📚 KS3 Maths Essentials: Compressed Revision of Year 8 Core Topics | KS3数学精华：八年级核心知识点压缩精讲
Mastering KS3 Mathematics requires a solid understanding of the foundational topics introduced in Year 8. This compressed revision guide covers the essential concepts from integers to probability, ensuring you are confident with operations, algebra, geometry and data handling. Each section presents clear explanations and worked examples to reinforce your learning, directly aligned with the UK National Curriculum for Key Stage 3.
精通KS3数学需要牢牢掌握八年级引入的基础主题。这份压缩复习指南涵盖了从整数到概率的核心概念，确保你能够轻松应对运算、代数、几何和数据处理。每个部分都提供清晰的解释和解析实例，以便巩固学习，完全符合英国国家课程关键阶段3的要求。
1. Integers and Order of Operations | 整数与运算法则
Integers include positive and negative whole numbers and zero. Adding a negative number is equivalent to subtraction: 5 + (-3) = 5 – 3 = 2. Subtracting a negative number becomes addition: 4 – (-2) = 4 + 2 = 6.
整数包括正整数、负整数和零。加上一个负数等同于减法：5 + (-3) = 5 – 3 = 2。减去一个负数变为加法：4 – (-2) = 4 + 2 = 6。
Multiplication and division with negatives follow a simple rule: same signs give a positive, different signs give a negative. For example, (-3) × (-4) = 12, while (-6) ÷ 2 = -3.
负数的乘法和除法遵循简单规则：同号得正，异号得负。例如，(-3) × (-4) = 12，而 (-6) ÷ 2 = -3。
The order of operations is remembered by BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction). Always work from left to right for equal priority operations. Evaluate: 3 + 4 × 2 = 3 + 8 = 11, not 7 × 2 = 14.
运算法则由BIDMAS (括号、指数、除法/乘法、加法/减法) 记忆。同等优先级的运算从左到右进行。计算：3 + 4 × 2 = 3 + 8 = 11，而不是7 × 2 = 14。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
To add or subtract fractions, they must have a common denominator. For 1/3 + 1/4, use denominator 12: 4/12 + 3/12 = 7/12.
加减分数时，需要有共同的分母。对于1/3 + 1/4，使用分母12：4/12 + 3/12 = 7/12。
Multiplying fractions: multiply numerators together, denominators together. 2/5 × 3/7 = 6/35. Dividing by a fraction: multiply by its reciprocal. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
分数乘法：分子相乘，分母相乘。2/5 × 3/7 = 6/35。除以一个分数等于乘以其倒数。3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。
Convert fractions to decimals by division, and to percentages by multiplying by 100. 3/8 = 0.375 = 37.5%. Common equivalents: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%.
分数转为小数用除法，转为百分数乘以100。3/8 = 0.375 = 37.5%。常见等价：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%。
| Fraction | Decimal | Percentage |
| 1/10 | 0.1 | 10% |
| 1/5 | 0.2 | 20% |
| 3/4 | 0.75 | 75% |
3. Ratio and Proportion | 比与比例
A ratio compares quantities. If a class has 12 boys and 8 girls, the ratio of boys to girls is 12:8, which simplifies to 3:2 by dividing both numbers by 4.
比用来比较数量。如果一个班有12名男生和8名女生，男生与女生的比是12:8，将两个数都除以4化简为3:2。
Proportion means parts of a whole. With ratio 3:2, the total parts = 5. So the proportion of boys is 3/5 and girls is 2/5.
比例指的是整体中的部分。比3:2，总份数=5。因此男生的比例是3/5，女生是2/5。
Direct proportion: as one quantity increases, the other increases at the same rate. If 5 apples cost £2, 10 apples cost £4.
正比：一个量增加，另一个量以相同速率增加。如果5个苹果2英镑，10个苹果4英镑。
4. Introduction to Algebra | 代数入门
Algebra uses letters to represent unknown numbers. An expression like 3a + 2b combines terms. Only like terms can be simplified: 5x + 2y – 2x = 3x + 2y.
代数使用字母表示未知数。表达式如3a + 2b组合各项。只有同类项可以化简：5x + 2y – 2x = 3x + 2y。
Substitution: replace letters with given values. If a=3, b=4, then 2a + 5b = 2×3 + 5×4 = 6+20=26.
代入法：用给定值替换字母。若a=3, b=4，那么2a + 5b = 2×3 + 5×4 = 6+20=26。
Expanding brackets: multiply each term inside by the term outside. 3(x+4) = 3x + 12. 2(3y-5) = 6y – 10.
展开括号：括号外项乘以括号内每一项。3(x+4) = 3x + 12。2(3y-5) = 6y – 10。
5. Solving Linear Equations | 解线性方程
An equation shows two expressions are equal. To solve, isolate the variable by performing inverse operations. Solve x + 7 = 15: subtract 7 from both sides → x = 8.
方程表示两个表达式相等。求解时，通过逆运算分离变量。解 x + 7 = 15：两边减7 → x = 8。
For 3x – 4 = 11, first add 4: 3x = 15, then divide by 3: x = 5.
对于3x – 4 = 11，先加4：3x = 15，然后除以3：x = 5。
Equations with variables on both sides: 5x – 2 = 2x + 7. Subtract 2x: 3x – 2 = 7. Add 2: 3x = 9, x = 3.
两边都有变量的方程：5x – 2 = 2x + 7。减去2x：3x – 2 = 7。加2：3x = 9, x = 3。
6. Angles and Parallel Lines | 角与平行线
Angles are measured in degrees. Acute angles < 90°, right angle = 90°, obtuse between 90° and 180°, reflex > 180°.
角度以度计量。锐角 < 90°，直角 = 90°，钝角在90°到180°之间，优角 > 180°。
On a straight line, angles sum to 180°. Vertically opposite angles are equal. Around a point sum to 360°.
在直线上，角度和为180°。对顶角相等。围绕一点的角度和为360°。
When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, co-interior (allied) angles sum to 180°. If ∠a is 70°, corresponding angle is 70°, alternate is 70°, co-interior is 110°.
当一条横截线切割平行线：同位角相等，内错角相等，同旁内角（共内角）和为180°。若∠a = 70°，同位角=70°，内错角=70°，同旁内角=110°。
7. Perimeter, Area and Volume | 周长、面积与体积
Perimeter of a rectangle: 2(l + w). Area of rectangle: l × w. Area of triangle: ½ × base × height.
长方形周长：2(长+宽)。长方形面积：长 × 宽。三角形面积：½ × 底 × 高。
Area of parallelogram: base × perpendicular height. Trapezium: ½(a + b)h, where a and b are parallel sides.
平行四边形面积：底 × 垂直高。梯形面积：½(a + b)h，其中a和b是平行边。
Volume of cuboid: length × width × height. Units: cm³, m³. Surface area: sum of areas of all faces. For circles: Circumference = 2πr, Area = πr² (π ≈ 3.14).
长方体体积：长 × 宽 × 高。单位：立方厘米、立方米。表面积：所有面的面积之和。对于圆：周长 = 2πr，面积 = πr² (π ≈ 3.14)。
8. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数与极差
Mean is the average: sum of values ÷ number of values. For data set 4, 7, 10, 10, 12: sum = 43, count = 5, mean = 8.6.
平均数是均值：数值总和 ÷ 数值个数。数据集4, 7, 10, 10, 12：总和=43，个数=5，平均数=8.6。
Median is the middle value when ordered. For 4, 7, 10, 10, 12, median = 10. If even count, median = average of two middle numbers.
中位数是排序后的中间值。对于4, 7, 10, 10, 12，中位数=10。如果个数为偶数，中位数取中间两个数的平均值。
Mode is the most frequent value (10 appears twice). Range = maximum – minimum = 12 – 4 = 8.
众数是出现最频繁的值（10出现两次）。极差 = 最大值 – 最小值 = 12 – 4 = 8。
9. Probability Basics | 概率基础
Probability is a number between 0 and 1, where 0 means impossible and 1 means certain. Probability = number of favourable outcomes / total number of outcomes.
概率是介于0和1之间的数，0表示不可能，1表示必然。概率 = 有利结果数 / 总结果数。
When rolling a fair six-sided die, probability of rolling a 3 = 1/6. Probability of an even number = 3/6 = 1/2.
掷一个均匀的六面骰子，掷出3的概率 = 1/6。掷出偶数的概率 = 3/6 = 1/2。
Complementary events: The probability of not A = 1 – P(A). If chance of rain is 0.3, chance of no rain is 0.7.
互补事件：非A的概率 = 1 – P(A)。如果下雨概率是0.3，不下雨概率是0.7。
10. Sequences and nth Term | 数列与第n项
A sequence follows a rule. In 3, 7, 11, 15, … the term-to-term rule is ‘add 4’. The position-to-term rule (nth term) = 4n – 1. Check: n=1 gives 4(1)-1=3, n=2 gives 7, etc.
数列遵循一定规则。在3, 7, 11, 15, …中，逐项规则是’加4’。位置到项的规则（第n项）= 4n – 1。验证：n=1得4(1)-1=3，n=2得7，等等。
For descending sequences: 10, 7, 4, 1, … common difference = -3, nth term = -3n + 13. Find the 20th term: -3×20+13 = -47.
对于递减数列：10, 7, 4, 1, …，公差 =
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