math research question examples

251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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181 Mathematics Research Topics From PhD Experts

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

• Roll two dice and calculate a probability
• Discuss ancient Greek mathematics
• Is math really important in school?
• Discuss the binomial theorem
• The math behind encryption
• Game theory and its real-life applications
• Analyze the Bernoulli scheme
• What are holomorphic functions and how do they work?
• Describe big numbers
• Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

• Methods to count discrete objects
• The origins of Greek symbols in mathematics
• Methods to solve simultaneous equations
• Real-world applications of the theorem of Pythagoras
• Discuss the limits of diffusion
• Use math to analyze the abortion data in the UK over the last 100 years
• Discuss the Knot theory
• Analyze predictive models (take meteorology as an example)
• In-depth analysis of the Monte Carlo methods for inverse problems
• Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

• Discuss the greatest common divisor
• Explain the extended Euclidean algorithm
• What are RSA numbers?
• Discuss Bézout’s lemma
• In-depth analysis of the square-free polynomial
• Discuss the Stern-Brocot tree
• Analyze Fermat’s little theorem
• What is a discrete logarithm?
• Gauss’s lemma in number theory
• Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

• Discuss Brun’s constant
• An in-depth look at the Brahmagupta–Fibonacci identity
• What is derivative algebra?
• Describe the Symmetric Boolean function
• Discuss orders of approximation in limits
• Solving Regiomontanus’ angle maximization problem
• What is a Quadratic integral?
• Define and describe complementary angles
• Analyze the incircle and excircles of a triangle
• Analyze the Bolyai–Gerwien theorem in geometry
• Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

• Mathematics and its appliance in Artificial Intelligence
• Try to solve an unsolved problem in math
• Discuss Kolmogorov’s zero-one law
• What is a discrete random variable?
• Analyze the Hewitt–Savage zero-one law
• What is a transferable belief model?
• Discuss 3 major mathematical theorems
• Describe and analyze the Dempster-Shafer theory
• An in-depth analysis of a continuous stochastic process
• Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

• Define the hyperbola
• Do we need to use a calculator during math class?
• The binomial theorem and its real-world applications
• What is a parabola in geometry?
• How do you calculate the slope of a curve?
• Define the Jacobian matrix
• Solving matrix problems effectively
• Why do we need differential equations?
• Should math be mandatory in all schools?
• What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

• Discuss the reductio ad absurdum approach
• Discuss Boolean algebra
• What is consistency proof?
• Analyze Trakhtenbrot’s theorem (the finite model theory)
• Discuss the Gödel completeness theorem
• An in-depth analysis of Morley’s categoricity theorem
• How does the Back-and-forth method work?
• Discuss the Ehrenfeucht–Fraïssé game technique
• Discuss Aleph numbers (Aleph-null and Aleph-one)
• Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

• Discuss the differential equation
• Analyze the Jacobson density theorem
• The 4 properties of a binary operation in algebra
• Analyze the unary operator in depth
• Analyze the Abel–Ruffini theorem
• Epimorphisms vs. monomorphisms: compare and contrast
• Discuss the Morita duality in algebraic structures
• Idempotent vs. nilpotent in Ring theory
• Discuss the Artin-Wedderburn theorem
• What is a commutative ring in algebra?
• Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

• What are the goals a mathematics professor should have?
• What is math anxiety in the classroom?
• Teaching math in UK schools: the difficulties
• Computer programming or math in high school?
• Is math education in Europe at a high enough level?
• Common Core Standards and their effects on math education
• Culture and math education in Africa
• What is dyscalculia and how does it manifest itself?
• When was algebra first thought in schools?
• Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

• What is a multiplication table?
• Analyze the Scholz conjecture
• Explain exponentiating by squaring
• Analyze the Myhill-Nerode theorem
• What is a tree automaton?
• Compare and contrast the Pushdown automaton and the Büchi automaton
• Discuss the Markov algorithm
• What is a Turing machine?
• Analyze the post correspondence problem
• Discuss the linear speedup theorem
• Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

• What is two-element Boolean algebra?
• The life of Gauss
• The life of Isaac Newton
• What is an orthodiagonal quadrilateral?
• Tessellation in Euclidean plane geometry
• Describe a hyperboloid in 3D geometry
• What is a sphericon?
• Discuss the peculiarities of Borel’s paradox
• Analyze the De Finetti theorem in statistics
• What are Martingales?
• The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

• Discuss Newton’s laws of motion
• Analyze the perpendicular axes rule
• How is a Galilean transformation done?
• The conservation of energy and its applications
• Discuss Liouville’s theorem in Hamiltonian mechanics
• Analyze the quantum field theory
• Discuss the main components of the Lorentz symmetry
• An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

• Most useful trigonometry functions in math
• The life of Archimedes and his achievements
• Trigonometry in computer graphics
• Using Vincenty’s formulae in geodesy
• Define and describe the Heronian tetrahedron
• The math behind the parabolic microphone
• Discuss the Japanese theorem for concyclic polygons
• Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

• Finding critical points in a graph
• The basics of calculus
• What makes a graph ultrahomogeneous?
• How do you calculate the area of different shapes?
• What contributions did Euclid have to the field of mathematics?
• What is Diophantine geometry?
• What makes a graph regular?
• Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

• What are extremal problems and how do you solve them?
• Discuss an unsolvable math problem
• How can supercomputers solve complex mathematical problems?
• An in-depth analysis of fractals
• Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
• Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
• An in-depth look at Einstein’s field equation
• The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

• When do we need to apply the L’Hôpital rule?
• Discuss the Leibniz integral rule
• Calculus in ancient Egypt
• Discuss and analyze linear approximations
• The applications of calculus in real life
• The many uses of Stokes’ theorem
• Discuss the Borel regular measure
• An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

• The life and work of the famous Pierre de Fermat
• What are limits and why are they useful in calculus?
• Explain the concept of congruency
• The life and work of the famous Jakob Bernoulli
• Analyze the rhombicosidodecahedron and its applications
• Calculus and the Egyptian pyramids
• The life and work of the famous Jean d’Alembert
• Discuss the hyperplane arrangement in combinatorial computational geometry
• The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

• Is paying a loan with another loan a good approach?
• Discuss the major causes of a stock market crash
• Best debt amortization methods in the US
• How do bank loans work in the UK?
• Calculating interest rates the easy way
• Discuss the pros and cons of annuities
• Basic business math skills everyone should possess
• Business math in United States schools
• Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

• What is the autoregressive conditional duration?
• Applying the ANOVA method to ranks
• Discuss the practical applications of the Bates distribution
• Explain the principle of maximum entropy
• Discuss Skorokhod’s representation theorem in random variables
• What is the Factorial moment in the Theory of Probability?
• Compare and contrast Cochran’s C test and his Q test
• Analyze the De Moivre-Laplace theorem
• What is a negative probability?

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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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I need a writer on algebra. I am a PhD student.Can i be helped by anybody/expert?

Please I want to do my MPhil research on algebra if you can help me

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Statistical Research Questions: Five Examples for Quantitative Analysis

Table of contents, introduction.

How are statistical research questions for quantitative analysis written? This article provides five examples of statistical research questions that will allow statistical analysis to take place.

In quantitative research projects, writing statistical research questions requires a good understanding and the ability to discern the type of data that you will analyze. This knowledge is elemental in framing research questions that shall guide you in identifying the appropriate statistical test to use in your research.

Once you feel confident that you can correctly identify the nature of your data, the following examples of statistical research questions will strengthen your understanding. Asking these questions can help you unravel unexpected outcomes or discoveries particularly while doing exploratory data analysis .

Five Examples of Statistical Research Questions

Topic 1: physical fitness and academic achievement.

A study was conducted to determine the relationship between physical fitness and academic achievement. The subjects of the study include school children in urban schools.

Statistical Research Question No. 1

Is there a significant relationship between physical fitness and academic achievement?

To allow statistical analysis to take place, there is a need to define what is physical fitness, as well as academic achievement. The researchers measured physical fitness in terms of  the number of physical fitness tests  that the students passed during their physical education class. It’s simply counting the ‘number of PE tests passed.’

On the other hand, the researchers measured academic achievement in terms of a passing score in Mathematics and English. The variable is the  number of passing scores  in both Mathematics and English.

Given the statistical research question, the appropriate statistical test can be applied to determine the relationship. A Pearson correlation coefficient test will test the significance and degree of the relationship. But the more sophisticated higher level statistical test can be applied if there is a need to correlate with other variables.

In the particular study mentioned, the researchers used  multivariate logistic regression analyses  to assess the probability of passing the tests, controlling for students’ weight status, ethnicity, gender, grade, and socioeconomic status. For the novice researcher, this requires further study of multivariate (or many variables) statistical tests. You may study it on your own.

Most of what I discuss in the statistics articles I wrote came from self-study. It’s easier to understand concepts now as there are a lot of resource materials available online. Videos and ebooks from places like Youtube, Veoh, The Internet Archives, among others, provide free educational materials. Online education will be the norm of the future. I describe this situation in my post about  Education 4.0 .

Topic 2: Climate Conditions and Consumption of Bottled Water

This study attempted to correlate climate conditions with the decision of people in Ecuador to consume bottled water, including the volume consumed. Specifically, the researchers investigated if the increase in average ambient temperature affects the consumption of bottled water.

Statistical Research Question No. 2

Now, it’s easy to identify the statistical test to analyze the relationship between the two variables. You may refer to my previous post titled  Parametric Statistics: Four Widely Used Parametric Tests and When to Use Them . Using the figure supplied in that article, the appropriate test to use is, again, Pearson’s Correlation Coefficient.

Source: Zapata (2021)

Topic 3: Nursing Home Staff Size and Number of COVID-19 Cases

Statistical research question no. 3.

Note that this study on COVID-19 looked into three variables, namely 1) number of unique employees working in skilled nursing homes, 2) number of weekly confirmed cases among residents and staff, and 3) number of weekly COVID-19 deaths among residents.

We call the variable  number of unique employees  the  independent variable , and the other two variables ( number of weekly confirmed cases among residents and staff  and  number of weekly COVID-19 deaths among residents ) as the  dependent variables .

Topic 4: Surrounding Greenness, Stress, and Memory

Scientific evidence has shown that surrounding greenness has multiple health-related benefits. Health benefits include better cognitive functioning or better intellectual activity such as thinking, reasoning, or remembering things. These findings, however, are not well understood. A study, therefore, analyzed the relationship between surrounding greenness and memory performance, with stress as a mediating variable.

Statistical Research Question No. 4

Is there a significant relationship between exposure to and use of natural environments, stress, and memory performance?

As you become more familiar and well-versed in identifying the variables you would like to investigate in your study, reading studies like this requires reading the method or methodology section. This section will tell you how the researchers measured the variables of their study. Knowing how those variables are quantified can help you design your research and formulate the appropriate statistical research questions.

Topic 5: Income and Happiness

Reading the abstract, we can readily identify one of the variables used in the study, i.e., money. It’s easy to count that. But for happiness, that is a largely subjective matter. Happiness varies between individuals. So how did the researcher measured happiness? As previously mentioned, we need to see the methodology portion to find out why.

If you click on the link to the full text of the paper on pages 10 and 11, you will read that the researcher measured happiness using a 10-point scale. The scale was categorized into three namely, 1) unhappy, 2) happy, and 3) very happy.

Statistical Research Question No. 5

Is there a significant relationship between income and happiness?

I do hope that upon reaching this part of the article, you are now well familiar on how to write statistical research questions. Practice makes perfect.

References:

Lega, C., Gidlow, C., Jones, M., Ellis, N., & Hurst, G. (2021). The relationship between surrounding greenness, stress and memory.  Urban Forestry & Urban Greening ,  59 , 126974.

Zapata, O. (2021). The relationship between climate conditions and consumption of bottled water: A potential link between climate change and plastic pollution. Ecological Economics, 187, 107090.

© P. A. Regoniel 12 October 2021 | Updated 08 January 2024

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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 Maths IA: 60 Examples and Guidance

Charles Whitehouse

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

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

What is the Mathematics IA?

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

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

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

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

What are the assessment criteria?

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

Source : IB Mathematics Applications and Interpretation Guide

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

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

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

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

Criterion E: Use of Mathematics (6 marks ) –

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

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

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

- IB Maths AI SL Study Notes

- IB Maths AI HL Study Notes

- IB Maths AA SL Study Notes

- IB Maths AA HL Study Notes

What are some example research questions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

How can I score highly?

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

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

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

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

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

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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 Extended Essay: Research Questions

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IB Command Terms

Command terms are the key terms and phrases used in examination questions. 

See the lists below for the terms and definitions for each IB subject

• Visual Arts command terms
• Biology command terms
• Chemistry command terms
• Math command terms
• Language and Literature command terms
• Economics command terms
• Physics command terms

Sample research questions

Source: IBO.org

Question Starters

Source: Oxford Course Companion, p.17

Research Questions: Class of 2020

Visual Arts: To what extent are the developments in American military aircraft nose designs from World War 2, Vietnam War and Gulf War more connected to individual crew preference than to the visual culture of America, 1940 - 1990?

English A: To what extent does William Shakespeare employ Elizabethan elements in Romeo and Juliet to build towards the eventual tragedy?

English A: To what extent did news media outlets, specifically CNN and Fox News, use different reporting methods to portray the candidates in the 2016 United States presidential election?

English A: How do aspects of real-life societies inform and shape A Clockwork Orange and The Handmaid’s Tale?

English A: How were African Americans portrayed in American sitcoms from 1980 to 2000?

English B: To what extent does 'The Fault in Our Stars' by John Green reflect 'The Hero's Journey'?

English B: To what extent does social media influence or have an effect on the english language usage?

English B: To what extent does the usage of AAVE affect an individual’s social status and mobility?

English B: To what extent does the development of the Super Bowl mirror specific aspects of American cultural, social, and economic trends?

Psychology: To what extent is the phenomenon of Karoshi, the overworking of office workers in Southeast Asia, interconnected with symptoms of depression?

Psychology: To What Extent is Cognitive Behavioral Therapy, as an addition to traditional pharmacotherapy necessary for the successful treatment of Schizophrenia?

Psychology: To what extent is Δ9-THC an effective antidepressant that can be used to treat patients with depression?

Chinese B: 论新时代网络语言对现代汉语有何种影响？

Chinese B: 中国嘻哈歌手如何运用歌词传播中华文化？How do Chinese Hip-hop Artists Promote Chinese Culture through Lyrics?

Economics: To what extent have increases in rental prices (2017-2018) shrunken the market for Korean restaurants in Chegongmiao, Shenzhen?

Economics: How does President Moon's minimum wage policy affect convenience store market in Seocho 1-dong, South Korea?

Business: To What Extent Does the WeChat Application in Tencent's Business Model Play a Role in its Success?

Business: To what extent has Samsung's acquisition of Harman Kardon helped in increasing revenue and access to new markets?

Research Questions: Class of 2019

World Studies (Biology & Economics): How does mountain gorilla (Gorilla beringei beringei) ecotourism support and strengthen local economies while ensuring the mountain gorillas’ health and survival?

English A, Category 3: How are skincare companies like: Neutrogena, Nivea, and Dove promoting white supremacy?

English B, Category 2B: To what extent do the changes in Cersei’s and Daenerys’s character development in the HBO series Game of Thrones show they are ultimately both motivated for and by power?

Visual Arts: To what extent did visual qualities in Coco Chanel’s Little Black Dress 1913, influence black dresses created by Chinese fashion designer Vivienne Tam and haute couture designer Yiqing Yin in regards to development in visual elements and societal acceptances in the 21st century?

Business Management: “To what extent was Apple Inc.’s acquisition of Beats Electronics, LLC an effective growth strategy?”

Math, Group 5: Investigating the Korean MERS outbreak using the SEIR model: How would hypothetical diseases be simulated if variables of the SEIR model were to be altered?

Business Management : To what extent has the benefits offered by Shekou International School helped them retain staff?

Economics: To what extent has the subsidy that was introduced in 2016 for hybrid electric vehicles led to a rise in demand for domestic car producers in Baden Württemberg?

Chemistry: How does the addition of salt (NaCl) which modifies the salinity affects the interfacial tension and stability of oil-in-water emulsion?

World Studies (History & Literature): What aspects of the anti-vaccination movement, and “The Crucible” relate to mass hysteria during the Salem Witch Trials?

Psychology: To What Extent Do Behavioural Addictions Fit The Criteria for ‘The Disease Model of Addiction’?

World Studies (Economics & Politics): To what extent has the political decision to host the 2016 Olympic Games in Rio de Janeiro affected Brazils’ economy and the financial welfare of its citizens?

Physics: To what extent does the volume of water affect the altitude gained by adding a constant pressure?

Psychology: To what extent are sociocultural and biological factors major causes of elderly people’s depressive behavior?

World Studies (History & Economics): To what extent did the LGBTQ community contribute to making attractive neighborhoods in the US and why are they pushed to leave them now?

Economics: To what extent is the cafe market in Shekou, Shenzhen, monopolistically competitive?

Economics: “Which is the most important factor that changes the real estate price rate in Magok-dong, Gangseo-gu, Seoul Korea?”

Business Management: To what extent has McDonald’s marketing strategies played a major role in becoming a prominent fast-food company in Korea?

World Studies (Physics & Geography): To What Extent is the Design of the Standard Houses that Gawak Kalinga Builds Able to Withstand the Seismic Hazards Present in Manila?

World Studies (Economics & Music): To what extent is electronic music’s economy expanding?

Visual Arts: To what extent did artistic influences and material sources impact the chair designs of Gerrit Thomas Rietveld Zig-Zag (1934), Verner Panton Panton S (1956), and Tom Dixon, Capellini, S Chair (1991)?

Business Management: To what extent has Apple Inc.’s prioritization of product innovation and advertising led to sales of iPhone X?

World Studies (Psychology & Geography): How do the combined effects of Socioeconomic Status and Diabetes increase prevalences of Alzheimer’s Disease in different regions within China (PRC)?

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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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Posing Researchable Questions in Mathematics and Science Education: Purposefully Questioning the Questions for Investigation

• Published: 07 April 2020
• Volume 18 , pages 1–7, ( 2020 )

Cite this article

• Jinfa Cai 1 &
• Rachel Mamlok-Naaman 2  

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A Correction to this article was published on 15 May 2020

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Perhaps the most obvious example is the set of 23 influential mathematical problems posed by David Hilbert that inspired a great deal of progress in the discipline of mathematics (Hilbert, 1901 -1902). Einstein and Infeld ( 1938 ) claimed that “to raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science” (p. 95). Both Cantor and Klamkin recommended that, in mathematics, the art of posing a question be held as high or higher in value than solving it. Similarly, in the history of science, formulating precise, answerable questions not only advances new discoveries but also gives scientists intellectual excitement (Kennedy, 2005 ; Mosteller, 1980 ).

In research related to mathematics and science education, there is no shortage of evidence for the impact of posing important and researchable questions: Posing new, researchable questions marks real advances in mathematics and science education (Cai et al., 2019a ). Although research in mathematics and science education begins with researchable questions, only recently have researchers begun to purposefully and systematically discuss the nature of researchable questions. To conduct research, we must have researchable questions, but what defines a researchable question? What are the sources of researchable questions? How can we purposefully discuss researchable questions?

This special issue marks effort for the field’s discussion of researchable questions. As the field of mathematics and science education matures, it is necessary to reflect on the field at such a metalevel (Inglis & Foster, 2018 ). Although the authors in this special issue discuss researchable questions from different angles, they all refer to researchable questions as those that can be investigated empirically. For any empirical study, one can discuss its design, its conduct, and how it can be written up for publication. Therefore, researchable questions in mathematics and science education can be discussed with respect to study design, the conduct of research, and the dissemination of that research.

Even though there are many lines of inquiry that we can explore with respect to researchable questions, each exploring the topic from a different angle, we have decided to focus on the following three aspects to introduce this special issue: (1) criteria for selecting researchable questions, (2) sources of researchable questions, and (3) alignment of researchable questions with the conceptual framework as well as appropriate research methods.

Criteria for Selecting Researchable Questions

It is clear that not all researchable questions are worth the effort to investigate. According to Cai et al. ( 2020 ), of all researchable questions in mathematics and science education, priority is given to those that are significant. Research questions are significant if they can advance the fields’ knowledge and understanding about the teaching and learning of science and mathematics. Through an analysis of peer reviews for a research journal, Cai et al. ( 2020 ) provide a window into the field’s frontiers related to significant researchable questions. In an earlier article, Cai et al. ( 2019a ) argued that

The significance of a research question cannot be determined just by reading it. Rather, its significance stands in relation to the knowledge of the field. The justification for the research question itself—why it is a significant question to investigate—must therefore be made clear through an explicit argument that ties the research question to what is and is not already known. (p. 118)

In their analysis, Cai et al. ( 2020 ) provide evidence that many reviews that highlighted issues with the research questions in rejected manuscripts specifically called for authors to make an argument to motivate the research questions, whereas none of the manuscripts that were ultimately accepted (pending revisions) received this kind of comment. Cai et al. ( 2020 ) provide a framework not only for analyzing peer reviews about research questions but also for how to communicate researchable questions in journal manuscript preparations.

Whereas Cai and his colleagues, as editors of a journal, discuss significant research questions from the perspective of peer review and publication, King, Ochsendorf, Solomon, and Sloane ( 2020 ), as program directors at the Directorate for Education and Human Resources at the U.S. National Science Foundation (NSF), discuss fundable research questions for research in mathematics and science education. King et al. ( 2020 ) situate their discussion of fundable research questions in the context of writing successful educational research grant proposals. For them, fundable research questions must be transformative and significant with specific and clear constructs. In addition, they present examples of STEM education research questions from different types of research (Institute of Education Sciences [IES] & NSF, 2013 ) and how the questions themselves direct specific design choices, methodologies, measures, study samples, and analytical models as well as how they can reflect the disciplinary orientations of the researchers.

Hjalmarson and Baker ( 2020 ) take a quite different approach to discussing researchable questions for teacher professional development. They argue for the need to include mathematics specialists (e.g. mathematics coaches or mathematics teacher leaders) for studying teacher learning and development. To Hjalmarson and Baker ( 2020 ), researchable questions related to teacher professional learning should be selected by including mathematics specialists because of their role in connecting research and practice.

Sanchez ( 2020 ) discuss, in particular, the importance of replication studies in mathematics and the kinds of researchable questions that would be productive to explore within this category. With the increased acknowledgement of the importance of replication studies (Cai et al., 2018 ), Sanchez Aguilar has provided a useful typology of fundamental questions that can guide a replication study in mathematics (and science) education.

Schoenfeld ( 2020 ) is very direct in suggesting that researchable questions must advance the field and that these research questions must be meaningful and generative: “What is meant by meaningful is that the answer to the questions posed should matter to either practice or theory in some important way. What is meant by generative is that working on the problem, whether it is ‘solved’ or not, is likely to provide valuable insights” (pp. XX). Schoenfeld calls for researchers to establish research programs—that is, one not only selects meaningful research questions to investigate but also continues in that area of research to produce ongoing insights and further meaningful questions.

Stylianides and Stylianides ( 2020 ) argue that, collectively, researchers can and need to pose new researchable questions. The new researchable questions are worth investigating if they reflect the field’s growing understanding of the web of potentially influential factors surrounding the investigation of a particular area. The argument that Stylianides and Stylianides ( 2020 ) use is very similar to Schoenfeld’s ( 2020 ) generative criteria, but Stylianides and Stylianides ( 2020 ) explicitly emphasize the collective nature of the field’s growing understanding of a particular phenomenon.

Sources of Researchable Questions

Research questions in science and mathematics education arise from multiple sources, including problems of practice, extensions of theory, and lacunae in existing areas of research. Therefore, through a research question’s connections to prior research, it should be clear how answering the question extends the field’s knowledge (Cai et al., 2019a ). Across the papers in this special issue lies a common theme that researchable questions arise from understanding the area under study. Cai et al. ( 2020 ) take the position that the significance of researchable questions must be justified in the context of prior research. In particular, reviewers of manuscripts submitted for publication will evaluate if the study is adequately motivated. In fact, posing significant researchable questions is an iterative process beginning with some broader, general sense of an idea which is potentially fruitful and leading, eventually, to a well-specified, stated research question (Cai et al., 2019a ). Similarly, King et al. ( 2020 ) argue that fundable research questions should be grounded in prior research and make explicit connections to what is known or not known in the given area of study.

Sanchez ( 2020 ) suggest that it is time for the field of mathematics and science education research to seriously consider replication studies. Researchable questions related to replication studies might arise from the examination of the following two questions: (1) Do the results of the original study hold true beyond the context in which it was developed? (2) Are there alternative ways to study and explain an identified phenomenon or finding? Similarly, Hjalmarson and Baker ( 2020 ) specifically suggest two needs related to mathematics specialists in studies of professional development that drive researchable questions: (1) defining practices and hidden players involved in systematic school change and (2) identifying the unit of analysis and scaling up professional development.

Schoenfeld ( 2020 ) uses various examples to illustrate the origin of researchable questions. One of his (perhaps most familiar) examples is his decade-long research on mathematical problem solving. He elaborates on how answering one specific research question leads to another and another. In the context of research on mathematical proof, Stylianides and Stylianides ( 2020 ) also illustrate how researchable questions arise from existing research in the area leading to new researchable questions in the dynamic process of educational research. The arguments and examples in both Schoenfeld ( 2020 ) and Stylianides and Stylianides ( 2020 ) are quite powerful in the sense that this source of researchable questions facilitates the accumulation of knowledge for the given areas of study.

A related source of researchable questions is not discussed in this set of papers—unexpected findings. A potentially powerful source of research questions is the discovery of an unexpected finding when conducting research (Cai et al., 2019b ). Many important advances in scientific research have their origins in serendipitous, unexpected findings. Researchers are often faced with unexpected and perhaps surprising results, even when they have developed a carefully crafted theoretical framework, posed research questions tightly connected to this framework, presented hypotheses about expected outcomes, and selected methods that should help answer the research questions. Indeed, unexpected findings can be the most interesting and valuable products of the study and a source of further researchable questions (Cai et al., 2019b ).

Of course, researchable questions can also arise from established scholars in a given field—those who are most familiar with the scope of the research that has been done. For example, in 2005, in celebrating the 125th anniversary of the publication of Science ’s first issue, the journal invited researchers from around the world to propose the 125 most important research questions in the scientific enterprise (Kennedy, 2005 ). A list of unanswered questions like this is a great source for researchable questions in science, just as the 23 great questions in mathematics by Hilbert ( 1901 -1902) spurred the field for decades. In mathematics and science education, one can look to research handbooks and compendiums. These volumes often include lists of unanswered research questions in the hopes of prompting further research in various areas (e.g. Cai, 2017 ; Clements, Bishop, Keitel, Kilpatrick, & Leung, 2013 ; Talbot-Smith, 2013 ).

Alignment of Researchable Questions with the Conceptual Framework and Appropriate Research Methods

Cai et al. ( 2020 ) and King et al. ( 2020 ) explicitly discuss the alignment of researchable questions with the conceptual framework and appropriate research methods. In writing journal publications or grant proposals, it is extremely important to justify the significance of the researchable questions based on the chosen theoretical framework and then determine robust methods to answer the research questions. According to Cai et al. ( 2019a ), justification for the significance of the research questions depends on a theoretical framework: “The theoretical framework shapes the researcher’s conception of the phenomenon of interest, provides insight into it, and defines the kinds of questions that can be asked about it” (p. 119). It is true that the notion of a theoretical framework can remain somewhat mysterious and confusing for researchers. However, it is clear that the theoretical framework links research questions to existing knowledge, thus helping to establish their significance; provides guidance and justification for methodological choices; and provides support for the coherence that is needed between research questions, methods, results, and interpretations of findings (Cai & Hwang, 2019 ; Cai et al., 2019c ).

Analyzing reviews for a research journal in mathematics education, Cai et al. ( 2020 ) found that the reviewers wanted manuscripts to be explicit about how the research questions, the theoretical framework, the methods, and the findings were connected. Even for manuscripts that were accepted (pending revisions), making explicit connections across all parts of the manuscript was a challenging proposition. Thus, in preparing manuscripts for publication, it is essential to communicate the significance of a study by developing a coherent chain of justification connecting researchable questions, the theoretical framework, and the research methods chosen to address the research questions.

The Long Journey Has Just Begun with a First Step

As the field of mathematics and science education matures, there is a need to take a step back and reflect on what has been done so that the field can continue to grow. This special issue represents a first step by reflecting on the posing of significant researchable questions to advance research in mathematics and science education. Such reflection is useful and necessary not only for the design of studies but also for the writing of research reports for publication. Most importantly, working on significant researchable questions cannot only contribute to theory generation about the teaching and learning of mathematics and science but also contribute to improving the impact of research on practice in mathematics and science classrooms.

To conclude, we want to draw readers’ attention to a parallel between this reflection on research in our field and a line of research that investigates the development of school students’ problem-posing and questioning skills in mathematics and science (Blonder, Rapp, Mamlok-Naaman, & Hofstein, 2015 ; Cai, Hwang, Jiang, & Silber, 2015 ; Cuccio-Schirripa & Steiner, 2000 ; Hofstein, Navon, Kipnis, & Mamlok-Naaman, 2005 ; Silver, 1994 ; Singer, Ellerton, & Cai, 2015 ). Posing researchable questions is critical for advancing research in mathematics and science education. Similarly, providing students opportunities to pose problems is critical for the development of their thinking and learning. With the first step in this journey made, perhaps we can dream of something bigger further on down the road.

Change history

15 may 2020.

The original version of this article unfortunately contains correction.

Blonder, R., Rapp, S., Mamlok-Naaman, R., & Hofstein, A. (2015). Questioning behavior of students in the inquiry chemistry laboratory: Differences between sectors and genders in the Israeli context. International Journal of Science and Mathematics Education, 13 (4), 705–732.

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Cai, J., Hwang, S., Hiebert, J., Hohensee, C., Morris, A., & Robison, V. (2020). Communicating the significance of research questions: Insights from peer review at a flagship journal. International Journal of Science and Mathematics Education . https://doi.org/10.1007/s10763-020-10073-x .

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Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., . . . Hiebert, J. (2019b). So what? Justifying conclusions and interpretations of data. Journal for Research in Mathematics Education , 50 (5), 470477. https://doi.org/10.5951/jresematheduc.50.5.0470 .

Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., . . . Hiebert, J. (2019c). Theoretical framing as justifying. Journal for Research in Mathematics Education , 50 (3), 218224. https://doi.org/10.5951/jresematheduc.50.3.0218 .

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Cai, J., Mamlok-Naaman, R. Posing Researchable Questions in Mathematics and Science Education: Purposefully Questioning the Questions for Investigation. Int J of Sci and Math Educ 18 (Suppl 1), 1–7 (2020). https://doi.org/10.1007/s10763-020-10079-5

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• Writing Strong Research Questions | Criteria & Examples

Writing Strong Research Questions | Criteria & Examples

Published on October 26, 2022 by Shona McCombes . Revised on November 21, 2023.

A research question pinpoints exactly what you want to find out in your work. A good research question is essential to guide your research paper , dissertation , or thesis .

All research questions should be:

• Focused on a single problem or issue
• Researchable using primary and/or secondary sources
• Feasible to answer within the timeframe and practical constraints
• Specific enough to answer thoroughly
• Complex enough to develop the answer over the space of a paper or thesis
• Relevant to your field of study and/or society more broadly

Table of contents

How to write a research question, what makes a strong research question, using sub-questions to strengthen your main research question, research questions quiz, other interesting articles, frequently asked questions about research questions.

You can follow these steps to develop a strong research question:

• Choose your topic
• Do some preliminary reading about the current state of the field
• Narrow your focus to a specific niche
• Identify the research problem that you will address

The way you frame your question depends on what your research aims to achieve. The table below shows some examples of how you might formulate questions for different purposes.

Using your research problem to develop your research question

Note that while most research questions can be answered with various types of research , the way you frame your question should help determine your choices.

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Research questions anchor your whole project, so it’s important to spend some time refining them. The criteria below can help you evaluate the strength of your research question.

Focused and researchable

Feasible and specific

Complex and arguable

Relevant and original

Chances are that your main research question likely can’t be answered all at once. That’s why sub-questions are important: they allow you to answer your main question in a step-by-step manner.

Good sub-questions should be:

• Less complex than the main question
• Focused only on 1 type of research
• Presented in a logical order

Here are a few examples of descriptive and framing questions:

• Descriptive: According to current government arguments, how should a European bank tax be implemented?
• Descriptive: Which countries have a bank tax/levy on financial transactions?
• Framing: How should a bank tax/levy on financial transactions look at a European level?

Keep in mind that sub-questions are by no means mandatory. They should only be asked if you need the findings to answer your main question. If your main question is simple enough to stand on its own, it’s okay to skip the sub-question part. As a rule of thumb, the more complex your subject, the more sub-questions you’ll need.

Try to limit yourself to 4 or 5 sub-questions, maximum. If you feel you need more than this, it may be indication that your main research question is not sufficiently specific. In this case, it’s is better to revisit your problem statement and try to tighten your main question up.

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If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Methodology

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

The way you present your research problem in your introduction varies depending on the nature of your research paper . A research paper that presents a sustained argument will usually encapsulate this argument in a thesis statement .

A research paper designed to present the results of empirical research tends to present a research question that it seeks to answer. It may also include a hypothesis —a prediction that will be confirmed or disproved by your research.

As you cannot possibly read every source related to your topic, it’s important to evaluate sources to assess their relevance. Use preliminary evaluation to determine whether a source is worth examining in more depth.

This involves:

• Reading abstracts , prefaces, introductions , and conclusions
• Looking at the table of contents to determine the scope of the work
• Consulting the index for key terms or the names of important scholars

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

Formulating a main research question can be a difficult task. Overall, your question should contribute to solving the problem that you have defined in your problem statement .

However, it should also fulfill criteria in three main areas:

• Researchability
• Feasibility and specificity
• Relevance and originality

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all the numbers are changing, but what doesn't change is the relationship between x and y: y is always one more than twice x. That is, y=2x+1. Finding what doesn't change "tames" the situation. So, you have tamed this problem! Yay. And if you want a fancy mathematical name for things that don’t vary, we call these things "invariants." The number of messed-up recruits is invariant, even though they are all wiggling back and forth, trying to figure out which way is right!

3) Encourage generalizations

So, of course, the next question that comes to my mind is how to generalize what you’ve already discovered: there are 15 ways that 2 mistakes can be arranged in a line of 6 recruits. What about a different number of mistakes? Or a different number of recruits? Is there some way to predict? Or, alternatively, is there some way to predict how these 15 ways of making mistakes will play out as the recruits try to settle themselves down? Which direction interests you?

4) Inquire about reasoning and rigor

The students were looking at the number of ways the recruits could line up with 2 out of n faced the wrong way: Anyway, I had a question of my own. It looks like the number of possibilities increases pretty fast, as the number of recruits increases. For example, I counted 15 possibilities in your last set (the line of six). What I wonder is this: when the numbers get that large, how you can possibly know that you've found all the possibilities? (For example, I noticed that >>>><< is missing.) The question "How do I know I've counted 'em all?" is actually quite a big deal in mathematics, as mathematicians are often called upon to find ways of counting things that nobody has ever listed (exactly like the example you are working on).

The students responded by finding a pattern for generating the lineups in a meaningful order: The way that we can prove that we have all the possibilities is that we can just add the number of places that the second wrong person could be in. For example, if 2 are wrong in a line of 6, then the first one doesn’t move and you count the space in which the second one can move in. So for the line of six, it would be 5+4+3+2+1=15. That is the way to make sure that we have all the ways. Thanks so much for giving challenges. We enjoyed thinking!

5) Work towards proof

a) The group wrote the following: When we found out that 6 recruits had 15 different starting arrangements, we needed more information. We needed to figure out how many starting positions are there for a different number of recruits.

By drawing out the arrangements for 5 recruits and 7 recruits we found out that the number of starting arrangements for the recruit number before plus that recruit number before it would equal the number of starting arrangements for that number of recruits.

We also found out that if you divide the starting arrangements by the number of recruits there is a pattern.

To which the mentor replied: Wow! I don't think (in all the years I've been hanging around mathematics) I've ever seen anyone describe this particular pattern before! Really nice! If you already knew me, you'd be able to predict what I'm about to ask, but you don't, so I have to ask it: "But why?" That is, why is this pattern (the 6, 10, 15, 21, 28…) the pattern that you find for this circumstance (two recruits wrong in lines of lengths, 4, 5, 6, 7, 8…)? Answering that—explaining why you should get those numbers and why the pattern must continue for longer lines—is doing the kind of thing that mathematics is really about.

b) Responding to students studying a circular variation of raw recruits that never settled down: This is a really interesting conclusion! How can you show that it will always continue forever and that it doesn’t matter what the original arrangement was? Have you got a reason or did you try all the cases or…? I look forward to hearing more from you.

6) Distinguish between examples and reasons

a) You have very thoroughly dealt with finding the answer to the problem you posed—it really does seem, as you put it, "safe to say" how many there will be. Is there a way that you can show that that pattern must continue? I guess I’d look for some reason why adding the new recruit adds exactly the number of additional cases that you predict. If you could say how the addition of one new recruit depends on how long the line already is, you’d have a complete proof. Want to give that a try?

b) A student, working on Amida Kuji and having provided an example, wrote the following as part of a proof: In like manner, to be given each relationship of objects in an arrangement, you can generate the arrangement itself, for no two different arrangements can have the same object relationships. The mentor response points out the gap and offers ways to structure the process of extrapolating from the specific to the general: This statement is the same as your conjecture, but this is not a proof. You repeat your claim and suggest that the example serves as a model for a proof. If that is so, it is up to you to make the connections explicit. How might you prove that a set of ordered pairs, one per pair of objects forces a unique arrangement for the entire list? Try thinking about a given object (e.g., C) and what each of its ordered pairs tells us? Try to generalize from your example. What must be true for the set of ordered pairs? Are all sets of n C2 ordered pairs legal? How many sets of n C2 ordered pairs are there? Do they all lead to a particular arrangement? Your answers to these questions should help you work toward a proof of your conjecture.

9) Encourage extensions

What you’ve done—finding the pattern, but far more important, finding the explanation (and stating it so clearly)—is really great! (Perhaps I should say "finding and stating explanations like this is real mathematics"!) Yet it almost sounded as if you put it down at the very end, when you concluded "making our project mostly an interesting coincidence." This is a truly nice piece of work!

The question, now, is "What next?" You really have completely solved the problem you set out to solve: found the answer, and proved that you’re right!

I began looking back at the examples you gave, and noticed patterns in them that I had never seen before. At first, I started coloring parts red, because they just "stuck out" as noticeable and I wanted to see them better. Then, it occurred to me that I was coloring the recruits that were back-to-back, and that maybe I should be paying attention to the ones who were facing each other, as they were "where the action was," so I started coloring them pink. (In one case, I recopied your example to do the pinks.) To be honest, I’m not sure what I’m looking for, but there was such a clear pattern of the "action spot" moving around that I thought it might tell me something new. Anything come to your minds?

10) Build a Mathematical Community

I just went back to another paper and then came back to yours to look again. There's another pattern in the table. Add the recruits and the corresponding starting arrangements (for example, add 6 and 15) and you get the next number of starting arrangements. I don't know whether this, or your 1.5, 2, 2.5, 3, 3.5… pattern will help you find out why 6, 10, 15… make sense as answers, but they might. Maybe you can work with [your classmates] who made the other observation to try to develop a complete understanding of the problem.

11) Highlight Connections

Your rule—the (n-1)+(n-2)+(n-3)+… +3+2+1 part—is interesting all by itself, as it counts the number of dots in a triangle of dots. See how?

12) Wrap Up

This is really a very nice and complete piece of work: you've stated a problem, found a solution, and given a proof (complete explanation of why that solution must be correct). To wrap it up and give it the polish of a good piece of mathematical research, I'd suggest two things.

The first thing is to extend the idea to account for all but two mistakes and the (slightly trivial) one mistake and all but one mistake. (If you felt like looking at 3 and all but 3, that'd be nice, too, but it's more work—though not a ton—and the ones that I suggested are really not more work.)

The second thing I'd suggest is to write it all up in a way that would be understandable by someone who did not know the problem or your class: clear statement of the problem, the solution, what you did to get the solution, and the proof.

I look forward to seeing your masterpiece!

Advice for Keeping a Formal Mathematics Research Logbook

As part of your mathematics research experience, you will keep a mathematics research logbook. In this logbook, keep a record of everything you do and everything you read that relates to this work. Write down questions that you have as you are reading or working on the project. Experiment. Make conjectures. Try to prove your conjectures. Your journal will become a record of your entire mathematics research experience. Don’t worry if your writing is not always perfect. Often journal pages look rough, with notes to yourself, false starts, and partial solutions. However, be sure that you can read your own notes later and try to organize your writing in ways that will facilitate your thinking. Your logbook will serve as a record of where you are in your work at any moment and will be an invaluable tool when you write reports about your research.

Ideally, your mathematics research logbook should have pre-numbered pages. You can often find numbered graph paper science logs at office supply stores. If you can not find a notebook that has the pages already numbered, then the first thing you should do is go through the entire book putting numbers on each page using pen.

• Date each entry.

• Work in pen.

• Don’t erase or white out mistakes. Instead, draw a single line through what you would like ignored. There are many reasons for using this approach:

– Your notebook will look a lot nicer if it doesn’t have scribbled messes in it.

– You can still see what you wrote at a later date if you decide that it wasn’t a mistake after all.

– It is sometimes useful to be able to go back and see where you ran into difficulties.

– You’ll be able to go back and see if you already tried something so you won’t spend time trying that same approach again if it didn’t work.

• When you do research using existing sources, be sure to list the bibliographic information at the start of each section of notes you take. It is a lot easier to write down the citation while it is in front of you than it is to try to find it at a later date.

• Never tear a page out of your notebook. The idea is to keep a record of everything you have done. One reason for pre-numbering the pages is to show that nothing has been removed.

• If you find an interesting article or picture that you would like to include in your notebook, you can staple or tape it onto a page.

Advice for Keeping a Loose-Leaf Mathematics Research Logbook

Get yourself a good loose-leaf binder, some lined paper for notes, some graph paper for graphs and some blank paper for pictures and diagrams. Be sure to keep everything that is related to your project in your binder.

– Your notebook will look a lot nicer if it does not have scribbled messes in it.

• Be sure to keep everything related to your project. The idea is to keep a record of everything you have done.

• If you find an interesting article or picture that you would like to include in your notebook, punch holes in it and insert it in an appropriate section in your binder.

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In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications.

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• Computational Biology
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• Computational Science & Numerical Analysis
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The Definitive Guide to Choosing the Best IB Mathematics IA Topics

by  Antony W

August 19, 2022

Imagine you were fascinated by the shape of nuclear reactor chimneys, called Natural Draft Cooling Towers. What an odd shape, you think (a hyperboloid, if you’re wondering). But how would you find out why they are so shaped? As it turns out, mathematics is a beautiful, fascinating language you can use to describe something like that. And when you explore such a personally fascinating subject within certain regulations of the IB, you will be writing your mathematics internal assessment.

As you can deduce, the IB math internal assessment is all about exploring mathematically a subject of personal interest. How well you do that and present your process determines how many marks out of the total 20 you will receive. However, most students live in mortal fear or both the IB math exam and the internal assessment.

If you are one such student, fear not. This detailed guide will teach you how you need to approach the math assessment by choosing the right topics, as well as how to make a top-grade essay. You can also get further IB guides in full detail here on our blog.

The team at Help for Assessments is also ready to give more than just passive “how to” assistance. Let us take the IB internal assessment nightmare away from you and write the essay for you. Our highly skilled and experienced team assures you of top-notch work, original and flawless in research and quality. All these at the best rates, so don’t hesitate to give us your order. We will even give you 25% off your first order to get you started.

Do you still want to push on? This article will give you all you need to choose suitable topics for your maths IA, explore what the test is about, and finish with some fresh sample idea topics for the math IA.

Struggling to pick your Math IA or AA topic? 

What the IB Maths Internal Assessment Topics All About

The mathematics internal assessment is about exploring the math behind a topic of interest, then presenting the whole thing in the form of a short thorough report. Ideally, the topic should be completely original and serve to prove that you have a firm grasp of mathematical concepts, principals, and knowledge.

According to the IA mathematics rubric, the IBO is looking for the following in a good essay:

• Communication : Communication measures how well you organize and explain your exploration. It must be logically developed and coherent. The essay generally should have three parts: an introduction, the rationale or explanation, and a conclusion.
• Mathematical presentation:  You need to use appropriate mathematical languages, with the right formulae, diagrams, tables, charts, models, and other figures as needed.
• Personal engagement:  This is a measure of the level to which the student interacts with the subject matter on a personal level. It is more than being original or authentic with the idea - it has to stem from some experience in your life or one you have direct links with.
• Reflection:  This criterion assesses how the student reviews and analyzes the problem and its solution in the exploration.
• Use of mathematics:  This is self-explanatory - how well does the student use the concepts, knowledge, and skills gained in their respective mathematical level so far?

Thus, the maths IA is about displaying your ability to apply abstract concepts and knowledge in math to a real-world situation and communicating clearly to your audience. In this case, the audience is your instructor/professor, and writing a good IA essay starts from picking the right topic.

Not sure whether to order Math IA or AA? Click below to see our latest samples! 

What Makes a Good Math IA Topic

A good topic for the IB mathematics internal assessment is one that allows you to fulfill all the above requirements. It will help you explore an area of interest deeply and exhaustively, while at the same time providing an avenue for you to put your acquired mathematical skills into good use.

If you are to be successful in these goals, your essay should have certain important qualities. These are the same qualities your topic should have:

• Simple and clear language: The rule of thumb is that everyone in your age group (your classmates) must be able to understand your essay. Thus, your topic needs to be simple. A complex subject will be hard for you to write, tough to break down into manageable bites, and ultimately cost you valuable marks.
• Concise and well-focused: The essay will be 6-12 pages in length, so you need to have a topic that can be exhaustively examined in that range. As is said in other IB assessments, the topic should be specific but not too narrow that it restrains or limits you. For example, our chosen problem of the nuclear reactor exhaust steam funnels could be confined to modeling the hyperboloid shape of the cooling towers and start from there.
• Interesting : The chosen topic needs to be exciting enough to elicit not only your interest but that of your audience. It needs to be one that is naturally intriguing to warrant a 12-page study into its underlying mathematical principles.
• Fresh : By fresh, we mean a relatively unexplored topic. Too many students do game theory, but that topic has been gone through by thousands of students in decades of consecutive years. Don't do that - go for something fresh, or seek to add an extra twist to something that already exists.
• Has clear connections with one or more mathematical fields: Your chosen topic should be based on a given mathematical area, or rely on a few of them. Drawing connections between different areas, e.g. Calculus and geometry will impress your instructor. However, always keep it within your attained academic level.

With the goal in mind, it's time to look into the process that precedes choosing a perfect math IA topic.

How to Choose a Mathematics IA Topic

Most math gurus insist that math is not just an abstract subject. It has real and effective links to the real world, and that is what the whole point of the internal assessment is.

To choose the topic, start with the real-life experiences to help you pinpoint an area of math you want to explore.

The following steps will help you:

• Brainstorm:  The first hurdle is to find a general area of interest, which ideally should be related or founded on your interests. For example, if you love medicine and want to study it in the future, you can start with a certain field that fascinates you. Future lawyers, businessmen, engineers, and IT enthusiasts can all find something intriguing to pursue.
• Narrow down: With the first few ideas, the next step is to find a topic that has a lot of mathematical potential. You will find that reading lots of journals, watching videos, and talking with friends gives you lots of seed ideas for this stage.
• Evaluate: Evaluate each of the ideas you have on the strength of the qualities above. Is it relevant? Simple enough for your level? Exciting? Will it be useful to you in your career or your life?
• General research: General research into the mathematics involved in your chosen topic will help you determine what you need to learn and determine how feasible your topic is. The point here is to find out if the topic is right for you, and whether you can find the right material to base your exploration on.
• Come up with a working research question: Every IA needs to have a research question to streamline the exploration and provide direction to the essay. As with the topic, the research question needs to be specific but just wide enough to give you enough material to fill the said 6-12 pages.

Or the other hand, perhaps this topic would be better suited to the Physics IA? It’s up to you!

Do you need help with your Math IA or AA? 

30+ Math IA Topics for SL and HL Levels

We’ve been writing IB Math IA assignments for over 5 years. From a professional academic writing assistance point of view, the number one challenge that many IB learners have is topic selection.

On the surface, Math IA is about investigating concepts within a topic of interest and presenting your findings in a 2,200-word report.

In practice, coming up with an original topic, which you can investigate to prove that you have a strong grip of mathematical concepts and principles can be somewhat challenging.

In this section, we put together a list of 20+ IB Math IA topic for SL and HL to help you understand what good topic looks like based on the selection criteria that we’ve shared in this guide.

• How accurate are mathematical predictions for events with a low likelihood but huge impact?
• Considering the risks involved in making decisions based on incomplete or conflicting information might make us more cautious.
• How do normal numbers fare when compared to random number generators?
• Create a virtual version of the disaster at Chernobyl and its subsequent effects on Japan.
• Do the numbers that make up a Pythagorean triple follow any kind of regularity?
• Does a high degree of association exist between BMI and GDP per capita?
• What kinds of character combinations are best for online safety in light of brute force attacks?
• To what extent does Bayesian probability work in a real-life setting and is it possible to add complexity to it?
• How do those involved in pyramid schemes or other forms of fast-paced fraud use mathematics to amass huge fortunes so quickly?
• Determine the climber's fall severity by measuring the distance from the final point of connection.
• Is it feasible to forecast the outcome of athletic events using a Poisson model and some familiarity with Excel?
• How long would it take someone to try to guess someone else’s password? What's the deal with adding symbols and digits to a password?
• How can we identify individual tones using sine waves of different frequencies?
• Is there a way to utilize arithmetic to predict how contagious diseases like measles will move across a population?
• Do large transfer window expenditures result in a higher victory percentage a sports season?
• With what method of voting can the most people be certain that their preferred candidate will win the election?
• A study of the geometric sequence's connection to the camera's aperture
• How well do you think integration would work to determine the gini coefficient for a sample of nations, allowing you to make reliable comparisons of the economic inequality between them?
• Find out if there's a correlation between music and fractions, or see if this series converges.
• If a fatal flu virus were to sweep the globe, how long do you think it would take humanity to perish?
• Examine the ratios of frequencies between notes to see why some do not sound good together while others do.
• What kinds of numbers have the most bearing on a basketball team's success?
• A look at how uncertainty about the future might make people more risk-averse and how asymmetric information plays a role in this phenomenon.
• The ideal amount of force and launch angle for a javelin or shot put world record throw.
• The gravitational attraction of other things in space causes the orbits of planets and comets to be elliptical. Look into the field of space exploration!
• Study the numbers behind the processes of heredity and natural selection.
• How can we utilize computers to learn more about sequences?
• Is it possible to employ computational methods in addition to pure mathematics to find an answer?
• Applying the mechanics of fast-moving spacecraft to the problem of interstellar travel
• Applying quantum mechanics, we may examine the universe's innate probabilistic character.
• If it takes the tortoise twice as long to cover the same distance as a runner, then there's no way the runner can catch up to it no matter how fast he runs.
• Using tools from probability and game theory, researchers investigate the most effective bluffing techniques for poker.
• Does the time it takes a kid to arrive to school depend on how far they live from the school?
• Check out the Guardian Stats to see if the top teams in the league are also the ones that commit the most fouls.

Tips to Help You Write the Best Math IA Assignment

The following tips can help you write a more comprehensive IB Math IA assignment:

1. Choose a Topic You’re Interested In

Since you will be working on your Math IA for a few months, it would be preferable if you choose a topic that genuinely interests you, rather than one that is simply required.

2. Use a Simple Language

Ensure that language is exact, clear, and succinct. Write your IA in a way that anybody of your age can read and understand.

Choosing a complicated topic may result in a disorganized and difficult-to-understand IA, so try to avoid doing so. Also, avoid writing lengthy accounts of your own experiences. This is the requirement for "Communication" on the math IA.

3. Use Appropriate Terms

Use appropriate mathematical notation and symbols throughout your IA report.

To properly format all mathematical symbols, use MathType or a comparable program for mathematical expressions. Doing so will enhance clarity and will get you easy points for "Mathematical Presentation"

IB also requires you to reflect on your findings in the report as part of your IA. Comment (thoughtfully), but avoid paraphrasing the results.

Describe some of the insights you've acquired from the IA mathematical conclusion. The greater the depth of your contemplation, the more points you will receive for the "Reflection" criterion.

Do You Need Help With Your IB Math Internal Assessment?

The mathematics internal assessment requires an astute mind to complete, and choosing the topic is the least of your worries. Many students are scared stiff on its account, but you don’t have to be. Help for Assessment is here to help you pass the internal assessment in maths, and not just because of this guide. The team here is made up of top IB experts who will do the internal assessment for you upon request. All you have to do is leave us your order here , and we guarantee you top grades and 100% original, impeccably researched, and fully proofed work. Of course, confidentiality is guaranteed, no matter where you come from. 

About the author 

Antony W is a professional writer and coach at Help for Assessment. He spends countless hours every day researching and writing great content filled with expert advice on how to write engaging essays, research papers, and assignments.

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I need help finding a math ia topic related to cars as its my interest but i am very confused and need some guidance

Hello Muhannad.

Thanks for reaching out. We’re happy to help. One of our representatives will get back to you via e-mail shortly. Thank you.

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Thanks for reaching out. We’re happy to help. One of our representatives will get back to you via your email for further assistance

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Kindly read this guide carefully again. It should help you choose an IA topic for your next assignment.

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IA idea check

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We’ve updated the guide to include example topics to help you get started.

Hello Everyone,

We've updated this guide with 30+ topic ideas in Math IA for SL and HL levels for inspiration. We hope this guide continues to be helpful.

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Choose Your Test

Sat / act prep online guides and tips, the complete ib extended essay guide: examples, topics, and ideas.

International Baccalaureate (IB)

IB students around the globe fear writing the Extended Essay, but it doesn't have to be a source of stress! In this article, I'll get you excited about writing your Extended Essay and provide you with the resources you need to get an A on it.

If you're reading this article, I'm going to assume you're an IB student getting ready to write your Extended Essay. If you're looking at this as a potential future IB student, I recommend reading our introductory IB articles first, including our guide to what the IB program is and our full coverage of the IB curriculum .

IB Extended Essay: Why Should You Trust My Advice?

I myself am a recipient of an IB Diploma, and I happened to receive an A on my IB Extended Essay. Don't believe me? The proof is in the IBO pudding:

If you're confused by what this report means, EE is short for Extended Essay , and English A1 is the subject that my Extended Essay topic coordinated with. In layman's terms, my IB Diploma was graded in May 2010, I wrote my Extended Essay in the English A1 category, and I received an A grade on it.

What Is the Extended Essay in the IB Diploma Programme?

The IB Extended Essay, or EE , is a mini-thesis you write under the supervision of an IB advisor (an IB teacher at your school), which counts toward your IB Diploma (learn more about the major IB Diploma requirements in our guide) . I will explain exactly how the EE affects your Diploma later in this article.

For the Extended Essay, you will choose a research question as a topic, conduct the research independently, then write an essay on your findings . The essay itself is a long one—although there's a cap of 4,000 words, most successful essays get very close to this limit.

Keep in mind that the IB requires this essay to be a "formal piece of academic writing," meaning you'll have to do outside research and cite additional sources.

The IB Extended Essay must include the following:

• A title page
• Contents page
• Introduction
• Body of the essay
• References and bibliography

Additionally, your research topic must fall into one of the six approved DP categories , or IB subject groups, which are as follows:

• Group 1: Studies in Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts

Once you figure out your category and have identified a potential research topic, it's time to pick your advisor, who is normally an IB teacher at your school (though you can also find one online ). This person will help direct your research, and they'll conduct the reflection sessions you'll have to do as part of your Extended Essay.

As of 2018, the IB requires a "reflection process" as part of your EE supervision process. To fulfill this requirement, you have to meet at least three times with your supervisor in what the IB calls "reflection sessions." These meetings are not only mandatory but are also part of the formal assessment of the EE and your research methods.

According to the IB, the purpose of these meetings is to "provide an opportunity for students to reflect on their engagement with the research process." Basically, these meetings give your supervisor the opportunity to offer feedback, push you to think differently, and encourage you to evaluate your research process.

The final reflection session is called the viva voce, and it's a short 10- to 15-minute interview between you and your advisor. This happens at the very end of the EE process, and it's designed to help your advisor write their report, which factors into your EE grade.

Here are the topics covered in your viva voce :

• A check on plagiarism and malpractice
• Your reflection on your project's successes and difficulties
• Your reflection on what you've learned during the EE process

Your completed Extended Essay, along with your supervisor's report, will then be sent to the IB to be graded. We'll cover the assessment criteria in just a moment.

We'll help you learn how to have those "lightbulb" moments...even on test day!  

What Should You Write About in Your IB Extended Essay?

You can technically write about anything, so long as it falls within one of the approved categories listed above.

It's best to choose a topic that matches one of the IB courses , (such as Theatre, Film, Spanish, French, Math, Biology, etc.), which shouldn't be difficult because there are so many class subjects.

Here is a range of sample topics with the attached extended essay:

• Biology: The Effect of Age and Gender on the Photoreceptor Cells in the Human Retina
• Chemistry: How Does Reflux Time Affect the Yield and Purity of Ethyl Aminobenzoate (Benzocaine), and How Effective is Recrystallisation as a Purification Technique for This Compound?
• English: An Exploration of Jane Austen's Use of the Outdoors in Emma
• Geography: The Effect of Location on the Educational Attainment of Indigenous Secondary Students in Queensland, Australia
• Math: Alhazen's Billiard Problem
• Visual Arts: Can Luc Tuymans Be Classified as a Political Painter?

You can see from how varied the topics are that you have a lot of freedom when it comes to picking a topic . So how do you pick when the options are limitless?

How to Write a Stellar IB Extended Essay: 6 Essential Tips

Below are six key tips to keep in mind as you work on your Extended Essay for the IB DP. Follow these and you're sure to get an A!

#1: Write About Something You Enjoy

You can't expect to write a compelling essay if you're not a fan of the topic on which you're writing. For example, I just love British theatre and ended up writing my Extended Essay on a revolution in post-WWII British theatre. (Yes, I'm definitely a #TheatreNerd.)

I really encourage anyone who pursues an IB Diploma to take the Extended Essay seriously. I was fortunate enough to receive a full-tuition merit scholarship to USC's School of Dramatic Arts program. In my interview for the scholarship, I spoke passionately about my Extended Essay; thus, I genuinely think my Extended Essay helped me get my scholarship.

But how do you find a topic you're passionate about? Start by thinking about which classes you enjoy the most and why . Do you like math classes because you like to solve problems? Or do you enjoy English because you like to analyze literary texts?

Keep in mind that there's no right or wrong answer when it comes to choosing your Extended Essay topic. You're not more likely to get high marks because you're writing about science, just like you're not doomed to failure because you've chosen to tackle the social sciences. The quality of what you produce—not the field you choose to research within—will determine your grade.

Once you've figured out your category, you should brainstorm more specific topics by putting pen to paper . What was your favorite chapter you learned in that class? Was it astrophysics or mechanics? What did you like about that specific chapter? Is there something you want to learn more about? I recommend spending a few hours on this type of brainstorming.

One last note: if you're truly stumped on what to research, pick a topic that will help you in your future major or career . That way you can use your Extended Essay as a talking point in your college essays (and it will prepare you for your studies to come too!).

#2: Select a Topic That Is Neither Too Broad nor Too Narrow

There's a fine line between broad and narrow. You need to write about something specific, but not so specific that you can't write 4,000 words on it.

You can't write about WWII because that would be a book's worth of material. You also don't want to write about what type of soup prisoners of war received behind enemy lines, because you probably won’t be able to come up with 4,000 words of material about it. However, you could possibly write about how the conditions in German POW camps—and the rations provided—were directly affected by the Nazis' successes and failures on the front, including the use of captured factories and prison labor in Eastern Europe to increase production. WWII military history might be a little overdone, but you get my point.

If you're really stuck trying to pinpoint a not-too-broad-or-too-narrow topic, I suggest trying to brainstorm a topic that uses a comparison. Once you begin looking through the list of sample essays below, you'll notice that many use comparisons to formulate their main arguments.

I also used a comparison in my EE, contrasting Harold Pinter's Party Time with John Osborne's Look Back in Anger in order to show a transition in British theatre. Topics with comparisons of two to three plays, books, and so on tend to be the sweet spot. You can analyze each item and then compare them with one another after doing some in-depth analysis of each individually. The ways these items compare and contrast will end up forming the thesis of your essay!

When choosing a comparative topic, the key is that the comparison should be significant. I compared two plays to illustrate the transition in British theatre, but you could compare the ways different regional dialects affect people's job prospects or how different temperatures may or may not affect the mating patterns of lightning bugs. The point here is that comparisons not only help you limit your topic, but they also help you build your argument.

Comparisons are not the only way to get a grade-A EE, though. If after brainstorming, you pick a non-comparison-based topic and are still unsure whether your topic is too broad or narrow, spend about 30 minutes doing some basic research and see how much material is out there.

If there are more than 1,000 books, articles, or documentaries out there on that exact topic, it may be too broad. But if there are only two books that have any connection to your topic, it may be too narrow. If you're still unsure, ask your advisor—it's what they're there for! Speaking of advisors...

Don't get stuck with a narrow topic!

#3: Choose an Advisor Who Is Familiar With Your Topic

If you're not certain of who you would like to be your advisor, create a list of your top three choices. Next, write down the pros and cons of each possibility (I know this sounds tedious, but it really helps!).

For example, Mr. Green is my favorite teacher and we get along really well, but he teaches English. For my EE, I want to conduct an experiment that compares the efficiency of American electric cars with foreign electric cars.

I had Ms. White a year ago. She teaches physics and enjoyed having me in her class. Unlike Mr. Green, Ms. White could help me design my experiment.

Based on my topic and what I need from my advisor, Ms. White would be a better fit for me than would Mr. Green (even though I like him a lot).

The moral of my story is this: do not just ask your favorite teacher to be your advisor . They might be a hindrance to you if they teach another subject. For example, I would not recommend asking your biology teacher to guide you in writing an English literature-based EE.

There can, of course, be exceptions to this rule. If you have a teacher who's passionate and knowledgeable about your topic (as my English teacher was about my theatre topic), you could ask that instructor. Consider all your options before you do this. There was no theatre teacher at my high school, so I couldn't find a theatre-specific advisor, but I chose the next best thing.

Before you approach a teacher to serve as your advisor, check with your high school to see what requirements they have for this process. Some IB high schools require your IB Extended Essay advisor to sign an Agreement Form , for instance.

Make sure that you ask your IB coordinator whether there is any required paperwork to fill out. If your school needs a specific form signed, bring it with you when you ask your teacher to be your EE advisor.

#4: Pick an Advisor Who Will Push You to Be Your Best

Some teachers might just take on students because they have to and aren't very passionate about reading drafts, only giving you minimal feedback. Choose a teacher who will take the time to read several drafts of your essay and give you extensive notes. I would not have gotten my A without being pushed to make my Extended Essay draft better.

Ask a teacher that you have experience with through class or an extracurricular activity. Do not ask a teacher that you have absolutely no connection to. If a teacher already knows you, that means they already know your strengths and weaknesses, so they know what to look for, where you need to improve, and how to encourage your best work.

Also, don't forget that your supervisor's assessment is part of your overall EE score . If you're meeting with someone who pushes you to do better—and you actually take their advice—they'll have more impressive things to say about you than a supervisor who doesn't know you well and isn't heavily involved in your research process.

Be aware that the IB only allows advisors to make suggestions and give constructive criticism. Your teacher cannot actually help you write your EE. The IB recommends that the supervisor spends approximately two to three hours in total with the candidate discussing the EE.

#5: Make Sure Your Essay Has a Clear Structure and Flow

The IB likes structure. Your EE needs a clear introduction (which should be one to two double-spaced pages), research question/focus (i.e., what you're investigating), a body, and a conclusion (about one double-spaced page). An essay with unclear organization will be graded poorly.

The body of your EE should make up the bulk of the essay. It should be about eight to 18 pages long (again, depending on your topic). Your body can be split into multiple parts. For example, if you were doing a comparison, you might have one third of your body as Novel A Analysis, another third as Novel B Analysis, and the final third as your comparison of Novels A and B.

If you're conducting an experiment or analyzing data, such as in this EE , your EE body should have a clear structure that aligns with the scientific method ; you should state the research question, discuss your method, present the data, analyze the data, explain any uncertainties, and draw a conclusion and/or evaluate the success of the experiment.

#6: Start Writing Sooner Rather Than Later!

You will not be able to crank out a 4,000-word essay in just a week and get an A on it. You'll be reading many, many articles (and, depending on your topic, possibly books and plays as well!). As such, it's imperative that you start your research as soon as possible.

Each school has a slightly different deadline for the Extended Essay. Some schools want them as soon as November of your senior year; others will take them as late as February. Your school will tell you what your deadline is. If they haven't mentioned it by February of your junior year, ask your IB coordinator about it.

Some high schools will provide you with a timeline of when you need to come up with a topic, when you need to meet with your advisor, and when certain drafts are due. Not all schools do this. Ask your IB coordinator if you are unsure whether you are on a specific timeline.

Below is my recommended EE timeline. While it's earlier than most schools, it'll save you a ton of heartache (trust me, I remember how hard this process was!):

• January/February of Junior Year: Come up with your final research topic (or at least your top three options).
• February of Junior Year: Approach a teacher about being your EE advisor. If they decline, keep asking others until you find one. See my notes above on how to pick an EE advisor.
• April/May of Junior Year: Submit an outline of your EE and a bibliography of potential research sources (I recommend at least seven to 10) to your EE advisor. Meet with your EE advisor to discuss your outline.
• Summer Between Junior and Senior Year: Complete your first full draft over the summer between your junior and senior year. I know, I know—no one wants to work during the summer, but trust me—this will save you so much stress come fall when you are busy with college applications and other internal assessments for your IB classes. You will want to have this first full draft done because you will want to complete a couple of draft cycles as you likely won't be able to get everything you want to say into 4,000 articulate words on the first attempt. Try to get this first draft into the best possible shape so you don't have to work on too many revisions during the school year on top of your homework, college applications, and extracurriculars.
• August/September of Senior Year: Turn in your first draft of your EE to your advisor and receive feedback. Work on incorporating their feedback into your essay. If they have a lot of suggestions for improvement, ask if they will read one more draft before the final draft.
• September/October of Senior Year: Submit the second draft of your EE to your advisor (if necessary) and look at their feedback. Work on creating the best possible final draft.
• November-February of Senior Year: Schedule your viva voce. Submit two copies of your final draft to your school to be sent off to the IB. You likely will not get your grade until after you graduate.

Remember that in the middle of these milestones, you'll need to schedule two other reflection sessions with your advisor . (Your teachers will actually take notes on these sessions on a form like this one , which then gets submitted to the IB.)

I recommend doing them when you get feedback on your drafts, but these meetings will ultimately be up to your supervisor. Just don't forget to do them!

The early bird DOES get the worm!

How Is the IB Extended Essay Graded?

Extended Essays are graded by examiners appointed by the IB on a scale of 0 to 34 . You'll be graded on five criteria, each with its own set of points. You can learn more about how EE scoring works by reading the IB guide to extended essays .

• Criterion A: Focus and Method (6 points maximum)
• Criterion B: Knowledge and Understanding (6 points maximum)
• Criterion C: Critical Thinking (12 points maximum)
• Criterion D: Presentation (4 points maximum)
• Criterion E: Engagement (6 points maximum)

How well you do on each of these criteria will determine the final letter grade you get for your EE. You must earn at least a D to be eligible to receive your IB Diploma.

Although each criterion has a point value, the IB explicitly states that graders are not converting point totals into grades; instead, they're using qualitative grade descriptors to determine the final grade of your Extended Essay . Grade descriptors are on pages 102-103 of this document .

Here's a rough estimate of how these different point values translate to letter grades based on previous scoring methods for the EE. This is just an estimate —you should read and understand the grade descriptors so you know exactly what the scorers are looking for.

Here is the breakdown of EE scores (from the May 2021 bulletin):

How Does the Extended Essay Grade Affect Your IB Diploma?

The Extended Essay grade is combined with your TOK (Theory of Knowledge) grade to determine how many points you get toward your IB Diploma.

To learn about Theory of Knowledge or how many points you need to receive an IB Diploma, read our complete guide to the IB program and our guide to the IB Diploma requirements .

This diagram shows how the two scores are combined to determine how many points you receive for your IB diploma (3 being the most, 0 being the least). In order to get your IB Diploma, you have to earn 24 points across both categories (the TOK and EE). The highest score anyone can earn is 45 points.

Let's say you get an A on your EE and a B on TOK. You will get 3 points toward your Diploma. As of 2014, a student who scores an E on either the extended essay or TOK essay will not be eligible to receive an IB Diploma .

Prior to the class of 2010, a Diploma candidate could receive a failing grade in either the Extended Essay or Theory of Knowledge and still be awarded a Diploma, but this is no longer true.

Figuring out how you're assessed can be a little tricky. Luckily, the IB breaks everything down here in this document . (The assessment information begins on page 219.)

40+ Sample Extended Essays for the IB Diploma Programme

In case you want a little more guidance on how to get an A on your EE, here are over 40 excellent (grade A) sample extended essays for your reading pleasure. Essays are grouped by IB subject.

• Business Management 1
• Chemistry 1
• Chemistry 2
• Chemistry 3
• Chemistry 4
• Chemistry 5
• Chemistry 6
• Chemistry 7
• Computer Science 1
• Economics 1
• Design Technology 1
• Design Technology 2
• Environmental Systems and Societies 1
• Geography 1
• Geography 2
• Geography 3
• Geography 4
• Geography 5
• Geography 6
• Literature and Performance 1
• Mathematics 1
• Mathematics 2
• Mathematics 3
• Mathematics 4
• Mathematics 5
• Philosophy 1
• Philosophy 2
• Philosophy 3
• Philosophy 4
• Philosophy 5
• Psychology 1
• Psychology 2
• Psychology 3
• Psychology 4
• Psychology 5
• Social and Cultural Anthropology 1
• Social and Cultural Anthropology 2
• Social and Cultural Anthropology 3
• Sports, Exercise and Health Science 1
• Sports, Exercise and Health Science 2
• Visual Arts 1
• Visual Arts 2
• Visual Arts 3
• Visual Arts 4
• Visual Arts 5
• World Religion 1
• World Religion 2
• World Religion 3

What's Next?

Trying to figure out what extracurriculars you should do? Learn more about participating in the Science Olympiad , starting a club , doing volunteer work , and joining Student Government .

Studying for the SAT? Check out our expert study guide to the SAT . Taking the SAT in a month or so? Learn how to cram effectively for this important test .

Not sure where you want to go to college? Read our guide to finding your target school . Also, determine your target SAT score or target ACT score .

As an SAT/ACT tutor, Dora has guided many students to test prep success. She loves watching students succeed and is committed to helping you get there. Dora received a full-tuition merit based scholarship to University of Southern California. She graduated magna cum laude and scored in the 99th percentile on the ACT. She is also passionate about acting, writing, and photography.

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Research sampler 5: examples in learning mathematics, by annie and john selden.

Successful Math Majors Generate Their Own Examples Being Asked For Examples Can Be Disconcerting Generating Counterexamples That Are Explanatory "If I Don't Know What It Says, How Can I Find an Example of It?" Coda

Examining examples and non-examples can help students understand definitions. While a square may be defined as a quadrilateral with four equal sides and one right angle, seeing concrete examples of squares of various sizes, as well as considering rectangular non-examples, can help children clarify the notion of square. When we teach linear algebra and introduce the concept of subspace, we often provide examples and non-examples for students. We may point out that the polynomials of degree less than or equal to two form a subspace of the space of all polynomials, whereas the polynomials of degree two do not. Is the provision of such examples always desirable? Would it perhaps be better to ask undergraduate students to provide their own examples and non-examples? Would they be able to? Given a false conjecture, would students be able to come up with counterexamples? Several studies shed light on these questions.

Successful Math Majors Generate Their Own Examples

In upper-division courses like abstract algebra and real analysis, students often encounter a host of formal definitions, many new to them. After presenting a few examples and non-examples along with a few proofs of theorems, we hope they will use these definitions to tackle problems, examine conjectures, and construct their own proofs. Is this the best way to proceed? How do such students deal with new definitions?

To answer this question, Randall P. Dahlberg and David L. Housman of Allegheny College conducted an in-depth study of eleven undergraduate students - ten seniors and one junior. All but one, who was in computer science, were math majors. The students had successfully completed introductory real analysis and algebra, as well as courses in linear algebra and foundations and a seminar covering set theory and the foundations of analysis. In individually conducted audio-taped interviews, the authors presented the students with a written definition of a "fine function," which they had made up to see how the students would deal with a formally defined concept. A function was called fine if it had a root (zero) at each integer. When interviewed, students were first asked to study this definition for five to ten minutes, saying or writing as much as possible of what they were thinking, after which they were asked to generate examples and non-examples of "fine functions." Subsequently, they were given functions, such as

and asked to determine whether these were examples and, if so, why. Next, they were asked to determine the truth of four conjectures, such as "No polynomial is a fine function." Finally they were asked about their perceptions of the interview.

Four basic learning strategies were used by the students on being presented with this new definition - example generation, reformulation, decomposition and synthesis, and memorization. Examples generated included the constant zero function and a sinusoidal graph with integer x -intercepts. Reformulations included

Decomposition and synthesis included underlining parts of the definition and asking about the meaning of "root." Two students simply read the definition - they could not provide examples without interviewer help and were the ones who most often misinterpreted the definition. They found the interview quite different from their usual mathematics classes, where examples and explanations were provided.

Of these four strategies, example generation (together with reflection) elicited the most powerful "learning events," i.e., instances where the authors thought students made real progress in understanding the newly introduced concept. Students who initially employed example generation as their learning strategy came up with a variety of discontinuous, periodic continuous, and non-periodic continuous examples and were able to use these in their explanations. Those who employed memorization or decomposition and synthesis as their learning strategies often misinterpreted the definition, e.g., interpreting the phrase "root at each integer" to mean a fine function must vanish at each integer in its domain, but that need not include all integers. Students who employed reformulation as their learning strategy developed algorithms to decide whether functions given them were fine, but had difficulty providing counterexamples to false conjectures. [Cf. "Facilitating Learning Events Through Example Generation," Educ. Studies in Math. 33, 283-299, 1997.]

Finally, Dahlberg and Housman note the relative ineffectualness of their attempted interventions. One student agreed, after a question and answer period with the interviewer, that the zero function was indeed a fine function, but immediately switched her attention to other ideas, not returning until much later when, through self-discovery, she actually realized the zero function was a fine function. Dahlberg and Housman suggest it might be beneficial to introduce students to new concepts by having them generate their own examples or having them decide whether teacher-provided candidates are examples or non-examples, before providing students examples and explanations. However, some of their students were reluctant to engage in either example generation or usage -- a not uncommon phenomenon in such circumstances.

Being Asked For Examples Can Be Disconcerting

Coming up with examples requires different cognitive skills from carrying out algorithms - one needs to look at mathematical objects in terms of their properties. To be asked for an example, whether of a "fine function" or something else, can be disconcerting. Students have no prelearned algorithms to show the "correct way." This is what Orit Hazzan and Rina Zazkis, of the Technion - Israel Institute of Technology, found when they asked three groups of preservice elementary teachers to provide examples of (1) a 6-digit number divisible first by 9, then by 17, (2) a function whose value at x = 3 is -2, and (3) a sample space and an event that has probability 2/7 in that space. In addition, they asked the students to explain how they generated their examples and to provide five additional examples.

The students used a variety of approaches to generate examples, beginning with trial and error, e.g., some simply picked a number at random and checked whether it was divisible by 9. Others picked a number N , and upon dividing by 17 and getting a remainder of 2, would use N -2 for their next trial. Students often found constructing examples and making the necessary choices difficult, e.g., they inquired of the interviewers whether the elements of the sample space were to be numbers, letters, or other objects. Some students designed their own algorithms for generating functions, e.g., one focused on y = ax + b , plugged in (3, -2) to get -2 = a *3 + b , chose a = 2 and solved for b = -8, finally declaring her function to be y = 2 x - 8.

Interestingly, very few students produced "trivial examples," such as 170,000 for a 6-digit number divisible by 17 or y = -2 as their function. Hazzan and Zazkis conjecture that these examples might not be seen as prototypical - a function is expected to involve x and a 6-digit number is seen as having a wider variety of digits. There was also a strong tendency to (directly) check the correctness of examples, e.g., some students who had created a number divisible by 17 by choosing a multiplier and performing the multiplication, verified the correctness of their example by division. Quite a number of students had difficulty dealing with "degrees of freedom," e.g., in order to find a number divisible by 9, one student who knew the sum of the digits needed to be divisible by 9, first chose 18, noted that 8 and 2 make 10, then broke 8 into the sum of 4, 3, and 1, and declared that 82431 should be divisible by 9. When asked for another strategy, she suggested something very similar -- making the initial sum 27, instead of 18.

Constructing examples proved to be more difficult for these students than checking the divisibility of a number, calculating the value of a function, or finding the probability of an event. They were often uncertain how to proceed and were especially troubled by having to make choices in mathematics. The authors suggest that teachers at all levels assign more "give an example" problems. [Cf. "Constructing Knowledge by Constructing Examples for Mathematical Concepts," Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education , Vol. 4, 299-306, 1997]

Furthermore, when students are allowed to discuss mathematical ideas and propose conjectures in class, teachers need to be able to evaluate student-generated examples, as well as to be able to propose counterexamples for their students' consideration. Students quite often fail to see a single counterexample as disproving a conjecture. This can happen when a counterexample is perceived as "the only" one that exists, rather than being seen as generic, e.g., sometimes the square root of 2 is considered the only irrational or | x | is perceived as the only continuous, nondifferentiable function.

Generating Counterexamples That Are Explanatory

Perhaps not surprisingly, experienced secondary mathematics teachers are better at generating explanatory counterexamples than preservice teachers. Irit Peled, University of Haifa, and Orit Zaslavsky, the Technion, asked some of each to generate at least one counterexample for each of the two following unfamiliar, false geometry statements supposedly given by a secondary student. (1) Two rectangles, having congruent diagonals, are congruent. (2) Two parallelograms, having one congruent side and one congruent diagonal, are congruent. They were also asked to explain how they came up with their counterexamples. None generated more than one counterexample for each task.

Two groups participated in the study -- 38 inservice teachers, most of whom had more than five years of teaching experience and a B.Sc. in mathematics and 45 third year student-teachers who had completed several advanced undergraduate mathematics courses. For the first conjecture (Task 1), 97% of the inservice teachers gave adequate counterexamples, i.e., ones that refuted the claim, but only 53% of the student-teachers did so. For the second conjecture (Task 2), 76% of the teachers and 42% of the student-teachers gave adequate counterexamples.

The counterexamples were analyzed for their explanatory power as specific, semi-general, and general. A specific counterexample is one which contradicts the claim, but gives no indication as to how one might construct similar or related counterexamples. For example, for Task 1 one subject carefully drew two rectangles of different dimensions, but with congruent diagonals. A counterexample was called semi-general if it provided some idea how one might generate similar or related counterexamples, but did not tell "the whole story" or did not cover "the whole space" of counterexamples. For instance, on Task 1, one subject drew two rectangles with congruent diagonals, but the angle between the two diagonals of second rectangle was indicated as twice that of the first rectangle. (Here it should be noted that, while some conjectures might not lend themselves to the generation of numerous counterexamples, i.e., they might be correct except for a small number of special "pathological" cases, these two conjectures were chosen to be far from "almost correct.") A general counterexample provides insight as to why a conjecture is false and suggests a way to generate an entire counterexample space. In response to Task 1, one subject specified that the angle between the diagonals could be arbitrary, rather than merely double that of the first rectangle.

Both teachers and student-teachers produced counterexamples of all the above types, but the former produced more semi-general and general counterexamples (92% vs. 38% on Task 1, and 61% vs. 33% on Task 2). Both of these types were labeled explanatory by the authors. The difficulty in suggesting only a specific counterexample lies in its potential for misleading students, whereas the pedagogical value of explanatory counterexamples lies in their ability to provide insight into why a conjecture fails. The authors suggest that both prospective and in-service mathematics teachers could benefit from an analysis and discussion of the pedagogical aspects of counterexamples. [Cf. "Counter-Examples That (Only) Prove and Counter-Examples That (Also) Explain," Focus on Learning Problems in Mathematics 19 (3), 49-61, 1997.]

"If I Don't Know What It Says, How Can I Find an Example of It?"

This hypothetical quote, illustrates the chicken-and-egg quandary some students might typically face when encountering a formal definition, whether of "fine function" or quotient group. A definition asserts the existence of something having certain properties. However, the student has often never seen or considered such a thing. To give an example or non-example, he/she would need at least some understanding of the concept. But how can he/she obtain such understanding? A good, and possibly the best, way seems to be through an examination of examples. Thus, the student is faced with an epistemological dilemma: Mathematical definitions, by themselves, supply few (psychological) meanings. Meanings derive from properties. Properties, in turn, depend on definitions. [This is a paraphrase from Richard Noss' plenary address to the September 1996 Research in Collegiate Mathematics Education Conference, as reported in Focus 17(1), 1&3, February 1997.] For mathematicians, this does not seem to be a dilemma. We suspect they view definitions differently than students - this allows them to search for examples in order to gain understandings of formal definitions.

Not only does such circularity play a role in students' failure to construct examples, so does their limited knowledge of concepts involved in a formal definition. When Zaslavsky and Peled asked 67 preservice and 36 inservice secondary teachers to provide examples of binary operations which were commutative and nonassociative, their subjects had great difficulty. Only 33% of the experienced teachers and 4% of the third-year undergraduate students came up with complete, correct, and well-justified examples. Just 56% of the experienced teachers and 31% of the student teachers were able to provide any kind of example (correct or incorrect). Upon investigating why this might be so, the authors found their subjects' underlying mathematical knowledge was deficient. For example, one subject defined a * b = | a + b | and claimed this was nonassociative because | a + b | + | c | does not equal | a | + | b + c |. Another proposed the operation of subtraction claiming it was commutative because -2 - 3 = -3 - 2, rather than 3 - (-2). Yet another proposed the unary operation

and tried to check commutativity using

The authors suggest their subjects tended to conflate commutativity and associativity due to the way the "issue of order" is treated in schools. For example, when a child is asked to calculate 6 + 7 + 4, he/she is usually encourage to do it more efficiently as (6 + 4 ) + 7 and told "order doesn't matter." [Cf. "Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operations," JRME 27(1), 67-78, 1996.]

Dahlberg and Housman also noted that their undergraduate subjects had trouble with the underlying concepts, e.g., function and root, making it hard to generate examples and non-examples of "fine functions." One student identified "root" with "continuity," three others initially thought the graph of the zero function was a point, and one did not believe the zero function was periodic. In addition, most students' initially thought in terms of functions which were nonconstant polynomials or continuous.

Since success in mathematics, especially at the advanced undergraduate and graduate levels appears to be associated with the ability to generate examples and counterexamples, what is the best way to develop this ability? One suggestion, given above, is to ask students at all levels to "give me an example of . . . ". Granted the inherent epistemological difficulties of finding examples for oneself, are we, in a well-intentioned attempt to help students understand newly defined concepts, ultimately hobbling them, by providing them with predigested examples of our own? Are we inadvertently denying students the opportunity to learn to generate examples for themselves? Difficulties with the strikingly simple idea of "fine function" suggest some students may be excessively dependent upon explicit instruction. Another in-between suggestion, given above, is to provide students with a list of potential examples (or counterexamples) and ask them to decide whether they are indeed examples (or counterexamples) and why. Are there other ways we might help students become example generators? Finally, a tendency to generate examples is not the same as an ability to do so -- it would be interesting to know how each of these relates to understanding and doing mathematics.

Dummy View - NOT TO BE DELETED

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