📚 IGCSE OCR Maths: Mind Map Quick Memorisation | IGCSE OCR 数学：思维导图速记
This revision guide organises key IGCSE OCR Mathematics topics into a visual ‘mind map’ skeleton, linking core definitions, formulas, and exam tricks. Read each section’s English statement first, then reinforce with the Chinese translation for bilingual retrieval.
本速记指南将 IGCSE OCR 数学核心主题整理成可视化“思维导图”框架，串联定义、公式与考试技巧。请先读英文要点，再以中文强化双语记忆。
1. Number Sets and Operations | 数集与运算
Recall the number hierarchy: Natural ℕ ⊂ Integers ℤ ⊂ Rational ℚ ⊂ Real ℝ. Irrational numbers like √2 or π fill the gaps in ℚ.
牢记数字层级：自然数 ℕ ⊂ 整数 ℤ ⊂ 有理数 ℚ ⊂ 实数 ℝ。无理数如 √2 或 π 填补 ℚ 的空隙。
Operations follow BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Wrong order is the top mistake in multi‑step arithmetic.
运算顺序遵循 BIDMAS：括号、指数、除、乘、加、减。顺序错误是多步计算中最常见的失分点。
HCF × LCM = a × b for two numbers. Use prime factor trees to find both quickly.
最大公因数 × 最小公倍数 = 两数之积。用质因数树可快速求出两者。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
To compare or combine, convert: fraction → decimal → percentage. Memorise key equivalents: ½ = 0.5 = 50%, ⅓ ≈ 0.333 = 33⅓%, ¼ = 0.25 = 25%, ⅕ = 0.2 = 20%.
比较或混合运算时请转换：分数 → 小数 → 百分数。熟记关键等价：½ = 0.5 = 50%，⅓ ≈ 0.333 = 33⅓%，¼ = 0.25 = 25%，⅕ = 0.2 = 20%。
Percentage increase: multiply by (1 + r/100). Decrease: multiply by (1 – r/100). Reverse percentages: divide by the multiplier to find the original amount.
百分比增长：乘以 (1 + r/100)。减少：乘以 (1 – r/100)。逆向百分数：除以乘数可得原值。
Recurring decimals: use algebraic method. For 0.3̅, let x = 0.333…, 10x = 3.333…, subtract to get 9x = 3, so x = ⅓.
循环小数：用代数法。设 x = 0.333…，10x = 3.333…，相减得 9x = 3，所以 x = ⅓。
3. Powers and Roots | 幂与根
Index laws: am × an = am+n, am ÷ an = am‑n, (am)n = amn. Negative exponents a⁻n = 1/an. Fractional exponents am/n = (n√a)m.
指数定律：am × an = am+n, am ÷ an = am‑n, (am)n = amn。负指数 a⁻n = 1/an。分数指数 am/n = (n√a)m。
Surds: simplify √12 = 2√3. Rationalise denominators: 1/√2 → √2/2. Never leave a surd in the denominator in final answers.
根号化简：√12 = 2√3。分母有理化：1/√2 → √2/2。最终答案中分母绝不能保留根号。
Standard form: A × 10n where 1 ≤ A < 10. Essential for very large or small numbers and calculator display.
科学记数法：A × 10n，其中 1 ≤ A < 10。处理极大或极小数字及计算器显示时必备。
4. Algebraic Manipulation | 代数运算
Expanding brackets: a(b + c) = ab + ac. Double brackets: (x + a)(x + b) = x² + (a+b)x + ab. Watch signs when a or b are negative.
展开括号：a(b + c) = ab + ac。双括号：(x + a)(x + b) = x² + (a+b)x + ab。注意 a 或 b 为负时的符号。
Factorising: look for common factors first. For quadratics, find two numbers that multiply to ac and add to b. Recognise difference of two squares: a² – b² = (a+b)(a–b).
因式分解：先提取公因数。对二次式，找出乘积为 ac 且和为 b 的两个数。识别平方差公式：a² – b² = (a+b)(a–b)。
Algebraic fractions: simplify by factorising numerator and denominator, then cancel common factors. For addition/subtraction, find a common denominator.
代数分式：将分子分母因式分解，然后约去公因式。加减运算时先通分。
5. Solving Equations and Inequalities | 解方程与不等式
Linear equations: isolate the unknown using inverse operations. Always do the same to both sides. Check solution by substitution.
线性方程：用逆运算分离未知数，两边同操作。代入验算。
Quadratic equations: factorise or use formula x = [–b ± √(b² – 4ac)] / 2a. The discriminant Δ = b² – 4ac tells the number of real roots: Δ > 0 → 2, Δ = 0 → 1, Δ < 0 → 0.
二次方程：因式分解或用公式 x = [–b ± √(b² – 4ac)] / 2a。判别式 Δ = b² – 4ac 指示实根个数：Δ > 0 → 2，Δ = 0 → 1，Δ < 0 → 0。
Simultaneous equations: elimination (add/subtract equations) or substitution. For one linear and one quadratic, substitute the linear expression into the quadratic, solve for the remaining variable, then back‑substitute.
联立方程：消元法（方程加减）或代入法。一次与二次联立时，将一次式代入二次，解出剩余变量再回代。
Inequalities: solve like equations but reverse the sign when multiplying/dividing by a negative number. Represent solution on a number line with open (strict) or closed (≤, ≥) circles.
不等式：解法类似方程，但当乘以或除以负数时不等号方向反转。用数轴表示解，空心圈表严格不等于，实心圈表含等号。
6. Sequences | 数列
Linear sequences: nth term = a + (n–1)d, where a = first term, d = common difference. Check by substituting n = 1,2,3.
等差线性数列：第 n 项 = a + (n–1)d，a 为首项，d 为公差。代入 n = 1,2,3 检验。
Quadratic sequences: second difference constant. nth term has form an² + bn + c. Find a = half the second difference, then use known terms to find b and c.
二次数列：二次差恒定。第 n 项形如 an² + bn + c。a = 二次差的一半，代入已知项求 b 和 c。
Special sequences: square numbers n², cube numbers n³, triangular numbers n(n+1)/2, Fibonacci (each term is sum of the two preceding).
特殊数列：平方数 n²，立方数 n³，三角形数 n(n+1)/2，斐波那契数列（每一项为前两项之和）。
7. Graphs of Functions | 函数图像
Straight line: y = mx + c. m = gradient (rise/run), c = y‑intercept. Parallel lines have equal m; perpendicular lines have gradients product –1.
直线：y = mx + c。m = 斜率（纵差/横差），c = y 截距。平行线 m 相等；垂直线斜率之积为 –1。
Quadratic graph: y = ax² + bx + c is a parabola. a > 0 gives U‑shape (minimum); a < 0 gives n‑shape (maximum). Vertex x = –b/(2a).
二次图像：y = ax² + bx + c 为抛物线。a > 0 开口向上（最小值）；a < 0 开口向下（最大值）。顶点 x = –b/(2a)。
Exponential, cubic, reciprocal graphs: y = kˣ (growth/decay), y = x³, y = 1/x. Know their general shapes for sketching and interpreting.
指数、三次、反比图像：y = kˣ（增长/衰减），y = x³，y = 1/x。掌握其大致形状以利绘图与解读。
8. Ratio, Proportion and Rates | 比、比例与变化率
Simplify ratios like fractions, dividing by HCF. Share a quantity in ratio a:b by finding total parts a+b and calculating each share.
化简比的方法如分数，除以最大公因数。按 a:b 分配数量时，先求总份数 a+b，再算各份。
Direct proportion: y ∝ x → y = kx. Inverse proportion: y ∝ 1/x → y = k/x. Always find constant k first using given data.
正比例：y ∝ x → y = kx。反比例：y ∝ 1/x → y = k/x。务必先用已知数据求出常数 k。
Rates: speed = distance/time, unit price, density = mass/volume. Use compound measures systematically; watch unit conversions (km to m, hours to seconds).
变化率：速度 = 距离/时间，单位价格，密度 = 质量/体积。系统化运用复合量纲；注意单位换算（千米转米，小时转秒）。
9. Geometry: Angles and Polygons | 几何：角与多边形
Angle rules: angles on a line sum to 180°, around a point 360°, vertically opposite angles equal. In triangles, sum of interior angles = 180°. Exterior angle = sum of two opposite interior angles.
角度规则：平角 180°，周角 360°，对顶角相等。三角形内角和 = 180°，外角等于两内对角之和。
Parallel lines: alternate angles equal, corresponding angles equal, co‑interior angles sum to 180°. Spot the ‘F’, ‘Z’, and ‘C’ shapes.
平行线：内错角相等，同位角相等，同旁内角和为 180°。识别 “F”、“Z”、“C” 形。
Polygons: interior angle sum = (n–2) × 180°. Exterior angle sum always 360° for any convex polygon. Regular polygon: each exterior = 360°/n, each interior = 180° – exterior.
多边形：内角和 = (n–2) × 180°。外角和恒为 360°（任何凸多边形）。正多边形：每个外角 = 360°/n，每个内角 = 180° – 外角。
10. Mensuration and Trigonometry | 测量与三角学
Perimeter and area: rectangle A = lw, triangle A = ½bh, circle C = 2πr, A = πr². Trapezium A = ½(a+b)h. Learn these formulas by heart.
周长与面积：矩形 A = lw，三角形 A = ½bh，圆 C = 2πr，A = πr²。梯形 A = ½(a+b)h。牢记公式。
Volume and surface area: cuboid V = lwh, prism V = cross‑section area × length, cylinder V = πr²h, sphere V = ⁴⁄₃πr³, SA = 4πr². Pyramid and cone: V = ⅓ base area × height.
体积与表面积：长方体 V = lwh，棱柱 V = 横截面积 × 长，圆柱 V = πr²h，球 V = ⁴⁄₃πr³，表面积 = 4πr²。棱锥与圆锥：V = ⅓ 底面积 × 高。
Right‑angled trigonometry: SOHCAHTOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. Use Pythagoras: a² + b² = c² for lengths.
直角三角形三角比：SOHCAHTOA：sin θ = 对/斜，cos θ = 邻/斜，tan θ = 对/邻。用毕达哥拉斯定理求边：a² + b² = c²。
Non‑right triangles: sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² – 2bc cos A. Use area formula ½ ab sin C.
非直角三角形：正弦定理 a/sin A = b/sin B = c/sin C；余弦定理 a² = b² + c² – 2bc cos A。面积公式 ½ ab sin C。
11. Transformations and Vectors | 变换与向量
Four transformations: translation (slide by vector), reflection (mirror line), rotation (centre, angle, direction), enlargement (centre, scale factor). Descriptions must be precise.
四种变换：平移（按向量滑动）、反射（镜线）、旋转（中心、角度、方向）、放大（中心、比例因子）。描述必须精确。
Enlargement: if scale factor k > 1, image is larger; 0 < k < 1, image smaller. Negative k gives an inverted image on the opposite side of centre.
放大：若比例因子 k > 1，像变大；0 < k < 1，像变小。k 为负时在与中心相反的一侧形成倒像。
Vectors: column vectors represent magnitude and direction. Addition: add components. Multiplication by scalar λ: multiply each component. Magnitude |v| = √(x² + y²).
向量：列向量表示大小和方向。加法：分量相加。标量乘法：各分量乘以 λ。模 |v| = √(x² + y²)。
12. Probability and Statistics | 概率与统计
Probability: P(A) = favorable outcomes / total outcomes. P(not A) = 1 – P(A). For combined events, use sample space diagrams or probability trees; multiply along branches, add between branches.
概率：P(A) = 有利结果数 / 总结果数。P(非 A) = 1 – P(A)。组合事件用样本空间图或概率树；沿分支相乘，分支间相加。
Statistics: mean = sum of data / number of data. Median = middle value when ordered. Mode = most frequent. Range = max – min. For grouped data, estimate mean using midpoints.
统计：平均数 = 数据和 / 数据个数。中位数 = 有序数居中值。众数 = 出现最多的值。极差 = 最大值 – 最小值。分组数据用组中值估算平均数。
Cumulative frequency and quartiles: plot cumulative frequency against upper class boundaries. Lower quartile at 25% of total frequency, median at 50%, upper quartile at 75%. Interquartile range (IQR) = UQ – LQ.
累积频数与四分位数：标绘累积频数对上组上限。下四分位数对应总频数的 25%，中位数 50%，上四分位数 75%。四分位距 IQR = UQ – LQ。
Probability distributions: sum of all probabilities = 1. For discrete variables, list or table. Understand expected frequency = probability × number of trials.
概率分布：所有概率之和为 1。离散变量用列表或表格。理解期望频数 = 概率 × 试验次数。
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