📚 Year 10 CIE Mathematics: Core Topics Overview | Year 10 CIE 数学：核心知识点梳理
Year 10 is a crucial stage in the CIE IGCSE Mathematics course, where students consolidate foundational concepts and prepare for the rigours of the final examination. This article provides a structured overview of the essential topics you will encounter, from number operations and algebra to geometry, statistics and probability. Mastering these fundamentals will not only boost your confidence but also build a solid bridge to Year 11 and the extended syllabus.
Year 10 是 CIE IGCSE 数学课程的关键阶段，学生将在此阶段巩固基础知识，为最终的考试做准备。本文系统梳理了从数的运算、代数，到几何、统计与概率的核心知识点。掌握这些基础内容不仅能增强你的信心，还能为你衔接 Year 11 和拓展课程打下坚实基础。
1. Number Types and Operations | 数的类型与运算
The number system is built from natural numbers (1, 2, 3, ...), integers ( ... , -2, -1, 0, 1, 2, ... ), rational numbers (fractions such as 3/4 or -7/2) and irrational numbers (like √2 or π). Understanding the hierarchy helps when simplifying expressions and solving problems. Operations follow the BIDMAS/PEMDAS order: Brackets, Indices, Division & Multiplication, Addition & Subtraction.
数系由自然数（1, 2, 3, ...）、整数（... , -2, -1, 0, 1, 2, ...）、有理数（如 3/4 或 -7/2）和无理数（如 √2 或 π）构成。理解这一层级有助于化简表达式和解决问题。运算遵循 BIDMAS/PEMDAS 顺序：括号、指数、乘除、加减。
ctorisation, e.g., 60 = 2² × 3 × 5. This is useful for finding highest common factors (HCF) and lowest common multiples (LCM).
质数恰好有两个不同的因数，即 1 和它本身（2, 3, 5, 7, 11, ...）。将一个数表示为质数的乘积就是它的质因数分解，例如 60 = 2² × 3 × 5。这在求最大公因数（HCF）和最小公倍数（LCM）时非常有用。
Working with directed numbers (positives and negatives) follows clear rules: adding a negative is equivalent to subtraction, and subtracting a negative becomes addition. Multiplying or dividing two numbers with the same sign gives a positive result; opposite signs give a negative result.
处理有向数（正数和负数）需遵循明确规则：加上一个负数等于减去它的绝对值，减去一个负数则变为加法。同号两数相乘或相除结果为正，异号结果为负。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
3/6, are created by multiplying or dividing the numerator and denominator by the same non‑ orward: multiply numerators together and denominators together. For division, multiply by the reciprocal of the divisor.
分数表示整体的一部分。等值分数如 1/2 = 2/4 = 3/6，可以通过将分子和分母同时乘以或除以同一个非零数得到。加减分数时需要先通分。乘法直接将分子相乘、分母相乘；除法则是乘以除数的倒数。
Decimals are another way of expressing fractions. Terminating decimals (e.g., 0.75) and recurring decimals (e.g., 0.3̅) can both be converted back to fractions. You should be comfortable rounding numbers to a given number of decimal places or significant figures, as required by CIE exam questions.
小数是分数的另一种表示方式。有限小数（如 0.75）和循环小数（如 0.3̅）都能转换回分数。你需要熟练掌握根据题目要求将数字四舍五入到指定的小数位数或有效数字。
Percentages mean 'out of 100'. Converting between fractions, decimals and percentages is a vital skill: 3/5 = 0.6 = 60% ease are calculated using the formula: percentage change = (change ÷ original value) × 100% a percentage change and need to find the original, are often tested in the core syllabus.
百分数意为“每一百份”。在分数、小数和百分数之间进行转换是一项关键技能：3/5 = 0.6 = 60%。百分数增减用公式计算：百分比变化 =（变化量 ÷ 原值）× 100%。反向百分数，即已知百分数变化后的最终值求原值，也是核心课程常考的内容。
3. Ratio, Proportion and Rates | 比、比例与速率
quantity is divided in a given ratio, the total number of parts is the sum of the ratio terms. For example, sharing £50 in the ratio 2:3 gives 2/5 × £50 = £20 and 3/5 × £50 = £30.
比用于比较两个或多个量的大小。化简比时将所有部分除以它们的最大公约数，例如 8:12 = 2:3。按给定比例分配一个量时，总份数等于比的各项之和。例如，将 £50 按 2:3 分配，得到 2/5 × £50 = £20 和 3/5 × £50 = £30。
ntities increase or decrease at the same rate; their ratio remains constant. If y is directly proportional to x, we write y ∝ es, maps (scales) and currency conversion are common applications of proportion and rates.
正比意味着两个量以相同比率增加或减少；它们的比值保持不变。如果 y 与 x 成正比，记作 y ∝ x 或 y = kx。涉及食谱、地图比例尺和货币换算的问题都是比例与速率的常见应用。
he value of one unit first, then scale to the required number. For example, if 5 pens cost £2, one pen costs £0.40, so 8 pens cost £3.20. Speed, density and other compound measures are also rates — make sure you can use formulas like speed = distance ÷ time.
归一法是一种强大的工具：先求出一个单位的量，再推算出所需的量。例如，若 5 支笔 £2，则一支笔 £0.40，因此 8 支笔 £3.20。速度、密度等复合量也是速率的一种——确保你能运用速度 = 距离 ÷ 时间等公式。
4. Algebraic Expressions and Manipulation | 代数表达式与化简
esent numbers. Terms are the building blocks of expressions (e.g., 3x, -2y, 5). Like terms — those with exactly the same variable part — can be collected by adding their coefficients. For instance, 4a + 3b - 2a + 5b = 2a + 8b.
代数用字母表示数。项是表达式的基本组成部分（如 3x、-2y、5）。同类项——变量部分完全相同的项——可以通过系数的加减进行合并。例如，4a + 3b - 2a + 5b = 2a + 8b。
erm inside the bracket is multiplied by the term outside: 5(x + 2) = 5x + 10. For double brackets, use FOIL or grid: (x + 3)(x - 2) = x² + x - 6.
项相乘：系数相乘，变量按指数律相乘。例如，3x² × 4x³ = 12x⁵。去括号时，将括号内的每一项与括号外的项相乘：5(x + 2) = 5x + 10。对于两个括号相乘，使用 FOIL 或表格法：(x + 3)(x - 2) = x² + x - 6。
, factorising gives (x + p)(x + q) where p + q = b and pq = c. This is a key skill for solving equations later.
因式分解是去括号的逆过程。首先寻找公因数，例如 6x + 9 = 3(2x + 3)。对于形如 x² + bx + c 的二次表达式，因式分解得到 (x + p)(x + q)，其中 p + q = b 且 pq = c。这是后续解方程的关键技能。
5. Solving Linear Equations and Inequalities | 解线性方程与不等式
inear equation, perform inverse operations to isolate the unknown. For example, 2x - 5 = 9 ⇒ 2x = 14 ⇒ Equations with unknowns on both sides require collecting like terms first: 3x + 2 = x + 10 ⇒ 2x = 8 ⇒ x = 4.
解线性方程时，使用逆运算分离出未知数。例如 2x - 5 = 9 ⇒ 2x = 14 ⇒ x = 7。务必代入检验答案。未知数出现在方程两边时，需要先合并同类项：3x + 2 = x + 10 ⇒ 2x = 8 ⇒ x = 4。
lying or dividing by a negative number reverses the inequality sign. For instance, -2x < 8 ⇒ utions are often shown on a number line with open (for <, >) or closed (for ≤, ≥) circles.
解不等式的方法类似，但要记住，乘以或除以一个负数时要反转不等号。例如 -2x < 8 ⇒ x > -4。解集通常用数轴表示，用空心圆（表示 <、>）或实心圆（表示 ≤、≥）标示。
d also be able to form equations from word problems. Translate the situation step by step: 'three more than twice a number is 11' becomes 2x + 3 = 11. For inequalities, phrases like 'at least' (≥) and 'at most' (≤) are important clues.
你还应具备根据应用题列出方程的能力。逐步翻译题意：“一个数的两倍加三等于 11” 变为 2x + 3 = 11。对于不等式，“至少”对应 ≥，“至多”对应 ≤ 是重要的线索。
6. Sequences and Patterns | 数列与模式
) sequences have a constant difference between terms. For example, 5, 8, 11, 14, ... has a be written as aₙ = dn + (a - d), where d is the common difference and a is the first term.
数列是一组按规律排列的数字。等差（线性）数列相邻两项的差为常数。例如，5, 8, 11, 14, ... 的公差为 +3。等差数列的第 n 项可写为 aₙ = dn + (a - d)，其中 d 为公差，a 为首项。
2. Recognising patterns in growing shapes or diagrams often leads to a linear sequence; the nth term then describes the number of tiles or matchsticks used.
对于数列 5, 8, 11, 14, ...，第 n 项为 3n + 2。由此可求出任意项，例如第 50 项：3×50 + 2 = 152。识别图形或火柴棍拼图中的增长模式常能导出线性数列，此时第 n 项可用于描述所需瓷砖或火柴棍的数量。
ences briefly. To generate terms from a given nth term, simply substitute n = 1, 2, 3, ...
虽然核心课程专注于等差数列，但也可能简单涉及二次数列。由给定的第 n 项公式生成各项时，只需代入 n = 1, 2, 3, ... 即可。
7. Straight Line Graphs | 直线图像
f a straight line is usually written as y = mx + c, where m is the gradient and c is the y‑intercept (the point where the line crosses the y‑ gradient m = (change in y) / (change in x) = Δy/Δx. Parallel lines have the same gradient.
直线方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距（直线与 y 轴的交点）。斜率 m =（y 的变化量）/（x 的变化量）= Δy/Δx。平行线具有相同的斜率。
To plot a straight line, choose at least three x‑values, calculate the corresponding y‑ vely, use the intercepts: find where the line crosses the axes. For 2x + 3y = 6, set x = 0 ⇒ y = 2 (y‑intercept); set y = 0 ⇒ x = 3 (x‑intercept).
绘制直线时，至少选取三个 x 值，计算对应的 y 值，然后描点并连成直线。另一种方法是利用截距：求直线与坐标轴的交点。对于 2x + 3y = 6，令 x = 0 ⇒ y = 2（y 轴截距）；令 y = 0 ⇒ x = 3（x 轴截距）。
Understanding these special cases is important for interpreting graphs and solving geometry problems.
水平线的方程为 y = k（斜率为零）；垂直线的方程为 x = h（斜率无定义）。理解这两种特殊情况对解读图像和求解几何问题十分重要。
8. Geometry and Angle Facts | 几何与角度关系
Basic angle facts are the foundation of geometry: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. When parallel lines are cut by a transversal, alternate angles are equal, corresponding angles are equal, and co‑interior (allied) angles sum to 180°.
基本角度定理是几何的基础：平角为 180°，周角为 360°，对顶角相等。当一条截线与两条平行线相交时，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, isosceles, scalene) and by angles (acute, right‑angled, obtuse). The sum of interior angles in any triangle is 180°. An exterior angle equals the sum of the two opposite interior angles. Scalene triangles have no equal sides or angles.
三角形可按边分类（等边、等腰、不等边）和按角分类（锐角、直角、钝角）。任意三角形的内角和为 180°。一个外角等于与它不相邻的两个内角之和。不等边三角形没有相等的边或角。
of sides. A regular polygon has all sides and angles equal; its exterior angle is 360° / n. The sum of exterior angles of any convex polygon is always 360°.
在多边形中，内角和 = (n - 2) × 180°，n 为边数。正多边形所有边和角均相等，其外角为 360° / n。任何一个凸多边形的外角和始终为 360°。
9. Pythagoras' Theorem and Trigonometry | 勾股定理与三角学
In a right‑angled triangle, Pythagoras' where c is the longest side. This is used to find a missing side length when two sides are known.
在直角三角形中，勾股定理指出，斜边的平方等于两直角边的平方和：a² + b² = c²，其中 c 为最长边。已知任意两边时，可用此定理求出第三边长度。
CIE core, θ is an acute angle (0° < θ < 90°). You will use these to find unknown sides or angles in right‑angled triangles, often with the help of a calculator.
三角比将角度与边长联系起来：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。在 CIE 核心课程中，θ 为锐角（0° < θ < 90°）。你需要运用这些比值，结合计算器，求出直角三角形中未知的边长或角度。
angle, use the inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹). A typical problem: In a right‑ (5/13) ≈ 22.6°. Always label the sides relative to the given angle before applying ratios.
求角度时使用反三角函数（sin⁻¹、cos⁻¹、tan⁻¹）。典型例题：直角三角形中对边长 5 cm，斜边 13 cm，则 sin θ = 5/13，所以 θ = sin⁻¹(5/13) ≈ 22.6°。在应用比值前务必根据给定角标出各边。
10. Perimeter, Area and Volume | 周长、面积与体积
t, parallelogram A = base × vertical height, trapezium A = ½(a + b)h, where a and b are the parallel sides.
周长是二维图形边界的总长度。矩形的周长 P = 2(l + w)。常见图形的面积公式：矩形 A = l × w，三角形 A = ½ × 底 × 高，平行四边形 A = 底 × 垂直高，梯形 A = ½(a + b)h，其中 a 和 b 为平行边。
.14 or the π button on your calculator. Questions may ask for the perimeter of compound shapes that combine rectangles and semicircles.
对于圆，需掌握含半径 r 和直径 d 的公式：周长 C = 2πr = πd，面积 A = πr²。π 取约 3.14 或使用计算器上的 π 键。考题可能要求计算由矩形和半圆组合而成的复合图形的周长。
sures the space occupied by a 3D solid. For a cuboid, V = length × width × height. A prism's volume is area of cross‑ le to convert between metric units of length, area and volume, e.g., 1 m³ = 1,000,000 cm³.
体积衡量三维立体所占空间的大小。长方体的体积 V = 长 × 宽 × 高。棱柱的体积 = 横截面积 × 长。你必须掌握长度、面积和体积的公制单位换算，例如 1 m³ = 1,000,000 cm³。
11. Statistics: Averages and Charts | 统计：平均数与图表
Three main averages are used to summarise data: the mean (sum of values ÷ number of values), the median (middle value when ordered) and the mode (most frequent value). The range (maximum − minimum) measures spread. For grouped data, the modal class has the highest frequency; the mean can be estimated using mid‑interval values.
描述数据集中趋势的三大平均数是：平均数（总和 ÷ 个数）、中位数（排序后中间的值）和众数（出现频率最高的值）。极差（最大值 − 最小值）衡量数据的离散程度。对于分组数据，众数组为频数最高的组；可用组中值估算平均数。
Data can be presented in various charts: bar charts, pie charts, line graphs, and stem‑and‑ show relationships between two variables; a line of best fit can be drawn to make predictions (correlation).
数据可用多种图表呈现：条形图、饼图、折线图和茎叶图。解读饼图时，扇区的圆心角 =（频数 ÷ 总数）× 360°。散点图展示两个变量之间的关系；可绘制最佳拟合线进行预测（相关性）。
For CIE core, you must be able to read and construct frequency tables, and compare data sets using averages and range. Always check for outliers — extreme values that can distort the mean — and decide which average best represents a data set.
在 CIE 核心课程中，你必须能阅读和构建频数表，并能运用平均数和极差比较数据集。要时刻留意极端值——可能扭曲平均数的异常数据——并判断哪种平均数最能代表一组数据。
12. Probability | 概率
ber of favourable outcomes) ÷ (total number of outcomes). The sum of probabilities of all possible outcomes is 1.
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。对于等可能结果，概率 =（有利结果的数量）÷（所有可能结果的总数）。所有可能结果的概率之和为 1。
You should be familiar with sample space diagrams, which list all possible outcomes of two events. For a coin and a fair six‑sided die, the sample space has 12 outcomes. The probability of getting a head and an even number is 1/4.
你应熟悉样本空间图，它列出两个事件的所有可能结果。以一枚硬币和一个均匀六面骰子为例，样本空间共有 12 个结果。掷出正面且点数为偶数的概率为 1/4。
umber of trials increases, relative frequency tends towards the theoretical probability. Expected frequency = probability × number of trials.
相对频率是通过实验对概率的估计：相对频率 =（事件发生的次数）÷（总试验次数）。随着试验次数的增加，相对频率会趋向于理论概率。期望频数 = 概率 × 试验次数。
events A and B, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
事件可以是互斥的（不能同时发生）或独立的（一个事件不影响另一个事件）。对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 且 B) = P(A) × P(B)。
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