docJak pokonać trudności z matematyką w szkole ponadgimnazjalnej
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Jak pokonać trudności z matematyką w szkole ponadgimnazjalnej
ŶŶĂWųŽŷƐŬĂ :ĂŬƉŽŬŽŶĂđƚƌƵĚŶŽƑĐŝnjŵĂƚĞŵĂƚLJŬČ w szkole ponadgimnazjalnej <ĂƌƚLJƉƌĂĐLJĚůĂƵĐnjŶŝſǁ OPOLE Wydawnictwo NOWIK Sp.j. 2016 1 ΞCopyright by Wydawnictwo Nowik Sp.j. 2016 Wydawnictwo Nowik Sp.j. 45-061 Opole, ul. Katowicka 39/104 Wydanie pierwsze, Opole 2016 ISBN: 978-83-62687-77-0 REDAKCJA: Wydawnictwo Nowik Sp.j. ^<B/BDE/͗Jolanta Brodziak WZK:<dK<B</͗Tomasz Fronckiewicz tƐnjĞůŬŝĞƉƌĂǁĂnjĂƐƚƌnjĞǏŽŶĞ͘ ZŽnjƉŽǁƐnjĞĐŚŶŝĂŶŝĞďĞnjnjŐŽĚLJtLJĚĂǁĐLJĐĂųŽƑĐŝƉƵďůŝŬĂĐũŝůƵďũĞũĨƌĂŐŵĞŶƚſǁ w jakiejkolwiek postaci jest zabronione. <ŽƉŝŽǁĂŶŝĞŵĞƚŽĚČŬƐĞƌŽŐƌĂĨŝĐnjŶČ͕ĨŽƚŽŐƌĂĨŝĐnjŶČ͕ƵŵŝĞƐnjĐnjĂŶŝĞ ŶĂŶŽƑŶŝŬĂĐŚŵĂŐŶĞƚLJĐnjŶLJĐŚŝŽƉƚLJĐnjŶLJĐŚŝŝŶŶLJĐŚŶĂƌƵƐnjĂƉƌĂǁĂĂƵƚŽƌƐŬŝĞŶŝŶŝejszej publikacji. <ƐĞƌŽǁĂŶŝĞnjĂďŝũĂŬƐŝČǏŬŝ͊ ^njĂŶŽǁŶLJnjLJƚĞůŶŝŬƵ͕ũĞǏĞůŝĐŚĐĞƐnjǁLJƌĂnjŝđƐǁŽũČŽƉŝŶŝħŶĂƚĞŵĂƚƚĞũƉƵďůŝŬĂĐũŝ͕ƉƌŽƐŝŵLJ o kontakt mailowy [email protected] ůƵďǁLJƉĞųŶŝĞŶŝĞĨŽƌŵƵůĂƌnjĂŶĂŶĂƐnjĞũƐƚƌonie www.nowik.com.pl Wydrukowane w Polsce ^njĐnjĞŐſųŽǁĞŝŶĨŽƌŵĂĐũĞŽŶĂƐnjLJĐŚƉƵďůŝŬĂĐũĂĐŚŶĂǁǁǁ͘ŶŽǁŝŬ͘ĐŽŵ͘Ɖů Dystrybucja: Wydawnictwo Nowik Sp.j. Biuro Handlowe: 45-061 Opole, ul. Katowicka 39/104 Tel./fax 77 454 36 04 http://www.nowik.com.pl e-mail: [email protected] 2 ^ƉŝƐƚƌĞƑĐŝ tƐƚħƉ ....................................................................................................................................... 5 KƉŝƐŬĂƌƚƉƌĂĐLJƉŽĚŬČƚĞŵŝĐŚƉƌnjLJĚĂƚŶŽƑĐŝĚŽƉƌĂĐLJnjƵĐnjŶŝĞŵ ŽƐƉĞĐLJĨŝĐnjŶLJĐŚƉƌŽďůĞŵĂĐŚǁƵĐnjĞŶŝƵƐŝħŵĂƚĞŵĂƚLJŬŝ ........................................................... 9 Karta pracy 1. Zbiory liczbowe .................................................................................................. 13 Karta pracy 2. WƌnjĞĚnjŝĂųLJůŝĐnjďŽǁĞ ............................................................................................ 18 Karta pracy 3. Obliczenia procentowe ...................................................................................... 22 Karta pracy 4. tnjŽƌLJƐŬƌſĐŽŶĞŐŽŵŶŽǏĞŶŝĂ ............................................................................ 30 Karta pracy 5. njŝĂųĂŶŝĂŶĂƉŝĞƌǁŝĂƐƚŬĂĐŚ ................................................................................ 32 Karta pracy 6. hƐƵǁĂŶŝĞŶŝĞǁLJŵŝĞƌŶŽƑĐŝnjŵŝĂŶŽǁŶŝŬĂ .......................................................... 39 Karta pracy 7͘njŝĂųĂŶŝĂŶĂƉŽƚħŐĂĐŚ........................................................................................ 41 Karta pracy 8͘njŝĂųĂŶŝĂŶĂůŽŐĂƌLJƚŵĂĐŚ................................................................................... 46 ௫ Karta pracy 9. tLJŬƌĞƐŝǁųĂƐŶŽƑĐŝĨƵŶŬĐũŝ ݕൌ ܽ .................................................................... 52 Karta pracy 10. tĂƌƚŽƑđďĞnjǁnjŐůħĚŶĂůŝĐnjďLJƌnjĞĐnjLJǁŝƐƚĞũ ....................................................... 58 Karta pracy 11. WƌnjLJďůŝǏĞŶŝĂ ...................................................................................................... 62 Karta pracy 12. tLJŬƌĞƐŝǁųĂƐŶŽƑĐŝĨƵŶŬĐũŝůŝŶŝŽǁĞũ ................................................................. 65 Karta pracy 13. Miejsce zerowe funkcji liniowej ....................................................................... 72 Karta pracy 14͘WƌŽƐƚĞƌſǁŶŽůĞŐųĞ ............................................................................................ 74 Karta pracy 15. WƌŽƐƚĞƉƌŽƐƚŽƉĂĚųĞ .......................................................................................... 76 Karta pracy 16͘ZſǁŶĂŶŝĞůŝŶŝŽǁĞ ............................................................................................ 78 Karta pracy 17. ĂĚĂŶŝĂƚĞŬƐƚŽǁĞƐƉƌŽǁĂĚnjĂůŶĞĚŽƌſǁŶĂŷůŝŶŝŽǁLJĐŚ ................................... 81 Karta pracy 18. EŝĞƌſǁŶŽƑđůŝŶŝŽǁĂ ......................................................................................... 84 Karta pracy 19. hŬųĂĚLJƌſǁŶĂŷůŝŶŝŽǁLJĐŚ ................................................................................ 87 Karta pracy 20. tLJŬƌĞƐŝǁųĂƐŶŽƑĐŝĨƵŶŬĐũŝ ݕൌ ܽ ݔଶ ................................................................ 95 ଶ ଶ Karta pracy 21͘tLJŬƌĞƐŝǁųĂƐŶŽƑĐŝĨƵŶŬĐũŝ ݕൌ ܽ ݔ ݍǡ ݕൌ ܽሺ ݔെ ሻ ............................. 99 Karta pracy 22. Miejsca zerowe funkcji kwadratowej .............................................................. 105 Karta pracy 23. WŽƐƚĂđŝůŽĐnjLJŶŽǁĂĨƵŶŬĐũŝŬǁĂĚƌĂƚŽǁĞũ .......................................................... 108 Karta pracy 24. WŽƐƚĂđŬĂŶŽŶŝĐnjŶĂĨƵŶŬĐũŝŬǁĂĚƌĂƚŽǁĞũ .......................................................... 111 Karta pracy 25. ZſǁŶĂŶŝĞŬǁĂĚƌĂƚŽǁĞ .................................................................................... 114 Karta pracy 26. EŝĞƌſǁŶŽƑđŬǁĂĚƌĂƚŽǁĂ ................................................................................. 118 Karta pracy 27. ZſǁŶĂŶŝĞƐƚŽƉŶŝĂǁŝħŬƐnjĞŐŽŶŝǏ2 .................................................................. 123 Karta pracy 28. ZſǁŶĂŶŝĞǁLJŵŝĞƌŶĞ ........................................................................................ 128 Karta pracy 29. tLJŬƌĞƐŝǁųĂƐŶŽƑĐŝĨƵŶŬĐũŝ ݕൌ ௫ ..................................................................... 131 Karta pracy 30. ^ƉŽƐŽďLJŽƉŝƐƵĐŝČŐƵ ........................................................................................ 137 Karta pracy 31. WŽũħĐŝĞŝǁnjſƌŽŐſůŶLJĐŝČŐƵĂƌLJƚŵĞƚLJĐnjŶĞŐŽ ................................................... 142 3 Karta pracy 32. ^ƵŵĂƉŽĐnjČƚŬŽǁLJĐŚǁLJƌĂnjſǁĐŝČŐƵĂƌLJƚŵĞƚLJĐnjŶĞŐŽ .................................... 147 Karta pracy 33. WŽũħĐŝĞŝǁnjſƌŽŐſůŶLJĐŝČŐƵŐĞŽŵĞƚƌLJĐnjŶĞŐŽ .................................................. 151 Karta pracy 34. ^ƵŵĂƉŽĐnjČƚŬŽǁLJĐŚǁLJƌĂnjſǁĐŝČŐƵŐĞŽŵĞƚƌLJĐnjŶĞŐŽ .................................... 156 Karta pracy 35. ĞĨŝŶŝĐũĞĨƵŶŬĐũŝƚƌLJŐŽŶŽŵĞƚƌLJĐnjŶLJĐŚŬČƚĂŽƐƚƌĞŐŽ ........................................ 160 Karta pracy 36. ĞĨŝŶŝĐũĞĨƵŶŬĐũŝƚƌLJŐŽŶŽŵĞƚƌLJĐnjŶLJĐŚŬČƚĂƌŽnjǁĂƌƚĞŐŽ .................................. 166 Karta pracy 37͘WŽĚƐƚĂǁŽǁĞƚŽǏƐĂŵŽƑĐŝŝǁnjŽƌLJƚƌLJŐŽŶŽŵĞƚƌLJĐnjŶĞ ...................................... 170 Karta pracy 38. KĚůĞŐųŽƑđŵŝħĚnjLJƉƵŶŬƚĂŵŝǁƵŬųĂĚnjŝĞǁƐƉſųƌnjħĚŶLJĐŚ͘ _ƌŽĚĞŬŽĚĐŝŶŬĂ................................................................................................. 173 Karta pracy 39. ZſǁŶĂŶŝĞƉƌŽƐƚĞũ ............................................................................................. 178 Karta pracy 40. WŽůĞƚƌſũŬČƚĂ .................................................................................................... 184 Karta pracy 41. WŽůĞƚƌſũŬČƚĂƌſǁŶŽďŽĐnjŶĞŐŽ ......................................................................... 189 Karta pracy 42. KŬƌČŐǁƉŝƐĂŶLJǁƚƌſũŬČƚ .................................................................................. 192 Karta pracy 43. KŬƌČŐŽƉŝƐĂŶLJŶĂƚƌſũŬČĐŝĞ .............................................................................. 195 Karta pracy 44. Pole kwadratu .................................................................................................. 198 Karta pracy 45. WŽůĞƌſǁŶŽůĞŐųŽďŽŬƵ....................................................................................... 202 Karta pracy 46. Pole rombu ...................................................................................................... 207 Karta pracy 47. Pole trapezu ..................................................................................................... 212 Karta pracy 48. <ČƚLJǁŽŬƌħŐƵ .................................................................................................. 215 Karta pracy 49. <ČƚLJǁŝĞůŽŬČƚĂĨŽƌĞŵŶĞŐŽ............................................................................... 218 Karta pracy 50. WŽůĂǁŝĞůŽŬČƚſǁƉŽĚŽďŶLJĐŚ ........................................................................... 221 Karta pracy 51. Kombinatoryka ................................................................................................ 225 Karta pracy 52. WƌnjĞƐƚƌnjĞŷnjĚĂƌnjĞŷĞůĞŵĞŶƚĂƌŶLJĐŚ͘ĚĂƌnjĞŶŝĂůŽƐŽǁĞ ................................... 228 Karta pracy 53. <ůĂƐLJĐnjŶĂĚĞĨŝŶŝĐũĂƉƌĂǁĚŽƉŽĚŽďŝĞŷƐƚǁĂ ...................................................... 231 Karta pracy 54. _ƌĞĚŶŝĂĂƌLJƚŵĞƚLJĐnjŶĂ ...................................................................................... 235 Karta pracy 55. Mediana ........................................................................................................... 240 Karta pracy 56. _ƌĞĚŶŝĂǁĂǏŽŶĂ................................................................................................ 243 Karta pracy 57. Wariancja i odchylenie standardowe .............................................................. 246 Karta pracy 58. ^njĞƑĐŝĂŶ............................................................................................................ 249 Karta pracy 59. WƌŽƐƚŽƉĂĚųŽƑĐŝĂŶ ............................................................................................. 251 Karta pracy 60. 'ƌĂŶŝĂƐƚŽƐųƵƉLJƉƌŽƐƚĞƚƌſũŬČƚŶĞ ...................................................................... 258 Karta pracy 61. 'ƌĂŶŝĂƐƚŽƐųƵƉLJƉƌŽƐƚĞĐnjǁŽƌŽŬČƚŶĞ ................................................................ 262 Karta pracy 62. njǁŽƌŽƑĐŝan foremny ...................................................................................... 266 Karta pracy 63. KƐƚƌŽƐųƵƉLJƉƌŽƐƚĞƚƌſũŬČƚŶĞ ............................................................................ 270 Karta pracy 64. KƐƚƌŽƐųƵƉLJƉƌŽƐƚĞĐnjǁŽƌŽŬČƚŶĞ ...................................................................... 274 Karta pracy 65. Walec ............................................................................................................... 278 Karta pracy 66. ^ƚŽǏĞŬ .............................................................................................................. 281 Karta pracy 67. Kula .................................................................................................................. 284 4 tƐnjLJƐƚŬŽŶĂůĞǏLJƵƉƌĂƐnjĐnjĂđũĂŬƚLJůŬŽŵŽǏŶĂ͕ ale nie bardziej Albert Einstein tƐƚħƉ WƌŽĐĞƐĚLJĚĂŬƚLJĐnjŶLJǁƐnjŬŽųĂĐŚŽďĞũŵƵũĞǁƐnjLJƐƚŬŝĐŚƵĐnjŶŝſǁ͕ƚĂŬǏĞƚLJĐŚ͕ŬƚſƌnjLJnjƚƌƵĚĞŵƌaĚnjČƐŽďŝĞnjŵĂƚĞŵĂƚLJŬČ͘njħƐƚŽƐČƚŽƵĐnjŶŝŽǁŝĞ͕ŬƚſƌnjLJĚŽƚĂƌůŝũƵǏĚŽĐnjǁĂƌƚĞŐŽĞƚĂƉƵĞĚƵŬĂĐLJũŶĞŐŽ͕ ĂůĞŵĂũČnjĂůĞŐųŽƑĐŝnjƉŽƉƌnjĞĚŶŝĐŚĞƚĂƉſǁďČĚǍnjĂŬŽƌnjĞŶŝųƐŝħǁŶŝĐŚůħŬƉƌnjĞĚŵĂƚĞŵĂƚLJŬČŝŶŝĞĐŚħđ ĚŽƚĞŐŽƉƌnjĞĚŵŝŽƚƵ͘WƌnjLJĐnjLJŶČŝĐŚŶŝĞƉŽǁŽĚnjĞŷǁŶĂƵĐĞŵŽǏĞďLJđůĞŶŝƐƚǁŽ͕ŶŝĞƐŬƵƚĞĐnjŶŽƑđĚnjŝĂųĂŷ ĚLJĚĂŬƚLJĐnjŶLJĐŚŝǁLJĐŚŽǁĂǁĐnjLJĐŚƐnjŬŽųLJĂůďŽnjĂŶŝĞĚďĂŶŝĂƌŽĚnjŝŶLJŝƑƌŽĚŽǁŝƐŬĂ͘ĞďLJŽǁĞŶŝĞƉŽǁodnjĞŶŝĂǁLJĞůŝŵŝŶŽǁĂđ͕ĐnjĂƐĂŵŝǁLJƐƚĂƌĐnjLJƚĂŬŝĐŚƵĐnjŶŝſǁnjŵŽƚLJǁŽǁĂđĚŽŶĂƵŬŝ͕ŶĂŬųŽŶŝđĚŽƐLJƐƚĞŵaƚLJĐnjŶĞũƉƌĂĐLJ͕ƉŽĚǁLJǏƐnjLJđŝĐŚƐĂŵŽŽĐĞŶħĐnjLJnjĂŝŶƚĞƌĞƐŽǁĂđŵĂƚĞŵĂƚLJŬČ͘njħƑĐŝĞũũĞĚŶĂŬǁŝĞůƵƐƉoƑƌſĚŶŝĐŚǁLJŵĂŐĂƐƉĞĐũĂůŶĞŐŽƉŽĚĞũƑĐŝĂǁƉƌŽĐĞƐŝĞĚLJĚĂŬƚLJĐnjŶLJŵ͕ƵnjĂƐadnionego w opiniach i orzeczeniach poradni psychologiczno-pedaŐŽŐŝĐnjŶLJĐŚƌſǏŶŽƌŽĚŶLJŵŝŽŐƌĂŶŝĐnjĞŶŝĂŵŝŝnjĂďƵƌnjĞŶŝĂƌŽnjǁoũŽǁLJŵŝ͘ůĂƚĞŐŽŵĂƚĞƌŝĂųnjĞďƌĂŶLJǁŶŝŶŝĞũƐnjĞũŬƐŝČǏĐĞƉƌnjĞnjŶĂĐnjŽŶLJũĞƐƚŶŝĞƚLJůŬŽĚůĂƵĐnjŶŝſǁ͕ƵŬƚſͲ ƌLJĐŚƚƌƵĚŶŽƑĐŝǁƵĐnjĞŶŝƵƐŝħŵĂƚĞŵĂƚLJŬŝƐČƐŬƵƚŬŝĞŵƌſǏŶĞŐŽƌŽĚnjĂũƵnjĂŶŝĞĚďĂŷ͕ĂůĞƚĂŬǏĞĚůĂƚLJĐŚ͕ u ŬƚſƌLJĐŚƚƌƵĚŶŽƑĐŝƚĞǁLJŶŝŬĂũČnjnjĂďƵƌnjĞŷƌŽnjǁŽũŽǁLJĐŚŽƌſǏŶLJŵĐŚĂƌĂŬƚĞƌnjĞŝŶĂƐŝůĞŶŝƵ͘ tƑƌſĚ njĂďƵƌnjĞŷ ƵƚƌƵĚŶŝĂũČĐLJĐŚ ƵĐnjŶŝŽŵ ŽĚŶĂůĞnjŝĞŶŝĞ Ɛŝħ ǁ ƐnjŬŽůĞ ŐųſǁŶĞ ŵŝĞũƐĐĞ njĂũŵƵũĞ autyzm wraz ze sƚĂŶŽǁŝČĐLJŵũĞŐŽųĂŐŽĚŶŝĞũƐnjČƉŽƐƚĂđnjĞƐƉŽųĞŵƐƉĞƌŐĞƌĂ. ĞǁnjŐůħĚƵŶĂƉƌŽďůĞŵLJ ǁƉŽƌŽnjƵŵŝĞǁĂŶŝƵƐŝħnjŽƚŽĐnjĞŶŝĞŵ͕ƐƚŽƉŝĞŷƌŽnjǁŽũƵƐƉŽųĞĐnjŶĞŐŽ͕ŽĚŵŝĞŶŶLJƐƉŽƐſďƉƌnjĞƚǁĂƌnjĂŶŝĂ ŝŶĨŽƌŵĂĐũŝŽƌĂnjƉŽƐƚƌnjĞŐĂŶŝĂƑǁŝĂƚĂ͕ĂƚĂŬǏĞƉƌŽďůĞŵLJnjĂďƐƚƌĂŬĐLJũŶLJŵŵLJƑůĞŶŝĞŵĚnjŝĞĐŝŝŵųŽĚnjŝĞǏ z ĂƵƚLJnjŵĞŵƐƚĂŶŽǁŝČǁŝĞůŬŝĞ ǁLJnjǁĂŶŝĞ ĚůĂƐLJƐƚĞŵƵĞĚƵŬĂĐũŝ͘EŝĞƐƚĞƚLJǁ ŶĂƐnjLJŵŬƌĂũƵŶŝĞƉŽǁƐƚĂų ĚŽƚĞũƉŽƌLJũĞĚŶŽůŝƚLJƐLJƐƚĞŵnjĂƉĞǁŶŝĂũČĐLJǁųĂƑĐŝǁĞǁƐƉĂƌĐŝĞĚůĂƵĐnjŶŝſǁnjnjĂďƵƌnjĞŶŝĂŵŝĂƵƚLJƐƚLJĐzŶLJŵŝ͘ŝČŐůĞďƌĂŬƵũĞ ĚůĂŶŝĐŚƉƌŽŐƌĂŵſǁ ĞĚƵŬĂĐLJũŶLJch, a njĞƐƚĂǁŵĂƚĞƌŝĂųſǁ ĚLJĚĂŬƚLJĐnjŶLJĐŚĚŽƐƚoƐŽǁĂŶLJĐŚĚŽƐƉĞĐLJĨŝĐnjŶLJĐŚƉŽƚƌnjĞďƚĂŬŝĐŚƵĐnjŶŝſǁũĞƐƚĚŽƑđƵďŽŐŝ;ĚŽƚLJĐnjLJƚŽnjǁųĂƐnjĐnjĂĐnjǁĂƌƚĞŐŽ etapu edukacyjnego). ƌĂŬƵũĞ ƚĞǏ ǁLJŬǁĂůŝĨŝŬŽǁĂŶĞũ ŬĂĚƌLJ ŶĂƵĐnjLJĐŝĞůƐŬŝĞũ ŵĂũČĐĞũ ĚŽƑǁŝĂĚĐnjĞŶŝĞ w pracy z uczniem autyƐƚLJĐnjŶLJŵ͘ KƉƌſĐnj ƚĞŐŽ ĚŽ ƉƌnjLJũħĐŝĂ ƵĐnjŶŝſǁ nj ƚĂŬŝŵŝ njĂďƵƌnjĞŶŝĂŵŝ ŶŝĞ ƐČ ƉƌnjLJŐŽƚŽǁĂŶĞƐnjŬŽųLJƉŽŶĂĚŐŝŵŶĂnjũĂůŶĞ͕ƉĂŶƵũĞǁŶŝĐŚďŽǁŝĞŵƉƌnjĞƉĞųŶŝĞŶŝĞŝŶŝĞŵĂƚĂŵŬůĂƐŝŶƚegracyjnych. AƵƚLJnjŵǁďƌĞǁƉŽǁƐnjĞĐŚŶĞŵƵƉƌnjĞŬŽŶĂŶŝƵŶŝĞnjĂǁƐnjĞǁLJƐƚħƉƵũĞnjƵƉŽƑůĞĚnjĞŶŝĞŵƵŵyƐųŽǁLJŵ͘dŽƚĞǏƵĐnjŶŝŽǁŝĞĚŽƚŬŶŝħĐŝĂƵƚLJnjŵĞŵŶŝĞŵƵƐnjČďLJđŶŝĞƐƉƌĂǁŶŝŝŶƚĞůĞŬƚƵĂůŶŝĞ;ũĞƐƚŝĐŚϰϬйͿ͘ njħƐƚŽ ĐŚĂƌĂŬƚĞƌLJnjƵũČ ƐŝħŽŶŝ ĚƵǏLJŵŝŵŽǏůŝǁŽƑĐŝĂŵŝ ƉŽnjŶĂǁĐnjLJŵŝ ŝ ďĂƌĚnjŽ ĚŽďƌČ ƉĂŵŝħĐŝČŵĞĐŚaŶŝĐnjŶČ͘njħƑđnj ŶŝĐŚ;ŽŬŽųŽϭϬйͿǁLJƌſǏŶŝĂƐŝħǁLJďŝƚŶLJŵŝnjĚŽůŶŽƑĐŝĂŵŝ͕ĂůĞnjǁLJŬůĞŽŐƌĂŶŝĐnjĂũČƐŝħŽŶĞ ĚŽ ǁČƐŬŝĞŐŽŽďƐnjĂƌƵ͘ t ƉƌnjLJƉĂĚŬƵ ŵĂƚĞŵĂƚLJŬŝ ƉƌnjĞũĂǁĞŵŽǁLJĐŚ njĚŽůŶŽƑĐŝŵŽǏĞ ďLJđ ŶĂ ƉƌnjLJŬųĂĚ ŶŝĞnjǁLJŬųĂ ƵŵŝĞũħƚŶŽƑđ ǁLJŬŽŶLJǁĂŶŝĂ ǁ ƉĂŵŝħĐŝ ĚnjŝĂųĂŷ ŶĂ ĚƵǏLJĐŚ ůŝĐnjďĂĐŚ͕ Ă njĂƌĂnjĞŵ ŶŝĞŵŽǏŶŽƑđ ƌŽnjǁŝČnjĂŶŝĂnjĂĚĂŶŝĂnjƚƌĞƑĐŝČĐnjLJĐĂųŬŽǁŝƚLJďƌĂŬǁLJŽďƌĂǍŶŝƉƌnjĞƐƚƌnjĞŶŶĞũ͘^njĐnjĞŐſůŶĞƉƌĞĚLJƐƉŽnjLJĐũĞ ƵĐnjŶŝĂ nj ĂƵƚLJnjŵĞŵ ŶĂũĐnjħƑĐŝĞũ ĚŽƚLJĐnjČ ũĞĚŶĞŐŽ ƐnjŬŽůŶĞŐŽ ƉƌnjĞĚŵŝŽƚƵ͘ dŽƚĞǏ njĂĚĂŶŝĞŵ ŶĂƵĐnjLJĐŝĞůĂ ũĞƐƚŝĐŚǁųĂƑĐŝǁĞǁLJŬŽƌnjLJƐƚĂŶŝĞŝǁnjŵŽĐŶŝĞŶŝĞǁĐĞůƵƉƌnjĞŶŝĞƐŝĞŶŝĂŶĂŝŶŶĞŽďƐnjĂƌLJ͕ďLJƵĐnjĞŷŵĂũČcy ƚĞnjĚŽůŶŽƑĐŝŵſŐųŽĚŶŽƐŝđƐƵŬĐĞƐLJǁƐnjŬŽůĞ͘ njħƐƚČƉƌnjLJĐnjLJŶČŶŝĞƉŽǁŽĚnjĞŷƵĐnjŶŝſǁǁƐnjŬŽůĞ ƐČƐƉĞĐLJĨŝĐnjŶĞnjĂďƵƌnjĞŶŝĂ funkcji poznawĐnjLJĐŚ͕ƵǁĂƌƵŶŬŽǁĂŶĞŶŝĞƉƌĂǁŝĚųŽǁLJŵĚnjŝĂųĂŶŝĞŵƵŬųĂĚƵŶĞƌǁŽǁĞŐŽ͘ĞƐƉſųƚLJĐŚnjĂďƵƌnjĞŷŶĂnjǁa- 5 ŶŽĚLJƐůĞŬƐũČƌŽnjǁŽũŽǁČ͘ ĂůŝĐnjĂƐŝħĚŽŶŝĞŐŽĚLJƐůĞŬƐũħ;ƚƌƵĚŶŽƑĐŝǁĐnjLJƚĂŶŝƵŝƉŝƐĂŶŝƵͿ͕ĚLJƐŽƌƚŽŐƌĂĨŝħ ;ƚƌƵĚŶŽƑĐŝǁŽƉĂŶŽǁĂŶŝƵnjĂƐĂĚƉŽƉƌĂǁŶĞũƉŝƐŽǁŶŝͿŽƌĂnjĚLJƐŐƌĂĨŝħ;ƚƌƵĚŶŽƑĐŝǁŽƉĂŶŽǁĂŶŝƵƉŝƐĂŶŝĂ kaligraficznego). Etiologia ĚLJƐůĞŬƐũŝŶŝĞnjŽƐƚĂųĂǁLJũĂƑŶŝŽŶĂ͕ĂůĞŶŝĞŬƚſƌnjLJďĂĚĂĐnjĞƵǁĂǏĂũČ͕ǏĞũĞƐƚƚŽ njĂďƵƌnjĞŶŝĞ Ž ƉŽĚųŽǏƵ ŐĞŶĞƚLJĐnjŶLJŵ͘ LJƐůĞŬƐũŝ ŶŝĞ ĚĂ Ɛŝħ ǁLJůĞĐnjLJđ͕ ůĞĐnj ĚnjŝħŬŝ ƐLJƐƚĞŵĂƚLJĐnjŶĞũ ƉƌĂĐLJ z ĚLJƐůĞŬƚLJŬŝĞŵ ŵŽǏŶĂ njųĂŐŽĚnjŝđ ũĞũ ŽďũĂǁLJ͘ ďĂĚĂŷǁLJŶŝŬĂ͕ ǏĞ njĂďƵƌnjĞŶŝĞ ƚŽ ŶŝĞ njĂǁƐnjĞ ǁLJǁŽųƵũĞ ƚƌƵĚŶŽƑĐŝnjŵĂƚĞŵĂƚLJŬČ͕ĂůďŽǁŝĞŵtylko ϲϬйĚLJƐůĞŬƚLJŬſǁŵĂƚƌƵĚŶŽƑĐŝnjŵĂƚĞŵĂƚLJŬČ, 29% ma poĚŽďŶĞ ƉƌĞĚLJƐƉŽnjLJĐũĞ ĚŽ ŵĂƚĞŵĂƚLJŬŝ ũĂŬ ƵĐnjŶŝŽǁŝĞ ďĞnj ĚLJƐůĞŬƐũŝ͕ ƌĞƐnjƚĂ njĂƑ͕ ĐnjLJůŝ ϭϭй͕ ŵĂ ǁLJƐŽĐĞ ƌŽnjǁŝŶŝħƚĞnjĚŽůŶŽƑĐŝŵĂƚĞŵĂƚLJĐnjŶĞ͘ĂďƵƌnjĞŶŝĂƌŽnjǁŽũŽǁĞŽĐŚĂƌĂŬƚĞƌnjĞĚLJƐůĞŬƚLJĐnjŶLJŵǁLJƐƚħƉƵũČ stosunkowo czħƐƚŽ͘EĂƉƌnjLJŬųĂĚǁWŽůƐĐĞǁLJŬƌLJƚŽũĞƵŽŬŽųŽϭϱйŽƐſďǁǁŝĞŬƵƐnjŬŽůŶLJŵ͘:ĂŬǁLJŶŝŬĂ njĚĂŶLJĐŚŽŐſůŶŽƉŽůƐŬŝĐŚ͕ŽĚƐĞƚĞŬƚĞŶƌŽƑŶŝĞ͕ŝƚŽǁnjĂƐŬĂŬƵũČĐŽƐnjLJďŬŝŵƚĞŵƉŝĞ͘KƉƌſĐnjƚĞŐŽŽďƐĞrǁƵũĞ Ɛŝħ ĚƵǏĞ ƌſǏŶŝĐĞ ǁ ũĞŐŽ ǁLJƐŽŬŽƑĐŝǁ ƉŽƐnjĐnjĞŐſůŶLJĐŚ ǁŽũĞǁſĚnjƚǁĂĐŚ ŝ ŵŝĂƐƚĂĐŚ͘ EĂ ƉƌnjLJŬųĂĚ ǁĞĚųƵŐ ĚĂŶLJĐŚ ĞŶƚƌĂůŶĞũ <ŽŵŝƐũŝ ŐnjĂŵŝŶĂĐLJũŶĞũ ǁ ǁŽũĞǁſĚnjƚǁŝĞ ƉŽŵŽƌƐŬŝŵ ŶĂ ĚLJƐůĞŬƐũħ ĐŝĞƌƉŝ ƉŽŶĂĚ ϭϳй ƵĐnjŶŝſǁ͕ ǁ ǁŽũĞǁſĚnjƚǁŝĞ ƑůČƐŬŝŵ ʹ 8,1%, a w opolskim ʹ tylko 7,3%. Wysoki odsetek ĚLJƐůĞŬƚLJŬſǁǁ ǁŽũĞǁſĚnjƚǁŝĞƉŽŵŽƌƐŬŝŵŵŽǏŶĂƚųƵŵĂĐnjLJđ ƉƌħǏŶČĚnjŝĂųĂůŶŽƑĐŝČWŽůƐŬŝĞŐŽdŽǁĂƌnjyƐƚǁĂLJƐůĞŬƐũŝ͕ĐŽnjǁŝħŬƐnjĂƑǁŝĂĚŽŵŽƑđƚĂŵƚĞũƐnjLJĐŚƌŽĚnjŝĐſǁŝŶĂƵĐnjLJĐŝĞůŝ͘t<ƌĂŬŽǁŝĞĚLJƐůĞŬƚLJŬŝĞŵ ũĞƐƚĐŽĐnjǁĂƌƚLJƵĐnjĞŷ͕ǁĞtƌŽĐųĂǁŝƵʹ ĐŽƉŝČƚLJ͕ĂǁtĂƌƐnjĂǁŝĞĚLJƐůĞŬƚLJĐLJƐƚĂŶŽǁŝČŽŬŽųŽϯϬйǁƐŬanjĂŶĞũƉŽƉƵůĂĐũŝ͘tŝĚĂđ ǁŝħĐ͕ǏĞŝŵǁŝħŬƐnjĞŵŝĂƐƚŽ͕ƚLJŵŽĚƐĞƚĞŬƵĐnjŶŝſǁnjĚLJƐůĞŬƐũČũĞƐƚǁLJǏƐnjLJ͘/ŶŶĞ ĚĂŶĞ ƉŽƚǁŝĞƌĚnjĂũČ njĂƑ ŵŶŝĞũƐnjČ ƑǁŝĂĚŽŵŽƑđ ƌŽĚnjŝĐſǁ nj ŵĂųLJĐŚ ŵŝĞũƐĐŽǁŽƑĐŝ ŽƌĂnj ŝĐŚ ŽŐƌĂŶŝĐnjŽŶLJ ĚŽƐƚħƉĚŽƉŽƌĂĚŶŝƉƐLJĐŚŽůŽŐŝĐnjŶŽ-pedagogicznych. WŽũĂǁŝĂũČƐŝħũĞĚŶĂŬŽƉŝŶŝĞ͕ǏĞůŝczba wydanych ŽƌnjĞĐnjĞŷŶŝĞŽĚnjǁŝĞƌĐŝĞĚůĂĨĂŬƚLJĐnjŶĞũůŝĐnjďLJƵĐnjŶŝſǁĐŝĞƌƉŝČĐLJĐŚŶĂĚLJƐůĞŬƐũħ͘EĂƉƌnjLJŬųĂĚŶŝĞŬƚſƌnjLJ ƐƉĞĐũĂůŝƑĐŝƐČnjĚĂŶŝĂ͕ǏĞĐnjħƑđŽǁLJĐŚŽƌnjĞĐnjĞŷǁLJĚĂǁĂŶĂũĞƐƚďĞnjƉŽĚƐƚĂǁŶŝĞŝnjďLJƚƉŽĐŚŽƉŶŝĞ͘/ŶaĐnjĞũŵſǁŝČĐ͕ƐƚĂƚLJƐƚLJŬŝŽĚnjǁŝĞƌĐŝĞĚůĂũČƐŬƵƚĞĐnjŶŽƑđŶŝĞƚLJůŬŽĚŝĂŐŶŽƐƚLJŬŝǁnjĂŬƌĞƐŝĞĚLJƐůĞŬƐũŝ͕ĂůĞƚĞǏ ͣůŽďďŝŶŐƵ͟ƌŽĚnjŝĐſǁĐŚĐČĐLJĐŚƵųĂƚǁŝđƐǁŽŝŵdzieciom ǏLJĐŝĞ͘ EŝĞnjŵŝĞŶŝĂƚŽũĞĚŶĂŬĨĂŬƚƵ͕ǏĞƉƌŽďůĞŵ ĚLJƐůĞŬƐũŝ ŝƐƚŶŝĞũĞ ŝ ǏĞ ŶŝĞ ŵŽǏŶĂ ŐŽ ůĞŬĐĞǁĂǏLJđ͘ ĂƚĞŵ ŽďŽǁŝČnjŬŝĞŵ ŶĂƵĐnjLJĐŝĞůŝ ũĞƐƚ ǁLJƉƌĂĐŽǁĂŶŝĞ skutecznych metod pracy z uczniem o ƚĂŬƐƉĞĐLJĨŝĐnjŶLJĐŚƉƌŽďůĞŵĂĐŚnjƵĐnjĞŶŝĞŵƐŝħŝĚŽƐƚŽƐŽǁĂŶŝĞ technik dydaktycznych do jego potrzeb. WƌnjĞũĂǁĞŵƐƉĞĐLJĨŝĐnjŶLJĐŚƚƌƵĚŶŽƑĐŝǁƵĐnjĞŶŝƵƐŝħŵĂƚĞŵĂƚLJŬŝũĞƐƚƚĞǏĐŽƌĂnjĐnjħƑĐŝĞũĚŝĂŐŶonjŽǁĂŶĂŝnjǁŝČnjĂŶĂǁǁŝĞůƵƉƌnjLJƉĂĚŬĂĐŚnjĚLJƐůĞŬƐũČƌŽnjǁŽũŽǁČĚLJƐŬĂůŬƵůŝĂ͘:ĞƐƚƚŽzaburzenie zdolnoƑĐŝŵĂƚĞŵĂƚLJĐnjŶLJĐŚ͕ŽďũĂǁŝĂũČĐĞƐŝħŬųŽƉŽƚĂŵŝǁǁLJŬŽŶLJǁĂŶŝƵƉƌŽƐƚLJĐŚĚnjŝĂųĂŷ͕Ă ƚĂŬǏĞǁƚǁŽƌnjeŶŝƵŵŶŝĞũůƵďďĂƌĚnjŝĞũnjųŽǏŽŶLJĐŚƵŬųĂĚſǁ ƉƌnjĞƐƚƌnjĞŶŶLJĐŚďČĚǍnjĞ njƌŽnjƵŵŝĞŶŝĞŵƉŽůĞĐĞŷĚŽnjĂĚĂŷ͘ DyƐŬĂůŬƵůŝĂŵŽǏĞďLJđƌŽnjƉŽnjŶĂŶĂƵĚnjŝĞĐŬĂ͕ŬƚſƌĞƵŬŽŷĐnjLJųŽϭϬůĂƚ͘K ƵĐnjŶŝĂĐŚĚŽƚŬŶŝħƚLJĐŚƚČƉƌnjyƉĂĚųŽƑĐŝČĐnjħƐƚŽŵſǁŝƐŝħ͕ǏĞƐČͣĐŚŽƌnjLJŶĂŵĂƚĞŵĂƚLJŬħ͘͟tWŽůƐĐĞnjƌŽďŝųŽƐŝħŽƚĞũͣĐŚŽƌŽďŝĞ͟ŐųŽƑŶŽ͕ŐĚLJǁƉƌŽǁĂĚnjŽŶŽŽďŽǁŝČnjĞŬnjĚĂǁĂŶŝĂŵĂƚƵƌLJnjŵĂƚĞŵĂƚLJŬŝ͕ĂůďŽǁŝĞŵǁŵĞĚŝĂĐŚnjĂĐnjħƚŽĚLJsŬƵƚŽǁĂđŽŬŽŶŝĞĐnjŶŽƑĐŝnjǁŽůŶŝĞŶŝĂĐŝĞƌƉŝČĐLJĐŚŶĂŶŝČƵĐnjŶŝſǁnjƚĞŐŽŽďŽǁŝČnjŬƵ͘ǁŽůĞŶŶŝĐLJŽǁĞŐŽ ƌŽnjǁŝČnjĂŶŝĂǁLJƐƚŽƐŽǁĂůŝŶĂǁĞƚŽĚƉŽǁŝĞĚŶŝǁŶŝŽƐĞŬĚŽŵŝŶŝƐƚĞƌƐƚǁĂĞĚƵŬĂĐũŝ͕ǁŬƚſƌLJŵĂƌŐƵŵĞnƚŽǁĂůŝ͕ǏĞųĂƚǁĂĚŽƉƌnjĞǁŝĚnjĞŶŝĂƉŽƌĂǏŬĂ ŶĂĞŐnjĂŵŝŶŝĞnjŵĂƚĞŵĂƚLJŬŝnjĂŵLJŬĂƚLJŵƵĐnjŶŝŽŵĚƌŽŐħĚŽ ƐnjŬſųǁLJǏƐnjLJĐŚ͕ŵŝŵŽǏĞŶŝĞũĞĚŶŽŬƌŽƚŶŝĞǁLJŬĂnjƵũČŽŶŝƵnjĚŽůŶŝĞŶŝĂǁƉƌnjĞĚŵŝŽƚĂĐŚŚƵŵĂŶŝƐƚLJĐnjŶLJĐŚ ŝ ĂƌƚLJƐƚLJĐnjŶLJĐŚ͘ DŝŶŝƐƚĞƌƐƚǁŽǁŶŝŽƐĞŬƚĞŶ ũĞĚŶĂŬ ŽĚƌnjƵĐŝųŽ͘ ǁƐŬƵƚĞŬ ĐnjĞŐŽ ǁϮϬϭϱ ƌŽŬƵƐƉƌĂǁĂƚĂ ƚƌĂĨŝųĂĚŽ,ĞůƐŝŷƐŬŝĞũ &ƵŶĚĂĐũŝWƌĂǁnjųŽǁŝĞŬĂ͘^ŬŝĞƌŽǁĂųĂ ŽŶĂ ƉŝƐŵŽĚŽZnjĞĐnjŶŝŬĂWƌĂǁKďLJǁĂƚĞlskich w Polsce, w ŬƚſƌLJŵ ƐƚǁŝĞƌĚnjŝųĂ͗ ͣWyŵſŐ njĚĂŶŝĂ ĞŐnjĂŵŝŶƵ ŵĂƚƵƌĂůŶĞŐŽ nj ŵĂƚĞŵĂƚLJŬŝ ƉƌnjĞnj ŽƐŽďħnjŶŝĞƉĞųŶŽƐƉƌĂǁŶŽƑĐŝČ͕ŬƚſƌĞũŶŝĞƉĞųŶŽƐƉƌĂǁŶŽƑđƵŶŝĞŵŽǏůŝǁŝĂnjĚĂŶŝĞƚĞŐŽĞŐnjĂŵŝŶƵ͕ǁƐLJƚƵĂĐũŝ gdy ŽƐŽďĂƚĂũĞƐƚǁƐƚĂŶŝĞnjĚĂđƉŽnjŽƐƚĂųĞŽďŽǁŝČnjŬŽǁĞĞŐnjĂŵŝŶLJŵĂƚƵƌĂůŶĞ͕ŶĂůĞǏLJƉŽƐƚƌnjĞŐĂđũĂŬŽ nieuzasaĚŶŝŽŶĞ ŝ ĚLJƐŬƌLJŵŝŶƵũČĐĞ ŽŐƌĂŶŝĐnjĞŶŝĞ ŵŽǏůŝǁŽƑĐŝ ŬŽƌnjLJƐƚĂŶŝĂ nj ƉƌĂǁĂ ĚŽ ŶĂƵŬŝ nj uwagi na 6 ŶŝĞƉĞųŶŽƐƉƌĂǁŶŽƑđ͘͟ tƚLJŵŵŝĞũƐĐƵŶĂůĞǏLJũĞĚŶĂŬnjĂƵǁĂǏLJđ͕ǏĞǁĞĚųƵŐǁŝĞůƵďĂĚĂĐnjLJͣĐnjLJƐƚĂĚLJsŬĂůŬƵůŝĂ͕͟ŵĂŶŝĨĞƐƚƵũČĐĂƐŝħǁLJųČĐnjŶŝĞƚƌƵĚŶŽƑĐŝĂŵŝǁnjĂŬƌĞƐŝĞŵĂƚĞŵĂƚLJŬŝƉƌnjLJƉƌĂǁŝĚųŽǁLJŵƌŽnjǁoũƵƉŽnjŶĂǁĐnjLJŵŝƉƌnjLJnjĚŽůŶŽƑĐŝĂĐŚũħnjLJŬŽǁLJĐŚ͕ǁLJƐƚħƉƵũĞĚŽƑđƌnjĂĚŬŽ͕ďŽƚLJůŬŽƵŽŬŽųŽϭйƵĐnjŶŝſǁ͘ dŽƚĞǏƉƌŽďůĞŵnjĞnjĚĂŶŝĞŵŵĂƚƵƌLJnjŵĂƚĞŵĂƚLJŬŝnjƉŽǁŽĚƵƚĞŐŽnjĂďƵƌnjĞŶŝĂĚŽƚLJĐnjLJƐƚŽƐƵŶŬŽǁŽŵaųĞũ ŐƌƵƉLJ ƵĐnjŶŝſǁ͘ EĂƐƵǁĂ Ɛŝħ ǁŝħĐ ƉLJƚĂŶŝĞ͕ ĐnjLJ ũĞƐƚĞƑŵLJ ǁ ƐƚĂŶŝĞ ŶĂƵĐnjLJđ ŝĐŚ ŵĂƚĞŵĂƚLJŬŝ ŶĂ ƚLJůĞ ĚŽďƌnjĞ͕ ďLJŶĂĞŐnjĂŵŝŶŝĞĚŽũƌnjĂųŽƑĐŝƵnjLJƐŬĂůŝnjƚĞŐŽƉƌnjĞĚŵŝŽƚƵ ƉƌnjLJŶĂũŵŶŝĞũǁLJŵĂŐĂŶĞ ϯϬйƉƵŶkƚſǁ͍ĚĞĐLJĚŽǁĂŶŝĞǁĂƌƚŽƉŽĚũČđƚĂŬČƉƌſďħ͕ůĞĐnjŶŝĞnjďħĚŶĞũĞƐƚĚŽƚĞŐŽǁųĂƑĐŝǁĞƉŽĚĞũƑĐŝĞŝŽĚƉowiednio dostosoǁĂŶLJŵĂƚĞƌŝĂųĚLJĚĂŬƚLJĐnjŶLJ͘ ĞƐƚĂǁϲϳŬĂƌƚƉƌĂĐLJnjŵĂƚĞŵĂƚLJŬŝnjĞďƌĂŶLJĐŚǁŶŝŶŝĞũƐnjĞũŬƐŝČǏĐĞƐƚĂŶŽǁŝƉƌſďħǁLJũƑĐŝĂŶĂ ƉƌnjĞĐŝǁ ƉŽƚƌnjĞďŽŵ ŶĂƵĐnjLJĐŝĞůŝ͕ ŬƚſƌnjLJ ŶĂ ĐŽ ĚnjŝĞŷ ƉƌĂĐƵũČ nj ƵĐnjniami o specyficznych potrzebach ĞĚƵŬĂĐLJũŶLJĐŚǁƐnjŬŽųĂĐŚƉŽŶĂĚŐŝŵŶĂnjũĂůŶLJĐŚ ŝŵŝĞƌnjČƐŝħnjƉƌŽďůĞŵĞŵƉƌnjLJŐŽƚŽǁĂŶŝĂƚLJĐŚƵĐnjŶŝſǁ ĚŽ ĞŐnjĂŵŝŶƵ ŵĂƚƵƌĂůŶĞŐŽ nj ŵĂƚĞŵĂƚLJŬŝ͘ dƌnjĞďĂ ũĞĚŶĂŬ njĂnjŶĂĐnjLJđ͕ ǏĞ njĞƐƚĂǁ ƚĞŶ ŶŝĞ ǁLJĐnjĞƌƉƵũĞ ǁƐnjLJƐƚŬŝĐŚ njĂŐĂĚŶŝĞŷ ƵũħƚLJĐŚ ǁ ƉŽĚƐƚĂǁŝĞ ƉƌŽŐƌĂŵŽǁĞũ nj ŵĂƚĞŵĂƚLJŬŝ ĚůĂ /s ĞƚĂƉƵ ĞĚƵŬĂĐLJũŶĞŐŽ͕ ŽďŽǁŝČnjƵũČĐĞũŽĚϮϬϭϮƌŽŬƵ͕ĂůĞĚĂũĞŶŝĞnjďħĚŶLJŝǁLJƌĂǍŶLJnjĂƌLJƐƚĞũƉŽĚƐƚĂǁLJ͘ŚŽĐŝĂǏƉƌŽƉŽŶŽǁĂŶĞ ƚƵ ŬĂƌƚLJ ƉƌĂĐLJ ŶŝĞ ŵŽŐČ ƉĞųŶŝđ ĨƵŶŬĐũŝ ƉƌnjĞǁŽĚŶŝŬĂ ƉŽ ƉŽĚƐƚĂǁŝĞ ƉƌŽŐƌĂŵŽǁĞũ nj ŵĂƚĞŵĂƚLJŬŝ͕ ƚŽ ũĞĚŶĂŬnjĂǁŝĞƌĂũČƉŽŶĂĚϱϱϬnjĂĚĂŷnjƌſǏŶLJĐŚĚnjŝĂųſǁŵĂƚĞŵĂƚLJŬŝŽŵŶŝĞũůƵďďĂƌĚnjŝĞũĚostosowanej ƐƚƌƵŬƚƵƌnjĞ͘<ĂƌƚLJƚĞŵŽǏŶĂǁLJŬŽƌnjLJƐƚĂđnjĂƌſǁŶŽĚŽƉƌĂĐLJŶĂůĞŬĐũŝnjĐĂųČŬůĂƐČ͕ũĂŬŝĚŽnjĂũħđǁLJƌſwŶĂǁĐnjLJĐŚĐnjLJƚĞǏƌĞǁĂůŝĚĂĐLJũŶLJĐŚ͘WƌnjLJĚĂĚnjČƐŝħŽŶĞƌſǁŶŝĞǏǁŶĂƵĐnjĂŶŝƵŝŶĚLJǁŝĚƵĂůŶLJŵ͘ĂǁŝĞƌĂũČ ƉĞųŶLJŽƉŝƐƚĞŽƌĞƚLJĐnjŶLJƉŽƌƵƐnjĂŶLJĐŚǁ nich zagadŶŝĞŷŵĂƚĞŵĂƚLJĐnjŶLJĐŚ;ƚũ͘ŽďũĂƑŶŝĞŶŝĂƉŽũħđ͕ǁnjŽƌLJ͕ ƌLJƐƵŶŬŝ͕ƌŽnjǁŝČnjĂŶŝĂƉƌnjLJŬųĂĚŽǁLJĐŚnjĂĚĂŷŝǁƐŬĂnjſǁŬŝͿ͘ĂƚĞŵŵŽŐČďLJđǁLJŬŽƌnjLJƐƚLJǁĂŶĞŶŝĞƚLJůŬŽ ĚŽƉƌĂĐLJƵĐnjŶŝĂƉŽĚŬŽŶƚƌŽůČŶĂƵĐnjLJĐŝĞůĂŶĂůĞŬĐũŝ͕ĂůĞƚĂŬǏĞĚŽƐĂŵŽĚnjŝĞůŶĞũƉƌĂĐLJǁĚŽŵƵ͘DŽŐČ ƐŝħŽŶĞŽŬĂnjĂđƉŽŵŽĐŶĞnjĂƌſǁŶŽƉƌnjLJǁƉƌŽǁĂĚnjĂŶŝƵŶŽǁĞũƉĂƌƚŝŝŵĂƚĞƌŝĂųƵŶĂůĞŬĐũŝ͕ũĂŬŝƉƌnjLJƉƌnjyŐŽƚŽǁLJǁĂŶŝƵĚŽŬůĂƐſǁŬŝ͘DŽǏŶĂƚĞǏƉŽŶŝĞƐŝħŐŶČđ͕ĐŚĐČĐƉƌnjLJƉŽŵŶŝĞđŵĂƚĞƌŝĂųŝƉŽđǁŝĐnjLJđƉƌnjĞĚ ĞŐnjĂŵŝŶĞŵĚŽũƌnjĂųŽƑĐŝnjŵĂƚĞŵĂƚLJŬŝ͕ƐƚĂŶŽǁŝČŽŶĞďŽǁŝĞŵƐŽůŝĚŶČďĂnjħǁŝĞĚnjLJ wymaganej na maƚƵƌnjĞ͘t ŶŝĞŬƚſƌLJĐŚŬĂƌƚĂĐŚ͕ŽƉƌſĐnjƚƌĞƑĐŝŽďŽǁŝČnjƵũČĐLJĐŚǁLJųČĐnjŶŝĞŶĂĐnjǁĂƌƚLJŵĞƚĂƉŝĞ ĞĚƵŬĂĐLJjŶLJŵ͕njŽƐƚĂųƉƌnjĞĚƐƚĂǁŝŽŶLJƌſǁŶŝĞǏŵĂƚĞƌŝĂųƌĞĂůŝnjŽǁĂŶLJǁ ŐŝŵŶĂnjũƵŵ͘ĂũĞŵLJǁŝħĐƵĐnjŶŝŽǁŝ ƐƉoƐŽďŶŽƑđŶĂĚƌŽďŝĞŶŝĂnjĂůĞŐųŽƑĐŝnjŐŝŵŶĂnjũƵŵǁƚƌĂŬĐŝĞƌĞĂůŝnjĂĐũŝŵĂƚĞƌŝĂųƵnjĐnjǁĂƌƚĞŐŽĞƚĂƉƵĞĚƵŬacyjnego. KŬŽůĞũŶŽƑĐŝƌĞĂůŝnjŽǁĂŶŝĂƚƌĞƑĐŝƉŽĚƐƚĂǁLJƉƌŽŐƌĂŵŽǁĞũĚĞĐLJĚƵũĞƉƌŽŐƌĂŵŶĂƵĐnjĂŶŝĂ͘<ĂƌƚLJ ƉƌĂĐLJŶŝĞnjŽƐƚĂųLJǁŝħĐƉŽŐƌƵƉŽǁĂŶĞǁĞĚųƵŐŬůĂƐ͕ǁŬƚſƌLJĐŚǁƉƌŽǁĂĚnjĂƐŝħĚĂŶLJŵĂƚĞƌŝĂų͕ůĞĐnjƵųoǏŽŶŽũĞƚĞŵĂƚLJĐnjŶŝĞ͘EŝĞƐƚŽƐƵũĞƐŝħƚƵƚĞǏƚƌĂĚLJĐLJũŶĞŐŽƉŽĚnjŝĂųƵŶĂĚnjŝĂųLJŵĂƚĞŵĂƚLJŬŝ͕ŬƚſƌLJƐƉŽƚyŬĂŵLJǁƐnjŬŽůŶLJĐŚƉŽĚƌħĐnjŶŝŬĂĐŚŝnjďŝŽƌĂĐŚnjĂĚĂŷ͘dŽ͕ǁ ũĂŬŝĞũ ŬŽůĞũŶŽƑĐŝŽǁĞŬĂƌƚLJďħĚČǁLJŬŽƌnjyƐƚĂŶĞ͕njĂůĞǏLJŽĚŝŶǁĞŶĐũŝŶĂƵĐnjLJĐŝĞůĂ͕ƉůĂŶƵŶĂƵĐnjĂŶŝĂŽƌĂnjŽĚƉŽƚƌnjĞďĞĚƵŬĂĐLJũŶLJĐŚƵĐnjŶŝĂůƵďĐĂųĞũ ŬůĂƐLJ͘ :ĞĚŶĂŬ ǁ ƐLJƚƵĂĐũŝ ŐĚLJ ƵĐnjĞŷ ƉƌĂĐƵũĞ nj ŬĂƌƚĂŵŝ ƐĂŵ͕ ďĞnj ŶĂƵĐnjLJĐŝĞůĂ͕ njĂůĞĐĂ Ɛŝħ njĂĐŚŽǁĂŶŝĞ ƵŬųĂĚƵnjĂƉƌŽƉŽŶŽǁĂŶĞŐŽƉƌnjĞnjĂƵƚŽƌĂŶŝŶŝĞũƐnjĞũŬƐŝČǏŬŝ͘ :ĂŬũƵǏǁƐƉŽŵŶŝĂŶŽ͕ŬĂƌƚLJŶŝĞƵǁnjŐůħĚŶŝĂũČǁƐnjLJƐƚŬŝĐŚƉĂƌƚŝŝŵĂƚĞƌŝĂųƵnjnjĂŬƌĞƐƵŵatematyki ŶĂ ĐnjǁĂƌƚLJŵ ĞƚĂƉŝĞ ĞĚƵŬĂĐLJũŶLJŵ͘ WŽŵŝŶŝħƚŽ ǁ ŶŝĐŚŵŝħĚnjLJ ŝŶŶLJŵŝ ƚĂŬŝĞ njĂŐĂĚŶŝĞŶŝĂ͕ ũĂŬ͗ ǁĂƌƚŽƑđ ŶĂũŵŶŝĞũƐnjĂŝŶĂũǁŝħŬƐnjĂĨƵŶŬĐũŝŬǁĂĚƌĂƚŽǁĞũǁ ƉƌnjĞĚnjŝĂůĞ ĚŽŵŬŶŝħƚLJŵ͕ƐƚLJĐnjŶĂĚŽŽŬƌħŐƵŝŽŬƌħŐŝ ƐƚLJĐnjŶĞ͕ ƐLJŵĞƚƌŝĂ ŽƐŝŽǁĂ ǁnjŐůħĚĞŵ ŽƐŝ ƵŬųĂĚƵ ǁƐƉſųƌnjħĚŶLJĐŚ ŝ ƐLJŵĞƚƌŝĂ ƑƌŽĚŬŽǁĂ ǁnjŐůħĚĞŵ ƉoĐnjČƚŬƵƵŬųĂĚƵǁƐƉſųƌnjħĚŶLJĐŚĐnjLJƉƌŽĐĞŶƚƐŬųĂĚĂŶLJ͘ƌĂŬƵũĞƌſǁŶŝĞǏnjĂŐĂĚŶŝĞŷnjǁŝČnjĂŶLJĐŚnjĞƐƉŽƐoďĂŵŝŽƉŝƐƵŝǁųĂƐŶŽƑĐŝĂŵŝƌſǏŶLJĐŚĨƵŶŬĐũŝŽƌĂnjƉƌnjĞŬƐnjƚĂųĐĂŶŝĞŵŝĐŚǁLJŬƌĞƐſǁ ʹ ƉŽũĂǁŝĂũČƐŝħ ŽŶĞ jedynie przy omawianiu konkretnych funkcji, takich jak funkcja liniowa, funkcja kwadratowa, funkcja ǁLJŬųĂĚŶŝĐnjĂĐnjLJĨƵŶŬĐũĂ͕ŬƚſƌĞũǁLJŬƌĞƐĞŵũĞƐƚŚŝƉĞƌďŽůĂ͘WŽnjǁŽůŝƚŽŶĂƵĐnjLJĐŝĞůŽǁŝŐĞŶĞƌŽǁĂđǁųĂƐŶĞ 7 ƉŽŵLJƐųLJƐųƵǏČĐĞƵnjƵƉĞųŶŝĞŶŝƵnjĂƉƌĞnjĞŶƚŽǁĂŶĞŐŽnjĞƐƚĂǁƵŬĂƌƚƉƌĂĐLJŽďƌĂŬƵũČĐĞƚĞŵĂƚLJ͘DŽǏŶĂƉƌnjLJ ƚLJŵǁnjŽƌŽǁĂđƐŝħŶĂŵĂƚĞƌŝĂůĞũƵǏƉƌnjLJŐŽƚŽǁĂŶLJŵ͕ĂůĞnjĂǁƐnjĞǁĂƌƚŽƐnjƵŬĂđŝŶŶLJĐŚĐŝĞŬĂǁLJĐŚƌŽzǁŝČnjĂŷ͘ <ĂƌƚLJ ƉƌĂĐLJ njĂŵŝĞƐnjĐnjŽŶĞ ǁ ƚĞũ ŬƐŝČǏĐĞ ƵųĂƚǁŝČ ŶĂƵĐnjLJĐŝĞůŽǁŝ ŵĂƚĞŵĂƚLJŬŝ ƉƌĂĐħ nj uczniem o ƐƉĞĐLJĨŝĐnjŶLJĐŚ ƉŽƚƌnjĞďĂĐŚ ĞĚƵŬĂĐLJũŶLJĐŚ ŽƌĂnj ƉŽnjǁŽůČ ŵƵ ͣďĞnjďŽůĞƑŶŝĞ͟ ƉƌnjĞƉƌŽǁĂĚnjŝđ ƚĂŬŝĞŐŽ ƵĐnjŶŝĂ ƉƌnjĞnj ĐnjǁĂƌƚLJ ĞƚĂƉĞĚƵŬĂĐLJũŶLJ͘ LJđŵŽǏĞ ĚnjŝħŬŝ ƚĞŵƵ ƵĐnjĞŷ ƚĞŶ njĚŽųĂ ŶĂǁĞƚ ƵnjLJƐŬĂđ ƉŽnjyƚLJǁŶLJ ǁLJŶŝŬ ŶĂ ĞŐnjĂŵŝŶŝĞ ŵĂƚƵƌĂůŶLJŵ͘ EŝĞǁLJŬůƵĐnjŽŶĞ͕ ǏĞ ǁƐŬƵƚĞŬ ǁLJŬŽƌnjLJƐƚĂŶŝĂ ŶŝŶŝĞũƐnjLJĐŚ ŬĂƌƚ ƉƌĂĐLJ ŶĂƵĐnjLJĐŝĞůŽǁŝ ƵĚĂ Ɛŝħ njŵŝĞŶŝđ ŶĞŐĂƚLJǁŶĞ ŶĂƐƚĂǁŝĞŶŝĞ ƵĐnjŶŝſǁ ĚŽ ŵĂƚĞŵĂƚLJŬŝ ŝ ƉŽĚǁĂǏLJđ ƉƌnjĞŬŽŶĂŶŝĞ͕ ǏĞ ũĞƐƚ ŽŶĂ ƚƌƵĚŶĂ͘ WƌĂǁĚnjŝǁĞ ũĞƐƚ ďŽǁŝĞŵ ƐƚǁŝĞƌĚnjĞŶŝĞ ǁħŐŝĞƌƐŬŝĞŐŽ ŵĂƚĞŵĂƚLJŬĂ :ŽŚŶĂ ǀŽŶ EĞƵŵĂŶŶĂ͗ ͣ:ĞƑůŝ ůƵĚnjŝĞ ŶŝĞ ǁŝĞƌnjČ͕ ǏĞ ŵĂƚĞŵĂƚLJŬĂ ũĞƐƚ ƉƌŽƐƚĂ͕ ƚŽ ƚLJůŬŽ ĚůĂƚĞŐŽ͕ ǏĞ ŶŝĞ njĚĂũČƐŽďŝĞƐƉƌĂǁLJũĂŬƐŬŽŵƉůŝŬŽǁĂŶĞũĞƐƚǏLJĐŝĞ͘͟ 8 KƉŝƐŬĂƌƚƉƌĂĐLJƉŽĚŬČƚĞŵŝĐŚƉƌnjLJĚĂƚŶŽƑĐŝ do pracy z uczniem o specyficznych problemach ǁƵĐnjĞŶŝƵƐŝħŵĂƚĞŵĂƚLJŬŝ ZŽnjǁſũƵŵŝĞũħƚŶŽƑĐŝŵĂƚĞŵĂƚLJĐnjŶLJĐŚƵĐnjŶŝĂnjĂůĞǏLJǁĚƵǏLJŵƐƚŽƉŶŝƵ od poziomu jego moƚLJǁĂĐũŝŽƌĂnjŽĚƐƉŽƐŽďƵŶĂƵĐnjĂŶŝĂĚŽƐƚŽƐŽǁĂŶĞŐŽĚŽũĞŐŽŝŶĚLJǁŝĚƵĂůŶLJĐŚƉŽƚƌnjĞď͘EŝĞƐƉŽƐſďƐŝħ ũĞĚŶĂŬŶŝĞnjŐŽĚnjŝđnjĞƐƚǁŝĞƌĚnjĞŶŝĞŵ͕ǏĞƐČƵĐnjŶŝŽǁŝĞďĂƌĚnjŝĞũůƵďŵŶŝĞũƵnjĚŽůŶŝĞŶŝǁ dziedzinie maƚĞŵĂƚLJŬŝ͕ĂŶŝĞŬƚſƌLJŵta matematyka po prostu ǁǏĂĚĞŶƐƉŽƐſďͣŶŝĞǁĐŚŽĚnjŝĚŽŐųŽǁLJ͘͟ KƚLJŵ͕ǏĞ z ŵĂƚĞŵĂƚLJŬČ ǁ ƉŽůƐŬŝĐŚ ƐnjŬŽųĂĐŚ ŶŝĞ ũĞƐƚ ĂǏ ƚĂŬ ǍůĞ ŽƌĂnj ǏĞ ŶŝĞ ŵĂŵLJ ŶĂ ƚLJŵ ƉŽůƵ ƉŽǁŽĚſǁ ĚŽ ǁƐƚLJĚƵ͕ŐĚLJǏĐĂųŬŝĞŵĚŽďƌnjĞǁLJƉĂĚĂŵLJŶĂƚůĞŝŶŶLJĐŚŬƌĂũſǁƵƌŽƉLJ͕ŵŽŐČƑǁŝĂĚĐnjLJđĐŚŽđďLJǁLJŶŝŬŝ ďĂĚĂŷ W/^ ;WƌŽŐƌamme for International Student Assessment ʹ WƌŽŐƌĂŵ DŝħĚnjLJŶĂƌŽĚŽǁĞũ KĐĞŶLJ hŵŝĞũħƚŶŽƑĐŝhĐnjŶŝſǁͿ͕ŬƚſƌĞƉƌnjĞƉƌŽǁĂĚnjŽŶŽǁϮϬϭϮƌŽŬƵ ǁƑƌſĚƉŝħƚŶĂƐƚŽůĂƚŬſǁ͘WůĂƐƵũČŶĂƐŽŶĞ ǁĐnjŽųſǁĐĞŬƌĂũſǁĞƵƌŽƉĞũƐŬŝĐŚ;njĂ,ŽůĂŶĚŝČ͕ƐƚŽŶŝČŝ &ŝŶůĂŶĚŝČͿ͘tŽƐƚĂƚŶŝĐŚůĂƚĂĐŚŽďƐĞƌǁƵũĞƐŝħ ǁLJƌĂǍŶČƉŽƉƌĂǁħǁƚLJŵǁnjŐůħĚnjŝĞ͘WƌnjLJƚŽĐnjŽŶĞǁLJŶŝŬŝŵŽŐČũĞĚŶĂŬƚLJůŬŽŶĂĐŚǁŝůħƉŽƉƌĂǁŝđŶĂŵ ƐĂŵŽƉŽĐnjƵĐŝĞŝnjǁŝħŬƐnjLJđƐĂŵŽŽĐĞŶħ͘ŽƐŬŽŶĂůĞnjĚĂũĞŵLJƐŽďŝĞďŽǁŝĞŵƐƉƌĂǁħnjƚĞŐŽ͕ǏĞƉƌnjĞĐŝħtŶĂ ƉŽůƐŬĂ ƐnjŬŽųĂ ƐŬƵƉŝĂ ĚƵǏČ ƌnjĞƐnjħ ƵĐnjŶŝſǁ͕ ĚůĂŬƚſƌLJĐŚ ƉƌŽďůĞŵLJ nj ŵĂƚĞŵĂƚLJŬČ ŶŝĞ ƐČŽďĐĞ͘ WoƚǁŝĞƌĚnjĂũČƚŽŵŝħĚnjLJŝŶŶLJŵŝƐƚĂƚLJƐƚLJŬŝĚŽƚLJĐnjČĐĞnjĚĂǁĂůŶŽƑĐŝŶĂĞŐnjĂŵŝŶŝĞŵĂƚƵƌĂůŶLJŵnjŵĂƚĞŵĂƚyŬŝ ŝ ŶŝĐ ŶĂ ƌĂnjŝĞ ŶŝĞ ǁƐŬĂnjƵũĞ ŶĂ ƚŽ͕ ďLJ ǁ ŶĂũďůŝǏƐnjLJŵ ĐnjĂƐŝĞ ƐLJƚƵĂĐũĂ ƚĂ ƵůĞŐųĂ njŶĂĐnjČĐĞũ ƉŽƉƌĂǁŝĞ͘ Wspomniane problemy majČƌſǏŶĞƉŽĚųŽǏĞ͘EĂũĐnjħƑĐŝĞũũĞƐƚƚŽ͗ x ŽŐƌĂŶŝĐnjŽŶĂƐƉƌĂǁŶŽƑđǁƉƌnjLJƉĂĚŬƵǁLJŬŽŶLJǁĂŶŝĂŽƉĞƌĂĐũŝŶĂůŝĐnjďĂĐŚ͕ x ŽŐƌĂŶŝĐnjŽŶĂnjĚŽůŶŽƑđĚŽƉŽƐųƵŐŝǁĂŶŝĂƐŝħƌĞƉƌĞnjĞŶƚĂĐũĂŵŝŽďƌĂnjŽǁLJŵŝŝƉŽũħĐŝĂŵŝ͕ x ŽŐƌĂŶŝĐnjŽŶĂnjĚŽůŶŽƑđĚŽnjĂƉĂŵŝħƚLJǁĂŶŝĂŝƐƚŽƐŽǁĂŶŝĂƉƌŽƐƚLJĐŚĂůŐŽƌLJƚŵſǁ͕ x ŽŐƌĂŶŝĐnjŽŶČnjĚŽůŶŽƑĐŝČĚĞĚƵŬĐũŝ͕ƵŽŐſůŶŝĂŶŝĂ͕ĂŶĂůŝnjŽǁĂŶŝĂŝǁLJĐŝČŐĂŶŝĂǁŶŝŽƐŬſǁ͘ x ŶŝƐŬŝƉŽnjŝŽŵǁLJŽďƌĂǍŶŝƉƌnjĞƐƚƌnjĞŶŶĞũ͕ x niski poziom koordynacji percepcyjno-motorycznej, x ŶŝƐŬŝƉŽnjŝŽŵŵŽƚLJǁĂĐũŝ͕ŬŽŶĐĞŶƚƌĂĐũŝŝĚŽũƌnjĂųŽƑĐŝĞŵŽĐũŽŶĂůŶĞũƉƌnjLJǁǁĂƌƵŶŬĂĐŚĚƵǏĞŐŽ ǁLJƐŝųŬƵŝŶƚĞůĞŬƚƵĂůŶĞŐŽ͘ _ƌŽĚŬŝĞŵnjĂƌĂĚĐnjLJŵĐnjħƑĐŝŽǁŽųĂŐŽĚnjČĐLJŵƐŬƵƚŬŝǁLJŵŝĞŶŝŽŶLJĐŚƉƌŽďůĞŵſǁŵŽŐČďLJđŽdpowiednio skonstruowane karty pracy, dostosowane do potrzeb ucznia o specyficznych potrzebach ĞĚƵŬĂĐLJũŶLJĐŚ͘<ĂƌƚLJƚĞƵųĂƚǁŝČƚĂŬŝŵƵĐnjŶŝŽŵŽƉĂŶŽǁĂŶŝĞŶĂǁĞƚŶĂũƚƌƵĚŶŝĞũƐnjLJĐŚnjĂŐĂĚŶŝĞŷŵĂƚeŵĂƚLJĐnjŶLJĐŚ͘ WŽŶĂĚƚŽ ƉŽnjǁŽůČ ŶĂ ǁLJŬŽƌnjLJƐƚĂŶŝĞ ǁ ƉƌŽĐĞƐŝĞ ĚLJĚĂŬƚLJĐnjŶLJŵ ŵŽĐŶLJĐŚ ƐƚƌŽŶ ƵĐnjŶŝĂ ŽƌĂnjŶĂǁnjŵŽĐŶŝĞŶŝĞƐƚƌŽŶƐųĂďƐnjLJĐŚ͘ DĂƚĞŵĂƚLJĐnjŶĞŬĂƌƚLJƉƌĂĐLJnjĂŵŝĞƐnjĐnjŽŶĞǁƚĞũŬƐŝČǏĐĞǁLJƌſǏŶŝĂũČƐŝħƐǁŽũČĐnjLJƚĞůŶŽƑĐŝČ;ĚuǏĂĐnjĐŝŽŶŬĂŝƉŽŐƌƵďŝĞŶŝĂͿŽƌĂnjŵĂũČƐƉſũŶČŝƉƌnjĞũƌnjLJƐƚČƐƚƌƵŬƚƵƌħ͘<ŽŵĨŽƌƚƉƌĂĐLJƵĐnjŶŝĂnjŽƐƚĂųnjǁŝħkƐnjŽŶLJƉƌnjĞnjŽŐƌĂŶŝĐnjĞŶŝĞůŝĐnjďLJnjĂĚĂŷŶĂƐƚƌŽŶŝĞŽƌĂnjǁLJƌſǏŶŝĞŶŝĞƉŽĐnjČƚŬƵŬĂǏĚĞŐŽnjĂĚĂŶŝĂ͘ŽĐnjħͲ ƑĐŝ ƚĞŽƌĞƚLJĐnjŶĞũ͕ njųŽǏŽŶĞũ nj ƉŽũħđ͕ ǁųĂƐŶŽƑĐŝ ŝ ǁnjŽƌſǁ͕ ĚŽųČĐnjŽŶŽ ŬŽŵĞŶƚĂƌnj nj ŶƵŵĞƌĂŵŝ njĂĚĂŷ͕ w ŬƚſƌLJĐŚƌŽnjǁŝČnjLJǁĂŶŝƵƉŽĚĂŶĞŝŶĨŽƌŵĂĐũĞďħĚČƉƌnjLJĚĂƚŶĞ͘EĞǁƌĂůŐŝĐnjŶĞŝ ǁLJŵĂŐĂũČĐĞƐnjĐnjĞŐſůŶĞũ 9 ƵǁĂŐŝŵŝĞũƐĐĂǁƚƌĞƑĐŝnjĂĚĂŷŝƉŽůĞĐĞŷnjŽƐƚĂųLJŽĚƉŽǁŝĞĚŶŝŽǁLJƌſǏŶŝŽŶĞ͗ƉŽĚŬƌĞƑůĞŶŝĞŵ͕ǁLJƚųƵƐnjĐnjeniem czcionki aůďŽŬŽŶƚƌĂƐƚŽǁLJŵƚųĞŵ͘ EĂƵǁĂŐħnjĂƐųƵŐƵũĞƐƉŽƐſďƉƌnjĞĚƐƚĂǁŝĞŶŝĂƉŽũħđ͕ǁnjŽƌſǁ͕ǁųĂƐŶŽƑĐŝ͕ĂůŐŽƌLJƚŵſǁŽƌĂnjƌŽzǁŝČnjĂŷƉƌnjLJŬųĂĚŽǁLJĐŚnjĂĚĂŷ͘WƌnjLJŝĐŚƉƌĞnjĞŶƚĂĐũŝƉŽƐųƵǏŽŶŽƐŝħƌſǏŶLJŵŝĨŽƌŵĂŵŝŐƌĂĨŝĐnjŶLJŵŝ͕ƚĂŬŝŵŝ jak: schematy blokowe, diagramy, tabele czy grafLJ͘WŽǁƐnjĞĐŚŶŝĞǁŝĂĚŽŵŽ͕ǏĞƚĂŬŝĞƌŽnjǁŝČnjĂŶŝĂƵųaƚǁŝĂũČ ƉƌnjLJƐǁĂũĂŶŝĞ ǁŝĞĚnjLJ ŵŝħĚnjLJ ŝŶŶLJŵŝ ƵĐnjŶŝŽŵ nj ĂƵƚLJnjŵĞŵ͕ ƐČ ƌſǁŶŝĞǏ ƉƌnjLJĚĂƚŶĞ ǁ ƉƌĂĐLJ z ƵĐnjŶŝĞŵĚLJƐůĞŬƚLJĐnjŶLJŵĐnjLJĐŝĞƌƉŝČĐLJŵŶĂĚLJƐŬĂůŬƵůŝħ͘tƑƌſĚůŝĐnjŶLJĐŚƉƌŽďůĞŵſǁƵĐnjŶŝſǁnjĚLJƐůĞkƐũČƌŽnjǁŽũŽǁČŽƌĂnjĚLJƐŬĂůŬƵůŝČŶĂůĞǏLJǁLJŵŝĞŶŝđƌſǁŶŝĞǏnjĂďƵƌnjĞŶŝĞƉŽĐnjƵĐŝĂĐnjĂƐƵ͕ŬƚſƌĞŵƵƚŽǁaƌnjLJƐnjLJnjĂďƵƌnjĞŶŝĞŽƌŝĞŶƚĂĐũŝǁnjƌŽŬŽǁĞũ͕ƐųƵĐŚŽǁĞũŝnjŵLJƐųƵƌſǁŶŽǁĂŐŝ͘ĂųĂŵĂƚĞŵĂƚLJŬĂ͕ƉŽĐnjLJŶĂũČĐ ŽĚƉƌŽƐƚĞũĂƌLJƚŵĞƚLJŬŝŝŶĂƐŬŽŵƉůŝŬŽǁĂŶLJĐŚŽƉĞƌĂĐũĂĐŚŬŽŷĐnjČĐ͕ǁLJŵĂŐĂƵƉŽƌnjČĚŬŽǁĂŶŝĂƉrzyczyn, ŶĂƐƚħƉƐƚǁ ŝ ĐnjĂƐƵ͘ WƌnjLJ njĂďƵƌnjĞŶŝƵ ǁĞǁŶħƚƌnjŶĞŐŽ ƉŽĐnjƵĐŝĂ ĐnjĂƐƵ njƌŽnjƵŵŝĞŶŝĞ ƉŽũħĐŝĂ ŶĂƐƚħƉƐƚǁĂ͕ ĐnjLJůŝ ƉƌĂǁŝĚųŽǁŽƑĐŝ ƌnjČĚnjČĐĞũ ŬŽůĞũŶŽƑĐŝČ ǁLJĚĂƌnjĞŷ͕ ũĞƐƚ ƵƚƌƵĚŶŝŽŶĞ͘ KƉĂƌƚĞ ŶĂ ƐĐŚĞŵĂƚĂĐŚ ďůŽŬoǁLJĐŚ ƉƌnjLJŬųĂĚŽǁĞ ƌŽnjǁŝČnjĂŶŝĂ ŶŝĞŬƚſƌLJĐŚ njĂĚĂŷ ŵŽŐČ ĐŚŽĐŝĂǏ ĐnjħƑĐŝŽǁŽ njųĂŐŽĚnjŝđ ƚĞ ƚƌƵĚŶŽƑĐŝ͘ hŵŽǏůŝǁŝĂũČŽŶĞƵƉŽƌnjČĚŬŽǁĂŶŝĞƉƌnjLJĐnjLJŶ͕ŶĂƐƚħƉƐƚǁŝĐnjĂƐƵǁƌŽnjƵŵŽǁĂŶŝƵŵĂƚĞŵĂƚLJĐnjŶLJŵ͘ WŽƉƌĂǁŶĞŝƐnjLJďŬŝĞƉƌnjĞƉŝƐĂŶŝĞƚƌĞƑĐŝnjƚĂďůŝĐLJĐnjħƐƚŽƐƚĂŶŽǁŝĚůĂƵĐnjŶŝĂŶŝĞůĂĚa wyzwanie. Problem ten spowodowany jest niskim poziomem koordynacji percepcyjno-motorycznej. Uczniowie ĐĂųČƐǁŽũČƵǁĂŐħƐŬƵƉŝĂũČŶĂƉŝƐĂŶŝƵʹ ƉƌĂĐƵũČƉŽĚƉƌĞƐũČĐnjĂƐƵ͕ŬƚſƌĞŐŽďƌĂŬƵũĞŝŵŶĂnjƌŽnjƵŵŝĞŶŝĞ ƉƌnjĞƉŝƐLJǁĂŶLJĐŚƚƌĞƑĐŝ͘<ĂƌƚLJnjŽƐƚĂųLJƐŬŽŶƐƚƌƵŽǁĂŶĞƚĂŬ͕ĂďLJŵŽŐųLJŝŵƐųƵǏLJđŶŝĞƚLJůŬŽũĂŬŽŵĂƚĞƌŝĂų đǁŝĐnjĞŶŝŽǁLJ͕ĂůĞƌſǁŶŝĞǏnjĂƐƚħƉŽǁĂđŶŽƚĂƚŬħnjůĞŬĐũŝ͘hųĂƚǁŝƚŽnjƉĞǁŶŽƑĐŝČƉƌĂĐħƵĐnjŶŝŽŵnjnjĂďurzeniami o charakterze dyslektycznym czy dysgraficznym. ŽƐƚŽƐŽǁĂŶŝĞŬĂƌƚĚŽƉƌĂĐLJnjƵĐnjŶŝĞŵŽƐƉĞĐLJĨŝĐnjŶLJĐŚƚƌƵĚŶŽƑĐŝĂĐŚǁƵĐnjĞŶŝƵƐŝħŵĂƚĞŵatyki ma swoje odzwierĐŝĞĚůĞŶŝĞƌſǁŶŝĞǏǁƌŽĚnjĂũĂĐŚnjĂĚĂŷ͕ŬƚſƌĞƚĞŬĂƌƚLJnjĂǁŝĞƌĂũČ͘KƉƌſĐnjƐƚĂŶĚĂrĚŽǁLJĐŚnjĂĚĂŷnjĂŵŬŶŝħƚLJĐŚŝŽƚǁĂƌƚLJĐŚŵŽǏĞŵLJŶĂŶŝĐŚnjŶĂůĞǍđƐĞƌŝħnjĂĚĂŷƚLJƉƵͣƉƌĂǁĚĂĐnjLJĨĂųƐnj͕͟ ͣĚŽƉĂƐƵũ ƉŽƐnjĐnjĞŐſůŶĞ ĞůĞŵĞŶƚLJ͕͟ ͣƵnjƵƉĞųŶŝũ ůƵŬŝ͕͟ ͣƵnjƵƉĞųŶŝũ ƉŽĚĂŶLJ ƐĐŚĞŵĂƚ ƌŽnjǁŝČnjĂŶŝĂ͕͟ ͣĚoŬŽŷĐnjƌŽnjǁŝČnjĂŶŝĞnjĂĚĂŶŝĂ͘͟EŝĞƉŽŵŝŶŝħƚŽƚĞǏnjĂĚĂŷƚLJƉƵ͗ͣƵĚŽǁŽĚŶŝũ͕ͣ͟ƵnjĂƐĂĚŶŝũ͕ͣ͟ǁLJŬĂǏ͕ǏĞ͕͟ĂůĞ ƉŽũĂǁŝĂũČ Ɛŝħ ŽŶĞ ƐƉŽƌĂĚLJĐnjŶŝĞ͘ EĂůĞǏLJ ƉĂŵŝħƚĂđ͕ ǏĞ njĂĚĂŶŝĂ ŶĂ ĚŽǁŽĚnjĞŶŝĞ ǁLJŵĂŐĂũČ ǁĞũƑĐŝĂ ŶĂ ƉŽnjŝŽŵĂďƐƚƌĂŬĐLJũŶĞŐŽŵLJƑůĞŶŝĂ͕ĐŽĚůĂƵĐnjŶŝſǁŽƐƉĞĐLJĨŝĐnjŶLJĐŚƉƌŽďůĞŵĂĐŚǁƵĐnjĞŶŝƵƐŝħ͕njǁųĂƐzĐnjĂ ĚůĂ ƚLJĐŚ nj ĂƵƚLJnjŵĞŵ͕ ũĞƐƚ ŶŝĞ ĚŽ ŽƐŝČŐŶŝħĐŝĂ͘ dĂŬŝĞ njĂĚĂŶŝĂ ƐČ ƵnjŶĂǁĂŶĞ njĂ ƚƌƵĚŶĞ ŶĂǁĞƚ ĚůĂ ucznia bez dysfunkcji rozwojowych. WƌŽďůĞŵĞŵƵĐnjŶŝſǁnjĚLJƐŬĂůŬƵůŝČũĞƐƚŵŝħĚnjLJŝŶŶLJŵŝnjĂƉŽŵŝŶĂŶŝĞǁnjŽƌſǁĐnjLJŬŽůĞũnych etaƉſǁũĂŬŝĞũƑŽƉĞƌĂĐũŝ͘WƌĂĐƵũČĐnjƚLJŵŝŬĂƌƚĂŵŝ͕ƵĐnjĞŷŵĂŶŝĐnjLJŵŶŝĞŽŐƌĂŶŝĐnjŽŶLJĚŽƐƚħƉĚŽǁnjŽƌſǁʹ ĐnjħƐƚŽ ƉŽũĂǁŝĂũČ Ɛŝħ ŽŶĞ ŶŝĞ ƚLJůŬŽ ǁ ĐnjħƑĐŝ ƚĞŽƌĞƚLJĐnjŶĞũ ŬĂƌƚLJ͕ ĂůĞ ƐČ ƌſǁŶŝĞǏ ƉƌnjLJƉŝƐĂŶĞ ĚŽ ƚƌĞƑĐŝ njĂĚĂŶŝĂ͘ŶƚŝĚŽƚƵŵŶĂƚƌƵĚŶŽƑĐŝǁnjĂƉĂŵŝħƚLJǁĂŶŝƵŬŽůĞũŶLJĐŚŬƌŽŬſǁǁŽƉĞƌĂĐũŝŵŽŐČďLJđnjƉĞǁŶoƑĐŝČnjĂĚĂŶŝĂƚLJƉƵͣƵnjƵƉĞųŶŝũůƵŬŝ͟ĐnjLJͣƵnjƵƉĞųŶŝũƉŽĚĂŶLJƐĐŚĞŵĂƚƌŽnjǁŝČnjĂŶŝĂ͕͟ǁŬƚſƌLJĐŚƉƌnjĞƉƌŽǁaĚnjĂŵLJƵĐnjŶŝĂŬƌŽŬƉŽŬƌŽŬƵƉƌnjĞnjŬŽůĞũŶĞĞƚĂƉLJ͘tŬĂƌƚĂĐŚƉƌnjLJƌŽnjǁŝČnjLJǁĂŶŝƵƉƌŽďůĞŵſǁŵĂƚĞŵatycznych stosowany jesƚǁLJƌĂǍŶLJƉŽĚnjŝĂųŶĂĞƚĂƉLJ͘hųĂƚǁŝƚŽďLJđŵŽǏĞƵĐnjŶŝŽǁŝŝĐŚnjĂƉĂŵŝħƚĂŶŝĞ͘ hĐnjŶŝŽǁŝĞnjĚLJƐůĞŬƐũČƌŽnjǁŽũŽǁČĐnjLJĚLJƐŬĂůŬƵůŝČƉŽƉĞųŶŝĂũČůŝĐnjŶĞďųħĚLJƉƌnjLJƉƌnjĞƉŝƐLJǁĂŶŝƵ ĚĂŶLJĐŚůŝĐnjďŽǁLJĐŚnjƚƌĞƑĐŝnjĂĚĂŶŝĂ͘ųħĚLJƚĞƉŽůĞŐĂũČŶĂůƵƐƚƌnjĂŶLJŵnjĂƉŝƐLJǁĂŶŝƵĐLJĨƌ͕ŽƉƵƐzczaniu ich ůƵď ƉƌnjĞƐƚĂǁŝĂŶŝƵ ĐnjLJ ƚĞǏ ŶĂ ƉƌnjĞƐƵŶŝħĐŝƵ ƉƌnjĞĐŝŶŬĂ ǁ ůŝĐnjďŝĞ ĚnjŝĞƐŝħƚŶĞũ͘ ĚĂƌnjĂ Ɛŝħ ƚŽ ƉƌnjĞǁĂǏŶŝĞ w njĂĚĂŶŝĂĐŚŽĚųƵǏƐnjĞũƚƌĞƑĐŝ͕ǁŬƚſƌLJĐŚĚĂŶĞůŝĐnjďŽǁĞƚƌnjĞďĂǁLJŽĚƌħďŶŝđnjƚĞŬƐƚƵ͘ďLJĐŚŽĐŝĂǏĐnjħͲ ƑĐŝŽǁŽƚĞŵƵnjĂƌĂĚnjŝđ͕ǁnjĂĚĂŶŝĂĐŚƚLJƉƵͣƵnjƵƉĞųŶŝũƉŽĚĂŶLJƐĐŚĞŵĂƚƌŽnjǁŝČnjĂŶŝĂ͟ĚĂŶĞůŝĐnjďŽǁĞƉoũĂǁŝĂũČƐŝħĚǁƵŬƌŽƚŶŝĞʹ ǁƚƌĞƑĐŝnjĂĚĂŶŝĂŽƌĂnjǁŽĚƌħďŶLJĐŚƉŽůĂĐŚ͘dĂŬŝƐƉŽƐſďƉƌĞnjĞŶƚĂĐũŝĚĂŶLJĐŚ njǁŝħŬƐnjĂŝĐŚĐnjLJƚĞůŶŽƑđŝŽŐƌĂŶŝĐnjĂǁnjŶĂĐnjŶLJŵƐƚŽƉŶŝƵƌŽnjƉƌŽƐnjĞŶŝĞƵǁĂŐŝƐƉŽǁŽĚŽǁĂŶĞŽďĞĐŶo- 10 ƑĐŝČƚĞŬƐƚƵǁƐČƐŝĞĚnjƚǁŝĞůŝĐnjďLJ͘tnjĂĚĂŶŝĂĐŚnjƉŽĚĂŶLJŵƐĐŚĞŵĂƚĞŵƌŽnjǁŝČnjĂŶŝĂnjĂĚďĂŶŽǁŝħĐŽƚŽ͕ ĂďLJŽƉĞƌĂĐũĞŵĂƚĞŵĂƚLJĐnjŶĞŵŽǏŶĂďLJųŽǁLJŬŽŶLJǁĂđǁďĞnjƉŽƑƌĞĚŶŝĞũďůŝƐŬŽƑĐŝnjǁĂƌƚŽƑĐŝĂŵŝůŝĐnjďoǁLJŵŝ͕ŬƚſƌĞƐČǁŶŝĐŚƐƚŽƐŽǁĂŶĞ͘tŬĂƌƚĂĐŚũĞƐƚƚĞǏĐĂųĂŵĂƐĂnjĂĚĂŷ͕ŬƚſƌĞƚĂŬŝĐŚƵĚŽŐŽĚŶŝĞŷũƵǏ ŶŝĞnjĂǁŝĞƌĂũČ͘ hĐnjŶŝſǁnjnjĂďƵƌnjĞŶŝĂŵŝǁųĂƑĐŝǁLJŵŝĚůĂĚLJƐůĞŬƐũŝƌŽnjǁŽũŽǁĞũǁLJƌſǏŶŝĂƚĞǏƚĞŶĚĞŶĐũĂĚŽŶŝeƉƌĂǁŝĚųŽǁĞŐŽ ƌŽnjŵŝĞƐnjĐnjĂŶŝĂ ŶĂ ŬĂƌƚĐĞ ƌŽnjǁŝČnjĂŶŝĂ njĂĚĂŶŝĂ ;ƚũ͘ ƚĞŬƐƚƵ͕ njĂƉŝƐƵ ŵĂƚĞŵĂƚLJĐnjnego i ƌLJƐƵŶŬſǁͿ͘WƌŽǁĂĚnjŝƚŽĚŽĐŚĂŽƐƵǁnjĂƉŝƐŝĞŝƐƉƌnjLJũĂƉŽƉĞųŶŝĂŶŝƵĚŽĚĂƚŬŽǁLJĐŚďųħĚſǁ͘<ĂƌƚLJ͕ŬƚſƌĞ ŽƉƵďůŝŬŽǁĂŶŽ ǁ ƚĞũ ŬƐŝČǏĐĞ͕ ƉŽnjǁĂůĂũČ ƚĞŐŽ ƵŶŝŬŶČđ ʹ ǁ ǁŝħŬƐnjŽƑĐŝ njĂĚĂŷ ǁƐŬĂnjƵũĞŵLJ ƵĐnjŶŝŽǁŝ ŵŝĞũƐĐĞ ŶĂ ŽďůŝĐnjĞŶŝĂ͕ ƐƉŽƌnjČĚnjĞŶŝĞ ǁLJŬƌĞƐƵ ĨƵŶŬĐũŝ͕ ŐƌĂĨŝĐnjŶČ ŝŶƚĞƌƉƌĞƚĂĐũħ ƉƌnjĞĚnjŝĂųƵ ůŝĐnjďŽǁĞŐŽ i zapisanie ŽĚƉŽǁŝĞĚnjŝ͘ t ƉƌnjLJƉĂĚŬƵ njĂĚĂŷ ŐĞŽŵĞƚƌLJĐnjŶLJĐŚ ŽŐƌĂŶŝĐnjŽŶŽ ĚŽ minimum sytuacje, w ŬƚſƌLJĐŚ ƵĐnjĞŷ njŽďůŝŐŽǁĂŶLJ ũĞƐƚ ĚŽ ƐĂŵŽĚnjŝĞůŶĞŐŽ ǁLJŬŽŶĂŶŝĂ ƌLJƐƵŶŬƵ͘ tŝĂĚŽŵŽ͕ ǏĞ Ƶ ƵĐnjŶŝĂ z ĚLJƐůĞŬƐũČƌŽnjǁŽũŽǁČƌLJƐƵŶŬŝƚĞƐČŶŝĞƉŽƌĂĚŶĞ͕ŶŝĞĐnjLJƚĞůŶĞŝĐnjħƐƚŽũĞƐƚǁŶŝĐŚnjĂďƵƌnjŽny kierunek, ĐŽƉƌŽǁĂĚnjŝĚŽďųħĚſǁǁƌŽnjǁŝČnjĂŶŝƵnjĂĚĂŶŝĂĂůďŽƚŽƌŽnjǁŝČnjĂŶŝĞƵŶŝĞŵŽǏůŝǁŝĂ͘ <ĂƌƚLJƉƌĂĐLJnjĂŵŝĞƐnjĐnjŽŶĞǁƚĞũŬƐŝČǏĐĞƐČnjŶĂŬŽŵŝƚLJŵŶĂƌnjħĚnjŝĞŵĚŽǁLJƉƌĂĐŽǁLJǁĂŶŝĂƉoǁƚĂƌnjĂůŶŽƑĐŝ ŝ ƐĐŚĞŵĂƚLJnjŵƵ ;ŽĚďLJǁĂ Ɛŝħ ƚŽ ŽĐnjLJǁŝƑĐŝĞ ŬŽƐnjƚĞŵ ƌſǏŶŽƌŽĚŶŽƑĐŝ njĂĚĂŷͿ. Stanowi to ĨƵŶĚĂŵĞŶƚ ƉƌĂĐLJ nj ƵĐnjŶŝĞŵ ĚLJƐůĞŬƚLJĐnjŶLJŵ͕ Ƶ ŬƚſƌĞŐŽ ŽĐnjĞŬŝǁĂŶLJ ĞĨĞŬƚ ŽƐŝČŐĂŵLJ ǁƐŬƵƚĞŬ ǁŝĞůoŬƌŽƚŶĞŐŽƉŽǁƚĂƌnjĂŶŝĂƚLJĐŚƐĂŵLJĐŚđǁŝĐnjĞŷ͕ŶĂƚŽŵŝĂƐƚǁƉƌnjLJƉĂĚŬƵƵĐnjŶŝĂnjĂƵƚLJnjŵĞŵ͕ŬƚſƌLJƐĐŚeŵĂƚĂŵŝŽƉĞƌƵũĞĚŽƑđƐƉƌĂǁŶŝĞ͕ƵŵŽǏůŝǁŝĂĚŽƐŬŽŶĂůĞŶŝĞũĞŐŽŵŽcnych ƐƚƌŽŶ͕ĐŽũĞƐƚƉƌnjĞĐŝĞǏũĞĚŶČ z ƉŽĚƐƚĂǁŽǁLJĐŚnjĂƐĂĚǁŶĂƵĐnjĂŶŝƵ͘tŝħŬƐnjŽƑđnjĂĚĂŷŽƚǁĂƌƚLJĐŚƉŽƉƌnjĞĚnjŽŶŽƌŽnjǁŝČnjĂŶLJŵƉƌnjLJŬųaĚĞŵ ĂůďŽ njĂĚĂŶŝĞŵ Ž njďůŝǏŽŶĞũ ƚƌĞƑĐŝ͕ ƵũħƚLJŵ ǁ ĨŽƌŵƵůĞ ͣƵnjƵƉĞųŶŝũ ƉŽĚĂŶLJ ƐĐŚĞŵĂƚ ƌŽnjǁŝČnjĂŶŝĂ͕ ĚŽŬŽŷĐnjƌŽnjǁŝČnjĂŶŝĞnjĂĚĂŶŝĂ͘͟ĂƚĞŵ ŬĂƌƚLJĚŽƐƚĂƌĐnjĂũČƵĐnjŶŝŽǁŝĐnjLJƚĞůŶLJŝƉƌnjĞũƌnjLJƐƚLJƐĐŚĞŵĂƚƌŽzǁŝČnjĂŶŝĂ ƐƚĂŶĚĂƌĚŽǁLJĐŚ njĂĚĂŷ͕ ũĞĚŶŽĐnjĞƑŶŝĞ ǁĚƌĂǏĂũČĐ ŐŽ ĚŽ ƐĂŵŽĚnjŝĞůŶĞũ ƉƌĂĐLJ nj ƚĞŬƐƚĞŵŵĂƚeŵĂƚLJĐnjŶLJŵ͘ ^ĐŚĞŵĂƚLJĐnjŶĞ ƉŽĚĞũƑĐŝĞ ĚŽ ŶĂƵĐnjĂŶŝĂ ŵĂƚĞŵĂƚLJŬŝ nie znajdzie zapewne uznania w oczach pewnej ŐƌƵƉLJŶĂƵĐnjLJĐŝĞůŝŝůƵĚnjŝnjǁŝČnjĂŶLJĐŚnjŽƑǁŝĂƚČ͘ƌĂƉŽƌƚƵ/ŶƐƚLJƚƵƚĂĚĂŷĚƵŬĂĐLJjŶLJĐŚ Ž ƐƚĂŶŝĞ ƉŽůƐŬŝĞũ ĞĚƵŬĂĐũŝ ǁ ϮϬϭϯ ƌŽŬƵ ǁLJŶŝŬĂ͕ ǏĞ ŶĂƵĐnjĂŶŝĞ ƐĐŚĞŵĂƚĂŵŝ ũĞƐƚ ŶĂũǁŝħŬƐnjLJŵ ͣŐƌnjĞĐŚĞŵ͟ŶĂƵĐnjLJĐŝĞůŝŵĂƚĞŵĂƚLJŬŝ͘tĂƌƚŽũĞĚŶĂŬ ƉĂŵŝħƚĂđ͕ ǏĞ ƚŽǁųĂƑŶŝĞƐĐŚĞŵĂƚLJ i proste algoƌLJƚŵLJƐČƉƵŶŬƚĞŵǁLJũƑĐŝĂĚŽŽƐŝČŐŶŝħĐŝĂǁLJǏƐnjĞŐŽƉŽnjŝŽŵƵǁŝĞĚnjLJŝƵŵŝĞũħƚŶŽƑĐŝŵĂƚĞŵĂƚLJĐnjŶLJĐŚ͘ ĞďLJ ƵĐnjĞŷ njŽƐƚĂų ŽĚŬƌLJǁĐČ ƉŽƐnjƵŬƵũČĐLJŵ ǁųĂƐŶĞũ ƐƚƌĂƚĞŐŝŝ ƌŽnjǁŝČnjĂŶŝĂ ƉƌŽďůĞŵƵ͕ ŵƵƐŝ ŶĂũƉŝĞƌǁ ŽƉĂŶŽǁĂđ ƉƌŽƐƚĞĂůŐŽƌLJƚŵLJ͘:ĞĚŶĂŬnjĚĞĐLJĚŽǁĂŶĂǁŝħŬƐnjŽƑđƚLJĐŚƵĐnjŶŝſǁ͕ ĚŽŬƚſƌLJĐŚĂĚƌĞƐŽǁĂŶĞ ƐČ ŽŵĂǁŝĂŶĞ ŬĂƌƚLJ͕ ŶĂ ƚĞŶ ǁLJǏƐnjLJ ƉŽnjŝŽŵ ŶŝŐĚLJ ŶŝĞ ǁĞũĚnjŝĞ͘ ^ŬŽƌŽ njĞ ǁnjŐůħĚƵ ŶĂ ŝĐŚ ƌſǏŶŽƌŽĚŶĞ ŽŐƌĂŶŝĐnjĞŶŝĂǁƉƌnjLJƐǁĂũĂŶŝƵǁŝĞĚnjLJŶŝĞŵŽǏĞŵLJŝĐŚŶĂƵĐnjLJđŬƌĞĂƚLJǁŶĞŐŽŝƚǁſƌĐnjĞŐŽƉŽĚĞũƑĐŝĂĚŽ ŵĂƚĞŵĂƚLJŬŝ͕ƚŽƉŽŬĂǏŵLJŝŵƉƌnjLJŶĂũŵŶŝĞũƐĐŚĞŵĂƚLJŝƉŽųſǏŵLJŶĂĐŝƐŬŶĂƉŽǁƚĂƌnjĂůŶŽƑđ ŬůƵĐnjŽǁLJĐŚ ƵŵŝĞũħƚŶŽƑĐŝ ŵĂƚĞŵĂƚLJĐnjŶLJĐŚ͘ WŽŶŽǁŶŝĞ ǁŝħĐ ŶĂ ƉŝĞƌǁƐnjLJ ƉůĂŶ ǁLJƐƵǁĂ Ɛŝħ ĚŽƐƚŽƐŽǁĂŶŝĞ ŵĞƚŽĚ ƉƌĂĐLJĚŽŵŽǏůŝǁŽƑĐŝƵĐnjŶŝĂŽƐƉĞĐLJĨŝĐnjŶLJĐŚƉŽƚƌnjĞďĂĐŚǁƵĐnjĞŶŝƵƐŝħŵĂƚĞŵĂƚLJŬŝ͘ Ž ŬůƵĐnjŽǁLJĐŚ ƵŵŝĞũħƚŶŽƑĐŝ͕ ŶĂ ŬƚſƌLJĐŚ ͣƐnjŬŽůŶĂ ŵĂƚĞŵĂƚLJŬĂ͟ Ɛŝħ ŽƉŝĞƌĂ͕ njĂůŝĐnjLJđ ŶĂůĞǏLJ ŵŝħĚnjLJ ŝŶŶLJŵŝ ƵŵŝĞũħƚŶŽƑđ ǁLJŬŽŶLJǁĂŶŝĂ ĚnjŝĂųĂŷ ŶĂ ůŝĐnjďĂĐŚ ŽƌĂnj ƌŽnjǁŝČnjLJǁĂŶŝĂ ƌſǏŶĞŐŽ ƌŽĚnjĂũƵ ƌſǁŶĂŷŝŶŝĞƌſǁŶŽƑĐŝ;ŝƐƚŽƚŶLJĐŚƌſǁŶŝĞǏǁŬŽŶƚĞŬƑĐŝĞĞŐnjĂŵŝŶƵŵĂƚƵƌĂůŶĞŐŽnjŵĂƚĞŵĂƚLJŬŝͿ͘tŬĂrtacŚƉŽųŽǏŽŶŽĚƵǏLJŶĂĐŝƐŬŶĂŬƐnjƚĂųĐĞŶŝĞƚLJĐŚǁųĂƑŶŝĞƵŵŝĞũħƚŶŽƑĐŝ͘EĂũƉƌŽƐƚƐnjĞŶĂǁĞƚnjĂĚĂŶŝĂ͕ŬƚſͲ ƌLJĐŚƌŽnjǁŝČnjĂŶŝĞŽƉĂƌƚĞũĞƐƚŶĂƉŽĚƐƚĂǁŝĞŶŝƵŽĚƉŽǁŝĞĚŶŝĐŚǁĂƌƚŽƑĐŝůŝĐnjďŽǁLJĐŚĚŽǁnjŽƌƵ͕njŵƵƐnjĂũČ ƵĐnjŶŝĂĚŽǁLJŬŽŶĂŶŝĂŵŶŝĞũůƵďďĂƌĚnjŝĞũnjųŽǏŽŶLJĐŚŽƉĞƌĂĐũŝŶĂůŝĐnjďĂĐŚ͘tŝĂĚŽŵŽ͕ǏĞƉƌŽĐĞƐƚĞŶũĞƐƚ njĂŬųſĐŽŶLJƵƵĐnjŶŝĂnjĚLJƐŬĂůŬƵůŝČĐnjLJ ĚLJƐůĞŬƐũČƌŽnjǁŽũŽǁČ͕ŵĂŵLJǁŝħĐŵŽǏůŝǁŽƑđ͕ ĂďLJĐŚŽĐŝĂǏĐnjħƑĐŝŽǁŽƚĞŶƉƌŽďůĞŵnjŶŝǁĞůŽǁĂđ͘EŝĞŵŽǏŶĂũĞĚŶĂŬnjĂƉŽŵŝŶĂđ͕ǏĞƵƵĐnjŶŝĂnjƚLJŵŝnjĂďƵƌnjĞŶŝĂŵŝďųħĚLJ ƌĂĐŚƵŶŬŽǁĞ ŵĂũČ ŵŶŝĞũƐnjČ ǁĂŐħ͕ ĚĞĐLJĚƵũČĐĂ ũĞƐƚ njĂƑ ŵĞƚŽĚĂ ƌŽnjǁŝČnjĂŶŝĂ njĂĚĂŶŝĂ͘ EŝĞ njǁĂůŶŝĂ ƚŽ 11 ũĞĚŶĂŬŶĂƵĐnjLJĐŝĞůĂnjŽďŽǁŝČnjŬƵƉƌĂĐLJŶĂĚƐƉƌĂǁŶŽƑĐŝČƌĂĐŚƵŶŬŽǁČƚLJĐŚƵĐnjŶŝſǁ͘hŵŝĞũħƚŶŽƑđƌŽzǁŝČnjLJǁĂŶŝĂ ƌſǁŶĂŷ ƌſǏŶĞŐŽ ƚLJƉƵ ŵŽǏŶĂ ĚŽƐŬŽŶĂůŝđ ŶŝĞ ƚLJůŬŽ njĂ ƉŽŵŽĐČ ŬĂƌƚ͕ ŬƚſƌĞ ďĞnjƉŽƑƌĞĚŶŝŽ tych ƌſǁŶĂŷĚŽƚLJĐnjČ;ƚũ͘ƌſǁŶĂŷůŝŶŝŽǁLJĐŚ͕ƌſǁŶĂŷŬǁĂĚƌĂƚŽǁLJĐŚ͕ƌſǁŶĂŷƐƚŽƉŶŝĂǁŝħŬƐnjĞŐŽŶŝǏϮ͕ ƌſǁŶĂŷǁLJŵŝĞƌŶLJĐŚ͕njĂĚĂŷƚĞŬƐƚŽǁLJĐŚƐƉƌŽǁĂĚnjĂůŶLJĐŚĚŽƌſǁŶĂŷůŝŶŝŽǁLJĐŚĐnjLJǁƌĞƐnjĐŝĞƉƌŽƐƚLJĐŚ ƌſǁŶĂŷnjǁĂƌƚŽƑĐŝČďĞnjǁnjŐůħĚŶČͿ͘ZſǁŶŝĞǏǁŝŶŶLJĐŚŬĂƌƚĂĐŚ͕njǁųĂƐnjĐnjĂƚLJĐŚĚŽƚLJĐnjČĐLJĐŚƉůĂŶŝŵĞƚƌŝŝ͕ ƐƚĞƌĞŽŵĞƚƌŝŝ͕ ĐŝČŐſǁ ĐnjLJ ƐƚĂƚLJƐƚLJŬŝ͕ njŶĂũĚnjŝĞŵLJ ǁŝĞůĞ njĂĚĂŷ͕ ŬƚſƌĞ njŵƵƐnjĂũČ ƵĐnjŶŝĂ ĚŽ ƌŽnjǁŝČnjĂŶŝĂ ŽĚƉŽǁŝĞĚŶŝĞŐŽƌſǁŶĂŶŝĂ͘tŶŝĞĐŽǁŵŶŝĞũƐnjLJŵƐƚŽƉŶŝƵƉƌĞnjĞŶƚŽǁĂŶĞƐČƚƵŶŝĞƌſǁŶŽƑĐŝ;ŶŝĞƌſǁŶoƑĐŝůŝŶŝŽǁĞŝŶŝĞƌſǁŶŽƑĐŝŬǁĂĚƌĂƚŽǁĞͿ͘ tŬĂƌƚĂĐŚnjĂĚďĂŶŽƚĞǏŽƚŽ͕ĂďLJǁŵŝĂƌħŵŽǏůŝǁŽƑĐŝ͕ŬŽŶĐĞŶƚƌƵũČĐƐŝħŶĂƵŽŐſůŶŝĂŶŝƵŝŶĚƵkĐLJũŶLJŵ͕ ƐƚǁĂƌnjĂđ ƐLJƚƵĂĐũĞ ĚŽ ŬƐnjƚĂųĐĞŶŝĂ ƵŵŝĞũħƚŶŽƑĐŝ ƵŽŐſůŶŝĂŶŝĂ͕ ŬƚſƌĞ ũĞƐƚ ĨƵŶĚĂŵĞŶƚĞŵ ƌŽnjuŵŽǁĂŶŝĂŵĂƚĞŵĂƚLJĐnjŶĞŐŽ͘hĐnjĞŷĐnjħƐƚŽƉŽǁLJŬŽŶĂŶŝƵĐŝČŐƵƉƌſďŵĂƚĞŵĂƚLJĐnjŶych musi dostrzec ƉƌĂǁŝĚųŽǁŽƑđ ǁ ƌĞnjƵůƚĂƚĂĐŚ ƚLJĐŚ Ɖƌſď ŝ ƐĨŽƌŵƵųŽǁĂđ ŽĚƉŽǁŝĞĚŶŝ ǁŶŝŽƐĞŬ͘ KĚŬƌLJǁĂŶŝĞ ƉƌŽƐƚLJĐŚ njĂůĞǏŶŽƑĐŝŝƉƌĂǁŝĚųŽǁŽƑĐŝǁŵĂƚĞŵĂƚLJĐĞŽĚďƵĚŽǁƵũĞƉŽĐnjƵĐŝĞǁŝĂƌLJǁĞǁųĂƐŶĞŵŽǏůŝǁŽƑĐŝ͕ŬƚſƌĞ ũĞƐƚ ǁLJƌĂǍŶŝĞ ŽƐųĂďŝŽŶĞ Ƶ ƵĐnjŶŝĂ njĞ ƐƉĞĐLJĨŝĐnjŶLJŵŝ ƉƌŽblemami w ƵĐnjĞŶŝƵ Ɛŝħ ŵĂƚĞŵĂƚLJŬŝ͘ tnjŽƌLJ i njĂůĞǏŶŽƑĐŝŵĂƚĞŵĂƚLJĐnjŶĞǁƉƌŽǁĂĚnjŽŶĞƚČŵĞƚŽĚČnjƌĞŐƵųLJƐČųĂƚǁŝĞũƐnjĞĚŽnjĂƉĂŵŝħƚĂŶŝĂ͘tŬĂrƚĂĐŚ͕ ƉƌŽĐĞƐ ƵŽŐſůŶŝĂŶŝĂ͕ ĚŽƑđ ƚƌƵĚŶLJ ĚůĂ ƵĐnjŶŝĂ nj njĂďƵƌnjĞŶŝĂŵŝ Ž ĐŚĂƌĂŬƚĞƌnjĞ ĚLJƐůĞŬƚLJĐnjŶLJŵ͕ ũĞƐƚ odpowiednio stymulowany i ŬŽŶƚƌŽůŽǁĂŶLJ͘ hĐnjĞŷ ŽƚƌnjLJŵƵũĞ ŬŽŵƉůĞƚŶLJ njĂŬƌĞƐ Ɖƌſď͕ Ă ũĞŐŽ ƌŽůĂ ƐƉƌŽǁĂĚnjĂƐŝħĚŽŝĐŚƉƌnjĞƉƌŽǁĂĚnjĞŶŝĂŝƉŽƐƚĂǁŝĞŶŝĂǁŶŝŽƐŬƵ͘dĞŶƐƉŽƐſďǁĚƌĂǏĂŶŝĂƵĐnjŶŝĂĚŽƉƌoƐƚĞŐŽƵŽŐſůŶŝĂŶŝĂƚLJƉƵŝŶĚƵŬĐLJũŶĞŐŽŵŽǏŶĂnjĂŽďƐĞƌǁŽǁĂđƉƌnjLJŽŵĂǁŝĂŶŝƵ͗ ଶ ଷ య ଷ య ସ x ƉŽƚħŐ൫ξܽ൯ ǡ ൫ξܽ൯ ǡ ൫ ξܽ ൯ ǡ ൫ ξܽ ൯ (Karta pracy 5), x ƉŽƌſǁŶLJǁĂŶŝĂƉŽƚħŐŽƚLJĐŚƐĂŵLJĐŚƉŽĚƐƚĂǁĂĐŚ(Karta pracy 7), x ǁųĂƐŶŽƑĐŝǁĂƌƚŽƑĐŝďĞnjǁnjŐůħĚŶĞũ (Karta pracy 10), x njĂƐĂĚƉŽƐƚħƉŽǁĂŶŝĂnjĞnjǁƌŽƚĞŵŶŝĞƌſǁŶŽƑĐŝŵŶŽǏŽŶĞũŽďƵƐƚƌŽŶŶŝĞƉƌnjĞnjůŝĐnjďħĚŽĚĂƚŶŝČ ĂůďŽƵũĞŵŶČ(Karta pracy 18), x graficznej ŵĞƚŽĚLJƌŽnjǁŝČnjLJǁĂŶŝĂƵŬųĂĚſǁƌſǁŶĂŷůŝŶŝŽǁLJĐŚ(Karta pracy 19), x ƉƌnjĞƐƵŶŝħĐŝĂǁLJŬƌĞƐƵĨƵŶŬĐũŝŬǁĂĚƌĂƚŽǁĞũǁnjĚųƵǏŽƐŝKyŝKz(Karta pracy 21), x ǁnjŽƌƵŽŐſůŶĞŐŽĐŝČŐƵĂƌLJƚŵĞƚLJĐnjŶĞŐŽ (Karta pracy 31), x ǁnjŽƌƵŽŐſůŶĞŐŽĐŝČŐƵŐĞŽŵĞƚƌLJĐnjŶĞŐŽ (Karta pracy 33), x wzoru ŶĂƐƵŵħƉŽĐnjČƚŬŽǁLJĐŚǁLJƌĂnjſǁĐŝČŐƵŐĞŽŵĞƚƌLJĐnjŶĞŐŽ;<ĂƌƚĂƉƌĂĐLJ 34), x ǁnjŽƌƵŶĂƐƚŽƐƵŶĞŬŽďǁŽĚſǁŝƉſůǁŝĞůŽŬČƚſǁƉŽĚŽďŶLJĐŚ(Karta pracy 50). WƌnjLJ ƉůĂŶŽǁĂŶŝƵ ƉƌŽĐĞƐƵ ĚLJĚĂŬƚLJĐnjŶĞŐŽ nj ƵĚnjŝĂųĞŵ ƵĐnjŶŝĂ Ž ƐƉĞĐLJĨŝĐnjŶLJĐŚ ƚƌƵĚŶŽƑĐŝĂĐŚ w ƵĐnjĞŶŝƵƐŝħƉŽǁŝŶŶŝƑŵLJďĂnjŽǁĂđŶĂƚLJŵ͕ĐŽƚĞŶƵĐnjĞŷƉŽƚƌĂĨŝŝǁLJŬŽŶƵũĞŶĂũůĞƉŝĞũ͕ŽƌĂnjƐnjƵŬĂđĚůĂ ŶŝĞŐŽƚĂŬŝĐŚŽďƐnjĂƌſǁŽĚĚnjŝĂųLJǁĂŷ͕ǁŬƚſƌLJĐŚŵĂŽŶǁŝħŬƐnjĞƐnjĂŶƐĞŶĂŽƐŝČŐŶŝħĐŝĞƐƵŬĐĞƐƵŝƉŽdǁLJǏƐnjĞŶŝĞǁųĂƐŶĞũƐĂŵŽŽĐĞŶLJ͘EĂƐnjĞĚnjŝĂųĂŶŝĂƉŽǁŝŶŶLJƉſũƑđǁŬŝĞƌƵŶŬƵƐƚǁĂƌnjĂŶŝĂƚĂŬŝĐŚƐLJƚƵĂcji, ŬƚſƌĞŵŝŶŝŵĂůŝnjƵũČƐųĂďĞƐƚƌŽŶLJƵĐnjŶŝĂŝŽŐƌĂŶŝĐnjĂũČŵŽǏůŝǁŽƑĐŝƉŽƉĞųŶŝĞŶŝĂƉƌnjĞnjŶŝĞŐŽďųħĚƵ͘tųĂƑŶŝĞƚĂŬĂŵLJƑůŬŝĞƌŽǁĂųĂautorem ƉƌnjLJŽƉƌĂĐŽǁLJǁĂŶŝƵŬĂƌƚƉƌĞnjĞŶƚŽǁĂŶLJĐŚǁƚĞũŬƐŝČǏĐĞ͘ 12 Zbiory liczbowe Karta pracy 1 /MZ>/Zzt/^dz, Zadanie 1. tLJƉŝƐnjĞůĞŵĞŶƚLJnjďŝŽƌſǁ. a) ܣെ njďŝſƌůŝĐnjďŶĂƚƵƌĂůŶLJĐŚŵŶŝĞjszych od 5; ܣൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ N ʹ liczby naturalne NW ʹ liczby niewymierne b) ܤെ njďŝſƌůŝĐnjďŶĂƚƵƌĂůŶLJĐŚݔ ƐƉĞųŶŝĂũČĐLJĐŚǁĂƌƵŶĞŬ͗ʹ ݔ൏ ͻ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤǤ C ʹ ůŝĐnjďLJĐĂųŬŽǁŝƚĞ W ʹ liczby wymierne c) ܥെ njďŝſƌůŝĐnjďĐĂųŬŽǁŝƚLJĐŚ ݔƐƉĞųŶŝaũČĐLJĐŚǁĂƌƵŶĞŬ͗െͶ ൏ ݔ ʹ ܥൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥ d) ܦെ njďŝſƌŶĂƚƵƌĂůŶLJĐŚĚnjŝĞůŶŝŬſǁ liczby 12 ܦൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤǤ ࡺ ൌ ሼǡ ǡ ǡ ǡ ǡ ǥ ሽ ൌ ሼǥ ǡ െǡ െǡ െǡ ǡ ǡ ǡ ǡ ǥ ሽ ࢃ െůŝĐnjďLJ͕ŬƚſƌĞŵŽǏŶĂnjĂƉŝƐĂđ w ƉŽƐƚĂĐŝƵųĂŵŬĂnjǁLJŬųĞŐŽ ࡺࢃ െ ůŝĐnjďLJ͕ŬƚſƌLJĐŚŶŝĞŵŽǏŶĂ njĂƉŝƐĂđǁƉŽƐƚĂĐŝƵųĂŵŬĂnjǁLJŬųĞŐŽ ďŝſƌ͕ŬƚſƌLJŶŝĞma ǏĂĚŶĞŐŽĞůĞŵĞŶƚƵ nazywamy zbiorem pustym ŝŽnjŶĂĐnjĂŵLJƐLJŵďŽůĞŵT͘ e) ܧെ njďŝſƌůŝĐnjďĐĂųŬŽǁŝƚLJĐŚ͕ŬƚſƌLJĐŚ kwadrat jest mniejszy od 5 ܧൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤǤ f) ܨെ njďŝſƌůŝĐnjďǁLJŵŝĞƌŶLJĐŚ͕ŬƚſƌLJĐŚ ŬǁĂĚƌĂƚũĞƐƚƌſǁŶLJϯ ܨൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ Zadanie 2. njLJnjďŝſƌ ܣjest podzbiorem zbioru ?ܤWpisz TAK lub NIE. W przypadku wskazania odpowiedzi NIE ƵnjĂƐĂĚŶŝũƐǁſũǁLJďſƌ͘ a) ܣൌ ሼʹǡ ͵ǡ Ͷǡ ͷሽǡ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ ďŝſƌ jest podzbiorem zbioru , ũĞǏĞli ŬĂǏĚLJĞůĞŵĞŶƚnjďŝŽƌƵ jest elementem zbioru . Zapisujemy to tak: ؿ NIE, ƉŽŶŝĞǁĂǏůŝĐnjďĂͷ jest elementem zbioru ܣi nie jest elementem zbioru ܤ. 13 c) ܣൌ ൛ξͶǡ ξͻൟǢ ܤൌ ሼʹǡ ͵ǡ Ͷሽ b) ܣൌ ሼെͳǡͲǡ ͳሽ Ǣ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ሽ ............................................................. ............................................................. ............................................................. ............................................................. ଵ ଷ ହ ଵ ଷ ଽ ସ ଽ e) ܣൌ ሼͳǡ ʹǡ ͵ሽǡ ܤൌ ቄͳǡ ǡ ǡ Ͷቅ ଶ ଷ d) ܣൌ ቄ ǡ ǡ ቅ ܤൌ ቄ ǡ ǡ ǡ ቅ ଶ ଶ ଶ ଶ ଶ ଶ ଶ ............................................................. ............................................................. ............................................................. ............................................................. Zadanie 3. ŽnjďŝŽƌƵnjƐnjĂƌĞŐŽƉŽůĂĚŽƉĂƐƵũnjďŝŽƌLJnjƉſůďŝĂųLJĐŚǁĞĚųƵŐnjĂƐaĚLJ͗njďŝſƌnjƉŽůĂďŝĂųĞŐŽũĞƐƚƉŽĚnjďŝŽƌĞŵnjďŝŽƌƵnjƉŽůĂƐnjĂƌĞŐŽ͘ ሼǡ ǡ ǡ ǡ ǡ ǡ ሽ ሼૠǡ ૡǡ ૢǡ ǡ ǡ ሽ kwadraty liczb naturalnych mniejszych od 4 naturalne dzielniki liczby 12 liczby naturalne mniejsze od 12 ሼെǡ ǡ ǡ ǡ ǡ ሽ ൛ξǡ ξૢǡ ξǡ ξૡൟ liczby pierwsze mniejsze od 13 ൛ξǡ ξǡ ξൟ Zadanie 4. ĚŝĂŐƌĂŵƵƉŽŶŝǏĞũŽĚĐnjLJtaj elementy njďŝŽƌſǁ ܣǡ ܤǡ ܤ ܣǤ a) ܣൌ ሼെͳǡ Ͳǡ ͳǡ ʹሽǢ ܤൌ ሼͲǡ ͳǡ ͵ǡ Ͷǡ ͷሽ ܤ ܣൌ ሼെͳǡ Ͳǡ ͳǡ ʹǡ ͵ǡ Ͷǡ ͷሽ ^hD/KZMt i െ ^ƵŵČnjďŝŽƌſǁ i ũĞƐƚnjďŝſƌƚLJĐŚ ĞůĞŵĞŶƚſǁ͕ŬƚſƌĞƐČĞůĞŵĞŶƚĂŵŝ zbioru lub zbioru . 14 b) ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥ c) ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤ ܤ ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܤ ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ െ െ Zadanie 5. ĂŶĞƐČnjďŝŽƌLJ ܣi ܤ. Wypisz elementy zbioru ܤ ܣ. a) ܣൌ ሼെ͵ǡ െʹǡ െͳሽ ܤൌ ሼെͳǡ Ͳǡ ͳǡ ʹǡ ͵ǡ Ͷሽ b) ܣൌ ሼͳǡ ʹǡ ͵ሽ ܤൌ ሼͳǡ ʹǡ ͵ሽ ൌ ሼെ͵ǡ െʹǡ െͳǡ Ͳǡ ͳǡ ʹǡ ͵ǡ Ͷሽ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ c) ܣൌ ሼെͶǡ െ͵ǡ െʹǡ െͳሽ ܤൌ ሼͳǡ ʹǡ ͵ǡ Ͷሽ d) ܣൌ ሼͲሽ ܤൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ͷሽ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ Zadanie 6. ĚŝĂŐƌĂŵƵƉŽŶŝǏĞũŽĚĐnjytaj elementy njďŝŽƌſǁ ܣǡ ܤǡ ܤ ת ܣ (o ile takie elementy istnieũČͿ͘ ILOCZYN /KZMt i ת a) ܣൌ ሼെ͵ǡ െʹǡ െͳǡ Ͳǡ ͳሽ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ ܤ ת ܣൌ ሼͲǡ ͳሽ A െ െ െ /ůŽĐnjLJŶĞŵnjďŝŽƌſǁ i ũĞƐƚnjďŝſƌ ƚĂŬŝĐŚĞůĞŵĞŶƚſǁ͕ŬƚſƌĞƐČ ƌſǁŶŽĐnjĞƑŶŝĞĞůĞŵĞŶƚĂŵŝ zbioru i zbioru Ǥ 15 b) ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥ c) ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤ ܤ ת ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܤ ת ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ െ ૡ ૠ Zadanie 7. ĂŶĞƐČ njďŝŽƌLJ ܣi ܤ. Wypisz elementy zbioru ( ܤ ת ܣo ile takie elementy ŝƐƚŶŝĞũČͿ͘ a) ܣൌ ሼെͷǡ െͶǡ െ͵ǡ െʹǡ െͳǡ Ͳሽ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ b) ܣൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ͷሽ ܤൌ ሼͳǡ ʹǡ ͵ǡ Ͷሽ תൌ ሼͲሽ תൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ c) ܣൌ ሼെͶǡ െ͵ǡ െʹǡ െͳሽ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ d) ܣൌ ሼ͵ǡ Ͷǡ ͷሽ ܤൌ ሼ͵ǡ Ͷǡ ͷሽ תൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ תൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ Zadanie 8. ĚŝĂŐƌĂŵƵƉŽŶŝǏĞũŽĚĐnjytaj elementy njďŝŽƌſǁ ܣǡ ܤǡ ܤ̳ܣǡ ܣ̳ܤ (o ile one istnieũČͿ͘ a) ܣൌ ሼെʹǡ െͳǡ Ͳǡ ͳሽ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ ܤ̳ܣൌ ሼെʹǡ െͳሽ ܣ̳ܤൌ ሼʹǡ ͵ǡ Ͷሽ ZME/ /KZMt i ̳ ̳ െ െ ZſǏŶŝĐČnjďŝŽƌſǁ i ũĞƐƚnjďŝſƌ ƚĂŬŝĐŚĞůĞŵĞŶƚſǁ͕ŬƚſƌĞƐČ elementami zbioru i ŶŝĞƐČ elementami zbioru Ǥ 16 b) ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥ c) ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤ ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤ ܤ̳ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܤ̳ܣൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ െ ܣ̳ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ܣ̳ܤൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ૡ Zadanie 9. ĂŶĞ ƐČ njďŝŽƌLJ ܣi ͘ܤtLJƉŝƐnj ĞůĞŵĞŶƚLJ njďŝŽƌſǁ ܤ̳ܣǢ ܣ̳ܤ (o ile one ŝƐƚŶŝĞũČͿ͘ a) ܣൌ ሼെǡ െͶǡ െʹǡ Ͳሽ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ b) ܣൌ ሼͳͲǡ ʹͲǡ ͵Ͳǡ ͶͲǡ ͷͲሽ ܤൌ ሼͳͲሽ ̳ ൌ ሼെǡ െͶǡ െʹሽ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ̳ ൌ ሼͳǡ ʹǡ ͵ǡ Ͷሽ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ c) ܣൌ ሼെͶǡ െ͵ǡ െʹǡ െͳሽ ܤൌ ሼͲǡ ͳǡ ʹǡ ͵ǡ Ͷሽ d) ܣൌ ሼ͵ǡ Ͷǡ ͷǡ ሽ ܤൌ ሼʹǡ ͵ǡ Ͷǡ ͷǡ ǡ ሽ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ Zadanie 10. ĂŶĞ ƐČ njďŝŽƌLJ͗ ܣെ njďŝſƌ ůŝĐnjď ŶĂƚƵƌĂůŶLJĐŚ ŵŶŝĞũƐnjLJĐŚ ŽĚ ϵ͖ ܤെ njďŝſƌ ŶĂƚƵƌĂůŶLJĐŚ ĚnjŝĞůŶŝŬſǁ ůŝĐnjďLJ ͳ͘ tLJƉŝƐnj ĞůĞŵĞŶƚLJ njďŝŽƌſǁ ܣǢ ܤǢ ܤ ܣǢ ܤ ת ܣǢ ܤ̳ܣǢ ܣ̳ܤ. ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ תൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤǤǤ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ ̳ ൌǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǥǤ 17
