Year 8 Edexcel Maths: Quick Reference Handbook of Formulas and Theorems | Year 8 Edexcel 数学：公式定理速查手册

📚 Year 8 Edexcel Maths: Quick Reference Handbook of Formulas and Theorems | Year 8 Edexcel 数学：公式定理速查手册

This quick reference handbook brings together all the essential formulas and theorems you need to master Year 8 Edexcel Mathematics. From prime factorisation and fraction conversions to solving equations, understanding angles, and calculating areas and volumes, each section is presented with clear explanations and paired bilingual content to reinforce your learning. Keep this guide handy for homework, revision, and exam preparation.

这本速查手册汇集了你掌握 Year 8 Edexcel 数学所需的所有基本公式和定理。从质因数分解、分数转换到解方程、理解角度、计算面积和体积，每个部分都配以清晰的解释和中英双语对照，以强化你的学习。将这本指南放在手边，用于作业、复习和备考。

1. Number Properties: Factors, Multiples, Primes | 数的性质：因数、倍数、质数

A prime number has exactly two distinct factors: 1 and itself. Composite numbers can be written as a unique product of prime factors; this is called prime factorisation. Use a factor tree to break a number down, then write the result using index (power) notation.

质数恰好有两个不同的因数：1 和它本身。合数可以唯一地写成质因数的乘积，这称为质因数分解。使用因子树将数字拆解，然后用指数（幂）记号写出结果。

Example: 60 = 2² × 3 × 5

示例：60 = 2² × 3 × 5

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides into them exactly. The Lowest Common Multiple (LCM) is the smallest number that is a multiple of each of them. A powerful relationship connects them:

两个或两个以上数字的最大公因数（HCF）是能整除它们的最大数。最小公倍数（LCM）是它们每一个倍数的最小公倍数。一个强有力的关系将它们联系起来：

a × b = HCF(a, b) × LCM(a, b)

Square numbers arise from multiplying a number by itself (1² = 1, 2² = 4, 3² = 9, …). Cube numbers come from multiplying three identical factors (2³ = 8). The square root √ returns the original factor; likewise the cube root ³√ undoes cubing.

平方数由数字自乘得出（1² = 1, 2² = 4, 3² = 9, …）。立方数来自三个相同因数的乘积（2³ = 8）。平方根 √ 返回原来的因数；类似地，立方根 ³√ 撤销立方的效果。

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

To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100. Percentages become fractions by writing them over 100 and simplifying.

将分数转换为小数，用分子除以分母。将小数转换为百分比，乘以 100。将百分比写成分母为 100 的分数并化简，即可得到分数形式。

3/8 = 0.375 → 0.375 × 100 = 37.5%

When finding a percentage of an amount, multiply the amount by the percentage as a decimal. For percentage increase or decrease, use the multiplier: 1 + r/100 for increase, 1 − r/100 for decrease.

求一个数的某百分比时，用该数乘以百分比的小数形式。对于百分比的增加或减少，使用乘数：增加用 1 + r/100，减少用 1 − r/100。

New value = Original × (1 ± r/100)

新值 = 原值 × (1 ± r/100)

To compare fractions, decimals and percentages, convert them all to the same form. Ordering them correctly is a key skill tested in exams.

比较分数、小数和百分比时，将它们全部转换为同一种形式。正确地排序它们是考试中测试的关键技能。

3. Ratio and Proportion | 比率和比例

A ratio compares two or more quantities. It can be simplified by dividing all parts by their highest common factor. When sharing an amount in a given ratio, work out the total number of parts first, then find the value of one part.

比率比较两个或更多的数量。可以通过将各部分除以它们的最大公因数来化简比率。当按给定比率分配数量时，先计算出总份数，再求出一份的值。

Share £60 in ratio 3 : 2 → Total parts = 5, 1 part = £12, shares = £36 and £24

按 3 : 2 分配 60 英镑 → 总份数 = 5，一份 = 12 英镑，分配为 36 英镑和 24 英镑

Proportion describes a multiplicative relationship. Two quantities are in direct proportion if they increase or decrease at the same rate, written as y ∝ x. This gives the formula y = kx, where k is the constant of proportionality.

比例描述一种乘法关系。如果两个量以相同的速率增加或减少，则它们成直接比例，记作 y ∝ x。由此得出公式 y = kx，其中 k 是比例常数。

y ∝ x ⇔ y = kx

If y = 12 when x = 4, then k = 3, so the equation is y = 3x.

如果 x = 4 时 y = 12，则 k = 3，因此方程为 y = 3x。

4. Algebraic Expressions: Simplifying and Expanding | 代数表达式：化简与展开

Like terms contain exactly the same variable letters and powers. To simplify, add or subtract the coefficients. Expanding means multiplying out brackets using the distributive law.

同类项包含完全相同的变量字母和指数。化简时，将系数相加或相减。展开就是利用分配律将括号乘开。

a(b + c) = ab + ac

Always watch out for negative signs: a(b − c) = ab − ac. Factorising is the reverse process — taking a common factor out of the expression, so ab + ac can be written as a(b + c).

要时刻注意负号：a(b − c) = ab − ac。因式分解是逆过程——将公因子提到表达式外面，所以 ab + ac 可以写成 a(b + c)。

When multiplying two binomials, use the FOIL method (First, Outer, Inner, Last) to ensure every term is paired correctly:

当两个二项式相乘时，使用 FOIL 方法（首、外、内、尾），确保每一项都正确配对：

(x + a)(x + b) = x² + (a + b)x + ab

This formula is extremely useful for quadratic expansions and will appear again later.

这个公式对二次式展开非常有用，今后还会反复出现。

5. Solving Linear Equations | 解线性方程

An equation states that two expressions are equal. The goal is to isolate the variable on one side, doing the same operation to both sides to keep the balance.

方程表明两个表达式相等。目标是把变量单独留在一侧，对等式两边进行相同的运算以保持平衡。

Use inverse operations: addition ↔ subtraction, multiplication ↔ division. For two‑step equations, undo the constant term first, then the coefficient of x.

使用逆运算：加法 ↔ 减法，乘法 ↔ 除法。对于两步方程，先消去常数项，再处理 x 的系数。

2x + 3 = 7 → 2x = 4 → x = 2

If the variable appears on both sides, collect all x‑terms on one side and numbers on the other. Always check your solution by substituting it back into the original equation.

如果变量出现在两侧，将所有含 x 的项移至一侧，数字移至另一侧。永远记得将解代回原方程进行验证。

6. Sequences: nth Term | 数列：第 n 项

A linear sequence increases or decreases by a constant difference, d. The position‑to‑term rule (nth term) allows you to find any term without listing the whole sequence.

线性数列以固定的差值 d 递增或递减。位置–项的规则（第 n 项）让你无需列出整个数列就能求出任一项。

uₙ = a + (n − 1)d

Here a is the first term, d is the common difference. For the sequence 5, 8, 11, 14, … the difference is 3, so uₙ = 5 + (n − 1)×3 = 3n + 2.

这里 a 是首项，d 是公差。对于数列 5, 8, 11, 14, …，差值为 3，因此 uₙ = 5 + (n − 1)×3 = 3n + 2。

To identify the nth term from a sequence, subtract the common difference from the first term to check the rule. The number before the first term gives the constant in uₙ = dn + c.

要从数列中识别第 n 项，用首项减去公差来检验规则。第一个项之前的数字给出 uₙ = dn + c 中的常数 c。

7. Linear Graphs: y = mx + c | 直线图：y = mx + c

A straight line graph can always be written in the slope–intercept form y = mx + c. The slope (gradient) m tells you how steep the line is, and the y‑intercept c is where the line crosses the y‑axis.

所有直线图像都可以写成斜截式 y = mx + c。斜率（梯度）m 告诉你直线的陡峭程度，y‑轴截距 c 是直线与 y 轴交点的纵坐标。

m = (y₂ − y₁) / (x₂ − x₁)

To plot a line, find the y‑intercept on the y‑axis, then use the slope as ‘rise over run’ to locate a second point. Horizontal lines have m = 0 (y = c); vertical lines have an undefined gradient (x = k).

画直线时，在 y 轴上确定截距，然后利用斜率 ‘垂直变幅 / 水平平移’ 找到第二个点。水平线的 m = 0（y = c）；垂直线梯度无定义（x = k）。

8. Angles: Parallel Lines and Triangles | 角：平行线与三角形

When a transversal crosses parallel lines, specific angle pairs are formed:

当一条横截线与平行线相交时，会形成特殊的角对：

Corresponding angles are equal. 同位角相等。
Alternate angles are equal. 内错角相等。
Co‑interior (supplementary) angles sum to 180°. 同旁内角之和为 180°。

Vertically opposite angles are always equal, whatever the lines.

无论什么直线，对顶角始终相等。

∠A + ∠B + ∠C = 180° (triangle angle sum)

∠A + ∠B + ∠C = 180°（三角形内角和）

The exterior angle of a triangle equals the sum of the two opposite interior angles. For any quadrilateral, all interior angles sum to 360°.

三角形的外角等于其两个不相邻内角的和。四边形的内角和为 360°。

9. Area and Perimeter | 面积和周长

Perimeter is the total length around a shape. Area measures the space inside, in square units. Every shape has its own formula, but many are linked.

周长是一个图形周围的总长度。面积测量内部的空间，单位为平方单位。每个图形都有自己的公式，但许多是相互关联的。

Rectangle: Area = lw, Perimeter = 2(l + w)

矩形：面积 = lw，周长 = 2(l + w)

Triangle: Area = ½ × base × height

三角形：面积 = ½ × 底 × 高

Parallelogram: Area = base × perpendicular height

平行四边形：面积 = 底 × 垂直高

Trapezium: Area = ½(a + b)h, where a and b are the parallel sides

梯形：面积 = ½(a + b)h，其中 a 和 b 为平行边

Circle: Area = πr², Circumference = 2πr or πd

圆：面积 = πr²，周长 = 2πr 或 πd

For compound shapes, split them into simpler parts, work out the area of each, and add or subtract as needed.

对于复合图形，将其拆分为简单的部分，分别计算面积，然后按需相加或相减。

10. Volume and Surface Area of Prisms | 棱柱的体积和表面积

Volume measures the space a 3D object occupies, in cubic units. A prism has a uniform cross‑section along its length. Its volume is simply the area of the cross‑section multiplied by the length (or height).

体积衡量三维物体占据的空间，单位是立方单位。棱柱沿长度方向具有均匀的横截面。其体积就是横截面积乘以长度（或高度）。

Volume of a prism = Area of cross‑section × length

棱柱的体积 = 横截面积 × 长度

For a cuboid (rectangular prism), the cross‑section is a rectangle, giving V = l × w × h. Surface area is the total area of all faces. For a cuboid:

对于长方体（矩形棱柱），横截面是矩形，得到 V = l × w × h。表面积是所有面的总面积。对于长方体：

Surface area = 2(lw + lh + wh)

表面积 = 2(lw + lh + wh)

For a cylinder, the cross‑section is a circle, so V = πr²h. Its curved surface area is 2πrh, and total surface area includes the two circular ends: 2πr² + 2πrh.

对于圆柱体，横截面是圆，所以 V = πr²h。其曲面积为 2πrh，总表面积包含两个圆形底面：2πr² + 2πrh。

11. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数和极差

The mean is calculated by adding all values and dividing by the number of values. It is sensitive to extreme values. The median is the middle value when data is ordered; it splits the data into two halves. If there are two middle numbers, take their mean.

平均数

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Year 8 Edexcel Maths: Quick Reference Handbook of Formulas and Theorems | Year 8 Edexcel 数学：公式定理速查手册