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

📚 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 comprehensive quick reference handbook covers essential formulas and theorems for the Year 8 CCEA Further Mathematics curriculum. It serves as a rapid revision tool designed to help you memorise and apply key mathematical principles efficiently. All rules and results are presented in a clear, structured format to aid last-minute study and long-term retention.

这本全面的速查手册涵盖了 Year 8 CCEA 进阶数学课程中的重要公式和定理。它是一个快速复习工具，旨在帮助你高效记忆并应用关键的数学原理。所有法则和结论都以清晰、结构化的形式呈现，便于考前冲刺和长期巩固。

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1. Number Operations & Properties | 数的运算与性质

c) = a × b + a × c. These properties hold for addition and multiplication of real numbers.

交换律：a + b = b + a 且 a × b = b × a。结合律：(a + b) + c = a + (b + c) 且 (a × b) × c = a × (b × c)。分配律：a × (b + c) = a × b + a × c。这些性质适用于实数的加法和乘法。

ves a positive: (-a) × (-b) = a × b. The product of a positive and a negative is negative.

负数运算法则：加上一个负数等于减去其相反数，a + (-b) = a - b。两个负数相乘或相除得正，(-a) × (-b) = a × b。正数与负数乘积为负。

A prime number has exactly two distinct factors, 1 and itself. The Highest Common Factor (HCF) is the largest factor shared by two or more numbers. The Lowest Common Multiple (LCM) is the smallest multiple common to them. For any two numbers a and b, the product of HCF and LCM equals the product of the numbers.

质数只有两个不同的因数，即 1 和它本身。最大公因数 (HCF) 是两个或多个数共有的最大因数。最小公倍数 (LCM) 是它们共有的最小倍数。对于任意两个数 a 和 b，有公式：

HCF(a, b) × LCM(a, b) = a × b

Factors are numbers that divide exactly into another number. Multiples are the results of multiplying a number by integers. Prime factorisation breaks a number into a product of primes.

因数是能整除另一个数的数。倍数是某数乘以整数得到的结果。质因数分解是将一个数写成质数的乘积。

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2. Fractions, Decimals & Percentages | 分数、小数和百分数

s: a/b × c/d = ac/bd. To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.

分数加减法须先通分：a/b + c/d = (ad + bc)/bd。乘法分子分母分别相乘：a/b × c/d = ac/bd。除以一个分数等于乘它的倒数：a/b ÷ c/d = a/b × d/c。

A decimal can be converted to a fraction by writing it over a power of 10 and simplifying. To turn a fraction into a decimal, divide the numerator by the denominator. To express a quantity as a percentage, multiply the fraction by 100%.

小数转分数可将小数写成分母为 10 的幂的分数再化简。分数转小数用分子除以分母。求百分数将分数乘以 100%。

rease: new value = original value × (1 ± percentage as a decimal). For example, increasing £50 by 20% gives £50 × 1.2 = £60. Finding a percentage of an amount: a% of b = (a/100) × b.

百分数增减关键公式：新值 = 原值 × (1 ± 百分数化为小数)。例如 £50 增加 20% 得 £50 × 1.2 = £60。求一个数的百分数：a% of b = (a/100) × b。

Percentage change = (difference / original) × 100%

Repeated percentage changes apply multipliers successively. Reverse percentages require dividing by the multiplier to find the original amount.

连续百分数变化依次乘以乘数。逆向百分数需除以乘数得到原值。

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3. Powers & Roots | 幂与方根

The laws of indices for any non-zero base a and integers m, n: a^m × aⁿ = a^m+n; a^m ÷ aⁿ = a^m-n; (a^m)ⁿ = a^mn; a⁰ = 1; and a^-n = 1/(aⁿ).

对于任意非零底数 a 及整数 m, n，指数法则为：a^m × aⁿ = a^m+n；a^m ÷ aⁿ = a^m-n；(a^m)ⁿ = a^mn；a⁰ = 1；以及 a^-n = 1/(aⁿ)。

a^m × aⁿ = a^m+n    |    a^m ÷ aⁿ = a^m-n

The square root of a number x is a value that, when multiplied by itself, gives x: √x · √x = x. The cube root is ∛ er: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144.

一个数 x 的平方根是自乘等于 x 的值：√x · √x = x。立方根记为 ∛x。需熟记的常见平方数：1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144。

Square roots of non-perfect squares can be estimated between two integers. For example, √50 lies between 7 and 8 because 7²=49 and 8²=64. Fractional indices represent roots: a^1/2 = √a, a^1/3 = ∛a.

非完全平方数的平方根可估算在两个整数之间，如 √50 介于 7 和 8 之间。分数指数表示根式：a^1/2 = √a，a^1/3 = ∛a。

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4. Algebraic Expressions & Simplifying | 代数表达式与化简

icients: 3x + 5x = 8x, and 2ab + 4ab = 6ab. Unlike terms cannot be simplified in this way.

同类项含有完全相同的变量且次数相同。可合并系数进行加减：3x + 5x = 8x，2ab + 4ab = 6ab。不同类项不能这样化简。

c. For two binomials, use the distributive law twice: (x + a)(x + b) = x² + (a + b)x + ab.

展开单项式括号，将外部项乘以内部每一项：a(b + c) = ab + ac。两个二项式相乘，连续使用分配律：(x + a)(x + b) = x² + (a + b)x + ab。

(x + a)(x + b) = x² + (a + b)x + ab

r: ax + ay = a(x + y). A difference of two squares factorises as a² - b² = (a + b)(a - b).

因式分解是展开的逆运算。先找最高公因式：ax + ay = a(x + y)。平方差公式为 a² - b² = (a + b)(a - b)。

Substituting values into an expression means replacing each variable with a given number and evaluating using BIDMAS/BODMAS. Always write the substituted expression clearly to avoid sign errors.

代入求值即用给定数字替换每个变量，并按照运算顺序 (BIDMAS/BODMAS) 计算。务必清楚写出替换后的表达式，避免符号错误。

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5. Solving Linear Equations | 解一元一次方程

l is to isolate the variable on one side. Perform the same operation on both sides to maintain balance.

一元一次方程可写成 ax + b = c 的形式。目标是使变量单独在等式一边。在等式两边同时进行相同运算以保持平衡。

x = (c - b)/a. Always check your solution by substituting back into the original equation.

步骤 1：两边同时减去 b，得 ax = c - b。步骤 2：两边除以 a (a ≠ 0)，得 x = (c - b)/a。务必将解代回原方程验算。

Equations may have brackets or variables on both sides. Expand brackets first, then collect like terms, and finally isolate the variable. Equations with fractions can be cleared by multiplying every term by the lowest common denominator.

方程可能含有括号或变量在两边。先去括号，再移项合并同类项，最后解出变量。含分数的方程可每项乘以最小公分母化简。

d, then ax - cx = d - b, giving (a - c)x = d - b, so x = (d - b)/(a - c), provided a ≠ c.

如果 ax + b = cx + d，则 ax - cx = d - b，得 (a - c)x = d - b，所以当 a ≠ c 时，x = (d - b)/(a - c)。

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6. Sequences & Patterns | 数列与规律

A sequence is a list of numbers following a rule. In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference d. The nth term formula allows you to find any term directly.

数列是按规律排列的一列数。在等差数列中，相邻项的差恒定，称为公差 d。通项公式能直接求出任意一项。

nth term of an arithmetic sequence: u_n = a + (n - 1)d

re a is the first term and d is the common difference. For example, sequence 3, 7, 11, 15, … has a = 3, d = 4. The 10th term is 3 + 9×4 = 39.

其中 a 为首项，d 为公差。例如数列 3, 7, 11, 15, … 中 a = 3, d = 4。第 10 项为 3 + 9×4 = 39。

Other sequences may follow geometric patterns, square numbers (1, 4, 9, 16, …), triangular numbers (1, 3, 6, 10, …), or Fibonacci-type rules where each term is the sum of the two preceding terms. Identify the pattern by looking at differences or ratios.

其他数列可能遵循几何模式、平方数 (1, 4, 9, 16, …)、三角形数 (1, 3, 6, 10, …) 或斐波那契类规则 (每一项是前两项之和)。通过观察差分或比值来识别规律。

To find a missing term or the rule, you can set up an equation using the nth term expression and solve for n. Always test your nth term rule on the first few terms.

求缺失项或找规律时，可利用通项表达式建立方程并解 n。通项公式必须用前几项检验。

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7. Ratio, Proportion & Rates | 比、比例和比率

mple, 8:12 simplifies to 2:3. Order is important: a:b is not the same as b:a unless a = b.

比用来比较两个或多个量。可同时除以最大公因数化简，如 8:12 简化为 2:3。顺序很重要：a:b 与 b:a 不同，除非 a = b。

tity in a given ratio, add the parts, then multiply the total by each fraction. If sharing £60 in the ratio 3:5, total parts = 8, first share = (3/8)×£60 = £22.50, second share = (5/8)×£60 = £37.50.

按比例分配时，先求总份数，再用总数量分别乘以各份分数。将 £60 按 3:5 分配，总份数 8，第一份为 (3/8)×£60 = £22.50，第二份为 (5/8)×£60 = £37.50。

. The constant k = y/x. Graphs of direct proportion are straight lines through the origin.

正比例指两变量以相同比率变化。如果 y 与 x 成正比，则 y = kx (k 为常数)。常数 k = y/x。正比例图像是过原点的直线。

ing are examples of ratios involving different units. Always include units when working with rates.

速率如速度 = 路程/时间、密度 = 质量/体积和单价都是涉及不同单位的比率。处理速率问题时务必要带单位。

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8. Geometry: Angles & Lines | 几何：角与线

Angle facts: angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal. A right angle is 90°.

角的性质：直线上的邻角之和为 180°。一点周围的角度和为 360°。对顶角相等。直角为 90°。

When a transversal crosses parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles add up to 180°. Recognise these in F, Z and C patterns.

当一直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°)。可通过 F 形、Z 形和 C 形来识别。

The angle sum of a triangle is 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles. In an isosceles triangle, base angles are equal. An equilateral triangle has three 60° angles.

三角形内角和为 180°。三角形的一个外角等于两个不相邻内角之和。等腰三角形两底角相等。等边三角形三个角均为 60°。

Sum of interior angles of an n-sided polygon = (n - 2) × 180°

xterior angle = 360° / n. The sum of exterior angles of any convex polygon is always 360°.

对于正多边形，每个内角 = (n - 2) × 180° / n，每个外角 = 360° / n。任何凸多边形的外角和均为 360°。

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9. Perimeter, Area & Volume | 周长、面积和体积

Perimeter is the total distance around a 2D shape. Area is measured in square units. Use the correct formula for each shape:

周长是二维图形一周的总长度。面积以平方单位度量。须根据不同图形使用正确公式：

Shape	Perimeter	Area
Rectangle	P = 2(l + w)	A = l × w
Triangle	Sum of all sides	A = ½ × b × h
Parallelogram	Sum of all sides	A = b × h
Trapezium	Sum of all sides	A = ½(a + b)h
Circle	C = 2πr or πd	A = πr²

s³. Volume is measured in cubic units, and capacity often in litres (1 litre = 1000 cm³).

常见棱柱的体积：体积 = 横截面积 × 高。长方体 V = l × w × h。正方体 V = s³。体积以立方单位度量，容积常用升表示 (1 升 = 1000 cm³)。

Composite shapes can be split into simpler shapes, their areas or volumes calculated separately and then summed. Always write the correct unit in your final answer.

组合图形可分割为简单图形，分别计算面积或体积，再相加。最终答案务必注明正确单位。

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10. Pythagoras' Theorem | 勾股定理

Pythagoras' theorem applies to right-angled triangles only. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

勾股定理仅适用于直角三角形。它表明斜边 (直角对边) 的平方等于另外两边的平方和。

a² + b² = c²

tenuse, and a and b are the lengths of the other two legs. To find the hypotenuse, use c = √a² + b². To find a shorter side, rearrange: a = √c² - b².

其中 c 为斜边长，a 和 b 为另外两直角边的长度。求斜边：c = √a² + b²。求直角边：a = √c² - b²。

Pythagorean triples are sets of three integers that satisfy the theorem, such as (3, 4, 5), (5, 12, 13) and (8, 15, 17). Multiples of these triples also form right-angled triangles.

勾股数是满足定理的三元整数，如 (3, 4, 5)、(5, 12, 13) 和 (8, 15, 17)。这些勾股数的倍数也能构成直角三角形。

Apply the theorem to solve problems involving distance in coordinate grids, rectangles split into right triangles, and real-world contexts such as ladders leaning against walls. Always identify the hypotenuse first.

应用该定理可解决坐标网格中的距离问题、分割矩形的直角三角形问题以及梯子靠墙等实际问题。解题时务必先确定斜边。

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11. Statistics: Averages & Charts | 统计：平均数与图表

n is calculated by summing all values and dividing by the number of values. Mean = Σx / n.

三种主要的平均数:均值、中位数和众数。均值为所有数值之和除以个数：Mean = Σx / n。

The median is the middle value when data is ordered. For an even number of data points, it is the mean of the two middle values. The mode is the value that appears most frequently. A data set can have no mode, one mode, or more than one mode.

中位数是排序后位于中间的值；当数据个数为偶数时，取中间两数的均值。众数是出现频率最高的值；数据集可能没有众数，也可能有一个或多个众数。

Range = maximum value - minimum value

elationships. In a pie chart, the angle for each category = (frequency / total frequency) × 360°.

统计图表包括离散数据的条形图、表示时间趋势的折线图、表示比例的饼图以及展现关系的散点图。饼图中，每个扇形的角度 = (频数 / 总数) × 360°。

When analysing data, compare averages and spread (range). A smaller range indicates more consistent data. Always read charts carefully, checking scales and labels.

分析数据时须比较平均数和离散程度 (极差)。极差越小表示数据越一致。阅读图表时务必仔细查看刻度和标签。

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12. Probability | 概率

Probability measures the chance of an event occurring. It can be expressed as a fraction, decimal or percentage. The probability scale ranges from 0 (impossible) to 1 (

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