📚 IGCSE Mathematics Revision Guide: Key Topics and Exam Strategies | IGCSE数学复习指南：核心考点与考试策略
This guide covers the essential topics in the IGCSE Mathematics syllabus, from number and algebra to geometry, statistics, and exam techniques. Each section is designed to help you consolidate your understanding and improve your performance in both Paper 1 and Paper 2.
本指南涵盖IGCSE数学课程的核心主题，包括数与算术、代数、几何、统计以及考试技巧。每一节内容旨在帮助你巩固理解，并在Paper 1和Paper 2中取得更好的表现。
1. Number and Arithmetic | 数与算术
Understanding the number system is the foundation of IGCSE Mathematics. You are expected to work confidently with integers, fractions, decimals, percentages, ratios, and standard form.
理解数系是IGCSE数学的基础。你需要熟练处理整数、分数、小数、百分数、比例和科学计数法。
Order of operations: Use BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) to simplify expressions correctly.
运算顺序：使用BIDMAS（括号、指数、乘除、加减）来正确化简表达式。
Standard form: Write very large or very small numbers as A × 10n — in this guide we write it as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
科学计数法：将非常大或非常小的数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。
Percentages: Convert between percentages, fractions, and decimals. Know how to calculate percentage increase and decrease.
百分数：在百分数、分数和小数之间转换。掌握计算百分数增加和减少的方法。
Percentage change = (new value − original value) ÷ original value × 100%
百分数变化 = （新值 − 原值）÷ 原值 × 100%
2. Algebra: Expressions and Equations | 代数：表达式与方程
Algebra is a major part of IGCSE Mathematics. You should be able to manipulate algebraic expressions, solve linear and quadratic equations, and rearrange formulas.
代数是IGCSE数学的重要部分。你需要能够运算代数表达式、解线性方程和二次方程，并重新排列公式。
Simplifying expressions: Collect like terms, expand brackets, and factorise common factors.
化简表达式：合并同类项、展开括号、提取公因式。
uations: Use inverse operations to isolate the variable. For example, 3x + 5 = 14 → x = 3.
解线性方程：使用逆运算分离变量。例如：3x + 5 = 14 → x = 3。
Solving quadratic equations: Factorise, use completing the square, or apply the quadratic formula.
解二次方程：使用因式分解、配方法或求根公式。
If ax² + bx + c = 0, then x = [−b ± √b² − 4ac] ÷ (2a)
若 ax² + bx + c = 0，则 x = [−b ± √b² − 4ac] ÷ (2a)
Remember: a2 - b2 is a special factorisation. It equals (a-b)(a+b). We write this as a² − b² = (a − b)(a + b).
注意： a2 - b2 是特殊因式分解。它等于 (a-b)(a+b)。我们用 a² − b² = (a − b)(a + b) 表示。
3. Coordinate Geometry | 坐标几何
Coordinate geometry connects algebra and geometry through graphs. You need to know how to plot points, find gradients, and write equations of straight lines.
坐标几何通过图像将代数与几何联系起来。你需要知道如何描点、求斜率以及写出直线方程。
ent formula: For two points (x₁, y₁) and (x₂, y₂), the gradient m = (y₂ − y₁) ÷ (x₂ − x₁).
斜率公式：对于两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m = (y₂ − y₁) ÷ (x₂ − x₁)。
a line: The general form is y = mx + c, where m is the gradient and c is the y-intercept.
直线方程：一般形式为 y = mx + c，其中 m 是斜率，c 是截距。
Parallel and perpendicular lines: Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1.
平行与垂直直线：平行线斜率相等。垂直线斜率相乘等于 −1。
Midpoint of (x₁, y₁) and (x₂, y₂) = ( (x₁ + x₂)÷2 , (y₁ + y₂)÷2 )
点 (x₁, y₁) 和 (x₂, y₂) 的中点 = ( (x₁ + x₂)÷2 , (y₁ + y₂)÷2 )
Distance between two points is found using Pythagoras: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
两点间距离使用勾股定理计算：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
4. Functions and Graphs | 函数与图像
Functions are a key concept in IGCSE Mathematics. You should know function notation, domain and range, and how to interpret different types of graphs.
函数是IGCSE数学的核心概念。你需要了解函数符号、定义域和值域，以及如何解读不同类型的图像。
nction notation: f(x) = 2x + 3 means that the input x is multiplied by 2, then 3 is added.
函数符号：f(x) = 2x + 3 表示输入 x 先乘以 2，再加 3。
ans apply g first, then f. For example, f(x) = x² and g(x) = x + 1, then fg(2) = f(3) = 9.
复合函数：fg(x) 表示先算 g，再算 f。例如 f(x) = x²，g(x) = x + 1，则 fg(2) = f(3) = 9。
Inverse functions: The inverse of f(x) is written f⁻¹(x). It reverses the operation of f.
反函数：f(x) 的反函数写作 f⁻¹(x)。它反转 f 的运算。
(straight line), quadratic (parabola), cubic, reciprocal (y = k/x), and exponential (y = aˣ).
你应该认识的图像：线性（直线）、二次（抛物线）、三次、反比例（y = k/x）以及指数（y = aˣ）。
The graph of f⁻¹(x) is the reflection of f(x) in the line y = x.
f⁻¹(x) 的图像是 f(x) 关于直线 y = x 的镜像。
5. Geometry and Trigonometry | 几何与三角学
Geometry requires reasoning about angles, shapes, and properties. Trigonometry deals with right-angled triangles and the sine/cosine rules for non-right triangles.
几何需要推理角度、形状和性质。三角学研究直角三角形以及非直角三角形的正弦定理和余弦定理。
Angle properties: Angles on a straight line sum to 180°, angles in a triangle sum to 180°, and angles in a quadrilateral sum to 360°.
角度性质：直线上的角之和为 180°，三角形内角和为 180°，四边形内角和为 360°。
Pythagoras theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
勾股定理：在直角三角形中，a² + b² = c²，其中 c 为斜边。
sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
三角比：sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边。
Sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C
余弦定理: c² = a² + b² − 2ab cos C
Use the sine rule when you know a side and its opposite angle. Use the cosine rule when you know two sides and the included angle, or three sides.
已知一条边和它的对角时使用正弦定理。已知两边及夹角或三边时使用余弦定理。
6. Vectors and Transformations | 向量与变换
Vectors describe magnitude and direction. Transformations include translation, reflection, rotation, and enlargement. Both topics are commonly tested in IGCSE.
向量描述大小和方向。变换包括平移、反射、旋转和放大。这两个主题在IGCSE考试中经常出现。
Vector notation: A vector can be written as a column vector binomab or using bold lowercase letters. In this guide we use the form [a b] or a = (a, b).
向量符号：向量可以写成列向量 binomab 或粗体小写字母。在本指南中我们用 [a b] 或 a = (a, b) 表示。
Adding and subtracting vectors: Add or subtract corresponding components.
向量加法与减法：对应分量相加或相减。
Enlargement: The scale factor k multiplies distances from the centre of enlargement. If k is negative, the image is on the opposite side of the centre.
放大：比例因子 k 乘以到放大中心的距离。如果 k 为负，像在中心的另一侧。
If vector p = (a, b) and vector q = (c, d), then p + q = (a + c, b + d)
若向量 p = (a, b)，q = (c, d)，则 p + q = (a + c, b + d)
Remember: The transformation that maps an object to its image must be described fully, including the equation of the mirror line, the angle and direction of rotation, or the centre and scale factor of enlargement.
注意：描述将原图形映射到像的变换时，必须完整说明，包括对称轴方程、旋转角度与方向，或放大的中心与比例因子。
7. Mensuration: Area, Perimeter and Volume | 测量：面积、周长与体积
Mensuration involves calculating lengths, areas, and volumes of 2D and 3D shapes. You must know the standard formulas and also how to apply them to composite shapes.
测量涉及计算二维和三维图形的长度、面积和体积。你需要掌握标准公式，并能够将其应用于组合图形。
mulas: Area = πr², circumference = 2πr. Also know the arc length and sector area formulas.
圆公式：面积 = πr²，周长 = 2πr。同时掌握弧长和扇形面积公式。
Prism volume: Volume = cross-sectional area × length.
棱柱体积：体积 = 横截面积 × 长度。
surface area = 2πr² + 2πrh. For a cone, surface area = πr² + πrl, where l is slant height.
表面积：圆柱表面积 = 2πr² + 2πrh。圆锥表面积 = πr² + πrl，其中 l 是斜高。
Arc length = (θ ÷ 360°) × 2πr, Area of sector = (θ ÷ 360°) × πr²
弧长 = (θ ÷ 360°) × 2πr，扇形面积 = (θ ÷ 360°) × πr²
Remember that 1 m³ = 1,000,000 cm³, and 1 litre = 1000 cm³.
注意：1 m³ = 1,000,000 cm³，1 升 = 1000 cm³。
8. Probability | 概率
Probability measures the likelihood of an event occurring. The IGCSE syllabus covers theoretical probability, tree diagrams, and conditional probability.
概率衡量事件发生的可能性。IGCSE课程涵盖理论概率、树状图和条件概率。
ility: P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.
基本概率：P(事件) = 有利结果数 ÷ 所有等可能结果总数。
The sum of probabilities of all possible outcomes is 1. The probability of an event not happening is 1 − P(event).
所有可能结果的概率之和等于 1。事件不发生的概率为 1 − P(事件)。
Tree diagrams: Multiply along branches for independent events, and add the probabilities of mutually exclusive paths.
树状图：独立事件沿支路相乘，互斥路径的概率相加。
P(A and B) = P(A) × P(B) if A and B are independent
若 A 与 B 独立，则 P(A 且 B) = P(A) × P(B)
For conditional probability, use P(A given B) = P(A and B) ÷ P(B).
对于条件概率，使用 P(A | B) = P(A 且 B) ÷ P(B)。
9. Statistics | 统计
Statistics includes data handling, averages, and measures of spread. You must be able to interpret and draw diagrams such as bar charts, histograms, and cumulative frequency graphs.
统计包括数据处理、平均值和离散程度度量。你需要能够解读和绘制条形图、直方图和累积频率图等图表。
Mean, median, and mode: The mean is the sum of all data divided by the number of values. The median is the middle value when data is ordered. The mode is the most frequent value.
平均数、中位数与众数：平均数等于所有数据之和除以数据个数。中位数是有序数据中的中间值。众数是出现次数最多的值。
Range: Range = largest value − smallest value. It measures spread.
极差：极差 = 最大值 − 最小值。它衡量离散程度。
Cumulative frequency: Plot cumulative frequency against upper boundaries. Use the resulting graph to estimate the median and quartiles.
累积频率：以累积频率对上边界作图。利用所得图估计中位数和四分位数。
Interquartile range = Upper quartile − Lower quartile
四分位距 = 上四分位数 − 下四分位数
When drawing a histogram, the area of each bar is proportional to the frequency, not the height.
绘制直方图时，每根条形的面积与频率成正比，而不是高度。
10. Exam Techniques and Common Mistakes | 考试技巧与常见错误
Many students lose marks not because they do not know the content, but because of simple errors. Learning good exam technique is essential to achieve a high grade.
许多学生失分不是因为他们不懂内容，而是因为简单错误。掌握良好的考试技巧对于取得高分至关重要。
Show all working: In IGCSE Mathematics, method marks are awarded even if the final answer is wrong. Write down each step clearly.
写出全部过程：在IGCSE数学中，即使最终答案错误，步骤分也会给分。请清晰写出每一步。
Check significant figures and decimal places: Read the question to see how you should round your answer. Often it says "give your answer to 3 significant figures".
检查有效数字和小数位：读题时注意要求如何四舍五入。通常题目会写明"精确到3位有效数字"。
Use the calculator effectively: Know how to use brackets, sine, cosine, tangent, and the standard form function on your calculator.
有效使用计算器：了解如何使用括号、正弦、余弦、正切和科学计数法功能。
A common mistake is confusing area with perimeter, or volume with surface area. Always check the units: cm² for area, cm³ for volume.
常见错误包括混淆面积与周长，或体积与表面积。务必检查单位：面积用 cm²，体积用 cm³。
Always read the question twice, and underline the key values and the required output.
务必读题两遍，并在关键数值和所求答案下划线。
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