KS3 Cambridge Further Mathematics: Vocabulary Quick-Memorisation Guide | KS3剑桥进阶数学：词汇术语速记指南

📚 KS3 Cambridge Further Mathematics: Vocabulary Quick-Memorisation Guide | KS3剑桥进阶数学：词汇术语速记指南

Mastering the vocabulary of further mathematics is the first step to excelling in KS3 Cambridge assessments. This guide presents key terms with clear bilingual explanations and memory-friendly tips to help you recall definitions quickly and accurately. Whether you are tackling algebra, functions, sets, or vectors, a solid command of terminology will boost both your problem-solving and communication skills.

掌握进阶数学的词汇是在KS3剑桥评估中取得优异成绩的第一步。本指南以清晰的双语解释和易于记忆的技巧呈现关键术语，帮助你快速准确地回忆定义。无论你是在应对代数、函数、集合还是向量，牢固掌握术语都将提升你的解题和沟通能力。

1. Algebraic Expressions | 代数表达式

In algebra, a variable (e.g., x, y) stands for an unknown value. A constant is a fixed number, like 5. A term is a product of numbers and variables; an expression is a combination of terms separated by + or – signs. When terms have powers that are whole numbers, we call the expression a polynomial, such as 3x² + 2x – 7. The leading coefficient is the number multiplying the term with the highest power. To help you remember: ‘coefficient’ comes from ‘co-‘ (together) and ‘efficient’ – the number that works together with a variable.

在代数中，变量（如 x, y）代表未知数。常数是固定数字，如5。项是数字与变量的乘积；表达式是由加减号连接的项的组合。当各项的幂次为非负整数时，我们称之为多项式，例如 3x² + 2x – 7。首项系数是最高次项的数字因子。助记：‘coefficient’ 可拆为 co-（一起）+ efficient，数字与变量“一同作用”。

Like terms share exactly the same variable parts, so 4ab and –ab can be combined. Simplifying an expression means adding or subtracting like terms. Expanding uses the distributive law to remove brackets: a(b + c) = ab + ac. Factorising reverses this by extracting common factors. For instance, 6x + 9 = 3(2x + 3). Visual clue: ‘factorise’ contains the word ‘factor’ – you are pulling a common factor out of each term.

同类项拥有完全相同的字母部分，因此 4ab 和 –ab 可以合并。化简表达式即加减同类项。展开运用分配律去除括号：a(b + c) = ab + ac。因式分解是其逆过程，提取公因子。例如 6x + 9 = 3(2x + 3)。助记：’factorise’ 里有 ‘factor’（因子），你正在从各项中“取出”公因子。

2. Equations and Inequalities | 方程与不等式

An equation states that two expressions are equal, shown by an equals sign =. To solve an equation means to find the value(s) of the variable that make the statement true. These values are called solutions or roots. The balance method keeps both sides equal by performing the same operation on each side, e.g., adding 4 to both sides of x – 4 = 10 gives x = 14. Visualise a set of scales: whatever you do to one side, you must do to the other.

方程表明两个表达式相等，用等号 = 连接。解方程就是找出使等式成立的变量取值，这些值称为解或根。天平法通过对方程两边执行相同操作来保持平衡，例如 x – 4 = 10 两边加4得 x = 14。想象一架天平：对一边做什么，另一边也必须同样处理。

An inequality compares expressions using symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). Strict inequalities (<, >) exclude the boundary, so an open circle is used on a number line. Non-strict inequalities (≤, ≥) include the boundary, shown with a closed circle. A crucial rule: when multiplying or dividing by a negative number, the inequality sign must be reversed. Think ‘negative flip’ to avoid mistakes.

不等式用符号比较表达式：<（小于）、>（大于）、≤（小于等于）、≥（大于等于）。严格不等式（<, >）不包含边界，在数轴上用空心圆表示。非严格不等式（≤, ≥）包含边界，用实心圆。一条关键规则：当乘以或除以负数时，必须反转不等号方向。记作“遇负则翻”。

3. Sequences and Patterns | 序列与规律

A sequence is an ordered list of numbers, each called a term. The position of a term is its index (1st, 2nd, 3rd, …). An arithmetic (linear) sequence has the same difference between consecutive terms, called the common difference, d. Its nth term formula is uₙ = a + (n – 1)d, where a is the first term. Example: 5, 8, 11, 14, … has a = 5, d = 3, so uₙ = 5 + 3(n – 1) = 3n + 2. Link the word ‘arithmetic’ to ‘additive’ – you keep adding d each time.

序列是一组有序排列的数字，每个数字称为一项。项的位置是其索引（第1项、第2项等）。等差数列相邻两项之差相等，这个差称为公差 d。其第 n 项公式为 uₙ = a + (n – 1)d，其中 a 为首项。例如序列 5, 8, 11, 14, … 中 a = 5, d = 3，因此 uₙ = 3n + 2。联想 ‘arithmetic’ 和 ‘additive’（加性的），每次都在加上公差。

A geometric sequence has a common ratio, r, between consecutive terms. Its nth term is uₙ = arⁿ⁻¹ (the first term times r to the power n–1). A Fibonacci-type sequence starts with two values and each term is the sum of the two before it, e.g., 1, 1, 2, 3, 5, 8, … Triangular numbers 1, 3, 6, 10, … are given by Tₙ = n(n+1)/2. Remember ‘geometric’ relates to ‘multiplying’ by a growth factor r, just as geometry often deals with ratios.

等比数列相邻两项之比相等，这个比称为公比 r。其第 n 项为 uₙ = arⁿ⁻¹（首项乘以 r 的 n–1 次方）。斐波那契型序列以前两个数字开始，每项是前两项之和，如 1, 1, 2, 3, 5, 8, …。三角形数 1, 3, 6, 10, … 的通项公式为 Tₙ = n(n+1)/2。“等比”（geometric）

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