📚 IGCSE OCR Mathematics: End-of-Term Revision Outline | IGCSE OCR 数学：期末复习提纲
This comprehensive revision outline covers the core topics prescribed by the OCR IGCSE Mathematics syllabus (J560). It is designed to help you consolidate essential skills, identify key formulas, and build confidence ahead of your end-of-term assessments. Each section highlights must-know concepts, common pitfalls, and practical tips for examination success.
这份全面的复习提纲覆盖了 OCR IGCSE 数学大纲 (J560) 的核心主题，旨在帮助你巩固基本技能、识别关键公式，并在期末考试前建立信心。每个小节都突出了必知概念、常见易错点以及应对考试的实用技巧。
1. Number and Operations | 数与运算
Understand place value, ordering integers, and the four operations. Apply BIDMAS/BODMAS correctly when evaluating expressions: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
理解位值、整数的排序及四则运算。在计算表达式时正确应用运算顺序：括号、指数、乘除、加减。
Work confidently with negative numbers: know that multiplying or dividing two negatives yields a positive result. Use a number line to visualise addition and subtraction of directed numbers.
熟练处理负数：知道两个负数相乘或相除得正数。借助数轴直观理解有向数的加减。
Identify prime numbers, factors, multiples, and the concepts of HCF (highest common factor) and LCM (lowest common multiple). Use prime factorisation to find HCF and LCM efficiently.
识别质数、因数、倍数，掌握最高公因数 (HCF) 和最小公倍数 (LCM) 的概念。利用质因数分解高效地寻找 HCF 和 LCM。
Round numbers to a given number of decimal places or significant figures. Estimate answers by rounding to one significant figure before calculating.
将数字四舍五入到指定的小数位数或有效数字。计算前先四舍五入到一位有效数字来估算结果。
2. Fractions, Decimals and Percentages | 分数、小数和百分比
Convert fluently between fractions, decimals and percentages. Memorise common equivalents: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333 = 33.3%.
熟练地在分数、小数和百分比之间转换。记住常见等值：1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333 = 33.3%。
Add, subtract, multiply and divide fractions by finding common denominators or using reciprocal for division. Simplify final answers and convert improper fractions to mixed numbers where appropriate.
通过通分或使用倒数进行分数的加减乘除。化简最终结果，并在适当情况下将假分数转化为带分数。
Calculate a percentage of a quantity, percentage increase/decrease, and express one quantity as a percentage of another. Recognise that percentage change = (change / original) × 100%.
计算一个数量的百分比、百分比的增减，以及一个数量占另一个数量的百分比。牢记百分比变化 = (变化量 / 原始值) × 100%。
Solve reverse percentage problems by setting up the equation: original amount after a 20% increase is 120% of the original. Use multipliers (e.g., ×1.2 for a 20% increase) to solve efficiently.
通过建立方程解决逆向百分比问题：如增加20%后的量是原量的120%。使用乘数（例如增加20%用 ×1.2）高效求解。
3. Ratio and Proportion | 比和比例
Simplify ratios by dividing all parts by their highest common factor. Write ratios in the form 1 : n or n : 1 where useful. Understand that the order of a ratio matches the order of quantities given.
通过除以最高公因数化简比。在需要时将比写成 1 : n 或 n : 1 的形式。理解比的顺序与所给量的顺序一致。
Divide a quantity in a given ratio by finding the total number of parts and then calculating the value of one part.
通过确定总份数，然后计算一份的值，来按给定的比分配一个量。
Recognise direct proportion: y ∝ x implies y = kx where k is the constant of proportionality. Graphically, direct proportion yields a straight line through the origin.
识别正比例关系：y ∝ x 意味着 y = kx，其中 k 是比例常数。图像上，正比例是一条通过原点的直线。
Understand inverse proportion: y ∝ 1/x gives y = k/x. As x increases, y decreases in a non-linear way, producing a curved graph.
理解反比例关系：y ∝ 1/x 得出 y = k/x。随着 x 增加，y 以非线性方式减少，图像为曲线。
Apply ratios to real-world contexts such as recipes, maps (scale drawings), and currency conversions. Always check units are consistent.
将比应用于实际情境，如配方、地图（比例图）和货币兑换。始终确保单位一致。
4. Algebraic Expressions | 代数表达式
Simplify expressions by collecting like terms: terms with the same variable and power can be added or subtracted. Keep signs carefully.
通过合并同类项化简表达式：具有相同变量和幂的项可以相加减。注意符号。
Multiply out brackets using the distributive law: a(b + c) = ab + ac. Then expand double brackets using FOIL (First, Outer, Inner, Last) and simplify:
(x + a)(x + b) = x² + (a + b)x + ab
使用分配律展开括号：a(b + c) = ab + ac。然后使用 FOIL （先外内后）展开双重括号并化简。
Factorise expressions by taking out the highest common factor (HCF). For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.
通过提取最大公因数 (HCF) 进行因式分解。对于形如 x² + bx + c 的二次式，找出两个相乘得 c、相加得 b 的数。
Recognise special expansions: difference of two squares a² – b² = (a + b)(a – b), and perfect square trinomials (a ± b)² = a² ± 2ab + b².
熟悉特殊展开：平方差公式 a² – b² = (a + b)(a – b)，以及完全平方式 (a ± b)² = a² ± 2ab + b²。
Manipulate algebraic fractions by factorising and cancelling common factors. Add or subtract algebraic fractions by finding a common denominator.
通过因式分解和约分来处理代数分式。通过同分母进行代数分式的加减。
Understand indices rules: am × an = am+n, am ÷ an = am–n, (am)n = amn, a0 = 1, and a–m = 1/am. Convert between fractional indices and roots: a½ = √a, a⅓ = ³√a.
掌握指数法则：am × an = am+n，am ÷ an = am–n，(am)n = amn，a0 = 1，以及 a–m = 1/am。在分数指数和根式间转换：a½ = √a，a⅓ = ³√a。
5. Equations and Inequalities | 方程与不等式
Solve linear equations by isolating the unknown using inverse operations. Always perform the same operation on both sides. Check your answer by substitution.
通过逆运算隔离未知数来解线性方程。始终在等式两边进行相同操作。用代入法检验答案。
Form and solve equations from word problems. Define the variable clearly, write an equation, and interpret the solution in context.
根据文字题建立并求解方程。清晰地定义变量，写出方程，并在上下文中解释解的意义。
Solve quadratic equations by factorising, then setting each bracket to zero. Alternatively, use the quadratic formula for harder cases:
x = [ –b ± √(b² – 4ac) ] / 2a
通过因式分解解二次方程，然后令每个括号等于零。对于较复杂的情况，可以使用二次公式。
Solve simultaneous linear equations either by elimination (adding/subtracting equations to remove one variable) or by substitution (replacing one variable with an expression).
通过消元法（方程相加减以消去一个变量）或代入法（用一个表达式替换一个变量）来解联立线性方程组。
Represent inequalities on a number line using open and closed circles, and solve linear inequalities remembering to reverse the inequality sign when multiplying or dividing by a negative.
在数轴上用空心和实心圆表示不等式，并解线性不等式。当乘以或除以负数时，记得反转不等号方向。
6. Graphs and Functions | 图形与函数
Plot straight-line graphs using the equation y = mx + c, where m is the gradient and c is the y-intercept. Find the gradient between two points (x₁, y₁) and (x₂, y₂) as (y₂ – y₁) / (x₂ – x₁).
使用方程 y = mx + c 绘制直线图，其中 m 为斜率，c 为 y 截距。计算两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率为 (y₂ – y₁) / (x₂ – x₁)。
Identify parallel lines (same gradient) and perpendicular lines (product of gradients = –1). Write equations of lines given a point and gradient, or two points.
识别平行线（斜率相同）和垂直线（斜率的乘积为 –1）。根据给定点和斜率，或两个点写出直线方程。
Plot quadratic graphs (y = x² + bx + c) by completing a table of values. Identify the turning point (vertex), roots (x-intercepts), and y-intercept. Recognise their symmetrical shape.
通过填写数值表绘制二次函数图 (y = x² + bx + c)。识别顶点、根（x 截距）和 y 截距。理解其对称形状。
Understand that a function is a special relationship where each input has exactly one output. Use function notation f(x) and evaluate f(a) by substitution. Find the inverse function f⁻¹(x) by rearranging.
理解函数是一种每个输入只有一个输出的特殊关系。使用函数符号 f(x)，并用代入法求 f(a)。通过重排求出反函数 f⁻¹(x)。
Apply transformations to graphs: y = f(x) + a (translation up/down), y = f(x + a) (translation left/right), y = –f(x) (reflection in x-axis), y = f(–x) (reflection in y-axis).
对图形进行变换：y = f(x) + a（上下平移），y = f(x + a)（左右平移），y = –f(x)（关于 x 轴对称），y = f(–x)（关于 y 轴对称）。
7. Geometry | 几何
Recall angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal, angles in a triangle sum to 180°.
复习角度定理：直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等，三角形内角和为 180°。
Learn properties of parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°.
学习平行线的性质：同位角相等，内错角相等，同旁内角之和为 180°。
Classify quadrilaterals by their properties: sum of interior angles is 360°. Know key types – squares, rectangles, parallelograms, rhombuses, trapeziums, kites – and their symmetry, side and diagonal properties.
按性质对四边形分类：内角和为 360°。知道关键类型——正方形、矩形、平行四边形、菱形、梯形、筝形——以及它们的对称性、边长和对角线性质。
Use congruence criteria (SSS, SAS, ASA, RHS) to prove triangles are identical, and similarity criteria (equal angles, proportional sides). Understand that similar shapes have the same shape but different sizes.
使用全等条件 (边边边、边角边、角边角、直角斜边) 证明三角形相等，以及相似条件（对应角相等，对应边成比例）。理解相似图形形状相同但大小不同。
Calculate interior and exterior angles of polygons. For an n-sided polygon, interior angle sum = (n – 2) × 180°; exterior angles sum to 360°. For a regular polygon, each exterior angle = 360/n.
计算多边形的内角和外角。对于 n 边形，内角和 = (n – 2) × 180°；外角和 = 360°。对于正多边形，每个外角 = 360/n。
8. Measures and Trigonometry | 测量与三角学
Convert between metric units of length, area and volume: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m; 1 cm² = 100 mm², 1 m² = 10,000 cm²; 1 cm³ = 1000 mm³, 1 litre = 1000 cm³.
在长度、面积和体积的公制单位之间转换：1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m；1 cm² = 100 mm², 1 m² = 10,000 cm²；1 cm³ = 1000 mm³, 1 升 = 1000 cm³。
Calculate perimeter and area of 2D shapes: rectangle (A = lw), triangle (A = ½bh), parallelogram (A = bh), trapezium (A = ½(a+b)h), and circle (C = 2πr, A = πr²). Memorise and apply the formula for compound shapes.
计算二维图形的周长和面积：矩形 (A = 长×宽)，三角形 (A = ½×底×高)，平行四边形 (A = 底×高)，梯形 (A = ½(a+b)h)，以及圆 (C = 2πr, A = πr²)。记忆并应用复合图形的公式。
Find volume and surface area of 3D shapes: cuboid (V = lwh), prism (V = area of cross-section × length), cylinder (V = πr²h, curved SA = 2πrh), cone (V = ⅓πr²h), sphere (V = ⁴⁄₃πr³, SA = 4πr²).
求立体图形的体积和表面积：长方体 (V = 长×宽×高)，棱柱 (V = 横截面积 × 长)，圆柱 (V = πr²h，侧面积 = 2πrh)，圆锥 (V = ⅓πr²h)，球 (V = ⁴⁄₃πr³，表面积 = 4πr²)。
Apply Pythagoras’ theorem in right-angled triangles:
a² + b² = c²
where c is the hypotenuse. Use it to find missing sides and check if a triangle is right-angled.
在直角三角形中应用勾股定理：a² + b² = c²，其中 c 是斜边。利用它求缺失的边长，并检验三角形是否为直角三角形。
Use trigonometric ratios (SOHCAHTOA) in right-angled triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Sides can be found by rearranging; angles are found using inverse trig functions sin⁻¹, cos⁻¹, tan⁻¹.
在直角三角形中使用三角比 (SOHCAHTOA)：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。通过重新排列求边长；利用反三角函数 sin⁻¹, cos⁻¹, tan⁻¹ 求角度。
Know the exact values for key angles: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. These are often tested without calculators.
记住关键角的精确值：sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2；cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2；tan 30° = 1/√3, tan 45° = 1, tan 60° = √3。这些常在非计算器题目中考到。
9. Statistics | 统计
Collect, organise and interpret data using frequency tables. Calculate the mean (sum of values ÷ frequency total), median (middle value), mode (most frequent) and range (largest – smallest).
使用频数表收集、整理和解读数据。计算平均数（数值总和 ÷ 频数总和）、中位数（中间值）、众数（出现次数最多）和极差（最大值 – 最小值）。
Construct and interpret pictograms, bar charts, pie charts, and line graphs. For pie charts, use the fact that total angle is 360° and angle of a sector = (frequency / total frequency) × 360°.
绘制并解读象形图、条形图、饼图和折线图。对于饼图，利用总角度为 360°，扇区角度 = (频数 / 总频数) × 360°。
Draw and analyse frequency diagrams for grouped data, including histograms with equal and unequal class widths. Understand that frequency density = frequency / class width.
绘制和分析分组数据的频数图，包括等宽和不等宽直方图。理解频数密度 = 频数 / 组距。
Calculate cumulative frequency and plot cumulative frequency curves (ogives). Use them to find the median, quartiles (lower quartile Q₁, upper quartile Q₃), and interquartile range (IQR = Q₃ – Q₁).
计算累积频数并绘制累积频数曲线 (ogive)。用它们求中位数、四分位数（下四分位数 Q₁，上四分位数 Q₃）和四分位距 (IQR = Q₃ – Q₁)。
Interpret scatter graphs: describe correlation (positive, negative, none) and draw a line of best fit to make predictions. Understand that correlation does not imply causation.
解读散点图：描述相关性（正、负、无），并画最佳拟合线进行预测。理解相关性不代表因果关系。
Compare data sets using averages and measures of spread. Choose the most appropriate average: mean for symmetrical data, median when outliers are present.
使用平均数和离散度量来比较数据集。选择最合适的平均数：对称数据用平均数，存在异常值时用中位数。
10. Probability | 概率
Understand that probability is measured on a scale from 0 (impossible) to 1 (certain). Express probabilities as fractions, decimals or percentages.
理解概率的度量范围从 0（不可能）到 1（必然）。将概率表示为分数、小数或百分比。
Apply the basic probability formula: P(event) = number of favourable outcomes / total number of outcomes. For equally likely outcomes, list all possibilities systematically.
应用基本概率公式：P(事件) = 有利结果的数量 / 所有可能结果的总数。对于等可能结果，系统列出所有可能性。
Use sample space diagrams and two-way tables to enumerate outcomes for two combined events. Calculate probabilities of single events, complementary events (P(not A) = 1 – P(A)).
使用样本空间图和双向表枚举两个组合事件的结果。计算单个事件的概率，以及互补事件的概率 (P(非 A) = 1 – P(A))。
Draw and interpret probability tree diagrams for independent and dependent events. Multiply along branches for combined probabilities, and add probabilities of mutually exclusive branches.
绘制并解读概率树状图，处理独立事件和相依事件。沿分枝相乘求组合概率，并加总互斥分枝的概率。
Understand experimental probability and expected frequency: expected frequency = probability × number of trials. Recognise that larger samples give more reliable estimates.
理解实验概率和预期频数：预期频数 = 概率 × 试验次数。认识到样本越大，估计越可靠。
Determine conditional probability using the formula P(A|B) = P(A and B) / P(B) and interpret it from Venn diagrams or tree diagrams.
使用公式 P(A|B) = P(A 且 B) / P(B) 确定条件概率，并从维恩图或树状图中解读。
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