📚 IGCSE CCEA Mathematics: Last-Minute Revision Notes | IGCSE CCEA 数学：考前冲刺笔记
As the IGCSE CCEA Mathematics exam approaches, a focused revision strategy is essential. These notes summarise the key concepts, formulas, and common pitfalls across the main topics: Number, Algebra, Geometry, Trigonometry, Statistics, and Probability. Use them to check your understanding and sharpen your problem-solving skills.
临近 IGCSE CCEA 数学考试，有重点的复习策略至关重要。本笔记总结了数与运算、代数、几何、三角学、统计和概率等主要板块的核心概念、公式和常见易错点，帮助你检查理解、提升解题能力。
1. Number Systems and Operations | 数系与运算
Classify numbers into natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), irrational numbers, and real numbers (ℝ). Recognise that π and √2 are irrational, while fractions and terminating or recurring decimals are rational.
将数字分类为自然数(ℕ)、整数(ℤ)、有理数(ℚ)、无理数和实数(ℝ)。注意 π 和 √2 是无理数，而分数与有限小数或循环小数都是有理数。
Prime factorisation is the foundation of LCM and HCF. Express a number as a product of primes, e.g. 60 = 2² × 3 × 5. The HCF is the product of the lowest powers of common primes, while the LCM uses the highest powers of all primes present.
质因数分解是求最小公倍数(LCM)和最大公因数(HCF)的基础。将数字写成质数乘积，如 60 = 2² × 3 × 5。HCF 取共有质因数的最低次幂之积，LCM 则取所有质因数的最高次幂之积。
Operations with fractions are tested frequently: addition/subtraction require a common denominator; multiplication multiplies numerators and denominators separately; division is multiplication by the reciprocal.
分数运算频繁考查：加减法需要通分，寻找公分母；乘法分子分母分别相乘；除法变为乘以倒数。
Convert between fractions, decimals and percentages efficiently. To change a recurring decimal to a fraction, set up an equation and multiply by a power of 10 to align the recurring part.
高效转换分数、小数和百分数。将循环小数化为分数时，设等式并乘以10的幂使循环部分对齐，再相减求解。
Standard form is used for very large or small numbers: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. When computing with standard form, handle the powers of 10 separately.
标准形式用于极大或极小数：a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。用标准形式计算时，先分别处理数字部分和10的指数部分。
Rounding and estimation: understand upper and lower bounds. For a measurement given to the nearest unit, the absolute error is half a unit. Upper bound = measured value + 0.5 × unit, lower bound = measured value − 0.5 × unit. Always consider bounds when calculating with rounded values.
近似与估计：理解上界与下界。对精确到某一单位的测量值，绝对误差为半个单位。上界 = 测量值 + 0.5 × 单位，下界 = 测量值 − 0.5 × 单位。使用近似值计算时一定要考虑误差界。
Surds can be simplified using √(ab) = √a × √b and rationalising denominators. Example: 1/√2 = √2/2.
根式化简运用 √(ab) = √a × √b 以及分母有理化。例如 1/√2 = √2/2。
2. Algebraic Expressions and Formulae | 代数表达式与公式
Simplify expressions by collecting like terms: terms with the same variable and power. Expand brackets using the distributive law, and factorise by taking out the highest common factor or by recognising quadratic trinomials.
通过合并同类项化简表达式：变量及其指数都相同的项才能合并。运用分配律展开括号，通过提取公因式或识别二次三项式进行因式分解。
Key expansion patterns: (a + b)(a − b) = a² − b²; (a ± b)² = a² ± 2ab + b².
重要展开模式：(a + b)(a − b) = a² − b²；(a ± b)² = a² ± 2ab + b²。
Factorising quadratics: for x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c, consider splitting the middle term or using the ‘ac’ method.
二次三项式因式分解：对 x² + bx + c，找到两数使其乘积为 c、和为 b。对 ax² + bx + c，考虑拆分中项或使用“ac 法”。
Substitute values into algebraic formulae, paying attention to negative numbers and the correct order of operations (BIDMAS/BODMAS). Rearranging formulae: treat the desired subject as the unknown and perform inverse operations step by step, just like solving equations.
将数值代入代数公式，注意负数与正确的运算次序（BIDMAS/BODMAS）。变换公式主项：把目标字母看作未知数，像解方程一样逐步进行逆运算。
Algebraic fractions: simplify by factorising numerator and denominator, then cancel common factors. Add or subtract by finding a common denominator.
代数分式：对分子分母因式分解后约去公因式，进行加减运算时先通分。
3. Equations and Inequalities | 方程与不等式
Solve linear equations by isolating the variable using inverse operations. Always perform the same operation on both sides. Check your solution by substituting it back into the original equation.
解线性方程时，用逆运算分离变量，每一步须在等号两边同时进行。将解代入原方程检验。
For quadratic equations, first set the equation to zero. Then factorise, or use the quadratic formula:
对于二次方程，先移项使右边为0，然后因式分解，或使用求根公式：
x = [−b ± √(b² − 4ac)] / (2a)
Remember that the discriminant b² − 4ac determines the number of real roots: positive → two distinct roots, zero → one repeated root, negative → no real roots.
记住判别式 b² − 4ac 决定实根个数：大于0 → 两个不等实根，等于0 → 一个重根，小于0 → 无实根。
Simultaneous equations can be solved by elimination, substitution, or graphically. For one linear and one quadratic, substitute the linear expression into the quadratic and solve.
联立方程组可用消元法、代入法或图像法求解。若一个是一次、一个是二次，将一次表达式代入二次方程求解。
Inequalities: solve similarly to equations, but if you multiply or divide by a negative number, reverse the inequality sign. Represent solutions on a number line and in set notation. Be careful with strict (<, >) and inclusive (≤, ≥) boundaries.
不等式：解法与方程类似，但若乘或除以负数，必须反转不等号。在数轴和集合符号中表示解，注意区分严格不等号(<, >)和含等号的不等号(≤, ≥)。
4. Sequences | 数列
Recognise and continue linear, quadratic, and simple geometric sequences. A linear sequence has a constant first difference; the nth term is an + b, where a is the common difference.
识别并延续线性、二次及简单等比数列。线性数列的一阶差为常数；第 n 项公式为 an + b，其中 a 为公差。
To find the nth term of a linear sequence, use the difference as the coefficient of n and adjust by finding the term when n = 1.
求线性数列的通项：把公差作为 n 的系数，再利用 n = 1 时的项求出常数部分。
Quadratic sequences have a constant second difference. The nth term is of the form an² + bn + c. The value a equals half the second difference.
二次数列的二阶差为常数，通项表达式为 an² + bn + c，其中 a 等于二阶差的一半。
For geometric sequences, each term is found by multiplying by a constant ratio r. The nth term is arⁿ⁻¹.
等比数列中，每一项乘以固定公比 r 得到下一项，第 n 项为 arⁿ⁻¹。
Other sequences include Fibonacci-type, where each term is the sum of the two preceding terms. Always check the rule provided and apply it systematically.
其他数列如斐波那契类型，每一项是前两项之和。务必根据给定规则系统化写出后续项。
5. Functions and Graphs | 函数与图像
Understand function notation such as f(x) = 2x + 1. To evaluate f(3), substitute x = 3. Composite functions fg(x) means applying g first, then f. Inverse functions f⁻¹(x) undo the effect of f(x); find by solving y = f(x) for x and swapping variables.
理解函数记号如 f(x) = 2x + 1。计算 f(3) 即将 x = 3 代入。复合函数 fg(x) 表示先作用 g 再作用 f。反函数 f⁻¹(x) 能撤销 f(x) 的效果，通过解 y = f(x) 并用 x, y 互换求得。
Graphs of common functions: y = mx + c (straight line), y = ax² + bx + c (parabola), y = a/x (rectangular hyperbola), y = aˣ (exponential), and y = sin x, y = cos x, y = tan x (trigonometric curves). Know their key shapes and intercepts.
常见函数图像：y = mx + c (直线), y = ax² + bx + c (抛物线), y = a/x (反比例双曲线), y = aˣ (指数曲线) 以及 y = sin x, cos x, tan x (三角函数曲线)。熟悉它们的基本形状与截距。
The vertex of a parabola y = a(x − h)² + k is (h, k). The line of symmetry is x = h. For y = ax² + bx + c, the vertex x-coordinate is −b/(2a).
抛物线 y = a(x − h)² + k 的顶点为 (h, k)，对称轴为 x = h。对于一般式 y = ax² + bx + c，顶点横坐标为 −b/(2a)。
Transformations of graphs: f(x) + a is vertical translation; f(x + a) is horizontal translation; −f(x) reflects in the x‑axis; f(−x) reflects in the y‑axis; af(x) stretches vertically by factor a.
图像变换：f(x) + a 为竖直平移，f(x + a) 为水平平移，−f(x) 关于 x 轴对称，f(−x) 关于 y 轴对称，af(x) 为竖直方向拉伸 a 倍。
6. Geometry | 几何
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, corresponding angles are equal, alternate angles are equal, and co‑interior angles sum to 180°.
角度基础：直线上的角之和为 180°，一点周围的角之和为 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°。
Properties of triangles: sum of interior angles = 180°. Know isosceles (two equal sides, two equal base angles), equilateral (all sides and angles 60°), and right‑angled triangles (apply Pythagoras’ theorem).
三角形性质：内角和为 180°。熟悉等腰三角形（两腰相等，两底角相等），等边三角形（三边相等，各角 60°），直角三角形（应用勾股定理）。
Pythagoras’ theorem: for any right‑angled triangle, a² + b² = c², where c is the hypotenuse. Recognise Pythagorean triples such as (3, 4, 5).
勾股定理：对于任何直角三角形，a² + b² = c²，其中 c 为斜边。识记勾股数组如 (3, 4, 5)。
Polygons: sum of interior angles = (n − 2) × 180°, sum of exterior angles = 360° always. For a regular polygon, each interior angle = (n − 2) × 180° / n.
多边形：内角和 = (n − 2) × 180°，外角和恒为 360°。正多边形每个内角 = (n − 2) × 180° / n。
Circles: know the definitions of radius, diameter, chord, tangent, arc, sector, segment. Tangents from a common external point are equal in length; the radius to the point of tangency is perpendicular to the tangent.
圆：理解半径、直径、弦、切线、弧、扇形、弓形等术语。同一点出发的两条切线长相等；过切点的半径垂直于切线。
Perimeter, area, volume formulas must be memorised:
周长、面积和体积公式必须熟记：
| Shape | Area/Volume |
|---|---|
| Rectangle | A = l × w |
| Triangle | A = ½ × b × h |
| Circle | A = πr², C = 2πr |
| Cuboid | V = l × w × h |
| Cylinder | V = πr²h, curved surface area = 2πrh |
| Sphere | V = 4/3 πr³, surface area = 4πr² |
7. Trigonometry | 三角学
Right‑angled triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use SOH CAH TOA to recall these. Always identify the sides relative to the given angle.
直角三角形中的比例：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。用 SOH CAH TOA 助记。务必先相对于已知角确定各边的角色。
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. The area of any triangle is ½ ab sin C.
对于非直角三角形，运用正弦定理：a/sin A = b/sin B = c/sin C，或余弦定理：a² = b² + c² − 2bc cos A。任意三角形面积 = ½ ab sin C。
Know the exact values for key angles (0°, 30°, 45°, 60°, 90°) without a calculator. For example, sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
熟记特殊角（0°, 30°, 45°, 60°, 90°）的精确值，如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
Angles of elevation and depression: measured from the horizontal. Draw a clear diagram, label the sides, and set up a trigonometric equation.
仰角与俯角：均从水平线起量。绘制清晰示意图，标出各边，建立三角方程求解。
Bearings are measured clockwise from North and given as three figures, e.g. 045°. Convert between bearings and right‑angled triangle settings reliably.
方位角从正北顺时针度量，以三位数表示，如 045°。熟练地在方位角与直角三角形情境间转换。
8. Statistics | 统计
Measures of central tendency: mean = sum of values ÷ number of values; median = middle value when ordered; mode = most frequent value. For grouped data, use the midpoint of the class interval to estimate the mean.
数据集中趋势度量：平均数 = 总和 ÷ 数据个数；中位数 = 排序后中间的值；众数 = 出现次数最多的值。对于分组数据，用组中点估计平均数。
Range = maximum − minimum. Interquartile range (IQR) = upper quartile (Q₃) − lower quartile (Q₁). IQR measures the spread of the middle 50% of data.
范围 = 最大值 − 最小值。四分位距 IQR = 上四分位数 (Q₃) − 下四分位数 (Q₁)。IQR 衡量中间50%数据的离散程度。
Represent data using bar charts, pie charts, stem‑and‑leaf diagrams, histograms (with unequal class widths: frequency density = frequency ÷ class width), and cumulative frequency curves. Use cumulative frequency graphs to find medians and quartiles.
用条形图、饼图、茎叶图、直方图（组距不同时，频率密度 = 频数 ÷ 组距）和累积频率曲线表示数据。利用累积频率图求中位数与四分位数。
Box plots display the minimum, Q₁, median, Q₃, and maximum. They are useful for comparing distributions and identifying outliers.
箱线图展示最小值、Q₁、中位数、Q₃ 和最大值，便于比较分布与识别异常值。
Scatter graphs show relationships between two variables. Add a line of best fit to identify correlation (positive, negative, or none) and make predictions.
散点图显示两变量关系，用最佳拟合线描述相关性（正相关、负相关、无相关）并进行预测。
9. Probability | 概率
Probability scale runs from 0 (impossible) to 1 (certain). The probability of an event not happening is 1 − P(event). For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率标度从 0（不可能）到 1（必然）。事件不发生的概率为 1 − P(事件)。等可能结果下，P(事件) = 有利结果数 / 总结果数。
For combined events, use sample space diagrams, two‑way tables, or tree diagrams. Multiply probabilities along branches for ‘and’; add probabilities of different branches for ‘or’.
对于组合事件，使用样本空间图、双向表或树状图。沿分支相乘计算“与”事件的概率；将不同分支的概率相加得到“或”事件的概率。
Conditional probability: P(A|B) = P(A ∩ B) / P(B). Tree diagrams often help clarify the situation by including changed probabilities on second branches.
条件概率：P(A|B) = P(A ∩ B) / P(B)。树状图中第二层分支的概率会根据条件改变，有助于理清思路。
Mutually exclusive events cannot happen simultaneously; P(A or B) = P(A) + P(B). Independent events do not affect each other; P(A and B) = P(A) × P(B). Verify independence by checking if P(A ∩ B) equals P(A) × P(B).
互斥事件不能同时发生，P(A 或 B) = P(A) + P(B)。独立事件相互无影响，P(A 与 B) = P(A) × P(B)。可通过检查 P(A ∩ B) 是否等于 P(A) × P(B) 来验证独立性。
Venn diagrams are helpful for visualising sets, unions (∪), intersections (∩), and complements (A’). They often simplify probability calculations with overlapping events.
文氏图有助于可视化集合、并集(∪)、交集(∩)与补集(A’)，常能简化带有重叠事件的概率计算。
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