📚 Year 9 AQA Mathematics: Core Knowledge Points | Year 9 AQA 数学：核心知识点梳理
Year 9 is the final year of Key Stage 3, consolidating all the foundation knowledge you will need before starting GCSE Maths. The AQA syllabus covers number, algebra, geometry, statistics and probability. This revision guide walks you through the most important topics, with clear explanations and examples.
九年级是 KS3 的最后一年，为你进入 GCSE 数学打好所有基础。AQA 课程涵盖数、代数、几何、统计和概率。这份复习指南将带你梳理最重要的知识点，并配以清晰的解释和例题。
1. Number and Place Value | 数与位值
Large numbers are read in groups of three digits separated by commas. For example, 4,500,000 is ‘four million, five hundred thousand’. Place value columns include units, tens, hundreds, thousands, ten thousands, hundred thousands, millions and so on.
大数按每三位一组用逗号分隔来读。例如 4,500,000 读作“四百五十万”。位值列包括个位、十位、百位、千位、万位、十万位、百万位等。
You must be able to round numbers to a given number of decimal places or significant figures. Rounding rules: if the next digit is 5 or more, round up; if less than 5, round down. For example, 3.456 rounded to 2 decimal places is 3.46; to 2 significant figures it is 3.5.
你必须能将数字四舍五入到指定的小数位数或有效数字。规则：下一位数大于或等于 5 则进位，小于 5 则舍去。例如 3.456 精确到小数点后两位是 3.46；精确到两位有效数字是 3.5。
Negative numbers are used in contexts such as temperature and bank balances. Operations with negatives: adding a negative is the same as subtracting its absolute value; subtracting a negative is the same as adding. For multiplication and division, two negatives make a positive.
负数用在温度和银行余额等情境中。负数运算：加上一个负数等于减去它的绝对值；减去一个负数等于加上一个正数。对于乘除法，负负得正。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Equivalent fractions are found by multiplying or dividing the numerator and denominator by the same non-zero number. Simplifying a fraction means reducing it to its lowest terms. For example, 12/16 simplifies to 3/4 by dividing both by 4.
等值分数可通过将分子和分母同时乘或除以同一个非零数得到。化简分数意味着将其约分到最简形式。例如 12/16 分子分母同除以 4 得到 3/4。
To add or subtract fractions, find a common denominator first. For 2/5 + 1/3, use 15: 6/15 + 5/15 = 11/15. Mixed numbers should be converted to improper fractions before multiplying or dividing. Multiply numerators together and denominators together; to divide, multiply by the reciprocal.
加减分数要先找到公分母。2/5 + 1/3，用 15 作分母：6/15 + 5/15 = 11/15。带分数在乘除前应先化为假分数。乘法是分子乘分子、分母乘分母；除法是乘以倒数。
Decimals are linked to fractions: 0.75 = 75/100 = 3/4. To convert a fraction to a decimal, divide the numerator by the denominator. Terminating decimals have a finite number of digits, while recurring decimals repeat a pattern infinitely.
小数与分数关联：0.75 = 75/100 = 3/4。分数转小数用分子除以分母。有限小数位数有限，循环小数则无限重复某一模式。
Percentages mean ‘out of 100’. To find a percentage of a quantity, multiply by the percentage over 100. Percentage increase or decrease uses the formula: new value = original × (1 ± percentage/100). Understand percentage change and reverse percentages.
百分数意为“每一百”。求一个数的百分之几，用它乘以百分数/100。增减百分比公式：新值 = 原值 × (1 ± 百分数/100)。理解百分数变化和逆向百分数。
3. Ratio and Proportion | 比与比例
A ratio compares parts of a whole, written as a:b. It can be simplified like fractions by dividing by common factors. For example, 8:12 simplifies to 2:3. Ratios can be scaled up to find equivalent amounts in real-life problems.
比用来比较整体的各部分，写作 a:b。它可以像分数一样通过除以公因数来化简。例如 8:12 化简为 2:3。在实际问题中，比可以通过放大来找到等值数量。
To divide a quantity in a given ratio, find the total number of parts, then work out the value of one part. For £50 in the ratio 2:3, total parts = 5, one part = £10, so 2 parts = £20 and 3 parts = £30.
按给定比分配数量：先求总份数，再求每份的值。如 £50 按 2:3 分配，总份数 5，每份 £10，故两份 £20，三份 £30。
Proportion describes a multiplicative relationship. Two quantities are in direct proportion if they increase or decrease at the same rate; y = kx. Inverse proportion means one increases as the other decreases; xy = k. Recognise proportional problems and solve using unitary method or constant of proportionality.
比例描述乘法关系。两个量若以相同速率增减则为正比例：y = kx。反比例则是一增一减：xy = k。识别比例问题并用单元法或比例常数求解。
4. Algebraic Expressions and Formulae | 代数表达式与公式
Algebra uses letters to represent unknown numbers. Terms are the building blocks of expressions, separated by + or − signs. Like terms have the same variable and power; they can be collected. For example, 3a + 2b − a + 5b = 2a + 7b.
代数用字母表示未知数。项是表达式的基本单位，由 + 或 − 分隔。同类项有相同的变量和指数，可以合并。如 3a + 2b − a + 5b = 2a + 7b。
Brackets are expanded using the distributive law: a(b + c) = ab + ac. Factorising is the reverse process: finding common factors and writing expressions as products, e.g., 6x + 9 = 3(2x + 3). Double brackets like (x + a)(x + b) expand to x² + (a+b)x + ab.
用乘法分配律展开括号：a(b + c) = ab + ac。因式分解是逆过程：提取公因式，将表达式写成乘积，如 6x + 9 = 3(2x + 3)。双括号如 (x + a)(x + b) 展开得 x² + (a+b)x + ab。
A formula shows the relationship between variables. You substitute known values to find unknowns. For example, in v = u + at, if u = 2, a = 3, t = 4, then v = 2 + 3×4 = 14. Rearranging formulae involves using inverse operations to change the subject.
公式表示变量之间的关系。代入已知值求未知数。如 v = u + at，若 u=2, a=3, t=4，则 v = 2 + 3×4 = 14。公式变形通过逆运算改变主项。
5. Equations and Inequalities | 方程与不等式
An equation states that two expressions are equal. Solving means finding the value of the unknown that makes it true. Keep the equation balanced by performing the same operation on both sides. For 2x + 3 = 11, subtract 3: 2x = 8, then x = 4.
方程表示两个表达式相等。求解即找到使等式成立的未知数值。保持方程平衡，两边同时进行相同运算。如 2x + 3 = 11，两边减 3：2x = 8，得 x = 4。
Inequalities use symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). Solve them like equations, but if you multiply or divide by a negative number, reverse the inequality sign. For −2x ≤ 6, divide by −2 gives x ≥ −3.
不等式使用符号：< (小于), > (大于), ≤ (小于等于), ≥ (大于等于)。解法类似方程，但若乘除负数，需反转不等号。如 −2x ≤ 6，除以 −2 得 x ≥ −3。
You can represent inequalities on a number line using open circles for strict inequalities and closed circles for inclusive ones. Forming equations from word problems involves identifying the unknown, writing expressions and setting up the equation.
你可以在数轴上表示不等式，严格不等用空心圆，包含等号用实心圆。从文字题建立方程需要设定未知数、写出表达式并建立等式。
6. Sequences | 序列
A sequence is a set of numbers following a rule. Arithmetic sequences have a common difference between terms. For 3, 7, 11, 15, …, the rule is ‘add 4’. The nth term of an arithmetic sequence is given by an + b, where a is the common difference and b is the zero term (value when n=0).
序列是按规则排列的一组数。等差数列相邻项之差恒定。如 3, 7, 11, 15, … 规则是“加 4”。等差数列的第 n 项为 an + b，其中 a 是公差，b 是第零项 (n=0 时的值)。
To find the nth term, calculate the difference and then work out what to add or subtract. For the sequence 5, 9, 13, 17, difference = 4, so nth term = 4n + 1. Check: n=1 gives 5.
求第 n 项：先求公差，再找出需要加减的数。序列 5, 9, 13, 17，公差 4，故第 n 项 = 4n + 1。验证：n=1 得 5。
Other sequences include geometric (multiplying by a constant) and special sequences like square numbers (1, 4, 9, 16, …) and triangular numbers (1, 3, 6, 10, …). Recognising patterns is key.
其他序列包括等比数列（乘以常数）和特殊序列如平方数 (1, 4, 9, 16, …) 和三角形数 (1, 3, 6, 10, …)。模式识别是关键。
7. Coordinates and Graphs | 坐标与图表
The Cartesian plane uses x (horizontal) and y (vertical) axes. A point is written as (x, y). You plot linear equations by calculating y-values for chosen x-values and joining the points with a straight line. The graph of y = mx + c has gradient m and y-intercept c.
笛卡尔坐标系使用 x 轴（水平）和 y 轴（垂直）。点写作 (x, y)。画线性方程时，选取 x 值计算 y 值，将点连成直线。y = mx + c 的图像斜率为 m，y 轴截距为 c。
The gradient is rise over run. Between two points (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁)/(x₂ − x₁). A positive gradient slopes upwards, negative slopes downwards. Parallel lines have equal gradients.
斜率是纵差除以横差。两点 (x₁, y₁) 和 (x₂, y₂) 之间，斜率 m = (y₂ − y₁)/(x₂ − x₁)。正斜率向上倾斜，负斜率向下倾斜。平行线斜率相等。
You can find the equation of a straight line given its graph or two points. Also, you need to interpret real-life graphs, such as distance-time graphs where the gradient is speed and a horizontal line means stationary.
根据图像或两点可求直线方程。此外，需理解实际情境图，如距离-时间图中斜率代表速度，水平线表示静止。
8. Angles and Shapes | 角与形状
Angle facts: angles on a straight line add to 180°, around a point to 360°. Vertically opposite angles are equal. In polygons, interior and exterior angles sum to specific values. The sum of interior angles of an n-sided polygon is (n−2)×180°. Each exterior angle of a regular polygon is 360°/n.
角的知识：直线上的角之和为 180°，一点周围的角之和为 360°。对顶角相等。多边形内角和与外交和有特定值。n 边形内角和 = (n−2)×180°。正多边形每个外角 = 360°/n。
Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). The sum of angles in a triangle is 180°. In an isosceles triangle, base angles are equal. Pythagoras’ theorem applies to right-angled triangles: a² + b² = c², where c is the hypotenuse.
三角形按边分为等边、等腰、不等边；按角分为锐角、直角、钝角。三角形内角和为 180°。等腰三角形底角相等。勾股定理适用于直角三角形：a² + b² = c²，c 为斜边。
Angles in parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°. Use these to find missing angles in diagrams.
平行线中的角：同位角相等，内错角相等，同旁内角之和为 180°。利用这些性质求图形中的未知角。
9. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around a shape. For rectangles, P = 2(l + w). For compound shapes, add all outer side lengths. Area of rectangle = length × width; triangle = ½ × base × height; parallelogram = base × vertical height; trapezium = ½(a + b)h.
周长是图形周边的距离。矩形周长 P = 2(l + w)。复合图形将所有外围边长相加。矩形面积 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 垂直高；梯形 = ½(a + b)h。
Area of a circle: A = π r², where r is the radius. Circumference: C = 2π r or π d. For composite shapes involving parts of circles, calculate areas and add or subtract as needed.
圆面积：A = π r²，r 为半径。圆周长：C = 2π r 或 π d。对于含部分圆的组合图形，分别计算面积再加减。
Volume measures the space inside a 3D shape. Volume of a cuboid = length × width × height. Prism volume = area of cross-section × length. Surface area is the total area of all faces. Use nets to visualise surfaces.
体积测量三维图形内部空间。长方体体积 = 长 × 宽 × 高。棱柱体积 = 横截面积 × 长度。表面积是所有面的总面积。可用展开图来想象表面。
10. Statistics and Charts | 统计与图表
Data can be displayed in various charts: bar charts for discrete data, line graphs for continuous trends, pie charts for proportions, and scatter graphs for relationships between two variables. A scatter graph can show positive, negative or no correlation.
数据可以用多种图表展示：条形图用于离散数据，线形图用于连续趋势，饼图用于比例，散点图用于两个变量的关系。散点图可以显示正相关、负相关或无相关。
Measures of central tendency: mean is the sum of data divided by the number of items; median is the middle value when ordered; mode is the most frequent value. Range shows spread: maximum minus minimum. Outliers can affect the mean significantly.
集中趋势量数：平均数是数据和除以项数；中位数是排序后中间的值；众数是出现频率最高的值。极差表示分散程度：最大值减最小值。异常值会显著影响平均数。
You should be able to construct and interpret frequency tables, including grouped frequency tables. From a frequency table, estimate the mean using midpoints of class intervals. Stem-and-leaf diagrams display data while preserving original values.
你应能编制和解读频数表，包括分组频数表。利用组中值可从频数表中估计平均数。茎叶图既展示数据又保留原始值。
11. Probability | 概率
Probability is the chance of an event happening, expressed as a fraction, decimal or percentage. It ranges from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率是事件发生的可能性，用分数、小数或百分数表示。范围从 0（不可能）到 1（必然）。对于等可能结果，P(事件) = 有利结果数 / 总结果数。
The probability that an event does not happen is 1 minus the probability that it does. Sample space diagrams list all possible outcomes of two events and can be used to calculate probabilities. For combined events, you can use tree diagrams.
事件不发生的概率等于 1 减去它发生的概率。样本空间图列出两个事件的所有可能结果，可用于计算概率。对于组合事件，可以用树状图。
Expected frequency = probability × number of trials. Understand that experimental probability may differ from theoretical probability due to randomness, but both approaches help predict outcomes.
期望频数 = 概率 × 试验次数。由于随机性，实验概率可能与理论概率不同，但两种方法都有助于预测结果。
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