📚 IGCSE Mathematics Grade 8: Core Revision Guide | IGCSE数学八年级：核心复习指南
This revision guide covers the essential topics for IGCSE Mathematics at Grade 8 level. It is designed to help students consolidate their understanding, practise key skills, and build confidence for examinations. Each section summarises the main ideas and provides worked examples where appropriate.
本复习指南涵盖IGCSE数学八年级的核心主题，旨在帮助学生巩固理解、练习关键技能并增强考试信心。每一节都总结了主要概念，并提供了相应的例题。
1. Number and Arithmetic | 数与算术
The foundation of all mathematics is a solid understanding of numbers. At Grade 8, students must be fluent in working with integers, fractions, decimals, and percentages. You should be able to perform calculations accurately using the correct order of operations: Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
数学的基础是对数字的深入理解。在八年级，学生必须熟练处理整数、分数、小数和百分比。你需要能使用正确的运算优先级（括号、指数、乘除、加减）准确地进行计算。
- Factors and multiples: know the difference, and find HCF and LCM using prime factorisation.
- 因数与倍数：了解区别，并通过质因数分解求最大公因数（HCF）和最小公倍数（LCM）。
- Fractions: add, subtract, multiply, and divide, including mixed numbers.
- 分数：加、减、乘、除，包括带分数。
- Percentages: convert between fractions, decimals, and percentages; calculate percentage change.
- 百分比：在分数、小数和百分数之间转换；计算百分比变化。
HCF(a,b) = product of common prime factors; LCM(a,b) = product of all prime factors (overlapping ones counted once).
HCF(a,b) = 公有质因数的乘积；LCM(a,b) = 所有质因数的乘积（重复的只算一次）。
When solving problems, always estimate first, then calculate exactly, and finally check your answer against the estimate.
解题时先估算，再精确计算，最后用估算检验答案。
2. Algebra: Simplifying and Solving | 代数：化简与求解
Algebra is a powerful tool for generalising patterns and solving problems. You should be able to simplify expressions by collecting like terms, expand brackets, and solve linear equations. Understanding the difference between expressions, equations, and identities is also essential.
代数是概括规律和解决问题的强大工具。你需要能通过合并同类项化简表达式、展开括号，并解线性方程。理解表达式、方程和恒等式之间的区别也很重要。
- Simplify: 3x + 2y – x + 5y becomes 2x + 7y.
- 化简：3x + 2y – x + 5y 变为 2x + 7y。
- Expand: (x + 3)(x – 2) = x² + x – 6.
- 展开：(x + 3)(x – 2) = x² + x – 6。
- Solve: 5x – 3 = 2x + 9 → x = 4.
- 解方程：5x – 3 = 2x + 9 → x = 4。
To solve a linear equation, isolate the variable by performing the same operation on both sides.
解线性方程时，通过两边执行相同运算来隔离未知数。
Also practise substituting values into formulas. For example, if y = 3x² + 2, then when x = 5, y = 3(25) + 2 = 77.
还要练习将数值代入公式。例如，若 y = 3x² + 2，当 x = 5 时，y = 3(25) + 2 = 77。
3. Simultaneous Equations | 联立方程
Simultaneous equations involve two or more unknown variables. For two linear equations, you can solve using either elimination or substitution. Graphically, the solution is the point where the two lines intersect.
联立方程涉及两个或更多未知数。对于两个线性方程，可以用消元法或代入法求解。在图像上，解就是两条直线的交点。
Elimination: add or subtract equations to remove one variable. Substitution: express one variable in terms of the other and replace.
消元法：通过加减方程消去一个未知数。代入法：用一个未知数表示另一个，然后代入。
- Example: y = 2x – 1 and y = x + 3. Subtract: (2x – 1) – (x + 3) = 0 → x = 4, y = 7.
- 例：y = 2x – 1 和 y = x + 3。相减：(2x – 1) – (x + 3) = 0 → x = 4, y = 7。
Check your solution by substituting both values into the original equations.
将两个值代入原方程来检验答案。
4. Linear Graphs and Gradients | 线性图像与斜率
A linear graph represents an equation of the form y = mx + c, where m is the gradient (slope) and c is the y-intercept (where the line crosses the y-axis). You should be able to plot a line from its equation, read the gradient and intercept from a graph, and write the equation of a line given two points.
线性图像表示形如 y = mx + c 的方程，其中 m 是梯度（斜率），c 是 y 截距（直线与 y 轴的交点）。你需要能根据方程作图、从图像中读取梯度和截距，并根据两点写出直线方程。
m = (y₂ – y₁) / (x₂ – x₁) for points (x₁, y₁) and (x₂, y₂).
对点 (x₁, y₁) 和 (x₂, y₂)，m = (y₂ – y₁) / (x₂ – x₁)。
If the graph is horizontal, m = 0. If it is vertical, the gradient is undefined.
若图像水平，则 m = 0；若垂直，则梯度不存在。
5. Angles and Polygons | 角与多边形
Geometry involves the study of shapes and their properties. You should know the angle rules for lines, triangles, and parallel lines. For polygons, the sum of interior angles is (n – 2) × 180°, where n is the number of sides. The sum of exterior angles is always 360°.
几何涉及图形及其性质的研究。你需要了解直线、三角形和平行线的角度规则。对于多边形，内角和为 (n – 2) × 180°，其中 n 是边数。外角和始终为 360°。
- Angles on a straight line: add to 180°.
- 直线上的角：和为 180°。
- Angles at a point: add to 360°.
- 同一点周围的角：和为 360°。
- Vertically opposite angles are equal.
- 对顶角相等。
- Corresponding and alternate angles on parallel lines are equal.
- 平行线上的同位角和内错角相等。
Interior angle of a regular n-gon = (n – 2) × 180° / n.
正 n 边形内角 = (n – 2) × 180° / n。
When calculating, always state the reason for each angle, e.g., ‘co-interior angles add to 180°’.
计算时，对每个角都要说明理由，例如“同旁内角之和为 180°”。
6. Mensuration: Area, Perimeter, Volume | 度量：面积、周长与体积
Mensuration deals with the measurement of geometric figures. You must know formulas for area and perimeter of common shapes, and for surface area and volume of 3D solids. Be careful with units: convert all lengths to the same unit before calculating.
度量涉及几何图形的测量。你必须掌握常见图形的面积和周长公式，以及三维体的表面积和体积公式。注意单位：计算前将所有长度转换成相同单位。
| Shape | Area Formula |
| Triangle | A = ½ × base × height |
| Rectangle | A = length × width |
| Circle | A = πr² |
| Trapezium | A = ½(a + b)h |
| Cylinder volume | V = πr²h |
| Prism volume | V = area of cross-section × length |
For composite shapes, split them into simple shapes, calculate each part, then add or subtract.
对于复合图形，先分割成简单图形，分别计算，然后相加或相减。
7. Trigonometry: Right-Angled Triangles | 三角学：直角三角形
Trigonometry links the sides and angles of right-angled triangles. Pythagoras’ theorem relates the lengths of the three sides, while the sine, cosine, and tangent ratios relate any side to another when an angle is known.
三角学将直角三角形的边与角联系起来。勾股定理描述三边关系，而正弦、余弦和正切比可将一个角与某两边联系起来。
a² + b² = c² (where c is the hypotenuse). sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
a² + b² = c²（c 为斜边）。sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Choose the correct ratio by identifying the known and required sides. Round to a sensible number of decimal places or significant figures.
通过识别已知边和目标边来选择正确的比。结果保留合理的位数或有效数字。
8. Statistics and Probability | 统计与概率
Statistics involves collecting, presenting, and interpreting data. Measures of central tendency (mean, median, mode) and spread (range) summarise data sets. Probability measures the chance of an event occurring, from 0 (impossible) to 1 (certain).
统计涉及数据的收集、呈现和解释。集中趋势（平均数、中位数、众数）和离散程度（极差）用于概括数据集。概率衡量事件发生的可能性，从 0（不可能）到 1（必然）。
- Mean = sum of all values ÷ number of values.
- 平均数 = 所有值之和 ÷ 值的个数。
- Median: middle value when data are in order.
- 中位数：数据排序后的中间值。
- Mode: most frequently occurring value.
- 众数：出现最多的值。
- Probability of an event = number of favourable outcomes ÷ total outcomes (if equally likely).
- 事件概率 = 有利结果数 ÷ 总结果数（等可能时）。
For combined events, list all outcomes or use a tree diagram to find probabilities.
对于复合事件，列出所有结果或使用树形图求概率。
9. Ratio, Proportion and Percentages | 比、比例与百分比
Ratios compare quantities, and proportions show how two ratios are equal. Direct proportion means that as one quantity increases, the other increases at the same rate. Inverse proportion means that as one increases, the other decreases. Percentages are a special type of ratio with denominator 100.
比是比较两个量，比例表示两个比相等。正比关系意味着一个量增加时，另一个量以相同速度增加。反比关系则是一个量增加时另一个量减少。百分比是分母为 100 的特殊比。
If y ∝ x, then y = kx. If y ∝ 1/x, then y = k/x. The constant k is called the constant of proportionality.
若 y ∝ x，则 y = kx；若 y ∝ 1/x，则 y = k/x。常数 k 称为比例常数。
When solving percentage problems, identify the original amount, the change, and express the change as a fraction of the original multiplied by 100.
解百分比问题时，确定原量、变化量，并将变化量表示为原量的分数乘以 100。
10. Exam Techniques and Common Mistakes | 考试技巧与常见错误
Success in mathematics exams comes from good preparation and avoiding careless errors. Always read the question carefully, show all working, and check your answers. Understand what the question asks: ‘calculate’, ‘simplify’, ‘prove’, or ‘estimate’ each have different expectations.
数学考试的成功来自于充分准备和避免粗心错误。仔细阅读题目，写出所有过程，并检查答案。理解题目的要求：“计算”、“化简”、“证明”或“估算”各有不同的期望。
- Do not forget units in final answers.
- 不要忘记最后答案的单位。
- When using a calculator, ensure it is in the correct mode (degrees for angles unless stated otherwise).
- 使用计算器时，确保处于正确模式（角度除非另有说明，否则用度）。
- Round only at the end, not during intermediate steps.
- 只在最后四舍五入，不要在中间步骤。
- Check if the answer is reasonable (e.g., a length cannot be negative).
- 检查答案是否合理（例如长度不能为负）。
Practise past papers under timed conditions, and review every mistake to understand its cause.
在限时条件下练习历年真题，并复习每个错误以理解其根源。
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