Edexcel International GCSE Mathematics A Student Book 1: Key Concepts Explained | Edexcel国际GCSE数学A学生用书1：知识点精讲

📚 Edexcel International GCSE Mathematics A Student Book 1: Key Concepts Explained | Edexcel国际GCSE数学A学生用书1：知识点精讲

This guide covers the essential topics from Edexcel International GCSE Mathematics A Student Book 1, providing clear explanations of key concepts, worked examples, and common exam techniques. It is designed to help you build a strong foundation in number, algebra, geometry, statistics, and probability, aligned with the Edexcel specification.

本指南涵盖Edexcel国际GCSE数学A学生用书1的核心主题，清晰讲解关键概念、典型例题与常见考试技巧，帮助你在数、代数、几何、统计和概率等方面打下扎实基础，完全贴合Edexcel考试大纲。

1. Number System and Operations | 数系与运算

Integers are whole numbers that can be positive, negative, or zero. You must be confident with the four operations (addition, subtraction, multiplication, division) and the order of operations (BIDMAS/BODMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction). For example, in 3 + 4 × 2, multiplication comes first, so the result is 11, not 14.

整数是正整数、负整数或零。你必须熟练掌握四则运算（加、减、乘、除）以及运算顺序（BIDMAS/BODMAS：括号、指数、乘除、加减）。例如，在3 + 4 × 2中，先乘后加，得11，而非14。

Factors are numbers that divide exactly into another number. The highest common factor (HCF) of two numbers is the largest factor they share. Multiples are the products of a number and an integer. The lowest common multiple (LCM) is the smallest multiple shared by two numbers. Prime numbers have exactly two factors: 1 and themselves.

因数是能整除另一个数的数。两数的最大公因数（HCF）是它们共有的最大因数。倍数是某个数与整数的乘积。最小公倍数（LCM）是两数共有的最小倍数。质数恰好有两个因数：1和它本身。

Example: Find HCF and LCM of 24 and 36. 24 = 2³ × 3, 36 = 2² × 3². HCF = 2² × 3 = 12. LCM = 2³ × 3² = 72.

例：求24和36的HCF和LCM。24 = 2³ × 3，36 = 2² × 3²。HCF = 2² × 3 = 12。LCM = 2³ × 3² = 72。

2. Fractions, Decimals and Percentages | 分数、小数和百分比

A fraction represents a part of a whole. To add or subtract fractions, you need a common denominator. Multiplying fractions: multiply numerators together and denominators together. Dividing by a fraction is the same as multiplying by its reciprocal.

分数表示整体的一部分。加减分数需要公分母。分数相乘：分子乘分子，分母乘分母。除以一个分数等于乘以它的倒数。

Decimals are another way of showing parts of a whole. Recurring decimals have a repeating pattern, indicated by a dot above the repeating digit(s). To convert a recurring decimal to a fraction, set up an equation and subtract to eliminate the repeating part.

小数是表示部分的另一种方式。循环小数有重复数字，用圆点标注。将循环小数化为分数时，列方程并相减消去循环部分。

Percentages are fractions out of 100. To increase an amount by 15%, multiply by 1.15; to decrease by 15%, multiply by 0.85. Reverse percentages find the original amount after a percentage change. Compound interest: A = P(1 + r/100)ⁿ.

百分数是以100为分母的分数。增加15%即乘以1.15；减少15%即乘以0.85。逆百分数求变化前的原值。复利公式：A = P(1 + r/100)ⁿ。

Convert 5/8 to a decimal and a percentage: 5 ÷ 8 = 0.625 = 62.5%.

将5/8化为小数和百分数：5 ÷ 8 = 0.625 = 62.5%。

3. Ratio and Proportion | 比与比例

A ratio compares two or more quantities. Simplify ratios by dividing all parts by their HCF. Divide a quantity in a given ratio by finding the total number of parts, then multiplying each ratio share by the value of one part.

比用于比较两个或多个量。通过除以所有项的HCF来化简比。按给定比分配数量时，先求总份数，再用一份的值乘各份数。

Proportion describes when two quantities change in relation to each other. Direct proportion means y = kx; as x doubles, y doubles. Inverse proportion means y = k/x; as x doubles, y halves. k is the constant of proportionality.

比例描述两个量之间的变化关系。正比例即y = kx，x翻倍则y翻倍。反比例即y = k/x，x翻倍则y减半。k是比例常数。

Divide £60 in the ratio 2:3. Total parts = 5, one part = £12. Shares: 2 × £12 = £24, 3 × £12 = £36.

按2:3分配60英镑。总份数=5，一份=12英镑。所得份额：2×12=24英镑，3×12=36英镑。

4. Algebraic Expressions and Formulae | 代数表达式与公式

Algebra uses letters to represent numbers. Terms can be combined if they have the same letter part (like terms). Expand brackets by multiplying each term inside by the term outside, paying attention to signs. For example, 3(x + 2) = 3x + 6.

代数用字母表示数。同类项可合并（字母部分相同）。去括号时，将括号内每一项乘以外面的项，注意符号。例如，3(x + 2) = 3x + 6。

Factorising is the reverse of expanding: rewrite an expression as a product. Take out the highest common factor. For quadratics like x² + 5x + 6, factorise into (x + 2)(x + 3) by finding two numbers that multiply to 6 and add to 5.

因式分解是展开的逆过程：将表达式写成乘积形式。提取最大公因数。对于二次式如x² + 5x + 6，分解为(x + 2)(x + 3)，找出乘积为6、和为5的两个数。

Substitution means replacing letters with given numbers. Always use brackets when substituting negative numbers. A formula is an equation showing the relationship between variables, e.g. A = πr².

代入法即将给定的数字替换字母。代入负数时务必使用括号。公式是表示变量关系的等式，如A = πr²。

Expand and simplify: 2(3a − 4) − (a + 5) = 6a − 8 − a − 5 = 5a − 13.

展开并化简：2(3a − 4) − (a + 5) = 6a − 8 − a − 5 = 5a − 13。

5. Linear Equations and Inequalities | 线性方程与不等式

To solve a linear equation, perform the same operation on both sides to isolate the unknown. Start by simplifying each side, then use inverse operations. Always check your solution by substituting it back.

解线性方程时，在等式两边施行相同运算以分离未知数。先化简每边，再用逆运算。最后代入原方程检验解。

Inequalities use symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal). Solving inequalities follows the same rules as equations, except that multiplying or dividing by a negative number reverses the inequality sign.

不等式符号：<（小于），>（大于），≤（小于等于），≥（大于等于）。解不等式与解方程规则相同，但乘或除以负数时，不等号方向要反转。

Represent inequalities on a number line with open circles for strict inequalities and closed circles for inclusive ones. Solve simultaneous linear equations by elimination (adding/subtracting equations) or substitution.

在数轴上表示不等式时，严格不等式用空心圆，包含等号用实心圆。解二元一次方程组可用消元法（加减方程）或代入法。

Solve: 2x − 5 = 3x + 1 → −5 − 1 = 3x − 2x → −6 = x. Solution: x = −6.

解方程：2x − 5 = 3x + 1 → −5 − 1 = 3x − 2x → −6 = x。解为x = −6。

6. Sequences and Patterns | 数列与规律

A sequence is a set of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference d. The nth term is given by a + (n − 1)d, where a is the first term.

数列是按规则排列的一组数。等差数列中，相邻两项之差为常数，称为公差d。第n项公式为a + (n − 1)d，其中a为首项。

Patterns from diagrams often lead to linear sequences. Find the nth term by linking the term number n to the number of shapes or dots. Use the formula to predict further terms. Check if a number is in the sequence by solving a + (n − 1)d = value; if n is a positive integer, it is a term.

图形规律常导出线性数列。将第n项与图形数量或点数联系起来，找出通项公式。用公式预测后面的项。判断某数是否在数列中：解a + (n − 1)d = 该值，若n为正整数则属数列。

Sequence: 5, 9, 13, 17, … d = 4, a = 5. nth term = 5 + (n − 1)4 = 4n + 1. The 20th term = 4(20) + 1 = 81.

数列：5, 9, 13, 17, … d = 4, a = 5。第n项 = 5 + (n − 1)4 = 4n + 1。第20项 = 4×20 + 1 = 81。

7. Graphs of Linear Functions | 线性函数图像

A linear function can be written as y = mx + c, where m is the gradient and c is the y-intercept. The gradient is the vertical change divided by the horizontal change between two points. A positive gradient slopes upward, a negative gradient slopes downward.

一次函数可写为y = mx + c，其中m为斜率，c为y轴截距。斜率等于两点间的垂直变化除以水平变化。正斜率向上倾斜，负斜率向下倾斜。

To plot a linear graph, choose at least three x-values, calculate the corresponding y-values, and plot the points. Draw a straight line through them. The x-intercept occurs when y = 0, and the y-intercept when x = 0.

绘制一次函数图像时，至少选取三个x值，计算对应的y值，描点并连成直线。x轴截距即y=0时，y轴截距即x=0时。

Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1 (m₁ × m₂ = −1). The equation of a line can be found from two points or from a point and a gradient.

平行线斜率相等。垂直线斜率之积为−1（m₁ × m₂ = −1）。可由两点或一点及斜率求直线方程。

Find equation of line through (2,5) with gradient 3. y − 5 = 3(x − 2) → y = 3x − 1.

求过点(2,5)且斜率为3的直线方程：y − 5 = 3(x − 2) → y = 3x − 1。

8. Geometry: Angles and Triangles | 几何：角与三角形

Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal. When a transversal crosses parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°.

平角（直线上邻角）和为180°。周角（绕点一周）和为360°。对顶角相等。当一条截线与平行线相交时，内错角相等，同位角相等，同旁内角互补（和为180°）。

Triangles are classified by sides (equilateral 3 equal sides, isosceles at least 2 equal sides, scalene no equal sides) and by angles (acute all < 90°, right-angled one 90°, obtuse one > 90°). The sum of interior angles in any triangle is 180°.

三角形按边分类：等边三角形（三边相等）、等腰三角形（至少两边相等）、不等边三角形。按角分类：锐角三角形（全<90°）、直角三角形（一个90°）、钝角三角形（一个>90°）。任何三角形内角和为180°。

The exterior angle of a triangle equals the sum of the two opposite interior angles. Pythagoras’ theorem: in a right-angled triangle, a² + b² = c², where c is the hypotenuse.

三角形外角等于与它不相邻的两个内角之和。勾股定理：直角三角形中，a² + b² = c²，c为斜边。

In triangle ABC, angle A = 50°, angle B = 60°. Find angle C: 180° − 50° − 60° = 70°.

三角形ABC中，角A = 50°，角B = 60°。求角C：180° − 50° − 60° = 70°。

9. Perimeter, Area and Volume | 周长、面积和体积

Perimeter is the distance around a shape. For a rectangle, P = 2(l + w); for a circle (circumference), C = 2πr or πd. Area measures the surface: rectangle A = l × w, triangle A = ½ × base × height, parallelogram A = b × h, trapezium A = ½(a + b)h, circle A = πr².

周长是图形边界的长度。矩形周长P = 2(l + w)；圆周长C = 2πr 或 πd。面积度量表面：矩形A = l × w，三角形A = ½ × 底 × 高，平行四边形A = b × h，梯形A = ½(a + b)h，圆A = πr²。

Volume of a prism = area of cross-section × length. For a cuboid, V = l × w × h. Volume of a cylinder = πr²h. Surface area is the total area of all faces. For compound shapes, split them into simpler shapes and sum the areas.

棱柱体积 = 横截面积 × 长度。长方体V = l × w × h。圆柱体V = πr²h。表面积是所有面的总面积。对于复合图形，拆分为简单图形后求和。

Find area of a trapezium with parallel sides 6 cm and 10 cm, height 4 cm: A = ½(6 + 10) × 4 = 32 cm².

求梯形面积：平行边6厘米和10厘米，高4厘米：A = ½(6 + 10) × 4 = 32 cm²。

10. Statistics: Averages and Range | 统计：平均数与范围

The mean is calculated by summing all data values and dividing by the number of items. The median is the middle value when data is ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values, measuring spread.

平均数（均值）由所有数据之和除以数据个数得到。中位数是排序后中间的值。众数是出现次数最多的值。范围（极差）是最大值与最小值之差，衡量数据的分散程度。

From a frequency table, estimate the mean by multiplying each midpoint by its frequency, summing, then dividing by the total frequency. The modal class is the group with the highest frequency. A cumulative frequency graph helps find the median and interquartile range.

由频数表估算平均数时，用每组的组中值乘以频数求和，再除以总频数。众数组是频数最高的组。累积频率图可用于求中位数和四分位距。

Data: 3, 5, 7, 7, 9. Mean = (3+5+7+7+9)/5 = 6.2. Median = 7. Mode = 7. Range = 9 − 3 = 6.

数据：3, 5, 7, 7, 9。平均数 = (3+5+7+7+9)/5 = 6.2。中位数 = 7。众数 = 7。范围 = 9 − 3 = 6。

11. Probability | 概率

Probability measures the chance of an event happening, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event) = (number of favourable outcomes) / (total number of outcomes). The sum of probabilities of all possible outcomes is 1.

概率度量事件发生的可能性，范围从0（不可能）到1（必然）。等可能结果下，P(事件) = （有利结果数）/（总结果数）。所有可能结果的概率之和为1。

Expected frequency = probability × number of trials. In probability trees, multiply along branches for combined events, and add probabilities for mutually exclusive events. Independent events: P(A and B) = P(A) × P(B).

期望频数 = 概率 × 试验次数。在概率树中，沿分支相乘求组合事件概率；互斥事件则相加。独立事件：P(A与B) = P(A) × P(B)。

Conditional probability considers events where one outcome affects another. Use two-way tables or Venn diagrams to organise information. P(A|B) = P(A and B) / P(B).

条件概率考虑一个结果影响另一个事件的情况。使用双向表或维恩图整理信息。P(A|B) = P(A与B) / P(B)。

A bag has 4 red and 6 blue balls. P(red) = 4/10 = 0.4. If a ball is taken and not replaced, P(two reds) = (4/10) × (3/9) = 12/90 = 2/15.

袋中有4红球6蓝球。P(红) = 4/10 = 0.4。若取后不放回，两次都取红球的概率 = (4/10) × (3/9) = 12/90 = 2/15。

12. Sets and Venn Diagrams | 集合与维恩图

A set is a collection of objects. The universal set ξ contains all elements under consideration. Basic notation: A ∪ B (union, elements in A or B), A ∩ B (intersection, elements in both A and B), A’ (complement, elements not in A).

集合是对象的汇集。全集ξ包含所考虑的全部元素。基本符号：A ∪ B（并集，在A或B中的元素），A ∩ B（交集，同时在A和B中的元素），A’（补集，不在A中的元素）。

Venn diagrams visually represent sets and their relationships. Place elements in overlapping regions for intersections. The number of elements in a set is denoted by n(A). Use Venn diagrams to solve problems involving categories, counting totals, and conditional probability.

维恩图直观表示集合及其关系。将元素填入重叠区域表示交集。集合元素个数记作n(A)。用维恩图解决涉及分类、计数总数和条件概率的问题。

For two sets, the addition rule is: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). If sets are disjoint, n(A ∩ B) = 0. Shade regions to represent compound set expressions.

两集合的加法规则：n(A ∪ B) = n(A) + n(B) − n(A ∩ B)。若集合不相交，则n(A ∩ B) = 0。可通过涂阴影表示复合集合表达式。

ξ = {1,2,3,4,5,6}, A = {1,2,4}, B = {2,3,4}. A ∪ B = {1,2,3,4}, A ∩ B = {2,4}, A’ = {3,5,6}.

ξ = {1,2,3,4,5,6}, A = {1,2,4}, B = {2,3,4}。A ∪ B = {1,2,3,4}，A ∩ B = {2,4}，A’ = {3,5,6}。

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Edexcel International GCSE Mathematics A Student Book 1: Key Concepts Explained | Edexcel国际GCSE数学A学生用书1：知识点精讲