📚 Year 8 CCEA Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 8 CCEA 进阶数学：公式定理速查手册
This quick reference handbook is designed to support Year 8 students following the CCEA Further Mathematics curriculum. It brings together the essential formulas, theorems, and key facts that you will use again and again throughout your studies. Keep this guide handy when revising, tackling homework, or preparing for assessments – it covers algebra, geometry, number, statistics, and beyond. Each section presents the core rules with brief explanations, followed by matching Chinese translations to help bilingual learners secure deep understanding.
本速查手册专为学习 CCEA 进阶数学课程的 Year 8 学生设计。它将你在学习过程中会反复使用的基本公式、定理和关键知识点集结在一起。复习、完成作业或备考时可以随时查阅——内容涵盖代数、几何、数论、统计等。每个部分都呈现核心规则并配以简要说明，并附有对应的中文翻译，帮助双语学习者牢固掌握知识。
1. Algebraic Expressions and Expansion | 代数式与展开
Algebraic terms are combined using the rules of arithmetic. Like terms have exactly the same variable part and can be added or subtracted. The distributive law allows us to expand brackets by multiplying the term outside by each term inside.
代数项按照算术规则进行合并。同类项具有完全相同的变量部分，可以相加或相减。分配律允许我们将括号展开，即用括号外的项分别乘以括号内的每一项。
Key rules:
核心规则：
erms: 3x + 5x = 8x; 7y − 2y = 5y. Unlike terms cannot be combined: 2a + 3b stays as it is.
同类项：3x + 5x = 8x；7y − 2y = 5y。不同类项不能合并：2a + 3b 保持不变。
Distributive property: a(b + c) = ab + ac, and (a + b)(c + d) = ac + ad + bc + bd.
分配律：a(b + c) = ab + ac，以及 (a + b)(c + d) = ac + ad + bc + bd。
Example: 2(x + 3) = 2x + 6; (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.
示例：2(x + 3) = 2x + 6；(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10。
2. Solving Linear Equations | 解一元一次方程
To solve an equation, we find the value of the unknown that makes the equality true. Isolate the variable by performing the same inverse operation on both sides of the equation. Check your solution by substituting it back into the original equation.
解方程时，我们找到使等式成立的未知数的值。通过在方程两边执行相同的逆运算来隔离变量。将解代入原方程检验可以验证结果。
General method: undo addition or subtraction first, then undo multiplication or division.
一般方法： 先处理加减，再处理乘除。
Example: 2x + 5 = 13 → 2x = 13 − 5 → 2x = 8 → x = 4.
示例：2x + 5 = 13 → 2x = 13 − 5 → 2x = 8 → x = 4。
If the equation contains brackets, expand first: 3(x − 2) = 9 → 3x − 6 = 9 → 3x = 15 → x = 5.
如果方程含有括号，先展开：3(x − 2) = 9 → 3x − 6 = 9 → 3x = 15 → x = 5。
r equations with variables on both sides, collect like terms on one side: 5x + 4 = 2x + 13 → 5x − 2x = 13 − 4 → 3x = 9 → x = 3.
对于变量在等号两边的方程，将同类项移到一边：5x + 4 = 2x + 13 → 5x − 2x = 13 − 4 → 3x = 9 → x = 3。
3. Sequences and Finding the nth Term | 数列与求第 n 项
A sequence is a list of numbers that follow a rule. An arithmetic sequence has a constant difference between consecutive terms. The nth term formula allows you to find any term in the sequence directly, without listing all preceding terms.
数列是一组按照某种规律排列的数字。等差数列的相邻两项之差是常数。第 n 项公式可以让你直接求出数列中的任意一项，而无需列出前面的所有项。
ence, nth term = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference.
对于等差数列，第 n 项 = a₁ + (n − 1)d，其中 a₁ 是首项，d 是公差。
Example: sequence 5, 9, 13, 17, … a₁ = 5, d = 4, so nth term = 5 + (n − 1)×4 = 4n + 1.
示例：数列 5, 9, 13, 17, … a₁ = 5，d = 4，因此第 n 项 = 5 + (n − 1)×4 = 4n + 1。
If the difference is not constant, check second differences. For quadratic sequences the nth term will involve n².
如果差不是常数，检查二次差。二次数列的第 n 项将包含 n²。
4. Powers and Roots | 幂与根
Powers (indices) show repeated multiplication. The small, raised number is the exponent. Roots are the inverse of powers. Understanding index laws helps simplify expressions efficiently.
幂表示重复相乘。上方的小数字是指数。根是幂的逆运算。理解指数法则有助于高效地化简表达式。
Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
指数法则： aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。
Example: 2³ × 2² = 2⁵ = 32; (x⁴)³ = x¹²; 5⁻² = 1/25.
示例：2³ × 2² = 2⁵ = 32；(x⁴)³ = x¹²；5⁻² = 1/25。
Square roots and cube roots: √a × √a = a; ∛a × ∛a × ∛ re root of a number is a value that, when multiplied by itself, gives the original number.
平方根与立方根：√a × √a = a；∛a × ∛a × ∛a = a。一个数的平方根是这样一个值，它与自身相乘得到原数。
5. Angle Facts and Parallel Lines | 角的基本事实与平行线
Angles are measured in degrees (°). When describing relationships, remember these essential angle facts. They form the foundation for solving geometric problems.
角度以度 (°) 为单位进行度量。在描述关系时，请记住这些基本的角的事实。它们是解决几何问题的基础。
Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
平角（直线上的角）之和为 180°。周角（绕一点的角）之和为 360°。对顶角相等。
When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, interior (co-interior) angles sum to 180°.
当一条截线与两条平行线相交时：同位角相等，内错角相等，同旁内角互补（和为 180°）。
Example: If two parallel lines are cut by a transversal and one alternate angle is 70°, the other alternate angle is also 70°, and the interior angle on the same side is 110°.
示例：如果两条平行线被一条截线所截，其中一个内错角为 70°，则另一个内错角也为 70°，而同旁内角为 110°。
6. Triangles and Quadrilaterals | 三角形与四边形
The sum of interior angles in any triangle is 180°. The sum of interior angles in any quadrilateral is 360°. Knowing these sums helps calculate missing angles.
任何三角形的内角和都是 180°。任何四边形的内角和都是 360°。知道这些和有助于计算缺失的角。
Triangles can be classified by sides: equilateral (3 equal sides, each angle 60°), isosceles (2 equal sides, base angles equal), scalene (no equal sides). By angles: acute, right, obtuse.
三角形可以按边分类：等边三角形（三条边相等，每个角 60°），等腰三角形（两条边相等，底角相等），不等边三角形（无边相等）。按角分类：锐角三角形，直角三角形，钝角三角形。
Quadrilaterals: square, rectangle, parallelogram, rhombus, trapezium, kite. For a parallelogram, opposite sides are equal and parallel, opposite angles are equal.
四边形：正方形，矩形，平行四边形，菱形，梯形，筝形。对于平行四边形，对边相等且平行，对角相等。
Exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形的外角等于与其不相邻的两个内角之和。
7. Perimeter, Area, and Volume | 周长、面积与体积
These measure the boundary length, surface covered, and space occupied by shapes. Use the correct formula for each shape and always include appropriate units.
这些分别度量边界的长度、覆盖的表面以及形状占据的空间。对每种形状使用正确的公式，并且始终附上合适的单位。
Rectangle: perimeter P = 2(l + w), area A = l × w.
矩形：周长 P = 2(l + w)，面积 A = l × w。
Triangle: area A = ½ × b × h.
三角形：面积 A = ½ × b × h。
Circle: circumference C = πd or C = 2πr; area A = πr². (π ≈ 3.14 or use calculator)
圆：周长 C = πd 或 C = 2πr；面积 A = πr²。（π ≈ 3.14 或使用计算器）
Cuboid: volume V = l × w × h, surface area = 2(lw + wh + lh).
长方体：体积 V = l × w × h，表面积 = 2(lw + wh + lh)。
Prism: volume V = area of cross-section × length.
棱柱：体积 V = 横截面积 × 长度。
8. Pythagoras’ Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is fundamental for finding missing lengths.
在直角三角形中，斜边（直角所对的边）的平方等于另外两条边的平方和。这一定理是求缺失边长的基本工具。
a² + b² = c², where c is the hypotenuse.
a² + b² = c²，其中 c 是斜边。
Example: If a = 3 cm, b = 4 cm, then c = √3² + 4² = √9+16 = √25 = 5 cm.
示例：如果 a = 3 cm，b = 4 cm，则 c = √3² + 4² = √9+16 = √25 = 5 cm。
To find a shorter side, rearrange: a² = c² − b², so a = √c² − b².
求直角边时，移项：a² = c² − b²，因此 a = √c² − b²。
sed to check if a triangle is right-angled: if a² + b² = c², the triangle is right-angled.
勾股定理也可用于检验三角形是否为直角三角形：若 a² + b² = c²，则该三角形是直角三角形。
9. Probability: Events and Expected Outcomes | 概率：事件与期望次数
Probability measures how likely an event is to happen. It is always a number between 0 (impossible) and 1 (certain), and can be expressed as a fraction, decimal, or percentage.
概率衡量一个事件发生的可能性。它总是一个介于 0（不可能）和 1（必然发生）之间的数，可以用分数、小数或百分数表示。
P(Event) = Number of favourable outcomes / Total number of equally likely outcomes
P(事件) = 有利结果的数量 / 等可能结果的总数
ided die, P(rolling a 4) = 1/6. The probability of an event not happening is 1 − P(event).
对于一个公平的六面骰子，P(掷出 4) = 1/6。事件不发生的概率为 1 − P(事件)。
ber of trials. If you flip a coin 100 times, the expected number of heads is ½ × 100 = 50.
期望频数 = 概率 × 试验次数。若抛硬币 100 次，正面的期望次数为 ½ × 100 = 50。
Sample space diagrams list all possible outcomes; they help when two events occur together (e.g., rolling two dice).
样本空间图列出所有可能的结果；在处理两个事件同时发生（例如掷两个骰子）时非常有帮助。
10. Ratio and Direct Proportion | 比与正比例
Ratios compare the sizes of two or more quantities. They can be simplified like fractions by dividing all parts by a common factor. To divide a quantity in a given ratio, find the total number of parts, then share accordingly.
比用来比较两个或多个量的大小。比可以像分数一样，将所有项除以一个公因数来化简。按给定比分配某个量时，先求出总份数，然后再按比例分配。
Simplify 12:18 by dividing by 6 → 2:3. To divide £60 in the ratio 2:3, total parts = 5, each part = £12, so amounts are £24 and £36.
化简 12:18，除以 6 得到 2:3。按 2:3 的比例分 60 英镑，总份数 = 5，每份 = 12 英镑，因此各得 24 英镑和 36 英镑。
ate; their ratio is constant. If y is directly proportional to x, then y/x = k, or y = kx.
如果两个量以相同的速率增加或减少，则它们成正比例；它们的比值恒定。若 y 与 x 成正比，则 y/x = k，或 y = kx。
itres of fuel. Miles per litre is 30/2 = 15, so for 5 litres it travels 5 × 15 = 75 miles.
示例：一辆汽车用 2 升燃油行驶 30 英里。每升英里数为 30/2 = 15，因此用 5 升可行驶 5 × 15 = 75 英里。
11. Percentages: Conversions and Calculations | 百分比：转换与计算
Percent means ‘out of 100’. Converting between fractions, decimals, and percentages is a key skill. Percentage change problems are common in real-life contexts.
百分数意思是“每一百”。在分数、小数和百分数之间进行转换是一项关键技能。百分比变化问题在现实生活中非常常见。
To convert a percentage to a decimal, divide by 100: 45% = 0.45. To convert a decimal to a percentage, multiply by 100: 0.7 = 70%.
将百分数转换为小数，除以 100：45% = 0.45。将小数转换为百分数，乘以 100：0.7 = 70%。
Finding a percentage of an amount: a% of B = (a/100) × B. Example: 20% of 80 = (20/100)×80 = 16.
求一个数量的百分之几：a% of B = (a/100) × B。示例：80 的 20% = (20/100)×80 = 16。
× (1 + percentage/100). Percentage decrease: new value = original × (1 − percentage/100).
百分比增加：新值 = 原值 × (1 + 百分比/100)。百分比减少：新值 = 原值 × (1 − 百分比/100)。
nt before a percentage change, reverse the process: original = new ÷ (1 ± percentage/100).
求百分比变化前的原值，逆向计算：原值 = 新值 ÷ (1 ± 百分比/100)。
12. Statistics: Mean, Median, Mode, and Range | 统计：平均数、中位数、众数和极差
These summary statistics help describe a data set. Always order the data first when finding the median. The mode is the most frequent value, while the range shows the spread.
这些概括统计量有助于描述一组数据。在求中位数时，总是先将数据排序。众数是出现最频繁的数值，而极差则显示数据的分布范围。
all values ÷ number of values. Example: data 3, 7, 8, 2, 5 → sum = 25, mean = 25 ÷ 5 = 5.
平均数 = 所有数值之和 ÷ 数值的个数。示例：数据 3, 7, 8, 2, 5 → 和 = 25，平均数 = 25 ÷ 5 = 5。
numbers. Data ordered: 2, 3, 5, 7, 8 → median is 5. For 2, 3, 5, 7 → median = (3+5)/2 = 4.
中位数 = 排序后中间的值。如果数值的个数为偶数，中位数是中间两个数的平均数。排序后：2, 3, 5, 7, 8 → 中位数为 5。若为 2, 3, 5, 7 → 中位数 = (3+5)/2 = 4。
data set can have one mode, more than one mode (bimodal), or no mode if all values appear once.
众数 = 出现频次最高的数值。一组数据可以有一个众数、多个众数（双峰），或者如果所有数值都只出现一次则无众数。
Range = highest value − lowest value. It measures how spread out the data are.
极差 = 最大值 − 最小值。它度量数据的离散程度。
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