📚 CIE IGCSE Mathematics (0580) Syllabus Overview | CIE IGCSE 数学课程大纲全面解析
The CIE IGCSE Mathematics (0580) syllabus is a rigorous two-year course typically starting in Year 10, building essential mathematical skills for further study and everyday life. This guide breaks down every major topic, assessment structure, and key tips for success, so you know exactly what to expect and how to prepare effectively.
CIE IGCSE 数学（0580）课程大纲是一门严格的两年制课程，通常从 Year 10 开始，为深入学习和日常生活培养必要的数学技能。本指南逐一解析每个主要主题、考试结构和成功的关键技巧，让你清楚预期、有效备考。
1. Overview and Assessment Structure | 大纲概览与考试结构
The syllabus is divided into two tiers: Core (grades C–G) and Extended (grades A*–E). Year 10 often covers the common content first, with later differentiation.
课程分为两个层级：核心（C–G 级）和扩展（A*–E 级）。Year 10 通常先覆盖共同内容，随后再进行区分。
Assessment consists of two written papers, each 2 hours (Core) or 2 hours 30 minutes (Extended). Paper 1 and Paper 2 both allow calculators; papers are equally weighted.
考试包括两份笔试，核心每份 2 小时，扩展每份 2 小时 30 分钟。卷一和卷二均允许使用计算器，各占 50% 权重。
Topics are grouped into Number, Algebra, Shape and Space, and Probability & Statistics. All are tested across both papers.
主题分为数、代数、图形与空间、概率与统计四大板块，两份试卷都会考查。
2. Number | 数与数系
Students must be comfortable with integers, fractions, decimals, percentages, and the conversion between them. Ordering and comparing rational numbers is fundamental.
学生需熟练使用整数、分数、小数和百分数及其互相转换。有理数的排序与比较是基础中的基础。
or very large or small numbers is required. Surds and their simplification, including rationalising denominators, appear in Extended.
要求掌握极大或极小数字的标准形式（a × 10ⁿ）表示。扩展部分涉及根式及其化简，包括分母有理化。
Topics also include ratios, proportions, map scales, percentage increase/decrease, and simple and compound interest calculations.
主题还包括比与比例、地图比例尺、百分数增减、单利与复利计算。
Set language and notation using ∪, ∩, ⊆, ∈, ∅, and universal sets are introduced, along with Venn diagrams.
引入集合语言与符号，如 ∪, ∩, ⊆, ∈, ∅ 和全集，并配合韦恩图使用。
Estimation and limits of accuracy (rounding to given decimal places or significant figures) are essential for checking answers.
估算与精确度界限（四舍五入到指定小数位或有效数字）对于验算答案至关重要。
3. Algebra | 代数
Basic operations include expanding brackets, factorising linear and quadratic expressions (including the difference of two squares), and simplifying algebraic fractions.
基本运算包括展开括号、因式分解线性和二次表达式（含平方差公式）以及化简代数分式。
Solving linear equations, simultaneous equations (by substitution or elimination), and quadratic equations by factorising, completing the square, or using the quadratic formula. For Core, quadratic solving may be limited.
解线性方程、联立方程（代入法或消元法）和二次方程（因式分解、配方法或求根公式）。核心层级可能对二次方程解法有所限制。
Inequalities are solved and represented on a number line. Extended candidates also handle linear programming and shading regions defined by inequalities.
求解不等式并在数轴上表示。扩展部分还涉及线性规划以及由不等式确定的区域着色。
Sequences: finding the nth term of linear and simple quadratic sequences, and using sequences in practical contexts.
数列：求线性数列和简单二次数列的第 n 项，并在实际问题中应用数列。
cluding the algebraic relationships y = kx and y = k/x, are used to model real situations.
正比与反比变化，包括 y = kx 和 y = k/x 的代数关系，用于模拟实际情形。
4. Functions | 函数
Students evaluate functions, find the inverse function f⁻¹(x), and form composite functions fg(x).
使用函数记法 f(x) = ...。学生计算函数值，求反函数 f⁻¹(x)，并构造复合函数 fg(x)。
The domain and range of a function are often restricted by real-world considerations. Graphs of functions are sketched, identifying key features like maxima, minima, and intercepts.
函数的定义域与值域常受实际情境限制。绘制函数图像，并识别极大值、极小值、截距等关键特征。
Exponential functions may be introduced, and transformation of graphs (translations, reflections) is an Extended requirement.
指数函数可能引入，扩展部分要求掌握图像的变换（平移、反射）。
5. Coordinate Geometry | 坐标几何
₁) and (x₂,y₂) is given by √[(x₂−x₁)² + (y₂−y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
两点 (x₁,y₁) 与 (x₂,y₂) 间的距离公式为 √[(x₂−x₁)² + (y₂−y₁)²]。中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
raight line is commonly expressed as y = mx + c, where m is gradient and c is y-intercept.
通过两点的直线斜率公式为 (y₂−y₁)/(x₂−x₁)。直线方程通常表示为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
× m₂ = −1. Students find equations of perpendicular lines and solve intersection problems.
平行线斜率相等；垂直线满足 m₁ × m₂ = −1。学生需求解垂直线方程并解决交点问题。
6. Geometry | 几何
Properties of angles at a point, on a straight line, and in triangles and quadrilaterals are used to solve problems. Circle theorems, including angles in a semicircle, tangent–radius, and cyclic quadrilaterals, are required for Extended tier.
利用点在点上、在直线上、在三角形和四边形中的角度性质解题。扩展层级要求掌握圆定理，包括半圆上的圆周角、切线与半径、圆内接四边形等。
Congruence and similarity conditions (SSS, SAS, ASA, RHS for congruence; RHS or AAA for similarity) are applied to lengths, areas, and volumes.
全等与相似条件（全等：SSS, SAS, ASA, RHS；相似：RHS 或 AAA）用于长度的计算以及面积和体积的比率。
Constructions using ruler and compasses, such as angle bisectors, perpendicular bisectors, and triangles given SSS, are examined on Paper 1 without calculator aids.
尺规作图，如角平分线、垂直平分线以及给定边边边作三角形，在非计算器卷上考查。
Symmetry: line symmetry and rotational symmetry for 2D shapes.
对称：二维图形的线对称与旋转对称。
7. Mensuration | 测量
mula for area of a sector and length of an arc are used: area = (θ/360)πr², arc length = (
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