About the Book

For an online preview, please email IB@haesemathematics.com.

Mathematics: Core Topics HL has been written for the IB Diploma Programme courses Mathematics: Analysis and Approaches HL, and Mathematics: Applications and Interpretation HL, for first teaching in August 2019, and first assessment in May 2021.

The book contains the content that is common to both courses. This material can all be taught first, giving the potential to teach all the HL students together from this book at the start of the course.

This is the first of two books students will require for the completion of their HL Mathematics course. Upon the completion of this book, students progress to the particular HL textbook for their course: either Mathematics: Analysis and Approaches HL, or Mathematics: Applications and Interpretation HL. This is expected to occur approximately 6-7 months into the two-year course.

This product has been developed independently from and is not endorsed by the International Baccalaureate Organization.  International Baccalaureate, Baccalaureát International, Bachillerato Internacional and IB are registered trademarks owned by the International Baccalaureate Organization.

Year Published: 2019
Page Count: 548
ISBN: 978-1-925489-58-3 (9781925489583)
Online ISBN: 978-1-925489-70-5 (9781925489705)

Table of Contents

Mathematics: Core Topics HL

1 STRAIGHT LINES 19
A Lines in the Cartesian plane 20
B Graphing a straight line 25
C Perpendicular bisectors 26
D Simultaneous equations 28
Review set 1A 30
Review set 1B 31
2 SETS AND VENN DIAGRAMS 33
A Sets 34
B Intersection and union 36
C Complement of a set 37
D Special number sets 38
E Interval notation 40
F Venn diagrams 43
G Venn diagram regions 46
H Problem solving with Venn diagrams 47
Review set 2A 50
Review set 2B 51
3 SURDS AND EXPONENTS 53
A Surds and other radicals 54
B Division by surds 58
C Exponents 59
D Laws of exponents 60
E Scientific notation 66
Review set 3A 69
Review set 3B 69
4 EQUATIONS 71
A Power equations 72
B Equations in factored form 74
C Quadratic equations 75
D Solving polynomial equations using technology 84
E Solving other equations using technology 86
Review set 4A 87
Review set 4B 88
5 SEQUENCES AND SERIES 89
A Number sequences 90
B Arithmetic sequences 92
C Geometric sequences 98
D Growth and decay 101
E Financial mathematics 103
F Series 111
G Arithmetic series 114
H Finite geometric series 119
I Infinite geometric series 123
Review set 5A 126
Review set 5B 128
6 MEASUREMENT 131
A Circles, arcs, and sectors 132
B Surface area 134
C Volume 140
D Capacity 150
Review set 6A 154
Review set 6B 155
7 RIGHT ANGLED TRIANGLE TRIGONOMETRY 157
A Trigonometric ratios 159
B Inverse trigonometric ratios 162
C Right angles in geometric figures 164
D Problem solving with trigonometry 169
E True bearings 174
F The angle between a line and a plane 176
Review set 7A 179
Review set 7B 181
8 THE UNIT CIRCLE AND RADIAN MEASURE 183
A Radian measure 184
B Arc length and sector area 186
C The unit circle 190
D Multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$ 196
E The Pythagorean identity 199
F Finding angles 201
G The equation of a straight line 203
Review set 8A 204
Review set 8B 205
9 NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY 207
A The area of a triangle 208
B The cosine rule 212
C The sine rule 216
D Problem solving with trigonometry 221
Review set 9A 228
Review set 9B 230
10 POINTS IN SPACE 233
A Points in space 234
B Measurement 236
C Trigonometry 240
Review set 10A 244
Review set 10B 245
11 PROBABILITY 247
A Experimental probability 249
B Two-way tables 253
C Sample space and events 255
D Theoretical probability 257
E Making predictions using probability 264
F The addition law of probability 265
G Independent events 267
H Dependent events 271
I Conditional probability 275
J Formal definition of independence 278
K Bayes' theorem 280
Review set 11A 284
Review set 11B 286
12 SAMPLING AND DATA 289
A Errors in sampling and data collection 292
B Sampling methods 294
C Writing surveys 300
D Types of data 302
E Simple discrete data 304
F Grouped discrete data 308
G Continuous data 309
Review set 12A 312
Review set 12B 313
13 STATISTICS 315
A Measuring the centre of data 316
B Choosing the appropriate measure 321
C Using frequency tables 323
D Grouped data 326
E Measuring the spread of data 328
F Box and whisker diagrams 332
G Outliers 335
H Parallel box and whisker diagrams 337
I Cumulative frequency graphs 340
J Variance and standard deviation 344
Review set 13A 353
Review set 13B 356
14 QUADRATIC FUNCTIONS 359
A Quadratic functions 361
B Graphs of quadratic functions 362
C Using the discriminant 369
D Finding a quadratic from its graph 372
E The intersection of graphs 376
F Problem solving with quadratics 379
G Optimisation with quadratics 381
H Quadratic inequalities 385
Review set 14A 389
Review set 14B 390
15 FUNCTIONS 393
A Relations and functions 394
B Function notation 397
C Domain and range 400
D Rational functions 405
E Composite functions 410
F Inverse functions 414
Review set 15A 421
Review set 15B 423
16 TRANSFORMATIONS OF FUNCTIONS 425
A Translations 426
B Stretches 429
C Reflections 435
D Miscellaneous transformations 438
E The graph of $y=\frac{1}{f(x)}$ 441
Review set 16A 443
Review set 16B 445
17 TRIGONOMETRIC FUNCTIONS 447
A Periodic behaviour 448
B The sine and cosine functions 452
C General sine and cosine functions 454
D Modelling periodic behaviour 459
E Fitting trigonometric models to data 461
F The tangent function 464
G Trigonometric equations 467
H Using trigonometric models 475
Review set 17A 477
Review set 17B 479
ANSWERS 483
INDEX 546

Authors

  • Michael Haese
  • Mark Humphries
  • Chris Sangwin
  • Ngoc Vo

Author

Michaelhaese

Michael Haese

Michael completed a Bachelor of Science at the University of Adelaide, majoring in Infection and Immunity, and Applied Mathematics. He studied laminar heat flow as part of his Honours in Applied Mathematics, and finished a PhD in high speed fluid flows in 2001. He has been the principal editor for Haese Mathematics since 2008.

What motivates you to write mathematics books?

My passion is for education as a whole, rather than just mathematics. In Australia I think it is too easy to take education for granted, because it is seen as a right but with too little appreciation for the responsibility that goes with it. But the more I travel to places where access to education is limited, the more I see children who treat it as a privilege, and the greater the difference it makes in their lives. But as far as mathematics goes, I grew up with mathematics textbooks in pieces on the kitchen table, and so I guess it continues a tradition.

What do you aim to achieve in writing?

I think a few things:

  • I want to write to the student directly, so they can learn as much as possible from the text directly. Their book is there even when their teacher isn't.
  • I therefore want to write using language which is easy to understand. Sure, mathematics has its big words, and these are important and we always use them. But the words around them should be as simple as possible, so the meaning of the terms can be properly explained to ESL (English as a Second Language) students.
  • I want to make the mathematics more alive and real, not by putting it in contrived "real-world" contexts which are actually over-simplified and fake, but rather through its history and its relationship with other subjects.

What interests you outside mathematics?

Lots of things! Horses, show jumping and course design, alpacas, badminton, running, art, history, faith, reading, hiking, photography ....

Author

Markhumphries

Mark Humphries

Mark has a Bachelor of Science (Honours), majoring in Pure Mathematics, and a Bachelor of Economics, both of which were completed at the University of Adelaide. He studied public key cryptography for his Honours in Pure Mathematics. He started with the company in 2006, and is currently the writing manager for Haese Mathematics.

What got you interested in mathematics? How did that lead to working at Haese Mathematics?

I have always enjoyed the structure and style of mathematics. It has a precision that I enjoy. I spend an inordinate amount of my leisure time reading about mathematics, in fact! To be fair, I tend to do more reading about the history of mathematics and how various mathematical and logic puzzles work, so it is somewhat different from what I do at work.

How did I end up at Haese Mathematics?

I was undertaking a PhD, and I realised that what I really wanted to do was put my knowledge to use. I wanted to pass on to others all this interesting stuff about mathematics. I emailed Haese Mathematics (Haese and Harris Publications as they were known back then), stating that I was interested in working for them. As it happened, their success with the first series of International Baccalaureate books meant that they were looking to hire more people at the time. I consider myself quite lucky!

What are some interesting things that you get to do at work?

On an everyday basis, it's a challenge (but a fun one!) to devise interesting questions for the books. I want students to have questions that pique their curiosity and get them thinking about mathematics in a different way. I prefer to write questions that require students to demonstrate that they understand a concept, rather than relying on rote memorisation.

When a new or revised syllabus is released for a curriculum that we write for, a lot of work goes into devising a structure for the book that addresses the syllabus. The process of identifying what concepts need to be taught, organising those concepts into an order that makes sense from a teaching standpoint, and finally sourcing and writing the material that addresses those concepts is very involved – but so rewarding when you hold the finished product in your hands, straight from the printer.

What interests you outside mathematics?

Apart from the aforementioned recreational mathematics activities, I play a little guitar, and I enjoy playing badminton and basketball on a social level.

Author

Chrissangwin

Chris Sangwin

Chris completed a BA in Mathematics at the University of Oxford, and an MSc and PhD in Mathematics at the University of Bath. He spent thirteen years in the Mathematics Department at the University of Birmingham, and from 2000-2011 was seconded half time to the UK Higher Education Academy "Maths Stats and OR Network" to promote learning and teaching of university mathematics. He was awarded a National Teaching Fellowship in 2006. Chris Sangwin joined the University of Edinburgh in 2015 as Professor of Technology Enhanced Science Education.

What are your learning and teaching interests in mathematics?

I teach mathematics at university but am particularly interested in core pure mathematics which starts in school and continues to be taught at university. Solving mathematical problems is at the heart of mathematics, and I enjoy teaching problem solving at university.

What interests you outside mathematics?

I really enjoy hill walking and mountaineering, particularly spending time with friends in the hills.

Why do you choose to collaborate with a small publisher on the other side of the world?

There is a unique team spirit in Haese which other publishers don't have. This makes authorship much more collaborative than my previous experiences, which is really enjoyable and I'm sure leads to much better quality books for students which are, after all, the whole point.

Author

Ngocvo

Ngoc Vo

Ngoc Vo completed a Bachelor of Mathematical Sciences at the University of Adelaide, majoring in Statistics and Applied Mathematics. Her Mathematical interests include regression analysis, Bayesian statistics, and statistical computing. Ngoc has been working at Haese Mathematics as a proof reader and writer since 2016.

What drew you to the field of mathematics?

Originally, I planned to study engineering at university, but after a few weeks I quickly realised that it wasn't for me. So I switched to a mathematics degree at the first available opportunity. I didn't really have a plan to major in statistics, but as I continued my studies I found myself growing more fond of the discipline. The mathematical rigor in proving distributional results and how they link to real-world data -- it all just seemed to click.

What are some interesting things that you get to do at work?

As the resident statistician here at Haese Mathematics, I get the pleasure of writing new statistics chapters and related material. Statistics has always been a challenging subject to both teach and learn, however it doesn't always have to be that way. To bridge that gap, I like to try and include as many historical notes, activities, and investigations as I can to make it as engaging as possible. The reasons why we do things, and the people behind them are often important things we forget to talk about. Statistics, and of course mathematics, doesn't just exist within the pages of your textbook or even the syllabus. There's so much breadth and depth to these disciplines, most of the time we just barely scratch the surface.

What interests you outside mathematics?

In my free time I like studying good typography and brushing up on my TeX skills to become the next TeXpert. On the less technical side of things, I also enjoy scrapbooking, painting, and making the occasional card.

Features

  • Snowflake (24 months)

    A complete electronic copy of the textbook, with interactive, animated, and/or printable extras.

  • Self Tutor

    Animated worked examples with step-by-step, voiced explanations.

  • Theory of Knowledge

    Activities to guide Theory of Knowledge projects.

  • Graphics Calculator Instructions

    For Casio fx-CG50, TI-84 Plus CE, TI-nspire, and HP Prime

Icon selftutor ib

This book offers SELF TUTOR for every worked example. On the electronic copy of the textbook, access SELF TUTOR by clicking anywhere on a worked example to hear a step-by-step explanation by a teacher. This is ideal for catch-up and revision, or for motivated students who want to do some independent study outside school hours.

Icon graphics calculator instructions%20%282%29

Graphics calculator instructions for Casio fx-CG50, TI-84 Plus CE, TI-nspire, and HP Prime are included with this textbook. The textbook will either have comprehensive instructions at the start of the book, specific instructions available from icons located throughout, or both. The extensive use of graphics calculators and computer packages throughout the book enables students to realise the importance, application, and appropriate use of technology.

Icon theory of knowledge

Theory of Knowledge is a core requirement in the International Baccalaureate Diploma Programme.

Students are encouraged to think critically and challenge the assumptions of knowledge. Students should be able to analyse different ways of knowing and kinds of knowledge, while considering different cultural and emotional perceptions, fostering an international understanding.

Snowflake

This book is available on electronic devices through our Snowflake learning platform. This book includes 24 months of Snowflake access, featuring a complete electronic copy of the textbook.

Where relevant, Snowflake features include interactive geometry, graphing, and statistics software, demonstrations, games, spreadsheets, and a range of printable worksheets, tables, and diagrams. Teachers are provided with a quick and easy way to demonstrate concepts, and students can discover for themselves and re-visit when necessary.

Support material

  • Errata

    Last updated - 29 Apr 2020