📚 Complete Mathematics for Cambridge Secondary 1 Book 2: Key Concepts Explained | 《剑桥初中数学第二册》知识点精讲
This guide covers the essential topics from Book 2 of the Cambridge Secondary 1 Mathematics series. Designed for learners aged 12–14, the book builds on foundational skills and introduces more advanced number work, algebra, geometry, and statistics. Each section revisits key ideas with clear explanations and practical examples to support revision and deeper understanding.
本指南涵盖了剑桥初中数学系列第二册的核心主题。该书为12-14岁的学习者设计，在基础技能之上引入更高级的数、代数、几何和统计内容。每个部分都会回顾关键概念，并提供清晰的解释和实用示例，以帮助复习和加深理解。
1. Integers and Order of Operations | 整数与运算顺序
Integers are whole numbers that can be positive, negative, or zero. When adding a negative integer, it is equivalent to subtracting its opposite. For multiplication and division, the product or quotient of two integers with the same sign is positive, while the result is negative if the signs differ.
整数是正整数、负整数和零的总称。加上一个负整数等于减去它的相反数。在乘法和除法中，同号两数相乘或相除得正，异号得负。
When several operations appear together, we follow the BIDMAS rule: Brackets, Indices (powers), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). For example, in 3 + 4 × 2², first calculate the power 2² = 4, then multiply 4 × 4 = 16, and finally add 3 to get 19.
当多个运算同时出现时，我们遵循 BIDMAS 法则：括号、指数（幂）、乘除（从左到右）、加减（从左到右）。例如，在 3 + 4 × 2² 中，先计算幂 2² = 4，再计算乘 4 × 4 = 16，最后加 3 得到 19。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Equivalent fractions represent the same part of a whole. To add or subtract fractions, first write them with a common denominator. For multiplication, multiply the numerators and multiply the denominators; for division, multiply by the reciprocal.
等值分数表示整体的相同部分。加减分数时，先将它们写成同分母的形式。乘法中，分子乘分子，分母乘分母；除法中，乘以倒数。
Decimals can be converted to fractions by using place value, and fractions to decimals by division. Percentages are fractions with denominator 100. To find a percentage of a quantity, change the percentage to a decimal and multiply. For instance, 15% of 200 is 0.15 × 200 = 30.
小数可以通过数位转化为分数，分数则通过除法转化为小数。百分比是分母为100的分数。求一个数量的百分比时，将百分比化成小数再相乘。例如，200的15%是0.15 × 200 = 30。
3. Algebraic Expressions and Simplification | 代数表达式与化简
An algebraic expression contains numbers, letters (variables) and operation symbols. Terms with exactly the same variable part are called like terms and can be combined. For example, 5a + 3a simplifies to 8a, but 5a + 3b cannot be added because the variables are different.
代数表达式包含数字、字母（变量）和运算符号。变量部分完全相同的项称为同类项，可以合并。例如，5a + 3a 简化为 8a，但 5a + 3b 不能相加，因为变量不同。
When expanding brackets, multiply each term inside by the term outside: 3(x + 2) becomes 3x + 6. Factorising is the reverse process, writing an expression as a product of its factors. The distributive law is fundamental in both directions.
去括号时，用括号外的项乘以括号内的每一项：3(x + 2) 变成 3x + 6。因式分解是逆过程，将表达式写成因式的乘积。分配律在两者中都是基础。
4. Solving Linear Equations | 解一元一次方程
A linear equation has the variable raised to the power of 1. To solve it, we perform inverse operations on both sides to isolate the variable. In an equation like 2x + 5 = 13, subtract 5 from both sides to get 2x = 8, then divide by 2 to find x = 4.
一元一次方程中变量的指数为1。解方程时，在等式两边进行逆运算以分离变量。对于 2x + 5 = 13，两边同时减去5得到 2x = 8，再除以2得 x = 4。
Equations may contain brackets or variables on both sides. Always aim to simplify first by expanding brackets and collecting like terms. Then move variable terms to one side and number terms to the other. Always check the solution by substitution.
方程可能含有括号或两边都有变量。始终先化简：去括号、合并同类项。然后将变量项移到一边，数字项移到另一边。最后务必用代入法检验解。
5. Sequences and the nth Term | 数列与第 n 项
A number sequence is a list of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference. For the sequence 3, 7, 11, 15, … the common difference is 4.
数列是按规则排列的一串数。在等差数列中，相邻两项的差是常数，称为公差。数列 3, 7, 11, 15, … 的公差是4。
The nth term gives any term’s value based on its position n. For an arithmetic sequence, it often has the form an + b, where a is the common difference. Here, nth term = 4n – 1. When n = 1, 4×1 – 1 = 3, which matches the first term.
第 n 项根据位置 n 给出该项的值。对于等差数列，它通常具有 an + b 的形式，其中 a 是公差。在此例中，第 n 项 = 4n – 1。当 n = 1 时，4×1 – 1 = 3，与第一项相符。
6. Ratio and Direct Proportion | 比与正比例
A ratio compares the sizes of two or more quantities. It can be simplified by dividing all parts by their highest common factor. For instance, the ratio 8:12 simplifies to 2:3. Ratios do not have units and can be expressed in the form 1:n or n:1.
比用来比较两个或多个数量的大小。可以通过将所有部分除以它们的最大公因数来化简。例如，比 8:12 化简为 2:3。比没有单位，可以写成 1:n 或 n:1 的形式。
Direct proportion means that as one quantity increases, the other increases at the same rate. Two quantities are in direct proportion if their ratio is constant. If 3 pens cost $1.50, then 5 pens cost $2.50 because the cost per pen is constant ($0.50 each).
正比例意味着一个量增加，另一个量以相同速率增加。如果两个量的比值恒定，则它们成正比例。如果3支笔的价格是1.50美元，那么5支笔的价格就是2.50美元，因为每支笔的单价是恒定的（0.50美元）。
7. Angles in Triangles and Quadrilaterals | 三角形和四边形的角
The sum of the interior angles in any triangle is 180°. This fact can be used to find a missing angle when two are known. For example, if a triangle has angles of 50° and 60°, the third angle is 180° – (50° + 60°) = 70°.
任何三角形的内角和都是180°。这一事实可用于已知两个角时求第三个角。例如，若一个三角形的两个角分别为50°和60°，则第三个角为 180° – (50° + 60°) = 70°。
The sum of the interior angles in a quadrilateral is 360°. Special quadrilaterals have additional properties: a parallelogram has opposite angles equal; a kite has one pair of opposite angles equal. Knowing these helps solve angle problems without measuring.
四边形的内角和为360°。特殊四边形还有其他性质：平行四边形的对角相等；风筝形有一组对角相等。了解这些性质有助于无需测量就能解决角度问题。
8. Perimeter, Area and Volume | 周长、面积和体积
Perimeter is the total distance around a 2D shape. For rectangles, it is 2(l + w), where l is length and w is width. Area measures the space inside a shape. The area of a rectangle is l × w, and the area of a triangle is ½ × base × height.
周长是二维图形一周的总长度。对于矩形，周长是 2(长 + 宽)。面积测量图形内部的空间大小。矩形的面积是长 × 宽，三角形的面积是 ½ × 底 × 高。
Volume measures the space inside a 3D solid. For a cuboid, volume = length × width × height. The metric units for area include cm² and m², while volume uses cm³ and m³. Be careful to use consistent units in calculations.
体积测量三维立体内部的空间。对于长方体，体积 = 长 × 宽 × 高。面积的公制单位包括 cm² 和 m²，而体积使用 cm³ 和 m³。计算时注意使用一致的单位。
9. Transformations: Reflections and Rotations | 变换：反射与旋转
A reflection flips a shape over a mirror line. Each point on the image is the same perpendicular distance from the mirror line as the corresponding point on the original shape. The line joining a point and its image is perpendicular to the mirror line.
反射将图形沿镜面线翻转。像上的每个点到镜面线的垂直距离与原图形上对应点到镜面线的距离相等。连接一个点与其像的线段垂直于镜面线。
A rotation turns a shape around a fixed centre through a given angle and direction (clockwise or anticlockwise). The distance from the centre to any point on the shape stays the same after rotation. Rotations are described by centre, angle and direction.
旋转使图形绕一个固定中心转动给定的角度和方向（顺时针或逆时针）。旋转后，从中心到图形上任一点的距离保持不变。描述旋转时需要指明中心、角度和方向。
10. Presenting Data: Charts and Averages | 数据展示：图表与平均数
Data can be displayed in bar charts, pie charts, and line graphs. A bar chart uses bars of equal width to show frequency; the height of each bar represents the frequency. Pie charts show proportions of a whole, with each sector’s angle representing its share.
数据可以用条形图、饼图和折线图展示。条形图用等宽的条形表示频数；条的高度代表频数。饼图展示部分与整体的比例关系，每个扇形的角度代表其份额。
The three common averages are mean, median and mode. The mean is the sum of all values divided by the number of values. The median is the middle value when data are ordered. The mode is the most frequent value. The range measures spread: largest value – smallest value.
三种常见的平均数是平均数、中位数和众数。平均数是所有数值之和除以数值的个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。极差衡量分散程度：最大值 – 最小值。
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