📚 IGCSE WJEC Maths: End-of-Term Revision Checklist | IGCSE WJEC 数学：期末复习提纲
This structured revision checklist covers every major topic in the IGCSE WJEC Mathematics syllabus. Use it to identify your strengths, close gaps and build confidence before your end-of-term assessment. Tick off each skill as you master it, and return regularly to those you find challenging.
这份系统性复习提纲涵盖了 IGCSE WJEC 数学考试大纲中的每一个主要课题。用它来识别你的优势、弥补薄弱环节，并在期末考试前建立信心。每掌握一项技能就在旁边打勾，并定期回顾你仍感到困难的部分。
1. Number and Arithmetic | 数与算术
Integer operations and order of operations (BIDMAS/BODMAS): You must be able to perform addition, subtraction, multiplication and division with positive and negative integers, and apply brackets, indices, division, multiplication, addition and subtraction in the correct sequence.
整数运算与运算法则（BIDMAS/BODMAS）：你必须能够对正负整数进行加减乘除，并按照括号、指数、除法、乘法、加法和减法的正确顺序进行计算。
Fractions and mixed numbers: Add, subtract, multiply and divide fractions and mixed numbers. Convert improper fractions to mixed numbers and vice versa. Simplify fractions fully and find fractions of quantities.
分数与带分数：进行分数和带分数的加减乘除。将假分数转换为带分数，反之亦然。将分数彻底化简，并求出一个数量的几分之几。
Decimals and significant figures: Round decimals to a given number of decimal places or significant figures. Multiply and divide decimals by powers of 10, and use written methods for multiplication and division of decimal numbers.
小数与有效数字：将小数四舍五入到指定的小数位数或有效数字。将小数乘以或除以 10 的幂，并使用笔算方法进行小数乘法与除法。
Percentages, increases and decreases: Express a quantity as a percentage of another; calculate percentage increase/decrease and find the original value after a percentage change (reverse percentages). Compare two quantities using percentages.
百分比、增加与减少：将一个数量表示为另一个数量的百分比；计算百分比增减，并求出百分比变化后的原值（反向百分比）。用百分比比较两个数量。
Indices and standard form: Use index notation with positive, negative and zero powers. Convert between ordinary numbers and standard form (A × 10ⁿ where 1 ≤ A < 10 and n is an integer). Perform calculations in standard form.
指数与标准形式：使用含有正整数、负整数和零指数的指数记法。在普通数字与标准形式（A × 10ⁿ，其中 1 ≤ A < 10，n 为整数）之间进行转换。以标准形式进行运算。
2. Ratio, Proportion and Rates of Change | 比、比例与变化率
Simplifying ratios and sharing in a ratio: Simplify a three-part ratio and divide a quantity into a given ratio. Use ratios in practical contexts such as recipes and maps.
化简比并按比例分配：化简三部分比，并将一个量按给定比例分配。在配方、地图等实际情境中使用比。
Direct and inverse proportion: recognise and solve problems involving direct proportion (y ∝ x) and inverse proportion (y ∝ 1/x). Write proportional formulae and use them to find unknown values.
正比与反比：识别并解决正比（y ∝ x）和反比（y ∝ 1/x）问题。写出比例关系式，并用它们求未知值。
Scale factors and scale diagrams: Interpret and use scale on maps and drawings. Enlarge or reduce quantities by a given scale factor in length, area and volume contexts.
比例因子与比例图：解读并使用地图和图纸上的比例。在长度、面积和体积情境中，按给定的比例因子放大或缩小数量。
Compound measures: Use and convert compound units such as speed (km/h or m/s), density (g/cm³) and pressure. Apply the formulae distance = speed × time, mass = density × volume.
复合量度：使用并换算复合单位，如速度（km/h 或 m/s）、密度（g/cm³）和压强。运用公式：距离 = 速度 × 时间，质量 = 密度 × 体积。
Percentage change as a multiplicative relationship: Interpret percentage change as a decimal multiplier, e.g. a 15% increase uses ×1.15. Apply repeated percentage change, including compound interest and depreciation.
将百分比变化理解为乘法关系：把百分比变化解读为小数乘数，例如增加 15% 即乘以 1.15。运用重复百分比变化，包括复利与折旧。
3. Algebraic Expressions and Formulae | 代数表达式与公式
Simplifying expressions: Collect like terms, multiply and divide algebraic terms, and simplify expressions involving powers (using the rules of indices).
化简表达式：合并同类项，乘除代数项，并运用指数法则化简含有幂的表达式。
Expanding brackets: Expand single brackets, double brackets and expressions with more than two brackets. Apply to squares of binomials, e.g. (x + a)² = x² + 2ax + a².
展开括号：展开单项式与多项式相乘，以及两个二项式相乘。应用于二项式的平方，如 (x + a)² = x² + 2ax + a²。
Factorising expressions: Factorise by taking out the highest common factor; factorise quadratics of the form x² + bx + c, and the difference of two squares x² − a².
因式分解：提取最高公因式进行因式分解；对形如 x² + bx + c 的二次式和平方差 x² − a² 进行因式分解。
Substitution: Substitute numbers into algebraic expressions and formulae, including those involving indices, fractions and brackets. Work with positive and negative values carefully.
代入：将数字代入代数表达式和公式，包括含有指数、分数和括号的式子。仔细处理正负数值。
Changing the subject of a formula: Rearrange formulae where the new subject appears once or twice. Use inverse operations systematically and deal with square roots, squares and fractions.
公式变形：将公式的主项变换为另一个变量，新主项可出现一次或两次。系统运用逆运算，并处理平方根、平方和分数。
4. Equations and Inequalities | 方程与不等式
Solving linear equations: Solve equations with the unknown on one or both sides, including those with brackets and fractions. Check solutions by substitution.
解线性方程：解未知数在一边或两边的方程，包括含有括号和分数的方程。用代入法检验解。
Solving quadratic equations: Solve by factorising, using the quadratic formula, or by completing the square. The quadratic formula for ax² + bx + c = 0 is x = (−b ± √(b² − 4ac)) / 2a. Recognise when a quadratic has no real roots.
解二次方程：通过因式分解、二次公式或配方法求解。对于 ax² + bx + c = 0，求根公式为 x = (−b ± √(b² − 4ac)) / 2a。识别二次方程何时没有实数根。
Simultaneous equations: Solve two linear equations in two unknowns by elimination or substitution. Extend to one linear and one quadratic equation, finding both points of intersection graphically or algebraically.
联立方程：通过消元法或代入法解两个二元一次方程。拓展到一线性一二次的方程组，通过作图或代数方法求两线的交点。
Inequalities: Solve linear inequalities and represent solutions on a number line. Solve quadratic inequalities by sketching graphs. Treat multiplying or dividing by a negative number with care (reverse the inequality sign).
不等式：解一元一次不等式，并将解表示在数轴上。通过草图解二次不等式。当不等式两边乘以或除以负数时，必须反转不等号方向。
Forming equations from words: Translate real-world problems into algebraic equations, solve them, and interpret the answer in context.
根据文字列方程：将现实问题转化为代数方程，求解并解释答案在实际情况下的意义。
5. Sequences and Graphs | 数列与图像
Generating sequences: Find the nth term of a linear sequence. Recognise and continue quadratic, geometric and other simple sequences. Use term-to-term and position-to-term rules.
生成数列：求线性数列的第 n 项。识别并延续二次、等比及其他简单数列。使用递推规则和通项公式。
Straight-line graphs: Plot graphs of linear functions; find and interpret gradient (m) and y-intercept (c) in y = mx + c. Work out the equation of a line given two points or one point and the gradient. Identify parallel and perpendicular lines.
直线图像：绘制一次函数的图像；在 y = mx + c 中求出并解释斜率（m）和 y 轴截距（c）。根据两点或一点与斜率求出直线方程。识别平行和垂直直线。
Quadratic and other graphs: Plot quadratic, cubic, reciprocal and exponential graphs. Identify key features: intercepts, turning points, asymptotes. Use graphs to solve equations approximately.
二次及其他函数图像：绘制二次、三次、反比例和指数函数的图像。识别关键特征：截距、拐点、渐近线。利用图像近似解方程。
Real-life graphs: Interpret distance-time, speed-time and conversion graphs. Calculate speed from a distance-time graph, acceleration and distance from a speed-time graph.
实际情境图像：解读距离-时间图、速度-时间图和换算图。从距离-时间图求速度，从速度-时间图求加速度与距离。
Gradient and area under graphs: Estimate gradient of a curve at a point by drawing a tangent. Understand that the area under a speed-time graph represents distance travelled.
曲线斜率与图下面积：通过画切线估计曲线上一点的斜率。理解速度-时间图下的面积代表所行驶的距离。
6. Geometry: Angles and Shapes | 几何：角度与图形
Angle facts and parallel lines: Use properties of angles on a straight line, at a point, vertically opposite angles, and angles in triangles. Apply corresponding, alternate and co-interior angle rules with parallel lines.
角度基本事实与平行线：运用平角、周角、对顶角以及三角形内角和的性质。运用平行线中的同位角、内错角和同旁内角法则。
Polygons: Calculate interior and exterior angles of regular and irregular polygons. Sum of interior angles = (n − 2) × 180°. Exterior angles sum to 360° irrespective of the number of sides.
多边形：计算正多边形和不规则多边形的内角与外角。内角和 = (n − 2) × 180°。外角和恒为 360°，与边数无关。
Circle theorems: Apply angle subtended by an arc at the centre and circumference, angles in a semicircle, angles in the same segment, and cyclic quadrilaterals. Use tangent-radius and alternate segment theorems.
圆定理：应用圆心角与圆周角关系、半圆上的圆周角、同弧上的圆周角相等以及圆内接四边形性质。使用切线与半径的垂直定理和弦切角定理。
Congruence and similarity: Prove triangles congruent using SSS, SAS, ASA and RHS. Use similarity to find missing lengths in overlapping and parallel triangles. Enlarge shapes by a fractional or negative scale factor.
全等与相似：用 SSS、SAS、ASA 和 RHS 证明三角形全等。利用相似性求重叠和平行三角形中的缺失边长。以分数或负数比例因子放大图形。
Constructions and loci: Construct perpendicular bisectors, angle bisectors, and triangles from given data. Draw loci to satisfy conditions involving distance from a point or line and equal distances from two points.
尺规作图与轨迹：作线段的垂直平分线、角平分线，并根据给定数据作三角形。画出满足到定点、定线距离相等或到两点等距条件的轨迹。
7. Mensuration: Length, Area and Volume | 测度：长度、面积与体积
Perimeter and area of 2D shapes: Find perimeter and area of triangles, parallelograms, trapezia, and composite shapes. Area of a triangle = ½ × base × height. Area of a trapezium = ½(a + b) × h.
平面图形的周长与面积：求三角形、平行四边形、梯形及组合图形的周长和面积。三角形面积 = ½ × 底 × 高；梯形面积 = ½(a + b) × 高。
Circles and sectors: Circumference = 2πr or πd, area = πr². Calculate arc length and sector area using fractions of 360°. Work with segments by subtracting triangles.
圆与扇形：周长 = 2πr 或 πd，面积 = πr²。用圆心角占 360° 的分数计算弧长与扇形面积。通过减去三角形来求弓形面积。
Volume of prisms and cylinders: Volume of a prism = area of cross-section × length. Volume of a cylinder = πr²h. Solve problems involving combinations of solids.
棱柱与圆柱的体积：棱柱体积 = 横截面面积 × 长。圆柱体积 = πr²h。解决组合体的体积问题。
Pyramids, cones and spheres: Volume of a pyramid = ⅓ × base area × height; cone volume = ⅓πr²h; sphere volume = 4/3 πr³. Surface area of a sphere = 4πr². Use curved surface area of a cone = πrl, where l is slant height.
棱锥、圆锥与球体：棱锥体积 = ⅓ × 底面积 × 高；圆锥体积 = ⅓πr²h；球体积 = 4/3 πr³。球的表面积 = 4πr²。圆锥侧面积 = πrl，其中 l 为母线长。
Metric units and conversions: Convert between mm, cm, m, km; mm², cm², m²; mm³, cm³, m³. Remember that 1 litre = 1000 cm³.
公制单位与换算：在毫米、厘米、米、千米之间进行换算；面积单位换算；体积单位换算。注意 1 升 = 1000 立方厘米。
8. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数
Pythagoras’ theorem: In a right-angled triangle, a² + b² = c² where c is the hypotenuse. Use it to find missing sides in 2D and 3D shapes, and to determine whether a triangle is right-angled.
勾股定理：在直角三角形中，a² + b² = c²，其中 c 为斜边。用该定理求平面和立体图形中的未知边长，并判断一个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. Use these ratios to find unknown sides and angles in right-angled triangles.
三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。运用这些比值求直角三角形中的未知边长与角度。
Exact trigonometric values: Know and use exact values for sin, cos and tan of 0°, 30°, 45°, 60° and 90° without a calculator.
精确三角值：在不使用计算器的情况下，熟记并运用 0°、30°、45°、60° 和 90° 的正弦、余弦和正切精确值。
Sine and cosine rules: For any triangle, sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: a² = b² + c² − 2bc cos A. Apply to find sides and angles in non-right-angled triangles.
正弦定理与余弦定理：对于任意三角形，正弦定理：a / sin A = b / sin B = c / sin C；余弦定理：a² = b² + c² − 2bc cos A。用于解非直角三角形的边与角。
Area of a triangle using sine: Area = ½ ab sin C. Use this formula when two sides and the included angle are known.
用正弦求三角形面积：面积 = ½ ab sin C。当已知两边及其夹角时使用该公式。
9. Statistics and Data Handling | 统计与数据处理
Types of data and sampling: Distinguish between discrete, continuous, qualitative and quantitative data. Understand random, stratified and systematic sampling techniques.
数据类型与抽样：区分离散数据、连续数据、定性数据与定量数据。理解随机抽样、分层抽样和系统抽样方法。
Averages and spread: Calculate mean, median, mode and range from a list, frequency table or grouped frequency table. Estimate the mean from grouped data using midpoints.
平均数与离散程度：从原始列表、频数表或分组频数表中计算平均数、中位数、众数和极差。利用组中值从分组数据估算平均数。
Charts and diagrams: Construct and interpret bar charts, pie charts, stem-and-leaf diagrams, frequency polygons and cumulative frequency curves. Find median, quartiles and interquartile range from cumulative frequency graphs.
图表：绘制并解读条形图、饼状图、茎叶图、频数多边形和累积频数曲线。从累积频数图中求中位数、四分位数和四分位距。
Box plots: Draw and interpret box-and-whisker plots. Use them to compare distributions, identify outliers and comment on skewness.
箱形图：绘制并解读箱形图。用它们比较数据分布、识别异常值并评论偏态。
Scatter graphs and correlation: Plot scatter diagrams, draw lines of best fit, and interpret correlation. Use interpolation to estimate unknown values; understand the dangers of extrapolation.
散点图与相关：绘制散点图，画出最佳拟合线并解释相关性。用内插法估算未知值；理解外推法的风险。
10. Probability | 概率
Basic probability: Understand that probability is on a scale from 0 to 1. Calculate theoretical probability as (number of favourable outcomes)/(total number of outcomes). Use the fact that probabilities of all outcomes sum to 1.
基本概率：理解概率的取值在 0 到 1 之间。计算理论概率为（有利结果数）/（总结果数）。运用所有结果的概率之和为 1 这一事实。
Relative frequency: Use experiments to estimate probabilities; relative frequency = (number of successful trials)/(total number of trials). Understand that more trials give a better estimate.
相对频率：通过实验估计概率；相对频率 = （成功试验次数）/（总试验次数）。理解试验次数越多，估计越准确。
Sample space diagrams: List all possible outcomes systematically using tables, lists or diagrams. Calculate probabilities of combined events.
样本空间图：用表格、列表或图示系统地列出所有可能结果。计算组合事件的概率。
Tree diagrams: Draw and use tree diagrams for independent and dependent events. Multiply along branches and add relevant outcomes. Understand conditional probability expressed as P(A|B).
树状图：为独立事件和相依事件绘制树状图。沿分支相乘并将相关结果相加。理解条件概率 P(A|B) 的含义。
Venn diagrams and set notation: Represent events with Venn diagrams; use union (A ∪ B), intersection (A ∩ B) and complement (A’). Calculate probabilities from Venn diagrams and two-way tables.
韦恩图与集合符号：用韦恩图表示事件；使用并集（A ∪ B）、交集（A ∩ B）和补集（A’）。从韦恩图和双向表中计算概率。
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