📚 Year 7 CIE Further Mathematics: Formula & Theorem Quick Reference Guide | 7年级 CIE 进阶数学：公式定理速查手册
This quick-reference guide brings together the essential formulas, theorems, and rules that Year 7 students meet in the CIE Further Mathematics course. Use it to revise key concepts, check your working, and build confidence before tests. Every section presents the core idea in a clear, example-driven style.
本速查手册汇总了7年级学生在CIE进阶数学课程中需要掌握的基本公式、定理与规则。你可以用它来复习重要概念、检查解题过程，并在考试前建立信心。每个小节都以清晰的示例方式呈现核心内容。
1. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
BIDMAS tells us the correct sequence for calculations: Brackets, Indices (powers), Division and Multiplication (working left to right), Addition and Subtraction (working left to right). Always follow this order to avoid mistakes.
BIDMAS 告诉我们正确的计算顺序：B括号，I指数（幂），D除法和M乘法（从左到右），A加法和S减法（从左到右）。一定要按照这个顺序计算，才不会出错。
inside brackets is done first, then the index, then multiplication, and finally addition.
例子：8 + 2 × (3² − 1) = 8 + 2 × (9 − 1) = 8 + 2 × 8 = 8 + 16 = 24。先算括号里的内容，再处理指数，接着算乘法，最后算加法。
2. Algebraic Expressions and Simplification | 代数表达式与化简
In algebra we collect like terms – a + 3a = 8a, but 5a + 3b cannot be simplified further because the variables are different.
在代数中，我们要合并同类项——也就是变量部分完全相同的项。例如，5a + 3a = 8a，但 5a + 3b 不能再化简，因为变量不同。
et: a(b + c) = ab + ac. For two brackets: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
展开括号时，要把括号外的项乘以括号里的每一项。单个括号：a(b + c) = ab + ac。两个括号的情况：(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6。
a(b + c) = ab + ac
3. Solving Linear Equations in One Variable | 一元一次方程的解法
st subtract 4 from both sides to get 3x = 15, then divide both sides by 3 to obtain x = 5.
像 3x + 4 = 19 这样的一元一次方程，我们可以通过对等式两边进行相同的运算来求解。先在两边减去4，得到 3x = 15，然后两边除以3，得出 x = 5。
Always check your solution by substituting it back into the original equation. If the left‑hand side equals the right‑hand side, the answer is correct. For 3(5) + 4 = 19, it works.
一定要把求得的解代回原方程进行检验。如果左边等于右边，答案就是正确的。对于 3(5) + 4 = 19，它是成立的。
ax + b = c ⇒ x = (c − b) ÷ a
4. Pythagoras' Theorem | 勾股定理
Pythagoras' theorem applies only to right‑angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。它指出，斜边（直角所对的边）的平方等于另外两条边的平方之和。
a² + b² = c²
If the legs are 3 cm and 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm. The theorem also helps to find a shorter side: b = √c² − a².
如果两条直角边分别为3 cm和4 cm，那么 c² = 3² + 4² = 9 + 16 = 25，因此 c = √25 = 5 cm。这个定理还可以用来求直角边的长度：b = √c² − a²。
5. Area and Perimeter of 2D Shapes | 二维图形的面积与周长
d width w: perimeter P = 2(l + w), area A = l × w. For a square of side a: P = 4a, A = a².
对于长为 l、宽为 w 的矩形：周长 P = 2(l + w)，面积 A = l × w。对于边长为 a 的正方形：周长 P = 4a，面积 A = a²。
e is A = ½ × base × height. Make sure the height is perpendicular to the base. For a right‑angled triangle, the legs can serve as base and height.
三角形的面积公式是 A = ½ × 底 × 高。注意高必须与底垂直。对于直角三角形，两条直角边可以直接作为底和高。
². (r = radius, d = diameter) Use π ≈ 3.14 or the π button on your calculator when needed.
圆：周长 C = 2πr 或 C = πd，面积 A = πr²。（r = 半径，d = 直径）需要时可以用 π ≈ 3.14 或计算器上的 π 键。
Triangle: A = ½bh | Circle: C = 2πr, A = πr²
6. Volume and Surface Area of 3D Shapes | 三维图形的体积与表面积
V = lwh and surface area SA = 2(lw + lh + wh). For a cube of edge a, V = a³ and SA = 6a².
一个长 l、宽 w、高 h 的长方体，体积 V = lwh，表面积 SA = 2(lw + lh + wh)。对于棱长为 a 的立方体，V = a³，表面积 SA = 6a²。
πr² + 2πrh. Remember to include the two circular ends when calculating total surface area.
对于底面半径为 r、高为 h 的圆柱体：体积 V = πr²h，侧面积 = 2πrh，总表面积 = 2πr² + 2πrh。计算总表面积时记得加上两个圆形底面的面积。
Cube: V = a³ | Cylinder: V = πr²h
7. Angles and Parallel Lines | 角与平行线
When a transversal cuts two parallel lines, several angle relationships appear. Vertically opposite angles are equal. Corresponding angles (F‑shape) are equal. Alternate angles (Z‑shape) are equal. Co‑interior (or allied) angles (C‑shape) add up to 180°.
当一条截线与两条平行线相交时，会产生几种角的关系。对顶角相等。同位角（F 形）相等。内错角（Z 形）相等。同旁内角（C 形）相加等于 180°。
The angles on a straight line sum to 180°, and angles around a point sum to 360°. In any triangle the three interior angles always sum to 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
直线上的角之和为 180°，绕一个点一周的角之和为 360°。任意三角形的三个内角之和始终为 180°。三角形的一个外角等于与它不相邻的两个内角之和。
Triangle: ∠A + ∠B + ∠C = 180°
8. Fractions, Decimals and Percentages | 分数、小数与百分数
nominator: ¾ = 0.75. To change a decimal to a percentage, multiply by 100: 0.75 × 100 = 75%. From a percentage to a decimal, divide by 100: 8% = 0.08.
将分数化为小数，用分子除以分母：¾ = 0.75。将小数化为百分数，乘以 100：0.75 × 100 = 75%。从百分数化为小数，除以 100：8% = 0.08。
Finding a percentage of an amount: 20% f 60 = 0.20 × 60 = 12. To increase or decrease by a percentage, use a multiplier. For a 15% increase, multiply by 1.15; for a 15% decrease, multiply by 0.85.
求一个数的百分之几：60 的 20% = 0.20 × 60 = 12。要按百分数增加或减少，使用乘数。增加 15%，乘以 1.15；减少 15%，乘以 0.85。
Fraction → Decimal → Percentage: multiply or divide by 100
9. Laws of Indices (Exponents) | 指数定律
exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
指数定律帮我们化简含有幂的表达式。同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。
When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. Any non‑ o the zero power equals 1: a⁰ = 1. A negative exponent means the reciprocal: a⁻ⁿ = 1 / aⁿ.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。任何非零底数的零次幂等于 1：a⁰ = 1。负指数代表倒数：a⁻ⁿ = 1 / aⁿ。
Example: 2³ × 2⁴ = 2⁷ = 128.
例子：2³ × 2⁴ = 2⁷ = 128。
Example: 5⁶ ÷ 5² = 5⁴ = 625.
例子：5⁶ ÷ 5² = 5⁴ = 625。
Example: (3²)³ = 3⁶ = 729, and 7⁰ = 1.
例子：(3²)³ = 3⁶ = 729，而 7⁰ = 1。
10. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. It can be simplified by dividing each part by their greatest common factor. The ratio 12 : 18 simplifies to 2 : 3 (dividing both by 6). The order of the numbers matters.
比用来比较两个或多个数量。可以通过将每部分除以它们的最大公因数来化简。比 12 : 18 简化为 2 : 3（两者都除以 6）。数字的顺序非常重要。
d the total number of parts. If the ratio is 3 : 5, there are 8 parts altogether. To share £64 in the ratio 3 : 5, one share = 3/8 × 64 = £24, the other = 5/8 × 64 = £40.
按给定比例分配一个量时，先求出总份数。如果比是 3 : 5，共有 8 份。将 64 英镑按 3 : 5 分配，一份是 3/8 × 64 = 24 英镑，另一份是 5/8 × 64 = 40 英镑。
oportion means that two quantities increase or decrease at the same rate. If 3 apples cost £1.50, then 7 apples cost (7/3) × 1.50 = £3.50.
正比例意味着两个量以相同的速率增加或减少。如果 3 个苹果花费 1.50 英镑，那么 7 个苹果花费 (7/3) × 1.50 = 3.50 英镑。
11. Sum of Interior Angles of a Polygon | 多边形内角和
2) × 180°. This formula works for any simple convex polygon, from triangles (n=3) upwards.
n 边形内角和公式为 (n − 2) × 180°。这个公式适用于任何简单凸多边形，从三角形 (n=3) 开始都适用。
Sum of interior angles = (n − 2) × 180°
× 180° = 540°. To find one interior angle of a regular polygon, divide the total sum by n.
对于四边形 (n=4)：内角和 = (4 − 2) × 180° = 360°。对于五边形 (n=5)：内角和 = (5 − 2) × 180° = 540°。求正多边形的一个内角，只需用总和除以 n。
12. Prime Factors, HCF and LCM | 质因数、最大公因数与最小公倍数
r example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. This is its prime factorisation in index form.
质数恰好有两个不同的因数：1 和它本身。用因子树可以将一个数写成质因数的乘积。例如，60 = 2 × 2 × 3 × 5 = 2² × 3 × 5。这就是它的指数形式的质因数分解。
) uses all prime factors raised to the highest power. For 36 (2² × 3²) and 48 (2⁴ × 3), HCF = 2² × 3 = 12, LCM = 2⁴ × 3² = 144.
两个数的最大公因数 (HCF) 是所有公共质因数取最小次幂的乘积。最小公倍数 (LCM) 则将所有质因数取最大次幂相乘。对于 36 (2² × 3²) 和 48 (2⁴ × 3)，HCF = 2² × 3 = 12，LCM = 2⁴ × 3² = 144。
Knowing prime factorisation makes it easy to work with fractions, simplify ratios, and solve problems involving multiples and factors.
掌握质因数分解有助于处理分数、化简比以及解决涉及倍数和因数的问题。
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