📚 Year 8 CAIE Advanced Mathematics: Core Knowledge Review | Year 8 CAIE 进阶数学：核心知识点梳理
This article provides a structured summary of the key topics covered in the Year 8 CAIE Advanced Mathematics syllabus. It is designed to help students consolidate their understanding and prepare for assessments by revisiting essential concepts in number, algebra, geometry, statistics and more. Each section presents concise explanations followed by the corresponding Chinese translation, making it suitable for bilingual learning environments.
本文系统梳理了Year 8 CAIE进阶数学课程的核心知识点，旨在帮助学生巩固理解并为考试做好准备。内容涵盖数字、代数、几何、统计等重点领域，每个要点先提供英文阐述，再给出对应的中文解释，适合双语学习场景。
1. Integers, Fractions, Decimals and Percentages | 整数、分数、小数和百分数
Students must be confident performing the four operations with integers, including negative numbers, and applying the correct order of operations (BIDMAS/BODMAS).
学生需要熟练进行包括负数在内的整数四则运算，并正确运用运算顺序（括号、指数、乘除、加减）。
ractions, decimals and percentages is a fundamental skill. For example, 3/8 = 0.375 = 37.5%. Learners should also be able to compare and order mixed sets of these forms.
分数、小数和百分数之间的互化是基本技能。例如 3/8 = 0.375 = 37.5%。学生还应能比较和排序包含这三种形式的混合集合。
Percentage increase and decrease problems, including finding the original amount after a percentage change, require a clear understanding of multipliers. An increase of 15% corresponds to multiplying by 1.15.
涉及百分比增减的问题，包括已知百分比变化后求原值，需要清楚乘数的概念。增加15%相当于乘以1.15。
2. Algebraic Expressions and Simplification | 代数表达式与化简
Algebraic terms are formed by numbers, variables and powers. In the term 7x²y, the coefficient is 7, and the variables are x and y with their respective powers.
代数项由数字、变量和幂组成。在 7x²y 中，系数是 7，变量是 x 和 y，并有各自的指数。
ed by adding or subtracting their coefficients. For instance, 5a + 3b − 2a + 4b = 3a + 7b.
同类项可以通过系数相加减来合并。例如 5a + 3b − 2a + 4b = 3a + 7b。
ab + ac. When a minus sign precedes a bracket, the signs of all terms inside are reversed.
去括号运用乘法分配律：a(b + c) = ab + ac。若括号前是负号，则括号内各项的符号都要变号。
factorisation involves taking out the highest common factor, e.g. 12x² − 8x = 4x(3x − 2).
简单的因式分解就是提取最大公因式，如 12x² − 8x = 4x(3x − 2)。
3. Solving Linear Equations | 解一元一次方程
quation 2x + 5 = 17 is solved by first subtracting 5 and then dividing by 2, giving x = 6.
解方程时通过在等式两边执行相同的运算来分离变量。方程 2x + 5 = 17 先减5再除以2，得 x = 6。
aining brackets should be expanded first: 3(2y − 4) = 18 → 6y − 12 = 18 → 6y = 30 → y = 5.
含有括号的方程应先去括号：3(2y − 4) = 18 → 6y − 12 = 18 → 6y = 30 → y = 5。
and the constants to the other. For example, 7x − 3 = 2x + 12 leads to 5x = 15, so x = 3.
当变量出现在方程两边时，把含变量项移到一边，常数项移到另一边。例如 7x − 3 = 2x + 12 得到 5x = 15，x = 3。
4. Ratio, Proportion and Rates | 比、比例和率
A ratio compares the sizes of two or more quantities. The ratio 3:5 can be scaled up or down by multiplying or dividing both parts by the same number.
比用于比较两个或多个量的大小。比 3:5 可以通过将两部分同时乘以或除以同一个数进行放大或缩小。
Dividing a quantity in a given ratio, such as sharing £120 in the ratio 2:3, involves finding the total number of parts (5) and calculating 2 parts (£48) and 3 parts (£72).
按给定比例分配数量，例如按 2:3 的比例分120英镑，先求出总份数（5份），再算出2份（48镑）和3份（72镑）。
Direct proportion means two quantities increase at the same rate. If 5 pens cost £2.50, then 8 pens cost (8/5) × £2.50 = £4.00. Rates such as speed (km/h) are treated similarly.
正比例指两个量以相同速率增加。若5支笔2.50镑，则8支笔花费 (8/5) × 2.50 = 4.00镑。速率如速度（公里/小时）也类似处理。
5. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°, and angles around a point total 360°. Vertically opposite angles are equal.
平角（直线上的角）之和为180°，绕一点一周的角之和为360°。对顶角相等。
um is 360°. This can be extended to any polygon using (n − 2) × 180° for an n-sided shape.
三角形内角和恒为180°。任意四边形的内角和为360°。这一规律可推广到任意n边形，内角和公式为 (n − 2) × 180°。
Regular polygons have all sides and angles equal. The exterior angle of a regular n-sided polygon is 360°/n, and the interior angle is 180° − exterior angle.
正多边形的所有边和角都相等。正n边形的外角为 360°/n，内角 = 180° − 外角。
6. Perimeter, Area and Volume | 周长、面积和体积
a shape. For a rectangle, P = 2(l + w). For a triangle, simply add the three side lengths.
周长是图形一周的总长度。矩形的周长 P = 2(l + w)。三角形的周长将三边长相加即可。
tangle A = l × w; Triangle A = ½ × b × h; Parallelogram A = b × h; Trapezium A = ½(a + b)h
面积公式：矩形 A = 长 × 宽；三角形 A = ½ × 底 × 高；平行四边形 A = 底 × 高；梯形 A = ½(上底 + 下底) × 高
and the circumference is C = 2πr, where π is taken as 3.14 or the button on a calculator.
圆的面积 A = πr²，周长 C = 2πr，其中 π 取值3.14或使用计算器上的π键。
r a cuboid, V = l × w × h. The surface area of a solid is the total area of all its faces.
棱柱的体积 = 横截面积 × 长度。长方体的体积 V = 长 × 宽 × 高。立体的表面积是所有面的总面积。
7. Statistics: Data Handling and Charts | 统计：数据处理与图表
The mean is calculated by summing all data values and dividing by the number of values. The median is the middle value when data are ordered; the mode is the most frequent value.
平均数（均值）通过所有数据值之和除以数据个数求得。中位数是将数据排序后的中间值；众数是出现频率最高的值。
ie chart shows proportions of a whole, where each sector angle = (frequency/total) × 360°.
条形图、饼图和折线图用于直观呈现数据。饼图显示整体中的比例，每个扇形的角度 = (频数/总数) × 360°。
Stem-and-leaf diagrams display data while preserving original values. They can be ordered to find medians and quartiles easily.
茎叶图可以展示数据并保留原始数值。将茎叶图排序后可方便地找到中位数和四分位数。
8. Probability | 概率
Probability is a measure of how likely an event is, expressed as a fraction, decimal or percentage between 0 (impossible) and 1 (certain).
概率衡量事件发生的可能性，用0（不可能）到1（必然）之间的分数、小数或百分数表示。
f favourable outcomes / total number of equally likely outcomes. The sum of probabilities of all mutually exclusive outcomes is 1.
某事件的概率 = 有利结果数 / 所有等可能结果总数。所有互斥结果的概率之和为1。
Experimental probability is based on trials or experiments, whereas theoretical probability is based on symmetry. As the number of trials increases, experimental probability tends to approach theoretical probability.
实验概率基于试验或实验，而理论概率基于对称性。当试验次数增加时，实验概率会趋近于理论概率。
9. Sequences and Patterns | 数列与规律
A number sequence follows a rule. In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference d.
数列按照一定的规则排列。在等差数列中，相邻两项的差恒定，这个差称为公差 d。
The nth term of an arithmetic sequence can be found using: nth term = a + (n − 1)d, where a is the first term. For the sequence 5, 9, 13, 17, …, the nth term is 5 + (n − 1)×4 = 4n + 1.
等差数列的第n项公式为：第n项 = a + (n − 1)d，其中a为首项。数列 5, 9, 13, 17, … 的第n项为 5 + (n − 1)×4 = 4n + 1。
Recognising and extending patterns using algebraic rules helps in problem solving. Students should be able to work with term-to-term rules and position-to-term rules.
用代数规则识别和延伸规律有助于解决问题。学生应能运用项到项的递推规则以及位置与项的对应规则。
10. Transformations and Symmetry | 变换与对称
Transformations include translations (slides), reflections (flips) and rotations (turns). Each is defined by specific parameters: translation by a vector, reflection in a mirror line, and rotation by an angle about a centre.
变换包括平移、反射和旋转。每种变换由其特定参数定义：平移用向量描述，反射需指定镜面线，旋转需指定中心和旋转角度。
Reflection symmetry occurs when a shape can be folded along a line so the two halves match. Rotational symmetry occurs when a shape fits onto itself during a rotation of less than 360° about its centre.
反射对称指图形可沿某条直线对折后完全重合。旋转对称指图形绕中心旋转小于360°后能与自身重合。
Enlargement changes the size of a shape by a scale factor from a centre of enlargement. A scale factor greater than 1 makes the shape larger; a fraction between 0 and 1 makes it smaller.
放大变换根据放大中心和一个比例因子改变图形大小。比例因子大于1时图形变大；介于0和1之间的分数使图形变小。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
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