Year 8 WJEC Maths: Quick Reference Guide to Formulas & Theorems | WJEC八年级数学：公式定理速查手册

📚 Year 8 WJEC Maths: Quick Reference Guide to Formulas & Theorems | WJEC八年级数学：公式定理速查手册

This handbook collects the essential formulas, rules and theorems you will meet in Year 8 WJEC Mathematics. Keep it close for quick revision, homework help and exam preparation.

本手册汇集了WJEC八年级数学中必备的公式、法则和定理，可用于快速复习、辅助作业和备考。

1. Number Properties | 数的性质

Square numbers are produced by multiplying an integer by itself. They are written with a small raised 2, called an exponent.

平方数是一个整数与自己相乘的结果，用上标小2表示，称为指数。

n² = n × n

Examples: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144.

示例：1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144。

Cube numbers are created by multiplying an integer by itself twice. The exponent 3 indicates cubing.

立方数是一个整数与自己相乘两次的结果，指数3表示立方。

n³ = n × n × n

Examples: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000.

示例：1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 10³ = 1000。

Square roots undo squaring. The symbol √ is used. A positive number has two square roots: a positive and a negative root, but the symbol √ usually gives the principal (non‑negative) root.

平方根是平方的逆运算，用符号 √ 表示。每个正数有两个平方根：一正一负，但符号 √ 通常指主平方根（非负的）。

√a × √a = a (for a ≥ 0)

For example, √25 = 5 because 5² = 25. Cube roots use the symbol ∛: ∛8 = 2 because 2³ = 8.

例如，√25 = 5 因为 5² = 25。立方根用符号 ∛：∛8 = 2 因为 2³ = 8。

A prime number is a whole number greater than 1 that has exactly two distinct factors: 1 and itself. The list starts 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

质数是大于1且只有两个不同因数（1和它自身）的整数，如 2, 3, 5, 7, 11, 13, 17, 19, 23, 29……

Factors are numbers that divide exactly into another number. Multiples are the products of a number and an integer.

因数是能整除给定数的数。倍数是某数与整数相乘的积。

Highest common factor (HCF) is the largest factor shared by two or more numbers. Lowest common multiple (LCM) is the smallest multiple they share.

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。

Every whole number greater than 1 can be written as a product of prime numbers; this is its prime factorisation. Use a factor tree to find it.

任何大于1的整数都可以写成质数的乘积，这就是质因数分解。可用因数树来找出。

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2. Fractions, Decimals and Percentages | 分数、小数与百分数

Fractions, decimals and percentages are three ways of representing parts of a whole. To convert between them, remember key equivalents.

分数、小数和百分数是表示整体的一部分的三种方式。记住重要等价关系即可相互转换。

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/3	0.333… (recurring)	33⅓%
1/10	0.1	10%
1/5	0.2	20%

To convert a fraction to a decimal, divide numerator by denominator. To convert a decimal to a percentage, multiply by 100 and add the % sign. To convert a percentage to a fraction, write it over 100 and simplify.

分数化小数：分子除以分母。小数化百分数：乘以 100 加上 % 号。百分数化分数：写成以 100 为分母的分数并化简。

To add or subtract fractions, first rewrite them with a common denominator. Multiply the numerator and denominator of each fraction so the denominators become the same.

加减分数，先通分使分母相同。对每个分数分子分母同乘，化成分母相同的分数。

a/b + c/d = (ad + bc) / bd

To multiply fractions, multiply the numerators and multiply the denominators. Simplify if possible.

分数相乘，分子乘分子，分母乘分母，能约分则约分。

a/b × c/d = (a×c) / (b×d)

To divide by a fraction, keep the first fraction, change division to multiplication and flip the second fraction (use its reciprocal).

除以一个分数时，保留第一个分数，除号变乘号，第二个分数分子分母颠倒（用它的倒数）。

a/b ÷ c/d = a/b × d/c

Finding a percentage of an amount: rewrite the percentage as a decimal and multiply. Percentage increase: new amount = original + (percentage × original). Percentage decrease: new amount = original − (percentage × original).

求一个数量的百分数：把百分数写成小数再相乘。增加百分数：新量 = 原量 + (百分数 × 原量)。减少百分数：新量 = 原量 − (百分数 × 原量)。

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3. Algebra Basics | 代数基础

In algebra, letters stand for unknown numbers. An expression contains numbers, letters and operation symbols. A term is a single part, e.g. 3x or −5.

代数中，字母代表未知数。代数式包含数字、字母和运算符号。项是单个部分，如 3x 或 −5。

A coefficient is the number part of a term that contains a variable. In 7y, 7 is the coefficient. A constant is a term without a variable.

系数是带有变量的项的数字部分，如 7y 中，7是系数。常数是不含变量的项。

Like terms have exactly the same variable and power; you can combine them by adding or subtracting their coefficients.

同类项具有完全相同的变量和指数，可对其系数加减加以合并。

5a + 3a = 8a

The distributive law lets you multiply a term across a bracket.

分配律允许将括号外的项乘入括号内每一项。

a(b + c) = ab + ac

Example: 2(x + 4) = 2x + 8. When expanding two brackets, multiply every term in the first by every term in the second: (x + 3)(x + 2) = x² + 5x + 6.

例如：2(x + 4) = 2x + 8。展开两个括号时，用第一个括号的每一项乘以第二个括号的每一项：(x + 3)(x + 2) = x² + 5x + 6。

Factorising means writing an expression as a product of factors by taking out the highest common factor.

因式分解是提出最大公因式，将代数式写成因式的乘积。

6x + 9 = 3(2x + 3)

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4. Solving Equations | 解方程

An equation shows two expressions are equal. To solve, find the value of the unknown that makes the equation true. Keep the equation balanced by doing the same operation to both sides.

方程表示两个代数式相等。解方程就是找出使等式成立的未知数的值。两边做相同运算以保持平衡。

For a one‑step equation such as x + 5 = 12, subtract 5 from both sides: x = 7. For 3x = 18, divide both sides by 3: x = 6.

如一步方程 x + 5 = 12，两边减5得 x = 7。又如 3x = 18，两边除以3得 x = 6。

Two‑step equations need two inverse operations. For 2x − 4 = 10, first add 4 to both sides (2x = 14), then divide by 2 (x = 7). Always check your solution by substituting it back into the original equation.

两步方程需要用两次逆运算。如 2x − 4 = 10，先两边加4得 2x = 14，再除以2得 x = 7。务必将解代回原方程检验。

Inequalities show that one expression is greater or less than another. The symbols are: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

不等式表示一个代数式大于或小于另一个。符号有：<（小于），>（大于），≤（小于或等于），≥（大于或等于）。

You solve inequalities just like equations, but if you multiply or divide by a negative number, you must reverse the inequality sign.

解不等式与解方程类似，但若乘以或除以一个负数，不等号必须反转。

−2x < 6 → x > −3

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5. Sequences and Patterns | 数列与规律

A number sequence is a list of numbers that follow a rule. An arithmetic sequence increases or decreases by the same amount each step; this amount is called the common difference.

数列是按某种规则排列的一列数。等差数列每一步增加或减少相同的数量，这个差称为公差。

Example: 4, 7, 10, 13, 16, … has a common difference of +3.

例如：4, 7, 10, 13, 16, … 公差为 +3。

The term‑to‑term rule tells you how to get from one term to the next. The position‑to‑term rule (nth term) lets you find any term directly from its position n.

项对项规则告诉你如何从前一项得到后一项。位置对项规则（第 n 项）可让你根据位置 n 直接求出任一项。

For an arithmetic sequence with first term a and common difference d, the nth term is:

对于首项为 a、公差为 d 的等差数列，第 n 项为：

nth term = a + (n − 1)d

Example: For 4, 7, 10, 13, … the nth term = 4 + (n − 1)×3 = 3n + 1. Check: when n = 1, 3×1 + 1 = 4; when n = 2, 3×2 + 1 = 7.

例如：数列 4, 7, 10, 13, … 的第 n 项 = 4 + (n − 1)×3 = 3n + 1。检验：n = 1 时，3×1 + 1 = 4；n = 2 时，3×2 + 1 = 7。

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6. Coordinates and Linear Graphs | 坐标与线性图

Coordinates are written as an ordered pair (x, y) on a Cartesian grid. The x‑coordinate gives the horizontal position, the y‑coordinate the vertical position.

坐标在笛卡尔网格上记作有序数对 (x, y)。x 坐标表示水平位置，y 坐标表示垂直位置。

The midpoint between two points (x₁, y₁) and (x₂, y₂) is found by averaging the x‑coordinates and averaging the y‑coordinates.

两点 (x₁, y₁) 和 (x₂, y₂) 的中点由 x 坐标的平均值和 y 坐标的平均值求得。

Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

A straight line graph can be described by the equation y = mx + c, where m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis).

直线图可用方程 y = mx + c 描述，其中 m 是斜率（倾斜程度），c 是 y 截距（直线与 y 轴的交点）。

y = mx + c

Gradient is calculated by picking two points on the line and dividing the vertical change by the horizontal change:

任取直线上两点，用纵差除以横差即可算出斜率：

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line rises; a negative gradient means it falls. If the equation is y = 2x + 3, the gradient is 2 and it crosses the y‑axis at (0, 3).

正斜率表示直线上升；负斜率表示直线下降。若方程为 y = 2x + 3，斜率为 2，y 截距为 (0, 3)。

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7. Angles and Lines | 角与线

Angles are measured in degrees (°). An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180°.

角以度 (°) 为单位。锐角小于90°，直角等于90°，钝角在90°到180°之间，优角大于180°。

Vertically opposite angles, formed when two lines cross, are equal. Angles on a straight line sum to 180°. Angles around a point sum to 360°.

两直线相交形成的对顶角相等。平角上的邻角之和为180°。环绕一个点的各角之和为360°。

When a transversal crosses two parallel lines, corresponding angles are equal, alternate angles are equal, and co‑interior (allied) angles sum to 180°.

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（共轭角）互补，和为180°。

The interior angles of a triangle always add to 180°.

三角形的内角和恒为180°。

a + b + c = 180°

The sum of the interior angles of a polygon with n sides is given by:

n 边形的内角和公式为：

Sum of interior angles = (n − 2) × 180°

For a regular polygon (all sides and angles equal), each interior angle = (n − 2) × 180° / n. Exterior angles of any polygon

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