Math 103 Contemporary Math Tuesday, January 25, 2005

The Sphere,The Torus & Flatland. How can one distinguish the sphere from a plane (Flatland) based solely on experiences on the surface? How can one distinguish the sphere from a torus based solely on experiences on the surface? –Use shadows?: look at shadows at the same time of day? This is a "local" feature of the surface. –Observe "curvature"? This is also a local property. Circumnavigate (Global)?: Go West -> return from the East, then go North-> return from the south. –On a sphere: –On a torus: –On a plane: Other issues: What about strange gravity? Finding an edge? How do you know when you start?

Measurement and the Pythagorean Theorem (PT) Do Pythagorean Activity Sheet Pythagorean Activity Sheet Virtual Manipulative for PT. Discuss Pythagorean Theorem and proofs.Pythagorean Theorem and proofs –Over 30 proofs of the Pythagorean theorem!Over 30 proofs of the Pythagorean theorem! –Many Java Applets that visualize proofs of the Pythagorean TheoremMany Java Applets that visualize proofs of the Pythagorean Theorem a 2 + b 2 = c 2

Outline of Video on PT [Put on reserve in library!] Background: Similar triangles –Area of triangles = 1/2 bh –Area of parallelogram = bh –Scaling: a linear scale change of r gives area change of factor r 2. 3 questions: running, moat, wind power... Proof of the PT: Similar right triangles: c = a 2 /c + b 2 /c applications and other proofs. Prop. 47 of Euclid. Dissection Proof. Prop 31 Book VI Similar shapes. Simple proof of PT using similar triangles of the triangle. Use in 3 dimensional space.

Puzzles and Polygons Measuring angles, lengths and areas. Squares, rectangles : 90 degree/ right angle triangles : add to 180 degrees- straight angle [Illustrated physically and with wingeometry] parallelograms: opposite angles are congruent, sum of consecutive angles =180 degrees Dissections, cut and paste methods of measurement. Cutting and reassembling polygons. The "Square Me" Puzzle

The triangle, quadrilateral, pentagon, and hexagon. More on measurements of angles and areas of polygons. A quadrilateral can be made from two triangles... so the sum of its interior angles is 2 * 180 = 360. A pentagon can be made from 3 triangles... so the sum of its interior angles is 3* 180 = ___. If the hexagon has all angles congruent( of equal measurement) then each angle will be ___/5 = ___ degrees! A hexagon can be made from 4 triangles... so the sum of its interior angles is 4* 180 = ___. If the hexagon has all angles congruent( of equal measurement) then each angle will be ___/6 = ___ degrees!

Measuring angles in Polygons # of sides# of Triangles Sum of Interior <‘s If equal, Measure of a single < n_______ ________

Tangrams Tangrams.Tangrams Virtual Tangram Puzzle More Tangram Activity Use templates to cut out pieces from larger (blue) sheets? [#Partners=3.] Cutting and reassembling polygons.