📚 Year 9 WJEC Further Maths: Vocabulary & Terminology Quick Reference Guide | WJEC Year 9 进阶数学：词汇术语速记指南
Mastering the language of mathematics is the first step towards confident problem-solving. This guide brings together the essential terms you will encounter in Year 9 WJEC Further Maths, explained clearly with paired English and Chinese explanations. Use it to revise, to spot connections between topics, and to build a solid foundation for GCSE-level studies.
掌握数学语言是自信解题的第一步。本指南汇集了你在 WJEC Year 9 进阶数学中会遇到的核心术语，并提供清晰的中英双语对照解释。用它来复习、发现知识点之间的联系，并为 GCSE 阶段的学习打下扎实基础。
1. Basic Operations and Indices | 基本运算与指数
At the heart of Further Maths lie the four basic operations and the laws of indices. Understanding these terms allows you to manipulate numbers and algebraic expressions fluently.
进阶数学的核心在于四则运算和指数定律。理解这些术语能让你熟练地处理数字和代数表达式。
Sum: The result of adding two or more numbers. For example, in 3 + 7 = 10, 10 is the sum.
和：两个或多个数相加的结果。例如，在 3 + 7 = 10 中，10 是和。
Product: The result of multiplying numbers or expressions. 6 × 4 = 24, so 24 is the product.
积：数或表达式相乘的结果。6 × 4 = 24，所以 24 是积。
Quotient: The result of division. In 15 ÷ 3 = 5, 5 is the quotient.
商：除法运算的结果。在 15 ÷ 3 = 5 中，5 是商。
Index (or exponent): In aⁿ, the small raised number n that tells you how many times a is multiplied by itself.
指数：在 aⁿ 中，上角标的小数字 n，表示 a 自乘的次数。
Standard form writing very large or very small numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
标准型：将极大或极小的数写成 A × 10ⁿ 的形式，其中 1 ≤ A < 10 且 n 为整数。
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
2. Algebraic Expressions and Equations | 代数表达式与方程
Algebra is about generalising number relationships. You need to be precise when naming the parts of expressions and when classifying equations.
代数是关于把数量关系一般化。在给表达式的各部分命名以及给方程分类时，必须精确。
Expression: A combination of terms with operations but no equality sign, such as 3x² + 2x − 5.
表达式：由项和运算符号组合而成，但没有等号，例如 3x² + 2x − 5。
Equation: A statement that two expressions are equal, containing an ‘=’ sign, such as 2x + 1 = 9.
方程：说明两个表达式相等且含有‘=’的陈述，如 2x + 1 = 9。
Term: A single number, variable, or product of numbers and variables. In 5xy + 3, 5xy and 3 are terms.
项：单个数字、变量或数字与变量的乘积。在 5xy + 3 中，5xy 和 3 都是项。
Coefficient: The number part of a term that multiplies the variable. In 7y, 7 is the coefficient.
系数：项中乘以变量的数字部分。在 7y 中，7 是系数。
Constant: A term that does not change its value; a number on its own, such as the 4 in x² − 4.
常数：值保持不变的项，即单独的数字，如 x² − 4 中的 4。
Linear equation: An equation whose highest power of the variable is 1, e.g. 3x + 2 = 11.
一次方程：变量最高次幂为 1 的方程，例如 3x + 2 = 11。
Simultaneous equations: A set of two or more equations that share the same variables and are solved together.
联立方程（方程组）：含有相同变量的两个或多个方程，需要一起求解。
3. Inequalities | 不等式
Inequalities express a range of possible solutions rather than a single value. The symbols and the idea of solution sets appear regularly in Further Maths problems.
不等式表示的是一组可能的解，而不是单一数值。不等号符号和解集的概念在进阶数学问题中经常出现。
Inequality n that compares two expressions using <, >, ≤ or ≥, showing one side is smaller or larger.
不等式：用 <, >, ≤ 或 ≥ 比较两个表达式的关系，表明一边小于或大于另一边。
Solution set: The set of all values that satisfy an inequality, often written using set notation like {x : x > 4} or shown on a number line.
解集：满足不等式的所有值的集合，常用集合符号表示（如 {x : x > 4}），或在数轴上表示。
Representing on a number line: An open circle marks strict inequalities (<, >); a closed circle marks ≤ or ≥.
数轴表示：空心圆点表示严格不等式（<, >）；实心圆点表示含等号的情形（≤, ≥）。
Integer solutions: Whole number values that satisfy an inequality, often listed explicitly in a given range.
整数解：满足不等式的整数值，在给定范围内常被明确列出。
4. Sequences | 序列与数列
Sequences build a bridge between algebra and patterns. You will learn to describe both arithmetic and quadratic sequences using nth term rules.
序列在代数与规律之间架起桥梁。你将学会用第 n 项公式来描述等差和二次序列。
Sequence: An ordered list of numbers following a rule, e.g. 3, 7, 11, 15, . . .
序列：按照某种规则排列的一列数，如 3, 7, 11, 15, . . .
Term-to-term rule: A rule that tells you how to go from one term to the next, such as 'add 4' for the sequence above.
递推规则：描述如何从一项得到下一项的规则，如上述序列中“加 4”。
nth term: A formula that gives the value of any term based on its position n. For an arithmetic sequence, it is often of the form an + b.
第 n 项（通项公式）：根据位置 n 给出该项值的公式。对于等差数列，常为 an + b 的形式。
Arithmetic sequence uence where the difference between consecutive terms is constant. nth term = a + (n − 1)d.
等差数列：相邻两项差为常数的序列。通项公式 = a + (n − 1)d。
Quadratic sequence: A sequence whose nth term includes n², and where the second difference between terms is constant.
二次序列：第 n 项包含 n² 且项之间的二阶差为常数的序列。
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
Working with points, gradients and equations of lines is a major topic. The vocabulary of this area links algebra directly to visual graphs.
处理点的坐标、坡度和直线方程是一个重要主题。这一部分的词汇将代数与直观图形直接联系起来。
Coordinates: An ordered pair (x, y) that gives the position of a point on a grid.
坐标：有序数对 (x, y)，表示点在网格上的位置。
Gradient (slope) f the steepness of a line, calculated as rise over run: m = (change in y) / (change in x).
坡度（斜率）：直线陡峭程度的度量，用垂直变化除以水平变化计算：m = (y 的变化) / (x 的变化)。
y-intercept: The point where a line crosses the y-axis, given by the constant c in y = mx + c.
y 轴截距：直线与 y 轴相交的点，即 y = mx + c 中的常数 c。
Equation of a straight line: Usually written as y = mx + c, where m is the gradient and c is the y-intercept.
直线方程：通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Midpoint: The point exactly halfway between two points; its coordinates are the averages of the x and y coordinates.
中点：两点正中间的点；其坐标为两点 x 坐标和 y 坐标的平均值。
6. Ratio and Proportion | 比与比例
Ratio, proportion and rates of change permeate both pure and applied mathematics. Year 9 Further Maths deepens your understanding of direct and inverse proportion.
比、比例和变化率贯穿纯数学和应用数学。Year 9 进阶数学加深了你对正比与反比的理解。
Ratio: A way of comparing two or more quantities, written as a : b or as a simplified fraction.
比：比较两个或多个数量的一种方式，写作 a : b 或化成最简分数。
Proportion: An equation stating that two ratios are equal, e.g. a/b = c/d.
比例：表明两个比相等的等式，例如 a/b = c/d。
Direct proportion crease or decrease by the same factor: y = kx, where k is the constant of proportionality.
正比：两个量按相同倍数增加或减少，即 y = kx，其中 k 为比例常数。
Inverse proportion e : When one quantity increases, the other decreases such that their product stays constant: y = k/x.
反比：一个量增加时，另一个量按使乘积保持不变的方式减小：y = k/x。
Unitary method: Finding the value of a single unit first, then scaling up or down to find the required amounts.
单位法：先求出单个单位的量，再按比例放大或缩小求出所需数量。
7. Pythagoras and Trigonometry | 勾股定理与三角学
Right-angled triangles introduce two fundamental tools: Pythagoras’ theorem and the three trigonometric ratios. Labelling sides correctly is crucial.
直角三角形引入了两个基本工具：勾股定理和三个三角比。正确标记各边至关重要。
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，总是对着直角。
Legs (shorter sides): The two sides that form the right angle; labelled as opposite and adjacent relative to a given acute angle.
直角边（短边）：构成直角的两条边；相对于某个锐角分别标记为对边和邻边。
Pythagoras' theorem: In a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
勾股定理：在直角三角形中，斜边的平方等于另外两边的平方之和。
a² + b² = c²
Sine (sin): Ratio of the opposite side to the hypotenuse: sin θ = opposite / hypotenuse.
正弦 (sin)：对边与斜边的比：sin θ = 对边 / 斜边。
Cosine (cos): Ratio of the adjacent side to the hypotenuse: cos θ = adjacent / hypotenuse.
余弦 (cos)：邻边与斜边的比：cos θ = 邻边 / 斜边。
Tangent (tan): Ratio of the opposite to the adjacent: tan θ = opposite / adjacent.
正切 (tan)：对边与邻边的比：tan θ = 对边 / 邻边。
8. Perimeter, Area and Volume | 周长、面积与体积
Measurement vocabulary extends from simple 2D shapes to 3D solids and composite figures. Surface area and volume of prisms are especially important in Further Maths.
测量词汇从简单的二维图形延伸到三维立体和组合图形。棱柱的表面积和体积在进阶数学中尤为重要。
Perimeter: The total distance around the outside of a 2D shape.
周长：二维图形外部一周的总长度。
Area: The amount of space inside a 2D shape, measured in square units.
面积：二维图形内部的空间大小，用平方单位度量。
Volume: The amount of space a 3D solid occupies, measured in cubic units.
体积：三维立体所占空间的大小，用立方单位度量。
Surface area: The total area of all the faces of a 3D solid.
表面积：三维立体所有面的总面积。
Prism e with a constant cross-section along its length. Volume = area of cross-section × length.
棱柱：横截面沿长度方向保持不变的立体。体积 = 横截面积 × 长度。
Compound shape: A shape made up of two or more simpler shapes; its area or volume is found by splitting it into known parts.
组合图形：由两个或多个简单图形组合而成的形状；通过拆分为已知部分来计算面积或体积。
9. Statistics and Charts | 统计与图表
Statistical terms help you summarise, represent and interpret data. Year 9 Further Maths often includes cumulative frequency and box plots alongside the classic averages.
统计术语帮助你概括、展示和解读数据。Year 9 进阶数学除经典平均数外，通常还包括累计频率和箱形图。
Mean: The arithmetic average, found by summing all data values and dividing by the number of values.
平均数（均值）：算术平均值，将所有数据相加再除以数据个数得到。
Median: The middle value when the data is arranged in order; resistant to extreme values.
中位数：数据按序排列后的中间值；不易受极端值影响。
Mode: The data value that appears most often; a set may have one mode, more than one, or none.
众数：出现次数最多的数据值；一组数据可以有一个、多个或没有众数。
Range: A measure of spread: range = maximum value − minimum value.
全距（极差）：衡量数据分散程度的量：全距 = 最大值 − 最小值。
Cumulative frequency: A running total of frequencies; helps to find medians and quartiles on a cumulative frequency graph.
累计频率：频率的累加总和；用于在累计频率图上求中位数和四分位数。
Box plot (box-and-whisker diagram): A graph that shows the minimum, lower quartile, median, upper quartile and maximum.
箱形图（箱线图）：显示最小值、下四分位数、中位数、上四分位数和最大值的图形。
10. Probability | 概率
Probability language quantifies chance. In Further Maths, you move beyond simple events to combined events, tree diagrams and the use of AND/OR rules.
概率语言将可能性量化。在进阶数学中，你将从简单事件过渡到复合事件、树状图以及“与”/“或”规则的使用。
Probability scale: A line from 0 to 1 where an impossible event has probability 0 and a certain event has probability 1.
概率尺度：从 0 到 1 的线段，不可能事件的概率为 0，必然事件的概率为 1。
Sample space: The set of all possible outcomes of an experiment, often listed in a sample space diagram.
样本空间：实验所有可能结果的集合，常以样本空间图列出。
Mutually exclusive events not happen at the same time, e.g. rolling a 3 and a 5 on one die. P(A or B) = P(A) + P(B).
互斥事件：不可能同时发生的事件，例如一枚骰子掷出 3 和 5。P(A 或 B) = P(A) + P(B)。
Independent events: Events where the outcome of one does not affect the other. P(A and B) = P(A) × P(B).
独立事件：一个事件的结果不影响另一个的事件。P(A 且 B) = P(A) × P(B)。
Tree diagram: A branching diagram that shows all possible outcomes of an event and their probabilities; especially useful for combined independent events.
树状图：一种分支图，显示事件所有可能的结果及其概率；对复合独立事件特别有用。
Relative frequency lity based on experimental data: relative frequency = frequency of outcome / total trials.
相对频率：基于实验数据的概率估计值：相对频率 = 某结果出现次数 / 试验总次数。
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