IGCSE Mathematics Revision Guide | IGCSE数学复习指南

📚 IGCSE Mathematics Revision Guide | IGCSE数学复习指南

This revision guide is designed for IGCSE Mathematics teachers and students. It covers the core topics that appear frequently in the examination, with concise explanations and key points for quick review.

本复习指南专为 IGCSE 数学教师与学生编写，涵盖考试中频繁出现的核心主题，以简明扼要的讲解和重点提示帮助快速复习。

1. Number and Algebra | 数与代数

Number work forms the foundation of IGCSE Mathematics. You must be confident with integers, fractions, decimals, percentages, ratios, and standard form.

数与代数是 IGCSE 数学的基础。你必须熟练掌握整数、分数、小数、百分比、比率和科学计数法。

Understand the order of operations (BIDMAS/BODMAS) and apply it correctly.

理解运算顺序（BIDMAS/BODMAS）并正确应用。
Convert between fractions, decimals, and percentages fluently.

熟练地进行分数、小数和百分比之间的转换。
Use standard form for very large or very small numbers, e.g., 3.2 × 10⁵.

使用科学计数法表示非常大或非常小的数，例如 3.2 × 10⁵。

a⁻ⁿ = 1/aⁿ, a⁰ = 1 (a ≠ 0)

Index laws are essential for simplifying expressions. Remember the rules for multiplying, dividing, and raising powers to powers.

指数法则对于化简表达式至关重要。记住相乘、相除以及幂的乘方的规则。

2. Equations and Inequalities | 方程与不等式

Solving linear equations is a required skill. You should also be able to rearrange formulas and solve simultaneous equations.

解线性方程是必备技能，你还应能够重新排列公式并解联立方程组。

For simultaneous equations, use substitution or elimination methods.

解联立方程时，可采用代入法或消元法。
Solve quadratic equations by factorisation, completing the square, or using the quadratic formula.

通过因式分解、配方法或求根公式解二次方程。

x = (−b ± √(b² − 4ac)) / 2a

Inequalities are solved similarly to equations, but remember that multiplying or dividing by a negative number reverses the inequality sign.

解不等式的方法与方程类似，但记得当乘以或除以负数时，不等号方向要改变。

3. Functions and Graphs | 函数与图像

Functions describe relationships between variables. You should understand function notation f(x), domain, and range.

函数描述变量之间的关系。你需要理解函数记号 f(x)、定义域和值域。

Know the shapes of linear, quadratic, cubic, reciprocal, and exponential graphs.

熟悉线性、二次、三次、反比例和指数函数图像的基本形状。
Sketch graphs by finding intercepts, turning points, and asymptotes.

通过求截距、驻点和渐近线来画函数草图。
Understand how transformations affect the graph: y = f(x) + a, y = f(x + a), y = −f(x), y = f(−x), and y = kf(x).

理解平移和伸缩变换对图像的影响：y = f(x) + a，y = f(x + a)，y = −f(x)，y = f(−x) 和 y = kf(x)。

y = f(x) + a moves the graph up by a units

4. Coordinate Geometry | 坐标几何

Coordinate geometry combines algebra and geometry. You must be able to find the gradient, midpoint, and length of a line segment.

坐标几何将代数与几何相结合。你必须能够求斜率、中点和线段的长度。

Gradient m = (y₂ − y₁) / (x₂ − x₁)

The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept.

直线方程可写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。

Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1.

平行直线斜率相等；垂直直线的斜率相乘等于 −1。
The distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²).

两点之间的距离为 √((x₂ − x₁)² + (y₂ − y₁)²)。

5. Geometry | 几何

Geometry includes angles, triangles, quadrilaterals, polygons, and circles. Know the angle properties and theorems.

几何包括角、三角形、四边形、多边形和圆。需要掌握角度性质与定理。

Angles on a straight line sum to 180°, angles at a point sum to 360°, and vertically opposite angles are equal.

直线上角的和为 180°，周角之和为 360°，对顶角相等。
The sum of interior angles of an n-sided polygon is (n − 2) × 180°.

n 边形的内角和为 (n − 2) × 180°。
Circle theorems: the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, and the angle in a semicircle is 90°.

圆定理：圆心角是圆周角的两倍，同弧上的圆周角相等，半圆内的圆周角为 90°。

6. Trigonometry | 三角学

Trigonometry is used to find missing sides and angles in right-angled triangles. The three basic ratios are sine, cosine, and tangent.

三角学用于求直角三角形中的未知边和角。三个基本比值为正弦、余弦和正切。

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent

For non-right-angled triangles, use the sine rule and cosine rule.

对于非直角三角形，使用正弦定理和余弦定理。

a/sin A = b/sin B = c/sin C, c² = a² + b² − 2ab cos C

You should also know the exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°.

你还应记住 0°、30°、45°、60°、90° 的正弦、余弦和正切精确值。

7. Vectors and Transformations | 向量与变换

Vectors represent quantities with magnitude and direction. You need to add, subtract, and multiply vectors by a scalar.

向量表示具有大小和方向的量。你需要进行向量的加法、减法以及数与向量的乘法。

A column vector ⎛x⎞ ⎝y⎠ describes movement right x and up y.

列向量 ⎛x⎞ ⎝y⎠ 表示向右移动 x 和向上移动 y。
Know the transformations: translation, reflection, rotation, and enlargement.

掌握四种变换：平移、反射、旋转和放缩。
Describe transformations using column vectors, mirror lines, angles, and scale factors.

使用列向量、对称轴、角度和比例因子描述变换。

8. Mensuration | 测量

Mensuration involves calculating lengths, areas, volumes, and surface areas. Memorise the standard formulas.

测量涉及长度、面积、体积和表面积的计算。请牢记标准公式。

Area of a triangle = ½ × base × height; area of a trapezium = ½(a + b)h.

三角形面积 = ½ × 底 × 高；梯形面积 = ½(a + b)h。
Circumference of a circle = 2πr; area of a circle = πr².

圆周长 = 2πr；圆面积 = πr²。
Volume of a prism = area of cross-section × length; volume of a cylinder = πr²h.

棱柱体积 = 横截面积 × 长度；圆柱体积 = πr²h。

For arcs and sectors, use the fraction of 360°: arc length = (θ/360) × 2πr, sector area = (θ/360) × πr².

对于弧和扇形，使用角度占 360° 的比例：弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。

9. Statistics | 统计

Statistics deals with data collection, representation, and interpretation. Understand mean, median, mode, and range.

统计学涉及数据的收集、表示和解释。理解平均数、中位数、众数和极差。

For grouped data, estimate the mean using mid-interval values.

对于分组数据，使用组中值估计平均数。
Calculate the interquartile range (IQR) = Q₃ − Q₁ to measure spread.

计算四分位距 IQR = Q₃ − Q₁ 来衡量离散程度。
Read and interpret stem-and-leaf diagrams, histograms, and cumulative frequency graphs.

读懂并解释茎叶图、直方图和累积频率图。

10. Probability | 概率

Probability measures the chance of an event occurring. It ranges from 0 to 1, or from 0% to 100%.

概率衡量事件发生的可能性，范围从 0 到 1，或从 0% 到 100%。

P(event) = (number of favourable outcomes) / (total number of outcomes)

For independent events, multiply probabilities: P(A and B) = P(A) × P(B). For mutually exclusive events, add probabilities: P(A or B) = P(A) + P(B).

对于独立事件，概率相乘：P(A 且 B) = P(A) × P(B)。对于互斥事件，概率相加：P(A 或 B) = P(A) + P(B)。

Use tree diagrams and Venn diagrams to solve compound probability problems.

使用树形图和韦恩图解决复合概率问题。

Mastering these core topics will give you a solid foundation for the IGCSE Mathematics examination. Regular practice and past paper review are the keys to success.

掌握这些核心主题将为你在 IGCSE 数学考试中打下坚实基础。定期练习和刷历年真题是成功的关键。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IGCSE Mathematics Revision Guide | IGCSE数学复习指南