Case Study Practice: Mastering Real-Life Applications in CAIE IGCSE Maths | 案例分析实战演练：攻克CAIE IGCSE数学中的应用题

📚 Case Study Practice: Mastering Real-Life Applications in CAIE IGCSE Maths | 案例分析实战演练：攻克CAIE IGCSE数学中的应用题

In the CAIE IGCSE Mathematics examinations, case study questions are designed to assess your ability to apply mathematical concepts to unfamiliar, real-world contexts. These multi-step problems often combine several topics—such as algebra, geometry, statistics, or number theory—into a single extended scenario. Mastering them requires not only solid subject knowledge but also a systematic approach to deconstructing the problem. This article offers a practical guide to tackling case study questions effectively, with detailed strategies and fully worked examples drawn from typical CAIE-style tasks.

在CAIE IGCSE数学考试中，案例分析题旨在评估你将数学概念应用于陌生的现实情境的能力。这类多步骤问题往往将代数、几何、统计或数论等多个主题整合在一个扩展情景中。攻克它们不仅需要扎实的学科知识，还需要系统拆解问题的方法。本文提供高效应对案例分析题的实用指南，并配有来自典型CAIE风格任务的详细策略与完整范例。

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1. Understanding the Problem Statement | 理解问题陈述

Begin by reading the entire case study carefully, including any diagrams, tables, or footnotes. Identify what the question is asking you to find—look for command words like ‘calculate’, ‘justify’, ‘compare’, or ‘determine’. Underline key numerical data, variables, and constraints mentioned in the text. Many students lose marks because they rush into calculations without fully grasping the context or the required final answer format, such as units, significant figures, or whether an explanation is needed.

开始时仔细阅读整个案例，包括图表、表格或脚注。确定题目要求你求解的内容——注意’calculate’、’justify’、’compare’或’determine’等指令词。划出文中提到的关键数值、变量和约束条件。许多学生因未充分理解上下文或最终答案格式（如单位、有效数字或是否需要解释）就匆忙计算而失分。

For example, a case study about a factory’s packaging might ask for the minimum surface area of a box given a fixed volume. The hidden constraint could be that dimensions must be positive integers, and the answer should be given in square centimetres to three significant figures. Noticing such details early saves time later.

例如，一个有关工厂包装的案例分析可能要求在给定体积下求盒子的最小表面积。隐藏的约束条件可能是尺寸必须为正整数，答案应以平方厘米表示并保留三位有效数字。提早注意这些细节可节省后续时间。

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2. Extracting and Organising Information | 提取并整理信息

List all the given quantities and assign variables to unknowns. A clear table or diagram often helps. For instance, if the case involves a journey with different speeds, create a distance-speed-time table. This prevents confusion and ensures no piece of data is overlooked.

列出所有已知量并为未知量分配变量。清晰的表格或示意图通常有帮助。例如，如果案例涉及不同速度的行程，可创建一个距离-速度-时间表。这能避免混淆并确保不遗漏任何数据。

Quantity (量)	Symbol (符号)	Value / Relationship (数值/关系)
Initial loan amount	P	£5000
Annual interest rate	r	4.5% = 0.045
Number of years	t	3
Final amount	A	A = P(1 + r)t

In geometry problems, sketch the shape and mark known lengths, angles, and right angles. If the problem describes a path or an angle of elevation, a labelled diagram is essential for applying trigonometry correctly.

在几何问题中，绘制形状并标记已知长度、角度和直角。如果问题描述了路径或仰角，标注清晰的示意图对于正确应用三角学至关重要。

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3. Building a Mathematical Model | 建立数学模型

Translate the real-world scenario into mathematical equations, inequalities, or functions. This is the core of case study success. Look for relationships described in words—’directly proportional to’, ‘varies with the square of’, ‘the sum of’, ‘the product of’—and express them with symbols.

将现实情景转化为数学方程、不等式或函数。这是案例分析成功的关键。寻找文字描述的关系——’与…成正比’、’随…的平方而变化’、’…之和’、’…的乘积’——并用符号表达。

For example, ‘the total cost consists of a fixed charge of $200 plus $15 per guest’ becomes C = 200 + 15n, where n is the number of guests. If a rectangular area uses 30 m of fencing with one side against a wall, the width w and length l might satisfy l + 2w = 30, and the area A = lw. The model then becomes A = w(30 – 2w).

例如，’总费用由200美元的固定费用加上每位客人15美元组成’变为 C = 200 + 15n，其中n为客人人数。若一块矩形区域使用30米围栏且一边靠墙，宽度w和长度l满足 l + 2w = 30，面积A = lw。于是模型变为 A = w(30 – 2w)。

Always state the domain of any variables. In the fencing example, w must be greater than 0 and less than 15 to keep dimensions positive.

务必说明变量的定义域。在围栏示例中，w必须大于0且小于15以保持尺寸为正。

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4. Selecting the Right Algebraic Tools | 选择合适的代数工具

Once the model is built, decide which algebraic technique to apply. Linear equations are straightforward, but many case studies lead to quadratic equations, simultaneous equations, or direct/inverse proportion.

一旦模型建立，决定使用哪种代数技巧。线性方程简单直接，但许多案例分析会引出二次方程、联立方程或正/反比例。

For quadratic models like A = -2w² + 30w, find the maximum by completing the square or using the vertex formula w = -b/(2a). Here a = -2, b = 30, so w = -30/(2 × -2) = 7.5. This gives the width for maximum area.

对于像 A = -2w² + 30w 这样的二次模型，通过配方法或使用顶点公式 w = -b/(2a) 求最大值。这里 a = -2，b = 30，因此 w = -30/(2 × -2) = 7.5。这给出了最大面积时的宽度。

Simultaneous equations often arise when two conditions must be met at once—like the cost of two different blends of coffee or the number of adult and child tickets sold. Substitution or elimination can be used; choose the method that keeps the working neat.

当两个条件必须同时满足时往往出现联立方程——比如两种不同咖啡混合的成本或售出的成人和儿童票数量。可使用代入法或消元法；选择能使计算过程整洁的方法。

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5. Interpreting Graphs and Charts | 解读图表

Case studies frequently include graphs—distance-time, speed-time, or conversion graphs. Be prepared to find gradients, areas under curves, and intercepts, and to relate them to real-life quantities. For a speed-time graph, the area represents distance travelled; the gradient gives acceleration.

案例分析经常包含图表——距离-时间图、速度-时间图或转换图。要准备好求梯度、曲线下方面积和截距，并将其与实际量关联。对于速度-时间图，面积代表行驶距离；梯度代表加速度。

When a graph is given, check the scale on each axis and the units. A common pitfall is misreading intervals. If the axis shows ‘time in hours since 08:00’, make sure to answer the question in the requested time format.

当给出图表时，检查各轴的比例和单位。常见陷阱是误读间隔。如果轴显示’自08:00起的小时数’，务必按要求的时间格式作答。

Statistical diagrams like cumulative frequency curves require you to estimate medians, quartiles, and interquartile ranges from the graph. Plot points accurately if you need to draw the curve yourself. Use a ruler for reading values.

统计图如累积频率曲线需要你从图中估算中位数、四分位数和四分位距。若需自己绘制曲线，请准确描点。使用直尺读取数值。

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6. Applying Geometry and Measurement | 活用几何与测量

Real-life case studies involving shapes call for knowledge of area, volume, Pythagoras’ theorem, and trigonometric ratios (SOH CAH TOA). When a problem mentions a ladder leaning against a wall or a cone-shaped container, draw the right-angled triangle or net immediately.

涉及形状的现实案例分析需要面积、体积、勾股定理和三角比（SOH CAH TOA）的知识。当问题提到梯子靠墙或圆锥形容器时，立即画出直角三角形或展开图。

Remember to use the correct formula for the situation. The volume of a cylinder is V = πr²h, while the curved surface area is 2πrh. For compound shapes, split them into rectangles, triangles, and sectors. If an angle is given in degrees, ensure your calculator is in degree mode. Answers should include units and appropriate degree of accuracy.

记住使用正确的公式。圆柱体积为 V = πr²h，侧面积为 2πrh。对于组合形状，将其拆分为矩形、三角形和扇形。若角度以度为单位，确保计算器处于度模式。答案应包含单位和适当的精确度。

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7. Handling Probability and Statistics in Context | 处理情境中的概率与统计

Case studies on probability might involve tree diagrams for successive events, or two-way tables for surveys. Read conditional probability phrases carefully: ‘given that’ tells you to restrict the sample space. If a manufacturing case discusses defective items, use replacement or non-replacement rules as stated.

案例分析中的概率可能涉及连续事件的树状图或调查的双向表。仔细阅读条件概率短语：’given that’ 告诉你要限制样本空间。如果制造案例讨论缺陷品，按照说明使用放回或不放回规则。

Statistical case studies often present raw data and ask for mean, median, mode, and range, or require you to choose an appropriate average. Justify your choice—for example, median is unaffected by outliers while mean uses all data values. Use grouped frequency tables for estimating the mean with midpoints.

统计案例分析常给出原始数据并要求平均数、中位数、众数和极差，或要求选择合适的平均数。证明你的选择——例如，中位数不受异常值影响，而平均数使用了所有数据值。使用分组频率表以组中值估算平均数。

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8. Checking and Validating Your Answers | 检查与验证答案

After obtaining a numerical answer, check it against the context. Does it make sense? For example, a negative length or a probability greater than 1 indicates an error. Substitute your solution back into the original equation or model. For a maximum problem, test a value either side of your optimum to confirm it gives a smaller result.

得到数值答案后，对照上下文检查。它合理吗？例如，负的长度或大于1的概率表明有误。将解代回原方程或模型。对于最值问题，在最优解两侧各取一个值测试，确认结果更小。

Also verify that you have used the required degree of accuracy—3 significant figures, or the nearest cent, for instance. If the question asks ‘give a reason for your answer’, write a concise sentence referencing the mathematics, not just a vague statement.

还要确认你使用了要求的精确度——例如3位有效数字或精确到分。如果问题要求’说明你的理由’，写一句简洁的话引用数学原理，而非模糊陈述。

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9. Worked Case Study 1: Maximising Enclosed Area | 实战案例1：最大化围合面积

Scenario: A farmer has 40 metres of fencing and wants to create a rectangular enclosure against a straight barn, using the barn as one side. Find the dimensions that give the maximum area, and calculate this maximum area.

情景： 一位农民有40米围栏，想利用笔直的谷仓作为一边，围一个矩形场地。求使面积最大的尺寸，并计算最大面积。

Let width perpendicular to barn = w metres. Then the side parallel to barn uses length l = 40 – 2w metres (since the barn provides one long side). Area A = l × w = w(40 – 2w) = 40w – 2w².

设垂直于谷仓的宽为 w 米。则平行于谷仓的一边使用长度 l = 40 – 2w 米（因为谷仓提供了一长边）。面积 A = l × w = w(40 – 2w) = 40w – 2w²。

This is a quadratic function with a = -2, b = 40, c = 0. The maximum occurs at w = -b/(2a) = -40/(2 × -2) = 10. So width = 10 m, length l = 40 – 2×10 = 20 m. The maximum area = 10 × 20 = 200 m².

这是二次函数，a = -2, b = 40, c = 0。最大值出现在 w = -b/(2a) = -40/(2 × -2) = 10。故宽 = 10 米，长 l = 40 – 2×10 = 20 米。最大面积 = 10 × 20 = 200 平方米。

Check: For w=9, A=9×22=198; w=11, A=11×18=198—both less than 200. Domain: w>0 and w<20 for positive dimensions. Answer is valid.

验证：w=9时，A=9×22=198；w=11时，A=11×18=198——都小于200。定义域：w>0且w<20以保证尺寸为正。答案有效。

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10. Worked Case Study 2: Compound Interest with Regular Savings | 实战案例2：复利与定期储蓄

Scenario: Maria invests $2000 in an account paying 3.2% interest compounded annually. She plans to add $500 at the end of each year. Calculate the total amount in the account after 4 years.

情景： 玛丽亚将2000美元投资于年利率3.2%按年复利的账户。她计划每年末追加500美元。计算4年后账户总金额。

Year 1: Starting amount = $2000. Interest = 2000 × 0.032 = $64. After interest = $2064. Then add $500, so end of Year 1 = $2564.

第1年：初始金额 = 2000美元。利息 = 2000 × 0.032 = 64美元。计息后 = 2064美元。然后追加500美元，故第1年末 = 2564美元。

Year 2: Start $2564. Interest = 2564 × 0.032 = $82.048 → $82.05 (to the nearest cent). After interest = $2646.05. Add $500 → $3146.05.

第2年：初始2564美元。利息 = 2564 × 0.032 = 82.048美元 → 82.05美元（精确到分）。计息后 = 2646.05美元。追加500 → 3146.05美元。

Year 3: Interest = 3146.05 × 0.032 = $100.6736 → $100.67. After interest = $3246.72. Add $500 → $3746.72.

第3年：利息 = 3146.05 × 0.032 = 100.6736 → 100.67美元。计息后 = 3246.72美元。追加500 → 3746.72美元。

Year 4: Interest = 3746.72 × 0.032 = $119.89504 → $119.90. After interest = $3866.62. Add $500 → $4366.62. Total after 4 years = $4366.62.

第4年：利息 = 3746.72 × 0.032 = 119.89504 → 119.90美元。计息后 = 3866.62美元。追加500 → 4366.62美元。4年后总额 = 4366.62美元。

Alternatively, use the future value formula for a regular payment, but step-by-step avoids formula errors and shows clear working, which gains method marks in CAIE exams.

或者，使用定期支付的终值公式，但逐步计算可避免公式错误并展示清晰演算过程，这在CAIE考试中可获得方法分。

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11. Worked Case Study 3: Angle of Elevation and Distance | 实战案例3：仰角与距离

Scenario: From a point A on level ground, the angle of elevation to the top of a tower is 28°. After walking 45 metres directly towards the tower to point B, the angle of elevation becomes 53°. Find the height of the tower to 3 significant figures.

情景： 从水平地面上的一点A，测得塔顶的仰角为28°。沿直线向塔走45米到达点B后，仰角变为53°。求塔的高度，保留3位有效数字。

Let the tower height be h metres, and the distance from B to the base of the tower be x metres. Then from A, distance to base = x + 45 m. Using tangent: tan 28° = h / (x + 45) and tan 53° = h / x.

设塔高为 h 米，B到塔底的距离为 x 米。则A到塔底的距离 = x + 45 米。利用正切：tan 28° = h / (x + 45) 且 tan 53° = h / x。

From the second equation, h = x tan 53°. Substitute into the first: tan 28° = (x tan 53°) / (x + 45). Cross-multiply: tan 28° (x + 45) = x tan 53°. Expand: x tan 28° + 45 tan 28° = x tan 53°.

由第二个方程，h = x tan 53°。代入第一个：tan 28° = (x tan 53°) / (x + 45)。交叉相乘：tan 28° (x + 45) = x tan 53°。展开：x tan 28° + 45 tan 28° = x tan 53°。

Rearrange to solve for x: 45 tan 28° = x (tan 53° – tan 28°). So x = 45 tan 28° / (tan 53° – tan 28°). Using a calculator, tan 28° ≈ 0.531709, tan 53° ≈ 1.32704. Difference ≈ 0.79533. Then x = 45 × 0.531709 / 0.79533 ≈ 23.928 / 0.79533 ≈ 30.08 m. Then h = 30.08 × tan 53° ≈ 30.08 × 1.32704 ≈ 39.9 m. To 3 s.f., height = 39.9 m.

整理求x：45 tan 28° = x (tan 53° – tan 28°)。故 x = 45 tan 28° / (tan 53° – tan 28°)。用计算器，tan 28° ≈ 0.531709，tan 53° ≈ 1.32704。差值 ≈ 0.79533。则 x = 45 × 0.531709 / 0.79533 ≈ 23.928 / 0.79533 ≈ 30.08 米。然后 h = 30.08 × tan 53° ≈ 30.08 × 1.32704 ≈ 39.9 米。保留3位有效数字，高度 = 39.9 米。

Alternative method: Use the fact that the angle of elevation increases as you approach; draw a clear diagram and label angles. Always store intermediate values in calculator memory to avoid rounding errors. This case illustrates how trigonometry and algebra merge in a practical measurement problem.

替代方法：利用靠近时仰角增大的事实；绘制清晰示意图并标注角度。始终在计算器内存中存储中间值以避免舍入误差。此案例展示了三角学与代数如何在实际测量问题中融合。

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12. Final Tips for Case Study Success | 案例分析成功要诀

Approach every case study with confidence: break it into manageable parts, use clear notation, and show all working. Even if your final answer is slightly off, logical steps earn most marks. Manage your time—don’t dwell on a single sub-question; move on and return later if needed.

充满自信地应对每个案例：将其拆分为可处理的部分，使用清晰的符号并展示所有解题步骤。即使最终答案略有偏差，合乎逻辑的步骤也可获得大部分分数。管理好时间——不要纠结于单个小问；先往下做，需要时再回头。

Practice past papers under timed conditions, and always review the mark scheme to understand what examiners expect. Remember, the skills you build here—logical reasoning, model building, and critical checking—are not just for the exam but for real-life problem solving.

在限时条件下练习历年试卷，并始终查看评分方案以了解考官期望。请记住，你在此建立的技能——逻辑推理、模型构建和批判性检查——不仅为考试，也为现实生活中的问题解决服务。

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