📚 Year 7 SQA Maths: Quick Reference Handbook of Formulas and Theorems | SQA 七年级数学：公式定理速查手册
This handbook offers a clear and concise summary of all the key formulas, theorems, and concepts you need to master in Year 7 SQA Mathematics. Keep it close for quick reviews and to boost your confidence before tests.
本手册清晰简明地总结了 SQA 七年级数学需要掌握的所有关键公式、定理和概念。随时翻阅，快速复习，增强考前信心。
1. Number Properties and Place Value | 数字性质与位值
Our number system uses place value: each digit’s worth depends on its position. In 72 654, the ‘2’ means 2000, the ‘5’ means 50, and the ‘7’ represents 70 000.
我们的数字系统使用位值：每个数字的价值取决于其所在位置。在 72 654 中，’2′ 表示 2000，’5′ 表示 50，’7′ 代表 70 000。
Addition and multiplication are commutative: a + b = b + a and a × b = b × a. Subtraction and division are not: a − b ≠ b − a, and a ÷ b ≠ b ÷ a.
加法和乘法满足交换律：a + b = b + a，a × b = b × a。减法和除法不满足：a − b ≠ b − a，a ÷ b ≠ b ÷ a。
With negative numbers: a + (−b) = a − b. Subtracting a negative gives addition: a − (−b) = a + b. Multiplying or dividing two negatives yields a positive: (−a) × (−b) = ab.
对于负数：a + (−b) = a − b。减去负数变为加法：a − (−b) = a + b。两个负数相乘或相除得正数：(−a) × (−b) = ab。
2. Fractions, Decimals and Percentages | 分数、小数和百分比
To change a fraction to a decimal, divide numerator by denominator: 3/4 = 0.75. To change a decimal to a percentage, multiply by 100%: 0.63 = 63%.
将分数化为小数，分子除以分母：3/4 = 0.75。将小数化为百分比，乘以 100%：0.63 = 63%。
Finding a fraction of an amount: multiply. 2/5 of 60 = (2/5) × 60 = 24. To add or subtract fractions, first rewrite them with a common denominator: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
求一个量的几分之几：乘法。60 的 2/5 = (2/5) × 60 = 24。分数的加减运算，先通分：1/3 + 1/4 = 4/12 + 3/12 = 7/12。
Multiply fractions straight across: a/b × c/d = (a×c) / (b×d). Divide by a fraction by multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.
分数相乘，分子分母分别相乘：a/b × c/d = (a×c) / (b×d)。除以分数等于乘以其倒数：a/b ÷ c/d = a/b × d/c。
Converting a percentage to a fraction: write the percentage over 100 and simplify. 25% = 25/100 = 1/4.
将百分比化成分数：写成分母为 100 的分数再化简。25% = 25/100 = 1/4。
3. Order of Operations (BODMAS/BIDMAS) | 运算顺序 (BODMAS/BIDMAS)
BODMAS tells the order: Brackets, Orders (powers/roots), Division & Multiplication (left to right), Addition & Subtraction (left to right). BIDMAS uses ‘Indices’ instead of ‘Orders’.
BODMAS 规定了运算顺序：括号、阶（幂/根）、除法和乘法（从左到右）、加法和减法（从左到右）。BIDMAS 用“指数”代替“阶”。
Without brackets, 8 + 2 × 5 = 8 + 10 = 18, not 50. With brackets, (8 + 2) × 5 = 10 × 5 = 50. Always work inside brackets first.
没有括号时，8 + 2 × 5 = 8 + 10 = 18，而不是 50。有括号时，(8 + 2) × 5 = 10 × 5 = 50。永远先算括号内的。
When powers appear, calculate them after brackets. 3 + 4² = 3 + 16 = 19. For a fraction like ¾ of 20, you can do 20 ÷ 4 × 3 following the left‑to‑right rules.
遇到幂时，括号之后计算。3 + 4² = 3 + 16 = 19。对于像 20 的 3/4，可按从左到右规则计算 20 ÷ 4 × 3。
4. Factors, Multiples and Primes | 因数、倍数与质数
A factor divides exactly into a number. Factors of 18: 1, 2, 3, 6, 9, 18. A multiple is the result of multiplying the number by an integer: multiples of 8 are 8, 16, 24, 32, …
因数能整除一个数。18 的因数：1, 2, 3, 6, 9, 18。倍数是数与整数相乘的结果：8 的倍数为 8, 16, 24, 32……
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19 are primes. 1 is not prime.
质数恰好有两个因数：1 和它本身。2, 3, 5, 7, 11, 13, 17, 19 都是质数。1 不是质数。
Prime factorisation writes a number as a product of primes. 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Use a factor tree to find it.
质因数分解是把一个数写成质数的乘积。60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。可用因数树来分解。
The Highest Common Factor (HCF) is the largest factor shared by two numbers. HCF of 24 and 36 is 12. The Lowest Common Multiple (LCM) is the smallest common multiple; LCM of 4 and 6 is 12.
最大公因数 (HCF) 是两个数共有的最大因数。24 和 36 的 HCF 是 12。最小公倍数 (LCM) 是最小的公有倍数；4 和 6 的 LCM 是 12。
5. Algebraic Notation and Simplifying | 代数记法与化简
Letters stand for unknown numbers. We write multiplication without the ‘×’ sign: 4 × n = 4n, 1 × y = y. A number times itself is written with an index: a × a = a².
字母代表未知数。乘法省略“×”号：4 × n = 4n，1 × y = y。一个数自乘用指数表示：a × a = a²。
Only like terms can be simplified: 5x + 2x = 7x, but 3a + 4b stays as it is. When multiplying, combine numbers and letters: 3a × 4b = 12ab.
只有同类项才能化简：5x + 2x = 7x，但 3a + 4b 不能合并。相乘时，数字与字母分别合并：3a × 4b = 12ab。
Expanding brackets means multiplying each term inside: a(b + c) = ab + ac. For example, 5(m − 2) = 5m − 10. Remember the sign rule.
去括号是指将括号外的数与括号内每一项相乘：a(b + c) = ab + ac。例如 5(m − 2) = 5m − 10。注意符号规则。
Substitution: replace letters with given values. If a = 3 and b = −2, then 4a − b = 4×3 − (−2) = 12 + 2 = 14.
代换：用给定数值代替字母。若 a = 3，b = −2，则 4a − b = 4×3 − (−2) = 12 + 2 = 14。
6. Solving Linear Equations | 解一次方程
An equation shows two equal expressions. To solve, keep the balance: do the same operation to both sides. The goal is to isolate the unknown on one side.
方程表示两个表达式相等。解方程要保持平衡：两边同时进行相同运算。目标是将未知数隔离在等式一侧。
One-step: x + 7 = 15 → x = 15 − 7 = 8. 4x = 20 → x = 20 ÷ 4 = 5. Two-step: 2x + 3 = 11 → 2x = 8 → x = 4.
一步方程：x + 7 = 15 → x = 15 − 7 = 8。4x = 20 → x = 20 ÷ 4 = 5。两步方程：2x + 3 = 11 → 2x = 8 → x = 4。
When the unknown appears on both sides: 5x − 2 = 3x + 6 → subtract 3x: 2x − 2 = 6 → add 2: 2x = 8 → divide: x = 4. Always check by substituting back.
当未知数在等式两边出现时：5x − 2 = 3x + 6 → 两边减 3x：2x − 2 = 6 → 加 2：2x = 8 → 除以 2：x = 4。解完务必代回检验。
Equations can involve fractions. Clear fractions by multiplying by the common denominator. x/3 = 4 → x = 12. For (2x)/5 = 6, multiply by 5 → 2x = 30 → x = 15.
方程中可含分数。通过乘以公分母消去分母。x/3 = 4 → x = 12。对于 (2x)/5 = 6，乘 5 得 2x = 30 → x = 15。
7. Sequences and the nth Term | 数列与第 n 项
A linear sequence increases or decreases by a constant amount. This common difference is the coefficient of n in the nth term formula.
线性数列以恒定的量递增或递减。这个公差就是第 n 项公式中 n 的系数。
The nth term of a linear sequence: T(n) = dn + (a − d), where d = common difference and a = first term. Often written as an + b, where a is the difference and b adjusts to give the first term.
线性数列的第 n 项：T(n) = dn + (a − d)，其中 d 为公差，a 为首项。通常写成 an + b，a 为公差，b 为使首项成立的调整数。
Example: 5, 9, 13, 17, … Here d = 4, a = 5. nth term = 4n + 1. Check: n=1 → 4(1)+1=5, n=2 → 9. The 20th term is 4×20+1 = 81.
示例：5, 9, 13, 17…… 此处 d = 4，a = 5。第 n 项 = 4n + 1。检验：n=1 → 5，n=2 → 9。第 20 项为 4×20+1 = 81。
Simple quadratic sequences (1, 4, 9, 16, …) follow the rule n². Recognising patterns helps to describe sequences in words.
简单的平方数列（1, 4, 9, 16……）遵循 n² 的规律。识别模式有助于用文字描述数列。
8. Measures: Length, Perimeter and Area | 度量：长度、周长和面积
Perimeter of a rectangle: P = 2(l + w) or 2l + 2w. For a square with side s: P = 4s. The perimeter of a regular polygon: P = number of sides × side length.
矩形周长：P = 2(l + w) 或 2l + 2w。边长为 s 的正方形：P = 4s。正多边形周长：P = 边数 × 边长。
Area of rectangle: A = l × w. Square: A = s². Triangle: A = ½ × base × height. The height must be perpendicular to the base.
矩形面积：A = l × w。正方形：A = s²。三角形：A = ½ × 底 × 高。高必须与底边垂直。
Compound shapes can be split into simpler shapes. Find each area then add or subtract. For example, an L‑shape can be divided into two rectangles.
组合图形可分割为简单图形。分别求面积后相加或相减。例如 L 形可分割为两个矩形。
Volume of a cuboid: V = length × width × height. Volume is measured in cubic units, e.g. cm³, m³. The volume of a cube: V = s³.
长方体体积：V = 长 × 宽 × 高。体积以立方单位计量，如 cm³, m³。立方体体积：V = s³。
Metric conversions: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m. For area, 1 m² = 10
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