Key Stage 1,2 and 4.

Teachers may refer to the “Remarks” column and the suggested lesson time of each
*Learning Unit in the Supplement, with the explanatory notes in this booklet being a *
supplementary reference, for planning the breadth and depth of treatment in learning
and teaching. Teachers are advised to teach the content of the junior secondary
Mathematics as a connected body of mathematical knowledge and develop in students
the capability for using mathematics to solve problems, reason and communicate.

Furthermore, it should be noted that the ordering of the Learning Units and Learning Objectives in the Supplement does not represent a prescribed sequence of learning and teaching. Teachers may arrange the learning content in any logical sequences which take account of the needs of their students.

Comments and suggestions on this booklet are most welcomed. They 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

Learning Unit Learning Objective Time
**Number and Algebra Strand **

1. Basic computation

1.1 recognise the tests of divisibility of 4, 6, 8 and 9 1.2 understand the concept of power

1.3 perform prime factorisation of positive integers

1.4 find the greatest common divisor and the least common multiple

1.5 perform mixed arithmetic operations of positive integers involving multiple levels of brackets

1.6 perform mixed arithmetic operations of fractions and decimals

8

**Explanatory Notes: **

The purpose of this Learning Unit is to improve the interface between the Mathematics curriculum of Key Stage 2 and Key Stage 3, so as to strengthen the vertical continuity of the curriculum. All Learning Objectives of this Learning Unit are basic knowledge of Key Stage 3, which extend students’ learning outcomes in Key Stage 2 and prepare students for learning the other Learning Units in Key Stage 3. Therefore, teachers are suggested to teach this Learning Unit at the start of Key Stage 3.

In Learning Objective 1.1, students are required to recognise tests of divisibility of 4, 6, 8 and 9. Tests of divisibility mean the methods of identifying whether a positive integer is divisible by the specified positive integer, which include both of the conditions of being divisible and not divisible. Students have recognised the tests of divisibility of 2, 3, 5, and 10 intuitively without proof in Learning Unit 4N2 “Division (II)” of the primary Mathematics curriculum. While in this Learning Objective, students are required to further recognise the tests of divisibility of 4, 6, 8 and 9. According to students’ abilities and interests, teachers may explain why the tests of divisibility hold, but the explanations are not required in the curriculum. The test of divisibility of 6 usually refers to whether a number can pass both tests of divisibility of 2 and 3. Teachers may point out this test of divisibility holds not solely because 6 = 2 3. Teachers may emphasise that the key factor for this test to hold lies on the fact that 2 and 3 do not have a common factor larger than 1. Teachers may make use of different examples such as 4, 12, 20, 28, 36, … which are divisible by both 2 and 4 but not divisible by 8, to help students recognise that not all tests of divisibility of composite numbers

may be constructed in similar sense as the test of divisibility of 6. However, the related
**explanation or proof is not required in the curriculum. This Learning Objective helps **
students handle the prime factorisation (or factorization) of positive integers in Learning
Objective 1.3, and find the greatest common divisor and the least common multiple in
Learning Objective 1.4.

**In the primary Mathematics curriculum, students are not required to learn the concept of **
power. In Learning Objective 1.2, the related concept in numerical computation is introduced
to help students manage the content in other Learning Objectives. This Learning Objective
only requires students to compute a given power of any positive integers. Students are also
required to recognise 3^{4} = 3333 = 81, and express 81 as 3^{4}. Computations involving
powers, such as 7^{2} 7^{3} = 7^{5}** are not required in this Learning Objective. Teachers may **
also introduce the concept of exponents.

Students recognised the concept of prime numbers and composite numbers in Learning Unit 4N3 “Multiples and factors” of the primary Mathematics curriculum. They also recognised that 1 is neither a prime number nor a composite number. In Learning Objective 1.3, through identifying prime numbers and composite numbers, students are required to decompose positive integers into the product of its prime factors, and represent the results using the notation of powers.

In Learning Unit 4N4 “Common multiples and common factors” of the primary Mathematics curriculum, students recognised how to find the least common multiples and highest common factors of two numbers by listing their multiples and factors, and by using short division.

They also recognised the short forms L.C.M. and H.C.F. In Learning Objective 1.4, students are required to apply prime factorisation in Learning Objective 1.3, and to extend the use of short division, to find the greatest common divisor and the least common multiple of two or more numbers. Teachers may consider whether or not the term “index notation” would be introduced during the discussion of prime factorisation. In this Learning Objective, students are also required to recognise H.C.F., gcd, etc. are short forms of the greatest common divisor. Learning prime factorisation helps students understand the concepts of the greatest common divisor and the least common multiple of polynomials in Learning Objective 4.4 of the Compulsory Part of senior secondary Mathematics.

Learning Objective 1.4 does not limit to finding the greatest common divisor and the least common multiple of two numbers. Hence, teachers are advised to use appropriate examples and counter examples to illustrate that in finding L.C.M. of more than two numbers, when the divisors are all prime numbers, correct results would be obtained regardless of the order

of division and whether the divisors are the common divisors of all numbers or just some of them. Since this Learning Objective serves to assist students in handling simplifications and operations of algebraic expressions in later stages, the exercises involving over-complicated operations or exceedingly large numbers should be avoided.

In Learning Unit 3N4 “Four arithmetic operations (I)” of the primary Mathematics
curriculum, students recognised and used brackets for mixed operations, in which more than
one pair of brackets may be involved. However, operations involving multiple levels of
**brackets, such as (4 − (2 − 1)) × 3, were not required. This restriction is no longer applicable **
to Key Stage 3. Hence in Learning Objective 1.5, students are required to perform mixed
arithmetic operations of positive integers involving multiple levels of brackets, such as 12 +
(7 – (5 – 2)), ((35 – 20) – (5 + 7)) × 2, etc. Teachers may introduce different types of brackets
for this Learning Objective to let students recognise the various notations of brackets, such
as ( ), [ ] and { }, etc.

In Learning Units 4N5 “Four arithmetic operations (II)”, 5N5 “Fractions (V)”, and 6N1

“Decimals (IV)” of the primary Mathematics curriculum, students performed mixed
arithmetic operations of three numbers (including integers, fractions and decimals). For
operations and comparison involving three fractions with different denominators, all
denominators should not exceed 12. The above restrictions are no longer applicable to Key
Stage 3. Through Learning Objectives 1.5 and 1.6, students should be able to perform mixed
arithmetic operations of integers, fractions and decimals involving multiple levels of
**brackets, but over-complicated operations are not required. **

Learning Unit Learning Objective Time
**Number and Algebra Strand **

2. Directed numbers

2.1 understand the concept of directed numbers

2.2 perform mixed arithmetic operations of directed numbers

2.3 solve problems involving directed numbers

9

**Explanatory Notes: **

This Learning Unit covers some basic concepts and knowledge of Key Stage 3, which are based on what students have learnt in primary Mathematics and prepares them for learning other Learning Units of Key Stage 3

In Learning Objective 2.1, teachers may use daily life examples such as a thermometer and a level indicator of a lift to introduce the concept of negative numbers, and discuss with students the meanings of negative numbers in daily life, such as debts, the temperature below zero, floor levels underground. All these examples carry the similar meaning of a measure less than or smaller than a certain reference point. Teachers may also use directed numbers as a kind of numerical representation in cases such as temperature, profit and loss, etc. to help students understand and accept the concept and applications of negative numbers, as well as the concept of directed numbers.

Teachers should use the number line to help students understand the concept of directed numbers. Teachers may guide students to understand that different points on a number line represent different numbers, with the point representing 0 on the number line may be regarded as the reference point of the number line. The distance between “0” and a point on the number line is the value of the number represented by the point. A number line may extend along the two opposite ends of “0” so that there exist a pair of points, each on one side of “0”, which are equidistant from “0”. The pair of numbers represented by this pair of points are opposite numbers to each other, and the number on the right side of “0” is usually known as a positive number, and the other one on left is usually known as a negative number. Teachers may use examples to illustrate the above abstract concept, such as the opposite number of 1 is –1, and the opposite number of –2 is 2. Moreover, 0 is the origin that it is neither a positive nor a negative number. Directed numbers include negative numbers, 0 and positive numbers. In this Learning Objective, students are required to represent directed numbers on a number line, and to compare the magnitude of directed numbers. Students are required to recognise

that, in general, a number on the right of another is larger. Teachers may guide students to use

“<” and “>” signs to represent relationships such as –7 < –5 and 7 > 5.

In Learning Objective 2.2, students are required to understand mixed arithmetic operations involving negative numbers, and perform mixed arithmetic operations of directed numbers involving multiple levels of brackets. Teachers may illustrate the addition and subtraction of directed numbers by moving the points on the number line or using other methods. Starting from moving points on the number line or other illustrative methods, students are also required to perform arithmetic operations gradually using common notations and expressions involving directed numbers, such as 3 + (–4), (–5) – (–7) and so on.

Teachers may use a multiplication table like below to help students construct the concepts of multiplication and division of directed numbers through observing patterns:

+3 +2 +1 0 −1 −2 −3

+3 +9 +6 +3 0 −3 −6 −9

+2 +6 +4 +2 0 −2 −4 −6

+1 +3 +2 +1 0 −1 −2 −3

0 0 0 0 0 0 0 0

−1 −3 −2 −1 0 1 2 3

−2 −6 −4 −2 0 2 4 6

−3 −9 −6 −3 0 3 6 9

Teachers may guide students to observe the pattern by filling in the blanks, for example, students may first fill in the products of positive numbers, observe the patterns in the rows and columns, and then fill in the products of a positive and a negative number and the products of two negative numbers. Similar tables of division may also be created.

While performing mixed arithmetic operations of directed numbers, students are required to extend the concept of mixed arithmetic operations of positive numbers learnt in Key Stage 2 and Learning Unit 1 “Basic computation to mixed arithmetic operations of directed numbers, including the rules of using brackets which are the basic computation skills students have to manage. Performing mixed arithmetic operations of directed numbers may familiarise students with the above skills, but over-complicated computations should be avoided.

In Learning Objective 2.3, students are required to solve problems involving directed
numbers, such as using directed numbers to describe real-life or mathematical problem
situations. The problems are suggested to be related to students’ daily-life experience or real-
**life scenarios. **

Learning Unit Learning Objective Time
**Number and Algebra Strand **

3. Approximate values and numerical estimation

3.1 recognise the concept of approximate values 3.2 understand the estimation strategies

3.3 solve related real-life problems

3.4 **design numerical estimation strategies according to the contexts and judge the reasonableness of the results obtained

6

**Explanatory Notes: **

This Learning Unit is formed by combining two Learning Units “Numerical estimation” and part of “Approximation and errors” in the original curriculum. In this Learning Unit, students are required to recognise the concept of approximate values, understand the estimation strategies, and solve related real-life problems. Designing numerical estimation strategies according to the contexts and judge the reasonableness of the results obtained is an enrichment topic. The concepts of maximum absolute errors, relative errors and percentage errors will be covered in Learning Objective 15.2 while the concept of scientific notations will be covered in Learning Objective 10.4.

In Learning Unit 4D1 “Bar charts (II)”, 5N1 “Multi-digit numbers” and 6N2 “Decimals (V)”

of the primary Mathematics curriculum, students recognised the concepts of rounding off a whole number to a certain place, and rounding off a decimal to the nearest tenth or hundredth.

In Learning Objective 3.1, students are required to recognise further the concept of approximate values, including rounding off a number to a certain number of significant figures, a certain place and a certain number of decimal places. Teachers may use daily-life examples, such as estimating the total numbers of teachers and students of a school, the distance between the school and an MTR station, and an area of a stadium, to illustrate the needs for approximate values, and consolidate the concept of approximate values. Teachers may discuss with students the reasons for taking approximate values. Students are required to recognise the concept of significant figures, such as the more number of significant figures an approximate value is correct to, the closer it is to the actual value. Similarly, teachers should discuss with students the concept of rounding off decimals to a certain number of decimal places. The concept of approximate values in Learning Objective 3.1 helps students

recognise the concept of errors in measurement in Learning Objective 15.1.

Learning Objective 3.2 includes three estimation strategies, namely rounding off, rounding up and rounding down. Students should understand the similarities and differences between these three estimation strategies, including the fact that for rounding up and rounding down, the approximated values obtained are respectively not less than and not larger than the actual value, but this does not hold for rounding off. Teachers may use vocabularies like “about”,

“near”, “a bit more than” or “a bit less than”, etc. to describe the result of estimation.

Building on Learning Objective 3.2, in Learning Objective 3.3, students are required to solve related real-life problems by using appropriate estimation strategies. Teachers may use a variety of real-life examples to help students identify under what daily life scenarios, it is suitable to use estimation, and determine which estimation strategies should be employed and how accurate the estimation should be under specific situations.

In Learning Objective 3.4, teachers may arrange suitable enrichment learning and teaching
activities according to students’ abilities and interests to discuss with students how to design
numerical estimation strategies according to the contexts. Teachers may discuss and analyse
**with students whether results obtained from the estimation strategies are reasonable. **

Learning Unit Learning Objective Time
**Number and Algebra Strand **

4. Rational and irrational numbers

*4.1 recognise the concept of nth root *

4.2 recognise the concepts of rational and irrational numbers 4.3 perform mixed arithmetic operations of simple quadratic

surds 𝑎√𝑏

4.4 **explore the relation between constructible numbers and rational and irrational numbers

7

**Explanatory Notes: **

This Learning Unit is an extension of students’ recognition of integers and fractions. It
introduces the concepts of rational and irrational numbers. In this Learning Unit, students are
required to recognise the definition of rational and irrational numbers as well as some of their
**examples. Students are not required to prove that a certain number is an irrational number, **
but they are required to recognise some common irrational numbers, such as 2, 3 and π.

In Learning Unit 1 “Basic computation”, students understood the concept of power, and
performed prime factorisation of positive integers with results presented in the form of
*powers. In Learning Objective 4.1, students are required to recognise the concept of nth root *
and its notation. Teachers may help students recognise the relationship between square and
square root by the examples such as: Given 2 squares, one of them with known length of
sides while the other with known area, and the area and length of sides are to be found
respectively. Similarly, students may recognise the relationship between cube and cube root
by calculating the length of side from its volume and the volume from its length of side of
*cubes, and consequently recognise the concept of nth root and its notation. Students are *
*required to calculate the nth root of a given number, such as √−8*^{3} , √81^{4} . However, this
Learning Objective does not require computations such as √2^{3} √4^{3} = √8^{3} ** . Students are not **
**required to use fractional power to represent nth root in this Learning Objective. **

*Students are required to recognise that a means the positive square root of a through *
examples such as 2^{2} = (–2)^{2} = 4 but 4 only equals 2. According to students’ abilities and
*interests, teachers may discuss with students why a in * *a has to be non-negative. *

Discussions related to complex numbers will be covered in Learning Unit 1 “Quadratic equations in one unknown” in the Compulsory Part of senior secondary Mathematics

curriculum.

In Learning Objective 4.2, students are required to recognise the concepts of rational and
irrational numbers, including that rational numbers can be written as fractions of numerators
and denominators both being integers (where denominators are non-zero), but irrational
numbers cannot be written as fractions in the aforementioned format. Students are only
required to recognise examples of irrational numbers such as 2, 3 and π. Students are
**not required to prove the above numbers as irrational numbers, but they are required to **
represent rational and irrational numbers on a number line by comparing the values, for
**example locating 2 between 1 and 2. Students are not required to locate irrational **
numbers on a number line by methods such as constructions using compasses and
straightedge. Students are only required to mark the approximate location of the numbers,
and not to mix up the order of different numbers with respect to their magnitude.

In Learning Objective 4.3, students are required to perform simplification and mixed
arithmetic operations of simple quadratic surds. In this Learning Objective, simple quadratic
*surds are limited to those in the form of a b, where a is a rational number and b is a positive *
*rational number. Students are required to use formulae √𝑐𝑑 = √𝑐√𝑑 (where c and d are *
non-negative rational numbers) and

*d*
*c*

*dc * * (where c is a non-negative rational number *
*and d is a positive rational number) to perform simplification and mixed arithmetic operations *
of simple quadratic surds, such as 12= 4×3=2 3 , 3+ 12= 3+2 3=3 3 and

8

3√2= ^{8×√2}

3√2×√2=^{4√2}

3 . However, more complicated mixed arithmetic operations such as

1

2+√3= ^{2−√3}

(2+√3)(2−√3)=^{2−√3}

4−3 = 2 − √3 will be covered in Module 2 of the Extended Part of
**the senior secondary Mathematics curriculum, and therefore are not required in this **
Learning Objective.

While performing mixed arithmetic operations of simple quadratic surds, students are required to recognise how to simplify the surds into the simplest form, and to operate with like surds. Teachers may help students understand the concept of like surds by recalling the concept of like terms. However, over-complicated operations should be avoided.

In Learning Objective 4.4, teachers may introduce appropriate enrichment learning and teaching activities on the exploration of the relation between constructible numbers and rational and irrational numbers according to students’ abilities and interests. Teachers may explore with students through related materials of mathematical history that the meaning of

the discovery of 2 being not a rational number in mathematical development, in order to raise students’ interest towards the development of the number system. Teachers may let students with higher ability briefly recognise the proof of 2 being an irrational number.

After students have recognised the basic knowledge of construction with compasses and straightedge in Learning Objective 20.5, and Pythagoras’ theorem in Learning Unit 25, teachers may also explore with students on the methods of constructing line segments with lengths of rational numbers or some irrational numbers with a given unit length using compasses and straightedge. Teachers may stimulate students to think about whether there are some lengths which are not constructible using similar methods, and thereby bring out the basic concept of constructible numbers. Students should be able to point out the square root of all rational numbers are constructible numbers. Teachers may let students further explore what numbers apart from the square root of rational numbers are constructible (such as 2, 1+ 2, etc. which are the square roots of constructible numbers).

Learning Unit Learning Objective Time
**Number and Algebra Strand **

5. Using percentages

5.1 understand the concept of percentage changes 5.2 solve related real-life problems

15

**Explanatory Notes: **

In Learning Unit 6N3 “Percentages (I)” and 6N4 “Percentages (II)” of the primary
Mathematics curriculum, students recognised the basic concept of percentages, performed
the interconversion between a percentage and a decimal, and the interconversion between a
percentage and a fraction. At the primary level, students were only required to solve simple
problems related to percentages and percentage changes. More complicated problems, such
**as those listed below, are not required: **

what percent is 100 more than 80

what is the percentage increase from 100 to 120

This Learning Unit aims at helping students understand further the concept of percentage changes.

In Learning Objective 5.1, students are required to calculate the percentage change from the given original value and new value. Students are required to understand that it is an increase of 20% (can be expressed as “the percentage change is +20%”) from 100 to 120, but not a decrease of 20% from 120 to 100. Students are required to master the relation amongst the original value, the new value and the percentage change. They are required to recognise that percentage change can also be called “percentage of change”.

Teachers may use the following diagrams to help students understand the concept of percentage change.

100% +20% 100% 20%

100% 20% 100% 20%

Students are required to make use of the following formulae flexibly to calculate the original value, the new value and the percentage change.

new value = original value × (1 + percentage change)

percentage change =new value−original value

original value × 100%

Students are required to note that percentage changes could be positive or negative. They are required to understand the meaning of positive and negative percentage changes in real-life situations.

The problems in Learning Objective 5.2 include those about discount and profit or loss, growth and depreciation, simple and compound interests, successive and component changes, and salary tax. As the problems of percentage may lead to many formulae, teachers should let students use the two formulae above flexibly in different situations to relieve students from the burden of reciting excessive formulae with similar meaning. When introducing the problem of discount and profit or loss, students are required to recognise the terms cost, marked price and selling price, and the relations of these terms. When calculating the problem of discount, students are required to recognise the meaning of common terms such as 20%

off. Teachers may also introduce the meaning of the Chinese terms “八折”, “八五折” used in daily life according to the needs and ability of students. When introducing the problem of simple and compound interests, students are required to recognise principal, interest rate, period, interest and amount, and the relations among them. Students are required to distinguish between simple interest and compound interest. Students are required to calculate salary tax which can be considered as a real-life application of percentage. However, problems involving complicated calculations, such as finding annual salary from a known amount of salary tax and tax allowance, should be avoided.

Teachers may consider using real-life examples or related learning elements in Science
Education or Technology Education KLAs, such as the growth rate of a certain species in
nature or depreciation rate of parts in machines, to design learning and teaching activities or
as class examples to let students recognise how percentage change is applied to solve real-
**life problems, and how percentage is used to describe real-life situations quantitatively. **

Learning Unit Learning Objective Time
**Number and Algebra Strand **

6. Rates, ratios and

proportions

6.1 understand the concepts of rates, ratios and proportions 6.2 solve problems involving rates, ratios and proportions

8

**Explanatory Notes: **

In Learning Unit 6M4 “Speed” of the primary Mathematics curriculum, students recognised the basic concept of rates. Students also recognised how to use the unitary method to solve problems involving direct proportion in Learning Units 4N5 “Four arithmetic operations (II)”

and 5N5 “Fractions (V)” of the primary Mathematics curriculum, but the term “direct proportion” was not introduced. This Learning Unit further discusses rates, ratios and proportions, including the concepts of direct and inverse proportion.

In Learning Objective 6.1, teachers may use daily-life examples, such as typing speed, ratio between the numbers of boys and girls in the class, to help students understand the meaning of rates, ratios and proportions, as well as their relations. When teachers introduce the concept of rate, teachers are advised to emphasise that rate represents the relation of the amount of one quantity per unit of the other quantity. Teachers may use examples such as speed (i.e. the distance travelled in each unit of time interval) to explain this concept. Teachers may let students understand how to perform the interconversion between different units of rates, such as the interconversion between km/h and m/s.

Students are required to understand the concept and notation of two-term ratios. In
*introducing a:b, it can be represented as * ^{𝑎}

𝑏* , where b0. Teachers may use daily-life *
examples, such as mixing household bleach and water in the ratio of 1:99, and the aspect ratio

“16:9” of movie screens and televisions, to introduce ratios. Teachers should also clarify that the concept of ratios in mathematics and the notation of ratios commonly used in daily-life are not necessarily the same. Teachers may cite non-examples such as the score of 1:0 in a football match as an illustration.

Students are required to understand the following properties of ratios:

*a:b and b:a are different. *

*a:b＝ 2:7 does not mean that a＝ 2 and b＝ 7. *

* a:b＝ ka:kb, where k is any non-zero real number. *

*Teachers may introduce to students the k-method (for example, if a:b＝ 2:7, one can let a *

*= 2k, b = 7k where k0) to solve the problems of ratio. *

Students are required to understand how to extend the two-term ratios to ratios of three-term
or more. In solving problems related to ratios, students would frequently face equations
involving fractions such as ^{𝑥}

3 =^{𝑥+1}

5 , etc. Teachers may help students revise the relevant techniques in solving these kinds of equations specifically.

Teachers may introduce direct proportion through recalling the unitary method to students.

Teachers may also use different daily-life examples to discuss direct and inverse proportions with students. As students have recognised speed in the Learning Unit 6M4 “Speed” of the primary Mathematics curriculum, teachers may consider to introduce the concept of inverse proportion by discussing with students that the time and the speed are in inverse proportion if the distance is fixed. Students may understand direct proportion from the concept of equal ratios. They may also understand the concept of inverse proportion through direct proportion.

*Teachers may introduce this concept by using a table to investigate the relation between x *
*and y if x and * ^{1}

𝑦 are in direct proportion.

Teachers should also use the concept of equal ratios to clarify some common
*misunderstandings of direct proportion and inverse proportion, such as “if y increases *
*(decreases) when x increases, then x and y must be in direct (inverse) proportion”. Teachers *
may use a counterexample to disprove the above assertions, for example,

*x * 1 2 3 4

*y * 1 4 9 16

*where x and y do not satisfy the relation of direct proportion. *

In Learning Objective 6.2, students are required to solve problems involving rates, ratios and proportions. Using direct and inverse proportion to solve related problems in different scenarios are also required. Teachers should note that this Learning Unit focuses on tackling problems of direct and inverse proportions by using ratios. Using variation relations (i.e.

direct and inverse variations) to solve direct and inverse proportions, and understanding direct

and inverse proportions through graphical representation belong to the learning content of Learning Unit 6 “Variations” of the Compulsory Part of the senior secondary Mathematics curriculum. Students are required to solve problems about plan diagrams involving scales.

While solving problems by direct and inverse proportions, the related equations should involve one unknown only. Teachers may consider using real-life examples or related learning elements in Science Education or Technology Education KLAs, such as common examples of maps and scale plans, discount, interest rate, exchange rate, density and concentration, to enhance learning and teaching. Teachers may also use the cell diagram under a microscope, online maps, or other real-life scenarios, to introduce the concept of scale, and to design classroom examples or exercises through these scenarios to enhance students’ ability and confidence in applying mathematical knowledge or skills in real-life situations or STEM related scenarios. This Learning Unit is connected with Learning Objectives 18.3 and 22.3 so that students may solve problems involving similar figures by the knowledge of ratios and proportions.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

7. Algebraic expressions

7.1 represent word phrases by algebraic expressions 7.2 represent algebraic expressions by word phrases 7.3 recognise the concept of sequences of numbers 7.4 recognise the preliminary idea of functions

7

**Explanatory Notes: **

At the secondary level, students are required to use algebraic symbols to represent more abstract mathematical concepts, in which algebraic expression is an important foundation of mathematical language. In this regard, teachers should select different examples to let students build up a solid foundation through this Learning Unit, so that they could learn the related topics afterward more effectively.

In Learning Unit 5A1 “Elementary algebra” of the primary Mathematics curriculum, students
recognised the use of letters to represent numbers, including recognising the meaning of the
*representations such as 3x, * ^{2𝑥}

3 (the algebraic expressions should involve only one unknown
*quantity), where 3x is 3x, x3 or x+x+x; * ^{𝑥}

3* is x3, * ^{1}

3× 𝑥 or 𝑥 ×^{1}

3. Students also used algebraic expressions to represent the operations of and relations between quantities that are described in words and involve unknown quantities. Students then adopt this as a foundation to solve problems of simple equations in Learning Unit 5A2 “Simple equations (I)” and 6A1 “Simple equations (II)” of the primary Mathematics curriculum. In Learning Unit 7 “Algebraic expressions”, students are required to further their learning in algebraic expressions and the related concepts, but the concepts of like terms and unlike terms would be dealt with in Learning Unit 11 “Polynomials”.

In Learning Objective 7.1, students learn how to represent word phrases by algebraic
expressions which are not confined to one unknown quantity, but are restricted to those
expressions which are combinations of addition, subtraction, multiplication, division, and
power of numbers and variables. Students should recognise the meanings of the
*representations such as ab being ab, and *

*b*

*a* * being ab, and teachers should remind *

*students the difference between the representation ab and the number such as 53 (that is 5 × *
10 + 3). Students should also recognise such as −^{2𝑥}

3 = ^{−2𝑥}

3 = ^{2𝑥}

−3 and that the expressions
*4(2a) should not be written as 42a. As students will come across some more complicated *
algebraic expressions in this Key Stage, to avoid ambiguity in presentation, students are
required to replace the use of the division sign “÷” gradually by fractional notations in
handling algebraic expressions.

The formulae of areas of 5M1 “Area (II)” at Key Stage 2 are described in words, for example,
the formula of area of a triangle is the base times the height and divided by 2. In this Learning
Unit, students are required to represent formulae by algebraic expressions. For instant, the
said formula of area of a triangle may be represented by 𝐴 =^{𝑏ℎ}

2. Students are required to
recognise some common word phrases in mathematics, including mathematical terms such
as “sum”, “product” and “square”. They are also required to recognise the importance of
bracket. For example, “(𝑎 + 𝑏)^{2}” and “𝑎^{2}+ 𝑏^{2}” represent different algebraic expressions
with different meanings. In order to let students recognise the algebraic expressions in-depth,
students are required to represent algebraic expressions by word phrases in Learning
objective 7.2.

Students are required to guess the next term of a sequence from some given terms and give explanations in Learning Objective 7.3. For example, in the sequence 1, 3, 5, ?, students may guess that the next term is 7 because the terms of the sequence are consecutive odd numbers;

or in the sequence 1, 2, 3, 5, 8, ?, students may guess that the next term is 13 because starting
from the third term of the sequence, the terms are the sum of the previous two terms. Teachers
should emphasise that the term is not unique when guessing merely from some given terms
of a sequence. Therefore, this Learning Objective stresses that students should present their
**guesses by explanation. However, students are not required to present their guesses and **
**explanations using algebraic method. In this Key Stage, teachers are also not required to **
introduce the algebraic expressions on recurrence relations between terms of sequences.

In this Learning Objective, students are required to find a particular term from the general
term of a sequence, such as finding the third term 𝑎_{3} from the general term of a sequence
𝑎_{𝑛} = 𝑛^{2}+ 1 . Sequences of odd numbers, even numbers, square numbers, and triangular
numbers are required. Teachers may discuss with students the difference between the
representations of a sequence with a given general term and that of a sequence with some
leading terms. Students are required to recognise that the values of all terms can be found
through the given general term, and the value of each term is unique.

Teachers may introduce Learning Objective 7.4 “recognise the preliminary idea of functions”

through the discussion of the general term of, for example, sequence of square numbers 𝑛^{2},
including the concept of input-processing-output. The input of this example is a positive
integer and the output is a square number. But the formula of the area of a square 𝑥^{2} is
different as the input is not confined to positive integers. The rigorous definition of functions,
including domains, co-domains, independent and dependent variables are dealt with in
Learning Unit 2 “Functions and graphs” of the Compulsory Part of the senior secondary
Mathematics curriculum.

The algebraic expressions discussed in this Leaning Unit are confined to expressions involving addition, subtraction, multiplication, division and powers of numbers and variables.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

8. Linear equations in one unknown

8.1 solve linear equations in one unknown

8.2 formulate linear equations in one unknown from a problem situation

8.3 solve problems involving linear equations in one unknown

7

**Explanatory Notes: **

In Learning Unit 5A2 “Simple equations (I)” and 6A1 “Simple equations (II)” of the primary Mathematics curriculum, students learnt solving special types of simple equations and related problems. Students also recognised the “balance principle” used in solving equations. This Learning Unit further requires students to solve general linear equations in one unknown.

In Learning Objective 8.1, students should understand the meaning of “solutions” .

As students have learnt representing word phrases by algebraic expressions in Learning Unit 7 “Algebraic expressions”, they should be equipped with sufficient foundation to formulate linear equations in one unknown from a problem situation in Learning Objective 8.2, and use linear equations in one unknown to solve problems in Learning Objective 8.3.

This Learning Unit discusses the linear equations in one unknown having one solution only.

Students may come across linear equations in one unknown having infinitely many solutions or no solutions in Learning Unit 9 “Linear equations in two unknowns” and Learning Unit 12 “Identities”.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

9. Linear equations in two

unknowns

9.1 understand the concept of linear equations in two unknowns and their graphs

9.2 solve simultaneous linear equations in two unknowns by the graphical method

9.3 solve simultaneous linear equations in two unknowns by the algebraic methods

9.4 formulate simultaneous linear equations in two unknowns from a problem situation

9.5 solve problems involving simultaneous linear equations in two unknowns

12

**Explanatory Notes: **

At junior secondary level, apart from mastering how to formulate and solve linear equations in one unknown in Learning Unit 8 “Linear equations in one unknown”, students are required to understand and solve linear equations in two unknowns in this Learning Unit. Using two unknowns to formulate equations and hence simultaneous linear equations in two unknowns helps students extend their understandings on equations, as well as solving more complex problems that could not be described by linear equation in one unknown easily. This Learning Unit also introduces the relation between algebra and graphs to students: After students learnt solving simultaneous linear equations in two unknowns by the graphical method and algebraic methods, they could extend the related concept to learn related topics in the senior secondary Mathematics curriculum, including Learning Unit 5 “More about equations”, Learning Unit 6 “Variations”, Learning Unit 9 “More about graphs of functions”, Learning Unit 10 “Equations of straight lines”, Learning Unit 13 “Equations of circles” of the Compulsory Part of the senior secondary Mathematics curriculum, and Learning Unit 14

“Systems of linear equations” of the Extended Part Module 2 of the senior secondary
Mathematics curriculum. This Learning Unit discusses linear equations in two unknowns in
*the form of ax + by = c, where a and b cannot be both 0. *

Students learnt using algebraic expressions to represent unknowns, and recognising the basic concept of input-processing-output of functions in Learning Unit 7 “Algebraic expressions”.

In Learning Objective 9.1, students are required to understand the concept that two algebraic

symbols representing two unknowns in the same equation. Students also learnt using substitution to find values of unknowns in formulae in Learning Unit 13 “Formulae”.

*Therefore, teachers may explain that a linear equation in two unknowns ax + by = c has *
infinitely many solutions using substitution after students have understood the concept of
**solutions of a linear equation in two unknowns. But students are not required to learn the **
concept of solution sets. Students should then understand the relation between the solutions
of a linear equation in two unknowns and its graph: Through the concept of coordinates of a
point in Learning Unit 26 “Rectangular coordinate system”, students should understand that
each solution of a linear equation in two unknowns could be regarded as the coordinates of a
point, and points corresponding to all the solutions constitute the graph of that equation.

Regarding the graph of a linear equation in two unknowns, students are required to understand that:

the graph of a linear equation in two unknowns is a straight line

all the coordinates of points lying on the straight line satisfy the linear equation in two unknowns

all the coordinates of points not lying on the straight line do not satisfy the linear equation in two unknowns

*Students are required to recognise that the graphs of the equations x = c and y = d are a *
vertical line and horizontal line respectively. Teachers may use Information Technology to
assist students in understanding the graphs of linear equations in two unknowns in-depth.

Nevertheless, students are required to plot the graphs of linear equations in two unknowns on graph papers by paper and pencil.

In Learning Objective 9.2, students are required to solve simultaneous linear equations in two unknowns by the graphical method. Students are required to understand that solving simultaneous linear equations in two unknowns means to find the solutions satisfying all the equations of simultaneous linear equations. Therefore, after students understood that the graph of a linear equation in one unknown is a straight line in Learning Objective 9.1, they should understand the solution of simultaneous linear equations in two unknowns is the coordinates of the point(s) lying on both straight lines. Appropriate use of Information Technology, including the feature of magnification of graphs, can increase the accuracy of the values of the solution obtained. Nevertheless, students should recognise that the exact values may not necessarily be found by the graphical method in Learning Objective 9.2. The simultaneous linear equations in two unknowns included in Learning Objective 9.2 are confined to those equations that have only one solution. The use of graphical methods in solving simultaneous linear equations in two unknowns that have no solutions or more than one solution are dealt with in the content of Learning Objective 10.2 of the Compulsory Part

of the senior secondary Mathematics curriculum.

In Learning Objective 9.3, students are required to solve simultaneous linear equations in two unknowns by algebraic methods which include substitution and elimination. Students are required to recognise how to tackle those simultaneous equations with no solutions, only one solution, and more than one solution by algebraic methods. Students are required to use “The equations have no solutions.” and “The equations have infinitely many solutions.” to describe the conclusion of the above two special cases “no solutions” and “more than one solutions”

**respectively. Students are not required to write down the general solutions of simultaneous **
linear equations in two unknowns that have infinitely many solutions.

After learning the methods of solving simultaneous linear equations in two unknowns in Learning Objectives 9.2 and 9.3, students are required to formulate and solve simultaneous linear equations in two unknowns from problem situations, and then solve problems in Learning Objectives 9.4 and 9.5. Teachers may emphasise that although some problem situations could be expressed by linear equations in one unknown only, it is in general clearer to describe the relations between variables in simultaneous linear equations in two unknowns.

Teachers may select problems related to students’ daily-life experience when teaching Learning Objective 9.5. They may discuss the meanings of solutions of equations from different scenarios to let students master the ways to solve problems involving simultaneous linear equations in two unknowns.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

10. Laws of integral indices

10.1 understand the laws of positive integral indices

10.2 understand the definitions of zero exponent and negative exponents

10.3 understand the laws of integral indices 10.4 understand scientific notations

10.5 understand the binary number system and the denary number system

10.6 **understand other numeral systems, such as the hexadecimal number system

11

**Explanatory Notes: **

In this Learning Unit, students start with the laws of positive integral indices and further understand that these laws are also applied to integral indices. Important applications of the laws of integral indices include scientific notations and the expanded form of different numeral systems. These applications strengthen the lateral coherence between Mathematics and other disciplines, including Science and Computer Literacy.

In Learning Objective 1.2, students only understand the concept of power and apply it to known numbers. In Learning Objective 10.1, students are required to understand the representations of the laws of positive integral indices by algebraic expressions and apply them on algebraic expressions, the laws 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*

The first three laws are related to computations with the same base, whereas the other laws are related to computations with the same index. In the calculation process, students are required to distinguish whether the case is of the same base or the same index, and then choose the appropriate law to reduce common mistakes such as:

* m*^{ 3 }*n*^{ 2}* = (mn)*^{ 3 +2}

2 4 2

4

3 6 3

6 ^{}

* (x*^{ 3})^{2}* = x*^{ 3+2}

Teachers may place emphasis on clarifying these mistakes in Learning Objective 10.1 to help students apply the laws of positive integral indices skillfully to build up a solid foundation to learn the laws of integral indices.

In Learning Objective 10.2, teachers may let students recognise that the laws of positive
integral indices could be extended to the laws of integral indices through the appropriate
definition of zero exponent and negative exponents. Students would further understand the
laws of rational indices in Learning Unit 3 “Exponential and logarithmic functions” of the
Compulsory Part of the senior secondary Mathematics curriculum. Teachers should
emphasise to students that 0^{0} is undefined. After understanding the definitions of zero
exponent and negative exponents, students are required to understand the laws of indices
listed in Learning Objective 10.1 are also applied to integral indices, and therefore these laws
are also the laws of integral indices, but the base involved should be non-zero.

In Learning Objective 10.3, students are required to extend Learning Objectives 10.1 and 10.2 to understand the laws of integral indices. Students are required to apply the laws to manipulate numerical and algebraic expressions involving integral indices. However, the focus of this Learning Objective is on students’ understanding of the laws of integral indices.

Over-complicated operations should therefore be avoided.

In Learning Objective 10.4, students are required to understand the advantage of scientific
notations, that is, to express a very large number or a number that is very close to zero in a
more concise form. Expressing these numbers in scientific notations is very common in
scientific calculators. Teachers may use real-life examples or related learning elements in
Science Education or Technology Education KLAs such as the distance between the sun and
the earth, micro measurements under microscopes, the speed of light (3×10^{8} m/s),
manipulation speed of a computer microprocessor, the amount of the greenhouse gas
emissions produced by power stations etc. to enhance students’ interest and facilitate learning

and teaching.

Learning Objective 10.5 involves only the understandings of non-negative integers in binary
and denary number system including the interconversion between binary numbers and denary
**numbers. Students are not required to learn the computations in different numeral systems **
except in denary number system. After mastering the concept of place value in Learning
Objective 10.5, teachers may discuss with students the enrichment topics of other numeral
systems in Learning Objective 10.6 according to the ability and interest of students.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

11. Polynomials 11.1 understand the concept of polynomials

11.2 perform addition, subtraction, multiplication and their mixed operations of polynomials

11.3 factorise polynomials

15

**Explanatory Notes: **

In Learning Unit 6A1 “Simple Equation (II)” of the primary Mathematics curriculum,
students recognised the basic concepts of like terms and unlike terms. Students learnt the
*rules of addition and subtraction of like terms to handle relation such as 8x+3x=11x, but *
**they were not required to learn the terms “like terms” and “unlike terms” at primary level. **

In this Learning Unit, students are further required to handle the operations of polynomials.

In Learning Objective 11.1, students are required to understand the concept of polynomials,
including the concepts of terms, monomials, binomials, orders, powers, constant terms, like
terms, unlike terms and coefficients. Teachers may clarify some common mistakes for
*students, for example, the terms of the polynomials x*^{3}*2x being mistakenly regarded as x*^{3 }
*and 2x. Besides, students are required to arrange the terms of polynomials in ascending order *
or descending order.

In Learning Unit 10 “Laws of integral indices”, students understand the laws of positive
*integral indices, such as x*^{2}*x*^{3}*=x*^{5}, etc., and recognised the distributive property of
multiplication in 4N5 “Four arithmetic operations (II)” of the primary Mathematics
curriculum. In Learning Objective 11.2, students are required to use the aforementioned
knowledge to perform addition, subtraction, multiplication of polynomials and their mixed
operations. They are required to perform operations of polynomials with more than one
variable. Performing division of polynomials belongs to the learning content of Learning Unit
4 “More about polynomials” of the Compulsory Part of the senior secondary Mathematics
curriculum. Students are required to understand the concept of expansion of polynomials
through multiplication of polynomials. Teachers may consider using the following area-
related diagrams to elucidate the related concept.

In Learning Objective 11.3, students are required to understand that factorisation can be regarded as a reverse process of expansion of polynomials. In order to understand the meaning of factorisation, students are required to understand the meaning of one polynomial being a factor of another polynomial. Students are required to perform factorisation by extracting common factors (and grouping of terms) and cross-method. Teachers may provide appropriate examples to let students recognise that not all quadratic polynomials can be factorised by the above methods. Students use identities to factorise (or factorize) polynomials in Learning Objective 12.3, and factor theorem in Learning Unit 4 “More about polynomials” in the Compulsory Part of the senior secondary Mathematics curriculum to factorise more general polynomials. Teachers may consider using the following diagrams to help students understand the meaning of factorisation of polynomials.

*x *

*x * *x*^{2} *2x *

*+ * *2 *

*+ *

*1 * *x * *2 *

*= * *+ *

*+ * *+ *

*x*^{2} *+ * *2x * *+ * *x * *+ * *2 *

*= *

*= * *x*^{2} *+ * *3x * *+ * *2 *

* a b a+b *
*x + x = x *

*xa + xb = x(a + b) *

*a b a+b *
*y + y = y *

*ya + yb = y( a + b) *

Teachers may combine the diagrams to demonstrate how to factorise by using the method of grouping of terms.

*xa + xb + ya + yb = x(a + b) + y( a + b) *
* = (x + y)(a + b) *

* a+b *
* x *

* a+b *
*y *

By allowing students to combine the diagrams in different ways, teachers may let students explore other ways of grouping of terms for factorisation.

* a b *

*x+y xa + xb + ya + yb = (xa + ya) + (xb + yb) *
* = (x + y)a + (x + y)b *

*= (x + y)(a + b) *

Factorisation by using the identities of difference of two squares or perfect squares is not included in this Learning Unit. The related content is covered in Learning Unit 12

“Identities”.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

12. Identities 12.1 understand the concept of identities

12.2 use identities to expand algebraic expressions 12.3 use identities to factorise polynomials

8

**Explanatory Notes: **

In Learning Objective 12.1, students are required to understand the difference between equations and identities as well as proving identities. An identity can be considered as an equation that the solutions are all numbers. Teachers may use a linear equation in one unknown with infinitely many solutions as an example of identities to explain the related concept.

Teachers are advised to help students differentiate the variables of identities with the
unknown coefficients, and introduce the method of comparing corresponding coefficients of
polynomials to find the unknown coefficients. For example, by comparing the coefficients
*of x and constant terms respectively in the identity A(2x+1)+B(x1)5x2, we have *
*2A+B = 5 and AB = 2, and the values A and B can then be solved. Students may also *
*solve A and B by substituting specific values of x, for example, putting 𝑥 = −*^{1}

2 and 𝑥 = 1.

Learning Objective 12.2 involves using identities to expand algebraic expressions, including
the identities of difference of squares (𝑎 − 𝑏)(𝑎 + 𝑏) ≡ 𝑎^{2}− 𝑏^{2} and perfect squares
(𝑎 ± 𝑏)^{2} ≡ 𝑎^{2}± 2𝑎𝑏 + 𝑏^{2} . The expansion of algebraic expressions are not confined to
polynomials, for example, ( 1)^{2} ^{2} 2 1_{2}

*a* *a*

*a* *a* .

In Learning Objective 12.3, students are required to perform factorisation by using the identities of Learning Objective 12.2, but the objects for factorisation are restricted to polynomials only.

**The identities of difference and sum of cubes are not required. Students may use factor **
*theorem to factorise polynomials such as x*^{3 } * a*^{3} in Learning Unit 4 “More about
polynomials” of the Compulsory Part of the senior secondary Mathematics curriculum.

This Learning Unit can also help students understand trigonometric identities to be introduced in Learning Objective 27.2.

Learning Unit Learning Objective Time
**Number and Algebra Strand **

13. Formulae 13.1 perform operations of algebraic fractions

13.2 use substitution to find the values of unknowns in the formulae

13.3 change the subject of formulae not involving radical signs

9

**Explanatory Notes: **

In Learning Unit 7 “Algebraic Expressions”, students are required to represent formulae (or formulas) by algebraic expressions. In Learning Unit 11 “Polynomials”, students are also required to perform addition, subtraction, multiplication and their mixed operations, and factorisation of polynomials. In this Learning Unit, students are required to further apply the above knowledge to algebraic expressions for the operations including moving terms, grouping terms and simplification, and finding values of unknowns by the method of substitution and solving equations.

In Learning Objective 13.1, students are required to simplify algebraic fractions through reduction of algebraic fractions. In order to avoid over complicated operations, algebraic fractions involved in this Learning Objective are confined to those having denominators as the product of linear factors with rational coefficients, for example:

*xy*
1 , _{2}

3

*xy*
*x* ,

) 1 (

1
*x*

*x* ,

)2

1
( *x*
*x*

*xy* , _{2}

) 2 (

6 3

*x*
*x*

*x* , etc. As such, even if students have not learnt how to factorise
polynomials in general and find the greatest common divisor of polynomials (which will be
handled only in the learning content of the Compulsory Part of the senior secondary
Mathematics curriculum), they can still reduce algebraic fractions to the lowest form.

Teachers may also compare the methods of simplifying algebraic fractions with reducing
fractions to help students clarify some common mistakes. For example, some students may
mistakenly simplify the algebraic fraction ^{3}

3
*x*

*x**y* to ^{1}

1 *y* by cancelling 3x in the
numerator and denominator.

In this Learning Objective, students are not required to do over complicated operations of
**algebraic expressions. They are not required to do operations involving finding the greatest **

common divisor in algebraic fractions with different denominators in this Key Stage. They
**are also not required to decompose fractions by partial fraction. **

Learning Objective 13.2 aims to help students understand that a formula is an algebraic equality to describe the relation between variables. By substituting different values into the same variables of a formula, the corresponding values of the unknown variable can be obtained through the same procedures of arithmetic operations. Teachers may consider using real-life examples or related learning elements in Science Education and Technology Education to strengthen the integration of students' knowledge, such as the density formula

*D* *m*

*v* or the formula for conversion of temperature units ^{9} 32
5
*F * *C* .

In Learning Objective 13.3, students are required to change the subject of formulae not involving radical signs by moving terms. In the process of changing subject of formulae, students may need to factorise part of the algebraic expressions in the formulae. Teachers may also summarise and compare the steps and sequences of mixed arithmetic operations, solving equations and change of subject of formulae to consolidate students' knowledge and skills in algebraic manipulations, so that students can be equipped to apply algebra to handle mathematical problems in more complex real-life scenarios, or in Science Education and Technology Education Key Learning Areas in the future.