📚 Year 7 Cambridge Mathematics Formula & Theorem Quick Reference | 7年级剑桥数学公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems, and key rules covered in Year 7 Cambridge Mathematics. It is designed to help you revise efficiently, complete homework with confidence, and prepare for assessments. Each section presents the content in a clear, bilingual format so you can learn the terminology in both English and Chinese.
本速查手册汇集了 7 年级剑桥数学的核心公式、定理和重要规则，旨在帮助你高效复习、自信地完成作业并做好评估准备。每一节都以清晰的中英对照形式呈现，让你同时掌握英文和中文的数学术语。
1. Number Properties & Operations | 数的性质与运算
BODMAS / BIDMAS: Brackets, Orders (powers/roots), Division & Multiplication (left to right), Addition & Subtraction (left to right)
运算顺序 (BODMAS/BIDMAS)：先计算括号，再计算乘方或根号，接着从左到右处理乘除，最后从左到右处理加减。
A factor is a whole number that divides exactly into another number with no remainder. For example, the factors of 18 are 1, 2, 3, 6, 9, 18.
因数是可以整除给定整数且没有余数的整数。例如，18 的因数有 1、2、3、6、9、18。
A multiple is the product of a number and any integer. The first five multiples of 6 are 6, 12, 18, 24, 30.
倍数是指一个数与任意整数的乘积。6 的前五个倍数是 6、12、18、24、30。
A prime number has exactly two distinct factors: 1 and itself. 2, 3, 5, 7, 11 are prime numbers; 2 is the only even prime.
质数是恰好有两个不同因数的数：1 和它本身。2、3、5、7、11 是质数；2 是唯一的偶质数。
The highest common factor (HCF) of two or more numbers is the largest number that is a factor of all of them. The lowest common multiple (LCM) is the smallest number that is a multiple of all of them.
两个或多个数的最大公因数 (HCF) 是这些数的公因数中最大的一个。最小公倍数 (LCM) 是这些数的公倍数中最小的一个。
re of 5 is 5² = 25. A cube number comes from multiplying a number by itself twice: 2³ = 8.
平方数是一个整数乘以自身的积。5 的平方是 5² = 25。立方数是一个数自身乘两次的积：2³ = 8。
√a is the square root: √25 = 5 | ³√a is the cube root: ³√27 = 3
√a 表示平方根：√25 = 5；³√a 表示立方根：³√27 = 3。
2. Fractions, Decimals & Percentages | 分数、小数和百分数
Fraction: write as fraction of 10, 100, 1000, then simplify. Example: 0.75 = 75/100 = 3/4
小数转分数：先写成分母为 10, 100, 1000 的分数，再约分。例如 0.75 = 75/100 = 3/4。
Fraction → Decimal: divide numerator by denominator. 3/8 = 3 ÷ 8 = 0.375
分数转小数：用分子除以分母。3/8 = 3 ÷ 8 = 0.375。
Percent → Decimal: divide by 100. 45% = 0.45 | Decimal → Percent: multiply by 100
百分数转小数：除以 100；小数转百分数：乘以 100。45% = 0.45。
To add or subtract fractions, first write them with a common denominator, then add or subtract the numerators. Keep the denominator the same.
分数加减时，先通分使其分母相同，再将分子加减，分母保持不变。
a/b × c/d = (a × c) / (b × d) | a/b ÷ c/d = a/b × d/c
分数乘法：分子乘分子，分母乘分母；分数除法：除以一个分数等于乘以它的倒数。
A percentage of a quantity: find (percentage ÷ 100) × quantity. For example, 20% of 50 = 0.20 × 50 = 10.
求一个数的百分之几：用 (百分数 ÷ 100) × 该数量。例如 50 的 20% = 0.20 × 50 = 10。
3. Ratios & Proportions | 比和比例
To simplify a ratio, divide all parts by their highest common factor. The ratio 8:12 simplifies to 2:3 by dividing both by 4.
化简比时，用所有项的最大公因数去除每一项。比 8:12 同除以 4 得到 2:3。
: b, the first share is (a / (a + b)) × total, the second share is (b / (a + b)) × total.
按比 a : b 分配一个总量时，第一份为 (a / (a + b)) × 总量，第二份为 (b / (a + b)) × 总量。
Two quantities are in direct proportion if their ratio stays constant. The unitary method finds the value of one unit first, then scales up.
若两个量的比值保持不变，则它们成正比例。归一法先求出一个单位的量，再按比例放大。
4. Algebraic Expressions | 代数表达式
terms can be added or subtracted. 3a + 2a = 5a, but 3a + 2b cannot be simplified further.
同类项含有完全相同的字母和指数；只有同类项才能相加减。3a + 2a = 5a，而 3a + 2b 不能再化简。
a × a = a² | a × a × a = a³ | a × b = ab
字母相乘简写：a × a 写作 a²；a × a × a 写作 a³；a × b 写作 ab。
aw: a(b + c) = ab + ac. Always multiply the term outside the bracket by every term inside.
乘法分配律：a(b + c) = ab + ac。括号外的项必须与括号内每一项相乘。
and following the order of operations. For example, if x = 3, then 2x + 1 = 2(3) + 1 = 7.
代入求值指的是把字母替换成给定的数，并遵循运算顺序计算。例如当 x = 3 时，2x + 1 = 2(3) + 1 = 7。
5. Linear Equations | 线性方程
To solve an equation, use inverse operations to isolate the unknown. Whatever you do to one side, do exactly the same to the other.
解方程时，用逆运算把未知数单独留在等号一边。在等号一边进行的任何运算，等号另一边也必须进行同样的运算。
x + a = b ⇒ x = b – a | ax = b ⇒ x = b ÷ a (a ≠ 0)
一步方程：x + a = b ⇒ x = b – a；ax = b ⇒ x = b ÷ a (a ≠ 0)。
For two-step equations such as 2x + 3 = 7, first subtract 3, then divide by 2: 2x = 4 ⇒ x = 2.
对于两步方程如 2x + 3 = 7，先减去 3 得到 2x = 4，再除以 2 得到 x = 2。
6. Sequences & Patterns | 数列与规律
A number sequence follows a rule. The term-to-term rule tells you how to get from one term to the next by adding, subtracting, multiplying or dividing each time.
数列按照一定规则排列。逐项法则告诉你如何通过每次加、减、乘或除从一项得到下一项。
An arithmetic sequence has a constant difference between consecutive terms. The nth term can be written as:
等差数列中相邻两项的差是常数。第 n 项可以写成：
nth term = a + (n – 1)d
第 n 项 = a + (n – 1)d，其中 a 是首项，d 是公差。
Example: sequence 5, 8, 11, 14,... has a = 5, d = 3, so the nth term is 5 + 3(n – 1) = 3n + 2.
例如数列 5, 8, 11, 14,... 的首项 a = 5，公差 d = 3，所以第 n 项为 5 + 3(n – 1) = 3n + 2。
7. Angles & 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°, a straight line is 180°, and a full turn is 360°.
角以度 (°) 为单位。锐角小于 90°，直角正好 90°，钝角介于 90° 到 180° 之间，平角为 180°，周角为 360°。
Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal.
直线上的邻角之和为 180°。围绕一点的所有角之和为 360°。对顶角相等。
When a transversal crosses parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°.
当一条截线与两条平行线相交时：同位角相等，内错角相等，同旁内角之和为 180°。
8. Triangles & Quadrilaterals | 三角形与四边形
The sum of the interior angles of any triangle is 180°. In an equilateral triangle all angles are 60°; an isosceles triangle has two equal base angles.
任何三角形的内角和均为 180°。等边三角形每个角都是 60°；等腰三角形的两个底角相等。
The sum of the interior angles of any quadrilateral is 360°. Special quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapeziums and kites.
任何四边形的内角和均为 360°。特殊的四边形包括正方形、矩形、平行四边形、菱形、梯形和筝形。
A parallelogram has opposite sides parallel and equal, opposite angles equal, and diagonals that bisect each other.
平行四边形的对边平行且相等，对角相等，对角线互相平分。
9. Perimeter, Area & Volume | 周长、面积和体积
Rectangle: P = 2(l + w) A = l × w
矩形周长 = 2(长 + 宽)，面积 = 长 × 宽。
Triangle: A = ½ × base × height
三角形面积 = ½ × 底 × 高。
Parallelogram: A = b × h | Trapezium: A = ½ (a + b) h
平行四边形面积 = 底 × 高；梯形面积 = ½ (上底 + 下底) × 高。
Circle: C = πd = 2πr | A = πr²
圆的周长 C = πd = 2πr，面积 A = πr²，其中 r 是半径，d 是直径。
Cuboid: V = l × w × h | Surface area = 2(lw + lh + wh)
长方体体积 V = 长 × 宽 × 高，表面积 = 2(长×宽 + 长×高 + 宽×高)。
Prism volume: V = area of cross-section × length
棱柱体积 = 横截面积 × 长度。
, 1 m = 100 cm, 1 cm = 10 mm. For area, square the conversion factor; for volume, cube it.
公制长度单位换算：1 km = 1000 m，1 m = 100 cm，1 cm = 10 mm。面积单位的进率是长度进率的平方；体积单位的进率是长度进率的立方。
10. Coordinates & Symmetry | 坐标与对称
A point is located on the coordinate grid using an ordered pair (x, y). The x-coordinate gives the horizontal position, the y-coordinate the vertical position.
一个点在坐标网格上的位置用有序数对 (x, y) 表示。x 坐标表示水平位置，y 坐标表示垂直位置。
A shape has line symmetry if it can be folded along a line so that both halves match exactly. The number of such lines is the order of line symmetry.
如果一个图形沿某条直线对折后两部分完全重合，则该图形具有线对称，这条直线即对称轴。
Rotational symmetry: a shape has rotational symmetry of order n if it looks exactly the same n times during a full 360° turn.
旋转对称：若一个图形在旋转一周 (360°) 的过程中有
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