Math 103 Contemporary Math
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Transcript of Math 103 Contemporary Math
Math 103 Contemporary Math
Thursday, January 20, 2005
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]
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?
Another example of "visual Math": • My Name: Martin Flashman • How to start a letter to me: Hello ___
________________• How many different openings are possible?
Professor Martin
Doctor Marty Flashman
Mister Flash
Omit Omit Omit
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 8*4 = 32 possibilities.
• This is an example of a visualization used to understand and solve a problem that initially is not connected to anything visual .
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?
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
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.