GCSE WJEC Maths: Last-Minute Revision Notes | GCSE WJEC 数学：考前冲刺笔记

📚 GCSE WJEC Maths: Last-Minute Revision Notes | GCSE WJEC 数学：考前冲刺笔记

This article brings together the most important facts, formulas and problem-solving strategies you need to revise before your WJEC GCSE Mathematics exam. Whether you are sitting the Foundation or Higher tier, these notes will help you focus on what really matters, avoid common traps and approach the paper with confidence.

本文汇总了 WJEC GCSE 数学考前必须复习的核心知识点、公式和解题策略。无论参加基础级还是高级考试，这份冲刺笔记都能帮助你抓住重点、避开常见陷阱，自信地面对试卷。

1. Number Basics and BIDMAS | 数字基础与运算顺序

Always apply the correct order of operations: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right). Many marks are lost when students ignore BIDMAS in multi-step calculations.

务必使用正确的运算顺序：括号、指数、除法与乘法（从左到右）、加法与减法（从左到右）。许多学生在多步运算中忽略 BIDMAS 而丢分。

Key terms: integer, factor, multiple, prime number, highest common factor (HCF), lowest common multiple (LCM). Remember that 1 is not a prime number.

关键术语：整数、因数、倍数、质数、最大公因数（HCF）、最小公倍数（LCM）。记住 1 不是质数。

HCF of 24 and 36 is 12.
LCM of 8 and 12 is 24.

24 和 36 的 HCF 是 12。
8 和 12 的 LCM 是 24。

For negative numbers: subtracting a negative is the same as adding a positive.

负数运算：减去一个负数等于加上它的相反数。

2. Fractions, Decimals and Percentages | 分数、小数与百分比

To add or subtract fractions, find a common denominator first. Multiply fractions by multiplying numerators and denominators separately. To divide by a fraction, multiply by its reciprocal.

分数加减先通分。分数乘法：分子乘分子，分母乘分母。除以分数等于乘它的倒数。

Convert between forms: ½ = 0.5 = 50%, ⅓ ≈ 0.333… = 33⅓%. Use the fact that a percentage is ‘out of 100’.

形式互化：½ = 0.5 = 50%，⅓ ≈ 0.333… = 33⅓%。记住百分数就是 “百分之几”。

Percentage increase/decrease: new value = original × (1 ± percentage as a decimal). Reverse percentages require division by the multiplier.

百分比增减：新值 = 原值 × (1 ± 百分数对应的小数)。逆向求原值要除以乘数。

Fraction	Decimal	Percentage
1/4	0.25	25%
3/5	0.6	60%

分数 | 小数 | 百分比

3. Powers, Roots and Standard Form | 幂、根与标准形式

Indices rules are essential: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Note that a⁰ = 1 and a⁻ⁿ = 1/aⁿ.

指数法则是核心：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。注意 a⁰ = 1，a⁻ⁿ = 1/aⁿ。

Fractional indices: the denominator gives the root, e.g. a½ = √a, a⅓ = ∛a. Negative indices mean reciprocal.

分数指数：分母表示根号，如 a½ = √a，a⅓ = ∛a。负指数表示倒数。

Standard form: A × 10ⁿ where 1 ≤ A < 10. Be careful with the direction of the decimal point move. On WJEC papers, writing a number as an ordinary number from standard form is a common question.

标准形式：A × 10ⁿ，其中 1 ≤ A < 10。注意小数点移动方向。WJEC 试卷常考将标准形式写回普通数。

2.4 × 10³ = 2400, 5.67 × 10⁻² = 0.0567

4. Algebraic Manipulation | 代数运算

Simplify expressions by collecting like terms: 3x + 2y – x + 5y = 2x + 7y. Expand brackets using the distributive law: a(b + c) = ab + ac.

合并同类项化简：3x + 2y – x + 5y = 2x + 7y。用分配律展开括号：a(b + c) = ab + ac。

Expand two brackets: (x + a)(x + b) = x² + (a+b)x + ab. For the difference of two squares: (a + b)(a – b) = a² – b².

展开两个括号：(x + a)(x + b) = x² + (a+b)x + ab。平方差公式：(a + b)(a – b) = a² – b²。

Factorisation is the opposite of expanding. Always look for a common factor first. For quadratics, find two numbers that multiply to give ac and add to give b.

因式分解是展开的逆运算。第一步始终寻找公因子。对于二次三项式，找两个数使其乘积为 ac，和为 b。

x² + 5x + 6 = (x + 2)(x + 3)

5. Solving Equations and Inequalities | 方程与不等式求解

Solve linear equations by performing the same operation on both sides to isolate the unknown. Always check your solution by substitution.

解线性方程时，两边同时进行相同运算以分离未知数。务必代入检验。

For quadratic equations, rearrange into the form ax² + bx + c = 0 and then factorise, or use the quadratic formula. WJEC Higher tier expects fluent use of the formula.

二次方程先整理成 ax² + bx + c = 0，然后因式分解或使用求根公式。WJEC 高级卷要求熟练应用公式。

x = [ -b ± √(b² – 4ac) ] / (2a)

Inequalities follow similar rules, but remember: multiplying or dividing by a negative number flips the inequality sign.

不等式规则类似，但记住：乘或除以负数时要改变不等号方向。

Represent solutions on a number line: open circle for < or >, closed circle for ≤ or ≥. Double inequalities: 3 < x ≤ 7.

在数轴上表示解：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。双重不等式如 3 < x ≤ 7。

6. Graphs and Functions | 图形与函数

Straight line: y = mx + c, where m is the gradient and c is the y-intercept. Gradient = change in y / change in x. Parallel lines have equal gradients.

直线：y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率 = y 的变化量 / x 的变化量。平行线斜率相等。

Quadratic graphs are parabolas. Roots are the x-intercepts, where the graph crosses the x-axis (y = 0). The turning point can be found by completing the square or using symmetry.

二次函数图像是抛物线。根是与 x 轴的交点 (y = 0)。顶点可通过配方法或利用对称性求得。

Understand function notation: f(x) = x² + 3. f(2) means substitute x = 2. Inverse functions and composite functions appear on Higher tier papers.

理解函数符号：f(x) = x² + 3。f(2) 表示代入 x = 2。高级卷会涉及反函数与复合函数。

Real-life graphs: distance-time graphs (gradient = speed) and velocity-time graphs (gradient = acceleration, area under graph = distance).

实际情境图：距离-时间图（斜率 = 速度），速度-时间图（斜率 = 加速度，图下方面积 = 距离）。

7. Ratio, Proportion and Rates of Change | 比、比例与变化率

Simplify ratio by dividing by common factors. Share a quantity in a ratio: add the parts to find the total number of shares, then divide.

化简比：除以公因数。按比例分配量：先加总份数得出总份额，再分配。

Direct proportion: y ∝ x means y = kx for a constant k. Inverse proportion: y ∝ 1/x means y = k/x. These are tested with graphs and equations.

正比例：y ∝ x 即 y = kx。反比例：y ∝ 1/x 即 y = k/x。常通过图像和方程考查。

Compound measures: speed = distance / time, density = mass / volume, pressure = force / area. Rearrange these confidently.

复合单位：速度 = 距离 / 时间，密度 = 质量 / 体积，压强 = 力 / 面积。要熟练变形这些公式。

Percentage change applied repeatedly (e.g. compound interest) uses multipliers: total = P × (1 + r/100)ⁿ. For depreciation, use (1 – r/100)ⁿ.

重复百分比变化（如复利）使用乘数：总额 = P × (1 + r/100)ⁿ。折旧用 (1 – r/100)ⁿ。

8. Geometry: Angles, Area and Volume | 几何：角度、面积与体积

Know angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal.

掌握角度知识：直线上的角之和为 180°，一点周围角之和为 360°，对顶角相等。

Parallel lines: corresponding angles are equal, alternate angles are equal, co-interior angles sum to 180°. Interior and exterior angles of polygons: sum of interior = (n-2)×180°, sum of exterior = 360°.

平行线：同位角相等，内错角相等，同旁内角互补（和为 180°）。多边形内角和 = (n-2)×180°，外角和始终为 360°。

Areas: triangle = ½ × base × height, parallelogram = base × vertical height, trapezium = ½(a + b)h, circle = πr², circumference = 2πr. WJEC provides a formula sheet but you must know when to apply each.

面积公式：三角形面积 = ½ × 底 × 高，平行四边形面积 = 底 × 垂直高，梯形面积 = ½(a+b)h，圆面积 = πr²，周长 = 2πr。WJEC 提供公式表，但你需要知道何时使用。

Volume: prism = area of cross-section × length, cylinder = πr²h, pyramid = ⅓ × base area × height, sphere = 4/3 πr³, cone = ⅓ πr²h.

体积：棱柱 = 横截面积 × 长，圆柱 = πr²h，棱锥 = ⅓ × 底面积 × 高，球体 = 4/3 πr³，圆锥 = ⅓ πr²h。

Volume of a cone = ⅓ πr²h

9. Pythagoras and Trigonometry | 毕达哥拉斯定理与三角函数

Pythagoras’ theorem: in a right-angled triangle, a² + b² = c² where c is the hypotenuse. Use it to find a missing side or to check if a triangle is right-angled.

毕达哥拉斯定理：直角三角形中，a² + b² = c² (c 为斜边)。用于求未知边或验证三角形是否为直角。

Trigonometry (SOH CAH TOA): sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Use these for right-angled triangles. Always label sides relative to the given angle.

三角函数 (SOH CAH TOA)：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。用于直角三角形。务必根据给定角标记各边。

In higher tier, know the sine rule: a/sin A = b/sin B = c/sin C, and cosine rule: a² = b² + c² – 2bc cos A. Use these for non-right-angled triangles. The area formula ½ ab sin C is also common.

高级卷需掌握正弦定理：a/sin A = b/sin B = c/sin C，和余弦定理：a² = b² + c² – 2bc cos A。用于非直角三角形。面积公式 ½ ab sin C 也常考。

Exact trigonometric values for 0°, 30°, 45°, 60°, 90° are expected in higher tier. For example, sin 30° = ½, cos 45° = √2/2, tan 60° = √3.

高级卷要求掌握 0°, 30°, 45°, 60°, 90° 的准确三角函数值，如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。

10. Statistics and Probability | 统计与概率

Calculate mean, median, mode and range. For grouped data, estimate the mean using midpoints. The interquartile range (IQR) measures spread and is preferred over range when comparing data sets.

计算平均数、中位数、众数和极差。对于分组数据，用组中点估算平均数。四分位距 (IQR) 用于衡量离散程度，比较数据时优于极差。

Probability: P(event) = number of favourable outcomes / total number of outcomes. Probabilities always between 0 and 1 inclusive. The sum of probabilities of all outcomes is 1.

概率：P(事件) = 有利结果数 / 总结果数。概率值始终在 0 到 1 之间。所有可能结果的概率和为 1。

Tree diagrams are vital for combined events. Multiply along branches and add probabilities for ‘or’ scenarios. Remember to adjust probabilities when items are not replaced.

树状图是处理复合事件的关键工具。沿着分支相乘，不同分支的概率相加（“或”的情况）。不放回抽取时要调整概率。

Venn diagrams and two-way tables help organise information for probability calculations. Use them to show intersections and unions: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

维恩图和双向表有助于整理概率信息。用于表示交集和并集：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。

11. Exam Techniques and Common Mistakes | 考试技巧与常见错误

Read every question twice. Underline key numbers and command words. Show all working — even if your final answer is wrong, you can earn method marks.

每道题读两遍。划出关键数字和指令词。展示所有解题步骤——即使最后答案有误，你仍可能拿到方法分。

Check units: convert cm to m, minutes to hours where necessary. Present monetary answers to two decimal places. Round only at the very end of a calculation.

检查单位：必要时将 cm 转为 m，分钟转为小时。货币答案保留两位小数。只在计算的最后一步四舍五入。

Use the formula sheet wisely; it is provided but understanding how to substitute values correctly is entirely up to you. Know which volume formula corresponds to a given solid.

明智使用公式表；虽然提供，但正确代入数值完全靠你自己。要清楚每个几何体对应哪个体积公式。

Common mistakes: confusing area and perimeter, misreading inequalities, forgetting to square the radius in circle formulas, incorrect angle labelling in trigonometry, and mixing up the sine and cosine rules. Finally, manage your time — aim for one mark per minute, and leave harder questions for a second pass.

常见错误：混淆面积与周长，看错不等号，圆公式中忘记半径平方，三角函数边角对应错误，混淆正弦与余弦定理。最后，合理分配时间——按每分一分钟的速度，较难题目留到第二轮再做。

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GCSE WJEC Maths: Last-Minute Revision Notes | GCSE WJEC 数学：考前冲刺笔记