دليل معلم مقرر رياضيات 1

‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬          

Original Title: ‫اﻟﺮﻳﺎﺿﻴﺎت‬ Geometry © 2010 ‫اﻟﺼﻒ ا ول اﻟﺜﺎﻧﻮي‬ By:   John A. Carter, Ph. D Gilbert J. Cuevas, Ph. D  Roger Day, Ph. D Carol E. Malloy, Ph. D  Contributing Authors  Jerry Cummins Dinah Zike   CONSULTANTS  Mathematical Content  Prof. Viken Hovsepian  Grant A. Fraser, Ph. D Arthur K. Wayman, Ph. D  Gifted and talented Shelbi K. Cole  College Readiness Robert Lee Kimball, Jr.  Graphing Calculator Ruth M. Casey  Mathematical Fluency Robert M. Capraro, Ph.D  Pre-AP Dixie Ross  Reading and Writing Releah Cossett Lent  Lynn T. Havens  www.macmillanmh.com          ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬                      www.obeikaneducation.com www.obeikaneducation.com English Edition Copyright © 2010 the McGrawHill CompaniesInc  ©  All rights reserved ©  Arabic Edition is published by Obeikan under agreement with e McGrawHill CompaniesInc© 2008        

‫‪ ‬‬ ‫ﻳﺴﺮﻧﺎ ﺃﻥ ﻧﻘ ﱢﺪﻡ ﺩﻟﻴﻞ ﺍﻟﻤﻌﻠﻢ ﻟﻤﺎﺩﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪ ،‬ﺁﻣﻠﻴﻦ ﺃﻥ ﻳﻜﻮﻥ ﻟﻜﻢ ﺍﻟﻤﺮﺷﺪ‬ ‫ﻓﻲ ﺗﺪﺭﻳﺲ ﺍﻟﻤﺎﺩﺓ‪ ،‬ﻭﺍﻟﺪﺍﻋﻢ ﻓﻲ ﺗﻘﻮﻳﻢ ﺍﻟﻄﻼﺏ‪ ،‬ﺑﻤﺎ ﻳﺤﻘﻖ ﺍﻷﻫﺪﺍﻑ ﺍﻟﻤﻨﺸﻮﺩﺓ‬ ‫ﻣﻦ ﺗﺪﺭﻳﺲ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪.‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫ﺗﻮﺿﺢ ﻫﺬﻩ ﺍﻟﻤﻘﺪﻣﺔ ﻛﻴﻔﻴﺔ ﺑﻨﺎﺀ ﺍﻟﺴﻠﺴﻠﺔ ﻋﻠﻤ ﹰﹼﻴﺎ ﻭﺗﺮﺑﻮ ﹼﹰﻳﺎ‪ ،‬ﻭﺗﺒﺮﺯ ﺍﻟﻨﻘﺎﻁ ﺍﻟﻤﺤﻮﺭﻳﺔ ﺍﻟﺘﻲ ﻳﺮﻛﺰ ﻋﻠﻴﻬﺎ‬ ‫ﺍﻟﻤﻨﻬﺞ ﻓﻲ ﻫﺬﺍ ﺍﻟﺼﻒ‪ ،‬ﻭﻓﻠﺴﻔﺔ ﺍﻟﺴﻠﺴﻠﺔ ﺍﻟﻤﺘﻮﺍﺯﻧﺔ ﺃﻓﻘ ﹰﹼﻴﺎ ﻭﺍﻟﻤﺘﺮﺍﺑﻄﺔ ﺭﺃﺳ ﹰﹼﻴﺎ‪ ،‬ﻭﺃﺳﺎﻟﻴﺐ ﺍﻟﺘﺪﺭﻳﺲ ﺍﻟﻤﺘﺒﻌﺔ‬ ‫ﻭﺍﻟﻤﺘﻨﻮﻋﺔ ﻓﻲ ﺍﻟﺪﻟﻴﻞ‪ ،‬ﻭﺃﻧﻮﺍﻉ ﺍﻟﺘﻘﻮﻳﻢ‪ ،‬ﻭﺃﺩﻭﺍﺗﻪ ﺍﻟﻤﻘﺘﺮﺣﺔ‪ ،‬ﺍﻟﺘﻲ ﺗﺮﺍﻋﻲ ﺍﻟﻔﺮﻭﻕ ﺍﻟﻔﺮﺩﻳﺔ ﺑﻴﻦ ﺍﻟﻄﻼﺏ‪.‬‬ ‫‪ ‬‬ ‫ﺗﻢ ﺗﻮﺯﻳﻊ ﺍﻟﻤﻘﺮﺭ ﺇﻟﻰ ﻓﺼﻮﻝ‪ .‬ﻭﻳﺒﺪﺃ ﺩﻟﻴﻞ ﺍﻟﻤﻌﻠﻢ ﻓﻲ ﻛﻞ ﻓﺼﻞ ﺑﺘﻘﺪﻳﻢ ﻧﻈﺮﺓ ﻋﺎﻣﺔ ﻋﻠﻴﻪ ﺗﺘﻀﻤﻦ ﻣﺨﻄ ﹰﻄﺎ ﻟﻠﺪﺭﻭﺱ‬ ‫ﻭﺃﻫﺪﺍﻓﻬﺎ‪ ،‬ﻭﻣﺼﺎﺩﺭ ﺗﺪﺭﻳﺴﻬﺎ‪ ،‬ﻭﺍﻟﺨﻄﺔ ﺍﻟﺰﻣﻨﻴﺔ ﺍﻟﻤﻘﺘﺮﺣﺔ ﻟﻠﺘﺪﺭﻳﺲ‪ .‬ﺛﻢ ﻳﻘ ﹼﺪﻡ ﺍﻟﺘﺮﺍﺑﻂ ﺍﻟﺮﺃﺳﻲ ﻟﻤﻮﺿﻮﻉ ﺍﻟﻔﺼﻞ ﺧﻼﻝ‬ ‫ﺍﻟﺼﻒ ﻭﺍﻟﺼﻔﻮﻑ ﺍﻷﺧﺮ￯‪ .‬ﻛﻤﺎ ﻳﻘﺘﺮﺡ ﺍﻟﺪﻟﻴﻞ ﺁﻟﻴﺔ ﻟﺘﻌﻠﻢ ﻣﻬﺎﺭﺍﺕ ﺍﻟﻔﺼﻞ ﻣﻦ ﺧﻼﻝ ﻣﻬﺎﺭﺓ ﺍﻟﺪﺭﺍﺳﺔ‪ .‬ﺛﻢ ﻳﻘﺪﻡ ﺩﻋﻤﺎ‬ ‫ﻟﻠﻤﻌﻠﻢ ﻣﻦ ﺧﻼﻝ ﺻﻔﺤﺔ ﺍﺳﺘﻬﻼﻝ ﺍﻟﻔﺼﻞ ﺍﻟﻤﻮﺟﻮﺩﺓ ﻓﻲ ﻛﺘﺎﺏ ﺍﻟﻄﺎﻟﺐ‪ ،‬ﻭﻛﻴﻔﻴﺔ ﺍﻻﺳﺘﻔﺎﺩﺓ ﻣﻨﻬﺎ ﻓﻲ ﺗﻘﺪﻳﻢ ﻣﻮﺿﻮﻉ‬ ‫ﺍﻟﻔﺼﻞ‪ ،‬ﻛﻤﺎ ﻳﺒﺮﺯ ﻏﺮﺽ ﺍﻟﻤﻄﻮﻳﺎﺕ ﻭﻭﻇﻴﻔﺘﻬﺎ ﻭﻭﻗﺖ ﺍﺳﺘﻌﻤﺎﻟﻬﺎ‪ .‬ﺛﻢ ﻳﻌﺮﺽ ﻣﺨﻄ ﹰﻄﺎ ﻟﻠﺘﻘﻮﻳﻢ ﺑﺄﻧﻮﺍﻋﻪ ﺍﻟﻤﺨﺘﻠﻔﺔ ﻭﺃﺩﻭﺍﺗﻪ‬ ‫ﺍﻟﻤﺘﻌﺪﺩﺓ‪.‬‬ ‫‪ ‬‬ ‫ﻳﻘ ﹼﺪﻡ ﺍﻟﺪﻟﻴﻞ ﺃﻧﺸﻄﺔ ﻣﻘﺘﺮﺣﺔ ﺗﺮﺍﻋﻲ ﺍﻟﻔﺮﻭﻕ ﺍﻟﻔﺮﺩﻳﺔ ﺑﻴﻦ ﺍﻟﻄﻼﺏ‪ ،‬ﻭﺑﺄﺳﺎﻟﻴﺐ ﺗﺪﺭﻳﺲ ﻣﺘﻨﻮﻋﺔ‪ ،‬ﺗﺴﺎﻋﺪ ﺍﻟﻤﻌﻠﻢ ﻓﻲ ﺗﺪﺭﻳﺲ ﻛﻞ‬ ‫ﺩﺭﺱ‪ .‬ﺑﻌﺪ ﺫﻟﻚ ﻳﻌﺮﺽ ﺍﻟﺪﻟﻴﻞ ﺍﻟﺪﺭﺱ ﺑﺨﻄﻮﺍﺕ ﻣﺤﺪﺩﺓ ﻫﻲ‪:‬‬ ‫‪ ‬ﻳﺒﻴﻦ ﺗﺮﺍﺑﻂ ﺍﻟﻤﻬﺎﺭﺍﺕ ﺍﻟﺮﺋﻴﺴﺔ ﻗﺒﻞ ﺍﻟﺪﺭﺱ ﻭﻓﻲ ﺃﺛﻨﺎﺋﻪ ﻭﺑﻌﺪﻩ‪.‬‬ ‫‪ ‬ﻳﻘ ﹼﺪﻡ ﻣﻘﺘﺮﺣﺎﺕ ﻟﻠﻤﻌﻠﻢ ﺣﻮﻝ ﻛﻴﻔﻴﺔ ﺗﺪﺭﻳﺲ ﺍﻟﺪﺭﺱ‪ ،‬ﺗﺘﻀﻤﻦ ﺃﺳﺌﻠﺔ ﺗﻌﺰﻳﺰ ﺣﻮﺍﺭﻳﺔ ﻭﺃﻧﺸﻄﺔ ﻣﻘﺘﺮﺣﺔ‪ ،‬ﻭﻳﺒﺮﺯ‬ ‫ﺍﻟﻤﺤﺘﻮ￯ ﺍﻟﺮﻳﺎﺿﻲ ﻟﻤﻮﺿﻮﻉ ﺍﻟﺪﺭﺱ‪ .‬ﻛﻤﺎ ﻳﻘ ﹼﺪﻡ ﺃﻣﺜﻠﺔ ﺇﺿﺎﻓﻴﺔ ﻟﻠﻤﻌﻠﻢ‪.‬‬ ‫‪ ‬ﻳﺘﻀﻤﻦ ﺗﺪﺭﻳﺒﺎﺕ ﻣﺘﻨﻮﻋﺔ ﺣﺴﺐ ﻣﺴﺘﻮﻳﺎﺕ ﺍﻟﻄﻼﺏ ﺗﺤﻘﻖ ﺃﻫﺪﺍﻑ ﺍﻟﺪﺭﺱ‪.‬‬ ‫‪ ‬ﻳﻘ ﹼﺪﻡ ﻣﻘﺘﺮﺣﺎﺕ ﻟﺘﻘﻮﻳﻢ ﺍﻟﺪﺭﺱ‪ ،‬ﻛﻤﺎ ﻳﺘﻀﻤﻦ ﻣﻘﺘﺮ ﹰﺣﺎ ﻟﻠﻤﻌﻠﻢ ﻟﻠﺘﺄﻛﺪ ﻣﻦ ﻣﺪ￯ ﺍﺳﺘﻴﻌﺎﺏ ﺍﻟﻄﻼﺏ ﻟﻠﻤﻔﺎﻫﻴﻢ‬ ‫ﻭﺇﺗﻘﺎﻧﻬﻢ ﺍﻟﻤﻬﺎﺭﺍﺕ ﺍﻟﻤﻘ ﹼﺪﻣﺔ ﻓﻲ ﺍﻟﺪﺭﺱ‪ ،‬ﻭﻳﻌﺮﺽ ﺍﻟﺪﻟﻴﻞ ﺁﻟﻴﺔ ﻟﻤﺘﺎﺑﻌﺔ ﺍﻟﻤﻄﻮﻳﺎﺕ‪.‬‬ ‫ﻛﻤﺎ ﻳﻘ ﹼﺪﻡ ﺍﻟﺪﻟﻴﻞ ﻓﻲ ﻛﻞ ﺩﺭﺱ ﺇﺟﺎﺑﺎﺕ ﻣﻔ ﹼﺼﻠﺔ ﻟﺒﻌﺾ ﺍﻷﺳﺌﻠﺔ ﻭﺍﻟﺘﻤﺎﺭﻳﻦ‪.‬‬ ‫‪  ‬‬ ‫ﺗﻘ ﹼﺪﻡ ﺍﻟﺴﻠﺴﻠﺔ ﺃﺳﺎﻟﻴﺐ ﻣﺘﻨﻮﻋﺔ ﻟﺘﻘﻮﻳﻢ ﺍﻟﻄﻼﺏ )ﺍﻟﺘﺸﺨﻴﺼﻲ ﻭﺍﻟﺘﻜﻮﻳﻨﻲ ﻭﺍﻟﺨﺘﺎﻣﻲ(‪ ،‬ﻭﺁﻟﻴﺎﺕ ﻟﻤﻌﺎﻟﺠﺔ ﺍﻷﺧﻄﺎﺀ‬ ‫ﻭﺍﻟﺼﻌﻮﺑﺎﺕ ﻟﺪ￯ ﺍﻟﻄﻼﺏ‪.‬‬ ‫ﻭﻧﺤﻦ ﺇﺫ ﻧﻘ ﹼﺪﻡ ﻫﺬﺍ ﺍﻟﺪﻟﻴﻞ ﻟﺰﻣﻼﺋﻨﺎ ﺍﻟﻤﻌﻠﻤﻴﻦ ﻭﺍﻟﻤﻌﻠﻤﺎﺕ‪ ،‬ﻟﻨﺄﻣﻞ ﺃﻥ ﻳﺤﻮﺫ ﺍﻫﺘﻤﺎﻣﻬﻢ‪ ،‬ﻭﻳﻠﺒﻲ ﻣﺘﻄﻠﺒﺎﺗﻬﻢ ﻟﺘﺪﺭﻳﺲ ﻫﺬﺍ‬ ‫ﺍﻟﻤﻘﺮﺭ‪ ،‬ﻭﻳﺴﺎﻋﺪﻫﻢ ﻓﻲ ﺃﺩﺍﺀ ﺭﺳﺎﻟﺘﻬﻢ‪.‬‬ ‫‪‬‬ ‫‪‬‬

 10A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 10C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 12  1 -1 19 1 -2 26 1 -3 36  1-3 37 1 -4 45  1 -5 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  53 1 -6 60 1 -7 66 1 -8 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  79 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   84A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 84C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 86  2 -1 92 2-2 94 2 -2 102 2 -3 108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  109 2 -4 117  2 -5 125  2-5 126  2 -6 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  143A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

 144A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 144C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  144E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 146 3 -1 153 3-2 154 3 -2 162 3 -3 170SSS , SAS 3 -4 178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  179ASA , AAS 3 -5 186  3-5 188  3 -6 196  3 -7 202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  211A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    212A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 212C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  212E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 214  4-1 215   4 -1 224 4-2 225  4 -2 233  4 -3 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  241 4 -4 248  4-5 250  4 -5 255 4 -6 263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  271A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  

          2  1 ‫ﺇﻥ ﺍﻟﺘﺮﺍﺑﻂ ﺍﻟﺮﺃﺳﻲ ﺍﻟﻘﻮﻱ ﺑﻴﻦ ﺍﻷﺳﺎﻟﻴﺐ ﺍﻟﺘﺪﺭﻳﺴﻴﺔ ﺑﺪ ﹰﺀﺍ ﻣﻦ ﺍﻟﺼﻒ ﺍﻷﻭﻝ‬ ‫ﻳﻌﺪ ﺍﻟﺘﺮﺍﺑﻂ ﺍﻟﺮﺃﺳﻲ ﻟﻠﻤﺤﺘﻮ￯ ﻋﻤﻠﻴﺔ ﻣﻬﻤﺔ ﺗﺴﺎﻋﺪ ﻃﻼﺑﻚ ﻋﻠﻰ ﺍﻟﺘﺤﻘﻖ‬ ،‫ﻳﺴﻬﻞ ﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺍﻻﻧﺘﻘﺎﻝ ﻣﻦ ﺍﻟﻤﺮﺣﻠﺔ ﺍﻻﺑﺘﺪﺍﺋﻴﺔ ﺇﻟﻰ ﺍﻟﻤﺘﻮﺳﻄﺔ‬ ‫ ﻭﻫﺬﺍ‬.‫ﻣﻦ ﺍﻟﺘﺴﻠﺴﻞ ﺍﻟﺪﻗﻴﻖ ﻟﻠﻤﺤﺘﻮ￯ ﻭﺗﺘﺎﺑﻌﻪ ﻣﻦ ﻣﺴﺘﻮ￯ ﺇﻟﻰ ﻣﺴﺘﻮ￯ ﺁﺧﺮ‬ ‫ ﻭﺍﻟﺘﻘﻨﻴﺎﺕ ﻭﺍﻟﻮﺳﺎﺋﻞ ﺍﻟﺤﺴﻴﺔ ﻭﺧﻄﺔ ﺍﻟﺪﺭﺱ‬،‫ ﺇﺫ ﺗﻌﻤﻞ ﺍﻟﻤﻔﺮﺩﺍﺕ‬.‫ﻓﺎﻟﺜﺎﻧﻮﻳﺔ‬ ‫ﻳﻤﻨﺤﻚ ﺍﻟﺜﻘﺔ ﺑﺄﻥ ﺍﻟﻤﺤﺘﻮ￯ ﻳﺘﻢ ﺗﻘﺪﻳﻤﻪ ﻭﺗﻌﺰﻳﺰﻩ ﻭﺗﻘﻮﻳﻤﻪ ﻓﻲ ﺍﻷﻭﻗﺎﺕ‬ ‫ ﻣﻤﺎ‬،‫ ﻛﻤﺎ ﻳﺴﺎﻋﺪ ﻋﻠﻰ ﺳﺪ ﺍﻟﺜﻐﺮﺍﺕ ﻭﺗﺠﻨﺐ ﺍﻟﺘﻜﺮﺍﺭ ﻏﻴﺮ ﺍﻟﻤﺒﺮﺭ‬،‫ﺍﻟﻤﻨﺎﺳﺒﺔ‬ ‫ﻭﺍﻟﻤﻌﺎﻟﺠﺔ ﻋﻠﻰ ﺍﻟﺘﻘﻠﻴﻞ ﻣﻦ ﻋﻮﺍﻣﻞ ﺍﻟﺼﻌﻮﺑﺔ ﻭﺍﻟﺘﺸﻮﻳﺶ ﺍﻟﺘﻲ ﻳﻮﺍﺟﻬﻬﺎ ﺑﻌﺾ‬ .‫ﻳﻤﻜﻨﻚ ﻣﻦ ﺗﻮﺟﻴﻪ ﺗﺪﺭﻳﺴﻚ ﻭﺗﻜﻴﻴﻔﻪ ﻟﻴﺘﻼﺋﻢ ﺣﺎﺟﺎﺕ ﺍﻟﻄﻼﺏ‬ .‫ﺍﻟﻄﻼﺏ ﻋﻨﺪﻣﺎ ﻳﻨﺘﻘﻠﻮﻥ ﻋﺒﺮ ﺍﻟﺼﻔﻮﻑ ﺍﻟﻤﺨﺘﻠﻔﺔ‬  3 ،‫ﺗﺸﺘﻤﻞ ﺻﻔﺤﺎﺕ ﺍﻟﺴﻠﺴﻠﺔ ﻋﻠﻰ ﺗﺼﺎﻣﻴﻢ ﺑﺼﺮﻳﺔ ﻣﺘﺴﻘﺔ ﻣﻦ ﺻﻒ ﻵﺧﺮ‬ ‫ ﻛﻤﺎ ﺗﺰﺩﺍﺩ‬،￯‫ﺗﺴﺎﻋﺪ ﺍﻟﻄﻼﺏ ﻋﻠﻰ ﺍﻻﻧﺘﻘﺎﻝ ﺑﺴﻼﺳﺔ ﻣﻦ ﻣﺮﺣﻠﺔ ﺇﻟﻰ ﺃﺧﺮ‬ ‫ﺩﺍﻓﻌﻴﺘﻬﻢ ﻟﻠﺘﻌﻠﻢ ﻭﺍﻟﻨﺠﺎﺡ ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﻃﺮﻳﻘﺔ ﺍﻟﺘﻌﺎﻣﻞ ﻣﻊ ﻫﺬﻩ ﺍﻟﺼﻔﺤﺎﺕ‬ .‫ﻣﺄﻟﻮﻓﺔ ﻟﺪﻳﻬﻢ‬ ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬ ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬   ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬     ‫ت‬‫ﺎ‬‫ﻳ‬‫ﻮ‬‫ﺘ‬‫ﺤ‬‫ﻤ‬‫ﻟ‬‫ا‬ ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬   ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬   ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬     www.obeikaneducation.com‫ت‬‫ﺎ‬‫ﻳ‬‫ﻮ‬‫ﺘ‬‫ﺤ‬‫ﻤ‬‫ﻟ‬‫ا‬      www.obeikaneducation.com   ‫اﻟﻤﺤﺘﻮﻳﺎت‬    www.obeikaneducation.com www.obeikaneducation.com ‫ت‬‫ﺎ‬‫ﻳ‬‫ﻮ‬‫ﺘ‬‫ﺤ‬‫ﻤ‬‫ﻟ‬‫ا‬ ٢           www.obeikaneducation.com       T1

 3  ،‫ ﻭﺧﺘﺎﻣﻴﺔ‬،‫ ﻭﺗﻜﻮﻳﻨﻴﺔ‬،‫ﺗﺘﻀﻤﻦ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ ﻣﺼﺎﺩﺭ ﻣﺘﻌﺪﺩﺓ ﻟﻠﺘﻘﻮﻳﻢ؛ ﺗﺸﺨﻴﺼﻴﺔ‬  1 .‫ ﻭﺇﺛﺮﺍﺋﻴﺔ‬،‫ﺇﺿﺎﻓﺔ ﺇﻟﻰ ﺧﻄﻂ ﻋﻼﺟﻴﺔ‬ ‫ ﻣﻦ ﺍﻟﻄﻠﺒﺔ ﺍﻟﺬﻳﻦ ﻳﻈﻬﺮﻭﻥ ﻧﺠﺎ ﹰﺣﺎ ﻓﻲ ﻣﺠﺎﻟﻲ‬٪٨٠ ‫ﺑﻴﻨﺖ ﻧﺘﺎﺋﺞ ﺍﻟﺒﺤﻮﺙ ﺃﻥ‬  4 ،‫ﺍﻟﺠﺒﺮ ﻭﺍﻟﻬﻨﺪﺳﺔ ﻓﻲ ﺍﻟﺼﻒ ﺍﻟﻌﺎﺷﺮ ﻳﻠﺘﺤﻘﻮﻥ ﺑﺎﻟﻜﻠﻴﺎﺕ ﺍﻟﺠﺎﻣﻌﻴﺔ ﺫﺍﺕ ﺍﻟﻌﻼﻗﺔ‬ ‫ ﻭﺃﺧﺮ￯ ﺇﺛﺮﺍﺋﻴﺔ‬،‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﻣﺼﺎﺩﺭ ﻣﺘﻨﻮﻋﺔ ﺗﺘﻀﻤﻦ ﺃﻧﺸﻄﺔ ﻭﺧﻄ ﹰﻄﺎ ﻋﻼﺟﻴﺔ‬ ‫ ﻭﺑﻨﺎ ﹰﺀ ﻋﻠﻰ ﺫﻟﻚ ﺍﻫﺘﻤﺖ ﺍﻟﺴﻠﺴﻠﺔ ﺑﺎﻟﺨﺮﺍﺋﻂ ﺍﻟﻤﻔﺎﻫﻴﻤﻴﺔ ﻟﻠﺨﺒﺮﺍﺕ‬.‫ﻭﻳﻨﺠﺤﻮﻥ‬ .‫ﻭﻓ ﹰﻘﺎ ﻟﻨﺘﺎﺋﺞ ﺍﻟﻄﻼﺏ ﻓﻲ ﺍﻟﺘﻘﻮﻳﻢ ﺍﻟﺘﺸﺨﻴﺼﻲ‬ .‫ﺍﻟﺴﺎﺑﻘﺔ ﻭﻃﻮﺭﺗﻬﺎ‬  2 ‫ﺗﻢ ﺗﻄﻮﻳﺮ ﺍﻟﺴﻠﺴﻠﺔ ﺑﺤﻴﺚ ﺗﺮﻛﺰ ﻋﻠﻰ ﺍﻟﻤﻬﺎﺭﺍﺕ ﻭﺍﻟﻤﻮﺿﻮﻋﺎﺕ ﺍﻟﺘﻲ ﻳﻮﺍﺟﻪ‬ .‫ﺍﻟﻄﻼﺏ ﺻﻌﻮﺑﺎﺕ ﻓﻴﻬﺎ؛ ﻣﺜﻞ ﺣﻞ ﺍﻟﻤﺴﺄﻟﺔ ﻓﻲ ﻛﻞ ﻣﺴﺘﻮ￯ ﺻﻔﻲ‬ ‫ ﻭﺗﺘﻀﻤﻦ ﺗﻌﺮﻑ ﺃﺧﻄﺎﺀ ﺍﻟﻄﻼﺏ‬ 1 3 - 5  2 1  ‫ﻭﻣﻌﺎﻟﺠﺘﻬﺎ؛ ﻭﺫﻟﻚ ﺑﻤﺮﺍﺟﻌﺔ ﺍﻟﻤﻔﺎﻫﻴﻢ ﻭﺍﻟﻤﻬﺎﺭﺍﺕ ﺍﻟﻤﺘﻌﻠﻘﺔ‬   (1   (1  (2 .‫ ﻗﺒﻞ ﺍﻻﻧﺘﻘﺎﻝ ﺇﻟﻰ ﺗﺪﺭﻳﺲ ﺍﻟﻤﻌﺮﻓﺔ ﺍﻟﺠﺪﻳﺪﺓ‬،‫ﺑﻬﺎ‬  (2  (3  (3  (4 ‫ ﻭﺗﺘﻀﻤﻦ ﺍﺳﺘﻌﻤﺎﻝ ﺑﺪﺍﺋﻞ‬ 2 ￯‫ﻭﺍﺳﺘﺮﺍﺗﻴﺠﻴﺎﺕ ﻣﺘﻨﻮﻋﺔ ﺗﻨﺎﺳﺐ ﺃﻧﻤﺎﻁ ﺍﻟﺘﻌﻠﻢ ﺍﻟﻤﺨﺘﻠﻔﺔ ﻟﺪ‬  (4  (5 .‫ﺍﻟﻄﻼﺏ‬  (5  (6  (6  5 6 – 8  9 – 1 2   (1 ‫ ﺑﻄﺮﻕ ﺗﻌﻠﻴﻢ ﺇﺿﺎﻓﻴﺔ؛‬،‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﻓﺮ ﹰﺻﺎ ﻋﺪﻳﺪﺓ ﻟﻠﻤﻌﻠﻢ ﻟﻴﻄ ﹼﻮﺭ ﺃﺩﺍﺀﻩ ﻣﻬﻨ ﹰﹼﻴﺎ‬  (1  (2 ‫ ﻭﺍﻟﻤﻮﺍﻗﻊ ﺍﻹﻟﻜﺘﺮﻭﻧﻴﺔ ﺍﻟﻤﺘﺮﺍﺑﻄﺔ‬،‫ ﻭﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﺍﻟﻤﺤﻮﺳﺒﺔ‬،‫ ﺍﻟﻔﻴﺪﻳﻮ‬:‫ﻣﺜﻞ‬  (3  (2  (4 .‫ﺗﺮﺍﺑﻄ ﹰـﺎ ﺭﺃﺳ ﹰﹼﻴﺎ ﻣﺘﻜﺎﻣ ﹰﻼ ﻣﻦ ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﺍﻻﺑﺘﺪﺍﺋﻲ ﺇﻟﻰ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻧﻲ ﻋﺸﺮ‬  (3  (5  (4  (5  (6 ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬  ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬  ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬       ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬      ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬              ‫اﻟﻤﻤﻠﻜﺔ اﻟﻌﺮﺑﻴﺔ اﻟﺴﻌﻮدﻳﺔ‬                        www.obeikaneducation.com   www.obeikaneducation.com      ‫ت‬‫ﺎ‬‫ﻳ‬‫ﻮ‬‫ﺘ‬‫ﺤ‬‫ﻤ‬‫ﻟ‬‫ا‬  www.obeikaneducation.com www.obeikaneducation.com      www.obeikaneducation.com  www.obeikaneducation.com    T2   

‫‪‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪    ‬‬ ‫‪ ‬‬ ‫• ﺍﺳﺘﻘﺼﺎﺀ ﺍﻟﻤﻔﺎﻫﻴﻢ ﻭﺑﻨﺎﺀ ﻓﻬﻢ ﺇﺩﺭﺍﻛﻲ‪.‬‬ ‫• ﺗﻄﻮﻳﺮ ﻣﻬﺎﺭﺍﺕ ﺇﺟﺮﺍﺋﻴﺔ ﻭﺣﺴﺎﺑﻴﺔ ﻭﺗﻌﺰﻳﺰﻫﺎ ﻭﺇﺗﻘﺎﻧﻬﺎ‪.‬‬ ‫• ﺗﻄﺒﻴﻖ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻓﻲ ﺣﻞ ﻣﺴﺎﺋﻞ ﻣﻦ ﻭﺍﻗﻊ ﺍﻟﺤﻴﺎﺓ‪.‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪          ‬‬ ‫‪‬ﻳﺴﺘﻌﻤﻞ ﻃﻼﺏ ‪‬‬ ‫ﹶﻳﺤﻮﻱ ﻛﻴ ﹲﺲ ﹶﻋﺪ ﹰﺩﺍ ﻣﻦ ﹶﺣ ﱠﺒﺎ ﹺﺕ ﺍﻟﺘﻔﺎ ﹺﺡ‪،‬‬ ‫‪      ‬‬ ‫ﻗﻄﻊ ﻋﺪ ﺑﻠﻮﻧﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻟﺘﻤﺜﻴﻞ ﺟﻤﻞ ﺍﻟﺠﻤﻊ‪.‬‬ ‫ﻭﺇﹺﻟﻰ ﺟﺎﻧ ﹺﺐ ﺍﻟﻜﻴ ﹺﺲ ﹸﺗﻔﺎﺣﺘﺎ ﹺﻥ؛ ﺇﺫ ﹾﻥ ﻋﺪ ﹸﺩ‬ ‫‪      ‬‬ ‫‪       ‬‬ ‫ﻭ ﹸﻳﻌﺪﱡ ﻫﺬﺍ ﺍﻟﻨﺸﺎﻁ ﺃﺳﺎ ﹰﺳﺎ ﻟﻠﻔﻬﻢ ﻭﺍﻟﻨﺠﺎﺡ ﻓﻲ ﺣﻞ‬ ‫ﺍﻟﺘﻔﺎ ﹺﺡ ﺍﻟ ﹸﻜﻠ ﱡﻲ ﹸﻳﺴﺎﻭﻱ ﻋ ﹶﺪ ﹶﺩ ﺍﻟﺘﻔﺎﺣﺎ ﹺﺕ‬ ‫ﻣﻌﺎﺩﻻﺕ ﺟﺒﺮﻳﺔ‪.‬‬ ‫‪      ‬‬ ‫ﻓﻲ ﺍﻟﻜﻴ ﹺﺲ ﺯﺍﺋ ﹶﺪ ‪.٢‬‬ ‫ﺍﻟ ﹸﻤﺘﻐ ﱢﻴ ﹸﺮ‬ ‫ﹸﻳﻤ ﹺﻜ ﹸﻦ ﺗﻤﺜﻴ ﹸﻞ ﺍﻟ ﹶﻌﺪ ﹺﺩ ﺍﻟ ﹶﻤﺠﻬﻮ ﹺﻝ ﻣﻦ ﺍﻟ ﱡﺘﻔﺎﺣﺎ ﹺﺕ ﺑﹺ ﹸﻤﺘﻐ ﱢﻴ ﹴﺮ‪ ،‬ﻭﺍﻟ ﹸﻤﺘﻐ ﱢﻴ ﹸﺮ ﺣﺮ ﹲﻑ ﺃﻭ ﺭﻣ ﹲﺰ ﹸﻳﻤﺜ ﹸﻞ‬ ‫ﺍﻟﻌﺒﺎﺭ ﹸﺓ ﺍﻟﺠﺒﺮ ﱠﻳ ﹸﺔ‬ ‫ﺣﺴﺎ ﹸﺏ ﻗﻴﻤ ﹴﺔ‬ ‫ﹶﻋﺪ ﹰﺩﺍ ﻣﺠﻬﻮ ﹰﻻ‪.‬‬ ‫‪www.obeikaneducation.com‬‬ ‫ﻋﺪ ﹸﺩ ﺍﻟﺘﻔﺎﺣﺎ ﹺﺕ ﺧﺎﺭ ﹶﺝ‬ ‫ﺱ‪٢+‬‬ ‫ﹶﻋﺪ ﹸﺩ ﺍﻟﺘﻔﺎﺣﺎ ﹺﺕ ﻓﻲ ﺍﻟﻜﻴ ﹺﺲ‬ ‫‪         ٧‬‬ ‫ﺍﻟﻜﻴ ﹺﺲ ﻗﻴﻤ ﹲﺔ ﹶﻣﻌﻠﻮﻣ ﹲﺔ‪.‬‬ ‫ﻗﻴﻤ ﹲﺔ ﹶﻣﺠﻬﻮﻟ ﹲﺔ‪.‬‬ ‫‪      ‬‬ ‫ﺍﻟﻌﺒﺎﺭ ﹸﺓ ﺍﻟﺠﺒﺮ ﱠﻳ ﹸﺔ ﹺﻣ ﹾﺜ ﹸﻞ ﺱ ‪ ،٢+‬ﹶﻣﺠﻤﻮﻋ ﹲﺔ ﻣﻦ ﺍﻟ ﹸﻤﺘﻐ ﱢﻴﺮﺍ ﹺﺕ ﻭﺍﻷﻋﺪﺍ ﹺﺩ ﹶﺗ ﹾﺮﺑﹺ ﹸﻄﻬﺎ ﹶﻋﻤﻠﻴ ﹲﺔ‬ ‫‪٧=٦ +١‬‬ ‫ﻭﺍ ﹺﺣﺪ ﹲﺓ ﻋﻠﻰ ﺍﻷ ﹶﻗ ﱢﻞ‪ .‬ﹺﻋﻨﺪﻣﺎ ﺗﺴﺘﺒﺪ ﹸﻝ ﺑﺎﻟﻤﺘﻐ ﱢﻴ ﹺﺮ ﻋﺪ ﹰﺩﺍ ﻓﻲ ﻋﺒﺎﺭ ﹴﺓ‪ ،‬ﹸﻳﻤﻜ ﹸﻨ ﹶﻚ ﺣﺴﺎ ﹸﺏ ﻗﻴﻤ ﹺﺔ‬ ‫‪٧=٥ +٢‬‬ ‫‪         ‬‬ ‫‪٧=٤ +٣‬‬ ‫‪       ‬‬ ‫ﺗﹺ ﹾﻠ ﹶﻚ ﺍﻟﻌﺒﺎﺭ ﹺﺓ‪.‬‬ ‫‪ ‬‬ ‫‪     ‬‬ ‫‪ ‬ﹶﺃﻭ ﹺﺟ ﹾﺪ ﻗﻴﻤ ﹶﺔ ﺍﻟﻌﺒﺎﺭ ﹺﺓ ﺱ ‪ ،٢+‬ﺇﺫﺍ ﻛﺎﻧ ﹾﺖ ﺱ = ‪٣‬‬ ‫ﺱ ‪ ٢ +‬ﺍ ﹾﻛ ﹸﺘ ﹺﺐﺍﻟﻌﺒﺎﺭ ﹶﺓ‪.‬ﺍﺳﺘﻌﻤ ﹾﻞ ﹸﻛﻮ ﹰﺑﺎﻭﻗﻄﻌ ﹶﺘﻲ‬ ‫ﹶﻋ ﱟﺪ ﻟﹺﺘﻤﺜﻴ ﹺﻞ ﺱ ‪٢+‬‬ ‫‪ ٢ + ٣‬ﻋ ﱢﻮ ﹾﺽ ﻋﻦ ﺱ ﺑﺎﻟ ﹶﻌﺪ ﹺﺩ ‪ ٣‬ﹶﺿ ﹾﻊ ‪ ٣‬ﹺﻗ ﹶﻄ ﹺﻊ‬ ‫ﻋ ﱟﺪ ﻓﻲ ﺍﻟ ﹸﻜﻮ ﹺﺏ‪.‬‬ ‫ﺍﺟﻤ ﹾﻊ ‪ ٣‬ﻭ ‪٢‬‬ ‫‪٥‬‬ ‫‪٧=٧+٠‬‬ ‫ﹶﺗ ﹾﻜ ﹺﻮﻳ ﹸﻦ ﺍ ﹾﻟ ﹶﻌ ﹶﺪ ﹺﺩ ‪٧‬‬ ‫ﺍﻟﻤﺠﻤﻮ ﹸﻉ‪٥‬‬ ‫‪٧=٦+١‬‬ ‫ﹸﻳ ﹶﺴﺎ ﹺﻭﻱ ﺍﻟ ﹼﻨﺎﺗﹺ ﹸﺞ‬ ‫ﺯﺍﺋﹺﺪ‬ ‫‪...‬‬ ‫‪...‬‬ ‫‪  ‬‬ ‫‪٧ = 7+‬‬ ‫‪0‬‬ ‫=‪٧‬‬ ‫‪+‬‬ ‫‪‬ﺃﻣﺎ ﻃﻼﺏ ‪ ‬ﻓﻴﺴﺘﻔﻴﺪﻭﻥ ﻣﻦ‬ ‫=‪٧‬‬ ‫‪+‬‬ ‫ﺧﺒﺮﺍﺗﻬﻢ ﻓﻲ ﺍﻟﺘﻌﺎﻣﻞ ﻣﻊ ﺍﻷﻛﻮﺍﺏ ﻭﻗﻄﻊ ﺍﻟﻌﺪ؛ ﻻﺳﺘﻌﻤﺎﻟﻬﺎ‬ ‫=‪٧‬‬ ‫‪+‬‬ ‫ﻓﻲ ﺗﻤﺜﻴﻞ ﻣﻌﺎﺩﻻﺕ ﺍﻟﺠﻤﻊ ﻭﺍﻟﻄﺮﺡ‪ ،‬ﻭﺣﻠﻬﺎ‪.‬‬ ‫=‪٧‬‬ ‫‪+‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫‪  ٥-١‬‬ ‫‪:٦ ١٢٨‬‬ ‫‪ T3‬‬

‫ﺗﺴﺘﻄﻴﻊ ﺃﺣﻴﺎ ﹰﻧﺎ ﺃﻥ ﺗﻜﺘﺐ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻓﻴﻬﺎ ﺍﻟﻤﺘﻐﻴﺮ ‪ x‬ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ‪ ، au2 + bu + c‬ﻓﻤﺜ ﹰﻼ ﺑﻔﺮﺽ ﺃﻥ ‪،u = x2‬‬ ‫ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ‪ x4 + 12x2 + 32‬ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ‪ (x2)2 + 12(x2) + 32‬ﺃﻭ ‪. u2 + 12u + 32‬‬ ‫ﻭﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﺍﻟﺠﺪﻳﺪﺓ ﻫﺬﻩ ﺗﻜﺎﻓﺊ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﺍﻷﺻﻠﻴﺔ‪ ،‬ﻭﻟﻜﻨﻬﺎ ﻣﻜﺘﻮﺑﺔ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ‪.‬‬ ‫‪    ‬‬ ‫‪ a,b,ca≠0au2+bu+c ‬‬ ‫‪x ‬‬ ‫‪xu‬‬ ‫‪12x6 + 8x3 + 1 = 3(2x3)2 + 4(2x3) + 1‬‬ ‫‪‬‬ ‫‪ 5‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺇﻥ ﺃﻣﻜﻦ ﺫﻟﻚ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪150n8 + 40n4 - 15 (a‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﺑﺤﺚ ﻋﻦ ﻋﺎﻣﻠﻴﻦ ﻟﻠﻌﺪﺩ ‪ 150‬؛ ﺃﺣﺪﻫﻤﺎ ﻣﺮﺑﻊ ﻛﺎﻣﻞ‪ ،‬ﻭﻋﻦ ﻋﺎﻣﻠﻴﻦ ﻟﻠﻌﺪﺩ ‪40‬؛ ﺃﺣﺪﻫﻤﺎ ﺍﻟﺠﺬﺭ ﺍﻟﺘﺮﺑﻴﻌﻲ ﻷﺣﺪ ﻋﺎﻣﻠﻲ ﺍﻟﻌﺪﺩ ‪.150‬‬ ‫‪ u‬‬ ‫‪‬‬ ‫‪150 = 6×25, 40 = 8×5‬‬ ‫‪150n8 + 40n4 - 15 = 6×25n8 + 8×5n4 - 15‬‬ ‫‪  ‬‬ ‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺑﻄﺎﻗﺎﺕ ﺍﻟﺠﺒﺮ ﻟﺤﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﻲ ﺗﺸﺘﻤﻞ ﻋﻠﻰ ﻣﺘﻐﻴﺮﺍﺕ ﻓﻲ ﻃﺮﻓﻴﻬﺎ‪.‬‬ ‫‪(5n4)2 = 25n8‬‬ ‫‪= 6(5n4)2 + 8(5n4) - 15‬‬ ‫‪ ‬‬ ‫‪y8 + 12y3 + 8 (b‬‬ ‫‪‬‬ ‫‪ ‬ﺍﺳﺘﻌﻤﻞ ﺑﻄﺎﻗﺎﺕ ﺍﻟﺠﺒﺮ ﻟﺤﻞ‪٣ :‬ﺱ ‪ = ١ +‬ﺱ ‪.٥ +‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻻ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺘﻬﺎ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ؛ ﻷﻥ ‪.y8 ≠ (y3)2‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪8x4 + 12x2 + 18 (5B‬‬ ‫‪x4 + 5x + 6 (5A‬‬ ‫‪‬‬ ‫‪  x ‬‬ ‫‪  ‬‬ ‫ﻳﻤﻜﻨﻚ ﻓﻲ ﺑﻌﺾ ﺍﻷﺣﻴﺎﻥ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺼﻮﺭﺓ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻟﺤﻞ ﻣﻌﺎﺩﻻﺕ ﻛﺜﻴﺮﺍﺕ ﺍﻟﺤﺪﻭﺩ ﺫﺍﺕ ﺩﺭﺟﺎﺕ ﺃﻛﺒﺮ ﻣﻦ‬ ‫ﻣ ﹼﺜﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪.‬‬ ‫‪‬‬ ‫ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ‪.‬‬ ‫‪+‬‬ ‫‪ 6‬‬ ‫ﺍﺣﺬﻑ ﻋﺪ ﹰﺩﺍ ﻣﺘﺴﺎﻭ ﹰﻳﺎ ﻣﻦ ﺑﻄﺎﻗﺎﺕ ﺱ‬ ‫‪‬‬ ‫‪‬‬ ‫ﻣﻦ ﻛﻞ ﻃﺮﻑ ﺇﻟﻰ ﺃﻥ ﺗﺼﺒﺢ ﺑﻄﺎﻗﺎﺕ ﺱ‬ ‫‪ ‬‬ ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪.18 x4 - 21x2 + 3 = 0 :‬‬ ‫‪‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫ﻓﻲ ﺃﺣﺪ ﺍﻟﻄﺮﻓﻴﻦ ﻓﻘﻂ‪.‬‬ ‫‪ +‬‬ ‫‪‬‬ ‫‪18x4 - 21x2 + 3 = 0‬‬ ‫ﺍﺳﺘﻌﻤﻠﻨﺎ ﺳﺎﺑ ﹰﻘﺎ ﻗﻄﻊ ﺍﻟﻌﺪ ﺍﻟﻤﻮﺟﺒﺔ ﻭﺍﻟ ﱠﺴﺎﻟﺒﺔ ﻟﺠﻤﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻟ ﱠﺼﺤﻴﺤﺔ ﻭﻃﺮﺣﻬﺎ‬ ‫‪ ‬‬ ‫ﺍﺣﺬﻑ ﻋﺪ ﹰﺩﺍ ﻣﺘﺴﺎﻭ ﹰﻳﺎ ﻣﻦ ﺑﻄﺎﻗﺎﺕ‬ ‫‪‬‬ ‫ﻭﺿﺮﺑﻬﺎ ﻭﻗﺴﻤﺘﻬﺎ‪ ،‬ﻛﺬﻟﻚ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺍﻷﻋﺪﺍﺩ ﺍﻟ ﱠﺼﺤﻴﺤﺔ ﺑﺒﻄﺎﻗﺎﺕ ﺍﻟﺠﺒﺮ‪.‬‬ ‫‪‬‬ ‫‪2(3x2)2 = 18x4‬‬ ‫‪2(3x2)2 - 7(3x2) + 3 = 0‬‬ ‫ﺍﻟﻌﺪﺩ )‪ (١‬ﻣﻦ ﻛﻞ ﻃﺮﻑ ﺇﻟﻰ ﺃﻥ ﺗﺼﺒﺢ‬ ‫‪‬‬ ‫ﻭﺍﻟﺠﺪﻭﻝ ﺍﻟ ﱠﺘﺎﻟﻲ ﻳﺒ ﱢﻴﻦ ﻫﺬﻳﻦ ﺍﻟﻨﻮﻋﻴﻦ ﻣﻦ ﺍﻟﻨﻤﺎﺫﺝ‪:‬‬ ‫ﺑﻄﺎﻗﺎﺕ ﺱ ﻭﺣﺪﻫﺎ ﻓﻲ ﺃﺣﺪ ﺍﻟﻄﺮﻓﻴﻦ‪.‬‬ ‫‪‬‬ ‫‪u = 3x2‬‬ ‫‪2u2 - 7u + 3 = 0‬‬ ‫‪ ‬‬ ‫‪(2u - 1)(u - 3) = 0‬‬ ‫‪‬‬ ‫‪u 3x2‬‬ ‫‪u=3‬‬ ‫ﺃﻭ‬ ‫‪u‬‬ ‫=‬ ‫‪_1‬‬ ‫‪ ‬‬ ‫ﺍﻟﻨﻤﻮﺫﺝ‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪_1‬‬ ‫‪ ‬‬ ‫‪3x2 = 3‬‬ ‫‪3x‬‬ ‫‪2‬‬ ‫=‬ ‫‪2‬‬ ‫‪-‬‬ ‫ﺍﻷﻛﻮﺍﺏ ﻭﻗﻄﻊ ﺍﻟﻌﺪ‬ ‫‪=   +‬‬ ‫‪x2 = 1‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫=‬ ‫‪_1‬‬ ‫‪6‬‬ ‫‪_√6‬‬ ‫‪x = ±1‬‬ ‫‪x‬‬ ‫=‬ ‫‪±‬‬ ‫‪6‬‬ ‫ﺑﻄﺎﻗﺎﺕ ﺍﻟﺠﺒﺮ ﺱ ‪١-‬‬ ‫‪.‬‬ ‫‪-‬‬ ‫‪_√6‬‬ ‫‪,‬‬ ‫‪_√6‬‬ ‫‪,1,‬‬ ‫‪-1‬‬ ‫ﻫﻲ‪:‬‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺣﻠﻮﻝ‬ ‫ﻭﺯﻉ ﺍﻟﺒﻄﺎﻗﺎﺕ ﺍﻟﻤﺘﺒﻘﻴﺔ ﻓﻲ ﻣﺠﻤﻮﻋﺘﻴﻦ‬ ‫‪  = x‬‬ ‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺃ ﱟﻱ ﻣﻦ ﻫﺬﻳﻦ ﺍﻟﻨﱠﻤﻮﺫ ﹶﺟﻴﻦ ﻟﺤ ﱢﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ‪.‬‬ ‫‪6‬‬ ‫‪6‬‬ ‫ﻣﺘﺴﺎﻭﻳﺘﻴﻦ‪.‬‬ ‫‪  x‬‬ ‫✓‬ ‫‪‬‬ ‫‪ = ‬‬ ‫‪8x4 + 10x2 - 12 = 0 (6B‬‬ ‫‪4x4 - 8x2 + 3 = 0 (6A‬‬ ‫ﻭﺑﻬﺬﺍ‪ ،‬ﺗﻜﻮﻥ ﻗﻴﻤﺔ ﺱ = ‪ ،٢‬ﻭﺑﻤﺎ ﺃﻥ‪ ، ٥ + ٢ = ١ + (٢)٣ :‬ﻓﺎﻟﺤﻞ ﺻﺤﻴﺢ‪.‬‬ ‫‪ ‬ﺍﺳ‪H‬ﺘﻌ‪C‬ﻤ‪TE‬ﻞ ﺍﻷﻛﻮﺍﺏ ﻭﻗﻄﻊ ﺍﻟﻌﺪ ﺃﻭ ﺍﻟ ﱠﺮﺳﻢ ﻟﹺ ﹶﺘ ﹸﺤ ﱠﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ :‬ﺱ ‪+‬‬ ‫‪146‬‬ ‫‪TECH‬‬ ‫‪TECH‬‬ ‫ﻭﻓﻲ ﺍﻟﻤﺮﺣﻠﺔ ﺍﻟﺜﺎﻧﻮﻳﺔ ﻳﺴﺘﻤﺮ ﺍﻟﻄﻠﺒﺔ ﺑﺎﺳﺘﻌﻤﺎﻝ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺑﻄﺎﻗﺎﺕ ﺍﻟﺠﺒﺮ ﻟﺤﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫‪+‬‬ ‫‪+x‬‬ ‫‪  ‬ﺱ ‪٢ = ٢ +‬ﺱ ‪٢  ١ +‬ﺱ ‪٣ = ٧ +‬ﺱ ‪٢  ٤ +‬ﺱ – ‪ = ٥‬ﺱ – ‪٧‬‬ ‫‪٢ ‬ﺱ ‪٤ = ٨ -‬ﺱ – ‪٢‬‬ ‫‪ + ٨ ‬ﺱ = ‪٣‬ﺱ ‪٤ ‬ﺱ = ﺱ – ‪٦‬‬ ‫ﻧﻤﻮﺫﺝ ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺍﻟﺒﻄﺎﻗﺎﺕ ﻻﺳﺘﻜﺸﺎﻑ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﻤﺘﻌﺪﺩﺓ‬ ‫‪ ‬‬ ‫)‪ec3kmark) (place checkmark) 3 (place checkmark‬ﺱ‪3 (plac٢e +ch‬‬ ‫ﺍﻟﺨﻄﻮﺍﺕ‪ ،‬ﻭﻳﻄﺒﻘﻮﻥ ﺍﻟﺨﻄﻮﺍﺕ ﺍﻟﺘﻲ ﻃﻮﺭﻭﻫﺎ ﻓﻲ‬ ‫‪ ‬ﺑ ﹼﻴﻦ ﺃ ﱡﻱ ﺧﺼﺎﺋﺺ ﺍﻟﺘﺴﺎﻭﻱ ﺗﺴﺘﻌﻤﻠﻬﺎ ﻟﻠﺘﺨﻠﺺ ﻣﻦ ﺍﻟﻌﺪﺩ ﻧﻔﺴﻪ ﻣﻦ ﺑﻄﺎﻗﺎﺕ ﺍﻟﺠﺒﺮ‬ ‫‪+‬‬ ‫ﺍﺣﺬﻑ ﺍﻟﻌﺪﺩ ﻧﻔﺴﻪ ﻣﻦ ﻗﻄﻣﻊ ﺍﻟﻦﻌﺪﻛ ﻣﻞﻦ ﻛﻃﱢﻞﺮﻃﺮﻑﻑﻋﻠﻰ ﻟﻮﺣﺔ ﺍﻟﻤﻌﺎﺩﻟﺔ‪.‬‬ ‫‪+‬‬ ‫ﺑﺤﻴﺚ ﻳﺼﺒﺢ ﺍﻟﻜﻮﺏ ﻭﺣﺪﻩ ﻓﻲ ﻃﺮﻑ‬ ‫ﻣﻌﻤﻞ ﺍﻟﺠﺒﺮ ﺇﻟﻰ ﺭﻣﻮﺯ ﺟﺒﺮﻳﺔ‪.‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﺱ ‪ ٢ - ٢ +‬‬ ‫ﻋﺪﺩ ﻗﻄﻊ ﺍﻟﻌﺪ ﺍﻟﻤﺘﺒ ﱢﻘﻴﺔ ﻓﻲ ﺍﻟﻄﺮﻑ ﺍﻷﻳﺴﺮ ﺗﻤ ﱢﺜﻞ‬ ‫‪+‬‬ ‫=‬ ‫ﻗﻴﻤﺔ ﺱ‬ ‫‪+‬‬ ‫‪+‬‬ ‫ﺱ= ‪٣‬‬ ‫ﺇﺫﻥ ﺱ = ‪ ،٣‬ﻭﺑﻤﺎ ﺃ ﱠﻥ ‪ ،٥ = ٢ + ٣‬ﻓﺎﻟﺤ ﱡﻞ ﺻﺤﻴﺢ‪.‬‬ ‫ﺍﺳﺘﻌﻤ ﹾﻞ ﺍﻷﻛﻮﺍﺏ ﻭﻗﻄﻊ ﺍﻟﻌﺪ ﺃﻭ ﺍﻟ ﱠﺮﺳﻢ ﻟﹺ ﹶﺘ ﹸﺤ ﱠﻞ ﻛ ﱠﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫ﺃ( ﺱ ‪ ٤ = ٤ +‬ﺏ( ‪ = ٥‬ﺱ ‪ ٤ +‬ﺟـ( ‪ + ١ = ٤‬ﺱ ﺩ( ‪ + ٢ = ٢‬ﺱ‬ ‫‪  ‬‬ ‫‪‬ﻳﻨﺘﻘﻞ ﻃﻼﺏ ‪ ‬ﺧﻼﻝ ﺍﻟﺘﻌﺎﻣﻞ ﻣﻊ ﺍﻟﺠﺒﺮ‪ ،‬ﻣﻦ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻷﻛﻮﺍﺏ ﻭﻗﻄﻊ ﺍﻟﻌﺪ‬ ‫ﺇﻟﻰ ﺍﺳﺘﻌﻤﺎﻝ ﻧﻤﺎﺫﺝ ﺟﺒﺮﻳﺔ ﺃﻛﺜﺮ ﺗﺠﺮﻳ ﹰﺪﺍ‪ .‬ﻭﻳﺤ ﹼﻞ ﺍﻟﻄﻼﺏ ﻓﻲ ﺍﻟﺪﺭﻭﺱ ﺍﻟﻼﺣﻘﺔ‪ ،‬ﻣﻌﺎﺩﻻﺕ ﺑﺴﻴﻄﺔ‬ ‫ﺗﺤﺘﻮﻱ ﻋﻠﻰ ﺭﻣﻮﺯ ﺟﺒﺮﻳﺔ‪.‬‬ ‫‪‬‬ ‫‪‬ﻳﻮ ﹼﺿﺢ ﺍﻟﺘﺴﻠﺴﻞ ﺍﻟﺘﻌﻠﻴﻤﻲ ﺍﻟﺬﻱ ﺗﻢ ﻭﺻﻔﻪ ﻗ ﹼﻮﺓ ﺍﻟﻤﻘﺎﺑﻠﺔ ﺑﻴﻦ ﺍﻟﻨﺘﻴﺠﺔ ﺍﻟﻤﺮﻏﻮﺏ ﻓﻴﻬﺎ ﻭﺍﻟﻨﺠﺎﺡ ﻓﻲ ﺍﻟﺠﺒﺮ‪.‬‬ ‫ﻭﺗﻌﻤﻞ ﻫﺬﻩ ﺍﻟﻌﻤﻠﻴﺔ ﺍﻟﺘﻄﻮﻳﺮﻳﺔ ﻋﻠﻰ ﺗﺠﻨﺐ ﻭﺟﻮﺩ ﻓﺠﻮﺍﺕ ﺃﻭ ﺗﺪﺍﺧﻼﺕ ﺑﻴﻦ ﻣﺴﺘﻮﻳﺎﺕ ﺍﻟﺼﻔﻮﻑ‪ ،‬ﻭﺗﺆ ﹼﻛﺪ‬ ‫ﻋﻠﻰ ﺃ ﹼﻥ ﻣﻔﺎﻫﻴﻢ ﻛﻞ ﺻﻒ ﻭﻣﻬﺎﺭﺍﺗﻪ ﻣﺒﻨﻴﺔ ﻋﻠﻰ ﺃﺳﺎﺱ ﻗﻮﻱ ﺗﻢ ﺗﻄﻮﻳﺮﻩ ﻓﻲ ﺻﻔﻮﻑ ﺳﺎﺑﻘﺔ‪ .‬ﻭﻳﺴﺘﻌﻤﻞ‬ ‫ﺍﻟﻤﻨﺤﻰ ﻧﻔﺴﻪ ﻋﺒﺮ ﺍﻟﻤﺴﺎﺭﺍﺕ ﺟﻤﻴﻌﻬﺎ ﺍﺑﺘﺪﺍ ﹰﺀ ﻣﻦ ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﺍﻻﺑﺘﺪﺍﺋﻲ ﺣﺘﻰ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻟﺚ ﺍﻟﺜﺎﻧﻮﻱ‪.‬‬ ‫‪T4 ‬‬

‫‪‬‬ ‫‪‬‬ ‫• ‪‬‬ ‫• ‪‬‬ ‫• ‪‬‬ ‫‪‬‬ ‫ﺗﺰ ﹼﻭﺩ ﺍﻟﺴﻠﺴﻠﺔ ﺍﻟﻄﻼﺏ ﺑﺨﻄﻂ ﻣﻼﺋﻤﺔ ﻟﺤﻞ ﺍﻟﻤﺴﺄﻟﺔ‪ ،‬ﻭﻣﻬﺎﺭﺍﺕ ﻭﺗﻄﺒﻴﻘﺎﺕ ﻋﻠﻴﻬﺎ ﺧﻼﻝ ﺍﻟﺼﻔﻮﻑ؛ ﺇﺫ ﻳﺘﻮﺍﻓﺮ ﻟﻬﻢ ﻓﺮﺹ ﻣﺴﺘﻤﺮﺓ ﻟﺘﻄﺒﻴﻖ ﻣﻬﺎﺭﺍﺕ‬ ‫ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪ ،‬ﻭﺣﻞ ﺍﻟﻤﺴﺎﺋﻞ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﺒﺼﺮﻱ‪ ،‬ﻭﺍﻻﺳﺘﺪﻻﻝ ﺍﻟﻤﻨﻄﻘﻲ‪ ،‬ﻭﺍﻟﺤﺲ ﺍﻟﻌﺪﺩﻱ‪ ،‬ﻭﺍﻟﺠﺒﺮ‪.‬‬ ‫‪ ‬‬ ‫ﺗﺬ ﹼﻛﺮ ﺃﻥ ﺟﻤﻴﻊ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺪﺍﺧﻠ ﹼﻴﺔ ﻟﻠﻤﻀﻠﻊ ﺍﻟﻤﻨﺘﻈﻢ ﻣﺘﻄﺎﺑﻘﺔ‪ .‬ﻭﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﻫﺬﻩ ﺍﻟﺤﻘﻴﻘﺔ ﻭﻧﻈﺮﻳﺔ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ‬ ‫‪‬‬ ‫ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺪﺍﺧﻠ ﹼﻴﺔ ﻟﻠﻤﻀﻠﻊ ﻹﻳﺠﺎﺩ ﻗﻴﺎﺱ ﺍﻟﺰﻭﺍﻳﺔ ﺍﻟﺪﺍﺧﻠﻴﺔ ﻷﻱ ﻣﻀﻠﻊ ﻣﻨﺘﻈﻢ‪.‬‬ ‫‪ ‬‬ ‫ﺗﺴﺎﻋﺪ ﺍﺳﺘﺮﺍﺗﻴﺠﻴﺎﺕ ﺣﻞ ﺍﻟﻤﺴﺄﻟﺔ ﺍﻟﻄﻼﺏ ﻋﻠﻰ‬ ‫‪ ‬‬ ‫ﺗﻌﻠﻢ ﻃﺮﺍﺋﻖ ﻣﺨﺘﻠﻔﺔ ﻟﻤﻮﺍﺟﻬﺔ ﺍﻟﻤﺴﺎﺋﻞ ﺍﻟﻜﻼﻣﻴﺔ‪.‬‬ ‫‪   2‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬ﻓﻲ ﺍﻟﻤﻨﻈﺮ ﺍﻟﻌﻠﻮﻱ ﻟﻠﻤﻈﻠﺔ ﺍﻟﻤﺠﺎﻭﺭﺓ‪ ،‬ﺗﺸ ﹼﻜﻞ ﺍﻷﻋﻤﺪﺓ ﺭﺅﻭﺱ ﻣﻀﻠﻊ‬ ‫ﺳﺪﺍﺳﻲ ﻣﻨﺘﻈﻢ‪ .‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻋﻨﺪ ﺃﻱ ﻣﻦ ﺃﺭﻛﺎﻥ ﺍﻟﻤﻈﻠﺔ‪.‬‬ ‫‪  ‬ﻣﻨﻈﺮ ﻋﻠﻮﻱ ﻟﻤﻈﻠﺔ ﺳﺪﺍﺳﻴﺔ ﻣﻨﺘﻈﻤﺔ ﺍﻟﺸﻜﻞ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬ﺇﻳﺠﺎﺩ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻲ ﺗﺸﻜﻞ ﻋﻨﺪ ﺃﻱ ﺭﻛﻦ ﻣﻦ‬ ‫‪‬‬ ‫ﺃﺭﻛﺎﻥ ﺍﻟﻤﻈﻠﺔ‪.‬‬ ‫ﺍﺭﺳﻢ ﺷﻜ ﹰﻼ ﻳﻤ ﱢﺜﻞ ﺍﻟﻤﻨﻈﺮ ﺍﻟﻌﻠﻮﻱ ﻟﻠﻤﻈﻠﺔ‪.‬‬ ‫‪2 ‬‬ ‫‪3 ‬‬ ‫‪1 ‬‬ ‫‪‬‬ ‫‪  (34‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺍﻟﺬﻱ ﻳﺒﻴﻦ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ‪‬‬ ‫‪3 ‬‬ ‫ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻲ ﺗﺘﺸ ﹼﻜﻞ ﻋﻨﺪ ﺃﻱ ﻣﻦ ﺃﺭﻛﺎﻥ ﺍﻟﻤﻈﻠﺔ ﻫﻲ ﺯﺍﻭﻳﺔ ﺩﺍﺧﻠ ﹼﻴﺔ ﻟﺴﺪﺍﺳﻲ ﻣﻨﺘﻈﻢ‪.‬‬ ‫‪190‬‬ ‫‪210‬‬ ‫ﻓﻲ ﻣﺪﺭﺳﺔ ﻣﺪﺓ ﺃﺭﺑﻊ ﺳﻨﻮﺍﺕ ﻣﺘﺘﺎﻟﻴﺔ‪1425 .‬‬ ‫‪ ‬ﺍﺳﺘﻌﻤﻞ ﻧﻈﺮﻳﺔ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺪﺍﺧﻠﻴﺔ ﻟﻠﻤﻀﻠﻊ ﻹﻳﺠﺎﺩ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ‬ ‫‪240‬‬ ‫ﺍﻟﺪﺍﺧﻠ ﹼﻴﺔ ﻟﻠﺴﺪﺍﺳﻲ‪ .‬ﻭﺑﻤﺎ ﺃﻥ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺪﺍﺧﻠﻴﺔ ﻟﻠﺴﺪﺍﺳﻲ ﺍﻟﻤﻨﺘﻈﻢ ﻣﺘﻄﺎﺑﻘﺔ‪ ،‬ﻓﺈﻥ ﻗﻴﺎﺱ ﻛﻞ ﺯﺍﻭﻳﺔ‬ ‫‪260‬‬ ‫ﺩﺍﺧﻠﻴﺔ ﻳﺴﺎﻭﻱ ﻧﺎﺗﺞ ﻗﺴﻤﺔ ﺍﻟﻤﺠﻤﻮﻉ ﻋﻠﻰ ﻋﺪﺩ ﺍﻟﺰﻭﺍﻳﺎ‪.‬‬ ‫‪1426‬‬ ‫‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪ ‬ﺃﻭ ﹰﻻ‪ :‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺪﺍﺧﻠ ﹼﻴﺔ‪.‬‬ ‫‪1427‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻣﻌﺘﻤ ﹰﺪﺍ ﻋﻠﻰ ﺑﻴﺎﻧﺎﺕ ﺍﻟﺠﺪﻭﻝ‪ ،‬ﻭﺍﺷﺮﺡ ﻛﻴﻒ ﻳﺆ ﱢﻳﺪ ﺗﻤﺜﻴﻠﻚ ‪1428‬‬ ‫‪‬‬ ‫˚‪S = (n - 2) 180‬‬ ‫ﺍﻟﺒﻴﺎﻧﻲ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪n=6‬‬ ‫˚‪= (6 - 2) 180‬‬ ‫ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺻﺤﻴ ﹰﺤﺎ ﺃﻭ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪4 ‬‬ ‫‪‬‬ ‫˚‪= 4 180˚ = 720‬‬ ‫‪ (35‬ﺇﺫﺍ ﻛﺎﻥ ‪ n‬ﻋﺪ ﹰﺩﺍ ﺃﻭﻟ ﹰﹼﻴﺎ‪ ،‬ﻓﺈﻥ ‪ n + 1‬ﻟﻴﺲ ﺃﻭﻟ ﹰﹼﻴﺎ‪.‬‬ ‫ﺛﺎﻧ ﹰﻴﺎ‪ :‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﻛﻞ ﺯﺍﻭﻳﺔ ﺩﺍﺧﻠ ﹼﻴﺔ‪.‬‬ ‫‪ (36‬ﺇﺫﺍ ﻛﺎﻥ ‪ x‬ﻋﺪ ﹰﺩﺍ ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﻓﺈﻥ ‪ –x‬ﻋﺪﺩ ﻣﻮﺟﺐ‪.‬‬ ‫‪ (37‬ﻓﻲ ﺍﻟﻤﺜﻠﺚ ‪ ABC‬ﺇﺫﺍ ﻛﺎﻥ‪ ، (AB)2 + (BC)2 = (AC)2 :‬ﻓﺈﻥ ‪ ABC‬ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻣﺠﻤﻮﻉ ﻗﻴ_ﺎﺳﺎﺕ ﺍﻟﺰﻭﺍ_ﻳﺎ ﺍﻟﺪﺍﺧﻠﻴﺔ_‬ ‫=‬ ‫‪7_20°‬‬ ‫‪ (38‬ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﻣﺴﺘﻄﻴﻞ ﺗﺴﺎﻭﻱ ‪ ،20 m2‬ﻓﺈﻥ ﻃﻮﻟﻪ ﻳﺴﺎﻭﻱ ‪ ، 10 m‬ﻭﻋﺮﺿﻪ ‪.2 m‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻋﺪﺩ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺪﺍﺧﻠﻴﺔ‬ ‫‪‬‬ ‫‪6‬‬ ‫‪  (39‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺩﻧﺎﻩ ﻟﺘﻌﻄﻲ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫˚‪= 120‬‬ ‫‪ ‬‬‫‪ ‬‬ ‫‪ ‬‬ ‫ﺇﺫﻥ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺘﻜ ﱢﻮﻧﺔ ﻋﻨﺪ ﻛﻞ ﺭﻛﻦ ﻳﺴﺎﻭﻱ ‪.120°‬‬ ‫‪‬‬ ‫‪120° 120°‬‬ ‫‪ ‬ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﺃ ﹼﻥ ﻫﺬﺍ ﺍﻟﻘﻴﺎﺱ ﺻﺤﻴﺢ‪ ،‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﻤﺴﻄﺮﺓ ﻭﺍﻟﻤﻨﻘﻠﺔ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪120°‬‬ ‫ﻟﺮﺳﻢ ﺳﺪﺍﺳﻲ ﻣﻨﺘﻈﻢ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺘﻪ ﺍﻟﺪﺍﺧﻠ ﹼﻴﺔ ‪.120°‬‬ ‫‪120°‬‬ ‫ﺳﻴﺮﺗﺒﻂ ﺍﻟﻀﻠﻊ ﺍﻷﺧﻴﺮ ﺑﻨﻘﻄﺔ ﺍﻟﺒﺪﺍﻳﺔ ﻷﻭﻝ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ ﹸﺭﺳﻤﺖ‪.‬‬ ‫‪25.0%‬‬ ‫‪6.8 ‬‬ ‫‪25.5%‬‬ ‫‪6.9 ‬‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫‪ ‬‬ ‫‪6.6% 1.8 ‬‬ ‫‪12‬‬ ‫‪  (2A‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺪﺍﺧﻠﻴﺔ ﻟﺴﺠﺎﺩﺓ ﻋﻠﻰ ﺷﻜﻞ ﺛﻤﺎﻧﻲ ﻣﻨﺘﻈﻢ‪.‬‬ ‫‪15.1%‬‬ ‫‪4.1 ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪1431 ‬‬ ‫‪  (2B‬ﺗﺰ ﱢﻳﻦ ﺍﻟﻨﻮﺍﻓﻴﺮ ﺍﻷﻣﺎﻛﻦ ﺍﻟﻌﺎﻣﺔ‪ ،‬ﻭﻳﻘﺎﻡ ﺑﻌﻀﻬﺎ ﻋﻠﻰ ﺷﻜﻞ ﻣﻀﻠﻌﺎﺕ ﻣﻨﺘﻈﻤﺔ‪ .‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ‬ ‫‪‬‬ ‫ﺍﻟﺪﺍﺧﻠﻴﺔ ﻟﻨﺎﻓﻮﺭﺓ ﻋﻠﻰ ﺷﻜﻞ ﺗﺴﺎﻋﻲ ﻣﻨﺘﻈﻢ‪.‬‬ ‫‪ (a‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻤﺠﻤﻮﻉ ﻋﺪﺩ ﺳﻜﺎﻥ ﺍﻟﻤﻨﺎﻃﻖ ﺍﻹﺩﺍﺭﻳﺔ ﺍﻷﺭﺑﻊ ﺍﻟﻮﺍﺭﺩﺓ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺃﻗﻞ ﻣﻦ ‪ 25%‬ﻣﻦ ﺳﻜﺎﻥ‬ ‫‪‬‬ ‫ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪.‬‬ ‫‪‬‬ ‫‪ 5  14‬‬ ‫‪ (b‬ﻳﺰﻳﺪ ﻋﺪﺩ ﺳﻜﺎﻥ ﺃ ﱟﻱ ﻣﻦ ﺍﻟﻤﻨﺎﻃﻖ ﺍﻹﺩﺍﺭﻳﺔ ﺍﻷﺭﺑﻊ ﻋﻠﻰ ﻣﻠﻴﻮ ﹶﻧﻲ ﻧﺴﻤ ﹴﺔ‪.‬‬ ‫‪‬‬ ‫‪  (40‬ﻳﻨﺺ ﺗﺨﻤﻴﻦ ﺟﻮﻟﺪ ﺑﺎﺥ ﻋﻠﻰ ﺃﻧﻪ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ﺃﻱ ﻋﺪﺩ ﺯﻭﺟﻲ ﺃﻛﺒﺮ ﻣﻦ ‪ 2‬ﻋﻠﻰ ﺻﻮﺭﺓ‬ ‫‪‬‬ ‫ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﺃﻭﻟﻴﻴﻦ‪ .‬ﻓﻌﻠﻰ ﺳﺒﻴﻞ ﺍﻟﻤﺜﺎﻝ‪.4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5 :‬‬ ‫‪ (a‬ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ ﻟﻸﻋﺪﺍﺩ ﺍﻟﺰﻭﺟﻴﺔ ﻣﻦ ‪ 10‬ﺇﻟﻰ ‪20‬‬ ‫‪ (b‬ﺇﺫﺍ ﹸﺃﻋﻄﻴﺖ ﺍﻟﺘﺨﻤﻴﻦ ﺍﻵﺗﻲ‪ :‬ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ﺃﻱ ﻋﺪﺩ ﻓﺮﺩﻱ ﺃﻛﺒﺮ ﻣﻦ ‪ 2‬ﻋﻠﻰ ﺻﻮﺭﺓ ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﺃﻭﻟﻴﻴﻦ‪.‬‬ ‫ﻓﻬﻞ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ ﺃﻡ ﺧﺎﻃﺊ؟ ﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪AB‬‬ ‫ﺇ‪‬ﺍﺫﺍﻟﻨ ﹸﺃﻘﻄﺿﺘﻴﺎﻔﻥ ﺍﺖﻟﻧﻮﻘﺍﻗﻄﻌﺔﺘﺎﺃﻥﺧﺮﻋﻠ￯ﻰ‪C‬ﻣﻋﺴﺘﻠﻘﻴﻰ ﺍﻢﻟﺗﻘﺸﻄ ﱢﻌﻜﺔﻼﺍﻟﻥﻤﻗﺴﻄﺘﻘﻌﻴﺔﻤﻣﺔﺴ_ﺘ‪_B‬ﻘﻴ‪_A‬ﻤ‪،‬ﺔ‪،‬‬ ‫‪___‬‬ ‫‪(41‬‬ ‫‪A CB‬‬ ‫ﻣﺜﻞ ‪. AB‬‬ ‫ﻓﺈﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ ﺗﺸ ﱢﻜﻞ ﺛﻼﺙ ﻗﻄﻊ ﻣﺴﺘﻘﻴﻤﺔ‪.‬‬ ‫‪‬‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺨﺘﻠﻔﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻣﻦ ﺃﺭﺑﻊ ﻧﻘﺎﻁ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ؟ ﻭﻣﻦ ﺧﻤﺲ ﻧﻘﺎﻁ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ؟‬ ‫ﺗﺘﻄﻠﺐ ﻫﺬﻩ ﺍﻟﻤﺴﺎﺋﻞ ﺍﺳﺘﻌﻤﺎﻝ ﻣﻬﺎﺭﺍﺕ ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻌﻠﻴﺎ‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻌﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺨﺘﻠﻔﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻣﻦ ‪ n‬ﻧﻘﻄﺔ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ‪.‬‬ ‫‪ (c‬ﺍﺧﺘﺒﺮ ﺗﺨﻤﻴﻨﻚ ﺑﺈﻳﺠﺎﺩ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺨﺘﻠﻔﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻣﻦ ‪ 6‬ﻧﻘﺎﻁ‪.‬‬ ‫)ﺍﻟﺘﺤﻠﻴﻞ‪ ،‬ﻭﺍﻟﺘﺮﻛﻴﺐ‪ ، ... ،‬ﺇﻟﺦ(‪.‬‬ ‫‪‬‬ ‫‪   (42‬ﻳﺘﻨﺎﻗﺶ ﺃﺣﻤﺪ ﻭﻋﻠﻲ ﻓﻲ ﻣﻮﺿﻮﻉ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ‪ .‬ﻓﻴﻘﻮﻝ ﺃﺣﻤﺪ‪ :‬ﺇﻥ ﺟﻤﻴﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ‬ ‫ﺃﻋﺪﺍﺩ ﻓﺮﺩﻳﺔ‪ .‬ﻓﻲ ﺣﻴﻦ ﻳﻘﻮﻝ ﻋﻠ ﱞﻲ‪ :‬ﻟﻴﺴﺖ ﺟﻤﻴﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ ﻓﺮﺩﻳﺔ‪ .‬ﻫﻞ ﻗﻮﻝ ﺃ ﱟﻱ ﻣﻨﻬﻤﺎ ﺻﺤﻴﺢ؟ ﻓ ﹼﺴﺮ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫‪17  1- 1‬‬ ‫‪ T5‬‬

‫‪‬‬ ‫ﺗﺴﺎﻋﺪ ﻣﺴﺎﺋﻞ ﺍﻟﺘﻤﺜﻴﻼﺕ ﺍﻟﻤﺘﻌﺪﺩﺓ ﺍﻟﻄﻼﺏ ﻋﻠﻰ ﺗﺼﻮﺭ ﺍﻟﻤﻔﺎﻫﻴﻢ ﻭﺗﻌﻤﻴﻖ ﺍﻟﻔﻬﻢ‪،‬‬ ‫ﻭﺗﺘﻀﻤﻦ‪ :‬ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻠﻔﻈﻴﺔ ﻭﺍﻟﻌﺪﺩﻳﺔ ﻭﺍﻟﺠﺒﺮﻳﺔ ﻭﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﻭﺍﻟﺠﺪﺍﻭﻝ ‪ ...‬ﺇﻟﺦ‪.‬‬ ‫‪ ‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺓ ﺃﺩﻧﺎﻩ ﻟﻜﺘﺎﺑﺔ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﻣﻌﻠﻮﻣﺎﺕ ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﺤﻴﺎﺓ‬ ‫ﻟﺘﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻨﻬﺎ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺃ ﱞﻱ ﻣﻨﻬﺎ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫”ﺍﻟﺤﻴﻮﺍﻥ ﺍﻟﺬﻱ ﺗﻈﻬﺮ ﻋﻠﻰ ﺟﺴﻤﻪ ﺧﻄﻮﻁ ﻫﻮ ﺍﻟﺤﻤﺎﺭ ﺍﻟﻮﺣﺸﻲ“‪.‬‬ ‫‪ (38‬ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪ (37‬ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ‬ ‫‪ (40‬ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪ (39‬ﻣﻌﻜﻮﺱ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺗﻴﻦ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﻗ ﱢﺮﺭ ﻫﻞ ﻫﻤﺎ ﻣﺘﻜﺎﻓﺌﺎﻥ ﻣﻨﻄﻘ ﹼﹰﻴﺎ ﺃﻡ ﻻ؟‬ ‫‪4‬‬ ‫‪∼(p → q) , ∼ p → ∼ q (41‬‬ ‫‪5‬‬ ‫‪∼(p → q) , ∼(∼ q → ∼ p) (42‬‬ ‫‪(p q) ∨ r , p (q ∨ r) (43‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺣ ﱢﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻨﻬﺎ‬ ‫ﺻﺎﺋ ﹰﺒﺎ ﺃﻡ ﺧﺎﻃ ﹰﺌﺎ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪ (44‬ﺇﺫﺍ ﻛﻨﺖ ﺗﻌﻴﺶ ﻓﻲ ﺍﻟﺪﻣﺎﻡ‪ ،‬ﻓﺈﻧﻚ ﺗﻌﻴﺶ ﻓﻲ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪.‬‬ ‫‪ (45‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻄﺎﺋﺮ ﻧﻌﺎﻣﺔ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﻄﻴﺮ‪.‬‬ ‫‪ (46‬ﺟﻤﻴﻊ ﺍﻟﻤﺮﺑﻌﺎﺕ ﻣﺴﺘﻄﻴﻼﺕ‪.‬‬ ‫‪ (47‬ﺟﻤﻴﻊ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺘﻄﺎﺑﻘﺔ ﻟﻬﺎ ﺍﻟﻄﻮﻝ ﻧﻔﺴﻪ‪.‬‬ ‫‪ (48‬ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻳﺤﻮﻱ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ‪90°‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺃﺷﻜﺎﻝ ﭬﻦ ﺃﺩﻧﺎﻩ؛ ﻟﺘﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ .‬ﻭﻓ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪.‬‬ ‫ﻣﺘﺴﺎﻗﻄﺔ‬ ‫ﺍﻟﺜﺪﻳﻴﺎﺕ‬ ‫ﺍﻟﺪﻭﺍﻝ‬ ‫‪‬‬ ‫ﺍﻷﻭﺭﺍﻕ‬ ‫ﺍﳊﻴﻮﺍﻧﺎﺕ ﺍﻟﺒﺤﺮﻳﺔ‬ ‫ﻏﻴﺮ ﺍﳋﻄﻴﺔ‬ ‫‪‬‬ ‫ﺩﺍﺋﻤﺔ ﺍﳋﻀﺮﺓ‬ ‫‪‬‬ ‫ﺍﻟﺪﻭﺍﻝ‬ ‫‪ ‬‬ ‫ﺍﻟﺘﺮﺑﻴﻌﻴﺔ‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ (49‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺪﺍﻟﺔ ﻏﻴﺮ ﺧﻄﻴﺔ‪ ،‬ﻓﺈﻧﻬﺎ ﺗﻜﻮﻥ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪ (50‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻜﻮﻥ ﺣﻴﻮﺍ ﹰﻧﺎ ﺑﺤﺮ ﹼﹰﻳﺎ‪.‬‬ ‫‪Angles and Parallel Lines‬‬ ‫‪ (51‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺸﺠﺮﺓ ﻣﺘﺴﺎﻗﻄﺔ ﺍﻷﻭﺭﺍﻕ‪ ،‬ﻓﺈﻧﻬﺎ ﻻ ﺗﻜﻮﻥ ﺩﺍﺋﻤﺔ ﺍﻟﺨﻀﺮﺓ‪.‬‬ ‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺒﻴﺎﻧﻴﺔ ‪TI - nspire‬؛ ﻟﺘﺴﺘﻜﺸﻒ ﻗﻴﺎﺳﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻦ ﻣﺴﺘﻘﻴﻤﻴﻦ ﻣﺘﻮﺍﺯﻳﻴﻦ ﻭﻗﺎﻃﻊ ﻟﻬﻤﺎ‪.‬‬ ‫‪  (52‬ﻓﻲ ﻫﺬﻩ ﺍﻟﻤﺴﺄﻟﺔ ﺳﻮﻑ ﺗﺴﺘﻘﺼﻲ ﺃﺣﺪ ﻗﻮﺍﻧﻴﻦ ﺍﻟﻤﻨﻄﻖ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫‪ ‬‬ ‫‪   (a‬ﺍﻛﺘﺐ ﺛﻼﺙ ﻋﺒﺎﺭﺍﺕ ﺷﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ‪ ،‬ﺑﺤﻴﺚ ﺗﻜﻮﻥ ﻧﺘﻴﺠﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻓﺮ ﹰﺿﺎ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺘﻲ ﺗﻠﻴﻬﺎ‪.‬‬ ‫‪ 3‬‬ ‫‪ 1‬‬ ‫‪   (b‬ﺍﺭﺳﻢ ﺷﻜﻞ ﭬﻦ ﻳﻮﺿﺢ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫• ﺍﺭﺳﻢ ﺍﻟﻨﻘﻄﺔ ‪ A‬ﻋﻠﻰ ‪ ، FG‬ﻭﺍﻟﻨﻘﻄﺔ ‪ B‬ﻋﻠﻰ ‪ ، JK‬ﻭﺫﻟﻚ ﺑﺎﻟﻀﻐﻂ‬ ‫‪   (c‬ﺍﻛﺘﺐ ﻋﺒﺎﺭ ﹰﺓ ﺷﺮﻃﻴ ﹰﺔ ﻣﺴﺘﻌﻤ ﹰﻼ ﻓﺮﺽ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﻟﻰ‪ ،‬ﻭﻧﺘﻴﺠﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺜﺎﻟﺜﺔ‪ .‬ﺇﺫﺍ ﻛﺎﻥ ﻓﺮﺽ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﻟﻰ ﺻﺎﺋ ﹰﺒﺎ‪ .‬ﻓﻬﻞ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻨﺎﺗﺠﺔ ﺻﺎﺋﺒ ﹰﺔ؟‬ ‫‪   (d‬ﺇﺫﺍ ﹸﺃﻋﻄﻴﺖ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﺸﺮﻃﻴﺘﻴﻦ ﺍﻟﺼﺎﺋﺒﺘﻴﻦ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، a‬ﻓﺈﻥ ‪ ، b‬ﻭﺇﺫﺍ ﻛﺎﻥ ‪ ،b‬ﻓﺈﻥ ‪ ،c‬ﻓﺎﻛﺘﺐ‬ ‫ﺗﺨﻤﻴﻨﹰﺎ ﺣﻮﻝ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ‪ c‬ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ‪ a‬ﺻﺎﺋﺒﺔ‪ .‬ﻓ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪.‬‬ ‫• ﺍﺭﺳﻢ ﻣﺴﺘﻘﻴ ﹰﻤﺎ ﻭﺳ ﱢﻢ ﺍﻟﻨﻘﻄﺘﻴﻦ ‪ F, G‬ﻋﻠﻴﻪ‪،‬‬ ‫‪33  1-3‬‬ ‫‪ ،‬ﺛﻢ ﺣ ﱢﺪﺩ ﻛ ﹰﹼﻼ ﻣﻦ‬ ‫ﻋﻠﻰ ﻭﺍﺧﺘﺮ‬ ‫ﺛﻢ ﺍﺧﺘﺮ‬ ‫ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ﺍﻟﻤﻔﺎﺗﻴﺢ‬ ‫ﺛﻢ ﺍﺧﺘﻴﺎﺭ‬ ‫ﺍﻟﻨﻘﻄﺘﻴﻦ ﻭﺗﺴﻤﻴﺘﻬﻤﺎ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ‬ ‫ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ ﺛﻢ‬ ‫‪ ،‬ﻭﺳ ﱢﻢ ﻛ ﹼﹰﻼ ﻣﻨﻬﻤﺎ‪.‬‬ ‫ﺍﺭﺳﻤﻪ‪ ،‬ﺛﻢ ﺍﺧﺘﺮ ﻧﻘﻄﺔ ﻋﻠﻴﻪ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ﻭﻣﻨﻬﺎ ﺍﺧﺘﺮ‬ ‫•‪ ‬ﹺﺻ ﹾﻞ ﺑﻴﻦ ﺍﻟﻨﻘﻄﺘﻴﻦ ‪ A, B‬ﻟﺮﺳﻢ ﺍﻟﻘﺎﻃﻊ ‪ ،AB‬ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ‬ ‫‪.‬‬ ‫‪ ،‬ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ‬ ‫ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ‬ ‫• ﺳ ﱢﻢ ﻛﻞ ﻣﻦ ﺍﻟﻨﻘﻄﺘﻴﻦ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ﺍﻟﻨﻘﻄﺔ‪ ،‬ﺛﻢ ﻋﻠﻰ‬ ‫ﺛﻢ ﺍﺿﻐﻂ ﻋﻠﻰ ﺍﻟﻨﻘﻄﺘﻴﻦ ‪A, B‬‬ ‫ﻭﺗﺴﻤﻴﺔ ﺍﻟﻨﻘﻄﺘﻴﻦ ﺑﺎﻟﺤﺮﻓﻴﻦ ‪FG‬‬ ‫ﻭﺍﺧﺘﻴﺎﺭ‬ ‫‪‬‬ ‫‪  4‬‬ ‫‪  2‬‬ ‫• ﺍﺭﺳﻢ ﻧﻘﻄﺘﻴﻦ ﻋﻠﻰ ‪ AB‬ﻭﺳ ﱢﻤﻬﻤﺎ ‪ C, D‬ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ‪،‬‬ ‫•‪ ‬ﺣ ﱢﺪﺩ ﻧﻘﻄ ﹰﺔ ﻻ ﺗﻘﻊ ﻋﻠﻰ ‪ FG‬ﻭﺳ ﱢﻤﻬﺎ ‪ J‬ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ‪ ،‬ﺛﻢ‬ ‫ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ ‪،‬‬ ‫ﺛﻢ ﺍﺿﻐﻂ ﻋﻠﻰ ﺍﻟﻤﺴﺘﻘﻴﻢ ‪AB‬‬ ‫ﻭﺍﺧﺘﺮ‬ ‫ﻭﺣ ﱢﺪﺩ ﻣﻜﺎﻥ ﺍﻟﻨﻘﻄﺘﻴﻦ ﻛﻤﺎ ﻓﻲ ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ‪.‬‬ ‫ﻭﺣﺪﺩ ﺍﻟﻨﻘﻄﺔ ﻭﺳ ﱢﻤﻬﺎ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ﺍﻟﻨﻘﻄﺔ ﺛﻢ ﻋﻠﻰ‬ ‫‪ ،‬ﺛﻢ ﺍﺧﺘﺮ‬ ‫• ﺳ ﱢﻢ ﻛ ﹰﹼﻼ ﻣﻨﻬﺎ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ‬ ‫ﻭﺗﺴﻤﻴﺔ ﺍﻟﻨﻘﻄﺔ ﺑﺎﻟﺤﺮﻑ‪J‬‬ ‫ﻭﺍﺧﺘﻴﺎﺭ‬ ‫ﻭﺳ ﹼﻤﻬﻤﺎ ﺑﹺـ ‪C, D‬‬ ‫ﻭﺍﺧﺘﻴﺎﺭ‬ ‫• ﺍﺭﺳﻢ ﻣﺴﺘﻘﻴﻤﺎ ﻳﻤ ﱡﺮ ﻓﻲ ‪ J‬ﻭﻳﻮﺍﺯﻱ ‪ FG‬ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ‬ ‫• ﻟﻘﻴﺎﺱ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺜﻤﺎﻧﻲ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻦ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﺜﻼﺛﺔ‪،‬‬ ‫ﺛﻢ ﺍﻟﻀﻐﻂ‬ ‫‪ ،‬ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ‬ ‫‪TI-nspire‬‬ ‫‪ ،‬ﺛﻢ ﺍﺧﺘﺮ ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﺍﺿﻐﻂ ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ‬ ‫ﻋﻠﻰﺍﻟﻨﻘﻄﺔ‪ J‬ﻭﺍﻟﻤﺴﺘﻘﻴﻢ‪،FG‬ﻓﻴﻨﺘﺞﻣﺴﺘﻘﻴﻢﻣﻮﺍ ﹴﺯ‪.‬‬ ‫ﺗﻮﻓﺮ ﻫﺬﻩ ﺍﻟﻤﻌﺎﻣﻞ ﻟﻠﻄﻼﺏ ﻓﺮﺻﺔ ﻟﻔﻬﻢ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻣﻦ‬ ‫ﻭﺍﺿﻐﻂ ﻋﻠﻰ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ ‪ J‬ﺛﻢ ‪ B‬ﺛﻢ ‪ ، D‬ﺳﻴﻈﻬﺮ ‪m∠JBD‬‬ ‫• ﺍﺧﺘﺮ ﻧﻘﻄ ﹰﺔ ﻋﻠﻴﻪ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ‪ ،‬ﻭﻣﻨﻬﺎ ﺍﺧﺘﺮ‪‬‬ ‫ﺧﻼﻝ ﺍﻟﺘﻤﺜﻴﻼﺕ ﺍﻟﺒﻴﺎﻧﻴﺔ‬ ‫ﻭﻟﻴﻜﻦ ‪78°‬‬ ‫ﺛﻢ ﺍﺿﻐﻂ ﻋﻠﻰ ﺍﻟﻤﺴﺘﻘﻴﻢ ﻭﺣ ﱢﺪﺩ ﺍﻟﻨﻘﻄﺔ‬ ‫• ﻛ ﱢﺮﺭ ﺫﻟﻚ ﻣﻊ ﺑﺎﻗﻲ ﺍﻟﺰﻭﺍﻳﺎ ﻹﻳﺠﺎﺩ ﻗﻴﺎﺳﺎﺗﻬﺎ‪.‬‬ ‫ﻭﺳ ﱢﻤﻬﺎ ﺑﺎﻟﻀﻐﻂ ﻋﻠﻰ ﺍﻟﻤﻔﺎﺗﻴﺢ ﻭﺍﺧﺘﺮ ﻣﻨﻬﺎ ‪‬‬ ‫ﻭﺳ ﱢﻤﻬﺎ ‪K‬‬ ‫‪ 2 92‬‬ ‫‪T6 ‬‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﺗﻘﻮﻳ ﹰﻤﺎ ﺻﺮﻳ ﹰﺤﺎ ﺫﺍ ﻣﻌﻨﻰ ﻟﻤﺪ￯ ﺗﻘﺪﻡ ﺍﻟﻄﻼﺏ ﻓﻲ ﺑﻨﻴﺔ ﺍﻟﻤﻨﻬﺞ ﻭﻓﻲ ﺍﻟﻤﻮﺍﺩ‬ ‫ﺍﻟﻤﺴﺎﻧﺪﺓ ﺍﻟﺘﻲ ﻳﺴﺘﻌﻴﻦ ﺑﻬﺎ ﺍﻟﻤﻌﻠﻢ‪.‬‬ ‫‪1 ‬‬ ‫‪3‬‬ ‫‪‬‬ ‫‪2 ‬‬ ‫‪1 ‬‬ ‫‪1‬‬ ‫‪  ‬‬ ‫‪1 ‬‬ ‫‪  ‬ﻗ ﱢﻮﻡ ﻣﻌﺮﻓﺔ ﻃﻼﺑﻚ ﻓﻲ ﺑﺪﺍﻳﺔ ﺍﻟﻌﺎﻡ ﺍﻟﺪﺭﺍﺳﻲ‬ ‫‪  ‬‬ ‫ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﺧﺘﺒﺎﺭﺍﺕ ﺗﺸﺨﻴﺼﻴﺔ ﻭﺍﺧﺘﺒﺎﺭﺍﺕ ﺗﺤﺪﻳﺪ ﺍﻟﻤﺴﺘﻮ￯‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻭﺳﻮﻑ ﻳﺴﺎﻋﺪﻙ ﻫﺬﺍ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﻣﺪ￯ ﺣﺎﺟﺔ ﻃﻼﺑﻚ ﻟﻤﻮﺍﺩ‬ ‫ﻭﻣﺼﺎﺩﺭ ﺗﻌﻠﻢ ﺇﺿﺎﻓﻴﺔ ﻟﻴﻜﻮﻧﻮﺍ ﻗﺎﺩﺭﻳﻦ ﻋﻠﻰ ﺍﻟﻤﻮﺍﺀﻣﺔ ﻣﻊ ﻣﻌﺎﻳﻴﺮ‬ ‫‪1 ‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻨﺪ ﻗﻴﻤﺔ ‪ x‬ﺍﻟ ﹸﻤﻌﻄﺎﺓ‪.‬‬ ‫ﻣﺴﺘﻮ￯ ﺍﻟﺼﻒ‪.‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ x2 – 2x + 11‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪. x = 6‬‬ ‫‪180 (x – 2) , x = 8 (2‬‬ ‫‪4x + 7 , x = 6 (1‬‬ ‫‪ ‬ﻗ ﱢﻮﻡ ﺍﻟﻤﻌﺎﺭﻑ ﺍﻟﺴﺎﺑﻘﺔ‬ ‫‪‬‬ ‫‪x2 – 2x + 11‬‬ ‫)‪_x(x - 3‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫=‬ ‫‪5‬‬ ‫‪(4‬‬ ‫‪5x2 – 3x , x = 2 (3‬‬ ‫‪x=6‬‬ ‫‪= (6)2 – 2(6) + 11‬‬ ‫ﻟﻄﻼﺑﻚ ﻓﻲ ﺑﺪﺍﻳﺔ ﺍﻟﻔﺼﻞ ﺃﻭ ﺍﻟﺪﺭﺱ‪ ،‬ﻣﻦ ﺧﻼﻝ‪:‬‬ ‫‪= 36 – 2(6) + 11‬‬ ‫‪2‬‬ ‫‪‬‬ ‫‪= 36 – 12 + 11‬‬ ‫ﻛﺘﺎﺏ ﺍﻟﻄﺎﻟﺐ‪:‬‬ ‫‪‬‬ ‫‪= 35‬‬ ‫‪x + (x + 1) + (x + 2) , x = 3 (5‬‬ ‫• ﺍﻟﺘﻬﻴﺌﺔ‬ ‫‪‬‬ ‫ﺍﻛﺘﺐ ﻛﻞ ﺗﻌﺒﻴﺮ ﻟﻔﻈﻲ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻠﻰ ﺻﻮﺭﺓ ﻋﺒﺎﺭﺓ ﺟﺒﺮﻳﺔ‪:‬‬ ‫• ﻓﻴﻤﺎ ﺳﺒﻖ‪ ،‬ﺍﻵﻥ‪ ،‬ﻟﻤﺎﺫﺍ‪.‬‬ ‫‪ (6‬ﺃﻗﻞ ﻣﻦ ﺧﻤﺴﺔ ﺃﻣﺜﺎﻝ ﻋﺪﺩ ﺑﺜﻤﺎﻧﻴﺔ‪.‬‬ ‫ﺩﻟﻴﻞ ﺍﻟﻤﻌﻠﻢ‪:‬‬ ‫• ﺑﺪﺍﺋﻞ ﺗﻨﻮﻳﻊ ﺍﻟﺘﻌﻠﻴﻢ‬ ‫‪ (7‬ﺃﻛﺜﺮ ﻣﻦ ﻣﺮﺑﻊ ﻋﺪﺩ ﺑﺜﻼﺛﺔ‪.‬‬ ‫ﺩﻟﻴﻞ ﺍﻟﺘﻘﻮﻳﻢ‬ ‫‪2 ‬‬ ‫ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ :‬‬ ‫• ﻧﻤﻮﺫﺝ ﺍﻟﺘﻮﻗﻊ‬ ‫‪8x – 10 = 6x (8‬‬ ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪. 36x – 14 = 16x + 58‬‬ ‫‪ T7‬‬ ‫‪‬‬ ‫‪36x – 14 = 16x + 58‬‬ ‫‪18 + 7x = 10x + 39 (9‬‬ ‫‪16x 36x – 14 – 16x = 16x + 58 –16x‬‬ ‫‪3(11x – 7) = 13x + 25 (10‬‬ ‫‪ 20x – 14 = 58‬‬ ‫‪_3‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫=‬ ‫‪5‬‬ ‫–‬ ‫‪2x‬‬ ‫‪(11‬‬ ‫‪14 20x – 14 + 14 = 58 + 14‬‬ ‫‪2‬‬ ‫‪  (12‬ﺍﺷﺘﺮﺕ ﻋﺎﺋﺸﺔ ‪ 4‬ﻛﺘﺐ ﺑﻘﻴﻤﺔ ‪ 52‬ﺭﻳﺎ ﹰﻻ؛ ﻟﺘﻘﺮﺃﻫﺎ‬ ‫‪‬‬ ‫‪20x = 72‬‬ ‫‪20‬‬ ‫‪_20x‬‬ ‫ﻓﻲ ﺃﺛﻨﺎﺀ ﺍﻹﺟﺎﺯﺓ ﺍﻟﺼﻴﻔﻴﺔ‪ .‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻜﺘﺐ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﺴﻌﺮ‪،‬‬ ‫‪_72‬‬ ‫‪20‬‬ ‫=‬ ‫ﻓﺎﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻹﻳﺠﺎﺩ ﺛﻤﻦ ﺍﻟﻜﺘﺎﺏ ﺍﻟﻮﺍﺣﺪ‪ ،‬ﺛﻢ ﹸﺣ ﱠﻠﻬﺎ‪.‬‬ ‫‪20‬‬ ‫‪ x = 3.6‬‬ ‫‪3 ‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻓﻲ ﻣﺜﺎﻝ ‪ 3‬ﻟﻺﺟﺎﺑﺔ ﻋﻤﺎ ﻳﺄﺗﻲ‪ :‬‬ ‫‪ (13‬ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﻨﻔﺮﺟﺘﻴﻦ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ‪C D ، m∠BXA = (3x + 5)° :‬‬ ‫‪ (14‬ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺘﺎﻣﺘﻴﻦ‪.‬‬ ‫‪B XE‬‬ ‫‪ ،m∠DXE = 56°‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪.x‬‬ ‫‪ (15‬ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺠﺎﻭﺭﺗﻴﻦ ﻣﺘﻜﺎﻣﻠﺘﻴﻦ ﻓﻲ ﺁﻥ ﻭﺍﺣﺪ‪.‬‬ ‫‪A‬‬ ‫‪m∠BXA = m∠DXE‬‬ ‫‪3x + 5 = 56‬‬ ‫‪ (16‬ﺇﺫﺍ ﻛﺎﻥ‪ m∠DXB = 116° :‬ﻭ ‪،m∠EXA = (3x + 2)°‬‬ ‫‪ ‬‬ ‫‪3x = 51‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪.x‬‬ ‫‪ ‬‬ ‫‪x = 17‬‬ ‫‪ (17‬ﺇﺫﺍ ﻛﺎﻥ‪m∠CXD = (6x – 13)° :‬‬ ‫‪5‬‬ ‫ﻭ ‪ ،m∠DXE = (10x + 7)°‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪.x‬‬ ‫‪3‬‬ ‫‪www.obeikaneducation.com  ‬‬ ‫‪2 ‬‬ ‫‪11 1 1‬‬

‫‪‬‬ ‫‪2‬‬ ‫‪1-5 1-1‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺃﺷﻜﺎﻝ ﭬﻦ ﺃﺩﻧﺎﻩ ﻟﺘﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ .‬ﻭﻓﺴﺮ ﺗﺒﺮﻳﺮﻙ ‪ 1-3 .‬‬ ‫ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ ‪ 1-1 .‬‬ ‫‪ ‬‬ ‫‪...... (2‬‬ ‫‪5, 5, 10, 15, 25, ......(1‬‬ ‫‪  ‬‬ ‫‪ (33‬ﺑ ﱢﻴﻦ ﺃ ﹼﹰﻳﺎ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺃﺍﻋﺕ ﺍﻂﻵﻣﺗﺜﻴﺎﺔ ﹰﺗﻨﻻﺘﻣﺞ ﻣﻀﻨﺎﻄ ﹼﹰﺩﻘﺍ ﹰﹼﻴﺎﻳﺒﻋﻴﻦﻦﺍﻟﺃﻌﺒﻥﺎﺭﺗﻛﻴﹼﹰﻼﻦ ﺍﻟﻣﺘﺎﻟﻦﻴﺘﺍﻴﻟﺘﻦ‪.‬ﺨﻤﻴﻨﻴﻦ ﺍﻵﺗﻴﻴﻦ‬ ‫‪ (14‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﻀﻠﻊ ﻣﺮﺑ ﹰﻌﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻣﺴﺘﻄﻴ ﹰﻼ‪.‬‬ ‫ﺇﺫﺍ ﺍﺷﺘﺮﻳﺖ ﻭﺟﺒﺘﻴﺧﺎﻦ‪،‬ﻃﻓﺈﺊﻧ‪:‬ﻚ ‪‬ﺳ‪‬ﺘ‪‬ﺤ‪‬ﺼ‪‬ﻞ‪1‬ﻋﻠ‪1-‬ﻰ‪‬ﻋﻠ‪‬ﺒﺔ‬ ‫‪.‬‬ ‫___‬ ‫ﻣﻨﺘﺼﻒ‬ ‫ﻣﺠﺎ ﹰﻧﺎ‪.‬‬ ‫ﻋﺼﻴﺮ‬ ‫ﺍﺷﺘﺮ￯ ﺧﻠﻴﻞ ﻭﺟﺒﺘ‪3‬ﻴ(ﻦ‪.‬ﺇﺫﺍ‪‬ﻛﺎﻥ ‪AB = BC‬‬ ‫‪ 4‬‬ ‫‪AC‬‬ ‫‪ B‬ﻧﻘﻄﺔ‬ ‫‪ ،‬ﻓﺈﻥ‬ ‫‪ ‬ﻳﻤﻜﻦ ﻟﻠﻄﻼﺏ ﺍﺳﺘﻌﻤﺎﻝ‬ ‫‪ (15‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻌﺎﻣﺪﻳﻦ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻻ ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻧﺎ ﻣﺘﻮﺍﺯﻳﻴﻦ‪.‬‬ ‫‪ A‬ﺍﺷﺘﺮ￯ ﺧﻠﻴﻞ‪4‬ﻭ(ﺟﺒﺇﺔﺫﺍﻭﺍﻛﺣﺎﺪﺓﻥ ﻓﻘ‪n‬ﻂﻋ‪.‬ﺪ ﹰﺩﺍ ﺣﻘﻴﻘ ﹼﹰﻴﺎ‪ ،‬ﻓﺈﻥ ‪.n3 > n‬‬ ‫ﺍﻷﺷﻜﺎﻝ ﻟﻌﻤﻞ ﻧﻤﻮﺫﺝ ﻟﻘﺎﻧﻮﻥ ﺍﻟﻔﺼﻞ‬ ‫‪ B‬ﺳﻴﺤﺼﻞ ﺧﻠﻴﻞ ﻋﻠﻰ ﻭﺟﺒﺔ ﻣﺠﺎﻧﻴﺔ‪.‬‬ ‫ﺍﻟﻤﻨﻄﻘﻲ ﻭﻗﺎﻧﻮﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﻤﻨﻄﻘﻲ‪ .‬ﺯ ﹼﻭﺩ‬ ‫‪  (16‬ﺗﻘﺎﺑﻞ ﻓﺮﻳﻘﺎ ﺍﻟﻔﺮﺳﺎﻥ ﻭﺍﻟﻔﻬﻮﺩ ﻓﻲ ﺍﻟﻤﺒﺎﺭﺍﺓ ﺍﻟﻨﻬﺎﺋﻴﺔ‪.‬‬ ‫ﺳﻴﺤﺼﻞ ﺧﺍﻠﻴﺳﺘﻞﻌﻋﻠﻤﻰﻞﻋﺍﻠﻟﺒ ﹶﺘﻌﺒﻲﺎﺭﻋﺍﺼﻴﺕﺮ‪r‬ﻣ ‪,‬ﺠﺎ ﹰﻧ‪q‬ﺎ‪ p,.‬ﻟﻜﺘﺎﺑﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻭﺻﻞ ﺃﻭ ﻓﺼﻞ ﺃﺩﻧﺎﻩ‪ ،‬ﺛﻢ‬ ‫‪C‬‬ ‫ﺍﻟﻄﻼﺏ ﺑﻮﺭﻗﺘﻴﻦ ﺻﻔﺮﺍﻭﻳﻦ ﻣﺮﺑﻌ ﹶﺘﻲ ﺍﻟﺸﻜﻞ‬ ‫ﻣﻌﺘﻤ ﹰﺪﺍ ﻋﻠﻰ ﺍﻟﻤﻌﻄﻴﺎﺕ‪ ،‬ﺣ ﱢﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﺘﻴﺠﺔ ﺻﺎﺋﺒﺔ ﺃﻡ ﻻ ﻓﻲ‬ ‫ﺣﺼﻞ ﺧﻠﻴﻞ ﻋﻠﻰ ﻋﻠﺒﺔ ﻋﺼﻴﺮ ﻣﺠﺎ ﹰﻧﺎ‪.‬‬ ‫‪D‬‬ ‫ﻣﻜﺘﻮﺏ ﻋﻠﻴﻬﻤﺎ ‪ ، p‬ﻭﻭﺭﻗﺘﻴﻦ ﺯﺭﻗﺎﻭﻳﻦ‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪ .‬ﻓ ﹼﺴﺮ ﺗﺒﺮﻳﺮﻙ‪1-2 .‬‬ ‫ﻣﺜﻠﺜ ﹶﺘﻲ ﺍﻟﺸﻜﻞ ﻣﻜﺘﻮﺏ ﻋﻠﻴﻬﻤﺎ ‪ ، q‬ﻭﻭﺭﻗﺘﻴﻦ‬ ‫ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ .‬ﻭﻓ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪1-4 .‬‬ ‫‪ :p‬ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ‪ 7‬ﺃﻳﺎﻡ‪.‬‬ ‫ﺣﻤﺮﺍﻭﻳﻦ ﺩﺍﺋﺮﻳ ﹶﺘﻲ ﺍﻟﺸﻜﻞ ﻣﻜﺘﻮﺏ ﻋﻠﻴﻬﻤﺎ ‪،r‬‬ ‫‪ ‬ﺍﻟﻔﺮﻳﻖ ﺍﻟﻔﺎﺋﺰ ﺑﺎﻟﻜﺄﺱ ﻫﻮ ﺍﻟﻔﺮﻳﻖ ﺍﻟﺬﻱ ﻳﺤﺮﺯ ﺃﻫﺪﺍ ﹰﻓﺎ‬ ‫‪ :q‬ﻓ‪‬ﻲ ﺍﻟﻴﻮﻡ ﺍﻟﻮﺍﺣﺪ ‪ 24‬ﺳﺎﻋﺔ‪.‬‬ ‫ﻭﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﻛﺘﺎﺑﺔ ﻃﺮﻳﻘﺔ ﺗﺮﺗﻴﺐ ﺍﻷﺷﻜﺎﻝ‬ ‫ﻗﺒﻞ ﺷﻬﺮ ﺍﻟﻤﺤﺮﻡ‪.‬‬ ‫‪ :r‬ﹶﺻ ﹶﻔﺮ ﻫﻮ ﺍﻟﺸﻬﺮ ﺍﻟﺬﻱ ﻳﺄﺗﻲ‬ ‫ﻟﺘﻤﺜﻴﻞ ﺍﻟﺼﻮﺭﺓ ﺍﻟﺮﻣﺰﻳﺔ ﻟﻠﻘﺎﻧﻮﻧﻴﻦ‪.‬‬ ‫ﺃﻛﺜﺮ ﻓﻲ ﻧﻬﺎﻳﺔ ﺍﻟﻤﺒﺎﺭﺍﺓ‪.‬‬ ‫‪1 - 3 ‬‬ ‫ﺍﻟﻤﻌﻠﻮﻣ‪5‬ﺎ(ﺕ ﺍ‪r‬ﻵﺗﻴﺔ ﻓ‪p‬ﻲ ﺣﻞ ﺍﻟﺴﺆﺍﻟﻴﻦ‬ ‫ﺍﺳﺘﻌﻤﻞ‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫ﺍﻟﻔﻬﻮﺩ‬ ‫ﻓﺮﻳﻖ‬ ‫ﺃﺣﺮﺯ‬ ‫ﺑﻴﻨﻤﺎ‬ ‫ﻣﺤﻼﺕ ﺻﻴﺎﻧﺔ ﺍﻟﺤﻮﺃﺍﺳﺣﻴﺮﺯﺐ ﻓﺮﻳﻖ ﺍﻟﻔﺮﺳﺎﻥ ‪ 3‬ﺃﻫﺪﺍﻑ‪،‬‬ ‫ﺇﺣﺪ￯‬ ‫ﻓﻲ‬ ‫ﺇﻋﻼﻥ‬ ‫ﻣ”ﺇﺪﻳﺫﺍﺮﻛﻭﻨﺍﻟﺘﺖﺴﺗﺒﻮﻳﺤﻖﺚﻋﺒ‪67‬ﺎﻋ((ﺭﺍﻦ ﺍﻟﺕ‪pr‬ﺴﻣ~ﺮﻭﻜﻋﺘﺔﻮ‪q‬ﺑﻭﺔﺍ‪p‬ﻋﻷﻠﻣﺎﻰﻥ ﻓﺻﻮﻲﺭﺓﺣﺎ)ﺇﺳﺫﺍﻮﺑ‪..‬ﻚ‪،.‬ﻓﺈﻓﻌﻥﻠﻴ‪...‬ﻚ(ﺑﻟﺘﻤﺮﺤﻭﻳﻞ ﺍﺞﻟﻨﺳﺠﻠﻌﻮﻬﻡ ﻟﻢ ﻭﺼﻴﺎﺧﻧﺪﺔﻣﺍﺎﻟﺗﻬﺤﻮﻢﺍ‪ .‬ﻳﺳﻴﻮﺟﺐﺪ‬ ‫ﻳﺴﺘﻌﻤﻞ‬ ‫ﻫﺪﻓﻴﻦ‪.‬‬ ‫ﺟﺎﺀ ﻓﻴﻪ‪:‬‬ ‫ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻢ ﺍﻟﻄﻼﺏ ﺍﻟﺪﺭﺳﻴﻦ ‪1-3, 1-4‬‬ ‫ﺑﺈﻋﻄﺎﺋﻬﻢ‪:‬‬ ‫‪ ‬ﺣ ﱢﺪﺩ ﺇﺫﺍ ﻛﺎﻥ ﻃﻼﺑﻚ ﻳﺤﺮﺯﻭﻥ ﺗﻘﺪ ﹰﻣﺎ ﻣﻨﺎﺳ ﹰﺒﺎ‬ ‫‪ ‬ﻓﺎﺯ ﻓﺮﻳﻖ ﺍﻟﻔﺮﺳﺎﻥ ﺑﺎﻟﻜﺄﺱ‪.‬‬ ‫‪ (35‬ﺍﻛﺘﺐ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫ﺍﻻﺧﺘﺒﺎﺭ ﺍﻟﻘﺼﻴﺮ ‪ ،2‬ﺹ )‪(11‬‬ ‫ﻓﻲ ﺃﺛﻨﺎﺀ ﺗﻌﻠﻤﻬﻢ ﻓﻲ ﻛﻞ ﺩﺭﺱ ﺃﻡ ﻻ‪ ،‬ﺑﺎﺳﺘﻌﻤﺎﻝ ﺃﻧﻮﺍﻉ ﺍﻟﺘﻘﻮﻳﻢ‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪ (36‬ﻣﺎ ﺍﻟﺮﺳﺎﻟﺔ ﺍﻟﺘﻲ ﻳﺮﻳﺪ ﺍ‪8‬ﻹ(ﻋ ﺃﻼﻛﻥ ﺇﻤﻳﻞﺼﺎﺍﻟﻟﻬﺎﺠﺇﻟﺪﻰﻭﺍﻟﻝﻨﺎﺍﻵﺱﺗﺣﻲﻮ‪.‬ﻝ‪‬ﻣ‪‬ﺤ‪‬ﻞ‪‬ﺍﻟ‪‬ﻨ‪‬ﺠ‪-2‬ﻮﻡ‪1‬؟‪‬‬ ‫ﺍﻵﺗﻴﺔ ﻟﺘﻨﻮﻳﻊ ﺍﻟﺘﺪﺭﻳﺲ ﻭﺍﻟﺘﺪﺭﻳﺒﺎﺕ‪:‬‬ ‫‪1-4‬‬ ‫‪p‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺻﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍ‪~q‬ﺕ ﺍﻟﻤﺮ‪p‬ﻛﺒﺔ ﺍﻵﺗﻴﺔ‪2q1-2~q:‬‬ ‫‪ a (37‬ﻭ ‪b‬‬ ‫‪(2-2 2-1) (1)  ‬‬ ‫‪‬‬ ‫‪‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ‬ ‫‪(41‬‬ ‫‪‬‬ ‫ﺍﻗﺮﺃ ﻛ ﹼﻞ ﺳﺆﺍﻝ ﺑﻌﻨﺎﻳﺔ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ ﺇﺟﺎﺑﺘﻚ ﻓﻲ ﺍﻟﻤﻜﺎﻥ ﺍﻟﻤﺨﺼﺺ ﻟﺬﻟﻚ‪:‬‬ ‫‪GH‬‬ ‫ﺣ ﱢﺪﺩ ﻛ ﹰﹼﻼ ﻣﻤﺎ ﻳﺄﺗﻲ ﻓﻲ ﺍﻟﺴﺆﺍﻟﻴﻦ ‪ 1‬ﻭ ‪ 2‬ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪2x°‬‬ ‫‪(37‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪L‬‬ ‫‪J‬‬ ‫‪1-3‬‬ ‫ﺣﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛﻞ‬ ‫‪a b ba‬‬ ‫‪F‬‬ ‫‪DM‬‬ ‫‪K‬‬ ‫‪________________(1‬‬ ‫‪ (9‬ﺇﺫﺍ ﻛﺎﻥ ﻟﻠﻤﻀﻠﻊ ﺧﻤﺴ‪1‬ﺔ( ﺃﻣﺿﺴﺘﻼﻮﻉ‪￯،‬ﻓﻳﺈﻧﻮﻪﺍﺯﺧﻤﻱﺎﺍﻟﺳﻤﻲ‪.‬ﺴﺘﻮ￯ ‪EGH‬‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪________________(2‬‬ ‫‪BA‬‬ ‫‪ (2‬ﺗﻘﺎﻃﻊ ﺍﻟﻤﺴﺘﻮﻳﻴﻦ ‪ ABC‬ﻭ ‪.EFB‬‬ ‫‪ (10‬ﺇ‪‬ﺫﺍ‪‬ﻛﺎﻥ ‪10‬‬ ‫‪TF‬‬ ‫‪F‬‬ ‫‪FT‬‬ ‫‪F‬‬ ‫ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﺍﻣﻟﻦﺰﺍﺍﻟﻭﻌﻳﺒﺎﺔﺭﺍﺍﻟﺘﺕ ﺍﻲﻵﻗﺗﻴﻴﺎﺔ ﺍﺳﻋﺘﻬﺎﻤﺎﺃﺃ ﹰﺩﺍﻗﺟﻋﻞﻠﻣﺐﻰ ﺍﻦﻟﻋﺸﻦﻜ ﺍﻞ ﺍﻟﻷﻤﺳﺠﺌﺎﻠﻭﺔﺭ؟‪8‬ﻓ ﹼﺴ‪-‬ﺮ‪3‬ﺇﻣﺟﺎﺑﺘﺴﺘﻚﻌ‪:‬ﻤ ﹰﻼ‬ ‫ﻳﻤﻜﻦ ﺍﻓﺘﺮﺍﺽ ﺻﻮﺍ ‪1‬ﺏ‪1‬ﺃ( ﱟﻱ‬ ‫ﻫﻞ‬ ‫‪FF‬‬ ‫‪F‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ∠DAB‬ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪ .‬‬ ‫‪(44‬‬ ‫‪________________ (3 a‬‬ ‫‪12‬‬ ‫‪63‬‬ ‫‪4‬‬ ‫‪(38‬‬ ‫• ﺑﺪﺍﺋﻞ ﺗﻨﻮﻳﻊ ﺍﻟﺘﻌﻠﻴﻢ‬ ‫• ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻤﻚ ﺑﻌﺪ ﻛﻞ ﻣﺜﺎﻝ‬ ‫‪87‬‬ ‫‪5‬‬ ‫ﺣ‪‬ﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞﺻ ﹼﻨﻒ ﻛ ﹼﻞ ﺯﻭﺝ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﻓﻲ ﺍﻷﺳﺌﻠﺔ ‪ 3-6‬ﺇﻟﻰ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺒﺎﺩﻟﺘﻴﻦ ﺩﺍﺧﻠ ﹼﹰﻴﺎ‪،‬‬ ‫‪∠DEC (45‬‬ ‫• ﺍﻟﺨﻄﻮﺓ ﺍﻟﺮﺍﺑﻌﺔ )ﺍﻟﺘﻘﻮﻳﻢ(‬ ‫• ﺗﺄﻛﺪ‬ ‫‪________________ (4‬‬ ‫‪b‬‬ ‫‪9 10‬‬ ‫‪11 12‬‬ ‫ﺃﻳﻬ‪‬ﻤﺎ ﺻﺎﺋﺒﺔ‪ ،‬ﻓﺒﺮﺭ ﺇﺟﺎﺑﺘ ﺃﻚﻭ‪.‬ﻣﺘ‪‬ﺒ‪‬ﺎ‪‬ﺩ‪‬ﻟ‪‬ﺘ‪‬ﻴ‪‬ﻦ‪-3‬ﺧ‪1‬ﺎ‪‬ﺭ‪‬ﺟ ﹰﹼﻴﺎ‪ ،‬ﺃﻭ ﻣﺘﻨﺎﻇﺮﺗﻴﻦ‪ ،‬ﺃﻭ ﻣﺘﺤﺎﻟﻔﺘﻴﻦ ‪:‬‬ ‫‪∠ADE (46‬‬ ‫‪p ~p q ~q ~q~p‬‬ ‫ﻓﻲ ﺧﻄﺔ ﺍﻟﺘﺪﺭﻳﺲ‬ ‫• ﻣﺴﺎﺋﻞ ﻣﻬﺎﺭﺍﺕ ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻌﻠﻴﺎ‬ ‫‪16 15‬‬ ‫‪14 13‬‬ ‫• ﻣﻌﺎﻟﺠﺔ ﺍﻷﺧﻄﺎﺀ‬ ‫‪________________ (5  1c-5d‬‬ ‫)ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ‪ ،‬ﺍﻛﺘﺐ(‬ ‫‪ ∠6 (4‬ﻭ ‪∠12‬‬ ‫‪ ∠2 (3‬ﻭ ‪∠10‬‬ ‫‪ ∠1 (12  AB ⊥ BC (47‬ﻭ ‪∠2‬‬ ‫‪TFTF‬‬ ‫‪F‬‬ ‫‪ ∠14 (6‬ﻭ ‪∠15‬‬ ‫‪ ∠1 (5‬ﻭ‪∠5‬‬ ‫• ﻣﺮﺍﺟﻌﺔ ﺗﺮﺍﻛﻤﻴﺔ‬ ‫‪________________ (6‬‬ ‫‪TFFT‬‬ ‫‪T‬‬ ‫• ﺍﺧﺘﺒﺎﺭ ﻣﻨﺘﺼﻒ ﺍﻟﻔﺼﻞ‬ ‫• ﺩﻟﻴﻞ ﺍﻟﺪﺭﺍﺳﺔ ﻭﺍﻟﻤﺮﺍﺟﻌﺔ‬ ‫‪FTTF‬‬ ‫‪T‬‬ ‫• ﺍﻟﻤﻄﻮﻳﺎﺕ‬ ‫‪________________ (7‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ‪ a b :‬ﹶﻭ ‪ ، m∠7 = 94°‬ﻓﺄﻭﺟﺪ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﺰﺍﻭﻳﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪:‬‬ ‫‪ ∠1 (13‬ﻭ ‪∠4‬‬ ‫‪FTFT‬‬ ‫‪T‬‬ ‫‪________________ (8‬‬ ‫‪ 52‬‬ ‫‪∠9 (8‬‬ ‫‪∠10 (7‬‬ ‫‪1 44‬‬ ‫‪(39‬‬ ‫‪a‬‬ ‫‪k m ~m ~mk‬‬ ‫‪________________ (9‬‬ ‫‪(5x - 7)° (4y + 3)°‬‬ ‫‪b‬‬ ‫‪ (9‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛ ﱟﻞ ﻣﻦ ‪ x, y‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪TTF‬‬ ‫‪F‬‬ ‫‪(3x + 17)°‬‬ ‫‪_______ (10‬‬ ‫‪‬‬ ‫‪TFT‬‬ ‫‪T‬‬ ‫‪U‬‬ ‫‪ (10‬ﺍﺧﺘﻴﺎﺭ ﻣﻦ ﻣﺘﻌﺪﺩ‪ :‬ﺃﻭﺟﺪ ‪ m∠UVW‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪FTF‬‬ ‫‪F‬‬ ‫‪138°‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺃﻥ‪A‬ﻳ(ﺘﻮ‪°‬ﺯ‪9‬ﻋ‪3‬ﻮﺍ ﻣﺠﻤﻮﻋﺎﺕ ﺻﻐﻴﺮﺓ‪ ،‬ﻭﻳﺘﻨﺎﻗﺸﻮﺍ ﺣﻮﻝ ﺃﻃﻌﻤﺘﻬﻢ‪81° (C‬‬ ‫‪FFT‬‬ ‫‪F‬‬ ‫‪V‬‬ ‫ﺍﻟﻤﻔﻀﻠﺔ ﻭﺍﻷﻃﻌﻤﺔ ﺍﻷﻛﺜﺮ ﺷﻌﺒﻴ ﹰﺔ‪ ،‬ﻭﻓﻲ ﺃﺛﻨﺎﺀ ﻣﻨﺎﻗﺸﺘﻬ‪B‬ﻢ( ﺍ‪°‬ﻃ‪2‬ﻠ‪4‬ﺐ ﺇﻟﻴﻬﻢ ﺗﺒﺮﻳ ﹰﺮﺍ ﻻﺳﺘﻨﺘﺎﺟﺎﺗﻬﻢ ﻋﻦ ﺃﻛﺜﺮ ﺍﻷﻃﻌﻤﺔ ﺍﻟﺘ‪(D‬ﻲ‪138°‬‬ ‫‪39°‬‬ ‫‪(40‬‬ ‫‪W‬‬ ‫ﻳﺤﺒﻮﻧﻬﺎ ﻭﻳﻔﻀﻠﻮﻧﻬﺎ‪ ،‬ﻭﺃﻥ ﻳﺼﻔﻮﺍ ﺃﻧﻮﺍﻉ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻟﺘﻲ ﺍﺳﺘﻌﻤﻠﻮﻫﺎ ﻟﻠﺘﻮﺻﻞ ﺇﻟﻰ ﺍﺳﺘﻨﺘﺎﺟﺎﺗﻬﻢ‪.‬‬ ‫‪z ~y‬‬ ‫‪‬‬ ‫‪(2-3) (2)  ‬‬ ‫‪‬‬ ‫‪y ~y z‬‬ ‫‪TFT‬‬ ‫‪T‬‬ ‫‪________________(1‬‬ ‫‪2‬‬ ‫‪TFF‬‬ ‫‪F‬‬ ‫‪________________(2‬‬ ‫‪FTT‬‬ ‫‪T‬‬ ‫‪________________(3‬‬ ‫‪FTF‬‬ ‫‪T‬‬ ‫‪________________(4‬‬ ‫ﺍﻗﺮﺃ ﻛ ﹼﻞ ﺳﺆﺍﻝ ﺑﻌﻨﺎﻳﺔ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ ﺇﺟﺎﺑﺘﻚ ﻓﻲ ﺍﻟﻤﻜﺎﻥ ﺍﻟﻤﺨﺼﺺ ﻟﺬﻟﻚ‪:‬‬ ‫‪g‬‬ ‫ﻫﻞ ﻳﻤﻜﻦ ﺇﺛﺒﺎﺕ ﺃﻥ ﺃ ﹰﹼﻳﺎ ﻣﻦ ﻣﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﺸﻜﻞ ﻣﺘﻮﺍﺯﻳﺔ‪ ،‬ﺍﻋﺘﻤﺎ ﹰﺩﺍ‬ ‫‪h‬‬ ‫‪1‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫ﻋﻠﻰ ﺍﻟﻤﻌﻄﻴﺎﺕ ﻓﻲ ﺍﻷﺳﺌﻠﺔ ‪ ،1-4‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺃ ﱡﻳﻬﺎ ﻣﺘﻮﺍﺯ ﹼﹰﻳﺎ‪،‬‬ ‫‪ 1 44‬‬ ‫‪2‬‬ ‫‪5‬‬ ‫ﻓﺎﺫﻛﺮ ﺍﻟﻤﺴ ﹼﻠﻤﺔ ﺃﻭ ﺍﻟﻨﻈﺮﻳﺔ ﺍﻟﺘﻲ ﺗﺒ ﹼﺮﺭ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫‪6‬‬ ‫‪78‬‬ ‫‪9‬‬ ‫‪p qj‬‬ ‫‪∠2 ∠3 (2‬‬ ‫‪∠1 ∠ 6 (1‬‬ ‫‪m∠ 7 + m∠ 6 = 180 (4‬‬ ‫‪∠4 ∠9 (3‬‬ ‫‪________________(5‬‬ ‫‪ (5‬ﺇﺫﺍ ﻛﺎﻥ‪ m∠3 = (5x - 17)° :‬ﹶﻭ ‪، m∠7 = (3x + 35)°‬‬ ‫‪ 2 ‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪ x‬ﺣﺘﻰ ﻳﻜﻮﻥ ‪. g h‬‬ ‫‪30  ‬‬ ‫‪ ‬‬ ‫• ﺍﻻﺧﺘﺒﺎﺭﺍﺕ ﺍﻟﻘﺼﻴﺮﺓ‬ ‫‪2 ‬‬ ‫‪‬‬ ‫• ﺍﺧﺘﺒﺎﺭ ﻣﻨﺘﺼﻒ ﺍﻟﻔﺼﻞ‪3‬‬ ‫؟‬ ‫___‬ ‫ﺗﺨﺎﻟﻒ‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‬ ‫ﺍﻟﻘﻄﻊ‬ ‫‪‬ﺃﻱ‬ ‫‪‬‬ ‫‪(17‬‬ ‫ﺻ ﱢﻨﻒ ﻛﻞ ﺯﻭﺝ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﻓﻴﻤﺎ ﻳﺄﺗﻲ ﺇﻟﻰ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺒﺎﺩﻟﺘﻴﻦ ﺩﺍﺧﻠ ﹰﹼﻴﺎ‪ ،‬ﺃﻭ‬ ‫‪‬‬ ‫‪CD‬‬ ‫ﻣﺘﺒﺎﺩﻟﺘﻴﻦ ﺧﺎﺭﺟ ﹰﹼﻴﺎ‪ ،‬ﺃﻭ ﻣﺘﻨﺎﻇﺮﺗﻴﻦ‪ ،‬ﺃﻭ ﻣﺘﺤﺎﻟﻔﺘﻴﻦ‪ ،‬ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ‪.‬‬ ‫‪A‬‬ ‫‪W‬‬ ‫‪∠6, ∠3 (1‬‬ ‫‪B‬‬ ‫‪V‬‬ ‫‪12‬‬ ‫‪56‬‬ ‫‪∠4, ∠7 (2‬‬ ‫‪E‬‬ ‫‪X‬‬ ‫‪34‬‬ ‫‪78‬‬ ‫‪∠5, ∠4 (3‬‬ ‫‪Z‬‬ ‫‪C‬‬ ‫‪D‬‬ ‫‪Y‬‬ ‫___‬ ‫___‬ ‫‪DE (C‬‬ ‫‪ZY (A‬‬ ‫___‬ ‫‪VZ (D‬‬ ‫ﺃﻭﺟﺪ ﻣﻴﻞ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺘﻴﻦ ‪ A B‬ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫ﺍﻟﺘﻲ ﺗﺠﻌﻞ ‪ .a b‬ﻭﺣ ﹼﺪﺩ ﺍﻟﻤﺴ ﹼﻠﻤﺔ ﺃﻭ ﺍﻟﻨﻈﺮﻳﺔ ﺍﻟﺘﻲ‬ ‫‪‬‬ ‫‪‬‬ ‫‪ab‬‬ ‫‪(2B)    ‬‬ ‫‪2‬‬ ‫‪ ‬ﻗ ﱢﻮﻡ ﻣﺪ￯ ﻧﺠﺎﺡ ﻃﻼﺑﻚ ﻓﻲ ﺗﻌﻠﻢ‬ ‫‪(4x + 11)° (8x + 1)°‬‬ ‫ﺍﻗﺮﺃ ﻛ ﹼﻞ ﺳﺆﺍ ﹴﻝ ﺑﻌﻨﺎﻳ ﹴﺔ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ ﺇﺟﺎﺑﺘﻚ ﻓﻲ ﺍﻟﻤﻜﺎﻥ ﺍﻟﻤﺨﺼﺺ ﻟﺬﻟﻚ‪:‬‬ ‫ﻣﻔﺎﻫﻴﻢ ﻛﻞ ﻓﺼﻞ ﺑﺎﺳﺘﻌﻤﺎﻝ ﻣﺎ ﻳﺄﺗﻲ‪:‬‬ ‫‪ ______________(1‬ﺃﻭﺟﺪﺍﻟﺒﻌﺪﺑﻴﻦﺍﻟﻨﻘﻄﺔ ‪P‬ﻭﺍﻟﻤﺴﺘﻘﻴﻢ ‪ ℓ‬ﻓﻲﻛ ﱟﻞﻣ ﱠﻤﺎﻳﺄﺗﻲ‪:‬‬ ‫‪ST‬‬ ‫ﻓﻲ ﺍﻟﺴﺆﺍﻟﻴﻦ ‪ 1‬ﻭ ‪ 2‬ﺣ ﱢﺪﺩ ﻛ ﹼﹰﻼ ﻣﻤﺎ ﻳﺄﺗﻲ ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪VU‬‬ ‫‪ (1‬ﺗﻘﺎﻃﻊ ﺍﻟﻤﺴﺘﻮ￯ ‪ SVX‬ﻭﺍﻟﻤﺴﺘﻮ￯ ‪. STU‬‬ ‫‪ ℓ‬ﺑﺎﻟﻨﻘﻄﺘﻴﻦ )‪ .(-4, 2) , (3, -5‬ﻭﺇﺣﺪﺍﺛﻴﺎ ﺍﻟﻨﻘﻄﺔ ‪P‬‬ ‫‪______________(2‬‬ ‫‪XW‬‬ ‫‪ (2‬ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ ﲣﺎﻟﻒ ‪. WY‬‬ ‫‪ ℓ‬ﺑﺎﻟﻨﻘﻄﺘﻴﻦ )‪ .(6, 5) , (2, 3‬ﻭﺇﺣﺪﺍﺛﻴﺎ ﺍﻟﻨﻘﻄﺔ ‪P‬‬ ‫‪ZY‬‬ ‫ﺃﺟﺐ ﻋﻦ ﺍﻷﺳﺌﻠﺔ ‪ ، 3-7‬ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫ﺻ ﹼﻨﻒ ﺯﻭﺝ ﺍﻟﺰﻭﺍﻳﺎ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻷﺳﺌﻠﺔ ‪ 3 – 5‬ﺇﻟﻰ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺒﺎﺩﻟﺘﻴﻦ ﺩﺍﺧﻠ ﹰﹼﻴﺎ‪،‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ ﻟﺘﺠﺪ ﻣﻴﻞ ﻛﻞ ﻣﺴﺘﻘﻴﻢ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫ﺃﻭ ﻣﺘﺒﺎﺩﻟﺘﻴﻦ ﺧﺎﺭﺟ ﹰﹼﻴﺎ‪،‬ﺃﻭ ﻣﺘﻨﺎﻇﺮﺗﻴﻦ‪ ،‬ﺃﻭ ﻣﺘﺤﺎﻟﻔﺘﻴﻦ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪y ______________(3‬‬ ‫‪1423‬‬ ‫‪115836671145mn ‬‬ ‫‪‬‬ ‫‪ ∠ 2 (3‬ﻭ‪∠12‬‬ ‫• ﻣﻌﺎﻟﺠﺔ ﺍﻷﺧﻄﺎﺀ‬ ‫• ﺍﺧﺘﺒﺎﺭ ﺍﻟﻔﺼﻞ‬ ‫‪9121011s‬‬ ‫‪t‬‬ ‫‪ ∠3 (4‬ﻭ‪∠5‬‬ ‫• ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫)‪(−4, 9‬‬ ‫‪______________(4‬‬ ‫‪______________(5‬‬ ‫• ﺍﻟﻤﻄﻮﻳﺎﺕ‬ ‫)‪(2, 6‬‬ ‫‪______________(6‬‬ ‫‪ ‬‬ ‫‪m‬‬ ‫‪O‬‬ ‫‪x‬‬ ‫‪∠7 (5‬ﻭ‪∠15‬‬ ‫)‪(10,−4‬‬ ‫• ﺍﺧﺘﺒﺎﺭ ﺍﻟﻤﻔﺮﺩﺍﺕ‬ ‫)‪(−10, 0‬‬ ‫• ﺍﺧﺘﺒﺎﺭ ﺍﻟﻔﺼﻞ )ﻧﻤﺎﺫﺝ ﻣﺘﻌﺪﺩﺓ(‬ ‫‪n‬‬ ‫‪ (6‬ﺇﺫﺍ ﻛﺎﻥ‪ m n :‬ﹶﻭ ‪ ،m∠8 = 86°‬ﻓﺄﻭﺟﺪ ‪.m∠13‬‬ ‫• ﺍﺧﺘﺒﺎﺭ ﺍﻹﺟﺎﺑﺎﺕ ﺍﻟﻤﻄﻮﻟﺔ‬ ‫• ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫‪______________(7‬‬ ‫‪ (7‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛ ﱟﻞ ﻣﻦ ‪ x‬ﻭ ‪ y‬ﺇﺫﺍ ﻛﺎﻥ‪، m∠4 = (6x – 5)° ,m n :‬‬ ‫ﻣﺴﺘﻘﻴﻢ ﻳﻮﺍﺯﻱ ‪. m‬‬ ‫‪.m∠9 = (3y – 10)° , m∠10 = (5x + 8)°‬‬ ‫‪.n‬‬ ‫ﺃﻭﺟﺪ ﻣﻴﻞ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺘﻴﻦ ﺍﻟﻤﺤ ﹼﺪﺩﺗﻴﻦ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻷﺳﺌﻠﺔ ‪:8-10‬‬ ‫‪  _ _____________(8‬ﻳﻌﻤﻞ ﻣﺤﻤﻮﺩ ﻣﻨﺪﻭﺏ ﻣﺒﻴﻌﺎﺕ‪ ،‬ﻭﻳﺘﻘﺎﺿﻰ ‪ 12‬ﺭﻳﺎ ﹰﻻ ﻋﻦ‬ ‫‪W(5, 5), V(-10, -4) (8‬‬ ‫ﻛﻞ ﺳﺎﻋﺔ ﻋﻤﻞ ﺯﺍﺋﺪ ﻋﻤﻮﻟﺔ ﻣﻘﺪﺍﺭﻫﺎ ‪ 15%‬ﻣﻦ ﻗﻴﻤﺔ ﻣﺒﻴﻌﺎﺗﻪ‪ .‬ﺍﻛﺘﺐ‬ ‫‪C(2, -15), A(-2, 9) (9‬‬ ‫‪______________(9‬‬ ‫‪L(-3, 9), G(-6, 14) (10‬‬ ‫ﻞ ﻣﺎ ﻳﺘﻘﺎﺿﺎﻩ ﻓﻲ ﺃﺣﺪ ﺍﻷﺳﺎﺑﻴﻊ ﺇﺫﺍ ﻛﺎﻧﺖ ﻗﻴﻤﺔ ﻣﺒﻴﻌﺎﺗﻪ‬ ‫ﻓﻲ ﺍﻷﺳﺌﻠﺔ ‪ ،11 – 13‬ﺣ ﹼﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ‪ CS‬ﹶﻭ ‪ KP‬ﻣﺘﻮﺍﺯﻳﻴﻦ ﺃﻭ ﻣﺘﻌﺎﻣﺪﻳﻦ‪ ،‬ﺃﻭ ﻏﻴﺮ ﺫﻟﻚ‪.‬‬ ‫‪______________(10‬‬ ‫‪P(6, -6), K(1, 9), S(5, 4),C(1,-12) (11‬‬ ‫‪139  2‬‬ ‫‪P(1, 4), K(-2, 10), S(-3, 2), C(-5, 6) (12‬‬ ‫‪______________(11‬‬ ‫‪P(9, 7), K(3, 3), S(-3, -5), C(-6, -7) (13‬‬ ‫‪______________(12‬‬ ‫‪______________(13‬‬ ‫‪ (14‬ﻳﺘﻘﺎ ﹶﺿﻰ ﻣﻜﺘﺐ ﺧﺪﻣﺎﺕ ﻃﻼﺑﻴﺔ ﻣﺒﻠﻎ ‪ 5.5‬ﺭﻳﺎ ﹰﻻ ﻋﻦ ﻛ ﹼﻞ ﺻﻔﺤ ﹴﺔ‪،‬‬ ‫ﻋﻨﺪ ﻃﺒﻊ ﺗﻘﺮﻳ ﹴﺮ ﻋﺪﺩ ﺻﻔﺤﺎﺗﻪ ‪ ، p‬ﻣﻀﺎ ﹰﻓﺎ ﺇﻟﻰ ﺫﻟﻚ ‪ 12‬ﺭﻳﺎ ﹰﻻ ﻟﺘﺠﻠﻴﺪﻩ‪.‬‬ ‫‪______________(14‬‬ ‫ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟ ﹰﺔ ﺗﻤ ﱢﺜﻞ ﺍﻟﺘﻜﻠﻔﺔ ﺍﻟﻜﻠﻴﺔ ‪ C‬ﻟﻄﺒﻊ ﻭﺗﺠﻠﻴﺪ ﺍﻟﺘﻘﺮﻳﺮ‪.‬‬ ‫‪ 2 ‬‬ ‫ﻣﺎ ﺗﻜﻠﻔﺔ ﻃﺒﻊ ﻭﺗﺠﻠﻴﺪ ﺗﻘﺮﻳ ﹴﺮ ﻣﻜ ﹼﻮ ﹴﻥ ﻣﻦ ‪ 50‬ﺻﻔﺤ ﹰﺔ؟‬ ‫‪38   ‬‬ ‫‪T8 ‬‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 3‬‬ ‫‪   1‬‬ ‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﺩﻋ ﹰﻤﺎ ﻭﺍﺳ ﹰﻌﺎ ﻳﺮﺍﻋﻲ ﺍﻟﻔﺮﻭﻕ ﺍﻟﻔﺮﺩﻳﺔ ﺑﻴﻦ ﺍﻟﻄﻼﺏ‪.‬‬ ‫ﺣﻴﺚ ﻳﺤﺘﻮﻱ ﻛﻞ ﻓﺼﻞ ﻭﻛﻞ ﺩﺭﺱ ﻋﻠﻰ ﺍﻗﺘﺮﺍﺣﺎﺕ ﻟﺘﺤﺪﻳﺪ ﺍﺣﺘﻴﺎﺟﺎﺕ ﻃﻼﺑﻚ ﻭﺗﻠﺒﻴﺘﻬﺎ‪.‬‬ ‫ﺍﻃﺮﺡ ﺍﻟﻤﺴﺄﻟﺔ ﺍﻵﺗﻴﺔ ﻋﻠﻰ ﺍﻟﻄﻼﺏ‪:‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺍﺳﺘﻜﺸﺎﻑ ﺟﻤﻊ ﺍﻟﻘﻄﻊ‬ ‫ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ ﻛﻞ ﺛﻼﺙ ﻧﻘﺎﻁ ﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ ﺗﺤﺪﺩ ﻣﺴﺘ ﹰﻮ￯ ﻭﺍﺣ ﹰﺪﺍ‪ ،‬ﻓﻤﺎ‬ ‫ﻛﻤﺎ ﺃﻥ ﺗﻨﻮﻳﻊ ﺍﻟﺘﻌﻠﻴﻢ ﻳﻠﺒﻲ ﺣﺎﺟﺎﺕ ﺍﻟﻔﺌﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ ‪:‬‬ ‫ﻋﺪﺩ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺘﻲ ﺗﺤ ﱢﺪﺩﻫﺎ ﺃﺭﺑﻊ ﻧﻘﺎﻁ ﻻ ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ ﻭﺍﺣﺪ؟ ﻭﻣﺎ‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻭﺍﻟﺰﻭﺍﻳﺎ‪ ،‬ﻭﺫﻟﻚ ﺑﻘﻴﺎﺱ ﺑﻌﺾ ﺍﻷﺷﻴﺎﺀ ﺍﻟﻤﻮﺟﻮﺩﺓ ﻓﻲ ﻏﺮﻓﺔ ﺍﻟﺼﻒ‪،‬‬ ‫ﺍﻟﻄﻼﺏ ﺩﻭﻥ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫ﻋﺪﺩ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺘﻲ ﺗﺤﺪﺩﻫﺎ ‪ 5‬ﻧﻘﺎﻁ ﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ؟‬ ‫ﻭﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻤﺘﺮ ﻹﻳﺠﺎﺩ ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ ﻏﺮﻓﺔ ﺍﻟﺼﻒ‪ ،‬ﻭﺍﻟﻤﻨﻘﻠﺔ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﺃﻥ‬ ‫ﺍﻟﻄﻼﺏ ﻓﻮﻕ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫ﺗﺤﺪﺩ ﺍﻟﻨﻘﺎﻁ ﺍﻷﺭﺑﻊ ﻣﺴﺘﻮ￯ ﻭﺍﺣ ﹰﺪﺍ ﻋﻠﻰ ﺍﻷﻗﻞ‪ ،‬ﻭ‪ 4‬ﻣﺴﺘﻮﻳﺎﺕ ﻋﻠﻰ ﺍﻷﻛﺜﺮ‪ ،‬ﻭﺗﺤﺪﺩ‬ ‫ﺯﺍﻭﻳﺘﻴﻦ ﻗﺎﺋﻤﺘﻴﻦ ﺗﺸ ﱢﻜﻼﻥ ﺧ ﹰﹼﻄﺎ ﻣﺴﺘﻘﻴ ﹰﻤﺎ‪.‬‬ ‫‪‬‬ ‫ﺍﻟﻨﻘﺎﻁ ﺍﻟﺨﻤﺲ ﻣﺴﺘﻮ￯ ﻭﺍﺣ ﹰﺪﺍ ﻋﻠﻰ ﺍﻷﻗﻞ‪ ،‬ﻭ ‪ 10‬ﻣﺴﺘﻮﻳﺎﺕ ﻋﻠﻰ ﺍﻷﻛﺜﺮ‪.‬‬ ‫‪  ‬ﻳﻤﻜﻦ ﻟﻠﻄﻼﺏ ﺃﻥ ﻳﺘﺪ ﱠﺭﺑﻮﺍ ﻋﻠﻰ ﺻﻴﺎﻏﺔ ﺗﺨﻤﻴﻨﺎﺕ‬ ‫ﺍﻟﺘﺴﺮﻳﻊ ﻭﺍﻹﺛﺮﺍﺀ‪ :‬ﻳﻤﻜﻦ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻤﺼﺎﺩﺭ ﻭﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ‪ ،‬ﺍﻟﺘﻲ ﺗﻢ ﺗﺼﻨﻴﻔﻬﺎ‬ ‫ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﻌﺼﻒ ﺍﻟﺬﻫﻨﻲ‪ ،‬ﻭﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ ﻣﻦ ﺍﻟﻄﺒﻴﻌﺔ‪ .‬ﻓﻤﺜ ﹰﻼ ﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ‬ ‫ﻟﻠﻄﻼﺏ ﻓﻮﻕ ﺍﻟﻤﺘﻮﺳﻂ‪ ،‬ﻣﻊ ﺍﻟﻄﻼﺏ ﺫﻭﻱ ﺍﻟﻤﺴﺘﻮ￯ ﺍﻟﺘﻌﻠﻴﻤﻲ ﺍﻟﻤﺘﻘﺪﻡ‪.‬‬ ‫ﻗﺮﺍﺀﺓ ﺍﻟﻌﺒﺎﺭﺓ ”ﺇﺫﺍ ﻟﻢ ﹸﺗ ﹾﺮ ﹶﻭ ﺍﻟﻨﺒﺎﺗﺎﺕ ﻛﻞ ﻳﻮﻡ ﻓﻠﻦ ﺗﺒﻘﻰ ﻋﻠﻰ ﻗﻴﺪ ﺍﻟﺤﻴﺎﺓ“‪ .‬ﻭﺍﻟﻤﺜﺎﻝ‬ ‫ﺍﻟﻤﻀﺎﺩ ﻟﻬﺎ ﺃﻥ ﻧﺒﺘﺔ ﺍﻟﺼﺒﺎﺭ ﻳﻤﻜﻦ ﺃﻥ ﺗﺒﻘﻰ ﺃﺳﺎﺑﻴﻊ ﻣﻦ ﺩﻭﻥ ﻣﺎﺀ‪.‬‬ ‫ﻭﻣﻮﺿﻮﻋﺎﺕ ﺍﻟﻄﺒﻴﻌﺔ ﻳﻤﻜﻦ ﺃﻥ ﺗﺸﻤﻞ ﺍﻟﻨﺒﺎﺗﺎﺕ ﻭﺍﻟﺤﻴﻮﺍﻧﺎﺕ ﻭﻋﻼﻗﺎﺕ ﺍﻟﺤﻴﻮﺍﻧﺎﺕ‬ ‫ﺍﻟﻤﻔﺘﺮﺳﺔ ﻭﺍﻟﻄﺮﺍﺋﺪ ﻭﺍﻟﺤﺸﺮﺍﺕ ﻭﺍﻟﻄﻘﺲ‪ ،‬ﻭﻫﻜﺬﺍ‪.‬‬ ‫‪ 2‬‬ ‫ﻭ ﹼﺿﺢ ﻟﻠﻄﻼﺏ ﻛﻴﻔﻴﺔ ﺍﻻﻧﺘﻘﺎﻝ ﻓﻲ ﺍﻟﺒﺮﻫﺎﻥ ﻣﻦ ﺍﻟﻔﺮﺽ ﺇﻟﻰ ﺍﻟﻨﺘﻴﺠﺔ ﺑﺎﺳﺘﻌﻤﺎﻝ‬ ‫ﻣﺨﻄﻂ ﺗﺴﻠﺴﻠﻲ‪ ،‬ﺑﺤﻴﺚ ﺗﻘﻮﺩ ﺍﻟﺸﺮﻭﻁ ﺍﻟﻤﻌﻄﺎﺓ ﺇﻟﻰ ﻋﺒﺎﺭﺍﺕ ﺍﻟﺒﺮﻫﺎﻥ ﻣﻊ ﺗﺒﺮﻳ ﹴﺮ ﻟﻜﻞ‬ ‫ﺧﻄﻮﺓ‪ ،‬ﻭﺗﻜﻮﻥ ﺍﻟﻨﺘﻴﺠﺔ ﻫﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻨﻬﺎﺋﻴﺔ ﻓﻲ ﺍﻟﺒﺮﻫﺎﻥ‪.‬‬ ‫‪  ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪1  ‬‬ ‫‪ 1-2‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫∼‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪p∧q‬‬ ‫‪“”‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪p∨q‬‬ ‫‪“”‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪10D  1‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪   3‬‬ ‫‪‬‬ ‫‪  (11 3‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺍﻟﺬﻱ ﻳﺒﻴﻦ ﻋﺪﺩ‬ ‫ﺍﻟﻘﻄﻊ ﺍﻟﻤﻨﺘﺠﺔ ﻓﻲ ﻣﺼﻨﻊ ﻟﺒﻌﺾ ﺍﻟﺴﻨﻮﺍﺕ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 3‬‬ ‫ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ ‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫✓ ‪‬‬ ‫ﺗﻢ ﺗﻨﻮﻳﻊ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻜﻞ ﺩﺭﺱ ﺣﺴﺐ ﻣﺴﺘﻮﻳﺎﺕ ﺍﻟﻄﻼﺏ‪:‬‬ ‫‪5 2007‬‬ ‫‪4 ‬‬ ‫‪7.2 2008‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻌﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﺳﻨﺔ ‪2017‬ﻡ ‪.‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪1-13‬؛ ﻟﻠﺘﺄﻛﺪ ﻣﻦ ﻓﻬﻢ‬ ‫‪9.2 2009‬‬ ‫ﺳﻴﻜﻮﻥ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﻋﺎﻡ ‪ 2017‬ﻧﺤﻮ ‪ 35‬ﻣﻠﻴﻮ ﹰﻧﺎ‪.‬‬ ‫ﺍﻟﻄﻠﺒﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺳﻔﻞ ﻫﺬﻩ‬ ‫‪14.1 2010‬‬ ‫ﺍﻟﺼﻔﺤﺔ؛ ﻟﺘﻌﻴﻴﻦ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻠﻄﻠﺒﺔ‬ ‫‪19.7 2011‬‬ ‫ﺑﺤﺴﺐ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫‪28.4 2012‬‬ ‫ﺃﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﻳﺒﻴﻦ ﺃﻥ ﻛ ﹼﹰﻼ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪4 ‬‬ ‫‪(12‬‬ ‫‪ (12‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠A‬ﻭ ‪ ∠B‬ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻟﻬﻤﺎ ﺿﻠ ﹰﻌﺎ ﻣﺸﺘﺮ ﹰﻛﺎ‪.‬‬ ‫ﺩﻭﻥ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (13‬ﺇﺫﺍ ﻗﻄﻊ ﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ ﻗﻄﻌ ﹰﺔ ﻣﺴﺘﻘﻴﻤ ﹰﺔ ﻋﻨﺪ ﻣﻨﺘﺼﻔﻬﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻌﺎﻣﺪﻫﺎ‪.‬‬ ‫‪45°‬‬ ‫‪A B 45°‬‬ ‫‪  ‬‬ ‫‪‬‬‫‪‬‬ ‫ﺿﻤﻦ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬‫‪‬‬ ‫‪‬‬ ‫‪‬‬‫‪ ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (14–19‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫‪1 ‬‬ ‫‪‬‬ ‫‪Proving Segments Relationships‬‬ ‫‪4, 8, 12, 16, 20 (16‬‬ ‫‪3, 6, 9, 12, 15 (15‬‬ ‫‪0, 2, 4, 6, 8 (14‬‬ ‫ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺃﻧﻪ ﻓﻲ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺘﻲ‬ ‫ﺗﺘﻀﻤﻦ ﺑﻴﺎﻧﺎﺕ ﻣﻦ ﻭﺍﻗﻊ ﺍﻟﺤﻴﺎﺓ‪ ،‬ﻟﻴﺲ‬ ‫‪1,‬‬ ‫‪_21 ,‬‬ ‫‪_14 ,‬‬ ‫‪_1‬‬ ‫‪(19‬‬ ‫‪1, 4, 9, 16 (18‬‬ ‫‪2, 22, 222, 2222 (17‬‬ ‫ﺑﺎﻟﻀﺮﻭﺭﺓ ﺃﻥ ﻳﻤ ﹼﺜﻞ ﺍﻟﺘﺨﻤﻴﻦ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ‬ ‫‪8‬‬ ‫ﺍﻟﻨﻤﻂ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻣﺎ ﻳﻤﻜﻦ ﺃﻥ ﻳﺤﺪﺙ‬ ‫‪‬‬ ‫ﻓﻮﻕ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫‪ (20‬ﻣﻮﺍﻋﻴﺪ ﺍﻟﻮﺻﻮﻝ‪ 10:00 :‬ﺻﺒﺎ ﹰﺣﺎ‪ 12:30 ،‬ﻣﺴﺎ ﹰﺀ ‪ 3:00 ،‬ﻣﺴﺎ ﹰﺀ‪ (20 ...... ،‬ﻳﺄﺗﻲ ﻛﻞ ﻣﻮﻋﺪ ﺑﻌﺪ ﺳﺎﻋﺘﻴﻦ ﻭﻧﺼﻒ‬ ‫ﻓﻲ ﺍﻟﻤﺴﺘﻘﺒﻞ‪.‬‬ ‫ﻳﻌﻤﻞ ﻋﺒﺪﺍﻟﻠﻪ ﻓﻲ ﻣﺤﻞ ﻟﺒﻴﻊ ﺍﻷﻗﻤﺸﺔ‪ ،‬ﻭﻳﻘﻴﺲ‬ ‫‪1 - 6 ‬‬ ‫‪ 1‬‬ ‫‪  ‬‬ ‫ﺍﻟﺴﺎﻋﺔ ﻣﻦ ﺍﻟﻤﻮﻋﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪ 5:30‬ﻣﺴﺎ ﹰﺀ‪.‬‬ ‫‪ (21‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻠﺮﻃﻮﺑﺔ‪100% , 93% , 86% , …… :‬‬ ‫‪ (21‬ﺗﻘﻞ ﻛﻞ ﻧﺴﺒﺔ ﻣﺌﻮﻳﺔ ﻋﻦ‬ ‫ﻓﻤﺜ ﹰﻼ‪ ،‬ﻗﺪ ﹸﺗﺸﻴﺮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫ﺍﻟﻘﻤﺎﺵ ﺑﻮﺿﻊ ﺣﺎﻓﺘﻪ ﻋﻨﺪ ﺣﺎﻓﺔ ﺗﺪﺭﻳﺞ ﺍﻟﻤﺴﻄﺮﺓ ﺍﻟﺘﻲ‬ ‫ﺍﻟﻨﺴﺒﺔ ﺍﻟﺴﺎﺑﻘﺔ ﺑﻤﻘﺪﺍﺭ‬ ‫ﺇﻟﻰ ﺗﺰﺍﻳﺪ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻓﻲ ﺃﺣﺪ‬ ‫ﻃﻮﻟﻬﺎ ﻣﺘﺮ ﻭﺍﺣﺪ‪ .‬ﻭﻟﻜﻲ ﻳﻘﻴﺲ ﺃﻃﻮﺍ ﹰﻻ ﻣﺜﻞ ‪،125 cm‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪ (22‬ﺃﻳﺎﻡ ﺍﻟﻌﻤﻞ‪ :‬ﺍﻷﺣﺪ‪ ،‬ﺍﻟﺜﻼﺛﺎﺀ‪ ،‬ﺍﻟﺨﻤﻴﺲ‪...... ،‬‬ ‫‪ 7%‬؛ ‪.79%‬‬ ‫ﺍﻷﺳﺎﺑﻴﻊ‪ ،‬ﺇ ﹼﻻ ﺃﻥ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻗﺪ‬ ‫ﻳﻘﻴﺲ ﻣﺘ ﹰﺮﺍ ﻣﻦ ﺍﻟﻘﻤﺎﺵ ﻭﻳﻀﻊ ﻋﻼﻣﺔ ﻋﻠﻴﻪ‪ ،‬ﺛﻢ ﻳﻘﻴﺲ‬ ‫‪1-7‬‬ ‫‪‬‬ ‫‪ (22‬ﻳﺄﺗﻲ ﻛﻞ ﻳﻮﻡ ﻋﻤﻞ ﺑﻌﺪ‬ ‫ﺗﻨﺨﻔﺾ ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﺬﻱ ﻳﻠﻴﻪ‪.‬‬ ‫‪ ‬‬ ‫ﻛﺘﺎﺑﺔ ﺑﺮﺍﻫﻴﻦ ﺟﺒﺮﻳﺔ ﻭﻫﻨﺪﺳﻴﺔ ﻋﻠﻰ‬ ‫‪ (23‬ﺍﺟﺘﻤﺎﻋﺎﺕ ﺍﻟﻨﺎﺩﻱ‪ :‬ﺍﻟﻤﺤ ﹼﺮﻡ‪ ،‬ﺭﺑﻴﻊ ﺃﻭﻝ‪ ،‬ﺟﻤﺎﺩ￯ ﺍﻷﻭﻟﻰ‪ (24–27 ...... ،‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺇﺟﺎﺑﺎﺕ‬ ‫ﻳﻮﻣﻴﻦ ﻣﻦ ﻳﻮﻡ ﺍﻟﻌﻤﻞ‬ ‫ﻣﻦ ﺗﻠﻚ ﺍﻟﻌﻼﻣﺔ ‪ 25 cm‬ﺃﺧﺮ￯‪.‬‬ ‫‪ ‬‬ ‫ﺻﻮﺭﺓ ﺍﻟﺒﺮﻫﺎﻥ ﺍﻟﺤﺮ ﻭﺍﻟﺒﺮﻫﺎﻥ ﺫﻱ‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺍﻟﺴﺒﺖ‪.‬‬ ‫ﻓﻴﺼﺒﺢ ﺍﻟﻄﻮﻝ‪100 cm + 25 cm = 125 cm :‬‬ ‫‪‬‬ ‫‪(25 (24‬‬ ‫‪ (23‬ﻳﻌﻘﺪ ﻛﻞ ﺍﺟﺘﻤﺎﻉ ﺑﻌﺪ‬ ‫ﺍﻟﻌﻤﻮﺩﻳﻦ‪.‬‬ ‫ﺷﻬﺮﻳﻦ ﻣﻦ ﺍﻻﺟﺘﻤﺎﻉ‬ ‫‪ ‬‬ ‫‪1-7‬‬ ‫‪......‬‬ ‫‪......‬‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺭﺟﺐ‪.‬‬ ‫‪‬‬ ‫ﻛﺘﺎﺑﺔ ﺑﺮﺍﻫﻴﻦ ﺗﺘﻀﻤﻦ ﺟﻤﻊ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ‬ ‫‪(27‬‬ ‫‪(26‬‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻭﺗﻄﺎﺑﻘﻬﺎ‪.‬‬ ‫‪   ‬ﻋﻠﻤﺖ ﻛﻴﻒ ﺗﻘﻴﺲ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﻤﺴﻄﺮﺓ‪ ،‬ﻭﺫﻟﻚ ﺑﻮﺿﻊ‬ ‫‪www.obeikaneducation.com‬‬ ‫‪1-7‬‬ ‫‪...... ......‬‬ ‫‪‬‬ ‫ﺻﻔﺮ ﺍﻟﻤﺴﻄﺮﺓ ﻋﻠﻰ ﺃﺣﺪ ﻃﺮ ﹶﻓﻲ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻭﻗﺮﺍﺀﺓ ﺍﻟﺘﺪﺭﻳﺞ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻠﻄﺮﻑ ﺍﻵﺧﺮ ﻣﻦ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‪،‬‬ ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻨﺘﺎﺟﻲ ﻹﺛﺒﺎﺕ‬ ‫ﺻﺤﺔ ﻋﺒﺎﺭﺍﺕ‪.‬‬ ‫‪  (28‬ﺑﺪﺃ ﻣﺎﺟﺪ ﺗﻤﺎﺭﻳﻦ ﺍﻟﺠﺮﻱ ﺍﻟﺴﺮﻳﻊ ﻗﺒﻞ ﺧﻤﺴﺔ ﺃﻳﺎﻡ‪ .‬ﻓﺮﻛﺾ ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻷﻭﻝ ‪ . 0.5 km‬ﻭﻓﻲ ﺍﻷﻳﺎﻡ‬ ‫‪‬‬ ‫‪(11a‬‬ ‫ﻓﻴﻤﺜﻞ ﻫﺬﺍ ﺍﻟﺘﺪﺭﻳﺞ ﻃﻮﻝ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‪ .‬ﻭﻫﺬﺍ ﻳﻮﺿﺢ ﻣﺴﻠﻤﺔ ﺍﻟﻤﺴﻄﺮﺓ‪.‬‬ ‫ﺍﻟﺜﻼﺛﺔ ﺍﻟﺘﺎﻟﻴﺔ ‪ . 0.75 km, 1 km, 1.25 km‬ﺇﺫﺍ ﺍﺳﺘﻤﺮ ﺗﻤﺮﻳﻨﻪ ﻋﻠﻰ ﻫﺬﺍ ﺍﻟﻨﻤﻂ‪ ،‬ﻓﻤﺎ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻲ ﻳﻘﻄﻌﻬﺎ ﻓﻲ‬ ‫‪30‬‬ ‫‪ ‬‬ ‫‪1 . 8 ‬‬ ‫ﺍﻟﻴﻮﻡ ﺍﻟﺴﺎﺑﻊ؟ ‪2 km‬‬ ‫‪ ‬‬ ‫‪25‬‬ ‫ﺿﻊ ﺗﺨﻤﻴ ﹰﻨﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫‪   ‬‬ ‫‪ (29‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪2 ‬‬ ‫‪20‬‬ ‫‪ (30‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩ ﻓﻲ ﺍﺛﻨﻴﻦ‪ ،‬ﻣﻀﺎ ﹰﻓﺎ ﺇﻟﻴﻪ ﻭﺍﺣﺪ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪ (31‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺪﺩﻳﻦ ‪ a‬ﻭ ‪ ، b‬ﺇﺫﺍ ﻛﺎﻥ ‪ .ab = 1‬ﻛ ﱞﻞ ﻣﻨﻬﻤﺎ ﻣﻘﻠﻮﺏ ﺍﻵﺧﺮ‬ ‫‪15‬‬ ‫___ ___‬ ‫‪ABA ‬‬ ‫‪ ‬‬ ‫‪ (32‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ AB‬ﻭﻣﺠﻤﻮﻋﺔ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﺒﻌﺪ ﻣﺴﺎﻓﺎﺕ ﻣﺘﺴﺎﻭﻳﺔ ﻋﻦ ‪ A‬ﻭ ‪ . B‬ﺗﺸﻜﻞ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻤﻨ ﱢﺼﻒ ﻟـ ‪. AB‬‬ ‫‪10‬‬ ‫‪  B‬‬ ‫‪ (33‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻭﺣﺠﻢ ﺍﻟﻬﺮﻡ ﺍﻟﻠﺬﻳﻦ ﻟﻬﻤﺎ ﺍﻟﻘﺎﻋﺪﺓ ﻧﻔﺴﻬﺎ ﻭﺍﻻﺭﺗﻔﺎﻉ ﻧﻔﺴﻪ‪.‬‬ ‫ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻳﺴﺎﻭﻱ ‪ 3‬ﺃﻣﺜﺎﻝ ﺣﺠﻢ ﺍﻟﻬﺮﻡ‪.‬‬ ‫‪5‬‬ ‫‪AB‬‬ ‫‪0‬‬ ‫‪00 07 08 09 10 11 12‬‬ ‫‪01234567‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﻣﺼﺎﺩﺭ ﻟﻜﻞ ﺩﺭﺱ‬ ‫ﺣﺴﺐ ﻣﺴﺘﻮﻳﺎﺕ ﺍﻟﻄﻼﺏ‪:‬‬ ‫‪ (14‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪2‬‬ ‫ﺩﻭﻥ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪10‬‬ ‫ﻳﻤﻜﻦ ﺍﻟﺘﻌﺒﻴﺮ ﻋﻦ ﻣﻌﻨﻰ ﻭﻗﻮﻉ ﻧﻘﻄﺔ ﺑﻴﻦ ﻧﻘﻄﺘﻴﻦ ﺃﺧﺮﻳﻴﻦ ﺑﻤﺴ ﹼﻠﻤﺔ ﺟﻤﻊ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‪.‬‬ ‫‪ 2‬‬ ‫ﺿﻤﻦ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫‪ 1 16‬‬ ‫‪ (15‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪3‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪18‬‬ ‫‪‬‬ ‫ﻓﻮﻕ ﺍﻟﻤﺘﻮﺳﻂ‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪1 . 9 ‬‬ ‫‪‬‬ ‫‪ (16‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪4‬‬ ‫ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻗﺮﺍﺀﺓ ﻓﻘﺮﺓ ”ﻟﻤﺎﺫﺍ ؟“‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪24‬‬ ‫‪‬‬ ‫‪BA, B , C ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (17‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺤﺘﻮﻱ ﻋﻠﻰ‬ ‫‪AB + BC = AC CA‬‬ ‫• ﻟﻤﺎﺫﺍ ﻳﺠﺐ ﻋﻠﻰ ﻋﺒﺪ ﺍﻟﻠﻪ ﻗﻴﺎﺱ ﺍﻟﻘﻤﺎﺵ‬ ‫ﺍﻟﺮﻗﻢ ‪ 2‬ﺯﻳﺎﺩﺓ ﻋﻠﻰ ﺃﺭﻗﺎﻡ ﺍﻟﺤﺪ ﺍﻟﺴﺎﺑﻖ‬ ‫‪AB BC ‬‬ ‫ﺑﻬﺬﻩ ﺍﻟﻄﺮﻳﻘﺔ؟ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻃﻮﻝ ﻗﻄﻌﺔ‬ ‫‪43 - 5614 - 38‬‬ ‫‪‬‬ ‫ﺍﻟﻘﻤﺎﺵ ﺍﻟﺘﻲ ﻳﺮﻳﺪ ﻗﻴﺎﺳﻬﺎ ﻳﺰﻳﺪ ﻋﻠﻰ ﻃﻮﻝ‬ ‫ﻟﻪ؛ ‪22222‬‬ ‫‪43 - 5639 - 4115 - 39‬‬ ‫‪‬‬ ‫‪A BC‬‬ ‫ﺍﻟﻤﺴﻄﺮﺓ‪.‬‬ ‫‪ (18‬ﻳﻨﺘﺞ ﻛﻞ ﺣﺪ ﻋﻦ ﺗﺮﺑﻴﻊ ﺍﻟﻌﺪﺩ ﺍﻟﻄﺒﻴﻌﻲ‬ ‫ﺍﻟﺬﻱ ﻳﻤ ﱢﺜﻞ ﺗﺮﺗﻴﺒﻪ؛ ‪25‬‬ ‫‪AC‬‬ ‫• ﹺﺻ ﹾﻒ ﻛﻴﻒ ﺃﻥ ﻗﻴﺎﺱ ‪ 100 cm‬ﺛﻢ ‪25 cm‬‬ ‫‪45 - 5639 - 53‬‬ ‫‪‬‬ ‫ﻭﻣﺴ ﹼﻠﻤﺔ ﺟﻤﻊ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺗﺴﺘﻌﻤﻞ ﺗﺒﺮﻳ ﹰﺮﺍ ﻓﻲ ﺍﻟﻌﺪﻳﺪ ﻣﻦ ﺍﻟﺒﺮﺍﻫﻴﻦ ﺍﻟﻬﻨﺪﺳﻴﺔ‪.‬‬ ‫ﹸﻳﻌﻄﻲ ﻃﻮﻝ ‪.125 cm‬‬ ‫ﺍﻟﺬﻱ‬ ‫ﺍﻟﺤﺪ‬ ‫ﻧﺼﻒ‬ ‫ﻳﺴ‪1‬ﺎ_ﻭﻱ‬ ‫ﻛﻞ ﺣﺪ‬ ‫‪(19‬‬ ‫ﺇﺫﺍ ﹸﺃﺿﻴﻒ ﺍﻟﻄﻮﻻﻥ ﺃﺣﺪﻫﻤﺎ ﺇﻟﻰ ﺍﻵﺧﺮ‬ ‫ﻳﺴﺒﻘﻪ؛‬ ‫‪16‬‬ ‫ﻓﺴﻴﻨﺘﺞ ﻋﻨﻬﻤﺎ ﺍﻟﻄﻮﻝ ﺍﻟﻜﻠﻲ‪.‬‬ ‫‪ 1 60‬‬ ‫• ﺇﺫﺍ ﺃﺭﺍﺩ ﻋﺒﺪ ﺍﻟﻠﻪ ﻗﻴﺎﺱ ‪ ،345 cm‬ﻓﻜﻢ ﻣﺮ ﹰﺓ‬ ‫‪1-7‬‬ ‫ﻳﻀﻊ ﻋﻼﻣﺔ ﻋﻠﻰ ﺍﻟﻘﻤﺎﺵ؟ ‪3‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 1 16‬‬ ‫‪(61) • ‬‬ ‫‪(65) •‬‬ ‫‪(61, 65) •‬‬ ‫‪(12) • ‬‬ ‫‪(12) •‬‬ ‫‪(12) •‬‬ ‫‪‬‬ ‫‪(39)  •‬‬ ‫‪(36) •‬‬ ‫‪ ‬‬ ‫‪(40)  •‬‬ ‫‪(36) •‬‬ ‫‪(38) •‬‬ ‫‪(38) •‬‬ ‫‪(39)  •‬‬ ‫‪(39)  •‬‬ ‫‪(40)  •‬‬ ‫‪ 1 60‬‬ ‫‪ T9‬‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﹸﻳﻘ ﱠﺪﻡ ﻓﻲ ﻛﻞ ﻓﺼﻞ ﻣﻦ ﻓﺼﻮﻝ ﻛﺘﺎﺏ ﺍﻟﻤﻌﻠﻢ ﻟﻤﺨﺘﻠﻒ ﺍﻟﺼﻔﻮﻑ ﻣﺪﺧﻞ‬ ‫‪‬‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫ﺷﺎﻣﻞ ﻟﻠﻤﻌﺎﻟﺠﺔ‪.‬‬ ‫‪(11)‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪111‬‬ ‫‪(8)‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫✓ ‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻳﺘﻀﻤﻦ ﻛﻞ ﻓﺼﻞ ﺍﻗﺘﺮﺍﺣﺎﺕ ﻟﻠﺘﺸﺨﻴﺺ ﻭﻣﺴﺘﻮﻳﺎﺕ ﺍﻟﻤﻌﺎﻟﺠﺔ‪.‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪ 1‬ﺍﺳﺘﻌﻤﺎﻝ ﻣﺠﻤﻮﻋﺎﺕ ﺃﺳﺌﻠﺔ‪.‬‬ ‫‪‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪ ‬‬ ‫‪2‬‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪!‬‬ ‫‪1‬‬ ‫‪4‬‬ ‫‪(11, 12)‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪‬‬ ‫‪ 2‬ﺍﺳﺘﻌﻤﺎﻝ ﺩﻟﻴﻞ ﺍﻟﺪﺭﺍﺳﺔ ﻭﺍﻟﻤﺮﺍﺟﻌﺔ‪ ،‬ﻭﺑﺪﺍﺋﻞ ﺗﻨﻮﻳﻊ ﺍﻟﺘﻌﻠﻴﻢ‪.‬‬ ‫‪1‬‬ ‫‪(52)‬‬ ‫‪1‬‬ ‫‪(13)‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪(74-78)‬‬ ‫‪1‬‬ ‫‪(79)‬‬ ‫‪‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪(82-83)‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﻧﺘﺎﺋﺞ ﺍﻻﺧﺘﺒﺎﺭ ﺍﻟﺴﺮﻳﻊ ﻭﻣﺨﻄﻂ‬ ‫ﺍﻟﻤﻌﺎﻟﺠﺔ؛ ﻟﻤﺴﺎﻋﺪﺗﻚ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﻣﺴﺘﻮ￯‬ ‫‪2‬‬ ‫‪www.obeikaneducation.com‬‬ ‫ﺍﻟﻤﻌﺎﻟﺠﺔ ﺍﻟﻤﻨﺎﺳﺐ‪ .‬ﻭﺍﻟﻌﺒﺎﺭﺓ \"ﺇﺫﺍ ‪ ...‬ﻓﻘﻢ\"‬ ‫‪1‬‬ ‫ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺗﺴﺎﻋﺪﻙ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﺍﻟﻤﺴﺘﻮ￯‬ ‫‪1 ‬‬ ‫‪1‬‬ ‫✓ ‪‬‬ ‫ﺍﻟﻤﻨﺎﺳﺐ ﻟﻠﻤﻌﺎﻟﺠﺔ‪ ،‬ﻭﺍﻗﺘﺮﺍﺡ ﻣﺼﺎﺩﺭ ﻟﻜﻞ‬ ‫‪  ‬‬ ‫‪1 ‬‬ ‫‪(15-20)1, 2A, 2B‬‬ ‫‪‬‬ ‫‪1 ‬‬ ‫ﻣﺴﺘﻮ￯‪.‬‬ ‫‪  ‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪(21-22)3‬‬ ‫‪(14)‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪(23)‬‬ ‫‪1 ‬‬ ‫)ﻳﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﺱ ‪(1-1‬‬ ‫‪(24-26)‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻨﺪ ﻗﻴﻤﺔ ‪ x‬ﺍﻟ ﹸﻤﻌﻄﺎﺓ‪.‬‬ ‫‪www.obeikaneducation.com‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ x2 – 2x + 11‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪. x = 6‬‬ ‫‪180 (x – 2) , x = 8 (2‬‬ ‫‪31 4x + 7 , x = 6 (1‬‬ ‫‪1080‬‬ ‫‪14 5x2 – 3x , x = 2 (3‬‬ ‫‪‬‬ ‫‪x2 – 2x + 11‬‬ ‫‪5‬‬ ‫)‪_x(x - 3‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫=‬ ‫‪5‬‬ ‫‪(4‬‬ ‫‪x=6‬‬ ‫‪= (6)2 – 2(6) + 11‬‬ ‫‪2‬‬ ‫‪= 36 – 2(6) + 11‬‬ ‫‪‬‬ ‫‪= 36 – 12 + 11‬‬ ‫‪12 x + (x + 1) + (x + 2) , x = 3 (5‬‬ ‫‪‬‬ ‫‪= 35‬‬ ‫‪‬‬ ‫ﺍﻛﺘﺐ ﻛﻞ ﺗﻌﺒﻴﺮ ﻟﻔﻈﻲ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻠﻰ ﺻﻮﺭﺓ ﻋﺒﺎﺭﺓ ﺟﺒﺮﻳﺔ‪:‬‬ ‫‪  ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪   ‬‬ ‫‪ 1 10C‬‬ ‫‪ (6‬ﺃﻗﻞ ﻣﻦ ﺧﻤﺴﺔ ﺃﻣﺜﺎﻝ ﻋﺪﺩ ﺑﺜﻤﺎﻧﻴﺔ‪5x - 8 .‬‬ ‫‪‬‬ ‫‪ (7‬ﺃﻛﺜﺮ ﻣﻦ ﻣﺮﺑﻊ ﻋﺪﺩ ﺑﺜﻼﺛﺔ‪x2+ 3 .‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪2 ‬‬ ‫ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪):‬ﻳ ‪‬ﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﻭﺱ ‪ 1-6‬ﺇﻟﻰ ‪(1-8‬‬ ‫‪1‬‬ ‫‪5 8x – 10 = 6x (8‬‬ ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪. 36x – 14 = 16x + 58‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪36x – 14 = 16x + 58‬‬ ‫‪-7 18 + 7x = 10x + 39 (9‬‬ ‫‪‬‬ ‫‪ 25%‬‬ ‫‪ ‬‬ ‫‪16x 36x – 14 – 16x = 16x + 58 –16x‬‬ ‫‪2.3 3(11x – 7) = 13x + 25 (10‬‬ ‫ﻳﻘﺪﻡ ﻣﺨﻄﻂ ﺍﻟﻤﻌﺎﻟﺠﺔ ﺍﻗﺘﺮﺍﺣﺎﺕ ﻟﻄﺮﺍﺋﻖ ﺍﻟﺘﻌﺎﻣﻞ ﻣﻊ ﺍﻟﻄﻼﺏ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ‬ ‫‪‬‬ ‫ﻧﺘﺎﺋﺞ ﺍﺧﺘﺒﺎﺭ \"ﺍﻟﺘﻬﻴﺌﺔ\" ﻓﻲ ﺑﺪﺍﻳﺔ ﻛﻞ ﻓﺼﻞ‪ .‬ﻭﺗﺴﺎﻋﺪﻙ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪ 20x – 14 = 58‬‬ ‫‪1.1‬‬ ‫‪_3‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫=‬ ‫‪5‬‬ ‫–‬ ‫‪2x‬‬ ‫‪(11‬‬ ‫ﺍﻟﺘﻲ ﻳﺘﻀﻤﻨﻬﺎ ﺍﻟﻤﺨﻄﻂ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﻣﺴﺘﻮ￯ ﺍﻟﻤﻌﺎﻟﺠﺔ ﺍﻟﺬﻱ ﺗﺴﺘﻌﻤﻠﻪ‪.‬‬ ‫‪2‬‬ ‫‪14 20x – 14 + 14 = 58 + 14‬‬ ‫‪  (12‬ﺍﺷﺘﺮﺕ ﻋﺎﺋﺸﺔ ‪ 4‬ﻛﺘﺐ ﺑﻘﻴﻤﺔ ‪ 52‬ﺭﻳﺎ ﹰﻻ؛ ﻟﺘﻘﺮﺃﻫﺎ‬ ‫‪‬‬ ‫‪20x = 72‬‬ ‫ﻓﻲ ﺃﺛﻨﺎﺀ ﺍﻹﺟﺎﺯﺓ ﺍﻟﺼﻴﻔﻴﺔ‪ .‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻜﺘﺐ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﺴﻌﺮ‪،‬‬ ‫‪20‬‬ ‫‪_20x‬‬ ‫‪(2)‬‬ ‫=‬ ‫‪_72‬‬ ‫ﻓﺎﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻹﻳﺠﺎﺩ ﺛﻤﻦ ﺍﻟﻜﺘﺎﺏ ﺍﻟﻮﺍﺣﺪ‪ ،‬ﺛﻢ ﹸﺣ ﱠﻠﻬﺎ‪.‬‬ ‫‪(10)‬‬ ‫‪20‬‬ ‫‪20‬‬ ‫‪ 4x = 52‬؛ ‪ 13‬ﺭﻳﺎ ﹰﻻ‬ ‫‪ x = 3.6‬‬ ‫‪3 ‬‬ ‫)ﻳﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﺱ ‪(1-8‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻓﻲ ﻣﺜﺎﻝ ‪ 3‬ﻟﻺﺟﺎﺑﺔ ﻋﻤﺎ ﻳﺄﺗﻲ‪ :‬‬ ‫‪www.obeikaneducation.com‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ‪C D ، m∠BXA = (3x + 5)° :‬‬ ‫‪ (13‬ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﻨﻔﺮﺟﺘﻴﻦ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫‪B XE‬‬ ‫‪ ،m∠DXE = 56°‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪.x‬‬ ‫‪∠BXD, ∠AXE‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪A‬‬ ‫‪m∠BXA = m∠DXE‬‬ ‫‪ (14‬ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺘﺎﻣﺘﻴﻦ‪∠CXD, ∠DXE .‬‬ ‫‪‬‬ ‫‪2‬‬ ‫‪3x + 5 = 56‬‬ ‫‪ 50%‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪3x = 51‬‬ ‫‪ (15‬ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺠﺎﻭﺭﺗﻴﻦ ﻣﺘﻜﺎﻣﻠﺘﻴﻦ ﻓﻲ ﺁﻥ ﻭﺍﺣﺪ‪.‬‬ ‫‪ ‬‬ ‫‪x = 17‬‬ ‫‪∠DXE, ∠EXA‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ (16‬ﺇﺫﺍ ﻛﺎﻥ‪ m∠DXB = 116° :‬ﻭ ‪،m∠EXA = (3x + 2)°‬‬ ‫‪5‬‬ ‫‪3‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪38 .x‬‬ ‫‪ (17‬ﺇﺫﺍ ﻛﺎﻥ‪m∠CXD = (6x – 13)° :‬‬ ‫ﻭ ‪ ،m∠DXE = (10x + 7)°‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪6 .x‬‬ ‫‪ ‬‬ ‫‪www.obeikaneducation.com ‬‬ ‫‪2 ‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪11 1 1‬‬ ‫‪‬‬ ‫‪1-5 1-1‬‬ ‫ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫ﻣﻦ‬ ‫ﻟﻜﻞ‬ ‫ﻟﺘﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ‬ ‫ﺃﺩﻧﺎﻩ‬ ‫ﭬﻦ‬ ‫ﺃﺷﻜﺎﻝ‬ ‫‪‬ﺍ‪‬ﺳ‪‬ﺘ‪‬ﻌﻤﻞ‬ ‫‪‬‬ ‫‪ 1-3‬‬ ‫‪ 1-1‬‬ ‫‪‬‬ ‫‪ 1-1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 1-1‬‬ ‫‪‬‬ ‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﻓﺮ ﹰﺻﺎ ﻣﺘﻌﺪﺩﺓ ﻟﻠﺘﻘﻮﻳﻢ ﺍﻟﺘﻜﻮﻳﻨﻲ ﻓﻲ ﻛﻞ ﻓﺼﻞ ﻟﻴﺤﺪﺩ‬ ‫ﻣﺴﺘﻄﻴﻼ‪.‬‬ ‫ﻳﻜﻮﻥ‬ ‫ﺎ‪ ،‬ﻓﺈﻧﻪ‬ ‫ﺃﻭ‬ ‫ﻧﺘﻴﺠﺔ‬ ‫ﺇﻟﻰ‬ ‫ﻟﻠﺘﻮﺻﻞ‬ ‫ﻧﻤ ﹰﻄﺎ؛‬ ‫ﺗﻤ ﱢﺜﻞ‬ ‫ﻣﺨﺘﻠﻔﺔ‬ ‫ﺃﻣﺜﻠﺔ‬ ‫ﻋﻦ‬ ‫ﻧﺘﺠﺖ‬ ‫ﻣﻌﻠﻮﻣﺎﺕ‬ ‫ﻋﻠﻰ‬ ‫ﻳﻌﺘﻤﺪ‬ ‫ﺍﻟﺬﻱ‬ ‫ﺍﻟﺘﺒﺮﻳﺮ‬ ‫‪‬ﻫﻮ‬ ‫ﺍﻟﻤﻌﻠﻢ ﺇﺫﺍ ﻛﺎﻧﺖ ﻫﻨﺎﻙ ﺿﺮﻭﺭﺓ ﻟﻠﻤﻌﺎﻟﺠﺔ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ ﻧﺘﺎﺋﺞ ﺍﻟﻄﻼﺏ‪.‬‬ ‫ﻋﺒﺎﺭﺓ‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻌﺎﻣﺪﻳﻦ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻻ ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻧﺎ ﻣﺘﻮﺍﺯﻳﻴﻦ‪.‬‬ ‫‪‬‬ ‫ﹸﺗﺴ ﹼﻤﻰ ﺗﺨﻤﻴﻨﹰﺎ‪.‬‬ ‫ﺗﻮﻓﺮ ﺍﻟﺴﻠﺴﻠﺔ ﺑﺪﺍﺋﻞ ﻣﺘﻌﺪﺩﺓ ﻟﻠﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻻ ﻳﺰﺍﻟﻮﻥ ﻳﻌﺎﻧﻮﻥ ﻣﻦ‬ ‫ﺍﻟﻔﺮﺳﺎﻥ ﻭﺍﻟﻔﻬﻮﺩ ﻓﻲ ﺍﻟﻤﺒﺎﺭﺍﺓ ﺍﻟﻨﻬﺎﺋﻴﺔ‪.‬‬ ‫ﺗﺨﻤﻴﻨﹰﺎ ﻳ‪‬ﺼ‪‬ﻒ‪ ‬ﺍ‪‬ﻟﻨ‪‬ﻤ‪‬ﻂ‪‬ﻓ ‪‬ﻲﺗﺍﻘﻟﺎﺑﻤﺘﻞﺘﺎﻓﺑﺮﻌﻳﺔﻘﺎ‬ ‫ﺍﻛﺘﺐ‬ ‫‪2‬‬ ‫‪‬ﺍ‪‬ﻵ‪‬ﺗ‪‬ﻴ‪‬ﺔ‪،‬‬ ‫ﺍﻛﺘـﺐ ﺗﺨﻤﻴﻨﹰـﺎ ﻳﺼـﻒ ﺍﻟﻨﻤـﻂ ﻓـﻲ‬ ‫‪1‬‬ ‫ﺻﻌﻮﺑﺎﺕ ﺑﻌﺪ ﺇﻧﻬﺎﺀ ﺍﻟﻔﺼﻞ ﺗﺴﺎﻋﺪﻫﻢ ﻋﻠﻰ ﺗﺤﺴﻴﻦ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫ﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﺘﻴﺠﺔ ﺻﺎﺋﺒﺔ ﺃﻡ ﻻ ﻓﻲ‬ ‫ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻤﺮﺑﻊ ﻓﻲ ﺍﻟﺸﻜﻞ‬ ‫ﺇﻳﺠﺎﺩ‬ ‫ﺛ‪-2‬ﻢ‪1‬ﺍ‪‬ﺳﺘﻌﻤﻠﻪ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﺘﺘﺎﺑﻌﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻓﻲ ﺇﻳﺠﺎﺩ ﺍﻟﺤ ﹼﺪ ﺍﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺘﺎﺑﻌﺔ‪:‬‬ ‫‪ ‬‬ ‫‪1-4‬‬ ‫ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫‪. 1, 3, 9, 27, 81 ,...‬‬ ‫ﻋﺰﺯ ﺍﻟﻤﻬﺎﺭﺍﺕ ﺍﻟﻀﺮﻭﺭﻳﺔ ﻣﻦ ﺧﻼﻝ ﺗﺪﺭﻳﺒﺎﺕ ﺇﻋﺎﺩﺓ ﺍﻟﺘﻌﻠﻴﻢ ﺑﺄﺳﻠﻮﺏ‬ ‫ﺗﺪﺭﻳﺴﻲ ﻭﻣﻌﺎﻟﺠﺔ ﻳﺨﺘﻠﻔﺎﻥ ﻋﻦ ﻛﺘﺎﺏ ﺍﻟﻄﺎﻟﺐ ﻭﻛﺘﺎﺏ ﺍﻟﺘﻤﺎﺭﻳﻦ‪.‬‬ ‫ﺍﻟﻔﺮﻳﻖ ﺍﻟﻔﺎﺋﺰ ﺑﺎﻟﻜﺄﺱ ﻫﻮ ﺍﻟﻔﺮﻳﻖ ﺍﻟﺬﻱ ﻳﺤﺮﺯ ﺃﻫﺪﺍ ﹰﻓﺎ‬ ‫‪1 ‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‪:‬‬ ‫‪T10 ‬‬ ‫ﺃﺣﺮﺯ ﻓﺮﻳﻖ ﺍﻟﻔﺮﺳﺎﻥ ‪ 3‬ﺃﻫﺪﺍﻑ‪ ،‬ﺑﻴﻨﻤﺎ ﺃﺣﺮﺯ ﻓﺮﻳﻖ ﺍﻟﻔﻬﻮﺩ‬ ‫‪1 3 9 27 81‬‬ ‫‪1 ‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‪:‬‬ ‫‪30 31 32 33 34‬‬ ‫ﺃﻃﻮﺍﻝ ﺃﺿﻼﻉ ﺍﳌﺮﺑﻌﺎﺕ ﻫﻲ‪ 1, 2, 3 :‬ﻭﺣﺪﺍ ﹴﺕ‪.‬‬ ‫‪  ‬ﺃﻱ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ ﺗﻨﺘﺞ ﻣﻨﻄﻘ ﹰﹼﻴﺎ ﻋﻦ‬ ‫‪2  1-2‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪:‬‬ ‫‪2 ‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪:‬‬ ‫ﻛﻞ ﻭﺍﺣﺪ ﻣﻦ ﻫﺬﻩ ﺍﻷﻋﺪ‪q‬ﺍ~ﺩ ﻫﻮ‪p‬ﻗﻮﺓ ﻟﻠﻌ‪q‬ﺪ~ﺩ ‪ p q .3‬ﺳﻴﻜﻮﻥ ﻃﻮﻝ ﺿﻠﻊ ﺍﳌﺮﺑﻊ ﰲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﱄ ‪ 4‬ﻭﺣﺪﺍ‪‬ﺕ‪1-4،‬‬ ‫ﺇﺫﻥ ﺳﻴﻜﻮﻥ ﰲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﱄ ‪16‬ﺇﺫﻣﺍﺮﺑﻛ ﹰﻨﻌﺎﺖ ﺃﺻﻐﺣ ﹰﺪﲑﺍ‪.‬ﻃﻼﺏ ﺍﻟﻤﺮﺣﻠﺔ ﺍﻟﺜﺎﻧﻮﻳﺔ‪ ،‬ﻓﺈﻥ ﻋﻤﺮﻙ ‪ 16‬ﺳﻨﺔ ﻋﻠﻰ‬ ‫ﺇﺫﻥ ﺳﻴﻜﻮﻥ ﺍﻟﻌﺪﺩ ﺍﻟﺘﺎﱄ ‪35‬؛ ﺃ ﹾﻱ ‪243‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﻋﻤﺮﻙ ‪ 16‬ﺳﻨﺔ ﻋﻠﻰ ﺍﻷﻗﻞ‪ ،‬ﻓﺈﻥ ﻋﻤﺮﻙ ﻳﺆ ﱢﻫﻠﻚ ﻟﻘﻴﺎﺩﺓ‬ ‫‪‬‬ ‫ﻠﻚ ﻟﻘﻴﺎﺩﺓ ﺍﻟﺴﻴﺎﺭﺓ‪ ،‬ﻓﺈﻧﻚ ﺃﺣﺪ ﻃﻼﺏ‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪:‬‬ ‫‪-80     -2           -5, 10, -20, 40,… (1‬‬ ‫ﻠﻚ ﻟﻘﻴﺎﺩﺓ ﺍﻟﺴﻴﺎﺭﺓ‪ ،‬ﻓﺄﻧﺖ ﻓﻲ ﺍﻟﻤﺮﺣﻠﺔ‬ ‫‪10000   1-3     10     1, 10, 100, 1000,… (2‬‬ ‫‪_95_    ‬‬ ‫‪   _51_     ‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫_‪_6‬‬ ‫‪,‬‬ ‫_‪_7‬‬ ‫‪,‬‬ ‫…‪_85_ ,‬‬ ‫‪(3‬‬ ‫‪5‬‬ ‫‪5‬‬ ‫ﺇﺫﺍ ﻛﻨﺖ ﺃﺣﺪ ﻃﻼﺏ ﺍﻟﻤﺮﺣﻠﺔ ﺍﻟﺜﺎﻧﻮﻳﺔ‪ ،‬ﻓﺈﻥ ﻋﻤﺮﻙ ﻳﺆ ﱢﻫﻠﻚ ﻟﻘﻴﺎﺩﺓ‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣ ﹼﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛ ﱠﻢ ﺃﻋﻂ ﺃﻣﺜﻠﺔ ﻋﺪﺩﻳﺔ‪ ،‬ﺃﻭ ﺍﺭﺳﻢ ﺃﺷﻜﺎ ﹰﻻ ﺗﺆﻳﺪ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪   4-7    ‬ﺇﺫ‪‬ﺍ‪‬ﻛ‪‬ﺎ‪‬ﻥ‪‬ﻋ‪‬ﻤ‪‬ﺮ‪‬ﻙ‪ 16‬ﺳﻨﺔ ﻋﻠﻰ ﺍﻷﻗﻞ‪ ،‬ﻓﺈﻧﻚ ﺃﺣﺪ ﻃﻼﺏ ﺍﻟﻤﺮﺣﻠﺔ‬ ‫‪ (5‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ‪∠ 1‬ﻭ ‪∠2‬ﺗﻜ ﹼﻮﻧﺎﻥ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪.‬‬ ‫‪A (-1, -1), B (2,2), C (4,4) (4‬‬ ‫ﺃﺣﻴﺎ ﹰﻧﺎ‬ ‫ﺻﺤﻴﺤﺔ‬ ‫ﺃﻭ‬ ‫ﺻﺤﻴﺤﺔ ﺩﺍﺋ ﹰﻤﺎ‬ ‫ﻳﺄﺗﻲ‬ ‫ﻣﻤﺎ‬ ‫ﺟﻤﻠﺔ‬ ‫ﻛﻞ‬ ‫ﻛﺎﻧﺖ‬ ‫ﺇ‪P‬ﺫﺍ‬ ‫ﻣﺎ‬ ‫ﺩ‬ ‫∠‪  2  ∠1‬‬ ‫‪      A, B, C ‬‬ ‫‪ 1-5‬‬ ‫‪ 1-3‬‬ ‫)‪y C(4, 4‬‬ ‫ﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪ ،‬ﻭﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﻓﻲ‬ ‫‪12‬‬ ‫‪R‬‬ ‫)‪B(2, 2‬‬ ‫‪.R , S‬‬ ‫‪T‬‬ ‫‪W‬‬ ‫‪A(–1, –1) O‬‬ ‫‪x‬‬ ‫‪ (7‬ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪.‬‬ ‫‪ ∠ABC (6‬ﻭ ‪ ∠DBE‬ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﺑﻠﺘﺎﻥ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫‪     ‬‬ ‫‪  ∠DBE  ∠ABC‬‬ ‫‪23 - 9 = 14, 15 - 7 = 8‬‬ ‫‪A BE‬‬ ‫‪1‬‬ ‫‪6‬‬ ‫‪CD‬‬ ‫‪  ‬‬

  ‫ ﻭﺑﺪﺍﺋﻞ‬،‫ ﻭﺇﻋﺎﺩﺓ ﺍﻟﺘﻌﻠﻴﻢ ﻭﺍﻟﺘﻌﺰﻳﺰ‬،‫ﺗﺘﻤ ﹼﻴﺰ ﺍﻟﺴﻠﺴﻠﺔ ﺑﺄﻧﻬﺎ ﻧﻤﻮﺫﺝ ﺗﻌﻠﻴﻢ ﻗﻮﻱ ﻳﺸﺘﻤﻞ ﻋﻠﻰ ﺑﺪﺍﺋﻞ ﺗﻨﻮﻳﻊ ﺍﻟﺘﻌﻠﻴﻢ‬ ،‫ ﻛﻤﺎ ﻳﺸﺘﻤﻞ ﻋﻠﻰ ﻧﺸﺎﻃﺎﺕ ﻗﺒﻠﻴﺔ ﻣﺘﻘﺪﻣﺔ‬،‫ ﻭﺇﺭﺷﺎﺩﺍﺕ ﻟﻠﻤﻌﻠﻢ ﺗﺴﺎﻋﺪﻩ ﻋﻠﻰ ﺗﻌ ﹼﺮﻑ ﻣﺴﺘﻮﻳﺎﺕ ﺍﻟﻄﻼﺏ‬،‫ﻟﻠﺘﻮﺳﻊ‬ .‫ﻭﺗﻘﻮﻳﻢ ﻣﺼﺎﺣﺐ ﻟﻠﺘﻌﻠﻴﻢ‬        2 1   1 7  4  ‫ﻳﺴﺎﻋﺪﻙ ﻣﺨﻄﻂ ﺍﻟﻔﺼﻞ ﻋﻠﻰ ﺍﻟﺘﺨﻄﻴﻂ‬ ‫ﻟﻠﺘﻌﻠﻴﻢ ﻣﻦ ﺧﻼﻝ ﺗﻮﺿﻴﺢ ﺍﻷﻫﺪﺍﻑ‬  1-8  1-7  1-6  1-5  1-4  1-3  1-3   ✓(11) ‫ ﻭﺍﻟﺘﻐﻄﻴﺔ‬،‫ﻭﺍﻟﺨﻄﺔ ﺍﻟﺰﻣﻨﻴﺔ ﺍﻟﻤﻘﺘﺮﺣﺔ‬    .‫ﺍﻟﺸﺎﻣﻠﺔ ﻟﻸﻓﻜﺎﺭ ﺍﻟﻤﺤﻮﺭﻳﺔ‬      1-2   1-1        •  •  •    •  •  •  •  •  •          •   •      •  •   •  •   •       •                                      73  64  58      33        •        (11)        •  •  •  •  •  (26)  (21)  (16) (13)  (6)   •  •   •  •   •  (41)  (36)  (31)  •  • (18) (14)  • (28) (23)   •  • (8)  •  •  • (19)  (15) (43) (38) (33)  •  •  •   • (29) (24)  (20) (9)   •   •  • (44) (39) (34)   •  •   •  (30)  (25)  (10)   •  •   •  (45)  (40)  (35)      (10) • (9) • (8) • (7) • (6) •    (13) • (12) • (11) • 6 8  6 2  5 5  4 6  4 0  2 8  2 0  1 3   70 , 72  61 , 65  58   49 , 51    39 , 44   33 , 34  23 , 25  14 , 15 , 18      ✓   ✓         • (52)   1 10A (74-78)  (79)  • 10B  1       1 ،‫ﹸﺑﻨﻴﺖ ﺍﻟﻤﻮﺍﺿﻴﻊ ﺍﻟﺪﺭﺍﺳﻴﺔ ﻋﻠﻰ ﺍﻟﻤﻔﺎﻫﻴﻢ ﻭﺍﻟﻤﻬﺎﺭﺍﺕ ﺍﻟﺴﺎﺑﻘﺔ ﻟﻠﺼﻒ ﺍﻟﻤﻌﻨﻲ‬  1-1 .‫ﻭﺗﺆﺳﺲ ﻟﻤﻮﺍﺿﻴﻊ ﻣﺴﺘﻘﺒﻠﻴﺔ‬ ‫ ﻭﺍﻟﺘﺒﺮﻳﺮ‬،‫ﺍﻟﺘﺨﻤﻴﻦ ﻫﻮ ﺗﻮ ﱡﻗﻊ ﻣﺪﺭﻭﺱ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ ﻣﻌﻠﻮﻣﺎﺕ ﻣﻌﺮﻭﻓﺔ‬ ‫ ﻭﻭﺣﺪﺍﺕ‬،‫• ﺍﻟﺘﻌﺒﻴﺮ ﻋﻦ ﺍﻷﻓﻜﺎﺭ ﺍﻟﺮﻳﺎﺿﻴﺔ ﻟﻐﻮ ﹰﹼﻳﺎ ﻭﺑﺄﺩﻭﺍﺕ ﻓ ﹼﻌﺎﻟﺔ‬ ‫ ﻭﺇﺫﺍ‬.‫ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻫﻮ ﺗﻔﺤﺺ ﻟﻌﺪﺓ ﺃﻭﺿﺎﻉ ﺧﺎﺻﺔ ﻟﻠﻮﺻﻮﻝ ﺇﻟﻰ ﺍﻟﺘﺨﻤﻴﻦ‬ ‫ ﻭﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﺒﻴﺎﻧﻴﺔ ﺃﻭ ﺍﻟﺮﻳﺎﺿﻴﺔ ﺃﻭ ﺍﻟﻌﺪﺩﻳﺔ ﺃﻭ ﺍﻟﻤﺎﺩﻳﺔ‬،‫ﻣﻨﺎﺳﺒﺔ‬ ‫ ﻭ ﹸﻳﺪﻋﻰ ﺍﻟﻤﺜﺎﻝ ﻓﻲ‬،‫ ﻓﺈﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃﺊ‬،‫ﻧﺎﻗﺾ ﻣﺜﺎﻝ ﻭﺍﺣﺪ ﺍﻟﺘﺨﻤﻴﻦ‬ .‫ﺃﻭ ﺍﻟﺠﺒﺮﻳﺔ‬ .‫ﻫﺬﻩ ﺍﻟﺤﺎﻟﺔ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‬ ‫• ﺇﺛﺒﺎﺕ ﺻﺤﺔ ﺍﻻﺳﺘﻨﺘﺎﺟﺎﺕ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺨﺼﺎﺋﺺ ﻭﺍﻟﻌﻼﻗﺎﺕ‬  1-2 .‫ﺍﻟﺮﻳﺎﺿﻴﺔ‬ ‫ ﻭﻻ‬،‫ﺍﻟﻌﺒﺎﺭﺓ ﻫﻲ ﺟﻤﻠﺔ ﺧﺒﺮﻳﺔ ﺇﻣﺎ ﺃﻥ ﺗﻜﻮﻥ ﺻﺤﻴﺤﺔ ﺃﻭ ﺗﻜﻮﻥ ﺧﺎﻃﺌﺔ‬ 1 ‫ ﻭ ﹸﺗﺴﻤﻰ ﺻﺤﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺃﻭ ﺧﻄﺆﻫﺎ ﻗﻴﻤ ﹶﺔ ﺍﻟﺼﻮﺍﺏ‬.￯‫ﺗﺤﺘﻤﻞ ﺃﻱ ﺣﺎﻟﺔ ﺃﺧﺮ‬ .‫• ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻟﻜﺘﺎﺑﺔ ﺗﺨﻤﻴﻦ‬ ‫ ﻭﻋﻜﺴﻬﺎ ﻭﻣﻌﻜﻮﺳﻬﺎ‬،‫• ﺗﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻌﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ‬ ‫ ﻭﻟﺬﻟﻚ ﻓﺈﻥ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻨﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ﻫﻮ ﻋﻜﺲ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ‬.‫ﻟﻬﺎ‬ ،‫\" ﻫﻮ ﻧﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ‬p ‫ ﻓﺈﻥ \"ﻟﻴﺲ‬، p ‫ ﻭﺇﺫﺍ ﺭﻣﺰﻧﺎ ﻟﻌﺒﺎﺭﺓ ﺑﺎﻟﺮﻣﺰ‬.‫ﻟﻠﻌﺒﺎﺭﺓ‬ .‫ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻬﺎ‬ .‫• ﺍﺳﺘﻌﻤﺎﻝ ﻗﺎﻧﻮ ﹶﻧﻲ ﺍﻟﻔﺼﻞ ﻭﺍﻟﻘﻴﺎﺱ ﺍﻟﻤﻨﻄﻘﻲ ﻟﻠﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻨﺘﺎﺟﻲ‬ . p ‫ﻭ ﹸﻳﺮﻣﺰ ﻟﻪ ﺑﺎﻟﺮﻣﺰ‬ ‫• ﺍﺳﺘﻌﻤﺎﻝ ﺗﻌﺮﻳﻔﺎﺕ ﺃﻭ ﺧﺼﺎﺋﺺ ﺟﺒﺮﻳﺔ ﺃﻭ ﻣﺴﻠﻤﺎﺕ ﺃﻭ ﻧﻈﺮﻳﺎﺕ‬ .‫ ﻭﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ ﻟﺘﻔﻨﻴﺪ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺨﻄﺄ‬،‫ﻹﺛﺒﺎﺕ ﺻﺤﺔ ﻋﺒﺎﺭﺍﺕ‬ ‫ ﻭﺇﺫﺍ ﺍﺳﺘﻌﻤﻠﺖ ﺃﺩﺍﺓ‬.‫ﻭﻳﻤﻜﻦ ﺭﺑﻂ ﻋﺒﺎﺭﺗﻴﻦ ﺃﻭ ﺃﻛﺜﺮ ﻟﺘﻜﻮﻳﻦ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ‬ ‫ ﻓﺈﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﻨﺎﺗﺠﺔ ﺗﺴﻤﻰ \"ﻋﺒﺎﺭﺓ‬،\"∧\" ‫ﺍﻟﺮﺑﻂ \"ﻭ\" ﻭﺭﻣﺰﻫﺎ‬ 1  ‫ ﻓﺈﻥ ﺍﻟﻌﺒﺎﺭﺓ‬،\" ∨\" ‫ ﺃﻣﺎ ﺇﺫﺍ ﺍﺳﺘﻌﻤﻠﺖ ﺃﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﺃﻭ\" ﻭﺭﻣﺰﻫﺎ‬.\"‫ﺍﻟﻮﺻﻞ‬ .‫• ﺍﻟﻤﻘﺎﺭﻧﺔ ﺑﻴﻦ ﺍﻟﺤﻠﻮﻝ ﺍﻟﺠﺒﺮﻳﺔ ﻭﺍﻟﺒﻴﺎﻧﻴﺔ ﻟﻤﻌﺎﺩﻻﺕ ﺗﺮﺑﻴﻌﻴﺔ ﻭﺗﻔﺴﻴﺮﻫﺎ‬ ‫ ﻭﻳﻤﻜﻦ ﺗﻮﺿﻴﺢ ﻋﺒﺎﺭ ﹶﺗﻲ ﺍﻟﻔﺼﻞ‬.\"‫ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﻨﺎﺗﺠﺔ ﹸﺗﺴﻤﻰ \"ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ‬ ‫ ﻭﺻﻴﺎﻏﺔ‬،‫• ﺗﺤﻠﻴﻞ ﻣﻮﺍﻗﻒ ﺭﻳﺎﺿﻴﺔ ﻣﻤﺜﻠﺔ ﺑﺪﻭﺍﻝ ﺍﻟﺠﺬﺭ ﺍﻟﺘﺮﺑﻴﻌﻲ‬ .‫ﻣﻌﺎﺩﻻﺕ ﺃﻭ ﻣﺘﺒﺎﻳﻨﺎﺕ ﻭﺍﺧﺘﻴﺎﺭ ﻃﺮﻳﻘﺔ ﻭﺣﻞ ﺍﻟﻤﺴﺎﺋﻞ‬ :‫ﻭﺍﻟﻮﺻﻞ ﺑﺄﺷﻜﺎﻝ ﭬﻦ ﻛﻤﺎ ﻳﻠﻲ‬  •   .‫ﻭﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻳﻤﻜﻦ ﺃﻥ ﺗﺴﺎﻋﺪ ﻋﻠﻰ ﺇﻳﺠﺎﺩ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ‬  • p ∼p pq p∨q pq p∧q TF TT T TT T FT TF T TF F FT T FT F FF F FF F ‫ ﺇﺫﺍ‬،‫ﻓﻲ ﺣﺎﻟﺔ ﺍﻟﻨﻔﻲ‬ ‫ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ‬ ‫ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ‬ ‫ ﻓﺈﻥ‬،‫ ﺻﺤﻴﺤﺔ‬p ‫ﻛﺎﻧﺖ‬ ‫ ﻋﻨﺪﻣﺎ‬،‫ﺧﺎﻃﺌﺔ ﻓﻘﻂ‬ ‫ ﻋﻨﺪﻣﺎ‬،‫ﺻﺤﻴﺤﺔ ﻓﻘﻂ‬ ‫ ﻭﺇﺫﺍ ﻛﺎﻧﺖ‬.‫∼ ﺧﺎﻃﺌﺔ‬p q ‫ ﻭ‬p ‫ﺗﻜﻮﻥ ﻛ ﱞﻞ ﻣﻦ‬ q ‫ ﻭ‬p ‫ﺗﻜﻮﻥ ﻛ ﱞﻞ ﻣﻦ‬ ∼p ‫ ﻓﺈﻥ‬،‫ ﺧﺎﻃﺌﺔ‬p .‫ﺧﺎﻃﺌﺔ‬ .‫ﺻﺤﻴﺤﺔ‬ .‫ﺻﺤﻴﺤﺔ‬ ‫ ﻋﻨﺪﻣﺎ‬،‫ﺗﺒﻴﻦ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺃﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﺗﻜﻮﻥ ﺻﺤﻴﺤﺔ ﻓﻘﻂ‬ ‫ ﺃﻣﺎ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ﻓﺘﻜﻮﻥ ﺻﺤﻴﺤﺔ ﺩﺍﺋ ﹰﻤﺎ ﺇ ﹼﻻ‬.‫ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ﺻﺤﻴﺤﺘﻴﻦ‬ .‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ﺧﺎﻃﺌﺘﻴﻦ‬  1 10E  T11

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪Inductive Reasoning and Conjection‬‬ ‫‪‬‬ ‫ﺗﻨﻈﻢ ﺧﻄﺔ ﺍﻟﺨﻄﻮﺍﺕ ﺍﻷﺭﺑﻌﺔ ﺗﺪﺭﻳﺴﻚ ﻭﺗﺘﻀﻤﻦ‪:‬‬ ‫‪   3‬‬ ‫‪‬‬ ‫‪  (11 3‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺍﻟﺬﻱ ﻳﺒﻴﻦ ﻋﺪﺩ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 1‬‬ ‫ﺍﻟﻘﻄﻊ ﺍﻟﻤﻨﺘﺠﺔ ﻓﻲ ﻣﺼﻨﻊ ﻟﺒﻌﺾ ﺍﻟﺴﻨﻮﺍﺕ‪.‬‬ ‫ﻓﻲ ﺃﺑﺤﺎﺙ ﺍﻟﺘﺴﻮﻳﻖ‪ ،‬ﻳﺘﻢ ﺗﺤﻠﻴﻞ ﺇﺟﺎﺑﺎﺕ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺷﺨﺎﺹ ﻋﻦ ﺃﺳﺌﻠﺔ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ ‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪‬‬ ‫ﻣﺤﺪﺩﺓ ﺣﻮﻝ ﺍﻟﻤﻨﺘﺞ‪ ،‬ﺛﻢ ﻳﺘﻢ ﺍﻟﺒﺤﺚ ﻋﻦ ﻧﻤﻄﻴﺔ ﻣﻌﻴﻨﺔ ﻓﻲ ﺍﻹﺟﺎﺑﺎﺕ ﺣﺘﻰ‬ ‫‪ ‬‬ ‫‪1-1‬‬ ‫ﺍﻟﻮﺻﻮﻝ ﺇﻟﻰ ﻧﺘﻴﺠﺔ‪ .‬ﻭﺗﺴﻤﻰ ﻫﺬﻩ ﺍﻟﻌﻤﻠﻴﺔ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ‪.‬‬ ‫ﺗﻤﺜﻴﻞ ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ ﺍﻟﻜﻤﻴﺎﺕ‪،‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻌﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﺳﻨﺔ ‪2017‬ﻡ ‪.‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺑﺎﺳﺘﻌﻤﺎﻝ ﻧﻤﺎﺫﺝ ﺣﺴﻴﺔ ﻭﺟﺪﺍﻭﻝ ‪،‬‬ ‫ﺳﻴﻜﻮﻥ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﻋﺎﻡ ‪ 2017‬ﻧﺤﻮ ‪ 35‬ﻣﻠﻴﻮ ﹰﻧﺎ‪.‬‬ ‫‪ ‬ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻫﻮ ﺗﺒﺮﻳﺮ ﹸﺗﺴﺘﻌﻤﻞ ﻓﻴﻪ ﺃﻣﺜﻠﺔ ﻣﺤﺪﺩﺓ ﻟﻠﻮﺻﻮﻝ‬ ‫‪‬‬ ‫ﻭﺗﻤﺜﻴﻼﺕ ﺑﻴﺎﻧﻴﺔ ﻭﻣﺨﻄﻄﺎﺕ‪ ،‬ﻭﻭﺻﻒ‬ ‫‪5 2007‬‬ ‫‪ 3‬‬ ‫‪‬‬ ‫ﺇﻟﻰ ﻧﺘﻴﺠﺔ‪ .‬ﻭﻋﻨﺪﻣﺎ ﺗﻔﺘﺮﺽ ﺍﺳﺘﻤﺮﺍﺭ ﻧﻤﻂ ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﻮﺗﻴﺮﺓ‪ ،‬ﻓﺈﻧﻚ ﺗﺴﺘﻌﻤﻞ‬ ‫ﻟﻔﻈﻲ ‪ ،‬ﻭﻣﻌﺎﺩﻻﺕ‪.‬‬ ‫✓ ‪‬‬ ‫‪ ‬‬ ‫ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ‪ ،‬ﻭ ﹸﺗﺴ ﹼﻤﻰ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻨﻬﺎﺋﻴﺔ ﺍﻟﺘﻲ ﺗﻮﺻﻠﺖ ﺇﻟﻴﻬﺎ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪1-1‬‬ ‫‪7.2 2008‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪1-13‬؛ ﻟﻠﺘﺄﻛﺪ ﻣﻦ ﻓﻬﻢ‬ ‫‪12345‬‬ ‫‪‬‬ ‫ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﺗﺨﻤﻴﻨﹰﺎ ‪.‬‬ ‫ﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ‬ ‫‪9.2 2009‬‬ ‫ﺍﻟﻄﻠﺒﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺳﻔﻞ ﻫﺬﻩ‬ ‫‪qqqqq‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻟﻜﺘﺎﺑﺔ‬ ‫‪14.1 2010‬‬ ‫ﺍﻟﺼﻔﺤﺔ؛ ﻟﺘﻌﻴﻴﻦ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻠﻄﻠﺒﺔ‬ ‫‪qqqqq‬‬ ‫‪‬‬ ‫‪inductive reasoning‬‬ ‫‪19.7 2011‬‬ ‫‪qqqqq‬‬ ‫‪‬‬ ‫ﺗﺨﻤﻴﻦ‪.‬‬ ‫‪28.4 2012‬‬ ‫ﺑﺤﺴﺐ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫‪qqqqq‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪1-1‬‬ ‫‪qqqqq‬‬ ‫‪‬‬ ‫‪conjecture‬‬ ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻟﻤﻨﻄﻘﻲ ﻹﺛﺒﺎﺕ ﺻﺤﺔ‬ ‫‪qqqqq‬‬ ‫‪‬‬ ‫ﻋﺒﺎﺭﺍﺕ ﻭﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ‪.‬‬ ‫‪counterexample‬‬ ‫‪4 ‬‬ ‫‪ q‬‬ ‫‪q‬‬ ‫‪‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪ 2‬‬ ‫‪ 3‬‬ ‫‪   1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 4‬‬ ‫‪   2‬‬ ‫‪‬‬ ‫ﺃﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ ﻳﺒﻴﻦ ﺃﻥ ﻛ ﹼﹰﻼ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻗﺮﺍﺀﺓ ﻓﻘﺮﺓ ”ﻟﻤﺎﺫﺍ؟“ ‪.‬‬ ‫‪ (12‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠A‬ﻭ ‪ ∠B‬ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻟﻬﻤﺎ ﺿﻠ ﹰﻌﺎ ﻣﺸﺘﺮ ﹰﻛﺎ‪.‬‬ ‫‪‬‬ ‫‪ (13‬ﺇﺫﺍ ﻗﻄﻊ ﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ ﻗﻄﻌ ﹰﺔ ﻣﺴﺘﻘﻴﻤ ﹰﺔ ﻋﻨﺪ ﻣﻨﺘﺼﻔﻬﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻌﺎﻣﺪﻫﺎ‪.‬‬ ‫‪4 ‬‬ ‫‪(12‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫• ﻣﺎ ﺍﻷﺷﻴﺎﺀ ﺍﻟﺘﻲ ﺗﻬﻢ ﺑﺎﺣﺚ ﺍﻟﺘﺴﻮﻳﻖ؟‬ ‫‪‬‬ ‫‪  ‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻣﺒﻴﻌﺎﺕ ﺍﻟﻤﻨﺘﺞ‪ ،‬ﻣﻘﺎﺭﻧﺘﻪ‬ ‫‪‬‬ ‫ﺑﺎﻟﻤﻨﺘﺠﺎﺕ ﺍﻟﻤﻨﺎﻓﺴﺔ‪.‬‬ ‫‪‬‬ ‫• ﻟﻤﺎﺫﺍ ﻳﻘﻮﻡ ﺍﻟﺒﺎﺣﺚ ﺑﺘﻮﺟﻴﻪ ﺍﻷﺳﺌﻠﺔ ﺇﻟﻰ‬ ‫‪‬‬ ‫‪‬‬ ‫‪45°‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺷﺨﺎﺹ ﻓﻘﻂ؟‬ ‫‪A B 45°‬‬ ‫‪‬‬‫‪‬‬ ‫‪  1‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻓﻲ ﻛﺜﻴﺮ ﻣﻦ ﺍﻷﺣﻴﺎﻥ‪،‬‬ ‫ﻳﺼﻌﺐ ﺗﻮﺟﻴﻪ ﺍﻷﺳﺌﻠﺔ ﺇﻟﻰ ﺟﻤﻴﻊ‬ ‫‪‬‬ ‫‪‬‬‫‪‬‬‫‪‬‬ ‫ﺍﻟﻤﺴﺘﻬﻠﻜﻴﻦ‪ ،‬ﻭﻟﺬﻟﻚ ﺗﻮ ﱠﺟﻪ ﺍﻷﺳﺌﻠﺔ ﺇﻟﻰ‬ ‫‪  ‬‬ ‫‪ ‬‬‫‪ ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﻣﻤﺜﻠﺔ‪.‬‬ ‫‪‬‬ ‫‪ (14–19‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪‬‬ ‫‪ (a‬ﻣﻮﺍﻋﻴﺪ ﻭﺻﻮﻝ ﺍﻟﺤﺎﻓﻼﺕ ﺇﻟﻰ ﻣﺤﻄﺔ ﺍﻟﺮﻛﻮﺏ ﻫﻲ‪ 8:30 :‬ﺻﺒﺎ ﹰﺣﺎ‪ 9:10 ،‬ﺻﺒﺎ ﹰﺣﺎ‪ 9:50 ،‬ﺻﺒﺎ ﹰﺣﺎ‪ 10:30 ،‬ﺻﺒﺎ ﹰﺣﺎ‪...... ،‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫‪1 ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ 1‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‪.‬‬ ‫‪4, 8, 12, 16, 20 (16‬‬ ‫‪3, 6, 9, 12, 15 (15‬‬ ‫‪0, 2, 4, 6, 8 (14‬‬ ‫ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺃﻧﻪ ﻓﻲ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺘﻲ‬ ‫ﺗﺘﻀﻤﻦ ﺑﻴﺎﻧﺎﺕ ﻣﻦ ﻭﺍﻗﻊ ﺍﻟﺤﻴﺎﺓ‪ ،‬ﻟﻴﺲ‬ ‫‪ 8:30‬ﺻﺒﺎ ﹰﺣﺎ‪ 9:10 ،‬ﺻﺒﺎ ﹰﺣﺎ‪ 9:50 ،‬ﺻﺒﺎ ﹰﺣﺎ‪ 10:30 ،‬ﺻﺒﺎ ﹰﺣﺎ ‪......‬‬ ‫‪‬‬ ‫‪_21 ,‬‬ ‫‪_41 ,‬‬ ‫‪_1‬‬ ‫ﺑﺎﻟﻀﺮﻭﺭﺓ ﺃﻥ ﻳﻤ ﹼﺜﻞ ﺍﻟﺘﺨﻤﻴﻦ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ‬ ‫‪1,‬‬ ‫‪8‬‬ ‫‪(19‬‬ ‫‪1, 4, 9, 16 (18‬‬ ‫‪2, 22, 222, 2222 (17‬‬ ‫ﺍﻟﻨﻤﻂ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻣﺎ ﻳﻤﻜﻦ ﺃﻥ ﻳﺤﺪﺙ‬ ‫‪40  40 40‬‬ ‫‪‬‬ ‫‪ (20‬ﻣﻮﺍﻋﻴﺪ ﺍﻟﻮﺻﻮﻝ‪ 10:00 :‬ﺻﺒﺎ ﹰﺣﺎ‪ 12:30 ،‬ﻣﺴﺎ ﹰﺀ ‪ 3:00 ،‬ﻣﺴﺎ ﹰﺀ‪ (20 ...... ،‬ﻳﺄﺗﻲ ﻛﻞ ﻣﻮﻋﺪ ﺑﻌﺪ ﺳﺎﻋﺘﻴﻦ ﻭﻧﺼﻒ‬ ‫ﻓﻲ ﺍﻟﻤﺴﺘﻘﺒﻞ‪.‬‬ ‫‪ 2‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪.‬‬ ‫ﺍﻟﺴﺎﻋﺔ ﻣﻦ ﺍﻟﻤﻮﻋﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪ 5:30‬ﻣﺴﺎ ﹰﺀ‪.‬‬ ‫‪ (21‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻠﺮﻃﻮﺑﺔ‪100% , 93% , 86% , …… :‬‬ ‫‪ (21‬ﺗﻘﻞ ﻛﻞ ﻧﺴﺒﺔ ﻣﺌﻮﻳﺔ ﻋﻦ‬ ‫ﻓﻤﺜ ﹰﻼ‪ ،‬ﻗﺪ ﹸﺗﺸﻴﺮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫ﻳﺰﻳﺪ ﻣﻮﻋﺪ ﻭﺻﻮﻝ ﺍﻟﺤﺎﻓﻠﺔ ‪ 40‬ﺩﻗﻴﻘﺔ ﻋﻦ ﻣﻮﻋﺪ ﻭﺻﻮﻝ ﺍﻟﺤﺎﻓﻠﺔ ﺍﻟﺘﻲ ﺳﺒﻘﺘﻬﺎ‪ .‬ﻣﻮﻋﺪ ﻭﺻﻮﻝ‬ ‫‪‬‬ ‫ﺍﻟﻨﺴﺒﺔ ﺍﻟﺴﺎﺑﻘﺔ ﺑﻤﻘﺪﺍﺭ‬ ‫ﺇﻟﻰ ﺗﺰﺍﻳﺪ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻓﻲ ﺃﺣﺪ‬ ‫‪ (22‬ﺃﻳﺎﻡ ﺍﻟﻌﻤﻞ‪ :‬ﺍﻷﺣﺪ‪ ،‬ﺍﻟﺜﻼﺛﺎﺀ‪ ،‬ﺍﻟﺨﻤﻴﺲ‪...... ،‬‬ ‫‪ 7%‬؛ ‪.79%‬‬ ‫ﺍﻷﺳﺎﺑﻴﻊ‪ ،‬ﺇ ﹼﻻ ﺃﻥ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻗﺪ‬ ‫ﺍﻟﺤﺎﻓﻠﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺳﻮﻑ ﻳﻜﻮﻥ ‪ 10:30‬ﺻﺒﺎ ﹰﺣﺎ ‪ 40 +‬ﺩﻗﻴﻘﺔ ﺃﻭ ‪ 11:10‬ﺻﺒﺎ ﹰﺣﺎ‪.‬‬ ‫‪ (22‬ﻳﺄﺗﻲ ﻛﻞ ﻳﻮﻡ ﻋﻤﻞ ﺑﻌﺪ‬ ‫ﺗﻨﺨﻔﺾ ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﺬﻱ ﻳﻠﻴﻪ‪.‬‬ ‫‪ (23‬ﺍﺟﺘﻤﺎﻋﺎﺕ ﺍﻟﻨﺎﺩﻱ‪ :‬ﺍﻟﻤﺤ ﹼﺮﻡ‪ ،‬ﺭﺑﻴﻊ ﺃﻭﻝ‪ ،‬ﺟﻤﺎﺩ￯ ﺍﻷﻭﻟﻰ‪ (24–27 ...... ،‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺇﺟﺎﺑﺎﺕ‬ ‫ﻳﻮﻣﻴﻦ ﻣﻦ ﻳﻮﻡ ﺍﻟﻌﻤﻞ‬ ‫‪(b‬‬ ‫‪(25 (24‬‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺍﻟﺴﺒﺖ‪.‬‬ ‫‪ (23‬ﻳﻌﻘﺪ ﻛﻞ ﺍﺟﺘﻤﺎﻉ ﺑﻌﺪ‬ ‫‪ ‬‬ ‫ﺷﻬﺮﻳﻦ ﻣﻦ ﺍﻻﺟﺘﻤﺎﻉ‬ ‫‪4 10‬‬ ‫‪18‬‬ ‫‪28‬‬ ‫‪40 . . . . . .‬‬ ‫‪  ‬‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺭﺟﺐ‪.‬‬ ‫‪......‬‬ ‫‪......‬‬ ‫‪ 1‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪(27‬‬ ‫‪(26‬‬ ‫‪4 10 18 28 40‬‬ ‫‪+6 +8 +10 +12‬‬ ‫‪...... ......‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪6, 8, 10, 12, ......‬‬ ‫‪ 354- 430‬‬ ‫‪  (28‬ﺑﺪﺃ ﻣﺎﺟﺪ ﺗﻤﺎﺭﻳﻦ ﺍﻟﺠﺮﻱ ﺍﻟﺴﺮﻳﻊ ﻗﺒﻞ ﺧﻤﺴﺔ ﺃﻳﺎﻡ‪ .‬ﻓﺮﻛﺾ ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻷﻭﻝ ‪ . 0.5 km‬ﻭﻓﻲ ﺍﻷﻳﺎﻡ‬ ‫‪‬‬ ‫‪(11a‬‬ ‫ﻳﻮﺿﺢ ﺍﻟﺘﺮﺍﺑﻂ ﺍﻟﺮﺃﺳﻲ ﻓﻲ ﺑﺪﺍﻳﺔ ﻛﻞ ﺩﺭﺱ ﺍﻷﻫﺪﺍﻑ‬ ‫ﺍﻟﺜﻼﺛﺔ ﺍﻟﺘﺎﻟﻴﺔ ‪ . 0.75 km, 1 km, 1.25 km‬ﺇﺫﺍ ﺍﺳﺘﻤﺮ ﺗﻤﺮﻳﻨﻪ ﻋﻠﻰ ﻫﺬﺍ ﺍﻟﻨﻤﻂ‪ ،‬ﻓﻤﺎ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻲ ﻳﻘﻄﻌﻬﺎ ﻓﻲ‬ ‫‪30‬‬ ‫‪ 2‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪ :‬ﺗﺰﺩﺍﺩ ﺃﻋﺪﺍﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﻤﻘﺪﺍﺭ ‪6, 8, 10, 12...‬؛ ﻟﺬﺍ ﺳﻴﺰﻳﺪ ﻋﺪﺩ ﺍﻟﻘﻄﻊ‬ ‫‪‬‬ ‫ﺍﻟﺘﻲ ﺗﺆﺩﻱ ﺇﻟﻰ ﻣﺤﺘﻮ￯ ﺍﻟﺪﺭﺱ ﺍﻟﺤﺎﻟﻲ ﻭﺍﻷﻫﺪﺍﻑ‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﻋﻠﻰ ﺳﺎﺑﻘﻪ ﺑﻤﻘﺪﺍﺭ ‪ 2 + 12‬ﺃﻭ ‪ 14‬ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ؛ ﻭﻋﻠﻴﻪ ﻓﺈﻥ‬ ‫‪‬‬ ‫ﺍﻟﺘﻲ ﺗﺘﺒﻌﻪ‪ ،‬ﻭﺍﻟﺬﻱ ﻳﺄﺗﻲ ﻓﻲ ﺇﻃﺎﺭ ﻭﺛﻴﻘﺔ ﺍﻟﻤﺪ￯ ﻭﺍﻟﺘﺘﺎﺑﻊ‬ ‫ﺍﻟﻴﻮﻡ ﺍﻟﺴﺎﺑﻊ؟ ‪2 km‬‬ ‫‪25‬‬ ‫ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﺳﻴﺤﺘﻮﻱ ﻋﻠﻰ ‪ 14 + 40‬ﺃﻭ ‪ 54‬ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ‪.‬‬ ‫‪‬‬ ‫ﻣﻦ ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﺍﻻﺑﺘﺪﺍﺋﻲ ﺇﻟﻰ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻟﺚ‬ ‫ﺿﻊ ﺗﺨﻤﻴ ﹰﻨﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫‪‬‬ ‫ﺍﻟﺜﺎﻧﻮﻱ‪.‬‬ ‫‪ (29‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪2 ‬‬ ‫‪20‬‬ ‫‪  ‬ﺍﺭﺳﻢ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ؛ ﻟﻜﻲ ﺗﺘﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺗﺨﻤﻴﻨﻚ‪ .‬‬ ‫‪ (30‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩ ﻓﻲ ﺍﺛﻨﻴﻦ‪ ،‬ﻣﻀﺎ ﹰﻓﺎ ﺇﻟﻴﻪ ﻭﺍﺣﺪ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪‬‬ ‫‪ (31‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺪﺩﻳﻦ ‪ a‬ﻭ ‪ ، b‬ﺇﺫﺍ ﻛﺎﻥ ‪ .ab = 1‬ﻛ ﱞﻞ ﻣﻨﻬﻤﺎ ﻣﻘﻠﻮﺏ ﺍﻵﺧﺮ‬ ‫‪15‬‬ ‫‪‬‬ ‫___ ___‬ ‫‪‬‬ ‫‪ (32‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ AB‬ﻭﻣﺠﻤﻮﻋﺔ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﺒﻌﺪ ﻣﺴﺎﻓﺎﺕ ﻣﺘﺴﺎﻭﻳﺔ ﻋﻦ ‪ A‬ﻭ ‪ . B‬ﺗﺸﻜﻞ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻤﻨ ﱢﺼﻒ ﻟـ ‪. AB‬‬ ‫‪10‬‬ ‫‪ (33‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻭﺣﺠﻢ ﺍﻟﻬﺮﻡ ﺍﻟﻠﺬﻳﻦ ﻟﻬﻤﺎ ﺍﻟﻘﺎﻋﺪﺓ ﻧﻔﺴﻬﺎ ﻭﺍﻻﺭﺗﻔﺎﻉ ﻧﻔﺴﻪ‪.‬‬ ‫ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻳﺴﺎﻭﻱ ‪ 3‬ﺃﻣﺜﺎﻝ ﺣﺠﻢ ﺍﻟﻬﺮﻡ‪.‬‬ ‫‪5‬‬ ‫‪0‬‬ ‫‪54‬‬ ‫‪00 07 08 09 10 11 12‬‬ ‫‪‬‬ ‫‪ 1 12‬‬ ‫‪ (14‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪2‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪10‬‬ ‫‪ 1 16‬‬ ‫‪ (15‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪3‬‬ ‫‪1-1‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪18‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪(14, 15) • ‬‬ ‫‪‬‬ ‫‪ (16‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪4‬‬ ‫‪(15, 18) •‬‬ ‫‪(14, 15, 18) •‬‬ ‫‪(6) • ‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪24‬‬ ‫‪(6) •‬‬ ‫‪(6) •‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ (17‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺤﺘﻮﻱ ﻋﻠﻰ‬ ‫‪(9)  •‬‬ ‫‪(6) •‬‬ ‫‪(6) •‬‬ ‫‪‬‬ ‫ﺍﻟﺮﻗﻢ ‪ 2‬ﺯﻳﺎﺩﺓ ﻋﻠﻰ ﺃﺭﻗﺎﻡ ﺍﻟﺤﺪ ﺍﻟﺴﺎﺑﻖ‬ ‫‪(10) •‬‬ ‫‪(8) •‬‬ ‫‪(8) •‬‬ ‫‪‬‬ ‫‪43 - 5614 - 38‬‬ ‫‪‬‬ ‫ﻟﻪ؛ ‪22222‬‬ ‫‪(9)  •‬‬ ‫‪(9)  •‬‬ ‫‪(10)  •‬‬ ‫‪43 - 5639 - 4115 - 39‬‬ ‫‪‬‬ ‫‪ (18‬ﻳﻨﺘﺞ ﻛﻞ ﺣﺪ ﻋﻦ ﺗﺮﺑﻴﻊ ﺍﻟﻌﺪﺩ ﺍﻟﻄﺒﻴﻌﻲ‬ ‫ﺍﻟﺬﻱ ﻳﻤ ﱢﺜﻞ ﺗﺮﺗﻴﺒﻪ؛ ‪25‬‬ ‫‪45 - 5639 - 53‬‬ ‫‪‬‬ ‫ﺍﻟﺬﻱ‬ ‫ﺍﻟﺤﺪ‬ ‫ﻧﺼﻒ‬ ‫ﻳﺴ‪1‬ﺎ_ﻭﻱ‬ ‫ﻛﻞ ﺣﺪ‬ ‫‪(19‬‬ ‫‪16‬‬ ‫ﻳﺴﺒﻘﻪ؛‬ ‫‪ 1 12‬‬ ‫‪ 1 16‬‬ ‫‪‬‬ ‫ﻳﺤﺘﻮﻱ ﻛﻞ ﺩﺭﺱ ﻋﻠﻰ ﺃﺳﺌﻠﺔ ﺍﻟﺘﻌﺰﻳﺰ ﻟﺘﺴﺘﻌﻤﻠﻬﺎ ﻓﻲ ﻣﺴﺎﻋﺪﺓ ﺍﻟﻄﻼﺏ ﻋﻠﻰ‬ ‫ﺍﺳﺘﻘﺼﺎﺀ ﺍﻷﻓﻜﺎﺭ ﺍﻟﺮﺋﻴﺴﺔ ﻟﻠﺪﺭﺱ ﻭﻓﻬﻤﻬﺎ‪.‬‬ ‫‪ ‬‬ ‫ﻳﻌ ﱡﺪ ﻛﻞ ﻣﺜﺎﻝ ﺇﺿﺎﻓﻲ ﺍﻧﻌﻜﺎ ﹰﺳﺎ ﻟﻤﺜﺎﻝ ﻓﻲ ﻛﺘﺎﺏ ﺍﻟﻄﺎﻟﺐ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪   3‬‬ ‫‪‬‬ ‫‪  (11 3‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺍﻟﺬﻱ ﻳﺒﻴﻦ ﻋﺪﺩ‬ ‫ﺍﻟﻘﻄﻊ ﺍﻟﻤﻨﺘﺠﺔ ﻓﻲ ﻣﺼﻨﻊ ﻟﺒﻌﺾ ﺍﻟﺴﻨﻮﺍﺕ‪.‬‬ ‫ﺑﻤﺎ ﺃﻥ ﻣﻌﻈﻢ ﺍﻟﺼﻔﻮﻑ ﺗﺸﻤﻞ ﻃﻼ ﹰﺑﺎ ﺫﻭﻱ ﻗﺪﺭﺍﺕ ﻣﺨﺘﻠﻔﺔ‪،‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 3‬‬ ‫ﻓﺈﻥ ﺑﺪﺍﺋﻞ ﺗﻨﻮﻳﻊ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻳﺴﻤﺢ ﻟﻚ ﺑﺘﻌﺪﻳﻞ ﺃﺳﺌﻠﺔ‬ ‫ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ ‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫✓ ‪‬‬ ‫‪5 2007‬‬ ‫ﺍﻟﻮﺍﺟﺐ ﺍﻟﻤﻨﺰﻟﻲ‪.‬‬ ‫‪7.2 2008‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻌﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﺳﻨﺔ ‪2017‬ﻡ ‪.‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪1-13‬؛ ﻟﻠﺘﺄﻛﺪ ﻣﻦ ﻓﻬﻢ‬ ‫‪9.2 2009‬‬ ‫ﺳﻴﻜﻮﻥ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﻋﺎﻡ ‪ 2017‬ﻧﺤﻮ ‪ 35‬ﻣﻠﻴﻮ ﹰﻧﺎ‪.‬‬ ‫ﺍﻟﻄﻠﺒﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺳﻔﻞ ﻫﺬﻩ‬ ‫‪‬‬ ‫ﺍﻟﺼﻔﺤﺔ؛ ﻟﺘﻌﻴﻴﻦ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻠﻄﻠﺒﺔ‬ ‫‪4 ‬‬ ‫‪14.1 2010‬‬ ‫ﺗﻮﻓﺮ ﻧﺸﺎﻃﺎﺕ ﺍﻟﺘﻘﻮﻳﻢ ﺍﻟﺘﻜﻮﻳﻨﻲ ﻃﺮﺍﺋﻖ ﺑﺪﻳﻠﺔ ﻟﺘﺤﺪﻳﺪ‬ ‫‪19.7 2011‬‬ ‫ﺑﺤﺴﺐ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫‪28.4 2012‬‬ ‫ﺍﺳﺘﻴﻌﺎﺏ ﺍﻟﻄﻼﺏ ﻓﻲ ﻧﻬﺎﻳﺔ ﻛﻞ ﺩﺭﺱ؛ ﻣﺜﻞ‪:‬‬ ‫ﺃﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﻳﺒﻴﻦ ﺃﻥ ﻛ ﹰﹼﻼ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪4 ‬‬ ‫‪(12‬‬ ‫• ‪ ‬ﻳﺮﺑﻂ ﺍﻟﻄﻼﺏ ﻣﺎ ﺗﻌﻠﻤﻮﻩ ﻓﻲ ﺍﻟﺪﺭﺱ‬ ‫‪ (12‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠A‬ﻭ ‪ ∠B‬ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻟﻬﻤﺎ ﺿﻠ ﹰﻌﺎ ﻣﺸﺘﺮ ﹰﻛﺎ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (13‬ﺇﺫﺍ ﻗﻄﻊ ﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ ﻗﻄﻌ ﹰﺔ ﻣﺴﺘﻘﻴﻤ ﹰﺔ ﻋﻨﺪ ﻣﻨﺘﺼﻔﻬﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻌﺎﻣﺪﻫﺎ‪.‬‬ ‫‪45°‬‬ ‫ﺍﻟﺤﺎﻟﻲ ﺑﻤﺎ ﺗﻌﻠﻤﻮﻩ ﺳﺎﺑ ﹰﻘﺎ‪.‬‬ ‫‪A B 45°‬‬ ‫‪ (43‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ .2, 4, 16, 256, 65536 :‬ﻳﻤﻜﻦ ﺇﻳﺠﺎﺩ ﻛﻞ ﺣﺪ ﺑﺘﺮﺑﻴﻊ ﺍﻟﺤﺪ ﺍﻟﺴﺎﺑﻖ ﻟﻪ‪،‬‬ ‫‪‬‬‫‪‬‬ ‫• ‪ ‬ﻳﺨ ﹼﻤﻦ ﺍﻟﻄﻼﺏ ﻛﻴﻔﻴﺔ ﺍﺭﺗﺒﺎﻁ ﺍﻟﺪﺭﺱ‬ ‫ﻛﻤﺎ ﻳﻤﻜﻦ ﺇﻳﺠﺎﺩ ﻛﻞ ﺣﺪ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺼﻴﻐﺔ ‪ ، 22n-1‬ﺣﻴﺚ ‪.n ≥ 1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬‫‪‬‬ ‫ﺍﻟﺤﺎﻟﻲ ﺑﺎﻟﺪﺭﺱ ﺍﻟﺘﺎﻟﻲ‪.‬‬ ‫‪ (44‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺧﻄﺄ؛ ﺇﺫﺍ‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫• ‪ ‬ﻳﺬﻛﺮ ﺍﻟﻄﻼﺏ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‬ ‫ﻛ ﱠﻮﻧﺖ ﺍﻟﻨﻘﺎﻁ ﺯﺍﻭﻳﺔ ﻣﺴﺘﻘﻴﻤﺔ ‪   (43‬ﺍﻛﺘﺐ ﻣﺘﺘﺎﺑﻌﺔ ﻋﺪﺩﻳﺔ ﺗﺘﺒﻊ ﺣﺪﻭﺩﻫﺎ ﻧﻤﻄﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ‪ ،‬ﻭﻭﺿﺢ ﺍﻟﻨﻤﻄﻴﻦ‪.‬‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪ ‬‬ ‫ﺍﻟﻤﺴﺘﻌﻤﻠﺔ ﻓﻲ ﺍﻟﻤﺴﺄﻟﺔ‪.‬‬ ‫‪ ‬ﺗﺄ ﹼﻣﻞ ﺍﻟﺘﺨﻤﻴﻦ‪” :‬ﺇﺫﺍ ﻛﺎﻧﺖ ﻧﻘﻄﺘﺎﻥ ﺗﺒ ﹸﻌﺪﺍﻥ ﺍﻟﻤﺴﺎﻓﺔ ﻧﻔﺴﻬﺎ ﻋﻦ ﻧﻘﻄﺔ ﺛﺎﻟﺜﺔ ﻣﻌﻠﻮﻣﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ‬ ‫‪(44‬‬ ‫ﻳﻜﻮﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﺃﻣﺎ‬ ‫‪ (14–19‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪‬‬ ‫ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ“‪ .‬ﻫﻞ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ ﺃﻡ ﺧﺎﻃﺊ؟ ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫ﺇﺫﺍ ﻟﻢ ﺗﻜﻦ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ‬ ‫‪ ‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫‪1 ‬‬ ‫• ‪ ‬ﻳﺠﺐ ﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﺠﻴﺒﻮﺍ‬ ‫‪‬‬ ‫ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪ ،‬ﻓﻴﻜﻮﻥ‬ ‫‪4, 8, 12, 16, 20 (16‬‬ ‫‪3, 6, 9, 12, 15 (15‬‬ ‫‪0, 2, 4, 6, 8 (14‬‬ ‫ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺃﻧﻪ ﻓﻲ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺘﻲ‬ ‫ﻋﻦ ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻄﻠﻮﺏ‪ ،‬ﻭﻳﺴﻠﻤﻮﺍ ﺍﻹﺟﺎﺑﺔ ﻟﻠﻤﻌﻠﻢ ﻗﺒﻞ‬ ‫ﺗﺘﻀﻤﻦ ﺑﻴﺎﻧﺎﺕ ﻣﻦ ﻭﺍﻗﻊ ﺍﻟﺤﻴﺎﺓ‪ ،‬ﻟﻴﺲ‬ ‫‪  (45‬ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﹸﺗﺠﺮﻱ ﻣﺴ ﹰﺤﺎ‪ .‬ﺍﺧﺘﺮ ﻣﻮﺿﻮ ﹰﻋﺎ ﻭﺍﻛﺘﺐ ﺛﻼﺛﺔ ﺃﺳﺌﻠﺔ ﻳﺘﻀﻤﻨﻬﺎ ﻣﺴ ﹸﺤﻚ‪ .‬ﻛﻴﻒ ﺗﺴﺘﻌﻤﻞ‬ ‫ﺍﻟﺘﺨﻤﻴﻦﺧﻄ ﹰﺄ‪.‬‬ ‫‪ 4‬‬ ‫‪1,‬‬ ‫‪_12 ,‬‬ ‫‪_14 ,‬‬ ‫‪_1‬‬ ‫‪(19‬‬ ‫‪1, 4, 9, 16 (18‬‬ ‫‪2, 22, 222, 2222 (17‬‬ ‫ﺑﺎﻟﻀﺮﻭﺭﺓ ﺃﻥ ﻳﻤ ﹼﺜﻞ ﺍﻟﺘﺨﻤﻴﻦ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ‬ ‫ﻣﻐﺎﺩﺭﺓ ﺍﻟﺼﻒ‪.‬‬ ‫ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻣﻊ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﺘﻲ ﺗﺤﺼﻞ ﻋﻠﻴﻬﺎ ﻣﻦ ﺧﻼﻝ ﻫﺬﺍ ﺍﻟﻤﺴﺢ؟ ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪C :‬‬ ‫‪8‬‬ ‫ﺍﻟﻨﻤﻂ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻣﺎ ﻳﻤﻜﻦ ﺃﻥ ﻳﺤﺪﺙ‬ ‫‪AB‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻛﺘﺎﺑﺔ‬ ‫‪ (20‬ﻣﻮﺍﻋﻴﺪ ﺍﻟﻮﺻﻮﻝ‪ 10:00 :‬ﺻﺒﺎ ﹰﺣﺎ‪ 12:30 ،‬ﻣﺴﺎ ﹰﺀ ‪ 3:00 ،‬ﻣﺴﺎ ﹰﺀ‪ (20 ...... ،‬ﻳﺄﺗﻲ ﻛﻞ ﻣﻮﻋﺪ ﺑﻌﺪ ﺳﺎﻋﺘﻴﻦ ﻭﻧﺼﻒ‬ ‫ﻓﻲ ﺍﻟﻤﺴﺘﻘﺒﻞ‪.‬‬ ‫ﺧﻤﺴﺔ ﺗﺨﻤﻴﻨﺎﺕ ﺣﻮﻝ ﻧﺸﺎﻃﺎﺕ ﻣﺪﺭﺳﺘﻬﻢ‬ ‫‪‬‬ ‫ﻭﺃﻧﻈﻤﺘﻬﺎ‪ ،‬ﺛﻢ ﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﺃﻥ ﺗﺒﺎﺩﻝ ﺍﻷﻭﺭﺍﻕ‪،‬‬ ‫ﺍﻟﺴﺎﻋﺔ ﻣﻦ ﺍﻟﻤﻮﻋﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪ 5:30‬ﻣﺴﺎ ﹰﺀ‪.‬‬ ‫‪ (21‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻠﺮﻃﻮﺑﺔ‪100% , 93% , 86% , …… :‬‬ ‫‪ (21‬ﺗﻘﻞ ﻛﻞ ﻧﺴﺒﺔ ﻣﺌﻮﻳﺔ ﻋﻦ‬ ‫ﻓﻤﺜ ﹰﻼ‪ ،‬ﻗﺪ ﹸﺗﺸﻴﺮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫ﻭﻣﺤﺎﻭﻟﺔ ﺇﻳﺠﺎﺩ ﻣﺜﺎ ﹴﻝ ﻣﻀﺎ ﱟﺩ ﻟﻜﻞ ﺗﺨﻤﻴﻦ‪.‬‬ ‫ﺍﻟﻨﺴﺒﺔ ﺍﻟﺴﺎﺑﻘﺔ ﺑﻤﻘﺪﺍﺭ‬ ‫ﺇﻟﻰ ﺗﺰﺍﻳﺪ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻓﻲ ﺃﺣﺪ‬ ‫‪ (22‬ﺃﻳﺎﻡ ﺍﻟﻌﻤﻞ‪ :‬ﺍﻷﺣﺪ‪ ،‬ﺍﻟﺜﻼﺛﺎﺀ‪ ،‬ﺍﻟﺨﻤﻴﺲ‪...... ،‬‬ ‫‪ 7%‬؛ ‪.79%‬‬ ‫ﺍﻷﺳﺎﺑﻴﻊ‪ ،‬ﺇ ﹼﻻ ﺃﻥ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻗﺪ‬ ‫‪\" ‬ﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻫﺎﺏ ﺇﻟﻰ ﺍﻟﻤﺪﺭﺳﺔ‬ ‫‪ (22‬ﻳﺄﺗﻲ ﻛﻞ ﻳﻮﻡ ﻋﻤﻞ ﺑﻌﺪ‬ ‫ﺗﻨﺨﻔﺾ ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﺬﻱ ﻳﻠﻴﻪ‪.‬‬ ‫‪ (47‬ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ ‪ ، a = 10 , b = 1‬ﻓﻤﺎ ﻗﻴﻤﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻵﺗﻴﺔ؟ ‪‬‬ ‫‪ (46‬ﺍﻧﻈﺮ ﺇﻟﻰ ﺍﻟﻨﻤﻂ ﺍﻵﺗﻲ‪:‬‬ ‫ﻣﻦ ﺍﻷﺣﺪ ﺇﻟﻰ ﺍﻟﺨﻤﻴﺲ“‪ .‬ﻭ‪‬ﺍﻟ‪‬ﻤﺜﺎ‪‬ﻝ‪‬ﺍ‪‬ﻟ‪‬ﻤ‪‬ﻀ‪‬ﺎ‪‬ﺩ‪ ‬‬ ‫‪ (23‬ﺍﺟﺘﻤﺎﻋﺎﺕ ﺍﻟﻨﺎﺩﻱ‪ :‬ﺍﻟﻤﺤ ﹼﺮﻡ‪ ،‬ﺭﺑﻴﻊ ﺃﻭﻝ‪ ،‬ﺟﻤﺎﺩ￯ ﺍﻷﻭﻟﻰ‪ (24–27 ...... ،‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺇﺟﺎﺑﺎﺕ‬ ‫ﻳﻮﻣﻴﻦ ﻣﻦ ﻳﻮﻡ ﺍﻟﻌﻤﻞ‬ ‫‪_32‬‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺍﻟﺴﺒﺖ‪.‬‬ ‫)‪2b + ab ÷ (a + b‬‬ ‫‪......‬‬ ‫ﻟﻬﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺃﻥ ﻳﻜﻮﻥ ﻳﻮﻡ ﻋﻴﺪ ﺍﻟﻔﻄﺮ ﻳﻮﻡ‬ ‫‪(25 (24‬‬ ‫‪ (23‬ﻳﻌﻘﺪ ﻛﻞ ﺍﺟﺘﻤﺎﻉ ﺑﻌﺪ‬ ‫‪11‬‬ ‫ﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﺍﻟﻨﻤﻂ؟ ‪B‬‬ ‫ﺷﻬﺮﻳﻦ ﻣﻦ ﺍﻻﺟﺘﻤﺎﻉ‬ ‫‪CA‬‬ ‫ﺍﻹﺛﻨﻴﻦ‪ ،‬ﺣﻴﺚ ﺇﺟﺎﺯﺓ ﺍﻟﻤﺪﺍﺭﺱ ﻓﻲ ﺫﻟﻚ‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺭﺟﺐ‪.‬‬ ‫‪ (48‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪،‬‬ ‫‪......‬‬ ‫‪......‬‬ ‫‪ AB‬ﻣﺤﻮﺭ ﺗﻨﺎﻇﺮ ‪ .∠DAC‬ﺃ ﱡﻱ ‪D B‬‬ ‫‪DB‬‬ ‫ﺍﻟﻴﻮﻡ‪.‬‬ ‫‪(27‬‬ ‫‪(26‬‬ ‫ﺍﻻﺳﺘﻨﺘﺎﺟﺎﺕ ﺍﻵﺗﻴﺔ ﻟﻴﺲ‬ ‫ﺻﺤﻴ ﹰﺤﺎ ﺑﺎﻟﻀﺮﻭﺭﺓ؟ ‪C B‬‬ ‫‪...... ......‬‬ ‫‪‬‬ ‫‪A ∠DAB ∠BAC A‬‬ ‫‪  (28‬ﺑﺪﺃ ﻣﺎﺟﺪ ﺗﻤﺎﺭﻳﻦ ﺍﻟﺠﺮﻱ ﺍﻟﺴﺮﻳﻊ ﻗﺒﻞ ﺧﻤﺴﺔ ﺃﻳﺎﻡ‪ .‬ﻓﺮﻛﺾ ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻷﻭﻝ ‪ . 0.5 km‬ﻭﻓﻲ ﺍﻷﻳﺎﻡ‬ ‫‪ ∠DAC B‬ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪.‬‬ ‫ﺍﺟﻤﻊ ﺍﻷﻭﺭﺍﻕ ﻣﻦ ﺍﻟﻄﻼﺏ ‪‬ﻋﻨ‪‬ﺪ‪‬ﺧ‪‬ﺮ‪‬ﻭ‪‬ﺟ‪‬ﻬﻢ‬ ‫ﺍﻟﺜﻼﺛﺔ ﺍﻟﺘﺎﻟﻴﺔ ‪ . 0.75 km, 1 km, 1.25 km‬ﺇﺫﺍ ﺍﺳﺘﻤﺮ ﺗﻤﺮﻳﻨﻪ ﻋﻠﻰ ﻫﺬﺍ ﺍﻟﻨﻤﻂ‪ ،‬ﻓﻤﺎ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻲ ﻳﻘﻄﻌﻬﺎ ﻓﻲ‬ ‫‪‬‬ ‫‪(11a‬‬ ‫ﻣﻦ ﺍﻟﻔﺼﻞ‪.‬‬ ‫‪30‬‬ ‫‪ A C‬ﻭ ‪ D‬ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫ﺍﻟﻴﻮﻡ ﺍﻟﺴﺎﺑﻊ؟ ‪2 km‬‬ ‫‪2(m∠BAC) = m∠DAC D‬‬ ‫ﺿﻊ ﺗﺨﻤﻴ ﹰﻨﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫‪25‬‬ ‫‪ (29‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪ (30‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩ ﻓﻲ ﺍﺛﻨﻴﻦ‪ ،‬ﻣﻀﺎ ﹰﻓﺎ ﺇﻟﻴﻪ ﻭﺍﺣﺪ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪2 ‬‬ ‫‪20‬‬ ‫‪ (31‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺪﺩﻳﻦ ‪ a‬ﻭ ‪ ، b‬ﺇﺫﺍ ﻛﺎﻥ ‪ .ab = 1‬ﻛ ﱞﻞ ﻣﻨﻬﻤﺎ ﻣﻘﻠﻮﺏ ﺍﻵﺧﺮ‬ ‫___ ___‬ ‫‪15‬‬ ‫‪ (32‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ AB‬ﻭﻣﺠﻤﻮﻋﺔ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﺒﻌﺪ ﻣﺴﺎﻓﺎﺕ ﻣﺘﺴﺎﻭﻳﺔ ﻋﻦ ‪ A‬ﻭ ‪ . B‬ﺗﺸﻜﻞ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻤﻨ ﱢﺼﻒ ﻟـ ‪. AB‬‬ ‫‪‬‬ ‫‪ (33‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻭﺣﺠﻢ ﺍﻟﻬﺮﻡ ﺍﻟﻠﺬﻳﻦ ﻟﻬﻤﺎ ﺍﻟﻘﺎﻋﺪﺓ ﻧﻔﺴﻬﺎ ﻭﺍﻻﺭﺗﻔﺎﻉ ﻧﻔﺴﻪ‪.‬‬ ‫‪10‬‬ ‫ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻳﺴﺎﻭﻱ ‪ 3‬ﺃﻣﺜﺎﻝ ﺣﺠﻢ ﺍﻟﻬﺮﻡ‪.‬‬ ‫‪5‬‬ ‫‪  (49‬ﺍﺷﺘﺮ￯ ﺑﺎﺳﻢ ﺣﻮ ﹶﺽ ﺳﻤ ﹴﻚ ﺻﻐﻴﺮ ﻋﻠﻰ ﺷﻜﻞ ﺃﺳﻄﻮﺍﻧﺔ ﺩﺍﺋﺮﻳﺔ ﻗﺎﺋﻤﺔ‪ ،‬ﻃﻮﻝ ﻗﻄﺮ ﻗﺎﻋﺪﺗﻬﺎ ‪ ، 25 cm‬ﻭﺍﺭﺗﻔﺎﻋﻬﺎ ‪،35 cm‬‬ ‫‪  ‬‬ ‫‪0‬‬ ‫ﺃﻭﺟﺪ ﺣﺠﻢ ﺍﻟﻤﺎﺀ ﺍﻟﻼﺯﻡ ﻟﹺﻤﻞ ﹺﺀ ﺍﻟﺤﻮﺽ‪17180.6 cm3  .‬‬ ‫‪00 07 08 09 10 11 12‬‬ ‫‪‬‬ ‫ﺃﻭﺟﺪ ﻣﺤﻴﻂ ‪ ABC‬ﺇﺫﺍ ﹸﺃﻋﻄﻴﺖ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺭﺅﻭﺳﻪ ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ :‬‬ ‫‪ (14‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪2‬‬ ‫‪26.69 A(–3, 2), B(2, –9), C(0, –10) (51‬‬ ‫‪10.47 A(1, 6), B(1, 2), C(3, 2) (50‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪10‬‬ ‫‪  (52‬ﻗﻴﺎﺱ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺘﺎﻣﺘﻴﻦ ﻳﺴﺎﻭﻱ ‪ (16z - 9)°‬ﻭ ‪ .(4z + 3)°‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ‪22.2 ;67.8  .‬‬ ‫‪ 1 16‬‬ ‫‪ (15‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪3‬‬ ‫‪  (53‬ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ‪ x = 3 :‬ﻭ ‪ y = -4‬ﻭ ‪ ،z = -5‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‪-16  . 5|x + y| - 3|2 - z| :‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪18‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (16‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪4‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪24‬‬ ‫‪‬ﺍﻛﺘﺐ ﻛﻠﻤﺔ \"ﺻﺢ\" ﺑﺠﻮﺍﺭ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺼﺤﻴﺤﺔ ﻭﻛﻠﻤﺔ \"ﺧﻄﺄ\" ﺑﺠﻮﺍﺭ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺨﺎﻃﺌﺔ‪.‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ (17‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺤﺘﻮﻱ ﻋﻠﻰ‬ ‫ﺍﻟﺮﻗﻢ ‪ 2‬ﺯﻳﺎﺩﺓ ﻋﻠﻰ ﺃﺭﻗﺎﻡ ﺍﻟﺤﺪ ﺍﻟﺴﺎﺑﻖ‬ ‫‪ (56‬ﺍﻟﻌﺪﺩ ‪ 9‬ﻋﺪﺩ ﺃﻭﻟﻲ ﺧﻄﺄ‬ ‫‪ 5 - 2 × 3 = 9 (55‬ﺧﻄ‪‬ﺄ‬ ‫‪ (54‬ﻛﻞ ﻣﺮﺑﻊ ﻫﻮ ﻣﺴﺘﻄﻴﻞ ﺻﺢ‬ ‫‪43 - 5614 - 38‬‬ ‫‪‬‬ ‫ﻟﻪ؛ ‪22222‬‬ ‫‪43 - 5639 - 4115 - 39‬‬ ‫‪‬‬ ‫‪ (18‬ﻳﻨﺘﺞ ﻛﻞ ﺣﺪ ﻋﻦ ﺗﺮﺑﻴﻊ ﺍﻟﻌﺪﺩ ﺍﻟﻄﺒﻴﻌﻲ‬ ‫‪ 1 18‬‬ ‫‪45 - 5639 - 53‬‬ ‫‪‬‬ ‫ﺍﻟﺬﻱ ﻳﻤ ﱢﺜﻞ ﺗﺮﺗﻴﺒﻪ؛ ‪25‬‬ ‫ﺍﻟﺬﻱ‬ ‫ﺍﻟﺤﺪ‬ ‫ﻧﺼﻒ‬ ‫ﻳﺴ‪1‬ﺎ_ﻭﻱ‬ ‫ﻛﻞ ﺣﺪ‬ ‫‪(19‬‬ ‫‪16‬‬ ‫ﻳﺴﺒﻘﻪ؛‬ ‫‪‬‬ ‫‪ ‬ﺍﻋﻤﻞ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻠﻌﺪﺩﻳﻦ ﺍﻟﺘﺎﻟﻴﻴﻦ ﻓﻲ ﺍﻟﻤﺘﺘﺎﺑﻌﺔ ﺍﻵﺗﻴﺔ‪9, 7, 10, 8, 11, 9, 12, . . . :‬‬ ‫‪ 1 16‬‬ ‫ﺍﻃﺮﺡ ‪ ،2‬ﺛﻢ ﺃﺿﻒ ‪3‬؛ ‪13 ،10‬‬ ‫‪‬‬ ‫‪ (45‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺃﻭﺩ ﺃﻥ ﺃﺟﺮﻱ ﻣﺴ ﹰﺤﺎ ﻷﻧﻮﺍﻉ ﺍﻷﻧﺸﻄﺔ ﺍﻟﺘﻲ ﻳﻤﺎﺭﺳﻬﺎ ﺍﻟﻨﺎﺱ ﻓﻲ ﻋﻄﻠﺔ ﻧﻬﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ‪ ،‬ﻭﺃﻃﺮﺡ‬ ‫ﺍﻷﺳﺌﻠﺔ ﺍﻵﺗﻴﺔ‪ :‬ﻣﺎ ﻋﻤﺮﻙ؟ ﻣﺎ ﻧﻮﻉ ﺍﻟﻨﺸﺎﻁ ﺍﻟﺬﻱ ﺗﻔﻀﻞ ﻣﻤﺎﺭﺳﺘﻪ ﻓﻲ ﻋﻄﻠﺔ ﻧﻬﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ؟‬ ‫ﻣﺎ ﻣﺪ￯ ﻣﻮﺍﻇﺒﺘﻚ ﻋﻠﻰ ﻣﻤﺎﺭﺳﺔ ﻫﺬﺍ ﺍﻟﻨﺸﺎﻁ؟ ﺛﻢ ﺑﻌﺪ ﺫﻟﻚ ﺃﺳﺘﻌﻤﻞ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻹﻳﺠﺎﺩ ﺃﻧﻤﺎ ﹴﻁ ﻓﻲ‬ ‫ﺍﻹﺟﺎﺑﺎﺕ ﻟﺘﺤﺪﻳﺪ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺍﻷﺷﺨﺎﺹ ﺍﻟﻤﺘﺴﺎﻭﻭﻥ ﻓﻲ ﺍﻟ ﹸﻌ ﹸﻤﺮ ﻳﻔﻀﻠﻮﻥ ﻣﻤﺎﺭﺳﺔ ﺍﻷﻧﺸﻄﺔ ﻧﻔﺴﻬﺎ ﺃﻡ ﻻ‪.‬‬ ‫‪ 1 18‬‬ ‫‪T12 ‬‬

‫‪‬‬ ‫ﺗﻌﻤﻞ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ ﻋﻠﻰ ﺍﻟﺮﺑﻂ ﺑﻴﻦ ﻣﺎ ﻳﺘﻌﻠﻤﻪ ﺍﻟﻄﻼﺏ ﻓﻲ ﺍﻟﻤﺪﺭﺳﺔ‬ ‫ﺍﻟﺜﺎﻧﻮﻳﺔ ﻭﻣﺎ ﻳﺘﻮﻗﻊ ﻣﻨﻬﻢ ﺃﻥ ﻳﻌﺮﻓﻮﻩ ﻋﻨﺪ ﺑﺪﺀ ﺩﺭﺍﺳﺘﻬﻢ ﺍﻟﺠﺎﻣﻌﻴﺔ‪.‬‬ ‫‪ ‬‬ ‫• ‪  ‬ﺗﺸﻤﻞ ﻣﻬﺎﺭﺍﺕ ﻣﺜﻞ‪ ،‬ﺍﻻﺳﺘﻴﻌﺎﺏ ﺍﻟﻘﺮﺍﺋﻲ‪ ،‬ﻭﺇﺩﺍﺭﺓ‬ ‫ﺇﻥ ﺍﻟﻤﻨﻬﺞ ﺍﻟﻘﻮﻱ ﻟﻠﻤﺪﺍﺭﺱ ﺍﻟﺜﺎﻧﻮﻳﺔ ﻣﺆﺷﺮ ﺟﻴﺪ ﻋﻠﻰ ﺍﻻﺳﺘﻌﺪﺍﺩ‬ ‫ﻟﻠﺪﺭﺍﺳﺔ ﺍﻟﺠﺎﻣﻌﻴﺔ )‪ .(Adelman 2006‬ﻓﺎﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻳﺪﺭﺳﻮﻥ‬ ‫ﺍﻟﻮﻗﺖ‪ ،‬ﻭﺗﺴﺠﻴﻞ ﺍﻟﻤﻼﺣﻈﺎﺕ‪ ... ،‬ﺇﻟﺦ‪ .‬ﻭﺗﻮﻓﺮ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ‬ ‫ﻛﺘﺐ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﺍﻟﻤﻌﺪﺓ ﻟﻠﻤﺮﺣﻠﺔ ﺍﻟﺜﺎﻧﻮﻳﺔ ﻣﻦ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ‬ ‫ﻓﺮ ﹰﺻﺎ ﻟﺘﻨﻤﻴﺔ ﻫﺬﻩ ﺍﻟﻤﻬﺎﺭﺍﺕ ﻣﻦ ﺧﻼﻝ ﺇﺭﺷﺎﺩﺍﺕ ﻗﺮﺍﺀﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‬ ‫ﻳﻜﻮﻧﻮﻥ ﺃﻛﺜﺮ ﺍﺳﺘﻌﺪﺍ ﹰﺩﺍ ﻟﻠﺪﺭﺍﺳﺔ ﺍﻟﺠﺎﻣﻌﻴﺔ ﻣﻦ ﺍﻟﺬﻳﻦ ﻟﻢ ﻳﺪﺭﺳﻮﻫﺎ‬ ‫ﻭﺭﻭﺍﺑﻂ ﺍﻟﻤﻔﺮﺩﺍﺕ‪ ،‬ﻭﺩﻟﻴﻞ ﺍﻟﺘﻮﻗﻊ ﻭﻏﻴﺮﻫﺎ‪.‬‬ ‫)‪.(Abraham & Crrech 2002‬‬ ‫‪‬‬ ‫ﻭﻓﻴﻤﺎ ﻳﺄﺗﻲ ﺑﻌﺾ ﻣﻨﺎﺣﻲ ﺍﻻﺳﺘﻌﺪﺍﺩ ﻟﻠﺪﺭﺍﺳﺔ ﺍﻟﺠﺎﻣﻌﻴﺔ ﺍﻟﺘﻲ ﻃﻮﺭﻫﺎ‪:‬‬ ‫ﻟﻢ ﺗﻌﺪ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻓﻲ ﻋﺎﻟﻢ ﺍﻟﺘﻘﻨﻴﺔ ﺍﻟﻤﻌﺎﺻﺮ ﻣﻘﺘﺼﺮﺓ ﻋﻠﻰ ﺍﻟﻄﻼﺏ‬ ‫‪David Conley at the University of Oregon‬‬ ‫ﺍﻟﺬﻳﻦ ﻳﻠﺘﺤﻘﻮﻥ ﺑﺎﻟﺠﺎﻣﻌﺎﺕ‪ .‬ﻓﻘﺪ ﺃﻇﻬﺮﺕ ﺇﺣﺪ￯ ﺍﻟﺪﺭﺍﺳﺎﺕ ﺃﻥ‬ ‫ﺍﻟﺒﺮﺍﻣﺞ ﺍﻟﺘﺪﺭﻳﺒﻴﺔ ﺍﻟﺘﻲ ﻳﺨﻀﻊ ﻟﻬﺎ ﺷﺨﺺ ﻳﺮﻳﺪ ﺍﻟﺤﺼﻮﻝ ﻋﻠﻰ ﻋﻤﻞ‬ ‫• ‪  ‬ﻭﻫﻲ ﻣﻬﺎﺭﺍﺕ ﺿﺮﻭﺭﻳﺔ ﻟﺘﻌﻠﻢ ﺍﻟﻤﺤﺘﻮ￯ ﻋﻠﻰ‬ ‫ﺗﺘﻄﻠﺐ ﺃﻥ ﻳﻜﻮﻥ ﻫﺬﺍ ﺍﻟﺸﺨﺺ ﻋﻠﻰ ﻣﺴﺘﻮ￯ ﻣﻌﻴﻦ ﻣﻦ ﺍﻟﺘﻌﻠﻴﻢ ﻓﻲ‬ ‫ﺍﻟﺠﺒﺮ ﻭﺍﻟﻬﻨﺪﺳﺔ ﻭﺗﺤﻠﻴﻞ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻭﺍﻹﺣﺼﺎﺀ ﻳﻤﺎﺛﻞ ﻣﺴﺘﻮ￯ ﺍﻟﻄﺎﻟﺐ‬ ‫ﺍﻟﻤﺴﺘﻮ￯ ﺍﻟﺠﺎﻣﻌﻲ‪ ،‬ﻭﺗﺸﻤﻞ‪ :‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ‪ ،‬ﻭﺣﻞ ﺍﻟﻤﺴﺄﻟﺔ‪،‬‬ ‫ﻭﺍﻟﺘﺒﺮﻳﺮ‪ ،‬ﻭﺗﺘﺎﺡ ﻓﻲ ﻛﻞ ﻳﻮﻡ ﻟﻠﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻳﺪﺭﺳﻮﻥ ﻫﺬﻩ‬ ‫ﺍﻟﺬﻱ ﻳﻠﺘﺤﻖ ﺑﺎﻟﺴﻨﺔ ﺍﻷﻭﻟﻰ ﻓﻲ ﺍﻟﺠﺎﻣﻌﺔ؛ ﺣﺘﻰ ﻳﻨﺠﺢ ﻓﻲ ﻋﻤﻠﻪ‪.‬‬ ‫ﺍﻟﺴﻠﺴﻠﺔ ﻓﺮﺹ ﻟﺘﻨﻤﻴﺔ ﻣﻬﺎﺭﺍﺕ ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻌﻠﻴﺎ ﻣﻦ ﺧﻼﻝ ﺍﻟﻤﺴﺎﺋﻞ‬ ‫ﺍﻟﺨﺎﺻﺔ ﺑﺬﻟﻚ‪.‬‬ ‫• ‪  ‬ﺇﻥ ﻛﺘﺐ ﺍﻟﻤﺮﺣﻠﺔ ﺍﻟﺜﺎﻧﻮﻳﺔ ﻣﻦ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﺘﺴﻘﺔ ﻣﻊ ﻣﻌﺎﻳﻴﺮ ﻋﺎﻟﻤﻴﺔ ﺩﻗﻴﻘﺔ ﺗﺸﻤﻞ ﻣﻌﺎﻳﻴﺮ ‪ NCTM‬ﻟﻠﺮﻳﺎﺿﻴﺎﺕ‬ ‫ﺍﻟﻤﺪﺭﺳﻴﺔ‪ ،‬ﻭﻏﻴﺮﻫﺎ‪.‬‬ ‫‪ T13‬‬

                                     T14 

  1-3  1-3   ✓(11)   1-2   1-1       •  •  •  •         •  •   •                              33     •     (11)     •  •   •  (16) (13)  (6)  •  • (18) (14)  •   •  • (8) (19)  (15)   •   •  (20) (9)  •  (10)    (8) • (7) • (6) • 2 8  2 0  1 3   33 , 34  23 , 25  14 , 15 , 18            1 10A

    2 1  1 7   4   1-8  1-7  1-6  1-5  1-4         •  •  •     •  •       •   •  •  •    •               73  64  58                 •   •  (26)  (21)   •   •  •  (41)  (36)  (31)  •  • (28) (23)  •  •  • (43) (38) (33)  •  • (29) (24)   •  •  • (44) (39) (34)  •   •  (30)  (25)  •  •   •  (45)  (40)  (35)   (10) • (9) •    (13) • (12) • (11) • 6 8  6 2  5 5  4 6  4 0  70 , 72  61 , 65  58   49 , 51    39 , 44     ✓   ✓  • (52)  (74-78)  (79)  • 10B  1

    ✓ (11) 1    111  ✓ (8) 1   1  www.obeikaneducation.com  2      1    1 ! 1 4 www.obeikaneducation.com (11, 12) 2 www.obeikaneducation.com 1  1 1 (52) (13) www.obeikaneducation.com www.obeikaneducation.com 2 1   1 (74-78) (79) www.obeikaneducation.com (82-83) www.obeikaneducation.com 1  ✓ (15-20)1, 2A, 2B  (21-22)3 (14) (23) (24-26) www.obeikaneducation.com              1 10C

  3    1 :‫ﺍﻃﺮﺡ ﺍﻟﻤﺴﺄﻟﺔ ﺍﻵﺗﻴﺔ ﻋﻠﻰ ﺍﻟﻄﻼﺏ‬ ‫ ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺍﺳﺘﻜﺸﺎﻑ ﺟﻤﻊ ﺍﻟﻘﻄﻊ‬ ‫ ﻓﻤﺎ‬،‫ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ ﻛﻞ ﺛﻼﺙ ﻧﻘﺎﻁ ﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ ﺗﺤﺪﺩ ﻣﺴﺘ ﹰﻮ￯ ﻭﺍﺣ ﹰﺪﺍ‬ ‫ﻋﺪﺩ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺘﻲ ﺗﺤ ﱢﺪﺩﻫﺎ ﺃﺭﺑﻊ ﻧﻘﺎﻁ ﻻ ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ ﻭﺍﺣﺪ؟ ﻭﻣﺎ‬ ،‫ ﻭﺫﻟﻚ ﺑﻘﻴﺎﺱ ﺑﻌﺾ ﺍﻷﺷﻴﺎﺀ ﺍﻟﻤﻮﺟﻮﺩﺓ ﻓﻲ ﻏﺮﻓﺔ ﺍﻟﺼﻒ‬،‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻭﺍﻟﺰﻭﺍﻳﺎ‬ ‫ ﻧﻘﺎﻁ ﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ؟‬5 ‫ﻋﺪﺩ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺘﻲ ﺗﺤﺪﺩﻫﺎ‬ ‫ ﻭﺍﻟﻤﻨﻘﻠﺔ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﺃﻥ‬،‫ﻭﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻤﺘﺮ ﻹﻳﺠﺎﺩ ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ ﻏﺮﻓﺔ ﺍﻟﺼﻒ‬ ‫ ﻭﺗﺤﺪﺩ‬،‫ ﻣﺴﺘﻮﻳﺎﺕ ﻋﻠﻰ ﺍﻷﻛﺜﺮ‬4‫ ﻭ‬،‫ﺗﺤﺪﺩ ﺍﻟﻨﻘﺎﻁ ﺍﻷﺭﺑﻊ ﻣﺴﺘﻮ￯ ﻭﺍﺣ ﹰﺪﺍ ﻋﻠﻰ ﺍﻷﻗﻞ‬ .‫ﺯﺍﻭﻳﺘﻴﻦ ﻗﺎﺋﻤﺘﻴﻦ ﺗﺸ ﱢﻜﻼﻥ ﺧ ﹼﹰﻄﺎ ﻣﺴﺘﻘﻴ ﹰﻤﺎ‬ .‫ ﻣﺴﺘﻮﻳﺎﺕ ﻋﻠﻰ ﺍﻷﻛﺜﺮ‬10 ‫ ﻭ‬،‫ﺍﻟﻨﻘﺎﻁ ﺍﻟﺨﻤﺲ ﻣﺴﺘﻮ￯ ﻭﺍﺣ ﹰﺪﺍ ﻋﻠﻰ ﺍﻷﻗﻞ‬ ‫ ﻳﻤﻜﻦ ﻟﻠﻄﻼﺏ ﺃﻥ ﻳﺘﺪ ﱠﺭﺑﻮﺍ ﻋﻠﻰ ﺻﻴﺎﻏﺔ ﺗﺨﻤﻴﻨﺎﺕ‬  ‫ ﻓﻤﺜ ﹰﻼ ﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ‬.‫ ﻭﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ ﻣﻦ ﺍﻟﻄﺒﻴﻌﺔ‬،‫ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﻌﺼﻒ ﺍﻟﺬﻫﻨﻲ‬ ‫ ﻭﺍﻟﻤﺜﺎﻝ‬.“‫ﻗﺮﺍﺀﺓ ﺍﻟﻌﺒﺎﺭﺓ ”ﺇﺫﺍ ﻟﻢ ﹸﺗ ﹾﺮ ﹶﻭ ﺍﻟﻨﺒﺎﺗﺎﺕ ﻛﻞ ﻳﻮﻡ ﻓﻠﻦ ﺗﺒﻘﻰ ﻋﻠﻰ ﻗﻴﺪ ﺍﻟﺤﻴﺎﺓ‬ .‫ﺍﻟﻤﻀﺎﺩ ﻟﻬﺎ ﺃﻥ ﻧﺒﺘﺔ ﺍﻟﺼﺒﺎﺭ ﻳﻤﻜﻦ ﺃﻥ ﺗﺒﻘﻰ ﺃﺳﺎﺑﻴﻊ ﻣﻦ ﺩﻭﻥ ﻣﺎﺀ‬ ‫ﻭﻣﻮﺿﻮﻋﺎﺕ ﺍﻟﻄﺒﻴﻌﺔ ﻳﻤﻜﻦ ﺃﻥ ﺗﺸﻤﻞ ﺍﻟﻨﺒﺎﺗﺎﺕ ﻭﺍﻟﺤﻴﻮﺍﻧﺎﺕ ﻭﻋﻼﻗﺎﺕ ﺍﻟﺤﻴﻮﺍﻧﺎﺕ‬ .‫ ﻭﻫﻜﺬﺍ‬،‫ﺍﻟﻤﻔﺘﺮﺳﺔ ﻭﺍﻟﻄﺮﺍﺋﺪ ﻭﺍﻟﺤﺸﺮﺍﺕ ﻭﺍﻟﻄﻘﺲ‬  2 ‫ﻭ ﹼﺿﺢ ﻟﻠﻄﻼﺏ ﻛﻴﻔﻴﺔ ﺍﻻﻧﺘﻘﺎﻝ ﻓﻲ ﺍﻟﺒﺮﻫﺎﻥ ﻣﻦ ﺍﻟﻔﺮﺽ ﺇﻟﻰ ﺍﻟﻨﺘﻴﺠﺔ ﺑﺎﺳﺘﻌﻤﺎﻝ‬ ‫ ﺑﺤﻴﺚ ﺗﻘﻮﺩ ﺍﻟﺸﺮﻭﻁ ﺍﻟﻤﻌﻄﺎﺓ ﺇﻟﻰ ﻋﺒﺎﺭﺍﺕ ﺍﻟﺒﺮﻫﺎﻥ ﻣﻊ ﺗﺒﺮﻳ ﹴﺮ ﻟﻜﻞ‬،‫ﻣﺨﻄﻂ ﺗﺴﻠﺴﻠﻲ‬ .‫ ﻭﺗﻜﻮﻥ ﺍﻟﻨﺘﻴﺠﺔ ﻫﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻨﻬﺎﺋﻴﺔ ﻓﻲ ﺍﻟﺒﺮﻫﺎﻥ‬،‫ﺧﻄﻮﺓ‬                   1   1-2  1      ∼    p∧q “”   p∨q “”     10D  1

‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ 1-1‬‬ ‫‪1‬‬ ‫ﺍﻟﺘﺨﻤﻴﻦ ﻫﻮ ﺗﻮ ﱡﻗﻊ ﻣﺪﺭﻭﺱ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ ﻣﻌﻠﻮﻣﺎﺕ ﻣﻌﺮﻭﻓﺔ‪ ،‬ﻭﺍﻟﺘﺒﺮﻳﺮ‬ ‫• ﺍﻟﺘﻌﺒﻴﺮ ﻋﻦ ﺍﻷﻓﻜﺎﺭ ﺍﻟﺮﻳﺎﺿﻴﺔ ﻟﻐﻮ ﹰﹼﻳﺎ ﻭﺑﺄﺩﻭﺍﺕ ﻓ ﹼﻌﺎﻟﺔ‪ ،‬ﻭﻭﺣﺪﺍﺕ‬ ‫ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻫﻮ ﺗﻔﺤﺺ ﻟﻌﺪﺓ ﺃﻭﺿﺎﻉ ﺧﺎﺻﺔ ﻟﻠﻮﺻﻮﻝ ﺇﻟﻰ ﺍﻟﺘﺨﻤﻴﻦ‪ .‬ﻭﺇﺫﺍ‬ ‫ﻣﻨﺎﺳﺒﺔ‪ ،‬ﻭﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﺒﻴﺎﻧﻴﺔ ﺃﻭ ﺍﻟﺮﻳﺎﺿﻴﺔ ﺃﻭ ﺍﻟﻌﺪﺩﻳﺔ ﺃﻭ ﺍﻟﻤﺎﺩﻳﺔ‬ ‫ﻧﺎﻗﺾ ﻣﺜﺎﻝ ﻭﺍﺣﺪ ﺍﻟﺘﺨﻤﻴﻦ‪ ،‬ﻓﺈﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃﺊ‪ ،‬ﻭ ﹸﻳﺪﻋﻰ ﺍﻟﻤﺜﺎﻝ ﻓﻲ‬ ‫ﺃﻭ ﺍﻟﺠﺒﺮﻳﺔ‪.‬‬ ‫ﻫﺬﻩ ﺍﻟﺤﺎﻟﺔ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫• ﺇﺛﺒﺎﺕ ﺻﺤﺔ ﺍﻻﺳﺘﻨﺘﺎﺟﺎﺕ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺨﺼﺎﺋﺺ ﻭﺍﻟﻌﻼﻗﺎﺕ‬ ‫‪ 1-2‬‬ ‫ﺍﻟﺮﻳﺎﺿﻴﺔ‪.‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﻫﻲ ﺟﻤﻠﺔ ﺧﺒﺮﻳﺔ ﺇﻣﺎ ﺃﻥ ﺗﻜﻮﻥ ﺻﺤﻴﺤﺔ ﺃﻭ ﺗﻜﻮﻥ ﺧﺎﻃﺌﺔ‪ ،‬ﻭﻻ‬ ‫‪1‬‬ ‫ﺗﺤﺘﻤﻞ ﺃﻱ ﺣﺎﻟﺔ ﺃﺧﺮ￯‪ .‬ﻭ ﹸﺗﺴﻤﻰ ﺻﺤﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺃﻭ ﺧﻄﺆﻫﺎ ﻗﻴﻤ ﹶﺔ ﺍﻟﺼﻮﺍﺏ‬ ‫• ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻟﻜﺘﺎﺑﺔ ﺗﺨﻤﻴﻦ‪.‬‬ ‫ﻟﻬﺎ‪ .‬ﻭﻟﺬﻟﻚ ﻓﺈﻥ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻨﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ﻫﻮ ﻋﻜﺲ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ‬ ‫• ﺗﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻌﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ‪ ،‬ﻭﻋﻜﺴﻬﺎ ﻭﻣﻌﻜﻮﺳﻬﺎ‬ ‫ﻟﻠﻌﺒﺎﺭﺓ‪ .‬ﻭﺇﺫﺍ ﺭﻣﺰﻧﺎ ﻟﻌﺒﺎﺭﺓ ﺑﺎﻟﺮﻣﺰ ‪ ، p‬ﻓﺈﻥ \"ﻟﻴﺲ ‪ \"p‬ﻫﻮ ﻧﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ‪،‬‬ ‫ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻬﺎ‪.‬‬ ‫ﻭ ﹸﻳﺮﻣﺰ ﻟﻪ ﺑﺎﻟﺮﻣﺰ ‪. p‬‬ ‫• ﺍﺳﺘﻌﻤﺎﻝ ﻗﺎﻧﻮ ﹶﻧﻲ ﺍﻟﻔﺼﻞ ﻭﺍﻟﻘﻴﺎﺱ ﺍﻟﻤﻨﻄﻘﻲ ﻟﻠﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻨﺘﺎﺟﻲ‪.‬‬ ‫• ﺍﺳﺘﻌﻤﺎﻝ ﺗﻌﺮﻳﻔﺎﺕ ﺃﻭ ﺧﺼﺎﺋﺺ ﺟﺒﺮﻳﺔ ﺃﻭ ﻣﺴﻠﻤﺎﺕ ﺃﻭ ﻧﻈﺮﻳﺎﺕ‬ ‫ﻭﻳﻤﻜﻦ ﺭﺑﻂ ﻋﺒﺎﺭﺗﻴﻦ ﺃﻭ ﺃﻛﺜﺮ ﻟﺘﻜﻮﻳﻦ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ‪ .‬ﻭﺇﺫﺍ ﺍﺳﺘﻌﻤﻠﺖ ﺃﺩﺍﺓ‬ ‫ﻹﺛﺒﺎﺕ ﺻﺤﺔ ﻋﺒﺎﺭﺍﺕ‪ ،‬ﻭﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ ﻟﺘﻔﻨﻴﺪ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺨﻄﺄ‪.‬‬ ‫ﺍﻟﺮﺑﻂ \"ﻭ\" ﻭﺭﻣﺰﻫﺎ \"∧\"‪ ،‬ﻓﺈﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﻨﺎﺗﺠﺔ ﺗﺴﻤﻰ \"ﻋﺒﺎﺭﺓ‬ ‫‪1‬‬ ‫ﺍﻟﻮﺻﻞ\"‪ .‬ﺃﻣﺎ ﺇﺫﺍ ﺍﺳﺘﻌﻤﻠﺖ ﺃﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﺃﻭ\" ﻭﺭﻣﺰﻫﺎ \"∨ \"‪ ،‬ﻓﺈﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﻨﺎﺗﺠﺔ ﹸﺗﺴﻤﻰ \"ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ\"‪ .‬ﻭﻳﻤﻜﻦ ﺗﻮﺿﻴﺢ ﻋﺒﺎﺭ ﹶﺗﻲ ﺍﻟﻔﺼﻞ‬ ‫‪‬‬ ‫ﻭﺍﻟﻮﺻﻞ ﺑﺄﺷﻜﺎﻝ ﭬﻦ ﻛﻤﺎ ﻳﻠﻲ‪:‬‬ ‫• ﺍﻟﻤﻘﺎﺭﻧﺔ ﺑﻴﻦ ﺍﻟﺤﻠﻮﻝ ﺍﻟﺠﺒﺮﻳﺔ ﻭﺍﻟﺒﻴﺎﻧﻴﺔ ﻟﻤﻌﺎﺩﻻﺕ ﺗﺮﺑﻴﻌﻴﺔ ﻭﺗﻔﺴﻴﺮﻫﺎ‪.‬‬ ‫• ﺗﺤﻠﻴﻞ ﻣﻮﺍﻗﻒ ﺭﻳﺎﺿﻴﺔ ﻣﻤﺜﻠﺔ ﺑﺪﻭﺍﻝ ﺍﻟﺠﺬﺭ ﺍﻟﺘﺮﺑﻴﻌﻲ‪ ،‬ﻭﺻﻴﺎﻏﺔ‬ ‫• ‪‬‬ ‫ﻣﻌﺎﺩﻻﺕ ﺃﻭ ﻣﺘﺒﺎﻳﻨﺎﺕ ﻭﺍﺧﺘﻴﺎﺭ ﻃﺮﻳﻘﺔ ﻭﺣﻞ ﺍﻟﻤﺴﺎﺋﻞ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻭﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻳﻤﻜﻦ ﺃﻥ ﺗﺴﺎﻋﺪ ﻋﻠﻰ ﺇﻳﺠﺎﺩ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ‪.‬‬ ‫• ‪‬‬ ‫‪p ∼p‬‬ ‫‪pq‬‬ ‫‪p∨q‬‬ ‫‪pq‬‬ ‫‪p∧q‬‬ ‫‪TF‬‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪FT‬‬ ‫‪TF‬‬ ‫‪T‬‬ ‫‪TF‬‬ ‫‪F‬‬ ‫‪FT‬‬ ‫‪T‬‬ ‫‪FT‬‬ ‫‪F‬‬ ‫‪FF‬‬ ‫‪F‬‬ ‫‪FF‬‬ ‫‪F‬‬ ‫ﻓﻲ ﺣﺎﻟﺔ ﺍﻟﻨﻔﻲ‪ ،‬ﺇﺫﺍ‬ ‫ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ‬ ‫ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ‬ ‫ﻛﺎﻧﺖ ‪ p‬ﺻﺤﻴﺤﺔ‪ ،‬ﻓﺈﻥ‬ ‫ﺧﺎﻃﺌﺔ ﻓﻘﻂ‪ ،‬ﻋﻨﺪﻣﺎ‬ ‫ﺻﺤﻴﺤﺔ ﻓﻘﻂ‪ ،‬ﻋﻨﺪﻣﺎ‬ ‫‪ ∼p‬ﺧﺎﻃﺌﺔ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫ﺗﻜﻮﻥ ﻛ ﱞﻞ ﻣﻦ ‪ p‬ﻭ ‪q‬‬ ‫ﺗﻜﻮﻥ ﻛ ﱞﻞ ﻣﻦ ‪ p‬ﻭ ‪q‬‬ ‫‪ p‬ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺈﻥ ‪∼p‬‬ ‫ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺗﺒﻴﻦ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺃﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﺗﻜﻮﻥ ﺻﺤﻴﺤﺔ ﻓﻘﻂ‪ ،‬ﻋﻨﺪﻣﺎ‬ ‫ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ﺻﺤﻴﺤﺘﻴﻦ‪ .‬ﺃﻣﺎ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ﻓﺘﻜﻮﻥ ﺻﺤﻴﺤﺔ ﺩﺍﺋ ﹰﻤﺎ ﺇ ﹼﻻ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ﺧﺎﻃﺌﺘﻴﻦ‪.‬‬ ‫‪ 1 10E‬‬

‫‪‬‬ ‫‪ 1-6‬‬ ‫‪ 1-3‬‬ ‫ﺍﺳ ﹸﺘﻌﻤﻠﺖ ﺧﺼﺎﺋﺺ ﺍﻟﻤﺴﺎﻭﺍﺓ ﻟﺤﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺠﺒﺮﻳﺔ ﻭﺍﻟﺘﺤﻘﻖ ﻣﻦ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻫﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺘﻲ ﻳﻤﻜﻦ ﺃﻥ ﹸﺗﻜﺘﺐ ﻋﻠﻰ ﺻﻮﺭﺓ‬ ‫ﺻﺤﺔ ﺍﻟﻌﻼﻗﺎﺕ ﺍﻟﺠﺒﺮﻳﺔ‪ .‬ﻭﺍﺳﺘﻌﻤﺎﻝ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻟﺨﻄﻮﺍﺕ ﺍﻟﺠﺒﺮﻳﺔ‬ ‫\"ﺇﺫﺍ‪ ...‬ﻓﺈ ﹼﻥ ‪ ،‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،p‬ﻓﺈﻥ ‪ .q‬ﺗﺴ ﹼﻤﻰ ﺍﻟﺠﻤﻠﺔ ﺍﻟﺘﻲ ﺗﻠﻲ ﻛﻠﻤﺔ \"ﺇﺫﺍ\"‬ ‫ﻟﺤﻞ ﻣﺴﺎﺋﻞ ﹸﻳﺸ ﱢﻜﻞ ﺑﺮﻫﺎ ﹰﻧﺎ ﺟﺒﺮ ﹼﹰﻳﺎ‪ .‬ﻭﻫﺬﺍ ﺍﻟﺒﺮﻫﺎﻥ ﻳﻤﻜﻦ ﺗﻨﻈﻴﻤﻪ ﺑﻜﺘﺎﺑﺔ‬ ‫ﺧﻄﻮﺍﺕ ﺍﻟﺤﻞ ﻟﻠﻤﻌﺎﺩﻟﺔ ﻓﻲ ﻋﻤﻮﺩ‪ ،‬ﻭﺍﻟﺨﺼﺎﺋﺺ ﺍﻟﺘﻲ ﺗﺒﺮﺭ ﻛﻞ ﺧﻄﻮﺓ‬ ‫ﻣﺒﺎﺷﺮﺓ ﺍﻟﻔﺮﺽ‪ ،‬ﻭﺍﻟﺘﻲ ﺗﻠﻲ ﻛﻠﻤﺔ \"ﻓﺈﻥ\" ﻣﺒﺎﺷﺮﺓ ﺍﻟﻨﺘﻴﺠﺔ‪ .‬ﻭﺑﺎﻟﺮﻣﻮﺯ‬ ‫ﻳﺴﺘﻌﻤﻞ ﺳﻬﻢ ﻣﺘﺠﻪ ﻣﻦ ﺍﻟﻔﺮﺽ ﺇﻟﻰ ﺍﻟﻨﺘﻴﺠﺔ‪ .‬ﻭﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫ﻓﻲ ﻋﻤﻮﺩ ﺁﺧﺮ‪ .‬ﻭﻳﺴﺘﻌﻤﻞ ﻓﻲ ﺍﻟﻬﻨﺪﺳﺔ ﻧﻤﻮﺫﺝ ﻣﺸﺎﺑﻪ ﻹﺛﺒﺎﺕ ﺻﺤﺔ‬ ‫ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﻭﺍﻟﻨﻈﺮﻳﺎﺕ‪ ،‬ﻭﻳﺘﻀﻤﻦ ﺍﻟﺒﺮﻫﺎﻥ ﺫﻭ ﺍﻟﻌﻤﻮﺩﻳﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫ﺻﺤﻴﺤﺔ ﺩﺍﺋ ﹰﻤﺎ‪ ،‬ﺇﻻ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻔﺮﺽ ﺻﺤﻴ ﹰﺤﺎ ﻭﺍﻟﻨﺘﻴﺠﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﻭﺍﻟﻤﺒﺮﺭﺍﺕ ﻣﻨﻈﻤ ﹰﺔ ﻓﻲ ﻋﻤﻮﺩﻳﻦ‪ ،‬ﺗﺴﻤﻰ ﻛﻞ ﺧﻄﻮﺓ ﻋﺒﺎﺭﺓ‪ ،‬ﻭﺗﺴﻤﻰ‬ ‫ﻭﺗﺸﺘﻖ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﺮﺗﺒﻄﺔ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﻌﻄﺎﺓ‪،‬‬ ‫ﺍﻟﺨﺎﺻﻴﺔ ﺗﺒﺮﻳ ﹰﺮﺍ‪.‬‬ ‫ﻓﻌﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻳﻨﺘﺞ ﻋﻦ ﺗﺒﺪﻳﻞ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،q‬ﻓﺈﻥ ‪ . p‬ﻭﻣﻌﻜﻮﺱ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻳﻨﺘﺞ ﻋﻦ‬ ‫‪ 1-7‬‬ ‫ﻧﻔﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،~p‬ﻓﺈﻥ‬ ‫‪ ،~q‬ﺃﻣﺎ ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻓﻴﻨﺘﺞ ﻋﻦ ﻧﻔﻲ ﻛ ﱟﻞ ﻣﻦ‬ ‫ﻳﻤﻜﻦ ﺇﻳﺠﺎﺩ ﻃﻮﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‪ ،‬ﻭﺍﺳﺘﻌﻤﺎﻝ ﻫﺬﻩ ﺍﻷﻃﻮﺍﻝ‬ ‫ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻟﻌﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،~q‬ﻓﺈﻥ ‪. ~p‬‬ ‫ﻓﻲ ﺍﻟﺤﺴﺎﺑﺎﺕ؛ ﻷﻧﻬﺎ ﺃﻋﺪﺍﺩ ﺣﻘﻴﻘﻴﺔ‪ .‬ﻭﺗﻨﺺ ﻣﺴﻠﻤﺔ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻋﻠﻰ ﺃﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﻮﺍﻗﻌﺔ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ ﺃﻭ ﻋﻠﻰ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ‬ ‫ﻭﻫﻨﺎﻙ ﺗﻜﺎﻓﺆ ﻣﻨﻄﻘﻲ ﺑﻴﻦ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ‪،‬‬ ‫ﻳﻤﻜﻦ ﺭﺑﻄﻬﺎ ﺑﺄﻋﺪﺍﺩ ﺣﻘﻴﻘﻴﺔ‪ ،‬ﺑﺤﻴﺚ ﺇﺫﺍ ﻭﻗﻌﺖ ﻧﻘﻄﺘﺎﻥ ‪ A‬ﻭ ‪ B‬ﻋﻠﻰ‬ ‫ﻭﻛﺬﻟﻚ ﺑﻴﻦ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﻜﻮﺳﻬﺎ‪.‬‬ ‫ﻣﺴﺘﻘﻴﻢ‪ ،‬ﻭﻛﺎﻧﺖ ﺍﻟﻨﻘﻄﺔ ‪ A‬ﺗ_ﻘ_ﺎﺑ_ﻞ ﺍﻟﻌﺪﺩ ﺻﻔ ﹰﺮﺍ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﻘﻄﺔ ‪ B‬ﺗﻘﺎﺑﻞ ﻋﺪ ﹰﺩﺍ‬ ‫ﻣﻮﺟ ﹰﺒﺎ‪ ،‬ﻳﻤﺜﻞ ﻃﻮﻝ ﺍﻟﻘﻄﻌﺔ ‪ . AB‬ﻭﻫﻨﺎﻙ ﻣﺴﻠﻤﺔ ﺃﺧﺮ￯ ﺗﻨﺺ ﻋﻠﻰ ﺃﻧﻪ‬ ‫‪ 1-4‬‬ ‫ﺇﺫﺍ ﻭﻗﻌﺖ ﺍﻟﻨﻘﻄﺔ ‪ B‬ﺑﻴﻦ ﺍﻟﻨﻘﻄﺘﻴﻦ ‪ A‬ﻭ ‪ C‬ﻋﻠﻰ ﺍﻟﻤﺴﺘﻘﻴﻢ ﻧﻔﺴﻪ‪ ،‬ﻓﺈﻥ‬ ‫ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻨﺘﺎﺟﻲ ﻳﺴﺘﻌﻤﻞ ﺍﻟﺤﻘﺎﺋﻖ ﺃﻭ ﺍﻟﻘﻮﺍﻋﺪ ﺃﻭ ﺍﻟﺘﻌﺎﺭﻳﻒ‬ ‫‪ ، AB + BC = AC‬ﻭﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺢ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫ﺃﻭ ﺍﻟﺨﺼﺎﺋﺺ ﻟﻠﻮﺻﻮﻝ ﺇﻟﻰ ﻧﺘﺎﺋﺞ ﻣﻨﻄﻘﻴﺔ‪ .‬ﻭﻣﻦ ﺃﺷﻜﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ‬ ‫ﺍﻻﺳﺘﻨﺘﺎﺟﻲ ﺃﻥ ﹸﻳﺴﺘﻌﻤﻞ ﻟﻠﺤﺼﻮﻝ ﻋﻠﻰ ﻧﺘﺎﺋﺞ ﻣﻦ ﻋﺒﺎﺭﺍﺕ ﺷﺮﻃﻴﺔ‪،‬‬ ‫ﻭﻳﻤﻜﻦ ﺍﺳﺘﻌﻤﺎﻝ ﺧﺼﺎﺋﺺ ﺍﻻﻧﻌﻜﺎﺱ ﻭﺍﻟﺘﻤﺎﺛﻞ ﻭﺍﻟﺘﻌﺪﻱ ﻟﻠﻤﺴﺎﻭﺍﺓ‬ ‫ﻭﻫﻮ ﻣﺎ ﹸﻳﺴ ﱠﻤﻰ ﻗﺎﻧﻮﻥ ﺍﻟﻔﺼﻞ ﺍﻟﻤﻨﻄﻘ ﹼﻲ‪ ،‬ﻭﺍﻟﺬﻱ ﻳﻨﺺ ﻋﻠﻰ ﺃﻧﻪ ﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫ﻓﻲ ﻛﺘﺎﺑﺔ ﺑﺮﺍﻫﻴﻦ ﺣﻮﻝ ﺗﻄﺎﺑﻖ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‪ .‬ﻭﺍﻟﻨﻈﺮﻳﺔ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻦ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ‪ p → q‬ﺻﺤﻴﺤﺔ‪ ،‬ﻭﻛﺎﻧﺖ ‪ p‬ﺻﺤﻴﺤﺔ‪ ،‬ﻓﺈﻥ ‪ q‬ﺗﻜﻮﻥ‬ ‫ﺻﺤﻴﺤﺔ‪ .‬ﻭﻣﻦ ﻗﻮﺍﻧﻴﻦ ﺍﻟﻤﻨﻄﻖ ﺍﻷﺧﺮ￯ ﻗﺎﻧﻮﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﻤﻨﻄﻘﻲ ﺍﻟﺬﻱ‬ ‫ﺍﻟﺒﺮﺍﻫﻴﻦ ﺗﻨﺺ ﻋﻠﻰ ﺃﻥ ﺗﻄﺎﺑﻖ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻫﻲ ﻋﻼﻗﺔ ﺍﻧﻌﻜﺎﺱ‬ ‫ﻳﻨﺺ ﻋﻠﻰ ﺃﻧﻪ ﺇﺫﺍ ﻛﺎﻧﺖ ‪ p → q‬ﺻﺤﻴﺤﺔ‪ ،‬ﻭﻛﺎﻧﺖ ‪ q → r‬ﺻﺤﻴﺤﺔ‪،‬‬ ‫ﻭﺗﻤﺎﺛﻞ ﻭﺗﻌ ﱟﺪ‪.‬‬ ‫ﻓﺈﻥ ‪ p → r‬ﺻﺤﻴﺤﺔ‪ .‬ﻭﻫﺬﺍ ﺍﻟﻘﺎﻧﻮﻥ ﻳﺸﺒﻪ ﻋﻼﻗﺔ ﺍﻟﺘﻌ ﱢﺪﻱ ﻟﻠﻤﺴﺎﻭﺍﺓ‪.‬‬ ‫‪ 1-8‬‬ ‫‪  1-5‬‬ ‫ﻳﻘﺪﻡ ﻫﺬﺍ ﺍﻟﺪﺭﺱ ﻣﺴ ﹼﻠﻤﺎﺕ ﻭﻧﻈﺮﻳﺎﺕ ﺣﻮﻝ ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ ﺍﻟﺰﻭﺍﻳﺎ‪،‬‬ ‫ﺍﻟﻤﺴ ﹼﻠﻤﺔ ﻓﻲ ﺍﻟﻬﻨﺪﺳﺔ ﻋﺒﺎﺭﺓ ﺗﻌﻄﻲ ﻭﺻ ﹰﻔﺎ ﻟﻌﻼﻗﺔ ﺃﺳﺎﺳﻴﺔ ﺑﻴﻦ ﺍﻟﻤﻔﺎﻫﻴﻢ‬ ‫ﺣﻴﺚ ﺗﻨﺺ ﻣﺴ ﹼﻠﻤﺔ ﺍﻟﻤﻨﻘﻠﺔ ﻋﻠﻰ ﺃﻧﻪ \"ﺗﺴﺘﻌﻤﻞ ﺍﻟﻤﻨﻘﻠﺔ ﻟﻠﺮﺑﻂ ﺑﻴﻦ ﻗﻴﺎﺱ‬ ‫ﺍﻟﻬﻨﺪﺳﻴﺔ ﺍﻷﻭﻟﻴﺔ‪ ،‬ﻭ ﹸﻳﺴ ﹼﻠﻢ ﺑﺼﺤﺘﻬﺎ ﺩﻭﻥ ﺑﺮﻫﺎﻥ‪ .‬ﻭﺣﺎﻟﻤﺎ ﻳﺘﻢ ﺑﻴﺎﻥ ﺻﺤﺔ‬ ‫ﺯﺍﻭﻳﺔ ﻭﻋﺪﺩ ﺣﻘﻴﻘﻲ ﻳﻘﻊ ﺑﻴﻦ ‪ 0°‬ﻭ ‪.\"180°‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺃﻭ ﺍﻟﺘﺨﻤﻴﻦ‪ ،‬ﻓﺈﻧﻬﺎ ﹸﺗﺴ ﹼﻤﻰ ﻧﻈﺮﻳﺔ‪ .‬ﻭﻳﻤﻜﻦ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻨﻈﺮﻳﺔ ‪-‬‬ ‫ﻭﺗﻨﺺ ﻣﺴ ﹼﻠﻤﺔ ﺟﻤﻊ ﺍﻟﺰﻭﺍﻳﺎ ﻋﻠﻰ ﺃﻧﻪ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﻘﻄﺔ ‪ R‬ﺩﺍﺧﻞ ‪∠PQS‬‬ ‫ﻣﺜﻠﻬﺎ ﻣﺜﻞ ﺍﻟﺘﻌﺮﻳﻔﺎﺕ ﺃﻭ ﺍﻟﻤﺴﻠﻤﺎﺕ ‪ -‬ﻟﺘﺒﺮﻳﺮ ﺻﺤﺔ ﻋﺒﺎﺭﺍﺕ ﺃﺧﺮ￯‪.‬‬ ‫ﻓﺈﻥ‪. m∠PQR + m∠RQS = m∠PQS :‬‬ ‫ﻭﺍﻟﺒﺮﻫﺎﻥ ﺩﻟﻴﻞ ﻣﻨﻄﻘﻲ‪ ،‬ﺣﻴﺚ ﺗﺒﺮﺭ ﺻﺤﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻓﻴﻪ ﺑﻌﺒﺎﺭﺓ ﺗﻢ ﻗﺒﻮﻟﻬﺎ‬ ‫ﻋﻠﻰ ﺃﻧﻬﺎ ﺻﺤﻴﺤﺔ‪ .‬ﻭﻣﻦ ﺃﺷﻜﺎﻝ ﺍﻟﺒﺮﻫﺎﻥ‪ :‬ﺍﻟﺒﺮﻫﺎ ﹸﻥ ﺍﻟﺤﺮ‪ ،‬ﻭﻫﻮ ﺗﺒﺮﻳﺮ‬ ‫ﻭﺍﻟﻌﻜﺲ ﺻﺤﻴﺢ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫ﻛﺘﺎﺑﻲ ﻟﺼﺤﺔ ﺗﺨﻤﻴﻦ‪ ،‬ﻭﻳﺒ ﹼﻴﻦ ﺍﻟﺒﺮﻫﺎﻥ ﺻﺤﺔ ﻣﺎ ﹸﻳﺮﺍﺩ ﺇﺛﺒﺎﺗﻪ ﻭﻳﻄ ﹼﻮﺭ ﻧﻈﺎ ﹰﻣﺎ‬ ‫ﻣﻦ ﺍﻟﺘﺒﺮﻳﺮﺍﺕ ﺍﻻﺳﺘﻨﺘﺎﺟﻴﺔ‪.‬‬ ‫‪10F  1‬‬

‫‪‬‬ ‫‪‬‬ ‫‪Reasoning and Proof‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫ﺳﺎﺑ ﹰﻘﺎ ﺗﻌ ﱠﻠﻢ ﺍﻟﻄﻼﺏ ﺍﻟﻄﺮﻳﻘﺔ ﺍﻟﻌﻠﻤﻴﺔ‪ ،‬ﻭﺳﻮﻑ‬ ‫‪‬‬ ‫ﻳﻀﻌﻮﻥ ﺗﺨﻤﻴﻨﺎﺕ ﻳﺮﺑﻄﻮﻥ ﻣﻦ ﺧﻼﻟﻬﺎ ﺑﻴﻦ ﻣﺎ‬ ‫‪ ‬‬ ‫ﺗﻌﻠﻤﻮﻩ‪ ،‬ﻭﻣﺎ ﺳﻴﺘﻌﻠﻤﻮﻧﻪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻔﺼﻞ‪.‬‬ ‫‪ ‬‬ ‫• ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺍﻟﺒﺤﺚ ﻋﻦ ﻣﻌﻨﻰ‬ ‫‪‬‬ ‫ﺍﻟﻄﺮﻳﻘﺔ ﺍﻟﻌﻠﻤﻴﺔ ﻭﻋﻨﺎﺻﺮﻫﺎ‪ ،‬ﻭﺷﺠﻌﻬﻢ‬ ‫ﻋﻠﻰ ﺍﺳﺘﻌﻤﺎﻝ ﻣﻼﺣﻈﺎﺗﻬﻢ ﺃﻭ ﺍﻟﺮﺟﻮﻉ ﺇﻟﻰ‬ ‫‪ ‬‬ ‫ﺍﻟﻜﺘﺐ ﺍﻟﺘﻲ ﺩﺭﺳﻮﻫﺎ ﻓﻲ ﺍﻟﺼﻔﻮﻑ ﺍﻟﺴﺎﺑﻘﺔ‪.‬‬ ‫‪ ‬‬ ‫‪  ‬‬ ‫• ﻭ ﱢﺯﻉ ﺍﻟﻄﻼﺏ ﻣﺠﻤﻮﻋﺎﺕ‪ ،‬ﺑﺤﻴﺚ ﺗﺨﺘﺎﺭ‬ ‫ﻛﻞ ﻣﺠﻤﻮﻋﺔ ﻧﻈﺮﻳﺔ ﻋﻠﻤﻴﺔ‪ ،‬ﻭﺗﺤﺪﺩ‬ ‫‪‬‬ ‫ﻓﺮﺿﻴﺎﺗﻬﺎ‪ ،‬ﻭﺍﻟﺘﺠﺎﺭﺏ ﺍﻟﻤﻤﻜﻨﺔ ﻟﻠﺘﺤﻘﻖ‬ ‫‪‬‬ ‫ﻣﻦ ﺻﺤﺘﻬﺎ‪ ،‬ﻭﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﺘﻲ ﻳﻤﻜﻦ ﺍﻟﺘﻮﺻﻞ‬ ‫‪ ‬‬ ‫ﺇﻟﻴﻬﺎ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫• ﺍﻃﻠﺐ ﺇﻟﻰ ﻛﻞ ﻣﺠﻤﻮﻋﺔ ﺷﺮﺡ ﺗﺼﻤﻴﻢ‬ ‫ﺗﺠﺮﺑﺔ ﻣﻤﻜﻨﺔ‪ ،‬ﻋﻠﻰ ﺃﻥ ﺗﺘﻀﻤﻦ ﺍﻟﻤﺘﻐﻴﺮﺍﺕ‬ ‫‪‬‬ ‫ﺍﻟﺘﻲ ﺳﺘﺆﺧﺬ ﻓﻲ ﺍﻟﺤﺴﺒﺎﻥ‪ ،‬ﻭﺍﻟﻄﺮﻳﻘﺔ ﺍﻟﺘﻲ‬ ‫‪‬‬ ‫ﺳ ﹸﺘﺠﻤﻊ ﺑﻬﺎ ﺍﻟﺒﻴﺎﻧﺎﺕ‪ ،‬ﻭﻃﺮﻳﻘﺔ ﺗﺪﻭﻳﻨﻬﺎ‪،‬‬ ‫‪1 ‬‬ ‫‪ ‬‬ ‫ﻭﻛﻴﻔﻴﺔ ﻣﻌﺎﻟﺠﺔ ﻧﺘﺎﺋﺞ ﻫﺬﻩ ﺍﻟﺘﺠﺮﺑﺔ‬ ‫‪ ‬‬ ‫‪  1‬‬ ‫ﻭﺗﺤﻠﻴﻠﻬﺎ‪.‬‬ ‫‪    3‬‬ ‫‪ 2‬‬ ‫‪‬‬ ‫• ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﺼﻔﻮﺍ ﻛﻴﻒ ﻳﻤﻜﻦ‬ ‫‪ ‬‬ ‫ﻟﻠﻨﻈﺮﻳﺎﺕ ﺍﻟﻌﻠﻤﻴﺔ ﺍﻟﺘﻲ ﺍﺧﺘﺎﺭﻭﻫﺎ ﺍﻟﺘﻨﺒﺆ‬ ‫‪¿ÉgÈdGh ôj ÈàdG‬‬ ‫‪äGOôØŸG‬‬ ‫ﺑﺴﻠﻮﻙ ﺃﻭ ﻋﻤﻠﻴﺔ ﻣﻌﻴﻨﺔ‪ ،‬ﻭﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﺃﻥ‬ ‫‪ôjÈàdG‬‬ ‫ﻳﻘﺎﺭﻧﻮﺍ ﺑﻴﻦ ﺃﻧﻮﺍﻉ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻟﻤﺨﺘﻠﻔﺔ‪ ،‬ﻭﺃﻥ‬ ‫‪≥£æŸG‬‬ ‫ﻳﻌﺮﻓﻮﺍ ﺃﻧﻬﻢ ﻋﻨﺪﻣﺎ ﻳﺘﻮﺻﻠﻮﻥ ﺇﻟﻰ ﺍﺳﺘﻨﺘﺎﺝ‬ ‫‪áä«GWQÉôÑ°©ûddGG‬‬ ‫‪¿ÉgÈdG‬‬ ‫ﻣﻦ ﺗﺠﺮﺑﺔ‪ ،‬ﻓﺈﻧﻬﻢ ﻳﻜ ﹼﻮﻧﻮﻥ ﺗﻌﻤﻴ ﹰﻤﺎ‪ ،‬ﻭﻋﻨﺪﻣﺎ‬ ‫‪ 1 10‬‬ ‫ﻳﺘﻨﺒﺆﻭﻥ ﺑﺴﻠﻮﻙ ﻣﻌﻴﻦ‪ ،‬ﻓﺈﻧﻬﻢ ﻳﺘﻮﺻﻠﻮﻥ‬ ‫ﺇﻟﻰ ﺍﺳﺘﻨﺘﺎﺝ ﺃﻛﺜﺮ ﺗﺤﺪﻳ ﹰﺪﺍ‪.‬‬ ‫‪‬‬ ‫‪ ‬ﻗ ﹼﺪﻡ ﻣﻔﺮﺩﺍﺕ ﺍﻟﻔﺼﻞ ﻣﺴﺘﻌﻤ ﹰﻼ‬ ‫‪ ‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﺮﻳﻂ ﺍﻟﻤﻨﺎﺳﺐ ﻋﻨﺪ‬ ‫‪ ‬ﺃﻥ ﻳﻜﺘﺐ ﺍﻟﻄﻼﺏ ﻋﻦ ﺍﻟﺘﺒﺮﻳﺮ ﻭﺍﻟﺒﺮﻫﺎﻥ‪.‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﺪ ﱢﻭﻧﻮﺍ ﻣﻼﺣﻈﺎﺗﻬﻢ‬ ‫ﺍﻟﻨﻤﻂ ﺍﻵﺗﻲ‪:‬‬ ‫ﺩﺭﺍﺳﺔ ﺍﻟﻄﻼﺏ ﻛﻞ ﺩﺭﺱ ﻓﻲ ﻫﺬﺍ ﺍﻟﻔﺼﻞ‪ ،‬ﻭﻋﻠﻰ ﺍﻟﻄﻼﺏ‬ ‫ﺃﻥ ﻳﻀﻴﻔﻮﺍ ﺍﻟﻤﻔﺮﺩﺍﺕ ﺍﻟﺠﺪﻳﺪﺓ ﺗﺤﺖ ﺷﺮﻳﻂ ﺍﻟﻤﻔﺮﺩﺍﺕ‬ ‫ﺗﺤﺖ ﻛﻞ ﺷﺮﻳﻂ ﻓﻲ ﻣﻄﻮﻳﺎﺗﻬﻢ ﺧﻼﻝ ﺩﺭﺍﺳﺘﻬﻢ‬ ‫ﺍﻟﺘﻌﺮﻳﻒ‪ :‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻫﻲ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﻓﻲ ﺃﺛﻨﺎﺀ ﺩﺭﺍﺳﺔ ﻛﻞ ﺩﺭﺱ‪.‬‬ ‫ﺍﻟﺘﻲ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺘﻬﺎ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ‬ ‫ﻟﻠﻔﺼﻞ ‪ .1‬ﻭ ﹼﺟﻪ ﺍﻟﻄﻼﺏ ﺇﻟﻰ ﻛﺘﺎﺑﺔ ﺍﻟﻤﻼﺣﻈﺎﺕ ﻓﻲ ﺃﺛﻨﺎﺀ‬ ‫‪‬‬ ‫ﻗﺮﺍﺀﺗﻬﻢ ﺍﻟﺪﺭﺱ ﺃﻭ ﺳﻤﺎﻋﻬﻢ ﺍﻟﺸﺮﺡ‪ ،‬ﻋﻠﻰ ﺃﻥ ﺗﺘﻀﻤﻦ‬ ‫\"ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪ ،\"...‬ﻭﺍﻟ ﹶﻔﺮﺽ ﻫﻮ ﺍﻟﺠﻤﻠﺔ‬ ‫ﺍﻟﺘﻲ ﺗﻠﻲ ﻛﻠﻤﺔ )ﺇﺫﺍ( ﻣﺒﺎﺷﺮ ﹰﺓ‪ .‬ﻭﺍﻟﻨﺘﻴﺠﺔ ﻫﻲ‬ ‫ﻧﻤﻮﺫﺝ ﺑﻨﺎﺀ ﺍﻟﻤﻔﺮﺩﺍﺕ‪ ،‬ﺹ )‪. (9‬‬ ‫ﻫﺬﻩ ﺍﻟﻤﻼﺣﻈﺎﺕ ﺗﻌﺮﻳﻔﺎﺕ ﺍﻟﻤﺼﻄﻠﺤﺎﺕ ﻭﺍﻟﻤﻔﺎﻫﻴﻢ‬ ‫ﻳﻜﻤﻞ ﺍﻟﻄﻼﺏ ﻫﺬﺍ ﺍﻟﻨﻤﻮﺫﺝ ﺑﻜﺘﺎﺑﺔ ﺗﻌﺮﻳﻒ ﻛﻞ ﻣﻔﺮﺩﺓ‬ ‫ﺍﻟﺠﻤﻠﺔ ﺍﻟﺘﻲ ﺗﻠﻲ ﻛﻠﻤﺔ )ﻓﺈﻥ( ﻣﺒﺎﺷﺮﺓ‪.‬‬ ‫ﺟﺪﻳﺪﺓ ﺗﻈﻬﺮ ﻟﻬﻢ ﻓﻲ ﺃﺛﻨﺎﺀ ﺩﺭﺍﺳﺔ ﺍﻟﻔﺼﻞ ﺃﻭ ﻣﺜﺎﻝ ﻋﻠﻴﻬﺎ‪،‬‬ ‫ﺍﻷﺳﺎﺳﻴﺔ‪ ،‬ﻭﺷ ﱢﺠﻌﻬﻢ ﻋﻠﻰ ﺍﻟﺒﺤﺚ ﻋﻦ ﺃﻣﺜﻠﺔ ﻋﻠﻰ ﻛﻞ ﻧﻮﻉ‬ ‫ﻭﻳﺴﺘﻔﻴﺪﻭﻥ ﻣﻦ ﺫﻟﻚ ﻓﻲ ﺃﺛﻨﺎﺀ ﺍﻟﻤﺮﺍﺟﻌﺔ ﻭﺍﻻﺳﺘﻌﺪﺍﺩ‬ ‫ﻣﻦ ﺃﻧﻮﺍﻉ ﺍﻟﺘﺒﺮﻳﺮﺍﺕ ﺍﻟﻤﻨﻄﻘﻴﺔ‪ ،‬ﻭﺗﺪﻭﻳﻨﻬﺎ ﺧﻠﻒ ﺻﻔﺤﺎﺕ‬ ‫ﻣﺜﺎﻝ‪ :‬ﺇﺫﺍ ﺃﻧﻬﻴﺖ ﻭﺍﺟﺒﺎﺗﻚ ﺍﻟﻤﻨﺰﻟﻴﺔ‪ ،‬ﻓﺈﻧﻪ‬ ‫ﻳﻤﻜﻨﻚ ﻣﺘﺎﺑﻌﺔ ﺑﺮﺍﻣﺞ ﺍﻟﺘﻠﻔﺎﺯ‪.‬‬ ‫ﻻﺧﺘﺒﺎﺭ ﺍﻟﻔﺼﻞ‪.‬‬ ‫ﻣﻄﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫ﺳﺆﺍﻝ‪ :‬ﻫﻞ ﻫﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﻋﻠﻰ ﺻﻮﺭﺓ‬ ‫\"ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪...‬؟ ﻧﻌﻢ‪ .‬ﻣﺎ ﺍﻟ ﹶﻔﺮﺽ؟ ﺃﻧﻬﻴﺖ‬ ‫ﻭﺍﺟﺒﺎﺗﻚ ﺍﻟﻤﻨﺰﻟﻴﺔ‪ .‬ﻣﺎ ﺍﻟﻨﺘﻴﺠﺔ؟ﻳﻤﻜﻨﻚ‬ ‫ﻣﺘﺎﺑﻌﺔ ﺑﺮﺍﻣﺞ ﺍﻟﺘﻠﻔﺎﺯ‪.‬‬ ‫‪ 1 10‬‬

1 1       1   ‫ﺍﺳﺘﻌﻤﻞ ﻧﺘﺎﺋﺞ ﺍﻻﺧﺘﺒﺎﺭ ﺍﻟﺴﺮﻳﻊ ﻭﻣﺨﻄﻂ‬ ￯‫ﺍﻟﻤﻌﺎﻟﺠﺔ؛ ﻟﻤﺴﺎﻋﺪﺗﻚ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﻣﺴﺘﻮ‬   \"‫ ﻓﻘﻢ‬... ‫ ﻭﺍﻟﻌﺒﺎﺭﺓ \"ﺇﺫﺍ‬.‫ﺍﻟﻤﻌﺎﻟﺠﺔ ﺍﻟﻤﻨﺎﺳﺐ‬ ￯‫ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺗﺴﺎﻋﺪﻙ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﺍﻟﻤﺴﺘﻮ‬ 1  (1-1 ‫)ﻳﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﺱ‬ ‫ ﻭﺍﻗﺘﺮﺍﺡ ﻣﺼﺎﺩﺭ ﻟﻜﻞ‬،‫ﺍﻟﻤﻨﺎﺳﺐ ﻟﻠﻤﻌﺎﻟﺠﺔ‬ .‫ ﺍﻟ ﹸﻤﻌﻄﺎﺓ‬x ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻨﺪ ﻗﻴﻤﺔ‬ .￯‫ﻣﺴﺘﻮ‬ . x = 6 ‫ ﺇﺫﺍ ﻛﺎﻧﺖ‬x2 – 2x + 11 ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ‬ 180 (x – 2) , x = 8 (2 31 4x + 7 , x = 6 (1 1080 14 5x2 – 3x , x = 2 (3  x2 – 2x + 11 5 _x(x - 3) , x = 5 (4 x=6 = (6)2 – 2(6) + 11 2  = 36 – 2(6) + 11 12 x + (x + 1) + (x + 2) , x = 3 (5  = 36 – 12 + 11  = 35 :‫ﺍﻛﺘﺐ ﻛﻞ ﺗﻌﺒﻴﺮ ﻟﻔﻈﻲ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻠﻰ ﺻﻮﺭﺓ ﻋﺒﺎﺭﺓ ﺟﺒﺮﻳﺔ‬ 5x - 8 .‫( ﺃﻗﻞ ﻣﻦ ﺧﻤﺴﺔ ﺃﻣﺜﺎﻝ ﻋﺪﺩ ﺑﺜﻤﺎﻧﻴﺔ‬6  x2+ 3 .‫( ﺃﻛﺜﺮ ﻣﻦ ﻣﺮﺑﻊ ﻋﺪﺩ ﺑﺜﻼﺛﺔ‬7   2  (1-8 ‫ ﺇﻟﻰ‬1-6 ‫ﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﻭﺱ‬ ‫)ﻳ‬:‫ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‬ 5 8x – 10 = 6x (8 1 . 36x – 14 = 16x + 58 ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‬     36x – 14 = 16x + 58 -7 18 + 7x = 10x + 39 (9   25%  16x 36x – 14 – 16x = 16x + 58 –16x 2.3 3(11x – 7) = 13x + 25 (10    20x – 14 = 58 1.1 _3 x + 1 = 5 – 2x (11 14 20x – 14 + 14 = 58 + 14 2 ‫ ﺭﻳﺎ ﹰﻻ؛ ﻟﺘﻘﺮﺃﻫﺎ‬52 ‫ ﻛﺘﺐ ﺑﻘﻴﻤﺔ‬4 ‫ﺍﺷﺘﺮﺕ ﻋﺎﺋﺸﺔ‬  (12  20x = 72 20 _20x ،‫ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻜﺘﺐ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﺴﻌﺮ‬.‫ﻓﻲ ﺃﺛﻨﺎﺀ ﺍﻹﺟﺎﺯﺓ ﺍﻟﺼﻴﻔﻴﺔ‬ _72 (2)  20 = .‫ ﺛﻢ ﹸﺣ ﱠﻠﻬﺎ‬،‫ﻓﺎﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻹﻳﺠﺎﺩ ﺛﻤﻦ ﺍﻟﻜﺘﺎﺏ ﺍﻟﻮﺍﺣﺪ‬ (10) 20 ‫ ﺭﻳﺎ ﹰﻻ‬13 ‫ ؛‬4x = 52 x = 3.6 3  (1-8 ‫)ﻳﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﺱ‬  :‫ ﻟﻺﺟﺎﺑﺔ ﻋﻤﺎ ﻳﺄﺗﻲ‬3 ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻓﻲ ﻣﺜﺎﻝ‬ www.obeikaneducation.com C D ، m∠BXA = (3x + 5)° :‫ﺇﺫﺍ ﻛﺎﻥ‬ .‫( ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﻨﻔﺮﺟﺘﻴﻦ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‬13 B XE .x ‫ ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬،m∠DXE = 56° ∠BXD, ∠AXE   A m∠BXA = m∠DXE 3x + 5 = 56 ∠CXD, ∠DXE .‫( ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺘﺎﻣﺘﻴﻦ‬14  2   3x = 51  50%   x = 17 .‫( ﻋ ﱢﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺠﺎﻭﺭﺗﻴﻦ ﻣﺘﻜﺎﻣﻠﺘﻴﻦ ﻓﻲ ﺁﻥ ﻭﺍﺣﺪ‬15    5 ∠DXE, ∠EXA   3 www.obeikaneducation.com ،m∠EXA = (3x + 2)° ‫ ﻭ‬m∠DXB = 116° :‫( ﺇﺫﺍ ﻛﺎﻥ‬16 38 .x ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬ m∠CXD = (6x – 13)° :‫( ﺇﺫﺍ ﻛﺎﻥ‬17 6 .x ‫ ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬،m∠DXE = (10x + 7)° ‫ﻭ‬ www.obeikaneducation.com   2  11 1 1 11 1 1

 1- 1  Inductive Reasoning and Conjection      1    ‫ ﻳﺘﻢ ﺗﺤﻠﻴﻞ ﺇﺟﺎﺑﺎﺕ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺷﺨﺎﺹ ﻋﻦ ﺃﺳﺌﻠﺔ‬،‫ﻓﻲ ﺃﺑﺤﺎﺙ ﺍﻟﺘﺴﻮﻳﻖ‬    ‫ ﺛﻢ ﻳﺘﻢ ﺍﻟﺒﺤﺚ ﻋﻦ ﻧﻤﻄﻴﺔ ﻣﻌﻴﻨﺔ ﻓﻲ ﺍﻹﺟﺎﺑﺎﺕ ﺣﺘﻰ‬،‫ﻣﺤﺪﺩﺓ ﺣﻮﻝ ﺍﻟﻤﻨﺘﺞ‬  .‫ ﻭﺗﺴﻤﻰ ﻫﺬﻩ ﺍﻟﻌﻤﻠﻴﺔ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ‬.‫ﺍﻟﻮﺻﻮﻝ ﺇﻟﻰ ﻧﺘﻴﺠﺔ‬  1-1   ،‫ﺗﻤﺜﻴﻞ ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ ﺍﻟﻜﻤﻴﺎﺕ‬ ‫ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻫﻮ ﺗﺒﺮﻳﺮ ﹸﺗﺴﺘﻌﻤﻞ ﻓﻴﻪ ﺃﻣﺜﻠﺔ ﻣﺤﺪﺩﺓ ﻟﻠﻮﺻﻮﻝ‬ ، ‫ﺑﺎﺳﺘﻌﻤﺎﻝ ﻧﻤﺎﺫﺝ ﺣﺴﻴﺔ ﻭﺟﺪﺍﻭﻝ‬    ‫ ﻭﻭﺻﻒ‬،‫ﻭﺗﻤﺜﻴﻼﺕ ﺑﻴﺎﻧﻴﺔ ﻭﻣﺨﻄﻄﺎﺕ‬ ‫ ﻓﺈﻧﻚ ﺗﺴﺘﻌﻤﻞ‬،‫ ﻭﻋﻨﺪﻣﺎ ﺗﻔﺘﺮﺽ ﺍﺳﺘﻤﺮﺍﺭ ﻧﻤﻂ ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﻮﺗﻴﺮﺓ‬.‫ﺇﻟﻰ ﻧﺘﻴﺠﺔ‬   ‫ ﻭ ﹸﺗﺴ ﹼﻤﻰ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻨﻬﺎﺋﻴﺔ ﺍﻟﺘﻲ ﺗﻮﺻﻠﺖ ﺇﻟﻴﻬﺎ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ‬،‫ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ‬   .‫ ﻭﻣﻌﺎﺩﻻﺕ‬، ‫ﻟﻔﻈﻲ‬  1-1 12345  . ‫ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﺗﺨﻤﻴ ﹰﻨﺎ‬ qqqqq     ‫ﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ‬ qqqqq  ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻟﻜﺘﺎﺑﺔ‬ qqqqq   qqqqq  .‫ﺗﺨﻤﻴﻦ‬ qqqqq   1-1 qqqqq ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻟﻤﻨﻄﻘﻲ ﻹﺛﺒﺎﺕ ﺻﺤﺔ‬ inductive reasoning .‫ﻋﺒﺎﺭﺍﺕ ﻭﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ‬  q q     2  conjecture    . “‫ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻗﺮﺍﺀﺓ ﻓﻘﺮﺓ ”ﻟﻤﺎﺫﺍ؟‬  counterexample  www.obeikaneducation.com  ‫• ﻣﺎ ﺍﻷﺷﻴﺎﺀ ﺍﻟﺘﻲ ﺗﻬﻢ ﺑﺎﺣﺚ ﺍﻟﺘﺴﻮﻳﻖ؟‬  ‫ ﻣﻘﺎﺭﻧﺘﻪ‬،‫ ﻣﺒﻴﻌﺎﺕ ﺍﻟﻤﻨﺘﺞ‬:‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‬    .‫ﺑﺎﻟﻤﻨﺘﺠﺎﺕ ﺍﻟﻤﻨﺎﻓﺴﺔ‬  1    ‫• ﻟﻤﺎﺫﺍ ﻳﻘﻮﻡ ﺍﻟﺒﺎﺣﺚ ﺑﺘﻮﺟﻴﻪ ﺍﻷﺳﺌﻠﺔ ﺇﻟﻰ‬ .‫ ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‬،‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ‬    ‫ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺷﺨﺎﺹ ﻓﻘﻂ؟‬ ...... ،‫ ﺻﺒﺎ ﹰﺣﺎ‬10:30 ،‫ ﺻﺒﺎ ﹰﺣﺎ‬9:50 ،‫ ﺻﺒﺎ ﹰﺣﺎ‬9:10 ،‫ ﺻﺒﺎ ﹰﺣﺎ‬8:30 :‫( ﻣﻮﺍﻋﻴﺪ ﻭﺻﻮﻝ ﺍﻟﺤﺎﻓﻼﺕ ﺇﻟﻰ ﻣﺤﻄﺔ ﺍﻟﺮﻛﻮﺏ ﻫﻲ‬a ،‫ ﻓﻲ ﻛﺜﻴﺮ ﻣﻦ ﺍﻷﺣﻴﺎﻥ‬:‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‬ ‫ﻳﺼﻌﺐ ﺗﻮﺟﻴﻪ ﺍﻷﺳﺌﻠﺔ ﺇﻟﻰ ﺟﻤﻴﻊ‬ .‫ ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‬1 ‫ ﻭﻟﺬﻟﻚ ﺗﻮ ﱠﺟﻪ ﺍﻷﺳﺌﻠﺔ ﺇﻟﻰ‬،‫ﺍﻟﻤﺴﺘﻬﻠﻜﻴﻦ‬ ...... ‫ ﺻﺒﺎ ﹰﺣﺎ‬10:30 ،‫ ﺻﺒﺎ ﹰﺣﺎ‬9:50 ،‫ ﺻﺒﺎ ﹰﺣﺎ‬9:10 ،‫ ﺻﺒﺎ ﹰﺣﺎ‬8:30  .‫ﻣﺠﻤﻮﻋﺔ ﻣﻤﺜﻠﺔ‬ 40  40  40  .‫ ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‬2 ‫ ﻣﻮﻋﺪ ﻭﺻﻮﻝ‬.‫ ﺩﻗﻴﻘﺔ ﻋﻦ ﻣﻮﻋﺪ ﻭﺻﻮﻝ ﺍﻟﺤﺎﻓﻠﺔ ﺍﻟﺘﻲ ﺳﺒﻘﺘﻬﺎ‬40 ‫ﻳﺰﻳﺪ ﻣﻮﻋﺪ ﻭﺻﻮﻝ ﺍﻟﺤﺎﻓﻠﺔ‬  .‫ ﺻﺒﺎ ﹰﺣﺎ‬11:10 ‫ ﺩﻗﻴﻘﺔ ﺃﻭ‬40 + ‫ ﺻﺒﺎ ﹰﺣﺎ‬10:30 ‫ﺍﻟﺤﺎﻓﻠﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺳﻮﻑ ﻳﻜﻮﻥ‬ (b 4 10 18 28 40 . . . . . . ‫ ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‬ 1  4 10 18 28 40 +6 +8 +10 +12      6, 8, 10, 12, ......  354- 430 ‫؛ ﻟﺬﺍ ﺳﻴﺰﻳﺪ ﻋﺪﺩ ﺍﻟﻘﻄﻊ‬6, 8, 10, 12... ‫ ﺗﺰﺩﺍﺩ ﺃﻋﺪﺍﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﻤﻘﺪﺍﺭ‬:‫ ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‬ 2 ‫ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ؛ ﻭﻋﻠﻴﻪ ﻓﺈﻥ‬14 ‫ ﺃﻭ‬2 + 12 ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﻋﻠﻰ ﺳﺎﺑﻘﻪ ﺑﻤﻘﺪﺍﺭ‬   .‫ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ‬54 ‫ ﺃﻭ‬14 + 40 ‫ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﺳﻴﺤﺘﻮﻱ ﻋﻠﻰ‬   .‫ ﺍﺭﺳﻢ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ؛ ﻟﻜﻲ ﺗﺘﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺗﺨﻤﻴﻨﻚ‬   54     1 12 1-1     (14, 15) •  (15, 18) • (14, 15, 18) • (6) •  (6) • (6) •  (6)  •  (9)  • (6) • (8) • (10)  • (8) • (9) • (9) • (10)  •  1 12

‫‪ (1C‬ﻳﻘﺴﻢ ﻛﻞ ﻣﺜﻠﺚ ﻣﻈﻠﻞ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﺴﺎﺑﻖ ﺇﻟﻰ‬ ‫✓ ‪‬‬ ‫ﺃﺭﺑﻌﺔ ﻣﺜﻠﺜﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﺍﻷﺿﻼﻉ ﻳﺘﻮﺳﻄﻬﺎ ﻣﺜﻠﺚ ﺃﺑﻴﺾ‪.‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫‪ (1A‬ﺍﻟﺸﻬﺮ ﺍﻟﺘﺎﻟﻲ ﻓﻲ‬ ‫‪‬‬ ‫‪ (1A‬ﻣﺘﺘﺎﺑﻌﺔ ﺃﺷﻬﺮ‪ :‬ﺻﻔﺮ‪ ،‬ﺭﺟﺐ‪ ،‬ﺫﻭ ﺍﻟﺤﺠﺔ‪ ،‬ﺟﻤﺎﺩ￯ ﺍﻷﻭﻟﻰ‪...... ،‬‬ ‫ﺍﻟﻤﺘﺘﺎﺑﻌﺔ ﻳﺄﺗﻲ ﺑﻌﺪ ﺧﻤﺴﺔ‬ ‫‪10, 4, -2, -8, ...... (1B‬‬ ‫ﺃﺷﻬﺮ ﻣﻦ ﺍﻟﺸﻬﺮ ﺍﻟﺴﺎﺑﻖ؛‬ ‫‪ 1, 3 ‬ﻳﺒ ﱢﻴﻨﺎﻥ ﻛﻴﻔﻴﺔ ﻭﺿﻊ ﺗﺨﻤﻴﻨﺎﺕ‬ ‫ﺷﻮﺍﻝ‪.‬‬ ‫ﺣﻮﻝ ﺃﻧﻤﺎﻁ ﻣﻌﻄﺎﺓ‪.‬‬ ‫‪ (1B‬ﺍﻟﻌﺪﺩ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﺍﻟﻤﺘﺘﺎﺑﻌﺔ ‪(1C‬‬ ‫‪ 2 ‬ﻳﺒﻴﻦ ﻛﻴﻔﻴﺔ ﻭﺿﻊ ﺗﺨﻤﻴﻦ ﺣﻮﻝ‬ ‫ﻳﻘﻞ ﺑﻤﻘﺪﺍﺭ ‪ 6‬ﻋﻦ ﺍﻟﻌﺪﺩ‬ ‫ﺷﻜﻞ ﻭﺍﺣﺪ‪.‬‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ‪-14‬‬ ‫✓ ‪‬‬ ‫‪139‬‬ ‫‪......‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺗﻤﺎﺭﻳﻦ \"ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻤﻚ\" ﺑﻌﺪ ﻛﻞ‬ ‫‪27‬‬ ‫ﻣﺜﺎﻝ؛ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻣﺪ￯ ﻓﻬﻢ ﺍﻟﻄﻠﺒﺔ ﺍﻟﻤﻔﺎﻫﻴﻢ‪.‬‬ ‫ﻟﻮﺿﻊ ﺗﺨﻤﻴﻨﺎﺕ ﺟﺒﺮﻳﺔ ﺃﻭ ﻫﻨﺪﺳﻴﺔ ﻳﺠﺐ ﺃﻥ ﺗﻘﺪﻡ ﺃﻣﺜﻠﺔ‪.‬‬ ‫‪ 2‬‬ ‫ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻟﻜ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺃﻋ ﹺﻂ ﺃﻣﺜﻠﺔ ﻋﺪﺩﻳﺔ ﺃﻭ ﺍﺭﺳﻢ ﺃﺷﻜﺎ ﹰﻻ ﺗﺴﺎﻋﺪ ﻋﻠﻰ‬ ‫ﺍﻟﻮﺻﻮﻝ ﻟﻬﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪‬‬ ‫‪ (a‬ﻧﺎﺗﺞ ﺟﻤﻊ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪.‬‬ ‫‪‬‬ ‫‪  1‬ﺍﻛﺘﺐ ﺃﻣﺜﻠﺔ‪.‬‬ ‫‪‬‬ ‫ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ‬ ‫‪1‬‬ ‫‪1 + 3 = 4 , 1 + 5 = 6 , 3 + 5 = 8 , 7 + 9 = 16‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ‬ ‫‪ ‬‬ ‫‪  2‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‪.‬‬ ‫ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫‪‬‬ ‫ﻻﺣﻆ ﺃﻥ ﺍﻷﻋﺪﺍﺩ ‪ 4, 6, 8, 16‬ﺟﻤﻴﻌﻬﺎ ﺯﻭﺟﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪2, 4, 12, 48, 240 (a‬‬ ‫‪ 3‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪.‬‬ ‫‪‬‬ ‫ﻧﺎﺗﺞ ﺟﻤﻊ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ ﻫﻮ ﻋﺪﺩ ﺯﻭﺟﻲ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺍﻟﺘﺨﻤﻴﻦ‪ :‬ﺍﺿﺮﺏ ﺍﻟﺤﺪ ‪ n‬ﻓﻲ‬ ‫‪ (b‬ﺍﻟﻘﻄﻌﺘﺎﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺘﺎﻥ ﺍﻟﻮﺍﺻﻠﺘﺎﻥ ﺑﻴﻦ ﻛﻞ ﺭﺃﺳﻴﻦ ﻣﺘﻘﺎﺑﻠﻴﻦ ﻓﻲ ﺍﻟﻤﺴﺘﻄﻴﻞ‪.‬‬ ‫‪ ‬‬ ‫ﺍﻟﻌﺪﺩ ‪ n + 1‬؛ ﻟﻠﺤﺼﻮﻝ ﻋﻠﻰ‬ ‫‪1 ‬‬ ‫‪‬‬ ‫‪1-5‬‬ ‫ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻟﻪ؛ ‪1440‬‬ ‫‪(b‬‬ ‫‪39‬‬ ‫‪...‬‬ ‫‪  2‬ﻻﺣﻆ ﺃﻥ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻮﺍﺻﻠﺔ ﺑﻴﻦ ﻛﻞ ﺭﺃﺳﻴﻦ ﻣﺘﻘﺎﺑﻠﻴﻦ ﻓﻲ ﻛﻞ ﻣﺴﺘﻄﻴﻞ ﺗﺒﺪﻭ‬ ‫ﻣﺘﺴﺎﻭﻳﺔ‪ .‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﻤﺴﻄﺮﺓ ﺃﻭ ﺍﻟﻔﺮﺟﺎﺭ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﺫﻟﻚ‪.‬‬ ‫‪18‬‬ ‫ﺍﻟﺘﺨﻤﻴﻦ‪ :‬ﺍﺟﻤﻊ ﺍﻟﻌﺪﺩ ‪3n + 3‬‬ ‫‪  3‬ﺍﻟﺘﺨﻤﻴﻦ‪ :‬ﺍﻟﻘﻄﻌﺘﺎﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺘﺎﻥ ﺍﻟﻮﺍﺻﻠﺘﺎﻥ ﺑﻴﻦ ﻛﻞ ﺭﺃﺳﻴﻦ ﻣﺘﻘﺎﺑﻠﻴﻦ ﻓﻲ ﺍﻟﻤﺴﺘﻄﻴﻞ ﻣﺘﻄﺎﺑﻘﺘﺎﻥ‪.‬‬ ‫ﺇﻟﻰ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻓﻲ‬ ‫ﺍﻟﺸﻜﻞ ﺫﻱ ﺍﻟﺘﺮﺗﻴﺐ ‪ n‬؛‬ ‫‪ (2A‬ﻧﺎﺗﺞ ﺟﻤﻊ ﻋﺪﺩﻳﻦ ﺯﻭﺟﻴﻴﻦ ﻋﺪﺩ ﺯﻭﺟﻲ؛ ﺃﻣﺜﻠﺔ‪:‬‬ ‫✓ ‪‬‬ ‫‪ (2C‬ﻣﺠﻤﻮﻉ ﻣﺮﺑ ﹶﻌﻲ ﻋﺪﺩﻳﻦ‬ ‫ﻟﻠﺤﺼﻮﻝ ﻋﻠﻰ ﻋﺪﺩ ﺍﻟﻘﻄﻊ‬ ‫ﻛﻠﻴﻴﻦ ﻣﺘﺘﺎﻟﻴﻴﻦ ﻋﺪﺩ ﻓﺮﺩﻱ؛‬ ‫‪2 + 4 = 6, 8 + 10 = 18, 20 + 16 = 36‬‬ ‫‪ (2A‬ﻧﺎﺗﺞ ﺟﻤﻊ ﻋﺪﺩﻳﻦ ﺯﻭﺟﻴﻴﻦ‪.‬‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ؛ ‪30‬‬ ‫ﺃﻣﺜﻠﺔ‪:‬‬ ‫‪ (2B‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ AB‬ﻭ ‪ ، EF‬ﺇﺫﺍ ﻛﺎﻧﺖ‪ AB = CD :‬ﻭ ‪ CD = EF‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫‪12 + 22 = 5,‬‬ ‫‪2‬‬ ‫‪ (2C‬ﻣﺠﻤﻮﻉ ﻣﺮﺑ ﹶﻌﻲ ﻋﺪﺩﻳﻦ ﻛﻠﻴﻴﻦ ﻣﺘﺘﺎﻟﻴﻴﻦ‪.‬‬ ‫‪22 + 32 = 13,‬‬ ‫‪52 + 62 = 61‬‬ ‫ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ‬ ‫‪13  1- 1‬‬ ‫ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺃﻋﻂ ﺃﻣﺜﻠﺔ‬ ‫‪‬‬ ‫ﻋﺪﺩﻳﺔ ﺃﻭ ﺍﺭﺳﻢ ﺃﺷﻜﺎ ﹰﻻ ﺗﺴﺎﻋﺪ ﻋﻠﻰ‬ ‫‪ AB = EF (2B‬؛ ﺃﻣﺜﻠﺔ‪:‬‬ ‫ﺍﻟﻮﺻﻮﻝ ﺇﻟﻰ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪ (a‬ﻣﺠﻤﻮﻉﻋﺪﺩﺯﻭﺟﻲﻭﻋﺪﺩﻓﺮﺩﻱ‪.‬‬ ‫‪ ‬ﺃﻋﻂ ﺍﻟﻄﻼﺏ‬ ‫‪F‬‬ ‫ﺗﺨﻤﻴﻦ‪ :‬ﻣﺠﻤﻮﻉ ﺍﻟﻌﺪﺩ ﺍﻟﺰﻭﺟﻲ‬ ‫ﻋﺪﺓ ﺃﻧﻤﺎﻁ‪ ،‬ﻭﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﺗﺨﻤﻴﻦ‬ ‫‪A‬‬ ‫‪C‬‬ ‫ﻭﺍﻟﻌﺪﺩ ﺍﻟﻔﺮﺩﻱ ﻳﻜﻮﻥ ﻓﺮﺩ ﹰﹼﻳﺎ؛‬ ‫ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪ ،‬ﺛﻢ ﺍﺧﺘﺮ‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪3 + 4 = 7 , 5 + 10 = 15‬‬ ‫ﺑﻌﺾ ﺍﻟﻄﻼﺏ‪ ،‬ﻭﺩﻋﻬﻢ ﻳﻌﺮﺿﻮﺍ‬ ‫‪D‬‬ ‫‪ (b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪: L, M, N‬‬ ‫ﺇﺟﺎﺑﺎﺗﻬﻢ‪ ،‬ﻭﻳﺸﺮﺣﻮﺍ ﺗﺒﺮﻳﺮﺍﺗﻬﻢ ﺃﻣﺎﻡ‬ ‫‪A‬‬ ‫‪C‬‬ ‫‪LM = 20, MN = 6, LN = 14‬‬ ‫ﺯﻣﻼﺋﻬﻢ‪.‬‬ ‫‪F BD‬‬ ‫‪B‬‬ ‫‪L NM‬‬ ‫‪14‬‬ ‫‪6‬‬ ‫‪20‬‬ ‫‪AB‬‬ ‫ﺍﻟﺘﺨﻤﻴﻦ‪ :‬ﺍﻟﻨﻘﺎﻁ ‪ L , M , N‬ﺗﻘﻊ‬ ‫‪CD‬‬ ‫ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫‪EF‬‬ ‫‪13  1-1‬‬

‫ﺗﻌﺘﻤﺪ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﻓﻲ ﺍﻟﻤﻮﺍﻗﻒ ﺍﻟﺤﻴﺎﺗﻴﺔ ﻋﻠﻰ ﺑﻴﺎﻧﺎﺕ ﻳﺘﻢ ﺟﻤﻌﻬﺎ ﺣﻮﻝ ﻣﻮﺿﻮﻉ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪ ‬‬ ‫‪  3‬‬ ‫‪ ‬ﺍﻟﺠﺪﻭﻝ ﺃﺩﻧﺎﻩ ﻳﺒ ﱢﻴﻦ‬ ‫‪3‬‬ ‫‪ ‬ﻗﺎﻡ ﺻﺎﺣﺐ ﺻﺎﻟﻮﻥ ﺣﻼﻗﺔ ﺑﺠﻤﻊ ﻣﻌﻠﻮﻣﺎﺕ ﺣﻮﻝ ﻋﺪﺩ ﺍﻟﺰﺑﺎﺋﻦ ﺍﻟﺬﻳﻦ ﻳﺮﺗﺎﺩﻭﻥ ﺍﻟﺼﺎﻟﻮﻥ ﺃﻳﺎﻡ ﺍﻷﺭﺑﻌﺎﺀ‬ ‫ﻣﺒﻴﻌﺎﺕ ﻣﺤﻞ ﺗﺠﺎﺭﻱ ﻟﻸﺷﻬﺮ ﺍﻟﺜﻼﺛﺔ‬ ‫ﻭﺍﻟﺨﻤﻴﺲ ﻭﺍﻟﺠﻤﻌﺔ ﻣﺪﺓ ﺳﺘﺔ ﺃﺷﻬﺮ؛ ﻛﻲ ﻳﻘﺮﺭ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻳﺠﺐ ﺯﻳﺎﺩﺓ ﻋﺪﺩ ﺍﻟﺤﻼﻗﻴﻦ ﺍﻟﻌﺎﻣﻠﻴﻦ ﻟﺪﻳﻪ ﻓﻲ ﺍﻷﻳﺎﻡ‬ ‫ﺍﻷﻭﻟﻰ ﻣﻦ ﺍﻓﺘﺘﺎﺣﻪ‪ ،‬ﻭﻳﺮﻳﺪ ﺻﺎﺣﺒﻪ ﺃﻥ‬ ‫ﺍﻟﺜﻼﺛﺔ ﺍﻷﺧﻴﺮﺓ ﻣﻦ ﻛﻞ ﺃﺳﺒﻮﻉ‪.‬‬ ‫ﻳﺘﻮﻗﻊ ﻣﻘﺪﺍﺭ ﻣﺒﻴﻌﺎﺗﻪ ﻓﻲ ﺍﻟﺸﻬﺮ ﺍﻟﺮﺍﺑﻊ‪.‬‬ ‫‪  ‬‬ ‫‪6‬‬ ‫‪5‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪450‬‬ ‫‪540‬‬ ‫‪406‬‬ ‫‪321‬‬ ‫‪255‬‬ ‫‪225‬‬ ‫‪‬‬ ‫‪705‬‬ ‫‪‬‬ ‫‪832‬‬ ‫‪685 692 642 635 552‬‬ ‫‪‬‬ ‫‪1987‬‬ ‫‪‬‬ ‫‪746 712 652 658 603‬‬ ‫‪1971‬‬ ‫‪1810‬‬ ‫‪1615‬‬ ‫‪1548‬‬ ‫‪1380‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪20000‬‬ ‫‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪    .‬‬ ‫‪‬‬ ‫‪60000‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪180000‬‬ ‫‪2000‬‬ ‫ﺑﻤﺎ ﺃﻧﻚ ﺗﺒﺤﺚ ﻋﻦ ﻧﻤﻂ ﻟﻪ ﻋﻼﻗﺔ ﺑﺎﻟﺰﻣﻦ‪ ،‬ﺇﺫﻥ‬ ‫‪2‬‬ ‫‪1800‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺷﻜﻞ ﺍﻻﻧﺘﺸﺎﺭ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪،‬‬ ‫‪‬‬ ‫‪1600‬‬ ‫ﺑﺠﻌﻞ ﺍﻟﻤﺤﻮﺭ ﺍﻷﻓﻘﻲ ﻳﻤﺜﻞ ﺍﻷﺷﻬﺮ ﻭﺍﻟﻤﺤﻮﺭ‬ ‫‪ ‬‬ ‫‪3‬‬ ‫‪1400‬‬ ‫ﺍﻟﺮﺃﺳﻲ ﻳﻤﺜﻞ ﻋﺪﺩ ﺍﻟﺰﺑﺎﺋﻦ‪ .‬ﺍﺭﺳﻢ ﻛﻞ ﻣﺠﻤﻮﻋﺔ ﻣﻦ‬ ‫‪‬‬‫‪1200‬‬ ‫‪ ‬‬ ‫ﺍﻟﺒﻴﺎﻧﺎﺕ ﺑﺎﺳﺘﻌﻤﺎﻝ ﻟﻮﻥ ﻣﺨﺘﻠﻒ‪ ،‬ﻭﺿﻊ ﻣﻔﺘﺎ ﹰﺣﺎ‬ ‫‪‬‬ ‫‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ‬ ‫‪1000‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪‬‬ ‫ﻟﻠﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ‪.‬‬ ‫‪800‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪600‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻳﻌﺘﻤﺪ ﻋﻠﻰ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪ ،‬ﻣﻔﺴ ﹰﺮﺍ ﻛﻴﻒ‬ ‫‪400‬‬ ‫‪1 23456‬‬ ‫ﻳﺆﻳﺪ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪‬‬‫‪200000‬‬ ‫‪1 234‬‬ ‫‪200‬‬ ‫‪160000‬‬ ‫‪‬‬ ‫‪120000‬‬ ‫‪‬‬ ‫‪0‬‬ ‫‪80000‬‬ ‫ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ ﻓﻲ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪ .‬ﻻﺣﻆ ﺃﻥ ﻋﺪﺩ‬ ‫‪40000‬‬ ‫ﺍﻟﺰﺑﺎﺋﻦ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻷﻳﺎﻡ ﺍﻟﺜﻼﺛﺔ ﻳﺒﺪﻭ ﺁﺧ ﹰﺬﺍ ﻓﻲ ﺍﻻﺯﺩﻳﺎﺩ‬ ‫‪0‬‬ ‫ﺑﻤﺮﻭﺭ ﺍﻷﺷﻬﺮ‪ ،‬ﻛﻤﺎ ﺃﻥ ﺍﻟﻤﺠﻤﻮﻉ ﺍﻟﻜﻠﻲ ﻳﺰﺩﺍﺩ ﻛﻞ ﺷﻬﺮ ﻋﻦ ﺍﻟﺸﻬﺮ ﺍﻟﺴﺎﺑﻖ‪.‬‬ ‫ـ ﺱ ﺃﻡ ﻕﻝ ﻩـ ‪1‬‬ ‫ﺑﻴﺎﻧﺎﺕ ﻫﺬﺍ ﺍﻟﻤﺴﺢ ﺗﺆﻳﺪ ﺗﺨﻤﻴﻦ ﺻﺎﺣﺐ ﺻﺎﻟﻮﻥ ﺍﻟﺤﻼﻗﺔ ﺑﺄﻥ ﺍﻟﻌﻤﻞ ﻓﻲ ﺍﻷﻳﺎﻡ ﺍﻟﺜﻼﺛﺔ ﺍﻷﺧﻴﺮﺓ ﻣﻦ ﻛﻞ‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻤﺒﻴﻌﺎﺕ ﺍﻷﺳﺒﻮﻉ‬ ‫ﺃﺳﺒﻮﻉ ﻳﺰﺩﺍﺩ؛ ﻣﻤﺎ ﻳﺘﻄﻠﺐ ﺯﻳﺎﺩﺓ ﻋﺪﺩ ﺍﻟﺤﻼﻗﻴﻦ ﺍﻟﻌﺎﻣﻠﻴﻦ ﻟﺪﻳﻪ ﻓﻲ ﻫﺬﻩ ﺍﻷﻳﺎﻡ‪.‬‬ ‫ﺍﻟﺮﺍﺑﻊ‪ ،‬ﻭﺑ ﹼﺮﺭ ﻫﺬﺍ ﺍﻟﺘﻨﺒﺆ ﺃﻭ ﺍﻻﺩﻋﺎﺀ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫‪‬‬ ‫ﺍﻟﺘﺨﻤﻴﻦ‪ :‬ﺍﻟﻤﺒﻴﻌﺎﺕ ﻓﻲ ﻛﻞ ﺷﻬﺮ‬ ‫‪20‬‬ ‫‪1402‬‬ ‫‪‬‬ ‫ﺗﺴﺎﻭﻱ ﺛﻼﺛﺔ ﺃﻣﺜﺎﻝ ﻣﺒﻴﻌﺎﺕ ﺍﻟﺸﻬﺮ‬ ‫‪22‬‬ ‫‪1407‬‬ ‫‪  (3‬ﻳﺒﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺳﻌﺮ‬ ‫‪‬‬ ‫‪29‬‬ ‫‪1412‬‬ ‫‪‬‬ ‫ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ﻟﺬﺍ ﻓﺈﻥ ﺍﻟﻤﺒﻴﻌﺎﺕ‬ ‫‪32‬‬ ‫‪1417‬‬ ‫ﻣﻨﺘﺞ ﺧﻼﻝ ﺍﻟﺴﻨﻮﺍﺕ ﻣﻦ ‪1402‬ﻫـ ﺇﻟﻰ ‪1427‬ﻫـ ‪.‬‬ ‫‪‬‬ ‫ﺳﺘﻜﻮﻥ ‪ 540000‬ﺭﻳﺎﻝ ﻓﻲ ﺍﻟﺸﻬﺮ‬ ‫‪37‬‬ ‫‪1422‬‬ ‫‪‬‬ ‫‪41‬‬ ‫‪1427‬‬ ‫‪ (A‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫ﺍﻟﺮﺍﺑﻊ ﺗﻘﺮﻳ ﹰﺒﺎ‪.‬‬ ‫‪‬‬ ‫‪ (B‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﺴﻌﺮ ﺍﻟﻤﻨﺘﺞ ﻋﺎﻡ ‪1432‬ﻫـ‪ 46 .‬ﺭﻳﺎ ﹰﻻ ﺗﻘﺮﻳ ﹰﺒﺎ‬ ‫‪‬‬ ‫‪ (C‬ﻫﻞ ﻣﻦ ﺍﻟﻤﻨﻄﻘﻲ ﺍﻟﻘﻮﻝ ﺑﺄﻥ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﺳﻴﺴﺘﻤﺮ ﺑﻤﺮﻭﺭ ﺍﻟﺰﻣﻦ؟‬ ‫ﻭﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﻛﺬﻟﻚ‪ ،‬ﻓﻜﻴﻒ ﺳﻴﺘﻐﻴﺮ؟ ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫‪ 1 14‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪(3A‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪50‬‬ ‫‪40‬‬ ‫‪30‬‬ ‫‪20‬‬ ‫‪10‬‬ ‫‪0‬‬ ‫‪1402 1407 1412 1417 1422 1427 1432‬‬ ‫‪‬‬ ‫‪ (3C‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻧﻌﻢ‪ ،‬ﻫﺬﺍ ﺍﻻﺗﺠﺎﻩ‬ ‫ﺍﻟﻤﺘﺰﺍﻳﺪ ﻣﻌﻘﻮﻝ؛ ﻷﻧﻪ ﻣﻦ ﺍﻟﻤﺤﺘﻤﻞ ﺃﻥ‬ ‫ﻳﺴﺘﻤﺮ ﺳﻌﺮ ﺍﻟﻤﻨﺘﺞ ﻓﻲ ﺍﻟﺰﻳﺎﺩﺓ ﻋﻠﻰ ﻣﺮ‬ ‫ﺍﻟﺴﻨﻴﻦ‪.‬‬ ‫‪ 1 14‬‬

‫‪  ‬ﺇﺛﺒﺎﺕ ﺻﺤﺔ ﺗﺨﻤﻴﻦ ﻣﻌﻴﻦ ﻟﻜﻞ ﺍﻟﺤﺎﻻﺕ‪ ،‬ﻳﺘﻄﻠﺐ ﺗﻘﺪﻳ ﹶﻢ ﺑﺮﻫﺎﻥ ﻟﺬﻟﻚ ﺍﻟﺘﺨﻤﻴﻦ‪ .‬ﺑﻴﻨﻤﺎ‬ ‫‪‬‬ ‫‪‬‬ ‫ﻹﺛﺒﺎﺕ ﻋﺪﻡ ﺻﺤﺔ ﺍﻟﺘﺨﻤﻴﻦ ﻳﻜﻔﻲ ﺗﻘﺪﻳﻢ ﻣﺜﺎﻝ ﻭﺍﺣﺪ ﻣﻌﺎﻛﺲ ﻟﻠﺘﺨﻤﻴﻦ‪ ،‬ﻭﻗﺪ ﻳﻜﻮﻥ ﻋﺪ ﹰﺩﺍ ﺃﻭ ﺭﺳ ﹰﻤﺎ ﺃﻭ ﻋﺒﺎﺭﺓ‪ ،‬ﻭﻫﺬﺍ‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﺍﻟﻤﺜﺎﻝ ﺍﻟﻤﻌﺎﻛﺲ ﹸﻳﺴﻤﻰ ﺍﻟﻤﺜﺎﻝ ﺍﻟﻤﻀﺎﺩ‪.‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪  4‬‬ ‫‪  ‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﺨﺘﺒﺮﻭﺍ‬ ‫ﺃﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﻳﺒ ﹼﻴﻦ ﺃﻥ ﻛ ﹰﹼﻼ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ ‬‬ ‫ﺟﻤﻴﻊ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﺤﺴﺎﺑﻴﺔ ﺍﻷﺳﺎﺳﻴﺔ ﺑﻤﺎ ﻓﻴﻬﺎ‬ ‫‪‬‬ ‫ﺍﻟﺠﺬﻭﺭ ﻭﺍﻟﻘﻮ￯‪ ،‬ﻋﻨﺪ ﺍﻟﺒﺤﺚ ﻋﻦ ﺍﻷﻧﻤﺎﻁ‬ ‫‪ (a‬ﺇﺫﺍ ﻛﺎﻥ ‪ n‬ﻋﺪ ﹰﺩﺍ ﺣﻘﻴﻘ ﹰﹼﻴﺎ‪ ،‬ﻓﺈﻥ ‪.n2 > n‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪ n‬ﻳﺴﺎﻭﻱ ‪ ،1‬ﻓﺈﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃﺊ؛ ﻷﻥ ‪12 ≯ 1‬‬ ‫‪A, B, C,...‬‬ ‫ﻓﻲ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻟﻌﺪﺩﻳﺔ‪ ،‬ﻭﺑ ﱢﻴﻦ ﻟﻬﻢ ﺃﻧﻪ ﻗﺪ‬ ‫‪‬‬ ‫ﻳﺘﻀﻤﻦ ﺍﻟﻨﻤﻂ ﺍﺳﺘﻌﻤﺎﻝ ﻋﻤﻠﻴﺘﻴﻦ ﺣﺴﺎﺑ ﱠﻴﺘﻴﻦ‪.‬‬ ‫__‬ ‫‪A, B‬‬ ‫‪‬‬ ‫‪ (b‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،JK = KL‬ﻓﺈﻥ ‪ K‬ﻣﻨﺘﺼﻒ ‪J L . JL‬‬ ‫‪AB BA‬‬ ‫ﻋﻨﺪﻣﺎ ﻻ ﺗﻘﻊ ‪ J, K, L‬ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪،‬‬ ‫‪ 4 ‬ﻳﺒ ﹼﻴﻦ ﻛﺘﺎﺑﺔ ﻣﺜﺎﻝ ﻣﻀﺎ ﱟﺩ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ‬ ‫‪‬‬ ‫ﻳﻜﻮﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃ ﹰﺌﺎ‪ .‬ﻓﻔﻲ ﺍﻟﺸ__ﻜ_ﻞ ﺍﻟﻤﺠﺎﻭﺭ ‪K ، JK = KL‬‬ ‫‪ABA, B‬‬ ‫ﺍﻟﻤﻌﻠﻮﻣﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ‪.‬‬ ‫ﻭﻟﻜﻦ ‪ K‬ﻟﻴﺴﺖ ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ ‪. JL‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،n = –4‬ﻓﺈﻥ ‪ ،–n = –(–4) = 4‬ﻭﻫﺬﺍ ﻋﺪﺩ ﻣﻮﺟﺐ‪.‬‬ ‫✓ ‪‬‬ ‫‪ (4A‬ﺇﺫﺍ ﻛﺎﻥ ‪ n‬ﻋﺪ ﹰﺩﺍ ﺣﻘﻴﻘ ﹰﹼﻴﺎ‪ ،‬ﻓﺈﻥ ‪ –n‬ﻳﻜﻮﻥ ﺳﺎﻟ ﹰﺒﺎ‪.‬‬ ‫‪ (4B‬ﺇﺫﺍ ﻛﺎﻥ‪ ، ∠ABE ∠DBC :‬ﻓﺈﻥ ‪ ∠ABE‬ﻭ ‪ ∠DBC‬ﻣﺘﻘﺎﺑﻠﺘﺎﻥ ﺑﺎﻟﺮﺃﺱ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫‪  4‬ﻳﺒ ﹼﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺃﺩﻧﺎﻩ ﻣﻌﺪﻻﺕ‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪:‬‬ ‫‪1 ‬‬ ‫ﺍﻟﺒﻄﺎﻟﺔ ﺑﻴﻦ ﺍﻟﻨﺴﺎﺀ ﺍﻟﺴﻌﻮﺩﻳﺎﺕ ﻓﻲ‬ ‫‪ (1‬ﺍﻟﺘﻜﻠﻔﺔ‪ 4.50 :‬ﺭﻳﺎﻻ ﹴﺕ‪ 6.75 ،‬ﺭﻳﺎﻻ ﹴﺕ‪ 9.00 ،‬ﺭﻳﺎﻻ ﹴﺕ‪...... ،‬‬ ‫‪ (1‬ﺗﺰﻳﺪ ﺍﻟﺘﻜﻠﻔﺔ ﻛﻞ ﻣﺮﺓ ﺑﻤﻘﺪﺍﺭ‬ ‫‪ 2.25‬ﺭﻳﺎﻝ ﻋﻦ ﺍﻟﻤﺮﺓ‬ ‫ﺑﻌﺾ ﺍﻟﻤﺪﻥ ﺍﻟﺴﻌﻮﺩﻳﺔ ﻭﻓﻖ ﺇﺣﺼﺎﺀﺍﺕ‬ ‫‪ (2‬ﻣﻮﺍﻋﻴﺪ ﺍﻧﻄﻼﻕ ﺍﻟﺤﺎﻓﻼﺕ‪ 10:15 :‬ﺻﺒﺎ ﹰﺣﺎ‪ 11:00 ،‬ﺻﺒﺎ ﹰﺣﺎ‪ 11:45 ،‬ﺻﺒﺎ ﹰﺣﺎ‪...... ،‬‬ ‫ﺍﻟﺴﺎﺑﻘﺔ؛ ‪ 11.25‬ﺭﻳﺎ ﹰﻻ‪.‬‬ ‫ﻳﻨﺘﻘﻞ ﺍﻟﺘﻈﻠﻴﻞ ﺇﻟﻰ ﺍﻟﺠﺰﺀ ﺍﻟﺘﺎﻟﻲ ﻛﻞ ﻣﺮﺓ ﻓﻲ ﺍﺗﺠﺎﻩ ﻋﻘﺎﺭﺏ ﺍﻟﺴﺎﻋﺔ‪.‬‬ ‫ﻋﺎﻡ ‪2004‬ﻡ‪ ،‬ﺃﻭﺟﺪ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‬ ‫‪ (2‬ﻳﺄﺗﻲ ﻛﻞ ﻣﻮﻋﺪ ﺑﻌﺪ ‪45‬‬ ‫ﻟﻠﻌﺒﺎﺭﺓ \"ﻣﻌﺪﻝ ﺍﻟﺒﻄﺎﻟﺔ ﺃﻋﻠﻰ ﻣﺎ ﻳﻜﻮﻥ‬ ‫‪(3‬‬ ‫ﺩﻗﻴﻘﺔ ﻣﻦ ﺍﻟﻤﻮﻋﺪ ﺍﻟﺴﺎﺑﻖ ﻟﻪ؛‬ ‫ﻓﻲ ﺍﻟﻤﺪﻥ ﺫﺍﺕ ﺍﻟﻌﺪﺩ ﺍﻷﻛﺒﺮ ﻣﻦ‬ ‫‪ 12:30‬ﻣﺴﺎ ﹰﺀ‪.‬‬ ‫ﺍﻟﺴﻜﺎﻥ\"‪.‬‬ ‫‪......................‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪(4‬‬ ‫‪19.3%‬‬ ‫‪4081152‬‬ ‫‪‬‬ ‫‪16.9%‬‬ ‫‪1294168‬‬ ‫‪‬‬ ‫ﻛﻞ ﺷﻜﻞ ﻓﻲ ﺍﻟﻨﻤﻂ ﻳﺤﻮﻱ‬ ‫‪‬‬ ‫‪38%‬‬ ‫‪100694‬‬ ‫‪‬‬ ‫ﺩﺍﺋﺮﺓ ﺇﺿﺎﻓﻴﺔ ﺧﺎﺭﺟﻴﺔ ﺯﻳﺎﺩﺓ‬ ‫‪25.6%‬‬ ‫‪744321‬‬ ‫‪‬‬ ‫‪16.0%‬‬ ‫‪378422‬‬ ‫‪‬‬ ‫ﻋﻠﻰ ﺩﻭﺍﺋﺮ ﺍﻟﺸﻜﻞ ﺍﻟﺴﺎﺑﻖ‪.......................‬‬ ‫‪40.6%‬‬ ‫‪85212‬‬ ‫‪3, 3, 6, 9, 15, ...... (5‬‬ ‫‪ (5‬ﻛﻞ ﺣ ﱟﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ‬ ‫‪ 2, 6, 14, 30, 62, ...... (6‬ﻳﺰﻳﺪ ﻛﻞ ﺣﺪ ﺑﻤﻘﺪﺍﺭ ‪ 2‬ﻋﻠﻰ ﻣﺜ ﹶﻠﻲ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪126‬‬ ‫ﻳﺴﺎﻭﻱ ﻣﺠﻤﻮﻉ ﺍﻟﺤﺪﻳﻦ‬ ‫ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫ﺍﻟﺴﺎﺑﻘﻴﻦ ﻟﻪ؛ ‪224‬‬ ‫‪ (7‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﺯﻭﺟﻴﻴﻦ‪ .‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﺯﻭﺟﻴﻴﻦ ﻫﻮ ﻋﺪﺩ ﺯﻭﺟﻲ‪.‬‬ ‫ﻣﺪﻳﻨﺔ ﺍﻟﺒﺎﺣﺔ ﻋﺪﺩ ﺳﻜﺎﻧﻬﺎ ‪، 85212‬‬ ‫‪ (8‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺪﺩﻳﻦ ‪ a‬ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻥ ‪ .a + b = 0‬ﻛﻞ ﻣﻦ ‪ a‬ﻭ ‪ b‬ﻣﻌﻜﻮﺱ ﻟﻶﺧﺮ‪.‬‬ ‫ﻭﻣﻌﺪﻝ ﺍﻟﺒﻄﺎﻟﺔ ﻓﻴﻬﺎ ﺃﻋﻠﻰ ﻣﻦ ﻣﻌﺪﻝ‬ ‫ﺍﻟﺒﻄﺎﻟﺔ ﻓﻲ ﺍﻟﺮﻳﺎﺽ ﺍﻟﺘﻲ ﻋﺪﺩ ﺳﻜﺎﻧﻬﺎ‬ ‫‪ (9‬ﻣﺠﻤﻮﻋﺔ ﺍﻟﻨﻘﺎﻁ ﻓﻲ ﺍﻟﻤﺴﺘﻮ￯ ‪ (9‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺠﻤﻮﻋﺔ ﺍﻟﻨﻘﺎﻁ ﻓﻲ ﺍﻟﻤﺴﺘﻮ￯ ﺍﻟﺘﻲ ﺗﺒﻌﺪ ﺍﻟﻤﺴﺎﻓﺔ ﻧﻔﺴﻬﺎ ﻋﻦ ﺍﻟﻨﻘﻄﺔ ‪. A‬‬ ‫ﺍﻟﺘﻲ ﺗﺒﻌﺪ ﺍﻟﺒﻌﺪ ﻧﻔﺴﻪ ﻋﻦ‬ ‫‪4087152‬‬ ‫____‬ ‫___‬ ‫___ ___‬ ‫‪ (10‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ AP‬ﻭ ‪ PB‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ M‬ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ ‪ AB‬ﻭﺍﻟﻨﻘﻄﺔ ‪ P‬ﻧﻘﻄ_ﺔ_ﻣ_ﻨﺘﺼﻒ ‪___ . AM‬‬ ‫ﻃﻮﻝ ‪ PB‬ﻳﺴﺎﻭﻱ ﺛﻼﺛﺔ ﺃﻣﺜﺎﻝ ﻃﻮﻝ ‪. AP‬‬ ‫ﺍﻟﻨﻘﻄﺔ ‪ A‬ﺗﻜ ﹼﻮﻥ ﺩﺍﺋﺮﺓ‪.‬‬ ‫‪15  1- 1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (4B‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﺍﻟﻨﻘﺎﻁ‬ ‫‪ ‬ﻭ ﱢﺯﻉ ﺍﻟﻄﻼﺏ ﻣﺠﻤﻮﻋﺎﺕ ﺻﻐﻴﺮﺓ‪ ،‬ﺛﻢ ﺍﻃﻠﺐ ﺇﻟﻰ ﻛﻞ ﻃﺎﻟﺐ ﺃﻥ ﻳﻜﺘﺐ ﻋﺒﺎﺭﺗﻴﻦ ﻏﻴﺮ‬ ‫‪ A, B, D‬ﻻ ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪،‬‬ ‫ﻭﺍﻟﻨﻘﺎﻁ ‪ E, B ,C‬ﻻ ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ‬ ‫ﺻﺤﻴﺤﺘﻴﻦ ﺩﺍﺋ ﹰﻤﺎ ﻋﻠﻰ ﺍﻷﻗﻞ‪ ،‬ﻭﻋﻠﻰ ﺑﻘﻴﺔ ﻃﻼﺏ ﻣﺠﻤﻮﻋﺘﻪ ﺇﻳﺠﺎﺩ ﻣﺜﺎﻝ ﻣﻀﺎﺩ ﻟﻜﻞ ﻋﺒﺎﺭﺓ‪.‬‬ ‫ﻭﺍﺣﺪﺓ‪ ،‬ﻳﻜﻮﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃ ﹰﺌﺎ‪ .‬ﻓﻲ‬ ‫ﺍﻟﺸﻜﻞ ﺍﻵﺗﻲ‪،∠ABE ∠DBC :‬‬ ‫ﻭﻟﻜﻦ ‪ ∠ABE‬ﻭ ‪ ∠DBC‬ﻏﻴﺮ‬ ‫ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫‪C‬‬ ‫‪B‬‬ ‫‪AD‬‬ ‫‪E‬‬ ‫‪15  1-1‬‬

‫‪‬‬ ‫‪  (11 3‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺍﻟﺬﻱ ﻳﺒﻴﻦ ﻋﺪﺩ‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﻟﻘﻄﻊ ﺍﻟﻤﻨﺘﺠﺔ ﻓﻲ ﻣﺼﻨﻊ ﻟﺒﻌﺾ ﺍﻟﺴﻨﻮﺍﺕ‪.‬‬ ‫‪ 3‬‬ ‫✓ ‪‬‬ ‫‪5 2007‬‬ ‫ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ ‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪7.2 2008‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪1-13‬؛ ﻟﻠﺘﺄﻛﺪ ﻣﻦ ﻓﻬﻢ‬ ‫‪9.2 2009‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻌﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﺳﻨﺔ ‪2017‬ﻡ ‪.‬‬ ‫ﺍﻟﻄﻠﺒﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺳﻔﻞ ﻫﺬﻩ‬ ‫‪14.1 2010‬‬ ‫ﺳﻴﻜﻮﻥ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﻓﻲ ﻋﺎﻡ ‪ 2017‬ﻧﺤﻮ ‪ 35‬ﻣﻠﻴﻮ ﹰﻧﺎ‪.‬‬ ‫ﺍﻟﺼﻔﺤﺔ؛ ﻟﺘﻌﻴﻴﻦ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻠﻄﻠﺒﺔ‬ ‫‪19.7 2011‬‬ ‫‪28.4 2012‬‬ ‫ﺑﺤﺴﺐ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫ﺃﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﻳﺒﻴﻦ ﺃﻥ ﻛ ﹼﹰﻼ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪4 ‬‬ ‫‪(12‬‬ ‫‪ (12‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠A‬ﻭ ‪ ∠B‬ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻟﻬﻤﺎ ﺿﻠ ﹰﻌﺎ ﻣﺸﺘﺮ ﹰﻛﺎ‪.‬‬ ‫‪ (13‬ﺇﺫﺍ ﻗﻄﻊ ﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ ﻗﻄﻌ ﹰﺔ ﻣﺴﺘﻘﻴﻤ ﹰﺔ ﻋﻨﺪ ﻣﻨﺘﺼﻔﻬﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻌﺎﻣﺪﻫﺎ‪.‬‬ ‫‪45°‬‬ ‫‪A B 45°‬‬ ‫‪‬‬ ‫‪ (14–19‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪1 ‬‬ ‫‪‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪.‬‬ ‫‪‬‬ ‫‪4, 8, 12, 16, 20 (16‬‬ ‫‪3, 6, 9, 12, 15 (15‬‬ ‫‪0, 2, 4, 6, 8 (14‬‬ ‫ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺃﻧﻪ ﻓﻲ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺘﻲ‬ ‫‪1,‬‬ ‫‪_21 ,‬‬ ‫‪_41 ,‬‬ ‫‪_1‬‬ ‫‪(19‬‬ ‫‪1, 4, 9, 16 (18‬‬ ‫‪2, 22, 222, 2222 (17‬‬ ‫ﺗﺘﻀﻤﻦ ﺑﻴﺎﻧﺎﺕ ﻣﻦ ﻭﺍﻗﻊ ﺍﻟﺤﻴﺎﺓ‪ ،‬ﻟﻴﺲ‬ ‫ﺑﺎﻟﻀﺮﻭﺭﺓ ﺃﻥ ﻳﻤ ﹼﺜﻞ ﺍﻟﺘﺨﻤﻴﻦ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ‬ ‫‪8‬‬ ‫ﺍﻟﻨﻤﻂ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻣﺎ ﻳﻤﻜﻦ ﺃﻥ ﻳﺤﺪﺙ‬ ‫‪ (20‬ﻣﻮﺍﻋﻴﺪ ﺍﻟﻮﺻﻮﻝ‪ 10:00 :‬ﺻﺒﺎ ﹰﺣﺎ‪ 12:30 ،‬ﻣﺴﺎ ﹰﺀ ‪ 3:00 ،‬ﻣﺴﺎ ﹰﺀ‪ (20 ...... ،‬ﻳﺄﺗﻲ ﻛﻞ ﻣﻮﻋﺪ ﺑﻌﺪ ﺳﺎﻋﺘﻴﻦ ﻭﻧﺼﻒ‬ ‫ﻓﻲ ﺍﻟﻤﺴﺘﻘﺒﻞ‪.‬‬ ‫ﺍﻟﺴﺎﻋﺔ ﻣﻦ ﺍﻟﻤﻮﻋﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪ 5:30‬ﻣﺴﺎ ﹰﺀ‪.‬‬ ‫‪ (21‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻠﺮﻃﻮﺑﺔ‪100% , 93% , 86% , …… :‬‬ ‫‪ (21‬ﺗﻘﻞ ﻛﻞ ﻧﺴﺒﺔ ﻣﺌﻮﻳﺔ ﻋﻦ‬ ‫ﻓﻤﺜ ﹰﻼ‪ ،‬ﻗﺪ ﹸﺗﺸﻴﺮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫ﺍﻟﻨﺴﺒﺔ ﺍﻟﺴﺎﺑﻘﺔ ﺑﻤﻘﺪﺍﺭ‬ ‫ﺇﻟﻰ ﺗﺰﺍﻳﺪ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻓﻲ ﺃﺣﺪ‬ ‫‪ (22‬ﺃﻳﺎﻡ ﺍﻟﻌﻤﻞ‪ :‬ﺍﻷﺣﺪ‪ ،‬ﺍﻟﺜﻼﺛﺎﺀ‪ ،‬ﺍﻟﺨﻤﻴﺲ‪...... ،‬‬ ‫‪ 7%‬؛ ‪.79%‬‬ ‫ﺍﻷﺳﺎﺑﻴﻊ‪ ،‬ﺇ ﹼﻻ ﺃﻥ ﺩﺭﺟﺎﺕ ﺍﻟﺤﺮﺍﺭﺓ ﻗﺪ‬ ‫‪ (22‬ﻳﺄﺗﻲ ﻛﻞ ﻳﻮﻡ ﻋﻤﻞ ﺑﻌﺪ‬ ‫ﺗﻨﺨﻔﺾ ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﺬﻱ ﻳﻠﻴﻪ‪.‬‬ ‫‪ (23‬ﺍﺟﺘﻤﺎﻋﺎﺕ ﺍﻟﻨﺎﺩﻱ‪ :‬ﺍﻟﻤﺤ ﹼﺮﻡ‪ ،‬ﺭﺑﻴﻊ ﺃﻭﻝ‪ ،‬ﺟﻤﺎﺩ￯ ﺍﻷﻭﻟﻰ‪ (24–27 ...... ،‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺇﺟﺎﺑﺎﺕ‬ ‫ﻳﻮﻣﻴﻦ ﻣﻦ ﻳﻮﻡ ﺍﻟﻌﻤﻞ‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺍﻟﺴﺒﺖ‪.‬‬ ‫‪(25 (24‬‬ ‫‪ (23‬ﻳﻌﻘﺪ ﻛﻞ ﺍﺟﺘﻤﺎﻉ ﺑﻌﺪ‬ ‫ﺷﻬﺮﻳﻦ ﻣﻦ ﺍﻻﺟﺘﻤﺎﻉ‬ ‫‪......‬‬ ‫‪......‬‬ ‫ﺍﻟﺴﺎﺑﻖ؛ ﺭﺟﺐ‪.‬‬ ‫‪(27‬‬ ‫‪(26‬‬ ‫‪...... ......‬‬ ‫‪‬‬ ‫‪  (28‬ﺑﺪﺃ ﻣﺎﺟﺪ ﺗﻤﺎﺭﻳﻦ ﺍﻟﺠﺮﻱ ﺍﻟﺴﺮﻳﻊ ﻗﺒﻞ ﺧﻤﺴﺔ ﺃﻳﺎﻡ‪ .‬ﻓﺮﻛﺾ ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻷﻭﻝ ‪ . 0.5 km‬ﻭﻓﻲ ﺍﻷﻳﺎﻡ‬ ‫‪‬‬ ‫‪(11a‬‬ ‫ﺍﻟﺜﻼﺛﺔ ﺍﻟﺘﺎﻟﻴﺔ ‪ . 0.75 km, 1 km, 1.25 km‬ﺇﺫﺍ ﺍﺳﺘﻤﺮ ﺗﻤﺮﻳﻨﻪ ﻋﻠﻰ ﻫﺬﺍ ﺍﻟﻨﻤﻂ‪ ،‬ﻓﻤﺎ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻲ ﻳﻘﻄﻌﻬﺎ ﻓﻲ‬ ‫‪30‬‬ ‫ﺍﻟﻴﻮﻡ ﺍﻟﺴﺎﺑﻊ؟ ‪2 km‬‬ ‫‪‬‬ ‫‪25‬‬ ‫ﺿﻊ ﺗﺨﻤﻴ ﹰﻨﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪:‬‬ ‫‪ (29‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪2 ‬‬ ‫‪20‬‬ ‫‪ (30‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩ ﻓﻲ ﺍﺛﻨﻴﻦ‪ ،‬ﻣﻀﺎ ﹰﻓﺎ ﺇﻟﻴﻪ ﻭﺍﺣﺪ‪ .‬ﺍﻟﻨﺎﺗﺞ ﻋﺪﺩ ﻓﺮﺩﻱ‬ ‫‪ (31‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺪﺩﻳﻦ ‪ a‬ﻭ ‪ ، b‬ﺇﺫﺍ ﻛﺎﻥ ‪ .ab = 1‬ﻛ ﱞﻞ ﻣﻨﻬﻤﺎ ﻣﻘﻠﻮﺏ ﺍﻵﺧﺮ‬ ‫‪15‬‬ ‫___ ___‬ ‫‪10‬‬ ‫‪ (32‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ AB‬ﻭﻣﺠﻤﻮﻋﺔ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﺒﻌﺪ ﻣﺴﺎﻓﺎﺕ ﻣﺘﺴﺎﻭﻳﺔ ﻋﻦ ‪ A‬ﻭ ‪ . B‬ﺗﺸﻜﻞ ﺍﻟﻌﻤﻮﺩ ﺍﻟﻤﻨ ﱢﺼﻒ ﻟـ ‪. AB‬‬ ‫‪5‬‬ ‫‪ (33‬ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻭﺣﺠﻢ ﺍﻟﻬﺮﻡ ﺍﻟﻠﺬﻳﻦ ﻟﻬﻤﺎ ﺍﻟﻘﺎﻋﺪﺓ ﻧﻔﺴﻬﺎ ﻭﺍﻻﺭﺗﻔﺎﻉ ﻧﻔﺴﻪ‪.‬‬ ‫‪0‬‬ ‫ﺣﺠﻢ ﺍﻟﻤﻨﺸﻮﺭ ﻳﺴﺎﻭﻱ ‪ 3‬ﺃﻣﺜﺎﻝ ﺣﺠﻢ ﺍﻟﻬﺮﻡ‪.‬‬ ‫‪00 07 08 09 10 11 12‬‬ ‫‪‬‬ ‫‪ (14‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪2‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪10‬‬ ‫‪ 1 16‬‬ ‫‪ (15‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪3‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪18‬‬ ‫‪‬‬ ‫‪ (16‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ‪4‬‬ ‫ﻋﻠﻰ ﺍﻟﺤﺪ ﺍﻟﺬﻱ ﻳﺴﺒﻘﻪ؛ ‪24‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (17‬ﻛﻞ ﺣﺪ ﻓﻲ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﻳﺤﺘﻮﻱ ﻋﻠﻰ‬ ‫ﺍﻟﺮﻗﻢ ‪ 2‬ﺯﻳﺎﺩﺓ ﻋﻠﻰ ﺃﺭﻗﺎﻡ ﺍﻟﺤﺪ ﺍﻟﺴﺎﺑﻖ‬ ‫‪43 - 5614 - 38‬‬ ‫‪‬‬ ‫ﻟﻪ؛ ‪22222‬‬ ‫‪43 - 5639 - 4115 - 39‬‬ ‫‪‬‬ ‫‪ (18‬ﻳﻨﺘﺞ ﻛﻞ ﺣﺪ ﻋﻦ ﺗﺮﺑﻴﻊ ﺍﻟﻌﺪﺩ ﺍﻟﻄﺒﻴﻌﻲ‬ ‫‪45 - 5639 - 53‬‬ ‫‪‬‬ ‫ﺍﻟﺬﻱ ﻳﻤ ﱢﺜﻞ ﺗﺮﺗﻴﺒﻪ؛ ‪25‬‬ ‫ﺍﻟﺬﻱ‬ ‫ﺍﻟﺤﺪ‬ ‫ﻧﺼﻒ‬ ‫ﻳﺴ‪1‬ﺎ_ﻭﻱ‬ ‫ﻛﻞ ﺣﺪ‬ ‫‪(19‬‬ ‫ﻳﺴﺒﻘﻪ؛‬ ‫‪16‬‬ ‫‪ 1 16‬‬

‫‪‬‬ ‫‪  (34 3‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ ﺍﻟﺬﻱ ﻳﺒﻴﻦ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ‪‬‬ ‫‪190‬‬ ‫ﻓﻲ ﻣﺪﺭﺳﺔ ﻣﺪﺓ ﺃﺭﺑﻊ ﺳﻨﻮﺍﺕ ﻣﺘﺘﺎﻟﻴﺔ‪ (a, b.‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ ‪1425‬‬ ‫‪210‬‬ ‫‪240‬‬ ‫‪1426‬‬ ‫‪ (a‬ﺃﻧﺸﺊ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﻧﺴﺐ ﻟﻌﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪1427‬‬ ‫‪260‬‬ ‫‪ (b‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻣﻌﺘﻤ ﹰﺪﺍ ﻋﻠﻰ ﺑﻴﺎﻧﺎﺕ ﺍﻟﺠﺪﻭﻝ‪ ،‬ﻭﺍﺷﺮﺡ ﻛﻴﻒ ﻳﺆ ﱢﻳﺪ ﺗﻤﺜﻴﻠﻚ ‪1428‬‬ ‫‪!‬‬ ‫ﺍﻟﺒﻴﺎﻧﻲ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪  ‬ﻓﻲ ﺍﻟﺴﺆﺍﻝ ‪،42‬‬ ‫‪ 4‬ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺻﺤﻴ ﹰﺤﺎ ﺃﻭ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫ﻳﺠﺐ ﺃﻥ ﻳﺘﺬﻛﺮ ﺍﻟﻄﻼﺏ ﺃﻥ ﺍﻟﻌﺪﺩ ‪2‬‬ ‫‪ (35‬ﺇﺫﺍ ﻛﺎﻥ ‪ n‬ﻋﺪ ﹰﺩﺍ ﺃﻭﻟ ﹼﹰﻴﺎ‪ ،‬ﻓﺈﻥ ‪ n + 1‬ﻟﻴﺲ ﺃﻭﻟ ﹼﹰﻴﺎ‪ .‬ﺧﺎﻃﺊ؛ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،n = 2‬ﻓﺈﻥ ‪ ،n + 1 = 3‬ﻭﻫﺬﺍ ﻋﺪﺩ ﺃﻭﻟﻲ‪.‬‬ ‫ﻫﻮ ﻋﺪﺩ ﺃﻭﻟﻲ‪ .‬ﻭﻣﻦ ﺍﻟﻤﻼﺣﻆ ﺃﻥ‬ ‫ﺃﺣﻤﺪ ﺃﻫﻤﻞ ﺍﻟﻌﺪﺩ ‪ 2‬ﻋﻨﺪﻣﺎ ﻋﻤﻞ‬ ‫‪ (36‬ﺇﺫﺍ ﻛﺎﻥ ‪ x‬ﻋﺪ ﹰﺩﺍ ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﻓﺈﻥ ‪ –x‬ﻋﺪﺩ ﻣﻮﺟﺐ‪ .‬ﺧﺎﻃﺊ؛ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، x = 2‬ﻓﺈﻥ ‪−x = −2‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺗﺨﻤﻴﻨﻪ ﺣﻮﻝ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪ (37‬ﻓﻲ ﺍﻟﻤﺜﻠﺚ ‪ ABC‬ﺇﺫﺍ ﻛﺎﻥ‪ ، (AB)2 + (BC)2 = (AC)2 :‬ﻓﺈﻥ ‪ ABC‬ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ‪ .‬ﺻﺤﻴﺢ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (39a‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻌﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻲ ﻣﻨﻄﻘﺔ‬ ‫ﻣﻜﺔ ﺍﻟﻤﻜﺮﻣﺔ ﻭﺣﺪﻩ ﻳﺴﺎﻭﻱ ‪25.5%‬‬ ‫‪ (38‬ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﻣﺴﺘﻄﻴﻞ ﺗﺴﺎﻭﻱ ‪ ،20 m2‬ﻓﺈﻥ ﻃﻮﻟﻪ ﻳﺴﺎﻭﻱ ‪ ، 10 m‬ﻭﻋﺮﺿﻪ ‪ .2 m‬ﺧﺎﻃﺊ؛ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻳﻤﻜﻦ‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫ﻣﻦ ﺳﻜﺎﻥ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪.‬‬ ‫‪‬‬ ‫ﺃﻥ ﻳﻜﻮﻥ ﺍﻟﻄﻮﻝ ‪، 5 m‬‬ ‫‪  (39‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺩﻧﺎﻩ ﻟﺘﻌﻄﻲ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪:‬‬ ‫‪ ‬‬ ‫‪ (39b‬ﻋﺪﺩ ﺳﻜﺎﻥ ﻣﻨﻄﻘﺔ ﺍﻟﻤﺪﻳﻨﺔ ﺍﻟﻤﻨﻮﺭﺓ‬ ‫ﻭﺍﻟﻌﺮﺽ ‪4 m‬‬ ‫‪ 1.8‬ﻣﻠﻴﻮﻥ ﻧﺴﻤﺔ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪10 = 5 + 5, 12 = 5 + 7, (40a‬‬ ‫‪25.0%‬‬ ‫‪6.8 ‬‬ ‫‪14 = 7 + 7, 16 = 5 + 11,‬‬ ‫‪18 = 7 + 11, 20 = 7+13‬‬ ‫‪25.5%‬‬ ‫‪6.9 ‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ (40b‬ﺧﺎﻃﺊ؛ ﻻ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ﺍﻟﻌﺪﺩ ‪ 3‬ﻋﻠﻰ‬ ‫‪6.6% 1.8 ‬‬ ‫‪12‬‬ ‫ﺻﻮﺭﺓ ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﺃﻭﻟ ﱠﻴﻴﻦ‪.‬‬ ‫‪15.1%‬‬ ‫‪4.1 ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪1431‬‬ ‫‪‬‬ ‫‪ (a‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻤﺠﻤﻮﻉ ﻋﺪﺩ ﺳﻜﺎﻥ ﺍﻟﻤﻨﺎﻃﻖ ﺍﻹﺩﺍﺭﻳﺔ ﺍﻷﺭﺑﻊ ﺍﻟﻮﺍﺭﺩﺓ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺃﻗﻞ ﻣﻦ ‪ 25%‬ﻣﻦ ﺳﻜﺎﻥ‬ ‫‪‬‬ ‫ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪ (a,b .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫‪‬‬ ‫‪ (b‬ﻳﺰﻳﺪ ﻋﺪﺩ ﺳﻜﺎﻥ ﺃ ﱟﻱ ﻣﻦ ﺍﻟﻤﻨﺎﻃﻖ ﺍﻹﺩﺍﺭﻳﺔ ﺍﻷﺭﺑﻊ ﻋﻠﻰ ﻣﻠﻴﻮ ﹶﻧﻲ ﻧﺴﻤ ﹴﺔ‪.‬‬ ‫‪‬‬ ‫‪  (40‬ﻳﻨﺺ ﺗﺨﻤﻴﻦ ﺟﻮﻟﺪ ﺑﺎﺥ ﻋﻠﻰ ﺃﻧﻪ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ﺃﻱ ﻋﺪﺩ ﺯﻭﺟﻲ ﺃﻛﺒﺮ ﻣﻦ ‪ 2‬ﻋﻠﻰ ﺻﻮﺭﺓ‬ ‫‪‬‬ ‫ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﺃﻭﻟﻴﻴﻦ‪ .‬ﻓﻌﻠﻰ ﺳﺒﻴﻞ ﺍﻟﻤﺜﺎﻝ‪ (a,b .4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5 :‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫‪ (a‬ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ ﻟﻸﻋﺪﺍﺩ ﺍﻟﺰﻭﺟﻴﺔ ﻣﻦ ‪ 10‬ﺇﻟﻰ ‪20‬‬ ‫‪ (b‬ﺇﺫﺍ ﹸﺃﻋﻄﻴﺖ ﺍﻟﺘﺨﻤﻴﻦ ﺍﻵﺗﻲ‪ :‬ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ﺃﻱ ﻋﺪﺩ ﻓﺮﺩﻱ ﺃﻛﺒﺮ ﻣﻦ ‪ 2‬ﻋﻠﻰ ﺻﻮﺭﺓ ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﺃﻭﻟﻴﻴﻦ‪.‬‬ ‫ﻓﻬﻞ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ ﺃﻡ ﺧﺎﻃﺊ؟ ﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪AB‬‬ ‫ﺇ‪‬ﺍﺫﺍﻟﻨ ﹸﺃﻘﻄﺿﺘﻴﺎﻔﻥ ﺍﺖﻟﻧﻮﻘﺍﻗﻄﻌﺔﺘﺎﺃﻥﺧﺮﻋﻠ￯ﻰ‪C‬ﻣﻋﺴﺘﻠﻘﻴﻰ ﺍﻢﻟﺗﻘﺸﻄ ﱢﻌﻜﺔﻼﺍﻟﻥﻤﻗﺴﻄﺘﻘﻌﻴﺔﻤﻣﺔﺴ_ﺘ‪_B‬ﻘﻴ‪_A‬ﻤ‪،‬ﺔ‪،‬‬ ‫‪___‬‬ ‫‪(41‬‬ ‫‪A CB‬‬ ‫ﻣﺜﻞ ‪. AB‬‬ ‫‪ (41b‬ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻳﺴﺎﻭﻱ‬ ‫ﻣﺠﻤﻮﻉ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻄﺒﻴﻌﻴﺔ ﺍﻷﻗﻞ ﻣﻦ ‪ .n‬ﻓﺈﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ ﺗﺸ ﱢﻜﻞ ﺛﻼﺙ ﻗﻄﻊ ﻣﺴﺘﻘﻴﻤﺔ‪.‬‬ ‫‪ (41c‬ﺗﺘﻜﻮﻥ ﺧﻤﺲ ﻋﺸﺮﺓ ﻗﻄﻌﺔ ‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺨﺘﻠﻔﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻣﻦ ﺃﺭﺑﻊ ﻧﻘﺎﻁ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ؟ ﻭﻣﻦ ﺧﻤﺲ ﻧﻘﺎﻁ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ؟‬ ‫ﻣﺴﺘﻘﻴﻤﺔ‪ .‬ﻓﺎﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ‪ (b .‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻌﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺨﺘﻠﻔﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻣﻦ ‪ n‬ﻧﻘﻄﺔ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ‪6; 10 .‬‬ ‫‪ (c‬ﺍﺧﺘﺒﺮ ﺗﺨﻤﻴﻨﻚ ﺑﺈﻳﺠﺎﺩ ﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺨﺘﻠﻔﺔ ﺍﻟﺘﻲ ﺗﺘﺸﻜﻞ ﻣﻦ ‪ 6‬ﻧﻘﺎﻁ‪.‬‬ ‫‪‬‬ ‫‪   (42‬ﻳﺘﻨﺎﻗﺶ ﺃﺣﻤﺪ ﻭﻋﻠﻲ ﻓﻲ ﻣﻮﺿﻮﻉ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ‪ .‬ﻓﻴﻘﻮﻝ ﺃﺣﻤﺪ‪ :‬ﺇﻥ ﺟﻤﻴﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ‬ ‫ﺃﻋﺪﺍﺩ ﻓﺮﺩﻳﺔ‪ .‬ﻓﻲ ﺣﻴﻦ ﻳﻘﻮﻝ ﻋﻠ ﱞﻲ‪ :‬ﻟﻴﺴﺖ ﺟﻤﻴﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ ﻓﺮﺩﻳﺔ‪ .‬ﻫﻞ ﻗﻮﻝ ﺃ ﱟﻱ ﻣﻨﻬﻤﺎ ﺻﺤﻴﺢ؟ ﻓ ﹼﺴﺮ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫ﻗﻮﻝ ﻋﻠﻲ ﺻﺤﻴﺢ؛ ﻷﻥ ﺍﻟﻌﺪﺩ ‪ 2‬ﻋﺪﺩ ﺃﻭﻟﻲ ﺯﻭﺟﻲ‬ ‫‪17  1- 1‬‬ ‫‪17  1-1‬‬

‫‪ (43‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ .2, 4, 16, 256, 65536 :‬ﻳﻤﻜﻦ ﺇﻳﺠﺎﺩ ﻛﻞ ﺣﺪ ﺑﺘﺮﺑﻴﻊ ﺍﻟﺤﺪ ﺍﻟﺴﺎﺑﻖ ﻟﻪ‪،‬‬ ‫ﻛﻤﺎ ﻳﻤﻜﻦ ﺇﻳﺠﺎﺩ ﻛﻞ ﺣﺪ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺼﻴﻐﺔ ‪ ، 22n-1‬ﺣﻴﺚ ‪.n ≥ 1‬‬ ‫‪ (44‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺧﻄﺄ؛ ﺇﺫﺍ‬ ‫ﻛ ﱠﻮﻧﺖ ﺍﻟﻨﻘﺎﻁ ﺯﺍﻭﻳﺔ ﻣﺴﺘﻘﻴﻤﺔ ‪  (43‬ﺍﻛﺘﺐ ﻣﺘﺘﺎﺑﻌﺔ ﻋﺪﺩﻳﺔ ﺗﺘﺒﻊ ﺣﺪﻭﺩﻫﺎ ﻧﻤﻄﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ‪ ،‬ﻭﻭﺿﺢ ﺍﻟﻨﻤﻄﻴﻦ‪.‬‬ ‫‪ ‬ﺗﺄ ﹼﻣﻞ ﺍﻟﺘﺨﻤﻴﻦ‪” :‬ﺇﺫﺍ ﻛﺎﻧﺖ ﻧﻘﻄﺘﺎﻥ ﺗﺒ ﹸﻌﺪﺍﻥ ﺍﻟﻤﺴﺎﻓﺔ ﻧﻔﺴﻬﺎ ﻋﻦ ﻧﻘﻄﺔ ﺛﺎﻟﺜﺔ ﻣﻌﻠﻮﻣﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ‬ ‫‪(44‬‬ ‫ﻳﻜﻮﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﺃﻣﺎ‬ ‫ﺇﺫﺍ ﻟﻢ ﺗﻜﻦ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ‬ ‫ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ“‪ .‬ﻫﻞ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ ﺻﺤﻴﺢ ﺃﻡ ﺧﺎﻃﺊ؟ ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪ ،‬ﻓﻴﻜﻮﻥ‬ ‫‪  (45‬ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﹸﺗﺠﺮﻱ ﻣﺴ ﹰﺤﺎ‪ .‬ﺍﺧﺘﺮ ﻣﻮﺿﻮ ﹰﻋﺎ ﻭﺍﻛﺘﺐ ﺛﻼﺛﺔ ﺃﺳﺌﻠﺔ ﻳﺘﻀﻤﻨﻬﺎ ﻣﺴ ﹸﺤﻚ‪ .‬ﻛﻴﻒ ﺗﺴﺘﻌﻤﻞ‬ ‫ﺍﻟﺘﺨﻤﻴﻦﺧﻄ ﹰﺄ‪.‬‬ ‫‪ 4‬‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪C :‬‬ ‫ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻣﻊ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﺘﻲ ﺗﺤﺼﻞ ﻋﻠﻴﻬﺎ ﻣﻦ ﺧﻼﻝ ﻫﺬﺍ ﺍﻟﻤﺴﺢ؟ ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫‪  ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻛﺘﺎﺑﺔ‬ ‫‪AB‬‬ ‫ﺧﻤﺴﺔ ﺗﺨﻤﻴﻨﺎﺕ ﺣﻮﻝ ﻧﺸﺎﻃﺎﺕ ﻣﺪﺭﺳﺘﻬﻢ‬ ‫ﻭﺃﻧﻈﻤﺘﻬﺎ‪ ،‬ﺛﻢ ﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﺃﻥ ﺗﺒﺎﺩﻝ ﺍﻷﻭﺭﺍﻕ‪،‬‬ ‫‪‬‬ ‫ﻭﻣﺤﺎﻭﻟﺔ ﺇﻳﺠﺎﺩ ﻣﺜﺎ ﹴﻝ ﻣﻀﺎ ﱟﺩ ﻟﻜﻞ ﺗﺨﻤﻴﻦ‪.‬‬ ‫‪ (47‬ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ ‪ ، a = 10 , b = 1‬ﻓﻤﺎ ﻗﻴﻤﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻵﺗﻴﺔ؟ ‪‬‬ ‫‪ (46‬ﺍﻧﻈﺮ ﺇﻟﻰ ﺍﻟﻨﻤﻂ ﺍﻵﺗﻲ‪:‬‬ ‫‪_32‬‬ ‫‪\" ‬ﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻫﺎﺏ ﺇﻟﻰ ﺍﻟﻤﺪﺭﺳﺔ‬ ‫)‪2b + ab ÷ (a + b‬‬ ‫‪......‬‬ ‫‪11‬‬ ‫ﻣﻦ ﺍﻷﺣﺪ ﺇﻟﻰ ﺍﻟﺨﻤﻴﺲ“‪ .‬ﻭﺍﻟﻤﺜﺎﻝ ﺍﻟﻤﻀﺎﺩ‬ ‫ﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﺍﻟﻨﻤﻂ؟ ‪B‬‬ ‫ﻟﻬﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺃﻥ ﻳﻜﻮﻥ ﻳﻮﻡ ﻋﻴﺪ ﺍﻟﻔﻄﺮ ﻳﻮﻡ‬ ‫‪ (48‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪،‬‬ ‫ﺍﻹﺛﻨﻴﻦ‪ ،‬ﺣﻴﺚ ﺇﺟﺎﺯﺓ ﺍﻟﻤﺪﺍﺭﺱ ﻓﻲ ﺫﻟﻚ‬ ‫‪ AB‬ﻣﺤﻮﺭ ﺗﻨﺎﻇﺮ ‪ .∠DAC‬ﺃ ﱡﻱ ‪D B‬‬ ‫‪CA‬‬ ‫ﺍﻟﻴﻮﻡ‪.‬‬ ‫ﺍﻻﺳﺘﻨﺘﺎﺟﺎﺕ ﺍﻵﺗﻴﺔ ﻟﻴﺲ‬ ‫‪DB‬‬ ‫ﺍﺟﻤﻊ ﺍﻷﻭﺭﺍﻕ ﻣﻦ ﺍﻟﻄﻼﺏ ﻋﻨﺪ ﺧﺮﻭﺟﻬﻢ‬ ‫ﻣﻦ ﺍﻟﻔﺼﻞ‪.‬‬ ‫ﺻﺤﻴ ﹰﺤﺎ ﺑﺎﻟﻀﺮﻭﺭﺓ؟ ‪C B‬‬ ‫‪A ∠DAB ∠BAC A‬‬ ‫‪ ∠DAC B‬ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪.‬‬ ‫‪ A C‬ﻭ ‪ D‬ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫‪2(m∠BAC) = m∠DAC D‬‬ ‫‪‬‬ ‫‪  (49‬ﺍﺷﺘﺮ￯ ﺑﺎﺳﻢ ﺣﻮ ﹶﺽ ﺳﻤ ﹴﻚ ﺻﻐﻴﺮ ﻋﻠﻰ ﺷﻜﻞ ﺃﺳﻄﻮﺍﻧﺔ ﺩﺍﺋﺮﻳﺔ ﻗﺎﺋﻤﺔ‪ ،‬ﻃﻮﻝ ﻗﻄﺮ ﻗﺎﻋﺪﺗﻬﺎ ‪ ، 25 cm‬ﻭﺍﺭﺗﻔﺎﻋﻬﺎ ‪،35 cm‬‬ ‫ﺃﻭﺟﺪ ﺣﺠﻢ ﺍﻟﻤﺎﺀ ﺍﻟﻼﺯﻡ ﻟﹺﻤﻞ ﹺﺀ ﺍﻟﺤﻮﺽ‪17180.6 cm3  .‬‬ ‫ﺃﻭﺟﺪ ﻣﺤﻴﻂ ‪ ABC‬ﺇﺫﺍ ﹸﺃﻋﻄﻴﺖ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺭﺅﻭﺳﻪ ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ :‬‬ ‫‪26.69 A(–3, 2), B(2, –9), C(0, –10) (51‬‬ ‫‪10.47 A(1, 6), B(1, 2), C(3, 2) (50‬‬ ‫‪  (52‬ﻗﻴﺎﺱ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺘﺎﻣﺘﻴﻦ ﻳﺴﺎﻭﻱ ‪ (16z - 9)°‬ﻭ ‪ .(4z + 3)°‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ‪22.2 ;67.8  .‬‬ ‫‪  (53‬ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ‪ x = 3 :‬ﻭ ‪ y = -4‬ﻭ ‪ ،z = -5‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‪-16  . 5|x + y| - 3|2 - z| :‬‬ ‫‪‬‬ ‫‪‬ﺍﻛﺘﺐ ﻛﻠﻤﺔ \"ﺻﺢ\" ﺑﺠﻮﺍﺭ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺼﺤﻴﺤﺔ ﻭﻛﻠﻤﺔ \"ﺧﻄﺄ\" ﺑﺠﻮﺍﺭ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺨﺎﻃﺌﺔ‪.‬‬ ‫‪ (56‬ﺍﻟﻌﺪﺩ ‪ 9‬ﻋﺪﺩ ﺃﻭﻟﻲ ﺧﻄﺄ‬ ‫‪ 5 - 2 × 3 = 9 (55‬ﺧﻄﺄ‪‬‬ ‫‪ (54‬ﻛﻞ ﻣﺮﺑﻊ ﻫﻮ ﻣﺴﺘﻄﻴﻞ ﺻﺢ‬ ‫‪ 1 18‬‬ ‫‪‬‬ ‫‪ ‬ﺍﻋﻤﻞ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻠﻌﺪﺩﻳﻦ ﺍﻟﺘﺎﻟﻴﻴﻦ ﻓﻲ ﺍﻟﻤﺘﺘﺎﺑﻌﺔ ﺍﻵﺗﻴﺔ‪9, 7, 10, 8, 11, 9, 12, . . . :‬‬ ‫ﺍﻃﺮﺡ ‪ ،2‬ﺛﻢ ﺃﺿﻒ ‪3‬؛ ‪13 ،10‬‬ ‫‪‬‬ ‫‪ (45‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺃﻭﺩ ﺃﻥ ﺃﺟﺮﻱ ﻣﺴ ﹰﺤﺎ ﻷﻧﻮﺍﻉ ﺍﻷﻧﺸﻄﺔ ﺍﻟﺘﻲ ﻳﻤﺎﺭﺳﻬﺎ ﺍﻟﻨﺎﺱ ﻓﻲ ﻋﻄﻠﺔ ﻧﻬﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ‪ ،‬ﻭﺃﻃﺮﺡ‬ ‫ﺍﻷﺳﺌﻠﺔ ﺍﻵﺗﻴﺔ‪ :‬ﻣﺎ ﻋﻤﺮﻙ؟ ﻣﺎ ﻧﻮﻉ ﺍﻟﻨﺸﺎﻁ ﺍﻟﺬﻱ ﺗﻔﻀﻞ ﻣﻤﺎﺭﺳﺘﻪ ﻓﻲ ﻋﻄﻠﺔ ﻧﻬﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ؟‬ ‫ﻣﺎ ﻣﺪ￯ ﻣﻮﺍﻇﺒﺘﻚ ﻋﻠﻰ ﻣﻤﺎﺭﺳﺔ ﻫﺬﺍ ﺍﻟﻨﺸﺎﻁ؟ ﺛﻢ ﺑﻌﺪ ﺫﻟﻚ ﺃﺳﺘﻌﻤﻞ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻹﻳﺠﺎﺩ ﺃﻧﻤﺎ ﹴﻁ ﻓﻲ‬ ‫ﺍﻹﺟﺎﺑﺎﺕ ﻟﺘﺤﺪﻳﺪ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺍﻷﺷﺨﺎﺹ ﺍﻟﻤﺘﺴﺎﻭﻭﻥ ﻓﻲ ﺍﻟ ﹸﻌ ﹸﻤﺮ ﻳﻔﻀﻠﻮﻥ ﻣﻤﺎﺭﺳﺔ ﺍﻷﻧﺸﻄﺔ ﻧﻔﺴﻬﺎ ﺃﻡ ﻻ‪.‬‬ ‫‪ 1 18‬‬

‫‪ ‬‬ ‫‪1 -1 ‬‬ ‫‪          ‬‬ ‫‪  (7)  ‬‬ ‫‪ (6)‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ 1-1‬‬ ‫‪  1-1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫ﻳﻜﻮ ﹸﻥ ﺍﻟﺘﺨﻤ ﹸﲔ ﺧﻄ ﹰﺄ‪ ،‬ﺇﺫﺍ ﹸﻭﺟ ﹶﺪ ﻣﺜﺎ ﹲﻝ ﻭﺍﺣ ﹲﺪ ﻳﺘﺒ ﱢ ﹸﲔ ﺃ ﹶﻥ ﺍﻟﺘﺨﻤ ﹶﲔ ﻓﻴﻪ ﻏ ﹸﲑ ﺻﺤﻴ ﹺﺢ‪ ،‬ﻭﻫﺬﺍ ﺍﳌﺜﺎ ﹸﻝ ﹸﻳﺴ ﱠﻤﻰ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪‬ﻫﻮ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻟﺬﻱ ﻳﻌﺘﻤﺪ ﻋﻠﻰ ﻣﻌﻠﻮﻣﺎﺕ ﻧﺘﺠﺖ ﻋﻦ ﺃﻣﺜﻠﺔ ﻣﺨﺘﻠﻔﺔ ﺗﻤ ﱢﺜﻞ ﻧﻤ ﹰﻄﺎ؛ ﻟﻠﺘﻮﺻﻞ ﺇﻟﻰ ﻧﺘﻴﺠﺔ ﺃﻭ ﻋﺒﺎﺭﺓ‬ ‫ﹸﺗﺴ ﹼﻤﻰ ﺗﺨﻤﻴ ﹰﻨﺎ‪.‬‬ ‫ﺃﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ ﻳﺒ ﱢﻴ ﹲﻦ ﻋﺪ ﹶﻡ ﺻﺤ ﹺﺔ ﺍﻟﺘﺨﻤﻴ ﹺﻦ ﺍﻵﺗﻲ‪.‬‬ ‫‪‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،AB BC‬ﻓﺈ ﱠﻥ ‪ B‬ﻧﻘﻄ ﹸﺔ ﻣﻨﺘﺼ ﹺﻒ ‪.AC‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﺍﻟﻤﺘﺘﺎﺑﻌﺔ‬ ‫ﺍﻛﺘـﺐ ﺗﺨﻤﻴﻨﹰـﺎ ﻳﺼـﻒ ﺍﻟﻨﻤـﻂ ﻓـﻲ‬ ‫ﻫﻞ ﻳﻤ ﹺﻜﻨﹸﻚ ﺃﻥ ﺗﺮﺳ ﹶﻢ ﺷﻜ ﹰﻼ ﺗﻜﻮ ﹸﻥ ﻓﻴﻪ ‪ ، AB BC‬ﻋﲆ ﺃ ﹼﻻ ﺗﻜﻮﻥ ‪ B‬ﻧﻘﻄﺔ ﻣﻨﺘﺼ ﹺﻒ ‪ AC‬؟‬ ‫‪C‬‬ ‫ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻓﻲ ﺇﻳﺠﺎﺩ ﻃﻮﻝ ﺿﻠﻊ ﺍﻟﻤﺮﺑﻊ ﻓﻲ ﺍﻟﺸﻜﻞ‬ ‫ﺍﻟﻤﺘﺘﺎﺑﻌﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻓﻲ ﺇﻳﺠﺎﺩ ﺍﻟﺤ ﹼﺪ ﺍﻟﺘﺎﻟﻲ ﻟﻠﻤﺘﺘﺎﺑﻌﺔ‪:‬‬ ‫‪3 cm‬‬ ‫‪A 3 cm B‬‬ ‫ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫‪. 1, 3, 9, 27, 81 ,...‬‬ ‫ﹸﻳﻌ ﹼﺪ ﺍﻟﺸﻜﻞ ﺍﳌﺠﺎﻭﺭ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ؛ ﻷﻥ ‪ B‬ﻟﻴﺴﺖ ﻭﺍﻗﻌ ﹰﺔ ﻋﲆ ‪ AC‬؛ ﺇﺫﻥ ﺍﻟﺘﺨﻤﲔ ﺧﺎﻃﺊ‪.‬‬ ‫‪1 ‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‪:‬‬ ‫‪1 3 9 27 81‬‬ ‫‪‬‬ ‫‪1 ‬ﺍﺑﺤﺚ ﻋﻦ ﻧﻤﻂ‪:‬‬ ‫ﺣ ﱢﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻛﻞ ﺗﺨﻤﻴﻦ ﻣ ﹼﻤﺎ ﻳﺄﺗﻲ ﺻﺤﻴ ﹰﺤﺎ ﺃﻡ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ‪:‬‬ ‫ﺃﻃﻮﺍﻝ ﺃﺿﻼﻉ ﺍﳌﺮﺑﻌﺎﺕ ﻫﻲ‪ 1, 2, 3 :‬ﻭﺣﺪﺍ ﹴﺕ‪.‬‬ ‫‪30 31 32 33 34‬‬ ‫‪ (1‬ﺇﺫﺍ ﻭﻗﻌﺖ ﺍﻟﻨﻘﺎﻁ ‪ A, B, C‬ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣ ﹴﺔ ﻭﺍﺣﺪ ﹴﺓ‪ ،‬ﻓﺈﻥ‬ ‫‪ (2‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠R‬ﻭ ‪ ∠S‬ﻣﺘﻜﺎﻣﻠﺘﲔ‪ ،‬ﻭ‪ ∠T‬ﻭ ‪ ∠R‬ﻣﺘﻜﺎﻣﻠﺘﻴﻦ ﺃﻳ ﹰﻀﺎ‪،‬‬ ‫‪2 ‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪:‬‬ ‫‪2 ‬ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ‪:‬‬ ‫ﺳﻴﻜﻮﻥ ﻃﻮﻝ ﺿﻠﻊ ﺍﳌﺮﺑﻊ ﰲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﱄ ‪ 4‬ﻭﺣﺪﺍﺕ‪،‬‬ ‫ﻛﻞ ﻭﺍﺣﺪ ﻣﻦ ﻫﺬﻩ ﺍﻷﻋﺪﺍﺩ ﻫﻮ ﻗﻮﺓ ﻟﻠﻌﺪﺩ ‪.3‬‬ ‫ﻓﺈﻥ ‪∠T ∠S‬‬ ‫‪.AC=BC+AB‬‬ ‫ﺇﺫﻥ ﺳﻴﻜﻮﻥ ﰲ ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﱄ ‪ 16‬ﻣﺮﺑ ﹰﻌﺎ ﺻﻐ ﹰﲑﺍ‪.‬‬ ‫ﺇﺫﻥ ﺳﻴﻜﻮﻥ ﺍﻟﻌﺪﺩ ﺍﻟﺘﺎﱄ ‪35‬؛ ﺃ ﹾﻱ ‪243‬‬ ‫‪ B A  C      ‬‬ ‫‪B‬‬ ‫‪‬‬ ‫‪AC‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪:‬‬ ‫‪-80     -2           -5, 10, -20, 40,… (1‬‬ ‫‪ (3‬ﺇﺫﺍ ﻛﺎﻧﺖ‪ ∠ABC‬ﹶﻭ ‪ ∠DEF‬ﻣﺘﻜﺎﻣﻠﺘﻴﻦ‪،‬‬ ‫‪10000         10     1, 10, 100, 1000,… (2‬‬ ‫‪ (4‬ﺇﺫﺍ ﻛﺎﻧﺖ‪ ،DE ⊥ EF‬ﻓﺈﻥ ‪ ∠DEF‬ﻗﺎﺋﻤﺔ‪.‬‬ ‫ﻓﺈﻧﻬﻤﺎ ﻣﺘﺠﺎﻭﺭﺗﺎﻥ ﻋﻠﻰ ﺧ ﱟﻂ ﻣﺴﺘﻘﻴ ﹴﻢ‪.‬‬ ‫‪_95_    ‬‬ ‫‪   _51_     ‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫_‪_6‬‬ ‫‪,‬‬ ‫_‪_7‬‬ ‫‪,‬‬ ‫…‪_85_ ,‬‬ ‫‪(3‬‬ ‫‪         ‬‬ ‫‪5‬‬ ‫‪5‬‬ ‫‪CD‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣ ﹼﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛ ﱠﻢ ﺃﻋﻂ ﺃﻣﺜﻠﺔ ﻋﺪﺩﻳﺔ‪ ،‬ﺃﻭ ﺍﺭﺳﻢ ﺃﺷﻜﺎ ﹰﻻ ﺗﺆﻳﺪ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪     4-7    ‬‬ ‫‪A BE F‬‬ ‫‪ (5‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ‪∠ 1‬ﻭ ‪∠2‬ﺗﻜ ﹼﻮﻧﺎﻥ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪.‬‬ ‫‪A (-1, -1), B (2,2), C (4,4) (4‬‬ ‫∠‪  2  ∠1‬‬ ‫‪      A, B, C ‬‬ ‫)‪P y C(4, 4‬‬ ‫‪12‬‬ ‫‪R‬‬ ‫)‪B(2, 2‬‬ ‫‪T‬‬ ‫‪W‬‬ ‫‪A(–1, –1) O‬‬ ‫‪x‬‬ ‫‪ (7‬ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﻋﺪﺩﻳﻦ ﻓﺮﺩﻳﻴﻦ‪.‬‬ ‫‪ ∠ABC (6‬ﻭ ‪ ∠DBE‬ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﺑﻠﺘﺎﻥ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫‪     ‬‬ ‫‪  ∠DBE  ∠ABC‬‬ ‫‪23 - 9 = 14, 15 - 7 = 8‬‬ ‫‪A BE‬‬ ‫‪1‬‬ ‫‪7‬‬ ‫‪ ‬‬ ‫‪1‬‬ ‫‪6‬‬ ‫‪CD‬‬ ‫‪  ‬‬ ‫‪ ‬‬ ‫‪ (9)  ‬‬ ‫‪ (8)‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ 1-1‬‬ ‫‪  1-1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (4‬ﺗﻢ ﺗﻜﻠﻴﻒ ﺻﺎﻟ ﹴﺢ ﺑﺘﻮﺯﻳﻊ ‪ 31‬ﻣﻴﺪﺍﻟﻴ ﹰﺔ ﻋﻠﻰ ﺃﻋﻀﺎﺀ‬ ‫‪  (1‬ﺩﺣﺮﺝ ﻋﻠ ﱞﻲ ﻛﺮﺓ ﺯﺟﺎﺟﻴﺔ ﻋﻠﻰ ﺳﻄﺢ ﻣﺎﺋﻞ‪،‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴﻨﹰﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﺍﻟﻤﺘﺘﺎﺑﻌﺔ‪:‬‬ ‫‪ 6‬ﻓﹺ ﹶﺮﻕ ﺭﻳﺎﺿﻴﺔ ﻣﺘﻨﺎﻓﺴﺔ‪ ،‬ﻓﺎﺳﺘﻨﺘﺞ ﺻﺎﻟﺢ ﺃﻥ ﻓﺮﻳ ﹰﻘﺎ ﻭﺍﺣ ﹰﺪﺍ‬ ‫ﻭﻛﺎﻥ ﻳﻘﻴﺲ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻲ ﺗﻘﻄﻌﻬﺎ ﺍﻟﻜﺮﺓ ﻛﻞ ﺛﺎﻧﻴﺔ‪ ،‬ﻭﺳ ﹼﺠﻞ‬ ‫ﻋﻠﻰ ﺍﻷﻗﻞ ﺳﻴﺤﺮﺯ ﺃﻛﺜﺮ ﻣﻦ ‪ 5‬ﻣﻴﺪﺍﻟﻴﺎﺕ‪ ،‬ﻓﻬﻞ ﺍﺳﺘﻨﺘﺎﺟﻪ‬ ‫‪              ‬‬ ‫‪(1‬‬ ‫ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺍﻵﺗﻲ‪:‬‬ ‫ﺻﺤﻴﺢ؟ ﺑ ﱢﺮﺭ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫ﺍﻷﻭﻟﻰ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺜﺎﻟﺜﺔ ﺍﻟﺮﺍﺑﻌﺔ‬ ‫‪‬‬ ‫‪5            ‬‬ ‫‪  30   6×5       ‬‬ ‫‪140 100 60 20 ‬‬ ‫‪-2 ,4 ,-8 ,16 ,-32 (4‬‬ ‫‪6‬‬ ‫‪,‬‬ ‫_‪_1_1‬‬ ‫‪,‬‬ ‫‪5,_9_2_,‬‬ ‫‪4‬‬ ‫‪(3‬‬ ‫‪-4 ,-1 ,2 ,5 ,8 (2‬‬ ‫‪2‬‬ ‫‪       ‬‬ ‫‪1_2_       ‬‬ ‫‪       ‬‬ ‫ﻣﺎ ﺍﳌﺴﺎﻓﺔ ﺍﻟﺘﻲ ﻳﻤﻜﻦ ﺃﻥ ﺗﻘﻄﻌﻬﺎ ﺍﻟﻜﺮﺓ ﰲ ﺍﻟﺜﺎﻧﻴﺔ ﺍﳋﺎﻣﺴﺔ‪.‬‬ ‫‪64 -2 ‬‬ ‫‪ 27    ‬‬ ‫‪11 3 ‬‬ ‫‪  (5‬ﻳﻮﺿﺢ ﺍﻟﺸﻜﻞ ﺍﻵﺗﻲ ﻣﺘﺘﺎﺑﻌﺔ ﻣﺮﺑﻌﺎﺕ‪ ،‬ﻛ ﱞﻞ ﻣﻨﻬﺎ‬ ‫‪180 cm‬‬ ‫ﻳﺘﻜﻮﻥ ﻣﻦ ﺑﻼﻃﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﻣﺮﺑﻌﺔ ﺍﻟﺸﻜﻞ‪.‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣ ﹼﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛ ﱠﻢ ﺃﻋﻂ ﺃﻣﺜﻠﺔ ﻋﺪﺩﻳﺔ‪ ،‬ﺃﻭ ﺍﺭﺳﻢ ﺃﺷﻜﺎ ﹰﻻ ﺗﺆﻳﺪ ﻫﺬﺍ ﺍﻟﺘﺨﻤﻴﻦ‪:‬‬ ‫‪      5–8    ‬‬ ‫‪ (5‬ﺗﻘﻊ ﺍﻟﻨﻘﺎﻁ ‪ A, B, C‬ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣ ﹴﺔ ﻭﺍﺣﺪ ﹴﺓ‪ ،‬ﻭﺗﻘﻊ ﺍﻟﻨﻘﻄﺔ ‪ D‬ﺑﻴﻦ ‪ (6 .B, C‬ﺍﻟﻨﻘﻄﺔ ‪ P‬ﻫﻲ ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ ‪.NQ‬‬ ‫‪  (2‬ﺍﻟﻌﺪﺩ ﺍﻷﻭﻟﻲ ﻫﻮ ﻋﺪﺩ ﺃﻛﺒﺮ ﻣﻦ ‪ ،1‬ﻭﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ‬ ‫‪NP = PQ‬‬ ‫‪      A, B, C, D ‬‬ ‫ﻋﻠﻰ ﻧﻔﺴﻪ ﻭﻋﻠﻰ ‪ 1‬ﻓﻘﻂ‪ ،‬ﺣﺎﻭﻝ ﺳﻌﺪ ﺃﻥ ﻳﺠﺪ ﻃﺮﻳﻘﺔ ﻣﻨﻬﺠﻴﺔ‬ ‫‪ (a‬ﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺍﺑﻊ ﻣﻦ ﻣﺘﺘﺎﺑﻌﺔ ﺍﳌﺮﺑﻌﺎﺕ؟‬ ‫ﻟﺘﺤﺪﻳﺪ ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ‪ ،‬ﻭﺑﻌﺪ ﻋﺪﺓ ﻣﺤﺎﻭﻻﺕ ﺗﻮﺻﻞ ﺇﻟﻰ ﺃﻥ‬ ‫‪NPQ‬‬ ‫‪A CD B‬‬ ‫‪ 2n - 1‬ﻳﻜﻮﻥ ﻋﺪ ﹰﺩﺍ ﺃﻭﻟ ﹼﹰﻴﺎ ﻷ ﱢﻱ ﻋﺪ ﹴﺩ ﻃﺒﻴﻌ ﱟﻲ ‪ ، n‬ﻓﻬﻞ ﺍﺳﺘﻨﺘﺎﺟﻪ‬ ‫‪ (8‬ﻧﺎﺗﺞ ﺿﺮﺏ ﻋﺪﺩﻳﻦ ﻓﺮﺩ ﱠﻳﻴﻦ‬ ‫‪ ∠2 ، ∠1 (7‬ﻣﺘﻜﺎﻣﻠﺘﺎﻥ ﹶﻭ ‪ ∠3 ، ∠1‬ﻣﺘﻜﺎﻣﻠﺘﺎﻥ‪.‬‬ ‫ﺻﺤﻴﺢ؟‬ ‫‪      ‬‬ ‫‪∠3 ∠2‬‬ ‫‪15 2n - 1 =15  n = 4     ‬‬ ‫‪3 × 5 = 15 , 9 × 7 = 63‬‬ ‫‪12‬‬ ‫‪5 × 3 = 15      ‬‬ ‫‪3‬‬ ‫‪ (b‬ﻋ ﹼﱪ ﻋﻦ ﻣﺘﺘﺎﺑﻌﺔ ﺍﳌﺮﺑﻌﺎﺕ‪ ،‬ﺑﻤﺘﺘﺎﺑﻌﺔ ﺃﻋﺪﺍﺩ؟‬ ‫‪  (3‬ﻭﺿﻌﺖ ﻓﺎﻃﻤﺔ ﻣﺨﻄ ﹰﻄﺎ ﻟﻨﹶﺴﺒﹺﻬﺎ‪ ،‬ﻣﻤ ﹼﺜ ﹰﻼ ﺑﺜﻼﺛﺔ ﺭﺳﻮﻡ‬ ‫ﺣ ﹼﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺻﺤﻴﺤﺔ ﺃﻡ ﺧﺎﻃﺌﺔ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪:‬‬ ‫‪12, 22, 32, 42, ‬‬ ‫ﻛﻤﺎ ﻫﻮ ﻣﻮﺿﺢ ﺃﺩﻧﺎﻩ‪ ،‬ﺣﻴﺚ ﺗﻤﺜﻞ ﺍﻟﻨﻘ ﹶﻄﺔ ﺍﻷﻭﻟﻰ ﻓﺎﻃﻤﺔ‪،‬‬ ‫‪ (9‬ﺇﺫﺍ ﻛﺎﻧﺖ‪∠ABC‬ﻭ‪∠CBD‬ﻣﺘﺠﺎﻭﺭﺗﻴﻦ ﻋﻠﻰ ﺧ ﱟﻂ ﻣﺴﺘﻘﻴ ﹴﻢ‪ ،‬ﻓﺈﻥ ‪.∠ABC ∠CBD‬‬ ‫ﻭﻳﻤﺜﻞ ﺍﻟﺮﺳﻢ ﺍﻟﺜﺎﻧﻲ ﻓﺎﻃﻤﺔ ﻭﻭﺍﻟ ﹶﺪﻳﻬﺎ‪ ،‬ﻭﻳﻤﺜﻞ ﺍﻟﺮﺳﻢ ﺍﻟﺜﺎﻟﺚ‬ ‫‪ (c‬ﺇﺫﺍ ﺍﺳﺘﻤﺮ ﺍﻟﻨﻤﻂ ﺑﻨﻔﺲ ﺍﻟﻄﺮﻳﻘﺔ‪ ،‬ﻓﻜﻢ ﻣﺮﺑ ﹰﻌﺎ ﺻﻐ ﹰﲑﺍ‬ ‫‪       ‬‬ ‫ﺳﻴﻜﻮﻥ ﰲ ﺍﻟﺸﻜﻞ ﺭﻗﻢ ‪10‬؟‬ ‫ﻓﺎﻃﻤﺔ ﻭﻭﺍﻟ ﹶﺪﻳﻬﺎ ﻭﺟ ﱠﺪﻳﻬﺎ ﻭﺟ ﱠﺪﺗﻴﻬﺎ‪.‬‬ ‫‪100  102‬‬ ‫‪ (10‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ AB, BC, AC‬ﻣﺘﻄﺎﺑﻘﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﻘﺎﻁ ‪ A, B, C‬ﺗﻘﻊ ﻋﻠﻰ ﺧ ﱟﻂ ﻣﺴﺘﻘﻴ ﹴﻢ‪.‬‬ ‫ﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺍﺑﻊ ﺍﻟﺬﻱ ﺳﱰﺳﻤﻪ ﻓﺎﻃﻤﺔ؟ ﻭﻣﺎﺫﺍ ﻳﻤﺜﻞ؟‬ ‫‪ AB, BC , AC   ‬‬ ‫‪AB‬‬ ‫‪ (11‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،AB + BC = AC‬ﻓﺈﻥ ‪.AB = BC‬‬ ‫‪C     ‬‬ ‫‪ (12‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠1‬ﻣﺘ ﱢﻤﻤﺔ ﻟ ﹺـ ‪ ،∠2‬ﻭﻛﺎﻧﺖ ‪ ∠1‬ﻣﺘﻤﻤﺔ ﻟ ﹺـ ‪ ∠3‬ﺃﻳ ﹰﻀﺎ‪ ،‬ﻓﺈﻥ ‪ .∠3 ∠2‬‬ ‫‪1‬‬ ‫‪8‬‬ ‫‪  ‬‬ ‫‪1‬‬ ‫‪9‬‬ ‫‪  ‬‬ ‫‪18A   1‬‬

‫‪  ‬‬ ‫‪‬‬ ‫‪1 -1 ‬‬ ‫‪         ‬‬ ‫‪ ‬‬ ‫‪( 6)‬‬ ‫‪( 1 0 )‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ 1 - 1‬‬ ‫‪   1-1‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻳﺼﻒ ﺍﻟﻨﻤﻂ ﻓﻲ ﻛﻞ ﻣﺘﺘﺎﺑﻌﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻠﻪ ﻹﻳﺠﺎﺩ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪:‬‬ ‫‪‬‬ ‫‪25 5, -10 , 15 , -20 ... (2‬‬ ‫‪... (1‬‬ ‫ﻋﻨﺪﻣﺎ ﺗﺘﻮﺻﻞ ﺇﱃ ﺍﺳﺘﻨﺘﺎﺝ ﺑﻌﺪ ﺍﺧﺘﺒﺎﺭﻙ ﻟﻌﺪﺓ ﺣﺎﻻﺕ ﻣﻌ ﱠﻴﻨﺔ‪ ،‬ﻓﺈﻧﻚ ﺗﺴﺘﻌﻤﻞ ﺍﻟﺘﱪﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ‪ .‬ﻭﻣﻊ ﺫﻟﻚ ﻛﻦ ﺣﺬ ﹰﺭﺍ ﺃﺛﻨﺎﺀ ﺍﺳﺘﻌﲈﻟﻚ‬ ‫‪0.375 12, 6, 3, 1.5, 0.75 ... (4‬‬ ‫ﻫﺬﺍ ﺍﻟﻨﻮﻉ ﻣﻦ ﺍﻟﺘﱪﻳﺮ؛ ﻷﻧﻪ ﰲ ﺣﺎﻟﺔ ﻭﺟﻮﺩ ﻣﺜﺎﻝ ﻣﻀﺎ ﱟﺩ ﻭﺍﺣﺪ‪ ،‬ﺳﻴﻜﻮﻥ ﻛﺎﻓ ﹰﻴﺎ ﻹﺛﺒﺎﺕ ﻋﺪﻡ ﺻﺤﺔ ﻫﺬﺍ ﺍﻻﺳﺘﻨﺘﺎﺝ‪.‬‬ ‫‪_1‬‬ ‫‪-2,‬‬ ‫‪1,‬‬ ‫‪-‬‬ ‫‪_1‬‬ ‫‪,‬‬ ‫‪_1‬‬ ‫‪,‬‬ ‫‪-‬‬ ‫‪_1‬‬ ‫‪...‬‬ ‫‪(3‬‬ ‫‪ ‬‬ ‫ﻫﻞ ﺍﻟﻌﺒﺎﺭﺓ ‪ x_1_ ≤1‬ﺻﺤﻴﺤﺔ ﻋﻨﺪ ﺍﻟﺘﻌﻮﻳﺾ ﻋﻦ ‪ x‬ﺑﺎﻷﻋﺪﺍﺩ ‪ 1, 2, 3‬؟ ﻭﻫﻞ ﻫﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﺃﻳ ﹰﻀﺎ ﻟﻜﻞ‬ ‫ﻣﺜﺎﻝ‬ ‫‪16‬‬ ‫‪2‬‬ ‫‪4‬‬ ‫‪8‬‬ ‫ﺍﻷﻋﺪﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ؟ ﺃﻭﺟﺪ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﺇﺫﺍ ﻛﺎﻥ ﺫﻟﻚ ﻣﻤﻜﻨﹰﺎ‪.‬‬ ‫ﺿﻊ ﺗﺨﻤﻴ ﹰﻨﺎ ﻟﻜﻞ ﻗﻴﻤﺔ ﺃﻭ ﻋﻼﻗﺔ ﻫﻨﺪﺳﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺃﻋﻂ ﺃﻣﺜﻠﺔ ﻋﺪﺩﻳﺔ ﺃﻭ ﺍﺭﺳﻢ ﺃﺷﻜﺎ ﹰﻻ ﺗﺴﺎﻋﺪ ﻋﻠﻰ ﺍﻟﻮﺻﻮﻝ ﺇﻟﻰ ﻫﺬﺍ‬ ‫‪ ،_x1_ = 2‬ﻭﻫﺬﺍ ﺍﳌﺜﺎﻝ ﺍﳌﻀﺎﺩ ﻳﺜﺒﺖ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﻟﻴﺴﺖ ﺻﺤﻴﺤ ﹰﺔ ﺩﺍﺋ ﹰﲈ‪.‬‬ ‫ﻓﺈﻥ‬ ‫‪،‬‬ ‫=‪x‬‬ ‫‪_1‬‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﻟﻜﻦ‬ ‫‪،‬‬ ‫_‪_1‬‬ ‫‪= 1,‬‬ ‫_‪_1‬‬ ‫<‬ ‫‪1‬‬ ‫‪,‬‬ ‫_‪_1‬‬ ‫<‬ ‫‪1‬‬ ‫ﺍﻟﺘﺨﻤﻴﻦ‪.‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪‬‬ ‫‪ (6‬ﺍﻟﻨﻘﺎﻁ ‪ R,S,T‬ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪،‬‬ ‫‪ ∠ ABC (5‬ﻗﺎﺋﻤﺔ‪.‬‬ ‫‪ (1‬ﻫﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ √ k 2 = k :‬ﺻﺤﻴﺤﺔ ﻓﻲ ﺣﺎﻟﺔ ﺍﻟﺘﻌﻮﻳﺾ ﻋﻦ ‪ k‬ﺑﺎﻷﻋﺪﺍﺩ ‪1 ,2 ,3‬؟ﻭﻫﻞ ﻫﺬﻩ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺻﺤﻴﺤﺔ ﻟﻜﻞ ﺍﻷﻋﺪﺍﺩ‬ ‫ﻭﺍﻟﻨﻘﻄﺔ ‪ S‬ﺗﻘﻊ ﺑﻴﻦ ‪ R‬ﻭ ‪.T‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪BA ⊥ BC :‬‬ ‫       ‪A‬‬ ‫ﺍﻟﺼﺤﻴﺤﺔ ﺃﻳ ﹰﻀﺎ؟ ﻫﺎﺕ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ ﺇﻥ ﺃﻣﻜﻦ‪.‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪RS + ST = RT :‬‬ ‫‪√(-2)2 ≠ -2        1 , 2, 3    ‬‬ ‫‪R ST‬‬ ‫‪BC‬‬ ‫ﺍﻷﻋﺪﺍﺩ‬ ‫ﻟﻜﻞ‬ ‫ﺻﺤﻴﺤﺔ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﻫﺬﻩ‬ ‫ﻭﻫﻞ‬ ‫‪0.7‬؟‬ ‫‪,4‬‬ ‫‪,‬‬ ‫_‪_1‬‬ ‫ﺑﺎﻷﻋﺪﺍﺩ‪:‬‬ ‫‪x‬‬ ‫ﻋﻦ‬ ‫ﺍﻟﺘﻌﻮﻳﺾ‬ ‫ﻋﻨﺪ‬ ‫ﺻﺤﻴﺤﺔ‬ ‫ﻫﻞ ﺍﻟﻌﺒﺎﺭﺓ‪2x = x + x :‬‬ ‫‪(2‬‬ ‫‪2‬‬ ‫‪ ABCD (8‬ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ‪.‬‬ ‫ﺍﻟﺤﻘﻴﻘ ﱠﻴﺔ ﺃﻳ ﹰﻀﺎ؟ ﻫﺎﺕ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ ﺇﻥ ﺃﻣﻜﻦ‪     .‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ AB = CD :‬ﻭ ‪.BC = AD‬‬ ‫‪_P__, Q,_R__, S (7‬ﻟﻴﺴ _ﺖ__ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪،‬‬ ‫ﻭ ‪PQ QR RS SP‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ (3‬ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﻋ ﹼﻴﻨﺖ ﺃﺭﺑﻊ ﻧﻘﺎﻁ‪ ، A, B, C, D :‬ﺛﻢ ﺭﺳﻤﺖ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤ‪‬ﺔ‪:‬‬ ‫‪AB‬‬ ‫‪C‬‬ ‫ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻫﺬﻩ ﺍﻟﻨﻘﺎﻁ ﺗﺸ ﹼﻜﻞ‬ ‫‪A‬‬ ‫‪DC‬‬ ‫ﺭﺅﻭﺱ ﻣﺮﺑﻊ ﺃﻭ ﻣﻌﻴﻦ‪.‬‬ ‫‪A‬‬ ‫‪ ، AB , BC, CD, DA‬ﻓﻬﻞ ﹸﺗﻌﻄﻲ ﻫﺬﻩ ﺍﻟﻄﺮﻳﻘﺔ ﺷﻜ ﹰﻼ ﺭﺑﺎﻋ ﹼﹰﻴﺎ ﺩﺍﺋ ﹰﻤﺎ ﺃﻡ ﺃﺣﻴﺎ ﹰﻧﺎ؟ ﻭﺿﺢ ‪B‬‬ ‫‪P QP‬‬ ‫‪Q‬‬ ‫‪B‬‬ ‫‪D‬‬ ‫ﺇﺟﺎﺑﺘﻚ ﺑﺎﻟﺮﺳﻢ‪  .‬‬ ‫‪C‬‬ ‫‪S RS‬‬ ‫‪R‬‬ ‫‪D‬‬ ‫ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻦ ﺍﻟﺘﺨﻤﻴﻨﺎﺕ ﺍﻵﺗﻴﺔ ﺻﺤﻴ ﹰﺤﺎ ﺃﻭ ﺧﺎﻃ ﹰﺌﺎ ‪ ،‬ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﻟﺘﺨﻤﻴﻦ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ ‪:‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ (4‬ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﺭﺳﻤﺖ ﺩﺍﺋﺮﺓ‪ ،‬ﻭﻭﺿﻌﺖ ﻋﻠﻴﻬﺎ ﺛﻼﺙ ﻧﻘﺎﻁ‪ ،‬ﺛﻢ ﻭﺻﻠﺖ ﺑﻴﻨﻬﺎ‪ ،‬ﻓﻬ‪‬ﻞ ‪  ‬‬ ‫ﺗﻜﻮﻥ ﺯﻭﺍﻳﺎ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻨﺎﺗﺞ ﺣﺎﺩﺓ ﺩﺍﺋ ﹰﻤﺎ ﺃﻡ ﺃﺣﻴﺎ ﹰﻧﺎ؟ ﻭ ﹼﺿﺢ ﺇﺟﺎﺑﺘﻚ ﺑﺎﻟﺮﺳﻢ‪  .‬‬ ‫‪ST‬‬ ‫=‬ ‫ﻭ‪TU‬‬ ‫ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‬ ‫‪_S, T, U‬ﻋﻠ__ﻰ‬ ‫‪ ‬ﺗﻘﻊ ﺍﻟﻨﻘﺎﻁ‬ ‫‪(9‬‬ ‫ﺻﺤﻴﺢ‬ ‫ﻣﻨﺘﺼﻒ ‪SU‬‬ ‫‪ ‬ﺍﻟﻨﻘﻄﺔ ‪ T‬ﻫﻲ‬ ‫‪ ∠1  (10‬ﻭ ‪ ∠2‬ﻣﺘﺠﺎﻭﺭﺗﺎﻥ‪.‬‬ ‫‪ ∠1 ‬ﻭ ‪ ∠2‬ﻣﺘﺠﺎﻭﺭﺗﺎﻥ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ‪ .‬ﺧﺎﻃﺊ ﻳﻤﻜﻦ ﺃﻥ ﺗﺘﺠﺎﻭﺭ ﺯﺍﻭﻳﺘﺎﻥ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ ‪60°‬‬ ‫‪ (5‬ﻳﻮﺿﺢ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻣﺘﺘﺎﺑﻌﺔ ﻣﺮﺑﻌﺎﺕ‪ ،‬ﻛ ﱞﻞ ﻣﻨﻬﺎ ﻳﺘﻜﻮﻥ ﻣﻦ ﺑﻼﻃﺎﺕ ﻣﺘﻄﺎﺑﻘﺔ ﻣﺮﺑﻌﺔ ﺍﻟﺸﻜﻞ‪.‬‬ ‫___ ___‬ ‫ﺗﺸﻜﻼﻥ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ ﻭﺗﺘﻘﺎﻃﻌﺎﻥ ﻓﻲ ﺍﻟﻨﻘﻄﺔ ‪.P‬‬ ‫ﻭ_‪_J_K‬‬ ‫‪GH‬‬ ‫‪‬‬ ‫‪(11‬‬ ‫ﺻﺤﻴﺢ‬ ‫___‬ ‫‪‬‬ ‫‪GH ⊥ JK‬‬ ‫‪   (12‬ﻳﺒﺪﺃ ﺭﺍﺷﺪ ﺑﺎﻟﻌﻄﺎﺱ ﻋﻨﺪﻣﺎ ﺗﺰﻫﺮ ﺍﻷﺷﺠﺎﺭ ﻓﻲ ﻓﺼﻞ ﺍﻟﺮﺑﻴﻊ‪ ،‬ﻭﻋﻨﺪﻣﺎ ﺗﻤﻄﺮ ﺍﻟﺴﻤﺎﺀ‪ ،‬ﻭﻗﺪ ﻋ ﹼﻠﻞ‬ ‫‪ (a‬ﺇﺫﺍ ﱂ ﻳﻜﻦ ﻟﺪﻳﻚ ﺃﻱ ﺑﻼﻃﺔ‪ ،‬ﻓﻜﻢ ﺑﻼﻃ ﹰﺔ ﺳﺘﺤﺘﺎﺝ ﻟﺘﻜﻮﻳﻦ ﺃﻭﻝ ﻣﺮﺑﻊ؟ ﻭﻛﻢ ﺑﻼﻃﺔ ﺗﻀﻴﻒ ﺇﱃ ﺍﳌﺮﺑﻊ ﺍﻷﻭﻝ ﻟﺘﻜﻮﻳﻦ ﺍﳌﺮﺑﻊ‬ ‫ﺭﺍﺷﺪ ﺃﺳﺒﺎﺏ ﺣﺴﺎﺳﻴﺘﻪ ﺑﺄﻧﻬﺎ ﻣﺮﺗﺒﻄﺔ ﺑﻔﺼﻞ ﺍﻟﺮﺑﻴﻊ‪ .‬ﻳﺘﺤﺴﺲ ﺭﺍﺷﺪ ﻋﻨﺪﻣﺎ ﺗﻤﻄﺮ ﺍﻟﺴﻤﺎﺀ‪ ،‬ﺇﺫﻥ ﻳﻤﻜﻦ ﺃﻥ ﻳﺘﺤﺴﺲ‬ ‫ﺍﻟﺜﺎﲏ؟ ﻭﻛﻢ ﺑﻼﻃﺔ ﺗﻀﻴﻒ ﺇﱃ ﺍﳌﺮﺑﻊ ﺍﻟﺜﺎﲏ ﻟﺘﻜﻮﻳﻦ ﺍﳌﺮﺑﻊ ﺍﻟﺜﺎﻟﺚ؟ ‪1, 3, 5‬‬ ‫ﻓﻲ ﻓﺼﻞ ﺍﻟﺸﺘﺎﺀ ﻭﺫﻟﻚ ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪.‬‬ ‫‪ (b‬ﻛ ﹼﻮﻥ ﲣﻤﻴﻨﹰﺎ ﺣﻮﻝ ﳎﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺘﻲ ﲢﺼﻞ ﻋﻠﻴﻬﺎ ﻣﻦ ﺇﺟﺎﺑﺘﻚ ﻟﻠﻔﺮﻉ ‪      .a‬‬ ‫‪ (c‬ﻛ ﹼﻮﻥ ﲣﻤﻴﻨﹰﺎ ﺣﻮﻝ ﳎﻤﻮﻉ ﺃﻭﻝ ‪ n‬ﻣﻦ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻔﺮﺩﻳﺔ‪n 2      n   .‬‬ ‫‪6‬‬ ‫‪1‬‬ ‫‪10‬‬ ‫‪  ‬‬ ‫‪ 1 18B‬‬

‫‪‬‬ ‫‪ 1- 2‬‬ ‫‪‬‬ ‫‪Logic‬‬ ‫‪ 1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﻋﻨﺪ ﺇﺟﺎﺑﺘﻚ ﻋﻦ »ﺃﺳﺌﻠﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺻﺢ ﺃﻭ ﺧﻄﺄ« ﻓﻲ ﺍﺧﺘﺒﺎﺭ‪،‬‬ ‫‪  ‬‬ ‫ﻓﺈﻧﻚ ﺗﺴﺘﻌﻤﻞ ﻣﺒﺪ ﹰﺃ ﺃﺳﺎﺳ ﹰﹼﻴﺎ ﻓﻲ ﺍﻟﻤﻨﻄﻖ‪.‬‬ ‫‪‬‬ ‫‪1-2‬‬ ‫‪1 - 1 ‬‬ ‫ﺇﻳﺠﺎﺩ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ ﻟﻠﺘﺨﻤﻴﻨﺎﺕ ﺍﻟﺨﺎﻃﺌﺔ‪.‬‬ ‫ﻓﻤﺜ ﹰﻼ ﺍﻧﻈﺮ ﺇﻟﻰ ﺧﺮﻳﻄﺔ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ ﻭﺃﺟﺐ ﻋﻦ‬ ‫‪ ‬‬ ‫‪1-2‬‬ ‫ﺍﻟﺨﺒﺮ ﺍﻟﺘﺎﻟﻲ ﺑﺼﺤﻴﺢ ﺃﻭ ﺧﺎﻃﺊ‪ :‬ﺃﺑﻬﺎ ﻣﺪﻳﻨﺔ ﺳﻌﻮﺩﻳﺔ‪.‬‬ ‫ﺗﻌﻴﻴﻦ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻌﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ‬ ‫‪  ‬‬ ‫ﻭﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ‪ ،‬ﻭﺗﻤﺜﻴﻞ ﻋﺒﺎﺭ ﹶﺗﻲ ﺍﻟﻮﺻﻞ‬ ‫ﺃﻧﺖ ﺗﻌﺮﻑ ﺃﻧﻪ ﻳﻮﺟﺪ ﺇﺟﺎﺑﺔ ﻭﺣﻴﺪﺓ ﺻﺎﺋﺒﺔ‪ ،‬ﺇﻣﺎ ﺻﺤﻴﺢ ﺃﻭ‬ ‫‪‬‬ ‫‪   ‬‬ ‫ﻭﺍﻟﻔﺼﻞ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺃﺷﻜﺎﻝ ﭬﻦ‪.‬‬ ‫ﺧﺎﻃﺊ‪.‬‬ ‫‪1-2‬‬ ‫‪‬‬ ‫‪ ‬ﺍﻟﻌﺒﺎﺭﺓ ﻫﻲ ﺟﻤﻠﺔ ﺧﺒﺮﻳﺔ ﻟﻬﺎ ﺣﺎﻟﺘﺎﻥ ﻓﻘﻂ ﺇﻣﺎ ﺃﻥ ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ ﺃﻭ ﺗﻜﻮﻥ ﺧﺎﻃﺌﺔ‪ ،‬ﻭﻻ ﺗﺤﺘﻤﻞ‬ ‫‪‬‬ ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻨﺘﺎﺟﻲ ﻹﺛﺒﺎﺕ‬ ‫ﺻﺤﺔ ﻋﺒﺎﺭﺓ‪.‬‬ ‫ﺃﻱ ﺣﺎﻟﺔ ﺃﺧﺮ￯‪ .‬ﻭﺻﻮﺍﺏ ﺍﻟﻌﺒﺎﺭﺓ )‪ (T‬ﺃﻭ ﺧﻄﺆﻫﺎ )‪ (F‬ﻳﺴﻤﻰ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪ ،‬ﻭﻳﺮﻣﺰ ﻟﻠﻌﺒﺎﺭﺓ ﺑﺮﻣﺰ ﻣﺜﻞ ‪ p‬ﺃﻭ ‪.q‬‬ ‫‪‬‬ ‫‪ 2‬‬ ‫‪T‬‬ ‫‪ : p‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﺷﻜﻞ ﺭﺑﺎﻋ ﹼﻲ‬ ‫‪‬‬ ‫‪‬‬ ‫ﻧﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ﻳﻔﻴﺪ ﻣﻌﻨﹰﻰ ﹸﻣﻀﺎ ﹼﹰﺩﺍ ﻟﻤﻌﻨﻰ ﺍﻟﻌﺒﺎﺭﺓ‪ .‬ﻭﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻪ ﻫﻮ ﻋﻜﺲ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻷﺻﻠﻴﺔ‪ ،‬ﻓﻤﺜ ﹰﻼ‪:‬‬ ‫‪statement‬‬ ‫ﻧﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ‪ p‬ﺃﻋﻼﻩ ﻫﻮ ‪ ، ~p‬ﺃﻭ \" ﻟﻴﺲ ‪ ، \" p‬ﺣﻴﺚ‪:‬‬ ‫ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻗﺮﺍﺀﺓ ﻓﻘﺮﺓ \"ﻟﻤﺎﺫﺍ؟\" ‪.‬‬ ‫‪‬‬ ‫‪F‬‬ ‫‪ : ~ p‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﻟﻴﺲ ﺷﻜ ﹰﻼ ﺭﺑﺎﻋ ﹰﹼﻴﺎ‬ ‫‪ ‬‬ ‫‪truth value‬‬ ‫ﻳﻤﻜﻨﻚ ﺭﺑﻂ ﻋﺒﺎﺭﺗﻴﻦ ﺃﻭ ﺃﻛﺜﺮ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺮﺍﺑﻂ )ﻭ( ‪ ،‬ﺃﻭ ﺍﻟﺮﺍﺑﻂ )ﺃﻭ( ﻟﺘﻜﻮﻳﻦ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ‪ .‬ﻭﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﺘﻲ‬ ‫• ﺍﺫﻛﺮ ﻋﺒﺎﺭ ﹰﺓ ﺻﺤﻴﺤ ﹰﺔ ﺣﻮﻝ ﺍﻟﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭﺓ‬ ‫‪‬‬ ‫ﻟﻠﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪ .‬ﻋﺒﺎﺭﺓ ﻣﻤﻜﻨﺔ‪:‬‬ ‫ﺗﺤﺘﻮﻱ )ﻭ( ﹸﺗﺴﻤﻰ ﻋﺒﺎﺭﺓ ﻭﺻﻞ‪ .‬ﻭﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﺻﺎﺋﺒﺔ ﻓﻘﻂ ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﺟﻤﻴﻊ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﻜﻮﻧﺔ ﻟﻬﺎ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﹶﺗ ﹸﺤ ﱡﺪ ﺍﻟﺠﻤﻬﻮﺭﻳ ﹸﺔ ﺍﻟﻴﻤﻨﻴﺔ ﺍﻟﻤﻤﻠﻜ ﹶﺔ ﻣﻦ‬ ‫‪negation‬‬ ‫ﺍﻟﺠﻨﻮﺏ‪.‬‬ ‫‪T‬‬ ‫‪ :p‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﺷﻜﻞ ﺭﺑﺎﻋ ﹼﻲ‬ ‫• ﺿﻊ ﺗﺨﻤﻴﻨﹰﺎ ﺣﻮﻝ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻫﻨﺎﻙ ﺑﺤﻴﺮﺍﺕ‬ ‫‪‬‬ ‫ﺃﻭ ﺑﺤﺮ ﻓﻲ ﻣﺪﻳﻨﺔ ﺟﺪﺓ‪ .‬ﺗﺨﻤﻴﻦ ﻣﻤﻜﻦ‪:‬‬ ‫‪T‬‬ ‫‪ :q‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﻣﻀ ﹼﻠﻊ ﻣﺤ ﹼﺪﺏ‬ ‫ﻣﺪﻳﻨﺔ ﺟﺪﺓ ﺗﻘﻊ ﻋﻠﻰ ﺳﺎﺣﻞ ﺍﻟﺒﺤﺮ ﺍﻷﺣﻤﺮ‪.‬‬ ‫‪compound statement‬‬ ‫‪ p‬ﻭ ‪ :q‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﺷﻜﻞ ﺭﺑﺎﻋﻲ ﻭﺍﻟﻤﺴﺘﻄﻴﻞ ﻣﻀﻠﻊ ﻣﺤ ﹼﺪﺏ‪.‬‬ ‫‪‬‬ ‫ﺑﻤﺎ ﺃﻥ ﻛﻠﺘﺎ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ‪ p‬ﻭ ‪ q‬ﺻﺎﺋﺒﺘﺎﻥ‪ ،‬ﻓﺈﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ‪ p‬ﻭ ‪ q‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺗﻜﺘﺐ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ‪ p‬ﻭ ‪ q‬ﺑﺎﻟﺮﻣﻮﺯ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ‪. p q‬‬ ‫‪conjunction‬‬ ‫‪ 1‬‬ ‫‪ ‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ‪ p, q, r‬ﻭﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻟﻜﺘﺎﺑﺔ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ .‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‬ ‫‪disjunction‬‬ ‫ﻣﺒﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪:‬‬ ‫‪‬‬ ‫‪ :p‬ﺍﻟﺸﻜﻞ ﻣﺜﻠﺚ‪.‬‬ ‫‪truth table‬‬ ‫‪ :q‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺿﻠﻌﺎﻥ ﻣﺘﻄﺎﺑﻘﺎﻥ‪.‬‬ ‫‪www.obeikaneducation.com‬‬ ‫‪ :r‬ﺟﻤﻴﻊ ﺯﻭﺍﻳﺎ ﺍﻟﺸﻜﻞ ﺣﺎﺩﺓ‪.‬‬ ‫‪‬‬ ‫‪ p (a‬ﻭ ‪r‬‬ ‫‪   ‬‬ ‫‪ p‬ﻭ ‪ :r‬ﺍﻟﺸﻜﻞ ﻣﺜﻠﺚ ﻭﺟﻤﻴﻊ ﺯﻭﺍﻳﺎ ﺍﻟﺸﻜﻞ ﺣﺎﺩﺓ‪.‬‬ ‫‪ ‬‬ ‫‪  ‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ‪ p‬ﺻﺎﺋﺒﺔ‪ ،‬ﻟﻜﻦ ﺍﻟﻌﺒﺎﺭﺓ ‪ r‬ﺧﺎﻃﺌﺔ‪ ،‬ﺇﺫﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ‪ p‬ﻭ ‪ r‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪   ‬‬ ‫‪q ~r (b‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ :q ~r‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺿﻠﻌﺎﻥ ﻣﺘﻄﺎﺑﻘﺎﻥ‪ ،‬ﻭﻟﻴﺲ ﺟﻤﻴﻊ ﺯﻭﺍﻳﺎ ﺍﻟﺸﻜﻞ ﺣﺎﺩﺓ‪.‬‬ ‫‪  ‬‬ ‫ﺑﻤﺎ ﺃﻥ ﻛﻼ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ‪ q‬ﻭ ‪ ~r‬ﺻﺎﺋﺒﺘﺎﻥ‪ ،‬ﻓﺈﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ‪ q ~r‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫✓ ‪ (1A, 1B ‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪ (1B‬ﻟﻴﺲ ‪ p‬ﻭ ﻟﻴﺲ ‪r‬‬ ‫‪p q (1A‬‬ ‫‪19  1- 2‬‬ ‫‪ ‬‬ ‫‪1-2‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪(23) • ‬‬ ‫‪(25) •‬‬ ‫‪(23, 25) •‬‬ ‫‪(7) • ‬‬ ‫‪(7) •‬‬ ‫‪(7) •‬‬ ‫‪‬‬ ‫‪(14)  •‬‬ ‫‪(11)  •‬‬ ‫‪‬‬ ‫‪(15) •‬‬ ‫‪(11) •‬‬ ‫‪(13) •‬‬ ‫‪(13) •‬‬ ‫‪(14)  •‬‬ ‫‪(14)  •‬‬ ‫‪(15) •‬‬ ‫‪19  1-2‬‬

‫ﺗﺴﻤﻰ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﺘﻲ ﺗﺤﺘﻮﻱ )ﺃﻭ( ﻋﺒﺎﺭﺓ ﻓﺼﻞ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ :p‬ﺩﺭﺱ ﻣﺎﻟﻚ ﺍﻟﻬﻨﺪﺳﺔ‪.‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ 3-1  ‬ﻛﻴﻔﻴﺔ ﺇﻳﺠﺎﺩ ﻗﻴﻤﺔ‬ ‫‪ :q‬ﺩﺭﺱ ﻣﺎﻟﻚ ﺍﻟﻜﻴﻤﻴﺎﺀ‪.‬‬ ‫‪ ‬‬ ‫‪  ‬‬ ‫ﺍﻟﺼﻮﺍﺏ ﻟﻌﺒﺎﺭﺍﺕ ﺍﻟ ﹶﻔ ﹾﺼﻞ ﺍﻟﻤﻨﻄﻘﻲ ﻭﺍﻟﻮﺻﻞ‬ ‫‪ p‬ﺃﻭ ‪ :q‬ﺩﺭﺱ ﻣﺎﻟﻚ ﺍﻟﻬﻨﺪﺳﺔ ﺃﻭ ﺩﺭﺱ ﻣﺎﻟﻚ ﺍﻟﻜﻴﻤﻴﺎﺀ‪.‬‬ ‫‪‬‬ ‫ﺍﻟﻤﻨﻄﻘﻲ‪.‬‬ ‫‪ ‬‬ ‫ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟ ﹶﻔ ﹾﺼﻞ ﺻﺎﺋﺒﺔ ﺇﺫﺍ ﻛﺎﻧﺖ ﺇﺣﺪ￯ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﻜﻮﻧﺔ ﻟﻬﺎ ﺻﺎﺋﺒﺔ‪ ،‬ﻭﺗﻜﻮﻥ ﺧﺎﻃﺌﺔ ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﻤﻴﻊ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫ﺍﻟﻤﻜﻮﻧﺔ ﻟﻬﺎ ﺧﺎﻃﺌﺔ‪ .‬ﻓﺈﺫﺍ ﺩﺭﺱ ﻣﺎﻟﻚ ﺍﻟﻬﻨﺪﺳﺔ ﺃﻭ ﺍﻟﻜﻴﻤﻴﺎﺀ ﺃﻭ ﻛﻠﻴﻬﻤﺎ‪ ،‬ﻓﺈﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ‪ p‬ﺃﻭ ‪ q‬ﺻﺎﺋﺒﺔ‪ .‬ﻭﺇﺫﺍ ﻟﻢ‬ ‫ﻳﺪﺭﺱ ﻣﺎﻟﻚ ﺃ ﹼﹰﻳﺎ ﻣﻦ ﺍﻟﻬﻨﺪﺳﺔ ﻭﺍﻟﻜﻴﻤﻴﺎﺀ‪ ،‬ﻓﺈﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ‪ p‬ﺃﻭ ‪ q‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ ‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺗﻤﺎﺭﻳﻦ \"ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻤﻚ\" ﺑﻌﺪ‬ ‫ﻛﻞ ﻣﺜﺎﻝ؛ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻣﺪ￯ ﻓﻬﻢ ﺍﻟﻄﻠﺒﺔ‬ ‫ﺗﻜﺘﺐ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ‪ p‬ﺃﻭ ‪ q‬ﺑﺎﻟﺮﻣﻮﺯ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ‪. p q‬‬ ‫ﺍﻟﻤﻔﺎﻫﻴﻢ‪.‬‬ ‫‪ 2‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ‪ p, q, r‬ﻭﺍﻟﺼﻮﺭﺓ ﺍﻟﻤﺠﺎﻭﺭﺓ؛ ﻟﻜﺘﺎﺑﺔ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ﻓﻲ‬ ‫ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ ﻣﺒﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫‪ :p‬ﻳﻨﺎﻳﺮ ﻣﻦ ﺃﺷﻬﺮ ﻓﺼﻞ ﺍﻟﺮﺑﻴﻊ‪.‬‬ ‫‪1‬‬ ‫‪ :q‬ﻋﺪﺩ ﺃﻳﺎﻡ ﺷﻬﺮ ﻳﻨﺎﻳﺮ ‪ 30‬ﻳﻮ ﹰﻣﺎ ﻓﻘﻂ‪.‬‬ ‫‪‬‬ ‫‪ :r‬ﻳﻨﺎﻳﺮ ﻫﻮ ﺃﻭﻝ ﺃﺷﻬﺮ ﺍﻟﺴﻨﺔ ﺍﻟﻤﻴﻼﺩﻳﺔ‪.‬‬ ‫‪        ‬‬ ‫‪2021‬‬ ‫‪1 2 345 6‬‬ ‫‪ q (a‬ﺃﻭ ‪r‬‬ ‫‪‬‬ ‫‪78‬‬ ‫‪9‬‬ ‫‪10 11 12‬‬ ‫‪13‬‬ ‫‪ q‬ﺃﻭ ‪ :r‬ﻋﺪﺩ ﺃﻳﺎﻡ ﺷﻬﺮ ﻳﻨﺎﻳﺮ ‪ 30‬ﻳﻮ ﹰﻣﺎ ﻓﻘﻂ ﺃﻭ ﻳﻨﺎﻳﺮ ﻫﻮ ﺃﻭﻝ ﺃﺷﻬﺮ‬ ‫‪2021‬‬ ‫ﺍﻟﺴﻨﺔ ﺍﻟﻤﻴﻼﺩﻳﺔ‪.‬‬ ‫‪2021‬‬ ‫‪14 15‬‬ ‫‪16 17 18 19‬‬ ‫‪20‬‬ ‫‪2021‬‬ ‫‪‬‬ ‫‪ q‬ﺃﻭ ‪ r‬ﺻﺎﺋﺒﺔ ﻷﻥ ﺍﻟﻌﺒﺎﺭﺓ ‪ r‬ﺻﺎﺋﺒﺔ‪ .‬ﻭﻛﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ‪ q‬ﺧﺎﻃﺌﺔ ﻻ ﻳﺆﺛﺮ‪.‬‬ ‫‪21 22 23‬‬ ‫‪24 25 26‬‬ ‫‪27‬‬ ‫‪28 29 30 31‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻟﻜﺘﺎﺑﺔ ﻋﺒﺎﺭﺓ‬ ‫‪1‬‬ ‫ﺍﻟﻮﺻﻞ ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ‬ ‫‪p q (b‬‬ ‫ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪ ،‬ﻣﺒﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫‪ :p q‬ﻳﻨﺎﻳﺮ ﻣﻦ ﺃﺷﻬﺮ ﻓﺼﻞ ﺍﻟﺮﺑﻴﻊ‪ ،‬ﺃﻭ ﻋﺪﺩ ﺃﻳﺎﻡ ﺷﻬﺮ ﻳﻨﺎﻳﺮ ‪ 30‬ﻳﻮ ﹰﻣﺎ ﻓﻘﻂ‪.‬‬ ‫‪ (2A‬ﻳﻨﺎﻳﺮ ﻫﻮ ﺃﻭﻝ ﺷﻬﺮ ﻓﻲ‬ ‫‪ :P‬ﺍﻟﻘﺪﻡ ﺗﻌﺎﺩﻝ ‪ 14‬ﺑﻮﺻ ﹰﺔ‪.‬‬ ‫ﺑﻤﺎ ﺃﻥ ﻛ ﹼﹰﻼ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺈﻥ ‪ p q‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﺍﻟﺴﻨﺔ ﺍﻟﻤﻴﻼﺩﻳﺔ ﺃﻭ ﻳﻨﺎﻳﺮ ﻣﻦ ﺃﺷﻬﺮ‬ ‫‪ :q‬ﺷﻬﺮ ﺭﻣﻀﺎﻥ ﻫﻮ ﺷﻬﺮ ﺍﻟﺼﻴﺎﻡ ﻋﻨﺪ‬ ‫‪~p r (c‬‬ ‫ﻓﺼﻞ ﺍﻟﺮﺑﻴﻊ‪ .‬ﺑﻤﺎ ﺃﻥ ‪ r‬ﺻﺎﺋﺒﺔ‬ ‫ﺍﻟﻤﺴﻠﻤﻴﻦ‪.‬‬ ‫ﻓﺈﻥ ‪ r‬ﺃﻭ ‪ p‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪ :~p r‬ﻳﻨﺎﻳﺮ ﻟﻴﺲ ﻣﻦ ﺃﺷﻬﺮ ﻓﺼﻞ ﺍﻟﺮﺑﻴﻊ ﺃﻭ ﻳﻨﺎﻳﺮ ﻫﻮ ﺃﻭﻝ ﺃﺷﻬﺮ ﺍﻟﺴﻨﺔ ﺍﻟﻤﻴﻼﺩﻳﺔ‪.‬‬ ‫‪ :r‬ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ ﻳﺴﺎﻭﻱ ‪°90‬‬ ‫‪ ~p r‬ﺻﺎﺋﺒﺔ؛ ﻷﻥ ‪ ~p‬ﺻﺎﺋﺒﺔ ﻭ ‪ r‬ﺻﺎﺋﺒﺔ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫‪p ∧ q (a‬‬ ‫‪ (2B‬ﻋﺪﺩ ﺃﻳﺎﻡ ﻳﻨﺎﻳﺮ ‪ 30‬ﻳﻮ ﹰﻣﺎ ✓ ‪‬‬ ‫‪p ~q (2C‬‬ ‫‪q ~r (2B‬‬ ‫ﻓﻘﻂ ﺃﻭ ﻳﻨﺎﻳﺮ ﻟﻴﺲ ﺃﻭﻝ ﺷﻬﺮ ﻓﻲ ‪ r (2A‬ﺃﻭ ‪p‬‬ ‫ﺍﻟﺴﻨﺔ ﺍﻟﻤﻴﻼﺩﻳﺔ‪ .‬ﺑﻤﺎ ﺃﻥ ﻛﻠﺘﺎ‬ ‫ﺍﻟﻘﺪﻡ ﺗﻌﺎﺩﻝ ‪ 14‬ﺑﻮﺻ ﹰﺔ‪ ،‬ﻭﺷﻬﺮ‬ ‫ﺭﻣﻀﺎﻥ ﻫﻮ ﺷﻬﺮ ﺍﻟﺼﻴﺎﻡ ﻋﻨﺪ‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ‪ ~r ، q‬ﺧﺎﻃﺌﺔ ﻓﺈﻥ‬ ‫‪ q ~r‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﺍﻟﻤﺴﻠﻤﻴﻦ‪) .‬ﺧﺎﻃﺌﺔ(‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (2C‬ﻳﻨﺎﻳﺮ ﻣﻦ ﺃﺷﻬﺮ ﻓﺼﻞ‬ ‫ﺑﻤﺎ ﺃﻥ ‪ p‬ﺧﺎﻃﺌﺔ ﺇﺫﻥ ‪ p ∧ q‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪p  ~p‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫ﺍﻟﺮﺑﻴﻊ‪ ،‬ﻭﻋﺪﺩ ﺃﻳﺎﻡ ﺷﻬﺮ ﻳﻨﺎﻳﺮ‬ ‫‪∼p r (b‬‬ ‫‪qp p q‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﻟﻴﺲ ‪ 30‬ﻳﻮ ﹰﻣﺎ‪ .‬ﺑﻤﺎ ﺃﻥ ‪ ~q‬ﺻﺎﺋﺒﺔ‬ ‫‪qpp q‬‬ ‫‪ ‬‬ ‫‪  ‬‬ ‫ﻓﺈﻥ ‪ p ~q‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻘﺪﻡ ﻻ ﺗﻌﺎﺩﻝ ‪ 14‬ﺑﻮﺻ ﹰﺔ‪،‬‬ ‫ﻭﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ ﻳﺴﺎﻭﻱ‬ ‫‪   ‬‬ ‫‪) °90‬ﺻﺎﺋﺒﺔ(‪.‬‬ ‫‪ 1 20‬‬ ‫ﺑﻤﺎ ﺃﻥ ‪ p‬ﺧﺎﻃﺌﺔ‪ ،‬ﺇﺫﻥ ‪ ∼p‬ﺻﺎﺋﺒﺔ ﻭ‪r‬‬ ‫ﺻﺎﺋﺒﺔ‪ ،‬ﻭﻋﻠﻴﻪ ﻓﺈﻥ ‪ ∼p ∧ r‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪ ‬ﺍﻛﺘﺐ ﻋﻠﻰ ﺍﻟﺴﺒﻮﺭﺓ ﻋﺒﺎﺭﺗﻴﻦ ﻣﻨﻄﻘﻴﺘﻴﻦ ‪ p‬ﻭ ‪ ، q‬ﻭﺍﻛﺘﺐ ﺃﻳ ﹰﻀﺎ ﺍﻟﺮﻣﻮﺯ ∨ ‪ ،∼, ∧,‬ﺛﻢ ﺿﻊ‬ ‫ﻫﺬﻩ ﺍﻟﺮﻣﻮﺯ ﺑﻴﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﻟﺘﻜﻮﻳﻦ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ‪ ،‬ﻭﻭﺿﺢ ﻟﻠﻄﻼﺏ ﻛﻴﻔﻴﺔ ﺇﻧﺸﺎﺀ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﻨﺎﺗﺠﺔ‪،‬‬ ‫ﺛﻢ ﺑﻌﺪ ﺫﻟﻚ ﻏ ﹼﻴﺮ ﻭﺿﻊ ﻫﺬﻩ ﺍﻟﺮﻣﻮﺯ ﻟﺘﻜﻮﻳﻦ ﻋﺒﺎﺭﺓ ﻣﻨﻄﻘﻴﺔ ﻣﺮﻛﺒﺔ ﺃﺧﺮ￯‪ ،‬ﻭﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪.‬‬ ‫‪ 1 20‬‬

‫ ﻭﻳﻤﻜﻦ ﺍﺳﺘﻌﻤﺎﻝ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ‬. ‫ﻳﻤﻜﻦ ﺗﻨﻈﻴﻢ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ﻓﻲ ﺟﺪﺍﻭﻝ ﺗﺴﻤﻰ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ‬  .‫ﻟﺘﺤﺪﻳﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻨﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ﻭﻟﻌﺒﺎﺭ ﹶﺗﻲ ﺍﻟﻮﺻﻞ ﻭﺍﻟﻔﺼﻞ‬             p q pq p q pq p ~p TTT TTT   TFT TFF TF  FTT FTF FFF FFF FT   ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻟﻜﺘﺎﺑﺔ ﻋﺒﺎﺭﺓ‬ 2  ‫ ﺛﻢ ﺃﻭﺟﺪ‬،‫ﺍﻟﻔﺼﻞ ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‬  .‫ ﻣﺒﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‬،‫ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‬ .‫ﻭﻛﺬﻟﻚ ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺃﻋﻼﻩ ﻹﻧﺸﺎﺀ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻷﻛﺜﺮ ﺗﻌﻘﻴ ﹰﺪﺍ‬   ___  3   ‫ ﺭﻣﺰ ﺧﺎﺹ ﻟﻠﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‬AB :p ~p q ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ‬ .AB .‫ ﺍﻟﺴﻨﺘﻤﺘﺮﺍﺕ ﻭﺣﺪﺍﺕ ﻗﻴﺎﺱ ﻣﺘﺮ ﹼﻳﺔ‬:q }1 p q ~p ~p q p, q, ~p, ~p q   1 TTFT p, q 2 .‫ ﻋﺪﺩ ﺃﻭﻟ ﱞﻲ‬9 :r TFFF FTTT p 3 q ‫ ﺃﻭ‬p (a FFTT ~p ‫ ﺭﻣﺰ ﺧﺎﺹ ﻟﻠﻘﻄﻌﺔ‬AB ‫ ﺃﻭ ﺍﻟﺴﻨﺘﻤﺘﺮﺍﺕ‬،AB ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‬ 2 34 ~p , q 4 (‫ )ﺻﺤﻴﺤﺔ‬.‫ﻭﺣﺪﺍﺕ ﻗﻴﺎﺱ ﻣﺘﺮﻳﺔ‬ } ~ p q } } q r (b  ✓  ‫ﺍﻟﺴﻨﺘﻤﺘﺮﺍﺕ ﻭﺣﺪﺍﺕ ﻗﻴﺎﺱ‬   .‫ ﻋﺪﺩ ﺃﻭﻟﻲ‬9 ‫ﻣﺘﺮﻳﺔ ﺃﻭ ﺍﻟﻌﺪﺩ‬ ‫ ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬.~p ~q ‫( ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ‬3  .(‫)ﺻﺤﻴﺤﺔ‬    ∼ p ___r (c .‫ ﹸﻋﺪ ﺇﻟﻰ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﻓﻲ ﺑﺪﺍﻳﺔ ﺍﻟﺪﺭﺱ‬.‫ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺃﺷﻜﺎﻝ ﭬﻦ‬  ‫ ﻟﻴﺲ ﺭﻣ ﹰﺰﺍ ﺧﺎ ﹼﹰﺻﺎ ﻟﻠﻘﻄﻌﺔ‬AB   .‫ ﻋﺪﺩ ﺃﻭﻟﻲ‬9 ‫ ﺃﻭ‬AB ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ‬ . ‫ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺷﻜﻞ ﺭﺑﺎﻋﻲ ﻭﺍﻟﻤﺴﺘﻄﻴﻞ ﻣﻀﻠﻊ ﻣﺤﺪﺏ‬:q ‫ ﻭ‬p ‫ ﺧﺎﻃﺌﺔ‬r ‫∼ ﻭ‬p ‫ﺑﻤﺎ ﺃﻥ ﻛ ﹰﹼﻼ ﻣﻦ‬   ‫ ﻭﻫﻲ ﺃﻳ ﹰﻀﺎ ﻣﻀﻠﻌﺎﺕ‬،‫ﺗﻌﻠﻢ ﺃﻥ ﺍﻟﻤﺴﺘﻄﻴﻼﺕ ﺃﺷﻜﺎﻝ ﺭﺑﺎﻋﻴﺔ‬   .‫ ∼ ﺧﺎﻃﺌﺔ ﺃﻳ ﹰﻀﺎ‬p r ‫ﻓﺈﻥ‬ ‫ ﻭﻳﺒ ﱢﻴﻦ ﺷﻜﻞ ﭬﻦ ﺃﻥ ﺍﻟﻤﺴﺘﻄﻴﻼﺕ ﺗﻘﻊ ﻓﻲ ﻣﻨﻄﻘﺔ ﺗﻘﺎﻃﻊ‬،‫ﻣﺤﺪﺑﺔ‬   ‫ ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺓ‬3    .‫ﻣﺠﻤﻮﻋﺔ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺮﺑﺎﻋﻴﺔ ﻭﻣﺠﻤﻮﻋﺔ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﻤﺤ ﹼﺪﺑﺔ‬   :‫ﻓﻴﻤﺎ ﻳﺄﺗﻲ‬  8  ‫ ﺗﻘﻊ ﺍﻟﻤﺴﺘﻄﻴﻼﺕ ﺿﻤﻦ ﻣﺠﻤﻮﻋﺔ ﺍﻷﺷﻜﺎﻝ‬:‫ﻭﺑﻤﻌﻨﻰ ﺁﺧﺮ‬   ∼ (∼ p q) (a .‫ ﻭﺃﻳ ﹰﻀﺎ ﺿﻤﻦ ﻣﺠﻤﻮﻋﺔ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﻤﺤﺪﺑﺔ‬،‫ﺍﻟﺮﺑﺎﻋﻴﺔ‬ n  p q ∼p ∼p ∧ q ∼(∼p ∧ q) 2n  TTF F T   TFF F T       FTT T F 21  1- 2 FFT F T p (∼ q r) (b !      p q r ∼q ∼q ∧ r p ∨(∼q ∧ r) ‫ ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺑﺄﻧﻪ ﺑﻤﻘﺪﻭﺭﻫﻢ ﺗﺒﺪﻳﻞ ﺃﻋﻤﺪﺓ‬  TTT F F T   ‫ ﺷﺮﻳﻄﺔ ﺃﻥ ﹸﻳﻤﻸ‬،3b ‫ ﻓﻲ ﺍﻟﻤﺜﺎﻝ ﺍﻹﺿﺎﻓﻲ‬p, q, r ‫ ﻋﻠﻰ ﺍﻟﻨﺤﻮ ﺍﻟﺼﺤﻴﺢ؛‬p ( q r), q r ‫ﺍﻟﻌﻤﻮﺩﺍﻥ‬ TTF F F T ‫ ﻭﺳﺘﻨﺘﻬﻲ ﺑﺨﻤﺲ ﺇﺟﺎﺑﺎﺕ‬،‫ﻷﻥ ﺍﻟﻨﺎﺗﺞ ﺍﻟﻨﻬﺎﺋﻲ ﻟﻦ ﻳﺘﻐﻴﺮ‬ TFT T T T ،‫ﺻﺎﺋﺒﺔ ﻭﺛﻼﺙ ﺇﺟﺎﺑﺎﺕ ﺧﺎﻃﺌﺔ ﻓﻲ ﺟﻤﻴﻊ ﺍﻟﺤﺎﻻﺕ‬ .‫ﻭﻟﻜﻦ ﺑﺘﺮﺗﻴﺐ ﻣﺨﺘﻠﻒ‬ TFF T F T FTT F F F FTF F F F FFT T T T FFF T F F 21  1-2

‫ﻳﻤﻜﻦ ﺃﻳ ﹰﻀﺎ ﺗﻤﺜﻴﻞ ﻋﺒﺎﺭﺓ ﺍﻟ ﹶﻔ ﹾﺼﻞ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺃﺷﻜﺎﻝ ﭬﻦ‪ .‬ﺇﻟﻴﻚ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ‪ :‬‬ ‫‪‬‬ ‫‪ :p‬ﺍﻟﺸﻜﻞ ﺳﺪﺍﺳﻲ‪.‬‬ ‫‪ ‬‬ ‫‪q p‬‬ ‫‪‬‬ ‫‪ :q‬ﺍﻟﺸﻜﻞ ﻣﻀ ﹼﻠﻊ ﻣﺤ ﹼﺪﺏ‪.‬‬ ‫‪‬‬ ‫‪ p‬ﺃﻭ ‪ :q‬ﺍﻟﺸﻜﻞ ﺳﺪﺍﺳﻲ ﺃﻭ ﻣﻀ ﹼﻠﻊ ﻣﺤ ﹼﺪﺏ‪.‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪~p ∧ q‬‬ ‫‪p ∧ q p ∧ ~q‬‬ ‫ﻓﻲ ﺷﻜﻞ ﭬﻦ ﺍﻟﻤﺠﺎﻭﺭ ﺗﻤﺜﻞ ﻋﺒﺎﺭﺓ ﺍﻟ ﹶﻔﺼﻞ ﺑﺎﺗﺤﺎﺩ‬ ‫‪ 4 ‬ﻳﺒ ﹼﻴﻦ ﻛﻴﻔﻴﺔ ﺍﺳﺘﻌﻤﺎﻝ ﺃﺷﻜﺎﻝ ﭬﻦ‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺘﻴﻦ‪ ،‬ﻭﻳﺤﻮﻱ ﺍﻻﺗﺤﺎﺩ ﺟﻤﻴﻊ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺘﻲ ﻫﻲ‬ ‫‪p ∨q‬‬ ‫ﻟﻮﺿﻊ ﺗﺨﻤﻴﻨﺎﺕ‪ ،‬ﻭﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﻜﻮﻧﻮﺍ‬ ‫ﺇﻣﺎ ﺳﺪﺍﺳﻴﺔ ﺃﻭ ﻣﺤﺪﺑﺔ ﺃﻭ ﻛﻼﻫﻤﺎ‪.‬‬ ‫ﺗﺘﻀﻤﻦ ﻋﺒﺎﺭﺓ ﺍﻟ ﹶﻔ ﹾﺼﻞ ﺍﻟﻤﻨﺎﻃﻖ ﺍﻟﺜﻼﺙ ﺍﻵﺗﻴﺔ‪:‬‬ ‫ﻗﺎﺩﺭﻳﻦ ﻋﻠﻰ ﻭﺿﻊ ﺗﺨﻤﻴﻦ‪ ،‬ﻭﻛﺘﺎﺑﺔ ﻋﺒﺎﺭﺓ‬ ‫‪ p ~q‬ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺴﺪﺍﺳﻴﺔ ﻏﻴﺮ ﺍﻟﻤﺤ ﹼﺪﺑﺔ‪.‬‬ ‫ﻣﺮﻛﺒﺔ‪ ،‬ﻭﺇﻳﺠﺎﺩ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪.‬‬ ‫‪ ~p q‬ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﻤﺤﺪﺑﺔ ﻏﻴﺮ ﺍﻟﺴﺪﺍﺳﻴﺔ‪.‬‬ ‫‪ p q‬ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺴﺪﺍﺳﻴﺔ ﺍﻟﻤﺤﺪﺑﺔ‪.‬‬ ‫‪‬‬ ‫‪4‬‬ ‫‪‬‬ ‫‪  ‬ﹸﻳﻈﻬﺮ ﺷﻜﻞ ﭬﻦ ﺃﺩﻧﺎﻩ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺷﺎﺭﻛﻮﺍ ﻓﻲ ﺣﻤﻠﺔ ﺑﻴﺌﻴﺔ ﻟﻠﺘﻮﻋﻴﺔ ﺑﺄﻫﻤﻴﺔ ﺍﻻﻗﺘﺼﺎﺩ ﻓﻲ‬ ‫‪ ‬ﺷﻜﻞ ﭬﻦ ﺍﻟﺘﺎﻟﻲ ﻳﺒ ﹼﻴﻦ ﻋﺪﺩ‬ ‫‪4‬‬ ‫‪‬‬ ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻮﺭﻕ ﺃﻗﻴﻤﺖ ﺧﻼﻝ ﺷﻬ ﹶﺮﻱ ﺭﺟﺐ ﻭﺷﻌﺒﺎﻥ‪.‬‬ ‫ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﺍﻟﺘﺤﻘﻮﺍ ﺑﺎﻷﻧﺸﻄﺔ‬ ‫‪ ‬‬ ‫‪ (a‬ﻛﻢ ﺷﺨ ﹰﺼﺎ ﺷﺎﺭﻙ ﻓﻲ ﺍﻟﺤﻤﻠﺔ ﻟﺸﻬﺮ ﺭﺟﺐ ﺃﻭ ﺷﻌﺒﺎﻥ؟‬ ‫ﺍﻟﺮﻳﺎﺿﻴﺔ‪.‬‬ ‫ﺍﺗﺤﺎﺩ ﺍﻟﻤﺠﻤﻮﻋﺘﻴﻦ ﻳﻤﺜﻞ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺷﺎﺭﻛﻮﺍ ﻓﻲ‬ ‫‪‬‬ ‫‪14 5‬‬ ‫ﺍﻟﺤﻤﻠﺔ ﺧﻼﻝ ﺷﻬﺮﻱ ﺭﺟﺐ ﺃﻭ ﺷﻌﺒﺎﻥ‪.‬‬ ‫‪‬‬ ‫‪6‬‬ ‫ﻓﻴﻜﻮﻥ ‪ 5 + 6 + 14‬ﺃﻭ ‪ 25‬ﺷﺨ ﹰﺼﺎ ﺷﺎﺭﻛﻮﺍ ﻓﻲ ﺍﻟﺤﻤﻠﺔ‬ ‫‪43‬‬ ‫ﺧﻼﻝ ﺍﻟﺸﻬﺮﻳﻦ‪.‬‬ ‫‪ 13‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪28‬‬ ‫‪‬‬ ‫‪ 20‬‬ ‫‪9‬‬ ‫‪‬‬ ‫‪ (b‬ﻛﻢ ﺷﺨ ﹰﺼﺎ ﺷﺎﺭﻙ ﻓﻲ ﺍﻟﺤﻤﻠﺔ ﺧﻼﻝ ﺷﻬ ﹶﺮﻱ ﺭﺟﺐ ﻭﺷﻌﺒﺎﻥ؟‬ ‫‪‬‬ ‫‪17 25‬‬ ‫ﺗﻘﺎﻃﻊ ﺍﻟﻤﺠﻤﻮﻋﺘﻴﻦ ﻳﻤﺜﻞ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺷﺎﺭﻛﻮﺍ ﻓﻲ ﺍﻟﺤﻤﻠﺔ ﺧﻼﻝ ﻛﻼ ﺍﻟﺸﻬﺮﻳﻦ‪ ،‬ﻟﺬﻟﻚ ﻫﻨﺎﻙ ‪6‬‬ ‫‪‬‬ ‫‪29‬‬ ‫ﺃﺷﺨﺎﺹ ﻓﻘﻂ ﺷﺎﺭﻛﻮﺍ ﻓﻲ ﺍﻟﺤﻤﻠﺔ ﺧﻼﻝ ﻛﻼ ﺍﻟﺸﻬﺮﻳﻦ‪.‬‬ ‫‪‬‬ ‫‪ (c‬ﻣﺎﺫﺍ ﻳﻤﺜﻞ ﺍﻟﻌﺪﺩ ‪ 14‬ﻓﻲ ﺍﻟﺸﻜﻞ؟‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﻤﺸﺎﺭﻛﻴﻦ ﻓﻲ‬ ‫ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺷﺎﺭﻛﻮﺍ ﻓﻲ ﺍﻟﺤﻤﻠﺔ ﺧﻼﻝ ﺷﻬﺮ ﺷﻌﺒﺎﻥ‪ ،‬ﻭﻟﻢ ﻳﺸﺎﺭﻛﻮﺍ ﺧﻼﻝ ﺷﻬﺮ ﺭﺟﺐ‪.‬‬ ‫ﺍﻷﻧﺸﻄﺔ ﺍﻟﺜﻼﺛﺔ؟ ‪9‬‬ ‫‪‬‬ ‫✓ ‪‬‬ ‫‪ (b‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﻤﺸﺎﺭﻛﻴﻦ ﻓﻲ‬ ‫ﻧﺸﺎﻁ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺃﻭ ﻧﺸﺎﻁ ﻛﺮﺓ‬ ‫‪‬‬ ‫‪‬‬ ‫‪  (4‬ﻳﺒﻴﻦ ﺷﻜﻞ ﭬﻦ ﺍﻟﻤﺠﺎﻭﺭ ﻋﺪﺩ ﻃﻼﺏ‬ ‫ﺍﻟﻴﺪ؟ ‪136‬‬ ‫ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﺍﻟﺜﺎﻧﻮﻱ ﺍﻟﺬﻳﻦ ﻧﺠﺤﻮﺍ ﻭﺍﻟﺬﻳﻦ ﻟﻢ ﻳﻨﺠﺤﻮﺍ ﻓﻲ‬ ‫‪ (c‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﻤﺸﺎﺭﻛﻴﻦ‬ ‫ﺍﺧﺘﺒﺎﺭﻱ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﺃﻭ ﺍﻟﻜﻴﻤﻴﺎﺀ‪.‬‬ ‫ﻓﻲ ﻧﺸﺎ ﹶﻃﻲ ﺍﻟﻜﺮﺓ ﺍﻟﻄﺎﺋﺮﺓ ﻭﻛﺮﺓ‬ ‫‪3 46 4‬‬ ‫‪ (A‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻧﺠﺤﻮﺍ ﻓﻲ ﺍﺧﺘﺒﺎﺭ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪ ،‬ﻭﻟﻢ‬ ‫ﺍﻟﻴﺪ‪ ،‬ﻭﻏﻴﺮ ﻣﺸﺎﺭﻛﻴﻦ ﻓﻲ ﻛﺮﺓ‬ ‫ﻳﻨﺠﺤﻮﺍ ﻓﻲ ﺍﺧﺘﺒﺎﺭ ﺍﻟﻜﻴﻤﻴﺎﺀ؟ ‪ 4‬ﻃﻼﺏ‬ ‫ﺍﻟﻘﺪﻡ؟ ‪17‬‬ ‫‪ (B‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻧﺠﺤﻮﺍ ﻓﻲ ﺍﺧﺘﺒﺎﺭ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‬ ‫ﻭﺍﺧﺘﺒﺎﺭ ﺍﻟﻜﻴﻤﻴﺎﺀ؟ ‪ 46‬ﻃﺎﻟ ﹰﺒﺎ ‪2‬‬ ‫‪ (C‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻟﻢ ﻳﻨﺠﺤﻮﺍ ﻓﻲ ﺃ ﱟﻱ ﻣﻦ ﺍﻻﺧﺘﺒﺎﺭﻳﻦ؟ ﻃﺎﻟﺒﺎﻥ‬ ‫‪ (D‬ﻣﺎ ﻋﺪﺩ ﻃﻼﺏ ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﺍﻟﺜﺎﻧﻮﻱ؟ ‪ 55‬ﻃﺎﻟ ﹰﺒﺎ‬ ‫‪ 1 22‬‬ ‫‪‬‬ ‫‪  ‬ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺃﻥ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻳﺠﺐ ﺃﻥ ﹸﺗﻈﻬﺮ ﻛﻞ ﺍﻟﺘﺮﺍﺗﻴﺐ ‪p q r‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫ﺍﻟﻤﻤﻜﻨﺔ ﻟﻘﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ﻛ ﹼﻠﻬﺎ ﻟﺘﺸﻤﻞ ﺟﻤﻴﻊ ﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﻤﻤﻜﻨﺔ‪ .‬ﻓﻔﻲ ﺍﻟﺒﺪﺍﻳﺔ‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫ﻧﺤﺪﺩ ﻋﺪﺩ ﺍﻷﺳﻄﺮ ﺍﻟﺘﻲ ﻧﺤﺘﺎﺟﻬﺎ‪ ،‬ﻓﻤﺜ ﹰﻼ؛ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺍﺕ ‪ p, q, r‬ﻓﺈﻥ ﻋﺪﺩ ﺍﻷﺳﻄﺮ‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪ ،23 = 8‬ﻭﺗﺮﺗﻴﺐ ﺍﻟﺠﺪﻭﻝ ﺑﺄﻥ ﻧﻀﻊ ﻛﻠﻤﺔ ﺻﻮﺍﺏ ﻓﻲ ﻧﺼﻒ ﺃﺳﻄﺮ ﻋﻤﻮﺩ ﺍﻟﻌﺒﺎﺭﺓ ‪T F F ،p‬‬ ‫ﻭﻛﻠﻤﺔ ﺧﻄﺄ ﻓﻲ ﻧﺼﻔﻬﺎ ﺍﻵﺧﺮ‪ ،‬ﻭﻓﻲ ﻋﻤﻮﺩ ﺍﻟﻌﺒﺎﺭﺓ ‪ q‬ﻧﺒﺎﺩﻝ ﺑﻴﻦ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﻛﻠﻤ ﹶﺘﻲ ﺻﻮﺍﺏ ‪F T T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫ﻣﻊ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﻛﻠﻤ ﹶﺘﻲ ﺧﻄﺄ‪ ،‬ﻭﻋﻤﻮﺩ ﺍﻟﻌﺒﺎﺭﺓ ‪ r‬ﻧﺒﺎﺩﻝ ﺑﻴﻦ ﻛﻠﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻭﺍﻟﺨﻄﺄ ﺳﻄ ﹰﺮﺍ‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﺑﺴﻄﺮ ﺣﺘﻰ ﻧﻬﺎﻳﺔ ﺍﻟﺠﺪﻭﻝ‪ ،‬ﻛﻤﺎ ﻫﻮ ﻣﺒﻴﻦ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫ﺑ ﹼﻴﻦ ﻟﻠﻄﻼﺏ ﺃﻧﻬﻢ ﻋﻨﺪﻣﺎ ﹸﻳﺘﻘﻨﻮﻥ ﺍﻟﺘﻌﻠﻴﻤﺎﺕ ﺍﻷﺳﺎﺳﻴﺔ ﺳﻴﺼﺒﺤﻮﻥ ﻗﺎﺩﺭﻳﻦ ﻋﻠﻰ ﺇﻛﻤﺎﻝ ﺍﻟﺠﺪﻭﻝ‪.‬‬ ‫‪ 1 22‬‬

‫✓ ‪ ‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ‪ p, q, r‬ﻟﻜﺘﺎﺑﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻭﺻﻞ ﺃﻭ ﻓﺼﻞ ﺃﺩﻧﺎﻩ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ ﻣﻔ ﱢﺴ ﹰﺮﺍ ﺗﺒﺮﻳﺮﻙ‪:‬‬ ‫‪1, 2‬‬ ‫‪ :p‬ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ﺳﺒﻌﺔ ﺃﻳﺎﻡ‪.‬‬ ‫‪‬‬ ‫‪ :q‬ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻟﻮﺍﺣﺪ ‪ 20‬ﺳﺎﻋﺔ‪.‬‬ ‫‪ (1‬ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ﺳﺒﻌﺔ ﺃﻳﺎﻡ‪ ،‬ﻭﻓﻲ‬ ‫‪ :r‬ﻓﻲ ﺍﻟﺴﺎﻋﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺩﻗﻴﻘﺔ‪ (1–6 .‬ﺍﻧﻈﺮ ﻫﺎﻣﺶ‬ ‫ﺍﻟﺴﺎﻋﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺩﻗﻴﻘﺔ‪ .‬ﺑﻤﺎ ﺃﻥ ﻛ ﹰﹼﻼ‬ ‫ﻣﻦ ‪ p‬ﻭ ‪ r‬ﺻﺤﻴﺤﺔ؛ ﺇﺫﻥ ﻛ ﱞﻞ ﻣﻦ ‪ p‬ﻭ ‪r‬‬ ‫‪q r (3‬‬ ‫‪p q (2‬‬ ‫‪ p (1‬ﻭ ‪r‬‬ ‫ﺻﺤﻴﺤﺔ‪.‬‬ ‫‪~p ~r (6‬‬ ‫‪p r (5‬‬ ‫‪ ~p (4‬ﺃﻭ ‪q‬‬ ‫‪p q ~q p ~q‬‬ ‫‪ (7‬ﺃﻛﻤﻞ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪3 ‬‬ ‫‪ (2‬ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ﺳﺒﻌﺔ ﺃﻳﺎﻡ ﻭﻓﻲ‬ ‫‪T T FT‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺻﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﻤﺮﻛﺒﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪:‬‬ ‫‪ (8, 9‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﺍﻟﻴﻮﻡ ﺍﻟﻮﺍﺣﺪ ‪ 20‬ﺳﺎﻋﺔ‪ p q .‬ﺧﺎﻃﺌﺔ؛‬ ‫‪T FTT‬‬ ‫‪FTFF‬‬ ‫‪~p ~q (9‬‬ ‫‪p q (8‬‬ ‫ﻷﻥ ‪ p‬ﺻﺤﻴﺤﺔ؛ ﻭ ‪ q‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪F FTT‬‬ ‫‪ (3‬ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻟﻮﺍﺣﺪ ‪ 20‬ﺳﺎﻋﺔ‪ ،‬ﺃﻭ ﻓﻲ‬ ‫‪‬‬ ‫‪  (10 4‬ﺍﺳﺘﻌﻤﻞ ﺷﻜﻞ ﭬﻦ ﺍﻟﻤﺠﺎﻭﺭ‪ ،‬ﻭﺍﻟﺬﻱ ﻳﻤﺜﻞ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ‬ ‫ﺍﻟﺴﺎﻋﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺩﻗﻴﻘﺔ‪q r .‬‬ ‫‪ ‬‬ ‫ﺻﺤﻴﺤﺔ؛ ﻷﻥ ‪ q‬ﺧﺎﻃﺌﺔ‪ ،‬ﻭ ‪ r‬ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﻳﺪﺭﺳﻮﻥ ﺍﻟﻠﻐﺘﻴﻦ ﺍﻟﻔﺮﻧﺴﻴﺔ ﻭﺍﻹﻳﻄﺎﻟﻴﺔ ﻓﻲ ﻣﻌﻬﺪ ﺍﻟﻠﻐﺎﺕ‪.‬‬ ‫‪8 3 11‬‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻳﺪﺭﺳﻮﻥ ﺍﻹﻳﻄﺎﻟﻴﺔ ﻓﻘﻂ؟ ‪8‬‬ ‫‪ (b‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻳﺪﺭﺳﻮﻥ ﺍﻹﻳﻄﺎﻟﻴﺔ ﻭﺍﻟﻔﺮﻧﺴﻴﺔ ﻣ ﹰﻌﺎ؟ ‪3‬‬ ‫‪ (4‬ﻟﻴﺲ ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ﺳﺒﻌﺔ ﺃﻳﺎﻡ‪ ،‬ﺃﻭ‬ ‫ﻓﻲ ﺍﻟﻴﻮﻡ ﺍﻟﻮﺍﺣﺪ ‪ 20‬ﺳﺎﻋﺔ‪ ~p .‬ﺃﻭ ‪q‬‬ ‫‪ (c‬ﻣﺎﺫﺍ ﻳﻤﺜﻞ ﺍﻟﻌﺪﺩ ‪ 11‬ﻓﻲ ﺍﻟﺸﻜﻞ؟‬ ‫ﺧﺎﻃﺌﺔ؛ ﻷﻥ ﻛ ﹼﹰﻼ ﻣﻦ ‪ ~p‬ﻭ ‪ q‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﻳﺪﺭﺳﻮﻥ ﺍﻟﻔﺮﻧﺴﻴﺔ ﻭﻻ ﻳﺪﺭﺳﻮﻥ ﺍﻹﻳﻄﺎﻟﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪ (5‬ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ﺳﺒﻌﺔ ﺃﻳﺎﻡ‪ ،‬ﺃﻭ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ‪ p, q, r, s‬ﻭﺍﻟﺨﺮﻳﻄﺔ ﺍﻟﻤﺠﺎﻭﺭﺓ؛ ﻟﻜﺘﺎﺑﺔ ﻛﻞ ﻋﺒﺎﺭﺓ‬ ‫‪1, 2‬‬ ‫ﻓﻲ ﺍﻟﺴﺎﻋﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺩﻗﻴﻘﺔ‪p r .‬‬ ‫ﻭ ﹾﺻ ﹴﻞ ﺃﻭ ﻓﺼ ﹴﻞ ﺃﺩﻧﺎﻩ‪ .‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ ﻣﻔ ﱢﺴ ﹰﺮﺍ ﺗﺒﺮﻳﺮﻙ‪:‬‬ ‫ﺻﺤﻴﺤﺔ؛ ﻷﻥ ﻛ ﹰﹼﻼ ﻣﻦ ‪ p‬ﻭ ‪ r‬ﺻﺤﻴﺤﺔ‪.‬‬ ‫‪ :p‬ﺍﻟﺮﻳﺎﺽ ﻋﺎﺻﻤﺔ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪ (11-16 .‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ‬ ‫ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪ :q‬ﺗﻘﻊ ﻣﻜﺔ ﺍﻟﻤﻜﺮﻣﺔ ﻋﻠﻰ ﺍﻟﺨﻠﻴﺞ ﺍﻟﻌﺮﺑﻲ‪.‬‬ ‫‪ (6‬ﻟﻴﺲ ﻓﻲ ﺍﻷﺳﺒﻮﻉ ﺍﻟﻮﺍﺣﺪ ﺳﺒﻌﺔ ﺃﻳﺎﻡ‪،‬‬ ‫‪ :r‬ﺗﻮﺟﺪ ﺣﺪﻭﺩ ﻣﺸﺘﺮﻛﺔ ﻟﻠﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ ﻣﻊ ﺍﻟﻌﺮﺍﻕ‪.‬‬ ‫ﻭﻟﻴﺲ ﻓﻲ ﺍﻟﺴﺎﻋﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺩﻗﻴﻘﺔ‪.‬‬ ‫‪ :s‬ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ ﺗﻘﻊ ﻏﺮﺑﻲ ﺍﻟﺒﺤﺮ ﺍﻷﺣﻤﺮ‪.‬‬ ‫‪ ~p ~r‬ﺧﺎﻃﺌﺔ؛ ﻷﻥ ‪ ~p‬ﺧﺎﻃﺌﺔ‪،‬‬ ‫ﻭ ‪ ~r‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ ~r (13‬ﺃﻭ ‪s‬‬ ‫‪p q (12‬‬ ‫‪ p (11‬ﻭ ‪r‬‬ ‫‪(8‬‬ ‫‪~s ~p (16‬‬ ‫‪ ~p (15‬ﻭ ‪~r‬‬ ‫‪r q (14‬‬ ‫‪p qp q‬‬ ‫‪p‬‬ ‫ﺃﻛﻤﻞ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺍﻵﺗﻲ‪:‬‬ ‫‪3 ‬‬ ‫‪TT T‬‬ ‫‪T‬‬ ‫‪TF F‬‬ ‫‪T‬‬ ‫‪q ~p ~p q (17‬‬ ‫‪FT F‬‬ ‫‪F‬‬ ‫‪FF F‬‬ ‫‪F‬‬ ‫‪TF‬‬ ‫‪F‬‬ ‫‪FF‬‬ ‫‪F‬‬ ‫‪TT T‬‬ ‫‪FT F‬‬ ‫‪(9‬‬ ‫‪p q ~p ~q ~p ~q‬‬ ‫‪23  1- 2‬‬ ‫‪TT F F‬‬ ‫‪F‬‬ ‫‪TF F T‬‬ ‫‪T‬‬ ‫‪FT T F‬‬ ‫‪T‬‬ ‫‪‬‬ ‫‪FF T T‬‬ ‫‪T‬‬ ‫‪   ‬‬ ‫‪.    ‬‬ ‫‪   ‬‬ ‫‪23  1-2‬‬

‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻵﺗﻴﺔ‪ (18-20 :‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪~p r (20‬‬ ‫‪~ (~ r q) (19‬‬ ‫‪~ (~ p) (18‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪   (21‬ﻗﺮﺭ ﻣﺪﺭﺱ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻣﻜﺎﻓﺄﺓ ﺍﻟﻄﻼﺏ‬ ‫‪ 3‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﻟﻤﺘﻔﻮﻗﻴﻦ ﺑﺎﺻﻄﺤﺎﺑﻬﻢ ﻓﻲ ﺭﺣﻠﺔ ﻣﺪﺭﺳﻴ ﹴﺔ‪ ،‬ﻭﻗﺮﺭ ﺃﻥ ﺗﻜﻮﻥ‬ ‫✓ ‪‬‬ ‫‪T‬‬ ‫‪ ‬‬ ‫ﺍﻟﻘﺎﻋﺪﺓ ﺃﻧﻪ \"ﺇﺫﺍ ﺗﻔ ﹼﻮﻕ ﺍﻟﻄﺎﻟﺐ ﻓﻲ ﺍﻻﺧﺘﺒﺎﺭ ﺍﻷﻭﻝ ﺃﻭ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪1–10‬؛ ﻟﻠﺘﺄﻛﺪ ﻣﻦ ﻓﻬﻢ‬ ‫‪T‬‬ ‫‪ ‬ﺗﻔﻮﻕ‬ ‫ﺍﻻﺧﺘﺒﺎﺭ ﺍﻟﺜﺎﻧﻲ ﻓﺈﻧﻪ ﺳﻴﺬﻫﺐ ﻓﻲ ﺍﻟﺮﺣﻠﺔ\"‪.‬‬ ‫ﺍﻟﻄﻠﺒﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺳﻔﻞ ﻫﺬﻩ‬ ‫ﺍﻟﺼﻔﺤﺔ؛ ﻟﺘﻌﻴﻴﻦ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻠﻄﻠﺒﺔ‬ ‫‪T‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (a‬ﺃﻛﻤﻞ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪F‬‬ ‫ﺑﺤﺴﺐ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫ﻟﻢ ﻳﺘﻔﻮﻕ ﺗﻔﻮﻕ‬ ‫ﻟﻢ ﻳﺘﻔﻮﻕ‬ ‫ﻟﻢ ﻳﺘﻔﻮﻕ‬ ‫‪ (b‬ﺇﺫﺍ ﺗﻔﻮﻕ ﺍﻟﻄﺎﻟﺐ ﻓﻲ ﺍﻻﺧﺘﺒﺎﺭﻳﻦ‪ ،‬ﻓﻬﻞ ﺳﻴﺬﻫﺐ ﻓﻲ ﻫﺬﻩ ﺍﻟﺮﺣﻠﺔ؟ ﻧﻌﻢ‬ ‫‪ (c‬ﺇﺫﺍ ﺗﻔﻮﻕ ﺍﻟﻄﺎﻟﺐ ﻓﻲ ﺍﻻﺧﺘﺒﺎﺭ ﺍﻷﻭﻝ ﻓﻘﻂ‪ ،‬ﻓﻬﻞ ﺳﻴﺬﻫﺐ ﻓﻲ ﻫﺬﻩ ﺍﻟﺮﺣﻠﺔ؟ ﻧﻌﻢ‬ ‫‪‬‬ ‫‪‬‬ ‫‪   (22‬ﹸﺳﺌﻞ ‪ 370‬ﺷﺨ ﹰﺼﺎ ﻣﻦ ﺍﻟﻔﺌﺔ ﺍﻟﻌﻤﺮﻳﺔ ﺑﻴﻦ‬ ‫‪4 ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 13-19‬ﺳﻨﺔ ﻋﻦ ﺍﻟﺠﻬﺎﺯ ﺍﻟﺬﻱ ﻳﺴﺘﻌﻤﻠﻮﻧﻪ ﻣﻦ ﺑﻴﻦ ﺍﻟﻬﺎﺗﻒ‬ ‫‪‬‬ ‫ﺍﻟﻤﺤﻤﻮﻝ ﻭﺍﻟﻘﺎﻣﻮﺱ ﺍﻹﻟﻜﺘﺮﻭﻧﻲ ﻭﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﻌﻠﻤﻴﺔ‪ ،‬ﻭ ﹸﻣ ﱢﺜﻠﺖ‬ ‫‪30 50‬‬ ‫‪80‬‬ ‫ﻧﺘﺎﺋﺞ ﺍﻻﺳﺘﻄﻼﻉ ﺑﺸﻜﻞ ﭬﻦ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫‪40‬‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﺬﻳﻦ ﻳﺴﺘﻌﻤﻠﻮﻥ ﺣﺎﺳﺒﺔ ﻋﻠﻤﻴﺔ ﻭﻗﺎﻣﻮ ﹰﺳﺎ‬ ‫‪30 20‬‬ ‫ﺇﻟﻜﺘﺮﻭﻧ ﹼﹰﻴﺎ ﻓﻘﻂ؟ ‪50‬‬ ‫‪110‬‬ ‫‪ (b‬ﻣﺎ ﻋﺪﺩ ﺍﻟﺬﻳﻦ ﻳﺴﺘﻌﻤﻠﻮﻥ ﺍﻷﺟﻬﺰﺓ ﺍﻟﺜﻼﺛﺔ؟ ‪40‬‬ ‫‪10‬‬ ‫‪‬‬ ‫‪ (c‬ﻣﺎ ﻋﺪﺩ ﺍﻟﺬﻳﻦ ﻳﺴﺘﻌﻤﻠﻮﻥ ﻫﺎﺗ ﹰﻔﺎ ﻣﺤﻤﻮ ﹰﻻ ﻓﻘﻂ؟ ‪110‬‬ ‫‪ (d‬ﻣﺎ ﻋﺪﺩ ﺍﻟﺬﻳﻦ ﻳﺴﺘﻌﻤﻠﻮﻥ ﻗﺎﻣﻮ ﹰﺳﺎ ﺇﻟﻜﺘﺮﻭﻧ ﹼﹰﻴﺎ ﻭﻫﺎﺗ ﹰﻔﺎ ﻣﺤﻤﻮ ﹰﻻ ﻓﻘﻂ؟ ‪20‬‬ ‫‪ (e‬ﻣﺎﺫﺍ ﻳﻤﺜﻞ ﺍﻟﻌﺪﺩ ‪ 10‬ﻓﻲ ﺍﻟﺸﻜﻞ؟ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﻻ ﻳﺴﺘﻌﻤﻠﻮﻥ ﺃ ﹼﹰﻳﺎ ﻣﻦ ﺍﻷﺟﻬﺰﺓ ﺍﻟﺜﻼﺛﺔ‪.‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻵﺗﻴﺔ‪ .‬ﺛﻢ ﻋ ﱢﻴﻦ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻨﻬﺎ‪ ،‬ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫ﺍﻟ ﹸﻤﻌﻄﺎﺓ ﺑﺠﺎﻧﺐ ﻛ ﱟﻞ ﻣﻨﻬﺎ ﺻﺎﺋﺒﺔ‪ (23-28 :‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪(~p q) r ; q, r (25‬‬ ‫‪p (~q r) ; p, r (24‬‬ ‫‪p (q r) ; p, q (23‬‬ ‫‪(~p q) ~r ; p, q (28 ~p (~q ~r) ; p, q, r (27 p (~q ~r) ; p, q, r (26‬‬ ‫‪‬‬ ‫‪  ‬ﻟﻨﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺘﻲ ﺗﺤﻮﻱ ﻛﻠﻤﺔ \"ﺟﻤﻴﻊ\" ﺃﻭ \"ﻛﻞ\"‪ ،‬ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺟﻤﻠﺔ \"ﻳﻮﺟﺪ ﻭﺍﺣﺪ ﻋﻠﻰ ﺍﻷﻗﻞ\" ﺃﻭ‬ ‫‪ (29‬ﻳﻮﺟﺪ ﻣﺮﺑﻊ ﻭﺍﺣﺪ ﻋﻠﻰ‬ ‫ﺍﻷﻗﻞ ﻟﻴﺲ ﻣﺴﺘﻄﻴ ﹰﻼ‪.‬‬ ‫\"ﻫﻨﺎﻙ ﻭﺍﺣﺪ ﻋﻠﻰ ﺍﻷﻗﻞ\"‪ .‬ﻭﻟﻨﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺘﻲ ﺗﺤﻮﻱ ﻛﻠﻤﺔ \"ﻳﻮﺟﺪ\"‪ ،‬ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﻛﻠﻤﺔ \"ﺟﻤﻴﻊ\" ﺃﻭ \"ﻛﻞ\"‪.‬‬ ‫‪ (30‬ﺟﻤﻴﻊ ﺍﻟﻄﻼﺏ ﻻ‬ ‫‪ :~p‬ﻳﻮﺟﺪ ﻣﻀﻠﻊ ﻭﺍﺣﺪ ﻋﻠﻰ ﺍﻷﻗﻞ ﻟﻴﺲ ﻣﺤﺪ ﹰﺑﺎ‪.‬‬ ‫‪ :p‬ﺟﻤﻴﻊ ﺍﻟﻤﻀﻠﻌﺎﺕ ﻣﺤﺪﺑﺔ‪.‬‬ ‫‪ :~q‬ﺟﻤﻴﻊ ﺍﻟﻤﺴﺎﺋﻞ ﻟﻬﺎ ﺣﻞ‪.‬‬ ‫‪ :q‬ﺗﻮﺟﺪ ﻣﺴﺄﻟﺔ ﻟﻴﺲ ﻟﻬﺎ ﺣﻞ‪.‬‬ ‫ﻳﺪﺭﺳﻮﻥ ﺍﻟﻠﻐﺔ ﺍﻟﻔﺮﻧﺴﻴﺔ‪.‬‬ ‫‪ (30‬ﻋﻠﻰ ﺍﻷﻗﻞ ﻳﻮﺟﺪ ﻃﺎﻟﺐ ﻭﺍﺣﺪ ﻳﺪﺭﺱ ﺍﻟﻠﻐﺔ ﺍﻟﻔﺮﻧﺴﻴﺔ‪.‬‬ ‫ﺍﻧ ﹺﻒ ﻛ ﹰﹼﻼ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ‪:‬‬ ‫‪ (31‬ﻳﻮﺟﺪ ﻋﻠﻰ ﺍﻷﻗﻞ ﻋﺪﺩ‬ ‫ﺣﻘﻴﻘﻲ ﻭﺍﺣﺪ ﻟﻴﺲ ﻟﻪ ﺟﺬﺭ‬ ‫‪ (29‬ﺟﻤﻴﻊ ﺍﻟﻤﺮﺑﻌﺎﺕ ﻣﺴﺘﻄﻴﻼﺕ‪.‬‬ ‫ﺗﺮﺑﻴﻌﻲ ﺣﻘﻴﻘﻲ‪.‬‬ ‫‪ (32‬ﺗﻮﺟﺪ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ ﻟﻴﺲ ﻟﻬﺎ ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ ‪.‬‬ ‫‪ (31‬ﻟﻜﻞ ﻋﺪﺩ ﺣﻘﻴﻘﻲ ﺟﺬﺭ ﺗﺮﺑﻴﻌﻲ ﺣﻘﻴﻘﻲ‪.‬‬ ‫ﻛﻞ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ ﻟﻬﺎ ﻧﻘﻄﺔ ﻣﻨﺘﺼﻒ‪.‬‬ ‫‪ 1 24‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪34 - 502911-20‬‬ ‫‪‬‬ ‫‪33 - 5024 - 28222111–20‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪48-5120-47‬‬ ‫‪ 1 24‬‬

‫‪R‬‬ ‫‪  (33‬ﺍﻷﻋﺪﺍﺩ ﻏﻴﺮ ﺍﻟﻨﺴﺒﻴﺔ )‪ ،(I‬ﻭﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺎﺋﺒﺔ )‪ (Z‬ﺗﻨﺘﻤﻲ ﺇﻟﻰ‬ ‫‪ (33‬ﻏﻴﺮ ﺻﺤﻴﺢ ﺃﺑ ﹰﺪﺍ‪.‬‬ ‫ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﻫﻲ ﺃﻋﺪﺍﺩ‬ ‫‪I‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ )‪ .(R‬ﻣﻌﺘﻤ ﹰﺪﺍ ﻋﻠﻰ ﺷﻜﻞ ﭬﻦ ﺍﻟﻤﺠﺎﻭﺭ‪ ،‬ﻫﻞ ﺻﺤﻴﺢ‬ ‫‪Z‬‬ ‫ﺃﺣﻴﺎ ﹰﻧﺎ ﺃﻡ ﺩﺍﺋ ﹰﻤﺎ‪ ،‬ﺃﻡ ﻏﻴﺮ ﺻﺤﻴﺢ ﺃﺑ ﹰﺪﺍ‪ ،‬ﺃﻥ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺎﺋﺒﺔ ﻫﻲ ﺃﻋﺪﺍﺩ ﻏﻴﺮ ﻧﺴﺒﻴﺔ؟‬ ‫ﻧﺴﺒﻴﺔ‪ ،‬ﻭﻟﻴﺴﺖ ﻏﻴﺮ ﻧﺴﺒﻴﺔ‪.‬‬ ‫‪ 4‬‬ ‫ﻓ ﹼﺴﺮ ﺗﺒﺮﻳﺮﻙ‪.‬‬ ‫‪  ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﺒ ﱢﻴﻨﻮﺍ‬ ‫‪   (34‬ﹺﺻ ﹾﻒ ﻣﻮﻗ ﹰﻔﺎ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻠﻪ ﺑﺸﻜﻞ ﭬﻦ ﺍﻵﺗﻲ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﻛﻴﻒ ﺳﺎﻋﺪﻫﻢ ﻣﻮﺿﻮﻉ ﺍﻟﺪﺭﺱ ﺍﻟﺴﺎﺑﻖ‬ ‫‪8‬‬ ‫ﺣﻮﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ﻋﻠﻰ ﺗﻌﻠﻢ ﺍﻟﻤﻨﻄﻖ‬ ‫‪25 48‬‬ ‫ﻭﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻓﻲ ﻫﺬﺍ ﺍﻟﺪﺭﺱ‪.‬‬ ‫‪ (35‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻟﻠﻤﺜﻠﺚ ‪19‬‬ ‫ﺛﻼﺛﺔ ﺃﺿﻼﻉ‪ ،‬ﻭﻟﻠﻤﺮﺑﻊ ﺃﺭﺑﻌﺔ‬ ‫✓ ‪‬‬ ‫‪‬‬ ‫ﺃﻭﻟﺿﺬﻟﻼﻉﻚ‪.‬ﺗﻛﻜﻠﻮﺘﺎﻥﺍﺍﻟﻟﻌﻌﺒﺒﺎﺎﺭﺭﺗﻴﺓ‪ ‬ﺍﻦﻟﻤﺮﺻﺎﻛﺋﺒﺒﺔﺔ‪،‬‬ ‫ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻢ ﺍﻟﻄﻼﺏ ﺍﻟﺪﺭﺳﻴﻦ ‪1-1, 1-2‬‬ ‫‪  (35‬ﺍﻛﺘﺐ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ ﺻﺎﺋﺒﺔ ﺗﺤﻮﻱ » ﻭ « ﻓﻘﻂ‪.‬‬ ‫ﺑﺈﻋﻄﺎﺋﻬﻢ‪:‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻻﺧﺘﺒﺎﺭ ﺍﻟﻘﺼﻴﺮ ‪ ،1‬ﺹ )‪(11‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪C‬‬ ‫‪.‬‬ ‫‪_1‬‬ ‫‪,‬‬ ‫‪1,‬‬ ‫‪_35 ,‬‬ ‫‪_37 ,‬‬ ‫‪3‬‬ ‫‪...‬‬ ‫ﺍﻟﻨﻤﻂ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﺤﺪ‬ ‫ﺧ ﱢﻤﻦ‬ ‫‪(37‬‬ ‫‪ (36‬ﺃ ﱡﻱ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ ﻟﻬﺎ ﻧﻔﺲ ﻗﻴﻤﺔ ‪B‬‬ ‫ﺻﻮﺍﺏ ﺍﻟﻌﺒﺎﺭﺓ ‪AB = BC‬؟ ‪A‬‬ ‫‪ (34‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﹸﺃﺟﺮﻱ ﺍﺳﺘﻄﻼﻉ ﺷﻤﻞ‬ ‫‪3‬‬ ‫‪ 100‬ﺷﺨﺺ؛ ﻟﻤﻌﺮﻓﺔ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﻮﺍ‬ ‫ﻳﻔﻀﻠﻮﻥ ﺍﻟﻤﺜﻠﺠﺎﺕ ﺑﻨﻜﻬﺔ ﺍﻟﻔﺎﻧﻴﻠﻴﺎ‬ ‫‪_11‬‬ ‫‪C‬‬ ‫‪_8‬‬ ‫‪A‬‬ ‫ﺃﻭ ﺍﻟﻔﺮﺍﻭﻟﺔ ﺃﻭ ﺍﻟﺸﻮﻛﻮﻻﺗﺔ‪ ،‬ﻓ ﹸﻮﺟﺪ ﺃﻥ‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪ 8‬ﺃﺷﺨﺎﺹ ﻳﻔﻀﻠﻮﻥ ﻧﻜﻬﺔ ﺍﻟﻔﺮﺍﻭﻟﺔ‬ ‫ﻓﻘﻂ‪ ،‬ﻭ‪ 25‬ﺷﺨ ﹰﺼﺎ ﻳﻔﻀﻠﻮﻥ ﻧﻜﻬ ﹶﺘﻲ‬ ‫‪_9‬‬ ‫‪D‬‬ ‫‪4B‬‬ ‫‪AC‬‬ ‫ﺍﻟﻔﺎﻧﻴﻠﻴﺎ ﻭﺍﻟﻔﺮﺍﻭﻟﺔ‪ ،‬ﻭ‪ 48‬ﺷﺨ ﹰﺼﺎ‬ ‫‪3‬‬ ‫‪AC = BC C‬‬ ‫‪m∠A = m∠C A‬‬ ‫ﻳﻔﻀﻠﻮﻥ ﻧﻜﻬﺔ ﺍﻟﻔﺎﻧﻴﻠﻴﺎ ﻓﻘﻂ‪ ،‬ﻭ‪19‬‬ ‫‪AB = AC D‬‬ ‫‪m∠A = m∠B B‬‬ ‫ﻳﻔﻀﻠﻮﻥ ﻧﻜﻬﺔ ﺍﻟﺸﻮﻛﻮﻻﺗﺔ ﻭﺍﻟﻔﺎﻧﻴﻠﻴﺎ‪.‬‬ ‫‪‬‬ ‫‪ (38‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﻻﺣﻆ ﺟﻤﻴﻞ ﺗﻘﺪﻳﻢ‬ ‫ﺳﻠﻄﺔ ﺍﻟﻔﻮﺍﻛﻪ ﻳﻮﻡ ﺍﻟﺜﻼﺛﺎﺀ‪ ،‬ﻭﺍﻓﺘﺮﺽ‬ ‫‪  (38‬ﻓﻲ ﻛﻞ ﻳﻮﻡ ﺛﻼﺛﺎﺀ ﻣﻦ ﺍﻷﺳﺎﺑﻴﻊ ﺍﻷﺭﺑﻌﺔ ﺍﻟﻤﺎﺿﻴﺔ‪ ،‬ﻗ ﱠﺪﻡ ﻣﻄﻌﻢ ﺳﻠﻄﺔ ﻓﻮﺍﻛﻪ ﻫﺪﻳﺔ ﺑﻌﺪ ﻛﻞ ﻭﺟﺒﺔ‪ .‬ﺍﻓﺘﺮﺽ ﺟﻤﻴﻞ‬ ‫ﺃﻥ ﻫﺬﺍ ﺍﻟﻨﻤﻂ ﺳﻮﻑ ﻳﺴﺘﻤﺮ؛ ﻟﺬﺍ‬ ‫ﺃﻧﻪ ﺳﻴﺘﻢ ﺗﻘﺪﻳﻢ ﺳﻠﻄﺔ ﻓﻮﺍﻛﻪ ﻳﻮﻡ ﺍﻟﺜﻼﺛﺎﺀ ﺍﻟﻘﺎﺩﻡ‪ .‬ﻣﺎ ﻧﻮﻉ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻟﺬﻱ ﺍﺳﺘﻌﻤﻠﻪ ﺟﻤﻴﻞ؟ ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ‪ 1-1 .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻘﺮﺍﺋﻲ ‪.‬‬ ‫‪_3‬‬ ‫ﺧ ﱢﻤﻦ ﺍﻟﺤﺪ ﺍﻟﺘﺎﻟﻲ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻤﺘﺘﺎﺑﻌﺎﺕ ﺍﻵﺗﻴﺔ ‪ .‬‬ ‫‪8‬‬ ‫‪6,‬‬ ‫‪3,‬‬ ‫‪_3‬‬ ‫‪,‬‬ ‫‪_3‬‬ ‫‪(41‬‬ ‫‪81 1, 3, 9, 27 (40‬‬ ‫‪11 3, 5, 7, 9 (39‬‬ ‫‪2‬‬ ‫‪4‬‬ ‫‪‬ﺣﻞ ﻛ ﹼﹰﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ‪ :‬‬ ‫‪8 4(m - 5) = 12 (44‬‬ ‫‪-1 3x + 9 = 6 (43‬‬ ‫‪24‬‬ ‫‪_y‬‬ ‫‪-‬‬ ‫‪7‬‬ ‫=‬ ‫‪5‬‬ ‫‪(42‬‬ ‫‪_y‬‬ ‫‪2‬‬ ‫‪25‬‬ ‫‪5‬‬ ‫‪+‬‬ ‫‪4‬‬ ‫=‬ ‫‪9‬‬ ‫‪(47‬‬ ‫‪9 2x - 7 = 11 (46‬‬ ‫‪-7 6(w + 7) = 0 (45‬‬ ‫‪‬‬ ‫‪ 4d - c (49‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪14 c = 2 , d = 4‬‬ ‫‪ ‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺠﺒﺮﻳﺔ ﺍﻵﺗﻴﺔ ﻟﻠﻘﻴﻢ ﺍﻟﻤﻌﻄﺎﺓ‪.‬‬ ‫‪ ab - 2a (51‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪10 a = -2 , b = -3‬‬ ‫‪ 2y + 3x (48‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪3 x =-1, y = 3‬‬ ‫‪ m2 + 7n (50‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪2 n = -2, m = 4‬‬ ‫‪25  1- 2‬‬ ‫‪‬‬ ‫‪ ‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ‪ p, q‬ﻟﻜﺘﺎﺑﺔ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ‪ ،‬ﻭﺑ ﹼﻴﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ \"ﻋﺒﺎﺭﺓ ﻭﺻﻞ\" ﺃﻡ \"ﻋﺒﺎﺭﺓ ﹶﻓ ﹾﺼﻞ\" ‪ ،‬ﺛﻢ‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪.‬‬ ‫‪ ABC p‬ﻣﺘﻄﺎﺑﻖ ﺍﻷﺿﻼﻉ‪.‬‬ ‫‪ ABC q‬ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ‪.‬‬ ‫‪ p‬ﺃﻭ ‪ ABC :q‬ﻣﺘﻄﺎﺑﻖ ﺍﻷﺿﻼﻉ ﺃﻭ ‪ ABC‬ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ‪ p .‬ﺃﻭ ‪ q‬ﻋﺒﺎﺭﺓ ﻓﺼﻞ‪.‬‬ ‫ﺑﻤﺎ ﺃﻧﻪ ﻻ ﺗﻮﺟﺪ ﺻﻮﺭﺓ ﻣﻌﻄﺎﺓ ﻟﻠﻤﺜﻠﺚ ‪ ،ABC‬ﺇﺫﻥ ﻻ ﻳﻤﻜﻦ ﺗﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ‪.‬‬ ‫‪25  1-2‬‬

‫‪  ‬‬ ‫‪1 -2 ‬‬ ‫‪          ‬‬ ‫‪ (12) ‬‬ ‫‪(11) ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪    1-2‬‬ ‫‪ 1-2‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪‬ﺇﺣﺪ￯ ﻃﺮﻕ ﺗﻨﻈﻴﻢ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ‪   ‬‬ ‫‪‬‬ ‫ﻫﻲ ﺇﻧﺸﺎﺀ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ‪ .‬ﻳﻈﻬﺮ ﺟﻬﺔ ﺍﻟﻴﺴﺎﺭ ﺟﺪﺍﻭﻝ ﺍﻟﺼﻮﺍﺏ ‪p q p ∧ q p q p ∨ q p ∼p‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﻫﻲ ﺟﻤﻠﺔ ﺧﺒﺮﻳﺔ ﺗﺤﺘﻤﻞ ﺍﻟﺼﻮﺍﺏ ﺃﻭ ﺍﻟﺨﻄﺄ ﻭﻻ ﺗﺤﺘﻤﻞ ﻏﻴﺮﻫﻤﺎ‪ .‬ﻭﻳﺮﻣﺰ ﺇﻟﻰ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﺑـ ) ‪ (T‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ‪،‬‬ ‫ﻭﺑﺎﻟﺮﻣﺰ ) ‪ (F‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ‪ ،‬ﻭﻳﻤﻜﻦ ﺃﻥ ﻧﺮﻣﺰ ﺇﻟﻰ ﺃﻱ ﻋﺒﺎﺭﺓ ﺑﺄﺣﺪ ﺍﻟﺤﺮﻭﻑ ﻭﻟﻴﻜﻦ ‪ . p‬ﻓﻤﺜ ﹰﻼ ﻳﻤﻜﻦ ﺃﻥ ﻧﺮﻣﺰ ﺇﻟﻰ ﺍﻟﻌﺒﺎﺭﺓ \"ﺍﻟﺮﻳﺎﺽ‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪TF‬‬ ‫ﻟﻜ ﱟﻞ ﻣﻦ ﻋﺒﺎﺭﺍﺕ ﺍﻟﻨﻔﻲ ﻭﺍﻟﻮﺻﻞ ﻭﺍﻟ ﹶﻔ ﹾﺼﻞ‪.‬‬ ‫‪TF‬‬ ‫‪F‬‬ ‫‪TF‬‬ ‫‪T‬‬ ‫‪FT‬‬ ‫ﻣﺪﻳﻨﺔ ﺳﻌﻮﺩﻳﺔ\" ﺑﺎﻟﺮﻣﺰ ‪ ،p‬ﻭﺗﻜﻮﻥ ﻫﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺎﺋﺒﺔ ‪.T‬‬ ‫‪FT F‬‬ ‫‪FT T‬‬ ‫ﻭﻳﻤﻜﻨﻨﺎ ﺭﺑﻂ ﻋﺒﺎﺭﺍﺕ ﻋﺪﺓ ﺑﻌﻀﻬﺎ ﺑﺒﻌﺾ ﻟﺘﻜﻮﻳﻦ ﻋﺒﺎﺭﺓ ﻣﺮﻛﺒﺔ‪.‬‬ ‫‪FF F‬‬ ‫‪FF F‬‬ ‫‪   ‬‬ ‫‪ ‬ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﻋﺒﺎﺭﺍﺕ ﺍﻟﻨﻔﻲ ﻭﺍﻟﻔﺼﻞ ﻭﺍﻟﻮﺻﻞ ﺑﺄﺷﻜﺎﻝ ﭬﻦ‪ ،‬ﻛﻤﺎ ﻫﻮ ﻣﺒﻴﻦ ﻓﻲ ﺍﻷﺷﻜﺎﻝ ﺃﺩﻧﺎﻩ‪.‬‬ ‫ﻫﻲ ﺍﻟﺮﺑﻂ ﺑﲔ ﺍﻟﻌﺒﺎﺭﺓ ‪ p‬ﻭﺍﻟﻌﺒﺎﺭﺓ ‪q‬‬ ‫ﻫﻲ ﺍﻟﺮﺑﻂ ﺑﲔ ﺍﻟﻌﺒﺎﺭﺓ ‪ p‬ﻭﺍﻟﻌﺒﺎﺭﺓ ‪q‬‬ ‫ﻧﻔﻲ ﺍﻟﻌﺒﺎﺭﺓ ‪ p‬ﻫﻮ ﻟﻴﺲ ‪.p‬‬ ‫ﺑﺄﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﺃﻭ\"‪.‬‬ ‫ﺑﺄﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﻭ\"‪.‬‬ ‫‪qp‬‬ ‫‪qp‬‬ ‫‪∼p‬‬ ‫∼‬ ‫‪‬‬ ‫‪p∨q‬‬ ‫‪p∧q‬‬ ‫‪p‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪  ‬ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺗﲔ ‪ p‬ﻭ ‪ ∼p‬ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ‪ p q‬ﺻﺎﺋﺒ ﹰﺔ ﻓﻘﻂ ﺗﻜﻮﻥ ﻋﺒﺎﺭﺓ ﺍﻟ ﹶﻔ ﹾﺼﻞ ‪ p q‬ﺻﺎﺋﺒ ﹰﺔ‪،‬‬ ‫ﺗﻮﻋﻴـﺔ‪ :‬ﺷـﻜﻞ ﭬـﻦ ‪ ‬‬ ‫‪2‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ‬ ‫‪1‬‬ ‫ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﻛ ﱞﻞ ﻣﻦ ‪ p‬ﻭ‪ q‬ﺻﺎﺋﺒﺔ‪ .‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺇﺣﺪ￯ ﺍﻟﻌﺒﺎﺭﺗﲔ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﻣﺘﻌﺎﻛﺴﺔ‪.‬‬ ‫‪r∨q‬‬ ‫ﺍﻟﻤﺠـﺎﻭﺭ ﻳﺒ ﱢﻴـﻦ ﻋـﺪﺩ ﺍﻟﻄـﻼﺏ ﺍﻟﺬﻳـﻦ ﻧﻔﺬﻭﺍ ‪4 2 3‬‬ ‫ﺍﻟﺘﺪﺭﻳﺒﻴـﻦ »ﺍﺳـﺘﺜﻤﺮ ﻭﻗﺘـﻚ« ﹶﻭ»ﺧﻄـﻂ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺟﺪﻭﻝ ﺻﻮﺍﺏ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ‪.‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ ﻟﻜﺘﺎﺑﺔ ﻋﺒﺎﺭ ﹶﺗﻲ‬ ‫‪2‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ ﻟﻜﺘﺎﺑﺔ‬ ‫‪1‬‬ ‫ﻟﻤﺴﺘﻘﺒﻠﻚ« ﻟﺰﻣﻼﺋﻬﻢ‪.‬‬ ‫ﺍﻟﻔﺼﻞ ﺍﻵﺗﻴﺘﻴﻦ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺻﻮﺍﺑﻬﺎ ﻣﺒ ﱢﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪:‬‬ ‫ﻋﺒﺎﺭ ﹶﺗﻲ ﺍﻟﻮﺻﻞ ﺍﻵﺗﻴﺘﻴﻦ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺻﻮﺍﺑﻬﺎ ﻣﺒﺮ ﹰﺭﺍ‬ ‫ﺇﺟﺎﺑﺘﻚ‪:‬‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﺷﺎﺭﻛﻮﺍ ﰲ ﺍﻟﺘﺪﺭﻳﺐ ﻋﲆ‬ ‫‪qr‬‬ ‫‪r∨q‬‬ ‫‪ :p‬ﻗﻄﺮ ﺍﻟﺪﺍﺋﺮﺓ ﻳﺴﺎﻭﻱ ﻣﺜ ﹶﻠﻲ ﻧﺼﻒ ﻗﻄﺮﻫﺎ‪.‬‬ ‫»ﺍﺳﺘﺜﻤﺮ ﻭﻗﺘﻚ« ﻭﱂ ﻳﺸﺎﺭﻛﻮﺍ ﰲ ﺍﻟﱪﻧﺎﻣﺞ ﺍﻟﺘﺪﺭﻳﺒﻲ‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪ :q‬ﻟﻠﻤﺴﺘﻄﻴﻞ ﺃﺭﺑﻌﺔ ﺃﺿﻼﻉ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ‪.‬‬ ‫‪ :p‬ﺍﻟﻔﻴﻞ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ‪.‬‬ ‫‪TF‬‬ ‫‪T‬‬ ‫»ﺧﻄﻂ ﳌﺴﺘﻘﺒﻠﻚ«؟ ‪3‬‬ ‫‪FT‬‬ ‫‪T‬‬ ‫‪p q (a‬‬ ‫‪q‬ﻟﻠﻤﺮﺑﻊ ﺃﺭﺑﻊ ﺯﻭﺍﻳﺎ ﻗﻮﺍﺋﻢ‪.‬‬ ‫‪FF‬‬ ‫‪F‬‬ ‫‪ (b‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻄﻼﺏ ﺍﻟﺬﻳﻦ ﺷﺎﺭﻛﻮﺍ ﰲ ﺍﻟﺘﺪﺭﻳﺒﲔ ﻣ ﹰﻌﺎ؟ ‪2‬‬ ‫ﺍﺭﺑﻂ ﺍﻟﻌﺒﺎﺭﺗﲔ ﺑﺄﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﺃﻭ\"‪.‬‬ ‫‪. p q (a‬‬ ‫ﻗﻄﺮ ﺍﻟﺪﺍﺋﺮﺓ ﻳﺴﺎﻭﻱ ﻣﺜ ﹶﲇ ﻧﺼﻒ ﻗﻄﺮﻫﺎ‪ ،‬ﺃﻭ ﻟﻠﻤﺴﺘﻄﻴﻞ ﺃﺭﺑﻌﺔ ﺃﺿﻼﻉ‬ ‫ﺍﺭﺑﻂ ﺍﻟﻌﺒﺎﺭﺗﲔ ﺑﺄﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﻭ\"‪:‬‬ ‫‪‬‬ ‫ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ‪ .‬ﺑﲈ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﱃ ﺻﺎﺋﺒﺔ‪ ،‬ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﳌﺮﻛﺒﺔ‬ ‫ﺍﻟﻔﻴﻞ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ‪ ،‬ﻭﻟﻠﻤﺮﺑﻊ ﺃﺭﺑﻊ ﺯﻭﺍﻳﺎ ﻗﻮﺍﺋﻢ‪.‬‬ ‫ﹼﳌﺎ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ‪ p, q‬ﺻﺎﺋﺒﺘﲔ‪ ،‬ﻓﺈﻥ ﻫﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺍﳌﺮﻛﺒﺔ‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺻﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻵﺗﻴﺔ‪:‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪∼( p ∧ ∼ r ) (2‬‬ ‫‪q ∧ ∼r (1‬‬ ‫‪∼p ∨ q (b‬‬ ‫‪∼p ∧ q (b‬‬ ‫ﺍﺭﺑﻂ ﺍﻟﻌﺒﺎﺭﺗﲔ ﺑﺄﺩﺍﺓ ﺍﻟﺮﺑﻂ \"ﺃﻭ\" ‪.‬‬ ‫‪ ∼p‬ﻫﻲ ﺍﻟﻌﺒﺎﺭﺓ \"ﺍﻟﻔﻴﻞ ﻟﻴﺲ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ\"‪ .‬ﺍﺭﺑﻂ ﺍﻟﻌﺒﺎﺭﺗﲔ‬ ‫)‪p r ∼r p ∧ ∼r ∼(p ∧∼r‬‬ ‫‪q r ∼r q ∧ ∼r‬‬ ‫ﻗﻄﺮ ﺍﻟﺪﺍﺋﺮﺓ ﻻ ﻳﺴﺎﻭﻱ ﻣﺜ ﹶﲇ ﻧﺼﻒ ﻗﻄﺮﻫﺎ‪ ،‬ﺃﻭ ﻟﻠﻤﺴﺘﻄﻴﻞ ﺃﺭﺑﻌﺔ‬ ‫‪ ∼p‬ﻭ‪ q‬ﺑﺎﻷﺩﺍﺓ \"ﻭ\" ‪.‬‬ ‫‪TT‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪TT‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﺃﺿﻼﻉ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻄﻮﻝ‪ .‬ﺑﲈ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺗﲔ ﺧﺎﻃﺌﺘﺎﻥ‪ ،‬ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺍﻟﻔﻴﻞ ﻟﻴﺲ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ ﻭﻟﻠﻤﺮﺑﻊ ﺃﺭﺑﻊ ﺯﻭﺍﻳﺎ ﻗﻮﺍﺋﻢ‪.‬‬ ‫‪TF‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪TF‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪FT‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪FT‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﺍﳌﺮﻛﺒﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﺍﳉﺰﺀ ﺍﻷﻭﻝ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺓ ﺍﳌﺮﻛﺒﺔ ‪ ∼p‬ﺧﻄﺄ‪ ،‬ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫‪FF‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪FF‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫ﺍﳌﺮﻛﺒﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ (3‬ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻳﺒ ﱢﻴﻦ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺣﻀﺮﻭﺍ ﺍﻟﻨﺪﻭﺗﻴﻦ ﺍﻟﺘﻮﻋﻮ ﱠﻳﺘﻴﻦ‬ ‫‪ ‬‬ ‫»ﻣﺮﺽ ﺍﻟﺴﻜﺮ« ﹶﻭ »ﻣﺮﺽ ﺍﻟﻀﻐﻂ«‪.‬‬ ‫‪‬‬ ‫‪50 20 75‬‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺣﴬﻭﺍ ﺍﻟﻨﺪﻭﺗﲔ؟ ‪20‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ ﻟﻜﺘﺎﺑﺔ ﻛﻞ \" ﻋﺒﺎﺭﺓ ﻭﺻﻞ\" ﺃﻭ \" ﻋﺒﺎﺭﺓ ﻓﺼﻞ\" ﻣ ﹼﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ ﻣﺒﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪:‬‬ ‫‪ (b‬ﻣﺎ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺣﴬﻭﺍ ﻧﺪﻭﺓ »ﻣﺮﺽ ﺍﻟﻀﻐﻂ«؟ ‪70‬‬ ‫‪ :r‬ﻟﻠﻤﺴﺘﻄﻴﻞ ﺃﺭﺑﻌﺔ ﺃﺿﻼﻉ‪.‬‬ ‫‪ :q‬ﻋﺪﺩ ﺃﻳﺎﻡ ﺷﻬﺮ ﺳﺒﺘﻤﺒﺮ ‪ 30‬ﻳﻮ ﹰﻣﺎ‪.‬‬ ‫‪10 + 8 = 18 :p‬‬ ‫‪ (c‬ﻣﺎ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﺣﴬﻭﺍ ﻧﺪﻭﺓ »ﻣﺮﺽ ﺍﻟﻀﻐﻂ« ﻭﱂ ﳛﴬﻭﺍ ﻧﺪﻭﺓ »ﻣﺮﺽ ﺍﻟﺴﻜﺮ«؟ ‪50‬‬ ‫‪     10 + 8 = 18‬‬ ‫‪p‬‬ ‫‪r (2‬‬ ‫‪  30     10 + 8 = 18‬‬ ‫‪p∧q‬‬ ‫‪(1‬‬ ‫‪  p, r  ‬‬ ‫‪ p, q   ‬‬ ‫‪     30     q∧ ∼r (4      30     q r (3‬‬ ‫‪ ∼r   ‬‬ ‫‪‬‬ ‫‪q,‬‬ ‫‪r‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪12‬‬ ‫‪  ‬‬ ‫‪1‬‬ ‫‪11‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪(14)  ‬‬ ‫‪( 1 3 )‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ (4‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﻓﻲ ﺍﻟﻤﺴﺄﻟﺔ ‪ 3‬ﻟﻺﺟﺎﺑﺔ ﻋﻦ ﺍﻷﺳﺌﻠﺔ‬ ‫‪ 1-2‬‬ ‫‪  1-2‬‬ ‫ﺍﻵﺗﻴﺔ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (a‬ﺇﺫﺍ ﺭ ﱠﺗﺐ ﻋﺎﻣﺮ ﻏﺮﻓﺘﻪ‪ ،‬ﻭﻧﻘﻞ ﺍﻟﻘﲈﻣﺔ‪ ،‬ﻭﱂ ﳛﻞ ﻭﺍﺟﺒﻪ‪ ،‬ﻓﻬﻞ‬ ‫‪ :  (1‬ﺳﺄﻝ ﺳﺎﻣﻲ ﺻﺪﻳﻘﻪ ﻳﻮﺳﻒ ﺇﻥ ﻛﺎﻥ ﻓﺮﻳﻖ ﻛﺮﺓ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ ﻟﻜﺘﺎﺑﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻭﺻﻞ ﺃﻭ ﻓﺼﻞ ﻣ ﹼﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪ ،‬ﻣﺒ ﱢﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪:‬‬ ‫ﻳﻤﻜﻨﻪ ﳑﺎﺭﺳﺔ ﺃﻟﻌﺎﺏ ﺍﻟﻔﻴﺪﻳﻮ؟ ‪‬‬ ‫ﺍﻟﻘﺪﻡ ﺍﻟﺬﻱ ﻳﺸﺎﺭﻙ ﻓﻴﻪ ﻗﺪ ﻓﺎﺯ ﻓﻲ ﺍﻟﻤﺒﺎﺭﺍﺓ ﻟﻴﻠﺔ ﺃﻣﺲ‪ ،‬ﻭﻫﻞ‬ ‫ﺳ ﱠﺠﻞ ﻫﺪ ﹰﻓﺎ‪ ،‬ﻓﺄﺟﺎﺏ ﻳﻮﺳﻒ \"ﻧﻌﻢ\"‪ .‬ﺛﻢ ﺳﺄﻝ ﺳﺎﻣﻲ ﻻﻋ ﹰﺒﺎ‬ ‫‪-3 - 2 = -5 :p‬‬ ‫‪ (b‬ﺇﺫﺍ ﱂ ﻳﺮ ﱢﺗﺐ ﻋﺎﻣﺮ ﻏﺮﻓﺘﻪ‪ ،‬ﻭﻧﻘﻞ ﺍﻟﻘﲈﻣﺔ‪ ،‬ﻭﺣﻞ ﻭﺍﺟﺒﻪ‪،‬‬ ‫ﺁﺧﺮ ﻓﻲ ﺍﻟﻔﺮﻳﻖ ﹸﻳﺪﻋﻰ ﺳﺎﻟ ﹰﻤﺎ‪ ،‬ﻫﻞ ﺳﺠﻞ ﻫﻮ ﺃﻭ ﻳﻮﺳﻒ ﻫﺪ ﹰﻓﺎ‬ ‫ﻓﻬﻞ ﻳﻤﻜﻨﻪ ﳑﺎﺭﺳﺔ ﺃﻟﻌﺎﺏ ﺍﻟﻔﻴﺪﻳﻮ؟ ‪‬‬ ‫ﻓﻲ ﺍﻟﻤﺒﺎﺭﺍﺓ؟ ﻓﺄﺟﺎﺏ ﺳﺎﻟﻢ ﺑـﹺ\"ﻧﻌﻢ\" ﺃﻳ ﹰﻀﺎ‪ .‬ﻣﺎ ﺍﻟﺬﻱ ﻳﻤﻜﻨﻚ‬ ‫‪ :q‬ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﺘﻘﺎﺑﻠﺔ ﺑﺎﻟﺮﺃﺱ ﻣﺘﻄﺎﺑﻘﺔ‪.‬‬ ‫‪ (c‬ﺇﺫﺍ ﱂ ﻳﺮ ﱢﺗﺐ ﻋﺎﻣﺮ ﻏﺮﻓﺘﻪ‪ ،‬ﻭﱂ ﻳﻨﻘﻞ ﺍﻟﻘﲈﻣﺔ‪ ،‬ﻭﺣ ﹼﻞ ﻭﺍﺟﺒﻪ‪،‬‬ ‫ﺍﺳﺘﻨﺘﺎﺟﻪ ﺣﻮﻝ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺳﺎﻟﻢ ﻗﺪ ﺳ ﱠﺠﻞ ﻫﺪ ﹰﻓﺎ ﺃﻡ ﻻ؟‬ ‫‪2 + 8 > 10 :r‬‬ ‫ﻓﻬﻞ ﻳﻤﻜﻨﻪ ﳑﺎﺭﺳﺔ ﺃﻟﻌﺎﺏ ﺍﻟﻔﻴﺪﻳﻮ؟ ‪‬‬ ‫‪ ‬‬ ‫‪ :s‬ﻣﺠﻤﻮﻉ ﻗﻴﺎ ﹶﺳﻲ ﺍﻟﺰﺍﻭﻳﺘﻴﻦ ﺍﻟﻤﺘﺘﺎﻣﺘﻴﻦ ﻳﺴﺎﻭﻱ ‪.90°‬‬ ‫‪  (5‬ﹸﺳﺌﻞ ‪ 200‬ﺷﺨ ﹴﺺ ﻋﻦ ﻧﻮﻉ ﺍﻟﻜﺘﺐ ﺍﻷﺩﺑﻴﺔ ﺍﻟﺘﻲ‬ ‫‪  (2‬ﻟﺪ￯ ﺳﺎﺭﺓ ﺻﻨﺪﻭﻕ ﻳﺤﻮﻱ ﻧﻮﻋﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻳﺤﺒﻮﻥ ﻗﺮﺍ ﹶﺀﺗﻬﺎ ﻣﻦ ﺑﻴﻦ ﺍﻟﺮﻭﺍﻳﺎﺕ ﻭﺍﻟﺸﻌﺮ ﻭﺍﻟﻤﺴﺮﺣﻴﺎﺕ‪،‬‬ ‫ﻣﻦ ﻗﻄﻊ ﺍﻟﺸﻮﻛﻮﻻﺗﺔ ﺍﻟﺼﻐﻴﺮﺓ ﻫﻤﺎ ﺍﻷﺑﻴﺾ ﻭﺍﻷﺳﻮﺩ‪،‬‬ ‫‪ p (1‬ﻭ ‪     -3 - 2 = -5 q‬‬ ‫ﻓﻜﺎﻧﺖ ﺍﻟﻨﺘﻴﺠﺔ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ﺍﻟﺘﻲ ﻳﻮﺿﺤﻬﺎ ﺷﻜﻞ ﭬﻦ ﺍﻵﺗﻲ‪.‬‬ ‫ﻭﻗﺪ ﺗﻨﺎﻭﻟﺖ ﻗﻄﻌﺔ ﺷﻮﻛﻮﻻﺗﺔ ﻣﻦ ﺍﻟﺼﻨﺪﻭﻕ‪ ،‬ﻓﻬﻞ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ )‪ (∼ p ∧ ∼q‬ﺻﺎﺋﺒﺔ ﺑﻨﺎ ﹰﺀ ﻋﻠﻰ ﺍﻟﻤﻌﻄﻴﺎﺕ ﺃﺩﻧﺎﻩ‪:‬‬ ‫‪  2 + 8 >10  -3 - 2 = -5 p ∧ r (2‬‬ ‫‪ :p‬ﺍﻟﺸﻮﻛﻮﻻﺗﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻷﺳﻮﺩ‪.‬‬ ‫‪ p (3‬ﺃﻭ ‪ 90°       -3 - 2 = -5 s‬‬ ‫‪ :q‬ﺍﻟﺸﻮﻛﻮﻻﺗﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻷﺑﻴﺾ‪.‬‬ ‫‪‬‬ ‫‪ 90°       2 + 8 > 10 r ∨ s (4‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪       -3 - 2 = -5 p ∧ ∼q (5‬‬ ‫‪5 72 38‬‬ ‫‪   (3‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻠﻌﺐ ﻋﺎﻣﺮ ﻟﻌﺒﺔ ﺍﻟﻔﻴﺪﻳﻮ ﺇﺫﺍ ﺭ ﹼﺗﺐ‬ ‫ﻏﺮﻓﺘﻪ ﺃﻭ ﻧﻘﻞ ﺍﻟ ﹸﻘﻤﺎﻣﺔ ﺇﻟﻰ ﺍﻟﺨﺎﺭﺝ‪ ،‬ﻭﻟﻜﻦ ﺇﺫﺍ ﻟﻢ ﻳﺤﻞ ﻭﺍﺟﺒﻪ‬ ‫‪ 2 + 8 ≤ 10       q ∨ ∼r (6‬‬ ‫‪‬‬ ‫‪8‬‬ ‫‪‬‬ ‫ﺍﻟﻤﻨﺰﻟﻲ ﻓﻠﻦ ﹸﻳﺴﻤﺢ ﻟﻪ ﺑﻤﻤﺎﺭﺳﺔ ﺃﻟﻌﺎﺏ ﺍﻟﻔﻴﺪﻳﻮ ﻣﻄﻠ ﹰﻘﺎ‪ .‬ﺃﻛﻤﻞ‬ ‫ﺃﻛﻤﻞ ﻛ ﹼﹰﻼ ﻣﻦ ﺟﺪﻭ ﹶﻟﻲ ﺍﻟﺼﻮﺍﺏ ﺍﻵﺗﻴﻴﻦ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﺃﺩﻧﺎﻩ ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ‪:‬‬ ‫‪ :p‬ﺭ ﹼﺗﺐ ﻋﺎﻣﺮ ﻏﺮﻓﺘﻪ‪.‬‬ ‫‪ :q‬ﻧﻘﻞ ﻋﺎﻣﺮ ﺍﻟﻘﲈﻣﺔ ﺇﱃ ﺍﳋﺎﺭﺝ‪.‬‬ ‫‪p q ∼q‬‬ ‫‪p ∨ ∼q‬‬ ‫‪(8‬‬ ‫)‪p q ∼p ∼p∧ q ∼(∼p ∧ q‬‬ ‫‪(7‬‬ ‫‪TT F‬‬ ‫‪T‬‬ ‫‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﳛ ﱡﺒﻮﻥ ﻗﺮﺍﺀﺓ ﺍﻷﻧﻮﺍﻉ ﺍﻟﺜﻼﺛﺔ‬ ‫‪TF T‬‬ ‫‪T‬‬ ‫‪TT‬‬ ‫‪FF‬‬ ‫‪T‬‬ ‫ﻣﻦ ﺍﻷﺩﺏ؟ ‪72‬‬ ‫‪FT F‬‬ ‫‪F‬‬ ‫‪ :r‬ﺣ ﹼﻞ ﻋﺎﻣﺮ ﻭﺍﺟﺒﻪ ﺍﳌﻨﺰﱄ‪.‬‬ ‫‪FF T‬‬ ‫‪T‬‬ ‫‪TF‬‬ ‫‪FF‬‬ ‫‪T‬‬ ‫‪ (b‬ﻣﺎ ﻋﺪﺩ ﺍﻷﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﳛ ﱡﺒﻮﻥ ﻗﺮﺍﺀﺓ ﺍﻟﺸﻌﺮ؟ ‪91‬‬ ‫‪ :s‬ﻳﻤﻜﻦ ﺃﻥ ﻳﲈﺭﺱ ﻋﺎﻣﺮ ﺃﻟﻌﺎﺏ ﺍﻟﻔﻴﺪﻳﻮ‪.‬‬ ‫‪FT‬‬ ‫‪TT‬‬ ‫‪F‬‬ ‫‪ (c‬ﻣﺎ ﺍﻟﻨﺴﺒﺔ ﺍﳌﺌﻮﻳﺔ ﻟﻸﺷﺨﺎﺹ ﺍﻟﺬﻳﻦ ﳛ ﱡﺒﻮﻥ ﻗﺮﺍﺀﺓ ﺍﻟﺸﻌﺮ‬ ‫ﻭﺍﻟﺮﻭﺍﻳﺎﺕ ﻣ ﹰﻌﺎ ﺑﺎﻟﻨﺴﺒﺔ ﻷﻭﻟﺌﻚ ﺍﻟﺬﻳﻦ ﳛﺒﻮﻥ ﻗﺮﺍﺀﺓ‬ ‫‪FF‬‬ ‫‪TF‬‬ ‫‪T‬‬ ‫ﺍﳌﴪﺣﻴﺎﺕ؟ ‪50%‬‬ ‫‪p‬‬ ‫‪q‬‬ ‫‪r p∨q‬‬ ‫‪s‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺻﻮﺍ ﹴﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﻤﺮﻛﺒﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪:‬‬ ‫‪TTTTT‬‬ ‫‪TTFTF‬‬ ‫‪∼p ∨ ∼r (10‬‬ ‫‪∼q ∧ r (9‬‬ ‫‪TFTTT‬‬ ‫‪p r ∼p ∼r‬‬ ‫‪p ∨∼r‬‬ ‫‪q r ∼q ∼q ∧ r‬‬ ‫‪TFFTF‬‬ ‫‪TT F F‬‬ ‫‪F‬‬ ‫‪TTF‬‬ ‫‪F‬‬ ‫‪FTTTT‬‬ ‫‪TF F T‬‬ ‫‪T‬‬ ‫‪TFF‬‬ ‫‪F‬‬ ‫‪FTFTF‬‬ ‫‪FT T F‬‬ ‫‪T‬‬ ‫‪FTT‬‬ ‫‪T‬‬ ‫‪FFTFF‬‬ ‫‪FF T T‬‬ ‫‪T‬‬ ‫‪FFT‬‬ ‫‪F‬‬ ‫‪FFFFF‬‬ ‫‪1‬‬ ‫‪13   ‬‬ ‫‪1‬‬ ‫‪14‬‬ ‫‪  ‬‬ ‫‪ 1 25A‬‬

‫‪  ‬‬ ‫‪‬‬ ‫‪1 -2 ‬‬ ‫‪         ‬‬ ‫‪ ‬‬ ‫‪( 7)‬‬ ‫‪( 1 5 )‬‬ ‫‪ 1 - 2‬‬ ‫‪ ‬‬ ‫‪  1-2‬‬ ‫‪ ‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ‪ p , q , r‬ﻟﻜﺘﺎﺑﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻭﺻﻞ ﺃﻭ ﻓﺼﻞ ﺃﺩﻧﺎﻩ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪ ،‬ﻣﺒﺮ ﹰﺭﺍ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫‪‬‬ ‫‪ :p‬ﻓﻲ ﺍﻟﺪﻗﻴﻘﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺛﺎﻧﻴﺔ‪.‬‬ ‫‪5 3961 82 74‬‬ ‫ﺍﻟﺴﻮﺩﻭﻛﻮ \"ﺗﻌﻨﻲ‪ :‬ﺍﻟﺮﻗﻢ ﺍﻟﻮﺣﻴﺪ\"‪ ،‬ﻭﻫﻲ ﹸﺃﺣﺠﻴﺔ‪ ،‬ﺭﻳﺎﺿﻴﺔ‪ ،‬ﻳﺘﻄﻠﺐ‬ ‫‪ :q‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﻜﺎﻣﻠﺘﺎﻥ ﺍﻟﻤﺘﻄﺎﺑﻘﺘﺎﻥ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ ‪90°‬‬ ‫‪1 7 43 25 9 68‬‬ ‫ﺣ ﱡﻠﻬﺎ ﺍﺳﺘﻌﲈﻝ ﺍﳌﻨﻄﻖ‪ .‬ﻭﺗﺘﺄﻟﻒ ﻋﺎﺩ ﹰﺓ ﻣﻦ ﺷﺒﻜ ﹴﺔ ﻣﺮﺑﻌ ﹴﺔ ﻣﻜﻮﻧ ﹴﺔ ﻣﻦ ‪9‬‬ ‫‪2 68 4 7 9 53 1‬‬ ‫‪-12 + 11 < -1 :r‬‬ ‫‪9 251 36 487‬‬ ‫ﺷﺒﻜﺎﺕ ﺟﺰﺋﻴﺔ؛ ﹸﻗ ﹼﺴﻤﺖ ﻛﻞ ﻭﺍﺣﺪ ﹴﺓ ﻣﻨﻬﺎ ﺇﱃ ﺗﺴﻌﺔ ﻣﺮﺑﻌﺎﺕ ﺻﻐﲑﺓ‪ .‬ﺗﺒﺪ ﹸﺃ‬ ‫‪ p ∧ q (1‬ﻓﻲ ﺍﻟﺪﻗﻴﻘﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺛﺎﻧﻴﺔ‪ ،‬ﻭﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﻜﺎﻣﻠﺘﺎﻥ ﺍﻟﻤﺘﻄﺎﺑﻘﺘﺎﻥ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ ‪90°‬؛‬ ‫ﺍﻷﺣﺠﻴﺔ ﺑﻮﺿﻊ ﺑﻀﻌﺔ ﺃﻋﺪﺍﺩ‪ ،‬ﻭ ﹸﻳﻄﻠﺐ ﺇﱃ ﺍﻟﻼﻋﺐ ﻣﻞﺀ ﺍﳌﺮﺑﻌﺎﺕ ﺍﳌﺘﺒﻘﻴﺔ‬ ‫ﺑﻤﺎ ﺃﻥ ﻛﻼ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺻﺤﻴﺤﺔ‪ ،‬ﻓﺈﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻮﺻﻞ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺑﺎﺳﺘﻌﲈﻝ ﺍﻟﻘﻮﺍﻋﺪ ﺍﻵﺗﻴﺔ‪:‬‬ ‫‪ q ∨ r (2‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﻜﺎﻣﻠﺘﺎﻥ ﺍﻟﻤﺘﻄﺎﺑﻘﺘﺎﻥ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ ‪ ،90°‬ﺃﻭ ‪ -12 + 11 < -1‬؛‬ ‫‪  ‬ﻳﺠﺐ ﺃﻥ ﻳﺤﻮﻱ ﻛﻞ ﺻﻒ ﻭﻛﻞ ﻋﻤﻮﺩ ﺍﻷﻋﺪﺍﺩ ﻣﻦ ‪4 8 6 9 5 7 1 2 3 1‬‬ ‫ﺑﻤﺎ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﻟﻰ ﺻﺤﻴﺤﺔ‪ ،‬ﺇﺫﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺇﻟﻰ ‪ 9‬ﻣﻦ ﺩﻭﻥ ﺗﻜﺮﺍﺭ ﺃ ﱟﻱ ﻣﻨﻬﺎ‪.‬‬ ‫‪713 84265 9‬‬ ‫‪ ∼p ∨ q (3‬ﻟﻴﺲ ﻓﻲ ﺍﻟﺪﻗﻴﻘﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺛﺎﻧﻴﺔ ﺃﻭ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﻜﺎﻣﻠﺘﺎﻥ ﺍﻟﻤﺘﻄﺎﺑﻘﺘﺎﻥ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻨﻬﻤﺎ ‪90°‬؛‬ ‫ﺑﻤﺎ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺜﺎﻧﻴﺔ ﺻﺤﻴﺤﺔ‪ ،‬ﺇﺫﻥ ﻋﺒﺎﺭﺓ ﺍﻟﻔﺼﻞ ﺻﺤﻴﺤﺔ‪.‬‬ ‫‪692 7 843 1 5‬‬ ‫‪ ‬ﳚﺐ ﺃﻥ ﲢﻮﻱ ﻛﻞ ﺷﺒﻜﺔ ﺟﺰﺋﻴﺔ ﺍﻷﻋﺪﺍﺩ ﻣﻦ ‪ 1‬ﺇﱃ ‪،9‬‬ ‫‪3 5 7291 846‬‬ ‫ﺩﻭﻥ ﺗﻜﺮﺍﺭ ﺃ ﱟﻱ ﻣﻨﻬﺎ‪.‬‬ ‫‪ ∼p ∧ ∼r (4‬ﻟﻴﺲ ﻓﻲ ﺍﻟﺪﻗﻴﻘﺔ ﺍﻟﻮﺍﺣﺪﺓ ‪ 60‬ﺛﺎﻧﻴﺔ ﻭ ‪ -12 + 11 ≥ -1‬؛ ﺑﻤﺎ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﻟﻰ ﺧﺎﻃﺌﺔ‪ ،‬ﺇﺫﻥ ﻋﺒﺎﺭﺓ‬ ‫ﺍﻟﻮﺻﻞ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪8 415 63 7 92‬‬ ‫ﺃﻛﻤﻞ ﺟﺪﻭ ﹶﻟﻲ ﺍﻟﺼﻮﺍﺏ ﺍﻵﺗﻴﻴﻦ‪:‬‬ ‫‪‬‬ ‫‪pq‬‬ ‫‪p p ∨ q q ∧ ( p ∨ q) (6‬‬ ‫‪p q p q p ∨ q p ∧ ( p ∨ q) (5‬‬ ‫‪ (1‬ﻣﺎ ﺍﻟﺒﺪﺍﻳﺔ ﺍﻟﺠﻴﺪﺓ ﻟﺤﻞ ﺍﻷﺣﺠﻴﺔ؟ ﻭﻟﻤﺎﺫﺍ؟‬ ‫‪TT‬‬ ‫‪FT‬‬ ‫‪T‬‬ ‫‪TTFF‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪   7, 8 ,9  4, 5, 6  1, 2 ,3           ‬‬ ‫‪TF‬‬ ‫‪FF‬‬ ‫‪F‬‬ ‫‪TFFT‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪                     ‬‬ ‫‪FT‬‬ ‫‪TT‬‬ ‫‪F‬‬ ‫‪FTTF‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪FF‬‬ ‫‪TT‬‬ ‫‪F‬‬ ‫‪FFTT‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺻﻮﺍﺏ ﻟﻜ ﹼﹰﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﻤﺮﻛﺒﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ ‪:‬‬ ‫‪ (2‬ﻭ ﹼﺿﺢ ﻛﻴﻒ ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺜﺎﻧﻴﺔ‪ ،‬ﻣﻦ ﺃﺟﻞ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻷﻋﺪﺍﺩ ﻛ ﱠﻠﻬﺎ ﻟﺤﻞ ﺍﻷﺣﺠﻴﺔ ﺍﻟﻜﺒﻴﺮﺓ‪.‬‬ ‫‪                    ‬‬ ‫‪pq p q‬‬ ‫‪∼q ∧ (∼p ∨ q) (8‬‬ ‫‪q‬‬ ‫‪q ∨ ( p ∧ ∼q) (7‬‬ ‫‪TT F F‬‬ ‫‪p ∨ q q∧( p ∨ q)    p‬‬ ‫)‪q p ∧ q q∨(p ∧ q‬‬ ‫‪                ‬‬ ‫‪TF F T‬‬ ‫‪FT T F‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪TT F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪ (3‬ﺃﻛﻤﻞ ﻫﺬﻩ ﺍﻷﹸ ﹾﺣﺠﻴﺔ‪.‬‬ ‫‪FF T T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪TF T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪      ‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪FT F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪FF T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﳖﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ ﺑﻌﺪ ﺍﻟﺪﻭﺍﻡ‬ ‫ﻳﺒ ﹼﻴﻦ ﺷﻜﻞ ﭬﻦ ﺍﻟﻤﺠﺎﻭﺭ ﻋﺪﺩ ﺍﻟﻤﻮﻇﻔﻴﻦ ﺍﻟﺬﻳﻦ ﻳﻌﻤﻠﻮﻥ ﻓﻲ ﺇﺟﺎﺯﺓ ﻧﻬﺎﻳﺔ‬ ‫‪5 3 17‬‬ ‫ﺍﻷﺳﺒﻮﻉ ﺃﻭ ﺑﻌﺪ ﻧﻬﺎﻳﺔ ﺍﻟﺪﻭﺍﻡ ﺍﻟﺮﺳﻤﻲ ﻓﻲ ﺇﺣﺪ￯ ﺍﻟﺸﺮﻛﺎﺕ‪.‬‬ ‫‪ (9‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻤﻮﻇﻔﻴﻦ ﺍﻟﺬﻳﻦ ﻳﻌﻤﻠﻮﻥ ﺑﻌﺪ ﺍﻟﺪﻭﺍﻡ ﻭﻓﻲ ﻧﻬﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ؟ ‪3‬‬ ‫‪ (10‬ﻣﺎ ﻋﺪﺩ ﺍﻟﻤﻮﻇﻔﻴﻦ ﺍﻟﺬﻳﻦ ﻳﻌﻤﻠﻮﻥ ﺑﻌﺪ ﺍﻟﺪﻭﺍﻡ ﺃﻭ ﻓﻲ ﻧﻬﺎﻳﺔ ﺍﻷﺳﺒﻮﻉ؟ ‪25‬‬ ‫‪7‬‬ ‫‪1‬‬ ‫‪15‬‬ ‫‪ ‬‬ ‫‪25B    1‬‬

‫ﺇﺫﺍ ﻛﻨﺖ ﺗﺮﻳﺪ ﺍﻟﺘﺤﺪﺙ‬  1- 3  ،‫ﺇﱃ ﻗﺴﻢ ﺧﺪﻣﺔ ﺍﻟﻌﻤﻼﺀ‬  Conditional Statements   2 ‫ﻓﺎﺿﻐﻂ ﺍﻟﺮﻗﻢ‬  1        ‫ ﻳﺤﻴﻠﻚ ﺟﻬﺎﺯ‬،‫ﻋﻨﺪ ﺇﺟﺮﺍﺀ ﻣﻜﺎﻟﻤﺔ ﻫﺎﺗﻔﻴﺔ ﻣﻊ ﺑﻌﺾ ﺍﻟﻤﺆﺳﺴﺎﺕ‬  ،‫ﺍﻟﺮﺩ ﺍﻵﻟﻲ ﺇﻟﻰ ﻗﺎﺋﻤﺔ ﻣﻦ ﺍﻟﺒﺪﺍﺋﻞ ﺗﺨﺘﺎﺭ ﻣﻨﻬﺎ ﺍﻟﻘﺴﻢ ﺍﻟﺬﻱ ﺗﺮﻳﺪ‬ 1-3  ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻤﻨﻄﻖ ﻭﺃﺷﻜﺎﻝ ﻓﻦ ﻟﺘﺤﺪﻳﺪ‬ .‫ﻭ ﹸﻳﺴﻤﻌﻚ ﺇﺭﺷﺎﺩﺍﺕ ﺑﺼﻴﻐﺔ ﻋﺒﺎﺭﺍﺕ ﺷﺮﻃﻴﺔ‬ 1 - 2  ،‫ ﻭﺍﻟﻮﺻﻞ‬،‫ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻨﻔﻲ‬ ‫ ﻭﺍﻹﺭﺷﺎﺩ ﺍﻟﻤﺒﻴﻦ‬.(...‫ ﻓﺈﻥ‬... ‫ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻫﻲ ﻋﺒﺎﺭﺓ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺘﻬﺎ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‬    ‫ ﺗﻘﺪﻳﻢ ﺃﻣﺜﻠﺔ ﻣﻀﺎﺩﺓ ﻟﺘﻔﻨﻴﺪ‬.‫ﻭﺍﻟﻔﺼﻞ‬ .‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺨﻄﺄ‬ .‫ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺃﻋﻼﻩ ﻣﺜﺎﻝ ﻋﻠﻰ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬   1-3     .(...‫ ﻓﺈﻥ‬...‫ﺗﺤﻠﻴﻞ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ )ﺇﺫﺍ‬   ‫ﻛﺘﺎﺑﺔ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ‬     .(...‫ ﻓﺈﻥ‬...‫ﺍﻹﻳﺠﺎﺑﻲ ﻟﻌﺒﺎﺭﺓ )ﺇﺫﺍ‬ 1-3   pq      ‫ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﺒﺮﻳﺮ ﺍﻻﺳﺘﻨﺘﺎﺟﻲ ﻹﺛﺒﺎﺕ‬  .‫ﺻﺤﺔ ﻋﺒﺎﺭﺓ ﻣﺎ‬ q p       qp      2   p     conditional statement  q . “‫ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻗﺮﺍﺀﺓ ﻓﻘﺮﺓ ”ﻟﻤﺎﺫﺍ؟‬   .‫ ﻳﻤﻜﻨﻚ ﺗﺤﺪﻳﺪ ﺍﻟ ﹶﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻴﻬﺎ ﺑﺴﻬﻮﻟﺔ‬،( ... ‫ ﻓﺈﻥ‬... ‫ﻋﻨﺪﻣﺎ ﺗﻜﺘﺐ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‬  hypothesis   1 ‫• ﻣﺎ ﺍﻟﻔﺎﺋﺪﺓ ﻣﻦ ﻗﺎﺋﻤﺔ ﺍﻟﺒﺪﺍﺋﻞ ﻓﻲ ﻧﻈﺎﻡ ﺍﻟﺮﺩ‬  ‫ ﺗﻤﻜﻴﻦ ﺍﻟﻤﺘﺼﻞ ﻣﻦ‬:‫ﺍﻵﻟﻲ؟ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‬ :‫ﺣ ﱢﺪﺩ ﺍﻟ ﹶﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‬ . ‫ ﻓﺴﻮﻑ ﺃﺳﺘﻌﻤﻞ ﺍﻟﻤﻈﻠﺔ‬، ‫( ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻄﻘﺲ ﻣﺎﻃ ﹰﺮﺍ‬a conclusion ‫ﺍﻟﺘﺤﺪﺙ ﺇﻟﻰ ﺍﻟﻘﺴﻢ ﺍﻟﺬﻱ ﻳﺮﻳﺪﻩ ﺑﺴﺮﻋﺔ‬ .‫ ﺍﻟﻄﻘﺲ ﻣﺎﻃﺮ‬:‫ﺍﻟ ﹶﻔﺮﺽ‬ .‫ﻭﺳﻬﻮﻟﺔ‬ .‫ ﺳﻮﻑ ﺃﺳﺘﻌﻤﻞ ﺍﻟﻤﻈﻠﺔ‬:‫ﺍﻟﻨﺘﻴﺠﺔ‬  .‫ ﺇﺫﺍ ﻛﺎﻥ ﺁﺣﺎﺩﻩ ﺻﻔ ﹰﺮﺍ‬10 ‫( ﻳﻘﺒﻞ ﺍﻟﻌﺪﺩ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ‬b  ‫• ﻣﺎ ﻗﺴﻤﺎ ﺍﻟﺠﻤﻠﺔ ﻓﻲ ﺍﻟﻤﺜﺎﻝ؟ ﺗﺮﻳﺪ‬ .‫ ﺁﺣﺎﺩ ﺍﻟﻌﺪﺩ ﺻﻔﺮ‬:‫ﺍﻟ ﹶﻔﺮﺽ‬ ‫ ﺍﺿﻐﻂ‬،‫ﺍﻟﺘﺤﺪﺙ ﺇﻟﻰ ﻗﺴﻢ ﺧﺪﻣﺔ ﺍﻟﻌﻤﻼﺀ‬ 10 ‫ ﻳﻘﺒﻞ ﺍﻟﻌﺪﺩ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ‬:‫ﺍﻟﻨﺘﻴﺠﺔ‬ related conditionals 2 ‫ﺍﻟﺮﻗﻢ‬  ‫• ﻣﺎ ﺍﻟﻤﺸﻜﻠﺔ ﺍﻟﺘﻲ ﻳﻤﻜﻦ ﺃﻥ ﺗﻨﺸﺄ ﻋﻦ‬ converse :‫ﺍﺳﺘﻌﻤﺎﻝ ﻧﻈﺎﻡ ﺍﻟﺮﺩ ﺍﻵﻟﻲ؟ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‬  ‫ﻗﺪ ﻻ ﻳﻤﻜﻦ ﺣﺼﺮ ﺟﻤﻴﻊ ﺍﻷﺳﺒﺎﺏ ﺍﻟﺘﻲ‬ .‫ﺗﺪﻋﻮ ﺍﻟﺸﺨﺺ ﺇﻟﻰ ﺍﻻﺗﺼﺎﻝ‬ inverse   contrapositive   logically equivalent www.obeikaneducation.com ‫( ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺇﺟﺎﺑﺎﺕ‬1A, 1B  ✓ .‫ ﻓﺈﻧﻪ ﺳﺪﺍﺳﻲ‬،‫( ﺇﺫﺍ ﻛﺎﻥ ﻟﻤﻀﻠﻊ ﺳﺘﺔ ﺃﺿﻼﻉ‬1A .‫ ﺇﺫﺍ ﺑﹺﻴﻌﺖ ﻧﺴﺦ ﺍﻟﻄﺒﻌﺔ ﺍﻷﻭﻟﻰ ﻛ ﹼﻠﻬﺎ‬،‫( ﺳﻴﺘﻢ ﺇﻧﺠﺎﺯ ﻃﺒﻌﺔ ﺛﺎﻧﻴﺔ ﻣﻦ ﺍﻟﻜﺘﺎﺏ‬1B  1 26 1-3     (34) •  (33, 34) • (33, 34) • (8) •  (8) • (8) •  (16)  •   (19 )  • (16)  • (18) • (20) • (18) • (19) • (19)  • (20) •  1 26

‫ﺗﻜﺘﺐ ﻛﺜﻴﺮ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺩﻭﻥ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻜﻠﻤﺘﻴﻦ )ﺇﺫﺍ‪ ،‬ﻓﺈﻥ(‪ ،‬ﻭﻟﻜﺘﺎﺑﺔ ﺗﻠﻚ ﺍﻟﻌﺒﺎﺭﺍﺕ ﻋﻠﻰ ﺻﻮﺭﺓ‬ ‫‪‬‬ ‫)ﺇﺫﺍ ‪ ...‬ﻓﺈﻥ‪ (...‬ﺣﺪﺩ ﺍﻟ ﹶﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ‪.‬‬ ‫‪ ‬‬ ‫‪  ‬‬ ‫ﻋﻨﺪ ﺷﺮﺍﺋﻚ ﺃ ﹼﹰﻳﺎ ﻣﻦ ﻣﻨﺘﺠﺎﺗﻨﺎ ﻗﺒﻞ ﻳﻮﻡ ﺍﻷﺭﺑﻌﺎﺀ‬ ‫ﺗﺤﺼﻞ ﻋﻠﻰ ﺧﺼﻢ ﺗﺸﺠﻴﻌﻲ‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ 1 ‬ﻳﺒ ﱢﻴﻦ ﻛﻴﻔﻴﺔ ﺗﺤﺪﻳﺪ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ‬ ‫‪‬‬ ‫‪‬‬ ‫‪  ‬‬ ‫ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 2 ‬ﻳﺒ ﱢﻴﻦ ﻛﻴﻔﻴﺔ ﻛﺘﺎﺑﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪‬‬ ‫ﻋﻠﻰ ﺻﻮﺭﺓ ”ﺇﺫﺍ ‪ ...‬ﻓﺈﻥ‪.“...‬‬ ‫‪‬‬ ‫‪ 3 ‬ﻳﺒﻴﻦ ﻛﻴﻔﻴﺔ ﺗﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫ﺇﺫﺍ ﺍﺷﺘﺮﻳﺖ ﺃ ﹼﹰﻳﺎ ﻣﻦ ﻣﻨﺘﺠﺎﺗﻨﺎ ﻗﺒﻞ ﻳﻮﻡ ﺍﻷﺭﺑﻌﺎﺀ ‪ ،‬ﻓﺈﻧﻚ ﺗﺤﺼﻞ ﻋﻠﻰ ﺧﺼﻢ ﺗﺸﺠﻴﻌﻲ‪.‬‬ ‫✓ ‪‬‬ ‫ﺗﺬﻛﺮ ﺃﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﺍﻟﻔﺮﺽ‪.‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺗﻤﺎﺭﻳﻦ ”ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻤﻚ“ ﺑﻌﺪ ﻛﻞ‬ ‫‪  2‬‬ ‫ﻣﺜﺎﻝ؛ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻣﺪ￯ ﻓﻬﻢ ﺍﻟﻄﻠﺒﺔ ﺍﻟﻤﻔﺎﻫﻴﻢ‪.‬‬ ‫ﺣ ﹼﺪﺩ ﺍﻟ ﹶﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﻛﺘﺒﻬﺎ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪:(...‬‬ ‫‪ (a‬ﺍﻟﺜﺪﻳﻴﺎﺕ ﺣﻴﻮﺍﻧﺎﺕ ﻣﻦ ﺫﻭﺍﺕ ﺍﻟﺪﻡ ﺍﻟﺤﺎﺭ‪.‬‬ ‫‪ (2A‬ﺍﻟﻔﺮﺽ‪ :‬ﻟﺪﻳﻚ ‪ 5‬ﺃﻭﺭﺍﻕ‬ ‫ﺍﻟ ﹶﻔﺮﺽ‪ :‬ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ‪.‬‬ ‫ﻧﻘﺪﻳﺔ ﻣﻦ ﻓﺌﺔ ﺍﻟﺮﻳﺎﻝ‪.‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻫﻮ ﻣﻦ ﺫﻭﺍﺕ ﺍﻟﺪﻡ ﺍﻟﺤﺎﺭ‪.‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻳﻤﻜﻦ ﺃﻥ ﺗﺒﺪﻟﻬﺎ ﺑﻮﺭﻗﺔ‬ ‫ﻭﺍﺣﺪﺓ ﻣﻦ ﻓﺌﺔ ‪ 5‬ﺭﻳﺎﻻﺕ‪.‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ‪ ،‬ﻓﺈﻧﻪ ﻣﻦ ﺫﻭﺍﺕ ﺍﻟﺪﻡ ﺍﻟﺤﺎﺭ‪.‬‬ ‫‪ (b‬ﺍﻟﻤﻨﺸﻮﺭ ﺍﻟﺬﻱ ﻗﺎﻋﺪﺗﺎﻩ ﻣﻀﻠﻌﺎﻥ ﻣﻨﺘﻈﻤﺎﻥ‪ ،‬ﻳﻜﻮﻥ ﻣﻨﺘﻈ ﹰﻤﺎ‪.‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﻟﺪﻳﻚ ‪ 5‬ﺃﻭﺭﺍﻕ ﻧﻘﺪﻳﺔ‬ ‫ﺍﻟ ﹶﻔﺮﺽ‪ :‬ﻗﺎﻋﺪﺗﺎ ﺍﻟﻤﻨﺸﻮﺭ ﻣﻀﻠﻌﺎﻥ ﻣﻨﺘﻈﻤﺎﻥ‪.‬‬ ‫ﻣﻦ ﻓﺌﺔ ﺍﻟﺮﻳﺎﻝ ﻓﺈﻧﻪ ﻳﻤﻜﻨﻚ ﺃﻥ‬ ‫‪‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻳﻜﻮﻥ ﺍﻟﻤﻨﺸﻮﺭ ﻣﻨﺘﻈ ﹰﻤﺎ‪.‬‬ ‫ﺗﺒﺪﻟﻬﺎ ﺑﻮﺭﻗﺔ ﻭﺍﺣﺪﺓ ﻣﻦ ﻓﺌﺔ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﻗﺎﻋﺪﺗﺎ ﺍﻟﻤﻨﺸﻮﺭ ﻣﻀﻠﻌﻴﻦ ﻣﻨﺘﻈﻤﻴﻦ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻣﻨﺘﻈ ﹰﻤﺎ‪.‬‬ ‫‪ 5‬ﺭﻳﺎﻻﺕ‪.‬‬ ‫ﺣ ﱢﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ‬ ‫‪1‬‬ ‫ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪2‬‬ ‫✓ ‪‬‬ ‫‪ (2B‬ﺍﻟﻔﺮﺽ‪ :‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬ ‫ﻣﺘﺘﺎﻣﺘﺎﻥ‪.‬‬ ‫‪ (a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺯﻭﺍﻳﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻲ‬ ‫‪ (2A‬ﻳﻤﻜﻦ ﺗﺒﺪﻳﻞ ‪ 5‬ﺃﻭﺭﺍﻕ ﻧﻘﺪﻳﺔ ﻣﻦ ﻓﺌﺔ ﺍﻟﺮﻳﺎﻝ ﺑﻮﺭﻗﺔ ﻧﻘﺪﻳ ﹴﺔ ﻭﺍﺣﺪ ﹴﺓ ﻣﻦ ﻓﺌﺔ ‪ 5‬ﺭﻳﺎﻻﺕ‪.‬‬ ‫ﻗﺎﺋﻤﺔ ﻓﺈﻧﻪ ﻣﺴﺘﻄﻴﻞ‪.‬‬ ‫‪ (2B‬ﻣﺠﻤﻮﻉ ﻗﻴﺎ ﹶﺳﻲ ﺍﻟﺰﺍﻭﻳﺘﻴﻦ ﺍﻟﻤﺘﺘﺎﻣﺘﻴﻦ ﻳﺴﺎﻭﻱ ‪90°‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ‬ ‫ﻳﺴﺎﻭﻱ ‪90°‬‬ ‫ﺍﻟﻔﺮﺽ‪ :‬ﺯﻭﺍﻳﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻲ‬ ‫ﻗﺎﺋﻤﺔ‪.‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ‪،‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻲ ﻣﺴﺘﻄﻴﻞ‪.‬‬ ‫ﺗﺬﻛﺮ ﺃﻥ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻭﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻧﻔﺴﻬﺎ ﺟﻤﻴﻌﻬﺎ ﻋﺒﺎﺭﺍﺕ ﻗﺪ ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ ﻭﻗﺪ ﺗﻜﻮﻥ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﻓﺈﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ﻳﺴﺎﻭﻱ‬ ‫ﻗﺎﻝ ﻋﻤﺮ ﻟﺰﻣﻼﺋﻪ‪ :‬ﺇﺫﺍ ﺃﻧﻬﻴﺖ ﻭﺍﺟﺒﻲ ﺍﻟﻤﻨﺰﻟﻲ‪ ،‬ﻓﺈﻧﻲ ﺳﻮﻑ ﺃﻟﻌﺐ ﺍﻟﻜﺮﺓ ﻣﻌﻜﻢ ‪.‬‬ ‫‪ (b‬ﺳﻴﺘﻘﺪﻡ ﻣﺤﻤﺪ ﺇﻟﻰ ﺍﻟﻤﺴﺘﻮ￯‬ ‫‪90°‬‬ ‫ﺍﻷﻋﻠﻰ ﻓﻲ ﺍﻟﺪﻭﺭﺓ ﺇﺫﺍ ﺃﻛﻤﻞ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﻟﻤﺴﺘﻮ￯ ﺍﻻﺑﺘﺪﺍﺋﻲ‪.‬‬ ‫‪ ‬‬ ‫ﺍﻟﻔﺮﺽ‪ :‬ﻣﺤﻤﺪ ﻳﻜﻤﻞ ﺍﻟﻤﺴﺘﻮ￯‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫ﺍﻻﺑﺘﺪﺍﺋﻲ‪.‬‬ ‫‪ ‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻣﺤﻤﺪ ﻳﺘﻘﺪﻡ ﺇﻟﻰ‬ ‫‪  ‬‬ ‫ﺍﻟﻤﺴﺘﻮ￯ ﺍﻷﻋﻠﻰ‪.‬‬ ‫‪‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪ ‬‬ ‫ﺣ ﱢﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛﻞ ﻋﺒﺎﺭﺓ‬ ‫ﻣﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﺍﻛﺘﺒﻬﺎ ﻋﻠﻰ ﺻﻮﺭﺓ‬ ‫‪   ‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪‬‬ ‫”ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪.“...‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ (a‬ﻗﻴﺎﺱ ﺍﻟﻤﺴﺎﻓﺔ ﻳﻜﻮﻥ ﻣﻮﺟ ﹰﺒﺎ‪.‬‬ ‫‪   ‬‬ ‫‪‬‬ ‫ﺍﻟﻔﺮﺽ‪ :‬ﹺﻗﻴ ﹶﺴ ﹺﺖ ﺍﻟﻤﺴﺎﻓﺔ‪،‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﺍﻟﻘﻴﺎﺱ ﻣﻮﺟﺐ‪.‬‬ ‫‪   ‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪‬‬ ‫ﺇﺫﺍ ﹺﻗﻴ ﹶﺴ ﹺﺖ ﺍﻟﻤﺴﺎﻓﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﻘﻴﺎﺱ‬ ‫‪ T‬‬ ‫ﻣﻮﺟﺐ‪.‬‬ ‫‪‬‬ ‫‪ (b‬ﺍﻟﻤﻀﻠﻊ ﺫﻭ ﺍﻷﺿﻼﻉ ﺍﻟﺨﻤﺴﺔ ﻫﻮ‬ ‫ﺷﻜﻞ ﺧﻤﺎﺳﻲ‪.‬‬ ‫ﻻﺣﻆ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ ﻓﻲ ﺟﻤﻴﻊ ﺍﻟﺤﺎﻻﺕ‪ ،‬ﺇ ﱠﻻ ﺃﻥ ﻳﻜﻮﻥ ﺍﻟﻔﺮﺽ ﺻﺎﺋ ﹰﺒﺎ ﻭﺍﻟﻨﺘﻴﺠﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﺍﻟﻔﺮﺽ‪ :‬ﺍﻟﻤﻀﻠﻊ ﻟﻪ ﺧﻤﺴﺔ‬ ‫‪27  1-3‬‬ ‫ﺃﺿﻼﻉ‪ .‬ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﺇﻧﻪ ﺷﻜﻞ‬ ‫‪ ‬‬ ‫ﺧﻤﺎﺳﻲ‪.‬‬ ‫‪ ‬ﻋﻨﺪ ﺗﺤﺪﻳﺪ‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﻟﻠﻤﻀﻠﻊ ﺧﻤﺴﺔ ﺃﺿﻼﻉ‬ ‫ﻓﺈﻧﻪ ﺧﻤﺎﺳﻲ‪.‬‬ ‫ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‪ ،‬ﺃﺧﺒﺮ ﺍﻟﻄﻼﺏ ﺃﻥ‬ ‫ﻳﺴﺘﻌﻤﻠﻮﺍ ﺍﻷﻗﻮﺍﺱ ﻟﺘﺤﺪﻳﺪ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ‬ ‫ﻛﻞ ﺣﺎﻟﺔ‪ .‬ﻭﻭ ﹼﺿﺢ ﻟﻬﻢ ﺃ ﹼﻧﻪ ﺇﺫﺍ ﺗﻄﺎﺑﻖ ﺍﻟﻔﺮﺽ ﻓﻲ ﺍﻟﺤﺎﻟﺔ‬ ‫ﻣﻊ ﺍﻟﻔﺮﺽ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﺻﻠﻴﺔ‪ ،‬ﻓﺈﻧﻪ ﻳﻤﻜﻦ ﻟﻠﻄﻼﺏ ﺃﻥ‬ ‫ﻳﻀﻌﻮﺍ ﺣﺮﻑ ‪ T‬ﻓﻮﻕ ﺍﻷﻗﻮﺍﺱ‪ ،‬ﻭﺇ ﹼﻻ ﻳﻤﻜﻨﻬﻢ ﺃﻥ ﻳﻀﻌﻮﺍ‬ ‫ﺣﺮﻑ ‪ ، F‬ﻭﻳﻤﻜﻨﻬﻢ ﻋﻤﻞ ﺍﻟﺸﻲﺀ ﻧﻔﺴﻪ ﻟﻠﻨﺘﻴﺠﺔ‪.‬‬ ‫‪27  1-3‬‬

‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﺴﺎﺑﻘﺔ ﻹﻧﺸﺎﺀ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪p q pq‬‬ ‫‪‬‬ ‫‪TT‬‬ ‫‪T‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪TF‬‬ ‫‪F‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪FT‬‬ ‫‪T‬‬ ‫‪‬‬ ‫‪FF‬‬ ‫‪T‬‬ ‫‪‬‬ ‫‪3‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺣ ﱢﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺓ‬ ‫‪‬‬ ‫‪ ‬‬ ‫ﺷﺮﻃﻴﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﺧﺻﺎﻃﺤﺌﻴﺔ‪،‬ﺤﺔﻓﺄﻓﻔﻋ ﱢ ﹺﺴﻂﺮﻣﺗﺜﺒﺎ ﹰﺮﻳﻻﺮﻣﻙ‪،‬ﻀﺎﺃ ﹼﹰﺩﻣﺍﺎ‪.‬ﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫‪‬‬ ‫ﻹﺛﺒﺎﺕ ﺻﺤﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ ،‬ﻳﺠﺐ ﻋﻠﻴﻚ ﺇﺛﺒﺎﺕ ﺃﻧﻪ ﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ ﺍﻟﻔﺮﺽ ﺻﺎﺋ ﹰﺒﺎ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺻﺎﺋﺒﺔ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫‪ (a‬ﺇﺫﺍ ﻃﺮﺣﺖ ﻋﺪ ﹰﺩﺍ ﻃﺒﻴﻌ ﹰﹼﻴﺎ ﻣﻦ ﻋﺪﺩ‬ ‫ﻭﻹﺛﺒﺎﺕ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ ﻳﻜﻔﻲ ﺃﻥ ﺗﻌﻄﻲ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ ‪.‬‬ ‫ﻃﺒﻴﻌﻲ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﺎﺗﺞ ﻳﻜﻮﻥ ﻋﺪ ﹰﺩﺍ‬ ‫‪ 3‬‬ ‫ﻃﺒﻴﻌ ﹰﹼﻴﺎ‪.‬‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ .2-7= -5 :‬ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺣ ﹼﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺻﺎﺋﺒﺔ‪ ،‬ﻓﻔ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪ ،‬ﺃﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ‪،‬‬ ‫ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪:‬‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ (a‬ﻋﻨﺪ ﻗﺴﻤﺔ ﻋﺪﺩ ﺻﺎﺋﺐ ﻋﻠﻰ ﻋﺪﺩ ﺻﺎﺋﺐ ﺁﺧﺮ‪ ،‬ﻳﻜﻮﻥ ﺍﻟﻨﺎﺗﺞ ﻋﺪ ﹰﺩﺍ ﺻﺎﺋ ﹰﺒﺎ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫‪ (b‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻬﺮ ﺍﻟﻤﺎﺿﻲ ﻫﻮ ﺷﻬﺮ‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ :‬ﻋﻨﺪ ﻗﺴﻤﺔ ‪ 1‬ﻋﻠﻰ ‪ ،2‬ﻳﻜﻮﻥ ﺍﻟﻨﺎﺗﺞ ‪0.5‬‬ ‫ﺭﺟﺐ‪ ،‬ﻓﺈﻥ ﻫﺬﺍ ﺍﻟﺸﻬﺮ ﻫﻮ ﺷﻌﺒﺎﻥ‪.‬‬ ‫ﺑﻤﺎ ﺃﻥ ‪ 0.5‬ﻟﻴﺲ ﻋﺪ ﹰﺩﺍ ﺻﺎﺋ ﹰﺒﺎ‪ ،‬ﻓﺈﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺧﺎﻃﺌﺔ‪ .‬ﻭﺑﻤﺎ ﺃﻧﻚ ﺍﺳﺘﻄﻌﺖ ﺇﻳﺠﺎﺩ ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ ،‬ﻓﺎﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﻛﻠﻤﺎ ﻛﺎﻥ ﺍﻟﻔﺮﺽ \"ﺍﻟﺸﻬﺮ‬ ‫ﺍﻟﻤﺎﺿﻲ ﺷﻬﺮ ﺭﺟﺐ\" ﺻﺤﻴ ﹰﺤﺎ‪،‬‬ ‫‪ (b‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻬﺮ ﺍﻟﻘﺎﺩﻡ ﻫﻮ ﺭﻣﻀﺎﻥ‪ ،‬ﻓﺈﻥ ﻫﺬﺍ ﺍﻟﺸﻬﺮ ﻫﻮ ﺷﻬﺮ ﺷﻌﺒﺎﻥ‪.‬‬ ‫‪ (3A‬ﺧﺎﻃﺌﺔ؛ ﺇﺫﺍ ﻛﺎﻥ‬ ‫ﻓﺈﻥ ﺍﻟﻨﺘﻴﺠﺔ \"ﻫﺬﺍ ﺍﻟﺸﻬﺮ ﻫﻮ ﺷﻬﺮ‬ ‫ﺭﻣﻀﺎﻥ ﻫﻮ ﺍﻟﺸﻬﺮ ﺍﻟﺬﻱ ﻳﻠﻲ ﺷﻬﺮ ﺷﻌﺒﺎﻥ؛ ﺇﺫﻥ ﻛﻠﻤﺎ ﻛﺎﻥ ﺍﻟﻔﺮﺽ )ﺍﻟﺸﻬﺮ ﺍﻟﻘﺎﺩﻡ ﺭﻣﻀﺎﻥ( ﺻﺎﺋ ﹰﺒﺎ‪،‬‬ ‫‪ ،m∠A=55°‬ﻓﺈﻥ ‪ ∠A‬ﺣﺎﺩﺓ‬ ‫ﺷﻌﺒﺎﻥ\" ﺳﺘﻜﻮﻥ ﺻﺤﻴﺤﺔ ﺃﻳ ﹰﻀﺎ؛‬ ‫ﻓﺈﻥ ﺍﻟﻨﺘﻴﺠﺔ )ﻫﺬﺍ ﺍﻟﺸﻬﺮ ﻫﻮ ﺷﻬﺮ ﺷﻌﺒﺎﻥ( ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ ﺃﻳ ﹰﻀﺎ؛ ﻭﻋﻠﻴﻪ ﻓﺈﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﻷﻥ ﺷﻬﺮ ﺷﻌﺒﺎﻥ ﻫﻮ ﺍﻟﺸﻬﺮ ﺍﻟﺬﻱ‬ ‫ﺃﻳ ﹰﻀﺎ‪ ،‬ﻭﻟﻜﻦ ﻗﻴﺎﺳﻬﺎ ﻟﻴﺲ ‪.35°‬‬ ‫‪ (c‬ﺇﺫﺍ ﻛﺎﻥ ﻟﻠﻤﺜﻠﺚ ﺃﺭﺑﻌﺔ ﺃﺿﻼﻉ‪ ،‬ﻓﺈﻧﻪ ﻣﻀﻠ ﹲﻊ ﻣﻘﻌ ﹲﺮ‪.‬‬ ‫ﻳﻠﻲ ﺷﻬﺮ ﺭﺟﺐ‪ ،‬ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﻻ ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ ﻟﻠﻤﺜﻠﺚ ﺃﺭﺑﻌﺔ ﺃﺿﻼﻉ؛ ﺇﺫﻥ ﺍﻟﻔﺮﺽ ﺧﺎﻃﺊ ﻭﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ ﺍﻟﻔﺮﺽ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺈﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫‪ (3B‬ﺻﺎﺋﺒﺔ؛ ﺍﻟﻔﺮﺽ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪ √x=-1‬ﺧﺎﻃﺊ؛ ﻷﻥ ﺍﻟﺠﺬﺭ‬ ‫‪ (c‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺇﺣﺪ￯ ﺯﻭﺍﻳﺎ ﺍﻟﻤﺴﺘﻄﻴﻞ‬ ‫ﻣﻨﻔﺮﺟﺔ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻣﺘﻮﺍﺯﻱ‬ ‫ﺍﻟﺘﺮﺑﻴﻌﻲ ﻻ ﻳﻜﻮﻥ ﺳﺎﻟ ﹰﺒﺎ ﻷﻱ‬ ‫ﻋﺪﺩ‪ ،‬ﻭﻋﻠﻴﻪ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺃﺿﻼﻉ‪ .‬ﺍﻟ ﹶﻔﺮﺽ ﺧﺎﻃﻲﺀ؛ ﻷﻧﻪ‬ ‫ﻻ ﻳﻤﻜﻦ ﺃﻥ ﺗﻜﻮﻥ ﺇﺣﺪ￯ ﺯﻭﺍﻳﺎ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻤﺴﺘﻄﻴﻞ ﻣﻨﻔﺮﺟﺔ‪ ،‬ﻭﺍﻟﻌﺒﺎﺭﺓ‬ ‫✓ ‪‬‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺘﻲ ﻳﻜﻮﻥ ﺍﻟ ﹶﻔﺮﺽ ﻓﻴﻬﺎ‬ ‫‪ (3A‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ∠A‬ﺣﺎﺩﺓ‪ ،‬ﻓﺈﻥ ‪m∠A = 35°‬‬ ‫ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﺗﻜﻮﻥ ﺻﺤﻴﺤ ﹰﺔ ﺩﺍﺋ ﹰﻤﺎ‪.‬‬ ‫‪ (3B‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، √x = −1‬ﻓﺈﻥ ‪(−1)2 = −1‬‬ ‫‪ 1 28‬‬ ‫‪ 1 28‬‬

‫‪ ‬ﻳﺮﺗﺒﻂ ﺑﺎﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﻌﻄﺎﺓ ﻋﺒﺎﺭﺍﺕ ﺷﺮﻃﻴﺔ ﺃﺧﺮ￯ ﺗﺴﻤﻰ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﺮﺗﺒﻄﺔ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 4 ‬ﻳﺒ ﱢﻴﻦ ﻛﻴﻔﻴﺔ ﺍﺳﺘﻌﻤﺎﻝ ﺟﺪﺍﻭﻝ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪p→q‬‬ ‫‪‬‬ ‫ﺍﻟﺼﻮﺍﺏ ﻹﺛﺒﺎﺕ ﺃﻥ ﻋﺒﺎﺭﺗﻴﻦ ﻣﺘﻜﺎﻓﺌﺘﺎﻥ‬ ‫‪،m∠A = 35°‬‬ ‫‪q→p‬‬ ‫‪∠A ‬‬ ‫‪~p → ~q‬‬ ‫‪ q ،p‬‬ ‫ﻣﻨﻄﻘ ﹼﹰﻴﺎ‪.‬‬ ‫‪~q → ~p‬‬ ‫‪‬‬ ‫‪،∠A‬‬ ‫‪‬‬ ‫‪m∠A = 35° ‬‬ ‫‪‬‬ ‫‪،m∠A ≠ 35°‬‬ ‫‪‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺗﻴﻦ‬ ‫‪4‬‬ ‫‪∠A‬‬ ‫‪ ∼(p∨q), ∼p∨q‬ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﺠﺪﻭﻝ‪،‬‬ ‫‪،∠A‬‬ ‫‪‬‬ ‫‪m∠A ≠ 35°‬‬ ‫‪‬‬ ‫ﺛﻢ ﻗﺮﺭ ﻫﻞ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ﻣﺘﻜﺎﻓﺌﺘﺎﻥ‬ ‫ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫‪‬‬ ‫‪p q ∼p ∼q (p ∨ q) ∼(p ∨ q) ∼p ∨ ∼q‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ‪ ،‬ﻓﻠﻴﺲ ﺑﺎﻟﻀﺮﻭﺭﺓ ﺃﻥ ﻳﻜﻮﻥ ﻋﻜﺴﻬﺎ ﻭﻣﻌﻜﻮﺳﻬﺎ ﺻﺎﺋﺒﻴﻦ‪ ،‬ﺑﻴﻨﻤﺎ ﻳﻜﻮﻥ ﺍﻟﻤﻌﺎﻛﺲ‬ ‫ﺍﻹﻳﺠﺎﺑﻲ ﺻﺎﺋ ﹰﺒﺎ‪ .‬ﻭﻳﻜﻮﻥ ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﺧﺎﻃ ﹰﺌﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪TTFF T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﻭﺑﺎﻟﻤﺜﻞ ﻓﺈﻥ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﻜﻮﺳﻬﺎ ﺇﻣﺎ ﺃﻥ ﻳﻜﻮﻧﺎ ﺻﺎﺋﺒﻴﻦ ﻣ ﹰﻌﺎ ﺃﻭ ﺧﺎﻃﺌﻴﻦ ﻣ ﹰﻌﺎ‪ .‬ﻭﺗﺴﻤﻰ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﻲ‬ ‫ﻟﻬﺎ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻧﻔﺴﻬﺎ ﻋﺒﺎﺭﺍﺕ ﻣﺘﻜﺎﻓﺌﺔ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫‪TFFT T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪  4‬‬ ‫‪FTTF T‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪‬ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻋﻜﺴﻬﺎ ﻭﻣﻌﻜﻮﺳﻬﺎ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﺠﺪﻭﻝ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ‬ ‫‪FFTT F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫ﻋﺒﺎﺭﺗﻴﻦ ﻣﺘﻜﺎﻓﺌﺘﻴﻦ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫ﻏﻴﺮ ﻣﺘﻜﺎﻓﺌﺘﻴﻦ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫‪p‬‬ ‫‪q‬‬ ‫‪∼p‬‬ ‫‪∼q‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪pq‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪∼q ∼p‬‬ ‫‪qp‬‬ ‫‪∼p ∼q‬‬ ‫‪ ‬‬ ‫‪TTFF‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪ ‬ﺯ ﹼﻭﺩ ﻛﻞ ﻃﺎﻟﺐ‬ ‫‪TFFT‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫ﺑﺒﻄﺎﻗﺎﺕ ﻣﻌﻨﻮﻧﺔ ﺑـ \"ﺍﻟﻔﺮﺽ\"‪\" ،‬ﺍﻟﻨﺘﻴﺠﺔ\"‬ ‫‪FTTF‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫\"ﻳﺆﺩﻱ ﺇﻟﻰ\" )ﺃﻭ ﺇﺷﺎﺭﺓ ﺳﻬﻢ ﻣﺘﺠﻪ ﻣﻦ‬ ‫ﺍﻟﻴﺴﺎﺭ ﺇﻟﻰ ﺍﻟﻴﻤﻴﻦ( ﺃﻋﻂ ﻛﻞ ﻃﺎﻟﺐ ﺑﻄﺎﻗﺘﻴﻦ‬ ‫‪FFTT‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫ﻛ ﱞﻞ ﻣﻨﻬﻤﺎ ﻣﻌﻨﻮﻧﺔ ﺑـ ”ﻟﻴﺲ“ ﺑﺎﻟﻠﻮﻥ ﺍﻷﺣﻤﺮ‪،‬‬ ‫ﺛﻢ ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺗﻜﻮﻳﻦ ﻋﺒﺎﺭﺍﺕ‬ ‫ﻣﻦﺧﻼﻝﺟﺪﻭﻝﺍﻟﺼﻮﺍﺏﻧﻼﺣﻆﺃﻧﻪﻟﻠﻌﺒﺎﺭﺗﻴﻦ‪ p → q‬ﹶﻭ ‪ ~q → ~p‬ﻗﻴﻢﺍﻟﺼﻮﺍﺏﻧﻔﺴﻬﺎﻟﺬﺍﻓﻬﻤﺎﻣﺘﻜﺎﻓﺌﺘﺎﻥ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫ﺷﺮﻃﻴﺔ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﺒﻄﺎﻗﺎﺕ‪ ،‬ﻭﺗﻜﻮﻳﻦ‬ ‫✓ ‪‬‬ ‫ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪.‬‬ ‫ﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﺴﺘﺠﻴﺒﻮﺍ ﻟﻠﻨﺸﺎﻁ ﺑﻮﺿﻊ‬ ‫‪ (4‬ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺍﺕ‪ ~(p q), ~p ~q, ~(p q), ~p ~q :‬ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﺠﺪﻭﻝ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ‬ ‫ﺍﻟﺒﻄﺎﻗﺎﺕ ﻓﻲ ﺍﻟﻮﺿﻊ ﻭﺍﻟﺘﺮﺗﻴﺐ ﺍﻟﺼﺤﻴﺤﻴﻦ‪،‬‬ ‫ﻭﻳﻤﻜﻨﻬﻢ ﺃﻳ ﹰﻀﺎ ﺃﻥ ﻳﺴﺘﻌﻤﻠﻮﺍ ﺍﻟﺒﻄﺎﻗﺎﺕ ﻟﺤﻞ‬ ‫ﺯﻭﺟﻴﻦ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺘﻜﺎﻓﺌﺔ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﺑﻌﺾ ﺍﻟﺘﺪﺭﻳﺒﺎﺕ ﺃﻭ ﺍﻷﻣﺜﻠﺔ ﻓﻲ ﻫﺬﺍ ﺍﻟﺪﺭﺱ‬ ‫ﺑﻜﺘﺎﺑﺔ ﺃﺟﺰﺍﺀ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻋﻠﻰ ﺍﻟﺒﻄﺎﻗﺎﺕ‬ ‫ﻣﻤﺎ ﺳﺒﻖ ﻧﻼﺣﻆ ﺃﻥ‪:‬‬ ‫ﺍﻟﻤﻨﺎﻇﺮﺓ‪.‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫• ‪  ‬‬ ‫• ‪ ‬‬ ‫• )‪~p ~q  ~(p q‬‬ ‫• )‪~p ~p  ~(p q‬‬ ‫‪29  1-3‬‬ ‫‪‬‬ ‫‪ ‬ﺍﻛﺘﺐ ﻋﺒﺎﺭﺓ‬ ‫‪p q ∼p ∼q p ∧ q p ∨ q ∼(p ∧ q) ∼p ∨ ∼q ∼(p ∨ q) ∼p ∧ ∼q (4‬‬ ‫ﺷﺮﻃﻴﺔ ﻋﻠﻰ ﺍﻟﺴﺒﻮﺭﺓ‪ ،‬ﺛﻢ ﺍﺳﺤﺐ ﻛ ﹼﹰﻼ‬ ‫‪TTFF T T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﻣﻦ ﺍﻟ ﹶﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ؛ ﻟﺘﺴﺎﻋﺪ ﻋﻠﻰ‬ ‫‪TFFT F T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﻛﺘﺎﺑﺔ ﺍﻟﻌﻜﺲ‪ ،‬ﻭﺍﻟﻤﻌﻜﻮﺱ‪ ،‬ﻭﺍﻟﻤﻌﺎﻛﺲ‬ ‫‪FTTF F T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻷﺻﻠﻴﺔ‪.‬‬ ‫‪FFTT F F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫)‪ ∼(p ∧ q‬ﺗﻜﺎﻓﺊ ﻣﻨﻄﻘ ﹰﹼﻴﺎ ‪∼p ∨ ∼q‬‬ ‫)‪ ∼(p ∨ q‬ﺗﻜﺎﻓﺊ ﻣﻨﻄﻘ ﹰﹼﻴﺎ ‪∼p ∧ ∼q‬‬ ‫‪29  1-3‬‬

‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﺘﻜﺎﻓﺆ ﺍﻟﻤﻨﻄﻘﻲ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻌﺒﺎﺭﺓ ﻣﺎ‪ .‬ﻓﻲ ﺍﻟﻤﺜﺎﻝ ‪ 5‬ﺃﺩﻧﺎﻩ‪ ،‬ﻻﺣﻆ ﺃﻥ ﻛ ﹼﹰﻼ ﻣﻦ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ ﺻﺎﺋﺒﺎﻥ‪ .‬ﻭﺃﻥ ﻛ ﹰﹼﻼ ﻣﻦ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﺧﺎﻃﺌﺎﻥ‪.‬‬ ‫‪‬‬ ‫‪5‬‬ ‫‪ ‬ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﻣﻌﻠﻮﻣﺎﺕ ﺍﻟﺮﺑﻂ‬ ‫‪ 3‬‬ ‫✓ ‪‬‬ ‫ﻣﻊ ﺍﻟﺤﻴﺎﺓ؛ ﻟﺘﺤﺪﻳﺪ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻨﻬﺎ ﺻﺎﺋ ﹰﺒﺎ ﺃﻡ ﺧﺎﻃ ﹰﺌﺎ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪18-1‬؛ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻓﻬﻢ‬ ‫ﺍﻷﺳﻮﺩ ﻫﻲ ﻗﻄﻂ ﺗﺴﺘﻄﻴﻊ ﺃﻥ ﺗﺰﺃﺭ‪.‬‬ ‫ﺍﻟﻄﻠﺒﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺃﺳﻔﻞ ﺍﻟﺼﻔﺤﺔ‬ ‫ﺃﻋﺪ ﻛﺘﺎﺑﺔ ﺍﻟﻌﺒﺎﺭﺓ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪. (...‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪:‬‬ ‫ﺍﻟﺘﺎﻟﻴﺔ؛ ﻟﺘﻌﻴﻴﻦ ﺍﻟﻮﺍﺟﺒﺎﺕ ﺍﻟﻤﻨﺰﻟﻴﺔ ﻟﻠﻄﻠﺒﺔ‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﺃﺳ ﹰﺪﺍ‪ ،‬ﻓﺈﻧﻪ ﻗ ﱞﻂ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﺰﺃﺭ‪.‬‬ ‫ﺑﺤﺴﺐ ﻣﺴﺘﻮﻳﺎﺗﻬﻢ‪.‬‬ ‫ﺍﻋﺘﻤﺎ ﹰﺩﺍ ﻋﻠﻰ ﺍﻟﻤﻌﻠﻮﻣﺎﺕ ﺍﻟﻤﺠﺎﻭﺭﺓ ﻋﻦ ﺍﻟﻴﻤﻴﻦ‪ ،‬ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻗ ﹰﹼﻄﺎ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﺰﺃﺭ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﺃﺳ ﹰﺪﺍ‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪:‬‬ ‫‪‬‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ :‬ﺍﻟﻨﻤﺮ ﻗﻂ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﺰﺃﺭ‪ ،‬ﻟﻜﻨﻪ ﻟﻴﺲ ﺃﺳ ﹰﺪﺍ‪.‬‬ ‫‪‬‬ ‫ﺇﺫﻥ ﻓﺎﻟﻌﻜﺲ ﺧﺎﻃﻲﺀ‪.‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﺤﻴﻮﺍﻥ ﺃﺳ ﹰﺪﺍ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻜﻮﻥ ﻗ ﹰﹼﻄﺎ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﺰﺃﺭ‪.‬‬ ‫ﺍﻟﻤﻌﻜﻮﺱ‪:‬‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ :‬ﺍﻟﻨﻤﺮ ﻟﻴﺲ ﺃﺳ ﹰﺪﺍ‪ ،‬ﻭﻟﻜﻨﻪ ﻗﻂ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﺰﺃﺭ‪.‬‬ ‫‪ 5 ‬ﻳﺒﻴﻦ ﻛﻴﻔﻴﺔ ﻛﺘﺎﺑﺔ ﺍﻟﻌﻜﺲ‬ ‫ﺇﺫﻥ ﺍﻟﻤﻌﻜﻮﺱ ﺧﺎﻃﻲﺀ‪.‬‬ ‫ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ‬ ‫ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﺤﻴﻮﺍﻥ ﻗ ﹰﹼﻄﺎ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﺰﺃﺭ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻜﻮﻥ ﺃﺳ ﹰﺪﺍ‪.‬‬ ‫ﺍﻋﺘﻤﺎ ﹰﺩﺍ ﻋﻠﻰ ﺍﻟﻤﻌﻠﻮﻣﺎﺕ ﺍﻟﺘﻲ ﻓﻲ ﺍﻟﻬﺎﻣﺶ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺎﺋﺒﺔ ‪.‬‬ ‫ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫ﺗﺤﻘﻖ ﻣﻦ ﺃﻥ ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺘﻜﺎﻓﺌﺔ ﻣﻨﻄﻘ ﹼﹰﻴﺎ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻧﻔﺴﻬﺎ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ ﻛﻼﻫﻤﺎ ﺻﺎﺋﺐ‪ .‬‬ ‫‪‬‬ ‫ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻛﻼﻫﻤﺎ ﺧﺎﻃﻲﺀ‪ .‬‬ ‫✓ ‪‬‬ ‫‪ ‬ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ‬ ‫‪5‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﺸﺮﻃﻴﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪ ،‬ﺛﻢ ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ‬ ‫ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫ﻣﻨﻬﺎ ﺻﺎﺋ ﹰﺒﺎ ﺃﻡ ﺧﺎﻃ ﹰﺌﺎ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪ (5A, 5B .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﺍﻵﺗﻴﺔ‪ ،‬ﻭﺣﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪ ،‬ﻭﺇﺫﺍ‬ ‫‪ (5A‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻠﺘﺎﻥ ﻟﻬﻤﺎ ﺍﻟﻘﻴﺎﺱ ﻧﻔﺴﻪ ﻣﺘﻄﺎﺑﻘﺘﺎﻥ‪.‬‬ ‫ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ‬ ‫ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪ (5B‬ﺍﻟﻔﺄﺭ ﻣﻦ ﺍﻟﻘﻮﺍﺭﺽ‪.‬‬ ‫ﺍﻟﺨﻔﺎﻓﻴﺶ ﺛﺪﻳﻴﺎﺕ ﺗﺴﺘﻄﻴﻊ ﺍﻟﻄﻴﺮﺍﻥ‪.‬‬ ‫ﺣ ﱢﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪:‬‬ ‫✓ ‪ ‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ‬ ‫‪1 ‬‬ ‫ﺧﻔﺎ ﹰﺷﺎ‪ ،‬ﻓﺈﻧﻪ ﺛﺪﻳ ﱞﻲ ﻳﺴﺘﻄﻴﻊ ﺍﻟﻄﻴﺮﺍﻥ‪.‬‬ ‫‪ (1‬ﻳﻮﻡ ﻏﺪ ﻫﻮ ﺍﻟﺴﺒﺖ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺠﻤﻌﺔ‪ .‬ﺍﻟﻔﺮﺽ‪ :‬ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺠﻤﻌﺔ؛ ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻳﻮﻡ ﻏﺪ ﻫﻮ ﺍﻟﺴﺒﺖ‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ‬ ‫ﺍﻟﺜﺪﻳﻴﺎﺕ ﺍﻟﺘﻲ ﺗﺴﺘﻄﻴﻊ ﺍﻟﻄﻴﺮﺍﻥ‪ ،‬ﻓﺈﻧﻪ‬ ‫‪ (2‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،2x + 5 > 7‬ﻓﺈﻥ ‪ . x > 1‬ﺍﻟﻔﺮﺽ ‪2x + 5 > 7‬؛ ﺍﻟﻨﺘﻴﺠﺔ‪x > 1 :‬‬ ‫‪ (3‬ﺍﻟﻔﺮﺽ‪ :‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬ ‫ﻳﻜﻮﻥ ﺧﻔﺎ ﹰﺷﺎ‪ .‬ﺧﺎﻃﺌﺔ؛ ﻫﻨﺎﻙ ﺛﺪﻳﻴﺎﺕ‬ ‫ﻣﺘﻜﺎﻣﻠﺘﺎﻥ‪ .‬ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﻣﺠﻤﻮﻉ‬ ‫ﺃﺧﺮ￯ ﺗﺴﺘﻄﻴﻊ ﺍﻟﻄﻴﺮﺍﻥ ﻣﺜﻞ ﺍﻟﻠﻴﻤﻮﺭ‪.‬‬ ‫‪ (3‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻜﺎﻣﻠﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ‪180°‬‬ ‫ﻗﻴﺎﺳﻲ ﺍﻟﺰﺍﻭﻳﺘﻴﻦ ﻳﺴﺎﻭﻱ ‪.180°‬‬ ‫ﺍﻟﻤﻌﻜﻮﺱ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﺤﻴﻮﺍﻥ‬ ‫‪ (4‬ﻳﻜﻮﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻌﺎﻣﺪﻳﻦ ﺇﺫﺍ ﻧﺘﺞ ﻋﻦ ﺗﻘﺎﻃﻌﻬﻤﺎ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪ .‬ﺍﻟﻔﺮﺽ‪ :‬ﻧﺘﺞ ﻋﻦ ﺗﻘﺎﻃﻊ ﻣﺴﺘﻘﻴﻤﻴﻦ ﺯﺍﻭﻳﺔ‬ ‫ﺧﻔﺎ ﹰﺷﺎ‪ ،‬ﻓﺈﻧﻪ ﻟﻴﺲ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ ﺍﻟﺘﻲ‬ ‫ﻗﺎﺋﻤﺔ؛ ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻌﺎﻣﺪﺍﻥ‪.‬‬ ‫ﺗﺴﺘﻄﻴﻊ ﺍﻟﻄﻴﺮﺍﻥ‪ .‬ﺧﺎﻃﺌﺔ؛ ﺍﻟﻠﻴﻤﻮﺭ‬ ‫‪ 1 30‬‬ ‫ﻟﻴﺲ ﺧﻔﺎ ﹰﺷﺎ‪ ،‬ﻭﻫﻮ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ ﺍﻟﺘﻲ‬ ‫‪‬‬ ‫ﺗﺴﺘﻄﻴﻊ ﺍﻟﻄﻴﺮﺍﻥ‪.‬‬ ‫ﻳﻤﻜﻦ ﺃﻥ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺗﺎﻥ ‪ p‬ﻭ ‪ q‬ﺑﺴﻴﻄﺘﻴﻦ‪ ،‬ﻭﻟﻜﻦ ﻟﻴﺲ ﺑﺎﻟﻀﺮﻭﺭﺓ ﺃﻥ ﺗﻜﻮﻧﺎ ﻣﺮﺗﺒﻄﺘﻴﻦ ﻣ ﹰﻌﺎ‪ .‬ﻭﻟﻜﻦ ﻓﻲ ﻫﺬﺍ ﺍﻟﺪﺭﺱ ﺳﺘﻜﻮﻥ‬ ‫ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ‬ ‫ﺇﺣﺪ￯ ﺍﻟﺠﻤﻞ ﻫﻲ ﺍﻟﻔﺮﺽ‪ ،‬ﻭﺍﻷﺧﺮ￯ ﻫﻲ ﺍﻟﻨﺘﻴﺠﺔ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ .‬ﺗﺄﻛﺪ ﻣﻦ ﺃﻥ ﺍﻟﻄﻼﺏ ﻳﻌﻠﻤﻮﻥ ﺃﻥ ﻫﺎﺗﻴﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ‬ ‫ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ ﺍﻟﺘﻲ ﺗﺴﺘﻄﻴﻊ‬ ‫ﺍﻟﻄﻴﺮﺍﻥ‪ ،‬ﻓﺈﻧﻪ ﻟﻴﺲ ﺧﻔﺎ ﹰﺷﺎ‪ .‬ﺻﺎﺋﺒﺔ‬ ‫ﺍﻟﻤﻨﻔﺼﻠﺘﻴﻦ ﻣﺎ ﺯﺍﻟﺘﺎ ﺑﺴﻴﻄﺘﻴﻦ ﻭﻏﻴﺮ ﻣﺮﺗﺒﻄﺘﻴﻦ‪ ،‬ﻭﻟﻜﻨﻬﻤﺎ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺑﻴﻨﻬﻤﺎ ﻋﻼﻗﺔ ﺍﺭﺗﺒﺎﻁ‪ .‬ﻭﻗﺒﻞ ﺍﻟ ﹸﻤﻀ ﱢﻲ ﹸﻗﺪ ﹰﻣﺎ ﻳﺠﺐ‬ ‫ﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺍﻟﺘﻌﺎﻣﻞ ﺑﺎﺭﺗﻴﺎﺡ ﻓﻲ ﺗﺤﺪﻳﺪ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻭﻣﻌﺮﻓﺔ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺓ ﻋﻠﻰ ﺣﺪﺓ‪ ،‬ﻭﻣﻌﺮﻓﺔ ﻗﻴﻤﺔ‬ ‫ﺍﻟﺼﻮﺍﺏ ﻓﻲ ﺣﺎﻝ ﺍﻟﺮﺑﻂ ﺑﻮﺍﺳﻄﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﺍﻟﺼﻴﻎ ﺍﻟﻤﺨﺘﻠﻔﺔ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪.‬‬ ‫‪ (5B‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻓﺄ ﹰﺭﺍ ﻓﺈﻧﻪ ﻣﻦ ﺍﻟﻘﻮﺍﺭﺽ‪.‬‬ ‫‪‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﻘﻮﺍﺭﺽ ﻓﺈﻧﻪ ﻓﺄﺭ‪ .‬ﺧﺎﻃﺌﺔ‪ ،‬ﺍﻟﺴﻨﺠﺎﺏ ﻣﻦ‬ ‫‪ (5A‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﻟﻠﺰﺍﻭﻳﺘﻴﻦ ﺍﻟﻘﻴﺎﺱ ﻧﻔﺴﻪ ﻓﺈﻧﻬﻤﺎ ﻣﺘﻄﺎﺑﻘﺘﺎﻥ‪.‬‬ ‫ﺍﻟﻘﻮﺍﺭﺽ‪ ،‬ﻟﻜﻨﺔ ﻟﻴﺲ ﻓﺄ ﹰﺭﺍ‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻟﻬﻤﺎ ﺍﻟﻘﻴﺎﺱ ﻧﻔﺴﻪ‪ .‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻤﻌﻜﻮﺱ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﺤﻴﻮﺍﻥ ﻓﺄ ﹰﺭﺍ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻜﻮﻥ ﻣﻦ ﺍﻟﻘﻮﺍﺭﺽ‪ .‬ﺧﺎﻃﺌﺔ‪،‬‬ ‫ﺍﻟﻤﻌﻜﻮﺱ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﻟﺰﺍﻭﻳﺘﻴﻦ ﺍﻟﻘﻴﺎﺱ ﻧﻔﺴﻪ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻏﻴﺮ ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪.‬‬ ‫ﺍﻟﺴﻨﺠﺎﺏ ﻟﻴﺲ ﻓﺄ ﹰﺭﺍ‪ ،‬ﻭﻟﻜﻨﻪ ﻣﻦ ﺍﻟﻘﻮﺍﺭﺽ‪.‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﻘﻮﺍﺭﺽ‪ ،‬ﻓﺈﻧﻪ ﻟﻴﺲ ﻓﺄ ﹰﺭﺍ‪ .‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ :‬ﺇﺫﺍ ﻟﻢ ﺗﻜﻦ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻜﻮﻥ ﻟﻬﻤﺎ‬ ‫ﺍﻟﻘﻴﺎﺱ ﻧﻔﺴﻪ‪ .‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪ 1 30‬‬