📚 Detailed Contents of IB Mathematics | IB 数学详细内容
The International Baccalaureate (IB) Mathematics curriculum provides a rigorous, coherent framework that develops students' mathematical knowledge, understanding, and skills. The two current courses—Analysis and Approaches (AA) and Applications and Interpretation (AI)—each offer standard level (SL) and higher level (HL) pathways, allowing students to tailor their studies to future aspirations in mathematics, science, engineering, social sciences, or the arts. This article presents a detailed breakdown of the syllabus contents, assessment components, and essential study strategies to help learners navigate the program with confidence.
国际文凭（IB）数学课程提供了一个严谨、连贯的框架，旨在发展学生的数学知识、理解与技能。当前的两门课程——分析与方法（AA）与应用和解释（AI）——均设有标准水平（SL）和高级水平（HL）路径，让学生能够根据未来在数学、科学、工程、社会科学或艺术等领域的志向量身定制学习。本文详细分解了教学大纲内容、评估组成部分以及关键学习策略，帮助学习者自信地驾驭这一课程。
1. IB Mathematics Curriculum Framework | IB 数学课程框架
The IB Mathematics curriculum is built around five core topic areas, though the depth and emphasis vary between the two courses. Both AA and AI share a common set of aims: to enable students to enjoy and appreciate mathematics, to develop logical and creative thinking, and to apply skills in real-world contexts. The AA course focuses on analytic approaches, formal proof, and abstract problem-solving, while the AI course emphasises technology, modelling, and statistical techniques. The framework divides content into SL (150 teaching hours) and HL (240 hours), with an additional internal assessment (IA) exploration component of 30 hours.
IB 数学课程围绕五个核心主题领域构建，但深度和侧重点因课程而异。AA 和 AI 课程都拥有一套共同的目标：使学生能够欣赏并喜爱数学，发展逻辑与创造性思维，以及在现实世界情境中应用技能。AA 课程侧重于分析方法、形式证明与抽象问题解决，而 AI 课程则强调技术、建模与统计技巧。该框架将内容划分为 SL（150 教学小时）和 HL（240 小时），另有 30 小时的内部评估（IA）探索部分。
2. Number and Algebra | 数与代数
This area forms the algebraic backbone for all IB Mathematics courses. At SL, both AA and AI cover scientific notation, arithmetic and geometric sequences and series, laws of exponents, logarithms (including common and natural logs), simple exponential equations, and the binomial theorem with positive integer exponents. AI SL additionally includes financial mathematics: compound interest, depreciation, and amortization schedules, often requiring the use of technology. AA SL extends into rational exponents, proofs by induction (HL only, but foundation is laid), and partial fractions (HL). At HL, both courses deepen algebraic manipulation. AA HL explores complex numbers, De Moivre's theorem, roots of unity, systems of linear equations, polynomial theorems (remainder, factor), and the binomial theorem for rational exponents. AI HL covers eigenvalues and eigenvectors, matrix algebra, transformations, and more advanced financial models.
此领域构成了所有 IB 数学课程的代数基础。在 SL 级别，AA 与 AI 均涵盖科学记数法、算术与几何数列及级数、指数律、对数（包括常用对数和自然对数）、简单指数方程以及正整数指数的二项式定理。AI SL 还额外包括金融数学：复利、折旧与摊还表，通常需要借助技术工具。AA SL 则延伸到有理指数、数学归纳法证明（仅 HL，但铺陈基础）和部分分式（HL）。在 HL 级别，两门课程都深化了代数操作。AA HL 探讨复数、棣莫弗定理、单位根、线性方程组、多项式定理（余式、因式）以及有理指数的二项式定理。AI HL 则覆盖特征值与特征向量、矩阵代数、变换以及更高级的金融模型。
3. Functions | 函数
xplored, along with their graphs and applications. AA SL includes rational functions of the form f(x)= (ax+b)/(cx+d) and introduces the concept of limits informally, while AI SL emphasises modelling with sinusoidal, quadratic, cubic, and logistic functions using graphing technology. At HL, AA advances significantly into reciprocal trigonometric functions, modulus functions, and detailed analysis of asymptotes and limits, culminating in continuity and differentiability. AI HL extends piecewise models, scaling, and non-linear regression to complex real-world data sets, and incorporates the use of matrices in transformations.
函数是整个大纲中一个统一的主题。两门课程的 SL 学生都学习函数的概念、定义域、值域、复合函数与反函数，以及图形的变换（平移、反射、伸缩）。他们还探讨二次函数、指数函数和对数函数及其图形和应用。AA SL 涵盖了形如 f(x)= (ax+b)/(cx+d) 的有理函数，并引入极限的非正式概念，而 AI SL 则强调使用图形技术对正弦、二次、三次和逻辑函数进行建模。在 HL 层级，AA 显著推进到倒数三角函数、含绝对值的函数，以及渐近线和极限的详细分析，最终导向连续性和可微性。AI HL 则将分段模型、缩放和非线性回归扩展到复杂的真实数据集，并将矩阵纳入变换中。
4. Geometry and Trigonometry | 几何与三角学
This topic includes both classical and analytical approaches. AA SL explores the distance between two points, midpoints, equations of lines, radian measure, the unit circle, trigonometric identities (Pythagorean, double angle), and solving trigonometric equations in a finite interval. Bearings and right-angled trigonometry appear in both courses, but AI SL also applies the sine and cosine rules, area of a triangle, and three-dimensional geometry—often within the context of surveying or navigation. AA HL extends into reciprocal trigonometric functions, compound angle identities, trigonometric proofs, and vector geometry: scalar and vector products, equations of planes, intersections of lines and planes, and distances. AI HL covers vector theory and uses it for kinematics and forces, adjacency matrices, and graph theory, including Eulerian trails, Hamiltonian paths, and algorithms on networks (Kruskal's, Dijkstra's).
该主题涵盖经典与解析两种方法。AA SL 探讨两点间的距离、中点、直线方程、弧度制、单位圆、三角恒等式（毕达哥拉斯恒等式、倍角公式）以及在有限区间内解三角方程。方位角和直角三角学在两门课程中都会出现，但 AI SL 还会应用正弦、余弦定理、三角形面积以及三维几何——通常结合测量或导航的情境。AA HL 延伸到倒数三角函数、复角恒等式、三角证明以及向量几何：标量积与向量积、平面方程、直线与平面的交点以及距离。AI HL 涵盖向量理论并将其用于运动学和力，邻接矩阵以及图论，包括欧拉路径、哈密顿路径和网络算法（克鲁斯卡尔算法、迪杰斯特拉算法）。
5. Statistics and Probability | 统计与概率
This is the most differentiating area between AA and AI. AA SL covers descriptive statistics (mean, median, mode, IQR), linear correlation, basic probability, Venn diagrams, tree diagrams, discrete random variables, and the binomial and normal distributions. AA HL adds Bayes' theorem, continuous random variables, and the Poisson distribution, with a focus on analytical derivations. By contrast, AI SL and HL both place heavy emphasis on technology-enabled statistics: chi-squared tests for independence and goodness of fit, t-tests, linear regression, and calculation of confidence intervals. AI HL incorporates multiple linear regression, ANOVA, non-linear modelling, and randomization tests. Throughout AI, students learn to interpret output from statistical software, critique sampling methods, and assess validity of models—skills directly applicable to social sciences and experimental research.
这是 AA 与 AI 之间差异最大的领域。AA SL 涵盖描述性统计（平均值、中位数、众数、四分位距）、线性相关、基础概率、文氏图、树形图、离散随机变量以及二项分布和正态分布。AA HL 增加了贝叶斯定理、连续随机变量和泊松分布，着重于解析推导。相比之下，AI SL 和 HL 都高度重视技术驱动的统计：用于独立性和拟合优度的卡方检验、t 检验、线性回归以及置信区间的计算。AI HL 则融合了多元线性回归、方差分析、非线性建模和随机化检验。在整个 AI 课程中，学生学习解释统计软件的输出、评价抽样方法并评估模型的有效性——这些技能直接适用于社会科学和实验研究。
6. Calculus | 微积分
Calculus is a cornerstone of both courses, though the approach differs. AA SL introduces limits conceptually, differentiation from first principles, rules of differentiation (power, product, quotient, chain), equations of tangents and normals, increasing/decreasing functions, and integration as anti-differentiation. Definite integrals and areas under curves are covered, along with kinematic applications. AA HL expands into implicit differentiation, optimisation, related rates, l'Hopital's rule, integration by substitution and parts, volumes of revolution, differential equations (separable and first-order linear), and Maclaurin series. AI SL develops calculus with a strong modelling emphasis: polynomial, exponential, and logistic growth models, rates of change in context, and numerical integration using technology. AI HL adds partial derivatives, gradient vectors, second derivatives for concavity, and solutions of coupled differential equations using phase portraits and Euler's method—tools essential for applied fields and engineering.
微积分是两门课程的基石，尽管处理方法不同。AA SL 从概念上引入极限，通过第一性原理求导，涵盖求导法则（幂法则、积法则、商法则、链式法则）、切线与法线方程、函数的增减性以及作为逆微分的积分。内容还包括定积分、曲线下的面积以及运动学应用。AA HL 则扩展到隐函数求导、最优化、相关变化率、洛必达法则、分部积分法和换元积分法、旋转体体积、微分方程（可分离和线性一阶）以及麦克劳林级数。AI SL 以强烈的建模视角发展微积分：多项式、指数和逻辑增长模型，情境中的变化率，以及使用技术进行数值积分。AI HL 增添了偏导数、梯度向量、判断凹凸性的二阶导数，以及利用相图和欧拉方法求解耦合微分方程组——这些工具对于应用领域和工程学至关重要。
7. Additional HL Topics for AA | 分析与方法高级补充主题
The extra 90 hours in AA HL bring depth and rigour. Proofs by induction, complex numbers in the complex plane, and further trigonometric identities join the core. Students study continuity and differentiability in formal terms, advanced applications of calculus (including related rates and optimisation), and entire sequences and series with absolute convergence tests. In vectors, lines and planes are extended to intersection calculations, and vector proofs are required. The syllabus also introduces discrete mathematics options such as proof techniques (direct, contradiction, contrapositive) and the basics of number theory. These demanding topics prepare students for university mathematics, physics, and engineering programs.
AA HL 中额外的 90 个小时赋予了课程深度和严谨性。归纳法证明、复平面上的复数以及更多三角恒等式被纳入核心。学生将用形式化术语学习连续性和可微性、微积分的高级应用（包括相关变化率和最优化），以及带有绝对收敛检验的完整数列与级数。在向量方面，直线和平面的内容扩展到交点计算，并要求进行向量证明。大纲还引入了离散数学选修主题，例如证明技巧（直接证明、反证法、逆否命题）和基础数论。这些要求很高的主题为学生进入大学数学、物理和工程学专业做好了准备。
8. Additional HL Topics for AI | 应用与解释高级补充主题
AI HL focuses on sophisticated modelling and computational tools. Content includes further matrix algebra (determinants, inverse of a 3x3 matrix, solution of linear systems), graph theory (trees, minimum spanning trees, Chinese postman problem, critical path analysis), and advanced statistics (non-linear regression, multiple linear regression with confidence intervals and hypothesis tests, ANOVA). Calculus extends to second-order differential equations, phase portraits, and numerical methods. The course often uses technology such as GeoGebra and statistical packages to explore large data sets, perform simulations, and validate models. These skills align directly with careers in data science, economics, and social research.
AI HL 侧重于复杂的建模和计算工具。内容包括进一步的矩阵代数（行列式、3×3 矩阵的逆、线性方程组的求解），图论（树、最小生成树、中国邮递员问题、关键路径分析），以及高级统计学（非线性回归、带置信区间和假设检验的多元线性回归、方差分析）。微积分扩展至二阶微分方程、相图和数值方法。课程常用 GeoGebra 和统计软件包等工具探索大型数据集、进行模拟并验证模型。这些技能与数据科学、经济学和社会研究等职业直接契合。
9. Internal Assessment (IA) Components | 内部评估组成部分
The mathematics exploration is a 12–20 page written report (SL/HL) accounting for 20% of the final grade. Students choose a topic of personal interest, apply mathematical concepts, and produce a coherent investigation. The IA is internally assessed by teachers and externally moderated. AA students tend to focus on theoretical aspects, proofs, or pure mathematical conjectures, while AI students often explore statistical modelling, financial analysis, or the mathematics behind games and design. Criteria include presentation, mathematical communication, personal engagement, reflection, and the use of mathematics. This component rewards creativity and genuine inquiry.
数学探索是一份 12–20 页的书面报告（SL/HL），占最终成绩的 20%。学生选择一个个人感兴趣的题目，应用数学概念并撰写连贯的调查研究。IA 由教师内部评分，外部进行审核。AA 学生倾向于围绕理论方面、证明或纯数学猜想进行研究，而 AI 学生则常探索统计建模、金融分析或游戏和设计背后的数学原理。评分标准包括表述、数学交流、个人投入、反思以及数学的运用。这一部分奖励创造力和真正的探究精神。
10. Assessment Overview and Tips | 评估概述与备考建议
Both SL and HL have three external papers, but their structure differs. For AA SL: Paper 1 (no calculator, 1.5h), Paper 2 (calculator allowed, 1.5h), and Paper 3 is not included; instead two papers only, but in fact AA SL has Papers 1 & 2. Wait, careful: AA SL: Paper 1 (no calculator, 1h30, 80 marks, 40%) and Paper 2 (GDC required, 1h30, 80 marks, 40%). AA HL: Paper 1 (no calc, 2h, 110m, 30%), Paper 2 (GDC, 2h, 110m, 30%), Paper 3 (no calc, 1h, 55m, 20%). AI SL: Paper 1 (GDC, 1h30, 80m, 40%), Paper 2 (GDC, 1h30, 80m, 40%). AI HL: Paper 1 (GDC, 2h, 110m, 30%), Paper 2 (GDC, 2h, 110m, 30%), Paper 3 (GDC, 1h, 55m, 20%). Tips: master the use of your graphing display calculator (GDC) for AI papers, practice non-calculator algebraic fluency for AA Paper 1, and complete past papers under timed conditions. Regularly review concepts using spaced repetition, and ensure you are comfortable writing clear mathematical explanations in English.
SL 和 HL 都有三份外部试卷，但结构不同。具体信息：AA SL：试卷一（不许用计算器，1 小时 30 分，80 分，占 40%）和试卷二（需用图形计算器，1 小时 30 分，80 分，40%）。AA HL：试卷一（无计算器，2 小时，110 分，30%），试卷二（图形计算器，2 小时，110 分，30%），试卷三（无计算器，1 小时，55 分，20%）。AI SL：试卷一（图形计算器，1 小时 30 分，80 分，40%），试卷二（图形计算器，1 小时 30 分，80 分，40%）。AI HL：试卷一（图形计算器，2 小时，110 分，30%），试卷二（图形计算器，2 小时，110 分，30%），试卷三（图形计算器，1 小时，55 分，20%）。备考建议：熟练掌握图形计算器（GDC）对 AI 试卷至关重要，针对 AA 试卷一练习无计算器下的代数流畅性，并在限时条件下完成往年真题。定期利用间隔重复复习概念，并确保能够用英语写出清晰的数学解释。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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