📚 IGCSE CIE Mathematics: End-of-Term Revision Checklist | IGCSE CIE 数学期末复习提纲
Preparing for your IGCSE CIE Mathematics exam requires a clear overview of the entire syllabus. This revision checklist breaks down core topics into manageable sections, from number operations to vectors and transformations. Use it to track your progress, sharpen your problem-solving skills, and enter the exam with confidence.
备战 IGCSE CIE 数学考试需要清晰地把握整个课程纲要。这份复习提纲将核心主题拆解为几个易于掌握的部分，从数字运算到向量与变换。用它来追踪你的进度、磨炼解题技巧，带着自信步入考场。
1. Number, Operations and Estimation | 数字、运算与估算
Work confidently with fractions, decimals and percentages. Be able to convert between them, calculate percentage increase or decrease, and solve reverse percentage problems using multipliers.
熟练处理分数、小数与百分数。能够相互转换，计算百分比增减，并利用乘数解决逆向百分数问题。
Master standard form (scientific notation) for very large and very small numbers. Practise rounding to a given number of decimal places or significant figures, and use estimation to check the reasonableness of answers.
掌握表示极大和极小数值的标准形（科学记数法）。练习按指定小数位数或有效数字进行舍入，并使用估算检验答案的合理性。
Apply index laws: am × an = am+n, am ÷ an = am−n, (am)n = amn. Simplify surds and work with numbers expressed in the form a√b.
运用指数律：am × an = am+n，am ÷ an = am−n，(am)n = amn。化简根式，并处理 a√b 形式的数。
2. Algebraic Expressions and Formulae | 代数表达式与公式
Expand products of binomials, e.g. (x + a)(x + b), and factorise quadratic expressions of the form x² + bx + c. For the extended syllabus, also factorise ax² + bx + c and complete the square.
展开二项式乘积，如 (x + a)(x + b)；将 x² + bx + c 形式的二次式因式分解。扩展课程还需分解 ax² + bx + c 并完成配方法。
Rearrange formulae to change the subject, including cases where the variable appears more than once. Substitute numerical values into expressions and evaluate correctly, paying attention to directed numbers.
变换公式的主项，包括变量出现多次的情况。将数值代入表达式并正确求值，注意有向数的处理。
Simplify algebraic fractions by cancelling common factors and perform operations with algebraic fractions. Understand the difference between an expression, an equation and an identity.
通过约去公因式化简代数分式，并进行代数分式的运算。理解表达式、方程与恒等式之间的区别。
3. Equations and Inequalities | 方程与不等式
Solve linear equations in one unknown, including those with brackets and fractional coefficients. Set up equations from word problems and interpret the solution in context.
解含有一个未知数的一元一次方程，包括含括号和分数系数的方程。根据文字题建立方程，并在情境中解释解的意义。
Solve quadratic equations by factorising, using the quadratic formula x = [−b ± √(b² − 4ac)] / 2a, or by completing the square. Recognise when the discriminant Δ = b² − 4ac determines the nature of the roots.
通过因式分解、使用求根公式 x = [−b ± √(b² − 4ac)] / 2a 或配方法解二次方程。识别判别式 Δ = b² − 4ac 如何决定根的性质。
Solve simultaneous linear equations in two unknowns by elimination, substitution or graphically. Solve linear inequalities and represent the solution set on a number line; use open and closed circles for strict and inclusive inequalities.
用消元法、代入法或图像法解二元一次联立方程组。解一元一次不等式并在数轴上表示解集；使用空心和实心圆点区分严格与包含不等号。
4. Ratio, Proportion and Rates of Change | 比例、比率与变化率
Simplify ratios and share quantities in a given ratio. Relate ratios to fractions and solve problems involving direct and inverse proportion, including the use of the proportionality constant k.
化简比率并按给定比例分配数量。将比率与分数建立联系；解决涉及正比和反比的问题，包括使用比例常数 k。
Calculate average speed, density and other compound measures by combining units. Convert between different units of speed, density and pressure, and solve problems involving rates of change.
通过组合单位计算平均速度、密度以及其他复合量度。在不同速度、密度和压力单位之间转换，并解决变化率问题。
Apply scale factors to lengths, areas and volumes in similar figures. Understand that for similar shapes, area scale factor is (length scale factor)² and volume scale factor is (length scale factor)³.
将长度、面积和体积的比例因子应用于相似图形。理解相似形中面积比例因子为（长度比例因子）²，体积比例因子为（长度比例因子）³。
5. Sets and Venn Diagrams | 集合与维恩图
Use set notation correctly: ∈ (element of), ∉ (not an element), ∩ (intersection), ∪ (union), A’ (complement). Describe sets by listing elements or by a rule.
正确运用集合符号：∈（属于），∉（不属于），∩（交集），∪（并集），A’（补集）。通过列举元素或描述规则来表示集合。
Draw and interpret Venn diagrams with up to three sets. Shade regions representing combinations of sets, and solve problems involving the number of elements in each region, including those given in word problems.
绘制并解读最多三个集合的维恩图。涂色表示集合的组合区域，并解决涉及各区域元素个数的问题，包括文字题。
Use the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) for two sets. For three sets, extend the principle and apply it to probability and counting problems.
使用公式 n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 处理两个集合。对于三个集合，延伸该原理并应用于概率与计数问题。
6. Functions and Graphs | 函数与图像
Understand function notation f(x) = … , evaluate f(a) for a given value of a, and determine the range of a simple function. For extended, find inverse functions f⁻¹(x) and form composite functions fg(x).
理解函数记号 f(x) = … ，对给定的 a 求值 f(a)，并确定简单函数的值域。扩展课程需求反函数 f⁻¹(x) 并构造复合函数 fg(x)。
Plot and sketch graphs of linear, quadratic, cubic and reciprocal functions. Identify key features: intercepts, turning points and asymptotes. Solve equations graphically by finding the intersection of two graphs.
绘制并描画一次函数、二次函数、三次函数与反比例函数的图像。识别关键特征：截距、转折点与渐近线。通过找出两图像的交点，以图像法解方程。
Calculate the gradient of a straight line from coordinates or from a graph, and find the equation of a line in the form y = mx + c. Determine whether lines are parallel (same gradient) or perpendicular (product of gradients = −1).
通过坐标或图像计算直线的斜率，并求出形如 y = mx + c 的直线方程。判断直线是否平行（斜率相等）或垂直（斜率乘积为 −1）。
7. Geometry and Mensuration | 几何与测量
Apply angle properties: angles on a straight line add to 180°, angles at a point add to 360°, vertically opposite angles are equal. Use parallel line angle facts (alternate, corresponding, co-interior) to find unknown angles.
应用角的性质：平角为 180°，周角为 360°，对顶角相等。运用平行线中的角（同位角、内错角、同旁内角）求未知角度。
Calculate interior and exterior angles of regular and irregular polygons using the sum of interior angles = (n − 2) × 180°. Solve problems involving triangles, quadrilaterals and circles, including circle theorems such as angles in the same segment and the alternate segment theorem.
利用内角和公式 (n − 2) × 180° 计算正多边形与不规则多边形的内角与外角。解决涉及三角形、四边形和圆的问题，包括弦切角定理、同弧上的圆周角相等等圆定理。
Find perimeter and area of 2D shapes, including triangles, parallelograms, trapeziums and circles. Use the formulas for arc length and sector area (θ/360 × 2πr and θ/360 × πr²).
计算二维图形的周长与面积，包括三角形、平行四边形、梯形与圆。运用弧长公式（θ/360 × 2πr）与扇形面积公式（θ/360 × πr²）。
Calculate surface area and volume of prisms, cylinders, pyramids, cones and spheres. Use the relationships between volume and capacity, and solve problems involving compound solids.
计算棱柱、圆柱、棱锥、圆锥与球的表面积和体积。运用体积与容积的关系，并解决组合立体的问题。
8. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
Apply Pythagoras’ theorem a² + b² = c² to find missing sides in right-angled triangles. Use the converse to determine whether a triangle is right-angled.
应用勾股定理 a² + b² = c² 求直角三角形中的未知边长。运用逆定理判断一个三角形是否为直角三角形。
Use sine, cosine and tangent ratios (SOH CAH TOA) to find sides and angles in right-angled triangles. Solve problems involving angles of elevation and depression, bearings, and simple 3D situations.
运用正弦、余弦和正切比（SOH CAH TOA）求直角三角形中的边与角。解决涉及仰角、俯角、方位角以及简单三维情境的问题。
For the extended syllabus, apply the sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A. Calculate the area of any triangle using ½ ab sin C.
扩展课程需运用正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A。使用公式 ½ ab sin C 计算任意三角形的面积。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° (π/6) | ½ | √3 / 2 | 1 / √3 |
| 45° (π/4) | √2 / 2 | √2 / 2 | 1 |
| 60° (π/3) | √3 / 2 | ½ | √3 |
Knowing these exact trigonometric values by heart speeds up work with non-calculator questions and helps when applying the sine and cosine rules.
熟记这些精确三角比数值能加快完成非计算器试题，并有助于应用正弦和余弦定理。
9. Statistics and Probability | 统计与概率
Calculate mean, median, mode and range from raw data and from frequency tables. Determine quartiles and interquartile range, and construct box-and-whisker plots to compare data sets.
从原始数据与频数表计算平均数、中位数、众数和极差。确定四分位数与四分位距，并绘制箱形图以比较数据集。
Draw and interpret cumulative frequency diagrams to estimate medians, quartiles and percentiles. Plot histograms for grouped data, using frequency density (frequency ÷ class width) where necessary.
绘制并解读累积频数图以估算中位数、四分位数和百分位数。在需要时为分组数据绘制直方图，使用频数密度（频数 ÷ 组距）。
Calculate probability using P(event) = (favourable outcomes) / (total outcomes). Use tree diagrams to handle successive events and ‘at least one’ scenarios. Apply the addition rule and, for independent events, the multiplication rule.
使用 P(事件) =（有利结果数）/（总结果数）计算概率。运用树形图处理连续事件和“至少有一次”的场景。应用加法法则，对于独立事件运用乘法法则。
For extended, work with conditional probability using the formula P(A|B) = P(A ∩ B) / P(B) and interpret statements involving ‘given that’.
扩展课程需运用条件概率公式 P(A|B) = P(A ∩ B) / P(B) 并解读含有“在…条件下”的陈述。
10. Vectors and Transformations | 向量与变换
Represent vectors as column vectors and in component form. Add and subtract vectors geometrically and algebraically, and multiply a vector by a scalar.
用列向量和分量形式表示向量。通过几何和代数方法对向量进行加减，以及将向量乘以标量。
Use vector methods to describe the position of points and to solve geometric problems, including proving that points are collinear or that a quadrilateral is a parallelogram.
使用向量方法描述点的位置并解决几何问题，包括证明点共线或四边形为平行四边形。
Describe and perform the four single transformations: reflection (in a mirror line), rotation (about a centre, by a given angle and direction), translation (by a column vector) and enlargement (with a centre and a scale factor). Recognise that rotations and reflections preserve lengths and angles, while enlargements preserve shape but not size.
描述并执行四种单一变换：反射（关于对称轴）、旋转（绕中心、按给定角度和方向）、平移（按列向量）以及放大（具有中心和比例因子）。认识旋转与反射保距保角，而放大则保形不保大小。
Find the transformation that maps one shape onto another. Describe fully, giving all necessary details. For enlargement, negative scale factors produce an image on the opposite side of the centre.
找出将一个图形映射到另一图形的变换。全面描述，给出所有必要细节。对于放大，负比例因子会在中心另一侧生成影像。
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