Sample · 00-Course Contents and Study Guide
Chapter 0 Overview
IB Mathematics: Analysis and Approaches HL · CHAPTER 0: Course Contents and Study Guide
This opening chapter is the map of the whole companion. Analysis and Approaches at Higher Level is a single course built from five syllabus topics — Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and Calculus — together with a mathematical toolkit and an individual exploration. The five topics are taught over 210 recommended hours, and the toolkit and exploration add a further 30, giving the 240 hours of a Higher Level course.
Every syllabus item in those five topics is covered here by exactly one unit, and the units are numbered with the same codes the IB uses, so unit 3.14 in this book is subtopic 3.14 of the guide. Sections 0.2 to 0.6 below list all 84 units topic by topic, flag which of them are Additional Higher Level (AHL) content, and explain how the ideas inside each topic connect. Section 0.1 explains the layout of a unit and the notation conventions, and section 0.7 sets out the assessment model and a workable order of study.
0.1 How To Use This Companion
0.1 How To Use This Companion
Each unit follows the same shape, so you always know where to look. Two or three teaching paragraphs develop the idea from what you already know; a reference box collects the results and formulae you are expected to recall or recognise; a trap box names the mistake that most often costs marks on that subtopic; and a worked example carries one full solution through to a boxed answer, written in the style an examiner expects to read.
Mathematics is set in LaTeX throughout, so a quadratic appears as \( ax^2 + bx + c = 0 \) rather than as plain text, and a displayed result such as the discriminant condition for two distinct real roots appears in its own block. Angles are in radians unless a degree sign is written explicitly, \( \ln x \) always means the natural logarithm, and intervals use the usual bracket conventions, so \( [a,b) \) includes \( a \) but not \( b \).
Units marked AHL are examined only at Higher Level. They are not an optional appendix: they are woven into the same papers as the core material and they carry the harder marks. Read them in sequence with the rest of the topic rather than saving them for the end.
📋 Reading the unit labels
| Label | Meaning | Level |
|---|---|---|
| SL | Core content, examined at both Standard and Higher Level | SL + HL |
| AHL | Additional Higher Level content, examined at HL only | HL |
| REV | Mixed review unit drawing on the whole topic | HL |
⚠️ The most expensive habit
Reading a worked example and recognising every step is not the same as being able to produce it. Recognition returns almost nothing under exam conditions. Work each example on paper with the solution covered, then compare; the gap you find is exactly the material worth revisiting.
0.2 Topic 1 — Number and Algebra (39 hours)
0.2 Topic 1 — Number and Algebra (39 hours)
Topic 1 begins with the machinery of number and grows into the algebra that the rest of the course quietly assumes. Sequences and series come first: arithmetic and geometric progressions, their sums, and the financial models built on them. Exponents and logarithms follow as inverse operations, and the laws \( a^m a^n = a^{m+n} \) and \( \log_b (xy) = \log_b x + \log_b y \) are two readings of the same structure.
Continue reading — 3 chapters, worked examples and practice.
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