Secondary Mathematics

Secondary

Mathematics

Learning and Teaching Resources

Learning and teaching materials for the transitional period of the revised junior

secondary Mathematics curriculum

Prepared by

Mathematics Education Section Curriculum Development Institute

Education Bureau

Hong Kong Special Administrative Region

2018

Table of Contents

Page

Preamble 1

I Introduction

1.1 Aims 2

1.2 Content 3

1.3 User Guide 4

II Concerns on interface

2.1 Implementation timeline for the revised Mathematics curriculum 5

2.2 Comparison of the content between the revised primary Mathematics curriculum and the primary Mathematics curriculum (2000)

6

III Examples on learning and teaching, and assessment

Example 1: The test of divisibility of 3 11 Example 2: Concepts of prime numbers and composite numbers 16

Example 3: Use short division to find the greatest common divisor and the least common multiple of two numbers

21 Example 4: The formula for areas of circles 28

Example 5: Angle (degree) 36

Example 6: Pie chart 41

Preamble

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 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 supplement booklets of the Mathematics Education Key Learning Area Curriculum Guide (Primary 1 - Secondary 6) (2017) was released on November 2017, aiming at providing a detailed account for the learning content of the revised Mathematics curriculum (Primary 1 - Secondary 6). The revision of the learning content of the revised Mathematics curriculum is based on the curriculum aims, and the guiding principles of curriculum design and assessment of mathematics education as stipulated in Mathematics Education Key Learning Area Curriculum Guide (Primary 1 - Secondary 6) (2017).

This booklet aims at providing learning and teaching resources to assist teachers to smoothen the interface between the original primary Mathematics curriculum and the revised junior secondary Mathematics curriculum during the first few years of the implementation of the revised junior secondary Mathematics curriculum.

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

I Introduction

1.1 Aims

The Mathematics Education Section of the Education Bureau released the revised Mathematics curriculum (Primary 1 - Secondary 6) on November 2017, and recommends secondary schools to implement the revised junior secondary Mathematics curriculum progressively from S1 with effect from the school year 2020/21, and then to implement the Compulsory Part of the revised senior secondary Mathematics curriculum from the school year 2023/24. The aforementioned arrangement have taken into account of the overall responses from teachers which proposed that the revised Mathematics curriculum should be implemented as soon as possible at the secondary level.

As the revised primary Mathematics curriculum at Key Stage 2 (i.e. P4 – P6) will be implemented progressively from P4 with effect from the school year 2020/21, three cohorts of S1 students (namely students entering S1 in school years 2020/21, 2021/22 or 2022/23) are going to study the revised junior curriculum after studying the curriculum of Mathematics Education Key Learning Area - Mathematics Curriculum Guide (P1-P6) (2000) (hereafter referred to as “the curriculum 2000”) instead of the revised primary Mathematics curriculum at the primary level. In this regard, schools need to design relevant and appropriate school-based curriculum for these three cohorts of students during this 3-year transitional period to enable a smooth interface for the revised junior secondary Mathematics curriculum.

This booklet is prepared with reference to the curriculum 2000, Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Primary Mathematics (2017), and Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Junior Secondary Mathematics (2017). It aims at providing the comparison between the revised primary Mathematics curriculum and the curriculum 2000, and examples of learning and teaching resources for the transitional period for teachers, so as to attain the following objectives:

(i) let teachers understand the implementation timeline for the revised Mathematics curriculum;

(ii) assist teachers to understand the differences between the revised primary Mathematics curriculum and the curriculum 2000; and

(iii) provide examples of learning and teaching resources of topics which are not covered in the curriculum 2000 but covered in the revised primary Mathematics curriculum, so as to assist teachers in designing school-based materials for the transitional period easily, and to ensure a smooth interface for students’ learning of the revised secondary Mathematics curriculum.

1.2 Content

The content of this booklet include:

A. Concerns on interface

In this part, teachers can make reference to the implementation timeline for the revised Mathematics curriculum, and the document Comparison of the content between the revised primary Mathematics curriculum and the primary Mathematics curriculum (2000), to design the school-based interface measures for the transitional period.

B. Examples on learning and teaching, and assessment

Examples of teaching and assessment activities are the key elements of this booklet:

Examples Topic

Example 1 The test of divisibility of 3

Example 2 Concepts of prime numbers and composite numbers Example 3 Use short division to find the greatest common divisor

and the least common multiple of two numbers Example 4 The formula for areas of circles

Example 5 Angle (degree)

Example 6 Pie chart

Each example on learning and teaching, and assessment includes basic information of the topic, suggested teaching content and exercise. The basic information of the topic includes the strand of the learning content, learning objective, prerequisite knowledge for students and suggested teaching plan. The suggested teaching content includes exploratory activities and the suggested answers, teaching resources and teachers’ points to note. The last one is the exercises and the suggested answers. Teachers may, according to students’ learning and their performances, modify the learning and teaching activities and the exercises to suit students’ learning needs, or design other appropriate learning, teaching and assessment tasks.

1.3 User Guide

1. This booklet is applicable for students entering S1 in school years 2020/21, 2021/22 or 2022/23.

Teachers may help students study related topics through the learning and teaching materials for the transitional period in this booklet.

2. The learning and teaching materials introduced in this booklet include three parts: "Basic information of the topic", "Suggested teaching content" and "Exercise". The "Basic information of the topic" provides a summary of the relevant topics, including: Strand, Learning objective, Prerequisite knowledge and Curriculum planning for teachers' reference. Teachers may refer to the prerequisite knowledge and the relevant learning objectives of the revised junior secondary Mathematics curriculum which are listed in the examples to integrate these materials into the learning and teaching of junior secondary Mathematics. In general, the learning and teaching materials for the transitional period contained in this booklet do not require teachers to teach independently according to the order of arrangement of this booklet. Teachers may also re-design or modify the exploratory activities / classroom activities and the exercises in the examples according to the learning progress of students.

3. Assessment is an integral part of learning and teaching. Teachers may adapt the curriculum planning, worksheets and assessment activities in the examples of this booklet according to students' interests, abilities and needs, as well as the school environment for students’ effective learning. Before designing the learning and teaching activities, teachers are suggested to analyse the potential difficulties and common mistakes students may encounter in the topic, and design appropriate activities for students to explore, construct new knowledge and develop their abilities in problem solving.

4. Regarding catering for learner diversity, teachers may adapt the learning and teaching materials in this booklet or re-design some appropriate materials with reference to the learning and teaching materials to cater for learner diversity of students according to their interests and abilities. In designing teaching activities, teachers may design more challenging activities for students with better learning ability to broaden their knowledge and stretch their potential. On the other hand, teachers may provide more specific guidelines for less able students, and provide for them sufficient time to think, discuss and explore. Teacher may also provide sufficient encouragement and support to help them master the basic knowledge and increase their interest in learning Mathematics.

5. The learning and teaching materials in this booklet and the related e-resources can be downloaded from the Mathematics Education Section’s website at the following address:

www.edb.gov.hk/tc/curriculum-development/kla/ma/res/js/transitional.html

II Concerns on interface

2.1 Implementation timeline for the revised Mathematics curriculum

Primary and secondary schools are advised to implement the curriculum according to the following timeline:

• Revised primary Mathematics curriculum for KS1 (i.e., Primary 1 – 3) will be implemented progressively with effect from the school year 2019/20.

• Revised primary Mathematics curriculum for KS2 (i.e., Primary 4 – 6) will be implemented progressively with effect from the school year 2020/21.

• Revised secondary Mathematics curriculum for KS3 (i.e., Secondary 1 – 3) will be implemented progressively with effect from the school year 2020/21.

• Revised senior secondary Mathematics curriculum for the Compulsory Part at KS4 (i.e., Secondary 4 – 6) will be implemented progressively with effect from the school year 2023/24.

• Revised senior secondary Mathematics curriculum for the Extended Part at KS4 (i.e., Secondary 4 – 6) will be implemented progressively with effect from the school year 2019/20.

2.2 Comparison of the Content between the Revised Primary Mathematics Curriculum (2017) and the Primary Mathematics Curriculum (2000)

The table below shows the comparison of the major revised content between the revised primary Mathematics curriculum (2017) and the primary Mathematics curriculum (2000). Teachers are advised to refer to the revised primary Mathematics curriculum (2017) when reading the table below to understand the details of the revision and the overall design of the revised Mathematics curriculum.

Learning Unit of Curriculum (2000)

Major revision Learning Unit of

revised curriculum Notes about revision

Delete Add Reorganise / Adjust

Move

Primary 1 1N5 Addition and

subtraction (I) 



 1N4 Addition and

subtraction (I)

- Add “recognise the associative property of addition”

- Delete “solve problems involving addition of three numbers”

- Delete “estimate the results of the calculations”

1M4 Time (I)  1M4 Time (I) - Add “measure and

compare the time intervals in hour” and “solve simple problems related to time intervals”

1S1 3-D shapes (I)  1S1 3-D shapes (I) - Delete “recognise prism and pyramid”

1S1 3-D shapes (I) 1S3 2-D shapes



 1S3 Directions and positions (I)

- Adjust the content of relative positions of objects and form a new Learning Unit 1S3

- Add relative position

“between”

1S2 Straight lines and curves 1S3 2-D shapes

 1S2 2-D shapes - Add “recognise the

intuitive concepts of points”

Primary 2 2N2 Addition and subtraction (II)

 2N2 Addition and

subtraction (II)

- Add “perform addition by using the commutative or associative properties of addition”

2M2 Time (II)  6M4 Speed - Move “find duration of

time using hours and minutes” to 6M4

2M4 Weight  3M5 Weight - Move to 3M5

Learning Unit of Curriculum (2000)

Major revision Learning Unit of

revised curriculum Notes about revision

Delete Add Reorganise / Adjust

Move

2S1 3-D shapes (II)  1S1 3-D shapes (1) - Move “recognise the intuitive concepts of prisms, cylinders, pyramids, cones” to 1S1

2S1 3-D shapes (II)  --- - Delete “make 3-D shapes”

2S4 Quadrilaterals (I)



 2S4 Quadrilaterals

(I)

- Add “recognise the concept of line segment”

and “recognise the concept of adjacent sides of quadrilaterals”

- Delete “recognise

trapeziums and rhombuses”

Primary 3 3N2 Addition and subtraction (IV)

 3N4 Four

arithmetic operations (I)

- Adjust the learning objectives on addition and subtraction of numbers within four digits and combine with those on mixed operations of addition and subtraction to form a new Learning Unit 3N4.

3N4 Division (I)  3N3 Division (I) - Delete “perform basic division by short division”

3N6 Fractions (I)  3N5 Fractions (I) - Add “recognise the

concept of equivalent fractions”

- Add “perform addition and subtraction of at most three fractions with the same denominator and the results are not greater than 1” and solve problems involving mainly graphical descriptions

3M2 Time (III) 



6M4 Speed - Delete “tell time in terms of o’clock, minutes and seconds”

- Move “find duration of time using minutes and seconds” to 6M4

3S1 Parallel and perpendicular

 2S1 Angles - Move the content of perpendicular to 2S1

3S3 Angles (II)  2S1 Angles - Move and combine with

Angles in P2 to form a new Learning Unit 2S1

Learning Unit of Curriculum (2000)

Major revision Learning Unit of

revised curriculum Notes about revision

Delete Add Reorganise / Adjust

Move

3S4 Triangles  3S2 Triangles - Add “recognise the

concept of isosceles right- angled triangles”,

“recognise the relations between different types of triangles” and “recognise that the sum of any two sides of a triangle is greater than the remaining side”

3D1 Block graphs  --- - Change to the new

enrichment topic 2E2 Block graphs

Primary 4

4N1 Multiplication (II)

 3N2 Multiplication (I)

- Move “recognise associative property of multiplication” to 3N2

4N-E1 Divisibility  4N2 Division (II) - Move “recognise the

divisibly test of 3” to 4N2 4N3 Acquaintance

with modern calculating devices

 --- - Delete the Learning Unit

4N-E2 Prime numbers and composite numbers

 4N3 Multiples and factors

- Move and combine with multiples and factors in P4 to form a new Learning Unit 4N3

4N5 Common multiples and common factors

 4N4 Common

multiples and common factors

- Add “use short division to find the highest common factors and the least common multiples of two numbers”

4N6 Mixed operations (II)

 4N5 Four

arithmetic operations (II)

- Add “recognise the distributive property of multiplication”

4S1 Quadrilaterals (III)

 3S1 Quadrilaterals (II)

- Move “recognise the concept and property of trapeziums” to 3S1

4S1 Quadrilaterals (III)

 4S1 Quadrilaterals

(III)

- Add “recognise the relations between different types of quadrilaterals”

4S3 Symmetry  6S1 Symmetry - Move to 6S1

4D1 Bar charts (I)  3D1 Bar charts (I) - Move “bar charts using the one-to-one, one-to-two and one-to-five

representations” to 3D1

Learning Unit of Curriculum (2000)

Major revision Learning Unit of

revised curriculum Notes about revision

Delete Add Reorganise / Adjust

Move

Primary 5

5N1 Large numbers  4D1 Bar charts (II) - Move “recognise

approximation of large numbers” to 4D1

5N4 Decimals (II)  4N8 Decimals (II) - Move to 4N8

5N6 Fractions (V)



 5N5 Fractions (V) - Add “the concept of

fractions can be regarded as the ratio of two whole numbers”

- Delete the problems involving finding the fraction of a number by which it is greater or less than another number, and finding the fractional change of a number when it changes to another number.

5M1 Area (II)  5M1 Area (II) - Add “recognise the

concept of height of quadrilaterals”

5M-E1 Angles (degree)

 6M1 Angles (degree)

- Change to a new Learning Unit 6M1

5S1 The eight compass points

 4S3 Directions and positions (III)

- Move to 4S3

5D2 Bar chart (II)  4D1 Bar charts (II) - Move “bar charts using the one-to-fifty and one-to- hundred representations”

to 4D1

5D1 Pictograms (II)  --- - Delete the Learning Unit

Primary 6

6N4 Percentages (II)  6N4 Percentages

(II)

- Delete the problems involving discount, complicated problems

related to percentages and percentage changes

6A1 Simple equations (II)

 5A2 Simple Equation (I)

- Move parts of “equations involving at most two steps” to 5A2 and only involve the calculation with whole numbers 6A1 Simple

equations (II)

 6A1 Simple

equations (II)

- Add two types of equations

“ax + bx = c” and “ax – bx

= c”

Learning Unit of Curriculum (2000)

Major revision Learning Unit of

revised curriculum Notes about revision

Delete Add Reorganise / Adjust

Move

6S1 3-D shapes (IV)



 5S2 3-D shape (III)

- Move “recognise the vertices and edges of 3-D shapes”, “recognise the cross sections parallel to the bases of prisms, cylinders, pyramids, cones” and “recognise the cross section of spheres” to 5S2

- Add “recognise the nets of cylinders”

6S1 3-D shapes (IV)  --- - Move “the relations

between the number of sides of the base, the number of faces, the number of edges and the number of vertices of a prism / a pyramid” to the new enrichment topic 5E2 Exploration of 3-D shapes

6S2 Circles  5S1 Circles - Move to 5S1

6D2 Bar charts (III)  5D1Bar charts (III) - Move “bar charts using the one-to-thousand, one-to- ten thousand and one-to-

hundred thousand representations” to 5D1

---  6M5Area (III) - New Learning Unit

(finding the area of a circle) and the learning contents were moved from secondary to 6M5

---  6D3 Pie charts - New Learning Unit and the

learning contents were moved from secondary to 6D3

---  6D4 Uses and

misuses of statistics

- New Learning Unit

Teachers may also download this comparison table from the Mathematics Education Section’s website for reference. The website address is as follow:

www.edb.gov.hk/attachment/en/curriculum-development/kla/ma/curr/CT_Pri_e.pdf

III Examples on learning and teaching, and assessment

Example 1: The test of divisibility of 3

A. Basic information of the topic Strand: Number and Algebra

Learning objective: Recognise the test of divisibility of 3, including the condition whether a positive integer is divisible by 3 or not. Students are not required to recognise the proof.

Prerequisite knowledge: Students recognised divisibility in Learning Unit 4N2 “Division (II)” of the primary Mathematics curriculum (2000) when the divisors are 2, 5 and 10.

[Note: Some students might have learned the test of divisibility of 3 in the Enrichment Topic 4N-E1 “Divisibility” at primary level.]

Curriculum planning: Teachers may introduce the test of divisibility of 3 before Learning Unit 1.1

“Recognise the tests of divisibility of 4, 6, 8 and 9”, or other appropriate topics, of the revised junior secondary Mathematics curriculum.

B. Suggested teaching content

1. Teachers may remind students of the knowledge of the tests of divisibility they learned at primary level.

 Divisibility: Students at primary level recognised the concept of divisibility by knowing that there is no remainder (i.e. the remainder is 0) when a number is divided by another number. For example,

32 ÷ 2 = 16 and there is no remainder. Therefore, 32 is divisible by 2. Besides, students also recognised the relation between multiples and divisibility. For example, as 25 is a multiple of 5, 25 is divisible by 5.

 At primary level, students learned the tests of divisibility of 2, 5 and 10:

Only the numbers with the unit digit being 2, 4, 6, 8 or 0 is divisible by 2.

Only the numbers with the unit digit being 5 or 0 is divisible by 5.

Only the numbers with the unit digit being 0 is divisible by 10.

2. Teachers may lead students to discover the common idea in the tests of divisibility of 2, 5 and 10, i.e.

If the unit digit of a positive integer is divisible by 2, 5 or 10, the number is divisible by 2, 5 or 10 respectively.

If the unit digit of a positive integer is not divisible by 2, 5 or 10, the number is not divisible by 2, 5 or 10 respectively.

However, whether the unit digit of a positive integer is divisible by 3 cannot be used to test whether the number is divisible by 3.

3. Teachers may conduct the following activity with students to explore the test of divisibility of 3.

Exploratory activity:

1. Open the file “The test of divisibility of 3.xlsx” and select the spreadsheet “2-digit number”, as shown below.

2. Input some positive integers which are less than 30 in the cells A2−A11.

3. Fill the cells of those 2-digit numbers in column A that are divisible by 3 (i.e. multiples of 3) with light blue colour. Fill the corresponding cells under the column “sum” with yellow colour as shown below.

4. Look at the column “sum”, which shows the sum of the tenth digit and the unit digit of an input number. What do you observe? Input more 2-digit numbers in cells A12−A16. Are there any similar results? Write down your findings in the following:

If a 2-digit number is divisible by 3, the sum of the tenth digit and the unit digit must ______________________.

If the sum of the tenth digit and the unit digit of a 2-digit number ______________________, the number is divisible by 3.

Look at the spreadsheet again.

If a 2-digit number is not divisible by 3, the sum of the tenth digit and the unit digit must ______________________.

If the sum of the tenth digit and the unit digit of a 2-digit number ______________________, the number is not divisible by 3.

5. Select the spreadsheet “3-digit number”. Input some 3-digit numbers in column A. Observe the sum of the hundredth digit, the tenth digit and the unit digit. Are there any similar results as that of 2-digit numbers?

6. Does the result still hold for 4-digit numbers or 5-digit numbers?

Conclusion:

Add all the digits of a positive number. If the sum is ______________, the number is divisible by 3;

and if the sum is ______________, the number is not divisible by 3.

Answers to Exploratory activity:

4. If a 2-digit number is divisible by 3, the sum of the tenth digit and the unit digit must be divisible by 3.

If the sum of the tenth digit and the unit digit of a 2-digit number is divisible by 3, the number is divisible by 3.

If a 2-digit number is not divisible by 3, the sum of the tenth digit and the unit digit must not be divisible by 3.

If the sum of the tenth digit and the unit digit of a 2-digit number is not divisible by 3, the number is not divisible by 3.

5. Yes 6. Yes

7. Conclusion:

Add all the digits of a positive number. If the sum is divisible by 3, the number is divisible by 3; and if the sum is not divisible by 3, the number is not divisible by 3.

Points to note:

1. The above exploratory activity may be conducted in paper and pencil. Using spreadsheet in computers can reduce students’ workload on calculation so that students may pay more attention in observing the patterns of outcomes, and the number of trials on different numbers can be increased.

2. The formulae for extracting the unit digit, the tenth digit, etc. from a number are shown below:

hundredth digit = MOD(INT(A2/100),10) tenth digit = MOD(INT(A2/10),10) unit digit = MOD(INT(A2),10)

3. According to students’ abilities, teachers may use examples to explain why the aforesaid test of divisibility works, but it should be noticed that such explanation is not required by the curriculum. For example,

852 = 800 + 50 + 2

= 8×100 + 5×10 + 2

= 8×(99 + 1) + 5×(9 + 1) + 2

= 8×99 + 8 + 5×9 + 5 + 2

= (8×99 + 5×9) + (8 + 5 + 2)

= (8×99 + 5×9) + 15

a multiple of 3 + a multiple of 3 = a multiple of 3

746 = 700 + 40 + 6

= 7×100 + 4×10 + 6

= 7×(99+1) + 4×(9+1) + 6

= 7×99 + 7 + 4×9 + 4 + 6

= (7×99 + 4×9) + (7 + 4 + 6)

= (7×99 + 4×9) + 17

a multiple of 3 + not a multiple of 3 = not a multiple of 3

C. Exercise

1. Which of the following numbers are divisible by 3? Circle the correct answer(s).

17 26 39 104 521 1077 2014

2. Write down three 3-digit numbers which are divisible by 3.

3. Write down three 4-digit numbers which are divisible by 3.

4. Write down three 5-digit numbers which are divisible by 3.

5. Write down the largest 4-digit number which is divisible by 3.

6. Write down the smallest 5-digit number which is divisible by 3.

7. The 3-digit number 2#5 is divisible by 3. Which digit(s) may # represent?

8. Let # represent one of the digits 0, 1, 2, …, 9. If 2, #, 5 is arranged to form the largest 3-digit number which is divisible by 3. Which digit does # represent?

Answers 1. 39, 1077

2. 369, 582, 960 or other suitable answers 3. 1296, 2574, 6888 or other suitable answers 4. 11145, 43125, 60594 or other suitable answers 5. 9999

6. 10002 7. 2, 5, 8

8. 8, as the largest 3-digit number required is 852.

Example 2: Concepts of prime numbers and composite numbers

A. Basic information of the topic

Strand: Number and Algebra

Learning objective: 1. Recognise prime numbers and composite numbers.

2. Find prime numbers within 100 by the Sieve of Eratosthenes.

Prerequisite knowledge: 1. Students recognised the concept of divisibility when the divisors are 2, 5 and 10 in Learning Unit 4N2 “Division (II)” of the primary Mathematics curriculum (2000).

2. Students developed the understanding of factors and find out all the factors of a number in Learning Unit 4N4 “Multiples and factors” of the primary Mathematics curriculum (2000).

[Note: Some students might have learned this topic in Enrichment Topic 4N-E2 “Prime numbers and composite numbers” at primary level.]

Curriculum planning: Teachers may introduce this topic before Learning Objective 1.3

“Perform prime factorisation of positive integers” in Learning Unit 1

“Basic computation”, or other topics whichever appropriate.

B. Suggested teaching content

1. Teachers may remind students of the concepts of factors, and the methods of finding all the factors of a positive integer. They may introduce the concepts of prime numbers and composite numbers through Activity 1 (see next pages). Students may categorise positive integers by their number of factors into the following categories:

(1) The prime numbers which have only 2 factors.

(2) The composite numbers which have 3 or more factors.

(3) The integer 1 which has only one factor. It is neither a prime number nor a composite number.

2. Teachers may conduct Activity 2 (see next pages) to guide students to find the prime numbers within 100 by the Sieve of Eratosthenes. In 250 BC, the ancient Greek mathematician Eratosthenes proposed the steps of finding the prime numbers between 2 and n:

List the integers 2 through n (for example, 100).

Step (1): Start with the smallest integer.

Step (2): Circle the smallest integer and then delete the other multiples of the integer.

Step (3): For all unmarked numbers, repeat step (2) until all numbers are marked.

Step (4): All circled numbers are prime numbers.

3. Teachers may refer to the ETV programme: 齊來找質數 (https://www.hkedcity.net/etv/resource/441554750).

Activity 1

1. List out all the factors of the integers from 1 to 20. Observe the number of factors of each integer. Complete the table below and categorise the integers by the number of factors:

Integer Factors Number

of factors Integer Factors Number

of factors

1 1 1 11

2 1, 2 2 12

3 13

4 1, 2, 4 3 14

5 15

6 16 1, 2, 4, 8, 16 5

7 17

8 18

9 19

10 20

2. Categorise the integers into three categories by the number of factors:

(1) Category 1: Only 2 factors: 2, .

(2) Category 2: 3 or more factors: 4, .

(3) Category 3: Only 1 factor: __________.

1. Prime numbers have only 2 factors. The factors of a prime number are 1 and the number itself.

2. Composite numbers have 3 or more factors.

3. The integer 1 has only one factor. It is neither a prime number nor a composite number.

3. Class exercise:

(a) Circle the prime numbers in the following: 23, 31, 39, 48, 51, 59 (b) Circle the composite numbers in the following: 26, 37, 41, 49, 53, 60

(c) Consider all even numbers, how many prime numbers are there? Please list out and explain your answer.

Suggested answer of Activity 1:

1. List out all the factors of the integers from 1 to 20. Observe the number of factors of each integer. Complete the table below and categorise the integers by the number of factors:

Integer Factors Number

of factors Integer Factors Number of factors

1 1 1 11 1, 11 2

2 1, 2 2 12 1, 2, 3, 4, 6, 12 6

3 1, 3 2 13 1, 13 2

4 1, 2, 4 3 14 1, 2, 7, 14 4

5 1, 5 2 15 1, 3, 5, 15 4

6 1, 2, 3, 6 4 16 1, 2, 4, 8, 16 5

7 1, 7 2 17 1, 17 2

8 1, 2, 4, 8 4 18 1, 2, 3, 6, 9, 18 6

9 1, 3, 9 3 19 1, 19 2

10 1, 2, 5, 10 4 20 1, 2, 4, 5, 10, 20 6

2. Categorise the integers into three categories by the number of factors:

(1) Category 1: Only 2 factors: 2, 3, 5, 7, 11, 13, 17, 19 .

(2) Category 2: 3 or more factors: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 . (3) Category 3: Only 1 factor: 1 .

3. Class exercise:

(a) Circle the prime numbers in the following: 23, 31, 39, 48, 51, 59 (b) Circle the composite numbers in the following: 26, 37, 41, 49, 53, 60

(c) Consider all even numbers, how many prime numbers are there? Please list out and explain your answer.

Ans: For all even numbers, only 2 is a prime number since all the other even numbers have factors including at least 1, 2 and the number itself, i.e., there are 3 or more factors.

Thus, all even numbers except 2 are composite numbers.

Points to note:

1. The discussion of prime numbers and composite numbers is limited to positive integers only.

2. As an enrichment of learning and teaching, and the introduction of the concept of prime factorisation of positive integers, teachers may explain to students according to their ability, that for each of the composite numbers, regardless of its number of factors, it contains factors other than 1 and itself, and hence it can be expressed as the multiple of two numbers other than 1 and itself, for example, 4 = 2 × 2, 12 = 3 × 4. Teacher may further explain that if the numbers that form the product contains a composite number, the number can further be expressed as the product of two numbers again, for example 12 = 3 × 4 = 3 × 2 × 2, until there are no composite numbers in the expression.

Activity 2 Sieve of Eratosthenes

1. Find the prime numbers within 100 by the Sieve of Eratosthenes.

To find the prime numbers between 2 and 100:

Step (1): Start with the smallest integer.

Step (2): Circle the smallest integer and then delete the other multiples of the integer.

Step (3): For all unmarked numbers, repeat step (2) until all numbers are marked.

Step (4): All circled numbers are prime numbers.

2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

2. Class exercise:

(a) How many prime numbers are there within 100?

What is the largest prime number within 100?

(b) Determine if the following statements are correct.

(i) prime number × prime number = prime number

(ii) prime number × composite number = composite number

Suggested answer of activity 2:

1. Find the prime numbers within 100 by the Sieve of Eratosthenes.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2. Class exercise:

(a) How many prime numbers are there within 100?

What is the largest prime number within 100?

Ans: 25 prime numbers. The largest prime number within 100 is 97.

(b) Determine if the following statements are correct.

(i) prime number × prime number = prime number

Ans: Incorrect (A prime number has only two factors, but the product obtained by multiplying two prime numbers must have more than two factors.)

(ii) prime number × composite number = composite number

Ans: Correct (Since a composite number itself has 3 or more factors, the product must also have 3 or more factors.)

C. Exercise

1. Write down all prime numbers within 100 – 130.

2. What is the largest prime number within 150?

3. Determine if the following statements are correct.

(i) prime number × 1 = composite number

(ii) composite number × composite number = composite number Suggested answer:

1. Write down all prime numbers within 100 – 130. Ans: 101, 103, 107, 109, 113, 127 2. What is the largest prime number within 150? Ans: 149

3. Determine if the following statements are correct.

(i) prime number × 1 = composite number

Ans: Incorrect (Since multiplying 1 to a number does not affect the number of factors of the number, there are only two factors after the prime number × 1).

(ii) composite number × composite number = composite number

Ans: Correct (Since the composite numbers on the left-hand side have 3 or more factors, the product must also have 3 or more factor.)

Example 3

Use short division to find

the greatest common divisor and the least common multiple of two numbers

A. Basic information of the topic Strand: Number and Algebra

Learning objective: Use short division to find the greatest common divisor and the least common multiple of two numbers. Students are not required to recognise the principle.

Prerequisite knowledge: 1. Students recognised the method of finding the common multiples and the least common multiple of two numbers by listing out their multiples; and that of finding the common divisors and the greatest common divisor of two numbers by listing out their factors in Learning Unit 4N5 “Common multiples and common factors” of the primary Mathematics curriculum (2000). Students recognised the terms “H.C.F.” and “L.C.M.” representing the greatest common divisor (or highest common factor) and the least common multiple respectively.

2. Besides, students recognised divisibility in Learning Unit 4N2

“Division (II)” of the primary Mathematics curriculum (2000) when the divisors are 2, 5 and 10.

3. Students recognised the tests of divisibility of 4, 6, 8 and 9 in Learning Objective 1.1 of Learning Unit 1 “Basic computation”

of the revised junior secondary Mathematics curriculum.

4. Students should have learned the test of divisibility of 3 through the learning and teaching materials for the transitional period of the revised junior secondary Mathematics curriculum.

5. Students recognised the prime factorisation of positive integers in Learning Objective 1.3 of Learning Unit 1 “Basic computation”

of the revised junior secondary Mathematics curriculum.

[Note: Some students might have learned to use short division to find the greatest common divisor and the least common multiple of two numbers at primary level.]

Curriculum planning: Teachers may introduce how to use short division to find the greatest common divisor and the least common multiple of two numbers at Learning Objective 1.4 “Find the greatest common divisor and the least common multiple”, or other appropriate topics, of the revised junior secondary Mathematics curriculum.

B. Suggested teaching content

1. Teachers may remind students of the listing method they learned at primary level.

 Example 1: Use the listing method to find the greatest common divisor of 12 and 18.

Arrange all the factors of 12 and 18 in the ascending order. Circle all the common factor(s) of 12 and 18.

Factors of 12: ○1^{E AA}○2^{E AA}○3^{E A} 4 ^A○6^{E A} 12 Factors of 18: ^A○1^{E AA}○2^{E AA}○3^{E A A}○6^{E A} 9 18

From the common factors circled, it is observed that 6 is the greatest common divisor of 12 and 18.

 Example 2: Use the listing method to find the least common multiple of 12 and 18.

Arrange the first few multiples of 12 and 18 in the ascending order. Circle the common multiples of 12 and 18.

Multiples of 12: 12 24 ^A○36^EA 48 60 ^A○72^EA ……

Multiples of 18: 18 ^A○36^EA 54 ^A○72^EA ……

From the common multiples circled, it is observed that 36 is the least common multiple of 12 and 18.

2. Teachers may discuss with students the pros and cons of the listing method, and then introduce the short division.

3. Teachers may conduct the following activity with students to explore how to use short division to find the greatest common divisor and the least common multiple of two numbers.

Exploratory activity:

1. Use the test of divisibility, list out all the prime common factors (i.e. the common factor that is also a prime) of 30 and 42. Guess the greatest common divisor of 30 and 42.

2. Browse the webpage https://www.geogebra.org/m/BKxeO3At . Input the values a = 30, b = 42.

Pull the red slider to the right one time, as shown in the figure below.

What is the relation between the number 2 circled in orange colour, and the pair of numbers 30, 42? _________________________________________

What is the relation between the two pairs of numbers 15, 21 and 30, 42?

_________________________________________

3. Pull the red slider to the right once again, as shown in the figure below.

What is the relation between the number 3 circled in green colour, and the pair of numbers 15, 21?

______________________________

4. When 30 and 42 are divided by the prime common factor ____ and ____ , the numbers____

and ____ circled in blue colour are obtained.

Thus, ____ ×____ = _____ is a common factor of 30 and 42.

Do the numbers ____ and ____ circled in blue colour have other prime common factor(s)?

_______

Can we find an even larger common factor of 30 and 42? (can / cannot) 5. Click the checkbox (HCF) in the figure above.

The greatest common divisor of 30 and 42 = ____________ = ________ . 6. Click the checkbox (LCM) in the figure above.

The least common multiple of 30 and 42 = ____________ = ________ .

7. Discuss with classmates how the webpage obtains the greatest common divisor and the least common multiple of 30 and 42.

8. Using the webpage above, try to find the greatest common divisor and the least common multiple of other two numbers.

Answers to exploratory activity:

1. The prime common factors of 30 and 42 are 2 and 3. The greatest common divisor of 30 and 42 is 2 × 3 = 6.

2. The number 2 circled in orange colour is a prime common factor of 30 and 42. The pair of numbers 15 and 21 is the result (quotient) when the pair of numbers 30 and 42 is divided by 2.

3. The number 3 circled in green colour is a prime common factor of 15 and 21.

4. When 30 and 42 are divided by the prime common factor 2 and 3 , the numbers 5 and 7 circled in blue colour are obtained.

Thus, 2 × 3 = 6 is a common factor of 30 and 42.

Do the numbers 5 and 7 circled in blue colour have other prime common factor(s)? No We cannot find an even larger common factor of 30 and 42.

5. The greatest common divisor of 30 and 42 = 2 × 3 = 6 .

6. The least common multiple of 30 and 42 = 2 × 3 × 5 × 7 = 210 .

Points to note:

1. The above explanatory activity can be conducted by paper and pencil. Using the online version of GeoGebra can reduce the time for calculation so that students may concentrate on recognising short division and the related relation, and study more exploratory cases. No matter what formats of exploratory activities are conducted, students are required to use the test of divisibility of numbers to conduct short division.

2. As students have learned the prime factorisation of a positive integer, teachers may represent the numbers 30 and 42 in the example as the multiples of their prime factors to demonstrate how the greatest common divisor and the least common multiple can be found.

2 2 × 3 × 5 2 × 3 × 7 2 30 42

3 3 ×5 3 × 7 3 15 21

5 7 5 7

3. Teachers should emphasise that the numbers should be divided by all common factor(s), namely 2 and 3 in the example, until the quotients do not have any common factors other than 1. Short division is completed for the numbers 5 and 7 in the example. Teachers may adopt more examples to illustrate this point.

4. According to students’ abilities, teachers may explain the principle of the short division in the above. The explanation is not required in the curriculum. Short division is an easy way to find the greatest common divisor and the least common multiple of two positive integers which are not relatively prime. Short division helps students find the greatest common divisor and the least common multiple of two numbers when they are not factorised to prime factors. The principle is to divide the two numbers by a common prime factor (i.e. 2 and 3 in the above example) until the numbers do not have any prime common factors (i.e. 5 and 7 in the above example). As 5 and 7 are relatively prime, the greatest common divisor of 30 and 42 is 2 × 3 = 6. Students may also observe that 2 × 3 is the greatest common divisor of 30 and 42 from 2 × 3 × 5 and 2 × 3 × 7. For 30 and 42, i.e. 2 × 3 × 5 and 2 × 3 × 7, the least common multiple is 2 × 3 × 5 × 7 = 210.

5. Teachers may explain to students the divisor is not necessarily a prime number. For example, the divisor may be 6 instead of 2 and 3 in the above example. The objective of using primes as the divisor is to extend the short division method to find the least common multiple of three or more numbers.

6. For more able students, teachers may discuss with students whether the order of divisions by 2 and 3 (i.e. the factors of 30 and 42) may affect the result. Teachers may ask students whether it may affect the result of calculation of the greatest common divisor and the least common multiple when the numbers 30 and 42 are divided by 6 directly (instead of 2 and 3).

7. Teachers may compare the pros and cons of the short division with the listing method.

C. Exercise

1. Using short division, find the greatest common divisor and the least common multiple of a and b.

a b greatest common divisor least common multiple

6 15

12 18

20 35

28 56

2. Randy used short division to find the greatest common divisor and the least common multiple of 60 and 90. Fill in the blanks with appropriate numbers.

2 60 90

10 15

The greatest common divisor 60 and 90 = 2 × ___ × ___ = ______.

The least common multiple of 60 and 90 = 2 × ___ × ___ × ___ × ___ = _______.

3. Fai used short division to find the least common multiple of 48 and 84. His steps are shown below:

2 48 84

2 24 42

12 21

He claimed that the least common multiple of 48 and 84 = 2 × 2 × 12 × 21 = 1008。

Is the calculation by Fai correct? If not, what is/are the mistake(s)? Correct Fai’s mistake(s).

4. Using short division, find the greatest common divisor and the least common multiple of a and b.

a b greatest common divisor least common multiple

30 45

48 60

72 120

110 165

Answers 1.

a b greatest common divisor least common multiple

6 15 3 30

12 18 6 36

20 35 5 140

28 56 28 56

2.

2 60 90

3 30 45

5 10 15

2 3

The greatest common divisor 60 and 90 = 2 × 3 × 5 = 30 .

The least common multiple of 60 and 90 = 2 × 3 × 5 × 2 × 3 = 180 .

3. Incorrect as 12 and 21 still have a common factor 3. Correct steps are shown below:

2 48 84

2 24 42

3 12 21

4 7

The least common multiple of 48 and 84 = 2 × 2 × 3 × 4 × 7 = 336。

4.

a b greatest common divisor least common multiple

30 45 15 90

48 60 12 240

72 120 24 360

110 165 55 330

Example 4: The formula for areas of circles

A. Basic information of the topic Strand: Measures, Shape and Space

Learning objective: 1. Recognise the formula for areas of circles 2. Apply the formula for areas of circles

Prerequisite knowledge: Students recognised and applied the formula for circumferences of circles in Learning Unit 6S2 “Circles” and 6M2 “Perimeter (II)” of the primary Mathematics curriculum (2000).

[Note: Teachers should note that “Find the diameter or radius of a circle from its area” is not required in the revised primary Mathematics curriculum. Thus, related content is not covered in this learning and teaching material. Teachers are required to include the related content in Learning Objective 16.2 “understand the formula for areas of sectors of circles” of the revised junior secondary Mathematics curriculum.]

Curriculum planning: Teachers may introduce the formula for areas of circles before Learning Objective 16.2 “understand the formula for areas of sectors of circles”, or other appropriate topics, of the revised junior secondary Mathematics curriculum.

B. Suggested teaching content

1. Teachers may remind students of the formula for circumferences of circles they learned at primary level.

2. Teachers may conduct one of the following activities with students to explore the formula for areas of circles.

Exploratory activity 1:

1. Go to the webpage http://m.geogebra.hk/?id=tmWgmahf&lang=2 . Click the checkbox “Dissect the circle into 8 sectors”. Pull the grey slider to the right for a suitable number of equal sectors, say 40. Pull the purple slider as shown in the figure below.

The blue shaded part is close to a parallelogram with the base ≈ , the height ≈ . (Give the answers in terms of r.)

The area of the blue shaded part ≈ . (Give the answer in terms of .r)

2. Re-click the checkbox “Dissect the circle into 8 sectors”. Select other number of equal sectors.

Is the estimated area of the blue shaded part the same as that in Question 1? _______

3. According to observation, the area of circle should be . (Give the answer in terms of r.)

Exploratory activity 2:

1. Go to the webpage https://www.geogebra.org/m/Hfk5waZN。

Press the button Start the show!. Press Pause when the animation finished as shown below.

2. The red shaded part is close to a triangle with the base ≈ , the height ≈ . (Give the answers in terms of r.)

The area of the red shaded part ≈ = . (Give the answer in terms of r.) 3. According to observation, the area of circle should be . (Give the answer in terms of .r)

Points to note:

1. The two exploratory activities above are to let students recognise the formula for areas of circles intuitively, but not a proof.

2. Teachers should notice that the proof is not required in the curriculum when introducing the formula A = πr².

3. Teachers should ask students to pay attention to which approximated values of π is used such as the approximated value of π from the calculator, or ²²

7 for easy calculation.

4. Answers to Exploratory activity 1

The blue shaded part is close to a parallelogram with the base ≈ πr , the height ≈ r .

The area of the blue shaded part ≈ πr² . The estimated area of the blue shaded part is the same as that in Question 1.

The area of circle should be πr² . 5. Answers to Exploratory activity 2

The red shaded part is close to a triangle with base ≈ 2πr , height ≈ r . The area of the red shaded part ≈ πr² 。

The area of circle should be πr² .

C. Exercise

1. Find the areas of the following circles. (Give the answers in terms of π.)

(a) radius = 10 cm (b) diameter = 24 m

2. Given that the circumference of a circle is 12π cm.

(a) Find its radius.

(b) Find its area. (Give the answer in terms of π.)

3. A circular garden with diameter 10 m is surrounded by a path with width 1 m. Find the area of the path. (Give the answer in terms of π.)

10 m

1 m

10 cm 24 m

4. Find the areas of the shaded region in the following figures. (Give the answers correct to 3 significant figures.)

(a) (b)

(c) length of a side of the square = 10 cm (d)

5. Wai Ming used a 2 m wire to form a shape of semicircle as shown in the following figure. Find the area enclosed by the semicircle. (Give the answers correct to 3 significant figures.)

6. A field consists of a rectangle PQRS and two semicircles as shown in the following figure. If the length PQ of the rectangle PQRS is 100 m and the perimeter of the field is 400 m, find the width PS of the rectangle PQRS, and the area of the field. (Give the answers correct to 3 significant figures.)

7. When two wires with equal length are used to form a square and a circle respectively, which figure has the larger area? Explain your answers.

4 cm

4 cm 3 cm

16 cm

P Q

S R

100 m

Answers

1. (a) 100π cm² (b) 144π m² 2. (a) 6 cm (b) 36π cm² 3. 11π m²

4. (a) 101 cm² (b) 18.8 cm² (c) 21.5 cm² (d) 9.42 cm² 5. 0.238 m²

6. PS = 63.7 m, Area = 9550 m² 7. circle

Example 5: Angle (degree)

A. Basic information of the topic

Strand: Measures, Shape and Space Learning objective: 1. Recognise “degree” (°).

2. Measure and compare the sizes of angles in degree.

- to name angles with the symbol “∠”, such as ∠A and ∠ABC.

- to recognise reflex angles, straight angles and round angles.

- to measure angles within 360° (0° and 360° are not required) using protractors.

3. Draw angles of given sizes.

Prerequisite knowledge: Students recognised the concepts of angles, right angles, acute angles and obtuse angles, and compared the sizes of angles in Learning Units 2S2

“Angles (I)” and 3S3 “Angles (II)” of the primary Mathematics curriculum (2000).

[Note: Some students might have learned this topic in Enrichment Topic 5M-E1 “Angle (degree)” at primary level. Learning Objectives include:

- Recognise “degree” (°).

- Measure angles up to 360° using a protractor.

- Draw angles of given sizes using a protractor.]

Curriculum planning: Teachers may introduce Angle (degree) in Learning Unit 19 “Angles and parallel lines”, or other learning units whichever appropriate.

B. Suggested teaching content

1. Teachers may remind students of the concept of angle, namely when two lines intersect at one point, angles at the point is formed; and the concepts of right angles, acute angles and obtuse angles. The symbol of arc, “╮”, for indicating a general angle and the symbol “ ” for indicating a right angle should be included.

2. Teachers may introduce the method of naming an angle, including the use of the symbol “∠”.

For example, ∠A and ∠BAC may be used to name the marked angle in the figure below.

A

B

C

3. Teachers may introduce the unit of angle as “degree” and it is expressed by the symbol “ ° ”.

Teachers may use a protractor to illustrate the concept: assume that the circumference of a circle can be divided into 360 equal parts, two adjacent radii will form an angle of 1 degree, and each of the angles formed at the centre of the circle is 1 degree, denoted “1° ”.

4. Teachers may describe the angles at centre of 1/3 circle, 1/4 circle and 1/6 circle by using “degree”

to strengthen students’ understanding of the unit.

5. Teachers may further guide students to describe acute angles, right angles and obtuse angles in degrees, and then introduce the names and concepts of straight angles, reflex angles and round angles.

acute angle right angle obtuse angle straight angle reflex angle round angle

Greater than 0º

and less than 90º 90º Greater than 90º

and less than 180º 180º Greater than 180º

and less than 360º 360º

6. Teacher may introduce the protractor and its use to students. Students are required to recognise the centre, the edge and the degree mark of the protractor.

Steps to measure an angle from 0° to 180° with a protractor:

(1) Place the centre of the protractor to the vertex of the angle;

(2) Align one arm of the angle with the edge of the protractor and measure the size of the angle using the set of degree mark which the arm on the edge of the angle points to “0”.

degree mark edge centre degree mark

(3) Read the degree mark where the other arm is pointing. The reading indicates the size of the angle in degree. For example, ∠BAC in the previous illustration is 38°.

[Note: Students may select the correct degree from the two sets of degree mark by recognising whether the measured angle is an acute or obtuse angle and hence inferring the range of degrees of the measured angle.

7. Teachers may conduct Activity 1 (see the next page) with students to measure the reflex angle with a semi-circular protractor.

8. Teachers may illustrate how to draw an angle of required size (from 0° to 180°) with a protractor:

(1) Draw a line segment (i.e. an arm of the angle).

(2) Place the edge of the protractor along the line segment and place the centre of the protractor at the vertex (i.e. one end of the arm).

(3) Find the required mark using the set of degree mark of which the line segment is pointing

“0”, and then mark a small dot at the circumference of the protractor at the required mark.

(4) Join the small dot to the vertex with a ruler to form the second arm of the angle.

(5) Label the angle with capital letters and mark the degree on the figure.

9. Teachers may conduct Activity 2 (see the next page) with students to draw the reflex angle with a semi-circular protractor.

A

B

C

Activity 1

Measure the size of the angles shown in the following questions with a protractor, write down the answer in the horizontal lines, and mark the degrees on the figures.

1.

∠ABC = ______________

2. (a) Measure ∠ABC

∠ABC = ______________

(b) reflex angle ABC = __________

3.

reflex angle PQR = __________

4.

reflex angle ABC = __________

Activity 2

Draw the required angles shown in the following questions with a protractor, and mark the degrees on the figures.

1. ∠ABC = 130º 2. (a) ∠PQR = 145º

(b) reflex angle PQR = 215º

3. reflex angle ABC = 250º 4. reflex angle PQR = 300º A

B C

A

B C

Q

P

R A

B C

C. Exercise

Measure the size of the angles with a protractor, mark the degrees on the figures, and write down the sizes and types of the angles.

Figure Size of angle Type of angle

1.

∠AOB = ______

acute angle / right angle / obtuse angle / straight angle / reflex angle / round angle

2.

∠ABC = ______

∠BAC = ______

∠ACB = ______

___________

___________

___________

3. Measure the sizes of angles of the triangle (label the vertices with your own choices of letters)

______________

______________

______________

___________

___________

___________

4. Measure the sizes of angles of the triangle (label the vertices with your own choices of letters)

Arranged by the sizes of angles from small to large

______________

______________

______________

______________

______________

______________

5. Draw an angle of the given size.

∠PQR = 210° _____________

A

B O

A

B C

Suggested answer:

Activity 1

Measure the size of the angles shown in the following questions with a protractor, write down the answer in the horizontal lines, and mark the degrees on the figures.

1.

∠ABC = _____ 110º _____

2. (a) Measure ∠ABC

∠ABC = _____ 80º _____

(b) reflex angle ABC = ___ 280º ___

3.

reflex angle PQR = _____ 240º _____

4.

reflex angle ABC = _____ 320º _____

Activity 2

Draw the required angles shown in the following questions with a protractor, and mark the degrees on the figures.

1. ∠ABC = 130º 2. (a) ∠PQR = 145º

(b) reflex angle PQR = 215º

3. reflex angle ABC = 250º 4. reflex angle PQR = 300º

80º

240º

250º

130º

300º 110º

A

B C

320º

215º A

B C

Q

P

R A

B C

A

B C

B

A

C

145º Q

P

R

Q

P

R

Exercise

Measure the size of the angles with a protractor, mark the degrees on the figures, and write down the sizes and types of the angles.

Figure Size of angle Type of angle

1.

∠AOB = 30°

acute angle / right angle / obtuse angle / straight angle / reflex angle / round angle 2.

∠ABC = 90°

∠BAC = 53°

∠ACB = 37°

right angle acute angle acute angle 3. Measure the sizes of angles of the triangle (label

the vertices with your own choices of letters)

∠ABC = 50°

∠BCA = 60°

∠CAB = 70°

acute angle acute angle acute angle

4. Measure the sizes of angles of the triangle (label

the vertices with your own choices of letters) Arranged by the sizes of angles from small to large

∠ABC = 120°

∠BCA = 20°

∠CAB = 40°

obtuse angle acute angle acute angle 5. Draw an angle of the given size.

∠PQR = 210° reflex angle A

B O

A

B C

A

B C

30°

A

C B

210°

P

Q R

90° 37°

53°

70°

60° 50°

40°

20° 120°

Example 6: Pie chart

A. Basic information of the topic

Strand: Data Handling

Learning objective: 1. Recognise pie charts.

2. Interpret pie charts.

- Students are not required to measure the angles at centre of a pie chart for calculations.

- Teachers may let students use IT to construct pie charts.

Prerequisite knowledge: Students should have learned angle (degree) through the learning and teaching materials for the transitional period of the revised junior secondary Mathematics curriculum.

Curriculum planning: Teachers may introduce this topic in Learning Unit 29 “Presentation of data”, or other learning units whichever appropriate.

[Note 1: The learning objective of pie charts in the revised primary Mathematics curriculum only requires students to interpret pie charts involving simple calculations, such as the case that the angle at centre of each sector is a multiple of 30º or 45º. In KS3, students should be able to handle more complex situations. Therefore, this learning and teaching materials does not follow the above restrictions, but teachers may start with simpler pie charts in teaching.]

[Note 2: In this topic, students understand the concept of sectors and angles at centre intuitively. Therefore, this learning and teaching material may not necessarily be arranged after Learning Objective 16.2.]

B. Suggested teaching content

1. Teachers may introduce that pie chart is one of the commonly used statistical charts to express the proportion of each of the item to the whole data set. The shape of a pie chat is circular and each sector represents a corresponding item. The name of items or its percentage are often indicated in or near the corresponding sectors (as shown in the figure below ^*).

*