📚 IGCSE Edexcel Maths: End-of-Year Revision Checklist | IGCSE Edexcel 数学：期末复习提纲
Preparing for the IGCSE Edexcel Maths exam can feel overwhelming, but a structured revision checklist helps you track your progress across every topic. This guide covers all major areas—Number, Algebra, Graphs, Geometry, Trigonometry, Probability, and Statistics—providing key facts and formulas you need to memorise. Use it to identify strengths, close gaps, and approach your exam with confidence.
准备 IGCSE Edexcel 数学考试可能会让人感到不知所措，但一份结构化的复习提纲能帮助你追踪每个主题的进展。这份指南涵盖所有主要领域——数字、代数、图形、几何、三角学、概率和统计——并提供了你需要记忆的关键事实和公式。用它来发现优势、弥补薄弱环节，从而自信地迎接考试。
1. Numbers and Operations | 数与运算
The number system includes natural numbers, integers, rational numbers (fractions, terminating or recurring decimals) and irrational numbers (e.g. √2, π). Understanding prime numbers, multiples, factors, highest common factor (HCF) and lowest common multiple (LCM) is essential for solving problems involving fractions and ratios.
数系包括自然数、整数、有理数（分数、有限小数或循环小数）和无理数（如 √2、π）。理解质数、倍数、因数、最大公因数(HCF)和最小公倍数(LCM)对于解决涉及分数和比率的题目至关重要。
Index laws: xᵐ × xⁿ = xᵐ⁺ⁿ, xᵐ ÷ xⁿ = xᵐ⁻ⁿ, (xᵐ)ⁿ = xᵐⁿ, x⁰ = 1, x⁻ⁿ = 1⁄xⁿ, x^(½) = √x, x^(p/q) = ᵠ√(xᵖ). Standard form writes large or small numbers as a × 10ⁿ with 1 ≤ a < 10, e.g. 3.2 × 10⁵, 4.1 × 10⁻³.
指数定律：xᵐ × xⁿ = xᵐ⁺ⁿ, xᵐ ÷ xⁿ = xᵐ⁻ⁿ, (xᵐ)ⁿ = xᵐⁿ, x⁰ = 1, x⁻ⁿ = 1⁄xⁿ, x^(½) = √x, x^(p/q) = ᵠ√(xᵖ)。标准形式将很大或很小的数写作 a × 10ⁿ，其中 1 ≤ a < 10，例如 3.2 × 10⁵，4.1 × 10⁻³。
Percentages: increase or decrease by a percentage using a multiplier (e.g., 15% increase → ×1.15). Reverse percentages find the original amount after a change. Ratio and proportion: share a quantity in a given ratio, solve direct and inverse proportion problems.
百分比：使用乘数进行百分比增减（例如 15% 增长 → 乘以 1.15）。逆向百分比求变化前的原始数量。比率与比例：按给定比率分配数量，解决正比和反比问题。
Upper and lower bounds: when a value is rounded, the true value lies within a bound, e.g. 5.6 to 1 d.p. lies between 5.55 and 5.65. Apply bounds to calculations to find maximum and minimum possible results.
上界与下界：当一个值被四舍五入时，真实值位于一个界限内，例如 5.6 保留 1 位小数介于 5.55 和 5.65 之间。在计算中应用界限以得出可能的最大和最小结果。
2. Algebraic Expressions | 代数式
Expanding brackets: a(b + c) = ab + ac, and double brackets (x + a)(x + b) = x² + (a+b)x + ab. Factorising reverses the process: find common factors, e.g. 3x² − 6x = 3x(x − 2). Factorise quadratic expressions like x² + 5x + 6 = (x + 2)(x + 3) and the difference of two squares: x² − a² = (x − a)(x + a).
展开括号：a(b + c) = ab + ac，以及双括号 (x + a)(x + b) = x² + (a+b)x + ab。因式分解是此过程的逆运算：找出公因式，例如 3x² − 6x = 3x(x − 2)。分解二次式如 x² + 5x + 6 = (x + 2)(x + 3)，以及平方差公式：x² − a² = (x − a)(x + a)。
Algebraic fractions: simplify by factorising first, multiply/divide just like numerical fractions, and add/subtract using a common denominator. Rearranging formulae: change the subject of a formula by using inverse operations while treating other variables as constants.
代数分式：先因式分解再化简，乘除运算与数值分式相同，加减运算则使用公分母。变换公式：利用逆运算并视其他变量为常数来改变公式的主项。
3. Equations and Inequalities | 方程与不等式
Solving linear equations: carry out inverse operations on both sides, e.g. 2x + 3 = 7 → x = 2. For quadratic equations x² + bx + c = 0, use factorising, completing the square, or the quadratic formula: x = [−b ± √(b² − 4ac)] / 2a. Always check for extraneous solutions.
解线性方程：对方程两边进行逆运算，例如 2x + 3 = 7 → x = 2。对于二次方程 x² + bx + c = 0，使用因式分解、配方法或二次求根公式：x = [−b ± √(b² − 4ac)] / 2a。始终检查是否有增根。
Simultaneous equations: solve two linear equations by substitution or by elimination. For one linear and one quadratic, substitute the linear expression. Inequalities: solve like equations but reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions on a number line with open or closed circles.
联立方程：通过代入法或消元法求解两个线性方程。对于一线性一二次的方程组，代入线性表达式求解。不等式：解法与方程类似，但当乘以或除以负数时要反转不等号方向。在数轴上用空心或实心圆点表示解集。
4. Sequences | 数列
Arithmetic (linear) sequences: the nth term is a + (n − 1)d where a is the first term and d is the common difference. d can be positive or negative. Quadratic sequences: the second difference is constant; the nth term has the form an² + bn + c. Find a, b, c by comparing with the sequence.
等差数列（线性数列）：第 n 项为 a + (n − 1)d，其中 a 为首项，d 为公差。d 可为正或负。二次数列：二阶差为常数；通项形式为 an² + bn + c。通过与数列比对求出 a、b、c。
Recognise special sequences: triangular numbers (1, 3, 6, 10, …), square numbers, cube numbers, and Fibonacci-style sequences where each term is the sum of the previous two.
识别特殊数列：三角形数（1, 3, 6, 10, …）、平方数、立方数，以及斐波那契型数列（每一项是前两项之和）。
5. Graphs | 图形
Straight-line graphs: equation y = mx + c, gradient m = rise/run, y-intercept c. Parallel lines share the same gradient; perpendicular lines satisfy m₁ × m₂ = −1. Midpoint of two points: ((x₁+x₂)/2, (y₁+y₂)/2). Distance between two points: √[(x₂−x₁)² + (y₂−y₁)²].
直线图：方程 y = mx + c，斜率 m = 垂直变化/水平变化，y 轴截距为 c。平行线有相同的斜率；垂线的斜率满足 m₁ × m₂ = −1。两点中点：((x₁+x₂)/2, (y₁+y₂)/2)。两点间距离：√[(x₂−x₁)² + (y₂−y₁)²]。
Plot quadratic, cubic and reciprocal graphs; identify roots, vertex and intercepts. Use a tangent to find the gradient at a point on a curve. Solve equations graphically by finding intersections. Interpret distance-time and speed-time graphs: area under a speed-time graph gives distance.
绘制二次、三次和反比函数图像；识别根、顶点和截距。利用切线求曲线上某点的梯度。通过寻找交点来利用图像解方程。解读距离-时间图和速度-时间图：速度-时间图下的面积代表距离。
6. Geometry | 几何
Angle facts: angles on a straight line sum to 180°, around a point 360°. Vertically opposite angles are equal. Parallel lines produce corresponding angles, alternate angles (equal) and co-interior angles (sum to 180°). Triangles add to 180°; an exterior angle equals the sum of the two opposite interior angles.
角度事实：直线上的角之和为 180°，绕一点的角之和为 360°。对顶角相等。平行线产生同位角、内错角（相等）和同旁内角（和为 180°）。三角形内角和为 180°；一个外角等于两个对内角之和。
Congruence conditions: SSS, SAS, ASA, RHS. Similar shapes have equal angles and proportional sides; length scale factor k → area factor k² → volume factor k³. Circle theorems: angle in a semicircle = 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; a tangent meets a radius at 90°; alternate segment theorem.
全等条件：SSS、SAS、ASA、RHS。相似形具有相等角和成比例的边；长度比例因子 k → 面积因子 k² → 体积因子 k³。圆定理：半圆上的圆周角等于 90°；同弧上的圆周角相等；圆内接四边形对角和为 180°；切线与过切点的半径互相垂直；弦切角定理。
7. Mensuration | 测量
Area formulas: rectangle A = l × w, triangle A = ½bh, parallelogram A = bh, trapezium A = ½(a+b)h, circle
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