📚 Year 11 WJEC Mathematics: Core Topics Summary | Year 11 WJEC 数学：核心知识点梳理
This article provides a structured summary of the essential topics for Year 11 WJEC Mathematics. It covers number, algebra, graphs, geometry, trigonometry, probability, statistics, ratio, and vectors, highlighting key formulas, methods, and exam tips to support your revision.
本文系统梳理了 Year 11 WJEC 数学的核心主题，涵盖数、代数、图像、几何、三角学、概率、统计、比例和向量等内容，提炼关键公式、解题方法和应试要点，助力你的复习备考。
1. Number and Operations | 数与运算
You should be confident working with integers, fractions, decimals and percentages, and can convert fluently between them. Recurring decimals can be expressed as fractions using algebraic methods.
你应当熟练进行整数、分数、小数和百分数的运算，并能在不同形式间灵活转换。循环小数可以通过代数方法化为分数。
Prime factorisation, highest common factor (HCF) and lowest common multiple (LCM) are fundamental for simplifying fractions and solving problems. Write numbers as products of prime factors using index notation.
质因数分解、最大公因数 (HCF) 和最小公倍数 (LCM) 是化简分数和解决问题的基础工具。用指数形式将数字写成质因数的乘积。
Standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Use laws of indices when multiplying or dividing numbers in standard form, and check bounds for rounded values.
标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。在进行标准形式数的乘除运算时运用指数律，并对经过四舍五入的数值注意上界和下界。
Surds: √(ab) = √a × √b. Rationalise denominators, e.g., 1/√a = √a/a, and simplify expressions like √(a²b) = a√b. Upper and lower bounds: a measurement given to the nearest unit has a true value within ±0.5 units.
二次根式：√(ab) = √a × √b。进行分母有理化，如 1/√a = √a/a，并化简类似 √(a²b) = a√b 的式子。上界和下界：若测量值按最近单位给出，真实值落在 ±0.5 单位范围内。
2. Algebraic Manipulation | 代数运算
Expanding brackets uses the distributive property: a(b + c) = ab + ac. For two linear expressions, (x + a)(x + b) = x² + (a + b)x + ab. Squaring a binomial gives (a ± b)² = a² ± 2ab + b².
展开括号运用分配律：a(b + c) = ab + ac。对于两个一次式，(x + a)(x + b) = x² + (a + b)x + ab。二项式的平方为 (a ± b)² = a² ± 2ab + b²。
Factorising reverses this process: take out common factors, recognise the difference of two squares a² − b² = (a + b)(a − b), and factorise quadratic trinomials x² + bx + c by finding two numbers that multiply to c and add to b.
因式分解是展开的逆过程：提取公因式，识别平方差公式 a² − b² = (a + b)(a − b)，并通过找到两个乘积为 c 且和为 b 的数来分解二次三项式 x² + bx + c。
Laws of indices are essential: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a. Apply these when simplifying algebraic fractions or solving exponential equations.
指数律是根本：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a^(1/n) = ⁿ√a。在化简代数分式或解指数方程时运用这些法则。
Changing the subject of a formula involves inverse operations. When the target variable appears more than once, factorise after grouping the terms that contain it.
公式变形涉及逆运算。当目标变量出现不止一次时，先把包含它的项集中并进行因式分解。
3. Equations, Inequalities and Sequences | 方程、不等式与数列
Solve linear equations by performing the same operation on both sides to isolate x. For simultaneous linear equations, use elimination or substitution; with one linear and one quadratic, substitution usually works best.
解一次方程时，两边进行相同的运算以分离 x。解联立一次方程组，使用消元法或代入法；若一个为一次、一个为二次，代入法通常最有效。
Quadratic equations can be solved by factorising, completing the square, or using the quadratic formula. The formula is centred below.
二次方程可通过因式分解、配方法或求根公式求解。求根公式居中展示如下。
x = (−b ± √(b² − 4ac)) / (2a)
Discriminant b² − 4ac tells you the nature of the roots: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.
判别式 b² − 4ac 揭示根的性质：正值表示两个不同实根，零表示一个重根，负值表示无实数根。
Inequalities are solved like equations, but reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions on a number line with open or closed circles.
解不等式的方法与解方程类似，但当乘或除以负数时，要反转不等号方向。用空心或实心圆点将解集表示在数轴上。
Sequences: find the nth term of a linear sequence as an + b. For quadratic sequences, the second difference is constant; the nth term has the form an² + bn + c.
数列：求线性数列的第 n 项，其形式为 an + b。对于二次数列，二阶差分为常数；第 n 项的形式为 an² + bn + c。
4. Graphs and Transformations | 图像与变换
Straight-line graphs have equation y = mx + c, where m is the gradient and c is the y-intercept. To find m, use m = Δy/Δx between two points. Parallel lines share the same gradient; perpendicular lines have gradients whose product is −1.
直线图像方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。求 m 时，用两点间的 Δy/Δx。平行直线斜率相等；互相垂直的直线，其斜率乘积为 −1。
Quadratic graphs y = ax² + bx + c are parabolas; the coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). The turning point can be found by completing the square. Cubic and reciprocal graphs have distinctive shapes that you should sketch confidently.
二次函数图像 y = ax² + bx + c 是抛物线；系数 a 决定开口向上 (a > 0) 还是向下 (a < 0)。可通过配方法求出转折点。三次函数和反比例函数的图像具有特殊形状，你应能熟练画出草图。
Transformations of graphs: f(x) + a translates vertically, f(x + a) translates horizontally, −f(x) reflects in the x-axis, f(−x) reflects in the y-axis, and a f(x) stretches vertically by scale factor a.
图像变换：f(x) + a 表示竖直平移，f(x + a) 表示水平平移，−f(x) 关于 x 轴对称，f(−x) 关于 y 轴对称，a f(x) 表示沿竖直方向拉伸 a 倍。
Solving equations graphically: the points of intersection of y = f(x) and y = g(x) give the solutions to f(x) = g(x). Use this to solve quadratic equations, and to estimate solutions from drawn graphs.
用图像法解方程：y = f(x) 与 y = g(x) 图像的交点即为方程 f(x) = g(x) 的解。可利用此法解二次方程，并由所绘图像估算解。
5. Geometry and Measures | 几何与测量
Angle facts: angles on a straight line sum to 180°, around a point sum to 360°, vertically opposite angles are equal, and angles in a triangle sum to 180°. Know alternate, corresponding and allied angles for parallel lines.
角度知识：平角为 180°，周角为 360°，对顶角相等，三角形内角和为 180°。掌握平行线中的内错角、同位角和同旁内角性质。
For polygons, sum of interior angles = (n − 2) × 180°; each exterior angle of a regular polygon = 360°/n. Circle theorems: angle at centre is twice angle at circumference, angle in a semicircle is 90°, angles in the same segment are equal.
多边形：n 边形内角和 = (n − 2) × 180°；正多边形每个外角 = 360°/n。圆定理：圆心角是圆周角的两倍，直径所对圆周角为 90°，同弧上的圆周角相等。
Circumference and area of a circle: C = 2πr, A = πr². Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr². For cylinders, volume = πr²h; curved surface area = 2πrh. For cones, volume = ⅓πr²h; curved surface area = πrl (where l is slant height).
圆的周长与面积：C = 2πr，A = πr²。弧长 = (θ/360) × 2πr；扇形面积 = (θ/360) × πr²。圆柱体积 = πr²h；侧面积 = 2πrh。圆锥体积 = ⅓πr²h；侧面积 = πrl（l 为斜高）。
Similar shapes: corresponding lengths are in proportion; area scale factor is (length factor)²; volume scale factor is (length factor)³. Use this in scale diagrams and enlargement problems.
相似形：对应边长成比例；面积比等于边长比的平方；体积比等于边长比的立方。在比例图和放大问题中应用这一点。
6. Pythagoras, Trigonometry and 3D Applications | 毕达哥拉斯定理、三角学与三维应用
Pythagoras’ theorem: in a right‑angled triangle, a² + b² = c², where c is the hypotenuse. Use it to find missing sides and to determine whether a triangle is right‑angled by checking if the sides satisfy the equation.
毕达哥拉斯定理：直角三角形中，a² + b² = c²，c 为斜边。用它求未知边长，并通过验证三边是否满足等式判断三角形是否为直角三角形。
Basic trigonometry: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Choose the correct ratio based on the known and unknown sides, and use inverse trig functions to find angles.
基本三角学：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。根据已知边和未知边选择恰当的比例，并用反三角函数求角度。
For non‑right‑angled triangles, the sine rule is a/sin A = b/sin B = c/sin C. The cosine rule is a² = b² + c² − 2bc cos A. Area of a triangle = ½ab sin C, utilising two sides and the included angle.
对于非直角三角形，正弦定理为 a/sin A = b/sin B = c/sin C。余弦定理为 a² = b² + c² − 2bc cos A。三角形面积 = ½ab sin C，利用两边及其夹角。
3D problems: identify right‑angled triangles within cuboids, pyramids or prisms. Often you will apply Pythagoras first to find a slant length, then use trigonometry to find an angle between a line and a plane.
三维问题：在长方体、棱锥或棱柱中识别直角三角形。通常先应用毕达哥拉斯定理求斜长，再用三角学求直线与平面的夹角。
7. Probability and Tree Diagrams | 概率与树状图
Probability is measured from 0 to 1. For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes. The sum of probabilities of all mutually exclusive events in a sample space is 1.
概率取值在 0 到 1 之间。对于等可能结果，P(事件) = 有利结果数 / 总结果数。样本空间中所有互斥事件的概率之和为 1。
Expected frequency = probability × number of trials. Tree diagrams help model successive independent events: multiply along branches for combined probabilities, and add the probabilities of different paths that lead to the same outcome.
期望频数 = 概率 × 试验次数。树状图可对相继发生的独立事件建模：沿分支相乘求组合概率，并将通向相同结果的不同路径概率相加。
For conditional probability, use P(A|B) = P(A ∩ B) / P(B). In a tree diagram, the probabilities on the second set of branches change according to the outcome of the first event.
对于条件概率，使用 P(A|B) = P(A ∩ B) / P(B)。在树状图中，第二层分支上的概率根据第一次事件的结果而改变。
When sampling without replacement, the total number of items decreases; adjust the denominators on the branches accordingly. This is a common exam scenario involving conditional probability.
当进行无放回抽样时，物品总数减少；相应调整分支上的分母。这在涉及条件概率的考试中经常出现。
8. Statistics and Data Handling | 统计与数据处理
Averages: mean, median, mode. From a frequency table, mean = Σ(fx)/Σf. The modal class is the group with the highest frequency; the median interval is found from cumulative frequency.
平均数：均值、中位数、众数。在频数表中，均值 = Σ(fx)/Σf。众数组是频数最高的组；中位数区间可从累积频数中找到。
Measures of spread: range, interquartile range (IQR = Q₃ − Q₁). Box plots display the minimum, Q₁, median, Q₃ and maximum, making comparisons between distributions straightforward.
离散度量：极差和四分位距 (IQR = Q₃ − Q₁)。箱形图展示最小值、第一四分位数、中位数、第三四分位数和最大值，使分布间的比较一目了然。
Cumulative frequency graphs can estimate medians, quartiles and percentiles. Plot the running total against the upper class boundary and draw a smooth curve.
累积频数图可估计中位数、四分位数和百分位数。将逐次累加总值对上组距上界描点，并绘制光滑曲线。
Histograms: area represents frequency, so frequency density = frequency / class width. Bar height is frequency density. Scatter graphs show correlation; a line of best fit can be drawn by eye to predict values.
直方图：面积代表频数，因此频数密度 = 频数 / 组距。条形高度为频数密度。散点图显示相关关系；可凭目测画出最佳拟合线以预测数值。
9. Ratio, Proportion and Rates of Change | 比、比例与变化率
Ratios compare quantities; simplify by dividing by common factors. To divide a quantity in a given ratio, find the total number of parts, calculate the value of one part, then multiply. Ratios can be linked to fractions and percentages.
比用于比较数量；除以公因数进行化简。将一数量按给定比例分配时，先求总份数，算出每份的值，再乘相应份数。比可与分数和百分数相关联。
Direct proportion: y = kx. Graph is a straight line through the origin. Inverse proportion: y = k/x. Graph is a rectangular hyperbola. From a table of values, check if the product xy or ratio y/x is constant.
正比例：y = kx，图像为过原点的直线。反比例：y = k/x，图像为双曲线。根据数值表，检查乘积 xy 或比值 y/x 是否为常数来确定比例关系。
Compound measures: speed = distance/time, density = mass/volume, pressure = force/area. Convert units carefully, e.g., km/h to m/s requires dividing by 3.6.
复合单位：速度 = 距离/时间，密度 = 质量/体积，压强 = 力/面积。注意单位换算，例如 km/h 换算为 m/s 需要除以 3.6。
Rates of change: in a distance‑time graph, gradient gives speed; in a velocity‑time graph, gradient gives acceleration, and area under the graph gives distance travelled.
变化率：距离‑时间图中，斜率表示速度；速度‑时间图中，斜率表示加速度，图下的面积表示行驶的距离。
10. Vectors and Vector Geometry | 向量与向量几何
Vectors represent both magnitude and direction. Write as column vectors [x over y] or as xi + yj. Multiply a vector by a scalar: k(xi + yj) = (kx)i + (ky)j.
向量表示大小和方向。可写作列向量 [x 在上 y 在下] 或 xi + yj 的形式。向量与标量相乘：k(xi + yj) = (kx)i + (ky)j。
Vector addition: (a i + b j) + (c i + d j) = (a + c)i + (b + d)j. Subtraction is similar. The magnitude of a vector v = xi + yj is |v| = √(x² + y²).
向量加法：(a i + b j) + (c i + d j) = (a + c)i + (b + d)j。减法类似。向量 v = xi + yj 的大小为 |v| = √(x² + y²)。
In geometry, vectors describe paths. For points A and B, AB = b − a (position vectors). To prove points A, B, C are collinear, show AB = kBC for some scalar k. To prove lines are parallel, show direction vectors are scalar multiples.
在几何中，向量描述路径。对于点 A 和 B，AB = b − a（位置向量）。要证明 A、B、C 三点共线，需证明存在某标量 k 使得 AB = kBC。要证两线平行，需证明方向向量成标量倍数。
Using vectors to solve ratio problems: express one vector as a combination of others. In a triangle, if M is the midpoint of AB, then OM = (OA + OB)/2. This extends to more complex vector proofs.
用向量解决比例问题：将一个向量表示为其他向量的组合。在三角形中，若 M 是 AB 的中点，则 OM = (OA + OB)/2。这一思路可推广到更复杂的向量证明中。
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