📚 Year 11 OCR Further Maths: Key Terminology Memorisation Guide | Year 11 OCR 进阶数学：关键术语速记指南
Mastering the specific vocabulary of OCR Level 2 Further Mathematics is just as important as solving equations. The right term can unlock the meaning of a question, prevent careless errors, and give you the confidence to explain your reasoning clearly. This guide breaks down the most critical terminology into bite-sized, memorable chunks — each term comes with a simple definition, a memory hook, and its Chinese equivalent to strengthen your dual-language understanding.
掌握 OCR 进阶数学课程中的特定词汇与掌握解题技巧同样重要。精准的术语能够帮你读懂题意、避免粗心错误，也能让你更自信地表达推理过程。本指南将最关键术语拆解成易记的小单元——每个术语都配有简洁定义、记忆钩子和对应的中文释义，帮助你强化双语理解，实现快速记忆。
1. Algebraic Essentials | 代数基础术语
Expression: A mathematical phrase without an equals sign, such as 3x² – 5x + 2. Think of it as a ‘fragment’ of a sentence rather than a full statement. Memory tip: ‘Expression has no equal tension.’
表达式：不含等号的数学式子，如 3x² – 5x + 2。可把它当作一个“短语”，而不是完整句子。记忆口诀：表达没有“等”分。
Equation: A statement that two expressions are equal, containing an equals sign. E.g., 2x + 1 = 9. Memory hook: ‘Equation has an equal relation.’
方程：两个表达式相等的陈述，含有等号。如 2x + 1 = 9。记忆法：方程二字都有“方”和“程”，但更简单——“Equal”中的equ联想到方程。
Identity: A relation that holds true for all values of the variable, often written with an ‘equals with three bars’ (≡) symbol. Example: (x+1)² ≡ x² + 2x + 1. Link: ‘Identity is identical, always true.’
恒等式：对所有变量取值都成立的关系，常用三横等号 (≡) 表示。例如 (x+1)² ≡ x² + 2x + 1。记忆：三横如铁轨，直直永不变。
Formula: A rule written in symbols, like v = u + at. Formulas are used to calculate specific quantities. Mnemonic: ‘Formula is a formal rule.’
公式：用符号表达的规则，如 v = u + at。公式用于计算特定量。记忆：公——公共的法则，式——式子。
2. Polynomials & Roots | 多项式与根
Polynomial: A sum of terms each consisting of a coefficient multiplied by a variable raised to a non-negative integer power. The degree is the highest power. Example: 4x³ – x + 7 has degree 3. Think: ‘Poly’ means many, many terms.
多项式：由系数与变量的非负整数次幂相乘构成的各项之和。次数指最高的指数。如 4x³ – x + 7 是三次多项式。记忆：多——许多项。
Factor Theorem: If f(a) = 0, then (x – a) is a factor of f(x). The reverse is also true. This is your go-to tool for splitting cubics. Hook: ‘If a is a root, (x – a) is a shoot (factor).’
因式定理：若 f(a) = 0，则 (x – a) 是 f(x) 的因式，反之亦然。这是分解三次式的核心工具。记忆：代入得零，因式诞生。
Remainder Theorem: When a polynomial f(x) is divided by (x – a), the remainder is f(a). Handy for finding remainders without long division. Memory: ‘Remainder is f(a)der.’
余式定理：多项式 f(x) 除以 (x – a) 时，余数为 f(a)。可免去长除法直接求余。记忆：余数就是 f(a) 数。
Discriminant (Δ): For quadratic ax² + bx + c = 0, Δ = b² – 4ac. If Δ > 0, two distinct real roots; Δ = 0, repeated root; Δ < 0, no real roots. Visual cue: Delta triangle (Δ) discriminates the root nature.
判别式 (Δ)：对于二次方程 ax² + bx + c = 0，Δ = b² – 4ac。Δ > 0 有两个不等实根；Δ = 0 有重根；Δ < 0 无实根。记忆：Δ 就像一把筛子，分辨根的性质。
Δ = b² – 4ac
3. Functions & Mappings | 函数与映射
Function: A rule that assigns each input exactly one output. Written as f(x). The vertical line test checks if a graph represents a function. Metaphor: a vending machine — one button gives one snack.
函数：将每个输入对应到唯一输出的规则，记作 f(x)。可用垂直线检验图像是否为函数。比喻：自动贩卖机——一个按钮只出一种零食。
Domain and Range: Domain is the set of possible inputs (x-values), range is the set of possible outputs (y-values). Memory: D before R in alphabet, x before y — Domain is x, Range is y.
定义域与值域：定义域是可能输入的集合（x值），值域是可能输出的集合（y值）。记忆：字母表中 D 在 R 前，坐标先 x 后 y——定义域管 x，值域管 y。
Inverse Function f⁻¹(x): Reverses the effect of f(x). The domain of f is the range of f⁻¹. Method: swap y and x, then solve for y. Mnemonic: inverse = reverse the journey.
反函数 f⁻¹(x)：逆转 f(x) 的映射。f 的定义域是 f⁻¹ 的值域。解法：交换 y 和 x，再解出 y。记忆：反函数就是“往回走”。
Composite Function fg(x): Apply g first, then f. fg(x) = f(g(x)). Not necessarily equal to gf(x). Think: ‘fg = first g, then f’ — alphabetical order?
复合函数 fg(x)：先执行 g，再执行 f。fg(x) 读作“f of g of x”。注意复合顺序！记忆：fg 读起来像 f 跟着 g，但实际是 g 先上车。
4. Differentiation Fundamentals | 微分基础
Derivative f'(x) or dy/dx: Represents the gradient of a curve at a point. It is the rate of change of y with respect to x. From first principles: limit of (f(x+h)-f(x))/h as h → 0. Hook: derivative = gradient function.
导数 f'(x) 或 dy/dx：表示曲线上某点的梯度，即 y 关于 x 的变化率。由第一原理定义：(f(x+h)-f(x))/h 当 h → 0 的极限。记忆：导数就是梯度函数。
Stationary Point: A point on a curve where dy/dx = 0. It can be a local maximum, minimum, or point of inflection. Use second derivative or gradient sign change to classify.
驻点：曲线上 dy/dx = 0 的点，可能是极大点、极小点或拐点。用二阶导数或梯度符号变化来判定类型。
Second Derivative f”(x): Rate of change of the gradient; tells you the concavity. If f”(x) > 0, the curve is convex (smile shape). If f”(x) < 0, it is concave (frown shape). Memory: positive second derivative = happy face ∪.
二阶导数 f”(x)：梯度的变化率，反映曲线的凹性。f”(x) > 0 则下凸（微笑形）；f”(x) < 0 则上凸（撅嘴形）。记忆：正二阶导数像笑脸 ∪。
Point of Inflection: Where the curve changes concavity, not necessarily a stationary point. The second derivative = 0 and changes sign. Mnemonic: ‘Inflection = flexing the bend.’
拐点：曲线改变凹凸性的点，不一定是驻点。需满足 f”(x) = 0 且符号改变。记忆：拐点就是弯曲方向的转折点。
5. Integration Basics | 积分初步
Indefinite Integral ∫ f(x) dx: The reverse of differentiation; it gives a family of functions plus a constant of integration +C. Think: ‘Integral is the anti-derivative.’
不定积分 ∫ f(x) dx：微分的逆运算，得到一族函数加上积分常数 +C。记忆：积分是反向的微分，每次都要加 C。
Definite Integral ∫ₐᵇ f(x) dx: Gives the signed area under the curve y = f(x) from x = a to x = b. Evaluated as F(b) – F(a). Memory: limits a to b give area between a and b.
定积分 ∫ₐᵇ f(x) dx：计算曲线 y = f(x) 下从 x = a 到 x = b 的有向面积。结果为 F(b) – F(a)。记忆：有上下限的积分求面积。
Constant of Integration +C: Essential for indefinite integrals because differentiation of a constant is zero. Always add +C unless finding a definite integral. Reminder: ‘Don’t forget the +C sea.’
积分常数 +C：不定积分必须加常数，因为常数的导数为零。除非计算定积分，否则永远不要忘记 +C。口诀：积分不加 C，考试泪垂垂。
6. Matrices & Transformations | 矩阵与变换
Matrix (plural: matrices): A rectangular array of numbers. In further maths, we focus on 2×2 matrices for geometric transformations. Think of a matrix as an operator that maps points.
矩阵：数字的矩形阵列。进阶数学主要利用 2×2 矩阵实现几何变换。可把矩阵看作一个映射点的“操作器”。
Identity Matrix I = [[1,0],[0,1]]: Leaves points unchanged when multiplied. Mirror: ‘I’ for do nothing.’
单位矩阵 I = [[1,0],[0,1]]：乘以任何点或矩阵都不改变它。记忆：I 代表“我”不动。
Determinant of A = ad – bc: For matrix A = [[a,b],[c,d]], det(A) = ad – bc. If determinant = 0, the matrix is singular (no inverse) and transformation collapses area to zero.
行列式 ad – bc：对于矩阵 A = [[a,b],[c,d]]，det = ad – bc。行列式为零则矩阵奇异（无逆），变换将面积压缩为零。
Inverse Matrix A⁻¹: A⁻¹A = I. Used to reverse a transformation. Exists only if determinant ≠ 0. Memory: inverse reversal, just like numbers.
逆矩阵 A⁻¹：满足 A⁻¹A = I。用于逆转变换。只有行列式非零才存在。
Transformation types: Common 2×2 matrices represent reflection (e.g., in x-axis [[1,0],[0,-1]]), rotation ([[cosθ,-sinθ],[sinθ,cosθ]]), enlargement (kI), and shear. Learn the patterns to recognise them instantly.
常见变换：2×2 矩阵可表示反射（如 x 轴反射 [[1,0],[0,-1]]）、旋转（[[cosθ,-sinθ],[sinθ,cosθ]]）、放大 (kI) 和剪切。牢记模式，快速辨认。
| Transformation | Matrix Pattern |
|---|---|
| Reflection in x-axis | [[1,0],[0,-1]] |
| Rotation 90° anticlockwise | [[0,-1],[1,0]] |
| Enlargement scale factor k | [[k,0],[0,k]] |
7. Vectors | 向量
Vector: A quantity with both magnitude and direction, often represented as a column vector or i, j notation. E.g., 3i + 4j. Contrast with scalar (magnitude only).
向量：既有大小又有方向的量，常用列向量或 i, j 表示，如 3i + 4j。标量只有大小。
Magnitude |v|: Length of the vector, found using Pythagoras: |3i+4j| = √(3²+4²) = 5. Mnemonic: magnitude = how ‘magni’ficent the distance is.
模 |v|：向量的长度，用勾股定理计算。例如 |3i+4j| = 5。记忆：模就是“有多长”。
Unit Vector: A vector of length 1, giving direction only. To find a unit vector in the direction of v, use v/|v|. Notation: often v̂. Hint: unit = ‘you knit’ a vector of length 1.
单位向量：长度为 1 的向量，仅指示方向。求法为 v/|v|。记忆：单位——长度为一。
Dot Product (Scalar Product) a·b: = |a||b|cosθ. For column vectors, sum of products of components. Used to find angles between vectors. If a·b = 0, vectors are perpendicular.
点积（标量积）a·b：= |a||b|cosθ。对列向量，为对应分量乘积之和。点积为零则向量垂直。
8. Trigonometry & Identities | 三角学与恒等式
Basic Ratios: sin, cos, tan: Defined on right-angled triangles (SOHCAHTOA) and on the unit circle. For an angle θ, sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent.
基本比：sin, cos, tan：在直角三角形和单位圆上定义。SOHCAHTOA 口诀记忆对边/斜边、邻边/斜边、对边/邻边。
Radians: Alternative angle measure in terms of π. π radians = 180°. Crucial for calculus. Memory: ‘Radius wrap’ — wrapping the radius around a circle creates one radian.
弧度：以 π 计量的另一种角度单位，π rad = 180°。在微积分中非常重要。记忆：弧长等于半径时的角度为 1 弧度。
Pythagorean Identity: sin²θ + cos²θ ≡ 1. Derived from the unit circle. Related forms: 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ (GCSE maybe not required, but useful). Core identity: ‘sin²+cos² equals one.’
勾股恒等式：sin²θ + cos²θ ≡ 1。源自单位圆。常用变形：1 + tan²θ = sec²θ。牢记基本版。
Exact Trigonometric Values: Know sin/cos/tan of 0°, 30°, 45°, 60°, 90° without calculator. Use the triangle trick: 45° right triangle legs 1,1,√2; 30°-60° triangle sides 1,√3,2.
特殊角精确值：熟记 0°, 30°, 45°, 60°, 90° 的三角函数值。巧用等腰直角三角形和 30-60 三角形。
sin 30° = ½, cos 30° = √3/2, tan 45° = 1
9. Sequences & Series | 数列与级数
Sequence: An ordered list of numbers following a rule. E.g., 3, 7, 11, 15, … is arithmetic. Notation: uₙ or aₙ for the nth term.
数列：按一定规则排列的一列数，如等差数列 3, 7, 11, 15 … 通项记作 uₙ。
Arithmetic Progression (AP): Each term is obtained by adding a constant difference d. nth term: a + (n-1)d. Sum of first n terms: Sₙ = n/2 (2a + (n-1)d). Memory: AP = Add Pattern.
等差数列 (AP)：每一项加固定公差 d。通项：a + (n-1)d。求和：Sₙ = n/2 (2a + (n-1)d)。记忆：等差就是“等差额”。
Geometric Progression (GP): Each term is multiplied by a constant ratio r. nth term: arⁿ⁻¹. Sum if |r|<1: S∞ = a/(1-r). Warning: ratio matters.
等比数列 (GP)：每一项乘固定公比 r。通项：arⁿ⁻¹。无穷等比收敛和 S∞ = a/(1-r)。
Sigma Notation Σ: Compact way to write sums. Σ (from i=1 to n) f(i) means sum of f(i) for i=1…n. Think: ‘Sigma sums it all.’
求和符号 Σ：紧凑地表示累加。Σ（i=1 到 n）f(i) 表示 f(1)+f(2)+…+f(n)。记忆：Σ 像个爪子抓总和。
10. Numerical Methods | 数值方法
Trapezium Rule: Approximate area under a curve using n strips. Area ≈ h/2 [y₀ + 2(y₁+y₂+…+yₙ₋₁) + yₙ], where h = (b-a)/n. Remember the pattern: ‘first and last once, others twice.’
梯形法则：用 n 个梯形条近似曲线下面积。面积 ≈ h/2 [y₀ + 2(y₁+…+yₙ₋₁) + yₙ]。记忆：首尾各一次，中间乘以二。
Sign Change Method: To locate roots of f(x)=0, find interval [a,b] where f(a) and f(b) have opposite signs. At least one root lies in between. Called a “change of sign” interval.
符号变化法：寻找 f(x)=0 的根时，若 f(a) 与 f(b) 异号，则 [a,b] 内至少有一个根。记忆：异号必有根。
Iterative Formula xₙ₊₁ = g(xₙ): Starting from x₀, repeatedly apply formula to converge to a root. Key: look for a “staircase” or “cobweb” diagram in your mind.
迭代公式 xₙ₊₁ = g(xₙ)：从初值 x₀ 开始，重复代入公式收敛到根。可联想阶梯图或蛛网图。
11. Coordinate Geometry | 坐标几何
Gradient m: (y₂ – y₁)/(x₂ – x₁). Tells you the steepness of a line. Positive m goes uphill; negative m goes downhill. Zero gradient = horizontal, infinite gradient = vertical.
梯度 m：(y₂ – y₁)/(x₂ – x₁)，表示直线的倾斜程度。正梯度上坡，负梯度下坡，零梯度水平。
Equation of a Straight Line: y – y₁ = m(x – x₁). Also y = mx + c
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