MATM

• Patterns in nature


  • Stars moving in circles across the sky each day
  • Weather season cycle each year with snowflakes containing sixfold symmetry
  • Fish patterns like spotted trunkfish, spotted puffer, etc.
  • Animals covered in patterns like stripes, spots, and blotches
  • Natural patterns like waves across oceans, sand dunes, formation of typhoon, water drops with ripples
• Leonardo Pisano Bogollo lived between 1170 and 1250 in Italy
• Characteristics of patterns

  • Regular
  • Repeated
  • Recurring forms or designs
  • Identify relationships
  • Find logical connections to form generalizations
• Mathematics


  • A useful way to think about nature and our world
  • A tool to quantify, organize, and control our world, predict phenomena, and make life easier
• Patterns in nature

  Visible regularities of form found in the natural world and universe
• Mathematics

  The study of pattern and structure, fundamental to physical and biological sciences, engineering, information technology, economics, and social sciences
• Fibonacci Sequence
• Fibonacci helped spread Hindu-Arabic Numerals through Europe in place of Roman Numerals
• Mathematics in the modern world

  • Helps organize patterns and regularities, predict behavior of nature and phenomena, control nature and occurrences, has numerous applications making it indispensable
• Caveman development of patterns

  Through experience, cavemen developed patterns (organization, classification) to survive
• Nature patterns are vital clues to the rules that govern natural processes
• Fibonacci Number is an integer in the infinite sequence 1, 1, 2, 3, 5, 8, 13...
• The golden ratio is (A + B)/A = golden ratio (5)
• The golden ratio can be expressed as the ratio between two numbers
• Leonardo Fibonacci helped spread Hindu-Arabic Numerals through Europe in place of Roman Numerals
• Mathematics language is precise, concise, and powerful
• The Golden Ratio is also known as the golden section, golden mean, divine proportion, and more
• Rabbit population growth


  Problem: Newly-born pair of rabbits can mate at one month, leading to exponential growth
• Symbols commonly used in Mathematics

  • English Alphabet, Numerals, Greek Letters, Grouping Symbols, Special Symbols
• The Fibonacci sequence was discovered after an investigation on the reproduction of rabbits
• Liber Abaci is a famous book by Leonardo Fibonacci
• The Fibonacci Sequence displays unique mathematical properties useful in fields like astronomy, botany, and financial markets
• When taking two successive Fibonacci numbers, their ratio is close to the Golden Ratio "φ" which is approximately 1.618034
• Fibonacci Sequence rule

  Add the last two terms to get the next term
• Fibonacci


  Nickname meaning "Son of Bonacci"
• Understanding the language of Math helps in grasping mathematical ideas and logic clearly
• Language of Mathematics

  A system of conventional symbols used to express mathematical concepts
• Sets are one of the most fundamental concepts in mathematics
• Uses of numbers

  • Cardinal numbers (used for counting)
  • Ordinal numbers (used for telling positions)
  • Nominal numbers (used only as a name to identify something)
• Translation: English to Math language

  Choose a quantity to be represented by a variable, then write the mathematical expression for each number
• Difficulties in the math language include different meanings or uses of words in Math and English
• Representation of a null set

  • Curly brackets { }
  • Slashed zero 
• Sets in mathematics

  A collection of well-defined and distinct objects, considered as an object in its own right
• Translation examples

  • Six less than twice a number is forty-five
  • A number minus seven yields ten
  • A total of six and some number
  • Twelve added to a number
  • Eight times a number is forty-eight
  • The produce of fourteen and a number
  • Twice a number minus eight
  • The quotient of a number and seven is two
  • Three-fourths of a number
  • Eight less than a number is five
  • How many times does five go into twenty?
• Compound Proposition

  • Conveys two or more ideas
• Propositional Logic


  • Washington, D.C., is the capital of the United States of America
  • Toronto is the capital of Canada
  • 1+1 = 2
  • 2+2 = 3
• Proposition


  A statement that is either true or false, expressed with symbols P, Q, R, or p, q, r
• Ways to represent a null set

  • Curly brackets { }
  • Slashed zero 
• Conjecture


  An educated guess based on repeated observations of a particular process or pattern
• Problem Solving using Deductive Reasoning


  Solving problems by applying premises, syllogism, and conclusions