📚 Year 8 CCEA Maths: Formula and Theorem Quick Reference Handbook | Year 8 CCEA 数学：公式定理速查手册
Welcome to the Year 8 CCEA Maths Formula and Theorem Quick Reference Handbook. This bilingual guide consolidates all the essential rules, formulas, and theorems that you will encounter in your Year 8 mathematics course. Whether you are revising for an exam or completing homework, keep this handbook handy to quickly recall key concepts.
欢迎使用 Year 8 CCEA 数学公式定理速查手册。这本双语指南汇集了你在 Year 8 数学课程中会学到的所有重要法则、公式和定理。无论是备考还是完成作业，你都可以随时查阅这本手册，快速回顾核心要点。
1. Number and Operations | 整数与运算
Place Value: In the decimal system, each digit’s value is ten times the value of the digit to its right. For example, in the number 4,372, the digit 4 represents 4 thousands.
位值：在十进制系统中，每位数字的值是其右边一位的十倍。例如在数字 4,372 中，4 代表 4 个千。
Order of Operations (BIDMAS): Always work out Brackets first, then Indices (powers and roots), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).
运算顺序 (BIDMAS)：先算括号，再算指数（乘方和根），然后从左到右算除法和乘法，最后从左到右算加法和减法。
Negative Numbers: Adding a negative number is the same as subtracting its positive. Subtracting a negative number is the same as adding its positive. For multiplication and division: same signs give a positive result, different signs give a negative result.
负数：加上一个负数等于减去它的相反数。减去一个负数等于加上它的相反数。乘除法：同号得正，异号得负。
Factors and Multiples: A factor divides exactly into a number. A multiple is the product of a number and an integer. The highest common factor (HCF) and lowest common multiple (LCM) are used in fraction and ratio problems.
因数与倍数：因数是能整除某数的整数。倍数是某数与整数的乘积。最大公因数 (HCF) 和最小公倍数 (LCM) 常用于分数和比率问题。
Squares, Cubes and Roots: 5² = 25, 5³ = 125. √25 = 5, ∛125 = 5. Know square numbers up to 15² and cube numbers up to 5³.
平方、立方与根：5² = 25，5³ = 125。√25 = 5，∛125 = 5。需掌握平方数（至15²）和立方数（至5³）。
Rounding: To round a number, look at the next digit to the right. If it is 5 or more, round up; if it is less than 5, round down. You can round to decimal places or significant figures.
舍入：舍入时看右边一位数字。若为 5 或以上则进一，若小于 5 则舍去。可以舍入到指定位数的小数或有效数字。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Equivalent Fractions: Multiply or divide the numerator and denominator by the same non-zero number to get an equivalent fraction. Simplify fractions by dividing by the HCF.
等值分数：将分子和分母同乘或同除以一个非零整数即得等值分数。通过除以最大公因数来化简分数。
Adding and Subtracting Fractions: Find a common denominator. For ⅓ + ¼, use denominator 12: (4/12) + (3/12) = 7/12.
分数加减法：先求公分母。例如 ⅓ + ¼，用公分母 12：(4/12) + (3/12) = 7/12。
Multiplying and Dividing Fractions: Multiply across: a/b × c/d = (a×c)/(b×d). Divide by multiplying by the reciprocal: a/b ÷ c/d = a/b × d/c.
分数乘除法：乘法：a/b × c/d = (a×c)/(b×d)。除法：乘以倒数，a/b ÷ c/d = a/b × d/c。
Fractions ↔ Decimals ↔ Percentages: ½ = 0.5 = 50%. To convert fraction to decimal, divide numerator by denominator. To convert decimal to percentage, multiply by 100.
分数、小数、百分数互化：½ = 0.5 = 50%。分数化小数：分子除以分母。小数化百分数：乘以100。
Percentage of an Amount: Find 1% by dividing by 100, then multiply by the required percentage. E.g., 30% of 80 = (80 ÷ 100) × 30 = 24.
求一个数的百分数：先除以100得1%，再乘以所求百分比。例如：80的30% = (80 ÷ 100) × 30 = 24。
Percentage Increase/Decrease: Percentage change = (change ÷ original amount) × 100. For an increase of 20% on £50, new amount = 50 × 1.20 = £60.
百分比增减：百分比变化 = (变化量 ÷ 原量) × 100。例如 £50 增加20%，新值为 50 × 1.20 = £60。
3. Ratio and Proportion | 比率与比例
Writing Ratios: Ratios compare quantities in the same order. Simplify by dividing all parts by their HCF. 12:8 simplifies to 3:2.
写比：比率按相同次序比较数量。将各项除以最大公因数来化简。12:8 化简为 3:2。
Sharing in a Ratio: To share £48 in the ratio 3:5, add the parts (3+5=8). One part = £48 ÷ 8 = £6. Then the shares are 3×6 = £18 and 5×6 = £30.
按比例分配：将 £48 按 3:5 分配，份数和为 3+5=8。每份 = £48 ÷ 8 = £6。分配额为 3×6 = £18 和 5×6 = £30。
Equivalent Ratios: Multiply or divide both sides by the same number. 2:3 is equivalent to 6:9 (multiplied by 3).
等价比：两边乘以或除以相同的数。2:3 等于 6:9（乘以3）。
Direct Proportion: Two quantities are in direct proportion if their ratio is constant. If y ∝ x, then y = kx for some constant k.
正比例：两个量的比值为常数时它们成正比例。若 y ∝ x，则 y = kx（k为常数）。
Scale Drawings: A scale of 1 : 50 means that 1 cm on the drawing represents 50 cm in real life. Use proportions to find actual lengths.
比例图：比例尺 1:50 表示图上 1 cm 代表实际 50 cm。利用比例关系求实际长度。
4. Basic Algebra: Expressions and Equations | 代数基础：表达式与方程
Using Letters in Algebra: Letters stand for unknown numbers. 3a means 3 × a. Do not write the multiplication sign.
代数用字母表示数：字母代表未知数。3a 表示 3 × a。乘号省略不写。
Collecting Like Terms: Only terms with exactly the same letters can be added or subtracted. 2x + 3x = 5x, but 2x + 3y cannot be simplified.
合并同类项：只有字母部分完全相同的项才能相加减。2x + 3x = 5x，但 2x + 3y 不能化简。
Expanding Brackets: Multiply everything inside the bracket by the term outside. a(b + c) = ab + ac. E.g., 3(x + 4) = 3x + 12.
去括号：将括号外的项乘以括号内的每一项。a(b + c) = ab + ac。例如 3(x + 4) = 3x + 12。
Substituting into Formulas: Replace letters with given numbers, then calculate using BIDMAS. If v = u + 10t and u = 2, t = 3, then v = 2 + 10×3 = 32.
代入公式求值：将数字代入字母，再按 BIDMAS 运算。若 v = u + 10t，u = 2，t = 3，则 v = 2 + 10×3 = 32。
Solving One-step Equations: Use inverse operations to isolate the variable. x + 5 = 12 → x = 12 – 5 = 7. 3x = 15 → x = 15 ÷ 3 = 5.
解一步方程：使用逆运算把变量单独留在等号一侧。x + 5 = 12 → x = 12 – 5 = 7。3x = 15 → x = 15 ÷ 3 = 5。
Solving Two-step Equations: 2x + 3 = 11. First subtract 3: 2x = 8, then divide by 2: x = 4.
解两步方程：2x + 3 = 11。先减3：2x = 8，再除以2：x = 4。
5. Sequences and Patterns | 序列与规律
Term-to-term Rule: This describes how to move from one term to the next. For the sequence 3, 7, 11, 15, … the rule is ‘add 4 each time’.
项间递推规则：描述如何从一项移动到下一项。例如序列 3, 7, 11, 15, … 的规则是“每次加4”。
Position-to-term Rule (nth term): For an arithmetic sequence, Tₙ = a + (n – 1)d, where a is the first term, d is the common difference, and n is the position. For 3, 7, 11, 15, … Tₙ = 3 + (n – 1)×4 = 4n – 1.
位置项公式（第n项）：对于等差序列，Tₙ = a + (n – 1)d，其中 a 为首项，d 为公差，n 为项位置。例如 3, 7, 11, 15,… Tₙ = 3 + (n – 1)×4 = 4n – 1。
Finding a Specific Term: To find the 20th term, substitute n = 20 into the nth term formula: T₂₀ = 4×20 – 1 = 79.
求特定项：求第20项，将 n = 20 代入通项公式：T₂₀ = 4×20 – 1 = 79。
Triangular Numbers: 1, 3, 6, 10, 15… The nth triangular number is given by Tₙ = ½ n(n+1).
三角形数：1, 3, 6, 10, 15… 第n个三角形数为 Tₙ = ½ n(n+1)。
6. Geometry: Angles and Lines
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