📚 IB Math Part 8: Projects – A Complete Guide to the Exploration | IB数学第八部分：项目 – 探索完全指南
The internal assessment (IA) in IB Mathematics, also known as the Mathematical Exploration, accounts for 20% of your final grade. It is a unique opportunity to investigate an area of mathematics that fascinates you, apply skills from the syllabus, and demonstrate personal engagement, mathematical communication, and reflection. Whether you are taking Analysis & Approaches (AA) or Applications & Interpretation (AI), the exploration project is your chance to go beyond textbook exercises and think like a mathematician.
IB 数学的内部评估（IA），又称数学探索，占最终成绩的 20%。这是一个独特的机会，让你研究自己感兴趣的数学领域，运用教学大纲中的技能，并展示个人参与、数学交流与反思。不论你选修的是分析与方法（AA）还是应用与解释（AI），探索项目都能让你超越课本练习，像数学家一样思考。
1. Understanding the IA Criteria | 理解内部评估标准
To succeed, you must know the five assessment criteria inside out. The exploration is marked out of 20 marks, distributed as follows: Criterion A (Presentation) – 4 marks, Criterion B (Mathematical Communication) – 3 marks, Criterion C (Personal Engagement) – 4 marks, Criterion D (Reflection) – 3 marks, and Criterion E (Use of Mathematics) – 6 marks. Every part of your project should be designed with these criteria in mind; leave nothing to chance.
要取得成功，你必须彻底理解五项评估标准。探索满分 20 分，分配如下：标准 A（表达）– 4 分，标准 B（数学交流）– 3 分，标准 C（个人参与）– 4 分，标准 D（反思）– 3 分，标准 E（数学运用）– 6 分。项目的每个部分都应围绕这些标准来设计，绝不能掉以轻心。
The table below summarises the weight and focus of each criterion. Use it as a checklist while you work.
下表总结了每条标准的权重与重点，请在工作时用作核对清单。
| Criterion | Marks | Key Focus |
|---|---|---|
| A: Presentation | 4 | Organisation, coherence, legibility |
| B: Mathematical Communication | 3 | Appropriate notation, terminology, graphs, diagrams |
| C: Personal Engagement | 4 | Personal interest, independent thinking, creativity |
| D: Reflection | 3 | Critical evaluation of results, limitations, extensions |
| E: Use of Mathematics | 6 | Sophistication, precision, relevance to the aim |
2. Choosing a Topic that Excites You | 选择让你感兴趣的题目
The best explorations grow out of genuine curiosity. Think about topics that connect with your hobbies, other subjects, or real‑world phenomena. For example, if you enjoy sports, you might model the flight of a basketball shot using projectile motion and quadratic functions. If you are fascinated by patterns, you could explore Fibonacci sequences in nature or the mathematics of some encryption methods. The key is to avoid copying a generic topic from a website – your passion will be rewarded under Criterion C.
最出色的探索源于真正的好奇心。想想与你的爱好、其他学科或现实世界现象相关的主题。例如，如果你热爱运动，可以用抛体运动和二次函数来模拟篮球投篮的轨迹。如果你对模式着迷，可以探讨自然界中的斐波那契数列或某些加密方法的数学原理。关键在于避免从网站照搬泛泛的题目——你的热情将在标准 C 中得到回报。
Students taking AA often lean towards topics in pure mathematics, such as an investigation of prime number distribution or an analysis of the irrationality of √2. AI students might prefer modelling population growth with differential equations or applying statistical tests to a survey dataset. Either pathway is acceptable, provided your exploration remains mathematically rich and personal.
选修 AA 的学生往往倾向于纯数学题目，例如质数分布探究或 √2 无理性的分析。AI 学生可能更喜欢用微分方程模拟人口增长，或将统计检验应用于问卷调查数据。两条路线均可接受，只要你的探索数学内容丰富且具有个人特色。
3. Developing a Clear Research Question | 制定明确的研究问题
Transform your broad idea into a focused, clear research question. The aim must be expressed in mathematical terms and should have scope for investigation. For instance, instead of "I want to look at music and maths," try "How can Fourier series be used to decompose a musical chord into its sine wave components?" A well‑defined aim sets the boundaries and guides your entire exploration.
将宽泛的想法转化为专注、明确的研究问题。目标必须用数学语言表达，并留有探究空间。例如，与其写“我想研究音乐与数学”，不如写成“如何用傅里叶级数将音乐和弦分解为正弦波分量？”一个定义明确的目标能划定边界，并指导整个探索过程。
Your research question should ideally appear in the introduction and be restated in the conclusion. Avoid questions that lead only to trivial computation or can be answered with a single formula. The best explorations involve multiple mathematical strands, such as calculus, probability, and geometry, woven together.
理想情况下，你的研究问题应出现在引言中，并在结论里重申。避免那些只会导致琐碎计算或能用单个公式回答的问题。最优秀的探索往往融合了微积分、概率、几何等多条数学主线。
4. Planning Your Mathematical Investigation | 规划数学调查
Once the topic is selected, identify the mathematical tools you will need. List the concepts, theorems, and techniques from your syllabus and beyond. For a project on the spread of a disease, you might need differential equations (the SIR model), Euler's method for numerical solutions, and descriptive statistics. Then, create a timeline: data collection or theoretical derivation, mathematical manipulation, graphing, analysis, and drafting.
选定主题后，明确你所需的数学工具。列出教学大纲内外的概念、定理和技巧。对于一个关于疾病传播的项目，你可能需要微分方程（SIR 模型）、欧拉数值解法以及描述性统计。接着制定时间线：数据收集或理论推导、数学运算、制图、分析以及撰写草稿。
It is wise to discuss your plan with your teacher early on. They can verify that the mathematics is at an appropriate level – complex enough to score well on Criterion E, but not so advanced that you cannot explain it clearly. Remember that unexplained, over‑complicated mathematics can hurt your Communication marks.
明智的做法是尽早与老师讨论你的计划。他们可以确认数学水平是否恰当——既要足够复杂以在标准 E 中得高分，又不要深奥到无法清晰解释。记住，未经解释且过于复杂的数学会损害你的交流得分。
5. Demonstrating Personal Engagement | 展示个人参与
Personal engagement means showing that you have independently thought about the topic. This could involve designing your own approach to data collection, making original observations, or connecting the mathematics to something personally meaningful. For example, if you investigate optimal queuing systems at your school canteen and collect your own timing data, that demonstrates initiative.
个人参与意味着展现你对主题的独立思考。这可以包括自己设计数据收集方法、提出独到的观察，或将数学与对你个人有意义的事物联系起来。例如，如果你研究学校食堂的最优排队系统并亲自收集时间数据，这就体现了主动性。
Mere descriptions of historical facts or textbook problems do not count as personal engagement. You should highlight moments when you made choices – such as selecting a particular mathematical method or modifying a model when initial assumptions failed. Indicate these moments explicitly using phrases like "I decided to..." or "I noticed that...".
仅仅描述史实或课本习题不算是个人参与。你应突出自己做选择的时刻——例如选用特定的数学方法，或在初始假设失败时调整模型。用“我决定……”“我注意到……”等措辞明确标示这些时刻。
6. Applying Mathematical Concepts beyond the Syllabus | 应用超出教学大纲的数学概念
To achieve top marks in Criterion E, your exploration must include mathematics that is commensurate with, and preferably goes beyond, the standard level of the course. You do not need to learn an entire new field, but exploring a well‑defined extension of a known topic often works well. For an AA student studying vectors, using matrices to represent transformations in three dimensions demonstrates extra sophistication.
若要在标准 E 中取得高分，你的探索必须包含与课程水平相当、最好超出课程标准的数学内容。你无需学习一整套新领域，但在已知主题上进行明确的拓展通常效果很好。对于学习向量的 AA 学生，使用矩阵表示三维变换便展现出额外的深度。
Examples of extension topics include: Markov chains, hypothesis testing for goodness of fit, logistic maps and chaos, Taylor series approximations, graph theory algorithms, or the mathematics of RSA encryption. Whatever you choose, ensure you understand it thoroughly and can explain it in your own words.
拓展主题的例子包括：马尔可夫链、拟合优度的假设检验、逻辑斯蒂映射与混沌、泰勒级数逼近、图论算法，或 RSA 加密的数学原理。无论选择什么，都要确保你透彻理解并能用自己的语言解释清楚。
7. Making Effective Use of Technology | 有效使用技术
Technology can elevate your exploration significantly. Use graphing software (GeoGebra, Desmos), a GDC, spreadsheets, or programming languages like Python to visualise functions, simulate data, or perform calculations that would be tedious by hand. For instance, running a Monte Carlo simulation to approximate π is both elegant and technologically rich.
技术可以显著提升你的探索水平。使用绘图软件（GeoGebra、Desmos）、图形计算器、电子表格或 Python 等编程语言来可视化函数、模拟数据或完成繁琐的手工计算。例如，运行蒙特卡罗模拟来逼近 π 既优雅又富含技术含量。
However, all technology use must be accompanied by clear explanations. Screenshots are not enough; you need to describe what you did, what the commands meant, and how the output relates to your aim. For a graph, label axes and indicate key points. Embedding relevant GDC screens or Python scripts in the body of the exploration demonstrates thorough documentation.
然而，所有技术使用都必须辅以清晰的解释。仅有屏幕截图是不够的；你需要描述你做了什么、命令的含义，以及输出如何与你的目标相关。对于图表，标注坐标轴并指明关键点。将相关的 GDC 屏幕或 Python 脚本嵌入探索正文，可展示详尽的记录。
8. Structuring Your Exploration | 构建探索的结构
A logically organised exploration is easier to follow and scores well on Presentation. A typical structure includes: an introduction with rationale and aim, a main body developing the mathematics, a conclusion that summarises findings and reflects, and a reference list. The body can be divided into sections with descriptive subheadings, such as "Derivation of the SIR model" or "Analysis of residuals".
逻辑清晰的探索更易阅读，并在表达方面获得高分。典型结构包括：包含理由与目标的引言、发展数学内容的主体、总结发现与反思的结论，以及参考文献列表。主体部分可用描述性小标题划分，如“SIR 模型的推导”或“残差分析”。
Keep your exploration concise; the recommended length is 6 to 12 pages, excluding appendices. Everything in the main body should be essential to your story. Place large tables, raw data, or lengthy code in an appendix, but never rely on them to convey key mathematical points. If it matters, it belongs in the body.
保持探索简洁；建议长度为 6 至 12 页，不含附录。主体中的所有内容都应对你的故事至关重要。将大型表格、原始数据或冗长代码放入附录，但绝不能依赖它们来传达关键数学要点。若内容重要，就应放在主体中。
9. Writing with Mathematical Clarity | 清晰撰写数学内容
tation like ∫ₐᵇ f(x) dx, and clear labelling of equations. Avoid ambiguous phrases such as "the function goes up fast" – instead say "the exponential function eˣ grows at a rate proportional to its current value".
你的探索必须在数学上精确且表达清晰。始终正确使用符号：变量用斜体、向量用粗体，使用适当的积分符号如 ∫ₐᵇ f(x) dx，并清晰标注方程。避免诸如“函数增长很快”等模糊表述——而应说“指数函数 eˣ 的增长速率与其当前值成正比”。
d b are sides of a triangle, and C is the included angle. Mathematical communication is about guiding the reader, not impressing them with obscure leaps.
每引入一个公式，都要定义每个符号。例如，若写 A = ½ab sin C，必须说明 a 和 b 是三角形的两边，C 是夹角。数学交流旨在引导读者，而非用晦涩的跳跃来震撼他们。
10. Reflecting Critically | 批判性反思
Reflection is the heart of Criterion D. Go beyond a simple "I learned a lot" and analyse the strengths and limitations of your model or method. Consider questions like: Were the assumptions realistic? How sensitive is the conclusion to changes in parameters? If you used a linear regression, discuss the r² value and potential outliers.
反思是标准 D 的核心。不要停留在简单的“我学到了很多”，而要分析模型或方法的优势与局限。思考以下问题：假设是否合乎实际？结论对参数变化的敏感度如何？如果你用了线性回归，请讨论 r² 值和潜在异常值。
You should also propose meaningful extensions. For instance, you might say: "Given more time, I would extend the project‑based percolation model to three dimensions, which would require a more advanced use of combinatorial probability." This shows you have thought beyond the current scope.
你还应提出有意义的拓展思路。例如，你可以说：“若有更多时间，我会将基于渗透的二维模型拓展至三维，这需要更深入地运用组合概率。”这表明你已经超越了当前范围进行了思考。
11. Common Mistakes to Avoid | 避免常见错误
Many explorations lose marks because of avoidable errors. A topic that is too broad (e.g., "the history of calculus") lacks focus and mathematics. Another common mistake is failing to involve enough original work: if your project simply repeats a known proof without personal input, engagement marks will suffer. Plagiarism, including copying from online exemplars, is a serious academic offence and can void your IA score entirely.
许多探索因可避免的错误而丢分。题目过于宽泛（如“微积分史”）会缺乏重点与数学内容。另一个常见错误是未包含足够的原创工作：若你的项目只是重复已知证明而无个人见解，参与分将大打折扣。抄袭，包括照搬在线范例，是严重的学术违规，可能导致 IA 成绩完全作废。
Don't neglect the word count and presentation rules. Examiners expect legible font, consistent layout, and proper citations. Overuse of screenshots without commentary, missing axis labels, and poorly formatted equations all count against Criterion A. Lastly, leaving the reflection to the very last sentence is a lost opportunity – reflect throughout, not just at the end.
不要忽视字数和汇报规则。考官希望看到清晰可读的字体、一致的布局和恰当的引用。过度使用无注释的截图、缺少坐标轴标签以及格式混乱的方程，都会影响标准 A 得分。最后，把反思留到最后一句是失去的机会——要贯穿全文进行反思，而不仅仅在结尾。
12. Final Checklist and Submission | 最终清单与提交
Before you submit, run through this checklist: is the aim clearly stated in the introduction? Does the mathematics match or exceed the syllabus level? Have you explained every formula and graph? Is personal engagement evident throughout, with explicit statements of independent thought? Are reflections present at multiple stages, not just the conclusion? Is the exploration between 6 and 12 pages, well structured, and free of spelling errors?
提交之前，请核对以下清单：引言中是否清晰陈述了目标？数学内容是否达到或超出教学大纲水平？是否解释了每一个公式和图表？全文是否体现了个人参与，并有明确的独立思考陈述？是否在多个阶段都进行了反思，而不仅限于结论？探索是否在 6 到 12 页之间，结构良好且无拼写错误？
Also ensure you have followed your school's specific submission guidelines regarding cover pages, electronic formats, and deadlines. A polished final draft that meets all administrative requirements conveys professionalism and respect for the assessment process. The exploration is a significant undertaking, but with careful planning, genuine interest, and rigorous mathematics, it can become the highlight of your IB Mathematics journey.
同时，确保你已遵守学校关于封面、电子格式和截止日期的具体提交要求。一份打磨完善的最终草稿，在行政要求上完全合规，能传达出专业素养与对评估过程的尊重。探索是一项重要任务，但通过周密计划、真正兴趣与严谨的数学，它可以成为你 IB 数学旅程中的亮点。
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