📚 Year 10 CAIE Maths: Case Study Practice | Year 10 CAIE 数学：案例分析实战演练
In Year 10 CAIE Mathematics, case study questions test your ability to apply mathematical concepts to real-world scenarios. You need to model a situation with equations, graphs, or statistics, then interpret your findings clearly. This article walks you through a complete case study on mobile phone plans, showing how to set up linear equations, draw graphs, make comparisons, and even use simple statistics for forecasting. Work through each step carefully to build confidence for your IGCSE-style assessments.
在 Year 10 的 CAIE 数学课程中，案例分析题考查你将数学概念应用到真实情境中的能力。你需要用方程、图像或统计来建立模型，然后清晰解读你的发现。本文通过一个完整的手机套餐案例，手把手教你如何建立线性方程、绘制图像、进行比较，甚至用简单统计来做预测。请仔细完成每一步，为 IGCSE 风格的评估积累信心。
1. Understanding the Scenario | 理解情境
Imagine two mobile network providers, FoneFast and DataPlus. FoneFast charges a fixed monthly rental of £10 plus 5p per minute of calls. DataPlus has no monthly rental but charges 8p per minute. You want to know which plan is cheaper depending on your monthly call time, and whether any other factors like data usage might change your decision. This is a typical CAIE case study where you must extract information, define variables, and build equations.
想象两家移动网络运营商，FoneFast 和 DataPlus。FoneFast 收取每月固定租金 10 英镑，外加每分钟通话费 5 便士。DataPlus 则没有月租，但每分钟通话费 8 便士。你想知道根据每月通话时长哪个套餐更便宜，以及数据用量等其他因素是否会改变你的决定。这是一个典型的 CAIE 案例研究，你需要提取信息、定义变量并建立方程。
2. Defining Variables and Forming Equations | 定义变量与建立方程
Let the number of minutes of calls per month be m. The cost in pence for FoneFast is: CostF = 1000 + 5m. (The £10 rental is 1000 pence.) The cost for DataPlus is: CostD = 8m. Notice that we choose pence to avoid decimals. Always define your variables clearly — examiners give marks for stating what each symbol represents.
设每月通话分钟数为 m。FoneFast 的费用（单位：便士）为：CostF = 1000 + 5m。（10 英镑月租即 1000 便士。）DataPlus 的费用为：CostD = 8m。注意我们使用便士以避开小数。一定要清晰定义变量——考官会因说明每个符号的含义而给分。
3. Finding the Break-Even Point | 寻找平衡点
To find when both plans cost the same, set the equations equal: 1000 + 5m = 8m. Solve: 1000 = 3m, so m = 1000 ÷ 3 ≈ 333.3 minutes. This means if you talk for about 333 minutes per month, the two plans cost the same. For fewer minutes, DataPlus is cheaper; for more minutes, FoneFast becomes the better deal.
为了找出两种套餐费用相同的时刻，令两个方程相等：1000 + 5m = 8m。求解得：1000 = 3m，因此 m = 1000 ÷ 3 ≈ 333.3 分钟。这意味着如果你每月通话约 333 分钟，两种套餐费用相同。通话时间更少，DataPlus 更便宜；通话时间更多，FoneFast 更划算。
4. Graphical Interpretation | 图像解读
Plot both cost equations on a graph with m on the horizontal axis (0 to 600 minutes) and cost in pence on the vertical axis. The line for DataPlus starts at the origin (0,0) with slope 8. The line for FoneFast starts at (0,1000) with slope 5. The intersection shows the break-even point. A sketch helps you see that below 333 minutes, the DataPlus line is lower; above it, the FoneFast line is lower. CAIE often requires a rough graph or interpretation of slopes and intercepts.
在一张图上画出两个费用方程，横轴表示 m（0 到 600 分钟），纵轴表示费用（便士）。DataPlus 的直线从原点 (0,0) 开始，斜率为 8。FoneFast 的直线从 (0,1000) 开始，斜率为 5。交点显示了平衡点。通过草图你可以看到，当通话时间低于 333 分钟时，DataPlus 的直线较低；超过该值后，FoneFast 的直线较低。CAIE 经常要求绘制粗略图形或解读斜率与截距的含义。
5. Cost Comparison for Specific Usage | 特定使用量下的费用比较
Suppose you talk for 200 minutes a month. FoneFast: 1000 + 5×200 = 2000 pence (£20.00). DataPlus: 8×200 = 1600 pence (£16.00). DataPlus saves £4.00. For 500 minutes: FoneFast = 1000 + 5×500 = 3500 pence (£35.00); DataPlus = 8×500 = 4000 pence (£40.00). FoneFast saves £5.00. Always calculate exact values to confirm your decision.
假设你每月通话 200 分钟。FoneFast：1000 + 5×200 = 2000 便士（20.00 英镑）。DataPlus：8×200 = 1600 便士（16.00 英镑）。DataPlus 节省 4.00 英镑。如果通话 500 分钟：FoneFast = 1000 + 5×500 = 3500 便士（35.00 英镑）；DataPlus = 8×500 = 4000 便士（40.00 英镑）。FoneFast 节省 5.00 英镑。始终计算准确数值来确认你的选择。
6. Introducing an Additional Factor: Texts and Data | 引入附加因素：短信和数据
Real-life plans often include free texts and data. Suppose FoneFast gives 500 free texts and 1 GB data within the £10 rental, while DataPlus charges 2p per text and £5 for each 500 MB data bolt-on. If you send 300 texts and use 800 MB per month, how does the comparison change? This turns the problem into a multi-variable case study, testing your ability to combine costs.
现实中的套餐常常包含免费短信和数据。假设 FoneFast 在 10 英镑月租中提供 500 条免费短信和 1 GB 数据，而 DataPlus 每条短信收费 2 便士，并且每增加 500 MB 数据需支付 5 英镑。如果你每月发送 300 条短信并使用 800 MB 数据，比较结果会如何变化？这将问题转化为多变量案例研究，考验你组合费用的能力。
7. Calculating Total Monthly Cost with Extras | 计算包含附加项目的月总费用
For FoneFast: 300 texts are within the 500 free allowance, so £0 extra. Data usage 800 MB exceeds the 1 GB (1000 MB) allowance? No, 800 MB is within 1 GB, so still free. Total cost = £10 + call charges as before. For DataPlus: 300 texts cost 300×2p = 600p = £6.00. Data: 800 MB needs two 500 MB bolt-ons costing 2×£5 = £10.00. Monthly cost = £6.00 + £10.00 + call charges. Now recalculate with 400 minutes: FoneFast = £10 + 5p×400 = £10 + £20 = £30. DataPlus = £6 + £10 + 8p×400 = £16 + £32 = £48. The gap widens further.
对于 FoneFast：300 条短信在 500 条免费额度内，因此额外费用为 £0。数据使用 800 MB 是否超过 1 GB（1000 MB）的额度？不，800 MB 在 1 GB 之内，所以仍然免费。总费用 = £10 + 之前的通话费用。对于 DataPlus：300 条短信费用为 300×2 便士 = 600 便士 = £6.00。数据：800 MB 需要两个 500 MB 的附加包，费用为 2×£5 = £10.00。月费 = £6.00 + £10.00 + 通话费用。现在以 400 分钟重新计算：FoneFast = £10 + 5p×400 = £10 + £20 = £30。DataPlus = £6 + £10 + 8p×400 = £16 + £32 = £48。差距进一步拉大。
8. Using Statistics to Predict Usage | 利用统计预测使用量
A case study might give past call records: e.g., the last six months’ minutes were 280, 310, 345, 370, 390, 420. Find the mean: (280+310+345+370+390+420) ÷ 6 = 2115 ÷ 6 = 352.5 minutes. The median is (345+370)÷2 = 357.5 minutes. Both suggest usage near the break-even point. If the trend is increasing, FoneFast will soon be cheaper consistently. Using moving averages or a line of best fit could strengthen your recommendation. CAIE expects you to connect statistical measures to decision-making.
案例研究可能给出过去的通话记录：例如，过去六个月的分钟数为 280、310、345、370、390、420。计算平均数：(280+310+345+370+390+420) ÷ 6 = 2115 ÷ 6 = 352.5 分钟。中位数为 (345+370)÷2 = 357.5 分钟。两者都显示使用量接近平衡点。如果趋势在上升，FoneFast 很快会稳定地更便宜。使用移动平均或最佳拟合线可以加强你的建议。CAIE 期望你将统计量度与决策联系起来。
9. Sensitivity Analysis: Changing Rates | 灵敏性分析：费率变化
What if FoneFast increases its per-minute rate to 6p? New equation: CostF = 1000 + 6m. New break-even: 1000 + 6m = 8m → 1000 = 2m → m = 500 minutes. Now the threshold is much higher. This shows how sensitive your recommendation is to small price changes. In exam case studies, you might be asked to consider ‘what-if’ scenarios and discuss their impact.
如果 FoneFast 将每分钟费率提高到 6 便士呢？新方程：CostF = 1000 + 6m。新的平衡点：1000 + 6m = 8m → 1000 = 2m → m = 500 分钟。现在门槛大大提高。这表明你的推荐对小幅价格变化有多敏感。在考试案例研究中，你可能会被要求考虑“如果……会怎样”的情景并讨论其影响。
10. Common Mistakes and How to Avoid Them | 常见错误与如何避免
Frequent errors include: mixing units (pounds and pence) without conversion, forgetting to define variables, misreading the break-even point as the cheaper option for all usage, and ignoring non-call costs. Always convert all amounts to the same unit, label axes clearly on graphs, and relate your answer back to the original context. Use a structured approach: extract data → model → solve → interpret → check.
常见错误包括：混合单位（英镑和便士）而未转换、忘记定义变量、误将平衡点当作所有使用量下的更便宜选项，以及忽略非通话费用。始终将所有金额转换为同一单位，在图上清楚标记坐标轴，并将答案联系回原始情境。采用结构化方法：提取数据 → 建模 → 求解 → 解读 → 检查。
11. Extending the Case Study: Recommendation Report | 案例扩展：撰写推荐报告
Many CAIE questions ask you to write a short recommendation. For example: ‘Based on an average of 350 minutes and growing, along with moderate text and data needs, FoneFast offers better overall value. However, if your usage stays reliably below 300 minutes and you use little data, DataPlus is cheaper.’ Support with numerical evidence, mention trends, and acknowledge assumptions. This demonstrates higher-order reasoning.
许多 CAIE 问题要求你写一份简短的推荐。例如：“基于平均 350 分钟且持续增长的通话量，以及适中的短信和数据需求，FoneFast 提供更好的整体价值。然而，如果你的使用量稳定保持在 300 分钟以下且数据用量很少，DataPlus 更便宜。”用数值证据支持，提及趋势，并承认假设。这展示了高阶推理能力。
12. Summary and Key Takeaways | 总结与关键要点
Case study practice bridges the gap between isolated skills and real problem-solving. Remember to: read the context carefully, define variables explicitly, set up accurate equations or graphs, perform break-even analysis, consider additional costs, use statistics for prediction, and communicate your findings clearly. The more you practise multi-step scenarios like this mobile plan comparison, the more confident you will become in tackling CAIE Year 10 case studies.
案例分析练习弥合了孤立技能与真实问题解决之间的差距。请记住：仔细阅读情境，明确定义变量，建立准确的方程或图像，进行平衡点分析，考虑附加成本，用统计做预测，并清晰地传达你的发现。你越多地练习像这个手机套餐对比这样的多步骤情景，你在挑战 CAIE Year 10 案例研究时就会越有信心。
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