Choosing your IA topic in IB Maths: 8 ideas that examiners love

Introduction: The IA Journey Begins

The Math Internal Assessment (IA) is a significant piece of coursework for all IB Diploma Programme students. Over my ten years teaching IB Maths, I've seen students approach the IA with a mix of excitement and apprehension. Choosing a topic is often the first major hurdle. A strong topic is the foundation of a good IA, and it can make the research and writing process much more enjoyable and productive. This article aims to guide you through selecting a topic that not only interests you but also has the potential to score well. My goal here is to provide practical advice based on what I’ve observed examiners appreciate. We'll explore various areas of mathematics and connect them to real-world applications or deeper mathematical explorations, giving you concrete ideas to consider for your own IA.

What Examiners Look For in an IA Topic

Before diving into specific ideas, it’s important to understand the criteria. Examiners want to see personal engagement, a clear mathematical aim, and a substantial mathematical investigation. This means your topic should allow you to demonstrate:

• Personal Engagement: A genuine interest in the topic, evident through your choices, reflections, and the depth of your investigation.
• Mathematical Communication: Clear, coherent, and concise use of mathematical language and notation.
• Mathematical Presentation: Well-organised work with appropriate use of diagrams, graphs, and tables.
• Mathematical Thinking: Evidence of critical thinking, problem-solving, and justification of methods.
• Reflection: Critical evaluation of your work, including limitations and extensions.
• Use of Mathematics: Appropriate and relevant mathematics at a level consistent with your course (SL or HL, AA or AI).

A good topic gives you the scope to hit all these points effectively. Avoid topics that are too simple, leading to a superficial investigation, or too complex, causing you to struggle with the underlying mathematics.

8 Ideas That Examiners Often Appreciate

Here are eight types of topics that, in my experience, tend to provide rich ground for investigation and allow students to demonstrate a wide range of mathematical skills. Remember to tailor these general ideas to your own interests and the specific mathematics of your course (AA SL, AA HL, AI SL, AI HL).

1. Optimization Problems

Optimization is a classic area of mathematics that involves finding the best solution from a set of alternatives. This can be applied to countless real-world scenarios.

• AA SL/HL, AI SL/HL: Investigating the optimal dimensions of packaging (e.g., a cylindrical can, a box with a specific volume) to minimize surface area or material cost. You can use calculus (differentiation) for continuous functions or consider discrete cases. My students often find this engaging when they pick a product they use daily.
• AA HL: Optimizing routes using graph theory (Dijkstra's algorithm, Minimum Spanning Tree) or linear programming for complex scenarios.
• AI SL/HL: Optimizing resource allocation in a business or production process, using linear programming techniques and exploring the constraints.

An example could be "Optimizing the shape of a dog food can to minimize material usage while maintaining a volume of $500 \text{ cm}^3$." This allows for differentiation, analysis of critical points, and a discussion of practical limitations.

2. Modeling Real-World Phenomena with Functions

Using mathematical functions to model and predict real-world phenomena is at the heart of applied mathematics. This is particularly strong for AI courses but also relevant for AA.

• AA SL/HL, AI SL/HL: Modeling population growth (e.g., bacteria, animal species) using exponential or logistic functions. You can collect real data, analyze it, and discuss the limitations of your model.
• AA SL/HL, AI SL/HL: Modeling the trajectory of a projectile (e.g., a thrown ball, a rocket) using quadratic functions and considering factors like air resistance.
• AA HL, AI HL: Modeling the spread of a disease (SIR models) using differential equations or discrete approximations. This involves more advanced calculus and can be very powerful.
• AI SL/HL: Modeling financial investments with compound interest, annuities, or loan repayments. This often involves sequences and series, which students find highly relevant.

Consider "Modeling the cooling of a hot cup of coffee using Newton's Law of Cooling, comparing theoretical predictions with experimental data." This allows for data collection, regression, and analysis of exponential decay.

3. Statistical Investigations and Hypothesis Testing

For students in AI, and relevant for AA SL/HL, statistical investigations offer a rich source of IA topics. These topics often involve collecting and analyzing real-world data.

• AI SL/HL, AA SL/HL: Investigating correlations between two variables (e.g., hours of study and exam scores, economic indicators and stock prices). This requires careful data collection, scatter plots, and calculating correlation coefficients ($r$ or Spearman's $r_s$).
• AI SL/HL, AA SL/HL: Comparing two populations using hypothesis tests (e.g., comparing the mean heights of students from two different schools, comparing proportions of success rates for different treatments). T-tests, chi-squared tests, and ANOVA are all possibilities depending on your course.

My students have often done well with topics like "Is there a correlation between daily screen time and academic performance among IB students?" This allows for surveys, data analysis, and a discussion of statistical significance. Remember to check out our AI SL Paper 2 guide for more on statistics.

4. Geometrical Explorations

Geometry provides a visual and often intuitive starting point for many interesting investigations.

• AA SL/HL, AI SL/HL: Investigating properties of specific geometric shapes, perhaps through transformations, or exploring non-Euclidean geometries for HL.
• AA SL/HL, AI SL/HL: Analyzing packing problems, such as how to pack circles in a square or spheres in a cube most efficiently. This can involve trigonometry and area/volume calculations.
• AA HL: Exploring fractals, their dimensions, and generating them using iterative processes. This can involve sequences, series, and advanced concepts of dimension.

A good example might be "An investigation into the area of polygons inscribed within a circle as the number of sides increases, leading to the approximation of $\pi$." This connects geometry to limits and numerical methods.

5. Applications of Sequences and Series

Sequences and series have wide-ranging applications, from financial mathematics to computer science.

• AA SL/HL, AI SL/HL: Modeling financial growth, annuities, or loan repayments using arithmetic and geometric sequences and series. This is especially strong for AI students.
• AA HL: Investigating convergence of series, Taylor series approximations, or Fourier series in specific contexts.
• AA SL/HL: Exploring patterns in natural phenomena like the Fibonacci sequence in plant structures or population dynamics.

"Modeling the depreciation of a car's value over time using a geometric progression and comparing it to real-world market data" is a practical and engaging topic.

6. Calculus in Physics or Engineering

For AA HL students, in particular, using calculus to model physical systems can be very rewarding.

• AA HL: Investigating projectile motion with air resistance, involving differential equations.
• AA HL: Analyzing harmonic motion (e.g., a mass on a spring, a pendulum) using second-order differential equations.
• AA HL: Exploring the concept of work done by a variable force using integration.

An IA on "Analyzing the motion of a damped pendulum using second-order differential equations and comparing different damping coefficients" would be very strong for an AA HL student.

7. Game Theory and Decision Making

Game theory, which involves mathematical models of strategic interaction among rational decision-makers, can be a fascinating area.

• AA SL/HL, AI SL/HL: Investigating simple game theory scenarios, such as the Prisoner's Dilemma, and analyzing optimal strategies using payoff matrices. This can be adapted for various levels.
• AI HL: Exploring more complex game theory models, perhaps involving probability and expected outcomes, or applying linear programming to find optimal mixed strategies.

A student could investigate "Optimal strategies in a simple board game like 'Rock, Paper, Scissors' using probability and expected value."

8. Cryptography and Number Theory

Number theory, the study of integers and their properties, forms the basis of modern cryptography. This is often more suited to AA courses, especially HL.

• AA SL/HL: Investigating properties of prime numbers, divisibility rules, or specific number sequences.
• AA HL: Exploring modular arithmetic and its applications in public-key cryptography (e.g., RSA algorithm in a simplified context).
• AA HL: Analyzing error-detecting codes based on number theory principles.

"An exploration of the Euclidean algorithm and its application in finding the greatest common divisor and solving linear Diophantine equations" could be a solid topic.

Next Steps: From Idea to Investigation

Once you have a few potential ideas, the next step is to research them more deeply. Try to find existing resources – textbooks, online articles, or even previous IAs (if available and used ethically for inspiration). Don't be afraid to tweak your topic as you learn more. The initial idea is just a starting point. Remember that your teacher is your best resource. Discuss your ideas with them to ensure they are viable for your specific course and have enough mathematical depth. They can help you refine your topic and guide you towards relevant mathematical concepts. I always encourage my students to come to me with at least three different ideas, even if they're just broad concepts. This helps us narrow down the best fit. For more general advice on preparing for the IB, you might find our pre-IB guide useful, and for specific resources, check out our study notes. The IA is your opportunity to explore a mathematical area that genuinely interests you. Choose a topic that you are curious about, and the process will be much more rewarding. Good luck!