📚 Year 7 CCEA Maths Formula & Theorem Quick Reference Handbook | Year 7 CCEA 数学：公式定理速查手册
This quick reference guide covers all the essential formulas, laws, and key facts you need for Year 7 CCEA Mathematics. Use it to revise number operations, algebra, geometry, measurement, and data handling. Each section presents clear statements in English, followed immediately by the corresponding Chinese translation, making it perfect for bilingual learners.
本速查手册涵盖 Year 7 CCEA 数学所需的所有基本公式、法则和关键知识点，适用于复习数的运算、代数、几何、测量和数据处理。每个要点先用英文陈述，紧接着提供对应的中文翻译，非常适合双语学习者使用。
1. Types of Numbers & Basic Operations | 数的类型与基本运算
Numbers can be natural numbers (1, 2, 3 …), whole numbers (0, 1, 2 …), integers (…, −2, −1, 0, 1, 2 …), fractions, decimals and percentages. The four basic operations are addition (+), subtraction (−), multiplication (×) and division (÷). When we multiply or divide two integers, the sign rules are: positive × positive = positive; negative × negative = positive; positive × negative = negative. For addition and subtraction, move left on the number line for subtraction and right for addition when dealing with negatives.
数可以分为自然数（1, 2, 3 …）、整数（0, 1, 2 …）、整数集合（…, −2, −1, 0, 1, 2 …）、分数、小数和百分数。四种基本运算是加法（+）、减法（−）、乘法（×）和除法（÷）。两个整数相乘或相除时，符号规则为：正 × 正 = 正；负 × 负 = 正；正 × 负 = 负。对于加法和减法，在数轴上进行负数运算时，减法向左移动，加法向右移动。
2. Order of Operations (BIDMAS) | 运算顺序（BIDMAS）
When a calculation involves more than one operation, we must follow a standard order: Brackets first, then Indices (powers, such as 2²), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right). The acronym BIDMAS helps remember this: B – Brackets, I – Indices, D – Division, M – Multiplication, A – Addition, S – Subtraction. For example, 3 + 4 × 2 = 3 + 8 = 11, not 14.
当一个计算涉及多个运算时，必须遵循标准顺序：先算括号，然后是指数（如 2²），接着是除法和乘法（从左到右），最后是加法和减法（从左到右）。助记词 BIDMAS 可以帮助记忆：B – 括号，I – 指数，D – 除法，M – 乘法，A – 加法，S – 减法。例如，3 + 4 × 2 = 3 + 8 = 11，而不是 14。
3. Fractions – Key Facts | 分数 – 关键知识点
A fraction represents a part of a whole. The top number is the numerator, the bottom number is the denominator. Equivalent fractions represent the same value, e.g. ½ = ²⁄₄. To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). To add or subtract fractions, they must have a common denominator. To multiply fractions, multiply the numerators together and the denominators together. To divide by a fraction, multiply by its reciprocal.
分数表示整体的一部分。上面的数叫分子，下面的数叫分母。等值分数代表相同的值，例如 ½ = ²⁄₄。化简分数时，将分子和分母同时除以它们的最大公因数（HCF）。分数加减法需要先通分，得到相同的分母。分数乘法：分子相乘作分子，分母相乘作分母。分数除法：除以一个分数等于乘以它的倒数。
4. Decimals and Percentages | 小数与百分数
A decimal is another way of writing a fraction with a denominator of 10, 100, 1000, etc. The digits after the decimal point represent tenths, hundredths, thousandths. To convert a fraction to a decimal, divide the numerator by the denominator. A percentage means ‘out of 100’. To change a decimal to a percentage, multiply by 100 and add the % sign. To change a percentage to a decimal, divide by 100. Common equivalents: ½ = 0.5 = 50%, ¼ = 0.25 = 25%, ¾ = 0.75 = 75%.
小数是分母为 10、100、1000 等的分数的另一种写法。小数点后的数字依次表示十分位、百分位、千分位。分数化小数：用分子除以分母。百分数表示“百分之几”。小数化百分数：乘以 100 并加上 % 号。百分数化小数：除以 100。常见的等值关系：½ = 0.5 = 50%，¼ = 0.25 = 25%，¾ = 0.75 = 75%。
5. Ratio and Proportion | 比与比例
A ratio compares the sizes of two or more quantities. It can be written as a:b or as a fraction a/b. To simplify a ratio, divide all parts by their HCF. When sharing a quantity in a given ratio, add the parts to find the total number of shares, then divide the quantity by this total, and multiply by each part. Proportion states that two ratios are equal. The unitary method involves finding the value of one unit first, then scaling up or down.
比用来比较两个或多个数量的大小。可以写成 a:b 或分数 a/b。化简比时，所有项都除以它们的最大公因数。按比例分配一个量时，先把各项相加得到总份数，再把总量除以总份数，然后乘以每项的份数。比例表示两个比相等。单位法是指先求出 1 个单位的值，再进行放大或缩小。
6. Algebraic Expressions | 代数表达式
Algebra uses letters (variables) to represent unknown numbers. An expression contains numbers, variables and operation signs but no equals sign. Like terms have exactly the same variable raised to the same power: they can be collected (simplified), e.g. 3a + 2a = 5a. Unlike terms cannot be combined. The distributive law says a(b + c) = ab + ac. When multiplying terms, multiply the numbers first, then the letters, e.g. 3a × 4b = 12ab.
代数使用字母（变量）表示未知数。表达式包含数字、变量和运算符号，但不含等号。同类项是指含有完全相同变量及其指数的项，可以合并化简，如 3a + 2a = 5a。不同类项不能合并。分配律为 a(b + c) = ab + ac。项相乘时，先乘系数，再乘字母部分，如 3a × 4b = 12ab。
7. Simple Equations | 简单方程
An equation states that two expressions are equal. To solve an equation, we find the value of the unknown that makes the statement true. Use inverse operations: for x + 3 = 7, subtract 3 from both sides to get x = 4. For 2x = 10, divide both sides by 2 to get x = 5. Always keep the equation balanced by doing the same thing to both sides. Check your answer by substituting it back into the original equation.
方程表示两个表达式相等。解方程就是求出使等式成立的未知数的值。使用逆运算：对于 x + 3 = 7，两边同时减去 3，得到 x = 4。对于 2x = 10，两边同时除以 2，得到 x = 5。始终保持方程平衡，即两边同时进行相同的操作。将答案代入原方程可检验是否正确。
8. Angles – Key Facts | 角 – 关键知识点
An angle is measured in degrees (°) using a protractor. Types of angles: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), straight (180°), reflex (between 180° and 360°). Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal. When two lines intersect, opposite angles are equal.
角用量角器以度（°）为单位测量。角的类型：锐角（小于 90°）、直角（正好 90°）、钝角（大于 90° 且小于 180°）、平角（180°）、优角（大于 180° 且小于 360°）。直线上的邻角之和为 180°。绕一点一周的角之和为 360°。对顶角相等。两条直线相交时，对顶角相等。
9. Triangles and Quadrilaterals | 三角形与四边形
The sum of the interior angles of a triangle is 180°. Triangles can be classified by sides: equilateral (all sides equal, all angles 60°), isosceles (two equal sides, two equal angles), scalene (no sides equal). By angles: acute-angled, right-angled, obtuse-angled. The sum of interior angles of a quadrilateral is 360°. Special quadrilaterals: square (4 equal sides, 4 right angles), rectangle (opposite sides equal, 4 right angles), parallelogram (opposite sides parallel and equal), rhombus (4 equal sides), trapezium (at least one pair of parallel sides).
三角形的内角和为 180°。按边分类：等边三角形（三边相等，每个角 60°）、等腰三角形（两条边相等，两个底角相等）、不等边三角形（三边均不相等）。按角分类：锐角三角形、直角三角形、钝角三角形。四边形的内角和为 360°。特殊四边形：正方形（四条等边，四个直角）、长方形（对边相等，四个直角）、平行四边形（对边平行且相等）、菱形（四条等边）、梯形（至少有一组对边平行）。
10. Perimeter | 周长
The perimeter of a shape is the total distance around its outer edge. For any polygon, add the lengths of all sides. For a rectangle with length l and width w, perimeter P = 2l + 2w or P = 2(l + w). For a square of side s, P = 4s. Always include the correct units (mm, cm, m, km) in your answer.
周长是图形外边线的总长度。对于任意多边形，将所有边长相加即可。长为 l、宽为 w 的长方形，周长 P = 2l + 2w 或 P = 2(l + w)。边长为 s 的正方形，P = 4s。答案中一定要注明正确的单位（毫米、厘米、米、千米）。
11. Area | 面积
Area is the amount of space inside a 2D shape, measured in square units (mm², cm², m²). Rectangle: A = l × w. Square: A = s². Triangle: A = ½ × base × height. For a triangle, the height must be perpendicular to the base. Parallelogram: A = base × perpendicular height. Compound shapes can be split into rectangles and triangles to find the total area.
面积是二维图形内部所占的空间大小，用平方单位（mm²、cm²、m²）表示。长方形面积：A = 长 × 宽。正方形面积：A = 边长²。三角形面积：A = ½ × 底 × 高。三角形的高必须垂直于底。平行四边形面积：A = 底 × 垂直高。对于组合图形，可将其分割成长方形和三角形再求总面积。
12. Statistics and Probability | 统计与概率
The mean (average) of a set of values is found by adding all the values and dividing by the number of values. The mode is the value that appears most often. The median is the middle value when the data are ordered. The range is the difference between the largest and smallest values. Probability is the chance that an event will happen, measured on a scale from 0 (impossible) to 1 (certain). Probability of an event = number of favourable outcomes ÷ total number of possible outcomes.
一组数据的平均数（均值）等于所有数据之和除以数据的个数。众数是出现次数最多的数据。中位数是将数据排序后位于中间的那个数。极差是最大值和最小值的差。概率表示事件发生的可能性，范围从 0（不可能）到 1（一定）。事件概率 = 有利结果的数量 ÷ 所有可能结果的总数。
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