📚 Year 11 CIE Mathematics: Core Topics Review | Year 11 CIE 数学：核心知识点梳理
This article provides a structured revision of the core topics covered in the Year 11 CIE IGCSE Mathematics syllabus (0580/0980). Whether you are taking the Core or Extended tier, a solid understanding of these fundamentals is essential for exam success. We break down key areas, highlighting formulas, definitions, and typical problem-solving approaches. Use this guide to consolidate your knowledge and identify any gaps before the final assessment.
本文旨在系统梳理 Year 11 CIE IGCSE 数学课程（0580/0980）的核心知识点。无论你选择的是 Core 还是 Extended 级别的考试，扎实掌握这些基础内容都是取得高分的关键。我们将逐一拆解重点领域，强调重要的公式、定义以及常见解题方法。通过这篇梳理，你可以有效地巩固知识，并在最终评估前查漏补缺。
1. Numbers, Operations and Percentages | 数、运算与百分数
Understand number sets: natural numbers (ℕ), integers (ℤ), rational numbers (ℚ), and real numbers (ℝ). You must be confident with directed numbers and the order of operations (BIDMAS/BODMAS). For fractions, decimals and percentages, be able to convert between them and apply percentage increase or decrease methods. Repeated percentage change uses a multiplier raised to a power.
掌握数字集合：自然数（ℕ）、整数（ℤ）、有理数（ℚ）和实数（ℝ）。你需要熟练运用有向数以及运算顺序（BIDMAS/BODMAS 法则）。对于分数、小数和百分数，要能够相互转换，并应用百分比增减的方法。重复百分比变化则需要将乘数自乘相应次数。
- Percentage increase: new value = original × (1 + r/100)
- Percent decrease: new value = original × (1 - r/100)
- Compound interest after n years: A = P(1 + r/100)ⁿ
百分比增加：新值 = 原值 × (1 + r/100)；百分比减少：新值 = 原值 × (1 - r/100)；n 年后的复利：A = P(1 + r/100)ⁿ。注意区分单利与复利。估算和上下界（limits of accuracy）也是常见考点，例如当测量精确到给定单位时，真值所在的范围。
2. Algebra Basics and Solving Equations | 代数基础与解方程
Algebraic manipulation includes expanding brackets, factorising expressions (common factors, difference of two squares, trinomials) and simplifying algebraic fractions. You should be able to rearrange formulae (change the subject) even when the variable appears twice. Linear equations and inequalities are solved by performing the same operation on both sides. For quadratic equations, use factorisation, completing the square, or the quadratic formula.
代数运算包括去括号、因式分解（提取公因式、平方差、三项式）以及简化的代数分式。你需要能对公式进行变换（改变主项），即使变量出现两次。解一元一次方程和不等式时，两边实行相同运算。对于二次方程，可采用因式分解、配方法来解，或使用求根公式。
x = [ -b ± √b² - 4ac ] / 2a
务必记住判别式 Δ = b² - 4ac 的意义：若 Δ > 0，两个不等实根；Δ = 0，一个实根（重根）；Δ < 0，无实根。解联立方程时，可以用代入法或消元法，含有一个线性和一个二次方程的方程组常出现在 Extended 试卷中。
3. Sequences | 数列
Recognise and continue linear, quadratic, cubic, and exponential sequences. For a linear sequence, the nth term is of the form an + b, where a is the common difference. Quadratic sequences have a constant second difference; their nth term is of the form an² + bn + c. You may also encounter patterns like triangle numbers, square numbers, and Fibonacci-type sequences.
识别并延续线性、二次、三次以及指数数列。线性数列的第 n 项公式为 an + b，其中 a 为公差。二次数列具有恒定的二次差，其第 n 项形式为 an² + bn + c。此外，你还可能遇到三角形数、平方数以及斐波那契型数列。
ays link the position-to-term rule to given terms and check your formula with n = 1, 2, 3.
对于几何数列（Extended 内容），第 n 项为 arⁿ⁻¹，根据需要也可以计算前 n 项之和。请务必将位置与项的规则关联到已知项，并用 n = 1, 2, 3 检验公式。
4. Functions and Graphs | 函数与图像
luate composite functions fg(x) and inverse functions f⁻¹(x). The graph of a function shows its key features: intercepts, gradient, turning points and asymptotes. You must sketch and interpret graphs of linear, quadratic, cubic, reciprocal, and exponential functions.
函数将每一个输入（x）唯一对应到一个输出（y）。使用函数记号，如 f(x) = 2x + 3。你需要会计算复合函数 fg(x) 和反函数 f⁻¹(x)。函数的图像展示其主要特征：截距、斜率、转折点和渐近线。必须会绘制并解读一次、二次、三次、反比例以及指数函数的图像。
m₁ × m₂ = -1. For distance-time and speed-time graphs, the gradient and area under the graph provide speed and distance travelled.
直线图像：y = mx + c，m 为斜率，c 为 y 轴截距。平行直线斜率相等，垂直直线满足 m₁ × m₂ = -1。在距离-时间和速度-时间图像中，图像斜率和图像下的面积分别代表速度和行驶距离。
5. Geometry: Angles, Polygons and Circles | 几何：角、多边形与圆
Know angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal. Parallel lines produce corresponding, alternate, and co-interior angle relationships. Triangle properties include angle sum 180°, exterior angle equals sum of two interior opposite angles, and Pythagoras' theorem for right-angled triangles.
熟悉角度基本事实：平角等于 180°，周角为 360°，对顶角相等。平行线产生同位角、内错角和同旁内角关系。三角形的性质包括内角和 180°、外角等于不相邻两内角之和，以及直角三角形的勾股定理。
c² = a² + b²
ular polygon has equal sides and equal angles. Circle theorems are a major topic: 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 and radius meet at 90°.
多边形角度：内角和 = (n - 2)×180°，任意凸多边形的外角和恒为 360°。正多边形边与角均相等。圆定理是重点：圆心角等于圆周角的两倍；同弧上的圆周角相等；半圆上的圆周角为 90°；圆内接四边形对角互补；切线与半径垂直。
6. Trigonometry and Pythagoras | 三角学与勾股定理
In right-angled triangles, use sine, cosine and tangent ratios: SOH CAH TOA.
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
在直角三角形中，运用正弦、余弦和正切比例：对边/斜边，邻边/斜边，对边/邻边。已知两边求角，或已知一边一角求其他边。Elevation and depression problems require drawing clear diagrams and choosing the correct ratio.
仰角和俯角问题要求画出清晰的示意图并选择正确比例。 in B = c/sin C. Cosine rule: a² = b² + c² - 2bc cos A. You need to decide which rule to apply based on given information.
Extended 级别还包括非直角三角形的正弦定理和余弦定理，以及面积公式 ½ab sin C。正弦定理：a/sin A = b/sin B = c/sin C；余弦定理：a² = b² + c² - 2bc cos A。必须根据已知条件判断使用哪个定理。
7. Mensuration: Area, Volume and Surface Area | 测量：面积、体积与表面积
is 2πr or πd. Arc length = (θ/360)×2πr, sector area = (θ/360)×πr². For composite shapes, break them into simpler parts.
熟记面积公式：三角形（½ × 底 × 高）、平行四边形（b × h）、梯形（½(a + b)h）和圆（πr²）。圆周长公式为 2πr 或 πd。弧长 = (θ/360)×2πr，扇形面积 = (θ/360)×πr²。针对组合图形，可将其拆解为简单图形进行计算。
lume = ⁴⁄₃πr³, surface area of sphere = 4πr². Always pay attention to units and remember that 1 m³ = 1 000 000 cm³, 1 litre = 1000 cm³.
棱柱体积 = 截面面积 × 长度。对于棱锥、圆锥和球体（Extended），使用指定公式：圆锥体积 = ⅓πr²h，球体积 = ⁴⁄₃πr³，球表面积 = 4πr²。注意单位换算，1 m³ = 1 000 000 cm³，1 升 = 1000 cm³。
8. Vectors and Transformations | 向量与变换
able to add and subtract vectors, multiply by a scalar, and find the magnitude of a vector |v| = √x² + y². Position vectors start from the origin. In geometry, vectors can prove collinearity and parallel lines.
向量用大小和方向描述平移，通常表示为列向量 (x y) 或 i, j 形式。需要掌握向量的加法、减法、数乘以及求模长 |v| = √x² + y²。位置向量以原点为起点。在几何中，可用向量证明共线和平行关系。
Transformations include translation, reflection, rotation, and enlargement. Descriptions must give full details: mirror line for reflection, centre, angle and direction for rotation, centre and scale factor for enlargement (and whether it is a fractional enlargement). Combined transformations are often read from right to left.
变换包括平移、反射、旋转和放大。描述必须完整：反射要写明镜像线；旋转要注明中心、角度和方向；放大要给出中心和比例因子（以及是否为分数放大）。复合变换通常从右向左解读。
9. Statistics: Data Representation and Averages | 统计：数据表示与平均数
Collect and organise data using frequency tables, bar charts, pie charts, histograms (Extended with frequency density), and cumulative frequency diagrams. The mean is sum of values divided by number of values; for grouped data, use the midpoint of each class. The median is the middle value when data is ordered; the mode is the most frequent value. The range gives a measure of spread.
用频数表、条形图、饼图、直方图（Extended 需使用频数密度）和累积频数图来收集和整理数据。平均数 = 所有数据之和 ÷ 数据个数；对于分组数据，使用每组的中点。中位数是排序后数据的中间值；众数是出现次数最多的值。极差是表现离散程度的一个量度。
, and construct box-and-whisker plots. Understand how to compare distributions using an average and a measure of spread. Scatter diagrams show correlation; a line of best fit can be drawn by eye and used for prediction.
在 Extended 内容中，计算四分位数和四分位距（IQR = Q3 - Q1），并绘制箱线图。学会通过平均数与离散量度比较分布。散点图显示相关性；可以用目测法画出最佳拟合线并进行预测。
10. Probability | 概率
en 0 and 1 inclusive. Use sample space diagrams and tree diagrams to list outcomes. For combined events, understand 'or' rule: P(A or B) = P(A) + P(B) - P(A and B) for non-mutually exclusive events, and 'and' rule for independent events: P(A and B) = P(A) × P(B).
事件的概率 = 有利结果数 / 等可能结果总数。概率值在 0 到 1 之间。使用样本空间图和树状图列出所有可能结果。对于组合事件，理解"或"规则：互不排斥事件 P(A 或 B) = P(A) + P(B) - P(A 且 B)；独立事件的"且"规则：P(A 且 B) = P(A) × P(B)。
l for successive events. Remember to multiply along branches and add probabilities for the required outcomes. Conditional probability (Extended) may be expressed using P(A| P(B). Always check that the sum of probabilities on branches from a single node equals 1.
树状图对于接连发生的事件尤其有用。沿分支相乘，再将所需结果的概率相加。条件概率（Extended）可以表示为 P(A|B) = P(A 且 B) / P(B)。务必检查从同一节点出发的各分支概率之和等于 1。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图