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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?
– 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!]
• 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.
• 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 Activity
Use templates to cut out pieces from
larger (blue) sheets?
[#Partners=3.]
• Cutting and reassembling polygons.
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