📚 Year 9 OCR Maths: Case Study Practical Exercises | 九年级 OCR 数学：案例分析实战演练
In Year 9 OCR Mathematics, applying your skills to real-world case studies is one of the most effective ways to build confidence and deepen understanding. This article walks you through a series of practical exercises based on a representative scenario: planning a school fundraising event. You will practise ratios, percentages, statistics, geometry, equations and data presentation, all within a single realistic problem. Each step introduces a new skill, mirrors the reasoning required by the OCR curriculum, and prepares you for assessment-style questions.
在九年级 OCR 数学中，将学到的技能应用于真实世界的案例分析是建立信心并加深理解的最有效方式之一。本文通过一个代表性场景——策划一场学校筹款活动，带你进行一系列实战练习。你将在一个贴近现实的问题中，练习比和比例、百分比、统计、几何、方程以及数据展示。每一步都引入一项新技能，反映 OCR 课程标准所要求的推理过程，并为考试风格的问题做好准备。
1. Understanding the Problem | 理解问题
The case study involves a Year 9 student council organising a charity fair. The team has a budget of £200, needs to set ticket prices, order supplies, arrange stall layouts, and analyse expected attendance. The first step is to read the brief carefully and identify what is being asked: calculate costs, work out profit margins, create scaled diagrams, and interpret survey results. This mirrors the AO3 problem-solving strand in OCR assessments, where you must dissect a written scenario into manageable mathematical tasks.
案例涉及九年级学生会组织一场慈善义卖会。团队预算为 200 英镑，需要设定票价、订购物资、安排摊位布局并分析预期参与人数。第一步是仔细阅读简介，明确具体要求：计算成本、计算利润空间、绘制比例图以及解读调查结果。这与 OCR 评价中 AO3 问题解决部分相呼应，你必须将文字场景分解为可操作的数学任务。
2. Identifying Key Information | 确定关键信息
From the brief: the fair will run for 3 hours; a survey of 150 students showed that 60% would attend if the ticket price is under £5; the hall dimensions are 18 m by 12 m; each stall requires a 3 m by 3 m area plus 1 m walkways; decorations cost £45, snacks £60, prizes £30. Highlighting numbers, units, and conditions is crucial. Organise these into a table to avoid missing anything.
从简介中可以得知：义卖会持续 3 小时；对 150 名学生的调查显示，若票价低于 5 英镑，60% 的人会参加；礼堂尺寸为 18 米乘 12 米；每个摊位需要 3 米 × 3 米区域外加 1 米通道；装饰花费 45 英镑，零食 60 英镑，奖品 30 英镑。突出数字、单位和条件是关键。将这些信息整理成表格，以免遗漏。
| Category | Detail | Value |
|---|---|---|
| Budget | Starting fund | £200 |
| Survey | Interested students (60% of 150) | 90 students |
| Venue | Hall area | 18 m × 12 m = 216 m² |
| Stall size | Including walkway | 4 m × 4 m footprint |
3. Ratio and Proportion in Budgeting | 预算中的比率与比例
The council decides to split the £200 budget between decorations, snacks and prizes in the ratio 3:4:1. Calculate each share by finding the total number of parts: 3 + 4 + 1 = 8. One part equals £200 ÷ 8 = £25. Therefore, decorations get £75, snacks get £100, and prizes get £25. However, the actual quotes are £45, £60 and £30, totalling £135. The remaining £65 can be used for emergency supplies. Understanding how ratios translate into monetary values ensures the spending stays within planned proportions.
学生会决定按 3:4:1 的比例将 200 英镑预算分配到装饰、零食和奖品上。通过求出总份数来计算每项的份额：3 + 4 + 1 = 8。一份等于 200 ÷ 8 = 25 英镑。因此，装饰得 75 英镑，零食得 100 英镑，奖品得 25 英镑。然而，实际报价为 45 英镑、60 英镑和 30 英镑，总计 135 英镑。剩余的 65 英镑可用于应急物资。理解比率如何转化为金额，可确保支出保持在计划比例之内。
4. Percentage Increases and Decreases | 百分比的增减
The team estimated 90 attendees from the survey, but they want to allow for a 20% increase in walk-in visitors. Increased attendance = 90 × 1.2 = 108. Tickets are priced at £4.00, but a promotional 10% discount is offered for buying in advance. The discounted price is £4.00 × 0.9 = £3.60. If 45 of the attendees buy early-bird tickets and the rest pay full price, the total ticket revenue = (45 × £3.60) + (63 × £4.00) = £162 + £252 = £414. These nested percentage calculations mirror the multi-step questions common in OCR papers.
团队根据调查预计有 90 名参与者，但他们想预留 20% 的临时到场人数增幅。增加后的人数 = 90 × 1.2 = 108。票价为 4.00 英镑，但提前购票可享受 10% 的促销折扣。折扣后价格为 4.00 × 0.9 = 3.60 英镑。若有 45 人购买早鸟票，其余人支付全价，则门票总收入 = (45 × 3.60) + (63 × 4.00) = 162 + 252 = 414 英镑。这些嵌套的百分比计算反映了 OCR 试卷中常见的多步问题。
5. Interpreting Statistical Data | 解读统计数据
The survey also recorded students’ favourite stall types: games 40%, food 35%, crafts 25%. A follow-up question asked how much they would spend inside the fair; the mean was £2.50 with a range of £4.00. The council must decide how many of each stall to commission. Using the percentages, for 108 visitors, expect about 43 for games, 38 for food, and 27 for crafts, but rounding sensibly gives 40, 40, 28 to fit stall capacities. The mean spend helps predict additional income: 108 × £2.50 = £270 from in-fair purchases. This reinforces the use of averages and proportions.
调查还记录了学生们最喜欢的摊位类型：游戏 40%，美食 35%，手工 25%。后续问题询问他们在义卖会内的消费额；平均值为 2.50 英镑，全距为 4.00 英镑。学生会必须决定每种摊位设置多少个。根据百分比，对于 108 名参与者，预计游戏摊位约 43 人，美食约 38 人，手工约 27 人，但明智地取整为 40、40、28 以匹配摊位容量。平均消费额有助于预测额外收入：108 × 2.50 = 270 英镑来自场内消费。这强化了平均数和比例的使用。
6. Using Graphs and Charts | 使用图表
Presenting the survey data visually aids decision-making. A pie chart for stall preferences uses angles: games 40% = 0.4 × 360° = 144°, food 35% = 126°, crafts 25% = 90°. A bar chart comparing expected numbers per stall with capacity limits (max 50 per game stall, 45 per food stall) quickly reveals if reallocation is needed. You may also plot a line graph showing how ticket revenue changes with price, using the formula revenue = number of attendees × ticket price. For instance, if price increases to £5 and attendance drops by 30%, the new revenue = (108 × 0.7) × £5 = 75.6 × £5 ≈ £378, lower than the original scenario. Graphs highlight such trade-offs clearly.
将调查数据可视化有助于决策。摊位偏好的饼图使用角度：游戏 40% = 0.4 × 360° = 144°，美食 35% = 126°，手工 25% = 90°。比较每个摊位预期人数与容量限制（每个游戏摊位最多 50 人，美食摊位 45 人）的条形图，可以迅速揭示是否需要重新分配。你还可以绘制折线图，显示门票收入如何随价格变化，使用公式 收入 = 参加人数 × 票价。例如，若价格涨至 5 英镑，参与人数下降 30%，则新收入 = (108 × 0.7) × 5 = 75.6 × 5 ≈ 378 英镑，低于原方案。图表能清晰展示这种权衡。
7. Geometry in Real-life Design | 实际设计中的几何
To fit the stalls into the hall, students must draw a scale plan. Using a scale of 1 cm to 2 m, the hall becomes 9 cm by 6 cm on paper. Each stall with its walkway is a 2 cm × 2 cm square. The available hall area is 216 m², and each stall requires 16 m² (4 m × 4 m). Maximum possible stalls = 216 ÷ 16 = 13.5, so up to 13 stalls could fit if no other space is needed. However, an entrance and stage consume 5 m × 4 m = 20 m². The remaining area is 196 m², allowing 196 ÷ 16 = 12.25, so 12 stalls practically. Geometry and area calculations directly impact the layout.
要将摊位安置在礼堂内，学生们必须绘制比例平面图。采用 1 厘米比 2 米的比例尺，礼堂在图纸上变成 9 厘米 × 6 厘米。每个摊位连同通道是一个 2 厘米 × 2 厘米的正方形。礼堂可用面积为 216 平方米，每个摊位占地 16 平方米（4 米 × 4 米）。最多可容纳摊位 = 216 ÷ 16 = 13.5，若不需其他空间，最多可放 13 个。然而，入口和舞台占用了 5 米 × 4 米 = 20 平方米。剩余面积为 196 平方米，可容纳 196 ÷ 16 = 12.25，因此实际可放 12 个摊位。几何与面积计算直接影响布局。
Total area = length × width = 18 × 12 = 216 m²
8. Solving Equations in Context | 情境中的方程求解
The council aims for a total profit of at least £500 after covering the £200 budget. If total revenue from tickets and in-fair sales is £414 + £270 = £684, then profit = £684 – £200 = £484, falling £16 short. To reach the target, they could increase ticket price or sell more snacks. Let the new ticket price be p pounds; if attendance stays at 108 and in-fair spend unchanged, the equation is 108p + 270 – 200 ≥ 500. Simplify: 108p + 70 ≥ 500 → 108p ≥ 430 → p ≥ 430 ÷ 108 ≈ 3.98. So a ticket price of £4.00 already meets the target approximately, but rounding and contingency suggest rounding up to £4.20 for a safe margin. Equation solving transforms a business goal into a precise mathematical condition.
学生会目标是在覆盖 200 英镑预算后，总利润至少达到 500 英镑。如果门票和场内销售总收入为 414 + 270 = 684 英镑，则利润 = 684 – 200 = 484 英镑，还差 16 英镑。为达成目标，他们可以提高票价或增加零食销量。假设新票价为 p 镑；若参与人数保持 108 且场内消费不变，方程为 108p + 270 – 200 ≥ 500。简化：108p + 70 ≥ 500 → 108p ≥ 430 → p ≥ 430 ÷ 108 ≈ 3.98。因此 4.00 英镑的票价已大致达到目标，但考虑到取整和应急，可将票价上调至 4.20 英镑以留出安全余地。方程求解将商业目标转化为精确的数学条件。
9. Making Predictions | 做出预测
Using the linear model, if the council runs the fair twice a year with 5% annual growth in attendance, the prediction for the second fair of next year becomes 108 × 1.05² = 108 × 1.1025 ≈ 119. This uses compound growth. They can also predict profit sensitivity by changing variables: what if snack costs rise 15%? Snack budget from £60 to £69; new total costs = £45 + £69 + £30 = £144; profit = £684 – £144 = £540, still exceeding target. Forecasting helps justify decisions and manage risks.
利用线性模型，如果每年举办两次义卖会，且参与人数年增长率为 5%，则预测明年第二次义卖会的参与人数为 108 × 1.05² = 108 × 1.1025 ≈ 119。这使用了复合增长。他们还可以通过更改变量来预测利润敏感性：假设零食成本上涨 15%？零食预算从 60 英镑涨至 69 英镑；新的总成本 = 45 + 69 + 30 = 144 英镑；利润 = 684 – 144 = 540 英镑，仍超过目标。预测有助于论证决策并管理风险。
10. Checking and Presenting Findings | 检查与展示发现
All calculations must be checked for order of magnitude and unit consistency. The final report includes an executive summary with key numbers (budget, expected profit 484–540, ticket price £4.00–£4.20), a scaled plan, a pie chart of stall preferences, a revenue sensitivity table, and recommendations. The mathematical reasoning is explained in simple language, linking each step to the initial brief. This mirrors the OCR expectation that students can communicate their process clearly, using appropriate mathematical vocabulary.
所有的计算都必须检查数量级和单位一致性。最终报告包括一份执行摘要，列出关键数字（预算、预期利润 484–540、票价 £4.00–£4.20），一份比例平面图，摊位偏好的饼图，一张收入敏感性表格以及建议。数学推理用简单的语言解释，并将每一步与初始需求联系起来。这与 OCR 的要求一致，即学生能够使用恰当的数学词汇清晰地交流其解题过程。
| Scenario | Ticket Price | Attendance | Profit (£) |
|---|---|---|---|
| Base case | £4.00 | 108 | 484 |
| Higher price | £4.20 | 108 | 505.60 |
| Attendance drop 30% | £5.00 | 75.6 | 378 |
11. Error Analysis and Rounding | 误差分析与取整
When rounding attendee numbers from percentages, you must handle remainders sensibly. 43.2 visitors to games stalls means 43 or 44 people; either choice affects queue lengths and stall staff allocation. In geometry, the 12.25 stalls result cannot be used as is—you must truncate to 12 whole stalls. Overestimating capacity could cause safety issues. Discussing these limitations shows evaluation skills. Where exact values are used in equations, maintain decimal answers until the final step, then round appropriately (p ≥ £3.981, but price must be in pence, so £4.00 is practical). This awareness is part of the mathematical maturity assessed in OCR problem-solving tasks.
当根据百分比对参与人数进行取整时，你必须合理处理余数。43.2 人去游戏摊位意味着 43 或 44 人；任一种选择都会影响排队长度和摊位人员配置。在几何中，12.25 个摊位的结果不能直接使用——你必须向下取整为 12 个整摊位。高估容量可能导致安全问题。讨论这些局限性展现了评估技巧。在方程中使用精确值时，保持小数答案直到最后一步，然后恰当取整（p ≥ 3.981，但价格必须以便士计，因此 £4.00 是实用的）。这种意识是 OCR 问题解决任务中评估的数学成熟度的一部分。
12. Cross-curricular Links and Real-life Application | 跨学科联系与实际应用
This case study blends mathematics with business studies, design technology, and citizenship. The budgeting and profit calculations require the same skills an entrepreneur uses; the scaled drawings tie into D&T; and the survey analysis supports democratic decision-making. OCR Mathematics encourages students to see these connections, preparing them for further study and employment. Repeating similar exercises with different scenarios—a sports tournament, a school trip, an environmental project—builds fluency in transferring mathematical skill to new situations.
该案例分析将数学与商业研究、设计技术及公民教育融为一体。预算与利润计算需要与创业者相同的技能；比例图与设计与技术课程相关；调查分析则为民主决策提供支持。OCR 数学鼓励学生看见这些关联，为他们的后续学习和就业做好准备。用不同的场景重复类似的练习——一场体育锦标赛、一次学校旅行、一个环保项目——可以培养将数学技能迁移至新情境的流畅性。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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