📚 Year 11 CAIE Mathematics: Core Concepts Revision | Year 11 CAIE 数学：核心知识点梳理
This comprehensive revision guide covers the essential topics in the CAIE IGCSE Mathematics (Core) syllabus for Year 11 students. Whether you are preparing for your final examinations or reinforcing classroom learning, a clear understanding of core concepts such as number operations, algebra, geometry, statistics and probability will build your confidence and problem-solving skills. Each section below presents key ideas with concise explanations, useful formulas and practical tips to help you master the material.
这本全面的复习指南涵盖了Year 11学生CAIE IGCSE数学（核心）课程的关键知识点。不论你是在准备期末考试还是巩固课堂所学，透彻理解数字运算、代数、几何、统计与概率等核心概念都将增强你的信心和解题能力。以下每个小节都通过简洁的解释、实用的公式和技巧，帮助你掌握这些内容。
1. Number Types and Operations | 数字类型与运算
You must be comfortable with natural numbers, integers, rational and irrational numbers, as well as prime, composite and square numbers. Remember that a prime number has exactly two distinct factors, 1 and itself.
你必须熟悉自然数、整数、有理数和无理数，以及质数、合数和平方数。请记住质数只有两个不同的因数：1和它本身。
Directed numbers (positive and negative) follow specific rules when adding, subtracting, multiplying or dividing. For example, multiplying two negative numbers gives a positive result: (−3) × (−4) = 12.
有向数（正数和负数）在加减乘除时遵循特定规则。例如，两个负数相乘得正数：(−3) × (−4) = 12。
The order of operations is remembered by the acronym BODMAS or BIDMAS (Brackets, Orders/Indices, Division and Multiplication, Addition and Subtraction). Always apply this hierarchy when evaluating expressions.
运算顺序可通过缩写BODMAS或BIDMAS（括号、指数、乘除、加减）来记忆。计算表达式时务必遵循这一层级。
Factors and multiples are fundamental: the highest common factor (HCF) and lowest common multiple (LCM) are often found using prime factor trees. Expressing a number as a product of prime factors in index form simplifies these calculations.
因数和倍数很基础：最大公因数（HCF）和最小公倍数（LCM）常用质因数树状图求出。将数字写成质因数乘积的指数形式能简化这些计算。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
Converting between fractions, decimals and percentages is a daily mathematical skill. To change a fraction to a decimal, divide the numerator by the denominator. Multiply a decimal by 100 to obtain a percentage.
分数、小数和百分数之间的转换是一项日常数学技能。将分数化为小数，用分子除以分母即可；小数乘以100即得百分数。
When adding or subtracting fractions, always find a common denominator first. For multiplication, multiply the numerators together and the denominators together, simplifying where possible. For division, flip the second fraction and multiply (reciprocal).
加减分数时，务必先找到公分母。乘法是将分子相乘、分母相乘，再约分。除法则是将第二个分数倒置（倒数）再相乘。
Percentage increase and decrease are calculated using the formula: Percentage change = (change ÷ original) × 100%. Understand the difference between simple and compound interest; compound interest is applied repeatedly on the new balance.
计算百分数增减的公式为：百分比变化 = (变化量 ÷ 原值) × 100%。要理解单利和复利的区别；复利是根据新余额反复计算利息。
s can be expressed as fractions using algebraic manipulation. For instance, to convert 0.3̇, let x = 0.333..., then 10x = 3.333..., subtract to get 9x = 3, so x = 1/3.
循环小数可以通过代数方法化为分数。例如，将0.3̇化成分数：设 x = 0.333...，则 10x = 3.333...，相减得 9x = 3，所以 x = 1/3。
3. Ratio and Proportion | 比和比例
A ratio compares the sizes of two or more quantities. It can be simplified just like a fraction. For example, a ratio of 8:12 simplifies to 2:3 by dividing both parts by their highest common factor, 4.
比用来比较两个或多个数量的大小，可以像分数一样化简。例如，比 8:12 除以它们的最大公因数 4，化简为 2:3。
the parts together to find the total number of shares, then multiply. For instance, divide £90 in the ratio 1:2:3 → total parts = 6, so each share = £15, giving £15, £30 and £45.
按给定比分配数量时，先将各份相加得到总份数，再相乘得出。例如，将90英镑按1:2:3分配，总份数 = 6，每份为15英镑，分别得到15、30和45英镑。
Direct proportion means that as one quantity increases, the other increases at the same rate; the graph is a straight line through the origin. Inverse proportion means one quantity increases as the other decreases, and their product remains constant.
正比例表示一个量增加，另一个量以相同速率增加；图像是一条过原点的直线。反比例表示一个量增加时另一个量减少，且它们的乘积保持不变。
Scale drawings and map scales require you to interpret ratios like 1 : 50 000, where 1 cm on the map represents 50 000 cm (0.5 km) in real life. Always convert to the same units before calculating.
比例图和地图比例尺要求解读如 1 : 50 000 这样的比，即图上 1 厘米代表实际 50000 厘米（0.5 公里）。计算前务必将单位统一。
4. Algebraic Expressions and Equations | 代数表达式与方程
An algebraic expression combines numbers, variables and operations. You should be able to collect like terms, expand brackets and factorise simple expressions. For example, expanding 3(x + 4) gives 3x + 12; factorising 5y − 15 gives 5(y − 3).
代数表达式由数字、变量和运算组合而成。你应该学会合并同类项、展开括号和因式分解简单表达式。例如，展开 3(x + 4) 得 3x + 12；因式分解 5y − 15 得 5(y − 3)。
Solving linear equations involves isolating the unknown. Always perform the same operation on both sides. For instance, solve 2x − 3 = 7 → add 3: 2x = 10 → divide by 2: x = 5.
解一元一次方程需要将未知数隔离，始终在方程两边进行相同操作。例如，解 2x − 3 = 7：加3得 2x = 10，除以2得 x = 5。
Quadratic equations of the form ax² + bx + c = 0 can be solved by factorising, completing the square or using the quadratic formula. In the core syllabus, you mainly need to factorise and then set each factor to zero.
形如 ax² + bx + c = 0 的二次方程可通过因式分解、配方法或求根公式求解。核心课程中，主要掌握因式分解后令每个因子为零的方法。
Simultaneous linear equations can be solved by elimination or substitution. Eliminate one variable by adding or subtracting the equations after making coefficients equal, then solve for the remaining variable.
联立一次方程组可用消元法或代入法求解。将其中一个变量的系数调整相等后，通过加减方程消去该变量，再求解另一变量。
5. Sequences and Inequalities | 数列与不等式
A sequence is a list of numbers following a rule. To find the nth term of a linear arithmetic sequence, use the formula: Tₙ = a + (n − 1)d, where a is the first term and d is the common difference.
数列是按规则排列的一串数字。求等差数列（线性数列）的第 n 项可用公式：Tₙ = a + (n − 1)d，其中 a 为首项，d 为公差。
For simple quadratic sequences, the second difference is constant. You can use the pattern of second differences to find the nth term expressed as an² + bn + c.
对简单的二次数列，其二阶差是常数。可利用二阶差的规律求出形如 an² + bn + c 的第 n 项表达式。
al to). Solve them just like equations, but remember: when multiplying or dividing by a negative number, the inequality sign reverses direction.
不等式用 <（小于）、>（大于）、≤（小于等于）和 ≥（大于等于）等符号表示取值范围。求解方法与方程类似，但要牢记：当乘以或除以一个负数时，不等号方向要改变。
strict inequalities (< or >) and closed circles for inclusive inequalities (≤ or ≥).
在数轴上表示不等式的解集时，严格不等式（< 或 >）用空心圆圈，包含等式的不等式（≤ 或 ≥）用实心圆圈。
6. Geometry – Angles and Polygons | 几何——角与多边形
Know your angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles sum to 180°.
掌握基本的角知识：直线上的角的和为180°，绕一点的角的和为360°，对顶角相等。当一条横线截两条平行线时，同位角相等、内错角相等、同旁内角互补（和为180°）。
The sum of interior angles of an n-sided polygon is (n − 2) × 180°. Each interior angle of a regular n-sided polygon is (n − 2) × 180° ÷ n. The sum of exterior angles of any convex polygon is always 360°.
n 边形内角和为 (n − 2) × 180°。正 n 边形的每个内角等于 (n − 2) × 180° ÷ n。任何凸多边形的外角和恒为 360°。
Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). Pythagoras' theorem applies only to right-angled triangles: a² + b² = c², where c is the hypotenuse.
三角形可按边（等边、等腰、不等边）和角（锐角、直角、钝角）分类。毕达哥拉斯定理仅适用于直角三角形：a² + b² = c²，其中 c 为斜边。
Congruent shapes are identical in size and shape; corresponding sides and angles are equal. Similar shapes have the same shape but different sizes; corresponding angles are equal and sides are in proportion.
全等图形大小和形状完全相同，对应边和对应角都相等。相似图形形状相同但大小不同；对应角相等且对应边成比例。
7. Measurement – Perimeter, Area and Volume | 测量——周长、面积和体积
Perimeter is the distance around a shape. For a rectangle: P = 2(l + w). Circumference of a circle: C = 2πr or C = πd.
周长是图形边界的总长度。矩形：P = 2(l + w)。圆的周长：C = 2πr 或 C = πd。
Key area formulas include: rectangle A = l × w, triangle A = ½ × base × height, parallelogram A = base × height, trapezium A = ½(a + b)h, circle A = πr².
关键的面积公式有：矩形 A = l × w，三角形 A = ½ × 底 × 高，平行四边形 A = 底 × 高，梯形 A = ½(a + b)h，圆 A = πr²。
Volume of a cuboid is V = l × w × h. For prisms, volume is area of cross-section × length. Cylinder: V = πr²h. Surface area of a cylinder: curved surface area = 2πrh, add areas of the two circular ends.
长方体的体积为 V = l × w × h。棱柱的体积是横截面积乘长度。圆柱：V = πr²h。圆柱表面积：侧面积 = 2πrh，再加上两个底面圆的面积。
and volume (cm³, m³) requires careful attention: 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³.
务必检查单位是否一致。长度单位（cm, m, km）、面积单位（cm², m²）和体积单位（cm³, m³）的换算需要格外注意：1 m² = 10 000 cm²，1 m³ = 1 000 000 cm³。
8. Trigonometry and Pythagoras | 三角学与毕达哥拉斯定理
θ = adjacent/hypotenuse, tangent θ = opposite/adjacent. Use SOH CAH TOA to remember them.
在直角三角形中，三角函数比定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用 SOH CAH TOA 辅助记忆。
To find an unknown side, choose the ratio that includes the known side and the unknown side, set up an equation and solve. To find an angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹).
求未知边长时，选择包含已知边和未知边的三角函数比，列方程求解。求角度时，使用反三角函数（sin⁻¹, cos⁻¹, tan⁻¹）。
Pythagoras' theorem is used to find a missing side when two sides are known: c² = a² + b² for the hypotenuse, or rearranged for a shorter side. Always identify the hypotenuse as the side opposite the right angle.
毕达哥拉斯定理用于已知两边求第三边：斜边 c² = a² + b²，或移项求直角边。务必确认斜边是直角所对的边。
In 3D problems, you may need to apply Pythagoras' theorem twice or use trigonometry. Draw the right-angled triangles you use, often on a separate sketch, to avoid confusion.
在三维问题中，可能需要两次运用毕达哥拉斯定理或结合三角函数。把你所用的直角三角形单独画出来，避免混淆。
9. Graphs of Linear and Quadratic Functions | 一次与二次函数图像
A linear function can be written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). To plot a line, find two points, such as the intercepts, and draw a straight line through them.
一次函数可写为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。画直线时，找出两个点（例如截距），然后过这两点连一条直线。
The gradient between two points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁). Parallel lines have the same gradient; perpendicular lines have gradients that multiply to −1.
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率 m = (y₂ − y₁) / (x₂ − x₁)。平行线斜率相同；垂直线斜率之积为 −1。
-intercept (c), the roots (x-intercepts by setting y=0) and the turning point (line of symmetry x = −b/(2a)).
二次函数的图像是抛物线。若 x² 的系数为正，开口向上；为负则开口向下。画草图时，要找出 y 轴截距 (c)、与 x 轴的交点（令 y=0 求根）以及顶点（对称轴 x = −b/(2a)）。
Solving equations graphically means looking for the intersection of two graphs. For example, the solution to f(x) = g(x) is the x-coordinate where y = f(x) and y = g(x) meet.
图像法解方程即找两个图像的交点。例如，方程 f(x) = g(x) 的解就是 y = f(x) 与 y = g(x) 交点的横坐标。
10. Statistics – Averages and Charts | 统计——平均数与图表
Three common measures of average: mode (most frequent value), median (middle value when data is ordered) and mean (sum of values divided by number of values). The range is the difference between the largest and smallest values.
三种常见的平均数：众数（出现次数最多的值）、中位数（数据排序后中间的数值）和平均数（总和除以数据个数）。极差是最大值与最小值的差。
For grouped data, use the midpoints of intervals to estimate the mean. A cumulative frequency table helps find the median and quartiles. The median is the value at the ½(n+1)th position; the lower quartile at ¼(n+1) and upper quartile at ¾(n+1).
对于分组数据，用组中值来估算平均数。累积频率表有助于求中位数和四分位数。中位数位于第 ½(n+1) 位，下四分位数位于 ¼(n+1) 位，上四分位数位于 ¾(n+1) 位。
Display data using bar charts (for discrete or categorical data), pie charts (showing proportions), histograms (for continuous data with frequency density on the y-axis), cumulative frequency curves (s-shaped) and scatter graphs (to show correlation).
用条形图（离散或分类数据）、饼图（显示比例）、直方图（连续数据，纵轴为频率密度）、累积频率曲线（S形）和散点图（显示相关关系）来展示数据。
11. Probability | 概率
Probability measures the chance of an event occurring, expressed as a fraction, decimal or percentage between 0 (impossible) and 1 (certain). For equally likely outcomes: Probability = number of favourable outcomes / total number of outcomes.
概率衡量事件发生的可能性，可用 0（不可能）到 1（必然）之间的分数、小数或百分数表示。等可能结果下：概率 = 有利结果数 / 总结果数。
or yellow ball from a bag containing only those colours: P(red) + P(blue) + P(yellow) = 1.
所有互斥（无重叠）结果的概率之和为 1。例如，从一个仅有红、蓝、黄三色球的袋子里摸球：P(红) + P(蓝) + P(黄) = 1。
In a tree diagram, multiply probabilities along branches for 'and' events, and add probabilities of separate branches for 'or' events. Replace the objects after each pick if the events are independent, otherwise do not replace (conditional probability topics appear in extended, but core may use simple tree diagrams without replacement).
在树状图中，“与”事件沿分支相乘概率，“或”事件将不同分支的概率相加。若事件相互独立，每次抽取后需放回；否则不放回（条件概率主要在扩展课程，但核心中可考察简单的不放回树状图）。
Expected frequency is the number of times you would expect an event to occur in a number of trials: Expected frequency = probability × number of trials.
期望频率是在多次试验中预期某事件发生的次数：期望频率 = 概率 × 试验次数。
12. Transformations and Symmetry | 变换与对称
Transformations change the position or size of a shape. Translation moves a shape by a column vector. Every point moves the same distance and direction. Describe it fully as a translation by vector (x y).
变换会改变图形的位置或大小。平移是用列向量移动图形，每个点移动相同的距离和方向。完整描述为通过向量 (x y) 的平移。
x=0), y=x and y=−x. After reflection, the shape is congruent to the original but reversed.
反射是将图形翻折到一条直线（镜面线）另一侧。常见镜面线有 x 轴 (y=0)、y 轴 (x=0)、y=x 和 y=−x。反射后，图形与原图全等但方向相反。
Rotation turns a shape around a fixed centre through a given angle (e.g. 90° clockwise, 180°). State the centre of rotation, angle and direction. A rotation of 180° about (0,0) maps (x, y) to (−x, −y).
旋转是图形绕一个固定中心按给定角度（如顺时针90°、180°）旋转。需说明旋转中心、角度和方向。关于原点 (0,0) 旋转180° 将 (x, y) 映射为 (−x, −y)。
Enlargement changes the size of a shape by a scale factor about a centre of enlargement. If the scale factor k > 1, the shape gets larger; if 0 < k < 1, it gets smaller. Start by drawing lines from the centre through vertices.
放大是以一个放大中心按比例因子改变图形大小。比例因子 k > 1 时图形变大；0 < k < 1 时图形变小。作图时从放大中心向各顶点连线。
Symmetry: A shape has line symmetry if it can be folded along a line so that both halves match; rotational symmetry if it looks the same after a rotation of less than 360° about its centre. The order of rotational symmetry is the number of times it matches in a full turn.
对称：如果一个图形可以沿一条直线对折后完全重合，则它具有线对称性；如果绕中心旋转小于360°后能与自身重合，则具有旋转对称性。旋转对称的阶是指在一个完整旋转中重合的次数。
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