Cambridge Lower Secondary Mathematics Stage 9: Essential Skills from Learner's Book 3 Sample | 剑桥初中数学第9阶段：学生用书3样章核心技能

📚 Cambridge Lower Secondary Mathematics Stage 9: Essential Skills from Learner's Book 3 Sample | 剑桥初中数学第9阶段：学生用书3样章核心技能

The Cambridge Lower Secondary Mathematics Stage 9 course builds on the skills developed in Stages 7 and 8, preparing learners for IGCSE or O Level Mathematics. This article reviews the essential topics that often appear in the Learner's Book 3 sample, including indices, standard form, algebraic manipulation, equations, Pythagoras' theorem, trigonometry, statistics and probability. Each section gives a clear method, a useful example and common exam tips.

剑桥初中数学第9阶段课程在第七和第八阶段的基础上进一步提升，为 IGCSE 或 O Level 数学做好准备。本文复习学生用书第3册样章中常见的重要主题，包括指数、标准形式、代数变形、方程、勾股定理、三角学、统计和概率。每一节都提供清晰的方法、有用的例题和常见考试提示。

1. Understanding Integer Indices | 理解整数指数

. The base is 3 and the index or exponent is 4. In Stage 9, you must be confident with positive, zero and negative integer indices.

指数表示重复相乘。例如，3⁴ 表示 3 × 3 × 3 × 3 = 81。底数是 3，指数是 4。在第九阶段，你必须熟练掌握正整数、零和负整数指数。

aᵐ × aⁿ = aᵐ⁺ⁿ

The multiplication law says that when multiplying powers with the same base, add the exponents. The division law subtracts exponents, and a zero exponent always gives 1. A negative exponent means the reciprocal of the positive power.

乘法法则表明，底数相同的幂相乘时，指数相加。除法法则将指数相减，零指数总是等于 1。负指数表示正幂的倒数。

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ

2. Laws of Indices in Action | 指数法则的应用

)⁴ ÷ 3⁵: first apply the power law to get 3⁸ ÷ 3⁵, then subtract exponents to get 3³ = 27.

化简 2⁻³ × 2⁵。使用乘法法则：2⁻³ × 2⁵ = 2⁻³⁺⁵ = 2² = 4。再如化简 (3²)⁴ ÷ 3⁵：先用幂的乘方法则得到 3⁸ ÷ 3⁵，再将指数相减得到 3³ = 27。

5² = 1/25. Never leave a negative index in a final answer unless the question asks for it.

考试提示：负数指数答案要写成分母中的正幂。例如 5⁻² = 1/5² = 1/25。除非题目要求，否则最终答案不要保留负指数。

3. Standard Form for Very Large and Small Numbers | 极大与极小数的标准形式

n is an integer. Large numbers have positive n, while very small numbers have negative n.

标准形式将一个数写成 a × 10ⁿ，其中 1 ≤ a < 10，n 是整数。很大的数 n 为正，很小的数 n 为负。

es to make a number between 1 and 10, and the number of places moved gives the value of n.

例如：0.00042 = 4.2 × 10⁻⁴；63000 = 6.3 × 10⁴。移动小数点使数字落在 1 到 10 之间，移动的位数就是 n 的值。

× 10⁻²) = 6 × 10¹ = 60. For division, divide the leading numbers and subtract the powers.

乘法时，把前面的数相乘，并把 10 的指数相加：(2 × 10³) × (3 × 10⁻²) = (2 × 3) × (10³ × 10⁻²) = 6 × 10¹ = 60。除法时，把前面的数相除，并把 10 的指数相减。

4. Expanding and Factorising Linear Expressions | 线性表达式的展开与因式分解

the reverse: find the highest common factor of all terms and write it outside the bracket.

展开指通过将括号内的每一项乘以外面的因式来去掉括号。例如 3(2x - 5) = 6x - 15。因式分解是展开的逆运算：找出所有项的最大公因数，把它写在括号外。

x + 3). Always check your factorisation by expanding mentally: 4 × 2x = 8x and 4 × 3 = 12.

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对于 8x + 12，最大公因数是 4，所以 8x + 12 = 4(2x + 3)。一定要通过心算展开来检验因式分解：4 × 2x = 8x，4 × 3 = 12。

Common mistake: forgetting to multiply the second term or losing negative signs. Write the common factor first, then the bracket, and check each sign carefully.

常见错误：忘记乘第二项或丢失负号。先写出公因式，再写括号，并仔细检查每一项的符号。

5. Solving Linear Equations | 解一元线性方程

For 3x + 2 = 14, subtract 2 from both sides: 3x = 12, then divide both sides by 3: x = 4.

解方程时，对方程两边进行相同运算以隔离未知数。对于 3x + 2 = 14，两边同时减 2：3x = 12，然后两边同时除以 3：x = 4。

ible. For 2(x - 3) = 10, divide both sides by 2 to get x - 3 = 5, then add 3 to get x = 8.

带括号时，先展开，或者如果可能先除以系数。对于 2(x - 3) = 10，两边同时除以 2 得到 x - 3 = 5，再加 3 得到 x = 8。

Always check your answer by substituting back into the original equation. If both sides are equal, the solution is correct.

一定要把答案代回原方程检验。如果两边相等，说明解是正确的。

6. Solving Simultaneous Equations by Elimination | 用消元法解联立方程

Simultaneous equations share two unknowns. Elimination works when the coefficients of one unknown match or can be made to match. Add or subtract the equations to remove that unknown.

联立方程含有两个未知数。当一个未知数的系数相同或可以化为相同时，可以使用消元法。将两个方程相加或相减，消去这个未知数。

3x = 9, so x = 3. Substitute into x - y = 2: 3 - y = 2, so y = 1. Solution: x = 3, y = 1.

例：2x + y = 7 和 x - y = 2。两式相加：3x = 9，所以 x = 3。代入 x - y = 2：3 - y = 2，所以 y = 1。解为：x = 3, y = 1。

If the coefficients are not matching, multiply one or both equations by a suitable number first. Then add or subtract to eliminate one variable.

如果系数不匹配，先给一个或两个方程乘上合适的数。然后再相加或相减，消去一个未知数。

7. Factorising Quadratic Expressions | 二次表达式的因式分解

umbers that multiply to c and add to b. These numbers go into the brackets (x + p)(x + q).

二次表达式形式为 ax² + bx + c。当 a = 1 时，找两个数 p 和 q，使它们的乘积等于 c，和等于 b。这两个数写进括号 (x + p)(x + q)。

Example: x² + 5x + 6 = (x + 2)(x + 3), because 2 × 3 = 6 and 2 + 3 = 5.

例：x² + 5x + 6 = (x + 2)(x + 3)，因为 2 × 3 = 6，2 + 3 = 5。

opposite signs. For x² - x - 12 = (x + 3)(x - 4), since 3 × (-4) = -12 and 3 + (-4) = -1.

当 c 为负数时，两个数异号。对于 x² - x - 12 = (x + 3)(x - 4)，因为 3 × (-4) = -12，

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