Case Study in Mathematics: Real-World Problem Solving | 数学案例分析实战演练

📚 Case Study in Mathematics: Real-World Problem Solving | 数学案例分析实战演练

In Year 7 SQA Mathematics, case studies help you apply your maths skills to real-life situations. This article will take you through a complete project—designing a school garden. You will practise measurement, area, budgeting, data handling, and problem-solving. Follow each step carefully and see how maths is used in everyday life.

在七年级 SQA 数学中，案例分析帮助你运用数学技能解决实际问题。本文将带你完成一个完整的项目——设计一个学校花园。你将练习测量、面积计算、预算制定、数据处理和问题解决。仔细跟随每一步，看看数学如何在日常生活中使用。

1. Understanding the Scenario | 理解场景

The school wants to create a rectangular garden of length 12 m and width 8 m. The garden will include a lawn, two flower beds, and a path. You need to plan the layout, calculate materials, survey classmates’ favourite flowers, and stay within a budget of £150.

学校想建一个长12米、宽8米的长方形花园。花园将包括一块草坪、两个花坛和一条小路。你需要规划布局、计算材料、调查同学们最喜欢的花，并且控制在150英镑的预算内。

Your task is broken into several stages. First, sketch a plan. Then handle measurements, costs, and data. Finally, present your findings.

你的任务分为几个阶段。首先，画一个平面图。然后处理测量、成本和数据。最后，展示你的发现。

2. Measurements and Units | 测量与单位

All measurements must be in consistent units. The garden dimensions are given in metres. For costs, you may need to convert centimetres to metres. Remember: 1 m = 100 cm, 1 m² = 10,000 cm².

所有测量值必须使用一致的单位。花园尺寸以米为单位。为了计算成本，你可能需要把厘米换算成米。记住：1 米 = 100 厘米，1 平方米 = 10,000 平方厘米。

Example: A flower bed edge is 50 cm wide. In metres, that is 50 ÷ 100 = 0.5 m.

示例：一个花坛边缘宽50厘米。换算成米，就是 50 ÷ 100 = 0.5 米。

When working with area, converting correctly is crucial. If a seed packet covers 2,000 cm², how many square metres is that? 2,000 ÷ 10,000 = 0.2 m².

计算面积时，正确换算至关重要。如果一包种子能覆盖2,000平方厘米，那是多少平方米？2,000 ÷ 10,000 = 0.2 平方米。

3. Calculating Perimeter and Area | 计算周长与面积

The garden is rectangular: length l = 12 m, width w = 8 m.

花园是长方形的：长 l = 12 米，宽 w = 8 米。

Perimeter = 2 × (l + w) = 2 × (12 + 8) = 2 × 20 = 40 m

Area = l × w = 12 × 8 = 96 m²

The total area is 96 m². You decide the lawn occupies 60% of the garden. The two flower beds together take up 25%, and the path the remaining 15%.

总面积是96平方米。你决定草坪占花园的60%，两个花坛共占25%，小路占剩下的15%。

Lawn area = 60% of 96 = 0.6 × 96 = 57.6 m²

草坪面积 = 96 的 60% = 0.6 × 96 = 57.6 平方米
Flower beds area = 25% of 96 = 0.25 × 96 = 24 m²

花坛面积 = 96 的 25% = 0.25 × 96 = 24 平方米
Path area = 15% of 96 = 0.15 × 96 = 14.4 m²

小路面积 = 96 的 15% = 0.15 × 96 = 14.4 平方米

Check: 57.6 + 24 + 14.4 = 96. ✓

检查：57.6 + 24 + 14.4 = 96。✓

4. Scale Drawing and Proportion | 比例图与比例

You draw a scale plan of the garden on A4 paper. A suitable scale is 1 cm : 1 m. So 12 m becomes 12 cm, and 8 m becomes 8 cm on paper.

你在A4纸上画花园的按比例缩小的平面图。合适的比例是 1厘米代表 1米。这样12米在纸上画12厘米，8米画8厘米。

The flower beds are identical squares, each of side 3.46 m (since 2 × 3.46² ≈ 24 m²). With the scale, each square side is 3.46 cm. You mark the path as a 1.2 m strip along one long side.

两个花坛是相同的正方形，边长均为3.46米（因为 2 × 3.46² ≈ 24 平方米）。按照比例，每个正方形的边长是3.46厘米。你将小路标记为沿着一条长边、宽1.2米的条形。

Check path area: 1.2 m × 12 m = 14.4 m² — correct.

检查小路面积：1.2 米 × 12 米 = 14.4 平方米——正确。

5. Budget and Material Costs | 预算与材料成本

You have £150. Costs are as follows:

你有150英镑。费用如下：

Item / 项目	Cost / 价格
Grass seed (1 kg covers 20 m², £5 per kg) / 草籽（1公斤覆盖20平方米，每公斤5英镑）	£5 per kg
Flower seeds (packet covers 2 m², £2 per packet) / 花种（每包覆盖2平方米，每包2英镑）	£2 per packet
Path gravel (1 bag covers 3 m², £6 per bag) / 小路碎石（每袋覆盖3平方米，每袋6英镑）	£6 per bag
Tools and misc. / 工具及其他	£30 fixed

Lawn seed needed: 57.6 m² ÷ 20 m²/kg = 2.88 kg. Round up to 3 kg. Cost = 3 × £5 = £15.

所需草籽：57.6 平方米 ÷ 20 平方米/公斤 = 2.88 公斤。向上取整到3公斤。成本 = 3 × 5 = 15英镑。

Flower seed: 24 m² ÷ 2 m²/packet = 12 packets. Cost = 12 × £2 = £24.

花种：24 平方米 ÷ 2 平方米/包 = 12包。成本 = 12 × 2 = 24英镑。

Gravel: 14.4 m² ÷ 3 m²/bag = 4.8 bags. Round up to 5 bags. Cost = 5 × £6 = £30.

碎石：14.4 平方米 ÷ 3 平方米/袋 = 4.8袋。向上取整到5袋。成本 = 5 × 6 = 30英镑。

Total cost = £15 + £24 + £30 + £30 = £99. You are £51 under budget. That money can be used for extra flowers or a bench.

总成本 = 15 + 24 + 30 + 30 = 99英镑。你还剩51英镑预算。这笔钱可以用来增加花卉或添置一条长凳。

6. Data Collection and Averages | 数据收集与平均数

To choose flower colours, you survey 30 classmates. They vote for their favourite: Red (R), Blue (B), Yellow (Y), Pink (P). Results: R: 8, B: 10, Y: 5, P: 7.

为了选择花的颜色，你调查了30位同学。他们投票选出最喜欢的颜色：红(R)、蓝(B)、黄(Y)、粉(P)。结果：R: 8, B: 10, Y: 5, P: 7。

The mode (most popular) is Blue with 10 votes.

众数（最受欢迎）是蓝色，10票。

Calculate the mean number of votes per colour: (8+10+5+7) ÷ 4 = 30 ÷ 4 = 7.5. The median order is 5, 7, 8, 10, so median = (7+8)/2 = 7.5.

计算每种颜色平均得票数：(8+10+5+7) ÷ 4 = 30 ÷ 4 = 7.5。按顺序排列是5, 7, 8, 10，所以中位数 = (7+8)/2 = 7.5。

You decide to plant flowers in proportion to votes: Blue 10/30 = ⅓, Red 8/30 = 4/15, Pink 7/30, Yellow 5/30 = ⅙. This affects how many packets of each colour to buy.

你决定按得票比例种植花卉：蓝色 10/30 = ⅓，红色 8/30 = 4/15，粉 7/30，黄 5/30 = ⅙。这会影响每种颜色需要买多少包花种。

7. Displaying Data with Charts | 用图表展示数据

A bar chart helps visualise the votes. Label x-axis with colours, y-axis with frequency. Draw bars of heights 8, 10, 5, 7.

条形图有助于直观显示投票情况。x轴标颜色，y轴标次数。画出高度为8、10、5、7的条形。

For a pie chart, calculate angles: total 30 votes. Each vote = 360° ÷ 30 = 12°. Blue: 10 × 12° = 120°, Red: 8 × 12° = 96°, Pink: 7 × 12° = 84°, Yellow: 5 × 12° = 60°.

画饼图时，计算角度：总共30票。每票 = 360° ÷ 30 = 12°。蓝色：10 × 12° = 120°，红色：8 × 12° = 96°，粉色：7 × 12° = 84°，黄色：5 × 12° = 60°。

Check: 120+96+84+60=360°. Use a protractor to draw the sectors.

检查：120+96+84+60=360°。用量角器画出扇形。

8. Fractions, Decimals and Percentages | 分数、小数和百分比

Converting between forms is essential for planning. Lawn fraction: 57.6/96 = 576/960 = 3/5 (simplify by ÷192: 57.6/96 = 0.6 exactly, so 3/5). Decimal: 0.6, percentage: 60%.

在规划中，形式间的转换必不可少。草坪面积占比：57.6/96 = 576/960 = 3/5（除以192化简：57.6/96 = 0.6，所以是3/5）。小数：0.6，百分比：60%。

Flower beds: 24/96 = 1/4 = 0.25 = 25%. Path: 14.4/96 = 144/960 = 3/20 = 0.15 = 15%.

花坛：24/96 = 1/4 = 0.25 = 25%。小路：14.4/96 = 144/960 = 3/20 = 0.15 = 15%。

When ordering seeds, you need ⅓ of the 12 flower packets for blue, i.e. 12 × ⅓ = 4 packets. Red: 4/15 of 12 = 3.2 packets — you must round up/down sensibly, perhaps 3 packets red and adjust other colours.

订购花种时，你需要12包中的⅓种蓝色，即 12 × ⅓ = 4包。红色：12 的 4/15 = 3.2包——你需要合理取整，也许3包红色，然后调整其他颜色。

9. Time Planning and Schedules | 时间规划与日程

The project runs over two weekends. Tasks and estimated times:

项目在两个周末内进行。任务和预计时间：

Clear site: 3 hours / 清理场地：3小时
Mark layout: 1 hour / 标记布局：1小时
Lay path gravel: 4 hours / 铺碎石小路：4小时
Sow lawn seed: 2 hours / 撒草籽：2小时
Plant flowers: 3 hours / 种花：3小时

Total time = 3+1+4+2+3 = 13 hours. If 4 volunteers work each day for 5 hours, total labour available over 2 days = 4 × 5 × 2 = 40 person-hours. Easily sufficient.

总时间 = 3+1+4+2+3 = 13小时。如果每天有4名志愿者工作5小时，两天可用总工时 = 4 × 5 × 2 = 40 人时。绰绰有余。

You also calculate that the lawn should be sown by the end of the first weekend, so it has time to germinate.

你还计算出草坪应该在第一个周末结束前播种完毕，这样有时间发芽。

10. Review and Reflect | 回顾与反思

You have used a wide range of Year 7 maths skills: measurement conversion, area and perimeter, ratio and proportion, fractions/decimals/percentages, data handling, averages, chart drawing, and budgeting.

你运用了七年级数学的许多技能：单位换算、面积与周长、比和比例、分数/小数/百分比、数据处理、平均数、图表绘制以及预算。

Check if any errors: The path area was 14.4 m², which is a strip 1.2 m × 12 m. The remaining width for lawn and flower beds is 8 − 1.2 = 6.8 m. Can you fit the lawn and two squares? Lawn area 57.6 m² requires length 57.6/6.8 ≈ 8.47 m—yes, fits within 12 m.

检查是否存在错误：小路面积14.4平方米，即1.2米 × 12米的长条。草坪和花坛剩下的宽度是 8 − 1.2 = 6.8米。你能放下草坪和两个正方形吗？草坪面积57.6平方米需要长度 57.6/6.8 ≈ 8.47米——可以的，12米长度足够。

You might present this case study as a written report and a poster. The project shows how maths helps make real decisions.

你可以将这个案例分析写成书面报告和一张海报。这个项目展示了数学如何帮助做出实际决定。

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Case Study in Mathematics: Real-World Problem Solving | 数学案例分析实战演练