📚 Complete Mathematics for Cambridge Secondary 1 Book 3: Key Concepts Explained | 剑桥初中数学第三册核心知识点精讲
The Complete Mathematics for Cambridge Secondary 1 Book 3 builds a strong foundation for IGCSE and beyond. This article distils the most important topics – from integers and algebra to geometry and probability – into clear, bilingual explanations. Work through each section to master essential skills and boost your confidence in problem‑solving.
《剑桥初中数学第三册》为 IGCSE 及更高阶段的学习打下坚实基础。本文将最重要的知识点——从整数和代数到几何和概率——提炼为清晰的双语解析。跟随每个章节掌握核心技能，提升解题信心。
1. Integers, Powers and Roots | 整数、幂与方根
Integers are the set of whole numbers and their negatives. When multiplying or dividing, remember: same signs give a positive result, different signs give a negative result. Powers, written as aⁿ, mean ‘a’ multiplied by itself ‘n’ times. Square roots, denoted by √, are the inverse of squaring; every positive number has two square roots (positive and negative). Cube roots find the number that was cubed.
整数包括正整数、负整数和零。乘除运算时记住：同号得正，异号得负。幂记作 aⁿ，表示 a 自乘 n 次。平方根用 √ 表示，是平方的逆运算；每个正数有两个平方根（一正一负）。立方根求的是被立方的数。
Example: (−3)² = 9, so √9 = ±3; 5³ = 125, ∛125 = 5
示例：(−3)² = 9，因此 √9 = ±3；5³ = 125，∛125 = 5
2. Fractions and Decimals | 分数与小数
Fractions express a part of a whole. Equivalent fractions have the same value but different numerators and denominators. To add or subtract fractions, first write them with a common denominator. Multiply fractions by multiplying numerators together and denominators together; to divide, multiply by the reciprocal. Decimals use place value (tenths, hundredths, thousandths). Recurring decimals can be converted to fractions using algebra.
分数表示整体的一部分。等值分数数值相同但分子分母不同。加减分数要先通分。分数相乘时分子相乘分母相乘；相除则乘以倒数。小数利用位值（十分位、百分位、千分位）。循环小数可通过代数方法转化为分数。
| Fraction | Decimal | Percentage |
|---|---|---|
| ¾ | 0.75 | 75% |
| ⅔ | 0.6̇ | 66.7% |
3. Percentages | 百分比
A percentage is a fraction with denominator 100. To find a percentage of a quantity, multiply by the percentage and divide by 100. Percentage increase or decrease is (change ÷ original amount) × 100%. Reverse percentage problems, where the final amount after an increase or decrease is known, require dividing by the multiplier (e.g. 1.15 for a 15% increase).
百分比是分母为 100 的分数。求一个数量的百分之几，先乘以百分数再除以 100。增减百分率用 (变化量 ÷ 原量) × 100% 计算。已知增减后结果求原值（逆向百分比），需除以乘数（如增长 15% 则乘数为 1.15）。
If a jacket costs £60 after a 25% discount, original price = 60 ÷ 0.75 = £80.
若一件夹克打七五折后售价 60 英镑，原价 = 60 ÷ 0.75 = 80 英镑。
4. Ratio and Proportion | 比与比例
Ratio compares quantities of the same kind. It can be simplified like fractions. Proportion tells us if two ratios are equal. Direct proportion means both quantities increase at the same rate (y = kx). Inverse proportion means one increases while the other decreases (y = k/x). Use the unitary method or cross multiplication to solve problems.
比是比较同种量的大小，可像分数一样化简。比例说明两个比是否相等。正比例表示两个量以相同速率增加 (y = kx)。反比例表示一个量增加另一个减少 (y = k/x)。可用归一法或交叉相乘法解题。
Example: 3 pens cost £2.10; 7 pens cost (2.10 ÷ 3) × 7 = £4.90.
示例：3 支笔 2.10 英镑；7 支笔花费 (2.10 ÷ 3) × 7 = 4.90 英镑。
5. Algebraic Expressions and Formulae | 代数表达式与公式
An algebraic expression contains numbers, variables, and operations. Like terms (e.g. 5x and 3x) can be combined. Expand brackets by multiplying each term inside by the term outside. Factorising is the reverse – writing an expression as a product of its factors. A formula expresses a relationship, like v = u + at. Substituting values into a formula yields a specific result.
代数表达式包含数字、变量和运算符号。同类项（如 5x 与 3x）可以合并。去括号时用括号外的项乘以括号内各项。因式分解是逆过程——把表达式写成因式乘积。公式表达关系，如 v = u + at。将数值代入公式可求得结果。
Expand: 2(3x + 4) = 6x + 8 Factorise: 15y − 10 = 5(3y − 2)
展开：2(3x + 4) = 6x + 8 因式分解：15y − 10 = 5(3y − 2)
6. Linear Equations | 一次方程
A linear equation has the variable raised to the power 1. Solve by isolating the variable using inverse operations. Always maintain balance: whatever you do to one side, do to the other. Equations with brackets should be expanded first. Equations with fractions are cleared by multiplying both sides by the LCM of the denominators.
一次方程中变量的最高次幂为 1。通过逆运算把变量分离出来求解。始终保持平衡：对方程一边做什么操作，另一边也要同样操作。有括号时先展开。含分数的方程，两边同乘各分母的最小公倍数以消去分母。
Solve: (x + 3)/2 = 5 → x + 3 = 10 → x = 7.
解方程：(x + 3)/2 = 5 → x + 3 = 10 → x = 7。
7. Sequences | 数列
A sequence is a list of numbers following a rule. An arithmetic sequence has a common difference (d). The nᵗʰ term of an arithmetic sequence is given by aₙ = a₁ + (n − 1)d, where a₁ is the first term. Recognising patterns in squares, cubes, and triangular numbers is also important. You can generate terms using the term‑to‑term rule or the position‑to‑term rule.
数列是按某种规律排列的一列数。等差数列有公差 (d)。其第 n 项公式为 aₙ = a₁ + (n − 1)d，其中 a₁ 为首项。掌握平方数、立方数和三角形数的模式也很重要。可以通过项间递推规则或通项公式生成各项。
Sequence: 4, 9, 14, 19, … a₁ = 4, d = 5 → nᵗʰ term = 5n − 1.
数列：4, 9, 14, 19, … 首项 = 4，公差 = 5 → 第 n 项 = 5n − 1。
8. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°, angles around a point sum to 360°. Vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and co‑interior angles sum to 180°. The sum of interior angles of an n‑sided polygon is (n − 2) × 180°. Each exterior angle of a regular n‑gon is 360°/n.
直线上的角之和为 180°，绕一点一周的角之和为 360°。对顶角相等。平行线中，内错角相等、同位角相等、同旁内角互补（和为 180°）。n 边形内角和为 (n − 2) × 180°。正 n 边形的每个外角为 360°/n。
For a regular pentagon: interior angle = 108°, exterior angle = 72°.
正五边形：内角 = 108°，外角 = 72°。
9. Perimeter and Area | 周长与面积
Perimeter is the distance around a shape. Circumference of a circle: C = 2πr or πd. Area formulae: rectangle A = lw, triangle A = ½bh, parallelogram A = bh, trapezium A = ½(a + b)h, circle A = πr². Composite shapes can be broken into simpler ones to find total area.
周长是形状边缘的总长度。圆的周长：C = 2πr 或 πd。面积公式：矩形 A = lw，三角形 A = ½bh，平行四边形 A = bh，梯形 A = ½(a + b)h，圆 A = πr²。组合图形可分割成简单图形求总面积。
Find area of a circle with radius 7 cm: A = π × 7² ≈ 154 cm² (using π ≈ 22/7).
求半径为 7 cm 的圆面积：A = π × 7² ≈ 154 cm² (取 π ≈ 22/7)。
10. Volume and Surface Area | 体积与表面积
Volume of a prism = area of cross‑section × length. For a cuboid, V = l × w × h. Surface area is the total area of all faces. Cylinder: volume = πr²h, curved surface area = 2πrh, total surface area = 2πr(r + h). Units must be consistent – converting between cm³, m³, and litres is a key skill.
棱柱体积 = 横截面积 × 长。长方体体积 V = l × w × h。表面积是所有面的总面积。圆柱：体积 = πr²h，侧面积 = 2πrh，总表面积 = 2πr(r + h)。单位必须一致——在 cm³、m³ 和升之间换算是关键技能。
Cube of side 5 cm: volume = 125 cm³, surface area = 150 cm².
边长为 5 cm 的立方体：体积 = 125 cm³，表面积 = 150 cm²。
11. Statistics | 统计
Key averages: mean (sum ÷ count), median (middle value when ordered), mode (most frequent). Range = maximum − minimum. Frequency tables help organise data; from a grouped frequency table, the modal class is the interval with highest frequency. Scatter graphs show correlation; a line of best fit can estimate values (interpolation).
主要平均数：均值（总和 ÷ 个数）、中位数（排序后中间值）、众数（出现次数最多）。极差 = 最大值 − 最小值。频数表有助于整理数据；在分组频数表中，众数所在组是频数最高的区间。散点图显示相关性；最佳拟合线可估算数值（内插法）。
Data: 3, 5, 5, 7, 9. Mean = 5.8, median = 5, mode = 5, range = 6.
数据：3, 5, 5, 7, 9。均值 = 5.8，中位数 = 5，众数 = 5，极差 = 6。
12. Probability | 概率
Probability of an event = number of favourable outcomes ÷ total number of equally likely outcomes. It is always between 0 and 1 (inclusive). For combined events, sample space diagrams or tree diagrams help enumerate possibilities. The sum of probabilities of all possible outcomes is 1. Expected frequency = probability × number of trials.
事件的概率 = 有利结果数 ÷ 所有等可能结果总数。概率值总是在 0 到 1 之间（含边界）。对于复合事件，可用样本空间图或树状图列出所有可能。所有可能结果的概率之和为 1。期望频数 = 概率 × 试验次数。
Rolling a fair die: P(even) = 3/6 = ½. In 300 rolls, expect about 150 even numbers.
掷一枚均匀骰子：P(偶数) = 3/6 = ½。掷 300 次，期望出现约 150 次偶数。
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