Πλατάρος-γιάννης-μικρές-μαθηματικές-εργασίες-2-από-6...

Click here to load reader

  • date post

    16-Feb-2017
  • Category

    Education

  • view

    20
  • download

    1

Embed Size (px)

Transcript of Πλατάρος-γιάννης-μικρές-μαθηματικές-εργασίες-2-από-6...

  • .

    2015

    , ,

    . .

    2 6

  • .

    . 37, 242 00 , ..

    [email protected]

    ,

    .

    , Bolzano,

    Rolle, , ,

    ,

    ,

    . ,

    , .

    , ,

    ,

    ,

    .

    ,

    . Gagne (1970) [8] - ,

    , -

    - .

    ,

    .

    (2011) :

    mailto:[email protected]

  • ...

    ,

    [1]

    , ,

    ( ) ,

    ,

    ,

    . [7] .,

    .

    ,

    ( , ) (t), s(t), a(t)

    [t1,t2]

    ( )

    .

    s(t) ,

    (= )

    , ()

    , , .

    , . ,

    , (

    ) .

    . Bolzano

    )

    . ( )

  • ) .

    ,

    .

    .

    ,

    .

    - ,

    ,

    ,

    .

    ,

    !

    ,

    , .

    ,

    ,

    ,

    .

    Bolzano (

    ) .

    )

    y=0

    f(x) .

    y=c,

    1: - .Bolzano, .. .. , , 0.

    2: . olzano, y=0, y=c, f()f()

  • f()f()
  • 1,2 3,

    4 . ,

    .

    ,

    .

    , .

    .

    S ( )

    . ,

    : ,

    . [2],[3]

    , : ,

    ,

    .

    . ,

    :

    4: 0 , , , , , S. , f(t). , ( ) , , .

  • f(t), [0,] f(0)=0 f(T)=S g(t), [0,],

    g(0)=S>0, g(T)=0. h(t)=g(t)-f(t) [0,]

    >,

    . Bolzano, h(0)=g(0)-f(0)=S. h(T)=g(T)-

    f(T)=g(T)-S, ( 4) g(T)-S

  • . ,

    .

    ,

    . ,

    .

    ,

    .

    ) , =300Km,

    100Km/h 3h .

    100Km/h

    3h,

    100Km/h. ,

    100Km/h.

    100Km/h,

    3h 100Km/h .

    (t), S .

    () S/T ,

    (S/T)T=S. .

    (t). ,

    ,

    =ct, (t)=t

    S(t)= t2

    ,

    ,

    .

    , .

  • ,

    . . Rolle

    , ; ( . ) .Bolzano,

    f(x), y=0.

    y=c, y=x (.Brower) y=x, y=f(x);

    , -

    ,

    .

    .

    .

    (

    ) [6] ( )

    ,

    ,

    ,

    ,

    .

    [1] ( 1168/2011 -

    . 59614/2) http://edu.klimaka.gr/nomothesia/fek/1377-

    fek-1168-2011-programma-spoudes-algebra-gewmetria-a-lykeiou.html ( 31/8/2011)

    [2] http://neadiastasi.blogspot.com/2009/10/blog-post.html ( 31/8/2011)

    [3] http://dorigo.wordpress.com/2006/02/15/solution-to-the-ubiquitous-

    monk-problem/ ( 31/8/2011)

    http://edu.klimaka.gr/nomothesia/fek/1377-fek-1168-2011-programma-spoudes-algebra-gewmetria-a-lykeiou.htmlhttp://edu.klimaka.gr/nomothesia/fek/1377-fek-1168-2011-programma-spoudes-algebra-gewmetria-a-lykeiou.htmlhttp://neadiastasi.blogspot.com/2009/10/blog-post.htmlhttp://dorigo.wordpress.com/2006/02/15/solution-to-the-ubiquitous-monk-problem/http://dorigo.wordpress.com/2006/02/15/solution-to-the-ubiquitous-monk-problem/

  • [4] : http://users.uoi.gr/kammas/economic_policy_2.pdf ( 31/8/2011)

    [5]

    :

    http://www.math.uoa.gr/me/dipl/dipl_kyriakopoulos.pdf ( 10/9/2011)

    [6] .. -

    : http://www.alfavita.gr/artra/art13_12_08_1439.php ( 31/8/2011)

    [7] Janvier, C. (1987). Translation Processes in Mathematics Education. In

    C. Janvier (Ed.), Problems of Representation in the Teaching and Learning

    of Mathematics (pp. 27-32). Hillsdale, NJ: Lawrence Erlbaum.

    [8] Gagn, R. M. (1970). The conditions of learning. 2nd edition. New

    York: Holt, Rinehart and Winston.

    http://users.uoi.gr/kammas/economic_policy_2.pdfhttp://www.math.uoa.gr/me/dipl/dipl_kyriakopoulos.pdfhttp://www.alfavita.gr/artra/art13_12_08_1439.php

  • , .

    3

    . , 37, 24200 ,

    [email protected]

    Abstract: The investigation of simple mathematical statements in a traditional

    way, in many cases, it is practically impossible. There are now accessible

    educational math software tools to rapidly explore mathematical statements that

    can not be performed in the traditional way of working. At the same time, shifts

    the focus on the discovery of mathematical knowledge in a playful way.

    :

    , , .

    ,

    . ,

    ,

    .

    : ,

    :

    , . ,

    ( )

    .

    . 1

    mailto:[email protected]

  • ,

    ,

    .

    , Sketchpad,

    ,

    .

    ,

    :

    1. ,

    . (. 1)

    ,

    .

    2.

    ,

    ,

    . (.2)

    ( )

    .

    ,

    , , ,, ,

    .

    3.

    .

    4. =//2

    AB =

    2

    AB ,

    .

    5. ,

    ( )

    . 2

  • 6.

    , ,

    .

    ,

    -. ,

    ,

    ( )

    ,

    ( 3)

    7.

    ,

    .

    . ( 4)

    8.

    , ( 5)

    ,

    ,

    .

    ,

    , .

    . 3

    . 4

    . 5

  • . ,

    ,

    .

    9. 6 :

    , ,,

    , , ,

    : ,

    ,

    .

    ,

    ,

    .

    ,

    ,

    .

    .

    ,

    ( 7)

    : ,

    . 7

    . 8

    . 6

  • .

    .

    ( 8)

    ,

    .

    10.

    ,

    ,

    . (.

    9)

    11.

    ,

    . (.10)

    12.

    .2,

    ,

    ,

    .11

    ,

    , .

    ,

    .

    ,

    ,

    ,

    . 9

    . 10

    . 11

  • .

    ,

    , .

    ,

    - , .

    : )

    . ,

    ,

    ,

    . ,

    , .

    , ,

    - ,

    , , .

    (

    , , ,

    .

    .

    ( , , ,

    ..) .

    ) ,

    ,

    , (,

    , ) .

    - , ,

    1.

    , ,

    ( ) ,

    1 Google

    , , , , , ! , . Albert von Szent-Gyorgyi, 1893-1986, . , , .

  • , ,

    , ,

    , ,

    -

    . , ,

    , 2

    (.. ) .

    ) ,

    , ,

    ,

    , ,

    ,

    (, , )

    , .

    .

    ) ( ) (

    ) ,

    , ,

    ()

    ) ,

    ( ) ,

    .

    3,

    2 .

    ! . 3 ( 83.17-28) , ,

    ,

    .

    .

    .

  • .

    () ( )

    () , () , .

    ()

    , .

    ,

    ( )

    . , ,

    .

    ( Mathematica, MathDad, MAPL, ..)

    ) , ,

    ,

    . ,

    .

    ( , )

    . ,

    , ,

    .

    ,

    , ( ,

    ) ,

    , , ,

    . - , -

    ,

    ( )

    ,

    . ,

    , .

    ,

    ( , , , )

    (, , ,

    , . . ) /

    ( ,

  • , , )

    .

    .

    ..

    ( ) ..

    ,

    .

    - :

    1) http://www.math.uoc.gr/~pamfilos/gGallery/problems/Pedal.html

    .

    2) http://mathworld.wolfram.com/PedalTriangle.html

    Wolfrarm Mathworld

    3) http://jwilson.coe.uga.edu/emt669/Student.Folders/Matteson.September/p

    edal/pedal.html Georgia

    4)

    1

    . 24-

    26/4/2009

    5)

    , ,

    2

    - 23-24-25

    2010

    6) , .

    25

    .

    7) Sketchpad . 27

    ,19-21 2010

    http://www.math.uoc.gr/~pamfilos/gGallery/problems/Pedal.htm