**Introduction:**

Mathematical exploration is an important part of the International Baccalaureate (IB) program, and writing a math Internal Assessment (IA) is a key requirement for students seeking to earn an IB diploma. The IA provides an opportunity for students to delve deeper into a mathematical concept of their choice, and to demonstrate their understanding and application of mathematical concepts and techniques.

Mathematics Analysis and Approaches (Math AA) is one of the two courses offered by the IB for Mathematics. Math AA focuses on developing students’ mathematical knowledge and skills in pure mathematics, including algebra, calculus, geometry, and trigonometry. Mathematics Applications and Interpretation (Math AI) is the other course that focuses on applying mathematics to real-world contexts, including statistics, probability, and modeling.

Both Math AA and Math AI require students to complete an IA by the end of the course and the IA is an extremely important part of a student’s math grade. However, writing a successful math IA requires more than just a solid grasp of **mathematical concepts**; it also requires careful planning and structure. In this blog post, we’ll explore the key elements that make up a strong math IA structure, including the introduction, mathematical background, exploration, and conclusion. We’ll discuss the purpose and content of each section, and provide tips and strategies for crafting a clear and effective IA. Whether you’re just getting started on your math IA, or are looking to refine and improve your existing work, this blog post will provide valuable insights and guidance to help you succeed.

**Structure of the Math IA:**

The IA for both Math AA and AI follows a similar structure. Students are required to select a topic from one of the four areas of study: Algebra, Functions and Equations, Circular Functions and Trigonometry, and Calculus. The IA should be approximately 12-20 pages long, excluding appendices, and must contain the following components:

**Introduction**:

In this section, you should introduce the topic you have chosen and explain why it is important or interesting. You should also include a clear statement of the aim and objectives of your IA.

**Bad Introduction:**

In this math IA, I will explore the topic of calculus. Calculus is a branch of mathematics that deals with rates of change and slopes of curves. I will look at some calculus concepts such as derivatives and integrals and try to explain them in simple terms. This will help readers understand the basics of calculus.

*Why it’s bad:*

*This introduction is too broad and does not provide a clear focus for the IA. It is also not engaging and doesn’t grab the reader’s attention.*

**Good Introduction:**

Have you ever wondered how the trajectory of a soccer ball is calculated after it’s been kicked? What about the math behind predicting the weather or designing roller coasters? All of these scenarios involve the application of calculus, a branch of mathematics that allows us to understand the behavior of curves and rates of change. In this IA, I will explore the concepts of calculus through real-world examples, demonstrating how it can be used to solve complex problems and enhance our understanding of the world around us.

*Why it’s good:*

*This introduction sets up the IA by engaging the reader with real-world scenarios and providing a clear focus on exploring calculus. It also hints at the importance and relevance of the topic, making the reader interested to learn more.*

**Mathematical Background**:

This section should include a brief overview of the mathematical concepts and techniques that you will be using in your IA. It is important to show a deep understanding of mathematical theory and to provide clear explanations of any equations or formulas you will be using.

**Bad Mathematical Background:**

- The background is too general and does not provide specific information on the mathematical concepts that will be used in the IA.
- It does not indicate the level of mathematical knowledge that is required to understand the IA.
- It does not provide any context or motivation for why these mathematical concepts are important.

**Good Mathematical Background:**

- The background provides specific information on the mathematical concepts that will be used in the IA.
- It indicates the level of mathematical knowledge that is required to understand the IA.
- It provides context and motivation for why these mathematical concepts are important.
- It is written in clear and concise language that is easy to understand.
- It may include references or additional resources that readers can use to refresh their knowledge of the relevant mathematical concepts.

**Exploration**:

This is the most substantial section of your IA and should demonstrate your ability to use mathematical tools to solve a real-world problem. You should explain the methodology used to investigate the problem and provide clear evidence of your calculations and reasoning. It is important to show your work clearly, including any graphs or diagrams you create.

**Bad Exploration Section:**

- The exploration is too simplistic and does not demonstrate a deep understanding of the mathematical concepts being discussed.
- The methodology used to explore the mathematical concepts is not clearly described, making it difficult for readers to follow the logic of the argument.
- There is no data or evidence provided to support the conclusions drawn in the exploration.
- The exploration does not connect the mathematical concepts to any real-world applications or examples.

**Good Exploration Section:**

- The exploration demonstrates a deep understanding of the mathematical concepts being discussed, and provides a clear explanation of how these concepts are being applied.
- The methodology used to explore the mathematical concepts is clearly described, making it easy for readers to follow the logic of the argument.
- The exploration is supported by data or evidence, which helps to strengthen the argument being made.
- The exploration connects the mathematical concepts to real-world applications or examples, which helps to provide context and motivation for why these concepts are important.
- The exploration may include diagrams, graphs, or other visual aids to help illustrate the mathematical concepts being discussed.
- The exploration may include examples of how the mathematical concepts have been used in previous research, or how they can be applied to solve real-world problems.

**Conclusion**:

In this section, you should summarize your findings and explain how they relate to the original aim and objectives of your IA. You should also reflect on any limitations of your investigation and suggest potential areas for further research.

**Bad Conclusion:**

In conclusion, we have explored the concepts of calculus and shown how they can be applied to solve problems. We have seen that calculus is a powerful tool for understanding rates of change and curves, and can be used to model a wide range of phenomena. We hope that this IA has given you a better understanding of calculus and its applications.

**Why it’s bad:**

*This conclusion is too general and does not summarize the specific findings of the IA. It also does not provide any insights or recommendations for further research or applications.*

**Good Conclusion:**

In conclusion, we have demonstrated how linear regression can be used to model real-world data and make predictions. Our analysis has shown that this technique can be used to effectively model and predict various phenomena, from the price of stocks to the spread of infectious diseases. However, there are still many areas where linear regression can be improved, such as accounting for nonlinear relationships or dealing with outliers. Future research in this area could explore these issues and develop new techniques for improving the accuracy and reliability of linear regression models.

**Why it’s good:**

*This conclusion provides a clear summary of the main findings of the IA, and suggests potential avenues for further research and development. It also highlights the relevance and importance of the topic, and indicates that there is still much to be learned and discovered in this field.*

**References**:

It is important to include a list of all sources that you have used in your IA. This includes any textbooks, articles, websites, or other resources that you have consulted. You should also provide clear citations within your IA to show where you have used information from these sources.

**Here are a few pointers on how to format your IA well:**

The title page should include the title of your IA, your name, your candidate number, the date, and the word count.*Title Page:*All pages of your IA should be numbered, including the title page and appendices.*Page Numbers:*Use a clear, legible font such as Times New Roman or Arial in size 12. Use a larger font for headings and subheadings.*Font and Size:*Use double line spacing throughout your IA, except in tables, equations, and diagrams.*Line Spacing:*Any additional material, such as raw data, should be included in appendices at the end of your IA. Make sure to refer to these appendices in the main body of your IA.*Appendices:*All graphs and diagrams should be labeled clearly and should have appropriate titles and axes. Use colors and shapes to distinguish between different data points or lines. Make sure that the labels and titles are legible and that the scales are appropriate for the data being presented.*Graphs and Diagrams:*All equations should be presented clearly, with variables and constants clearly labeled. Use appropriate notation and make sure that equations are properly formatted, with fractions, exponents, and other mathematical symbols clearly presented.*Equations:*All tables should be labeled clearly and should have appropriate column headings. Make sure that the tables are presented in a logical order and that any units of measurement are clearly indicated.*Tables:*Use clear and concise language throughout your IA. Avoid using jargon or technical terms unless they are necessary. Make sure that your writing is grammatically correct and that you use appropriate punctuation.*Language:*Finally, make sure that you proofread your IA carefully before submitting it. Check for spelling and grammar errors, and make sure that all of your equations, graphs, and tables are properly labeled and formatted. It can be helpful to have someone else read through your IA to check for errors or inconsistencies.*Proofreading:*

**Here are a few additional tips to help you write a successful IB Math AA and AI IA:**

You will be spending a lot of time on your IA, so it is important to choose a topic that you find engaging and challenging. This will help you stay motivated and focused throughout the process.*Choose a topic that you are interested in:*In order to demonstrate a deep understanding of the mathematical concepts and techniques used in your IA, it is important to use a variety of sources and to cite them properly. This demonstrates that you have researched your topic thoroughly and are able to apply the knowledge you have gained from a range of different sources. Textbooks and academic journals can provide a strong foundation for your research, as they often present complex mathematical concepts in a clear and concise manner. Reputable websites, such as those associated with educational institutions or professional organizations, can also provide useful information and insights.*Use a variety of sources:*it is important to not only arrive at the correct answer to a problem but also to show the steps you took to reach that answer. This process is referred to as “showing your work.” By including all of your calculations, graphs, and diagrams in your solution, you are demonstrating your thought process and reasoning. Additionally, including visual aids such as graphs and diagrams can make your solution more clear and effective. They can help to illustrate complex mathematical concepts and make them easier to understand. It is important to label all of your graphs and diagrams and to include a caption or explanation that describes their significance. This will help your audience understand the context of your work and the purpose of your visual aids.*Show your working:*The IA is an opportunity to apply your mathematical knowledge and skills to real-world problems. Whenever possible, use examples that are relevant to your own life or that demonstrate the practical applications of the mathematical concepts you are studying. By selecting examples that are relevant and meaningful to you, you can not only create a more engaging project but also deepen your understanding of the mathematical concepts you are studying. The IA provides an opportunity to demonstrate the practical applications of mathematics and showcase the importance of this field in solving real-world problems.*Use real-world examples:*Don’t be afraid to ask your teacher or classmates for feedback on your IA. They may be able to provide helpful suggestions for improving your work or catching any errors that you may have missed. It is important to approach feedback with an open mind and a willingness to make changes to your work. Be receptive to constructive criticism and take the time to carefully consider any suggestions or comments that are made. Use this feedback to refine your ideas, improve your analysis, and make your IA more effective overall.*Seek feedback:*

In conclusion, the structure of your IB Math AI and AA IA is crucial to the success of your project. Your IA should be well-organized, clearly written, and contain all necessary components, such as an introduction, background information, data collection and analysis, and a conclusion. By following the guidelines provided by the IB, selecting an appropriate topic, and using effective mathematical tools, you can create a compelling IA that showcases your skills and knowledge in mathematics. Remember to plan ahead, manage your time effectively, and seek help if needed. With these tips in mind, you can excel in your IB Math IA and AA IA and achieve your academic goals.