📚 Case Study Mastery: Real-World Problem Solving | 案例分析实战演练
In Year 8 OCR Mathematics, case studies bring together all the skills you have learned – from percentages to algebra, geometry to statistics. This article walks you through a realistic scenario: planning a school fair. You will see how to extract information, set up equations, calculate areas, interpret graphs, and make sensible predictions, just as you would in an exam-style case study question.
在 Year 8 OCR 数学中，案例分析将你学过的所有技能汇聚一堂——从百分比到代数，从几何到统计。这篇文章将通过一个真实场景：策划学校游园会，带你一步步实践如何提取信息、建立方程、计算面积、解读图表并做出合理的预测，就像考试中的案例分析题一样。
1. Understanding the Scenario | 理解情境
Our case study is about organising a school summer fair. You are part of the planning committee, and you need to manage the budget, design the layout, run a snack stall, and predict attendance. First, read the problem statement carefully and identify what is being asked.
我们的案例是组织一次学校夏季游园会。你是策划委员会的一员，需要管理预算、设计场地布局、经营一个小吃摊位并预测参与人数。首先要仔细阅读问题陈述，明确要求你做什么。
Typical tasks might include: calculating profit after costs, finding the area of a stall, working out the best value when buying supplies, and estimating probabilities of sunny weather.
典型任务可能包括：计算扣除成本后的利润、计算摊位面积、在采购物资时计算最佳性价比，以及估算晴天的概率。
2. Extracting Key Information | 提取关键信息
Highlight numbers, units, and relationships. For example, the school has a field measuring 50 m by 30 m. Each stall requires a space of 4 m by 3 m. The budget for supplies is £150. Entrance tickets cost £2 per person, but there is a 25% discount for families of four or more.
标出数字、单位和关系。例如，学校有一块 50 米 × 30 米的场地。每个摊位需要 4 米 × 3 米的空间。采购物资的预算是 150 英镑。入场票价为每人 2 英镑，但四人及以上的家庭可享受 25% 折扣。
Write down what you know in a structured way. Use a table to organise costs or dimensions.
用结构化的方式写下已知信息。可以用表格来整理成本或尺寸。
| Item | Details |
|---|---|
| Field size | 50 m × 30 m |
| Stall dimensions | 4 m × 3 m |
| Budget | £150 |
| Ticket price | £2 (25% discount for families 4+) |
这种结构化的方法能防止遗漏关键信息。
3. Translating Words into Mathematics | 将文字转化为数学
Many problems require you to form an expression or equation. Suppose you want to find the profit P from the snack stall if you sell n cupcakes. Cupcakes cost £0.80 each to make and are sold for £1.50. The stall also has a fixed hire cost of £10.
许多问题需要你列出表达式或方程。假设你想计算卖出 n 个纸杯蛋糕时小吃摊的利润 P。每个蛋糕的成本是 0.80 英镑，售价为 1.50 英镑。摊位还有 10 英镑的固定租金。
Profit = revenue – costs. Revenue = 1.50n, costs = 0.80n + 10. So the equation is:
利润 = 收入 – 成本。收入 = 1.50n，成本 = 0.80n + 10。因此方程为：
P = 1.50n – (0.80n + 10) = 0.70n – 10
This linear equation shows that you need to sell more than 14 cupcakes to make a profit (since 0.70 × 15 – 10 = £0.50 > 0).
这个线性方程显示，你需要卖出超过 14 个蛋糕才能盈利（因为 0.70 × 15 – 10 = 0.50 英镑 > 0）。
4. Working with Percentages and Discounts | 处理百分比与折扣
The family discount on tickets is a classic percentage problem. If a family of 5 buys tickets, the full price would be 5 × £2 = £10. With a 25% discount, they pay:
家庭门票折扣是一个典型的百分比问题。如果一个五口之家买票，原价为 5 × 2 = 10 英镑。享受 25% 折扣后，他们付：
£10 × (1 – 0.25) = £7.50
The saving is £2.50. Alternatively, you can calculate the discount as 0.25 × £10 = £2.50 and subtract.
节省了 2.50 英镑。或者，你也可以计算折扣为 0.25 × 10 = 2.50 英镑再相减。
Now, if 30 families of 4, 15 families of 3, and 20 individuals attend, calculate total ticket revenue.
现在，如果有 30 个四口之家、15 个三口之家和 20 个单人参加，计算总门票收入。
- Families of 4: each family price = 4 × £2 × 0.75 = £6. Total = 30 × £6 = £180.
- 四口之家：每家票价 = 4 × 2 × 0.75 = 6 英镑。总计 = 30 × 6 = 180 英镑。
- Families of 3: no discount (only 4+ qualifies). Price = 3 × £2 = £6. Total = 15 × £6 = £90.
- 三口之家：无折扣（仅 4 人及以上符合条件）。票价 = 3 × 2 = 6 英镑。总计 = 15 × 6 = 90 英镑。
- Individuals: 20 × £2 = £40.
- 单人：20 × 2 = 40 英镑。
Grand total = £180 + £90 + £40 = £310.
总收入 = 180 + 90 + 40 = 310 英镑。
5. Using Ratio and Proportion | 运用比例与比率
You need to mix orange squash and water in the ratio 1 : 4 to make 15 litres of drink. To find the amount of squash needed, the ratio parts total 1 + 4 = 5. Squash = (1/5) × 15 L = 3 L. Water = (4/5) × 15 L = 12 L.
你需要按 1 : 4 的比例混合浓缩橙汁和水，制作 15 升饮料。求所需浓缩液量：比例份数总和为 1 + 4 = 5。浓缩液 = (1/5) × 15 L = 3 L。水 = (4/5) × 15 L = 12 L。
Another example: the ratio of boys to girls helping at the fair is 3 : 5. If there are 24 boys, how many girls? Set up the proportion: boys/girls = 3/5 = 24/x. Cross-multiply: 3x = 120, x = 40. So there are 40 girls.
另一个例子：帮忙游园会的男生和女生人数比是 3 : 5。如果有 24 名男生，女生有多少？列出比例：男生/女生 = 3/5 = 24/x。交叉相乘得 3x = 120，x = 40。所以有 40 名女生。
Proportion is also used when scaling recipes for the bake sale. If a cookie recipe uses 200 g of flour for 12 cookies, how much flour is needed for 30 cookies? (200 g/12 cookies) × 30 = 500 g.
比例还用于调整义卖烘焙食谱。如果一个饼干配方用 200 克面粉做 12 块饼干，那么做 30 块需要多少面粉？ (200 g / 12) × 30 = 500 g。
6. Interpreting Graphs and Charts | 解读图表
A bar chart shows the number of visitors attending the fair over the last four years: 2019: 200, 2020: 180, 2021: 250, 2022: 300. You are asked to predict the attendance for this year. Describe the trend: generally increasing, with a dip in 2020. Use this to estimate maybe 320–350 people, but justify.
条形图显示了过去四年游园会的参观人数：2019 年 200 人，2020 年 180 人，2021 年 250 人，2022 年 300 人。要求预测今年的参与人数。描述趋势：总体增长，2020 年有所下降。据此估计可能在 320–350 人之间，但需说明理由。
Scatter graphs might link ice cream sales to temperature. If the line of best fit suggests sales = 2 × temperature (°C) – 10, then at 25°C predicted sales = 40 ice creams
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