Math Toolkit and Writing Tips for the IB Mathematics Exploration | 数学工具包与探究论文写作要点

📚 Math Toolkit and Writing Tips for the IB Mathematics Exploration | 数学工具包与探究论文写作要点

The IB Mathematics Internal Assessment (IA), often called the “Exploration,” is a unique opportunity to demonstrate curiosity, creativity, and mathematical reasoning. Unlike a typical exam, the Exploration is a written paper where you choose your own topic, apply a suitable toolkit of mathematical tools, and communicate your findings clearly. In this article, we will discuss essential mathematical tools and practical writing strategies to help you produce a high-scoring Exploration.

IB 数学内部评估（IA），通常称为“探究论文”，是一个展示好奇心、创造力和数学推理能力的独特机会。与常规考试不同，探究论文是一篇由你自主选题、运用合适的数学工具包、并清晰传达研究结果的书面文章。在本文中，我们将讨论重要的数学工具和实用的写作策略，帮助你写出一篇高分探究论文。

1. Understanding the IB Exploration Criteria | 理解 IB 探究论文的评分标准

Before choosing tools and writing, you must understand how the Exploration is assessed. The IB uses five criteria: Presentation (4 marks), Mathematical Communication (4 marks), Personal Engagement (3 marks), Reflection (3 marks), and Use of Mathematics (6 marks). The total is 20 marks, and the IA contributes 20% of your final grade in both Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI).

在开始选择工具和写作之前，你必须理解探究论文如何被评分。IB 使用五项标准：展示（4 分）、数学交流（4 分）、个人参与（3 分）、反思（3 分）和数学应用（6 分）。总分为 20 分，在数学分析与方法（AA）以及数学应用与解释（AI）中，这一内部评估均占最终成绩的 20%。

2. Choosing the Right Mathematical Toolkit | 选择合适的数学工具包

A “toolkit” in the IB Exploration means all the resources you use to explore, compute, visualize, and verify your mathematics. This includes graphing calculators, dynamic geometry software, spreadsheets, programming languages, and online computational tools. The key is not to use all of them, but to use the ones that genuinely support your investigation.

IB 探究论文中的“工具包”是指你用于探索、计算、可视化和验证数学的所有资源，包括图形计算器、动态几何软件、电子表格、编程语言和在线计算工具。关键不是使用所有工具，而是选择真正支持你研究的那些工具。

Graphing calculators (e.g., TI-Nspire, Casio fx-CG) | 图形计算器（如 TI-Nspire、Casio fx-CG）
Dynamic geometry software (e.g., GeoGebra, Desmos) | 动态几何软件（如 GeoGebra、Desmos）
Computational engines (e.g., Wolfram Alpha, Python) | 计算引擎（如 Wolfram Alpha、Python）
Spreadsheets (e.g., Excel, Google Sheets) | 电子表格（如 Excel、Google Sheets）

3. Using Desmos and GeoGebra for Visualization | 使用 Desmos 和 GeoGebra 进行可视化

Desmos is excellent for plotting functions, exploring transformations, and performing regressions. You can insert screenshots of graphs in your Exploration to illustrate key ideas. GeoGebra is more powerful for geometry, 3D graphs, vectors, and dynamic sliders. Both tools help you test conjectures before formal proof.

Desmos 非常适合绘制函数图像、探索变换和进行回归分析。你可以在探究论文中插入图形截图，以说明关键思想。GeoGebra 在几何、3D 图形、向量和动态滑块方面更强大。这两种工具都能帮助你在正式证明之前检验猜想。

y = a·sin(bx + c) + d

For example, by using sliders for parameters a, b, c, d in GeoGebra, you can visually show how amplitude, period, phase shift, and vertical shift affect a sine function.

例如，在 GeoGebra 中使用参数 a、b、c、d 的滑块，你可以直观地展示振幅、周期、相位移和垂直位移如何影响正弦函数。

4. Leveraging Graphing Calculators for Numerical Analysis | 利用图形计算器进行数值分析

A graphing calculator is not just for drawing curves; it can compute numerical derivatives, definite integrals, matrices, statistics, and probability distributions. In your Exploration, mention the calculator functions you used and why they are appropriate. For instance, using the “nDeriv” function to approximate a derivative can support a numerical exploration of rates of change.

图形计算器不仅仅是用来画曲线的；它可以计算数值导数、定积分、矩阵、统计量和概率分布。在你的探究论文中，说明你使用了哪些计算器功能以及为什么它们合适。例如，使用“nDeriv”函数来逼近导数，可以支持关于变化率的数值探索。

5. Using Python for Simulation and Data Modelling | 使用 Python 进行模拟和数据建模

For AI students, Python is especially valuable. You can run Monte Carlo simulations, generate random samples, fit curves, and perform statistical tests. Even a short Python script can show personal engagement and a deeper level of mathematics. Always include the code or a clear description of the algorithm in the appendix, and interpret the output in the main body.

对于 AI 学生，Python 尤其有价值。你可以运行蒙特卡洛模拟、生成随机样本、拟合曲线以及执行统计检验。即使是一个简短的 Python 脚本也能展示个人参与和更深层次的数学。记得在附录中包含代码或算法的清晰描述，并在正文中解释输出结果。

import random
x = [random.uniform(0, 1) for _ in range(1000)]

The code above generates 1000 uniform random numbers, which can be used in a simulation of the Monte Carlo method for approximating π.

上面的代码生成 1000 个均匀随机数，可用于蒙特卡洛方法近似 π 的模拟。

6. Excel and Spreadsheets for Statistical Analysis | 使用 Excel 和电子表格进行统计分析

Spreadsheets are powerful when your topic involves large datasets, linear regression, chi-square tests, or normal distributions. You can use Excel’s built-in functions such as AVERAGE, STDEV, CORREL, and LINEST to compute statistical values quickly. When presenting results, include tables with clear headings and units.

当你的主题涉及大型数据集、线性回归、卡方检验或正态分布时，电子表格非常强大。你可以使用 Excel 内置函数，如 AVERAGE、STDEV、CORREL 和 LINEST，快速计算统计值。在展示结果时，请包含带有清晰标题和单位的表格。

Dataset	Mean	Standard Deviation
Sample A	12.5	3.2
Sample B	15.8	4.1

7. Structuring the Exploration Paper | 构建探究论文的结构

A well-structured Exploration typically follows this sequence: title page, introduction, methodology, mathematical development, results, reflection, and conclusion. Use clear headings and subheadings. The exploration should read as a story — from a question, through mathematical work, to an answer and critical reflection.

一篇结构良好的探究论文通常遵循以下顺序：标题页、引言、方法、数学发展、结果、反思和结论。使用清晰的标题和副标题。探究应像一个故事一样展开——从一个问题出发，经历数学工作，到达答案和批判性反思。

8. Writing the Introduction and Defining the Aim | 撰写引言并明确研究目标

Your introduction must state the research question clearly and explain why you are interested in it. This is where personal engagement begins. Avoid vague statements like “I want to learn about math.” Instead, say something specific: “This exploration aims to determine the optimal angle for a basketball free throw using projectile motion equations, because I play basketball and have observed that release angle significantly affects accuracy.”

你的引言必须清楚地陈述研究问题，并解释你为什么对它感兴趣。这是个人参与的开始。避免这样的模糊陈述：“我想了解数学。”而应该这样说：“本探究旨在使用抛体运动方程确定篮球罚球的最佳出手角度，因为我打篮球，并观察到出手角度显著影响命中率。”

9. Presenting the Mathematical Development | 展示数学发展过程

This is the core of your Exploration. Present definitions, formulas, and derivations step by step. Use equations sparingly but accurately. For example, if you are calculating compound interest, show the general formula and then substitute your values.

这是你探究论文的核心。请逐步展示定义、公式和推导。适量且准确地使用方程。例如，如果你在计算复利，先展示通用公式，然后代入你的数值。

A = P × (1 + r/n)ⁿᵗ

where A is the final amount, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years.

其中 A 是最终金额，P 是本金，r 是年利率，n 是每年复利次数，t 是年数。

10. Demonstrating Personal Engagement and Reflection | 展示个人参与和反思

Personal engagement is shown not by using “I” a lot, but by making decisions and reflecting on them. Explain why you chose a particular method, what you expected to find, and how you felt when results differed. Reflection should be continuous, not just a final paragraph. Compare your results with real-world values, discuss limitations, and suggest extensions.

个人参与不是通过大量使用“我”这个词来展示，而是通过做出决定并对它们进行反思来体现。解释你为什么选择某种方法，你期望发现什么，以及当结果与预期不同时你的感受。反思应该是持续性的，而不仅仅是最后一段。将你的结果与现实世界值进行比较，讨论局限性，并提出扩展方向。

11. Avoiding Common Pitfalls | 避免常见错误

Many students lose marks by spending too much time on description rather than mathematics, using tools without justification, or overusing technology to produce impressive visuals that do not support the argument. Another common mistake is including massive tables of raw data in the main body; instead, place them in an appendix. Also, do not simply quote an exact textbook proof — add your own commentary and interpretation.

许多学生因为花了太多时间在描述而非数学上、使用工具却没有说明理由、或过度使用技术来制作华而不实却不支持论点的视觉内容而失分。另一个常见错误是在正文中放入大量原始数据表；相反，应将它们放在附录中。另外，不要简单地照搬教科书中的证明——要加入你自己的评论和解释。

12. Final Checklist for a High-Scoring IA | 高分 IA 的最终检查清单

Before submission, check that your Exploration has a clear title, a focused research question, defined variables, correct units, logical flow, a variety of mathematical skills appropriate to your level, genuine reflection, and a list of references. Verify that all graphs and tables are labelled and explained in the text. Finally, read the IB criteria again and self-assess your draft against each criterion.

在提交之前，请检查你的探究论文是否具有清晰的标题、聚焦的研究问题、明确的变量、正确的单位、逻辑流程、适合你水平的多样数学技能、真实的反思和参考文献列表。确认所有图形和表格都有标签并在正文中加以解释。最后，再次阅读 IB 评分标准，并根据每项标准自我评估你的草稿。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Math Toolkit and Writing Tips for the IB Mathematics Exploration | 数学工具包与探究论文写作要点