Transcript Math 103 Contemporary Math - Humboldt State University

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...
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numeral: a symbol for representing a number
– such as V, 5, five, cinq, chamesh, cinco
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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"
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– Is
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
3
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Here the numerals are connected to points, so the points are considered to visualize
the corresponding numbers.
Use Wingeometry?
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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
Mister
Omit
Marty
Flash
Omit
Flashman
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.
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Squares, rectangles, parallelograms and triangles.
Dissections, cut and paste methods of measurement.
Cutting and reassembling polygons.
The Square Me Puzzle.
• Do Pythagorean Activity Sheet
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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?
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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.