Year 8 WJEC Maths: How to Structure and Write a Mathematical Investigation (Paper) | WJEC 八年级数学：数学探究论文写作框架与范文

📚 Year 8 WJEC Maths: How to Structure and Write a Mathematical Investigation (Paper) | WJEC 八年级数学：数学探究论文写作框架与范文

In Year 8 WJEC Mathematics, you will sometimes be asked to complete a written investigation or a short paper where you explore a mathematical problem, explain your reasoning, and present your findings clearly. This type of task is not just about getting the right answer — it is about showing how you think, how you test ideas, and how you communicate mathematically. A well-structured paper makes your work easier to follow and can help you achieve higher marks. Below is a clear framework you can use, together with a full worked example.

在 WJEC 八年级数学中，你有时需要完成一篇书面探究或短论文，你需要探索一个数学问题，解释你的推理，并清晰地展示你的发现。这类任务不仅仅是得到正确答案，更在于展示你如何思考、如何检验想法以及如何进行数学交流。结构清晰的论文能让你的作业更易于理解，有助于你获得更高分数。下面提供一个你可以使用的清晰框架，并附上一个完整的范例。

1. Understanding the Task (Reading and Decoding) | 理解题目（阅读与解码）

Before you write anything, read the investigation prompt carefully. Highlight the key mathematical words, the question you need to explore, and any conditions or constraints. Ask yourself: 'What is this problem really asking me to find out?' If the prompt gives a scenario, rephrase it in your own words. This step prevents you from rushing off in the wrong direction.

在动笔之前，仔细阅读探究提示。标出关键的数学词汇、你需要探索的问题以及任何条件或限制。问自己：“这道题真正要我发现什么？”如果提示给出了一个情景，用你自己的话重新表述一遍。这一步能避免你一开始就走错方向。

2. Stating the Aim and Hypothesis | 阐明目标与假设

Begin your paper with a short introduction that states what you are investigating and what you predict might happen. In mathematics, a hypothesis does not have to be a formal guess — it can be a statement like 'I think the pattern will increase by 3 each time' or 'I predict the total area will always be a multiple of 4'. This shows you are thinking ahead and gives your investigation direction.

以简短引言开篇，说明你要探究的内容以及你预测可能发生的情况。在数学中，假设不一定是正式的猜测——它可以是一句“我认为模式每次会增加 3”或“我预测总面积总是 4 的倍数”。这显示你在提前思考，并为你的探究指明了方向。

3. Defining Variables and Notation | 定义变量与符号

Introduce any letters or symbols you will use. For example, 'Let n be the term number' or 'Let l stand for the length of the rectangle'. Keep your notation simple and consistent. A table of definitions at the start can make your paper look organised and professional.

引入你要使用的任何字母或符号。例如，“设 n 为项数”或“设 l 代表长方形的长”。保持符号简单一致。在开头列出定义表格会让你的论文看起来有条理又专业。

4. Gathering and Organising Data | 收集与整理数据

Generate some initial data by trying small cases or simple examples. Draw a table to record your results. In Year 8, this often means working out the first few terms of a sequence, measuring diagrams, or listing combinations systematically. A well-labelled table with clear headings (e.g. 'Term number (n)' and 'Number of squares') makes patterns easier to spot.

通过尝试小数值或简单例子生成一些初步数据。画一个表格来记录结果。在八年级，这通常意味着算出数列的前几项、测量图形或系统地列出组合。带有清晰标题（如“项数（n）”和“正方形数量”）的表格能让模式更容易被发现。

5. Looking for Patterns and Relationships | 寻找模式与关系

Examine your table and describe any patterns you notice in words first. Are the results increasing by a constant amount? Is there a multiplying factor? Could there be a square or triangular number pattern? Write down your observations, then try to express the pattern as a rule. Use phrases like 'each time n increases by 1, the total goes up by…' to build towards a formula.

检查你的表格，先用文字描述你注意到的任何模式。结果是以恒定的数量增加吗？有没有乘数因子？会不会是平方数或三角形数模式？写下你的观察，然后尝试将模式表达为一条规则。使用“每增加 1，总数就增加……”这样的短语来逐渐推导出公式。

6. Forming a General Rule (the nth Term or Formula) | 形成一般规则（第 n 项或公式）

This is the heart of a mathematical investigation. Use the pattern you found to write an algebraic rule. For a sequence, write an expression for the nth term, such as '3n + 2'. For a shape investigation, you might write a formula for perimeter or area in terms of a variable length. Show the steps of how you derived your rule: for example, 'I noticed the sequence went 4, 7, 10, 13… the difference is 3, so the rule starts with 3n, and to get the first term 4 I need to add 1, giving 3n + 1.'

这是数学探究的核心。利用你发现的模式写出一条代数规则。对于数列，写出第 n 项的表达式，例如“3n + 2”。对于图形探究，你可能需要写出以可变长度为变量的周长或面积公式。展示你是如何推导出规则的步骤：例如，“我注意到数列是 4、7、10、13……差为 3，所以规则以 3n 开始，而要得到第一项 4 我需要加 1，因此得到 3n + 1。”

7. Testing and Verifying Your Rule | 检验与验证你的规则

small values you already know, then try a larger one like n = 10 or n = 20. Ask yourself: 'Does my formula give the right answer for my early examples? Does it give a sensible answer for a new case?' If your rule fails, go back and refine it — mistakes are part of the process and should be discussed in your paper.

切勿未经检验就相信一条规则。将数值代回原问题或表格。先用你已经知道的小数值检验，再用较大的数值如 n = 10 或 n = 20 检验。问自己：“我的公式对我之前的例子给出了正确答案吗？对新情况它给出了合理的答案吗？”如果规则失败了，就回头修正——错误是整个过程的一部分，应在论文中讨论。

8. Justifying Your Rule with Reasoning | 用推理论证你的规则

A top-grade investigation does not just present a formula — it explains why the formula works. Use diagrams, words, or structural arguments. For example, if you found that the number of matchsticks for a row of squares is 3n + 1, you might say: 'Each square needs 3 new matchsticks, and the first square needs one extra to start the row.' This deepens your mathematical communication.

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一篇高分的探究不仅呈现公式，还会解释公式为何成立。使用图表、文字或结构性论证。例如，如果你发现一排正方形的火柴棍数量是 3n + 1，你可以说：“每个正方形需要 3 根新火柴棍，而第一个正方形需要额外一根来起始整排。”这能深化你的数学交流。

9. Extending the Investigation | 拓展探究

Once you have a solid rule, try to go further. Ask 'What if…?' questions: What if the pattern started from a different number? What if the shape was a rectangle instead of a square? What if I added another dimension? Even a small extension shows curiosity and a deeper understanding. You do not need to solve everything perfectly — simply showing that you have thought about related problems is valuable.

一旦你有了可靠的规则，尝试更进一步。提出“如果……会怎样？”的问题：如果模式从一个不同的数字开始呢？如果形状是长方形而不是正方形呢？如果我增加一个维度呢？即使是一个小小的拓展也能展现好奇心和更深的理解。你不需要完美解决所有问题——仅仅展示你思考过相关问题就已经很有价值。

10. Conclusion: Summarising What You Learned | 结论：总结所学

End your paper with a short concluding paragraph. State clearly what you found, restate your final rule, and mention whether your hypothesis was correct. Reflect on any surprises or difficulties you encountered. A strong conclusion ties everything together and shows that you have completed the investigation thoughtfully.

用一段简短的结论段落结束你的论文。清晰地陈述你发现了什么，重申你的最终规则，并提及你的假设是否正确。反思你遇到的任何意外或困难。一个有力的结论能把所有内容串联起来，表明你已深思熟虑地完成了探究。

11. Worked Example: The Growing Staircase | 完整范例：增长的楼梯

Task: A staircase is made of square blocks. Step 1 has 1 block, Step 2 has 3 blocks (2 in the bottom row and 1 on top), Step 3 has 6 blocks (3 at the bottom, 2 in the middle, 1 on top). Investigate the number of blocks needed for any step number n.

题目：一个楼梯由正方形积木搭成。第 1 级有 1 块积木，第 2 级有 3 块（底层 2 块，上层 1 块），第 3 级有 6 块（底层 3 块，中层 2 块，上层 1 块）。探究任意级数 n 所需的积木数量。

Writing the paper:

撰写论文：

Introduction: I am investigating the total number of blocks in a staircase that grows by one row each time. My hypothesis is that the total will be the sum of the first n whole numbers.

引言：我正在探究每次增加一排的楼梯的积木总数。我的假设是，总数将是前 n 个整数的和。

Data table:

Step (n)	Blocks (total)
1	1
2	3
3	6
4	10
5	15

Pattern spotting: The differences are 2, 3, 4, 5… so not linear. The totals are triangular numbers: 1, 3, 6, 10, 15… I recognise this as the sum of the integers from 1 to n.

模式识别：差值是 2、3、4、5……所以不是线性的。总数是三角形数：1、3、6、10、15……我认出这是从 1 到 n 的整数之和。

Deriving the rule: The formula for the sum 1 + 2 + … + 1)/2. I checked: for n = 4, 4 × 5 ÷ 2 = 10, correct. For n = 5, 5 × 6 ÷ 2 = 15, correct.

推导规则：和 1 + 2 + … + n 的公式是 n(n + 1)/2。检验：当 n = 4 时，4 × 5 ÷ 2 = 10，正确。当 n = 5 时，5 × 6 ÷ 2 = 15，正确。

Justification: If I put two identical staircases together, they form a rectangle of size n by (n + 1). So one staircase is half of that, giving n(n + 1)/2. I drew a diagram to show this.

论证：如果把两个相同的楼梯拼在一起，它们会形成一个 n 行 (n + 1) 列的长方形。所以一个楼梯是它的一半，即 n(n + 1)/2。我画了一个图来展示这一点。

Extension: What if the staircase started with 2 blocks in the first row? The sequence would be 2, 5, 9, 14… I found the rule n(n + 3)/2 and tested it. I also explored the number of blocks visible from the front, which is just n.

拓展：如果第一层有 2 块积木呢？数列将是 2、5、9、14……我找到了规则 n(n + 3)/2 并进行了检验。我还探讨了从前面看到的积木数量，那正是 n。

Conclusion: sis was correct. I found the geometric proof particularly satisfying. This investigation helped me link algebra with shapes.

结论：高度为 n 的楼梯积木数量由三角形数公式 T = n(n + 1)/2 给出。我的假设是正确的。我发现几何证明特别令人满意。这次探究帮我将代数与形状联系了起来。

12. Final Presentation Tips | 最终呈现技巧

Hand in a neat, well-spaced document. Use headings to separate sections, include clear tables and labelled diagrams. Write in full sentences and check your spelling. Number your pages if it is longer than one side. If you use ICT, keep the formatting simple — a clear font like Arial or Times New Roman at size 12 is ideal. Always include your name, date, and the investigation title. A well-presented investigation shows you care about your mathematical communication.

提交整洁、行距适中的文档。使用标题分隔各部分，包含清晰的表格和带标注的图表。用完整句子书写并检查拼写。如果超过一面，就给页面编号。如果使用计算机，保持格式简单——像 Arial 或 Times New Roman 这样的清晰字体 12 号是理想选择。始终写明姓名、日期和探究标题。展示良好的呈现效果表明你重视数学交流。

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