📚 GCSE OCR Maths: End-of-Term Revision Checklist | GCSE OCR 数学：期末复习提纲
This end-of-term revision checklist covers the key topics from the OCR GCSE Mathematics syllabus. Use it to identify gaps in your knowledge and practise the essential skills needed for your exams. Each section below summarises what you need to know and provides quick revision prompts.
这份期末复习提纲涵盖了 OCR GCSE 数学大纲的核心主题。用它来发现自己的知识薄弱点，并练习考试所需的关键技能。下面的每个小节都概括了你需要掌握的内容，并提供了快速复习提示。
1. Number Operations and Integers | 数的运算与整数
Be confident with the four operations (addition, subtraction, multiplication, division) applied to integers, decimals and negative numbers. Know the correct order of operations (BIDMAS/BODMAS). Understand place value, and be able to round numbers to a given number of decimal places or significant figures. You should be able to use inequality symbols <, >, ≤, ≥ correctly and list integers that satisfy a given condition.
要熟练掌握整数、小数和负数的四则运算（加、减、乘、除）。牢记运算顺序（BIDMAS/BODMAS 规则）。理解位值，并能够将数字四舍五入到指定的小数位数或有效数字。还应会正确使用不等号 <, >, ≤, ≥，并能列出满足给定条件的整数。
Prime numbers, factors, multiples, highest common factor (HCF) and lowest common multiple (LCM) are essential tools. You must be able to express a number as a product of prime factors using index notation. Being fluent with these concepts helps when working with fractions and algebraic manipulation.
质数、因数、倍数、最大公因数（HCF）和最小公倍数（LCM）是基础工具。你必须能够用质因数乘积的形式（指数表示法）表示一个数。熟练运用这些概念有助于处理分数和代数运算。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
You need to be able to convert freely between fractions, decimals and percentages. For fractions, practise finding equivalent fractions, simplifying, and converting between mixed numbers and improper fractions. Know how to add, subtract, multiply and divide fractions, including mixed numbers, and apply these skills in problem-solving contexts.
你需要能在分数、小数和百分数之间自由转换。对于分数，练习寻找等值分数、约分，以及带分数与假分数的互化。掌握分数的加、减、乘、除（包括带分数），并能在解决问题时运用这些技能。
Percentages should be second nature: calculate a percentage of a quantity, express one quantity as a percentage of another, and work with percentage increase and decrease. Understand the difference between simple and compound interest, and be prepared to solve reverse percentage problems (finding the original amount). Recurring decimals and their fractional equivalents also appear regularly.
百分数应做到烂熟于心：计算一个量的百分之几，用一个量表示另一个量的百分数，处理百分数增减。理解单利与复利的区别，并准备好解决反向百分数问题（求原值）。循环小数及其对应的分数形式也经常出现。
3. Algebraic Expressions and Manipulation | 代数表达式与运算
Algebra underpins most of GCSE maths. You must be comfortable collecting like terms, expanding single and double brackets, and factorising expressions by taking out common factors and by grouping. Factorising quadratics of the form x² + bx + c is a core skill; more able students should also tackle quadratics where the coefficient of x² is greater than 1.
代数是 GCSE 数学的基础。你必须能熟练合并同类项、展开单项和双项括号，并通过提取公因式和分组进行因式分解。对形如 x² + bx + c 的二次式进行因式分解是核心技能；能力强的学生还应能处理 x² 系数大于 1 的二次式。
Manipulate formulas by changing the subject, and substitute positive and negative numbers into expressions correctly. Know the index laws for multiplication and division of powers, and understand the meaning of zero and negative indices. You should also be familiar with algebraic fractions and simplifying them by factorising and cancelling.
通过转换公式主体来变形公式，并能正确地将正负数代入表达式。掌握幂的乘除法指数运算律，理解零指数和负指数的意义。还应熟悉代数分式，并会通过因式分解和约分进行化简。
4. Equations and Inequalities | 方程与不等式
Solving linear equations with unknowns on one or both sides is fundamental. Move on to setting up equations from worded problems. For quadratic equations (x² + bx + c = 0), you will normally solve by factorising; be aware that some may require the quadratic formula. Know that the solutions of an equation correspond to the x-intercepts of its graph.
解答有一侧或两侧含未知数的一次方程是基本功。进一步要从文字题建立方程。对于二次方程 (x² + bx + c = 0)，通常通过因式分解求解；注意有些题可能需用到求根公式。要知道方程的解对应其图像的 x 轴截距。
Solving linear inequalities follows the same rules as equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions on a number line using open and closed circles. Solve pairs of simultaneous linear equations either by elimination or substitution, and interpret the solution graphically as the point of intersection of two lines.
解一次不等式遵循与方程相同的法则，但要记住当乘以或除以一个负数时，不等号方向要改变。用数轴上的空心圆和实心圆表示解集。用消元法或代入法解联立一次方程组，并能将解解释为图形上两条直线的交点。
5. Graphs | 图像
Plot straight-line graphs from a table of values or directly from an equation in the form y = mx + c. You must be able to identify and interpret gradient (m) and y-intercept (c), and find the equation of a line given its graph or two points. Understand the meaning of parallel and perpendicular gradients.
从数值表或直接根据 y = mx + c 形式的方程绘制直线图像。你必须能够识别和解释斜率（m）与 y 轴截距（c），并能根据图像或两点坐标求出直线方程。理解平行和垂直斜率的意义。
Quadratic graphs (y = x², etc.) produce a parabola; cubic and reciprocal graphs have distinctive shapes. You should be able to sketch these, identify turning points and intercepts, and use graphs to solve equations by finding intersection points. Distance-time and velocity-time graphs also appear, so revise how to interpret and calculate speed, acceleration and distance from them.
二次函数图像（如 y = x²）呈抛物线形；三次函数和反比例函数图像具有独特形状。你应能画出这些图像的草图，识别转折点和截距，并能利用图像通过交点来解方程。距离—时间图和速度—时间图也会出现，因此要复习如何从中解读并计算速度、加速度和距离。
6. Ratio, Proportion and Rates of Change | 比、比例与变化率
Ratios can be simplified and written in the form 1 : n or n : 1. You need to divide quantities in a given ratio and solve problems involving ratios when the total or a difference is known. Direct proportion means that two quantities increase at the same rate; inverse proportion means one increases as the other decreases. Represent direct and inverse proportion algebraically (y ∝ x, y ∝ 1/x).
比可以化简并写成 1 : n 或 n : 1 的形式。你需要按给定比例分配量，并在已知总量或差值时解决比例问题。正比例意味着两个量以相同速率增加；反比例则是一个增加时另一个减少。用代数方法表示正比和反比关系（y ∝ x, y ∝ 1/x）。
Rates of change connect closely with gradients. Understand compound measures such as speed (distance/time), density (mass/volume) and pressure (force/area). Be able to convert between related compound units, for example km/h to m/s. Scale factors for lengths, areas and volumes are a key topic, as is the concept of similarity in shapes.
变化率与斜率密切相关。理解复合度量，如速度（距离/时间）、密度（质量/体积）和压强（力/面积）。能在相关的复合单位之间进行换算，例如 km/h 与 m/s 的转换。长度、面积和体积的比例因子是重点，图形的相似概念同样重要。
7. Geometry and Measures | 几何与测量
Know the properties of angles: on a straight line, around a point, vertically opposite, alternate, corresponding and co-interior angles when parallel lines are involved. Be able to calculate interior and exterior angles of regular and irregular polygons. Angle facts for triangles and quadrilaterals are essential. Understand symmetry – line symmetry and rotational symmetry.
掌握角的基本性质：平角、周角、对顶角，以及平行线中的内错角、同位角和同旁内角。能计算规则和不规则多边形的内角与外角。三角形和四边形的角度关系是基本要求。理解对称——线对称和旋转对称。
Use formulas for area (rectangle, triangle, parallelogram, trapezium, circle) and volume (cube, cuboid, prism, cylinder, sphere, pyramid, cone). For circles, know and use the formulas for circumference (C = 2πr) and area (A = πr²), leaving answers in terms of π where appropriate. Be comfortable using Pythagoras’ theorem and simple trigonometry (SOHCAHTOA) in right-angled triangles to find missing lengths and angles.
运用面积公式（矩形、三角形、平行四边形、梯形、圆）和体积公式（正方体、长方体、棱柱、圆柱、球、棱锥、圆锥）。对于圆，掌握并运用周长公式（C = 2πr）和面积公式（A = πr²），并根据要求保留 π 的表达形式。熟练运用勾股定理和直角三角形中的基础三角比（SOHCAHTOA）来求未知边和角。
8. Pythagoras and Trigonometry | 勾股定理与三角学
In a right-angled triangle, a² + b² = c², where c is the hypotenuse. Use this to find a missing side or to check if a triangle is right-angled. For trigonometry, the three ratios sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent are the core of finding sides and angles. Extend this to 3D problems where you apply Pythagoras or trigonometry twice.
在直角三角形中，a² + b² = c²，其中 c 为斜边。用该定理求未知边长或检验一个三角形是否为直角三角形。对于三角学，正弦 = 对边/斜边、余弦 = 邻边/斜边、正切 = 对边/邻边这三个比值是求边和角的核心。将此扩展到需要两次应用勾股定理或三角比的立体几何问题。
Be familiar with exact trigonometric values for angles of 0°, 30°, 45°, 60° and 90°. These often appear in non-calculator papers. You may also be asked to apply the sine rule, cosine rule and area formula (½ ab sin C) for non-right-angled triangles, though only at Higher tier – check your tier requirements.
熟悉 0°、30°、45°、60° 和 90° 这些角度的精确三角值。它们常出现在非计算器试卷中。你还可能被要求对非直角三角形应用正弦定理、余弦定理和面积公式（½ ab sin C），不过这仅适用于 Higher 级别——请根据自己的考试级别确认要求。
9. Transformations and Vectors | 变换与向量
Understand and perform the four geometric transformations: translation (describe using column vectors), reflection (identify mirror lines), rotation (state centre, angle and direction) and enlargement (give centre and scale factor). For enlargement, note the effect of a fractional scale factor (reduction) and negative scale factor (which reverses direction).
理解并执行四种几何变换：平移（用列向量描述）、反射（指出镜面线）、旋转（说明中心、角度和方向）和位似（给出中心和比例因子）。对于位似，注意分数比例因子（缩小）和负比例因子（反向）的影响。
Vectors describe movement; they can be added, subtracted and multiplied by a scalar. Use vector notation for translations and solve geometric problems involving parallel vectors and collinear points. You should be able to write one vector as a multiple of another to show parallelism.
向量用于描述位移；它们可以相加、相减以及与标量相乘。在平移中使用向量记法，并解决涉及平行向量和共线点的几何问题。你应该能将一个向量写成另一个向量的数乘，以说明它们平行。
10. Probability | 概率
Probability measures the chance of an event occurring, and is always between 0 and 1 (or 0% and 100%). Learn to list all outcomes systematically using sample space diagrams or two-way tables. For combined events, use tree diagrams to show independent and dependent events. The probabilities on each set of branches must sum to 1.
概率衡量事件发生的可能性，始终介于 0 和 1 之间（或 0% 到 100%）。学会用系统的列举法（如样本空间图或双向表）列出所有结果。对于组合事件，使用树状图来展示独立事件和相关事件。每组分支上的概率之和必须为 1。
The AND rule (multiply) and OR rule (add) must be applied correctly for independent and mutually exclusive events. Understand relative frequency and the idea that experimental probabilities tend towards theoretical probabilities as the number of trials increases. Know how to read and construct Venn diagrams for sets and probability.
对独立事件和互斥事件正确运用“与”规则（相乘）和“或”规则（相加）。理解相对频率，以及随着试验次数增加，实验概率趋近于理论概率的概念。知道如何解读和绘制用于集合和概率的文氏图。
11. Statistics | 统计学
Be able to collect and organise data using frequency tables, grouped frequency tables, and two-way tables. Represent data with bar charts, pie charts, pictograms, line graphs and frequency polygons. For grouped data, the histogram (with frequency density on the vertical axis if class widths are unequal) may be tested at Higher tier.
能够使用频数表、分组频数表及双向表来收集和整理数据。用条形图、饼图、象形图、折线图和频数多边形来表示数据。对于分组数据，直方图（若组距不等则纵轴为频数密度）可能在 Higher 级别考查。
Calculate averages: mean, median, mode and range for discrete data. For grouped data, find an estimate of the mean and identify the modal class and the interval containing the median. Scatter graphs show correlation; draw a line of best fit to make predictions. Interpret these statistical measures to compare data sets and identify outliers.
计算离散数据的平均数：平均数、中位数、众数和极差。对于分组数据，估算平均数，并找出众数组和中位数所在区间。散点图显示相关性；绘制最佳拟合线进行预测。解读这些统计量以比较数据集并识别异常值。
12. Exam-Style Problem Solving | 考试型问题解决
OCR GCSE Maths places a strong emphasis on problem solving. You will often meet multi-step questions that combine two or more topic areas, such as using ratio with algebra or geometry with probability. Practise reading questions carefully, identifying the mathematics required, and showing clear, logical working. When using a calculator, always double-check your inputs and consider whether the answer is reasonable.
OCR GCSE 数学非常注重问题解决能力。你经常会遇到融合两个或更多知识领域的多步问题，比如将比与代数结合，或几何与概率结合。练习仔细审题，找出所需的数学方法，并展示清晰、有逻辑的解题步骤。使用计算器时，务必核对输入，并思考答案是否合理。
Regular timed practice of past papers is the best way to improve. Use the mark scheme to understand how marks are awarded – for correct method even if the final answer is wrong. Learn from your mistakes and revisit the related sections in this checklist. A calm, organised approach on exam day will make a huge difference.
定期限时练习历年真题是提高的最佳途径。用评分方案了解分值分配——即使最终答案有误，正确的方法也能得分。从错误中学习，并重新回顾本提纲中的相关章节。考试当天保持冷静、有条不紊的答题方式，会带来巨大帮助。
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