Cambridge Lower Secondary Mathematics Book 7: Key Topics Explained | 剑桥初中数学第七册知识点精讲

📚 Cambridge Lower Secondary Mathematics Book 7: Key Topics Explained | 剑桥初中数学第七册知识点精讲

Cambridge Lower Secondary Mathematics Book 7 lays the foundation for secondary-level mathematical thinking. This article provides a clear, bilingual walkthrough of the essential concepts covered in the course, helping learners reinforce their understanding and prepare for assessments with confidence.

剑桥初中数学第七册为中学阶段的数学思维奠定基础。本文对该册的核心概念进行了清晰的双语精讲，帮助学习者巩固理解，自信备考。

1. Integers and Place Value | 整数与位值

Understanding place value up to one million is central to reading, writing, and comparing large numbers. We also extend this knowledge to decimals and learn to multiply and divide whole numbers by 10, 100, or 1000.

理解到百万级的位值是读写和比较大数据的关键。我们还要将这一知识拓展到小数，并学会将整数乘除以 10、100 或 1000。

Negative numbers are introduced through real-life contexts such as temperature and bank balances. Ordering negatives, adding, and subtracting on a number line are key skills.

通过温度和银行余额等现实情境引入负数。在数线上给负数排序、进行加减运算是核心技能。

Place value columns help represent numbers as Thousands, Hundreds, Tens, and Units.

位值列帮助将数字表示为千、百、十、个。
When multiplying by 100, digits shift two places to the left; when dividing by 1000, three places to the right.

乘以 100 时，数字向左移动两位；除以 1000 时，向右移动三位。

2. Factors, Multiples and Primes | 因数、倍数与质数

A factor is a whole number that divides exactly into another number. Identifying factor pairs helps with mental calculation and later with simplifying fractions.

因数是能整除另一个数的整数。找出因数对有助于心算以及后续的分数化简。

Multiples are the products of a number and any integer. The lowest common multiple (LCM) and highest common factor (HCF) are used to solve problems involving fractions and ratios.

倍数是数与任何整数的乘积。最小公倍数 (LCM) 和最大公因数 (HCF) 用来解决分数和比例相关问题。

A prime number has exactly two distinct factors: 1 and itself. The prime numbers below 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Prime factorisation breaks a composite number into a product of primes, e.g., 60 = 2² × 3 × 5.

质数恰好有两个不同的因数：1 和它本身。小于 30 的质数有 2、3、5、7、11、13、17、19、23 和 29。质因数分解是将合数拆分为质数的乘积，例如 60 = 2² × 3 × 5。

3. Fractions, Decimals and Percentages | 分数、小数与百分比

Equivalent fractions represent the same proportion. We compare and order fractions by converting them to a common denominator. Adding and subtracting fractions with different denominators relies on this skill.

等值分数表示相同的比例。通过通分将分数化为同分母，再进行比较和排序。异分母分数的加减法依赖这一技能。

Decimals are fractions with denominators that are powers of ten. Rounding decimals to a given place value, such as hundredths (0.01), is a practical skill used in money and measurement.

小数是分母为 10 的幂的分数。将小数四舍五入到指定位值（如百分位 0.01）是货币和测量中的实用技能。

A percentage is a fraction with denominator 100. Converting between fractions, decimals, and percentages is essential. Key equivalences like ½ = 0.5 = 50% must be memorised. Finding a percentage of a quantity, e.g., 15% of £80, uses multiplication by a decimal or fraction.

百分比是分母为 100 的分数。分数、小数和百分比的互化至关重要。必须熟记如 ½ = 0.5 = 50% 等关键等价关系。求一个数的百分之几（如 £80 的 15%）需使用小数或分数乘法。

4. Algebraic Expressions and Substitution | 代数表达式与代入

Algebra uses letters to represent unknown numbers or variables. An algebraic expression combines numbers, variables, and operation symbols without an equals sign, e.g., 3n + 5.

代数用字母表示未知数或变量。代数表达式是数、变量和运算符号的组合，不含等号，例如 3n + 5。

Like terms have the same variable raised to the same power. They can be collected to simplify expressions: 2x + 5x = 7x, and 3a + 2b – a = 2a + 2b.

同类项具有相同的变量和指数。收集同类项可化简表达式：2x + 5x = 7x，以及 3a + 2b – a = 2a + 2b。

Substitution means replacing a variable with a specific value. When n = 4, the expression 2n + 3 becomes 2 × 4 + 3 = 11. It is vital to follow the correct order of operations (BIDMAS/BODMAS).

代入是指用具体的值替换变量。当 n = 4 时，表达式 2n + 3 变为 2 × 4 + 3 = 11。遵循正确的运算顺序（BIDMAS / BODMAS）至关重要。

If a = 3 and b = –2, find 2a² – b: 2 × 3² – (–2) = 2 × 9 + 2 = 20

设 a = 3, b = –2，求 2a² – b：2 × 3² – (–2) = 2 × 9 + 2 = 20

5. Solving Linear Equations | 解一元一次方程

An equation states that two expressions are equal. The unknown is often represented by a letter like x. To solve, we isolate the unknown on one side using inverse operations.

方程表示两个表达式相等。未知数通常用 x 等字母表示。求解时，通过逆运算在一边孤立未知数。

One-step equations: x + 7 = 15 → x = 8; 4x = 20 → x = 5. Two-step equations: 3x – 2 = 13 → add 2 → 3x = 15 → x = 5.

一步方程：x + 7 = 15 → x = 8；4x = 20 → x = 5。两步方程：3x – 2 = 13 → 加 2 → 3x = 15 → x = 5。

Equations with unknowns on both sides require collecting terms: 5x + 4 = 2x + 13 → 3x + 4 = 13 → 3x = 9 → x = 3. Always check the solution by substitution.

未知数在等号两边的方程需要移项：5x + 4 = 2x + 13 → 3x + 4 = 13 → 3x = 9 → x = 3。最后一定要代入检验。

6. Sequences and Functions | 数列与函数

A number sequence follows a pattern or term-to-term rule. For the sequence 3, 7, 11, 15, …, the rule is ‘add 4’. The position-to-term rule gives a formula for the nth term.

数列按一定规律或项间递推规则排列。对于数列 3, 7, 11, 15, …，规则是“加 4”。位置–项规则则给出第 n 项的公式。

For a linear sequence, the nth term is of the form an + b. In the sequence above, the nth term is 4n – 1. This allows finding any term, e.g., the 100th term is 4(100) – 1 = 399.

对于线性数列，第 n 项具有 an + b 的形式。上述数列的第 n 项是 4n – 1。由此可求任意项，如第 100 项是 4(100) – 1 = 399。

A function is an input-output machine. It can be shown as a mapping or an equation, e.g., y = 2x + 1. In a table of values, we can plot points to form a straight-line graph.

函数是一种输入–输出机器，可用映射或方程表示，如 y = 2x + 1。通过数值表描点可以得到直线图形。

7. Angles and Constructions | 角度与作图

Angles are measured in degrees (°). Key angle facts include: angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal.

角度以度数 (°) 计量。基本角度事实包括：一直线上的角度之和为 180°，绕一点的角度之和为 360°，对顶角相等。

In a triangle, the sum of interior angles is 180°. An equilateral triangle has three 60° angles; an isosceles triangle has base angles equal. In a quadrilateral, the sum is 360°.

三角形内角和为 180°。等边三角形三个角都是 60°；等腰三角形底角相等。四边形的内角和为 360°。

Compass and protractor constructions include drawing a perpendicular bisector, an angle bisector, and constructing triangles given SSS, SAS, or ASA information.

圆规和量角器作图包括画垂直平分线、角平分线，以及根据 SSS、SAS 或 ASA 条件构造三角形。

8. Perimeter, Area and Volume | 周长、面积与体积

Perimeter is the distance around a shape. For a rectangle, P = 2(l + w). For composite shapes, add all side lengths after finding missing lengths.

周长是图形一周的长度。矩形周长 P = 2(l + w)。对于组合图形，先找出缺失的边长再全部相加。

Area is measured in square units. Rectangle: A = l × w. Triangle: A = ½ × base × height. Parallelogram: A = base × perpendicular height. Compound shapes can be split into known figures.

面积以平方单位计量。矩形面积 A = l × w。三角形面积 A = ½ × 底 × 高。平行四边形面积 A = 底 × 垂直高。组合图形可拆分为已知图形。

Volume of a cuboid: V = length × width × height. Units are cubic, e.g., cm³. Nets of 3D solids help visualise surface area.

长方体体积 V = 长 × 宽 × 高。单位为立方，如 cm³。立体图形的展开图有助于理解表面积。

9. Ratio and Proportion | 比与比例

A ratio compares parts of a whole, written as 3 : 2 or 3 to 2. It can be simplified like a fraction by dividing by common factors.

比用来比较整体的各个部分，写作 3 : 2 或 3 比 2。可像分数一样除以公因数进行化简。

Sharing in a ratio: for a total of £80 shared in the ratio 3 : 5, add parts → 8 parts, one part = £10, so the shares are £30 and £50.

按比例分配：将 £80 按 3 : 5 分配，总份数 → 8 份，每份 = £10，因此分别得到 £30 和 £50。

Proportion describes a multiplicative relationship. Direct proportion means as one quantity doubles, another doubles. The unitary method finds the value of one unit first, e.g., 5 pens cost £3.50, so 1 pen costs £0.70, then 8 pens cost £5.60.

比例描述乘性关系。正比例意味着一量翻倍，另一量也翻倍。单一法先求单位量的值，例如 5 支笔 £3.50，一支笔 £0.70，那么 8 支笔 £5.60。

10. Probability | 概率

Probability measures the chance of an event, on a scale from 0 (impossible) to 1 (certain). It is written as a fraction, decimal, or percentage.

概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。它可用分数、小数或百分比表示。

The probability of an event = number of favourable outcomes ÷ total number of equally likely outcomes. For a fair six-sided die, P(rolling a 3) = ⅙.

事件概率 = 有利结果的数量 ÷ 所有等可能结果的总数。对于一个公平的六面骰子，P(掷出 3 点) = ⅙。

Probabilities of all possible outcomes sum to 1. Complementary events: P(not A) = 1 – P(A). Experimental probability comes from trials, e.g., tossing a coin 100 times.

所有可能结果的概率之和为 1。互补事件：P(非 A) = 1 – P(A)。实验概率来自反复试验，例如抛硬币 100 次。

11. Data Handling and Averages | 数据处理与平均数

Data can be collected through surveys and organised into frequency tables. Bar charts, pictograms, and line graphs display discrete data clearly.

数据可通过调查收集并整理成频数表。条形图、象形图和折线图能清晰地展示离散数据。

The three main averages are: mode (most frequent), median (middle value when ordered), and mean (sum of values ÷ number of values). The range measures spread: highest minus lowest.

三种主要的平均数是：众数（出现次数最多的值）、中位数（排序后中间的值）和平均数（总和 ÷ 数据个数）。极差用来衡量离散程度：最大值减最小值。

For grouped frequency, estimate the mean using midpoints. A stem-and-leaf diagram orders data and preserves original values, making the median easy to locate.

对于分组频数，用组中点估算平均数。茎叶图既能排序数据又保留原始值，方便找出中位数。

Set: 2, 3, 3, 5, 7	Mode = 3, Median = 3, Mean = (2+3+3+5+7)/5 = 4
集合: 2, 3, 3, 5, 7	众数 = 3, 中位数 = 3, 平均数 = (2+3+3+5+7)/5 = 4

12. Transformations and Symmetry | 变换与对称

Line symmetry exists when a shape can be folded along a mirror line so that both halves match. The number of lines of symmetry depends on the shape: a square has 4; a rectangle has 2.

线对称是指图形能沿镜线对折后两半完全重合。对称线的数量因图形而异：正方形有 4 条，长方形有 2 条。

Rotational symmetry is the number of times a shape fits onto itself in a full 360° turn. A regular pentagon has rotational symmetry of order 5.

旋转对称是指图形旋转一周（360°）过程中与自身重合的次数。正五边形的旋转对称阶数为 5。

Transformations on a coordinate grid: reflection flips a shape over a mirror line such as the x-axis or y-axis. Translation slides a shape by a given vector, e.g., (3, –2) means 3 right and 2 down.

坐标网格上的变换：反射是将图形沿 x 轴或 y 轴等镜线翻转。平移是按给定向量滑动图形，例如 (3, –2) 表示右移 3，下移 2。

We can describe a reflection by giving the line equation (x = 0, y = –1, etc.) and a translation by the column vector. Combining transformations follows a specific order.

描述反射时需给出镜线方程（如 x = 0、y = –1 等），平移则给出列向量。组合变换应遵循特定顺序。

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Cambridge Lower Secondary Mathematics Book 7: Key Topics Explained | 剑桥初中数学第七册知识点精讲