📚 KS3 AQA Further Maths: Formula & Theorem Quick Reference Handbook | KS3 AQA 进阶数学：公式定理速查手册
Mastering Further Maths at KS3 means building a solid toolkit of formulas and theorems that you can recall instantly. This quick reference handbook brings together all the essential rules, identities, and relationships you will need for the AQA syllabus – from simplifying algebraic expressions to applying circle theorems. Each entry is explained clearly and paired with practical tips so you can use them confidently in classwork, homework, and assessments.
掌握 KS3 进阶数学意味着构建一个能够随时调用的公式与定理工具箱。本速查手册汇集了 AQA 教学大纲所需的所有基本规则、恒等式和关系式——从化简代数表达式到应用圆定理。每个条目都配有清晰的解释和实用提示，帮助你在课堂练习、作业和考试中自信运用。
1. Algebraic Expressions and Simplification | 代数表达式与化简
When working with algebra, like terms can be combined by adding or subtracting their coefficients. For example, 3x + 5x simplifies to 8x, while 2x² + 3x² gives 5x².
在处理代数时，同类项可以通过加减它们的系数来合并。例如，3x + 5x 化简为 8x，而 2x² + 3x² 得到 5x²。
multiplying terms, multiply the numbers and the variables separately, e.g. 4a × 3b = 12ab.
乘法和除法遵循指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ。进行单项式乘法时，数字和字母分别相乘，例如 4a × 3b = 12ab。
Double brackets are expanded using the FOIL technique: (a + b)(c + d) = ac + ad + bc + bd.
展开单项式括号时，将括号内的每一项乘以括号外的项：a(b + c) = ab + ac。双括号展开使用乘法分配律：(a + b)(c + d) = ac + ad + bc + bd。
For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆过程。首先找出公因式，例如 6x² + 9x = 3x(2x + 3)。对于形如 x² + bx + c 的二次式，找到两个相乘得 c、相加得 b 的数。
2. Solving Linear Equations and Inequalities | 解线性方程与不等式
e 2x + 5 = 13, subtract 5 from both sides to get 2x = 8, then divide by 2 to obtain x = 4.
解线性方程可以通过在等号两边执行相同的运算来隔离未知数。例如，解 2x + 5 = 13，两边同时减5得 2x = 8，再除以2得 x = 4。
ntains brackets, expand them first: 3(x - 2) = 9 becomes 3x - 6 = 9, so 3x = 15 and x = 5.
若方程含有括号，先展开：3(x - 2) = 9 变成 3x - 6 = 9，因此 3x = 15，x = 5。
we multiply or divide by a negative number. Solve −2x ≤ 6 by dividing by −2 to get x ≥ −3.
解不等式时步骤相同，但如果乘以或除以一个负数，不等号方向需改变。求解 −2x ≤ 6，两边除以 −2 得 x ≥ −3。
Solutions can be represented on a number line with open or closed circles. Solving a pair of simultaneous equations can be done by elimination or substitution.
解集可以用数轴上的空心或实心圆点表示。解联立方程组可以使用消元法或代入法。
3. Quadratic Expressions and Factorising | 二次表达式与因式分解
wo numbers whose product is +10 and sum is +7: 2 and 5. So, x² + 7x + 10 = (x + 2)(x + 5).
二次表达式的一般形式为 ax² + bx + c。要将 x² + 7x + 10 因式分解，需要找到两个乘积为 +10 且和为 +7 的数：2和5。因此，x² + 7x + 10 = (x + 2)(x + 5)。
When the coefficient of x² is not 1, we can use the 'splitting the middle term' method or trial and improvement. For example, 2x² + 5x + 3 factorises to (2x + 3)(x + 1).
当 x² 的系数不为1时，可采用“中间项拆分”法或试凑法。例如，2x² + 5x + 3 因式分解为 (2x + 3)(x + 1)。
n: a² − b² = (a + b)(a − b). So x² − 9 = (x + 3)(x − 3). Recognising this rule saves time.
平方差是一种特殊模式：a² − b² = (a + b)(a − b)。所以 x² − 9 = (x + 3)(x − 3)。识别这一规则能节省时间。
The quadratic formula gives solutions to ax² + bx + c = 0: x = [−b ± √b² − 4ac] / 2a. The discriminant b² − 4ac tells us how many real solutions exist.
二次公式给出 ax² + bx + c = 0 的解：x = [−b ± √b² − 4ac] / 2a。判别式 b² − 4ac 告诉我们存在多少个实数解。
4. Sequences and the nth Term | 数列与第n项
the formula aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference.
等差数列的相邻项之差为常数。使用公式 aₙ = a₁ + (n − 1)d 可求出第n项，其中 a₁ 为首项，d 为公差。
the difference is 2, the nth term is 3 + (n − 1)×2 = 2n + 1. The sequence goes 3, 5, 7, 9, …
若首项为3，公差为2，则第n项为 3 + (n − 1)×2 = 2n + 1。该数列为 3, 5, 7, 9, …
For quadratic sequences where the second difference is constant, the nth term has the form an² + bn + c. Work out the second difference, halve it to find a, and then solve for b and c.
对于二次差为常数的二次数列，第n项具有 an² + bn + c 的形式。先求出二次差，除以2得到 a，再解出 b 和 c。
. The nth term is a₁ × rⁿ⁻¹. Recognising pattern types helps you choose the right formula.
等比数列每步乘以固定的公比 r。第n项为 a₁ × rⁿ⁻¹。识别数列类型有助于选择正确的公式。
5. Indices and Standard Form | 指数与标准形式
for simplifying expressions: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a¹/ⁿ = ⁿ√a.
指数法则是化简表达式的关键：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a¹/ⁿ = ⁿ√a。
dices indicate reciprocals: 5⁻² = 1/5² = 1/25. Fractional indices link to roots: 27¹/³ = ³√27 = 3. You can combine them: 8²/³ = (³√8)² = 2² = 4.
负指数表示倒数：5⁻² = 1/5² = 1/25。分数指数与方根相关：27¹/³ = ³√27 = 3。可以组合使用：8²/³ = (³√8)² = 2² = 4。
4500 = 4.5 × 10³; 0.0032 = 3.2 × 10⁻³. It is useful for very large or very small numbers.
标准形式将一个数写为 a × 10ᵐ，其中 1 ≤ a < 10，m 为整数。4500 = 4.5 × 10³；0.0032 = 3.2 × 10⁻³。这适用于极大或极小的数。
When adding or subtracting in standard form, convert to the same power of 10 first. Multiplying: multiply the a values and add the exponents. Dividing: divide the a values and subtract the exponents.
在标准形式下进行加减运算，需先转换为相同的10的幂。乘法：将 a 值相乘并指数相加。除法：将 a 值相除并指数相减。
6. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数
ls the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse.
在直角三角形中，勾股定理指出斜边的平方等于两直角边的平方和：a² + b² = c²，其中 c 为斜边。
To find a missing hypotenuse, use c = √a² + b². To find a shorter side, use a = √c² − b². Always identify the hypotenuse first – it is opposite the right angle.
求缺失的斜边，用 c = √a² + b²。求一条直角边，用 a = √c² − b²。务必先确认斜边——它正对着直角。
e/hypotenuse (SOH), cosine = adjacent/hypotenuse (CAH), tangent = opposite/adjacent (TOA).
三角函数比值联系边与角：正弦 = 对边/斜边 (SOH)，余弦 = 邻边/斜边 (CAH)，正切 = 对边/邻边 (TOA)。
so if the hypotenuse is 10, the opposite side is 5. Inverse trig functions find the angle.
使用计算器可以求未知边或角度。例如，sin 30° = 0.5，所以若斜边为10，则对边为5。反三角函数用于求角度。
7. Area, Perimeter and Volume | 面积、周长与体积
lar height. The area of a trapezium is ½(a + b) × h, where a and b are the parallel sides.
三角形面积 = ½ × 底 × 高。平行四边形面积 = 底 × 垂直高。梯形面积 = ½(a + b) × h，其中 a 和 b 为平行边。
r is the radius and d is the diameter. Leave answers in terms of π unless asked otherwise.
对于圆，周长 = 2πr 或 πd，面积 = πr²。记住 r 是半径，d 是直径。除非特别要求，可保留 π 的形式作答。
me = length × width × height. A cylinder’s volume is πr²h and a sphere’s volume is ⁴/₃πr³.
棱柱的体积 = 横截面积 × 长。长方体的体积 = 长 × 宽 × 高。圆柱体积为 πr²h，球体积为 ⁴/₃πr³。
h + 2πr². Know the compound measures like speed = distance/time and density = mass/volume.
表面积需计算所有面的总面积。圆柱的表面积为 2πrh + 2πr²。掌握复合量度，如速度 = 距离/时间，密度 = 质量/体积。
8. Angles and Circle Theorems | 角与圆定理
Angles on a straight line sum to 180°, around a point sum to 360°. Vertically opposite angles are equal. The sum of angles in a triangle is 180°, and in a quadrilateral is 360°.
直线上的角之和为180°，绕一点一周的角之和为360°。对顶角相等。三角形的内角和为180°，四边形的内角和为360°。
In parallel lines, alternate angles are equal, corresponding angles are equal, and interior angles sum to 180°. Use these to find missing angles in diagrams.
在平行线中，内错角相等，同位角相等，同旁内角之和为180°。利用这些性质可求图形中的未知角。
Circle theorems: the angle in a semicircle is 90°; the angle at the centre is twice the angle at the circumference; angles in the same segment are equal.
圆定理：半圆内的圆周角为90°；圆心角等于两倍圆周角；同弧上的圆周角相等。
A tangent is perpendicular to the radius at the point of contact. The alternate segment theorem says the angle between a tangent and a chord equals the angle in the alternate segment.
切线与过切点的半径垂直。弦切角定理表明切线与弦的夹角等于弦所对的圆周角（在交互弧段上）。
9. Probability and Statistics | 概率与统计
tcomes. Probabilities add up to 1. For mutually exclusive events, P(A or B) = P(A) + P(B).
事件的概率 = 有利结果的数量 / 所有等可能结果的总数。概率之和为1。对于互斥事件，P(A 或 B) = P(A) + P(B)。
Tree diagrams help show combined events: multiply along branches and add across outcomes.
对于独立事件，P(A 和 B) = P(A) × P(B)。树状图有助于展示组合事件：沿着分支相乘，再将不同路径的结果相加。
The mean of a set of data is sum of values ÷ number of values. The median is the middle value; the mode is the most frequent; the range is highest minus lowest.
一组数据的平均数 = 总和 ÷ 数据个数。中位数是中间值；众数是出现频率最高的值；极差 = 最大值 − 最小值。
For grouped data, estimate the mean by using midpoints. A cumulative frequency graph helps find the median and quartiles. Box plots display the five-number summary.
对于分组数据，使用组中值来估算平均数。累积频率图有助于找到中位数和四分位数。箱线图展示五数概况。
10. Ratio, Proportion and Percentage | 比、比例与百分比
A ratio compares parts, e.g. 2:3. To share £50 in the ratio 2:3, add the parts (2+3=5), then one part is £10, so the shares are £20 and £30.
比用于比较部分，例如 2:3。按 2:3 分配50英镑，部分之和为2+3=5，每份10英镑，因此分配为20英镑和30英镑。
roportion means y = k/x; as x increases, y decreases. Use the constant k to find unknowns.
正比例意味着 y = kx；如果 x 翻倍，y 也翻倍。反比例意味着 y = k/x；x 增加时 y 减少。利用常数 k 求未知量。
Percentages: to find a percentage of an amount, multiply by the percentage divided by 100. To increase by 15%, multiply by 1.15; to decrease by 15%, multiply by 0.85.
百分比：求一个数的百分之几，乘以该百分比除以100。增加15%即乘以1.15；减少15%即乘以0.85。
Percentage change = (change/original) × 100% , R is rate, T is time in years. Compound interest multiplies by (1 + r/100)ⁿ each period.
变化百分比 = (变化量/原值) × 100%。单利使用 I = PRT/100，其中 P 为本金，R 为年利率，T 为年数。复利每期乘以 (1 + r/100)ⁿ。
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