📚 Year 7 OCR Maths: Framework and Model for Writing Mathematical Investigations | Year 7 OCR 数学：数学探究写作框架与范文
In Year 7, OCR Mathematics encourages students not just to find answers but to explain their thinking and structure their work like a mini investigation. A well-written mathematical paper shows clear reasoning and helps you secure top marks in extended tasks.
在七年级OCR数学中，不仅要求得出答案，更鼓励学生解释自己的思考过程，像小探究一样组织写作。一篇结构清晰的数学论文能展现你的推理能力，帮助你在长篇任务中获得高分。
1. Understanding the Investigation Brief | 理解探究任务要求
The first step is to read the brief several times. Underline the command words such as 'investigate', 'compare', 'explain' or 'justify'. Identify what you need to find out and which mathematical topics are involved, such as number patterns, shapes or data handling.
第一步是反复阅读任务说明。在“探究”、“比较”、“解释”或“证明”等指令词下划线。明确你需要发现什么，以及涉及哪些数学主题，如数字规律、图形或数据处理。
Write down the main question in your own words. For example, if the brief says 'Investigate how the area of a rectangle changes when the perimeter is fixed', rephrase it as 'I will explore how many different rectangles I can make with a perimeter of 24 cm and find the one with the largest area'. This clarifies your goal.
用自己的话写下主要问题。例如，如果任务说“探究当周长固定时，长方形的面积如何变化”，可以转述为“我要探究周长为24厘米的长方形可以有多少种不同形状，并找出面积最大的那个”。这会明确你的目标。
2. Planning Your Investigation | 规划你的探究
Before you start writing, create a simple plan. List the steps: gather tools, collect data, organise findings, look for patterns, and draw conclusions. A plan keeps your work focused and prevents you from jumping ahead too quickly.
在动笔之前，先做一个简单计划。列出步骤：准备工具，收集数据，整理发现，寻找模式，得出结论。有计划能让你保持专注，避免急于求成。
Decide what you will record in a table. For the rectangle investigation, you might list length, width and calculated area. Having a clear recording method from the start saves time later.
决定你要在表格中记录什么。在长方形探究中，你可以列出长、宽和计算出的面积。从一开始就有清晰的记录方法可以节省后续时间。
3. Writing an Engaging Introduction | 撰写引人入胜的引言
Your introduction should state the aim of the investigation and your initial thoughts. Begin with a sentence like 'In this investigation, I aim to find out...'. Then explain why the topic is interesting or link it to a real-life context, such as designing a garden with a fixed amount of fencing.
引言部分应当说明探究的目标以及你的初步想法。用“在本探究中，我旨在发现……”这样的句子开头。然后解释为什么这个主题有趣，或将其与现实生活联系起来，比如用固定长度的围栏设计花园。
Avoid copying the brief word for word. Instead, show that you understand the problem. For instance: 'When a farmer has a limited amount of fencing, the shape of the field can affect how much land he can use. I will explore this idea using rectangles with a fixed perimeter.'
避免逐字照抄任务原文。要体现出你对问题的理解。例如：“当农民的围栏长度有限时，田地的形状会影响可使用土地的面积。我将利用周长固定的长方形来探究这个想法。”
4. Describing Your Method Clearly | 清晰描述你的方法
he investigation. Use the past tense because you are describing what you did. For example: ' nd calculated the corresponding widths using the formula width = (perimeter ÷ 2) − length.'
用简单步骤解释你是如何进行探究的。使用过去时，因为你是在描述已经做过的事情。例如：“首先，我选择了24厘米的周长。然后列出可能的整数长度，并用公式宽 = (周长 ÷ 2) − 长计算出相应的宽度。”
If you used any equipment like graph paper, rulers or spreadsheets, mention them. Always justify your choices: 'I selected whole numbers to keep the investigation manageable and to spot patterns easily.'
如果你使用了方格纸、直尺或电子表格等工具，要提到它们。始终要说明你选择的理由：“我选择整数是为了让探究可控，更容易发现规律。”
5. Presenting Results with Tables and Lists | 用表格和列表展示结果
Organise your findings in a neat table. Here is an example for the rectangle investigation with perimeter 24 cm:
将你的发现整理在一个整洁的表格中。以下是周长为24厘米的长方形探究示例：
| Length (cm) | Width (cm) | Area (cm²) |
|---|---|---|
| 1 | 11 | 11 |
| 2 | 10 | 20 |
| 3 | 9 | 27 |
| 4 | 8 | 32 |
| 5 | 7 | 35 |
| 6 | 6 | 36 |
Always label your table clearly and include units. You can also add a column for observations, such as how the area changes.
始终清晰地标注表格并包括单位。你还可以添加一列用于观察结果，比如面积如何变化。
6. Analysing Patterns and Making Conjectures | 分析规律并提出猜想
Look at the numbers in your table. What do you notice? s and then decreases. The maximum area occurs when length = width = 6 cm, giving a square.
观察表格中的数字。你注意到了什么？在长方形的例子中，你可能会发现随着长度增加，面积先增加后减少。当长 = 宽 = 6厘米时面积最大，也就是正方形。
Write down any patterns you spot: 'The areas are symmetric around the square. The differences between consecutive areas grow smaller as I approach the square.' This shows deeper thinking.
写下你发现的任何规律：“面积以正方形为中心对称分布。当我接近正方形时，相邻面积之间的差值越来越小。”这展示出更深入的思考。
Form a conjecture: 'I conjecture that for a fixed perimeter, the rectangle with the largest area is the square.' A conjecture is an educated guess that you have not yet proven but is supported by your data.
形成一个猜想：“我猜想对于固定周长的长方形来说，面积最大的是正方形。”猜想是一个有根据的推测，虽然尚未证明，但得到了数据的支持。
7. Including Visual Representations | 包含可视化表示
A graph or chart can make your pattern clearer. For the rectangle investigation, you could draw a bar chart or line graph showing length on the horizontal axis and area on the vertical axis. Describe what the graph shows in words.
图表可以使规律更加清晰。在这个长方形探究中，你可以画一个条形图或折线图，横轴表示长，纵轴表示面积。用文字描述图表所显示的内容。
neat graph on graph paper. Point out the peak of the graph and link it to your conjecture: 'e graph clearly peaks at length = 6 cm, confirming that the square gives the maximum area.'
即使没有电脑，你也可以在方格纸上画一个整洁的草图。指出图形的最高点，并将其与你的猜想联系起来：“图形在长 = 6厘米处明显达到峰值，证实了正方形面积最大。”
8. Drawing a Conclusion | 得出结论
Your conclusion should answer the original question and summarise your main findings. State clearly: 'I found that when the perimeter is 24 cm, the rectangle with dimensions 6 cm by 6 cm (a square) has the greatest area, 36 cm².'
你的结论应该回答最初的问题，并总结主要发现。明确表述：“我发现当周长为24厘米时，尺寸为6厘米乘6厘米的长方形（正方形）面积最大，为36平方厘米。”
Explain whether your conjecture was correct and if there were any surprises. If you extended the investigation to other perimeters, mention whether the same rule held: 'I tested perimeters of 20 cm and 28 cm and found the square always had the largest area, so my conjecture holds.'
解释你的猜想是否正确，是否有任何意外发现。如果你将探究扩展到其他周长，要说明是否符合同样规律：“我测试了20厘米和28厘米的周长，发现正方形始终面积最大，因此我的猜想成立。”
9. Evaluating Your Work and Suggesting Extensions | 评估你的工作并提出扩展建议
ation includes reflection. Discuss what went well and what you would improve. For example: ' ole numbers. If I used decimals like length = 5.5 cm, I might find an even closer pattern.'
一项强有力的探究包括反思。讨论哪些做得好，哪些可以改进。例如：“我的表格很清晰，但我只使用了整数。如果我使用小数，比如长 = 5.5厘米，可能会发现更精细的规律。”
Suggest next steps: 'I could investigate other shapes like triangles or circles with a fixed perimeter. Another extension would be to explore fixed area and varying perimeter.' This shows curiosity and deeper mathematical thinking.
提出后续步骤：“我可以研究其他形状，比如周长固定的三角形或圆形。另一个拓展是探究面积固定而周长变化的情况。”这体现了好奇心和更深入的数学思维。
10. A Complete Model Response (Excerpt) | 完整范文示例（节选）
Below is a short model of how an investigation introduction and first data section might look. Note the clear structure and use of mathematical language.
下面是一个简短的探究引言和第一部分数据的范文。注意结构的清晰和数学语言的使用。
Model:
Investigating Rectangles with a Fixed Perimeter
In this investigation, I wanted to find out which rectangle has the maximum area when the perimeter is fixed at 24 cm. This relates to real life, for example when building a pen with limited fencing. I predicted that a square would give the largest area because it is the most balanced shape.
范文：
《探究周长固定的长方形》
在本探究中，我想找出当周长固定为24厘米时，哪个长方形的面积最大。这与现实生活有关，比如用有限的围栏建造一个围栏。我预测正方形会给出最大的面积，因为它是最均衡的形状。
Method and Data
I listed whole-number lengths from 1 cm to 11 cm. For each length, I found the width by subtracting the length from half the perimeter (12 cm). I calculated the area by multiplying length and width. I recorded my results in a table.
方法与数据
我列出了从1厘米到11厘米的整数边长。对于每个长度，我通过从半周长（12厘米）中减去长度得到宽度。我通过长乘宽计算出面积，并将结果记录在表格中。
| Length (cm) | Area (cm²) | Published by TutorHao | Year 7 Mathematics Revision Series | aleveler.com
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