Maths IA – Maths Exploration?Topics

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Maths IA – Maths Exploration Topics

A list with over 300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework. Suitable for?Applications and Interpretations students (SL and HL) and also analysis and Approaches students (SL and HL).

Algebra and number

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?

IB Maths Super Exploration Guide

I’ve written a Super Exploration Guide (168 pages) to help you get excellent marks on your coursework. Get advice from an IB Examiner! Please note this is not an automatic download – but will be emailed to you the same day.

Geometry

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.

IB Maths Revision with?Revision Village

There’s a really?fantastic website?been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.?I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both analysis and Applications.

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!