Mathematical Requirements for IGCSE Mathematics | IGCSE 数学课程的数学要求

📚 Mathematical Requirements for IGCSE Mathematics | IGCSE 数学课程的数学要求

In the IGCSE Mathematics syllabus, “mathematical requirements” are the essential skills and techniques that you are expected to use across all topics. These include careful arithmetic, algebraic manipulation, geometric reasoning, statistical interpretation and the effective use of a scientific calculator. Mastering these requirements helps you solve exam problems accurately, efficiently and with clear written communication.

在 IGCSE 数学大纲中，“数学要求”是指你在所有专题中都需要运用的基本技能与技巧。这包括细心计算、代数变换、几何推理、统计解释以及高效使用科学计算器。掌握这些要求能帮助你准确、高效地解决考试问题，并写出清晰的过程。

1. Number and Arithmetic | 数与算术

One key requirement is fluency with the number system. You must be able to work with integers, fractions, decimals, percentages, ratios and proportions, and switch between them confidently.

一个关键要求是熟练掌握数系。你必须能够处理整数、分数、小数、百分数、比和比例，并自信地在它们之间转换。

You also need to use standard form, indices and roots. For example, calculate (2.5 × 10⁻¹) × (4 × 10²) and write the answer in standard form. Using index laws, you must remember that a³ × a² = a⁵ and (a³)² = a⁶.

你还需要使用标准形式、指数和根。例如，计算 (2.5 × 10⁻¹) × (4 × 10²) 并用标准形式写出答案。在运用指数法则时，你必须记住 a³ × a² = a⁵，且 (a³)² = a⁶。

(2.5 × 10⁻¹) × (4 × 10²) = 10 × 10¹ = 1.0 × 10²

In addition, you must apply the correct order of operations and round results to a required degree of accuracy, such as 3 significant figures or 2 decimal places.

此外，你必须运用正确的运算顺序，并将结果四舍五入到所需精度，例如保留 3 位有效数字或 2 位小数。

2. Algebra | 代数

Algebra is a central requirement. You must expand brackets, factorise expressions and simplify algebraic fractions. For instance, expand (x + 3)(x − 4) to obtain x² − x − 12.

代数是一项核心要求。你必须能够展开括号、分解因式并化简代数分数。例如，展开 (x + 3)(x − 4) 得到 x² − x − 12。

Solving linear equations, simultaneous equations and quadratic equations is essential. For quadratics, you may use factorisation, completing the square or the quadratic formula:

解线性方程、联立方程和二次方程至关重要。对于二次方程，你可以使用因式分解、配方法或求根公式：

x = (−b ± √(b² − 4ac)) / 2a

Inequalities and number lines, sequences and functions also form part of this area. You should be comfortable with notation such as f(x) = x² + 3x − 2 and with finding the inverse of a function.

不等式与数轴、数列和函数也是该领域的一部分。你应该熟悉 f(x) = x² + 3x − 2 这样的记号，并能够求一个函数的反函数。

3. Geometry and Trigonometry | 几何与三角

Geometric requirements include properties of angles, triangles, quadrilaterals, circles and polygons. You must solve problems involving parallel lines, bearings and circle theorems such as the angle in a semicircle being 90°.

几何要求包括角、三角形、四边形、圆和多边形的性质。你必须解决涉及平行线、方位角和圆定理的问题，例如半圆内的圆周角为 90°。

You also need to calculate perimeter, area, volume and surface area of 2D and 3D shapes, including compound shapes. For a right-angled triangle, Pythagoras’ theorem states that a² + b² = c², where c is the hypotenuse.

你还需要计算二维和三维图形的周长、面积、体积和表面积，包括组合图形。对于直角三角形，勾股定理表明 a² + b² = c²，其中 c 是斜边。

Trigonometry includes the sine, cosine and tangent ratios. In a right-angled triangle, sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent. You must use these to find unknown sides and angles.

三角学包括正弦、余弦和正切比。在直角三角形中，sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。你必须用它们求未知边和角。

4. Coordinate Geometry | 坐标几何

You must be able to plot points and draw straight-line graphs. The equation y = mx + c is central, where m is the gradient and c is the y-intercept. To find the equation of a line, you may be given two points or one point and the gradient.

你必须能够描点并绘制直线图。方程 y = mx + c 是核心，其中 m 是斜率，c 是 y 轴截距。要求一条直线的方程时，你可以已知两点或一个点与斜率。

You should also find the gradient between two points, the midpoint and the distance between two points. Parallel lines have the same gradient, while perpendicular lines have gradients whose product is −1.

你还应能求两点间的斜率、中点和距离。平行线的斜率相同，而垂直线的斜率乘积为 −1。

m = (y₂ − y₁) / (x₂ − x₁), midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

5. Mensuration and Measurement | 测量

Measurement requirements include converting between metric units, time units and currency. You should also know how to apply scale factors on maps and diagrams. For example, a scale of 1 : 100 means 1 cm on the map represents 100 cm in real life.

测量要求包括在公制单位、时间单位和货币之间进行换算。你还应知道如何在地图和图形上应用比例因子。例如，比例尺 1 : 100 表示地图上 1 cm 代表实际 100 cm。

Formulas for area and volume are frequently needed, such as the area of a trapezium ½(a + b)h, the volume of a cylinder πr²h and the surface area of a cone.

面积和体积公式经常用到，例如梯形面积 ½(a + b)h、圆柱体积 πr²h 以及圆锥的表面积。

You must also understand bounds and accuracy. If a length is measured as 12 cm to the nearest cm, the lower bound is 11.5 cm and the upper bound is 12.5 cm. Repeating calculations with bounds can change the final result.

你还必须理解误差界限与精确度。如果长度测量为 12 cm（精确到最近 cm），则下界为 11.5 cm，上界为 12.5 cm。使用界限重复计算可能会改变最终结果。

6. Statistics and Probability | 统计与概率

In statistics, you need to collect, organise and interpret data using charts such as bar charts, line graphs, pie charts, stem-and-leaf diagrams and histograms. You should also be able to compare data sets using the mean, median, mode and range.

在统计方面，你需要使用柱状图、折线图、饼图、茎叶图和直方图等图表来收集、整理和解释数据。你还应能使用平均数、中位数、众数和极差来比较数据集。

Probability requirements include understanding mutually exclusive and independent events. For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).

概率要求包括理解互斥事件和独立事件。对于互斥事件，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 且 B) = P(A) × P(B)。

You may also need to use tree diagrams and Venn diagrams to organise information and solve multi-step probability questions.

你可能还需要使用树形图和维恩图来整理信息，并解决多步骤概率问题。

7. Graphs and Functions | 函数与图像

This section requires you to work with function notation, including composite functions and inverse functions. For example, if f(x) = 2x + 3, then f(2) = 7 and fg(x) means apply g first, then f.

本部分要求你掌握函数记号，包括复合函数和反函数。例如，若 f(x) = 2x + 3，则 f(2) = 7；fg(x) 表示先应用 g，再应用 f。

You must be able to draw and interpret quadratic, cubic, reciprocal and exponential graphs, and know the basic shapes of these curves. For example, y = 1/x has two branches and y = aˣ is always increasing if a > 1.

你必须能够绘制和解释二次、三次、反比例和指数函数图像，并知道这些曲线的基本形状。例如，y = 1/x 有两条分支，而 y = aˣ 在 a > 1 时始终递增。

Transformation of graphs by translation or reflection is also required. A graph y = f(x) + a moves up by a units; y = f(x − a) moves right by a units. You should also interpret gradient and area under speed-time graphs.

还要求通过平移或反射来变换图像。图像 y = f(x) + a 向上移动 a 个单位；y = f(x − a) 向右移动 a 个单位。你还应能解释速度-时间图像中的斜率与面积。

8. Using Calculators and Precision | 计算器使用与精确度

A scientific calculator is essential for IGCSE Mathematics. You must use it correctly for powers, roots, trigonometric functions, logarithms and standard form. Always check that your calculator is in the correct mode for angles, usually degrees.

科学计算器对于 IGCSE 数学是必不可少的。你必须正确使用它进行幂、根、三角函数、对数和标准形式的计算。始终检查计算器角度模式是否正确，通常使用度数。

You should show appropriate rounding using decimal places or significant figures, and always check that your answers are reasonable. For example, if you are calculating the height of a building, an answer of 0.6 m is likely wrong.

你应该按小数位或有效数字进行适当舍入，并始终检查答案是否合理。例如，如果你计算一栋建筑物的高度，那么 0.6 m 的答案很可能出错。

Understand the difference between exact values and approximations. For example, √2 is exact, while 1.414 is an approximation to 3 decimal places. In many questions, leaving your answer as √2 is acceptable unless a decimal is requested.

理解精确值与近似值之间的区别。例如，√2 是精确值，而 1.414 是保留 3 位小数的近似值。在许多题目中，除非题目要求小数，否则把答案保留为 √2 是允许的。

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Mathematical Requirements for IGCSE Mathematics | IGCSE 数学课程的数学要求