📚 GCSE AQA Maths: Final Revision Checklist | GCSE AQA 数学：期末复习提纲
This revision checklist covers all the key topics in the AQA GCSE Mathematics syllabus. Use it to structure your final review, identify weak areas, and boost your confidence before the exam. Each section includes the most common question types and the essential facts you need to know.
这份复习提纲涵盖了 AQA GCSE 数学教学大纲中的所有关键主题。用它来安排你的最终复习，找出薄弱环节，并在考试前增强信心。每一节都包含最常见的题型和你必须掌握的核心知识点。
1. Number Operations and Integers | 数字运算与整数
You must be able to add, subtract, multiply, and divide positive and negative integers correctly. Remember that the product of two negative numbers is positive, and a negative divided by a positive gives a negative. Place value, ordering, and rounding to a given number of decimal places or significant figures are also essential.
你必须能正确进行正负整数的加减乘除。记住两个负数相乘得正，负数除以正数得负。位值、大小排序以及按指定的小数位数或有效数字进行四舍五入也是基本功。
Using the order of operations (BIDMAS/BODMAS) is frequently tested. Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right) must be applied strictly. In multi-step problems, show all steps clearly to avoid errors.
运算顺序（BIDMAS/BODMAS）是常考内容。必须严格按照括号、指数、乘除（从左到右）、加减（从左到右）的顺序进行计算。在多步骤问题中，清楚展示每一步以避免错误。
2. Fractions, Decimals, and Percentages | 分数、小数与百分数
You should be confident converting between fractions, decimals, and percentages. To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100. Conversely, a percentage becomes a decimal by dividing by 100.
你应当能熟练地在分数、小数和百分数之间进行转换。分数转小数用分子除以分母；小数转百分数乘以 100；反过来，百分数除以 100 就变成小数。
For fraction calculations, find common denominators when adding or subtracting. When multiplying fractions, multiply the numerators and multiply the denominators. To divide by a fraction, multiply by its reciprocal. Percentages often appear as increase/decrease problems or finding a percentage of a quantity; using a multiplier (e.g., 1.15 for a 15% increase) is efficient.
进行分数运算时，加减法要先通分；乘法是分子乘分子、分母乘分母；除以一个分数等于乘它的倒数。百分数常以增减问题或求一个量的百分之几的形式出现；使用乘数（例如 15% 的增长用 1.15）会更高效。
3. Ratio and Proportion | 比与比例
Ratios can be simplified by dividing all parts by a common factor. When sharing a quantity in a given ratio, add the parts to find the total number of shares, then divide the quantity accordingly. Scales on maps and diagrams use ratios; understand how to use a scale factor to find real lengths.
比可以通过将各部分除以公因数来化简。按给定比例分配一个量时，先把比的各项相加得到总份数，再按比例分配。地图和图纸上的比例尺用比表示，要懂得如何用比例因子求实际长度。
Direct proportion means as one quantity increases, the other increases at the same rate; it is often solved using the unitary method or a formula y = kx. Inverse proportion describes a relationship where one quantity increases as the other decreases, given by y = k/x. Exchange rates and best-buy problems are practical applications of proportion.
正比例是指一个量增加时，另一个量以相同速率增加；常用单位法或公式 y = kx 求解。反比例描述一个量增加而另一个量减少的关系，表达式为 y = k/x。汇率和最佳购买问题是比例的实际应用。
4. Algebraic Expressions and Simplification | 代数表达式与化简
You must be able to simplify expressions by collecting like terms. Terms with exactly the same variable parts can be added or subtracted. Expanding brackets involves multiplying each term inside the bracket by the term outside: a(b + c) = ab + ac. Factorising is the reverse process – taking out the highest common factor.
你必须能够通过合并同类项来化简表达式。变量部分完全相同的项可以相加或相减。展开括号就是用括号外的项乘以括号内的每一项：a(b + c) = ab + ac。因式分解是逆过程——提出最高公因式。
When multiplying two binomials, use a grid or FOIL (First, Outer, Inner, Last). For example, (x + 2)(x + 3) = x² + 5x + 6. Recognise special products like (a + b)² = a² + 2ab + b² and the difference of two squares a² − b² = (a − b)(a + b). These are very common in GCSE exam questions.
进行两个二项式相乘时，可用网格法或 FOIL 法。例如 (x + 2)(x + 3) = x² + 5x + 6。要能识别特殊乘法，如 (a + b)² = a² + 2ab + b² 和平方差 a² − b² = (a − b)(a + b)。这些在 GCSE 考题中非常常见。
5. Equations and Inequalities | 方程与不等式
Solving linear equations involves isolating the unknown using inverse operations. Always do the same to both sides. For example, 3x + 5 = 20 → 3x = 15 → x = 5. Equations with unknowns on both sides require collecting terms on one side first.
解线性方程就是用逆运算求出未知数。始终对方程两边进行相同的操作。例如 3x + 5 = 20 → 3x = 15 → x = 5。若未知数在两边，需要先把含未知数的项移到一边。
Quadratic equations are solved by factorising, completing the square, or using the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Remember to set the equation to zero first. Inequalities work much like equations, but when multiplying or dividing by a negative number, reverse the inequality sign. Represent solutions on a number line using open or closed circles.
二次方程用因式分解、配方法或求根公式求解：x = (−b ± √(b² − 4ac)) / 2a。记住先将方程置零。不等式与方程类似，但乘以或除以负数时，不等号要反向。在数轴上表示解集时用空心或实心圆点。
6. Graphs and Functions | 图形与函数
You must be able to plot and interpret straight-line graphs. The equation y = mx + c has gradient m and y-intercept c. Parallel lines have the same gradient; perpendicular lines have gradients that multiply to −1. Draw graphs accurately using a table of values.
你必须能够绘制和解读直线图形。方程 y = mx + c 中 m 是斜率，c 是 y 轴截距。平行线斜率相等；垂直线斜率之积为 −1。利用数值表精确绘制图形。
Quadratic graphs are parabolas; their turning points can be found by completing the square or by using symmetry. Cubic and reciprocal graphs also appear. Understand how to solve equations graphically by finding the intersection points of two graphs. Real-life graphs (distance-time, velocity-time) require you to interpret gradients and areas under the graph.
二次函数的图像是抛物线；其顶点可用配方法或对称性求得。三次函数图像和反比例图像也会出现。要理解如何通过求两个图像的交点来解方程。现实生活中的图像（如距离-时间图、速度-时间图）需要解释斜率和图像下的面积。
7. Geometry Basics and Transformations | 几何基础与变换
Know the properties of triangles, quadrilaterals, polygons, and circles. The sum of interior angles of a polygon with n sides is (n − 2) × 180°. Exterior angles sum to 360° for any convex polygon. Congruent shapes are identical in size and shape; similar shapes have the same angles and proportional sides.
了解三角形、四边形、多边形和圆的性质。n 边形的内角和为 (n − 2) × 180°。任何凸多边形的外角和为 360°。全等图形大小和形状完全相同；相似图形角度相等，对应边成比例。
Transformations include translation, reflection, rotation, and enlargement. Describe transformations fully by giving the necessary details: vector for translation, mirror line for reflection, centre and angle for rotation, scale factor and centre for enlargement. Negative scale factors produce an enlargement on the opposite side of the centre.
变换包括平移、反射、旋转和放大。完整描述一个变换需要给出所有细节：平移用向量，反射用对称轴，旋转用中心和角度，放大用比例因子和中心。负比例因子会在中心另一侧形成放大的像。
8. Angles and Bearings | 角与方位角
Angle facts form the foundation of many geometry problems. Angles on a straight line sum to 180°; angles around a point sum to 360°. Vertically opposite angles are equal. Alternate angles, corresponding angles, and co-interior angles apply to parallel lines.
角度知识是许多几何问题的基础。直线上各角之和为 180°；围绕一点各角之和为 360°。对顶角相等。在平行线中，内错角、同位角和同旁内角各有其关系。
Bearings are measured clockwise from North and are always given as three figures (e.g., 045°). To find a bearing, draw the North line at the starting point and measure clockwise. Use angle facts and trigonometry to calculate unknown bearings in diagrams.
方位角从北按顺时针方向测量，始终用三位数字表示（如 045°）。求方位角时，先在起点画出指北线，然后顺时针度量。利用角度知识和三角学来计算图中的未知方位角。
9. Perimeter, Area, and Volume | 周长、面积与体积
Learn the formulas for area: triangle (½×base×height), rectangle, parallelogram (base×height), trapezium (½(a+b)×h), and circle (πr²). Circumference of a circle is πd or 2πr. For compound shapes, split them into simpler shapes, calculate areas separately, then add or subtract as needed.
记住面积公式：三角形（½×底×高）、矩形、平行四边形（底×高）、梯形（½(a+b)×h）、圆（πr²）。圆周长为 πd 或 2πr。组合图形的面积可拆分成简单图形，分别计算再加减。
Volume of prisms is area of cross-section × length. Cylinder volume is πr²h. For pyramids and cones, volume = ⅓ × base area × height. Surface area is the total area of all faces; for curved solids, know formulas for curved surface area. Density, mass, and volume problems use the formula density = mass/volume.
棱柱体积 = 横截面积 × 长度。圆柱体积为 πr²h。棱锥和圆锥的体积 = ⅓ × 底面积 × 高。表面积是所有面的总面积；对于曲面立体，需掌握侧面积公式。密度、质量、体积问题用公式密度 = 质量/体积。
10. Pythagoras and Trigonometry | 勾股定理与三角学
Pythagoras’ theorem states that in a right-angled triangle, a² + b² = c², where c is the hypotenuse. Use it to find a missing side length. To check if a triangle is right-angled, verify whether the sides satisfy the theorem.
勾股定理指出在直角三角形中 a² + b² = c²，其中 c 是斜边。用来求未知边长。判断一个三角形是否为直角三角形，只需检验三边是否满足这个关系即可。
Trigonometry ratios (SOH CAH TOA) relate angles to side lengths: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. These ratios can find missing lengths and angles. In non-right-angled triangles, use the sine rule (a/sin A = b/sin B = c/sin C) and cosine rule (a² = b² + c² − 2bc cos A). Use the cosine rule when given two sides and the included angle, or three sides.
三角比 (SOH CAH TOA) 将角度与边长联系起来：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。这些比可求未知边长或角度。对于非直角三角形，使用正弦定理 (a/sin A = b/sin B = c/sin C) 和余弦定理 (a² = b² + c² − 2bc cos A)。已知两边及其夹角或三边时用余弦定理。
11. Probability | 概率
Probability is measured on a scale from 0 to 1. The probability of an event not happening is 1 minus the probability that it does happen. For mutually exclusive events, probabilities can be added. For independent events, multiply probabilities.
概率的取值范围在 0 到 1 之间。事件不发生的概率等于 1 减去它发生的概率。对于互斥事件，概率可以相加。对于独立事件，概率应相乘。
Tree diagrams are invaluable for combined events. Label branches with probabilities (which must sum to 1 at each branching point). Multiply along branches for sequential events, and add relevant branch probabilities for overall outcomes. Conditional probability considers how the probability of an event changes given that another event has already occurred; these questions often use a frequency table or a tree diagram with second-branch probabilities changed.
树形图对组合事件极有帮助。在分支上标出概率（每个分支点上的概率之和必须为 1）。沿着分支相乘得到连续事件的概率，把相关分支的概率相加得到最终结果。条件概率考虑的是在另一事件已经发生的情况下概率如何改变；这类题目常使用频数表或二阶段概率变化的树形图。
12. Statistics and Data Handling | 统计与数据处理
You need to be able to calculate mean, median, mode, and range from a list of numbers, a frequency table, or a grouped frequency table. The mean from a frequency table is Σ(fx)/Σf, where x is the data value and f is its frequency. For grouped data, use the mid-interval value.
你必须能够从一组数字、频数表或分组频数表中计算平均数、中位数、众数和极差。由频数表求平均数的公式为 Σ(fx)/Σf，其中 x 是数据值，f 是它的频数。对于分组数据，使用组中值。
Statistical diagrams include pie charts, bar charts, histograms (for continuous data with unequal class widths, frequency = frequency density × class width), cumulative frequency graphs (used to find medians and quartiles), and box plots. Scatter graphs show correlation; a line of best fit can be used to make predictions. Be careful with interpreting graphs and identifying misleading representation.
统计图表包括饼图、条形图、直方图（用于连续数据，组距不等时，频数 = 频数密度 × 组距）、累积频数图（用来求中位数和四分位数）以及箱形图。散点图展示相关关系；最佳拟合线可用来进行预测。解读图表时要仔细，并能识别误导性的呈现方式。
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