📚 Year 10 Cambridge Mathematics: Core Topic Overview | 剑桥10年级数学核心知识点梳理
As students progress into Year 10 of the Cambridge IGCSE pathway, they encounter a wide range of mathematical topics that form the foundation for the final examination. This article provides a structured overview of the core areas you need to master, from number and algebra to geometry and statistics. Understanding these building blocks is essential for success in both Core and Extended tiers.
当学生进入剑桥IGCSE 10年级时，他们将接触广泛的数学主题，这些内容构成了最终考试的基础。本文系统梳理了需要掌握的核心领域，从数字和代数到几何和统计。理解这些基本模块对于Core和Extended阶段的学习都至关重要。
1. Number and Operations | 数字与运算
en these forms and understand the relationship between them. For instance, 0.75 = 3/4 = 75%.
在10年级，你必须熟练掌握整数、分数、小数和百分数的性质。在不同形式之间流畅转换，并理解它们之间的关系。例如，0.75 = 3/4 = 75%。
Prime numbers, multiples, factors, highest common factor (HCF) and lowest common multiple (LCM) are fundamental. Use prime factor decomposition to find HCF and LCM efficiently.
质数、倍数、因数、最大公因数 (HCF) 和最小公倍数 (LCM) 是基础内容。使用质因数分解可以高效地找到 HCF 和 LCM。
It is used for very large or very small numbers, such as the speed of light 3.0 × 10⁸ m/s.
标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。它用于非常大或非常小的数字，如光速 3.0 × 10⁸ m/s。
Rounding to a given number of decimal places or significant figures, and estimating answers by approximating numbers, are key skills for checking calculations.
四舍五入到指定的小数位数或有效数字，并通过近似数字来估算答案，是检查计算的关键技能。
a × 10ⁿ (1 ≤ a < 10)
2. Algebraic Expressions and Equations | 代数表达式与方程
, expanding brackets, and factorising. The distributive law a(b + c) = ab + ac is central.
代数操作包括通过合并同类项、展开括号和因式分解来简化表达式。分配律 a(b + c) = ab + ac 是核心。
ch as (x + a)(x + b) = x² + (a+b)x + ab, and factorise quadratics of the form x² + bx + c.
你应该能够展开两个二项式的乘积，如 (x + a)(x + b) = x² + (a+b)x + ab，并对形如 x² + bx + c 的二次式进行因式分解。
he unknown. Remember to perform the same operation on both sides. For example, 3x + 5 = 20 ⇒ 3x = 15 ⇒ x = 5.
解一元一次方程包括隔离未知数。记住在等式两边进行相同的操作。例如，3x + 5 = 20 ⇒ 3x = 15 ⇒ x = 5。
Simultaneous linear equations can be solved by elimination or substitution. Interpret the solution graphically as the point of intersection of two lines.
联立一次方程可以通过消元法或代入法求解。将解图形化解释为两条直线的交点。
ax + by = c, dx + ey = f
3. Inequalities | 不等式
ows the same rules as equations, except multiplying or dividing by a negative number reverses the inequality sign.
不等式使用符号 <, >, ≤, ≥。解线性不等式遵循与方程相同的规则，但当乘以或除以负数时，不等号方向改变。
closed circle for ≤ or ≥. For compound inequalities such as 2 < x ≤ 5, show the interval.
在数轴上表示不等式：< 或 > 用空心圆圈，≤ 或 ≥ 用实心圆圈。对于复合不等式如 2 < x ≤ 5，需标出区间。
You can also list integer solutions that satisfy inequalities, which is common in problem-solving contexts.
你还可以列出满足不等式的整数解，这在解题中很常见。
-3 < 2x + 1 ≤ 5
4. Functions and Graphs | 函数与图像
tput. Use function notation f(x) = 2x + 3. Evaluate f(2) = 7. Understand domain and range.
函数将输入与输出关联。使用函数记号 f(x) = 2x + 3。计算 f(2) = 7。理解定义域和值域。
he gradient and c is the y-intercept. Find the gradient as rise/run = (y₂ - y₁)/(x₂ - x₁).
绘制线性图像 y = mx + c，其中 m 是斜率，c 是 y 轴截距。通过 rise/run = (y₂ - y₁)/(x₂ - x₁) 计算斜率。
e parabolas. Identify the vertex and intercepts. Use graphs to solve equations approximately.
二次函数图像 y = ax² + bx + c 形成抛物线。确定顶点和截距。利用图像近似求解方程。
Recognise the shapes of cubic, reciprocal and exponential graphs for the Extended tier.
对于 Extended 层次，需识别三次、倒数和指数函数图像的形状。
m = (y₂ - y₁)/(x₂ - x₁)
5. Geometry of Lines and Angles | 直线与角的几何
Know angle facts: angles on a straight line sum to 180°, vertically opposite angles are equal, and angles at a point sum to 360°.
掌握角的性质：直线上的角之和为180°，对顶角相等，一点周角之和为360°。
Parallel lines cut by a transversal produce corresponding, alternate and interior angles. Use these to find unknown angles.
平行线被截线切割产生同位角、内错角和同旁内角。利用这些角求未知角。
Properties of triangles: angle sum 180°, exterior angle equals sum of two opposite interior angles. In isosceles triangles, base angles are equal.
三角形的性质：内角和180°，外角等于两个不相邻内角之和。等腰三角形底角相等。
gons: interior angle sum = (n-2) × 180°, exterior angle sum = 360° for any convex polygon.
多边形：内角和 = (n-2) × 180°，任意凸多边形的外角和为360°。
6. Perimeter, Area and Volume | 周长、面积与体积
Calculate perimeter of rectilinear shapes and circumference of a circle: C = 2πr or πd.
计算直线围成形状的周长和圆的周长：C = 2πr 或 πd。
ea formulas: rectangle A = l × w, triangle A = ½bh, circle A = πr², trapezium A = ½(a+b)h.
面积公式：长方形 A = l × w，三角形 A = ½bh，圆 A = πr²，梯形 A = ½(a+b)h。
For compound shapes, split into simpler parts, calculate each area and sum them.
对于组合形状，分解成简单部分，计算各部分面积再求和。
of prisms and cylinders: Volume = area of cross-section × length. Cylinder volume = πr²h.
棱柱和圆柱的体积与表面积：体积 = 横截面积 × 长度。圆柱体积 = πr²h。
A = πr²
7. Pythagoras and Trigonometry | 毕达哥拉斯定理与三角学
Pythagoras' right-angled triangle, a² + b² = c², where c is the hypotenuse. Use to find missing sides.
毕达哥拉斯定理：在直角三角形中，a² + b² = c²，其中 c 是斜边。用于求未知边长。
, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. SOH CAH TOA is a helpful mnemonic.
三角比：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。记忆口诀 SOH CAH TOA 很有帮助。
Use inverse trig functions to find angles. Solve right-angled triangle problems including angles of elevation and depression.
利用反三角函数求角。解决实际问题中的直角三角形问题，包括仰角和俯角。
c² = a² + b²
8. Statistics | 统计
Collect, organise and represent data using bar charts, pie charts, histograms with equal class intervals, and frequency polygons.
收集、整理并使用条形图、饼图、等组距直方图和频数多边形表示数据。
sum of values / number of values, median is the middle value, mode is the most frequent value.
集中趋势的度量：均值 = 值的总和 / 值的个数，中位数是中间的值，众数是最频繁出现的值。
m - minimum gives a measure of spread. Interquartile range is used for more robust spread.
极差 = 最大值 - 最小值，给出分散程度的度量。四分位距用于更稳健的离散度量。
Scatter graphs show correlation. Draw a line of best fit and use it to make predictions.
散点图显示相关性。画出最佳拟合线并用其进行预测。
9. Probability | 概率
ally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率是可能性的一种度量，范围从0到1。对于等可能结果，P(事件) = 有利结果的数量 / 总结果数量。
and tree diagrams to enumerate outcomes. For independent events, P(A and B) = P(A) × P(B).
使用样本空间图和树状图列举结果。对于独立事件，P(A 且 B
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