📚 Year 7 CIE Maths: Core Knowledge Overview | Year 7 CIE 数学：核心知识点梳理
Building a strong mathematical foundation in Year 7 is essential for success in the CIE curriculum. This article brings together the key topics you will encounter, explaining each concept clearly so you can feel confident with numbers, algebra, geometry, and data handling. Let’s revisit the core ideas that underpin all your future learning.
在 Year 7 阶段打下扎实的数学基础对 CIE 课程的学习至关重要。本文梳理了你将接触到的核心知识点，清晰地解释每一个概念，帮助你自信地应对数、代数、几何和数据处理。让我们一起回顾那些支撑未来学习的关键思想。
1. Understanding Place Value and Integers | 理解位值和整数
Integers are the set of whole numbers, including zero, and their negatives. The place value system lets us represent any integer using digits 0–9, where the position of a digit determines its value: units, tens, hundreds, thousands, and so on. For large numbers, we group digits using spaces or commas. The value of each column is ten times the column to its right, so 10³ = 1000, 10⁴ = 10 000, and 10⁶ = 1 000 000.
整数是包括零及其负数的所有正整数和负整数。位值系统让我们能用 0–9 的数字表示任何整数，数字的位置决定其值：个位、十位、百位、千位等。对于大数，我们用空格或逗号分组。每一列的值是其右边一列的十倍，因此 10³ = 1000，10⁴ = 10000，10⁶ = 1000000。
2. Operating with Whole Numbers | 整数运算
The four operations – addition, subtraction, multiplication and division – are the building blocks of arithmetic. When you add or subtract, always align digits by place value. For multiplication, you can use grid methods or column multiplication. Division can be done using short or long division. A crucial rule is the order of operations, remembered by BODMAS or BIDMAS: Brackets, Orders (powers and roots), Division and Multiplication (working from left to right), Addition and Subtraction (left to right). Ignoring this order leads to wrong answers.
四种基本运算——加、减、乘、除——是算术的基石。做加减法时，一定要按位值对齐数位。乘法可以使用网格法或竖式乘法。除法可用短除法或长除法。一条至关重要的规则是运算顺序，用 BODMAS 或 BIDMAS 记忆：括号、阶（幂和方根）、除法和乘法（从左往右计算）、加法和减法（从左往右）。忽略这个顺序会得到错误答案。
3. Introducing Negative Numbers | 负数的引入
Negative numbers appear when we go below zero, such as temperature readings or bank debts. On a number line, negatives extend to the left of zero. Adding a negative number is the same as subtracting its positive: 5 + (−3) = 5 − 3 = 2. Subtracting a negative is like adding: 5 − (−3) = 5 + 3 = 8. Multiplying two negative numbers gives a positive product: (−2) × (−4) = 8, while a positive times a negative stays negative.
负数出现在低于零的情形，比如温度读数或银行欠款。在数轴上，负数向零的左侧延伸。加上一个负数等于减去它的相反数：5 + (−3) = 5 − 3 = 2。减去一个负数等于加上正数：5 − (−3) = 5 + 3 = 8。两个负数相乘得正数：(−2) × (−4) = 8，而正数乘负数仍得负数。
4. Fractions: Concepts and Equivalence | 分数：概念与等值
A fraction represents a part of a whole, written as numerator over denominator. Equivalent fractions name the same amount: ½ = ²⁄₄ = ⁴⁄₈. You can create equivalent fractions by multiplying or dividing both numerator and denominator by the same non‑zero number. Simplifying a fraction means writing it in its lowest terms by dividing by the highest common factor (HCF). Mixed numbers combine a whole number and a proper fraction, while improper fractions have a numerator larger than the denominator.
分数表示整体的一部分，写作分子/分母。等值分数表示相同的大小：½ = ²⁄₄ = ⁴⁄₈。你可以通过将分子和分母同时乘以或除以同一个非零数来得到等值分数。化简分数意味着除以分子和分母的最大公因数 (HCF)，将其写成最简形式。带分数由一个整数和一个真分数组成，而假分数的分子大于分母。
5. Operations with Fractions | 分数运算
To add or subtract fractions, make the denominators the same by finding equivalent fractions, then add or subtract the numerators. For multiplication, simply multiply the numerators together and the denominators together. To divide by a fraction, flip the second fraction (find its reciprocal) and multiply. Always give your final answer in its simplest form and, where appropriate, convert improper fractions to mixed numbers.
要加减分数，先通分使分母相同，得到等值分数，再将分子相加或相减。乘法直接分子乘分子、分母乘分母即可。除以一个分数时，把第二个分数倒过来（求倒数），再乘以它。最终答案一定要化到最简，并在合适时将假分数转化为带分数。
Example: ²⁄₃ ÷ ⁴⁄₅ = ²⁄₃ × ⁵⁄₄ = 10⁄12 = ⁵⁄₆
6. Decimals and Percentages | 小数与百分数
Decimals are another way to write fractions with denominators 10, 100, 1000, etc. Each decimal place has a value: tenths (0.1), hundredths (0.01), thousandths (0.001). Percent means ‘out of 100’, so to convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply the decimal by 100%. For example, ¾ = 0.75 = 75%. You also need to convert percentages back to decimals by dividing by 100, which means moving the decimal point two places left.
小数是分母为 10、100、1000 等的分数的另一种写法。每个小数位都有位值：十分位 (0.1)、百分位 (0.01)、千分位 (0.001)。百分数表示‘每一百份中的几份’，所以要将分数转化为百分数，可以将其化为分母 100 的等值分数，或者将小数乘以 100%。例如，¾ = 0.75 = 75%。你还需要会将百分数化回小数，即除以 100，也就是把小数点向左移两位。
7. Introduction to Algebra | 代数入门
Algebra uses letters to stand for unknown numbers or quantities that can vary. An expression like 3a + 5 is a combination of terms. We simplify expressions by collecting like terms: 2x + 3x = 5x, and 4y − y = 3y. Multiplication involving letters is written without the times sign: m × n = mn, and 3 × k = 3k. A formula shows the relationship between variables, such as P = 4s for the perimeter of a square. Substituting numbers into an expression or formula means replacing the letters with given values.
代数用字母表示未知数或可变化的量。像 3a + 5 这样的式子是由若干个项组合而成的。我们通过合并同类项来化简式子：2x + 3x = 5x，4y − y = 3y。涉及字母的乘法省略乘号：m × n = mn，3 × k = 3k。公式表示变量之间的关系，例如正方形的周长公式 P = 4s。把数字代入式子或公式，就是用给定数值替换字母。
8. Sequences and Patterns | 数列与规律
A sequence is a list of numbers following a rule. In Year 7 you mainly work with linear sequences where the difference between consecutive terms is constant. You learn to describe the term‑to‑term rule (e.g. ‘add 5 each time’) and to find missing terms. You also start to use the position‑to‑term rule, which links the term number (n) to the term itself. For the sequence 3, 7, 11, 15, …, the nth term rule is 4n − 1. Understanding how patterns grow helps you make predictions.
数列是一组按规律排列的数。在 Year 7 你主要学习线性数列，即相邻两项的差保持不变的数列。你要学会描述项与项之间的变化规律（如‘每次加 5’），并求出缺失的项。你也会开始使用位置与项的对应规律，将项数 (n) 和该项的值联系起来。对于数列 3, 7, 11, 15, …，第 n 项公式为 4n − 1。理解规律如何增长有助于做出预测。
9. Measuring Geometry: Perimeter, Area and Volume | 几何测量：周长、面积与体积
Perimeter is the distance around a shape. For a rectangle, P = 2(l + w). Area measures the surface inside a 2D shape in square units. The area of a rectangle is length × width, while the area of a triangle is half the base times the height. The volume of a cuboid (rectangular prism) is length × width × height, measured in cubic units. You also learn to find areas of compound shapes by breaking them into rectangles and triangles.
周长是图形一周的长度。矩形的周长为 P = 2(l + w)。面积衡量二维图形内部表面的大小，单位为平方单位。矩形的面积是长乘宽，三角形的面积是底乘高的一半。长方体（直角棱柱）的体积为长 × 宽 × 高，用立方单位度量。你还要学习将组合图形拆分成矩形和三角形来求面积。
Triangle area: A = ½ × b × h
Volume of cuboid: V = l × w × h
10. Angles and Lines | 角与线
An angle measures the turn between two rays at a vertex. Angles are measured in degrees (°). Key facts for Year 7 include: angles on a straight line add to 180°; angles around a point add to 360°; vertically opposite angles are equal. You also investigate angles in a triangle, which always sum to 180°. Identifying acute (0°–90°), right (90°), obtuse (90°–180°) and reflex (180°–360°) angles supports problem solving. Complementary and supplementary angles are also useful: two angles that add to 90° are complementary, to 180° are supplementary.
角衡量从一点出发的两条射线之间的旋转量，以度 (°) 为单位。Year 7 需要掌握的关键事实：直线上的角之和为 180°；一点周围的角之和为 360°；对顶角相等。你还会研究三角形内角和，总是等于 180°。识别锐角 (0°–90°)、直角 (90°)、钝角 (90°–180°) 和优角 (180°–360°) 有助于解决问题。互余和互补的概念也很实用：相加为 90° 的两角互余，相加为 180° 的两角互补。
11. Collecting and Representing Data | 数据的收集与表示
Handling data starts with collecting information and organising it into frequency tables. You then display data using bar charts (for categorical data), pictograms, line graphs (to show changes over time), and pie charts (to show proportions of a whole). When creating a bar chart, label both axes clearly and use equal gaps between bars. A pie chart uses angles to represent frequencies, where the total 360° corresponds to the whole data set. Reading and interpreting these diagrams accurately is essential.
数据处理从收集信息并将其整理成频数表开始。接着你可以用条形图（用于分类数据）、象形图、折线图（显示随时间的变化）和饼图（显示各部分占总体的比例）来展示数据。绘制条形图时，要清楚标注坐标轴，并让条形之间间隔相等。饼图用角度代表频数，整个圆 360° 对应全部数据。准确读取和解读这些图表非常重要。
12. Averages and Range | 平均数与范围
The three main averages are the mode (most frequent value), median (middle value when ordered), and mean (sum of all values divided by the number of values). The range is the difference between the largest and smallest values, describing how spread out the data are. To find the median of an even number of data points, calculate the mean of the two middle numbers. The mean can be affected by outliers, making the median sometimes a better measure of centre. Practising these calculations with small datasets builds confidence.
三种主要的平均数是众数（出现次数最多的值）、中位数（排序后的中间值）和平均数（所有值之和除以值的个数）。范围是最大值与最小值之差，描述数据的离散程度。当数据个数为偶数时，中位数需取中间两个数的平均数。平均数易受异常值影响，因此中位数有时更能代表数据的中心。用小型数据集反复练习能增强信心。
Mean = (Sum of values) ÷ (Number of values)
Range = Largest value − Smallest value
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