📚 Cambridge Lower Secondary Mathematics Book 9 Core Concepts | 剑桥初中数学第九册核心知识点精讲
Welcome to this comprehensive revision guide covering the essential topics from the Cambridge Lower Secondary Mathematics Learner’s Book 9 (2nd Edition). The content is structured to help you revisit key ideas, master methods, and feel confident when solving problems. Each section pairs an English explanation with its Chinese equivalent, allowing you to learn bilingually while grasping the underlying mathematics.
欢迎阅读本指南，它涵盖了剑桥初中数学第九册（第二版）的核心知识点。本文以中英双语对照的形式梳理关键概念与解题方法，帮助你稳步复习、增强解题信心，同时深入理解数学原理。
1. Integers, Powers and Roots | 整数、幂与根
Integers include positive and negative whole numbers. When you multiply or divide integers, the result follows the sign rules: same signs produce a positive, different signs produce a negative.
整数包含正整数、负整数和零。乘除运算的符号规则为：同号得正，异号得负。
You can express repeated multiplication using exponent notation. For example, 2 × 2 × 2 = 2³, where 2 is the base and 3 is the exponent. The exponent tells you how many times the base is used as a factor.
您可以用指数表示重复相乘。例如 2 × 2 × 2 = 2³，其中 2 是底数，3 是指数，表示底数相乘的次数。
The laws of exponents apply when multiplying or dividing powers with the same base: aᵐ × aⁿ = aᵐ⁺ⁿ, and aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
同底数幂的乘除法则为：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ。
3⁴ × 3² = 3⁶ = 729
A square root undoes squaring: √(x²) = x for non‑negative x. A cube root undoes cubing: ∛(x³) = x. For example, √49 = 7 and ∛27 = 3.
平方根是平分的逆运算：√(x²) = x（x ≥ 0）。立方根是立方的逆运算：∛(x³) = x。例如 √49 = 7，∛27 = 3。
You can also work with negative exponents: a⁻ⁿ = 1/(aⁿ). And any non‑zero number raised to the power 0 equals 1: a⁰ = 1 (a ≠ 0).
负指数表示倒数：a⁻ⁿ = 1/(aⁿ)。任何非零数的零次幂等于 1：a⁰ = 1 (a ≠ 0)。
2. Algebraic Expressions and Equations | 代数式与方程
An algebraic expression contains variables, numbers and operation symbols. You simplify expressions by collecting like terms – terms that have exactly the same variable part.
代数式包含变量、数字和运算符号。通过合并同类项（含有完全相同的变量部分的项）来化简表达式。
When expanding brackets, multiply each term inside by the factor outside. For example, 3(x + 4) = 3x + 12. With two brackets, use the distributive law twice.
去括号时，用括号外的因数乘括号内的每一项。例如 3(x + 4) = 3x + 12。若有两个括号相乘，则需要应用两次分配律。
(x + 2)(x – 5) = x² – 3x – 10
Factoring is the reverse of expanding. Look for a common factor or recognise patterns such as the difference of two squares: a² – b² = (a – b)(a + b).
因式分解是展开的逆过程。可以提取公因式，或者识别平方差公式：a² – b² = (a – b)(a + b)。
An equation states that two expressions are equal. To solve it, carry out inverse operations while keeping the equation balanced. Always check your solution by substituting it back.
方程表示两个式子相等。求解时，通过逆运算保持等式平衡，并将解代回原方程检验。
You will also encounter inequalities, which use symbols like <, >, ≤, ≥. Solving them follows similar steps, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
您还会碰到不等式，使用符号 <、>、≤、≥。求解步骤与方程相似，但要记得：当乘或除以负数时，不等号方向要改变。
3. Fractions, Decimals and Percentages | 分数、小数和百分数
Equivalent fractions represent the same portion of a whole. You can create equivalent fractions by multiplying or dividing the numerator and denominator by the same non‑zero number.
等值分数表示相同的部分。将分子和分母同时乘或除以同一个非零数，就能得到等值分数。
When adding or subtracting fractions, find a common denominator first. For multiplication, multiply numerators together and denominators together. For division, multiply by the reciprocal.
分数加减时，先化成同分母。乘法时，分子相乘作分子，分母相乘作分母。除法时，乘以除数的倒数。
Decimals are another way to write fractions whose denominators are powers of ten. You can convert a fraction to a decimal by division. Terminating decimals stop after a finite number of digits; recurring decimals have a repeating pattern.
小数是分母为 10 的幂的另一种写法。可以用除法将分数化为小数。有限小数会在有限位后停止，循环小数则出现重复的数码模式。
Percent means ‘out of 100’. To convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply by 100%. For a decimal, multiply by 100%.
百分数表示“每一百”。将分数化百分数，可化为分母 100 的等值分数，或乘以 100%。小数则直接乘以 100%。
Finding a percentage of an amount: first convert the percentage to a decimal or fraction, then multiply. You can also use the unitary method or the formula: part = (percentage ÷ 100) × whole.
求一个数的百分之几：先将百分数化为小数或分数，然后相乘。也可使用归归一法或公式：部分 = (百分数 ÷ 100) × 整体。
4. Ratio, Rate and Proportion | 比、率与比例
A ratio compares quantities of the same kind, written as a : b. You simplify ratios by dividing both sides by the greatest common divisor, just like simplifying a fraction.
比用于比较同类量，写作 a : b。化简比的方式与约分类似，两边同除以最大公约数。
When sharing a quantity in a given ratio, add the parts to find the total number of shares, then divide. For example, to share £56 in the ratio 3 : 5, total shares = 8; one share = £56 ÷ 8 = £7, then 3 shares = £21, 5 shares = £35.
按比分配时，先求总份数，再将总量除以总份数得到一份的量。例如按 3 : 5 分 56 英镑，总份数 = 8，一份 = 7 英镑，所以三份为 21 英镑，五份为 35 英镑。
A rate compares two quantities of different kinds, e.g. speed (km/h) or cost per kg. A unit rate makes the second quantity 1, which helps with comparison.
率（比率）比较两个不同类的量，如速度（千米/时）或价格（元/千克）。单位率将第二个量化为 1，便于比较。
Proportion describes the equality of two ratios. Direct proportion means that 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。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。
Inverse proportion means that as one quantity increases, the other decreases so that the product remains constant. If y is inversely proportional to x, then xy = k (constant).
反比例指的是当一个量增大时，另一个量以乘积不变的方式减小。若 y 与 x 成反比，则 xy = k（常数）。
5. Sequences and Functions | 数列与函数
A sequence is a list of numbers that often follows a rule. The term-to-term rule tells you how to get from one term to the next; the position-to-term rule (nth term) lets you find any term directly.
数列是按规律排列的一列数。递推规则描述前后项之间的关系；通项规则（第 n 项）让您直接求出任意项。
For an arithmetic sequence, the difference between consecutive terms is constant. The nth term is: a + (n − 1)d, where a is the first term and d is the common difference.
等差数列中，相邻两项的差为常数。第 n 项为：a + (n − 1)d，其中 a 为首项，d 为公差。
You can describe a function using words, an algebraic rule, a table of values, or a graph. The input (x) is mapped to the output (y) by the function rule, often written as y = f(x).
函数可以用文字描述、代数式、数值表或图像表示。输入 (x) 通过函数规则映射到输出 (y)，常写作 y = f(x)。
A linear function has the form y = mx + c. Its graph is a straight line. m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y-axis).
线性函数形如 y = mx + c，图像为直线。m 是斜率（坡度），c 是 y 轴截距（直线与 y 轴交点的纵坐标）。
To find the equation of a line from two points, first calculate the gradient m = (y₂ − y₁)/(x₂ − x₁), then substitute one point to find c.
已知两点求直线方程：先求斜率 m = (y₂ − y₁)/(x₂ − x₁)，再代入其中一个点的坐标求出 c。
6. Angles and Geometrical Reasoning | 角与几何推理
Angles are measured in degrees (°). You need to know names for different types: acute (0–90°), right (90°), obtuse (90–180°), straight (180°) and reflex (180–360°).
角以度（°）为单位。需要掌握角的分类：锐角（0–90°）、直角（90°）、钝角（90–180°）、平角（180°）、优角（180–360°）。
When two lines intersect, vertically opposite angles are equal. Angles on a straight line sum to 180°, and angles around a point sum to 360°.
两直线相交形成对顶角，对顶角相等。共线角之和为 180°，绕点一周的角之和为 360°。
Parallel lines create special angle relationships: corresponding angles are equal, alternate interior angles are equal, and co‑interior angles sum to 180°.
平行线产生特殊的角关系：同位角相等，内错角相等，同旁内角互补（和为 180°）。
In a triangle, the interior angles sum to 180°, and the exterior angle equals the sum of the two opposite interior angles. These facts help you find missing angles.
三角形内角和为 180°，外角等于不相邻的两个内角之和。这些性质常用于计算未知角。
7. Transformations | 图形变换
A translation slides a shape without turning or flipping. It is described by a column vector (x above y), where x is the horizontal movement and y is the vertical movement.
平移将图形滑移，不旋转也不翻转。它可以用列向量 (x 在上，y 在下) 描述，x 为横向移动分量，y 为纵向移动分量。
A reflection flips a shape over a mirror line. The line of reflection can be vertical, horizontal, or diagonal (e.g. y = x). The image is the same distance from the mirror line as the object.
反射将图形沿镜线翻转。反射轴可以是竖直、水平或斜线（如 y = x）。像与图形到镜线的距离相等。
A rotation turns a shape about a centre through a given angle and direction (clockwise or anticlockwise). You need to specify the centre, angle, and direction of rotation.
旋转让图形绕某个中心转动给定角度和方向（顺时针或逆时针）。描述旋转时必须指明中心、角度和方向。
An enlargement changes the size of a shape by a scale factor. If the scale factor is greater than 1, the shape gets larger; if between 0 and 1, it gets smaller. A negative scale factor also causes a reversal of direction.
位似变换通过比例因子改变图形大小。比例因子大于 1 时图形放大，介于 0 和 1 之间时缩小。负比例因子还会让图形反向。
8. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around a shape. For regular shapes, you can use formulas; for compound shapes, add the lengths of all sides. Make sure all lengths are in the same units.
周长是围绕图形一周的长度。规则图形可用公式计算；组合图形则将所有边长相加。确保单位一致。
Area of a rectangle = length × width; area of a triangle = ½ × base × height; area of a parallelogram = base × vertical height.
矩形面积 = 长 × 宽；三角形面积 = ½ × 底 × 高；平行四边形面积 = 底 × 垂直高。
Area of a trapezium = ½ × (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular height.
梯形面积 = ½ × (a + b) × h，其中 a 和 b 为上底和下底的长度，h 为垂直高。
For circles, circumference = 2πr or πd, and area = πr². When π is needed, you may be asked to leave answers in terms of π or use 3.14.
圆的周长 = 2πr 或 πd，面积 = πr²。根据题目要求，可用 π 表示结果或取 3.14 近似计算。
Volume measures the space inside a 3D shape. Volume of a prism = area of cross‑section × length. For a cylinder, volume = πr²h. Remember to include cubic units.
体积衡量立体内部的空间大小。棱柱的体积 = 底面积 × 高；圆柱的体积 = πr²h。记得带上立方单位。
9. Statistics | 统计
Data can be collected and organised into frequency tables. A frequency table shows how often each value or group of values occurs. It is the basis for many diagrams.
数据可收集并整理成频数表。频数表显示每个数值（或每组数值）出现的次数。它是绘制多种统计图的基础。
The mean (average) is calculated by summing all data values and dividing by the number of values. The median is the middle value when data are ordered; the mode is the most frequent value; the range is the difference between the largest and smallest values.
平均数（均值）等于所有数据之和除以数据个数。中位数是将数据排序后位于中间的数值；众数是出现次数最多的值；极差是最大值与最小值之差。
For grouped data, you can estimate the mean by using midpoints of intervals. The modal class is the interval with the highest frequency. The median class interval can be identified using cumulative frequency.
对于分组数据，可用组中点估算平均数。众数组是频数最高的区间。借助累积频数可以确定中位数所在的区间。
Bar charts display discrete data, while histograms display grouped continuous data (with no gaps between bars). A pie chart shows proportions; each sector angle = (frequency ÷ total) × 360°.
条形图展示离散数据；直方图展示分组连续数据（条间无缝隙）。饼图显示比例关系，每个扇形的圆心角 = (频数 ÷ 总数) × 360°。
Scatter graphs show the relationship between two variables. If the points roughly follow a straight line, there is correlation: positive, negative or none. A line of best fit can be used to make predictions.
散点图展示两个变量之间的关系。若点大致沿直线分布，则存在相关：正相关、负相关或不相关。最佳拟合线可用于预测。
10. Probability | 概率
Probability measures how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). The theoretical probability of an event = (number of favourable outcomes) ÷ (total number of equally likely outcomes).
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。事件的理论概率 =（有利结果数）÷（所有等可能结果总数）。
When outcomes are combined, you can list all possibilities in a sample space. A two‑way table or a tree diagram helps organise outcomes for two events, especially when events are independent.
当结果组合时，可以列出所有可能性的样本空间。二维表或树状图有助于整理两个事件的结果，尤其是当事件相互独立时。
The probability of an event not occurring = 1 – probability that it does occur. This is called the complement rule.
事件不发生的概率 = 1 – 事件发生的概率。这称为互补规则。
Experimental probability (relative frequency) = (number of times an event occurs) ÷ (total number of trials). With more trials, the experimental probability tends to get closer to the theoretical probability.
实验概率（相对频率）=（事件发生次数）÷（试验总次数）。试验次数越多，实验概率越接近理论概率。
For independent events A and B, P(A and B) = P(A) × P(B). This multiplication rule is essential when finding the probability of two events both happening.
对于独立事件 A 和 B，P(A 且 B) = P(A) × P(B)。在求两事件同时发生的概率时，这一乘法规则至关重要。
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