📚 Cambridge IGCSE Mathematics Core and Extended Full Syllabus Revision | 剑桥 IGCSE 数学核心与扩展全模块复习
This guide is designed for Cambridge IGCSE Mathematics candidates using the Core and Extended Coursebook (ISBN 9781108746342). It breaks the syllabus into ten manageable sections, covering essential formulas, common misconceptions, and exam tips. Use it alongside past papers for the best results.
本指南专为使用核心与扩展教材（ISBN 9781108746342）的剑桥 IGCSE 数学考生设计。它将考试大纲分为十个易于掌握的小节，涵盖必备公式、常见误区与应试技巧。建议与历年真题配合使用，效果最佳。
1. Number and Operations | 数与运算
Master the number system: integers, fractions, decimals, percentages, ratios, and directed numbers. Core candidates must perform operations without a calculator for small numbers; Extended candidates must also simplify surds and use standard form.
掌握数系：整数、分数、小数、百分数、比和带符号数。核心考生需要在不使用计算器的情况下完成小数字运算；扩展考生还必须化简根式并使用标准形式。
Key skills include converting recurring decimals to fractions, finding percentage increase or decrease, and calculating compound interest using the multiplier method.
关键技能包括将循环小数转化为分数、计算百分比增减，以及使用乘数法计算复利。
Extended candidates should know the rules of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ⁿ = 1/aⁿ. They must also estimate upper and lower bounds after rounding.
扩展考生应掌握指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ、(aᵐ)ⁿ = aᵐⁿ 和 a⁻ⁿ = 1/aⁿ。他们还必须在四舍五入后估算上界和下界。
2. Algebra and Equations | 代数与方程
Simplify expressions by collecting like terms and expanding brackets. Factorising is the reverse process: for example, x² + 5x + 6 = (x + 2)(x + 3).
通过合并同类项和去括号化简代数式。因式分解是逆过程，例如 x² + 5x + 6 = (x + 2)(x + 3)。
Solve linear equations by isolating the variable. For simultaneous equations, use substitution or elimination. Extended candidates must solve one linear and one quadratic pair.
通过移项求解一元一次方程。对于联立方程组，使用代入法或消元法。扩展考生还需要求解一个一次方程与一个二次方程的组合。
For quadratics, use factorising, completing the square, or the quadratic formula x = (-b ± √(b² – 4ac)) / 2a. Check the discriminant Δ = b² – 4ac to determine the number of real roots.
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。检验判别式 Δ = b² – 4ac 可判断实根的个数。
3. Graphs and Functions | 函数与图像
Plot graphs by constructing a table of values. Recognise the shapes of linear, quadratic, cubic, reciprocal, and exponential graphs.
通过列表取值描点绘制图像。识别一次、二次、三次、反比例和指数函数的图像形状。
The gradient of a straight line is m = (y₂ – y₁)/(x₂ – x₁), and the y-intercept is c in y = mx + c. Parallel lines have equal gradients; perpendicular lines have gradients multiplying to -1.
直线的斜率 m = (y₂ – y₁)/(x₂ – x₁)，y 轴截距为 y = mx + c 中的 c。平行线斜率相等；垂直线的斜率乘积为 -1。
Extended candidates must understand function notation f(x), composite functions fg(x), and inverse functions f⁻¹(x). To find an inverse, swap x and y and rearrange.
扩展考生需要理解函数记号 f(x)、复合函数 fg(x) 和反函数 f⁻¹(x)。求反函数时，交换 x 与 y 后整理即可。
4. Coordinate Geometry | 坐标几何
Find the midpoint of two points (x₁, y₁) and (x₂, y₂) as ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between them is √((x₂ – x₁)² + (y₂ – y₁)²).
两点 (x₁, y₁) 与 (x₂, y₂) 的中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √((x₂ – x₁)² + (y₂ – y₁)²)。
The equation of a straight line can be written as y – y₁ = m(x – x₁). Use two known points to find m, then substitute one point to find c.
直线方程可写为 y – y₁ = m(x – x₁)。先用两个已知点求出 m，再代入其中一个点求出 c。
Extended candidates may need to find the equation of a perpendicular bisector or solve geometry problems involving straight-line intersections.
扩展考生可能需要求垂直平分线的方程，或解决涉及直线交点的几何问题。
5. Geometry and Circle Theorems | 几何与圆定理
Angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, and angles in a triangle sum to 180°. Learn polygon angle sums: (n – 2) × 180°.
基本角度：直线上的角之和为 180°，一点周围的角之和为 360°，三角形内角和为 180°。掌握多边形内角和公式：(n – 2) × 180°。
Circle theorems: angle in a semicircle is 90°; angle at centre is twice angle at circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent meets radius at 90°.
圆定理：半圆内的圆周角为 90°；圆心角是圆周角的两倍；同弧上的圆周角相等；圆内接四边形对角互补；切线与半径垂直。
Congruent triangles match exactly (SSS, SAS, ASA, RHS). Similar triangles have equal angles and proportional sides; scale factor k changes lengths by k, areas by k², and volumes by k³.
全等三角形完全重合（SSS、SAS、ASA、RHS）。相似三角形对应角相等、对应边成比例；比例因子 k 使长度变为 k 倍，面积变为 k² 倍，体积变为 k³ 倍。
6. Mensuration | 测量
Know area formulas: rectangle A = lw, triangle A = ½bh, circle A = πr², trapezium A = ½(a + b)h. Circumference C = 2πr.
掌握面积公式：矩形 A = lw、三角形 A = ½bh、圆 A = πr²、梯形 A = ½(a + b)h。圆的周长 C = 2πr。
Volume formulas: cuboid V = lwh, prism V = cross-sectional area × length, cylinder V = πr²h, sphere V = 4/3πr³, cone V = 1/3πr²h, pyramid V = 1/3 base area × height.
体积公式：长方体 V = lwh、棱柱 V = 底面积 × 长、圆柱 V = πr²h、球 V = 4/3πr³、圆锥 V = 1/3πr²h、棱锥 V = 1/3 底面积 × 高。
For sectors, arc length = θ/360 × 2πr and sector area = θ/360 × πr², where θ is measured in degrees.
对扇形，弧长 = θ/360 × 2πr，扇形面积 = θ/360 × πr²，其中 θ 以度为单位。
7. Trigonometry | 三角学
In right-angled triangles, use SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Know exact values for 0°, 30°, 45°, 60°, 90°.
在直角三角形中，使用 SOHCAHTOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。熟记 0°、30°、45°、60°、90° 的精确值。
For non-right-angled triangles, use the Sine Rule a/sin A = b/sin B = c/sin C and the Cosine Rule a² = b² + c² – 2bc cos A. The area formula is ½ab sin C.
对于非直角三角形，使用正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² – 2bc cos A。面积公式为 ½ab sin C。
Extended candidates must solve trigonometric equations within a given range and sketch graphs of y = sin x, y = cos x, and y = tan x.
扩展考生需要在给定范围内解三角方程，并绘制 y = sin x、y = cos x 和 y = tan x 的图像。
8. Vectors and Transformations | 向量与变换
A vector has magnitude and direction. Column vectors represent displacement: (x y). Add vectors by adding components; multiply by scalar k: k(x y) = (kx ky). The magnitude is √(x² + y²).
向量既有大小又有方向。列向量表示位移：(x y)。向量相加即对应分量相加；乘以标量 k：k(x y) = (kx ky)。模长为 √(x² + y²)。
Transformations: translation moves a shape by a vector; reflection flips over a mirror line; rotation turns about a centre; enlargement scales from a centre by factor k.
变换：平移按向量移动图形；反射沿镜面翻折；旋转绕中心转动；放大以中心为基准按比例因子 k 缩放。
Describe transformations fully: give type, centre, angle and direction for rotation, mirror line for reflection, vector for translation, and centre plus scale factor for enlargement.
完整描述变换：旋转要给出类型
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