📚 KS3 Maths: Essential Maths Book 7 Compressed – Key Concepts Explained | KS3 数学：Essential Maths Book 7.compressed 知识点精讲
This article distils the most important topics from a typical Year 7 mathematics textbook into a clear, bilingual revision guide. Whether you are preparing for end-of-year tests or simply consolidating your understanding, the explanations and examples below will help you master the key concepts. Each section pairs an English explanation with a matching Chinese version, making it easier to follow and learn the terminology in both languages.
本文提炼了七年级数学教材中最核心的知识点，整理成清晰的双语复习指南。无论是准备期末考试，还是巩固已学内容，下面的讲解与例题都能帮助你掌握关键概念。每个部分都用英文和中文对照讲解，方便你同时熟悉两种语言的数学术语。
1. Number Operations and Place Value | 整数运算与位值
In Year 7 you work with whole numbers up to millions, understanding the value of each digit. You add, subtract, multiply and divide large numbers, and learn to use written methods such as column addition and long multiplication confidently. Place value means that in the number 3 456 782, the digit 3 stands for three million, while the 6 stands for six thousand.
七年级会处理大到百万的整数，理解每位数字的值。你需要自信地运用竖式加法、长乘法等书面方法进行大数的加减乘除。位值意味着在数字 3 456 782 中，数字 3 代表三百万，而 6 代表六千。
The four basic operations are revised with an emphasis on order of operations (BIDMAS/BODMAS). Brackets first, then Indices (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right). For example, 3 + 4 × 2 = 3 + 8 = 11, not 14.
四则运算的复习重点是运算顺序（括号－指数－乘除－加减）。先算括号，再算指数（幂），然后乘除（从左到右），最后加减（从左到右）。例如 3 + 4 × 2 = 3 + 8 = 11，而不是 14。
Negative numbers are introduced. Adding a negative is the same as subtracting, and subtracting a negative is the same as adding. On a number line, moving left means decreasing, moving right means increasing. So −5 + 3 = −2 and 4 − (−2) = 4 + 2 = 6.
负数也被引入。加一个负数等于减去其绝对值，减去一个负数等于加上其绝对值。在数轴上，向左移动表示减小，向右移动表示增大。因此 −5 + 3 = −2，而 4 − (−2) = 4 + 2 = 6。
Factors, multiples and prime numbers are explored. A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11, …). The highest common factor (HCF) and lowest common multiple (LCM) are used to simplify fractions and solve problems.
还探究了因数、倍数和质数。质数只有两个因数，即 1 和它本身（2, 3, 5, 7, 11, …）。最大公因数（HCF）和最小公倍数（LCM）用于化简分数和解决问题。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Understanding that fractions, decimals and percentages are three ways of representing parts of a whole is a central theme. To convert a fraction to a decimal, divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75. To change a decimal to a percentage, multiply by 100: 0.75 × 100% = 75%.
理解分数、小数和百分数是表示整体部分量的三种方式，这是核心主题。把分数化小数，用分子除以分母：3/4 = 3 ÷ 4 = 0.75。把小数化成百分数，乘以 100%：0.75 × 100% = 75%。
Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same non‑zero number. Simplifying a fraction means dividing until the numerator and denominator have no common factor other than 1. For example, 8/12 = 2/3 after dividing by 4.
等值分数通过将分子和分母同时乘或除以同一个非零数得到。化简分数就是除以公因数，直到分子分母只有公因数 1。例如 8/12 除以 4 后得到 2/3。
Adding and subtracting fractions requires a common denominator. To add 1/3 + 1/4, use the denominator 12: 4/12 + 3/12 = 7/12. Multiplying fractions is straightforward: multiply numerators and denominators. Dividing by a fraction means multiplying by its reciprocal.
分数加减需要公分母。计算 1/3 + 1/4，用分母 12：4/12 + 3/12 = 7/12。分数乘法直接分子乘分子、分母乘分母。除以一个分数等于乘以它的倒数。
Comparing fractions, decimals and percentages is often done by converting them all to the same form. Ordering 0.6, 55% and 3/5 is easy once you know 3/5 = 0.6 = 60%, so 55% is the smallest.
比较分数、小数和百分数时，通常先将它们化为同一种形式。给 0.6、55% 和 3/5 排序，只要知道 3/5 = 0.6 = 60%，即可得出 55% 最小。
3. Introduction to Algebra | 代数基础
Algebra uses letters to stand for unknown numbers or variables. An expression like 3a + 2b combines numbers and letters with operations. You learn to collect like terms: 5x + 2x simplifies to 7x, but 3x + 4y cannot be simplified further because the letters are different.
代数用字母表示未知数或变量。像 3a + 2b 这样的表达式将数字和字母用运算符号连接起来。你需要学会合并同类项：5x + 2x 化简为 7x，但 3x + 4y 不能进一步化简，因为字母不同。
Substitution means replacing letters with given numbers. If a = 3 and b = 5, then 2a + b = 2×3 + 5 = 11. Brackets are expanded using the distributive law: 3(x + 2) = 3x + 6. This is often modelled with area diagrams.
代入是指用具体数值替换字母。若 a = 3 且 b = 5，则 2a + b = 2×3 + 5 = 11。运用分配律展开括号：3(x + 2) = 3x + 6，常用面积模型来演示。
Writing simple formulas from words is an essential skill. “The total cost C of n apples at 30p each” becomes C = 30n. You also learn to use function machines that show input → rule → output, helping to understand the idea of a mapping.
根据文字编写简单公式是一项基本技能。“n 个苹果，每个 30 便士的总费用 C”可表示为 C = 30n。你还会学习函数机器，展示输入 → 规则 → 输出，以帮助理解映射的概念。
The equals sign represents balance. Whatever operation you perform on one side of an equation must also be done to the other. This idea is reinforced with balancing scales and is the foundation for solving equations.
等号代表平衡。对方程一边进行的任何运算，另一边也必须同样进行。借助天平模型强化这一思想，并为解方程奠定基础。
4. Solving Simple Equations | 解简单方程
An equation states that two expressions are equal, and the goal is to find the value of the unknown. For a one‑step equation like x + 5 = 12, subtract 5 from both sides to get x = 7. For x − 3 = 9, add 3 to both sides, giving x = 12.
方程表示两个表达式相等，目标是找出未知数的值。对于一步方程，如 x + 5 = 12，两边同时减去5，得到 x = 7。对于 x − 3 = 9，两边加3，得出 x = 12。
Multiplication and division equations are handled similarly. If 4x = 20, divide both sides by 4: x = 5. If x/6 = 3, multiply both sides by 6: x = 18. The key rule is to perform the inverse operation.
涉及乘除的方程处理方法类似。若 4x = 20，两边除以4，得 x = 5。若 x/6 = 3，两边乘6，得 x = 18。关键原则是使用逆运算。
Two‑step equations combine operations. Solve 2x + 3 = 11 by first subtracting 3 from both sides (2x = 8), then dividing by 2 (x = 4). Always check your answer by substituting it back into the original equation.
两步方程结合了多种运算。解 2x + 3 = 11，先两边减3（2x = 8），再除以2（x = 4）。务必把答案代入原方程检验。
Forming equations from word problems is practised. “I think of a number, double it and add 7. The result is 23.” Let the number be n, then 2n + 7 = 23, leading to n = 8. This bridges the gap between arithmetic and algebra.
根据文字题建立方程需要练习。“我想一个数，把它加倍再加7，结果是23。”设该数为 n，则 2n + 7 = 23，解得 n = 8。这连接了算术与代数。
5. Angles and Shapes | 角与形状
Angles are measured in degrees using a protractor. Acute angles are less than 90°, right angles are exactly 90°, obtuse angles are between 90° and 180°, and reflex angles are between 180° and 360°. Angles on a straight line add up to 180°, and angles around a point sum to 360°.
角用量角器以度为单位测量。锐角小于 90°，直角等于 90°，钝角介于 90° 和 180° 之间，优角（反角）介于 180° 和 360° 之间。直线上的角之和为 180°，绕一点的角之和为 360°。
Vertically opposite angles are formed when two lines cross; they are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and co‑interior (allied) angles sum to 180°. These rules are used to find missing angles in diagrams.
两直线相交产生对顶角，它们相等。在平行线中，内错角相等，同位角相等，同旁内角之和为 180°。这些规则用于求图形中的未知角。
Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). The sum of interior angles in any triangle is 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形可按边分类（等边、等腰、不等边）或按角分类（锐角、直角、钝角）。任意三角形内角和为 180°。三角形的一个外角等于与它不相邻的两个内角之和。
Properties of quadrilaterals are studied: square, rectangle, parallelogram, rhombus, trapezium and kite. For example, a parallelogram has opposite sides equal and parallel, and opposite angles equal. The sum of interior angles in any quadrilateral is 360°.
学习四边形的性质：正方形、长方形、平行四边形、菱形、梯形和风筝形。例如，平行四边形对边平行且相等，对角相等。任意四边形内角和为 360°。
6. Coordinates and Graphs | 坐标与图形
Coordinates are written as ordered pairs (x, y). The x‑axis is horizontal, the y‑axis vertical, and they cross at the origin (0, 0). The first number tells you how far to move right (positive) or left (negative), the second how far to move up (positive) or down (negative).
坐标写成有序对 (x, y)。x 轴水平，y 轴垂直，它们在原点 (0, 0) 相交。第一个数字表示向右（正）或向左（负）移动的距离，第二个数字表示向上（正）或向下（负）移动的距离。
Plotting points and reading coordinates from a grid is a basic skill. You also learn to draw and interpret line graphs that show how one quantity changes in relation to another. A straight‑line graph with equation y = mx + c is introduced, where m is the gradient and c is the y‑intercept.
在网格上描点和读取坐标是基本技能。你还将学习绘制和解读显示两个量如何变化的折线图。引入直线方程 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Real‑life graphs such as distance–time graphs are examined. A horizontal line on a distance–time graph means the object is stationary; a steeper line means a faster speed. The slope (gradient) represents speed.
研究实际生活中的图形，如距离–时间图。距离–时间图中的水平线段表示物体静止；线段越陡，速度越快。斜率代表速度。
Midpoint of a line segment can be found by averaging the coordinates of the end points. If A(2, 3) and B(6, 7), the midpoint is ((2+6)/2, (3+7)/2) = (4, 5). This skill supports later work in geometry.
线段的中点可以通过平均端点坐标求得。若 A(2, 3) 和 B(6, 7)，中点为 ((2+6)/2, (3+7)/2) = (4, 5)。这一技能为以后的几何学习提供支持。
7. Data Handling and Statistics | 数据处理与统计
Data is collected, organised and displayed in different ways. Tally charts and frequency tables are used to record raw data. From these you can calculate the mode (most frequent value), median (middle value when ordered), mean (sum divided by count) and range (largest minus smallest).
数据可以通过不同方式收集、整理和展示。划记表和频数表用于记录原始数据。由此可以计算众数（最常见值）、中位数（按顺序排列后的中间值）、平均数（总和除以数量）和极差（最大值减最小值）。
Bar charts show frequencies with bars of equal width; the height of each bar represents the frequency. Dual bar charts allow comparison between two sets of data. Pictograms use symbols to represent a certain number of items, and a key is essential.
条形图用等宽的长条表示频数，每个长条的高度代表频数。双条形图可以对两组数据进行比较。象形图用符号代表一定数量的项目，必须配有图例。
Pie charts display proportions of a whole. The total angle of 360° is divided according to the frequencies. To find the angle for a category, calculate (frequency ÷ total frequency) × 360°. Interpreting pie charts involves estimating fractions and percentages.
饼图展示整体中各部分的比例。总角度 360° 按频数分配。求某一类别的角度，计算为 (频数 ÷ 总频数) × 360°。解读饼图需要估算分数和百分数。
Line graphs are used to show trends over time. The horizontal axis usually represents time, and the vertical axis the quantity being measured. Plotting points and joining them with straight lines helps to visualise increases, decreases or plateaus.
折线图用于显示随时间变化的趋势。水平轴通常表示时间，垂直轴表示被测量。描点并用直线连接有助于直观地看出上升、下降或平稳的变化。
8. Measurement and Units | 测量与单位
Length, mass and capacity are measured using metric units: millimetres (mm), centimetres (cm), metres (m), kilometres (km); grams (g), kilograms (kg); millilitres (ml), litres (l). Conversions within the metric system rely on multiplying or dividing by powers of 10. For example, 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g.
长度、质量和容量使用公制单位：毫米（mm）、厘米（cm）、米（m）、千米（km）；克（g）、千克（kg）；毫升（ml）、升（l）。公制单位间的换算基于乘或除以10的幂。例如，1 km = 1000 m，1 m = 100 cm，1 kg = 1000 g。
Perimeter is the distance around the outside of a shape. For a rectangle, perimeter = 2 × (length + width). For a regular polygon, multiply the side length by the number of sides. You solve problems involving missing sides when the perimeter is known.
周长是形状外部边界的总长度。矩形的周长 = 2 × (长 + 宽)。对于正多边形，用边长乘以边数即可。已知周长时，可求解缺失的边长。
Area is measured in square units. The area of a rectangle = length × width. The area of a triangle = (base × height) ÷ 2. The area of a parallelogram is base × perpendicular height, and the area of a trapezium is ½ × (a + b) × h, where a and b are the parallel sides.
面积以平方单位计量。矩形面积 = 长 × 宽。三角形面积 = (底 × 高) ÷ 2。平行四边形面积 = 底 × 垂直高，梯形面积 = ½ × (a + b) × h，其中 a 和 b 为平行边。
Volume for cuboids is length × width × height, measured in cubic units such as cm³. You also learn to convert between units of area (1 m² = 10 000 cm²) and volume (1 m³ = 1 000 000 cm³), understanding the scaling effect of square and cubic units.
长方体的体积 = 长 × 宽 × 高，单位是立方单位，如 cm³。你还要学习面积单位换算（1 m² = 10 000 cm²）和体积单位换算（1 m³ = 1 000 000 cm³），理解平方单位和立方单位的缩放效应。
9. Ratio and Proportion | 比与比例
Ratio compares the sizes of two or more quantities. It can be written in the form a:b or as a fraction. Simplifying a ratio is like simplifying a fraction: divide both sides by their highest common factor. 8:12 simplifies to 2:3 after dividing by 4.
比用来比较两个或更多量的大小。可以写成 a:b 的形式或分数形式。化简比就像化简分数：两边除以它们的最大公因数。8:12 除以 4 后化简为 2:3。
Sharing in a given ratio is done by finding the total number of parts and working out the value of one part. To split £60 in the ratio 2:3, total parts = 5, so one part = £60 ÷ 5 = £12. The shares are 2 × £12 = £24 and 3 × £12 = £36.
按给定比分配的方法是求出总份数，再算出一份的值。按 2:3 分配 £60，总份数 = 5，一份 = £60 ÷ 5 = £12。两人分得 2 × £12 = £24 和 3 × £12 = £36。
Proportion tells you if two ratios are equivalent. Direct proportion means as one quantity doubles, the other also doubles. You can use the unitary method: find the value for one unit first, then scale to the required amount. Tables and graphs of direct proportion produce straight lines through the origin.
比例说明两个比是否相等。正比例意味着一个量加倍，另一个量也加倍。可以使用归一法：先求一个单位的值，再扩展到所需数量。正比例的表格和图形会产生过原点的直线。
Scale drawing and maps use ratios. A scale of 1:50 000 on a map means 1 cm represents 50 000 cm in real life, which is 500 m or 0.5 km. Converting between map distances and real distances involves multiplication or division by the scale factor.
比例图与地图使用比。地图上的比例尺 1:50 000 表示 1 cm 代表实际 50 000 cm，即 500 m 或 0.5 km。进行地图距离与实际距离的转换要用比例因子乘或除。
10. Sequences and Patterns | 序列与规律
A number sequence is an ordered list of numbers following a rule. The rule might be “add 4 each time” or “multiply by 2”. The numbers in a sequence are called terms. Finding the term‑to‑term rule lets you continue the sequence: 5, 9, 13, 17, … (rule: +4).
数列是按某种规则排列的一串数。规则可能是“每次加4”或“每次乘以2”。数列中的数称为项。找出项与项之间的变化规则，就能继续写出数列：5, 9, 13, 17, …（规则：+4）。
Linear sequences have a constant difference between consecutive terms. To find the nth term of a linear sequence, relate it to the multiplication table. For 3, 7, 11, 15, … the difference is 4, so the nth term is 4n − 1 (when n = 1, 4×1 − 1 = 3).
线性序列相邻项的差是常数。求线性序列的第 n 项时，把它与乘法表联系起来。对于 3, 7, 11, 15, …，差为 4，所以第 n 项为 4n − 1（n = 1 时，4×1 − 1 = 3）。
Patterns in shapes can be described using sequences. For matchstick patterns, count the number of sticks needed for each diagram and find the rule. A row of n squares needs 3n + 1 matches. Expressing this algebraically links visual patterns to formulas.
图形中的规律可以用数列来描述。对于火柴棒图案，数出每个图形所需的火柴数，找到规则。一排 n 个正方形需要 3n + 1 根火柴。用代数式表达可以将视觉规律与公式联系起来。
Square numbers (1, 4, 9, 16, …), triangular numbers (1, 3, 6, 10, …) and Fibonacci‑type sequences are introduced. Understanding these special sequences enriches pattern‑spotting and prepares for higher‑level algebra.
还介绍平方数（1, 4, 9, 16, …）、三角形数（1, 3, 6, 10, …）以及斐波那契型数列。理解这些特殊序列可以丰富规律发现能力，并为更高阶代数做准备。
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