IGCSE Edexcel Maths Mind Map Speedy Revision | IGCSE Edexcel 数学：思维导图速记

📚 IGCSE Edexcel Maths Mind Map Speedy Revision | IGCSE Edexcel 数学：思维导图速记

When preparing for IGCSE Edexcel Mathematics, the sheer volume of topics can feel overwhelming. A mind map turns linear notes into a visual web of connections, allowing you to see how concepts relate and to trigger memory quickly. This article serves as a structured mind map walkthrough, covering every major topic area in the syllabus. As you read, imagine a central ‘IGCSE Maths’ node branching into Number, Algebra, Graphs, and beyond, with each branch further splitting into key formulas, methods, and exam tips.

在准备 IGCSE Edexcel 数学考试时，庞大的知识体系常令人感到无从下手。思维导图能把线性的笔记变成一张可视化的关联网络，让你看清概念之间的联系并快速触发记忆。本文相当于一份结构化的思维导图导览，涵盖了考纲中每一个主要领域。阅读时，请想象一个中心节点“IGCSE 数学”，向外分出数、代数、函数图像等分支，每个分支再继续分裂为关键公式、解题方法和考试技巧。

1. Number System Mind Map Branch | 数系思维导图分支

The Number branch splits into types of numbers, operations, and conventions. Start with natural numbers, then integers, rationals, irrationals, and real numbers. Remember that surds like √2 are irrational, while recurring decimals are rational.

数系分支首先分为数字类型、运算法则和数学约定。从自然数出发，延伸到整数、有理数、无理数和实数。记住像 √2 这样的根式是无理数，而循环小数则是有理数。

Prime factorisation sits at the heart of this branch: every composite number can be written as a product of primes using a factor tree. From here, you branch to Highest Common Factor (HCF) and Lowest Common Multiple (LCM).

质因数分解是这个分支的核心：每个合数都可以用因子树写成质数的乘积。由此延伸出最大公因数 (HCF) 和最小公倍数 (LCM)。

To find HCF, multiply the lowest powers of common prime factors. To find LCM, multiply the highest powers of all prime factors present.

求 HCF 时，将公共质因数的最小次幂相乘；求 LCM 时，将所有质因数的最高次幂相乘。

Standard form (A × 10ⁿ, 1 ≤ A < 10) is essential for very large or small numbers. Operations in standard form link to laws of indices: xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ.

标准形式 (A × 10ⁿ，1 ≤ A < 10) 是处理极端数值的关键。标准形式的运算与指数律紧密相连：xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ。

Fractions, decimals and percentages conversions form a triangular sub-branch. Use the multiplier method for percentage increase/decrease. Rounding to decimal places and significant figures, plus upper and lower bounds, complete this branch.

分数、小数和百分数的互化构成一个三角形子分支。使用乘数法处理百分数的增减。保留小数位数与有效数字的约分，以及上界和下界的计算，完善了整个数系分支。

Upper bound = measured value + ½ × precision

2. Algebra Essentials Mind Map Branch | 代数核心思维导图分支

Algebra branch begins with simplifying expressions, expanding brackets, and factorising. Key identities include (a + b)² = a² + 2ab + b² and the difference of two squares a² – b² = (a – b)(a + b).

代数分支从化简表达式、展开括号和因式分解开始。核心恒等式包括 (a + b)² = a² + 2ab + b² 以及平方差 a² – b² = (a – b)(a + b)。

Solving linear equations links to unknown on both sides, brackets, and fractions. Always perform inverse operations in the correct order. For quadratic equations, first try factorisation, then quadratic formula.

解一元一次方程会涉及未知数在等号两边、含括号和分数的情况。务必按照正确顺序进行逆运算。对于二次方程，先尝试因式分解，再使用求根公式。

If ax² + bx + c = 0 cannot be factorised, use the formula: x = [ –b ± √(b² – 4ac) ] / 2a.

如果 ax² + bx + c = 0 无法因式分解，则使用公式：x = [ –b ± √(b² – 4ac) ] / 2a。

Simultaneous equations can be solved by elimination or substitution. When one is quadratic, substitution is often the only way. Always check your solutions by substituting back.

联立方程组可通过消元法或代入法求解。当其中一个方程为二次时，通常只能使用代入法。务必回代检验解的正确性。

Inequalities are solved like equations but remember to flip the inequality sign when multiplying or dividing by a negative number. Show solutions on a number line with open or closed circles.

不等式的解法与等式类似，但要记住当乘或除以一个负数时，不等号方向要改变。在数轴上表示解集时注意使用空心或实心圆点。

3. Graphs and Functions Mind Map Branch | 图像与函数思维导图分支

The Graphs branch covers plotting straight lines, quadratics, cubics, reciprocals, and exponentials. For y = mx + c, m is the gradient and c is the y-intercept. To find the gradient between two points, use (y₂ – y₁) / (x₂ – x₁).

图像分支涵盖绘制直线、二次函数、三次函数、反比例函数和指数函数的图像。对于 y = mx + c，m 是斜率，c 是 y 轴截距。两点间斜率公式为 (y₂ – y₁) / (x₂ – x₁)。

Quadratic graphs are parabolas. The turning point can be found by completing the square. If the coefficient of x² is positive, the parabola smiles; if negative, it frowns. Roots are the x-intercepts where y = 0.

二次函数的图像为抛物线。其顶点可通过配方法求得。若 x² 的系数为正，抛物线开口向上；为负则向下。方程的根是图像与 x 轴的交点，即 y = 0 时的横坐标。

Graph transformations follow a clear set of rules: f(x) + a moves the graph up by a, f(x + a) moves it left by a, –f(x) reflects in the x-axis, f(–x) reflects in the y-axis. Stretches multiply the coordinates accordingly.

函数图像变换有一套清晰的规则：f(x) + a 使图像向上平移 a 个单位，f(x + a) 向左平移 a 个单位，–f(x) 关于 x 轴对称，f(–x) 关于 y 轴对称。拉伸变换则会按比例改变坐标。

Real-life graphs: distance–time graphs show speed as gradient; velocity–time graphs show acceleration as gradient and distance as area under the graph. This visual link is a powerful exam tool.

实际应用图像：距离–时间图中，斜率代表速度；速度–时间图中，斜率代表加速度，曲线下面积代表位移。这种可视化联系是解题的有力工具。

4. Geometry and Angles Mind Map Branch | 几何与角度思维导图分支

Angles branch starts with basic facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal. Parallel lines create alternate, corresponding, and co-interior angle relationships.

角度分支从基本事实开始：直线上的邻角之和为 180°，同顶点的周角之和为 360°，对顶角相等。平行线则带来内错角、同位角和同旁内角的关系。

Polygons: sum of interior angles = (n – 2) × 180°, sum of exterior angles always = 360°. For a regular polygon, each interior angle = [(n – 2) × 180°] / n. Circle theorems form a dense sub-branch.

多边形：内角和 = (n – 2) × 180°，外角和恒等于 360°。正多边形的每个内角 = [(n – 2) × 180°] / n。圆的定理构成了一个密集的子分支。

Angle at centre is twice angle at circumference; angles in the same segment are equal; angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°.

圆心角是圆周角的两倍；同弧上的圆周角相等；半圆内的圆周角是直角；圆内接四边形的对角之和为 180°。

Congruence and similarity link to side ratios. For similar triangles, corresponding sides are in proportion. Use scale factors to find missing lengths, areas, and volumes.

全等与相似与边长比例相连。相似三角形的对应边成比例。运用比例因子可以求出未知长度、面积和体积。

Area scale factor = (length scale factor)²
Volume scale factor = (length scale factor)³

5. Measures and Mensuration Mind Map Branch | 测量与求积思维导图分支

This branch stores all perimeter, area, and volume formulas. For a circle: circumference = 2πr or πd, area = πr². A sector area is (θ/360) × πr², and arc length = (θ/360) × 2πr.

这个分支储存了所有周长、面积和体积公式。圆：周长 = 2πr 或 πd，面积 = πr²。扇形面积 = (θ/360) × πr²，弧长 = (θ/360) × 2πr。

Triangles: area = ½ × base × height. Use Heron’s formula or ½ab sin C when given two sides and included angle. Pythagoras’ theorem a² + b² = c² is the gateway to right‑angled triangle problems.

三角形：面积 = ½ × 底 × 高。已知两边及夹角时，可用 ½ab sin C 公式。勾股定理 a² + b² = c² 是解决直角三角形问题的大门。

3D shapes: prism volume = area of cross-section × length. Pyramid volume = ⅓ × base area × height. Sphere surface area = 4πr², volume = ⁴⁄₃πr³. Cone curved surface area = πrl, volume = ⅓πr²h.

立体图形：棱柱体积 = 横截面积 × 长度。棱锥体积 = ⅓ × 底面积 × 高。球体表面积 = 4πr²，体积 = ⁴⁄₃πr³。圆锥侧面积 = πrl，体积 = ⅓πr²h。

Units conversion: 1 cm³ = 1 ml, 1 m³ = 1000 litres. Speed = distance/time, density = mass/volume, pressure = force/area. Use triangle diagrams to rearrange.

单位换算：1 立方厘米 = 1 毫升，1 立方米 = 1000 升。速度 = 距离/时间，密度 = 质量/体积，压强 = 力/面积。使用三角形图示轻松地变形式子。

6. Trigonometry Mind Map Branch | 三角学思维导图分支

Right‑angled trigonometry uses SOH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Always label the sides relative to the given angle.

直角三角学使用 SOH CAH TOA：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。始终要根据已知角度标示各边。

For non‑right triangles, use the sine rule: a/sin A = b/sin B = c/sin C, or the cosine rule: a² = b² + c² – 2bc cos A. Cosine rule links to finding an angle or a side when you have two sides and the included angle.

对于非直角三角形，使用正弦定理：a/sin A = b/sin B = c/sin C，或余弦定理：a² = b² + c² – 2bc cos A。当已知两边及其夹角时，余弦定理可用于求边或角。

Bearings are measured clockwise from North, always three digits. Trigonometry with bearings often creates right‑angled triangles with parallel line properties.

方位角以正北为基准顺时针测量，始终用三位数字表示。与方位角结合的三角学问题常借助平行线性质构造直角三角形。

3D Pythagoras and trigonometry: find a right‑angled triangle inside the solid. The longest diagonal in a cuboid is √(l² + w² + h²). Practice visualising hidden triangles.

三维勾股与三角学：在立体内部找到直角三角形。长方体的最长对角线为 √(长² + 宽² + 高²)。要勤于练习想象隐藏的三角形。

7. Vectors and Transformations Mind Map Branch | 向量与变换思维导图分支

Vectors describe both magnitude and direction. Column vectors (ᵃᵦ) represent movement. Adding vectors: add components separately. Multiplying by a scalar changes magnitude and possibly direction.

向量同时描述大小和方向。列向量 (ᵃᵦ) 表示位移。向量相加需将对应分量分别相加。数乘会改变向量的大小，并可能改变方向。

Vector geometry uses parallel vectors and collinearity. If AB = k×BC, then A, B, C are collinear. Express routes like AB = b – a where a and b are position vectors of A and B.

向量几何涉及平行向量与共线问题。若 AB = k×BC，则 A、B、C 三点共线。路径向量可表示为 AB = b – a，其中 a 和 b 是点 A 和 B 的位置向量。

Transformations: translations by a vector, reflections in lines (y = x, axes, etc.), rotations by 90°, 180°, 270° around a centre, and enlargements by a scale factor (including negative scale factors).

变换：按向量平移、关于直线（如 y = x，坐标轴）的反射、绕中心旋转 90°、180°、270° 的旋转，以及按比例因子进行的放大（包括负比例因子）。

Describing a single transformation: two reflections in parallel lines give a translation; two reflections in intersecting lines give a rotation. This is a high‑order link in the mind map.

描述单一变换：两次关于平行线的反射得到一个平移；两次关于相交线的反射得到一个旋转。这是思维导图中较高层级的关联。

8. Statistics and Data Handling Mind Map Branch | 统计与数据处理思维导图分支

Collecting data: qualitative (non‑numerical) or quantitative (discrete/continuous). Sampling methods include random, stratified, systematic, and quota. Stratified sampling ensures fair representation: (stratum size/population) × sample size.

数据收集：定性数据（非数值）或定量数据（离散/连续）。抽样方法包括随机、分层、系统及配额抽样。分层抽样确保公平代表性：(层规模/总体) × 样本容量。

Charts: bar charts, pie charts, histograms (with frequency density = frequency/class width), cumulative frequency curves, and box plots. The interquartile range (IQR) = Q3 – Q1.

图表：条形图、饼图、直方图（频率密度 = 频数/组距）、累积频率曲线和箱形图。四分位距 (IQR) = Q3 – Q1。

Averages: mean (sum of values/total number), median (middle value), mode (most frequent). From a grouped table, use mid‑interval values to estimate mean. Compare distributions using median and IQR.

平均数：均值（总和/总数）、中位数（中间值）、众数（出现最多的值）。从分组频数表中，用组中值估算均值。可使用中位数和四分位距比较分布特征。

Scatter graphs show correlation. A line of best fit lets you make predictions. The equation of the line of best fit can be used for estimation, but beware of extrapolation beyond the data range.

散点图显示相关性。最佳拟合线可用于预测。最佳拟合线的方程可用于估算，但要小心不要超出数据范围进行外推。

9. Probability Mind Map Branch | 概率思维导图分支

Basic probability: P(event) = number of favourable outcomes / total number of outcomes. Probabilities always lie between 0 and 1. The sum of probabilities of all mutually exclusive events is 1.

基本概率：P(事件) = 有利结果数 / 总结果数。概率始终介于 0 与 1 之间。所有互斥事件的概率之和为 1。

Tree diagrams: multiply along branches for ‘and’, add probabilities for ‘or’. For conditional probability, the second set of branches depends on the first outcome. P(A|B) = P(A and B)/P(B).

树状图：沿分支相乘表示“且”，将不同路径概率相加表示“或”。条件概率中，第二层分支的概率取决于第一次的结果。P(A|B) = P(A 且 B)/P(B)。

Venn diagrams: the intersection represents A and B, union is A or B. Use the addition rule: P(A or B) = P(A) + P(B) – P(A and B). Frequency trees offer an alternative approach.

韦恩图：交集表示 A 且 B，并集表示 A 或 B。使用加法规则：P(A 或 B) = P(A) + P(B) – P(A 且 B)。频率树提供了另一种解题思路。

Independent events: P(A and B) = P(A) × P(B). Watch out for ‘without replacement’ scenarios, which make events dependent and require updated probabilities.

独立事件：P(A 且 B) = P(A) × P(B)。注意“不放回”情形，会让事件变得相关，此时需要更新概率值。

10. Sequences and Series Mind Map Branch | 数列与级数思维导图分支

Linear sequences have a common difference. The nth term of a linear sequence is an + b, where a is the difference and a + b is the first term. Quadratic sequences need a second layer of differences.

线性数列有固定的公差。线性数列的第 n 项公式为 an + b，其中 a 是公差，a + b 为首项。二次数列则需要用到二阶差分。

For quadratic sequences, the second difference is constant (2a). The nth term formula is an² + bn + c. Find a, then set up simultaneous equations to find b and c.

对于二次方数列，二阶差分是常数 (2a)。第 n 项公式为 an² + bn + c。先求出 a，再通过联立方程求出 b 和 c。

Other sequences include geometric progressions (common ratio), Fibonacci‑type (each term sum of two previous), and patterns based on diagrams. Always list the first few terms to spot the rule.

其他数列包括等比数列（有公比）、斐波那契型数列（每一项为前两项之和）以及基于图形的模式。始终先列出前几项来发现规律。

11. Ratio, Proportion and Rates Mind Map Branch | 比、比例与速率思维导图分支

Ratios are simplified like fractions. Divide a quantity in a given ratio by finding the total number of parts. Direct proportion: y = kx, graph is a straight line through the origin. Inverse proportion: y = k/x, graph is a rectangular hyperbola.

比可像分数一样化简。按给定比例分配数量时先求出总份数。正比例：y = kx，图像为过原点的直线。反比例：y = k/x，图像为双曲线。

Best buy problems compare value by finding price per unit. Compound measures like speed and density link back to the Measures branch. Unitary method is the key: find the value of 1 first.

最佳购买问题通过计算单位价格来比较价值。复合单位如速度和密度又回到测量分支。单位法是关键：先求出 1 对应的值。

Percentage profit/loss and simple/compound interest are vital applications. Compound interest: A = P(1 + r/100)ⁿ. Depreciation uses the same structure with minus sign.

利润/亏损百分比以及单利、复利是重要应用。复利公式：A = P(1 + r/100)ⁿ。折旧公式结构相同，只是将加号变为减号。

12. Building Your Own Mind Map and Revision Tips | 构建你自己的思维导图与复习技巧

Start with a blank sheet. Write ‘IGCSE Edexcel Maths’ in the centre. Draw thick branches for the main topic areas listed above. Use colours and small sketches to reinforce memory. Then, add thinner sub‑branches for each formula and rule. The act of creating the map is itself a powerful revision activity.

从一张白纸开始，中央写下“IGCSE Edexcel 数学”。用粗线画出上述各主题领域的主要分支。使用不同颜色和小图标来强化记忆。接着，为每一个公式和法则添加更细的子分支。绘制思维导图的过程本身就是一种强效的复习活动。

On the day before the exam, condense your map to a one‑page key formula sheet, then mentally reconstruct the full map without looking. This tests your recall and reveals any weak links.

考试前一天，将你的思维导图浓缩成一页关键公式表，然后在不看原图的情况下在脑中重建全图。这能检验你的记忆并暴露出薄弱环节。

Always link new problem types back to your map. For example, a bearings question involves Trigonometry, Angles, and possibly Vectors. Seeing these connections speeds up problem solving and reduces panic.

始终将新题型与你的思维导图联系起来。例如，一道方位角题可能同时涉及三角学、角度，或许还有向量。看到这些联系能加快解题速度，减少紧张情绪。

Remember, the mind map is a living document. Update it with tricky questions from past papers, common mistakes, and examiner tips. By exam day, your brain will have a robust mental blueprint of the entire syllabus.

请记住，思维导图是一份活的文档。随时用历年试卷中的难题、常见错误和考官提示来更新它。到考试那天，你的大脑将拥有一幅涵盖整个考纲的可靠心理蓝图。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IGCSE Edexcel Maths Mind Map Speedy Revision | IGCSE Edexcel 数学：思维导图速记