📚 Year 8 CIE Further Mathematics: Formula & Theorem Quick Reference | Year 8 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 you will encounter in Year 8 CIE Further Mathematics. Use it to revise efficiently, check your work, and deepen your understanding of key concepts. Every entry is paired with a clear explanation in both English and Chinese to support bilingual learners.
这份速查手册汇集了 Year 8 CIE 进阶数学中必学的公式、定理和法则。你可以用它高效复习、检查作业、加深对核心概念的理解。每一条目都配有清晰的中英文解释，帮助双语学生更好地掌握内容。
1. Algebraic Expressions and Simplification | 代数表达式的化简
Like terms can be combined by adding or subtracting their coefficients. The algebraic conventions of multiplication and division must be followed when simplifying expressions.
同类项可以通过相加减系数来合并。化简表达式时须遵循乘法和除法的代数书写惯例。
Combining like terms: ax + bx = (a + b)x, 3y - 2y = y
合并同类项：ax + bx = (a + b)x，3y - 2y = y
Multiplying terms: a × b = ab, 2x × 3y = 6xy, x × x = x²
乘法：a × b = ab，2x × 3y = 6xy，x × x = x²
Dividing terms: a ÷ b = a/b, 12x² ÷ 4x = 3x
除法：a ÷ b = a/b，12x² ÷ 4x = 3x
anding brackets: a(b + c) = ab + ac; always multiply each term inside by the term outside.
去括号展开：a(b + c) = ab + ac；必须将括号外的项乘以括号内的每一项。
2. Solving Linear Equations | 解一次方程
To solve a linear equation, perform inverse operations in reverse BIDMAS order, keeping the equation balanced by doing the same to both sides.
解一次方程时，要按照逆 BIDMAS 顺序进行逆运算，每一步都要在方程两边同时操作以保持平衡。
Addition/subtraction first: For x + 5 = 12, subtract 5 from both sides → x = 7.
先加减：对于 x + 5 = 12，两边同时减去 5 → x = 7。
Multiplication/division next: For 3x = 18, divide both sides by 3 → x = 6.
再乘除：对于 3x = 18，两边同时除以 3 → x = 6。
Two-step equation: 2x - 4 = 10 → add 4 → 2x = 14 → divide by 2 → x = 7.
两步方程：2x - 4 = 10 → 两边加 4 → 2x = 14 → 两边除以 2 → x = 7。
ons with brackets: Expand first, then solve. 2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4.
带括号的方程：先展开，再求解。2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4。
3. Linear Inequalities | 一次不等式
Linear inequalities are solved similarly to equations, but multiplying or dividing by a negative number reverses the inequality sign.
一次不等式的解法与方程类似，但当两边乘以或除以一个负数时，不等号方向要改变。
Solving: x + 3 > 7 → x > 4.
求解：x + 3 > 7 → x > 4。
Negative coefficient: -2x ≤ 10 → divide by -2, reverse sign → x ≥ -5.
负系数处理：-2x ≤ 10 → 除以 -2，不等号变向 → x ≥ -5。
Representing on a number line: Open circle for <, >; closed circle for ≤, ≥.
在数轴上表示：用空心点表示 <, >；实心点表示 ≤, ≥。
Consecutive inequalities: -1 < 2x + 1 ≤ 5 solve by splitting or isolating the middle part.
连不等式：-1 < 2x + 1 ≤ 5，可拆分求解或整体解出中间部分。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
Understanding interchange between fractions, decimals and percentages is essential. A percentage is a fraction out of 100.
掌握分数、小数和百分数之间的互相转换至关重要。百分数就是分母为 100 的分数。
Fraction to decimal: a/b = a ÷ b.
分数化小数：a/b = a ÷ b。
Decimal to percentage: Multiply by 100%. 0.45 = 45%.
小数化百分数：乘以 100%。0.45 = 45%。
Percentage of an amount: p% of Q = (p/100) × Q.
求一个量的百分之几：p% of Q = (p/100) × Q。
Percentage increase/decrease: New = Original × (1 ± r/100).
百分数增减：新值 = 原值 × (1 ± r/100)。
5. Ratio and Proportion | 比与比例
A ratio compares parts of a whole; a proportion shows that two ratios are equal. Unitary method is often used for direct proportion problems.
比表示部分与部分的比较；比例表示两个比相等。解决正比例问题时常用到单位量法。
Simplifying ratios: Divide all parts by their HCF. 12:8 = 3:2.
化简比：所有部分同时除以它们的最大公因数。12:8 = 3:2。
Sharing in a ratio: For a ratio a:b, the first share = (a/(a+b)) × total.
按比例分配：对于 a:b，第一份 = (a/(a+b)) × 总数。
Direct proportion: y ∝ x → y = kx. If 4 pens cost £3, 10 pens cost £7.50.
正比例：y ∝ x → y = kx。若 4 支笔 £3，则 10 支笔 £7.50。
Inverse proportion: y ∝ 1/x → y = k/x. Time to complete a job decreases as more people work.
反比例：y ∝ 1/x → y = k/x。工作人数越多，完成工作的时间越短。
6. Sequences and Patterns | 数列与规律
A sequence is a list of numbers following a rule. Term-to-term rules describe how to move from one term to the next; position-to-term rules give the nth term directly.
数列是按照某种规则排列的一列数字。项与项之间的规则描述如何从一项得到下一项；位置与项的规则直接给出第 n 项表达式。
Arithmetic sequence: nth term = a + (n - 1)d, where a = first term, d = common difference.
等差数列：第 n 项 = a + (n - 1)d，a 为首项，d 为公差。
Linear nth term: For sequence 5, 8, 11, 14,..., nth term = 3n + 2.
线性第 n 项：对于数列 5, 8, 11, 14,...，第 n 项 = 3n + 2。
Quadratic sequences: Second difference constant. nth term involves n², often found by comparing with square numbers.
二次数列：二阶差为常数。第 n 项含有 n²，常通过与平方数对比求得。
Other patterns: Fibonacci, triangular numbers. Recognise patterns and describe them algebraically.
其他规律：斐波那契数列、三角形数等。能识别规律并用代数描述。
7. Perimeter, Area and Volume Formulas | 周长、面积与体积公式
Know the formulas for standard 2D and 3D shapes. Always use consistent units and remember that area is measured in square units, volume in cubic units.
熟记常见二维和三维图形的公式。始终使用一致的单位，并记住面积用平方单位，体积用立方单位。
Rectangle: Perimeter = 2(l + w), Area = l × w.
矩形：周长 = 2(长 + 宽)，面积 = 长 × 宽。
Triangle: Area = ½ × base × height.
三角形：面积 = ½ × 底 × 高。
Circle: Circumference = 2πr or πd, Area = πr². (Use π ≈ 3.14 or calculator π.)
圆：周长 = 2πr 或 πd，面积 = πr²。（使用 π ≈ 3.14 或计算器上的 π。）
Parallelogram: Area = base × perpendicular height.
平行四边形：面积 = 底 × 高（垂直高度）。
Trapezium: Area = ½ × (a + b) × h, where a and b are parallel sides.
梯形：面积 = ½ × (上底 + 下底) × 高，其中 a 和 b 为平行边。
Cuboid: Volume = l × w × h, Surface area = 2(lw + wh + lh).
长方体：体积 = 长 × 宽 × 高，表面积 = 2(lw + wh + lh)。
Prism: Volume = area of cross-section × length.
棱柱：体积 = 横截面积 × 长。
8. Pythagoras’ Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem allows you to find a missing side length.
在直角三角形中，斜边的平方等于两直角边的平方和。利用该定理可以求出未知的边长。
a² + b² = c²
where c is the hypotenuse (斜边), a and b are the legs (直角边).
Finding the hypotenuse: If legs 3 and 4, c = √3² + 4² = √25 = 5.
求斜边：两直角边为 3 和 4，则 c = √3² + 4² = √25 = 5。
Finding a leg: If c = 13, a = 5, then b = √13² - 5² = √144 = 12.
求直角边：c = 13，a = 5，则 b = √13² - 5² = √144 = 12。
Checking for a right angle: If a² + b² = c², the triangle is right-angled.
判断直角：如果 a² + b² = c²，则该三角形为直角三角形。
9. Coordinate Geometry and Graphs | 坐标几何与函数图像
The Cartesian plane uses ordered pairs (x, y). Straight-line graphs represent linear relationships and can be analysed through their gradients and intercepts.
平面直角坐标系使用有序数对 (x, y)。直线图表示线性关系，可通过斜率和截距进行分析。
Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂)/2, (y₁ + y₂)/2).
中点坐标：(x₁, y₁) 与 (x₂, y₂) 的中点 = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Gradient (slope) = (y₂ - y₁)/(x₂ - x₁).
斜率（梯度）= (y₂ - y₁)/(x₂ - x₁)。
Equation of a straight line: y = mx + c, where m is gradient, c is y-intercept.
直线方程：y = mx + c，m 为斜率，c 为 y 轴截距。
ines have the same gradient; perpendicular lines have gradients product -1 (m₁ × m₂ = -1).
平行直线斜率相等；垂直直线斜率乘积为 -1 (m₁ × m₂ = -1)。
Finding equation given one point and gradient: y - y₁ = m(x - x₁).
已知一点和斜率求方程：y - y₁ = m(x - x₁)。
10. Transformations and Symmetry | 变换与对称
Transformations move or change a shape. Reflection, rotation, translation and enlargement are the four basic types. Understand how to describe each fully.
变换可以移动或改变图形。反射、旋转、平移和放大是四种基本变换类型。要能完整描述每一种变换。
Reflection: Mirror line is the perpendicular bisector of lines joining corresponding points.
反射：对称轴（镜像线）是对应点连线的垂直平分线。
Rotation: Requires centre of rotation, angle and direction (clockwise/anticlockwise).
旋转：需要指明旋转中心、旋转角度及方向（顺时针/逆时针）。
Translation: Described by a column vector (x above y).
平移：用列向量（上方为 x 位移，下方为 y 位移）描述。
Enlargement: Requires centre of enlargement and scale factor k. Coordinates multiply by k from the centre.
放大：需要放大中心和比例因子 k。各点到放大中心的距离乘以 k。
Symmetry: Line symmetry (reflective) and rotational symmetry. Order of rotational symmetry is how many times a shape maps onto itself in a full turn.
对称：线对称（反射对称）与旋转对称。旋转对称的阶数是指图形在旋转一周的过程中与自身重合的次数。
11. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数及极差
These measures summarise data sets. The mean uses all values; the median gives the middle; the mode is the most frequent; range shows spread.
这些统计量可以概括一组数据。平均数用到所有数值；中位数是中间值；众数是出现次数最多的值；极差表示离散程度。
Mean = (sum of all values) ÷ (number of values).
算术平均数 = 所有数据之和 ÷ 数据个数。
Order data, then pick the middle. If even number, median = mean of the two middle values.
中位数：将数据排序后，取中间的那个数。若数据个数为偶数，中位数为中间两数的平均数。
Mode: Value(s) that appear most often; a set can have one mode, more than one mode, or no mode.
众数：出现次数最多的数值；一组数据可有一个众数、多个众数，或没有众数。
Range = maximum value – minimum value.
极差 = 最大值 – 最小值。
Frequency tables: Mean = Σ(f×x) / Σf, where x is data value, f is frequency.
频数表：平均数 = Σ(频数×数据值) / 总频数。
12. Probability | 概率
ally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率用来度量事件发生的可能性，范围从 0（不可能）到 1（必然）。对于等可能结果，概率 P(事件) = 有利结果数 / 所有可能结果数。
Basic formula: P(A) = n(A) / n(S), where S is the sample space.
基本公式：P(A) = 事件 A 的结果数 / 样本空间的结果总数。
Complement: P(not A) = 1 - P(A).
互补事件：P(非 A) = 1 - P(A)。
Expected frequency: Expected number of times = P(event) × number of trials.
期望频数：预期出现次数 = 事件概率 × 试验次数。
Mutually exclusive events: P(A or B) = P(A) + P(B) if A and B cannot happen together.
互斥事件：若 A 和 B 不可能同时发生，则 P(A 或 B) = P(A) + P(B)。
Sample space diagrams and tree diagrams help list outcomes systematically.
样本空间图和树形图有助于系统地列出所有可能结果。
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