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 Course1
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 Course1
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. TheTopic summary
and Your sayprovide 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
