📚 Year 9 AQA Mathematics: Case Study Practice in Action | Year 9 AQA 数学：案例分析实战演练
Case study questions in the AQA Year 9 syllabus require you to apply multiple mathematical skills to a single real‑world scenario. This article walks you through a complete energy‑consumption investigation, showing how to organise data, calculate statistics, use algebra, apply geometry and reason with probability — just as you would in an exam.
AQA 九年级大纲中的案例分析题要求你将多种数学技能应用于一个真实的场景。本文将带你完整地进行一次家庭能源消耗调查，展示如何整理数据、计算统计量、运用代数、应用几何以及进行概率推理——正如你在考试中需要做的那样。
1. Case Study Overview: Energy Consumption at Home | 案例概述：家庭能源消耗
Sam wants to reduce his family's electricity bill and evaluate whether solar panels would be a good investment. He records the monthly electricity usage and cost for one year. The table below shows his data.
Sam 想要降低家里的电费开支，并评估安装太阳能电池板是否划算。他记录了一整年每个月的用电量和费用。下表是他的数据。
| Month / 月份 | Electricity Usage (kWh) / 用电量 (kWh) | Cost (£) / 费用 (£) |
|---|---|---|
| Jan | 320 | 49.60 |
| Feb | 280 | 44.40 |
| Mar | 260 | 41.80 |
| Apr | 240 | 39.20 |
| May | 220 | 36.60 |
| Jun | 200 | 34.00 |
| Jul | 210 | 35.30 |
| Aug | 230 | 37.90 |
| Sep | 250 | 40.50 |
| Oct | 270 | 43.10 |
| Nov | 300 | 47.00 |
| Dec | 330 | 50.90 |
The cost is calculated using a fixed charge plus a rate per kWh. We will explore this relationship later. Throughout the case study you will be asked to interpret, analyse and make decisions using only the mathematical tools expected at Year 9.
费用按固定服务费加上每度电的单价计算，我们稍后会探究这一关系。在整个案例分析中，你需要仅使用九年级所学的数学工具来解释、分析并作出决策。
2. Collecting Data: Monthly Electricity Usage Table | 数据收集：月度用电量表
The raw data in the table allows us to see the monthly variation. Before performing any calculations, it is good practice to check that the data is reasonable and complete. Notice that usage is highest in winter and lowest in summer, which matches what we would expect.
表格中的原始数据使我们能够看到每月的波动。在进行任何计算之前，检查数据是否合理和完整是一个好习惯。注意冬季用电量最高、夏季最低，这符合我们的预期。
AQA exam questions often ask you to comment on any trends you notice. You could describe the pattern: 'Electricity usage decreases from January to June, then increases again from July to December.'
AQA 考试题目常常要求你评论所发现的趋势。你可以描述这一模式：'用电量从一月到六月下降，然后从七月到十二月再次上升。'
3. Frequency Table for Grouped Data | 分组数据频数表
To summarise the usage more compactly, Sam groups the data into intervals of 40 kWh. He creates the following frequency table.
为了更简洁地概括用电量，Sam 将数据按 40 kWh 的区间进行分组，并制作了下面的频数表。
| Electricity usage, u (kWh) / 用电量 u (kWh) | Tally / 划记 | Frequency / 频数 |
|---|---|---|
| 200 ≤ u < 240 | |||| | 4 |
| 240 ≤ u < 280 | |||| | 4 |
| 280 ≤ u < 320 | || | 2 |
| 320 ≤ u < 360 | || | 2 |
Grouping the data sacrifices some detail but makes it easier to identify which usage bands are most common. In an exam you may be asked to construct a similar table from a list of numbers.
分组会牺牲一些细节，但可以更容易地看出哪些用电区间最常见。考试中可能要求你根据一串数字构建类似的表格。
4. Drawing a Bar Chart | 绘制条形图
Using the frequency table, you can draw a bar chart where each bar represents a class interval. The height of each bar corresponds to the frequency. Remember that for grouped data the bars should touch, as the data is continuous.
利用频数表，你可以绘制条形图，每个条形代表一个组距。条形的高度对应频数。请记住，对于分组数据，条形应紧挨在一起，因为数据是连续的。
When plotting, label both axes clearly: the horizontal axis shows 'Electricity usage (kWh)' and the vertical axis shows 'Frequency'. Always give your chart a title, such as 'Monthly electricity usage over one year'.
绘制时，要清晰地标注两条轴：横轴显示 '用电量 (kWh)'，纵轴显示 '频数'。始终给图表加上标题，例如 '一年间每月用电量'。
5. Calculating the Mean | 计算平均数
To find the mean monthly electricity usage, add all 12 monthly values and divide by 12.
要计算平均每月用电量，将所有 12 个月的数值相加，然后除以 12。
Sum = 320 + 280 + 260 + 240 + 220 + 200 + 210 + 230 + 250 + 270 + 300 + 330 = 3110 kWh
总和 = 320 + 280 + 260 + 240 + 220 + 200 + 210 + 230 + 250 + 270 + 300 + 330 = 3110 kWh
Mean = 3110 ÷ 12 ≈ 259.2 kWh (to 1 d.p.)
平均数 = 3110 ÷ 12 ≈ 259.2 kWh (保留一位小数)
The mean gives us the typical monthly consumption. However, it is affected by the high winter values, so we should also look at other measures.
平均数给出了典型的月消耗量。但它受到冬季高数值的影响，因此我们还应考察其他度量指标。
6. Median and Range | 中位数与范围
First, list the usage figures in ascending order: 200, 210, 220, 230, 240, 250, 260, 270, 280, 300, 320, 330. Since there are 12 values (an even number), the median is the average of the 6th and 7th values.
首先，将用电量按升序排列：200, 210, 220, 230, 240, 250, 260, 270, 280, 300, 320, 330。因为有 12 个数值（偶数个），中位数是第 6 个和第 7 个数值的平均值。
Median = (250 + 260) ÷ 2 = 255 kWh
中位数 = (250 + 260) ÷ 2 = 255 kWh
uses around 255 kWh. The range is found by subtracting the smallest from the largest: 330 – 200 = 130 kWh, showing considerable variability across the year.
中位数受极端值影响较小，并证实典型月份用电量约为 255 kWh。范围通过最大值减最小值得到：330 – 200 = 130 kWh，表明全年存在较大波动。
7. Percentages and Proportional Reasoning | 百分比与比例推理
Sam learns that the electricity price will rise by 5% ... + 50.90 = 500.30. Divide by 12 to get the current mean cost: 500.30 ÷ 12 ≈ 41.69. A 5% increase makes the new mean cost 41.69 × 1.05 ≈ 43.78.
Sam 得知明年电价将上涨 5%。为了估算新的月费用，先从表格求出平均月费用。将费用栏相加：49.60 + 44.40 + ... + 50.90 = 500.30。除以 12 得到当前平均费用：500.30 ÷ 12 ≈ 41.69。上涨 5% 后，新的平均费用为 41.69 × 1.05 ≈ 43.78。
30 + 320 + 280 = 930 kWh. As a percentage of the 3110 kWh total: (930 ÷ 3110) × 100 ≈ 29.9%. This shows nearly 30% of electricity is used in the three coldest months.
你还可以问冬季用电量占全年总用量的百分比。以十二月到二月为例：330 + 320 + 280 = 930 kWh。占总量 3110 kWh 的百分比为：(930 ÷ 3110) × 100 ≈ 29.9%。这表明将近 30% 的电量是在最冷的三个月消耗的。
8. Algebraic Cost Model | 代数成本模型
From the data, the cost £ age in kWh. Check with January: 0.13 × 320 + 8 = 41.6 + 8 = 49.6, which matches the table.
从数据来看，费用 £C 似乎遵循一个线性规则：C = 0.13u + 8，其中 u 为用电量 (kWh)。用一月份验证：0.13 × 320 + 8 = 41.6 + 8 = 49.6，与表格相符。
Sam wants to keep his monthly bill strictly below £50. Set up an inequality and solve it.
Sam 希望每月电费严格控制在 £50 以下。列出一个不等式并求解。
0.13u + 8 < 50
0.13u < 42
u < 42 ÷ 0.13 ≈ 323.08
Therefore, Sam must use less than about 323 kWh in a month to keep the bill under £50. In practice, he would need to reduce consumption in the high‑usage winter months.
因此，Sam 必须使用少于约 323 kWh 才能使账单低于 £50。实际上，他需要在用电量高的冬季减少消耗。
9. Geometry: Solar Panel Area | 几何：太阳能电池板面积
Sam's roof is rectangular, measuring 8.5 m by 5.2 m. Each solar panel is a rectangle of 1.6 m by 1.0 m. Calculate the roof area and the area of one panel.
Sam 的屋顶是矩形，尺寸为 8.5 m × 5.2 m。每块太阳能板是 1.6 m × 1.0 m 的矩形。计算屋顶面积和一块板的面积。
Roof area = 8.5 × 5.2 = 44.2 m²
Panel area = 1.6 × 1.0 = 1.6 m²
ould be arranged on the roof. In reality, some space is needed for mounting, so Sam decides a layout of 20 panels is practical.
理论上最多可安装的板数（忽略间隙和朝向）= 44.2 ÷ 1.6 ≈ 27.6。因此最多可放置 27 块板。实际上，安装需要预留空间，因此 Sam 认为布置 20 块板是可行的。
erage of 1.2 kWh per day. Twenty panels would produce 20 × 1.2 = 24 kWh per day. Over a 30‑day month, that is 720 kWh — more than double the average monthly usage, but output varies with weather.
每块板平均每天发电 1.2 kWh。20 块板每天可发电 20 × 1.2 = 24 kWh。按一个月 30 天计算，总发电量为 720 kWh——是平均月用电量的两倍多，但发电量会随天气变化。
10. Probability and Expected Savings | 概率与预期节省
Sam assumes that on a sunny day the panels generate the full 24 kWh, but on a cloudy day production falls to half. In his area, the probability of a sunny day is 0.4 and cloudy 0.6. The expected daily output is:
Sam 假设晴天时电池板满发 24 kWh，阴天发电量减半。在他所在地区，晴天的概率为 0.4，阴天为 0.6。预期的日发电量为：
Expected output = (0.4 × 24) + (0.6 × 12) = 9.6 + 7.2 = 16.8 kWh
预期发电量 = (0.4 × 24) + (0.6 × 12) = 9.6 + 7.2 = 16.8 kWh
Over 30 days, expected monthly generation = 30 × 16.8 = 504 kWh. Sam' enerate a surplus. This surplus could be sold back to the grid, further reducing his bill.
30 天中，预期月发电量 = 30 × 16.8 = 504 kWh。Sam 平均每月用电 259 kWh，因此平均而言电池板将产生余电。余电可卖回电网，进一步降低他的账单。
Using the cost model, 504 kWh generated would otherwise cost 0.13 × 504 + 8 = £ g is therefore substantial, though the exact amount depends on the actual weather pattern.
利用成本模型，如果这 504 kWh 是从电力公司购买，费用将是 0.13 × 504 + 8 = £73.52。因此预期节省是相当可观的，尽管确切数额取决于实际的天气情况。
11. Conclusion and Reflection | 结论与反思
This case study shows how a single situation can link many maths topics: handling data, averages, percentages, algebra, area and probability. AQA often designs questions in this integrated way to test your ability to choose appropriate methods and communicate your reasoning clearly.
这个案例研究表明，一个简单的情境可以串联起许多数学专题：数据处理、平均数、百分比、代数、面积和概率。AQA 常常以这种综合的方式来设计题目，考察你选择合适方法并清晰表达推理过程的能力。
When approaching any case study, read the whole scenario first, extract the key numbers, and decide which mathematical skill fits each part. Show all your working step by step, just as we have done here.
在处理任何案例分析时，先通读整个场景，提取关键数字，并决定哪部分用哪种数学技能。分步骤展示所有解题过程，就像我们在这里所做的一样。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图