📚 Year 10 CAIE Mathematics: Core Topics Review | Year 10 CAIE 数学：核心知识点梳理
Year 10 CAIE Mathematics builds the foundational knowledge required for the IGCSE 0580 examination. This article reviews the core topics that students must master, including number operations, algebra, geometry, and statistics. Understanding these fundamentals will ensure success in year 10 and beyond.
Year 10 CAIE 数学为 IGCSE 0580 考试奠定基础。本文梳理了学生必须掌握的核心知识点，涵盖数的运算、代数、几何和统计等内容。掌握这些基础将确保在 10 年级及以后取得成功。
1. Number and Arithmetic | 数与算术
Numbers form the language of mathematics. Students need to recognise natural numbers, integers, rational and irrational numbers, as well as prime and composite numbers. Operations with fractions, decimals and percentages must be fluent, along with ratio, proportion and standard form (scientific notation).
数是数学的语言。学生需要识别自然数、整数、有理数和无理数、质数与合数。必须熟练进行分数、小数和百分数的运算，同时掌握比、比例和标准形式（科学记数法）。
Key index laws are tested regularly:
指数定律是常考内容：
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
1 ≤ a < 10. Being able to convert between ordinary numbers and standard form is essential.
标准形式将极大或极小的数表示为 a × 10ⁿ，其中 1 ≤ a < 10。能熟练在普通数字和标准形式之间转换非常关键。
2. Algebra Basics | 代数基础
Algebra is the core of IGCSE mathematics. Students collect like terms, expand brackets and factorise expressions. They must understand that letters represent variable quantities and work comfortably with algebraic fractions.
代数是 IGCSE 数学的核心。学生需要合并同类项、展开括号和因式分解。必须理解字母代表变量，并能熟练处理代数分式。
Common expansions and factorisations include:
常见的展开与因式分解包括：
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b² = (a + b)(a – b)
Expanding products of two binomials, such as (x + 3)(2x – 5), and factorising quadratics like x² + 7x + 12, are essential skills. Rearranging formulas to make a given variable the subject is also tested.
展开两个二项式的乘积，例如 (x + 3)(2x – 5)，以及对 x² + 7x + 12 这类二次式进行因式分解，是必备技能。公式变形以将指定变量作为主题也是考查重点。
3. Equations and Inequalities | 方程与不等式
Solving linear equations in one variable and setting up equations from word problems are fundamental. Year 10 students also solve simple quadratic equations by factorisation and learn to represent solutions on a number line.
解一元一次方程以及根据文字题建立方程是基础内容。10 年级学生还要会用因式分解法解简单的二次方程，并学会在数轴上表示解。
ax² + bx + c = 0 can be solved when the expression factorises. For inequalities such as 3x – . Remember that multiplying or dividing by a negative number reverses the inequality sign.
当二次式可因式分解时，即可求解 ax² + bx + c = 0。对于 3x – 4 ≤ 11 这样的不等式，解集表示为 x ≤ 5。切记乘或除以负数时要反转不等号方向。
Simultaneous linear equations are solved by elimination or substitution. Graphical interpretation of equations and inequalities is introduced through shading regions on coordinate axes.
线性方程组通过消元法或代入法求解。通过坐标轴上的区域涂色，引入方程和不等式的图形解释。
4. Sequences | 数列
Recognising and generating sequences helps build pattern-recognition skills. Arithmetic sequences (also called linear sequences) have a constant difference between consecutive terms. The nth term formula is:
识别和生成数列有助于培养模式识别能力。等差数列（也叫线性数列）相邻两项的差为常数。第 n 项公式为：
un = a + (n – 1)d
where a is the first term and d is the common difference. Students also work with quadratic sequences whose second difference is constant, and learn to find the nth term of a quadratic sequence.
其中 a 为首项，d 为公差。学生还要处理二阶差为常数的二次数列，并学会求二次数列的第 n 项。
Geometric sequences involve a constant ratio r, but for Year 10 the focus is mainly on identifying the multiplier and generating the next few terms.
等比数列涉及恒定比值 r，但 10 年级的重点主要是识别乘数并生成后续几项。
5. Functions | 函数
f(x) is introduced. Students evaluate functions for given inputs, e.g. f(3) when f(x) = 2x – ctions f⁻¹(x) are key concepts. The inverse reverses the mapping of the original function.
引入函数记号 f(x)。学生需计算给定输入下的函数值，例如 f(x) = 2x – 7 时的 f(3)。复合函数 f(g(x)) 和反函数 f⁻¹(x) 是核心概念。反函数逆向了原函数的映射关系。
functions. Understanding that f(x) = 0 finds the x-intercept is important for graph work.
定义域和值域以非正式的方式讨论；学生需会代入数值并解涉及函数的方程。理解 f(x) = 0 可求得 x 轴截距对绘图形非常重要。
6. Graphs and Coordinate Geometry | 图形与坐标几何
fundamental. m represents the gradient, c the y-intercept. Students find the equation from two points or from one point and the gradient. They also explore parallel and perpendicular lines: parallel lines share the same gradient, while the product of gradients for perpendicular lines is –1.
直线方程 y = mx + c 是基础。m 表示斜率，c 表示 y 轴截距。学生需根据两点或一点与斜率求直线方程。他们还要探索平行线和垂直线：平行线的斜率相等，而垂直线斜率的乘积为 –1。
The distance between two points (x₁, y₁) and (x₂, y₂) is:
两点之间的距离公式为：
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). Quadratic graphs (parabolas) are drawn by completing a table of values, and the turning point is identified.
中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。通过列表描点绘制二次函数图像（抛物线），并识别转折点。
7. Geometry | 几何
terior angles of polygons are core. The sum of interior angles of an n-sided polygon is (n–2) × 180°. Congruence and similarity criteria are introduced informally.
平行线上的角度法则（内错角、同位角、同旁内角）、三角形与四边形的内角和以及多边形的外角是核心知识点。n 边形内角和为 (n–2) × 180°。非正式引入全等和相似的判定条件。
Circle geometry in Year 10 covers the parts of a circle (radius, diameter, chord, tangent, arc) and the relationship between angle at the centre and angle at circumference. The angle in a semicircle is 90°.
10 年级的圆几何涵盖圆各部分的名称（半径、直径、弦、切线、弧），以及圆心角与圆周角的关系。半圆上的圆周角为 90°。
| Shape | Angle Properties |
|---|---|
| Triangle | Sum of interior angles = 180° |
| Quadrilateral | Sum = 360° |
| Exterior angle | Equals sum of two opposite interior angles |
8. Mensuration | 测量
Calculating perimeter, area and volume is applied mathematics. Students must know and apply formulas for common shapes, including circles, triangles, parallelograms and trapezia. Composite shapes are broken into simpler parts.
计算周长、面积和体积属于应用数学。学生必须掌握并应用常见图形的公式，包括圆、三角形、平行四边形和梯形。复合图形需分解为基本图形进行计算。
Area of circle: A = πr²
Circumference: C = 2πr
Area of triangle: A = ½bh
a cylinder is 2πr² + 2πrh. Use of π in exact form (e.g. 16π) or with π ≈ 3.14 is expected.
棱柱（包括圆柱）的体积为横截面积 × 长。圆柱表面积为 2πr² + 2πrh。需要能用精确形式（如 16π）或使用 π ≈ 3.14 进行计算。
9. Trigonometry | 三角学
Right-angled triangle trigonometry is introduced using the ratios sine, cosine and tangent. The SOH CAH TOA memory aid helps recall:
直角三角形三角学引入正弦、余弦和正切比。SOH CAH TOA 助记法有助于记忆：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Students use these ratios to find missing sides and angles in right-angled triangles. Applications include angles of elevation and depression, bearings, and solving real-world problems with diagrams.
学生利用这些比求直角三角形中的缺失边长和角度。应用包括仰角和俯角、方位角，以及通过图示解决实际问题。
The exact values for 0°, 30°, 45°, 60°, 90° should be memorised:
需要记住 0°、30°、45°、60°、90° 的精确值：
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
10. Probability | 概率
Probability quantifies the chance of an event occurring. It is expressed as a fraction, decimal or percentage between 0 and 1. Students calculate the probability of single events, use the probability scale, and understand that the probabilities of all mutually exclusive outcomes sum to 1.
概率量化事件发生的可能性。以 0 至 1 之间的分数、小数或百分数表示。学生需计算单一事件的概率，使用概率标尺，并理解所有互斥结果的概率之和为 1。
For combined events, sample space diagrams and tree diagrams are used. When drawing a tree diagram, probabilities are multiplied along branches for 'and' combinations and added for 'or' combinations. Replacement or non-replacement is crucial.
对于组合事件，使用样本空间图和树状图。画树状图时，'且'的情况沿分支乘概率，'或'的情况则加概率。是否放回是关键。
B) applies for independent events. For mutually exclusive events, P(A or B) = P(A) + P(B).
对于独立事件，P(A 且 B) = P(A) × P(B)。对于互斥事件，P(A 或 B) = P(A) + P(B)。
11. Statistics | 统计
Statistics involves collecting, presenting and interpreting data. Measures of central tendency—mean, median, mode—are calculated. The range is a simple measure of spread: highest value minus lowest value.
统计学涉及数据的收集、呈现和解读。集中趋势的度量——平均数、中位数、众数——需要计算。极差是简单的离散度量：最大值减最小值。
Data is represented using bar charts, pie charts, frequency tables, histograms (for continuous data) and cumulative frequency diagrams. Students should be able to read and interpret these charts, as well as calculate the mean from a frequency table.
数据用条形图、饼图、频数表、直方图（用于连续数据）和累积频率图表示。学生应能读懂并解释这些图表，并能从频数表中计算平均数。
Cumulative frequency graphs help estimate the median, quartiles and interquartile range. These are used to construct box-and-whisker plots, which provide a visual summary of data spread.
累积频率图有助于估算中位数、四分位数和四分位间距。这些用于构建箱线图，从而提供数据分布的直观概括。
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