📚 IGCSE CIE Maths: Pre-Exam Essential Revision Checklist | IGCSE CIE 数学考前重点复习纲要
This revision checklist covers the core and extended topics of CIE IGCSE Mathematics 0580. Work through each section to recall key formulae, common pitfalls, and efficient problem-solving strategies. Focus on understanding concepts and practising past paper questions.
本考前复习纲要涵盖 CIE IGCSE 数学 0580 核心与扩展内容。逐节回顾关键公式、常见易错点与高效解题思路，注重理解概念并结合历年真题练习，助你锁定高分。
1. Number Operations and Estimation | 数字运算与估算
Revise BIDMAS/BODMAS for order of operations. Work confidently with fractions, decimals and percentages, including conversions. Rounding to a given number of significant figures or decimal places is essential. Express large and small numbers in standard form a × 10ⁿ where 1 ≤ a < 10.
重温运算顺序 BIDMAS/BODMAS。熟练进行分数、小数和百分数的运算与互化。掌握按要求保留有效数字或小数位数。用标准形式 a × 10ⁿ（1 ≤ a < 10）表示极大或极小数字。
Standard form: a × 10ⁿ, 1 ≤ a < 10
2. Ratio, Proportion and Percentages | 比、比例与百分比
Share quantities in a given ratio, solve direct and inverse proportion problems (including algebraic approaches for extension). Calculate percentage increase/decrease, reverse percentages, and compound interest. The compound interest formula is crucial for multi-year growth.
按给定比例分配数量，解决正比和反比问题（扩展层次要求代数法）。计算百分比增减、逆推百分比以及复利。复利公式是多年期增长的关键。
Compound interest: A = P(1 + r/100)^t
3. Algebraic Expressions and Equations | 代数表达式与方程
Expand and simplify brackets, factorise linear and quadratic expressions (including difference of two squares). Solve linear equations, quadratic equations by factorising, completing the square (extension) or using the quadratic formula. Solve simultaneous linear equations by elimination or substitution.
展开并化简括号，分解因式（包括平方差）。解一元一次方程，用因式分解、配方法（扩展）或求根公式解二次方程。用消元法或代入法解联立一次方程组。
Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a)
4. Inequalities and Graphing | 不等式与图像
Solve linear inequalities, represent solutions on a number line using open/closed circles. Remember to reverse the inequality sign when multiplying or dividing by a negative number. For extension, shade regions defined by multiple linear inequalities on the coordinate plane and find the feasible region.
解一元一次不等式，用空心/实心圆点在数轴上表示解集。注意乘或除以负数时需反转不等号。扩展层次要求在坐标平面上给出一组线性不等式并标示可行域。
5. Sequences | 数列
Find the nth term of a linear (arithmetic) sequence using the formula a + (n−1)d. For quadratic sequences (extension), use the second difference method. Recognise geometric sequences and know the formula for the sum of the first n terms if required. Always check subsequent terms.
运用公式 a + (n−1)d 求线性（等差）数列的第 n 项。扩展层次的二次数列利用二阶差分法求解。识别等比数列，需要时能写出前 n 项和的公式。务必验证后续项。
nth term of linear sequence: a + (n−1)d
6. Graphs of Functions | 函数图像
Plot straight line graphs, quadratic, cubic, reciprocal and exponential functions. Know how to find gradient and intercept from y = mx + c. Use graphs to solve equations (e.g., intersection points) and estimate gradients of tangent curves. Identify turning points and their nature.
绘制直线、二次、三次、倒数和指数函数图像。会从 y = mx + c 求斜率和截距。利用图像解方程（如交点法）和估算曲线切线梯度。识别拐点及其为极大或极小点。
7. Angles and Polygons | 角度与多边形
Use angle facts at a point, on a straight line, vertically opposite angles, angles in parallel lines (alternate, corresponding, co-interior). Calculate interior and exterior angles of regular and irregular polygons: sum of interior angles = (n−2)×180°, exterior angles sum to 360°.
应用周角、平角、对顶角以及平行线中的同位角、内错角、同旁内角性质。计算正多边形和不规则多边形的内角与外角：内角和 = (n−2)×180°，外角和恒等于 360°。
Sum of interior angles = (n−2)×180°
8. Circle Theorems | 圆定理
Learn key theorems: angle at centre is twice angle at circumference; angles in the same segment are equal; angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°. Tangent is perpendicular to radius; alternate segment theorem. Apply to find missing angles.
掌握核心定理：圆心角等于两倍圆周角；同弧上的圆周角相等；半圆所对圆周角为 90°；圆内接四边形对角互补（和为 180°）。切线与半径垂直；弦切角定理。综合运用求未知角。
9. Mensuration – Area and Volume | 测量 – 面积与体积
Memorise formulae for area of triangle (½bh), parallelogram (bh), trapezium (½(a+b)h), circle (πr²), sector area (θ/360 × πr²) and arc length (θ/360 × 2πr). Volume and surface area of prisms, cylinder (πr²h), cone (⅓πr²h), sphere (V=4/3πr³, SA=4πr²). Use Pythagoras to find missing dimensions in 3D shapes.
熟记三角形、平行四边形、梯形、圆面积公式，扇形面积和弧长公式。掌握棱柱、圆柱、圆锥、球的体积与表面积公式。利用毕达哥拉斯定理求立体图形中的缺失边长。
Circle: C = 2πr, A = πr² | Sector: Arc = (θ/360)×2πr, Area = (θ/360)×πr²
Cylinder: V = πr²h | Cone: V = ⅓πr²h | Sphere: V = ⁴⁄₃πr³, SA = 4πr²
10. Trigonometry and Pythagoras | 三角学与毕达哥拉斯定理
Apply Pythagoras’ theorem a² + b² = c² for right-angled triangles. Use sine, cosine and tangent ratios (SOH CAH TOA) to find missing sides and angles. For non-right-angled triangles, employ the sine rule and cosine rule; remember the ambiguous case for the sine rule. Extension includes 3D Pythagoras and trigonometry in three dimensions.
对直角三角形使用毕达哥拉斯定理 a² + b² = c²。利用正弦、余弦、正切比 (SOH CAH TOA) 求未知边和角。对于任意三角形，运用正弦定理和余弦定理；注意正弦定理可能出现的双解问题。扩展部分涉及三维毕达哥拉斯及空间三角学。
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b² + c² − 2bc cos A
11. Statistics | 统计
Calculate mean, median, mode and range; work with grouped frequency tables to estimate mean. Construct and interpret cumulative frequency curves to find quartiles and interquartile range, and draw box plots. For histograms, use frequency density = frequency ÷ class width. Compare data sets using appropriate measures.
计算平均数、众数、中位数和极差；用分组频数表估算均值。绘制并解读累积频数曲线以求得四分位数和四分位距，继而作出箱线图。直方图需牢记频率密度 = 频数 ÷ 组距。选择合适的度量比较数据集。
Frequency density = frequency ÷ class width
12. Probability | 概率
Probability = number of favourable outcomes / total number of outcomes. For mutually exclusive events, add probabilities; for independent events, multiply probabilities. Use tree diagrams to handle combined events; remember probabilities on branches must sum to 1. For conditional probability, consider the reduced sample space.
概率 = 有利结果数 / 总结果数。互斥事件概率相加，独立事件概率相乘。运用树状图处理组合事件；留意各分支概率之和须为 1。条件概率问题中，注意样本空间的缩减。
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