IB Maths IA: 60 Examples and Guidance (2025)

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.

The International Baccalaureate (IB) made several changes to the curriculum and assessment methods, particularly in Mathematics. As of 2025, the IB has updated the syllabi for Mathematics Analysis and Approaches (AA) and Mathematics Applications and Interpretation (AI). The IAs now include a greater emphasis on real-world applications and data analysis.

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. A top-rated Singapore IB tutor recommended that the exploration should be 10 to 15 pages long and students will spend 8 to 12 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”. Use of Mathematics has been updated to emphasize the importance of not only correctness but also the innovative application of mathematical concepts.

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 of IB Maths AA SL, IB Maths AA HL, IB Maths AI SL and IB Maths AI HL Resources, developed by expert IB teachers and examiners:

📚 View IB Maths AA SL Practice Questions
📚 View IB Maths AA HL Practice Questions
📚 View IB Maths AI SL Practice Questions
📚 View IB Maths AI HL Practice Questions

📚 View IB Maths AA SL Study Notes
📚 View IB Maths AA HL Study Notes
📚 View IB Maths AI SL Study Notes
📚 View IB Maths AI HL Study Notes

📚 View IB Maths AA SL Past Papers
📚 View IB Maths AI HL Past Papers
📚 View IB Maths AI SL Past Papers
📚 View IB Maths AI HL Past Papers

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.

Fractals are fascinating mathematical structures that demonstrate self-similarity and infinite complexity. This investigation could involve generating and analyzing fractals like the Mandelbrot set, exploring their relationship with chaos theory, and understanding how they model real-world chaotic systems.

• Use software or equations to create fractals like the Mandelbrot or Julia sets.
• Explore fractal properties: self-similarity, fractal dimensions, and complexity.
• Analyze how small parameter changes affect fractal behavior, linking to chaos theory.
• Apply fractal models to real-world systems like fluid dynamics and signal processing.

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

Recursive sequences are a fundamental tool in mathematics with widespread applications in algorithms and cryptography. This project could focus on generating sequences, analyzing their patterns, and exploring their use in real-world scenarios like data sorting and encryption.

• Generate sequences like Fibonacci and analyze growth behavior and patterns.
• Derive closed-form expressions and study convergence.
• Explore applications in computer science (sorting/searching algorithms).
• Investigate cryptographic uses like Fibonacci-based encryption or pseudorandom number generation.

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

Differential equations are essential for modeling dynamic systems in physics and engineering. This study could focus on classifying equations, solving them using analytical or numerical methods, and examining their applications in areas like heat transfer and harmonic motion.

• Study classifications: ODEs, PDEs, linear vs. non-linear equations.
• Solve equations using separation of variables or numerical techniques.
• Investigate applications in physics (oscillations, circuits) and engineering (fluid dynamics, heat transfer).
• Present findings through derivations, graphs, and real-world examples.

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

Chaotic systems are unpredictable yet governed by deterministic rules. This project could involve simulating systems like the Lorenz attractor to understand features such as sensitivity to initial conditions and strange attractors, while also exploring applications in physics and biology.

• Model systems like the logistic map or Lorenz attractor using simulations.
• Analyze chaos features: bifurcations, sensitivity to initial conditions, and strange attractors.
• Apply chaos theory to real-world scenarios like turbulence and population dynamics.
• Use tools like phase plots and bifurcation diagrams to visualize results.

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

Optimizing delivery routes involves solving logistical problems with mathematical algorithms. This study could explore the use of optimization techniques to minimize travel time and fuel consumption while accounting for real-world constraints like traffic and package priorities.

• Use algorithms like TSP or Dijkstra’s to optimize delivery routes.
• Incorporate constraints: traffic patterns, package weight, and delivery windows.
• Calculate outcomes to reduce fuel costs and improve efficiency.
• Present findings with annotated maps, cost analyses, and simulations.

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

Mathematical models are essential for understanding and predicting the spread of diseases. This project could focus on simulating disease transmission dynamics using real-world data, while accounting for factors like vaccination and population density, and testing the model’s accuracy against observed outcomes.

• Collect data on infection rate, transmission rate, and population demographics.
• Develop a mathematical model incorporating population density, age distribution, and vaccination rates.
• Test the model’s accuracy by comparing predictions to actual disease spread data.
• Simulate scenarios such as vaccination campaigns or quarantine measures to analyze their effectiveness.

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

Geometric shapes exhibit unique relationships between their properties like volume, surface area, and weight. This study could involve creating and analyzing 3D models or physical objects to uncover trends and understand how shape influences these attributes.

• Test various shapes for volume, surface area, and weight using 3D modeling or laboratory tools.
• Analyze the relationship between these properties and shape characteristics.
• Present findings through graphs or charts highlighting trends and patterns.
• Explore real-world applications such as optimizing packaging or architectural designs.

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

Projectile motion follows precise physical laws, influenced by factors like angle, velocity, and environmental conditions. This investigation could involve conducting controlled experiments to track trajectories and applying findings to practical scenarios like sports or engineering.

• Conduct experiments tracking projectile motion at various angles and velocities using high-speed cameras or sensors.
• Analyze motion properties: velocity, acceleration, and range.
• Explore effects of air resistance and vacuum conditions on motion.
• Apply findings to real-world contexts, such as rocket design or sports physics.

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

The movement of celestial bodies is governed by gravitational interactions. This project could involve using Newton’s law of gravitation to model a planet’s trajectory and comparing predictions with observational data to refine the model’s accuracy.

• Collect data on mass, position, and velocity of the planet and influencing celestial bodies.
• Use the law of gravitation to calculate forces and predict changes in velocity and direction.
• Simulate the planet’s path over time and compare with observed trajectories.
• Explore the model’s implications for understanding orbital mechanics and space exploration.

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

Conic sections like circles, ellipses, parabolas, and hyperbolas have diverse properties and applications. This project could explore their mathematical foundations and real-world uses, from satellite dishes to planetary orbits.

• Analyze equations defining conic sections and their key properties.
• Study applications in geometry (e.g., reflective properties of parabolas) and physics (e.g., elliptical orbits).
• Develop simulations to demonstrate these applications in real-world contexts.
• Present findings on how conic sections solve practical problems in technology and science.

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

Mathematical models of virus transmission are vital for understanding outbreaks and planning interventions. This project could involve simulating the spread of a virus while evaluating the impact of strategies like vaccination and social distancing, helping to predict outcomes under various scenarios.

• Account for factors such as infection rate, transmission probability, and intervention effectiveness.
• Simulate scenarios like social distancing, vaccination campaigns, or quarantines.
• Compare predicted outcomes of different strategies using data visualizations.
• Use real-world data to validate the model and predict potential future outbreaks.

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

The structure of a network can significantly influence how information or diseases spread. This investigation could involve simulating the spread through various network topologies, analyzing how the structure impacts the rate and extent of spread.

• Consider key factors like transmission probability, recovery rate, and decay.
• Test different network structures: random, scale-free, and small-world networks.
• Visualize results with graphs or heat maps showing the progression over time.
• Highlight real-world implications, such as information dissemination or controlling outbreaks.

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

Population growth models provide insights into demographic changes and help predict future trends. This project could involve creating a model based on historical data and identifying factors influencing growth rates.

• Collect data on population size, birth rate, death rate, and migration.
• Develop a mathematical model using exponential or logistic growth equations.
• Validate predictions by comparing them with historical growth data.
• Explore applications, such as urban planning or environmental impact analysis.

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

Exponential functions describe growth and decay processes that are critical in finance and economics. This project could focus on analyzing their properties and applying them to scenarios like compound interest or economic trends.

• Define key properties: domain, range, growth/decay rates, and asymptotes.
• Apply exponential models to real-world contexts, such as:
  • Compound interest calculations.
  • Population growth projections.
  • Stock market trends.
• Graph exponential functions and interpret the results in practical terms.
• Explore how small parameter changes significantly impact long-term outcomes.

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

Sports predictions rely on statistical modeling of historical data and team performance metrics. This project could involve analyzing trends and developing a model to forecast game outcomes with a measure of accuracy.

• Collect data on win-loss records, player statistics, and team performance trends.
• Develop a statistical model incorporating factors like player form and head-to-head history.
• Validate the model using additional historical data and refine it based on expert feedback.
• Present predictions with an assessment of accuracy, limitations, and uncertainties.

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

Sequences play a fundamental role in mathematics, especially in understanding limits and patterns. This project could involve analyzing various sequences to study their convergence or divergence and exploring practical applications in fields like computer science and physics.

• Test sequences such as arithmetic, geometric, and recursive for convergence/divergence.
• Use computer simulations to visualize sequence behavior under different conditions.
• Investigate applications in algorithms, signal processing, or physical systems.
• Present findings with graphs and explore potential advancements in analyzing sequences.

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

Angles are central to geometry and trigonometry, forming the basis of many mathematical relationships. This investigation could involve analyzing various angle types and their roles in solving geometric and trigonometric problems.

• Explore angle types (acute, obtuse, right, straight) and their properties.
• Study relationships between angles in polygons, triangles, and circles.
• Create visual aids, such as diagrams and graphs, to demonstrate concepts.
• Highlight applications in construction, navigation, and physics.

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

Probability theory provides powerful tools for predicting outcomes in games. This project could involve creating a statistical model that incorporates key variables to predict game results with greater accuracy.

• Collect historical game data, including scores, team statistics, and external factors (e.g., weather, injuries).
• Calculate probabilities of outcomes based on relevant variables.
• Develop and refine a predictive model using statistical analysis.
• Test the model’s reliability and identify potential limitations or improvements.

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

Inequalities are essential in understanding constraints and optimization problems in mathematics. This project could focus on analyzing their behavior and applying them to practical scenarios in algebra and calculus.

• Graph inequalities, such as linear, quadratic, and exponential, to observe behavior.
• Apply inequalities in calculus to find maxima and minima under constraints.
• Explore real-world applications like optimizing resources in production or predicting trends in data.
• Use examples to connect theoretical concepts with practical uses.

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

Graphs are versatile tools for modeling relationships in various fields. This study could involve exploring their types and evaluating their effectiveness in conveying insights in computer science and social sciences.

• Conduct a literature review on types of graphs (e.g., bar, line, scatter, network).
• Develop criteria to evaluate graph effectiveness in visualizing data.
• Compare graphs used in computer science (e.g., network graphs) and social science (e.g., demographic trends).
• Create guidelines for selecting and designing effective graphs for specific research purposes.

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

Matrices play a crucial role in linear algebra and quantum mechanics, serving as tools to solve equations and represent quantum states. This project could involve analyzing the behavior of various matrix types through operations and exploring their practical applications in both fields.

• Perform experiments with matrices (e.g., inverses, eigenvalues, and determinants) in linear algebra.
• Analyze how matrices are used in quantum mechanics to represent states and operators.
• Explore applications such as solving systems of linear equations or modeling quantum systems.
• Evaluate results through computational tools and theoretical explanations.

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

Predictive models help businesses make informed investment decisions by analyzing trends and data. This project could involve using historical and market-specific variables to create a model for evaluating potential investment outcomes.

• Collect data on market trends, industry growth rates, and financial statements.
• Develop a predictive model incorporating variables like consumer demand and competitive performance.
• Test and refine the model using historical data and adjust based on new information.
• Present the model’s predictions, highlighting potential risks and returns.

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

Forest fire modeling helps understand fire dynamics and evaluate containment methods. This project could involve simulating fire spread using environmental data and comparing the effectiveness of various strategies.

• Collect data on factors like wind direction, temperature, humidity, and fuel load.
• Develop a computer model to simulate fire spread calibrated with historical fire data.
• Test containment strategies such as firebreaks, water deployment, and fire retardants.
• Evaluate effectiveness by comparing fire spread with and without specific strategies.

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

Optimization is essential in engineering and computer science for solving complex problems efficiently. This project could focus on analyzing different types of optimization problems and comparing algorithms to identify best practices.

• Conduct a literature review on types of optimization problems (e.g., linear, nonlinear, combinatorial).
• Develop a framework to analyze problem size, solution space complexity, and algorithm efficiency.
• Apply the framework to case studies, testing algorithms like gradient descent or genetic algorithms.
• Compare algorithm performance and recommend best practices for different scenarios.

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

Geometric transformations like translation and scaling are foundational in creating computer graphics and animations. This project could involve analyzing transformations to understand their applications and limitations.

• Review properties of transformations (e.g., translation, rotation, scaling, shearing).
• Develop test cases to demonstrate transformations in graphic creation and animation.
• Compare the effectiveness of transformations for different graphic applications.
• Identify challenges and limitations, such as distortion or computational cost, in complex transformations.

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

Predicting election outcomes requires analyzing polling data alongside historical trends and demographic factors. This project could involve building a model that identifies voting patterns, predicts results, and tests scenarios such as changes in voter turnout or opinion shifts.

• Collect polling data and analyze it using regression analysis or machine learning algorithms.
• Train the model on historical election data to improve accuracy.
• Use the model to identify key factors driving voter behavior.
• Test scenarios like varying voter turnout or shifts in public opinion to refine predictions.

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

Integrals are fundamental tools in mathematics, with applications in calculating areas, volumes, and physical quantities. This project could involve analyzing definite and indefinite integrals to understand their behavior and practical uses in fields like physics.

• Compare definite and indefinite integrals by calculating areas under curves.
• Analyze motion by applying integrals to calculate velocity and acceleration over time.
• Explore applications in physics, such as work done by a force or energy stored in a system.
• Use visual representations and real-world examples to connect theory to practical applications.

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

Probability distributions help model uncertainties in data and are widely applied in fields like finance and statistics. This project could involve analyzing their properties and simulating data to evaluate their effectiveness in real-world contexts.

• Review types of distributions, such as normal, binomial, and Poisson, and their properties.
• Develop a framework to analyze distribution relevance in different scenarios.
• Simulate data using statistical software to test the framework.
• Evaluate applications in finance (e.g., risk modeling) and statistics (e.g., hypothesis testing).

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

Predictive models enable businesses to evaluate the success of marketing strategies using consumer data. This project could involve analyzing patterns in purchasing behavior and demographics to forecast campaign outcomes.

• Collect consumer data on demographics, purchasing habits, and social media activity.
• Train the model on historical marketing campaign data to identify patterns and trends.
• Use the model to predict outcomes of future campaigns based on input data.
• Test model accuracy by comparing predictions with actual campaign results.

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

Symmetry is a key concept in geometry and physics, influencing everything from pattern design to the study of natural structures. This project could explore different types of symmetry and their applications in creating patterns and understanding physical phenomena.

• Study bilateral, radial, and rotational symmetry by analyzing shapes and their axes of symmetry.
• Create diagrams or models to demonstrate symmetrical properties and patterns.
• Explore symmetry’s role in physics, such as in crystal structures or particle interactions.
• Highlight applications in art, architecture, and scientific modeling.

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

The spread of rumours and news stories can be understood through mathematical modelling, considering factors like sharing rates, belief likelihood, and social dynamics. This project could also explore how such propagation affects behaviour, attitudes, or the spread of related misinformation.

• Develop a model accounting for factors like initial spreaders, sharing rates, and belief likelihood.
• Analyse behavioural changes, attitude shifts, or related misinformation.
• Use real-world data to refine the model and validate its predictions.
• Explore applications in media influence, marketing strategies, or public information campaigns.

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

Exponential growth and decay functions describe many real-world processes, from population dynamics to radioactive decay. This study could involve exploring their mathematical properties, visualising them graphically, and applying them to fields such as biology, physics, and economics.

• Study equations defining exponential growth and decay functions and their properties.
• Graph functions to visualise their behaviour and identify trends.
• Explore applications such as population growth, disease spread, or radioactive decay.
• Use findings to inform decisions and develop predictive models for future trends.

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

Pandemic modelling is essential for predicting case numbers and evaluating intervention strategies. This project could involve simulating the pandemic’s spread while assessing the impact of measures like vaccination, social distancing, and mask-wearing.

• Develop a model incorporating factors like transmission rate, incubation period, and recovery rate.
• Simulate and compare intervention strategies (e.g., mask usage, vaccination).
• Validate the model using real-world data and refine it based on new information.
• Apply results to inform public health policies and control measures.

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

Functions such as linear, quadratic, and exponential are widely used to model real-world phenomena. This project could explore their properties and applications by analysing data sets and identifying the best-fitting functions for various scenarios.

• Study types of functions (linear, quadratic, exponential, logarithmic) and their properties.
• Analyse real-world data sets to determine which function best fits the data.
• Explore applications in physics, chemistry, economics, and engineering.
• Use findings to demonstrate how functions help predict trends and solve problems.

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

Numerical methods for solving differential equations are crucial in modelling complex systems. This project could involve comparing methods like Euler’s and Runge-Kutta in terms of accuracy and efficiency when applied to phenomena such as heat transfer or chemical reactions.

• Simulate differential equations using methods such as Euler’s and Runge-Kutta.
• Model physical phenomena like fluid flow, heat transfer, or chemical reactions.
• Compare methods based on accuracy, computational time, and error analysis.
• Optimise numerical methods for specific scientific and engineering applications.

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

Predictive models in healthcare help improve patient outcomes by leveraging medical history and demographic data. This project could involve developing a machine learning model to forecast treatment effectiveness and guide decision-making.

• Collect patient data, including demographics, medical history, and treatment outcomes.
• Use statistical analysis and machine learning to develop the predictive model.
• Validate the model with a separate patient data set for accuracy and reliability.
• Explore applications in personalised medicine and optimising treatment plans.

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

Linear regression models are powerful tools for identifying trends in data. This project could explore the application of regression techniques to public opinion polls, comparing models to find the best fit for predicting patterns.

• Collect public opinion poll data on a specific topic, such as political preferences.
• Apply models like simple, multiple, and logistic regression to analyse trends.
• Compare model performance to determine the most suitable one for the dataset.
• Use findings to make predictions or provide insights for policy decisions.

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

Startups’ growth potential can be predicted by analysing market trends and financial indicators. This project could focus on creating a model that identifies patterns and uses them to forecast success.

• Collect data on market trends, competition, funding, and startup performance.
• Identify patterns and correlations using statistical analysis.
• Develop and test a predictive model on data from existing startups.
• Refine the model to predict the growth potential of new startups based on key factors.

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

Statistical distributions are essential for analysing and interpreting public health data. This project could involve exploring distributions like normal, Poisson, and binomial to identify trends and draw conclusions about health outcomes.

• Analyse public health data using normal, Poisson, and binomial distributions.
• Understand the properties of each distribution and select the most suitable one for the data.
• Use statistical software to plot data and identify trends and patterns.
• Present results through graphs, tables, and statistical summaries for better understanding.

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

Mathematical series exhibit unique behaviours depending on their type. This project could involve testing the convergence or divergence of various series and exploring their properties visually and analytically.

• Test geometric, arithmetic, and harmonic series for convergence or divergence.
• Use formulas and calculations to determine the behaviour of each series.
• Represent results using graphs and charts for visual comparison.
• Analyse conditions that affect the convergence or divergence of different series.

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

Understanding how functions behave near their limits provides critical insights into their properties and applications. This project could involve exploring asymptotes, continuity, and inflection points across various function types.

• Graph functions to study their behaviour near limits, focusing on asymptotes and discontinuities.
• Determine continuity and identify points of inflection where the function changes concavity.
• Compare and contrast the similarities and differences in limit behaviour across function types.
• Present findings with visual aids and detailed explanations in a report or presentation.

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

Set theory is foundational in mathematics, and this project could explore properties such as cardinality and operations like union and intersection. Analysing relationships between different sets can deepen understanding of their applications.

• Analyse finite, infinite, empty sets, and subsets, focusing on their properties.
• Explore operations like intersection, union, complement, and Cartesian product.
• Use Venn diagrams and examples to visually represent relationships between sets.
• Highlight real-world applications in fields like computer science and logic.

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

Different number systems, such as real, complex, and p-adic numbers, exhibit unique properties and applications. This project could compare their characteristics and analyse their role in solving mathematical problems.

• Conduct a literature review to understand the properties of each number system.
• Design mathematical problems to test properties, such as graphing functions or solving equations.
• Compare key features like completeness, algebraic structure, and visual representation.
• Analyse how these systems apply to real-world problems in physics, engineering, or cryptography.

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

Calculus of variations is a powerful tool for modelling physical systems. This project could involve using it to predict system behaviour over time and testing the model against real-world data.

• Collect data on the system’s initial state and external influencing factors.
• Use the calculus of variations to derive equations describing the system’s behaviour.
• Test the model’s accuracy by comparing predictions to real-world observations.
• Refine the model for reliability and explore applications in physics and engineering.

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

Topology studies the properties of spaces that remain invariant under continuous transformations. This project could involve exploring different types of topological spaces and analysing their fundamental properties.

• Study Euclidean spaces, metric spaces, and topological manifolds, focusing on compactness, connectedness, and continuity.
• Use diagrams and models to visualise the relationships between different spaces.
• Analyse mathematical tools used to explore topological properties and invariants.
• Highlight applications of topology in physics, computer science, and robotics.

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

Integrals like line and surface integrals play a vital role in solving problems related to fields like electromagnetism and fluid dynamics. This project could involve studying their properties and testing their applications through experiments or simulations.

• Research the use of line integrals for calculating work in force fields.
• Investigate surface integrals for determining flux through surfaces.
• Design and conduct simulations or experiments to test specific hypotheses.
• Analyse results to evaluate the behaviour and practical applications of integrals.

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

Chemical kinetics provides insights into how reactions progress over time. This project could focus on creating a predictive model based on reaction data and testing its accuracy against experimental results.

• Collect data on initial reactant concentrations, temperature, and other factors.
• Develop a mathematical model using chemical kinetics equations.
• Test the model by comparing predictions to experimental data.
• Refine the model to optimise accuracy and predict reaction behaviour under varying conditions.

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

Algebraic structures form the backbone of abstract algebra and have wide-ranging applications. This project could explore the properties of a specific structure and compare them with others to highlight their unique characteristics.

• Conduct a literature review on groups, rings, fields, and their key properties.
• Select a specific algebraic structure to investigate in depth.
• Perform calculations and analyse examples to illustrate key concepts.
• Compare and contrast the chosen structure with other algebraic structures.

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

Functions like trigonometric, logarithmic, and hyperbolic functions are foundational in modelling real-world phenomena. This project could analyse their behaviour and applications across various scientific and engineering disciplines.

• Study the properties of trigonometric, logarithmic, and hyperbolic functions.
• Analyse real-world data sets to identify which functions best fit specific scenarios.
• Develop models using these functions to predict future outcomes.
• Explore practical applications in engineering, physics, or finance.

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

Financial markets are influenced by numerous variables, making prediction complex yet achievable with advanced mathematical techniques. This project could involve using models to forecast market behaviour and inform investment strategies.

• Collect data on stock prices, interest rates, and economic indicators.
• Use mathematical tools like stochastic calculus and differential equations to develop a model.
• Test and refine the model using historical and real-time data.
• Present a validated model that predicts market behaviour and supports 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.

Complex systems, such as networks or agent-based models, exhibit intricate behaviours that depend on interactions between components. This project could involve developing simulations to study these systems and analyse their dynamics under varying conditions.

• Build simulation models for network dynamics, agent-based systems, or game-theoretic interactions.
• Incorporate relevant variables and agent behaviours into the models.
• Analyse system behaviour under different scenarios to identify patterns and dynamics.
• Explore real-world applications, such as social networks, ecosystems, or economic systems.

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

Partial differential equations (PDEs) are fundamental in modelling physical phenomena. This project could explore different types of PDEs, their numerical solutions, and applications in fields like fluid dynamics and heat transfer.

• Conduct a literature review on the types of PDEs, such as elliptic, parabolic, and hyperbolic equations.
• Develop mathematical models to simulate PDE behaviour using numerical methods.
• Analyse applications in physics and engineering, such as heat transfer or electromagnetic fields.
• Use simulations to visualise solutions and validate findings.

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

Fluid dynamics allows for predicting how fluids behave under various conditions. This project could involve creating and validating a computational model to optimise fluid systems and improve efficiency in real-world applications.

• Use computational fluid dynamics (CFD) software to model fluid systems.
• Simulate fluid behaviour under varying conditions, such as flow rate or temperature.
• Validate the model by comparing predictions with experimental data.
• Optimise system design by testing geometric changes or chemical additives.

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

Geometric objects like manifolds and curves are vital in understanding spatial relationships and physical theories. This project could involve analysing their properties, testing theories, and exploring applications in fields like relativity or mechanics.

• Research the properties of geometric objects, such as manifolds, curves, and surfaces.
• Conduct experiments or simulations to test existing theories and models.
• Analyse and synthesise findings to draw conclusions about geometric properties.
• Explore applications in physics, such as general relativity or structural mechanics.

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.

Stochastic processes model systems influenced by randomness and uncertainty. This project could involve simulating processes like random walks and analysing their applications in statistics and real-world scenarios.

• Simulate stochastic processes using software like Python or R.
• Compare simulation results with theoretical predictions to validate models.
• Analyse real-world examples, such as stock price movements or weather patterns.
• Explore applications of stochastic processes in probability theory and statistical modelling.

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

Mathematical biology allows for modelling complex biological systems and predicting their future behaviour. This project could focus on creating a model to study aspects such as population growth, disease spread, or environmental impacts.

• Collect data on variables like population size, birth and death rates, and environmental factors.
• Develop a mathematical model to predict the system’s behaviour over time.
• Test and refine the model using real-world data and observations.
• Explore applications, such as predicting disease outbreaks or ecological changes.

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

Wave phenomena underpin many technologies in physics and engineering. This project could involve studying their properties through experiments and analysing their applications in fields like communication and medical imaging.

• Conduct experiments to measure wave properties: frequency, wavelength, amplitude, and speed.
• Use instruments like oscilloscopes, microphones, and antennas to collect data.
• Analyse applications in designing communication systems, medical imaging, and musical instruments.
• Present findings with graphs and visual aids to demonstrate key insights.

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

Optimisation in dynamic environments helps solve problems that evolve over time. This project could focus on testing various algorithms to identify the most effective strategies for different scenarios.

• Simulate optimisation algorithms like dynamic programming or genetic algorithms.
• Test algorithm performance under different scenarios and parameters.
• Analyse which algorithms work best for specific types of dynamic problems.
• Use findings to develop efficient optimisation 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.

Social network analysis provides insights into the dynamics of interactions and influence. This project could involve modelling a social network to predict changes in behaviour or structure over time.

• Collect data on connections, interactions, and changes in the social network.
• Apply techniques like centrality measures, community detection, and opinion dynamics.
• Develop a model to predict future network behaviour or structural changes.
• Validate the model with additional data or real-world comparisons.

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.

Algebraic curves and surfaces are integral to many advanced mathematical theories and applications. This project could focus on analysing their properties and exploring uses in areas like cryptography or coding theory.

• Conduct a literature review on the properties of algebraic curves and surfaces.
• Use mathematical software to generate and analyse examples of these curves and surfaces.
• Explore applications in algebraic geometry, such as cryptographic protocols or error-correcting codes.
• Present findings in a research paper or presentation with visual illustrations.

What are Interdisciplinary Approaches in Mathematics IA?

With the increasing focus on interdisciplinary learning, students are encouraged to link their Mathematics IA with subjects such as economics, environmental science, and computer science. This approach not only enriches the mathematical exploration but also allows students to showcase their ability to connect concepts across different fields.

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.

An IB tutor in Hong Kong said:

"One important aspect that students ignore is that they 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.