math 103 contemporary math
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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? - PowerPoint PPT PresentationTRANSCRIPT
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
• 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
Outline of Video on PT
[Put on reserve in library!]
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= a2 /c + b2 /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.
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 <
3 1 180 60
4 2 360 90
5 3 540 108
6 4 720 120
n _______ _______ ________
Tangrams
• Tangrams.
• Virtual Tangram Puzzle
• More
• Tangram ActivityUse templates to cut out pieces from larger (blue) sheets? [#Partners=3.]
• Cutting and reassembling polygons.