📚 Brief Contents of IB Mathematics | IB数学简要内容
The International Baccalaureate (IB) Diploma Programme offers two distinct mathematics courses that equip students with the analytical skills and conceptual understanding needed for university study and beyond. This article provides a concise yet comprehensive overview of the syllabus contents, highlighting the key topics, assessment structure, and the differences between the courses.
国际文凭（IB）大学预科项目提供两门不同的数学课程，旨在培养学生进入大学及未来所需的各项分析技能与概念理解。本文对教学大纲内容进行简明而全面的介绍，重点梳理关键主题、评估结构以及两门课程之间的差异。
1. Introduction to the IB Mathematics Framework | IB数学课程框架简介
The IB mathematics curriculum is designed to cater to a wide range of student interests and abilities. It focuses on developing mathematical knowledge, logical reasoning, and problem-solving skills through real-world applications and theoretical exploration.
IB数学课程旨在满足不同学生的兴趣与能力需求。课程着重通过现实世界应用与理论探索来发展数学知识、逻辑推理能力以及问题解决技能。
Students must choose one mathematics course from two available options: Analysis and Approaches (AA) or Applications and Interpretation (AI). Both courses are offered at Standard Level (SL) and Higher Level (HL), giving four possible pathways.
学生必须从两门可用课程中选择一门：分析与方法（AA）或应用与解释（AI）。两门课都设有标准水平（SL）和高级水平（HL），因此共有四条可选路径。
2. Two Distinct Courses: AA and AI | 两种不同课程：分析与方法（AA）与应用和解释（AI）
Analysis and Approaches (AA) places a strong emphasis on algebraic manipulation, formal proof, and calculus. It is ideal for students who enjoy abstract reasoning and intend to pursue mathematics, physics, or engineering at university.
分析与方法（AA）高度重视代数运算、形式化证明以及微积分。该课程非常适合喜欢抽象推理并计划在大学攻读数学、物理或工程专业的学生。
Applications and Interpretation (AI) focuses on modelling, statistics, and the use of technology to solve real-world problems. It suits learners who are interested in applying mathematics to economics, social sciences, or natural sciences with a data-driven approach.
应用与解释（AI）侧重于建模、统计以及运用技术解决现实问题。它适合那些有兴趣将数学以数据驱动的方式应用于经济学、社会科学或自然科学的学生。
3. Levels: Standard Level and Higher Level | 水平：标准水平（SL）与高级水平（HL）
SL courses cover foundational mathematical concepts and require 150 teaching hours. They provide a solid grounding for university programmes that do not demand a heavy mathematical specialisation.
标准水平（SL）课程覆盖基础的数学概念，需要150学时。它们为不要求深厚数学专门化的大学课程提供扎实的基础。
HL courses extend the SL content with additional depth and breadth, requiring 240 teaching hours. HL students study more advanced topics such as complex numbers, vectors in three dimensions, or further calculus techniques, depending on the course chosen.
高级水平（HL）课程在SL基础上加深加广，需要240学时。HL学生会学习更进阶的主题，例如复数、三维向量或进一步的微积分技巧，具体内容取决于所选课程。
4. Core Syllabus Topics Overview | 核心教学大纲主题概述
Both AA and AI share five overarching syllabus strands: Number and Algebra, Functions, Geometry and Trigonometry, Statistics and Probability, and Calculus. The relative weighting and depth of treatment differ significantly between courses and levels.
分析与方法和应用与解释共享五个大纲主题：数与代数、函数、几何与三角、统计与概率以及微积分。各主题的相对权重和处理深度在不同课程和水平之间有显著差异。
The table below summarises the recommended teaching hours allocated to each strand for SL and HL in both courses. These allocations guide the depth of study and influence the style of exam questions.
下表汇总了在两门课程的SL和HL中每个主题的推荐学时分配。这些分配指导着学习的深度并影响考题风格。
| Strand / 主题 | AA SL | AA HL | AI SL | AI HL |
|---|---|---|---|---|
| Number and Algebra / 数与代数 | 19 | 39 | 16 | 29 |
| Functions / 函数 | 21 | 32 | 31 | 42 |
| Geometry and Trigonometry / 几何与三角 | 25 | 51 | 18 | 46 |
| Statistics and Probability / 统计与概率 | 27 | 33 | 39 | 52 |
| Calculus / 微积分 | 28 | 55 | 19 | 41 |
5. Number and Algebra | 数与代数
Arithmetic and geometric sequences and series form a core part of both courses. In AA SL and HL, students learn to apply sigma notation and derive sum formulas; AI places these concepts in financial contexts such as compound interest and loan repayments.
等差数列和等比数列及其级数是两门课程的核心内容。在AA SL和HL中，学生学会使用求和符号并推导求和公式；AI课程则将这些概念置于复利和贷款偿还等金融情境中。
Exponents and logarithms are treated algebraically in AA, with emphasis on solving equations analytically. In AI, the focus shifts to modelling exponential growth and decay using technology. Complex numbers (AA HL) and matrices (AI HL) are exclusive extension topics.
指数与对数在AA中以代数方式处理，强调解析求解方程；AI则将重点转移到用技术建模指数增长与衰减。复数（AA HL）和矩阵（AI HL）是各自独有的拓展主题。
6. Functions | 函数
The study of functions includes linear, quadratic, cubic, exponential, logarithmic, and trigonometric types. AA students often explore transformations, domain restrictions, and the formation of composite and inverse functions through algebraic manipulation.
函数的学习包括线性、二次、三次、指数、对数和三角函数等类型。AA学生通常通过代数操作探究变换、定义域限制以及复合函数和反函数的构造。
AI students concentrate on graphical interpretation and modelling: extracting parameters from data, fitting models using regression, and evaluating the appropriateness of a chosen function. Rational functions appear more prominently in AA HL.
AI学生集中于图形解释与建模：从数据中提取参数、运用回归拟合模型并评估所选函数的适宜性。有理函数在AA HL中更加突出。
7. Geometry and Trigonometry | 几何与三角学
Circular functions, trigonometric identities, and the solution of triangles are covered in all courses. The unit circle approach and radian measure are fundamental. AA extends this to proving identities such as compound angle formulas and double angle formulas.
圆函数、三角恒等式和三角形求解在所有课程中均有涉及。单位圆方法和弧度制是基础。AA课程在此基础上延伸到证明诸如复合角公式和二倍角公式等恒等式。
Vectors are introduced in both SL courses and expanded at HL. AA HL includes vector equations of lines and planes, scalar and vector products, and geometric applications. AI HL uses vectors to model motion and solve problems on neighbouring topics like kinematics and forces.
向量在两门SL课程中都会介绍，并在HL中扩展。AA HL包括直线和平面的向量方程、标量积和向量积以及几何应用。AI HL则利用向量对运动进行建模并解决运动学和力等相关问题。
8. Statistics and Probability | 统计与概率
Descriptive statistics, basic probability rules, and discrete random variables are common starting points. Both courses use probability distributions; the normal and binomial distributions feature in SL, while AA HL adds the Poisson distribution.
描述性统计、基本概率规则和离散随机变量是共同的起点。两门课程都使用概率分布；正态分布和二项分布出现在SL中，AA HL则增加了泊松分布。
AI places a much heavier emphasis on statistical techniques: hypothesis testing, chi-squared tests, t-tests, and the use of p-values are explored in depth. AI HL further covers non-linear regression, Markov chains, and the central limit theorem in a practical setting.
AI极其强调统计技巧：深入探究假设检验、卡方检验、t检验和p值的运用。AI HL还进一步以实用方式涵盖非线性回归、马尔可夫链和中心极限定理。
9. Calculus | 微积分
Differentiation and integration are introduced conceptually through limits of gradients and areas. AA students learn to differentiate from first principles and master a wide range of derivative and antiderivative rules, including the chain rule, product rule, and integration by substitution.
微分和积分通过梯度极限和面积概念进行引入。AA学生学会从基本原理求导并掌握一系列求导和反导法则，包括链式法则、乘积法则和换元积分法。
HL calculus further develops techniques such as implicit differentiation, related rates, integration by parts, and solving separable differential equations. Maclaurin series are studied to approximate functions. AI HL focuses on numerical approaches, Euler’s method for differential equations, and the interpretation of phase portraits.
HL微积分进一步发展隐函数求导、相关变化率、分部积分以及求解可分离变量微分方程等技巧。学习麦克劳林级数以逼近函数。AI HL则侧重于数值方法、微分方程的欧拉方法以及对相图的解释。
10. The Internal Assessment (IA) | 内部评估（IA）
The IA is an independent mathematical exploration that accounts for 20% of the final grade. Students choose a topic of personal interest, apply their mathematical skills, and produce a written report of approximately 12-20 pages.
内部评估是一项独立的数学探究，占最终成绩的20%。学生选择一个个人感兴趣的主题，运用数学技能，并撰写一份大约12至20页的书面报告。
The exploration should demonstrate logical reasoning, reflection, and mathematical communication. It is assessed against five criteria: Presentation, Mathematical Communication, Personal Engagement, Reflection, and Use of Mathematics.
该探究应展示逻辑推理、反思和数学交流。它依据五个标准进行评估：呈现、数学交流、个人投入、反思和数学运用。
11. External Assessment Structure | 外部评估结构
AA SL has two examination papers: Paper 1 (no calculator) and Paper 2 (calculator allowed). AI SL also has two papers, but both allow technology. HL courses have three papers, with Paper 3 consisting of extended problem-solving tasks.
AA SL有两份试卷：试卷一（禁用计算器）和试卷二（允许使用计算器）。AI SL也有两份试卷，但两份均允许使用技术工具。HL课程有三份试卷，试卷三由拓展性问题解决任务组成。
Question styles vary: AA papers often feature short, structured algebraic problems, whereas AI includes longer contextual questions requiring interpretation of data and models. The grade boundaries and weightings differ, but all paths assess the same command terms and cognitive levels.
题目风格各异：AA试卷通常包含简短的、结构化的代数问题，而AI包含需要解释数据和模型的长情境题。等级边界和权重各不相同，但所有路径都评估相同的指令术语和认知层次。
12. Key Concepts and Skills | 关键概念与技能
Across all courses, students develop skills in inquiry, logical deduction, and abstraction. The fundamental concepts of relationships, approximation, representation, and systems permeate every topic and encourage students to see connections across disciplines.
通过所有课程，学生培养探究、逻辑演绎和抽象化的技能。关系、近似、表示和系统等基本概念贯穿每个主题，促使学生看到跨学科的联系。
Technology plays a distinct role: in AI, graphical display calculators and software are essential for exploration and modelling, while in AA, the emphasis is more on analytical reasoning, though technology is still used to check results and visualise graphs.
技术发挥着独特的作用：在AI中，图形显示计算器和软件对探索与建模至关重要；而在AA中，更强调分析推理，但仍然使用技术来核对结果和可视化图形。
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