📚 Year 7 CIE Further Mathematics: Core Concepts Review | Year 7 CIE 进阶数学：核心知识点梳理
Year 7 Further Mathematics builds on primary school knowledge and introduces more abstract concepts. This article summarises the key topics covered in the CIE curriculum, providing a clear revision guide for students.
Year 7 进阶数学在小学知识的基础上进一步引入更抽象的概念。本文梳理 CIE 课程涵盖的核心知识点，为学生提供一份清晰的复习指南。
1. Integers and Negative Numbers | 整数与负数
Adding a negative number is equivalent to subtracting its positive counterpart. For example, 5 + (−3) = 5 − 3 = 2. Subtracting a negative number changes to addition: 5 − (−3) = 5 + 3 = 8.
整数包括正整数、负整数和零。在数轴上，数字向右增大，向左减小。加上一个负数等于减去它的相反数，例如 5 + (−3) = 5 − 3 = 2。减去一个负数变为加法：5 − (−3) = 5 + 3 = 8。
are different, the result is negative. For instance, (−4) × (−3) = 12, but (−4) × 3 = −12.
两个整数相乘或相除时，同号得正，异号得负。例如 (−4) × (−3) = 12，但 (−4) × 3 = −12。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
mplify a fraction, divide the numerator and denominator by their highest common factor. Adding or subtracting fractions requires a common denominator.
等值分数表示相同的值，例如 1/2 = 2/4 = 3/6。化简分数时，分子分母同时除以它们的最大公因数。分数加减需要先通分找到公分母。
al. Decimals can be converted to percentages by multiplying by 100. For example, 0.75 = 75%.
分数相乘时，分子乘分子，分母乘分母。除以一个分数等于乘以它的倒数。小数转换为百分数只需乘以 100，例如 0.75 = 75%。
3. Algebraic Expressions | 代数表达式
o the same power; only like terms can be combined. For example, 4a + 3a = 7a, but 4a + 3b cannot be simplified further.
代数中，字母表示未知数或变量。如 3x + 2y − 5 这样的表达式包含多个项。同类项具有相同的变量和相同的指数，只有同类项才能合并。例如 4a + 3a = 7a，但 4a + 3b 不能进一步化简。
brackets: 2(x + 3) = 2x + 6. It also works in reverse for factorising: 2x + 6 = 2(x + 3).
分配律为 a(b + c) = ab + ac，用于展开括号：2(x + 3) = 2x + 6。反过来也可用于因式分解：2x + 6 = 2(x + 3)。
4. Simple Equations | 简单方程
des to isolate the variable. For example, x + 5 = 12 → subtract 5 from both sides → x = 7.
方程表示两个表达式相等。求解未知数时，等式两边同时进行相同的运算以分离变量。例如 x + 5 = 12，两边同时减去 5 得 x = 7。
uations can also involve two steps: 2x + 3 = 11 → subtract 3: 2x = 8 → divide by 2: x = 4.
当变量有系数时，两边同除。3x = 15 → x = 5。方程也可能需要两步：2x + 3 = 11，先减 3 得 2x = 8，再除以 2 得 x = 4。
5. Angles and Lines | 角与线
Angles are measured in degrees (°). Acute angles are less than 90°, right angles exactly 90°, obtuse between 90° and 180°, and reflex angles greater than 180°. Angles on a straight line add up to 180°, and angles around a point sum to 360°.
角以度 (°) 为单位。锐角小于 90°，直角等于 90°，钝角介于 90° 和 180° 之间，优角大于 180°。直线上的角之和为 180°，绕一点一周的角之和为 360°。
Vertically opposite angles are equal. When a transversal cuts parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles on the same side sum to 180°.
对顶角相等。当一条截线与两条平行线相交时，同位角相等，内错角相等，同旁内角互补（和为 180°）。
6. Triangles and Quadrilaterals | 三角形与四边形
Triangles can be classified by sides: equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal); or by angles: acute, right, obtuse. The sum of interior angles in any triangle is 180°.
三角形可按边分类：等边三角形（三边相等），等腰三角形（两边相等），不等边三角形；也可按角分类：锐角三角形、直角三角形、钝角三角形。任意三角形的内角和为 180°。
Quadrilaterals have four sides. Squares, rectangles, parallelograms, rhombuses, trapeziums and kites each have specific angle and side properties. The interior angle sum of any quadrilateral is 360°.
四边形有四条边。正方形、长方形、平行四边形、菱形、梯形和筝形各有特定的角与边性质。任意四边形的内角和为 360°。
7. Perimeter, Area and Volume | 周长、面积与体积
½ × base × height, parallelogram area = base × height. All formulas use consistent units.
周长是图形一周的长度。对于长方形，周长 P = 2(长 + 宽)。面积度量表面大小；长方形面积 = 长 × 宽，三角形面积 = ½ × 底 × 高，平行四边形面积 = 底 × 高。所有公式使用一致的单位。
are cubic, e.g., cm³. Surface area can also be calculated by finding the sum of the areas of all faces.
长方体体积 = 长 × 宽 × 高。单位为立方单位，如 cm³。表面积可通过计算所有面的面积总和得到。
8. Data Handling and Averages | 数据处理与平均数
Data can be summarised using averages: mean (sum of values ÷ number of values), median (middle value when ordered), mode (most frequent value). The range is the difference between the largest and smallest values, indicating spread.
数据可以用平均数来概括：平均数（总和 ÷ 个数）、中位数（排序后中间的值）、众数（出现频率最高的值）。极差是最大值与最小值的差，反映数据的离散程度。
hart shows proportions of a whole; the angle for each sector = (frequency ÷ total) × 360°.
数据常用条形图、象形图、折线图和饼图表示。饼图显示整体中各部分的比例；每个扇形的角度 = (频数 ÷ 总数) × 360°。
9. Probability Basics | 概率基础
le outcomes ÷ total number of possible outcomes, assuming all outcomes are equally likely.
概率衡量事件发生的可能性，从 0（不可能）到 1（必然）。理论概率 = 有利结果数 ÷ 所有可能结果总数，前提是每个结果等可能。
Experimental probability is based on actual trials: number of times event occurs ÷ total number of trials. As the number of trials increases, experimental probability tends to approach theoretical probability.
实验概率基于实际试验：事件发生次数 ÷ 试验总次数。随着试验次数增加，实验概率会趋近理论概率。
10. Number Patterns and Sequences | 数字规律与数列
A sequence is an ordered list of numbers. Each number is a term. Recognising the pattern helps to find the next term. In an arithmetic sequence, the difference between consecutive terms is constant (common difference).
数列是一组有序排列的数字。每个数字称为一项。识别规律有助于找出下一项。在等差数列中，相邻两项的差为常数（公差）。
nth term = a + (n − 1)d
− 1)d, where a is the first term and d is the common difference. For example, 3, 7, 11, 15 … has nth term = 3 + (n − 1)×4 = 4n − 1.
等差数列的第 n 项可表示为：a + (n − 1)d，其中 a 是首项，d 是公差。例如 3, 7, 11, 15 … 的第 n 项为 3 + (n − 1)×4 = 4n − 1。
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