📚 Year 11 CCEA Maths: Case Study Practical Exercises | CCEA 数学案例分析实战演练
Case study questions in CCEA GCSE Mathematics integrate real-world contexts with multiple mathematical skills. They require careful reading, strategic planning, and accurate calculations. This article provides a hands-on approach to mastering such exam questions, with worked examples and practical tips.
在 CCEA GCSE 数学考试中，案例分析题将真实情境与多种数学技能相结合。这类题目需要仔细阅读、策略性规划和准确计算。本文将通过实例和实用技巧，帮助考生掌握案例分析题的解答方法。
1. Understanding the Case Study Approach | 理解案例分析方法
A case study question presents a scenario—planning an event, managing finances, or designing a structure—followed by several sub-questions. These often connect number, algebra, geometry, and statistics.
案例分析题会给出一个场景——如活动策划、财务管理或结构设计——然后提出若干子问题。这些问题通常将数字、代数、几何和统计知识结合起来。
In CCEA exams, marks are awarded not only for the final answer but also for clear working, correct units, and sensible interpretations. You must demonstrate problem-solving skills across different mathematical strands.
在 CCEA 考试中，得分不仅取决于最终答案，还取决于清晰的解题步骤、正确的单位以及合理的解释。你必须展示跨不同数学范畴的问题解决能力。
2. Reading and Extracting Key Information | 阅读并提取关键信息
Begin by reading the entire scenario once, then highlight or underline numbers, units, conditions (e.g. 'at most', 'per person', 'before 10% discount'), and what the question asks you to find.
首先通读整个场景，然后高亮或下划线标出数字、单位、条件（如“最多”、“每人”、“10%折扣前”）以及题目要求你求解的内容。
Create a small table or list of known values. For example, if a trip involves 120 students, 8 teachers, and a minibus that seats 16, write these figures clearly. This prevents confusion later.
把已知数据制成一个小表格或清单。例如，一次旅行有 120 名学生、8 名教师、每辆小巴可坐 16 人，把这些数字清晰记录下来，可以避免后续混淆。
3. Breaking Down the Problem into Steps | 将问题分解为步骤
A case study often contains a series of connected parts. Tackle them in order, as later parts usually depend on earlier ones. For each step, decide which mathematical topic applies—proportion, area, averages, etc.
案例分析通常包含一连串相互关联的子问题。按顺序解答，因为后面的部分往往依赖于前面的结果。每一步都要确定所需使用的数学主题——比例、面积、平均数等。
Write your working in a logical sequence. If a calculation produces an answer that seems unreasonable (e.g. a negative cost), check your method and units before moving on.
把解题过程按逻辑顺序写出。如果计算结果看起来不合理（例如出现负成本），在继续下一步前检查你的方法和单位。
4. Case Study Scenario 1: Planning a School Trip | 案例场景一：规划学校旅行
A school plans a trip to a science museum for 120 students and 8 teachers. Minibuses seat 16 people each. Each minibus costs £80 to hire plus £0.30 per km. The museum is 45 km from the school. Museum entry costs £5.50 per student; teachers are free. The total budget is £1500.
一所学校计划组织 120 名学生和 8 名教师前往科学博物馆。小巴每辆可坐 16 人，租金为每辆 80 英镑，外加每公里 0.30 英镑。博物馆距学校 45 公里。博物馆门票每名学生 5.50 英镑，教师免费。总预算为 1500 英镑。
ber of minibuses needed. Total people = 120 + 8 = 128. Number of minibuses = 128 ÷ 16 = 8.
步骤 1：计算所需小巴数量。总人数 = 120 + 8 = 128。小巴数量 = 128 ÷ 16 = 8。
0 km. Total distance for 8 minibuses is not needed because cost is calculated per minibus.
步骤 2：计算总行驶距离。需要往返，因此每辆小巴的行驶距离 = 45 × 2 = 90 公里。无需计算 8 辆车的总距离，因为费用是按车计算的。
Step 3: Compute the travel cost per minibus. Fuel charge = 90 × £0.30 = £27. Cost per minibus = £80 + £27 = £107. Total travel cost = 8 × £107 = £856.
步骤 3：计算每辆小巴的交通费用。燃油费 = 90 × 0.30 英镑 = 27 英镑。每辆车费用 = 80 英镑 + 27 英镑 = 107 英镑。交通总费用 = 8 × 107 英镑 = 856 英镑。
Step 4: Work out the total museum entry cost. 120 students × £5.50 = £660.
步骤 4：计算博物馆门票总费用。120 名学生 × 5.50 英镑 = 660 英镑。
Step 5: Find the total spending and compare with the budget. Total cost = £856 + £660 = £1516. Budget = £1500. Overspend = £1516 - £1500 = £16.
步骤 5：计算总支出并与预算比较。总费用 = 856 英镑 + 660 英镑 = 1516 英镑。预算 = 1500 英镑。超支 = 1516 英镑 - 1500 英镑 = 16 英镑。
Step 6: The school is £16 over budget. A suggestion could be to ask for a 2% = 7.9375 → still 8 minibuses. Alternatively, negotiate a group discount on museum tickets.
步骤 6：学校超预算 16 英镑。可以建议要求小巴租金打 2% 折扣，或将教师人数减少到 7 人（若学校政策允许），但 127 ÷ 16 = 7.9375，仍需 8 辆小巴。另一种方法是与博物馆协商团体门票折扣。
5. Case Study Scenario 2: Analysing Sales Data | 案例场景二：分析销售数据
A school tuck shop records the number of fruit pots sold over five days: Monday 24, Tuesday 30, Wednesday 18, Thursday 36, Friday 42. The shop wants to find the average sales and predict the stock needed for the following week.
学校小卖部记录了五天水果杯的销售量：周一 24 杯，周二 30 杯，周三 18 杯，周四 36 杯，周五 42 杯。小卖部希望计算平均销售量，并预测下周所需库存。
| Day | Sales |
|---|---|
| Monday | 24 |
| Tuesday | 30 |
| Wednesday | 18 |
| Thursday | 36 |
| Friday | 42 |
Calculate the mean: (24+30+18+36+42) ÷ 5 = 150 ÷ 5 = 30 fruit pots.
计算平均数：(24+30+18+36+42) ÷ 5 = 150 ÷ 5 = 30 杯。
Find the median: ordered data 18, 24, 30, 36, 42 → median = 30.
求中位数：排序后数据为 18, 24, 30, 36, 42 → 中位数 = 30。
Determine the range: 42 - 18 = 24. This shows spread of sales.
计算极差：42 - 18 = 24。这反映了销售量的离散程度。
k. However, as Friday sales are highest, extra stock may be needed at the end of the week.
对于下周，根据平均数，小卖部应至少准备 150 杯（5 × 30）以维持库存。但由于周五销量最高，周末可能需要额外备货。
6. Case Study Scenario 3: Designing a Garden | 案例场景三：设计花园
A rectangular garden measures 8 m by 5 m. A semicircular flower bed of diameter 3 m is located at one end. The rest of the area will be covered with turf costing £6.50 per m². A wooden fence is to be installed around the garden, except for a 1 m wide gate.
一个矩形花园尺寸为 8 米 × 5 米。一端有一个直径为 3 米的半圆形花坛。其余区域将铺设草皮，草皮价格为每平方米 6.50 英镑。花园四周将安装木栅栏，但留一个 1 米宽的门。
Step 1: Total area of rectangle = 8 × 5 = 40 m².
步骤 1：矩形总面积 = 8 × 5 = 40 平方米。
s r = 3 ÷ 2 = 1.5 m. Using π ≈ 3.14, area = ½ × 3.14 × (1.5)² = ½ × 3.14 × 2.25 ≈ 3.53 m².
步骤 2：半圆形面积 = ½ × π × r²。半径 r = 3 ÷ 2 = 1.5 米。使用 π ≈ 3.14，面积 = ½ × 3.14 × (1.5)² = ½ × 3.14 × 2.25 ≈ 3.53 平方米。
Area of turf = 40 - 3.53 = 36.47 m²
Step 3: Cost of turf = 36.47 × £6.50 = £237.06 (rounded to nearest penny).
步骤 3：草皮费用 = 36.47 × 6.50 英镑 = 237.06 英镑（四舍五入到分）。
f rectangle = 2 × (8 + 5) = 26 m. Length to fence = 26 - 1 (gate) = 25 m. If fencing costs £12 per metre, total fence cost = 25 × 12 = £300.
步骤 4：矩形周长 = 2 × (8 + 5) = 26 米。需装栅栏长度 = 26 - 1（门）= 25 米。若栅栏每米 12 英镑，则栅栏总费用 = 25 × 12 = 300 英镑。
Step 5: Total project cost = £237.06 + £300 = £537.06.
步骤 5：项目总费用 = 237.06 英镑 + 300 英镑 = 537.06 英镑。
7. Handling Measurement and Scale Drawings | 处理测量与比例绘图
Case studies often involve maps or scaled plans. For example, a map has a scale 1 : 25 000. The distance between two landmarks on the map is 8 cm. To find the actual distance in km, use the scale ratio.
案例分析常涉及地图或按比例绘制的平面图。例如，一张地图的比例尺为 1:25 000。两处地标在地图上的距离为 8 厘米。求实际距离（公里）需运用比例尺。
00 000 cm. Convert to metres: 200 000 ÷ 100 = 2 000 m. Convert to km: 2 000 ÷ 1000 = 2 km.
地图距离（厘米）→ 实际距离（厘米）：8 × 25 000 = 200 000 厘米。换算为米：200 000 ÷ 100 = 2 000 米。换算为公里：2 000 ÷ 1 000 = 2 公里。
tual area = (3.5 × 25 000) × (2.4 × 25 000) cm², then convert carefully to m² or hectares.
如果同一地图上一块矩形田地长 3.5 厘米、宽 2.4 厘米，则其实际面积 = (3.5 × 25 000) × (2.4 × 25 000) 平方厘米，然后仔细换算为平方米或公顷。
Actual area = 87 500 cm × 60 000 cm = 5 250 000 000 cm² = 525 000 m² = 52.5 hectares
Always check that units are consistent and conversions are handled step by step to avoid errors.
务必检查单位是否一致，并逐步进行单位换算，以避免错误。
8. Financial Mathematics in Context | 情境中的金融数学
Finance topics such as simple and compound interest, discounts, VAT, and profit/loss appear regularly in case studies. Consider a car worth £12 000 which depreciates by 15% per year. Its value after 2 years can be found using a multiplier of 0.85 each year.
单利、复利、折扣、增值税及盈亏等金融话题经常出现在案例分析中。假设一辆车价值 12 000 英镑，每年贬值 15%。两年后的价值可用每年乘数 0.85 来计算。
Value after 2 years = 12 000 × (0.85)² = 12 000 × 0.7225 = £8670
In a shopping scenario, an item priced at £80 excluding VAT (20%) requires the final price: £80 × 1.2 = £96. If a 10% discount is then applied, the amount to pay becomes £96 × 0.9 = £86.40.
在购物场景中，一件商品标价 80 英镑（不含 20% 增值税），最终价格为：80 英镑 × 1.2 = 96 英镑。若再打九折，应付金额为 96 英镑 × 0.9 = 86.40 英镑。
When approaching a financial case study, identify the order of operations carefully—discount before tax or after? The wording of the problem determines the correct sequence.
解答金融类案例分析时，需仔细识别运算顺序——折扣在税前还是税后？题目的措辞决定了正确的计算顺序。
9. Checking and Interpreting Results | 检查与解释结果
After obtaining a numerical answer, always ask: Does this answer make sense in the given context? If a trip cost exceeds the budget, the solution is not complete without suggesting adjustments.
在得到数值答案后，要始终问自己：这个答案在给定情境中是否合理？如果旅行费用超出预算，未提出调整建议的解答是不完整的。
Explain findings in words. For the garden case, you might conclude: "The turf area is 36.47 m², which will cost £237.06. The fence requires 25 m of material at £12/m, giving a total of £537.06." This written interpretation can earn communication marks in CCEA exams.
用文字解释结果。以花园案例为例，你可以总结：“草皮面积为 36.47 平方米，费用为 237.06 英镑。栅栏需 25 米材料，每米 12 英镑，总计 537.06 英镑。” 这类文字解释可在 CCEA 考试中获得交流分。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One frequent error is mixing units, such as using centimetres with metres without conversion. Another is misreading percentages—remember that '20% off' means multiply by 0.8, not 0.2.
一个常见错误是单位混淆，比如厘米和米混用而未换算。另一个是误读百分比——请记住，“打八折”意味着乘以 0.8，而非 0.2。
Forgetting to double the journey for a return trip, like in minibus example, can lead to underestimating travel cost. Also, rounding too early can distort final answers; keep intermediate values to sufficient precision.
像小巴例子中忘记往返双倍路程，会导致低估交通费用。此外，过早四舍五入可能使最终答案失真；中间数值应保留足够精度。
Always re-read the question after you finish to ensure all parts have been answered, including 'give a reason' or 'state your assumption' prompts.
完成后务必重读题目，确保所有部分都已作答，包括“给出理由”或“陈述你的假设”等提示。
11. Time Management and Exam Tips | 时间管理与考试技巧
Case study questions are often high-mark items. Allocate enough time to plan and check. A rough guide is 1 minute per mark, but spend 2–3 minutes initially reading and annotating the text.
案例分析题通常分值较高。要分配足够时间进行规划和检查。粗略分配为每分钟 1 分，但最初应花 2-3 分钟阅读并标注文本。
Present working clearly with bullet points or numbered steps. Even if a final answer is slightly off, correct method marks can be gained. If stuck, move to the next sub-part, as some may be independent.
用项目符号或编号步骤清晰展示解题过程。即使最终答案略有偏差，仍可获得方法分。如果卡住，可以跳到下一个子问题，因为有些部分是相对独立的。
12. Conclusion and Practice Encouragement | 结论与练习鼓励
Mastering case study questions strengthens your overall mathematical fluency. Regular practice with CCEA-style scenarios builds confidence, enabling you to move seamlessly between arithmetic, measures, and statistics.
掌握案例分析题能增强你的整体数学流利度。经常练习 CCEA 风格的场景题可以建立自信，使你能在算术、测量和统计之间自如切换。
Seek out past paper problems and create your own mini case studies from everyday situations—budgeting pocket money, planning a party, or redesigning your bedroom. The more you apply maths, the more natural it becomes.
寻找历年真题并从日常情境中创建自己的小型案例分析——预算零花钱、策划聚会或重新设计卧室。越多应用数学，它就越变得自然。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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