Artificial Neural Networks and Function Approximations
Economic Applications of Lagrange Multipliers
Generating Fractals using Matrices
How to make a world map?
以下是一些IB数学AA和AI内部评估的研究主题（建议篇幅为12-20页）：
Analyzing the Role of Vaccination using SVIR Models
Estimating the Gini Coefficient
Benford's Law and Fraud detection
Solving equations using Origami
Calculating the value of Pi
The Mathematics of Plinko Boards
Do mobile games lie about the probability of getting rare items or characters?
Is the flight path of airplanes always the shortest between the destinations?
Choosing a research question
With IB Maths IAs, everyone knows that the most difficult part is choosing a question. Your teachers usually assign questions in class, and so it might seem strange at first to come up with your own. To figure out what you would like to research, let’s first look at what counts as a good research question.
A good research question:
Is clear in its aim and direction
Is specific in what it wants to find out
Gives you a problem that you can answer within your Maths IA
Suggests possible methodology and approaches
Here’s a bad example. If I want to do a Maths IA on football and write “The Mathematics behind Football” as my research question, then my reader has no idea of what I want to research at first glance. This research topic is too general and it isn’t even technically a question!
Here’s a better approach. I write “What position, direction, and angle should a football be kicked in order to maximise probability of it hitting the net?” This is a lot more wordy, but it is a much clearer and therefore better research question. From the research question alone, your reader can already get a sense of your methodology and what kind of approaches you’ll take in your experiments. How far should you stand from the net? What direction should you shoot? Should you kick from the right, the left, or from front and centre? What angle and trajectory will give you the highest probability of getting the ball in the net? Your reader can easily foresee that you will conduct these kinds of experiments to answer your overall question.
What is personal engagement, and how do I fulfil it?
Another requirement on the IB syllabus is the fulfilment of personal engagement in your Maths IA. Readers will not be interested in an IA if you, too, are not interested in it and just consider it as another piece of homework. Ideally, you will use the mathematics and theories that you studied in class and apply it to study the people around you, conducting experiments linked to real life. To fulfil personal engagement, we recommend that students consider their hobbies, extra-curricular activities, or sports that they enjoy. Are there any potential maths experiments that you can conduct that are relevant to your interests?
For example, if you really like cameras and taking pictures in your spare time, you might research the following in your Maths IA: “What angle and height of the camera will result in the clearest/most beautiful pictures?” In this case, you’d also consider: What is my definition of “most clear” and “most beautiful”? Does it mean that the main subject is perfectly in the middle? How high should my tripod legs be? How shall I test the angles of the camera?
All these aspects are interesting for your reader, and excellent to use as part of your topic. This topic is something you’re passionate about, it is probably not well-researched, and it truly belongs to you and your life experiences. You benefit from the experiment’s outcome because you will improve your photo-taking skills through maths! This proves that you are personally invested in the results and fulfils personal engagement.
Where to start
If you’re still feeling conflicted and directionless about your Maths IA topic, a popular format to follow is the Curve-fitting IA. In a curve-fitting IA, you will collect data and use it to coordinate points on a graph, making a scatter plot. Then, you can think back on the functions and curves you learnt in class, e.g. quadratics, exponentials, logarithms, straight lines, etc. Which curve fits your coordinated points most closely? The curve of best fit will become your model, the equation that best describes the relationship between your x and y.
Then, you can explore why the relationship between x and y is this way. Do some evaluations and write down your theories to show that you have thought deeply about your results. Lastly, remember that your mathematics model is probably an oversimplification of a complex reality. You are limited by what you have learnt so far in high school, so your model will contain some massive assumptions and limitations. Examine these issues too!
One benefit of a curve-fitting IA is that the format and structure is very straightforward. If you follow this format, you should have no problem knowing what to focus on for each section. Another great benefit is that you can use a curve-fitting IA to fit virtually every topic that allows for data collection. If it’s a topic where you can see correlations and potential relationships between different factors, you can use it as your IA.
For example, if your passion is movies, then perhaps you will want to investigate the relationship between how much money is invested in a movie, and how much money it actually makes at the box office. Is there a relationship between the two? Is it an exponential relationship, a straight line, or quadratics, etc? Or perhaps, against your expectations, you find that there is no correlation between the amount of money invested and the amount made through ticket sales at all!
Overall, a curve-fitting IA is a great place to start because of its clarity, straightforwardness, and potential variety. Even if many different students use this format, the topic and experiments they come out with will look very different. If you are at a loss for what to do, look into a curve-fitting IA first.
*当你开始你的第一个数学 IA 时，你可能对解决问题和方程充满信心，但你不确定如何展示你的研究成果。你的读者会期望你使用特定的语言和格式。在你进行调查时，请记住以下一些秘诀和技巧！
例如，假设你决定在数学 IA 中研究投篮。你的目标是找出如何更准确地射击，理想的飞行路径等。这种 IA 不需要收集数据，但需要使用一些物理定理知识。请注意，如果你想研究本文中提到的曲线拟合 IA，那么你将不得不收集数据。
2. 一手数据优于二手数据
在数学 IA 中，你自己收集的主要数据比你从其他人的实验中获得的次要数据更可取。主要数据指的是你亲自设计调查、撰写问题、采访同学或上街询问公众得到的数据。这比谷歌搜索以前的实验并在线查找数据要好得多，因为这是次要数据。IA偏好学生使用一手数据是因为这有助于满足教学大纲中的个人参与的评分标准。如果你自己设计并进行调查，那么你在 IA 中的参与度自然要高得多。
当然，某些研究课题是不可能收集到一手数据的。例如，如果你的 IA 研究电影投资金额和票房之间可能存在的相关性，你就不可能设计自己的调查。相反，你可以上网下载每部电影的预算与票房数据，然后根据这些辅助数据进行研究。虽然这样进行研究调查也可以，但记住二手数据不应是你的首选！
3. 以收集超过30个数据为目标
这点非常简洁明了。如果你能自行选择要收集多少数据，一个好的经验法则是收集不少于30 个数据。如果超过这个数字，就可以大大降低抽样的变异性，你的实验结果受到的影响会更小，也会更可靠！虽然数学 IA 没有严格规定要收集多少个数据，但你应尽量收集30 个数据或以上。
如果我们回顾一下 IB 数学的课程大纲，很明显，考官只是希望你把在课堂上学到的东西应用到你周围的现实生活中。 实际上有一整个数学领域叫做应用数学。你的 IA 应该展示你的创造性思维和对你已经知道的数学知识的有效运用。
如果你选择一个超难的大学程度的数学问题，这并不能保证你能在 IA 中取得高分。假设你决定在你的 IA 中研究和求解麦克斯韦的电磁方程。这不算是一个很好的数学应用，因为麦克斯韦电磁方程太深奥了，它主要侧重于理论，所需要用到的多变量微积分甚至连数学 HL 学生都没有学过。
当然，某些数学 IA 课题可能需要超出课程大纲的数学知识——这取决于你的 IA 研究目标。如果你真的决定使用课程大纲外的数学知识，我们建议你只用一点就够了，或者尽量少用。此外，你必须一开始就在你的 IA 中明确构建这些全新的陌生数学概念并作出充分解释。如果你没有在课堂上学过这些概念，你必须详细地解释清楚你是如何从初步的探索到最后得出最终结果的。简单来说，你班上不知道这些概念的朋友应该要能读懂你的 IA 并明白你的解释。
5. 演示和表达至关重要
数学 IA 经常被忽视的一个方面是演示和表达。然而，这非常重要！你不仅要理解你的 IA，并且要确保你的读者也理解。
A. 使用明确定义和适当的数学符号
明确定义你将使用的术语。如果你使用 x 和 y，请说明它们在你的研究中代表什么，例如 y 代表年收入，x 代表接受教育的年数。如果定义不清楚，考官可能会扣很多分！
此外，尝试使用正确的数学符号。由于现在很多学生在数学课上都用电脑做笔记，因此他们可能会养成使用斜线 / 表示除法、向上箭头 ^ 表示上升幂和 INT 表示积分的习惯。这不是标准的数学符号，所以在 IA 中你应该尽量使用正确的符号，就像你手写出来的一样。例如，x 的 3 次方应该写成 x3，函数 f(x) 的积分应该写成 ∫ f(x) dx 而不是 int f(x) dx。
要在计算机上正确设置格式，请使用 Microsoft Word 的公式编辑器。在右上角的工具栏上，你会找到对你的 IA 有用的常用数学符号和格式。例如。微积分、微分、积分、求和符号等。方程编辑器不难使用，你只需通过试错法来慢慢探索。
B. 保持清晰的思维逻辑
不要忘记在IA中列明你的运算步骤，即使这些运算步骤看起来非常简单和明显。当你在家庭作业或课堂完成一道数学题时，你可能会想跳过一些步骤，而你的老师无论如何可能都会给你分。但是在一份像样的数学 IA 中，你必须清楚展示和表达每个步骤背后的想法和逻辑，以便你的读者理解。所有细节都不能忽视！
