📚 GCSE CCEA Maths Final Revision Checklist | GCSE CCEA 数学期末复习提纲
As the GCSE CCEA Mathematics exam approaches, a clear revision checklist can help you organise your study, track your progress, and focus on the topics that matter most. This guide covers the essential knowledge and skills for both Foundation and Higher tiers, presented in a side‑by‑side bilingual format to support your learning. Use it to identify strengths, close gaps, and walk into the exam with confidence.
随着 GCSE CCEA 数学考试临近，一份清晰的复习提纲能帮助你合理安排学习计划、跟踪进度并聚焦核心考点。本文涵盖 Foundation 与 Higher 层级所需的关键知识与技能，采用双语对照形式，助你查漏补缺。利用这份提纲找出薄弱环节，自信应考。
1. Number and Operations | 数与运算
Build fluency with the four operations across integers, fractions, decimals and percentages. You must be able to convert freely between these forms and apply them in practical contexts such as best buys, compound interest, and reverse percentages. For Higher tier, include recurring decimals and their fraction equivalents.
熟练掌握整数、分数、小数和百分数的四则运算，并能自如地在不同形式间转换。会在最佳购买、复利、逆推百分数等现实情境中应用。Higher 层还需处理循环小数与其分数形式。
Work confidently with primes, multiples, factors, LCM and HCF. Use prime factor decomposition (product of prime factors) to solve problems and apply index laws for multiplication, division and powers. Estimation and rounding are vital for checking answers and for questions on upper and lower bounds.
熟练运用质数、倍数、因数、最小公倍数和最大公因数。利用质因数分解解决问题，并掌握乘法、除法及幂的指数运算法则。估算与舍入是检验答案及处理上下界问题的重要工具。
Higher candidates must simplify surds, such as √18 = 3√2, and rationalise denominators like 1/(√5 + 1). Be able to manipulate standard form (a × 10ⁿ) without a calculator, and understand the effect of negative powers. Use approximation to determine error intervals.
Higher 考生必须能化简根式，如 √18 = 3√2，并进行分母有理化，如 1/(√5 + 1)。能不用计算器处理标准形式 (a × 10ⁿ)，并理解负指数幂的影响。使用近似值确定误差区间。
2. Algebra Fundamentals | 代数基础
Manipulate algebraic expressions by collecting like terms, expanding single and double brackets, and factorising into single brackets. For quadratic expressions, factorise x² + bx + c and, on Higher, ax² + bx + c where a > 1. Recognise and use the difference of two squares: a² − b² = (a + b)(a − b).
通过合并同类项、展开单项与双项括号、提取公因式来化简代数式。能因式分解二次式 x² + bx + c；Higher 层还需分解形如 ax² + bx + c (a > 1)的式子。识别并运用平方差公式：a² − b² = (a + b)(a − b)。
Solve linear equations with unknowns on both sides, including those with brackets and fractions. Form and solve equations from word problems. For quadratic equations, use factorising, completing the square, or the quadratic formula. On Higher, be prepared to use the formula:
x = [−b ± √(b² − 4ac)] / 2a
解未知数在两侧的线性方程，包括含括号和分数的方程。能从文字题列方程并求解。对于二次方程，会用因式分解法、配方法或求根公式。Higher 层需运用公式：
x = [−b ± √(b² − 4ac)] / 2a
Handle inequalities: solve linear inequalities and represent solution sets on a number line. Know how to solve simultaneous equations by elimination and substitution, and interpret solutions graphically. Work with sequences: find the nth term of linear and quadratic sequences, and use term‑to‑term rules. Higher: proof and geometric sequences.
处理不等式：解线性不等式并在数轴上表示解集。会用消元法和代入法解联立方程，并能用图像解释解。掌握数列：求线性与二次数列的第 n 项，使用项间递推规则。Higher：代数证明与等比数列。
3. Graphs and Functions | 图形与函数
Plot straight line graphs using y = mx + c, identifying gradient and intercept. Recognise shapes of quadratic, cubic, reciprocal and exponential graphs. Solve equations graphically, finding roots and intersection points. For Higher, understand exponential growth and decay models.
用 y = mx + c 绘制直线图，辨别斜率与截距。识别二次、三次、反比例和指数函数的图像形状。用图像解方程，找出根与交点。Higher 层理解指数增长与衰减模型。
Use function notation: f(x) = 2x + 3. Find inverse functions f⁻¹(x) and composite functions fg(x). Interpret transformations of graphs: translations, reflections and stretches. Higher: apply these to trigonometric and other functions.
使用函数记号：f(x) = 2x + 3。求反函数 f⁻¹(x) 和复合函数 fg(x)。解读图像的变换：平移、反射与拉伸。Higher：将这些变换应用于三角函数等。
Calculate rate of change as gradient of a chord or tangent. For Higher, be able to differentiate xⁿ to find gradients and apply to kinematics (velocity and acceleration). Use graphs to find approximate areas under curves.
将变化率视为弦或切线的斜率。Higher 层能对 xⁿ 求导以计算梯度，并应用于运动学（速度与加速度）。用图像估算曲线下方面积。
4. Ratio, Proportion and Rates of Change | 比、比例与变化率
Simplify ratios and divide quantities in a given ratio, linking to fractions and linear functions. Understand scale factors in diagrams and maps. Solve problems involving direct and inverse proportion, setting up proportionality equations and identifying the constant k.
化简比并按比例分配量，将比与分数和线性函数关联。理解图形与地图中的比例尺。解决正比与反比问题，建立比例方程并确定常数 k。
Interpret real‑life graphs: distance–time (gradient = speed) and velocity–time (gradient = acceleration, area = distance). Use compound measures such as speed, density and pressure, converting between units where necessary. For Higher, work with rates of change in calculus contexts.
解读实际生活图表：距离‑时间图（斜率 = 速度）和速度‑时间图（斜率 = 加速度，面积 = 距离）。使用速度、密度、压力等复合量度，必要时进行单位换算。Higher 层在微积分背景中处理变化率。
5. Geometry and Measures | 几何与测量
Recall angle facts: angles on a straight line sum to 180°, vertically opposite angles are equal, and angles around a point add to 360°. Apply angle properties of parallel lines, triangles, quadrilaterals and polygons (interior and exterior angle sums). Use these facts in multi‑step angle problems and proofs.
熟记角度事实：直线上的角之和为 180°，对顶角相等，周角为 360°。运用平行线、三角形、四边形和多边形的角度性质（内角和外角和）。在多步骤角度问题和证明中运用这些事实。
Calculate area and perimeter of rectangles, triangles, parallelograms, trapezia, and circles. Find arc length and sector area using fractions of 360°. Compute surface area and volume of prisms and cylinders; Higher must also handle cones, spheres and frustums with relevant formulas.
计算矩形、三角形、平行四边形、梯形和圆的面积与周长。利用 360° 的分数求弧长和扇形面积。计算棱柱和圆柱的表面积与体积；Higher 还需掌握圆锥、球及平截体的相关公式。
Understand similarity and congruence. Use SSS, SAS, ASA, RHS rules to prove triangles congruent. For similar shapes, apply linear scale factor k to get area scale factor k² and volume scale factor k³. Solve problems involving bearings and scale drawings.
理解相似与全等。运用 SSS、SAS、ASA、RHS 规则证明三角形全等。对于相似形，应用线尺度因子 k 得出面积尺度因子 k² 和体积尺度因子 k³。解决涉及方位角和比例图的问题。
6. Pythagoras and Trigonometry | 勾股定理与三角学
Use Pythagoras’ theorem a² + b² = c² to find missing sides in right‑angled triangles, including in 3D problems where you need to find diagonal lengths of cuboids. Apply trigonometric ratios sin θ, cos θ, tan θ to find sides and angles, and solve problems involving angles of elevation and depression.
运用勾股定理 a² + b² = c² 求直角三角形中的未知边，包括在三维问题中求长方体的对角线长。应用正弦 sin θ、余弦 cos θ、正切 tan θ 求边长和角度，解决仰角和俯角问题。
Know exact trigonometric values for 0°, 30°, 45°, 60°, 90° without a calculator. For Higher: use the sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A to solve non‑right‑angled triangles. Apply the area formula ½ ab sin C.
不借助计算器记住 0°、30°、45°、60°、90° 的精确三角值。Higher 层：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 解非直角三角形。运用面积公式 ½ ab sin C。
7. Probability | 概率
Calculate the probability of a single event and list outcomes using sample space diagrams. For combined events, use frequency trees, two‑way tables and tree diagrams. Apply the AND rule (multiply along branches) and OR rule (add probabilities of mutually exclusive events).
计算单一事件的概率，并用样本空间图列出结果。对于组合事件，使用频率树、双向表和树形图。运用 “且” 法则（沿分支相乘）和 “或” 法则（互斥事件概率相加）。
For conditional probability (Higher), interpret questions where events are dependent, using tree diagrams with conditional branches. Use Venn diagrams to represent events and apply the formula P(A|B) = P(A ∩ B) / P(B). Understand independence formally.
处理条件概率（Higher）：解读事件相关的情景，使用含条件分支的树形图。利用文氏图表示事件，并运用公式 P(A|B) = P(A ∩ B) / P(B)。正式理解独立性的概念。
8. Statistics | 统计
Calculate and compare the mean, median, mode and range for small data sets and from frequency tables. Understand the advantages of each average. Construct and interpret bar charts, pie charts, box plots, and cumulative frequency diagrams; use them to find median, quartiles and interquartile range.
计算并比较小数据集及频数表中的平均数、中位数、众数和极差。理解各平均数的适用情形。绘制并解读条形图、饼图、箱线图和累积频数图；利用它们求中位数、四分位数和四分位距。
For grouped data, estimate the mean and draw histograms with equal (or unequal) class widths using frequency density = frequency ÷ class width. Interpret scatter graphs: describe correlation, draw a line of best fit, and make predictions. Higher: understand sampling methods, including stratified sampling from proportional groups.
对于分组数据，估算平均数并绘制组距相等（或不等）的直方图，使用频数密度 = 频数 ÷ 组距。解读散点图：描述相关性，绘制最佳拟合线并进行预测。Higher：理解抽样方法，包括按比例分层抽样。
9. Vectors and Transformations | 向量与变换
Represent vectors as column vectors and arrow diagrams. Add and subtract vectors, and multiply by a scalar. Solve geometric vector problems, showing that points are collinear by proving one vector is a scalar multiple of another. Express parallel vectors and understand resultant vectors.
用列向量和箭头图表示向量。进行向量的加、减及标量乘法。解决几何向量问题，通过证明一个向量是另一向量的标量倍数来显示点共线。表示平行向量并理解合向量。
Describe and perform single and combined transformations: translations, reflections (in line x = c, y = k, y = x, y = −x), rotations (by 90°, 180°, 270° about a point), and enlargements (including negative and fractional scale factors). Use transformations to map one shape onto another.
描述并执行单一及组合变换：平移、反射（关于直线 x = c、y = k、y = x、y = −x）、旋转（绕一点转 90°、180°、270°）和放大（含负数和分数尺度因子）。利用变换将图形映射到另一图形上。
10. Key Formulas and Exam Tips | 关键公式与考试贴士
Memorise these essential formulas (many are not on the formula sheet):
牢记以下核心公式（多不在提供的公式表上）：
| Formula (English) | 公式 (中文) |
|---|---|
| Area of trapezium = ½(a + b)h | 梯形面积 = ½(a + b)h |
| Volume of prism = area of cross‑section × length | 棱柱体积 = 截面积 × 长 |
| Circle circumference = πd or 2πr, area = πr² | Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信） CommentsMore posts |
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导