📚 Year 7 WJEC Further Maths: Winter Intensive Revision Plan | 7年级WJEC进阶数学：寒假强化复习计划
The winter break is the perfect opportunity for Year 7 students studying WJEC Further Maths to consolidate core skills, address any gaps and build confidence ahead of the spring term. A structured, daily revision plan that revisits number work, algebra, geometry and data handling will help you return to school feeling prepared and motivated. This guide breaks down an effective 10-session intensive revision programme, with clear explanations and practice tips for each topic.
寒假是学习 WJEC 进阶数学的 7 年级学生巩固核心技能、弥补知识漏洞并为春季学期建立信心的理想时机。一份结构化的每日复习计划，重新回顾数论、代数、几何和数据处理等内容，能让你在返校时感到从容和积极。本指南将一个高效的 10 模块强化复习方案进行拆解，为每个主题提供清晰的解释和练习建议。
1. Number Sense and Operations | 数感与运算
Begin your revision by strengthening mental arithmetic with integers. Practice addition, subtraction, multiplication and division including negative numbers. Remember that subtracting a negative is the same as adding the positive value.
从强化整数的口算开始。练习加、减、乘、除运算，包括负数的运算。记住，减去一个负数等于加上对应的正数。
6 - 2)². Always handle brackets first, then indices, division/multiplication, and finally addition/subtraction.
练习包含运算顺序（BIDMAS/BODMAS）的题目，例如 3 + 4 × (6 - 2)²。始终先处理括号，然后是指数，接着除法和乘法，最后做加法和减法。
Revise factors, multiples and prime numbers. List all factor pairs of 48 and identify its prime factorisation using a factor tree.
复习因数、倍数和质数。列出 48 的所有因数对，并用因子树写出它的质因数分解形式。
- Daily drill: spend 15 minutes on quick mental calculations with negative numbers and powers.
- 每日训练：花15分钟快速进行含负数和幂的口算练习。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
subtract fractions by finding a common denominator, e.g. 1/3 + 2/5 = 5/15 + 6/15 = 11/15.
复习等值分数、约分，以及带分数与假分数之间的转换。确保能通过通分进行分数的加减运算，例如 1/3 + 2/5 = 5/15 + 6/15 = 11/15。
Multiply fractions by multiplying numerators and denominators; divide by multiplying by the reciprocal. For decimals, practise the four operations and rounding to decimal places and significant figures.
分数相乘时，分子相乘、分母相乘；分数相除时，乘以倒数。对于小数，练习四则运算，以及将结果保留到指定小数位和有效数字。
een fractions, decimals and percentages. Memorise key equivalences such as 1/4 = 0.25 = 25% and apply them to comparison problems.
在分数、小数和百分数之间熟练转换。熟记如 1/4 = 0.25 = 25% 等关键等价关系，并将其运用到比较题中。
Fraction to decimal: a/b = a ÷ b. Percent means 'out of 100'.
分数化小数：a/b = a ÷ b。百分数意为“每一百份中的”。
3. Algebraic Expressions and Simplification | 代数表达式与化简
Algebra is a cornerstone of Further Maths. Start by collecting like terms: 3a + 5b - a + 2b simplifies to 2a + 7b. Treat letters as objects and combine only identical variable terms.
代数是进阶数学的基石。从合并同类项开始：3a + 5b - a + 2b 化简为 2a + 7b。把字母看作对象，只合并完全相同变量的项。
negative sign sits outside the bracket, remember to change the sign of every term inside.
使用分配律展开括号，例如 4(2x - 3) = 8x - 12。当括号外有负号时，切记括号内每一项的符号都要改变。
HCF is 3x, giving 3x(2x + 3). Practise writing expressions from word phrases too, such as 'three more than twice a number' → 2n + 3.
通过提取最大公因数进行因式分解。对于 6x² + 9x，最大公因数为 3x，因此分解为 3x(2x + 3)。同时练习根据文字描述列出表达式，例如“一个数的两倍再加三” → 2n + 3。
4. Solving Linear Equations | 解一元一次方程
isolate the unknown. Solve 2x + 5 = 17 by subtracting 5, then dividing by 2, giving x = 6.
扎实的解方程能力至关重要。反复操练天平法：两边进行相同运算以分离未知数。解 2x + 5 = 17，先减去5，再除以2，得到 x = 6。
bles on both sides. For 3(x - 4) = 2x + 1, expand first, then collect x terms on one side.
拓展到含有括号或变量在等式两侧的方程。对于 3(x - 4) = 2x + 1，先展开，再将含 x 的项集中到一边。
Always check your solution by substituting back into the original equation. In Further Maths you may also meet simple fractional equations; multiply through by the denominator to clear fractions.
务必将解代回原方程进行检验。在进阶数学中，你还会遇见简单的分式方程；可以在等式两边同时乘以分母来消去分母。
If ax + b = c, then x = (c - b)/a.
如果 ax + b = c，则 x = (c - b)/a。
5. Geometry: Angles and Lines | 几何：角与直线
Angles on a straight line add up to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. Use these facts to find missing angles in diagrams.
直线上的角之和为 180°，绕一点一周的角之和为 360°，对顶角相等。运用这些事实求出图形中的未知角。
Revise angle rules for parallel lines traversed by a transversal: corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°. Practice identifying these pairs quickly.
复习平行线与截线的角度法则：同位角相等，内错角相等，同旁内角之和为 180°。练习快速识别这些角对。
Learn to write short geometric proofs using these angle relationships. For instance, show that two given lines are parallel by demonstrating that alternate angles are equal.
学会使用这些角度关系书写简短的几何证明。例如，通过证明内错角相等来表明给定的两条直线平行。
| Angle type | Relationship |
| Corresponding | Equal |
| Alternate | Equal |
| Co-interior | Sum to 180° |
角度类型对应关系
(表格形式：同位角相等，内错角相等，同旁内角互补)
6. Properties of 2D Shapes | 二维图形的性质
Review classification of triangles by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). Recall that the sum of interior angles in any triangle is 180°.
复习三角形按边分类（等边、等腰、不等边）和按角分类（锐角、直角、钝角）。记住任何三角形的内角和都是 180°。
For quadrilaterals, memorise the properties of squares, rectangles, parallelograms, rhombuses, trapeziums and kites. Highlight symmetry, equal sides, parallel sides and diagonal properties.
对于四边形，熟记正方形、矩形、平行四边形、菱形、梯形和鸢形的性质。突出对称性、等边、平行边以及对角线的特性。
Draw and label diagrams accurately. Practice constructing triangles with compasses and ruler to strengthen understanding of geometric constraints.
精确绘制并标注图形。练习用圆规和直尺作三角形，以加深对几何约束条件的理解。
- Key fact: the sum of interior angles of an n-sided polygon = (n - 2) × 180°.
- 关键事实：n 边形的内角和 = (n - 2) × 180°。
7. Perimeter, Area and Volume | 周长、面积与体积
= ½ × b × h for triangles, A = b × h for parallelograms, and A = ½(a + b)h for trapeziums.
计算直线图形的周长，并使用 C = πd 或 C = 2πr 求圆的周长。对于面积，掌握矩形的 A = 长 × 宽，三角形的 A = ½ × 底 × 高，平行四边形的 A = 底 × 高，以及梯形的 A = ½(a + b)h。
he diameter, halve it first. Pay attention to units; convert all lengths to the same unit before calculating.
圆的面积公式为 A = πr²。给出直径时要先取半。注意单位；计算前将所有长度转换为相同单位。
d that volume is measured in cubic units and be able to find the volume by counting cubes.
对于体积，从长方体开始：V = 长 × 宽 × 高。理解体积以立方为单位，并能够通过数立方块的方法求出体积。
trapezium = (a + b)h / 2, where a and b are the parallel sides, h is perpendicular height.
梯形面积 = (a + b)h / 2，其中 a 和 b 是平行边，h 是垂直高度。
8. Data Handling and Averages | 数据处理与平均值
Revisit building and interpreting frequency tables, bar charts, pictograms and line graphs. Be able to read data with precision, including grouped frequency tables.
重新练习构建和解读频数表、柱状图、象形图和折线图。要求能精确读取数据，包括分组频数表。
Calculate mean, median, mode and range. The mean is the sum of values divided by the number of data points; the median is the middle value when ordered; the mode is the most frequent value.
计算平均数、中位数、众数和极差。平均数是用数据总和除以数据个数；中位数是将数据排序后居于中间的值；众数是出现次数最多的值。
Discuss which average best represents a data set. For example, the median is more robust when outliers are present. Compare two distributions using range and mean.
讨论哪种平均数最能代表一组数据。例如，当存在异常值时中位数更具鲁棒性。使用极差和平均数对两组分布进行比较。
ing angles: category frequency ÷ total frequency × 360°. Check your angle sum totals 360°.
练习通过计算角度来绘制饼图：类别频数 ÷ 总频数 × 360°。检验所有度数之和为 360°。
9. Problem-Solving and Word Problems | 问题解决与应用题
Applied questions bridge separate topics and develop reasoning. Break down word problems into steps: identify what you are asked to find, list known information, and translate into mathematical statements.
应用题将不同主题联系在一起，并锻炼推理能力。将文字题分解为步骤：明确求解目标，列出已知信息，并转化为数学语言。
Common contexts include money, measurement, ratios and age puzzles. For ratio problems, use bar models or set up equations. When a problem says 'twice as many', assign variables carefully.
常见情境包括货币、测量、比例和年龄谜题。对比例问题可以使用条形模型或建立方程。当题目出现“两倍多”的说法时，要仔细设定变量。
Practise past paper style multi-step questions. Show all working clearly; even if a final answer is wrong, method marks can be gained in the real exam.
练习真题风格的多步骤题目。清晰展示所有步骤；即使最终答案错误，在真实考试中也能获得过程分。
- Tip: highlight key numbers and words in the question before solving.
- 窍门：解题前先将题目中的关键数字和词语高亮标注。
10. Mental Maths and Exam Techniques | 心算与应试技巧
on times tables up to 12 × 12, squares up to 15², and simple fraction-decimal conversions.
提高心算能力可以节省时间并减少粗心错误。每天花短暂时间反复练习 12 × 12 以内的乘法表、15² 以内的平方数以及简单的分数小数转换。
Develop effective exam habits: read the entire question, underline command words, and manage your time per mark. If a question is worth 3 marks, aim to spend about 3 minutes on it.
培养高效的应试习惯：阅读整个问题，划出指令词，并按题目的分值管理时间。如果一道题值3分，就计划花费大约3分钟去做。
Always leave time to check answers. Re-do calculations in a different order and ask 'Is my answer reasonable?' Estimation skills are crucial here.
务必留出时间检查答案。用不同的顺序重算一遍，并问自己“我的答案合理吗？” 估算能力在此处非常关键。
Use the holiday to create a revision resource, such as a fold-out summary sheet of key formulas for number, algebra and geometry. Writing things down in your own words strengthens memory.
利用假期制作复习资料，例如一张折叠式的摘要表，汇总数论、代数和几何的关键公式。用自己的语言书写可以增强记忆。
Success = consistent daily practice + understanding, not just memorising.
成功 = 坚持不懈的每日练习 + 理解，而不仅仅是记忆。
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