📚 Year 8 AQA Further Maths: Quick Guide to Key Vocabulary | Year 8 AQA 进阶数学：词汇术语速记指南
This guide provides concise definitions for essential mathematical terms covered in the AQA Year 8 Further Maths course. Mastering these words will sharpen your reasoning, help you decode exam questions and build a stronger foundation for higher-level topics.
本指南为 AQA Year 8 进阶数学课程中的核心数学术语提供了简明定义。掌握这些词汇能提升你的逻辑推理能力，帮助你精准拆解题意，并为高阶内容打下坚实基础。
1. Algebraic Expressions and Equations | 代数表达式与方程
Variable: A variable is a symbol, often a letter such as x or y, that represents an unknown or a value that can change within a problem.
变量：变量是一个符号，通常使用 x 或 y 这样的字母，用来表示未知数或在问题中可以变化的数值。
Coefficient: The coefficient is the number multiplying a variable. In the expression 5x, 5 is the coefficient of x.
系数：系数是乘以变量的数字。在表达式 5x 中，5 就是 x 的系数。
Term: A term is a single mathematical element, which can be a constant, a variable or a product of numbers and variables. For example, 3y, –2 and 4xy are all terms.
项：项是单一的数学元素，可以是一个常数、一个变量或者数字与变量的乘积。例如 3y、–2 和 4xy 都是项。
Expression: An expression is a combination of terms linked by operation symbols such as addition or subtraction, but it does not contain an equals sign. Example: 2a + 3b – 7.
表达式：表达式是由运算符号（如加号或减号）连接起来的项的集合，不含等号。例如 2a + 3b – 7。
Equation: An equation is a mathematical statement that shows two expressions are equal, connected by an equals sign. Solving equations finds the unknown value. Example: x + 5 = 12.
方程：方程是用等号表示两个表达式相等的数学陈述。解方程就是求出未知数的值。例如 x + 5 = 12。
Simplify: To simplify an expression means to combine like terms and reduce it to its simplest form. 3x + 2x simplifies to 5x.
化简：化简表达式是指合并同类项，将其写成最简形式。3x + 2x 化简后为 5x。
Expand: Expanding involves removing brackets by multiplying each term inside by the factor outside. Example: 4(x + 3) expands to 4x + 12.
展开：展开是指通过乘法分配去掉括号，将每一项乘以括号外的因式。例如 4(x + 3) 展开后得到 4x + 12。
Factorise: Factorising is the reverse of expanding; it writes an expression as a product of its factors. 6x + 9 factorises to 3(2x + 3).
因式分解：因式分解是展开的逆运算，把一个表达式写成因式乘积的形式。6x + 9 因式分解为 3(2x + 3)。
Substitute: Substitution means replacing a variable with a given number and then calculating the result. In the expression 2n + 5, substituting n = 3 gives 11.
代入：代入是指用给定的数字替换变量，然后计算出结果。对于表达式 2n + 5，代入 n = 3 得到 11。
2. Number Systems and Operations | 数系与运算
Integer: An integer is any whole number, positive, negative or zero, without fractional or decimal parts. –4, 0 and 9 are integers.
整数：整数是任何没有分数或小数部分的正数、负数或零。–4、0 和 9 都是整数。
Prime Number: A prime number is a whole number greater than 1 that has exactly two distinct factors: 1 and itself. 2, 3, 5, 7 and 11 are primes.
质数（素数）：质数是大于 1 且恰好只有两个不同因数的正整数：1 和它本身。2、3、5、7 和 11 都是质数。
Composite Number: A composite number is a positive integer with more than two factors. 12 is composite because its factors are 1, 2, 3, 4, 6 and 12.
合数：合数是拥有超过两个因数的正整数。12 是合数，因为它的因数包含 1、2、3、4、6 和 12。
Factor: A factor of a number divides exactly into that number without leaving a remainder. The factors of 18 are 1, 2, 3, 6, 9 and 18.
因数：因数是可以整除某个数而不留余数的数字。18 的因数有 1、2、3、6、9 和 18。
Multiple: A multiple is the result of multiplying a number by an integer. The multiples of 5 include 5, 10, 15, 20 and so on.
倍数：倍数是一个数与整数相乘的结果。5 的倍数包括 5、10、15、20 等。
Highest Common Factor (HCF): The HCF of two or more numbers is the largest factor shared by all of them. The HCF of 24 and 36 is 12.
最大公因数（HCF）：两个或多个数字的最大公因数是它们共有的最大因数。24 和 36 的 HCF 是 12。
Lowest Common Multiple (LCM): The LCM is the smallest number that is a multiple of each number in a set. The LCM of 4 and 6 is 12.
最小公倍数（LCM）：LCM 是一组数字共有的最小倍数。4 和 6 的 LCM 是 12。
Reciprocal: The reciprocal of a number is 1 divided by that number. The reciprocal of 5 is 1/5, and the reciprocal of a fraction a/b is b/a.
倒数：一个数的倒数是用 1 除以这个数。5 的倒数是 1/5，分数 a/b 的倒数是 b/a。
3. Powers and Roots | 幂与根
Base and Index: In a power aⁿ, a is the base and n is the index or exponent. The index tells how many times the base is multiplied by itself.
底数和指数：在幂 aⁿ 中，a 是底数，n 是指数（或幂指数）。指数表示底数自乘的次数。
Square: Squaring a number means multiplying it by itself. 7² = 7 × 7 = 49.
平方：对一个数求平方意味着这个数乘以它本身。7² = 7 × 7 = 49。
Cube: Cubing a number means raising it to the power of 3. 4³ = 4 × 4 × 4 = 64.
立方：对一个数求立方就是将它升到 3 次幂。4³ = 4 × 4 × 4 = 64。
Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol is √. √81 = 9 because 9 × 9 = 81.
平方根：一个数的平方根是自乘后得到原数的值，符号为 √。√81 = 9，因为 9 × 9 = 81。
Cube Root: The cube root of a number is a value that, when cubed, equals the original number. The symbol is ∛. ∛125 = 5 because 5³ = 125.
立方根：一个数的立方根是立方后等于原数的值，符号为 ∛。∛125 = 5，因为 5³ = 125。
Powers of 10: Powers of 10 are numbers like 10² = 100, 10³ = 1000, or 10⁻¹ = 0.1. They are widely used in standard form.
10 的幂：10 的幂例如 10² = 100、10³ = 1000 或 10⁻¹ = 0.1。这些数字广泛用于科学记数法。
4. Geometry and Measures | 几何与度量
Angles: An acute angle is less than 90°; a right angle equals exactly 90°; an obtuse angle is greater than 90° but less than 180°; a reflex angle is greater than 180°.
角：锐角小于 90°；直角恰好等于 90°；钝角大于 90°且小于 180°；优角大于 180°。
Perpendicular: Two lines are perpendicular if they meet at a right angle. The symbol ⊥ is often used to denote perpendicular lines.
垂直：如果两条直线相交成直角，则它们互相垂直。常用符号 ⊥ 表示垂直线。
Parallel: Parallel lines are always the same distance apart and never meet, no matter how far they are extended. They are marked with arrow symbols.
平行：平行线始终保持相同的距离，无论怎样延伸都不会相交。通常用箭头符号标注。
Polygon: A polygon is a closed 2D shape with straight sides. Triangles, quadrilaterals, pentagons and hexagons are all polygons.
多边形：多边形是由直边组成的闭合二维图形。三角形、四边形、五边形和六边形都属于多边形。
Perimeter: The perimeter is the total distance around the edge of a 2D shape, found by adding the lengths of all sides.
周长：周长是二维图形边缘的总距离，通过将所有边长相加得到。
Area: Area measures the amount of space inside a 2D shape, expressed in square units such as cm² or m². Rectangle area = length × width.
面积：面积衡量二维图形内部的空间大小，用平方单位表示，如 cm² 或 m²。矩形面积 = 长 × 宽。
Volume: Volume measures the space inside a 3D object, in cubic units. Volume of a cuboid = length × width × height.
体积：体积衡量三维物体内部的空间，以立方单位表示。长方体体积 = 长 × 宽 × 高。
Pythagoras’ Theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
a² + b² = c²
勾股定理：在直角三角形中，斜边的平方等于两直角边的平方之和。
a² + b² = c²
5. Transformations and Symmetry | 变换与对称
Translation: A translation slides a shape from one position to another without turning, flipping or changing size. Every point moves the same distance in the same direction.
平移：平移是将图形从一个位置滑动到另一个位置，不旋转、不翻转、不改变大小。每个点都按照相同的方向移动相同的距离。
Reflection: A reflection flips a shape over a mirror line, producing a mirror image. The reflected shape is congruent to the original.
反射：反射是将图形沿一条镜像线翻转，产生一个镜像。反射后的图形与原图形全等。
Rotation: Rotation turns a shape about a fixed point, called the centre of rotation, through a given angle and direction. Size and shape stay the same.
旋转：旋转是围绕一个定点（旋转中心）按给定角度和方向转动图形。大小和形状保持不变。
Enlargement: An enlargement changes the size of a shape by a scale factor about a centre of enlargement. A scale factor greater than 1 makes it larger; a factor between 0 and 1 makes it smaller.
放大：放大是按比例因子相对于放大中心改变图形大小。比例因子大于 1 会使图形变大；介于 0 和 1 之间会使图形缩小。
Line Symmetry: A shape has line symmetry if it can be folded along a line so that both halves match exactly. Regular polygons have many lines of symmetry.
线对称：如果一个图形可以沿某条直线对折后两边完全重合，则该图形具有线对称。正多边形有多条对称轴。
Rotational Symmetry: A shape has rotational symmetry if it looks the same more than once during a full 360° turn. The order of rotational symmetry tells how many times the shape matches itself in one complete rotation.
旋转对称：若图形在旋转一周的过程中有不止一次与自身重合，则具有旋转对称。旋转对称的阶数表示图形在完整旋转一圈中与自身重合的次数。
Congruence and Similarity: Congruent shapes are identical in size and shape. Similar shapes have the same shape but may differ in size, with corresponding lengths in the same ratio.
全等与相似：全等图形的大小和形状完全相同。相似图形形状相同但大小可以不同，对应边成相同的比例。
6. Sequences and Patterns | 数列与规律
Sequence: A sequence is an ordered list of numbers following a rule. Each number in the list is called a term.
数列：数列是一组按照某种规则排列的数字。数列中的每一个数字叫作项。
Term-to-Term Rule: This rule connects one term to the next. In the sequence 3, 7, 11, 15, … the term-to-term rule is ‘add 4’.
递推规则：该规则将一项与下一项联系起来。在数列 3, 7, 11, 15, … 中，递推规则是 “加 4”。
nth term (Position-to-Term Rule): The nth term formula allows you to calculate any term directly from its position n. For the linear sequence 4, 7, 10, 13, … the nth term is 3n + 1.
第 n 项（通项公式）：第 n 项公式让你能够根据位置 n 直接计算任意一项。对于线性数列 4, 7, 10, 13, …，第 n 项是 3n + 1。
Arithmetic Sequence: An arithmetic sequence is one where the difference between consecutive terms is constant. This difference is called the common difference.
等差数列：等差数列是指相邻两项之差恒定的数列。这个差值称为公差。
Geometric Sequence: In a geometric sequence, each term is found by multiplying the previous term by a fixed number, called the common ratio. For example, 2, 6, 18, 54, … has a common ratio of 3.
等比数列：在等比数列中，每一项都是前一项乘以一个固定的数，这个数称为公比。例如 2, 6, 18, 54, … 的公比为 3。
Pattern Recognition: Spotting patterns helps you predict future terms or write general rules. Patterns can appear as shapes or numbers and are fundamental to algebraic thinking.
图形规律识别：发现规律有助于预测后续项或写出通项规则。规律可以表现为图形或数字，是代数思维的基础。
7. Ratio, Proportion and Rates | 比、比例和比率
Ratio: A ratio compares the sizes of two or more quantities, showing how many times one value contains another. Ratios can be written with a colon, like 3 : 4.
比：比用于比较两个或多个量的大小，表示一个值包含另一个值的倍数。比可以用冒号书写，如 3 : 4。
Simplifying Ratios: Ratios are simplified by dividing all parts by their highest common factor. 8 : 12 simplifies to 2 : 3.
化简比：通过将所有部分除以它们的最大公因数来化简比。8 : 12 化简后为 2 : 3。
Proportion: Proportion describes how one quantity changes in relation to another. Direct proportion means that as one quantity increases, the other increases at the same rate.
比例：比例描述了一个量如何随另一个量变化。正比例意味着一个量增加时，另一个量以相同的速率增加。
Direct Proportion Formula: When two quantities y and x are in direct proportion, y = kx, where k is the constant of proportionality.
y = kx
正比例公式：当两个量 y 和 x 成正比例时，y = kx，其中 k 是比例常数。
y = kx
Inverse Proportion: Inverse proportion means that when one quantity increases, the other decreases in such a way that their product remains constant. In basic form, y = k/x.
反比例：反比例意味着一个量增加时，另一个量以乘积保持恒定的方式减少。基本形式为 y = k/x。
Scale Factor: A scale factor is the number used to multiply the dimensions of a shape to produce an enlarged or reduced similar shape. A scale factor of 2 doubles all lengths.
比例因子：比例因子是用来乘以图形尺寸以产生放大或缩小的相似图形的数字。比例因子为 2 会使所有长度翻倍。
Rate: A rate is a ratio comparing two quantities of different units, such as speed (km/h) or density (kg/m³). Unit rate gives the amount per single unit of the second quantity.
比率：比率是比较两种不同单位量的比，如速度（km/h）或密度（kg/m³）。单位比率表示每单位第二个量所对应的量。
8. Fractions, Decimals and Percentages | 分数、小数和百分比
Equivalent Fractions: Equivalent fractions have the same value even though they look different. 1/2, 2/4 and 3/6 are all equivalent.
等值分数：等值分数尽管看起来不同，但数值相等。1/2、2/4 和 3/6 都是等值分数。
Simplifying Fractions: To simplify a fraction, divide numerator and denominator by their highest common factor. 12/16 simplifies to 3/4.
约分：约分是将分子和分母除以它们的最大公因数。12/16 约分为 3/4。
Converting Between Fractions, Decimals and Percentages: You can convert between the three forms. 1/4 = 0.25 = 25%. To change a decimal to a percentage, multiply by 100.
分数、小数与百分比的互换：三种形式可以互相转换。1/4 = 0.25 = 25%。将小数转为百分数只需乘以100。
Percentage Increase and Decrease: To increase by a percentage, add the percentage of the amount to the original. To decrease, subtract it. A 20% increase on 50 gives 60.
百分比的增减：增加百分比时，将原值的百分部分加到原值上；减少时则减去。50 增加 20% 后变成 60。
Recurring Decimal: A recurring decimal has a digit or block of digits that repeat forever. 1/3 = 0.333… is written with a dot above the repeating digit.
循环小数：循环小数中含有无限重复的一位或一组数字。1/3 = 0.333… 需要在重复数字上方加点表示。
Fraction of an Amount: To find a fraction of a quantity, multiply the quantity by the numerator and then divide by the denominator. 3/5 of 200 is (200 × 3) ÷ 5 = 120.
求一个量的几分之几：求一个量的几分之几，先将量乘以分子，再除以分母。200 的 3/5 是 (200 × 3) ÷ 5 = 120。
9. Data Handling and Statistics | 数据处理与统计
Mean: The mean is the sum of all values divided by the number of values. It is often called the average.
平均数：平均数等于所有数值之和除以数值的个数，通常简称为均值。
Median: The median is the middle value when data is arranged in order. For an even number of values, it is the mean of the two middle numbers.
中位数：中位数是将数据按顺序排列后位于中间的值。当数据个数为偶数时，中位数是中间两个数的平均值。
Mode: The mode is the value that appears most frequently in a data set. A data set can have more than one mode or no mode at all.
众数：众数是数据集中出现次数最多的数值。一个数据集可以有一个以上的众数，也可能没有众数。
Range: The range measures how spread out the data is. It is the difference between the largest and smallest values.
极差：极差衡量数据的分散程度，即最大值与最小值之差。
Frequency Table: A frequency table records how often each value occurs. Tally marks help to count frequencies easily.
频数表：频数表记录每个数值出现的次数。画“正”字计数有助于轻松统计频数。
Graphs and Charts: Key diagrams include bar charts (comparing quantities), pie charts (showing proportions of a whole), line graphs (showing change over time) and scatter graphs (showing relationships between two variables).
统计图：关键图形包括条形图（比较数量）、饼图（展示整体的比例）、折线图（展示随时间的变化）和散点图（展示两个变量之间的关系）。
10. Probability | 概率
Probability Scale: Probability ranges from 0 (impossible) to 1 (certain). A probability of 0.5 means an event is equally likely to happen or not happen.
概率标度：概率范围从 0（不可能）到 1（一定）。概率为 0.5 表示发生与不发生的机会相等。
Event and Outcome: An outcome is a possible result of a trial. An event is a set of one or more outcomes. Rolling a 3 is an outcome; rolling an even number is an event.
事件与结果：结果是一次试验的可能得到的结果。事件是一个或多个结果的集合。掷出3点是一个结果；掷出偶数是事件。
Equally Likely: Outcomes are equally likely when they have the same chance of occurring. A fair coin has equally likely outcomes of heads and tails.
等可能：当所有结果发生的可能性相同时，称它们是等可能的。一枚匀质硬币掷出正面和反面的结果是等可能的。
Sample Space: The sample space is the set of all possible outcomes. For a dice roll, the sample space is {1, 2, 3, 4, 5, 6}.
样本空间：样本空间是所有可能结果的集合。对于掷骰子，样本空间为 {1, 2, 3, 4, 5, 6}。
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