📚 Year 9 WJEC Further Mathematics: Essential Terminology Quick-Memorisation Guide | Year 9 WJEC 进阶数学：词汇术语速记指南
Building a strong mathematical vocabulary is essential for success in the WJEC Further Mathematics course. Understanding precise definitions allows you to follow reasoning, communicate solutions, and tackle problems confidently. This guide provides a quick-memorisation overview of key terms grouped by topic, with English explanations followed by their Chinese equivalents, designed to support bilingual learners in Year 9.
建立扎实的数学词汇基础对于在 WJEC 进阶数学课程中取得成功至关重要。理解精确的定义能让你跟上推理、交流解题过程并自信地应对题目。本指南按主题汇总关键术语，提供英文解释并紧跟中文对应，旨在帮助9年级的双语学习者快速记忆。
1. Why Terminology Matters in Further Mathematics | 为什么术语在进阶数学中如此重要
A term is not just a word: it is a compressed instruction. When an exam question says "state the coefficient" or "give your answer in exact form", the marker is testing whether you recognise the exact mathematical object being named. In Further Mathematics, the gap between a correct answer and a lost mark is often one word - for example, writing "solve" when the question asked you to "prove".
一个术语不只是单词，而是一组压缩的指令。当考题要求 "state the coefficient"（写出系数）或 "give your answer in exact form"（以精确值作答）时，阅卷人考查的正是你是否识别出被点名的数学对象。在进阶数学中，正确答案与失分之间常常只差一个词——例如题目要求 "prove"（证明），你却写成了 "solve"（求解）。
Terminology also makes learning faster. Once you can name a structure, you can search for it, revise it, and link it to a method. A student who knows the phrase "difference of two squares" can retrieve an entire factorisation routine in a second; a student who only sees "a² − b²" has to rediscover it every time.
术语还能让学习提速。一旦你能为一个结构命名，你就能检索它、复习它，并把它与解题方法挂钩。知道 "difference of two squares"（平方差）这个说法的学生，一秒钟就能调出整套因式分解流程；而只看到 "a² − b²" 的学生，每次都要重新推导一遍。
- Precision: definitions tell you exactly what is allowed and what is not. | 精确性：定义告诉你什么可以做、什么不可以做。
- Speed: named methods are recalled faster than unnamed ones. | 速度：有名字的方法比没名字的方法回忆得更快。
- Communication: clear wording earns method marks even when arithmetic slips. | 表达：即使算错，清晰的表述也能拿到方法分。
- Bilingual transfer: matching English and Chinese terms locks meaning in two ways. | 双语迁移：英中术语配对，用两条通路锁住含义。
2. Core Algebra Vocabulary | 核心代数词汇
Term: a single number, a single variable, or a product of numbers and variables, for example 5, x, or 3xy. Coefficient: the numerical factor multiplying the variable or variables in a term - in 7x² the coefficient is 7. Constant: a term with no variable part, such as −4 in 3x − 4.
项（Term）：单独的一个数、单独的一个字母，或数与字母的乘积，例如 5、x 或 3xy。系数（Coefficient）：项中乘在字母前面的数字因子——在 7x² 中系数是 7。常数项（Constant）：不含字母的项，例如 3x − 4 中的 −4。
Expression: a combination of terms linked by operators (+, −) with no equals sign, such as 4x + 2y − 7. Equation: a statement that two expressions are equal, containing an "=" sign, such as 2x + 3 = 11. Formula: a rule expressed as an equation that links quantities, such as A = πr².
代数式（Expression）：由运算符（+、−）连接的若干项的组合，不含等号，例如 4x + 2y − 7。方程（Equation）：用等号连接两个代数式的陈述，例如 2x + 3 = 11。公式（Formula）：用等式表达的、联系若干量的规律，例如 A = πr²。
The difference between an expression and an equation is not cosmetic. You simplify an expression but you solve an equation. Using the wrong verb in a written answer is one of the most common avoidable errors in Year 9.
代数式与方程的区别绝非形式问题：你化简（simplify）代数式，但你解（solve）方程。在书面作答中使用错误的动词，是9年级最常见的可避免失分点之一。
| English term | 中文术语 | Key idea |
| Identity | 恒等式 | True for every value of the variable, shown with ≡ |
| Inequality | 不等式 | Uses <, >, ≤, ≥ instead of = |
| Like terms | 同类项 | Same variable part, so they can be collected |
| Subject of a formula | 公式的主元 | The single variable written on its own, e.g. y in y = 3x + 1 |
3. Expanding, Factorising and Algebraic Fractions | 展开、因式分解与代数分式
Expand: remove brackets by multiplying out, so 3(x + 4) becomes 3x + 12. Factorise: the reverse process - write an expression as a product of factors. Common factor: a factor shared by every term, such as 5x in 5x² + 10x. Quadratic expression: an expression of the form ax² + bx + c where a ≠ 0.
展开（Expand）：通过相乘去掉括号，例如 3(x + 4) 变为 3x + 12。因式分解（Factorise）：相反的过程——把代数式写成几个因式的乘积。公因式（Common factor）：每一项都含有的因式，例如 5x² + 10x 中的 5x。二次式（Quadratic expression）：形如 ax² + bx + c 的代数式，其中 a ≠ 0。
− b). A perfect square states that a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)².
有两个标准模式必须做到不假思索。平方差（difference of two squares）告诉我们 a² − b² = (a + b)(a − b)；完全平方（perfect square）告诉我们 a² + 2ab + b² = (a + b)²，以及 a² − 2ab + b² = (a − b)²。
(a + b)² = a² + 2ab + b² a² − b² = (a + b)(a − b)
In algebraic fractions, the numerator is the top line and the denominator is the bottom line. To simplify a fraction you cancel a common factor from the numerator and the denominator. You may cancel factors only - never individual terms of a sum.
在代数分式中，分子（numerator）在上面，分母（denominator）在下面。要化简（simplify）一个分式，就要从分子与分母中约分（cancel）公因式。你只能约去因式——绝不能约去和式中的单独项。
- Factorise first, then cancel. | 先因式分解，再约分。
- Multiplying fractions: multiply numerators together and denominators together. | 分式相乘：分子乘分子，分母乘分母。
- Dividing fractions: multiply by the reciprocal (flip the second fraction). | 分式相除：乘以倒数（把第二个分式上下颠倒）。
- Adding fractions: use a common denominator first. | 分式相加：先通分。
4. Indices, Surds and Standard Form | 指数、根式与标准形式
In the expression aⁿ, the base is a and the index (or power, or exponent) is n. The index laws allow you to combine powers quickly, and they are used in almost every later topic in Further Mathematics.
在表达式 aⁿ 中，底数（base）是 a，指数（index / power / exponent）是 n。指数律（index laws）让你能快速合并幂，它们在进阶数学后续几乎所有主题中都会用到。
bsp; aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1 ÷ aⁿ
a1/n = ⁿ√a am/n = ⁿ√aᵐ
A surd is an irrational root left in exact form, such as √2 or 3 + √5. Rationalising the denominator means rewriting a fraction so that no surd remains in the denominator; the conjugate of 3 + √5 is 3 − √5, and multiplying by it removes the surd.
根式（surd）是以精确形式保留的无理根，例如 √2 或 3 + √5。分母有理化（rationalising the denominator）是指改写分式，使分母中不再含根式；3 + √5 的共轭式（conjugate）是 3 − √5，乘以它即可消去根式。
Standard form scientific notation) writes a number as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. The order of magnitude is the power of ten, used to compare very large or very small quantities.
标准形式（standard form）（也称科学计数法）把一个数写成 A × 10ⁿ 的形式，其中 1 ≤ A < 10，n 为整数。数量级（order of magnitude）就是那个 10 的幂，用来比较极大或极小的量。
5. Sequences, Series and Proof Language | 数列、级数与证明用语
A sequence is an ordered list of numbers called terms. The nth term is a rule that generates any term from its position n. An arithmetic sequence increases by a fixed common difference d; a geometric sequence multiplies by a fixed common ratio r. A recurrence relation defines each term using previous terms, such as uₙ₊₁ = 2uₙ + 3.
数列（sequence）是按顺序排列的一列数，其中的数称为项（terms）。第 n 项（nth term）是由位置 n 生成任意一项的规则。等差数列（arithmetic sequence）按固定的公差（common difference）d 递增；等比数列（geometric sequence）按固定的公比（common ratio）r 相乘。递推关系（recurrence relation）用前几项定义后一项，例如 uₙ₊₁ = 2uₙ + 3。
uₙ = a + (n − 1)d uₙ = arⁿ⁻¹ Sₙ = n ÷ 2 × [2a + (n − 1)d]
S∞ = a ÷ (1 − r) for |r| < 1
A series is the sum of the terms of a sequence, written with sigma notation as Σ. A sequence is convergent if its terms approach a fixed limit; otherwise it is divergent. In proof, you may be asked to prove, show that, or deduce - each has a slightly different expectation.
级数（series）是数列各项之和，用求和符号（sigma notation）Σ 表示。若数列的项趋近于某个固定的极限（limit），该数列收敛（convergent）；否则发散（divergent）。在证明题中，你可能被要求 prove（证明）、show that（说明）或 deduce（推出）——三者的作答要求略有不同。
- Prove: give a complete logical argument from known facts. | 证明：从已知事实出发给出完整逻辑论证。
- Show that: reach a stated result, showing every essential step. | 说明：推导到给定结果，关键步骤必须呈现。
- Deduce / hence: use the previous part's result, not a fresh method. | 推出／由此：使用上一问的结果，而非另起炉灶。
6. Coordinate Geometry Terminology | 坐标几何术语
Coordinates (x, y) fix a point relative to the origin. The gradient m measures steepness, calculated as the change in y divided by the change in x. The y-intercept c is where the line crosses the y-axis, and the x-intercept is where it crosses the x-axis.
坐标（coordinates）(x, y) 确定一个点相对于原点的位置。斜率（gradient）m 衡量陡峭程度，等于 y 的变化量除以 x 的变化量。y 截距（y-intercept）c 是直线与 y 轴的交点，x 截距（x-intercept）是与 x 轴的交点。
m = (y₂ − y₁) ÷ (x₂ − x₁) y = mx + c
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is −1, so m₁ × m₂ = −1. The midpoint of two points averages their coordinates, and the distance between them comes from Pythagoras' theorem.
平行直线斜率相等。垂直（perpendicular）直线的斜率之积为 −1，即 m₁ × m₂ = −1。两点的中点（midpoint）由其坐标的平均值给出，两点间的距离（distance）由勾股定理得出。
Midpoint: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2) Distance: √[(x₂ − x₁)² + (y₂ − y₁)²]
A tangent touches a curve at exactly one point; a normal is perpendicular to the tangent there. A locus is the set of all points satisfying a given condition - for example, the set of points a fixed distance from a centre is a circle.
切线（tangent）与曲线恰好只有一个公共点；法线（normal）在该点与切线垂直。轨迹（locus）是满足给定条件的全部点的集合——例如，到某定点距离恒定的点的集合是一个圆。
7. Functions, Domain and Transformations | 函数、定义域与图像变换
A function maps each input to exactly one output. The domain is the set of allowed inputs, and the range is the set of resulting outputs. A mapping may be one-to-one (each output comes from one input) or many-to-one (several inputs share an output). Only one-to-one functions have an inverse function.
函数（function）把每个输入映射到恰好一个输出。定义域（domain）是允许输入的集合，值域（range）是相应输出的集合。映射（mapping）可以是一对一（每个输出只对应一个输入），也可以是多对一（多个输入共用一个输出）。只有一对一函数才存在反函数（inverse function）。
The composite function fg(x) means apply g first, then f. The modulus function |x| gives the non-negative size of x. An asymptote is a line that a curve approaches but never reaches.
复合函数（composite function）fg(x) 表示先作用 g，再作用 f。绝对值函数（modulus function）|x| 给出 x 的非负大小。渐近线（asymptote）是曲线无限接近却永不相交的直线。
- Translation: y = f(x) + a shifts up by a; y = f(x − a) shifts right by a. | 平移：y = f(x) + a 上移 a；y = f(x − a) 右移 a。
- Stretch: af(x) stretches vertically by factor a; y = f(kx) stretches horizontally by factor 1 ÷ k. | 伸缩：y = af(x) 纵向伸缩 a 倍；y = f(kx) 横向伸缩 1 ÷ k 倍。
- Reflection: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis. | 反射：y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。
Careful ordering matters: transformations are applied in a specific sequence, and a horizontal change inside the bracket behaves in the opposite direction to intuition.
顺序非常重要：图像变换按特定次序作用，而括号内部针对 x 的变化方向往往与直觉相反。
8. Trigonometry and Radian Measure | 三角学与弧度制
In a right-angled triangle, the hypotenuse is the longest side, and the opposite and adjacent sides are named relative to the angle being used. The three primary ratios are sine, cosine and tangent, remembered as SOH-CAH-TOA.
在直角三角形中，斜边（hypotenuse）是最长边，对边（opposite）与邻边（adjacent）是相对于所用角来命名的。三个基本比值是正弦（sine）、余弦（cosine）和正切（tangent），可用 SOH-CAH-TOA 记忆。
For non-right-angled triangles, use the sine rule and cosine rule. The area formula with a sine term works when you know two sides and the included angle.
对于非直角三角形，使用正弦定理（sine rule）与余弦定理（cosine rule）。含正弦的面积公式适用于已知两边及其夹角的情形。
a ÷ sin A = b ÷ sin B = c ÷ sin C
a² = b² + c² − 2bc cos A Area = ½ ab sin C
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Radians connect angle to length directly, which is why they dominate later calculus work.
弧度（radian）是圆中一段长度等于半径的弧在圆心处所对的角。弧度把角与长度直接联系起来，这也是它在后续微积分中占主导地位的原因。
π rad = 180° arc length = rθ sector area = ½ r²θ
The amplitude of a trigonometric graph is half the distance between its maximum and minimum values, and the period is the length of one complete cycle. An identity is true for every value of θ, unlike an equation which is true only for particular values.
三角函数图像的振幅（amplitude）是最大值与最小值之差的一半，周期（period）是完成一个完整循环的长度。像 sin²θ + cos²θ = 1 这样的恒等式（identity）对任意 θ 都成立，而方程只对特定值成立。
9. Calculus Preliminaries: Limits, Gradients and Derivatives | 微积分预备：极限、梯度与导数
A limit describes the value a function approaches as the input approaches a chosen value. The gradient of a curve at a point is the gradient of the tangent there, found by letting the chord shrink towards the point. This limiting gradient is the derivative, written dy ÷ dx, and the process of finding it is differentiation.
极限（limit）描述当自变量趋近某值时函数所趋近的值。曲线在某点的梯度就是该点切线的斜率，可通过让弦不断收缩到该点求得。这个极限梯度就是导数（derivative），记作 dy ÷ dx，求它的过程称为微分（differentiation）。
Key vocabulary includes rate of change (what a derivative measures in context), increasing and decreasing functions, stationary points where dy ÷ dx = 0, and maximum or minimum points classified with the second derivative.
核心词汇包括变化率（
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