📚 KS3 CAIE Advanced Mathematics: Core Topic Summary | KS3 CAIE 进阶数学：核心知识点梳理
KS3 CAIE Advanced Mathematics builds a strong foundation in algebra, geometry, statistics and problem-solving skills essential for IGCSE and beyond.
KS3 CAIE 进阶数学为代数、几何、统计和解题技巧打下坚实基础，对 IGCSE 及更深层次的学习至关重要。
1. Algebraic Expressions and Simplification | 代数表达式与化简
Algebra uses symbols, usually letters like x or y, to stand for unknown numbers or variables.
代数使用符号，通常是像 x 或 y 的字母，代表未知数或变量。
Terms with the same letter and power are called like terms; for example, 3x and -5x are like terms.
具有相同字母和幂的项称为同类项；例如，3x 和 -5x 是同类项。
You simplify expressions by collecting like terms: 2a + 3b – a + 4b = a + 7b.
通过合并同类项来化简表达式：2a + 3b – a + 4b = a + 7b。
The rules of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ (where a ≠ 0).
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ（其中 a ≠ 0）。
brackets follows the distributive law: a(b + c) = ab + ac; be careful with negative signs.
去括号遵循分配律：a(b + c) = ab + ac；要注意负号的处理。
2. Linear Equations and Inequalities | 线性方程与不等式
An equation states that two expressions are equal, such as 2x + 3 = 11.
方程表明两个表达式相等，例如 2x + 3 = 11。
the variable by performing inverse operations: subtract 3, then divide by 2, giving x = 4.
解方程时，通过逆运算隔离变量：先减 3，再除以 2，得到 x = 4。
is similar to equations, but if you multiply or divide by a negative number, reverse the inequality sign.
不等式使用符号 <, >, ≤, ≥ 来比较数值；解不等式与方程类似，但如果乘以或除以负数，需反转不等号。
Solutions can be shown on a number line with open or closed circles.
解集可用数轴上的空心或实心圆点表示。
Linear equations with unknowns on both sides and those involving fractions are also covered.
也涉及未知数在两边以及含有分数的线性方程。
3. Sequences and the nth Term | 数列与第 n 项
A sequence is an ordered list of numbers; in an arithmetic sequence, the difference between consecutive terms is constant, called the common difference d.
数列是一组有序排列的数；在等差数列中，相邻两项的差恒定，称为公差 d。
The nth term of an arithmetic sequence is given by Tₙ = a + (n – 1)d, where a is the first term.
等差数列的第 n 项公式为 Tₙ = a + (n – 1)d，其中 a 为首项。
For example, the sequence 3, 7, 11, 15, … has nth term 4n – 1.
例如，数列 3, 7, 11, 15, … 的第 n 项为 4n – 1。
given number belongs to the sequence, or sum a series using the formula Sₙ = n/2 [2a + (n – 1)d].
你还可以求出特定项，判断一个给定的数是否属于该数列，或利用公式 Sₙ = n/2 [2a + (n – 1)d] 求和。
Recognising patterns in quadratic or geometric sequences is also introduced.
课程还引进了二次数列或等比数列的规律识别。
4. Coordinates and Linear Graphs | 坐标与直线图
Points in the Cartesian plane are written as (x, y); axes intersect at the origin (0, 0).
平面直角坐标系中的点表示为 (x, y)；坐标轴相交于原点 (0, 0)。
f a straight line is y = mx + c, where m is the gradient (slope) and c is the y-intercept.
直线方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Gradient is calculated by rise over run: m = (y₂ – y₁) / (x₂ – x₁).
斜率通过垂直变化除以水平变化计算：m = (y₂ – y₁) / (x₂ – x₁)。
Parallel lines have the same gradient; perpendicular lines have gradients that multiply to –1.
平行线斜率相同；互相垂直的直线斜率乘积为 –1。
Reading and plotting distance–time graphs helps understanding of speed and motion.
阅读和绘制距离-时间图有助于理解速度与运动。
5. Ratio, Proportion and Rates | 比、比例与变化率
Ratio compares two or more quantities; it can be simplified by dividing by a common factor.
比用来比较两个或多个数量；可以通过除以公因数来化简。
A proportion is a statement that two ratios are equal, e.g. a : b = c : d.
比例是两个比相等的陈述，例如 a : b = c : d。
Direct proportion: y = kx; inverse proportion: y = k/x, where k is a constant.
正比例：y = kx；反比例：y = k/x，其中 k 为常数。
ates involve one quantity changing with respect to another, such as speed = distance/time.
变化率指一个量相对于另一个量的变化，例如速度 = 距离/时间。
Scale drawings and maps use a ratio to represent actual distances.
比例尺图和地图使用比来表示实际距离。
6. Geometry of Shapes and Angles | 图形与角度几何
Angles on a straight line sum to 180°, around a point sum to 360°; vertically opposite angles are equal.
直线上的角之和为 180°，绕一点的角之和为 360°；对顶角相等。
The sum of interior angles in a triangle is 180°; in an n-sided polygon it is (n – 2) × 180°.
三角形内角和为 180°；n 边形的内角和为 (n – 2) × 180°。
l × w, triangle = ½ × base × height, circle = πr², parallelogram = base × vertical height.
面积公式：矩形 = 长 × 宽，三角形 = ½ × 底 × 高，圆 = πr²，平行四边形 = 底 × 垂直高。
Volume of a cuboid = l × w × h; volume of a prism = area of cross-section × length.
长方体体积 = 长 × 宽 × 高；棱柱体积 = 横截面积 × 长度。
Properties of quadrilaterals, symmetry and constructions are also covered.
此外还涉及四边形的性质、对称性与尺规作图。
7. Pythagoras’ Theorem | 勾股定理
uare of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
在直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²。
You can use Pythagoras' theorem to find a missing side or to check if a triangle is right-angled.
可利用勾股定理求未知边长，或检验一个三角形是否为直角三角形。
Applications include finding distances in coordinate geometry and real-life problems.
应用包括在坐标几何中求距离以及解决实际问题。
The Pythagorean triples like (3, 4, 5) and (5, 12, 13) are special sets that satisfy the theorem.
勾股数组如 (3, 4, 5) 和 (5, 12, 13) 是满足定理的特殊整数组。
8. Introduction to Trigonometry | 三角学入门
Trigonometry relates the angles and sides of right-angled triangles using sine, cosine and tangent.
三角学利用正弦、余弦和正切将直角三角形的边与角联系起来。
ions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
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