📚 Year 8 CCEA Maths: Formula and Theorem Quick Reference Handbook | Year 8 CCEA 数学：公式定理速查手册
This handbook collects the most important formulas, rules and theorems you will meet in Year 8 CCEA Mathematics. Use it to check facts quickly while doing homework or revising for tests. Each section presents key ideas in English and Chinese, with clearly displayed formulas.
本手册汇集了你在 Year 8 CCEA 数学课程中将遇到的最重要的公式、法则和定理。可用于做作业或备考时快速查阅。每个小节都用中英双语呈现核心要点，公式展示清晰明了。
1. Number Properties and Arithmetic | 数字性质与算术
A prime number is a whole number greater than 1 that can only be divided exactly by 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
质数是大于 1 且只能被 1 和自身整除的整数。前十个质数依次为 2, 3, 5, 7, 11, 13, 17, 19, 23 和 29。
Factors of a number divide it exactly; multiples are the results of multiplying the number by whole numbers. The highest common factor (HCF) and lowest common multiple (LCM) are useful when working with fractions and ratios.
一个数的因数能整除它；倍数则是将该数乘以整数得到的结果。在处理分数和比时，最大公因数 (HCF) 和最小公倍数 (LCM) 非常有用。
Square numbers come from multiplying a number by itself. The first twelve squares are:
平方数是由一个数自身相乘得到的。前十二个平方数如下：
| 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 produced by multiplying a number by itself twice, e.g. 2³ = 8, 3³ = 27.
立方数是一个数自乘两次得到的结果，例如 2³ = 8，3³ = 27。
The order of operations (BIDMAS) must be followed: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).
必须遵循运算顺序 (BIDMAS)：括号，指数，除法和乘法（从左到右），加法和减法（从左到右）。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
To change a fraction to a decimal, divide the numerator by the denominator.
要将分数化为小数，用分子除以分母。
Decimal = Numerator ÷ Denominator
小数 = 分子 ÷ 分母
To convert a decimal to a percentage, multiply by 100 and add the % sign.
要将小数化为百分比，乘以 100 并加上百分号。
Percentage = Decimal × 100%
To find a percentage of a quantity, multiply the quantity by the percentage written as a decimal.
要求一个量的百分比，用该量乘以写成小数形式的百分比。
Part = (Percentage ÷ 100) × Whole
部分 = (百分比 ÷ 100) × 整体
ons, decimals and percentages should be memorised for common values such as 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25% and 3/4 = 0.75 = 75%.
常见的分数、小数和百分比的等价关系应该熟记，如 1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%。
3. Ratio and Proportion | 比与比例
A ratio compares two or more quantities in the same units. It can be simplified by dividing all parts by their highest common factor.
比用于比较两个或多个相同单位的量。可以通过将所有部分除以它们的最大公因数来化简比。
For example, the ratio 8:12 simplifies to 2:3 (divide both by 4).
例如，比 8:12 化简为 2:3（两者均除以 4）。
In direct proportion, two quantities increase or decrease at the same rate. The statement 'a is directly proportional to b' means a/b is constant.
在正比例中，两个量以相同的速率增加或减少。“a 与 b 成正比”意味着 a/b 是常数。
When two ratios are equal, you can cross-multiply:
当两个比相等时，可以交叉相乘：
If a : b = c : d, then a × d = b × c
如果 a : b = c : d，则 a × d = b × c
This property is useful for solving missing values in proportional tables.
这一性质对于求解比例表中的未知值非常有用。
4. Algebraic Expressions and Simplifying | 代数表达式与化简
Terms with the same letter and power are called like terms and can be combined by adding or subtracting their coefficients.
含有相同字母且相同指数的项称为同类项，可以通过加减其系数来合并。
3a + 5a = 8a
The distributive law allows us to expand brackets: multiply each term inside the bracket by the term outside.
分配律允许我们展开括号：将括号外的项乘以括号内的每一项。
a(b + c) = a × b + a × c
a(b + c) = ab + ac
The reverse process is factorising: taking a common factor out of two or more terms.
逆向过程是因式分解：从两项或多项中提取公因数。
ab + ac = a(b + c)
Remember that a letter on its own like x has a coefficient of 1, and x × x = x².
记住单独的字母如 x 的系数为 1，且 x × x = x²。
5. Solving Linear Equations | 解线性方程
To solve an equation, carry out the same operation on both sides to keep it balanced.
要解方程，需在等号两边执行相同的运算，以保持等式平衡。
For one-step equations:
对于一步方程：
x + a = b ⇒ x = b − a
x − a = b ⇒ x = b + a
a × x = b ⇒ x = b ÷ a
x ÷ a = b ⇒ x = a × b
For two-step equations, first undo the addition or subtraction, then undo the multiplication or division.
对于两步方程，先去掉加或减，再去掉乘或除。
2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 4
Always check your solution by substituting it back into the original equation.
始终将解代回原方程进行检验。
6. Coordinates and Graphs | 坐标与图像
A point on a grid is given by its coordinates (x, y), where x is the horizontal position and y is the vertical position.
网格上的点用坐标 (x, y) 表示，其中 x 是水平位置，y 是垂直位置。
The midpoint of the line segment joining (x₁, y₁) and (x₂, y₂) is:
连接 (x₁, y₁) 和 (x₂, y₂) 的线段的中点坐标为：
Midpoint = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )
中点 = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )
Horizontal lines have equations of the form y = c (a constant); vertical lines have x = k.
水平线的方程形如 y = c（常数）；垂直线形如 x = k。
The line y = x passes through the origin at 45°; y = −x is its reflection in the x-axis.
直线 y = x 以 45° 角通过原点；y = −x 是它在 x 轴上的反射。
at least three x-values, calculate the y-values, plot the points and join them with a ruler.
当根据方程如 y = 2x + 1 绘制直线图像时，至少选取三个 x 值，计算对应的 y 值，描点后用直尺连成线。
7. Perimeter, Area and Volume | 周长、面积与体积
Formulas for perimeter (P) and area (A) of common 2D shapes are summarised below. Lengths are measured in units such as cm or m; areas are in square units (cm², m²).
常见二维图形的周长 (P) 和面积 (A) 公式总结如下。长度以 cm 或 m 等为单位；面积以平方单位 (cm²， m²) 表示。
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle (l = length, w = width) | P = 2(l + w) | A = l × w |
| Square (s = side) | P = 4s | A = s² |
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