📚 IB AQA Maths: Mind Map Quick Revision | IB AQA 数学：思维导图速记
Mind maps turn scattered facts into a web of connected ideas, making revision faster and deeper. By arranging topics like algebra, functions, trigonometry and calculus around a central concept, you visualise how core principles link together, which is exactly what IB Diploma and AQA A-level mathematics demand. This article provides a ready-to-use mind map walkthrough, with each branch explained in clear English and Chinese so you can review efficiently and retain key relationships for the exam.
思维导图能将零散的知识点编织成相互联系的知识网络，让复习更快、理解更深。围绕核心概念，把代数、函数、三角和微积分等主题分层展开，你就能直观地看到基本原理之间的关联——这正是 IB 文凭课程与 AQA A-level 数学所要求的综合能力。本文提供了一套可直接运用的思维导图讲解，每个分支都用清晰的英文和中文加以说明，帮助你高效复习，牢牢记住考试中的关键纽带。
1. Building a Maths Mind Map | 构建数学思维导图
A maths mind map starts with a central node – the main topic, for example 'Functions' and use symbols such as f(x), √, ∫, θ to compress information. Always add a small summary box at the edge of a branch to hold a critical formula or a common mistake.
一张数学思维导图从中心节点开始——比如“函数”。从中心向外辐射出主要分支：定义、类型、性质、图像、变换以及关键方程。用颜色标注每个分支（例如微分规则用红色，积分规则用蓝色）能够激发视觉记忆。每个分支上只写一个词或短语，利用 f(x)、√、∫、θ 等符号压缩信息。在分支的末端加上一个总结小框，记录一条关键公式或常见错误。
- Central topic: Function families, Calculus, Probability, etc.
- 中心主题：函数族、微积分、概率等
- First-level branches: definition, notation, graphical features, algebraic rules.
- 一级分支：定义、符号、图像特征、代数规则
2. Number and Algebra Core | 数与代数核心
Number systems scaffold all of mathematics. The mind map should separate natural, integer, rational, real and complex numbers, each with their own closure properties. Indices, surds and logarithms form a parallel branch because they connect multiplication and division to addition and subtraction. Laws of exponents are compact enough to sit in a single formula box: am × an = am+n, (am)n = amn, a−n = 1/an.
数系是整个数学的支架。思维导图应将自然数、整数、有理数、实数与复数分开，每种数都注明封闭性质。指数、根式和对数构成平行的分支，因为它们把乘除与加减联系起来。指数律非常简洁，适合放入一个公式框：am × an = am+n，(am)n = amn，a−n = 1/an。
heir own sub-branch: factorisation, completing the square, the quadratic formula x = [−b ± √b² − 4ac . Linear inequalities and modulus inequalities are best placed alongside equations, with clear rules for reversing the sign when multiplying by a negative.
二次方程应拥有独立的分支：因式分解、配方法、求根公式 x = [−b ± √b² − 4ac] ⁄ 2a，以及判别式 Δ = b² − 4ac。线性不等式和绝对值不等式最好与方程并列，并注明乘以负数时不等号要掉头的规则。
| Number sets | ℝ (reals) ⊃ ℚ (rationals) ⊃ ℤ (integers) ⊃ ℕ (naturals) |
| 数集 | 实数 ⊃ 有理数 ⊃ 整数 ⊃ 自然数 |
| Key surd rule | √ab = √a × √b, but √a+b ≠ √a + √b |
| 根式关键规则 | √ab = √a × √b，但 √a+b ≠ √a + √b |
3. Functions and Their Graphs | 函数与其图像
unctions f(g(x)) and inverse functions f⁻¹(x) are linked by the reflection property in the line y = x.
先从定义开始：一个映射 f: X → Y，每个输入刚好对应一个输出。接着思维导图分支出单射、多对一、满射与普通函数。定义域和值域占据中心位置，周围标注常见限制，如分母 ≠ 0 和被开方数 ≥ 0。复合函数 f(g(x)) 与反函数 f⁻¹(x) 通过关于直线 y = x 的对称性联系在一起。
d reflections y = −f(x), y = f(−x). A quick reference table pins these patterns to memory.
图像变换构成一个形象的分支：y = f(x) + a（上下平移），y = f(x + a)（左右平移），y = a f(x)（纵向伸缩），y = f(ax)（横向伸缩），以及反射 y = −f(x)、y = f(−x)。一张速查表能把这些模式固定在记忆中。
y = 2f(3x − 1) + 4 orizontal stretch scale factor 1/3, shift right 1/3, then vertical stretch ×2, shift up 4.
y = 2f(3x − 1) + 4 → 先处理内部：横向伸缩 1/3 倍，右移 1/3，然后纵向拉伸 2 倍，上移 4。
4. Geometry and Trigonometry | 几何与三角学
− y₁ = m(x − x₁). Parallel and perpendicular lines add the rules m₁ = m₂ and m₁ × m₂ = −1.
解析几何为距离、中点和斜率搭建了一条分支。距离公式 √[(x₂−x₁)² + (y₂−y₁)²] 与斜率 m = (y₂−y₁)/(x₂−x₁) 进一步导出直线方程：y − y₁ = m(x − x₁)。平行与垂直直线的规则 m₁ = m₂ 和 m₁ × m₂ = −1 紧挨着摆放。
ules extend the map to any triangle: a² = b² + c² − 2bc cos A, and a/sin A = b/sin B = c/sin C.
三角学分成直角三角形定义（sin = 对/斜，cos = 邻/斜，tan = 对/邻）和单位圆两部分。三个恒等式 sin²θ + cos²θ = 1、tanθ = sinθ/cosθ 和 1 + tan²θ = sec²θ 放进高亮框里。弧度制将弧长 s = rθ 与扇形面积 ½r²θ 连接起来。正弦定理和余弦定理将导图拓展到任意三角形：a² = b² + c² − 2bc cos A，a/sin A = b/sin B = c/sin C。
5. Differentiation Roadmap | 微分路线图
instantaneous rate of change. The mind map begins with the first-principles definition: f'(x) = limh→0 t around this core. Each rule comes with a micro-example: d/dx (xⁿ) = nxⁿ⁻¹, d/dx (uv) = u'v + uv', d/dx (u/v) = (u'v − uv')/v², and d/dx f(g(x)) = f'(g(x)) g'(x).
微分是测量瞬时变化率的分支。思维导图从第一性原理定义开始：f'(x) = limh→0 [f(x+h) − f(x)]/h。幂法则、乘积法则、商法则和链式法则环绕着这个核心。每个法则附带一个微型例子：d/dx (xⁿ) = nxⁿ⁻¹，d/dx (uv) = u'v + uv'，d/dx (u/v) = (u'v − uv')/v²，以及 d/dx f(g(x)) = f'(g(x)) g'(x)。
Special derivatives of ex nd normals, increasing/decreasing functions and stationary points. The second derivative f'' nd point of inflection. Optimisation problems sit at the end of this branch, summarised by 'set f'(x) = 0, solve, check f''(x)'.
ex、ln x、sin x、cos x、tan x 的特殊导数占据一个专属气泡。应用部分分支到切线法线、单调性与驻点。二阶导数 f''(x) 决定凹凸性和拐点。最优化问题坐落在这一分支的末端，用一句总结概括：“令 f'(x) = 0，求解，检验 f''(x)”。
| Function | Derivative |
| ex | ex |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
6. Integration Techniques | 积分技巧
tion and calculates area under a curve. The mind map anchors on the fundamental theorem: ∫ab ndefinite integration rules mirror differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ ex dx = ex + C, ∫ 1/x dx = ln|x| + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.
积分是微分的逆运算，用于计算曲线下的面积。思维导图以微积分基本定理为核心：∫ab f(x) dx = F(b) − F(a)。不定积分的规则与微分镜像对称：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，∫ ex dx = ex + C，∫ 1/x dx = ln|x| + C，∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。
for rational expressions. Definite integrals add the concept of area between curves: A = ∫ab ications link displacement, velocity and acceleration: s(t) = ∫ v(t) dt, v(t) = ∫ a(t) dt.
积分技巧以子分支的形式生长：换元法（选取 u = 内部函数），分部积分法 ∫ u dv = uv − ∫ v du，以及有理式所用部分分式。定积分引入曲线间的面积概念：A = ∫ab [f(x) − g(x)] dx。运动学应用将位移、速度和加速度串起来：s(t) = ∫ v(t) dt，v(t) = ∫ a(t) dt。
7. Sequences and Series | 数列与级数
Arithmetic and geometric sequences form two primary arms. For an arithmetic sequence, un = a + (n−1)d, Sn = n/2 [2a + (n−1)d]. For geometric, un = arn−1, Sn = a(1 − rⁿ)/(1 − r) and the infinite sum S∞ = a/(1 − r) for |r| < 1. The mind map then connects to sigma notation Σ and the binomial expansion (a + b)n = Σnk=0 (n choose k) an−k bk.
等差数列和等比数列构成两条主要臂膀。等差数列：un = a + (n−1)d，Sn = n/2 [2a + (n−1)d]。等比数列：un = arn−1，Sn = a(1 − rⁿ)/(1 − r)，当 |r| < 1 时无穷和 S∞ = a/(1 − r)。思维导图接着连接求和符号 Σ 和二项式展开 (a + b)n = Σnk=0 (n k) an−k bk。
Convergence and divergence tests are summarised by a quick checklist: ratio test, comparison test and the necessary condition limn→∞ an s work often intertwines with proof by induction, which merits a linked bubble on the map.
收敛与发散检验汇总为一份快速清单：比值判别法、比较判别法、以及必要条件 limn→∞ an = 0。在 IB 和 AQA 背景下，级数专题常常与数学归纳法证明交织，因此导图上需要一个连线气泡。
8. Statistics and Data Handling | 统计与数据处理
The statistics mind map places measures of central tendency (mean, median, mode) and measures of spread (range, interquartile range, variance, standard deviation) at its heart. Frequency tables and cumulative frequency graphs support the median and quartile calculations. Box-and‑whisker plots condense the five-number summary visually.
统计思维导图把中心趋势度量（均值、中位数、众数）和离散程度度量（极差、四分位距、方差、标准差）放在核心位置。频数表和累积频率图支撑着中位数和四分位数的计算。箱线图将五数概括形象地浓缩呈现。
th the residual = observed − predicted. Remember that correlation does not imply causation – a caution node should be added.
双变量数据分支到散点图、相关性（皮尔逊 r）和线性回归。回归直线 y = a + bx 通过最小二乘法获得，残差 = 观测值 − 预测值。别忘了“相关不蕴涵因果”的警示节点，一定要加上去。
Probability distributions – the binomial B(n, p) and the normal N(μ, σ²) – CD, and inverse normal for finding boundaries. The central limit theorem underpins sampling, bridging data to probability.
概率分布——二项分布 B(n, p) 和正态分布 N(μ, σ²)——作为独立的叶片连接上来。正态分布技能包含标准化 z = (x − μ)/σ、正态 CD 的使用，以及用逆正态求边界。中心极限定理是抽样的基石，把数据和概率桥接起来。
9. Probability Concepts | 概率概念
ability starts with the basic formula P(A) = n(A) / n(S), leading to the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Conditional probability and tree diagrams form the next layer: P(A|B) = P(A ∩ B) / P(B). The multiplication rule for independent events P(A ∩ B) = P(A) × P(B) is placed right next to the conditional rule to highlight the contrast.
概率从基本公式 P(A) = n(A) / n(S) 开始，延伸出加法法则 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。条件概率与树状图构成下一层：P(A|B) = P(A ∩ B) / P(B)。独立事件的乘法法则 P(A ∩ B) = P(A) × P(B) 紧挨着条件法则放置，以突出差异。
Mutually exclusive events (P(A ∩ (X²) − [E(X)]² appear as a practical sub-branch often tested alongside probability distributions.
互斥事件（P(A ∩ B) = 0）与独立事件相对而立。文氏图和双向表是思维导图上不可或缺的可视化工具。就 IB 和 AQA 而言，离散随机变量及其期望 E(X) = Σ x·P(X=x) 与方差 Var(X) = E(X²) − [E(X)]² 构成一条实用子分支，常与概率分布联合考查。
10. Vectors and Matrices | 向量与矩阵
Vectors branch into magnitude |a| = √x² + y², direction, and operations: addition, scalar multiplication, dot product a·b = |a||b| cos θ. The angle between vectors is derived from cos θ = (a·b) / (|a||b| tion vectors and the vector equation of a line r = a + λd are fundamental for 3D geometry.
向量分支到大小 |a| = √x² + y²、方向、以及运算：加法、标量乘法、点积 a·b = |a||b| cos θ。向量间的夹角由 cos θ = (a·b) / (|a||b|) 求得。位置向量、方向向量与直线向量方程 r = a + λd 是三维几何的基础。
g simultaneous linear equations using inverses or row reduction (Gaussian elimination) bridges algebra and matrices.
矩阵出现在一个关联节点上：阶数（行 × 列）、加法、减法、标量乘法与矩阵乘法（行乘列）。2×2 矩阵的行列式 det(A) 和逆矩阵 A⁻¹ 放在一起储存：若 A = [a b; c d]，则 det = ad − bc，A⁻¹ = (1/det) [d −b; −c a]。用逆矩阵或行化简（高斯消元法）解联立线性方程，将代数与矩阵桥接起来。
11. Exponentials and Logarithms | 指数与对数
Exponentials ax and logarithms loga inverse functions, so they share a reflection link across y = x. The natural exponential ex and natural logarithm ln x merit special treatment because of their calculus properties. Key log laws — loga(MN) = loga M + loga N, loga(M/N) = loga M − loga N, loga Mk = k loga M — are displayed in a single block.
指数 ax 和对数 loga x 互为反函数，因此它们在 y = x 两侧形成反射链接。自然指数 ex 和自然对数 ln x 因其微积分性质而值得特别对待。重要对数律——loga(MN) = loga M + loga N，loga(M/N) = loga M − loga N，loga Mk = k loga M——被醒目地放在一个信息块里。
Solving exponential equations usually involves taking logs: 2x = 5 → x = log2 ations, linking back to differential equations in the calculus section. The graph of y = ex ntal asymptote y = 0, while y = ln x passes through (1,0) with a vertical asymptote x = 0.
求解指数方程通常需要取对数：2x = 5 → x = log25 = ln5/ln2。逻辑增长与衰减模型展示了现实应用，与微积分部分的微分方程相连接。y = ex 的图像恒过 (0,1) 且有一条水平渐近线 y = 0，而 y = ln x 过 (1,0) 并有一条垂直渐近线 x = 0。
12. Putting It All Together: The Revision Mind Map | 综合运用：复习思维导图
Effective revision with mind maps requires active recall. Cover a branch and try to reproduce its sub-topics from memory, then check against the map. Use the English–Chinese paired notes in this article to build dual-language retrieval. The crossing links between branches — for instance, how differentiation of sin x yields cos x which then appears in integration — should be highlighted with arrows and a short explanatory phrase.
利用思维导图有效复习需要主动回忆。遮住一个分支，试着凭记忆复述它的子主题，再对照导图检查。用本文的英文–中文配对笔记来建立双语检索。分支之间交叉的链接——例如对 sin x 求导得到 cos x，而 cos x 又出现在积分中——应当用箭头和简短的说明句加以高亮。
Finally, customise your map with personal memory hooks, such as a silly cartoon next to 'product rule' or a colour burst around 'integration by parts'. Revise in short, timed sessions — 25 minutes map study followed by 5 minutes free recall — and watch how scattered topics merge into a powerful network of mathematical understanding, ready for any IB or AQA exam paper.
最后，用你自己的记忆钩子个性化你的导图，比如在“乘积法则”旁画个滑稽的简笔画，或在“分部积分法”周围喷一抹亮色。以定时短时段进行复习——25 分钟导图学习加 5 分钟自由回忆——你会发现零散的主题如何融合成一张强有力的数学理解网络，为任何 IB 或 AQA 试卷做好充分准备。
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