Maths IA – Exploration Topics: 200 ideas for investigations.
The authors of the latest Pearson Mathematics SL and HL books – (which look really good) have come up with 200 ideas for students doing their maths explorations. These topics touch a really large cross-section of mathematics. A lot of these ideas would need the students to go away and research independently to find out about them – which in itself would be a nice activity. There’s also a lot of potential for integrating some of this content into maths lessons and ToK discussions. I’ve posted the full list here, and included some links and additional information about some of the topics – so visit the full page for more details. Here are just an edited selection below:
Algebra and number
1) Modular arithmetic
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
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) Odd 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.
Geometry
1) 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.
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.
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.
Statistics and modelling
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
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.
Games and game theory
1) The prisoner’s dilemma: The use of game theory in psychology and economics.
2) Sudoku
3) Gambler’s fallacy: A good chance to investigate misconceptions in probability and probabilities in gambling. Why does the house always win?
4) Poker and other card games
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
Topology and networks
1) Knots
2) Steiner problem
3) Chinese postman problem
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
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
If these are not enough have a look at the full 200 topics here.
