Year 8 CIE Advanced Mathematics Formula & Theorem Quick Reference Handbook | Year 8 CIE 进阶数学：公式定理速查手册

📚 Year 8 CIE Advanced Mathematics Formula & Theorem Quick Reference Handbook | Year 8 CIE 进阶数学：公式定理速查手册

This handbook gathers all the essential formulas, theorems and key concepts you are expected to know for Year 8 CIE Advanced Mathematics. Use it regularly to reinforce your understanding, check facts quickly and build confidence before tests.

本手册汇集了 Year 8 CIE 进阶数学所需掌握的所有核心公式、定理和重要概念。勤加使用能帮助你巩固理解、快速查证事实，并在考试前建立信心。

1. Algebraic Operations & Expressions | 代数运算与表达式

Algebraic expressions combine numbers, letters and operation symbols. The basic principle is that only like terms can be added or subtracted. When multiplying or dividing terms, coefficients are worked numerically and variables follow exponent rules.

代数表达式将数字、字母和运算符号组合在一起。基本原则是只有同类项才能相加或相减。当乘除各项时，系数进行数值运算，变量遵循指数法则。

a + a = 2a (adding like terms) | 同类项相加：a + a = 2a
a × a = a² (same base, add exponents) | 同底数幂相乘，指数相加：a × a = a²
(am)(an) = am+n (product of powers) | 幂的乘积：(am)(an) = am+n
am ÷ an = am−n (quotient of powers) | 幂的除法：am ÷ an = am−n
(am)n = amn (power of a power) | 幂的乘方：(am)n = amn
a0 = 1 (any non‑zero base raised to 0 equals 1) | 任何非零底数的 0 次幂等于 1
a⁻¹ = 1/a (negative exponent means reciprocal) | 负指数表示倒数：a⁻¹ = 1/a
(ab)n = an bn (power of a product) | 积的乘方：(ab)n = an bn

When expanding brackets, always multiply every term inside by the factor outside: a(b + c) = ab + ac. Some more advanced expansions include the difference of two squares: (a + b)(a − b) = a² − b².

去括号时，总是用括号外的因数乘以括号内的每一项：a(b + c) = ab + ac。一些更高级的展开包括平方差公式：(a + b)(a − b) = a² − b²。

2. Solving Linear Equations | 解一元一次方程

A linear equation in one variable can be written in the form ax + b = c. The goal is to isolate the variable by performing the same operation on both sides of the equation. Always keep the balance – what you do to one side, do to the other.

一元一次方程可写成 ax + b = c 的形式。目标是对方程两边进行相同的运算，从而将变量隔离出来。始终保持平衡——对一边所做的，另一边也必须做。

Addition/subtraction steps: x + 5 = 12 ⇒ x = 12 − 5 = 7
乘法/除法步骤： 3x = 15 ⇒ x = 15 ÷ 3 = 5
方程式包含括号： 先展开括号，如 2(x + 3) = 10 ⇒ 2x + 6 = 10 ⇒ x = 2
变量出现在方程两边： 将含变量的项移到同一侧，常数移到另一侧。例如：5x + 2 = 3x + 8 ⇒ 5x − 3x = 8 − 2 ⇒ 2x = 6 ⇒ x = 3

Always check your solution by substituting it back into the original equation. If both sides give the same value, the solution is correct.

始终将求得的解代入原方程检验。如果两边得出的值相同，则解正确。

3. Inequalities | 不等式

An inequality compares two expressions using signs such as <, >, ≤, ≥. Solving an inequality is similar to solving an equation, but remember: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

不等式使用 <、>、≤、≥ 等符号来比较两个表达式。解不等式与解方程类似，但要记住：如果两边同时乘以或除以一个负数，必须反转不等号方向。

x + 4 > 9 ⇒ x > 5 (subtracting 4 from both sides does not change the sign)
2x ≤ 10 ⇒ x ≤ 5 (dividing by positive 2, sign unchanged)
−3x < 12 ⇒ x > −4 (dividing by −3 reverses the < to >)

Inequalities can be shown on a number line: an open circle means the endpoint is not included (< or >), a closed circle means it is included (≤ or ≥). Also, double inequalities like −2 < x ≤ 3 represent a range of values.

不等式可以在数轴上表示：空心圆表示端点不包含在内（< 或 >），实心圆表示端点包含在内（≤ 或 ≥）。此外，像 −2 < x ≤ 3 这样的双重不等式表示一个取值范围。

4. Pythagoras’ Theorem | 毕达哥拉斯定理

In a right‑angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This is written as c² = a² + b², where c is the hypotenuse and a, b are the legs.

在直角三角形中，斜边（直角对边）的平方等于另外两条边的平方和。这写作 c² = a² + b²，其中 c 为斜边，a、b 为直角边。

To find the length of the hypotenuse: c = √(a² + b²). To find a shorter side: a = √(c² − b²) or b = √(c² − a²). Always check that the triangle is right‑angled before using the theorem.

求斜边长度：c = √(a² + b²)。求一条直角边：a = √(c² − b²) 或 b = √(c² − a²)。使用定理前务必确认该三角形是直角三角形。

Example: For a triangle with legs 3 cm and 4 cm, the hypotenuse is √(3² + 4²) = √(9 + 16) = √25 = 5 cm. This is the classic 3‑4‑5 Pythagorean triple.

示例： 直角边为 3 cm 和 4 cm 的三角形，斜边为 √(3² + 4²) = √(9 + 16) = √25 = 5 cm。这是经典的 3‑4‑5 毕达哥拉斯三元组。

5. Area and Perimeter of Plane Figures | 平面图形的面积与周长

Knowing how to calculate area and perimeter is essential for solving measurement problems. The area is the amount of surface covered, measured in square units; the perimeter is the total distance around the shape.

懂得如何计算面积和周长对于解决测量问题至关重要。面积是覆盖表面的量度，以平方单位表示；周长是围绕图形一周的总距离。

Shape / 图形	Perimeter / 周长	Area / 面积
Rectangle (length l, width w) / 长方形	P = 2(l + w)	A = lw
Square (side s) / 正方形	P = 4s	A = s²
Triangle (base b, height h) / 三角形	Sum of three sides	A = ½ bh
Parallelogram (base b, height h) / 平行四边形	2(a + b) where a, b are adjacent sides	A = bh
Trapezium (parallel sides a, b; height h) / 梯形	Sum of all four sides	A = ½ (a + b)h
Circle (radius r) / 圆	Circumference C = 2πr	A = πr²

For composite shapes, divide the figure into simpler shapes, calculate individual areas and add them. For the circle, π is approximately 3.14 or 22/7 unless a more precise value is requested.

对于组合图形，可将图形拆分成简单图形，分别计算面积再求和。对于圆，π 通常近似取 3.14 或 22/7，除非题目要求更精确的值。

6. Surface Area and Volume of Solids | 立体图形的表面积与体积

Three‑dimensional shapes have surface area (total area of all faces) and volume (space occupied). Know these key formulas for prisms and cylinders.

三维图形有表面积（所有面的总面积）和体积（所占空间大小）。牢记以下关于棱柱和圆柱的关键公式。

Cube of side s: Volume V = s³, Surface Area SA = 6s²
正方体（边长 s）： 体积 V = s³，表面积 SA = 6s²
Cuboid (length l, width w, height h): V = lwh, SA = 2(lw + lh + wh)
长方体（长 l，宽 w，高 h）： V = lwh，SA = 2(lw + lh + wh)
Prism (any uniform cross‑section): V = area of cross‑section × length
棱柱（任何均匀截面）： 体积 = 截面积 × 长度
Cylinder (radius r, height h): V = πr²h, Curved Surface Area = 2πrh, Total SA = 2πr(r + h)
圆柱体（半径 r，高 h）： 体积 V = πr²h，侧面积 = 2πrh，总表面积 = 2πr(r + h)

Remember that volume is measured in cubic units (e.g., cm³, m³) and surface area in square units (e.g., cm², m²). Always use consistent units before calculating.

记住体积以立方单位量度（如 cm³、m³），表面积以平方单位量度（如 cm²、m²）。在计算之前始终统一单位。

7. Ratio, Proportion and Scale | 比、比例与比例尺

A ratio compares the sizes of two or more quantities. It is written as a : b and can be simplified like fractions. A proportion is an equation stating two ratios are equal: a/b = c/d.

比用于比较两个或多个量的大小，写作 a : b，可以像分数一样化简。比例是一个说明两个比相等的等式：a/b = c/d。

Simplifying a ratio: 8 : 12 = 2 : 3 (divide by HCF 4)
化简比： 8 : 12 = 2 : 3（除以最大公因数 4）
Dividing a quantity in a given ratio: Share £50 in the ratio 2:3 ⇒ total parts = 5, one part = £10, so £20 and £30.
按给定比分配： 按 2:3 分配 50 英镑 ⇒ 总份数 5，每份 10 英镑，故为 20 英镑和 30 英镑。
Direct proportion: y ∝ x means y = kx for a constant k. If x doubles, y doubles.
正比例： y ∝ x 表示 y = kx（k 为常数）。若 x 翻倍，y 也翻倍。
Inverse proportion: y ∝ 1/x means y = k/x. If x doubles, y halves.
反比例： y ∝ 1/x 表示 y = k/x。若 x 翻倍，y 减半。

Scale drawings use a scale factor. A scale of 1 : n means that 1 unit on the drawing represents n units in real life. Area scale factor = (linear scale factor)².

比例图使用比例尺。1 : n 的比例尺表示图上 1 单位代表实际 n 单位。面积比例系数 =（线性比例系数）²。

8. Percentages | 百分数

A percentage is a fraction with denominator 100. To convert a percentage to a decimal, divide by 100. To find a percentage of an amount, multiply the amount by the percentage as a decimal.

百分数是分母为 100 的分数。将百分数转换为小数，需除以 100。求一个数的百分之多少，用该数乘以百分数的小数形式。

Percentage increase: New value = original × (1 + percentage/100)
增加百分数： 新值 = 原值 × (1 + 百分数/100)
Percentage decrease: New value = original × (1 − percentage/100)
减少百分数： 新值 = 原值 × (1 − 百分数/100)
Reverse percentage: If a price after a 20% discount is £80, original price = £80 ÷ 0.8 = £100.
逆推百分数： 若打八折后价格为 80 英镑，原价 = 80 ÷ 0.8 = 100 英镑。
Simple interest: I = P × r × t, where P is principal, r is rate per year (decimal), t is time in years.
单利： I = P × r × t，其中 P 为本金，r 为年利率（小数），t 为时间（年）。

Be comfortable switching between fractions, decimals and percentages. For example, 3/5 = 0.6 = 60%. This fluency helps in solving proportion and probability problems.

要能熟练地在分数、小数和百分数之间进行转换。例如，3/5 = 0.6 = 60%。这种熟练度有助于解决比例和概率问题。

9. Statistics: Averages and Charts | 统计：平均数与图表

Statistics involves collecting, organizing and interpreting data. The three common measures of central tendency are mean, median and mode. The range gives an idea of spread.

统计学涉及数据的收集、整理和解读。三种常见的集中趋势量数是平均数、中位数和众数。极差反映数据的分散程度。

Mean: sum of all values ÷ number of values
平均数： 总和 ÷ 数值的个数
Median: middle value when data is ordered; if even number of values, median = mean of the two middle numbers
中位数： 将数据排序后的中间值；若数据个数为偶数，则中位数为中间两数的平均数
Mode: the value that occurs most often (there may be more than one mode or none)
众数： 出现次数最多的数值（众数可能不止一个，也可能没有）
Range: largest value − smallest value
极差： 最大值 − 最小值

Data can be displayed in bar charts, pie charts, line graphs and stem‑and‑leaf diagrams. For grouped data, the modal class is the interval with the highest frequency. The mean of grouped data is estimated using the midpoint of each interval times its frequency, divided by total frequency.

数据可以用条形图、饼图、折线图和茎叶图表示。对于分组数据，众数所在的组是频数最高的区间。分组数据的平均数需要用每个区间的中间值乘以频数再除以总频数来估算。

10. Probability Basics | 概率基础

Probability measures how likely an event is to happen. It is a number between 0 (impossible) and 1 (certain). The probability of an event A is P(A) = number of favorable outcomes / total number of possible outcomes, provided all outcomes are equally likely.

概率衡量某一事件发生的可能性大小，是一个介于 0（不可能）和 1（必然）之间的数。若所有结果等可能发生，事件 A 的概率 P(A) = 有利结果数 / 所有可能结果的总数。

Sum of probabilities: The probabilities of all possible outcomes add up to 1.
概率之和： 所有可能结果的概率总和为 1。
Complement: P(not A) = 1 − P(A)
互补事件： P(非 A) = 1 − P(A)
Sample space: A list or table of all possible outcomes helps count favorable ones.
样本空间： 列出所有可能结果的列表或表格，有助于计算有利结果的数量。
Experimental probability: Estimated from trials: relative frequency = number of times event occurs / total number of trials.
实验概率： 通过试验估算：相对频率 = 事件发生次数 / 试验总次数。

For two independent events, the probability of both occurring is P(A and B) = P(A) × P(B). This multiplication rule does not apply if the events affect each other.

对于两个独立事件，两者同时发生的概率为 P(A 且 B) = P(A) × P(B)。如果事件互相影响，则不适用于此乘法法则。

11. Sequences and the nth Term | 数列与第 n 项公式

A sequence is an ordered list of numbers following a pattern. Arithmetic sequences have a constant difference between consecutive terms. The nth term formula allows you to find any term in the sequence without listing all previous ones.

数列是按一定规律排列的一列数。等差数列中相邻两项的差为常数。第 n 项公式能让你无需列出前面所有项即可直接求出数列的任意一项。

Arithmetic sequence nth term: T(n) = a + (n − 1)d, where a = first term, d = common difference.
等差数列第 n 项： T(n) = a + (n − 1)d，其中 a 为首项，d 为公差。
Example: 5, 8, 11, 14, … a = 5, d = 3, so T(n) = 5 + (n − 1)×3 = 3n + 2.
示例： 5, 8, 11, 14, … a = 5，d = 3，因此 T(n) = 5 + (n − 1)×3 = 3n + 2。
Finding the term: the 20th term is T(20) = 3×20 + 2 = 62.
求指定项： 第 20 项 T(20) = 3×20 + 2 = 62。

Other simple sequences include square numbers (1, 4, 9, 16, …), triangular numbers (1, 3, 6, 10, …) and Fibonacci‑type sequences where each term is the sum of the two previous ones.

其他简单数列包括平方数（1, 4, 9, 16, …）、三角形数（1, 3, 6, 10, …）以及类似斐波那契数列那样每一项是前两项之和的数列。

12. Angles and Geometric Reasoning | 角与几何推理

Understanding angle facts is crucial for solving geometric problems. Angles are measured in degrees. Key rules include angles on a straight line, around a point, and in triangles and quadrilaterals.

理解角的性质是解决几何问题的关键。角以度为单位。关键法则包括直线上的角、绕一点一周的角以及三角形和四边形中的角。

Angles on a straight line sum to 180°.
平角等于 180°（直线上相邻角之和为 180°）。
Angles around a point sum to 360°.
周角为 360°（绕一点一周的角总和为 360°）。
Vertically opposite angles are equal.
对顶角相等。
Angles in a triangle sum to 180°. In a right‑angled triangle, the two acute angles sum to 90°.
三角形内角和 为 180°。在直角三角形中，两个锐角互余，和为 90°。
Angles in a quadrilateral sum to 360°.
四边形内角和 为 360°。
Parallel lines: When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, interior angles sum to 180°.
平行线： 当一条截线与平行线相交时：同位角相等，内错角相等，同旁内角互补（和为 180°）。

These rules can be combined to find unknown angles in complex diagrams. Always look for lines with arrow marks to identify parallel lines.

这些法则可以组合使用，以求出复杂图形中的未知角。务必寻找标有箭头的直线以识别平行线。

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Year 8 CIE Advanced Mathematics Formula & Theorem Quick Reference Handbook | Year 8 CIE 进阶数学：公式定理速查手册