📚 Year 10 CCEA Further Mathematics: Key Terminology Quick Reference Guide | Year 10 CCEA 进阶数学：词汇术语速记指南
Mastering the vocabulary of Further Mathematics is the first step towards confident problem‑solving. This guide covers the essential terms you will meet in the CCEA Year 10 specification, organised by topic. Each term is paired with a plain‑English definition, a memory tip where useful, and a Chinese mirror to help you build a bilingual mathematical toolkit.
攻克进阶数学词汇是自信解题的第一步。这份指南按照主题整理了 CCEA Year 10 考纲中的核心术语。每个词条都配有通俗的英文定义、实用的记忆提示以及对应的中文释义，帮助同学们构建中英双语数学思维工具箱。
1. Algebraic Foundations | 代数基础
Variable – A symbol, usually a letter, that stands for an unknown or changeable number. Think of it as a ‘placeholder’ that can take different values. Memory tip: ‘Vary’ + ‘able’.
变量 – 一个通常用字母表示的符号，代表未知或可变的数。可以把它看作是一个可以取不同值的“占位符”。记忆窍门：可“变”的量。
Coefficient – The number multiplying a variable in a term. In 5x², 5 is the coefficient. Memory trick: ‘co‑’ means together; the coefficient works together with the variable.
系数 – 代数项中与变量相乘的数字。例如 5x² 中，5 是系数。记忆窍门：系数和变量“协同”出现。
Expression – A mathematical phrase made up of numbers, variables and operations (no equals sign). Example: 3a + 2b. Contrast with equation, which has an equals sign.
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 3a + 2b。与含有等号的方程区分开来。
Term – A single number, variable or product of the two, separated by ‘+’ or ‘−’. In 4xy − 7, both 4xy and −7 are terms.
项 – 单个数字、变量或二者的乘积，由加号或减号分隔。在 4xy − 7 中，4xy 和 −7 都是项。
Polynomial – An algebraic expression with one or more terms, with non‑negative integer exponents. x² + 2x + 1 is a polynomial; 1/x is not.
多项式 – 含有一个或多个项、指数为非负整数的代数式。如 x² + 2x + 1 是多项式；1/x 则不是。
Like terms – Terms that have exactly the same variable part (same letters and powers). 3x² and −5x² are like; 3x² and 3x are not. You can only add/subtract like terms.
同类项 – 变量的次数和字母部分完全相同的项。例如 3x² 和 −5x² 是同类项；而 3x² 和 3x 不是。只能对同类项进行加减合并。
2. Functions and Graphs | 函数与图像
Function – A rule that links each input to exactly one output. Written as f(x) = … . The ‘vertical line test’ on a graph shows if a relation is a function.
函数 – 将每个输入对应到唯一输出的规则，记作 f(x) = …。图像上的“竖线检验”可以判断一个关系是否为函数。
Domain – The set of all possible input values (x‑values) for a function. For f(x) = √x, the domain is x ≥ 0.
定义域 – 函数所有可能输入值（x 值）的集合。例如 f(x) = √x 的定义域是 x ≥ 0。
Range – The set of all possible output values (y‑values) that the function produces.
值域 – 函数所有可能输出值（y 值）的集合。
Turning point – A point on a graph where the curve changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum). At a turning point the gradient is zero.
拐点/转向点 – 图像上曲线由上升转为下降（局部极大值）或由下降转为上升（局部极小值）的点。在拐点处斜率为零。
Asymptote – A line that a graph approaches but never touches. Hyperbolas often have horizontal and vertical asymptotes.
渐近线 – 图像无限接近但永远不触及的一条直线。双曲线常带有水平和垂直渐近线。
Composite function – A function made by applying one function to the output of another. Written as f(g(x)), pronounced ‘f of g of x’. Work from the inside out.
复合函数 – 将一个函数的输出作为另一个函数的输入而生成的函数，记作 f(g(x))。计算时由内向外进行。
3. Introduction to Calculus | 微积分初步
Derivative – The instantaneous rate of change of a function, also called the gradient of the curve. Notation: dy/dx or f ‘(x). For y = xⁿ, the derivative is nxⁿ⁻¹.
导数 – 函数的瞬时变化率，也称作曲线的斜率。符号为 dy/dx 或 f ‘(x)。对于 y = xⁿ，其导数为 nxⁿ⁻¹。
Differentiation – The process of finding the derivative. Often used to find gradients, velocities and turning points.
微分 – 求导的过程。常用于求梯度、速度和拐点。
Second derivative – The derivative of the derivative, written d²y/dx² or f ”(x). It tells you the rate of change of the gradient and helps classify turning points (min/max).
二阶导数 – 导数的导数，记作 d²y/dx² 或 f ”(x)。它表示斜率的变化率，用于判断拐点的性质（极小或极大）。
Stationary point – Any point where the first derivative equals zero (dy/dx = 0). Includes local maxima, minima and points of inflection.
驻点/稳定点 – 一阶导数为零（dy/dx = 0）的点，包括局部极大点、极小点和拐点。
Indefinite integral – The reverse of differentiation, also called antiderivative. Notation: ∫ f(x) dx. Always includes a constant of integration +C.
不定积分 – 微分的逆运算，又称原函数。符号为 ∫ f(x) dx，结果总要加上积分常数 +C。
Definite integral – The integral with limits, giving a numerical value. It represents the area under a curve between two x‑values.
定积分 – 带有上下限的积分，结果是一个数值。它的几何意义是曲线下两 x 值之间的面积。
4. Trigonometry and Circular Functions | 三角学与圆函数
Sine, Cosine, Tangent – The three primary trigonometric ratios. For an angle θ in a right‑angled triangle: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. Memory aid: SOH CAH TOA.
正弦、余弦、正切 – 三个基本三角比。对于直角三角形中的角θ：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。记忆口诀：SOH CAH TOA（对斜、邻斜、对邻）。
Radian – An alternative unit of angle measure. 1 radian is the angle subtended by an arc equal in length to the radius. π radians = 180°.
弧度 – 角度的一种度量单位。1 弧度是弧长等于半径时所对的圆心角。π 弧度 = 180°。
Period – The interval over which a trigonometric function’s shape repeats. The period of sinx and cosx is 2π (360°); tanx has period π (180°).
周期 – 三角函数图形重复出现的最小间隔。sinx 和 cosx 的周期为 2π（360°）；tanx 的周期为 π（180°）。
Amplitude – Half the vertical distance between the maximum and minimum values of a wave. For y = a sinx, the amplitude is |a|.
振幅 – 波形最大值与最小值之间的垂直距离的一半。对于 y = a sinx，振幅为 |a|。
Trigonometric identity – An equation involving trig functions that holds true for all allowed angles. Key one: sin²θ + cos²θ = 1.
三角恒等式 – 涉及三角函数的、对所有允许的角度都成立的等式。最重要的一个是 sin²θ + cos²θ = 1。
5. Vectors and Matrices | 向量与矩阵
Vector – A quantity that has both magnitude (size) and direction. Written as a column matrix or as a bold/low‑case letter with arrow. Contrast with scalar, which has only magnitude.
向量 – 既有大小又有方向的量。可用列矩阵或带箭头的粗体/小写字母表示。与只有大小的标量相对。
Magnitude – The length of a vector. For a vector (a, b), magnitude = √(a² + b²), found by Pythagoras.
模/大小 – 向量的长度。对于向量 (a, b)，模 = √(a² + b²)，用勾股定理求得。
Unit vector – A vector with magnitude 1. Often used to show direction. In 2D, the standard unit vectors are i (1,0) and j (0,1).
单位向量 – 模为 1 的向量，常用来表示方向。二维中的标准单位向量是 i (1,0) 和 j (0,1)。
Matrix – A rectangular array of numbers, organised in rows and columns. Used to store data and to perform transformations (rotation, reflection, enlargement).
矩阵 – 按行和列排列的数字矩形阵列。用于存储数据和表示几何变换（旋转、反射、放大）。
Determinant – A single number calculated from a square matrix. For 2×2 matrix (a b; c d), det = ad − bc. Used to find inverse and area scale factor.
行列式 – 由方阵算出的一个数值。对于 2×2 矩阵 (a b; c d)，行列式 = ad − bc。用于求逆矩阵和面积比例因子。
Identity matrix – The square matrix that acts like ‘1’ in matrix multiplication. 2×2 identity: (1 0; 0 1). Multiplying by it leaves the original matrix unchanged.
单位矩阵 – 在矩阵乘法中起“1”作用的方阵。2×2 单位矩阵为 (1 0; 0 1)。任何矩阵与之相乘保持不变。
6. Probability and Statistics | 概率与统计
Sample space – The set of all possible outcomes of an experiment. When rolling a fair die, the sample space is {1,2,3,4,5,6}.
样本空间 – 一个实验所有可能结果的集合。例如掷一粒公正的骰子，样本空间为 {1,2,3,4,5,6}。
Mutually exclusive – Two events that cannot happen at the same time. P(A and B) = 0. Flipping a coin cannot be both heads and tails simultaneously.
互斥事件 – 不可能同时发生的两个事件。P(A 且 B) = 0。抛硬币不能同时是正面和反面。
Independent events – Events where the outcome of one does not affect the other. For independent events, P(A and B) = P(A) × P(B).
独立事件 – 一个事件的发生不影响另一个事件发生的概率。对于独立事件，P(A 且 B) = P(A) × P(B)。
Conditional probability – The probability of an event happening given that another event has already occurred. Notation: P(A|B). Use the formula P(A|B) = P(A and B) / P(B).
条件概率 – 在已知另一事件已经发生的条件下，某事件发生的概率。符号为 P(A|B)，公式为 P(A|B) = P(A 且 B) / P(B)。
Mean, Median, Mode – Measures of central tendency. Mean is the average (Σx/n). Median is the middle value. Mode is the most frequent value. For grouped data, use mid‑interval values.
平均数、中位数、众数 – 集中趋势的量度。平均数是算术平均（Σx/n），中位数是中间值，众数是出现频率最高的值。对分组数据，用组中值计算。
Interquartile range (IQR) – A measure of spread: IQR = upper quartile − lower quartile. It captures the middle 50% of data, ignoring extreme values.
四分位距 (IQR) – 离散程度的量度：IQR = 上四分位数 − 下四分位数。它覆盖中间 50% 的数据，不受极端值影响。
7. Mechanics Vocabulary | 力学词汇
Displacement – The straight‑line distance from a starting point in a specified direction (a vector). Different from distance, which is a scalar path length.
位移 – 从起点沿特定方向的直线距离（向量）。与路程不同，路程是标量，指实际路径的长度。
Velocity – The rate of change of displacement with time; it is a vector. Speed is the scalar magnitude of velocity.
速度 – 位移随时间的变化率，是向量。速率 是速度的大小，为标量。
Acceleration – The rate of change of velocity with time. Can be positive (speeding up) or negative (slowing down). Unit: m/s².
加速度 – 速度随时间的变化率。可以是正值（加速）或负值（减速）。单位：米/秒²。
Resultant force – The single force that has the same effect as all the original forces acting together. Found by vector addition.
合力 – 与所有原作用力总效果相同的单一力，由向量加法求出。
Equilibrium – A state where the resultant force on an object is zero. The object stays at rest or moves with constant velocity.
平衡状态 – 物体所受合力为零的状态。物体保持静止或匀速直线运动。
SUVAT equations – Five equations linking displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t) for motion in a straight line with constant acceleration.
SUVAT 方程 – 描述匀加速直线运动中位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）之间关系的五个方程。
8. Sequences, Series and Logarithms | 数列、级数与对数
Arithmetic sequence – A number list where the difference between consecutive terms is constant, called the common difference (d). 2, 5, 8, 11,… is arithmetic.
等差数列 – 相邻两项的差为常数的数列，这个常数叫作公差（d）。例如 2, 5, 8, 11,…。
Geometric sequence – A sequence where the ratio between consecutive terms is constant, called the common ratio (r). 3, 6, 12, 24,… is geometric with r = 2.
等比数列 – 相邻两项的比值为常数的数列，这个常数叫作公比（r）。例如 3, 6, 12, 24,… 是公比为 2 的等比数列。
Limits and convergence – As a sequence’s term number increases, if the terms approach a fixed value L, the sequence converges to L. Geometric series with |r|<1 converge.
极限与收敛 – 随着项数增大，数列的项趋于某个固定值 L，则称该数列收敛于 L。公比的绝对值小于 1 的等比数列收敛。
Logarithm – The inverse of exponentiation. logₐb = c means aᶜ = b. Common bases: base 10 (log) and base e (ln). Key law: loga(xy) = loga x + loga y.
对数 – 指数的逆运算。logₐb = c 表示 aᶜ = b。常用底数：以 10 为底（记作 log）和以 e 为底（记作 ln）。主要法则：loga(xy) = loga x + loga y。
Exponential growth/decay – A model where a quantity increases/decreases at a rate proportional to its current value, often written as N = N₀ekt (growth for k > 0, decay for k < 0).
指数增长/衰减 – 一个量的变化速率与其当前值成正比，通常写作 N = N₀ekt（k > 0 为增长，k < 0 为衰减）。
9. Coordinate Geometry and Inequalities | 坐标几何与不等式
Gradient – The steepness of a line, found by (change in y)/(change in x) or m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1.
斜率/梯度 – 直线的倾斜程度，计算公式为 (y 的变化量)/(x 的变化量) 或 m = (y₂ − y₁)/(x₂ − x₁)。平行线斜率相等，垂直线的斜率乘积为 −1。
Midpoint – The point exactly halfway between two given points. Coordinates: ((x₁+x₂)/2, (y₁+y₂)/2).
中点 – 两点连线的正中间点。坐标公式：((x₁+x₂)/2, (y₁+y₂)/2)。
Distance formula – Derived from Pythagoras: distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²].
距离公式 – 由勾股定理导出：两点 (x₁,y₁) 和 (x₂,y₂) 之间的距离为 √[(x₂−x₁)² + (y₂−y₁)²]。
Equation of a circle – In standard form: (x − a)² + (y − b)² = r², where (a,b) is the centre and r is the radius. Complete the square to convert from general form.
圆的方程 – 标准形式：(x − a)² + (y − b)² = r²，其中 (a,b) 为圆心，r 为半径。可通过配方法将一般式化为标准式。
Linear inequality – Like an equation but with <, >, ≤ or ≥. Solution is a region. Graphing: use a dashed line for strict inequalities and shade the correct side.
线性不等式 – 形如方程，但使用 <、>、≤ 或 ≥。解为一个区域。作图时严格不等式用虚线，并正确涂阴影。
Feasible region – In linear programming, the set of all points that satisfy all given inequalities (constraints). The optimal solution lies at a vertex of this region.
可行域 – 在线性规划中，满足所有约束不等式的点的集合。最优解出现在可行域的某一个顶点上。
10. Memory Strategies and Exam Tips | 记忆策略与应试技巧
Active recall – Instead of rereading notes, close the book and write down everything you remember about a topic. Then check for gaps. This strengthens long‑term memory much more effectively than passive reading.
主动回忆 – 与其反复阅读笔记，不如合上书本，写下某个主题你能记住的所有内容，然后再查漏补缺。这种方法比被动阅读更能有效强化长期记忆。
Spaced repetition – Review new vocabulary after one day, then after three days, a week, and a month. Short, frequent reviews embed terms into long‑term memory.
间隔重复 – 学习新词汇后，分别在一天后、三天后、一周后和一个月后复习。短时高频的复习能将术语牢固地嵌入长期记忆中。
Chunking and grouping – Group related terms together. For example, all differentiation vocabulary can be learned as a family: derivative, stationary point, second derivative, increasing/decreasing. Creating mind maps helps visual learners.
组块与归类 – 将相关术语分组学习。例如，把所有微分词汇作为一个家族：导数、驻点、二阶导数、递增/递减。思维导图对视觉型学习者尤其有效。
Dual coding – Combine words with simple sketches, graphs or symbols. Drawing a quick gradient triangle next to the word ‘derivative’ reinforces the connection between the term and its visual meaning.
双重编码 – 把文字与简单的示意图、图像或符号结合起来。在“导数”一词旁边画一个斜率三角形，能强化术语与其视觉含义之间的联结。
Exam‑style practice – Use past paper questions to see how terminology appears in context. When a question asks for ‘a stationary point and its nature’, you must 1) set dy/dx = 0, 2) find x and y coordinates, 3) use the second derivative to classify it. Linking words to procedures builds automaticity.
真题练习 – 通过历年真题观察术语在题目中的实际应用。当题目要求“求驻点并判断其性质”时，你需要 1) 令 dy/dx = 0, 2) 求 x 和 y 坐标, 3) 用二阶导数判断极大还是极小。把词汇与解题流程挂钩，能培养条件反射式的解题能力。
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