📚 WJEC Year 7 Further Mathematics: Case Study Practical Exercises | WJEC 七年级进阶数学：案例分析实战演练
In the real world, mathematics is rarely presented as a list of isolated problems. Instead, it appears embedded in situations that require thoughtful analysis, selection of appropriate methods, and clear communication of results. This article explores how Year 7 pupils following the WJEC Further Mathematics curriculum can develop these vital skills through case study practical exercises. By working through realistic scenarios such as organizing a school fair, analysing zoo attendance data, and planning a garden, you will see how number, algebra, geometry, and statistics come together to solve meaningful problems.
在现实世界中，数学很少以孤立的题目列表形式出现。它往往嵌入在需要深思熟虑的分析、适当方法的选择以及清晰结果沟通的情境中。本文探讨了学习 WJEC 进阶数学课程的七年级学生如何通过案例分析的实战演练来培养这些重要技能。通过解决组织学校博览会、分析动物园游客数据、规划花园等真实场景，你将看到数字、代数、几何和统计如何协同作用解决有意义的问题。
1. Understanding Case Studies in Further Mathematics | 理解进阶数学中的案例分析
A case study in mathematics is more than a word problem. It presents a real-life situation that requires you to identify the mathematical questions hidden within it. In WJEC Year 7 Further Mathematics, case studies encourage you to read carefully, extract numerical information, decide what calculations are needed, and then interpret your answers in context. This approach builds the problem-solving and reasoning skills essential for success at GCSE and beyond.
数学中的案例分析不仅仅是一道应用题。它呈现了一个现实生活情境，要求你识别其中隐藏的数学问题。在 WJEC 七年级进阶数学中，案例分析鼓励你仔细阅读、提取数字信息、决定需要进行哪些计算，然后在上下文中解释你的答案。这种方法能够培养对 GCSE 及以后学习至关重要的解决问题和推理能力。
2. Case Study 1: Planning the School Fair | 案例一：策划学校博览会
Imagine your school is holding a summer fair to raise funds for a new library. You have been asked to help plan the budget, layout, and activities. The initial information states that the school will provide £300 as a starting fund. You plan to spend part of this on materials, games, and refreshments, and you need to ensure that total costs do not exceed the fund and that the fair can make a surplus. This scenario involves proportional reasoning, area calculation, and basic financial mathematics.
假设你的学校正在举办一次夏季博览会，为新建图书馆筹集资金。你被邀请帮忙规划预算、布局和活动。初步信息表明，学校将提供 300 英镑作为启动资金。你计划将其中一部分用于材料、游戏和茶点，并且需要确保总成本不超过资金，同时博览会能产生盈余。这个情境涉及比例推理、面积计算和基本的财务数学。
3. Budget Allocation and Proportional Reasoning | 预算分配与比例推理
The school wants to divide the £300 so that 40% goes on materials and decorations, 35% on food and drinks, and the remaining 25% on prizes. To find the exact amounts, we calculate: 40% of £300 = 0.40 × 300 = £120; 35% = 0.35 × 300 = £105; 25% = 0.25 × 300 = £75. As a check, £120 + £105 + £75 = £ and decimals is fundamental to managing budgets. When you need to adjust the budget later—for example, if decorations are cheaper by ½—you can apply proportional reasoning quickly.
学校希望将 300 英镑按以下比例分配：40% 用于材料和装饰，35% 用于食品和饮料，剩余的 25% 用于奖品。要找到具体金额，我们计算：300 英镑的 40% = 0.40 × 300 = 120 英镑；35% = 0.35 × 300 = 105 英镑；25% = 0.25 × 300 = 75 英镑。作为验算，120 + 105 + 75 = 300 英镑。百分比、分数和小数的这种运用是管理预算的基础。当后续需要调整预算时——例如，如果装饰品便宜了一半——你可以迅速应用比例推理。
4. Designing the Fairground Layout | 设计博览会场地布局
quires a 2 m by 1.5 m rectangular area. If you plan to have 8 stalls, the total area needed is 8 × (2 × 1.5) = 8 × 3 = 24 m². The stage is a square with side length 5 m, giving an area of 5 × 5 = 25 m². The remaining area will be for standing and seating. The total playground area is 30 × 20 = 600 m². After stalls and stage, 600 – (24 + 25) = 551 m² remains, which is about 92% of the total area— tanding perimeter and area in such a context shows how geometry is used in event planning.
博览会将在长 30 米、宽 20 米的矩形操场上举行。你需要为摊位、舞台和座位区分配空间。每个游戏摊位需要一个 2 米乘 1.5 米的矩形区域。如果你计划设置 8 个摊位，需要的总面积为 8 × (2 × 1.5) = 8 × 3 = 24 平方米。舞台是边长为 5 米的正方形，面积为 5 × 5 = 25 平方米。剩余区域将用于站立和座位。操场总面积为 30 × 20 = 600 平方米。除去摊位和舞台，剩余 600 – (24 + 25) = 551 平方米，约占整个面积的 92%——有足够空间供访客使用。在这样的背景下理解周长和面积，说明了几何知识如何在活动策划中应用。
5. Case Study 2: The Local Zoo Attendance | 案例二：当地动物园游客数据
A local zoo has provided monthly attendance data for one year. The data, in thousands of visitors, is: Jan 12, Feb 10, Mar 15, Apr 18, May 22, Jun 28, Jul 35, Aug 38, Sep 25, Oct 20, Nov 14, Dec 13. Your task is to analyse this data, find averages, and advise the zoo on when to run special events. This case study draws on statistical skills from the WJEC Further Mathematics syllabus, including mean, median, mode, and range, as well as bar charts.
一个当地动物园提供了一年的月度游客数据。数据（以千人为单位）如下：一月 12，二月 10，三月 15，四月 18，五月 22，六月 28，七月 35，八月 38，九月 25，十月 20，十一月 14，十二月 13。你的任务是分析这些数据，找出平均值，并向动物园建议在何时举办特别活动。本案例运用了 WJEC 进阶数学大纲中的统计技能，包括平均数、中位数、众数和极差，以及条形图。
6. Interpreting Bar Charts and Averages | 解读条形图与平均值
(35) and August (38), showing a clear summer peak. To find the mean, we add all 12 values: 12+10+15+18+22+28+35+38+25+20+14+13 = 250. Then divide by 12: 250 ÷ 12 ≈ 20.8 thousand visitors per month. The median is the middle value when ordered; the data is already in months but we can sort: 10,12,13,14,15,18,20,22,25,28,35,38. With 12 values, the median is the average of the 6th and 7th: (18+20)/2 = 19 thousand. The mode is the most frequent, but no value repeats, so there is no mode. The range is 38 – 10 = 28 thousand, indicating a large variation between months.
首先，我们可以绘制一个条形图，横轴为月份，纵轴为游客数（千人）。最高的条形是七月（35）和八月（38），显示出明显的夏季高峰。为了求平均数，我们将 12 个值相加：12+10+15+18+22+28+35+38+25+20+14+13 = 250。然后除以 12：250 ÷ 12 ≈ 20.8 千人，即每月平均约 20,800 名游客。中位数是将数据排序后的中间值；数据已按月份排列，但我们可以排序为：10,12,13,14,15,18,20,22,25,28,35,38。有 12 个值，中位数是第 6 和第 7 个值的平均数：(18+20)/2 = 19 千人。众数是最常出现的值，但这里没有重复，因此没有众数。极差为 38 – 10 = 28 千人，表明各月之间差异很大。
7. Making Predictions with Trends | 利用趋势进行预测
Using the averages and the shape of the bar chart, we can advise the zoo. Because attendance is highest in July and August, the zoo should schedule major events like “Family Fun Days” or evening safaris during these months. The low attendance in winter suggests that the zoo might close some sections or offer discounted tickets to boost numbers. Further analysis could involve calculating the ratio of summer to winter visitors. For example, comparing the average of July and August (36.5) with January and February (11) gives a ratio of about 3.3 : 1. This simple comparison helps managers plan staffing. Remember: in WJEC Year 7 Further Mathematics, you are expected to describe trends in your own words and use mathematical reasoning to support recommendations.
利用平均值和条形图的形状，我们可以给动物园提供建议。由于七月和八月的游客量最高，动物园应该在这些月份安排“家庭欢乐日”或夜间游猎等主要活动。冬季的低游客量表明，动物园或许可以关闭部分区域或提供折扣票以提高人数。进一步分析可能涉及计算夏季与冬季游客的比率。例如，比较七月和八月的平均数（36.5）与一月和二月（11），得到比率约为 3.3∶1。这种简单的比较有助于管理人员规划人员安排。请记住：在 WJEC 七年级进阶数学中，要求你用自己语言描述趋势，并使用数学推理来支持建议。
8. Case Study 3: Designing a Rectangular Garden | 案例三：设计矩形花园
A gardening club has a rectangular plot of land with a length that is 3 metres more than its width. The width is unknown — let's call it w metres. The club plans to plant flowers along the perimeter and lay turf over the whole area. This case study involves forming algebraic expressions and using them to solve for dimensions when the perimeter or area is known, a key skill in the algebra section of the Further Mathematics curriculum.
一个园艺俱乐部有一块矩形土地，其长度比宽度多 3 米。宽度未知——我们设其为 w 米。俱乐部计划沿周界种花，并在整个区域内铺设草皮。这个案例涉及建立代数表达式，并在已知周长或面积的情况下用来求解尺寸，这是进阶数学课程代数部分的一项关键技能。
9. Algebraic Patterns in Garden Beds | 花园苗床中的代数模式
3w square metres. Suppose the club knows the perimeter is 30 m. We can write an equation: 4w + 6 = 30. Subtracting 6 from both sides gives 4w = 24, so w = 6 m. Then length = 9 m. The area is then 54 m². You could also work backwards: if the area is 54 m², solve w(w+3)=54, which leads to w²+3w– etters to represent unknowns and forming equations is a cornerstone of algebraic thinking.
如果宽度为 w，那么长度为 w + 3。周长 P = 2 × (宽 + 长) = 2(w + w + 3) = 2(2w + 3) = 4w + 6 米。面积 A = 宽 × 长 = w(w + 3) = w² + 3w 平方米。假设俱乐部知道周长为 30 米。我们可以列出方程：4w + 6 = 30。两边减去 6 得到 4w = 24，因此 w = 6 米。那么长度为 9 米。面积则为 54 平方米。你也可以反向推算：如果面积为 54 平方米，解 w(w+3)=54，会得到 w²+3w–54=0；但在七年级，我们可以尝试整数值，发现 w=6 适用。这个用字母表示未知数并建立方程的过程是代数思维的基石。
10. Reflection and Key Skills | 反思与关键技能
These case studies illustrate that mathematics is a set of connected tools rather than separate topics. In the school fair study, you used percentages, multiplication of decimals, area, and subtraction. In the zoo data analysis, you applied the four averages (mean, median, mode, range) and graphical representation. The garden design required algebraic notation, substitution, and solving simple linear equations. WJEC Year 7 Further Mathematics expects you to confidently move between these areas and to communicate your reasoning clearly. When you encounter a new problem, ask: 'What is the question really asking? Which mathematical concept applies? How can I check my answer?'
这些案例分析说明，数学是一套相互联系的工具，而非孤立的知识点。在学校博览会的研究中，你运用了百分比、小数乘法、面积和减法。在动物园数据分析中，你应用了四个平均值（平均数、中位数、众数和极差）以及图形表示。花园设计则需要代数符号、代入法和解简单的一次方程。WJEC 七年级进阶数学期望你自信地穿梭于这些领域之间，并清晰地表达你的推理过程。当你遇到新问题时，问问自己：“问题真正在问什么？哪个数学概念适用？我怎样才能检验答案？”
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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