📚 Year 11 WJEC Mathematics: Core Knowledge Points Summary | Year 11 WJEC 数学：核心知识点梳理
Mastering Year 11 WJEC Mathematics means having a solid grasp of the fundamental building blocks that will be tested in both calculator and non-calculator papers. This guide brings together the essential topics across number, algebra, geometry, statistics and probability, giving you a clear pathway to reinforce your understanding and tackle exam questions with confidence.
掌握 Year 11 WJEC 数学意味着对数、代数、几何、统计和概率等基础模块有扎实的掌握，这些内容都会出现在可用计算器和不可用计算器的试卷中。本文梳理了这些核心主题，为你提供一条清晰的复习路径，帮助你巩固理解，自信应对考试题目。
1. Number Concepts | 数的概念
Understanding the types of numbers and their properties underpins many topics. You need to be comfortable with integers, fractions, decimals, percentages, and directed numbers, as well as applying the four operations in the correct order using BIDMAS/BODMAS.
理解数字的类型及其性质是许多专题的基础。你需要熟悉整数、分数、小数、百分数和有向数，并能够运用 BIDMAS/BODMAS 规则按照正确顺序进行四种运算。
using prime factor decomposition. For example, the prime factor form of 60 is 2² × 3 × 5.
关键技能包括利用质因数分解求两个或更多数的最大公因数（HCF）和最小公倍数（LCM）。例如，60 的质因数分解形式为 2² × 3 × 5。
. You must be able to convert between ordinary and standard form, and to calculate with numbers in standard form.
标准形式用于简洁地表示非常大或非常小的数：a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。你必须能够在普通形式和标准形式之间转换，并能用标准形式进行计算。
Rounding to decimal places (d.p.) and significant figures (s.f.), and estimating answers by rounding, are essential checking tools. For example, 4.876 to 2 d.p. is 4.88; to 2 s.f. it is 4.9.
将数值舍入到指定小数位数（d.p.）和有效数字位数（s.f.），以及通过舍入进行估算，是重要的验证工具。例如，4.876 保留 2 位小数为 4.88，保留 2 位有效数字为 4.9。
2. Algebra Foundations | 代数基础
Algebraic manipulation forms the backbone of problem solving. You must be proficient in simplifying expressions by collecting like terms, expanding brackets, and factorising. For instance, 3x + 5y – 2x + y simplifies to x + 6y.
代数运算是解决问题的基础。你必须熟练地通过合并同类项、展开括号和因式分解来化简表达式。例如，3x + 5y – 2x + y 可化简为 x + 6y。
Expanding double brackets: (x + 2)(x – 3) = x² – 3x + 2x – 6 = x² – x – 6. Factorising a quadratic like x² – x – 6 back into (x + 2)(x – 3) requires finding two numbers that multiply to –6 and add to –1.
展开双括号：(x + 2)(x – 3) = x² – 3x + 2x – 6 = x² – x – 6。将二次式如 x² – x – 6 分解回 (x + 2)(x – 3) 需要找到两个数，其乘积为 –6，其和为 –1。
must be memorised: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a1/n = ⁿ√a. Fractional and negative indices often appear in higher-tier questions.
必须熟记指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及 a1/n = ⁿ√a。分数指数和负指数经常出现在高阶试题中。
3. Solving Equations and Inequalities | 解方程与不等式
ing the unknown. Always perform the same operation on both sides. For example, to solve 2x – 5 = 13, add 5 to get 2x = 18, then divide by 2 to give x = 9.
线性方程要求将未知数单独解出，始终在等式两边进行相同的运算。例如，解 2x – 5 = 13，两边加 5 得 2x = 18，再除以 2 得 x = 9。
then apply the principle that if the product is zero, at least one factor is zero. For x² – 5x + 6 = 0, factorise to (x – 2)(x – 3) = 0 → x = 2 or x = 3.
用因式分解法解二次方程时，先将方程设为零，进行因式分解，然后运用“乘积为零则至少有一个因式为零”的原理。对于 x² – 5x + 6 = 0，分解为 (x – 2)(x – 3) = 0 → x = 2 或 x = 3。
For inequalities, treat them like equations but remember to flip the inequality sign when multiplying or dividing by a negative number. For example, –2x < 8 becomes x > –4.
对于不等式，解法与方程类似，但要记住当乘以或除以一个负数时，不等号方向要改变。例如，–2x < 8 变为 x > –4。
ubstitution method is particularly useful when one equation is linear and the other is non‑linear, for example y = x² and y = 2x + 3.
联立方程可以通过消元法或代入法求解。当一个方程为线性而另一个为非线性时，代入法特别有用，例如 y = x² 和 y = 2x + 3。
4. Sequences and nth Term | 数列与第 n 项
terms. To find the nth term of an arithmetic sequence, use the formula: nth term = a + (n – s the first term and d is the common difference. In WJEC, it is often expressed as dn + (a – d).
线性数列的相邻两项之间的差为常数。求等差数列第 n 项的公式为：第 n 项 = a + (n – 1)d，其中 a 是首项，d 是公差。在 WJEC 中，常表示为 dn + (a – d)。
For example, the sequence 5, 8, 11, 14, … has d = 3, a = 5, so the nth term is 3n + 2.
例如，数列 5, 8, 11, 14, … 中 d = 3，a = 5，因此第 n 项为 3n + 2。
Quadratic sequences have a second difference that is constant. The nth term takes the form an² + bn + c. You find a by halving the second difference, then set up equations to find b and c.
二次数列的二次差为常数。第 n 项的形式为 an² + bn + c。先将二次差除以 2 得到 a，然后建立方程求出 b 和 c。
Recognising special sequences such as triangular numbers, square numbers and Fibonacci‑type sequences is also examined.
识别特殊数列，如三角形数、平方数和斐波那契型数列，也会出现在考试中。
5. Ratio, Proportion and Rates of Change | 比例、比率与变化率
Working with ratios often involves sharing a quantity in a given ratio. To share £ , first find the total number of parts (3 + 5 = 8), then calculate the value of one part (£60 ÷ 8 = £7.50), so the shares are 3 × £7.50 = £22.50 and 5 × £7.50 = £37.50.
处理比率问题经常涉及按给定比例分配数量。按 3:5 的比例分配 £60，先求总份数（3 + 5 = 8），再求每份的价值（£60 ÷ 8 = £7.50），因此分配额为 3 × £7.50 = £22.50 和 5 × £7.50 = £37.50。
Graphs of direct proportion are straight lines through the origin; inverse proportion produces a hyperbola.
正比例：两个量以相同的速率增加或减少，因此 y = kx。反比例：y = k/x。正比例的图像是过原点的直线；反比例产生双曲线。
Compound measures like speed (distance ÷ time), density (mass ÷ volume) and pressure (force ÷ area) require correct unit conversions. For example, converting 50 km/h into m/s involves multiplying by 1000 and dividing by 3600.
复合量度如速度（距离 ÷ 时间）、密度（质量 ÷ 体积）和压强（力 ÷ 面积）要求正确的单位换算。例如，将 50 km/h 转换成 m/s，需乘以 1000 再除以 3600。
Percentage change, repeated percentage change (compound interest and depreciation), and reverse percentages are core applications. The multiplier method is essential: for a 15% increase, multiply by 1.15; for a 15% decrease, multiply by 0.85.
百分比变化、重复百分比变化（复利和折旧）以及逆向百分数是核心应用。乘数法至关重要：增加 15% 则乘以 1.15；减少 15% 则乘以 0.85。
6. Geometry: Angles and Properties of Shapes | 几何：角度与图形的性质
Angle rules form the basis of geometric reasoning: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal, and angles in a triangle sum to 180°.
角度法则是几何推理的基础：直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等，三角形内角之和为 180°。
Parallel lines create equal corresponding angles, equal alternate angles, and co‑interior angles summing to 180°. These are frequently tested in multi‑step angle problems.
平行线产生相等的同位角、相等的内错角以及和为 180° 的同旁内角。这些在多步角度问题中经常考查。
Polygon angle properties: sum of interior angles of an n‑sided polygon = (n – × 180°. Each exterior angle of a regular polygon = 360° ÷ n, and the interior angle = 180° – exterior angle.
多边形的角度性质：n 边形内角和 = (n – 2) × 180°。正多边形的每个外角 = 360° ÷ n，内角 = 180° – 外角。
Circle theorems are a higher‑tier staple: angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°.
圆定理是更高层级的常考内容：圆心角是同一弧上圆周角的二倍；半圆内的圆周角为 90°；同弧上的圆周角相等；圆内接四边形的对角互补（和为 180°）。
7. Perimeter, Area and Volume | 周长、面积与体积
perpendicular height; trapezium = ½(a + b)h; circle area = πr², circumference = 2πr or πd.
标准面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 垂直高；梯形 = ½(a + b)h；圆面积 = πr²，周长 = 2πr 或 πd。
For composite shapes, split the shape into simpler parts, calculate their areas, then add or subtract as required. Always check for missing lengths by using properties of rectangles and symmetry.
对于组合图形，将图形分割成简单部分，分别计算其面积，然后根据需要进行加减。始终利用矩形的性质及对称性推导缺失的边长。
Volume of prisms = area of cross‑ lume of a pyramid = ⅓ × base area × height; cone volume = ⅓ πr²h; sphere volume = ⁴⁄₃ πr³.
棱柱的体积 = 横截面积 × 长度。对于圆柱体，这就是 πr²h。棱锥的体积 = ⅓ × 底面积 × 高；圆锥体积 = ⅓ πr²h；球体积 = ⁴⁄₃ πr³。
here = 4πr²; surface area of a cone = πrl + πr², where l is the slant height. These higher‑tier formulas must be applied accurately, including correct substitution and rounding.
球的表面积 = 4πr²；圆锥表面积 = πrl + πr²，其中 l 是斜高。这些高阶公式必须准确应用，包括正确代入和按要求取近似值。
8. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数
Pythagoras’ theorem applies only to right‑ ere c is the hypotenuse. Use it to find a missing side, or to check if a triangle is right‑angled by verifying the equality.
勾股定理仅适用于直角三角形：a² + b² = c²，其中 c 为斜边。可用于求未知边长，或通过验证等式是否成立来判断三角形是否为直角三角形。
Trigonometric ratios in a right‑ acent/hypotenuse, tan θ = opposite/adjacent. To recall them, use the mnemonic SOH CAH TOA.
直角三角形的三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用助记符 SOH CAH TOA 来记忆。
When finding an angle, use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹). In non‑right‑ ngles, apply the sine rule: a/sin A = b/sin B = c/sin C, and the cosine rule: a² = b² + c² – 2bc cos A.
求角度时使用反三角函数（sin⁻¹，cos⁻¹，tan⁻¹）。对于非直角三角形，应用正弦定理：a/sin A = b/sin B = c/sin C，以及余弦定理：a² = b² + c² – 2bc cos A。
own. Exact trigonometric values for 0°, 30°, 45°, 60° and 90° (e.g. sin 30° = ½, cos 30° = √3/2) should be memorised.
当知道两边及其夹角时，可用三角形面积公式 ½ ab sin C。需要记住 0°、30°、45°、60° 和 90° 的精确三角函数值（例如 sin 30° = ½，cos 30° = √3/2）。
9. Data Handling and Statistics | 数据处理与统计
sed to represent data visually. For continuous data, histograms with equal class intervals are typical, while unequal intervals require frequency density = frequency ÷ class width.
数据收集与整理从设计合适的表格和图表开始。条形图、饼图、折线图和散点图均可用于直观展示数据。对于连续数据，通常使用组距相等的直方图，而组距不等时则需要计算频率密度 = 频数 ÷ 组宽。
= middle value when data is ordered; mode = most frequent value. The range = highest value – lowest value, representing spread.
平均数：均值 = 数值总和 ÷ 数值个数；中位数 = 数据排序后的中间值；众数 = 出现频率最高的值。极差 = 最大值 – 最小值，表示数据的离散程度。
ative frequency diagrams help find the median, quartiles and interquartile range (IQR = Q3 – Q1). Box plots (box‑and‑whisker diagrams) give a visual summary of median, quartiles and extremes.
累积频数图用于查找中位数、四分位数和四分位距（IQR = Q3 – Q1）。箱线图则以可视化方式概括中位数、四分位数和极端值。
Scatter graphs show correlation between two variables. A line of best fit can be drawn by eye, passing through the mean point, and used to make predictions. Be aware that correlation does not imply causation.
散点图显示两个变量之间的相关性。可通过目测绘制一条最佳拟合线，通常经过均值点，并用于预测。要注意相关关系并不暗示因果关系。
10. Probability | 概率
bability is measured on a scale from 0 to 1. The probability of an event not happening = 1 – ally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率的度量范围为 0 到 1。事件不发生的概率 = 1 – 事件发生的概率。对于等可能结果，P(事件) = 有利结果的数量 / 总结果数量。
umber of trials. This helps to predict how many times an outcome will occur in repeated experiments.
期望频数 = 概率 × 试验次数。这有助于预测在重复试验中某个结果将发生多少次。
exclusive events: P(A or B) = P(A) + P(B). Conditional probability (higher tier) uses P(A|B) = P(A and B) / P(B).
维恩图和树状图是强大的工具。对于独立事件：P(A 与 B) = P(A) × P(B)。对于互斥事件：P(A 或 B) = P(A) + P(B)。条件概率（高阶）使用 P(A|B) = P(A 与 B) / P(B)。
When using tree diagrams, multiply along branches for ‘and’, add the probabilities of different branches for ‘or’. Always check that probabilities at each branch point sum to 1.
使用树状图时，沿分支相乘表示“与”，将不同分支的概率相加表示“或”。务必检查每个分支点的概率之和为 1。
11. Graphs of Functions | 函数图像
Straight‑line graphs are defined by y = mx + c, where m is the gradient and c is the y‑ allel lines have equal gradients while perpendicular lines have gradients that multiply to –1.
直线图像的方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。你应当能通过两点求直线方程，并理解平行线斜率相等，而垂直线的斜率乘积为 –1。
arabolas. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0). Key features are the turning point and the roots (x‑intercepts), which can be found by factorising or using the quadratic formula.
二次函数图像（y = ax² + bx + c）为抛物线。a 的正负决定了抛物线开口向上（a > 0）还是向下（a < 0）。关键特性是转折点和与 x 轴的交点（根），可通过因式分解或求根公式求得。
Other important graphs include cubic, reciprocal (y = k/x), and exponential (y = aˣ o know trigonometric graphs (y = sin x, cos x, tan x) and recognise their periodic nature.
其他重要图像包括三次函数、反比例函数（y = k/x）和指数函数（y = aˣ）。高阶学生还需了解三角函数图像（y = sin x，cos x，tan x）并认识其周期性。
phs: y = f(x) + a is a vertical translation; y = f(x + a) is a horizontal translation; y = –f(x) is a reflection in the x‑axis; y = f(–x) is a reflection in the y‑axis.
图像变换：y = f(x) + a 为沿 y 轴平移；y = f(x + a) 为沿 x 轴平移；y = –f(x) 为关于 x 轴的反射；y = f(–x) 为关于 y 轴的反射。
12. Key Formulae and Revision Tips | 关键公式与复习建议
Below is a compact summary of essential formulas you must know for the WJEC examination. Some will be given on the formula sheet, but familiarity speeds up your work and reduces errors.
以下是 WJEC 考试必须掌握的关键公式简明汇总。部分公式会印在公式表上，但熟练掌握能加快解题速度并减少错误。
| Topic | Formula | 中文说明 |
|---|---|---|
| Area of a circle | A = πr² | 圆面积 |
| Circumference | C = 2πr | 圆周长 |
| Pythagoras | a² + b² = c² | 勾股定理 |
| Sine rule | a/sin A = b/sin B = c/sin C | 正弦定理 |
| Cosine rule | a² = b² + c² – 2bc cos A | 余弦定理 |
| Area (triangle) | ½ ab sin C | 三角形面积（已知两边夹角） |
| Volume of prism | Area of cross‑section × length | 棱柱体积 |
| Compound interest | Amount = P(1 + r/100)ⁿ | 复利计算 |
Effective revision involves more than just reading notes. Practise past papers under timed conditions, identify weak areas using a topic checklist, and review mistakes carefully. For problems involving multiple steps, write clear workings — many marks are awarded for method even if the final answer is incorrect.
有效复习不仅仅是阅读笔记。要在计时条件下练习历年真题，借助主题清单找出薄弱环节，并认真分析错误。对于多步问题，写出清晰的过程——即使最终答案有误，正确的解题方法也能获得大量分数。
During the exam, manage your time: spend roughly one minute per mark. If stuck on a question, move on and return later. Always check your answers, especially for negative signs and unit conversions, and round final answers only at the last step. Keep your working neat so that you can follow your own reasoning when back‑checking.
考试时要管理好时间：大致按每分一分钟分配时间。如果卡在一道题上，先跳过，稍后再回来做。始终检查答案，尤其注意负号和单位换算，只在最后一步才取近似值。保持过程书写整洁，以便回顾时能清楚看懂自己的推理。
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