KS3 Maths: Essential Maths Book 7i Compressed – Key Concepts Explained | KS3 数学：精华数学 7i 压缩版知识点精讲

📚 KS3 Maths: Essential Maths Book 7i Compressed – Key Concepts Explained | KS3 数学：精华数学 7i 压缩版知识点精讲

Welcome to this focused revision guide covering the essential topics from the compressed KS3 Year 7 maths curriculum, often referred to as Book 7i. This resource is designed to help you master core concepts such as number operations, fractions, algebra, geometry, and data handling. Each section provides clear explanations in both English and Chinese to support bilingual learning and revision.

欢迎阅读这份聚焦 KS3 七年级数学压缩版（常被称为 7i 书）的复习指南，涵盖所有核心知识点。文章针对数字运算、分数、代数、几何与数据处理等内容，以中英双语配对讲解，帮助你系统巩固基础、提升双语理解能力。

1. Integers and the Four Operations | 整数与四则运算

Integers are whole numbers that can be positive, negative, or zero. Mastery of addition, subtraction, multiplication, and division with integers is fundamental at KS3. Remember that subtracting a negative number is equivalent to adding its positive value: a − (−b) = a + b.

整数包括正整数、负整数和零。熟练掌握整数的加、减、乘、除是 KS3 阶段的基础。特别要记住，减去一个负数等于加上它的相反数：a − (−b) = a + b。

When multiplying or dividing integers, the sign rules are crucial: positive × positive = positive, negative × negative = positive, and positive × negative = negative. The same applies to division. Always perform operations inside brackets first, then powers, then multiplication/division (left to right), and finally addition/subtraction (BIDMAS).

进行乘除运算时，符号规则非常重要：正 × 正 = 正，负 × 负 = 正，正 × 负 = 负。除法规则相同。永远遵循运算顺序：先算括号，再算乘方，接着从左到右计算乘除，最后计算加减（即 BIDMAS 法则）。

Estimation and rounding are also covered in this topic. Rounding to the nearest 10, 100, or to a given number of decimal places helps check whether an answer is reasonable. Use the digit immediately to the right of the rounding digit: if it is 5 or more, round up.

估算与四舍五入也是本主题的重要内容。将数值四舍五入到最近的 10、100 或指定的小数位数，有助于判断结果是否合理。观察保留位后一位数字，如果大于或等于 5，则向前一位进一。

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

Fractions represent parts of a whole. Equivalent fractions have the same value despite different numerators and denominators. Simplifying a fraction means dividing both numerator and denominator by their highest common factor (HCF). For example, ⁴/₈ simplifies to ½ because both can be divided by 4.

分数表示整体的一部分。等价分数虽然分子和分母不同，但数值相同。化简分数的方法是同时除以分子和分母的最大公因数（HCF）。例如 ⁴/₈ 可以化简为 ½，因为它们都能被 4 整除。

Converting between fractions, decimals, and percentages is a key skill. To change a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. The table below shows common equivalents.

能在分数、小数和百分数之间相互转换是一项核心技能。分数化小数，用分子除以分母；小数化百分数，乘以 100。下表给出了一些常用的等价形式。

Fraction	Decimal	Percentage
½	0.5	50%
¼	0.25	25%
³/₄	0.75	75%
⅕	0.2	20%
¹/₁₀	0.1	10%

Percentages are used to compare quantities and to solve problems involving increases and decreases. To find a percentage of an amount, convert the percentage to a decimal and multiply. For example, 15% of 200 = 0.15 × 200 = 30.

百分数用于比较数量以及解决增长和减少问题。求一个数的百分之几，先将百分数化为小数，再乘以这个数。例如，200 的 15% 等于 0.15 × 200 = 30。

3. Algebraic Expressions | 代数表达式

Algebra uses letters (variables) to represent unknown numbers. An algebraic expression combines numbers, variables, and operation signs. For instance, 3x + 5 is an expression where ‘3’ is the coefficient of x. Only like terms can be simplified by addition or subtraction.

代数用字母（变量）表示未知数。代数表达式由数字、变量和运算符号组成。例如，3x + 5 是一个表达式，其中 3 是 x 的系数。只有同类项才能通过加减进行合并。

When simplifying expressions, group like terms together. For example, 2a + 3b + 4a − b simplifies to 6a + 2b. The distributive law is used to expand brackets: a(b + c) = ab + ac. Always multiply each term inside the bracket by the term outside.

化简表达式时，先将同类项分组。例如，2a + 3b + 4a − b 可化简为 6a + 2b。运用分配律去括号：a(b + c) = ab + ac。务必用括号外的项乘以括号内的每一项。

Factorising is the reverse of expanding: it means taking out a common factor. For example, 6x + 9 can be written as 3(2x + 3) because 3 is a factor of both 6 and 9. This skill is essential for solving equations later on.

因式分解是去括号的逆运算，即提取公因数。例如，6x + 9 可以写成 3(2x + 3)，因为 3 是 6 和 9 的公因数。这项技能对后续解方程至关重要。

4. Solving Simple Equations | 解简单方程

An equation states that two expressions are equal. The goal is to find the value of the unknown that makes the equation true. Use inverse operations to isolate the variable. Always do the same to both sides to maintain balance.

方程表示两个表达式相等。我们的目标是找到使方程成立的未知数的值。运用逆运算分离变量，始终在等号两边进行相同的操作以保持平衡。

For one-step equations like x + 5 = 12, subtract 5 from both sides to get x = 7. For two-step equations such as 3x − 4 = 11, first add 4 to both sides (3x = 15), then divide both sides by 3 to obtain x = 5.

对于一步方程，如 x + 5 = 12，两边同时减去 5 得到 x = 7。对于两步方程，如 3x − 4 = 11，先两边加 4 得 3x = 15，再将两边除以 3，解得 x = 5。

Equations may involve brackets or negative terms. Always simplify first if needed. The solution can be checked by substituting the value back into the original equation. Word problems require translating a written scenario into an equation before solving.

方程可能含有括号或负数项。如有需要，先进行化简。求得的解应带回原方程验算。文字题需要先将情景翻译成方程，再求解。

5. Angles and Straight Lines | 角度与直线

Angles are measured in degrees (°). A right angle is 90°, an acute angle is between 0° and 90°, an obtuse angle between 90° and 180°, and a reflex angle between 180° and 360°. Angles on a straight line add up to 180°.

角度以度（°）为单位。直角为 90°，锐角在 0° 到 90° 之间，钝角在 90° 到 180° 之间，优角在 180° 到 360° 之间。同一直线上的角之和为 180°。

Angles around a point add up to 360°. Vertically opposite angles are equal when two lines intersect. Knowing these basic angle facts allows you to calculate missing angles without measuring, often by forming and solving simple equations.

围绕一点的角之和为 360°。两直线相交时，对顶角相等。掌握这些基本角度知识后，你就能在不测量的情况下计算未知角，常常通过设立并求解简单方程来完成。

Parallel lines have special angle properties: corresponding angles are equal, alternate angles are equal, and interior (co-interior) angles sum to 180°. Using these relationships, you can find angles in diagrams involving parallel lines and a transversal.

平行线具有特殊的角性质：同位角相等，内错角相等，同旁内角互补（和为 180°）。利用这些关系，可以求出含有平行线和截线图形中的各个角度。

6. Perimeter and Area | 周长与面积

Perimeter is the total distance around the outside of a 2D shape. It is found by adding all side lengths. For a rectangle, perimeter P = 2(l + w) where l is length and w is width. For regular polygons, multiply one side length by the number of sides.

周长是指二维图形外边界的总长度，通过将所有边长相加求得。矩形的周长公式为 P = 2(l + w)，其中 l 为长，w 为宽。正多边形的周长等于一条边长乘以边数。

Area measures the surface inside a shape. The area of a rectangle is calculated as length × width. The area of a triangle is half the base times the vertical height. These formulas can be written as:

面积衡量图形内部的表面大小。矩形的面积为长 × 宽。三角形的面积为底乘高的一半。这些公式可表示为：

Area of rectangle = length × width

矩形面积 = 长 × 宽

Area of triangle = ½ × base × height

三角形面积 = ½ × 底 × 高

Compound shapes can be split into simpler rectangles and triangles. Work out the area of each part and then add them together. Always use the same units for all measurements and include unit squares (e.g., cm²) in your answer.

组合图形可以拆分成简单的矩形和三角形，分别计算各部分面积再相加。所有测量必须使用相同单位，答案要带上平方单位（如 cm²）。

7. Coordinates and Straight-Line Graphs | 坐标与直线图

Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position on a Cartesian grid. The x-axis is horizontal, the y-axis is vertical, and they intersect at the origin (0,0). Points in the first quadrant have both coordinates positive.

坐标写作 (x, y)，x 表示横轴位置，y 表示纵轴位置。直角坐标系中，横轴为 x 轴，纵轴为 y 轴，它们相交于原点 (0,0)。第一象限内的点两个坐标均为正。

Plotting points accurately is the first step to drawing graphs. A straight-line graph is produced by an equation of the form y = mx + c, where m is the gradient and c is the y‑intercept. For simple cases, you can create a table of values.

准确描点是绘制图像的第一步。形如 y = mx + c 的方程会生成一条直线图像，其中 m 是斜率，c 是 y 轴截距。对于简单情形，可先列出数值表。

For example, the line y = 2x + 1 passes through points such as (0,1), (1,3), and (2,5). Plot these points and join them with a ruler. Understanding coordinates also helps in solving geometric problems, such as finding midpoints or missing vertices of shapes.

例如，直线 y = 2x + 1 会经过 (0,1)、(1,3) 和 (2,5) 等点。描出这些点并用直尺连线。理解坐标同样有助于解决几何问题，比如求中点或图形缺失的顶点。

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

Data can be collected, organised, and displayed in various ways, including bar charts, pictograms, and line graphs. Choosing an appropriate representation helps to interpret the data clearly. Frequency tables make it easier to summarise large amounts of raw data.

数据可以收集、整理并以多种方式呈现，包括条形图、象形图和折线图。选择合适的图表有助于清晰解读数据。频数表能够让大量原始数据的汇总变得更为简便。

The three common averages are the mode (the most frequent value), the median (the middle value when ordered), and the mean (the sum of all values divided by the number of values). The range measures spread and is calculated as highest value minus lowest value.

三种常见的平均数是众数（出现次数最多的值）、中位数（按顺序排列后中间的值）和平均数（所有数值之和除以个数）。极差描述数据的分散程度，等于最大值减最小值。

When working with a frequency table, the mean can be found by multiplying each value by its frequency, summing these products, and then dividing by the total frequency. The median position is the (n+1)/2 th value, where n is the total frequency.

若使用频数表求平均数，需要将每个值乘以其频数，求和后再除以总频数。中位数的位置是第 (n+1)/2 个数据，其中 n 为总频数。

Interpreting charts and tables often involves comparing two data sets. A dual bar chart or a comparative line graph can illustrate differences clearly. Reasonable conclusions must be supported by data evidence, not assumptions.

解读图表和表格时常需比较两组数据。复式条形图或对比折线图能够清晰地展现差异。任何合理结论都必须有数据证据支撑，而非依靠假设。

9. Metric Unit Conversions | 公制单位转换

The metric system is used for measuring length, mass, and capacity. Common units include millimetre (mm), centimetre (cm), metre (m), kilometre (km) for length; gram (g) and kilogram (kg) for mass; millilitre (ml) and litre (L) for capacity.

公制广泛用于长度、质量和容积的度量。长度常用毫米 (mm)、厘米 (cm)、米 (m)、千米 (km)；质量常用克 (g) 和千克 (kg)；容积常用毫升 (ml) 和升 (L)。

Conversions rely on multiplying or dividing by powers of 10. For example, 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m. Similarly, 1 kg = 1000 g and 1 L = 1000 ml. A conversion table can help you remember key relationships.

单位转换基于 10 的倍数进行乘除。例如，1 cm = 10 mm，1 m = 100 cm，1 km = 1000 m。同样，1 kg = 1000 g，1 L = 1000 ml。记住这些关键换算关系非常重要。

When solving problems, always ensure units are consistent before performing calculations. For instance, to find the area in cm² when dimensions are given in metres, convert to cm first. Metric conversions also appear in real-world contexts, such as measuring ingredients or distances.

解题时务必在计算前统一单位。比如，已知尺寸单位为米，而要求以 cm² 为面积单位，则需先将长度转换为厘米。单位转换也出现在生活场景中，如测量食材用量或两地距离。

10. Introduction to Probability | 概率初步

Probability measures the chance that an event will happen. It is given on a scale from 0 (impossible) to 1 (certain), often written as a fraction, decimal, or percentage. For example, the probability of flipping a fair coin and getting heads is ½ or 0.5.

概率用来衡量事件发生的可能性，取值范围从 0（不可能）到 1（一定），常用分数、小数或百分数表示。例如，掷一枚均匀硬币，正面朝上的概率是 ½ 或 0.5。

The probability of an event can be calculated by the formula: number of favourable outcomes divided by total number of possible outcomes, provided all outcomes are equally likely. Outcomes can be listed using sample space diagrams or two-way tables.

如果所有可能性出现的机会均等，概率可以用公式计算：事件发生的可能结果数 ÷ 所有可能结果总数。可以用样本空间图或双向表格列出所有结果。

Experimental probability is based on actual trials, while theoretical probability is what should happen in theory. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability. The sum of probabilities of all mutually exclusive outcomes is 1.

实验概率基于实际试验，理论概率则是理论上应得的结果。随着试验次数的增加，实验概率会趋近于理论概率。所有互斥事件结果的概率之和为 1。

Simple combined events can be analysed by listing all possibilities. For two independent events, the probability of both happening is the product of their individual probabilities. Understanding probability helps make informed predictions in games and daily situations.

简单的组合事件可以通过列举所有可能性进行分析。对于两个独立事件，二者同时发生的概率是各自概率的乘积。理解概率有助于在游戏和日常生活情境中作出合理预测。

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KS3 Maths: Essential Maths Book 7i Compressed – Key Concepts Explained | KS3 数学：精华数学 7i 压缩版 知识点精讲