📚 Year 8 CIE Further Mathematics: Vocabulary and Terminology Quick Memorization Guide | Year 8 CIE 进阶数学：词汇术语速记指南
Mastering the vocabulary of Further Mathematics is the first step toward solving problems with confidence. This guide provides clear definitions, memorable shortcuts, and bilingual explanations for the key terms you will encounter in the Year 8 CIE syllabus. Use these pairings to internalise the language of mathematics and boost your exam performance.
掌握进阶数学的词汇是自信解题的第一步。本指南针对 Year 8 CIE 大纲中的关键术语，提供了清晰的定义、巧记捷径和中英双语解释。借助这些配对内容，你可以内化数学语言，显著提升考试表现。
1. Algebra Foundations: Variables, Constants, and Coefficients | 代数基础：变量、常数与系数
A variable is a symbol, usually a letter like x or y, that stands for an unknown number. Think of it as an empty box waiting for a value.
变量是一个符号，通常用 x 或 y 这样的字母表示，代表一个未知数。可以把它想象成一个等待填值的空盒子。
A constant is a fixed number that does not change, such as 5, −3, or π. The word ‘constant’ literally means ‘standing firm’.
常数是一个固定不变的数，例如 5、−3 或 π。’constant’ 一词的字面意思就是 ‘站稳不变’。
The coefficient is the number multiplying a variable. In 7x², 7 is the coefficient. Remember: ‘coefficient’ comes from ‘co-‘ (together) and ‘efficient’ (working), so it works together with the variable.
系数是与变量相乘的数。在 7x² 中，7 就是系数。巧记：’coefficient’ 由 ‘co-‘（共同）和 ‘efficient’（起作用）构成，它和变量共同起作用。
2. Expressions, Equations, and Identities: Knowing the Difference | 表达式、方程与恒等式：区别何在
An expression is a combination of variables, numbers, and operations without an equals sign, for example 3a + 2b − 5. It is a phrase, not a full sentence.
表达式是由变量、数字和运算符号组成的式子，没有等号，例如 3a + 2b − 5。它是一个短语，不是完整的句子。
An equation states that two expressions are equal by using an equals sign, such as 2x + 3 = 11. It acts like a balance scale: both sides must have the same value.
方程通过等号表明两个表达式相等，例如 2x + 3 = 11。它就像一个天平：两边必须等值。
An identity is an equation that is true for all values of the variable, often written with the ‘≡’ symbol. For example, (x + 1)² ≡ x² + 2x + 1. Identities are always true, not just for specific solutions.
恒等式是对变量的所有取值都成立的等式，常用 ‘≡’ 符号表示。例如 (x + 1)² ≡ x² + 2x + 1。恒等式永远成立，不限于特定解。
Like terms are terms that contain exactly the same variable part. 5x and −2x are like terms, but 3x² and 3x are not. Only like terms can be added or subtracted directly.
同类项是含有完全相同的变量部分的项。5x 和 −2x 是同类项，但 3x² 和 3x 不是。只有同类项才能直接相加或相减。
3. Expanding and Factorising: Brackets and Common Factors | 展开与因式分解：括号与公因子
To expand means to multiply out brackets. Using the distributive law, a(b + c) becomes ab + ac. Picture the term outside ‘inviting’ everyone inside to a multiplication party.
展开就是去括号，做乘法。运用分配律，a(b + c) 变成 ab + ac。形象记忆：括号外的项 ‘邀请’ 里面的每一项参加乘法派对。
To factorise is the reverse process: writing an expression as a product of factors. The first step is always to take out the highest common factor (HCF). For 6x + 9, the HCF is 3, giving 3(2x + 3).
因式分解是逆过程：把一个表达式写成因式的乘积。第一步永远是提取最大公因数 (HCF)。对于 6x + 9，HCF 是 3，得到 3(2x + 3)。
A quadratic expression has the form ax² + bx + c, where a ≠ 0. Quad means four, referring to the square (second power) rather than the number of terms. Visualise a square whose area is x².
二次表达式形式为 ax² + bx + c，其中 a ≠ 0。’Quad’ 指四，但这里是指平方（二次幂），而不是项数。可以想象一个面积为 x² 的正方形。
4. Inequalities and Their Symbols | 不等式及其符号
An inequality compares two values using symbols: ‘>’ means ‘greater than’, ‘<' means 'less than', '≥' means 'greater than or equal to', and '≤' means 'less than or equal to'. Think of the open side of the symbol as a hungry crocodile always wanting to eat the larger number.
不等式用符号比较两个值：’>’ 表示 ‘大于’，’<' 表示 '小于'，'≥' 表示 '大于或等于'，'≤' 表示 '小于或等于'。可以把符号的开口想象成饥饿的鳄鱼嘴，总是想吃较大的数。
When graphing inequalities on a number line, an open circle (○) shows the endpoint is not included, while a closed circle (●) shows the endpoint is included. ‘Open’ means the door is shut for that number.
在数轴上画不等式时，空心圆 (○) 表示端点不包含在内，实心圆 (●) 表示端点包含在内。’空心’ 意味着那个数的大门关着。
A compound inequality joins two inequalities with ‘and’ or ‘or’, for instance −1 < x ≤ 4. This tells you x is sandwiched between −1 and 4, where 4 is included but −1 is not.
复合不等式用 ‘and’ 或 ‘or’ 连接两个不等式，例如 −1 < x ≤ 4。这意味着 x 夹在 −1 和 4 之间，其中 4 包含在内，−1 不包含。
5. Sequences: Arithmetic and Geometric Patterns | 序列：等差与等比模式
A sequence is an ordered list of numbers following a rule. Each number is called a term. The rule is the ‘recipe’ that generates the list.
序列是按照某种规则排列的一列数。每个数叫做一个项。规则就是生成这列数的 ‘配方’。
An arithmetic sequence changes by adding or subtracting the same value each time. This fixed difference is the common difference, d. For 5, 9, 13, 17, … the common difference is +4.
等差数列每次增加或减少相同的值。这个固定的差叫做公差 d。对于 5, 9, 13, 17, …，公差为 +4。
A geometric sequence changes by multiplying or dividing by the same value each time. This multiplier is the common ratio, r. In 3, 6, 12, 24, … the common ratio is 2.
等比数列每次乘以或除以相同的值。这个乘数叫做公比 r。在 3, 6, 12, 24, … 中，公比为 2。
The nth term formula lets you find any term without listing all the previous ones. For arithmetic sequences: a + (n−1)d, where a is the first term. Think of it as a telescope that jumps straight to term number n.
第 n 项公式可以让你直接求出任意一项，而不必列出前面所有项。等差数列公式：a + (n−1)d，其中 a 是首项。它就像一个望远镜，直接跳到第 n 项。
6. Ratio, Rate, and Proportion: Key Distinctions | 比、比率与比例：关键区别
A ratio compares two or more quantities of the same unit, written as a : b. It shows relative size, like mixing juice with water in the ratio 1 : 4. Remember the colon separates the parts.
比比较两个或多个同类量，写作 a : b。它表示相对大小，比如按 1 : 4 的比例混合果汁和水。记住冒号把各个部分隔开。
A rate compares two quantities with different units, such as speed (km/h) or price per litre (£/L). The word ‘per’ often signals a rate.
比率比较两个不同单位的量，例如速度 (km/h) 或每升价格 (£/L)。’per’ 这个单词通常暗示着比率。
Proportion means two ratios are equal, forming a proportion equation, e.g. a/b = c/d. Direct proportion means as one quantity increases, the other increases at the same rate; the graph is a straight line through the origin.
比例表示两个比相等，构成比例方程，例如 a/b = c/d。正比例意味着一个量增加时，另一个量以相同速率增加；其图像是一条过原点的直线。
7. Geometry: Angles, Polygons, and Circle Vocabulary | 几何：角、多边形与圆的词汇
An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180°. ‘Acute’ sounds like ‘a cute’ small angle.
锐角小于 90°，直角恰好是 90°，钝角介于 90° 到 180° 之间，优角大于 180°。’Acute’ 听起来像是 ‘a cute’ (可爱的) 小角。
Complementary angles add up to 90°, while supplementary angles add up to 180°. To remember: ‘C’ comes before ‘S’, and 90 comes before 180; or think ‘Complementary = Corner of a rectangle (90°), Supplementary = Straight line (180°).’
互余角之和为 90°，互补角之和为 180°。巧记：’C’ 在 ‘S’ 之前，90 在 180 之前；或者 ‘Complementary = Corner (90°)，Supplementary = Straight line (180°)’。
A polygon is a closed shape with straight sides. A regular polygon has all sides and all angles equal. The interior angle sum of an n-sided polygon is (n−2) × 180°. Visualise triangles inside: each triangle contributes 180°.
多边形是由直线段围成的封闭图形。正多边形的所有边和所有角都相等。n 边形的内角和为 (n−2) × 180°。想象多边形内部被分成三角形：每个三角形贡献 180°。
Circle terms: the radius (r) is the distance from centre to edge; the diameter (d = 2r) cuts through the centre; the circumference is the perimeter (C = 2πr); a chord joins two points on the circle; an arc is part of the circumference.
圆的术语：半径 (r) 是从圆心到圆周的距离；直径 (d = 2r) 通过圆心；周长是圆的边界长 (C = 2πr)；弦连接圆上两点；弧是圆周的一部分。
8. Perimeter, Area, and Volume: Measures and Formulae | 周长、面积与体积：度量与公式
Perimeter is the total distance around a 2D shape. For a rectangle, P = 2(l + w). Think of walking around a park: the perimeter is how far your feet travel.
周长是二维图形一周的总长度。对于矩形，P = 2(l + w)。想象绕公园散步：周长就是你双脚走过的距离。
Area measures the space inside a 2D shape, expressed in square units. Rectangle area: A = l × w. Triangle area: A = ½ × base × height. The formula for a triangle is half a rectangle, which is easy to remember.
面积度量二维图形内部的空间，用平方单位表示。矩形面积：A = l × w。三角形面积：A = ½ × 底 × 高。三角形面积公式就是矩形面积的一半，这样很好记。
Volume measures the space inside a 3D solid, in cubic units. For a cuboid, V = l × w × h. For prisms, V = area of cross-section × length. Volume is all about ‘filling’ a shape.
体积度量三维立体内部的空间，用立方单位表示。长方体体积：V = l × w × h。棱柱体积：V = 截面积 × 长。体积就是关于 ‘填充’ 一个形状。
Surface area is the total area of all the faces of a 3D solid. Think of it as the amount of wrapping paper needed to cover a gift.
表面积是三维立体所有面的总面积。可以想象成包装礼物所需的包装纸面积。
9. Pythagoras and Trigonometry: Right-Angled Triangle Vocabulary | 毕达哥拉斯定理与三角学：直角三角形词汇
In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle. The word ‘hypotenuse’ comes from Greek ‘hypoteinousa’, meaning ‘stretching under’. It sits under the right angle.
在直角三角形中，斜边是最长的边，对着直角。’hypotenuse’ 这个词源自希腊语 ‘hypoteinousa’，意为 ‘在下方伸展’。它就在直角下方。
The opposite side is the side facing the angle you are considering (θ). The adjacent side is the side next to the angle, but not the hypotenuse. Label these by standing at the angle and looking across: opposite is ‘across’, adjacent is ‘next to’.
对边就是你正在考虑的角 (θ) 所对的边。邻边是挨着该角的边，但不是斜边。标记时，站在角的位置看过去：对边在 ‘对面’，邻边在 ‘旁边’。
Pythagoras’ theorem states: a² + b² = c², where c is the hypotenuse. This only works for right-angled triangles. Mnemonic: ‘The square on the hypotenuse equals the sum of the squares on the other two sides.’
毕达哥拉斯定理指出：a² + b² = c²，其中 c 是斜边。这仅适用于直角三角形。记忆口诀：’斜边的平方等于两直角边的平方和’。
Sine, cosine, and tangent (sin, cos, tan) are ratios of sides in a right-angled triangle. SOH CAH TOA is the golden rule: Sin θ = Opposite/Hypotenuse, Cos θ = Adjacent/Hypotenuse, Tan θ = Opposite/Adjacent.
正弦、余弦和正切 (sin, cos, tan) 是直角三角形中边的比值。SOH CAH TOA 是黄金法则：Sin θ = 对边/斜边，Cos θ = 邻边/斜边，Tan θ = 对边/邻边。
10. Coordinate Geometry: Gradient, Intercept, and Linear Graphs | 坐标几何：斜率、截距与线性图
The gradient (slope) measures how steep a line is. It is calculated as rise ÷ run, or (y₂ − y₁) / (x₂ − x₁). A positive gradient slopes uphill, a negative gradient slopes downhill. ‘Gradient’ and ‘grade’ share a root – think of a road’s steepness grade.
斜率 (梯度) 度量一条直线的陡峭程度。计算公式为 上升 ÷ 水平距离，或 (y₂ − y₁) / (x₂ − x₁)。正斜率上坡，负斜率下坡。’Gradient’ 和 ‘grade’ (坡度) 同源——联想道路的坡度标记。
The y-intercept is the point where the line crosses the y-axis (where x = 0). In the equation y = mx + c, c is the y-intercept. Think of c as the ‘starting point’ on the vertical axis.
y 轴截距是直线与 y 轴相交的点（此时 x = 0）。在方程 y = mx + c 中，c 就是 y 轴截距。把 c 想象成纵轴上的 ‘起点’。
A linear equation graphs as a straight line. Its standard form is y = mx + c, where m is the gradient and c is the y-intercept. Any equation whose highest power is 1 will be linear.
线性方程的图像是一条直线。其标准式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。最高次幂为 1 的方程都是线性的。
Parallel lines have the same gradient but different intercepts. Perpendicular lines have gradients that are negative reciprocals: m₁ × m₂ = −1. If one gradient is 2, the perpendicular gradient is −½.
平行线有相同的斜率，但截距不同。垂直线的斜率互为负倒数：m₁ × m₂ = −1。如果一个斜率是 2，那么垂直线的斜率就是 −½。
11. Statistics: Mean, Median, Mode, and Range | 统计：平均数、中位数、众数与范围
The mean is the arithmetic average. Add all the data values and divide by the number of values. It is the ‘balancing point’ of the data set but can be affected by extreme outliers.
平均数 (均值) 是算术平均值。把所有数据值相加后除以数据的个数。它是数据集的 ‘平衡点’，但容易受极端异常值的影响。
The median is the middle value when data are ordered. If there are two middle numbers, take their mean. The median splits the data into two halves and is robust, meaning it is not swayed by outliers.
中位数是将数据排序后位于中间的值。如果有两个中间数，则取它们的平均值。中位数将数据分成两半，具有稳健性，即不会因异常值而偏移。
The mode is the most frequently occurring value. A data set can have no mode, one mode (unimodal), or more than one mode (bimodal). Think of ‘mode’ as ‘most often’.
众数是出现频率最高的值。一组数据可能没有众数，有一个众数（单峰），或多个众数（双峰）。可以把 ‘mode’ 联想成 ‘most often’。
The range is the difference between the largest and smallest values: max − min. It gives a quick measure of spread but ignores the middle 50%.
范围 (极差) 是最大值与最小值的差：最大值 − 最小值。它快速度量了数据的分散程度，但忽略了中间 50% 的情况。
12. Probability and Sets: Events, Outcomes, and Venn Diagrams | 概率与集合：事件、结果与维恩图
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). Probability = (number of favourable outcomes) / (total number of outcomes). It is often written as a fraction, decimal, or percentage.
概率度量事件发生的可能性，范围从 0（不可能）到 1（必然）。概率 = (有利结果数) / (总结果数)。通常用分数、小数或百分数表示。
An outcome is a possible result of an experiment. An event is a set of one or more outcomes. For rolling a die, getting a 5 is an outcome; getting an even number is an event containing outcomes 2, 4, and 6.
结果是试验可能产生的一种情况。事件是一个或多个结果的集合。掷骰子时，得到 5 是一个结果；得到偶数是一个事件，包含 2、4、6 这些结果。
The sample space is the set of all possible outcomes. Listing or drawing a table helps find all possibilities. Remember: sample space = complete menu of outcomes.
样本空间是所有可能结果的集合。用列表或画表格有助于找出所有可能。记住：样本空间 = 结果的完整菜单。
Mutually exclusive events cannot happen at the same time. Flipping a coin cannot show both heads and tails. If A and B are mutually exclusive, P(A and B) = 0.
互斥事件不能同时发生。抛一枚硬币不可能同时出现正面和反面。如果 A 和 B 互斥，则 P(A 且 B) = 0。
A Venn diagram uses overlapping circles to show relationships between sets. The union (A ∪ B) contains everything in A or B. The intersection (A ∩ B) contains only what is in both. The complementPublished by TutorHao | Year 8 进阶数学 Revision Series | aleveler.com
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