📚 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。
d 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
lgebraic 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)。
ic 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
te 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))。
ing 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
₁, 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)。
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
m 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‑ tenuse, 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。
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。
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 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。
epresentations include bar charts, pie charts, histograms (with frequency density on the y‑axis), cumulative frequency curves, and box‑and‑whisker plots. A histogram' ty = 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' arios 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 ∩ ndependent 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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