📚 IGCSE Maths: Last-Minute Revision Notes | IGCSE 数学：考前冲刺笔记
These concise revision notes cover the core topics, essential formulas, and common pitfalls you need to know for your IGCSE Mathematics exam. Each section pairs an English explanation with its Chinese equivalent, helping you review efficiently in the final days before the test.
这份精简的冲刺笔记涵盖了IGCSE数学考试的核心主题、必备公式和常见易错点。每部分都以英文和中文配对讲解，帮助你在考前最后几天高效复习。
1. Number and Operations | 数与运算
Understand place value, fractions, decimals, percentages, and directed numbers. Remember the order of operations: BIDMAS/BODMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction).
理解位值、分数、小数、百分数和有向数。牢记运算顺序：括号、指数、乘除、加减（BIDMAS/BODMAS）。
Key conversions: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333. Multiplying by 100 converts a decimal to a percentage; dividing by 100 does the reverse.
关键转换：½ = 0.5 = 50%，¼ = 0.25 = 25%，¾ = 0.75 = 75%，⅓ ≈ 0.333。乘以100将小数化为百分数，除以100则反之。
For standard form, write large or small numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Example: 45,000 = 4.5 × 10⁴.
科学记数法将很大或很小的数写成 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n 为整数。例如：45 000 = 4.5 × 10⁴。
2. Algebraic Foundations | 代数基础
Simplify expressions by collecting like terms: 3x + 2y – x + 5y = 2x + 7y. Expand brackets using the distributive law: a(b + c) = ab + ac. For double brackets, use FOIL: (x + a)(x + b) = x² + (a+b)x + ab.
通过合并同类项化简表达式：3x + 2y – x + 5y = 2x + 7y。使用分配律展开括号：a(b + c) = ab + ac。对于两项括号相乘，使用FOIL法则：(x + a)(x + b) = x² + (a+b)x + ab。
Factorising is the reverse of expanding. Look for a common factor: 6x² + 9x = 3x(2x + 3). For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b: x² + 5x + 6 = (x + 2)(x + 3).
因式分解是展开的逆运算。先提取公因子：6x² + 9x = 3x(2x + 3)。对于形如 x² + bx + c 的二次式，找出两个乘积为 c、和为 b 的数：x² + 5x + 6 = (x + 2)(x + 3)。
The difference of two squares: a² – b² = (a – b)(a + b). Recognise it instantly, e.g., x² – 16 = (x – 4)(x + 4).
平方差公式：a² – b² = (a – b)(a + b)。快速识别，如 x² – 16 = (x – 4)(x + 4)。
3. Equations and Inequalities | 方程与不等式
Solve linear equations by performing the same operation on both sides. Example: 2x + 3 = 11 → subtract 3: 2x = 8 → divide by 2: x = 4.
解一次方程时，两边同时进行相同运算。例如：2x + 3 = 11 → 减3：2x = 8 → 除以2：x = 4。
For quadratic equations, set equal to zero, factorise, then set each factor to zero. If x² + 5x + 6 = 0, then (x+2)(x+3)=0 gives x = -2 or x = -3. If it doesn’t factorise, use the quadratic formula:
对于二次方程，先使其等于零，因式分解，再令每个因式等于零。如果 x² + 5x + 6 = 0，则 (x+2)(x+3)=0 得 x = -2 或 x = -3。若无法因式分解，使用求根公式：
x = (-b ± √(b² – 4ac)) / 2a
Inequalities work like equations, except when multiplying or dividing by a negative number, reverse the sign. E.g., -2x > 6 → x < -3. On a number line, use an open circle for < or >, closed circle for ≤ or ≥.
不等式的解法与方程类似，但当乘或除以负数时，要改变不等号方向。例如：-2x > 6 → x < -3。在数轴上，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
4. Graphs and Functions | 图像与函数
The equation y = mx + c represents a straight line, where m is the gradient (rise/run) and c is the y-intercept. Parallel lines have equal gradients; perpendicular lines have gradients that multiply to -1.
直线方程为 y = mx + c，其中 m 是斜率（垂直变化/水平变化），c 是 y 轴截距。平行线斜率相等；互相垂直的直线斜率乘积为 -1。
Plot quadratic graphs (y = ax² + bx + c) as smooth parabolas. Find the turning point by completing the square or averaging the roots. For y = (x – p)² + q, the vertex is (p, q).
二次函数图像 y = ax² + bx + c 为光滑的抛物线。通过配方法或求根的平均值找到转折点。对于 y = (x – p)² + q，顶点为 (p, q)。
Functions map inputs to outputs. f(x) notation: if f(x) = 2x + 1, then f(3) = 7. The inverse function f⁻¹(x) reverses the mapping. Composite functions like f(g(x)) apply g first, then f.
函数将输入映射到输出。f(x) 记号：若 f(x) = 2x + 1，则 f(3) = 7。反函数 f⁻¹(x) 逆转映射。复合函数如 f(g(x)) 先应用 g，再应用 f。
5. Geometry and Mensuration | 几何与测量
Know angle facts: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and interior angles sum to 180°.
掌握角度性质：平角为 180°；周角为 360°；对顶角相等。平行线中，内错角相等，同位角相等，同旁内角之和为 180°。
Area and perimeter: rectangle A = l × w, triangle A = ½ × b × h, circle C = 2πr or πd, A = πr². Composite shapes: split into simpler parts, calculate separately, then add or subtract.
面积与周长：矩形 A = l × w，三角形 A = ½ × b × h，圆周长 C = 2πr 或 πd，面积 A = πr²。组合图形：分割成简单部分，分别计算，再相加或相减。
Volume and surface area: cuboid V = lwh, prism V = area of cross-section × length, cylinder V = πr²h, curved surface area = 2πrh. Remember density = mass/volume.
体积与表面积：长方体 V = lwh，棱柱 V = 横截面积 × 长度，圆柱 V = πr²h，侧面积 = 2πrh。记住密度 = 质量 / 体积。
6. Trigonometry | 三角学
In a right-angled triangle, SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use these to find missing sides or angles by rearranging.
在直角三角形中，使用 SOH CAH TOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。通过变形求未知边或角。
Sine rule: a/sin A = b/sin B = c/sin C, used for non-right-angled triangles when you know two angles and a side, or two sides and a non-included angle (be careful of the ambiguous case).
正弦定理：a/sin A = b/sin B = c/sin C，用于非直角三角形，当已知两角一边，或两边及一个非夹角时（注意可能存在的歧义情况）。
Cosine rule: a² = b² + c² – 2bc cos A. Use when given two sides and the included angle, or three sides. Area of a triangle: ½ ab sin C.
余弦定理：a² = b² + c² – 2bc cos A。当已知两边及其夹角，或已知三边时使用。三角形面积公式：½ ab sin C。
Exact trigonometric values for 0°, 30°, 45°, 60°, 90° must be memorised, e.g., sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
牢记 0°、30°、45°、60°、90° 的精确三角函数值，如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
7. Probability | 概率
Probability ranges from 0 (impossible) to 1 (certain). P(event) = number of favourable outcomes / total number of outcomes. For mutually exclusive events, P(A or B) = P(A) + P(B).
概率范围从 0（不可能）到 1（必然）。P(事件) = 有利结果数 / 总结果数。对于互斥事件，P(A 或 B) = P(A) + P(B)。
For independent events, P(A and B) = P(A) × P(B). Tree diagrams help visualise sequences of events: multiply along branches for combined probabilities, add probabilities of different branches for the same outcome.
对于独立事件，P(A 且 B) = P(A) × P(B)。树状图有助于可视化事件序列：沿分支相乘得到组合概率，将相同结果的不同分支概率相加。
Conditional probability: P(A given B) = P(A and B)/P(B). Always check whether events are independent or dependent before applying rules.
条件概率：P(A|B) = P(A 且 B) / P(B)。使用规则前务必检查事件是独立还是相关的。
8. Statistics | 统计
Mean = sum of values / number of values. Median is the middle value when ordered; if there are two middle values, take their mean. Mode is the most frequent value. Range = maximum – minimum.
平均数 = 总和 / 数据个数。中位数是排序后中间的值；若有两个中间数，取它们的平均值。众数是出现频率最高的值。极差 = 最大值 – 最小值。
For grouped data, estimate the mean using midpoints. Cumulative frequency graphs help estimate the median, quartiles, and interquartile range (IQR = Q₃ – Q₁).
对于分组数据，使用组中值估算平均数。累积频数图用于估算中位数、四分位数及四分位距（IQR = Q₃ – Q₁）。
Scatter graphs: if points follow a trend, describe the correlation as positive, negative, or none. A line of best fit can be drawn by eye and used to make predictions.
散点图：若点呈现趋势，用正相关、负相关或无相关来描述。最佳拟合线可凭视觉画出，用于预测。
9. Vectors and Transformations | 向量与变换
A vector has magnitude and direction, written as a column vector (x above y) or with bold letters. Add vectors by adding components: (a,b) + (c,d) = (a+c, b+d). Multiply by a scalar multiplies each component.
向量有大小和方向，可以写成列向量（x在上、y在下）或用粗体字母表示。向量相加即各分量相加：(a,b) + (c,d) = (a+c, b+d)。标量乘法是将每个分量相乘。
Translation moves every point by a vector. Reflection flips a shape over a mirror line (e.g., x = 0 or y = x). Rotation turns a shape by a given angle about a centre. Enlargement scales distances from a centre by a scale factor; negative scale factors produce an image on the opposite side.
平移是按向量移动每个点。反射是将图形翻折到镜线另一侧（如 x = 0 或 y = x）。旋转是绕中心按给定角度转动图形。放大是以中心为基准，按比例因子缩放距离；负比例因子产生的像在中心的另一侧。
For combined transformations, apply the one closest to the object first. Describe fully: type, mirror line or centre and angle/direction, or centre and scale factor.
对于复合变换，先运用最靠近原图形的变换。完整描述变换类型：对于反射要说明镜线，对于旋转要说明中心、角度和方向，对于放大要说明中心和比例因子。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Double-check units: convert all measurements to the same unit before calculating. In graph questions, label axes, use appropriate scales, and plot points carefully.
仔细检查单位：计算前将所有测量值转换为相同单位。在图形问题中，标明坐标轴，选用合适的比例，仔细描点。
When solving equations, always substitute your answer back to verify. For inequality questions, remember to flip the sign when dividing by a negative.
解方程时，务必将答案代回验证。对于不等式问题，记住除以负数时要改变不等号方向。
Show full working; marks are awarded for method even if the final answer is wrong. Read the question carefully to see if a specific degree of accuracy (e.g., 3 significant figures) is required.
写出完整的解题步骤；即便最终答案有误，方法正确也能得分。仔细审题，看是否要求特定的精确度（如保留3位有效数字）。
Manage your time: spend roughly one minute per mark. If stuck on a question, move on and return later. Bring a calculator with fresh batteries and know how to use its key functions (fractions, squares, trig, statistics mode).
合理分配时间：大约每分钟完成一分值的题目。如果卡在某一题，先往后做，之后再回来。携带电量充足的计算器，并熟悉其关键功能（分数、平方、三角、统计模式）。
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