📚 IGCSE AQA Maths: Ultimate Final Revision Guide | IGCSE AQA 数学：终极期末复习指南
This guide is your one‑stop checklist for the IGCSE AQA Mathematics syllabus. It pulls together all the major topics, must‑know formulas, and exam tips to help you focus your final revision efficiently.
本指南是你的 IGCSE AQA 数学一站式自查清单。它汇集了所有主要知识点、必会公式和考试技巧，帮助你高效进行期末复习。
1. Number and Arithmetic | 数与算术
Master the four operations with fractions, decimals, and percentages. Remember to find a common denominator when adding or subtracting fractions, and to flip the second fraction for division.
掌握分数、小数和百分数的四则运算。记住分数加减时要通分，分数除法时要将第二个分数取倒数。
Understand place value and the order of operations (BIDMAS/BODMAS). Always evaluate brackets first, then indices, followed by division and multiplication, and finally addition and subtraction.
理解位值和运算顺序 (BIDMAS/BODMAS)。先算括号，再算指数，接着乘除，最后加减。
Standard form expresses numbers as a × 10ⁿ, where 1 ≤ a < 10. Practise converting between ordinary numbers and standard form, and using it for multiplication and division.
标准形式将数字表示为 a × 10ⁿ，其中 1 ≤ a < 10。练习普通数字与标准形式之间的转换，以及用它进行乘除运算。
Be comfortable with upper and lower bounds after rounding. When a length is given as 6.4 cm to 1 decimal place, the lower bound is 6.35 cm and the upper bound is 6.45 cm.
熟悉四舍五入后的上限和下限。若某长度四舍五入到一位小数为 6.4 cm，则下限为 6.35 cm，上限为 6.45 cm。
Key Formula: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
关键公式：aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
2. Algebraic Manipulation | 代数运算
Simplify expressions by collecting like terms: 3x + 5y – 2x + y = x + 6y. Always keep the sign with the term that follows it.
通过合并同类项化简表达式：3x + 5y – 2x + y = x + 6y。符号始终跟随其后面的项。
Expand brackets carefully using the distributive law. For double brackets, remember FOIL (First, Outer, Inner, Last) or the grid method.
运用分配律小心展开括号。对于双括号，记住 FOIL 法（首项、外项、内项、末项）或网格法。
Factorise by taking out the highest common factor, then recognise special cases like the difference of two squares: a² – b² = (a + b)(a – b).
提取公因式进行因式分解，然后识别特殊情况如平方差：a² – b² = (a + b)(a – b)。
For quadratic expressions, factorise into two binomials or use the quadratic formula when needed. Always check by expanding.
对于二次表达式，分解成两个一次因式，或在需要时使用求根公式。之后务必通过展开检验。
Quadratic Formula: x = [-b ± √(b² – 4ac)] / (2a)
求根公式：x = [-b ± √(b² – 4ac)] / (2a)
3. Equations and Inequalities | 方程与不等式
Solve linear equations by performing inverse operations on both sides. Always do the same action to keep the equation balanced.
通过两边执行逆运算求解线性方程。始终执行相同操作以保持等式平衡。
For simultaneous equations, use elimination or substitution. Draw the graphs if asked to find a graphical solution.
解联立方程组时，使用消元法或代入法。如果要求图像解法，则绘制图像。
Quadratic equations can be solved by factorising, completing the square, or the quadratic formula. The discriminant b² – 4ac tells you the nature of the roots.
二次方程可通过因式分解、配方法或求根公式求解。判别式 b² – 4ac 能说明根的性质。
Inequalities are solved similarly to equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
不等式的解法与方程类似，但当乘或除以负数时，记得将不等号方向反转。
Shade regions on a graph to show inequality solutions. A solid line is used for ≤ or ≥, a dashed line for < or >.
在图像中用阴影表示不等式解。实线用于 ≤ 或 ≥，虚线用于 < 或 >。
4. Sequences, Functions and Graphs | 数列、函数与图像
Find the nth term of a linear sequence: an + b, where a is the common difference. For quadratic sequences, the second difference is constant and you need to find an² + bn + c.
求线性数列的第 n 项：an + b，其中 a 是公差。对于二次数列，二阶差为常数，需要求出 an² + bn + c。
Understand function notation: f(x) = 2x + 3. Composite functions fg(x) means apply g first, then f. Inverse functions reverse the process.
理解函数记号：f(x) = 2x + 3。复合函数 fg(x) 表示先执行 g 再执行 f。反函数则逆转此过程。
Plot and recognise the shapes of graph types: linear y = mx + c, quadratic y = ax² + bx + c (a parabola), cubic y = ax³ + bx² + cx + d, reciprocal y = a/x, exponential y = aˣ.
绘制并识别各种图像类型：线性 y = mx + c，二次 y = ax² + bx + c（抛物线），三次 y = ax³ + bx² + cx + d，反比例 y = a/x，指数 y = aˣ。
Interpret gradients and y‑intercepts. In distance–time graphs, gradient equals speed; in velocity–time graphs, gradient equals acceleration and area under the graph equals distance.
解读斜率和 y 轴截距。在距离–时间图中，斜率等于速度；在速度–时间图中，斜率等于加速度，图像下的面积等于距离。
5. Ratio, Proportion and Rates of Change | 比、比例与变化率
Simplify ratios by dividing both sides by common factors. Use ratios to share quantities in a given ratio and to compare unit rates.
通过除以公因子化简比。利用比例按给定比例分配量，并比较单位比率。
Direct proportion means y = kx, and the graph is a straight line through the origin. Inverse proportion means y = k/x, and the graph is a rectangular hyperbola.
正比例指 y = kx，其图像是一条过原点的直线。反比例指 y = k/x，其图像是等轴双曲线。
Percentage change, profit, loss, simple interest and compound interest are key. The compound interest formula is A = P(1 + r/100)ⁿ.
百分比变化、盈亏、单利和复利都是重点。复利公式为 A = P(1 + r/100)ⁿ。
Work with speed, density, and pressure using the triangle relationships: Speed = Distance÷Time, Density = Mass÷Volume, Pressure = Force÷Area.
运用速度、密度和压强的三角形关系：速度 = 距离÷时间，密度 = 质量÷体积，压强 = 力÷面积。
6. Geometry: Angles and Shapes | 几何：角与形
Know the angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal, angles in a triangle sum to 180°.
掌握角度定理：直线上角之和为 180°，一点周围角之和为 360°，对顶角相等，三角形内角和为 180°。
In parallel lines, corresponding angles are equal, alternate angles are equal, and co‑interior angles sum to 180°. Use these to find missing angles.
平行线中，同位角相等，内错角相等，同旁内角互补（和为 180°）。利用这些性质求未知角。
Polygons: sum of interior angles = (n – 2) × 180°, sum of exterior angles = 360° for any convex polygon. The interior angle of a regular polygon = 180° – exterior angle.
多边形：内角和 = (n – 2) × 180°，任何凸多边形的外角和 = 360°。正多边形的内角 = 180° – 外角。
Circle theorems: angle at centre is twice angle at circumference, angle in a semicircle is 90°, angles in the same segment are equal, opposite angles in a cyclic quadrilateral sum to 180°.
圆的定理：圆心角是圆周角的两倍，半圆上的圆周角是 90°，同弧上的圆周角相等，圆内接四边形的对角互补。
7. Pythagoras and Trigonometry | 勾股定理与三角学
Pythagoras’ theorem: a² + b² = c², where c is the hypotenuse. Use it to find missing sides in right‑angled triangles and to check if a triangle is right‑angled.
勾股定理：a² + b² = c²，其中 c 为斜边。用它求直角三角形中的未知边，也用于判断三角形是否为直角三角形。
Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Label the sides relative to the given angle.
三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。相对于给定角标好各边。
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 of a triangle = ½ ab 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)。三角形面积 = ½ ab sin C。
Always check if your calculator is in degree mode when working with angles. Round final answers sensibly based on the context.
进行角度计算时，始终检查计算器是否处于度数模式。根据题意合理四舍五入最终答案。
8. Perimeter, Area and Volume | 周长、面积与体积
Learn the formulas for area of a rectangle, triangle, parallelogram, trapezium (½(a + b)h), and circle (πr²). Circumference of a circle = 2πr or πd.
熟记矩形、三角形、平行四边形、梯形 (½(a + b)h) 和圆 (πr²) 的面积公式。圆周长 = 2πr 或 πd。
Surface area and volume of prisms: volume = area of cross‑section × length. For a cylinder, volume = πr²h, curved surface area = 2πrh.
棱柱的表面积和体积：体积 = 底面积 × 高。圆柱体的体积 = πr²h，侧面积 = 2πrh。
For cones and spheres: cone volume = ⅓ πr²h, sphere volume = ⁴⁄₃ πr³, sphere surface area = 4πr². Make sure you can use these in context.
圆锥和球体：圆锥体积 = ⅓ πr²h，球体积 = ⁴⁄₃ πr³，球表面积 = 4πr²。确保能在实际题目中应用。
Understand how to convert between units of area (e.g. 1 m² = 10000 cm²) and volume (1 m³ = 1000000 cm³). Use conversion factors carefully.
理解面积单位换算（如 1 m² = 10000 cm²）和体积单位换算（1 m³ = 1000000 cm³）。小心使用换算因子。
9. Probability | 概率
Probability is always a number between 0 and 1. The sum of probabilities of all mutually exclusive outcomes is 1. P(A’) = 1 – P(A).
概率始终是 0 到 1 之间的一个数。所有互斥结果的概率和为 1。P(A’) = 1 – P(A)。
For combined events, use sample space diagrams, tree diagrams (with AND rules: multiply along branches; OR rule: add probabilities for final outcomes).
对于组合事件，使用样本空间图、树形图（AND 规则：沿分枝相乘；OR 规则：将最终结果的概率相加）。
Conditional probability considers how the probability changes after an event has occurred. Look for key words such as ‘given that’ or ‘if’.
条件概率考虑事件发生后概率如何变化。注意题目中的关键词，如”已知”或”如果”。
Venn diagrams help organise sets and probabilities. The intersection represents outcomes in both sets, the union represents outcomes in either set.
维恩图有助于整理集合和概率。交集表示同时属于两个集合的成员，并集表示属于任意一个集合的成员。
10. Statistics | 统计
Measures of central tendency: mean (average, sum ÷ count), median (middle value), mode (most frequent). Choose the best measure based on the data distribution.
集中趋势度量：平均数（平均值，总和÷个数）、中位数（中间值）、众数（最常见值）。根据数据分布选择最佳度量。
Measures of spread: range (max – min), interquartile range (IQR = Q₃ – Q₁). IQR is less affected by outliers, so often preferred.
离散度量：极差（最大值–最小值）、四分位距 (IQR = Q₃ – Q₁)。IQR 受异常值影响较小，因此常被优先选用。
Display data appropriately: bar charts for categorical data, pie charts for proportions, histograms for grouped continuous data (frequency density = frequency ÷ class width).
恰当地展示数据：条形图用于分类数据，饼图用于比例，直方图用于分组连续数据（频数密度 = 频数÷组距）。
In scatter graphs, correlation describes the relationship. Draw the line of best fit by eye and use it for prediction (interpolation is safer than extrapolation).
散点图中，相关性描述关系的方向和强度。通过目测画出最佳拟合线，并用于预测（内插比外推更可靠）。
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