📚 IGCSE Edexcel Mathematics Revision Guide | IGCSE Edexcel 数学复习指南
The Edexcel IGCSE Mathematics syllabus may feel broad, but it is also highly predictable. A focused revision plan that combines conceptual understanding with rapid recall can turn a mountain of content into a set of manageable tools. This guide breaks the course into the key skills you need, shows the classic question styles, and gives you a clear way to prepare for both papers.
Edexcel IGCSE 数学的考纲虽然看似覆盖面广，但题型其实高度可预测。将概念理解与快速记忆相结合的复习计划，可以把庞大的知识体系化为一套易于掌握的工具。本指南将课程内容拆解为必备技能，展示经典题型，并为你提供两套试卷的清晰备考路径。
1. Understand the Exam Structure | 了解考试结构
Before you revise content, you must know the shape of the exam. Edexcel IGCSE Mathematics A has two papers, each worth 50% of your final grade. Paper 1 is a non-calculator paper and Paper 2 allows a calculator. Both papers are two hours long and each has 100 marks.
在复习具体内容之前，你必须了解考试结构。Edexcel IGCSE 数学 A 包含两套试卷，各占总成绩的 50%。试卷一不允许使用计算器，试卷二允许使用计算器。两套试卷时长均为两小时，满分各为 100 分。
Paper 1: Pure Mathematics and Statistics topics without a calculator. You must be fluent with arithmetic, indices, surds, and algebraic manipulation by hand.
试卷一：不使用计算器完成纯数学和统计内容。你必须熟练掌握手算算术、指数、根式以及代数变形。
Paper 2: The same syllabus but with a calculator. Questions often have less friendly numbers, and technology supports accurate computation.
试卷二：考查相同大纲内容，但允许使用计算器。题目中的数据往往不那么“友好”，需要借助计算器进行精确计算。
Your final grade is awarded on the 9 to 1 scale. Because the grade boundaries can vary, aim to score well above the boundary for your target grade rather than close to it.
最终成绩采用 9 至 1 的评分等级。由于分数线可能浮动，目标应当是明显高于你所希望等级的分数线，而不是刚好压线。
2. Master Number and Indices | 掌握数与指数
The foundation of IGCSE Mathematics is number sense. You need to work confidently with integers, fractions, decimals, percentages, standard form, and ratios. A common theme is converting between forms without losing accuracy.
IGCSE 数学的基础是数感。你需要自信地处理整数、分数、小数、百分比、标准形式和比。常见的考法是在不同形式之间转换且不失精度。
Index laws appear almost every year. You should be able to simplify expressions like these instantly:
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
You also need negative and fractional indices. A negative index means a reciprocal, and a fractional index involves a root. For example:
a⁻ⁿ = 1/aⁿ, a1/n = ⁿ√a, am/n = (ⁿ√a)ᵐ
Surds: simplify √12 as 2√3. Rationalise a denominator such as 1/√5 by multiplying numerator and denominator by √5.
根式：将 √12 化简为 2√3。将 1/√5 这类分母有理化时，可上下同乘 √5。
Standard form: write 5,800,000 as 5.8 × 10⁶ and 0.00072 as 7.2 × 10⁻⁴.
标准形式：5,800,000 写作 5.8 × 10⁶，0.00072 写作 7.2 × 10⁻⁴。
When working without a calculator, practise mental arithmetic and efficient written methods. Do not rely on the calculator in Paper 2 for simple checks; it can hide gaps in understanding.
在不使用计算器时，应多练习心算和高效的手算方法。在试卷二中，即使是简单检查也不要完全依赖计算器，否则会掩盖概念上的漏洞。
3. Algebra and Equations | 代数与方程
Algebra is the largest single strand of the course. You must be able to expand brackets, factorise expressions, solve linear and quadratic equations, and manipulate algebraic fractions.
代数是本课程中占比最大的分支。你必须能够展开括号、分解因式、解线性方程和二次方程，并处理代数分式。
A classic skill is solving a quadratic equation by factorising. You should also know the quadratic formula for cases where factorisation is not possible:
x = (−b ± √b² − 4ac) / 2a
Completing the square is also in the syllabus. It helps you find turning points and solve equations without using a formula.
配方法也在考纲范围内。它有助于求抛物线顶点，且能在不套用公式的情况下解方程。
r inequalities: solve −3 ≤ 2x − 1 < 5 by performing the same operation on all three parts.
线性不等式：解 −3 ≤ 2x − 1 < 5 时，对三个部分同时执行相同运算。
Simultaneous equations: practise both elimination and substitution. For a quadratic linear pair, use substitution and solve the resulting quadratic.
联立方程：练习消元法和代入法。对于“二次+一次”方程组，使用代入法解得到的二次方程。
Sequences: recognise common difference and common ratio. Use nth term notation such as 3n + 2 for an arithmetic sequence.
数列：识别公差与公比。等差数列的通项可写作 3n + 2 等形式。
Always check your solutions by substituting them back into the original equation. This is especially important after squaring both sides or multiplying by an unknown denominator.
解出答案后务必代回原方程验证。尤其在两边平方或乘以含未知数的分母后，这一步尤为重要。
4. Coordinate Geometry and Graphs | 坐标几何与函数图像
Coordinate geometry connects algebra with shape. You need the equation of a straight line, the gradient formula, and the ability to find midpoints and lengths of line segments.
坐标几何将代数与图形联系起来。你需要掌握直线方程、斜率公式，并能求中点和线段长度。
For two points (x₁, y₁) and (x₂, y₂), the gradient is:
m = (y₂ − y₁) / (x₂ − x₁)
en as y = mx + c. Parallel lines have equal gradients; perpendicular lines multiply to −1.
直线方程通常写作 y = mx + c。平行线的斜率相等；互相垂直的直线斜率相乘等于 −1。
roots, turning point, and line of symmetry. The graph y = (x − p)² + q has vertex (p, q).
二次函数图像：确认根、顶点和对称轴。y = (x − p)² + q 的顶点为 (p, q)。
learn the general shape of y = x³, and y = a/x, where branches lie in opposite quadrants.
三次函数与反比例函数图像：记住 y = x³ 和 y = a/x 的基本形状，反比例函数的两个分支位于相对的象限。
Exponential graphs: understand how y = kˣ grows or decays depending on whether k is greater or less than 1.
指数函数图像：理解 y = kˣ 的增长或衰减取决于 k 大于还是小于 1。
When reading graphs questions, pay attention to axes and scales. Many errors come from misreading a graph, not from missing a formula.
做图像题时，请留意坐标轴和刻度。很多错误并非因为公式不会，而是因为读错了图。
5. Trigonometry and Bearings | 三角学与方位角
Trigonometry in IGCSE covers right-angled triangles, the sine rule, the cosine rule, and the area of a triangle. You also need to apply these to three-figure bearings.
IGCSE 三角学涵盖直角三角形、正弦定理、余弦定理以及三角形面积。你还需将这些知识应用于三方位角。
For a right-angled triangle, the three key ratios are:
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
For non-right-angled triangles, use the sine rule and cosine rule:
a/sin A = b/sin B = c/sin C
a² = b² + c² − 2bc cos A
Bearings are always measured clockwise from north and written as three digits, such as 047°.
方位角总是从正北方向顺时针测量，并写成三位数，如 047°。
When solving bearing problems, draw a clear diagram and mark the north line at each corner.
解方位角问题时，务必画出清晰图形，并在每个顶点处标出北方参考线。
Use the sine rule for two known angles and one opposite side, or two sides and one opposite angle. Use the cosine rule for two sides and the included angle, or three given sides.
已知两角和一条对边，或两边和一角时用正弦定理。已知两边及其夹角，或三边时用余弦定理。
Remember to set your calculator to degree mode. A single wrong mode setting can make every trig answer incorrect.
记住将计算器设为角度制。一次模式错设就可能让所有三角答案全部出错。
6. Mensuration: Area, Volume, and Surface Area | 几何测量：面积、体积与表面积
Mensuration asks you to calculate lengths, angles, areas, and volumes. You need a strong memory of standard formulae and the ability to combine them in composite shapes.
几何测量要求你计算长度、角度、面积和体积。你需要牢记标准公式，并能在组合图形中灵活运用。
For a circle, the key facts are:
Area = πr², Circumference = 2πr
For a sector with angle θ, the arc length and sector area are fractions of the full circle:
Arc length = (θ/360°) × 2πr, Sector area = (θ/360°) × πr²
rea of cross-section × length. Practise triangular prisms, cylinders, and compound solids.
棱柱体：体积 = 截面面积 × 长度。多练习三棱柱、圆柱和组合体。
s and cones: volume = (1/3) × base area × height. A cone also has curved surface area πrl.
棱锥和圆锥：体积 = (1/3) × 底面积 × 高。圆锥的侧面积为 πrl。
rea = 4πr². These often appear in problems that ask you to convert one solid into another.
球体：体积 = (4/3)πr³，表面积 = 4πr²。这类公式常出现在将一个立体转化为另一个立体的题目中。
Always match units before calculating. Convert all lengths to the same unit, especially in volume and density problems.
计算前务必统一单位。特别是在体积和密度问题中，要将所有长度转换成同一单位。
7. Probability and Statistics | 概率与统计
Statistics and probability questions test your ability to interpret data and calculate likelihood. The key is to use tables and tree diagrams carefully.
统计与概率题考查你解读数据和计算可能性的能力。关键是谨慎使用表格和树状图。
For probability, remember the fundamental rules:
P(not A) = 1 − P(A)
P(A and B) = P(A) × P(B) for independent events
Tree diagrams: write probabilities on every branch. Multiply along branches for combined outcomes, and add across separate branches for mutually exclusive events.
树状图：在每条分支上写清概率。求复合事件时沿分支相乘；求互斥事件时跨分支相加。
Mean, median, mode, and range: practise with grouped data and use midpoints to estimate the mean from frequency tables.
平均数、中位数、众数和极差：练习分组数据，并使用组中值从频数表估算平均数。
Histograms: remember that frequency is given by the area of the bar, not the height, when class widths differ.
直方图：当组距不同时，频数由矩形的面积决定，而不是高度。
In conditional probability questions, read the wording carefully. Phrases such as “given that” or “if” change the denominator of the probability fraction.
在条件概率题中，仔细阅读题干。“已知……”或“如果……”这类说法会改变概率分数的分母。
8. Vectors and Transformations | 向量与几何变换
Vectors combine movement and algebra. You need to understand column vectors, vector addition, scalar multiplication, and parallel vectors.
向量将运动与代数结合起来。你需要理解列向量、向量加法、标量乘法以及平行向量。
If a = (2₁? Actually vector notation in Unicode: column vector (2 above 3) can be written with matrix? Need no LaTex. Use (2, 3) or (2 3)? Need show column vector maybe use . But no LaTex. Could use "2 above 3" as text? We can write vector notation in HTML with ? Let's simplify: "Vector a = (2, 3) in column form" or "a = (2↕3)"? Better: " as (a b), with the top value being the horizontal movement and the bottom value vertical." But need exact. Use Unicode? There is not column vector easy. Could use "a = [2 above 3]" but okay. The instructions say use Unicode, not LaTex. We can use matrix? HTML table? Overkill. Let's write "a = (2, 3)" se column form perhaps in brackets with semicolon: a = (2; 3) meaning top then bottom. Let's use "a = (2; 3)" to represent column vector? Actually in many text-based contexts, column vector can be (2, 3)ᵀ (transpose). Since no LaTex, use "a = (2, 3)ᵀ". That's correct transpose convention. Need maybe explain "T indicates vertical layout." Let's craft. Let's write: For a column vector, we often write (x, y)ᵀ, where x is the horizontal component and y is the vertical component. Translation vector T = (3, −2)ᵀ means move 3 units right and 2 units down. Vector operations: a + b = (x₁ + x₂, y₁ + y₂)ᵀ 2a = (2x₁, 2y₁)ᵀ Need pair Chinese. rection/centre of rotation, vector of translation, scale factor and centre of enlargement.
Need bullet points. Need not overdo. Let's craft.9. Problem Solving and Reasoning | 问题解决与推理
Many students understand individual topics but fail to combine them. The IGCSE exam often places a number skill inside a geometry problem, or an algebraic result inside a statistics question. Train yourself to recognise which mathematical tool is needed before performing calculations.
许多学生能理解单个知识点，却不会综合运用。IGCSE 考试常把数字技能嵌在几何题中，或把代数结论放在统计题里。训练自己在动手计算前先识别需要哪种数学工具。
Read the final line of the question first. It tells you the required answer form: exact, to 3 significant figures, or in terms of π.
先读题目的最后一行。它会告诉你所需答案的形式：精确值、保留 3 位有效数字，还是含 π。
Show every step of working. Method marks can rescue you when the final answer is wrong.
写出每一步过程。即使最终答案错误，过程分仍可能救回你。
Use estimation before doing a difficult calculation. If your final value is far from the estimate, check your method.
在复杂计算前先估算。如果最终值与估算相差很远，就应回头检查方法。
A useful habit is to turn word problems into algebraic sentences. Let the unknown be x, then build equations from the relationships described in the text.
一个有效的习惯是把应用题转化为代数语句。设未知数为 x，再根据题干中的关系建立方程。
10. Final Revision Strategy | 考前复习策略
In the last weeks before the exam, move from passive reading to active testing. Your brain learns best when you retrieve a formula or method from memory without looking at a textbook.
在考前最后几周，要从被动阅读转向主动测试。在不看课本的情况下从记忆中提取公式或方法，学习效果最好。
Use past papers on a strict timer. Sit exactly two hours and practise presenting complete answers.
用历年真题并按严格时间计时。完整练习两小时，并训练规范作答。
Make a one-page summary for each unit. Write down only the formulas and methods you forget often.
为每个单元制作一页纸总结。只记录你经常忘记的公式和方法。
Review mistakes in grouped clusters. If you lose marks on fractions, solve ten fraction-heavy questions in one session.
按错误类型集中复习。如果分数老是扣在分数计算上，就用整段时间集中做十道涉及分数的题。
Practise non-calculator arithmetic daily. Fluency in Paper 1 builds confidence for the whole exam.
每天练习不使用计算器的算术。试卷一的熟练度会为你整个考试带来自信。
Do not try to memorise every question. Instead, memorise patterns: “two equations and two unknowns means simultaneous equations”, “right-angled triangle and an angle means trigonometry”.
不要试图背下每一道题。应记住模式：“两个方程两个未知数 → 联立方程”，“直角三角形加一个角 → 三角学”。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图