📚 IGCSE Maths High-Frequency Topic Summary | IGCSE 数学：高频考点总结
This article summarises the most frequently examined topics in IGCSE Mathematics, helping you focus your revision on areas that consistently appear in papers. Mastery of these core concepts will build a solid foundation and boost your confidence for the exam. Each section pairs key English explanations with Chinese translations to support bilingual learners.
本文总结了 IGCSE 数学中最常考的主题，帮助你将复习集中在试卷中反复出现的领域。掌握这些核心概念将打下坚实基础，并提升你应考的信心。每个部分均提供英文要点与中文翻译的配对，以支持双语学习者。
1. Number Operations and Bounds | 数的运算与界限
IGCSE candidates must be comfortable with four operations on integers, fractions, decimals and percentages, including correct order of operations (BIDMAS/BODMAS). Upper and lower bounds are tested when numbers are rounded or truncated; remember to add or subtract half the degree of accuracy.
IGCSE 考生必须熟练掌握整数、分数、小数和百分比的四则运算，包括正确的运算顺序（括号、指数、乘除、加减）。当数字被四舍五入或截断时，常考上限和下限；记住要加减一半的精确度。
Bounds rule: For a measurement given to the nearest unit, the lower bound = value − 0.5×unit, upper bound = value + 0.5×unit. For calculations, use bounds to find maximum and minimum possible results.
界限规则： 对于给出到最接近单位的测量值，下限 = 值 − 0.5×单位，上限 = 值 + 0.5×单位。在计算中，使用界限来求可能的最大和最小结果。
2. Algebraic Manipulation | 代数式化简
Expanding brackets, factorising expressions (including quadratics) and simplifying algebraic fractions are essential skills. Pay special attention to the difference of two squares: a² − b² = (a + b)(a − b).
展开括号、分解因式（包括二次式）以及简化代数分式是基本技能。特别要注意平方差公式：a² − b² = (a + b)(a − b)。
When factorising quadratics of the form x² + bx + c, look for two numbers that multiply to c and add to b. For harder quadratics like ax² + bx + c, use the grouping method or inspection.
对形如 x² + bx + c 的二次式进行因式分解时，寻找两个数相乘得 c、相加得 b。对较难的二次式如 ax² + bx + c，使用分组法或观察法。
Example: Expand and simplify (2x − 3)(x + 5) = 2x² + 10x − 3x − 15 = 2x² + 7x − 15.
例： 展开并化简 (2x − 3)(x + 5) = 2x² + 10x − 3x − 15 = 2x² + 7x − 15。
3. Linear and Quadratic Equations | 线性与二次方程
Solving linear equations often involves rearranging terms, collecting like terms and isolating the unknown. Always check your answer by substitution. Quadratic equations are solved by factorising, completing the square or using the quadratic formula.
解线性方程通常涉及移项、合并同类项以及隔离未知数。始终通过代入检验答案。二次方程可通过因式分解、配方法或二次根公式求解。
Quadratic formula: If ax² + bx + c = 0, then x = [−b ± √(b² − 4ac)] / (2a). The discriminant b² − 4ac determines the nature of the roots.
二次根公式： 若 ax² + bx + c = 0，则 x = [−b ± √(b² − 4ac)] / (2a)。判别式 b² − 4ac 决定根的性质。
Simultaneous equations, both linear-linear and one linear one quadratic, are common. Use elimination or substitution for linear pairs; substitute the linear into the quadratic for mixed pairs.
联立方程，既包括线性–线性，也包括一个线性一个二次的组，都很常见。线性方程组使用消元法或代入法；混合方程组则将线性方程代入二次方程。
4. Functions and Graphs | 函数与图像
Understand function notation f(x), domain and range, composite functions fg(x) and inverse functions f⁻¹(x). Sketch and interpret graphs of linear, quadratic, cubic, reciprocal and exponential functions.
理解函数符号 f(x)、定义域和值域、复合函数 fg(x) 和反函数 f⁻¹(x)。绘制并解读线性、二次、三次、倒数和指数函数的图像。
Transformations of graphs: y = f(x) + a (vertical translation), y = f(x + a) (horizontal translation), y = −f(x) (reflection in x-axis), y = f(−x) (reflection in y-axis), y = af(x) (vertical stretch).
图像的变换：y = f(x) + a（垂直平移），y = f(x + a)（水平平移），y = −f(x)（关于 x 轴对称），y = f(−x)（关于 y 轴对称），y = af(x)（垂直拉伸）。
Plotting quadratics accurately requires identifying the vertex, axis of symmetry and intercepts. The turning point can be found by completing the square.
准确绘制二次函数图像需要确定顶点、对称轴和截距。通过配方法可以找到转折点。
5. Geometry: Angles and Polygons | 几何：角与多边形
Angle properties on a straight line (sum to 180°), around a point (360°), vertically opposite angles (equal), alternate and corresponding angles on parallel lines are tested frequently. Interior angle sum of an n-sided polygon = (n − 2) × 180°.
直线上的角（和为 180°）、点周角（360°）、对顶角（相等）、平行线中的内错角和同位角是常考内容。n 边形的内角和 = (n − 2) × 180°。
Exterior angle of a regular polygon = 360° / n. Bearings and three-figure bearings appear regularly; remember bearings are measured clockwise from North.
正多边形的外角 = 360° / n。方位角与三位数方位角经常出现；记住方位角是从正北顺时针测量的。
Circle theorems: angle at centre is twice angle at circumference; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius.
圆定理：圆心角是圆周角的两倍；同弓形内的角相等；圆内接四边形的对角互补；切线与半径垂直。
- Angle between tangent and chord = angle in the alternate segment.
- 切线与弦的夹角等于弦切角定理中的另一弓形角。
6. Trigonometry | 三角学
SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Apply to right-angled triangles for finding missing sides and angles.
三角函数口诀：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。应用于直角三角形求解缺失的边和角。
Sine rule: a/sin A = b/sin B = c/sin C or sin A / a = sin B / b = sin C / c. Cosine rule: a² = b² + c² − 2bc cos A. Use when given two sides and the included angle or three sides.
正弦定理：a/sin A = b/sin B = c/sin C，或 sin A / a = sin B / b = sin C / c。余弦定理：a² = b² + c² − 2bc cos A。当已知两边及其夹角或三边时使用。
Area of a triangle using trigonometry: Area = ½ ab sin C. Exact trigonometric values for 0°, 30°, 45°, 60°, 90° must be memorised.
用三角学求三角形面积：面积 = ½ ab sin C。必须牢记 0°、30°、45°、60°、90° 的精确三角函数值。
| Angle θ | sin θ | cos θ | tan θ |
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
7. Mensuration: Area, Volume and Surface Area | 测量：面积、体积与表面积
Key formulas: area of triangle = ½ × base × height; area of circle = πr²; circumference = 2πr. Volume of prism = area of cross-section × length. Volume of cylinder = πr²h; cone = ⅓πr²h; sphere = 4/3 πr³.
关键公式：三角形面积 = ½ × 底 × 高；圆面积 = πr²；圆周长 = 2πr。棱柱体积 = 截面面积 × 长。圆柱体积 = πr²h；圆锥 = ⅓πr²h；球体 = 4/3 πr³。
Surface area of a cylinder: 2πr² + 2πrh. Surface area of a cone: πrl + πr² where l is slant height. Units conversion is vital: 1 m³ = 1 000 000 cm³, 1 litre = 1000 cm³.
圆柱表面积：2πr² + 2πrh。圆锥表面积：πrl + πr²，其中 l 为斜高。单位换算至关重要：1 m³ = 1 000 000 cm³，1 升 = 1000 cm³。
8. Statistics: Averages and Charts | 统计：平均数和图表
Mean = sum of values / number of values. Median = middle value when ordered. Mode = most frequent value. Range = largest − smallest. From grouped frequency, use midpoints to estimate the mean.
平均数 = 值的总和 / 值的个数。中位数 = 按序排列后的中间值。众数 = 出现最频繁的值。范围 = 最大值 − 最小值。对分组频数表，用组中点来估计平均数。
Interpret and construct bar charts, pie charts, histograms (frequency density = frequency / class width), cumulative frequency graphs and box-and-whisker plots (lower quartile, median, upper quartile, interquartile range).
解读并绘制条形图、饼图、直方图（频数密度 = 频数 / 组距）、累积频数曲线图和箱线图（下四分位数、中位数、上四分位数、四分位距）。
Scatter graphs show correlation; a line of best fit can be used for prediction. Only interpolate within the range of data.
散点图显示相关性；最佳拟合线可用于预测。仅限在数据范围内内插。
9. Probability | 概率
Probability of an event = (number of favourable outcomes) / (total number of outcomes). Probabilities can be expressed as fractions, decimals or percentages and must sum to 1 for exhaustive mutually exclusive events.
事件的概率 = （有利结果的数量）/（总结果的数量）。概率可以用分数、小数或百分比表示，且对于穷尽的互斥事件，其概率之和必为 1。
Tree diagrams help with combined events: multiply along branches for “and”, add probabilities of different branches for “or”. Conditional probability modifies the probability on the second branch based on the first outcome.
树状图有助于组合事件：沿着分支相乘表示“和”，将不同分支的概率相加表示“或”。条件概率根据第一个事件的结果修改第二条分支上的概率。
Venn diagrams and two-way tables are used to organise outcomes. The complement rule: P(A’) = 1 − P(A). Union and intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
文氏图和二元表格用于组织结果。补集法则：P(A’) = 1 − P(A)。并集与交集：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
10. Vectors and Transformations | 向量与变换
Vectors represent magnitude and direction; they are written as column vectors or with i, j notation. Addition, scalar multiplication and finding the magnitude |a| = √(x² + y²) are common tasks.
向量表示大小和方向；可写成列向量或使用 i、j 符号表示。向量的加法、标量乘法以及求模 |a| = √(x² + y²) 是常见任务。
Transformations: reflection (mirror line), rotation (centre, angle, direction), translation (column vector), enlargement (centre, scale factor). Negative scale factors produce an inverted image. Combinations of transformations require careful mapping.
变换：反射（对称轴）、旋转（旋转中心、角度、方向）、平移（列向量）、放大（中心、比例因子）。负比例因子产生倒像。变换的组合需要仔细作图。
For enlargement with scale factor k, side lengths multiply by k, area multiplies by k². For similar shapes, area ratio = (linear ratio)², volume ratio = (linear ratio)³.
在放大变换中，边长为 k 倍，面积变为 k² 倍。对于相似形，面积比 = (线性比)²，体积比 = (线性比)³。
11. Ratio, Proportion and Percentages | 比、比例与百分比
Writing ratios in simplest form, dividing a quantity in a given ratio, and solving proportion problems (direct and inverse) are essential. For direct proportion, y = kx; for inverse, y = k/x.
将比写成最简形式、按给定比例分配数量，以及解决正反比例问题至关重要。正比例：y = kx；反比例：y = k/x。
Percentage increase = (change / original) × 100%. Compound interest formula: Amount = P(1 + r/100)ⁿ. Depreciation works similarly with a minus sign.
增长百分比 = （变化量 / 原值）× 100%。复利公式：总金额 = P(1 + r/100)ⁿ。折旧同理，但使用减号。
Reverse percentages require working back to the original amount after a percentage change. Often tested in sale and tax contexts.
反向百分比需要从百分比变化后的金额回推原始金额。常在打折和税务情境中考查。
12. Sequences and Series | 数列
Finding the nth term of linear and quadratic sequences is a high-frequency topic. Linear nth term = dn + (a − d), where d is common difference and a is the first term.
求线性数列和二次数列的第 n 项是高频话题。线性第 n 项 = dn + (a − d)，其中 d 为公差，a 为首项。
Quadratic sequences have second difference constant. The nth term is of the form an² + bn + c. Set up equations using first few terms to find a, b, c.
二次数列具有恒定的二阶差。第 n 项形式为 an² + bn + c。利用前几项建立方程求出 a、b、c。
Arithmetic sequences from the extended syllabus: nth term = a + (n − 1)d, sum = n/2 [2a + (n − 1)d]. Geometric sequences are tested in the extended tier.
拓展课程中的等差数列：第 n 项 = a + (n − 1)d，和 = n/2 [2a + (n − 1)d]。等比数列在拓展级别中考查。
Published by TutorHao | IGCSE Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导