The School Garden Project: Case Study in Action | 校园花园项目案例实战演练

📚 The School Garden Project: Case Study in Action | 校园花园项目案例实战演练

In this case study, you will step into the role of a project planner for a new school garden. You will use mathematics to design the layout, calculate materials, manage a budget, and present your findings. This exercise brings together skills from number, algebra, geometry, ratio, and statistics – just like in a real-world scenario. Let’s get started!

在这个案例研究中，你将担任新校园花园的项目规划师。你将运用数学来设计布局、计算材料、管理预算并展示你的发现。这个练习整合了数字、代数、几何、比例和统计等技能，就像真实场景一样。让我们开始吧！

1. Understanding the Problem | 理解问题

The school has allocated a rectangular plot of land measuring 15 metres in length and 10 metres in width for the garden. Your task is to design a garden that includes a vegetable patch, a flower border, a small pond, and a path. You must work within a budget of £450 and ensure the design is practical for students to maintain.

学校分配了一块长 15 米、宽 10 米的长方形地块用于花园。你的任务是设计一个包含菜地、花坛、小池塘和小径的花园。你必须在 450 英镑的预算内完成，并确保设计方便学生维护。

Before starting any calculations, list the key information given: the dimensions of the plot, the budget, and the four main features. Identify what needs to be found: areas, material quantities, costs, and a final plan drawn to scale.

在开始计算之前，列出给出的关键信息：地块尺寸、预算和四个主要元素。确定需要求解的内容：面积、材料数量、成本以及按比例绘制的最终平面图。

2. Collecting Data and Measurements | 收集数据和测量

Measure the total area of the plot first. Since it is a rectangle, area = length × width. So, total area = 15 m × 10 m = 150 m². This is your working space.

首先测量地块的总面积。因为它是长方形，面积 = 长 × 宽。所以，总面积 = 15 米 × 10 米 = 150 平方米。这就是你的工作空间。

You decide to allocate spaces as follows: vegetable patch 40% of total area, flower border 25%, pond 10%, and the remaining area for the path. Record these percentages in a table.

你决定按以下方式分配空间：菜地占总面积的 40%，花坛占 25%，池塘占 10%，剩余面积作为小径。将这些百分比记录在表格中。

Feature	Percentage of Total Area	Area (m²)
Vegetable Patch	40%	60
Flower Border	25%	37.5
Pond	10%	15
Path	25%	37.5

Check: 60 + 37.5 + 15 + 37.5 = 150 m², which matches the total area. Good data collection ensures no mistakes later.

检查：60 + 37.5 + 15 + 37.5 = 150 平方米，与总面积相符。良好的数据收集确保后期不会出错。

3. Area Calculations and Unit Conversions | 面积计算与单位换算

The vegetable patch is to be divided into square plots of side length 1 m. How many such squares fit? Since the area is 60 m², you need 60 squares. However, you plan to use wooden planks to border each square. Each square has a perimeter of 4 m, but shared edges reduce the total wood needed. For simplicity, you decide to border the entire patch as one rectangle of dimensions 10 m by 6 m (since 10×6=60). The perimeter = 2 × (10+6) = 32 m.

菜地将被划分为边长 1 米的正方形小块。能放下多少个这样的小块？因为面积为 60 平方米，所以需要 60 个正方形。然而，你计划用木板给每个正方形围边。每个正方形周长 4 米，但共用边会减少所需木材总量。为简化，你决定将整个菜地作为一个 10 米乘 6 米的长方形来围边（因为 10×6=60）。周长 = 2 × (10+6) = 32 米。

Wood planks are sold in lengths of 2 m at £3.50 per plank. Number of planks needed = 32 ÷ 2 = 16 planks. Cost = 16 × £3.50 = £56. Always include units when recording.

木板以 2 米长出售，每块 3.50 英镑。所需木板数量 = 32 ÷ 2 = 16 块。成本 = 16 × 3.50 英镑 = 56 英镑。记录时务必包含单位。

For the flower border, you choose a curved design, so you approximate the length of edging needed as 1.2 times the perimeter of a square of equivalent area. A square of area 37.5 m² has side √37.5 ≈ 6.12 m, perimeter ≈ 24.5 m. Edging ≈ 24.5 × 1.2 = 29.4 m. Round up to 30 m to allow for wastage.

对于花坛，你选择曲线设计，因此将所需边缘材料的长度近似为等面积正方形周长的 1.2 倍。面积为 37.5 平方米的正方形边长为 √37.5 ≈ 6.12 米，周长 ≈ 24.5 米。边缘材料长度 ≈ 24.5 × 1.2 = 29.4 米。向上取整至 30 米，以预留损耗。

4. Ratio and Proportion in Design | 设计中的比例与比率

The pond is circular. Given its area of 15 m², the radius r can be found using A = πr². So r² = 15 ÷ π ≈ 15 ÷ 3.14 = 4.78, thus r ≈ 2.19 m. You want a safety border of gravel around the pond with a width of 0.5 m. This increases the radius of the total pond-plus-border circle to 2.69 m.

池塘是圆形的。给定面积 15 平方米，半径 r 可由 A = πr² 求出。因此 r² = 15 ÷ π ≈ 15 ÷ 3.14 = 4.78，所以 r ≈ 2.19 米。你想在池塘周围设置一条 0.5 米宽的砾石安全边。这将池塘加边缘的总圆半径增加到 2.69 米。

The ratio of the pond area to the total plot area is 15:150, which simplifies to 1:10. This means the pond takes up one-tenth of the garden. Understanding ratios helps you visualise the space distribution.

池塘面积与地块总面积的比是 15:150，简化为 1:10。这意味着池塘占据了花园的十分之一。理解比率有助于你可视化空间分布。

Gravel is sold in bags covering 2 m² each at £4.80 per bag. The border area = π × (2.69² – 2.19²) = π × (7.2361 – 4.7961) ≈ 3.14 × 2.44 = 7.66 m². You need 7.66 ÷ 2 ≈ 3.83, so buy 4 bags. Cost = 4 × £4.80 = £19.20.

砾石按袋出售，每袋覆盖 2 平方米，每袋 4.80 英镑。边缘面积 = π × (2.69² – 2.19²) = π × (7.2361 – 4.7961) ≈ 3.14 × 2.44 = 7.66 平方米。你需要 7.66 ÷ 2 ≈ 3.83，因此购买 4 袋。成本 = 4 × 4.80 英镑 = 19.20 英镑。

5. Budgeting with Decimals and Percentages | 小数与百分比预算

Now track your spending. You have already allocated £56 for the vegetable border and £19.20 for gravel. The path requires paving slabs. The path area is 37.5 m². Slabs come in 0.5 m by 0.5 m squares, so each slab area = 0.25 m². Number of slabs = 37.5 ÷ 0.25 = 150 slabs.

现在追踪你的支出。你已经分配了 56 英镑用于菜地围边，19.20 英镑用于砾石。小径需要铺路板。小径面积为 37.5 平方米。路板是 0.5 米 × 0.5 米的正方形，因此每块面积 = 0.25 平方米。路板数量 = 37.5 ÷ 0.25 = 150 块。

Each slab costs £1.85. Cost = 150 × £1.85 = £277.50. To find the remaining budget, calculate total spent so far: £56 + £19.20 + £277.50 = £352.70. Budget left = £450 – £352.70 = £97.30.

每块路板 1.85 英镑。成本 = 150 × 1.85 英镑 = 277.50 英镑。要计算剩余预算，算出目前总支出：56 英镑 + 19.20 英镑 + 277.50 英镑 = 352.70 英镑。剩余预算 = 450 英镑 – 352.70 英镑 = 97.30 英镑。

You want to spend 15% of the total budget on plants and seeds. 15% of £450 = 0.15 × 450 = £67.50. Check if you have enough: £97.30 > £67.50, so yes, and you can even add a small bench with the surplus.

你希望将总预算的 15% 用于植物和种子。450 英镑的 15% = 0.15 × 450 = 67.50 英镑。检查是否足够：97.30 英镑 > 67.50 英镑，所以足够，你甚至可以用剩余的钱添置一个小长椅。

6. Statistical Representation of the Garden Layout | 花园布局的统计表示

To present the area distribution clearly, draw a pie chart. The vegetable patch corresponds to 40% which is 40/100 × 360° = 144°. The flower border: 25% → 90°. The pond: 10% → 36°. The path: 25% → 90°. Check: 144+90+36+90 = 360°.

为了清晰地展示面积分配，绘制一个饼状图。菜地对应 40%，即 40/100 × 360° = 144°。花坛：25% → 90°。池塘：10% → 36°。小径：25% → 90°。检验：144+90+36+90 = 360°。

You can also represent the area data using a bar chart where the x-axis lists the features and the y-axis shows the area in m². This helps compare the sizes visually. The bar for vegetable patch would be 60 units high, flower border 37.5, pond 15, and path 37.5.

你还可以使用条形图来表示面积数据，其中 x 轴列出各个元素，y 轴显示面积（平方米）。这有助于直观地比较大小。菜地的条形高度为 60，花坛为 37.5，池塘为 15，小径为 37.5。

Calculate the mean area of the four features: (60 + 37.5 + 15 + 37.5) ÷ 4 = 150 ÷ 4 = 37.5 m². The median: arrange areas in order: 15, 37.5, 37.5, 60. Median = (37.5+37.5)/2 = 37.5 m². The mode is 37.5 m² (appears twice). These statistics summarise the layout.

计算这四个元素的平均面积：(60 + 37.5 + 15 + 37.5) ÷ 4 = 150 ÷ 4 = 37.5 平方米。中位数：将面积按顺序排列：15, 37.5, 37.5, 60。中位数 = (37.5+37.5)/2 = 37.5 平方米。众数是 37.5 平方米（出现两次）。这些统计量概括了布局。

7. Algebraic Expressions for Cost Analysis | 成本分析的代数表达

You can create algebraic formulas to quickly adjust costs if prices change. Let the number of wood planks = w, cost per plank = c. Total wood cost = w × c. If you later extend the plot, the perimeter P = 2(l + w) where l and w are lengths of the patch sides. Then planks needed = P/2.

你可以创建代数公式，以便在价格变化时快速调整成本。设木板数量 = w，每块成本 = c。木材总成本 = w × c。如果你后来延长地块，周长 P = 2(l + w)，其中 l 和 w 是菜地边长。然后所需木板量 = P/2。

For the paving slabs, if the slab side length is s metres, then area per slab = s². For a path area A, number of slabs n = A ÷ s². If s = 0.5, then n = A ÷ 0.25 = 4A. So if the path area changes to 40 m², you’d need 160 slabs instantly without recalculating from scratch.

对于铺路板，如果路板边长为 s 米，则每块面积 = s²。对于小径面积 A，路板数量 n = A ÷ s²。如果 s = 0.5，那么 n = A ÷ 0.25 = 4A。因此，如果小径面积变为 40 平方米，你立即知道需要 160 块，无需从头计算。

The total budget formula: B = cost_veg + cost_flower + cost_pond + cost_path + cost_plants. Substitute variables to see how changes in one feature affect the total. This builds early algebraic reasoning.

总预算公式：B = 菜地成本 + 花坛成本 + 池塘成本 + 小径成本 + 植物成本。代入变量以观察某个元素的变化如何影响总额。这培养了初步的代数推理能力。

8. Final Decision and Communication | 最终决策与沟通

After completing all calculations, create a scale drawing of the garden on A4 paper. Use a scale of 1:50, so 1 cm represents 0.5 m. The plot dimensions become 30 cm by 20 cm. Mark each feature with its dimensions and label the materials needed.

完成所有计算后，在 A4 纸上按比例绘制花园图。使用 1:50 的比例，即 1 厘米代表 0.5 米。地块尺寸变为 30 厘米乘 20 厘米。标出每个元素的尺寸并注明所需材料。

Write a short report summarising the design, the mathematical methods used, the budget breakdown, and any assumptions made (e.g., rounding up quantities, approximating π as 3.14). Justify why your design is within budget and suitable for students.

写一份简短报告，总结设计、所用的数学方法、预算明细以及所做的假设（例如，数量向上取整，π 取 3.14 近似）。说明你的设计为何符合预算且适合学生。

Finally, reflect on what you learned. Which skill was most challenging – area calculation, percentage, or drawing to scale? Case studies like this show how mathematics is not just about numbers on a page, but a tool for solving real problems.

最后，反思你学到了什么。哪项技能最具挑战性——面积计算、百分比还是按比例绘图？像这样的案例研究表明，数学不仅仅是纸上的数字，更是解决实际问题的工具。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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