📚 IGCSE Math 0607 Key Topics Explained | IGCSE 数学 0607 知识点精讲
The Cambridge IGCSE International Mathematics (0607) syllabus is designed to build a strong foundation in mathematical knowledge and skills, with an emphasis on applying concepts to real-world contexts and using graphical calculators effectively. This article provides a focused revision of the core topics, from number operations to probability, highlighting key definitions, formulae, and common problem-solving strategies. Whether you are preparing for the Core or Extended tier, mastering these essentials will boost your confidence and performance.
剑桥 IGCSE 国际数学（0607）大纲旨在为数学知识和技能打下坚实基础，强调将概念应用于实际情境并有效使用图形计算器。本文聚焦核心主题，从数的运算到概率，重点梳理关键定义、公式以及常见解题策略。无论你备考的是核心卷还是扩展卷，掌握这些要点都将增强你的信心和表现。
1. Number and Set Language | 数与集合语言
Understand the classification of numbers: natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), and irrational numbers such as √2 or π. Be able to work with prime factors, highest common factor (HCF), lowest common multiple (LCM), and directed numbers. Set notation, including A ∪ B (union), A ∩ B (intersection), and A’ (complement), is essential for logical reasoning.
理解数的分类：自然数(ℕ)、整数(ℤ)、有理数(ℚ)和无理数，如√2或π。能够处理质因数、最大公因数(HCF)、最小公倍数(LCM)以及有向数。集合符号，包括 A ∪ B（并集）、A ∩ B（交集）和 A’（补集），对逻辑推理至关重要。
For fractions, decimals, and percentages, converting between forms and applying them to increase/decrease problems is fundamental. Standard form (scientific notation) is written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Calculations with surds involve rationalising the denominator, e.g., 1/√2 = √2/2.
对于分数、小数和百分比，相互转换并应用于增减问题是基础。标准形式（科学记数法）写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。根式运算涉及分母有理化，例如 1/√2 = √2/2。
- Prime factorisation: 216 = 2³ × 3³
- 质因数分解: 216 = 2³ × 3³
- Upper and lower bounds: if a length is given as 8.4 cm to 1 decimal place, bounds are 8.35 cm and 8.45 cm
- 上界和下界: 若长度给出 8.4 cm（精确至 1 位小数），上下界为 8.35 cm 和 8.45 cm
2. Algebra: Manipulation and Equations | 代数：运算与方程
Algebraic manipulation is at the heart of IGCSE Mathematics. You must be confident in expanding brackets, factorising expressions (including quadratics), and simplifying algebraic fractions. Key identities: (a + b)² = a² + 2ab + b² and a² − b² = (a − b)(a + b).
代数运算是 IGCSE 数学的核心。你必须熟练掌握去括号、因式分解（包括二次式）和化简代数分式。关键恒等式：(a + b)² = a² + 2ab + b² 以及 a² − b² = (a − b)(a + b)。
Solving linear equations may involve fractions and unknowns on both sides. For quadratic equations, use factorisation, completing the square, or the quadratic formula x = [−b ± √(b² − 4ac)] / 2a. Simultaneous equations can be solved by substitution, elimination, or graphically.
解线性方程可能涉及分数和含未知数的两边。对于二次方程，可使用因式分解、配方法或求根公式 x = [−b ± √(b² − 4ac)] / 2a。联立方程可通过代入法、消元法或图解法求解。
Quadratic Formula: x = (−b ± √(b² − 4ac)) / 2a
Inequalities are solved similarly to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions on a number line.
不等式的解法与方程类似，但当乘以或除以负数时，切记要反转不等号。在数轴上表示解的区间。
3. Functions | 函数
A function relates each input (x) to exactly one output f(x). The domain is the set of all possible inputs, and the range is the set of all possible outputs. For IGCSE 0607, you must be able to evaluate functions, find inverse functions f⁻¹(x), and form composite functions fg(x) = f(g(x)).
函数将每个输入(x)对应到恰好一个输出 f(x)。定义域是所有可能输入的集合，值域是所有可能输出的集合。在 0607 中，你必须能够求函数值、求反函数 f⁻¹(x) 以及构成复合函数 fg(x) = f(g(x))。
The inverse function is found by writing y = f(x), swapping x and y, and solving for y. The domain of f⁻¹ is the range of f. A function only has an inverse if it is one-to-one.
求反函数的方法是写出 y = f(x)，交换 x 和 y，再解出 y。f⁻¹ 的定义域是 f 的值域。函数只有一一对应时才有反函数。
Graphs of functions such as linear, quadratic, cubic, reciprocal, exponential, and trigonometric must be recognised. Transformations of graphs – translations, reflections, and stretches – change the equation systematically.
必须认识函数图像，如一次、二次、三次、反比例、指数和三角函数图像。图像的变换——平移、反射和伸缩——会系统地改变函数式。
| Transformation | Effect on y = f(x) |
|---|---|
| Translation k units right | y = f(x – k) |
| Reflection in x-axis | y = -f(x) |
| Stretch vertically by factor a | y = a f(x) |
4. Coordinate Geometry | 坐标几何
Working with points on the Cartesian plane involves calculating distance, midpoint, and gradient. The distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
在笛卡尔坐标系中处理点需要计算距离、中点和斜率。(x₁, y₁) 和 (x₂, y₂) 之间的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The gradient of a line is m = (y₂ − y₁)/(x₂ − x₁). The equation of a straight line is often written as y = mx + c or y − y₁ = m(x − x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
直线的斜率是 m = (y₂ − y₁)/(x₂ − x₁)。直线方程常写作 y = mx + c 或 y − y₁ = m(x − x₁)。平行线斜率相等；垂直线的斜率满足 m₁ × m₂ = −1。
To find the equation of a perpendicular bisector, first find the midpoint of the segment and then use the negative reciprocal of the original gradient. Interpret the y-intercept and x-intercept from the equation or graph.
求垂直平分线的方程，先求线段中点，然后使用原斜率的负倒数。从方程或图像中解读 y 轴截距和 x 轴截距。
5. Geometry and Circle Theorems | 几何与圆定理
Geometry topics include angles in parallel lines, triangles, quadrilaterals, and polygons. The sum of interior angles of an n-sided polygon is (n − 2) × 180°. Exterior angles sum to 360°.
几何主题包括平行线中的角、三角形、四边形和多边形。n 边多边形的内角和为 (n − 2) × 180°。外角和恒为 360°。
Circle theorems are a major part of the syllabus. You need to know and apply: angle at centre is twice angle at circumference; angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°; the tangent is perpendicular to the radius at the point of contact; and the alternate segment theorem.
圆定理是大纲的重要部分。你需要知道并应用：圆心角等于圆周角的两倍；同弧上的圆周角相等；半圆上的圆周角为 90°；圆内接四边形对角互补；切线与过切点的半径垂直；以及弦切角定理。
Similarity and congruence (SSS, SAS, ASA, RHS) are used to prove relationships in length and area. The scale factor for areas of similar shapes is k², and for volumes is k³.
相似和全等（SSS、SAS、ASA、RHS）用于证明长度和面积关系。相似图形面积的比尺因子为 k²，体积的比尺因子为 k³。
6. Trigonometry | 三角学
Trigonometry begins with right‑angled triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Exact values for 30°, 45°, and 60° should be memorised.
三角学从直角三角形开始：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。应记住 30°、45° 和 60° 的准确值。
The sine rule (a/sin A = b/sin B = c/sin C) and cosine rule (a² = b² + c² − 2bc cos A) allow you to solve any triangle. The area formula (1/2)ab sin C is used for non‑right triangles.
正弦定理 (a/sin A = b/sin B = c/sin C) 和余弦定理 (a² = b² + c² − 2bc cos A) 可用于解任意三角形。面积公式 (1/2)ab sin C 用于非直角三角形。
Trigonometric graphs of y = sin x, y = cos x, y = tan x have specific shapes and periods. Transformations of these graphs follow the same rules as functions. Solving trigonometric equations within a given domain often requires considering the CAST diagram.
三角函数的图像 y = sin x、y = cos x、y = tan x 有特定的形状和周期。这些图像的变换遵守与一般函数相同的规则。在给定范围内解三角方程常需使用 CAST 图。
7. Vectors | 向量
A vector has both magnitude and direction, represented as a column vector v = (x, y) or using i, j notation. Vector operations include addition, subtraction, and multiplication by a scalar. The magnitude of vector v = (x, y) is |v| = √(x² + y²).
向量既有大小又有方向，可用列向量 v = (x, y) 或 i、j 符号表示。向量运算包括加法、减法和数乘。向量 v = (x, y) 的大小为 |v| = √(x² + y²)。
Parallel vectors are scalar multiples of each other. Position vectors describe the location of a point relative to the origin O. AB = OB − OA is used to find the vector between two points. Collinearity can be proven by showing that vectors between points are parallel.
平行向量互为标量的倍数。位置向量描述某点相对于原点 O 的位置。AB = OB − OA 用于求两点间的向量。通过证明点间向量平行，可推断共线。
In geometry, vectors solve ratio problems and prove that lines are parallel or points divide a line segment in a given ratio. The midpoint of AB is (1/2)(a + b).
在几何中，向量可解决比例问题，并证明直线平行或点按给定比例分割线段。AB 的中点为 (1/2)(a + b)。
8. Transformations | 变换
Transformations change the position, shape, or size of a figure. They include translation (sliding), reflection (flipping over a mirror line), rotation (turning about a centre), and enlargement (rescaling about a centre with a scale factor).
变换改变图形的位置、形状或大小。包括平移（滑动）、反射（沿镜线翻转）、旋转（绕中心转动）和放大（绕中心按比例缩放）。
To describe a transformation fully, state the type and all necessary details: for reflection, the equation of the mirror line; for rotation, the centre, angle (90° clockwise/anticlockwise), and direction; for enlargement, the centre and scale factor. Negative scale factors produce an inversion.
要完整描述一个变换，需说明类型及所有必要细节：反射给出镜线方程；旋转给出中心、角度（90° 顺时针/逆时针）和方向；放大给出中心和比例因子。负比例因子产生倒转。
Combining transformations leads to more complex mappings. The inverse transformation reverses the effect. Matrices can represent transformations such as reflections and rotations. For example, the matrix [0 -1; 1 0] represents a 90° rotation anticlockwise about the origin.
组合变换会产生更复杂的映射。逆变换可逆转效果。矩阵可以表示诸如反射和旋转的变换。例如，矩阵 [0 -1; 1 0] 表示绕原点逆时针旋转 90°。
9. Statistics | 统计
Statistics involves collecting, organising, and interpreting data. Measures of central tendency – mean, median, mode – and measures of spread – range, interquartile range (IQR), and standard deviation – summarise data sets. For grouped data, use mid-interval values to estimate the mean.
统计学涉及收集、整理和解读数据。集中趋势的度量——平均数、中位数、众数——和离散度的度量——极差、四分位距(IQR)和标准差——用于概括数据集。对于分组数据，使用组中值来估算平均数。
Graphical representations include bar charts, pie charts, histograms (with frequency density on the y‑axis), cumulative frequency curves, and box‑and‑whisker plots. A histogram’s frequency density = frequency ÷ class width. Cumulative frequency graphs help find medians and quartiles.
图形表示包括条形图、饼图、直方图（y 轴为频数密度）、累积频率曲线和箱线图。直方图的频数密度 = 频数 ÷ 组距。累积频率图有助于求中位数和四分位数。
Scatter diagrams show correlation between two variables. A line of best fit (by eye) can be used to make predictions. Understand the difference between correlation and causation.
散点图显示两个变量之间的相关性。最佳拟合线（目测画出）可用于预测。理解相关与因果之间的区别。
10. Probability | 概率
Probability is measured on a scale from 0 (impossible) to 1 (certain). The probability of an event not occurring is 1 − P(event). For combined events, use tree diagrams to list outcomes and multiply probabilities along branches. Remember to sum probabilities for mutually exclusive events.
概率用从 0（不可能）到 1（必然）的尺度衡量。事件不发生的概率为 1 − P(事件)。对于组合事件，使用树形图列出结果，并沿分支相乘概率。互斥事件的概率要相加。
Conditional probability is expressed as P(A|B) = P(A ∩ B) / P(B). Tree diagrams with ‘without replacement’ scenarios require careful updating of probabilities for the second pick. Venn diagrams and two‑way tables also help organise information for probability calculations.
条件概率表示为 P(A|B) = P(A ∩ B) / P(B)。涉及“不放回”情景的树形图需要仔细更新第二次选择的概率。维恩图和双向表也有助于整理信息进行概率计算。
For independent events, P(A ∩ B) = P(A) × P(B). The product rule for independent events is a common exam trap; ensure events genuinely have no influence on each other.
对于独立事件，P(A ∩ B) = P(A) × P(B)。独立事件的乘法定理是考试中常见的陷阱；要确保事件确实互不影响。
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