📚 Year 11 CIE Mathematics: Summer Preview and Bridging Course | Year 11 CIE 数学：暑期预习与衔接课程
Welcome to your summer bridging course for Year 11 CIE IGCSE Mathematics. This programme is built to strengthen the foundations laid in Year 10 and to preview the most challenging topics you will meet in Year 11. We focus on the Extended syllabus (0580/0980), targeting Grades A* to C. Over the next few weeks, you will revisit algebra, trigonometry, geometry, functions, and statistics, while also encountering new ideas such as advanced graph transformations, vectors in 2D, and conditional probability. Each section below pairs an English explanation with a Chinese summary, helping you build subject knowledge and academic language at the same time.
欢迎参加 Year 11 CIE IGCSE 数学暑期衔接课程。本课程旨在夯实 Year 10 基础，并提前接触 Year 11 中最具挑战性的主题。我们紧扣 Extended 考纲（0580/0980），目标等级为 A* 到 C。接下来的几周里，你将重温代数、三角比、几何、函数和统计，同时学习更深层次的图像变换、二维向量和条件概率等新内容。下文的每个要点均以英文阐释，再附中文总结，帮助你在掌握知识的同时提升学科英语能力。
1. Number, Ratio and Proportion | 数、比和比例
We begin by revising surds and applying percentages to compound changes. You are expected to simplify expressions such as √48 − √27 and to solve problems involving repeated percentage increase or decrease, including depreciation and compound interest.
我们从根式和百分比应用开始复习。你需要能够化简像 √48 − √27 这样的式子，并解决涉及重复百分比增减的问题，包括折旧和复利计算。
e able to find the constant of proportionality k and use the relationship to calculate unknown values.
正比例和反比例常出现在情境题中。例如，若 y 与 x² 成反比，则 y = k/x²。你需要求出比例常数 k，并用该关系式计算未知值。
2. Algebraic Manipulation and Equations | 代数运算与方程
Factorising quadratics is a core skill. You should be confident with expressions like 6x² − 13x + 6, splitting the middle term or using inspection to write it as (2x − 3)(3x − 2). Remember to always check by expanding.
二次三项式的因式分解是一项核心技能。你需要熟练处理诸如 6x² − 13x + 6 的式子，通过拆分中项或观察法将其写成 (2x − 3)(3x − 2)。记得始终用展开来检验。
Simultaneous equations now extend to one linear and one quadratic. Substitute the linear expression into the quadratic, solve the resulting quadratic in one variable, then find the corresponding y values. Always present your solutions as coordinate pairs.
联立方程现在扩展到一个是线性、一个是二次的形式。把线性表达式代入二次式，求解得到的一元二次方程，然后求出相应的 y 值。始终用坐标对的形式呈现你的解。
b)/(c − d), you may need to make c the subject. Cross-multiply, expand, collect terms containing c on one side, factorise c, and divide.
公式主项变换常常需要进行因式分解。例如，给定 a = (2b)/(c − d)，你需要让 c 成为主项。交叉相乘，展开，将含 c 的项移到一边，对 c 因式分解，然后相除。
3. Functions and Transformations | 函数与图像变换
(−2), solve f(x) = 1, and determine inverse functions where the original function is one-to-one.
函数接收一个输入并产生唯一一个输出。记号 f(x) = x² − 3 意味着对于任意 x，输出为 x² − 3。你需要能够求出 f(−2)，求解 f(x) = 1，并在原函数是一一对应时确定它的反函数。
Graph transformations follow clear rules: f(x) + a shifts the graph up by a units; f(x + a) shifts it left by a units; −f(x) reflects it in the x-axis; f(−x) reflects it in the y-axis. For stretches, af(x) stretches vertically by scale factor a, while f(ax) stretches horizontally by factor 1/a.
图像变换有清晰规律：f(x) + a 将图像向上平移 a 个单位；f(x + a) 将其向左平移 a 个单位；−f(x) 使图像关于 x 轴对称；f(−x) 关于 y 轴对称。拉伸方面，af(x) 沿竖直方向以比例系数 a 拉伸，f(ax) 沿水平方向以比例系数 1/a 拉伸。
4. Coordinate Geometry and Linear Graphs | 坐标几何与线性图像
The gradient of a straight line between (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). Parallel lines share the same gradient; perpendicular lines have gradients whose product is −1. You must be able to find the equation of a line given a point and a gradient, or given two points.
两点 (x₁, y₁) 与 (x₂, y₂) 之间直线的斜率为 (y₂ − y₁)/(x₂ − x₁)。平行线斜率相等；垂直线的斜率乘积为 −1。你需要能够根据一个点与斜率，或两个点，求出直线的方程。
m: √[(x₂ − x₁)² + (y₂ − y₁)²]. These skills underpin many geometry problems in the CIE exam.
求线段中点十分直接：((x₁ + x₂)/2, (y₁ + y₂)/2)。两点之间的距离可用勾股定理求得：√[(x₂ − x₁)² + (y₂ − y₁)²]。这些技能是 CIE 考试中许多几何问题的基础。
5. Trigonometry in Right Triangles and Beyond | 直角三角形三角比及其延伸
tive to the given angle. For non-right-angled triangles, the sine rule a/sin A = b/sin B = c/sin C and cosine rule a² = b² + c² − 2bc cos A become essential.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。要学习根据给定角识别各边。对于非直角三角形，正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A 变得至关重要。
Know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°. These are easily derived from the two special triangles: an isosceles right triangle (45°–45°–90°) and half an equilateral triangle (30°–60°–90°). Using these, you can handle exact surd forms without a calculator.
牢记 0°、30°、45°、60° 和 90° 的 sin、cos 和 tan 的精确值。它们可轻松从两个特殊三角形推得：等腰直角三角形（45°–45°–90°）和等边三角形的一半（30°–60°–90°）。借助它们，你可以手算含有根式的精确结果。
6. Mensuration and Circle Theorems | 测量与圆定理
ence = 2πr. An arc length with angle θ at the centre is (θ/360) × 2πr. Sector area = (θ/360) × πr².
你需要求圆的面积和周长、扇形的面积以及弧长。对于半径为 r 的圆，面积 = πr²，周长 = 2πr。圆心角为 θ 的弧长为 (θ/360)×2πr。扇形面积 = (θ/360)×πr²。
Circle theorems must be learnt and applied. Key ones: the angle in a semicircle is 90°; angles in the same segment are equal; the angle at the centre is twice the angle at the circumference; and opposite angles in a cyclic quadrilateral sum to 180°.
必须学习并应用圆定理。关键定理有：半圆内的圆周角为 90°；同弦上的圆周角相等；圆心角是圆周角的两倍；圆内接四边形的对角互补（和为 180°）。
7. Vectors in Two Dimensions | 二维向量
A vector describes both magnitude and direction, written as a column vector (xy) or using unit vectors i and j. Vector addition, subtraction, and multiplication by a scalar are foundational. Geometrically, a vector can represent a translation or a directed line segment.
向量表示了大小和方向，可写成列向量 (xy) 的形式，或使用单位向量 i 和 j 表示。向量的加法、减法和数乘是基础。几何上，向量可代表一次平移或有向线段。
l, and B lies on AC. Often you will need to prove that points are collinear by showing vectors are scalar multiples.
你需要能够求两点间的向量，并处理平行和共线向量。如果 AB = 2b 且 BC = 6b，那么 AB 与 BC 平行，且 B 在 AC 上。常见考题要求你通过证明向量成标量倍数关系来证明点共线。
8. Data Handling and Statistics | 数据处理与统计
histograms, the key is frequency density = frequency / class width. Use the area of each bar to represent frequency.
你必须能够绘制并解读累积频率图、箱线图以及组距不等的直方图。对于直方图，关键概念是频率密度 = 频数 / 组距。用每个直条的面积来代表频数。
Calculating mean, median, mode and range for grouped and ungrouped data is routine. For grouped data, use mid-interval values to estimate the mean. The interquartile range, found from a cumulative frequency curve or by ordering data, measures spread effectively.
计算分组和未分组数据的平均数、中位数、众数和极差是常规要求。对于分组数据，利用组中值来估计平均数。四分位距可通过累积频率曲线或数据排序求得，它能有效度量离散程度。
9. Probability and Tree Diagrams | 概率与树形图
Probability always lies between 0 and 1. For combined events, you can list outcomes systematically or use a probability tree diagram, especially when events are sequential and without replacement. Multiply along branches for ‘and’ and add probabilities for ‘or’ where mutually exclusive.
概率始终介于 0 和 1 之间。对于组合事件，你可以系统列出所有结果，或使用概率树形图，尤其当事件顺序发生且无放回时。沿分支相乘得“且”的概率；对于互斥事件，相加得“或”的概率。
Conditional probability appears in Year 11 and often surprises students. The notation P(A|B) means “probability of A given B has occurred”. The formula is P(A|B) = P(A and B) / P(B). Tree diagrams can make such questions much clearer.
条件概率出现在 Year 11 课程中，常令学生意外。记号 P(A|B) 表示“在 B 已发生的条件下 A 发生的概率”。公式为 P(A|B) = P(A 且 B)/P(B)。树形图可以使此类问题清晰得多。
10. Sequences and Variation | 数列与变分
You will study arithmetic sequences where a common difference d is added each time, and geometric sequences with a common ratio r. Be ready to find the nth term: a + (n − 1)d for arithmetic, and arⁿ⁻¹ for geometric. Understand when a sequence converges (|r| < 1).
你将学习等差数列（每次加上公差 d）和等比数列（公比为 r）。准备好求第 n 项：等差数列为 a + (n − 1)d，等比数列为 arⁿ⁻¹。理解数列何时收敛（|r| < 1）。
ften give a set of values to find k, then use the equation to find unknowns. Real-world contexts like speed–time or cost–people relationships make this topic practical.
正变分意味着 y = kx，逆变分意味着 y = k/x。题目通常给出一组值以求 k，然后利用方程求未知数。速度—时间或费用—人数等实际情境使这个主题变得很实用。
11. Advanced Graphs and Inequalities | 高级图像与不等式
procal graphs is expected. You should know the shape of y = x², y = x³, y = 1/x, and y = ax for a > 1. Transformations applied to these are common exam material, so connect this with the functions section.
要求能够画出二次、三次和倒数函数的简图。你应熟知 y = x²、y = x³、y = 1/x 和 y = ax（a > 1）的形状。对这些图像进行变换是常见考题，因此要将这部分与函数篇章联系起来。
Solving quadratic inequalities such as x² − 5x + 6 < 0 requires a sign diagram or graph. Factorise to (x − 2)(x − 3) < 0, identify the roots, and test intervals. The solution is 2 < x < 3. Always express the final answer using set notation or on a number line.
解二次不等式如 x² − 5x + 6 < 0 需要运用符号图或图像。因式分解为 (x − 2)(x − 3) < 0，找到根，然后检验区间。解为 2 < x < 3。始终用集合记号或在数轴上表示最终答案。
12. Problem-Solving and Exam Technique | 解题与应试技巧
Top-scoring candidates do not just know the facts; they can read a word problem, extract the mathematical model, and choose the correct strategy. Always highlight key numbers and units, draw a diagram if possible, and check whether your answer makes sense in context.
高分考生不仅知道知识点，还能阅读文字题，提取数学模型，并选择正确的策略。要始终标出关键数字和单位，如果可能就画出示意图，并检查答案在题目情境下是否合理。
Manage your time carefully: the CIE Extended paper 2 (non-calculator) and paper 4 (calculator) both require brisk working. For non-calculator papers, practise arithmetic with fractions, decimals and surds. For calculator papers, double-check your mode and brackets. Show clear working to earn method marks even if the final answer is wrong.
仔细管理时间：CIE Extended 的试卷二（不可用计算器）和试卷四（可用计算器）都需要快速答题。对于不可用计算器的试卷，练习分数、小数和根式运算。对于可用计算器的试卷，再次确认模式和括号。即使最终答案有误，清晰的解题步骤也能帮助你获得方法分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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