📚 KS3 Advanced Maths: Pre-exam Revision Notes | KS3 进阶数学：考前冲刺笔记
This revision guide covers essential advanced topics for Key Stage 3 Mathematics. It is designed to help you consolidate your knowledge, practise key skills, and approach exams with confidence. Each section presents concepts in clear English explanations followed by Chinese translations, making it ideal for bilingual learners.
这份复习指南涵盖了KS3数学进阶阶段的核心内容，旨在帮助你巩固知识、练习关键技能，并自信地应对考试。每一部分都以清晰的英文解释配合中文翻译，适合双语学习者使用。
1. Number Systems and Operations | 数系与运算
Understand the real number system, including integers, rational numbers, and irrational numbers. Rational numbers can be expressed as fractions a/b where a and b are integers (b ≠ 0). Irrational numbers, such as π and √2, cannot be written as exact fractions.
理解实数系统，包括整数、有理数和无理数。有理数可以表示为分数 a/b，其中 a 和 b 为整数且 b ≠ 0。无理数如 π 和 √2 无法写成精确的分数。
Master operations with negative numbers: adding a negative is subtracting, subtracting a negative is adding. Multiplication and division follow sign rules: same signs give positive, different signs give negative.
掌握负数的运算：加上一个负数等于减去其绝对值，减去一个负数等于加上其绝对值。乘法和除法遵循符号规则：同号得正，异号得负。
Explore powers and roots. For any non-zero a, a⁰ = 1. The index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Negative indices represent reciprocals: a⁻ⁿ = 1/aⁿ.
探索幂与根。对于任何非零的 a，a⁰ = 1。指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。负指数表示倒数：a⁻ⁿ = 1/aⁿ。
Standard form writes large or small numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. For example, 45 000 = 4.5 × 10⁴, and 0.00032 = 3.2 × 10⁻⁴.
标准形式将很大或很小的数写为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。例如，45000 = 4.5 × 10⁴，0.00032 = 3.2 × 10⁻⁴。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Convert between fractions, decimals and percentages fluently. To change a fraction to a decimal, divide numerator by denominator. To change a decimal to a percentage, multiply by 100. To express a percentage as a fraction, write it over 100 and simplify.
熟练进行分数、小数与百分比之间的转换。将分数化为小数，用分子除以分母。将小数化为百分数，乘以 100。将百分数写成分数，以 100 为分母并化简。
Perform fraction operations: for addition/subtraction, find a common denominator; for multiplication, multiply numerators and denominators; for division, multiply by the reciprocal.
进行分数运算：加减法时，先求公分母；乘法时，分子相乘、分母相乘；除法时，乘以除数的倒数。
Solve percentage increase and decrease problems using multipliers. A 15% increase corresponds to a multiplier of 1.15; a 10% decrease uses 0.9. Compound changes apply multipliers sequentially.
用乘数法解决百分比增减问题。增长 15% 对应的乘数为 1.15；减少 10% 则使用 0.9。复合变化需要依次应用乘数。
Express one quantity as a percentage of another: (part/whole) × 100%. For example, 18 out of 60 is (18/60)×100% = 30%.
将一个量表示为另一个量的百分比：(部分/整体) × 100%。例如，18 占 60 的百分之几？(18/60)×100% = 30%。
3. Ratio and Proportion | 比与比例
Write ratios in simplest form, dividing all parts by common factors. A ratio 12:8 simplifies to 3:2. Ratios can compare more than two quantities, e.g., 2:3:5.
将比写成最简形式，各部分除以公因数。比例 12:8 化简为 3:2。比例可以比较两个以上的量，例如 2:3:5。
Share a quantity in a given ratio. For a total of £90 split in ratio 2:3, the total number of parts is 5. One part = £90 ÷ 5 = £18, so shares are 2×£18 = £36 and 3×£18 = £54.
按给定比例分配量。将 £90 按 2:3 分配，总份数为 5。一份为 £90 ÷ 5 = £18，因此分配额为 2×£18 = £36 和 3×£18 = £54。
Understand direct proportion: y is directly proportional to x if y = kx for a constant k. Graphs of direct proportion are straight lines through the origin. Inverse proportion: y = k/x, giving a curve that never touches axes.
理解正比例：如果 y = kx（k 为常数），则 y 与 x 成正比。正比例图形是过原点的直线。反比例：y = k/x，形成不与坐标轴接触的曲线。
Solve ratio problems involving scaling up recipes or maps. Use the unitary method: find the value of one unit first, then scale to the required amount.
解决涉及配方或地图缩放的比例问题。使用归一法：先求出一份的量，再缩放到所需数量。
4. Algebraic Expressions and Simplification | 代数表达式与化简
Combine like terms: terms with identical variable parts can be added or subtracted. For example, 3x + 5x = 8x, but 3x + 5y cannot be combined further.
合并同类项：变量部分相同的项可以相加减。例如，3x + 5x = 8x，但 3x + 5y 不能进一步合并。
Expand brackets using the distributive law: a(b + c) = ab + ac. For double brackets, like (x + 3)(x + 2), multiply each term: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6.
使用分配律展开括号：a(b + c) = ab + ac。对于双括号，如 (x + 3)(x + 2)，逐项相乘：(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6。
Factorise expressions by identifying the highest common factor (HCF). For 4x² + 12x, HCF is 4x, so 4x(x + 3). For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.
通过找出最高公因式 (HCF) 进行因式分解。对于 4x² + 12x，HCF 为 4x，故 4x(x + 3)。对于形如 x² + bx + c 的二次式，寻找两个数使其积为 c，和为 b。
Simplify algebraic fractions by factorising and cancelling common factors. (x²−9)/(x+3) = (x−3)(x+3)/(x+3) = x−3, provided x ≠ −3.
通过因式分解并约去公因式来化简代数分式。(x²−9)/(x+3) = (x−3)(x+3)/(x+3) = x−3，但要求 x ≠ −3。
5. Solving Equations and Inequalities | 解方程与不等式
Solve linear equations by performing the same operation on both sides to isolate the variable. Always check your answer by substitution. Example: 2x + 5 = 13 ⇒ 2x = 8 ⇒ x = 4.
解一元一次方程，两边同时进行相同运算以分离变量。务必代入答案检验。示例：2x + 5 = 13 ⇒ 2x = 8 ⇒ x = 4。
Tackle equations with brackets and fractions: expand brackets first, then eliminate fractions by multiplying through by the common denominator.
解决带括号和分数的方程：先展开括号，然后两边乘以公分母消去分数。
Solve simultaneous linear equations using elimination or substitution. For 2x + y = 7 and x − y = 2, add to eliminate y: 3x = 9 ⇒ x = 3, then y = 1.
使用消元法或代入法解联立方程组。对于 2x + y = 7 和 x − y = 2，相加消去 y：3x = 9 ⇒ x = 3，再得 y = 1。
Solve inequalities, remembering to reverse the sign when multiplying or dividing by a negative. 3 − 2x < 7 ⇒ −2x < 4 ⇒ x > −2. Represent solutions on a number line.
解不等式，注意乘或除以负数时要反转不等号。3 − 2x < 7 ⇒ −2x < 4 ⇒ x > −2。用数轴表示解集。
6. Sequences and Patterns | 数列与规律
Find the nth term of arithmetic sequences. The nth term formula is of the form an + b, where a is the common difference. For sequence 5, 8, 11, 14,…, difference = 3, so nth term = 3n + 2.
求等差数列的第 n 项。第 n 项公式形如 an + b，其中 a 为公差。对于数列 5, 8, 11, 14,…，公差为 3，故第 n 项 = 3n + 2。
Recognise quadratic sequences where the second difference is constant. For sequence 2, 6, 12, 20,…, first differences: 4,6,8; second difference = 2. The nth term is n² + n.
识别二次差为常数的二阶等差数列。若数列 2, 6, 12, 20,…，一次差：4,6,8；二次差为 2。第 n 项为 n² + n。
Use term-to-term rules, such as Fibonacci-like sequences: next term = sum of previous two. For 1, 1, 2, 3, 5, …
使用递推规则，如类斐波那契数列：下一项等于前两项之和。例如 1, 1, 2, 3, 5, …
Find whether a number is in a sequence by setting the nth term equal to it and solving for n. n must be a positive integer.
判断一个数是否在某数列中，可令第 n 项等于该数并解出 n，n 必须为正整数。
7. Geometry: Angles and Polygons | 几何：角与多边形
Recall angle facts: angles on a straight line sum to 180°, around a point sum to 360°, vertically opposite angles are equal. Use these to find missing angles.
回顾角度关系：直线上的夹角和为 180°，一点周角为 360°，对顶角相等。利用这些求未知角。
In parallel lines, corresponding angles are equal, alternate angles are equal, and allied (co-interior) angles sum to 180°. These appear in ‘Z’, ‘F’ and ‘C’ patterns.
在平行线中，同位角相等，内错角相等，同旁内角互补（和为 180°）。它们分别呈 ‘Z’、’F’ 和 ‘C’ 字形。
Interior angles of polygons: sum of interior angles of an n-sided polygon = (n−2)×180°. Each interior angle of a regular polygon = (n−2)×180°/n. Exterior angles of any convex polygon sum to 360°.
多边形的内角：n 边形内角和 = (n−2)×180°。正多边形每个内角 = (n−2)×180°/n。任何凸多边形的外角和均为 360°。
Use Pythagoras’ theorem in right-angled triangles: a² + b² = c², where c is the hypotenuse. Use it to find missing sides and to prove if a triangle is right-angled.
在直角三角形中使用勾股定理：a² + b² = c²，其中 c 为斜边。用于求未知边长，或证明三角形是否为直角三角形。
8. Perimeter, Area and Volume | 周长、面积与体积
Calculate perimeter of complex shapes by adding all side lengths. For circles, the circumference C = 2πr or πd, where r is radius and d is diameter.
计算组合图形的周长，将各边长相加。对于圆，周长 C = 2πr 或 πd，其中 r 为半径，d 为直径。
Area formulas: rectangle = length × width; triangle = ½ × base × vertical height; parallelogram = base × perpendicular height; trapezium = ½(a+b)h; circle area = πr².
面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 高；梯形 = ½(a+b)h；圆面积 = πr²。
Calculate areas of compound shapes by splitting into standard figures, finding individual areas and summing. For parts of circles, use fractions of πr² based on the sector angle.
计算组合图形的面积，可将其分割为标准图形，分别求面积再相加。对于部分圆，根据圆心角分数使用 πr² 的比例部分。
Volume of prisms: Volume = area
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