📚 KS3 Maths: Essential Maths Book 9F Key Concepts | KS3 数学：Essential Maths Book 9F 知识点精讲
This revision guide covers the most important topics from the Essential Maths Book 9F, designed for Key Stage 3 students. You will revisit core number skills, algebra, geometry, and statistics, with clear explanations and examples to help you master each concept. Whether you are preparing for an end-of-year test or building a strong foundation for GCSE, these notes will support your learning.
本复习指南涵盖 Essential Maths Book 9F 中的重要主题，专为 KS3 学生设计。你将回顾核心的算术、代数、几何和统计技能，通过清晰的解释和示例来掌握每个概念。无论你是在准备年终考试，还是为 GCSE 打下坚实的基础，这些笔记都会助力你的学习。
1. Number Types and Place Value | 数的类型与位值
Whole numbers, negative numbers, decimals and large numbers all rely on a solid understanding of place value. Each digit’s position tells us its value: units, tens, hundreds, tenths, hundredths, and so on. Negative numbers appear to the left of zero on the number line and are used in contexts like temperature or bank balances.
整数、负数、小数和大数都依赖于对位值的牢固理解。每个数字的位置决定了它的值：个、十、百、十分位、百分位等。负数出现在数轴上零的左边，常用于温度或银行余额等情境。
When ordering negative numbers, remember that -5 is less than -2 because it lies further left. The symbols < and > help compare values: -7 < -3 and 0.2 > 0.15. Practice writing numbers in words and figures, and understand what each digit represents in a number like 3.406 (3 units, 4 tenths, 0 hundredths, 6 thousandths).
在排列负数时，记住 -5 小于 -2，因为它位于更左的位置。符号 < 和 > 用于比较大小：-7 < -3 且 0.2 > 0.15。练习用文字和数字书写数值，并理解像 3.406 中每个数字的含义（3 个一，4 个十分之一，0 个百分之一，6 个千分之一）。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Fractions, decimals and percentages are three different ways of expressing parts of a whole. A fraction like 3/4 means 3 parts out of 4. As a decimal it is 0.75, and as a percentage it is 75%. To convert between them, remember: fraction to decimal – divide numerator by denominator; decimal to percentage – multiply by 100; percentage to fraction – write over 100 and simplify.
分数、小数和百分比是表示整体的一部分的三种不同方式。像 3/4 这样的分数表示 4 份中的 3 份。作为小数是 0.75，作为百分比是 75%。在它们之间转换时记住：分数转小数——分子除以分母；小数转百分比——乘以 100；百分比转分数——写成 100 分之几并化简。
When adding or subtracting fractions, you must first find a common denominator. For example, 1/3 + 1/4 = 4/12 + 3/12 = 7/12. Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same number. Simplifying a fraction means dividing until the numerator and denominator have no common factor other than 1.
在加减分数时，必须首先找到公分母。例如，1/3 + 1/4 = 4/12 + 3/12 = 7/12。等值分数通过将分子和分母同时乘以或除以同一个数得到。化简分数意味着一直除以公因数，直到分子和分母除 1 以外没有其他公因数。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
| 7/10 | 0.7 | 70% |
Fractions, decimals and percentages often appear in real-life problems, such as calculating discounts or interpreting data.
分数、小数和百分比经常出现在现实问题中，例如计算折扣或解读数据。
3. Ratio and Proportion | 比和比例
Ratio compares the sizes of two or more quantities. It can be written in the form a : b (e.g., 2 : 3). If the ratio of boys to girls in a class is 2 : 3, then for every 2 boys there are 3 girls. The total number of parts is 2 + 3 = 5. To find individual amounts from a total, divide the total by the sum of the parts and multiply accordingly.
比用于比较两个或多个数量的大小。它可以写成 a : b 的形式（例如 2 : 3）。如果一个班级男女生的比是 2 : 3，那么每 2 个男生对应 3 个女生。总份数为 2 + 3 = 5。要从总数中求出各部分量，用总数除以总份数，再相应乘以份数。
Proportion refers to the relationship that says two ratios are equal. For example, if 4 pens cost £1.20, then 10 pens cost £3.00. This is direct proportion: as the number of pens increases, the cost increases at the same rate. You can use the unitary method (find the value of one item) or set up equivalent fractions to solve proportion problems.
比例指的是两个比相等的这种关系。例如，如果 4 支笔花费 1.20 英镑，那么 10 支笔花费 3.00 英镑。这是正比例：笔的数量增加，费用以同样速率增加。你可以使用归一法（先求一个的量）或建立等值分数来解决比例问题。
Simplifying ratios is like simplifying fractions: divide both sides by their highest common factor. The ratio 12 : 18 simplifies to 2 : 3 by dividing by 6.
化简比类似于化简分数：将两边同时除以它们的最大公因数。比 12 : 18 可化简为 2 : 3，除以 6 即可。
4. Introduction to Algebra | 代数入门
Algebra uses letters to represent unknown numbers or variables. An algebraic expression is a combination of numbers, letters and operation symbols, such as 3a + 2b – 5. A term is a part of an expression separated by + or – signs. Like terms (e.g., 2x and 5x) can be combined by adding or subtracting the coefficients.
代数用字母表示未知数或变量。代数表达式是数字、字母和运算符号的组合，如 3a + 2b – 5。项是用 + 或 – 分隔的表达式的一部分。同类项（如 2x 和 5x）可以通过相加或相减系数来合并。
You can substitute numbers into expressions. If a = 4 and b = -1, then 3a + 2b = 3(4) + 2(-1) = 12 – 2 = 10. Expanding brackets means multiplying each term inside the bracket by the term outside: 3(x + 2) = 3x + 6. The reverse process is factorising: taking out a common factor, e.g., 4x + 8 = 4(x + 2).
你可以将数字代入表达式。如果 a = 4，b = -1，那么 3a + 2b = 3(4) + 2(-1) = 12 – 2 = 10。展开括号意味着将括号外的项乘以括号内的每一项：3(x + 2) = 3x + 6。逆过程是因式分解：提取公因式，例如 4x + 8 = 4(x + 2)。
A formula is a special equation that shows the relationship between different quantities, such as the area of a rectangle: A = l × w. You can rearrange a formula to make a different variable the subject.
公式是一种特殊的方程，用来表示不同量之间的关系，例如矩形的面积：A = l × w。你可以对公式进行变形，将另一个变量作为主体。
5. Solving Equations | 解方程
An equation states that two expressions are equal, and it contains an equals sign. The goal is to find the value of the unknown letter that makes the equation true. To solve an equation, you must keep the equation balanced – whatever you do to one side, you must do to the other.
方程表明两个表达式相等，并包含等号。目标是找到使方程成立的未知字母的值。解方程时，必须保持等式的平衡——对一边做什么运算，对另一边也要做同样的运算。
Simple one-step equations: x + 5 = 12 → subtract 5 from both sides, x = 7. For 3x = 21, divide both sides by 3 to get x = 7. Two-step equations involve an extra operation: 2x + 3 = 11 → first subtract 3 from both sides to get 2x = 8, then divide by 2 to find x = 4.
简单的一步方程：x + 5 = 12 → 两边同时减 5，得 x = 7。对于 3x = 21，两边同时除以 3，得 x = 7。两步方程包含额外的运算：2x + 3 = 11 → 首先两边减 3，得 2x = 8，然后除以 2，得 x = 4。
Equations with unknowns on both sides require you to collect like terms. Example: 5x – 2 = 2x + 7. Subtract 2x from both sides: 3x – 2 = 7. Add 2 to both sides: 3x = 9, so x = 3. Always check your solution by substituting back into the original equation.
带有未知数在两边的方程要求你合并同类项。例如：5x – 2 = 2x + 7。两边减 2x：3x – 2 = 7。两边加 2：3x = 9，因此 x = 3。始终将解代回原方程进行检验。
6. Sequences | 数列
A sequence is an ordered list of numbers following a rule. The numbers in a sequence are called terms. An arithmetic sequence changes by adding or subtracting the same amount each time – this is the common difference. For example, in the sequence 3, 7, 11, 15, … the term-to-term rule is ‘add 4’.
数列是按照一定规则排列的一列数。数列中的数称为项。等差数列每次都通过加上或减去相同的量来变化——这是公差。例如，数列 3, 7, 11, 15, … 的项间规律是“加 4”。
You can find the position-to-term rule (nth term) of an arithmetic sequence. If the common difference is d, the nth term is of the form d × n + something. For the sequence 3, 7, 11, 15, the nth term is 4n – 1 because when n = 1, 4×1 – 1 = 3; n = 2, 4×2 – 1 = 7, etc.
你可以找到等差数列的序号与项的关系（第 n 项）。如果公差为 d，第 n 项的形式为 d × n + 某数。对于数列 3, 7, 11, 15，第 n 项为 4n – 1，因为当 n = 1 时，4×1 – 1 = 3；n = 2，4×2 – 1 = 7，等等。
Sequences can also be geometric, where each term is multiplied by the same factor, or follow other patterns like square numbers (1, 4, 9, 16…) or triangular numbers. Understanding sequences helps develop logical thinking and prepares for later work on functions.
数列也可以是等比数列（每一项乘以相同的因子），或遵循其他规律，如平方数（1, 4, 9, 16…）或三角形数。理解数列有助于发展逻辑思维，并为以后的函数学习做准备。
7. Angles and Polygons | 角与多边形
Angles are measured in degrees (°) and classified by size: acute (0–90°), right angle (90°), obtuse (90–180°), and reflex (180–360°). On a straight line, angles sum to 180°. Around a point, they sum to 360°. Vertically opposite angles are equal.
角以度（°）为单位测量，并按大小分类：锐角（0–90°）、直角（90°）、钝角（90–180°）和优角（180–360°）。在一条直线上，角之和为 180°。围绕一个点，角之和为 360°。对顶角相等。
When parallel lines are crossed by a transversal, special angle pairs are formed: corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°. Recognising these allows you to find missing angles without measuring.
当平行线被一条截线所切时，形成特殊的角对：同位角相等，内错角相等，同旁内角之和为 180°。识别这些关系可以让你不经测量直接求出缺失的角。
Polygons are closed shapes with straight sides. A regular polygon has all sides and angles equal. The sum of interior angles of an n-sided polygon is (n – 2) × 180°. For example, a pentagon (5 sides) has interior angles summing to (5-2)×180° = 540°. Each exterior angle of a regular polygon is 360° ÷ n.
多边形是由直线边围成的封闭图形。正多边形的所有边和角都相等。一个 n 边形的内角和为 (n – 2) × 180°。例如，五边形（5 条边）的内角和为 (5-2)×180° = 540°。正多边形的每个外角为 360° ÷ n。
8. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the total distance around the edge of a 2D shape. For a rectangle, P = 2(l + w). For compound shapes, add the lengths of all outer sides. Area is the amount of space inside a 2D shape, measured in square units (cm², m²). The area of a rectangle is length × width.
周长是二维图形边界线的总长度。对于矩形，P = 2(长 + 宽)。对于组合图形，将所有外围边长的长度相加。面积是二维图形内部空间的大小，以平方单位（cm², m²）计量。矩形的面积是长 × 宽。
Triangle area: ½ × base × height. Parallelogram area: base × perpendicular height. Trapezium area: ½ × (sum of parallel sides) × height. Always use the perpendicular height, not the slant height. For compound areas, split the shape into simpler rectangles or triangles, work out each area, then add them together.
三角形面积：½ × 底 × 高。平行四边形面积：底 × 垂直高。梯形面积：½ × (上底 + 下底) × 高。一定要使用垂直高，而不是斜高。对于组合图形面积，将图形拆分成更简单的矩形或三角形，分别计算面积，然后相加。
Volume measures the space inside a 3D solid. The volume of a cuboid is length × width × height, measured in cubic units (cm³, m³). A prism’s volume is area of cross-section × length. Capacity is often measured in litres (1 litre = 1000 cm³).
体积测量三维立体内部的空间。长方体的体积为长 × 宽 × 高，以立方单位（cm³, m³）计量。棱柱的体积为横截面积 × 长度。容量常以升为单位（1 升 = 1000 cm³）。
9. Statistics and Averages | 统计与平均数
Statistics involves collecting, organising and interpreting data. Data can be displayed in bar charts, pie charts, line graphs, and frequency tables. The mode is the most frequent value, the median is the middle value when data is ordered, the mean is the sum of all values divided by the number of values, and the range is the difference between the largest and smallest.
统计学涉及收集、整理和解读数据。数据可以用条形图、饼图、折线图和频率表来展示。众数是最频值，中位数是数据排序后的中间值，平均数是所有数据之和除以数据个数，极差是最大值与最小值之差。
To find the median with an odd number of data values, order them and pick the middle one. With an even number, the median is halfway between the two middle numbers. For example, the median of 3, 7, 9, 12 is (7+9)/2 = 8. The mean can be affected by outliers, while the median is more robust.
数据个数为奇数时求中位数，先排序再取正中间的值。个数为偶数时，中位数是中间两个数的平均值。例如，3, 7, 9, 12 的中位数为 (7+9)/2 = 8。平均数可能受极端值影响，而中位数更为稳健。
Pie charts show proportions: the angle for each sector = (frequency ÷ total frequency) × 360°. In a frequency table, you can estimate the mean by using the midpoint of each class interval multiplied by the frequency.
饼图显示比例：每个扇区的角度 = （频数 ÷ 总频数）× 360°。在频率表中，你可以用每组区间的中点乘以频数来估算平均数。
10. Probability | 概率
Probability measures how likely an event is to happen. It ranges from 0 (impossible) to 1 (certain) and can be expressed as a fraction, decimal or percentage. The probability of an event = number of favourable outcomes ÷ total number of possible outcomes.
概率衡量一个事件发生的可能性。其值从 0（不可能）到 1（必然），可以用分数、小数或百分比表示。事件的概率 = 有利结果数 ÷ 所有可能结果总数。
If you roll a fair six-sided dice, the probability of rolling a 4 is 1/6. The probability of rolling an even number is 3/6 = 1/2. The sum of probabilities of all possible outcomes is 1. Events are mutually exclusive if they cannot happen at the same time.
如果你掷一个均匀的六面骰子，掷出 4 的概率是 1/6。掷出偶数的概率是 3/6 = 1/2。所有可能结果的概率之和为 1。如果两个事件不能同时发生，它们就是互斥的。
To find the probability of two independent events both happening, multiply their individual probabilities. For example, probability of getting a head on a coin and a 6 on a dice = 1/2 × 1/6 = 1/12. In probability experiments, the relative frequency tends towards the theoretical probability as the number of trials increases.
求两个独立事件同时发生的概率，将各自的概率相乘。例如，抛硬币得到正面且掷骰子得 6 的概率 = 1/2 × 1/6 = 1/12。在概率实验中，随着试验次数增加，相对频率会趋近于理论概率。
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