Mathematics Internal Assessment (IA) is a crucial component of the IB Diploma Programme. It is a student-led exploration that enables the learner to delve deeper into a mathematical concept or problem of their interest. The Math IA is a 12-20 page report that showcases the student’s ability to conduct independent research, analyze data, apply mathematical concepts, and present their findings in a clear and coherent manner.

*These are the reasons why the IA is invaluable:*

1.Holistic assessment: The IA gives an opportunity to students to showcase their research skills, and is a more holistic assessment of their mathematical abilities.

2.Developement of critical skills: Problem-solving, logical reasoning, data analysis, and communication are crucial for academic and professional settings.

3.Application of Mathematics in real-world contexts: This allows students to have a more relatable experience with the abstract Mathematical knowledge they have gleaned in their coursework.

4.Development of research and analytical skills that can be used in the university and research environments.

5.A solid IA serves as a good indication to universities that a student is well-prepared to undertake independent study or research. Writing a Math IA can be daunting, but with proper planning and execution, it can be a rewarding experience that helps you understand the beauty and power of mathematics.

In this guide, we will discuss how to organize your Mathematics IA.

## IA Cover Page

It should have the following elements:

A)Title: The title of the Math IA should be clear, concise, and reflective of the exploration’s focus. It should give the reader an idea of what the IA is about.

B) IB Candidate Number: This is your unique identification number provided by the International Baccalaureate organization. It typically consists of three letters followed by three digits (e.g., ABC123). The candidate number helps identify your work during assessment and ensures anonymity.

C) Session: Indicate the examination session during which the IA was submitted. For example, it could be “May 2023” or “November 2023,” depending on the IB exam session.

D) 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) Course and Level: Specify whether the Math IA is for Standard Level (SL) or Higher Level (HL) Mathematics. This is crucial as the level of sophistication and depth of mathematical exploration may vary between SL and HL.

F) Date of Submission: Include the date when you submitted the Math IA. This helps track the timeline of your work.

G) School Name and Code: Mention the name of your school and the school code. This information helps identify the institution where the IA was completed.

H) Candidate Name: Include your full name on the cover page. This ensures that your IA is properly identified and credited to you.

I) Abstract (Optional): Some schools may require an abstract to be included on the cover page. An abstract is a concise summary of the IA, typically consisting of 100-150 words. It should provide an overview of the research question, the methods used, and the main findings.

J) Declaration of Authenticity (If Required): Some schools or examiners may ask you to sign a declaration of authenticity on the cover page, stating that the work is entirely your own and has not been plagiarized or copied from other sources.

## IA Important Rubric Requirements

(A) Mathematical Presentation (0-4)

In order to get the full 4 marks for this criterion, there must be a coherent structure, accompanied with clear explanations. A general guideline for this is to break the IA into:

### 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?** This is where you define the objective for your IA, and state what you want to achieve with this aim. 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.

Some examples of previous IA topics are listed below (these are basic topics and not finalised research questions) and in the appendix. However, remember that there ought to be some personal engagement within the topic-choosing process:

“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.

Then how: This means your plan of action. The level of difficulty must be appropriate for the your level (SL or HL).

### 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.

Note that the theory section should include not only the explanation of relevant theories but also the related background information. Likewise, ensure that the calculation section is comprehensive and includes all necessary formulas and mathematical steps.

### 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.”

**Bibliography**

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

**Mathematical communication **

- This rubric assess the student’s ability to organize his research and findings in a coherent manner. The fluency of the mathematical language used is assessed via the following ways:Notation
- Symbols (choose universally accepted ones)
- Terminology (All key terms and variables must be defined appropriately when first introduced)

**Personal Engagement**

A unique part of this IA is the personal engagement. This rubric assess a student’s ability to make the Mathematics his/her own – Infusing creativity in the exploration of ideas from multiple angles. The student must develop his or her own independent understanding of the topic he/she is exploring. For example, you could write in the first person why you have decided to pursue your interest in Fourier series. You could link it with your interest in understanding heat conduction. You discuss the challenges you encountered during your research, discuss how you felt, and emphasize the mathematical concepts you have learned during the investigative process.

**Use of Mathematics**

This rubric assesses how well a student uses the Mathematics in the exploration. Simplicity is encouraged, and the level of difficulty should be pegged at the level of the course. The Mathematics must be precise, and exhibiting a clear logical structure. 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. Students must show the mastery and use of complex concepts, be able to see the Mathematical problem from various viewpoints, and see the ways to link different areas of Mathematics.

E.g., use of mathematics carries a weighting of up to 6/20 while reflection carries a weighting of up to 3/20. Hence, you should expect to spend more pages on calculations than your reflection.

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 |

In conclusion, a well-structured layout is crucial for a successful Math IA, including an introduction, main body, analysis, and conclusion. Meeting the rubric requirements is also essential for a high grade, which includes criteria such as presentation, communication, and reflection. By following these guidelines, you can write an engaging and informative Math IA.

If in doubt, reach out to experienced tutors at Quintessential Education for extra help and guidance. Start your journey towards academic success today!