📚 Mind Map Memory Hacks for IGCSE CIE Mathematics | 思维导图速记法：IGCSE CIE 数学
Unlock the power of visual learning with mind maps to master every topic in IGCSE CIE Mathematics. This guide breaks the syllabus into key branches, each containing essential formulas, quick mnemonics and exam-ready reminders. Use these structured branches to build your own revision maps and recall concepts instantly under pressure.
利用思维导图的视觉学习力量，全面掌握 IGCSE CIE 数学的每一个主题。本指南将考纲分解为核心分支，每个分支都包含必备公式、快速记忆术和备考提醒。用这些结构化的分支来搭建你自己的复习导图，在考试压力下也能立即回忆起关键概念。
1. Number | 数
Begin the Number branch with number sets: natural numbers ℕ = {1,2,3,…}, integers ℤ = {…-2,-1,0,1,2,…}, rational numbers ℚ (fractions and recurring decimals), irrational numbers (non-repeating decimals such as π and √2), and real numbers ℝ which combine rational and irrational.
在“数”分支中先从数集开始：自然数 ℕ = {1,2,3,…}，整数 ℤ = {…-2,-1,0,1,2,…}，有理数 ℚ（分数和循环小数），无理数（不循环小数，如 π 和 √2），以及由有理数和无理数组成的实数 ℝ。
Fraction-decimal-percentage conversions form the backbone of many questions: 1/2 = 0.5 = 50%, 1/3 ≈ 0.333… = 33.3%, 1/4 = 0.25 = 25%. The percentage change formula is (change ÷ original) × 100%. For reverse percentages, divide by (1 ± rate) after identifying the original amount.
分数-小数-百分数转换是许多题目的基础：1/2 = 0.5 = 50%，1/3 ≈ 0.333… = 33.3%，1/4 = 0.25 = 25%。百分数变化公式为 (变化量 ÷ 原值) × 100%。处理逆向百分数时，先确定原值，再除以 (1 ± 变化率)。
Standard form is written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Operations: multiply the a values and add the exponents; for division, divide a values and subtract exponents. Always check your calculator display mode.
标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。运算规则：相乘时将数字部分相乘，指数相加；相除时将数字部分相除，指数相减。务必检查计算器显示模式。
Surds: √a × √b = √(ab). To rationalise a denominator like 1/√a, multiply top and bottom by √a. For a denominator of a + √b, use its conjugate a – √b. Upper and lower bounds: for a measurement given to the nearest unit, the bounds are ±0.5 of that unit.
根式：√a × √b = √(ab)。要有理化形如 1/√a 的分母，分子分母同乘 √a。对于分母为 a + √b 时，使用共轭根式 a – √b。界限值：对于给出至最近单位的测量值，上下界为 ±0.5 个单位。
Percentage change = (change/original) × 100%
百分数变化 = (变化量/原值) × 100%
2. Algebra | 代数
Expanding brackets: use FOIL (First, Outer, Inner, Last) for two binomials. Perfect squares: (a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b². Difference of squares: (a + b)(a – b) = a² – b².
展开括号：两个二项式相乘使用 FOIL（首项、外项、内项、末项）。完全平方：(a + b)² = a² + 2ab + b²，(a – b)² = a² – 2ab + b²。平方差公式：(a + b)(a – b) = a² – b²。
Factorising steps: first take out the highest common factor. For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c, use the ‘ac method’ or grouping.
因式分解步骤：先提取最大公因数。对于形如 x² + bx + c 的二次式，找出两个数使其乘积为 c、和为 b。对于 ax² + bx + c 的形式，使用“ac 方法”或分组分解法。
Solving quadratic equations: try factorising first; if not possible, use the quadratic formula or complete the square. The quadratic formula works for any ax²+bx+c=0.
解二次方程：优先尝试因式分解；若无法分解，使用二次公式或配方法。二次公式适用于任何 ax²+bx+c=0。
x = (-b ± √(b² – 4ac)) / (2a)
二次公式：x = (-b ± √(b² – 4ac)) / (2a)
Simultaneous equations: elimination works by adding/subtracting to remove one variable. Substitution replaces one variable with an expression. Always check your solutions in both original equations. Inequalities: solve like equations, but reverse the inequality sign when multiplying or dividing by a negative number.
联立方程组：消元法通过加减消去一个变量；代入法将一个变量用表达式替代。务必代入原方程组检验解。不等式：解法与方程相同，但乘或除以负数时需将不等号反向。
Sequences: for linear (arithmetic) sequences, nth term = a + (n-1)d, where a is first term and d is common difference. For quadratic sequences, the second difference is constant; use the formula an² + bn + c. Direct proportion: y ∝ x means y = kx; inverse proportion: y ∝ 1/x means y = k/x.
数列：线性（等差）数列的第 n 项公式为 a + (n-1)d，其中 a 为首项，d 为公差。二次数列的二次差为常数；使用通项形式 an² + bn + c。正比例：y ∝ x 即 y = kx；反比例：y ∝ 1/x 即 y = k/x。
3. Graphs | 图像
Straight line graphs: gradient m = rise/run, y-intercept c. Equation y = mx + c. Parallel lines have equal gradients; perpendicular lines have gradients that multiply to -1 (m₁ × m₂ = -1). Finding a midpoint: average of x and y coordinates.
直线图像：斜率 m = 纵向变化/横向变化，y 截距为 c。方程为 y = mx + c。平行线斜率相等；垂线斜率乘积为 -1 (m₁ × m₂ = -1)。中点坐标求法：x 和 y 坐标分别取平均值。
Quadratic graphs produce a parabola. The sign of x² determines U-shape (positive) or n-shape (negative). The vertex gives the maximum or minimum point. Cubic, reciprocal (y = k/x) and exponential graphs each have distinctive shapes you must recognise and sketch.
二次函数图像为抛物线。x² 系数的正负决定开口向上（正）还是向下（负）。顶点给出最大值或最小值。三次函数、倒数函数 (y = k/x) 和指数函数的图像各有独特形状，必须能够识别并速写。
Solving equations graphically: plot both functions and find intersection points. For a tangent to a curve, draw a right-angled triangle to estimate gradient. Distance-time graphs: gradient = speed; flat sections mean stationary. Speed-time graphs: gradient = acceleration; area under graph = distance travelled.
图像法解方程：绘制两个函数图像并找出交点。对于曲线的切线，构造直角三角形来估算斜率。距离-时间图：斜率 = 速度；水平段表示静止。速度-时间图：斜率 = 加速度；图下面积 = 行进距离。
Transformations of graphs: y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by -a; y = -f(x) reflects in the x-axis; y = f(-x) reflects in the y-axis; y = af(x) stretches vertically by scale factor a.
图像变换：y = f(x) + a 纵向平移 a；y = f(x + a) 横向平移 -a；y = -f(x) 关于 x 轴对称反射；y = f(-x) 关于 y 轴对称反射；y = af(x) 纵向拉伸 a 倍。
4. Geometry & Angles | 几何与角
Core angle facts: angles on a straight line sum to 180°, around a point sum to 360°, vertically opposite angles are equal. In parallel lines: alternate angles are equal, corresponding angles are equal, co-interior angles sum to 180°.
基础角度事实：直线上的角之和为 180°，绕一点一周的角之和为 360°，对顶角相等。平行线中：内错角相等，同位角相等，同旁内角之和为 180°。
Angles in polygons: sum of interior angles = (n – 2) × 180°; sum of exterior angles = 360° always. For a regular polygon, each interior angle = (n-2)×180°/n, each exterior = 360°/n. Triangles: angle sum 180°, exterior angle equals sum of two opposite interior angles.
多边形内角：内角和 = (n – 2) × 180°；外角和恒为 360°。对于正多边形，每个内角 = (n-2)×180°/n，每个外角 = 360°/n。三角形：内角和 180°，外角等于两个不相邻内角之和。
Circle theorems: angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; a tangent is perpendicular to the radius at the point of contact; the alternate segment theorem states that the angle between a tangent and a chord equals the angle in the alternate segment.
圆定理：圆心角是圆周角的两倍；半圆上的圆周角为 90°；同弓形上的圆周角相等；圆内接四边形对角之和为 180°；切线与过切点的半径垂直；弦切角定理指出切线与弦之间的夹角等于交替弓形上的圆周角。
5. Mensuration | 测量
Area formulas: rectangle = length × width; triangle = ½ × base × height; parallelogram = base × perpendicular height; trapezium = ½(a + b)h; circle = πr². Circumference = πd = 2πr.
面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 垂直高度；梯形 = ½(a + b)h；圆 = πr²。周长 = πd = 2πr。
Volume of prisms = area of cross-section × length. Cylinder volume = πr²h; sphere volume = 4/3 πr³; cone volume = 1/3 πr²h. Surface area of a cylinder = 2πr² + 2πrh (both ends). Sphere surface area = 4πr²; curved surface of cone = πrl, where l is slant height.
棱柱体积 = 横截面积 × 长度。圆柱体积 = πr²h；球体积 = 4/3 πr³；圆锥体积 = 1/3 πr²h。圆柱表面积 = 2πr² + 2πrh（含两端）。球表面积 = 4πr²；圆锥侧面积 = πrl，l 为斜高。
Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr². When dealing with compound shapes, split the figure into standard shapes, find individual areas or volumes, then add or subtract appropriately.
弧长 = (θ/360) × 2πr；扇形面积 = (θ/360) × πr²。处理组合图形时，将图形拆分为标准形状，分别求面积或体积，再相应加减。
Arc length = θ/360 × 2πr
弧长 = θ/360 × 2πr
6. Trigonometry | 三角学
Right-angled trigonometry: SOH CAH TOA — sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. Exact values: sin30°=1/2, sin45°=√2/2, sin60°=√3/2; cos30°=√3/2, cos45°=√2/2, cos60°=1/2; tan30°=1/√3, tan45°=1, tan60°=√3.
直角三角形三角学：SOH CAH TOA — sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。精确值：sin30°=1/2，sin45°=√2/2，sin60°=√3/2；cos30°=√3/2，cos45°=√2/2，cos60°=1/2；tan30°=1/√3，tan45°=1，tan60°=√3。
Sine rule: a/sinA = b/sinB = c/sinC — used when you have two angles and a side or two sides and a non-included angle. Cosine rule: a² = b² + c² – 2bc cosA — used for two sides and the included angle or three sides. Area of triangle = ½ ab sinC.
正弦定理：a/sinA = b/sinB = c/sinC — 当已知两角一边或两边及其一对角时使用。余弦定理：a² = b² + c² – 2bc cosA — 用于已知两边及其夹角或三边。三角形面积 = ½ ab sinC。
Bearings are measured clockwise from North, always written as three digits, e.g. 045° or 310°. 3D trigonometry: break down into right-angled triangles, often finding a base diagonal first, then the required angle or length using Pythagoras and basic trig.
方位角从正北方向顺时针测量，始终用三位数字表示，例如 045° 或 310°。三维三角学：将问题拆解为直角三角形，通常先求底面对角线，再利用勾股定理和基本三角函数求所需角度或边长。
7. Vectors & Transformations | 向量与变换
Vectors are represented as column vectors or in bold type. The magnitude (length) of vector v = (x, y) is √(x² + y²). Vector addition: add the x components and y components separately. Scalar multiplication multiplies both components by the scalar.
向量用列向量或黑体字母表示。向量 v = (x, y) 的模（长度）为 √(x² + y²)。向量加法：分别对 x 分量和 y 分量相加。数乘将两个分量各自乘以该数值。
Translations are described by a column vector. Reflections: mirror line can be x-axis, y-axis, y = x, y = -x, or any line given. Rotations require centre, angle and direction (clockwise/anticlockwise). Enlargements are defined by centre and scale factor; negative scale factors also cause a 180° rotation.
平移用列向量描述。反射：镜面线可以是 x 轴、y 轴、y = x、y = -x 或任何给定直线。旋转需指定中心、角度和方向（顺时针/逆时针）。放大由中心和比例因子定义；负比例因子还会产生 180° 的旋转效果。
When describing a single transformation, give full details: for rotation, state ‘rotation, centre ( , ), angle, direction’; for enlargement, ‘enlargement, centre ( , ), scale factor’. Check if the transformation is a combination by trying a simple sequence.
在描述单一变换时需提供完整细节：旋转需说明“旋转，中心 ( , )，角度，方向”；放大需说明“放大，中心 ( , )，比例因子”。若疑似复合变换，可尝试简单顺序进行验证。
8. Statistics | 统计
Averages: mean = sum of data / number of data; median = middle value when ordered; mode = most frequent value; range = maximum – minimum. For grouped frequency, use the midpoint of each class to estimate the mean.
平均数：均值 = 数据总和 / 数据个数；中位数为排序后最中间的数值；众数为出现频率最高的值；极差 = 最大值 – 最小值。对于分组频率，使用每组的组中值来估算均值。
Cumulative frequency graphs plot the running total against the upper class boundary. Use these to find the median (50th percentile), quartiles (Q1 at 25%, Q3 at 75%) and interquartile range (IQR = Q3 – Q1). Box plots display these five-number summaries visually.
累积频率图将累计频数相对于上限值描点。用于求中位数（第50百分位数）、四分位数（Q1 在25%，Q3 在75%）和四分位距（IQR = Q3 – Q1）。箱形图将这些五数概括直观呈现。
Histograms: when class widths are unequal, plot frequency density = frequency / class width. The area of each bar represents frequency. Scatter diagrams show correlation — positive, negative or none. A line of best fit can be used for prediction, but avoid extrapolation beyond the data range.
直方图：当组距不相等时，绘制频率密度 = 频率 / 组距。每个柱条的面积代表频数。散点图显示相关性——正相关、负相关或无相关。最佳拟合线可用于预测，但避免超出数据范围的外推。
9. Probability | 概率
Probability of an event = (number of favourable outcomes) / (total number of outcomes), always between 0 and 1. Use sample space diagrams for two combined events. ‘And’ rule: for independent events, P(A and B) = P(A) × P(B). ‘Or’ rule: for mutually exclusive events, P(A or B) = P(A) + P(B).
事件的概率 = (有利结果的数目) / (所有可能结果的总数)，始终介于 0 和 1 之间。两件组合事件使用样本空间图。乘法法则：对于独立事件，P(A 和 B) = P(A) × P(B)。加法法则：对于互斥事件，P(A 或 B) = P(A) + P(B)。
Tree diagrams help with sequential events. Multiply along branches for combined probabilities; add relevant branch outcomes for ‘at least’ scenarios. Conditional probability: P(A|B) = P(A and
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