IGCSE Edexcel Mathematics: Complete Revision Guide | IGCSE Edexcel 数学：完整复习指南

📚 IGCSE Edexcel Mathematics: Complete Revision Guide | IGCSE Edexcel 数学：完整复习指南

This comprehensive revision guide covers the essential topics for the IGCSE Edexcel Mathematics syllabus. Whether you are preparing for the Higher or Foundation tier, mastering these core areas will significantly boost your confidence and exam performance. We have organised the content into ten focused sections, each aligned with the key assessment objectives of the Edexcel specification.

这份完整复习指南涵盖了 IGCSE Edexcel 数学大纲中的核心主题。无论你准备的是 Higher 还是 Foundation 级别，掌握这些核心领域都将显著提升你的信心和考试成绩。我们将内容分为十个重点章节，每个章节均与 Edexcel 考纲的关键评估目标保持一致。

1. Number Systems and Surds | 数系与根式

The IGCSE Edexcel syllabus requires a solid understanding of the number system, including natural numbers, integers, rational numbers, and irrational numbers. You must be able to classify numbers correctly and perform calculations involving surds (irrational square roots) with confidence.

IGCSE Edexcel 考纲要求学生对数系有扎实的理解，包括自然数、整数、有理数和无理数。你必须能够正确分类数字，并自信地进行涉及根式（无理平方根）的运算。

Key skills include simplifying surds such as √48 = √(16 × 3) = 4√3, rationalising denominators, and expanding expressions like (√2 + 1)². For example:

关键技能包括化简根式，例如 √48 = √(16 × 3) = 4√3，有理化分母，以及展开类似 (√2 + 1)² 的表达式。例如：

(√2 + 1)² = 2 + 2√2 + 1 = 3 + 2√2

When rationalising a denominator, multiply both numerator and denominator by the conjugate. For 1/(√5 − 2), multiply by (√5 + 2) to get (√5 + 2)/(5 − 4) = √5 + 2. Always simplify your final answer completely.

对分母进行有理化时，将分子和分母同时乘以共轭式。对于 1/(√5 − 2)，乘以 (√5 + 2) 可得 (√5 + 2)/(5 − 4) = √5 + 2。始终将最终答案化简到最简形式。

Understand the difference between rational and irrational numbers / 理解有理数与无理数的区别
Simplify surds by identifying square factors / 通过识别平方因数来化简根式
Rationalise denominators using the conjugate / 使用共轭式对分母进行有理化
Expand brackets involving surds accurately / 准确展开涉及根式的括号

2. Algebra: Expanding and Factorising | 代数：展开与因式分解

Algebra forms the backbone of IGCSE Mathematics. You must be fluent in expanding single and double brackets, including perfect squares and the difference of two squares. Factorising is the reverse process and requires careful attention to common factors and grouping techniques.

代数是 IGCSE 数学的基石。你必须熟练掌握展开单括号和双括号，包括完全平方和平方差公式。因式分解是相反的过程，需要仔细关注公因数和分组技巧。

Memorise these essential identities:

请牢记这些基本恒等式：

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

a² − b² = (a + b)(a − b)

For quadratic expressions of the form ax² + bx + c, when a = 1, find two numbers that multiply to give c and add to give b. For example, x² + 5x + 6 = (x + 2)(x + 3). When a ≠ 1, use the grouping method or the ac-method to split the middle term.

对于形如 ax² + bx + c 的二次表达式，当 a = 1 时，找到两个数，使其乘积为 c 且和为 b。例如，x² + 5x + 6 = (x + 2)(x + 3)。当 a ≠ 1 时，使用分组法或 ac 法来拆分中间项。

Expand single brackets by distributing each term / 通过逐项分配展开单括号
Use FOIL or grid methods for double brackets / 使用 FOIL 或网格法展开双括号
Recognise and apply the difference of two squares / 识别并应用平方差公式
Factorise quadratics completely before solving / 在求解前完成二次式的因式分解

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

Solving linear equations is a fundamental skill tested extensively in IGCSE Edexcel papers. You should be able to solve equations with unknowns on both sides, equations involving fractions, and equations with brackets. Inequalities follow similar rules, but you must remember that multiplying or dividing by a negative number reverses the inequality sign.

求解线性方程是 IGCSE Edexcel 试卷中广泛考查的基本技能。你应该能够求解未知数在两侧的方程、涉及分数的方程以及带括号的方程。不等式的规则类似，但必须记住：乘以或除以负数时，不等号方向要反转。

Consider the equation: 3(x − 2) = 2x + 5. First expand: 3x − 6 = 2x + 5. Then subtract 2x from both sides: x − 6 = 5. Finally add 6: x = 11. Always check your answer by substituting back into the original equation.

考虑方程：3(x − 2) = 2x + 5。首先展开：3x − 6 = 2x + 5。然后两边减去 2x：x − 6 = 5。最后加 6：x = 11。始终将答案代回原方程进行验证。

For inequalities, represent solutions on a number line using open circles for < and >, and closed circles for ≤ and ≥. When writing the solution set, use set notation or interval notation as required.

对于不等式，在数轴上表示解集时，使用空心圆表示 < 和 >，使用实心圆表示 ≤ 和 ≥。书写解集时，根据要求使用集合记号或区间记号。

−2 ≤ x + 3 < 7

To solve this compound inequality, subtract 3 from all parts: −5 ≤ x < 4. The solution includes all real numbers from −5 (inclusive) to 4 (exclusive).

要求解这个复合不等式，所有部分都减去 3：−5 ≤ x < 4。解集包含从 −5（包含）到 4（不包含）的所有实数。

Collect like terms systematically on both sides / 系统地在两侧合并同类项
Clear fractions by multiplying through by the common denominator / 乘以公分母以消去分数
Reverse the inequality sign when multiplying or dividing by a negative / 乘以或除以负数时反转不等号
Represent inequality solutions graphically on number lines / 在数轴上图形化表示不等式的解

4. Simultaneous Equations | 联立方程

Simultaneous equations involve finding values of two or more variables that satisfy multiple equations at once. The two main methods are elimination and substitution. The elimination method requires aligning the coefficients of one variable and then adding or subtracting the equations to eliminate that variable.

联立方程是指同时满足多个方程的变量的解。两种主要方法是消元法和代入法。消元法需要对齐某个变量的系数，然后通过相加或相减方程来消去该变量。

Example: Solve 2x + 3y = 12 and 4x − y = 10. Multiply the second equation by 3 to align y coefficients: 12x − 3y = 30. Now add both equations: 14x = 42, so x = 3. Substitute x = 3 into 2x + 3y = 12: 6 + 3y = 12, giving y = 2.

示例：求解 2x + 3y = 12 和 4x − y = 10。将第二个方程乘以 3 以对齐 y 的系数：12x − 3y = 30。现在将两个方程相加：14x = 42，所以 x = 3。将 x = 3 代入 2x + 3y = 12：6 + 3y = 12，得 y = 2。

When one equation is linear and the other is quadratic, the substitution method is preferred. Substitute the linear expression into the quadratic equation, solve the resulting quadratic, then find the corresponding y values. This may yield two pairs of solutions.

当一个方程是线性而另一个是二次时，优选代入法。将线性表达式代入二次方程，求解所得的二次方程，然后找到对应的 y 值。这可能会产生两对解。

Align coefficients carefully before adding or subtracting / 在加减前仔细对齐系数
Check solutions by substituting into both original equations / 通过代入两个原方程来验证解
Use substitution when one equation is linear and one is quadratic / 当一线性一二次时使用代入法
Interpret simultaneous equations graphically as intersection points / 将联立方程图形化解释为交点

5. Quadratic Equations and the Formula | 二次方程与公式法

Quadratic equations can be solved by factorising, completing the square, using the quadratic formula, or drawing graphs. You need to know when each method is most appropriate. Factorising is quickest when the expression factorises neatly, but the quadratic formula always works.

二次方程可以通过因式分解、配方法、使用二次公式或绘制图形来求解。你需要知道每种方法最适用的场景。当表达式可以整齐地因式分解时，因式分解最快，但二次公式始终有效。

The quadratic formula for ax² + bx + c = 0 is:

对于 ax² + bx + c = 0，二次公式为：

x = (−b ± √(b² − 4ac)) / 2a

The discriminant b² − 4ac determines the nature of the roots. If b² − 4ac > 0, there are two distinct real roots. If b² − 4ac = 0, there is one repeated real root. If b² − 4ac < 0, there are no real roots.

判别式 b² − 4ac 决定了根的性质。如果 b² − 4ac > 0，则有两个不同的实数根。如果 b² − 4ac = 0，则有一个重根。如果 b² − 4ac < 0，则没有实数根。

Completing the square transforms ax² + bx + c into a(x + p)² + q. This form is useful for finding turning points. For the equation x² + 6x + 5 = 0, complete the square: (x + 3)² − 9 + 5 = 0, so (x + 3)² = 4, giving x = −3 ± 2, i.e. x = −1 or x = −5.

配方法将 ax² + bx + c 转换为 a(x + p)² + q。这种形式有助于找到顶点坐标。对于方程 x² + 6x + 5 = 0，配方：(x + 3)² − 9 + 5 = 0，所以 (x + 3)² = 4，得 x = −3 ± 2，即 x = −1 或 x = −5。

Attempt factorising first before resorting to the formula / 先尝试因式分解，再使用公式法
Use the discriminant to determine the number of real roots / 使用判别式确定实数根的个数
Write the quadratic formula exactly as given on the formula sheet / 严格按照公式表书写二次公式
Complete the square to find the vertex of a parabola / 通过配方法找到抛物线的顶点

6. Graphs and Functions | 图形与函数

Understanding the graphs of linear, quadratic, cubic, reciprocal, and exponential functions is essential. You should be able to sketch these graphs, identify key features such as intercepts and turning points, and interpret transformations including translations and reflections.

理解线性、二次、三次、反比例和指数函数的图形至关重要。你应该能够绘制这些图形的草图，识别截距和顶点等关键特征，并理解包括平移和反射在内的变换。

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1. For example, if line A has gradient 2, a perpendicular line has gradient −1/2.

直线方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。平行线具有相等的斜率。垂线的斜率相乘为 −1。例如，如果直线 A 的斜率为 2，则垂线的斜率为 −1/2。

For quadratic graphs y = ax² + bx + c, the graph is a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward. The y-intercept is at (0, c), and the axis of symmetry is x = −b/(2a).

对于二次函数图像 y = ax² + bx + c，图像是抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。y 截距在 (0, c)，对称轴为 x = −b/(2a)。

Graph transformations follow specific rules. Translating y = f(x) by the vector (a, b) gives y − b = f(x − a), meaning the graph shifts a units right and b units up. A reflection in the x-axis changes y to −y, giving y = −f(x).

图形变换遵循特定规则。将 y = f(x) 平移向量 (a, b) 得到 y − b = f(x − a)，表示图形向右移动 a 个单位并向上移动 b 个单位。关于 x 轴的反射将 y 变为 −y，得到 y = −f(x)。

Identify the gradient and y-intercept from y = mx + c / 从 y = mx + c 中识别斜率和 y 截距
Sketch parabolas showing intercepts and vertex / 绘制抛物线草图，标出截距和顶点
Apply translations using vector notation / 使用向量记号进行平移变换
Recognise reciprocal graphs y = k/x and their asymptotes / 识别反比例函数 y = k/x 及其渐近线

7. Geometry and Circle Theorems | 几何与圆定理

Geometry questions require precise reasoning and the ability to justify each step. Key topics include angle properties of parallel lines, triangles, quadrilaterals, polygons, and especially circle theorems. You must be able to apply these theorems to find unknown angles.

几何题需要精确推理和逐步论证的能力。关键主题包括平行线、三角形、四边形、多边形的角性质，尤其是圆定理。你必须能够应用这些定理求出未知角度。

The essential circle theorems are: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; the tangent at any point is perpendicular to the radius; and the angle between a tangent and a chord equals the angle in the alternate segment.

基本圆定理包括：圆心角是圆周角的两倍；半圆内的圆周角为 90°；同一弓形内的角相等；圆内接四边形对角之和为 180°；任意点的切线与半径垂直；切线与弦之间的夹角等于弦切角定理中的圆周角。

Circle theorems are often combined with algebraic expressions. For example, if the angle at the centre is (4x + 10)° and the angle at the circumference is (2x − 5)°, then:

圆定理常与代数表达式结合。例如，若圆心角为 (4x + 10)°，圆周角为 (2x − 5)°，则：

4x + 10 = 2(2x − 5)

Solving gives 4x + 10 = 4x − 10, which leads to no solution — so always check whether the given angles are consistent. In valid problems, the relationship will yield a finite solution.

求解得 4x + 10 = 4x − 10，导致无解——因此始终检查给定角度是否一致。在有效的问题中，关系式会产生有限解。

State the circle theorem used when working out each angle / 在计算每个角度时注明所使用的圆定理
Add dynamic markings (right angles, equal sides) to diagrams / 在图形上标注直角、等边等标记
Use the property that the exterior angle of a cyclic quadrilateral equals the interior opposite angle / 利用圆内接四边形外角等于内对角
Combine algebra with geometry to solve multi-step problems / 将代数与几何结合以解决多步问题

8. Trigonometry | 三角学

Trigonometry in IGCSE covers right-angled triangles using sine, cosine, and tangent, as well as the sine rule, cosine rule, and the area of a triangle formula for non-right-angled triangles. You must also know the exact values of trigonometric ratios for 0°, 30°, 45°, 60°, and 90°.

IGCSE 三角学涵盖直角三角形的正弦、余弦和正切，以及非直角三角形的正弦定理、余弦定理和三角形面积公式。你还必须知道 0°、30°、45°、60° 和 90° 三角比的精确值。

For a right-angled triangle with angle θ, remember SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. For non-right-angled triangles, the sine rule is:

对于含角 θ 的直角三角形，记住 SOHCAHTOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。对于非直角三角形，正弦定理为：

a/sin A = b/sin B = c/sin C

The cosine rule: c² = a² + b² − 2ab cos C. Use the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS). The area formula is Area = ½ab sin C.

余弦定理：c² = a² + b² − 2ab cos C。当已知两边及其夹角（SAS）或三边（SSS）时使用余弦定理。面积公式为 Area = ½ab sin C。

Exact values you must memorise: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3, sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1, sin 60° = √3/2, cos 60° = ½, tan 60° = √3. Also note sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0.

你必须记忆的精确值：sin 30° = ½，cos 30° = √3/2，tan 30° = 1/√3，sin 45° = √2/2，cos 45° = √2/2，tan 45° = 1，sin 60° = √3/2，cos 60° = ½，tan 60° = √3。还要注意 sin 0° = 0，cos 0° = 1，sin 90° = 1，cos 90° = 0。

Identify which trigonometric rule applies before calculating / 在计算前确定适用哪个三角规则
Round answers to the required degree of accuracy, usually 3 significant figures / 按要求精度四舍五入，通常为 3 位有效数字
Check whether the ambiguous case of the sine rule applies / 检查正弦定理的模糊情形是否适用
Use the correct units for angles (degrees) and side lengths / 使用正确的单位（角度用度，边长用长度单位）

9. Vectors and Transformations | 向量与几何变换

Vectors represent quantities with both magnitude and direction. In IGCSE mathematics, you must understand vector notation, addition, subtraction, scalar multiplication, and finding the magnitude of a vector. Geometric transformations include translation, reflection, rotation, and enlargement.

向量表示既有大小又有方向的量。在 IGCSE 数学中，你必须理解向量记号、加法、减法、标量乘法以及求向量的模。几何变换包括平移、反射、旋转和放缩。

A vector written as column notation ⌈a⌉ / ⌊b⌋ represents a displacement of a units horizontally and b units vertically. The magnitude of this vector is √(a² + b²). For example, the vector ⌈3⌉ / ⌊4⌋ has magnitude √(9 + 16) = 5.

以列向量 ⌈a⌉ / ⌊b⌋ 书写的向量表示水平位移 a 个单位、垂直位移 b 个单位。该向量的模为 √(a² + b²)。例如，向量 ⌈3⌉ / ⌊4⌋ 的模为 √(9 + 16) = 5。

For transformations, you must describe each type precisely. A translation is described by a column vector. A reflection requires the equation of the mirror line. A rotation requires a centre, angle, and direction. An enlargement requires a scale factor and centre of enlargement.

对于变换，你必须精确描述每种类型。平移用列向量描述。反射需要镜面直线方程。旋转需要中心、角度和方向。放缩需要比例因子和放缩中心。

Negative scale factors for enlargement produce an image on the opposite side of the centre, with the shape rotated 180°. For example, an enlargement of scale factor −2 doubles the distance from the centre and flips the shape across the centre point.

负数比例因子的放缩会在中心的另一侧产生图像，图形旋转 180°。例如，比例因子为 −2 的放缩将距中心的距离加倍，并将图形绕中心点翻转。

Add vectors component-wise and multiply scalars correctly / 按分量相加向量并正确计算标量乘法
Use the column vector format in your answers / 在答案中使用列向量格式
Give complete descriptions for each transformation / 为每种变换给出完整描述
Remember that the magnitude is always a positive length / 记住模始终为正值长度

10. Probability and Statistics | 概率与统计

Probability and statistics questions count for a substantial portion of the IGCSE Edexcel examination. Key skills include calculating probabilities from equally likely outcomes, using tree diagrams for multi-stage events, and interpreting statistical measures such as mean, median, mode, and range.

概率与统计题在 IGCSE Edexcel 考试中占有相当大的比重。关键技能包括从等可能结果计算概率、使用树形图处理多阶段事件，以及解释均值、中位数、众数和极差等统计量。

The fundamental rule is that P(event) = favourable outcomes / total possible outcomes. Probabilities always lie between 0 and 1 inclusive. The sum of probabilities of all mutually exclusive outcomes equals 1. For complementary events, P(A’) = 1 − P(A).

基本规则是 P(事件) = 有利结果数 / 总可能结果数。概率始终在 0 和 1（含）之间。所有互斥结果的概率之和为 1。对于互补事件，P(A’) = 1 − P(A)。

For independent events A and B, P(A and B) = P(A) × P(B). For mutually exclusive events A and B, P(A or B) = P(A) + P(B). Tree diagrams are particularly useful for representing sequential events with multiple branches.

对于独立事件 A 和 B，P(A 且 B) = P(A) × P(B)。对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。树形图特别适合表示多分支的顺序事件。

When calculating the mean from a frequency table, use the formula Σ(fx)/Σf. For grouped data, use the midpoint of each interval as the representative x value. The median is the middle value when data is ordered; for even counts, average the two middle values.

从频数表计算均值时，使用公式 Σ(fx)/Σf。对于分组数据，使用每个区间的中点为代表值 x。中位数是有序数据中的中间值；对于偶数个数据，取两个中间值的平均数。

Write probabilities as simplified fractions or decimals / 将概率写为化简分数或小数
Multiply along branches and add across branches in tree diagrams / 树形图中沿分支相乘、跨分支相加
Identify whether events are independent or mutually exclusive / 识别事件是独立的还是互斥的
Use midpoints for grouped data when calculating the mean / 计算分组数据均值时使用组中值

11. Sequences and Series Basics | 数列与级数基础

Sequences appear frequently in the non-calculator paper. You must identify patterns, generate terms, and find the nth term of linear and quadratic sequences. Linear sequences have a constant common difference, giving the form an + b. Quadratic sequences have constant second differences, giving the form an² + bn + c.

数列在非计算器试卷中频繁出现。你必须识别规律、生成项，并求线性数列和二次数列的通项。线性数列有常数公差，形式为 an + b。二次数列有常数二阶差，形式为 an² + bn + c。

To find the nth term of a linear sequence, first find the common difference, which becomes the coefficient of n. Then determine the constant term by substituting n = 1. For example, the sequence 5, 8, 11, 14 has a common difference of 3, so the nth term is 3n + 2.

要求线性数列的通项，首先找到公差，它成为 n 的系数。然后通过代入 n = 1 确定常数项。例如，数列 5, 8, 11, 14 的公差为 3，因此通项为 3n + 2。

For geometric sequences, each term is found by multiplying the previous term by a constant ratio r. The nth term is a × rⁿ⁻¹. For example, the sequence 2, 6, 18, 54 has a = 2 and r = 3, so the nth term is 2 × 3ⁿ⁻¹.

对于等比数列，每一项由前一项乘以常数比 r 获得。通项为 a × rⁿ⁻¹。例如，数列 2, 6, 18, 54 中 a = 2，r = 3，因此通项为 2 × 3ⁿ⁻¹。

Check the difference between consecutive terms to classify a sequence / 检查相邻项之差对数列分类
Use second differences to confirm quadratic sequences / 使用二阶差确认二次数列
Substitute n = 1, 2, 3 to verify your nth term formula / 代入 n = 1, 2, 3 验证通项公式
Remember geometric sequences multiply, not add / 记住等比数列是相乘而非相加

12. Exam Strategy and Common Mistakes | 考试策略与常见错误

Success in IGCSE Edexcel Mathematics depends not only on knowing the content but also on exam technique. Proper time management, clear working, and careful checking are essential for maximising your score. Many students lose marks due to careless errors rather than lack of understanding.

在 IGCSE Edexcel 数学中取得成功不仅取决于掌握内容，还取决于考试技巧。合理的时间管理、清晰的书写和仔细的检查对于最大化得分至关重要。许多学生因粗心错误而非不理解而失分。

Always show your full working, as method marks are awarded even when the final answer is incorrect. Round answers to the specified degree of accuracy at the end of the calculation, not earlier, to avoid accumulating rounding errors. Use a sharp pencil for diagrams and a ruler for straight lines.

始终展示完整的解题过程，因为即使最终答案错误，方法分也会被授予。在计算结束时按指定精度四舍五入，而不是提前，以避免累积舍入误差。绘图使用削尖的铅笔，直线用直尺。

Common mistakes include: forgetting to reverse the inequality sign when dividing by a negative; incorrectly applying the quadratic formula due to sign errors; confusing the sine rule with the cosine rule; omitting units in the final answer; and misreading the scale on graphs.

常见错误包括：除以负数时忘记反转不等号；因符号错误而错误应用二次公式；混淆正弦定理与余弦定理；最终答案遗漏单位；以及误读图形上的刻度。

Allocate about one minute per mark and move on when stuck / 每题约一分钟一分，卡住时先跳过
Always substitute solutions back into the original equation / 始终将解代回原方程验证
Draw large, clear diagrams and label all key points / 绘制大而清晰的图形并标注所有关键点
Read the question twice to avoid misinterpreting the requirement / 读题两遍以避免误解要求
Review your paper in the last ten minutes for arithmetic slips / 最后十分钟检查试卷中的算术失误

By mastering these twelve core areas, practising past papers, and refining your exam technique, you will be well-prepared for the IGCSE Edexcel Mathematics examination. Consistent practice is the single most effective way to improve both speed and accuracy. Remember to review the official formula booklet provided by Edexcel before your exam, as understanding exactly which formulas are supplied can save valuable time.

通过掌握这十二个核心领域、练习历年试卷并完善考试技巧，你将为 IGCSE Edexcel 数学考试做好充分准备。持续练习是提高速度和准确性的最有效方法。记得在考试前复习 Edexcel 提供的官方公式手册，因为准确了解哪些公式已被提供可以节省宝贵时间。

Published by TutorHao | IGCSE Edexcel Mathematics Revision Series | aleveler.com

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IGCSE Edexcel Mathematics: Complete Revision Guide | IGCSE Edexcel 数学：完整复习指南