📚 Year 9 Essential Maths Book 9i: Core Skills | Year 9 基础数学核心技能
This revision guide reviews the key topics in the Year 9 Essential Maths course, with a focus on the skills students need for confident problem solving. Each section includes a short explanation and a worked example.
本复习指南回顾 Year 9 基础数学课程的关键主题，重点放在学生解题所需的核心技能。每节都包含简短说明和解题示例。
1. Number Skills and BIDMAS | 数的运算与 BIDMAS 规则
Always apply operations in the correct order: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
始终按照正确顺序进行运算：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。
Example: 3 + 4 × 2 = 3 + 8 = 11, not 14.
示例：3 + 4 × 2 = 3 + 8 = 11，而不是 14。
Negative numbers follow clear rules: adding a negative is subtraction; subtracting a negative is addition; multiplying or dividing two negatives gives a positive.
负数遵循明确规则：加上负数等于减法；减去负数等于加法；两个负数相乘或相除得正数。
Worked example: (-2) × (-3) + 5 = 6 + 5 = 11.
解答示例：(-2) × (-3) + 5 = 6 + 5 = 11。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.
将分数转换为小数，用分子除以分母；将小数转换为百分数，乘以 100。
Example: 3/4 = 0.75 = 75%. In reverse, 45% = 45/100 = 9/20.
示例：3/4 = 0.75 = 75%。反过来，45% = 45/100 = 9/20。
When adding or subtracting fractions, find a common denominator first. For multiplication, multiply the numerators and denominators separately.
分数加减时，先找到公分母；分数相乘时，分子乘分子，分母乘分母。
Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
示例：2/3 + 1/4 = 8/12 + 3/12 = 11/12。
3. Ratio and Proportion | 比与比例
A ratio compares parts of a whole. To share an amount in a ratio, add the parts to find the total number of shares, then divide the amount by this total.
比用于比较整体的各部分。按比例分配数量时，先将各部分相加得到总份数，再用总量除以总份数。
Example: Divide £120 in the ratio 3:5. Total parts = 8, one part = £15, so the shares are £45 and £75.
示例：将 £120 按 3:5 分配。总份数 = 8，每份 = £15，因此份额为 £45 和 £75。
Proportion problems often involve scaling up or down. If 3 pens cost £2.10, one pen costs £0.70 and 5 pens cost £3.50.
比例问题常涉及放大或缩小。如果 3 支笔售价 £2.10，一支笔为 £0.70，5 支笔为 £3.50。
4. Algebra: Simplifying and Expanding | 代数：化简与展开
Collect like terms by combining terms that have the same variable and power. Constants combine separately.
合并同类项时，将具有相同变量和幂的项合并，常数项单独合并。
Example: 4a + 3b - 2a + 5b simplifies to 2a + 8b.
示例：4a + 3b - 2a + 5b 化简为 2a + 8b。
tive law: a(b + c) = ab + ac. Double brackets follow: (x + a)(x + b) = x² + (a + b)x + ab.
用分配律展开括号：a(b + c) = ab + ac。双括号展开遵循：(x + a)(x + b) = x² + (a + b)x + ab。
Example: 3(x + 4) = 3x + 12; (x + 2)(x + 5) = x² + 7x + 10.
示例：3(x + 4) = 3x + 12；(x + 2)(x + 5) = x² + 7x + 10。
5. Solving Linear Equations | 解一元一次方程
Use inverse operations to isolate the unknown. Always keep the equation balanced by doing the same operation to both sides.
使用逆运算求未知数。始终对方程两边进行相同操作以保持平衡。
Example: 2x + 5 = 13. Subtract 5: 2x = 8. Divide by 2: x = 4.
示例：2x + 5 = 13。两边减 5：2x = 8。除以 2：x = 4。
appears on both sides, collect like terms first. 5x - 3 = 2x + 6 becomes 3x = 9, so x = 3.
当未知数出现在方程两边时，先合并同类项。5x - 3 = 2x + 6 变为 3x = 9，所以 x = 3。
6. Sequences and the nth Term | 数列与第 n 项
An arithmetic sequence has a constant difference between consecutive terms. The nth term can be written as an + b, where a is the common difference.
等差数列的相邻项之差为常数。第 n 项可以写成 an + b，其中 a 是公差。
, 10, 13 has common difference 3. The nth term is 3n + 1 because when n = 1, 3(1) + 1 = 4.
示例：数列 4, 7, 10, 13 的公差为 3。第 n 项为 3n + 1，因为当 n = 1 时，3(1) + 1 = 4。
Use the nth term to find any term, such as the 20th term: 3(20) + 1 = 61.
利用第 n 项可求任意项，如第 20 项：3(20) + 1 = 61。
7. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°, angles around a point sum to 360°, and the three angles in a triangle sum to 180°.
直线上的角之和为 180°，围绕一点的角之和为 360°，三角形内角和为 180°。
(n - 2) × 180°. For example, a pentagon has 5 sides, so its interior angles sum to (5 - 2)
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