📚 Year 9 AQA Further Maths: Key Terminology Memory Guide | Year 9 AQA 进阶数学：词汇术语速记指南
Mastering key terminology is the first step towards excelling in AQA Further Maths at Year 9. This bilingual guide breaks down essential terms, offers clever memory tricks, and pairs every English definition with its Chinese equivalent, helping you build a solid vocabulary foundation that will carry you through the entire course.
掌握核心术语是 Year 9 AQA 进阶数学拔尖的第一步。这本双语指南拆解关键术语，提供巧妙的记忆技巧，并将每条英文定义与中文对应配对，帮助你打下牢固的词汇基础，贯穿整个课程。
1. Algebra Essentials | 代数基础术语
Variable: A symbol, usually a letter, that represents an unknown or changeable number. Think of 'vary' – its value can change. Example: in 3x + 2, x is the variable.
变量：代表未知或可变值的符号，通常是字母。联想“变化”——它的值可以改变。例如 3x + 2 中，x 是变量。
Coefficient: The number multiplying a variable. Memory trick: 'co-' means together, and it works efficiently with the variable. In 5y, 5 is the coefficient.
系数：乘以变量的数字。记忆法：“co-”表示共同，它与变量一起起作用。在 5y 中，5 是系数。
Constant: A fixed number that stands alone. It does not ‘vary’. In m + 7, the number 7 is the constant.
常数：固定的数字，不“变化”。在 m + 7 中，数字 7 是常数。
Expression: A collection of terms connected by + or −, with no equals sign. For instance, 4a − 2b + 1. It expresses a calculation but does not assert equality.
表达式：由 + 或 − 连接的项的组合，没有等号。例如 4a − 2b + 1。它表达了一种运算，但并不声明相等关系。
Equation equal, always containing an equals sign. Example: 2x + 3 = 11. Solve to find the unknown.
方程：表明两个表达式相等的数学陈述，始终包含等号。例：2x + 3 = 11。求解未知数。
Identity: An equation that is true for all values of the variable, often denoted by the triple bar ≡. For example, (x+1)² ≡ x² + 2x + 1. Memory: 'identical' for all inputs.
恒等式：对所有变量值都成立的等式，常用三横线 ≡ 表示。例：(x+1)² ≡ x² + 2x + 1。记忆法：“恒”意味着始终相同。
2. Equations and Inequalities | 方程与不等式
Linear equation ower 1, forming a straight line when graphed. General form: ax + b = 0. Simplicity is key: 'linear = line'.
线性方程：变量为一次方的方程，图像是一条直线。一般式：ax + b = 0。关键点：“线性即直线”。
Quadratic equation: An equation of degree 2, containing x². Standard form: ax² + bx + c = 0. The word ‘quad’ suggests four, but here it comes from ‘quadratus’ meaning square.
二次方程：次数为 2 的方程，包含 x²。标准型：ax² + bx + c = 0。“quad”通常指四，但此处源自拉丁语“quadratus”意为平方。
Root / Solution variable. Often called the x-intercept on a graph. If 2² − 5×2 + 6 = 0, then 2 is a root.
根 / 解：代入变量后使方程成立的值。在图像上常称为 x 截距。如果 2² − 5×2 + 6 = 0，那么 2 是一个根。
Discriminant: For ax²+bx+c=0, the discriminant Δ = b² − 4ac. It ‘ sitive Δ gives two real roots, zero gives one repeated root, negative gives no real roots.
判别式：对于 ax²+bx+c=0，判别式 Δ = b² − 4ac。它“判别”根的性质：Δ 大于 0 有两个实根，等于 0 一个重根，小于 0 无实根。
Inequality notation lity; non-strict (≤, ≥) include equality. Memory: the open side faces the larger quantity.
不等号记法：符号 <，>，≤，≥ 表示两边不相等的关系。严格不等号 (<, >) 不包括相等；非严格 (≤, ≥) 包括相等。记忆：开口朝向较大的量。
3. Polynomials and Factorisation | 多项式与因式分解
Polynomial: An expression with multiple terms made of variables raised to non‑negative integer powers. ‘Poly’ means many, ‘nomial’ means term. Example: 3x⁴ − 2x² + x − 7.
多项式：由多个项组成的表达式，变量指数为非负整数。“多项”意指多个项。例：3x⁴ − 2x² + x − 7。
Degree: The highest power of the variable in a polynomial. In 5x³ + 2x − 1, the degree is 3. It tells us how many roots to potentially expect.
次数：多项式中变量的最高幂。在 5x³ + 2x − 1 中，次数为 3。它提示我们最多可能有多少个根。
Factorisation: Breaking a polynomial into a product of simpler expressions. It’s like ‘reverse expanding’. Factorise x² − 9 into (x−3)(x+3). A factor is a building block.
因式分解：把多项式拆成更简单表达式的乘积。类似“展开的逆运算”。将 x² − 9 分解为 (x−3)(x+3)。因式就像积木块。
Difference of two squares as (a−b)(a+b). Recognising this pattern speeds up working. Example: 4x² − 25 = (2x)² − 5².
平方差公式：特殊形式 a² − b² 可分解为 (a−b)(a+b)。识别这一模式能加快运算。例：4x² − 25 = (2x)² − 5²。
Completing the square: Rewriting a quadratic x² + bx as (x + b/2)² − (b/2)². This technique reveals the vertex of a parabola and is essential for solving and sketching.
配方法：将二次式 x² + bx 改写为 (x + b/2)² − (b/2)²。该技巧揭示抛物线的顶点，是求解和画图的关键方法。
4. Sequences and Series | 序列与级数
Sequence: An ordered list of numbers following a rule. Each number is a term. The rule may be explicit (nth term) or recursive (using previous terms).
序列：按规则排列的一列数。每个数称为一项。规则可以是用 n 表示的显式公式，也可以是依赖前一项的递推关系。
nth term: A formula that generates any term of a sequence by substituting the position number n. For the sequence 3, 7, 11, 15..., the nth term is 4n − 1.
通项公式：代入位置编号 n 就能生成序列中任意项的公式。对于序列 3, 7, 11, 15...，通项为 4n − 1。
Arithmetic progression cutive terms is constant (common difference d). Example: 2, 5, 8, 11,... with d = 3. Think ‘add the same amount’.
算术序列：相邻项之差为常数（公差 d）的序列。例：2, 5, 8, 11,...，d = 3。记住“每次都加同样的数”。
Geometric progression tant (common ratio r). Example: 3, 6, 12, 24,... with r = 2. The growth is multiplicative.
几何序列：每一项由前一项乘一个常数（公比 r）得到的序列。例：3, 6, 12, 24,...，r = 2。增长是倍数关系。
Fibonacci sequence: A famous recursive sequence starting 1, 1, 2, 3, 5, 8,... where each term is the sum of the two preceding. It appears in nature, art and architecture.
斐波那契数列：著名的递推序列，前两项为 1, 1，之后每一项是前两项之和。出现在自然界、艺术和建筑中。
5. Introduction to Trigonometry | 三角函数入门
Hypotenuse: The longest side of a right‑angled triangle, opposite the right angle. Memory: 'hypo' means under, but it stretches across – it is ‘subjected’ to both legs.
斜边：直角三角形中最长的边，直角的对边。记忆法：斜边像是“斜躺”在直角对面。
Opposite: The side facing the angle of interest (θ). It varies depending on which angle you choose. If θ changes, the opposite side changes.
对边：正对所考察角 (θ) 的边。选择不同的角，对边随之改变。
Adjacent: The side next to the angle θ (excluding the hypotenuse). Together with the opposite, it forms the arms of the angle.
邻边：与角 θ 相邻的边（不含斜边）。与对边共同构成角的两边。
Sine, Cosine, Tangent cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The classic mnemonic is SOHCAHTOA.
正弦，余弦，正切：基本比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。经典记忆口诀：SOHCAHTOA。
SOHCAHTOA osine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. Chant it rhythmically.
SOHCAHTOA 口诀：三角比值的记忆法：Sin = 对/斜，Cos = 邻/斜，Tan = 对/邻。有节奏地念出来就能记住。
6. Trigonometric Ratios and Identities | 三角比值与恒等式
Exact values for special angles: 30°, 45° and 60° appear so often that their sin, cos and tan values must be memorised. Use the table below as a quick reference.
特殊角的精确值：30°、45° 和 60° 出现频繁，必须记住它们的正弦、余弦和正切值。下表可供快速查阅。
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Pythagorean Identity for any angle θ. It derives from the unit circle and the Theorem of Pythagoras. Memory: it's the trig version of a² + b² = c².
毕达哥拉斯恒等式：对任意角 θ，sin²θ + cos²θ = 1 恒成立。它源自单位圆和勾股定理。记忆：这是 a² + b² = c² 的三角版本。
Tangent Identity s you find tan from sine and cosine, and it helps simplify many trigonometric expressions.
正切恒等式：tan θ = sin θ / cos θ。该关系可用正弦和余弦求出正切，并帮助简化许多三角表达式。
7. Vectors | 向量
Vector: A quantity that has both magnitude (size) and direction. Represented by a bold letter v or an arrow above. Displacement and velocity are vectors; speed is not.
向量：既有大小又有方向的量。用粗体 v 或上加箭头表示。位移和速度是向量；速率则不是。
Scalar: A quantity with only magnitude, no direction. Examples: distance, mass, temperature. Vectors can be multiplied by scalars to change their length.
标量：只有大小没有方向的量。如距离、质量、温度。向量可乘以标量以改变其长度。
Magnitude or : The length or size of a vector, calculated using Pythagoras. For a column vector (x, y), magnitude = √x² + y². It's always a non‑negative number.
模：向量的长度或大小，用勾股定理计算。对于列向量 (x, y)，模 = √x² + y²。总是非负数。
Column vector: A way to write a vector as two numbers stacked vertically, representing horizontal and vertical components. e.g. (3, 4) written as [3 over 4] means 3 right, 4 up.
列向量：将向量写成上下排列的两个数字，表示水平分量和垂直分量。如 (3,4) 写成 [3 上 4] 表示向右 3，向上 4。
Resultant vector: The single vector that has the same effect as two or more vectors added together. Found by adding component‑wise. It's the 'combined effect'.
合向量：与两个或多个向量相加后作用相同的单个向量。通过分量相加得到。即“总效果”。
8. Matrices | 矩阵
Matrix: A rectangular array of numbers arranged in rows and columns. Used to store data, perform transformations and solve systems. Individual numbers are called elements.
矩阵：按行和列排列的数字矩形阵列。用于存储数据、进行变换和求解方程组。其中的每个数叫做元素。
Order of a matrix: Described as rows × columns.
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