## Preface

*New Century Maths General: HSC Course *continues and extends the mathematics introduced in our
first book,* New Century Maths General: Preliminary Course*,again covering the five strands of General
Mathematics: Financial mathematics, Data analysis, Measurement, Probability and Algebraic
modelling. It retains the useful and popular features of the *Preliminary* text and of the *New Century *
*Maths* series in general:

plenty of worked examples and well-graded exercises, including extension work carefully sequenced chapters covering the scope and depth of the course outcomes presented at the start of every chapter

investigations, group and modelling activities

the use of technology such as spreadsheets, graphics calculators and the Internet

**Just for the record **containing interesting facts and mathematical applications related to the topic
comprehensive chapter reviews summarising and revising each chapter

glossary and index.

To prepare students for their HSC exams (trial and external), four practice papers containing HSC-style
questions have been included in this book as well as a practice HSC exam. The promotion of effective
study skills continues in the form of **Study tips** in each chapter, while the final chapter of the book
(Chapter 11) provides exercises that allow the systematic revision of all syllabus areas. We have
endeavoured to write a useful and relevant textbook encompassing the spirit of the General Mathematics
course.

Writing these two books for the General Mathematics course has been a challenging but rewarding experience for both of us. We acknowledge and thank the following people for their support over the past 18 months: the review team for their constructive comments on our early drafts, David Kellock and Nancy Gugliotti of Nelson Thomson Learning for overseeing the project, Rhonda Idczak for

transforming our manuscript into the book you see before you, and the authors of the *New Century *
*Maths 7–10* series, whose pioneering efforts have made our job a little easier.

Our thanks go to our families, friends and colleagues for their understanding during our writing efforts. Margaret thanks her husband John for his patience and support, her daughters Kara and Shelley for their encouragement, and her grandson Joshua for his inspiration. Robert thanks all his colleagues and teachers past and present for their wisdom and positive influences, especially his parents, John and Nancy Yen, whom he considers the greatest teachers of all.

To the teachers and students using this book for the first time, we wish you success in completing the General Mathematics course, and look forward to your comments and suggestions.

Margaret Willard and Robert Yen

## The review team

Mark Adamson, Holsworthy High School

Judy Binns, Mulwaree High School and co-author of *New Century Maths 7* and *8*
John Dillon, Hurlstone Agricultural High School

Christine Doyle, James Ruse Agricultural High School

The author team

Margaret Willard and Robert Yen are both practising teachers and active members of the Mathematical Association of NSW (MANSW). Margaret is the Senior Head Teacher of Mathematics (TAFE) at OTEN as well as the manager of MANSW’s post-secondary programs. She has taught at numerous

## Syllabus reference grid

### Syllabus Strand

**Syllabus**
**Substrand**

**New Century Maths General: ****HSC Course**

## FINANCIAL MATHEMATICS

FM4 Credit and borrowing Chapter 3 Credit and loans

FM5 Annuities and loan repayments Chapter 8 Long term investing

FM6 Depreciation Chapter 8 Long term investing

**DATA ANALYSIS**

DA5 Interpreting data sets Chapter 4 Statistical distributions

DA6 The normal distribution Chapter 10 The normal distribution and correlation

DA7 Correlation Chapter 10 The normal distribution and correlation

## MEASUREMENT

M5 Further applications of area and

volume Chapter 2 Area and volume

M6 Applications of trigonometry Chapter 5 The sine and cosine rules

M7 Spherical geometry Chapter 7 Geometry of the Earth

**PROBABILITY**

PB3 Multistage events Chapter 6 Probability

PB4 Applications of probability Chapter 6 Probability

**ALGEBRAIC**
**MODELLING**

AM3 Algebraic skills and techniques Chapter 1 Equations and functions

AM4 Modelling linear and non-linear relationships

Chapter 1 Equations and functions Chapter 9 Functions and graphs

Preface iii

Syllabus reference grid iv

New Century General Maths teaching schedule viii

How to use this book x

How to study for Maths xi

Algebraic expressions 2

Scientific notation 4

Formulas 5

Solving equations 9

Equations involving powers and roots 11

Changing the subject of a formula 14

Equations and formulas 18

Linear functions 22

Intersection of lines 30

Chapter review 35

## Contents

1

Equations and functions

(Algebraic modelling)

1

Composite areas 40

Parts of a circle 45

Area of an ellipse 50

Simpson’s rule for approximating areas

and volumes 53

Surface areas of prisms and pyramids 57

Surface area of a cylinder 61

Surface area and volume of a sphere 63

Volumes of composite solids 66

Error in measurement and calculation 71

Chapter review 75

2

## Area and volume

(Measurement)

39

Flat rate loans 80

Buying on terms 84

Reducing balance loans 88

Using published loan repayment tables 90

Using technology to compare home loans 93

Credit card payments 97

Chapter review 101

3

Credit and loans

(Financial Mathematics)

79

Collecting and displaying data 110

Summary statistics 112

Features of a statistical display 118

Investigating outliers 122

Comparing data sets using charts 129

Two-way tables 133

Using multiple displays to compare

data sets 136

4

## Statistical distributions

(Data analysis)

109

Practice paper one

105

Right-angled triangle trigonometry 148

Bearings 152

Trigonometry with obtuse angles 157

The sine rule 160

Using the sine rule to find a missing angle 165

The cosine rule 168

Using the cosine rule to find a missing angle 171

Area of a triangle 174

Applications of the sine and cosine rules 177

Surveying 182

Chapter review 189

5

## The sine and cosine rules

(Measurement)

147

The meaning of probability 194

Tree diagrams and tables 199

The multiplication principle for counting 202

Counting arrangements 205

Counting unordered selections 209

Ordered and unordered selections 213

Probability tree diagrams 216

Expectation 220

Probability simulations 227

Probability in testing 231

Chapter review 237

6

## Probability

(Probability)

193

Latitude and longitude 246

Great circle distances 253

Nautical miles and knots 258

Longitude and time differences 262

International time zones 267

Chapter review 275

7

Geometry of the Earth

(Measurement)

245

Interest calculations 280

Annuities 284

Future value of an annuity 284

Present value of an annuity 290

Using tables for annuity calculations 294

Loan repayments 300

Depreciation 305

Straight line method of depreciation 305 Declining balance method of depreciation 310

Calculating tax deductions 318

Chapter review 322

8

## Long term investing

(Financial mathematics)

279

Practice paper three

373

The quadratic function 328

Maximum and minimum problems 334

The cubic function 339

The exponential function 343

Exponential growth and decay 347

The hyperbolic function 352

More applications of functions 356

Direct variation 360

Inverse variation 365

Chapter review 369

9

## Functions and graphs

(Algebraic modelling)

327

Practice paper two

241

Formulas list 413

Area and volume 414

The sine and cosine rules 417

Geometry of the Earth 420

Equations and functions 421

Functions and graphs 423

Credit and loans 425

Long term investing 430

Statistical distributions 433

The normal distribution and correlation 437

Probability 440

11

## Revision 413

Practice paper four

443

Practice HSC exam

447

Glossary

455

Index

498

Answers

465

The normal distribution 378

*Z*-scores 383

Comparing normal distributions 388

Scatterplots 393

Correlation 396

Regression lines 400

Chapter review 409

10

The normal distribution and correlation

(Data analysis)

377

New Century General Maths

## teaching schedule

Term 1

**Preliminary Course**

1

2

3

4

5

6

7

8

9

10

1 Linear equations and functions

(AM)

2 Calculations and spreadsheets

(FM, M)

3 Measurement, area and volume

(M)

4 Statistical samples and displays

(DA)

Term 2

1

2

3

4

5

6

7

8

9

10

4 cont’d 5 Ratios and

similar figures

(M)

6 Earning and taxation

(FM)

7 Trigonometry

(M)

Term 3

1

2

3

4

5

6

7

8

9

10

8 Rates and linear modelling

(AM)

9 Probability

(PB)

10 Savings and investments

(FM)

Term 4

**HSC Course**

1

2

3

4

5

6

7

8

9

10

11 Statistical measurement

(DA)

1 Equations and functions

(AM)

2 Area and volume

(M)

Term 5

1

2

3

4

5

6

7

8

9

10

3 Credit and loans

(FM)

4 Statistical distributions

(DA)

5 The sine and cosine rules

(M)

6 Probability

(PB)

Term 6

1

2

3

4

5

6

7

8

9

10

6 cont’d 7 Geometry of

the Earth

(M)

8 Long term investing

(FM)

9 Functions and graphs

(AM)

Term 7

1

2

3

4

5

6

7

8

9

10

9 cont’d

10 The normal distribution and correlation

(DA)

11 Revision of General Mathematics

## How to use this book

RECORD highlights important facts, rules and formulas that students need to remember and

note in their workbooks and topic summaries.

FAST FORWARD indicates extension questions: challenging problems for advanced

students and opportunities for further investigation.

## REWIND–P indicates review of Preliminary coursework that can comprise up to 30% of the

final HSC exam. However, some content is part of

both

the Preliminary and HSC courses,

and in these cases this symbol is

not

used.

INVESTIGATIONS, MODELLING and GROUP ACTIVITIES.

THINK questions for discussion and consideration and IDEAS for teaching and learning.

## TECHNOLOGY activities involving calculators, spreadsheets, graphics calculators,

graphing software and the Internet.

PRACTICE PAPERS generally after every 3 chapters are mixed review exercises, revision

tests written in a style similar to that of HSC exams.

CHAPTER REVIEWS summarise and conclude each chapter. They contain a listing of

## chapter sub-headings, a mind map/summary exercise,

Your say

and a

Chapter assignment

.

TOPIC SUMMARIES contain overviews of chapters and mind map exercises.

YOUR SAY is a checklist of questions for students to reflect on what they have (or have not)

learned in each chapter. It encourages them to think and write about their learning as well as

## providing direction for further study and revision.

CHAPTER ASSIGNMENTS are review exercises covering the skills learned in each

chapter.

STUDY TIPS and hints are presented throughout this book, generally two per chapter.

JUST FOR THE RECORD contains interesting facts, unusual trivia and mathematical

## applications relating to the content of the topic being covered.

The GLOSSARY and INDEX are provided for ready reference and to encourage students to

understand and use the language of mathematics.

The SYLLABUS REFERENCE GRID at the front of the book matches the syllabus structure

to the relevant chapters in the book.

**P**

How to study for Maths

‘A journey of a thousand miles begins with a single step.’
*Lao Tzu (c. 570–490 BC), ancient Chinese philosopher*

Any type of study or endeavour is a long journey—challenging at times but not unachievable. The learning of mathematics is the same. Sometimes when studying and revising Maths, it’s difficult to know where or how to begin. This introductory part of the textbook is like a road map, providing a framework of ideas and pointers on how to make that first step. Good luck!

## Four principles of learning mathematics

1. Mathematics is about mastering a collection of skills

Mathematics is unlike other subjects. It has more to do with skills than knowledge or content. You don't get better at it by using the library or Internet to find out more about the subject or by reading and researching a topic in greater detail. Instead, you get better by doing it more. Maths is about mastering a collection of skills, and applying these skills to solve problems.

2. Homework and study are not the same thing

Homework is ‘practice’ work set by your teacher each day, reinforcing your learning of a particular skill. Study, on the other hand, is your own ‘revision’ work for strengthening and improving your understanding of the subject. Study is the homework set by yourself.

## 3. You cannot learn Maths by halves—you’ve got to know it all!

Unlike most subjects, all mathematical knowledge is interconnected. This means the theory and skills of one Maths topic are related to those of another. Mathematics is also sequential, becoming more complex at each stage. You have to learn it in a particular order, so that if you miss out on an important step, then it becomes more difficult to progress. Consequently, many people have trouble understanding mathematics because of ‘gaps’ in their knowledge—they don't have the full picture. And people who only know half their Maths don’t get 50% in a Maths test—they get less. Just as you cannot run until after you have learned to walk, you cannot learn about Algebra until after you have mastered Number.

So mathematics is not a subject you can learn by halves. You must know it all in order to understand it properly. You need to be thorough in your study. This means covering all gaps in your knowledge and strengthening any weak areas you may have.

4. Maths questions are often predictable... really!

However, once you have mastered your Maths skills, that’s it! You’re done! You don't need to do any further reading or research to gain a deeper understanding of the subject. Compared to other subjects, the types of questions asked in Maths exams are more predictable and conventional, similar to the types of exercises you have seen in class.

## Four stages in studying for mathematics

With the above principles in mind, there are four stages in studying for Maths.

1. Practise your Maths

This is the simplest stage. Do your homework. You need good training before you can become a Maths master. As with any type of skill, the only way of strengthening your abilities is through practice, practice, practice. Aim to achieve a high level of understanding. You need to master the basics thoroughly before you can move on to more complex work.

2. Rewrite your Maths

Take possession of the Maths by rewriting the theory and examples in your own words. Make

Rewriting and summarising your work allow you to see the full picture. A **Topic summary**

gives you a view of the whole topic in 2 to 3 pages, while a collection of topic summaries gives you a view of the whole course. Both also provide direction for further revision because they often highlight your own strengths and weaknesses.

## 3. Attack your Maths

Find your areas of weakness and work on them. Fill in any gaps in your mathematical knowledge. Most of your study time should be spent on attacking your weaknesses. Use your topic summaries for general revision, but spend longer periods on in-depth work with your problem areas. Don’t spend too much time on work you already know well, unless it boosts your self-confidence. During this stage, you may need to rewrite or restructure your topic summaries to clarify some points.

4. Test your Maths

Finally, test yourself using mixed revision exercises, assignments and past exams. After all, if you’re preparing for a big test, you need to go back and practise your Maths skills on a variety of questions. Studying past exam papers is an excellent way of adapting to the format of the questions and the level of difficulty expected. It will also help you develop and improve your exam technique. Remember, mathematics is one of the few subjects in which the exam questions are fairly predictable. The examiners are not going to pose many unusual questions or unconventional problems.

Using this book to study for mathematics

Many features of this book have been designed to help you enhance your Maths study.

## Important facts and formulas highlighted in boxes:

Review sections:

A **Chapter review** at the end of every chapter contains:

a list of the chapter’s subheadings

a **Topic summary**

**Your say**: questions for thinking and reflecting about the topic

a **Chapter assignment**

Make an effort to summarise each chapter as soon as you have completed it. The**Topic summary**

and **Your say**provide suggestions, ideas and advice for rewriting your Maths, either in written
form or as a mind map. The **Chapter assignment** will help you test your Maths. It revises all of
the important concepts and skills learned in that chapter.

The four **Practice papers** and the **Practice HSC exam** consist of mixed review questions. These
are presented in the style of an HSC exam.

## Within each chapter, all important facts and formulas are highlighted in a box like

this, labelled by a ‘Record’ button. Be sure to include these facts and formulas in

your topic summaries.

Useful study tips have been included throughout this book to help you learn, remember and revise your mathematics. These are little hints for effective study, especially handy for senior students. Some of these tips are general by nature and not necessarily restricted to the learning of mathematics. The study tips in each chapter are presented in a box like this.

Study tips