📚 Year 11 Eduqas Maths: Core Topics Review | Year 11 Eduqas 数学：核心知识点梳理
This revision guide brings together the essential GCSE Mathematics topics for Year 11 students following the Eduqas specification. Each section distils key facts, formulas and methods you need to master before the final exams. Work through the content systematically, practise past-paper questions and pay close attention to the paired English–Chinese explanations to strengthen both your mathematical fluency and your exam confidence.
本复习指南整理了Eduqas考试局Year 11学生必须掌握的GCSE数学核心主题。每个部分凝练了考试前需要熟练的关键事实、公式和方法。请系统学习这些内容，练习历年真题，并仔细阅读中英对照的讲解，以提升数学流畅度和考试信心。
1. Number Operations and Properties | 数与运算性质
Understand place value, rounding to decimal places and significant figures, and the order of operations (BIDMAS/BODMAS). Be able to work with negative numbers confidently, including in context problems involving temperature or debt.
理解位值，会按小数位数和有效数字四舍五入，并掌握运算顺序（BIDMAS/BODMAS）。能够熟练处理负数，包括涉及温度或负债的实际问题。
Factors, multiples, primes, HCF and LCM are core tools. Use prime factor trees to express numbers as a product of prime factors. The highest common factor (HCF) and lowest common multiple (LCM) can then be found systematically, often required for fraction arithmetic.
因数、倍数、质数、最大公因数 (HCF) 和最小公倍数 (LCM) 是核心工具。利用质因数树将数字表示为质因数的乘积，然后系统地求出 HCF 和 LCM，这常用于分数运算。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Convert fluently between fractions, decimals and percentages. For a fractional multiplier, divide by the denominator and multiply by the numerator. To increase by a percentage, use a decimal multiplier greater than 1; for a decrease, use a multiplier less than 1.
熟练进行分数、小数和百分数之间的转换。求一个数的几分之几时，除以分母再乘以分子。增加一个百分数时，使用大于1的小数乘数；减少时，使用小于1的乘数。
Reverse percentage problems require you to divide by the multiplier. If a price includes 20% VAT, the original price is the final amount divided by 1.20. Compound interest and depreciation are modelled with repeated percentage change, often using the formula A = P(1 ± r/100)ⁿ.
逆向百分数问题需要用乘数去除。若价格包含20%增值税，原价等于最终金额÷1.20。复利与折旧用重复百分比变化建模，常使用公式 A = P(1 ± r/100)ⁿ。
3. Powers, Roots and Standard Form | 幂、根与标准形式
Apply index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Negative indices produce reciprocals, and fractional indices represent roots: a^{1/n} = n√a, a^{m/n} = (n√a)ᵐ.
运用指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。负指数产生倒数，分数指数表示根：a^{1/n} = n√a，a^{m/n} = (n√a)ᵐ。
Standard form is written as A × 10ⁿ where 1 ≤ A < 10. You must be able to add, subtract, multiply and divide numbers in standard form, often adjusting powers of ten so that the decimal parts can be combined.
标准形式写作 A × 10ⁿ，其中 1 ≤ A < 10。你需要能够对标准形式的数进行加、减、乘、除，通常需要调整10的幂次以使小数部分可合并。
4. Algebraic Expressions and Equations | 代数表达式与方程
Simplify expressions by collecting like terms, expanding brackets using the distributive law, and factorising by extracting common factors. For quadratic expressions, factorise into two binomials where possible, and solve by setting each factor to zero.
通过合并同类项、使用分配律展开括号、提取公因式进行因式分解来化简表达式。对于二次表达式，尽可能因式分解为两个一次因式，并令每个因式为零来解方程。
Linear equations are solved using inverse operations. When an equation contains fractions, multiply every term by the common denominator. Simultaneous linear equations can be solved by elimination or substitution, and the solution should be checked in both original equations.
使用逆运算求解线性方程。当方程含有分数时，每项乘以公分母。线性联立方程可用消元法或代入法求解，解需代入原方程检验。
The quadratic formula x = [-b ± √(b² – 4ac)] / 2a solves any quadratic equation ax² + bx + c = 0. Completing the square is also required, and its link to turning points of quadratic graphs must be understood.
二次公式 x = [-b ± √(b² – 4ac)] / 2a 可求解任意 ax² + bx + c = 0。也需要掌握配方法，并理解它与二次函数图像顶点之间的联系。
5. Sequences, Functions and Graphs | 数列、函数与图像
Find the nth term of linear sequences (an + b) and recognise quadratic sequences where the second difference is constant. Use the nth term to generate any term and to decide whether a given number belongs to the sequence.
找出线性数列的第 n 项 (an + b)，并识别二阶差为常数的二次数列。用第 n 项生成任意项，或判断某个数是否属于该数列。
Plot straight-line graphs from y = mx + c, where m is the gradient and c is the y-intercept. Parallel lines share the same gradient; perpendicular lines satisfy m₁ × m₂ = –1. Quadratic, cubic and reciprocal graphs should be sketched and interpreted.
由 y = mx + c 绘制直线图，其中 m 是斜率，c 是 y 轴截距。平行直线斜率相等；互相垂直直线的斜率满足 m₁ × m₂ = –1。需要会描绘并解读二次、三次和反比例函数图像。
For simultaneous equations, the point of intersection of two graphs gives the solution. Functions are written as f(x) and can be composed and inverse functions found for one-to-one functions.
对于联立方程，两条图像的交点坐标即为解。函数写作 f(x)，可进行复合运算，并求单射函数的反函数。
6. Ratio, Proportion and Rates of Change | 比、比例与变化率
Simplify ratios and share quantities in a given ratio by dividing by the total number of parts. Use the unitary method to find the value of one share. Ratio problems often appear in recipes, maps and scale diagrams.
化简比，并按给定比例分配数量，即除以总份数。使用归一法求一份的量。比例问题常出现在食谱、地图和比例图中。
Direct proportion: y = kx; inverse proportion: y = k/x. Recognise graphs of proportional relationships, and use given data pairs to find the constant k before answering further questions.
正比例：y = kx；反比例：y = k/x。识别比例关系的图像，并使用给出的数据对求出常数 k 后再回答问题。
Speed, density and pressure are compound measures. Use formula triangles or rearranged equations: Speed = Distance ÷ Time, Density = Mass ÷ Volume, Pressure = Force ÷ Area. Convert between units carefully.
速度、密度和压强是复合量度。可使用公式三角形或变形公式：速度=路程÷时间，密度=质量÷体积，压强=压力÷面积。注意单位换算。
7. Geometry: Angles, Shapes and Area | 几何：角、形状与面积
Know angle facts: angles on a straight line sum to 180°, angles in a triangle sum to 180°, vertically opposite angles are equal, and angles in polygons follow (n–2)×180° for the sum of interior angles.
掌握角的基本事实：平角180°，三角形内角和180°，对顶角相等，多边形内角和公式为 (n–2)×180°。
Area formulas: triangle = ½ × base × height, trapezium = ½ (a + b)h, circle = πr². Circumference = 2πr. For arcs and sectors, use the fraction angle/360 of the full circle.
面积公式：三角形=½×底×高，梯形=½(a+b)h，圆=πr²。周长=2πr。弧长和扇形面积用圆心角/360的分数计算。
Volume of prisms = area of cross-section × length. Cylinder volume = πr²h. Surface area requires careful visualisation of nets. For cones and spheres, you will be given the formulas, but must apply them accurately.
棱柱体积=横截面积×长。圆柱体积=πr²h。表面积需要仔细想象展开图。圆锥和球体的体积公式会提供，但需准确应用。
8. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
In any right-angled triangle, a² + b² = c², where c is the hypotenuse. Use this to find missing sides and to determine whether a triangle is right-angled. Leave answers in surd form where exact values are required.
任意直角三角形中，a² + b² = c²，其中 c 为斜边。用该定理求未知边长并判断三角形是否为直角三角形。需要精确值时保留根号形式。
Sine, cosine and tangent ratios relate the angles to side lengths in a right-angled triangle. SOHCAHTOA helps: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. For non-right-angled triangles, use the sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² – 2bc cos A.
正弦、余弦和正切比将直角三角形中的角与边长联系起来。用 SOHCAHTOA 记忆：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。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。
Know the exact trigonometric values for 0°, 30°, 45°, 60° and 90° – for example sin 30° = ½, cos 45° = √2/2. These are frequently tested without a calculator.
熟记 0°、30°、45°、60° 和 90° 的精确三角函数值，例如 sin 30° = ½，cos 45° = √2/2。这些常在不允许使用计算器的题目中考查。
9. Transformations and Vectors | 变换与向量
Four types of transformation: translation (vector), reflection (mirror line), rotation (centre, angle, direction) and enlargement (centre, scale factor). Negative scale factors produce an inversion through the centre of enlargement.
变换有四种：平移（用向量表示）、反射（指定对称轴）、旋转（中心、角度、方向）和放大（中心、比例因子）。负比例因子会产生通过放大中心的反演。
Describe transformations fully and use column vectors to represent translations. For combined transformations, apply them in the given order. Congruence and similarity links: two shapes are similar if one is an enlargement of the other.
完整描述变换，并用列向量表示平移。组合变换需按给定顺序施行。全等与相似的联系：若一个图形是另一个的放大，则两者相似。
Vectors represent both magnitude and direction. Add, subtract and multiply by a scalar. Use column vectors or bold letters. The vector between two points can be found by subtracting position vectors.
向量表示大小和方向。可进行加减和乘以标量。使用列向量或加粗字母。两点间的向量可通过位置向量相减得到。
10. Probability | 概率
Probability is measured on a scale from 0 to 1. For mutually exclusive outcomes, sum the probabilities to find the probability of either/or. The sum of probabilities of all possible outcomes is 1.
概率用0到1之间的数值度量。对于互斥事件，可通过相加求“或”的概率。所有可能结果的概率之和为1。
Use sample space diagrams, two-way tables and tree diagrams to list outcomes systematically. For independent events, P(A and B) = P(A) × P(B). For conditional probability, tree diagrams show changed probabilities on the second branches.
使用样本空间图、双向表和树状图系统地列出结果。独立事件中，P(A 且 B) = P(A) × P(B)。对于条件概率，树状图的第二分支显示变化的概率。
Understand expected frequency: multiply the probability by the number of trials. Pure frequency trees and Venn diagrams also appear, with notation A ∪ B, A ∩ B and A’.
理解期望频数：概率乘以试验次数。纯频数树和文氏图也会出现，使用符号 A ∪ B、A ∩ B 和 A’。
11. Statistics | 统计
Calculate averages: mean, median, mode and range. Distinguish between discrete and continuous data. For grouped frequency tables, estimate the mean using midpoints and identify the modal class and the interval containing the median.
计算平均数：均值、中位数、众数和极差。区分离散数据与连续数据。对于分组频数表，用组中值估算均值，并识别众数组及中位数所在区间。
Construct and interpret statistical diagrams: bar charts, pie charts, frequency polygons, cumulative frequency curves and histograms. For histograms with unequal class widths, frequency density = frequency ÷ class width.
绘制并解读统计图：条形图、饼图、频数多边形、累积频数曲线和直方图。对于组距不等的直方图，频数密度 = 频数 ÷ 组距。
Box plots display the five-number summary: minimum, lower quartile, median, upper quartile, maximum. Compare distributions using measures of central tendency and spread. Scatter graphs reveal correlation, and lines of best fit can make predictions.
箱线图展示五数概括：最小值、下四分位数、中位数、上四分位数、最大值。用集中趋势和离散程度比较分布。散点图揭示相关性，并可用最佳拟合线进行预测。
12. Problem Solving and Mathematical Reasoning | 问题解决与数学推理
Many exam questions require linking multiple topics. Break down the problem into smaller steps, represent information diagrammatically where possible, and always interpret your answer in the original context. Check for reasonableness.
许多考题需要联系多个主题。将问题分解为小步骤，尽可能用图表表示信息，并始终将答案放回原情境中解释。检查答案的合理性。
Algebraic proof uses logical sequences of equalities to show that a statement holds for all numbers. For example, prove that the sum of three consecutive integers is a multiple of 3 by letting them be n, n+1, n+2.
代数证明利用一系列等式的逻辑链条来证明一个命题对所有数成立。例如，设三个连续整数为 n、n+1、n+2，证明它们的和是3的倍数。
Estimation, rounding and using exact values (surd or π) are skills embedded across the syllabus. In geometry proofs, you may be asked to show a result using angle properties. Practise expressing reasoning clearly using mathematical language.
估算、四舍五入和使用精确值（根号或π）是贯穿整个大纲的技能。在几何证明中，可能要求运用角的性质推导结果。练习用数学语言清晰地表达推理过程。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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