📚 Year 8 CAIE Further Mathematics: Vocabulary Terminology Quick Memorisation Guide | Year 8 CAIE 进阶数学：词汇术语速记指南
Mastering mathematical vocabulary is the first step towards solving problems fluently. This quick memorisation guide brings together all the essential terms you will meet in the CAIE Year 8 Further Mathematics course. Each term is explained in clear English, with a paired Chinese translation to help you build confidence and recall quickly. Use this reference to strengthen your foundation in algebra, geometry, statistics and more.
掌握数学词汇是流畅解决问题的第一步。这份速记指南汇集了CAIE Year 8进阶数学课程中的所有核心术语。每个术语都有清晰的英文解释和对应的中文翻译，帮助你快速建立信心并牢固记忆。用这份参考资料为代数、几何、统计等模块打下坚实的基础。
1. Algebraic Expressions and Equations | 代数表达式与方程
Term: A term is a single number, a variable, or numbers and variables multiplied together. For example, 5, -3x, and 2ab are all terms.
项：项是一个单独的数、一个变量，或者是数与变量相乘的积。例如 5、-3x 和 2ab 都是项。
Variable: A variable is a symbol, usually a letter, that stands for an unknown number or a number that can change.
变量：变量是一个符号，通常是字母，用来表示未知的数或可以改变的数。
Coefficient: The coefficient is the number part that multiplies a variable. In 7y, the coefficient is 7.
系数：系数是与变量相乘的数字部分。在 7y 中，系数是 7。
Constant: A constant is a fixed number that does not change. In the expression 4x + 9, the number 9 is a constant.
常数：常数是一个固定不变的数。在表达式 4x + 9 中，9 是常数。
Expression: An expression is a combination of terms that are added or subtracted. It does not contain an equals sign, e.g. 3a + 2b - 5.
表达式：表达式是由项通过加减组合而成的式子，不含等号，例如 3a + 2b - 5。
Equation: An equation shows that two expressions are equal. It always has an '=' sign, e.g. 2x + 3 = 11.
方程：方程表示两个表达式相等，总是含有等号 '='，例如 2x + 3 = 11。
Formula a relationship between different quantities. For example, A = l × w is a formula for area.
公式：公式是表示不同量之间关系的方程。例如 A = l × w 是面积公式。
Inequality o : An inequality compares two values using signs like <, >, ≤ or ≥. It tells us which side is larger or smaller.
不等式：不等式使用 <、>、≤ 或 ≥ 比较两个值，说明哪边更大或更小。
2. Expanding and Factorising | 展开与因式分解
Expand: To expand brackets means to multiply each term inside the bracket by the term outside. For example, 3(x + 4) expands to 3x + 12.
展开：展开括号意味着将括号外的项乘以括号内的每一项。例如 3(x + 4) 展开得到 3x + 12。
Factorise: Factorising is the reverse of expanding. You rewrite an expression as a product of its factors. For instance, 6x + 9 factorises to 3(2x + 3).
因式分解：因式分解是展开的逆运算，将表达式写成因子的乘积。例如 6x + 9 因式分解得到 3(2x + 3)。
Factor: A factor is a number or expression that divides another exactly without leaving a remainder. Factors of 10 are 1, 2, 5 and 10.
因子：因子是能整除另一个数或表达式而没有余数的数或式子。10 的因子有 1、2、5 和 10。
Common factor: A common factor is a factor that two or more terms share. In 4x and 8, the common factor is 4.
公因子：公因子是两个或多个项共有的因子。在 4x 和 8 中，公因子是 4。
Linear expression: A linear expression contains variables raised only to the power of 1, like 5x + 2 or y - 7.
线性表达式：线性表达式中变量的指数仅为 1，例如 5x + 2 或 y - 7。
Quadratic expression: A quadratic expression includes a variable squared, such as x² + 3x - 4. Its highest power is 2.
二次表达式：二次表达式包含变量的平方，例如 x² + 3x - 4，最高次数为 2。
3. Sequences and the nth Term | 数列与第 n 项
Sequence: A sequence is an ordered list of numbers that follows a rule. Each number in the list is called a term of the sequence.
数列：数列是按照某种规则排列的一列有序的数。数列中的每一个数称为数列的项。
nth term rule: The nth term rule is a formula that allows you to calculate any term in a sequence directly from its position n. For the sequence 3, 7, 11, 15, ... the nth term is 4n - 1.
第 n 项公式：第 n 项公式是一个根据位置 n 直接求出数列中任意一项的公式。数列 3, 7, 11, 15, ... 的第 n 项公式为 4n - 1。
Arithmetic sequence: An arithmetic sequence is one where the difference between consecutive terms stays the same. This difference is called the common difference.
等差数列：等差数列是相邻两项的差保持不变的数列。这个差称为公差。
Common difference: The common difference is the constant value you add (or subtract) to go from one term to the next. In 2, 5, 8, 11, the common difference is +3.
公差：公差是从一项到下一项所要加（或减）的固定值。在 2, 5, 8, 11 中，公差为 +3。
Generate terms a sequence using a given rule. You substitute n = 1, 2, 3, ... into the nth term formula.
生成项：生成项是指用给定的规则产出数列中的数。你将 n = 1, 2, 3, ... 代入第 n 项公式即可。
Square, cube and triangular numbers: Square numbers are 1, 4, 9, 16, ... (n²). Cube numbers are 1, 8, 27, ... (n³). Triangular numbers are 1, 3, 6, 10, ... (½ n(n+1)).
平方数、立方数和三角形数：平方数是 1, 4, 9, 16, ... (n²)。立方数是 1, 8, 27, ... (n³)。三角形数是 1, 3, 6, 10, ... (½ n(n+1))。
4. Ratio, Proportion and Rates | 比率、比例与速率
Ratio: A ratio compares two or more quantities of the same kind. It is written using a colon, like 3 : 5, or as a fraction.
比：比用来比较两个或多个同类量，用冒号表示，如 3 : 5，也可写成分数形式。
Proportion: A proportion states that two ratios are equal. For example, 1/2 = 2/4 is a proportion.
比例：比例说明两个比相等。例如 1/2 = 2/4 就是一个比例。
Scale factor: A scale factor is the number that all lengths are multiplied by when an object is enlarged or reduced. A scale factor of 2 doubles every length.
比例因子：比例因子是图形放大或缩小时每条长度所乘的数。比例因子为 2 时，所有长度变为原来的两倍。
Direct proportion: Two quantities are in direct proportion if one multiplies by the same factor as the other. As one doubles, the other doubles too.
正比例：如果两个量按相同的倍数变化，它们就成正比例。一个量翻倍，另一个量也翻倍。
Inverse proportion: In inverse proportion, when one quantity is multiplied, the other is divided. For instance, as speed increases, travel time decreases for a fixed distance.
反比例：在反比例中，一个量增加时另一个量按比例减少。例如，在固定距离下，速度越快，所需时间越短。
Rate: A rate compares two different quantities. Common examples are kilometres per hour (km/h) or price per kilogram ($/kg).
速率：速率比较两个不同类的量，常见的例子如千米每小时 (km/h) 或每千克价格 ($/kg)。
Unitary method e : The unitary method involves finding the value of one unit first, then scaling up or down. For example, to find the cost of 7 pens when 5 cost $10, first find 1 pen = $2.
归一法：归一法是先求出一个单位的值，再推算所求的数量。例如，5 支笔 10 美元，先求 1 支笔 2 美元，再算 7 支笔。
5. Percentages and Finance | 百分比与金融
Percent: Percent means 'out of 100'. The symbol % is used. 45% is 45 per hundred or 45/100.
百分比：百分比表示 '每一百份中的几份'，用符号 % 表示。45% 即每百份中的 45 份，或 45/100。
Percentage increase a percentage of the original amount. New value = original × (1 + percentage increase/100).
百分比增加：百分比增加是数值的增长量占原始量的百分比。新值 = 原始值 × (1 + 增加百分比/100)。
Percentage decrease Percentage decrease works similarly. New value = original × (1 - percentage decrease/100).
百分比减少：百分比减少类似。新值 = 原始值 × (1 - 减少百分比/100)。
Simple interest n : Simple interest is calculated only on the original sum of money (principal). I = P × R × T, where R is the rate and T is time.
单利：单利仅根据原始本金计算利息。I = P × R × T，R 是利率，T 是时间。
Compound interest: Compound interest is earned on both the original principal and on any interest already added. This makes money grow faster over time.
复利：复利既计算本金产生的利息，也计算已累积的利息产生的利息，使得资金随时间增长更快。
Profit and loss: Profit is the positive difference between selling price and cost price. Loss occurs when selling price is lower than cost price.
利润与亏损：利润是售价高于成本价的差额，亏损是售价低于成本价的差额。
Discount n the marked price, often expressed as a percentage. Sale price = marked price - discount.
折扣：折扣是对标价给予的减价，通常以百分比表示。成交价 = 标价 - 折扣。
6. Angles and Lines | 角与线
Here is a quick reference table for angle types and properties:
以下是角度类型及性质的快速参考表：
| Term (English) | Explanation | 中文术语 | 说明 |
|---|---|---|---|
| Acute angle | Between 0° and 90° | 锐角 | 大于 0° 且小于 90° |
| Right angle | Exactly 90° | 直角 | 恰好 90° |
| Obtuse angle | Greater than 90° and less than 180° | 钝角 | 大于 90° 且小于 180° |
| Straight angle | Exactly 180° | 平角 | 恰好 180° |
| Reflex angle | Greater than 180° but less than 360° | 优角 | 大于 180° 且小于 360° |
Complementary angles: Two angles that add up to 90° are called complementary angles.
余角：两个角的和为 90° 时，它们互为余角。
Supplementary angles: Two angles that add up to 180° are supplementary angles.
补角：两个角的和为 180° 时，它们互为补角。
Vertically opposite angles: When two straight lines cross, the angles opposite each other are equal. They are called vertically opposite angles.
对顶角：两条直线相交时，相对的两个角相等，这两个角称为对顶角。
Angles on a straight line: Angles that lie on a straight line add up to 180°.
直线上的角：在一条直线上的角之和为 180°。
Angles around a point: All angles around a single point sum to 360°.
绕一点的角度：绕一个点的所有角之和为 360°。
Parallel lines and transversals: When a line cuts a pair of parallel lines, special angle pairs are formed: corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.
平行线与截线：一条截线与两条平行线相交时，会形成特殊的角对：同位角相等，内错角相等，同旁内角之和为 180°。
7. 2D Shapes and Transformations | 二维图形与变换
Polygon: A polygon is a closed 2D shape made up of straight line segments. Triangles, quadrilaterals and pentagons are all polygons.
多边形：多边形是由直线段围成的封闭二维图形，如三角形、四边形和五边形都是多边形。
Regular polygon: A regular polygon has all sides the same length and all interior angles equal, like a square or an equilateral triangle.
正多边形：
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