📚 IGCSE AQA Mathematics: Mind Map Quick Revision | IGCSE AQA 数学：思维导图速记
Revising for IGCSE AQA Mathematics can feel like navigating a maze of formulas and topics. A mind map transforms scattered facts into a clear, visual network, helping you see connections and recall information quickly. This article breaks down the entire syllabus into key branches, providing a structured summary to build your own mind map and master every topic efficiently.
复习 IGCSE AQA 数学就像在公式和主题的迷宫中穿行。思维导图能将零散的知识点转化为清晰、可视化的网络，帮助你看到联系并快速回忆信息。本文把整个课程大纲分解为几个关键分支，提供结构化的总结，让你构建自己的思维导图，高效掌握每个主题。
1. The Big Picture: Building Your Mind Map | 整体框架：构建你的思维导图
Start with a central node labelled ‘IGCSE AQA Mathematics’. From it, draw six main branches: Number, Algebra, Ratio & Proportion, Geometry & Measures, Probability, and Statistics. Use colour-coding for each branch. Then add sub-branches for each sub-topic, such as ‘Fractions’ under Number, or ‘Linear Graphs’ under Algebra. Write key formulas, rules, and examples directly on the branches. Add small icons or symbols to trigger visual memory. Revisit your mind map regularly and test yourself by blanking out sections and trying to recall them.
从中心节点“IGCSE AQA 数学”开始，画出六大主分支：数、代数、比与比例、几何与测量、概率和统计。为每个分支使用不同的颜色。然后为每个子主题添加次级分支，比如“数”下的“分数”，或“代数”下的“直线图像”。将关键公式、法则和示例直接写在分支上。添加小图标或符号来触发视觉记忆。定期回顾你的思维导图，并通过遮盖部分内容自我测试来强化记忆。
2. Number Essentials | 数的要点
BIDMAS: The order of operations is Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Failure to follow this leads to wrong answers in calculator and non-calculator questions.
BIDMAS：运算顺序为括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。不遵守这个顺序会在计算器题和非计算器题中都得出错误答案。
Fractions: To add or subtract, find a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal. Always simplify your answer and convert improper fractions to mixed numbers where appropriate.
分数：加减时先找公分母；乘法是分子乘分子、分母乘分母；除法乘以倒数。始终化简答案，适当情况下将假分数化为带分数。
Percentages: To increase by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 − r/100). For reverse percentages, divide the given amount by the multiplier to find the original value. Percentage change = (difference / original) × 100%.
百分数：增加 r% 即乘以 (1 + r/100)；减少 r% 乘以 (1 − r/100)。反向百分数时，用给定金额除以乘数得到原值。百分数变化率 = (差值 / 原值) × 100%。
Standard form: A number in standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. For example, 4 600 = 4.6 × 10³ and 0.0032 = 3.2 × 10⁻³. When multiplying or dividing, deal with the numbers and powers of 10 separately.
标准形式：标准形式的数字写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。例如 4 600 = 4.6 × 10³，0.0032 = 3.2 × 10⁻³。乘法或除法时，分别处理数字部分和 10 的幂次。
Indices rules: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. For fractional indices: a^{1/n} = ⁿ√a, and a^{m/n} = (ⁿ√a)ᵐ. These are essential for Extended tier.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。分数指数：a^{1/n} = ⁿ√a，a^{m/n} = (ⁿ√a)ᵐ。这些是 Extended 层级的核心内容。
Surds (Extended): Simplify surds by taking out square factors: √12 = 2√3. Rationalise denominators, e.g. 1/√2 → √2/2. Do not leave surds in the denominator of your final answer.
根式 (Extended)：通过提取平方因子化简根式，例如 √12 = 2√3。分母有理化，例如 1/√2 化为 √2/2。最终答案中分母不能保留根式。
HCF and LCM: Use prime factorisation. For HCF, multiply the lowest powers of common prime factors. For LCM, multiply the highest powers of all prime factors present. Applying this to worded problems is common.
最大公因数与最小公倍数：使用质因数分解。HCF 取共同质因数的最小幂次相乘，LCM 取所有质因数的最高幂次相乘。此类应用题很常见。
3. Algebra: Expressions and Equations | 代数：表达式与方程
Expanding brackets: a(b + c) = ab + ac. For double brackets, (x + a)(x + b) = x² + (a+b)x + ab. For triple brackets (Extended), expand two first, then multiply by the third.
去括号：a(b + c) = ab + ac。双括号展开：(x + a)(x + b) = x² + (a+b)x + ab。三重括号（Extended）先展开两个，再与第三个相乘。
Factorising: Take out common factors first. For quadratics x² + bx + c, find two numbers that multiply to c and add to b. For difference of two squares: a² − b² = (a+b)(a−b).
因式分解：先提取公因式。对于二次式 x² + bx + c，找两个数使其积为 c、和为 b。平方差公式：a² − b² = (a+b)(a−b)。
Solving linear equations: Rearrange by performing the same operation on both sides. If the equation contains fractions, multiply every term by the common denominator to clear them.
解一元一次方程：通过两边同时进行相同运算来移项。如果方程含有分数，将每一项乘以公分母消去分母。
Solving quadratics: Factorise and set each bracket equal to zero, or use the quadratic formula. The formula is given in the exam, but you must know how to substitute correctly: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / 2a.
解二次方程：因式分解后令每个括号为0，或使用求根公式。公式会在试卷中给出，但你必须能正确代入：对于 ax² + bx + c = 0，x = [−b ± √(b² − 4ac)] / 2a。
Simultaneous equations: Solve by elimination (add or subtract equations to remove a variable) or substitution. For non-linear simultaneous equations (Extended), substitute one equation into the other.
联立方程：用消元法（加减方程消去一个变量）或代入法求解。非线性联立方程（Extended）则将一个方程代入另一个。
Inequalities: Solve like equations, but reverse the inequality sign if multiplying or dividing by a negative number. Represent solutions on a number line with open (for < or >) and closed (for ≤ or ≥) circles.
不等式：解法同方程，但当乘以或除以负数时，不等号方向要反转。在数轴上表示解集时，< 或 > 用空心圆圈，≤ 或 ≥ 用实心圆圈。
4. Algebra: Sequences and Graphs | 代数：数列与图像
Arithmetic sequences: The nth term is a + (n−1)d, where a is the first term and d is the common difference. Finding d from any two terms is a core skill.
等差数列：第 n 项公式为 a + (n−1)d，其中 a 是首项，d 是公差。从任意两项求公差是核心技能。
Linear graphs: y = mx + c. m is the gradient (rise over run), c is the y-intercept. Parallel lines have equal gradients. Perpendicular lines satisfy m₁ × m₂ = −1.
直线图像：y = mx + c，m 为斜率（纵向变化 / 横向变化），c 为 y 截距。平行线斜率相等，垂直线斜率乘积为 −1。
Quadratic graphs: y = ax² + bx + c forms a parabola. If a > 0, it is U-shaped; if a < 0, it is ∩-shaped. Roots are where y = 0, and the turning point can be found by completing the square (Extended).
二次函数图像：y = ax² + bx + c 是抛物线。a > 0 时开口向上呈 U 形，a < 0 时开口向下呈 ∩ 形。根是 y = 0 的点，极值点可通过配方法求得（Extended）。
Other graphs (Extended): Recognise cubic y = x³, reciprocal y = 1/x, and exponential y = aˣ. Use graphs to solve equations, e.g. find approximate solutions to x² − 2 = 0 by plotting y = x² − 2.
其他函数图像 (Extended)：识别三次函数 y = x³、反比例函数 y = 1/x、指数函数 y = aˣ。利用图像解方程，例如绘制 y = x² − 2 求 x² − 2 = 0 的近似解。
5. Ratio, Proportion and Rates of Change | 比、比例与变化率
Simplifying ratios: Divide all parts by their highest common factor. Ratios can be scaled up by multiplying all parts by the same number.
化简比：所有部分除以其最大公因数。比可以通过所有部分同乘以一个数来放大。
Dividing in a ratio: To share £1200 in the ratio 3:5, add the parts (3+5=8), then £1200 ÷ 8 × 3 and £1200 ÷ 8 × 5. Always check the sum of the amounts equals the total.
按比例分配：如将 £1200 按 3:5 分配，先加总份数 (3+5=8)，然后 £1200 ÷ 8 × 3 和 £1200 ÷ 8 × 5。务必检查各部分之和等于总量。
Direct and inverse proportion (Extended): If y is directly proportional to x, y = kx. If y is inversely proportional to x, y = k/x. Find k from given values, then use it to solve problems.
正比例与反比例 (Extended)：y 与 x 成正比例，则 y = kx。y 与 x 成反比例，则 y = k/x。由已知值求出 k，再用于解题。
Compound measures: Speed = distance ÷ time, Density = mass ÷ volume, Pressure = force ÷ area. Learn the units and how to convert between them (e.g. km/h to m/s: ÷ 3.6).
复合量度：速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 力 ÷ 面积。记住单位及其换算（例如 km/h 转 m/s：除以 3.6）。
Growth and decay (Extended): Repeated percentage change formula: Final amount = Initial amount × (1 ± r)ⁿ, where r is the rate per interval and n is the number of intervals.
增长与衰减 (Extended)：重复百分数变化公式：最终量 = 初始量 × (1 ± r)ⁿ，其中 r 为每期变化率，n 为期数。
6. Geometry: Angles and Shapes | 几何：角与形状
Angle facts: Angles on a straight line add to 180°. Angles around a point add to 360°. Vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles sum to 180°.
角度事实：直线上角之和为 180°，点周围角之和为 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 of an n-sided polygon = (n−2) × 180°. Each exterior angle of a regular polygon = 360° ÷ n. Interior + exterior = 180°.
多边形内角和外角：n 边形内角和 = (n−2) × 180°。正多边形每个外角 = 360° ÷ n。内角 + 外角 = 180°。
Triangles and quadrilaterals: Know properties of equilateral, isosceles, right-angled triangles. For quadrilaterals: square, rectangle, parallelogram, rhombus, trapezium and kite – learn side, angle and diagonal properties.
三角形与四边形：熟记等边三角形、等腰三角形、直角三角形的性质。四边形：正方形、矩形、平行四边形、菱形、梯形、风筝形——掌握边、角和对角线的性质。
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