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Kokeellinen matematiikan opetus. B. Zimmermann Jenan yliopisto Joensuu , 23.03.2010. Sisältö. Johdanto Kuutioiden tutkiminen Laskeminen Monikulmioluvut Avaruuden tutkiminen Pelit ja palapelit Escher- kolmiot ja muiden projektien kehittäminen Kahvitauko

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kokeellinen matematiikan opetus

Kokeellinenmatematiikanopetus

B. Zimmermann

Jenanyliopisto

Joensuu, 23.03.2010

sis lt
Sisältö
  • Johdanto
  • Kuutioidentutkiminen
    • Laskeminen
    • Monikulmioluvut
    • Avaruudentutkiminen
    • Pelitjapalapelit
    • Escher-kolmiotjamuidenprojektienkehittäminen
  • Kahvitauko
  • Muidenongelmakenttientutkiminen
    • Jatkoakuutioidentutkimiselle
    • Polyndromin (jamuidenkohteiden) tutkiminen
    • Paperintaittamisongelmajamuutongelmat
    • (Geometrinen) todennäköisyyslaskenta
    • Tulitikkuongelmat

Experimental Mathematics - Joensuu - Maaliskuu2010 - Bernd Zimmermann

cubes plugged and unplugged
Cubes – plugged and unplugged

The system:

Experimental Mathematics - Joensuu - maaliskuu 2010 - Bernd Zimmermann

investigations by cubes
Investigations by cubes
  • http://www.dicksystem.com/

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

how many
How many?

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

construct figures with the corresponding number of cubes
Construct figures with the corresponding number of cubes

5 7 8 13 14

(change of representation)

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

addition and partition
Addition and partition…

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

counting and patterns
Counting and patterns

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

laskentamenetelm termit laskentamenetelm
Laskentamenetelmä → termit → laskentamenetelmä
  • 3*3; 3*4; … 3*(n-1)
  • 3 + 3*2; 3 + 3*3; … 3 + 3*(n-2)
  • 3(n-1) = 3n – 3 ja

3 + 3(n-2) = 3 + 3n – 3*2 = 3n - 3

3n – 3

Muunnatermiä...

=

counting and patterns1
Counting and patterns

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

counting and patterns triangular numbers
Counting and patterns: triangular numbers

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

counting and patterns figured numbers
Counting and patterns: figured numbers
  • What about quadrilateral numbers?
  • ….

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

counting and patterns in space
Counting and patterns in space

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

counting and patterns in space pyramidal numbers
Counting and patterns in space: pyramidal numbers
  • What is the number of consecutive triangular numbers?

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

slide16

Counting cubes in cubes

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

further problems on cubes
Further problems on cubes

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

games and puzzles
Games and puzzles
  • Polyominos:

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

games and puzzles1
Games and puzzles

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

computer science unplugged sorting

?

Computer-scienceunplugged: Sorting

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

space excursions soma
Space excursions:Soma

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

space excursions soma ready
Space excursions:Soma - ready

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

escher impossible figures modeled by plugged cubes
Escher – impossible figures -modeled by plugged cubes

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

geometry tessellations
Geometry: Tessellations

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

tessellation of the plane with regular polygons
Tessellation of the plane with regular polygons
  • Try to tessellate the plane with many exemplars of regular polygons of a specific type (triangles, squares,…)
  • Which yield a complete covering of the plane?
  • Why?
  • Which polygons are not appropriate and why?

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

checking your tessellation by using polydron
Checking your tessellation by using polydron
  • http://www.polydron.co.uk/cgi-bin/index.cgi?page=/index&userid=49171106&currency=pounds

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

which regular polygons are appropriate to tessellate the plane
Which regular polygons are appropriate to tessellate the plane

Of course with squares and…

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

which regular polygons are appropriate to tessellate the plane1

Which regular polygons are appropriate to tessellate the plane

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

tessellation by an arbitrary triangle
Tessellationby an arbitrary triangle

Use a molding tool!!

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

creamagnetic platonic solids how many
Creamagnetic: Platonic solids – how many?
  • Strong magnets
  • http://www.creamagnetic-spiele.de/

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

euler s polyhedron theorem

Euler‘s polyhedron theorem

Relation between S; K; P??

geometric probability
Geometric Probability
  • Toss a coin on a grid…

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

geometric probability1
Geometric probability

Hits by random????

  • Circle: Throws (dart).

Zufallsversuch???

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

geometric probability liitu sota luokassa
Geometric probability: Liitu-sotaluokassa
  • Circle: picture of the blackboard; every pupil has 10 throws

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

taittamisongelma
Taittamisongelma
  • Ota A4-paperi. Taita yksi kerta keskeltä ja leikkaa kaikki nurkat pois.
  • Miltä paperi nyt näyttää? Millaisia struktuureja liität tilanteeseen?
  • Taita paperi uudelleen toisessa suunnassa ja palaa vaiheeseen 1. Älä avaa! Löytääksesi struktuureja, toista niin monta kertaa kuin haluat tai voit.

1. Taitto ja leikkaus

2. Taitto ja leikkaus

4. Taitto ja leikkaus

3. Taitto ja leikkaus

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

paperfolding1
Paperfolding
  • Rectangle of the same perimeter with maximal area (isoperimetric problem)
  • Doubling the square
  • Pentagon-folding

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

geoboard

1

Geoboard

B

I

A

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

how many matches counting and structuring
How many matches?Counting and structuring

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

further games to be invented
Further games to be invented…

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

references
References

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

slide45

Thank you for your attention!

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann

examples
Examples
  • Pythagoras in Sweden
  • Fermat-point in triangles
  • Fermat-tours in triangles

Experimental Mathematics - Helsinki - helmikuu 2009 - Bernd Zimmermann