📚 Year 7 CAIE Advanced Mathematics: Core Knowledge Compilation | Year 7 CAIE 进阶数学：核心知识点梳理
This article provides a comprehensive revision guide for Year 7 CAIE Advanced Mathematics, covering the essential topics that build a strong foundation for future studies. The key areas include number theory, algebra, geometry, statistics, and an introduction to probability. Each section is carefully explained to help students master core concepts and examination techniques.
本文为 Year 7 CAIE 进阶数学提供一份全面的复习指南，涵盖为未来学习打下坚实基础的核心主题。关键领域包括数论、代数、几何、统计以及概率入门。每个部分都经过精心讲解，帮助学生掌握核心概念与考试技巧。
1. Integers and Operations | 整数及其运算
In Year 7 Advanced Mathematics, a solid understanding of integers—including negative numbers—is essential. Students learn to perform addition, subtraction, multiplication, and division with integers, applying rules such as the order of operations (BODMAS/BIDMAS). Mastering these operations allows students to handle more complex algebraic and numerical problems with confidence.
在七年级进阶数学中，扎实掌握整数（包括负数）是基础。学生要学会整数的加减乘除，并应用运算顺序（如括号、指数、乘除、加减）。掌握这些运算能让学生自信地应对更复杂的代数与数值问题。
- Integer definition: whole numbers including negatives, zero and positives — 整数定义：包括负整数、零和正整数的整数。
- Adding a negative is equivalent to subtraction; subtracting a negative is equivalent to addition — 加上负数等于做减法；减去负数等于做加法。
- Multiplication/Division sign rules tive × positive = positive; negative × negative = positive; positive × negative = negative — 乘除符号法则：正正得正，负负得正，正负得负。
- Order of operations: Brackets, Orders (powers/indices), Division and Multiplication (left to right), Addition and Subtraction (left to right) — 运算顺序：括号、指数、乘除（从左到右）、加减（从左到右）。
(-3) × (-4) = 12 and (-8) ÷ 2 = -4
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Fluency in converting between fractions, decimals and percentages is a core skill. Students must be able to simplify fractions, find equivalent fractions, and perform all four operations with mixed numbers. Understanding how to compare and order these forms is equally important for data interpretation and problem solving.
熟练地在分数、小数和百分数之间进行转换是一项核心技能。学生必须能够约分、寻找等值分数，并能对带分数进行四则运算。理解如何比较和排序这些形式对于数据解读和问题求解同样重要。
- Fraction basics: numerator/denominator, proper, improper, mixed numbers — 分数基础：分子/分母，真分数，假分数，带分数。
- Equivalence: 1/2 = 0.5 = 50% — 等值关系：1/2 = 0.5 = 50%。
- Operations with fractions: addition/subtraction require common denominators; multiplication: multiply numerators and denominators; division: multiply by the reciprocal — 分数运算：加减需通分；乘法：分子乘分子，分母乘分母；除法：乘以倒数。
- Percentage increase/decrease: find the change and express as a percentage of the original — 百分比增减：找出变化量并表示为原值的百分数。
2/3 ÷ 5/6 = 2/3 × 6/5 = 12/15 = 4/5
3. Algebraic Expressions and Formulae | 代数表达式与公式
Algebra is the language of advanced mathematics. Year 7 students learn to use letters to represent variables, simplify expressions by collecting like terms, and substitute numbers into formulae. The distributive law is introduced to expand brackets, which is fundamental for solving equations later on.
代数是进阶数学的语言。七年级学生学习使用字母表示变量、通过合并同类项简化表达式，以及将数字代入公式。分配律被引入用于展开括号，这对日后解方程至关重要。
- Variables and constants: a letter representing an unknown, e.g. x, y — 变量与常量：字母代表未知数，如 x, y。
- Collecting like terms: 5a + 3b - 2a = 3a + 3b — 合并同类项：5a + 3b - 2a = 3a + 3b。
- Substitution: if a=3 and b=4, then 2a + b = 10 — 代入求值：若 a=3, b=4，则 2a + b = 10。
- Expanding brackets: a(b + c) = ab + ac — 展开括号：a(b + c) = ab + ac。
3(x + 4) = 3x + 12 and 2x(x - 5) = 2x² - 10x
4. Linear Equations and Inequalities | 线性方程与不等式
he statement true. Students use inverse operations and balance methods, often presented as 'do the same to both sides' end this idea by using symbols >, <, ≥ and ≤, and solutions can be shown on a number line.
解线性方程是找出使等式成立的未知数值。学生使用逆运算和平衡方法，通常表述为“等式两边进行相同操作”。不等式通过使用 >, <, ≥ 和 ≤ 符号延伸了这一概念，解集可在数轴上表示。
- Equation solving steps: isolate the variable — 解方程步骤：隔离变量。
- Example: 2x + 3 = 11 → 2x = 8 → x = 4 — 示例：2x + 3 = 11 → 2x = 8 → x = 4。
- Inequality notation: x ≥ 2 means x is greater than or equal to 2; open/closed circles on number line — 不等式符号：x ≥ 2 表示 x 大于或等于 2；数轴上用空心/实心圆。
- Multiplying/dividing by a negative reverses the inequality sign — 乘以或除以负数时，不等号方向改变。
If -3y < 9, then y > -3
5. Sequences and Patterns | 数列与规律
Sequences are ordered lists of numbers governed by a rule. Students learn to recognise arithmetic sequences (constant difference), geometric sequences (constant ratio), and special sequences like square and triangular numbers. Finding the nth term allows prediction of any term in a linear sequence without listing all previous terms.
数列是按一定规则排列的一列数。学生学习识别等差数列（公差固定）、等比数列（公比固定），以及特殊数列如平方数和三角数。寻找第 n 项可以在不列出前面所有项的情况下预测线性数列中任意一项。
- Arithmetic sequence: 3, 7, 11, 15, ... (add 4) — 等差数列：3, 7, 11, 15, ...（每次加4）。
- Geometric sequence: 2, 6, 18, 54, ... (multiply by 3) — 等比数列：2, 6, 18, 54, ...（每次乘3）。
- Triangular numbers: 1, 3, 6, 10, ... — 三角数：1, 3, 6, 10, ...。
- nth term of a linear sequence: for 5, 8, 11, 14, ..., nth term = 3n + 2 — 线性数列的第 n 项：对于 5, 8, 11, 14, ..., 第 n 项 = 3n + 2。
Term-to-term rule: start at 5, add 3 each time → nth term = 3n + 2
6. Angles and Shape Properties | 角与形状性质
Geometry in Year 7 focuses on classifying angles, understanding properties of triangles and quadrilaterals, and applying facts about angles on a straight line, at a point, and in parallel lines. Symmetry (line and rotational) is also explored, along with basic constructions using a compass and protractor.
七年级几何的重点是角的分类、理解三角形和四边形的性质，并应用直线上的角、点周围的角以及平行线中的角的相关结论。此外还探究对称性（线对称和旋转对称），以及使用圆规和量角器进行基本作图。
- Angle types: acute (0°–90°), right (90°), obtuse (90°–180°), reflex (180°–360°) — 角的种类：锐角(0°–90°)、直角(90°)、钝角(90°–180°)、优角(180°–360°)。
- Angle facts: sum of angles on a straight line = 180°; angles around a point = 360° — 角的基本事实：直线上的角之和 = 180°；一点周围的角之和 = 360°。
- Parallel line angles: alternate angles are equal, corresponding angles are equal, co-interior angles sum to 180° — 平行线中的角：内错角相等、同位角相等、同旁内角互补(和为180°)。
- Triangles = 180°; types by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse) — 三角形：内角和 = 180°；按边分（等边、等腰、不等边）和按角分（锐角、直角、钝角）。
If ∠A = 40° and ∠B = 60° in a triangle, then ∠C = 180° - (40° + 60°) = 80°
7. Coordinates and Linear Graphs | 坐标与线性图形
ar graphs by generating a table of values. Understanding the equation of a straight line in the form y = mx + c is introduced at a basic level, with a focus on interpreting the gradient and intercept.
在所有四个象限中运用坐标是一项基本技能。学生描点、确定点的坐标，并通过生成数值表绘制简单线性图形。理解形如 y = mx + c 的直线方程被初步引入，重点在于解读斜率和截距。
- Coordinates: (x, y), negative values for quadrants II, III, IV — 坐标：(x, y)，在第二、三、四象限会出现负值。
- Plotting points: 'along the corridor, up the stairs' — 描点：“先横后纵”。
- Table of values: for y = 2x + 1, choose x = -2, -1, 0, 1, 2 and calculate y — 数值表：对于 y = 2x + 1，选取 x = -2, -1, 0, 1, 2 并计算 y。
- Recognising slope: positive gradient goes uphill, negative gradient goes downhill — 识别斜率：正斜率向上倾斜，负斜率向下倾斜。
Line y = 2x + 1: when x = 0, y = 1; when x = 1, y = 3 → passes through (0,1) and (1,3)
8. Perimeter, Area and Volume | 周长、面积与体积
Measurement topics extend to calculating perimeters of rectilinear shapes, areas of triangles and parallelograms, and volumes of cuboids. Students learn to use formulae, convert between units (mm, cm, m; ml, l; g, kg), and solve problems involving compound shapes. The circle is introduced with its circumference and area.
测量部分拓展到计算直线形图形的周长、三角形和平行四边形的面积以及长方体的体积。学生学习使用公式，进行单位换算（毫米、厘米、米；毫升、升；克、千克），并解决涉及组合图形的问题。圆也连同其周长和面积共同引入。
- Perimeter: total distance around a shape — 周长：图形边界的总长度。
- Area formulae: rectangle = l × w; triangle = ½ × b × h; parallelogram = b × h — 面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 高。
- Circle: circumference C = 2πr or πd; area A = πr² — 圆：周长 C = 2πr 或 πd；面积 A = πr²。
- Volume: cuboid = l × w × h — 体积：长方体 = 长 × 宽 × 高。
- Unit conversions: 1 m = 100 cm, 1 cm² = 100 mm², 1 litre = 1000 cm³ — 单位换算：1米 = 100厘米，1平方厘米 = 100平方毫米，1升 = 1000立方厘米。
For a rectangle 5 cm by 3 cm, perimeter = 16 cm, area = 15 cm²
9. Data Collection and Representation | 数据收集与表示
Statistics in Year 7 covers designing data collection methods, organising raw data into frequency tables, and presenting data using bar charts, pie charts, and line graphs. Students calculate the mean, median, mode, and range, interpreting what these measures tell us about a data set. Spotting outliers and making comparisons are also practised.
七年级统计涉及设计数据收集方法、将原始数据整理为频数表，以及使用条形图、饼图和折线图展示数据。学生计算平均数、中位数、众数和极差，解读这些统计量关于数据集的含义。也会练习发现异常值并进行比较。
- Frequency tables: tally charts to record data — 频数表：用计数符号记录数据的表格。
- Bar chart: bars of equal width, gaps between bars; height represents frequency — 条形图：等宽长条，条间有间隔；高度代表频数。
- Pie chart: circles divided into sectors; angle = (frequency / total) × 360° — 饼图：圆被划分为扇形；扇形角度 = (频数 / 总数) × 360°。
- Averages sum of values ÷ number of values; median = middle value when ordered; mode = most frequent — 平均数：平均数 = 数值之和 ÷ 数值个数；中位数 = 排序后中间的值；众数 = 出现最多的值。
- Range = highest value - lowest value — 极差 = 最大值 - 最小值。
Data: 2, 3, 5, 5, 8 → mean = 4.6, median = 5, mode = 5, range = 6
10. Introduction to Probability | 概率入门
Probability is the measure of how likely an event is to happen, expressed as a fraction, decimal, or percentage between 0 (impossible) and 1 (certain). Students learn to calculate theoretical probability for equally likely outcomes, list sample spaces systematically, and use probability to make predictions. Experimental probability from simple experiments is also compared with theoretical expectations.
概率是对事件发生可能性的度量，用分数、小数或百分数表示，介于 0（不可能）和 1（必然）之间。学生学习计算等可能结果的理论概率、系统列举样本空间，并使用概率进行预测。来自简单实验的实验概率也会与理论期望进行比较。
- Probability scale: 0 → impossible, 0.5 → even chance, 1 → certain — 概率尺度：0→不可能，0.5→等可能，1→必然发生。
- Basic formula: P(Event) = Number of favourable outcomes / Total number of equally likely outcomes — 基本公式：P(事件) = 有利结果数 / 等可能结果总数。
- Sample space: list all possible outcomes, e.g. tossing a coin: {Head, Tail}; rolling a die: {1,2,3,4,5,6} — 样本空间：列举所有可能结果，例如抛硬币：{正面, 反面}；滚骰子：{1,2,3,4,5,6}。
- Probability of an event not happening = 1 - P(event happening) — 事件不发生的概率 = 1 - P(事件发生)。
Rolling an even number on a fair die: P(even) = 3/6 = 1/2
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