IA Cover Page

Your title page should include

    • Title
    • The IB Number (In the format “ABC123”)
    • Session (i.e. May 2021)

IA Important Rubric Requirements

Page Count

12-20 Pages in length with double spacing. The page length per subsection is not set, but one can imagine it should correspond to the marking rubric. E.g., use of mathematics carries a weighting of upto 6/20 while reflection carries a weighting of upto 3/20. Hence, you should expect to spend more pages on calculations than your reflection.

Personal Engagement

A unique part of this IA is the personal engagement.

Bibliography

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

IA Layout

Section 1: Introduction

Introduction – Why, what, then how.

Why?

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

What?

Picking a topic – specifically an aim – should be considered carefully and in conjunction with your tutor/teacher to ensure there is sufficient depth to your topic (as this depends on whether you’re taking SL or HL Math). Make your aim explicitly – this is important.

Note the difference between receiving a 6/6 for the use of mathematics rubric for HL/SL according to the IB:
SL – “Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Thorough knowledge and understanding are demonstrated.”
HL – “Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and demonstrates sophistication and rigour. Thorough knowledge and understanding are demonstrated.”

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

Some examples of previous IA topics are listed below:
“Why planes travel a curved route and not a seemingly direct route”
“Does the stock market’s returns warrant its variance?”
“Projectile motion”
“L’Hôptal’s rule and evaluating limits”
“Image rotations using rotational matrices”

How?

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

Section 2 (Body): Theory & Calculation

Theory

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

Calculation

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

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

Section 3: Reflection, Conclusion, and Bibliography of Mathematical Exploration

Conclusion

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

Reflection

Your reflection should occur throughout your IA; however, you may also include a separate section depending on the layout of your IA. Here’s what you should do:

Consider limitations and extensions of your conclusion.
Similarly consider strengths and weaknesses.
Relate the mathematics within the exploration to your personal knowledge (or personal engagement).
Raise future research questions.
The IB states your reflection must be “crucial, deciding or deeply insightful. It will often develop the exploration by addressing the mathematical results and their impact on the student’s understanding of the topic.”
Consider limitations and extensions of your conclusion.
Similarly consider strengths and weaknesses.
Relate the mathematics within the exploration to your personal knowledge (or personal engagement).
Raise future research questions.
The IB states your reflection must be “crucial, deciding or deeply insightful. It will often develop the exploration by addressing the mathematical results and their impact on the student’s understanding of the topic.”

Bibliography

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

Appendix

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

Topics to spur interest (note these topics are not developed but are included to stimulate your own more developed aims):

Algebra and number theory Geometry Calculus/analysis and functions
Modular arithmetic 

Goldbach’s conjecture 

Probabilistic number theory

Applications of complex numbers 

Diophantine equations 

Continued fractions

General solution of a cubic equation 

Applications of logarithms 

Polar equations

Patterns in Pascal’s triangle 

Finding prime numbers 

Random numbers

Mersenne primes

Magic squares and cubes

Loci and complex numbers 

Matrices and Cramer’s rule

Matrices and coding/decoding

Image rotations using rotational matrices 

Divisibility tests

Egyptian fractions

Complex numbers and transformations 

Euler’s identity: ei_ + 1 = 0

Chinese remainder theorem 

Fermat’s last theorem 

Natural logarithms of complex numbers

Twin primes problem 

Hypercomplex numbers 

Diophantine application: Cole numbers

Odd perfect numbers 

Euclidean algorithm for GCF 

Palindrome numbers

Factorable sets of integers of the form ak + b

Algebraic congruences 

Inequalities related to Fibonacci numbers

Combinatorics – art of counting 

Boolean algebra

Graphical representation of roots of complex numbers

Roots of unity 

Fermat’s little theorem 

Prime number sieves

Recurrence expressions for phi (golden ratio)

Physical, biological and social sciences

Radiocarbon dating 

Gravity, orbits and escape velocity 

Mathematical methods in economics

Mathematical modelling in Biology

Modelling in Geography: Gini’s coefficient

Modelling of average monthly temperature 

Biostatistics 

Genetics 

Crystallography

Computing centres of mass 

Elliptical orbits 

Logarithmic scales – decibel, Richter, etc.

Fibonacci sequence and spirals in nature

Predicting an eclipse 

Concepts of equilibrium in economics 

Mathematics of the ‘credit crunch’

Branching patterns of plants

Column buckling – Euler theory

Non-Euclidean geometries 

Cavalieri’s principle 

Packing 2D and 3D shapes

Ptolemy’s theorem 

Hexaflexagons 

Minimal surfaces and soap bubbles

Tesseract – a 4D cube 

Map projections 

Tiling the plane – tessellations

Penrose tiles 

Morley’s theorem

Cycloid curve

Symmetries of spider webs 

Fractal tilings 

Euler line of a triangle

Fermat point for polygons and polyhedra

Pick’s theorem and lattices

Properties of a regular pentagon

Conic sections 

Nine-point circle 

Geometry of the catenary curve

Regular polyhedra 

Euler’s formula for polyhedra 

Eratosthenes – measuring earth’s circumference

Stacking cannon balls 

Ceva’s theorem for triangles 

Constructing a cone from a circle

Conic sections as loci of points 

Consecutive integral triangles 

Area of an ellipse

Mandelbrot set and fractal shapes 

Curves of constant width 

Sierpinksi triangle

Squaring the circle

Polyominoes Reuleaux triangle

Architecture and trigonometry 

Spherical geometry 

Gyroid – a minimal surface

Geometric structure of the universe 

Rigid and non-rigid geometric structures

Tangrams

Numerical analysis

Linear programming 

Fixed-point iteration 

Methods of approximating π

Applications of iteration 

Numerical methods solving of equations

Numerical methods of Integration

Estimating size of large crowds

Generating the number e 

Descartes’ rule of signs 

Numerical Methods for solving differential equations

Mean value theorem Torricelli’s trumpet (Gabriel’s horn) 

Integrating to infinity

Applications of power series 

Newton’s law of cooling 

Fundamental theorem of calculus

Brachistochrone (minimum time) problem

Second order differential equations 

L’Hôpital’s rule and evaluating limits

Hyperbolic functions 

The harmonic series

Time series 

Torus – solid of revolution

Projectile motion 

Why e is base of natural logarithm function

Statistics and modelling

Traffic flow 

Logistic function and constrained growth

Modelling growth of tumours

Modelling epidemics/spread of a virus 

Modelling the shape of a bird’s egg 

Correlation coefficients

Central limit theorem 

Modelling change in record performances for a sport

Hypothesis testing

Modelling radioactive decay 

Least squares regression 

Modelling the carrying

Modelling the carrying capacity of the earth

Regression to the mean 

Modelling growth of computer power past few decades

Modelling of time series 

Probability and probability distributions

Monte Carlo simulations 

Random walks

Insurance and calculating risks 

Poisson distribution and queues 

Determination of π by probability

Bayes’ theorem 

Normal distribution and natural phenomena

Medical tests and probability 

Probability and expectation

Logic and sets

Codes and ciphers 

Set theory and different ‘size’ infinities 

Mathematical induction (strong)

Proof by contradiction 

Zeno’s paradox of Achilles and the tortoise