Module 1 gmath

• Mathematics

  The development of all types of formal deductive reasoning
• Mathematics


  The science of calculation
• Mathematics


  The science of numbers and space
• Mathematics


  The science of measurement, quantity, and magnitude
• Mathematics


  A way to settle in the mind of children a habit of reasoning
• Mathematics


  A discipline investigating "formal structures"
• Mathematics


  The "science of orders"
• Mathematics


  The "science of order in progression"
• Mathematics


  A logical construct based on many axioms of either set theory or number theory
• Mathematics


  • Presented deductively at school
  • Well structured
  • Problems are algorithmically approached
• Mathematics


  The science of numbers and their operations, interrelations, combinations, generalizations, abstractions, and space configurations of their structure, measurement, transformations, and generalizations
• Mathematics


  A desire for a particular kind of knowing, that is self-contained on the individual or may be seen as autonomous thinking
• Mathematics


  • A study of patterns and relations
  • A way of thinking
• Mathematics


  • An art characterized by order and internal consistency
  • A language that uses carefully defined terms and symbols
  • A tool
• Essential characteristics of mathematics


  • Precision
  • Definition
  • Reasoning
  • Coherence
  • Purposefulness
• Precision


  Mathematical statements are clear and unambiguous
• Definition


  The bedrock of mathematical structure and the platform that supports reasoning
• Reasoning


  The lifeblood of mathematics, the engine that drives proving and problem-solving
• Coherence


  Concepts and skills are interwoven in mathematics
• Purposefulness


  Mathematics is goal-oriented, and for every concept or skill, there is a purpose for it
• Mathematics


  The science of order, patterns, structure, and logical relationships
• Mathematics


  The language of science
• Euclid said "The laws of nature are but the mathematical thoughts of God"
• Galileo said "Mathematics is the language in which God has written the Universe"
• Mathematics is everywhere, seen anywhere in the universe
• Formal system of thought for recognizing, classifying, and exploiting patterns allows systematizing and organizing ideas of patterns, discovering great secrets of nature's patterns
• Majority of our knowledge of mathematics and modern science is strictly based and supported by our environmental observations
• Geometry


  The branch of mathematics that describes shapes and establishes the relationships between them
• Polygons


  Figures with regular shapes
• Spatial patterns can be represented by a fairly small collection of fundamental geometrical shapes and relationships that have corresponding symbolic representations
• The human mind relies heavily on its perception of shapes and patterns to make sense of the world
• Real objects never perfectly match a geometric figure, but more or less approximate them
• Some ideas and terms of geometry


  • Points
  • Lines
  • Planes
  • Triangles
  • Rectangles
  • Squares
  • Circles
  • Ellipses
  • Rectangular solids
  • Spheres
  • Relationships of similarity and congruence
  • Relationships of convex, concave, intersecting, and tangent
  • Angles between lines or planes
  • Parallel and perpendicular relationships between lines and planes
  • Forms of symmetry such as displacement, reflection, and rotation
  • Pythagorean theorem
• Shape and measurement (magnitude) or scale can have important consequences for the performance of systems
• Changing the size of objects while keeping the same shape can have profound effects due to scaling geometry: Area varies as the square of linear dimensions, and volume varies as the cube
• Some natural phenomena (such as the shapes of clouds, mountains, and coastlines) seem to exhibit fractal-like patterns that look very similar at any scale
• Sphere


  • The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator
• Hexagons


  • The shape of a beehive
• Fractals


  • Look very similar to one another when observed at any scale
  • Basic components are similar to the whole
  • Can find similar shapes even when zoomed in
  • Involve a complex process that goes through an infinite number of iterations
• Fractals


  • Have self-similarity
  • Have fractional dimension
  • Formed by iteration