📚 IGCSE CCEA Maths: Final Exam Revision Checklist | IGCSE CCEA 数学：期末复习提纲
This comprehensive revision checklist is designed for IGCSE CCEA Mathematics students preparing for their final exams. It covers the essential topics, key formulas, common pitfalls and exam tips across the entire syllabus. Use it to structure your revision, identify weak areas and build confidence in working through problems systematically.
这份全面的复习提纲专为备考 IGCSE CCEA 数学期末考试的学生设计。它涵盖了整个教学大纲中的必考主题、关键公式、常见易错点和应试技巧。用它来规划你的复习、找出薄弱环节，并建立系统解题的信心。
1. Number Basics and Operations | 数字基础与运算
Ensure you can confidently work with directed numbers (positive and negative integers) for all four operations. Remember that subtracting a negative is equivalent to adding a positive, e.g. −3 − (−5) = −3 + 5 = 2.
确保你能熟练运用正负数进行四则运算。记住减去一个负数等于加上对应的正数，例如 −3 − (−5) = −3 + 5 = 2。
Apply the correct order of operations: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Use BIDMAS or BODMAS to avoid errors in multi‑step calculations.
使用正确的运算顺序：括号、指数、乘除（从左到右）、加减（从左到右）。运用 BIDMAS 或 BODMAS 规则避免多步计算中的错误。
Know how to round numbers to a given number of decimal places or significant figures, and use estimation to check the reasonableness of answers. For example, 46.7 × 0.53 ≈ 50 × 0.5 = 25.
知道如何将数字四舍五入到指定的小数位数或有效数字，并用估算检验答案的合理性。例如 46.7 × 0.53 ≈ 50 × 0.5 = 25。
Express numbers as products of prime factors using factor trees, and use these to find the highest common factor (HCF) and lowest common multiple (LCM) efficiently.
用质因数树将数字表示为质因数的乘积，并利用它们高效地求最高公因数（HCF）和最低公倍数（LCM）。
2. Fractions, Decimals, and Percentages | 分数、小数和百分数
Convert fluently between fractions, decimals and percentages. Key equivalents to memorise: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 33.3%, 2/3 ≈ 66.7%, 1/8 = 0.125 = 12.5%.
熟练地在分数、小数和百分数之间转换。需要记住的关键等价关系：1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 33.3%, 2/3 ≈ 66.7%, 1/8 = 0.125 = 12.5%。
Perform addition and subtraction of fractions by finding a common denominator. For multiplication, multiply numerators and denominators directly; for division, multiply by the reciprocal.
通过通分进行分数的加减运算。乘法直接分子乘分子、分母乘分母；除法乘以除数的倒数。
Calculate a percentage of a quantity, increase or decrease by a percentage, and find the percentage change using the formula: percentage change = (difference ÷ original) × 100%.
计算一个量的百分比、增加或减少一个百分比，以及用公式计算百分比变化：百分比变化 = (差值 ÷ 原值) × 100%。
Understand reverse percentages: when given a value after a percentage change, find the original amount by dividing by the appropriate multiplier (e.g. after a 15% increase, divide by 1.15).
理解逆运算百分比：当已知百分比变化后的值，通过除以相应的乘数求出原值（例如增加 15% 后，除以 1.15）。
Work with simple interest and compound interest, including depreciation. Compound interest formula: A = P(1 + r/100)ⁿ, where P is principal, r is rate per period, n is number of periods.
掌握单利和复利（包括贬值）。复利公式：A = P(1 + r/100)ⁿ，其中 P 为本金，r 为每期利率，n 为期数。
3. Ratio and Proportion | 比率与比例
Write ratios in their simplest form and divide quantities into a given ratio, e.g. splitting £120 in the ratio 2:3:5 gives parts of £24, £36 and £60.
以最简形式书写比率，并按给定比率分配数量，例如将 120 英镑按 2:3:5 分配，各部分为 24 英镑、36 英镑和 60 英镑。
Understand the difference between ratio and proportion. Recognise direct and inverse proportion in real‑life contexts and use the unitary method to solve problems.
理解比率与比例的区别。在实际情境中识别正比例与反比例，并运用归一法解决问题。
For direct proportion, y = kx; for inverse proportion, y = k/x (where k is constant). Use given data to find k, then answer subsequent parts of the question.
正比例关系：y = kx；反比例关系：y = k/x（k 为常数）。利用已知数据求出 k，再解答后续问题。
Use scale factors for maps, plans and similar figures. A scale of 1 : 25 000 on a map means 1 cm represents 25 000 cm (0.25 km) on the ground.
运用比例尺处理地图、平面图和相似图形。地图上 1 : 25 000 的比例尺表示 1 cm 代表实际 25 000 cm（0.25 km）。
4. Algebraic Expressions and Equations | 代数表达式与方程
Simplify expressions by collecting like terms and using the laws of indices: xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ. Handle negative and zero indices correctly: x⁻ⁿ = 1/xⁿ, x⁰ = 1.
通过合并同类项并运用指数法则化简表达式：xᵃ × xᵇ = xᵃ⁺ᵇ，xᵃ ÷ xᵇ = xᵃ⁻ᵇ，(xᵃ)ᵇ = xᵃᵇ。正确处理负指数和零指数：x⁻ⁿ = 1/xⁿ，x⁰ = 1。
Expand brackets accurately, including double brackets: (a + b)(c + d) = ac + ad + bc + bd. Factorise expressions by taking out common factors, and factorise quadratics of the form x² + bx + c into double brackets.
准确展开括号，包括双重括号：(a + b)(c + d) = ac + ad + bc + bd。通过提取公因式进行因式分解，并将 x² + bx + c 型的二次式分解为两个一次式乘积。
Solve linear equations with unknowns on one or both sides. Always perform the same operation on both sides. For equations containing fractions, multiply every term by the common denominator first.
解未知数位于一侧或两侧的线性方程。等式两边始终保持同一种运算。含有分数的方程，先乘以最简公分母。
Solve quadratic equations by factorising, using the quadratic formula, or completing the square. The quadratic formula is:
通过因式分解、二次公式或配方法解二次方程。二次公式为：
x = [−b ± √(b² − 4ac)] / (2a)
Always set the quadratic to zero first, and check solutions by substitution.
务必先将二次方程设为零，并代入原式检验。
5. Inequalities and Sequences | 不等式与数列
Represent inequalities on a number line: open circle for < or >, closed circle for ≤ or ≥. Solve linear inequalities in a similar way to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
用数轴表示不等式：< 或 > 用空心圈，≤ 或 ≥ 用实心圈。解线性不等式的方法与方程类似，但乘以或除以负数时务必反向改变不等号方向。
Generate terms of a sequence from the nth term rule, e.g. n² + 3 or 2n − 5. Recognise linear sequences (common difference) and quadratic sequences (second difference constant).
根据第 n 项公式生成数列的项，例如 n² + 3 或 2n − 5。识别等差数列（公差恒定）和二次数列（二级差恒定）。
Find the nth term of a linear sequence: write as an + b, where a is the common difference and b is adjusted using the first term. For quadratic sequences, the nth term is of the form an² + bn + c, where a is half the second difference.
求出等差数列的第 n 项：写成 an + b 形式，其中 a 为公差，b 利用首项调整得到。二次数列的第 n 项为 an² + bn + c 形式，其中 a 为二级差的一半。
6. Graphs and Functions | 图像与函数
Plot straight‑line graphs from tables of values or using the gradient‑intercept form y = mx + c, where m is the gradient and c is the y‑intercept. Gradient between two points = (y₂ − y₁)/(x₂ − x₁).
根据表格数值绘制直线图像，或运用斜截式 y = mx + c，其中 m 为斜率，c 为 y 轴截距。两点间的斜率 = (y₂ − y₁)/(x₂ − x₁)。
Interpret speed‑time, distance‑time and conversion graphs. In a distance‑time graph, the gradient gives speed; in a speed‑time graph, the gradient gives acceleration and the area under the line gives distance travelled.
解读速度‑时间、距离‑时间及转换图像。在距离‑时间图中，斜率给出速度；在速度‑时间图中，斜率给出加速度，线下面积给出行经的距离。
Plot quadratic graphs (parabolas), reciprocal graphs (y = k/x) and other functions. Identify key features: roots (x‑intercepts), vertex (turning point), and asymptotes for reciprocal graphs.
绘制二次图像（抛物线）、反比例图像（y = k/x）及其他函数图像。识别关键特征：根（与 x 轴交点）、顶点（转折点）以及反比例图像的渐近线。
Solve simultaneous equations graphically by finding the intersection of two lines. Use algebraic methods (substitution and elimination) for exact solutions, particularly when one equation is quadratic.
通过寻找两直线交点，用图像法解联立方程组。使用代数法（代入法和加减消元法）求得精确解，尤其当其中一个方程为二次方程时。
7. Geometry: Angles and Polygons | 几何：角与多边形
Apply angle facts on straight lines (sum to 180°), around a point (360°), vertically opposite angles (equal), and angles in parallel lines (alternate, corresponding, co‑interior). Co‑interior angles sum to 180°.
运用直线上的角（和为 180°）、绕一点的周角（360°）、对顶角（相等）及平行线中的角（内错角、同位角、同旁内角）等定理。同旁内角互补，和为 180°。
Calculate interior and exterior angles of regular polygons. Sum of interior angles = (n − 2) × 180°; each interior angle = [(n − 2) × 180°] / n. Exterior angle = 360° / n.
计算正多边形的内角和外角。内角和 = (n − 2) × 180°；每个内角 = [(n − 2) × 180°] / n。外角 = 360° / n。
Use properties of triangles: sum of angles 180°, isosceles (two equal sides and base angles), equilateral (all 60°). In right‑angled triangles apply Pythagoras’ theorem: a² + b² = c² where c is the hypotenuse.
运用三角形的性质：内角和 180°，等腰三角形（两腰相等，底角相等），等边三角形（每个角 60°）。在直角三角形中应用勾股定理：a² + b² = c²，c 为斜边。
Know circle terminology (radius, diameter, chord, tangent, arc, sector, segment) and use angle properties: angle at centre is twice angle at circumference, angle in a semicircle is 90°, angles in the same segment are equal.
了解圆的术语（半径、直径、弦、切线、弧、扇形、弓形）并运用角的性质：圆心角等于圆周角的两倍，半圆内的圆周角为 90°，同弧上的圆周角相等。
8. Mensuration: Area, Volume, and Trigonometry | 测量：面积、体积与三角函数
Calculate perimeters and areas of common shapes: rectangle (A = l × w), triangle (A = ½ × base × height), parallelogram (A = b × h), trapezium (A = ½ (a + b)h), circle (A = πr², C = 2πr). Use π ≈ 3.14 or the π button on a calculator.
计算常见图形的周长和面积：矩形（A = 长 × 宽），三角形（A = ½ × 底 × 高），平行四边形（A = 底 × 高），梯形（A = ½ (上底+下底) × 高），圆（A = πr², C = 2πr）。使用 π ≈ 3.14 或计算器上的 π 键。
Find volume and surface area of 3D shapes: cuboid (V = lwh, SA = 2(lw + lh + wh)), prism (V = area of cross‑section × length), cylinder (V = πr²h, SA = 2πrh + 2πr²), sphere (V = ⁴/₃πr³, SA = 4πr²), cone (V = ⅓πr²h).
求立体图形的体积和表面积：长方体（V = 长×宽×高，SA = 2(长宽+长高+宽高)），棱柱（V = 底面积 × 高），圆柱（V = πr²h，SA = 2πrh + 2πr²），球体（V = ⁴/₃πr³，SA = 4πr²），圆锥（V = ⅓πr²h）。
Apply trigonometric ratios (SOH CAH TOA) in right‑angled triangles: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. Use inverse trig functions to find angles.
在直角三角形中应用三角比（SOH CAH TOA）：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。使用反三角函数求角。
Use the sine rule and cosine rule for non‑right‑angled triangles. Sine rule: a/sinA = b/sinB = c/sinC. Cosine rule: a² = b² + c² − 2bc·cosA. Know when to use each rule (given two angles and a side, or two sides and a non‑included angle → sine rule; given three sides or two sides and the included angle → cosine rule).
对非直角三角形使用正弦定理和余弦定理。正弦定理：a/sinA = b/sinB = c/sinC。余弦定理：a² = b² + c² − 2bc·cosA。知道何时使用：已知两角一边或两边及非夹角 → 正弦定理；已知三边或两边及其夹角 → 余弦定理。
9. Vectors and Transformations | 向量与变换
Represent vectors as column vectors or labelled letters. Add and subtract vectors, and multiply a vector by a scalar. The magnitude of vector (a b) is √(a² + b²).
用列向量或带箭头的字母表示向量。进行向量的加减以及数乘。向量 (a b) 的模为 √(a² + b²)。
Use vectors to describe translations (e.g. translation by vector (3 −2) moves a shape 3 units right and 2 units down). Solve geometry problems using parallel and equal vectors – e.g. AB = k·CD implies AB and CD are parallel.
运用向量描述平移变换（例如向量 (3 −2) 的平移将图形向右移动 3 个单位、向下移动 2 个单位）。利用平行且相等的向量解决几何问题——如 AB = k·CD 表明 AB 与 CD 平行。
Perform and combine transformations: reflection (mirror line), rotation (centre, angle, direction), translation, and enlargement (centre, scale factor). Understand that the order of transformations matters.
进行并组合变换：反射（镜面线）、旋转（中心、角度、方向）、平移以及放大（中心、比例因子）。理解变换的顺序会影响最终结果。
Describe a fully an enlargement with negative or fractional scale factors. When the scale factor is negative, the image is on the opposite side of the centre of enlargement and is inverted.
完整描述带负比例因子或分数比例因子的放大。当比例因子为负时，像出现在放大中心的另一侧且呈倒置。
10. Statistics and Probability | 统计与概率
Calculate mean, median, mode and range from lists, frequency tables and grouped frequency tables. For grouped data, use the midpoint of each class interval to estimate the mean.
从数据列表、频数表和分组频数表中计算平均数、中位数、众数和极差。对于分组数据，用每组组中值估算平均数。
Draw and interpret bar charts, pie charts, pictograms, line graphs, stem‑and‑leaf diagrams, box plots and cumulative frequency graphs. A cumulative frequency graph can be used to find medians, quartiles and interquartile range.
绘制并解释条形图、饼图、象形图、折线图、茎叶图、箱线图和累积频率图。累积频率图可用于求中位数、四分位数和四分位距。
Determine the probability of a single event: P(A) = number of favourable outcomes / total number of outcomes. For combined events, use sample space diagrams, two‑way tables or tree diagrams. Remember that probabilities on a tree diagram multiply along branches and add across branches.
计算单个事件的概率：P(A) = 有利结果数 / 总结果数。对于复合事件，使用样本空间图、双向表格或树状图。记住树状图中沿分支相乘，跨分支相加。
Understand conditional probability (probability of an event given that another has occurred) and how it appears in tree diagrams with second branches having different probabilities depending on the first outcome.
理解条件概率（已知另一事件发生的情况下某事件的概率），以及它在树状图中的表示：第二级分支的概率会因第一级结果不同而变化。
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