📚 Year 9 Cambridge Further Maths: Key Terms Memorisation Guide | Year 9 剑桥进阶数学：词汇术语速记指南
In Year 9 Cambridge Further Mathematics, building a solid foundation in mathematical vocabulary is essential. This guide provides effective memorisation techniques and clear explanations for key terms across every major topic, helping you to recognise, understand, and apply them with confidence in class tests, end-of-year exams and future IGCSE studies.
在 Year 9 剑桥进阶数学中，打牢数学词汇的基础至关重要。本指南针对各个主要专题的关键术语，提供有效的速记方法和清晰的解释，帮助你在课堂测验、学年考试以及未来的 IGCSE 学习中辨识、理解并自信地应用这些术语。
1. Why Vocabulary Decides Your Further Maths Grade | 为什么词汇决定你的进阶数学成绩
Most marks lost in Year 9 Further Mathematics are not lost because the algebra is too difficult, but because a single word was misread. A question that says 'write down the exact value' is not asking for a decimal, and a question that says 'hence' expects you to use the previous part of your working.
Year 9 进阶数学中丢掉的分数，大多不是因为代数太难，而是因为看错了一个词。题目说 'write down the exact value'（写出精确值），就不是要小数；题目说 'hence'（由此），就要求你使用上一小问的结果。
Precise vocabulary also speeds up your thinking. When you instantly know that 'integer' excludes fractions and that 'prime' excludes 1, you stop hesitating in the middle of a solution and you stop making careless errors.
精确的词汇还能加快你的思考速度。当你立刻知道 'integer'（整数）不包含分数、'prime'（质数）不包含 1 时，你在解题过程中就不会犹豫，也不会犯粗心错误。
Finally, terminology is the shared language of the Cambridge syllabus. The same words reappear in IGCSE and A Level, so every term you master now pays interest for years.
最后，术语是剑桥考纲的共同语言。同样的词会在 IGCSE 和 A Level 反复出现，所以你现在掌握的每一个术语，都会在未来多年持续产生回报。
2. A Four-Step Routine for Memorising Terms | 术语记忆四步法
Step one is to say the word aloud and read its definition in a full sentence. Reading silently often hides gaps in understanding, while speaking forces your brain to process the word as a sound pattern as well as a symbol.
第一步是把单词大声读出来，并用完整句子读一遍定义。默读常常会掩盖理解上的漏洞，而朗读会迫使大脑把单词同时当作声音模式来处理。
Step two is to rewrite the definition in your own words in fewer than fifteen words. If you cannot compress it, you do not yet understand it.
第二步是用自己的话、在十五个词以内重写定义。如果你压缩不了，说明你还没有真正理解。
Step three is to produce one example and one non-example. For instance, 7 is prime, but 9 is not prime because it has three factors: 1, 3 and 9.
第三步是给出一个正例和一个反例。例如，7 是质数，但 9 不是质数，因为它有三个因数：1、3 和 9。
Step four is spaced retrieval: test yourself after one day, three days, one week and one month, using either paper flashcards or a Leitner box. Retrieval practice beats re-reading by a wide margin.
第四步是间隔提取练习：在一天、三天、一周和一个月后自测，可用纸质卡片或莱特纳盒。提取练习的效果远胜于反复阅读。
Learn → Compress → Example & Non-example → Space It Out
3. Number Types and Sets | 数的类型与集合
A natural number is a positive counting number starting from 1 (1, 2, 3, ...). Some textbooks include 0, so always check the convention in your own course book.
自然数是从 1 开始的正整数（1, 2, 3, ...）。有些教材把 0 也算作自然数，所以务必确认你自己课本的约定。
An integer is any whole number that can be positive, negative, or zero, such as -2, 0, 5. The symbol for the integers is ℤ.
整数是包括正整数、负整数和零的整数，例如 -2, 0, 5。整数集的符号是 ℤ。
A rational number can be written as a fraction a/b where a and b are integers and b ≠ 0; for example, ³/₄ and -2.5 are rational. Every terminating or recurring decimal is rational.
有理数可以写成分数 a/b 的形式，其中 a 和 b 是整数且 b ≠ 0；例如 ³/₄ 和 -2.5 是有理数。所有有限小数和循环小数都是有理数。
An irrational number cannot be expressed as a simple fraction; its decimal goes on forever without repeating, like π and √2.
无理数不能表示为简单分数；小数形式无限不循环，如 π 和 √2。
Real numbers include all rational and irrational numbers, covering every point on the number line.
实数包含所有有理数和无理数，对应数轴上的每一点。
A prime number has exactly two distinct positive factors: 1 and itself, e.g. 2, 3, 5, 7. 1 is not prime, and 2 is the only even prime.
质数（素数）恰好有两个不同的正因数：1 和它本身，例如 2, 3, 5, 7。1 不是质数，而 2 是唯一的偶质数。
| Term | Symbol | 中文 |
| Natural numbers | ℕ | 自然数 |
| Integers | ℤ | 整数 |
| Rational numbers | ℚ | 有理数 |
| Real numbers | ℝ | 实数 |
Set notation is examined almost every year: '∈' means 'is an element of', '∉' means 'is not an element of', '⊂' means 'is a subset of', '∪' is union and '∩' is intersection. The empty set is written ∅.
集合符号几乎每年都考：'∈' 表示"属于"，'∉' 表示"不属于"，'⊂' 表示"包含于"，'∪' 是并集，'∩' 是交集。空集写作 ∅。
4. Factors, Multiples, Primes and Indices | 因数、倍数、质数与指数
A factor (or divisor) divides a number exactly with no remainder; a multiple is what you get when you multiply that number by an integer. The HCF is the highest common factor and the LCM is the lowest common multiple.
因数（约数）能整除某个数、没有余数；倍数则是把这个数乘以一个整数所得的结果。HCF 是最大公因数，LCM 是最小公倍数。
ted division: 360 = 2³ × 3² × 5. This is the safest route to HCF and LCM of large numbers.
质因数分解把一个数写成质数的乘积，可用因数树或短除法：360 = 2³ × 3² × 5。这是求大数 HCF 和 LCM 最稳妥的方法。
such as 49 = 7²; a cube number is the product of three identical factors, such as 64 = 4³.
平方数是一个整数与自身相乘的积，如 49 = 7²；立方数是三个相同因数相乘的积，如 64 = 4³。
A surd is an irrational root left in exact form, such as √3 or 5√2. To rationalise the denominator you multiply top and bottom by the surd, or by the conjugate for expressions like 1/(3 + √2).
根式（surd）是以精确形式保留的无理数根，例如 √3 或 5√2。分母有理化时，分子分母同乘该根式；形如 1/(3 + √2) 的式子则乘以共轭式。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 a⁻ⁿ = 1/aⁿ a1/n = ⁿ√a
5. Algebraic Vocabulary: Terms, Expressions and Identities | 代数词汇：项、表达式与恒等式
A term is a single number or a product of numbers and letters, such as 7x²; a coefficient is the numerical factor in front of a letter, so in -4ab the coefficient is -4.
项是单个数字，或数字与字母的乘积，例如 7x²；系数是字母前面的数字因数，所以在 -4ab 中系数是 -4。
A constant does not change value, while a variable can. Like terms have identical letter parts and can be collected; 3x and 5x are like terms, but 3x and 3x² are not.
常数的值不变，而变量可以变化。同类项的字母部分完全相同，可以合并；3x 与 5x 是同类项，但 3x 与 3x² 不是。
An expression has no equals sign, an equation contains '=' nly true for particular values, and an identity is true for all values and uses the symbol '≡'.
表达式没有等号；方程包含 '='，只对特定值成立；恒等式对所有值都成立，使用符号 '≡'。
To expand means to remove brackets; to factorise means to write an expression as a product of factors. These are inverse processes, and the acronym FOIL helps with double brackets, though grid multiplication is safer.
展开是去掉括号；因式分解是把表达式写成因数的乘积。二者是互逆过程，缩写 FOIL 有助于计算双括号，但表格乘法更稳妥。
b², (a - b)² = a² - 2ab + b², and the difference of two squares a² - b² = (a + b)(a - b).
需要背熟的恒等式模式包括：(a + b)² = a² + 2ab + b²，(a - b)² = a² - 2ab + b²，以及平方差 a² - b² = (a + b)(a - b)。
6. Solving Equations and Inequalities | 方程与不等式的求解
To solve an equation means to find every value of the variable that makes it true; these values are the roots or solutions, and they form the solution set.
解方程就是找出使等式成立的所有变量取值；这些值称为根或解，它们构成解集。
Rearranging, or changing the subject, means isolating one variable by performing the same inverse operations to both sides. Simultaneous equations in two unknowns can be solved by elimination, by substitution, or graphically.
移项（改变主元）是指对等式两边做相同的逆运算，从而把某个变量单独分离出来。含两个未知数的联立方程可用消元法、代入法或图像法求解。
The inequality symbols are '<' (less than), '>' (greater than), '≤' (less than or equal to) and '≥' . Remember that multiplying or dividing by a negative number reverses the inequality sign.
不等号有 '<'（小于）、'>'（大于）、'≤'（小于等于）和 '≥'（大于等于）。切记：乘以或除以负数时，不等号方向要反转。
x = (-b ± √b² - 4ac) / 2a
The expression b² - 4ac is the discriminant: if it is positive there are two distinct real roots, if it is zero there is one repeated root, and if it is negative there are no real roots. Completing the square rewrites ax² + bx + c in the form a(x + p)² + q.
表达式 b² - 4ac 叫做判别式：为正时有两个不同的实根，为零时有一个重根，为负时没有实根。配方则把 ax² + bx + c 改写成 a(x + p)² + q 的形式。
7. Sequences and Series Terminology | 数列与级数术语
A sequence is an ordered list of numbers called terms, written u₁, u₂, u₃, ... A series is what you get when you add the terms of a sequence together.
数列是有序排列的一列数，其中的数称为项，记作 u₁, u₂, u₃, ... 级数则是把数列各项相加所得的结果。
A term-to-term rule tells you how to get from one term to the next, such as 'add 4' nth term (or general term) is a formula that gives any term directly, such as uₙ = 3n + 2.
递推法则告诉你如何从一项得到下一项，例如"加 4"。第 n 项（通项）是能直接给出任意一项的公式，例如 uₙ = 3n + 2。
is the first term. In a geometric sequence the common ratio r is constant, so uₙ = arⁿ⁻¹.
等差数列中公差 d 恒定，故 uₙ = a + (n - 1)d，其中 a 是首项。等比数列中公比 r 恒定，故 uₙ = arⁿ⁻¹。
e relation expresses each term using earlier ones, for example uₙ₊₁ = 2uₙ - 1 with u₁ = 3.
斐波那契型数列把前两项相加得到下一项。递推关系用前面的项表示每一项，例如 uₙ₊₁ = 2uₙ - 1，且 u₁ = 3。
Key words also include increasing, decreasing, periodic and convergent, the last meaning the terms approach a fixed limiting value.
关键词还包括递增、递减、周期性和收敛，最后一个指各项趋近于某个固定的极限值。
8. Coordinates, Graphs and Functions | 坐标、图像与函数
The x-axis is horizontal and the y-axis is vertical; they meet at the origin (0, 0). An ordered pair (x, y) gives a point, and the four regions of the plane are called quadrants.
x 轴水平、y 轴竖直，它们相交于原点 (0, 0)。有序数对 (x, y) 表示一个点，坐标平面被分成的四个区域称为象限。
nge in y) ÷ (change in x). A line in the form y = mx + c has gradient m and y-intercept c.
直线的斜率（梯度）衡量其陡峭程度：斜率 = (y 的变化量) ÷ (x 的变化量)。形如 y = mx + c 的直线，斜率为 m，y 轴截距为 c。
se product is -1. A line through (x₁, y₁) with gradient m has equation y - y₁ = m(x - x₁).
平行直线的斜率相等；垂直直线的斜率之积为 -1。过点 (x₁, y₁) 且斜率为 m 的直线方程为 y - y₁ = m(x - x₁)。
A function is a rule that assigns exactly one output to each input; f(x) is function notation. The domain is the set of allowed inputs and the range is the set of possible outputs.
函数是一种对应规则，对每个输入给出唯一输出；f(x) 是函数记号。定义域是允许的输入集合，值域是可能的输出集合。
An inverse function f⁻¹(x) reverses the effect of f(x), and a composite function fg(x) means applying g first and then f. Graphs to recognise include linear, quadratic, cubic, reciprocal, exponential and trigonometric curves, together with asymptotes they approach but never touch.
反函数 f⁻¹(x) 逆转 f(x) 的作用；复合函数 fg(x) 表示先作用 g 再作用 f。需要辨认的图像包括直线、抛物线、三次曲线、反比例曲线、指数曲线和三角函数曲线，以及它们趋近却永不相交的渐近线。
9. Geometry and Mensuration Vocabulary | 几何与度量术语
A polygon is a closed plane shape with straight sides; its corners are vertices. Triangles may be equilateral, isosceles or scalene, and quadrilaterals include squares, rectangles, parallelograms, trapeziums, rhombuses and kites.
多边形是由直线边围成的封闭平面图形，其角顶称为顶点。三角形可以是等边、等腰或不等边；四边形包括正方形、矩形、平行四边形、梯形、菱形和筝形。
Angles are acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), straight (180°) or reflex (greater than 180°). Lines are parallel if they never meet, and perpendicular if they meet at 90°.
角可分为锐角（小于 90°）、直角（恰为 90°）、钝角（90° 至 180°）、平角（180°）和优角（大于 180°）。永不相交的直线是平行线，相交成 90° 的是垂直直线。
Congruent shapes are identical in shape and size; similar shapes have equal angles and proportional sides, linked by a scale factor. The area scale factor is the square of the length scale factor, and the volume scale factor is its cube.
全等图形形状和大小完全相同；相似图形对应角相等、对应边成比例，由相似比（比例因子）联系。面积比是长度比的平方，体积比是长度比的立方。
area is the total area of its faces. For a prism, volume = cross-sectional area × length.
周长是图形边界长度；面积是图形内部大小；体积是立体所占空间，表面积是所有面的面积之和。棱柱的体积 = 截面积 × 长度。
For a circle, the radius is the distance from the centre to the edge, the diameter is twice the radius, a chord joins two points on the circle, a tangent touches it at exactly one point, an arc is part of the circumference, and a sector is the region bounded by two radii and an arc.
对圆而言，半径是圆心到圆上一点的距离，直径是半径的两倍，弦连接圆上两点，切线只与圆相切于一点，弧是圆周的一部分，扇形由两条半径和一段弧围成。
erence = 2πr Area of circle = πr² Area of sector = (θ/360) × πr²
10. Pythagoras, Trigonometry and Similarity | 勾股定理、三角学与相似
s the longest side, opposite the right angle; the other two sides are the legs. Pythagoras' theorem states that a² + b² = c², where c is the hypotenuse.
在直角三角形中，斜边是最长边，位于直角对面；另外两边是直角边。勾股定理为 a² + b² = c²，其中 c 是斜边。
se, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. The mnemonic is SOHCAHTOA.
三个三角比涉及对边、邻边和斜边：sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边。记忆口诀是 SOHCAHTOA。
Exact values to memorise are sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 45° = 1/√2 and sin 60° = √3/2. A bearing is measured clockwise from north and written with three digits, such as 045°.
需要背熟的精确值有 sin 30° = ½，cos 60° = ½，tan 45° = 1，sin 45° = 1/√2，sin 60° = √3/2。方位角从正北顺时针量度，用三位数字书写，例如 045°。
The angle of elevation is measured upwards from the horizontal, and the angle of depression downwards. In non-right-angled triangles, the sine rule and cosine rule apply, together with the area formula below.
仰角从水平线向上量度，俯角向下量度。在非直角三角形中，要使用正弦定理和余弦定理，以及下面的面积公式。
Area = ½ ab sin C a/sin A = b/sin B a² = b² + c² - 2bc cos A
11. Probability and Statistics Terms | 概率与统计术语
An experiment produces outcomes, and the sample space is the set of all possible outcomes. An event is a subset of the sample space, and its probability lies between 0 and 1 inclusive.
试验会产生结果，样本空间是所有可能结果的集合。事件是样本空间的子集，其概率介于 0 与 1 之间（含 0 和 1）。
Mutually exclusive events cannot happen at the same time, so P(A ∪ B) = P(A) + P(B). Independent events do not affect each other, so P(A ∩ rams show successive events, while Venn diagrams and two-way tables show overlapping ones.
互斥事件不能同时发生，故 P(A ∪ B) = P(A) + P(B)。独立事件互不影响，故 P(A ∩ B) = P(A) × P(B)。树状图展示连续发生的事件，韦恩图和双向表展示互相重叠的事件。
The complementary event 'not A' has probability 1 - P(A). Conditional probability is the probability of A given that B has already happened, written P(A | B).
补事件"非 A"的概率为 1 - P(A)。条件概率是在 B 已发生的前提下 A 发生的概率，写作 P(A | B)。
For a data set, the mean is the total divided by the count, the median is the middle value when ordered, the mode is the most frequent value, and the range is the largest value minus the smallest.
对一组数据，平均数（均值）是总和除以个数，中位数是排序后位于中间的值，众数是出现次数最多的值，极差是最大值减最小值。
le) measures spread while ignoring outliers. A frequency table records how often each value occurs.
四分位数把有序数据分成四部分，四分位距（IQR = 上四分位数 - 下四分位数）用于衡量离散程度，同时排除异常值的影响。频数表记录每个取值出现的次数。
On a scatter graph, positive correlation means both variables rise together, negative correlation means one falls as the other rises, and a line of best fit estimates the trend. Always distinguish correlation from causation.
在散点图中，正相关表示两个变量同增，负相关表示一个增大时另一个减小，最佳拟合线用于估计趋势。务必区分相关性与因果关系。
12. Exam Command Words and Final Checklist | 考试指令词与最终检查清单
Command words carry marks. Learn to match each one with the correct action before you ever enter the exam hall.
指令词直接对应得分点。在进考场之前，就要学会把每个指令词与正确的作答动作对应起来。
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