Thursday, January 20 (ppt)
- More Introduction to course organization.
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What is Visual Mathematics?
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Mathematics that studies topics related to visual experience. [Geometry, Topology, Motion]
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Visualization of mathematics that is not inherently visual. [Visualizing Counting]
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Example: Numbers...
- numeral: a symbol for representing a number
- Number: a form of universal language to describe
anything/ physical things/ concepts related to measurement
- such as V, 5, five, cinq, chamesh, cinco
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Frege distinguished numerals from numbers in the late 19th century.
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We can compare numbers... for instance we say" 3 is less than 5"
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Is 3 smaller than 5?
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Numerals are symbols (visual or linguistic) that we use to represent
numbers.
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We use numbers to measure (lengths) and put things in order (which was
first).
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Another common visual representaion of numbers uses the
number
line.
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___.___.___.___.___.___.___.______
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1
2 3
4
5 6 7
Here there 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":
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How to start a letter to me: Hello ___________
Professor | Martin |
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Doctor | Marty | Flashman |
Mister | Flash | omit |
Omit | Omit |
|
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How many different openings are possible?
- We can visualize this problem
with a "tree"
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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 .
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Miscellaneous: Some topics we will study.
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The film lists as a guide to the course topics.
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The color problem.
- The motion problem.
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The Sphere and the Torus.
Who first showed the earth was a sphere?
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Measurement and the Pythagorean Theorem (PT)
a2 + b2 = c2
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Measuring angles, lengths and areas.
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Squares, rectangles, parallelograms and triangles.
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Dissections, cut and paste methods of measurement.
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Cutting and reassembling polygons.
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The Square Me Puzzle.
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Do Pythagorean
Activity Sheet
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Show video on PT
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Puzzles and Polygons
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Flatland and the plane
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The triangle, quadrilateral, pentagon, and hexagon.
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More on measurements of angles and areas of polyons.
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Activity: Measuring angles in regular polygons.