📚 GCSE AQA Maths: Last-Minute Revision Notes | GCSE AQA 数学：考前冲刺笔记
As your AQA GCSE Maths exam draws near, these last‑minute revision notes focus on the most frequently assessed topics, essential formulas, common pitfalls, and efficient problem‑solving strategies. Use this guide for quick recall, targeted practice, and confidence building in the final days before your papers.
随着AQA GCSE数学考试临近，这份考前冲刺笔记聚焦高频考点、必背公式、常见陷阱和高效解题策略。在考前最后几天，用这份指南快速回顾、针对性练习，提升信心。
1. Key Number Skills | 关键数字技能
Always double‑check place value when multiplying or dividing by powers of 10. A common error is moving the decimal point the wrong way, especially with numbers less than 1.
乘除10的幂时务必检查数位。常见错误是小数点方向移反，尤其是处理小于1的数字时。
For fraction calculations, convert mixed numbers to improper fractions first. Remember that dividing by a fraction is the same as multiplying by its reciprocal.
进行分数运算时，先把带分数化为假分数。记住除以一个分数等于乘以它的倒数。
The product of prime factors is a favourite exam topic. Write numbers in index form, e.g., 72 = 2³ × 3². This links directly to finding HCF and LCM using Venn diagrams.
质因数分解是必考题。用指数形式书写，如72 = 2³ × 3²。这直接关联用维恩图求最大公因数和最小公倍数。
When rounding, underline the digit to which you are rounding and look at the next digit. For significant figures, start counting from the first non‑zero digit.
四舍五入时，在舍入位下划线并看下一位。有效数字则从第一个非零数字开始计数。
| Bounds | Upper and lower bounds: ± half of the accuracy. If a length is 6.4 cm to 1 d.p., the bounds are 6.35 cm ≤ L < 6.45 cm. |
误差界：上下界 = ± 精度的一半。若长度6.4 cm精确到1位小数，则6.35 cm ≤ L < 6.45 cm。
2. Algebraic Essentials | 代数基础要点
Expanding brackets: multiply every term inside the bracket by the term outside. Watch negative signs carefully. (x + 3)(x – 2) expands to x² + x – 6.
去括号：用外面的项乘括号内每一项。注意负号。(x + 3)(x – 2) 展开得 x² + x – 6。
Factorising a quadratic x² + bx + c: find two numbers that multiply to c and add to b. For 2x² + 7x + 3, split the middle term and factor by grouping.
因式分解二次式 x² + bx + c：找两个数乘积为c，和为b。对于 2x² + 7x + 3，分裂中项并分组分解。
Laws of indices are critical: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
指数法则极其重要：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。
Manipulating formulas: rearrange by doing the same to both sides. If y = mx + c, then m = (y – c)/x. Always check your subject appears only once on one side.
公式变形：等式两边同时操作。若 y = mx + c，则 m = (y – c)/x。务必检查所求变量只在等式一边出现一次。
3. Ratio, Proportion & Rates of Change | 比、比例与变化率
Simplify ratios by dividing by the HCF. A ratio of ¼ : ¾ becomes 1 : 3 after multiplying by 4. Always express ratios in their simplest integer form.
用最大公因数化简比。比 ¼ : ¾ 乘以4后变为 1 : 3。始终以最简整数比表示。
For direct proportion, y = kx; for inverse proportion, y = k/x. Use given values to find k, then answer the question. Link graphs: direct proportion is a straight line through the origin.
正比例关系 y = kx，反比例关系 y = k/x。用已知值求常数 k，再解答问题。关联图像：正比例图像是过原点的直线。
Compound measures such as speed (distance ÷ time), density (mass ÷ volume), and pressure (force ÷ area) require consistent units. Convert minutes to hours where needed.
复合量度如速度（距离 ÷ 时间）、密度（质量 ÷ 体积）和压强（力 ÷ 面积）须统一单位。必要时将分钟化为小时。
Percentage increase: multiply by (1 + percentage as decimal). A 15% increase ×1.15. Reverse percentages: divide by the multiplier.
百分比增加：乘以 (1 + 以小数值表示的百分数)。15% 增长即乘1.15。逆向百分数：除以该乘数。
4. Geometry & Measures | 几何与测量
Angle facts: angles on a straight line sum to 180°, around a point 360°. Vertically opposite angles are equal. Alternate and corresponding angles are equal only when lines are parallel.
角度定理：直线上的角之和为180°，绕一点和为360°。对顶角相等。同位角和内错角相等的前提是两直线平行。
Area of a trapezium = ½(a+b)h. Circle area = πr², circumference = πd or 2πr. For a sector, area = (θ/360) × πr², arc length = (θ/360) × 2πr.
梯形面积 = ½(a+b)h。圆面积 = πr²，周长 = πd 或 2πr。扇形面积 = (θ/360) × πr²，弧长 = (θ/360) × 2πr。
Volume of prisms = area of cross‑section × length. For a cylinder, V = πr²h. Surface area of a cylinder = 2πr² + 2πrh. For spheres, volume = ⁴⁄₃πr³, surface area = 4πr².
棱柱体积 = 横截面积 × 长度。圆柱体 V = πr²h。圆柱体表面积 = 2πr² + 2πrh。球体体积 = ⁴⁄₃πr³，表面积 = 4πr²。
Units: 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³. It is a common mistake to treat area conversion as linear.
单位换算：1 m² = 10 000 cm²，1 m³ = 1 000 000 cm³。常见错误是将面积换算当作线性换算。
5. Pythagoras & Trigonometry | 毕达哥拉斯与三角学
Pythagoras’ theorem: a² + b² = c², where c is the hypotenuse. Use it only in right‑angled triangles. Always label the hypotenuse as the longest side opposite the right angle.
毕达哥拉斯定理：a² + b² = c²，c为斜边。仅用于直角三角形。务必把最长边（直角的对边）标为斜边。
Trigonometry: SOH CAH TOA – sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. For non‑right triangles, use sine rule a/sin A = b/sin B or cosine rule a² = b² + c² – 2bc cos A.
三角学：SOH CAH TOA——sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。非直角三角形用正弦定理 a/sin A = b/sin B 或余弦定理 a² = b² + c² – 2bc cos A。
When solving for an angle, use the inverse function. Check that your calculator is in degree mode. Sketch the triangle first.
求角度时用反函数。确保计算器处于角度制模式。画出示意图。
Exact trigonometric values for 0°, 30°, 45°, 60°, 90° are expected. For example, sin 30° = ½, tan 45° = 1. Memorising these saves time.
需要牢记 0°、30°、45°、60°、90° 的精确三角值。例如 sin 30° = ½，tan 45° = 1。熟记它们可以节省时间。
6. Probability | 概率
Probability = number of favourable outcomes / total number of outcomes. It is always between 0 and 1 inclusive.
概率 = 有利结果数 / 总结果数。总是在0到1之间（含）。
For independent events, multiply probabilities: P(A and B) = P(A) × P(B). For mutually exclusive events, add: P(A or B) = P(A) + P(B).
独立事件，概率相乘：P(A 且 B) = P(A) × P(B)。互斥事件，概率相加：P(A 或 B) = P(A) + P(B)。
Tree diagrams handle successive events. Multiply along branches and add across outcomes. Do not cancel probabilities across different branches.
树状图处理相继事件。沿分支相乘，把不同结果相加。不要在不同分支间抵消概率。
Conditional probability: if the question says “given that”, reduce your sample space. The formula P(A|B) = P(A∩B)/P(B) is useful but often a table or diagram is faster.
条件概率：题目中出现“已知”时需缩小样本空间。公式 P(A|B) = P(A∩B)/P(B) 有用，但通常表格或图表更快。
7. Statistics | 统计
Mean = sum of values ÷ number of values. Median is the middle value when ordered; mode is most frequent. Range = maximum – minimum.
平均数 = 总和 ÷ 个数。中位数是排序后的中间值；众数是出现最多的值。范围 = 最大减最小。
For grouped frequency tables, find estimated mean using midpoints. The modal class is the class with highest frequency.
分组频率表，用组中点估算平均数。众数组是频数最高的组。
Box plots show minimum, lower quartile, median, upper quartile, maximum. IQR = UQ – LQ. Outliers are usually values beyond 1.5×IQR from the quartiles.
箱线图显示最小值、下四分位数、中位数、上四分位数、最大值。四分位距 IQR = UQ – LQ。离群值通常指超出四分位数1.5×IQR的值。
Scatter graphs: correlation is not causation. Draw a line of best fit that passes through the mean point (x̄, ȳ) to make predictions.
散点图：相关性不代表因果性。画一条穿过均值点(x̄, ȳ)的最佳拟合线进行预测。
8. Equations & Inequalities | 方程与不等式
Solve linear equations by doing inverse operations. Always check your solution by substituting it back into the original equation.
通过逆运算解线性方程。一定要把解代回原方程检验。
Quadratic equations: set to zero and factorise, or use the quadratic formula x = [–b ± √(b² – 4ac)] / 2a. The formula is given on the formula sheet, but you must know when to use it.
二次方程：设为零后因式分解，或使用求根公式 x = [–b ± √(b² – 4ac)] / 2a。公式表会提供，但必须知道何时使用。
Simultaneous equations can be solved by elimination or substitution. If one equation is quadratic, substitution is usually best.
联立方程可用消元法或代入法求解。若其中一个方程为二次，通常用代入法最佳。
Inequalities: treat like equations, but flip the sign when multiplying or dividing by a negative number. Represent solution sets on a number line with open or closed circles.
不等式：解法如方程，但乘或除以负数时要反向不等号。用数轴表示解集，空心圆圈或实心圆点。
9. Graphs & Functions | 图形与函数
Key graphs: linear (y = mx + c), quadratic (parabola), cubic, reciprocal (y = k/x), exponential. Be able to sketch them and identify intercepts.
关键函数图像：线性（y = mx + c）、二次（抛物线）、三次、反比例（y = k/x）、指数。应能草绘并确定截距。
Gradient = (change in y)/(change in x). Parallel lines have equal gradients. Perpendicular lines have gradients whose product is –1.
斜率 = (y变化量)/(x变化量)。平行线斜率相等。垂直线斜率乘积为 –1。
Transformations: f(x) + a moves the graph up, f(x + a) moves left. f(–x) reflects in y‑axis, –f(x) reflects in x‑axis.
图像变换：f(x) + a 向上平移，f(x + a) 向左平移。f(–x) 关于 y 轴反射，–f(x) 关于 x 轴反射。
For real‑life graphs, the gradient represents a rate (speed, cost per item). The area under a speed‑time graph gives distance.
实际生活图像中，斜率表示变化率（速度、单价）。速度-时间图像下的面积表示距离。
10. Exam Tips & Common Mistakes | 考试技巧与常见错误
Always show your method. Even if the final answer is wrong, you can earn marks for a correct process. Write down the formula you are using before substituting.
务必展示解题过程。即使最终答案错误，正确步骤也能得分。先写出所用公式，再代入数值。
Check that your answer is reasonable. If a probability is 1.2 or a length is negative, you have made a mistake. Estimation helps catch errors.
检查答案是否合理。若概率为1.2或长度为负值，你肯定出错了。估算有助于发现错误。
Units matter: if a question gives lengths in cm and asks for volume in litres, convert systematically. 1 litre = 1000 cm³.
单位非常重要：若题目单位是厘米，要求以升为单位给出体积，需系统换算。1升 = 1000 cm³。
Read the question carefully. Underline “give your answer in terms of π”, “to the nearest penny”, or “simplify fully”. Many marks are lost by not following the final instruction.
仔细审题。在“答案用 π 表示”、“精确到便士”、“完全化简”等指令下划线。许多人因为没注意到最后的作答要求而失分。
Manage your time: do not spend too long on one question. If stuck, move on and return later. Practice papers under timed conditions are the best way to prepare.
管理时间：不要在一道题上耗费太久。卡住就先跳过去，稍后再回来。限时模拟训练是最佳备考方式。
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