No title

‫ﺩﻭﺭﻩﻱ ﺷﺎﻧﺰﺩﻫﻢ‬

‫‪/‬‬

‫ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺷﻤﺎﺭﻩﻱ ‪4‬‬

‫‪/‬‬

‫‪/‬‬

‫‪ 48‬ﺻﻔﺤﻪ‬

‫‪58‬‬

‫ﺭﻳﺎﺿﯽ‬ ‫ﺩﻭﺭﺓ ﺭﺍﻫﻨﻤﺎﻳﻰ ﺗﺤﺼﻴﻠﻰ‬

‫ﻭﺯﺍﺭﺕ ﺁﻣﻮﺯﺵ ﻭ ﭘﺮﻭﺭﺵ‬ ‫ﺳﺎﺯﻣﺎﻥ ﭘﮋﻭﻫﺶ ﻭ ﺑﺮﻧﺎﻣﻪﺭﻳﺰﻯ ﺁﻣﻮﺯﺷﻰ‬ ‫ﺩﻓﺘﺮ ﺍﻧﺘﺸﺎﺭﺍﺕ ﻛﻤﻚﺁﻣﻮﺯﺷﻰ‬

‫ﻣﺪﻳﺮ ﻣﺴﺆﻝ ‪ :‬ﻣﺤﻤﺪ ﻧﺎﺻﺮﻯ ﺳﺮﺩﺑﻴﺮ‪ :‬ﺣﻤﻴﺪﺭﺿﺎ ﺍﻣﻴﺮﻯ ﻣﺪﻳﺮ ﺩﺍﺧﻠﻰ ‪ :‬ﺣﺴﻴﻦ ﻧﺎﻣﻰ ﺳﺎﻋﻰ‬ ‫ﺍﻋﻀﺎﻯ ﻫﻴﺌﺖ ﺗﺤﺮﻳﺮﻳﻪ‪ :‬ﺣﺴﻦ ﺍﺣﻤﺪﻯ‪ ،‬ﺣﻤﻴﺪﺭﺿﺎ ﺍﻣﻴﺮﻯ‪ ،‬ﺯﻫﺮﻩ ﭘﻨﺪﻯ‪،‬‬ ‫ﺳﭙﻴﺪﻩ ﭼﻤﻦﺁﺭﺍ‪ ،‬ﺧﺴﺮﻭ ﺩﺍﻭﺩﻯ‪ ،‬ﻣﻴﺮﺷﻬﺮﺍﻡ ﺻﺪﺭ‪ ،‬ﺣﺴﻴﻦ ﻧﺎﻣﻰ ﺳﺎﻋﻰ‪ ،‬ﺳﻴﺪ ﻣﺤﻤﺪﺭﺿﺎ ﻫﺎﺷﻤﻰ ﻣﻮﺳﻮﻯ‬ ‫ﻭﻳﺮﺍﺳﺘﺎﺭ‪ :‬ﻣﺮﺗﻀﻰ ﺣﺎﺟﻌﻠﻰﻓﺮﺩ‬ ‫ﻃﺮﺍﺡ ﮔﺮﺍﻓﻴﻚ ‪ :‬ﻋﻠﻰ ﺩﺍﻧﺸﻮﺭ ﺗﺼﻮﻳﺮﮔﺮ‪ :‬ﺳﺎﻡ ﺳﻠﻤﺎﺳﻰ‬ ‫ﻧﺸﺎﻧﻰ ﺩﻓﺘﺮ ﻣﺠﻠﻪ ‪:‬ﺗﻬﺮﺍﻥ‪ ،‬ﺍﻳﺮﺍﻧﺸﻬﺮ ﺷﻤﺎﻟﻰ‪ ،‬ﭘﻼﻙ ‪ ،266‬ﺻﻨﺪﻭﻕ ﭘﺴﺘﻰ ‪ 6585‬ـ ‪15875‬‬ ‫ﻧﻤﺎﺑﺮ ‪88301478 :‬‬ ‫ﺗﻠﻔﻦ ‪9 :‬ـ‪ 8 8831161‬ـ‪ 021‬ﺩﺍﺧﻠﻰ‪374:‬‬ ‫ﺭﺍﻳﺎﻧﺎﻣﻪ‪[email protected] :‬‬ ‫ﭘﺎﻳﮕﺎﻩ ﺍﻃﻼﻉ ﺭﺳﺎﻧﻰ ‪www.roshdmag.ir :‬‬ ‫ﺗﻠﻔﻦ ﭘﻴﺎﻡﮔﻴﺮ ﻧﺸﺮﻳﺎﺕ ﺭﺷﺪ ‪88301482:‬‬ ‫ﻛﺪ ﻣﺪﻳﺮ ﻣﺴﺌﻮﻝ ‪ 102:‬ﻛﺪ ﺩﻓﺘﺮ ﻣﺠﻠﻪ ‪ 113 :‬ﻛﺪ ﻣﺸﺘﺮﻛﻴﻦ ‪102 :‬‬ ‫ﻧﺸﺎﻧﻰ ﺍﻣﻮﺭ ﻣﺸﺘﺮﻛﻴﻦ ‪ :‬ﺗﻬﺮﺍﻥ‪ ،‬ﺻﻨﺪﻭﻕ ﭘﺴﺘﻰ‪16595 / 111:‬‬ ‫ﺗﻠﻔﻦ ﺍﻣﻮﺭ ﻣﺸﺘﺮﻛﻴﻦ ‪77336656 :‬‬ ‫ﭼﺎپ ‪ :‬ﺷﺮﻛﺖ ﺍﻓﺴﺖ )ﺳﻬﺎﻣﻰ ﻋﺎﻡ(‬ ‫ﺷﻤﺎﺭﮔﺎﻥ ‪ 19000:‬ﻧﺴﺨﻪ‬

‫ﻓﻬﺮﺳﺖ‬ ‫ﺣﺮﻑ ﺍﻭﻝ ﺯﻛﺎﺕ ﺩﺍﻧﺶ‪ /‬ﺣﻤﻴﺪﺭﺿﺎ ﺍﻣﻴﺮﻱ‪2/‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫ﮔﻔﺖ ﻭ ﮔﻮ‬

‫ﺭﻳﺎﺿﻴﺎﺕ ﺭﻗﺎﺑﺘﻲ‪ /‬ﺍﺣﺴﺎﻥ ﻳﺎﺭﻣﺤﻤﺪﻱ‪3 /‬‬

‫ﻣﺴﺘﻄﻴﻞ ﻃﻮﻝ ﻛﺪﺍﻡ ﺍﺳﺖ؟‪ /‬ﺯﻳﻨﺐ ﻣﺮﺍﺩﺧﺎﻧﻲ‪7 /‬‬

‫ﺩﺭ‬

‫ﺑﺨﺶﭘﺬﻳﺮﻱ‪/‬‬

‫ﻣﺤﻤﻮﺩ ﺩﺍﻭﺭﺯﻧﻲ‪ 8 /‬ﻧﻘﺎﻁ ﺍﻣﻦ‪ /‬ﺣﺴﻦ ﺍﺣﻤﺪﻱ‪ 19 /‬ﺧﻮﺍﻧﺪﻧﻲﻫﺎﻳﻲ ﺍﺯ‬ ‫ﺭﻳﺎﺿﻴﺎﺕ‪ /‬ﺯﻳﻨﺐ ﮔﻠﺒﺮﺍﺭﻱ‪ 22 /‬ﻣﺴﺌﻠﻪﻫﺎﻱ ﻭﺍﻗﻌﻲ‪ /‬ﺍﻋﻈﻢ ﭘﻮﺭﭘﺮﻭﻳﻦ‪/‬‬ ‫‪25‬‬

‫ﻭﺍژﻩﻫﺎﻱ ﺭﻳﺎﺿﻲ »ﺳﺎﺩﻩﻛﺮﺩﻥ ﻋﺒﺎﺭﺕ«‪» ،‬ﺗﺴﺎﻭﻱ=«‪ /‬ﺳﭙﻴﺪﻩ‬

‫ﭼﻤﻦﺁﺭﺍ‪30 /‬‬

‫ﺩﻭ ﻣﺴﺌﻠﻪﻱ ﺟﺎﻟﺐ‪ /‬ﺷﺎﺩﻱ ﺑﻬﺎﺭﻱ‪ 34 /‬ﺣﻞ ﻣﺴﺌﻠﻪ‬

‫ﻗﺪﻡ ﺑﻪ ﻗﺪﻡ‪ /‬ﺳﺎﻳﻪ ﻣﻬﺮﺑﺎﻥ‪36 /‬‬ ‫ﺭﻳﺎﺿﻲ ﻭ ﺑﺎﺯﻱ ﺑﺎﺯﻱ ﺩﻭ ﻧﻔﺮﻩ‪ /‬ﺯﻫﺮﻩ ﭘﻨﺪﻱ‪6 /‬‬ ‫ﺟﺪﻭﻝ ﻭ ﺳﺮﮔﺮﻣﻲ ﺟﺪﻭﻝ‪ /‬ﻣﺤﻤّ ﺪ ﻋﺰﻳﺰﻱﭘﻮﺭ‪ 11 /‬ﺟﺪﻭﻝ ‪1‬‬

‫ﻣﻲﺗﻮﺍﻥ ﻫﺮ ﻣﺴﺌﻠﻪﺍﻱ ﺭﺍ ﺣﻞ ﻛﺮﺩ! ‪ /‬ﺁﺯﺍﺩﻩ‬

‫ﺷﺎﻛﺮﻱ‪14 /‬‬ ‫ﻫﻤﺮﺍﻩ ﺑﺎ ﻛﺘﺎﺏ ﻧﮕﺎﻫﻲ ﻧﻮ ﺑﻪ ﻣﻘﺴﻮﻡﻋﻠﻴﻪ‪ /‬ﻣﺠﻴﺪ ﻣﻨﺸﻮﺭﻱ‪/‬‬ ‫‪27‬‬

‫ﻣﻌﻤﺎ ﻭ ﺳﺮﮔﺮﻣﻲ‬

‫ﻣﻌﻤﺎﻫﺎﻳﻲ ﺑﻜﺮ ﺑﺮﺍﻱ ﺗﺎﺑﺴﺘﺎﻥ‪ /‬ﻋﻠﻴﺮﺿﺎ‬

‫ﻳﻮﺳﻔﻲ‪38 /‬‬ ‫ﺳﺆﺍﻝﻫﺎﻯ ﻣﺴﺎﺑﻘﻪﺍﻯ ﻣﺴﺎﺑﻘﻪﻱ ﺭﻳﺎﺿﻲ ﺍﺳﺘﺮﺍﻟﻴﺎ )‪/(2010‬‬ ‫ﺗﺮﺟﻤﻪﻱ ﺳﭙﻴﺪﻩ ﭼﻤﻦﺁﺭﺍ‪45 /‬‬

‫ﻣﻌﺮﻓﻰ ﻛﺘﺎﺏ‬

‫ﺭﻳﺎﺿﻴﺎﺕ ﺯﻳﺒﺎ ﻭ ﺩﻭﺳﺖﺩﺍﺷﺘﻨﻲ‪ /‬ﺟﻌﻔﺮ‬

‫ﺭﺑﺎﻧﻲ‪48 /‬‬

‫ﺗﺎ ‪ /100‬ﻧﺴﺮﻳﻦ ﺷﺮﻳﻔﻴﺎﻥ‪13/‬‬

‫ﺍﻧﺪﻳﺸﻪﻭﺭﺯﻱ‬

‫ﻣﺨﺎﻃﺮﺍﺕ ﺳﻔﺮ ﺩﺭ ﺳﻴﺎﺭﻩﻱ ﻧﺎﻟﻮﻣﺮ‪ /‬ﺗﺮﺟﻤﻪﻱ‬

‫ﺣﺴﻦ ﻳﺎﻭﺭﺗﺒﺎﺭ‪12 /‬‬

‫ﻗﺎﺑﻞ ﺗﻮﺟﻪ ﻧﻮﻳﺴﻨﺪﮔﺎﻥ ﻭ ﻣﺘﺮﺟﻤﺎﻥ‪:‬‬ ‫ﻼ ﺩﺭ ﺟﺎﻯ ﺩﻳﮕﺮﻯ ﭼﺎپ ﻧﺸﺪﻩ ﺑﺎﺷﺪ‪ .‬ـ ﻣﻘﺎﻟﻪﻫﺎﻯ ﺗﺮﺟﻤﻪ ﺷﺪﻩ ﺑﺎﻳﺪ ﺑﺎ ﻣﺘﻦ ﺍﺻﻠﻰ ﻫﻤﺨﻮﺍﻧﻰ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ ﻭ ﻣﺘﻦ ﺍﺻﻠﻰ ﻧﻴﺰ ﻫﻤﺮﺍﻩ ﺁﻥ‬ ‫ـ ﻣﻘﺎﻟﻪﻫﺎﻳﻰ ﻛﻪ ﺑﺮﺍﻯ ﺩﺭﺝ ﺩﺭ ﻣﺠﻠﻪ ﻣﻰﻓﺮﺳﺘﻴﺪ‪ ،‬ﺑﺎﻳﺪ ﺑﺎ ﺍﻫﺪﺍﻑ ﻭ ﺳﺎﺧﺘﺎﺭ ﺍﻳﻦ ﻣﺠﻠﻪ ﻣﺮﺗﺒﻂ ﺑﺎﺷﺪ ﻭ ﻗﺒ ً‬ ‫ﺑﺎﺷﺪ‪ .‬ﭼﻨﺎﻥﭼﻪ ﻣﻘﺎﻟﻪ ﺭ ﺍ ﺧﻼﺻﻪ ﻣﻰﻛﻨﻴﺪ‪ ،‬ﺍﻳﻦ ﻣﻮﺿﻮﻉ ﺭﺍ ﻗﻴﺪ ﺑﻔﺮﻣﺎﻳﻴﺪ‪ .‬ـ ﻣﻘﺎﻟﻪ ﻳﻚ ﺧﻂ ﺩﺭ ﻣﻴﺎﻥ‪ ،‬ﺩﺭ ﻳﻚ ﺭﻭﻯ ﻛﺎﻏﺬ ﻭ ﺑﺎ ﺧﻂ ﺧﻮﺍﻧﺎ ﻧﻮﺷﺘﻪ ﻳﺎ ﺗﺎﻳﭗ ﺷﻮﺩ‪ .‬ﻣﻘﺎﻟﻪﻫﺎ ﻣﻰﺗﻮﺍﻧﻨﺪ ﺑﺎ ﻧﺮﻡﺍﻓﺰﺍﺭ ‪ word‬ﻭ ﺑﺮ ﺭﻭﻯ ‪ CD‬ﻳﺎ ﻓﻼﭘﻰ ﻭ ﻳﺎ ﺍﺯ ﻃﺮﻳﻖ ﺭﺍﻳﺎﻧﺎﻣﻪ‬ ‫ﻣﺠﻠﻪ ﺍﺭﺳﺎﻝ ﺷﻮﻧﺪ‪ .‬ـ ﻧﺜﺮ ﻣﻘﺎﻟﻪ ﺑﺎﻳﺪ ﺭﻭﺍﻥ ﻭ ﺍﺯ ﻧﻈﺮ ﺩﺳﺘﻮﺭ ﺯﺑﺎﻥ ﻓﺎﺭﺳﻰ ﺩﺭﺳﺖ ﺑﺎﺷﺪ ﻭ ﺩﺭ ﺍﻧﺘﺨﺎﺏ ﻭﺍژﻩﻫﺎﻯ ﻋﻠﻤﻰ ﻭ ﻓﻨﻰ ﺩﻗﺖ ﻻﺯﻡ ﻣﺒﺬﻭﻝ ﺷﻮﺩ‪ .‬ـ ﻣﺤﻞ ﻗﺮﺍﺭ ﺩﺍﺩﻥ ﺟﺪﻭﻝﻫﺎ‪ ،‬ﺷﻜﻞﻫﺎ ﻭ ﻋﻜﺲﻫﺎ ﺩﺭ ﻣﺘﻦ ﻣﺸﺨﺺ ﺷﻮﺩ‪.‬‬ ‫ـ ﻣﻘﺎﻟﻪ ﺑﺎﻳﺪ ﺩﺍﺭﺍﻯ ﭼﻜﻴﺪﻩ ﺑﺎﺷﺪ ﻭ ﺩﺭ ﺁﻥ ﻫﺪﻑﻫﺎ ﻭ ﭘﻴﺎﻡ ﻧﻮﺷﺘﺎﺭ ﺩﺭ ﭼﻨﺪ ﺳﻄﺮ ﺗﻨﻈﻴﻢ ﺷﻮﺩ‪ .‬ـ ﻛﻠﻤﺎﺕ ﺣﺎﻭﻯ ﻣﻔﺎﻫﻴﻢ ﻧﻤﺎﻳﻪ )ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ( ﺍﺯ ﻣﺘﻦ ﺍﺳﺘﺨﺮﺍﺝ ﻭ ﺭﻭﻯ ﺻﻔﺤﻪﺍﻯ ﺟﺪﺍﮔﺎﻧﻪ ﻧﻮﺷﺘﻪ ﺷﻮﻧﺪ‪ .‬ـ ﻣﻘﺎﻟﻪ ﺑﺎﻳﺪ ﺩﺍﺭﺍﻯ ﺗﻴﺘﺮ ﺍﺻﻠﻰ‪ ،‬ﺗﻴﺘﺮﻫﺎﻯ‬ ‫ﻓﺮﻋﻰ ﺩﺭ ﻣﺘﻦ ﻭ ﺳﻮﺗﻴﺘﺮ ﺑﺎﺷﺪ‪ .‬ـ ﻣﻌﺮﻓﻰﻧﺎﻣﻪﻯ ﻛﻮﺗﺎﻫﻰ ﺍﺯ ﻧﻮﻳﺴﻨﺪﻩ ﭘﻴﻮﺳﺖ ﺷﻮﺩ‪ .‬ـ ﻣﺠﻠﻪ ﺩﺭ ﺭﺩ‪ ،‬ﻗﺒﻮﻝ‪ ،‬ﻭﻳﺮﺍﻳﺶ ﻭ ﺗﻠﺨﻴﺺ ﻣﻘﺎﻟﻪﻫﺎﻯ ﺭﺳﻴﺪﻩ ﺁﺯﺍﺩ ﺍﺳﺖ‪ .‬ـ ﻣﻘﺎﻻﺕ ﺩﺭﻳﺎﻓﺘﻰ ﺑﺎﺯﮔﺮﺩﺍﻧﺪﻩ ﻧﻤﻰﺷﻮﻧﺪ‪ .‬ـ ﺁﺭﺍﻯ ﻣﻨﺪﺭﺝ ﺩﺭ ﻣﻘﺎﻟﻪ ﺿﺮﻭﺭﺗﺎً ﻣﺒﻴﻦ‬ ‫ﺭﺃﻯ ﻭ ﻧﻈﺮ ﻣﺴﺌﻮﻻﻥ ﻣﺠﻠﻪ ﻧﻴﺴﺖ‪.‬‬

‫ﺣﺮﻑ ﺍﻭﻝ‬

‫ز﹋︀ت دا﹡︩‬ ‫ﺍﻟﻌﻠﻢ َﺗﻌﻠﻴﻤ ُﻪ َﻣﻦ ٰﻻ َﻳ ْﻌ َﻠ ُﻢ‪.‬‬ ‫ﮐـﺎ ُﺉ ْ ِ ْ‬ ‫ﻛﺴﻲ ﻛﻪ ﻧﻤﻲﺩﺍﻧﺪ‪.‬‬ ‫ﻗﺎﻝ ﺭﺳﻮﻝ ﺍﷲ )ﺹ(‪َ :‬ﺯ ٰ‬ ‫‪1‬‬

‫ﺶ‪ ،‬ﻳﺎﺩ ﺩﺍﺩﻥ ﺁﻥ ﺍﺳﺖ ﺑﻪ‬

‫ﺹ( ﻓﺮﻣﻮﺩﻧﺪ‪ :‬ﺯﻛﺎﺕ‪ 2‬ﺩﺍﻧ‬

‫ﭘﻴﺎﻣﺒﺮ )‬

‫ﺩﻫﻲ ﻭ ﻳﺎﺩﮔﻴﺮﻱ ﺭﻳﺎﺿﻲ‪،‬‬

‫ﻳﺎﺩﮔﻴﺮﻱ‪ ،‬ﺑﻪﻭﻳـﮋﻩ ﻳﺎﺩ‬

‫ﻦ ﺭﻭﺵﻫﺎﻱ ﻳﺎﺩﺩﻫﻲ ﻭ‬ ‫ﺍﺯ ﺑﻬﺘﺮﻳـ‬ ‫ﺩﻩ ﺍﺯ ﺧﺮﺩﺟﻤﻌﻲ ﺍﺳﺖ‪.‬‬ ‫ﺵ ﻛﺎﺭ ﮔﺮﻭﻫﻲ ﻭ ﺍﺳﺘﻔﺎ‬ ‫ﺮﻳﻦ ﺭﺍﻩ ﺗﺜﺒﻴﺖ ﻭ ﺗﻌﻤﻴﻖ‬ ‫ﺭﻭ ِ‬ ‫ﺿﻴﺎﺕ ﻳﺎﺩ ﮔﺮﻓﺘﻪﺍﻳﺪ‪ ،‬ﺑﻬﺘ‬ ‫ﺭﻳﺎ‬ ‫ﺩﺭ‬ ‫ﺭﺍ‬ ‫ﻲ‬ ‫ﺤﺜ‬ ‫ﻣﺒ‬ ‫ﻳﺎ‬ ‫ﻲ‬ ‫ﺩﺗﺎﻥ ﺍﺳﺖ‪ ،‬ﺯﻳﺮﺍ ﻫﻨﮕﺎﻡ‬ ‫ﺍﮔﺮ ﻣﻄﻠﺒ‬ ‫ﺎﻥ ﻭ ﻫﻤﻜﻼﺳﻲﻫﺎﻱ ﺧﻮ‬ ‫ﻳﺎﺩ ﺩﺍﺩﻥ ﺁﻥ ﺑﻪ ﺩﻭﺳـﺘ‬ ‫ﻧﻘﺎﻁ ﻣﺒﻬﻢ ﺁﻥ ﻣﻲﺑﺮﻳﺪ‬ ‫ﺁﻥ ﻣﻮﺿﻮﻉ‪،‬‬ ‫ﮕﺮﻱ‪ ،‬ﭘﻲ ﺑﻪ ﺍﺷﻜﺎﻻﺕ ﻭ‬ ‫ﻌﻠﻴﻢ ﻳﻚ ﻣﻮﺿﻮﻉ ﺑﻪ ﺩﻳ‬ ‫ﻓﺮﺍﻧﮕﺮﻓﺘﻪﺍﻳﺪ‪ .‬ﻫﻢﭼﻨﻴﻦ‬ ‫ﺗﺪﺭﻳﺲ ﻭ ﺗ‬ ‫ﺍﺯ ﺁﻥ ﺭﺍ ﺑﻪﻃﻮﺭ ﻛﺎﻣـﻞ‬ ‫ﻲﺷـﻮﻳﺪ ﻛﻪ ﺑﺨﺶﻫﺎﻳﻲ‬ ‫ﻛﻤﻚ ﻛﻨﺪ ﺗﺎ ﺁﻥ ﻣﻄﻠﺐ‬ ‫ﻭ ﻣﺘﻮﺟﻪ ﻣ‬ ‫ﺕ ﻭ ﻧﻈﺮﺍﺕ ﺧﻮﺩ ﺑﻪ ﺷﻤﺎ‬ ‫ﺍﻻ‬ ‫ﺳﺆ‬ ‫ﻭﺳﺖ ﺷﻤﺎ ﻧﻴﺰ ﻣﻲﺗﻮﺍﻧﺪ ﺑﺎ‬ ‫ﻞ ﻣﺮﺑﻮﻁ ﺑﻪ ﺁﻥ ﻣﻮﺿﻮﻉ‬ ‫ﺩ‬ ‫ﺑﻪ ﻛﻤﻚ ﻳﻜﺪﻳﮕﺮ ﻣﺴﺎﺋ‬ ‫ﻛﺎﻣﻞ ﺟﺎ ﺑﻴﻔﺘﺪ ﻭ ﺑﺘﻮﺍﻧﻴﺪ‬ ‫ﺑﺮﺍﻱ ﺷﻤﺎ‬ ‫ﺭﺍ ﺣﻞ ﻛﻨﻴﺪ‪.‬‬ ‫ﺹ( ﻧﻘﻞ ﺷـﺪ‪ ،‬ﺻﺤﺒﺖ ﺍﺯ‬ ‫ﻱ ﻛﻼﻡ ﺍﺯ ﭘﻴﺎﻣﺒﺮ ﺍﻛﺮﻡ )‬ ‫ﺪﺍ‬ ‫ﺍﺑﺘ‬ ‫ﺩﺭ ﺣﺪﻳﺚ ﻧﺒﻮﻱ ﻛﻪ ﺩﺭ‬ ‫ﻌﻠﻴﻢ ﺑﻪ ﻛﺴﺎﻧﻲ ﺍﺳﺖ ﻛﻪ‬ ‫ﺍﻛﺮﻡ )ﺹ( ﺯﻛﺎﺕ ﻋﻠﻢ‪ ،‬ﺗ‬ ‫ﺒﺮ‬ ‫ﻋﻠﻢ ﺍﺳـﺖ‪ .‬ﺍﺯ ﻧﮕﺎﻩ ﭘﻴﺎﻣ‬ ‫ﻋﻠﻢ ﺧﻮﺩ ﺑﻪ ﺩﻳﮕﺮﺍﻥ ﺑﻪ‬ ‫ﺯﻛﺎ ِ‬ ‫ﺕ‬ ‫ﻤﺎ ﺑـﺎ ﺗﻌﻠﻴﻢ ﻭ ﻳﺎﺩﺩﻫﻲ‬ ‫ﻲ ﺑﻬﺮﻩﺍﻧﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺷـ‬ ‫ﺍﺯ ﺁﻥ ﻋﻠﻢ ﺑ‬ ‫ﻲﺩﻫﻴﺪ ﻭ ﻫﻤﺎﻥﻃﻮﺭ ﻛﻪ‬ ‫ﻛﻪ ﺯﻛﺎﺕ ﻋﻠﻢ ﺧـﻮﺩ ﺭﺍ ﻣ‬ ‫ﺁﻥ‬ ‫ﻭﻝ‬ ‫ﺍ‬ ‫ﺪ‪:‬‬ ‫ﺩﻭ ﻣﻬﻢ ﺩﺳـﺖ ﻳﺎﻓﺘﻪﺍﻳ‬ ‫ﻠﻢ ﺷـﻤﺎ ﻧﻴﺰ ﺑﺎ ﻳﺎﺩﺩﻫﻲ‬ ‫ﻮﺩ ﺑﺎ ﺑﺮﻛﺖ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ ،‬ﻋ‬ ‫ﺷـ‬ ‫ﺖ‬ ‫ﻣﺎﻟﻲ ﻛﻪ ﺯﻛﺎﺗﺶ ﭘﺮﺩﺍﺧ‬ ‫ﺧﻮﺩﺗﺎﻥ ﻣﻮﺿﻮﻉ ﺭﺍ ﺑﻪﻃﻮﺭ‬ ‫ﻫﺮ‬ ‫ﺧﻮﺍﻫﺪ ﺷـﺪ‪ .‬ﺩﻭﻡ ﺁﻥﻛﻪ‬ ‫ﺖ ﻭ ﺍ ِﻥﺷـﺎءﺍ‪ ...‬ﻛﺎﺭﺑﺮﺩﻱ‬ ‫ﭘﺮﺑﺮﻛ‬ ‫ﻧﻴﺰ ﺑﻬﺮﻩﻣﻨﺪ ﻣﻲﺷﻮﻳﺪ‪.‬‬ ‫ﻴﺪ ﻭ ﺍﺯ ﻧﻈﺮﺍﺕ ﺩﻳﮕﺮﺍﻥ‬ ‫ﻛﺎﻣﻞ ﺩﺭﻙ ﻣﻲﻛﻨ‬ ‫ﺩﺭﺱ ﻭ ﺗﻤﺮﻳﻦ ﻭ ﺗﻼﺵ‪،‬‬ ‫ﺁﻳﻨﺪﻩ ﺑﺎ ﺗﻤﺮﻛﺰ ﺩﺭ ﻛﻼﺱ‬ ‫ﻲ‬ ‫ﺼﻴﻠ‬ ‫ﺗﺤ‬ ‫ﭘﺲ ﺑﻴﺎﻳﻴﺪ ﺩﺭ ﺳﺎﻝ‬ ‫ﺳﺘﺎﻥ ﻭ ﻫﻤﻜﻼﺳﻲﻫﺎﻱ‬ ‫ﻳﺪ ﻭ ﺳـﭙﺲ ﺁﻥ ﺭﺍ ﺑﻪ ﺩﻭ‬ ‫ﻴﺮ‬ ‫ﺑﮕ‬ ‫ﺐ ﺩﺭﺳـﻲ ﺭﺍ ﺧﻮﺏ ﻳﺎﺩ‬ ‫ﺯﻣﻴﻨﻪ ﺑﻪ ﻧﺘﻴﺠﻪﻱ ﺧﻮﺏ‬ ‫ﻣﻄﺎﻟ‬ ‫ﻳﺎﺩ ﺑﺪﻫﻴﺪ‪ .‬ﺍﮔﺮ ﺩﺭ ﺍﻳﻦ‬ ‫ﺁﻥ ﺩﺭﺱ ﺿﻌﻒ ﺩﺍﺭﻧﺪ‪،‬‬ ‫ﺧﻮﺩ ﻛﻪ ﺩﺭ‬ ‫ﻭ ﺑﺮﺍﻱ ﻣﺎ ﺍﺭﺳﺎﻝ ﻛﻨﻴﺪ‪.‬‬ ‫ﻛﺮﺩﻳﺪ ﺁﻥ ﺭﺍ ﺑﻨﻮﻳﺴﻴﺪ‬ ‫ﺮﻩﻱ ﺟﺎﻟﺒﻲ ﺩﺳﺖ ﭘﻴﺪﺍ‬ ‫ﻳﺎ ﺧﺎﻃ‬ ‫‪ ،2‬ﺣﺪﻳﺚ ‪.10‬‬ ‫‪ ،2‬ﺻﻔﺤﻪﻱ ‪5‬‬ ‫ﭘﻲﻧﻮﺷﺖ‬ ‫ﭼﺎپ ‪ ،(88‬ﺟﻠﺪ‬ ‫ﻣﻌﻨ ِﻲ ﭘﺎﻛﻴﺰﻩﺷ‬ ‫ﺠﻠﺴﻲ )ﺭﻩ(‪) ،‬‬ ‫ﻭ ﻫﻤﭽﻨﻴﻦ ﺑﻪ‬ ‫ﻻﻧﻮﺍﺭ‪ ،‬ﻋﻼﻣﻪ ﻣ‬ ‫ﻮ ﻭ ﻗﺪ ﻛﺸﻴﺪﻥ‬ ‫‪ .1‬ﺑﺤﺎﺭﺍ‬ ‫ﺮﺁﻧﻲ ﺑﻪ ﻣﻔﻬﻮﻡ ﻧﻤ ّ‬ ‫ﺕ ﺩﺭ ﻓﺮﻫﻨﮓ ﻗ‬ ‫‪ .2‬ﺯﻛﺎ‬

‫ﺪﻥ ﺁﻣﺪﻩ ﺍﺳﺖ‪.‬‬

‫‪2‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫ر﹬︀︲﹫︀ت ر﹇︀︋︐‪﹩‬‬ ‫ﺍﺣﺴﺎﻥ ﻳﺎﺭﻣﺤﻤﺪﻱ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﺭﻗﺎﺑﺖ‪ ،‬ﺗﻴﺰﻫﻮﺷﺎﻥ‪ ،‬ﺁﺯﻣﻮﻥ ﻭﺭﻭﺩﻱ‪ ،‬ﻣﻌﺎﺩﻟﻪﻱ ﺧﻄﻲ‪ ،‬ﺩﺳﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﺩﻭ ﻣﺠﻬﻮﻝ‪.‬‬

‫ﻣﻘﺪﻣﻪ‬

‫ﺭﻳﺎﺿﻲﭘﮋﻭﻩ ﺑﺮﺍﻱ ﭘﻴﺮﻭﺯﻱ ﻭ ﻛﺎﻣﻴﺎﺑﻲ ﺩﺭ ﺍﻳﻦﮔﻮﻧﻪ ﺭﻗﺎﺑﺖﻫﺎﻱ ﺭﻳﺎﺿﻲ‬

‫ﺑـﺎ ﺗﻮﺟﻪ ﺑﻪ ﻋﻼﻗـﻪﻱ ﺭﻭﺯﺍﻓـﺰﻭﻥ ﻧﻮﺟﻮﺍﻧﺎﻥ ﻣﻤﻠﻜـﺖ ﻋﺰﻳﺰﻣﺎﻥ‬

‫ﺑﺎﻳﺪ ﺑﻪ ﺁﻥ ﻣﺠﻬﺰ ﺑﺎﺷﺪ‪ .‬ﺩﺭ ﺍﻧﺘﻬﺎ ﻣﺘﺬﻛﺮ ﻣﻲﺷﻮﻡ ﺍﺯ ﺁﻥﺟﺎ ﻛﻪ ﻣﺴﺎﺋﻞ‬

‫ﺑﻪ ﺷـﺮﻛﺖ ﺩﺭ ﻣﺴـﺎﺑﻘﺎﺕ ﻭ ﺭﻗﺎﺑﺖﻫـﺎﻱ ﺭﻳﺎﺿﻲ ﻣﺎﻧﻨـﺪ ﺍﻟﻤﭙﻴﺎﺩﻫﺎﻱ‬

‫ﻣﻄﺮﺡ ﺷـﺪﻩ ﺩﺭ ﺍﻟﻤﭙﻴﺎﺩﻫﺎ‪ ،‬ﻣﺴـﺎﺑﻘﺎﺕ ﻭ ﺁﺯﻣﻮﻥﻫﺎﻱ ﺭﻳﺎﺿﻲ ﻣﺨﺘﺺ‬

‫ﺭﻳﺎﺿـﻲ‪ ،‬ﺁﺯﻣﻮﻥﻫﺎﻱ ﻭﺭﻭﺩﻱ ﻣﺪﺍﺭﺱ ﺗﻴﺰﻫﻮﺷـﺎﻥ‪ ،‬ﻣـﺪﺍﺭﺱ ﻧﻤﻮﻧﻪ‬

‫ﻧﻮﺟﻮﺍﻧﺎﻥ ﻧﻴﺰ ﺷﺎﻣﻞ ﻣﺒﺎﺣﺜﻲ ﺩﺭ ﻫﻨﺪﺳﻪ ‪ ،‬ﻧﻈﺮﻳﻪﻱ ﻋﺪﺩﻫﺎ ‪ ،‬ﺟﺒﺮ ﻭ‬

‫ﻭ ‪ ...‬ﻭ ﻧﻴﺰ ﺍﺳـﺘﻘﺒﺎﻝ ﺁﻥﻫﺎ ﺍﺯ ﻣﺴـﺎﺋﻞ ﭼﺎﻟﺶﭘﺬﻳﺮ ﺑﻪ ﻣﻨﻈﻮﺭ ﻛﺴـﺐ‬

‫ﺣﺴﺎﺏ ﻭ ﻫﻮﺵ ﺍﺳﺖ‪ ،‬ﺑﻪ ﺷﻤﺎ ﻧﻮﺟﻮﺍﻧﺎﻥ ﮔﺮﺍﻥﻗﺪﺭ ﭘﻴﺸﻨﻬﺎﺩ ﻣﻲﺷﻮﺩ‬

‫ﺁﮔﺎﻫـﻲ ﻭ ﺩﺍﻧـﺶ ﺑﻴﺶﺗـﺮ ﻭ ﺩﺭ ﻧﻬﺎﻳﺖ ﺍﺭﺗﻘﺎﻱ ﺳـﻄﺢ ﻛﻴﻔﻲ ﻭ ﻛﻤﻲ‬

‫ﻛﻪ ﺩﺭﺑﺎﺭﻩﻱ ﺍﺛﺒﺎﺕ ﻗﻀﺎﻳﺎ‪ ،‬ﺭﺍﻩﺣﻞﻫﺎ ﻭ ﭘﺎﺳـﺦﻫﺎﻱ ﻣﺴﺎﺋﻞ ﺍﺭﺍﺋﻪ ﺷﺪﻩ‪،‬‬

‫ﺍﻃﻼﻋـﺎﺕ ﻭ ﻣﻬﺎﺭﺕﻫﺎﻱ ﭘﻴﺮﺍﻣﻮﻥ ﻣﻮﺿﻮﻋﺎﺕ ﮔﻮﻧﺎﮔﻮﻥ ﺭﻳﺎﺿﻲ‪ ،‬ﺑﺮﺍﻱ‬

‫ﺍﺑﺘﺪﺍ ﺗﻔﻜﺮ ﻭ ﺗﻌﻤﻖ ﻣﻨﺎﺳـﺐ ﺭﺍ ﺩﺍﺷـﺘﻪ ﺑﺎﺷﻴﺪ ﻭ ﺑﻪ ﺻﻮﺭﺕ ﻣﺴﺘﻘﻴﻢ‬

‫ﺁﺷـﻨﺎﻳﻲ ﻫﺮﭼﻪ ﺑﻴﺶﺗـﺮ ﻧﻮﺟﻮﺍﻧﺎﻥ ﺍﻳﺮﺍﻥ ﻫﻤﻴﺸـﻪ ﺳـﺮﻓﺮﺍﺯ ﻛﻪ ﺩﺭ‬

‫ﺑﻪ ﺳـﺮﺍﻍ ﺑﺮﻫﺎﻥ ﻗﻀﺎﻳﺎ‪ ،‬ﺭﺍﻩﺣﻞﻫﺎ ﻭ ﭘﺎﺳـﺦﻫﺎﻱ ﺍﺭﺍﻳﻪ ﺷﺪﻩ ﺍﺯ ﺳﻮﻱ‬

‫ﺩﻭﺭﻩﻱ ﺭﺍﻫﻨﻤﺎﻳﻲ )ﺳﻨﻴﻦ ﺩﻭﺍﺯﺩﻩ‪ ،‬ﺳﻴﺰﺩﻩ ﻭ ﭼﻬﺎﺭﺩﻩ ﺳﺎﻟﮕﻲ( ﺑﻪ ﺳﺮ‬

‫ﻧﮕﺎﺭﻧـﺪﻩﻱ ﻣﻘﺎﻟﻪ ﻧﺮﻭﻳﺪ ﺗﺎ ﺑﺘﻮﺍﻧﻴﺪ ﺑﻪ ﻣﻬـﺎﺭﺕ ﻭ ﺗﻜﻨﻴﻚﻫﺎﻱ ﻻﺯﻡ ﻭ‬

‫ﻣﻲﺑﺮﻧﺪ‪ ،‬ﺗﺼﻤﻴﻢ ﺑﺮ ﺁﻥ ﺷـﺪ ﻛﻪ ﺳﻠﺴـﻠﻪ ﻣﻘﺎﻻﺗﻲ ﭘﻴﺮﺍﻣﻮﻥ ﻣﻮﺿﻮﻉ‬

‫ﻛﺎﻓﻲ ﺩﺭ ﺍﻳﻦ ﺯﻣﻴﻨﻪ ﺑﺮﺳﻴﺪ‪) .‬ﺍﻟﺒﺘﻪ ﺑﻪ ﺍﻳﻦ ﻋﻠﺖ ﻛﻪ ﺍﺭﺍﻳﻪﻱ ﺭﺍﻩﺣﻞ ﺩﺭ‬

‫ﻳﺎﺩﺷـﺪﻩ ﺑﺎ ﻋﻨﺎﻭﻳﻦ ﻣﻔﻴﺪ ﻭ ﻣﻮﺛﺮ ﺑـﺮﺍﻱ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﻧﻮﺟﻮﺍﻥ ﺗﻬﻴﻪ‬

‫ﻣﻮﺿﻮﻋﺎﺕ ﻫﻨﺪﺳﻪ ﻭ ﻧﻈﺮﻳﻪﻱ ﻋﺪﺩﻫﺎ ﻧﻴﺎﺯ ﺑﻪ ﺩﺭﻙ ﺷﻬﻮﺩﻱ ﻣﺘﻌﺎﻟﻲ‬

‫ﺷﻮﺩ ﻛﻪ ﺍﻳﻦ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺑﺎ ﺳﺒﻚ ﻭ ﺳﻴﺎﻕ ﻗﻀﺎﻳﺎ‪ ،1‬ﻧﻜﺎﺕ ﻣﻮﺭﺩ ﻧﻴﺎﺯ‪،‬‬

‫ﻭ ﻗـﻮﻩﻱ ﺧﻼﻗﻴﺖ ﺑﺎﻻ ﺩﺍﺭﻧﺪ ﻭ ﺣﻞ ﻛﺮﺩﻥ ﻣﺴـﺎﺋﻞ ﺩﺭ ﺍﻳﻦ ﻋﻨﺎﻭﻳﻦ ﺍﺯ‬

‫ﻣﺴـﺎﺋﻠﻲ ﻛﻪ ﺩﺭ ﺍﻳﻦ ﺯﻣﻴﻨﻪ ﻣﻄﺮﺡ ﻣﻲﺷﻮﺩ‪ :‬ﺑﻪ ﻫﻤﺮﺍﻩ ﺑﺮﻫﺎﻥ‪ ،2‬ﭘﺎﺳﺦ‬

‫ﻳﻚ ﺍﺳـﻠﻮﺏ ﻭ ﺭﻭﺵ ﺧﺎﺹ ﭘﻴﺮﻭﻱ ﻧﻤﻲﻛﻨﻨـﺪ ﻭ ﺑﺮﺍﻱ ﺣﻞ ﻫﺮﻳﻚ ﺍﺯ‬

‫ﻭ ﺭﻭﺵ ﺣﻞ ﺁﻥﻫﺎ ﺁﺷﻨﺎ ﺷﻮﻧﺪ‪ .‬ﺍﻟﺒﺘﻪ ﺑﺎ ﺑﺮﺭﺳﻲ ﺩﻗﻴﻖ ﻣﺴﺎﺋﻠﻲ ﻛﻪ ﺩﺭ‬

‫ﺁﻥﻫﺎ ﻧﻴﺎﺯ ﺑﻪ ﺩﺍﺷـﺘﻦ ﺧﻼﻗﻴﺘﻲ ﻣﺨﺘﺺ ﺑﻪ ﺧﻮﺩ ﺍﺳﺖ‪ ،‬ﺩﺭ ﺍﻳﻦ ﻣﻮﺍﺭﺩ‪،‬‬

‫ﺍﻟﻤﭙﻴﺎﺩﻫﺎ‪ ،‬ﻣﺴﺎﺑﻘﺎﺕ ﻭ ﺁﺯﻣﻮﻥﻫﺎﻱ ﺭﻳﺎﺿﻲ‪ 3‬ﻛﻪ ﺑﺮﺍﻱ ﺳﻨﻴﻦ ﻧﻮﺟﻮﺍﻧﺎﻥ‬

‫ﻫﻤﺖ ﻭ ﻣﻤﺎﺭﺳﺖ ﺑﻴﺶﺗﺮﻱ ﺑﻪ ﺧﺮﺝ ﺩﻫﻴﺪ(‪.‬‬

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‫ﭼﻪ ﺩﺭ ﺩﺍﺧﻞ ﻛﺸـﻮﺭ ﻭ ﭼﻪ ﺩﺭ ﺧﺎﺭﺝ ﻛﺸﻮﺭ ﻃﺮﺍﺣﻲ ﻭ ﺍﺭﺍﺋﻪ ﻣﻲﺷﻮﺩ‪.‬‬ ‫ﺭﻭﺷـﻦ ﻭ ﻣﺸﺨﺺ ﻣﻲﺷـﻮﺩ ﻛﻪ ﻫﺮ ﺭﻳﺎﺿﻲﭘﮋﻭﻩ ﻧﻮﺟﻮﺍﻧﻲ ﻛﻪ ﺁﺭﺯﻭﻱ‬ ‫ﻣﻮﻓﻘﻴﺖ ﺩﺭ ﭼﻨﻴﻦ ﺭﻗﺎﺑﺖﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺭﺍ ﺑﻪ ﺳـﺮ ﻣﻲﭘﺮﻭﺭﺍﻧﺪ‪ ،‬ﺑﺎﻳﺪ ﺍﺯ‬ ‫ﺩﺍﻧﺸﻲ ﻓﺮﺍﺗﺮ ﺍﺯ ﻣﻮﻓﻘﻴﺖ ﺯﻣﺎﻧﻲ ﻭ ﻣﻜﺎﻧﻲ ﻛﻪ ﺳﻦ ﻭ ﺳﺎﻝ ﺍﻭ ﻣﻲﻃﻠﺒﺪ‬ ‫ﺑﺮﺧﻮﺭﺩﺍﺭ ﺑﺎﺷـﺪ‪ .‬ﺑﻪ ﻫﻤﻴﻦ ﻋﻠﺖ ﺩﺭ ﺍﻳـﻦ ﻣﺠﻤﻮﻋﻪ ﻣﻘﺎﻻﺕ ﻋﻼﻭﻩ ﺑﺮ‬ ‫ﻣﻄﺮﺡ ﻛﺮﺩﻥ ﻋﻨﺎﻭﻳﻦ ﭘﻴﻜﺎﺭﺟﻮ ﺩﺭﺑﺎﺭﻩﻱ ﻣﻄﺎﻟﺐ ﺩﺭﺳﻲ ﻛﻪ ﻣﺘﻨﺎﺳﺐ‬ ‫ﺑـﺎ ﺩﻭﺭﻩﻱ ﺗﺤﺼﻴﻠـﻲ ﺍﻳﻦ ﻧﻮﺟﻮﺍﻧﺎﻥ ﺍﺳـﺖ‪ ،‬ﺑﻪ ﻣﻨﻈﻮﺭ ﺍﻳﺠﺎﺩ ﺳـﻮﺍﺩ‬ ‫ﺭﻳﺎﺿـﻲ ﺍﻓﺰﻭﻥﺗﺮ‪ ،‬ﻛﻪ ﺑﺎﻋﺚ ﻋﻤﻠﻲ ﺳـﺎﺧﺘﻦ ﺍﺳـﺘﻌﺪﺍﺩﻫﺎﻱ ﺑﺎﻟﻘﻮﻩﻱ‬ ‫ﺭﻳﺎﺿﻲﭘﮋﻭﻫﺎﻥ ﻧﻮﺑﺎﻭﻩ ﻣﻲﺷـﻮﺩ‪ ،‬ﺩﺭ ﭘﺎﺭﻩﺍﻱ ﻣـﻮﺍﺭﺩ ﺑﻪ ﺍﺭﺍﺋﻪﻱ ﻗﻀﺎﻳﺎ‬ ‫ﻭ ﻧـﻜﺎﺕ ﺍﺭﺯﻧﺪﻩ ﻭ ﭘﻮﻳﺎ ﺩﺭ ﻛﻨﺎﺭ ﻣﺴـﺎﺋﻞ ﻣﺮﺗﺒﻂ ﺑﻪ ﺁﻥﻫﺎ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪.‬‬ ‫ﻛﻪ ﺍﻳﻦ ﻣﻄﺎﻟﺐ ﺑﺎ ﻋﻨﺎﻭﻳﻦ ﻭ ﻣﻮﺍﺭﺩﻱ ﻣﻨﺎﺳـﺐ ﺍﺳـﺖ ﻛﻪ ﻫﺮ ﻧﻮﺑﺎﻭﻩﻱ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

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‫ﺭﺍﻫﺒﺮﺩﻫـﺎﻱ ﺗﺸـﻜﻴﻞ ﻣﻌﺎﺩﻟـﻪﻱ ﺧﻄﻲ ﻭ‬ ‫ﺗﺸﻜﻴﻞ ﺩﺳﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ‬ ‫ﺩﺭ ﺯﻳﺮ ﭘﺮﺳﺶﻫﺎﻱ ﭼﻬﺎﺭﮔﺰﻳﻨﻪﺍﻱ ﮔﻮﻧﺎﮔﻮﻥ ﻭ ﭘﻴﻜﺎﺭﺟﻮ ﺭﺍ ﺑﻪ ﻫﻤﺮﺍﻩ‬ ‫ﭘﺎﺳﺦﻫﺎﻱ ﺗﺸــﺮﻳﺤﻲ ﺁﻥﻫﺎ ﻛﻪ ﺟﻨﺒﻪﻱ ﺭﻗﺎﺑﺘﻲ ﺑﺮﺍﻱ ﺭﻳﺎﺿﻲﭘﮋﻭﻫﺎﻥ‬ ‫ﻧﻮﺟــﻮﺍﻥ ﺩﺍﺭﻧﺪ ﻭ ﺩﺭ ﺁﺷــﻨﺎﻳﻲ ﻭ ﺍﻋﺘــﻼﻱ ﺗﻮﺍﻥ ﺫﻫﻨﻲ ﺍﻳﺸــﺎﻥ ﺑﺮﺍﻱ‬ ‫ﻣﻮﻓﻘﻴﺖ ﺩﺭﺁﺯﻣﻮﻥﻫﺎﻱ ﺍﻟﻤﭙﻴﺎﺩﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺩﺍﺧﻞ ﻭ ﺧﺎﺭﺝ ﺍﺯ ﻛﺸــﻮﺭ‪،‬‬ ‫ﺁﺯﻣﻮﻥﻫﺎﻱ ﻭﺭﻭﺩﻱ ﻣﺪﺍﺭﺱ ﺗﻴﺰﻫﻮﺷﺎﻥ ﻭ ﻣﺪﺍﺭﺱ ﻧﻤﻮﻧﻪ ﻭ ﻛﺎﺭﺑﺮﺩﻱ ﻭ‬ ‫‪ ...‬ﻛﺎﺭﺑﺮﺩﻱ ﺍﺭﺯﻧﺪﻩ ﺩﺍﺭﻧﺪ‪ ،‬ﺍﺭﺍﻳﻪ ﻣﻲﺩﻫﻴﻢ‪ .‬ﺍﻟﺒﺘﻪ ﺩﺭ ﭘﺎﺭﻩﺍﻱ ﺍﺯ ﻣﻮﺍﺭﺩ ﻛﻪ‬ ‫ﺫﻛﺮ ﻧﻜﺎﺕ ﻳﺎ ﻣﻄﺎﻟﺒﻲ ﺑﺮﺍﻱ ﺩﺭﻙ ﺑﻬﺘﺮ ﺍﻳﻦ ﭘﺮﺳﺶﻫﺎﻱ ﭼﻬﺎﺭﮔﺰﻳﻨﻪﺍﻱ‬ ‫ﻻﺯﻡ ﺑﻪ ﻧﻈﺮ ﻣﻲﺭﺳﺪ‪ ،‬ﺍﺯ ﺑﻴﺎﻥ ﻭ ﺍﺭﺍﻳﻪﻱ ﺁﻥ ﺩﺭﻳﻎ ﻧﻜﺮﺩﻩ ﻭ ﺁﻥ ﺭﺍ ﻫﻤﺮﺍﻩ‬ ‫ﺑﺎ ﭘﺎﺳﺦﻫﺎﻱ ﺗﺸﺮﻳﺤﻲ ﺁﻭﺭﺩﻩﺍﻳﻢ‪.‬‬ ‫‪ .1‬ﺩﺭ ﻳﻚ ﻗﻠﻚ ‪ 625‬ﺭﻳﺎﻝ ﺍﺯ ﺳــﻜﻪﻫﺎﻱ ‪ 5‬ﺭﻳﺎﻟﻲ ﻭ ‪ 20‬ﺭﻳﺎﻟﻲ ﻭﺟﻮﺩ‬ ‫ﺩﺍﺭﺩ‪ .‬ﺍﮔﺮ ﻣﺠﻤﻮﻋﺎ ‪ 35‬ﺳــﻜﻪ ﺩﺭ ﺍﻳﻦ ﻗﻠﻚ ﻭﺟﻮﺩ ﺩﺍﺷــﺘﻪ ﺑﺎﺷﺪ‪،‬‬ ‫ﭼﻨﺪ ﺳﻜﻪﻱ ‪ 20‬ﺭﻳﺎﻟﻲ ﺩﺭ ﺍﻳﻦ ﻗﻠﻚ ﻭﺟﻮﺩ ﺩﺍﺭﺩ؟‬ ‫‪15 (2‬‬ ‫‪30 (1‬‬ ‫‪5 (3‬‬ ‫‪10 (4‬‬ ‫‪ .2‬ﻣﻘﺪﺍﺭ ﻛﺴــﺮﻱ ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 2‬ﺍﺳــﺖ‪ .‬ﺍﮔﺮ ﺍﺧﺘــﻼﻑ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ‬ ‫‪3‬‬ ‫ﻛﺴــﺮ ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 7‬ﺑﺎﺷﺪ‪ ،‬ﻣﺠﻤﻮﻉ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﻛﺴﺮ ﻛﺪﺍﻡﻳﻚ ﺍﺯ‬ ‫ﮔﺰﻳﻨﻪﻫﺎﻱ ﺯﻳﺮ ﺍﺳﺖ؟‬ ‫‪35 (2‬‬ ‫‪30 (1‬‬ ‫‪45 (3‬‬ ‫‪40 (4‬‬ ‫‪ .3‬ﻣﺠﻤﻮﻉ ﺳﻪ ﻋﺪﺩ ﻓﺮﺩ ﻣﺘﻮﺍﻟﻲ ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 87‬ﺍﺳﺖ‪ ،‬ﻣﺠﻤﻮﻉ ﻳﻜﺎﻥﻫﺎﻱ‬ ‫ﺁﻥ ﺳﻪ ﻋﺪﺩ ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﮔﺰﻳﻨﻪﻫﺎﻱ ﺯﻳﺮ ﺍﺳﺖ؟‬ ‫‪17 (2‬‬ ‫‪7 (1‬‬ ‫‪8 (3‬‬ ‫‪18 (4‬‬

‫‪ .4‬ﻧﻴﻤﺎ ﻭ ﺳﻴﻨﺎ ﺭﻭﻱ ﻫﻢ ‪ 10500‬ﺭﻳﺎﻝ ﭘﻮﻝ ﺩﺍﺭﻧﺪ‪ .‬ﺑﻌﺪ ﺍﺯ ﺁﻥ ﻛﻪ ﻧﻴﻤﺎ‬ ‫‪ 1‬ﭘﻮﻝ ﺧﻮﺩ ﻭ ﺳــﻴﻨﺎ ‪ 1‬ﭘﻮﻝ ﺧﻮﺩ ﺭﺍ ﺧﺮﺝ ﻛﺮﺩﻧﺪ‪ ،‬ﻣﻘﺪﺍﺭ ﭘﻮﻝ‬ ‫‪6‬‬ ‫‪3‬‬ ‫ﻧﻴﻤﺎ ﺩﻭ ﺑﺮﺍﺑﺮ ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﺳــﻴﻨﺎ ﺷــﺪ‪ .‬ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﻫﺮﻳﻚ ﺍﺯ ﺁﻥﻫﺎ‬ ‫ﻛﺪﺍﻡ ﮔﺰﻳﻨﻪ ﺍﺳﺖ؟‬ ‫‪ (1‬ﻧﻴﻤﺎ ‪ 3000‬ﺭﻳﺎﻝ ﻭ ﺳﻴﻨﺎ ‪ 7500‬ﺭﻳﺎﻝ‬ ‫‪ (2‬ﻧﻴﻤﺎ ‪ 7500‬ﺭﻳﺎﻝ ﻭ ﺳﻴﻨﺎ ‪ 3000‬ﺭﻳﺎﻝ‬ ‫‪ (3‬ﻧﻴﻤﺎ ‪ 3500‬ﺭﻳﺎﻝ ﻭ ﺳﻴﻨﺎ ‪ 7000‬ﺭﻳﺎﻝ‬ ‫‪ (4‬ﻧﻴﻤﺎ ‪ 7000‬ﺭﻳﺎﻝ ﻭ ﺳﻴﻨﺎ ‪ 3500‬ﺭﻳﺎﻝ‬ ‫‪ .5‬ﺍﮔﺮ ﺑﻪ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﻛﺴﺮﻱ ‪ 3‬ﻭﺍﺣﺪ ﺍﺿﺎﻓﻪ ﺷﻮﺩ‪ ،‬ﺁﻥ ﻛﺴﺮ ﺑﺮﺍﺑﺮ‬ ‫‪4‬‬ ‫ﺑﺎ ‪ 5‬ﻭ ﺍﮔﺮ ﺍﺯ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﺁﻥ ﻛﺴــﺮ ‪ 3‬ﻭﺍﺣﺪ ﻛﻢ ﺷــﻮﺩ‪ ،‬ﺁﻥ‬ ‫‪1‬‬ ‫ﻛﺴــﺮ ﺑﺮﺍﺑﺮ ﺑﺎ ﺧﻮﺍﻫﺪ ﺷﺪ‪ ،‬ﻣﺠﻤﻮﻉ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﺁﻥ ﻛﺴﺮ‬ ‫⎧‬ ‫‪2‬‬ ‫ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﮔﺰﻳﻨﻪﻫﺎﻱ ﺯﻳﺮ ﺍﺳﺖ؟‬ ‫‪34 (2‬‬ ‫‪43 (1‬‬ ‫‪12 (4‬‬ ‫‪21 (3‬‬ ‫‪ .6‬ﺩﺭ ﺷــﻜﻞ ﺯﻳﺮ ﻣﺠﻤﻮﻉ ﻣﺴــﺎﺣﺖﻫﺎﻱ ‪ s2 ،s1‬ﻭ ‪ s3‬ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 37‬ﻭ‬ ‫ﻣﺴﺎﺣﺖ ﻗﺴﻤﺖ ﻫﺎﺷﻮﺭﺯﺩﻩ ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 6‬ﺍﺳﺖ‪ ،‬ﻣﺤﻴﻂ ﻣﺴﺘﻄﻴﻞ ‪s3‬‬ ‫ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﮔﺰﻳﻨﻪﻫﺎﻱ ﺯﻳﺮ ﺍﺳﺖ؟‬

‫‪S1‬‬ ‫‪S2‬‬ ‫‪21 (1‬‬ ‫‪14 (3‬‬

‫‪S3‬‬ ‫‪7 (2‬‬ ‫‪12 (4‬‬

‫‪ .7‬ﺳــﺎﻧﺎﺯ ﺍﺯ ﮔﻠﻨﺎﺭ ﭘﺮﺳﻴﺪ‪ :‬ﭼﻨﺪ ﺳﺎﻝ ﺩﺍﺭﻱ؟ ﮔﻠﻨﺎﺭ ﺟﻮﺍﺏ ﺩﺍﺩ‪ :‬ﻭﻗﺘﻲ‬ ‫ﺗﻮ ﺑﻪ ﺳﻦ ﺍﻣﺮﻭﺯ ﻣﻦ ﺑﺮﺳﻲ‪ ،‬ﻣﻦ ﺩﻭ ﺑﺮﺍﺑﺮ ﺳﻦ ﺍﻵﻥ ﺗﻮ ﺳﻦ ﺧﻮﺍﻫﻢ‬ ‫ﺩﺍﺷﺖ‪ .‬ﺍﮔﺮ ﻣﺠﻤﻮﻉ ﺳﻦ ﺩﻭ ﻧﻔﺮ ﺁﻥﻫﺎ ‪ 30‬ﺳﺎﻝ ﺑﺎﺷﺪ‪ ،‬ﮔﻠﻨﺎﺭ ﭼﻨﺪ‬ ‫ﺳﺎﻝ ﺩﺍﺭﺩ؟‬ ‫‪14 (2‬‬ ‫‪12 (1‬‬ ‫‪18 (3‬‬ ‫‪20 (4‬‬ ‫‪ .8‬ﻣﺪﺕ ﺯﻣﺎﻧﻲ ﻛﻪ ﺍﺯ ﺳــﺎﺧﺖ ﻳﻚ ﻛﺸــﺘﻲ ﻣﻲﮔــﺬﺭﺩ‪ ،‬ﺑﺮﺍﺑﺮ ﻣﺪﺕ‬ ‫ﺯﻣﺎﻧﻲ ﺍﺳﺖ ﻛﻪ ﺍﺯ ﺳﺎﺧﺖ ﺩﻳﮓ ﺑﺨﺎﺭ ﺁﻥ ﺗﺎ ﺯﻣﺎﻧﻲ ﻛﻪ ﻋﻤﺮ ﻛﺸﺘﻲ‬ ‫ﺑــﻪ ﺍﻧﺪﺍﺯﻩﻱ ﻋﻤﺮ ﻓﻌﻠﻲ ﺩﻳﮓ ﺑﺨﺎﺭ ﺁﻥ ﺷــﻮﺩ‪ ،‬ﻣﻲﮔﺬﺭﺩ‪ .‬ﻣﺠﻤﻮﻉ‬ ‫‪4‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﻋﻤﺮ ﻓﻌﻠﻲ ﻛﺸــﺘﻲ ﻭ ﺩﻳﮓ ﺑﺨﺎﺭ ﺁﻥ ‪ 49‬ﺳــﺎﻝ ﺍﺳــﺖ‪ .‬ﺍﺯ ﺳﺎﺧﺖ‬ ‫ﻛﺸﺘﻲ ﭼﻨﺪ ﺳﺎﻝ ﻣﻲﮔﺬﺭﺩ؟‬ ‫‪21 (2‬‬ ‫‪25 (1‬‬ ‫‪24 (4‬‬ ‫‪28 (3‬‬ ‫‪ .9‬ﺍﮔﺮ ﺣﺴــﻦ ‪ 3‬ﺩﻓﺘﺮ ﻭ ‪ 5‬ﺧﻮﺩﻛﺎﺭ ﺑﺨﺮﺩ‪ 10 ،‬ﺗﻮﻣﺎﻥ ﺍﺯ ﭘﻮﻟﺶ ﺑﺎﻗﻲ‬ ‫ﻣﻲﻣﺎﻧــﺪ‪ .‬ﺍﮔﺮ ﺍﻭ ‪ 2‬ﺩﻓﺘﺮ ﻭ ‪ 8‬ﺧﻮﺩﻛﺎﺭ ﺑﺨــﺮﺩ‪ ،‬ﭘﻮﻟﻲ ﺑﺮﺍﻱ ﺍﻭ ﺑﺎﻗﻲ‬ ‫ﻧﻤﻲﻣﺎﻧﺪ ﻭ ﻗﻴﻤﺖ ‪ 2‬ﺩﻓﺘﺮ ﺑﺎ ﻗﻴﻤﺖ ‪ 5‬ﺧﻮﺩﻛﺎﺭ ﺑﺮﺍﺑﺮ ﺍﺳــﺖ‪ .‬ﻣﻘﺪﺍﺭ‬ ‫ﭘﻮﻝ ﺣﺴﻦ ﻛﺪﺍﻡ ﮔﺰﻳﻨﻪ ﺍﺳﺖ؟‬ ‫‪200 (1‬‬ ‫‪240 (2‬‬ ‫‪260 (3‬‬ ‫‪280 (4‬‬ ‫‪ .10‬ﺩﻭ ﺷــﻤﻊ ﻫﻢﻃﻮﻝ ﺭﺍ ﺑﺎ ﻫﻢ ﺭﻭﺷــﻦ ﻣﻲﻛﻨﻴﻢ‪ .‬ﺷــﻤﻊ ﺍﻭﻝ ﺩﺭ ‪4‬‬ ‫ﺳــﺎﻋﺖ ﻭ ﺷﻤﻊ ﺩﻭﻡ ﺩﺭ ‪ 3‬ﺳﺎﻋﺖ ﻣﻲﺳــﻮﺯﺩ‪ .‬ﺑﺎ ﻓﺮﺽ ﺁﻥﻛﻪ ﻫﺮ‬ ‫ﺷﻤﻊ ﺑﻪ ﻣﻴﺰﺍﻥ ﺛﺎﺑﺘﻲ ﺑﺴﻮﺯﺩ‪ ،‬ﭘﺲ ﺍﺯ ﮔﺬﺷﺖ ﭼﻨﺪ ﺳﺎﻋﺖ‪ ،‬ﻃﻮﻝ‬ ‫ﺷﻤﻊ ﺍﻭﻝ ﺩﻭ ﺑﺮﺍﺑﺮ ﺷﻤﻊ ﺩﻭﻡ ﻣﻲﺷﻮﺩ؟‬ ‫‪3‬‬ ‫‪4‬‬

‫‪1/5 (2‬‬

‫‪2 (3‬‬

‫‪2/4 (4‬‬

‫‪(1‬‬

‫‪ .11‬ﻣﻘﺪﺍﺭﻱ ﭘﻮﻝ ﺩﺭ ﺻﻨﺪﻭﻕ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ .‬ﻗﺮﺍﺭ ﺑﺮ ﺍﻳﻦ ﺍﺳــﺖ ﻛﻪ ﺳﻪ‬ ‫ﻧﻔــﺮ ﺑﻪ ﺗﺮﺗﻴﺐ ﻫﺮﻳﻚ ﺑﻪ ﺍﻧــﺪﺍﺯﻩﻱ ﻣﺒﻠﻐﻲ ﻛﻪ ﺩﺭ ﺻﻨﺪﻭﻕ ﻭﺟﻮﺩ‬ ‫ﺩﺍﺭﺩ‪ ،‬ﭘــﻮﻝ ﺑﻪ ﺻﻨﺪﻭﻕ ﺍﺿﺎﻓﻪ ﻛﻨﻨﺪ ﻭ ﺳــﭙﺲ ﻣﺒﻠﻎ ‪ 40‬ﺗﻮﻣﺎﻥ ﺍﺯ‬ ‫ﺁﻥ ﺑﺮﺩﺍﺭﻧﺪ‪ .‬ﺑﻌﺪ ﺍﺯ ﺍﻳﻦﻛﻪ ﻫﺮ ﺳﻪ ﻧﻔﺮ ﺍﻳﻦ ﻛﺎﺭ ﺭﺍ ﺍﻧﺠﺎﻡ ﺩﺍﺩﻧﺪ‪ ،‬ﺩﺭ‬ ‫ﺻﻨــﺪﻭﻕ ﭘﻮﻟﻲ ﻧﻤﻲﻣﺎﻧﺪ‪ .‬ﺩﺭ ﺍﺑﺘــﺪﺍ ﭼﻪﻗﺪﺭ ﭘﻮﻝ ﺩﺭ ﺻﻨﺪﻭﻕ ﺑﻮﺩﻩ‬ ‫ﺍﺳﺖ؟‬ ‫‪ 25 (2‬ﺗﻮﻣﺎﻥ‬ ‫‪ 55 (1‬ﺗﻮﻣﺎﻥ‬ ‫‪ 35 (4‬ﺗﻮﻣﺎﻥ‬ ‫‪ 45 (3‬ﺗﻮﻣﺎﻥ‬

‫‪ .12‬ﻃﻮﻝ ﻣﺴــﺘﻄﻴﻠﻲ ﺍﺯ ‪ 2‬ﺑﺮﺍﺑﺮ ﻋﺮﺽ ﺁﻥ ‪ 4‬ﻣﺘﺮ ﻛﻢﺗﺮ ﺍﺳﺖ‪ .‬ﺍﮔﺮ ‪6‬‬ ‫ﻣﺘﺮ ﺍﺯ ﻃﻮﻝ ﻛﻢ ﻛﻨﻴﻢ ﻭ ‪ 2‬ﻣﺘﺮ ﺑﻪ ﻋﺮﺽ ﺍﺿﺎﻓﻪ ﻛﻨﻴﻢ‪ ،‬ﻣﺴــﺘﻄﻴﻞ‬ ‫ﺗﺒﺪﻳﻞ ﺑﻪ ﻣﺮﺑﻊ ﻣﻲﺷﻮﺩ‪ .‬ﺍﺧﺘﻼﻑ ﻣﺴﺎﺣﺖ ﺍﻳﻦ ﻣﺴﺘﻄﻴﻞ ﺑﺎ ﻣﺮﺑﻊ‬ ‫ﺑﻪ ﻭﺟﻮﺩ ﺁﻣﺪﻩ‪ ،‬ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﮔﺰﻳﻨﻪﻫﺎﻱ ﺯﻳﺮ ﺍﺳﺖ؟‬ ‫‪12 (2‬‬ ‫‪24 (1‬‬ ‫‪44 (4‬‬ ‫‪48 (3‬‬ ‫‪ .13‬ﺳﻪ ﭘﺴــﺮ ﺑﭽﻪ ﺗﻮﺍﻓﻖ ﻣﻲﻛﻨﻨﺪ ﻛﻪ ﺗﻌﺪﺍﺩﻱ ﻣﻬﺮﻩﻱ ﺩﺍﺧﻞ ﻳﻚ‬ ‫ﻛﻴﺴــﻪ ﺭﺍ ﺑﻪ ﺭﻭﺵ ﺯﻳﺮ ﺑﻴﻦ ﺧﻮﺩ ﺗﻘﺴﻴﻢ ﻛﻨﻨﺪ‪ .‬ﺁﻥﭼﻪ ﭘﺴﺮﺑﭽﻪﻱ‬ ‫ﺍﻭﻝ ﺑﺮﻣﻲﺩﺍﺭﺩ‪ ،‬ﻳﻚ ﻣﻬﺮﻩ ﺑﻴﺶﺗﺮ ﺍﺯ ﻧﺼﻒ ﻣﻬﺮﻩﻫﺎﺳﺖ‪ .‬ﭘﺴﺮﺑﭽﻪﻱ‬ ‫‪1‬‬ ‫ﺩﻭﻡ‪ 3 ،‬ﺑﺎﻗﻲﻣﺎﻧــﺪﻩ ﺭﺍ ﺑﺮﻣــﻲﺩﺍﺭﺩ ﻭ ‪ 4‬ﻣﻬــﺮﻩﻱ ﺑﺎﻗﻲﻣﺎﻧــﺪﻩ ﺭﺍ‬ ‫ﭘﺴﺮﺑﭽﻪﻱ ﺳــﻮﻡ ﺑﺮﻣﻲﺩﺍﺭﺩ‪ .‬ﺗﻌﺪﺍﺩ ﻣﻬﺮﻩﻫﺎﻳﻲ ﻛﻪ ﺑﻪ ﭘﺴﺮﺑﭽﻪﻱ‬ ‫ﺩﻭﻡ ﺭﺳﻴﺪﻩ‪ ،‬ﻛﺪﺍﻡ ﮔﺰﻳﻨﻪ ﺍﺳﺖ؟‬ ‫‪3 (2‬‬ ‫‪2 (1‬‬ ‫‪6 (4‬‬ ‫‪4 (3‬‬ ‫‪ .14‬ﺩﺭ ﻳﻚ ﺳﺎﻟﻦ‪ n ،‬ﺻﻨﺪﻟﻲ ﺑﻪ ﮔﻮﻧﻪﺍﻱ ﭼﻴﺪﻩ ﺷﺪﻩ ﺍﺳﺖ ﻛﻪ ﺗﻌﺪﺍﺩ‬ ‫ﺻﻨﺪﻟﻲﻫــﺎ ﺩﺭ ﻫــﺮ ﺭﺩﻳﻒ ﻭ ﻫﺮ ﺳــﺘﻮﻥ ﺑﺎ ﻫﻢ ﺑﺮﺍﺑﺮ ﺍﺳــﺖ ﻭ ﺍﮔﺮ‬ ‫ﺑﺨﻮﺍﻫﻴــﻢ ﺍﺯ ﻫﺮ ﺭﺩﻳﻒ ‪ 3‬ﺻﻨﺪﻟﻲ ﻛﻢ ﻛﻨﻴﻢ‪ ،‬ﻳﻚ ﺳــﺘﻮﻥ ﺍﺿﺎﻓﻪ‬ ‫ﻣﻲﺷﻮﺩ ﻭ ﺍﮔﺮ ﺍﺯ ﻫﺮ ﺭﺩﻳﻒ ‪ 5‬ﺻﻨﺪﻟﻲ ﻛﻢ ﻛﻨﻴﻢ‪ ،‬ﺩﻭ ﺳﺘﻮﻥ ﺍﺿﺎﻓﻪ‬ ‫ﻣﻲﺷﻮﺩ‪ n.‬ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﮔﺰﻳﻨﻪﻫﺎﻱ ﺯﻳﺮ ﺍﺳﺖ؟‬ ‫‪60 (1‬‬ ‫‪40 (2‬‬ ‫‪80 (3‬‬ ‫‪50 (4‬‬ ‫ﭘﻲﻧﻮﺷﺖ‪:‬‬

‫‪1. Tbcorens‬‬ ‫‪2. Proof‬‬ ‫‪3. Matbcmatics‬‬ ‫‪4. Goomctryy‬‬ ‫‪5. Numberr Thoory‬‬ ‫‪6. Algcbra‬‬ ‫‪7. Arithmetic‬‬ ‫‪etic‬‬ ‫‪8. Intuitivee‬‬

‫ﺗﻮﺟﻪ ‪ :‬ﭘﺎﺳﺦ‬ ‫ﺍﻳﻦ ﭘﺮﺳﺶﻫﺎ ﺭﺍ ﺩﺭ‬ ‫ﺻﻔﺤـﺔ ‪ 42‬ﻣﺠﻠﻪ‬ ‫ﻣﻄﺎﻟﻌـﻪ ﻛﻨﻴــﺪ‪.‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪5‬‬

‫ﺭﻳﺎﺿﻲ ﻭ ﺑﺎﺯﻱ‬

‫︋︀زی دو ﹡﹀︣ه‬ ‫ﺯﻫﺮﻩ ﭘﻨﺪﻱ‬

‫ﻓﻜﺮ ﻭ ﺑﻜﺮ ﺑﺎ ﻋﺪﺩﻫﺎ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﺑﺎﺯﻱ ﻭ ﺭﻳﺎﺿﻲ‪ ،‬ﺣﺪﺱ ﻣﻨﻄﻘﻲ‪ ،‬ﻋﺪﺩ‪ ،‬ﺑﺮﻧﺪﻩ‪ ،‬ﻛﺎﻏﺬ‪.‬‬

‫ﺍﻳﻦ ﺑﺎﺯﻱ ﻳﻚ ﺑﺎﺯﻱ ﺩﻭ ﻧﻔﺮﻩ ﺍﺳﺖ ﻛﻪ ﺑﺮﺍﻱ ﺍﻧﺠﺎﻡ ﺁﻥ‪ ،‬ﻓﻘﻂ ﺑﻪ ﻛﺎﻏﺬ‬ ‫ﻭ ﻣﺪﺍﺩ ﺍﺣﺘﻴﺎﺝ ﺩﺍﺭﻳﺪ‪.‬‬ ‫• ﻫﺮ ﻛﺪﺍﻡ ﻳﻚ ﺑﺮﮔﻪﻱ ﻛﺎﻏﺬ ﺭﺍ ﺑﻪ ﺷﻜﻞ ﺯﻳﺮ ﺧﻂﻛﺸﻲ ﻛﻨﻴﺪ ﻭ‬ ‫ﺻﻔﺤﻪﻱ ﺑﺎﺯﻱ ﺧﻮﺩﺗﺎﻥ ﺭﺍ ﺑﺴﺎﺯﻳﺪ‪.‬‬ ‫ﻋﺪﺩ ﺣﺪﺱ ﺯﺩﻩ ﺷﺪﻩ‬

‫ﻗﻀﺎﻭﺕ‬

‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪5‬‬ ‫‪6‬‬ ‫‪7‬‬

‫ﺷﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ﻋﻼﻣﺖ × ﺑﺮﺍﻱ ﻫﺮ ﺭﻗﻤﻲ ﻛﻪ ﺩﺭﺳﺖ ﺍ ّﻣﺎ ﻧﺎﺑﻪﺟﺎ ﺣﺪﺱ ﺯﺩﻩ ﺷﺪﻩ‬ ‫ﺍﺳﺖ‪.‬‬ ‫ﺑﺮﺍﻱ ﻣﺜﺎﻝ‪ ،‬ﺍﮔﺮ ﻋﺪﺩ ﺷﻤﺎ ‪ 274‬ﺍﺳﺖ‪ ،‬ﺑﺮﺍﻱ ﺣﺪﺱ ‪ ،235‬ﻗﻀﺎﻭﺕ‬ ‫‪ ،‬ﺑﺮﺍﻱ ﺣﺪﺱ ‪ 452‬ﻗﻀﺎﻭﺕ × × ﻭ ﺑﺮﺍﻱ ﺣﺪﺱ ‪ 247‬ﻗﻀﺎﻭﺕ × ×‬ ‫ﺭﺍ ﺍﻧﺠﺎﻡ ﺩﻫﻴﺪ‪.‬‬ ‫ﺳﻌﻲ ﻛﻨﻴﺪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻗﻀﺎﻭﺕ ﺍﻧﺠﺎﻡ ﺷﺪﻩ ﺩﺭ ﺣﺪﺱﻫﺎﻱ ﻗﺒﻠﻲ ﺩﺭ ﻫﺮ‬ ‫ﻣﺮﺣﻠﻪ ﺣﺪﺱ ﺑﻬﺘﺮ ﻳﺎ ﻣﺆﺛﺮﺗﺮﻱ ﺑﺰﻧﻴﺪ!‬ ‫• ﺑﺮﻧﺪﻩﻱ ﺑﺎﺯﻱ ﻛﺴﻲ ﺍﺳﺖ ﻛﻪ ﺯﻭﺩﺗﺮ ﻋﺪﺩ ﺑﺎﺯﻳﻜﻦ ﻣﻘﺎﺑﻠﺶ ﺭﺍ‬ ‫ﺣﺪﺱ ﺑﺰﻧﺪ‪.‬‬ ‫ﺑﻪ ﺟﺪﻭﻝ ﭘﺮ ﺷﺪﻩﻱ ﺯﻳﺮ ﻧﮕﺎﻩ ﻛﻨﻴﺪ‪ .‬ﺁﻳﺎ ﻣﻲﺗﻮﺍﻧﻴﺪ ﻋﺪﺩ ﺩﺭﺳﺖ ﺭﺍ‬ ‫ﺣﺪﺱ ﺑﺰﻧﻴﺪ؟‬

‫‪8‬‬

‫ﻗﻀﺎﻭﺕ‬

‫ﻋﺪﺩ ﺣﺪﺱ ﺯﺩﻩ ﺷﺪﻩ‬

‫‪9‬‬

‫××‬

‫‪762‬‬

‫‪1‬‬

‫×‬

‫‪862‬‬

‫‪2‬‬

‫‪516‬‬

‫‪3‬‬

‫‪942‬‬

‫‪4‬‬

‫‪10‬‬ ‫‪11‬‬ ‫‪12‬‬

‫• ﻫﺮﻳﻚ‪ ،‬ﻋﺪﺩﻱ ﺳﻪ ﺭﻗﻤﻲ ﺍﻧﺘﺨﺎﺏ ﻛﻨﻴﺪ ﻭ ﻃﻮﺭﻱ ﻛﻪ ﺑﺎﺯﻳﻜﻦ‬ ‫ﻣﻘﺎﺑﻞ ﻧﺒﻴﻨﺪ‪ ،‬ﺁﻥ ﺭﺍ ﭘﺸﺖ ﺻﻔﺤﻪﻱ ﺑﺎﺯﻱ ﺧﻮﺩﺗﺎﻥ ﺑﻨﻮﻳﺴﻴﺪ‪.‬‬ ‫• ﺩﺭ ﻃﻮﻝ ﺑﺎﺯﻱ ﻗﺮﺍﺭ ﺍﺳﺖ ﻫﺮﻳﻚ ﺍﺯ ﺷﻤﺎ ﺑﺎ ﺣﺪﺱﻫﺎﻱ ﻣﻨﻄﻘﻲ‪،‬‬ ‫ﻋﺪﺩ ﺍﻧﺘﺨﺎﺑﻲ ﺑﺎﺯﻳﻜﻦ ﺣﺮﻳﻒ ﺭﺍ ﺣﺪﺱ ﺑﺰﻧﻴﺪ‪ .‬ﺑﺎﺯﻱ ﺭﺍ ﺑﺎ ﻳﻚ‬ ‫ﺣﺪﺱ ﺷﺮﻭﻉ ﻛﻨﻴﺪ ﻭ ﻫﺮ ﺩﻭ ﻳﻚ ﻋﺪﺩ ﺑﮕﻮﻳﻴﺪ‪.‬‬ ‫• ﻋﺪﺩﻱ ﺭﺍ ﻛﻪ ﻣﻲﮔﻮﻳﻴﺪ‪ ،‬ﺩﺭ ﺳﺘﻮﻥ ﻋﺪﺩ ﺣﺪﺱ ﺯﺩﻩ ﺷﺪﻩ‬ ‫ﺑﻨﻮﻳﺴﻴﺪ ﻭ ﻗﻀﺎﻭﺕ ﺑﺎﺯﻳﻜﻦ ﻣﻘﺎﺑﻞ ﺭﺍ ﺭﻭﺑﻪﺭﻭﻱ ﺁﻥ ﺩﺭ ﺳﺘﻮﻥ‬ ‫ﻗﻀﺎﻭﺕ ﻭﺍﺭﺩ ﻛﻨﻴﺪ‪.‬‬ ‫• ﺧﻮﺩﺗﺎﻥ ﻧﻴﺰ ﻋﺪﺩ ﺣﺪﺱ ﺯﺩﻩ ﺷﺪﻩ ﺗﻮﺳﻂ ﺑﺎﺯﻳﻜﻦ ﻣﻘﺎﺑﻞ ﺭﺍ ﺑﺎ‬ ‫ﻋﺪﺩ ﺍﻧﺘﺨﺎﺑﻲ ﺧﻮﺩﺗﺎﻥ ﻣﻘﺎﻳﺴﻪ ﻛﻨﻴﺪ ﻭ ﺁﻥ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ‬ ‫ﻗﻀﺎﻭﺕ ﻛﻨﻴﺪ‪.‬‬ ‫ﻋﻼﻣﺖ ﺑﺮﺍﻱ ﻫﺮ ﺭﻗﻤﻲ ﻛﻪ ﺩﺭﺳﺖ ﻭ ﺩﺭ ﺟﺎﻱ ﺧﻮﺩ ﺣﺪﺱ ﺯﺩﻩ‬ ‫‪6‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺣﺪﺱ ‪ 1‬ﻭ ‪ 2‬ﺭﺍ ﺑﺎ ﻫﻢ ﻣﻘﺎﻳﺴﻪ ﻛﻨﻴﺪ‪ .‬ﺁﻳﺎ ﻣﻲﺗﻮﺍﻥ ﻣﻄﻤﺌﻦ ﺑﻮﺩ ﻛﻪ‬ ‫ﺭﻗﻢ ﺻﺪﮔﺎﻥ ‪ 8‬ﺍﺳﺖ؟‬ ‫ﺣﺎﻻ ﺑﻪ ﺣﺪﺱ ‪ 3‬ﻧﮕﺎﻩ ﻛﻨﻴﺪ‪ 5 .‬ﻛﻪ ﺩﺭ ﺟﺎﻱ ﺧﻮﺩﺵ ﻧﻴﺴﺖ! ﻓﻜﺮ‬ ‫ﻣﻲﻛﻨﻴﺪ ‪ 1‬ﺩﺭ ﺟﺎﻱ ﺧﻮﺩﺵ ﺍﺳﺖ ﻳﺎ ‪6‬؟ ﺍﮔﺮ ‪ 1‬ﺩﺭ ﺟﺎﻱ ﺧﻮﺩﺵ ﺑﺎﺷﺪ‪،‬‬ ‫ﺩﺭ ﺣﺪﺱ ‪ ،2‬ﺭﻗﻢ ‪ 2‬ﺩﺭﺳﺖ ﺑﻪ ﻛﺎﺭ ﺭﻓﺘﻪ ﺍﺳﺖ ﻳﺎ ‪6‬؟ ﺍﮔﺮ ‪ 6‬ﺩﺭ ﺣﺪﺱ‬ ‫‪ 3‬ﺩﺭ ﺟﺎﻱ ﺧﻮﺩﺵ ﺑﺎﺷﺪ‪ ،‬ﭼﻪﻃﻮﺭ؟ ﺁﻳﺎ ﻣﻲﺗﻮﺍﻥ ﻣﻄﻤﺌﻦ ﺑﻮﺩ ﻛﻪ ﺭﻗﻢ‬ ‫ﻳﻜﺎﻥ ‪ 6‬ﺍﺳﺖ؟ ﺑﻪ ﺣﺪﺱ ‪ 4‬ﺗﻮﺟﻪ ﻛﻨﻴﺪ! ﻓﻜﺮ ﻣﻲﻛﻨﻴﺪ ﻛﺪﺍﻡ ﺭﻗﻢ ﺩﺭ‬ ‫ﺟﺎﻱ ﺧﻮﺩ ﻗﺮﺍﺭ ﺩﺍﺭﺩ؟ ﺁﻳﺎ ﻣﻲﺗﻮﺍﻥ ﻣﻄﻤﺌﻦ ﺑﻮﺩ ﻛﻪ ﻋﺪﺩ ﺍﻧﺘﺨﺎﺑﻲ ‪846‬‬ ‫ﺑﻮﺩﻩ ﺍﺳﺖ؟‬ ‫ﻓﻜﺮ ﻣﻲﻛﻨﻴﺪ ﻗﻀﺎﻭﺕﻫﺎﻱ‬ ‫ﺑﺮﺍﻱ ﺣﺪﺱﻫﺎﻱ ﺑﻬﺘﺮ ﻛﻤﻚ‬ ‫ﺑﻴﺶﺗﺮﻱ ﻣﻲﻛﻨﻨﺪ ﻳﺎ ﻗﻀﺎﻭﺕﻫﺎﻱ × ؟‬

‫؟‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫در ﹞︧︐︴﹫﹏ ‪،‬‬ ‫︵‪﹢‬ل ﹋︡ام ا︨️؟ !‬ ‫ﺯﻳﻨﺐ ﻣﺮﺍﺩﺧﺎﻧﻰ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻃﻮﻝ‪ ،‬ﻋﺮﺽ‪ ،‬ﻣﺴﺘﻄﻴﻞ‪ ،‬ﺩﺳﺘﮕﺎﻩ ﻣﺨﺘﺼﺎﺕ‪.‬‬ ‫ﺩﺭ ﻣﺴﺘﻄﻴﻞﻫﺎﻯ ﻣﻘﺎﺑﻞ ﻃﻮﻝ ﻭ ﻋﺮﺽ ﺭﺍ ﻣﺸﺨﺺ ﻛﻨﻴﺪ ‪:‬‬

‫ﺩﺭ ﺯﻧﺪﮔﻰ ﺭﻭﺯﻣﺮﻩ ﻣﺮﺳــﻮﻡ ﺍﺳﺖ ﻛﻪ ﺩﺭ ﻫﺮ ﻣﺴﺘﻄﻴﻞ‪ ،‬ﺑﻠﻨﺪﺗﺮﻳﻦ‬ ‫ﺿﻠﻊ ﺭﺍ ﻃﻮﻝ ﻭ ﻛﻮﺗﺎﻩﺗﺮﻳﻦ ﺿﻠﻊ ﺭﺍ ﻋﺮﺽ ﻣﻰﮔﻴﺮﻧﺪ ‪ .‬ﺁﻳﺎ ﭼﻨﻴﻦ ﻣﻄﻠﺒﻰ‬ ‫ﻣﻰﺗﻮﺍﻧﺪ ﻭﺍﻗﻌﻴﺖ ﺩﺍﺷــﺘﻪ ﺑﺎﺷــﺪ؟ ﺑﺎ ﻣﺮﻭﺭﻯ ﺑﺮ ﻳــﻚ ﻛﺮﻩ ﺟﻐﺮﺍﻓﻴﺎﻳﻰ‪،‬‬ ‫ﻣﻰﺗــﻮﺍﻥ ﺑﻪ ﺍﻳﻦ ﻣﻄﻠﺐ ﺭﺳــﻴﺪ ﻛــﻪ ﻣﺪﺍﺭﻫﺎ ﺧﻂﻫــﺎﻯ ﺍﻓﻘﻰ ﻭ ﻧﺼﻒ‬ ‫ﺍﻟﻨﻬﺎﺭﻫــﺎ ﺧﻄﻬﺎﻯ ﻋﻤﻮﺩﻯ ﻫﺴــﺘﻨﺪ‪ ،‬ﻛﻪ ﺍﻳﻦ ﻣﻮﺿــﻮﻉ ﺭﺍ ﻣﻰﺗﻮﺍﻥ ﺑﻪ‬ ‫ﺩﺳــﺘﮕﺎﻩ ﻣﺨﺘﺼﺎﺕ ﻧﺴﺒﺖ ﺩﺍﺩ ﻭ ﺁﻥ )ﺧﻂﻫﺎﻱ ﺍﻓﻘﻲ ﻭ ﻧﺼﻒﺍﻟﻨﻬﺎﺭﻫﺎ(‬ ‫ﺭﺍ ﻣﺤﻮﺭ ﻃﻮﻝ ﻭ ﻣﺤﻮﺭ ﻋﻤﻮﺩﻯ ﺭﺍ ﻣﺤﻮﺭ ﻋﺮﺽ ﻧﺎﻣﻴﺪ‪.‬‬ ‫ﺣﺎﻝ ﺍﮔﺮ ﺩﻭ ﻣﺴــﺘﻄﻴﻞ ‪ a‬ﻭ ‪ b‬ﺭﺍ ﺭﻭﻯ ﺩﺳﺘﮕﺎﻩ ﻣﺨﺘﺼﺎﺕ ﻧﻤﺎﻳﺶ‬ ‫ﺩﻫﻴﻢ‪ ،‬ﺑﺎﻋﺚ ﺍﻳﻦ ﻛﺸــﻒ ﻣﻰ ﺷﻮﻳﻢ ﻛﻪ ﻃﻮﻝ ﻣﻰﺗﻮﺍﻧﺪ ﺍﺯ ﻋﺮﺽ ﻛﻢﺗﺮ‬ ‫ﻳﺎ ﺑﻴﺶﺗﺮ ﺑﺎﺷﺪ‪.‬‬

‫‪b‬‬

‫‪a‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪7‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫︋︟︩︎︢﹬︣ی‬ ‫ﻣﺤﻤﻮﺩ ﺩﺍﻭﺭﺯﻧﻲ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﺑﺨﺶﭘﺬﻳﺮﻱ‪ ،‬ﺗﻘﺴﻴﻢ‪ ،‬ﺍﻋﺪﺍﺩ ‪ 1‬ﺗﺎ ‪ ،30‬ﺍﻋﺪﺍﺩ ﺯﻭﺝ‪ ،‬ﺍﻋﺪﺍﺩ ﻓﺮﺩ‪ ،‬ﻣﻀﺎﺭﺏ‪.‬‬ ‫ﺁﻳﺎ ﺗﺎ ﺑﻪ ﺣﺎﻝ ﺑﺎ ﺍﻳﻦ ﺳﺆﺍﻻﺕ ﺑﺮﺧﻮﺭﺩ ﻛﺮﺩﻩﺍﻳﺪ؟‬ ‫ﺁﻳﺎ ‪ 11253‬ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ؟‬ ‫ﺁﻳﺎ ‪ 4591‬ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ؟ ﺑﺮ ‪ 11‬ﭼﻪﻃﻮﺭ؟‬ ‫ﺍﮔﺮ ﻣﺘﻦ ﺯﻳﺮ ﺭﺍ ﻣﻄﺎﻟﻌﻪ ﻛﻨﻴﺪ‪ ،‬ﻣﻲﺗﻮﺍﻧﻴﺪ ﺑﺪﻭﻥ ﺍﻧﺠﺎﻡ ﻋﻤﻞ ﺗﻘﺴﻴﻢ‪،‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﻫﺮ ﻋﺪﺩﻱ ﺭﺍ ﺑﺮ ﺑﻌﻀﻲ ﺍﺯ ﺍﻋﺪﺍﺩ ‪ 1‬ﺗﺎ ‪ 30‬ﺑﺮﺭﺳــﻲ ﻛﻨﻴﺪ‪.‬‬ ‫ﺑﻌﻀــﻲ ﺍﺯ ﻗﺴــﻤﺖﻫﺎ ﺑﻪ ﺗﻮﺿﻴﺢ ﺑﻴﺸــﺘﺮﻱ ﻧﻴﺎﺯ ﺩﺍﺭﻧﺪ ﻛــﻪ ﺁﻥﻫﺎ ﺭﺍ ﺑﺎ‬ ‫ﻋﻼﻣﺖ * ﻣﺸﺨﺺ ﻛﺮﺩﻩﺍﻳﻢ ﻭ ﺩﺭ ﭘﺎﻳﺎﻥ ﺑﻪ ﺁﻥﻫﺎ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪ :1‬ﻫﻤﺔ ﺍﻋﺪﺍﺩ ﺑﺮ ﻋﺪﺩ ‪ 1‬ﺑﺨﺶﭘﺬﻳﺮﻧﺪ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑﺮ ‪ :2‬ﻫﻤﺔ ﺍﻋﺪﺍﺩ ﺯﻭﺝ ﺑــﺮ ﻋﺪﺩ ‪ 2‬ﺑﺨﺶﭘﺬﻳﺮﻧﺪ‬ ‫)ﺑــﻪ ﻋﺪﺩﻱ ﺯﻭﺝ ﻣﻲﮔﻮﻳﻴﻢ ﻛﻪ ﺭﻗﻢ ﻳﻜﺎﻥ ﺁﻥ ﻳﻜﻲ ﺍﺯ‬ ‫ﺍﻋﺪﺍﺩ ‪ ،8 ،6 ،4، 2 ،0‬ﺑﺎﺷﺪ(‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑــﺮ ‪ :3‬ﺍﮔــﺮ ﻣﺠﻤــﻮﻉ ﺍﺭﻗــﺎﻡ ﻳﻚ ﻋــﺪﺩ ﺑﺮ ‪3‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‪ ،‬ﺁﻥ ﻋﺪﺩ ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪.‬‬ ‫ﻣﺜ ً‬ ‫ﻼ ‪ 78‬ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ ،‬ﺯﻳــﺮﺍ ‪ 8+7 =15‬ﻭ‬ ‫‪ 15‬ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﻭﻟﻲ ﻋﺪﺩ ‪ 259204‬ﺑﺮ ‪3‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﻧﻴﺴــﺖ‪ ،‬ﺯﻳﺮﺍ ‪ 2+5+9+2+0+4=22‬ﻭ ‪22‬‬ ‫ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﻧﻴﺴﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪ :4‬ﺍﮔﺮ ﺁﺧﺮﻳﻦ ﺩﻭ ﺭﻗﻢ ﻳﻚ ﻋﺪﺩ ﺑﺮ ‪ 4‬ﺑﺨﺶﭘﺬﻳﺮ‬ ‫ﺑﺎﺷــﻨﺪ‪ ،‬ﺁﻥ ﻋﺪﺩ ﺑﺮ ‪ 4‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﻣﺜ ً‬ ‫ﻼ ‪ 716‬ﺑﺮ‬ ‫‪ 4‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﺯﻳﺮﺍ ‪ 16‬ﺑﺮ ‪ 4‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑﺮ ‪ :5‬ﺍﮔــﺮ ﺭﻗﻢ ﻳﻜﺎﻥ ﻳﻚ ﻋﺪﺩ ‪ 0‬ﻳﺎ ‪ 5‬ﺑﺎﺷــﺪ‪،‬‬ ‫ﺁﻥ ﻋﺪﺩ ﺑﺮ ‪ 5‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ .‬ﻣﺜــ ً‬ ‫ﻼ ‪ 4995‬ﺑﺮ ‪5‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑﺮ ‪ :6‬ﺍﮔﺮ ﻳــﻚ ﻋﺪﺩ ﺯﻭﺝ ﺑﺎﺷــﺪ ﻭ ﻫﻢﭼﻨﻴﻦ ﺑﺮ ‪ 3‬ﻧﻴﺰ‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‪ ،‬ﺣﺘﻤﺎً ﺑﺮ ‪ 6‬ﻧﻴﺰ ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪.‬‬ ‫ﻣﺜ ً‬ ‫ﻼ ‪ 73452‬ﺑﺮ ‪ 2‬ﻭ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ ،‬ﭘﺲ ﺑﺮ ‪6‬‬ ‫ﻧﻴﺰ ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫‪8‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪* :7‬‬

‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑــﺮ ‪ :8‬ﺍﮔﺮ ﺁﺧﺮﻳﻦ ﺳــﻪ ﺭﻗﻢ ﻳﻚ ﻋــﺪﺩ ﺑﺮ ‪8‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‪ ،‬ﺁﻥ ﻋﺪﺩ ﺑﺮ ‪ 8‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪.‬‬ ‫ﻣﺜ ً‬ ‫ﻼ ‪ 3128‬ﺑﺮ ‪ 8‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ ،‬ﺯﻳﺮﺍ ‪ 128‬ﺑﺮ ‪8‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ )‪(128=8×16‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑــﺮ ‪ :9‬ﺍﮔﺮ ﻣﺠﻤﻮﻉ ﻳﻚ ﻋﺪﺩ ﺑــﺮ ‪ 9‬ﺑﺨﺶﭘﺬﻳﺮ‬ ‫ﺑﺎﺷﺪ‪ ،‬ﺁﻥ ﻋﺪﺩ ﺑﺮ ‪ 9‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﻣﺜ ً‬ ‫ﻼ ‪ 5148‬ﺑﺮ‬ ‫‪ 9‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ ،‬ﺯﻳﺮﺍ ‪ 5+1+4+8=18‬ﻭ ‪ 18‬ﺑﺮ ‪9‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑﺮ ‪ :10‬ﺍﮔﺮ ﻳﻚ ﻋﺪﺩ ﺑﻪ ﺻﻔﺮ ﺧﺘﻢ ﺷــﻮﺩ‪ ،‬ﺁﻥ‬ ‫ﻋﺪﺩ ﺑﺮ ‪ 10‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ ،‬ﻣﺎﻧﻨــﺪ ﺍﻋﺪﺍﺩ ‪ 790‬ﻭ‬ ‫‪.23350‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑﺮ ‪ :11‬ﺍﮔﺮ ﺍﺧﺘﻼﻑ ﺑﻴﻦ ﻣﺠﻤﻮﻉ ﺭﻗﻢﻫﺎﻳﻲ ﻛﻪ‬ ‫ﺩﺭ ﻣــﻜﺎﻥ ﺯﻭﺝ ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ ﺭﺍ ﺍﺯ ﻣﺠﻤﻮﻉ ﺭﻗﻢﻫﺎﻳﻲ ﻛﻪ‬ ‫ﺩﺭ ﻣــﻜﺎﻥ ﻓﺮﺩ ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ‪ ،‬ﻣﺤﺎﺳــﺒﻪ ﻛﻨﻴﻢ ﻭ ﺣﺎﺻﻞ‬ ‫ﺻﻔﺮ ﺑﺎﺷــﺪ ﻭ ﻳــﺎ ﺑﺮ ‪ 11‬ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‪ ،‬ﺁﻥ ﻋﺪﺩ‬ ‫ﺑﺮ ‪ 11‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ .‬ﻣﺜــ ً‬ ‫ﻼ ﻋﺪﺩ ‪ 54934‬ﺑﺮ ‪11‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﺯﻳﺮﺍ ‪ (4+9+5) - (3+4) =11‬ﻭ ‪11‬‬ ‫ﺑﺮ ‪ 11‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑــﺮ ‪ :12‬ﺍﮔﺮ ﻣﺠﻤــﻮﻉ ﺍﺭﻗﺎﻡ ﻳﻚ ﻋــﺪﺩ ﺑﺮ ‪3‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ ﻭ ﺁﺧﺮﻳﻦ ﺩﻭ ﺭﻗﻢ ﺁﻥ ﻋﺪﺩ ﻧﻴﺰ ﺑﺮ ‪12‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‪ ،‬ﺁﻥ ﻋﺪﺩ ﺑﺮ ‪ 12‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﻣﺜ ً‬ ‫ﻼ ﻋﺪﺩ ‪ 62532‬ﺑﺮ ‪ 12‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﺯﻳﺮﺍ ‪=18‬‬ ‫‪ 6+2+5+3+2‬ﻭ ‪ 18‬ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﻫﻢﭼﻨﻴﻦ‪،‬‬ ‫ﺁﺧﺮﻳﻦ ﺩﻭ ﺭﻗﻢ ﺁﻥ ﻳﻌﻨﻲ ‪ 32‬ﺑﺮ ‪ 4‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪،‬‬ ‫ﭘﺲ ﺍﻳﻦ ﻋﺪﺩ ﺑﺮ ‪ 4‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪* :13‬‬

‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪* :17‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪* :19‬‬

‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑﺮ ‪ :20‬ﺍﮔﺮ ﺭﻗﻢ ﺩﻫﮕﺎﻥ ﻳﻚ ﻋﺪﺩ ﺯﻭﺝ ﺑﺎﺷــﺪ‬ ‫ﻭ ﺭﻗــﻢ ﻳــﻜﺎﻥ ﺁﻥ ﻧﻴﺰ ﺻﻔﺮ ﺑﺎﺷــﺪ‪ ،‬ﺁﻥ ﻋــﺪﺩ ﺑﺮ ‪20‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﻣﺎﻧﻨﺪ ﻋﺪﺩ ‪.7960‬‬ ‫ﺑﺨﺶﭘﺬﻳــﺮﻱ ﺑــﺮ ‪ :21‬ﺍﮔﺮ ﻳﻚ ﻋﺪﺩ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‬ ‫ﻭ ﻣﺠﻤــﻮﻉ ﺍﺭﻗﺎﻡ ﺁﻥ ﻧﻴﺰ ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷــﺪ‪ ،‬ﺁﻥ‬ ‫ﻋﺪﺩ ﺑﺮ ‪ 21‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ‪ .‬ﻣﺜ ً‬ ‫ﻼ ﻋﺪﺩ ‪ 1638‬ﺑﺮ ‪7‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ ﻭ ﻫﻢﭼﻨﻴﻦ ﻣﺠﻤﻮﻉ ﺍﺭﻗﺎﻡ ﺁﻥ ﻳﻌﻨﻲ‬ ‫‪ 1+6+3+8=18‬ﺑﺮ ‪ 3‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﭘﺲ ﺍﻳﻦ ﻋﺪﺩ‬ ‫ﺑﺮ ‪ 21‬ﻧﻴﺰ ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﻪ ‪* :23‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪* :29‬‬ ‫ﺑﺮﺍﻱ ﺑﺮﺭﺳــﻲ ﺑﺨﺶﭘﺬﻳﺮﻱﻫﺎﻳﻲ ﻛﻪ ﺩﺭ ﺑﺎﻻ ﺑﺎ ﻋﻼﻣﺖ * ﻣﺸﺨﺺ‬ ‫ﺷﺪﻩﺍﻧﺪ‪ ،‬ﺭﻭﺵ ﺳﺎﺩﻩﺍﻱ ﺑﻪ ﻧﺎﻡ ﺭﻭﺵ ﻣﻀﺎﺭﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﺓ ﺑﺨﺶﭘﺬﻳﺮﻱ‬ ‫ﻭ ﻳــﺎ ﺑﻪ ﺍﺧﺘﺼﺎﺭ »ﻣﻀﺎﺭﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ« ﻭﺟــﻮﺩ ﺩﺍﺭﺩ ﻛﻪ ﺑﺮﺍﻱ ﻫﺮ‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﻪ ﻛﻤﻚ ﻋﺪﺩﻱ ﺧﺎﺹ ﺑﻪ ﻧﺎﻡ »ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ«‬ ‫ﺑﻪ ﺭﺍﺣﺘﻲ ﻣﻲﺗﻮﺍﻧﻴﻢ ﺑﺨﺶﭘﺬﻳﺮﻱ ﻫﺮ ﻋﺪﺩﻱ ﺍﺳﺘﻔﺎﺩﻩ ﻛﺮﺩ‪.‬‬ ‫ﺩﺭ ﺯﻳﺮ ﻧﺸﺎﻥ ﻣﻲﺩﻫﻴﻢ ﻛﻪ ﭼﻪﻃﻮﺭ ﺍﺯ »ﻣﻀﺮﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ« ﺩﺭ‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﻳﻚ ﻋﺪﺩ ﺍﺳــﺘﻔﺎﺩﻩ ﻣﻲﺷــﻮﺩ ﻭ ﺑﻌﺪ ﺍﺯ ﺭﻭﺵ ﭘﻴﺪﺍ ﻛﺮﺩﻥ‬ ‫ﺍﻳﻦ ﻣﻀﺮﺏ ﺭﺍ ﺗﻮﺿﻴﺢ ﻣﻲﺩﻫﻴﻢ‪.‬‬

‫ﺭﻭﺵ »ﻣﻀﺎﺭﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ«‬

‫ﺑﺮﺍﻱ ﺑﺮﺭﺳــﻲ ﻛﺮﺩﻥ ﺍﻳﻦﻛﻪ ﻋﺪﺩﻱ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﻳﺎ ﻧﻪ‪،‬‬ ‫ﻋﺪﺩ ‪ 5‬ﺭﺍ ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ ﺍﻧﺘﺨﺎﺏ ﻣﻲﻛﻨﻴﻢ‪ 5 .‬ﺭﺍ ﺩﺭ‬ ‫ﺭﻗﻢ ﻳﻜﺎﻥ ﻋﺪﺩ ﺩﺍﺩﻩ ﺷﺪﻩ ﺿﺮﺏ ﻣﻲﻛﻨﻴﻢ ﻭ ﺣﺎﺻﻞ ﺭﺍ ﺑﻪ ﻋﺪﺩﻱ ﻛﻪ ﺍﺯ‬ ‫ﺣﺬﻑ ﺭﻗﻢ ﻳﻜﺎﻥ ﺑﻪ ﺩﺳــﺖ ﺁﻣﺪﻩ ﺍﺳﺖ‪ ،‬ﻣﻲﺍﻓﺰﺍﻳﻴﻢ‪ .‬ﺍﮔﺮ ﺍﻳﻦ ﻋﺪﺩ ﺑﺮ ‪7‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ‪ ،‬ﻋﺪﺩ ﺍﻭﻟﻴﻪ ﻧﻴﺰ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ ﻭ ﺍﮔﺮ ﺍﻳﻦ ﻋﺪﺩ‬ ‫ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﻧﺒﺎﺷﺪ‪ ،‬ﻋﺪﺩ ﺍﻭﻟﻴﻪ ﻧﻴﺰ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬ ‫ﺑــﺮﺍﻱ ﺍﻭﻟﻴﻦ ﻣﺜﺎﻝ‪ ،‬ﻣﻲﺧﻮﺍﻫﻴﻢ ﺑﺮﺭﺳــﻲ ﻛﻨﻴﻢ ﻛــﻪ ﺁﻳﺎ ‪ 91‬ﺑﺮ ‪7‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ ﻳﺎ ﻧﻪ؟‬ ‫ﺭﻗﻢ ﻳﻜﺎﻥ ﺍﻳﻦ ﻋﺪﺩ ﻳﻌﻨﻲ ‪ 1‬ﺭﺍ ﺩﺭ ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ ﻳﻌﻨﻲ ‪5‬‬ ‫ﺿﺮﺏ ﻣﻲﻛﻨﻴﻢ‪ .‬ﺟﻮﺍﺏ ‪ 5‬ﺍﺳــﺖ‪ .‬ﺍﮔــﺮ ‪ 5‬ﺭﺍ ﺑﻪ ‪ 9‬ﺍﺿﺎﻓﻪ ﻛﻨﻴﻢ‪ ،‬ﺣﺎﺻﻞ‬ ‫‪ 14‬ﻣﻲﺷــﻮﺩ ﻛﻪ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ ،‬ﭘﺲ ‪ 91‬ﻧﻴﺰ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ‬ ‫ﺍﺳﺖ‪.‬‬ ‫ﺩﺭ ﻣﺜﺎﻟﻲ ﺩﻳﮕﺮ‪ ،‬ﻣﻲﺧﻮﺍﻫﻴﻢ ﺑﺒﻴﻨﻴﻢ ﻛﻪ ﺁﻳﺎ ‪ 123‬ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ‬ ‫ﺍﺳﺖ ﻳﺎ ﻧﻪ؟‬ ‫ﺭﻗــﻢ ﻳﻜﺎﻥ ﺍﻳﻦ ﻋﺪﺩ ﻳﻌﻨــﻲ ‪ 3‬ﺭﺍ ﺩﺭ ‪) 5‬ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ(‬ ‫ﺿــﺮﺏ ﻣﻲﻛﻨﻴﻢ‪ .‬ﺟﻮﺍﺏ ﺑﻪ ﺩﺳــﺖ ﺁﻣــﺪﻩ ﻳﻌﻨــﻲ ‪ 15‬ﺭﺍ ﺑﺎ ‪ 12‬ﺟﻤﻊ‬ ‫ﻣﻲﻛﻨﻴﻢ ﻛﻪ ‪ 27‬ﻣﻲﺷــﻮﺩ ﻭ ﭼﻮﻥ ‪ 27‬ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﻧﻴﺴــﺖ‪ ،‬ﭘﺲ‬ ‫‪ 123‬ﻧﻴﺰ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﻧﻴﺴﺖ‪.‬‬ ‫ﺑﺮﺍﻱ ﺁﺧﺮﻳﻦ ﻣﺜﺎﻝ‪ ،‬ﺁﻳﺎ ‪ 1638‬ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ ﻳﺎ ﻧﻪ؟‬ ‫ﺭﻗــﻢ ﻳﻜﺎﻥ ﺍﻳﻦ ﻋــﺪﺩ ﻳﻌﻨﻲ ‪ 8‬ﺭﺍ ﺩﺭ ‪ 5‬ﺿــﺮﺏ ﻣﻲﻛﻨﻴﻢ‪ .‬ﺣﺎﺻﻞ‬ ‫‪ 40‬ﻣﻲﺷــﻮﺩ‪ .‬ﺍﮔﺮ ‪ 40‬ﺭﺍ ﺑــﻪ ‪ 163‬ﺍﺿﺎﻓﻪ ﻛﻨﻴﻢ‪ ،‬ﺟﻮﺍﺏ ‪ 203‬ﺍﺳــﺖ‪.‬‬ ‫ﺑﺮﺍﻱ ﺍﻳﻦﻛﻪ ﺑﺒﻴﻨﻴﻢ ‪ 203‬ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳــﺖ ﻳﺎ ﻧﻪ‪ ،‬ﺩﻭﺑﺎﺭﻩ ﻫﻤﻴﻦ‬ ‫ﻗﺎﻋﺪﻩ ﺭﺍ ﺑﻪ ﻛﺎﺭ ﻣﻲﺑﺮﻳﻢ‪ .‬ﺭﻗﻢ ﻳﻜﺎﻥ ﻋﺪﺩ ‪ 203‬ﻳﻌﻨﻲ ‪ 3‬ﺭﺍ ﺩﺭ ‪ 5‬ﺿﺮﺏ‬ ‫ﻣﻲﻛﻨﻴﻢ‪ ،‬ﺣﺎﺻﻞ ‪ 15‬ﻣﻲﺷــﻮﺩ‪ ،‬ﺣﺎﺻﻞ ‪ 15‬ﻣﻲﺷﻮﺩ‪ .‬ﺍﮔﺮ ‪ 15‬ﺭﺍ ﺑﺎ ‪20‬‬ ‫ﺟﻤﻊ ﻛﻨﻴﻢ‪ ،‬ﺟﻮﺍﺏ ‪ 35‬ﻣﻲﺷﻮﺩ ﻛﻪ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﭘﺲ ‪203‬‬ ‫ﻭ ﺩﺭ ﻧﺘﻴﺠﻪ ‪ 1638‬ﻧﻴﺰ ﺑﺮ ‪ 7‬ﺑﺨﺶﭘﺬﻳﺮﻧﺪ‪.‬‬ ‫ﺳﺆﺍﻟﻲ ﻛﻪ ﻣﻤﻜﻦ ﺍﺳﺖ ﺍﻳﻦﺟﺎ ﻣﻄﺮﺡ ﺷﻮﺩ‪ ،‬ﺍﻳﻦ ﺍﺳﺖ ﻛﻪ ﻣﻀﺮﺏ‬ ‫ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ ﭼﻪﻃﻮﺭ ﻣﺸﺨﺺ ﻣﻲﺷﻮﺩ؟‬ ‫ﺩﺭ ﺯﻳﺮ ﺑﻪ ﺟﻮﺍﺏ ﺍﻳﻦ ﺳﺆﺍﻝ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪.‬‬

‫ﭼﻪﻃﻮﺭ ﻣﻀﺎﺭﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ‬ ‫ﺭﺍ ﭘﻴﺪﺍ ﻛﻨﻴﻢ؟‬

‫ﺍﮔﺮ ﺑﺨﻮﺍﻫﻴــﻢ ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ ﺭﺍ ﺑــﺮﺍﻱ ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ‬ ‫ﻋﺪﺩﻱ ﻣﺎﻧﻨﺪ ‪ a‬ﭘﻴﺪﺍ ﻛﻨﻴﻢ‪ a ،‬ﺭﺍ ﻋﺪﺩﻱ ﺿﺮﺏ ﻣﻲﻛﻨﻴﻢ ﻛﻪ ﺭﻗﻢ ﻳﻜﺎﻥ‬ ‫ﻋﺪﺩ ﺣﺎﺻﻞ ‪ 9‬ﺑﺎﺷــﺪ‪ .‬ﺍﻛﻨــﻮﻥ ‪ 9‬ﺭﺍ ﺍﺯ ﺭﻗﻢ ﻳﻜﺎﻥ ﺣﺬﻑ ﻣﻲﻛﻨﻴﻢ ﻭ ﺑﻪ‬ ‫ﻋﺪﺩ ﺑﻪ ﺩﺳﺖ ﺁﻣﺪﻩ ﻳﻚ ﻭﺍﺣﺪ ﺍﺿﺎﻓﻪ ﻣﻲﻛﻨﻴﻢ‪ .‬ﺍﻳﻦ ﻋﺪﺩ ﻫﻤﺎﻥ ﻣﻀﺮﺏ‬ ‫ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

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‫ﺑﺮﺍﻱ ﻣﺜﺎﻝ‪ ،‬ﺍﮔﺮ ﺑﺨﻮﺍﻫﻴﻢ ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ ﺭﺍ ﺑﺮ ﻋﺪﺩ ‪ 7‬ﭘﻴﺪﺍ‬ ‫ﻛﻨﻴﻢ‪ ،‬ﻛﺎﻓﻲ ﺍﺳــﺖ ‪ 7‬ﺭﺍ ﺩﺭ ‪ 7‬ﺿﺮﺏ ﻛﻨﻴﻢ ﺗﺎ ﺭﻗﻢ ﻳﻜﺎﻥ ﻋﺪﺩ ﺣﺎﺻﻞ‬ ‫‪ 9‬ﺷــﻮﺩ‪ .‬ﺍﻛﻨــﻮﻥ ﺟﻮﺍﺏ ﺍﻳــﻦ ﺣﺎﺻﻞﺿﺮﺏ ﺭﺍ ﻛﻪ ‪ 49‬ﺍﺳــﺖ ﺩﺭ ﻧﻈﺮ‬ ‫ﻣﻲﮔﻴﺮﻳﻢ ﻭ ‪ 9‬ﺭﺍ ﺍﺯ ﺭﻗﻢ ﻳﻜﺎﻥ ﺣﺬﻑ ﻭ ﻳﻚ ﻭﺍﺣﺪ ﺑﻪ ﻗﻴﻤﺖ ﺑﺎﻗﻲﻣﺎﻧﺪﻩ‪،‬‬ ‫ﻳﻌﻨﻲ ‪ 4‬ﺍﺿﺎﻓﻪ ﻣﻲﻛﻨﻴﻢ‪ .‬ﻋﺪﺩ ﺑﻪﺩﺳــﺖﺁﻣﺪﻩ‪ ،‬ﻳﻌﻨﻲ ‪ ،5‬ﻫﻤﺎﻥ ﻣﻀﺮﺏ‬ ‫ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ ﺑﺮﺍﻱ ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ﻋﺪﺩ ‪ 7‬ﺍﺳﺖ‪ .‬ﺩﺭ ﺑﺎﻻ ﺩﻳﺪﻳﻢ ﻛﻪ‬ ‫ﭼﻪﻃــﻮﺭ ﺑﻪ ﻛﻤــﻚ ﺍﻳﻦ ﻋﺪﺩ ﻣﻲﺗﻮﺍﻥ ﺑﺨﺶﭘﺬﻳــﺮﻱ ﻫﺮ ﻋﺪﺩ ﺑﺮ ‪ 7‬ﺭﺍ‬ ‫ﺑﺮﺭﺳﻲ ﻛﺮﺩ‪.‬‬ ‫ﺍﻛﻨﻮﻥ ﻣﻀﺎﺭﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ ﺭﺍ ﺑﺮﺍﻱ ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ﺍﻋﺪﺍﺩ ‪،13‬‬ ‫‪ 23 ،19 ،17‬ﻭ ‪ 29‬ﭘﻴﺪﺍ ﻣﻲﻛﻨﻴﻢ‪.‬‬ ‫‪ .1‬ﺑﺮﺍﻱ ﻋﺪﺩ ‪ 13‬ﺩﺍﺭﻳﻢ‪ .13×3=39 :‬ﺭﻗﻢ ﻳﻜﺎﻥ ‪ 9‬ﺍﺳــﺖ ﻛﻪ ﺍﮔﺮ‬ ‫ﻳﻚ ﻭﺍﺣﺪ ﺑﻪ ﺩﻫﮕﺎﻥ ﻳﻌﻨﻲ ﻫﻤﺎﻥ ﻗﺴــﻤﺖ ﺑﺎﻗﻲﻣﺎﻧــﺪﻩ ﺍﺿﺎﻓﻪ ﻛﻨﻴﻢ‪،‬‬ ‫ﻋﺪﺩ ‪ 4‬ﺑﻪﺩﺳــﺖ ﻣﻲﺁﻳــﺪ ﻛﻪ ﺍﻳﻦ ﻋﺪﺩ ﻫﻤﺎﻥ ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﺓ‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪ 13‬ﺍﺳﺖ‪.‬‬ ‫‪ .2‬ﺑﺮﺍﻱ ﻋﺪﺩ ‪ 17‬ﺩﺍﺭﻳﻢ‪ .17×7=119 :‬ﺍﮔﺮ ﻳﻚ ﻭﺍﺣﺪ ﺑﻪ ‪ 11‬ﺍﺿﺎﻓﻪ‬ ‫ﻛﻨﻴﻢ‪ ،‬ﻋﺪﺩ ‪ 12‬ﺑﻪﺩﺳﺖ ﻣﻲﺁﻳﺪ ﻛﻪ ﻣﻀﺮﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ ﺍﺳﺖ‪.‬‬ ‫‪ .3‬ﺑــﺮﺍﻱ ﻋﺪﺩ ‪ 19‬ﺩﺍﺭﻳﻢ‪ .19×1=19 :‬ﺍﮔــﺮ ﻳﻚ ﻭﺍﺣﺪ ﺑﻪ ﺩﻫﮕﺎﻥ‬ ‫ﺍﺿﺎﻓﻪ ﻛﻨﻴﻢ‪ ،‬ﻋﺪﺩ ‪ 2‬ﺑﻪﺩﺳــﺖ ﻣﻲﺁﻳﺪ ﻛﻪ ﻫﻤﺎﻥ ﻣﻀﺮﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ‬ ‫ﺍﺳﺖ‪.‬‬

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‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪ .4‬ﺑﺮﺍﻱ ﻋﺪﺩ ‪ 23‬ﺩﺍﺭﻳﻢ‪ .23×3=69 :‬ﺑﺎ ﺍﺿﺎﻓﻪ ﻛﺮﺩﻥ ﻳﻚ ﻭﺍﺣﺪ ﺑﻪ‬ ‫ﺩﻫﮕﺎﻥ‪ ،‬ﻋﺪﺩ ‪ 7‬ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﻀﺮﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ ﺑﻪﺩﺳﺖ ﻣﻲﺁﻳﺪ‪.‬‬ ‫‪ .5‬ﺑﺮﺍﻱ ﻋﺪﺩ ‪ 29‬ﻧﻴﺰ ﺩﺍﺭﻳﻢ‪ .29×1=29 :‬ﺍﮔﺮ ﻳﻚ ﻭﺍﺣﺪ ﺑﻪ ﺩﻫﮕﺎﻥ‬ ‫ﺍﺿﺎﻓﻪ ﺷﻮﺩ‪ ،‬ﻋﺪﺩ ‪ 3‬ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﻀﺮﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﻩ ﺑﻪ ﺩﺳﺖ ﻣﻲﺁﻳﺪ‪.‬‬ ‫ﺑــﻪ ﻋﻨــﻮﺍﻥ ﻣﺜﺎﻝ‪ ،‬ﻣﻲﺧﻮﺍﻫﻴــﻢ ﺗﻘﺴــﻴﻢﭘﺬﻳﺮﻱ ‪ 351‬ﺑﺮ ‪ 132‬ﺭﺍ‬ ‫ﺑﺮﺭﺳــﻲ ﻛﻨﻴﻢ‪ .‬ﻫﻤﺎﻥﻃﻮﺭ ﻛﻪ ﺩﺭ ﺑﺎﻻ ﺩﻳﺪﻳﻢ‪ ،‬ﻣﻀﺮﺏ ﺑﺮﺭﺳــﻲﻛﻨﻨﺪﻩ‬ ‫ﺑﺮﺍﻱ ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪ ،13‬ﻋﺪﺩ ‪ 4‬ﺍﺳــﺖ‪ .‬ﺍﮔﺮ ‪ 4‬ﺭﺍ ﺩﺭ ﺭﻗﻢ ﻳﻜﺎﻥ ﺍﻳﻦ‬ ‫ﻋﺪﺩ ﺿﺮﺏ ﻛﻨﻴﻢ‪ 4×1=4 ،‬ﺑﻪ ﺩﺳــﺖ ﻣﻲﺁﻳﺪ ﻭ ﺍﮔﺮ ﺁﻥ ﺭﺍ ﺑﻪ ﺑﻘﻴﺔ ﺍﻳﻦ‬ ‫ﻋــﺪﺩ ﻳﻌﻨﻲ ‪ 39‬ﺍﺿﺎﻓﻪ ﻛﻨﻴﻢ‪ 35+4=39 ،‬ﺣﺎﺻﻞ ﻣﻲﺷــﻮﺩ ﻛﻪ ﺑﺮ ‪13‬‬ ‫ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﭘﺲ ‪ 351‬ﻧﻴﺰ ﺑﺮ ‪ 13‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬ ‫ﺩﺭ ﻣﺜﺎﻟﻲ ﺩﻳﮕﺮ‪ ،‬ﻣﻲﺧﻮﺍﻫﻴﻢ ﺑﺪﺍﻧﻴﻢ ﻛﻪ ﺁﻳﺎ ‪ 578‬ﺑﺮ ‪ 17‬ﺑﺨﺶﭘﺬﻳﺮ‬ ‫ﺍﺳﺖ ﻳﺎ ﺧﻴﺮ‪.‬‬ ‫ﺭﻗﻢ ﻳﻜﺎﻥ ﺍﻳﻦ ﻋﺪﺩ ﻳﻌﻨﻲ ‪ 8‬ﺭﺍ ﺩﺭ ﻋﺪﺩ ‪ 12‬ﻛﻪ ﻣﻀﺮﺏ ﺑﺮﺭﺳﻲﻛﻨﻨﺪﺓ‬ ‫ﺑﺨﺶﭘﺬﻳﺮﻱ ﺑﺮ ‪ 17‬ﺍﺳﺖ‪ ،‬ﺿﺮﺏ ﻣﻲﻛﻨﻴﻢ‪ 8×12=96 :‬ﻭ ﺣﺎﺻﻞ ﺭﺍ ﺑﻪ‬ ‫ﺑﻘﻴﺔ ﻋﺪﺩ ﺍﺿﺎﻓﻪ ﻣﻲﻛﻨﻴﻢ‪ .57+96=153 :‬ﺑﺮﺍﻱ ﺑﺨﺶﭘﺬﻳﺮﻱ ‪ 153‬ﺑﺮ‬ ‫‪ ،17‬ﺩﻭﺑــﺎﺭﻩ ﻫﻤﻴﻦ ﺭﻭﺵ ﺭﺍ ﺑﻪ ﻛﺎﺭ ﻣﻲﺑﺮﻳﻢ‪ .‬ﺭﻗﻢ ﻳﻜﺎﻥ ﻳﻌﻨﻲ ‪ 3‬ﺭﺍ ﺩﺭ‬ ‫‪ 12‬ﺿــﺮﺏ ﻣﻲﻛﻨﻴﻢ‪ 3×12=36 :‬ﻭ ﺣﺎﺻــﻞ ﺭﺍ ﺑﺎ ﺑﻘﻴﺔ ﻋﺪﺩ ﻳﻌﻨﻲ ‪15‬‬ ‫ﺟﻤﻊ ﻣﻲﻛﻨﻴﻢ‪ 36+15=51 :‬ﻭ ﭼﻮﻥ ‪ 51‬ﻣﻀﺮﺏ ‪ 17‬ﺍﺳﺖ‪ ،‬ﭘﺲ ‪153‬‬ ‫ﻭ ﺩﺭ ﻧﺘﻴﺠﻪ ‪ 578‬ﻧﻴﺰ ﺑﺮ ‪ 17‬ﺑﺨﺶﭘﺬﻳﺮﻧﺪ‪.‬‬

‫ﺟﺪﻭﻝ ﻭ ﺳﺮﮔﺮﻣﻲ‬

‫︗︡ول‬ ‫ﻣ ّ‬ ‫ﺤﻤﺪ ﻋﺰﻳﺰﻱﭘﻮﺭ‬

‫ﺍﻓﻘﻲ‬ ‫‪ 29 .1‬ﺍﻓﻘــﻲ ﺑﻪﻋﻼﻭﻩﻱ ‪ 9 .5 14963‬ﺍﻓﻘﻲ ﺑﻪﻋﻼﻭﻩﻱ ‪.6 152‬‬ ‫ﻳــﺎﺯﺩﻩ ﺑﺮﺍﺑــﺮ ‪ 28‬ﻋﻤﻮﺩﻱ ‪ 15 .8‬ﺍﻓﻘﻲ ﺗﻘﺴــﻴﻢ ﺑﺮ ﻳﺎﺯﺩﻩ ‪14 .9‬‬ ‫ﻋﻤــﻮﺩﻱ ﺑﻪﻋﻼﻭﻩﻱ ﻫﺸــﺘﺎﺩ ﻭ ﭘﻨﺞ ‪ 4 .11‬ﻋﻤــﻮﺩﻱ ﺑﻪﻋﻼﻭﻩﻱ ﺩﻭ‬ ‫‪ .13‬ﻣﺮﺑــﻊ ﻳﻚ ﻋــﺪﺩ ‪ .15‬ﺗﻌﺪﺍﺩ ﺛﺎﻧﻴﻪﻫﺎ ﺩﺭ ﺷــﺶ ﺩﻗﻴﻘﻪ ‪.16‬‬ ‫‪ 10‬ﻋﻤــﻮﺩﻱ ﺑﻪﻋــﻼﻭﻩﻱ ‪ 15 .17 10019‬ﻋﻤﻮﺩﻱ ﻣﻨﻬﺎﻱ ﻧﻮﺩ ﻭ‬ ‫ﻧﻪ ‪ 22 .19‬ﻋﻤﻮﺩﻱ ﺑﻪﻋﻼﻭﻩﻱ ‪ 15 .21 118‬ﺍﻓﻘﻲ ﺗﻘﺴﻴﻢ ﺑﺮ‬ ‫ﺷــﺶ ‪ .22‬ﺳــﻪ ﺑﺮﺍﺑﺮ ‪ 18‬ﻋﻤﻮﺩﻱ ‪ 21 .24‬ﺍﻓﻘﻲ ﻣﻨﻬﺎﻱ ﺳﻲ ﻭ‬ ‫ﭘﻨﺞ ‪ 7 .25‬ﻋﻤﻮﺩﻱ ﺑﻪﻋﻼﻭﻩﻱ ‪ .27 243‬ﺗﻌﺪﺍﺩ ﺛﺎﻧﻴﻪﻫﺎ ﺩﺭ ﻧﻪ‬ ‫ﺩﻗﻴﻘﻪ ‪ 16 .29‬ﺍﻓﻘﻲ ﺑﻪﻋﻼﻭﻩﻱ ‪8984‬‬ ‫‪4‬‬ ‫‪7‬‬ ‫‪12‬‬

‫‪2‬‬

‫‪3‬‬

‫‪1‬‬

‫‪6‬‬ ‫‪10‬‬

‫‪11‬‬

‫︎︀︨︞ ︗︡ول ︫﹝︀رهی ﹇︊﹏‬

‫‪5‬‬ ‫‪9‬‬

‫‪15‬‬

‫‪8‬‬ ‫‪13‬‬

‫‪14‬‬

‫ﻋﻤﻮﺩﻱ‬ ‫‪ .1‬ﻳﻚ ﻋﺪﺩ ﺍ ّﻭﻝ ‪ .2‬ﺗﻌﺪﺍﺩ ﻣﺎﻩﻫﺎ ﺩﺭ ﻳﺎﺯﺩﻩ ﺳــﺎﻝ ‪ 17 .3‬ﺍﻓﻘﻲ‬ ‫ﻣﻨﻬﺎﻱ ﻫﻔﺘﺎﺩ ﻭ ﭘﻨﺞ ‪ 5 .4‬ﻋﻤﻮﺩﻱ ﺗﻘﺴــﻴﻢ ﺑﺮ ﭼﻬﺎﺭ ‪ .5‬ﭼﻬﺎﺭ‬ ‫ﺑﺮﺍﺑــﺮ ‪ 1‬ﻋﻤــﻮﺩﻱ ‪ 6 .7‬ﺍﻓﻘﻲ ﺑﻪﻋﻼﻭﻩﻱ ‪ 10 .8 101‬ﻋﻤﻮﺩﻱ‬ ‫ﻣﻨﻬــﺎﻱ ‪ 13 .10 4026‬ﺍﻓﻘــﻲ ﺿــﺮﺏ ﺩﺭ ‪ 2‬ﻋﻤﻮﺩﻱ ‪25 .12‬‬ ‫ﺍﻓﻘﻲ ﺑﻪﻋﻼﻭﻩﻱ ‪ 2 .14 9806‬ﻋﻤﻮﺩﻱ ﺑﻪﻋﻼﻭﻩﻱ ﺑﻴﺴﺖ ﻭ ﭼﻬﺎﺭ‬ ‫‪ 8 .15‬ﺍﻓﻘﻲ ﺿﺮﺏ ﺩﺭ ﻫﻔﺖ ‪ .18‬ﺗﻌﺪﺍﺩ ﻣﺎﻩﻫﺎ ﺩﺭ ﻧﻪ ﺳﺎﻝ ‪.20‬‬ ‫ﺗﻌﺪﺍﺩ ﻣﺎﻩﻫﺎ ﺩﺭ ﺩﻩ ﺳﺎﻝ ‪ 26 .22‬ﻋﻤﻮﺩﻱ ﺿﺮﺏ ﺩﺭ ﻳﺎﺯﺩﻩ ‪.23‬‬ ‫‪ 11‬ﺍﻓﻘﻲ ﺿﺮﺏ ﺩﺭ ﭘﻨﺞ ‪ 22 .26‬ﺍﻓﻘﻲ ﺗﻘﺴــﻴﻢ ﺑﺮ ﻧﻪ ‪27 .28‬‬ ‫ﺍﻓﻘﻲ ﺗﻘﺴﻴﻢ ﺑﺮ ﺩﻭﺍﺯﺩﻩ‬

‫‪3‬‬

‫‪20‬‬

‫‪18‬‬

‫‪19‬‬

‫‪2‬‬

‫‪23‬‬

‫‪24‬‬ ‫‪28‬‬

‫‪27‬‬

‫‪3‬‬

‫‪21‬‬

‫‪22‬‬ ‫‪26‬‬ ‫‪29‬‬

‫‪25‬‬

‫‪7‬‬

‫‪9‬‬

‫‪16‬‬ ‫‪17‬‬

‫‪O‬‬

‫‪7‬‬

‫‪O‬‬

‫‪8‬‬

‫‪6‬‬

‫‪9‬‬

‫‪O‬‬

‫‪4‬‬

‫‪2‬‬ ‫‪8‬‬ ‫‪6‬‬

‫‪4‬‬ ‫‪9‬‬ ‫‪7‬‬

‫‪3‬‬

‫‪1‬‬ ‫‪2‬‬

‫‪6‬‬

‫‪1‬‬

‫‪8‬‬

‫‪O‬‬

‫‪9‬‬

‫‪6‬‬

‫‪5‬‬

‫‪O‬‬

‫‪3‬‬

‫‪O‬‬

‫‪8‬‬ ‫‪6‬‬

‫‪5‬‬

‫‪2‬‬

‫‪4‬‬

‫‪O‬‬

‫‪8‬‬

‫‪5‬‬

‫‪4‬‬

‫‪3‬‬

‫‪1‬‬ ‫‪4‬‬ ‫‪1‬‬

‫‪5‬‬

‫‪O‬‬

‫‪O‬‬

‫‪4‬‬

‫‪5‬‬ ‫‪1‬‬

‫‪6‬‬ ‫‪O‬‬

‫‪4‬‬

‫‪6‬‬

‫‪4‬‬

‫‪5‬‬ ‫‪3‬‬

‫‪O‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪6‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪11‬‬

‫ﺍﻧﺪﻳﺸﻪﻭﺭﺯﻱ‬

‫﹞︟︀︵︣ات ︨﹀︣ در ︨﹫︀رهی ﹡︀﹛‪︣﹞﹢‬‬ ‫ﺗﺮﺟﻤﻪﻱ‬

‫‪ :‬ﺣﺴﻦ ﻳﺎﻭﺭ ﺗﺒﺎﺭ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻣﻌﻤﺎ‪ ،‬ﺁﺯﻣﻮﻥ ﻭ ﺧﻄﺎ‪.‬‬ ‫ﺍﺗﻮﺑﻴﺎ‬

‫ﺳﺎﻝ ‪ 2988‬ﻣﻴﻼﺩﻯ ﺍﺳﺖ ﻭ ﺳﺎﻛﻨﺎﻥ ﺳﻴﺎﺭﻩﻯ ))ﻧﺎﻟﻮﻣﺮ(( ﺩﺭ ﺑﺮﺍﺑﺮ‬ ‫))ﺍﻣﭙﺮﺍﺗﻮﺭﻯ ﺟﺎﺑﺮﺍﻧﻪﻯ ﺯﻣﻴﻦ (( ﻋﻠﻢ ﻃﻐﻴﺎﻥ ﺑﺮﺍﻓﺮﺍﺷــﺘﻪﺍﻧﺪ‪ .‬ﺷﻬﺮﺕ ﻭ‬ ‫ﺭﻳﻮ‬ ‫‪1‬‬ ‫ﺍﻋﺘﺒﺎﺭ ﺷــﻤﺎ ﺑﻪ ﻋﻨﻮﺍﻥ ﻫﻨﺮﻣﻨﺪﻯ ﭘﺮﻃﺮﻓﺪﺍﺭ ﺳﺒﺐ ﺷﺪﻩ ﺍﺳﺖ ﻛﻪ ﺑﺮﺍﻯ‬ ‫‪ 3‬ﺯﻧﮕﺒﺎﺭ‬ ‫ﺳــﺮﮔﺮﻡ ﻛﺮﺩﻥ ﻧﻴﺮﻭﻫﺎﻯ ﻣﺤﺎﻓــﻆ‪ ،‬ﺩﻭﺍﺯﺩﻩ ﭘﺎﻳﮕﺎﻩ ﻧﻈﺎﻣﻰ ﺯﻣﻴﻨﻴﺎﻥ ﺩﺭ‬ ‫ﺍﻳﻦ ﻛﺮﻩﻯ ﺟﻨﮓﺯﺩﻩ‪ ،‬ﺑﻪ ﺁﻥﺟﺎ ﻓﺮﺍ ﺧﻮﺍﻧﺪﻩ ﺷﻮﻳﺪ‪ .‬ﺍﻳﻦ ﭘﺎﻳﮕﺎﻩﻫﺎ ﻛﻪ ﺑﻪ‬ ‫ﻳﺎﻟﻲ ‪4‬‬ ‫ﺳﻨﮕﺎﭘﻮﺭ ‪o‬‬ ‫ﻓﺎﺻﻠــﻪﻯ ﻳﻚ ﻛﻴﻠﻮﻣﺘﺮﻯ ﺍﺯ ﻳﻜﺪﻳﮕﺮ ﺩﺭ ﻃﻮﻝ ﻳﻚ ﺑﺰﺭﮔﺮﺍﻩ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ‬ ‫ﺍﻧﺪ‪ ،‬ﭼﻨﺎﻥ ﺳــﺎﺧﺘﻪ ﺷــﺪﻩﺍﻧﺪ ﻛﻪ ﺭﻭﻯ ﻫﻢ ﻳﻚ ﻣﺴــﻴﺮ ﺑﺴﺘﻪﻯ ﺣﻠﻘﻪ‬ ‫‪1‬‬ ‫ﻣﺎﻧﻨﺪ ﺭﺍ ﺗﺸﻜﻴﻞ ﻣﻰﺩﻫﻨﺪ‪.‬‬ ‫ﻫﻨﮓ‬ ‫‪ُ o‬ﺭﻡ‬ ‫ﺷــﻤﺎ ﻫﻤﺮﺍﻩ ﻳﻚ ﺍﺗﻮﺑﻮﺳــﻰ ﻣﺨﺼﻮﺹ ﺭﺍﻫﭙﻴﻤﺎﻳﻰ ﺩﺭ ﻧﺎﻟﻮﻣﺮ‪ ،‬ﺩﺭ‬ ‫ﻛﻨﮓ‬ ‫‪o‬‬ ‫ﺭﻳﺠﺰ‬ ‫ﻳﻜﻰ ﺍﺯ ﭘﺎﻳﮕﺎﻫﻬﺎﻯ ﺩﻭﺍﺯﺩﻩﮔﺎﻧﻪ ﻳﺎﺩ ﺷﺪﻩ ﻓﺮﻭﺩ ﻣﻰﺁﻳﻴﺪ‪ .‬ﻣﺎﻣﻮﺭﻳﺖ ﺷﻤﺎ‬ ‫ﻫﻠﻴﺲﺑﻴﻮﺩ ‪o‬‬ ‫ﺁﻥ ﺍﺳــﺖ ﻛــﻪ ﺣﺮﻛﺖ ﺧــﻮﺩ ﺭﺍ ﺍﺯ ﻳﻚ ﭘﺎﻳــﮕﺎﻩ ﺁﻏﺎﺯ ﻛﻨﻴــﺪ ﻭ ﭘﺲ ﺍﺯ‬ ‫‪1‬‬ ‫ﮔﺬﺷﺘﻦ ﺍﺯ ﺳﺎﻳﺮ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺑﻪ ﭘﺎﻳﮕﺎﻩ ﻣﺒﺪﺍ ﺑﺎﺯ ﮔﺮﺩﻳﺪ‪ .‬ﺍﻳﻦ ﻳﻚ ﻣﺎﻣﻮﺭﻳﺖ‬ ‫ﺭﻭﻳﻴﻦ‬ ‫‪1‬‬ ‫ﺧﻄﺮﻧﺎﻙ ﺍﺳﺖ‪ ،‬ﺯﻳﺮﺍ ﺩﺭ ﺻﻮﺭﺕ ﻭﻗﻮﻉ ﻫﺮﮔﻮﻧﻪ ﺍﺷﺘﺒﺎﻩ ﻳﺎ ﻟﻐﺰﺵ ﻣﻤﻜﻦ‬ ‫ﻧﻮﻭﺭ‬ ‫ﺍﺳﺖ ﺩﺭ ﻣﻴﺎﻧﻪﻯ ﺭﺍﻩ ﺑﻪ ﺩﺳﺖ ﺷﻮﺭﺷﻴﺎﻥ ﻧﺎﻟﻮﻣﺮ ﺩﺳﺘﮕﻴﺮ ﺷﻮﻳﺪ ﻭ ﻣﻮﺭﺩ‬ ‫ﺷﻜﻨﺠﻪﻯ ﺭﻭﺍﻧﻰ ﻗﺮﺍﺭ ﮔﻴﺮﻳﺪ‪.‬‬ ‫ﺍﻳﻦ ﻭﺳﻴﻠﻪﻯ ﻧﻘﻠﻴﻪﻯ ﺭﺍﺣﺖ ﺷﻤﺎ ﺩﺭ ﻫﺮ ﻛﻴﻠﻮﻣﺘﺮ ﻳﻚ ﻟﻴﺘﺮ ﺑﻨﺰﻳﻦ‬ ‫ﻣﺼــﺮﻑ ﻣﻰﻛﻨﺪ ﻭ ﺑﺎﻙ ﺑﻨﺰﻳﻦ ﺁﻥ ﻫﻨﮕﺎﻡ ﻓﺮﻭﺩ ﺑﺮ ﺳــﻴﺎﺭﻩ ﺑﺎﻳﺪ ﺧﺎﻟﻰ ﺣﺠﻢ ﺑﺎﻙ ﺑﻨﺰﻳﻦ ﺍﺗﻮﺑﻮﺱ ﺑﻪ ﺷــﻤﺎ ﺍﻣﻜﺎﻥ ﻣﻰﺩﻫﺪ ﻛﻪ ﺑﺘﻮﺍﻧﻴﺪ ﻫﻤﻪﻯ‬ ‫ﺑﻨﺰﻳﻦ ﻫﺮ ﻳﻚ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺭﺍ ﺩﺭ ﺁﻥ ﺟﺎﻯ ﺩﻫﻴﺪ‪.‬‬ ‫ﺷﺪﻩ ﺑﺎﺷﺪ‪.‬‬ ‫ﺑﻪ ﺍﻳﻦ ﺗﺮﺗﻴﺐ‪ ،‬ﺁﻳﺎ ﻣﻰﺗﻮﺍﻧﻴﺪ ﺑﻰ ﺁﻥﻛﻪ ﺑﻪ ﻋﻠﺖ ﺗﻤﺎﻡ ﺷﺪﻥ ﺳﻮﺧﺖ‬ ‫ﺧﻮﺷﺒﺨﺘﺎﻧﻪ ﺩﺭ ﺑﺮﺧﻰ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎﻯ ﻧﻈﺎﻣﻰ ﻣﻘﺪﺍﺭﻯ ﺑﻨﺰﻳﻦ ﻭﺟﻮﺩ‬ ‫ﺩﺍﺭﺩ‪ .‬ﺩﺭ ﻭﺍﻗﻊ ﻣﺠﻤﻮﻉ ﺑﻨﺰﻳﻦ ﻣﻮﺟﻮﺩ ﺩﺭ ﺗﻤﺎﻣﻰ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺁﻥﻗﺪﺭ ﻫﺴﺖ ﺩﺭ ﻣﻴﺎﻥ ﺭﺍﻩ ﺑﻤﺎﻧﻴﺪ‪ ،‬ﻳﻚ ﺩﻭﺭ ﻛﺎﻣﻞ ﻃﻮﻝ ﺑﺰﺭﮔﺮﺍﻩ ﺭﺍ ﺑﭙﻴﻤﺎﻳﻴﺪ؟ ﺑﺎ ﺍﻳﻦ‬ ‫ﻛﻪ ﺷــﻤﺎ ﺑﺘﻮﺍﻧﻴﺪ ﺑﺎ ﺍﺳــﺘﻔﺎﺩﻩ ﺍﺯ ﺁﻥ‪ ،‬ﻳﻚ ﺩﻭﺭ ﻛﺎﻣﻞ ﻣﺴﻴﺮ ﺑﺰﺭﮔﺮﺍﻩ ﺭﺍ ﺣﺴﺎﺏ ﺑﺎﻳﺪ ﺩﺭ ﻛﺪﺍﻡ ﻳﻚ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎ ﻓﺮﻭﺩ ﺁﻳﻴﺪ؟‬ ‫‪ .2‬ﻧﺨﺴﺘﻴﻦ ﻣﻌﻤﺎ ﺭﺍ ﻣﻰﺗﻮﺍﻥ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺭﻭﺵ ))ﺁﺯﻣﻮﻥ ﻭ ﺧﻄﺎ((‬ ‫ﺑﭙﻴﻤﺎﻳﻴﺪ‪ .‬ﺍﻣﺎ ﺷــﺮﺍﻳﻂ ﺟﻨﮕﻰ ﺍﻳﺠﺎﺏ ﻣﻰﻛﻨﺪ ﻛﻪ ﺩﺭ ﺑﻌﻀﻰ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎ‬ ‫ﻫﻴﭻ ﺑﻨﺰﻳﻨﻰ ﻧﺒﺎﺷﺪ‪ .‬ﺍﺯ ﺍﻳﻦ ﺭﻭ‪ ،‬ﺑﻬﺘﺮ ﺍﺳﺖ ﺗﺮﺗﻴﺒﻰ ﺑﺪﻫﻴﺪ ﻛﻪ ﺑﻪ ﻫﻨﮕﺎﻡ ﺣﻞ ﻛﺮﺩ‪ .‬ﺍﻣﺎ ﺍﮔﺮ ﻣﺠﺒﻮﺭ ﺷﻮﻳﺪ ﻳﻚ ﭼﻨﻴﻦ ﻣﺴﻴﺮﻯ ﺭﺍ ﺩﺭ ﻳﻚ ﺑﺰﺭﮔﺮﺍﻩ‬ ‫ﺭﺳــﻴﺪﻥ ﺑﻪ ﭘﺎﻳﮕﺎﻩﻫﺎﻯ ﻓﺎﻗﺪ ﺑﻨﺰﻳﻦ‪ ،‬ﻣﻘﺪﺍﺭﻯ ﺳﻮﺧﺖ ﺩﺭ ﺑﺎﻙ ﺩﺍﺷﺘﻪ ﺣﻠﻘﻮﻯ ﺷﻜﻞ ﺩﺍﺭﺍﻯ ﺻﺪﻫﺎ ﭘﺎﻳﮕﺎﻩ ﻧﻈﺎﻣﻰ ﺑﭙﻴﻤﺎﻳﻴﺪ ﭼﻪ ﺧﻮﺍﻫﻴﺪ ﻛﺮﺩ؟‬ ‫ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ‪ ،‬ﺑﻰﺗﺮﺩﻳﺪ ﺑﺎﻳﺪ ﺷــﻴﻮﻩﺍﻯ ﻣﺼﻮﻥ ﺍﺯ ﺧﻄﺎ ﺑﺮﮔﺰﻳﻨﻴﺪ ﻛﻪ‬ ‫ﺑﺎﺷﻴﺪ‪ .‬ﺣﺎﻝ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺍﻳﻦ ﻧﻜﺎﺕ ﺑﻪ ﺩﻭ ﻣﻌﻤﺎﻯ ﺯﻳﺮ ﺗﻮﺟﻪ ﻛﻨﻴﺪ‪:‬‬ ‫‪ .1‬ﻫﻤﺎﻥﮔﻮﻧﻪ ﻛﻪ ﺩﺭ ﺗﺼﻮﻳﺮ ﺩﻳﺪﻩ ﻣﻰﺷــﻮﺩ‪ ،‬ﻣﺤﻞ ﻭ ﻧﺎﻡ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺷــﻤﺎ ﺭﺍ ﺩﺭ ﺍﺗﺨﺎﺫ ﺗﺼﻤﻴﻢ ﺑﺮﺍﻯ ﻳﺎﻓﺘﻦ ﻧﻘﻄﻪﻯ ﺁﻏﺎﺯ ﺣﺮﻛﺖ )ﺍﮔﺮ ﻳﻚ‬ ‫ﻭ ﻣﻘــﺪﺍﺭ ﺑﻨﺰﻳﻦ ﻣﻮﺟﻮﺩ ﺩﺭ ﻫﺮ ﻳﻚ ﺍﺯ ﺁﻥﻫﺎ ﻣﺸــﺨﺺ ﺷــﺪﻩ ﺍﺳــﺖ‪ .‬ﭼﻨﻴﻦ ﻧﻘﻄﻪﻯ ﺁﻏﺎﺯﻯ ﺍﻣﻜﺎﻥ ﭘﺬﻳﺮ ﺑﺎﺷــﺪ( ﻳﺎﺭﻯ ﻛﻨــﺪ‪ .‬ﺁﻳﺎ ﻣﻰﺗﻮﺍﻧﻴﺪ‬ ‫)ﺷﻤﺎﺭﻩﻫﺎ ﻧﻤﺎﻳﺸﮕﺮ ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ ﻣﻮﺟﻮﺩ ﺩﺭ ﻫﺮ ﭘﺎﻳﮕﺎﻩ ﺍﺳﺖ( ﺩﺭ ﺿﻤﻦ ﭼﻨﻴﻦ ﺷﻴﻮﻩﺍﻯ ﺭﺍ ﺑﻴﺎﺑﻴﺪ؟‬ ‫‪1‬‬

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‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

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‫ﺟﺪﻭﻝ ﻭ ﺳﺮﮔﺮﻣﻲ‬

‫︗︡ول ‪١٠٠ ︀︑ ١‬‬ ‫ﻧﺴﺮﻳﻦ ﺷﺮﻳﻔﻴﺎﻥ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﺟﺪﻭﻝ ﺍﻋﺪﺍﺩ‪ ،‬ﺍﻋﺪﺍﺩ ‪ 1‬ﺗﺎ ‪ ،100‬ﺣﺪﺱ‪.‬‬

‫ﺟﺪﻭﻝ ﺭﻭﺑﻪ ﺭﻭ ﺭﺍ ﺧﻴﻠﻰ ﺩﻳﺪﻩ ﺍﻳﺪ! ﺍﻳﻦ ﺑﺎﺭ‬ ‫ﻛﻤﻰ ﺩﻗﻴﻖﺗﺮ ﻧﮕﺎﻩ ﻛﻨﻴﺪ‪.‬‬ ‫ﻓﺮﺽ ﻛﻨﻴﺪ ﺩﻭ ﺗﺎ ﺍﺯ ﺍﻳﻦ ﺟﺪﻭﻝﻫﺎ ﺩﺭﺳﺖ‬ ‫ﻛﺮﺩﻩﺍﻳﻢ‪ ،‬ﺩﻗﻴﻘﺎ ﻫﻢﺍﻧﺪﺍﺯﻩ ﻭ ﻫﻢﺷﻜﻞ‪.‬‬ ‫ﺩﻭ ﺗــﺎ ﺟــﺪﻭﻝ ﺭﺍ ﻃﻮﺭﻯ ﺭﻭﻯ ﻫــﻢ ﻗﺮﺍﺭ‬ ‫ﻣﻰﺩﻫﻴــﻢ ﻛﻪ ﻋﺪﺩﻫﺎ ﺭﻭﻯ ﻫﻢ ﻗﺮﺍﺭ ﺑﮕﻴﺮﻧﺪ! ﺑﻪ‬ ‫ﺟﻬﺖ ﻓﻠﺶﻫﺎ ﺩﺭ ﺷﻜﻞﻫﺎﻯ ﺯﻳﺮ ﺩﻗﺖ ﻛﻨﻴﺪ‪:‬‬ ‫ﺩﺭ ﻫــﺮ ﻳﻚ ﺍﺯ ﺣﺎﻟﺖ ﻫﺎﻯ ﺑﺎﻻ‪ ،‬ﻛﺪﺍﻡ ﻋﺪﺩ‬ ‫ﺟــﺪﻭﻝ ﺯﻳﺮﻯ‪ ،‬ﺯﻳﺮ ﻋــﺪﺩ ‪ 24‬ﺍﺯ ﺟﺪﻭﻝ ﺭﻭﻳﻰ‬ ‫ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﺳﺖ؟‬ ‫ﻣﻰﺗﻮﺍﻧﻴــﺪ ﺩﻭ ﺗﺎ ﺟﺪﻭﻝ ﺩﺭﺳــﺖ ﻛﻨﻴﺪ ﻭ‬ ‫ﺁﻥﻫــﺎ ﺭﺍ ﺩﺭ ﻫــﺮ ﭼﻬﺎﺭ ﺣﺎﻟــﺖ ﺭﻭﻯ ﻫﻢ ﻗﺮﺍﺭ‬ ‫ﺩﻫﻴﺪ‪ .‬ﺳــﭙﺲ ﺍﺑﺘﺪﺍ ﺣﺪﺱ ﺑﺰﻧﻴﺪ ﺯﻳﺮ ‪ 24‬ﭼﻪ‬ ‫ﻋﺪﺩﻯ ﺍﺳــﺖ‪ ،‬ﺳﭙﺲ ﺁﺭﺍﻡ ﺩﻭ ﺟﺪﻭﻝ ﺭﺍ ﺍﺯ ﻫﻢ‬ ‫ﺟﺪﺍ ﺳــﺎﺯﻳﺪ ﻭ ﺩﺭﺳﺘﻰ ﭘﺎﺳــﺨﺘﺎﻥ ﺭﺍ ﺑﺮﺭﺳﻰ‬ ‫ﻛﻨﻴﺪ‪.‬‬

‫ﭘﺸﺖ ﺟﺪﻭﻝ‬ ‫‪ 1‬ﺗﺎ ‪100‬‬ ‫ﺭﻭﻳﻲ‬

‫ﭘﺸﺖ ﺟﺪﻭﻝ‬ ‫‪ 1‬ﺗﺎ ‪100‬‬ ‫ﺭﻭﻳﻲ‬

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‫‪99 100‬‬

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‫ﭘﺸﺖ ﺟﺪﻭﻝ‬ ‫‪ 1‬ﺗﺎ ‪100‬‬ ‫ﺭﻭﻳﻲ‬

‫ﭘﺸﺖ ﺟﺪﻭﻝ‬ ‫‪ 1‬ﺗﺎ ‪100‬‬ ‫ﺭﻭﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪13‬‬

‫ﮔﻔﺖ ﻭ ﮔﻮ‬

‫﹞‪﹢︑﹩‬ان ﹨︣‬ ‫﹞︧﹮﹚﹤ای را ︝﹏ ﹋︣د!‬ ‫﹎﹀️و ﹎‪ ︀︋ ﹢‬دا﹡︩آ﹞‪﹢‬زان و ﹞︺﹚﹜ ر﹬︀︲‪﹩‬‬ ‫﹞︡ر︨﹤ی را﹨﹠﹝︀﹬‪﹩︊︨︀﹝﹧︵ ︡﹫﹧︫ ﹩‬‬ ‫ﺁﺯﺍﺩﻩ ﺷﺎﻛﺮﻱ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﮔﻔﺖﻭ ﮔﻮ‪ ،‬ﻣﺪﺭﺳﻪﻱ ﺭﺍﻫﻨﻤﺎﻳﻲ‪ ،‬ﺷﻬﻴﺪ ﻃﻬﻤﺎﺳﺒﻲ‪ ،‬ﻣﻌﻠﻢ ﺭﻳﺎﺿﻲ‪ ،‬ﺩﺭﺱ ﺭﻳﺎﺿﻲ‪.‬‬

‫ﺍﺷﺎﺭﻩ‬ ‫ﻭﻗﺘﻲ ﻛﺎﺭ ﺑﺎ ﺍﻋﺪﺍﺩ ﺭﺍ ﺷـﺮﻭﻉ ﻣﻲﻛﻨـﻲ‪ ،‬ﺫﻫﻨﺖ ﻣﺜﻞ ﻳﺦ ﻣﻨﺠﻤﺪ‬ ‫ﺍﺳـﺖ‪ ،‬ﻫﻤﻴﻦﻃﻮﺭ ﻛﻪ ﺑﺎ ﺍﻋﺪﺍﺩ ﺑﺎﺯﻱ ﻣﻲﻛﻨﻲ‪ ،‬ﺍﻳﻦ ﻳﺦ ﺁﺏ ﻣﻲﺷـﻮﺩ‪،‬‬ ‫ﻳﻌﻨـﻲ ﺫﻫﻨـﺖ ﻧـﺮﻡ ﻣﻲﺷـﻮﺩ ﻭ ﺩﺭ ﺁﻥ ﻟﺤﻈـﻪ ﺗﻘﺮﻳﺒ ًﺎ ﻣﻲﺗـﻮﺍﻥ ﻫﺮ‬ ‫ﻣﺴﺌﻠﻪﺍﻱ ﺭﺍ ﺣﻞ ﻛﺮﺩ!‬ ‫ﺧﻴﻠﻲ ﺑﻪ ﺫﻫﻨﺘﺎﻥ ﻓﺸﺎﺭ ﻧﻴﺎﻭﺭﻳﺪ‪ .‬ﺍﻳﻦ ﺟﻤﻼﺕ‪ ،‬ﺣﺮﻑﻫﺎﻱ ﻫﻴﭻﻳﻚ‬ ‫ﺍﺯ ﻧﻮﺍﺑﻎ ﺭﻳﺎﺿﻲ ﻛﻪ ﺷـﻤﺎ ﻣﻲﺷﻨﺎﺳﻴﺪ‪ ،‬ﻧﻴﺴـﺖ! ﺍﻳﻦ ﺣﺮﻑﻫﺎﻱ ﻳﻚ‬ ‫ﺷـﺎﮔﺮﺩ ﻛﻼﺱ ﺳﻮﻡ ﻣﺪﺭﺳـﻪﻱ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺷـﻬﻴﺪ ﻃﻬﻤﺎﺳﺒﻲ ﺍﺳﺖ‬ ‫ﻛﻪ ﭘـﺎﻱ ﺻﺤﺒﺖ ﺍﻭ ﻭ ﺩﻭﺍﺯﺩﻩ ﻧﻔﺮ ﺍﺯ ﺩﻭﺳـﺘﺎﻥ ﻭ ﻣﻌﻠﻢ ﺭﻳﺎﺿﻲﺷـﺎﻥ‬ ‫ﻧﺸﺴـﺘﻴﻢ‪ .‬ﺁﻥﻫﺎ ﺍﺯ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﺣـﺮﻑ ﺯﺩﻧﺪ ﻭ ﮔﻔﺘﻨﺪ ﻛﻪ ﺑﻌﻀﻲﻫﺎ‬ ‫ﻋﺎﺷﻘﺎﻧﻪ ﺩﻭﺳﺘﺶ ﺩﺍﺭﻧﺪ ﻭ ﻋﺪﻩﺍﻱ ﺍﺯ ﺗﻪ ﺩﻝ ﺍﺯ ﺁﻥ ﻣﺘﻨﻔﺮﻧﺪ‪.‬‬ ‫ﺧﺎﻧﻢ ﺍﻣﺎﻣﻲ ﻫﻢ‪ ،‬ﻣﻌﻠﻢ ﺑﭽﻪﻫﺎ ﻭ ﻣﺪﺍﻓﻊ ﺳﺮﺳﺨﺖ ﺭﻳﺎﺿﻲ ﺍﺳﺖ ﻭ‬ ‫ﺭﺍﻩﺣﻞﻫﺎﻱ ﺧﻮﺑﻲ ﺑﺮﺍﻱ ﻣﻮﻓﻖ ﺑـﻮﺩﻥ ﺩﺭ ﺭﻳﺎﺿﻲ ﻣﻲﺩﺍﻧﺪ ﻛﻪ ﺑﻪ ﺩﺭﺩ‬ ‫ﻫﻤﻪﻱ ﺑﭽﻪﻫﺎﻱ ﺩﻭﺭﻩ ﺭﺍﻫﻨﻤﺎﻳﻲ ﻣﻲﺧﻮﺭﺩ‪.‬‬ ‫ﺁﻥﭼـﻪ ﺩﺭ ﺍﺩﺍﻣـﻪ ﻣﻲﺧﻮﺍﻧﻴﺪ ﺣﺮﻑﻫﺎﻱ ﺑﭽﻪﻫـﺎ ﻭ ﻣﻌﻠﻢ ﺭﻳﺎﺿﻲ‬ ‫ﻣﺪﺭﺳﻪﻱ ﺷﻬﻴﺪ ﻃﻬﻤﺎﺳﺒﻲ ﺍﺳﺖ‪.‬‬

‫‪14‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺑﭽﻪﻫﺎ ﺩﻭ ﮔﺮﻭﻩ ﺷﺪﻧﺪ‪ .‬ﻫﺮ ﺩﻭ ﮔﺮﻭﻩ ﻧﻤﺮﻩﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺧﻮﺑﻲ‬ ‫ﮔﺮﻓﺘﻪﺍﻧﺪ‪ .‬ﻳﻚ ﮔﺮﻭﻩ ﺑﺎ ﺷـﻮﺭ ﻭ ﻫﻴﺠﺎﻥ ﺍﺯ ﻋﻼﻗﻪﺷﺎﻥ ﺑﻪ ﺭﻳﺎﺿﻲ‬ ‫ﻣﻲﮔﻮﻳﻨﺪ ﻭ ﮔـﺮﻭﻩ ﺩﻳﮕﺮ ﺍﺯ ﺑﻲﻋﻼﻗﮕﻲﺷـﺎﻥ ﺑﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ‬ ‫ﺣـﺮﻑ ﻣﻲﺯﻧﻨﺪ‪ .‬ﺍﺯ ﺁﻥﻫﺎ ﻣﻲﺧﻮﺍﻫـﻢ ﻳﻜﻲ ﻳﻜﻲ ﺩﻻﻳﻞ ﻋﻼﻗﻪ ﻭ‬ ‫ﺑﻲﻋﻼﻗﮕﻲﺷﺎﻥ ﺭﺍ ﺑﻪ ﺍﻳﻦ ﺩﺭﺱ ﺑﮕﻮﻳﻨﺪ‪.‬‬ ‫ﺷـﺒﻨﻢ ﺍﺑﻮﺍﻟﻔﻀﻠﻲ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﺑــﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﻋﻼﻗﻪ ﺩﺍﺭﻡ‪،‬‬ ‫ﭼــﻮﻥ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﻳــﺎﺩ ﮔﺮﻓﺘﻦ ﺭﻳﺎﺿــﻲ ﺩﺭ ﺯﻧﺪﮔــﻲ ﺧﻴﻠﻲ ﻛﻤﻜﻢ‬ ‫ﻣﻲﻛﻨﺪ‪.‬‬ ‫ﻳﺎﺩ ﮔﺮﻓﺘﻦ ﺭﻳﺎﺿﻲ ﺩﺭ ﺁﻳﻨﺪﻩ ﻭ ﺍﻧﺘﺨﺎﺏ ﺷﻐﻞ ﻫﻢ ﻛﻤﻚ ﺯﻳﺎﺩﻱ ﺑﻪ‬ ‫ﻣﻦ ﻣﻲﻛﻨﺪ‪ .‬ﻋﻼﻭﻩ ﺑﺮ ﺍﻳﻦﻫﺎ ﻣﻦ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﺭﻳﺎﺿﻲ ﺩﺭﺱ ﺷــﻨﻴﺪﻧﻲ‬ ‫ﺍﺳﺖ ﻛﻪ ﺩﺭ ﺑﻴﺶﺗﺮ ﺭﺷﺘﻪﻫﺎ ﺣﺘﻲ ﺍﺩﺑﻴﺎﺕ ﻧﻘﺶ ﺩﺍﺭﺩ‪.‬‬ ‫ﻧﺮﺟﺲ ﺳﻌﻴﺪﻱ ﺗﻮﺿﻴﺢ ﻣﻲﺩﻫﺪ‪ :‬ﺭﻳﺎﺿﻲ ﺳﺨﺖ ﺍﺳﺖ‪ .‬ﻣﻔﺎﻫﻴﻢ‬ ‫ﮔﻴﺞﻛﻨﻨﺪﻩﺍﻱ ﺩﺍﺭﺩ‪ .‬ﺍﮔﺮ ﻣﻲﺷــﺪ ﺭﻳﺎﺿﻲ ﺭﺍ ﺑﺎ ﺷــﻜﻞ ﻳﺎ ﺭﻭﺷﻲ ﻏﻴﺮ ﺍﺯ‬ ‫ﺍﻳﻦ ﻳﺎﺩ ﺩﺍﺩ‪ ،‬ﻫﻤﻪ ﺩﻭﺳﺘﺶ ﺩﺍﺷﺘﻨﺪ‪.‬‬ ‫ﺷـﺒﻨﻢ ﺍﺑﻮﺍﻟﻔﻀﻠﻲ ﻣﻌﺘﻘﺪ ﺍﺳﺖ ﻛﻪ ﺭﻳﺎﺿﻲ ﺳﺨﺖ ﺍﺳﺖ‪ ،‬ﺍﻣﺎ ﺍﮔﺮ‬ ‫ﺁﺩﻡ ﺩﻧﺒﺎﻝ ﺩﺭﺱ ﺭﺍ ﺑﮕﻴﺮﺩ ﻳــﺎ ﺧﻮﺩﺵ ﺑﻴﻦ ﻋﻼﺋﻢ ﺭﻳﺎﺿﻲ ﺭﺍﺑﻄﻪﻫﺎﻳﻲ‬ ‫ﺑﺮﻗﺮﺍﺭ ﻛﻨﺪ‪ ،‬ﺩﺭﻛﺶ ﺁﺳﺎﻥﺗﺮ ﻣﻲﺷﻮﺩ‪.‬‬

‫ﻧﺮﺟﺲ ﺳـﻌﻴﺪﻱ ﺍﻋﺘﻘﺎﺩ ﺩﻳﮕﺮﻱ ﺩﺍﺭﺩ‪ :‬ﺍﻳﻦ ﺩﺭﺳــﺖ ﺍﺳــﺖ ﻛﻪ‬ ‫ﺩﺭﻙ ﺭﻳﺎﺿــﻲ ﺑﺎ ﺍﻳﻦ ﺭﻭﺵ ﺭﺍﺣﺖﺗﺮ ﻣﻲﺷــﻮﺩ‪ ،‬ﺍﻣــﺎ ﻫﺮﻛﺲ ﻇﺮﻓﻴﺘﻲ‬ ‫ﺩﺍﺭﺩ ﻭ ﺭﺍﻩﺣﻠﻲ ﻛﻪ ﺑﻪ ﺫﻫﻦ ﻳﻚ ﻧﻔﺮ ﻣﻲﺭﺳﺪ‪ ،‬ﺷﺎﻳﺪ ﺑﻪ ﺫﻫﻦ ﻧﻔﺮ ﺩﻳﮕﺮ‬ ‫ﻧﺮﺳﺪ‪.‬‬ ‫ﺑﻪ ﻧﺮﺟﺲ ﻣﻲﮔﻮﻳﻢ ﺗﻮ ﻧﻤــﺮﻩﻱ ﺭﻳﺎﺿﻲﺍﺕ ﺭﺍ ‪ 20‬ﮔﺮﻓﺘﻪﺍﻱ‪ ،‬ﭘﺲ‬ ‫ﺑﻪ ﺧﺎﻃﺮ ﺳــﺨﺖ ﺑﻮﺩﻥ ﺭﻳﺎﺿﻲ ﻧﻴﺴــﺖ ﻛﻪ ﻧﺴﺒﺖ ﺑﻪ ﺁﻥ ﺑﻲﻋﻼﻗﻪﺍﻱ‪،‬‬ ‫ﺩﺭﺳﺖ ﺍﺳﺖ؟‬ ‫ﻧﺮﺟﺲ ﺳﻌﻴﺪﻱ ﺟﻮﺍﺏ ﻣﻲﺩﻫﺪ‪ :‬ﻣﻔﺎﻫﻴﻢ ﺭﻳﺎﺿﻲ ﺧﻴﻠﻲ ﺷﺒﻴﻪ ﺑﻪ‬ ‫ﻫﻢ ﻫﺴﺘﻨﺪ ﻭ ﺑﻪ ﻫﻤﻴﻦ ﺩﻟﻴﻞ ﮔﻴﺞﻛﻨﻨﺪﻩﺍﻧﺪ‪.‬‬ ‫ﺍﺯ ﺍﻭ ﻣﻲﭘﺮﺳﻢ‪ :‬ﺗﻮ ﺍﻳﻦ ﻣﺸﻜﻞ ﺭﺍ ﭼﮕﻮﻧﻪ ﺣﻞ ﻛﺮﺩﻱ؟‬ ‫ﻧﺮﺟﺲ ﺳﻌﻴﺪﻱ ﭘﺎﺳﺦ ﻣﻲﺩﻫﺪ‪ :‬ﻣﻦ ﺁﻥﻗﺪﺭ ﺭﻳﺎﺿﻲ ﺭﺍ ﺧﻮﺍﻧﺪﻩﺍﻡ‬ ‫ﺗﺎ ‪ 20‬ﮔﺮﻓﺘﻢ‪ ،‬ﺍﻣﺎ ﺑﻪ ﺁﻥ ﻋﻼﻗﻪﻣﻨﺪ ﻧﺸﺪﻡ‪.‬‬ ‫ﻓﺎﻃﻤـﻪ ﻧﺼﻴﺮﻱ ﻫــﻢ ﻭﺍﺭﺩ ﺑﺤﺚ ﻣﻲﺷــﻮﺩ ﻭ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﻓﻜﺮ‬ ‫ﻣﻲﻛﻨﻢ ﻛﺴــﻲ ﻛﻪ ﺭﻳﺎﺿــﻲ ﺭﺍ ‪ 20‬ﻣﻲﮔﻴﺮﺩ‪ ،‬ﻭﻟﻲ ﺑــﻪ ﺁﻥ ﻋﻼﻗﻪﻣﻨﺪ‬ ‫ﻧﻴﺴﺖ‪ ،‬ﺩﭼﺎﺭ ﺩﻭﮔﺎﻧﮕﻲ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ﻣــﻦ ﺧﻮﺩﻡ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﺭﻳﺎﺿــﻲ ﺩﺭ ﺭﺃﺱ ﺯﻧﺪﮔﻲﺍﻡ ﻗﺮﺍﺭ ﺩﺍﺭﺩ‪ .‬ﺑﺎ‬ ‫ﺭﻳﺎﺿﻲ ﺧﻮ ﮔﺮﻓﺘﻪﺍﻡ‪.‬ﺍﺣﺴــﺎﺱ ﻣﻲﻛﻨﻢ ﺩﺭ ﺯﻧﺪﮔﻲﺍﻡ ﺑﺎ ﺭﻳﺎﺿﻲ ﺧﻴﻠﻲ‬ ‫ﻛﺎﺭ ﺩﺍﺭﻡ‪ .‬ﻳﻜﻲ ﺍﺯ ﺳﺎﺩﻩﺗﺮﻳﻦ ﻛﺎﺭﺑﺮﺩﻫﺎﻱ ﺭﻳﺎﺿﻲ‪ ،‬ﺧﺮﻳﺪ ﻛﺮﺩﻥﻫﺎﻱ ﻫﺮ‬ ‫ﺭﻭﺯﻩﻱ ﻣﺎﺳﺖ‪ .‬ﻛﺎﺭﺑﺮﺩ ﺩﻳﮕﺮ‪ ،‬ﺣﻞ ﭼﻴﺴﺘﺎﻥﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺍﺳﺖ‪.‬‬

‫‪1390‬‬ ‫‪139‬‬ ‫ﺎﻥ ‪0‬‬ ‫ﻥ‬ ‫ﺗﺎﺑﺴﺘﺎﻥ‬ ‫ﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘ‬ ‫ﻢ‪،‬‬ ‫ﺷﺎﻧﺰﺩﻫﻢ‪،‬‬ ‫ﺷﺎﻧﺰﺩﻫ‬ ‫ﺩﻭﺭﺓ ﺷ ﻧ‬ ‫ﺩﺩﻭﺭﺓ‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬ ‫ﺎﻳﻲ‬ ‫ﻫﻨﻤﺎ‬ ‫ﺭﺍﻫﻨﻤ‬ ‫ﺭﺭﺍ‬

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‫ﺭﻳﺎﺿﻲ‪ ،‬ﻛﺎﺭ ﻛﺮﺩﻥ ﻳﺎ ﺑﺎﺯﻱ ﻛﺮﺩﻥ ﺑﺎ ﺍﻋﺪﺍﺩ ﺍﺳﺖ‪ .‬ﻣﻦ ﻭﻗﺘﻲ ﺣﺎﺻﻞ‬ ‫ﺟﻤﻴﻠﻪ ﺍﻣﺎﻣﻲ )ﻣﻌﻠﻢ ﺑﭽﻪﻫﺎ( ﭘﺎﺳــﺦ ﺑﺮﺧﻲ ﺍﺯ ﺣﺮﻑﻫﺎﻱ ﺑﭽﻪﻫﺎ‬ ‫ﻛﺎﺭ ﻛﺮﺩﻥ ﺑﺎ ﺍﻋﺪﺍﺩ ﺭﺍ ﻣﻲﺑﻴﻨﻢ‪ ،‬ﺣﺲ ﻣﻲﻛﻨﻢ ﻛﺎﺭ ﺧﻮﺑﻲ ﺍﻧﺠﺎﻡ ﺩﺍﺩﻩﺍﻡ ﺭﺍ ﻣﻲﺩﻫــﺪ ﻭ ﻣﻲﮔﻮﻳــﺪ‪ :‬ﺑﭽﻪﻫﺎ ﺷــﻤﺎ ﺑــﻪ ﺍﻳﻦ ﺩﻟﻴﻞ ﻓﻜــﺮ ﻣﻲﻛﻨﻴﺪ‬ ‫ﻭ ﻧﺘﻴﺠﻪﺍﺵ ﺭﺍ ﻣﺸﺎﻫﺪﻩ ﻣﻲﻛﻨﻢ‪.‬‬ ‫ﺑﻌﻀﻲ ﺍﺯ ﻣﻄﺎﻟﺐ ﺭﻳﺎﺿﻲ ﮔﻴﺞﻛﻨﻨﺪﻩ ﺍﺳــﺖ ﻛــﻪ ﺍﺛﺒﺎﺕ ﺍﻳﻦ ﻣﻄﺎﻟﺐ ﺭﺍ‬ ‫ﻣﻦ ﺍﺣﺴــﺎﺱ ﻣﻲﻛﻨﻢ ﻫﺮ ﻣﻌﺎﺩﻟﻪﺍﻱ ﺭﺍ ﻛﻪ ﺣﻞ ﻣﻲﻛﻨﻢ‪ ،‬ﻧﺘﻴﺠﻪﻱ ﻳــﺎﺩ ﻧﮕﺮﻓﺘﻪﺍﻳــﺪ‪ .‬ﺍﮔﺮ ﺑﻪ ﻣﺒﺎﺣﺚ ﺩﺭﺱﻫﺎﻱ ﺭﻳﺎﺿﻲﺗــﺎﻥ ﺩﺭ ﺍﺑﺘﺪﺍﻳﻲ ﻭ‬ ‫ﺗﻼﺷﻢ ﺭﺍ ﻣﻲﺑﻴﻨﻢ‪ .‬ﺑﻪ ﻫﻤﻴﻦ ﺩﻟﻴﻞ ﻋﺎﺷﻖ ﺭﻳﺎﺿﻲ ﻫﺴﺘﻢ‪.‬‬ ‫ﺭﺍﻫﻨﻤﺎﻳــﻲ ﺩﻗﺖ ﻛﻨﻴــﺪ‪ ،‬ﻣﻲﺑﻴﻨﻴﺪ ﻛﻪ ﺑﻪ ﺑﻌﻀﻲ ﺍﺯ ﻣﺴــﺎﺋﻞ ﺩﺭﻣﻘﻄﻊ‬ ‫ﺷـﻘﺎﻳﻖ ﺻﺎﺩﻗﻲ ﻫﻢ ﺣﺮﻑﻫﺎﻳﻲ ﺩﺍﺭﺩ‪ :‬ﻣﻤﻜﻦ ﺍﺳــﺖ ﺭﻳﺎﺿﻲ ﺩﺭ ﺍﺑﺘﺪﺍﻳﻲ ﺍﺷــﺎﺭﻩ ﺷﺪﻩ ﺑﻮﺩ ﺗﺎ ﺫﻫﻦ ﺷــﻤﺎ ﺑﺎ ﺁﻥ ﺁﺷﻨﺎ ﺑﺎﺷﺪ‪ ،‬ﻭﻟﻲ ﻣﻄﺎﻟﺐ‬ ‫ﺯﻧﺪﮔﻲ ﻛﺎﺭﺑﺮﺩ ﺯﻳﺎﺩﻱ ﺩﺍﺷــﺘﻪ ﺑﺎﺷــﺪ‪ ،‬ﺍﻣﺎ ﻣﻦ ﺭﺍﺑﻄــﻪﻱ ﺑﻴﻦ ﺍﻋﺪﺍﺩ ﺭﺍ ﺗﻜﻤﻴﻠﻲﺗﺮ ﺭﺍ ﺑﻌﺪﻫﺎ ﺩﺭ ﻣﻘﻄﻊ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺧﻮﺍﻧﺪﻳﺪ‪.‬‬ ‫ﺩﺭﻙ ﻧﻤﻲﻛﻨﻢ‪ .‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ‪ ،‬ﭼﺮﺍ ‪ 2+2‬ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 4‬ﺍﺳــﺖ‪ .‬ﺭﻭﺍﺑﻄﻲ ﻣﺜﻞ‬ ‫ﺩﺭﺱﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺭﺍﻫﻨﻤﺎﻳﻲ ﻫﻢ ﺑﻪ ﻫﻤﻴﻦ ﺷــﻜﻞ ﺍﺳــﺖ‪ .‬ﺍﺛﺒﺎﺕ‬ ‫ﺍﻳﻦ ﺑﺎﻋﺚ ﺳــﺮﺩﺭﮔﻤﻲ ﻣﻦ ﻣﻲﺷﻮﺩ‪ .‬ﻣﻦ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﺑﻌﻀﻲ ﺍﺯ ﺭﻭﺍﺑﻂ ﺑﺮﺧــﻲ ﺍﺯ ﻣﻔﺎﻫﻴﻢ ﺭﺍ ﺑﻌﺪﻫﺎ ﺩﺭ ﺩﺑﻴﺮﺳــﺘﺎﻥ ﻭ ﻣﻘﺎﻃــﻊ ﺑﺎﻻﺗﺮ ﺧﻮﺍﻫﻴﺪ‬ ‫ﺭﻳﺎﺿﻲ ﺩﻟﻴﻠﻲ ﻣﻨﻄﻘﻲ ﻧﺪﺍﺭﻧﺪ‪.‬‬ ‫ﺧﻮﺍﻧــﺪ‪ .‬ﺍﻵﻥ ﺗﺪﺭﻳﺲ ﺍﻳــﻦ ﻣﻔﺎﻫﻴﻢ ﺑﺮﺍﻱ‬ ‫ﻓﺎﻃﻤﻪ ﺧﺪﺍﻳﻲ ﻣﻌﺘﻘﺪ ﺍﺳــﺖ‪ :‬ﻣﻦ‬ ‫ﺷﻤﺎ ﻛﺎﺑﺮﺩﻱ ﻧﺪﺍﺭﺩ ﻭ ﺳﻨﮕﻴﻦ ﺍﺳﺖ‪.‬‬ ‫ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﺍﻧﺴــﺎﻥ ﺍﮔــﺮ ﺧﻮﺩﺵ ﻫﻢ‬ ‫ﺷـﺒﻨﻢ ﺍﺑﻮﺍﻟﻔﻀﻠـﻲ ﺩﻟﻴــﻞ ﺩﻳﮕــﺮ‬ ‫ﻧﺨﻮﺍﻫﺪ‪ ،‬ﺭﻳﺎﺿﻲ ﺭﺍ ﻳﺎﺩ ﻣﻲﮔﻴﺮﺩ‪ .‬ﺷــﻤﺎ‬ ‫ﻋﻼﻗﻪﻣﻨﺪﻱﺍﺵ ﺑﻪ ﺍﻳــﻦ ﺩﺭﺱ ﺭﺍ ﻣﻄﺮﺡ‬ ‫ﺑﻪ ﺁﺩﻡﻫﺎﻱ ﺑﻲﺳﻮﺍﺩ ﺗﻮﺟﻪ ﻛﻨﻴﺪ‪ .‬ﺁﻥﻫﺎ‬ ‫ﻣﻲﻛﻨﺪ‪ :‬ﭘﻴﭽﻴﺪﮔﻲ ﺭﻳﺎﺿﻲ ﻳﻜﻲ ﺍﺯ ﺩﻻﻳﻞ‬ ‫ﻫﻢ ﺑﺎ ﺍﻳﻦﻛﻪ ﺩﺭﺳــﻲ ﺭﺍ ﺑﻪ ﻧﺎﻡ ﺭﻳﺎﺿﻲ‬ ‫ﻋﻼﻗﻪﻣﻨﺪﻱ ﻣﻦ ﺑﻪ ﺍﻳﻦ ﺩﺭﺱ ﺍﺳــﺖ‪ .‬ﻣﻦ‬ ‫ﻧﺨﻮﺍﻧﺪﻩﺍﻧــﺪ‪ ،‬ﺍﻣﺎ ﺭﻳﺎﺿــﻲ ﻣﻲﺩﺍﻧﻨﺪ ﻭ ﺍﺯ‬ ‫ﻋﺎﺷــﻖ ﭼﻴﺰﻫﺎﻱ ﻣﺠﻬﻮﻟﻢ‪ .‬ﭼﻴﺰﻫﺎﻳﻲ ﻛﻪ‬ ‫ﺁﻥ ﺩﺭ ﺯﻧﺪﮔﻲﺷــﺎﻥ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻨﺪ‪.‬‬ ‫ﺧﻮﺩﻣﺎﻥ ﺑﺎﻳﺪ ﺑﮕﺮﺩﻳﻢ ﺗﺎ ﻭﺍﻗﻌﻴﺘﺶ ﺭﺍ ﭘﻴﺪﺍ‬ ‫ﺑــﺮﺍﻱ ﻣﺜــﺎﻝ‪ ،‬ﻣﻴﻮﻩﻓﺮﻭﺷــﻲ ﺭﺍ ﺩﺭ ﻧﻈﺮ‬ ‫ﻛﻨﻴﻢ‪.‬‬ ‫ﺑﮕﻴﺮﻳﺪ ﻛﻪ ﺍﺻ ً‬ ‫ﻼ ﺭﻳﺎﺿﻲ ﻧﺨﻮﺍﻧﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ﻓﺎﻃﻤﻪ ﻧﺼﻴﺮﻱ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻭﻗﺘﻲ ﻛﺎﺭ‬ ‫ﺑﺮﺍﻱ ﺣﺴﺎﺏ ﻭ ﻛﺘﺎﺏ ﻛﺎﺭ ﻭ ﺯﻧﺪﮔﻲﺍﺵ‬ ‫ﺑﺎ ﺍﻋﺪﺍﺩ ﺭﺍ ﺷــﺮﻭﻉ ﻣﻲﻛﻨﻲ‪ ،‬ﺫﻫﻨﺖ ﻣﺜﻞ‬ ‫ﻣﺠﺒﻮﺭ ﺍﺳﺖ ﺍﺯ ﺭﻳﺎﺿﻲ ﺍﺳﺘﻔﺎﺩﻩ ﻛﻨﺪ ﻭ‬ ‫ﻳﺦ ﻣﻨﺠﻤﺪ ﺍﺳﺖ ﻭ ﻫﻤﻴﻦﻃﻮﺭ ﻛﻪ ﺑﺎ ﺍﻋﺪﺍﺩ‬ ‫ﺣﺘﻲ ﺑﺪﻭﻥ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻣﺎﺷﻴﻦ ﺣﺴﺎﺏ‪،‬‬ ‫ﺑﺎﺯﻱ ﻣﻲﻛﻨﻲ‪ ،‬ﺍﻳﻦ ﻳﺦ ﺁﺏ ﻣﻲﺷﻮﺩ‪ ،‬ﻳﻌﻨﻲ‬ ‫ﺫﻫﻨﺖ ﻧﺮﻡ ﻣﻲﺷﻮﺩ ﻭ ﺩﺭ ﺁﻥ ﻟﺤﻈﻪ ﺗﻘﺮﻳﺒﺎً‬ ‫ﻣﺤﺎﺳﺒﺎﺕ ﻛﺎﺭﻱﺍﺵ ﺭﺍ ﺍﻧﺠﺎﻡ ﻣﻲﺩﻫﺪ‪.‬‬ ‫ﻓﺎﻃﻤﻪ ﻧﺼﻴﺮﻱ ﻣﻲﮔﻮﻳﺪ‪ :‬ﺭﻳﺎﺿﻲ‬ ‫ﻣﻲﺗﻮﺍﻥ ﻫﺮ ﻣﺴﺌﻠﻪﺍﻱ ﺭﺍ ﺣﻞ ﻛﺮﺩ!‬ ‫ﺩﺭ ﺗﻤﺎﻡ ﺭﺷﺘﻪﻫﺎ ﻛﺎﺭﺑﺮﺩ ﺩﺍﺭﺩ ﻭ ﺑﺎ ﺗﻤﺎﻡ‬ ‫ﻳﻜــﻲ ﺍﺯ ﺩﻻﻳﻠﻲ ﻛﻪ ﺑﺎﻋﺚ ﻣﻲﺷــﻮﺩ‬ ‫ﺁﻥﻫﺎ ﺩﺭ ﺍﺭﺗﺒﺎﻁ ﺍﺳﺖ‪ .‬ﺩﻭﺳﺘﻢ ﻣﻲﮔﻔﺖ ﻛﻪ ﺑﻌﻀﻲ‬ ‫ﺑﭽﻪﻫﺎ ﺭﻳﺎﺿﻲ ﺭﺍ ﺩﻭﺳــﺖ ﻧﺪﺍﺷــﺘﻪ ﺑﺎﺷــﻨﺪ ﻳﺎ ﺁﻥ‬ ‫ﭘﻴﭽﻴﺪﮔﻲ‬ ‫ﺍﺯ ﻣﻔﺎﻫﻴﻢ ﺭﻳﺎﺿﻲ ﻣﻨﻄﻘﻲ ﻧﻴﺴﺘﻨﺪ ﻭ ﺍﻭ ﺍﻳﻤﺎﻥ ﻧﺪﺍﺭﺩ‬ ‫ﺭﺍ ﻳﺎﺩ ﻧﮕﻴﺮﻧﺪ‪ ،‬ﺍﻳﻦ ﺍﺳــﺖ ﻛﻪ ﺩﺭﺱ ﺭﺍ ﺳــﺮ ﻛﻼﺱ‬ ‫ﻛﻪ ﺩﺭﺳــﺖ ﺑﺎﺷــﻨﺪ‪ .‬ﻣﻦ ﻣﻲﮔﻮﻳﻢ ﻛﻪ ﺗﻮ ﻣﻲﺗﻮﺍﻧﻲ‬ ‫ﻳﺎﺩ ﻣﻲﮔﻴﺮﻧﺪ ﻭ ﺑﺎ ﺧﻮﺩﺷــﺎﻥ ﻣﻲﮔﻮﻳﻨﺪ ﺍﻳﻦ ﺩﺭﺱ‬ ‫ﺭﻳﺎﺿﻲ ﻳﻜﻲ ﺍﺯ ﺩﻻﻳﻞ‬ ‫ﺍﻳﻦ ﻣﻔﺎﻫﻴــﻢ ﺭﺍ ﺑﺮﺍﻱ ﺧﻮﺩﺕ ﺍﺛﺒﺎﺕ ﻛﻨﻲ ﻳﺎ ﺍﺯ ﻳﻚ‬ ‫ﭼﻪﻗﺪﺭ ﺁﺳــﺎﻥ ﺑــﻮﺩ! ﺍﻣﺎ ﺗﻮﺟﻪ ﻧﻤﻲﻛﻨﻨــﺪ ﻛﻪ ﺍﻳﻦ‬ ‫ﻋﻼﻗﻪﻣﻨﺪﻱ ﻣﻦ ﺑﻪ‬ ‫ﻣﺘﺨﺼﺺ ﺭﻳﺎﺿﻲ ﺑﺨﻮﺍﻫﻲ ﺁﻥ ﺭﺍ ﺑﺮﺍﻳﺖ ﺛﺎﺑﺖ ﻛﻨﺪ‪.‬‬ ‫ﺩﺭﺱ ﺩﺭ ﺣﺎﻓﻈــﻪﻱ ﻛﻮﺗﺎﻩﻣﺪﺕ ﺁﻥﻫﺎ ﺟﺎﻱ ﮔﺮﻓﺘﻪ‬ ‫ﺁﻥ ﺍﺳﺖ‪ .‬ﻣﻦ ﻋﺎﺷﻖ‬ ‫ﻛﺴــﻲ ﻛﻪ ﺍﻳﻦ ﻣﻔﺎﻫﻴﻢ ﺭﺍ ﺑﻪ ﺧﻮﺑﻲ ﺑﺪﺍﻧﺪ ﻣﻲﺗﻮﺍﻧﺪ‬ ‫ﺍﺳــﺖ ﻭ ﺍﮔﺮ ﺗﻤﺮﻳــﻦ ﺑﻴﺶﺗﺮﻱ ﺭﻭﻱ ﺁﻥ ﻧﺪﺍﺷــﺘﻪ‬ ‫ﭼﻴﺰﻫﺎﻱ ﻣﺠﻬﻮﻟﻢ‬ ‫ﺑﺎ ﻳﻚ ﻧﻤﺎﻳﺶ ﺳــﺎﺩﻩ ﺁﻥ ﺭﺍ ﺍﺛﺒــﺎﺕ ﻛﻨﺪ‪ .‬ﻣﻦ ﻓﻜﺮ‬ ‫ﺑﺎﺷــﻨﺪ‪ ،‬ﺁﻥ ﺭﺍ ﻓﺮﺍﻣﻮﺵ ﻣﻲﻛﻨﻨﺪ‪ .‬ﺍﮔﺮ ﺑﺨﻮﺍﻫﻴﻢ ﻳﻚ‬ ‫ﻣﻲﻛﻨﻢ ﺍﻓــﺮﺍﺩﻱ ﻛﻪ ﺭﻳﺎﺿﻲ ﺭﺍ ﺩﻭﺳــﺖ ﻧﺪﺍﺭﻧﺪ ﺑﻪ‬ ‫ﻣﺴﺌﻠﻪﻱ ﺭﻳﺎﺿﻲ ﻭﺍﺭﺩ ﺣﺎﻓﻈﻪﻱ ﺑﻠﻨﺪ ﻣﺪﺕ ﻣﺎ ﺷﻮﺩ‪،‬‬ ‫ﻛﻤﻚ ﺍﻓﺮﺍﺩ ﺑﺎﺗﺠﺮﺑﻪ ﻧﻴﺎﺯ ﺩﺍﺭﻧﺪ ﻛﻪ ﺑﻪ ﺁﻥﭼﻪ ﻣﻲﮔﻮﻳﻨﺪ ﺍﻳﻤﺎﻥ ﺩﺍﺷــﺘﻪ ﺑﺎﻳــﺪ ﺑﻴﺶﺗــﺮ ﺭﻭﻱ ﺁﻥ ﻛﺎﺭ ﻛﻨﻴﻢ ﻭ ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﺍﺳــﺖ ﻛﻪ ﺑﻪ ﺍﻳﻦ‬ ‫ﺑﺎﺷﻨﺪ ﻭ ﺑﺘﻮﺍﻧﻨﺪ ﺩﺭﺳــﺘﻲ ﻣﻄﺎﻟﺒﻲ ﺭﺍ ﻛﻪ ﺩﺭﺑﺎﺭﻩﺍﺵ ﺣﺮﻑ ﻣﻲﺯﻧﻨﺪ ﺑﻪ ﺩﺭﺱ ﻋﻼﻗﻪﻣﻨﺪ ﻣﻲﺷــﻮﻳﻢ‪ ،‬ﻭﻟﻲ ﺍﮔﺮ ﺁﻥ ﺭﺍ ﻛﻨﺎﺭ ﺑﮕﺬﺍﺭﻳﻢ‪ ،‬ﻧﺴــﺒﺖ ﺑﻪ‬ ‫ﺩﻳﮕﺮﺍﻥ ﺛﺎﺑﺖ ﻛﻨﻨﺪ‪.‬‬ ‫ﺁﻥ ﺳﺮﺩ ﻣﻲﺷﻮﻳﻢ ﻭ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﻛﻢﻛﻢ ﺍﺯ ﺯﻧﺪﮔﻲﻣﺎﻥ ﻛﻨﺎﺭ ﮔﺬﺍﺷﺘﻪ‬ ‫ﻧﺮﺟﺲ ﺳـﻌﻴﺪﻱ ﺑﺎﺯ ﻫﻢ ﮔﻼﻳﻪﺍﻱ ﺩﺍﺭﺩ‪ :‬ﻣﻦ ﻳﻚ ﻣﺸــﻜﻞ ﺩﻳﮕﺮ ﻣﻲﺷﻮﺩ‪.‬‬ ‫ﻫﻢ ﺑﺎ ﺩﺭﺱ ﺭﻳﺎﺿــﻲ ﺩﺍﺭﻡ‪ .‬ﺑﻪ ﻧﻈﺮ ﻣﻦ ﻓﻘﻂ ‪ 45‬ﺩﻗﻴﻘﻪﻱ ﺍﻭﻝ ﻛﻼﺱ‬ ‫ﻧﺮﺟﺲ ﺳـﻌﻴﺪﻱ ﺍﺯ ﻋﻘﺎﻳﺪ ﻗﺒﻠﻲﺍﺵ ﺩﻓــﺎﻉ ﻣﻲﻛﻨﺪ ﻭ ﻣﻲﮔﻮﻳﺪ‪:‬‬ ‫ﺭﻳﺎﺿﻲ ﺧﻮﺏ ﺍﺳﺖ‪.‬‬ ‫ﺣﺮﻑ ﻣﺎ ﺍﻳﻦ ﻧﻴﺴــﺖ ﻛﻪ ﺭﻳﺎﺿﻲ ﺍﺯ ﺯﻧﺪﮔﻲﻣﺎﻥ ﻛﻨﺎﺭ ﮔﺬﺍﺷﺘﻪ ﺷﻮﺩ‪ ،‬ﻣﺎ‬ ‫‪16‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﻣﻲﮔﻮﻳﻴــﻢ ﺭﻳﺎﺿﻲ ﺭﺍ ﻓﻘﻂ ﺑﻪ ﺁﻥ ﺍﻧﺪﺍﺯﻩﺍﻱ ﺑﻪ ﻣﺎ ﺑﺪﻫﻨﺪ ﻛﻪ ﺩﺭ ﺯﻧﺪﮔﻲ‬ ‫ﻣﺎ ﻛﺎﺭﺑﺮﺩ ﺩﺍﺭﺩ‪.‬‬ ‫ﺟﻤﻴﻠﻪ ﺍﻣﺎﻣﻲ ﺍﺯ ﻛﺎﺭﺑﺮﺩﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺻﺤﺒﺖ ﻣﻲﻛﻨﺪ‪:‬‬ ‫ﻣﻦ ﻫﻤﻴﺸــﻪ ﺑﻪ ﺷــﺎﮔﺮﺩﺍﻧﻢ ﮔﻔﺘﻪﺍﻡ ﻛﻪ ﺭﻳﺎﺿﻲ ﺩﺭ ﺗﻤﺎﻡ ﻣﺴــﺎﺋﻞ‬ ‫ﺯﻧﺪﮔﻲ ﻧﻘــﺶ ﺩﺍﺭﺩ‪ .‬ﻛﺎﻓﻲ ﺍﺳــﺖ ﺩﻗﻴﻖﺗﺮ ﺑﻪ ﺍﻃﺮﺍﻓﻤــﺎﻥ ﻧﮕﺎﻩ ﻛﻨﻴﻢ‪.‬‬ ‫ﺗﻮﻧﻞﻫﺎ‪ ،‬ﭘﻞﻫﺎ ﻭ ﺣﺘﻲ ﺳــﺎﺧﺘﻤﺎﻥﻫﺎﻳﻲ ﻛﻪ ﻣﺎ ﺩﺭ ﺁﻥ ﺯﻧﺪﮔﻲ ﻣﻲﻛﻨﻴﻢ‬ ‫ﻳﺎ ﻫﺮ ﺭﻭﺯ ﺍﺯ ﺁﻥﻫﺎ ﺍﺳــﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻣﺤﺎﺳﺒﺎﺕ ﺭﻳﺎﺿﻲ‬ ‫ﺳــﺎﺧﺘﻪ ﺷــﺪﻩﺍﻧﺪ‪ ،‬ﺍﺻ ً‬ ‫ﻼ ﻧﻤﻲﺷــﻮﺩ ﺗﺼﻮﺭ ﻛﺮﺩ ﻛﻪ ﺭﻳﺎﺿﻲ ﺭﺍ ﺍﺯ ﺳﺎﻳﺮ‬ ‫ﺭﺷﺘﻪﻫﺎ ﺣﺬﻑ ﻛﻨﻨﺪ‪.‬‬ ‫ﻳﻚ ﺳﺆﺍﻝ ﺣﺴﺎﺱ ﺍﺯ ﺑﭽﻪﻫﺎ ﻣﻲﭘﺮﺳﻢ‪:‬‬ ‫ﭼﻪﻗـﺪﺭ ﻣﻌﻠﻢﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺩﺭ ﻋﻼﻗﻪﻣﻨﺪ‬ ‫ﻛـﺮﺩﻥ ﺷـﻤﺎ ﺑـﻪ ﺩﺭﺱ ﺭﻳﺎﺿـﻲ ﺗﺄﺛﻴـﺮ‬ ‫ﺩﺍﺷﺘﻪﺍﻧﺪ؟‬ ‫ﺷـﻘﺎﻳﻖ ﺻﺎﺩﻗﻲ ﺩﺭ ﭘﺎﺳــﺦ ﺑﻪ ﺍﻳﻦ ﺳــﺆﺍﻝ‬ ‫ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻌﻠﻤﻲ ﻛﻪ ﺑﺸﺎﺵ ﻭ ﺧﻨﺪﻩﺭﻭ ﺑﺎﺷﺪ ﻭ ﺑﺎ‬ ‫ﺑﭽﻪﻫﺎ ﺻﻤﻴﻤﺎﻧﻪ ﺣﺮﻑ ﺑﺰﻧﺪ ﻭ ﺑﺨﻨﺪﺩ ﻭ ﺭﻳﺎﺿﻲ ﺭﺍ‬ ‫ﺑﺎ ﺷﻴﻮﻩﻫﺎﻳﻲ ﻣﺜﻞ ﻧﻤﺎﻳﺶ ﺩﺍﺩﻥ ﻳﺎ ﺗﻌﺮﻳﻒ ﻛﺮﺩﻥ‬ ‫ﺩﺍﺳــﺘﺎﻥ ﺩﺭﺱ ﺑﺪﻫﺪ‪ ،‬ﻣﺴﻠﻤﺎً ﺩﺭ ﺗﺪﺭﻳﺲ ﺭﻳﺎﺿﻲ‬ ‫ﻣﻮﻓﻖﺗــﺮ ﺍﺳــﺖ ﻭ ﺑﭽﻪﻫــﺎ ﺩﺭﺱ ﺍﻭ ﺭﺍ ﺑﻬﺘــﺮ ﻳﺎﺩ‬ ‫ﻣﻲﮔﻴﺮﻧﺪ‪ .‬ﻣﻦ ﺩﺭ ﺳــﺎﻝ ﮔﺬﺷــﺘﻪ ﻣﻌﻠﻤﻲ ﺩﺍﺷﺘﻢ‬ ‫ﻛﻪ ﻣﺮﺍ ﺑﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﻋﻼﻗﻪﻣﻨﺪ ﻛﺮﺩ‪ .‬ﺍﻳﻦ ﻣﻌﻠﻢ‬ ‫ﺩﺭﺱ ﻫﻨﺪﺳﻪ ﺭﺍ ﺑﺎ ﻧﻤﺎﻳﺶ ﻳﺎ ﺗﻌﺮﻳﻒ ﻳﻚ ﺩﺍﺳﺘﺎﻥ‬ ‫ﺑﻪ ﻣــﺎ ﺩﺭﺱ ﻣﻲﺩﺍﺩ‪ .‬ﺍﻣﺎ ﺍﻳﻦﻫــﺎ ﻫﻢ ﻓﻘﻂ ﺩﺭ ﻳﺎﺩ‬ ‫ﮔﺮﻓﺘﻦ ﺩﺭﺱ ﺗﺄﺛﻴﺮ ﺩﺍﺷﺖ ﻧﻪ ﻋﻼﻗﻪﻣﻨﺪﻱ ﺑﻪ ﺁﻥ‪.‬‬ ‫ﻣﺮﻳـﻢ ﺟﺒﺎﺭﻱ ﺟــﻮﺍﺏ ﻣﻲﺩﻫــﺪ‪ :‬ﻣﻦ ﻓﻜﺮ ﻣﻲﻛﻨــﻢ ﺭﻳﺎﺿــﻲ‬ ‫ﻫﺮﺟﻮﺭ ﻫﻢ ﻛﻪ ﺗﺪﺭﻳﺲ ﺷﻮﺩ‪ ،‬ﺍﺻ ً‬ ‫ﻼ ﺟﺎﻟﺐ ﻧﻴﺴﺖ‪ .‬ﺭﻳﺎﺿﻲ ﻣﺠﻤﻮﻋﻪﺍﻱ‬ ‫ﺍﺳﺖ ﻛﻪ ﺑﻪ ﺷﻤﺎ ﻣﻲﺩﻫﻨﺪ ﻭ ﻓﻘﻂ ﺍﺯ ﺷﻤﺎ ﺟﻮﺍﺏ ﻣﻲﺧﻮﺍﻫﻨﺪ‪ .‬ﻓﻘﻂ ﻫﻢ‬ ‫ﺑﺎ ﺍﻋﺪﺍﺩ ﺳﺮ ﻭ ﻛﺎﺭ ﺩﺍﺭﻱ‪.‬‬ ‫ﻧﺮﺟﺲ ﺳـﻌﻴﺪﻱ ﻣﻲﮔﻮﻳــﺪ‪ :‬ﻳﻜﻲ ﺍﺯ ﻣﻌﻠﻢﻫﺎﻱ ﻣﺎ ﺩﺭﺳــﻲ ﺭﺍ ﺑﺎ‬ ‫ﺷــﻜﻞ ﻳﺎﺩ ﻣﻲﺩﺍﺩ‪ .‬ﻣﻦ ﻫﺮ ﭼﻴﺰﻱ ﺭﺍ ﻣﻤﻜﻦ ﺑﻮﺩ ﺳﺮ ﺍﻣﺘﺤﺎﻥ ﻓﺮﺍﻣﻮﺵ‬ ‫ﻛﻨﻢ ﺍﻣﺎ ﺁﻥ ﻣﺜﺎﻝ ﺭﺍ ﻛﻪ ﺷــﻜﻠﻲ ﺍﺯ ﻧﺎﻥ ﺑﺮﺑﺮﻱ ﺑﻮﺩ ﻫﻴﭻﻭﻗﺖ ﻓﺮﺍﻣﻮﺵ‬ ‫ﻧﻜﺮﺩﻡ‪.‬‬ ‫ﺯﻳﻨـﺐ ﺍﻃﻬﺮﻱ ﺟــﻮﺍﺏ ﻣﻲﺩﻫﺪ‪ :‬ﻣــﻦ ﺑــﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﺍﺻ ً‬ ‫ﻼ‬ ‫ﻋﻼﻗﻪﺍﻱ ﻧﺪﺍﺭﻡ‪ .‬ﺭﻳﺎﺿﻲ ﺩﺭﺱ ﻛﺴﻞﻛﻨﻨﺪﻩﺍﻱ ﺍﺳﺖ‪ .‬ﺩﻭ ﻋﺪﺩ ﺩﺍﺭﻳﻢ ﻛﻪ‬ ‫ﺑﻌﺪ ﺍﺯ ﺍﻧﺠﺎﻡ ﻣﺤﺎﺳﺒﺎﺗﻲ ﺭﻭﻱ ﺁﻥﻫﺎ ﻳﻚ ﻋﺪﺩ ﺑﻪ ﺩﺳﺖ ﻣﻲﺁﻭﺭﻳﻢ‪.‬‬ ‫ﺭﻋﻨﺎ ﺍﻣﺮﺍﻳﻲ ﻣﻌﺘﻘﺪ ﺍﺳــﺖ‪ :‬ﺣﺠﻢ ﻣﻄﺎﻟﺒﻲ ﻛﻪ ﺩﺭ ﺩﺭﺱ ﺭﻳﺎﺿﻲ‬ ‫ﺑﺎﻳــﺪ ﻳﺎﺩ ﺑﮕﻴﺮﻳﻢ ﺧﻴﻠﻲ ﺯﻳﺎﺩ ﺍﺳــﺖ‪ .‬ﺧﻴﻠﻲ ﺍﺯ ﺍﻳــﻦ ﻣﻄﺎﻟﺐ ﻫﻴﭻﻭﻗﺖ‬ ‫ﻛﺎﺭﺑــﺮﺩﻱ ﺩﺭ ﺯﻧﺪﮔﻲ ﻣﺎ ﻧﺪﺍﺭﻧــﺪ‪ .‬ﻣﻐﺰ‪ ،‬ﺭﺍ ﺁﻥﻗﺪﺭ ﭘﺮ ﻣﻲﻛﻨﻨﺪ ﻛﻪ ﺩﻳﮕﺮ‬ ‫ﻫﻴــﭻ ﻓﺎﻳﻞ ﺧﺎﻟﻲ ﺩﺭ ﺁﻥ ﺑﺎﻗﻲ ﻧﻤﻲﻣﺎﻧﺪ ﻭ ﭼﻴﺰ ﺟﺪﻳﺪﻱ ﺭﺍ ﻧﻤﻲﺗﻮﺍﻧﻴﻢ‬

‫ﻳﺎﺩ ﺑﮕﻴﺮﻳﻢ‪.‬‬ ‫ﺩﻟــﻢ ﻣﻲﺧﻮﺍﻫــﺪ ﺑﺪﺍﻧﻢ ﺍﻳﻦ ﺩﻭ ﮔــﺮﻭﻩ ﻋﻼﻗﻪﻣﻨــﺪ ﻭ ﺑﻲﻋﻼﻗﻪ ﺑﻪ‬ ‫ﺭﻳﺎﺿﻲ ﻓﻜﺮ ﻛﺮﺩﻩﺍﻧﺪ ﺩﺭ ﺁﻳﻨﺪﻩ ﭼﻪ ﺷﻐﻠﻲ ﺭﺍ ﺑﺮ ﻋﻬﺪﻩ ﺑﮕﻴﺮﻧﺪ‪ .‬ﺍﺯ ﺑﭽﻪﻫﺎ‬ ‫ﻣﻲﭘﺮﺳــﻢ‪ :‬ﺩﻭﺳــﺖ ﺩﺍﺭﻳﺪ ﺩﺭ ﺁﻳﻨﺪﻩ ﭼﻪﻛﺎﺭﻱ ﺷــﻮﻳﺪ ﻭ ﺁﻳﺎ ﻣﻲﺩﺍﻧﻴﺪ‬ ‫ﺷــﻐﻠﻲ ﺭﺍ ﻛﻪ ﻣﻲﺧﻮﺍﻫﻴﺪ ﺩﺭ ﺁﻳﻨﺪﻩ ﺑﺮ ﻋﻬﺪﻩ ﺑﮕﻴﺮﻳﺪ ﭼﻪﻗﺪﺭ ﺑﺎ ﺭﻳﺎﺿﻲ‬ ‫ﺳﺮﻭ ﻛﺎﺭ ﺩﺍﺭﺩ؟‬ ‫ﺷـﻘﺎﻳﻖ ﺻﺎﺩﻗﻲ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﺩﻭﺳﺖ ﺩﺍﺭﻡ ﺩﺍﺭﻭﺳﺎﺯ ﺷﻮﻡ‪ .‬ﺍﻟﺒﺘﻪ‬ ‫ﺩﺍﺭﻭﺳﺎﺯﻱ ﺷﻐﻠﻲ ﺍﺳﺖ ﻛﻪ ﺑﻪ ﺭﻳﺎﺿﻴﺎﺕ ﺍﺣﺘﻴﺎﺝ ﺩﺍﺭﺩ‪ ،‬ﺍﻣﺎ ﺑﻪ ﺭﻳﺎﺿﻴﺎﺕ‬ ‫ﭘﻴﺸﺮﻓﺘﻪﺍﻱ ﻧﻴﺎﺯ ﻧﺪﺍﺭﺩ‪.‬‬ ‫ﻣﺮﻳﻢ ﺟﺒﺎﺭﻱ ﺟﻮﺍﺏ ﻣﻲﺩﻫﺪ‪ :‬ﻣﻦ ﻣﻲﺧﻮﺍﻫﻢ‬ ‫ﺩﻧﺪﺍﻥﭘﺰﺷــﻚ ﺷــﻮﻡ‪ .‬ﺭﻳﺎﺿﻴﺎﺕ ﺗﺎ ﺣﺪﻱ ﺩﺭ ﺍﻳﻦ‬ ‫ﺷــﻐﻞ ﻣﻮﺭﺩ ﻧﻴﺎﺯ ﺍﺳــﺖ ﺍﻣﺎ ﺟﺰء ﺩﺭﺱﻫﺎﻱ ﺍﺻﻠﻲ‬ ‫ﺍﻳﻦ ﺭﺷــﺘﻪ ﻧﻴﺴﺖ‪ .‬ﺩﻧﺪﺍﻥﭘﺰﺷــﻚ ﺑﻴﺶﺗﺮ ﺑﺎﻳﺪ ﺑﺎ‬ ‫ﻋﻠﻮﻡ ﺗﺠﺮﺑﻲ ﺁﺷﻨﺎ ﺑﺎﺷﺪ‪.‬‬ ‫ﺯﻳﻨﺐ ﺍﻃﻬﺮﻱ ﭘﺎﺳــﺦ ﻣﻲﺩﻫﺪ‪ :‬ﻣﻦ ﺩﻭﺳــﺖ‬ ‫ﺩﺍﺭﻡ ﻣﻌﻠــﻢ ﺟﻐﺮﺍﻓﻲ ﺷــﻮﻡ‪ .‬ﺩﻭﺳــﺖ ﺩﺍﺭﻡ ﺑﺪﺍﻧﻢ‬ ‫ﭘﺪﻳﺪﻩﻫﺎﻳﻲ ﻣﺜﻞ ﺟﻠﮕﻪ ﻭ ‪ ...‬ﭼﮕﻮﻧﻪ ﺍﻳﺠﺎﺩ ﻣﻲﺷﻮﻧﺪ‪.‬‬ ‫ﺍﻳﻦ ﺭﺷﺘﻪ ﺧﻴﻠﻲ ﺑﺎ ﺭﻳﺎﺿﻲ ﺳﺮ ﻭ ﻛﺎﺭ ﻧﺪﺍﺭﺩ‪.‬‬ ‫ﻧﺮﺟﺲ ﺳﻌﻴﺪﻱ ﺑﺮﻧﺎﻣﻪﻱ ﺩﻳﮕﺮﻱ ﺩﺍﺭﺩ‪ :‬ﻣﻦ‬ ‫ﺩﻟــﻢ ﻣﻲﺧﻮﺍﻫﺪ ﻃﻠﺒﮕﻲ ﻭ ﻭﻛﺎﻟــﺖ ﺭﺍ ﺍﺩﺍﻣﻪ ﺩﻫﻢ‪.‬‬ ‫ﻫﻴﭽﻜﺪﺍﻡ ﺑﻪ ﺭﻳﺎﺿﻴﺎﺕ ﺍﺭﺗﺒﺎﻃﻲ ﻧﺪﺍﺭﻧﺪ‪.‬‬ ‫ﺭﻋﻨﺎ ﺍﻣﺮﺍﻳﻲ ﻫﻢ ﺍﺯ ﻋﻼﻗﻪﻣﻨﺪﻱﺍﺵ ﻣﻲﮔﻮﻳﺪ‪:‬‬ ‫ﻣﻦ ﺑﻪ ﺩﻭ ﺭﺷــﺘﻪﻱ ﺧﻴﻠﻲ ﻣﺘﻔــﺎﻭﺕ ﻋﻼﻗﻪﻣﻨﺪﻡ‪.‬‬ ‫ﻋﻜﺎﺳﻲ ﻭ ﺩﺍﺭﻭﺳــﺎﺯﻱ؛ ﻛﻪ ﺧﻴﻠﻲ ﻫﻢ ﺑﺎ ﺭﻳﺎﺿﻴﺎﺕ‬ ‫ﻛﺎﺭﻱ ﻧﺪﺍﺭﻧﺪ‪.‬‬ ‫ﺁﺭﺯﻭ ﻋﺎﺭﻓﻲ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﻫﻢ ﺩﻭﺳــﺖ ﺩﺍﺭﻡ ﺷــﺎﻋﺮ ﻭ ﻧﻮﻳﺴﻨﺪﻩ‬ ‫ﺷﻮﻡ‪ .‬ﺍﻟﺒﺘﻪ ﺍﮔﺮ ﻧﻮﻳﺴﻨﺪﻩ ﻧﺸﻮﻡ‪ ،‬ﻧﻘﺎﺵ ﺧﻮﺍﻫﻢ ﺷﺪ‪.‬‬ ‫ﻓﺎﻃﻤﻪ ﺟﻴﺮﺍﻧﻲ ﺩﺭ ﭘﺎﺳﺦ ﺑﻪ ﺍﻳﻦ ﺳﺆﺍﻝ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﻫﻢ ﺩﻭﺳﺖ‬ ‫ﺩﺍﺭﻡ ﺩﻛﺘﺮﺍﻱ ﺭﻳﺎﺿﻲ ﺑﮕﻴﺮﻡ ﻭ ﺍﺳﺘﺎﺩ ﺩﺍﻧﺸﮕﺎﻩ ﺷﻮﻡ‪.‬‬ ‫ﺷـﺒﻨﻢ ﺍﺑﻮﺍﻟﻔﻀﻠﻲ ﺭﻳﺎﺿﻲ ﺭﺍ ﺩﻭﺳﺖ ﺩﺍﺭﺩ‪ ،‬ﺍﻣﺎ ﺷﻐﻠﺶ ﺭﺑﻄﻲ ﺑﻪ‬ ‫ﺍﻳﻦ ﺭﺷﺘﻪ ﻧﺪﺍﺭﺩ‪ .‬ﺍﻭ ﺩﺭﺍﻳﻦﺑﺎﺭﻩ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﺩﻟﻢ ﻣﻲﺧﻮﺍﻫﺪ ﻣﺎﻣﺎ ﺷﻮﻡ‪.‬‬ ‫ﺍﻟﺒﺘﻪ ﺍﻳﻦ ﺷﻐﻞ ﺭﺑﻄﻲ ﺑﻪ ﺭﻳﺎﺿﻲ ﻧﺪﺍﺭﺩ‪.‬‬ ‫ﺍﻗﻠﻴﻤﺎ ﺍﻟﺴـﺎﺩﺍﺕ ﺳﻴﺪﺯﺍﺩﻩ ﻣﻌﺘﻘﺪ ﺍﺳﺖ‪ :‬ﺑﻪ ﻧﻈﺮ ﻣﻦ ﻛﻞ ﺟﻬﺎﻥ‬ ‫ﻫﺴــﺘﻲ ﺑﺮ ﭘﺎﻳﻪﻱ ﺭﻳﺎﺿﻲ ﺑﺮﭘﺎﺳﺖ ﻭ ﻫﻤﻪﻱ ﺭﺷﺘﻪﻫﺎ ﺑﺎ ﺭﻳﺎﺿﻲ ﺍﺭﺗﺒﺎﻁ‬ ‫ﺩﺍﺭﻧﺪ‪ .‬ﺑﺮﺍﻱ ﻣﺜــﺎﻝ‪ ،‬ﻣﻌﻠﻢ ﺟﻐﺮﺍﻓﻲ ﺑﺪﻭﻥ ﺩﺍﻧﺴــﺘﻦ ﺭﻳﺎﺿﻲ ﻧﻤﻲﺗﻮﺍﻧﺪ‬ ‫ﻣﻮﻓﻖ ﺑﺎﺷﺪ‪.‬‬ ‫ﻣﻦ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﺑﭽﻪﻫﺎﻳﻲ ﻛﻪ ﻣﻲﮔﻮﻳﻨﺪ ﺭﻳﺎﺿﻲ ﺭﺍ ﺑﺎﻳﺪ ﺑﺎ ﻧﻤﺎﻳﺶ‬ ‫ﻭ ﺩﺍﺳﺘﺎﻥ ﺗﺪﺭﻳﺲ ﻛﺮﺩ‪ ،‬ﺩﺭ ﺍﺷﺘﺒﺎﻩﺍﻧﺪ‪ .‬ﻓﺮﺽ ﻛﻨﻴﺪ ﻫﺮ ﺳﺎﻝ ﺭﻳﺎﺿﻲ ﺭﺍ‬ ‫ﺑﺎ ﻧﻤﺎﻳﺶ ﺑﻪ ﺷــﻤﺎ ﻳﺎﺩ ﺩﺍﺩﻧﺪ ﻭ ﺑﻪ ﺧﺎﻃﺮ ﺍﻳﻦ ﺭﻭﺵ ﺗﺪﺭﻳﺲ ﺑﻪ ﺭﻳﺎﺿﻲ‬ ‫ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬ ‫ﺷﻤﺎﺭﻩ ﻯ‬ ‫ﺷﺎﻧﺰﺩﻫﻢ‪،‬‬ ‫ﺷﺎﻧﺰﺩﻫﻢ‪،‬‬ ‫ﺩﻭﺭﺓ‬ ‫ﺩﻭﺭﻩﻯ‬

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‫ﻋﻼﻗﻪﻣﻨﺪ ﺷﺪﻳﺪ ﻭ ﺭﺷﺘﻪﻱ ﺭﻳﺎﺿﻲ ﺭﺍ ﺍﺩﺍﻣﻪ ﺩﺍﺩﻳﺪ‪ .‬ﺁﻳﺎ ﺑﺎﻳﺪ ﺩﺭ ﺩﺍﻧﺸﮕﺎﻩ ﻛﻼﺳﻲ ﺑﺎﺷﻢ ﻗﺒﻞ ﺍﺯ ﺗﺪﺭﻳﺲ ﻫﺮ ﺩﺭﺳﻲ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﻭ ﺭﻭﺷﻲ ﺭﺍ ﺑﺮﺍﻱ‬ ‫ﻫﻢ ﺑﺮﺍﻱ ﺷــﻤﺎ ﺑﺎ ﻧﻤﺎﻳﺶ ﺩﺭﺱ ﺩﺍﺩﻩ ﺷــﻮﺩ؟ ﻣــﻦ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﺫﻫﻦ ﺗﺪﺭﻳﺲ ﺍﻧﺘﺨﺎﺏ ﻣﻲﻛﻨﻢ ﻛﻪ ﺑﭽﻪﻫﺎ ﺭﺍ ﻧﺴــﺒﺖ ﺑﻪ ﺩﺭﺱ ﻛﻨﺠﻜﺎﻭ ﻛﻨﺪ‪.‬‬ ‫ﺍﻧﺴﺎﻥ ﺧﻮﺍﻩﻧﺎﺧﻮﺍﻩ ﻛﻨﺠﻜﺎﻭ ﺍﺳــﺖ ﻭ ﺩﻭﺳﺖ ﺩﺍﺭﺩ ﭼﻴﺰﻫﺎﻱ ﺑﻴﺶﺗﺮﻱ ﺑــﺎ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻧﻢ ﺣﺮﻑ ﻣﻲﺯﻧﻢ ﻭ ﺣﺮﻑﻫﺎﻱ ﺁﻥﻫﺎ ﺭﺍ ﮔﻮﺵ ﻣﻲﻛﻨﻢ ﺗﺎ‬ ‫ﺑﺪﺍﻧﺪ‪ .‬ﻣﻦ ﺩﻭﺳــﺖ ﺩﺍﺭﻡ ﺟﻮﺍﺏ ﻣﺴــﺌﻠﻪﺍﻱ ﺭﺍ ﻛﻪ ﻧﻤﻲﺩﺍﻧﻢ ﭘﻴﺪﺍ ﻛﻨﻢ ﺩﻟﻴﻞ ﺑﻲﻋﻼﻗﮕﻲﺷــﺎﻥ ﺭﺍ ﺑﻪ ﺩﺭﺱ ﺑﻔﻬﻤﻢ‪ .‬ﺑﺎ ﺑﭽﻪﻫﺎ ﺩﻭﺳــﺘﺎﻧﻪ ﺣﺮﻑ‬ ‫ﻣﻦ ﺑﻪ ﺍﻳﻦ ﺩﻟﻴﻞ ﺑﻪ ﺭﻳﺎﺿﻲ ﻋﻼﻗﻪﻣﻨﺪ ﺷﺪﻡ‪ .‬ﺭﻳﺎﺿﻲ ﺍﻳﻦ ﻓﺮﺻﺖ ﺭﺍ ﺑﻪ ﻣﻲﺯﻧﻢ ﻭ ﻛﺎﺭﻱ ﻣﻲﻛﻨﻢ ﺗﺎ ﺑﭽﻪﻫﺎ ﻋﺎﺷﻖ ﻣﻌﻠﻢ ﺭﻳﺎﺿﻲﺷﺎﻥ ﺷﻮﻧﺪ‪ .‬ﺍﮔﺮ‬ ‫ﺁﺩﻡ ﻣﻲﺩﻫﺪ ﺗﺎ ﺩﺭﺑﺎﺭﻩﻱ ﺧﻮﺩﺵ ﻭ ﺟﻬﺎﻧﻲ ﻛﻪ ﺩﺭ ﺁﻥ ﺯﻧﺪﮔﻲ ﻣﻲﻛﻨﺪ‪ ،‬ﺑﭽﻪﻫــﺎ ﺑﺎ ﺁﻣﺪﻥ ﻣﻌﻠﻢ ﺭﻳﺎﺿﻲ ﺑﻪ ﺳــﺮ ﻛﻼﺱ ﺑﮕﻮﻳﻨﺪ‪ :‬ﻭﺍﻱ! ﺍﻳﻦ ﻣﻌﻠﻢ‬ ‫ﺩﻭﺑﺎﺭﻩ ﺁﻣﺪ! ﻳﺎ ﺍﺣﺴــﺎﺱ ﻛﻨﻨﺪ ﺍﻳﻦ ﻣﻌﻠﻢ ﻓﻘﻂ ﻗﺼﺪ ﺍﺫﻳﺖ ﻛﺮﺩﻧﺸﺎﻥ ﺭﺍ‬ ‫ﺍﻃﻼﻋﺎﺗﻲ ﻛﺴﺐ ﻛﻨﺪ‪.‬‬ ‫ﻣﻌﺼﻮﻣﻪ ﺳـﻴﺪ ﻣﺤﻤﺪﺯﺍﺩﻩ ﻫﻢ ﺍﺯ ﺁﺭﺯﻭﻫﺎﻳﺶ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﺑﻪ ﺩﺍﺭﺩ‪ ،‬ﻣﻄﻤﺌﻨﺎً ﺩﺭﺱ ﺭﺍ ﻳﺎﺩ ﻧﻤﻲﮔﻴﺮﻧﺪ‪.‬‬ ‫ﺍﻗﻠﻴﻤﺎ ﺍﻟﺴﺎﺩﺍﺕ ﺳﻴﺪﺯﺍﺩﻩ‪ :‬ﻣﻦ ﺍﮔﺮ ﺑﺨﻮﺍﻫﻢ ﺑﭽﻪﻫﺎ ﺭﺍ ﺑﻪ ﺭﻳﺎﺿﻲ‬ ‫ﺩﻭ ﺷــﻐﻞ ﻣﻬﻨﺪﺳﻲ ﺳﺎﺧﺘﻤﺎﻥ ﻭ ﻧﻮﻳﺴــﻨﺪﮔﻲ ﻋﻼﻗﻪﻣﻨﺪﻡ‪ .‬ﺭﻳﺎﺿﻴﺎﺕ‬ ‫ﻋﻼﻗﻪﻣﻨﺪ ﻛﻨﻢ ﺑﺎﻳﺪ ﺑﺎ ﺷﻮﺭ ﻭ ﺍﺷﺘﻴﺎﻕ ﭘﺎﻱ ﺗﺨﺘﻪ‬ ‫ﻛﺎﺭﺑﺮﺩ ﺯﻳﺎﺩﻱ ﺩﺭ ﻣﻬﻨﺪﺳﻲ ﺳﺎﺧﺘﻤﺎﻥ ﺩﺍﺭﺩ‪.‬‬ ‫ﺭﻳﺎﺿﻲ ﺍﻳﻦ ﻓﺮﺻﺖ‬ ‫ﺳــﻴﺎﻩ ﺑــﺮﻭﻡ ﻭ ﺩﺭﺱ ﺭﺍ ﺑﺎ ﺷــﻮﺧﻲﻫﺎﻱ ﺭﻳﺎﺿﻲ‬ ‫ﺍﻗﻠﻴﻤـﺎ ﺍﻟﺴـﺎﺩﺍﺕ ﺳـﻴﺪﺯﺍﺩﻩ ﻫــﻢ ﺷــﻐﻞ‬ ‫ﻫﻤﺮﺍﻩ ﻛﻨﻢ‪.‬‬ ‫ﺁﻳﻨﺪﻩﺍﺵ ﺭﺍ ﺍﻧﺘﺨﺎﺏ ﻛﺮﺩﻩ ﺍﺳﺖ‪ :‬ﻣﻦ ﺑﻪ ﺍﺧﺘﺮﺷﻨﺎﺳﻲ‬ ‫ﺭﺍ ﺑﻪ ﺁﺩﻡ ﻣﻲﺩﻫﺪ ﺗﺎ‬ ‫ﺯﻫﺮﺍ ﻓﺘﺤﻲ ﭘﺎﺳـﺦ ﻣﻲﺩﻫﺪ‪ :‬ﻣﻦ ﺩﻭﺳﺖ‬ ‫ﻋﻼﻗﻪﻣﻨﺪﻡ ﻭ ﺍﻳﻦ ﺭﺷﺘﻪ ﺍﺭﺗﺒﺎﻁ ﻧﺰﺩﻳﻜﻲ ﺑﺎ ﺭﻳﺎﺿﻴﺎﺕ‬ ‫ﺩﺭﺑﺎﺭﻩﻱ ﺧﻮﺩﺵ ﻭ‬ ‫ﺩﺍﺭﻡ ﺑﺎ ﺷﺎﮔﺮﺩﺍﻧﻢ ﺣﺮﻑ ﺑﺰﻧﻢ ﻭ ﺁﻥﻫﺎ ﺭﺍ ﻗﺎﻧﻊ ﻛﻨﻢ‬ ‫ﺩﺍﺭﺩ‪.‬‬ ‫ﺯﻧﺪﮔﻲ‬ ‫ﺁﻥ‬ ‫ﺩﺭ‬ ‫ﻛﻪ‬ ‫ﺟﻬﺎﻧﻲ‬ ‫ﻛﻪ ﺭﻳﺎﺿﻲ ﺩﺭﺱ ﺟﺎﻟﺐ ﻭ ﺷﻨﻴﺪﻧﻲ ﺍﺳﺖ‪.‬‬ ‫ﻓﺎﻃﻤﻪ ﻧﺼﻴﺮﻱ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﺩﻭﺳﺖ ﺩﺍﺭﻡ ﺩﺭ‬ ‫ﺳـﻴﺪﻩ ﺯﻫـﺮﺍ ﺩﺭﻳﺎﺑﺎﺭﻱ ﺍﻳــﺪﻩﻱ ﺩﻳﮕﺮﻱ‬ ‫ﺁﻳﻨــﺪﻩ ﺩﻛﺘﺮ ﻣﺘﺨﺼﺺ ﺩﻳﺎﺑﺖ ﺷــﻮﻡ‪ .‬ﭼﻮﻥ ﺧﻮﺩﻡ‬ ‫ﻣﻲﻛﻨﺪ‪ ،‬ﺍﻃﻼﻋﺎﺗﻲ‬ ‫ﺩﺍﺭﺩ‪ :‬ﻣﻦ ﻫﻢ ﺳــﻌﻲ ﻣﻲﻛﻨﻢ ﺑﭽﻪﻫﺎ ﺭﺍ ﺑﻪ ﺩﺭﺱ‬ ‫ﺩﻳﺎﺑﺖ ﺩﺍﺭﻡ ﺑﻴﻤﺎﺭﻡ ﺭﺍ ﻣﻲﺗﻮﺍﻧﻢ ﺩﺭﻙ ﻛﻨﻢ ﻭ ﺑﻔﻬﻤﻢ‬ ‫ﻛﺴﺐ ﻛﻨﺪ‬ ‫ﺟﺬﺏ ﻛﻨﻢ‪ .‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ ﺑﺎ ﺗﺸــﻮﻳﻖ ﻳﺎ ﻧﻤﺮﻩ ﺩﺍﺩﻥ‬ ‫ﺑﻴﻤــﺎﺭ ﻣﺒﺘﻼ ﺑــﻪ ﺩﻳﺎﺑﺖ ﺍﺯ ﺑﻴﻤــﺎﺭﻱﺍﺵ ﭼﻪ ﺭﻧﺠﻲ‬ ‫ﺑﺮﺍﻱ ﭘﺎﺳــﺦ ﺑﻪ ﻳﻚ ﺳــﺆﺍﻝ ﺧﺎﺹ‪ .‬ﺑﺎ ﺍﻳﻦ ﺭﻭﺵ‬ ‫ﻣﻲﺑﺮﺩ‪.‬‬ ‫ﻣﻤﻜﻦ ﺍﺳﺖ ﺩﺭ ﺍﺑﺘﺪﺍ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺑﺮﺍﻱ ﺗﺸﻮﻳﻖ‬ ‫ﻣﻦ ﻓﻜﺮ ﻣﻲﻛﻨﻢ ﻳﺎﺩ ﺩﺍﺩﻥ ﻭ ﻳﺎﺩ ﮔﺮﻓﺘﻦ ﺭﻳﺎﺿﻲ‬ ‫ﻫــﻢ ﺑﻪ ﺍﻳــﻦ ﺩﺭﻙ ﻣﺘﻘﺎﺑﻞ ﻧﻴﺎﺯ ﺩﺍﺭﺩ‪ ،‬ﻳﻌﻨﻲ ﺗــﻮ ﺍﺯ ﻳﺎﺩ ﮔﺮﻓﺘﻦ ﺭﻳﺎﺿﻲ ﺷــﺪﻥ ﻳﺎ ﻧﻤﺮﻩ ﮔﺮﻓﺘﻦ ﺩﺭﺱ ﺑﺨﻮﺍﻧﻨــﺪ‪ ،‬ﺍﻣﺎ ﻛﻢﻛﻢ ﺑﻪ ﺩﺭﺱ ﻋﻼﻗﻪﻣﻨﺪ‬ ‫ﻣﻲﺷﻮﻧﺪ‪.‬‬ ‫ﺑﻬﺮﻩﺍﻱ ﺑﺮﺩﻩ ﺑﺎﺷﻲ ﺑﺘﻮﺍﻧﻲ ﺁﻥ ﺭﺍ ﺑﻪ ﺩﻳﮕﺮﺍﻥ ﻧﻴﺰ ﻳﺎﺩ ﺑﺪﻫﻲ‪.‬‬ ‫ﺷﺒﻨﻢ ﺍﺑﻮﺍﻟﻔﻀﻠﻲ ﺗﺤﻠﻴﻞ ﺟﺎﻟﺒﻲ ﺩﺍﺭﺩ‪ :‬ﺧﻴﻠﻲ ﺍﺯ ﻣﻌﻠﻢﻫﺎ ﺧﻮﺩﺷﺎﻥ‬ ‫ﻣﻦ ﻋﺎﺷﻖ ﻧﻮﻳﺴﻨﺪﮔﻲ ﻫﻢ ﻫﺴﺘﻢ‪.‬‬ ‫ﺟﻤﻴﻠـﻪ ﺍﻣﺎﻣﻲ )ﻣﻌﻠﻢ ﺑﭽﻪﻫــﺎ( ﺭﺍﻩﺣﻞﻫﺎﻳﻲ ﺍﺭﺍﺋــﻪ ﻣﻲﺩﻫﺪ ﻛﻪ ﻧﺴﺒﺖ ﺑﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﻋﻼﻗﻪﺍﻱ ﻧﺪﺍﺭﻧﺪ‪ .‬ﻣﻦ ﺍﮔﺮ ﺑﺨﻮﺍﻫﻢ ﺗﺪﺭﻳﺲ ﻛﻨﻢ‬ ‫ﺧﻴﻠﻲ ﺍﺯ ﻣﺸــﻜﻼﺕ ﺑﭽﻪﻫﺎ ﺩﺭ ﺩﺭﺱ ﺭﻳﺎﺿــﻲ ﺭﺍ ﺣﻞ ﻣﻲﻛﻨﺪ‪ :‬ﻣﻦ ﺑﻪ ﺍﻭﻝ ﺩﺭ ﺧﻮﺩﻡ ﻋﻼﻗﻪ ﺍﻳﺠﺎﺩ ﻣﻲﻛﻨﻢ‪ .‬ﺑﻌﺪ ﺍﺯ ﺁﻥ ﺩﺍﻧﺶﺁﻣﻮﺯ ﻫﻢ ﺩﺭﺱ ﺭﺍ‬ ‫ﺑﭽﻪﻫﺎ ﺗﻮﺻﻴﻪ ﻣﻲﻛﻨﻢ ﻛﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﺭﺍ ﺩﺭ ﻫﻤﺎﻥ ﺭﻭﺯ ﺑﺨﻮﺍﻧﻨﺪ‪ .‬ﺍﮔﺮ ﻳﺎﺩ ﻣﻲﮔﻴﺮﺩ ﻭ ﺑﻪ ﺁﻥ ﻋﻼﻗﻪﻣﻨﺪ ﻣﻲﺷﻮﺩ‪.‬‬ ‫ﻓﺎﻃﻤﻪ ﺧﺪﺍﻳﻲ ﻣﻲﮔﻮﻳﺪ‪ :‬ﻣﻦ ﺍﮔﺮ ﻣﻌﻠﻢ ﺭﻳﺎﺿﻲ ﺷــﻮﻡ ﻫﻴﭻﻭﻗﺖ‬ ‫ﺑﭽﻪﻫﺎ ﻋﺼﺮ ﻫﻤــﺎﻥ ﺭﻭﺯﻱ ﻛﻪ ﻣﻌﻠﻢ ﻣﻄﻠﺐ ﺟﺪﻳــﺪﻱ ﺑﻪ ﺁﻥﻫﺎ ﮔﻔﺘﻪ‬ ‫ﺍﺳــﺖ‪ ،‬ﺁﻥ ﺭﺍ ﻣﺮﻭﺭ ﻛﻨﻨﺪ‪ ،‬ﺩﺭﺱ ﺭﺍ ﺑﻪ ﺳﺎﺩﮔﻲ ﻳﺎﺩ ﻣﻲﮔﻴﺮﻧﺪ ﻭ ﻓﺮﺍﻭﺵ ﻛﺴــﻲ ﺭﺍ ﺗﻨﺒﻴﻪ ﻧﻤﻲﻛﻨﻢ‪ ،‬ﭼﻮﻥ ﺗﻨﺒﻴﻪ ﺩﺍﻧﺶﺁﻣــﻮﺯ ﺭﺍ ﺍﺯ ﺩﺭﺱ ﻣﺘﻨﻔﺮ‬ ‫ﻣﻲﻛﻨﺪ‪ .‬ﻋﻼﻭﻩ ﺑﺮ ﺍﻳﻦ‪ ،‬ﺑﺎ ﺷﺎﮔﺮﺩﺍﻧﻢ ﺻﺤﺒﺖ ﻣﻲﻛﻨﻢ ﻭ ﺑﻪ ﺁﻥﻫﺎ ﻛﻤﻚ‬ ‫ﻧﻤﻲﻛﻨﻨﺪ‪ .‬ﺍﻳﻦ ﺑﺎﻋﺚ ﻣﻲﺷﻮﺩ ﺑﻪ ﺩﺭﺱ ﻋﻼﻗﻪﻣﻨﺪ ﺷﻮﻧﺪ‪.‬‬ ‫ﺑﻪ ﻧﻈــﺮ ﻣﻦ‪ ،‬ﻫﻤﻪﻱ ﺍﻓﺮﺍﺩ ﻇﺮﻓﻴﺖ ﻳﺎﺩﮔﻴــﺮﻱ ﻫﻤﻪﭼﻴﺰ ﺭﺍ ﺩﺍﺭﻧﺪ‪ .‬ﻣﻲﻛﻨﻢ ﺑﺮﺍﻱ ﺩﺭﺱ ﺧﻮﺍﻧﺪﻥﺷــﺎﻥ ﺑﺮﻧﺎﻣﻪ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪ .‬ﺑﺮﺍﻱ ﺍﻳﻦ ﻛﺎﺭ‬ ‫ﺍﻳﻦﻛﻪ ﺑﻌﻀﻲﻫﺎ ﻣﻲﮔﻮﻳﻨﺪ ﻣﺎ ﺍﺳــﺘﻌﺪﺍﺩ ﻳﺎﺩﮔﻴــﺮﻱ ﺭﻳﺎﺿﻲ ﺭﺍ ﻧﺪﺍﺭﻳﻢ‪ ،‬ﺑﻪ ﻫﺮﻳﻚ ﺍﺯ ﺷــﺎﮔﺮﺩﺍﻧﻢ ﺯﻣﺎﻥ ﻣﺸﺨﺼﻲ ﻣﻲﺩﻫﻢ ﺗﺎ ﺑﺘﻮﺍﻧﻢ ﺑﺎ ﻣﺸﻜﻼﺕ‬ ‫ﺣﺮﻑ ﺩﺭﺳﺘﻲ ﻧﻴﺴﺖ‪ .‬ﺑﭽﻪﻫﺎ ﻛﻨﺠﻜﺎﻭ ﺑﺎﺷﻴﺪ ﻭ ﺳﻌﻲ ﻛﻨﻴﺪ ﺍﺳﺘﻌﺪﺍﺩﻫﺎﻱ ﻭ ﺩﻻﻳﻞ ﺑﻲﻋﻼﻗﮕﻲﺷﺎﻥ ﺑﻪ ﺩﺭﺱ ﺁﺷﻨﺎ ﺷﻮﻡ‪.‬‬ ‫ﺣــﺮﻑ ﭘﺎﻳﺎﻧﻲ ﺟﻤﻴﻠﻪ ﺍﻣﺎﻣﻲ‪ ،‬ﻧﺼﻴﺤﺘﻲ ﺑﻪ ﺑﭽﻪﻫﺎﻳﻲ ﺍﺳــﺖ ﻛﻪ ﺍﺯ‬ ‫ﻣﺨﺘﻠﻒ ﺍﺯ ﺟﻤﻠﻪ ﺍﺳﺘﻌﺪﺍﺩ ﺭﻳﺎﺿﻲ ﺭﺍ ﺩﺭ ﻭﺟﻮﺩﺗﺎﻥ ﺷﻜﻮﻓﺎ ﻛﻨﻴﺪ‪.‬‬ ‫ﺍﺯ ﺑﭽﻪﻫﺎﻳﻲ ﻛﻪ ﺭﻳﺎﺿﻲ ﺭﺍ ﺩﻭﺳـﺖ ﺩﺍﺭﻧﺪ ﻣﻲﺧﻮﺍﻫﻢ ﺗﺼﻮﺭ ﻭﺿﻊ ﻛﻼﺱ ﻭ ﻧﺤﻮﻩﻱ ﺗﺪﺭﻳــﺲ ﺭﻳﺎﺿﻲ ﮔﻼﻳﻪ ﻣﻲﻛﻨﻨﺪ‪ .‬ﺍﻭ ﻣﻲﮔﻮﻳﺪ‪:‬‬ ‫ﻛﻨﻨﺪ ﻣﻌﻠﻢ ﺷﺪﻩﺍﻧﺪ ﻭ ﺍﺯ ﺁﻥﻫﺎ ﺧﻮﺍﺳﺘﻪﺍﻧﺪ ﻛﻼﺳﻲ ﺭﺍ ﺍﺩﺍﺭﻩ ﻛﻨﻨﺪ ﻣﻴﺰﺍﻥ ﻓﻌﺎﻟﻴﺖ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺩﺭ ﻛﻼﺱ‪ ،‬ﻳﻜﻲ ﺍﺯ ﻋﻮﺍﻣﻠﻲ ﺍﺳﺖ ﻛﻪ ﺁﻥﻫﺎ‬ ‫ﻛﻪ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻧـﺶ ﻫﻴﭻ ﻋﻼﻗﻪﺍﻱ ﺑﻪ ﺩﺭﺱ ﺭﻳﺎﺿﻲ ﻧﺪﺍﺭﻧﺪ‪ .‬ﺍﺯ ﺭﺍ ﺑﻪ ﺩﺭﺱ ﻋﻼﻗﻪﻣﻨﺪ ﻣﻲﻛﻨﺪ‪ .‬ﺑﭽﻪﻫﺎ ﺑﺎﻳﺪ ﺧﻮﺩﺷــﺎﻥ ﺩﺭﺑﺎﺭﻩﻱ ﺷﻜﻞ‬ ‫ﺗﺪﺭﻳﺲ ﺭﻳﺎﺿﻲ ﻧﻈﺮ ﺑﺪﻫﻨﺪ‪ .‬ﺩﺭ ﻛﻼﺱ ﻣﺸﺎﺭﻛﺖ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ ﻭ ﺳﻌﻲ‬ ‫ﺁﻥﻫﺎ ﻣﻲﭘﺮﺳﻢ‪ :‬ﺍﻳﻦ ﻛﻼﺱ ﺭﺍ ﭼﻪﻃﻮﺭ ﺍﺩﺍﺭﻩ ﻣﻲﻛﻨﻴﺪ؟‬ ‫ﻓﺎﻃﻤـﻪ ﻧﺼﻴﺮﻱ ﺟﻮﺍﺏ ﻣﻲﺩﻫﺪ‪ :‬ﺑﻪ ﻧﻈﺮ ﻣﻦ ﻣﻌﻠﻢ ﻓﻘﻂ ﺗﺎ ﭘﻨﺞ ﻛﻨﻨﺪ ﺍﺯ ﺯﻣﺎﻧﺸﺎﻥ ﺑﻴﺶﺗﺮﻳﻦ ﺍﺳﺘﻔﺎﺩﻩ ﺭﺍ ﺑﻜﻨﻨﺪ‪.‬‬ ‫ﺩﺭﺻــﺪ ﺑﺮ ﻳﺎﺩﮔﻴــﺮﻱ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺗﺄﺛﻴﺮ ﺩﺍﺭﺩ‪ .‬ﺍﮔــﺮ ﻣﻦ ﻣﻌﻠﻢ ﭼﻨﻴﻦ‬ ‫‪18‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫﹡﹆︀ط ا﹞‪﹟‬‬ ‫﹇︧﹝️ آ︠︣‬ ‫ﺣﺴﻦ ﺍﺣﻤﺪﻱ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻧﻘﺎﻁ‪ ،‬ﺩﺍﻳﺮﻩ‪ ،‬ﻣﺴﻴﺮ ﺍﻣﻦ‪ ،‬ﺧﻂ ﺭﺍﺳﺖ‪.‬‬ ‫ﻓــﺮﺽ ﻛﻨﻴﺪ ﺩﻭ ﺭﺷــﺘﻪ ﻛﻮﻩ )ﺧﻂ( ﻧﺎﺍﻣــﻦ ‪ L1‬ﻭ ‪ L2‬ﺩﺍﺭﻳﻢ ﻛﻪ ﺑﻪ ﺻﻮﺭﺕ ﻣﻮﺍﺯﻱ ﺑﺎ ﻫﻢ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪﺍﻧﺪ‪ .‬ﻣﻲﺧﻮﺍﻫﻴﻢ ﺑﻴﻦ ﺍﻳﻦ ﺩﻭ ﺭﺷــﺘﻪ ﻛﻮﻩ‪،‬‬ ‫ﻣﺴــﻴﺮﻱ ﺗﻌﻴﻴﻦ ﻛﻨﻴﻢ ﻛﻪ ﺷــﻬﺮﻫﺎﻱ )ﻧﻘﺎﻁ( ‪ A‬ﻭ ‪ B‬ﺭﺍ ﺑﻪ ﻳﻜﺪﻳﮕﺮ ﻣﺘﺼﻞ ﻛﻨﺪ‪ .‬ﺑﻪ ﻧﻈﺮ ﺷــﻤﺎ ﭼﮕﻮﻧﻪ ﻣﻲﺗﻮﺍﻥ ﺍﻣﻦﺗﺮﻳﻦ ﻣﺴﻴﺮ ﺭﺍ ﺍﻧﺘﺨﺎﺏ ﻛﺮﺩ؟‬ ‫)ﻓﺮﺽ ﻛﻨﻴﺪ ﺷﺪﺕ ﻧﺎﺍﻣﻨﻲ ﺩﺭ ﻫﺮ ﺩﻭ ﺭﺷﺘﻪ ﺑﺮﺍﺑﺮ ﺍﺳﺖ(‪.‬‬

‫‪L1‬‬

‫‪B‬‬

‫‪A‬‬

‫‪L2‬‬

‫ﺩﺭ ﺣﻞ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﻫﻢ ﺍﺯ ﺍﻳﺪﻩﻱ ﺩﺍﻳﺮﻩﻫﺎﻱ ﻣﻤﺎﺱ )ﺷﺒﻴﻪ ﻣﺴﺌﻠﻪﻱ ﻗﺒﻞ( ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ‪ .‬ﺑﻪ ﺗﻌﺪﺍﺩ ﺩﻟﺨﻮﺍﻩ ﺩﺍﻳﺮﻩ ﺭﺳﻢ ﻣﻲﻛﻨﻴﻢ ﺑﻪﻃﻮﺭﻱ‬ ‫ﻛﻪ ﺑﺮ ﺍﻳﻦ ﺩﻭ ﻣﻨﺤﻨﻲ ﻣﻤﺎﺱ ﺑﺎﺷــﻨﺪ‪ .‬ﺣﺎﻝ ﺍﮔﺮ ﻣﺮﻛﺰ ﺍﻳﻦ ﺩﺍﻳﺮﻩ ﺭﺍ ﺑﻪ ﻫﻢ ﻭﺻﻞ ﻛﻨﻴﻢ ﻣﺴــﻴﺮ ﺍﻣﻦ ﻣﻮﺭﺩ ﻧﻈﺮ ﺑﻪﺩﺳــﺖ ﺧﻮﺍﻫﺪ ﺁﻣﺪ‪ .‬ﻫﺮ ﭼﻪ ﺗﻌﺪﺍﺩ‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪19‬‬

‫ﺩﺍﻳﺮﻩﻫﺎ ﺑﻴﺸﺘﺮ ﺑﺎﺷﺪ ﻣﺴﻴﺮ ﺩﻗﻴﻖﺗﺮﻱ ﺭﺍ ﻧﺸﺎﻥ ﺧﻮﺍﻫﺪ ﺩﺍﺩ‪ .‬ﭼﺮﺍ؟ ) ﺑﺮﺍﻱ ﺭﺳﻢ ﻣﻲﺗﻮﺍﻧﻴﺪ ﺍﺯ ﭘﺮﮔﺎﺭ ﻳﺎ ﺷﺎﺑﻠﻮﻥ ﺍﺳﺘﻔﺎﺩﻩ ﻛﻨﻴﺪ‪(.‬‬

‫‪L1‬‬

‫‪B‬‬

‫‪A‬‬

‫‪L2‬‬

‫ﺑﻪ ﻧﻈﺮ ﺷﻤﺎ ﭼﺮﺍ ﻣﺴﻴﺮ ﺑﻪ ﺩﺳﺖ ﺁﻣﺪﻩ ﺍﻣﻦﺗﺮﻳﻦ ﻣﺴﻴﺮ ﺍﺳﺖ؟‬ ‫ﺩﺭ ﺟﻬﺎﻥ ﻭﺍﻗﻌﻲ ﺁﻳﺎ ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﺍﻳﻦ ﺭﻭﺵ ﺍﺳﺘﻔﺎﺩﻩ ﻛﺮﺩ؟ ﭼﻪ ﻣﻮﺍﻧﻊ ﻭﺟﻮﺩ ﺧﻮﺍﻫﺪ ﺩﺍﺷﺖ؟‬ ‫ﻳﻜﻲ ﺍﺯ ﭘﻴﭽﻴﺪﮔﻲﻫﺎﻱ ﻣﺴــﺎﺋﻞ ﻭﺍﻗﻌﻲ ﺍﻳﻦ ﺍﺳــﺖ ﻛﻪ ﺷﺪﺕ ﻧﺎ ﺍﻣﻨﻲ ﺩﺭ ﺩﻭ ﺭﺷــﺘﻪ ﻛﻮﻩ ﺑﺮﺍﺑﺮ ﻧﺒﺎﺷﺪ‪ .‬ﺑﺮﺍﻱ ﺣﻞ ﺍﻳﻦ ﻣﺸﻜﻞ ﻣﻲﺗﻮﺍﻧﻴﺪ ﺑﺎﺯ ﻫﻢ‬ ‫ﻣﺴــﺌﻠﻪ ﺭﺍ ﺩﺭ ﺣﺎﻟﺖﻫﺎﻱ ﺳــﺎﺩﻩ ﺷــﺪﻩ ) ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻦ ﺧﻂ ﺭﺍﺳﺖ( ﺣﻞ ﻛﻨﻴﺪ ﻭ ﺑﻌﺪ ﺍﺯ ﺍﻳﻨﻜﻪ ﺍﻳﺪﻩﻫﺎﻱ ﺍﺻﻠﻲ ﺭﺍ ﭘﻴﺪﺍ ﻛﺮﺩﻳﺪ‪ ،‬ﺁﻥﻫﺎ ﺭﺍ ﺩﺭ ﺣﺎﻟﺖ‬ ‫ﻛﻠﻲ ﻣﻄﺮﺡ ﻛﻨﻴﺪ‪.‬‬ ‫ﺑﺮﺍﻱ ﻣﺜﺎﻝ ﺭﻭﻱ ﺣﺎﻟﺘﻲ ﻛﺎﺭ ﻛﻨﻴﺪ ﻛﻪ ﺷــﺪﺕ ﻧﺎ ﺍﻣﻨﻲ ﺩﺭ ﻳﻜﻲ ﺍﺯ ﺭﺷــﺘﻪ ﻛﻮﻩﻫﺎ ﺩﻭ ﺑﺮﺍﺩﺭ ﺩﻳﮕﺮﻱ ﺑﺎﺷــﺪ‪ .‬ﭘﺎﺳﺦﻫﺎﻱ ﺟﺎﻟﺐﺗﺎﻥ ﺭﺍ ﺑﺮﺍﻱ ﻣﺠﻠﻪ‬ ‫ﺍﺭﺳﺎﻝ ﻛﻨﻴﺪ‪.‬‬

‫‪20‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫︎︀︨︞‬ ‫﹞︟︀︵︣ات ︨﹀︣ در ︨﹫︀رهی ﹡︀﹛‪︣﹞﹢‬‬

‫ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ‬

‫ﺍﺗﻮﭘﻴﺎ‬

‫ﻣﺮﺍﻛﺶ‬ ‫ﺯﻧﮕﺒﺎﺭ‬ ‫ﺳﻨﮕﺎﭘﻮﺭ‬

‫ﻳﺎﻟﻲ‬ ‫ﺭﻳﻮ‬

‫ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ‬

‫ﺭﻡ‬

‫ﺳﻨﮕﺎﭘﻮﺭ‬

‫ﻫﻠﻴﺲﺑﻴﻮﺩ‬ ‫ﺭﻭﻳﻴﻦ‬

‫ﻳﺎﻟﻲ‬ ‫ﻫﻨﮓﻛﻨﮓ‬ ‫ﺭﻳﺠﺰ‬ ‫ﻧﻮﻭﺭ‬

‫ﺯﻧﮕﺒﺎﺭ‬

‫ﻣﺮﺍﻛﺶ‬ ‫ﺍﺗﻮﭘﻴﺎ‬ ‫ﺭﻳﻮ‬

‫ﺯﻧﮕﺒﺎﺭ‬

‫ﺭﻡ‬ ‫ﺳﻨﮕﺎﭘﻮﺭ‬

‫ﺭﻭﻳﻴﻦ‬

‫ﻧﻮﻭﺭ‬ ‫ﻫﻠﻴﺲﺑﻴﻮﺩ‬

‫ﺳﻨﮕﺎﭘﻮﺭ‬

‫‪11 12 13 14 15 16‬‬

‫ﺭﻡ‬ ‫ﺭﻭﻳﻴﻦ‬ ‫ﻫﻠﻴﺲﺑﻴﻮﺩ‬ ‫ﻧﻮﻭﺭ‬ ‫ﺭﻳﺠﺰ‬ ‫ﻫﻨﮓﻛﻨﮓ‬

‫‪4 5‬‬ ‫‪0 1 2 3‬‬

‫ﺯﻧﮕﺒﺎﺭ‬ ‫ﻣﺮﺍﻛﺶ‬ ‫ﺍﺗﻮﭘﻴﺎ‬ ‫ﺭﻳﻮ‬ ‫ﻳﺎﻟﻲ‬ ‫ﻫﻨﮓﻛﻨﮓ‬ ‫ﺭﻳﺠﺰ‬

‫ﺣﺪ ﻗﺮﺍﺭ ﺩﺍﺭﺩ‪ .‬ﺣﺮﻛﺖ ﺍﺗﻮﺑﻮﺱ ﺑﺎ ﺑﺎﻙ ﺧﺎﻟﻰ ﻭ ﺍﻓﺰﺍﻳﺶﻫﺎ ﻭ ﻛﺎﻫﺶﻫﺎﻯ‬ ‫ﻣﻘﺪﺍﺭ ﺳﻮﺧﺖ ﺩﺭ ﻫﺮ ﻳﻚ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺩﺭﺳﺖ ﺑﻪ ﻣﺜﺎﺑﻪ ﺣﺮﻛﺖ ﺍﺗﻮﺑﻮﺱ‬ ‫ﺩﺭ ﺍﻳﻦ ﺳــﻔﺮ ﻓﺮﺿﻰ ﺍﺳﺖ ﺑﻨﺎﺑﺮﺍﻳﻦ‪ ،‬ﺗﻔﺎﻭﺕ ﻣﻴﺎﻥ ﺑﻨﺰﻳﻦ ﺑﺎﻙ ﺍﺗﻮﺑﻮﺱ‬ ‫ﺷــﻤﺎ )ﻧﻮﺳﺎﻥﻫﺎﻯ ﺣﺠﻢ ﺑﻨﺰﻳﻦ ﺩﺭ ﺑﺎﻙ( ﻭ ﻣﻘﺪﺍﺭ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺩﺭ‬ ‫ﻧﻤﻮﺩﺍﺭ ‪ ،‬ﺑﺎﻳﺪ ﻳﻜﺴــﺎﻥ ﺑﺎﺷــﺪ ﻭ ﺁﻥ ﻫﻤﻮﺍﺭﻩ ﺑﺮﺍﺑﺮ ﺍﺳــﺖ ﺑﺎ ﭘﺎﻳﻴﻦﺗﺮﻳﻦ‬ ‫ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳــﻦ ﺩﺭ ﻧﻤﻮﺩﺍﺭ‪ .‬ﭼﻮﻥ ﻧﻤﻮﺩﺍﺭ ﻫﺮﮔﺰ ﺑﻪ ﺯﻳﺮ ﭘﺎﻳﻴﻦﺗﺮﻳﻦ ﻧﻘﻄﻪ‬ ‫ﺧﻮﺩ ﻧﻤﻰﺁﻳﺪ‪ ،‬ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ ﺑﺎﻙ ﺷﻤﺎ ﻫﺮﮔﺰ ﺑﻪ ﺯﻳﺮ ﺻﻔﺮ ﻧﺨﻮﺍﻫﺪ ﺭﻓﺖ؛‬ ‫ﺣﺘــﻰ ﺩﺭ ﻣﻮﺍﺭﺩﻯ ﻛــﻪ ﺩﺭ ﻧﻤﻮﺩﺍﺭ ﭼﻨﺪ ﻧﻘﻄﻪ ﺑــﺎ ﭘﺎﻳﻴﻦﺗﺮﻳﻦ ﻧﻘﻄﻪﻯ‬ ‫ﻧﻤﻮﺩﺍﺭ ﻫﻤﺴــﺎﻥ ﺑﺸﻮﻧﺪ‪ .‬ﺍﻳﻦ ﺑﺪﺍﻥ ﻣﻌﻨﺎﺳــﺖ ﻛﻪ ﻭﺳﻴﻠﻪﻯ ﻧﻘﻠﻴﻪ ﺷﻤﺎ‬ ‫ﺑﻪ ﻫﻨﮕﺎﻡ ﺭﺳﻴﺪﻥ ﺑﻪ ﺁﻥ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺑﺎﻙ ﺑﻨﺰﻳﻨﺶ ﺧﺎﻟﻰ ﻣﻰﺷﻮﺩ ﻭ ﺷﻤﺎ‬ ‫ﻣﻰﺗﻮﺍﻧﻴﺪ ﺩﺭ ﺁﻥ ﭘﺎﻳﮕﺎﻩ ﺑﺮﺍﻯ ﺍﺩﺍﻣﻪ ﺣﺮﻛﺖ ﺑﻨﺰﻳﻦ ﺑﮕﻴﺮﻳﺪ‪.‬‬ ‫ﻧﻤﻮﺩﺍﺭ ‪ 2‬ﻧﺸــﺎﻥ ﺩﻫﻨﺪﻩﻯ ﺳــﻄﺢ ﺑﻨﺰﻳﻦ ﺍﺗﻮﺑﻮﺱ ﺷــﻤﺎ ﺩﺭ ﻃﻰ‬ ‫ﺳﻔﺮﻯ ﺍﺳﺖ ﻛﻪ ﺍﺯ ﺯﻧﮕﺒﺎﺭ ﻭ ﺑﺎ ﺑﺎﻙ ﺧﺎﻟﻰ ﺁﻏﺎﺯ ﻣﻰﺷﻮﺩ )ﺗﻮﺟﻪ ﺩﺍﺷﺘﻪ‬ ‫ﺑﺎﺷﻴﺪ ﻛﻪ ﻫﺮ ﻧﻤﻮﺩﺍﺭ ﺩﺭ ﻭﺍﻗﻊ ﻳﻚ ﺧﻂ ﭘﻴﻮﺳﺘﻪ ﺭﺍ ﻧﺸﺎﻥ ﻣﻰﺩﻫﺪ‪ ،‬ﺯﻳﺮﺍ‬ ‫ﻧﺨﺴﺘﻴﻦ ﻭ ﺁﺧﺮﻳﻦ ﻧﻘﻄﻪ ﺭﻭﻯ ﻧﻤﻮﺩﺍﺭ ﻳﻜﻰ ﻫﺴﺘﻨﺪ(‪.‬‬

‫ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ‬

‫‪... ..‬‬ ‫‪.‬‬ ‫‪.‬‬ ‫‪. . .. .‬‬ ‫‪.‬‬ ‫‪.‬‬ ‫‪..‬‬

‫‪1‬‬

‫‪. . . .. .. . . 3‬‬ ‫‪.‬‬ ‫‪. .. .‬‬ ‫‪..‬‬ ‫‪.. .‬‬ ‫‪2‬‬ ‫‪. . . . ..‬‬ ‫‪. ...‬‬

‫‪8 9 10 11 12 13‬‬ ‫ﺳﻨﮕﺎﭘﻮﺭ‬

‫‪ .١‬ﺍﮔــﺮ ﺩﺭ ﭘﺎﻳﮕﺎﻩ ﺯﻧﮕﺒﺎﺭ ﻓﺮﻭﺩ ﺁﻳﻴــﺪ ﻭ ﺍﺯ ﺁﻥ ﻧﻘﻄﻪ ﺩﺭ ﺟﻬﺖ ﺣﺮﻛﺖ‬ ‫ﻋﻘﺮﺑﻪﻫﺎﻯ ﺳﺎﻋﺖ‪ ،‬ﻣﺎﻣﻮﺭﻳﺖ ﺧﻮﺩ ﺭﺍ ﺁﻏﺎﺯ ﻛﻨﻴﺪ‪ ،‬ﺧﻮﺍﻫﻴﺪ ﺗﻮﺍﻧﺴﺖ‬ ‫ﻣﺴــﻴﺮ ﺑﺰﺭﮔﺮﺍﻩ ﺭﺍ ﻳﻚ ﺩﻭﺭ ﻛﺎﻣﻞ ﺑﭙﻴﻤﺎﻳﺪ‪ .‬ﺍﻣﺎ ﭼﻨﺎﻥﭼﻪ ﺑﺨﻮﺍﻫﻴﺪ‬ ‫ﺩﺭ ﺟﻬﺖ ﻋﻜﺲ ﺣﺮﻛﺖ ﻋﻘﺮﺑﻪﻫﺎﻯ ﺳﺎﻋﺖ ﺭﺍﻧﻨﺪﮔﻰ ﻛﻨﻴﺪ‪ ،‬ﺑﺎﻳﺪ ﺍﺯ‬ ‫ﻳﻜﻰ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎﻯ ﻫﻨﮓﻛﻨﮓ ﻳﺎ ﺑﺎﻟﻰ ﺭﺍﻩ ﺑﻴﻔﺘﻴﺪ‪.‬‬ ‫‪ .٢‬ﻓﺮﺽ ﻛﻨﻴﺪ ﺷــﻤﺎ ﺩﺭ ﻳﻚ ﭘﺎﻳﮕﺎﻩ ﺑﻪﺧﺼﻮﺹ ﻣﺎﻧﻨﺪ ﺳﻨﮕﺎﭘﻮﺭ ﻓﺮﻭﺩ‬ ‫ﺁﻣﺪﻩﺍﻳــﺪ ﻭ ﺑﺮﺍﻯ ﭘﻴﻤﻮﺩﻥ ﻃــﻮﻝ ﺭﺍﻩ‪ ،‬ﺑﻨﺰﻳﻦ ﻛﺎﻓﻰ ﺩﺭ ﺑﺎﻙ ﺩﺍﺭﻳﺪ‪.‬‬ ‫ﺣﺎﻝ ﻧﻤﻮﺩﺍﺭﻯ ﺭﺳــﻢ ﻛﻨﻴﺪ ﻛﻪ ﻧﻤﺎﻳﺎﻧﮕﺮ ﻣﺤﺘﻮﺍﻯ ﺑﻨﺰﻳﻦ ﺑﺎﻙ ﺷﻤﺎ‬ ‫ﺩﺭ ﺗﻤﺎﻡ ﻣﺴــﻴﺮ ﺑﺎﺷﺪ ﻭ ﺩﺭ ﻋﻴﻦ ﺣﺎﻝ ﻧﺸــﺎﻥ ﺑﺪﻫﺪ ﻛﻪ ﺑﻪ ﻫﻨﮕﺎﻡ‬ ‫ﻋﺒــﻮﺭ ﺍﺯ ﭘﺎﻳﮕﺎﻩﻫﺎ ﺩﺭ ﻛﺪﺍﻡﻳﻚ ﺍﺯ ﺁﻥﻫﺎ ﺳــﻮﺧﺖﮔﻴﺮﻯ ﻣﻰﻛﻨﻴﺪ‪.‬‬ ‫ﺩﺭ ﺗﺼﻮﻳــﺮ ‪ 1‬ﻧﻤﻮﺩﺍﺭ ﻣﺮﺑﻮﻁ ﺑﻪ ﻣﻌﻤﺎﻯ ‪ 1‬ﺭﺍ ﻣﻰﺑﻴﻨﻴﺪ ﻛﻪ ﻣﺴــﻴﺮ‬ ‫ﺣﺮﻛﺖ ﺭﺍ ﻧﺸﺎﻥ ﻣﻰﺩﻫﺪ‪ .‬ﺧﻂﻫﺎﻯ ﻋﻤﻮﺩﻯ ﻧﻤﺎﻳﺎﻧﮕﺮ ﺟﺎﻳﮕﺎﻩﻫﺎﻯ‬ ‫ﺑﻨﺰﻳﻦ ﻭ ﻋﺪﺩﻫﺎ ﻧﻤﺎﻳﻨﺪﻩ ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ ﻣﻮﺟﻮﺩ ﺩﺭ ﺑﺎﻙ ﺍﺳﺖ‪.‬‬ ‫ﻣﻘﺪﺍﺭ ﺑﻨﺰﻳﻦ ﺍﺗﻮﺑﻮﺱ ﺩﺭ ﻧﻘﻄﻪﻯ ﺁﻏﺎﺯ ﺣﺮﻛﺖ ﻫﻤﻮﺍﺭﻩ ﺩﺭ ﭘﺎﻳﻴﻦﺗﺮﻳﻦ‬

‫ﺍﻣﺎ ﺗﻮﺟﻪ ﺩﺍﺷــﺘﻪ ﺑﺎﺷــﻴﺪ ﻣﻪ ﻧﻤﻮﺩﺍﺭ ﻓﻮﻕ ﻣﺴﻴﺮ ﺣﺮﻛﺘﻰ ﺭﺍ ﻧﺸﺎﻥ‬ ‫ﻣﻰﺩﻫﺪ ﻛﻪ ﺩﺭ ﺟﻬﺖ ﻋﻘﺮﺑﻪﻫﺎﻯ ﺳﺎﻋﺖ ﺻﻮﺭﺕ ﮔﺮﻓﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﺑﺮﺍﻯ ﻧﺸﺎﻥ ﺩﺍﺩﻥ ﻳﻚ ﻣﺴﻴﺮ ﭘﻴﻤﻮﺩﻩ ﺷﺪﻩ ﺩﺭ ﺟﻬﺖ ﻋﻜﺲ ﺣﺮﻛﺖ‬ ‫ﻋﻘﺮﺑﻪﻫﺎﻯ ﺳــﺎﻋﺖ‪ ،‬ﺑﺎﻳﺪ ﻧﻤﻮﺩﺍﺭ ﺟﺪﻳﺪﻯ ﺑﻜﺸــﻴﺪ‪ .‬ﺷــﻜﻞ ‪ 3‬ﭼﻨﻴﻦ‬ ‫ﻧﻤﻮﺩﺍﺭﻯ ﺑﺮﺍﻯ ﻣﻌﻤﺎﻯ ‪ 1‬ﺭﺍ ﻧﺸــﺎﻥ ﻣﻰﺩﻫﺪ ‪ .‬ﻫﻤﺎﻥﻃﻮﺭ ﻛﻪ ﻣﻰﺑﻴﻨﻴﺪ‪،‬‬ ‫ﺩﺭ ﺍﻳﻦ ﻧﻤﻮﺩﺍﺭ ﺩﻭ ﻧﻘﻄﻪ ﺩﺭ ﭘﺎﻳﻴﻦﺗﺮﻳﻦ ﺳﻄﺢ ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ ﻛﻪ ﻣﻰﺗﻮﺍﻧﻴﺪ‬ ‫ﺳﻔﺮ ﺭﺍ ﺍﺯ ﻫﺮ ﻳﻚ ﺍﺯ ﺍﻳﻦ ﺩﻭ ﻧﻘﻄﻪ ﺁﻏﺎﺯ ﻛﻨﻴﺪ‪.‬‬ ‫ﭼــﻮﻥ ﺩﺭ ﻫﺮ ﻧﻤﻮﺩﺍﺭ ﺑﺎﻳﺪ ﺩﺳــﺖ ﻛﻢ ﻳﻚ ﻧﻘﻄــﻪ ﺩﺭ ﭘﺎﻳﻴﻦﺗﺮﻳﻦ‬ ‫ﺳــﻄﺢ ﻭﺍﻗﻊ ﺷﺪﻩ ﺑﺎﺷــﺪ‪ ،‬ﻫﺮ ﻣﺴــﻴﺮ ﺣﺮﻛﺘﻰ ﻧﺎﮔﺰﻳﺮ ﻳﻚ ﺣﻠﻘﻪ )ﻳﻚ‬ ‫ﻣﺴﻴﺮ ﺣﻠﻘﻪ ﻣﺎﻧﻨﺪ ﻣﺴﺪﻭﺩ( ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫︠‪﹢‬ا﹡︡﹡‪︀﹨﹩‬﹬‪ ﹩‬از ر﹬︀︲﹫︀ت‬ ‫ﺯﻳﻨﺐ ﮔﻠﺒﺮﺍﺭﻯ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﺷﺎﺧﻪﻫﺎﻱ ﺭﻳﺎﺿﻲ‪ ،‬ﺭﻳﺎﺿﻴﺎﺕ ﻣﺤﺾ‪ ،‬ﻫﻨﺪﺳﻪ‪ ،‬ﺍﻣﻜﺎﻥ‬

‫ﺭﻳﺎﺿﻴﺎﺕ ﻭ ﺷﺎﺧﻪﻫﺎﻱ ﺁﻥ‬

‫ﺭﻳﺎﺿﻴﺎﺕ‪ ،‬ﻋﻠﻢ ﻧﻈﻢ ﺍﺳﺖ ﻭ ﻣﻮﺿﻮﻉ ﺁﻥ ﻳﺎﻓﺘﻦ‪ ،‬ﺗﻮﺻﻴﻒ ﻭ ﺩﺭﻙ‬ ‫ﻧﻈﻤﻰ ﺍﺳـﺖ ﻛﻪ ﺩﺭ ﻭﺿﻌﻴﺖﻫـﺎﻯ ﺑﻪ ﻇﺎﻫﺮ ﭘﻴﭽﻴﺪﻩ ﻧﻬﻔﺘﻪ ﺍﺳـﺖ ﻭ‬ ‫ﺍﺑﺰﺍﺭﻫـﺎﻯ ﺍﺻﻮﻟﻰ ﺍﻳﻦ ﻋﻠﻢ‪ ،‬ﻣﻔﺎﻫﻴﻤﻰ ﻫﺴـﺘﻨﺪ ﻛﻪ ﺑـﻪ ﻛﻤﻚ ﺁﻥﻫﺎ‬ ‫ﻣﻰﺗﻮﺍﻧﻴـﻢ ﺍﻳﻦ ﻧﻈﻢ ﺭﺍ ﺗﻮﺻﻴﻒ ﻛﻨﻴﻢ‪ .‬ﻋﻠـﻢ ﺭﻳﺎﺿﻰ ﻗﺎﻧﻮﻧﻤﻨﺪ ﻛﺮﺩﻥ‬ ‫ﺗﺠﺮﺑﻴﺎﺕ ﻃﺒﻴﻌﻰ ﺍﺳـﺖ ﻛﻪ ﺩﺭ ﮔﻴﺎﻫﺎﻥ ﻭ ﺑﻘﻴﻪﻯ ﻣﺨﻠﻮﻗﺎﺕ ﻣﺸﺎﻫﺪﻩ‬ ‫ﻣﻰﻛﻨﻴـﻢ‪ .‬ﻋﻠﻮﻡ ﺭﻳﺎﺿﻴﺎﺕ ﺍﻳﻦ ﺗﺠﺮﺑﻴﺎﺕ ﺭﺍ ﺩﺳـﺘﻪﺑﻨﺪﻯ ﻭ ﻗﺎﻧﻮﻧﻤﻨﺪ‬ ‫ﻣﻰﻛﻨﻨﺪ ﻭ ﺗﻮﺳﻌﻪ ﻣﻰﺩﻫﻨﺪ‪.‬‬ ‫ﺩﻛﺘﺮ ﺭﻳﺎﺿﻰ‪ ،‬ﺍﺳـﺘﺎﺩ ﺭﻳﺎﺿﻰ‪ ،‬ﻧﻴﺰ ﺩﺭ ﻣﻌﺮﻓﻰ ﺍﻳﻦ ﻋﻠﻢ ﻣﻰﮔﻮﻳﺪ‪:‬‬ ‫))ﺭﻳﺎﺿﻴﺎﺕ ﻋﻠﻢ ﻣﺪﻝﺩﻫﻰ ﺑﻪ ﺳﺎﻳﺮ ﻋﻠﻮﻡ ﺍﺳﺖ‪ ،‬ﻳﻌﻨﻰ ﺯﺑﺎﻥ ﻣﺸﺘﺮﻙ‬ ‫ﻧﻈﺮﻳﺎﺕ ﻋﻠﻤﻰ ﺳـﺎﻳﺮ ﻋﻠﻮﻡ‪ ،‬ﻋﻠﻢ ﺭﻳﺎﺿﻰ ﺍﺳﺖ‪ .‬ﺍﻣﺮﻭﺯﻩ ﺍﮔﺮ ﻋﻠﻤﻰ ﺭﺍ‬ ‫ﻧﺘﻮﺍﻥ ﺑﻪ ﺯﺑﺎﻥ ﺭﻳﺎﺿﻰ ﺑﻴﺎﻥ ﻛﺮﺩ ﻋﻠﻢ ﻧﻴﺴﺖ‪((.‬‬

‫ﻣﺎﻫﻴﺖ ﺭﻳﺎﺿﻴﺎﺕ‬ ‫ﺭﻳﺎﺿﻴــﺎﺕ ﺑﺮ ﺧــﻼﻑ ﺗﺼﻮﺭ ﺑﺮﺧــﻰ ﺍﻓﺮﺍﺩ‪ ،‬ﻳﻚ ﺳــﺮﻯ ﻓﺮﻣﻮﻝ ﻭ‬ ‫ﻗﻮﺍﻋﺪ ﻧﻴﺴــﺖ ﻛﻪ ﻫﻤﻴﺸــﻪ ﻭ ﺩﺭ ﻫﻤﻪ ﺟﺎ ﺑﺘﻮﺍﻥ ﺍﺯ ﺁﻥ ﺍﺳﺘﻔﺎﺩﻩ ﻛﺮﺩ‪،‬‬ ‫ﺑﻠﻜﻪ ﺭﻳﺎﺿﻴﺎﺕ ﺩﺭﺳــﺖ ﻓﻬﻤﻴﺪﻥ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﻭ ﺩﺭﺳﺖ ﻓﻜﺮ ﻛﺮﺩﻥ‬ ‫ﺑﺮﺍﻯ ﺭﺳــﻴﺪﻥ ﺑﻪ ﺟﻮﺍﺏ ﺍﺳــﺖ‪ .‬ﺑﺮﺍﻯ ﺑﻪ ﺩﺳــﺖ ﺁﻭﺭﺩﻥ ﺍﻳﻦ ﺗﻮﺍﻧﺎﻳﻰ‪،‬‬ ‫ﺩﺍﻧﺸــﺠﻮ ﺑﺎﻳﺪ ﺻﺒﺮ ﻭ ﭘﺸــﺘﻜﺎﺭ ﻻﺯﻡ ﺭﺍ ﺩﺍﺷــﺘﻪ ﺑﺎﺷــﺪ ﺗﺎ ﺑﺘﻮﺍﻧﺪ ﺣﺘﻰ‬ ‫ﺑﻪ ﻣﺪﺕ ﭼﻨﺪﻳﻦ ﺳــﺎﻋﺖ ﺩﺭ ﺑﺎﺭﻩﻯ ﻳﻚ ﻣﺴــﺌﻠﻪﻯ ﺭﻳﺎﺿﻰ ﺑﻴﻨﺪﻳﺸﺪ‬ ‫ﻭ ﺩﺭ ﻧﻬﺎﻳــﺖ ﺑــﺎ ﺍﺑﺘﻜﺎﺭ ﻭ ﺧﻼﻗﻴﺖ ﺁﻥ ﺭﺍ ﺣﻞ ﻛﻨــﺪ‪ .‬ﻓﺎﺭﻍ ﺍﻟﺘﺤﺼﻴﻼﻥ‬ ‫ﺍﻳــﻦ ﺭﺷــﺘﻪ ﻣﻰﺗﻮﺍﻧﻨﺪ ﭘﺲ ﺍﺯ ﭘﺎﻳــﺎﻥ ﺗﺤﺼﻴــﻼﺕ‪ ،‬ﺩﺭ ﺍﺩﺍﺭﺍﺕ ﺩﻭﻟﺘﻰ‬ ‫ﺑﺮﺍﻯ ﻣﺴــﺌﻮﻟﻴﺖﻫﺎﻳﻰ ﻛﻪ ﺑﻪ ﻧﻮﻋﻰ ﺑﺎ ﺗﺠﺰﻳﻪ ﻭ ﺗﺤﻠﻴﻞ ﻣﺴﺎﺋﻞ ﺳﺮﻭﻛﺎﺭ‬ ‫ﺩﺍﺭﻧﺪ‪ ،‬ﺩﺭ ﺑﺨﺶ ﺧﺼﻮﺻﻰ ﺩﺭ ﺍﻣﻮﺭﻯ ﻫﻤﺎﻧﻨﺪ ﻃﺮﺍﺣﻰ ﺳﻴﺴــﺘﻢﻫﺎ ﺩﺭ‬ ‫‪22‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺍﻣﺮ ﺑﻬﻴﻨﻪﺳــﺎﺯﻯ ﻭ ﺑﻬﺮﻩﻭﺭﻯ‪ ،‬ﺩﺭ ﺑﺨــﺶ ﺻﻨﻌﺖ ﺑﺮﺍﻯ ﺍﻣﻮﺭﻯ ﻫﻤﺎﻧﻨﺪ‬ ‫ﻣﺪﻝﺳــﺎﺯﻯﻫﺎﻯ ﺭﻳﺎﺿﻰ ﻭ ﺩﺭ ﺁﻣﻮﺯﺵ ﻭ ﭘﺮﻭﺭﺵ ﻭ ‪ ، ...‬ﻣﺴﺌﻮﻟﻴﺖﻫﺎﻯ‬ ‫ﻣﺘﻔﺎﻭﺗﻰ ﺭﺍ ﺑﺮ ﻋﻬﺪﻩ ﮔﻴﺮﻧﺪ‪ .‬ﺭﺋﻴﺲ ﺍﺗﺤﺎﺩﻳﻪﻯ ﺑﻴﻦﺍﻟﻤﻠﻠﻰ ﺭﻳﺎﺿﻰﺩﺍﻧﺎﻥ‬ ‫ﺟﻬــﺎﻥ ﺩﺭ ﻳﺎﺯﺩﻫﻤﻴﻦ ﺍﺟﻼﺱ ﺁﻛﺎﺩﻣﻰ ﺟﻬﺎﻥ ﺳــﻮﻡ ﻛﻪ ﺑﻪ ﺗﺎﺯﮔﻰ ﺩﺭ‬ ‫ﺗﻬﺮﺍﻥ ﺑﺮﮔﺰﺍﺭ ﺷــﺪ‪ ،‬ﻋﻨﻮﺍﻥ ﻛﺮﺩ ﻛﻪ ﺑﻬﺘﺮ ﺍﺳــﺖ ﺑﮕﻮﻳﻴﻢ ))ﺭﻳﺎﺿﻴﺎﺕ ﻭ‬ ‫ﻛﺎﺭﺑﺮﺩﻫﺎﻯ ﺁﻥ(( ﻧﻪ ﺍﻳﻦﻛﻪ ﺭﻳﺎﺿﻴﺎﺕ ﺭﺍ ﺑﻪ ﻣﺤﺾ ﻭ ﻛﺎﺭﺑﺮﺩﻯ ﺗﻔﻜﻴﻚ‬ ‫ﻛﻨﻴﻢ‪ ،‬ﺯﻳﺮﺍ ﺑﻪ ﺍﻋﺘﻘﺎﺩ ﺭﻳﺎﺿﻰﺩﺍﻥﻫﺎ ﻫﻴﭻ ﻣﻘﻮﻟﻪﻯ ﺭﻳﺎﺿﻰ ﻧﻴﺴــﺖ ﻛﻪ‬ ‫ﺭﻭﺯﻯ ﻛﺎﺭﺑﺮﺩﻯ ﺑﺮﺍﻯ ﺁﻥ ﭘﻴﺪﺍ ﻧﺸﻮﺩ‪.‬‬ ‫ﺭﻳﺎﺿﻴﺎﺕ ﻣﺤﺾ ﺑﻴﺶﺗﺮ ﺑﻪ ﻗﻀﺎﻳﺎ ﻭ ﺍﺳــﺘﺪﻻﻝﻫﺎ‪ ،‬ﻣﻨﻄﻖ ﻣﻮﺟﻮﺩ‬ ‫ﺩﺭ ﺁﻥﻫﺎ ﻭ ﭼﮕﻮﻧﮕﻰ ﺍﺛﺒﺎﺗﺸــﺎﻥ ﻣﻰﭘﺮﺩﺍﺯﺩ‪ .‬ﺍﻣﺎ ﺩﺭ ﺭﻳﺎﺿﻴﺎﺕ ﻛﺎﺭﺑﺮﺩﻯ‬

‫ﭼﮕﻮﻧﻪ ﺍﺳﺘﻔﺎﺩﻩ ﻛﺮﺩﻥ ﻭ ﺑﻪ ﻛﺎﺭ ﮔﺮﻓﺘﻦ ﻗﻀﺎﻳﺎ‪ ،‬ﺁﻣﻮﺯﺵ ﺩﺍﺩﻩ ﻣﻰﺷﻮﺩ‪.‬‬ ‫ﺑﻪ ﻋﺒــﺎﺭﺕ ﺩﻳﮕﺮ‪ ،‬ﺩﺭ ﺍﻳﻦ ﺷــﺎﺧﻪ ‪،‬ﻛﺎﺭﺑﺮﺩ ﺭﻳﺎﺿﻴﺎﺕ ﺩﺭ ﻣﺴــﺎﺋﻞ‬ ‫ﻣﻮﺟــﻮﺩ ﺩﺭ ﺟﺎﻣﻌﻪ ﺑﻴﺎﻥ ﻣﻰﺷــﻮﺩ‪ .‬ﻭﻗﺘﻰ ﺻﺤﺒــﺖ ﺍﺯ ﺭﻳﺎﺿﻰ ﻣﺤﺾ‬ ‫ﻣﻰﺷــﻮﺩ‪ ،‬ﻧﺒﺎﻳﺪ ﺗﺼﻮﺭ ﻛﺮﺩ ﻛﻪ ﺗﻨﻬﺎ ﺑﺎﻳﺪ ﺩﺭ ﮔﻮﺷــﻪﺍﻯ ﻧﺸﺴــﺖ ﻭ ﺑﻪ‬ ‫ﺣﻞ ﻣﺴــﺎﺋﻞ ﺭﻳﺎﺿﻰ ﭘﺮﺩﺍﺧﺖ ﺑﻠﻜﻪ ﺍﻳــﻦ ﻋﻠﻢ‪ ،‬ﺑﻪ ﻭﻳﮋﻩ ﺩﺭ ﻣﺪﺍﺭﺝ ﺑﺎﻻ‪،‬‬ ‫ﺍﺭﺗﺒــﺎﻁ ﻧﺰﺩﻳﻜﻰ ﺑــﺎ ﻃﺒﻴﻌﺖ ﺩﺍﺭﺩ ﺑﻪ ﻋﺒﺎﺭﺕ ﺩﻳﮕــﺮ ﺍﻳﺪﻩﻫﺎﻯ ﺭﻳﺎﺿﻰ‬ ‫ﺍﺯ ﺫﻫﻦ ﭘﮋﻭﻫﺸــﮕﺮﺍﻥ ﻧﻤﻰﺭﻭﻳﺪ‪ ،‬ﺑﻠﻜﻪ ﺭﻳﺎﺿﻰﺩﺍﻥﻫﺎ ﺍﻏﻠﺐ ﺍﻟﻬﺎﻡ ﺧﻮﺩ‬ ‫ﺭﺍ ﺍﺯ ﻃﺒﻴﻌــﺖ ﻣﻰﮔﻴﺮﻧﺪ ﻭ ﺑﻪ ﻗــﻮﻝ ﺯﺍﻥ ﺑﺎﭘﺘﻴﺖ ﻓﻮﺭﻳــﻪ‪ ،‬ﺭﻳﺎﺿﻰﺩﺍﻥ‬ ‫ﻣﺸﻬﻮﺭ ﻗﺮﻥ ﻧﻮﺯﺩﻫﻢ ﻓﺮﺍﻧﺴــﻪ ))ﺗﻌﻤﻖ ﺩﺭ ﻃﺒﻴﻌﺖ‪ ،‬ﭘﺮﺑﺎﺭ ﺗﺮﻳﻦ ﻣﻨﺎﺑﻊ‬ ‫ﺍﻛﺘﺸــﺎﻓﺎﺕ ﺭﻳﺎﺿﻰ ﺍﺳﺖ((‪ .‬ﺭﻳﺎﺿﻴﺎﺕ ﻛﺎﺭﺑﺮﺩﻯ ﺑﻪ ﺷﺎﺧﻪﺍﻯ ﺍﺯ ﺭﻳﺎﺿﻰ‬ ‫ﮔﻔﺘﻪ ﻣﻰﺷــﻮﺩ ﻛﻪ ﻛﺎﺭﺑﺮﺩ ﻋﻠﻤﻰ ﻣﺸــﺨﺼﻰ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ؛ ﺑﺮﺍﻯ ﻣﺜﺎﻝ‬ ‫ﺩﺭ ﺍﻗﺘﺼﺎﺩ‪ ،‬ﻛﺎﻣﭙﻴﻮﺗﺮ‪ ،‬ﻓﻴﺰﻳﻚ ﻳﺎ ﺁﻣﺎﺭ ﺍﺣﺘﻤﺎﻝ ﻛﺎﺭﺑﺮﺩ ﺩﺍﺷــﺘﻪ ﺑﺎﺷﺪ ﻭ‬ ‫ﺭﻳﺎﺿﻰ ﻣﺤﺾ ﻧﻴﺰ ﺷﺎﺧﻪﺍﻯ ﮔﻔﺘﻪ ﻣﻰﺷﻮﺩ ﻛﻪ ﺑﻪ ﻧﻈﺮﻳﻪﭘﺮﺩﺍﺯﻯ ﺭﻳﺎﺿﻰ‬ ‫ﻣﻰﭘﺮﺩﺍﺯﺩ‪ ،‬ﺍﻣﺎ ﺑﺎﻳﺪ ﺗﻮﺟﻪ ﺩﺍﺷــﺖ ﻛﻪ ﺍﻣﺮﻭﺯﻩ ﺍﻳﻦ ﺩﻭ ﮔﺮﺍﻳﺶ ﺁﻥﭼﻨﺎﻥ‬ ‫ﺩﺭ ﻫﻢ ﺍﺩﻏﺎﻡ ﺷﺪﻩﺍﻧﺪ؛ ﻛﻪ ﻣﺮﺯﻯ ﺭﺍ ﻧﻤﻰﺗﻮﺍﻥ ﺑﻴﻦ ﺁﻥﻫﺎ ﻣﺸﺨﺺ ﻛﺮﺩ‪.‬‬ ‫ﮔﺎﻩ ﻳــﻚ ﺗﺌﻮﺭﻯ ﻛﺎﻣــ ً‬ ‫ﻼ ﻣﺤﺾ ﺑﺎ ﻭﺭﻭﺩ ﺑﻪ ﻣﺮﺣﻠــﻪﻯ ﻛﺎﺭﺑﺮﺩﻯ ﭼﻮﻥ‬ ‫ﺩﺭ ﻋﻤﻞ ﺑﺎ ﻣﺸــﻜﻞ ﺭﻭﺑﻪ ﺭﻭ ﻣﻰﺷــﻮﺩ‪ ،‬ﺑﺎﺭ ﺩﻳﮕﺮ ﺑــﻪ ﺣﻮﺯﻩﻯ ﺗﺌﻮﺭﻯ‬ ‫ﺑﺮﻣﻰﮔــﺮﺩﺩ ﻭ ﺩﺭ ﻧﻬﺎﻳــﺖ ﭘﺲ ﺍﺯ ﺭﻓﻊ ﻧﻘﺎﻳﺺ‪ ،‬ﺩﻭﺑــﺎﺭﻩ ﻭﺍﺭﺩ ﻣﺮﺣﻠﻪﻯ‬ ‫ﻛﺎﺭﺑﺮﺩﻯ ﺧﻮﺍﻫﺪ ﺷﺪ‪ .‬ﻳﻌﻨﻰ ﻳﻚ ﺗﻌﺎﻣﻞ ﻭ ﺍﺭﺗﺒﺎﻁ ﺩﻭﺟﺎﻧﺒﻪ ﺑﻴﻦ ﺭﻳﺎﺿﻰ‬ ‫ﻛﺎﺭﺑﺮﺩﻯ ﻭ ﻣﺤﺾ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪.‬‬ ‫ﻛﺎﺭﺑﺮﺩ ﺭﻳﺎﺿﻰ ﺩﺭ ﻋﻠﻮﻡ ﻣﺨﺘﻠﻒ ﺍﻧﻜﺎﺭﻧﺎﭘﺬﻳﺮ ﺍﺳﺖ‪ .‬ﺑﺮﺍﻯ ﻣﺜﺎﻝ‪ ،‬ﺩﺭ‬ ‫ﺟﺎﻣﻌﻪﺷﻨﺎﺳﻰ ﻧﻈﺮﻳﻪﻯ ﺍﺣﺘﻤﺎﻝ ﻭ ﻧﻈﺮﻳﻪﻯ ﮔﺮﻭﻩﻫﺎ ﻧﻘﺶ ﺑﺴﻴﺎﺭ ﻣﻬﻤﻰ‬ ‫ﺍﻳﻔﺎ ﻣﻰﻛﻨﺪ‪ .‬ﺩﺭ ﻛﻞ ﺑﺎﻳﺪ ﮔﻔﺖ ﻛﻪ ﻫﻤﻪﻯ ﺻﻨﺎﻳﻊ ﺯﻳﺮﺳــﺎﺧﺖ ﺭﻳﺎﺿﻰ‬ ‫ﺩﺍﺭﻧﺪ ﻭ ﺑﻪ ﻫﻤﻴﻦ ﺩﻟﻴــﻞ ﺩﺭ ﻫﻤﻪﻯ ﻣﺮﺍﻛﺰ ﺻﻨﻌﺘﻰ ﻭ ﺗﺤﻘﻴﻘﺎﺗﻰ ﺩﻧﻴﺎ‪،‬‬ ‫ﺭﻳﺎﺿﻰﺩﺍﻥﻫﺎ ﺩﺭ ﻛﻨﺎﺭ ﻣﻬﻨﺪﺳــﺎﻥ ﻭ ﺩﺍﻧﺸﻤﻨﺪﺍﻥ ﺳﺎﻳﺮ ﻋﻠﻮﻡ ﺣﻀﻮﺭﻯ‬ ‫ﻓﻌــﺎﻝ ﺩﺍﺭﻧﺪ ﻭ ﺁﻥ ﭼﻪ ﺩﺭ ﻧﻬﺎﻳﺖ ﺍﺭﺍﺋﻪ ﻣﻰﺷــﻮﺩ ﻧﺘﻴﺠﻪ ﻛﺎﺭ ﺗﻴﻤﻰ ﺁﻥ‬ ‫ﻫﺎﺳــﺖ‪ .‬ﺍﮔﺮ ﺩﺭ ﺟﺎﻣﻌﻪﻯ ﻣﺎ ﻣﺸــﺎﻏﻞ ﺟﻨﺒﻪﻯ ﻋﻠﻤﻰ ﺩﺍﺷــﺘﻪ ﺑﺎﺷﻨﺪ‪،‬‬ ‫ﺑﻰﮔﻤﺎﻥ ﺑﻪ ﺗﻌﺪﺍﺩ ﻗﺎﺑﻞ ﺗﻮﺟﻬﻰ ﺭﻳﺎﺿﻰﺩﺍﻥ ﻧﻴﺎﺯ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷــﺖ‪ ،‬ﺯﻳﺮﺍ‬ ‫ﻳﻚ ﺭﻳﺎﺿﻰﺩﺍﻥ ﻣﻲﺗﻮﺍﻧﺪ ﻣﺸــﻜﻼﺕ ﺭﺍ ﺑﻪ ﺭﻭﺵ ﻋﻠﻤﻰ ﺣﻞ ﻛﻨﺪ‪ .‬ﺍﻟﺒﺘﻪ‬ ‫ﺍﻳﻦ ﺑﻪ ﺁﻥ ﻣﻌﻨﺎ ﻧﻴﺴــﺖ ﻛﻪ ﺩﺭ ﺣﺎﻝ ﺣﺎﺿﺮ ﻫﻴﭻ ﻓﺮﺻﺖ ﺷــﻐﻠﻰ ﺑﺮﺍﻯ‬ ‫ﻳﻚ ﺭﻳﺎﺿﻰﺩﺍﻥ ﻭﺟﻮﺩ ﻧــﺪﺍﺭﺩ‪ ،‬ﺍﻣﺎ ﺑﺎﻳﺪ ﺣﻀﻮﺭ ﺭﻳﺎﺿﻰﺩﺍﻥﻫﺎ ﺩﺭ ﻣﺮﺍﻛﺰ‬ ‫ﺗﺤﻘﻴﻘﺎﺗﻰ ﻭ ﺻﻨﻌﺘﻰ ﭘﺮﺭﻧﮓﺗﺮ ﺑﺎﺷﺪ‪ .‬ﻫﺮ ﻗﺪﺭ ﺷﻐﻞ ﻳﻚ ﻓﺮﺩ ﺗﺨﺼﺼﻰﺗﺮ‬ ‫ﺷﻮﺩ‪ ،‬ﻣﻴﺰﺍﻥ ﺭﻳﺎﺿﻴﺎﺗﻰ ﻛﻪ ﻻﺯﻡ ﺩﺍﺭﺩ‪ ،‬ﺑﻴﺶﺗﺮ ﻣﻰﺷﻮﺩ‪ .‬ﺑﺮﺍﻯ ﻣﺜﺎﻝ ﻳﻚ‬ ‫ﻣﻬﻨﺪﺱ‪ ،‬ﺍﻟﻜﺘﺮﻭﻧﻴﻚ ﺍﺯ ﺁﻧﺎﻟﻴﺰ ﺗﺎﺑﻌﻰ ﻭ ﻓﺮﺁﻳﻨﺪﻫﺎﻯ ﺗﺼﺎﺩﻓﻰ ﺍﺳــﺘﻔﺎﺩﻩ‬ ‫ﻣﻰﻛﻨﺪ ﻳﺎ ﻳﻚ ﺑﺮﻧﺎﻣﻪﺭﻳﺰ ﭘﺮﻭژﻩﻫﺎﻯ ﺍﻗﺘﺼﺎﺩﻯ ﺍﺯ ﻣﻄﺎﻟﺐ ﭘﻴﺸــﺮﻓﺘﻪﻯ‬ ‫ﺁﻣﺎﺭﻯ ﻣﺎﻧﻨﺪ ﺳــﺮﻯﻫﺎﻯ ﺯﻣﺎﻧﻰ ﺑﻪ ﻋﻨﻮﺍﻥ ﺍﺑﺰﺍﺭ ﻛﺎﺭ ﻳﺎﺭﻯ ﻣﻰﮔﻴﺮﺩ‪ .‬ﺑﻪ‬ ‫ﻫﻤﻴﻦ ﺩﻟﻴﻞ‪ ،‬ﺍﻣﺮﻭﺯﻩ ﺗﺮﺑﻴﺖ ﻣﺘﺨﺼﺼﺎﻥ ﻋﻠﻢ ﺭﻳﺎﺿﻰ‪ ،‬ﻳﻌﻨﻰ ﺍﻓﺮﺍﺩﻯ ﻛﻪ‬ ‫ﺑﺘﻮﺍﻧﻨــﺪ ﺭﻳﺎﺿﻴﺎﺕ ﻣﻮﺭﺩ ﻧﻴﺎﺯ ﺭﺍ ﺁﻣﻮﺯﺵ ﺩﻫﻨﺪ ﻳﺎ ﺗﻮﻟﻴﺪ ﻛﻨﻨﺪ‪ ،‬ﺍﻫﻤﻴﺖ‬ ‫ﺑﺴــﻴﺎﺭ ﺯﻳﺎﺩﻯ ﺩﺍﺭﺩ‪ ،‬ﺯﻳﺮﺍ ﻻﺯﻣﻪﻯ ﭘﻴﺸــﺮﻓﺖ ﺩﺭ ﺗﻜﻨﻮﻟﻮژﻯ‪ ،‬ﺗﻮﺟﻪ ﺑﻪ‬

‫ﺩﺍﻧﺶ ﺭﻳﺎﺿﻰ ﺍﺳﺖ‪ .‬ﺩﺭﺳﺖ ﺍﺳﺖ ﻛﻪ ﺩﺭ ﺟﺎﻣﻌﻪﻯ ﻣﺎ ﻣﻜﺎﻥ ﻣﺸﺨﺼﻰ‬ ‫ﺑــﺮﺍﻯ ﺟﺬﺏ ﻓﺎﺭﻍ ﺍﻟﺘﺤﺼﻴﻼﻥ ﺭﻳﺎﺿﻰ ﻭﺟــﻮﺩ ﻧﺪﺍﺭﺩ‪ ،‬ﺭﻳﺎﺿﻰ ﺑﻪ ﺩﻟﻴﻞ‬ ‫ﻧﻈﻢ ﻓﻜﺮﻯ ﻭ ﺑﻴﻨﺶ ﻋﻤﻴﻘﻰ ﻛﻪ ﺩﺭ ﺩﻭﺭﺍﻥ ﺗﺤﺼﻴﻞ ﺑﻪ ﺩﺳﺖ ﻣﻰﺁﻭﺭﺩ‪،‬‬ ‫ﻣﻰﺗﻮﺍﻧﺪ ﺑﺎ ﻣﻄﺎﻟﻌﻪ ﻭ ﺗﻼﺵ ﺷــﺨﺼﻰ ﺩﺭ ﺑﺴــﻴﺎﺭﻯ ﺍﺯ ﺷــﻐﻞﻫﺎ ﺣﺘﻰ‬ ‫ﺷﻐﻞﻫﺎﻳﻰ ﻛﻪ ﺩﺭ ﻇﺎﻫﺮ ﺍﺭﺗﺒﺎﻃﻰ ﺑﺎ ﺭﻳﺎﺿﻰ ﻧﺪﺍﺭﻧﺪ‪ ،‬ﻣﻮﻓﻖ ﻣﻰﺷﻮﺩ ‪.‬‬

‫ﺗﻮﺍﻧﺎﻳﻰﻫﺎﻯ ﻣﻮﺭﺩ ﻧﻴﺎﺯ ﻭ ﻗﺎﺑﻞ ﺗﻮﺻﻴﻪ‬ ‫ﺷــﺎﻳﺪ ﻣﻬﻢﺗﺮﻳﻦ ﺗﻮﺍﻧﺎﻳﻰ ﻋﻠﻤﻰ ﻳﻚ ﺩﺍﻧﺸﺠﻮﻯ ﺭﻳﺎﺿﻰ‪ ،‬ﺗﺴﻠﻂ ﺑﺮ‬ ‫ﺩﺭﻭﺱ ﺭﻳﺎﺿﻰ ﺭﺍﻫﻨﻤﺎﻳﻰ ﻭ ﺩﺑﻴﺮﺳﺘﺎﻥ ﺑﺎﺷﺪ ﻛﻪ ﺍﻳﻦ ﺍﻣﺮ ﺗﻨﻬﺎ ﺯﺍﻳﻴﺪﻩﻯ‬ ‫ﻋﻼﻗﻪﻯ ﺷﺨﺼﻰ ﺑﻪ ﺍﻳﻦ ﺩﺭﺱ ﺍﺳﺖ‪ .‬ﺍﻳﻦ ﺭﺷﺘﻪ ﻧﻴﺎﺯﻣﻨﺪ ﺩﺍﻧﺸﺠﻮﻳﺎﻧﻰ‬ ‫ﺍﺳﺖ ﻛﻪ ﺍﺯ ﻧﻈﺮ ﺫﻫﻨﻰ ﺁﻣﺎﺩﮔﻰ ﺟﺬﺏ ﺍﻳﺪﻩﻫﺎﻯ ﺟﺪﻳﺪ ﺭﺍ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‬ ‫ﻭ ﺑﺘﻮﺍﻧﻨﺪ ﺍﻟﮕﻮﻫﺎ ﻭ ﻧﻈﻢ ﺭﺍ ﺗﺸــﺨﻴﺺ ﺩﻫﻨﺪ ﻭ ﻣﺴــﺎﺋﻞ ﻏﻴﺮ ﻣﺘﻌﺎﺭﻑ ﺭﺍ‬ ‫ﺣــﻞ ﻛﻨﻨﺪ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺩﻳﮕﺮ‪ ،‬ﺭﻭﺣﻴﻪﻯ ﻋﻠﻤﻰ‪ ،‬ﺗﻔﻜﺮ ﺍﻧﺘﻘﺎﺩﻯ ﻭ ﺗﻮﺍﻧﺎﻳﻰ‬ ‫ﺗﺠﺰﻳﻪ ﻭ ﺗﺤﻠﻴﻞ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪.‬‬ ‫ﺍﺯ ﺁﻥﺟــﺎ ﻛﻪ ﺭﻳﺎﺿﻴﺎﺕ ﻭﺭﻭﺩ ﺑﻪ ﻋﺮﺻﻪﻫﺎﻯ ﻧﺎﺷــﻨﺎﺧﺘﻪ ﻭ ﻛﺸــﻒ‬ ‫ﻗﻮﺍﻧﻴﻦ ﺁﻥ ﺍﺳــﺖ‪ ،‬ﻋﻼﻗﻪﻣﻨــﺪﻯ ﺑﻪ ﻣﺒﺎﺣﺚ ﺭﻳﺎﺿــﻰ ﺍﺯ ﻫﻤﺎﻥ ﺩﻭﺭﺍﻥ‬ ‫ﺗﺤﺼﻴﻞ ﺩﺭ ﻣﺪﺍﺭﺱ ﻣﺸــﺨﺺ ﻣﻰﺷﻮﺩ‪ .‬ﻫﻤﻴﻦ ﻋﻼﻗﻪﻣﻨﺪﻯ ﺍﺳﺖ ﻛﻪ‬ ‫ﻣﻰﺗﻮﺍﻧﺪ ﺭﺍﻩﻫﺎﻯ ﺑﺴــﻴﺎﺭ ﺳﺨﺖ ﺭﺍ ﺑﺮﺍﻯ ﻋﻼﻗﻪﻣﻨﺪﺍﻥ ﺍﻳﻦ ﺭﺷﺘﻪ ﻫﻤﻮﺍﺭ‬ ‫ﺳــﺎﺯﺩ‪ .‬ﻳﻚ ﺭﻳﺎﺿــﻰﺩﺍﻥ ﻗﺒﻞ ﺍﺯ ﻫﺮ ﭼﻴﺰ ﺑﺎﻳﺪ ﺟــﺮﺃﺕ ﻗﺪﻡﮔﺬﺍﺭﻯ ﺩﺭ‬ ‫ﻭﺍﺩﻯ ﻧﺎﺷﻨﺎﺧﺘﻪﻫﺎ ﺭﺍ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪ .‬ﺑﻪ ﻃﻮﺭ ﻛﻠﻰ‪ ،‬ﺩﻗﺖ‪ ،‬ﺗﺠﺰﻳﻪ ﻭ ﺗﺤﻠﻴﻞ‬ ‫ﺻﺤﻴﺢ ﻭ ﺻﺒﺮ ﻭ ﭘﺸﺘﻜﺎﺭ‪ ،‬ﺳﻪ ﻋﺎﻣﻞ ﺍﺻﻠﻰ ﺩﺭ ﺗﻮﻓﻴﻖ ﺩﺍﻭﻃﻠﺐ ﺩﺭ ﺍﻳﻦ‬ ‫ﺭﺷــﺘﻪ ﺍﺳﺖ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻭﺿﻌﻴﺖ ﻧﻴﺎﺯ ﻛﺸــﻮﺭ ﺑﻪ ﺍﻳﻦ ﺭﺷﺘﻪ‪ ،‬ﺩﺭ ﺣﺎﻝ‬ ‫ﺣﺎﺿﺮ ﻫﺮ ﻭﺯﺍﺭﺗﺨﺎﻧﻪ ﻳﺎ ﺷــﺮﻛﺖ ﻫﺎ ﻣﻰﺗﻮﺍﻧﻨﺪ ﻓﺎﺭﻍ ﺍﻟﺘﺤﺼﻴﻼﻥ ﺭﻳﺎﺿﻰ‬ ‫ﻣﺤﺾ ﻳﺎ ﻛﺎﺭﺑﺮﺩﻯ ﺭﺍ ﺟﺬﺏ ﻛﻨﻨﺪ‪ .‬ﺭﺷﺘﻪﻫﺎﻯ ﻣﺨﺘﻠﻒ ﺭﻳﺎﺿﻰ ﺟﺎﻳﮕﺎﻩ‬ ‫ﻭﺳــﻴﻌﻰ ﺩﺭ ﺟﺎﻣﻌﻪ ﺩﺍﺭﻧﺪ‪ ،‬ﺍﺯ ﺁﻥ ﺟﻤﻠﻪ‪ :‬ﺗﻤﺎﻡ ﺭﺷــﺘﻪﻫﺎﻯ ﻣﻬﻨﺪﺳﻰ‪،‬‬ ‫ﺭﺷــﺘﻪﻫﺎﻯ ﻣﺨﺘﻠــﻒ ﻋﻠﻮﻡ ﭘﺎﻳﻪ )ﻓﻴﺰﻳﻚ‪ ،‬ﺷــﻴﻤﻰ‪ ،‬ﺯﻳﺴﺖﺷﻨﺎﺳــﻰ‪،‬‬ ‫ﺯﻣﻴﻦﺷﻨﺎﺳﻰ(‪ ،‬ﭘﺰﺷﻜﻰ‪ ،‬ﻋﻠﻮﻡ ﻛﺎﻣﭙﻴﻮﺗﺮ‪ ،‬ﺍﻛﺘﺸﺎﻓﺎﺕ ﻓﻀﺎﻳﻰ‪ ،‬ﺑﺎﺯﺭﮔﺎﻧﻰ‪،‬‬ ‫ﺑﺮﻧﺎﻣﻪﺭﻳﺰﻯﻫﺎﻯ ﺩﻭﻟﺘﻰ‪ .‬ﺍﻏﻠﺐ ﺭﺷﺘﻪﻫﺎﻯ ﻭﺍﺑﺴﺘﻪ ﺑﻪ ﺻﻨﻌﺖ‪ ،‬ﻣﺪﻳﺮﻳﺖ‬ ‫ﻭ ﺭﺷــﺘﻪﻫﺎﻯ ﻣﺨﺘﻠﻒ ﻛﺸﺎﻭﺭﺯﻯ ﺑﻪ ﺭﺷــﺘﻪﻯ ﺭﻳﺎﺿﻰ ﻭﺍﺑﺴﺘﻪ ﺍﻧﺪ ﻭ ﺍﺯ‬ ‫ﺁﻥ ﺑﻪ ﻃﻮﺭ ﻣﺴــﺘﻘﻴﻢ ﺍﺳــﺘﻔﺎﺩﻩ ﻣﻰﻛﻨﻨﺪ‪ .‬ﻫﻢﭼﻨﻴﻦ ﺑﺨﺶ ﺑﺰﺭﮔﻰ ﺍﺯ‬ ‫ﻓﻌﺎﻟﻴﺖﻫﺎﻯ ﺍﻗﺘﺼﺎﺩﻯ ﻭ ﺗﻮﻟﻴﺪﻯ ﻛﺸــﻮﺭ ﺩﺭ ﻃﺮﺡﻫﺎﻯ ﻣﺨﺘﻠﻒ‪ ،‬ﻣﺎﻧﻨﺪ‬ ‫ﻧﻔﺖ‪ ،‬ﭘﺘﺮﻭﺷــﻴﻤﻰ‪ ،‬ﺣﻤﻞ ﻭ ﻧﻘﻞ ﻭ ‪ ،...‬ﻣﺴــﺘﻘﻴﻢ ﻳﺎ ﻏﻴﺮ ﻣﺴــﺘﻘﻴﻢ ﺍﺯ‬ ‫ﺭﻳﺎﺿﻰ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻰﻛﻨﻨﺪ‪.‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

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‫ﻫﻨﺪﺳﻪ ﭼﻄﻮﺭ ﻭ ﭼﮕﻮﻧﻪ ﺁﻏﺎﺯ ﺷﺪ؟‬ ‫ﭼﻪ ﻛﺴــﻰ ﺑﺮﺍﻯ ﻧﺨﺴــﺘﻴﻦ ﺑﺎﺭ ﺧﻂ‪ ،‬ﺍﻧﺤﻨﺎ ﻭ ﺍﺷﻜﺎﻟﻰ ﺭﺍ ﻛﻪ ﻣﺎ ))ﺷﻜﻞﻫﺎﻯ ﻫﻨﺪﺳﻰ((ﻣﻰﻧﺎﻣﻴﻢ ﻛﺸﻒ ﻛﺮﺩ؟ ﺍﻳﻦ ﺷﻜﻞ ﺭﺍ ﺍﻧﺴﺎﻥﻫﺎﻯ ﺍﻭﻟﻴﻪﺍﻯ‬ ‫ﻛﺸــﻒ ﻛﺮﺩﻧﺪ‪ ،‬ﺯﻳﺮﺍ ﺍﻳﻦ ﺷــﻜﻞ ﺩﺭ ﺟﺎﻯ ﺟــﺎﻯ ﻃﺒﻴﻌﺖ‪ ،‬ﺍﻳﻦ ﻣﻮﺯﻩﻯ‬ ‫ﻫﻨﺮﻯ ﻋﻈﻴﻢ ﺧﺪﺍﺩﺍﺩﻯ ﻳﺎﻓﺖ ﻣﻰﺷﻮﻧﺪ‪ ،‬ﺑﻴﺎﻳﻴﺪ ﺑﻪ ﺩﻩﻫﺎ ﻫﺰﺍﺭ ﺳﺎﻝ ﻗﺒﻞ‬ ‫ﺑﺮﮔﺮﺩﻳﻢ ﻭ ﺯﻣﺎﻧﻰ ﺭﺍ ﻣﺠﺴﻢ ﻛﻨﻴﻢ ﻛﻪ ﻧﺨﺴﺘﻴﻦ ﺍﻧﺴﺎﻥﻫﺎ ﺗﻚﺗﻚ ﻳﺎ ﺑﻪ‬ ‫ﺻﻮﺭﺕ ﮔﺮﻭﻫﻰ ﺭﻭﻯ ﺯﻣﻴــﻦ ﺁﺯﺍﺩ ﺑﻮﺩﻧﺪ‪ .‬ﺗﻤﺎﻣﻰ ﺭﺍﺯﻫﺎﻯ ﺑﺰﺭگ ﻧﻬﻔﺘﻪ‬ ‫ﺑﺰﺭگ ﻭ ﻣﻨﺎﺑﻊ ﺍﻋﺠﺎﺏﺁﻭﺭ ﺳﺮ ﺑﺴﺘﻪ ﺑﻮﺩ ﻭ ﺑﻪ ﻛﺸﻒ ﻧﻴﺎﺯ ﺩﺍﺷﺖ‪.‬‬ ‫ﺍﻧﺴــﺎﻥ ﻫﺎﻯ ﺍﻭﻟﻴﻪ ﺍﺯ ﺗﺮﺱ ﺭﻋﺪ ﻭ ﺑﺮﻕ‪ ،‬ﺧﻮﺩ ﺭﺍ ﭘﻨﻬﺎﻥ ﻣﻰﻛﺮﺩﻧﺪ‪،‬‬ ‫ﺍﺯ ﻧﻴﺮﻭﻫــﺎﻯ ﭘﺮ ﺭﻣــﺰ ﻭ ﺭﺍﺯ ﺟﻬﺎﻥ ﺁﻓﺮﻳﻨﺶ ﺩﺭ ﻫﺮﺍﺱ ﺑﻮﺩﻧﺪ ﻭ ﺑﺎ ﻛﻮﺗﺎﻩ‬ ‫ﺷﺪﻥ ﺭﻭﺯﻫﺎ ﻭ ﻏﺮﻭﺏ ﺧﻮﺭﺷــﻴﺪ ﻓﻜﺮ ﻣﻰﻛﺮﺩﻧﺪ ﻛﻪ ﺭﻭﺯ ﺑﺮﺍﻯ ﻫﻤﻴﺸﻪ‬ ‫ﺍﺯ ﺑﻴــﻦ ﻣﻰﺭﻭﺩ ﻭ ﺁﻧﺎﻥ ﺩﺭ ﺗﺎﺭﻳﻜﻰ ﺳــﺮﺩ ﻭ ﻣﻄﻠــﻖ ﺗﻨﻬﺎ ﺑﺎﻗﻰ ﺧﻮﺍﻫﻨﺪ‬ ‫ﻣﺎﻧــﺪ‪ .‬ﺑــﻪ ﻫﻤﻴﻦ ﺩﻟﻴﻞ ﺩﺭ ﻛﻨــﺎﺭ ﺁﺗﺶ ﭘﺮﺍﺭﺯﺵ ﮔﺮﺩ ﻫــﻢ ﻣﻰﺁﻣﺪﻧﺪ‪.‬‬ ‫ﺁﺗﺶ ﻧﺨﺴــﺘﻴﻦ ﺭﺍﺯ ﺑﺰﺭﮔﻰ ﺑــﻮﺩ ﻛﻪ ﺍﺯ ﺩﻝ ﻃﺒﻴﻌﺖ ﺑﻴﺮﻭﻥ ﻛﺸــﻴﺪﻩ‬ ‫ﺷــﺪ‪ .‬ﺍﻧﺴﺎﻥﻫﺎﻯ ﻣﺎﻗﺒﻞ ﺗﺎﺭﻳﺦ ﺁﺗﺶ ﺭﺍ ﺑﺮﺍﻯ ﺍﻭﻟﻴﻦ ﺑﺎﺭ ﺍﺯ ﺻﺎﻋﻘﻪﺍﻯ ﻛﻪ‬ ‫ﺩﺭﺧﺘﺎﻥ ﺭﺍ ﻣﻰﺳﻮﺯﺍﻧﺪ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩﻧﺪ ﻭ ﺳﭙﺲ ﺁﻣﻮﺧﺘﻨﺪ ﻛﻪ ﭼﮕﻮﻧﻪ‬ ‫ﺁﺗﺶ ﺭﺍ ﺗﻬﻴﻪ ﻛﻨﻨﺪ‪.‬‬ ‫ﺍﻣﺎ ﺍﻳﻦ ﻫﻢ ﻧﺘﻮﺍﻧﺴــﺖ ﺗﺮﺱ ﺍﺯ ﺩﺳﺖ ﺩﺍﺩﻥ ﺧﻮﺭﺷﻴﺪ ﺭﺍ ﺍﺯ ﺩﻟﺸﺎﻥ‬ ‫ﺑﺰﺩﺍﻳﺪ‪ .‬ﺁﻥﻫﺎ ﺑﻪ ﺩﻟﻴﻞ ﺍﻳﻦ ﻛﻪ ﺗﺮﺳﺸــﺎﻥ ﺭﺍ ﺑﺎ ﻳﻜﺪﻳﮕﺮ ﺗﻘﺴﻴﻢ ﻛﻨﻨﺪ ﻭ‬ ‫ﻧﻴﺰ ﺑﺮﺍﻯ ﻛﻤﻚ ﺑﻪ ﺑﺎﺯﮔﺸﺖ ﺧﻮﺭﺷﻴﺪ ﺑﻪ ﻫﻨﮕﺎﻡ ﺍﺑﺮﻯ ﺷﺪﻥ ﻳﺎ ﺧﻮﺭﺷﻴﺪ‬ ‫ﮔﺮﻓﺘﮕــﻰ ﻳﺎ ﺩﻳﮕﺮ ﻭﻗﺎﻳﻊ ﻃﺒﻴﻌﻰ‪ ،‬ﻣﺮﺍﺳــﻢ ﻭﻳﮋﻩﺍﻯ ﺑﺮﮔﺰﺍﺭ ﻣﻰﻛﺮﺩﻧﺪ ﻭ‬ ‫ﺑــﻪ ﺧﻮﺍﻧﺪﻥ ﺩﻋﺎ ﻭ ﻗﺮﺑﺎﻧﻰ ﺣﻴﻮﺍﻧﺎﺕ ﺑﺮﺍﻯ ﺁﻓﺮﻳﻨﻨﺪﻩﻯ ﺑﺰﺭگ ﻭ ﻣﻬﺮﺑﺎﻥ‬ ‫ﻣﻰﭘﺮﺩﺍﺧﺘﻨﺪ ﻭ ﺳــﭙﺲ ﺭﻓﺘﻪﺭﻓﺘﻪ ﺑﺎ ﺑﺎﺯﮔﺸــﺖ ﮔﺮﻣﺎ ﻭ ﻧﻮﺭ‪ ،‬ﺭﻭﺣﻴﻪﻯ‬ ‫ﺧــﻮﺩ ﺭﺍ ﺑﺎﺯ ﻣﻰﻳﺎﻓﺘﻨــﺪ‪ .‬ﺁﻳﺎ ﺗﺎ ﺑﻪ ﺣﺎﻝ ﺑﻠﻮﺭ ﻛﻮﺍﺭﺗــﺰ ﺭﺍ ﺩﻳﺪﻩ ﺍﻳﺪ؟ ﺍﻳﻦ‬

‫ﺑﻠﻮﺭﻫﺎ ﻣﻨﺸﻮﺭﻯ ﺷﺶ ﻭﺟﻬﻰ ﻫﺴﺘﻨﺪ ﻛﻪ ﻳﻚ ﻫﺮﻡ ﺷﺶ ﻭﺟﻬﻰ ﺭﻭﻯ‬ ‫ﺁﻥﻫﺎ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﺳﺖ‪.‬‬ ‫ﺁﻳﺎ ﻫﺮﮔــﺰ ﺍﻭﺍﻳﻞ ﺑﻬﺎﺭ ﺩﺭ ﺟﻨﮕﻞ ﻳﺎ ﺩﺷــﺖ ﺑﻮﺩﻩﺍﻳــﺪ؟ ﺩﺭﺧﺘﺎﻥ ﻭ‬ ‫ﮔﻴﺎﻫﺎﻥ ﺑﻪ ﻳﻜﺒﺎﺭﻩ ﺷــﻜﻮﻓﻪ ﻣﻰﻛﻨﻨﺪ‪ ،‬ﺑﻌﻀﻰ ﺑﺎ ﺳﻪ ﮔﻠﺒﺮگ ﻭ ﺑﻌﻀﻰ ﺑﺎ‬ ‫ﭼﻬﺎﺭ ﮔﻠﺒﺮگ ﻭ ﺷﻜﻮﻓﻪﻫﺎﻯ ﺑﻌﻀﻰ ﭘﻨﺞ ﺿﻠﻌﻰ ﻫﺴﺘﻨﺪ ﺍﮔﺮ ﺧﻴﺎﺭﻯ ﺭﺍ‬ ‫ﺣﻠﻘﻪﺣﻠﻘﻪ ﻛﻨﻴﺪ ﺩﺍﻧﻪﻫﺎﻯ ﺁﻥ ﺭﺍ ﺩﺭ ﺳﻪ ﻗﺴﻤﺖ ﻭ ﺍﮔﺮ ﻓﻠﻔﻞ ﺳﺒﺰﻯ ﺭﺍ‬ ‫ﺍﺯ ﻫﻢ ﺑﮕﺸــﺎﻳﻴﺪ ﺩﺍﻧﻪﻫﺎﻯ ﺁﻥ ﺭﺍ ﺩﺭ ﭼﻬﺎﺭ ﻗﺴــﻤﺖ ﺧﻮﺍﻫﻴﺪ ﺩﻳﺪ‪ .‬ﺣﺎﻝ‬ ‫ﭘﻴﺎﺯﻯ ﺭﺍ ﺣﻠﻘﻪﺣﻠﻘﻪ ﻛﻨﻴﺪ‪ ،‬ﻣﺸــﺎﻫﺪﻩ ﺧﻮﺍﻫﻴﺪ ﻛﺮﺩ ﻛﻪ ﭘﻴﺎﺯ ﺑﻪ ﺻﻮﺭﺕ‬ ‫ﺩﻭﺍﻳﺮﻯ ﻣﻨﻈﻢ ﺍﺯﻫﻢ ﺟﺪﺍ ﻣﻰﺷﻮﺩ‪.‬‬ ‫ﺍﮔﺮ ﺑﻪ ﻳﻚ ﺳــﺘﺎﺭﻩﻯ ﺩﺭﻳﺎﻳﻰ ﻛﻪ ﻫﻤﺮﺍﻩ ﺑــﺎ ﺍﻣﻮﺍﺝ ﺁﺏ ﺑﻪ ﻣﻨﺎﻃﻖ‬ ‫ﻛﻢﻋﻤﻖ ﺩﺭﻳﺎ ﺁﻣﺪﻩ ﺍﺳﺖ ﺑﺮﺧﻮﺭﺩ ﻛﻨﻴﺪ‪ ،‬ﻣﺘﻮﺟﻪ ﺧﻮﺍﻫﻴﺪ ﺷﺪ ﻛﻪ ﺍﻏﻠﺐ‬ ‫ﺁﻥﻫﺎ ﭘﻨﺞ ﮔﻮﺷﻪ ﻫﺴﺘﻨﺪ‪ .‬ﻫﻤﻪ ﺟﺎﻯ ﻃﺒﻴﻌﺖ ﭘﺮ ﺍﺯ ﺗﺮﻛﻴﺐﻫﺎﻳﻰ ﺍﺳﺖ‬ ‫ﻛﻪ ﻣﺎ ﺁﻥﻫﺎ ﺭﺍ ﺍﺷــﻜﺎﻝ ﺳﺎﺩﻩ ﻫﻨﺪﺳــﻰ ﻣﻰﻧﺎﻣﻴﻢ‪ .‬ﺑﺎ ﻭﺟﻮﺩ ﺗﻔﺎﻭﺕﻫﺎﻯ‬ ‫ﻇﺎﻫﺮﻯ ﺩﺭ ﻃﺒﻴﻌﺖ ﻭ ﺩﺭ ﺗﻤﺎﻣﻰ ﻋﺎﻟﻢ ﻭ ﺩﺭ ﺟﺰﺋﻴﺎﺕ ﻳﮕﺎﻧﮕﻰ ﻭ ﻭﺣﺪﺕ‬ ‫ﻣﻮﺝ ﻣﻰﺯﻧﺪ‪.‬‬ ‫ﺑﻪ ﻳﻚ ﺩﺍﻧﻪﻯ ﺑﺮﻑ ﺑﻴﻨﺪﻳﺸﻴﺪ ﺍﻳﻦ ﮔﻞﻫﺎﻯ ﻳﺨﻰ ﺷﺶ ﺿﻠﻌﻰ ﻛﻪ‬ ‫ﺩﺭ ﺍﺭﺗﻔﺎﻉ ﺑﺎﻻ ﺑﺮ ﺍﺛﺮ ﻧﻴﺮﻭﻯ ﺑﺎﺩ ﻭ ﺳﺮﻣﺎ ﻓﺸﺮﺩﻩ ﻣﻰﺷﻮﻧﺪ ﻭ ﻳﺦ ﻣﻰﺯﻧﻨﺪ‬ ‫ﻭ ﺑﻪ ﺻﻮﺭﺕ ﺷــﺶﺿﻠﻌﻰ ﺑﺎﻗﻰ ﻣﻰﻣﺎﻧﻨﺪ‪ .‬ﻣﻄﺎﻟﻌــﻪ ﺩﺭ ﻗﺎﻧﻮﻥ ﻃﺒﻴﻌﺖ‬ ‫ﺍﺳﺖ ﻛﻪ ﺭﻳﺎﺿﻴﺎﺕ ﺭﺍ ﺟﺬﺍﺏ ﻣﻰﻛﻨﻨﺪ‪.‬‬ ‫ﺍﻧﺴﺎﻥﻫﺎﻯ ﻫﻮﻟﻨﺎﻙ ﻭ ﺣﺘﻰ ﻫﺮﺍﺱﺁﻭﺭﻯ ﺑﻪ ﻃﺒﻴﻌﺖ ﻭ ﻧﻴﺮﻭﻫﺎﻯ ﺁﻥ‬ ‫ﻧﺰﺩﻳﻚ ﺑﻮﺩﻧﺪ ﺁﻥﻫﺎ ﺍﻋﺠﺎﺯ ﻣﻮﺟﻮﺩ ﺩﺭ ﻃﺒﻴﻌﺖ ﺭﺍ ﺑﻪ ﺷــﺪﺕ ﻣﻰﺩﻳﺪﻧﺪ‬ ‫ﻭ ﻟﻤﺲ ﻣﻰﻛﺮﺩﻧﺪ‪ .‬ﺑﻪ ﻫﻤﻴﻦ ﺟﻬﺖ ﺑﻮﺩ ﻛﻪ ﺍﻧﺴــﺎﻥ ﺍﻭﻟﻴﻪ ﺍﺯ ﻧﻤﺎﻳﺸﮕﺎﻩ‬ ‫ﺁﻓﺮﻳﻨﺶ ﻫﻨﺮﻯ ﺟﻬﺎﻥ‪ ،‬ﻫﻨﺪﺳﻪ ﺁﻣﻮﺧﺖ‪.‬‬

‫ﺭﻳﺎﺿﻴﺎﺕ ﺍﻣﻜﺎﻥ‬ ‫ﺷﺎﻧﺲ ﺑﺮﻧﺪﻩ ﺷﺪﻥ ﻛﻢ ﺍﺳﺖ ﻳﺎ ﺯﻳﺎﺩ؟ ﺁﻳﺎ ﺷﻤﺎ ﻣﻰﺗﻮﺍﻧﻴﺪ ﺍﺯ ﻗﻮﺍﻧﻴﻦ‬ ‫ﺷــﺎﻧﺲ ﭘﻴﺶﺑﻴﻨﻰ ﺍﺳــﺘﻔﺎﺩﻩ ﻛﻨﻴﺪ؟ ﺁﻳﺎ ﻣﻰﺩﺍﻧﻴﺪ ﻛــﻪ ﻭﻗﺘﻰ ﺩﺭ ﻳﻚ‬

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‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫﹞︧﹮﹚﹤﹨︀ى وا﹇︺‪﹩‬‬ ‫ﺍﻋﻈﻢ ﭘﻮﺭﭘﺮﻭﻳﻦ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻣﺴﺌﻠﻪﻫﺎﻱ ﻭﺍﻗﻌﻲ‪ ،‬ﺳﺮﻋﺖ‪ ،‬ﻣﺴﺎﺣﺖ‪.‬‬ ‫ﺩﺭ ﺍﻳﻦ ﺳـﺘﻮﻥ ﻗﺼـﺪ ﺩﺍﺭﻳﻢ ﻣﺴـﺌﻠﻪﻫﺎﻱ ﻭﺍﻗﻌﻲ ﻃـﺮﺡ ﻛﻨﻴﻢ؛‬ ‫ﻣﺴـﺌﻠﻪﻫﺎﻳﻲ ﻛﻪ ﺍﻳﻦ ﻃـﺮﻑ ﻭ ﺁﻥ ﻃﺮﻑ ﺷـﻨﻴﺪﻩﺍﻳﻢ ﻭ ﺩﻳﺪﻩﺍﻳﻢ ﻛﻪ‬ ‫ﭼﮕﻮﻧﻪ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺗﻮﺍﻧﺴﺘﻪﺍﻧﺪ ﺑﺎ ﺍﻃﻼﻋﺎﺕ ﻭ ﻣﻬﺎﺭﺕﻫﺎﻱ‬ ‫ﺭﻳﺎﺿﻲ ﺧﻮﺩ‪ ،‬ﺁﻥﻫﺎ ﺭﺍ ﺣﻞ ﻛﻨﻨﺪ‪ .‬ﺷﻤﺎ ﻫﻢ ﻣﺴﺌﻠﻪﻫﺎﻱ ﺭﻳﺎﺿﻲ ﻭﺍﻗﻌﻲ‬

‫ﺗﺎﺑﻠﻮﻳﻲ ﺩﺭ ﻛﻨﺎﺭ ﺟﺎﺩﻩ ﻧﺼﺐ ﺷــﺪﻩ ﺍﺳﺖ ﻛﻪ ﻓﺎﺻﻠﻪﻱ ﻣﺎ ﺗﺎ ﻛﺎﺷﺎﻥ ﺭﺍ‬ ‫ﻧﺸــﺎﻥ ﻣﻲﺩﻫﺪ‪ .‬ﻣﻦ ﻫﻢ ﺳﺎﻋﺖ ﺩﺍﺭﻡ‪ .‬ﺳــﺮﻋﺖ ﺭﺍ ﻣﺤﺎﺳﺒﻪ ﻣﻲﻛﻨﻢ‪«.‬‬ ‫ﺑﻌﺪ ﺍﺯ ﭼﻨﺪ ﺩﻗﻴﻘﻪ ﺻﺎﻟﺤﻪ ﮔﻔﺖ ﻛﻪ ‪ 5‬ﻛﻴﻠﻮﻣﺘﺮ ﺭﺍ ﺩﺭ ‪ 2‬ﺩﻗﻴﻘﻪ ﻭ ‪40‬‬ ‫ﺛﺎﻧﻴﻪ ﻃﻲ ﻛﺮﺩﻩﺍﻳﻢ ﻭ ﺗﻮﺍﻧﺴﺖ ﺳﺮﻋﺖ ﻣﺎﺷﻴﻦ ﺭﺍ ﺣﺴﺎﺏ ﻛﻨﺪ‪.‬‬

‫ﺭﺍ ﻛـﻪ ﺑﺎ ﺁﻥﻫﺎ ﻣﻮﺍﺟﻪ ﺷـﺪﻩﺍﻳﺪ ﻳﺎ ﺩﺭ ﺍﻃﺮﺍﻑ ﺧـﻮﺩ ﺩﻳﺪﻩﺍﻳﺪ‪ ،‬ﺑﺮﺍﻱ‬ ‫ﻣﺠﻠﻪ ﺑﻔﺮﺳـﺘﻴﺪ ﺗﺎ ﺑﺎ ﻧﺎﻡ ﺧﻮﺩﺗﺎﻥ ﺩﺭ ﺍﻳﻦ ﺳـﺘﻮﻥ ﻃﺮﺡ ﺷـﻮﻧﺪ‪ .‬ﺩﺭ‬ ‫ﺿﻤﻦ‪ ،‬ﻣﻲﺗﻮﺍﻧﻴﺪ ﻣﺴـﺌﻠﻪﻫﺎﻱ ﻃﺮﺡﺷـﺪﻩ ﺩﺭ ﺍﻳﻦ ﺷﻤﺎﺭﻩ ﺭﺍ ﻫﻢ ﺣﻞ‬ ‫ﻛﻨﻴﺪ ﻭ ﺭﺍﻩﺣﻞﻫﺎﻳﺘﺎﻥ ﺭﺍ ﺑﺮﺍﻱ ﻣﺠﻠﻪ ﺑﻔﺮﺳﺘﻴﺪ‪.‬‬

‫ﻣﺴﺌﻠﻪﻱ ‪1‬‬

‫ﻧﻮﻳﺴﻨﺪﻩ‪ :‬ﺷﻴﺮﻳﻦ ﺣﺠﺎﺯﻱ‬

‫ﻣﺎﺩﺭﻡ ﻳﻚ ﺗﺎﺑﻠﻮ ﻓﺮﺵ ﺯﻳﺒﺎ ﺧﺮﻳﺪﻩ ﺑﻮﺩ ﻭ ﻣﻲﺧﻮﺍﺳﺖ ﺁﻥ ﺭﺍ ﻭﺳﻂ‬ ‫ﺩﻳــﻮﺍﺭ ﺍﻃﺎﻕ ﭘﺬﻳﺮﺍﻳﻲ ﻧﺼﺐ ﻛﻨﺪ‪ .‬ﺍﻭ ﻣﻲﺧﻮﺍﺳــﺖ ﺗﺎﺑﻠﻮ ﺩﻗﻴﻘﺎً ﻭﺳــﻂ‬ ‫ﺩﻳﻮﺍﺭ ﻗﺮﺍﺭ ﺑﮕﻴﺮﺩ ﻭ ﺍﺯ ﻣﻦ ﺧﻮﺍﺳﺖ ﺗﺎ ﻓﺎﺻﻠﻪﻱ ﺗﺎﺑﻠﻮ ﺗﺎ ﻫﺮ ﻃﺮﻑ ﺩﻳﻮﺍﺭ‬ ‫ﺭﺍ ﭘﻴــﺪﺍ ﻛﻨﻢ‪ .‬ﻣﻦ ﺍﻳﻦ ﻣﺴــﺌﻠﻪ ﺭﺍ ﺑﺎ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﻃــﻮﻝ ﺗﺎﺑﻠﻮ ﻭ ﻃﻮﻝ‬ ‫ﺩﻳﻮﺍﺭ‪ ،‬ﺣﻞ ﻛﺮﺩﻡ‪.‬‬

‫ﻣﺴﺌﻠﻪﻱ ‪3‬‬

‫ﻧﻮﻳﺴﻨﺪﻩ‪ :‬ﺁﻗﺎﻱ ﺣﺒﻴﺒﻲ‬

‫ﻣﻦ ﺩﺭ ﺭﻭﺳــﺘﺎ ﺑﻪ ﺩﻧﻴﺎ ﺁﻣﺪﻩﺍﻡ‪ .‬ﺍﻣﺎ ﺣﺎﻻ ﺳﺎﻝﻫﺎﺳــﺖ ﻛﻪ ﺩﺭ ﺷﻬﺮ‬ ‫ﺯﻧﺪﮔﻲ ﻣﻲﻛﻨﻢ‪ .‬ﻫﻤﻴﺸﻪ ﺩﻟﻢ ﻣﻲﺧﻮﺍﺳﺖ ﺑﻪ ﻳﺎﺩ ﻛﻮﺩﻛﻲﻫﺎﻳﻢ ﺩﺭ ﺧﺎﻧﻪ‬ ‫ﻣﺮﻍ ﻭ ﺧﺮﻭﺱ ﺩﺍﺷــﺘﻪ ﺑﺎﺷــﻢ‪ ،‬ﺍﻣﺎ ﺯﻧﺪﮔﻲ ﺩﺭ ﺁﭘﺎﺭﺗﻤﺎﻥ ﺍﻳﻦ ﺍﻣﻜﺎﻥ ﺭﺍ‬ ‫ﺑﻪ ﻣﻦ ﻧﻤﻲﺩﺍﺩ‪ .‬ﭼﻨﺪﻱ ﭘﻴﺶ ﺗﻮﺍﻧﺴﺘﻢ ﻳﻚ ﺧﺎﻧﻪﻱ ﻛﻮﭼﻚ ﺣﻴﺎﻁﺩﺍﺭ‬ ‫ﺑﺨﺮﻡ ﻭ ﺑﺎ ﺧﺎﻧﻮﺍﺩﻩﺍﻡ ﺩﺭ ﺍﻳﻦ ﺧﺎﻧﻪ ﺯﻧﺪﮔﻲ ﻛﻨﻢ‪ .‬ﺭﻭﺯﻱ ﺗﺼﻤﻴﻢ ﮔﺮﻓﺘﻢ‬ ‫ﻛﻪ ﺩﺭ ﻛﻨﺎﺭ ﺣﻴﺎﻁ ﺍﻳﻦ ﺧﺎﻧﻪ ﻳﻚ ﺣﺼﺎﺭ ﺑﺮﺍﻱ ﻧﮕﻪﺩﺍﺭﻱ ﻣﺮﻍ ﻭ ﺧﺮﻭﺱ‬ ‫ﺩﺭﺳــﺖ ﻛﻨﻢ‪ 10 .‬ﻣﺘﺮ ﺣﺼﺎﺭ ﺧﺮﻳﺪﻡ ﺗﺎ ﺩﺭ ﻛﻨــﺎﺭ ﻳﻜﻲ ﺍﺯ ﺩﻳﻮﺍﺭﻫﺎﻱ‬ ‫ﺣﻴﺎﻁ ﺳــﻪ ﺩﻳﻮﺍﺭ ﺑﺴــﺎﺯﻡ ﻭ ﻳﻚ ﭼﻬﺎﺭ ﺩﻳﻮﺍﺭﻱ ﻣﺴﺘﻄﻴﻞ ﺷﻜﻞ ﺑﺮﺍﻱ‬ ‫ﻣﺮﻍ ﻭ ﺧﺮﻭﺱﻫﺎ ﺁﻣﺎﺩﻩ ﻛﻨﻢ‪.‬‬

‫‪41‬‬ ‫‪4‬‬

‫‪11‬‬ ‫‪2‬‬

‫ﻣﺴﺌﻠﻪﻱ ‪2‬‬

‫ﻧﻮﻳﺴﻨﺪﻩ‪ :‬ﺁﻗﺎﻱ ﺻﺎﺩﻗﻲ‬

‫ﻫﻨﮕﺎﻡ ﺳــﻔﺮ ﺑﻪ ﻛﺎﺷﺎﻥ ﺳﺮﻋﺖ ﺳﻨﺞ ﻣﺎﺷــﻴﻨﻢ ﺧﺮﺍﺏ ﺷﺪﻩ ﺑﻮﺩ‪.‬‬ ‫ﺩﺧﺘﺮﻡ ﺻﺎﻟﺤﻪ ﻛﻪ ﺩﺍﻧﺶﺁﻣﻮﺯ ﺍﻭﻝ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺍﺳــﺖ ﺩﺭ ﺻﻨﺪﻟﻲ ﻋﻘﺐ‬ ‫ﻣﺎﺷﻴﻦ ﻧﺸﺴﺘﻪ ﺑﻮﺩ‪ .‬ﺍﺯ ﺍﻭ ﭘﺮﺳﻴﺪﻡ‪ ،‬ﺁﻳﺎ ﻣﻲﺗﻮﺍﻧﻲ ﺳﺮﻋﺖ ﺣﺮﻛﺘﻤﺎﻥ ﺭﺍ‬ ‫ﻣﺤﺎﺳﺒﻪ ﻛﻨﻲ؟ ﺍﻭ ﻧﮕﺎﻫﻲ ﺑﻪ ﻛﻨﺎﺭ ﺟﺎﺩﻩ ﻛﺮﺩ ﻭ ﮔﻔﺖ‪» :‬ﻫﺮ ‪ 5‬ﻛﻴﻠﻮﻣﺘﺮ‬

‫ﻣﺴــﺌﻠﻪ ﺍﻳﻦ ﺑﻮﺩ ﻛﻪ ﻧﻤﻲﺩﺍﻧﺴــﺘﻢ ﻃﻮﻝ ﻭ ﻋــﺮﺽ ﻻﻧﻪﻱ ﻣﺮﻍ ﻭ‬ ‫ﺧﺮﻭﺱﻫﺎ ﺭﺍ ﭼﻪﻗﺪﺭ ﺑﺎﻳﺪ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻡ ﺗﺎ ﺑﺘﻮﺍﻧﻢ ﺑﺎ ‪ 10‬ﻣﺘﺮ ﺣﺼﺎﺭﻱ‬ ‫ﻛﻪ ﺩﺍﺷﺘﻢ ﺑﻴﺶﺗﺮﻳﻦ ﻣﺴﺎﺣﺖ ﻣﻤﻜﻦ ﺭﺍ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﻡ‪ .‬ﭘﺴﺮﻡ ﺣﺎﻣﺪ‬ ‫ﻛﻪ ﺩﺍﻧﺶﺁﻣﻮﺯ ﺩﻭﻡ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺍﺳﺖ‪ ،‬ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺭﺍ ﺣﻞ ﻛﺮﺩ‪.‬‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪25‬‬

‫ﻣﺴﺌﻠﻪﻱ ‪4‬‬ ‫ﺑــﺮﺍﻱ‬ ‫ﺧﻮﺍﺳــﺖ ﺑ ﺮ ﻱ‬ ‫ﺖ‬ ‫ﻣﺎﻫــﺎﻥ ﻣﻲ‬ ‫ﻼﻩ ﺍﻳﻤﻨﻲ‬ ‫ﺩﻭﭼﺮﺧﻪﺍﺵ‪ ،‬ﻗﻔﻞ‪ ،‬ﺯﻧﮓ ﻭ ﻛﻼﻩ‬ ‫ﻗﻴﻤﺖﻫﺎﻳﺸﺎﻥ‬ ‫ﺑﺨﺮﺩ‪ .‬ﺑﺴــﺘﻪﻫﺎﻱ ﺯﻳﺮ ﺑﺎ ﻤﺖ‬ ‫ﺷــﺪ‪ .‬ﻣﺎﻫﺎﻥ‬ ‫ﺩﺭ ﻓﺮﻭﺷــﮕﺎﻩ ﺩﻳﺪﻩ ﻣﻲ ﺪ‪.‬‬ ‫ﻚ ﻗﻔﻞ‪،‬‬ ‫ﻣﻲﺧﻮﺍﺳــﺖ ﺑﺪﺍﻧﺪ ﺑﺮﺍﻱ ﻳﻚ‬ ‫ﺍﻳﻤﻨﻲ ﺭﻭﻱ‬ ‫ﻳﻚ ﺯﻧﮓ ﻭ ﻳــﻚ ﻛﻼﻩ ﻲ‬ ‫ﻫﻢ ﭼﻪﻗﺪﺭ ﺑﺎﻳــﺪ ﭘﺮﺩﺍﺧﺖ ﻛﻨﺪ‪ .‬ﺍﻭ‬ ‫ﺭﺍﻫﻨﻤﺎﻳﻲ ﺩﺭﺱ‬ ‫ﻲ‬ ‫ﻛﻪ ﺩﺭ ﻛﻼﺱ ﺳــﻮﻡ‬ ‫ﺌﻠﻪ ﺭﺍ ﺣﻞ‬ ‫ﻣﻲﺧﻮﺍﻧﺪ ﺗﻮﺍﻧﺴــﺖ ﺍﻳﻦ ﻣﺴﺌﻠﻪ‬ ‫ﻛﻨﺪ‪.‬‬ ‫‪ 4400‬ﺗﻮﻣﺎﻥ‬ ‫‪ 5300‬ﺗﻮﻣﺎﻥ‬ ‫‪ 3300‬ﺗﻮﻣﺎﻥ‬

‫ﻣﺴﺌﻠﻪﻱ ‪5‬‬ ‫ﭘــﺲ ﺍﺯ ﻳــﻚ ﻣﻬﻤﺎﻧﻲ ﺍﺯ ﻳﻚ ﺟﻌﺒﻪ ﺷــﻜﻼﺕ ‪ 72‬ﺗﺎﻳﻲ ﻓﻘﻂ ‪25‬‬ ‫ﺷﻜﻼﺕ ﺯﻳﺮ ﺑﺎﻗﻲﻣﺎﻧﺪﻩ ﺑﻮﺩ‪.‬‬ ‫ﻣــﺎﺩﺭ ﺗﻴﻨﺎ ﺭﻭﻱ ﺟﻌﺒﻪ ﺷــﻜﻼﺕ ﺭﺍ ﺧﻮﺍﻧﺪ؛ ﺭﻭﻱ ﺁﻥ ﻧﻮﺷــﺘﻪ ﺑﻮﺩ‬ ‫»ﺗﻌﺪﺍﺩ ﺷﻜﻼﺕ ﻗﻬﻮﻩﺍﻱ ﺩﺭ ﺍﻳﻦ ﺑﺴﺘﻪ ﺩﻭ ﺑﺮﺍﺑﺮ ﺷﻜﻼﺕﻫﺎﻱ ﺷﻴﺸﻪﺍﻱ‬ ‫ﺍﺳــﺖ‪ «.‬ﺍﻭ ﭘﺮﺳﻴﺪ‪» :‬ﺗﻴﻨﺎ‬ ‫ﻓﻜﺮ ﻣﻲﻛﻨــﻲ ﻣﻬﻤﺎﻥﻫﺎ‬ ‫ﺑﻴﺶﺗﺮ ﺷــﻜﻼﺕ ﺷﻴﺮﻱ‬ ‫ﺩﻭﺳــﺖ ﺩﺍﺷــﺘﻪﺍﻧﺪ ﻳــﺎ‬ ‫ﻗﻬــﻮﻩﺍﻱ؟« ﺗﻴﻨﺎ ﺑﺎ ﺗﻮﺟﻪ‬ ‫ﺑــﻪ ﻫﻤــﻪﻱ ﺍﻃﻼﻋــﺎﺕ‪،‬‬ ‫ﺳــﺆﺍﻝ ﻣــﺎﺩﺭ ﺭﺍ ﭘﺎﺳــﺦ‬ ‫ﺩﺍﺩ‪ .‬ﺷــﻤﺎ ﻫﻢ ﻣﻲﺗﻮﺍﻧﻴﺪ‬ ‫ﺳﺆﺍﻝ ﻣﺎﺩﺭ ﺗﻴﻨﺎ ﺭﺍ ﭘﺎﺳﺦ‬ ‫ﺩﻫﻴﺪ؟‬

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‫‪1390‬‬ ‫‪1390‬‬ ‫ﺗﺎﺑﺴﺘﺎﻥ‬ ‫ﺗﺎﺑﺴﺘﺎﻥ‬ ‫ﺷﻤﺎﺭﺓ ‪،4‬‬ ‫ﺷﻤﺎﺭﺓ ‪،4‬‬ ‫ﺷﺎﻧﺰﺩﻫﻢ‪،‬‬ ‫ﺷﺎﻧﺰﺩﻫﻢ‪،‬‬ ‫ﺩﻭﺭﺓ ﺩﻭﺭﺓ‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬ ‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﻣﺴﺌﻠﻪﻱ ‪6‬‬ ‫ﭘﺪﺭ ﺳــﻴﻨﺎ ﺍﺯ ﻣﺸــﺘﺮﻳﺎﻥ ﺩﺍﺋﻤﻲ ﻳﻚ ﻛﻔﺶﻓﺮﻭﺷــﻲ ﺍﺳــﺖ‪ .‬ﺑﻪ‬ ‫ﻫﻤﻴﻦ ﺟﻬﺖ‪ ،‬ﺍﻳﻦ ﻛﻔﺶﻓﺮﻭﺷــﻲ ﺩﺭ ﺭﻭﺯ ﺗﻮﻟﺪ ﭘﺪﺭ ﺳــﻴﻨﺎ ﻳﻚ ﻛﺎﺭﺕ‬ ‫ﺗﺨﻔﻴﻒ ‪ 40‬ﺩﺭﺻﺪﻱ ﺑﺮﺍﻱ ﺍﻭ ﻓﺮﺳﺘﺎﺩ‪ .‬ﭘﺪﺭ ﻫﻤﺮﺍﻩ ﺳﻴﻨﺎ ﺑﺮﺍﻱ ﺧﺮﻳﺪ ﺑﻪ‬ ‫ﻛﻔﺶﻓﺮﻭﺷﻲ ﺭﻓﺘﻨﺪ ﻭ ﻣﺘﻮﺟﻪ ﺷﺪﻧﺪ ﻛﻪ ﻫﻢﺯﻣﺎﻥ ﺩﺭ ﺍﻳﻦ ﻛﻔﺶﻓﺮﻭﺷﻲ‬ ‫ﻫﻤﻪﻱ ﻛﻔﺶﻫﺎ ﺑﺎ ‪ ٪15‬ﺗﺨﻔﻴﻒ ﺑﺮﺍﻱ ﻫﻤﻪ ﻓﺮﻭﺧﺘﻪ ﻣﻲﺷــﻮﺩ‪ .‬ﭘﺪﺭ ﺍﺯ‬ ‫ﺳﻴﻨﺎ ﭘﺮﺳﻴﺪ ﻓﻜﺮ ﻣﻲﻛﻨﻲ ﺑﻬﺘﺮ ﺍﺳﺖ ﺍﻭﻝ ‪ ٪ 15‬ﺧﺮﻳﺪﻣﺎﻥ ﺭﺍ ﺑﮕﻴﺮﻳﻢ‬ ‫ﻭ ﺑﻌــﺪ ﻛﺎﺭﺕ ﺗﺨﻔﻴــﻒ ‪ 40‬ﺩﺭﺻﺪﻱ ﺭﺍ ﻧﺸــﺎﻥ ﺑﺪﻫﻴﻢ ﻭ ﺍﺯ ﻗﻴﻤﺘﻲ ﻛﻪ‬ ‫ﺑﺎﻳﺪ ﭘﺮﺩﺍﺧﺖ ﻛﻨﻴﻢ‪ 40 ،‬ﺩﺭﺻﺪ ﻛﻢ ﻛﻨﻴﻢ ﻳﺎ ﺍﻳﻦﻛﻪ ﺍﻭﻝ ‪ ٪40‬ﺗﺨﻔﻴﻒ‬ ‫ﺑﮕﻴﺮﻳﻢ ﻭ ﺑﻌﺪ ‪ 15‬ﺩﺭﺻﺪ؟ ﺳــﻴﻨﺎ ﻛﻤﻲ ﻓﻜﺮ ﻛﺮﺩ ﻭ ﭘﺎﺳﺦ ﺳﺆﺍﻝ ﭘﺪﺭ ﺭﺍ‬ ‫ﺩﺍﺩ‪ .‬ﺁﻳﺎ ﺷﻤﺎ ﻫﻢ ﻣﻲﺗﻮﺍﻧﻴﺪ ﺳﺆﺍﻝ ﭘﺪﺭ ﺳﻴﻨﺎ ﺭﺍ ﭘﺎﺳﺦ ﺩﻫﻴﺪ؟‬

‫ﻫﻤﺮﺍﻩ ﺑﺎ ﻛﺘﺎﺏ‬

‫﹡﹍︀﹨‪﹢︧﹆﹞ ﹤︋ ﹢﹡ ﹩‬م ︻﹚﹫﹤‬ ‫ﻣﺠﻴﺪ ﻣﻨﺸﻮﺭﻱ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻣﻘﺴﻮﻡﻋﻠﻴﻪ‪ ،‬ﻣﻘﺴﻮﻡﻋﻠﻴﻪﻫﺎﻱ ﺩﻳﮕﺮ‪ ،‬ﻋﺪﺩ ‪ ،1‬ﺍﻋﺪﺍﺩ ﺍﻭﻝ‪ ،‬ﻧﻤﻮﺩﺍﺭ ﺩﺭﺧﺘﻲ‪.‬‬

‫ﻛﻞ ﻣﺎﺟﺮﺍﻳﻲ ﻛﻪ ﻣﻲﺧﻮﺍﻫﻢ ﺑﺮﺍﻳﺘﺎﻥ ﺗﻌﺮﻳﻒ ﻛﻨﻢ ﻣﺮﺑﻮﻁ ﻣﻲﺷــﻮﺩ‬ ‫ﺑﻪ ﭘﺮﺳﺸﻲ ﻛﻪ ﺩﺭ ﺯﻧﮓ ﺣﺴﺎﺏ‪ ،‬ﻣﻌﻠﻢ ﺭﻭﻱ ﺗﺎﺑﻠﻮ ﻧﻮﺷﺖ‪:‬‬ ‫»ﺍﮔﺮ ‪ 14‬ﻭ ‪ 15‬ﺩﻭ ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪ ﻳﻚ ﻋﺪﺩ ﺑﺎﺷﻨﺪ‪ ،‬ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ‬ ‫ﺩﻳﮕﺮ ﺁﻥ ﻋﺪﺩ ﺭﺍ ﺑﻨﻮﻳﺴﻴﺪ‪«.‬‬ ‫ً‬ ‫ﺑﻌﺪ ﺍﺯ ﻛﻤﻲ ﻫﻤﻬﻤﻪ ﺗﻘﺮﻳﺒﺎ ﺗﻤﺎﻡ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﻣﺸــﻐﻮﻝ ﻧﻮﺷــﺘﻦ‬ ‫ﺷــﺪﻧﺪ‪ .‬ﻣﻌﻠﻢ ﺣﺴــﺎﺏ‪ ،‬ﺑﭽﻪﻫــﺎﻱ ﻛﻼﺱ ﺭﺍ ﺑﻪ ﮔﺮﻭﻩﻫﺎﻱ ﺳــﻪ ﻧﻔﺮﻱ‬ ‫ﺩﺳﺘﻪﺑﻨﺪﻱ ﻛﺮﺩﻩ ﺑﻮﺩ ﻭ ﻣﻦ ﻭ ﺳﻌﻴﺪ ﻭ ﻣﺤﻤﺪ ﻫﻢﮔﺮﻭﻩ ﺑﻮﺩﻳﻢ‪.‬‬ ‫ﻣﺤﻤــﺪ ﺳــﺮﺵ ﺭﺍ ﺍﺯ ﺭﻭﻱ ﻛﺎﻏﺬ ﺑﻠﻨــﺪ ﻛﺮﺩ ﻭ ﮔﻔــﺖ‪» :‬ﺍﻭﻝ ﺑﺎﻳﺪ‬ ‫‪ 14‬ﻭ ‪ 15‬ﺭﺍ ﺩﺭ ﻫــﻢ ﺿــﺮﺏ ﻛﻨﻴﻢ ﻛﻪ ﻣﻲﺷــﻮﺩ ‪ 210‬ﻋﺪﺩ ﻣﻮﺭﺩ ﻧﻈﺮ‬ ‫ﺑﺎﺷﻪ؟ ‪ 420‬ﻫﻢ ﻣﻲﺗﻮﻧﻪ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﺑﺎﺷﻪ‪ ،‬ﭼﻮﻥ ‪ 14‬ﻭ ‪ 15‬ﻣﻘﺴﻮﻡ‬ ‫ﻋﻠﻴﻪﻫﺎﻱ ‪ 420‬ﻫﻢ ﻫﺴــﺘﻨﺪ‪ .‬ﻳﺎ ‪ 630‬ﻳﺎ ‪ 840‬ﻳﺎ ‪ « ....‬ﻗﺮﺍﺭ ﺷﺪ ﺩﻭﺑﺎﺭﻩ‬ ‫ﺍﺯ ﺍﻭﻝ ﺷــﺮﻭﻉ ﻛﻨﻴﻢ‪ .‬ﻣﻌﻠــﻢ ﺑﻪ ﻣﺎ ﻳﺎﺩ ﺩﺍﺩﻩ ﺑﻮﺩ ﻛﻪ ﻫﺮﺟﺎ ﺑﻪ ﻣﺸــﻜﻞ‬ ‫ﺑﺮﺧﻮﺭﺩﻳﻢ ﻳﻚ ﻗﺪﻡ ﺑﻪ ﻋﻘﺐ ﺑﺮﮔﺮﺩﻳﻢ ﻭ ﺍﺯ ﻧﻮ ﺷــﺮﻭﻉ ﻛﻨﻴﻢ )ﺑﺎﺯﮔﺸﺖ‬ ‫ﺑﻪ ﻋﻘﺐ ﻭ ﺷﺮﻭﻉ ﺩﻭﺑﺎﺭﻩ!(‬ ‫ﻣﻦ ﮔﻔﺘﻢ‪» :‬ﻣﮕﻪ ‪ 1‬ﻣﻘﺴــﻮﻡ ﻋﻠﻴﻪ ﻫﻤﻪﻱ ﺍﻋﺪﺍﺩ ﻧﻴﺴﺖ؛ ﺧﻮﺏ ‪1‬‬

‫ﺭﺍ ﺑﻨﻮﻳﺴﻴﻢ‪«.‬‬ ‫ﺳــﻌﻴﺪ ﮔﻔﺖ‪ ،15=3×5 ، 14=2×7» :‬ﭘﺲ ﻣﻌﻠﻮﻡ ﻣﻴﺸﻪ ‪،3 ،2‬‬ ‫‪ 5‬ﻭ ‪ 7‬ﻫﻢ ﻣﻘﺴﻮﻡﻋﻠﻴﻪﻫﺎﻱ ﺁﻥ ﻋﺪﺩ ﻫﺴﺘﻨﺪ‪«.‬‬ ‫ﻣﺤﻤﺪ ﺑﺎ ﻧﺸﺎﻥ ﺩﺍﺩﻥ ﻧﻤﻮﺩﺍﺭ ﺩﺭﺧﺘﻲ ﺍﻋﺪﺍﺩ ‪ 14‬ﻭ ‪ 15‬ﮔﻔﺖ‪» :‬ﻣﻦ‬ ‫ﻫﻢ ﺑﻪ ﺍﻳﻦ ﻧﺘﻴﺠﻪ ﺭﺳﻴﺪﻡ ﻛﻪ ‪ 5 ،3 ،2‬ﻭ ‪ 7‬ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ﺁﻥ ﻋﺪﺩ‬ ‫ﻫﺴﺘﻨﺪ‪«.‬‬ ‫‪15‬‬ ‫‪14‬‬ ‫‪3‬‬

‫‪5‬‬

‫‪2‬‬

‫‪7‬‬

‫ﺗﺎ ﺍﻳﻦﺟﺎ ﻫﻤﻪﭼﻴﺰ ﺩﺍﺷﺖ ﺧﻮﺏ ﭘﻴﺶ ﻣﻲﺭﻓﺖ‪ .‬ﻣﺎ ﺗﻮﺍﻧﺴﺘﻪ ﺑﻮﺩﻳﻢ‬ ‫ﭼﻬــﺎﺭ ﺗﺎ ﻋﺪﺩ ﺍﻭﻝ ﭘﻴﺪﺍ ﻛﻨﻴﻢ ﻛﻪ ﺩﺭ ﺳــﺎﺧﺘﻦ ﻋــﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﺑﻪ ﻛﺎﺭ‬ ‫ﺭﻓﺘﻪ ﺑﻮﺩﻧﺪ‪.‬‬ ‫ﻣــﻦ ﻳﺎﺩﻡ ﺁﻣﺪ ﻛﻪ ﻣﻌﻠﻢ ﻳﻚ ﺭﻭﺯ ﺩﺭ ﻣﻮﺭﺩ ﻛﺎﺭﺧﺎﻧﻪﻱ ﻋﺪﺩﺳــﺎﺯﻱ‬ ‫ﺑﺮﺍﻱ ﻣﺎ ﺻﺤﺒﺖ ﻛﺮﺩﻩ ﺑﻮﺩ‪.‬‬ ‫ﺑﺮﺍﻱ ﻣﺜﺎﻝ‪ ،‬ﺍﮔﺮ ﻣﺎﺩﻩﻱ ﺍﻭﻟﻴﻪﻱ ﻛﺎﺭﺧﺎﻧﻪﻱ ﻋﺪﺩﺳﺎﺯﻱ‪ ،‬ﻋﺪﺩ ﺍﻭﻝ ‪2‬‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪27‬‬

‫ﺑﺎﺷﺪ ﻣﺤﺼﻮﻻﺕ ﺍﻳﻦ ﻛﺎﺭﺧﺎﻧﻪ ﺑﻪ ﺷﻜﻞ ﺯﻳﺮ ﺗﻮﻟﻴﺪ ﻣﻲﺷﻮﻧﺪ‪:‬‬ ‫‪2‬‬ ‫‪2×2‬‬ ‫‪2×2×2‬‬ ‫‪2×2×2×2‬‬ ‫‪...‬‬

‫‪2 ، 4 ، 8 ،16 ، ...‬‬

‫ﻭ ﺍﮔــﺮ ﻣــﺎﺩﻩﻱ ﺍﻭﻟﻴﻪﻱ ﻛﺎﺭﺧﺎﻧﻪﻱ ﻋﺪﺩﺳــﺎﺯﻱ‪ ،‬ﺍﻋﺪﺍﺩ ﺍﻭﻝ ‪ 2‬ﻭ ‪3‬‬ ‫ﺑﺎﺷﻨﺪ‪ ،‬ﻣﺤﺼﻮﻻﺕ ﻛﺎﺭﺧﺎﻧﻪ ﺑﻪ ﺷﻜﻞ ﺯﻳﺮ ﺗﻮﻟﻴﺪ ﻣﻲﺷﻮﻧﺪ‪:‬‬

‫‪2×3‬‬ ‫‪2×3×3‬‬ ‫‪2×3×3×3‬‬ ‫‪2×2×3‬‬ ‫‪2×2×3×3‬‬ ‫‪2×2×3×3×3‬‬ ‫‪2×3×2×3 2×2×2×3×3 2×2×2×3×3×3‬‬ ‫‪...‬‬

‫‪...‬‬

‫‪...‬‬

‫‪...‬‬ ‫‪...‬‬ ‫‪...‬‬

‫‪6 ،12 ،24 ، . . . ، 18 ،36 ، 72، . . .‬‬

‫ﺑﻌﺪ ﺍﺯ ﺍﻳﻦ ﻳﺎﺩﺁﻭﺭﻱ ﺑﺮﻣﻲﮔﺮﺩﻳﻢ ﺑﻪ ﺣﻞ ﻣﺴﺌﻠﻪﻱ ﺧﻮﺩﻣﺎﻥ‪.‬‬ ‫ﻣﺎ ﺑﺎ ﻳﻚ ﻛﺎﺭﺧﺎﻧﻪﺍﻱ ﺳﺮ ﻭ ﻛﺎﺭ ﺩﺍﺭﻳﻢ ﻛﻪ ﻣﻮﺍﺩ ﺍﻭﻟﻴﻪﻱ ﺁﻥ ﺍﻋﺪﺍﺩ ﺍﻭﻝ‬ ‫‪ 5 ،3 ،2‬ﻭ ‪ 7‬ﻫﺴﺘﻨﺪ ﻭ ﻣﺤﺼﻮﻻﺕ ﺁﻥ ﺑﻪ ﺷﻜﻞ ﺯﻳﺮ ﺗﻮﻟﻴﺪ ﻣﻲﺷﻮﻧﺪ‪:‬‬

‫‪...‬‬

‫‪...‬‬

‫‪7‬‬

‫‪28‬‬

‫‪5‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪2‬‬

‫‪9‬‬ ‫‪3‬‬

‫ﺩﺭ ﺍﺩﺍﻣــﻪﻱ ﺻﺤﺒﺖﻫــﺎﻱ ﻣــﻦ ﻣﺤﻤﺪ ﮔﻔﺖ ﻛــﻪ ﺩﺭ ﺗﻮﻟﻴﺪ ﻋﺪﺩ‬ ‫ﻣﻮﺭﺩﻧﻈــﺮ ﺍﺯ ﻣﻘﺴــﻮﻡ ﻋﻠﻴﻪﻫــﺎﻱ ﺍﻭﻝ ‪ 5 ،3 ،2‬ﻭ ‪ 7‬ﺣﺪﺍﻗــﻞ ﻳﻚ ﺑﺎﺭ‬ ‫ﺍﺳﺘﻔﺎﺩﻩ ﺷﺪﻩ ﺍﺳﺖ ﻭ ﺑﺎ ﻧﺸﺎﻥ ﺩﺍﺩﻥ ﺷﻜﻞ ﺯﻳﺮ ﺑﺮﺍﻱ ﻣﺎ ﺗﻮﺿﻴﺢ ﺩﺍﺩ ﻛﻪ‬ ‫ﭼﮕﻮﻧﻪ ﺗﻮﺍﻧﺴﺘﻪ ﺍﺳﺖ ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ﺩﻳﮕ ِﺮ ﻋﺪﺩ ﺭﺍ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩ‪:‬‬

‫‪3‬‬

‫‪27‬‬ ‫‪3‬‬

‫‪...‬‬

‫‪....‬‬ ‫‪....‬‬ ‫‪....‬‬ ‫‪....‬‬

‫‪2×2×2×3×5×7‬‬ ‫‪2×3×3×3×5×7‬‬ ‫‪2×3×5×5×5×7‬‬ ‫‪2×3×5×7×7×7‬‬

‫‪2×2×3×5×7‬‬ ‫‪2×3×3×5×7‬‬ ‫‪2×3×5×5×7‬‬ ‫‪2×3×5×7×7‬‬

‫‪2×3×5×7‬‬

‫ﺳــﻌﻴﺪ ﻫﻢ ﻛﻪ ﺑﻴﻜﺎﺭ ﻧﺸﺴﺘﻪ ﺑﻮﺩ‪ ،‬ﻧﺸﺎﻥ ﺩﺍﺩ ﻛﻪ ﺑﻪ ﻛﻤﻚ ﺟﺪﻭﻝ‬ ‫ﻧﻈﺎﻡﺩﺍﺭ‪ ،‬ﻫﻤﻴﻦ ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎ ﺭﺍ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﺩﻩ ﺍﺳﺖ‪:‬‬ ‫‪2‬‬ ‫‪2×3‬‬ ‫‪2×3×5 2×3×5×7‬‬ ‫‪3‬‬ ‫‪2×5‬‬ ‫‪2×3×7‬‬ ‫‪5‬‬ ‫‪2×7‬‬ ‫‪2×5×7‬‬ ‫‪7‬‬ ‫‪3×5‬‬ ‫‪3×5×7‬‬ ‫‪3×7‬‬ ‫‪5×7‬‬ ‫ﻭ ﺍﻟﺒﺘﻪ ﻋﺪﺩ ‪ 1‬ﻛﻪ ﺩﺭ ﺷﺮﻭﻉ ﺣﻞ ﻣﺴﺌﻠﻪ ﺁﻥﺭﺍ ﻧﻮﺷﺘﻪ ﺑﻮﺩﻳﻢ‪.‬‬ ‫ﺍﻳﻦﺟﺎ ﺑﻮﺩ ﻛﻪ ﻣﻦ ﺑﺎ ﺑﻠﻨﺪ ﻛﺮﺩﻥ ﺩﺳــﺖ ﺍﺯ ﻣﻌﻠﻢ ﺧﻮﺍﺳﺘﻢ ﺩﺭﺳﺘﻲ‬ ‫ﺟﻮﺍﺏ ﻣﺎ ﺭﺍ ﺑﺮﺭﺳﻲ ﻛﻨﺪ‪.‬‬ ‫ﺁﻗﺎﻱ ﻣﻌﻠﻢ ﺑﻌﺪﺍﺯ ﺍﻳﻦﻛﻪ ﻣﺎ ﺭﺍ ﺗﺸــﻮﻳﻖ ﻛﺮﺩ‪ ،‬ﮔﻔﺖ‪ :‬ﺣﺎﻻ ﺑﻪ ﻋﻨﻮﺍﻥ‬ ‫ﺟﺎﻳﺰﻩ ﺑﻪ ﺍﻳﻦ ﺳﺆﺍﻝ ﭘﺎﺳﺦ ﺩﻫﻴﺪ‪:‬‬ ‫»ﺍﮔﺮ ‪ 6‬ﻭ ‪ 9‬ﺩﻭ ﻣﻘﺴــﻮﻡ ﻋﻠﻴﻪ ﻳﻚ ﻋﺪﺩ ﺑﺎﺷﻨﺪ‪ ،‬ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ‬ ‫ﺩﻳﮕﺮ ﺁﻥ ﺭﺍ ﺑﻨﻮﻳﺴﻴﺪ‪«.‬‬ ‫ﻣﺎ ﺩﻭﺑﺎﺭﻩ ﻣﺸﻐﻮﻝ ﻧﻮﺷﺘﻦ ﺟﻮﺍﺏ ﺷﺪﻳﻢ‪.‬‬ ‫ﺑﻌــﺪ ﺍﺯ ﻣﺪﺕ ﻛﻮﺗﺎﻫﻲ ﻣﺤﻤﺪ ﺟﻮﺍﺑﺶ ﺭﺍ ﺑﺎ ﻣﺎ ﻧﺸــﺎﻥ ﺩﺍﺩ‪ .‬ﺍﻭ ﺍﻭﻝ‬ ‫‪ 6‬ﻭ ‪ 9‬ﺭﺍ ﺩﺭ ﻫــﻢ ﺿﺮﺏ ﻛﺮﺩ‪ (6×9=54) .‬ﻭ ﺳــﭙﺲ ﺑﻪ ﻛﻤﻚ ﻧﻤﻮﺩﺍﺭ‬ ‫ﺩﺭﺧﺘﻲ‪ ،‬ﻣﻘﺴــﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ﺍﻭﻝ ﻋﺪﺩ ‪ 54‬ﺭﺍ ﺑﻪ ﺩﺳــﺖ ﺁﻭﺭﺩﻩ ﺑﻮﺩ ﻛﻪ‬ ‫ﻋﺒﺎﺭﺕ ﺑﻮﺩﻧﺪ ﺍﺯ ‪ 2‬ﻭ ‪.3‬‬ ‫‪54‬‬

‫‪2‬‬

‫‪3‬‬

‫ﺑﻌﺪ ﺑﻪ ﻛﻤﻚ ﻛﺎﺭﺧﺎﻧﻪﻱ ﻋﺪﺩﺳــﺎﺯﻱ ﻭ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺍﻳﻦﻛﻪ ﺍﺯ ﻋﺪﺩ‬ ‫‪ 2‬ﺑﻴﺶ ﺍﺯ ﻳﻚ ﺑﺎﺭ ﻭ ﺍﺯ ﻋﺪﺩ ‪ 3‬ﺑﻴﺶ ﺍﺯ ﺳﻪ ﺑﺎﺭ ﺍﺳﺘﻔﺎﺩﻩ ﻧﺸﻮﺩ‪ ،‬ﻣﻘﺴﻮﻡ‬ ‫ﻋﻠﻴﻪﻫﺎﻱ ﺩﻳﮕﺮ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﺭﺍ ﺑﻪ ﺷﻜﻞ ﺯﻳﺮ ﺗﻮﻟﻴﺪ ﻛﺮﺩﻩ ﺑﻮﺩ‪:‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2×3‬‬ ‫‪3×3‬‬ ‫‪2×3×3‬‬ ‫‪3×3×3‬‬ ‫‪2×3×3×3‬‬ ‫ﻭ ﺍﻟﺒﺘﻪ ﻋﺪﺩ ﻳﻚ ﺭﺍ ﻫﻢ ﻓﺮﺍﻣﻮﺵ ﻧﻜﺮﺩﻩ ﺑﻮﺩ!‬ ‫ﻣﻦ ﻫﻨﮕﺎﻡ ﺩﻳﺪﻥ ﺟﻮﺍﺏ ﻣﺤﻤﺪ ﺑﻪ ﻳﺎﺩ ﺣﺮﻑ ﺳــﻌﻴﺪ ﺩﺭ ﻣﻮﺭﺩ ‪14‬‬ ‫ﻭ ‪ 15‬ﺩﺭ ﻣﺴــﺌﻠﻪﻱ ﻗﺒﻞ ﺍﻓﺘﺎﺩﻡ ﻭ ﺑﻪ ﻣﺤﻤــﺪ ﮔﻔﺘﻢ‪» :‬ﻣﺤﻤﺪ ﺟﺎﻥ‪ ،‬ﺍﺯ‬ ‫ﻛﺠﺎ ﻣﻌﻠﻮﻡ ﻛﻪ ‪ 54‬ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﺑﺎﺷــﺪ؟ ﺷــﺎﻳﺪ ﻋــﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ‪18‬‬

‫ﺑﺎﺷــﺪ‪ ،‬ﭼﻮﻥ ‪ 6‬ﻭ ‪ 9‬ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ‪ 18‬ﻫﺴﺘﻨﺪ ﻳﺎ ﺷﺎﻳﺪ ‪ 36‬ﻳﺎ ‪54‬‬ ‫ﻳﺎ ‪ 72‬ﻳﺎ ‪« ....‬‬ ‫ﺳــﻌﻴﺪ ﺑﺎ ﺗﻜﺎﻥ ﺩﺍﺩﻥ ﺳــﺮﺵ ﺩﺭ ﺗﺄﻳﻴﺪ ﺣﺮﻑ ﻣﻦ‪ ،‬ﻧﻮﺷﺘﻪﻫﺎﻳﺶ‬ ‫ﺭﺍ ﺑﻪ ﻣﺎ ﻧﺸــﺎﻥ ﺩﺍﺩ ﻭ ﮔﻔﺖ ﻛﻪ ﭼﺮﺍ ‪ 54‬ﻣﻤﻜﻦ ﺍﺳــﺖ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ‬ ‫ﻧﺒﺎﺷﺪ‪.‬‬ ‫‪6‬‬ ‫‪9‬‬ ‫‪3‬‬

‫‪3‬‬

‫‪3‬‬

‫‪2‬‬

‫ﺍﻭ ﺍﺑﺘﺪﺍ ﻧﻤﻮﺩﺍﺭ ﺩﺭﺧﺘﻲ ‪ 6‬ﻭ ‪ 9‬ﺭﺍ ﺭﺳﻢ ﻛﺮﺩﻩ ﻭ ﻣﻌﻠﻮﻡ ﺷﺪﻩ ﺑﻮﺩ ﻛﻪ‬ ‫‪ 3‬ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪ ﻣﺸﺘﺮﻙ ‪ 6‬ﻭ ‪ 9‬ﺍﺳﺖ ﻭ ﺑﻪ ﻫﻤﻴﻦ ﺩﻟﻴﻞ ﺣﺎﺻﻞ ﺿﺮﺏ‬ ‫‪ 6‬ﻭ ‪ 9‬ﻳﻌﻨﻲ ‪ 54‬ﻣﻤﻜﻦ ﺍﺳﺖ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﻧﺒﺎﺷﺪ‪.‬‬ ‫ﺑﻌﺪ ﺍﺯ ﺍﻳﻦ ﺗﻮﺿﻴﺢ ﻛﻮﺗﺎﻩ ﺳــﻌﻴﺪ ﻣﻦ ﺍﻳﻦﻃﻮﺭ ﺍﺩﺍﻣﻪ ﺩﺍﺩﻡ ﻛﻪ ﺍﻭﻝ‬ ‫ﻧﻤﻮﺩﺍﺭ ﺩﺭﺧﺘﻲ ‪ 6‬ﻭ ‪ 9‬ﺭﺍ ﺭﺳــﻢ ﻛﺮﺩﻩﺍﻡ ﻭ ﺑﻌﺪ ﺍﺯ ﺭﻭﻱ ﻧﻤﻮﺩﺍﺭ ﻣﺘﻮﺟﻪ‬ ‫ﺷــﺪﻡ ﻛﻪ ‪ 3‬ﺩﺭ ﺍﻳﻦ ﺩﻭ ﺩﺭﺧﺖ ﻣﻴﻮﻩﻱ ﻣﺸﺘﺮﻙ ﺍﺳﺖ‪ ،‬ﭘﺲ ﺩﺭ ﺗﻮﻟﻴﺪ‬ ‫ﻋــﺪﺩ ﻣﻮﺭﺩﻧﻈــﺮ ﺍﺯ ﺑﻴﺶ ﺍﺯ ﻳــﻚ ‪ 2‬ﻭ ﺩﻭ ﺗﺎ ‪ 3‬ﺍﺳــﺘﻔﺎﺩﻩ ﻧﻤﻲﻛﻨﻢ ﻭ‬ ‫ﺑﻪ ﻛﻤﻚ ﻛﺎﺭﺧﺎﻧﻪﻱ ﻋﺪﺩﺳــﺎﺯﻱ ﻭ ﺟــﺪﻭﻝ ﻧﻈﺎﻡﺩﺍﺭ ﺑﻪ ﺟﻮﺍﺏ‪ ،‬ﻳﻌﻨﻲ‬ ‫ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ﺩﻳﮕﺮ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ‪ ،‬ﺭﺳﻴﺪﻩﺍﻡ‪.‬‬ ‫‪9‬‬ ‫‪3‬‬

‫‪6‬‬ ‫‪3‬‬

‫ﻋﻠﻴﻪ ﻣﺸــﺘﺮﻙ ‪ 6‬ﻭ ‪ 9‬ﺑﺎ ﺭﺳﻢ ﻧﻤﻮﺩﺍﺭ ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪ ﺑﻪ ﺟﻮﺍﺏ ﺭﺳﻴﺪﻩ‬ ‫ﺑﻮﺩ!‬ ‫ﻣﺎ ﻫﻤﻴﺸــﻪ ﺩﺭ ﺭﺳﻢ ﻧﻤﻮﺩﺍﺭ ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪ ﻳﻚ ﻋﺪﺩ ﺍﺯ ﺧﻮﺩ ﻋﺪﺩ‬ ‫‪3‬‬

‫‪6‬‬

‫‪12‬‬

‫‪1‬‬

‫‪2‬‬

‫‪4‬‬

‫‪÷2‬‬ ‫‪÷3‬‬

‫ﺷــﺮﻭﻉ ﻣﻲﻛﻨﻴﻢ ﺗﺎ ﺑﻪ ﻳﻚ ﺑﺮﺳﻴﻢ‪ .‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ ﺩﺭ ﻣﻮﺭﺩ ﻋﺪﺩ ‪ 12‬ﺑﻌﺪ‬ ‫ﺍﺯ ﺍﻳﻦﻛﻪ ﻓﻬﻤﻴﺪﻳﻢ ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ﺍﻭﻝ ‪ ،12‬ﺍﻋﺪﺍﺩ ‪ 2‬ﻭ ‪ 3‬ﻫﺴﺘﻨﺪ‪،‬‬ ‫ﻧﻤﻮﺩﺍﺭ ﺭﺍ ﺍﺯ ﮔﻮﺷﻪﻱ ﺑﺎﻻ ﺳﻤﺖ ﭼﭗ ﻭ ﺍﺯ ﺧﻮﺩ ﻋﺪﺩ ﻳﻌﻨﻲ ‪ 12‬ﺷﺮﻭﻉ‬ ‫ﻭ ﺑﻪ ﮔﻮﺷﻪﻱ ﭘﺎﻳﻴﻦ ﺳﻤﺖ ﺭﺍﺳﺖ ﻣﻲﺭﻭﻳﻢ ﺗﺎ ﺑﻪ ﻋﺪﺩ ‪ 1‬ﺑﺮﺳﻴﻢ‪.‬‬ ‫ﻭ ﻣﻲﺩﺍﻧﻴﻢ ﺍﻋﺪﺍﺩ ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺩﺭ ﻧﻤﻮﺩﺍﺭ‪ ،‬ﻳﻌﻨﻲ ‪ 6 ،3 ،4 ،2 ،1‬ﻭ‬ ‫‪ 12‬ﻣﻘﺴﻮﻡ ﻋﻠﻴﻪﻫﺎﻱ ﻋﺪﺩ ‪ 12‬ﻫﺴﺘﻨﺪ‪.‬‬ ‫ﺍﻣﺎ ﺩﺭ ﺣﻞ ﺍﻳﻦ ﻣﺴﺌﻠﻪ‪ ،‬ﺳﻌﻴﺪ ﺍﺯ ﮔﻮﺷﻪﻱ ﭘﺎﻳﻴﻦ ﺳﻤﺖ ﭼﭗ ﻭ ﺍﺯ‬ ‫ﻋﺪﺩ ‪ 1‬ﺷﺮﻭﻉ ﻛﺮﺩﻩ ﻭ ﺑﻪ ﺳﻤﺖ ﺑﺎﻻ ﻭ ﺑﻪ ﺳﻤﺖ ﭼﭗ ﺭﻓﺘﻪ ﺑﻮﺩ‪.‬‬ ‫ﺑﺎ ﻧﮕﺎﻩ ﻛﺮﺩﻥ ﺑﻪ ﻧﻤﻮﺩﺍﺭ ﻣﻌﻠﻮﻡ ﻣﻲﺷﺪ ﻛﻪ ﻫﺮﻳﻚ ﺍﺯ ﺍﻋﺪﺍﺩ ﺑﻌﺪ ﺍﺯ‬ ‫‪ 18‬ﭼﻪ ﺍﺯ ﺳــﻤﺖ ﭼﭗ ﻭ ﭼﻪ ﺭﻭﺑﻪ ﺑﺎﻻ ﻣﻲﺗﻮﺍﻧﺪ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﺑﺎﺷﺪ‪،‬‬ ‫ﻭﻟﻲ ‪ 18‬ﻛﻮﭼﻚﺗﺮﻳﻦ ﻭ ﻟﺬﺍ ﻣﻄﻤﺌﻦﺗﺮﻳﻦ ﺟﻮﺍﺏ ﺑﺮﺍﻱ ﺳﺆﺍﻝ ﺍﺳﺖ‪.‬‬

‫‪3‬‬

‫‪2‬‬ ‫‪3‬‬

‫‪2‬‬

‫‪2×3‬‬ ‫‪2×3×3‬‬ ‫‪3×3 1‬‬ ‫ﺳﻌﻴﺪ ﻭ ﻣﺤﻤﺪ‪ ،‬ﻫﺮ ﺩﻭ ﺣﺮﻑ ﻣﺮﺍ ﺗﺄﻳﻴﺪ ﻛﺮﺩﻧﺪ‪.‬‬ ‫ﺳــﻌﻴﺪ ﺑﺎ ﻟﺒﺨﻨﺪ ﻣﻌﻨﺎﺩﺍﺭﻱ ﻛﻪ ﻧﺸــﺎﻥ ﻣــﻲﺩﺍﺩ ﺭﺍﻩ ﺟﺪﻳﺪﻱ ﺑﻪ‬ ‫ﺫﻫﻨﺶ ﺭﺳــﻴﺪﻩ ﺍﺳــﺖ‪ ،‬ﺑﻌﺪ ﺍﺯ ﻣﺪﺕ ﻛﻮﺗﺎﻫﻲ ﻧﻮﺷﺘﻪﻱ ﺧﻮﺩ ﺭﺍ ﺑﻪ ﻣﺎ‬ ‫ﻧﺸﺎﻥ ﺩﺍﺩ‪.‬‬ ‫ﺍﻭ ﭘﺲ ﺍﺯ ﺭﺳﻢ ﻧﻤﻮﺩﺍﺭ ﺩﺭﺧﺘﻲ ﻭ ﻭﺟﻮﺩ ﻋﺪﺩ ‪ 3‬ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﻘﺴﻮﻡ‬

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‫‪54‬‬

‫‪108‬‬

‫‪....‬‬

‫‪....‬‬

‫‪9‬‬

‫‪18‬‬

‫‪36‬‬

‫‪72‬‬

‫‪....‬‬

‫‪3‬‬

‫‪6‬‬

‫‪12‬‬

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‫‪....‬‬

‫‪1‬‬

‫‪2‬‬

‫‪4‬‬

‫‪8‬‬

‫‪....‬‬

‫ﻭ ﺍﺯ ﺭﻭﻱ ﻧﻤﻮﺩﺍﺭ‪ ،‬ﻣﻘﺴﻮﻡﻋﻠﻴﻪﻫﺎﻱ ‪ 18‬ﻳﺎ ﻫﻤﺎﻥ ﻋﺪﺩ ﻣﻮﺭﺩﻧﻈﺮ ﺭﺍ‬ ‫ﻣﻲﺗﻮﺍﻥ ﻧﻮﺷﺖ‪ ،‬ﻳﻌﻨﻲ‪:‬‬ ‫‪ 18‬ﻭ ‪ 9‬ﻭ ‪ 6‬ﻭ ‪ 3‬ﻭ ‪ 2‬ﻭ ‪1‬‬ ‫ﺍﻳﻦﺑﺎﺭ ﻣﻦ ﺩﻭﺑﺎﺭﻩ ﺩﺳﺘﻢ ﺭﺍ ﺑﻠﻨﺪ ﻛﺮﺩﻡ ﻭ ﺑﻪ ﺁﻗﺎﻱ ﻣﻌﻠﻢ ﮔﻔﺘﻢ ﻛﻪ‬ ‫ﻣﺎ ﺑﻪ ﺟﻮﺍﺏ ﺭﺳﻴﺪﻩﺍﻳﻢ‪.‬‬ ‫ﻟﺒﺨﻨﺪ ﺭﺿﺎﻳﺖ ﻣﻌﻠﻢ ﻧﺸــﺎﻥ ﻣﻲﺩﺍﺩ ﻛﻪ ﻫﺮﺩﻭ ﺭﺍﻩ ﻣﺎ ﺩﺭﺳﺖ ﺍﺳﺖ‬ ‫ﻭ ﺑﺮﻕ ﺷﺎﺩﻱ ﺩﺭ ﭼﺸﻤﺎﻥ ﻣﺎ ﺩﻳﺪﻩ ﻣﻲﺷﺪ‪.‬‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

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‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫واژه﹨︀ی ر﹬︀︲‪﹩‬‬ ‫»︨︀ده ﹋︣دن ︻︊︀رت«‪︀︧︑» ،‬وی =«‬ ‫ﺳﭙﻴﺪﻩ ﭼﻤﻦﺁﺭﺍ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻭﺍژﻩﻫﺎﻱ ﺭﻳﺎﺿﻲ‪ ،‬ﺳﺎﺩﻩ ﻛﺮﺩﻥ ﻋﺒﺎﺭﺕ‪ ،‬ﺗﺴﺎﻭﻱ = ‪.‬‬ ‫ﺩﺭ ﺍﻳــﻦ ﺑﺨﺶ ﺍﺯ ﻭﺍژﻩﻫﺎﻱ ﺭﻳﺎﺿﻲ ﺑﻪ ﺩﻭ ﻣﻮﺿﻮﻉ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪ :‬ﻳﻜﻲ‬ ‫ﺍﺻﻄﻼﺡ »ﺳــﺎﺩﻩ ﻛﺮﺩﻥ ﻋﺒــﺎﺭﺕ« ﻭ ﺩﻳﮕﺮﻱ ﻣﻌﻨﺎﻫﺎﻱ ﻣﺨﺘﻠﻒ‬ ‫ﻣﻌﻨﺎﻱ‬ ‫ِ‬ ‫ﻋﻼﻣﺖ‪.« = » :‬‬ ‫ﻫﻤﻪﻱ ﻣﺎ ﺍﻭﻟﻴﻦ ﺑﺎﺭ ﺍﺻﻄﻼﺡ »ﺳــﺎﺩﻩ ﻛﺮﺩﻥ« ﺭﺍ ﺩﺭ ﺩﺑﺴــﺘﺎﻥ ﻭ ﺩﺭ‬ ‫ﻛﺎﺭ ﺑﺎ ﻛﺴﺮﻫﺎﻱ ﻣﺴﺎﻭﻱ ﺷﻨﻴﺪﻩﺍﻳﻢ‪:‬‬ ‫»ﻛﺴﺮﻫﺎﻱ ﺯﻳﺮ ﺭﺍ ﺗﺎ ﺣﺪ ﺍﻣﻜﺎﻥ ﺳﺎﺩﻩ ﻛﻨﻴﺪ‪:‬‬ ‫‪4‬‬ ‫‪8‬‬

‫‪,‬‬

‫‪3‬‬ ‫‪15‬‬

‫‪,‬‬

‫‪21‬‬ ‫‪56‬‬

‫‪.... ,‬‬

‫ﺗﻮﺟــﻪ ﻛﻨﻴﻢ ﻛﻪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻣﺤﻞ » =«‪ ،‬ﺧﻂ ﻛﺴــﺮﻱ ﻣﻘﺎﺑﻞ ﻋﻼﻣﺖ‬ ‫=‪ ،‬ﺧﻂ ﻛﺴــﺮﻱ ﺍﺻﻠﻲ ﺍﺳــﺖ؛ ﻭﻟــﻲ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﺁﻥ‪ ،‬ﺧﻮﺩﺷــﺎﻥ‬ ‫ﻋﺒﺎﺭﺕﻫﺎﻱ ﺭﻳﺎﺿﻲ ﻫﺴــﺘﻨﺪ ﻭ ﻳﻚ ﻋﺪﺩ ﻃﺒﻴﻌﻲ ﻧﻴﺴــﺘﻨﺪ ﻛﻪ »ﻛﺴﺮ‬ ‫ﺭﺍ ﺳــﺎﺩﻩ ﻛﻨﻴﻢ«‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎﻳﺪ ﻧﺨﺴــﺖ ﻋﺒﺎﺭﺕِ )‪ (1‬ﺭﺍ ﻣﺤﺎﺳﺒﻪ ﻛﻨﻴﻢ‬ ‫ﻭ ﺣﺎﺻــﻞ ﺁﻥ ﺭﺍ ﻛــﻪ ﺍﺣﺘﻤﺎﻻً ﻳﻚ ﻋﺪﺩ ﻛﺴــﺮﻱ ﺍﺳــﺖ‪ ،‬ﺑﻴﺎﺑﻴﻢ ﻭ ﺁﻥ‬ ‫ﻛﺴــﺮ ﺭﺍ ﺗﺎ ﺣﺪ ﺍﻣﻜﺎﻥ ﺳـﺎﺩﻩ ﻛﻨﻴﻢ‪ .‬ﭘﺲ ﺩﺭ ﺍﻳﻦﺟﺎ‪ ،‬ﺳـﺎﺩﻩ ﻛﺮﺩﻥ‪،‬‬ ‫ﺷــﺎﻣﻞ ﻣﺤﺎﺳﺒﺎﺕ ﻭ ﺳﺎﺩﻩ ﻛﺮﺩﻥ ﻛﺴﺮ ﺣﺎﺻﻞ ﻣﻲﺷﻮﺩ ﻛﻪ ﻗﺪﺭﻱ‬ ‫ﺑﺎ ﻣﻌﻨﺎﻱ ﻗﺒﻠﻲ ﺗﻔﺎﻭﺕ ﺩﺍﺭﺩ‪ .‬ﺑﮕﺬﺍﺭﻳﺪ ﻋﻤﻠﻴﺎﺕ ﺯﻳﺮ ﺭﺍ ﺍﻧﺠﺎﻡ ﺩﻫﻴﻢ‪:‬‬

‫ﻭ ﻫﻤﻪ ﺑﻪ ﺧﻮﺑﻲ ﻣﻲﺩﺍﻧﻴﺪ ﻛﻪ »ﺳــﺎﺩﻩ ﻛــﺮﺩﻥ« ﺩﺭ ﺍﻳﻦ ﺟﺎ‪ ،‬ﻳﻌﻨﻲ‬ ‫ﻳﺎﻓﺘﻦ ﻛﺴـﺮﻱ ﻣﺴـﺎﻭﻱ ﺑﺎ ﻛﺴـﺮ ﻣﻮﺭﺩﻧﻈﺮ ﻛﻪ ﺑﺰﺭگﺗﺮﻳﻦ ﻣﻘﺴﻮﻡ‬ ‫ﻋﻠﻴﻪ ﻣﺸﺘﺮﻙ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﺁﻥ‪ ،‬ﻋﺪﺩ ‪ 1‬ﺑﺎﺷﺪ‪ .‬ﻳﻌﻨﻲ ﺻﻮﺭﺕ ﻣﺨﺮﺝ‬ ‫ﺁﻥ ﻛﺴــﺮ ﻫﻢﺯﻣﺎﻥ ﺑﻪ ﻫﻴﭻ ﻋﺪﺩﻱ ﺟﺰ ‪ 1‬ﺗﻘﺴﻴﻢﭘﺬﻳﺮ ﻧﺒﺎﺷﻨﺪ‪ .‬ﺣﺘﻲ ﺩﺭ‬ ‫ﺗﺎ ﺍﻳﻦﺟﺎ ﺑﺎ ﺍﻧﺠﺎﻡ ﻣﺤﺎﺳــﺒﺎﺕ ﻭ ﭼﻬﺎﺭ ﻋﻤﻞ ﺍﺻﻠﻲ‪ ،‬ﺣﺎﺻﻞ ﻋﺒﺎﺭﺕ‬ ‫ﻛﺘﺎﺏ ﺭﻳﺎﺿﻲ ﭘﺎﻳﻪﻱ ﺍﻭﻝ ﺭﺍﻫﻨﻤﺎﻳﻲ‪ ،‬ﺗﻤﺮﻳﻦﻫﺎﻳﻲ ﺑﻪ ﺍﻳﻦ ﺻﻮﺭﺕ ﺩﺍﺭﻳﻢ‬ ‫‪28‬‬ ‫ﺍﺳﺖ‪ .‬ﺣﺎﻝ ﺍﻳﻦ ﻛﺴﺮ ﺭﺍ )ﺑﻪ ﻣﻌﻨﺎﻱ ﺍﻭﻝ(‬ ‫ﺭﺍ ﭘﻴﺪﺍ ﻛﺮﺩﻳﻢ ﻛﻪ ﻛﺴ ِﺮ‬ ‫ﻛﻪ ﺑﻪ ﻃﻮﺭ ﻣﺴﺘﻘﻴﻢ ﺑﻪ ﺍﻳﻦ ﻣﻮﺿﻮﻉ ﺍﺷﺎﺭﻩ ﻣﻲﻛﻨﺪ‪:‬‬ ‫‪30‬‬ ‫ﻛﺴــﺮﻫﺎﻱ ﺯﻳــﺮ ﺭﺍ ﺑﺎ ﺍﺳــﺘﻔﺎﺩﻩ ﺍﺯ ﺏ‪.‬ﻡ‪ِ .‬ﻡ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ‪ ،‬ﺳــﺎﺩﻩ‬ ‫‪28 14‬‬ ‫ﺳﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ‪:‬‬ ‫=‬ ‫‪30 15‬‬ ‫ﻛﻨﻴﺪ‪:‬‬ ‫‪6 3 4 7‬‬ ‫‪1 2‬‬ ‫‪1− +‬‬ ‫‪− +‬‬ ‫‪2 3 = 6 6 6 = 6 = 7 ÷ 5 = 7 × 4 = 28‬‬ ‫‪1‬‬ ‫‪5‬‬ ‫‪3‬‬ ‫‪6 4 6 5 30‬‬ ‫‪1‬‬ ‫‪2−‬‬ ‫‪4‬‬ ‫‪4‬‬ ‫‪4‬‬

‫‪51‬‬

‫‪121‬‬

‫‪14‬‬

‫‪.... ,‬‬ ‫‪,‬‬ ‫ﺍﺳﺖ‪.‬‬ ‫ﭘﺲ ﭘﺎﺳﺦ ﺳﺆﺍﻝ ﻓﻮﻕ‪،‬‬ ‫‪99‬‬ ‫‪68‬‬ ‫‪15‬‬ ‫ﺍ ّﻣــﺎ ﻫﻨﻮﺯ ﻣﺎﺟﺮﺍ ﺗﻤﺎﻡ ﻧﺸــﺪﻩ ﺍﺳــﺖ‪ .‬ﺯﻣﺎﻧﻲ ﻛــﻪ ﺩﺭ ﭘﺎﻳﻪﻱ ﺩﻭﻡ‬ ‫ﺍ ّﻣﺎ ﺳـﺎﺩﻩ ﻛﺮﺩﻥ ﻛﺴـﺮ ﻭ ﻳﺎﻓﺘﻦ ﻛﺴﺮﻫﺎﻱ ﻣﺴــﺎﻭﻱ‪ ،‬ﺗﻨﻬﺎ ﺟﺎﻳﻲ‬ ‫ﺭﺍﻫﻨﻤﺎﻳﻲ ﺑﺎ ﻋﺒﺎﺭﺕﻫﺎﻱ ﺟﺒﺮﻱ ﺁﺷــﻨﺎ ﻣﻲﺷﻮﻳﻢ ﺍﺯ ﻣﺎ ﻣﻲﺧﻮﺍﻫﻨﺪ ﻛﻪ‬ ‫ﻧﻴﺴﺖ ﻛﻪ ﺑﻪ ﻣﺎ ﮔﻔﺘﻪ ﻣﻲﺷﻮﺩ‪» :‬ﺳﺎﺩﻩ ﻛﻨﻴﺪ« !‬ ‫ﻋﺒﺎﺭﺕﻫﺎﻳﻲ ﻣﺎﻧﻨﺪ ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﺭﺍ ﺳﺎﺩﻩ ﻛﻨﻴﻢ‪:‬‬ ‫ﺑﻪ ﻣﺜﺎﻝ ﺯﻳﺮ ﺗﻮﺟﻪ ﻛﻨﻴﺪ‪:‬‬ ‫= ‪) x + 2y − 1 + 3 x − y‬ﺍﻟﻒ‬ ‫»ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﺭﺍ ﺗﺎ ﺣﺪ ﺍﻣﻜﺎﻥ ﺳﺎﺩﻩ ﻛﻨﻴﺪ‪:‬‬ ‫= ‪) 3aa − 4b + 2a + b − 7‬ﺏ‬ ‫‪1 2‬‬ ‫‪1− +‬‬ ‫ﺍ ّﻣﺎ ﺩﻳﮕﺮ ﻧﻪ ﻛﺴﺮﻱ ﺍﺳﺖ‪ ،‬ﻧﻪ ﺧﻂ ﻛﺴﺮﻱ! ﺩﺭ ﭼﻨﻴﻦ ﻋﺒﺎﺭﺕﻫﺎﻳﻲ‪،‬‬ ‫= ‪2 3‬‬ ‫)‪(1‬‬ ‫‪3‬‬ ‫ﺗﻔﺮﻳﻖ ﺟﻤﻠﻪﻫﺎﻱ ﻣﺸﺎﺑﻪ ﺑﺎ ﻳﻜﺪﻳﮕﺮ‬ ‫ﻣﻨﻈﻮﺭ ﺍﺯ ﺳـﺎﺩﻩ ﻛﺮﺩﻥ‪ ،‬ﺟﻤﻊ ﻳﺎ‬ ‫ِ‬ ‫‪2−‬‬ ‫‪4‬‬ ‫ﺍﺳﺖ‪ .‬ﺣﺎﻝ ﺑﻪ ﻣﺜﺎﻝ ﺯﻳﺮ ﺗﻮﺟﻪ ﻛﻨﻴﺪ‪:‬‬ ‫ﺑﻪ ﻇﺎﻫﺮ ﺧﻂﻫﺎﻱ ﻛﺴــﺮﻱ ﺯﻳﺎﺩﻱ ﺩﺭ ﺍﻳﻦ ﻋﺒﺎﺭﺕ ﻫﺴﺖ‪ ،‬ﻭﻟﻲ ﺑﺎﻳﺪ‬ ‫ﻋﺒﺎﺭﺕﻫﺎﻱ ﺯﻳﺮ ﺭﺍ ﺳﺎﺩﻩ ﻛﻨﻴﺪ‪:‬‬ ‫‪30‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪1 3‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪x + 3y − 5 + − y + x‬‬ ‫‪2 2‬‬ ‫‪3‬‬ ‫‪2‬‬

‫ﺍ ّﻣﺎ ﺑﺎﺯ ﺑﻪ ﺳــﺮﺍﻍ ﻋﺒﺎﺭﺕﻫﺎﻱ ﺟﺒﺮﻱ ﻣﻲﺭﻭﻳﻢ ﻭ ﺍﺯ ﻫﻤﺎﻥ ﻣﺜﺎﻝﻫﺎﻱ‬ ‫ﺑﺨﺶ ﻗﺒﻞ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﻛﻨﻴﻢ‪:‬‬

‫ﺩﺭ ﺍﻳﻦ ﻋﺒﺎﺭﺕ‪ ،‬ﻛﺴــﺮﻫﺎﻳﻲ ﻭﺟــﻮﺩ ﺩﺍﺭﺩ )ﻣﺜﻞ ‪ ( 2 ، 3 ، 1‬ﺍ ّﻣﺎ‬ ‫‪2‬‬

‫‪2‬‬

‫‪3‬‬

‫ﺍﻳﻦﺟﺎ ﻣﻨﻈﻮﺭ ﺍﺯ ﺳــﺎﺩﻩ ﻛﺮﺩﻥ‪ ،‬ﺳــﺎﺩﻩ ﻛﺮﺩﻥ ﻛﺴــﺮ ﻧﻴﺴﺖ‪ ،‬ﺯﻳﺮﺍ ﻳﻚ‬ ‫ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﺍﺳﺖ )‪ (1‬ﻭ ﻟﺬﺍ ﺑﺎﻳﺪ ﺟﻤﻼﺕ ﻣﺸﺎﺑﻪ ﺭﺍ ﺗﺸﺨﻴﺺ ﺩﻫﻴﻢ‬ ‫ﻭ ﺁﻥﻫﺎ ﺭﺍ ﺑﺎ ﻫﻢ ﺟﻤﻊ ﻳﺎ ﺗﻔﺮﻳﻖ ﻛﻨﻴﻢ‪.‬‬ ‫‪1 3‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪12‬‬ ‫‪9‬‬ ‫‪9‬‬ ‫‪x + 3y − 5 + − y + x = x + y −‬‬ ‫‪2 4‬‬ ‫‪15‬‬ ‫‪3‬‬ ‫‪15‬‬ ‫‪4‬‬ ‫‪2‬‬

‫ﺧﻮﺏ‪ ،‬ﺑﻪ ﻧﻈﺮ ﻣﻲﺭﺳــﺪ ﻛﻪ ﻛﺎﺭ ﺗﻤﺎﻡ ﺷﺪﻩ ﺍﺳــﺖ‪ ،‬ﺍ ّﻣﺎ ﺍﮔﺮ ﺧﻮﺏ‬ ‫ﺩﻗﺖ ﻛﻨﻴﺪ‪ ،‬ﺿﺮﻳﺐ ‪ x‬ﻳﻌﻨﻲ ‪ 12‬ﻛﺴﺮﻱ ﺍﺳﺖ ﻛﻪ ﺳﺎﺩﻩ ﻣﻲﺷﻮﺩ‪ :‬ﭘﺲ‬ ‫‪15‬‬ ‫ﺣﺎﺻﻞ ﻋﺒﺎﺭﺕ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﻧﻴﺰ ﻣﻲﺗﻮﺍﻥ ﻧﻮﺷﺖ‪:‬‬ ‫‪4‬‬ ‫‪9‬‬ ‫‪9‬‬ ‫‪x+ y−‬‬ ‫‪5‬‬ ‫‪4‬‬ ‫‪2‬‬

‫ﻟــﺬﺍ ﺩﺭ ﺍﻳﻦ ﻣﺜــﺎﻝ‪ ،‬ﻋﻼﻭﻩ ﺑﺮ ﺳــﺎﺩﻩ ﻛــﺮﺩﻥ ﻋﺒﺎﺭﺕﻫﺎﻱ ﺟﺒﺮﻱ‬ ‫)ﺑﻪ ﻣﻌﻨﺎﻱ ﺟﻤﻊ ﻳﺎ ﺗﻔﺮﻳﻖ ﺟﻤﻼﺕ ﻣﺸــﺎﺑﻪ(‪ ،‬ﺳــﺎﺩﻩ ﻛﺮﺩﻥ ﻛﺴﺮ ﻧﻴﺰ‬ ‫ﺩﺍﺷﺘﻴﻢ!‬ ‫ﺣﺎﻝ ﺷــﻤﺎ ﺑﮕﻮﻳﻴﺪ ﻛﻪ ﺩﺭ ﺳــﺆﺍﻝ ﺯﻳﺮ‪ ،‬ﻣﻨﻈﻮﺭ ﺍﺯ »ﺳــﺎﺩﻩ ﻛﻨﻴﺪ«‬ ‫ﭼﻴﺴﺖ؟‬ ‫ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﺭﺍ ﺗﺎ ﺣﺪ ﺍﻣﻜﺎﻥ ﺳﺎﺩﻩ ﻛﻨﻴﺪ‪:‬‬ ‫‪21× ( −14) × 55 × 4‬‬ ‫‪( −8) × ( −49) × 11‬‬

‫ﭘﺎﺳــﺦ‪ :‬ﺑﻪ ﻣﻌﻨﺎﻱ ﺳﺎﺩﻩ ﻛﺮﺩﻥ ﻛﺴﺮ ﺍﺳﺖ‪ ،‬ﺯﻳﺮﺍ ﻋﺒﺎﺭﺕ ﺩﺍﺩﻩ ﺷﺪﻩ‬ ‫ﺩﺭ ﻭﺍﻗﻊ ﻳﻚ ﻛﺴــﺮ ﺍﺳــﺖ ﻛﻪ ﺍﻋﺪﺍﺩ ﺻﻮﺭﺕ ﻭ ﻣﺨﺮﺝ ﺑﻪ ﺣﺎﺻﻞ ﺿﺮﺏ‬ ‫ﺍﻋﺪﺍﺩﻱ ﺩﻳﮕﺮ ﺗﺠﺰﻳﻪ ﺷــﺪﻩﺍﻧﺪ‪ .‬ﺍﮔﺮ ﺑﺨﻮﺍﻫﻴﻢ ﺩﻗﻴﻖﺗﺮ ﺑﮕﻮﻳﻴﻢ‪ ،‬ﺩﺭ ﻭﺍﻗﻊ‬ ‫ﻣﻌﻨﺎﻱ ﻣﺤﺎﺳﺒﻪ ﻭ ﺳﭙﺲ ﺳﺎﺩﻩ ﻛﺮﺩﻥ ﻛﺴﺮ ﺣﺎﺻﻞ ﺍﺳﺖ‪.‬‬

‫ﻋﻼﻣﺖ » =«‬

‫ﺣﺘﻤﺎً ﻫﻤﻪﻱ ﺷــﻤﺎ ﻋﻼﻣﺖ » =« ﺭﺍ ﻣﻲﺷﻨﺎﺳــﻴﺪ‪ .‬ﺍﻳﻦ ﻋﻼﻣﺖ ﺭﺍ‬ ‫ﺍﻭﻟﻴﻦ ﺑﺎﺭ ﺩﺭ ﺭﻳﺎﺿﻲ ﻛﻼﺱ ﺍﻭﻝ ﺩﺑﺴﺘﺎﻥ ﺩﻳﺪﻳﻢ‪:‬‬ ‫? =‪1+1‬‬ ‫ﺩﺭ ﻋﺒﺎﺭﺕِ =‪ ،1+1‬ﺑﺎﻳﺪ ﺣﺎﺻﻞ ﺟﻤﻊ ﺳﻤﺖ ﭼﭗ ﺗﺴﺎﻭﻱ ﺭﺍ ﺑﻴﺎﺑﻴﻢ‬ ‫ﻭ ﺩﺭ ﺳــﻤﺖ ﺭﺍﺳﺖ ﺗﺴﺎﻭﻱ ﺑﻨﻮﻳﺴﻴﻢ‪ .‬ﺍﻳﻦ ﻣﻌﻨﺎ ﺑﺮﺍﻱ ﺗﻤﺎﻡ ﻋﺒﺎﺭﺕﻫﺎﻱ‬ ‫ﻋﺪﺩﻱ ﻛﻪ ﻋﻤﻠﻴﺎﺕ ﻣﺸﺎﺑﻪ ﺩﺭ ﺁﻥ ﺭﺍ ﺑﺪﺍﻧﻴﻢ‪ ،‬ﻳﻜﺴﺎﻥ ﺍﺳﺖ‪:‬‬ ‫‪1‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫= ) ‪−(2 − 3 + 7× 5 − (4 − ( −8)) ) ÷ (1 −‬‬ ‫‪8‬‬

‫ﺧــﺪﺍﻱ ﻣﻦ! ﻋﺒﺎﺭﺕ ﺧﻴﻠﻲ ﻃﻮﻻﻧﻲ ﺷــﺪ! ﺣﺘﻲ ﺣﺴــﺎﺏ ﻧﻜﺮﺩﻳﻢ‬ ‫ﺟﻮﺍﺑﺶ ﭼﻪ ﻋﺪﺩﻱ ﻣﻲﺷﻮﺩ؟ ﺍ ّﻣﺎ ﺑﻪ ﻫﺮ ﺣﺎﻝ ﻣﻌﻨﺎﻱ = ﺩﺭ ﻋﺒﺎﺭﺕ ﺍﺧﻴﺮ‪،‬‬ ‫ﻫﻤﺎﻥ ﻳﺎﻓﺘﻦ ﺣﺎﺻﻞ ﻋﺒﺎﺭﺕِ ﺳــﻤﺖ ﭼﭗ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﻮﺍﺭﺩ‪ ،‬ﺁﻥﭼﻪ‬ ‫ﺳــﻤﺖ ﺭﺍﺳﺖ = ﻣﻲﻧﻮﻳﺴﻴﻢ‪ ،‬ﻳﻚ ﻋﺪﺩ ﺍﺳــﺖ )ﮔﻮﻳﺎ‪ ،‬ﮔﻨﮓ ﻳﺎ ﺻﺤﻴﺢ‬ ‫)ﻣﻨﻔﻲ ﻳﺎ ﻣﺜﺒﺖ(‪ ،‬ﻓﺮﻗﻲ ﻧﻤﻲﻛﻨﺪ‪ ،‬ﺑﻪ ﻫﺮ ﺣﺎﻝ ﻳﻚ ﻋﺪﺩ ﺍﺳﺖ(‪.‬‬

‫? = ‪x + 2y − 1 + 3 x − y‬‬

‫ﺣﺎﺻــﻞ ﻋﺪﺩﻱ ﺑﺮﺍﻱ‬ ‫ﺩﺭ ﺍﻳﻦﺟــﺎ‪ ،‬ﻣﻌﻨﺎﻱ = ﺍﻳﻦ ﻧﻴﺴــﺖ ﻛﻪ ﻳﻚ‬ ‫ِ‬ ‫ﻋﺒﺎﺭﺕِ ﺳﻤﺖ ﭼﭗ ﺑﻪ ﺩﺳــﺖ ﺁﻭﺭﻳﻢ‪ ،‬ﺑﻠﻜﻪ ﺍﻳﻦﺟﺎ ﺑﺎﻳﺪ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ‬ ‫ﺭﺍ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺟﻤﻊ ﻳﺎ ﺗﻔﺮﻳﻖ ﺟﻤﻼﺕ ﻣﺸﺎﺑﻪ‪ ،‬ﺳﺎﺩﻩﺗﺮ ﻛﻨﻴﻢ‪ .‬ﭘﺲ =‬ ‫ﺩﺭ ﺍﻳﻦ ﻋﺒﺎﺭﺕ ﺑﻪ ﻣﻌﻨﺎﻱ ﺩﻳﮕﺮﻱ ﺁﻣﺪﻩ ﺍﺳــﺖ ﻭ ﺩﺭ ﻭﺍﻗﻊ ﺍﺯ ﺁﻥﺟﺎ ﻛﻪ‬ ‫ﻋﺒﺎﺭﺕ ‪ x‬ﻭ ‪ y‬ﻭ ﻋﺪﺩﻱ ﺩﺭ ﺍﻳﻦ ﻋﺒﺎﺭﺕ ﻣﺸــﺎﺑﻪ ﻧﻴﺴــﺘﻨﺪ‪ ،‬ﻫﺮﮔﺰ ﺳﻤﺖ‬ ‫ﭼﭗ ﺗﺴﺎﻭﻱ ﺑﺎ ﻳﻚ ﻋﺪﺩ ﺑﺮﺍﺑﺮ ﻧﻴﺴﺖ‪.‬‬ ‫‪x + 2y − 1 + 3 x − y = 4x + y − 1‬‬ ‫ﺳﻤﺖ ﺭﺍﺳﺖ ﺗﺴﺎﻭﻱ ﻛﻪ ﻳﻚ‬ ‫ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﺟﺪﻳﺪ ﺍﺳﺖ‪.‬‬

‫ﺩﺭ ﻋﺒﺎﺭﺕ ﺯﻳﺮ‪ ،‬ﻣﻌﻨﺎﻱ = ﭼﻴﺴﺖ؟‬

‫? = ‪a + 2b‬‬

‫ﺩﺭ ﺍﻳﻦﺟﺎ ﻧﻴﺰ ﺑﻪ ﻣﻌﻨﺎﻱ ﻳﺎﻓﺘﻦ ﻳﻚ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﺳــﺎﺩﻩﺗﺮ ﺍﺳﺖ‬ ‫)ﻛﻪ ﺟﻤﻠﻪﻫﺎﻱ ﻣﺸﺎﺑﻪ ﻧﺪﺍﺷﺘﻪ ﺑﺎﺷﺪ(‪ .‬ﻭﻟﻲ ﺍﺯ ﺁﻥﺟﺎ ﻛﻪ ‪ a‬ﻭ ‪ 2b‬ﻣﺸﺎﺑﻪ‬ ‫ﻧﻴﺴــﺘﻨﺪ‪ ،‬ﭘﺲ ﺩﺭ ﻭﺍﻗﻊ ﻫﻴﭻ ﻋﺒﺎﺭﺕ ﺟﺪﻳﺪﻱ ﺟﻠﻮ ﻣﺴﺎﻭﻱ ﻧﻤﻲﺗﻮﺍﻧﻴﻢ‬ ‫ﺑﻨﻮﻳﺴﻴﻢ! ﺍﻳﻦ‪ ،‬ﺍﺯ ﺁﻥ =ﻫﺎﻳﻲ ﺍﺳﺖ ﻛﻪ ﺳﻤﺖ ﺭﺍﺳﺘﺶ ﻋﺒﺎﺭﺕ ﺳﺎﺩﻩﺗﺮﻱ‬ ‫ﻧﻤﻲﺗﻮﺍﻥ ﻧﻮﺷﺖ‪.‬‬ ‫ﺩﺭ ﺗﺴﺎﻭﻱ ﺯﻳﺮ‪ = ،‬ﺑﻪ ﭼﻪ ﻣﻌﻨﺎﺳﺖ؟‬ ‫‪2‬‬

‫? = ) ‪2x ( x − 4 y‬‬

‫ﺩﺭ ﺍﻳﻦﺟــﺎ ﺑﺎﻳﺪ ‪ 2x‬ﺭﺍ ﺩﺭ ﻋﺒــﺎﺭﺕ ﺩﻭﺟﻤﻠﻪﺍﻱ ‪ ، x 2 − 4 y‬ﭘﺨﺶ‬ ‫ِ‬ ‫ﺧﺎﺻﻴﺖ ﭘﺨﺸﻲ‪ ،‬ﺿﺮﺏ ﻛﻨﻴﻢ( ﻭ ﻳﻚ ﺩﻭ‬ ‫ﻛﻨﻴﻢ )ﻳﻌﻨﻲ ﺑﺎ ﺍﺳــﺘﻔﺎﺩﻩ ﺍﺯ‬ ‫ﺟﻤﻠﻪﺍﻱ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﻳﻢ‪:‬‬ ‫‪3‬‬

‫‪2‬‬

‫‪2x ( x − 4 y) = 2x − 8 xy‬‬

‫ﻭ ﻋﺒــﺎﺭﺕِ ‪ ، 2x 3 − 8 xy‬ﺩﻭ ﺟﻤﻠﻪﻱ ﻏﻴﺮ ﻣﺸــﺎﺑﻪ ﺩﺍﺭﺩ ﻛﻪ ﺑﺎ ﻫﻢ‬ ‫ﺟﻤﻊ )ﻳﺎ ﺗﻔﺮﻳﻖ( ﻧﻤﻲﺷﻮﻧﺪ‪ .‬ﭼﻨﻴﻦ ﺣﺎﻟﺘﻲ ﺑﻪ ﺍﻳﻦ ﺩﻟﻴﻞ ﭘﻴﺶ ﻣﻲﺁﻳﺪ‬ ‫ﻛﻪ ‪) x 2 − 4 y‬ﻋﺒﺎﺭﺕ ﺩﺍﺧﻞ ﭘﺮﺍﻧﺘﺰ( ﻧﻴﺰ ﺍﺯ ﺩﻭ ﺟﻤﻠﻪﻱ ‪ x2‬ﻭ ‪ -4y‬ﻛﻪ‬ ‫ﻣﺸﺎﺑﻪ ﻧﺒﻮﺩﻧﺪ‪ ،‬ﺗﺸﻜﻴﻞ ﺷﺪﻩ ﺑﻮﺩ‪.‬‬ ‫ﭘﺲ ﺩﺭ ﻣﺜﺎﻝ ﺍﺧﻴﺮ‪ = ،‬ﺑﻪ ﻣﻌﻨﺎﻱ ﻳﺎﻓﺘﻦ ﺣﺎﺻﻞ ﺿﺮﺏِ ﺗﻚﺟﻤﻠﻪﺍﻱ‬ ‫‪ 2x‬ﺩﺭ ﺟﻤﻠﻪﻱ ‪ x 2 − 4 y‬ﺍﺳــﺖ‪ .‬ﺩﺭ ﻣﺜــﺎﻝ ﺯﻳﺮ‪ ،‬ﺑﺎﺯ ﻫﻢ = ﺑﻪ ﻣﻌﻨﺎﻱ‬ ‫ﻳﺎﻓﺘﻦ ﺣﺎﺻﻞ ﺿﺮﺏ ﺍﺳــﺖ‪ ،‬ﺍﻣﺎ ﺣﺎﺻﻞ ﺿﺮﺏِ ﺩﻭ ﺟﻤﻠﻪﻱ ‪ x-y‬ﺩﺭ ﺩﻭ‬ ‫ِ‬ ‫ﺟﻤﻠﻪﻱ ‪x+y‬؛ ﻭ ﺍﻟﺒﺘﻪ ﺩﺭ ﺍﻳﻦ ﻣﺜﺎﻝ ﻭ ﻣﺜﺎﻝ ﻗﺒﻠﻲ‪ ،‬ﻫﻴﭻ ﻳﻚ ﺍﺯ ﺣﺎﺻﻞ‬ ‫ﺿﺮﺏﻫﺎ ﻋﺪﺩ ﻧﻴﺴــﺘﻨﺪ‪ ،‬ﺑﻠﻜــﻪ ﺑﺎﺯ ﻫﻢ ﻳﻚ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﺑﻪ ﺩﺳــﺖ‬ ‫ﻣﻲﺁﻳﺪ‪:‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪31‬‬

‫‪2‬‬

‫‪2‬‬

‫‪( x − y)( x + y) = x − xy + xy − y‬‬

‫ﺧﻮﺏ‪ ،‬ﺑﻪ ﻧﻈﺮ ﻣﻲﺭﺳــﺪ ﻛﺎﺭﻣﺎﻥ ﺗﻤﺎﻡ ﺷــﺪﻩ ﺍﺳﺖ؛ ﺍ ّﻣﺎ ﻳﻚ ﺩﻗﻴﻘﻪ‬ ‫ﺻﺒﺮ ﻛﻨﻴﺪ! ﺩﺭ ﻋﺒﺎﺭﺕ ﺳﻤﺖ ﺭﺍﺳﺖ‪ ،‬ﺟﻤﻼﺕ ‪ xy‬ﻭ ‪ -xy‬ﺑﺎ ﻫﻢ ﻣﺸﺎﺑﻪ‬ ‫ﻫﺴﺘﻨﺪ ﻭ ﺣﺎﺻﻞ ﺟﻤﻊ ﺁﻥﻫﺎ‪ ،‬ﺻﻔﺮ ﺍﺳﺖ‪ xy − xy = 0 :‬ﭘﺲ ﻣﻲﺗﻮﺍﻥ‬ ‫ﻳﻚ = ﺩﻳﮕﺮ ﮔﺬﺍﺷﺖ ﻭ ﻋﺒﺎﺭﺕ ﺭﺍ ﺳﺎﺩﻩﺗﺮ ﻛﺮﺩ‪:‬‬ ‫‪2‬‬

‫‪2‬‬

‫‪2‬‬

‫‪2‬‬

‫‪(xx − y )( x + y ) = x − xy + xy − y = x − y‬‬

‫ﻭ ﻋﺒﺎﺭﺕ ﺳــﺎﺩﻩﺗﺮﻱ ﻧﻤﻲﺗﻮﺍﻥ ﺟﻠﻮ ﺁﻥ ﻧﻮﺷﺖ )ﻣﺜﻞ ﻳﻜﻲ ﺍﺯ ﻣﺜﺎﻝﻫﺎﻱ‬ ‫ﻗﺒــﻞ(‪ .‬ﺍ ّﻣﺎ ﺍﮔﺮ ﺩﺍﻧﺶﺁﻣﻮﺯ ﺳــﻮﻡ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺑﺎﺷــﻴﺪ ﻭ ﻓﺎﻛﺘﻮﺭﮔﻴﺮﻱ ﻳﺎ‬ ‫ﺗﺒﺪﻳﻞ ﭼﻨﺪ ﺟﻤﻠﻪﺍﻱ ﺑﻪ ﺣﺎﺻﻞ ﺿﺮﺏ ﺭﺍ ﻳﺎﺩ ﮔﺮﻓﺘﻪ ﺑﺎﺷــﻴﺪ‪ ،‬ﻣﻲﺑﻴﻨﻴﺪ‬ ‫ﺍﺯ ‪ 2xy‬ﻣﻲﺗﻮﺍﻥ ﺩﺭ ﺍﻳﻦ ﺳﻪ ﺟﻤﻠﻪ ﻓﺎﻛﺘﻮﺭ ﮔﺮﻓﺖ ﻭ ﻋﺒﺎﺭﺕ ﺳﻤﺖ ﭼﭗ‬ ‫ﻣﺜﺎﻝ‬ ‫ﺭﺍ ﺑﻪ ﺣﺎﺻﻞ ﺿﺮﺏ ﺗﺒﺪﻳﻞ ﻛﺮﺩ‪ ،‬ﻳﻌﻨﻲ ﺑﺮﻋﻜﺲ ﺁﻥ ﻛﺎﺭﻱ ﻛﻪ ﺩﺭ‬ ‫ِ‬ ‫)‪ 2x ( x 2 − 4 y‬ﻛﺮﺩﻳﻢ!‬ ‫ﭘﺲ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪2‬‬

‫ﺍﻳﻦ ﻣﺴﺎﻭﻱ ﺑﻪ ﻣﻌﻨﺎﻱ ﺳﺎﺩﻩ‬ ‫ﻛﺮﺩﻥ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﺍﺳﺖ‬

‫ﺍﻳﻦ ﻣﺴﺎﻭﻱ ﺑﻪ ﻣﻌﻨﺎﻱ ﺿﺮﺏ ﻛﺮﺩﻥ ﺩﻭ‬ ‫ﻋﺒﺎﺭﺕ ﺳﻤﺖ ﭼﭗ ﺩﺭ ﻳﻜﺪﻳﮕﺮ ﺍﺳﺖ‬ ‫)ﺍﻟﺒﺘﻪ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺧﺎﺻﻴﺖ ﭘﺨﺸﻲ(‬

‫ﺧﻮﺏ‪ ،‬ﺑﻪ ﻧﻈﺮ ﻣﻲﺭﺳﺪ » =« ﺩﺭ ﻋﺒﺎﺭﺕﻫﺎﻱ ﻣﺨﺘﻠﻒ‪ ،‬ﻧﺸﺎﻥﺩﻫﻨﺪﻩﻱ‬ ‫»ﺩﺭﺧﻮﺍﺳــﺖﻫﺎﻱ« ﻣﺨﺘﻠﻒ ﺍﺳﺖ! ﻣﺎ ﺑﺎﻳﺪ ﺍﺯ ﺷــﻜﻞ ﻭ ﺍﺟﺰﺍﻱ ﻋﺒﺎﺭﺕ‬ ‫ﺩﺍﺩﻩ ﺷﺪﻩ‪ ،‬ﻣﺘﻮﺟﻪ ﺷﻮﻳﻢ ﻛﻪ ﭼﻪ ﻧﻮﻉ ﻋﻤﻠﻴﺎﺗﻲ ﺭﺍ ﺍﻧﺠﺎﻡ ﺩﻫﻴﻢ ﻭ ﺳﻤﺖ‬ ‫ﺭﺍﺳــﺖ ﺗﺴﺎﻭﻱ‪ ،‬ﭼﻪ ﺑﻨﻮﻳﺴﻴﻢ‪ .‬ﺍﮔﺮ ﻓﻜﺮ ﻣﻲﻛﻨﻴﺪ ﻣﻌﻨﺎﻱ ﺩﻳﮕﺮﻱ ﺑﺮﺍﻱ‬ ‫= ﺩﺭ ﻋﺒﺎﺭﺕﻫﺎﻳﻤﺎﻥ ﻧﺪﺍﺭﻳﻢ‪ ،‬ﺑﻪ ﻣﺜﺎﻝ ﺯﻳﺮ ﺗﻮﺟﻪ ﻛﻨﻴﺪ ﻭ ﺗﻮﺿﻴﺢ ﺩﻫﻴﺪ‬ ‫ﭼﻪ ﺍﺗﻔﺎﻗﻲ ﺍﻓﺘﺎﺩﻩ ﻭ ﻫﺮ = ﺑﻪ ﻣﻌﻨﺎﻱ ﭼﻴﺴﺖ؟‬ ‫)‪(3‬‬

‫)‪( 2‬‬

‫)‪(1‬‬

‫‪3 x − 1 = 3 ( −2) − 1 = − 6 − 1 = − 7‬‬

‫ﺑﻠﻪ‪ ،‬ﺩﺭﺳﺖ ﺍﺳﺖ‪ = :‬ﺷﻤﺎﺭﻩ )‪ ،(1‬ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ ﻋﺒﺎﺭﺕ ‪ ،3x-1‬ﺑﻪ‬ ‫ﺍﺯﺍﻱ )ﺍﻳﻦ ﻣﺴﺎﻭﻱ ‪ x=-2‬ﻳﻌﻨﻲ ‪ x‬ﻫﻤﺎﻥ ﻋﺪﺩ ‪ -2‬ﺍﺳﺖ ﻭ ﺑﺎﺯ ﻣﻌﻨﻲﺍﺵ‬ ‫ﻓﺮﻕ ﻣﻲﻛﻨﺪ! ﻭﺍﻱ ﺧﺪﺍﻱ ﻣﻦ ! ! ! ( ﺑﺎ ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ‬ ‫‪3 ( −2) − 1‬‬ ‫ﺑﺮﺍﺑﺮ ﺍﺳﺖ‪.‬‬ ‫= ﺷﻤﺎﺭﻩﻱ )‪ (2‬ﻳﻌﻨﻲ ﺣﺎﺻﻞ ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ ‪ 3( −2) − 1‬ﺑﺎ ﺣﺎﺻﻞ‬ ‫ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ ‪ -6-1‬ﺑﺮﺍﺑﺮ ﺍﺳــﺖ ﻭ ﺑﺎﻻﺧﺮﻩ = ﺷﻤﺎﺭﻩﻱ ‪ (3‬ﺑﻪ ﻣﻌﻨﺎﻱ‬ ‫ﺍﻳﻦ ﺍﺳــﺖ ﻛﻪ ﺣﺎﺻﻞ ﻋﺒﺎﺭﺕ ‪ ،-6-1‬ﻋﺪﺩ ‪ -7‬ﺍﺳﺖ‪ .‬ﭘﺲ = ﺷﻤﺎﺭﻩﻱ‬ ‫)‪ (1‬ﺩﺭ ﺍﻳﻦﺟﺎ ﺑﻪ ﻣﻌﻨﺎﻱ ﺟﺎﮔﺬﺍﺭﻱ ﻳﻚ ﻋﺪﺩ ﺩﺭ ﻋﺒﺎﺭﺕ ﺳﻤﺖ ﭼﭗ ﺑﻪ‬ ‫ﺟﺎﻱ ﻣﺠﻬﻮﻝ )ﻳﺎ ﻣﺘﻐﻴﺮ( ﺁﻥ ﻋﺒﺎﺭﺕ ﻭ ﻳﺎﻓﺘﻦ ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ ﺑﻮﺩ ﻭ ﺍﻳﻦ‬ ‫ﻣﻌﻨﺎ ﺑﺎ ﺳﺎﻳﺮ ﻣﻌﺎﻧﻲ ﻛﻪ ﺑﺮﺭﺳﻲ ﻛﺮﺩﻳﻢ‪ ،‬ﻣﺘﻔﺎﻭﺕ ﺍﺳﺖ‪.‬‬ ‫ﺣﺎﻝ ﺷــﻤﺎ ﺑﮕﻮﻳﻴﺪ ﻛﻪ ﻣﻌﻨــﺎﻱ = ﺩﺭ ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﭼﻴﺴــﺖ؟ )ﺩﺭ‬ ‫ﻭﺍﻗﻊ‪» ،‬ﺩﺭﺧﻮﺍﺳــﺖ« ﭼﻴﺴــﺖ ﻭ ﭼﻪ ﭼﻴﺰﻱ ﺑﺎﻳﺪ ﺩﺭ ﺳــﻤﺖ ﺭﺍﺳﺖ =‬ ‫ﺑﻨﻮﻳﺴﻴﻢ؟(‬ ‫‪2‬‬

‫ﻓﻘﻂ ﺣﻮﺍﺳــﺘﺎﻥ ﺑﺎﺷــﺪ ﻛﻪ ﺩﻭﺑﺎﺭﻩ ﻳﻚ ﻣﺴــﺎﻭﻱ ﺩﺭ ﺳﻤﺖ ﺭﺍﺳﺖ‬ ‫ﻧﮕﺬﺍﺭﻳﺪ ﻭ ‪ 2xy‬ﺭﺍ ﺩﻭﺑﺎﺭﻩ ﺩﺭ ‪ 2x − 3 + y‬ﭘﺨﺶ ﻧﻜﻨﻴﺪ‪ ،‬ﭼﻮﻥ ﺩﻭﺑﺎﺭﻩ‬ ‫ﻫﻤــﺎﻥ ﻋﺒﺎﺭﺕِ ‪ 4x 2 y − 6 xy + 2xy2‬ﺑﻪ ﺩﺳــﺖ ﻣﻲﺁﻳــﺪ! )ﺧﻴﻠﻲ ﺍﺯ‬ ‫ﺑﭽﻪﻫﺎ ﺑﻪ ﻋﺎﺩﺕ‪ ،‬ﺍﻳﻦ ﻛﺎﺭ ﺭﺍ ﻣﻲﻛﻨﻨﺪ(‪.‬‬ ‫ﻧﻜﻨﺪ ﺍﺯ ﺍﻳﻦ ﻫﻤﻪ ﺗﻨﻮﻉ ﺩﺭ ﻧﻮﻉ ﺩﺭﺧﻮﺍﺳــﺖﻫﺎ ﺑﺮﺍﻱ ﻧﻮﺷﺘﻦ ﭼﻴﺰﻱ‬ ‫ﺩﺭ ﺳــﻤﺖ ﺭﺍﺳﺖ = ﮔﻴﺞ ﺷﺪﻩ ﺑﺎﺷﻴﺪ؟ ! ﺑﺮﺍﻱ ﺍﻳﻦ ﻛﻪ ﻳﻚ ﺟﻤﻊﺑﻨﺪﻱ‬ ‫ﻛﻨﻴﻢ‪ ،‬ﻳﻚ ﺩﻭﺭ ﺩﻳﮕﺮ ﻣﻄﻠﺐ ﺭﺍ ﺍﺯ ﺁﻥﺟﺎ ﻛﻪ ﺩﺭﺑﺎﺭﻩﻱ ﻋﻼﻣﺖ = ﻧﻮﺷﺘﻪ‬ ‫ﺷﺪﻩ ﺍﺳﺖ ﺑﺨﻮﺍﻧﻴﺪ ﻭ ﺳﭙﺲ ﺍﺯ ﺍﻳﻦﺟﺎ ﺑﻪ ﺑﻌﺪ ِ ﻣﻄﻠﺐ ﺭﺍ ﺍﺩﺍﻣﻪ ﺩﻫﻴﺪ‪.‬‬ ‫ﺑﺎ ﻣﺮﻭﺭ ﻛﻠﻲ ﻣﺜﺎﻝﻫﺎ‪ ،‬ﻣﺘﻮﺟﻪ ﻣﻲﺷﻮﻳﻢ ﻛﻪ ﺩﺭ ﻛﻞ ﻋﻼﻣﺖ ﺗﺴﺎﻭﻱ‬ ‫) =( ﺩﺭ ﻣﻮﺍﺭﺩ ﺯﻳﺮ ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﻣﻲﮔﻴﺮﺩ‪:‬‬ ‫‪ (1‬ﺑﻴــﻦ ﺣﺎﺻﻞ ﻋﺪﺩﻱ ﻳــﻚ ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ ﻭ ﺧــﻮ ِﺩ ﺁﻥ ﻋﺒﺎﺭﺕ‬ ‫ﻋﺪﺩﻱ‪ ،‬ﻣﺜﻞ‬ ‫‪1 3 1‬‬ ‫‪3‬‬ ‫‪+ − =1‬‬ ‫‪2 2 4‬‬ ‫‪4‬‬

‫‪ (2‬ﺑﻴﻦ ﺩﻭ ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ ﻛﻪ ﺣﺎﺻﻞ ﻳﻜﺴﺎﻥ ﺩﺍﺭﻧﺪ‪ ،‬ﻣﺜﻞ‬

‫‪1+22 +4×5 = 1+4+2 0‬‬

‫‪ (3‬ﺑﻴﻦ ﺩﻭ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﻛﻪ ﻳﻜﻲ ﺑﺎ ﻋﻤﻠﻴﺎﺕ ُﻣﺠﺎﺯ ﺍﺯ ﺩﻳﮕﺮﻱ ﺑﻪ‬ ‫ﻣﺜﻞ ﺍﻧﻮﺍﻉ ﻧﻤﻮﻧﻪﻫﺎﻱ ﺯﻳﺮ‪:‬‬ ‫ﺩﺳﺖ ﺁﻣﺪﻩ ﺍﺳﺖ‪ِ ،‬‬ ‫)ﺟﻤﻊ ﺟﻤﻼﺕ ﻣﺸﺎﺑﻪ(‬ ‫‪4a − 6 b + a = 5a − 6 b‬‬ ‫‪2( x − 5) + 3 (2 − x ) = 2x − 10 + 6 − 3 x = − x − 4‬‬

‫‪2‬‬

‫? = ‪4 x y − 6 xy + 2xy‬‬

‫ﺍﮔﺮ ﻫﻨﻮﺯ ﺣﺪﺱ ﻧﺰﺩﻩﺍﻳﺪ ﻛﻪ ﺑﺎﻳﺪ ﭼﻪ ﻛﺎﺭ ﻛﻨﻴﺪ‪ ،‬ﺑﻪ ﺍﻳﻦ ﺗﻮﺟﻪ ﻛﻨﻴﺪ‬ ‫ﻛﻪ ﻋﺒﺎﺭﺕ ﺳــﻤﺖ ﭼﭗ‪ ،‬ﻳﻚ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﺍﺳــﺖ‪ ،‬ﻭﻟﻲ ﺩﺭﺧﻮﺍﺳﺖ‬ ‫ﺿــﺮﺏ ﺩﺭ ﺁﻥ ﻭﺟﻮﺩ ﻧﺪﺍﺭﺩ‪ ،‬ﺯﻳﺮﺍ ﻓﻘﻂ ﺳــﻪ ﺟﻤﻠﻪﻱ ‪ 4 x 2 y‬ﻭ ‪− 6 xy‬‬ ‫ﻭ ‪ + 2xy2‬ﺭﺍ ﺩﺍﺭﺩ‪ .‬ﭘﺲ ﺩﺭ ﻭﺍﻗﻊ ﻳﻚ ﺳــﻪ ﺟﻤﻠﻪﺍﻱ ﺍﺳــﺖ ﻭ ﺍﻳﻦ ﺳﻪ‬ ‫ﺟﻤﻠﻪ ﺑﺎ ﻫﻢ ﻣﺸﺎﺑﻪ ﻧﻴﺴﺘﻨﺪ‪ .‬ﭘﺲ ﭼﻪ ﻛﺎﺭ ﻛﻨﻴﻢ؟ ﻗﻄﻌﺎً ﺑﻪ ﻧﻈﺮ ﻣﻲﺭﺳﺪ‬ ‫ﻛﻪ ﺍﻳﻦ = ﺍﺯ ﺁﻥ ﻣﺴــﺎﻭﻱﻫﺎﻳﻲ ﺑﺎﺷﺪ ﻛﻪ ﺳﻤﺖ ﺭﺍﺳﺘﺶ ﺧﺎﻟﻲ ﻣﻲﻣﺎﻧﺪ‬

‫‪2‬‬

‫) ‪4 x y − 6 xy + 2xy = 2xy (2x − 3 + y‬‬

‫ﺳﻤﺖ ﺭﺍﺳﺖ‪ ،‬ﺑﺎ ﺟﻤﻊ ﻣﺸﺎﺑﻪ ﺑﻪ ﺩﺳﺖ ﻣﻲ ﺁﻳﺪ‬

‫ﺳﻤﺖ ﺭﺍﺳﺖ‪ ،‬ﺑﺎ ﭘﺨﺸﻲ ﻋﺪﺩ ﺩﺭ‬ ‫ﻋﺒﺎﺭﺕﻫﺎﻱ ﺩﺍﺧﻞ ﭘﺮﺍﻧﺘﺰ ﺑﻪ ﺩﺳﺖ ﻣﻲﺁﻳﺪ‪.‬‬

‫)ﺑﺎ ﻓﺎﻛﺘﻮﺭﻱ ﺳﻤﺖ ﺭﺍﺳﺖ ﺑﻪ ﺩﺳﺖ ﻣﻲﺁﻳﺪ(‬ ‫‪2‬‬

‫)‪6 x − 3 x = 3 x((2x − 1‬‬

‫‪ (4‬ﺩﺭ ﻋﺒﺎﺭﺕﻫــﺎﻱ ﺟﺒﺮﻱ‪ ،‬ﻭﻗﺘﻲ ﻣﺘﻐﻴﺮﻫــﺎﻱ ﺁﻥ ﻋﺒﺎﺭﺕ‪ ،‬ﺍﻋﺪﺍﺩ‬ ‫ﻣﻮﺭﺩﻧﻈﺮ ﺭﺍ ﻣﻲﮔﺰﺍﺭﻳﻢ ﻭ ﺁﻥ ﺭﺍ ﺑﻪ ﻳﻚ ﻋﺒﺎﺭﺕ ﻋﺪﺩﻱ ﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ‪،‬‬ ‫ﻣﺜﻞ‬ ‫ِ‬ ‫‪2‬‬

‫‪2‬‬

‫)‪x − 8 + y = 3 − 8 + ( −5‬‬ ‫‪32‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﻛﻪ ﺩﺭ ﺁﻥ ﺑﻪ ﺟﺎﻱ ‪ ،x‬ﻋﺪﺩ ‪ 3‬ﻭ ﺑﻪ ﺟﺎﻱ ‪ ،y‬ﻋﺪﺩ ‪ -5‬ﺭﺍ ﮔﺬﺍﺷﺘﻪﺍﻳﻢ‪،‬‬ ‫ﻳﻌﻨﻲ ‪ x=3‬ﻭ ‪ ،y=-5‬ﻛﻪ ﺍﻳﻦ ﺩﻭ ﺗﺴﺎﻭﻱ ﺍﺧﻴﺮ ﺑﻪ ﻣﻌﻨﺎﻱ ﺑﺮﺍﺑﺮﻱ ﻣﻘﺎﺩﻳﺮ‬ ‫‪ x‬ﻭ ‪ y‬ﺑﺎ ﺍﻋﺪﺍﺩ ﻣﻌﻴﻨﻲ ﻫﺴﺘﻨﺪ‪ ،‬ﻳﻌﻨﻲ‪:‬‬ ‫‪ (5‬ﺑﺮﺍﺑﺮﻱ ﺩﻭ ﺷﻲء ﺍﺯ ﻳﻚ ﺟﻨﺲ ﺑﺎ ﻫﻢ‪ ،‬ﻣﺜﻞ ﻫﻤﺎﻥ ‪ x=3‬ﻭ ‪y=-5‬‬ ‫ﺩﺭ ﻣﺜﺎﻝ ﻗﺒﻞ‪.‬‬ ‫ﺍﻟﺒﺘﻪ ﺗﻮﺟﻪ ﻛﻨﻴﺪ ﻛﻪ ﺑﺎﻳﺪ ﺟﻨﺲ ﺩﻭ ﺷــﻲء ﺩﻭ ﻃﺮﻑ ﺗﺴﺎﻭﻱ ﻣﺜﻞ‬ ‫ﻫﻢ ﺑﺎﺷﺪ‪ .‬ﺩﺭ ﻣﺠﻤﻮﻋﻪﻫﺎ ﻳﺎ ﺩﺭ ﺑﺤﺚ ﺑﺮﺩﺍﺭﻫﺎ ﻧﻴﺰ ﺗﺴﺎﻭﻱﻫﺎﻳﻴﻲ ﺍﺯ ﺍﻳﻦ‬ ‫ﺩﺳﺖ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪0‬‬ ‫ﺯﻭﺝ ﻛﻢﺗﺮ ﺍﺯ ‪1‬‬ ‫ﻣﺜــﺎﻝ ‪ (1‬ﺍﮔﺮ ‪ =A‬ﻣﺠﻤﻮﻋﻪﻱ ﻋﺪﺩﻫﺎﻱ ﻃﺒﻴﻌﻲ ِ‬ ‫ﻭ‬ ‫‪ =B‬ﻣﺠﻤﻮﻋﻪﻱ ﻣﻀﺎﺭﺏ ﻃﺒﻴﻌﻲ ‪ 2‬ﻛﻪ ﻳﻚ ﺭﻗﻤﻲ ﺑﺎﺷﻨﺪ‪،‬‬ ‫ﺁﻥﮔﺎﻩ ‪A=B‬‬ ‫‬ ‫‬ ‫ﻣﺜﺎﻝ ‪ (2‬ﺩﺭ ﺷﻜﻞ ﺯﻳﺮ‪ ،‬ﺑﺮﺩﺍﺭﻫﺎﻱ ‪ a‬ﻭ ‪ b‬ﺑﺮﺍﺑﺮﻧﺪ‪ ،‬ﻳﻌﻨﻲ‬ ‫‬

‫‬

‫‬

‫‪a=b‬‬

‫ﺩﻓﺘﺮ ﺍﻧﺘﺸﺎﺭﺍﺕ ﻛﻤﻚ ﺁﻣﻮﺯﺷﻰ‬

‫ﺑﺎ ﻣﺠﻠﻪﻫﺎﻯ ﺭﺷﺪﺁﺷﻨﺎ ﺷﻮﻳﺪ‬ ‫ﻣﺠﻠﻪﻫﺎﻯ ﺭﺷـﺪ ﺗﻮﺳـﻂ ﺩﻓﺘﺮ ﺍﻧﺘﺸـﺎﺭﺍﺕ ﻛﻤﻚﺁﻣﻮﺯﺷﻰ‬ ‫ﺳـﺎﺯﻣﺎﻥ ﭘﮋﻭﻫـﺶ ﻭ ﺑﺮﻧﺎﻣﻪﺭﻳﺰﻯ ﺁﻣﻮﺯﺷـﻰ ﻭﺍﺑﺴـﺘﻪ ﺑﻪ‬ ‫ﻭﺯﺍﺭﺕ ﺁﻣـﻮﺯﺵ ﻭ ﭘـﺮﻭﺭﺵ ﺗﻬﻴـﻪ ﻭ ﻣﻨﺘﺸـﺮ ﻣﻰﺷـﻮﻧﺪ‪:‬‬

‫ﻣﺠﻠﻪ ﻫﺎﯼ ﺩﺍﻧﺶ ﺁﻣﻮﺯﯼ‬ ‫) ﺑﻪ ﺻﻮﺭﺕ ﻣﺎﻫﻨﺎﻣﻪ ﻭ ‪ 8‬ﺷﻤﺎﺭﻩ ﺩﺭ ﻫﺮ ﺳﺎﻝ ﺗﺤﺼﻴﻠﻰ ﻣﻨﺘﺸﺮ ﻣﻰﺷﻮﻧﺪ ( ‪:‬‬

‫)ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺁﻣﺎﺩﮔﻰ ﻭ ﭘﺎﻳﻪﻯ ﺍﻭﻝ ﺩﻭﺭﻩﻯ ﺩﺑﺴﺘﺎﻥ(‬

‫‬

‫‪a=b‬‬ ‫‬

‫‪b‬‬

‫)ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﭘﺎﻳﻪﻫﺎﻯ ﺩﻭﻡ ﻭ ﺳﻮﻡ ﺩﻭﺭﻩﻯ ﺩﺑﺴﺘﺎﻥ(‬

‫‬

‫)ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﭘﺎﻳﻪﻫﺎﻯ ﭼﻬﺎﺭﻡ ﻭ ﭘﻨﺠﻢ ﺩﻭﺭﻩﻯ ﺩﺑﺴﺘﺎﻥ(‬

‫‪a‬‬

‫)ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺩﻭﺭﻩﻯ ﺭﺍﻫﻨﻤﺎﻳﻰ ﺗﺤﺼﻴﻠﻰ(‬

‫ﻣﺜﻞ‬ ‫‪ (6‬ﺑﻴﻦ ﻧﺎﻡ ﺷﻲء ﻭ ﺧﻮﺩ ﺷﻲء ﻗﺮﺍﺭ ﻣﻲﮔﻴﺮﺩ‪ِ ،‬‬ ‫ﻳﺎ‬

‫)ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺩﻭﺭﻩﻯ ﻣﺘﻮﺳﻄﻪﻭﭘﻴﺶﺩﺍﻧﺸﮕﺎﻫﻰ(‬

‫}‪ = { 1 , 2 , 3 , 4 , ....‬‬

‫⎤‪ ⎡1‬‬

‫⎥ ⎢ =¡‬ ‫⎦‪⎣0‬‬

‫‪ (7‬ﺩﺭ ﻫﻨﺪﺳــﻪ‪ ،‬ﺑﻴﻦ ﻧﺎﻡ ﻳﻚ ﺷــﻲء ﻭ ﺍﻧﺪﺍﺯﻩﻱ ﺁﻥ ﻗﺮﺍﺭ ﻣﻲﮔﻴﺮﺩ‪،‬‬ ‫ﻣﺜﻞ‬ ‫ِ‬ ‫‬ ‫‪= 45‬‬ ‫‪ AB = 3cm‬ﻳﺎ‬ ‫‪AB‬‬ ‫‪ (8‬ﺁﻳﺎ ﺑﺎﺯ ﻫﻢ ﻣﻮﺍﺭﺩ ﺩﻳﮕﺮﻱ ﻫﺴــﺖ ﻛﻪ ﻣﻦ ﻓﺮﺍﻣﻮﺵ ﻛﺮﺩﻩﺍﻡ ﻧﺎﻡ‬ ‫ﺑﺒــﺮﻡ؟ ﺑﻪ ﻫــﺮ ﺣﺎﻝ ﺑﻪ ﺍﻳﻦ ﺟﻤﻊﺑﻨﺪﻱ ﺭﺳــﻴﺪﻳﻢ ﻛﻪ ﻭﻗﺘﻲ ﻋﻼﻣﺖ =‬ ‫ﺭﺍ ﺩﻳﺪﻳﻢ ﻭ ﺳــﻤﺖ ﺭﺍﺳــﺖ ﺁﻥ ﺧﺎﻟﻲ ﺑﻮﺩ‪ ،‬ﺣﺘﻤﺎً ﻧﺒﺎﻳﺪ ﻳﻚ ﻋﻤﻠﻴﺎﺕ ﻳﺎ‬ ‫ﻣﺤﺎﺳــﺒﺎﺗﻲ ﺍﻧﺠﺎﻡ ﺩﻫﻴﻢ ﺗﺎ ﺩﺭ ﺳﻤﺖ ﺭﺍﺳﺖ ﺗﺴﺎﻭﻱ ﭼﻴﺰﻱ ﺑﻨﻮﻳﺴﻴﻢ‪.‬‬ ‫ﭼﮕﻮﻧﮕﻲ ﻭ ﻧﻮﻉ ﻧﻮﺷــﺘﻦ ﭼﻴﺰﻱ ﺩﺭ ﺳﻤﺖ ﺭﺍﺳﺖ ﺗﺴﺎﻭﻱ ﺑﺴﺘﮕﻲ ﺩﺍﺭﺩ‬ ‫ﺑﻪ ﻣﻌﻨﺎﻳﻲ ﻛﻪ ﺁﻥ ﻋﺒﺎﺭﺕﻫﺎ ﻭ ﺗﺴﺎﻭﻱﻫﺎ ﻣﻲﺩﻫﻨﺪ‪.‬‬ ‫ﺣﺎﻝ ﺁﻳﺎ ﻣﻲﺗﻮﺍﻧﻴﺪ ﺑﮕﻮﻳﻴﺪ ﺗﺴﺎﻭﻱ ﺯﻳﺮ ﭼﻴﺴﺖ؟ ‪4x-6=12‬‬ ‫ﺑﻠﻪ‪ ،‬ﺩﺭﺳــﺖ ﺍﺳﺖ؛ ﺍﻳﻦ ﺗﺴــﺎﻭﻱ ﻳﻚ ﻣﻌﺎﺩﻟﻪ ﺍﺳــﺖ ﻛﻪ ﻣﺎ ﺑﺮﺍﻱ‬ ‫ﺗﻜﻤﻴــﻞ ﺗﺴــﺎﻭﻱ ﻛﺎﺭﻱ ﺍﻧﺠﺎﻡ ﻧﻤﻲﺩﻫﻴﻢ‪ ،‬ﺑﻠﻜﻪ ﺗﺴــﺎﻭﻱ ﺭﺍ ﺷــﺨﺺ‬ ‫ﺩﻳﮕﺮﻱ ﭘﻴﺪﺍ ﻛﺮﺩﻩ ﺍﺳــﺖ ﻭ ﻣﺎ ﺑﻪ ﻛﻤﻚ ﺁﻥ‪ ،‬ﻣﻘﺪﺍﺭ ﻣﺠﻬﻮﻝ )ﻳﻌﻨﻲ ‪(x‬‬ ‫ﺭﺍ ﭘﻴﺪﺍ ﻣﻲﻛﻨﻴﻢ‪ .‬ﭘﺲ‪:‬‬ ‫‪ (9‬ﺑﻴﻦ ﺩﻭ ﻋﺒﺎﺭﺕ ﺟﺒﺮﻱ ﻛﻪ ﺩﺳﺖ ﻛﻢ ﻳﻜﻲ ﺍﺯ ﺁﻥﻫﺎ ﺣﺪﺍﻗﻞ ﻳﻚ‬ ‫ﻣﺠﻬﻮﻝ ﺩﺍﺭﺩ ﻭ ﺑﻪ ﺁﻥ ﺗﺴﺎﻭﻱ‪ ،‬ﻣﻌﺎﺩﻟﻪ ﮔﻮﻳﻴﻢ‪.‬‬

‫ﻣﺠﻠﻪ ﻫﺎﯼ ﺑﺰﺭﮔﺴﺎﻝ ﻋﻤﻮﻣﯽ‬ ‫)ﺑﻪ ﺻﻮﺭﺕ ﻣﺎﻫﻨﺎﻣﻪ ﻭ ‪ 8‬ﺷﻤﺎﺭﻩ ﺩﺭ ﻫﺮ ﺳﺎﻝ ﺗﺤﺼﻴﻠﻰ ﻣﻨﺘﺸﺮ ﻣﻰﺷﻮﻧﺪ(‪:‬‬ ‫ﺭﺷﺪ ﺁﻣـﻮﺯﺵ ﺍﺑﺘــﺪﺍﻳﯽ‬ ‫ﺁﻣﻮﺯﺷﯽ‬

‫ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﺭﺍﻫﻨﻤـﺎﻳﯽ ﺗﺤﺼﻴﻠﯽ‬

‫ﺭﺷﺪ ﻣﺪﺭﺳﻪ ﻓﺮﺩﺍ‬

‫ﺭﺷﺪ ﻣﺪﻳﺮﻳﺖ ﻣﺪﺭﺳﻪ‬

‫ﺭﺷﺪ ﺗﻜﻨﻮﻟﻮﮊﯼ‬

‫ﺭﺷﺪ ﻣﻌﻠﻢ‬

‫ﻣﺠﻠﻪ ﻫﺎﯼ ﺑﺰﺭﮔﺴﺎﻝ ﺍﺧﺘﺼﺎﺻﯽ‬

‫)ﺑﻪ ﺻﻮﺭﺕ ﻓﺼﻠﻨﺎﻣﻪ ﻭ ‪ 4‬ﺷﻤﺎﺭﻩ ﺩﺭ ﻫﺮ ﺳﺎﻝ ﺗﺤﺼﻴﻠﻰ ﻣﻨﺘﺸﺮ ﻣﻰﺷﻮﻧﺪ(‪:‬‬

‫ﺭﺷﺪ ﺑﺮﻫﺎﻥ ﺭﺍﻫﻨﻤﺎﻳﻰ )ﻣﺠﻠﻪ ﺭﻳﺎﺿﻰ ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺩﻭﺭﻩﻯ ﺭﺍﻫﻨﻤﺎﻳﻰ ﺗﺤﺼﻴﻠﻰ(‬ ‫ﺭﺷﺪ ﺑﺮﻫﺎﻥ ﻣﺘﻮﺳﻄﻪ )ﻣﺠﻠﻪ ﺭﻳﺎﺿﻰ ﺑﺮﺍﻯ ﺩﺍﻧﺶﺁﻣﻮﺯﺍﻥ ﺩﻭﺭﻩﻯ ﻣﺘﻮﺳﻄﻪ(‬

‫ﺭﺷﺪ ﺁﻣﻮﺯﺵ‬

‫ﻗﺮﺁﻥ ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﻣﻌﺎﺭﻑ ﺍﺳﻼﻣﻰ ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﺯﺑﺎﻥ ﻭ ﺍﺩﺏ ﻓﺎﺭﺳﻰ ﺭﺷﺪ ﺁﻣﻮﺯﺵ‬ ‫ﻫﻨﺮ ﺭﺷـﺪ ﻣﺸـﺎﻭﺭ ﻣﺪﺭﺳﻪ ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﺗﺮﺑﻴﺖﺑﺪﻧﻰ ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﻋﻠﻮﻡ ﺍﺟﺘﻤﺎﻋﻰ‬ ‫ﺭﺷـﺪ ﺁﻣﻮﺯﺵ ﺗﺎﺭﻳﺦ‬ ‫ﺭﻳﺎﺿﻰ‬

‫ﺭﺷـﺪ ﺁﻣﻮﺯﺵ ﺟﻐﺮﺍﻓﻴﺎ ﺭﺷـﺪ ﺁﻣﻮﺯﺵ ﺯﺑﺎﻥ ﺭﺷﺪ ﺁﻣﻮﺯﺵ‬

‫ﺭﺷـﺪ ﺁﻣﻮﺯﺵ ﻓﻴﺰﻳﻚ‬

‫ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﺷﻴﻤﻰ‬

‫ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﺯﻳﺴﺖﺷﻨﺎﺳﻰ‬

‫ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﺯﻣﻴﻦﺷﻨﺎﺳﻰ ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﻓﻨﻰﻭﺣﺮﻓﻪﺍﻯ ﺭﺷﺪ ﺁﻣﻮﺯﺵ ﭘﻴﺶ ﺩﺑﺴﺘﺎﻧﻰ‬

‫ﻣﺠﻠﻪﻫﺎﻯ ﺭﺷــﺪ ﻋﻤﻮﻣــﻰ ﻭ ﺍﺧﺘﺼﺎﺻﻰ ﺑــﺮﺍﻯ ﺁﻣﻮﺯﮔﺎﺭﺍﻥ‪ ،‬ﻣﻌﻠﻤــﺎﻥ‪ ،‬ﻣﺪﻳﺮﺍﻥ ﻭ‬ ‫ﻛﺎﺭﻛﻨــﺎﻥ ﺍﺟﺮﺍﻳــﻰ ﻣــﺪﺍﺭﺱ‪ ،‬ﺩﺍﻧﺶﺟﻮﻳﺎﻥ ﻣﺮﺍﻛــﺰ ﺗﺮﺑﻴﺖﻣﻌﻠﻢ ﻭ ﺭﺷــﺘﻪﻫﺎﻯ‬ ‫ﺩﺑﻴﺮﻯ ﺩﺍﻧﺸــﮕﺎﻩﻫﺎ ﻭ ﻛﺎﺭﺷﻨﺎﺳــﺎﻥ ﺗﻌﻠﻴﻢ ﻭ ﺗﺮﺑﻴﺖ ﺗﻬﻴﻪ ﻭ ﻣﻨﺘﺸــﺮ ﻣﻰﺷﻮﻧﺪ‪.‬‬

‫ﻧﺸـﺎﻧﻰ‪ :‬ﺗﻬــﺮﺍﻥ‪ ،‬ﺧﻴﺎﺑــﺎﻥ ﺍﻳﺮﺍﻧﺸــﻬﺮ ﺷﻤﺎﻟﻰ‪،‬ﺳــﺎﺧﺘﻤﺎﻥ ﺷــﻤﺎﺭﻩﻯ‪4‬‬ ‫ﺁﻣﻮﺯﺵﻭﭘﺮﻭﺭﺵ ‪ ،‬ﭘﻼﻙ ‪ ،266‬ﺩﻓﺘﺮ ﺍﻧﺘﺸﺎﺭﺍﺕ ﻛﻤﻚﺁﻣﻮﺯﺷﻰ‪.‬‬ ‫ﺗﻠﻔﻦ ﻭ ﻧﻤﺎﺑﺮ‪ 88301478 :‬ـ ‪021‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪33‬‬

‫ﺷﺮﺍﻳﻂ‪:‬‬ ‫‪ .1‬ﭘﺮﺩﺍﺧﺖ ﻣﺒﻠﻎ ‪ 70/000‬ﺭﻳﺎﻝ ﺑﻪ ﺍﺯﺍﻯ ﻳﻚ ﺩﻭﺭﻩ ﻳﻚ ﺳﺎﻟﻪ ﻣﺠﻠﻪﻯ ﺩﺭﺧﻮﺍﺳﺘﻰ‪،‬‬ ‫ﺑﻪ ﺻﻮﺭﺕ ﻋﻠﻰﺍﻟﺤﺴﺎﺏ ﺑﻪ ﺣﺴﺎﺏ ﺷﻤﺎﺭﻩﻯ ‪ 39662000‬ﺑﺎﻧﻚ ﺗﺠﺎﺭﺕ ﺷﻌﺒﻪﻯ‬ ‫ﺳﻪ ﺭﺍﻩ ﺁﺯﻣﺎﻳﺶ )ﺳﺮﺧﻪﺣﺼﺎﺭ( ﻛﺪ ‪ 395‬ﺩﺭ ﻭﺟﻪ ﺷﺮﻛﺖ ﺍﻓﺴﺖ‪.‬‬ ‫‪ .2‬ﺍﺭﺳﺎﻝ ﺍﺻﻞ ﻓﻴﺶ ﺑﺎﻧﻜﻰ ﺑﻪ ﻫﻤﺮﺍﻩ ﺑﺮگ ﺗﻜﻤﻴﻞ ﺷﺪﻩ ﻯ ﺍﺷﺘﺮﺍﻙ‬ ‫ﺑﺎﭘﺴﺖﺳﻔﺎﺭﺷﻰ‪) .‬ﻛﭙﻰﻓﻴﺶﺭﺍﻧﺰﺩﺧﻮﺩﻧﮕﻪ ﺩﺍﺭﻳﺪ‪(.‬‬ ‫ﻧﺎﻡ ﻣﺠﻠﻪ ﻫﺎﻯﺩﺭﺧﻮﺍﺳﺘﻰ‪:‬‬ ‫‪......................................................................................‬‬ ‫‪......................................................................................‬‬ ‫‪......................................................................................‬‬

‫ﻧﺎﻡ ﻭ‬

‫ﻧﺎﻡﺧﺎﻧﻮﺍﺩﮔﻰ‪. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .:‬‬

‫ﺗﺎﺭﻳﺦ‬

‫ﺗﻮﻟﺪ‪. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .:‬‬

‫ﻣﻴﺰﺍﻥ ﺗﺤﺼﻴﻼﺕ‪:‬‬

‫‪.................................................................‬‬

‫ﺗﻠﻔﻦ‪. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .:‬‬

‫ﻧﺸﺎﻧﻰ ﻛﺎﻣﻞ ﭘﺴﺘﻰ‪:‬‬

‫ﺍﺳﺘﺎﻥ‪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . :‬ﺷﻬﺮﺳﺘﺎﻥ‪:‬‬ ‫ﺧﻴﺎﺑﺎﻥ‪. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :‬‬ ‫ﻛﺪﺍﺷﺘﺮﺍﻙ‪. . . . . . . . . . . . . . . . . . . . . . . . :‬‬ ‫ﭘﻼﻙ‪ . . . . . . . . . . . . . . . . . . . :‬ﺷﻤﺎﺭﻩﻯ ﭘﺴﺘﻰ‪. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :‬‬ ‫‪.............................‬‬

‫ﺩﺭ ﺻﻮﺭﺗﻰ ﻛﻪ ﻗﺒ ًﻼ ﻣﺸﺘﺮﻙ ﻣﺠﻠﻪ ﺑﻮﺩﻩﺍﻳﺪ‪ ،‬ﺷﻤﺎﺭﻩﻯ ﺍﺷﺘﺮﺍﻙ ﺧﻮﺩ ﺭﺍ ﺑﻨﻮﻳﺴﻴﺪ‪:‬‬

‫ﺍﻣﻀﺎ‪:‬‬

‫ﺻﻨـﺪﻭﻕ ﭘﺴﺘﻰ ﻣﺮﻛﺰﺑﺮﺭﺳﻰﺁﺛﺎﺭ‪:‬‬ ‫ﺻﻨـﺪﻭﻕ ﭘﺴﺘﻰ ﺍﻣﻮﺭﻣﺸﺘﺮﻛﻴﻦ‪:‬‬

‫‪15875/6567‬‬ ‫‪16595/111‬‬

‫ﻧﺸﺎﻧﻰ ﺍﻳﻨﺘﺮﻧﺘﻰ‪:‬‬

‫‪www.roshdmag.ir‬‬

‫ﺍﻣﻮﺭ ﻣﺸﺘﺮﻛﻴﻦ‪:‬‬

‫‪ 77335110‬ـ ‪77336656‬ـ‪021‬‬

‫ﭘﻴﺎﻡﮔﻴﺮ ﻣﺠﻠﻪ ﻫﺎﻯ ﺭﺷﺪ‪:‬‬

‫‪ 88301482‬ـ‪021‬‬

‫ﻳﺎﺩﺁﻭﺭﻯ‪:‬‬

‫ﻫﺰﻳﻨﻪﻯ ﺑﺮﮔﺸﺖ ﻣﺠﻠﻪ ﺩﺭ ﺻﻮﺭﺕ ﺧﻮﺍﻧﺎ ﻭ ﻛﺎﻣﻞ ﻧﺒﻮﺩﻥ ﻧﺸﺎﻧﻰ ﻭ ﻋﺪﻡ ﺣﻀﻮﺭ‬ ‫ﮔﻴﺮﻧﺪﻩ‪ ،‬ﺑﺮﻋﻬﺪﻩﻯ ﻣﺸﺘﺮﻙ ﺍﺳﺖ‪.‬‬ ‫ﻣﺒﻨﺎﻯ ﺷﺮﻭﻉ ﺍﺷﺘﺮﺍﻙ ﻣﺠﻠﻪ ﺍﺯ ﺯﻣﺎﻥ ﺩﺭﻳﺎﻓﺖ ﺑﺮگ ﺍﺷﺘﺮﺍﻙ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬

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‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫ﺑﺮگ ﺍﺷﺘﺮﺍﻙ ﻣﺠﻠﻪﻫﺎﻯ ﺭﺷﺪ‬

‫ﺷﺎﺩﻱ ﺑﻬﺎﺭﻱ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫـﺎ‪ :‬ﺗﻘﺴــﻴﻢ‪ ،‬ﺑﺎﻗﻲﻣﺎﻧﺪﻩ‪ ،‬ﺗﻘﺴــﻴﻢ‬ ‫ﺷﻜﻼﺕ‪ ،‬ﺳﻬﻢ ﻓﺮﺯﻧﺪﺍﻥ‪.‬‬

‫دو‬ ‫﹞︧﹮﹚﹤ی ︗︀﹛︉‬ ‫ﻣﺴﺌﻠﻪﻱ ﺍﻭﻝ‪ :‬ﻣﺎﺩﺭﻱ ‪ 25‬ﺷــﻜﻼﺕ ﺩﺍﺷﺖ‪ .‬ﺍﻭ ﻣﻰﺧﻮﺍﺳﺖ ﺍﻳﻦ‬ ‫‪ 25‬ﺷــﻜﻼﺕ ﺭﺍ ﺑﻴﻦ ‪ 5‬ﻓﺮﺯﻧﺪﺵ ﺗﻘﺴــﻴﻢ ﻛﻨﺪ‪ .‬ﺍﻭ ﺑﻪ ﺗﺮﺗﻴﺐ ﺯﻳﺮ ﻋﻤﻞ‬ ‫ﻛﺮﺩ‪:‬‬ ‫‪1‬‬ ‫ﻳﻚ ﺷــﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ﺷﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪6‬‬ ‫ﺍﻭﻝ‪.‬‬ ‫‪1‬‬ ‫ﺩﻭ ﺷــﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ﺷــﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪6‬‬ ‫ﺩﻭﻡ‪.‬‬ ‫‪1‬‬ ‫ﺳﻪ ﺷﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ﺷــﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪6‬‬ ‫ﺳﻮﻡ‪.‬‬ ‫‪1‬‬ ‫ﭼﻬﺎﺭ ﺷﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ﺷﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪6‬‬ ‫ﭼﻬﺎﺭﻡ‪.‬‬ ‫‪1‬‬ ‫ﭘﻨﺞ ﺷــﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ﺷﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪6‬‬ ‫ﭘﻨﺠﻢ‪.‬‬ ‫ﺍﺑﺘﺪﺍ ﺑﺪﻭﻥ ﻣﺤﺎﺳﺒﻪ ﺣﺪﺱ ﺑﺰﻧﻴﺪ ﭼﻪ ﻛﺴﻰ ﺷﻜﻼﺕﻫﺎﻯ ﺑﻴﺶﺗﺮﻯ‬ ‫ﮔﺮﻓﺘﻪ ﺍﺳــﺖ‪ .‬ﺳــﭙﺲ ﺑﺎ ﻣﺤﺎﺳﺒﻪ‪ ،‬ﺩﺭﺳﺘﻰ ﻳﺎ ﻧﺎﺩﺭﺳــﺘﻰ ﺣﺪﺳﺘﺎﻥ ﺭﺍ‬ ‫ﺑﺮﺭﺳﻰ ﻛﻨﻴﺪ‪.‬‬ ‫ﻣﺴـﺌﻠﻪﻱ ﺩﻭﻡ‪ :‬ﭘﺪﺭﻱ ﺗﻌﺪﺍﺩﻯ ﺷﻜﻼﺕ ﺩﺍﺷﺖ‪ .‬ﺍﻭ ﺷﻜﻼﺕﻫﺎ ﺭﺍ‬ ‫ﺑﻪ ﺗﺮﺗﻴﺐ ﺯﻳﺮ ﺑﻴﻦ ﻓﺮﺯﻧﺪﺍﺵ ﺗﻘﺴﻴﻢ ﻛﺮﺩ‪.‬‬ ‫ﻳﻚ ﺷــﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ‪ 1‬ﺷﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪5‬‬ ‫ﺍﻭﻝ‪.‬‬

‫ﺩﻭ ﺷــﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ‬ ‫ﺩﻭﻡ‪.‬‬ ‫ﺳﻪ ﺷــﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ‬ ‫ﺳﻮﻡ‪.‬‬

‫‪1‬‬ ‫‪5‬‬

‫ﺷــﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬ ‫‪1‬‬ ‫‪5‬‬

‫ﺷﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪ‬

‫ﻭ ﺑﻪ ﻫﻤﻴﻦ ﺗﺮﺗﻴﺐ ﺑﺮﺍﻯ ﻓﺮﺯﻧﺪﺍﻥ ﺑﻌﺪﻯ!‬ ‫ﺩﺭ ﭘﺎﻳﺎﻥ‪ ،‬ﺗﻌﺪﺍﺩ ﺷــﻜﻼﺕﻫﺎﻳﻰ ﻛﻪ ﻫﺮ ﻳــﻚ ﺍﺯ ﻓﺮﺯﻧﺪﺍﻧﺶ ﮔﺮﻓﺘﻪ‬ ‫ﺑﻮﺩﻧﺪ ﺑﺎ ﺑﻘﻴﻪ ﻣﺴﺎﻭﻯ ﺑﻮﺩ!!!‬ ‫ﻓﻜﺮ ﻣﻰﻛﻨﻴﺪ ﺍﻭ ﭼﻨﺪ ﺷﻜﻼﺕ ﺭﺍ ﺑﻴﻦ ﭼﻨﺪ ﻓﺮﺯﻧﺪﺵ ﺗﻘﺴﻴﻢ ﻛﺮﺩﻩ‬ ‫ﺍﺳﺖ؟‬ ‫ﭘﺎﺳﺦ ﻣﺴﺌﻠﻪﻱ ﺍﻭﻝ‪:‬‬ ‫ﺑﺎﻗﻰﻣﺎﻧﺪﻩ‬

‫‪⇒ 25 − 5 = 20‬‬

‫‪24‬‬ ‫‪=5‬‬ ‫‪6‬‬

‫‪1+‬‬

‫ﺑﺎﻗﻰﻣﺎﻧﺪﻩ‬

‫‪20 − 5 = 15‬‬

‫⇒‬

‫‪18‬‬ ‫‪=5‬‬ ‫‪6‬‬

‫‪2+‬‬

‫ﻓﺮﺯﻧﺪ ﺩﻭﻡ‬

‫ﺑﺎﻗﻰﻣﺎﻧﺪﻩ‬

‫‪⇒ 15 − 5 = 10‬‬

‫‪12‬‬ ‫‪=5‬‬ ‫‪6‬‬

‫‪3+‬‬

‫ﻓﺮﺯﻧﺪ ﺳﻮﻡ‬

‫ﺑﺎﻗﻰﻣﺎﻧﺪﻩ‬

‫‪10 − 5 = 5‬‬

‫‪4+‬‬

‫ﻓﺮﺯﻧﺪ ﭼﻬﺎﺭﻡ‬

‫ﺑﺎﻗﻰﻣﺎﻧﺪﻩ‬

‫‪=5‬‬

‫‪0‬‬ ‫‪6‬‬

‫ﻧﺘﻴﺠﻪ ﺳﻬﻢ‬

‫ﻗﺎﺑﻞ ﻣﺤﺎﺳﺒﻪ‬ ‫ﻧﻴﺴﺖ‬ ‫×‬

‫ﻓﺮﺯﻧﺪ ﺍﻭﻝ‬

‫‪5+‬‬

‫ﭘﺎﺳﺦ ﻣﺴﺌﻠﻪﻱ ﺩﻭﻡ‪:‬‬ ‫‪1‬‬ ‫ﺑﻪ ﻓﺮﺯﻧﺪ ﺍﻭﻟﺶ ﻳﻚ ﺷﻜﻼﺕ ﺑﻪ ﻫﻤﺮﺍﻩ ﺷﻜﻼﺕﻫﺎﻯ ﺑﺎﻗﻰﻣﺎﻧﺪﻩ‬ ‫‪5‬‬ ‫ﺩﺍﺩﻩ ﺍﺳــﺖ‪ ،‬ﭘﺲ ﭘﺎﺳــﺦ ﺍﺯ ﻳﻜﻰ ﺍﺯ ﻣﻀﺮﺏﻫﺎﻯ ‪ 5‬ﻳﻚ ﻭﺍﺣﺪ ﺑﻴﺶﺗﺮ‬ ‫ﺍﺳﺖ‪.‬‬ ‫ﺩﺭ ﺿﻤﻦ‪ ،‬ﺗﻌﺪﺍﺩ ﻛﻞ ﺷــﻜﻼﺕﻫﺎ ﺑﺎﻳﺪ ﻣﻀﺮﺑﻰ ﺍﺯ ﻓﺮﺯﻧﺪ ﺍﻭﻝ ﺑﺎﺷﺪ‪،‬‬ ‫ﺯﻳﺮﺍ ﺳﻬﻢ ﻫﻤﻪﻯ ﻓﺮﺯﻧﺪﺍﻥ ﻣﺴﺎﻭﻯ ﺑﻮﺩﻩ ﺍﺳﺖ‪.‬‬ ‫ﺑﻪ ﺟﺪﻭﻝ ﺯﻳﺮ ﻭ ﺣﺪﺱﻫﺎﻳﻰ ﻛﻪ ﺯﺩﻩﺍﻳﻢ ﻧﮕﺎﻩ ﻛﻨﻴﺪ‪:‬‬

‫⇒‬

‫‪6‬‬ ‫‪=5‬‬ ‫‪6‬‬

‫ﻓﺮﺯﻧﺪ ﺩﻭﻡ ﻧﺘﻴﺠﻪ‬

‫‪2‬‬ ‫‪5‬‬

‫‪2+‬‬ ‫×‬

‫ﺳﻬﻢ ﻓﺮﺯﻧﺪﺍﻥ‬ ‫‪10‬‬ ‫‪2+‬‬ ‫ﺍﻭﻝ ﻭ ﺩﻭﻡ ‪= 4‬‬ ‫‪5‬‬ ‫ﻣﺴﺎﻭﻱ ﺍﺳﺖ‬

‫ﺳﻬﻢ ﻓﺮﺯﻧﺪ ﺍﻭﻝ‬

‫‪ 6‬ﻣﻀﺮﺏ ‪2‬‬ ‫ﺍﺳﺖ‬

‫‪5‬‬ ‫‪=2‬‬ ‫‪5‬‬

‫‪ 11‬ﻣﻀﺮﺏ‬ ‫‪ 3‬ﻧﻴﺴﺖ‬

‫‪10‬‬ ‫‪=3‬‬ ‫‪5‬‬

‫‪ 16‬ﻣﻀﺮﺏ‬ ‫‪ 4‬ﻫﺴﺖ‬

‫ﺗﻌﺪﺍﺩ ﺷﻜﻼﺕﻫﺎ‬

‫‪1+‬‬

‫‪6‬‬

‫‪1+‬‬

‫‪11‬‬

‫‪15‬‬ ‫‪1+‬‬ ‫‪=4‬‬ ‫‪5‬‬

‫‪16‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺟﺪﻭﻝ ﺑﺎﻻ ﺑﻪ ﻧﻈﺮ ﻣﻰﺭﺳﺪ ﺣﺪﺱ ‪ 16‬ﺣﺪﺱ ﺩﺭﺳﺘﻰ‬ ‫ﺑﺎﺷﺪ‪.‬‬ ‫‪5‬‬ ‫‪0‬‬ ‫‪4+‬‬ ‫ﺳﻬﻢ ﻓﺮﺯﻧﺪ ﺳﻮﻡ ﻧﻴﺰ ‪ 3 + = 4‬ﻭ ﺳﻬﻢ ﻓﺮﺯﻧﺪ ﭼﻬﺎﺭﻡ ‪= 4‬‬ ‫‪5‬‬ ‫‪5‬‬ ‫ﺷﻜﻼﺕ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬

‫ﻓﺮﺯﻧﺪ ﭘﻨﺠﻢ‬

‫ﺟﺎﻟﺐ ﻧﻴﺴﺖ ‪ .‬ﻫﻤﻪﻯ ﺁﻥﻫﺎ ﺑﻪ ﺗﻌﺪﺍﺩ ﻣﺴﺎﻭﻯ ﺷﻜﻼﺕ‬ ‫ﮔﺮﻓﺘﻪﺍﻧﺪ!!!‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪35‬‬

‫ﺩﺍﻧﺶﺍﻓﺰﺍﻳﻲ‬

‫︝﹏ ﹞︧﹮﹚﹤ ﹇︡م ︋﹤ ﹇︡م‬ ‫ﺳﺎﻳﻪ ﻣﻬﺮﺑﺎﻥ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻣﺴﺎﺣﺖ‪ ،‬ﺷﻜﻞﻫﺎﻱ ﻣﻌﺎﺩﻝ‪ ،‬ﻧﺴﺒﺖ ﻣﺴﺎﺣﺖﻫﺎ‪.‬‬

‫ﭼﻜﻴﺪﻩ‪ :‬ﻳﻚ ﻣﺴــﺌﻠﻪ ﺩﺭ ﻣﻮﺿﻮﻉ ﻣﺴــﺎﺣﺖ ﻣﺜﻠﺚ ﻗﺪﻡ ﺑﻪ ﻗﺪﻡ‪،‬‬ ‫ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺭﺍﻫﺒﺮﺩ ﺣﻞ ﻣﺴﺌﻠﻪ ﻭ ﺑﻪ ﻛﺎﺭﮔﻴﺮﻱ ﻋﺒﺎﺭﺕﻫﺎﻱ ﺟﺒﺮﻱ ﺣﻞ‬ ‫ﺷﺪﻩ ﺍﺳﺖ‪.‬‬

‫ﺑﺮﺍﻱ ﺷــﺮﻭﻉ ﻣﺤﻞ ﺑﺮﺧــﻮﺭﺩ ‪ BQ‬ﻭ ‪ CP‬ﺭﺍ ‪ O‬ﺑﻨﺎﻣﻴﺪ ﻭ ‪ AO‬ﺭﺍ‬ ‫ﺭﺳــﻢ ﻛﻨﻴﺪ‪ .‬ﻣﺴﺎﺣﺖ ‪ AOQ‬ﺭﺍ ﺑﺮﺍﺑﺮ ‪ a‬ﻭ ﻣﺴﺎﺣﺖ ‪ AOP‬ﺭﺍ ﺑﺮﺍﺑﺮ ‪b‬‬ ‫‪A‬‬ ‫ﻓﺮﺽ ﻛﻨﻴﺪ‪.‬‬

‫ﺑﻪ ﺷﻜﻞ ﺯﻳﺮ ﻧﮕﺎﻩ ﻛﻨﻴﺪ‪.‬‬

‫‪a‬‬

‫‪Q‬‬

‫‪A‬‬

‫‪8‬‬

‫?‬

‫‪Q‬‬

‫‪10‬‬

‫‪P‬‬

‫‪10‬‬

‫‪5‬‬

‫‪O‬‬

‫‪B‬‬

‫‪C‬‬

‫‪5‬‬

‫‪8‬‬

‫‪b‬‬

‫‪P‬‬

‫‪B‬‬

‫‪ -1‬ﺷﻜﻞ ﺯﻳﺮ ﻗﺴــﻤﺘﻲ ﺍﺯ ﺷﻜﻞ ﺑﺎﻻﺳﺖ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺁﻥ‪ ،‬ﺟﺎﻫﺎﻱ‬ ‫ﺧﺎﻟﻲ ﺭﺍ ﺩﺭ ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﭘﺮ ﻛﻨﻴﺪ‪:‬‬ ‫‪H‬‬

‫‪C‬‬

‫‪Q‬‬ ‫ﺩﺭ ﻣﺜﻠﺚ ‪ ABC‬ﺩﻭ ﭘﺎﺭﻩ ﺧﻂ ‪ BQ‬ﻭ ‪ CP‬ﺭﺳﻢ ﺷﺪﻩﺍﻧﺪ ﻭ ﻣﺜﻠﺚ‬ ‫ﺑﻪ ﭼﻬﺎﺭ ﺗﻜﻪ ﺗﻘﺴﻴﻢ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﻣﺴﺎﺣﺖ ﺳﻪ ﺗﺎ ﺍﺯ ﺗﻜﻪﻫﺎ ﺩﺭﻭﻥ ﺁﻥﻫﺎ‬ ‫ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﻣﺴﺎﺣﺖ ﺗﻜﻪ ﭼﻬﺎﺭﻡ ﺭﺍ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﻳﺪ‪.‬‬ ‫ﻣﺴــﺌﻠﻪ ﺑﻪ ﻧﻈﺮ ﻣﺸﻜﻞ ﺍﺳــﺖ! ﺍﻳﻦﻃﻮﺭ ﻧﻴﺴﺖ؟ ﻗﺪﻡ ﺑﻪ ﻗﺪﻡ ﺑﺎ ﻣﺎ‬ ‫ﭘﻴﺶ ﺑﻴﺎﻳﻴﺪ ﺗﺎ ﺑﺘﻮﺍﻧﻴﺪ ﺍﻳﻦ ﻣﺴﺌﻠﻪ ﺭﺍ ﺣﻞ ﻛﻨﻴﺪ‪.‬‬ ‫‪36‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪O‬‬

‫‪8‬‬ ‫‪10‬‬ ‫‪C‬‬

‫‪B‬‬

‫‪8‬‬ ‫‪10‬‬

‫=‬

‫‪QO‬‬ ‫‪.......‬‬

‫=‬

‫‪QO × ...........‬‬

‫=‬

‫‪BO × ...........‬‬

‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬

‫‪OCQ‬‬

‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬

‫‪OCB‬‬

‫‪ -5‬ﺷــﻜﻞ ﺭﻭﺑﺮﻭ ﻗﺴﻤﺖ ﺩﻳﮕﺮﻱ ﺍﺯ ﺷــﻜﻞ ﺍﺳﺖ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺁﻥ‪،‬‬ ‫ﺟﺎﻫﺎﻱ ﺧﺎﻟﻲ ﺭﺍ ﺩﺭ ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﭘﺮ ﻛﻨﻴﺪ‪A :‬‬

‫‪ -2‬ﺷــﻜﻞ ﺭﻭﺑﺮﻭ ﻗﺴﻤﺖ ﺩﻳﮕﺮﻱ ﺍﺯ ﺷــﻜﻞ ﺍﺳﺖ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺁﻥ‪،‬‬ ‫ﺟﺎﻫﺎﻱ ﺧﺎﻟﻲ ﺭﺍ ﺩﺭ ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﭘﺮ ﻛﻨﻴﺪ‪:‬‬

‫‪P‬‬

‫‪A‬‬

‫‪O‬‬

‫‪Q‬‬ ‫‪H‬‬

‫‪H‬‬

‫‪C‬‬ ‫‪O‬‬

‫‪QO×.............‬‬ ‫‪QO‬‬ ‫‪a‬‬ ‫=‬ ‫=‬ ‫‪BO×............‬‬ ‫‪....... 5+ b‬‬

‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬

‫‪ -6‬ﺗﺴــﺎﻭﻱ ﺯﻳﺮ ﺍﺯ ﺩﻭ ﺗﺴﺎﻭﻱ ﺣﺎﺻﻞ ﺍﺯ ﺳﺆﺍﻻﺕ ‪ 4‬ﻭ ‪ 5‬ﺑﻪ ﺩﺳﺖ‬ ‫ﺁﻣﺪﻩ ﺍﺳﺖ‪:‬‬ ‫‪10 a + 8‬‬

‫‪B‬‬

‫=‬

‫‪OC × ........... OC a +8‬‬ ‫=‬ ‫=‬ ‫=‬ ‫‪OP ×........... ...... ......‬‬

‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬

‫‪AOC‬‬ ‫‪AOP‬‬

‫‪b‬‬

‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬ ‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬

‫‪OAQ‬‬ ‫‪OAB‬‬

‫‪ -3‬ﺗﺴــﺎﻭﻱ ﺯﻳﺮ ﺍﺯ ﺩﻭ ﺗﺴﺎﻭﻱ ﺣﺎﺻﻞ ﺍﺯ ﺳﺆﺍﻻﺕ ‪ 1‬ﻭ ‪ 2‬ﺑﻪ ﺩﺳﺖ‬ ‫ﺁﻣﺪﻩ ﺍﺳﺖ‪:‬‬ ‫‪8‬‬ ‫‪a‬‬ ‫=‬ ‫‪10 5 + b‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺭﺍﺑﻄﻪﻱ ﺑﺎﻻ‪ ،‬ﺗﺴﺎﻭﻱ ﺭﻭﺑﺮﻭ ﺭﺍ ﻛﺎﻣﻞ ﻛﻨﻴﺪ‪:‬‬ ‫‪10a=40+.......‬‬

‫=‬

‫‪5‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺍﻳﻦ ﺭﺍﺑﻄﻪ‪ ،‬ﺗﺴﺎﻭﻱ ﺭﻭﺑﺮﻭ ﺭﺍ ﻛﺎﻣﻞ ﻛﻨﻴﺪ‪:‬‬ ‫‪10b=5a+...‬‬ ‫‪ -7‬ﺩﻭ ﺗﺴــﺎﻭﻱ ﺑﻪ ﺩﺳــﺖ ﺁﻣﺪﻩ ﺩﺭ ﺳــﺆﺍﻻﺕ ‪ 3‬ﻭ ‪ 6‬ﺭﺍ ﻛﻨﺎﺭ ﻫﻢ‬ ‫ﺑﮕﺬﺍﺭﻳﺪ ﻭ ﺳــﻌﻲ ﻛﻨﻴﺪ ﻣﻘﺎﺩﻳﺮﻱ ﺑــﺮﺍﻱ ‪ a‬ﻭ ‪ b‬ﭘﻴﺪﺍ ﻛﻨﻴﺪ ﻛﻪ ﻫﺮ ﺩﻭ‬ ‫ﺗﺴﺎﻭﻱ ﺑﺮﻗﺮﺍﺭ ﺑﺎﺷﻨﺪ‪.‬‬ ‫‪ -8‬ﺍﮔﺮ ﺑﻪ ‪ a=12‬ﻭ ‪ b=10‬ﺭﺳــﻴﺪﻩﺍﻳﺪ‪ ،‬ﻛﺎﺭﺗﺎﻥ ﺩﺭﺳﺖ ﺍﺳﺖ! ﺁﻳﺎ‬ ‫ﻣﻲﺗﻮﺍﻧﻴﺪ ﺭﻭﺵ ﺩﻳﮕﺮﻱ ﺑﺮﺍﻱ ﺣﻞ ﻣﺴــﺌﻠﻪﻱ ﺍﻭﻟﻴﻪ ﭘﻴﺪﺍ ﻛﻨﻴﺪ؟ ﺳﻌﻲ‬ ‫ﻛﻨﻴﺪ!‬

‫‪ -4‬ﺷﻜﻞ ﺯﻳﺮ ﻗﺴﻤﺖ ﺩﻳﮕﺮﻱ ﺍﺯ ﺷﻜﻞ ﻗﺒﻠﻲ ﺍﺳﺖ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺁﻥ‪،‬‬ ‫ﺟﺎﻫﺎﻱ ﺧﺎﻟﻲ ﺭﺍ ﺩﺭ ﻋﺒﺎﺭﺕ ﺯﻳﺮ ﭘﺮ ﻛﻨﻴﺪ‪:‬‬

‫‪P‬‬

‫‪O‬‬

‫‪C‬‬

‫‪H‬‬

‫‪B‬‬

‫‪OC × ............. OC‬‬ ‫=‬ ‫=‬ ‫‪O‬‬ ‫‪OP ×............. ......‬‬

‫=‬

‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬ ‫ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ‬

‫‪BOC‬‬ ‫‪BOP‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪37‬‬

‫ﻣﻌﻤﺎ ﻭ ﺳﺮﮔﺮﻣﻲ‬

‫﹞︺﹝︀﹨︀﹬‪︣︋ ︣﹊︋ ﹩‬ای ︑︀︋︧︐︀ن‬ ‫ﻋﻠﻴﺮﺿﺎ ﻳﻮﺳﻔﻲ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﻣﻌﻤﺎ ﻭ ﺳﺮﮔﺮﻣﻲ‪ ،‬ﻣﺜﻠﺚ‪ ،‬ﺧﻂ ﻣﺴﺘﻘﻴﻢ‪ ،‬ﻗﻄﺎﺭ‪ ،‬ﺳﺎﻋﺖ‪.‬‬ ‫‪ .1‬ﻣﺜﻠﺚ ﺟﺎﺩﻭﻳﻲ؛ ﺳــﻪ ﻋﺪﺩ ‪ 2 ،1‬ﻭ ‪ 3‬ﺩﺭ ﺭﺃﺱﻫﺎﻱ ﻳﻚ ﻣﺜﻠﺚ‬ ‫ﻗﺮﺍﺭ ﺩﺍﺩﻩ ﺷــﺪﻩ ﺍﺳﺖ‪ .‬ﺣﺎﻝ ﺍﻋﺪﺍﺩ ‪ 8 ،7 ،6 ،5 ،4‬ﻭ ‪ 9‬ﺭﺍ ﺩﺭ ﺍﺿﻼﻉ ﺁﻥ‬ ‫ﻃﻮﺭﻱ ﻗﺮﺍﺭ ﺩﻫﻴﺪ ﻛﻪ ﺟﻤﻊ ﻫﺮ ﺿﻠﻊ )ﺑﺎ ﺭﺃﺱﻫﺎﻱ ﺁﻥ( ‪ 17‬ﺷﻮﺩ‪.‬‬ ‫ﻣﺴــﺌﻠﻪﻱ ﻣﺸــﻜﻞﺗﺮ‪ :‬ﺣﺎﻝ ﺍﻋﺪﺍﺩ ‪ 1‬ﺗﺎ ‪ 9‬ﺭﺍ ﺑــﺪﻭﻥ ﺍﻳﻦﻛﻪ ﺑﺪﺍﻧﻴﻢ‬ ‫ﻛﺪﺍﻡ ﻳــﻚ ﺩﺭ ﺭﺃﺱﻫﺎﻱ ﻣﺜﻠﺚ ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ‪ ،‬ﻃــﻮﺭﻱ ﺩﺭ ﺍﻃﺮﺍﻑ ﻣﺜﻠﺚ‬ ‫)ﺭﺃﺱﻫﺎ ﻭ ﺍﺿﻼﻉ( ﻗﺮﺍﺭ ﺩﻫﻴﺪ ﻛﻪ ﺟﻤﻊ ﺁﻥ ‪ 20‬ﺷﻮﺩ‪.‬‬ ‫‪ .2‬ﺗـﻮپ ﺑﺎﺯﻱ ﺩﺧﺘﺮ ﺑﭽﻪﻫﺎ؛ ‪ 12‬ﺩﺧﺘﺮ ﺑﭽﻪ ﺩﺭ ﺩﺍﺧﻞ ﺩﺍﻳﺮﻩﺍﻱ‬ ‫ﺗﻮپ ﺭﺍ ﺑﺮﺍﻱ ﻳﻜﺪﻳﮕﺮ ﭘﺮﺗﺎﺏ ﻣﻲﻛﻨﻨﺪ؛ ﻫﺮﻳﻚ ﺑﺮﺍﻱ ﻧﻔﺮ ﺳــﻤﺖ ﭼﭙﻲ‬ ‫ﺧــﻮﺩ‪ .‬ﺯﻣﺎﻧﻲ ﻛﻪ ﺗﻮپ ﻳــﻚ ﺩﻭﺭ ﻛﺎﻣﻞ ﺭﺍ ﺑﻪ ﺩﻭﺭ ﺩﺍﻳــﺮﻩ ﭼﺮﺧﻴﺪ‪ ،‬ﺩﺭ‬ ‫ﺟﻬﺖ ﻋﻜﺲ ﺗــﻮپ ﺑﻪ ﻧﻘﻄﻪﻱ ﺍﻭﻝ ﺑﻪ ﻫﻤﺎﻥ ﺗﺮﺗﻴﺐ ﺑﺮ ﻣﻲﮔﺮﺩﺩ‪ .‬ﺑﻌﺪ‬ ‫ﺍﺯ ﻣــﺪﺕ ﻛﻮﺗﺎﻫﻲ ﻳﻜﻲ ﺍﺯ ﺩﺧﺘﺮ ﺑﭽﻪﻫﺎ ﭘﻴﺸــﻨﻬﺎﺩ ﺩﺍﺩ‪» :‬ﺣﺎﻻ ﺗﻮپ ﺭﺍ‬ ‫ﻳــﻚ ﺩﺭ ﻣﻴــﺎﻥ ﭘﺮﺗﺎﺏ ﻛﻨﻴﻢ«‪ .‬ﺍﻣﺎ ﺩﻳﮕﺮﻱ ﮔﻔــﺖ‪» :‬ﺍﮔﺮ ﻣﺎ ﺍﻳﻦ ﻛﺎﺭ ﺭﺍ‬ ‫ﺍﻧﺠــﺎﻡ ﺩﻫﻴﻢ ﺗﺎ ﺯﻣﺎﻧﻲ ﻛﻪ ‪ 12‬ﻧﻔﺮ ﻫﺴــﺘﻴﻢ ﻧﺼﻒ ﺩﺧﺘﺮﻫﺎ ﻧﻤﻲﺗﻮﺍﻧﻨﺪ‬ ‫ﺑﺎﺯﻱ ﻛﻨﻨﺪ‪«.‬‬ ‫ﺍﻭﻟﻲ‪» :‬ﺑﻨﺎﺑﺮﺍﻳﻦ ‪ 2‬ﺩﺭ ﻣﻴﺎﻥ ﺍﻳﻦ ﻛﺎﺭ ﺭﺍ ﺍﻧﺠﺎﻡ ﺩﻫﻴﻢ‪«.‬‬ ‫ﺩﻭﻣــﻲ‪» :‬ﺍﻳﻦﻛﻪ ﺑﺪﺗﺮﻩ‪ ،‬ﺑــﺎ ﺍﻳﻦ ﻛﺎﺭ ﺗﻨﻬﺎ ‪ 4‬ﻧﻔــﺮ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺑﺎﺯﻱ‬ ‫ﻛﻨﻨــﺪ؛ ﻣﺎ ﺑﺎﻳﺪ ‪ 4‬ﺩﺭﻣﻴﺎﻥ ﺗــﻮپ ﺭﺍ ﭘﺮﺗﺎﺏ ﻛﻨﻴﻢ ﺗﺎ ﻧﻔــﺮ ﭘﻨﺠﻢ ﺁﻥ ﺭﺍ‬ ‫ﺑﮕﻴﺮﺩ؛ ﺗﺮﻛﻴﺐ ﺩﻳﮕﺮﻱ ﻭﺟﻮﺩ ﻧﺪﺍﺭﺩ‪«.‬‬ ‫ﺍﻭﻟﻲ‪» :‬ﻭ ﺍﮔﺮ ﻣﺎ ‪ 6‬ﻧﻔﺮ ﺭﺍ ﺟﺎ ﺑﮕﺬﺍﺭﻳﻢ؟«‬ ‫ﺩﻭﻣﻲ‪» :‬ﺍﻳﻦ ﻛﺎﺭ ﻣﺎﻧﻨﺪ ﺍﻳﻦ ﺍﺳﺖ ﻛﻪ ﻣﺎ ‪ 4‬ﻧﻔﺮ ﺭﺍ ﺟﺎ ﺑﮕﺬﺍﺭﻳﻢ‪ ،‬ﺗﻨﻬﺎ‬ ‫ﺗﻮپ ﺩﺭ ﺟﻬﺖ ﻣﺨﺎﻟﻒ ﭘﺮﺗﺎﺏ ﻣﻲﺷﻮﺩ‪«.‬‬ ‫ﺍﻭﻟــﻲ‪» :‬ﻭ ﺍﮔــﺮ ﻣﺎ ﻫﺮ ‪ 10‬ﻧﻔــﺮ ﺭﺍ ﺟﺎ ﺑﮕﺬﺍﺭﻳﻢ‪ ،‬ﺑــﻪ ﻃﻮﺭﻱﻛﻪ ﻧﻔﺮ‬ ‫ﻳﺎﺯﺩﻫﻢ ﺗﻮپ ﺭﺍ ﺑﮕﻴﺮﺩ ﭼﻪﻃﻮﺭ؟«‬ ‫ﺩﻭﻣﻲ‪» :‬ﺧﻮﺏ ﻫﻤﻴﻦ ﺣﺎﻻ ﻫﻢ ﺩﺍﺭﻳﻢ ﻫﻤﻴﻦ ﻛﺎﺭ ﺭﺍ ﻣﻲﻛﻨﻴﻢ‪«.‬‬ ‫ﺁﻥﻫﺎ ﺷــﺮﻭﻉ ﻛﺮﺩﻧﺪ ﺑﻪ ﻛﺸﻴﺪﻥ ﺷــﻜﻞﻫﺎﻱ ﻣﺨﺘﻠﻒ ﭘﺮﺗﺎﺏ ﺗﻮپ‬ ‫ﺑــﻪ ﻳﻜﺪﻳﮕﺮ ﻭ ﭘﺲ ﺍﺯ ﻣﺪﺕ ﻛﻮﺗﺎﻫﻲ ﻣﺘﻮﺟﻪ ﺷــﺪﻧﺪ ﻛﻪ ﺣﻖ ﺑﺎ ﺩﻭﻣﻲ‬ ‫ﻳﻌﻨﻲ ﺳﺎﺭﺍ ﺑﻮﺩ‪ .‬ﻳﻌﻨﻲ ﺑﺮﺍﻱ ﺍﻳﻦﻛﻪ ﻫﻤﻪ ﺩﺭ ﺑﺎﺯﻱ ﺷﺮﻛﺖ ﻛﻨﻨﺪ‪ ،‬ﻋﻼﻭﻩ‬ ‫ﺑــﺮ ﺣﺎﻟﺖ ﺍﻭﻝ ﻳﻌﻨﻲ ﺑﻪ ﺗﺮﺗﻴﺐ ﺑﻪ ﻫﻤﻪ ﺗﻮپ ﺭﺍ ﭘﺮﺗﺎﺏ ﻛﺮﺩﻥ‪ ،‬ﺗﻨﻬﺎ ﺩﺭ‬ ‫‪38‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺣﺎﻟﺖ ‪ 4‬ﺩﺭ ﻣﻴﺎﻥ ﻳﺎ ﺣﺎﻟﺖ ﻣﻜﻤﻞ ﺁﻥ ﻳﻌﻨﻲ ‪ 6‬ﺩﺭ ﻣﻴﺎﻥ ﺍﺳﺖ ﻛﻪ ﺗﻮپ‬ ‫ﺑﻪ ﻫﻤﻪ ﻣﻲﺭﺳﺪ ﻭ ﻫﻤﻪ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺩﺭ ﺑﺎﺯﻱ ﺷﺮﻛﺖ ﻛﻨﻨﺪ )ﺷﻜﻞ ﺍﻟﻒ(‪.‬‬ ‫ﺍﻛﻨﻮﻥ ﺍﮔﺮ ﺗﻌﺪﺍﺩ ﻧﻔﺮﺍﺕ ‪ 13‬ﻧﻔﺮ ﺑﺎﺷــﺪ‪ ،‬ﺗﻮپ ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﻳﻚ ﺩﺭ ﻣﻴﺎﻥ‬ ‫)ﺷﻜﻞ ﺏ(‪ 2 ،‬ﺩﺭ ﻣﻴﺎﻥ )ﺷﻜﻞ ﺝ(‪ 3 ،‬ﺩﺭﻣﻴﺎﻥ )ﺷﻜﻞ چ( ﻳﺎ ‪ 4‬ﺩﺭﻣﻴﺎﻥ‬ ‫)ﺷــﻜﻞ ﺡ( ﺑﻪ ﻳﻜﺪﻳﮕﺮ ﭘﺮﺗﺎﺏ ﻛﺮﺩ ﺑﺪﻭﻥ ﺍﻳﻦﻛﻪ ﻛﺴﻲ ﺟﺎ ﺑﻤﺎﻧﺪ‪ .‬ﺣﺎﻻ‬ ‫ﺍﮔﺮ ﺗﻮپ ‪ 5‬ﻳﺎ ‪ 6‬ﺩﺭ ﻣﻴﺎﻥ ﭘﺮﺗﺎﺏ ﺷﻮﺩ ﭼﻪﻃﻮﺭ؟ ﺷﻜﻞ ﺁﻥ ﺭﺍ ﺑﻜﺸﻴﺪ‪.‬‬ ‫‪5‬‬

‫‪13 9‬‬

‫‪4‬‬

‫‪7‬‬

‫‪8‬‬

‫‪1‬‬ ‫‪10‬‬ ‫‪6‬‬

‫‪2‬‬

‫‪8‬‬

‫‪9‬‬ ‫‪13‬‬

‫‪1‬‬

‫‪10‬‬

‫‪10‬‬

‫‪11‬‬

‫‪9‬‬

‫‪4‬‬

‫‪12‬‬

‫‪8‬‬

‫‪4‬‬ ‫‪1‬‬

‫‪13‬‬

‫‪10‬‬

‫‪6‬‬

‫‪9‬‬

‫‪4‬‬

‫‪5‬‬

‫‪3‬‬

‫‪2‬‬

‫‪3‬‬

‫‪11‬‬

‫‪2‬‬

‫‪7‬‬

‫‪10‬‬

‫‪5‬‬

‫‪8‬‬

‫‪3‬‬

‫‪5‬‬

‫‪12‬‬

‫‪1‬‬

‫‪12‬‬

‫‪11‬‬

‫‪6 13‬‬

‫‪12‬‬

‫‪6‬‬

‫‪11‬‬

‫‪3‬‬ ‫‪6‬‬

‫‪7‬‬

‫‪2‬‬

‫‪4‬‬

‫‪9‬‬

‫‪7‬‬

‫‪1‬‬

‫‪12‬‬ ‫‪11‬‬

‫‪7‬‬

‫‪8‬‬

‫‪5‬‬

‫‪2‬‬

‫‪ .3‬ﭼﻬﺎﺭ ﺧﻂ ﻣﺴـﺘﻘﻴﻢ )ﺭﺍﺳـﺖ(؛ ﻣﺮﺑﻌﻲ ﺑﺎ ‪ 9‬ﻧﻘﻄﻪ ﺑﻪ ﻣﺎﻧﻨﺪ‬ ‫ﺷﻜﻞ ﺯﻳﺮ ﺑﺴﺎﺯﻳﺪ‪ .‬ﺑﺪﻭﻥ ﺍﻳﻦﻛﻪ ﻣﺪﺍﺩ ﺭﺍ ﺍﺯ ﺭﻭﻱ ﻛﺎﻏﺬ ﺑﺮﺩﺍﺭﻳﺪ‪ ،‬ﺗﻨﻬﺎ ﺑﺎ‬ ‫ﭼﻬﺎﺭ ﺧﻂ ﺭﺍﺳﺖ ﺗﻤﺎﻡ ﻧﻘﺎﻁ ﺭﺍ ﺑﻪ ﻳﻜﺪﻳﮕﺮ ﻣﺘﺼﻞ ﻛﻨﻴﺪ‪.‬‬

‫‪ .4‬ﺟـﺪﺍ ﻛﺮﺩﻥ ﺑﺰﻫﺎ ﺍﺯ ﻛﻠﻢﻫﺎ؛ ﺍﻛﻨــﻮﻥ ﺑﻪ ﺟﺎﻱ ﻣﺘﺼﻞ ﻛﺮﺩﻥ‬ ‫ﻧﻘﺎﻁ‪ ،‬ﺗﻨﻬﺎ ﺑﺎ ‪ 3‬ﺧﻂ ﺭﺍﺳــﺖ ﺗﻤﺎﻡ ﺑﺰﻫﺎ ﺭﺍ ﺍﺯ ﻛﻠﻢﻫﺎ ﺩﺭ ﺷﻜﻞ ﺯﻳﺮ ﺟﺪﺍ‬ ‫ﻛﻨﻴﺪ‪.‬‬

‫‪ .7‬ﺻﻔﺤﻪﻱ ﺳـﺎﻋﺖ ﺟﻴﺒﻲ؛ ﺁﻳﺎ ﻣﻲﺗﻮﺍﻧﻴﺪ ﻛﺎﺭﻱ ﻛﻨﻴﺪ ﺗﺎ ﺑﺎ ‪2‬‬ ‫ﺧﻂ ﺭﺍﺳــﺖ‪ ،‬ﺻﻔﺤﻪﻱ ﺳﺎﻋﺖ ﺟﻴﺒﻲ ﺭﺍ ﻃﻮﺭﻱ ﺗﻘﺴﻴﻢ ﻛﻨﻴﺪ ﺗﺎ ﺟﻤﻊ‬ ‫ﺍﻋﺪﺍﺩ ﻫﺮ ﻗﺴﻤﺖ ﺑﺎ ﻳﻜﺪﻳﮕﺮ ﺑﺮﺍﺑﺮ ﺑﺎﺷﻨﺪ؟‬ ‫ﺁﻳﺎ ﻣﻲﺗﻮﺍﻧﻴﺪ ﺁﻥ ﺭﺍ ﺑﻪ ‪ 6‬ﻗﺴــﻤﺖ ﺗﻘﺴــﻴﻢ ﻛﻨﻴﺪ ﺑﻪ ﻃﻮﺭﻱﻛﻪ ﻫﺮ‬ ‫ﻗﺴــﻤﺖ ﺷﺎﻣﻞ ﺩﻭ ﻋﺪﺩ ﺑﺎﺷــﺪ ﻛﻪ ﺟﻤﻊ ﻫﺮ ﺩﻭ ﻋﺪﺩ ﺩﺭ ﻫﺮ ﻗﺴﻤﺖ ﺑﺎ‬ ‫ﻳﻜﺪﻳﮕﺮ ﺑﺮﺍﺑﺮ ﺑﺎﺷﻨﺪ‪.‬‬

‫‪11 12 1‬‬ ‫‪2‬‬

‫‪10‬‬

‫‪3‬‬

‫‪9‬‬ ‫‪4‬‬

‫‪ .5‬ﺩﻭ ﻗﻄـﺎﺭ؛ ﻗﻄﺎﺭﻱ ﺑــﺪﻭﻥ ﺗﻮﻗﻒ ﺑﺎ ﺳــﺮﻋﺖ ‪ 60‬ﻛﻴﻠﻮﻣﺘﺮ ﺩﺭ‬ ‫ﺳــﺎﻋﺖ ﺍﺯ ﺗﻬﺮﺍﻥ ﺑﻪ ﻣﻘﺼﺪ ﻣﺸﻬﺪ ﺣﺮﻛﺖ ﻣﻲﻛﻨﺪ ﻭ ﻗﻄﺎﺭ ﺩﻳﮕﺮﻱ ﺩﺭ‬ ‫ﻫﻤﺎﻥ ﺯﻣﺎﻥ ﺑﺎ ﺳــﺮﻋﺖ ‪ 40‬ﻛﻴﻠﻮﻣﺘﺮ ﺩﺭ ﺳﺎﻋﺖ ﺑﺪﻭﻥ ﺗﻮﻗﻒ ﺍﺯ ﻣﺸﻬﺪ‬ ‫ﺑﻪ ﻣﻘﺼﺪ ﺗﻬﺮﺍﻥ ﺣﺮﻛﺖ ﻣﻲﻛﻨﺪ‪ .‬ﺣﺎﻝ ﻫﺮﻳﻚ ﺍﺯ ﺩﻭ ﻗﻄﺎﺭ ﻳﻚ ﺳــﺎﻋﺖ‬ ‫ﻗﺒﻞ ﺍﺯ ﻋﺒﻮﺭ ﺍﺯ ﻳﻜﺪﻳﮕﺮ‪ ،‬ﭼﻪ ﻣﺴﺎﻓﺘﻲ ﭘﻴﻤﻮﺩﻩﺍﻧﺪ‪.‬‬ ‫‪ .6‬ﺟﺰﺭ ﻭ ﻣﺪ ﺩﺭﻳﺎ؛ ﻛﺸــﺘﻲ ﺩﺭ ﻛﻨﺎﺭ ﺳــﺎﺣﻞ ﻟﻨﮕــﺮ ﺍﻧﺪﺍﺧﺘﻪ ﻭ‬ ‫ﻧﺮﺩﺑﺎﻧﻲ ﻛﻪ ﺍﺯ ﻃﻨﺎﺏ ﺳﺎﺧﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﺍﺯ ﻛﺸﺘﻲ ﺗﺎ ﺳﻄﺢ ﺩﺭﻳﺎ ﭘﺎﻳﻴﻦ‬ ‫ﺍﻧﺪﺍﺧﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺍﻳﻦ ﻧﺮﺩﺑﺎﻥ ﺩﺍﺭﺍﻱ ‪ 10‬ﭘﻠﻪ ﺍﺳﺖ ﻛﻪ ﺍﺯ ﻳﻜﺪﻳﮕﺮ ﺑﻪ‬ ‫ﺍﻧﺪﺍﺯﻩﻱ ‪ 12‬ﺍﻳﻨﭻ )‪ 30‬ﺳﺎﻧﺘﻲﻣﺘﺮ( ﻓﺎﺻﻠﻪ ﺩﺍﺭﻧﺪ‪ ،‬ﺁﺧﺮﻳﻦ ﭘﻠﻪ ﺑﻪ ﺳﻄﺢ‬ ‫ﺁﺏ ﺩﺭﻳﺎ ﺭﺳــﻴﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭﻳﺎ ﺩﺭ ﺣﺎﻝ ﺣﺎﺿﺮ ﺁﺭﺍﻡ ﺍﺳﺖ‪ .‬ﺑﻪ ﺩﻟﻴﻞ ﺟﺰﺭ‬ ‫ﻭ ﻣﺪ ﺩﺭﻳﺎ‪ ،‬ﺁﺏ ﺑﺎ ﺳــﺮﻋﺖ ‪ 4‬ﺍﻳﻨﭻ )‪ 10‬ﺳــﺎﻧﺘﻴﻤﺘﺮ( ﺩﺭ ﺳﺎﻋﺖ ﺑﺎﻻ ﻣﻲ‬ ‫ﺁﻳﺪ‪ .‬ﺣﺎﻝ ﺑﮕﻮﻳﻴﺪ ﭼﻪ ﻣﺪﺕ ﻃﻮﻝ ﻣﻲﻛﺸــﺪ ﺗﺎ ﺁﺏ ﺑﻪ ﺳــﻮﻣﻴﻦ ﭘﻠﻪ ﺍﺯ‬ ‫ﺑﺎﻻﻱ ﻧﺮﺩﺑﺎﻥ ﺑﺮﺳﺪ‪.‬‬

‫‪5‬‬

‫‪6‬‬

‫‪7‬‬

‫‪8‬‬

‫‪ .8‬ﺻﻔﺤﻪﻱ ﺳـﺎﻋﺖ ﺷﻜﺴﺘﻪ ﺷﺪﻩ؛ ﺩﺭ ﻳﻚ ﻣﻮﺯﻩﻱ ﻗﺪﻳﻤﻲ‪،‬‬ ‫ﺳﺎﻋﺖ ﻗﺪﻳﻤﻲ ﺭﺍ ﺑﺎ ﺍﻋﺪﺍﺩ ﺭﻭﻣﻲ ﻗﺪﻳﻤﻲ ﺩﻳﺪﻡ ﻛﻪ ﺑﻪ ﺟﺎﻱ ﻋﺪﺩ ﺭﻭﻣﻲ‬ ‫‪ ،(4) IV‬ﻋﺪﺩ ﻗﺪﻳﻤﻲ ‪ IIII‬ﻗﺮﺍﺭ ﺩﺍﺷــﺖ‪ .‬ﺗﺮﻛﻲ ﺩﺭ ﺻﻔﺤﻪﻱ ﺳــﺎﻋﺖ‬ ‫ﺁﻥ ﺭﺍ ﺑﻪ ‪ 4‬ﻗﺴــﻤﺖ ﺗﻘﺴﻴﻢ ﻛﺮﺩﻩ ﺑﻮﺩ‪ .‬ﻫﻤﺎﻥﻃﻮﺭ ﻛﻪ ﺩﺭ ﺗﺼﻮﻳﺮ ﻧﺸﺎﻥ‬ ‫ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺟﻤﻊ ﺍﻋﺪﺍﺩ ﻫﺮ ﻗﺴﻤﺖ ﻛﻪ ﺍﺯ ‪ 17‬ﺗﺎ ‪ 21‬ﺍﺳﺖ ﺑﺎ ﻫﻢ‬ ‫ﺑﺮﺍﺑﺮ ﻧﻴﺴــﺘﻨﺪ‪ .‬ﺁﻳﺎ ﻣﻲﺗﻮﺍﻧﻴﺪ ﺗﻨﻬﺎ ﺑــﺎ ﺗﻐﻴﻴﺮ ﻳﻚ ﺗﺮﻙ ﻛﺎﺭﻱ ﻛﻨﻴﺪ ﻛﻪ‬ ‫ﺟﻤﻊ ﻫﺮ ‪ 4‬ﻗﺴﻤﺖ ‪ 20‬ﺷﻮﺩ؟‬ ‫)ﺭﺍﻫﻨﻤﺎﻳﻲ‪ :‬ﻫﺮ ﺗﺮﻛﻲ ﻛﻪ ﺗﻐﻴﻴﺮ ﻛﻨﺪ ﻧﺒﺎﻳﺪ ﺍﺯ ﻭﺳــﻂ ﺳﺎﻋﺖ ﻋﺒﻮﺭ‬ ‫ﻛﻨﺪ‪(.‬‬

‫‪ .9‬ﺳـﺎﻋﺖ ﺷـﮕﻔﺖﺍﻧﮕﻴﺰ؛ ﺳﺎﻋﺖﺳﺎﺯﻱ ﺷــﺎﮔﺮﺩ ﺧﻮﺩ ﺭ ﺍﺑﺮﺍﻱ‬ ‫ﺗﻌﻮﻳﺾ ﻋﻘﺮﺑﻪﻫﺎﻱ ﺷﻜﺴــﺘﻪﻱ ﻳﻚ ﺳﺎﻋﺖ ﺩﻳﻮﺍﺭﻱ ﺩﺭ ﺧﺎﻧﻪﺍﻱ ﺑﺰﺭگ‬ ‫ﻭ ﻗﺪﻳﻤﻲ ﻣﻲﻓﺮﺳــﺘﺪ‪ .‬ﺷــﺎﮔﺮﺩ ﺍﻭ ﺩﺭ ﺗﺎﺭﻳﻜﻲ ﺷﺐ ﺑﺎ ﻋﺠﻠﻪ ﻋﻘﺮﺑﻪﻫﺎﻱ‬ ‫ﺁﻥ ﺳــﺎﻋﺖ ﺭﺍ ﺗﻌﻮﻳﺾ ﻭ ﺁﻥ ﺭﺍ ﺑﺎ ﺳــﺎﻋﺖ ﺟﻴﺒﻲ ﺧﻮﺩ ﺗﻨﻈﻴﻢ ﻣﻲﻛﻨﺪ‬ ‫ﻭ ﻋﻘﺮﺑــﻪﻱ ﺑﺰﺭگ ﺭﺍ ﺭﻭﻱ ‪ 12‬ﻭ ﻋﻘﺮﺑﻪﻱ ﻛﻮﭼﻚ ﺭﺍ ﺭﻭﻱ ﻋﺪﺩ ‪ 6‬ﻗﺮﺍﺭ‬ ‫ﻣﻲﺩﻫﺪ ﺗﺎ ﺯﻣﺎﻥ ‪ 6‬ﺷــﺐ ﺭﺍ ﻧﺸــﺎﻥ ﺩﻫﺪ ﻭ ﺑﺎ ﻋﺠﻠﻪ ﺑﺮﻣﻲﮔﺮﺩﺩ‪ .‬ﻫﻨﻮﺯ‬ ‫ﺍﺯ ﺭﺳــﻴﺪﻥ ﺍﻭ ﭼﻨﺪ ﺩﻗﻴﻘﻪﺍﻱ ﻧﻤﻲﮔﺬﺭﺩ ﻛﻪ ﺗﻠﻔﻦ ﺑﻪ ﺻﺪﺍ ﺩﺭﻣﻲﺁﻳﺪ ﻭ‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪39‬‬

‫ﺻﺎﺣــﺐ ﺁﻥ ﺧﺎﻧﻪﻱ ﺑﺰﺭگ ﺑﺎ ﻋﺼﺒﺎﻧﻴﺖ ﻣﻲﮔﻮﻳﺪ ﻛﻪ ﺳــﺎﻋﺖ ﺩﻳﻮﺍﺭﻱ‬ ‫ﺍﻭ ﻛﺎﻣﻞ ﺩﺭﺳــﺖ ﻧﺸﺪﻩ ﺍﺳــﺖ ﻭ ﺯﻣﺎﻥ ﺭﺍ ﺍﺷﺘﺒﺎﻩ ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‪ .‬ﻭﻗﺘﻲ‬ ‫ﺷﺎﮔﺮﺩ ﺑﻪ ﺧﺎﻧﻪﻱ ﺁﻥ ﻣﺮﺩ ﻣﻲﺭﻭﺩ ﻭ ﻣﻲﺑﻴﻨﺪ ﻛﻪ ﺳﺎﻋﺖ ﭼﻨﺪ ﺛﺎﻧﻴﻪﺍﻱ‬ ‫ﺍﺯ ‪ 8‬ﮔﺬﺷــﺘﻪ ﺍﺳﺖ ﻭ ﺳﺎﻋﺖ ﺟﻴﺒﻲ ﺧﻮﺩ ﺭﺍ ﺑﻪ ﺁﻥ ﻣﺮﺩ ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‬ ‫ﻭ ﻣﻲﮔﻮﻳﺪ‪» :‬ﺑﺒﻴﻨﻴﺪ ﺳﺎﻋﺖ ﺷﻤﺎ ﺧﻮﺍﺏ ﻧﺮﻓﺘﻪ ﺍﺳﺖ ﻭ ﺯﻣﺎﻥ ﺭﺍ ﺩﺭﺳﺖ‬ ‫ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‪«.‬‬ ‫ﻣﺮﺩ ﺣﺮﻑ ﺍﻭ ﺭﺍ ﺗﺄﻳﻴﺪ ﻣﻲﻛﻨﺪ ﻭ ﺷــﺎﮔﺮﺩ ﺑــﻪ ﻛﺎﺭﮔﺎﻩ ﺑﺮﻣﻲﮔﺮﺩﺩ‪.‬‬ ‫ﺻﺒــﺢ ﻓــﺮﺩﺍﺭﻱ ﺁﻥ ﺭﻭﺯ ﺩﻭﺑﺎﺭﻩ ﺁﻥ ﻣﺮﺩ ﺯﻧﮓ ﻣﻲﺯﻧــﺪ ﻭ ﺑﺎ ﻧﺎﺭﺍﺣﺘﻲ ﻭ‬ ‫ﺷــﮕﻔﺘﻲ ﻣﻲﮔﻮﻳﺪ ﻛﻪ ﺳــﺎﻋﺖ ﺍﻭ ﺩﻳﻮﺍﻧﻪ ﺷﺪﻩ ﻭ ﻫﺮ ﺯﻣﺎﻧﻲ ﺭﺍ ﻛﻪ ﺩﻟﺶ‬ ‫ﺑﺨﻮﺍﻫﺪ ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ!‬ ‫ﺷــﺎﮔﺮ ﺑﻪ ﻣﻨﺰﻝ ﺁﻥ ﻣﺮﺩ ﻣﻲﺭﻭﺩ ﻭ ﺳــﺎﻋﺖ ﺍﻭ ﺭﺍ ﺑﺮﺭﺳﻲ ﻣﻲﻛﻨﺪ ﻭ‬ ‫ﭘﺲ ﺍﺯ ﭼﻚ ﻛﺮﺩﻥ ﺁﻥ ﺑﺎ ﺳــﺎﻋﺖ ﺧﻮﺩ ﺑﺎ ﻛﻤــﺎﻝ ﺗﻌﺠﺐ ﻣﻲﺑﻴﻨﺪ ﻛﻪ‬ ‫ﺳــﺎﻋﺖ ﺯﻣﺎﻥ ﺩﺭﺳﺖ ﺭﺍ ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‪ .‬ﺑﻠﻪ ﺳﺎﻋﺖ ﻛﻤﻲ ﺍﺯ ‪ 7‬ﮔﺬﺷﺘﻪ‬ ‫ﺭﺍ ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‪.‬‬ ‫ﺷﺎﮔﺮﺩ ﺳﺎﻋﺖﺳﺎﺯ ﺑﺎ ﻋﺼﺒﺎﻧﻴﺖ ﺑﻪ ﺁﻥ ﻣﺮﺩ ﻣﻲﮔﻮﻳﺪ‪» :‬ﺳﺎﻋﺖ ﺷﻤﺎ‬ ‫ﻛﻪ ﺩﺭﺳﺖ ﻛﺎﺭ ﻣﻲﻛﻨﺪ‪ ،‬ﻣﻦ ﺭﺍ ﺳﺮ ﻛﺎﺭ ﻣﻲﮔﺬﺍﺭﻱ؟ !«‬

‫‪C‬‬ ‫‪E‬‬ ‫‪B‬‬

‫‪A‬‬

‫‪F‬‬

‫‪D‬‬

‫ﺣﺎﻝ ‪ 3‬ﺩﮔﻤــﻪ ﺭﺍ ﺑﺮﺩﺍﺭﻳﺪ ﻭ ‪ 6‬ﺩﮔﻤﻪﻱ ﺑﺎﻗﻲﻣﺎﻧﺪﻩ ﺭﺍ ﺩﺭ ‪ 3‬ﺭﺩﻳﻒ‬ ‫ﻃــﻮﺭﻱ ﻗــﺮﺍﺭ ﺩﻫﻴﺪ ﻛﻪ ﻫﺮ ﺭﺩﻳــﻒ ﺩﺍﺭﺍﻱ ‪ 3‬ﺩﮔﻤﻪ ﺑﺎﺷــﺪ )ﺍﻳﻦﺑﺎﺭ ﺍﺯ‬ ‫ﺭﺩﻳﻒﻫﺎﻱ ﻓﺮﻋﻲ ‪ 2‬ﺩﮔﻤﻪﺍﻱ ﺻﺮﻑﻧﻈﺮ ﻛﻨﻴﺪ(‪.‬‬ ‫‪ .11‬ﭼﻴﺪﻥ ‪ 16‬ﻣﻬﺮﻩ ﺩﺭ ﻳﻚ ﺩﻭﺯ ‪ 10‬ﺭﺩﻳﻔﻲ ﻛﻪ ﺩﺭ ﻫﺮ ﻛﺪﺍﻡ‬ ‫‪ 4‬ﻣﻬﺮﻩ ﻗﺮﺍﺭ ﺩﺍﺷـﺘﻪ ﺑﺎﺷـﺪ‪ ،‬ﻛﺎﺭ ﺁﺳﺎﻧﻲ ﺍﺳــﺖ‪ ،‬ﻭﻟﻲ ﺳﺨﺖﺗﺮ ﺍﻳﻦ‬ ‫ﺍﺳﺖ ﻛﻪ ‪ 9‬ﻣﻬﺮﻩ ﺭﺍ ﺩﺭ ﻳﻚ ﺩﻭﺯ ‪ 6‬ﺭﺩﻳﻔﻲ ﻛﻪ ﻫﺮ ﻛﺪﺍﻡ ﺩﺍﺭﺍﻱ ‪ 3‬ﻣﻬﺮﻩ‬ ‫ﺍﺳﺖ ﻗﺮﺍﺭ ﺩﻫﻴﺪ‪.‬‬ ‫‪ .12‬ﺍﻟﮕﻮ ﻭ ﻃﺮﺡ ﭼﻴﻨﺶ ﺳﻜﻪﻫﺎ؛ ﺑﺮگ ﻛﺎﻏﺬﻱ ﺑﺮﺩﺍﺭﻳﺪ ﻭ ﺷﻜﻞ‬ ‫ﺯﻳﺮ ﺭﺍ ﺩﺭ ﺁﻥ ﻛﭙﻲ ﻛﻨﻴﺪ ﻭ ﺁﻥﺭﺍ ﺗﺎ ‪ 3‬ﺑﺮﺍﺑﺮ ﺑﺰﺭگ ﻛﻨﻴﺪ ﻭ ‪ 17‬ﺳﻜﻪ ﺑﻪ‬ ‫ﺷﻜﻞ ﺯﻳﺮ ﺁﻣﺎﺩﻩ ﻛﻨﻴﺪ‪:‬‬ ‫‪ 5‬ﻋﺪﺩ‬ ‫ﺳﻜﻪﻱ ‪ 20‬ﺭﻳﺎﻟﻲ‬ ‫‪ 3‬ﻋﺪﺩ‬ ‫ﺳﻜﻪﻱ ‪ 15‬ﺭﻳﺎﻟﻲ‬ ‫‪ 3‬ﻋﺪﺩ‬ ‫ﺳﻜﻪﻱ ‪ 10‬ﺭﻳﺎﻟﻲ‬ ‫‪ 6‬ﻋﺪﺩ‬ ‫ﺳﻜﻪﻱ ‪ 5‬ﺭﻳﺎﻟﻲ‬ ‫ﺩﺭﻫﺮ ﺧﺎﻧﻪ ﺳــﻜﻪﻫﺎ ﺭﺍ ﻃﻮﺭﻱ ﻗﺮﺍﺭ ﺩﻫﻴﺪ )ﻫﺮ ﺧﺎﻧﻪ ﻳﻚ ﺳﻜﻪ( ﻛﻪ‬ ‫ﺟﻤﻊ ﺁﻥﻫﺎ ﺩﺭ ﻫﺮ ﺧﻂ ﺭﺍﺳﺖ ‪ 55‬ﺭﻳﺎﻝ ﺷﻮﺩ‪.‬‬

‫ﺣﺎﻝ‪ ،‬ﺷﻤﺎ ﺑﮕﻮﻳﻴﺪ ﻛﻪ ﭼﻪ ﺍﺗﻔﺎﻗﻲ ﺍﻓﺘﺎﺩﻩ ﺍﺳﺖ‪.‬‬ ‫‪ .10‬ﺳـﻪ ﺗﺎ ﺩﺭ ﻳﻚ ﺭﺩﻳﻒ؛ ﺩﺭ ﻳﻚ ﺟﺪﻭﻝ ‪ 9‬ﺩﮔﻤﻪ ﺭﺍ ﺑﻪ ﺷﻜﻞ‬ ‫ﻣﺮﺑﻊ ﺳــﻪ ﺩﺭ ﺳــﻪ ﻗﺮﺍﺭ ﺩﻫﻴــﺪ‪ .‬ﺯﻣﺎﻧﻲ ﻛﻪ ‪ 2‬ﺩﮔﻤﻪ ﻳــﺎ ﺑﻴﺶﺗﺮ ﺍﺯ ﺁﻥ‬ ‫ﺩﺭ ﻳﻚ ﺧﻂ ﺭﺍﺳــﺖ ﻗﺮﺍﺭ ﮔﺮﻓﺖ ﻣﻲﮔﻮﻳﻴــﻢ ﺁﻥﻫﺎ ﺩﺭ ﻳﻚ ﺭﺩﻳﻒ ﻗﺮﺍﺭ‬ ‫ﮔﺮﻓﺘﻨﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ‪ ،‬ﺭﺩﻳﻒﻫﺎﻱ ‪ AB‬ﻭ ‪ CD‬ﻫﺮ ﻛﺪﺍﻡ ﺩﺍﺭﺍﻱ ﺩﻭ ﺩﮔﻤﻪ ﻭ‬ ‫ﺭﺩﻳﻒ ‪ EF‬ﺩﺍﺭﺍﻱ ‪ 2‬ﺩﮔﻤﻪ ﺍﺳﺖ‪.‬‬ ‫ﭼﻪ ﺗﻌﺪﺍﺩ ﺭﺩﻳﻒﻫﺎﻱ ‪ 2‬ﻭ ‪ 3‬ﺩﮔﻤﻪﺍﻱ ﺩﺭ ﺷﻜﻞ ﻭﺟﻮﺩ ﺩﺍﺭﺩ؟‬ ‫‪40‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪ .13‬ﺍﺯ ﻳـﻚ ﺗـﺎ ‪19‬؛ ﺍﻋﺪﺍﺩ ‪ 1‬ﺗﺎ ‪ 19‬ﺭﺍ ﺩﺭ ﺧﺎﻧﻪﻫﺎﻱ ﺷــﻜﻞ ﺯﻳﺮ‬ ‫ﻃﻮﺭﻱ ﻗﺮﺍﺭ ﺩﻫﻴﺪ ﺗﺎ ﺟﻤﻊ ﺳﻪ ﻋﺪﺩ ﻭﺍﻗﻊ ﺩﺭ ﻳﻚ ﺩﺍﻳﺮﻩ ‪ 30‬ﺷﻮﺩ‪.‬‬

‫‪ .14‬ﺑﺎ ﺳـﺮﻋﺖ ﻭ ﺩﺭﻋﻴﻦ ﺣﺎﻝ ﻫﻮﺷﻤﻨﺪﺍﻧﻪ؛ ﻋﻨﻮﺍﻥ ﻣﺴﺌﻠﻪ ﺑﻪ‬ ‫ﺷﻤﺎ ﻣﻲﮔﻮﻳﺪ ﻛﻪ ﭼﻪﻃﻮﺭ ﻣﺴﺌﻠﻪ ﺭﺍ ﺣﻞ ﻛﻨﻴﺪ‪.‬‬ ‫ﺍﻟﻒ( ﺍﺗﻮﺑﻮﺳﻲ ﺍﺯ ﺗﻬﺮﺍﻥ ﺑﻪ ﺳﻮﻱ ﻗﻢ ﺣﺮﻛﺖ ﻣﻲﻛﻨﺪ ﻭ ﻳﻚ ﺳﺎﻋﺖ‬ ‫ﺑﻌﺪ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭﻱ ﺍﺯ ﻗﻢ ﻭ ﺍﻟﺒﺘﻪ ﺑﺎ ﺳﺮﻋﺖ ﻛﻢﺗﺮ ﺍﺯ ﺍﺗﻮﺑﻮﺱ ﺑﻪ ﺳﻮﻱ‬ ‫ﺗﻬﺮﺍﻥ ﺣﺮﻛﺖ ﻣﻲﻛﻨﺪ‪ .‬ﺯﻣﺎﻧﻲ ﻛﻪ ﻫﺮ ﺩﻭ ﺑﻪ ﻫﻢ ﻣﻲﺭﺳــﻨﺪ ﻛﺪﺍﻡ ﻳﻚ‬ ‫ﺍﺯ ﺗﻬﺮﺍﻥ ﺩﻭﺭﺗﺮ ﻫﺴﺘﻨﺪ‪.‬‬ ‫‪10‬‬ ‫ﺏ( ﻛﺪﺍﻡﻳﻚ ﺑﺎ ﺍﺭﺯﺵﺗﺮ ﺍﺳــﺖ‪ .‬ﻳﻚ ﻛﻴﻠﻮ ﻃﻼﻱ ﻫﺰﺍﺭ ﺗﻮﻣﺎﻧﻲ‬ ‫ﻳﺎ ﻧﻴﻢﻛﻴﻠﻮ ﻃﻼﻱ ‪ 20‬ﻫﺰﺍﺭ ﺗﻮﻣﺎﻧﻲ؟‬ ‫ﺝ( ﺳﺎﻋﺖ ‪ 6‬ﺯﻧﮓ ﺳﺎﻋﺖ ﺩﻳﻮﺍﺭﻱ ‪ 6‬ﺑﺎﺭ ﺑﻪ ﺻﺪﺍ ﺩﺭﻣﻲﺁﻳﺪ‪ .‬ﺑﺎ ﻧﮕﺎﻩ‬ ‫ﺑﻪ ﺳــﺎﻋﺖ ﻣﭽﻲ ﺧﻮﺩ ﻣﺘﻮﺟﻪ ﺷــﺪﻡ ﻛﻪ ﺑﻴﻦ ﻫﺮ ﺑﺎﺭ ﺑﻪ ﺻﺪﺍ ﺩﺭﺁﻣﺪﻥ‬ ‫ﺯﻧﮓ ﺳﺎﻋﺖ ‪ 30‬ﺛﺎﻧﻴﻪ ﻃﻮﻝ ﻣﻲﻛﺸﺪ ﺗﺎ ﺳﺎﻋﺖ ﺩﻳﻮﺍﺭﻱ ﺩﺭ ﻧﻴﻤﻪ ﺷﺐ‬ ‫‪ 12‬ﺑﺎﺭ ﺑﻪ ﺻﺪﺍ ﺩﺭﺁﻳﺪ؟‬ ‫ﺩ( ﺳﻪ ﭘﺮﺳــﺘﻮ ﺍﺯ ﻳﻚ ﻧﻘﻄﻪ ﺑﻪ ﺑﻴﺮﻭﻥ ﭘﺮﻭﺍﺯ ﻣﻲﻛﻨﻨﺪ‪ .‬ﭼﻪ ﺯﻣﺎﻧﻲ‬ ‫ﺁﻥﻫﺎ ﺩﺭ ﻳﻚ ﺳﻄﺢ ﺩﺭ ﻓﻀﺎ ﻗﺮﺍﺭ ﻣﻲﮔﻴﺮﻧﺪ؟‬ ‫‪ .15‬ﺧﺮﭼﻨـﮓ ﭘﺮ ﺍﺯ ﺍﺷـﻜﺎﻝ ﻣﺨﺘﻠﻒ؛ ﺧﺮﭼﻨــﮓ ﺯﻳﺮ ﺍﺯ ‪17‬‬ ‫ﻗﻄﻌﻪﻱ ﺷﻤﺎﺭﻩﮔﺬﺍﺭﻱ ﺷﺪﻩ ﺗﺸﻜﻴﻞ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺁﻥ ﺭﺍ ﺭﻭﻱ ﻳﻚ ﻛﺎﻏﺬ‬ ‫ﻛﭙﻲ ﻛﻨﻴﺪ ﻭ ﺑﺎ ﻗﻴﭽﻲ ﺁﻥﻫﺎ ﺭﺍ ﺍﺯ ﻫﻢ ﺟﺪﺍ ﻛﻨﻴﺪ‪ .‬ﺣﺎﻝ ﺑﺎ ﺍﺳــﺘﻔﺎﺩﻩ ﺍﺯ‬ ‫ﺗﻤﺎﻡ ﻧﻘﺎﻁ ﻗﻄﻌﺎﺕ ﺁﻥ‪ ،‬ﻳﻚ ﺩﺍﻳﺮﻩ ﺑﺴــﺎﺯﻳﺪ ﻭ ﺑﺎ ﻛﻨﺎﺭﻩﻫﺎﻱ ﺁﻥﻫﺎ ﻳﻚ‬ ‫ﻣﺮﺑﻊ‪.‬‬

‫‪3‬‬ ‫‪1‬‬

‫‪2‬‬ ‫‪2‬‬

‫‪3‬‬ ‫‪4‬‬

‫‪1‬‬

‫‪3‬‬

‫‪3‬‬

‫‪5‬‬ ‫‪5‬‬

‫‪6‬‬ ‫‪6‬‬

‫‪7‬‬ ‫‪8‬‬

‫‪7‬‬

‫‪ .17‬ﻣﮕﺲ ﺑﻲﻗﺮﺍﺭ؛ ﺩﻭ ﺩﻭﭼﺮﺧﻪﺳــﻮﺍﺭ ﻳﻜﻲ ﺍﺯ ﺗﻬﺮﺍﻥ ﺑﻪ ﺳﻤﺖ‬ ‫ﺳﻤﻨﺎﻥ ﻭ ﺩﻳﮕﺮﻱ ﺍﺯ ﺳﻤﻨﺎﻥ ﺑﻪ ﻃﺮﻑ ﺗﻬﺮﺍﻥ ﺑﻪ ﻃﻮﺭ ﻫﻤﺰﻣﺎﻥ ﺷﺮﻭﻉ ﺑﻪ‬ ‫ﺣﺮﻛﺖ ﻛﺮﺩﻧﺪ‪ .‬ﻭﻗﺘﻲ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭﺍﻥ ‪ 180‬ﻛﻴﻠﻮﻣﺘﺮ ﺍﺯ ﻳﻜﺪﻳﮕﺮ ﻓﺎﺻﻠﻪ‬ ‫ﺩﺍﺷﺘﻨﺪ‪ ،‬ﻣﺎﺟﺮﺍﺟﻮﻳﻲ ﻣﮕﺲ ﺷﺮﻭﻉ ﺷﺪ‪ .‬ﺍﺯ ﺷﺎﻧﻪﻱ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭ ﺍﻭﻟﻲ‬ ‫ﺷــﺮﻭﻉ ﺑﻪ ﭘﺮﻭﺍﺯ ﻛﺮﺩ ﺗﺎ ﺑﻪ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭ ﺩﻭﻣﻲ ﺑﺮﺳﺪ‪ .‬ﻭﻗﺘﻲ ﻣﮕﺲ ﺑﻪ‬ ‫ﺩﻭﻣﻲ ﺭﺳﻴﺪ‪ ،‬ﺳﺮﻳﻊ ﺑﺪﻭﻥ ﺗﻮﻗﻒ ﺑﺮﮔﺸﺖ‪ .‬ﻣﮕﺲ ﻫﻤﻴﻦﻃﻮﺭ ﺑﻪ ﺣﺮﻛﺖ‬ ‫ﺭﻓﺖ ﻭ ﺁﻣﺪ ﺧﻮﺩ ﺍﺩﺍﻣﻪ ﺩﺍﺩ ﺗﺎ ﺩﻭ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭ ﺑﻪ ﻳﻜﺪﻳﮕﺮ ﺭﺳﻴﺪﻧﺪ‪ .‬ﺩﺭ‬ ‫ﺍﻳﻦ ﻫﻨﮕﺎﻡ ﻣﮕﺲ ﺭﻭﻱ ﺑﻴﻨﻲ ﻳﻜﻲ ﺍﺯ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭﺍﻥ ﻧﺸﺴﺖ‪.‬‬

‫ﺳــﺮﻋﺖ ﻣﮕﺲ ‪ 30‬ﻛﻴﻠﻮﻣﺘﺮ ﺩﺭ ﺳﺎﻋﺖ ﻭ ﺳﺮﻋﺖ ﺩﻭﭼﺮﺧﻪﺳﻮﺍﺭﺍﻥ‬ ‫ﻧﻴﺰ ‪ 15‬ﻛﻴﻠﻮﻣﺘﺮ ﺩﺭ ﺳــﺎﻋﺖ ﺍﺳﺖ‪ .‬ﺣﺎﻝ ﺑﮕﻮﻳﻴﺪ ﻣﮕﺲ ﭼﻪ ﻣﺴﺎﻓﺘﻲ ﺭﺍ‬ ‫ﭘﻴﻤﻮﺩﻩ ﺍﺳﺖ؟‬ ‫‪ .18‬ﺳﺎﻝ ﻭﺍﺭﻭﻧﻪ؛ ﺁﺧﺮﻳﻦ ﺳﺎﻟﻲ ﻛﻪ ﺗﺎﻛﻨﻮﻥ ﻭﺍﺭﻭﻧﻪﻱ ﺁﻥ ﺭﺍ ﻣﺎﻧﻨﺪ‬ ‫ﺧﻮﺩﺵ ﺍﺳﺖ‪ ،‬ﻛﺪﺍﻡ ﺍﺳﺖ؟ )ﺑﻪ ﻫﺠﺮﻱ ﻗﻤﺮﻱ‪ ،‬ﺷﻤﺴﻲ ﻭ ﻣﻴﻼﺩﻱ(‬ ‫‪ .19‬ﺩﻭ ﻟﻄﻴﻔﻪ؛‬ ‫ﺍﻟﻒ( ﻣﺮﺩﻱ ﺑﻪ ﭘﺴﺮﺵ ﺗﻠﻔﻦ ﻣﻲﻛﻨﺪ ﻭ ﺍﺯ ﺍﻭ ﻣﻲﺧﻮﺍﻫﺪ ﺗﺎ ﻣﻘﺪﺍﺭﻱ‬ ‫ﻭﺳﺎﻳﻞ ﻻﺯﻡ ﺑﺮﺍﻱ ﺍﻭ ﺑﺨﺮﺩ ﺗﺎ ﺑﺮﺍﻱ ﻣﺴﺎﻓﺮﺕ ﺁﻣﺎﺩﻩ ﺷﻮﺩ‪ .‬ﭘﺪﺭ ﺑﻪ ﭘﺴﺮﺵ‬ ‫ﻣﻲﮔﻮﻳــﺪ ﻛﻪ ﭘﻮﻝ ﻛﺎﻓﻲ ﺑــﺮﺍﻱ ﺍﻭ ﺭﻭﻱ ﻣﻴﺰ ﺩﺭ ﺩﺍﺧﻞ ﭘﺎﻛﺖ ﻗﺮﺍﺭ ﺩﺍﺩﻩ‬ ‫ﺍﺳــﺖ‪ .‬ﻭﻗﺘﻲ ﭘﺴﺮ ﺑﻪ ﺍﺗﺎﻕ ﭘﺪﺭ ﻣﻲﺭﻭﺩ ﭘﺎﻛﺘﻲ ﺭﺍ ﭘﻴﺪﺍ ﻣﻲﻛﻨﺪ ﻛﻪ ﺭﻭﻱ‬ ‫ﺁﻥ ﻋﺪﺩ ‪ 87‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﻓﺮﻭﺷﮕﺎﻩ ﭘﺴﺮ ﺑﻪ ﺍﻧﺪﺍﺯﻩﻱ ‪ 80‬ﻫﺰﺍﺭ‬ ‫ﺗﻮﻣﺎﻥ ﺧﺮﻳﺪ ﻣﻲﻛﻨﺪ‪ ،‬ﻭﻟﻲ ﻣﻮﻗﻊ ﭘﺮﺩﺍﺧﺖ ﻧﻪ ﺗﻨﻬﺎ ‪ 7‬ﻫﺰﺍﺭ ﺗﻮﻣﺎﻥ ﺍﺿﺎﻓﻪ‬ ‫ﻧﻤــﻲﺁﻭﺭﺩ‪ ،‬ﺑﻠﻜﻪ ﺑﺪﻫﻜﺎﺭ ﻧﻴﺰ ﻣﻲﺷــﻮﺩ! ﺣﺎﻝ ﺑﮕﻮﻳﻴــﺪ ﭼﻪﻗﺪﺭ ﺑﺪﻫﻜﺎﺭ‬ ‫ﻣﻲﺷﻮﺩ ﻭ ﭼﺮﺍ؟‬ ‫ﺏ( ﺍﻋﺪﺍﺩ ‪ 1‬ﺗﺎ ‪ 9‬ﺭﺍ ﺭﻭﻱ ﺗﻜﻪ ﻛﺎﻏﺬ ﻳﺎﺩﺩﺍﺷﺖ ﻛﺮﺩﻩ ﻭ ﺁﻥﻫﺎ ﺭﺍ ﺩﺭ‬ ‫ﺩﻭ ﺭﺩﻳﻒ ﻣﺎﻧﻨﺪ ﺷــﻜﻞ ﺯﻳﺮ ﻗﺮﺍﺭ ﺩﻫﻴﺪ‪ .‬ﺣﺎﻝ ﺩﻭ ﺗﻜﻪ ﻛﺎﻏﺬ ﺭﺍ ﻃﻮﺭﻱ‬ ‫ﺟﺎﺑﻪﺟﺎ ﻛﻨﻴﺪ ﺗﺎ ﺟﻤﻊ ﺍﻋﺪﺍﺩ ﻫﺮ ﺩﻭ ﺳﺘﻮﻥ ﺑﺎ ﻫﻢ ﺑﺮﺍﺑﺮ ﺷﻮﺩ‪.‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪41‬‬

‫ﭘﺎﺳﺦﻫﺎ ﺩﺭ ﺷﻤﺎﺭﻩﻱ ﺁﻳﻨﺪﻩﻱ ﻣﺠﻠﻪ‬

‫‪4‬‬

‫‪ .16‬ﻗﻴﻤـﺖ ﻳﻚ ﻛﺘـﺎﺏ؛ ﻗﻴﻤﺖ ﻳﻚ ﻛﺘﺎﺏ ﺑﺮﺍﺑﺮ ﺍﺳــﺖ ﺑﺎ ﻫﺰﺍﺭ‬ ‫ﺗﻮﻣﺎﻥ ﺑﻪﻋﻼﻭﻩﻱ ﻧﺼﻒ ﻗﻴﻤﺖ ﺁﻥ‪ .‬ﺣﺎﻝ ﻗﻴﻤﺖ ﻛﺘﺎﺏ ﭼﻪﻗﺪﺭ ﺍﺳﺖ؟‬

‫︎︀︨︞ ر﹬︀︲﹫︀ت ر﹇︀︋︐‪﹩‬‬ ‫ﭘﺎﺳﺦ ‪ : 1‬ﮔﺰﻳﻨﻪﻱ ‪ 1‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔﺮ ﺗﻌﺪﺍﺩ ﺳﻜﻪﻫﺎﻱ ‪ 5‬ﺭﻳﺎﻟﻲ ﺭﺍ ‪ x‬ﻭ ﺗﻌﺪﺍﺩ ﺳﻜﻪﻫﺎﻱ ‪ 20‬ﺭﻳﺎﻟﻲ ﺭﺍ‬ ‫‪ y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪− 20 y = −700‬‬

‫‪×( −2 0 ) ⎧−20 x‬‬

‫⎨‬ ‫‪⎩5 x + 20 y = 625‬‬

‫‪− 15 x = −75 ⇒ x = 5‬‬

‫‪⎧x + y = 35‬‬ ‫⎨‬ ‫‪⎩5 x + 20 y = 625‬‬

‫⇒‬

‫ﭘﺎﺳﺦ ‪ : 4‬ﮔﺰﻳﻨﻪﻱ ‪ 2‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔﺮ ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﻧﻴﻤﺎ ﺭﺍ ‪ x‬ﻭ ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﺳﻴﻨﺎ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪،‬‬ ‫ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪⎧ x + y =105 00‬‬ ‫‪⎧ x + y =1 0 500‬‬ ‫‪⎧⎪ x + y =1 0 5 00‬‬ ‫⎪‬ ‫⎪‬ ‫⎨⇒‬ ‫⇒‬ ‫‪⎨ 1‬‬ ‫⎨‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪5‬‬ ‫‪⎩⎪2x −5 y = 0‬‬ ‫) ‪⎪ x − x =2( y − y‬‬ ‫‪⎪ x =2× y‬‬ ‫‪6‬‬ ‫‪6‬‬ ‫‪⎩ 3‬‬ ‫‪⎩3‬‬

‫‪⇒ y = 30‬‬

‫‪⎧−2x −2 y =−2100‬‬ ‫⎨‬ ‫‪⎩2x −5 y = 0‬‬ ‫‪−7 y=−21000⇒ y=3 000‬‬

‫⇒‬

‫ﭘﺎﺳﺦ ‪ : 2‬ﮔﺰﻳﻨﻪﻱ ‪ 2‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔﺮ ﻛﺴﺮ ﻣﻮﺭﺩﻧﻈﺮ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ‪ x‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ‬ ‫‪y‬‬ ‫ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪⎪⎧3 x −2 y= 0‬‬ ‫‪⎧3 x −2 y= 0‬‬ ‫⎨ ⇒‬ ‫⎨‬ ‫‪1‬‬ ‫‪⎪⎩− x + y =7‬‬ ‫‪⎩−3 x + 3 y =21‬‬

‫‪y=21‬‬ ‫‪⇒ x =14‬‬ ‫‪⇒ x + y=35‬‬

‫‪⎧⎪ x + y =1 0 5 00‬‬ ‫⎨⇒‬ ‫‪⎪⎩2x −5 y = 0‬‬

‫‪⇒ x = 7500‬‬

‫‪⎧x 2‬‬ ‫= ⎪‬ ‫⇒ ‪⎨y 3‬‬ ‫‪⎪ y − x =7‬‬ ‫‪×3‬‬ ‫⎩‬

‫ﭘﺎﺳﺦ ‪ : 3‬ﮔﺰﻳﻨﻪﻱ ‪ 2‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔــﺮ ﻋﺪﺩ ﻃﺒﻴﻌﻲ ﻣﻮﺭﺩﻧﻴــﺎﺯ ﺭﺍ ‪ x‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳــﻢ‪ ،‬ﺍﻋﺪﺍﺩ ﺑﻌﺪﻱ‬ ‫ﻋﺒﺎﺭﺗﻨﺪ ﺍﺯ‪:‬‬ ‫‪x , x + 1 , x + 2 , x + 3 , x + 4 , ...‬‬

‫ﺑﻨﺎﺑﺮﺍﻳــﻦ ﺍﮔﺮ ﺳــﻪ ﻋﺪﺩ ﻓــﺮﺩ ﺭﺍ ‪ x+2 ،x‬ﻭ ‪ x+3‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪،‬‬ ‫ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪x + ( x + 2) + ( x + 4) = 87 ⇒ 3 x + 6 = 87‬‬ ‫‪⎪⎧ x +2=29‬‬

‫⎨ ⇒ ‪⇒ 3 x = 81 ⇒ x = 27‬‬

‫‪⎪⎩ x +4=31‬‬

‫‪⎧ x =27‬‬ ‫⎪‬ ‫‪⇒ ⎨ x +2=29 ⇒ 7+9+1=17‬‬ ‫‪⎪ x +4=31‬‬ ‫⎩‬

‫‪42‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﭘﺎﺳﺦ ‪ : 5‬ﮔﺰﻳﻨﻪﻱ ‪ 4‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫‪x‬‬ ‫ﺍﮔﺮ ﻛﺴﺮ ﻣﻮﺭﺩﻧﻈﺮ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ‬ ‫‪4‬‬

‫‪x +3‬‬

‫ﺩﺍﺭﻳﻢ‪ y+3 = 5 :‬ﻭ‬

‫‪x −3 1‬‬ ‫=‬ ‫‪y −3 2‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎﻳﺪ ﺩﺳــﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟــﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ‬ ‫ﺣﻞ ﻛﻨﻴﻢ‪.‬‬

‫‪⎧ x +3 4‬‬ ‫=‬ ‫⎪‬ ‫‪⎪ y+3 5‬‬ ‫⎨‬ ‫‪⎪ x −3 = 1‬‬ ‫‪⎪⎩ y−3 2‬‬

‫ﺭﺍ‬

‫‪⎧ x +3 4‬‬ ‫=‬ ‫⎪‬ ‫‪⎪⎧5 x −4 y =−3‬‬ ‫) ‪⎪⎧5( x +3 )=4( y +3‬‬ ‫‪⎪ y+3 5‬‬ ‫⎨ ⇒‬ ‫⎨ ⇒‬ ‫⎨‬ ‫‪x‬‬ ‫‪−‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪⎩⎪2x − y =3‬‬ ‫‪⎩⎪2( x −3 )= y −3‬‬ ‫⎪‬ ‫=‬ ‫‪⎪⎩ y −3 2‬‬ ‫‪⎧5 x −4 y =−3‬‬ ‫‪⎧5 x −4 y =−3‬‬ ‫⎨⎪ ‪⇒ ×3−4‬‬ ‫⎨⇒‬ ‫‪⎩⎪2x − y =3‬‬ ‫‪⎩−8 x +4 y =−12‬‬

‫‪−3 x =−15 ⇒ x =5‬‬

‫‪x 5‬‬ ‫‪= ⇒ x + y = 12‬‬ ‫‪y 7‬‬

‫⇒‪⇒ y = 7‬‬

‫ﭘﺎﺳﺦ ‪ : 6‬ﮔﺰﻳﻨﻪﻱ ‪ 3‬ﺻﺤﻴﺢ ﺍﺳﺖ؟‬ ‫ﺍﮔﺮ ﺍﻧﺪﺍﺯﻩﻱ ﺿﻠﻊ ﻣﺮﺑﻊ ‪ s1‬ﺭﺍ ‪ a‬ﻭ ﺍﻧﺪﺍﺯﻩﻱ ﻃﻮﻝ ﻣﺴــﺘﻄﻴﻞ ﺍﻳﺠﺎﺩ‬ ‫ﺷــﺪﻩ ﺗﻮﺳﻂ ﻣﺴــﺘﻄﻴﻞ ‪ s1‬ﻭ ﻣﺮﺑﻊ ‪ s2‬ﺭﺍ ‪ b‬ﺑﻨﺎﻣﻴﻢ‪ ،‬ﺍﻧﺪﺍﺯﻩﻱ ﻣﺴﺎﺣﺖ‬ ‫‪1‬‬ ‫ﻫﺎﺷــﻮﺭﺯﺩﻩ ﺑﺮﺍﺑﺮ ﺑﺎ ) ‪ s A = × a ( b − a‬ﻭ ﻣﺤﻴﻂ ﻣﺴﺘﻄﻴﻞ ‪ s3‬ﺑﺮﺍﺑﺮ‬ ‫‪2‬‬ ‫ﺑﺎ ‪ 2b‬ﺧﻮﺍﻫﺪ ﺷﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ‪:‬‬ ‫‪⎧⎪s1+s2 +s3 =37‬‬ ‫⎨ ⇒‬ ‫‪2‬‬ ‫‪⎩⎪ab−a =12‬‬

‫ﺍﺯ ﻃﺮﻓﻲ‪ ،‬ﭼﻮﻥ ﺩﺍﺭﻳﻢ‪، s1 = a 2 :‬‬ ‫ﺑﻨﺎﺑﺮﺍﻳﻦ‪:‬‬

‫) ‪s 3 = a ( b − a ) ، s2 = ( b − a‬‬

‫‪⎧ 2‬‬ ‫‪2‬‬ ‫‪⎪a +( b −a ) +a ( b −a )=37‬‬ ‫⎨‬ ‫‪⎪⎩ab −a 2 =6‬‬

‫⇒‬

‫‪x = 2( y − ( x − y )) = 2y − 2x + 2y‬‬ ‫‪⇒ 3 x = 4y ⇒ 3 x − 4y = 0‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎﻳﺪ ﺩﺳــﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ ‪ ⎪⎧⎨ x + y=49‬ﺭﺍ‬ ‫‪⎩⎪3 x −4 y = 0‬‬ ‫ﺣﻞ ﻛﻨﻴﻢ‪ .‬ﭘﺲ‪:‬‬ ‫‪⎧⎪4 x +4 y =196‬‬ ‫⎨ ⇒‬ ‫⎨‬ ‫‪⎪⎩3 x −4 y = 0‬‬ ‫‪⎪⎩3 x −4 y = 0‬‬

‫‪⎪ x + y =49‬‬ ‫⎧ ‪×4‬‬

‫‪7 x = 196 ⇒ x = 28‬‬

‫‪⎧s1+s2 +s3 =37‬‬ ‫⎪‬ ‫‪⎨1‬‬ ‫‪⎪ ×a ( b−a )=6‬‬ ‫‪⎩2‬‬ ‫‪2‬‬

‫ﭘﺎﺳﺦ ‪ : 8‬ﮔﺰﻳﻨﻪﻱ ‪ 3‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔﺮ ﺳﻦ ﻛﺸﺘﻲ ﺭﺍ ‪ x‬ﻭ ﻋﻤﺮ ﺩﻳﮓ ﺑﺨﺎﺭ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﺑﻨﺎﺑﺮ‬ ‫ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪.x+y=49 :‬‬ ‫ﺩﺭ ﺿﻤﻦ ‪ x-y‬ﺳــﺎﻝ ﻗﺒﻞ ﻋﻤﺮ ﻛﺸــﺘﻲ ﺑﺮﺍﺑﺮ ﺑﺎ ﻋﻤﺮ ﻓﻌﻠﻲ ﺩﻳﮓ‬ ‫ﺑﺨﺎﺭ ﺑﻮﺩﻩ ﺍﺳﺖ‪ ،‬ﭘﺲ ﺩﺍﺭﻳﻢ‪:‬‬

‫ﭘﺎﺳﺦ ‪ : 9‬ﮔﺰﻳﻨﻪﻱ ‪ 3‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔﺮ ﻗﻴﻤﺖ ﺩﻓﺘﺮ ﺭﺍ ‪ x‬ﻭ ﻗﻴﻤﺖ ﺧﻮﺩﻛﺎﺭ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﺑﻨﺎﺑﺮ‬ ‫ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪ 2x = 5 y :‬ﻭ ‪. 3 x + 5 y + 10 = 2x + 8 y‬‬ ‫ﺑﻨﺎﺑﺮﺍﻳــﻦ ﺑﺎﻳﺪ ﺩﺳــﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ ‪⎪⎧ x −3 y =−10‬‬ ‫⎨‬ ‫‪⎩⎪2x −5 y = 0‬‬

‫ﺭﺍ ﺣﻞ ﻛﻨﻴﻢ‪ .‬ﭘﺲ‬

‫‪⎧ 2 2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪⎪a + b −2ab +a +ab −a =37‬‬

‫⎨ ⇒‬

‫‪⎪⎩ab −a 2 =12‬‬

‫ﺍﺯ ﺩﺳﺘﮕﺎﻩ ﺑﺎﻻ ﻧﺘﻴﺠﻪ ﻣﻲﮔﻴﺮﻳﻢ ﻛﻪ‬ ‫ﺟﺎﻳﮕﺰﻳﻨﻲ ﺍﻳﻦ ﻣﻘﺎﺩﻳﺮ ﺩﺭ‬ ‫ﺩﺍﺭﻳﻢ‪:‬‬

‫‪⎪⎧−2x +6 y =20‬‬ ‫⎨ ⇒‬ ‫‪⎩⎪2x −5 y = 0‬‬

‫⎧‬ ‫‪2‬‬ ‫‪ . ⎪⎨ab−a =12‬ﻛﻪ ﺍﺯ‬

‫‪⎪⎩a 2 −ab=−12‬‬ ‫‪2‬‬

‫‪2‬‬

‫‪⇒ x = 50‬‬ ‫‪2‬‬

‫‪2‬‬

‫‪a + b − 2ab + a + ab − a = 37‬‬ ‫‪2‬‬

‫‪2‬‬

‫⎪ )‪×( −2‬‬ ‫‪⎧ x −3 y =−10‬‬

‫⎨‬ ‫‪⎩⎪2x −5 y = 0‬‬

‫‪y = 20‬‬ ‫‪⇒ 2x + 8 y = 260‬‬

‫‪2‬‬

‫‪2(a − ab ) + b + (ab − a ) = 37‬‬ ‫‪2‬‬

‫‪2‬‬

‫‪⇒ 2 × ( −12) + b + 12 = 37 ⇒ b = 49 ⇒ b = 7‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ‪ ،‬ﻣﺤﻴﻂ ﻣﺴﺘﻄﻴﻞ ‪ s3‬ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 2b=14‬ﺍﺳﺖ‪.‬‬ ‫ﭘﺎﺳﺦ ‪ : 7‬ﮔﺰﻳﻨﻪﻱ ‪ 4‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺍﮔﺮ ﺳــﻦ ﺳــﺎﻧﺎﺯ ﺭﺍ ‪ x‬ﻭ ﺳــﻦ ﮔﻠﻨﺎﺭ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈــﺮ ﺑﮕﻴﺮﻳﻢ‪ ،‬ﺑﻨﺎﺑﺮ‬ ‫ﺻﻮﺭﺕ ﻣﺴــﺌﻠﻪ ﺩﺍﺭﻳﻢ‪ . x + y = 30 ، y = 2x :‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎﻳﺪ ﺩﺳﺘﮕﺎﻩ‬ ‫ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ ‪ ⎧⎪⎨2x − y= 0‬ﺭﺍ ﺣﻞ ﻛﻨﻴﻢ‪ .‬ﭘﺲ‪:‬‬ ‫‪⎩⎪ x + y =30‬‬

‫‪⎧2x − y = 0‬‬ ‫⎨‬ ‫‪⎩ x + y =30‬‬ ‫‪⇒ y = 20‬‬ ‫‪3 x =30 ⇒ x =10‬‬ ‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪43‬‬

‫ﭘﺎﺳﺦ ‪ : 10‬ﮔﺰﻳﻨﻪﻱ ‪ 4‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﻃﻮﻝ ﺷﻤﻊ ﺭﺍ ‪ L‬ﻭ ﻣﺪﺕ ﺯﻣﺎﻥ ﻣﻮﺭﺩ ﻧﻈﺮ ﺭﺍ ‪ x‬ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪.‬‬ ‫ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ‪ ،‬ﭼﻮﻥ ﺩﺭ ﻫﺮ ﺳﺎﻋﺖ ‪ 1‬ﺷﻤﻊ ﺍﻭﻝ ﻣﻲﺳﻮﺯﺩ‪ ،‬ﭘﺲ‬ ‫‪4‬‬

‫ﺩﺭ ‪ x‬ﺳﺎﻋﺖ ‪ x‬ﺁﻥ ﻣﻲﺳﻮﺯﺩ ﻭ ﭼﻮﻥ ﻃﻮﻝ ﺁﻥ ﺭﺍ ‪ L‬ﻓﺮﺽ ﻛﺮﺩﻩﺍﻳﻢ‪،‬‬ ‫‪4‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻃﻮﻝ ﺷﻤﻊ ﺍﻭﻝ ﺑﻌﺪ ﺍﺯ ‪ x‬ﺳﺎﻋﺖ ﺑﺮﺍﺑﺮ ﺑﺎ‬ ‫ﺩﻭﻡ ﺑﺮﺍﺑﺮ ﺑﺎ‬

‫‪x‬‬ ‫‪L −3 L‬‬

‫‪x‬‬ ‫×‪L − L‬‬ ‫‪4‬‬

‫ﻭ ﺷﻤﻊ‬

‫ﻣﻲﺷﻮﻧﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬

‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫) ‪xL = 2( L − xL ) ⇒ L (1 − x ) = 2L(1 − x‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪⇒ 1− x = 2 − x ⇒ x = 2 / 4‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪L−‬‬

‫ﭘﺎﺳﺦ ‪ : 11‬ﮔﺰﻳﻨﻪﻱ ‪ 4‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﺍﻭﻟﻴﻪ ﺩﺭ ﺻﻨﺪﻭﻕ ﺭﺍ ‪ x‬ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ‬ ‫ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﺻﻨﺪﻭﻕ ﭘﺲ ﺍﺯ ﻣﺮﺍﺟﻌﻪﻱ ﻧﻔﺮ ﺍﻭﻝ‬ ‫‪x + x − 40 = 2x − 40‬‬

‫ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﺻﻨﺪﻭﻕ ﭘﺲ ﺍﺯ ﻣﺮﺍﺟﻌﻪﻱ ﻧﻔﺮ ﺩﻭﻡ‬ ‫‪(2x − 4 0 ) + (2x − 40 ) − 4 0 = 4 x − 120‬‬

‫ﻣﻘﺪﺍﺭ ﭘﻮﻝ ﺻﻨﺪﻭﻕ ﭘﺲ ﺍﺯ ﻣﺮﺍﺟﻌﻪﻱ ﻧﻔﺮ ﺳﻮﻡ‬ ‫‪(2x − 120 ) + (2x − 12 0 ) − 40 = 8 x − 280‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ‪:‬‬

‫‪8 x − 288 0 = 0 ⇒ 8 x = 28 0 ⇒ x = 35‬‬

‫ﭘﺎﺳﺦ ‪ : 12‬ﮔﺰﻳﻨﻪﻱ ‪ 4‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﻃﻮﻝ ﻣﺴــﺘﻄﻴﻞ ﺭﺍ ‪ x‬ﻭ ﻋﺮﺽ ﺁﻥ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈــﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﺑﻨﺎﺑﺮ‬ ‫ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪ x = 2y − 4‬ﻭ ‪x − 6 = y + 2‬‬ ‫‪⎧ x −2 y=−4‬‬ ‫⎪‬ ‫⎨ ﺭﺍ‬ ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎﻳﺪ ﺩﺳــﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ‬ ‫‪⎩⎪ x − y=8‬‬ ‫ﺣﻞ ﻛﻨﻴﻢ‪.‬‬ ‫ﭘﺲ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪⎧⎪ x −2 y =−4‬‬

‫⎨ ⇒‬

‫‪⎩⎪− x + y =−8‬‬

‫‪⎧⎪ x −2 y=−4‬‬ ‫⎨‬ ‫⎩ )‪×( −1‬‬ ‫‪⎪ x − y =8‬‬

‫‪− y = −12 ⇒ y = 12‬‬ ‫‪⇒ x = 20‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ‪ ،‬ﻣﺴﺎﺣﺖ ﻣﺴﺘﻄﻴﻞ ﻣﺰﺑﻮﺭ ﺑﻪ ﻃﻮﻝ ‪ x=20‬ﻭ ﻋﺮﺽ ‪y=40‬‬

‫‪44‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺑﺮﺍﺑﺮ ﺑﺎ ‪ 20×12=240‬ﻭ ﻣﺴﺎﺣﺖ ﻣﺮﺑﻊ ﻣﻮﺭﺩﻧﻈﺮ ﺑﺮﺍﺑﺮ ﺑﺎ ‪14×14=196‬‬ ‫ﺍﺳﺖ‪ .‬ﭘﺲ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬ ‫‪240-196=44‬‬ ‫ﭘﺎﺳﺦ ‪ : 13‬ﮔﺰﻳﻨﻪﻱ ‪ 1‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺗﻌــﺪﺍﺩ ﻣﻬﺮﻩﻫــﺎﻱ ﺩﺍﺧﻞ ﻛﻴﺴــﻪ ﺭﺍ ‪ x‬ﺩﺭ ﻧﻈــﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﺑﻨﺎﺑﺮ‬ ‫ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫ﺗﻌﺪﺍﺩ ﻣﻬﺮﻩﻫﺎﻳﻲ ﻛﻪ ﭘﺴﺮ ﺑﭽﻪﻱ ﺍﻭﻝ ﺑﺮﻣﻲﺩﺍﺭﺩ‪:‬‬ ‫‪1‬‬ ‫‪x +1‬‬ ‫‪2‬‬

‫ﺗﻌﺪﺍﺩ ﻣﻬﺮﻩﻫﺎﻳﻲ ﻛﻪ ﭘﺴﺮﺑﭽﻪﻱ ﺩﻭﻡ ﺑﺮﻣﻲﺩﺍﺭﺩ‪:‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪( x − ( x + 1)) = x −‬‬ ‫‪2‬‬ ‫‪6‬‬ ‫‪3‬‬ ‫‪3‬‬

‫ﺑﻨﺎﺑﺮﺍﻳــﻦ ﺑﺎﻳﺪ ﻣﻌﺎﺩﻟــﻪﻱ ‪ 1 x + 1 + 1 x − 1 + 4 = x‬ﺭﺍ ﺣﻞ‬ ‫‪2‬‬ ‫‪6‬‬ ‫‪3‬‬ ‫ﻛﻨﻴﻢ‪ .‬ﭘﺲ‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪14‬‬ ‫‪x +1+ x − + 4 = x ⇒ x + = x‬‬ ‫‪2‬‬ ‫‪6‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪14‬‬ ‫=‪⇒ x‬‬ ‫‪⇒ x = 14‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺍﮔﺮ ﺩﺭ ﻋﺒﺎﺭﺕ ‪ 1 x − 1‬ﺭﺍ ﻗﺮﺍﺭ ﺩﻫﻴﻢ‪ ،‬ﺗﻌﺪﺍﺩ ﻣﻬﺮﻩﻫﺎﻳﻲ‬ ‫‪6‬‬ ‫‪3‬‬

‫ﻛﻪ ﺑﻪ ﭘﺴــﺮﺑﭽﻪﻱ ﺩﻭﻡ ﺭﺳــﻴﺪﻩ ﺍﺳــﺖ ﺑﻪ ﺩﺳــﺖ ﻣﻲﺁﻳﺪ ﻭ ﺧﻮﺍﻫﻴﻢ‬ ‫ﺩﺍﺷﺖ‪:‬‬ ‫‪1‬‬ ‫‪1 14 1 7 1 6‬‬ ‫‪× 14 − = − = − = = 2‬‬ ‫‪6‬‬ ‫‪3 6 3 3 3 3‬‬

‫ﭘﺎﺳﺦ ‪ : 14‬ﮔﺰﻳﻨﻪﻱ ‪ 1‬ﺻﺤﻴﺢ ﺍﺳﺖ‪.‬‬ ‫ﺗﻌــﺪﺍﺩ ﺻﻨﺪﻟﻲﻫﺎ ﺩﺭ ﻫــﺮ ﺭﺩﻳﻒ ﺭﺍ ‪ x‬ﻭ ﺗﻌــﺪﺍﺩ ﺻﻨﺪﻟﻲﻫﺎ ﺩﺭ ﻫﺮ‬ ‫ﺳﺘﻮﻥ ﺭﺍ ‪ y‬ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﺑﻨﺎﺑﺮ ﺻﻮﺭﺕ ﻣﺴﺌﻠﻪ ﺩﺍﺭﻳﻢ‪:‬‬ ‫‪ ( x − 3 )( y + 1) = xy‬ﻭ ‪(xx − 5)( y + 2) = xy‬‬ ‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺎﻳﺪ ﺩﺳﺘﮕﺎﻩ ﺩﻭ ﻣﻌﺎﺩﻟﻪ ﻭ ﺩﻭ ﻣﺠﻬﻮﻟﻲ ‪⎪⎧( x −3 )( y +1) = xy‬‬ ‫⎨‬ ‫‪⎩⎪( x −5)( y +2) = xy‬‬ ‫ﺭﺍ ﺣﻞ ﻛﻨﻴﻢ‪ .‬ﭘﺲ‪:‬‬ ‫‪⎧⎪ x −3 y =3‬‬ ‫⎨⇒‬ ‫‪⎪⎩2x −5 y =10‬‬

‫‪⎧⎪( x −3 )( y +1) = xy‬‬ ‫⎨‬ ‫‪⎪⎩( x −5)( y +2) = xy‬‬

‫‪⎧⎪−2x +6 y=−6‬‬ ‫⎨ ⇒‬ ‫⎨‬ ‫‪⎪⎩2x −5 y=10‬‬ ‫‪⎩⎪2x −5 y=10‬‬

‫⎧ )‪×( −2‬‬ ‫‪⎪ x −3 y =3‬‬

‫‪⇒ x = 15‬‬

‫‪y=4‬‬

‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬ ‫‪n = xy = 4 × 15 = 6 0‬‬

‫⇒‬

‫︨‪﹣‬ال﹨︀ی ︎﹠︕﹎︤﹬﹠﹤ای ︎︀﹬﹤ی اول را﹨﹠﹝︀﹬‪﹩‬‬ ‫ﺗﺮﺟﻤﻪ‬

‫ﻛﻠﻴﺪﻭﺍژﻩﻫﺎ‪ :‬ﺳــﺆﺍﻝﻫﺎﻱ ﻣﺴــﺎﺑﻘﻪﺍﻱ‪،‬‬ ‫ﺭﻳﺎﺿــﻲ ﺍﺳــﺘﺮﺍﻟﻴﺎ‪ ،‬ﭘﻨﺞ ﮔﺰﻳﻨــﻪﺍﻱ‪ ،‬ﺩﻭﺭﻩﻱ‬ ‫ﺭﺍﻫﻨﻤﺎﻳﻲ‪.‬‬

‫ﻱ ﺳﭙﻴﺪﻩ ﭼﻤﻦﺁﺭﺍ‬

‫‪ .1‬ﻋﺪﺩ ﻳﻚ ﻫﺰﺍﺭ ﻭ ﺑﻴﺴﺖ ﻭ ﻫﻔﺖ ﻛﺪﺍﻡ‬ ‫ﺍﺳﺖ؟‬ ‫ﺍﻟﻒ( ‪ 100027‬ﺏ( ‪10027‬‬ ‫ﺙ( ‪27‬‬ ‫ﺕ( ‪127‬‬ ‫پ( ‪1027‬‬ ‫‪ .2‬ﺳــﺎﺭﺍ ﺩﺭﻭﻥ ﻓﺮﻭﺷــﮕﺎﻩ ﺣﻴﻮﺍﻧــﺎﺕ‬ ‫ﺧﺎﻧﮕﻲ ﺍﻳﺴــﺘﺎﺩﻩ ﻭ ﺍﺯ ﭘﻨﺠــﺮﻩ ﺑﻪ ﺑﻴﺮﻭﻥ ﻧﮕﺎﻩ‬ ‫ﻣﻲﻛﻨﺪ‪ .‬ﺳﺎﺭﺍ ﻧﻮﺷﺘﻪﻱ ﺭﻭﻱ ﭘﻨﺠﺮﻩ ﺭﺍ ﭼﮕﻮﻧﻪ‬ ‫ﻣﻲﺑﻴﻨﺪ؟‬ ‫ﺍﻟﻒ( ‪POHS T E P‬‬ ‫ﺏ( ‪POH S TEP‬‬

‫پ( ‪T E P POH S‬‬ ‫ﺕ( ‪POH S T E P‬‬ ‫ﺙ( ‪P OH S T EP‬‬

‫‪PET SHOP‬‬

‫ﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬ ‫ﺷﺎﻧﺰﺩﻫﻢ‪،‬‬ ‫ﺰﺩﻫﻢ‬ ‫ﺷﺎﻧﻧﺰﺰﺩﺩﻫ‬ ‫ﺩﻭﺭﺓ ﺷ‬ ‫ﺩﻭﺭﺭﺓ‬ ‫ﺩﻭ‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪45‬‬

‫ﺳﺆﺍﻝﻫﺎﻱ ﻣﺴﺎﺑﻘﻪﺍﻱ‬

‫)‪(٢٠١٠‬‬ ‫)‪٢٠١٠‬‬

‫‪ .3‬ﻓﺮﻫﺎﺩ ‪ 14‬ﺳﺎﻟﻪ ﺍﺳﺖ‪ .‬ﻓﺮﻳﺒﺎ ‪ 10‬ﺳﺎﻟﻪ‬ ‫ﺳﻦ ﭘﺪﺭ ﻓﺮﻫﺎﺩ ﻭ ﻓﺮﻳﺒﺎ ﺩﻭ ﺑﺮﺍﺑﺮ ﻣﺠﻤﻮﻉ‬ ‫ﺍﺳﺖ‪ .‬ﱢ‬ ‫ﺳﻦ ﺁﻥ ﺩﻭ ﺍﺳﺖ‪ .‬ﭘﺪﺭ ﭼﻨﺪ ﺳﺎﻟﻪ ﺍﺳﺖ؟‬ ‫ﱢ‬ ‫ﺏ( ‪48‬‬ ‫ﺍﻟﻒ( ‪46‬‬ ‫ﺙ( ‪54‬‬ ‫ﺕ( ‪52‬‬ ‫پ( ‪50‬‬ ‫‪ .4‬ﻧﻘﻄــﻪﻱ ﻭﺳــﻂ ﻫــﺮ ﺿﻠــﻊ ﻣﺮﺑﻊ ﺭﺍ‬ ‫ﻣﻄﺎﺑﻖ ﺷﻜﻞ ﺑﻪ ﻫﻢ ﻭﺻﻞ ﻛﺮﺩﻩﺍﻳﻢ‪ ،‬ﻗﺴﻤﺘﻲ‬ ‫ﺍﺯ ﺷــﻜﻞ ﺭﻧﮓ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﻗﺴﻤﺖ ﺭﻧﮓﺷﺪﻩ‬ ‫ﭼﻪ ﻛﺴﺮﻱ ﺍﺯ ﻣﺮﺑﻊ ﺑﺰﺭگ ﺍﺻﻠﻲ ﺍﺳﺖ؟‬

‫ﺍﻟﻒ( ‪7‬‬ ‫پ( ‪10‬‬ ‫ﺙ( ‪18‬‬

‫ﭘﺮﺳﺶﻫﺎﻱ ‪ 11‬ﺗﺎ ‪20‬‬ ‫ﻫﺮ ﻛﺪﺍﻡ ‪ 4‬ﺍﻣﺘﻴﺎﺯ ﺩﺍﺭﺩ‪.‬‬ ‫‪ .11‬ﺑﺮﻧﺎﻣﻪﻱ ﺭﻭﺯﺍﻧﻪﻱ ﺩﺑﺴــﺘﺎﻥ ﺑﻪ ﺷﻜﻞ‬ ‫ﺯﻳﺮ ﺍﺳــﺖ‪ .‬ﺩﺭ ﻫــﺮ ﺭﻭﺯ ﭼﻨــﺪ ﺩﻗﻴﻘﻪ ﺑﺮﺍﻱ‬ ‫ﻛﻼﺱ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﺍﺳﺖ؟‬

‫ﺏ( ‪8‬‬ ‫ﺕ( ‪12‬‬

‫ﺍﻟﻒ(‬ ‫ﺕ(‬

‫‪1‬‬ ‫‪4‬‬ ‫‪1‬‬

‫ﺏ(‬ ‫ﺙ(‬

‫‪3‬‬

‫‪1‬‬ ‫‪6‬‬

‫پ(‬

‫‪3‬‬

‫‪1‬‬ ‫‪5‬‬

‫‪ .5‬ﺩﺭ ﺣﻴﺎﻁ ﻣﺪﺭﺳﻪ ﺩﺭ ﻳﻚ ﺻﻒ‪ ،‬ﺳﺎﺭﺍ‬ ‫ﭘﺸــﺖ ﺳﺮ ﻣﺮﻳﻢ ﺍﻳﺴﺘﺎﺩﻩ ﻭ ﺳﻤﻴﺮﺍ ﺑﻴﻦ ﺳﺎﺭﺍ‬ ‫ﻭ ﻣﺮﻳﻢ ﺍﺳــﺖ‪ .‬ﺳــﺎﺭﺍ ﺟﻠﻮﻱ ﭘﺮﻭﺍﻧﻪ ﺍﺳﺖ ﻛﻪ‬ ‫ﭘﺮﻭﺍﻧﻪ ﺧﻮﺩﺵ ﺟﻠﻮﻱ ﭘﺮﺳﺘﻮ ﺍﺳﺖ‪ .‬ﭼﻬﺎﺭﻣﻴﻦ‬ ‫ﻧﻔﺮ ﺩﺭ ﺍﻳﻦ ﺻﻒ ﻛﻴﺴﺖ؟‬ ‫ﺏ( ﻣﺮﻳﻢ‬ ‫ﺍﻟﻒ( ﺳﺎﺭﺍ‬ ‫ﺕ( ﭘﺮﻭﺍﻧﻪ‬ ‫پ( ﺳﻤﻴﺮﺍ‬ ‫ﺙ( ﭘﺮﺳﺘﻮ‬ ‫‪ .6‬ﻣﺠﻤﻮﻉ ﭘﻨﺞ ﻋﺪﺩ ‪ 2010‬ﺷﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ﻳﻜﻲ ﺍﺯ ﺍﻳــﻦ ﺍﻋﺪﺍﺩ ﺍﺯ ‪ 235‬ﺑــﻪ ‪ 532‬ﺗﻐﻴﻴﺮ‬ ‫ﻣﻲﻛﻨﺪ‪ .‬ﻣﺠﻤﻮﻉ ﺟﺪﻳﺪ ﭼﻪ ﻋﺪﺩﻱ ﺍﺳﺖ؟‬ ‫ﺏ( ‪2542‬‬ ‫ﺍﻟﻒ( ‪1723‬‬ ‫ﺕ( ‪1896‬‬ ‫پ( ‪2360‬‬ ‫ﺙ( ‪2307‬‬ ‫‪ .7‬ﻫﺸــﺖ ﻣﻜﻌــﺐ ﺭﺍ ﺑﻪ ﺷــﻜﻞ ﺯﻳﺮ ﺑﻪ‬ ‫ﻫــﻢ ﭼﺴــﺒﺎﻧﺪﻩﺍﻳﻢ‪ ،‬ﭼﻨﺪ ﻭﺟــﻪ ﺍﺯ ﻣﻜﻌﺐﻫﺎ‬ ‫ﭼﺴﺐﻛﺎﺭﻱ ﺷﺪﻩﺍﻧﺪ؟‬ ‫‪46‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺑﺮﻧﺎﻣﻪﻱ ﺻﺒﺤﮕﺎﻩ‬

‫‪9-9:10‬‬

‫ﻛﻼﺱ‬

‫‪9:10-11:00‬‬

‫ﺯﻧﮓ ﺗﻔﺮﻳﺢ‬

‫‪11:00-11:30‬‬

‫ﻛﻼﺱ‬

‫‪11:30-13:00‬‬

‫ﻭﻗﺖ ﻧﻬﺎﺭ‬

‫‪13:00-13:50‬‬

‫ﻛﻼﺱ‬

‫‪13:50-15:00‬‬

‫ﭘﺎﻳﺎﻥ ﻛﺎﺭ ﺩﺑﺴﺘﺎﻥ‬

‫‪15:00‬‬

‫‪ .8‬ﺑﺎ ﺍﺳــﺘﻔﺎﺩﻩ ﺍﺯ ﺳــﻪ ﻛﺎﺭﺕ ﺯﻳــﺮ‪ ،‬ﺍﻋﺪﺍﺩ‬ ‫ﺳﻪﺭﻗﻤﻲ ﻣﻲﺳﺎﺯﻳﻢ ﺑﻪ ﻃﻮﺭﻱ ﻛﻪ ﺍﺯ ﻫﺮ ﻛﺎﺭﺕ‬ ‫ﺩﺭ ﻫﺮ ﻋﺪﺩ ﻓﻘﻂ ﻳﻚ ﺑﺎﺭ ﺍﺳﺘﻔﺎﺩﻩ ﻛﻨﻴﻢ‪ .‬ﺍﺧﺘﻼﻑ‬ ‫ﺑﻴــﻦ ﺑﺰﺭگﺗﺮﻳﻦ ﻭ ﻛﻮﭼﻚﺗﺮﻳﻦ ﻋﺪﺩﻱ ﻛﻪ ﺑﻪ‬ ‫ﺍﻳﻦ ﺗﺮﺗﻴﺐ ﻣﻲﺳﺎﺯﻳﻢ ﭼﻴﺴﺖ؟‬

‫‪2 7 5‬‬ ‫‪2‬‬

‫ﻓﻌﺎﻟﻴﺖ‬

‫ﺯﻣﺎﻥ‬

‫ﺍﻟﻒ( ‪477‬‬ ‫ﺕ( ‪1009‬‬ ‫ﺙ( ‪555‬‬

‫ﺍﻟﻒ( ‪300‬‬ ‫پ( ‪500‬‬ ‫ﺙ( ‪240‬‬

‫ﺏ( ‪495‬‬ ‫ﺕ( ‪468‬‬

‫‪ .9‬ﺩﺭ ﻳــﻚ ﻗﺮﻋﻪﻛﺸــﻲ‪ ،‬ﭘــﺪﺭﻡ ‪1000‬‬ ‫ﺗﻮﻣﺎﻥ ﺑﺮﻧﺪﻩ ﺷﺪ‪ .‬ﺍﻭ ﺧﻤﺲ‬ ‫‪1‬‬ ‫‪1‬‬ ‫) ( ﺁﻥ ﺭﺍ ﺩﺭ ﺑﺎﻧﻚ ﮔﺬﺍﺷﺖ ﻭ ﺭﺑﻊ ) (‬ ‫‪4‬‬ ‫‪5‬‬ ‫ﻭ ﺑﻘﻴﻪﻱ ﺁﻥ ﺭﺍ ﺑﻪ ﻣﻦ ﺩﺍﺩ ﻭ ﻫﺮﭼﻪ ﻣﺎﻧﺪﻩ ﺑﻮﺩ‬ ‫ﺑﻪ ﻣﺎﺩﺭﻡ ﺩﺍﺩ‪ .‬ﭼﻪ ﻣﺒﻠﻐﻲ ﺑﻪ ﻣﺎﺩﺭﻡ ﺭﺳﻴﺪ؟‬ ‫ﺍﻟﻒ( ‪ 400‬ﺗﻮﻣﺎﻥ ﺏ( ‪ 888‬ﺗﻮﻣﺎﻥ‬ ‫پ( ‪ 450‬ﺗﻮﻣﺎﻥ‬ ‫ﺕ( ‪ 550‬ﺗﻮﻣﺎﻥ‬ ‫ﺙ( ‪ 600‬ﺗﻮﻣﺎﻥ‬

‫ﺏ( ‪250‬‬ ‫ﺕ( ‪270‬‬

‫‪ .12‬ﻣﻴﺎﻧﮕﻴﻦ ﺩﻭ ﻋﺪﺩ ‪ 11‬ﺍﺳﺖ‪ .‬ﺍﮔﺮ ﻳﻜﻲ‬ ‫ﺍﺯ ﻋﺪﺩﻫﺎ ‪ 6‬ﺗﺎ ﺍﺯ ﺩﻳﮕﺮﻱ ﺑﻴﺶﺗﺮ ﺑﺎﺷــﺪ‪ ،‬ﻋﺪﺩ‬ ‫ﺑﺰﺭگﺗﺮ ﻛﺪﺍﻡ ﺍﺳﺖ؟‬ ‫ﺏ( ‪8‬‬ ‫ﺍﻟﻒ( ‪6‬‬ ‫پ( ‪11‬‬ ‫ﺙ( ‪17‬‬ ‫ﺕ( ‪14‬‬ ‫‪ .13‬ﭼﻪ ﻛﺴــﺮﻱ ﺍﺯ ﻣﺴــﺘﻄﻴﻞ ﺯﻳﺮ ﺭﻧﮓ‬ ‫ﺷﺪﻩ ﺍﺳﺖ؟‬ ‫‪3‬‬

‫‪ .10‬ﺍﮔــﺮ ﺩﺭ ﺷــﻜﻞ ﺯﻳــﺮ ﺍﺯ ﻧﻘﻄﻪﻱ ‪A‬‬ ‫ﺣﺮﻛﺖ ﻛﻨﻴــﻢ ﻭ ﺩﻭﺑﺎﺭﻩ ﺑﻪ ‪ A‬ﺑﺎﺯﮔﺮﺩﻳﻢ‪ ،‬ﭼﻪ‬ ‫ﻣﺴﺎﻓﺘﻲ ﭘﻴﻤﻮﺩﻩﺍﻳﻢ؟‬

‫‪2‬‬

‫‪4‬‬ ‫‪12m‬‬

‫‪13m‬‬

‫ﺍﻟﻒ( ‪ 52‬ﻣﺘﺮ‬ ‫پ( ‪ 52‬ﻣﺘﺮ‬ ‫ﺙ( ‪ 50‬ﻣﺘﺮ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫‪A‬‬

‫ﺏ( ‪ 48‬ﻣﺘﺮ‬ ‫ﺕ( ‪ 50‬ﻣﺘﺮ‬

‫ﺍﻟﻒ(‬ ‫پ(‬

‫‪1‬‬ ‫‪2‬‬

‫‪1‬‬ ‫‪3‬‬

‫ﺏ(‬ ‫ﺕ(‬

‫‪5‬‬ ‫‪12‬‬ ‫‪2‬‬ ‫‪7‬‬

‫ﺙ(‬

‫‪3‬‬ ‫‪8‬‬

‫‪ .14‬ﺩﺭ ﮔﺮﻭﻫــﻲ ﺍﺯ ‪ 55‬ﺩﺍﻧﺶﺁﻣــﻮﺯ‪39 ،‬‬ ‫ﻧﻔﺮ ﺩﺭ ﻛﻼﺱ ﺭﻳﺎﺿﻲ ﺛﺒﺖﻧﺎﻡ ﻛﺮﺩﻩﺍﻧﺪ ﻭ ‪35‬‬

‫ﺏ‬

‫‪16‬‬

‫ﺏ‬

‫‪15‬‬

‫پ‬

‫‪14‬‬

‫ﺏ‬

‫‪13‬‬

‫ﺕ‬

‫‪12‬‬

‫ﺕ‬

‫‪11‬‬

‫ﺕ‬

‫‪10‬‬

‫ﺙ‬

‫‪9‬‬

‫ﺏ‬

‫‪8‬‬

‫ﺙ‬

‫‪7‬‬

‫ﺙ‬

‫‪6‬‬

‫ﺕ‬

‫‪5‬‬

‫ﺍﻟﻒ‬

‫‪4‬‬

‫ﺏ‬

‫‪3‬‬

‫ﺙ‬

‫‪2‬‬

‫پ‬

‫‪1‬‬ ‫ﺳﺆﺍﻝ‬

‫ﮔﺰﻳﻨﻪﻱ ﺻﺤﻴﺢ‬ ‫ﭘﺎﺳﺦ ﭘﺮﺳﺶﻫﺎ‪:‬‬

‫‪ .20‬ﺩﺭ ﺷﻜﻞ ﺯﻳﺮ‪ ،‬ﻣﺴﺎﺣﺖ ﺳﻪ ﻣﺴﺘﻄﻴﻞ‬ ‫ﺑﺮ ﺣﺴﺐ ﺳــﺎﻧﺘﻲﻣﺘﺮ ﻣﺮﺑﻊ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ﻣﺴﺎﺣﺖ ﻗﺴﻤﺖ ﺭﻧﮕﻲ ﭼﻨﺪ ﺳﺎﻧﺘﻲﻣﺘﺮ ﻣﺮﺑﻊ‬ ‫ﺍﺳﺖ؟‬

‫‪17‬‬

‫‪ .17‬ﻫــﺮ ﻳﻚ ﺍﺯ ﺍﻋــﺪﺍﺩ ‪ 5 ،4 ،3 ،2 ،1‬ﺭﺍ‬ ‫ﺩﺭ ﻳﻜﻲ ﺍﺯ ﺩﺍﻳﺮﻩﻫﺎﻱ ﺷﻜﻞ ﺯﻳﺮ ﻗﺮﺍﺭ ﻣﻲﺩﻫﻴﻢ‬ ‫ﺗــﺎ ﺍﻋﺪﺍﺩﻱ ﻛــﻪ ﺑﺎ ﻳﻚ ﺧﻂ ﺑــﻪ ﻫﻢ ﻣﺘﺼﻞ‬ ‫ﻣﻲﺷﻮﻧﺪ‪ ،‬ﭘﺸﺖ ﺳــﺮ ﻫﻢ ﺑﺎﺷﻨﺪ‪ .‬ﻣﺠﻤﻮﻉ ‪X‬‬ ‫ﻭ ‪ Y‬ﭼﻨﺪ ﻣﻲﺗﻮﺍﻧﺪ ﺑﺎﺷﺪ؟‬

‫‪ .19‬ﻛﻼﻩﻓﺮﻭﺷــﻲ ﺩﺭ ﻳــﻚ ﺭﻭﺯ ﺣﺮﺍﺟﻲ‪،‬‬ ‫ﻛﻼﻩﻫــﺎﻱ ‪ 1200‬ﺗﻮﻣﺎﻧﻲ ﺭﺍ ﺻﺒﺢ ﻓﺮﻭﺧﺖ ﻭ‬ ‫ﻛ ً‬ ‫ﻼ ‪ 72000‬ﺗﻮﻣــﺎﻥ ﻓﺮﻭﺵ ﻛﺮﺩ‪ .‬ﺑﻌﺪ ﺍﺯﻇﻬﺮ‪،‬‬ ‫ﻗﻴﻤﺖ ﺑﺎﻗﻲﻣﺎﻧــﺪﻩﻱ ﻛﻼﻩﻫﺎ ﺭﺍ ‪ 900‬ﻛﺎﻫﺶ‬ ‫ﺩﺍﺩ ﻭ ﺑــﻪ ﺗﻌﺪﺍﺩ ﺩﻭ ﺑﺮﺍﺑــﺮ ﻓﺮﻭﺵ ﺻﺒﺢ‪ ،‬ﻛﻼﻩ‬ ‫ﻓﺮﻭﺧﺘﻪ ﺷﺪ‪ .‬ﻛﻞ ﺩﺭﺁﻣﺪ ﺍﻳﻦ ﻛﻼﻩﻓﺮﻭﺷﻲ ﺩﺭ‬ ‫ﺁﻥ ﺭﻭﺯ ﭼﻨﺪﺗﻮﻣﺎﻥ ﺑﻮﺩﻩ ﺍﺳﺖ؟‬ ‫ﺍﻟﻒ( ‪180000‬‬ ‫ﺏ( ‪90000‬‬ ‫پ( ‪126000‬‬ ‫ﺕ( ‪144000‬‬ ‫ﺙ( ‪288000‬‬

‫ﺕ‬

‫ﺍﻟﻒ( ‪ 100‬ﻭ ‪60‬‬ ‫پ( ‪ 90‬ﻭ ‪70‬‬ ‫ﺙ( ‪ 75‬ﻭ ‪50‬‬

‫ﺏ( ‪ 60‬ﻭ ‪90‬‬ ‫ﺕ( ‪ 86‬ﻭ ‪36‬‬

‫ﺍﻟﻒ( ‪216‬‬ ‫ﺕ( ‪207‬‬

‫ﺏ( ‪54‬‬ ‫ﺙ( ‪200‬‬

‫پ( ‪181‬‬

‫‪18‬‬

‫‪ .16‬ﺑﺮﺍﻱ ﺳــﺎﺧﺖ ﻳﻚ ﭘﺘﻮﻱ ﻣﺴﺘﻄﻴﻞ‬ ‫ﺷــﻜﻞ ﺑــﻪ ﺍﺑﻌــﺎﺩ ‪ 120‬ﺳــﺎﻧﺘﻲﻣﺘﺮ ﺩﺭ ‪90‬‬ ‫ﺳــﺎﻧﺘﻲﻣﺘﺮ‪ ،‬ﺣﺎﺷــﻴﻪﻫﺎﻳﻲ ﻣﺎﻧﻨﺪ ﺷــﻜﻞ ﺑﻪ‬ ‫ﻳــﻚ ﺗﻜﻪ ﭘﺘــﻮﻱ ﻛﻮﭼﻚﺗﺮ ﺍﺿﺎﻓــﻪ ﻛﺮﺩﻳﻢ‪.‬‬ ‫ﺍﮔﺮ ﺿﺨﺎﻣﺖ ﺣﺎﺷــﻴﻪﻫﺎ ﺩﻭﺭ ﺗﺎ ﺩﻭﺭ ﭘﺘﻮ ﻳﻚ‬ ‫ﺍﻧﺪﺍﺯﻩ ﺑﺎﺷــﺪ‪ ،‬ﺍﺑﻌﺎﺩ ﭘﺘﻮﻱ ﻣﺮﻛﺰﻱ )ﺑﺮ ﺣﺴﺐ‬ ‫ﺳﺎﻧﺘﻲﻣﺘﺮ( ﭼﻪ ﺍﻧﺪﺍﺯﻩﺍﻱ ﻣﻲﺗﻮﺍﻧﺪ ﺑﺎﺷﺪ؟‬

‫‪ .18‬ﺷﻜﻞ ﺯﻳﺮ‪ ،‬ﻧﻤﻮﺩﺍﺭ ﻳﻚ ﺑﺎﻏﭽﻪ ﺍﺳﺖ‪.‬‬ ‫ﻗﺴــﻤﺘﻲ ﺍﺯ ﺑﺎﻏﭽﻪ‪ ،‬ﭼﻤﻦ ﻛﺎﺷــﺘﻪ ﺷــﺪﻩ ﻭ‬ ‫ﻗﺴــﻤﺘﻲ ﺍﺯ ﺁﻥ ﺑﺎ ﺳــﻨﮓﻫﺎﻱ ﻣﺮﺑﻊ ﺷــﻜﻞ‪،‬‬ ‫ﺳﻨﮓﻓﺮﺵ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬ ‫ِ‬ ‫ﻗﺴﻤﺖ ﭼﻤﻦ ﻛﺎﺷﺘﻪ ﺷﺪﻩ‪،‬‬ ‫ﻣﺴــﺎﺣﺖ ﻛﻞ‬ ‫‪ 108‬ﻣﺘﺮ ﻣﺮﺑﻊ ﺍﺳﺖ‪.‬‬ ‫ﻣﺴــﺎﺣﺖ ﻗﺴﻤﺖ ﺳــﻨﮓﻓﺮﺵ ﭼﻨﺪ ﻣﺘﺮ‬ ‫ﻣﺮﺑﻊ ﺍﺳﺖ؟‬

‫ﺕ‬

‫‪2‬‬

‫ﺙ( ‪3‬‬

‫پ( ‪6‬‬

‫ﺏ( ‪48‬‬ ‫ﺙ( ‪70‬‬

‫‪19‬‬

‫ﺕ(‬

‫‪1‬‬

‫‪4‬‬

‫‪6‬‬

‫‪3‬‬

‫ﺍﻟﻒ( ‪3‬‬ ‫ﺕ( ‪7‬‬

‫ﺏ( ‪4‬‬ ‫ﺙ( ‪8‬‬

‫ﺍﻟﻒ( ‪36‬‬ ‫ﺕ( ‪60‬‬

‫پ( ‪56‬‬

‫ﺍﻟﻒ‬

‫ﺍﻟﻒ( ‪4‬‬

‫ﺏ(‬

‫‪1‬‬

‫‪4‬‬

‫پ(‬

‫‪1‬‬

‫‪4‬‬

‫‪X‬‬

‫‪٢٠‬‬

‫‪20‬‬

‫‪ .15‬ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ ﺑﺮ ﺣﺴﺐ ﺷﺶﺿﻠﻌﻲ‬ ‫ﻭﺍﺣﺪ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺩﺭ ﺷﻜﻞ ﭼﻴﺴﺖ؟‬

‫‪٢٥‬‬

‫پ‬

‫ﻧﻔــﺮ ﺩﺭ ﻛﻼﺱ ﻋﻠﻮﻡ‪ .‬ﭼﻨﺪ ﺩﺍﻧﺶﺁﻣﻮﺯ ﺩﺭ ﻫﺮ‬ ‫ﺩﻭ ﻛﻼﺱ ﺭﻳﺎﺿﻲ ﻭ ﻋﻠﻮﻡ ﺛﺒﺖﻧﺎﻡ ﻛﺮﺩﻩﺍﻧﺪ؟‬ ‫پ( ‪19‬‬ ‫ﺏ( ‪16‬‬ ‫ﺍﻟﻒ( ‪20‬‬ ‫ﺙ( ‪55‬‬ ‫ﺕ( ‪4‬‬

‫‪Y‬‬

‫‪٧٠‬‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫‪47‬‬

‫ﻣﻌﺮﻓﻲ ﻛﺘﺎﺏ‬

‫ر﹬︀︲﹫︀ت ز﹬︊︀‬ ‫و دو︨️دا︫︐﹠‪﹩‬‬ ‫ﺟﻌﻔﺮ ﺭﺑﺎﻧﻲ‬

‫ﻧﻮﻳﺴﻨﺪﻩ‪ :‬ﻣﺎﺭﺗﻴﻦ ﻟﻮ‬ ‫ﺗﺮﺟﻤﻪ‪ :‬ﻛﺎﻇﻢ ﻓﺎﺋﻘﻲ‬

‫ﻧﺎﺷﺮ‪ :‬ﻟﻮﺡ ﺩﺍﻧﺶ‪1379 ،‬‬ ‫]ﺗﻬﺮﺍﻥ‪66403512 :‬ـ‪[021‬‬

‫ﻣﺎ ﺗﺎﻛﻨﻮﻥ ﭼﻨﺪ ﻛﺘﺎﺏ ﻣﺨﺘﻠﻒ ﺭﺍ ﻛﻪ ﺭﻳﺎﺿﻲ ﺭﺍ ﺍﺯ ﻃﺮﻳﻖ ﺑﺎﺯﻱ‪،‬‬ ‫ﻣﻌﻤﺎ‪ ،‬ﻟﻄﻴﻔﻪ ﻭ ﺳــﺮﮔﺮﻣﻲ ﺑﺮﺍﻱ ﺷﻤﺎ ﺁﺳﺎﻥ ﻣﻲﻛﻨﺪ ﻭ ﻣﻬﻢﺗﺮ‬ ‫ّ‬ ‫ﺍﺯ ﺍﻳﻦ‪ ،‬ﺑﺎﻋﺚ ﻣﻲﺷــﻮﺩ ﺷــﻤﺎ ﺑﻪ ﺍﻳﻦ ﺩﺭﺱ ﻋﻼﻗﻤﻨﺪ ﺷﻮﻳﺪ‬ ‫ﻣﻌﺮﻓﻲ ﻛﺮﺩﻩﺍﻳﻢ‪ .‬ﻛﺘﺎﺏ »ﺭﻳﺎﺿﻴﺎﺕ ﺯﻳﺒﺎ ﻭ ﺩﻭﺳﺖﺩﺍﺷﺘﻨﻲ«‬ ‫ﻧﻴﺰ ﻳﻜﻲ ﺍﺯ ﺍﻳﻦ ﻛﺘﺎﺏﻫﺎﺳﺖ‪ .‬ﺍﻳﻦ ﻛﺘﺎﺏ ﺷﺎﻣﻞ ‪ 145‬ﻣﺴﺌﻠﻪ‬ ‫ﺩﺭ ﺍﻧﻮﺍﻉ ﻣﺨﺘﻠﻒ ﺍﺳــﺖ ﻛﻪ ﺑﺪﻭﻥ ﻧﻈﻢ ﻭ ﺗﺮﺗﻴﺐ ﻣﺸﺨﺼﻲ‪،‬‬ ‫ﺑﻪ ﺩﻧﺒﺎﻝ ﻳﻜﺪﻳﮕﺮ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪﺍﻧﺪ‪ .‬ﺩﺭ ﭘﺎﻳﺎﻥ ﻛﺘﺎﺏ ﻧﻴﺰ ﭘﺎﺳﺦ‬ ‫ﻫﻤﻪ ﻣﺴﺎﺋﻞ ﺁﻣﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻨﺠﺎ ﺩﻭ ﻧﻤﻮﻧﻪ ﺍﺯ ﺍﻳﻦ ﻣﺴﺎﺋﻞ‬ ‫ﺭﺍ ﺑﺮﺍﻱ ﺷــﻤﺎ ﻣﻲﺁﻭﺭﻳﻢ‪ .‬ﻳﻜﻲ ﺑــﺪﻭﻥ ﺟﻮﺍﺏ ﻭ ﺩﻳﮕﺮﻱ ﺑﺎ‬ ‫ﺟﻮﺍﺏ‪.‬‬ ‫• ‪ 6‬ﮔﺮﺑــﻪ ‪ 6‬ﻣــﻮﺵ ﺭﺍ ﺩﺭ ‪ 6‬ﺩﻗﻴﻘﻪ ﻣﻲﺧﻮﺭﻧﺪ‪ .‬ﭼﻨﺪ‬ ‫ﮔﺮﺑﻪ ‪ 60‬ﻣﻮﺵ ﺭﺍ ﺩﺭ ‪ 60‬ﺩﻗﻴﻘﻪ ﻣﻲﺧﻮﺭﻧﺪ؟‬ ‫• ﺍﺯ ﻳﻚ ﻧﻔﺮ ﭘﺮﺳﻴﺪﻧﺪ‪ :‬ﭼﻬﻞ ﺳﺎﻝ ﭘﻴﺶ ﭼﻨﺪ ﺑﻬﺎﺭ‬ ‫ﺍﺯ ﻋﻤﺮ ﺗﻮ ﻣﻲﮔﺬﺷﺖ؟ ﻭ ﺍﻭ ﭼﻨﻴﻦ ﭘﺎﺳﺦ ﺩﺍﺩ‪ :‬ﺍﮔﺮ‬ ‫ﺑﻪ ﺳــﻦ ﻣﻦ ﺩﺭ ﺁﻥ ﺯﻣﺎﻥ ﺣﺎﺻﻠﻀﺮﺏ ‪ 5‬ﺩﺭ ‪ 7‬ﻭ‬ ‫ﻫﻢﭼﻨﻴــﻦ‪ 7‬ﺩﺭ ‪ 3‬ﺭﺍ ﻣﻲﺍﻓﺰﻭﺩﻳﺪ ﻭ ﺣﺎﺻﻠﻀﺮﺏ‬ ‫‪ 6‬ﺩﺭ ‪ 9‬ﺑــﻪ ﺍﺿﺎﻓــﻪ ‪ 4‬ﺭﺍ ﺍﺯ ﺁﻥ ﻛﻢ ﻣﻲﻛﺮﺩﻳﺪ‪،‬‬ ‫ﺳــﻦ ﺁﻥ ﺯﻣﺎﻥ ﻣﻦ‬ ‫ﺣﺎﺻﻞ ﻣﺴــﺎﻭﻱ ﺑﺎ ﺩﻭﺑﺮﺍﺑﺮ‬ ‫ّ‬ ‫ﻣﻨﻬــﺎﻱ ‪ 20‬ﺑﻮﺩ‪ .‬ﺍﮔﺮ ﺍﻳــﻦ ﮔﻔﺘﻪ ﻭﻱ ﺻﺤﻴﺢ‬ ‫ﺑﺎﺷﺪ‪ ،‬ﺍﻭ ﺩﺭ ﺣﺎﻝ ﺣﺎﺿﺮ ﭼﻨﺪ ﺳﺎﻝ ﺩﺍﺭﺩ؟‬ ‫ﺟﻮﺍﺏ ﺭﺍ ﺍﺯ ﺍﻳﻦ ﻣﻌﺎﺩﻟﻪ ﭘﻴﺪﺍ ﻛﻨﻴﺪ‪:‬‬ ‫‪x+(5×7)+(7×3)-(6×9+4)=2x-20‬‬ ‫‪48‬‬

‫ﺭﺍﻫﻨﻤﺎﻳﻲ‬

‫ﺩﻭﺭﺓ ﺷﺎﻧﺰﺩﻫﻢ‪ ،‬ﺷﻤﺎﺭﺓ ‪ ،4‬ﺗﺎﺑﺴﺘﺎﻥ ‪1390‬‬