how to write conclusion for math ia

IB Math IA Guide - Math IA Shenanigans That No One Will Tell Yeh!

Ace Your IB Math IA with our ultimate guide for 2023! Get top marks and ace your IA with ease. Discover proven tips, tricks and strategies to nail your Math IA today!

Table of content

Criterion a - mathematical presentation: (levels- 0, 1, 2, 3, 4), criterion b - mathematical communication: (levels- 0, 1, 2, 3, 4), criterion c - personal engagement: (levels - 0, 1, 2, 3), criterion d - reflection: (levels - 0, 1, 2, 3), criterion e - use of mathematics: (levels - 0, 1, 2, 3, 4, 5, 6), introduction, body of your exploration.

IB Math students will tell you how they’re always on the edge of their seats for some help, but IB Math IA takes that anxiety to an entirely different level. The reality is far from frightening; nonetheless, IB Math IA can be handled well with a unique IB Math IA topic in hand and lots of coffee! 

But does that guarantee a dependable 7?

It takes more than just a perfect IB Math IA topic to ace.

How’s that, you’d ask.

From researching several IB Math IA examples to planning the mathematical working of your exploration, your IB Math IA structure will get you into trouble if you don’t give it the time it demands. With all the varied content available in bulk online, the process is bound to become anything but easy.

But worry not!

You are at the right place - The Ultimate Guide to IB Math IA!

This article covers IB Math IA rubrics, process key pointers, the structure of the investigation, and interesting IB Math IA topics that will stimulate your mind and help you begin your exploration!

You should also know about the updated course structure of IB Mathematics. Students are allowed to opt for any one of the following four courses in Math:

• Analysis and Approaches ( AA HL ) - Higher Level
• Analysis and Approaches  (AA SL)  - Standard Level
• Applications and Interpretation  (AI HL)  - Higher Level
• Applications and Interpretation  (AI SL)  - Standard Level

For more information about choosing your course, check out our latest article:  Confused about IB Math AA & IB Math AI?

Also, here’s a great surprise for all students!

Patrick Jones , the creator of PatrickJMT Math Videos, acknowledged as the best Math teacher globally with over 1.2 million subscribers on YouTube, has gotten on board with our team at Nail IB! How great is that! He is already working on creating an entire Nail IB video course, and it will prove to be a wholesome guide for you as you tread on your IB Math journey! You should check out his excellent, world-renowned content  here !

Before moving any further, we insist you check out our Free IB Resources for  IB Mathematics SL and IB Mathematics HL.  These are specially assembled for your benefit and will surely assist you on your IB Math journey!

For an absolute hold on IB Math, check out our premium notes designed and curated specially for you, be it  IB Mathematics SL  or  IB Mathematics HL . These bundles are not just limited to messages but offer past year papers and How-to Guides for Extended Essays, Internal Assessments, and more; examples included! You’re in for a smooth ride with these by your side;)

You can also stream our webinar on  How to Write an IB Math Internal Assessment in under 30 minutes  and hear directly from a recent IB graduate to understand the fundamental pointers and some fantastic hacks to lay the foundation of your IB Math IA. Getting the proper guidance ensures you a 7 in the subject you have feared for too long. Click  here  to watch it now!

First things first, let’s understand the criteria. Unless we acknowledge the requirements against which our exploration is scored, it’ll be equivalent to a shot in the dark. The conditions, irrespective of whether you opt for SL, HL (AA or AI), are as follows:

Assessment is done on the conciseness, brevity, and clarity/coherence of your investigation. The proper structure must be given to your IA. As per IB guidelines,

a coherent exploration is,

• Logically developed
• Easy to follow and,
• Meets the Aim.

Also, a well-organized exploration,

• Includes Introduction
• Describes the Aim of the investigation and
• Has a Conclusion

Assessment is done on the appropriateness of the mathematical terminology, notation, and symbols used to progress the exploration. Marks and notes should be correctly used as are used in IB textbooks. For example, x2 should not be written as x^2.

If used, different mathematical representation tools such as tables, graphs, and diagrams must be relevant to the working and be commented on/explained well. Avoid inconsistent use of Mathematical terminology. Applying ICT Tools(for example-  GeoGebra and Desmos ) should be made wisely. For Calculations, Graphic Display Calculators can also be used, but that doesn't outdo math formulas' importance.

Assessment is done on the personal involvement shown. The sure shot way to ensure Engagement is, first and foremost, by going ahead with a topic that interests you (something unique or that affects real-life situations). Personal Engagement is seen throughout the exploration by:

• Independent thinking and creativity showed by the student
• Making the Math Idea your own
• Investigating the idea from varied perspectives
• Exploring different possibilities

Avoid portraying superficial interest. Opportunities for demonstration of personal Engagement should be noticed.

Assessment is done on the evaluation and analysis of the investigation. Mentioning the significance of your exploration results, discussing possible limitations, and justifying why you chose the procedure you did can portray a fair reflection of the IA. Merely explaining your results will get you only a score of 1 out of 3. According to IB guidelines, a review should be meaningful and critical.

A meaningful reflection includes

• Considering limitations in the work
• Comparing different Mathematical Approaches
• Linking to the Aim
• Commenting on the Learning

A critical reflection entails,

• Considering  What Next
• Discussing the implications of the results
• Discussing the strengths and weaknesses of the approaches
• Considering different perspectives

Reflection is an analysis of the student's work, seen throughout the exploration, not just the Conclusion.

Assessment is done on the implementation of Mathematics in the IA. It is essential to understand that the Math used should be on par with the course, nothing too simple and nothing you need help understanding. Also, the Mathematics used should be fully understood and engrained by you. Unfamiliar Mathematics, if used, should be explained well by giving personal examples. As per IB guidelines, students are expected to produce work that is,

• Commensurate  with the level of the course(should either be part of the syllabus at a similar level or slightly beyond)
• Relevant  Mathematics used means Math which supports the development of the exploration towards the completion of its Aim

To score higher levels in Criterion E, it is crucial to understand the meaning of the following terms:

• Precise  Mathematics is error-free and uses an appropriate level of accuracy at all times
• Sophistication  means the Math should be commensurate with the HL syllabus
• Rigour  involves clarity of logic and language when making Mathematical arguments and calculations.

Here  is a fully annotated sample IB Math IA, going through which you can gain a lot of insight  (Read the annotations properly).

Another example,  the Breaking the Code  investigation(fully annotated)- is given here for IB Mathematics HL. The IA explores encryption and decryption in the context of Mathematics.

• Going through the report, we see that the document needs more structure in the beginning since the Introduction does not mention the Rationale or Aim. You don't want to be committing such a blunder.
• Moving further, we see that the body encompasses Math, which is well explained, thereby excelling on criterion E when graded on the SL scale. To excel in HL, the math should have been more rigorous than descriptive.
• Toward the end, the report needs to include a Reflection on the results obtained, which doesn't fare well for the IA. Potential implications of the topic need to be included, and the Conclusion seems bland.

Another interesting annotated sample- for IB Mathematics SL- Regularisation of Irregular Verbs: When can I use the words swimming and know correctly? is given for reference  here . Understand which key points have been missed and which have been taken care of. The more you go through sample IAs, the better your chances of preparing an investigation that'll be scored well.

Like any other exploration, your IB Math IA expects you to give it a fair shot of effort and interest. If you see it as a burden, it will undoubtedly become one. Equally important is to draw an analogy between the topic of your choice and the math involved. Taking care of these points in general, let's understand all that goes into making one's IB Math IA (a brief outline):

• It can have a personal story attached; mention it in your Introduction. If not, explain how the topic underhand impacts real-world situations and motivates you to land on it. Your passion for the IA idea shows in your work, and you don't want to be doing your investigation just for the sake of it.
• f you're curious about  Fibonacci numbers, the Golden ratio , and nature alike, try looking for relationships among them on the Internet. This will entail going through many research papers, publications, and journals and finally settling on mathematical findings and proofs that will help you investigate that particular something you wish to explore. For example, if you want to study how  Traffic Jams have math running in the background , research it in detail since Traffic snarls are an imminent pain for us all. Only you will have to pick up parts you think are relevant and understandable to you.
• ​​​​​​​​​​​​​​ The cycle of Inquiry, Action, and Reflection in learning is vital. Learning the implication of  Plagiarism, Collusion, and Duplication of Work  is essential to keep one's IA transparent and impressive.
• ​​​​​​​​​​​​​​ Besides citing references in the bibliography section, ensure you include it in the body as a footnote or in the exploration itself. Citing credible sources shows how transparent your work is and helps examiners cross-check for correctness. Acknowledging the author's work is essential to the IA-making process.

With this, let's discuss the Structure/Layout of the Investigation. There are numerous guidelines available all over the Internet. Regarding the IA length, though you should keep 6 -12 pages as the prescribed length, your focus should be on including all that pertains to your idea and ruling out everything that's not. So don't set out with a mindset to refrain from exceeding six pages; set out to include everything you know needs to be. Similarly, it is advised to make sure the Math used is suitable for SL and HL levels.

Without any further adieu, let's highlight what the layout of the IA should look like:

• Sets the background of your exploration and gives an argument for your topic choice. Your Rationale tells why you chose so and so topic.
• ​​​​​​​​​​​​​​ This is where you define your investigation's objective and tell what you wish to achieve with this idea.
• ​​​​​​​​​​​​​​ Let's say, for instance, you have opted for Math HL; for a simple mathematical investigation that scores six on the Math SL grade scale, you might end up with a mere four on the HL. The difficulty of your opted Math subject should reflect in your Internal Assessment. It would help if you also outlined the areas of mathematics you will cover in your investigation.
• Elaborate on the method you used for the exploration and justify why you chose to proceed with that particular method.
• Use relevant mathematical tools like labelled graphs, charts, etc., for your mathematical work and explain them in the IA context.
• State your results relating them to the Aim of your Internal Assessment. The significance and impact of these results should be highlighted, as well. In addition, briefly tell how the exploration was helpful to you and all you have gained from it. Possibilities of extension should be mentioned. The bibliography is for you to cite the sources used by you in the making of the IA. We suggest you use the  Citation Machine  for additional guidance in the bibliography.

Now that we're comfortable with the IB Math IA structure let's look at some interesting IB Math IA topics that will get your creative juices flowing and help kickstart your Math IA journey today!

• Simulating models to study and forecast weather patterns. (You could come up with a personal account that led you to land on something like this)
• Exploring the different probabilities associated with a game of your choice; for example, Solitaire(if you're a game buff).
• Investigating the  Math associated with the Global Positioning System(GPS)  and the intricacies of the technology involved.
• Exploring  Fermat's little theorem  or  Goldbach's conjecture (one of the most significant unsolved problems in Mathematics).
• Finding the volume and surface area of an egg, apple, mango or any other real-world object using Calculus's power (Simulation could be used). A good IA on  modelling  manages to score an easy 15-16 marks out of 20.
• Investigating the structural designs of bridges that prevent collapse under loading.
• Studying complex roots graphically.
• Exploring how guitar frets are arranged in Pythagoras Ratios.
• Comparing which will prove beneficial: lump-sum payment of a lottery prize or fee done in instalments?
• Understanding how ISBN codes and Credit Card Codes can be cracked.

And that's a wrap!

Just like any other IA, IB Mathematics IA needs to be started early so that you don't end up compiling just anything at the last minute!

Give it the time it needs, and it will surely pay off. It might seem heavy, but once you decide to pursue an idea of your liking, there will be no turning back! Keep in mind the essential pointers and win the battle courageously!

Want some A-quality guidance? Look no further; at Nail IB, we have assembled premium content for you to ace your IBs, and you should check out our resources for a smooth IB experience. Click  here  for top-notch IB resources or to assess how your prep is going! Our exclusive Nail IB course, created by Patrick Jones, will be out soon too, so stay tuned, as there is no way you would want to miss the holy grail every Math IB student wants!!

This article will serve as a solid foundation for your Math IB Internal Assessment.

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An Ultimate Handbook for IB Math Internal Assessment (IA)

Whenever it comes to IB Maths Internal assessment students are mostly seen as tense about how to do well in their IB Maths IAs.

Blen tells you that you can handle the dreadful dilemma of IB Maths IA in the savviest by picking the right Math IA topic.

Our expert tutor at Blen always advises students to give sufficient time to their IB IA structure because it surely needs good mathematical work and tremendous effort.

Before moving on to the quick guide, we would like to share the current course structure of IB Mathematics with you. From the four courses provided below students are required to pick one.

AA HL, Analysis and Approaches, Higher level.

Aa sl, analysis and approaches, standard level., ai hl, applications and interpretation, higher level, ai sl, applications and interpretation, standard level.

Here is a quick guide to the criteria of IB Maths Internal assessment regardless of AA or AI (SL, HL).

A- Mathematical Presentation ( Levels 0 to 4)

IBDP Math IA is based on how clear, concise and brief your investigation is with a proper structure.

According to IB guidelines,

a coherent exploration is,

following a logic

Comprehensible, fulfilling the aim.

Also, a well-organized investigation,

have introduction 

Specifies the aim, ends with a conclusion,  b - mathematical communication: (levels 0 to 4).

This is based on the right use of mathematical terminology, symbols, and notations. Such as avoiding using x2 like x^2. If you are adding any graphs, tables, or flow sheets, they must be relevant to your work.

 C - Personal Engagement: (Levels 0 to 3)

Assessment must show your Engagement based on:

thinking and creativity

Using your idea, exploration from a different perspective, considering all the possibilities, d - reflection: (levels 0 to 3)  .

Assessment must be based on meaningful reflections such as:

use of different mathematical approaches

Connection to the aim, including strengths and weaknesses of the approaches, considering various perspectives, e - use of mathematics (levels - 0 to 6).

As per IB guidelines, students are expected to produce work that is,

According to IB guidelines, your work should be:

Proportional

Following clarity and logic.

Some significant points to be taken into notice before making an IB Math IA:

Brainstorm the topic.

Do well research., understand and follow academic honesty., don't leave the referencing..

Here is a sample layout of a Math IA:

  Introduction:

It must be the rationale, explains the objective and plan of action used in exploration.

Explaining the methodology used with proper justification, and use of relevant graphs with an explanation.

Conclusion:

End your assessment by relating the results to the aim, and explain how this exploration is beneficial for you. Also, don't forget to cite the sources.

Blen hopes this article will be useful for your IB Math IA .

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How to Structure & Format Your Maths IA

• March 23, 2023

Table of Contents

• 1 Introduction:
• 2.1 Introduction:
• 2.2 Bad Introduction:
• 2.3 Good Introduction:
• 2.4 Mathematical Background:
• 2.5 Exploration:
• 2.6 Conclusion:
• 2.7 Good Conclusion:
• 2.8 References:

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 . Math IA examples can serve as valuable references when embarking on your IA journey.

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 . A well-crafted introduction, like in the following maths IA examples , can engage your readers and set the stage for your exploration:

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. You can find that in most Math IA examples or sample IAs often begin with a solid mathematical foundation:

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 look at some of t hese math ia exploration examples

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 . A poorly-crafted conclusion, like in the following math IA conclusion  examples , can leave a bad impression:

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.

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.

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:

• Title Page: The title page should include the title of your IA, your name, your candidate number, the date, and the word count.
• Page Numbers: All pages of your IA should be numbered, including the title page and appendices.
• Font and Size: Use a clear, legible font such as Times New Roman or Arial in size 12. Use a larger font for headings and subheadings.
• Line Spacing: Use double line spacing throughout your IA, except in tables, equations, and diagrams.
• Appendices: 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.
• Graphs and Diagrams: 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.
• Equations: 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.
• Tables: 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.
• Language: 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.
• Proofreading: 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.

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

• Choose a topic that you are interested in: 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.
• Use a variety of sources: 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.
• Show your working: 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.
• Use real-world examples: 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.
• Seek feedback: 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.

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.

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IB Maths IA: 60 Examples and Guidance

Charles Whitehouse

The International Baccalaureate Diploma Programme offers a variety of assessments for students, including Internal Assessments (IAs), which are pieces of coursework marked by students’ teachers. The Mathematics Internal Assessment follows the same assessment criteria across Mathematics Analysis and Approaches (AA) and Mathematics Application and Interpretation (AI). It forms 20% of a student’s Mathematics grade.

In this article, we will cover everything you need to know about the IB Mathematics IA, including the structure, assessment criteria, and some tips for success.

What is the Mathematics IA?

The Maths IA is an individual exploration of an area of mathematics, based on the student’s own work with guidance from their teacher. Mathematical communication is an important part of the IA, which should be demonstrated through both effective written communication and use of formulae, diagrams, tables, and graphs. The exploration should be 12 to 20 pages long and students will spend 10 to 15 hours on the work.

Even A-Level Maths tutors and A-Level Further Maths tutors have found the concept of an IA-like component in IB Maths to be both challenging and rewarding, enriching the learning experience.

To learn more about the IB Maths Internal Assessment, you can have a look at the IB Maths AA resources as well as the IB Maths AI resources :

• IB Maths AA Past Papers
• IB Maths AI Past Papers

What are the assessment criteria?

Like most IB IAs, the IB Maths IA is marked on a group of 5 criteria which add up to 20 marks. Online Maths tutors recommend to look through these carefully before and during your investigation, to ensure that you are hitting the criteria to maximise your mark.

Source : IB Mathematics Applications and Interpretation Guide

Criterion A: Communication (4 marks) – This refers to the organisation and coherence of your work, and the clarity of your explanations. The investigation should be coherent, well-organized, and concise.

Criterion B: Mathematical Presentation (4 marks) – This refers to how well you use mathematical language, including notation, symbols and terminology. Your notation should be accurate, sophisticated, and consistent. Define your key terms and present your data in a varied but proper way (including labelling those graphs).

Criterion C: Personal Engagement (3 marks) – There should be evidence of outstanding personal engagement in the IA. This is primarily demonstrated through showing unique thinking, not just repeating analysis found in textbooks. This can be evidenced through analysing independently or creatively, presenting mathematical ideas in their own way, exploring the topic from different perspectives, making and testing predictions.

Criterion D: Reflection (3 marks) – This refers to how you evaluate both your sources and the strengths and weaknesses of any methodology you use. There should be “substantial evidence of critical reflection”. This could be demonstrated by considering what another stage of investigation could be, discussing implications of results, discussing strengths and weaknesses of approaches, and considering different perspectives.

Criterion E: Use of Mathematics (6 marks ) –

Note that only 6 marks are available for the actual use of mathematics! The focus of the investigation is on explaining well and analysing with genuine, personal curiosity. The level of mathematics expected also depends on the level the subject is studied at: Standard Level students’ maths is expected to be “correct”, while Higher Level students’ maths is expected to be “precise” and demonstrate “sophistication and rigour”.

Examiners are primarily looking for thorough understanding, which also requires clear communication of the principles behind the mathematics used - not just coming to the right answer.

Have a look at our comprehensive set resources for IB Maths developed by expert IB teachers and examiners!

- IB Maths AI SL Study Notes

- IB Maths AI HL Study Notes

- IB Maths AA SL Study Notes

- IB Maths AA HL Study Notes

What are some example research questions?

Students should choose a research area that they are interested in and have a comprehensive understanding of. Often, student may choose to consult with an expert IB Maths tutor to help them decide a good question. It should have a link to something of personal interest, as indicated by Criterion C. Popular topics include Calculus, Algebra and Number (proof), Geometry, Statistics, and Probability, or Physics. Some students make links between Math and other subjects – a good way to combine knowledge from your other IB courses!

Here are examples with details of potential research questions that could inspire your Mathematics IA:

1 - Investigating the properties of fractals and their relationship to chaos theory.

Use computer software or mathematical equations to generate and analyze fractals. Explore the patterns and properties of the fractals, such as self-similarity and complexity. Investigate how changes in the initial conditions or parameters affect the resulting fractals. Analyze the relationship between fractals and chaos theory, and how fractals can be used to model chaotic systems. Present findings through visual representations and data analysis.

2 - Analyzing the behavior of recursive sequences and their applications in computer science and cryptography.

Use mathematical formulas to generate recursive sequences and analyze their behavior. This could involve plotting the sequences and observing patterns, finding closed-form expressions for the sequences, and exploring their applications in computer science and cryptography. For example, recursive sequences can be used in algorithms for sorting and searching data, and in encryption methods such as the Fibonacci cipher. The results of the analysis could be presented in a research paper or presentation.

3 - Exploring the properties of different types of differential equations and their applications in physics and engineering.

Conduct research on the different types of differential equations and their applications in physics and engineering. This could involve studying examples of differential equations used in fields such as fluid dynamics, electromagnetism, and quantum mechanics. The properties of each type of differential equation could be analyzed, such as their order, linearity, and homogeneity. The applications of each type of differential equation could also be explored, such as how they are used to model physical systems and solve engineering problems. The findings could be presented in a report or presentation.

4 - Investigating the properties of chaotic dynamical systems and their applications in physics and biology.

Use computer simulations to model chaotic dynamical systems and explore their behavior. This could involve studying the Lorenz attractor, the logistic map, or other well-known examples of chaotic systems. The simulations could be used to investigate the sensitivity of the systems to initial conditions, the presence of strange attractors, and other key features of chaotic dynamics. The results could then be applied to real-world systems in physics and biology, such as weather patterns, population dynamics, or chemical reactions.

5 - Designing an optimized route for a delivery service to minimize travel time and fuel costs.

Use a computer program or algorithm to analyze data on the locations of delivery destinations and the most efficient routes to reach them. The program would need to take into account factors such as traffic patterns, road conditions, and the size and weight of the packages being delivered. The output would be a map or list of optimized delivery routes that minimize travel time and fuel costs. This could be used to improve the efficiency and profitability of the delivery service.

6 - Developing a model to predict the spread of infectious diseases in a population.

Collect data on the population size, infection rate, and transmission rate of the disease in question. Use this data to create a mathematical model that simulates the spread of the disease over time. The model should take into account factors such as population density, age distribution, and vaccination rates. The accuracy of the model can be tested by comparing its predictions to real-world data on the spread of the disease. The model can be used to explore different scenarios, such as the impact of different vaccination strategies or the effectiveness of quarantine measures.

7 - Investigating the relationship between different geometric shapes and their properties.

Conduct a series of experiments in which different geometric shapes are tested for various properties such as volume, surface area, and weight. The data collected could then be analyzed to determine if there is a relationship between the shape of an object and its properties. This could involve creating 3D models of the shapes using computer software, or physically measuring the shapes using laboratory equipment. The results could be presented in a graph or chart to illustrate any trends or patterns that emerge.

8 - Analyzing the behavior of projectile motion and its applications in physics.

Conduct experiments in which a projectile is launched at different angles and velocities, and its trajectory is tracked using high-speed cameras or other measurement devices. The data collected can be used to analyze the motion of the projectile and determine its velocity, acceleration, and other physical properties. This information can then be applied to real-world scenarios, such as designing rockets or calculating the trajectory of a ball in sports. Additionally, the behavior of projectile motion can be studied in different environments, such as in the presence of air resistance or in a vacuum, to better understand its applications in physics.

9 - Developing a model to predict the path of a planet based on gravitational forces.

Collect data on the mass, position, and velocity of the planet at a given time. Use the law of gravitation to calculate the gravitational forces acting on the planet from other celestial bodies in the system. Use this information to predict the path of the planet over time, taking into account any changes in velocity or direction caused by gravitational forces. The accuracy of the model could be tested by comparing its predictions to observations of the planet's actual path.

10 - Investigating the properties of conic sections and their applications in geometry and physics.

Use mathematical equations to explore the properties of conic sections such as circles, ellipses, parabolas, and hyperbolas. Investigate their applications in geometry, such as in the construction of satellite dishes and reflectors, and in physics, such as in the orbits of planets and comets. Develop models and simulations to demonstrate these applications and their impact on real-world scenarios.

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11 - Modeling the spread of a virus through a population and analyzing the effectiveness of different intervention strategies.

Develop a mathematical model that simulates the spread of the virus through a population. The model would need to take into account factors such as the infectiousness of the virus, the rate of transmission between individuals, and the effectiveness of different intervention strategies such as social distancing or vaccination. The model could then be used to analyze the effectiveness of different intervention strategies and predict the potential impact of future outbreaks. The output of the model would be a set of data and visualizations that show the predicted spread of the virus and the effectiveness of different intervention strategies.

12 - Modeling the spread of a rumor or disease through a network and analyzing the impact of network topology.

Develop a mathematical model that simulates the spread of the rumor or disease through a network. The model should take into account factors such as the probability of transmission between individuals, the rate of recovery or decay of the rumor or disease, and the structure of the network. The impact of network topology could be analyzed by comparing the spread of the rumor or disease in different types of networks, such as random, scale-free, or small-world networks. The results of the simulation could be visualized using graphs or heat maps to show the spread of the rumor or disease over time.

13 - Developing a model to predict the growth of a population over time.

Collect data on the current population size and growth rate of the population over a period of time. Use this data to develop a mathematical model that predicts the population growth rate over time. The model could be tested by comparing its predictions to actual population growth data from previous years. The model could also be used to predict future population growth and to identify factors that may affect the population's growth rate.

14 - Investigating the properties of exponential functions and their applications in finance and economics.

Develop a mathematical model for an exponential function, including its domain and range, growth/decay rate, and asymptotes. Use this model to analyze real-world scenarios in finance and economics, such as compound interest, population growth, or stock market trends. Graph the function and interpret the results in terms of the original problem.

15 - Developing a model to predict the outcomes of a sporting event based on historical data and team statistics.

Collect historical data on the two teams playing in the sporting event, including their win-loss records, player statistics, and any relevant trends or patterns. Use this data to develop a statistical model that predicts the outcome of the game based on these factors. The model can then be tested and refined using additional data and feedback from experts in the field. The final output would be a prediction of the outcome of the game, along with a measure of the model's accuracy and any potential limitations or uncertainties.

16 - Analyzing the behavior of different types of sequences and their convergence or divergence.

Use mathematical models and computer simulations to analyze the behavior of different types of sequences. This would involve testing various sequences for convergence or divergence, and comparing their behavior under different conditions. The results of these simulations could be used to develop new mathematical theories and algorithms for analyzing sequences, and could have applications in fields such as computer science, physics, and engineering.

17 -Investigating the properties of different types of angles and their relationship to geometry and trigonometry.

Conduct a study of different types of angles, including acute, obtuse, right, and straight angles. Explore their properties, such as their degree measurements, relationships to other angles, and their use in geometry and trigonometry. This could involve creating visual aids, such as diagrams or graphs, to illustrate the concepts being studied. The results of the study could be presented in a report or presentation format, highlighting the key findings and insights gained from the investigation.

18 - Developing a model to predict the outcomes of a game based on probability theory.

Collect data on the outcomes of previous games, including the teams playing, the score, and any relevant factors such as weather conditions or injuries. Use this data to calculate the probability of each team winning based on various factors. Develop a model that takes into account these probabilities and predicts the outcome of future games. The model would need to be tested and refined using additional data and statistical analysis. The final output would be a reliable model for predicting the outcomes of games based on probability theory.

19 - Analyzing the behavior of different types of inequalities and their applications in algebra and calculus.

Create a graph to visually represent the behavior of different types of inequalities, such as linear, quadratic, and exponential inequalities. Use examples to demonstrate how these inequalities can be applied in algebra and calculus, such as finding the maximum or minimum value of a function subject to certain constraints. Additionally, provide real-world applications of these concepts, such as optimizing production processes or predicting population growth.

20 - Investigating the properties of different types of graphs and their applications in computer science and social science.

Conduct a literature review to identify the different types of graphs and their applications in computer science and social science. Develop a set of criteria for evaluating the effectiveness of different types of graphs in conveying information and insights. Use these criteria to analyze and compare several examples of graphs from each field. Based on the analysis, identify the most effective types of graphs for different types of data and research questions in each field. Develop guidelines for selecting and creating effective graphs in computer science and social science research.

21 - Analyzing the behavior of different types of matrices and their applications in linear algebra and quantum mechanics.

Conduct experiments to test the behavior of different types of matrices in linear algebra and quantum mechanics. For example, in linear algebra, the inverse of a matrix can be calculated and used to solve systems of linear equations. In quantum mechanics, matrices are used to represent quantum states and operators. The behavior of these matrices can be analyzed by performing matrix operations and observing the resulting changes in the system. The applications of these matrices in various fields can also be explored and analyzed.

22 - Developing a model to predict the outcomes of a business investment based on market trends and financial data.

Collect and analyze market trends and financial data relevant to the business investment. This could include factors such as industry growth rates, consumer demand, and financial statements of similar companies. Using this data, develop a predictive model that takes into account various variables and their potential impact on the investment. The model could be tested and refined using historical data and adjusted as new information becomes available. The output would be a prediction of the potential outcomes of the investment based on the model's calculations.

23 - Modeling the spread of a forest fire and analyzing the effectiveness of different containment strategies.

Develop a computer model of the forest fire spread using data on wind direction, temperature, humidity, and fuel load. The model could be calibrated using historical data on past forest fires to ensure its accuracy. Different containment strategies could then be simulated in the model, such as creating fire breaks or using water or fire retardant chemicals to slow the spread of the fire. The effectiveness of each strategy could be evaluated by comparing the simulated fire spread with and without the strategy in place.

24 - Analyzing the behavior of different types of optimization problems and their applications in engineering and computer science.

Conduct a literature review to identify different types of optimization problems and their applications in engineering and computer science. Develop a framework for analyzing the behavior of these problems, taking into account factors such as the size of the problem, the complexity of the solution space, and the type of optimization algorithm used. Apply this framework to a set of case studies, comparing the performance of different optimization algorithms and identifying best practices for solving different types of optimization problems.

25 - Investigating the properties of different types of geometric transformations and their applications in computer graphics and animation.

Conduct a literature review to gather information on the properties of different geometric transformations and their applications in computer graphics and animation. This could include translations, rotations, scaling, and shearing. Develop a set of test cases to demonstrate the use of these transformations in creating different types of graphics and animations. The results of these tests could be used to compare the effectiveness of different types of transformations for different applications. Additionally, the limitations and challenges associated with each transformation could be identified and discussed.

26 - Developing a model to predict the outcomes of an election based on polling data.

Collect polling data from a representative sample of the population and analyze it using statistical methods such as regression analysis or machine learning algorithms. The model would need to be trained on historical election data to ensure its accuracy. The output of the model would be a prediction of the likely outcome of the election based on the polling data and the historical trends. The model could also be used to identify key factors that are driving voter behavior and to test different scenarios, such as changes in voter turnout or shifts in public opinion.

27 - Analyzing the behavior of different types of integrals and their applications in calculus and physics.

Conduct a series of experiments to analyze the behavior of different types of integrals, such as definite and indefinite integrals, and their applications in calculus and physics. For example, one experiment could involve calculating the area under a curve using both definite and indefinite integrals and comparing the results. Another experiment could involve analyzing the motion of an object using calculus and determining its velocity and acceleration at different points in time. The results of these experiments could be used to develop a deeper understanding of the behavior of integrals and their applications in various fields.

28 - Studying the properties of different types of probability distributions and their applications in statistics and finance.

Conduct a literature review to gather information on different types of probability distributions and their applications in statistics and finance. Develop a theoretical framework to analyze the properties of these distributions and their relevance in different contexts. Use statistical software to simulate data and test the theoretical framework. Analyze the results and draw conclusions about the usefulness of different probability distributions in various applications.

29 - Developing a model to predict the outcomes of a marketing campaign based on consumer data.

Collect consumer data such as demographics, purchasing habits, and social media activity. Use this data to identify patterns and trends that can be used to develop a predictive model. The model would need to be trained using historical data on marketing campaigns and their outcomes. Once the model is trained, it can be used to predict the outcomes of future marketing campaigns based on the input data. The accuracy of the model can be tested by comparing its predictions to the actual outcomes of the campaigns.

30 - Investigating the properties of different types of symmetry and their relationship to geometry and physics.

Conduct a study of different types of symmetry, such as bilateral, radial, and rotational symmetry. This could involve creating models or diagrams of different symmetrical shapes and analyzing their properties, such as the number of axes of symmetry and the angles of rotation. The relationship between symmetry and geometry could be explored by examining how different symmetrical shapes can be used to create geometric patterns. The relationship between symmetry and physics could be investigated by exploring how symmetrical structures are used in physics, such as in the design of crystals or the study of particle physics.

31 - Modeling the spread of a rumor or news story through a population and analyzing its impact.

Develop a mathematical model that simulates the spread of the rumor or news story through a population. This model could take into account factors such as the initial number of people who hear the rumor, the rate at which they share it with others, and the likelihood that each person will believe and share the rumor. The impact of the rumor could be analyzed by looking at factors such as changes in people's behavior or attitudes, or the spread of related rumors or misinformation. The model could be refined and tested using data from real-world examples of rumor or news story propagation.

32 - Analyzing the behavior of different types of exponential growth and decay functions and their applications in science and engineering.

Use mathematical models to analyze the behavior of exponential growth and decay functions. This could involve studying the equations that describe these functions, graphing them to visualize their behavior, and analyzing how they are used in various fields such as biology, economics, and physics. Applications could include modeling population growth, decay of radioactive materials, and the spread of diseases. The results of this analysis could be used to inform decision-making in these fields and to develop more accurate models for predicting future trends.

33 - Modeling the spread of a pandemic through a population and analyzing the effectiveness of different intervention strategies.

Develop a mathematical model that simulates the spread of the pandemic through a population, taking into account factors such as the transmission rate, incubation period, and recovery rate. The model could be used to predict the number of cases over time and the effectiveness of different intervention strategies, such as social distancing, mask-wearing, and vaccination. The model would need to be validated using real-world data and adjusted as new information becomes available. The results of the analysis could be used to inform public health policies and interventions to control the spread of the pandemic.

34 - Analyzing the behavior of different types of functions and their applications in science and engineering.

Conduct a study of different types of functions, such as linear, quadratic, exponential, and logarithmic functions, and their applications in science and engineering. This could involve analyzing real-world data sets and modeling them using different types of functions to determine which function best fits the data. The study could also explore the use of functions in fields such as physics, chemistry, and economics, and how they are used to make predictions and solve problems. The results of the study could be presented in a report or presentation, highlighting the importance of understanding the behavior of different types of functions in various fields.

35 - Analyzing the behavior of different types of numerical methods for solving differential equations and their applications in science and engineering.

Conduct a series of simulations using different numerical methods for solving differential equations, such as Euler's method, Runge-Kutta methods, and finite difference methods. The simulations could involve modeling physical phenomena such as fluid flow, heat transfer, or chemical reactions. The accuracy and efficiency of each method could be compared by analyzing the error and computational time for each simulation. The results could be applied to optimize numerical methods for solving differential equations in various scientific and engineering applications.

36 - Developing a model to predict the outcomes of a medical treatment based on patient data and medical history.

Collect patient data and medical history, including demographic information, medical conditions, medications, and treatment outcomes. Use statistical analysis and machine learning algorithms to develop a predictive model that can accurately predict the outcomes of a medical treatment based on patient data and medical history. The model would need to be validated using a separate set of patient data to ensure its accuracy and reliability. The model could then be used to inform medical decision-making and improve patient outcomes.

37 - Analyzing the behavior of different types of linear regression models and their applications in analyzing trends in public opinion polls.

Collect data from public opinion polls on a particular topic of interest, such as political preferences or social attitudes. Use different types of linear regression models, such as simple linear regression, multiple linear regression, and logistic regression, to analyze the data and identify trends and patterns. Compare the performance of the different models and determine which one is most appropriate for the specific data set and research question. The results of the analysis could be used to make predictions or inform policy decisions.

38 - Developing a model to predict the growth of a startup company based on market trends and financial data.

Collect market trend data and financial data for a range of startup companies. Use statistical analysis to identify patterns and correlations between the data. Develop a predictive model based on these patterns and correlations, taking into account factors such as industry trends, competition, funding, and management. The model could be tested and refined using data from existing startups, and could be used to make predictions about the growth potential of new startups based on their characteristics and market conditions.

39 - Studying the properties of different types of statistical distributions and their applications in analyzing public health data.

Analyze public health data using different statistical distributions such as normal, Poisson, and binomial distributions. This would involve understanding the properties and characteristics of each distribution and selecting the appropriate one based on the nature of the data being analyzed. The data could then be plotted and analyzed using statistical software to identify trends and patterns, and to draw conclusions about the health outcomes being studied. The results could be presented in the form of graphs, tables, and statistical summaries.

40 - Investigating the properties of different types of series and their convergence or divergence.

Conduct a series of tests on different types of series, such as geometric, arithmetic, and harmonic series. Use mathematical formulas and calculations to determine their convergence or divergence. Graphs and charts could be used to visually represent the data and make comparisons between the different types of series. The results of the tests could be analyzed to draw conclusions about the properties of each type of series and their behavior under different conditions.

41 - Analyzing the behavior of different types of functions and their limits.

Graph the different types of functions and analyze their behavior as the input values approach certain limits. This could involve finding the asymptotes, determining if the function is continuous or discontinuous at certain points, and identifying any points of inflection. The results could be presented in a report or presentation, highlighting the similarities and differences between the different types of functions and their limits.

42 - Investigating the properties of different types of sets and their relationships in set theory.

Conduct a comparative analysis of different types of sets, such as finite and infinite sets, empty sets, and subsets. Investigate their properties, such as cardinality, intersection, union, and complement. Use diagrams and examples to illustrate the relationships between the different types of sets. This analysis could be used to develop a deeper understanding of set theory and its applications in various fields.

43 - Exploring the properties of different types of number systems, such as real, complex, or p-adic numbers.

Conduct a literature review of the properties of different number systems and compile a list of key characteristics and equations. Then, design a series of mathematical problems that test these properties for each type of number system. These problems could include solving equations, graphing functions, and analyzing patterns. The results of these problems could be used to compare and contrast the properties of each number system.

44 - Developing a model to predict the behavior of a physical system using calculus of variations.

Collect data on the physical system being studied, such as its initial state and any external factors that may affect its behavior. Use the calculus of variations to develop a mathematical model that predicts the system's behavior over time. The model can then be tested against real-world observations to determine its accuracy and refine the model as needed. The final output would be a reliable model that accurately predicts the behavior of the physical system.

45 - Investigating the properties of different types of topological spaces and their relationships in topology.

Conduct a study of the different types of topological spaces, including Euclidean spaces, metric spaces, and topological manifolds. Analyze their properties, such as compactness, connectedness, and continuity, and explore how they are related to each other. This could involve creating visual representations of the spaces, such as diagrams or models, and using mathematical tools to analyze their properties. The results of the study could be used to better understand the fundamental principles of topology and their applications in various fields.

46 - Analyzing the behavior of different types of integrals, such as line integrals or surface integrals, and their applications in physics and engineering.

Conduct a literature review on the different types of integrals and their applications in physics and engineering. This could include researching the use of line integrals in calculating work done by a force field or the use of surface integrals in calculating flux through a surface. Based on the findings, develop a research question or hypothesis related to the behavior of a specific type of integral and its application in a particular field. Design and conduct an experiment or simulation to test the hypothesis and analyze the results to draw conclusions about the behavior of the integral and its practical applications.

47 - Developing a model to predict the behavior of a chemical reaction using chemical kinetics.

Collect data on the initial concentrations of reactants, temperature, and other relevant factors for the chemical reaction being studied. Use this data to develop a mathematical model that predicts the behavior of the reaction over time. The model could be tested by comparing its predictions to actual experimental data collected during the reaction. Adjustments could be made to the model as needed to improve its accuracy. The final model could be used to predict the behavior of the reaction under different conditions or to optimize reaction conditions for maximum efficiency.

48 - Investigating the properties of different types of algebraic structures, such as groups, rings, or fields.

Conduct a thorough literature review to gather information on the properties of different algebraic structures. Develop a clear research question or hypothesis to guide the investigation. Choose a specific algebraic structure to focus on and collect data by performing calculations and analyzing examples. Compare and contrast the properties of the chosen algebraic structure with other types of algebraic structures to draw conclusions about their similarities and differences. Present findings in a clear and organized manner, using appropriate mathematical language and notation.

49 - Analyzing the behavior of different types of functions, such as trigonometric, logarithmic, or hyperbolic functions, and their applications in science and engineering.

Conduct a study of the behavior of different types of functions, such as trigonometric, logarithmic, or hyperbolic functions, and their applications in science and engineering. This study could involve analyzing real-world data sets and identifying which type of function best fits the data. The study could also involve creating models using different types of functions to predict future outcomes or behavior. The results of this study could be used to inform decision-making in fields such as engineering, finance, or physics.

50 - Developing a model to predict the behavior of a financial market using mathematical finance.

Collect data on the financial market, such as stock prices, interest rates, and economic indicators. Use mathematical models, such as stochastic calculus and differential equations, to analyze the data and develop a predictive model. The model could be tested and refined using historical data and validated using real-time data. The output would be a model that can be used to predict the behavior of the financial market and inform investment decisions.

51 - Investigating the properties of different types of complex systems and their behavior, such as network dynamics, agent-based models, or game theory.

Develop a simulation model for each type of complex system being investigated. The model would need to incorporate the relevant variables and interactions between agents or components of the system. The behavior of the system could then be observed and analyzed under different conditions or scenarios. This would allow for a better understanding of the properties and dynamics of each type of complex system and how they may behave in real-world situations.

52 - Analyzing the behavior of different types of partial differential equations and their applications in physics and engineering.

Conduct a literature review to identify different types of partial differential equations and their applications in physics and engineering. Develop mathematical models to simulate the behavior of these equations and analyze their solutions using numerical methods. The results of the analysis could be used to gain insights into the behavior of physical systems and to develop new technologies or improve existing ones. Examples of applications could include fluid dynamics, heat transfer, and electromagnetic fields.

53 - Developing a model to predict the behavior of a fluid using fluid dynamics.

Use computational fluid dynamics software to create a model of the fluid system being studied. The software would simulate the behavior of the fluid under different conditions, such as changes in flow rate or temperature. The model could be validated by comparing its predictions to experimental data. Once validated, the model could be used to predict the behavior of the fluid under different conditions, such as changes in the geometry of the system or the addition of different chemicals. These predictions could be used to optimize the design and operation of the fluid system.

54 - Investigating the properties of different types of geometric objects, such as manifolds or curves, and their applications in geometry and physics.

Conduct a literature review to gather information on the properties of different geometric objects and their applications in geometry and physics. This could involve researching existing theories and models, as well as conducting experiments or simulations to test these theories. The findings could then be analyzed and synthesized to draw conclusions about the properties of different geometric objects and their potential applications in various fields. This could also involve developing new theories or models based on the findings.

55 - Analyzing the behavior of different types of stochastic processes, such as random walks or Markov chains, and their applications in probability theory and statistics.

Conduct simulations of different stochastic processes using software such as R or Python. Analyze the behavior of the simulations and compare them to theoretical predictions. Use the results to draw conclusions about the properties of the different stochastic processes and their applications in probability theory and statistics. Additionally, explore real-world examples of stochastic processes, such as stock prices or weather patterns, and analyze their behavior using the concepts learned from the simulations.

56 - Developing a model to predict the behavior of a biological system using mathematical biology, such as population dynamics, epidemiology, or ecology.

Collect data on the biological system being studied, such as population size, birth and death rates, and environmental factors. Use this data to develop a mathematical model that can predict the behavior of the system over time. The model can be tested and refined using additional data and compared to real-world observations to ensure its accuracy. This model could be used to make predictions about the future behavior of the system, such as the spread of a disease or the impact of environmental changes on a population.

57 - Investigating the properties of different types of wave phenomena, such as sound waves or electromagnetic waves, and their applications in physics and engineering.

Conduct experiments to study the properties of different types of wave phenomena, such as frequency, wavelength, amplitude, and speed. These experiments could involve using instruments such as oscilloscopes, microphones, and antennas to measure and analyze the waves. Applications of these wave phenomena could include designing communication systems, medical imaging technologies, and musical instruments. The results of these experiments could be presented in a report or presentation, highlighting the key findings and their significance in physics and engineering.

58 - Analyzing the behavior of different types of optimization problems in dynamic environments, such as optimal control or dynamic programming.

Conduct simulations of different optimization algorithms in dynamic environments, using various scenarios and parameters to test their performance. The results could be analyzed to determine which algorithms are most effective in different types of dynamic environments and under what conditions. This information could be used to develop more efficient and effective optimization strategies for real-world applications.

59 - Developing a model to predict the behavior of a social network using social network analysis, such as centrality measures, community detection, or opinion dynamics.

Collect data on the social network, such as the number of connections between individuals, the frequency and content of interactions, and any changes in the network over time. Use social network analysis techniques to identify patterns and trends in the data, such as the most influential individuals, the formation of subgroups or communities, and the spread of opinions or behaviors. Develop a model based on these findings that can predict future behavior or changes in the network. The model could be tested and refined using additional data or by comparing its predictions to real-world outcomes.

60 - Investigating the properties of different types of algebraic curves and surfaces, such as elliptic curves or algebraic varieties, and their applications in algebraic geometry.

Conduct a literature review to gather information on the properties of different types of algebraic curves and surfaces. Use mathematical software to generate and analyze examples of these curves and surfaces. Explore their applications in algebraic geometry, such as in cryptography or coding theory. Present findings in a research paper or presentation.

How can I score highly?

Scoring highly in the mathematics internal assessment in the IB requires a combination of a thorough understanding of mathematical concepts and techniques, effective problem-solving skills, and clear and effective communication.

To achieve a high score, students should start by choosing a topic that interests them and that they can explore in depth. They should also take the time to plan and organize their report, making sure to include a clear introduction, a thorough development, and a thoughtful conclusion. The introduction in particular should demonstrate students’ genuine personal engagement with the topics.

Students should pay attention to the formal presentation and mathematical communication, making sure to use proper mathematical notation, correct grammar and spelling, and appropriate use of headings and subheadings.

Finally, students should make sure to engage with the problem and reflect on their own learning, and also make connections between different mathematical concepts and techniques. If they feel difficulty in these, then taking the help of an IB tutor can prove to be quite beneficial.

By following these steps, students can increase their chances of scoring highly on their mathematics internal assessment and contribute positively to their overall grade in the IB Mathematics course.

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Written by: Charles Whitehouse

Charles scored 45/45 on the International Baccalaureate and has six years' experience tutoring IB and IGCSE students and advising them with their university applications. He studied a double integrated Masters at Magdalen College Oxford and has worked as a research scientist and strategy consultant.

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The Full Guide to the IB Math IA Rubric

Hello, future IB scholars! When I first encountered the IB Math IA rubric, I was filled with questions. Now, with extensive IB experience under my belt, I’m here to share insights and tips on the Math IA format to streamline your path. Let’s get started!

Understanding the IB Math IA Rubric

Throughout my many interactions with the IB curriculum and students, it’s become undeniable that thoroughly comprehending the IB Math IA rubric isn’t just beneficial – it’s vital. When one fully understands the criteria and expectations, crafting an excellent Internal Assessment becomes more straightforward. Not only does it provide structure, but it also gives a clearer vision of what the IB examiners are actively seeking.

The Components of the IB Math IA Rubric

The IB Math Internal Assessment rubric has multiple components, each designed to assess a specific skill or area of knowledge.

• Personal Engagement . It evaluates the personal connection and significance of your chosen Math topic .
• Exploration . Here, the depth and breadth of your investigation are assessed, including the clarity of your aim, the context, and the methods employed.
• Analysis . The robustness of your mathematical processes, the accuracy of results, and the validity of conclusions drawn fall under this component.
• Evaluation . It focuses on the critical review of your approach and results, allowing for reflections on the validity and significance of your conclusions.
• Communication . As with any academic work, communication is critical. Your IA’s clarity, structure, and coherence will be judged here.

From my experience, one often overlooked aspect is the subtle differences between the IB Math HL IA rubric and the IB Math SL IA rubric. While both levels seek depth and clarity, the Higher Level demands a slightly more complex exploration and a deeper mathematical understanding.

Importance of Adhering to the IB Math Internal Assessment Rubric

Sticking to the rubric isn’t just about ticking off boxes. It provides a roadmap to excellence. By aligning your IA with the rubric’s criteria, you ensure that every effort you make is directed toward enhancing the quality of your work. This alignment often results in a higher grade, showcasing your mathematical prowess and commitment to the IB’s rigorous standards.

Yet, striking a balance in the quest for perfection is essential. Over-optimizing or excessively tweaking your IA to meet every minute detail can sometimes lead to a loss of personal touch and authenticity. So, while the rubric is an invaluable tool, it’s equally essential to let your genuine interest and passion for the subject shine through.

The Proper IB Math IA Format and Structure

Throughout my tenure with the IB, I’ve realized that a well-formulated Math IA format is akin to the skeleton of a building – it provides the necessary framework and structure. Ensuring your IA adheres to the correct Math IA structure lays the groundwork for a coherent and logically sequenced document.

Beginning with the Math IA Introduction

A compelling Math IA introduction not only introduces your topic but also captures the essence of your investigation. It isn’t merely an entry point; it’s your first handshake with the evaluator. But how to write a Math IA introduction? You need to demonstrate the relevance and significance of your chosen topic here. When you master writing a Math IA introduction, you set clear expectations and create a strong foundation for the following sections.

Building a Strong Math IA Outline

Think of your Math IA outline as the blueprint of your investigation. It provides a bird’s eye view of your approach, methods, and anticipated outcomes. By referring to a comprehensive Math IA guide, you can ensure your work mirrors the expectations the Math internal assessment structure sets. Here are some tips for creating IB Math IA outline:

• While it gives Maths IA structure, it’s not set in stone. As your research and writing progresses, you might find new avenues to explore or certain sections that need reworking. An outline allows for these adjustments while ensuring you remain on track.
• Your outline will also help you make seamless transitions between sections so the narrative flows smoothly. It ensures that readers can easily follow along.
• Once you have an outline, sharing your ideas with mentors, teachers, or peers for feedback becomes easier. Early feedback can be invaluable, offering insights or pointing out pitfalls you might have yet to consider.

In essence, your outline should serve as a roadmap, guiding readers seamlessly through the different stages of your investigation.

The Main Content and Word Count

Managing the Math IA word count can initially seem challenging, especially when you have so much to convey. However, from my extensive experience with the IB, I see this limitation as a genuinely hidden advantage. It prompts us to be concise, ensuring that every word and every sentence is purposeful.

Instead of perceiving the Math IA page limit as a restrictive boundary, it’s more beneficial to regard it as a tool that refines our focus. This constraint encourages students to highlight their most significant points, ensuring that irrelevant details are set aside. Effective communication, after all, is about getting to the point, not about length.

As you learn more about how to write a Math IA, several core guidelines emerge:

• Not every piece of information is of equal importance. Identify the essential aspects of your investigation and give them the attention they deserve.
• Avoid over-complicating your sentences. Aim for clear and concise expressions.
• Ensure every detail and every example aligns with your central research question or the purpose of your investigation.
• Your initial draft is just a starting point. Go back, enhance your content, clarify your expressions, and remove any redundancies to improve the quality of your IA .

The goal is not merely to fit within the word count but to create an engaging, informative document that showcases your profound understanding. So what to do in this case? Using the word count as a guiding principle , you can produce a Math IA that excels in content and clarity.

Topics to Read:

• IB Math IA Topics | 13 Different Options
• Comparing IB Math IA and Math AA: An In-Depth Look at the Differences and Similarities
• Math IA Grading Boundarie
• Breaking Down the Costs of Writing a Math IA: A Budgeting Guide for IB Students
• Mastering the IB Math IA Deadline for 2023: Time Management Tips and Tricks
• A Complete Guide to Comparing IB Math IA and Math AA
• IB Math IA vs Math AA

Designing the Math IA Cover Page

Though often overlooked, a cover page is pivotal in setting the tone. Your Math IA cover page should blend professionalism and personal touch, giving evaluators a glimpse of what lies within. Incorporating essential details such as your name, candidate number, and topic while ensuring visual appeal can make a notable difference.

Concluding the Math IA

Drawing your IA to a close is not just about ending your investigation. A meticulously written Math IA conclusion ties all your findings together, providing clear takeaways for the evaluator. It’s an opportunity to revisit the significance of your work, reflect on the process, and highlight its broader implications in mathematics.

Additional Tips and Information for Students

The world of IB is vast, and within it, the Math IA represents a critical component that assesses your mathematical skills and your ability to research, analyze, and present. Let’s get into some nuances and fun trivia!

IB Math SL IA Format vs. IB Math HL IA Format

The difference between the Standard Level (SL) and Higher Level (HL) formats isn’t merely about depth but approach, complexity, and topics chosen. While both demand clarity and rigorous investigation:

• SL IA . Typically, this format focuses more on real-world applications of mathematics. It’s about applying mathematical concepts to everyday scenarios or problems.
• HL IA . This format leans towards more profound theoretical concepts. Here, you’re expected to apply and explore abstract mathematical ideas, perhaps venturing into topics not directly covered in the syllabus.

For both formats, always remember that your choice of topic can set the tone for your entire investigation. Thus, choose wisely and pick something that genuinely interests you.

Don’t let the stress of choosing an IA topic hold you back.

Are you struggling to come up with topic suggestions for your IB Internal Assessment?

Our experienced writers can help you choose the perfect topic for your IA

Tailored to your specific subject and requirements.

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Understanding the Math IA Criteria

The Math IA criteria serve as the backbone of your assessment. They outline what the IB is looking for:

• Personal Engagement . It is where your passion for the topic should shine through. Why did you choose it? What personal or real-world relevance does it have?
• Mathematical Presentation . How well do you communicate mathematically? It includes everything from the correct use of notation to clear and logical structuring of your mathematical arguments.
• Reflection . Here, the IB wants to see that you’ve thought deeply about your topic. What did you learn? Were there any limitations in your approach?

Familiarize yourself with these criteria and constantly refer to them, as you can differentiate between a good IA and a great one.

Fun Facts and Trivia Related to Math IA

The Math IA has been a staple of the IB Diploma for over thirty years! Over this time, it has evolved to meet the changing demands of mathematics as a discipline and the world. For instance:

• The idea behind the Math IA was to provide students with a platform to showcase their understanding of mathematical concepts in a real-world context. It is like a bridge between theoretical knowledge and practical application.
• Earlier versions of the Math IA had different criteria and were assessed differently.
• The word count and format have been tweaked multiple times to make the assessment more streamlined and focused.

As you work on your IA, know you’re participating in a rich tradition of inquiry and exploration that spans decades!

As we wrap up, remember that the IB program offers immense value despite its challenges. Use this guide, soak in the insights, and approach your Math IA with the zest and confidence it warrants. And remember, our experts at IB Writing Service are here to help! Best of luck!

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How to Write Your Math IA

Aug 19, 2021 | IB subjects

1. What Exactly is the Math IA?

1.1 about the math ia.

The Math IA is an internal assessment that makes up 20% of your final grade. Also known as the mathematical exploration component, students are expected to create a 6-12 page research paper with a topic of their choice.

Although the IB Mathematics curriculum includes two subjects, Analysis & Approaches (AA) and Applications & Interpretations (AI), the IA itself is the same for both of them. For SL students, the content is based on what you have learned in class. On the other hand, HL students will be expected to write something that is not from the IB syllabus.

1.2 Distribution of Marks

The criteria for both Math AA and AI are the same. The distribution of marks is also the same for both SL and HL students. The criteria includes the following five components: Presentation, Mathematical Communication, Personal Engagement, Reflection, and Use of Mathematics.

As you can see, the IA puts a lot of focus on the Use of Mathematics as it makes up 6 marks of your entire IA (20 marks in total) . So don’t forget the importance of this component!

1.3 The Process

Here is the process of crafting your IA:

• Deciding the topic
• Doing the calculations
• Analysing the results
• Constructing the IA

The Math IA is mainly researched based unlike the Science IA.

After deciding your topic, you will have to conduct some research on the formulas and concepts that you would like to use and see if they can be applied to real life situations. When you managed to fully understand the use of the formulas, use them to do your calculations. Your aim is to find a solution to your investigation through this process.

The next step of this is to analyse your process of calculations and gain feedback from your teachers. Finally, construct your IA using all the information mentioned above.

It is said that the process of deciding the topic is the hardest part of the Math IA . Your IA directly links to your final score for maths, so having a good understanding of how to structure your IA is very important . Don’t worry! In this article, you will find tons of advices for writing your IA!

2. How to Decide Your Math IA Topic

2.1 show the examiner how math can be applied to our everyday lives.

Many students have trouble deciding what to write for the IA. As a result, many would try to see if they can link their hobbies such as music and geography to mathematical concepts. However, it is actually better to choose a mathematical concept first and then think about how you can apply that to our everyday lives .

In simple words, your Math IA should be something that can show the examiner how mathematical concepts can be used in the real world. So, without doubt, the main spice is the use of mathematical formulas and concepts .

If you choose your topic through your hobbies first, then there is a possibility that your focus might go off the road. As a result, you might end up talking more about your hobby instead of math . However, if you choose your topic through the mathematical concepts first, you will most likely end up going into the right direction.

Therefore, it is recommended that you make a list of mathematical formulas and concepts first . Then, think about which ones are suitable for applying to real life.

2.2 How Difficult Should the Math be?

The next question that many of you might ask is, “how difficult should the math be?”. The truth is, it doesn’t have to be more difficult than the curriculum itself . For SL students, choose something that is the same level as the curriculum. The same goes for HL students.

SL students shouldn’t be too worried as you can choose anything from the syllabus. However, for HL students, many of you might be quite frustrated as you can’t choose something that is from the curriculum. It is also difficult to judge how hard a mathematical concept is. For those of you who are struggling, here are some links that can give you some inspiration!

• 50 IB Math IA Topic Ideas

https://www.lanternaeducation.com/ib-blog/50-ib-maths-ia-topic-ideas/

• 20 Math Internal Assessment Topic Ideas for IB Standard Level

https://writersperhour.com/blog/20-math-internal-assessment-topic-ideas-for-ib-standard-level

• IB HL Math IA Topics

https://coggle.it/diagram/WQCrik-6SAABUsF1/t/ib-hl-math-ia-topics

3. How Should You Write Your Math IA?

As mentioned above, the IA is based on five components. So how can you score high on these components?

3.1 Understanding the Components

<Presentation>

The way you present the data and the formulas will be evaluated. Your research content and results should also be effectively communicated throughout the IA.

Things that you should consider

• Make your IA easy to read
• Use the same font for all the mathematical formulas
• Colour your graphs and tables
• Number your graphs and figures (refer back to the numbers when talking about them)
• Don’t go over the page limit
• Show your IA to a third person (your family or friends) to see if it is easy to read

<Mathematical Communication>

Your use of mathematical symbols and equations will be evaluated. You should also define the mathematical terms when necessary.

• Use mathematical equations for your explanations

(or use tools such as screenshots of your graphing calculator, data, or graphs)

• Define terms that you haven’t learned in class

<Personal Engagement>

The personal engagement component is where you should show your personal interest in the topic as well as originality.

• Talk about your personal experience and explain why you chose to research on the topic
• Talk about the challenges that you have faced and how you have solved them
• Show how your IA has led to your personal growth
• Use first person pronouns when explaining your thoughts and personal experiences
• Explain your feelings when faced with challenges

(ex. I thought of _______, therefore I tried to __________)

<Reflection>

This component evaluates whether students have effectively reflected on their research.

• Explain anomalies and give future solutions for the problems
• If you have conducted your research in a wrong manner and managed to fix it in the end, explain how you have corrected your mistakes.

<Use of Mathematics>

This component is used to see whether the math you have used matches the level of your course.

• The most important thing is to show whether you have fully understood the topic rather than how difficult your topic is
• Choose a simple topic, but dive deep into the topic and analyse it thoroughly

3.2 The Structure

There isn’t a specific answer to how you should structure your Math IA. There are IAs that score high without having labels for the Introduction and the Conclusion.

However, as you have seen in the “Presentation” section, the effectiveness of your presentation is one of the important points that you should consider while writing your IA. Therefore, it is recommended that you divide your IA into different sections . Again, there isn’t a rule for this so as long as it is easy to read, it should be fine!

Here is one of the ways that you can structure your IA:

• Introduction
• Background Information
• Calculation

So how should you write each section? Read the section below to find out more!

3.3 How Should You Write Each Section?

<Introduction>

Your introduction should talk about what the IA is about , why you chose that particular topic , and the purpose of your IA . Remember to write the following in your introduction!

• Research question

Describe what you are investigating in a brief manner.

Write about why you chose the topic and how you are going to conduct your research . Remember to include your personal experiences when explaining because this will help you score better in the Personal Engagement component!

Be explicit about what you are trying to find out through this research.

<Background Information>

In this section, you should talk about what kind of knowledge (mathematical concepts, formulas, ways of researching) is needed in order to conduct an investigation on your research question.

The most important thing is to make your IA understandable to someone who has 0 knowledge about the mathematical concepts . Therefore, explain the mathematical concepts, ways of researching & collecting data, and all the steps involved clearly.

<Calculation>

This is the practical part of your IA where you have to give an answer to your research question . It is the place where you should use the mathematical concepts explained in you background information to further explore the topic.

The ways of your investigation depends on your topic. However, you are most likely going to collect data and do calculations based on it. The Mathematical Communication component is based on this section so use the mathematical formulas to explain your processes rather than using words.

Because there will be a lot of information in this section, the key is to make sure that everything is presented neatly and concisely . If you have collected a set of data, use visuals representations such as graphs to present it. That way, you can also gain points int he Presentation component too.

In order to make everything easy to understand, you should also include every single mathematical formula regardless of how simple it is .

<Conclusion>

For the conclusion, you should reflect on your entire IA instead of simply stating what happened. Here are some things that you should keep in mind when writing the IA.

• Evaluation of results

By now, you have probably answered the research question through your calculations. Using this answer, refer back to your rationale. Reflect about why you chose that topic, whether you have managed to find out what you wanted to know, and whether the answer was the same as your hypothesis .

• Limitation and challenges faced

Write about the strengths and weaknesses of your research. Again, the most important thing is your reflection. Reflect on the strengths and weaknesses. Then, explain why it went well or why it didn’t go well . Also mention your solution to certain problems that you faced during your research .

• Improvements and further exploration

State improvements that you will make in the future if you were to conduct this research again . Also talk about how you research can lead to other possible research topics . Writing about how your IA leads to the future can help you conclude it perfectly.

<Citations>

Include all citations with the correct style and format.

4. What You Should Remember

After looking at how you should write your IA, you probably have a good grasp of how everything goes now. Lastly, this article will introduce some important points that you should never forget while writing your IA!

4.1 What’s the Point of Your Research?

The main point of your research is the fact that it is a piece of research conducted by students, rather than it being a full explanation about the logic behind a mathematical concept. The examiner wants to know how the student has thought about the research process and how he/ she has analysed the results .

There is a page limit so always remember to include your own thoughts into your IA rather than a whole bunch of explanations.

4.2 Know the Importance of Your Teacher’s Feedback

Your math teacher is going to be your IA examiner so the IB limits the amount of time your teacher help you in order to protect your originality.

As a result, your feedback time is limited to only once or twice . Use this time wisely and give your best shot. Your draft should be a completed IA so that your teacher can give you advice on how to bring it to a higher level.

4.3 Handy Softwares That You Can Use

There are several softwares that can help draw graphs and figures. It is important to have clear graphical representations or tables in your IA so here are some handy softwares that you can use!

This can be used for geometry, algebra, statistics and calculus application.

• Microsoft Excel

This software can help you with collecting data and making graphs from your data.

• Microsoft Word

You can insert mathematical formulas using this software.

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Ultimate Guide to IB Math IA: Tips, Tricks, and Expert Support

Navigating the world of the International Baccalaureate (IB) Math Internal Assessment (IA) can be challenging. This guide is designed to provide you with essential tips and tricks to excel in your IA and introduce you to expert support that can make a significant difference in your academic journey.

Understanding the IB Math IA

The IB Math IA is a cornerstone of the IB program, allowing students to delve deep into a mathematical concept of their choice. Whether you're enrolled in Mathematics Analysis and Approaches (Math AA) or Mathematics Applications and Interpretation (Math AI), the IA is a pivotal component of your grade. It's not just about understanding mathematical concepts; it's about meticulous planning, structure, and application.

Key Elements of a Strong Math IA Structure:

Introduction: Engage the reader with real-world scenarios and provide a clear focus on your chosen topic.

Mathematical Background: Offer a concise overview of the mathematical concepts and techniques you'll be using.

Exploration: Demonstrate your ability to apply mathematical tools to real-world problems, showcasing your calculations and reasoning.

Conclusion: Summarize your findings, reflect on any limitations, and suggest potential areas for further research.

References: Cite all sources you've used, ensuring academic integrity.

Crafting Your Math IA: Tips and Tricks

Choose an Engaging Topic: Select a topic that resonates with you. Your passion will shine through in your work.

Diversify Your Sources: Use a mix of textbooks, academic journals, and reputable websites to showcase a well-researched IA.

Show Your Work: Detail your thought process, calculations, and reasoning. Visual aids like graphs and diagrams can be invaluable.

Incorporate Real-World Examples: Connect mathematical concepts to real-life scenarios to make your IA more relatable and engaging.

Seek Feedback: Regularly consult with peers, teachers, or experts to refine your IA.

Expert IB Math IA Support with Rajat Sir

If you're seeking specialized "IB Math IA help", Rajat Sir offers tailored "IB Maths IA support" to guide you through the process.

Personalized Feedback on IAs:

Rajat Sir has a proven track record of helping students secure perfect 7s on their Math IAs. The unique approach includes:

Detailed Review: Send your IA/EE/TOK Essay, and Rajat Sir will provide comments and suggestions.

Interactive Sessions: Engage in voice or Zoom calls to discuss feedback and areas of improvement.

Iterative Process: Work on the suggestions and repeat the process until your IA reaches its highest potential.

If you're at the initial stages or struggling with writing, Rajat Sir can assist in topic selection and the writing phase.

Online Personal Tutoring:

Rajat Sir provides the best IB Math IA teacher support through online personal tutoring sessions. With firsthand experience of the IB process, Rajat Sir focuses on ensuring students grasp the core concepts and develop the skills needed to excel in their exams.

Tutoring Offered For:

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The IB Math IA journey can be demanding, but with the right strategies and expert support, success is within reach. Whether you're starting your IA or refining it, this guide and Rajat Sir's expertise can be your roadmap to achieving a perfect score.

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IB Math Internal Assessment Solved: A Guide for Math AA and AI Students

Wait so you’re telling me I have to write an assignment…for Maths?! How do I even come up with a topic? Do I need to ‘invent’ new math? How on earth am I supposed to score well on this assessment?

In this guide, we will share our tried-and-tested winning formula for both Math Analysis and Approaches (Math AA) students and Math Applications and Interpretations (Math AI) students to succeed in your Math Internal Assessment.

Topic Selection:

When selecting a topic, it is always great to start with an area of interest to make it easier to add elements of personal engagement (this will become apparent later, when we take a look at the marking criteria).

This may be a particular sport, hobby or extracurricular activity which you participate in. Some popular examples include basketball, football, swimming, business/economics, music or even mangoes (yes, I have seen a Math IA on mangoes).

Now you may be thinking “this is the most generic and useless advice I have ever received” but from here, we can get more specific based on which Math course you are undertaking. For Math AI students, your course is heavily focused on statistics and statistical testing, while Math AA students have far more content on modelling, functions, trigonometry and calculus.

So the trick from here is to take your chosen field, let’s say for example we choose football, and try to connect the mathematical topics to this area of interest:

But how do I get from ‘Football’ to these ideas?!

Well the starting point is to consider what kind of questions do we often ask in real life? In the case of football, one might ask whether teams should be spending more money on new transfers or developing young talent? Or perhaps whether players should shoot ‘low and hard’ or ‘high with placement’ on penalties? Starting with these questions, we can then ask whether we can use any of our Mathematical knowledge to investigate an answer!

Finally, the process of narrowing in on a topic is reliant on two factors – applicability of Mathematical tools and availability of data. You must be able to apply the Mathematical tools that you have learned throughout the IB in your IA, and you must also consider whether the data you need for your exploration is publicly available or can be personally collected.

Writing the Internal Assessment:

It can often be difficult to start writing the Math IA. What do I write first? What calculations should be done in advance? How do I even know if the Math is going to work?!

First, plan out the IA, particularly all the tools, tests and Mathematical calculations to be completed. We then recommend you complete all the ‘Math work’ in the IA before commencing any substantial writing, as you may find that the tools which you originally planned are not relevant or that the data cannot be found or collected accurately. This means that data collection (whether this is through online research or directly) is the first step, followed by performing the calculations for the IA. After this is completed and you are satisfied that the calculations have been performed well and the conclusions from your tests provide enough content for an interesting discussion, we can then start writing the IA itself.

While the structure of the IA varies between course and even between particular topics, a broad framework would be:

Introduction and Rationale – Introduce the topic and your rationale for writing the IA

Aim/Approach – Outline the aim of your IA and how you will be approaching your topic

Data Collection – Explain how you collected your data, define any key terms/assumptions and present your data

Main Body – The main body of the Math IA will depend on the focus questions built within your topic and the particular tools which you will employ

Conclusion and Evaluation – Present the conclusions of your IA and evaluate strengths/limitations

The IB Marking Criteria – Writing for Success:

Wait, hold on a minute. So should my IA be just a presentation of calculations? How much working do I need to show? Do I need to explain the tools or is it assumed knowledge?

To answer these questions, let’s delve into the marking criteria and take a look at what they really want from you.

Criterion A: Presentation (4 Marks)

Your IA needs to look clean, coherent and well-organised. It must have well-constructed diagrams to assist clarity when necessary. Importantly, to hit the top band, it must not only be coherent and well-organised but concise.

This means that when presenting calculations, do not repeat all steps each time you do a new calculation, if the steps have previously been explained. For example, if you are calculating the mean, median and mode for a series of datasets, show an example calculation for the first value but present the remaining findings in a table without showing working for all of them.

Criterion B: Mathematical Communication (4 Marks)

Communicating mathematics is very difficult. Your biggest asset in the Math IA will be diagrams, tables and charts which can accompany your text.

Other elements of this criterion to note include defining all key terms and variables (and remaining consistent throughout the IA), using the correct notations, symbols and terminology and showing your line-by-line working to set out your proofs.

Criterion C: Personal Engagement (3 Marks)

You can easily score 1 or 2 marks in this criteria by merely relating your chosen topic to your own personal experiences and motivations. For example, the conclusions of an IA about football may be useful to implement in your own footballing.

However, to maximise your marks in this criterion, you need to demonstrate personal engagement in the way you approach the task. Implementing creative ways of collecting data, utilising tools or even applying your data can all contribute to this criterion.

Criterion D: Reflection (3 Marks)

One common misconception about this criterion is that you can ‘throw in some reflection at the end in the evaluation section’. This approach will not allow you to score more than 1/3, as this criterion requires both ‘substantial evidence’ (i.e. throughout the entire IA) of ‘critical reflection’.

Some ways to integrate this include continuously linking back to the aims of the exploration (e.g. explaining how each conclusion you draw can be applied to your aim/the real world), commenting on new discoveries that you make as you are making them and commenting on not only the results but the tools, calculations and approaches you have taken.

One tip from us is to always write as though you are discovering everything for the first time, almost like a journal/logbook; for example, you may complete the first section and note that the result appears to support a particular hypothesis, although you will need to complete further calculations to be sure.

Essentially, reflect on your results as they appear, their meaning to your aim, the strengths and limitations of the tools you are using, and your own ‘thoughts’ which pop up throughout the writing process.

Criterion E: Use of Mathematics (6 Marks)

In my opinion, this criterion is very poorly named as it is not only your ‘Use of Mathematics’ which is being awarded marks but your understanding of the math which you are using. For example, compare the wording for a 3/6 and a 6/6 in this criteria:

3/6: Relevant mathematics commensurate with the level of the course is used. Limited understanding is demonstrated.

6/6: Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Thorough knowledge and understanding are demonstrated.

‘Relevant’ mathematics refers to tools which support the development of the exploration toward the completion of its aim – this also indicates that we should be justifying WHY we are using particular tools prior to using them and only using a tool if it contributes to our aims. However, you’ll notice that using relevant mathematical tools commensurate with the course is merely a prerequisite to scoring half marks in this criterion. So where do the other marks come from?

Well firstly, you need to use it correctly. It does not need to be perfect to score in the top bracket, however it must not detract from the flow of the mathematics or lead to an unreasonable outcome.

More importantly, you need to show knowledge and understanding (and this must be thorough to score 6/6). So what exactly does this involve? You must:

Justify the use of each mathematical tool (e.g. Why are we using this tool and how will it help us investigate the aim)

Explain how the tool works (e.g. If calculating Pearson’s Correlation Coefficient, not only explaining the process of calculating the coefficient using the formula but explaining how the formula works)

Interpreting what our results mean (e.g. What is the formal conclusion and what is the relevance to our aims)

In essence, to score highly in this criterion, you need to dive back into your Mathematics textbook or class notes and read that one section that no one ever pays much attention to – the part that explains why formulas work, how they are derived and what you are actually doing when you are substituting values into these formulas. Ultimately, this IA was designed by the IB to assess two overarching principles: firstly, your ability to take the mathematical concepts, tools and theories and apply them to a real-world problem and secondly, to prompt students to reflect upon and delve into the intricacies of the mathematical content which they are learning.

And that’s it, everything you need to know to succeed in your IB Math Internal Assessment for both Math AI and Math AA students! If you are still struggling to get started, write or generally unsure about how exactly to succeed in this task, reach out! Our team of successful IB tutors are well-equipped to assist with the IB Math IA and are able to work with you from topic selection to final completion of this task.

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March 15, 2023

A Complete Guide to Excelling in Math IA

Mathematics Internal Assessment (IA) is a crucial component of the IB Diploma Programme. It is a student-led exploration that enables the learner to delve deeper into a mathematical concept or problem of their interest. The Math IA is a 6-12 page report that showcases the student’s ability to conduct independent research, analyze data, apply mathematical concepts, and present their findings in a clear and coherent manner. Writing a Math IA can be daunting, but with proper planning and execution, it can be a rewarding experience that helps you understand the beauty and power of mathematics. In this guide, we will discuss how to organize your Mathematics IA.

IA Cover Page

Title page should include: Your Title The IB Number (In the format “ABC123”) Session (i.e. May 2021)

IA Important Rubric Requirements

Page Count: 12-20 Pages in length with double spacing. The page length per subsection is not set, but one can imagine it should correspond to the marking rubric.

E.g., use of mathematics carries a weighting of upto 6/20 while reflection carries a weighting of upto 3/20. Hence, you should expect to spend more pages on calculations than your reflection.

Personal Engagement: A unique part of this IA is the personal engagement.

Bibliography: A detailed bibliography is required so you must keep all sources which you utilise throughout your IA process.

Section 1: Introduction Introduction – Why, what, then how. Why? Your IA introduction should include a rationale for why you have chosen your topic for your Mathematical Exploration (the name of this IA). You should find some personal way to engage with your chosen topic to satisfy this requirement. Choose a topic you’re genuinely interested in, state said interest explicitly and use your own personal examples where possible.

What? Picking a topic – specifically an aim – should be considered carefully and in conjunction with your tutor/teacher to ensure there is sufficient depth to your topic (as this depends on whether you’re taking SL or HL Math). Make your aim explicitly – this is important. Note the difference between receiving a 6/6 for the use of mathematics rubric for HL/SL according to the IB:

SL – “Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Thorough knowledge and understanding are demonstrated.” HL – “Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and demonstrates sophistication and rigour. Thorough knowledge and understanding are demonstrated.”

In both cases you should use mathematics of a similar level to what you are studying in your respective studies. However, in HL, the mathematics that is explored must be precise and shows sophistication and rigour.

Some examples of previous IA topics are listed below (these are basic topics and not finalised research questions) and in the appendix. However, remember that there ought to be some personal engagement within the topic-choosing process: “Why planes travel a curved route and not a seemingly direct route” “Does the stock market’s returns warrant its variance?” “Projectile motion” “L’Hôptal’s rule and evaluating limits” “Image rotations using rotational matrices”

How? You must outline how your exploration topic relates to your specific curriculum, how you’ve completed the exploration, and provide any necessary background information – your classmates should be able to understand your IA if they were to read it.

Section 2 (Body): Theory & Calculation

Theory Provide only the relevant theory needed to reach a conclusion/understanding of your aim. If there is a particular method (in mathematics, there are often numerous ways to reach the same answer) that you’ve used you should explain the method and why you’ve used this method.

Calculation For this section you must include all formulae and assumptions (i.e., the actual numbers) used to make your calculations and the mathematical steps that you took to reach your aim. Note assumptions’ pertinence if someone wants to repeat your exploration. After going through your mathematical work you must explain how they relate to your exploration topic. Depending on the type of exploration in which you are partaking you should use appropriate graphs, tables, x-y-z planes, or other methods of presenting your results. See below.

As can be seen from the figures above, figures are labelled appropriately. Calculations come with brief explanations and connect the earlier theory with the specific scenario in your exploration.

Section 3: Reflection, Conclusion, and Bibliography of Mathematical Exploration Conclusion Your conclusion is a continuation of section 1 and 2. You are answering your aim from your introduction (section 1) with the theory and calculations in from section 2. This should be done in a clear, concise, and coherent manner. Not only should you explain the results and implications of your calculations, but you ought to relate this to the aim raised in your introduction. You may also include much of the reflection in your conclusion if you prefer a more integrated approach. Note the IB says the following regarding where the reflection should be placed: “Substantial evidence means that the critical reflection is present throughout the exploration. If it appears at the end of the exploration it must be of high quality and demonstrate how it developed the exploration in order to achieve a level 3.” This implies a preference for integration but it does not mean you are excluding yourself from a level 3/3 grade for the reflection rubric.

Reflection Your reflection should occur throughout your IA; however, you may also include a separate section depending on the layout of your IA. Here’s what you should do: Consider limitations and extensions of your conclusion. Similarly consider strengths and weaknesses. Relate the mathematics within the exploration to your personal knowledge (or personal engagement). Raise future research questions. The IB states your reflection must be “crucial, deciding or deeply insightful. It will often develop the exploration by addressing the mathematical results and their impact on the student’s understanding of the topic.”

Bibliography You should include a thorough bibliography to support your introduction, background, theory, and perhaps calculations. Types of relevant sources include online databases, your school textbook, or specific theories found both online and physically.

Appendix Updated (2021) grading rubric: Criterion A Presentation: 0-4 Criterion B Mathematical communication: 0-4 Criterion C Personal engagement: 0-3 Criterion D Reflection: 0-3 Criterion E Use of mathematics: 0-6 Total: 0-20

In conclusion, a well-structured layout is crucial for a successful Math IA, including an introduction, main body, analysis, and conclusion. Meeting the rubric requirements is also essential for a high grade, which includes criteria such as presentation, communication, and reflection. By following these guidelines, you can write an engaging and informative Math IA.

If in doubt, reach out to experienced tutors at Quintessential Education for extra help and guidance. Start your journey towards academic success today!

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A Comprehensive Guide: IB Math Applications and Interpretations SL IA

Making your internal assessment (IA) in Math AI can be tricky, especially if you think that math is not your strong suit. Don’t worry, we’re here to help! Here is a comprehensive guide to the Mathematics: Applications and Interpretations IA.

What is the Math AI IA?

The Math AI IA is a report of how you can apply your learnings from the Math AI course into exploring a topic of your choice. Cooking, sports, architecture, finance — this topic can be anything that interests you, and it is a chance to explore the math behind it in a guided way. 

The Math AI IA is worth 20% of your total course grade in Math AI. As with all IAs, the International Baccalaureate Organization (IBO) sets a standard criteria* for marking IAs, outlined here below:

*For a more comprehensive look at the criteria, kindly refer to the Mathematics: applications and interpretations guide (2019) from the IBO.

For HL students, Criterion E explicitly states that students must either use concepts only found in the HL syllabus or use concepts from the SL syllabus in a “complex way that is beyond what could reasonably be expected of an SL student” (IBO, 2019). A good way to check for this is through looking at your Math AI formula booklet. If you are an HL student, your IA should include at least one formula marked as Additional Higher Level (AHL). Examples of these formulas include the logistic function, transformation matrices, and the Poisson distribution. 

SL students may choose to use either SL or AHL formulas, but HL students are strongly recommended to apply as many AHL concepts as possible in their IA. Other than this, there are no differences between how IAs are graded across SL and HL.

Choosing your topic

Because math has various real-world applications, this is an incredibly flexible IA to pick a topic for. To figure out where to start, there are generally two ways to find a topic: (1) Find a topic you are interested in, then apply it to at least one Math AI lesson; or (2) find a Math AI lesson you enjoy, then identify one practical application of that topic.

Many of my classmates found their IA topics the first way, starting off with something they were passionate about and then looking through the syllabus for concepts they could apply. For example, one of them liked baking, so they decided to explore the relationship between the ingredients in the recipe and how their cupcakes turned out. 

In my experience, I chose my topic the second way. I had no idea what to do for my IA topic, but I did enjoy making Voronoi diagrams when we covered them in class. I found the concepts easy to understand, and I liked working with something I can visually see. From there, I connected Voronoi diagrams with things that interested me such as healthcare.

Regardless of how you choose your topic though, a sturdy Math IA topic should enable you to:

• Be genuinely interested about it
• Apply at least one lesson from the Math AI SL/HL course
• Have at least one set of data you can do calculations with
• Produce results you can evaluate (e.g. Are my calculated answers realistic or not?

Structuring your IA

Now that you have your topic, it is time to structure your IA. Although there are no strict rules for how to assemble your IA, Math IAs generally follow this flow:

• Plan of action

Methodology

Evaluation and extensions.

It may look like a lot, but don’t worry! It is simpler than you think, and these steps will be further explained in order below. I will use my own IA in Math AI SL as an example, but feel free to cross-reference this with any sample IAs provided by your teacher.

The Rationale is essentially the introduction of your IA. It outlines what you want to explore (i.e. your research question), the reason why you are interested in it, and the background behind it.

As a whole, your Rationale should only be about one page long. In 2-3 paragraphs, you should be able to answer the following questions:

Although this is an academic paper, using personal pronouns like ‘I’ and ‘we’ is not discouraged in the Rationale as these are used to convey personal engagement (Criterion C). That being said, you must ensure that you balance your personal experiences with research to support your research question. 

Plan of Action

The Plan of Action is all about how you accomplish your research aim, including the mathematical concepts you will use to do so. In 1-2 pages, this is what you must do with a Plan of Action:

Given the flexibility of the Math AI IA, some students may struggle with figuring out the steps for their methodology. Topic choice is key to figuring out the structure of your IA. While all IAs in Math AI will have elements in common, certain topics may involve sections that may not be necessary in others. 

Some topics may involve collecting primary data through a survey or an experiment. For example, my friend who did baking for her IA had to vary the amount of an ingredient she used and then measure the resulting height of her cupcakes. As such, IAs like these may require sections for procedures and results.

Other topics may use secondary data from databases. For example, some IAs I have seen look at mental health statistics and compare them with statistics for economic growth (e.g. GDP). Since these IAs typically work with large databases, they must set a range for the data they will use (e.g. Real Madrid scores from 2010-2020). These IAs also usually have sections for making and explaining graphs.

Still, other topics may instead assign their own data points through modeling. What this usually means is that they place images on modeling software (e.g. Desmos, GeoGebra) and then mark points of interest on the graph. This is common for IAs using Voronoi diagrams and calculus. For example, this was my set-up in GeoGebra:

A sample set-up for modeling IAs. In this case, I placed an image of the city map onto GeoGebra and adjusted it so that a unit of 1 on the graph represented 1 kilometer. I then marked the locations of hospitals in the photo. For example, Hospital A had the coordinates (1.11, 4.38) on the graph.

The beauty of the Math IA is that you can drive it in any direction you want. In reality most IAs use a combination of methods and data sources, adding sections for each part as needed. Whatever methods you use though, be sure that they follow a logical order (Criterion A) and show your thought process in terms of math (Criteria C and D).

Typically the longest section of the IA, the Methodology is where you carry out the methods to achieve your research aim. How this looks would vary with your Plan of Action, but this is where mathematical communication (Criterion B) and use of mathematics (Criterion E) are highly emphasized.

This is a general guide you can follow for your methodology:

• Introduce the step
• State why you are doing the step
• Present the formula
• Do a sample calculation with the formula for one data point
• Present the calculated results with a diagram or a table

This is an example of a sample calculation I did in my IA:

Here are some methodology tips that may be useful:

• Label your diagrams and calculations so that you can refer to them throughout the methodology.
• IAs can be handwritten or typed. If you are typing your IA, use the Equation function to ensure that you express proper mathematical notation. 
• Note: Computer notation is acceptable if it is software generated. For example, if you are including a table from a database where the exponents in numbers are marked with ^, it is acceptable to have the numbers as they are as long as you properly cite the database.
• For larger sets of data, check with your teacher if it is acceptable to place the full list of calculated results in an appendix. If you do choose to include an appendix, don’t forget to reference it when presenting your data.
• Keep a checklist of the steps you have done so that you do not accidentally repeat calculations. 
• If you have to use a formula again, simply cite it (e.g. “Refer to Calculation 8”) instead of doing another sample calculation with it. Repeated calculations are penalized under Criterion A.
• Explain the implications of your decisions whenever possible. For example, will running this statistical test account for one variable but not another? This will help the examiner understand the decisions behind your process (Criterion D).

Methodology usually takes up more than 10 pages in an IA, owing to the diagrams and calculations in it. Regardless of the length, what is essential is that the reader can follow your train of thought. You have learned a lot of math over the course, so take the time to show it!

The Evaluation and Extensions section is where you describe the strengths and weaknesses of your methodology. It is also where you outline future opportunities for research based on your work. Sometimes it is placed right after the Conclusion, but it might be helpful to have this section before the Conclusion instead so that you can refer to it as you summarize your work.

This section is pretty straightforward, but many students miss marks because they simply evaluate and forget to include extensions. Pro tip: Use the weaknesses in your evaluation as chances for future research! You can do this in three steps:

The Evaluation and Extensions section is key to scoring marks in Criterion D, which is about reflecting on the quality of your work. It is typically only 1-2 pages long.

Now that you have done the math, it is time to share your results. Your Conclusion should:  

Formatting your IA

Before you submit your Math AI IA though, there are a few formatting checkpoints it must pass:

• Does it have a cover page? The cover page must contain the IA title, subject (Mathematics: applications and interpretations SL/HL), page count, and candidate code. 
• Does it have a table of contents and page numbers? The table of contents must follow the citation format required by your school (e.g. APA, MLA, Chicago). Page numbers usually do not include the cover page.
• Are the paragraphs double spaced? The IB requires double spacing in Math AI IAs to allow examiners to annotate IAs while marking.
• Are the tables and diagrams labeled? All tables and diagrams must be labeled with a figure legend, including those at the appendix.
• Do you have a bibliography or reference page? This is typically placed at the very end of the IA, after the Evaluations and Extensions section and before the appendix (if any).

Does your IA tick all the marks? If yes, then congratulations! You have finally finished your IA in Math AI.

Personally I learned a lot while doing my IA in Math AI SL. It was tough figuring out what to research about and how to do it, but once I had my methodology set the momentum continued. Before I knew it, I had completed my IA in math — a subject I once thought I was hopeless in! I hope you will find your Math AI IA journey incredibly rewarding, building your confidence in math one step at a time.

You may also like…

• An Approachable Guide for Analytical Students: IB Mathematics AA by Mateusz Konat and Wiktoria Wojnarowicz
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• Jun 2, 2021

How I got a 7 in the IB Maths IA 🙌

This is a complete guide to the Mathematics AA and AI Internal Assessment. Follow these steps to ensure your IA meets a high standard.

Table of Contents

Big takeaway: methodology first, topic second, how do i find and choose my mathematical method, plan purpose personal engagement, make the right start: thoroughly scope and plan, the first section is easy tick-boxing, don't over-engineer your personal engagement, use of mathematics, how much math is enough math, demonstrating understanding vs getting a correct answer, reflection: more than just a conclusion.

With very little practical information available online and teachers barely helping us with IAs, the process is such a struggle now. On top of that, the Maths IA has become a VERY significant part of your final grade in the last year (especially non-exam route).

So I put together a complete guide to the Math IA mixed in with a bunch of personal advice to make this entire project a lot easier. These are the steps you need to ensure your IA meets a high standard.

I'm going to presume you know the basics but, if not, go properly read the IA section in the Mathematics AA or AI guides . You are being assessed on 5 criteria for the Internal Assessment. If you can truly understand how the whole assignments graded, you won't do useless things like including irrelevant maths or creating a fictional story about your love for linear regression (please don't...)

1. Big Takeaway: Methodology First, Topic Second

Find the mathematical method that will lead you to a final number or equation and THEN think about the topic you want to apply that method to.

Contrary to what everyone says, the way you start the IA is very important. Most of us just use a topic like football shots or a piano composition and make some maths up about it. But this only forces irrelevant math into your IA and weakens your entire exploration.

Remember the reason the IB wants you to do an IA is so that you can "develop a wider appreciation for mathematical concepts and processes". So let's do exactly that.

Instead of thinking about the topic first, find and deeply understand a mathematical technique or method that is an extension of the concepts in the maths course. Learn about this topic and its common applications in the world. Then, brainstorm the ways you can apply this method to investigate a different topic/application that you're interested in.

For me, I knew wanted to explore ordinary differential equations and after learning advanced ways to use them I decided to focus on modelling tumor growth in the brain. Another friend of mine used integration and La Grange multipliers to find the most efficient way to lose calories on a treadmill.

Both of us scored 7s and as you can see we started off by researching calculus-based techniques and took time to understand all of the math behind them before thinking about the topic that we'd want to investigate using mathematical method/s.

Other successful mathematical concepts include the volume of revolution (calculus) based IAs because you can learn to plot multiple curves for difficult objects with a large potential for further application. Moreover, statistical methods such as correlation + regression or testing for normality (eg. in certain fashion or food products) can also score highly if you go in-depth with the techniques and analysis.

Start with Khan Academy's Mathematics page and read through the chapters and topics for calculus and statistics (those are the main ones). On top of all the IB-related topics, there is a nice compilation of different techniques and applications beyond the scope of the course that you can use for your IA too.

Research the chosen mathematical method further and actually try to fully understand and appreciate how it is used. Use books, PDFs, and find obscure YouTube videos to help with exploring the topic. Ensure that you only pick a topic once you are 100% sure you can understand and apply it.

Shortlist 2-3 topics and go through how your researched mathematical technique will be applied and how you will actually explore it. Flesh each topic out as much as possible and scope what your IA will look like before deciding on one. I, personally, shortlisted both modelling the growth of the GBM tumor and modelling the spread of different infectious diseases (different topics using similar mathematical methods and concepts).

2. Plan? Purpose? Personal Engagement?

Before starting, it is crucial that detailed notes are made on a) the mathematical approach specific to your topic b) the analysis of the maths and c) your overarching aim.

The trend the IB examiner reports mention is that students present difficult math concepts and equations without knowing how to properly apply them or without even having a clear purpose for their investigation. You can easily avoid this by properly scoping out the investigation before you start typing.

Research and plan the important details of the method you're using and the analysis you want to conduct. Ensure that you connect every part of the whole exploration to the purpose.

Just starting after finding a topic isn't wise for this IA. I made this mistake and had to restart after writing 3 full pages on one topic without realising that my entire mathematical approach would be better suited to the topic I previously ruled out. It is so important to design the right framework for the idea, you shouldn't just wing it because a lot of people do and get stuck in the middle.

There is no template and no perfect structure for your IA. But I'd suggest dividing the first section up into an Introduction, Rationale, and Aim/Plan of Action for clarity. Always make it a habit to clearly "show" the examiner the ways in which you are fulfilling the criteria. Make it easy for them to give you marks.

These are a few important questions to ask yourself as you write the first section up:

Is the aim of the exploration clear and explicit for anyone reading?

Are the key terms, main variables, and constraints defined (where required)?

Have you given a step-by-step plan of action and explanation of how you will explore your question/aim? (incl. description of data collection)

Avoid cheesy background stories. Please. Students believe that the 3 marks for PE are only awarded if they talk about the overwhelming curiosity they've had about Pascal's triangle their whole life. Do not add blatant lies like these, examiners see right through them and it undermines your investigation.

The PE criterion C is most misunderstood. Instead of creating an unnatural backstory, show the examiner authentic personal engagement. Show them that you have "driven the exploration forward in a creative way". Make sure the topic has some personal importance to you but add layers of complexity to your investigation to exhibit organic engagement. Every bit of extra effort you put to further your investigation is evidence of your personal involvement.

In my IA, there was no experimental data for certain variables in my tumor growth model so I learned how to use the Matlab software to optimise and find estimations of the unknown parameters myself. Learning a new skill and finding a creative solution for the missing variables was a very strong example of my personal engagement. Keep in mind this is just what I did and there are much simpler ways to achieve the PE marks too.

Did you do preliminary experiments to ensure your exploration makes sense? After you accomplished the purpose, did you further your aim by testing the equation against other variables too? How did you manipulate the final results to further analyse them? Adding new and unique levels of complexity is the key.

3. Use of Mathematics

It is better to do a few things well than a lot of things not so well.

"If the mathematics used is a) relevant to the topic being explored, b) commensurate with the level of the course and c) understood by the student, then it can achieve a high level in criterion E."

The mathematics used needs to be relevant to a topic in the course obviously and we shouldn't be using any maths that we don't understand. Usually, students use "overly abstract and sophisticated concepts beyond their course" without truly understanding the math. Examiner reports reveal that a lot of IAs force these complicated methods in the hopes of gaining extra marks but end up losing marks because they couldn't clearly link every part of the exploration to their investigation. This is the official feedback given by the IB:

"Students should be discouraged from using difficult mathematics beyond the HL syllabus if this cannot lead to some creativity or personalised problem."

A cascade of formulas taken from some online journal without citation isn't the way you score in criteria E. Using mathematics beyond the HL syllabus will often lack thorough understanding and will make it "difficult for students to demonstrate Personal Engagement or Reflection too". You need to ensure that you deeply understand the mathematical methods being used in your IA. Everything included should be thoroughly justified and should be helping you build towards your aim.

These are two very different things. Obtaining the correct answer isn't even enough to demonstrate "some understanding" and won't even guarantee getting a 2/6 in Criteria E. Clearly, demonstrating understanding is a big deal.

It's one of the reasons I've constantly emphasised the fact that you need to gain a deep understanding of the mathematical methods and concepts before even starting. Instead of just adding calculations onto your IA, explain every step and justify everything you do to solve the problem.

"Illustrating with examples or practical applications" within your explanations is the strongest way to demonstrate your knowledge of the mathematics. Be explicit, excruciatingly clear, and don't shorten this part of your IA.

The biggest mistake people make with their "reflection" is believing this criterion can be fulfilled with a few paragraphs summarising their exploration at the end of their IA. The IB examiner's report for the IA confirms this: Many students just discuss "the scope and limitations of the work done and include no meaningful or critical reflection."

The IB wants to see your growth throughout the IA. They assess how the student reviews, analyses, and evaluates their exploration at different stages. Reflective elements should be present throughout.

For example, in the introduction itself, you can briefly reflect on the method you've chosen. Why is this the best way to address the aim of your investigation? During data collection, you can question the credibility and room for errors in the data and results. Discuss ways in which you can address or account for that. Continuously examine the strength of your investigation and include reflective elements throughout.

For the higher achievement levels make sure you do these:

I mplications of the results: linking your results to the aim of the exploration and the real impact these conclusions can make.

Consider further explorations: there are always better methods or more advanced maths that can be used to solve the problem you are addressing. Evaluate and compare them against your own approach in 1 or 2 paragraphs.

Compare strengths and weaknesses of the mathematical methods and data analyses you conducted.

Bonus: Brainstorm the topic from a different perspective (use technology such as software for model development).

Throughout IB we're literally trained to critically evaluate and reflect on all of our work and progress. It's an inherent advantage we have compared to most other students. Let's use it well.

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Hey. Your tips are very useful. I am interested in doing tumour growth modelling as well but I am having difficulty finding data of tumour growth in people. Could you share where you accessed your data for your modelling?

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Decoding The IB Mathematics Internal Assessment

What is the purpose of the ib internal assessment.

The IB Mathematics Internal Assessment comprises a 12 to 15-page report on an area of Mathematics which is of interest to students. The purpose of the IB Mathematics internal Assessment is to give students the opportunity to apply Mathematics to the real world. 

The primary goal of the IB Programme is to help individuals develop into curious, knowledgeable, communicative, principled, open-minded, compassionate, risk-taking, reflective, and balanced learners. When you attempt the IB Maths Internal Assessment, students can further incorporate their traits as IB learners into their learning. 

Common misconceptions of the IB Mathematics Internal Assessment in Singapore

• Students should explore Mathematical topics beyond IB Higher Level Mathematics in order to score a 7

This is not true. Unlike IB Mathematics Extended Essay, students doing IB Mathematics Internal Assessment are not required to explore beyond the IB Mathematics curriculum. In fact, some IB students are overly ambitious and write an exploration beyond what they can handle, such as Fourier series and Laplace transformations. Instead of impressing the IB examiner, students ended up with mediocre scores as they were unable to explain the mathematical thinking behind it well. 

• The IB Mathematics Internal Assessment is an academic paper which involves regurgitating Maths concepts gleaned from Math journals, chapters from textbooks, and online resources such as Math journals and youtube videos.

The IB Maths IA is meant to help students appreciate Mathematics in a real-world context. Students need to apply Mathematics to any data collected or real-world scenarios and show how Mathematics solves problems in that setting. Rehashing information or maths problems from journals or textbooks will not help students to score high marks.

• The IB Mathematics Internal Assessment requires students to come up with new Mathematical formulae.

Do not worry. Students are not required to produce new maths formulas or theories. It is unrealistic to expect high school students to do that as they are not Maths professors. Students are only required to apply the mathematics they know in real-world contexts.

What are the assessment criteria for the Math IA ?

Students are assessed on five criteria; Communication, Mathematical presentation, Personal Engagement, Reflection and use of Mathematics.

Criterion A: Communication 

This criterion assesses the overall organisation and structure of the exploration. A well-organised exploration includes an introduction, a rationale and an aim. Additionally, the rationale explains why this topic was chosen. The rationale is the ‘fun’ part of the IA, as most students use ideas which connect to their childhood. The aim of the exploration shows what the exploration hopes to achieve. The exploration needs a conclusion to summarise the ideas presented.

A thorough investigative approach is developed with the aid of logic and should not be overly complicated. Any argument you make must be presented with efficiency, and all the terminology used should be well defined. Get your peers to review your IA to ensure the concepts utilised will suit the essay. Use simple and concise explanations for any written maths so that anyone reading it would be able to comprehend the argument. If they express any confusion or falter when trying to make sense of your argument, it is a sign that your presentation should be reviewed and amended accordingly.

Criterion B: Mathematical presentation

This criterion assesses how mathematical languages (symbols, terms, and notations, for example) are used in the IA. Individuals need to use the correct terminology, language, and representations (diagrams, formulae, charts, graphs, and tables.) Employing the proper ICT tools like spreadsheets, graphs, and screenshots of graphic display calculators will also be checked. 

Criterion C: Personal engagement

This criterion assesses “ownership of the exploration”. The student is required to show independent thinking and examine mathematical ideas through their own lens. Many of the maths formulae and explanations can be found in textbooks and online resources. By personalising mathematical ideas, students are able to present their own views and analysis.

Criterion D: Reflection 

This criterion assesses a student’s ability to analyse, review, and evaluate the investigation. Reflection is found throughout the investigation, along with its conclusion. It can also be exemplified by taking its limitations or expansions of maths concepts into account. Queries regarding the significance of your lessons and meditations on how they can be extended will then be addressed. 

Criterion E: Use of Mathematics

This criterion assesses how adeptly students utilised maths in their investigation. They should be able to do work that is comparable to the course’s depth and difficulty. The mathematical concepts employed should be of the syllabus or something approximately equivalent. If it doesn’t compare to the course’s contents, you will only receive a maximum of two marks for this criterion.

Maths can be seen as accurate even if there are discrepancies, provided it doesn’t impede the flow of the maths. Optimising mathematical applications will involve the utilisation of difficult maths ideas, finding a different angle to view the problem, and identifying underlying structures that would connect to different mathematical concepts.

How do you structure your IB Maths IA?

Let us show you a template to help you write your Math IA.

Step 1: Research a suitable topic.

This is the most challenging step. Students are required to find a suitable topic with an appropriate level of maths and real-world application. It is not uncommon for students to face countless rejections from their teacher before a Math IA topic finally gets approved.

Spend some time searching for a topic and reading outside of your school textbook. Many students may not have read Math academic journals before. However, it is good to get into the habit of doing so. These academic journals are a rich source of ideas for students. Some of our students obtained many good ideas from these journals and aced their Math Internal Assessment.

Some topics to avoid:

• a typical textbook solution 
• Topics commonly attempted by previous students (for example, the Monty Hall game, the spread of Covid)

Step 2: Submit a proposal to your teacher

Before commencing your Internal Assessment, submit a detailed proposal to your teacher. In your proposal, you should outline the type of Mathematics formulas and working used. Explain to your teacher how your IA can be applied in the real world.

Step 3: Writing your IA

As soon as your teacher approves your IA, you should start writing! The IB program is like a marathon, and students are inundated with many assignments. Starting early leaves you more time to make amendments or even change your topic if the topic does not work out. 

The best way to write an IA is in the first person’s point of view. This is because personal engagement is one of the criteria for Math IA, and writing in the first person makes it easier for the reader to connect with you.

Most students I know find writing an essay for Math daunting. Many students have not had the experience of writing a Math essay throughout their student life, so they do not know how to get started!

Don’t worry. We’ve got you covered. Through our esteemed IB Maths tuition , we have a step-by-step approach to help you through the Math Internal assessment in a stress-free manner. 

Refer to the table below for a quick overview of the sections to include in your IA.

IB Mathematics Internal Assessment Topic Ideas

For students who are facing a dearth of Math Internal Assessment topic ideas, our IB Maths tutors in Singapore have thought of a list of possible ideas to help you begin.

Some possible ideas for Math IA include:

• Calculus: Finding the volume or surface area of an odd-shaped object
• Graph theory: Finding the shortest distance between two cities
• Differential equations: Tumour growth prediction
• Statistics: Modelling data using statistical tools
• Voronoi Diagrams: Finding the best location to open a restaurant
• Calculus: Topology
• Trigonometry: Applications to music
• Statistics: How much to price each game at a school fun fair
• Permutations and Combinations: Winning at Tic Tac Toe
• Trigonometry: Dissonance in music
• Differential equations: Newton’s Law of cooling

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IB Maths Resources from Intermathematics

IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides

Maths IA – 300 Maths Exploration Topics

Maths ia – 300 maths exploration topics:.

Scroll down this page to find over  300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework.  Topics include Algebra and Number (proof), Geometry, Calculus, Statistics and Probability, Physics, and links with other subjects.  Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL).

New online Maths IA Course!

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Gain the inside track on what makes a good coursework piece from an IB Maths Examiner as you learn all the skills necessary to produce something outstanding.  This course is written for current IB Mathematics students.  There is more than 240 minutes of video tutorial content as well as a number of multiple choice quizzes to aid understanding.  There are also a number of pdf downloads to support the lesson content.  I think this will be really useful – check it out!

Modelling allows us to predict real world events using mathematical functions.  

Why is this topic a good idea?

This topic is a nice combination of graphical skills, regression and potentially calculus.  It easily links to the real world and so is easy to find engaging ideas.

Some suggested ideas:

Calculus and Physics

Calculus allows us to understand rates of change and therefore motion over time.  It’s one of the most powerful tools ever invented.

This topic allows a nice demonstration of calculus skills, often links with graphical ideas and is easy to create real world links.

Data and Probability

The modern currency of the internet is data – and data collection and data interpretation skills are essential.

This topic if done well can bring in ideas of probability, statistics and other branches of mathematics.

Statistics and data analysis are important skills in business and science.  There are many different tests which help us understand the significance of results

This topic if done well can allow students to do some experiments and investigations

Geometry connects us with mathematics done for over 2000 years by the likes of Euclid done with compasses and rulers.

This topic is a nice combination of graphical skills and the ability to apply to new situations.

Pure Mathematics

Pure mathematics allows us to experience ideas of proof and gets us closer to what “real” mathematicians do.  

This topic is a nice chance to explore ideas in proof, number theory and complex numbers.

Matrices and computing

Here are some more interesting topic ideas spanning a variety of mathematical fields – linking to matrices and computational ideas

Using matrices to make fractals

Google page rank – billion dollar maths!

Further ideas

If the ideas above aren’t enough I’ve also added even more ideas and links below.  Please explore!

1) Modular arithmetic – This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3.

2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics.

3) Probabilistic number theory

4) Applications of complex numbers : The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.

5) Diophantine equations : These are polynomials which have integer solutions. Fermat’s Last Theorem is one of the most famous such equations.

6) Continued fractions : These are fractions which continue to infinity. The great Indian mathematician Ramanujan discovered some amazing examples of these.

7) Patterns in Pascal’s triangle : There are a large number of patterns to discover – including the Fibonacci sequence.

8) Finding prime numbers : The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. There is a $1 million prize for solving the Riemann Hypothesis and $250,000 available for anyone who discovers a new, really big prime number.

9) Random numbers

10) Pythagorean triples : A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle).

11) Mersenne primes : These are primes that can be written as 2^n -1.

12) Magic squares and cubes : Investigate magic tricks that use mathematics. Why do magic squares work?

13) Loci and complex numbers

14) Egyptian fractions : Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 2/3 could be written as 1/6 + 1/2. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?

15) Complex numbers and transformations

16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics.

17) Chinese remainder theorem . This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. It involves understanding the modulo operation.

18) Fermat’s last theorem : A problem that puzzled mathematicians for centuries – and one that has only recently been solved.

19) Natural logarithms of complex numbers

20) Twin primes problem : The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. The twin prime conjecture states that there are infinitely many consecutive primes ( eg. 5 and 7 are consecutive primes). There has been a recent breakthrough in this problem.

21) Hypercomplex numbers

22) Diophantine application: Cole numbers

23) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). ie 6 is a perfect number because 1 + 2 + 3 = 6.

24) Euclidean algorithm for GCF

25) Palindrome numbers: Palindrome numbers are the same backwards as forwards.

26) Fermat’s little theorem : If p is a prime number then a^p – a is a multiple of p.

27) Prime number sieves

28) Recurrence expressions for phi (golden ratio): Phi appears with remarkable consistency in nature and appears to shape our understanding of beauty and symmetry.

29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth $1million to anyone who solves it (not for the faint hearted!)

30) Time travel to the future : Investigate how traveling close to the speed of light allows people to travel “forward” in time relative to someone on Earth. Why does the twin paradox work?

31) Graham’s Number – a number so big that thinking about it could literally collapse your brain into a black hole.

32) RSA code – the most important code in the world? How all our digital communications are kept safe through the properties of primes.

33) The Chinese Remainder Theorem : This is a method developed by a Chinese mathematician Sun Zi over 1500 years ago to solve a numerical puzzle. An interesting insight into the mathematical field of Number Theory.

34) Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2? . A post which looks at the maths behind this particularly troublesome series.

35) Fermat’s Theorem on the sum of 2 squares – An example of how to use mathematical proof to solve problems in number theory.

36) Can we prove that 1 + 2 + 3 + 4 …. = -1/12 ? How strange things happen when we start to manipulate divergent series.

37) Mathematical proof and paradox – a good opportunity to explore some methods of proof and to show how logical errors occur.

38) Friendly numbers, Solitary numbers, perfect numbers. Investigate what makes a number happy or sad, or sociable! Can you find the loop of infinite sadness?

39) Zeno’s Paradox – Achilles and the Tortoise – A look at the classic paradox from ancient Greece – the philosopher “proved” a runner could never catch a tortoise – no matter how fast he ran.

40) Stellar Numbers – This is an excellent example of a pattern  sequence investigation. Choose your own pattern investigation for the exploration.

41) Arithmetic number puzzle – It could be interesting to do an exploration where you solve number problems – like this one.

42)  Normal Numbers – and random number generators  – what is a normal number – and how are they connected to random number generators?

43)  Narcissistic Numbers  – what makes a number narcissistic – and how can we find them all?

44)  Modelling Chaos  – how we can use grahical software to understand the behavior of sequences

45) The Mordell Equation .  What is the Mordell equation and how does it help us solve mathematical problems in number theory?

46) Ramanujan’s Taxi Cab and the Sum of 2 Cubes .  Explore this famous number theory puzzle.

47) Hollow cubes and hypercubes investigation.  Explore number theory in higher dimensions!

48) When do 2 squares equal 2 cubes?  A classic problem in number theory which can be solved through computational power.

49) Rational approximations to irrational numbers.  How accurately can be approximate irrationals?

50) Square triangular numbers.  When do we have a square number which is also a triangular number?

51) Complex numbers as matrices – Euler’s identity.  We can use a matrix representation of complex numbers to test whether Euler’s identity still holds.

52) Have you got a Super Brain?  How many different ways can we use to solve a number theory problem?

1a)  Non-Euclidean geometries:  This allows us to “break” the rules of conventional geometry – for example, angles in a triangle no longer add up to 180 degrees. In some geometries triangles add up to more than 180 degrees, in others less than 180 degrees.

1b)  The shape of the universe  – non-Euclidean Geometry is at the heart of Einstein’s theories on General Relativity and essential to understanding the shape and behavior of the universe.

2)  Hexaflexagons:  These are origami style shapes that through folding can reveal extra faces.

3)  Minimal surfaces and soap bubbles : Soap bubbles assume the minimum possible surface area to contain a given volume.

4)  Tesseract – a 4D cube : How we can use maths to imagine higher dimensions.

5)  Stacking cannon balls:  An investigation into the patterns formed from stacking canon balls in different ways.

6)  Mandelbrot set and fractal shapes : Explore the world of infinitely generated pictures and fractional dimensions.

7)  Sierpinksi triangle : a fractal design that continues forever.

8)  Squaring the circle : This is a puzzle from ancient times – which was to find out whether a square could be created that had the same area as a given circle. It is now used as a saying to represent something impossible.

9)  Polyominoes : These are shapes made from squares. The challenge is to see how many different shapes can be made with a given number of squares – and how can they fit together?

10)  Tangrams:  Investigate how many different ways different size shapes can be fitted together.

11)  Understanding the fourth dimension:  How we can use mathematics to imagine (and test for) extra dimensions.

12)  The Riemann Sphere  – an exploration of some non-Euclidean geometry. Straight lines are not straight, parallel lines meet and angles in a triangle don’t add up to 180 degrees.

13)  Graphically understanding complex roots  – have you ever wondered what the complex root of a quadratic actually means graphically? Find out!

14)  Circular inversion  – what does it mean to reflect in a circle? A great introduction to some of the ideas behind non-euclidean geometry.

15)  Julia Sets and Mandelbrot Sets  – We can use complex numbers to create beautiful patterns of infinitely repeating fractals. Find out how!

16)  Graphing polygons investigation.   Can we find a function that plots a square?  Are there functions which plot any polygons?  Use computer graphing to investigate.

17)  Graphing Stewie from Family Guy.  How to use graphic software to make art from equations.

18)  Hyperbolic geometry  – how we can map the infinite hyperbolic plane onto the unit circle, and how this inspired the art of Escher.

19)  Elliptical Curves – how this class of curves have importance in solving Fermat’s Last Theorem and in cryptography.

20)  The Coastline Paradox  – how we can measure the lengths of coastlines, and uses the idea of fractals to arrive at fractional dimensions.

21)  Projective geometry  –  the development of geometric proofs based on points at infinity.

22)  The Folium of Descartes . This is a nice way to link some maths history with studying an interesting function.

23)  Measuring the Distance to the Stars . Maths is closely connected with astronomy – see how we can work out the distance to the stars.

24)  A geometric proof for the arithmetic and geometric mean . Proof doesn’t always have to be algebraic. Here is a geometric proof.

25)  Euler’s 9 Point Circle . This is a lovely construction using just compasses and a ruler.

26)  Plotting the Mandelbrot Set  – using Geogebra to graphically generate the Mandelbrot Set.

27)  Volume optimization of a cuboid   – how to use calculus and graphical solutions to optimize the volume of a cuboid.

28)  Ford Circles – how to generate Ford circles and their links with fractions.

29)  Classical Geometry Puzzle: Finding the Radius . This is a nice geometry puzzle solved using a variety of methods.

30)  Can you solve Oxford University’s Interview Question? .  Try to plot the locus of a sliding ladder.

31)  The Shoelace Algorithm to find areas of polygons .  How can we find the area of any polygon?

32)  Soap Bubbles, Wormholes and Catenoids . What is the geometric shape of soap bubbles?

33)  Can you solve an Oxford entrance question?   This problem asks you to explore a sliding ladder.

34)  The Tusi circle  – how to create a circle rolling inside another circle using parametric equations.

35)  Sphere packing  – how to fit spheres into a package to minimize waste.

36)  Sierpinski triangle  – an infinitely repeating fractal pattern generated by code.

37)  Generating e through probability and hypercubes .  This amazing result can generate e through considering hyper-dimensional shapes.

38)  Find the average distance between 2 points on a square .  If any points are chosen at random in a square what is the expected distance between them?

39)  Finding the average distance between 2 points on a hypercube .  Can we extend our investigation above to a multi-dimensional cube?

40)  Finding focus with Archimedes.   The Greeks used a very different approach to understanding quadratics – and as a result had a deeper understanding of their physical properties linked to light and reflection.

41)  Chaos and strange Attractors: Henon’s map .  Gain a deeper understanding of chaos theory with this investigation.

Calculus/analysis and functions

1)  The harmonic series:  Investigate the relationship between fractions and music, or investigate whether this series converges.

2)  Torus – solid of revolution : A torus is a donut shape which introduces some interesting topological ideas.

3)  Projectile motion:  Studying the motion of projectiles like cannon balls is an essential part of the mathematics of war. You can also model everything from Angry Birds to stunt bike jumping. A good use of your calculus skills.

4)  Why e is base of natural logarithm function:  A chance to investigate the amazing number e.

5)  Fourier Transforms – the most important tool in mathematics?  Fourier transforms have an essential part to play in modern life – and are one of the keys to understanding the world around us. This mathematical equation has been described as the most important in all of physics. Find out more! (This topic is only suitable for IB HL students).

6)  Batman and Superman maths  – how to use Wolfram Alpha to plot graphs of the Batman and Superman logo

7)  Explore the Si(x) function  – a special function in calculus that can’t be integrated into an elementary function.

8)  The Remarkable Dirac Delta Function . This is a function which is used in Quantum mechanics – it describes a peak of zero width but with area 1.

9)  Optimization of area – an investigation . This is an nice example of how you can investigation optimization of the area of different polygons.

10)  Envelope of projectile motion .  This investigates a generalized version of projectile motion – discover what shape is created.

11)  Projectile Motion Investigation II . This takes the usual projectile motion ideas and generalises them to investigate equations of ellipses formed.

12)  Projectile Motion III: Varying gravity .  What would projectile motion look like on different planets?

13)  The Tusi couple – A circle rolling inside a circle . This is a lovely result which uses parametric functions to create a beautiful example of mathematical art.

14)  Galileo’s Inclined Planes .  How did Galileo achieve his breakthrough understanding of gravity?  Follow in the footsteps of a genius!

Statistics and modelling 1 [topics could be studied in-depth]

1) Traffic flow : How maths can model traffic on the roads.

2) Logistic function and constrained growth

3)  Benford’s Law  – using statistics to catch criminals by making use of a surprising distribution.

4)  Bad maths in court  – how a misuse of statistics in the courtroom can lead to devastating miscarriages of justice.

5)  The mathematics of cons  – how con artists use pyramid schemes to get rich quick.

6)  Impact Earth  – what would happen if an asteroid or meteorite hit the Earth?

7)  Black Swan events  – how usefully can mathematics predict small probability high impact events?

8)  Modelling happiness  – how understanding utility value can make you happier.

9)  Does finger length predict mathematical ability?  Investigate the surprising correlation between finger ratios and all sorts of abilities and traits.

10) Modelling epidemics/spread of a virus

11)  The Monty Hall problem  – this video will show why statistics often lead you to unintuitive results.

12) Monte Carlo simulations

13) Lotteries

14)  Bayes’ theorem : How understanding probability is essential to our legal system.

15)  Birthday paradox:  The birthday paradox shows how intuitive ideas on probability can often be wrong. How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday? Find out!

16)  Are we living in a computer simulation?  Look at the Bayesian logic behind the argument that we are living in a computer simulation.

17)  Does sacking a football manager affect results ? A chance to look at some statistics with surprising results.

18)  Which times tables do students find most difficult?  A good example of how to conduct a statistical investigation in mathematics.

19)  Introduction to Modelling.  This is a fantastic 70 page booklet explaining different modelling methods from Moody’s Mega Maths Challenge.

20)  Modelling infectious diseases  – how we can use mathematics to predict how diseases like measles will spread through a population

21)  Using Chi Squared to crack codes  – Chi squared can be used to crack Vigenere codes which for hundreds of years were thought to be unbreakable. Unleash your inner spy!

22)  Modelling Zombies  – How do zombies spread? What is your best way of surviving the zombie apocalypse? Surprisingly maths can help!

23)  Modelling music with sine waves  – how we can understand different notes by sine waves of different frequencies. Listen to the sounds that different sine waves make.

24)  Are you psychic?  Use the binomial distribution to test your ESP abilities.

25)  Reaction times  – are you above or below average? Model your data using a normal distribution.

26)  Modelling volcanoes  – look at how the Poisson distribution can predict volcanic eruptions, and perhaps explore some more advanced statistical tests.

27)  Could Trump win the next election ? How the normal distribution is used to predict elections.

28)  How to avoid a Troll  – an example of a problem solving based investigation

29)  The Gini Coefficient  – How to model economic inequality

30)  Maths of Global Warming – Modeling Climate Change  – Using Desmos to model the change in atmospheric Carbon Dioxide.

31)  Modelling radioactive decay   – the mathematics behind radioactivity decay, used extensively in science.

32)  Circular Motion: Modelling a Ferris wheel . Use Tracker software to create a Sine wave.

33)  Spotting Asset Bubbles . How to use modeling to predict booms and busts.

34)  The Rise of Bitcoin . Is Bitcoin going to keep rising or crash?

35)  Fun with Functions! . Some nice examples of using polar coordinates to create interesting designs.

36)  Predicting the UK election using linear regression . The use of regression in polling predictions.

37)  Modelling a Nuclear War . What would happen to the climate in the event of a nuclear war?

38)  Modelling a football season .  We can use a Poisson model and some Excel expertise to predict the outcome of sports matches – a technique used by gambling firms.

39) Modeling hours of daylight  – using Desmos to plot the changing hours of daylight in different countries.

40)  Modelling the spread of Coronavirus (COVID-19) . Using the SIR model to understand epidemics.

41)  Finding the volume of a rugby ball (or American football) .  Use modeling and volume of revolutions.

42)  The Martingale system paradox.   Explore a curious betting system still used in currency trading today.

Statistics and modelling 2 [more simplistic topics: correlation, normal, Chi squared]

1)  Is there a correlation between hours of sleep and exam grades? Studies have shown that a good night’s sleep raises academic attainment.

2)  Is there a correlation between height and weight?  (pdf).  The NHS use a chart to decide what someone should weigh depending on their height. Does this mean that height is a good indicator of weight?

3)  Is there a correlation between arm span and foot height?   This is also a potential opportunity to discuss the  Golden Ratio  in nature.

4) Is there a correlation between smoking and lung capacity?

5)  Is there a correlation between GDP and life expectancy?  Run the Gapminder graph to show the changing relationship between GDP and life expectancy over the past few decades.

7)  Is there a correlation between numbers of yellow cards a game and league position? Use the Guardian Stats data to find out if teams which commit the most fouls also do the best in the league.

8)  Is there a correlation between Olympic 100m sprint times and Olympic 15000m times? Use the Olympic database to find out if the 1500m times have got faster in the same way the 100m times have got quicker over the past few decades.

9) Is there a correlation between time taken getting to school and the distance a student lives from school?

10) Does eating breakfast affect your grades?

11) Is there a correlation between  stock prices of different companies?  Use Google Finance to collect data on company share prices.

12) Is there a correlation between  blood alcohol laws and traffic accidents ?

13) Is there a correlation between  height and basketball ability?   Look at some stats for NBA players to find out.

14) Is there a correlation between  stress and blood pressure ?

15) Is there a correlation between  Premier League wages and league positions ?

16) Are a sample of student heights  normally distributed?  We know that adult population heights are normally distributed – what about student heights?

17) Are a sample of flower heights normally distributed?

18) Are a sample of student weights normally distributed?

19)  Are the IB maths test scores normally distributed?  (pdf). IB test scores are designed to fit a bell curve. Investigate how the scores from different IB subjects compare.

20) Are the weights of “1kg” bags of sugar normally distributed?

21)  Does gender affect hours playing sport?  A UK study showed that primary school girls play much less sport than boys.

22) Investigation into the distribution of  word lengths in different languages . The English language has an average word length of 5.1 words.  How does that compare with other languages?

23)  Do bilingual students have a greater memory recall than non-bilingual students? Studies have shown that bilingual students have better “working memory” – does this include memory recall?

Games and game theory

1) The prisoner’s dilemma : The use of game theory in psychology and economics.

3)  Gambler’s fallacy:  A good chance to investigate misconceptions in probability and probabilities in gambling. Why does the house always win?

4)  Bluffing in Poker:  How probability and game theory can be used to explore the the best strategies for bluffing in poker.

5)  Knight’s tour in chess:  This chess puzzle asks how many moves a knight must make to visit all squares on a chess board.

6) Billiards and snooker

7) Zero sum games

8)  How to “Solve” Noughts and Crossess  (Tic Tac Toe) – using game theory. This topics provides a fascinating introduction to both combinatorial Game Theory and Group Theory.

9)  Maths and football  – Do managerial sackings really lead to an improvement in results? We can analyse the data to find out. Also look at the  finances  behind Premier league teams

10) Is there a correlation between  Premier League wages and league position?  Also look at how the  Championship  compares to the Premier League.

11)  The One Time Pad  – an uncrackable code? Explore the maths behind code making and breaking.

12)  How to win at Rock Paper Scissors . Look at some of the maths (and psychology behind winning this game.

13)  The Watson Selection Task  – a puzzle which tests logical reasoning.  Are maths students better than history students?

Topology and networks

2) Steiner problem

3)  Chinese postman problem  – This is a problem from graph theory – how can a postman deliver letters to every house on his streets in the shortest time possible?

4) Travelling salesman problem

5)  Königsberg bridge problem : The use of networks to solve problems. This particular problem was solved by Euler.

6)  Handshake problem : With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else?

7)  Möbius strip : An amazing shape which is a loop with only 1 side and 1 edge.

8) Klein bottle

9) Logic and sets

10)  Codes and ciphers : ISBN codes and credit card codes are just some examples of how codes are essential to modern life. Maths can be used to both make these codes and break them.

11)  Zeno’s paradox of Achilles and the tortoise : How can a running Achilles ever catch the tortoise if in the time taken to halve the distance, the tortoise has moved yet further away?

12)  Four colour map theorem  – a puzzle that requires that a map can be coloured in so that every neighbouring country is in a different colour. What is the minimum number of colours needed for any map?

13)  Telephone Numbers  – these are numbers with special properties which grow very large very quickly. This topic links to graph theory.

14) The Poincare Conjecture and Grigori Perelman  – Learn about the reclusive Russian mathematician who turned down $1 million for solving one of the world’s most difficult maths problems.

Mathematics and Physics

1)  The Monkey and the Hunter – How to Shoot a Monkey  – Using Newtonian mathematics to  decide where to aim when shooting a monkey in a tree.

2)  How to Design a Parachute  – looking at the physics behind parachute design to ensure a safe landing!

3)  Galileo: Throwing cannonballs off The Leaning Tower of Pisa  – Recreating Galileo’s classic experiment, and using maths to understand the surprising result.

4)  Rocket Science and Lagrange Points  – how clever mathematics is used to keep satellites in just the right place.

5)  Fourier Transforms – the most important tool in mathematics?  – An essential component of JPEG, DNA analysis, WIFI signals, MRI scans, guitar amps – find out about the maths behind these essential technologies.

6)  Bullet projectile motion experiment  – using Tracker software to model the motion of a bullet.

7)  Quantum Mechanics – a statistical universe?   Look at the inherent probabilistic nature of the universe with some quantum mechanics.

8)  Log Graphs to Plot Planetary Patterns . The planets follow a surprising pattern when measuring their distances.

9)  Modeling with springs and weights . Some classic physics – which generates some nice mathematical graphs.

10)  Is Intergalactic space travel possible?  Using the physics of travel near the speed of light to see how we could travel to other stars.

Maths and computing

1)  The Van Eck Sequence  – The Van Eck Sequence is a sequence that we still don’t fully understand – we can use programing to help!

2)  Solving maths problems using computers  – computers are really useful in solving mathematical problems.  Here are some examples solved using Python.

3)  Stacking cannonballs – solving maths with code  – how to stack cannonballs in different configurations.

4)  What’s so special about 277777788888899?  – Playing around with multiplicative persistence – can you break the world record?

5)  Project Euler: Coding to Solve Maths Problems . A nice starting point for students good at coding – who want to put these skills to the test mathematically.

6)  Square Triangular Numbers .  Can we use a mixture of pure maths and computing to solve this problem?

7)  When do 2 squares equal 2 cubes?  Can we use a mixture of pure maths and computing to solve this problem?

8)  Hollow Cubes and Hypercubes investigation .  More computing led investigations

9)  Coding Hailstone Numbers .  How can we use computers to gain a deeper understanding of sequences?

Further ideas:

1)  Radiocarbon dating  – understanding radioactive decay allows scientists and historians to accurately work out something’s age – whether it be from thousands or even millions of years ago.

2)  Gravity, orbits and escape velocity  – Escape velocity is the speed required to break free from a body’s gravitational pull. Essential knowledge for future astronauts.

3)  Mathematical methods in economics  – maths is essential in both business and economics – explore some economics based maths problems.

4)  Genetics  – Look at the mathematics behind genetic inheritance and natural selection.

5)  Elliptical orbits  – Planets and comets have elliptical orbits as they are influenced by the gravitational pull of other bodies in space. Investigate some rocket science!

6)  Logarithmic scales  – Decibel, Richter, etc. are examples of log scales – investigate how these scales are used and what they mean.

7)  Fibonacci sequence and spirals in nature  – There are lots of examples of the Fibonacci sequence in real life – from pine cones to petals to modelling populations and the stock market.

8)  Change in a person’s BMI over time  – There are lots of examples of BMI stats investigations online – see if you can think of an interesting twist.

9)  Designing bridges  – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse.

10)  Mathematical card tricks  – investigate some maths magic.

11)  Flatland by Edwin Abbott  – This famous book helps understand how to imagine extra dimension. You can watch a short video on it  here

12)  Towers of Hanoi puzzle  – This famous puzzle requires logic and patience. Can you find the pattern behind it?

13)  Different number systems  – Learn how to add, subtract, multiply and divide in Binary. Investigate how binary is used – link to codes and computing.

14)  Methods for solving differential equations  – Differential equations are amazingly powerful at modelling real life – from population growth to to pendulum motion. Investigate how to solve them.

15)  Modelling epidemics/spread of a virus  – what is the mathematics behind understanding how epidemics occur? Look at  how infectious Ebola really is .

16)  Hyperbolic functions  – These are linked to the normal trigonometric functions but with notable differences. They are useful for modelling more complex shapes.

17)  Medical data mining  – Explore the use and misuse of statistics in medicine and science.

18) Waging war with maths: Hollow squares .  How mathematical formations were used to fight wars.

19)  The Barnsley Fern: Mathematical Art  – how can we use iterative processes to create mathematical art?

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IB Math SL Internal Assessment: Directions

• Statistical Analysis

Length:  12-20 pages - no word count; logic, precision and clarity count more than length. 

Method: 

• An introduction that states your rationale/purpose for the topic
• Cite any references or direct quotes using MLA format
• State all definitions and explanations of concepts
• Use proper notation, applicable graphs, and mathematical computations
• Personal engagement
• A conclusion that ties up the major ideas, including whether the results are reasonable. What went wrong? Why? Think TOK
• Use the rubric to guide you

Introduction

Can an examiner clearly identify your topic?

How did you collect your data? What questions did you ask? Did you do a survey? Who answered it? Did you give it personally or over social media?

Do you have well defined processes and well defined parameters?

Step by step explanations.

What are your variables?

What do you need to find to complete your process?

What else do you need to know?

All processes should be in the order that you will complete them. Correlation must be completed before Linear Regression, so it should be described in the introduction before linear regression. 

Make sure your pages are numbered and that your work is double spaced. IB prefers Arial font.

Data Analysis

1. If creating a graph, make sure everything is labeled

2.  Show  ALL  work, every step

3. Include all equations - use the format from class, do not Google the equations

4. After each process, give a brief explanation to the results of the process and how they connect to your topic

5. Create your model, then use technology to compare

6. Analyze for extrapolating and interpolating data

1. Data should be collected through survey, observation, or research

2. Data should be relevant

3. Data should be sufficient in both quantity and quality: 30-50 for a good model

4. Data should be organized in a form that is appropriate for analysis

5. If you are using a survey, you must include the survey - it can be in the paper or added as an appendix

6. Data taken from another source must be cited and must be raw, unanalyzed data; do not use percentages for Chi Squared

1. Create a meaningful conclusion, bringing back together all of your processes and summarizing the results

2. Explain how each process connects to bring you to the conclusion of either proving or disproving your hypothesis

3. Indicate validity - did the processes you used properly help you get to the conclusion you wanted to achieve? Is there any other process that could have been added or anything you would have done differently?

4. Discuss reliability of your model if using statistics

5. Think of real world applications, do outside factors affect the outcome?

6. Could this project lead you into more research or analysis? How?

7. BE REFLECTIVE!

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Mathematics

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