Pre-U AQA Mathematics: Terminology Fast-Track Guide | Pre-U AQA 数学：词汇术语速记指南

📚 Pre-U AQA Mathematics: Terminology Fast-Track Guide | Pre-U AQA 数学：词汇术语速记指南

Mastering the language of mathematics is the first critical step towards success in the AQA Pre-U (A-Level) Mathematics course. Whether you are simplifying algebraic expressions, applying the chain rule, or interpreting a statistical hypothesis test, a clear command of terminology will help you understand exam questions and communicate your solutions precisely. This guide provides a curated set of key terms, presented in English alongside their Chinese equivalents, together with memory-friendly explanations and quick revision tips. Use it as a daily warm‑up or a last‑minute check before the examination.

掌握数学语言是在 AQA Pre‑U（A‑Level）数学课程中取得成功的关键第一步。无论你是在化简代数表达式、应用链式法则，还是解读统计假设检验，对术语的清晰把握都能帮你理解考题并准确表达解题思路。本指南精选了核心术语，以英文和中文对照的形式呈现，并配有便于记忆的解释和复习小贴士。你可以将其用作每日热身，也可以作为考前最后的查漏补缺。

1. Algebraic Foundations | 代数基础

Expression: A mathematical phrase that can contain numbers, variables, and operation symbols, but does not have an equals sign. For example, 3x² − 5x + 7 is an expression.

表达式： 由数字、变量和运算符号组成的数学短语，不含等号。例如，3x² − 5x + 7 是一个表达式。

Equation: A statement that two expressions are equal, always containing an equals sign. Solving an equation means finding the value(s) of the variable that make the statement true.

方程： 陈述两个表达式相等的关系式，总含有等号。解方程就是求出使该陈述成立的变量值。

Term: A single number, variable, or the product of numbers and variables separated by addition or subtraction signs. In 2x³ + 4y, the terms are 2x³ and 4y.

项： 由一个数字、变量或它们的乘积组成的单元，通过加号或减号分隔。在 2x³ + 4y 中，项为 2x³ 和 4y。

Coefficient: The numerical factor multiplying a variable in a term. In −7a²b, the coefficient is −7.

系数： 项中乘以变量的数字因数。在 −7a²b 中，系数为 −7。

Constant: A fixed value that does not change. Numbers such as 8, π, or −3 are constants.

常数： 固定不变的值。像 8、π 或 −3 这样的数字都是常数。

Like terms: Terms that have exactly the same variable parts raised to the same powers. Only like terms can be combined by addition or subtraction.

同类项： 含有完全相同变量部分且指数相同的项。只有同类项才能通过加减合并。

• Memory trick: Think of like terms as “same shape” blocks – you can stack them, while unlike terms must stay separate.
• 记忆技巧：把同类项想象成“相同形状”的积木——你可以把它们叠起来，不同形状的必须分开摆放。

2. Functions and Graphs | 函数与图像

Function: A special relationship where each input (x) is assigned exactly one output f(x). Often written as f: x → 2x + 1 or simply f(x) = 2x + 1.

函数： 一种特殊的关系，每个输入 (x) 都恰好对应一个输出 f(x)。常写作 f: x → 2x + 1 或简写为 f(x) = 2x + 1。

Domain: The complete set of possible input values (x-values) for which the function is defined.

定义域： 使得函数有意义的所有可能输入值（x 值）的集合。

Range: The set of all output values (f(x)-values) that the function actually produces.

值域： 函数实际输出的所有值（f(x) 值）的集合。

Inverse function: The function that reverses the effect of the original function, denoted by f⁻¹(x). It swaps the roles of x and y.

反函数： 逆转原函数作用的函数，记为 f⁻¹(x)。它将 x 和 y 的角色互换。

Composite function: A function created by applying one function to the result of another, written as f(g(x)) or (f ∘ g)(x).

复合函数： 将第一个函数的结果代入第二个函数而成的函数，写作 f(g(x)) 或 (f ∘ g)(x)。

One-to-one function: A function where each output comes from exactly one input, essential for having an inverse over its whole domain.

一一对应函数： 每个输出值都恰好来自一个输入值的函数，这是它在整个定义域上存在反函数的前提。

3. Coordinate Geometry | 坐标几何

Gradient (slope): The measure of steepness of a line, calculated as (change in y) / (change in x). Often denoted by m.

斜率（坡度）： 直线倾斜程度的度量，计算为 (y 的变化量) / (x 的变化量)。常用 m 表示。

Intercept: The point where a line or curve crosses an axis. The y-intercept is where x = 0, and the x-intercept is where y = 0.

截距： 直线或曲线与坐标轴相交的点。y 轴截距是 x = 0 时的点，x 轴截距是 y = 0 时的点。

Midpoint: The point exactly halfway between two given points A(x₁, y₁) and B(x₂, y₂); its coordinates are ((x₁+x₂)/2, (y₁+y₂)/2).

中点： 位于已知两点 A(x₁, y₁) 和 B(x₂, y₂) 正中间的点，坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。

Distance formula: The length between two points is found using the Pythagorean theorem: √[(x₂ − x₁)² + (y₂ − y₁)²].

距离公式： 两点间的距离用勾股定理求得：√[(x₂ − x₁)² + (y₂ − y₁)²]。

Equation of a circle: For a circle with centre (a, b) and radius r, the equation is (x − a)² + (y − b)² = r².

圆的方程： 以 (a, b) 为圆心、r 为半径的圆的方程为 (x − a)² + (y − b)² = r²。

4. Sequences and Series | 数列与级数

Arithmetic sequence: A list of numbers with a constant difference d between consecutive terms. The nth term is uₙ = a + (n − 1)d.

等差数列： 相邻两项之差为常数 d 的一列数字。第 n 项公式为 uₙ = a + (n − 1)d。

Geometric sequence: A sequence where each term is found by multiplying the previous term by a constant ratio r. The nth term is uₙ = arⁿ⁻¹.

等比数列： 每一项都等于前一项乘以常数比值 r 而得到的数列。第 n 项公式为 uₙ = arⁿ⁻¹。

Series: The sum of the terms of a sequence. An arithmetic series adds the terms of an arithmetic sequence.

级数： 数列各项之和。等差数列各项之和称为等差级数（算术级数）。

Summation notation (Σ): A compact way to write the sum of terms, e.g. Σ (from n=1 to N) of uₙ.

求和符号 (Σ)： 表示项之和的简洁写法，例如 Σ (从 n=1 到 N) uₙ。

Convergent series: An infinite series that approaches a finite limit as the number of terms increases. Geometric series converge when |r| < 1.

收敛级数： 项数无限增加时无限趋近于一个有限极限的无穷级数。等比级数在 |r| < 1 时收敛。

5. Trigonometry | 三角学

Sine, Cosine, Tangent: The three primary trigonometric ratios defined for a right‑angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

正弦、余弦、正切： 针对直角三角形定义的三个基本三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

CAST diagram: A memory aid showing which trigonometric ratios are positive in each quadrant: All positive, Sin positive, Tan positive, Cos positive going anticlockwise from the first quadrant.

CAST 图： 显示每一象限哪些三角比为正的记忆工具：从第一象限逆时针依次为 All（全正）、Sin 正、Tan 正、Cos 正。

Radian: A unit of angular measure based on arc length; π radians = 180°. One radian is the angle subtended by an arc equal to the radius.

弧度： 基于弧长的一种角度度量单位；π 弧度 = 180°。1 弧度是弧长等于半径所对的圆心角。

Trigonometric identities: Fundamental relationships such as tan θ = sin θ / cos θ and sin²θ + cos²θ = 1. They are used to simplify expressions and solve equations.

三角恒等式： 基本关系式，如 tan θ = sin θ / cos θ 和 sin²θ + cos²θ = 1。用于化简表达式和求解方程。

Period: The horizontal length of one complete cycle of a periodic function. sin θ and cos θ have period 2π; tan θ has period π.

周期： 周期函数完成一个完整循环所需的水平长度。sin θ 和 cos θ 的周期是 2π，tan θ 的周期是 π。

6. Calculus: Differentiation | 微积分：微分

Derivative: The rate of change of a function with respect to its variable, denoted by f'(x) or dy/dx. It gives the gradient of the tangent at a point.

导数： 函数关于其变量的变化率，记作 f'(x) 或 dy/dx。它表示一点处切线的斜率。

Power rule: For f(x) = xⁿ, the derivative is f'(x) = n xⁿ⁻¹. This is the fundamental rule for differentiating polynomials.

幂法则： 对于 f(x) = xⁿ，其导数为 f'(x) = n xⁿ⁻¹。这是多项式求导的基本法则。

Stationary point: A point on a curve where the gradient is zero (f'(x) = 0). It can be a maximum, minimum, or point of inflection.

驻点： 曲线上梯度为零 (f'(x) = 0) 的点。它可能是极大值点、极小值点或拐点。

Second derivative: The derivative of the derivative, written as f”(x) or d²y/dx². It tells you how the gradient is changing and helps classify stationary points.

二阶导数： 导数的导数，写作 f”(x) 或 d²y/dx²。它反映斜率如何变化，并用于判断驻点类型。

Chain rule: Used to differentiate compositions of functions: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x).

链式法则： 用于复合函数求导：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。

7. Calculus: Integration | 微积分：积分

Integral: The reverse process of differentiation, also called antiderivative. The indefinite integral ∫ f(x) dx gives a family of functions differing by a constant C.

积分： 微分的逆过程，也称原函数（反导数）。不定积分 ∫ f(x) dx 得到一族相差常数 C 的函数。

Power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. This is the counterpart to differentiation’s power rule.

积分幂法则： ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。这与微分的幂法则相对应。

Definite integral: An integral with upper and lower limits, giving the signed area under the curve between those limits. The constant C cancels out.

定积分： 具有上下限的积分，给出曲线在该区间下的有号面积。常数 C 会被消去。

Area under a curve: For a function y = f(x) that lies above the x‑axis, the area between x = a and x = b is exactly the definite integral ∫ₐᵇ f(x) dx.

曲线下方面积： 对于在 x 轴上方的函数 y = f(x)，x = a 到 x = b 之间的面积恰为定积分 ∫ₐᵇ f(x) dx。

Fundamental Theorem of Calculus: It links differentiation and integration, stating that if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

微积分基本定理： 将微分与积分联系起来，如果 F 是 f 的原函数，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。

8. Exponentials and Logarithms | 指数与对数

Exponential function: A function of the form f(x) = aˣ or specifically f(x) = eˣ, where e ≈ 2.71828. The base e appears frequently in natural growth and decay.

指数函数： 形如 f(x) = aˣ 或特别地 f(x) = eˣ 的函数，其中 e ≈ 2.71828。底数 e 在自然增长和衰减模型中频繁出现。

Logarithm: The inverse of exponentiation; logₐ b = c means aᶜ = b. Natural log (ln) has base e; common log (log₁₀) has base 10.

对数： 指数运算的逆运算；logₐ b = c 表示 aᶜ = b。自然对数 (ln) 以 e 为底，常用对数 (log₁₀) 以 10 为底。

Log laws (product, quotient, power): logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x − logₐ y; logₐ (xᵏ) = k logₐ x.

对数运算法则： logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x − logₐ y；logₐ (xᵏ) = k logₐ x。

Derivative of eˣ: The function f(x) = eˣ is its own derivative, f'(x) = eˣ. This unique property underpins exponential growth models.

eˣ 的导数： 函数 f(x) = eˣ 的导数等于它本身，f'(x) = eˣ。这一独特性质是指数增长模型的基础。

Derivative of ln x: d/dx (ln x) = 1/x, for x > 0. Combined with the chain rule, it helps integrate 1/x and differentiate logarithmic expressions.

ln x 的导数： d/dx (ln x) = 1/x（x > 0）。结合链式法则，可用来对 1/x 积分以及求对数表达式的导数。

9. Vectors | 向量

Vector: A quantity that has both magnitude (size) and direction, often represented by a directed line segment or column notation. Scalars have magnitude only.

向量： 既有大小（模）又有方向的量，常用有向线段或列矢量表示。标量只有大小。

Magnitude of a vector: For a vector v = (x, y), the magnitude is |v| = √(x² + y²). It is the length of the vector.

向量的模： 对于向量 v = (x, y)，其模为 |v| = √(x² + y²)。它就是向量的长度。

Unit vector: A vector whose magnitude is 1. The standard basis vectors are i (1, 0) and j (0, 1).

单位向量： 模为 1 的向量。标准基向量为 i (1, 0) 和 j (0, 1)。

Dot product (scalar product): For vectors a and b, the dot product is a · b = |a||b| cos θ. It produces a scalar and is used to find angles and test perpendicularity.

点积（标量积）： 对于向量 a 和 b，点积为 a · b = |a||b| cos θ。结果是一个标量，用于求夹角和判断垂直。

Vector equation of a line: r = a + t d, where a is a position vector on the line, d is the direction vector, and t is a scalar parameter.

直线的向量方程： r = a + t d，其中 a 是直线上一点的位置向量，d 是方向向量，t 为标量参数。

10. Statistics: Data and Probability | 统计：数据与概率

Mean (average): The sum of all data values divided by the number of data points. Sensitive to extreme values.

平均数（均值）： 所有数据值之和除以数据个数。易受极端值影响。

Median: The middle value when data are ordered. It is robust and not affected by outliers.

中位数： 数据排序后位于中间的值。它稳健，不受异常值影响。

Standard deviation: A measure of how spread out the data is around the mean. A low standard deviation means data points are clustered close to the mean.

标准差： 衡量数据围绕均值的分散程度。标准差小意味着数据点聚集在均值附近。

Probability distribution: A description of all possible outcomes of a random variable and their probabilities. The sum of all probabilities must be 1.

概率分布： 对随机变量所有可能结果及其概率的描述。所有概率之和必须为 1。

Normal distribution: A continuous, bell‑shaped distribution defined by mean μ and standard deviation σ. Notation: X ~ N(μ, σ²).

正态分布： 一种连续的钟形曲线分布，由均值 μ 和标准差 σ 定义。记为 X ~ N(μ, σ²)。

Hypothesis testing: A statistical method to decide whether there is enough evidence to reject a null hypothesis H₀ in favour of an alternative H₁. p‑values help make the decision.

假设检验： 一种统计方法，用于判断是否有足够证据拒绝原假设 H₀ 而支持备择假设 H₁。p 值辅助做出决定。

11. Mechanics: Kinematics | 力学：运动学

Displacement: The straight‑line distance in a given direction from a fixed origin. It is a vector quantity, distinct from distance which is a scalar.

位移： 从固定原点沿给定方向的直线距离。它是矢量，有别于标量距离。

Velocity: The rate of change of displacement with respect to time, a vector. Speed is the magnitude of velocity, a scalar.

速度： 位移对时间的变化率，为矢量。速率是速度的大小，为标量。

Acceleration: The rate of change of velocity with respect to time. It can be caused by a change in speed or direction.

加速度： 速度对时间的变化率。可由速率或方向的改变而引起。

SUVAT equations: Five equations of motion under constant acceleration relating variables s, u, v, a, t. Example: v = u + at, s = ut + ½at².

SUVAT 方程： 匀加速度下涉及变量 s、u、v、a、t 的五个运动方程。例如：v = u + at，s = ut + ½at²。

Gradient and area relationships: In velocity‑time graphs, the gradient gives acceleration, and the area under the graph gives displacement.

斜率与面积关系： 在速度‑时间图中，斜率表示加速度，图线下面积表示位移。

12. Exam Tips and Mnemonics | 考试技巧与速记法

BIDMAS/BODMAS: The order of operations reminder – Brackets, Indices (or Orders), Division, Multiplication, Addition, Subtraction. Always follow this sequence when simplifying expressions.

运算顺序口诀： 括号、指数（阶乘）、乘除、加减。代数化简时务必遵循这一顺序。

Differentiate first, then integrate: A common exam mistake is integrating a derivative without adding the constant. Remember: ∫ f'(x) dx = f(x) + C, and whenever you get a general solution, check for boundary conditions.

先微后积，常数别忘： 常见错误是对导数积分时忘记加常数。记住：∫ f'(x) dx = f(x) + C，得到通解后一定要用边界条件求特解。

Trig values trick: Use a 0°, 30°, 45°, 60°, 90° table and remember √ over 2 pattern for sin and cos: sin 0° = √0/2, sin 30° = √1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = √4/2. Cos values run in reverse.

特殊角三角函数速记： 利用 0°, 30°, 45°, 60°, 90° 的表格，记住 sin 的规律是 √0/2, √1/2, √2/2, √3/2, √4/2。cos 值正好倒过来。

Sketch graphs first: In calculus and coordinate geometry, quickly sketching the function helps visualize stationary points, intercepts, and area signs. It can prevent sign errors in definite integrals.

先画草图： 在微积分和坐标几何题中，快速画出函数草图有助于直观看出驻点、截距和面积符号，避免定积分中出现正负号错误。

Check units and domain: Mechanics problems often mix units (metres, seconds). Statistically, ensure probabilities are between 0 and 1 and that the domain of a function matches the context.

检查单位和定义域： 力学题常混用单位（米、秒）。统计学上，确保概率在 0 到 1 之间，函数的定义域要符合实际情境。

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