📚 Year 8 CIE Mathematics: Core Topics Overview | Year 8 CIE 数学：核心知识点梳理
Welcome to a comprehensive overview of the core topics in Year 8 CIE Mathematics. This guide breaks down each major area of the curriculum, highlighting key concepts, methods, and examples. Whether you are revising for an end-of-year exam or strengthening your essential skills, these notes will help you build a solid foundation for future studies.
欢迎阅读 Year 8 CIE 数学核心知识点梳理。本文分解了课程中的每一个重要领域，重点介绍关键概念、解题方法和实例。无论你是在准备年终考试还是巩固基本功，这些笔记都将帮你为以后的学习打下坚实的基础。
1. Number Skills | 数字技能
Number skills form the backbone of Year 8 mathematics. You must be confident with integers, order of operations, factors, multiples, primes, and indices. Always follow BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) to evaluate expressions correctly.
数字技能是 Year 8 数学的基础。你需要熟练掌握整数、运算顺序、因数、倍数、质数和指数。计算表达式时务必遵循 BIDMAS 法则（括号、指数、乘除、加减）。
A prime number has exactly two distinct factors, 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13. Remember that 1 is not prime. The highest common factor (HCF) and lowest common multiple (LCM) are found using prime factorisation or listing methods.
质数有且只有两个不同的因数，即 1 和它本身。前几个质数是 2、3、5、7、11、13。要记住 1 不是质数。求最大公因数（HCF）和最小公倍数（LCM）可以使用质因数分解法或列举法。
Indices tell us how many times a base number is multiplied by itself. For example, 2³ means 2 × 2 × 2 = 8. You should also handle negative integer indices and the zero index: any non-zero number to the power of 0 equals 1.
指数表示底数自乘的次数。例如 2³ 表示 2 × 2 × 2 = 8。你还需要处理负整数指数和零指数：任何非零数的 0 次幂都等于 1。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Understanding the links between fractions, decimals and percentages is crucial. A fraction like ¾ is equivalent to 0.75, which is 75%. To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100.
理解分数、小数与百分比之间的联系至关重要。¾ 这种分数等于 0.75，也就是 75%。把分数化成小数时，用分子除以分母；把小数化成百分比时，乘以 100。
When adding or subtracting fractions, first find a common denominator. For multiplication, multiply numerators together and denominators together. Division is performed by multiplying by the reciprocal. Always simplify your answers to the lowest terms.
加减分数时，先通分，用公分母进行计算。乘法直接用分子相乘、分母相乘。除法通过乘以倒数来完成。最后记得把答案约分成最简分数。
In Year 8 you also work with percentage increase and decrease. To calculate 15% of 200, multiply 0.15 × 200 = 30. A price reduced by 20% becomes 80% of the original amount. Reverse percentages help find the original quantity before a change.
在 Year 8 你还会学习百分比增减问题。例如，200 的 15% 就用 0.15 × 200 = 30。降价 20% 后，价格变为原价的 80%。逆向百分比则用来求变化前的初始量。
3. Ratio and Proportion | 比和比例
Ratio compares two or more quantities. If a fruit bowl has apples and bananas in the ratio 3 : 5, for every 3 apples there are 5 bananas. Ratios can be simplified by dividing each part by a common factor, just like fractions.
比用来比较两个或多个量。如果水果碗里苹果与香蕉的比是 3 : 5，就意味着每 3 个苹果对应 5 个香蕉。和分数一样，比可以通过除以公因数来进行化简。
Proportion problems often involve direct and inverse relationships. In direct proportion, as one quantity doubles, the other doubles. In inverse proportion, one quantity increases while the other decreases at the same rate. The unitary method is a powerful tool for solving proportion questions.
比例问题常涉及正比和反比关系。正比例中，一个量翻倍时，另一个量也跟着翻倍。反比例中，一个量增大，另一个量以同样速率减小。单位法（先求一个单位的值）是解决比例问题的利器。
Sharing in a given ratio is a key skill. To share £120 in the ratio 2:3, first add the parts to get 5 parts total, then each part is £120 ÷ 5 = £24, so the shares are 2 × £24 = £48 and 3 × £24 = £72.
按给定比例分配是一项关键技能。将 120 英镑按 2:3 分配，先算总份数 2+3=5 份，每份是 £120 ÷ 5 = £24，再算出两笔金额分别是 2 × £24 = £48 和 3 × £24 = £72。
4. Algebraic Expressions | 代数表达式
In algebra, letters represent unknown numbers. The expression 3a + 5a simplifies to 8a because they are like terms. Terms with different variable letters or powers, such as 3x and 3x², cannot be added together.
代数学里，字母表示未知数。表达式 3a + 5a 可以简化为 8a，因为它们是同类项。字母或指数不同的项，比如 3x 和 3x²，则不能直接相加。
Expanding brackets uses the distributive law: a(b + c) = ab + ac. For example, 4(2x + 3) = 8x + 12. When a negative sign is outside the bracket, remember to reverse the sign of every term inside.
去括号使用乘法对加法的分配律：a(b + c) = ab + ac。例如 4(2x + 3) = 8x + 12。如果括号外面有负号，记得括号内每一项都要变号。
Factorising is the reverse of expanding. Take out the highest common factor of all terms. The expression 6x + 9 factorises to 3(2x + 3). Always check your factorisation by expanding back to the original expression.
因式分解是去括号的逆过程。找出所有项的公因式并将其提到括号外。6x + 9 分解成 3(2x + 3)。分解后务必展开回原来形式进行检查。
5. Equations and Sequences | 方程与数列
Solving equations means finding the unknown value that makes the statement true. Use inverse operations to isolate the variable. For 2x + 5 = 13, subtract 5 from both sides to get 2x = 8, then divide by 2 to find x = 4.
解方程就是求出使等式成立的未知数值。使用逆运算将变量单独放在等式一边。对于 2x + 5 = 13，两边先减去 5 得到 2x = 8，再同时除以 2，解得 x = 4。
Equations with unknowns on both sides require collecting like terms. For 5x – 2 = 3x + 6, subtract 3x from both sides to get 2x – 2 = 6, then solve. Always keep the equation balanced.
未知数出现在等式两边的方程需要先合并同类项。例如 5x – 2 = 3x + 6，两边同减 3x 得到 2x – 2 = 6，然后再求解。操作时要始终保持等式平衡。
Sequences follow a rule to generate terms. A linear sequence has a constant difference between terms. For the sequence 3, 7, 11, 15… the term-to-term rule is ‘add 4’, and the position-to-term rule (nth term) is 4n – 1.
数列按照一定的规则生成各项。线性数列中，相邻两项的差为常数。对于数列 3, 7, 11, 15…，项间递推关系是 “加 4”，用位置 n 表示的通项公式则为 4n – 1。
6. Geometry: Angles and Shapes | 几何：角与图形
Angles are measured in degrees. A right angle is 90°, an acute angle is less than 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180°. The sum of angles on a straight line is 180°, and angles around a point add to 360°.
角的大小以度为单位。直角等于 90°，锐角小于 90°，钝角介于 90° 到 180° 之间，优角大于 180°。直线上的角之和为 180°，围绕一个点的角之和为 360°。
When parallel lines are crossed by a transversal, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°. These facts help you find unknown angles in diagrams without measuring.
当平行线被一条横截线所截时，同位角相等，内错角相等，同旁内角之和为 180°。利用这些性质，不用量角器就能在图中求出未知角。
Triangles are classified by sides and angles. The sum of interior angles in any triangle is 180°. An equilateral triangle has three 60° angles, an isosceles triangle has two equal base angles, and a right-angled triangle has one 90° angle.
三角形可按照边和角进行分类。任何三角形的内角和都是 180°。等边三角形的三个角都是 60°，等腰三角形有两个相等的底角，直角三角形则有一个 90° 角。
7. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around a shape. For a rectangle, P = 2(l + w). Area measures the surface covered. Common formulas: rectangle area A = l × w, triangle area A = ½ × b × h, parallelogram area A = b × h, and trapezium area A = ½ (a + b) h.
周长是图形一周的总长度。矩形周长为 P = 2(l + w)。面积测量的是表面覆盖的大小。常用公式有：矩形面积 A = l × w，三角形面积 A = ½ × b × h，平行四边形面积 A = b × h，梯形面积 A = ½ (a + b) h。
Compound shapes are split into simpler figures to find total area. Remember to use consistent units; if lengths are in cm, area is in cm². For circles, circumference C = 2πr and area A = πr², where π ≈ 3.14.
复合图形可以拆分成简单图形来求总面积。注意单位要一致，若长度使用 cm，面积单位就是 cm²。对于圆，周长 C = 2πr，面积 A = πr²，其中 π 约等于 3.14。
Volume measures the space inside a 3D shape. A cuboid’s volume V = l × w × h, and is given in cubic units. The surface area of a solid is the total area of all its faces. Prism volume equals area of cross-section × length.
体积用来度量立体图形内部空间的大小。长方体的体积 V = l × w × h，用立方单位表示。立体图形的表面积是所有面的面积总和。棱柱的体积等于横截面积乘以长度。
8. Transformations and Symmetry | 图形变换与对称
Transformations map a shape to a new position. In a translation, every point moves the same distance in the same direction, described by a column vector. Reflection flips a shape over a mirror line, keeping corresponding points equidistant from the mirror.
图形变换将图形移动到新的位置。平移变换中，每个点都向同一方向移动相同的距离，用列向量描述。反射变换将图形沿对称轴翻转，对应点到对称轴的距离相等。
Rotation turns a shape around a centre of rotation by a given angle and direction (clockwise or anticlockwise). You must state the centre, angle, and direction. Enlargement changes the size of a shape by a scale factor from a centre of enlargement, preserving shape but not length.
旋转变换是图形绕某个旋转中心，按给定的角度和方向（顺时针或逆时针）转动。描述旋转时必须给出中心、角度和方向。位似放大是以放大中心为基准，按比例因子改变图形的大小，形状不变但长度改变。
Symmetry is closely linked. A shape has line symmetry if it can be folded along a line so both halves match. Rotational symmetry occurs if a shape looks the same when rotated through less than 360° about its centre. The order of rotational symmetry tells how many times it matches in a full turn.
对称性与图形变换关系密切。若一个图形沿着某条直线对折后两部分完全重合，它就具有该线对称。如果图形绕其中心旋转小于 360° 后能与自身重合，它就具有旋转对称。旋转对称的阶数等于旋转一周中与自身重合的次数。
9. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides: c² = a² + b², where c is the longest side opposite the right angle.
勾股定理适用于直角三角形。定理描述为：斜边的平方等于另外两条直角边的平方和，即 c² = a² + b²，其中 c 是直角所对的最长边（斜边）。
To find a missing hypotenuse, square the two shorter sides, add them, and take the square root. For example, if a = 3 cm and b = 4 cm, then c = √(3² + 4²) = √25 = 5 cm. To find a shorter side, rearrange the formula: a² = c² – b².
求未知斜边时，将两条直角边的长度分别平方、相加，再开平方根。例如，已知 a = 3 cm，b = 4 cm，则 c = √(3² + 4²) = √25 = 5 cm。求直角边时，将公式变形为 a² = c² – b²。
Pythagoras’ theorem is widely used to solve problems involving ladders against walls, distances between points on a grid, and determining whether a triangle is right-angled. Always check that the units are the same before calculating.
勾股定理广泛应用于解决梯子靠墙问题、网格上两点间的距离问题，以及判断一个三角形是否为直角三角形。计算前要确保所有单位统一。
10. Statistics and Probability | 统计与概率
Statistics involves collecting, organising, and interpreting data. The mean is calculated 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, and the range is the difference between the largest and smallest values.
统计涉及数据的收集、整理和解读。平均数（均值）等于所有数据之和除以数据个数。中位数是将数据排序后最中间的值。众数是出现次数最多的值，极差则是最大值与最小值之差。
Data can be displayed in bar charts, pie charts, and line graphs. A bar chart shows frequencies for discrete categories, while a stem-and-leaf diagram keeps original data visible. You should be able to read information from these charts and construct them neatly.
数据可以用条形图、饼图和折线图来展示。条形图显示离散类别的频数，茎叶图则能同时保留原始数据。你需要能够从这些图表中读取信息，并整洁地绘制它们。
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). The probability of a fair coin landing heads is ½. For equally likely outcomes, probability = number of favourable outcomes / total number of outcomes.
概率用来衡量事件发生的可能性大小，取值范围从 0（不可能）到 1（必然）。抛一枚均匀硬币出现正面的概率是 ½。当所有结果等可能时，概率 = 有利结果数 / 所有可能结果总数。
In Year 8 you also explore experimental probability and the idea that relative frequency approaches theoretical probability with many trials. Probability trees are introduced for two successive events, helping to find probabilities of combined events.
在 Year 8，你还会探索实验概率，并理解随着试验次数增多，相对频数会趋近理论概率。对于两个连续事件，将引入概率树形图，用来计算复合事件的概率。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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