📚 Year 7 WJEC Chinese: Quick Reference Guide to Formulas and Theorems | 七年级WJEC中文：公式定理速查手册
This bilingual quick reference guide is designed for Year 7 students following the WJEC Chinese curriculum. It presents essential mathematical formulas, theorems, and procedures using clear English and Chinese explanations, helping learners become confident in using mathematical language in both languages. From basic arithmetic laws to geometry formulas and introductory algebra, every concept is broken down into practical, easy-to-remember points that reinforce classroom learning and support independent revision.
这本双语速查手册专为学习WJEC中文课程的七年级学生编写。它用清晰的英文和中文解释呈现了重要的数学公式、定理和运算步骤，帮助学习者熟练地运用两种语言表达数学知识。从基本的算术运算律到几何公式和代数入门，每一个概念都被分解为实用、易记的要点，既能强化课堂所学，也便于学生独立复习。
1. Commutative Law of Addition | 加法交换律
The commutative law of addition states that changing the order of addends does not change the sum. In other words, for any numbers a and b, a + b = b + a. This property holds for all real numbers and is fundamental in mental arithmetic and simplifying expressions.
加法交换律指出，改变加数的顺序不会改变和。换言之，对于任意数字a和b，a + b = b + a。这个性质对所有实数都成立，是心算和化简表达式的基础。
a + b = b + a
Example: If you have 7 + 5, it gives the same result as 5 + 7, both equal to 12. This law allows us to rearrange sums to make calculations easier, especially when adding several numbers.
示例：7 + 5 的结果与 5 + 7 相同，都等于 12。这条定律允许我们重新排列加法项，以便更轻松地进行计算，特别是在连加多个数字时。
2. Commutative Law of Multiplication | 乘法交换律
Multiplication is also commutative: the order of factors does not affect the product. For any numbers a and b, a × b = b × a. This principle is used frequently when rearranging factors to make multiplication simpler or when verifying answers.
乘法同样满足交换律：因数的顺序不影响乘积。对于任意数字a和b，a × b = b × a。这一原理常用于重新排列因数以简化乘法，或在验证答案时使用。
a × b = b × a
Example: 4 × 9 = 36 and 9 × 4 = 36. In everyday language, we might say ‘four nines’ or ‘nine fours’ but the total remains unchanged.
示例：4 × 9 = 36，9 × 4 = 36。在日常表达中，我们可能会说“四个九”或“九个四”，但总数保持不变。
3. Associative Law of Addition and Multiplication | 加法与乘法结合律
The associative law states that when adding or multiplying three or more numbers, the way in which the numbers are grouped does not change the result. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c). This allows us to choose the most convenient grouping to calculate step by step.
结合律指出，当三个或更多数字相加或相乘时，数字的分组方式不会改变最终结果。加法：(a + b) + c = a + (b + c)；乘法：(a × b) × c = a × (b × c)。这使我们能够选择最方便的分组方式，逐步进行计算。
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
Example: (2 + 3) + 4 = 5 + 4 = 9, and 2 + (3 + 4) = 2 + 7 = 9. Similarly, (2 × 3) × 4 = 6 × 4 = 24, and 2 × (3 × 4) = 2 × 12 = 24.
示例：(2 + 3) + 4 = 5 + 4 = 9，而 2 + (3 + 4) = 2 + 7 = 9。同样，(2 × 3) × 4 = 6 × 4 = 24，而 2 × (3 × 4) = 2 × 12 = 24。
4. Distributive Law | 分配律
The distributive law connects multiplication and addition. It states that multiplying a sum by a number gives the same result as multiplying each addend individually and then adding the products. Algebraically: a × (b + c) = a × b + a × c. This law is essential for expanding brackets and simplifying algebraic expressions.
分配律将乘法与加法联系起来。它指出，一个数乘以一个和，等于将这个数分别乘以每一个加数，然后再将乘积相加。代数表达式为：a × (b + c) = a × b + a × c。此定律对于展开括号和化简代数表达式至关重要。
a(b + c) = ab + ac
Example: 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27. It also works in reverse – we can factor out a common factor from a sum.
示例：3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27。该定律也可以反向使用——从和中提取公因数。
5. Perimeter of Rectangles and Squares | 长方形和正方形的周长
The perimeter of a shape is the total distance around its outer edge. For a rectangle with length l and width w, the perimeter P is given by the formula P = 2(l + w) or P = 2l + 2w. For a square of side length s, the perimeter is P = 4s. These formulas help us calculate fencing, framing, or border lengths.
图形的周长是指围成该图形外边缘的总长度。对于长为l、宽为w的长方形，周长P的公式为 P = 2(l + w) 或 P = 2l + 2w。对于边长为s的正方形，周长 P = 4s。这些公式帮助我们计算围栏、边框或镶边的长度。
Rectangle: P = 2(l + w)
Square: P = 4s
Example: A rectangle with length 8 cm and width 5 cm has perimeter 2 × (8 + 5) = 2 × 13 = 26 cm. A square of side 6 cm has perimeter 4 × 6 = 24 cm.
示例：一个长8厘米、宽5厘米的长方形，周长为 2 × (8 + 5) = 2 × 13 = 26 厘米。边长为6厘米的正方形，周长为 4 × 6 = 24 厘米。
6. Area of Rectangles, Triangles and Parallelograms | 长方形、三角形和平行四边形的面积
Area measures the surface inside a 2D shape. The area of a rectangle is A = l × w. The area of a triangle is half the base times the height: A = ½ × b × h. A parallelogram’s area is base times vertical height: A = b × h. In each case, height must be perpendicular to the base.
面积衡量的是二维图形内部的表面大小。长方形的面积 A = l × w。三角形的面积是底乘以高的一半：A = ½ × b × h。平行四边形的面积是底乘以垂直高度：A = b × h。在每种情况下，高必须与底边垂直。
Rectangle: A = l × w
Triangle: A = ½ × b × h
Parallelogram: A = b × h
Example: A triangle with base 10 cm and height 6 cm has area ½ × 10 × 6 = 30 cm². A parallelogram with base 8 cm and perpendicular height 5 cm has area 8 × 5 = 40 cm².
示例：底为10厘米、高为6厘米的三角形，面积是 ½ × 10 × 6 = 30 平方厘米。底为8厘米、垂直高为5厘米的平行四边形，面积是 8 × 5 = 40 平方厘米。
7. Volume of Cubes and Cuboids | 立方体和长方体的体积
Volume measures the space inside a 3D object. A cuboid (rectangular prism) with length l, width w and height h has volume V = l × w × h. A cube is a special cuboid where all edges are equal, so volume V = s³, where s is the side length. Volume is expressed in cubic units.
体积衡量的是三维物体内部的空间大小。长为l、宽为w、高为h的长方体，体积 V = l × w × h。立方体是一种特殊的长方体，所有棱长相等，因此体积 V = s³，其中s为棱长。体积用立方单位表示。
Cuboid: V = l × w × h
Cube: V = s³
Example: A cuboid measuring 4 cm by 3 cm by 2 cm has volume 4 × 3 × 2 = 24 cm³. A cube with side 5 cm has volume 5³ = 125 cm³.
示例：一个尺寸为4厘米×3厘米×2厘米的长方体，体积是 4 × 3 × 2 = 24 立方厘米。一个棱长5厘米的立方体，体积是 5³ = 125 立方厘米。
8. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies to right-angled triangles. It states that 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. If the sides are a and b, and the hypotenuse is c, then the relationship is written as a² + b² = c². This theorem is widely used to find missing side lengths.
勾股定理适用于直角三角形。它指出，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之和。如果两条直角边分别为a和b，斜边为c，那么关系式写作 a² + b² = c²。该定理广泛用于求解未知边长。
a² + b² = c²
Example: If a right-angled triangle has shorter sides of 3 cm and 4 cm, then the hypotenuse c satisfies 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm.
示例：如果一个直角三角形的两条较短边分别为3厘米和4厘米，那么斜边c满足 3² + 4² = 9 + 16 = 25，因此 c = √25 = 5 厘米。
9. Calculating the Mean | 计算平均数
The mean (average) of a set of numbers is found by adding all the values together and dividing by the number of values. The formula is: Mean = Sum of all data values ÷ Number of data values. This measure of central tendency helps describe a typical value in a data set, though it can be affected by very high or very low outliers.
一组数据的平均数（均值）是将所有数值相加，再除以数值的个数求得。公式为：平均数 = 所有数据值之和 ÷ 数据值的个数。这种集中趋势的度量有助于描述数据集中的典型值，但它可能会受到极高或极低异常值的影响。
Mean = (x₁ + x₂ + … + xₙ) ÷ n
Example: The mean of 5, 8, 12, 6 and 9 is (5+8+12+6+9) ÷ 5 = 40 ÷ 5 = 8.
示例：5，8，12，6 和 9 的平均数是 (5+8+12+6+9) ÷ 5 = 40 ÷ 5 = 8。
10. Converting Fractions, Decimals and Percentages | 分数、小数和百分比的转换
A percentage means ‘out of 100’. To convert a fraction to a percentage, multiply by 100. To convert a decimal to a percentage, multiply by 100 and add the % sign. To convert a percentage to a decimal, divide by 100. These conversions are used in problems involving discounts, interest, and proportions.
百分比表示“每一百份中的多少”。将分数转换为百分比，乘以100即可。将小数转换为百分比，乘以100并加上%符号。将百分比转换为小数，除以100即可。这些转换用于涉及折扣、利息和比例的问题。
Fraction → Percentage: (a/b) × 100%
Decimal → Percentage: d × 100%
Percentage → Decimal: p% ÷ 100
Example: 3/5 as a percentage is (3 ÷ 5) × 100% = 0.6 × 100% = 60%. The decimal 0.25 is equivalent to 25%. To find 30% of a quantity, multiply by 0.30.
示例：3/5 化为百分比是 (3 ÷ 5) × 100% = 0.6 × 100% = 60%。小数 0.25 相当于 25%。求一个量的30%，只需乘以 0.30。
11. Simple Algebraic Expressions and Equations | 简单代数表达式和方程
In algebra, letters represent unknown numbers. An expression like 3x + 2 combines numbers and variables. An equation states that two expressions are equal, e.g., 2x + 5 = 13. To solve an equation, perform the same operation on both sides to isolate the variable, keeping the balance. The goal is to find the value of x that makes the statement true.
在代数中，字母表示未知数。像 3x + 2 这样的表达式结合了数字和变量。方程表示两个表达式相等，例如 2x + 5 = 13。解方程时，在等式两边进行相同的运算，以隔离变量，保持平衡。目标是求出使等式成立的x的值。
Solve: 2x + 5 = 13 → 2x = 8 → x = 4
Example: To solve x/3 – 2 = 4, first add 2 to both sides: x/3 = 6. Then multiply both sides by 3: x = 18. Always check by substituting back into the original equation.
示例：解方程 x/3 – 2 = 4，首先两边加2：x/3 = 6。然后两边乘以3：x = 18。记住代回原方程检验。
12. Order of Operations (BIDMAS/BODMAS) | 运算顺序（括号/指数–乘除–加减）
To evaluate a mathematical expression correctly, we follow a standard order: Brackets, Indices (or Orders), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). The acronyms BIDMAS or BODMAS help remember this sequence. Without this rule, the same expression could give different answers.
为正确计算数学表达式，我们遵循标准顺序：括号、指数（或幂）、除法和乘法（从左到右）、加法和减法（从左到右）。助记词 BIDMAS 或 BODMAS 有助于记住这一顺序。没有这一规则，同一个表达式可能得出不同答案。
B – Brackets, I – Indices, DM – Division & Multiplication, AS – Addition & Subtraction
Example: Calculate 6 + 4 × (3 – 1)² ÷ 2. First bracket: 3 – 1 = 2; indices: 2² = 4; then multiplication/division from left: 4 × 4 = 16, 16 ÷ 2 = 8; finally addition: 6 + 8 = 14.
示例：计算 6 + 4 × (3 – 1)² ÷ 2。先算括号：3 – 1 = 2；指数：2² = 4；然后从左乘除：4 × 4 = 16，16 ÷ 2 = 8；最后加法：6 + 8 = 14。
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