Dr Toh Tin LamReviewer: Stephen Chew

MATHS 360NORMAL TECHNICAL

•AP

PRO

VE

DBY

MINISTRY OF EDUCATION

•

for use from 2016–2

020

Hypotenuse

Adjacent

0r

0 45

1 0

20

30

4 0

50 55 60 65 70

Length of side adjacent to

cosine of = Length of hypotenuse

4Volume of sphere = πr 3

3

0r

0 45

1 0

20

30

4 0

50 55 60 65 70

Length of side adjacent to

cosine of = Length of hypotenuse

4Volume of sphere = πr 3

3Dr To

h T

in Lam

• Review

er: Step

hen

Ch

ewM

AT

HS

36

0 MATHS 360NORMAL TECHNICAL

Dr Toh Tin LamReviewer: Stephen Chew

Maths 360 Normal (Technical) is written in line with the latest GCE Normal (Technical) Level syllabus from the Ministry of Education, Singapore. The series emphasises the acquisition and mastery of mathematical concepts and skills, and the development of thinking, reasoning and meta-cognitive skills through a mathematical approach to problem-solving. Concepts are developed in a logical and coherent manner, while promoting a positive attitude towards learning Mathematics.

Stimulates Learning• Interesting real-world chapter openers arouse interest in the topics covered.• Discuss and alternative methods stimulate students’ creativity and

thinking skills.• Real-life questions help students to appreciate the value of the concepts

taught and make informed decisions in daily life.• Comic strips present and teach concepts to make the learning of

Mathematics interesting.

Enhances Understanding• Concepts are presented in a systematic way — from the fundamentals to

more advanced principles — to help students appreciate and internalise mathematical concepts.

• Activity fosters students’ learning experiences and enriches their ICT journey through the exploration, investigation and discussion of mathematical ideas.

• Example and its Solution facilitate students’ understanding and application of concepts. Process skills are highlighted at appropriate junctures to help students make connections among mathematical ideas.

• Sidebar features provide students with pointers to key concepts, misconceptions and assess their understanding.

• Key idea summarises key learning outcomes and highlights important concepts and/or formulae.

Builds Examination Confi dence• Practice questions are graded to cater to students with different abilities.

The questions reinforce concepts covered and help students to identify their areas of weakness and misconception. Exam-like questions help students to build confi dence for the examinations.

• Chapter revision at the end of each chapter helps students to self-assess what they have learnt.

• Revision topics are included for students to recall and revise all the topics learnt throughout the four years.

This book has been reviewed by an experienced educator and has been found to meet the learning needs of students.

ISBN 978-981-01-9945-6

9 789810 199456

C01(M)M360NT4TB_Cover.indd 1 4/13/15 9:16 AM

01_(M)M360NTTB4_TP.indd 1 6/4/15 10:13 AM

Maths 360 Normal (Technical) is a new series of mathematics learning materials, designed primarily for students preparing for the Singapore GCE ‘N’ level examination. The series adopts the latest Mathematics Syllabus for Normal (Technical) from the Ministry of Education, Singapore.

The series consists of a textbook and a workbook for each level. The textbook examines mathematics from a real-world perspective, adopting a real-world approach that will help students appreciate the various applications of Mathematics. National Education (NE), fi nancial literacy concepts and 21st century competencies are infused in the series, where appropriate.

There are a variety of learning experiences that include hands-on activities and the use of ICT. Cartoons and comics have been infused in the textbook and workbook where appropriate. They help students acquire new knowledge, as they depict what goes on in our daily lives and the decisions we make to achieve optimal results.

Specially for the Secondary 4 students, revision topics are included to help them revise what they have learnt from Secondary 1 to Secondary 4. In addition, two review papers are provided in the workbook to help students familiarise with the examination format.

Students going through this series will fi nd the learning of Mathematics an interesting and rewarding experience.

Preface

02(M)M360NTTB4_FM.indd 3 6/10/15 5:32 PM

PREFACE

CHAPTER 1 Cumulative Frequency Distribution 1 Revisit 1.1 Data Analysis Involving Percentage and Probability 2 1.2 Cumulative Frequency Table 7 1.3 Cumulative Frequency Graph 12 1.4 Quartiles and Percentiles 19 Chapter 1 Key Ideas 29 Chapter 1 Revision 30

CHAPTER 2 Mensuration 35 Revisit 2.1 Conversion of Units 36 2.2 Length of an Arc 39 2.3 Area of a Sector 45 2.4 Surface Area and Volume of a Pyramid 50 2.5 Surface Area and Volume of a Cone 57 2.6 Surface Area and Volume of a Sphere 62 Chapter 2 Key Ideas 68 Chapter 2 Revision 69

CHAPTER 3 Trigonometry 71 Revisit 3.1 Pythagoras’ Theorem 72 3.2 The Sine Ratio 76 3.3 The Cosine Ratio 82 3.4 The Tangent Ratio 88 3.5 Trigonometric Ratios in Word Problems 93 3.6 Angles of Elevation and Depression 97 Chapter 3 Key Ideas 103 Chapter 3 Revision 104

Contents

02(M)M360NTTB4_FM.indd 4 6/3/15 3:03 PM

REVISION 1 Arithmetic 107

REVISION 2 Algebra 133

REVISION 3 Graphs 149

REVISION 4 Geometry 163

REVISION 5 Mensuration 181

REVISION 6 Trigonometry 193

REVISION 7 Probability and Statistics 203

ANSWERS 223

02(M)M360NTTB4_FM.indd 5 6/5/15 4:59 PM

Photo

p. 3, savings jar, © Pogonici / Dreamstime.com; p. 17, heat wave high temperatures, © Lucidwaters / Dreamstime.com; p. 24, marathon running race, © Jays / Dreamstime.com; p. 25, mortar and degree, © Pixelrobot / Dreamstime.com; p. 35, p. 50, pyramid in Egypt, © Hypermania37 / Dreamstime.com; p. 35, paper cone cups, © Design56 / Dreamstime.com; p. 35, piece of pizza, © Drti / Dreamstime.com; p. 35, globe, © Braendan Yong / Dreamstime.com; p. 57, party hat 6, © Stacy Barnett / Dreamstime.com; p. 62, soccer ball, © Bolygomaki / Dreamstime.com; p. 63, red glass marble, © Oblachko / Dreamstime.com; p. 71, single tree on green fi eld, © Keerati / Dreamstime.com; p. 101, building inspector looking up and pointing at the building under construction, © Kiriill Ryzhov / Dreamstime.com; p. 106, night lighthouse beam, © Joseph Mercier / Dreamstime.com; p. 119, pink hairdryer in hand, © Vladimirkim3722 / Dreamstime.com; p. 129, cow herd in farm pasture, © Ken Cole / Dreamstime.com; p. 133, satay or sate, © Szefei / Dreamstime.com; p. 157, currency exchange, © Shuo Wang / Dreamstime.com; p. 175, stationary set, © Christian Bridgwater / Dreamstime.com; p. 220, students 018, © Enriquecalvoal / Dreamstime.com

While every effort has been made to contact copyright holders of materials reproduced, we have been unsuccessful in some instances. To these copyright holders, we offer our sincere apologies and hope they will take our liberty in good faith. We would welcome any information which would enable us to contact the copyright holders/owners involved.

Acknowledgements

02(M)M360NTTB4_FM.indd 6 7/23/15 5:45 PM

1CHAPTER Cumulative

Frequency DistributionTo � nd out whether you have performed well in an examination, knowing your examination mark is not enough. The marks obtained by the other students taking the examination are also important. You can use cumulative frequency to help you � nd out your performance in the examination in comparison to other students.

In this chapter, you will learn to:• interpret and analyse cumulative frequency diagrams• fi nd the quartiles, interquartile range, percentiles and range

Show me your report card.

Daddy, I scored only 59 marks in the examination.

I see. You performed better than 90% of your schoolmates. The examination must be very diffi cult.

Oh! You only performed better than 30% of your schoolmates. The examination must be very easy. Show me your

report card.

Daddy, I did very well in the examination. I scored 82 marks.

03(M)M360NTTB4_01.indd 1 6/30/15 5:44 PM

1-1Data Analysis Involving Percentage and Probability

RevisitLesson

PercentageA percentage is expressed as a number out of 100. The symbol ‘%’ is used to indicate a percentage.

1Example

Express each of the following as a fraction in its lowest terms.(a) 70% (b) 26%

(a) 710

(b) 1350

Try

Express each of the following as a fraction in its lowest terms.(a) 25% (b) 52%

Solution

(a) 25% = 25100

(b) 52% = 52100

= 14

= 1325

Expressing a quantity A as a percentage of another quantity B means AB

× 100%.

2Example

In a class of 40 students, 30 of them participated in a charity walk. Find the percentage of students who(a) participated in the charity walk,(b) did not participate in the charity walk.

2 Chapter 1

03(M)M360NTTB4_01.indd 2 6/30/15 5:44 PM

There are 50 shops in a shopping centre. 12 of the shops are restaurants. Find the percentage of shops that are(a) restaurants,(b) not restaurants.

(a) 24% (b) 76%

Try

Solution

(a) Percentage of students who participated in the charity walk = 30

40 × 100%

= 75%

(b) Number of students who did not participate in the charity walk = 40 – 30 = 10

Percentage of students who did not participate in the charity walk = 10

40 × 100%

= 25%

We often need to calculate the value of a given percentage of a quantity.

3Example

(a) Find 70% of $200.(b) Christine saves 30% of her monthly allowance of $800.

How much money does she save?

Solution

(a) 70% of $200 = 70100

× $200

= $140

(b) 30% of $800 = 30100

× $800

= $240

She saves $240.

3Cumulative Frequency Distribution

03(M)M360NTTB4_01.indd 3 6/30/15 5:45 PM

(a) Find 20% of 500 m.(b) James spent 60% of the last 24 hours playing computer games. How many hours did he spend

playing computer games?

(a) 100 m (b) 14.4 h

Try

ProbabilityIf all outcomes are equally likely to happen,

Probability of an event = number of possible outcomes of eventtotal numberr of possible outcomes

4Example

Jenny throws an ordinary unbiased die. What is the probability that the die shows a number that is at most a 4?

Solution

All possible outcomes = {1, 2, 3, 4, 5, 6}Total number of possible outcomes = 6

Event (the die shows a number that is at most a 4) = {1, 2, 3, 4}Number of possible outcomes of event = 4

Probability that the die shows a number that is at most a 4 = 46

= 23

Maths NoteAt most 4 means not more than 4, i.e. � 4.

Choon Heng throws an ordinary unbiased die. What is the probability that he obtains a number that is less than 4?

12

Try

Maths NoteLess than 4 means smaller than 4 (cannot be 4), i.e. � 4.

Data AnalysisWe have learnt to represent and analyse data using different forms of statistical representations. Let us look at an example involving percentage and probability.

4 Chapter 1

03(M)M360NTTB4_01.indd 4 8/19/15 4:29 PM