GCSE CIE Mathematics: Final Revision Checklist | GCSE CIE 数学：期末复习提纲

📚 GCSE CIE Mathematics: Final Revision Checklist | GCSE CIE 数学：期末复习提纲

This revision checklist covers the essential topics for the CIE IGCSE Mathematics (0580) examination. Work through each section methodically, and use past paper questions to test your understanding.

本复习提纲涵盖了 CIE IGCSE 数学（0580）考试的核心主题。请有条理地复习每个部分，并使用历年真题来检测你的掌握程度。

1. Number and Arithmetic | 数与算术

Make sure you can classify numbers as natural numbers, integers, rational (fractions, terminating/recurring decimals), irrational (surds, π), and real numbers. Understand the difference between prime and composite numbers.

确保你能将数分类为自然数、整数、有理数（分数、有限或循环小数）、无理数（根式、π）和实数。理解质数与合数的区别。

Be able to express a number as a product of its prime factors using a factor tree or repeated division, and use this to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers.

能够用因数树或重复除法将数字表示为其质因数的乘积，并用此方法求两个或多个数的最大公约数（HCF）和最小公倍数（LCM）。

Apply the correct order of operations (brackets, indices, division/multiplication, addition/subtraction). Know how to round numbers to a given number of decimal places or significant figures, and determine upper and lower bounds for measurements.

运用正确的运算顺序（括号、指数、乘除、加减）。知道如何将数字舍入到指定的小数位或有效数字，并确定测量值的上限和下限。

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2. Fractions, Decimals and Percentages | 分数、小数与百分数

Convert fluently between fractions, decimals and percentages. For example, 0.35 = 35% = 7/20. Recurring decimals can be converted to fractions by solving an equation.

熟练地在分数、小数和百分数之间进行转换。例如 0.35 = 35% = 7/20。循环小数可以通过解方程的方法转换为分数。

Perform addition, subtraction, multiplication and division of fractions, including mixed numbers. When dividing, remember to multiply by the reciprocal.

进行分数的加、减、乘、除运算，包括带分数。进行除法时，记住要乘以倒数。

Calculate percentage increase and decrease, reverse percentages, and simple interest (I = PRT/100). For compound interest, use the multiplier method: Amount = P × (1 + r/100)ⁿ.

计算百分数的增加和减少、逆向百分数以及简单利息 (I = PRT/100)。对于复利，使用乘数法：A = P × (1 + r/100)ⁿ。

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3. Ratio and Proportion | 比与比例

Simplify ratios to their lowest terms and divide a quantity in a given ratio. Solve problems involving direct proportion (y ∝ x) and inverse proportion (y ∝ 1/x), and find proportionality constants.

将比化为最简形式，并按照给定的比分配数量。解决涉及正比例 (y ∝ x) 和反比例 (y ∝ 1/x) 的问题，并求出比例常数。

Understand and use compound measures such as speed = distance ÷ time, density = mass ÷ volume, and pressure = force ÷ area. Convert between related units, e.g. km/h to m/s.

理解并运用复合量度，如速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 压力 ÷ 面积。在相关单位之间进行转换，例如 km/h 与 m/s。

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4. Algebraic Expressions and Equations | 代数表达式与方程

Simplify algebraic expressions by collecting like terms, expand brackets (including double brackets), and factorise using common factors, difference of two squares (a² – b² = (a–b)(a+b)), and factorising quadratics x² + bx + c.

通过合并同类项化简代数表达式，展开括号（包括双重括号），并使用公因式、平方差公式 (a² – b² = (a–b)(a+b)) 以及二次三项式因式分解 (x² + bx + c)。

Solve linear equations and inequalities, represent solutions on a number line. Solve simultaneous equations, both linear and one linear with one quadratic (by substitution).

解线性方程和不等式，在数轴上表示解集。解联立方程组，包括两个线性方程，以及一个线性与一个二次方程（使用代入法）。

Solve quadratic equations by factorising, completing the square, or using the quadratic formula x = [–b ± √(b² – 4ac)] / 2a. Be aware of the discriminant (b² – 4ac) for determining the number of roots.

通过因式分解、配方法或使用二次公式 x = [–b ± √(b² – 4ac)] / 2a 解二次方程。了解判别式 (b² – 4ac) 用来判断根的数量。

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5. Graphs and Functions | 图像与函数

Interpret straight line graphs in the form y = mx + c, where m is the gradient and c is the y-intercept. Find the gradient between two points, and the equation of a line given a point and gradient or two points.

解读形式为 y = mx + c 的直线图像，其中 m 为梯度，c 为 y 轴截距。计算两点之间的梯度，并根据一点和梯度或两点求出直线方程。

Sketch quadratic graphs, identifying the vertex (turning point) and roots. Recognise cubic, reciprocal (y = 1/x), and exponential graphs. Use function notation to evaluate f(x), find composite functions fg(x), and inverse functions f⁻¹(x).

绘制二次函数的图像，识别顶点（转折点）和根。识别三次函数、倒数函数 (y = 1/x) 和指数函数的图像。使用函数符号计算 f(x)，求复合函数 fg(x) 和反函数 f⁻¹(x)。

Interpret real-life graphs: distance–time (gradient = speed) and speed–time (area = distance, gradient = acceleration). Also be able to apply simple transformations to graphs: y = f(x) + a, y = f(x + a), y = –f(x), etc.

解读实际生活图像：距离–时间图（梯度 = 速度）和速度–时间图（面积 = 距离，梯度 = 加速度）。还要能够对图像进行简单变换：y = f(x) + a，y = f(x + a)，y = –f(x) 等。

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6. Inequalities and Sequences | 不等式与数列

Solve linear inequalities and represent the solution set on a number line. Solve two inequalities simultaneously and shade the feasible region on a graph.

解线性不等式，并在数轴上表示解集。解两个不等式组成的联立不等式，并在图像上绘制可行域。

Find the nth term of a linear sequence (arithmetic progression) using the formula aₙ = a₁ + (n–1)d, where a₁ is the first term and d is the common difference. Determine terms from given patterns and recognise quadratic sequences.

使用公式 aₙ = a₁ + (n–1)d 求线性数列（等差数列）的第 n 项，其中 a₁ 是首项，d 是公差。根据给出的规律确定各项，并辨认二次数列。

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7. Geometry: Angles and Polygons | 几何：角与多边形

Recall basic angle facts: angles on a straight line sum to 180°, angles around a point 360°, vertically opposite angles are equal. Identify corresponding, alternate and co-interior angles on parallel lines.

回忆基本角度事实：直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。识别平行线上的同位角、内错角和同旁内角。

Calculate the sum of interior angles of an n-sided polygon: (n – 2) × 180°. The sum of exterior angles is always 360°. Use these to find missing angles in regular and irregular polygons.

计算 n 边形内角和：(n – 2) × 180°。外角和始终为 360°。利用这些性质求正多边形和不规则多边形中的未知角度。

Apply circle theorems: the angle at the centre is twice the angle at the circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius; and use the alternate segment theorem.

运用圆定理：圆心角是同一弧上圆周角的两倍；半圆上的圆周角是直角；同弧上的圆周角相等；圆内接四边形对角互补（和为 180°）；切线垂直于半径；并运用弦切角定理（交替弦

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