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.