📚 Cambridge KS3 Mathematics: Comprehensive Topic Revision for Stages 7–9 | 剑桥 KS3 数学：阶段 7–9 全面主题复习
2.pdf. Each section below pairs key ideas with worked examples to help you revise efficiently.
本复习指南涵盖剑桥初中数学 KS3 阶段所需的核心技能。你可以在完成 p301_2.pdf 等混合练习页时将其作为快速参考。以下每一节都将关键知识点与示例配对，帮助你高效复习。
1. Number Operations and Place Value | 数的运算与位值
10² the digit 3 has a value of 300, not 3, because the power of ten moves the decimal point two places to the right.
理解位值、运算顺序和有向数在 KS3 阶段至关重要。例如，在 3.0 × 10² 中，数字 3 的值是 300，而不是 3，因为 10 的幂将小数点向右移动了两位。
- BIDMAS order: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
- 中文：BIDMAS 运算顺序：括号、指数、乘除、加减。
- Directed numbers: A negative sign in front of a bracket changes the sign of every term inside.
- 中文：有向数：括号前的负号会改变括号内每一项的符号。
a × (b + c) = a × b + a × c
2. Fractions, Decimals and Percentages | 分数、小数和百分比
etween fractions, decimals and percentages is a key KS3 skill. To compare 2/5, 0.41 and 43%, write them all as percentages. Since 2/5 = 40% and 0.41 = 41%, the order is 40% < 41% < 43%.
在分数、小数和百分比之间进行转换是 KS3 的关键技能。要比较 2/5、0.41 和 43%，把它们都写成百分比。因为 2/5 = 40%，0.41 = 41%，所以顺序是 40% < 41% < 43%。
- Fraction to decimal: Divide the numerator by the denominator.
- 中文：分数转小数：用分子除以分母。
- Decimal to percentage: Multiply by 100 and add the % sign.
- 中文：小数转百分比：乘以 100 并加上 % 符号。
2/5 = 0.4 = 40%
3. Ratio and Proportion | 比与比例
If the ratio of boys to girls is 3 : 2, the total number of parts is 3 + 2 = 5. To divide £40 in this ratio, one share is £40 ÷ 5 = £8, so boys get £24 and girls get £16.
比用于比较具有相同单位的数量。如果男生与女生的比是 3 : 2，那么总份数是 3 + 2 = 5。将 40 英镑按这个比例分配，一份是 40 ÷ 5 = 8 英镑，所以男生得 24 英镑，女生得 16 英镑。
- Divide in a ratio: Find the value of one share, then multiply by each ratio part.
- 中文：按比例分配：先求出一份的值，再乘以比的每一部分。
- Proportion: Direct proportion means both quantities increase or decrease at the same rate.
- 中文：比例：正比例意味着两个量以相同速率增加或减少。
part = (ratio part ÷ total parts) × total amount
4. Algebraic Expressions and Simplification | 代数表达式与化简
Algebra uses letters to stand for unknown numbers. Simplify expressions by collecting like terms. For example, 3a + 2b + 5a − b can be rearranged as 3a + 5a + 2b − b, which simplifies to 8a + b.
代数用字母表示未知数。通过合并同类项来化简表达式。例如，3a + 2b + 5a − b 可以整理为 3a + 5a + 2b − b，化简后得到 8a + b。
- Like terms: Terms with exactly the same letters and powers, such as 4x and −2x.
- 中文：同类项：具有完全相同的字母和指数的项，例如 4x 和 −2x。
- Expanding brackets: Multiply the term outside by every term inside.
- 中文：去括号：用括号外的项乘以括号内的每一项。
3a + 2b + 5a − b = 8a + b
5. Solving Linear Equations | 解一元一次方程
first subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to find x = 4.
解方程时，对方程两边进行相同的运算以分离未知数。例如，解方程 2x + 3 = 11，先两边同时减去 3，得到 2x = 8，然后两边同时除以 2，求出 x = 4。
- Check solutions: Substitute your answer back into the original equation.
- 中文：检验解：将你的答案代回原方程。
- Equations with brackets: Expand first, then simplify and solve.
- 中文：含括号的方程：先去括号，再化简并求解。
2x + 3 = 11 → 2x = 8 → x = 4
6. Sequences and the nth Term | 数列与第 n 项
A linear sequence has a constant difference between consecutive terms. The nth term formula allows you to find any term without listing all previous ones. For the sequence 5, 8, 11, 14, ... the common difference is 3, so the nth term is 3n + 2.
线性数列的相邻项差恒定。第 n 项公式使你不必列出前面所有项就能求出任意项。对于数列 5、8、11、14、……，公差是 3，所以第 n 项是 3n + 2。
- Common difference d: Subtract any term from the next term.
- 中文：公差 d：用后一项减去前一项。
- nth term: Use dn + (a − d), where a is the first term.
- 中文：第 n 项：使用 dn + (a − d)，其中 a 是首项。
nth term = dn + (a − d)
7. Angles and Properties of Shapes | 角度与图形性质
Know angle facts for points, lines, triangles and quadrilaterals. Angles on a straight line add to 180°, angles around a point add to 360°, and angles in a triangle add to 180°.
掌握点、线、三角形和四边形的角度性质。直线上的角相加为 180°，一点周围的角相加为 360°，三角形内角和为 180°。
- Parallel lines: Alternate angles are equal, corresponding angles are equal.
- 中文：平行线：内错角相等，同位角相等。
- Polygon angle sum: For an n-sided polygon, the interior angle sum is (n − 2) × 180°.
- 中文：多边形内角和：n 边形的内角和为 (n − 2) × 180°。
interior angle sum = (n − 2) × 180°
8. Perimeter, Area and Volume | 周长、面积与体积
Use the correct units and formulas for perimeter, area and volume. For compound shapes, split them into simpler parts and add or subtract the relevant areas.
使用正确的单位和公式计算周长、面积和体积。对于组合图形，将其拆分为更简单的部分，然后相加或相减相应的面积。
- Trapezium area: Half the sum of the parallel sides times the height.
- 中文：梯形面积：两底之和的一半乘以高。
- Cuboid volume: Length times width times height.
- 中文：长方体体积：长乘以宽乘以高。
Area of trapezium = ½ × (a + b) × h
9. Transformations | 变换
Transformations move or change shapes. Reflection, rotation, translation and enlargement each have specific rules. A translation is described by a column vector showing horizontal and vertical movement.
变换使图形移动或改变。反射、旋转、平移和放大各有特定规则。平移用列向量表示水平和垂直移动。
- Reflection: Each point is the same perpendicular distance from the mirror line.
- 中文：反射：每个点到镜线的垂直距离相等。
- Enlargement: A scale factor k multiplies each side length by k.
- 中文：放大：比例系数 k 使每条边长度乘以 k。
scale factor k → new length = k × original length
10. Statistics and Probability | 统计与概率
Collect, represent and interpret data using bar charts, pie charts and frequency tables. Probability can be expressed as a fraction, decimal or percentage. The probabilities of all possible outcomes add to 1.
使用条形图、饼图和频数表收集、表示和解释数据。概率可以表示为分数、小数或百分比。所有可能结果的概率之和为 1。
- Mean: Sum of values divided by the number of values.
- 中文：平均数：所有数值之和除以数值的个数。
- Probability: Favourable outcomes divided by total outcomes.
- 中文：概率：有利结果数除以总结果数。
Probability = favourable outcomes ÷ total outcomes
11. Coordinates and Linear Graphs | 坐标与线性图
intercept. The gradient m measures steepness, while c is where the line crosses the y-axis.
在四个象限中描点并绘制直线图。方程 y = mx + c 给出斜率和 y 轴截距。斜率 m 衡量倾斜程度，而 c 是直线与 y 轴的交点。
- Gradient: Rise over run, or change in y divided by change in x.
- 中文：斜率：纵向变化量除以横向变化量。
- Parallel lines: Lines with the same gradient are parallel.
- 中文：平行线：斜率相同的直线互相平行。
gradient m = Δy ÷ Δx
12. Problem Solving and Reasoning | 问题解决与推理
mixed practice such as p301_2.pdf, you may need to combine several topics in one question.
将你的技能应用于多步骤问题。仔细读题，确定已知量和未知量，选择策略并检验答案。在 p301_2.pdf 等混合练习中，你可能需要在一道题中综合多个主题。
- Estimate first: Use estimation and inverse operations to check results.
- 中文：先估算：使用估算和逆运算检验结果。
- Show working: Clear steps earn method marks even if the final answer is wrong.
- 中文：展示过程：即使最终答案错误，清晰的步骤也能获得方法分。
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