📚 Mind Map Memorisation for GCSE CIE Maths | GCSE CIE 数学：思维导图速记
Creating visual mind maps is one of the most effective ways to lock in the huge range of topics covered in GCSE CIE Mathematics. Instead of reading notes passively, you actively organise formulas, properties and problem-solving strategies around central ideas. This article shows you how to build a set of interconnected mind maps for every major area of the CIE syllabus – Core and Extended – so you can revise faster and recall more accurately in the exam.
画视觉思维导图是牢固掌握 GCSE CIE 数学庞杂知识点最有效的方法之一。与其被动翻看笔记，不如围绕核心概念主动梳理公式、性质与解题策略。本文为你展示如何为 CIE 考纲（Core 与 Extended）的每个主要领域构建一套相互关联的思维导图，让你复习更快、考试时回想更准。
1. Number and Operations: Core Concepts | 数与运算：核心概念
Start your number mind map with a central bubble labelled “Number Systems”. Branch out into natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), irrational numbers, and real numbers (ℝ). On the rational branch, link to terminating and recurring decimals. Add a sub-branch for prime numbers, listing the first few primes and the concept of prime factorisation. Include square numbers, cube numbers, and their roots. Under operations, draw branches for BIDMAS (order of operations), directed numbers rules, and standard form (a × 10ⁿ where 1 ≤ a < 10). Use colour coding: green for number types, blue for operations, red for powers and roots.
先画一个中心气泡 “Number Systems”。分支为自然数 (ℕ)、整数 (ℤ)、有理数 (ℚ)、无理数和实数 (ℝ)。在有理数分支下，连接有限小数和循环小数。另设质数分支，列出前几个质数和质因数分解概念。纳入平方数、立方数及其方根。在运算分支画出 BIDMAS（运算顺序）、有向数规则和标准形式 (a × 10ⁿ, 1 ≤ a < 10)。用颜色区分：绿色表示数类，蓝色表示运算，红色表示幂与方根。
2. Fractions, Decimals and Percentages Mind Map | 分数、小数和百分比思维导图
From a central “FDP” node, draw three main branches: Fractions, Decimals, Percentages. On each branch write conversion arrows between them: fraction to decimal (divide numerator by denominator), decimal to percentage (multiply by 100), percentage to fraction (write over 100 and simplify). Under fractions, add sub-branches for adding/subtracting (common denominator), multiplying (multiply tops and bottoms), dividing (multiply by reciprocal), and simplifying fractions. Include mixed numbers and improper fractions conversion. Under decimals, put rounding to decimal places and significant figures. Under percentages, show percentage increase/decrease (multiplier method: × (1 ± r/100)), reverse percentages, and simple/compound interest formula: A = P(1 + r/100)ⁿ for compound interest.
中心节点 “FDP” 分出三大分支：分数、小数、百分比。每个分支上写出相互转换箭头：分数转小数（分子除以分母）、小数转百分比（×100）、百分比转分数（写成分母 100 再约分）。在分数分支下，增加加减（通分）、乘法（分子乘分子、分母乘分母）、除法（乘以倒数）和约分子分支。纳入带分数与假分数互化。小数分支列出四舍五入到小数位和有效数字。百分比分支展示百分比增减（乘数法：× (1 ± r/100)）、逆推百分比、单利和复利公式：复利 A = P(1 + r/100)ⁿ。
3. Algebraic Expressions and Equations | 代数表达式与方程
Centre your map on “Algebraic Manipulation”. Branch into expanding brackets (distributive law), factorising (common factor, difference of two squares x² – y² = (x–y)(x+y), trinomials). Another main branch: solving linear equations (isolate x, balancing method) and quadratic equations (factorising, quadratic formula x = [–b ± √(b²–4ac)] / 2a, completing the square). Add a branch for simultaneous equations (substitution, elimination) and inequalities (flip sign when multiplying/dividing by a negative). For Extended students, include fractional equations, algebraic fractions, and how to factorise by grouping. A “Formulae” sub-branch can collect important rearrangements like changing the subject of a formula.
中心写 “Algebraic Manipulation”。分支出展开括号（分配律）、因式分解（公因数、平方差 x² – y² = (x–y)(x+y)、三项式）。另一大分支：解一元一次方程（移项、平衡法）和二次方程（因式分解、求根公式 x = [–b ± √(b²–4ac)] / 2a、配方法）。增加联立方程组分支（代入法、消元法）和不等式分支（乘除负数时方向反转）。对 Extended 学生，加入分式方程、代数分式和分组分解法。可设 “Formulae” 子分支，收集重要变形式如公式主项变换。
4. Sequences, Functions and Graphs | 数列、函数与图像
Create a central idea “Sequences & Functions”. From sequences, branch into term-to-term rules and position-to-term rules (nth term). Linear sequences: nth term = an + b, where a is common difference. Quadratic sequences: nth term = an² + bn + c, second difference constant = 2a. For functions, define f(x), domain and range. Show function notation and flowchart method for inverse functions f⁻¹(x). Then graph branches: plotting linear graphs (y = mx + c, m = gradient = Δy/Δx, c = y-intercept), quadratic graphs (parabola shape, turning point), reciprocal graphs y = k/x, exponential graphs y = aˣ. Add recognition of gradient and intercept from equation, parallel lines (same gradient), perpendicular lines (product of gradients = –1). Use sketched mini graphs on the mind map.
中心为 “Sequences & Functions”。数列分支分为项间规则和通项规则（第 n 项）。线性数列：第 n 项 = an + b，a 为公差。二次数列：第 n 项 = an² + bn + c，二阶差恒定 = 2a。函数部分定义 f(x)、定义域和值域。展示函数记号和逆函数 f⁻¹(x) 的流程法。图像分支：绘制直线图 (y = mx + c, m = 斜率 = Δy/Δx, c = y 截距)、二次抛物线（形状、顶点）、反比例图像 y = k/x、指数图像 y = aˣ。加入从方程识别斜率和截距、平行线（斜率相等）、垂直线（斜率乘积 = –1）。在导图上画简图。
5. Geometry: Angles, Lines and Polygons | 几何：角、线及多边形
Start with “Angle Rules” in the centre. Radiate out: angles on a straight line sum to 180°, angles at a point sum to 360°, vertically opposite angles are equal. Then parallel lines branch: alternate angles equal, corresponding angles equal, interior (co-interior) angles sum to 180°. Next, triangle properties: sum of interior angles = 180°, exterior angle = sum of opposite interior angles. Quadrilateral angle sum = 360°. Polygon branches: interior angle = (n–2)×180°/n, exterior angle = 360°/n, sum of interior angles = (n–2)×180°. For circle geometry (Extended): angle at centre = 2×angle at circumference, angle in a semicircle = 90°, opposite angles in cyclic quadrilateral sum to 180°, tangent perpendicular to radius. Add symmetry: line symmetry and rotational symmetry for common polygons.
中心写 “Angle Rules”。辐射出：直线上的角之和 180°、点周角之和 360°、对顶角相等。平行线分支：内错角相等、同位角相等、同旁内角之和 180°。然后是三角形性质：内角和 180°、外角等于两对内角之和。四边形内角和 360°。多边形分支：内角 = (n–2)×180°/n，外角 = 360°/n，内角和 = (n–2)×180°。圆几何（Extended）：圆心角 = 2×圆周角、半圆上的圆周角 = 90°、圆内接四边形对角互补、切线与半径垂直。加上对称性：常见多边形的线对称与旋转对称。
6. Mensuration: Perimeter, Area and Volume | 测量：周长、面积与体积
Build a mind map with “Mensuration” at the centre. Split into 2D and 3D. For 2D, list formulas: rectangle area = l × w, triangle area = ½bh, parallelogram area = bh, trapezium area = ½(a+b)h, circle circumference = 2πr or πd, area = πr². Add sector arc length = θ/360 × 2πr, sector area = θ/360 × πr². For 3D: prism volume = area of cross-section × length; cylinder volume = πr²h, surface area = 2πrh + 2πr²; cone volume = ⅓πr²h, curved surface area = πrl; sphere volume = 4/3πr³, surface area = 4πr²; pyramid volume = ⅓ base area × height. Highlight relationship: sphere surface area is derivative of volume. Use a table on the map to compare formulas. Also note unit conversions: 1 m² = 10,000 cm², 1 m³ = 1,000,000 cm³, 1 litre = 1000 cm³.
中心写 “Mensuration”。分为二维与三维。二维列出公式：矩形面积 = l × w，三角形面积 = ½bh，平行四边形面积 = bh，梯形面积 = ½(a+b)h，圆周长 = 2πr 或 πd，面积 = πr²。加入扇形弧长 = θ/360 × 2πr，扇形面积 = θ/360 × πr²。三维：棱柱体积 = 横截面积 × 长；圆柱体积 = πr²h，表面积 = 2πrh + 2πr²；圆锥体积 = ⅓πr²h，曲面面积 = πrl；球体积 = 4/3πr³，表面积 = 4πr²；棱锥体积 = ⅓ 底面积 × 高。强调关系：球体积导数为表面积。在导图上用表格比较公式。还注意单位换算：1 m² = 10,000 cm²，1 m³ = 1,000,000 cm³，1 L = 1000 cm³。
7. Trigonometry and Pythagoras’ Theorem | 三角学与勾股定理
Center “Trigonometry”. First branch: Pythagoras’ theorem – for right-angled triangles, a² + b² = c² where c is the hypotenuse. Add the converse: if a² + b² = c² then triangle is right. Next, trigonometric ratios: SOH CAH TOA. Label a triangle with opposite (O), adjacent (A), hypotenuse (H). sin θ = O/H, cos θ = A/H, tan θ = O/A. Include exact trig values for 0°, 30°, 45°, 60°, 90° in a small table. For non‑right triangles (Extended): sine rule a/sin A = b/sin B = c/sin C, and cosine rule a² = b² + c² – 2bc cos A (or cos A = (b²+c²–a²)/2bc). Area formula: ½ab sin C. Show how to choose the right rule: right‑angled → Pythagoras/SOHCAHTOA; two angles and a side → sine rule; two sides and included angle → cosine rule or area. Include bearings and 3D trigonometry (angle between line and plane).
中心 “Trigonometry”。第一分支：勾股定理——直角三角形中 a² + b² = c²（c 为斜边），加上逆定理：若 a² + b² = c² 则是直角。然后三角比：SOH CAH TOA。标记对边 (O)、邻边 (A)、斜边 (H)。sin θ = O/H, cos θ = A/H, tan θ = O/A。用一个表格列出 0°, 30°, 45°, 60°, 90° 的精确值。对非直角三角形（Extended）：正弦定理 a/sin A = b/sin B = c/sin C，余弦定理 a² = b² + c² – 2bc cos A（或 cos A = (b²+c²–a²)/2bc）。面积公式：½ab sin C。给出选择法则：直角 → 勾股/SOHCAHTOA；两角一边 → 正弦定理；两边夹角 → 余弦定理或面积。加入方位角和三维三角（线与平面夹角）。
8. Vectors and Transformations | 向量与变换
The mind map has two large branches: vectors and transformations. For vectors, remember notation: column vector (x y) and component form xi + yj. Include magnitude |v| = √(x²+y²), addition, subtraction, scalar multiplication. Show geometric meaning: translation by vector. Parallel vectors are scalar multiples; collinear points share a common vector relationship. Vector geometry problems often require path finding: AB = OB – OA. For transformations: central bubble “Single Transformations” with branches – translation (by a vector), reflection (specify mirror line, e.g., x = a, y = b, y = x, y = –x), rotation (centre, angle, direction), enlargement (centre, scale factor k; area scale factor = k²). (Extended) Stretch (factor parallel to x or y) and shear. Add a branch for describing transformations fully: “An enlargement by scale factor –½ about (1,2)” means both reflection and size change because negative scale factor. Combine transformations: use inverses to reverse effects.
思维导图有两大分支：向量与变换。向量记忆：列向量 (x y) 和分量式 xi + yj。包括模 |v| = √(x²+y²)，加减与数乘。几何意义：按向量平移。平行向量是倍数关系；共线点满足共同的向量关系。向量几何题常需路径法：AB = OB – OA。变换分支：中心气泡 “Single Transformations”，分支——平移（按某向量）、反射（指定镜面线，如 x = a, y = b, y = x, y = –x）、旋转（中心、角度、方向）、放大（中心、比例因子 k；面积比例因子 = k²）。（Extended）拉伸（平行于 x 或 y 轴因子）和错切。添加描述变换的分支：”绕 (1,2) 放大比例因子 –½” 意味着既反射又改变大小，因为负比例因子。复合变换：用逆变换还原效果。
9. Statistics and Data Representation | 统计与数据表示
Draw a central “Statistics” node. Branch into data collection types: discrete, continuous, categorical. Then data representation: bar charts, pie charts, histograms (Extended: frequency density = frequency / class width), cumulative frequency curves (median, quartiles, interquartile range), scatter graphs (correlation, line of best fit, interpolation). Averages branch: mean = sum of values ÷ number, median (middle), mode (most frequent), range. For grouped data: estimate mean using midpoints, modal class, median from cumulative frequency. Include stem-and-leaf diagrams and box plots (five-number summary: min, Q1, median, Q3, max). Think about comparing distributions: comment on median and IQR or mean and range. Add a small note on sampling: random, stratified.
中心 “Statistics”。分支出数据收集类型：离散、连续、分类。然后数据表示：条形图、饼图、直方图（Extended：频率密度 = 频数 / 组距）、累积频率曲线（中位数、四分位数、四分位距）、散点图（相关、最佳拟合线、内插）。平均数分支：均值 = 总和 ÷ 个数，中位数（最中间），众数（最多），极差。分组数据：用组中点估算均值、众数组、由累积频率求中位数。包含茎叶图和箱形图（五数概括：最小、下四分位、中位、上四分位、最大）。比较分布时评价中位数与四分位距或均值与极差。加一注释：随机抽样、分层抽样。
10. Probability Tree and Combined Events | 概率树与复合事件
Create a central “Probability” bubble. First branch: basic rules – probability = number of favourable outcomes / total outcomes, 0 ≤ P ≤ 1, P(not A) = 1 – P(A). Represent sample space by listing, two‑way tables. Next, independent events: P(A and B) = P(A) × P(B). Mutually exclusive: P(A or B) = P(A) + P(B). For combined events that are not mutually exclusive: P(A or B) = P(A) + P(B) – P(A and B). Then focus on tree diagrams: label branches with probabilities, multiply along branches, add probabilities of relevant outcomes. Include replacement (probabilities constant) and without replacement (probabilities change, denominators reduce). For conditional probability (Extended): P(A|B) = P(A and B) / P(B). Use Venn diagrams (union, intersection, complement) to visualise sets. Draw a mini tree diagram directly on the mind map to reinforce structure.
中心气泡 “Probability”。第一分支：基本规则——概率 = 有利结果数 / 总结果数，0 ≤ P ≤ 1，P(非 A) = 1 – P(A)。样本空间用列举法、双向表格。独立事件：P(A 和 B) = P(A) × P(B)。互斥事件：P(A 或 B) = P(A) + P(B)。非互斥复合事件：P(A 或 B) = P(A) + P(B) – P(A 和 B)。重点放在树状图：在分支上标概率，沿分支乘起来，把相关结果概率相加。区分放回（概率不变）和不放回（概率变化，分母减小）。条件概率（Extended）：P(A|B) = P(A 和 B) / P(B)。用维恩图（并、交、补）直观表示集合。直接在导图上画一棵迷你树状图加强结构印象。
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