📚 Year 10 WJEC Further Mathematics: Vocabulary & Terminology Quick Memorisation Guide | WJEC 进阶数学词汇术语速记指南
Mastering the specific vocabulary of Year 10 WJEC Further Mathematics is the first step towards confident problem-solving. This guide breaks down essential terms from algebra, calculus, vectors, matrices and more into clear English-Chinese pairs, complete with memory hooks so you can recall them quickly in your exam. Use these bite-sized explanations to strengthen your mathematical literacy and bridge the language gap between classroom English and Chinese understanding.
掌握 Year 10 WJEC 进阶数学的专业词汇是自信解题的第一步。本指南将代数、微积分、向量、矩阵等主题的核心术语分解成清晰的中英对照，并附上记忆技巧，帮助你在考试中快速回忆。用这些简洁易懂的解释来强化你的数学语言能力，打通英文课堂术语与中文理解之间的通道。
1. Algebraic Expressions & Core Terms | 代数表达式与核心术语
Variable – A symbol, often a letter like x or y, that represents an unknown number which may change.
变量 – 通常用字母（如 x 或 y）表示一个未知且可变的数。
Constant – A fixed numerical value that does not change, such as 5 or -3.
常数 – 一个固定不变的数值，例如 5 或 -3。
Term – A single number, variable or product of numbers and variables separated by '+' or '-' in an expression, like 3x² or -2y.
项 – 表达式中由 '+' 或 '-' 分隔的一个数字、变量或数字与变量的乘积，如 3x² 或 -2y。
Coefficient – The number multiplying a variable in a term. Think of it as the 'co-worker' of the variable: in 7y, 7 is the coefficient.
系数 – 项中与变量相乘的数字。可以联想为变量的 '合作伙伴'：在 7y 中，7 就是系数。
Like terms – Terms that contain exactly the same variable parts raised to the same powers; only like terms can be combined by addition or subtraction.
同类项 – 所含的字母部分及其指数完全相同的项；只有同类项才能通过加减合并。
2. Equations and Inequalities | 方程与不等式
Equation – mathematical statement showing that two expressions are equal, connected by an equals sign '=', such as 2x + 3 = 7.
方程 – 含有等号 '=' 的数学陈述，表示两个表达式相等，如 2x + 3 = 7。
Identity – An equation that is true for all values of the variable, often written with a triple bar '≡'. For example, (x + 2)² ≡ x² + 4x + 4.
恒等式 – 对变量的所有取值都成立的等式，常用三条横线 '≡' 表示，如 (x + 2)² ≡ x² + 4x + 4。
Solution (or root) – The value of the variable that makes an equation true. Graphically, it is where a curve crosses the x-axis.
解（或根） – 使方程成立的变量的值。在图像上表现为曲线与 x 轴的交点。
Inequality – A relation using signs '<', '>', '≤' or '≥' to compare two expressions, defining a range of values rather than a single number.
不等式 – 使用 '<'、'>'、'≤' 或 '≥' 连接的关系，比较两个表达式并定义出一个取值范围。
Simultaneous equations – A set of two or more equations with the same variables; solving them finds values that satisfy all equations at once.
联立方程组 – 一组包含相同变量的方程，求解联立方程就是找到同时满足所有方程的公共解。
3. Functions and Graphs | 函数与图像
Function (f(x)) – A rule that assigns exactly one output to each input. Notation: f(x) read as 'f of x'.
函数 (f(x)) – 一种规则，对每个输入值给出唯一的一个输出值。记号 f(x) 读作 'f of x'。
Domain – The complete set of all possible input values (x) that a function can accept.
定义域 – 一个函数能够接受的所有可能输入值 (x) 的集合。
Range – The set of all possible output values (y or f(x)) that a function can produce.
值域 – 一个函数能够产生的所有可能输出值 (y 或 f(x)) 的集合。
Inverse function (f⁻¹(x)) – A function that reverses the effect of the original function, swapping inputs and outputs. Only exists if the function is one-to-one.
反函数 (f⁻¹(x)) – 逆转原函数效果的函数，将输入与输出互换。只有一一对应的函数才具有反函数。
Composite function (fg(x)) – Applying one function to the result of another; read as 'f of g of x'. Always work from the inside out.
复合函数 (fg(x)) – 将一个函数的结果代入另一个函数；读作 'f of g of x'。计算时始终从最内层向外计算。
4. Sequences and Series | 数列与级数
Arithmetic sequence – A list of numbers where the difference between consecutive terms is constant, called the common difference.
等差数列 – 相邻两项的差为常数的数列，这个常数差称为公差。
Geometric sequence – A sequence where each term is multiplied by a fixed number (common ratio) to get the next term.
等比数列 – 每一项乘以一个固定数（公比）得到下一项的数列。
Series – The sum of the terms of a sequence. For example, 1 + 3 + 5 + 7 is a finite series.
级数 – 数列各项的和。例如 1 + 3 + 5 + 7 就是一个有限级数。
Sigma notation (Σ) – Greek capital letter sigma. For example, Σ (from n=1 to 4) of n² means 1² + 2² + 3² + 4².
求和符号 (Σ) – 用希腊大写字母 sigma 简洁地表示多项之和。例如 Σ (n=1 到 4) n² 表示 1² + 2² + 3² + 4²。
5. Coordinate Geometry | 坐标几何
Gradient (m) – A measure of the steepness of a line, calculated as change in y divided by change in x. Positive gradient slopes upward.
斜率 (m) – 衡量直线倾斜程度的量，等于 y 的变化量除以 x 的变化量。正斜率表示直线向上倾斜。
Midpoint – The point exactly halfway between two given points. Its coordinates are the averages of the x- and y-coordinates of the endpoints.
中点 – 位于两点正中间的点，其坐标为两点对应坐标的平均值。
Equation of a straight line – + c, where m is the gradient and c is the y-intercept (where the line crosses the y-axis).
直线方程 – 通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距（直线与 y 轴的交点）。
Perpendicular lines – Two lines that intersect at right angles (90°). Their gradients multiply to give -1.
垂直线 – 相交成直角 (90°) 的两条直线，它们的斜率乘积为 -1。
6. Trigonometry | 三角学
Sine (sin), Cosine (cos), Tangent (tan) – iangles: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent.
正弦 (sin), 余弦 (cos), 正切 (tan) – 直角三角形中定义的三角比：sinθ = 对边/斜边, cosθ = 邻边/斜边, tanθ = 对边/邻边。
Hypotenuse – The longest side of a right-angled triangle, always opposite the right angle.
斜边 – 直角三角形中最长的边，总是直角的对边。
Angle of elevation – The angle measured upwards from the horizontal line of sight.
仰角 – 从水平视线向上测量的角度。
Trigonometric identity – An equation involving trig ratios that is true for all angles, such as
三角恒等式 – 涉及三角比且对所有角度都成立的等式，例如
sin²θ + cos²θ = 1
This identity is the gateway to solving many trigonometric equations in WJEC Further Maths.
这一恒等式是解决 WJEC 进阶数学中许多三角方程问题的钥匙。
7. Vectors | 向量
Vector – A quantity that has both magnitude (size) and direction, often represented by a directed line segment or a column vector like (3, 4)ᵀ.
向量 – 既有大小又有方向的量，通常用有向线段或列向量如 (3, 4)ᵀ 表示。
Magnitude – The length or size of a vector, found using Pythagoras' theorem. For vector v = (x, y), magnitude = √x² + y².
模（大小） – 向量的长度或大小，通过勾股定理求得。对于向量 v = (x, y)，模为 √x² + y²。
Scalar – A quantity that has only magnitude, no direction, like an ordinary number. Multiplying a vector by a scalar changes its magnitude but keeps its direction (if scalar is positive).
标量 – 只有大小而没有方向的量，例如普通的数字。向量乘以标量会改变其大小但保持方向（若标量为正）。
Resultant vector – The single vector that represents the sum of two or more vectors; the 'overall' effect.
合向量 – 表示两个或多个向量之和的单个向量，即 '总' 效果。
8. Matrices | 矩阵
Matrix – der of a matrix is given by rows × columns, e.g., a 2 × 2 matrix has 2 rows and 2 columns.
矩阵 – 由数字按行和列排列成的矩形阵列。矩阵的阶用 行数 × 列数 表示，例如 2 × 2 矩阵有 2 行 2 列。
Element – Each individual number inside a matrix; elements are often labelled by their row and column position.
元素 – 矩阵内部的每一个数字；元素通常按照其所在的行与列位置来标记。
Identity matrix (I) – A square matrix with 1s on the main diagonal and 0s elsewhere. It acts like the number 1 in matrix multiplication.
单位矩阵 (I) – 主对角线上元素为 1、其余为 0 的方阵，在矩阵乘法中相当于数字 1 的作用。
Determinant – the determinant is ad - bc. A zero determinant means the matrix is singular (no inverse).
行列式 – 对于 2 × 2 矩阵 [a, b; c, d]，行列式为 ad - bc。行列式为零意味着该矩阵是奇异的（不可逆）。
9. Differentiation | 微分
Derivative – The instantaneous rate of change of a function, found by differentiating. Notation: dy/dx or f'(x).
导数 – 函数瞬时变化率，通过微分求得。记作 dy/dx 或 f'(x)。
Differentiation from first principles – Using the limit definition of a derivative to find the gradient of a curve at a point, often expressed as
第一原理求导 – 利用导数的极限定义来求曲线上某点的斜率，通常表示为
f'(x) = limit (h→0) of [f(x+h) - f(x)] / h
Power rule – differentiating xⁿ: multiply by the power and reduce the power by 1. If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹.
幂法则 – 求导 xⁿ 的快捷方法：乘以指数再将指数减 1。若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。
Stationary point – ere the gradient is zero (dy/dx = 0). It can be a maximum, minimum or point of inflection.
驻点 – 曲线上梯度为零（dy/dx = 0）的点，可以是极大值、极小值或拐点。
10. Integration | 积分
Integral – The reverse process of differentiation; an indefinite integral finds the family of antiderivatives, while a definite integral computes the accumulated area under a curve.
积分 – 微分的逆运算；不定积分求出一族原函数，而定积分计算曲线下的累积面积。
Constant of integration (+C) – Whenever an indefinite integral is found, an arbitrary constant must be added because differentiation loses constant terms.
积分常数 (+C) – 求不定积分时，必须加上任意常数，因为求导过程中常数项会消失。
Definite integral – ted by ∫ₐᵇ f(x) dx, giving a numerical value equal to the net area under f(x) from a to b.
定积分 – 带有上限和下限的积分，记作 ∫ₐᵇ f(x) dx，其值为 f(x) 从 a 到 b 的净面积。
Power rule for integration – crease the power by 1 and divide by the new power. ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ -1.
积分幂法则 – 微分幂法则的逆过程：指数加 1 再除以新指数。∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n ≠ -1。
11. Numerical Methods | 数值方法
Trial and improvement – A systematic method of finding an approximate solution to an equation by testing different values and narrowing the interval until the desired accuracy is achieved.
试位法（试错逼近） – 一种通过逐次代入测试值并不断缩小区间来寻找方程近似解的系统性方法，直到达到所需精度。
Iteration – A process that uses a formula to generate a sequence of approximations, using each result to compute the next one until the values settle down (converge) to a solution.
迭代法 – 用一个公式生成一串近似值的过程，将每一个结果代入计算下一个近似值，直到数值稳定（收敛）于一个解。
Convergence – When an iterative sequence tends towards a specific limit or root, meaning the approximations get closer and closer to the true answer.
收敛 – 当一个迭代序列趋近于特定的极限或根时，意味着近似值越来越接近真实答案。
12. Essential Mathematical Verbs | 核心数学动词
Simplify – To reduce an expression to its shortest, clearest form by collecting like terms or cancelling common factors.
化简 – 通过合并同类项或约去公因子，将表达式化为最简形式。
Expand – To multiply out brackets, e.g., expanding 3(x + 2) gives 3x + 6.
展开 – 将括号乘开，例如展开 3(x + 2) 得到 3x + 6。
Factorise – The reverse of expanding; writing an expression as a product of its factors, such as turning x² - 9 into (x + 3)(x - 3).
因式分解 – 展开的逆过程；将表达式写成其因式的乘积，例如将 x² - 9 化为 (x + 3)(x - 3)。
Solve – To find the value(s) of the variable that satisfy an equation, often by isolating the unknown.
解（求解） – 求出满足方程的变量值，通常通过隔离未知数实现。
Evaluate – To calculate the numerical value of an expression or function by substituting numbers in place of variables.
求值 – 通过代入数字替代表达式或函数中的变量，计算出数值结果。
Prove – To show that a mathematical statement is true using logical reasoning, often by building a chain of equalities or using identities.
证明 – 用逻辑推理展示一个数学命题为真，常通过构造等式的链条或运用恒等式完成。
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