📚 Foreword | 前言
Welcome to the IB Mathematics revision series. Whether you are starting your Diploma Programme journey or preparing for the final examinations, this resource has been designed to support every student in mastering the concepts, applying problem-solving strategies, and achieving the highest possible grade. The International Baccalaureate Mathematics curriculum is rigorous, diverse, and intellectually rewarding. This foreword sets out the structure of the series, explains the key features of the two IB Mathematics courses, and provides practical guidance on how to use the revision materials effectively.
欢迎来到IB数学复习系列。不论你刚刚开始文凭课程之旅，还是在为最终考试做准备，这份资源旨在帮助每一位学生掌握概念、运用解题策略并取得尽可能高的等级。国际文凭数学课程严格、多样且富有思维挑战。这篇前言将说明本系列的结构，介绍两种IB数学课程的主要特点，并提供关于如何高效使用复习材料的实用指南。
1. Welcome to the IB Mathematics Series | 欢迎来到IB数学系列
This series has been crafted to serve the entire IB Mathematics community – both Analysis and Approaches (AA) and Applications and Interpretation (AI), at Standard Level (SL) and Higher Level (HL). Each article in the collection focuses on a specific syllabus topic, breaking it down into clear explanations, worked examples, and practice questions aligned with the latest IB assessments. The goal is to make abstract ideas concrete, to build confidence, and to help you think like a mathematician.
本系列专为整个IB数学社区打造——涵盖分析与方法(AA)和应用与解释(AI)，包括标准水平(SL)和高级水平(HL)。系列中的每篇文章聚焦一个具体的教学大纲主题，将其分解为清晰的解释、范例和与最新IB评估一致的练习题。目标是让抽象的概念变得具体，建立信心，并帮助你像数学家一样思考。
2. Understanding the IB Mathematics Curriculum | 理解IB数学课程
Since the 2019 curriculum update, IB Mathematics has been organised into two distinct routes. Analysis and Approaches (AA) places a strong emphasis on algebraic manipulation, formal proof, and the development of a deep theoretical understanding. Applications and Interpretation (AI) emphasises the use of mathematics to model real-world phenomena, with a greater reliance on technology and statistical analysis. Choosing the right pathway depends on your university plans, personal strengths, and interest in pure versus applied mathematics.
自2019年课程更新以来，IB数学分为两条不同的路线。分析与方法(AA)着重强调代数运算、形式化证明以及深刻理论理解的培养。应用与解释(AI)则强调利用数学对现实世界现象进行建模，更依赖技术和统计分析。选择哪条路径取决于你的大学规划、个人优势以及对纯数学与应用数学的兴趣。
3. Analysis and Approaches (AA) vs Applications and Interpretation (AI) | 分析与方法(AA)与应用与解释(AI)
The table below summarises the key differences between the two courses. It is essential to recognise that neither route is inherently easier; they simply serve different academic purposes.
下表总结了两门课程的关键区别。必须认识到，没有哪条路线天然更容易；它们只是服务于不同的学术目标。
| Feature | AA (Analysis and Approaches) | AI (Applications and Interpretation) |
|---|---|---|
| Focus | Algebraic rigour, proofs, calculus theory | Modelling, statistics, technology integration |
| Calculus content | Deep, including limits, advanced integration, differential equations (HL) | Applied calculus for modelling, less formal theory |
| Statistics & Probability | Standard distributions, hypothesis testing (HL) | Extensive: large data sets, chi-squared tests, linear regression, probabilistic models |
| Graphic Display Calculator | Used mainly in Paper 2; analytical methods dominate | Central to both papers; used for data handling, graphing, numerical solving |
| Ideal for | Mathematics, engineering, physics, computer science degrees | Social sciences, economics, natural sciences, business, medicine |
无论你选择哪条路径，本系列都会用清晰的标记(AA-SL、AA-HL、AI-SL、AI-HL)让你找到对应的内容。这样你就能轻松定位到与自己课程完全相关的章节。
4. Standard Level (SL) vs Higher Level (HL) | 标准水平(SL)与高级水平(HL)
The difference between SL and HL is primarily one of depth and breadth. At HL, students encounter additional topics such as complex numbers, vectors, and more advanced calculus or statistical techniques, depending on the route. HL students also complete a third examination paper (Paper 3) that demands extended reasoning and problem-solving skills. The recommended teaching hours are 150 for SL and 240 for HL, which reflects the increased pace and workload.
SL与HL的区别主要在于深度和广度。HL学生需要学习额外的主题，如复数、向量以及更高级的微积分或统计技术（取决于路线）。HL学生还要参加第三张试卷（Paper 3），这要求具备拓展推理和问题解决能力。建议的教学时数SL为150小时，HL为240小时，这反映了节奏和工作量的增加。
5. Assessment Overview | 评估概览
IB Mathematics assessment consists of an External Assessment (written examinations) and an Internal Assessment (the mathematical exploration). The external component is always worth 80% of the final grade, while the IA accounts for the remaining 20%. This consistent weighting across SL and HL means that both the exams and the exploration are critical to success. It is vital to manage your time so that you can prepare thoroughly for both components.
IB数学评估由外部评估（书面考试）和内部评估（数学探索）组成。外部部分始终占总成绩的80%，而IA占剩下的20%。这一权重在SL和HL中保持一致，意味着考试和探索都对成功至关重要。合理分配时间，为两部分都做好充分准备非常重要。
6. The Internal Assessment (IA) | 内部评估(IA)
The IA is an independent investigation into a mathematical topic of your choice. It should demonstrate personal engagement, clear mathematical communication, and appropriate use of technology. The exploration is assessed against five criteria: Presentation, Mathematical Communication, Personal Engagement, Reflection, and Use of Mathematics. A well-chosen topic – something you are genuinely curious about – often leads to a stronger, more enjoyable piece of work. Throughout this series, you will find IA inspiration boxes that suggest possible exploration ideas linked to the topic at hand.
IA是一项关于你自选数学主题的独立调查。它应展现个人参与性、清晰的数学交流能力以及恰当的技术运用。探索报告依据五项标准进行评判：表达、数学交流、个人参与、反思以及数学运用。一个精心选择的主题——你真正感到好奇的东西——往往能带来更有深度、更令人享受的作品。在本系列中，你会看到IA灵感框，它们会结合当前主题提出可能的探索思路。
7. Exam Papers Structure | 考试试卷结构
The examination structure varies between SL and HL. SL candidates sit two papers: Paper 1, where the use of a calculator is not permitted, and Paper 2, where a Graphic Display Calculator (GDC) is required. HL candidates take three papers: Paper 1 (no calculator), Paper 2 (calculator required) and Paper 3 (calculator required), which focuses on two extended problem-solving contexts. Understanding the format, timing, and command terms (e.g., 'find', 'prove', 'hence') is essential for effective revision.
考试结构因SL和HL而有所不同。SL考生参加两张试卷：Paper 1不得使用计算器，Paper 2必须使用图形显示器(GDC)。HL考生需参加三张试卷：Paper 1（不得使用计算器）、Paper 2（需使用计算器）和Paper 3（需使用计算器），Paper 3侧重两个拓展的问题解决情境。理解试卷格式、时间安排和指令词（如“find”、“prove”、“hence”）对于高效复习至关重要。
8. Core Topics in IB Mathematics | IB数学核心主题
Despite the differences between the two routes, there is considerable overlap in the core topics. The table below highlights the main areas covered in each course, allowing you to see at a glance what is common and what is specific to your pathway.
尽管两条路线存在差异，但核心主题有相当多的重叠。下表列出了每门课程涵盖的主要领域，让你能够一目了然地看到哪些是共通的，哪些是你所选路径特有的。
| Topic Area | AA content highlights | AI content highlights |
|---|---|---|
| Number and Algebra | Sequences, series, binomial theorem, complex numbers (HL), proofs by induction (HL) | Logarithms, exponentials, loans and amortisation, matrices (HL) |
| Functions | Domain, range, transformations, inverse functions, modulus, rational functions | Modelling with functions, sinusoidal models, logistic functions |
| Geometry and Trigonometry | Radian measure, trig identities, vector geometry (HL), equations of lines and planes (HL) | Voronoi diagrams, graph theory, 3D geometry with GDC |
| Statistics and Probability | Descriptive statistics, probability distributions, binomial and normal distributions, hypothesis testing (HL) | Chi-squared tests, t-tests, correlation, regression, data analysis with large data sets |
| Calculus | Limits, first principles, differentiation and integration techniques, differential equations (HL) | Numerical differentiation, trapezoidal rule, optimisation, differential equations for modelling |
9. Mathematical Toolkit for Success | 数学学习工具箱
To thrive in IB Mathematics, you need more than just a textbook. Your Graphic Display Calculator (such as the TI-Nspire or Casio fx-CG50) is a vital tool for exploration, checking answers, and tackling exam questions on calculator papers. However, you must also become fluent in analytical techniques for the non-calculator paper. The IB formula booklet, provided during exams, should be your constant companion during revision – learn what is in it and where each formula can be found so that you do not waste time in the assessment.
要在IB数学中脱颖而出，你需要的不仅仅是教科书。图形显示器（如TI-Nspire或Casio fx-CG50）是探索、检验答案以及应对允许使用计算器试题的重要工具。然而，你还必须熟练掌握分析技巧以应对不可使用计算器的试卷。IB公式手册在考试期间提供，应始终陪伴你的复习过程——熟悉其中的内容以及每个公式的位置，这样在评估中就不会浪费时间。
10. How to Use This Revision Series | 如何使用本复习系列
Each article in this series is structured around a single topic and follows the same format: a concise summary of key concepts, worked examples with step-by-step reasoning, examiner tips, and checkpoint questions. We recommend that you first read through the summary to consolidate your understanding, then attempt the checkpoint questions without referring to the solutions. After self-assessment, review your mistakes and then try the additional practice problems provided at the end. Repeat this cycle regularly, and make sure to cross-reference with the IB syllabus guide.
本系列中的每篇文章都围绕一个主题构建，并遵循相同的格式：关键概念的简明总结、配有逐步推理的范例、考官提示以及检查点问题。我们建议你先通读总结以巩固理解，然后在不看答案的情况下尝试检查点问题。自我评估后，回顾错误，再尝试末尾提供的额外练习题。定期重复这一循环，并确保经常与IB教学大纲指南对照。
11. Common Challenges and Tips | 常见挑战与建议
Many students find the jump from pre-IB to IB Mathematics demanding. One common pitfall is memorising procedures without understanding the underlying concepts – exam questions are designed to test flexibility, not rote learning. Another challenge is time pressure in Paper 1, where you must perform all algebraic manipulations by hand. To overcome this, practise writing clear, logical solutions under timed conditions. Also, do not neglect your IA; start early and keep a journal of ideas. Consistent, active revision over two years is far more effective than last-minute cramming.
许多学生发现从IB预备课程到IB数学的跳跃要求很高。常见的误区之一是只记步骤而不理解背后的概念——试题旨在考查思维的灵活性，而非死记硬背。另一个挑战是Paper 1的时间压力，你必须徒手完成所有代数运算。为了克服这一点，要在限时条件下练习书写清晰、有逻辑的解题过程。同时，不要忽视IA；尽早开始并保持记录想法的日志。在两年的时间里进行持续、主动的复习远比最后时刻突击有效得多。
12. Final Words of Encouragement | 最后的鼓励
IB Mathematics is challenging, but it is also a journey that develops thinking skills you will carry for life. Every mistake is a learning opportunity; every difficult problem solved strengthens your mathematical intuition. This revision series is here to guide you step by step. Work steadily, ask questions, and believe in your ability to improve. We are confident that with dedication and the right resources, you can achieve an excellent result and, more importantly, a genuine appreciation for the beauty of mathematics.
IB数学充满挑战，但它也是一段发展思维技能的旅程，会让你受益终身。每个错误都是一个学习的机会；每解出一道难题都会增强你的数学直觉。这份复习系列将一步一步引导你。稳步前进，提出问题，并相信自己有进步的能力。我们相信，凭借投入和正确的资源，你不仅能取得优异的成绩，更重要的是，能够真正欣赏数学之美。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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