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.