📚 Year 7 CAIE Maths: Quick Reference Handbook of Formulas and Theorems | Year 7 CAIE 数学：公式定理速查手册
This quick-reference handbook brings together every essential formula, rule, and theorem you’ll encounter in the Year 7 Cambridge International (CAIE) Mathematics curriculum. Use it to check facts while doing homework, to prepare for tests, or to build a solid foundation for later years. From whole-number arithmetic to introductory algebra, geometry, and statistics, each entry is presented bilingually so you can master both the English terminology and the Chinese equivalent.
这本速查手册汇集了 Year 7 剑桥国际（CAIE）数学课程涉及的每一个关键公式、规则和定理。你可以用它来核对作业中的知识点、备考小测验，或是为以后的学习打下扎实的基础。内容覆盖整数运算、代数入门、几何和统计，每个条目都用中英双语呈现，帮助你同时掌握英文术语和中文表达。
1. Place Value and Rounding | 数位值与四舍五入
In our decimal number system, each digit has a place value ten times larger than the digit to its right. The places are units (10⁰), tens (10¹), hundreds (10²), thousands (10³), and so on. After the decimal point, the places are tenths (10⁻¹), hundredths (10⁻²), and thousandths (10⁻³).
在我们的十进制数系统中，每个数字的数位值是其右边数字的十倍。数位依次是个位（10⁰）、十位（10¹）、百位（10²）、千位（10³）等。小数点后是十分位（10⁻¹）、百分位（10⁻²）和千分位（10⁻³）。
To round a number, look at the digit to the right of the place you are rounding to. If that digit is 5 or more, round up; if it is 4 or less, leave the rounding digit unchanged. When rounding to a required number of decimal places or significant figures, all less significant digits become zero or are dropped.
四舍五入时，看要保留数位右边的那一位数字。如果该数字大于或等于 5，则进位；如果小于或等于 4，则舍去。当要求保留一定的小数位数或有效数字时，其余较不重要的数位变为零或被省略。
Example: 37 648 rounded to the nearest hundred is 37 600 (since the tens digit is 4). 37 648 rounded to 3 significant figures is 37 600, written as 3.76 × 10⁴.
例子：37 648 四舍五入到最近的百位是 37 600（因为十位数字是 4）。37 648 保留 3 个有效数字是 37 600，写作 3.76 × 10⁴。
- Place value: units, tens, hundreds, thousands — 个、十、百、千
- Decimal places: tenths, hundredths, thousandths — 十分位、百分位、千分位
- Rounding rule: ≥ 5 round up, ≤ 4 round down — 四舍五入规则
2. Addition, Subtraction, Multiplication and Division | 加减乘除
The four basic operations obey several key properties. Addition and multiplication are commutative (order does not matter): a + b = b + a and a × b = b × a. They are also associative (grouping does not matter): (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
四种基本运算遵循几个重要性质。加法和乘法满足交换律（顺序无关紧要）：a + b = b + a，a × b = b × a。它们也满足结合律（分组无关紧要）：(a + b) + c = a + (b + c)，(a × b) × c = a × (b × c)。
Subtraction and division are neither commutative nor associative. For division, remember that any number divided by 1 equals itself, and a number divided by itself equals 1 (except for 0). Division by zero is undefined.
减法和除法既不满足交换律也不满足结合律。对于除法，请记住任何数除以 1 等于它本身，而一个非零数除以它本身等于 1。零不能做除数。
In Year 7, you expand column addition, subtraction, and long multiplication. Long division is often presented using the standard bus-stop method. Key vocabulary: sum (addition result), difference (subtraction result), product (multiplication result), quotient (division result), remainder.
在 Year 7，你会进一步学习列竖式加法、减法和长乘法。长除法通常用“公交站”法书写。关键术语：和（加法结果）、差（减法结果）、积（乘法结果）、商（除法结果）、余数。
3. Order of Operations (BIDMAS/PEMDAS) | 运算顺序
To avoid confusion when an expression contains several operations, we follow the BIDMAS rule: Brackets first, then Indices (powers or roots), then Division and Multiplication (working left to right), and finally Addition and Subtraction (working left to right). The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is used in some regions.
当一个表达式包含多种运算时，为避免混淆，我们遵循 BIDMAS 规则：先算括号，再算指数（乘方或开方），然后算除法和乘法（从左到右），最后算加法和减法（从左到右）。有些地区使用 PEMDAS（括号、指数、乘除、加减）的首字母缩写。
Important: Multiplication does not always come before division; they are done in the order they appear from left to right. The same applies to addition and subtraction.
重要提示：乘法并不总是在除法之前计算，而是按照从左到右出现的顺序计算。加法和减法同理。
Example: 7 + 3 × (10 − 2)² ÷ 4. Brackets first: 10 − 2 = 8. Indices: 8² = 64. Then division and multiplication left to right: 3 × 64 = 192, then 192 ÷ 4 = 48. Finally addition: 7 + 48 = 55.
例子：7 + 3 × (10 − 2)² ÷ 4。先算括号：10 − 2 = 8。指数：8² = 64。然后从左到右计算乘除：3 × 64 = 192，然后 192 ÷ 4 = 48。最后加法：7 + 48 = 55。
4. Factors, Multiples and Primes | 因数、倍数与质数
A factor of a number divides that number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, 12. A multiple of a number is the result of multiplying that number by an integer; the multiples of 5 are 5, 10, 15, 20, …
一个数的因数就是能整除这个数且没有余数的数。例如，12 的因数有 1, 2, 3, 4, 6, 12。一个数的倍数则是该数与整数相乘的结果；5 的倍数有 5, 10, 15, 20 …
A prime number is a whole number greater than 1 that has exactly two distinct factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19. Note that 1 is not prime. A composite number has more than two factors.
质数是大于 1 且恰好只有两个不同因数的整数：1 和它本身。前几个质数是 2, 3, 5, 7, 11, 13, 17, 19。注意 1 不是质数。合数则拥有多于两个因数。
Prime factorisation means writing a number as a product of prime factors. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. The highest common factor (HCF) of two or more numbers is the largest factor they share; the lowest common multiple (LCM) is the smallest multiple they share.
质因数分解是指把一个数写成质因数相乘的形式。例如，60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。两个或更多数的最大公因数（HCF）是它们共有的最大因数；最小公倍数（LCM）是它们共有的最小倍数。
- Factor – 因数
- Multiple – 倍数
- Prime number – 质数（素数）
- Prime factorisation – 质因数分解
- HCF (Highest Common Factor) – 最大公因数 (HCF)
- LCM (Lowest Common Multiple) – 最小公倍数 (LCM)
5. Fractions, Decimals and Percentages | 分数、小数与百分数
A fraction represents a part of a whole and is written as numerator/denominator. Equivalent fractions have the same value but different numerators and denominators; you find them by multiplying or dividing both numerator and denominator by the same non‑zero number. To add or subtract fractions, make the denominators the same first, then add or subtract the numerators.
分数表示整体的一部分，写作 分子/分母。等值分数数值相同但分子分母不同，可以将分子分母同时乘或除以同一个非零数来得到。进行分数加减法时，先通分使分母相同，再对分子进行加减。
To multiply fractions, multiply the numerators together and the denominators together. To divide by a fraction, multiply by its reciprocal (flip the second fraction). Mixed numbers should be converted to improper fractions before multiplying or dividing.
分数相乘，分子乘分子，分母乘分母。除以一个分数，等于乘以它的倒数（将第二个分数上下颠倒）。带分数在乘除之前要先化成假分数。
A decimal is another way of showing a fraction whose denominator is a power of ten. To convert a fraction to a decimal, divide the numerator by the denominator. Terminating decimals end after a finite number of digits; recurring decimals repeat a pattern. Percent means ‘out of 100’, so to convert a decimal to a percentage multiply by 100; to convert a percentage to a decimal divide by 100.
小数是分母为 10 的幂的分数的另一种表示形式。将分数转化为小数，用分子除以分母。有限小数在有限位数后终止；循环小数则重复某个模式。百分数表示“每一百份”，因此将小数转化为百分数乘以 100；将百分数转化为小数则除以 100。
Common equivalences: 1/2 = 0.5 = 50%; 1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%; 1/10 = 0.1 = 10%; 1/5 = 0.2 = 20%.
常见等值关系：1/2 = 0.5 = 50%；1/4 = 0.25 = 25%；3/4 = 0.75 = 75%；1/10 = 0.1 = 10%；1/5 = 0.2 = 20%。
6. Negative Numbers and Number Lines | 负数与数轴
Negative numbers are numbers less than zero. On a horizontal number line, negative numbers lie to the left of zero and positive numbers to the right. The further left a negative number, the smaller its value: −10 is less than −4.
负数是小于零的数。在水平的数轴上，负数位于零的左边，正数位于右边。负数越往左数值越小：−10 小于 −4。
Adding a negative number is the same as subtracting its positive: 5 + (−3) = 5 − 3 = 2. Subtracting a negative number is the same as adding its positive: 5 − (−3) = 5 + 3 = 8. When multiplying or dividing two numbers with the same sign, the result is positive; if the signs are different, the result is negative.
加上一个负数等于减去它的正数：5 + (−3) = 5 − 3 = 2。减去一个负数等于加上它的正数：5 − (−3) = 5 + 3 = 8。两数相乘或相除时，同号得正，异号得负。
Examples: (−4) × (−5) = 20; (−24) ÷ 6 = −4; 8 × (−3) = −24.
例如：(−4) × (−5) = 20；( −24 ) ÷ 6 = −4；8 × (−3) = −24。
7. Algebraic Expressions and Simplifying | 代数式与化简
In algebra, letters (variables) represent unknown numbers. A term is a combination of numbers and letters multiplied together, e.g. 3xy. Like terms have exactly the same letter part; you add or subtract like terms by adding or subtracting their coefficients (the number part).
在代数中，字母（变量）代表未知数。项是数字和字母相乘的组合，例如 3xy。同类项具有完全相同的字母部分；你要通过加减它们的系数（数字部分）来合并同类项。
Examples: 5a + 2a = 7a; 3x + 2y + 5x = 8x + 2y. You cannot simplify 3a + 2b further because a and b are unlike terms. When multiplying, you can multiply the coefficients and combine the letters: 3x × 4y = 12xy.
例子：5a + 2a = 7a；3x + 2y + 5x = 8x + 2y。3a + 2b 无法再化简，因为 a 和 b 不是同类项。乘法时，系数相乘，字母可以合并：3x × 4y = 12xy。
Expand brackets by multiplying each term inside the bracket by the term outside: 3(a + 4) = 3a + 12; 2x( x + 5) = 2x² + 10x. Factorising is the reverse process: look for a common factor and write it outside the bracket, e.g. 6y + 9 = 3(2y + 3).
去括号（展开）时，将括号外的项乘括号内的每一项：3(a + 4) = 3a + 12；2x( x + 5) = 2x² + 10x。因式分解是逆过程：寻找公因数并把它写到括号外面，例如 6y + 9 = 3(2y + 3)。
8. Simple Equations and Solving | 简易方程与求解
An equation states that two expressions are equal. To solve an equation, find the value of the variable that makes the statement true. The golden rule is to keep the equation balanced — whatever you do to one side, you must do to the other.
方程表明两个表达式相等。解方程就是找出使等式成立的变量的值。黄金法则是保持方程平衡 — 你对一边做了什么操作，另一边也必须做同样的操作。
Use inverse operations: to undo an addition, subtract; to undo a subtraction, add; to undo a multiplication, divide; to undo a division, multiply. For a two‑step equation like 2x + 3 = 11, subtract 3 first (2x = 8), then divide by 2 (x = 4).
使用逆运算：抵消加法用减法，抵消减法用加法，抵消乘法用除法，抵消除法用乘法。对于像 2x + 3 = 11 这样的两步方程，先减 3（2x = 8），再除以 2（x = 4）。
Always check your answer by substituting it back into the original equation. Bracket equations may require expansion before solving.
始终要将答案代回原方程检验。含有括号的方程可能需要先去括号再求解。
- Equation – 方程
- Variable – 变量
- Inverse operation – 逆运算
- Balancing – 保持平衡
9. Angles and Lines | 角与线
An angle is formed when two straight lines meet at a point (the vertex). Angles are measured in degrees (°). A full turn is 360°, a straight line is 180°, a right angle is 90°.
角由两条直线相交于一点（顶点）形成。角度以度（°）为单位。一整圈是 360°，一条直线是 180°，直角是 90°。
Key angle facts: angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles (formed when two lines cross) are equal.
关键角度知识：直线上的角之和为 180°。绕一点一周的角之和为 360°。对顶角（两直线交叉形成的角）相等。
Types of angles: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), reflex (between 180° and 360°). When a line crosses two parallel lines, alternate angles are equal, corresponding angles are equal, and interior angles on the same side of the transversal add up to 180°.
角的分类：锐角（小于 90°），直角（正好 90°），钝角（90°到 180°之间），优角（180°到 360°之间）。当一条直线与两条平行线相交时，内错角相等，同位角相等，同旁内角之和为 180°。
- Acute, right, obtuse, reflex – 锐角、直角、钝角、优角
- Straight line: 180° – 平角 180°
- Angles at a point: 360° – 周角 360°
- Vertically opposite angles – 对顶角
- Alternate, corresponding, interior angles – 内错角、同位角、同旁内角
10. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around a 2D shape; add all the side lengths. For a rectangle, perimeter = 2 × (length + width). For a square, perimeter = 4 × side length.
周长是一个二维图形边界的总长度；把所有边长相加即可。对于长方形，周长 = 2 ×（长 + 宽）。对于正方形，周长 = 4 × 边长。
Area is the amount of space inside a 2D shape, measured in square units (cm², m², etc.). Rectangle area = length × width; square area = side²; triangle area = (base × height) ÷ 2. For a parallelogram, area = base × perpendicular height. Area of a trapezium = (a + b)/2 × height, where a and b are the lengths of the parallel sides.
面积是二维图形内部的空间大小，以平方单位（cm², m² 等）计量。长方形面积 = 长 × 宽；正方形面积 = 边长²；三角形面积 =（底 × 高）÷ 2。对于平行四边形，面积 = 底 × 垂直高。梯形面积 = (a + b)/2 × 高，其中 a 和 b 是两条平行边的长度。
Volume measures the space inside a 3D solid, in cubic units (cm³, m³). Volume of a cuboid = length × width × height; volume of a cube = side³.
体积测量三维立体内部的空间大小，以立方单位（cm³、m³）计量。长方体体积 = 长 × 宽 × 高；正方体体积 = 边长³。
- Perimeter – 周长
- Area – 面积
- Volume – 体积
- Cuboid, cube – 长方体、正方体
- Trapezium – 梯形
11. Averages, Range and Data Handling | 平均数、极差与数据处理
Three measures of central tendency are used to summarise data: the mean (average) is found by adding all the values and dividing by the number of values. The median is the middle value when the data are arranged in order. The mode is the value that appears most often. A data set can have one mode, more than one mode (bimodal or multimodal), or no mode.
有三种集中趋势的度量可用于概括数据：平均数（均值）是将所有数值相加再除以数值的个数得到。中位数是将数据按顺序排列后中间的那个数。众数是出现次数最多的数值。一个数据集可能有一个众数、多个众数（双众数或多众数），或者没有众数。
The range measures spread and is calculated as largest value − smallest value. In Year 7, you work with frequency tables, bar charts, pie charts and line graphs. For a pie chart, the angle of each sector = (frequency ÷ total frequency) × 360°.
极差用于衡量分散程度，计算公式为：最大值 – 最小值。在 Year 7，你将接触到频数表、条形图、饼图和折线图。绘制饼图时，每个扇形的角度 =（频数 ÷ 总频数）× 360°。
When interpreting graphs, read the title, axis labels and scales carefully. Calculate statistics only from the data given, not guesses.
解读图表时，要仔细阅读标题、轴标签和刻度。只能根据给出的数据计算统计量，不能凭猜测。
12. Powers, Roots and Standard Form | 乘方、方根与科学记数法
An exponent (or index, power) tells you how many times to multiply a number by itself. For example, 5³ = 5 × 5 × 5 = 125. Any number to the power of 1 is the number itself. Any positive number to the power of 0 equals 1.
指数（或称乘方）告诉你一个数自身相乘的次数。例如，5³ = 5 × 5 × 5 = 125。任何数的 1 次方等于它本身。任何正数的 0 次方等于 1。
The square root of a number finds what number multiplied by itself gives the original number. √25 = 5 because 5 × 5 = 25. Cube root ∛64 = 4 because 4³ = 64. You should know squares up to 15² = 225 and cubes up to 5³ = 125 and 10³ = 1000.
平方根是找出哪个数自乘得到原数。√25 = 5，因为 5 × 5 = 25。立方根 ∛64 = 4，因为 4³ = 64。你应该熟记 1–15 的平方（到 15² = 225）以及 1–5 和 10 的立方（5³ = 125，10³ = 1000）。
Standard form (scientific notation) writes a number as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Large numbers: 45 000 = 4.5 × 10⁴. Small numbers: 0.0032 = 3.2 × 10⁻³. Standard form is useful for very large or very small measurements.
科学记数法将一个数写成 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n 为整数。大数：45 000 = 4.5 × 10⁴。小数：0.0032 = 3.2 × 10⁻³。科学记数法在表示非常大或非常小的量时很有用。
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