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Maths SL & HL

202 Exploration Ideas/Topics

Algebra & Number Theory

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

Pythagorean triples

Mersenne primes

Magic squares & cubes

Loci and complex numbers

Matrices and Cramer’s rule

Divisibility tests

Egyptian fractions

Complex numbers & transformations

Euler’s identity: 1 0ie

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

Statistics & 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

Regression to the mean

Modelling growth of computer power

Geometry

Non-Euclidean geometries

Cavalieri’s principle

Packing 2D and 3D shapes

Ptolemy’s theorem

Hexaflexagons

Heron’s formula

Geodesic domes

Proofs of Pythagorean theorem

Minimal surfaces & 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 & polyhedra

Pick’s theorem & 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

Biorhythms

Wi-Fi Triangulation

Calculus/Analysis & Functions

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 (min.time) problem

Second order differential equations

l’Hopital’s rule and evaluating limits

Hyperbolic functions

Maths SL & HL

202 Exploration Ideas/Topics

Algebra & Number Theory

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

Pythagorean triples

Mersenne primes

Magic squares & cubes

Loci and complex numbers

Matrices and Cramer’s rule

Divisibility tests

Egyptian fractions

Complex numbers & transformations

Euler’s identity: 1 0ie

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

Statistics & 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

Regression to the mean

Modelling growth of computer power

Geometry

Non-Euclidean geometries

Cavalieri’s principle

Packing 2D and 3D shapes

Ptolemy’s theorem

Hexaflexagons

Heron’s formula

Geodesic domes

Proofs of Pythagorean theorem

Minimal surfaces & 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 & polyhedra

Pick’s theorem & 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

Biorhythms

Wi-Fi Triangulation

Calculus/Analysis & Functions

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 (min.time) problem

Second order differential equations

l’Hopital’s rule and evaluating limits

Hyperbolic functions