IGCSE Edexcel Maths: Last-Minute Revision Notes | IGCSE Edexcel 数学：考前冲刺笔记

📚 IGCSE Edexcel Maths: Last-Minute Revision Notes | IGCSE Edexcel 数学：考前冲刺笔记

This revision guide compiles the most essential topics for the IGCSE Edexcel Mathematics exam, providing last-minute reminders and key strategies. Use it to reinforce core concepts, avoid common pitfalls, and boost your confidence before the test.

本考前冲刺笔记汇总了 IGCSE Edexcel 数学考试中最关键的知识点，提供临考提醒和重要策略。用此笔记巩固核心概念、避开常见错误，并在考试前提升信心。

1. Number Skills and BIDMAS | 数字运算与运算顺序

Always follow the correct order of operations: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). A common mistake is misplacing brackets with negative numbers: (-3)² = 9, but -3² = -9.

始终遵循正确的运算顺序：括号、指数、乘除（从左至右）、加减（从左至右）。一个常见的错误是在负数中忘记加括号：(-3)² = 9，而 -3² = -9。

Product of prime factors helps find HCF and LCM. Write numbers as products of primes using index form, then for HCF take the lowest power of common primes, and for LCM take the highest power of all primes.

用质因数分解求最大公因数（HCF）和最小公倍数（LCM）很有效。将数字写成质数的乘积（含指数形式），求 HCF 时取各公共质数的最小指数，求 LCM 时取所有质数的最高指数。

2. Fractions, Decimals and Percentages | 分数、小数和百分比

To convert a fraction to a decimal, divide the numerator by the denominator. For a percentage, multiply the decimal by 100%. Converting a percentage to a decimal: divide by 100. E.g. 0.45 = 45%, 3/5 = 0.6 = 60%.

分数转小数：分子除以分母。转百分比：小数乘以 100%。百分比转小数：除以 100。例如 0.45 = 45%，3/5 = 0.6 = 60%。

Percentage increase: new value = original × (1 + r/100). Percentage decrease: new value = original × (1 – r/100). For reverse percentages divide by the multiplier, e.g. a price after 20% increase is original × 1.20, so original = sale price ÷ 1.20.

百分比增加：新值 = 原值 × (1 + r/100)。百分比减少：新值 = 原值 × (1 – r/100)。反向百分比问题要除以乘数，例如涨价 20% 后的价格是原价 × 1.20，因此原价 = 售价 ÷ 1.20。

Compound interest: Amount = P(1 + r/100)ⁿ for n years. For depreciation (decay) use a minus sign. Half-life problems are solved similarly by using multiplier 0.5 raised to number of half-lives.

复利公式：本息和 = P(1 + r/100)ⁿ，n 为年数。折旧（衰减）用减号。半衰期问题类似，乘数为 0.5 的半衰期次方。

3. Ratio and Proportion | 比率与比例

Simplify a ratio by dividing all parts by their HCF. When sharing a quantity in a given ratio, find the total number of parts, divide the quantity by that total, then multiply by each part number.

化简比率：将所有部分除以它们的 HCF。按给定比率分配数量时，求总份数，用总量除以总份数，再乘以各部分的份数。

Direct proportion: y = kx. If x doubles, y doubles. Inverse proportion: y = k/x. If x doubles, y halves. Look for phrases like ‘directly proportional’ or ‘varies inversely’. Set up an equation to find constant k first.

正比例关系：y = kx。x 加倍，y 也加倍。反比例关系：y = k/x。x 加倍，y 减半。注意题目中 ‘directly proportional’ 或 ‘varies inversely’ 的字眼，先建立方程求常数 k。

4. Algebraic Expressions and Equations | 代数表达式与方程

When expanding brackets, multiply each term inside by the term outside. Be careful with signs: a(b – c) = ab – ac. Expand double brackets using FOIL: (x + a)(x + b) = x² + (a+b)x + ab. For (a+b)² = a² + 2ab + b².

展开括号时，将括号外的每一项乘以括号内的每一项。注意符号：a(b – c) = ab – ac。用 FOIL 法则展开双括号：(x + a)(x + b) = x² + (a+b)x + ab。完全平方公式 (a+b)² = a² + 2ab + b²。

Factorising is the reverse of expanding. Look for common factors first: 3x + 6 = 3(x + 2). For quadratics of form x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c, use splitting the middle term or factorising by grouping.

因式分解是展开的逆运算。先找公因式：3x + 6 = 3(x + 2)。对于二次项 x² + bx + c，找两个数乘积为 c、和为 b。对于 ax² + bx + c 的二次式，可用拆分中项或分组分解法。

Solve linear equations by doing the same operation to both sides. To solve a quadratic equation x² + bx + c = 0, factorise and set each bracket to zero. If it does not factorise, use the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a.

解线性方程时对两边同时进行相同运算。解二次方程 x² + bx + c = 0 时，先因式分解，再令各括号为 0。若无法分解，使用求根公式：x = [-b ± √(b² – 4ac)] / 2a。

5. Sequences and nth Term | 数列与通项公式

For an arithmetic (linear) sequence, the nth term is given by a + (n-1)d, where a is the first term and d is the common difference. Write it as an + b form: find the difference d, then adjust to match the first term.

等差数列（线性数列）的第 n 项公式为 a + (n-1)d，其中 a 是首项，d 是公差。写成 an + b 的形式：先找出公差 d，再调整常数项以匹配首项。

For a quadratic sequence, work out the first and second differences. The coefficient of n² is half the second difference. Subtract the quadratic part from the original terms to find a linear remainder, then combine.

二次数列需要求一阶差和二阶差。n² 的系数是二阶差的一半。从原数列各项中减去 n² 部分，得到的余项为线性部分，再合并得到通项。

6. Graphs and Functions | 图形与函数

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. Gradient = (change in y) / (change in x). Parallel lines have the same gradient; perpendicular lines have gradients that multiply to -1.

直线方程为 y = mx + c，m 是斜率，c 是 y 轴截距。斜率 = (y 的变化) / (x 的变化)。平行线斜率相同；垂直线的斜率积为 -1。

To plot a quadratic graph y = ax² + bx + c, complete a table of values. The turning point occurs at x = -b/(2a). For cubic and reciprocal graphs, know their basic shapes. Transformations: f(x) + a shifts up by a; f(x + a) shifts left by a.

绘制二次函数图像 y = ax² + bx + c 需先列表。其顶点在 x = -b/(2a) 处。熟悉三次函数和反比例函数的基本形状。图形变换：f(x) + a 向上移 a 单位；f(x + a) 向左移 a 单位。

Distance-time graphs: gradient gives speed; horizontal lines mean stationary. Speed-time graphs: gradient gives acceleration; area under graph gives distance travelled.

距离-时间图：斜率表示速度；水平线代表静止。速度-时间图：斜率表示加速度；图下面积代表所行距离。

7. Geometry: Angles and Triangles | 几何：角与三角形

Angles on a straight line sum to 180°, angles around a point sum to 360°. Vertically opposite angles are equal. In parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles sum to 180°.

直线上的角之和为 180°，一点周角之和为 360°。对顶角相等。平行线中：同位角相等，内错角相等，同旁内角之和为 180°。

The sum of interior angles of a polygon with n sides is (n-2) × 180°. Each exterior angle of a regular polygon is 360°/n. The exterior angle is the supplementary angle to the interior angle.

n 边形内角和为 (n-2) × 180°。正多边形的每个外角为 360°/n。外角与内角互补。

In a triangle, the largest angle is opposite the longest side. Pythagoras’ theorem: a² + b² = c² for right-angled triangles. Congruent triangles tests: SSS, SAS, ASA, RHS. Similar triangles have equal angles and proportional sides.

三角形中大角对大边。勾股定理：直角三角形中 a² + b² = c²。全等判定：SSS、SAS、ASA、RHS。相似三角形对应角相等，对应边成比例。

8. Perimeter, Area and Volume | 周长、面积和体积

Key formulas (use consistently):

重要公式（需熟练应用）：

Shape	Area / Volume formula
Triangle	Area = ½ × base × height
Parallelogram	Area = base × height
Trapezium	Area = ½(a + b)h
Circle	Circumference = 2πr or πd; Area = πr²
Cylinder	Volume = πr²h; Curved surface area = 2πrh
Cone	Volume = ⅓πr²h; Curved surface area = πrl
Sphere	Volume = ⁴⁄₃πr³; Surface area = 4πr²

When dealing with composite shapes, break them into simpler parts. For prisms, Volume = area of cross-section × length. Cone and sphere formulas are given on the formula sheet, but you must know how to use them.

组合图形可拆分为基本形状。棱柱体积 = 横截面积 × 长度。圆锥和球体公式会在试卷公式表中给出，但务必知道如何正确使用。

9. Statistics: Averages and Charts | 统计：平均值与图表

Mean = sum of values ÷ number of values. For grouped data, use the midpoint of each class. Median is the middle value; mode is the most frequent. Range = max – min. Interquartile range = Q₃ – Q₁.

平均数 = 总和 ÷ 个数。分组数据用组中点。中位数是中间值；众数是出现最多的值。极差 = 最大值 – 最小值。四分位距 = Q₃ – Q₁。

Cumulative frequency graphs show the running total. Use them to estimate the median and quartiles. Histograms: area of bar = frequency, not height. Frequency density = frequency / class width, used for unequal intervals.

累积频率图显示累计总数，可用来估算中位数和四分位数。直方图：条形面积代表频数，而不是高度。频数密度 = 频数 ÷ 组距，用于不等组距。

Scatter graphs show correlation: positive, negative or none. A line of best fit can be used for prediction. Be aware that extrapolation beyond the data range is unreliable.

散点图显示相关性：正相关、负相关或无相关。最佳拟合线可用于预测。注意在数据范围之外的外推是不可靠的。

10. Probability | 概率

Probability = number of favourable outcomes / total number of outcomes. Always between 0 and 1. Two events are mutually exclusive if they cannot happen at the same time: P(A or B) = P(A) + P(B).

概率 = 有利结果数 ÷ 所有可能结果数。概率值始终在 0 到 1 之间。互斥事件不能同时发生：P(A 或 B) = P(A) + P(B)。

For independent events, P(A and B) = P(A) × P(B). Tree diagrams are essential for combined events: multiply along branches, add probabilities of different branches. Conditional probability uses P(A|B) = P(A and B) / P(B).

独立事件中 P(A 且 B) = P(A) × P(B)。树状图是解决组合概率的关键：沿分支相乘，不同分支的概率相加。条件概率公式 P(A|B) = P(A 且 B) / P(B)。

11. Trigonometry and Pythagoras | 三角学与勾股定理

In right-angled triangles, label sides: opposite (opp), adjacent (adj) and hypotenuse (hyp). SOHCAHTOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. Use the inverse functions to find an angle.

在直角三角形中标记各边：对边、邻边和斜边。SOHCAHTOA 记忆法：sin θ = 对/斜，cos θ = 邻/斜，tan θ = 对/邻。求角时使用反三角函数。

For non-right-angled triangles, use the sine rule: a/sin A = b/sin B = c/sin C, or the cosine rule: a² = b² + c² – 2bc cos A. Area = ½ ab sin C. Always check if the triangle has an ambiguous case with the sine rule.

非直角三角形使用正弦定理：a/sin A = b/sin B = c/sin C，或余弦定理：a² = b² + c² – 2bc cos A。三角形面积 = ½ ab sin C。使用正弦定理时注意是否存在多解情况。

3D Pythagoras: for a cuboid with sides a, b, c, the space diagonal is √(a² + b² + c²). Trigonometry in 3D often requires identifying right triangles within the solid.

三维勾股定理：长方体棱长为 a、b、c，体对角线长为 √(a² + b² + c²)。三维三角学问题通常需要找出立体图形中的直角三角形。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Read each question carefully, underlining command words. Show all working clearly; method marks are often awarded even if the final answer is incorrect. Always write units where applicable.

仔细读题，圈划指令词。清晰展示所有解题步骤；即便最终答案错误，方法也可能得分。凡涉及单位处必须标明单位。

Avoid rounding until the final answer. Use exact values when possible. Check your answers with estimation. Manage time: spend roughly 1 minute per mark, leaving time to review.

在得出最后答案前不要过早四舍五入，尽量保留精确值。用估算检查答案的合理性。合理分配时间：大致每分对应一分钟，留出检查时间。

Common pitfalls: forgetting to include ± when square rooting, incorrect cancellation in algebra, mixing up area and circumference formulas, and misreading histogram axes. Double-check constructions with a protractor and compass.

常见失分点：开方时忘记 ± 号，代数约分出错，混淆面积与周长公式，误读直方图坐标轴。用尺规作图后用量角器和圆规复核。

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IGCSE Edexcel Maths: Last-Minute Revision Notes | IGCSE Edexcel 数学：考前冲刺笔记