📚 GCSE AQA Maths: Mind Map Quick Revision | GCSE AQA 数学：思维导图速记
This mind map summary breaks down the entire GCSE AQA Maths syllabus into bite-sized, interconnected branches. Each section highlights the key concepts, formulas, and tips you need to memorise quickly and effectively for your exams.
这个思维导图摘要将GCSE AQA数学大纲分解成易于记忆的、相互关联的分支。每个部分都突出了你需要快速有效记忆的关键概念、公式和考试技巧。
1. Number Basics | 数字基础
Understand the types of numbers and basic operations that underpin many GCSE topics.
理解支撑许多GCSE主题的数字类型和基本运算。
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Integers, decimals, and negative numbers: rules for addition, subtraction, multiplication, division with negatives.
整数、小数和负数：负数加减乘除的规则。
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Factors, multiples, primes, HCF and LCM: use prime factor trees.
因数、倍数、质数、最高公因数与最小公倍数：使用质因数树。
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Powers (indices) and roots: laws of indices (aᵐ × aⁿ = aᵐ⁺ⁿ). Standard form (A × 10ⁿ).
幂（指数）与根：指数定律 (aᵐ × aⁿ = aᵐ⁺ⁿ)。标准形式 (A × 10ⁿ)。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Converting fluently between these three forms and performing calculations is essential.
在三种形式之间流畅转换并进行计算至关重要。
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Add, subtract, multiply, divide fractions: find common denominators, use ‘Keep, Change, Flip’ for division.
分数加减乘除：找公分母，除法使用“保、变、倒”。
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Convert fractions to decimals to percentages: e.g., ½ = 0.5 = 50%.
分数转小数转百分比：例如 ½ = 0.5 = 50%。
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Percentage increase/decrease, reverse percentages, compound interest.
百分比增减、逆向百分比、复利。
3. Algebraic Expressions | 代数表达式
Simplify, expand, and factorise algebraic expressions confidently.
自信地化简、展开和因式分解代数表达式。
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Collecting like terms, multiplying terms, using brackets: a(b + c) = ab + ac.
合并同类项、项相乘、使用括号：a(b + c) = ab + ac。
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Expand double brackets: (x + a)(x + b) = x² + (a+b)x + ab. Expand (a + b)² = a² + 2ab + b².
展开双括号：(x + a)(x + b) = x² + (a+b)x + ab。展开 (a + b)² = a² + 2ab + b²。
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Factorise quadratic expressions, difference of two squares: a² − b² = (a − b)(a + b).
因式分解二次表达式、平方差：a² − b² = (a − b)(a + b)。
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Algebraic fractions, rearrange formulas (change the subject).
代数分式、公式变形（改变主项）。
4. Equations and Inequalities | 方程与不等式
Solve linear and quadratic equations, simultaneous equations, and represent inequalities.
解线性方程、二次方程、方程组，并表示不等式。
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Solve linear equations with unknowns on both sides, including brackets and fractions.
解两边含未知数的线性方程，包括括号和分数。
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Solve quadratic equations: factorising, using the quadratic formula (x = [−b ± √(b²−4ac)] / 2a).
解二次方程：因式分解、使用求根公式 (x = [−b ± √(b²−4ac)] / 2a)。
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Simultaneous equations: elimination or substitution method, also graphical solution.
方程组：消元法或代入法，也可图解。
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Inequalities on a number line, solving linear inequalities, regions represented by inequalities.
数轴上的不等式，解线性不等式，由不等式表示的区域。
5. Graphs | 图形
Plot and interpret straight-line graphs, quadratic curves, and real-life graphs.
绘制并理解直线图、二次曲线和实际应用图形。
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Straight-line graphs: y = mx + c, gradient m, y-intercept c. Parallel and perpendicular lines.
直线图：y = mx + c，斜率 m，y轴截距 c。平行与垂直线。
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Quadratic graphs: y = x², interpreting roots and turning points.
二次曲线：y = x²，理解根和转折点。
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Distance-time and velocity-time graphs: gradient gives speed or acceleration, area under velocity-time gives distance.
距离-时间图和速度-时间图：斜率表示速度或加速度，速度-时间图下的面积表示距离。
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Transformations of graphs: y = f(x) + a, y = f(x + a), etc.
图形变换：y = f(x) + a，y = f(x + a) 等。
6. Ratio and Proportion | 比率与比例
Use ratios to compare quantities, share amounts, and solve proportion problems.
使用比率比较量、分配金额并解决比例问题。
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Simplify ratios, divide a quantity in a given ratio. Use ratio notation.
化简比率，按给定比率分配数量。使用比率符号。
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Direct and inverse proportion: y ∝ x (y = kx), y ∝ 1/x (y = k/x).
正比和反比：y ∝ x (y = kx)，y ∝ 1/x (y = k/x)。
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Scale drawings, maps, scale factors. Best buy and unitary method.
比例图、地图、比例尺。最优购买和单位方法。
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Compound measures: speed = distance/time, density = mass/volume, pressure = force/area.
复合单位：速度 = 距离/时间，密度 = 质量/体积，压强 = 力/面积。
7. Geometry Fundamentals | 几何基础
Master angle rules, properties of shapes, and circle theorems.
掌握角度规则、图形性质和圆定理。
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Angles on a straight line sum to 180°, around a point 360°. Vertically opposite angles are equal.
直线上的角和为180°，点周角和为360°。对顶角相等。
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Angles in polygons: sum of interior angles = (n−2)×180°, exterior angles sum to 360°.
多边形内角：内角和 = (n−2)×180°，外角和为360°。
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Circle theorems: angle at centre is twice angle at circumference, angles in same segment equal, angle in a semicircle is 90°, opposite angles in cyclic quadrilateral sum to 180°, tangent-radius perpendicular, alternate segment theorem.
圆定理：圆心角是圆周角的两倍，同弧上的圆周角相等，半圆上的圆周角是90°，圆内接四边形对角互补，切线与半径垂直，弦切角定理。
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Congruence (SSS, SAS, ASA, RHS) and similarity.
全等（SSS, SAS, ASA, RHS）和相似。
8. Pythagoras and Trigonometry | 毕达哥拉斯与三角学
Apply Pythagoras’ theorem and trigonometric ratios in right-angled triangles and beyond.
在直角三角形及其他情境中应用毕达哥拉斯定理和三角比。
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Pythagoras: a² + b² = c² (c is hypotenuse). Use to find missing sides or check right angles.
毕达哥拉斯定理：a² + b² = c²（c为斜边）。用于求缺失边长或检查直角。
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Trigonometry: SOH CAH TOA — sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
三角学：SOH CAH TOA — sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
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Exact trig values for 0°, 30°, 45°, 60°, 90°.
0°、30°、45°、60°、90°的精确三角值。
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Sine and cosine rules for non-right-angled triangles, area = ½ ab sin C.
非直角三角形的正弦定理和余弦定理，面积 = ½ ab sin C。
9. Mensuration and Units | 测量与单位
Calculate perimeter, area, volume, and convert between units accurately.
准确计算周长、面积、体积并进行单位转换。
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Perimeter and area of rectangles, triangles, parallelograms, trapeziums, circles (A = πr², C = 2πr or πd).
矩形、三角形、平行四边形、梯形、圆的周长和面积 (A = πr², C = 2πr 或 πd)。
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Volume and surface area of prisms, cylinders, pyramids, cones, spheres. Volume of prism = area of cross-section × length.
棱柱、圆柱、棱锥、圆锥、球体的体积和表面积。棱柱体积 = 横截面积 × 长度。
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Metric unit conversions: length (mm, cm, m, km), mass (g, kg), capacity (ml, l). Imperial units maybe.
公制单位换算：长度 (mm, cm, m, km)，质量 (g, kg)，容积 (ml, l)。可能涉及英制单位。
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Area conversion: 1 m² = 10,000 cm², volume: 1 m³ = 1,000,000 cm³.
面积换算：1 m² = 10,000 cm²，体积：1 m³ = 1,000,000 cm³。
10. Probability | 概率
Calculate probabilities for single and combined events, using tree diagrams and Venn diagrams.
使用树状图和维恩图计算单一事件和组合事件的概率。
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Probability scale from 0 to 1. Expected frequency = probability × number of trials.
概率范围从0到1。期望频数 = 概率 × 试验次数。
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Mutually exclusive events: P(A or B) = P(A) + P(B). Independent events: P(A and B) = P(A) × P(B).
互斥事件：P(A
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