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

📚 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

This revision guide covers the core skills you need for Cambridge Lower Secondary Mathematics at KS3. Use it as a quick reference when working through mixed practice pages such as p301_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

Understanding place value, order of operations and directed numbers is essential at KS3. For example, in 3.0 × 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

Converting between 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

Ratios compare quantities sharing a common unit. 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

To solve an equation, do the same operation to both sides to isolate the unknown. For example, to solve 2x + 3 = 11, 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

Plot points in all four quadrants and draw straight-line graphs. The equation y = mx + c gives the gradient and y-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

Apply your skills to multi-step problems. Read carefully, identify knowns and unknowns, choose a strategy and check your answer. In 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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