📚 GCSE Maths: End-of-Term Revision Checklist | GCSE 数学：期末复习提纲
As the term draws to a close, it's time to consolidate your GCSE Maths knowledge with a structured revision plan. This checklist covers all major topic areas, from number skills to statistics, providing a bilingual summary of key concepts and formulas that commonly appear on GCSE exams. Use this guide to identify strengths and areas for improvement, and practise plenty of past paper questions to build confidence.
学期接近尾声，是时候用一份结构化的复习计划来巩固你的 GCSE 数学知识了。这份复习提纲涵盖了从数字技能到统计学的所有主要专题，用中英双语总结了考试中常见的核心概念和公式。请利用这份指南找出自己的强项和薄弱环节，并通过大量历年真题练习来增强信心。
1. Number Skills and Operations | 数字技能与运算
Understand the place value system for whole numbers and decimals, including reading and writing numbers up to millions and beyond. It is essential to order positive and negative integers, decimals and fractions correctly on a number line.
理解整数和小数的位值系统，包括读写百万及以上的数字。能够在数轴上正确排列正负整数、小数和分数是基础中的基础。
Master the four operations (addition, subtraction, multiplication and division) with negative numbers. Remember that subtracting a negative number is the same as adding its positive counterpart.
熟练掌握负数的四则运算（加、减、乘、除）。记住减去一个负数等于加上它的相反数。
Be confident with factors, multiples, prime numbers, highest common factor (HCF) and lowest common multiple (LCM). Use prime factor trees to express any integer as a product of primes, which helps when finding HCF and LCM of larger numbers.
熟练因数、倍数、质数、最大公因数 (HCF) 和最小公倍数 (LCM)。用质因数树将任意整数写成质数乘积的形式，这在求较大数的 HCF 和 LCM 时尤其有用。
powers: xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ. Understand that x⁰ = 1 and x⁻¹ = 1/x.
应用正指数、负指数和零指数的指数运算律：xᵃ × xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ。理解 x⁰ = 1 以及 x⁻¹ = 1/x。
e 1 ≤ a < 10 and n is an integer. Practise converting between ordinary numbers and standard form, and performing calculations using standard form.
熟练处理表示极大或极小数字的标准形式（科学记数法）。标准形式写为 a × 10ⁿ，其中 1 ≤ a < 10 且 n 为整数。练习普通记数与标准形式间的互化，以及用标准形式进行运算。
| English Term | 中文术语 | Example |
|---|---|---|
| Prime factorisation | 质因数分解 | 60 = 2² × 3 × 5 |
| Standard form | 标准形式 | 4.5 × 10⁻³ |
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Simplify fractions by dividing numerator and denominator by their highest common factor. Convert between mixed numbers and improper fractions fluently, as both forms appear in exam questions.
通过分子分母同除以最大公因数来化简分数。熟练在带分数与假分数之间互化，两种形式都会在考题中出现。
Add, subtract, multiply and divide fractions confidently. For addition and subtraction, find a common denominator first. For division, multiply by the reciprocal (flip the second fraction).
自信地进行分数的加减乘除。加减运算要先通分求公分母；除法则是乘以倒数（将第二个分数上下颠倒）。
ween fractions, decimals and percentages. Memorise key equivalences such as 1/2 = 0.5 = 50%, 1/3 ≈ 0.333..., 1/4 = 0.25 = 25%, and so on. This speed is valuable in calculator and non-calculator papers.
在分数、小数和百分数之间无缝转换。牢记关键等值关系，如 1/2 = 0.5 = 50%，1/3 ≈ 0.333...，1/4 = 0.25 = 25% 等。这种速度在计算器与非计算器卷面上都很有价值。
Calculate percentage increase and decrease using a multiplier. For an increase of r%, multiply the original amount by (1 + r/100); for a decrease, multiply by (1 - r/100). Reverse percentage problems require dividing by the original multiplier to find the amount before the change.
使用乘数计算百分比的增减。增加 r% 时，将原数乘以 (1 + r/100)；减少时乘以 (1 - r/100)。反推百分数问题则需要除以原来的乘数，以求出变化前的量。
Find a fraction of an amount by multiplying, and a percentage of an amount by converting the percentage to a fraction or decimal first. These skills underpin ratio, proportion and probability questions throughout the GCSE syllabus.
求一个数的几分之几用乘法；求一个数的百分之几先将百分数转换为分数或小数再乘。这些技能是整个 GCSE 大纲中比、比例和概率问题的基础。
3. Algebra Fundamentals | 代数基础
ab, and y × y as y². Understand terms, coefficients, expressions, equations and identities – precise vocabulary helps avoid confusion.
使用正确的代数记法：将 3 × x 写为 3x，a × b 写为 ab，y × y 写为 y²。理解项、系数、表达式、方程和恒等式——精确的术语有助于避免混淆。
a + 2b. Always check that terms have exactly the same variable part before combining them.
通过合并同类项化简表达式，如 2a + 3b + 5a - b = 7a + 2b。合并前务必检查各项是否具有完全相同的变量部分。
8x - 12. For double brackets, use the FOIL method or grid: (x + 5)(x - 2) = x² + 3x - 10.
准确展开括号。单项括号时，将括号外的项乘入括号内每一项：4(2x - 3) = 8x - 12。双项括号则使用 FOIL 法或网格法：(x + 5)(x - 2) = x² + 3x - 10。
ssions of the form x² + bx + c, factorise into two brackets by finding two numbers that multiply to c and add to b.
通过提取最大公因式进行因式分解，如 6x² + 9x = 3x(2x + 3)。对形如 x² + bx + c 的二次式，找出乘积为 c、和为 b 的两个数，分解为两个一次因式的乘积。
ules are fundamental when working with powers, roots and later with exponential functions.
将指数运算律应用于代数式：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。这些规则在处理幂、根以及后续的指数函数时至关重要。
Substitute values into formulas and expressions, including those with brackets and fractions. Pay careful attention to negative numbers and the order of operations (BIDMAS/BODMAS) when evaluating.
将数值代入公式和表达式，包括含有括号和分数的式子。代入求值时特别注意负号与运算次序（先括号、再指数、再乘除、最后加减）。
4. Solving Equations and Inequalities | 解方程与不等式
Solve linear equations where the unknown appears on one or both sides. Use inverse operations to isolate the variable, and always check your solution by substituting it back into the original equation.
解未知数在一边或两边的线性方程。使用逆运算分离变量，并始终将解代回原方程进行检验。
Tackle quadratic equations by factorising, setting each bracket equal to zero, and solving for x. When the quadratic does not factorise neatly, apply the quadratic formula:
通过因式分解、令每个括号为零并解出 x 来处理二次方程。若二次式不易分解，则使用二次公式：
x = [ -b ± √b² - 4ac ] / 2a
For foundation tier, focus primarily on factorising; higher tier candidates must be fluent with the formula and with completing the square.
基础层级主要关注因式分解；高阶考生还需熟练运用公式法与配方法。
then add or subtract the equations. Always present solutions in the form x = ... , y = ...
用消元法或代入法解二元一次方程组。消元时，使某一变量的系数相等，然后两方程相加减。答案始终写成 x = ... , y = ... 的形式。
Represent inequalities on a number line using open and closed circles. Solve linear inequalities in the same way as equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
在数轴上用空心和实心圆表示不等式。解一元一次不等式时与解方程类似，但乘或除以负数时必须反转不等号方向。
Form and solve equations from word problems. Translate real-life scenarios into algebraic sentences, clearly defining the variable and constructing the equation before solving.
从文字题中建立并求解方程。将实际情景翻译为代数语句，解题前明确定义变量并构建方程。
5. Graphs and Functions | 图像与函数
g the x-coordinates and y-coordinates of the endpoints: midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
在四个象限中描点、读坐标。求线段中点时，将两端点的 x 坐标与 y 坐标分别取平均数：中点 = ((x₁+x₂)/2, (y₁+y₂)/2)。
e gradient and c is the y-intercept. Calculate gradient as rise / run, and be able to find the equation of a line given two points or a point and the gradient.
理解直线方程的斜截式 y = mx + c，其中 m 为斜率，c 为 y 轴截距。用纵向增量除以横向增量计算斜率，并能够根据两点或一点与斜率求直线方程。
creating a table of values. Recognise their characteristic shape (a parabola) and identify roots, the vertex and the line of symmetry. For higher tier, also plot cubic and reciprocal graphs.
通过列取值表绘制二次函数 y = ax² + bx + c 的图像。识别其典型形状（抛物线），找出根、顶点与对称轴。高阶学生还需绘制三次函数与反比例函数的图像。
Interpret real-life graphs such as distance–time and velocity–time graphs. The gradient of a distance–time graph gives speed; the area under a velocity–time graph gives distance.
解读实际情境中的图像，如距离–时间图与速度–时间图。距离–时间图的斜率表示速度；速度–时间图下的面积表示距离。
Use function notation f(x) and evaluate functions for given inputs. For higher tier, apply simple transformations: f(x + a) translates the graph -a units horizontally, and f(x) + a translates it a units vertically.
使用函数记法 f(x) 并对给定输入求值。高阶学生还需应用简单变换：f(x + a) 使图像水平平移 -a 个单位，f(x) + a 使图像垂直平移 a 个单位。
6. Ratio, Proportion and Rates of Change | 比、比例与变化率
Write ratios in their simplest form, using whole numbers. Divide a quantity into a given ratio by finding the total number of parts and calculating the value of one part first.
用最简整数比表示比例关系。按给定比例分配数量时，先求总份数并算出一份的值，再求各部分的量。
line through the origin. For inverse proportion, y = k/x, and the graph is a curve that never touches the axes.
解决正比例问题：y = kx，其中 k 为比例常数。正比例图像是一条过原点的直线。反比例关系 y = k/x 的图像则是永不到达坐标轴的曲线。
Use the unitary method to find best buys. Reduce quantities to a common unit (e.g., price per 100 g or per litre) and compare. This skill also helps with converting currencies and interpreting recipes.
用单位法寻找最划算的选择。将数量归一到统一单位（如每 100 克或每升的价格）进行比较。这一技能在货币兑换和食谱调整中同样有用。
ea. Be able to rearrange these formulas using formula triangles or algebraic manipulation.
处理复合量度：速度 = 距离 ÷ 时间，密度 = 质量 ÷ 体积，压强 = 力 ÷ 面积。要能运用公式三角形或代数变形来对公式进行移项转化。
Apply scale factors to maps and scale drawings. A scale of 1 : n means that real distances are n times larger than those on the plan. Link scale factors to linear, area and volume enlargements for similar shapes.
运用比例尺解读地图与比例图。比例 1 : n 表示实际距离是图上距离的 n 倍。将比例因子与相似图形的长度、面积和体积放大联系起来。
7. Geometry and Measures | 几何与测量
Calculate missing angles using basic angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and interior angles sum to 180°.
运用基本角度性质计算未知角：平角为 180°，周角为 360°，对顶角相等。平行线中，内错角相等、同位角相等、同旁内角互补（和为 180°）。
or and exterior angles of polygons. The sum of interior angles of an n-sided polygon is (n – 2) × 180°. The sum of exterior angles of any convex polygon is always 360°.
理解多边形的内角与外角。n 边形内角和为 (n – 2) × 180°。任意凸多边形外角和恒为 360°。
| Shape | Area Formula | 形状 | 面积公式 |
|---|---|---|---|
| Rectangle | A = l × w | 长方形 | A = 长 × 宽 |
| Triangle | A = ½ × b × h | 三角形 | A = ½ × 底 × 高 |
| Parallelogram | A = b × h | 平行四边形 | A = 底 × 高 |
| Trapezium | A = ½ (a + b)h | 梯形 | A = ½ (上底+下底) × 高 |
| Circle | A = πr² | 圆 | A = π × 半径² |
s, break them down into simpler shapes, calculate individual areas, and add or subtract as needed.
求直线组成的图形的周长，以及圆的周长（C = 2πr 或 C = πd）。对于组合图形，先拆分为简单图形，分别计算面积，
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