mathematical analysis essay topics

260 Interesting Math Topics for Essays & Research Papers

Mathematics is the science of numbers and shapes. Writing about it can give you a fresh perspective and help to clarify difficult concepts. You can even use mathematical writing as a tool in problem-solving.

In this article, you will find plenty of interesting math topics. Besides, you will learn about branches of mathematics that you can choose from. And if the thought of letters and numbers makes your head swim, try our custom writing service . Our professionals will craft a paper for you in no time!

And now, let’s proceed to math essay topics and tips.

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

• 🏫 Math Topics for High School
• 🎓 College Math Topics
• 🤔 Advanced Math
• 📚 Math Research
• ✏️ Math Education
• 💵 Business Math

🔍 References

• Number theory in everyday life.
• Logicist definitions of mathematics.
• Multivariable vs. vector calculus.
• 4 conditions of functional analysis.
• Random variable in probability theory.
• How is math used in cryptography?
• The purpose of homological algebra.
• Concave vs. convex in geometry.
• The philosophical problem of foundations.
• Is numerical analysis useful for machine learning?

What exactly is mathematics ? First and foremost, it is very old. Ancient Greeks and Persians were already utilizing mathematical tools. Nowadays, we consider it an interdisciplinary language.

Biologists, linguists, and sociologists alike use math in their work. And not only that, we all deal with it in our daily lives. For instance, it manifests in the measurement of time. We often need it to calculate how much our groceries cost and how much paint we need to buy to cover a wall.

Simply put, mathematics is a universal instrument for problem-solving. We can divide pure math into three branches: geometry, arithmetic, and algebra. Let’s take a closer look:

• Geometry By studying geometry, we try to comprehend our physical surroundings. Geometric shapes can be simple, like a triangle. Or, they can form complicated figures, like a rhombicosidodecahedron.
• Arithmetic Arithmetic deals with numbers and simple operations: subtraction, addition, division, and multiplication.
• Algebra Algebra is used when the exact numbers are unclear. Instead, they are replaced with letters. Businesses often need algebra to predict their sales.

It’s true that most high school students don’t like math. However, that doesn’t mean it can’t be a fun and compelling subject. In the following section, you will find plenty of enthralling mathematical topics for your paper.

If you’re struggling to start working on your essay, we have some fun and cool math topics to offer. They will definitely engage you and make the writing process enjoyable. Besides, fun math topics can show everyone that even math can be entertaining or even a bit silly.

• The link between mathematics and art – analyzing the Golden Ratio in Renaissance-era paintings.
• An evaluation of Georg Cantor’s set theory.
• The best approaches to learning math facts and developing number sense.
• Different approaches to probability as explored through analyzing card tricks. 
• Chess and checkers – the use of mathematics in recreational activities.
• The five types of math used in computer science.
• Real-life applications of the Pythagorean Theorem. 
• A study of the different theories of mathematical logic.
• The use of game theory in social science.
• Mathematical definitions of infinity and how to measure it.
• What is the logic behind unsolvable math problems?
• An explanation of mean, mode, and median using classroom math grades.
• The properties and geometry of a Möbius strip.
• Using truth tables to present the logical validity of a propositional expression.
• The relationship between Pascal’s Triangle and The Binomial Theorem. 
• The use of different number types: the history.
• The application of differential geometry in modern architecture.
• A mathematical approach to the solution of a Rubik’s Cube.
• Comparison of predictive and prescriptive statistical analyses.
• Explaining the iterations of the Koch snowflake.
• The importance of limits in calculus.
• Hexagons as the most balanced shape in the universe.
• The emergence of patterns in chaos theory.
• What were Euclid’s contributions to the field of mathematics?
• The difference between universal algebra and abstract algebra.

🏫 Math Essay Topics for High School

When writing a math paper, you want to demonstrate that you understand a concept. It can be helpful if you need to prepare for an exam. Choose a topic from this section and decide what you want to discuss.

• Explain what we need Pythagoras’ theorem for.
• What is a hyperbola?
• Describe the difference between algebra and arithmetic.
• When is it unnecessary to use a calculator ?
• Find a connection between math and the arts.
• How do you solve a linear equation?
• Discuss how to determine the probability of rolling two dice.
• Is there a link between philosophy and math?
• What types of math do you use in your everyday life?
• What is the numerical data?
• Explain how to use the binomial theorem.
• What is the distributive property of multiplication?
• Discuss the major concepts in ancient Egyptian mathematics. 
• Why do so many students dislike math?
• Should math be required in school?
• How do you do an equivalent transformation?
• Why do we need imaginary numbers?
• How can you calculate the slope of a curve?
• What is the difference between sine, cosine, and tangent?
• How do you define the cross product of two vectors?
• What do we use differential equations for?
• Investigate how to calculate the mean value.
• Define linear growth.
• Give examples of different number types.
• How can you solve a matrix?

🎓 College Math Topics for a Paper

Sometimes you need more than just formulas to explain a complex idea. That’s why knowing how to express yourself is crucial. It is especially true for college-level mathematics. Consider the following ideas for your next research project:

• What do we need n-dimensional spaces for?
• Explain how card counting works.
• Discuss the difference between a discrete and a continuous probability distribution. 
• How does encryption work? 
• Describe extremal problems in discrete geometry.
• What can make a math problem unsolvable?
• Examine the topology of a Möbius strip.

• What is K-theory? 
• Discuss the core problems of computational geometry.
• Explain the use of set theory .
• What do we need Boolean functions for?
• Describe the main topological concepts in modern mathematics.
• Investigate the properties of a rotation matrix.
• Analyze the practical applications of game theory.
• How can you solve a Rubik’s cube mathematically?
• Explain the math behind the Koch snowflake.
• Describe the paradox of Gabriel’s Horn.
• How do fractals form?
• Find a way to solve Sudoku using math.
• Why is the Riemann hypothesis still unsolved?
• Discuss the Millennium Prize Problems.
• How can you divide complex numbers?
• Analyze the degrees in polynomial functions.
• What are the most important concepts in number theory?
• Compare the different types of statistical methods.

🤔 Advanced Topics in Math to Write a Paper on

Once you have passed the trials of basic math, you can move on to the advanced section. This area includes topology, combinatorics, logic, and computational mathematics. Check out the list below for enticing topics to write about:

• What is an abelian group?
• Explain the orbit-stabilizer theorem.
• Discuss what makes the Burnside problem influential.
• What fundamental properties do holomorphic functions have?
• How does Cauchy’s integral theorem lead to Cauchy’s integral formula?
• How do the two Picard theorems relate to each other?
• When is a trigonometric series called a Fourier series?
• Give an example of an algorithm used for machine learning.
• Compare the different types of knapsack problems.
• What is the minimum overlap problem?
• Describe the Bernoulli scheme.
• Give a formal definition of the Chinese restaurant process.
• Discuss the logistic map in relation to chaos.
• What do we need the Feigenbaum constants for?
• Define a difference equation.
• Explain the uses of the Fibonacci sequence.
• What is an oblivious transfer?
• Compare the Riemann and the Ruelle zeta functions.
• How can you use elementary embeddings in model theory?
• Analyze the problem with the wholeness axiom and Kunen’s inconsistency theorem.
• How is Lie algebra used in physics ?
• Define various cases of algebraic cycles.
• Why do we need étale cohomology groups to calculate algebraic curves?
• What does non-Euclidean geometry consist of?
• How can two lines be ultraparallel?

📚 Math Research Topics for a Paper

Choosing the right topic is crucial for a successful research paper in math. It should be hard enough to be compelling, but not exceeding your level of competence. If possible, stick to your area of knowledge. This way your task will become more manageable. Here are some ideas:

• Write about the history of calculus.
• Why are unsolved math problems significant?
• Find reasons for the gender gap in math students.
• What are the toughest mathematical questions asked today?
• Examine the notion of operator spaces.
• How can we design a train schedule for a whole country?
• What makes a number big?

• How can infinities have various sizes?
• What is the best mathematical strategy to win a game of Go?
• Analyze natural occurrences of random walks in biology.
• Explain what kind of mathematics was used in ancient Persia.
• Discuss how the Iwasawa theory relates to modular forms.
• What role do prime numbers play in encryption?
• How did the study of mathematics evolve?
• Investigate the different Tower of Hanoi solutions.
• Research Napier’s bones. How can you use them?
• What is the best mathematical way to find someone who is lost in a maze?
• Examine the Traveling Salesman Problem. Can you find a new strategy?
• Describe how barcodes function.
• Study some real-life examples of chaos theory. How do you define them mathematically?
• Compare the impact of various ground-breaking mathematical equations .
• Research the Seven Bridges of Königsberg. Relate the problem to the city of your choice.
• Discuss Fisher’s fundamental theorem of natural selection.
• How does quantum computing work?
• Pick an unsolved math problem and say what makes it so difficult.

✏️ Math Education Research Topics

For many teachers, the hardest part is to keep the students interested. When it comes to math, it can be especially challenging. It’s crucial to make complicated concepts easy to understand. That’s why we need research on math education.

• Compare traditional methods of teaching math with unconventional ones.
• How can you improve mathematical education in the U.S.?
• Describe ways of encouraging girls to pursue careers in STEM fields.
• Should computer programming be taught in high school?
• Define the goals of mathematics education .
• Research how to make math more accessible to students with learning disabilities. 
• At what age should children begin to practice simple equations?
• Investigate the effectiveness of gamification in algebra classes. 
• What do students gain from taking part in mathematics competitions?
• What are the benefits of moving away from standardized testing ?
• Describe the causes of “ math anxiety .” How can you overcome it?
• Explain the social and political relevance of mathematics education.
• Define the most significant issues in public school math teaching.
• What is the best way to get children interested in geometry?
• How can students hone their mathematical thinking outside the classroom?
• Discuss the benefits of using technology in math class. 
• In what way does culture influence your mathematical education?
• Explore the history of teaching algebra.
• Compare math education in various countries.

• How does dyscalculia affect a student’s daily life?
• Into which school subjects can math be integrated?
• Has a mathematics degree increased in value over the last few years?
• What are the disadvantages of the Common Core Standards?
• What are the advantages of following an integrated curriculum in math?
• Discuss the benefits of Mathcamp.

🧮 Algebra Topics for a Paper

The elegance of algebra stems from its simplicity. It gives us the ability to express complex problems in short equations. The world was changed forever when Einstein wrote down the simple formula E=mc². Now, if your algebra seminar requires you to write a paper, look no further! Here are some brilliant prompts:

• Give an example of an induction proof.
• What are F-algebras used for?
• What are number problems?
• Show the importance of abstract algebraic thinking. 
• Investigate the peculiarities of Fermat’s last theorem.
• What are the essentials of Boolean algebra?
• Explore the relationship between algebra and geometry.
• Compare the differences between commutative and noncommutative algebra.
• Why is Brun’s constant relevant?
• How do you factor quadratics?
• Explain Descartes’ Rule of Signs.
• What is the quadratic formula?
• Compare the four types of sequences and define them.
• Explain how partial fractions work.
• What are logarithms used for?
• Describe the Gaussian elimination.
• What does Cramer’s rule state?
• Explore the difference between eigenvectors and eigenvalues.
• Analyze the Gram-Schmidt process in two dimensions.
• Explain what is meant by “range” and “domain” in algebra.
• What can you do with determinants?
• Learn about the origin of the distance formula.
• Find the best way to solve math word problems.
• Compare the relationships between different systems of equations.
• Explore how the Rubik’s cube relates to group theory.

📏 Geometry Topics for a Research Paper

Shapes and space are the two staples of geometry. Since its appearance in ancient times, it has evolved into a major field of study. Geometry’s most recent addition, topology, explores what happens to an object if you stretch, shrink, and fold it. Things can get pretty crazy from here! The following list contains 25 interesting geometry topics:

• What are the Archimedean solids?
• Find real-life uses for a rhombicosidodecahedron.
• What is studied in projective geometry?
• Compare the most common types of transformations.
• Explain how acute square triangulation works.
• Discuss the Borromean ring configuration.
• Investigate the solutions to Buffon’s needle problem.
• What is unique about right triangles?

• Describe the notion of Dirac manifolds.
• Compare the various relationships between lines.
• What is the Klein bottle?
• How does geometry translate into other disciplines, such as chemistry and physics?
• Explore Riemannian manifolds in Euclidean space.
• How can you prove the angle bisector theorem?
• Do a research on M.C. Escher’s use of geometry.
• Find applications for the golden ratio .
• Describe the importance of circles.
• Investigate what the ancient Greeks knew about geometry.
• What does congruency mean?
• Study the uses of Euler’s formula.
• How do CT scans relate to geometry?
• Why do we need n-dimensional vectors?
• How can you solve Heesch’s problem?
• What are hypercubes?
• Analyze the use of geometry in Picasso’s paintings.

➗ Calculus Topics to Write a Paper on

You can describe calculus as a more complicated algebra. It’s a study of change over time that provides useful insights into everyday problems. Applied calculus is required in a variety of fields such as sociology, engineering, or business. Consult this list of compelling topics on a calculus paper:

• What are the differences between trigonometry, algebra, and calculus?
• Explain the concept of limits.
• Describe the standard formulas needed for derivatives.
• How can you find critical points in a graph?
• Evaluate the application of L’Hôpital’s rule.
• How do you define the area between curves?
• What is the foundation of calculus?

• How does multivariate calculus work?
• Discuss the use of Stokes’ theorem.
• What does Leibniz’s integral rule state?
• What is the Itô stochastic integral?
• Explore the influence of nonstandard analysis on probability theory.
• Research the origins of calculus.
• Who was Maria Gaetana Agnesi?
• Define a continuous function.
• What is the fundamental theorem of calculus?
• How do you calculate the Taylor series of a function?
• Discuss the ways to resolve Runge’s phenomenon.
• Explain the extreme value theorem.
• What do we need predicate calculus for?
• What are linear approximations?
• When does an integral become improper?
• Describe the Ratio and Root Tests.
• How does the method of rings work?
• Where do we apply calculus in real-life situations?

💵 Business Math Topics to Write About

You don’t have to own a company to appreciate business math. Its topics range from credits and loans to insurance, taxes, and investment. Even if you’re not a mathematician, you can use it to handle your finances. Sounds interesting? Then have a look at the following list:

• What are the essential skills needed for business math?
• How do you calculate interest rates?
• Compare business and consumer math.
• What is a discount factor?
• How do you know that an investment is reasonable?
• When does it make sense to pay a loan with another loan?
• Find useful financing techniques that everyone can use.
• How does critical path analysis work?
• Explain how loans work.
• Which areas of work utilize operations research?
• How do businesses use statistics?
• What is the economic lot scheduling problem?
• Compare the uses of different chart types.
• What causes a stock market crash?
• How can you calculate the net present value?
• Explore the history of revenue management.
• When do you use multi-period models?
• Explain the consequences of depreciation.
• Are annuities a good investment?
• Would the U.S. financially benefit from discontinuing the penny?
• What caused the United States housing crash in 2008?
• How do you calculate sales tax?
• Describe the notions of markups and markdowns. 
• Investigate the math behind debt amortization.
• What is the difference between a loan and a mortgage?

With all these ideas, you are perfectly equipped for your next math paper. Good luck!

• What Is Calculus?: Southern State Community College
• What Is Mathematics?: Tennessee Tech University
• What Is Geometry?: University of Waterloo
• What Is Algebra?: BBC
• Ten Simple Rules for Mathematical Writing: Ohio State University
• Practical Algebra Lessons: Purplemath
• Topics in Geometry: Massachusetts Institute of Technology
• The Geometry Junkyard: All Topics: Donald Bren School of Information and Computer Sciences
• Calculus I: Lamar University
• Business Math for Financial Management: The Balance Small Business
• What Is Mathematics: Life Science
• What Is Mathematics Education?: University of California, Berkeley
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Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

• The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

• The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

• The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

• Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

• The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

• Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

• The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

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Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

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Mathematics Extended Essay Topics for IB

Table of contents

• Writing Metier

Ever felt the exhilaration that comes with cracking a complex math problem? That’s the sheer beauty of mathematics – it’s a world where numbers dance, patterns emerge, and solutions await the keen observer. Felt that? Hah?

Now, imagine channeling that same enthusiasm and zest into your Math Extended essay. It might sound like a challenge, but isn’t that what we thrive on? With the right Math EE topic, you can showcase your mathematical prowess and leave an indelible mark on your IB evaluators following the general criteria. 

To help you with that, I’ve curated a list of Math extended essay ideas that span the spectrum from the accessible to the intricate. 

So, let’s begin.

IB Math extended essay ideas

These aren’t just topics; they’re gateways to exploration, understanding, and innovation. 

Ready to get started?

Easy-to-approach topics

For those looking for a gentler introduction to the Math Extended essay, below is a list of 10 engaging and manageable IB Math EE topic ideas, each with a research question and a brief description on how to approach it:

Patterns in Pascal’s Triangle

Research Question: 

How do patterns emerge in Pascal’s Triangle?

Description: 

Begin with the construction of Pascal’s Triangle, then delve into the exploration of its patterns, such as triangular numbers and the Fibonacci sequence.

Mathematics of Voting Systems

How do different voting systems weigh individual votes mathematically?

Introduce various voting systems, then analyze the mathematical principles behind each, discussing fairness and potential biases.

Geometry in Art

Research Question:  

How is geometry used to create visual appeal in art?

Explore various art pieces, highlighting the geometric principles employed, and discuss their impact on aesthetics.

The Math Behind Baking

How do ratios in baking recipes affect the final product?

Investigate the importance of maintaining specific ratios in baking recipes and experiment with variations to observe changes in outcomes.

Probability in Card Games

How does understanding probability enhance success in card games like Poker?

Delve into the rules of Poker and then analyze various hands and scenarios using probability principles.

Symmetry in Nature

How is symmetry exhibited in natural formations?

Explore the concept of symmetry mathematically and then identify and analyze its presence in various natural formations like leaves, flowers, and animals.

Mathematics of Music

How do mathematical patterns influence musical scales and rhythms?

Investigate the structure of musical scales, rhythms, and time signatures, highlighting the underlying mathematical patterns.

Simple Interest vs. Compound Interest

How do simple and compound interest methods impact savings over time?

Define both interest methods and then use mathematical modeling to compare their effects on savings over various periods.

The Golden Ratio in Architecture

How is the Golden Ratio applied in iconic architectural designs?

Introduce the concept of the Golden Ratio and then analyze its application in various famous architectural structures.

Mathematics of Juggling

How do mathematical patterns and sequences dictate juggling techniques?

Explore basic juggling techniques and patterns, analyzing the mathematical sequences that jugglers use to maintain rhythm and control.

Each of these topic ideas and RQs is approachable yet offers ample scope for exploration and analysis within the realm of mathematics.

Intermediate topics

For those seeking a bit more challenge without diving into the deep end, the topics below offer a balanced blend of complexity and accessibility.

Here’s a list of 10 average-difficulty IB Math Extended essay topics, each with a research question and a brief description of how to approach it:

Game Theory in Economics

How does game theory provide insights into competitive market behaviors?

Introduce game theory principles and use business world case studies to show strategic decision-making influenced by game theory.

Fractals and Nature

How do fractal patterns manifest in natural phenomena?

Description:  

Delve into the mathematical concept of fractals and explore their appearance in various natural settings, such as coastlines, mountains, and plants.

Mathematics of Population Growth Models

How do mathematical models accurately predict population growth in urban areas?

Introduce population growth models and use real urban area data to validate these models, discussing influencing factors.

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Cryptography and Internet Security

How do modern cryptographic techniques ensure data security on the internet?

Explore the evolution of cryptography, focusing on modern techniques, and discuss their role in ensuring data security in online transactions.

Mathematical Modeling of Epidemics

How can mathematical models predict the spread of infectious diseases?

Introduce basic epidemiological models and analyze their effectiveness in predicting disease spread using real-world data.

Topology and Coffee Cups

How does topology explain the mathematical similarity between a coffee cup and a donut?

Dive into the basics of topology and use it to explain the often-cited comparison between coffee cups and donuts, emphasizing the concept of homeomorphism.

Chaos Theory and Weather Forecasting

How does chaos theory impact the accuracy of long-term weather forecasts?

Introduce chaos theory, then delve into its application in meteorology, discussing the challenges and limitations in predicting weather.

Mathematics Behind Neural Networks

How do mathematical algorithms drive the functioning of neural networks in artificial intelligence?

Explore the structure of neural networks and delve into the mathematical algorithms that enable their learning and functioning.

Optimization Problems in Logistics

How can mathematical optimization improve efficiency in supply chain logistics?

Investigate real-world logistics challenges and demonstrate how mathematical optimization techniques can offer solutions.

Number Theory in Modern Cryptography

How does advanced number theory underpin modern cryptographic techniques?

Delve into number theory concepts and their application in modern cryptographic methods, emphasizing their role in ensuring data security.

These topics and research questions offer a balanced blend of complexity and accessibility, making them suitable for students seeking a moderate challenge in their Math Extended essay.

Advanced topics (with potential for higher rewards)

For the brave hearts ready to dive deep, these topics are challenging but can yield impressive results when tackled effectively.

The list of 10 challenging IB Math Extended essay ideas, each with a research question and a brief description:

Navier–Stokes Existence and Smoothness

What are the mathematical implications of the unsolved Navier–Stokes existence and smoothness problem in fluid dynamics?

Delve deep into the Navier–Stokes equations, exploring their significance in fluid dynamics and the challenges surrounding their unsolved problems.

Riemann Hypothesis and Prime Numbers

How does the Riemann Hypothesis relate to the distribution of prime numbers?

Investigate the Riemann Hypothesis, its historical context, and its profound implications for the distribution of prime numbers.

Quantum Cryptography and Key Distribution

How does quantum cryptography enhance security in key distribution?

Explore the principles of quantum mechanics applied to cryptography, focusing on the advantages and challenges of quantum key distribution.

Elliptic Curves and Cryptography

How do elliptic curves underpin modern cryptographic techniques?

Dive into the mathematics of elliptic curves and their pivotal role in modern cryptographic algorithms.

Four Color Theorem and Graph Theory

How does the Four Color Theorem demonstrate the complexities of graph coloring?

Investigate the history and proof of the Four Color Theorem, discussing its implications and challenges in graph theory.

Complex Analysis in Quantum Mechanics

How does complex analysis influence the mathematical foundations of quantum mechanics?

Delve into the principles of complex analysis and explore their applications and significance in quantum mechanics.

Twin Prime Conjecture

What are the implications and challenges of the Twin Prime Conjecture in number theory?

Investigate the Twin Prime Conjecture, its history, and its significance in the realm of number theory.

Knot Theory and DNA Replication

How does knot theory provide insights into the process of DNA replication?

Explore the mathematical intricacies of knot theory and its applications in understanding the complexities of DNA replication.

Ergodic Theory and Thermodynamics

How does ergodic theory connect with the principles of statistical thermodynamics?

Dive deep into ergodic theory, exploring its foundational concepts and its connections to statistical thermodynamics.

Langlands Program and Number Theory

What is the Langlands Program, and how does it seek to unify different areas of mathematics?

Investigate the Langlands Program, its objectives, and its profound implications in unifying disparate areas of mathematics, particularly number theory.

These topics are undoubtedly challenging but can yield impressive results when tackled effectively. They offer a deep dive into complex mathematical concepts and theories, making them ideal for students aiming for the highest grades in their Math Extended essay.

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Overused Math EE topics (proceed with caution)

While these topics are popular, they’ve been explored extensively. If you choose one, ensure you bring a fresh perspective:

• The Fibonacci Sequence in Nature.
• Pi and its Mysteries.
• The Monty Hall Problem.
• The Birthday Paradox.
• Exploring Pascal’s Triangle.
• The Mathematics of the Pyramids.
• The Tower of Hanoi Problem.
• Benford’s Law.
• The Four Color Theorem.
• The Mathematics of the Rubik’s Cube.

Keep in mind that selecting such a topic will require hard work to find a new custom angle of research and sometimes might not lead to higher grades .

In Conclusion

Alright, IB champs, that wraps up our curated selection of Math EE topics, each with its unique flavor and challenge. The topic is just the starting point. The real magic happens when you infuse it with your insights, analysis, and unique perspective. 

Mathematics isn’t just about numbers; it’s about stories and trips within the confines of equations and graphs. As you gear up to tackle your Extended essay, keep that passion alive, keep questioning, and most importantly, enjoy the process. 

Don’t forget to reserve extra space because it may take time to write an extended essay . And if you’re ever in doubt, remember that Writing Metier is here to guide you through every step of the process.

Free topic suggestions

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Extended Essay: Group 5: Mathematics

• General Timeline
• Group 1: English Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts
• Interdisciplinary essays
• Six sub-categories for WSEE
• IB Interdisciplinary EE Assessment Guide
• Brainstorming
• Pre-Writing
• Research Techniques
• The Research Question
• Paraphrasing, Summarising and Quotations
• Writing an EE Introduction
• Writing the main body of your EE
• Writing your EE Conclusion
• Sources: Finding, Organising and Evaluating Them
• Conducting Interviews and Surveys
• Citing and Referencing
• Check-in Sessions
• First Formal Reflection
• Second Formal Reflection
• Final Reflection (Viva Voce)
• Researcher's Reflection Space (RRS) Examples
• Information for Supervisors
• How is the EE Graded?
• EE Online Resources
• Stavanger Public Library
• Exemplar Essays
• Extended Essay Presentations
• ISS High School Academic Honesty Policy

Mathematics

An extended essay (EE) in mathematics is intended for students who are writing on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Essays in this group are divided into six categories:

• the applicability of mathematics to solve both real and abstract problems
• the beauty of mathematics—eg geometry or fractal theory
• the elegance of mathematics in the proving of theorems—eg number theory
• the history of mathematics: the origin and subsequent development of a branch of mathematics over a period of time, measured in tens, hundreds or thousands of years
• the effect of technology on mathematics:
• in forging links between different branches of mathematics,
• or in bringing about a new branch of mathematics, or causing a particular branch to flourish.

These are just some of the many different ways that mathematics can be enjoyable or useful, or, as in many cases, both.

For an Introduction in a Mathematics EE look HERE . 

Choice of topic

The EE may be written on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Students may choose mathematical topics from fields such as engineering, the sciences or the social sciences, as well as from mathematics itself.

Statistical analyses of experimental results taken from other subject areas are also acceptable, provided that they focus on the modeling process and discuss the limitations of the results; such essays should not include extensive non-mathematical detail.

A topic selected from the history of mathematics may also be appropriate, provided that a clear line of mathematical development is demonstrated. Concentration on the lives of, or personal rivalries between, mathematicians would be irrelevant and would not score highly on the assessment criteria.

It should be noted that the assessment criteria give credit for the nature of the investigation and for the extent that reasoned arguments are applied to an appropriate research question.

Students should avoid choosing a topic that gives rise to a trivial research question or one that is not sufficiently focused to allow appropriate treatment within the requirements of the EE.

Students will normally be expected either to extend their knowledge beyond that encountered in the Diploma Programme mathematics course they are studying or to apply techniques used in their mathematics course to modeling in an appropriately chosen topic.

However, it is very important to remember that it is an essay that is being written, not a research paper for a journal of advanced mathematics, and no result, however impressive, should be quoted without evidence of the student’s real understanding of it.

Example and Treatment of Topic

Examples of topics

These examples are just for guidance. Students must ensure their choice of topic is focused (left-hand column) rather than broad (right-hand column

Treatment of the topic

Whatever the title of the EE, students must apply good mathematical practice that is relevant to the

chosen topic, including:

• data analysed using appropriate techniques

• arguments correctly reasoned

• situations modeled using correct methodology

• problems clearly stated and techniques at the correct level of sophistication applied to their solution.

Research methods

Students must be advised that mathematical research is a long-term and open-ended exploration of a set of related mathematical problems that are based on personal observations. 

The answers to these problems connect to and build upon each other over time.

Students’ research should be guided by analysis of primary and secondary sources.

A primary source for research in mathematics involves:

• data-gathering

• visualization

• abstraction

• conjecturing

• proof.

A secondary source of research refers to a comprehensive review of scholarly work, including books, journal articles or essays in an edited collection.

A literature review for mathematics might not be as extensive as in other subjects, but students are expected to demonstrate their knowledge and understanding of the mathematics they are using in the context of the broader discipline, for example how the mathematics they are using has been applied before, or in a different area to the one they are investigating.

Writing the essay

Throughout the EE students should communicate mathematically:

• describing their way of thinking

• writing definitions and conjectures

• using symbols, theorems, graphs and diagrams

• justifying their conclusions.

There must be sufficient explanation and commentary throughout the essay to ensure that the reader does not lose sight of its purpose in a mass of mathematical symbols, formulae and analysis.

The unique disciplines of mathematics must be respected throughout. Relevant graphs and diagrams are often important and should be incorporated in the body of the essay, not relegated to an appendix.

However, lengthy printouts, tables of results and computer programs should not be allowed to interrupt the development of the essay, and should appear separately as footnotes or in an appendix. Proofs of key results may be included, but proofs of standard results should be either omitted or, if they illustrate an important point, included in an appendix.

Examples of topics, research questions and suggested approaches

Once students have identified their topic and written their research question, they can decide how to

research their answer. They may find it helpful to write a statement outlining their broad approach. These

examples are for guidance only.

An important note on “double-dipping”

Students must ensure that their EE does not duplicate other work they are submitting for the Diploma Programme. For example, students are not permitted to repeat any of the mathematics in their IA in their EE, or vice versa.

The mathematics EE and internal assessment

An EE in mathematics is not an extension of the internal assessment (IA) task. Students must ensure that they understand the differences between the two.

• The EE is a more substantial piece of work that requires formal research
• The IA is an exploration of an idea in mathematics.

It is not appropriate for a student to choose the same topic for an EE as the IA. There would be too much danger of duplication and it must therefore be discouraged.

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IB Maths EE examples

Filter exemplars, ¿en qué medida se puede aplicar el análisis de fourier para calcular una partitura musical a partir de una grabación, to what extent can random walk theory and chaos theory effectively explain the sek/eur exchange rate market behaviour, want to get full marks for your ee allow us to review it for you 🎯, what is the probability distribution of the number of successful ball selections in the ball box battle game, where players aim to pick five balls out of 36 balls, utilizing the probability mass function (pmf), cumulative distribution function (cdf), and expected value e(x) of the hypergeometric probability distribution, is the richards growth model suitable for predicting the future growth of passenger vehicles on jakarta's roads, how effective are logistic regression models in predicting short-term stock market performance, fast track your coursework with mark schemes moderated by ib examiners. upgrade now 🚀, how can an artificial neural network be used to identify handwritten digits, to what length does the lotka-volterra model help us understand the predator-prey relation in an ecosystem, how can laplace transformation be used to solve differential equations in undamped vibration and kirchhoff's laws of current and voltage, how can the exit velocity vector of a spacecraft performing a hyperbolic flyby manoeuvre be calculated, how does the implementation of pell's equation in rsa cryptography affect its efficiency, and what is the security of the prime fake modulus variant against fermat's factorization and wiener's attack, to what extent is the insight into the concepts of consonance and dissonance in music theory provided by the mathematical descriptions of the string motions within stringed instruments described by the fourier series, to what extent prime counting function and the riemann zeta function are similar, and what is the importance of their similarities, how are elliptic curves used in cryptography to make information secure in today’s world, how does elliptic curve cryptography ensure secure communication of information on the internet, to what extent is the naive bayes classifier effective in classifying iris plants data set, how do laplace transformations and complex exponential substitution enable the solution of a first order differential equation to obtain the sinusoidal current in a basic rl circuit, the analysis of approaches and extension of a combinatorial geometry problem, to what extent, statistically, are the causal effects of parenting programs on rural chinese toddlers’ cognition different between genders, to what extent is the mathematical proof of ‘ac ohm’s law’ applicable to ac circuits, how can matrix analysis of game theory be used to guide foreign policy in the ongoing us-iran nuclear tensions, how does the fourier series is used as a musical signal for chord structure, how can the surface area of a violin's top plate be calculated, what is the height and number of towers in a suspension bridge between tuas and jurong island such that it will bear the heaviest possible load and have the lowest cost, how does the idea of transpositions from group theory apply itself to the solution of the futurama problem, how trigonomotry applied in astronomy, to what extent do trigonometry, elliptical geometry and calculus aid forensic analysts in blood spatter analysis and the estimation of the time of death of a corpse, dans quelle mesure, l’utilisation de pi (π) est-il devenu un facteur important dans la vie quotidienne et l’évolution du monde, what is modular arithmetic, and how can it be applied to solve problems in the field of number theory, to what extent do the motion of strings within stringed instruments described by fourier series give insight into consonance and dissonance in music theory, we will investigate whether the supply and demand affect the performance of a stock in the us equity market, to what extent the areas of mathematics such as differ- ential geometry and calculus of variations can be used to generalize the brachis- tochrone problem at planes to curved surfaces embedded in three dimensions, to what extent does the distance between the path that goes through the gold nucleus and one of the asymptotes of the α particle’s hyperbolic trajectory affect the trajectory of the α particle in rutherford's gold foil experiment, what is the most optimal point on football field to score a goal, (35/36) hyperbolic flyby trajectories in orbital mechanicss, generating arbitrary uniform probability distributions, how does abstract mathematics apply to physical phenomena in the process of the development of schrodinger’s equations, what are some ways the riemann zeta function is connected to the prime numbers, that helps us investigate and better understand them, the rsa algorithm and its vulnerabilities.

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How to Apply a Mathematical Approach to Essay Writing

The use of mathematical strategies has long gone beyond solving complex levels and problems, making it easier and faster to complete tasks from other disciplines. One example of an academic task where the math approach will come into handy is writing an essay. With various techniques, your academic grades will increase, and your writing style will prosper.

Beneficial Impact on Your Writing

As a queen of science, mathematics promotes the development of various skills applicable to solving diverse problems and the formation of true-and-tried strategies for completing multiple academic assignments. With accuracy, objectivity, and logical rigor as a few fundamental principles, it provides students with a powerful tool to optimize many learning processes.

Not the least of them is writing essays that combine diverse techniques and approaches, the alternation of which will allow students to pump a range of skills from different angles. Using various mathematical models contributes to a more in-depth understanding of multiple cases and concepts while delegating tasks to professionals will save time to develop extra skills, requiring to visit ScamFighter for honest reviews preliminarily to avoid making a mistake in the choice.

Regular application of mathematical techniques, accompanied by developing strategies based on your experience and specific secrets, helps significantly expand the list of abilities and enhance those you already have. Among them are:

• logical reasoning;
• analytical abilities;
• mathematical literacy;
• visualization skills;
• critical thinking;

Many of the above may be helpful to you outside of academics, boosting personal and professional growth. These include activities such as communication strengthened through developed argumentation skills, deep understanding of concepts, etc.

Subsequently, you will quickly and qualitatively analyze large volumes of complex materials, making informed conclusions. It will help you figure out is MyAssignmentHelp.com safe and answer similar issues to see through suspicious companies in the shortest possible time, eliminating the risk of twisting you around someone’s finger by contacting only trusted services.

Where It Will Come in Very Handy

The mathematical approach can become the core of many strategies for writing academic papers in multiple fields of knowledge. Narrowing the latter’s focus, we can highlight economics in studying complex processes and concepts that mathematical methods can help with. In addition to in-depth analysis of large volumes of data, they will help you predict the further development of economic processes and identify key trends.

It also applies to essays studying scientific and high-tech phenomena where math strategies will cost modeling and interpretation. In addition, the use of mathematical approaches when writing essays on philosophy strengthens the argumentation and evidence provided. The well-known approaches of deduction and induction will allow you to shape the paper’s logical sequence while maintaining the structure’s integrity.

Specialists with many years of experience are always ready to help you with any topic, regardless of academic discipline, while the paperhelp.org promo codes on reddit and other available gifts enhance your experience.

Math Techniques to Apply in Writing Papers

Many ways to make your academic paper better and more detailed while at the same time infusing your writing style with new skills are based on multiple math approaches. Among them is a statistical method that strengthens your arguments and thoughts by involving various facts, surveys, and analyses. Another way to improve your paper would be to conduct a unique survey and then implement the results and processed information.

The use of math models, which facilitate the study of complex concepts and phenomena, will be no less valuable. Its beneficial effect lies in predicting trends, making effective comparisons, and identifying correlations between several concepts. Boosting visualization with graphs, figures, and diagrams will help achieve the latter. Probability theory can help analyze and assess the probability of a particular event. It will be especially effective if your essay topic explores random phenomena and considers potential risks.

Beyond academic writing, climate change approaches help develop valuable skills for solving various problems. You will not notice how you start reading your favorite book, article, or EssayBot review with increased attention to detail and in-depth analysis of multiple statements.

A Few More Things to Consider

Writing an essay using mathematical approaches can make your paper richer, making it easier to complete various tasks at specific workflow stages and presenting the materials you have mastered in the best light. However, it is necessary to remember some nuances to avoid the opposite effect, which manifests in various shortcomings that worsen the quality of the essay and confuse readers.

One of the primary reasons for the latter is an overabundance of formal vocabulary, turning your paper into a treasure trove of mathematical concepts. It makes it necessary to maintain maximum clarity, providing all relevant information where required. The same goes for introducing a variety of visual components, moving the tracking of their relevance to the top of your to-do list.

It is also necessary to familiarize yourself with the central requirements and extra recommendations to find a mathematical approach to develop your topic quickly. At the same time, answers to questions like is EssayBot legit will bring you closer to successfully writing a paper without unnecessary investments. Another mandatory task is carefully proofreading and checking the essay after finishing writing, eliminating all the shortcomings and unnecessary things.

Final Thoughts

Using mathematical approaches to write essays can be advantageous from different angles, contributing to the development of a wide range of skills. However, adhering to a few points is essential to achieve maximum effect and avoid obstacles.

by: Effortless Math Team about 4 months ago (category: Blog )

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GED Mathematical Reasoning Prep 2020-2021 The Most Comprehensive Review and Ultimate Guide to the GED Math Test

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Essays on Mathematical Analysis

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To many learners, crafting Mathematical Analysis papers comes easy; others require the help of various types. The WowEssays.com collection includes professionally crafted sample essays on Mathematical Analysis and related issues. Most definitely, among all those Mathematical Analysis essay examples, you will find a piece that resonates with what you see as a worthy paper. You can be sure that literally every Mathematical Analysis work presented here can be used as a vivid example to follow in terms of overall structure and composing different parts of a paper – introduction, main body, or conclusion.

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Engineering {type) To Use As A Writing Model

Importance of Mathematics in our Daily Life

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Bilal Haq, a scientist from the National Science Foundation Arlingtion, Virginia encountered foraminifera on his trip to Qingdao in China. Foraminifera are small creatures that grow shells on their bodies. Their shells are varied and very diverse, as if they are created by visionary artists. Haq thought the foraminifera would make beautiful pieces of art and should be put on display. Haq’s fellow colleague from China, Zheng, agreed and put things into action by involving local craft workers. It resulted in a park for foraminifera since the end of 2009 and already a million people have visited (Larkins, 2012).

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Introduction to Mathematical Analysis I - Second Edition

(0 reviews)

Beatriz Lafferriere, Portland State University

Gerardo Lafferriere, Portland State University

Mau Nam Nguyen, Portland State University

Copyright Year: 2016

ISBN 13: 9781365605529

Publisher: Portland State University Library

Language: English

Formats Available

Conditions of use.

Table of Contents

• 1 Tools for Analysis
• 2 Sequences
• 3 Limits and Continuity
• 4 Differentiation
• 5 Solutions and Hints for Selected Exercises

Ancillary Material

• Portland State University Library

About the Book

Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.

The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds.

The second edition includes a number of improvements based on recommendations from students and colleagues and on our own experience teaching the course over the last several years.

In this edition we streamlined the narrative in several sections, added more proofs, many examples worked out in detail, and numerous new exercises. In all we added over 50 examples in the main text and 100 exercises (counting parts).

About the Contributors

Beatriz Lafferriere , Assistant Professor, Fariborz Maseeh Department of Mathematics and Statistics, Assistant Chair for Undergraduate Program, Director of Undergraduate Advising, Portland State University. PhD Rutgers University.

Gerardo Lafferriere , Professor, Fariborz Maseeh Department of Mathematics and Statistics, Portland State University. PhD Rutgers University. Area of Specialty: Mathematical Control Theory, Hybrid Systems, Mathematical Biology, Robotics

Mau Nam Nguyen , Associate Professor, Fariborz Maseeh Department of Mathematics and Statistics, Portland State University. Ph.D. 2007 Wayne State University. Area of Specialty: Variational & Convex Analysis, Mathematical Optimization, Non-Linear & Functional Analysis.

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Analytical Essay Topics: 50+ Ideas and Examples to Consider

by  Antony W

June 9, 2024

Every analytical essay assignment starts with topic selection. Following this, you have to conduct research to find relevant materials for reference, and then structure, write, and edit the essay before handing it over for review and grading.

Unfortunately, you can’t come by the best topics fast. Sometimes you have to look at pre-written ideas to speed up the brainstorming process.

So in this post, we give you 50+ topic ideas that you may find interesting enough to explore in an analyti c al essay . Whether you’re in high school or pursing a course in college, you just might find a good topic from this list to research and write about.

Key Takeaways

• In an analytical essay, you identify an argument and then use relevant claims and evidence to do an in-depth analysis.
• Place more focus on the theme of an author’s work, particularly on how use they use literary devicesto communicate their message.
• Remember that analysis isn’t similar to arguing . Your goal isn’t to present which side of an issue is right. Rather, you should focus on presenting analysis with examples that strengthen your argument based on a written piece of work.

50+ Best Analytical Essay Topics

The following is a list of at least 50 topics that may be a good fit for your analysis essay if your teacher hasn’t given you one yet: 

Critical Analysis Essay Topics

• Analyze how social media platforms have shaped communication, relationships, information dissemination, and societal norms.
• Examine the trajectory of a specific celebrity’s career, exploring the factors contributing to their rise to fame and subsequent decline.
• How technological advancements over the past decade have transformed various aspects of our lives.
• Assess how religious beliefs and institutions shape human values and moral frameworks.
• Analyze the systemic and societal impacts of racial discrimination in the United States of America.
• The multifaceted effects of climate change on the environment, ecosystems, communities, and global sustainability efforts.
• How media censorship, intentional or unintentional, shapes public perceptions, influences narratives, and impacts access to information and freedom of expression.

Rhetorical Analysis Essay Topics

The following are some of the best rhetorical analysis essay topics:

• How an author’s choice of language, including diction and syntax, influences and shapes their argument or narrative.
• The effect of different tones, such as solemn, humorous, or nostalgic, on the overall impact and emotional resonance of a story or poem
• How you can employ humor as a tool to effectively convey messages, provoke thought, or analyze societal norms.
• Assess the strategies, messaging, and visual elements utilized in a political ad campaign and their impact on audience perception and behavior.
• How filmmakers manipulate sound, visuals, and cinematic techniques to convey specific messages or evoke emotions in their audiences.
• Examine how cultural assumptions and biases influence an author’s argument or viewpoint and how they convey these within a text.
• Evaluate the effects of irony, including situational, verbal, or dramatic, in classic literature.

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High School Analytical Essay Topics

• What are the secondary effects of living with phobias?
• The recommended amount of sleep for students to maintain academic performance
• How do phobias develop and manifest psychologically and what are the associated physiological responses?
• Strategies and interventions that one can implement to prevent and mitigate the development of gambling addiction
• How technological advancements have influenced the creation, distribution, and reception of art and literature in contemporary society
• Reasons behind the prevalence of music as a form of expression, identification, and emotional connection among teenagers
• The role of education in shaping individuals to become more informed, empathetic, and engaged global citizens
• How do various forms of entertainment perpetuate stereotypes about women and minorities?

Analytical Essay Topics on Education

• Evaluate the pros and cons of enforcing school uniforms to students of all academic levels.
• Discuss strategies and policies that college student unions can implement to address and prevent bullying in schools.
• Investigate the reasons behind disparities in school funding.
• Discuss whether acquiring knowledge is synonymous with receiving an education.
• Analyze approaches and reforms that could elevate the quality and effectiveness of education.
• The role and impact of examinations in students’ academic careers

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Persuasive Analysis Essay Topics

• Investigate the therapeutic effects of music on mental health. 
• Challenge the boundaries of freedom of expression versus offensive language in music.
• Evaluate the effectiveness and potential consequences of imposing more tariffs as a solution to the financial issues faced by the United States.
• Discuss the feasibility and ethical implications of abandoning nuclear power by all states.
• Compare and contrast the feasibility, efficiency, and environmental impact of solar energy as a substitute for hydroelectric energy.
• Discuss various strategies and their ethical implications for addressing illegal immigration.
• Investigate the correlation between hobbies and personality development and show how one’s interests and pursuits can shape character and traits.

Sports Analytical Essay Topics

• Analyze how Title IX legislation has shaped and transformed the landscape of college sports.
• Examine how gender norms and stereotypes influence behavior, perception, and opportunities within professional sports.
• Discuss how media coverage shapes public perception of athletes.
• Investigate the factors contributing to certain countries excelling in particular sports.
• Assess the elements that contribute to some athletes’ popularity over others, considering factors like skill, personality, marketability, and media presence.
• Analyze how advancements in technology have changed sports over time.
• Evaluate the effects of financial considerations, sponsorships, and revenue on college sports programs.

Science Analysis Essay Topics

• Analyze the diverse effects of climate change on ecosystems worldwide
• Examine how technological innovations, such as AI, big data, and advanced instruments, have revolutionized and accelerated scientific research across disciplines.
• Discuss the socio-economic, environmental, and healthcare factors contributing to the prevalence of certain diseases in developing nations compared to developed ones.
• Investigate the implications of genetic engineering techniques in agriculture.
• Assess the potential implications of recent advancements in medical technology on healthcare delivery and outcomes.
• Evaluate the efficiency, scalability, and environmental impact of alternative energy sources such as solar, wind, or geothermal energy in mitigating reliance on fossil fuels.

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25 Interesting Math Topics: How to Write a Good Math Essay

writing good math essay

Mathematics is a fascinating world of numbers, shapes, and patterns. 

Whether you are a student looking to grasp math concepts or someone who finds math intriguing, these topics will spark your curiosity and help you discover the beauty of mathematics straightforwardly and engagingly.

In this article, I will explore interesting math topics that make this subject not only understandable but also enjoyable.

Why Write About Mathematics

First, it helps demystify a subject that many find intimidating. By breaking down complex mathematical concepts into simple, understandable language, we can make math accessible to a wider audience, fostering greater understanding and appreciation.

Second, writing about mathematics allows us to showcase the practical applications of math in everyday life, from managing personal finances to solving real-world problems.

This helps readers recognize the relevance of math and its role in various fields and industries.

Additionally, writing about mathematics can inspire curiosity and a love for learning.

It encourages critical thinking and problem-solving skills, promoting intellectual growth and academic success.

Finally, mathematics is a universal language that transcends cultural and linguistic barriers.

After discussing math topics, we can connect with a global audience, fostering a sense of unity and collaboration in the pursuit of knowledge

 25 Interesting Math Topics to Write On

 Mathematics is a vast and intriguing field, offering a multitude of interesting topics to explore and write about.

Here are 25 such topics that promise to engage both math enthusiasts and those seeking a deeper understanding of this fascinating subject.

1. Fibonacci Sequence: Delve into the mesmerizing world of numbers with this sequence, where each number is the sum of the two preceding ones.

2. Golden Ratio: Explore the ubiquity of the golden ratio in art, architecture, and nature.

3. Prime Numbers: Investigate the mysterious properties of prime numbers and their role in cryptography.

4. Chaos Theory: Understand the unpredictability of chaotic systems and how small changes can lead to drastically different outcomes.

5. Game Theory: Examine the strategies and decision-making processes behind games and real-world situations.

6. Cryptography: Uncover the mathematical principles behind secure communication and encryption.

7. Fractals: Discover the self-replicating geometric patterns that occur in nature and mathematics.

8. Probability Theory: Dive into the world of uncertainty and randomness, where math helps us make informed predictions.

9. Number Theory: Explore the properties and relationships of integers, including divisibility and congruence.

10. Geometry of Art: Analyze how geometry and math principles influence art and design.

11. Topology: Study the properties of space that remain unchanged under continuous transformations, leading to the concept of “rubber-sheet geometry.”

12. Knot Theory: Investigate the mathematical study of knots and their applications in various fields.

13. Number Systems: Learn about different number bases, such as binary and hexadecimal, and their significance in computer science.

14. Graph Theory: Explore networks, relationships, and the mathematics of connections.

15. The Monty Hall Problem: Delight in this famous probability puzzle based on a game show scenario.

16. Calculus: Examine the principles of differentiation and integration that underlie a wide range of scientific and engineering applications.

17. The Riemann Hypothesis: Consider one of the most famous unsolved problems in mathematics involving the distribution of prime numbers.

18. Euler’s Identity: Marvel at the beauty of Euler’s equation, often described as the most elegant mathematical formula.

19. The Four-Color Theorem: Uncover the fascinating problem of coloring maps with only four colors without adjacent regions sharing the same color.

20. P vs. NP Problem: Delve into one of the most critical unsolved problems in computer science, addressing the efficiency of algorithms.

21. The Bridges of Konigsberg: Explore a classic problem in graph theory that inspired the development of topology.

22. The Birthday Paradox: Understand the surprising likelihood of shared birthdays in a group.

23. Non-Euclidean Geometry: Step into the world of geometries where Euclid’s parallel postulate doesn’t hold, leading to intriguing alternatives like hyperbolic and elliptic geometry.

24. Perfect Numbers: Learn about the properties of numbers that are the sum of their proper divisors.

25. Zero: The History of Nothing: Trace the historical and mathematical significance of the number zero and its role in the development of mathematics.

How to Write a Good Math Essay

Mathematics essays , though often perceived as daunting, can be a rewarding way to delve into the world of mathematical concepts, problem-solving, and critical thinking.

Whether you are a student assigned to write a math essay or someone who wants to explore math topics in-depth, this guide will provide you with the key steps to write a good math essay that is clear, concise, and engaging.

1. Understanding the Essay Prompt

Before you begin writing, it’s crucial to understand the essay prompt or question.

Analyze the specific topic, the scope of the essay, and any guidelines or requirements provided by your instructor.

Mostly, this initial step sets the direction for your essay and ensures you stay on topic.

2. Research and Gather Information

You need to gather relevant information and resources to write a strong math essay. This includes textbooks, academic papers, and reputable websites.

Make sure to cite your sources properly using a recognized citation style such as APA, MLA, or Chicago.

3. Structuring Your Math Essay

Start with a clear introduction that provides an overview of the topic and the main thesis or argument of your essay. This section should capture the reader’s attention and present a roadmap for what to expect.

The body of your essay is where you present your arguments, explanations, and evidence. Use clear subheadings to organize your ideas. Ensure that your arguments are logical and well-structured.

Begin by defining any important mathematical concepts or terms necessary to understand your topic.

Clearly state your main arguments or theorems. Please support them with evidence, equations, diagrams, or examples.

Explain the logical steps or mathematical reasoning behind your arguments. This can include proofs, derivations, or calculations.

Ensure your writing is clear and free from jargon that might confuse the reader. Explain complex ideas in a way that’s accessible to a broader audience.

Whenever applicable, include diagrams, graphs, or visual aids to illustrate your points. Visual representations can enhance the clarity of your essay.

Summarize your main arguments, restate your thesis, and offer a concise conclusion. Address the significance of your findings and the implications of your research or discussion.

4. Proofreading and Editing

Once you’ve written your math essay, take the time to proofread and edit it. Pay attention to grammar, spelling, punctuation, and the overall flow of your writing.

Ensure that your essay is well-organized and free from errors.

Consider seeking feedback from peers or an instructor to gain a fresh perspective.

5. Presentation and Formatting

A well-presented essay is more likely to engage the reader. Follow these formatting guidelines:

• Use a legible font (e.g., Times New Roman or Arial) in a standard size (12-point).
• Double-space your essay and include page numbers if required.
• Create a title page with your name, essay title, course information, and date.
• Use section headings and subheadings for clarity.
• Include a reference page to cite your sources appropriately.

6. Mathematical Notation and Symbols

Mathematics relies heavily on notation and symbols. Ensure that you use mathematical notation correctly and consistently.

If you introduce new symbols or terminology, define them clearly for the reader’s understanding.

7. Seek Clarification

If you encounter difficulties or ambiguities in your math essay, don’t hesitate to seek clarification from your instructor or peers.

Discussing complex mathematical concepts with others can help you refine your understanding and improve your essay.

8. Practice and Feedback

Writing math essays, like any skill, improves with practice. The more you write and receive feedback, the better you’ll become.

Take your time with initial challenges. Instead, view them as opportunities for growth and learning.

With dedication and attention to detail, you can craft a math essay that not only conveys your mathematical knowledge but also engages and informs your readers.

Josh Jasen or JJ as we fondly call him, is a senior academic editor at Grade Bees in charge of the writing department. When not managing complex essays and academic writing tasks, Josh is busy advising students on how to pass assignments. In his spare time, he loves playing football or walking with his dog around the park.

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Mathematics: Analysis and Approaches IA Topics

Hello, IB scholars! As a seasoned IB writer, I’m here to guide you through the IB Mathematics Analysis and Approaches (IB Math AA) course, mainly focusing on the Internal Assessment (IA). In my experience, the IA is a crucial part of your IB Math AA, blending theoretical knowledge with practical application. Writing IB Math AA Internal Assessment is also challenging, so I will give you some IB Math AA IA ideas.

What Is IB Mathematics Analysis and Approaches?

IB Mathematics Analysis and Approaches (IB Math AA) is a course within the International Baccalaureate (IB) Diploma Programme, designed for students interested in developing their mathematics skills, particularly in the context of analysis and approaches. By the way, you can also read about the differences between IB Math IA and Math AA in our blog.

As I know from my experience, this course is well-suited for those who enjoy the challenges of solving complex problems and are interested in pursuing mathematics or related disciplines in higher education and their future careers.

IB Math AA emphasizes a thorough understanding of mathematical concepts, theories, and techniques. It includes topics like algebra, functions, calculus, and geometry, which are in-depth explored. The course is offered at two levels:

• Higher Level (HL) is more rigorous and includes additional topics and greater depth, making it suitable for students who might pursue mathematics, engineering, physical sciences, or economics in university.
• Standard Level (SL), while still challenging, covers fewer topics and is aimed at students who require substantial mathematics but not at the same intensity as HL.

The course places a strong emphasis on developing analytical and critical thinking skills. Students learn to approach complex mathematical problems systematically and develop and apply different problem-solving strategies.

A significant component of the course is the Internal Assessment , which is an independent, student-led project. The IA differs from Extended Essay and allows students to research an area of mathematics of personal interest, encouraging creativity and applying skills learned in the course.

IB Math AA prepares students for higher education, particularly in subjects that require strong analytical and mathematical skills. It’s recognized by universities for its rigor and depth, providing a solid foundation for further academic pursuits.

So, IB Mathematics Analysis and Approaches is a comprehensive, challenging course for students passionate about mathematics and its applications. It fosters a deep understanding of mathematical concepts and prepares students for further studies and careers where high-level math skills are essential.

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IB Math AA IA Ideas and Topics

According to general IB criteria, an ideal topic should be challenging and accessible. It means balancing complexity with your skills and resources. Also, remember the IB Math AA IA topic should allow for thorough mathematical research. Below are some IB Math AA IA ideas you can consider.

Standard Level (SL) IB Math AA IA Ideas

For Standard Level (SL) students, real-world applications are a goldmine. Topics like sports statistics or architecture geometry can be engaging and informative. Also, remember creativity is your ally. Even standard topics can become intriguing with a unique angle.

• The Mathematics of Credit Card Interest . How does compound interest affect the long-term cost of credit card debt?
• Statistics in Sports: Analyzing Player Performance . Can statistical models accurately predict a basketball player’s performance based on past game data?
• Geometry in Architecture . How are geometric principles used in the design of modern buildings?
• The Fibonacci Sequence in Nature . How accurately does the Fibonacci sequence predict natural patterns, like the arrangement of leaves or seeds?
• Calculus in Medicine: Drug Concentration Over Time . How does calculus model the decreasing concentration of a drug in the bloodstream over time?
• Mathematics of Music: Frequencies and Ratios . What is the relationship between frequency ratios and musical harmony?
• Projectile Motion in Physics . How does calculus help in predicting the trajectory of a projectile?
• Statistics and Climate Change . Can a statistical analysis of temperature data over the past 50 years predict future climate trends?
• Golden Ratio in Art and Design . How is the golden ratio applied in famous artworks and designs?
• The Math Behind Voting Systems . How do different voting systems affect the outcome of elections?
• Probability in Games of Chance . What is the probability of winning in a specific card game, and how can strategies be optimized?
• Mathematics of Population Growth . How can mathematical models predict population growth in a specific region?
• Optimization in Business: Minimizing Costs . How can calculus be used to minimize production costs in a manufacturing business?
• Mathematics of Loan Amortization . How does an amortization schedule break down loan payments into interest and principal?
• Statistical Analysis of Consumer Behavior Trends . Can statistical methods predict future consumer behavior based on past purchasing data?
• Mathematical Exploration of Stock Market Trends . Can mathematical models identify patterns in stock market fluctuations?
• Calculus in Environmental Science: Carbon Footprint Analysis . How can calculus be used to model the change in a city’s carbon footprint over time?
• Probability in Genetics: Mendelian Inheritance . What is the probability of inheriting specific genetic traits according to Mendel’s laws?
• The Math Behind Cryptocurrencies . How does the underlying mathematics of blockchain technology ensure security in cryptocurrencies?
• Ratios in Photographic Composition . How do ratios, such as the rule of thirds, enhance composition in photography?
• Mathematical Patterns in Historical Architecture . What mathematical patterns are evident in the architecture of ancient civilizations?
• Statistics in Healthcare: Analyzing Patient Data . Can statistical analysis of patient data predict disease outbreaks?
• Applications of Logarithms in Astronomical Calculations . How are logarithms used in calculating distances of celestial bodies?
• Mathematical Models in Sports Strategy . How can mathematical models improve decision-making in team sports strategies?
• Geometry of Shadows: Understanding Eclipses . How does geometry explain the occurrence and patterns of solar and lunar eclipses?
• Application of Probability in Insurance . How do insurance companies use probability to calculate premiums?
• Mathematics in Cooking: Ratios and Proportions . How are ratios and proportions critical in creating culinary recipes?

These IB Math AA IA ideas and research questions aim to inspire your exploration, combining theoretical knowledge with practical application in diverse fields.

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Higher level (sl) ib math aa ia ideas.

For Higher Level (HL) students, the challenge intensifies. Your IA should reflect a deeper understanding of complex concepts. Think about integrating advanced calculus or algebra in real-world scenarios or theoretical problems.

• Non-Euclidean Geometry: Exploring Hyperbolic Space . How does hyperbolic geometry differ from Euclidean geometry in the representation of space?
• Advanced Calculus in Economics: Market Equilibrium . How can differential calculus be used to find the market equilibrium point?
• Fractal Geometry in Nature . How do fractal dimensions provide a measure of complexity in natural patterns?
• Complex Numbers in Electrical Engineering . How do complex numbers model alternating current circuits?
• Game Theory in Economics . How can game theory explain decision-making in oligopolistic markets?
• Cryptography: RSA Encryption . How does the RSA algorithm use prime numbers for secure communication?
• Fourier Series in Signal Processing . How does the Fourier series decompose complex signals into constituent frequencies?
• Chaos Theory in Meteorology . How does chaos theory explain the unpredictability of weather patterns?
• Monte Carlo Simulations in Risk Analysis . How can Monte Carlo simulations predict risk in financial investments?
• Topology and Its Applications . How does topology explain properties of space that are preserved under continuous deformations?
• Linear Algebra in Computer Graphics . How is linear algebra used in creating 3D computer graphics?
• Mathematical Modeling of Epidemics . How can mathematical models predict the spread of an infectious disease?
• Calculus of Variations in Physics . How does the calculus of variations contribute to understanding the principles of classical mechanics?
• Numerical Methods in Engineering Problems . How are numerical methods used to solve non-linear equations in engineering?
• Statistics: Regression Analysis in Market Research . How can regression analysis predict consumer behavior based on historical data?
• Advanced Probability in Genetics: Beyond Mendelian Inheritance . How can advanced probability models explain genetic inheritance patterns that deviate from Mendel’s laws?
• Graph Theory in Optimizing Transportation Networks . How can graph theory be applied to optimize public transportation networks?
• Differential Equations in Population Dynamics . How do differential equations model predator-prey dynamics in an ecosystem?
• Number Theory in Cryptography Beyond RSA . What are the mathematical principles behind modern cryptographic algorithms beyond RSA?
• Vector Calculus in Physics: Electromagnetic Fields . How does vector calculus describe the behavior of electromagnetic fields?
• Mathematical Logic and Its Applications in Computer Science . How does mathematical logic underpin the algorithms used in artificial intelligence?
• Advanced Statistics in Psychological Research . How can advanced statistical methods improve the reliability of psychological studies?
• Linear Programming in Resource Allocation . How does linear programming solve complex resource allocation problems in industries?
• Quantum Mechanics: Mathematical Foundations . What are the key mathematical principles underlying quantum mechanics?
• Mathematical Analysis of Renewable Energy Systems . How can mathematics optimize the efficiency of renewable energy systems?
• Game Theory in Political Science . How does game theory apply to decision-making processes in political science?
• Analyzing Financial Markets with Stochastic Processes . How do stochastic processes model the unpredictable nature of financial markets?

These IB Math AA IA ideas provide many opportunities for HL students to explore mathematical concepts in practical and theoretical contexts.

The Bottom Line

In conclusion, dear students, consider your IB Math AA IA an exceptional platform to display your proficiency and passion for mathematics. Approach this task with eagerness and thoroughness. Based on my extensive experience, a carefully planned IA can fulfill your IB experience. I wish you the best of luck, and may your mathematical investigations bring remarkable findings! Also, remember that our  IB IA Writing Service  experts are here to help you.

Nick Radlinsky

Nick Radlinsky is a devoted educator, marketing specialist, and management expert with more than 15 years of experience in the education sector. After obtaining his business degree in 2016, Nick embarked on a quest to achieve his PhD, driven by his commitment to enhancing education for students worldwide. His vast experience, starting in 2008, has established him as a reputable authority in the field.

Nick's article, featured in Routledge's " Entrepreneurship in Central and Eastern Europe: Development through Internationalization ," highlights his sharp insights and unwavering dedication to advancing the educational landscape. Inspired by his personal motto, "Make education better," Nick's mission is to streamline students' lives and foster efficient learning. His inventive ideas and leadership have contributed to the transformation of numerous educational experiences, distinguishing him as a true innovator in his field.

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As an experienced IB writer, I’ve compiled this complete guide to help you feel strong as you take on this critical part of the IB Diploma Programme. This article details the methods and skills you need to ace Paper 1, from understanding how the test is set up and choosing the right texts.

IB Economics IA Article Suggestions 2024/2025

When IB students start their Economics Internal Assessment (IA), it’s important for them to pick an interesting topic. For the school years 2024/2025, we will consider many different areas of economics, ranging from the rise of inflation to the changing nature of global trade.

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IB Math SL Internal Assessment (IA) Example Topics

What’s covered:, what is the ib internal assessment (ia), ib math ia examples and ideas, how is the ib math ia scored, how does the ib math ia affect my admissions chances.

The International Baccalaureate Diploma Program , or IBDP for short, is the most common alternative to an upper-level highschool education to an AP class. It is not only challenging, but also highly recognized for college admissions around the world, making its rigor and academic difficulty worthwhile. 

One of the best parts about the IBDP is that the program takes a holistic approach to schooling, as opposed to a system solely based on examinations. An important component of this is the internal assessment (IA). An IA is any component of IB that is internally graded (i.e not graded by IB). Usually the biggest one of these is a research-based essay, often called an investigation at most schools, that will allow students to take concepts learned in their 6 classes, and write a university-level paper to showcase their extents to learn above the required amount. In this blog, we will be talking about the IA for IB Math SL, one of IB’s most popular courses. 

The IA is more simple than it sounds. Any piece of work that is internally graded at your school, be it oral, a presentation, or any form for media, is considered an IA. An externally graded assessment such as the official examinations at the end of the two-year program, is an external assessment. Understanding this difference is important to better prepare for your numerous IAs. 

Most schools will require students to submit research papers for an IA on top of presentations and other mediums, with this research paper constituting a bulk of the IA portion of the total grade. This research paper is often intended to be investigative in nature, allowing the student to question a concept and use this curiosity to show higher-level thinking.

In the case of IB Math SL , most students will take concepts taught in class and find practical ways to use these concepts in real life. This allows students to prove understanding of a concept and its uses. 

You might be struggling with getting started on your IB Math IA, but don’t worry because this is very common. Most students struggle to come up with original ideas that are interesting and enjoyable to do. Hopefully the following ideas can provide some inspiration for your IAs. 

IA Idea 1 – Birthday Paradox 

The idea tackles the paradoxical concept of how probability dictates that those in a room full of people are very likely to share a birthday.

This idea uses concepts from probability and data coursework in Math SL. While solving the paradox would be nothing new, perhaps taking the information and applying it into other mathematical concepts, or testing the data with a sample population of people would be an interesting direction to take. Presentation of the data is as important too – powerpoints are the go to method, so try something new! Maybe you could develop a simple application to showcase the paradox in the form of a game. The possibilities are endless, and just goes to show how creative you can be with the IA.

IA Idea 2 – Modeling Statistics

Modeling IAs can generally refer to any project where a certain change in statistics is being presented in a visual manner. One example is modeling the radioactive decay of chernobyl, and determining if conditions are suitable for living. Another example can be modeling the population of a country and determining the time left until the area is deemed overpopulated. 

Modeling is in itself an easy task but trying to make the information digestible and interesting is the hard part.

IA Idea 3 – Analyzing the Math of Monopoly

A fun IA idea could be analyzing the probability and statistics behind the game Monopoly! Is there a certain strategy to win or is the game luck-based? Try collecting data on how the game behaves based on numerous trials, conveniently collected through playing the game with friends! Maybe you find that the game isn’t so based on luck, and probability shows a consistent and reliable tactic to win? This concept can be done with almost any board game, making the application of the idea boundless!

This IA idea brings the fun into Mathematics SL, and would be as fun to write a report on!

The IA is internally scored, meaning the IB isn’t responsible for any grade associated with your IA assignments. Your teachers however, are heavily associated with the process of creating your IAs for all your classes. They are the ones responsible for your IA grades, so it’s always best to refer to your teacher and request any sort of rubric that you could work with! 

Each class will require a different method of grading IAs. For example, a practical science like Chemistry HL would need a more investigative approach in an IA through the scientific method as compared to an IA in Spanish SL. This means you must face each IA with a different mindset!

The IAs in Math SL make a little more than 10% of the total grade, which is a sizable chunk! You should aim to take advantage of this, because the IAs are definitely scored more leniently than the examinations and other external assessments. This 10% can be the difference between a 6 and 7 in the final grade of the class!

Don’t Overdo It

While the IAs are important, they are not intended to be as massive of a research project as the extended essay (EE), the most important piece of research work you’ll be doing as an IB student. If you find that an IA takes up time from working on your EE, you should change the IA topic to make it easier. The EE is worth much more than the IA, to the point where failing the EE immediately disqualifies a student from gaining the diploma!

Save a Great Idea

If you notice your IA idea is too difficult for the scope of an IA, then maybe it could make a great EE idea. Perhaps with more fleshing out, you could create a great EE topic, which would be beneficial in making it easier to get the diploma!

Make it Interesting

The IAs are a great opportunity to be more fun and creative compared to other projects that require more serious tones and formats. Create media and forms of presentations that are enjoyable and fun! 

The IAs have no direct effects on your chances of admissions, but they do help you become a more holistic student, a quality that schools find attractive. Amongst holistic mindsets, there are numerous other factors admissions officers seek out when accepting students into their schools. To get a better understanding of your shots into a university, try CollegeVine’s admission calculator ! This intuitive tool accounts for many factors, from GPA to extracurriculars, and tells you your shots of getting into your dream school!

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50 IB Maths IA Topic Ideas

IB Maths is a struggle for most people going through their diploma. To make matters worse, on top of just doing the dreaded maths exam, we’re also expected to write a Maths IA exploration into a topic of our choice! Where do you even begin such a task? How do you even choose a topic? To make things easier, we have plenty of free Maths resources! Firstly though, we’ve compiled 50 common Maths IA topics that may spark some creative juices and set you on your way to conquering one of the hardest assignments of the diploma!

Once you have chosen your topic, you may want to check out our other posts on how to structure and format your Maths IA or how to write your IA .

NOTE: These topics are purely meant as inspiration and are not to be chosen blindly. Even though many of these topics led to high scores for some of our graduates in the past, it is important that you listen to the advice of your subject teacher before choosing any topic!

• Pascal’s triangle : Discovering patterns within this famous array of numbers
• Pythagorean triples : Can you find patterns in what numbers form a pythagorean triple?
• Monty Hall problem : How does Bayesian probability work in this real-life example, and can you add a layer of complexity to it?
• The Chinese Remainder Theorem : An insight into the mathematics of number theory
• Sum of all positive integers is -1/12? Explore this fascinating physics phenomenon through the world of sequences and series
• Birthday paradox : Why is it that in a room of people probability dictates that people are very likely to share a birthday? 
• Harmonic series : Explore why certain notes/chords in music sound dissonant, and others consonant, by looking at the ratios of frequencies between the notes.
• Optimizing areas : Optimizing the area of a rectangle is easy, but can you find a way to do it for any polygon?
• Optimizing volumes : Explore the mathematics of finding a maximum volume of a cuboid subject to some constraint
• Flow of traffic : How does mathematics feed into our traffic jams that we endure every morning?
• Football statistics : Does spending a lot of cash during the transfer window translate to more points the following year? Or is there a better predictor of a team’s success like wages, historic performance, or player valuation?
• Football statistics #2 : How does a manager sacking affect results? 
• Gini coefficient : Can you use integration to derive the gini coefficient for a few countries, allowing you to accurately compare their levels of economic inequality?
• Linear regressions : Run linear regressions using OLS to predict and estimate the effect of one variable on another.
• The Prisoner’s Dilemma : Use game theory in order to deduce the optimal strategy in this famous situation
• Tic Tac Toe : What is the optimal strategy in this legendary game? Will my probability of winning drastically increase by some move that I can make?
• Monopoly : Is there a strategy that dominates all others? Which properties should I be most excited to land on?
• Rock Paper Scissors : If I played and won with rock already, should I make sure to change what I play this time? Or is it better to switch? 
• The Toast problem : If there is a room of some number of people, how many toasts are necessary for everyone to have toasted with everyone?
• Cracking a Password : How long would it take to be able to correctly guess a password? How much safer does a password get by adding symbols or numbers?
• Stacking Balls : Suppose you want to place balls in a cardboard box, what is the optimal way to do this to use your space most effectively?
• The Wobbly Table : Many tables are wobbly because of uneven ground, but is there a way to orient the tables to make sure they are always stable?
• The Stable Marriage Problem : Is there a matching algorithm that ensures each person in society ends up with their one true love? What is the next best alternative if this is not viable?
• Mathematical Card Tricks : Look at the probabilities at play in the famous 3 card monte scam. 
• Modelling the Spread of a Virus : How long would it take for us all to be wiped out if a deadly influenza spreads throughout the population?
• The Tragedy of the Common s : Our population of fish is dwindling, but how much do we need to reduce our production by in order to ensure the fish can replenish faster than we kill?
• The Risk of Insurance : An investigation into asymmetric information and how being unsure about the future state of the world may lead us to be risk-averse
• Gabriel’s Horn : This figure has an infinite surface area but a finite volume, can you p rove this?
• Modelling the Shape of an Egg : Although it may sound easy, finding the surface area or volume of this common shape requires some in-depth mathematical investigation
• Voting Systems : What voting system ensures that the largest amount of people get the official that they would prefer? With 2 candidates this is logical, but what if they have more than 2?
• Probability : Are Oxford and Cambridge biased against state-school applicants?
• Statistics : With Tokyo 2020 around the corner, how aboutmodelling change in record performances for a particular discipline?
• Analysing Data : In the 200 meter dash, is there an advantage to a particular lane in track? 
• Coverage : Calculation of rate of deforestation, and afforestation. How long will our forests last?
• Friendly numbers, Solitary numbers, perfect numbers : Investigate what changes the condition of numbers
• Force : Calculating the intensity of a climber’s fall based upon their distance above where they last clamped in
• Königsberg bridge problem : Using networks to solve problems. 
• Handshake problem : How many handshakes are required so that everyone shakes hands with all the other people in the room? 
• The mathematics of deceit : How con artists use pyramid schemes to get rich quick!
• Modelling radioactive decay : The maths of Chernobyl – when will it be safe to live there?
• Mathematics and photography : Exploring the relationship between the aperture of a camera and a geometric sequence
• Normal Distribution : Using distributions to examine the 2008 financial crisis
• Mechanics : Body Proportions for Track and Field events
• Modelling : How does a cup of Tea cool?
• Relationships : Do BMI ratings and country wealth share a significant relationship?
• Modelling : Can we mathematically model musical chords and concepts like dissonance?
• Evaluating limits : Exploring L’Hôpital’s rule
• Chinese postman problem : How do we calculate shortest possible routes?
• Maths and Time : Exploring ideas regarding time dilation
• Plotting Planets : Using log functions to track planets!

So there we have it: 50 IB Maths IA topic ideas to give you a head-start for attacking this piece of IB coursework ! We also have similar ideas for Biology , Chemistry , Economics , History , Physics , TOK … and many many more tips and tricks on securing those top marks on our free resources page – just click the ‘Maths resources’ button!

Still feeling confused, or want some personalised help? We offer online private tuition from experienced IB graduates who got top marks in their Maths IA. 

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Topics in Mathematical Analysis and Applications

• © 2014
• Themistocles M. Rassias 0 ,
• László Tóth 1

Department of Mathematics, National Technical University of Athens, Athens, Greece

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Department of Mathematics, University of Pécs, Pécs, Hungary

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• Proves results in detail and discusses several examples rendering the presentation theory accessible to a wide audience
• Content is useful for both pure and applied mathematicians as well as scientists and engineers oriented in interdisciplinary research
• Includes supplementary material: sn.pub/extras

Part of the book series: Springer Optimization and Its Applications (SOIA, volume 94)

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This volume presents significant advances in a number of theories and problems of Mathematical Analysis and its applications in disciplines such as Analytic Inequalities, Operator Theory, Functional Analysis, Approximation Theory, Functional Equations, Differential Equations, Wavelets, Discrete Mathematics and Mechanics. The contributions focus on recent developments and are written by eminent scientists from the international mathematical community. Special emphasis is given to new results that have been obtained in the above mentioned disciplines in which Nonlinear Analysis plays a central role.

Some review papers published in this volume will be particularly useful for a broader readership in Mathematical Analysis, as well as for graduate students. An attempt is given to present all subjects in this volume in a unified and self-contained manner, to be particularly useful to the mathematical community.

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An Overview on Tools from Functional Analysis

Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

Harmonic Analysis Methods for Function Spaces, Differential Equations and Data Science

• control theory
• network analysis
• operational research
• real functions

Table of contents (34 chapters)

Front matter, simple proofs of some bernstein–mordell type inequalities.

• Vandanjav Adiyasuren, Tserendorj Batbold

Hilbert-Type Inequalities Including Some Operators, the Best Possible Constants and Applications: A Survey

• Vandanjav Adiyasuren, Tserendorj Batbold, Mario Krnić

A Fixed Point Approach to Stability of the Quadratic Equation

• M. Almahalebi, A. Charifi, S. Kabbaj, E. Elqorachi

Aspects of Global Analysis of Circle-Valued Mappings

• Dorin Andrica, Dana Mangra, Cornel Pintea

A Remark on Some Simultaneous Functional Inequalities in Riesz Spaces

• Bogdan Batko, Janusz Brzdȩk

Elliptic Problems on the Sierpinski Gasket

• Brigitte E. Breckner, Csaba Varga

Initial Value Problems in Linear Integral Operator Equations

• L. P. Castro, M. M. Rodrigues, S. Saitoh

Extension Operators that Preserve Geometric and Analytic Properties of Biholomorphic Mappings

• Teodora Chirilă

Normal Cones and Thompson Metric

• Ştefan Cobzaş, Mircea-Dan Rus

Functional Operators and Approximate Solutions of Functional Equations

• Stefan Czerwik, Krzysztof Król

Markov-Type Inequalities with Applications in Multivariate Approximation Theory

• Nicholas J. Daras

The Number of Prime Factors Function on Shifted Primes and Normal Numbers

• Jean-Marie De Koninck, Imre Kátai

Imbedding Inequalities for Composition of Green’s and Potential Operators

• Shusen Ding, Yuming Xing

On Approximation Properties of q -King Operators

• Zoltán Finta

Certain Szász-Mirakyan-Beta Operators

• N. K. Govil, Vijay Gupta, Danyal Soybaş

Extremal Problems and g -Loewner Chains in \(\mathbb{C}^{n}\) and Reflexive Complex Banach Spaces

• Ian Graham, Hidetaka Hamada, Gabriela Kohr

Different Durrmeyer Variants of Baskakov Operators

• Vijay Gupta

Hypergeometric Representation of Certain Summation–Integral Operators

• Vijay Gupta, Themistocles M. Rassias

On a Hybrid Fourth Moment Involving the Riemann Zeta-Function

• Aleksandar Ivić, Wenguang Zhai

Editors and Affiliations

Themistocles M. Rassias

László Tóth

Bibliographic Information

Book Title : Topics in Mathematical Analysis and Applications

Editors : Themistocles M. Rassias, László Tóth

Series Title : Springer Optimization and Its Applications

DOI : https://doi.org/10.1007/978-3-319-06554-0

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer International Publishing Switzerland 2014

Hardcover ISBN : 978-3-319-06553-3 Published: 27 October 2014

Softcover ISBN : 978-3-319-34409-6 Published: 10 September 2016

eBook ISBN : 978-3-319-06554-0 Published: 13 October 2014

Series ISSN : 1931-6828

Series E-ISSN : 1931-6836

Edition Number : 1

Number of Pages : IX, 814

Number of Illustrations : 8 b/w illustrations, 20 illustrations in colour

Topics : Optimization , Real Functions , Numerical Analysis , Functions of a Complex Variable , Operator Theory , Special Functions

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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!

I have just made a comprehensive online course:  Getting a 7 in IB Maths Coursework . 

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 EE - Propel Your Mind To A 34/34

Unlock the secrets to acing your IB Math EE with a guaranteed 34/34 score! Propel your mind to success and make the most of your potential. 💯

Table of content

Extended essay & what is it❓, ib math ee vs math ia 🥊 , wth is an ib math extended essay 🌋, a brief about ib math ee 🩲, sexy pointers you can't-miss 🤤, 4 ways to find the right math ee topic ⭕, formulate your research question 🔎, planning your research & essay 📖, structuring your essay 🏗️, ib math ee reflection, what to do 🪞, ib math ee criteria 🧮, ib math ee examples 🛹.

It's time we address the elephant in the room.

IB Mathematics Extended Essay.

There, we said it.

If you've taken up IB Math EE, we're here to tell you that it isn't over, contrary to what you might have heard.

Researching, planning, and finally writing a good IB Math EE is no walk in the park, but rest assured that you'll feel different once you've gone through this valuable document.

Nail IB presents the only comprehensive guide you need to nail your IB Math EE.

Let's keep it accurate.

IB Math EE is a rigorous, independent, academic research essay that calls for,

• a keen interest in Mathematics,
• strong mathematical aptitude & reasoning,
• a focused and disciplined approach, and most importantly,
• essential tips, techniques, and quality guidance.

When one opts for IB Math EE, one should ensure a strong passion for Mathematics and a solid mathematical foundation. Not to mention being willing to devote their time and effort to this self-directed research.

Disclaimer -  If you have second thoughts, you should reconsider and re-evaluate your choice because it's no use aiming in the dark.

There needs to be more!

Invaluable tips and techniques are the last missing piece of the IB Math EE puzzle.

Given the panic and stigma around Math Extended Essay, what students need, is not some more panic; instead, expert guidance to keep them pumped to write their EE papers with skill, precision, and utmost zeal. And so, true to our word, we're here to help you.

Say it out loud for me - I got this!

Before diving any further, we are thrilled to share our premium IB Math resources. Check out the  IB Math AI SL  bundle and the  IB Math AI HL  bundle, including everything from comprehensive, up-to-date video lecture courses by Adam Nazha to curated IB Resource Libraries. All your IB DP Math queries are now answered!

While you're at it, check out our  IB Math AA SL  repository, where Patrick Jones (yep, you got that right, our very own  PatrickJMT  - the best Mathematics instructor in the world) will guide you with a brilliant video lecture course he has created to help you nail your concepts and exams alike! Happy learning!

Returning to our one-stop guide for all things IB Math EE, let us see what this article will cover.

• Extended Essay & What is it ❓
• IB Math EE Vs Math IA  🥊
• WTH Is An IB Math Extended Essay?  🌋
• A Brief About IB Math EE ?  🩲
• Sexy Pointers You Can't Miss  🤤
• 4 Ways To Find The Right Math EE topic  ⭕
• Formulate Your Research Question  🔎
• Planning Your Research and Essay  📖
• Structuring Your Essay  🏗️
• Writing Your Essay Writing Your Reflection  🪞
• IB Math EE Criteria  🧮
• IB Math EE Examples  🛹

First things first, let’s briefly discuss what an Extended Essay is.

As the IB defines it, the Extended Essay is a  4000-word in-depth  study, part of the IB Diploma Programme core mandatory for all students.

A  500-word  reflection component-inclusive of 3 reflections - is submitted with the academic writing.

This, of course, is done under the supervision of your IB advisor. The critical thing to note is that you can choose to write an Extended Essay on any subject offered by the IB in your session, though it is highly recommended you settle on a topic you are studying as a part of your IB curriculum.

The purpose of an Extended Essay is to motivate students to conduct formal research on a subject area of interest, independently work on the chosen research question and gain guidance from their respective IB mentors every step of the way.

The IB highlights the significance of the Extended Essay:

• Hands-on learning for undergraduate research
• An opportunity for students to research an area of interest relating to one of the 6 DP subjects opted for by the student.

The skills the Extended Essay aims to develop in students are:

• Coming up with a relevant Research Question
• Personally engaging with a topic of exploration
• A coherent communication of ideas
• Strong Argument building

For better understanding, go through  this document on Extended Essays   by the IB. It will make things more transparent.

Students often confuse their Internal Assessments with Extended Essays, which is a grave mistake. Gaining clarity on both these components of the IB DP core is essential. Let us understand the significant difference between the two.

• While Internal Assessments require students to investigate and explore an idea, Extended Essays are more formal. Students need to perform primary and secondary research on a research question they choose based on a diverse topic about the subject. Since EE is an independently researched essay, your mentor may only be able to give you some of the answers you wanted, unlike the IA.
• Students must write one Extended Essay on a subject of their choice for their IB Diploma Programme, but I must be prepared for all the topics.
• Internal Assessments are graded internally by the subject instructor and then moderated externally. Extended Essays are marked externally by examiners appointed by the IB. Your grade on your Extended Essay is combined with your Theory of Knowledge Essay/Exhibition score to assign you your overall score/points.

Word of caution:  Do not choose the same investigation topic/research question for your Internal Assessment and the Extended Essay.

For a much better breakdown of the EE document, you must go through our article on the  IB Extended Essay . It contains an IB EE checklist you must attend to and a detailed overview of the structure and planning of the EE.

Now that we’ve got our basics to, let’s get down to business. The fear around IB Math Extended Essay makes the most confident students hesitate and doubt their choices. But we aren’t here to scare you. The myth about the horrors of the IB Math EE isn’t all false; it’s a caution. Because unless you’re prepared with ammunition, strategy, and the will to put in the effort, there’s a slim chance you can survive the Math Extended Essay battle.

If you’ve opted for Math Extended Essay, we assume you have a strong passion for Mathematics. When the going gets tough while working on one’s Extended Essay-which, which is usually the case-your only hope at the end of the tunnel is your drive for the subject you’ve chosen. Being a rigorous subject for writing one’s Extended Essay, Mathematics can be incredibly demanding. So unless you want to play it risky, we suggest you reassess your liking for Mathematics. Once you know for sure, that’s when it begins. Lots of research. Since it is an independently written document, you want to be doing your homework well.

We will further break down the IB Math Extended Essay, from planning to execution, so that you can organize your thoughts and make appropriate decisions.

The Math Extended Essay is an in-depth research document covering any topic with a mathematical focus, i.e., not limited to the mathematics theory (shouldn’t be entirely descriptive). IB has divided Math Extended Essays into six categories:

• the applicability of mathematics to solve both real and abstract problems
• The beauty of mathematics examples includes geometry or fractal theory.
• the application of mathematics in the proving of theorems, for example, number theory
• the history of mathematics: the origin and subsequent development of a branch of mathematics over some time, measured in tens, hundreds, or thousands of years
• the effect of technology on mathematics
• in creating links between different branches of mathematics,
• Or in bringing about a new branch of mathematics or causing a particular unit to flourish.

Your Math EE topic should be an extension of your IB Maths syllabus or the practical application of the knowledge you’ve gained in class.

Before moving on to the planning and structure of the Math Extended Essay, here are some pointers you absolutely cannot afford to miss:

• Attempting a Math EE requires a decent amount of hard work. While conducting preliminary research, ensure you land on a topic you can personally engage with and are familiar with. Going for a complicated subject, you can’t comprehend and interpret futile.
• The mathematical work you put in will have to meet a certain level of sophistication, so avoid going for a topic that gives rise to a trivial research question. Your research question shouldn’t be too generic or broad on similar lines. Ensure you can sufficiently narrow down your area of investigation.
• Another thing to remember is that the topic you finally choose should have a good research foundation and room for a novel contribution.
• Incorporate Primary and Secondary research wherever appropriate.
• Proper Referencing should be done for the essay.
• For Reflection, emphasize the Planning, Research, and Writing Processes.

Spitball ideas -  You begin here. Write down any interesting ideas you can come up with. This is your brainstorming session.

Do your Research -  You could not, would not want to meddle with this step, let alone skip it. Research. Read reports, articles, and documentation. Keep jotting important points down simultaneously. The study will and must go on throughout your search for the right math EE topic.

Approach professors/professionals/experts -  Contact concerned lecturers, professors, and industry professionals. They can provide you with invaluable insights. It would be best if you also discussed ideas with your mentor.

Point to Note -  Your goal should be to find a topic you understand and contribute to. It is always recommended to refer to your Math syllabus; something there inspires you to get the ball rolling!

Your essay will be structured as a document answering the Research Question you’re working on. It should be focused, clear and arguable. You will build your essay around your Research Question; hence pay close attention to it.

All you will include in your essay must be mentioned in your Research Question. This means the focused area you will be exploring and the methods you will use to achieve a plausible outcome.

Honing your Research Question is a continuous process. If it appears relatively easy or complex, please rephrase it. Good Research Questions may start with: “How can we model…”, “To what extent…” etc.

Collect Primary and Secondary data and incorporate it wherever appropriate.

Primary sources for Math research include 

• Data-gathering
• Visualization
• Abstraction
• Conjecturing

A few sources to gather Secondary data include Google Scholar, textbooks, databases such as JSTOR, etc. For your essay, know your aims and devise a step-by-step plan to answer your research question while maintaining a mathematical focus.

• Though you can approach your essay in whichever way you like, there should be a consistent logical structure.
• The work done in your essay should develop to answer your research question.
• Subheadings will provide your work with much-needed coherence and viability.
• Describe your way of thinking
• Write definitions and Conjectures.
• Use symbols, theorems, graphs, and diagrams
• Justify your conclusions.
• Visual aids such as graphs and diagrams should be included in the body, while tables and computer programs should be mentioned in the footnotes or the appendix.
• Data must be analyzed appropriately using relevant techniques, correctly reasoned arguments, and adequate methodology to achieve a result.
• Provide the necessary explanation and commentary for all your work.
• Throughout the writing, you may need to re-evaluate the method, approach, and structure to fit the word count adequately.

Once you’re finished writing, you will have to reassess everything from scratch- the equations, the theory, etc., to ensure the coherence of your essay.

• The three reflections should highlight how the process of writing your essay changed in due course of work and what you’ve learned from it.
• The Reflection should emphasize the planning, research, and writing process.
• You should discuss your choices, the re-evaluations you had to make, the limitations of your work, and how you overcame them, if at all.
• The three reflections are written at the beginning, mid, and towards the end of your essay and should have a word limit of approximately 150, 150, and 200, respectively.

Extended Essays are assessed externally by IB-appointed examiners and are marked on a scale of 0 to 34. The score obtained by a student corresponds to a particular band. The bands are:

A -  refers to work of an excellent standard (Level 6)

B -  refers to the creation of a good standard (Level 6)

C -  refers to work of a satisfactory standard (Level-12)

D -  refers to the product of a mediocre standard (Level 4)

E -  refers to the design of an elementary average (Level 6)

Anything below D will be considered a failing grade, and the student will not be awarded the IB diploma.  As we said, your EE is not a component you can mess up.

The Extended Essay Rubric includes the following Assessment Criteria:

Criterion A -   Focus and Method (Covers the topic, research question, and methodology)

Criterion B -  Knowledge and Understanding (Covers context, subject-specific terminology, and concepts)

Criterion C -  Critical Thinking (Covers research, analysis, discussion, and evaluation)

Criterion D -  Presentation (Covers structure, layout)

Criterion E -  Engagement (Covers reflection on planning and progress)

Now that we’ve discovered and learned about the process of writing a Math EE, here are a couple of ideas to give you an idea about how your Math EE questions should be framed:

Topic -  Archimedes’ calculation of areas

Research Question -  What is the legacy of Archimedes’ calculations of circular and parabolic areas in today’s integration methods?

Topic -  Approximation of irrational numbers by rational numbers

Research Question -  How well can π, e, √2, and other irrationals be approximated by rational numbers?

Topic -   The geometry of navigation

Research Question -  What was the role of mathematics and geometry in navigation when we relied on the stars? Does it still play a part now we have manufactured satellites?

You should also check out this  sample band, A Math EE, published by the IB . This will give you an honest insight into making a band-A essay. Also, go through  band B, C, and D extended essays  to understand the loopholes to avoid.

And that’s a wrap!

We hope this comprehensive IB Math EE guide answers your queries regarding the EE. We also suggest you go through  this IB Extended Essay guide  to clarify the Extended Essay component of your IB DP curriculum. You should also try Nail IB’s Extended Essay guide; check it out  here !

At Nail IB, we’re committed to delivering A-quality guidance so that your IB DP journey is smooth and successful. You can explore  our take-test  series here and access   premium guides  here.

Happy learning!

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