📚 Edexcel International GCSE (9-1) Mathematics A Student Book 1: Core Topics and Exam Skills | 爱德思国际 GCSE (9-1) 数学 A 学生用书 1：核心主题与考试技巧
This revision guide covers the essential content from Edexcel International GCSE (9-1) Mathematics A Student Book 1, focusing on the core topics, common question types and the key skills you need to build confidence. The book is organised into units on number, algebra, graphs, shape and space, and data handling, and this article summarises the most exam-relevant ideas from each area.
本复习指南涵盖爱德思国际 GCSE (9-1) 数学 A 学生用书 1 的核心内容，重点讲解基础主题、常见题型以及建立解题信心所需的关键技能。该书按数、代数、图形、空间与形状以及数据处理等单元编排，本文汇总了每个领域中最具考试价值的概念。
1. Number Skills and Place Value | 数字技能与位值
Place value tells us the value of each digit in a number, and it underpins ordering decimals, multiplying by powers of 10 and rounding. For example, in 0.0468 the digit 4 is in the hundredths place and the digit 8 is in the ten-thousandths place.
位值告诉我们一个数字中每个数位的数值，它是小数排序、乘以 10 的幂以及四舍五入的基础。例如，在 0.0468 中，数字 4 位于百分位，数字 8 位于万分位。
Rounding to decimal places counts digits after the decimal point, while rounding to significant figures counts from the first non-zero digit. When estimating, round each number to one significant figure first, then calculate the approximate answer.
四舍五入到小数位是从小数点后开始计数，而四舍五入到有效数字是从第一个非零数字开始计数。估算时，应先将每个数四舍五入到一位有效数字，再计算近似答案。
- 0.0468 to 2 significant figures = 0.047 | 0.0468 保留 2 位有效数字 = 0.047
- Estimate 19.8 × 3.1 ≈ 20 × 3 = 60 | 估算 19.8 × 3.1 ≈ 20 × 3 = 60
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Fractions, decimals and percentages are three ways of expressing the same part of a whole. To convert a fraction to a decimal, divide the numerator by the denominator; to convert a decimal to a percentage, multiply by 100.
分数、小数和百分比是表示同一个整体的部分的三种方式。要将分数转换为小数，用分子除以分母；要将小数转换为百分比，则乘以 100。
For percentage increase or decrease, use a multiplier. An increase of 15% means multiplying by 1.15, and a decrease of 15% means multiplying by 0.85. This is much quicker than finding 15% and adding or subtracting it separately.
对于百分比的增加或减少，使用乘法因子。增加 15% 意味着乘以 1.15，减少 15% 意味着乘以 0.85。这比先求出 15% 再加或减要快得多。
a/b = a ÷ b, x × 1.15 = 115% of x, x × 0.85 = 85% of x
3. Ratio and Proportion | 比与比例
A ratio compares the sizes of two or more quantities. Ratios can be simplified by dividing every part by a common factor, just like simplifying a fraction. For example, 12:8 simplifies to 3:2 by dividing both parts by 4.
比用于比较两个或多个量的大小。比可以通过将每一部分同时除以一个公因数来化简，就像化简分数一样。例如，12:8 每一部分都除以 4，化简为 3:2。
To divide a quantity in a given ratio, first add the parts of the ratio to find the total number of shares. Then divide the quantity by this total to find the value of one share, and multiply by each part.
按给定比例分配一个量时，先将比的各部分相加得到总份数。然后用总量除以总份数得到一份的值，再分别乘以各部分。
- Divide £60 in the ratio 2:3 → total parts = 5, one part = £12, so amounts are £24 and £36. | 将 £60 按 2:3 分配 → 总份数 = 5，一份 = £12，因此分别为 £24 和 £36。
4. Algebraic Expressions and Simplification | 代数表达式与化简
An algebraic expression is a combination of numbers, letters and operations. Terms with exactly the same variable part, such as 3x and −5x, are called like terms and can be added or subtracted by combining their coefficients.
代数表达式是数字、字母和运算符号的组合。变量部分完全相同的项（如 3x 和 −5x）称为同类项，它们可以通过合并系数来相加或相减。
Expanding a single bracket uses the distributive law: a(b + c) = ab + ac. Factorising is the reverse process: look for the highest common factor of all terms and write it outside the bracket.
展开单项括号使用分配律：a(b + c) = ab + ac。因式分解是相反的过程：找出所有项的最高公因式并将其写在括号外。
3(2x + 5) = 6x + 15, 4x + 6 = 2(2x + 3)
5. Solving Linear Equations | 解一元一次方程
A linear equation contains an unknown, usually x, and no powers higher than 1. The goal is to isolate the unknown on one side by using inverse operations in the correct order, keeping the equation balanced by doing the same thing to both sides.
一元一次方程含有一个未知数（通常是 x），且未知数的最高次数不超过 1。解题目标是利用逆运算按正确顺序将未知数单独留在等式一边，同时对等式两边进行相同操作以保持平衡。
If an equation contains brackets, expand them first. If it contains fractions, multiply every term by the lowest common denominator to clear the fractions before solving.
如果方程含有括号，应先展开括号。如果方程含有分数，则先将每一项乘以最小公分母以消去分母，再求解。
- Solve 2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4. | 解方程 2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4。
- Solve x/3 + 2 = 5 → x/3 = 3 → x = 9. | 解方程 x/3 + 2 = 5 → x/3 = 3 → x = 9。
6. Inequalities | 不等式
Inequalities compare expressions using the symbols < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). They can be shown on a number line with an open circle for strict inequalities and a closed circle for inclusive ones.
不等式使用符号 <（小于）、>（大于）、≤（小于或等于）和 ≥（大于或等于）来比较表达式。它们可以在数轴上表示，严格不等式用空心圆，包含等号的不等式用实心圆。
Solving an inequality is very similar to solving an equation, but there is one critical rule: if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign.
解不等式与解方程非常相似，但有一条关键规则：如果将两边同时乘以或除以一个负数，必须改变不等号的方向。
−2x < 8 → x > −4
7. Sequences and Patterns | 数列与规律
A sequence is a list of numbers that follows a rule. The term-to-term rule tells you how to move from one term to the next, while the position-to-term rule (nth term) lets you find any term directly from its position n.
数列是按一定规则排列的一组数。项与项之间的规则告诉你如何从一项得到下一项，而位置到项规则（第 n 项）让你直接根据位置 n 求出任意一项。
For an arithmetic sequence, the difference between consecutive terms is constant. The nth term of an arithmetic sequence with first term a and common difference d is given by a + (n − 1)d, which can also be written as dn + (a − d).
对于等差数列，相邻两项之间的差是常数。首项为 a、公差为 d 的等差数列的第 n 项公式为 a + (n − 1)d，也可以写成 dn + (a − d)。
- Sequence 3, 7, 11, 15… has d = 4 and a = 3, so nth term = 4n − 1. | 数列 3, 7, 11, 15… 的公差 d = 4，首项 a = 3，所以第 n 项 = 4n − 1。
8. Coordinates and Straight-Line Graphs | 坐标与直线图
Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. Plotting points accurately on the Cartesian plane is the first step in drawing and interpreting straight-line graphs.
坐标写成 (x, y)，其中 x 是水平位置，y 是垂直位置。在笛卡尔平面上准确描点是绘制和解读直线图的第一步。
The equation of a straight line is usually written as y = mx + c, where m is the gradient and c is the y-intercept. The gradient is calculated as change in y divided by change in x, often written as (y₂ − y₁)/(x₂ − x₁).
直线方程通常写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率等于 y 的变化量除以 x 的变化量，常写作 (y₂ − y₁)/(x₂ − x₁)。
gradient = Δy/Δx = (y₂ − y₁)/(x₂ − x₁)
9. Angles and Polygons | 角与多边形
Basic angle facts are essential for solving geometry problems: angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal. These facts help you find missing angles in diagrams without measuring.
基本角度事实是解决几何问题的基础：直线上相邻角之和为 180°，一点周围的角度之和为 360°，对顶角相等。这些事实帮助你在图中不借助测量求出未知角。
The sum of the interior angles of a triangle is always 180°, and for a quadrilateral it is 360°. For a polygon with n sides, the sum of the interior angles is (n − 2) × 180°. The sum of the exterior angles of any convex polygon is always 360°.
三角形的内角和恒为 180°，四边形的内角和为 360°。对于一个 n 边形，其内角和为 (n − 2) × 180°。任意凸多边形的外角和恒为 360°。
Interior angle of a regular n-gon = (n − 2) × 180° ÷ n
10. Averages, Range and Probability | 平均数、极差与概率
The three common averages are the mode (most frequent value), the median (middle value when data are ordered) and the mean (sum of values divided by the number of values). The range is the difference between the largest and smallest values and measures spread.
三种常见的平均数是众数（出现频率最高的值）、中位数（数据按顺序排列后的中间值）和平均数（数值总和除以数值个数）。极差是最大值与最小值之差，用于衡量数据的离散程度。
Probability is measured on a scale from 0 to 1, where 0 means impossible and 1 means certain. The theoretical probability of an event is the number of favourable outcomes divided by the total number of possible outcomes, assuming all outcomes are equally likely.
概率用 0 到 1 的尺度来衡量，其中 0 表示不可能发生，1 表示必然发生。事件的理论概率等于有利结果的数量除以所有可能结果的总数，前提是所有结果等可能。
For mutually exclusive events, the probabilities of all possible outcomes add up to 1. So if the probability of an event is p, the probability it does not happen is 1 − p.
对于互斥事件，所有可能结果的概率之和为 1。因此，如果某事件发生的概率为 p，那么它不发生的概率就是 1 − p。
11. Exam Techniques for Student Book 1 Topics | 学生用书 1 主题的考试技巧
Always show your working in structured questions because method marks can be awarded even if the final answer is incorrect. Write down each step clearly and keep numbers in their exact form until the final line, unless the question asks for rounding.
在结构化题目中一定要展示解题步骤，因为即使最终答案错误，也可以获得方法分。清晰地写下每一步，除非题目要求四舍五入，否则在最后一步之前保持数字的精确形式。
Include units in final answers where the question involves measurement, money or time. Check your answer by substituting it back into the original equation or by asking whether the result is reasonable in the context of the problem.
当题目涉及测量、货币或时间时，最终答案要写明单位。通过将答案代回原方程或结合题目背景判断结果是否合理来检查答案。
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