Math 103 Contemporary Math

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Math 103 Contemporary Math

• Thursday, January 20, 2005

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More Introduction to Course Organization

• Web Materials
• What is Visual Mathematics?
• Mathematics that studies topics related to visual
  experience. Geometry, Topology, Motion
• Visualization of mathematics that is not
  inherently visual. Visualizing Counting,
  numbers 

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Example Numbers...

• numeral a symbol for representing a number 
• such as V, 5, five, cinq, chamesh, cinco
• Number a form of universal language to describe
  anything/ physical things/ concepts related to
  measurement
• Frege distinguished numerals from numbers in the
  late 19th century.
• We can compare numbers... for instance we say" 3
  is less than 5"
• Is 3 smaller than 5?
• Numerals are symbols (visual or linguistic) that
  we use to represent numbers.
• We use numbers to measure (lengths) and put
  things in order (which was first).
• Another common visual representation of numbers
  uses the number line.
• ___.___.___.___.___.___.___.___
• 1     2     3     4     5     6     7
• Here the numerals are connected to points, so the
  points are considered to visualize the
  corresponding numbers. Use Wingeometry?
• We visualize equations that  give relations
  between numbers with graphs in the coordinate
  plane.3x 2y 6 is visualized by the graph of
  a line ... Use Wingeometry? or Winplot?

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Another example of "visual Math"

• My Name Martin Flashman
• How to start a letter to me  Hello ___
  ________________
• How many different openings are possible?

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Tree for counting

• We can visualize this problem with a "tree"
• This visualization allows us to count the
  possibilities easily...
• seeing there are 8 possibilities for each of 4
  title branches so that the total is 84 32
  possibilities.
• This is an example of a visualization used to
  understand and solve a problem that initially is
  not connected to anything visual .

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Miscellaneous Some topics we will study

• The film lists as a guide to the course topics.
• The color problem.
• The motion problem.
• The Sphere  and the Torus. Who first showed the
  earth was a sphere?

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Measurement and the Pythagorean Theorem (PT)

• Measuring angles, lengths and areas.
• Squares, rectangles, parallelograms and
  triangles.
• Dissections, cut and paste methods of
  measurement.
• Cutting and reassembling polygons.
• The Square Me Puzzle.
• Do Pythagorean Activity Sheet
• Virtual Manipulative for PT.
• Discuss Pythagorean Theorem and proofs.
• Over 30 proofs of the Pythagorean theorem!
• Many Java Applets that visualize proofs of the
  Pythagorean Theorem

a2 b2 c2
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Next Time?

• Show video on PT
• Puzzles and Polygons
• Flatland and the plane
• The triangle, quadrilateral, pentagon, and
  hexagon.
• More on measurements of angles and areas of
  polyons.
• Activity Measuring angles in regular polygons.