📚 Year 8 CAIE Further Mathematics: Core Knowledge Review | Year 8 CAIE 进阶数学：核心知识点梳理
Year 8 Further Mathematics builds on the foundation laid in earlier stages and introduces more advanced concepts that are essential for success in the CAIE curriculum. This review covers the core knowledge you need to master, from integers and powers to probability and statistics. Each topic is presented with clear English explanations alongside Chinese translations to support bilingual learners. By working through these sections, you will strengthen your problem-solving skills and deepen your mathematical understanding.
八年级进阶数学建立在早期阶段的基础上，引入了对CAIE课程成功至关重要的更高级概念。这篇梳理涵盖了你需要掌握的核心知识，从整数和幂到概率与统计。每个主题都配有清晰的英文解释和对应的中文翻译，以支持双语学习者。通过学习这些章节，你将增强解决问题的能力并加深对数学的理解。
1. Integers, Powers and Roots | 整数、幂与方根
Integers are whole numbers that can be positive, negative or zero. They do not include fractions or decimals. When working with integers, you must be confident with the four operations and the order of operations (BIDMAS/BODMAS).
整数是可以为正、负或零的完整数字，不包括分数或小数。在处理整数时，你必须熟练掌握四种运算以及运算顺序（括号、指数、乘除、加减）。
t. 5³ = 5 × 5 × 5 = 125. Any number to the power of 1 is itself, and any non-zero number to the power of 0 equals 1.
幂，或指数，告诉我们将基数自身相乘的次数。例如，4² 表示 4 × 4 = 16。右上角的小数字是指数。5³ = 5 × 5 × 5 = 125。任何数的1次幂都是它本身，任何非零数的0次幂都等于1。
when multiplied by itself, gives the original number. The symbol for square root is √. So √25 = 5 because 5² = 25. Cube roots use the symbol ∛: ∛27 = 3 because 3³ = 27.
方根是幂的逆运算。一个数的平方根是一个值，当它自身相乘时得到原数。平方根符号是 √。因此 √25 = 5，因为 5² = 25。立方根使用符号 ∛：∛27 = 3，因为 3³ = 27。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
Fractions represent parts of a whole. A fraction has a numerator (top) and a denominator (bottom). You need to be able to simplify fractions, find equivalent fractions, and convert between mixed numbers and improper fractions. When adding or subtracting fractions, they must have the same denominator.
分数代表整体的一部分。分数有分子（上）和分母（下）。你需要能够化简分数、找到等值分数，以及在带分数和假分数之间进行转换。加减分数时，它们必须有相同的分母。
ple, 3/4 = 0.75. To convert a decimal to a percentage, multiply by 100. So 0.75 × 100 = 75%.
小数是表达部分的另一种方式。将分数转换为小数，用分子除以分母。例如，3/4 = 0.75。将小数转换为百分数，乘以100。所以 0.75 × 100 = 75%.
Percentages mean 'out of 100'. You should be able to find a percentage of a quantity, increase or decrease an amount by a given percentage, and express one quantity as a percentage of another. These skills are vital for dealing with real-life problems such as discounts, interest and statistics.
百分数意为“每百分之”。你应能求一个数量的百分比、按给定百分比增加或减少一个量，以及将一个量表示为另一个量的百分比。这些技能对于处理折扣、利息和统计等现实问题至关重要。
3. Ratio and Proportion | 比和比例
A ratio compares two or more quantities, showing how much of one thing there is compared to another. Ratios can be written in the form a : b or a : b : c. They can be simplified by dividing each part by a common factor. For example, the ratio 8 : 12 simplifies to 2 : 3 by dividing both sides by 4.
比用于比较两个或多个数量，显示一个事物相对于另一个有多少。比可以写成 a : b 或 a : b : c 的形式。它们可以通过将各部分除以公因数来化简。例如，比 8 : 12 通过两边除以4简化为 2 : 3。
hat total to find the value of one share, and multiply by each part. For example, to share £60 in the ratio 3 : 2, the total shares are 5. One share is £60 ÷ 5 = £12, so the amounts are 3 × £12 = £36 and 2 × £12 = £24.
要将一个量按给定比分配，先把比的各部分相加得到总份数。然后用总量除以总份数得到一份的值，再乘以各部分。例如，按 3 : 2 分配 £60，总份数是5，一份是 £60 ÷ 5 = £12，所以数量为 3 × £12 = £36 和 2 × £12 = £24。
Proportion describes a relationship where two quantities change at the same rate. If one quantity doubles, the other doubles too; this is direct proportion. You can use the unitary method or multiply by a scale factor to solve proportion problems.
比例描述了两个量以相同速率变化的关系。如果一个量翻倍，另一个也翻倍；这就是正比例。你可以使用归一法或乘以一个比例因子来解决比例问题。
4. Algebraic Expressions | 代数表达式
Algebra uses letters (variables) to represent unknown numbers. An expression is a combination of variables, numbers and operations, such as 3x + 2 or 5a - 7b. You should be able to collect like terms: for instance, 2x + 3x simplifies to 5x, and 4y - y + 2 simplifies to 3y + 2.
代数用字母（变量）表示未知数。表达式是变量、数字和运算的组合，例如 3x + 2 或 5a - 7b。你应该能够合并同类项：例如，2x + 3x 简化为 5x，4y - y + 2 简化为 3y + 2。
there is a negative sign outside, be careful with the signs inside: -2(3x - 5) = -6x + 10.
展开括号意味着将括号外的项乘以括号内的每一项。例如，4(x + 3) = 4x + 12。如果外面有负号，要注意里面的符号：-2(3x - 5) = -6x + 10。
Factorising is the reverse of expanding. You look for the highest common factor of the terms and write it outside brackets. For example, 6x + 9 factorises to 3(2x + 3). Being able to switch between expanded and factorised forms is a key algebra skill.
因式分解是展开的逆运算。你需要找出各项的最高公因数并写在括号外面。例如，6x + 9 因式分解为 3(2x + 3)。能够在展开式和分解式之间转换是关键的代数技能。
5. Linear Equations | 线性方程
r 8, you solve linear equations with one unknown. Start with one-step equations like x + 5 = 12, where x = 7, or 3x = 18, so x = 6.
方程表明两个表达式相等。解方程意味着找到使等式成立的变量的值。在八年级，你解含有一个未知数的线性方程。从一步方程开始，如 x + 5 = 12，得 x = 7；或 3x = 18，得 x = 6。
ns with brackets must be expanded first. Equations with unknowns on both sides require collecting like terms on one side.
两步方程涉及两种运算。例如，2x + 3 = 11：先将两边减3得到 2x = 8，然后除以2得 x = 4。有括号的方程必须先展开。未知数在等号两边的方程需要将同类项移到同一边。
Always check your solution by substituting it back into the original equation. A common mistake is forgetting to apply inverse operations to both sides equally.
始终通过将解代回原方程来检查。一个常见错误是忘记对两边同等应用逆运算。
6. Sequences | 数列
A sequence is a list of numbers that follow a pattern. Each number in the sequence is called a term. You need to recognise and continue sequences using a term-to-term rule, such as 'add 4' or 'multiply by 2'.
数列是一串按规律排列的数字。数列中的每个数字称为一项。你需要识别并使用项到项的规则延续数列，如“加4”或“乘以2”。
e written as an + b, where a is the common difference and b is the value when n = 0. For example, the sequence 5, 8, 11, 14... has a common difference of 3, so the nth term is 3n + 2. (Check: when n = 1, 3×1 + 2 = 5.)
数列的第n项给出了一个无需列出所有前项就能找到任一项的公式。对于线性等差数列，第n项可以写成 an + b，其中 a 是公差，b 是当 n = 0 时的值。例如，数列 5, 8, 11, 14... 的公差是3，所以第n项是 3n + 2。（检验：当 n = 1 时，3×1 + 2 = 5。）
You should also be able to find sequences from patterns of shapes and to solve problems involving sequences in real-world contexts.
你还应该能够从形状模式中找到数列，并解决涉及实际背景的数列问题。
7. Geometry: Angles and Shapes | 几何：角与形状
Angles are measured in degrees. Key angle facts include: angles on a straight line add up to 180°, angles around a point add up to 360°, and vertically opposite angles are equal. In a triangle, the sum of interior angles is always 180°; in a quadrilateral it is 360°.
角以度为单位。关键的角知识包括：直线上的角之和为180°，绕一点的角之和为360°，对顶角相等。三角形内角和始终为180°；四边形内角和为360°。
Parallel lines create special angle relationships. Alternate angles are equal, corresponding angles are equal, and co-interior (or allied) angles add up to 180°. You must be able to identify these and use them to find missing angles.
平行线产生特殊的角关系。内错角相等，同位角相等，同旁内角之和为180°。你必须能够识别这些关系并用它们求出缺失的角。
You will also work with properties of triangles (equilateral, isosceles, scalene) and quadrilaterals (square, rectangle, parallelogram, rhombus, trapezium, kite). Knowing their symmetry and side or angle properties helps in solving geometric problems.
你还将学习三角形（等边、等腰、不等边）和四边形（正方形、矩形、平行四边形、菱形、梯形、风筝形）的性质。了解它们的对称性以及边或角的性质有助于解决几何问题。
8. Perimeter, Area and Volume | 周长、面积和体积
or 2πr, where r is the radius and d is the diameter. π (pi) is approximately 3.14 or 22/7.
周长是二维形状一周的总距离。对于多边形，只需将所有边长相加。圆的周长 C = π × d 或 2πr，其中 r 是半径，d 是直径。π 约等于 3.14 或 22/7。
Area measures the space inside a shape. Common area formulas include:
面积测量形状内部的空间。常用面积公式包括：
| Shape | Area Formula |
|---|---|
| Rectangle | A = l × w |
| Triangle | A = ½ × b × h |
| Parallelogram | A = b × h |
| Trapezium | A = ½ (a + b) × h |
| Circle | A = π × r² |
. For a prism, volume is area of cross-section × length. Units are always cubed, e.g. cm³.
体积测量三维立体内部的空间。长方体的体积 V = 长 × 宽 × 高。对于棱柱，体积 = 横截面积 × 长度。单位总是立方，如 cm³。
9. Transformations | 变换
Transformations change the position or size of a shape on a coordinate grid. The four types you need to know are reflection, rotation, translation and enlargement.
变换改变形状在坐标格上的位置或大小。你需要知道的四种类型是反射、旋转、平移和放大。
n the opposite side. Common mirror lines include the x-axis, y-axis, and lines like y = x.
反射将形状翻转到镜线另一侧。反射后的形状与原来到镜线的距离相同，但在另一侧。常见的镜线包括 x 轴、y 轴以及直线 y = x 等。
A rotation turns a shape around a fixed centre of rotation by a given angle and direction (clockwise or anticlockwise). You need to be able to rotate shapes 90°, 180° or 270° about a point, using tracing paper or by counting squares.
旋转是让形状绕一个固定的旋转中心按给定角度和方向（顺时针或逆时针）转动。你需要能够绕一点将形状旋转 90°、180° 或 270°，可以使用描图纸或数方格的方法。
A translation slides a shape without turning or flipping. It is described by a column vector, for example (3/−2) meaning move 3 units right and 2 units down. The shape remains congruent (same size and shape).
平移是让形状滑动而不转动或翻转。它由列向量描述，例如 (3/−2) 表示向右移动3格，向下移动2格。形状保持全等（大小和形状相同）。
An enlargement changes the size of a shape by a scale factor from a centre of enlargement. If the scale factor is greater than 1, the shape gets bigger; between 0 and 1, it gets smaller. A negative scale factor also reverses direction.
放大通过一个放大中心按比例因子改变形状的大小。如果比例因子大于1，形状变大；比例因子在0和1之间则变小。负比例因子还会使方向反转。
10. Pythagoras' Theorem | 勾股定理
Pythagoras' theorem applies to right-angled triangles only. It states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. The formula is:
勾股定理仅适用于直角三角形。它表明斜边（最长边）的平方等于其他两条边的平方和。公式为：
a² + b² = c²
then take the square root. For example, if a = 3 cm and b = 4 cm, c² = 9 + 16 = 25, so c = √25 = 5 cm.
其中 c 是斜边，a 和 b 是较短的直角边。要计算斜边，将 a 和 b 的平方相加，然后开平方根。例如，如果 a = 3 cm，b = 4 cm，c² = 9 + 16 = 25，所以 c = √25 = 5 cm。
are root. You can use this to solve problems involving ladders leaning against walls, distances on maps, and many real-life situations.
要求直角边，需将公式变形：a² = c² - b²，然后开平方根。你可以用它解决如靠在墙上的梯子、地图上的距离以及许多现实情形的问题。
11. Statistics and Charts | 统计与图表
Statistics involves collecting, organising and interpreting data. The three common averages are mean, median and mode. The mean is found by adding all values and dividing by the number of values. The median is the middle value when data is ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values; it measures spread.
统计涉及收集、整理和解释数据。三种常见的平均数是均值、中位数和众数。均值通过将所有数值相加再除以数值的个数得出。中位数是数据排序后的中间值。众数是出现频率最高的数值。极差是最大值与最小值的差；它衡量数据的分散程度。
ies; the bars must have equal widths and gaps between them. Pie charts represent proportions of a whole; each sector' s over time. Scatter graphs display the relationship between two variables and may show positive, negative or no correlation.
数据可以多种方式展示。条形图显示类别的频数；条形的宽度必须相等且彼此之间有间隙。饼图表示整体的比例；每个扇区的角度计算为（频数 ÷ 总数）× 360°。折线图展示随时间变化的趋势。散点图显示两个变量之间的关系，可能呈现正相关、负相关或无相关。
12. Probability | 概率
Probability measures how likely an event is to happen. It is expressed as a number between 0 and 1, or as a fraction, decimal or percentage. 0 means impossible, 1 means certain. For a fair six-sided dice, the probability of rolling a 3 is 1/6.
概率衡量一个事件发生的可能性。它用一个0到1之间的数字表示，也可用分数、小数或百分数表示。0 表示不可能，1 表示必然发生。对于一个均匀的六面骰子，掷出3的概率是 1/6。
The probability of an event not happening is 1 − probability of the event happening. If the probability of rain is 0.3, the probability of no rain is 0.7. When all outcomes are equally likely, theoretical probability can be found by:
一个事件不发生的概率是 1 − 该事件发生的概率。如果下雨的概率是 0.3，不下雨的概率就是 0.7。当所有结果等可能时，理论概率可通过下式计算：
P(event) = number of favourable outcomes ÷ total number of possible outcomes
cy of event ÷ total number of trials. The more trials you do, the closer experimental probability generally gets to theoretical probability.
实验概率基于试验或调查的结果。它由以下方式得出：P(事件) = 事件发生的频数 ÷ 总试验次数。进行的试验次数越多，实验概率通常越接近理论概率。
You should be able to list all possible outcomes using a sample space (such as a table) and use this to find probabilities for combined events, like rolling two dice.
你应该能够使用样本空间（例如表格）列出所有可能的结果，并利用它来计算复合事件的概率，如掷两个骰子。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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