📚 IGCSE Maths: End-of-Term Revision Outline | IGCSE 数学：期末复习提纲
This revision guide condenses the entire IGCSE Mathematics syllabus into ten focused sections. Use it to check your understanding, practise worked examples, and identify any topics needing extra attention before the exam. Each section pairs an English explanation with a Chinese translation so you can study in either language.
这份复习提纲将 IGCSE 数学全部内容浓缩为十个核心板块。你可以用它来检查理解程度、练习例题，并在考前找出需要额外强化的知识点。每个板块都提供了中英双语解释，方便你选择任一语言进行复习。
1. Number | 数与运算
Master directed numbers, fractions, decimals and percentages. Be able to convert freely between fractions, decimals and percentages, and apply them in real‑life contexts such as percentage increase, decrease and reverse percentage problems.
掌握有向数、分数、小数和百分数。能够自如地在分数、小数和百分数之间进行转换，并应用在实际场景中，例如百分数的增减和逆向百分数问题。
The laws of indices are fundamental: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Know how to handle zero, negative and fractional indices, and how to express numbers in standard form A × 10ⁿ.
指数法则是基础：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。要理解零指数、负指数和分数指数的含义，并会使用科学记数法 A × 10ⁿ。
Work with surds and upper/lower bounds. Simplify expressions like √50 = 5√2 and calculate bounds when values are given to a certain degree of accuracy. Understand ratio, proportion and direct/inverse variation.
处理根式与上下界。化简如 √50 = 5√2 的表达式，并在数值为给定精度时计算上下界。理解比、比例以及正比和反比关系。
2. Algebra Basics | 代数基础
Simplify algebraic expressions by collecting like terms, expanding brackets and factorising. Recognise common factorisations: difference of two squares a² – b² = (a – b)(a + b), and trinomials such as x² + bx + c.
通过合并同类项、展开括号和因式分解化简单数式。记住常用因式分解：平方差 a² – b² = (a – b)(a + b)，以及二次三项式如 x² + bx + c。
Manipulate formulae by changing the subject. Rearrange equations step by step, treating the subject as a variable to be isolated. Use function notation f(x) and find inverse functions f⁻¹(x) as well as composite functions fg(x).
会变换公式主项。逐步整理方程，将目标字母视为需要分离的变量。使用函数记号 f(x)，会求反函数 f⁻¹(x) 和复合函数 fg(x)。
Sequences: find the nth term of linear and simple quadratic sequences. A linear sequence has nth term an + b; determine a (common difference) and b by substitution.
数列：会求一次和简单二次数列的通项公式。一次数列的通项为 an + b，通过公差求 a，再代入求 b。
3. Equations and Inequalities | 方程与不等式
Solve linear equations, including those with brackets and fractions. Always perform the same operation on both sides. For simultaneous equations, use elimination or substitution confidently.
解一元一次方程，包括带括号和分数的方程。始终在等式两边进行相同运算。对于联立方程组，熟练使用消元法或代入法。
Quadratic equations are solved by factorising first; if factorisation is messy, use the quadratic formula:
x = [-b ± √(b² – 4ac)] / 2a
二次方程优先尝试因式分解；若因式分解困难，则使用求根公式：x = [-b ± √(b² – 4ac)] / 2a。
For inequalities, remember that multiplying or dividing by a negative number reverses the sign. Represent solutions on a number line, using open or closed circles, and write them in set notation or as intervals.
解不等式时注意，乘或除以负数要反转不等号。在数轴上表示解集，用空心或实心圆点，并用集合记号或区间表示。
Graphical inequalities: shade the unwanted region when dealing with multiple linear inequalities; the feasible region is left unshaded, bounded by solid or dashed lines.
图解不等式组：处理多个一次不等式时，习惯涂掉不需要的区域，可行域保持不涂色，边界线用实线（含等号）或虚线。
4. Graphs of Functions | 函数图像
Plot and sketch linear, quadratic, cubic, reciprocal and exponential graphs. Recognise the shapes: straight line for y = mx + c, parabola for y = x², hyperbola for y = 1/x, and steady growth for y = aˣ.
能绘制并勾勒一次、二次、三次、反比例和指数函数的图像。记住形状：y = mx + c 是直线，y = x² 是抛物线，y = 1/x 是双曲线，y = aˣ 表示稳定增长或衰减。
Gradients and intercepts: the gradient m is the vertical change divided by horizontal change. The y‑intercept is the point where the graph crosses the y‑axis. For any curve, the gradient at a point can be estimated by drawing a tangent.
斜率与截距：斜率 m 定义为纵坐标的变化除以横坐标的变化。y 轴截距是图像与 y 轴交点的纵坐标。对于任何曲线，某点的斜率可以通过画切线来估算。
Understand graph transformations: f(x) + a moves the graph up; f(x + a) moves it left; –f(x) reflects in the x‑axis; f(–x) reflects in the y‑axis; a f(x) stretches vertically.
理解图像变换：f(x) + a 向上平移；f(x + a) 向左平移；–f(x) 关于 x 轴对称；f(–x) 关于 y 轴对称；a f(x) 纵向拉伸。
5. Geometry | 几何
Know angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, vertically opposite angles are equal, angles in a triangle sum to 180°, and the exterior angle of a triangle equals the sum of the two opposite interior angles.
掌握角度定理：平角等于 180°，周角等于 360°，对顶角相等，三角形内角和 180°，三角形的外角等于两个不相邻内角之和。
Parallel lines: corresponding angles are equal, alternate angles are equal, and interior (co‑interior) angles sum to 180°. Apply these to find missing angles in diagrams with several intersections.
平行线性质：同位角相等，内错角相等，同旁内角互补（180°）。在多线交汇的图形中综合运用这些性质求未知角。
Circle theorems (for Extended): 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°; tangent meets radius at 90°. Prove simple geometric statements.
圆定理（Extended 课程）：圆心角是圆周角的两倍；同弧上的圆周角相等；直径所对的圆周角是直角；圆内接四边形对角互补；切线与半径垂直。能证明简单的几何命题。
6. Mensuration | 测量
Be fluent with area and perimeter for rectangles, triangles, parallelograms, trapeziums and circles. Use the table below for quick reference.
熟练掌握矩形、三角形、平行四边形、梯形和圆的面积与周长。速查公式见下表。
| Shape | Perimeter / Circumference | Area |
|---|---|---|
| Rectangle | 2(l + w) | l × w |
| Triangle | a + b + c | ½ × b × h |
| Parallelogram | 2(a + b) | b × h |
| Trapezium | Sum of all sides | ½(a + b) × h |
| Circle | πd or 2πr | πr² |
For 3D shapes, calculate surface area and volume. Key formulas: prism volume = area of cross‑section × length; cylinder volume = πr²h; cone volume = ⅓πr²h; sphere volume = ⁴⁄₃πr³. Arc length = θ/360 × 2πr, sector area = θ/360 × πr².
三维图形的表面积与体积：棱柱体积 = 截面积 × 长；圆柱体积 = πr²h；圆锥体积 = ⅓πr²h；球体体积 = ⁴⁄₃πr³。弧长公式：θ/360 × 2πr；扇形面积：θ/360 × πr²。
Work with compound measures: speed = distance / time, density = mass / volume, pressure = force / area. Convert units, e.g. km/h to m/s by multiplying by ⁵⁄₁₈.
处理复合量：速度 = 距离/时间，密度 = 质量/体积，压强 = 压力/面积。注意单位换算，例如 km/h 转 m/s 需要乘以 ⁵⁄₁₈。
7. Trigonometry | 三角学
In right‑angled triangles, use SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Remember exact values for 0°, 30°, 45°, 60° and 90° derived from special triangles.
在直角三角形中使用 SOH CAH TOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。记住 0°、30°、45°、60°、90° 特殊角的精确值，可由特殊三角形推导。
For non‑right‑angled triangles, apply the sine rule a/sin A = b/sin B = c/sin C (used when given two angles and a side, or two sides and a non‑included angle) and the cosine rule a² = b² + c² – 2bc cos A (used for two sides and the included angle, or three sides).
对于非直角三角形，使用正弦定理 a/sin A = b/sin B = c/sin C（适用于已知两角一边或两边一对角），以及余弦定理 a² = b² + c² – 2bc cos A（适用于已知两边及其夹角或三边）。
Area of a triangle can be found using ½ ab sin C when two sides and the included angle are known. Bearings are measured clockwise from north, always given as three digits.
已知两边及其夹角时，三角形面积可用公式 ½ ab sin C 求得。方位角从正北按顺时针方向测量，总以三位数表示。
3‑D trigonometry: identify right‑angled triangles within a solid, often using Pythagoras first to find a missing length, then apply trigonometric ratios.
立体三角学：在立体图形中找出直角三角形，经常需要先用勾股定理求出未知边长，再使用三角比。
8. Vectors and Transformations | 向量与变换
A vector has both magnitude and direction. It can be written as a column vector (x above y) or as a combination of i and j. Add and subtract vectors, and multiply by a scalar. Find the magnitude of a vector (x, y) as √(x² + y²).
向量既有大小又有方向。可表示为列向量（x 在上 y 在下）或 i、j 的组合形式。能进行向量的加减和数乘。向量 (x, y) 的模为 √(x² + y²)。
Understand transformations: translation (described by a vector), reflection (state the mirror line), rotation (give centre, angle and direction), and enlargement (specify centre and scale factor). For a fractional or negative scale factor, the image size and orientation change accordingly.
理解四种变换：平移（由向量描述）、反射（指明镜面线）、旋转（给出旋转中心、角度和方向）和放大（指明中心和比例因子）。当比例因子为分数或负数时，像的大小和朝向会相应变化。
Describe combined transformations as a single transformation where possible. Recognise that the order of transformations affects the result.
尽可能将组合变换描述为单一变换。要注意变换的顺序影响最终结果。
9. Probability | 概率
Probability is a measure between 0 and 1. For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes. The sum of probabilities of all possible outcomes is 1.
概率是介于 0 和 1 之间的度量。对于等可能结果，P(事件) = 有利结果数 / 总结果数。所有可能结果的概率之和为 1。
Use sample space diagrams, Venn diagrams and tree diagrams to organise outcomes. On a probability tree, multiply along branches and add probabilities of mutually exclusive events. Remember that for independent events, P(A and B) = P(A) × P(B).
会使用样本空间图、韦恩图和树状图整理结果。在树状图上，沿分支相乘，互斥事件概率相加。记住独立事件满足 P(A 且 B) = P(A) × P(B)。
Conditional probability: P(A given B) = P(A and B) / P(B). Read questions carefully to decide whether events are independent or conditional.
条件概率：P(A|B) = P(A 且 B) / P(B)。仔细审题，判断事件是独立还是条件相关。
10. Statistics | 统计
Collect and represent data using bar charts, pie charts, histograms, frequency polygons and cumulative frequency graphs. For grouped data, the histogram uses frequency density = frequency / class width on the vertical axis.
收集数据并用条形图、饼图、直方图、频数多边形和累积频数图表示。对于分组数据，直方图的纵轴为频数密度 = 频数 / 组距。
Averages: mean, median, mode and range. The median is the (n + 1)/2th value in an ordered list; in grouped data, find the median class from cumulative frequency. The modal class has the highest frequency. Estimate the mean of grouped data using midpoints × frequency.
平均数：平均数、中位数、众数和极差。中位数是有序数列中第 (n + 1)/2 个数值；对于分组数据，从累积频数图中确定中位数组。众数类是频数最高的组。用组中值 × 频数估计分组数据的平均数。
Measures of spread: interquartile range IQR = upper quartile – lower quartile. Draw and interpret box‑and‑whisker plots. Scatter graphs show correlation; add a line of best fit to make predictions.
离散程度：四分位距 IQR = 上四分位数 – 下四分位数。会绘制并解释盒须图。散点图展示相关性，添加最佳拟合线可进行预测。
Always check for misleading graphs: axes that do not start at zero, inconsistent scales, or exaggerated pictograms can distort the message.
随时检查被错误呈现的统计图：坐标轴不是从零开始、比例不一致或夸大的象形图都可能扭曲信息。
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