Μαθηματικά Α' Δημοτικού - eBooks4Greeks.gr

ebooks4greeks.gr
  • No tags were found...

Μαθηματικά Α' Δημοτικού - eBooks4Greeks.gr

Y°°PAºEIKPITE-A•IO§O°HTE÷ڿϷÌappleÔ˜ §ÂÌÔÓ›‰Ë˜, ∫·ıËÁËÙ‹˜ ÙÔ˘ ·ÓÂappleÈÛÙËÌ›Ô˘ ¢˘ÙÈ΋˜ª·Î‰ÔÓ›·˜Aı·Ó¿ÛÈÔ˜ £ÂÔ‰ÒÚÔ˘, ∂Îapple·È‰Â˘ÙÈÎfi˜A¯ÈÏϤ·˜ K·„¿Ï˘, K·ıËÁËÙ‹˜ ÙÔ˘ ·ÓÂappleÈÛÙËÌ›Ô˘ M·Î‰ÔÓ›·˜¢ËÌ‹ÙÚÈÔ˜ ÓÂ˘Ì·ÙÈÎfi˜, §¤ÎÙÔÚ·˜ ÙÔ˘ ·ÓÂappleÈÛÙËÌ›Ô˘ ¢˘ÙÈ΋˜M·Î‰ÔÓ›·˜£ÂÔ‰fiÛÈÔ˜ Z·¯·ÚÈ¿‰Ë˜, AÓ·appleÏËÚˆÙ‹˜ K·ıËÁËÙ‹˜ ÙÔ˘ ·ÓÂappleÈÛÙËÌ›Ô˘AıËÓÒÓM·Ú›· KÔÙÛ·ÎÒÛÙ·, ¯ÔÏÈ΋ ‡Ì‚Ô˘ÏÔ˜£ÂfiÊÈÏÔ˜ T˙ÒÚÙ˙˘, ∂Îapple·È‰Â˘ÙÈÎfi˜EIKONO°PAºHHºI§O§O°IKH EIME§EIAYEY£YNO TOY MA£HMATOKATA TH Y°°PAºHYEY£YNH TOY YOEP°OYE•øºY§§OPOEKTYøTIKEEP°AIE∫ˆÓÛÙ·ÓÙ›ÓÔ˜ ∞ÚÒÓ˘, ÎÈÙÛÔÁÚ¿ÊÔ˜-∂ÈÎÔÓÔÁÚ¿ÊÔ˜ºÚfiÛˆ •ÈÍ‹, ºÈÏfiÏÔÁÔ˜°ÂÒÚÁÈÔ˜ ∆‡apple·˜, ªfiÓÈÌÔ˜ ¿Ú‰ÚÔ˜ ÙÔ˘ ·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘M·Ú›· XÈÔÓ›‰Ô˘-MÔÛÎÔÊfiÁÏÔ˘, Eapple›ÎÔ˘ÚÔ˜ K·ıËÁ‹ÙÚÈ·ÙÔ˘ ·ÓÂappleÈÛÙËÌ›Ô˘ AÈÁ·›Ô˘AÓ‰Ú¤·˜ °ÎÔÏÊÈÓfiappleÔ˘ÏÔ˜, ∂ÈηÛÙÈÎfi˜ ∫·ÏÏÈÙ¤¯Ó˘ACCESS °Ú·ÊÈΤ˜ T¤¯Ó˜ A.E.°ã ∫... / ∂∂∞∂∫ ππ / ∂Ó¤ÚÁÂÈ· 2.2.1. / ∫·ÙËÁÔÚ›· Ú¿ÍÂˆÓ 2.2.1.·:«∞Ó·ÌfiÚʈÛË ÙˆÓ appleÚÔÁÚ·ÌÌ¿ÙˆÓ ÛappleÔ˘‰ÒÓ Î·È Û˘ÁÁÚ·Ê‹ Ó¤ˆÓ ÂÎapple·È‰Â˘ÙÈÎÒÓ apple·Î¤ÙˆÓ»∞π¢∞°ø°π∫√ π¡∆π∆√À∆√ªÈ¯¿Ï˘ ∞Á. ·apple·‰fiappleÔ˘ÏÔ˜√ÌfiÙÈÌÔ˜ ∫·ıËÁËÙ‹˜ ÙÔ˘ ∞..£.Úfi‰ÚÔ˜ ÙÔ˘ ·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘Ú¿ÍË Ì ٛÙÏÔ:«˘ÁÁÚ·Ê‹ Ó¤ˆÓ ‚È‚Ï›ˆÓ Î·È apple·Ú·ÁˆÁ‹˘appleÔÛÙËÚÈÎÙÈÎÔ‡ ÂÎapple·È‰Â˘ÙÈÎÔ‡ ˘ÏÈÎÔ‡ Ì ‚¿ÛËÙÔ ¢∂ Î·È Ù· ∞ ÁÈ· ÙÔ ¢ËÌÔÙÈÎfi Î·È ÙÔ NËappleÈ·ÁˆÁ›Ի∂appleÈÛÙËÌÔÓÈÎfi˜ Àapple‡ı˘ÓÔ˜ ŒÚÁÔ˘°ÂÒÚÁÈÔ˜ ∆‡apple·˜MfiÓÈÌÔ˜ ¿Ú‰ÚÔ˜ ÙÔ˘ ·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘∞Ó·appleÏËÚˆÙ‹˜ ∂appleÈÛÙËÌÔÓÈÎfi˜ Àapple‡ı˘ÓÔ˜ ŒÚÁÔ˘°ÂÒÚÁÈÔ˜ √ÈÎÔÓfiÌÔ˘MfiÓÈÌÔ˜ ¿Ú‰ÚÔ˜ ÙÔ˘ ·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘ŒÚÁÔ Û˘Á¯ÚËÌ·ÙÔ‰ÔÙÔ‡ÌÂÓÔ 75% ·applefi ÙÔ ∂˘Úˆapple·˚Îfi ∫ÔÈÓˆÓÈÎfi ∆·ÌÂ›Ô Î·È 25% ·applefi ÂıÓÈÎÔ‡˜ applefiÚÔ˘˜.


À√Àƒ°∂π√ ∂£¡π∫∏ ∞π¢∂π∞ ∫∞𠣃∏∫∂Àª∞∆ø¡∞π¢∞°ø°π∫√ π¡∆π∆√À∆√÷ڿϷÌappleÔ˜ §ÂÌÔÓ›‰Ë˜ Aı·Ó¿ÛÈÔ˜ £ÂÔ‰ÒÚÔ˘ A¯ÈÏϤ·˜ K·„¿Ï˘¢ËÌ‹ÙÚÈÔ˜ ÓÂ˘Ì·ÙÈÎfi˜ANA¢OXO Y°°PAºH: ∂§§∏¡π∫∞ °ƒ∞ªª∞∆∞ ∞.∂.M·ıËÌ·ÙÈο Aã¢ËÌÔÙÈÎÔ‡M·ıËÌ·ÙÈο Ù˘ º‡Û˘ Î·È Ù˘ Zˆ‹˜¢EYTEPO TEYXOOP°ANIMO EK¢OEø ¢I¢AKTIKøN BIB§IøNA£HNA


¢ÔÌ‹ ÙÔ˘ ‚È‚Ï›Ô˘Xڈ̷ÙÈο ۇ̂ÔÏ·K¿ı ÎÂÊ¿Ï·ÈÔ, ·Ó¿ÏÔÁ· Ì ÙËıÂÌ·ÙÈ΋ appleÂÚÈÔ¯‹ ÛÙËÓ ÔappleÔ›··Ó·Ê¤ÚÂÙ·È, ¤¯ÂÈ ¤Ó· ¯ÚÒÌ·.OÈ appleÂÚÈÔ¯¤˜ Â›Ó·È ÔÈ ÂÍ‹˜:AÚÈıÌÔ›Ú¿ÍÂȘ°ÂˆÌÂÙÚ›·MÂÙÚ‹ÛÂȘÚÔ‚Ï‹Ì·Ù·‡Ì‚ÔÏÔ - ÎÏÂȉ› ÁÈ·ÙÔ Â›‰Ô˜ Ù˘ ÂÚÁ·Û›·˜appleÔ˘ ·ÎÔÏÔ˘ı› *AÚÈıÌfi˜ÎÂÊ·Ï·›Ô˘T›ÙÏÔ˜ÎÂÊ·Ï·›Ô˘Eapple·Ó¿ÏË„ËEÈÎÔÓ›‰È· (ۇ̂ÔÏ· ÎÏÂȉȿ)ÙËÓ Âapple¿Óˆ ·ÚÈÛÙÂÚ‹ ÁˆÓ›· οı ‰Ú·ÛÙËÚÈfiÙËÙ·˜˘apple¿Ú¯ÂÈ ¤Ó· ·applefi Ù· ·ÎfiÏÔ˘ı· ۇ̂ÔÏ·:O ˘ı·ÁfiÚ·˜ appleÔ˘ ÛΤÊÙÂÙ·È- ‡Ì‚ÔÏÔ ÛΤ„˘:: EÌÊ·Ó›˙ÂÙ·È Û‰ڷÛÙËÚÈfiÙËÙ˜ ÓÔÂÚÒÓ ˘appleÔÏÔÁÈÛÌÒÓ.H ̤ÏÈÛÛ· - ‡Ì‚ÔÏÔ ÂÚÁ·ÙÈÎfiÙËÙ·˜:EÌÊ·Ó›˙ÂÙ·È Û ‰Ú·ÛÙËÚÈfiÙËÙ˜ ÂÊ·ÚÌÔÁ‹˜Î·È ÂÌapple¤‰ˆÛ˘.O Û·ÏÔ˜ ȯÓËÏ¿Ù˘ - ‡Ì‚ÔÏÔ ·Ó·Î¿Ï˘„˘ :EÌÊ·Ó›˙ÂÙ·È ÛÙȘ ‰Ú·ÛÙËÚÈfiÙËÙ˜ appleÔ˘ÂÈÛ¿ÁÔ˘Ó ÙÔ˘˜ Ì·ıËÙ¤˜ ÛÙË Ó¤· ÁÓÒÛË.O ÂϤʷÓÙ·˜ - ‡Ì‚ÔÏÔ ÌÓ‹Ì˘:EÌÊ·Ó›˙ÂÙ·È ÛÙȘ ‰Ú·ÛÙËÚÈfiÙËÙ˜Âapple·Ó¿Ï˄˘.OÌ¿‰· Ì·ıËÙÒÓ - ‡Ì‚ÔÏÔ ÔÌ·‰ÈÎfiÙËÙ·˜:EÌÊ·Ó›˙ÂÙ·È Û ‰Ú·ÛÙËÚÈfiÙËÙ˜ appleÔ˘ ›ӷȉ˘Ó·Ùfi Ó· Á›ÓÔ˘Ó Û ÔÌ¿‰Â˜.4AÚÈıÌfi˜ ÛÂÏ›‰·˜


¢ÔÌ‹ ÙÔ˘ ‚È‚Ï›Ô˘AÚÈıÌfi˜‰Ú·ÛÙËÚÈfiÙËÙ·˜¢È‰·ÎÙÈÎÔ› ÛÙfi¯ÔÈÙÔ˘ ÎÂÊ·Ï·›Ô˘ËÌ›ˆÛË ÁÈ· ÙÔ ‰¿ÛηÏÔÛÙÔ˘˜ ÓÔÂÚÔ‡˜ ˘appleÔÏÔÁÈÛÌÔ‡˜5


OÈ ‹ÚˆÂ˜ ÙÔ˘ ‚È‚Ï›Ô˘ı·ÁfiÚ·˜ Ô ¿ÌÈÔ˜ (appleÂÚ›appleÔ˘ 600 apple.Ã.)O ˘ı·ÁfiÚ·˜ ‹Ù·Ó ¤Ó·˜ ÛappleÔ˘‰·›Ô˜ Ì·ıËÌ·ÙÈÎfi˜ Ù˘·Ú¯·ÈfiÙËÙ·˜ appleÔ˘ ÁÂÓÓ‹ıËΠÛÙË ¿ÌÔ. ÿ‰Ú˘Û ÌÈ· Û¯ÔÏ‹, ÙËÛ¯ÔÏ‹ ÙˆÓ ˘ı·ÁÔÚ›ˆÓ, ÔÈ ÔappleÔ›ÔÈ ÌÂÏÂÙÔ‡Û·Ó ÙË ÊÈÏÔÛÔÊ›·, Ù·Ì·ıËÌ·ÙÈο Î·È ÙȘ ÂappleÈÛً̘. ∂›¯Â ‰·ÛοÏÔ˘˜ ÌÂÁ¿ÏÔ˘˜ÛÔÊÔ‡˜ Ù˘ ·Ú¯·ÈfiÙËÙ·˜ Î·È Ù·Í›‰Â„ ÛÙËÓ ∞Û›· Î·È ÙËÓ ∞›Á˘appleÙÔfiappleÔ˘ ÌÂϤÙËÛ ÙËÓ ·ÈÁ˘appleÙȷ΋ ÊÈÏÔÛÔÊ›·, Ù· Ì·ıËÌ·ÙÈο, ÙËÓ·ÛÙÚÔÓÔÌ›· Î·È ÙËÓ È·ÙÚÈ΋.O ˘ı·ÁfiÚ·˜ ¤ÌÂÈÓ ÁÓˆÛÙfi˜ ˆ˜ Ô ¿ÓıÚˆappleÔ˜ appleÔ˘ ¤‚ÏÂappleÂapple·ÓÙÔ‡ ·ÚÈıÌÔ‡˜.O ˘ı·ÁfiÚ·˜H KÔÚ›Ó·6


˘OÈ ‹ÚˆÂ˜ ÙÔ˘ ‚È‚Ï›Ô˘Àapple·Ù›· Ë ∞ÏÂÍ·Ó‰ÚÈÓ‹ (370-415 Ì.Ã.)∏ Àapple·Ù›· ‹Ù·Ó Ë appleÚÒÙË Á˘Ó·›Î· Ì·ıËÌ·ÙÈÎfi˜ ÛÙËÓ πÛÙÔÚ›·. °ÂÓÓ‹ıËÎÂÛÙËÓ ∞ÏÂÍ¿Ó‰ÚÂÈ·.◊Ù·Ó ÎfiÚË ÙÔ˘ ÊÈÏfiÛÔÊÔ˘ £¤ˆÓ·, ‰È¢ı˘ÓÙ‹ ÙÔ˘ ·ÓÂappleÈÛÙËÌ›Ô˘ Ù˘∞ÏÂÍ¿Ó‰ÚÂÈ·˜. °È· ÙÔ ÏfiÁÔ ·˘Ùfi ›¯Â ÙËÓ Ù‡¯Ë Ó· ·appleÔÎÙ‹ÛÂÈ Ûapple¿ÓÈ·ÌfiÚʈÛË, Û ÌÈ· ÂappleÔ¯‹ appleÔ˘ Ë ı¤ÛË Ù˘ Á˘Ó·›Î·˜ ÛÙËÓ ÎÔÈÓˆÓ›· ‹Ù·ÓappleÔχ ‰È·ÊÔÚÂÙÈ΋ ·applefi fi,ÙÈ Û‹ÌÂÚ·. ˘Ó¤¯ÈÛ ÙȘ ÛappleÔ˘‰¤˜ Ù˘ ÛÙËÓ∞ı‹Ó· Î·È ÙË ƒÒÌË ÂÓÙ˘appleˆÛÈ¿˙ÔÓÙ·˜ fiÏÔ˘˜ fiÛÔÈ ÙËÓÛ˘Ó·Ó·ÛÙÚ¤ÊÔÓÙ·Ó Ì ÙÔ appleÓ‡̷, ÙË ÛÂÌÓfiÙËÙ·, ÙËÓ ÔÌÔÚÊÈ¿ Î·È ÙËÓ¢ÁψÙÙ›· Ù˘. ∂appleÈÛÙÚ¤ÊÔÓÙ·˜ ÛÙËÓ ∞ÏÂÍ¿Ó‰ÚÂÈ· appleÔχ Û‡ÓÙÔÌ··Ó·‰Â›¯ıËΠ۠ÌÂÁ¿ÏË ‰·ÛοϷ Ù˘ ÊÈÏÔÛÔÊ›·˜ Î·È ÙˆÓÌ·ıËÌ·ÙÈÎÒÓ.H Yapple·Ù›·O MÂϤÙ˘H B¿ÛˆH ÿÏÓÙ·7


ÂÚȯfiÌÂÓ·Xڈ̷ÙÈο ۇ̂ÔÏ·Eapple·Ó¿ÏË„ËAÚÈıÌÔ›Ú¿ÍÂȘ°ÂˆÌÂÙÚ›·MÂÙÚ‹ÛÂȘÚÔ‚Ï‹Ì·Ù·¢ÔÌ‹ ÙÔ˘ ‚È‚Ï›Ô˘ 4-5OÈ ‹ÚˆÂ˜ ÙÔ˘ ‚È‚Ï›Ô˘ 6-7ÂÚȯfiÌÂÓ· 8-98


ÂÚȯfiÌÂÓ·Bã ÂÚ›Ô‰Ô˜∞ÚÈıÌÔ›: √È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50 – ‡ÛÙËÌ· ·Ú›ıÌË-Û˘, ÌÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜.Ú¿ÍÂȘ: ∞Ê·ÈÚ¤ÛÂȘ Ì ·ÚÈıÌÔ‡˜ ̤¯ÚÈ ÙÔ 10 –∞ıÚÔ›ÛÌ·Ù· Ì appleÔÏÏÔ‡˜ fiÚÔ˘˜ – ÚÔÛı¤-ÛÂȘ Ì ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜.°ÂˆÌÂÙÚ›·: ÿڷÍË ÁÚ·ÌÌÒÓ – ∫›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈ-Ṳ̂ÓÔ ¯·ÚÙ› – °ÂˆÌÂÙÚÈο Û¯‹Ì·Ù·.ªÂÙÚ‹ÛÂȘ: ªÔÙ›‚· – √ ¯ÚfiÓÔ˜.EÓfiÙËÙ· 5Ë: √π ∞ƒπ£ª√π ª∂Ãπ ∆√ 50,ª√¡∞¢∂ ∫∞π ¢∂∫∞¢∂ – ∆∂∆ƒ∞°ø¡πª∂¡√ Ã∞ƒ∆π3334353637383940414243∫ÂÊ¿Ï·ÈÔ 33Ô:√ÚÁ¿ÓˆÛË Û˘ÏÏÔÁÒÓ – √È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50 12-13∫ÂÊ¿Ï·ÈÔ 34Ô:ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (π) 14-15∫ÂÊ¿Ï·ÈÔ 35Ô:∞ıÚÔ›ÛÌ·Ù· Ì appleÔÏÏÔ‡˜ fiÚÔ˘˜ 16-17∫ÂÊ¿Ï·ÈÔ 36Ô:∫›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ› 18-19∫ÂÊ¿Ï·ÈÔ 37Ô:ÚÔ‚Ï‹Ì·Ù· 20-21∫ÂÊ¿Ï·ÈÔ 38Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ· 22-23EÓfiÙËÙ· 6Ë:ª√¡∞¢∂ ∫∞π ¢∂∫∞¢∂ – °∂øª∂∆ƒπ∫∞ Ã∏ª∞∆∞ – Ã√¡√∫ÂÊ¿Ï·ÈÔ 39Ô:ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (ππ) 26-27∫ÂÊ¿Ï·ÈÔ 40fi:°ÂˆÌÂÙÚÈο Û¯‹Ì·Ù· 28-29∫ÂÊ¿Ï·ÈÔ 41Ô:√ ¯ÚfiÓÔ˜ 30-31∫ÂÊ¿Ï·ÈÔ 42Ô:ÚÔÛı¤ÛÂȘ Ì ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ 32-33∫ÂÊ¿Ï·ÈÔ 43Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ· 34-35EÓfiÙËÙ· 7Ë: Ã∞ƒ∞•∂π KAI ∞∑§ – ƒ√£∂∏∫∞π ∞º∞πƒ∂∏ – ∏ À∂ƒµ∞∏ ∆∏ ¢∂∫∞¢∞45464748495051∫ÂÊ¿Ï·ÈÔ 45Ô:÷ڿÍÂȘ, apple·˙Ï Î·È ÌˆÛ·˚ο 38-39∫ÂÊ¿Ï·ÈÔ 46Ô:ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ‰È„‹ÊÈˆÓ Î·È ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ 40-41∫ÂÊ¿Ï·ÈÔ 47Ô:∏ appleÚfiÛıÂÛË Î·È Ë ·Ê·›ÚÂÛË ˆ˜·ÓÙ›ÛÙÚÔʘ appleÚ¿ÍÂȘ – ∏ ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ 42-43∫ÂÊ¿Ï·ÈÔ 48Ô:ÀappleÔÏÔÁÈÛÌÔ› – ∂appleÈÛÙÚÔÊ‹ ÛÙËÓ appleÂÓÙ¿‰· 44-45∫ÂÊ¿Ï·ÈÔ 49Ô:ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË –¢È„‹ÊÈÔÈ Î·È ÌÔÓÔ„‹ÊÈÔÈ ·ÚÈıÌÔ› 46-47∫ÂÊ¿Ï·ÈÔ 50fi:ÚÔ‚Ï‹Ì·Ù· 48-49∫ÂÊ¿Ï·ÈÔ 51Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ· 50-51EÓfiÙËÙ· 8Ë: √π ∞ƒπ£ª√π ª∂Ãπ ∆√ 70 – ƒ∞•∂π– ª∂∆ƒ∏∏ – Àªª∂∆ƒπ∞525354555657∫ÂÊ¿Ï·ÈÔ 52Ô:√È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 70 54-55∫ÂÊ¿Ï·ÈÔ 53Ô:∂ÈÛ·ÁˆÁ‹ ÛÙÔÓ appleÔÏÏ·appleÏ·ÛÈ·ÛÌfi 56-57∫ÂÊ¿Ï·ÈÔ 54Ô:ª¤ÙÚËÛË ÌÂÁÂıÒÓ 58-59∫ÂÊ¿Ï·ÈÔ 55Ô:ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË ‰È„‹ÊÈˆÓ ·ÚÈıÌÒÓ 60-61∫ÂÊ¿Ï·ÈÔ 56Ô:∂ÈÛ·ÁˆÁ‹ ÛÙË Û˘ÌÌÂÙÚ›· 62-63∫ÂÊ¿Ï·ÈÔ 57Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ· 64-65EÓfiÙËÙ· 9Ë: √π ∞ƒπ£ª√π ª∂Ãπ ∆√ 100 – ƒ∞•∂π– µ∞ƒ√ – °∂øª∂∆ƒπ∫∞ Ã∏ª∞∆∞°ã ÂÚ›Ô‰Ô˜∞ÚÈıÌÔ›: √È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 100.Ú¿ÍÂȘ: ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ‰È„‹ÊÈˆÓ Î·ÈÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ – ÚÔÛı¤ÛÂȘ ηȷʷÈÚ¤ÛÂȘ Ì ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ –ÔÏÏ·appleÏ·ÛÈ·ÛÌfi˜.°ÂˆÌÂÙÚ›·: ÷ڿÍÂȘ, apple·˙Ï, appleÏ·ÎfiÛÙÚˆÙÔ Î·È ÌˆÛ·˚ο– °ÂˆÌÂÙÚÈο Û¯‹Ì·Ù· – ˘ÌÌÂÙÚ›·.ªÂÙÚ‹ÛÂȘ: ª¤ÙÚËÛË Û˘Ó¯ÒÓ ÌÂÁÂıÒÓ – µ¿ÚÔ˜ –¡ÔÌ›ÛÌ·Ù·.585960616263∫ÂÊ¿Ï·ÈÔ 58Ô:√È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 100 – ÃÚ‹Ì· 68-69∫ÂÊ¿Ï·ÈÔ 59Ô:ÔÏÏ·appleÏ·ÛÈ·ÛÌfi˜ Î·È ‰È·›ÚÂÛË 70-71∫ÂÊ¿Ï·ÈÔ 60fi:µ¿ÚÔ˜ – §ÂÈÙÔ˘ÚÁ›· ˙˘Á·ÚÈ¿˜ 72-73∫ÂÊ¿Ï·ÈÔ 61Ô:÷ڿÍÂȘ Û¯ËÌ¿ÙˆÓ – ·˙Ï Î·È appleÏ·ÎfiÛÙÚˆÙÔ 74-75∫ÂÊ¿Ï·ÈÔ 62Ô:ÚÔ‚Ï‹Ì·Ù· 76-77∫ÂÊ¿Ï·ÈÔ 63Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ· 78-799


EÓfiÙËÙ· 5Ë:API£MOI MEXPI TO 50, MONA¢E KAI¢EKA¢E – TETPA°øNIMENO XAPTI6538-72x15842333435363738∫ÂÊ¿Ï·ÈÔ 33Ô:√ÚÁ¿ÓˆÛË Û˘ÏÏÔÁÒÓ– AÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50∫ÂÊ¿Ï·ÈÔ 34Ô:ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (π)∫ÂÊ¿Ï·ÈÔ 35Ô:∞ıÚÔ›ÛÌ·Ù· Ì appleÔÏÏÔ‡˜ fiÚÔ˘˜∫ÂÊ¿Ï·ÈÔ 36Ô:∫›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ›∫ÂÊ¿Ï·ÈÔ 37Ô:ÚÔ‚Ï‹Ì·Ù·∫ÂÊ¿Ï·ÈÔ 38Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·EÓfiÙËÙ· 5ËÙ· ∫ÂÊ¿Ï·È· 33 Î·È 34 ı· ¯ÚËÛÈÌÔappleÔÈ‹ÛÔ˘-Ì ÙÔ ·ÚÈıÌËÙ‹ÚÈÔ, Ù· ˙¿ÚÈ·, Ù· ÓÔÌ›ÛÌ·Ù· ηȿÏÏ· ˘ÏÈο Ù¤ÙÔÈ·, appleÔ˘ apple·ÚÔ˘ÛÈ¿˙Ô˘Ó ÙÔ˘˜·ÚÈıÌÔ‡˜ Ì ÔÚÁ·ÓˆÌ¤ÓË ‰ÔÌ‹ Ì ‚¿ÛË ÙˉÂο‰· Î·È ÙËÓ appleÂÓÙ¿‰·. ÙÔ ∫ÂÊ¿Ï·ÈÔ 35 ı·apple·›ÍÔ˘Ì ·Ú¯Èο ÛÙËÓ Ù¿ÍË ÙÔ apple·È¯Ó›‰È «O·ÚÈıÌfi˜-ÛÙfi¯Ô˜» appleÚÔÎÂÈ̤ÓÔ˘ Ó· ·ÛÎËıÔ‡-Ì Û ·ıÚÔ›ÛÌ·Ù· Ì appleÂÚÈÛÛfiÙÂÚÔ˘˜ ·applefi ‰‡ÔfiÚÔ˘˜.∫·ÙfiappleÈÓ ÛÙÔ ∫ÂÊ¿Ï·ÈÔ 36, appleÚÔÎÂÈ̤ÓÔ˘ Ó·ÂÍÔÈÎÂȈıԇ̠̠ÙËÓ Î›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈ-Ṳ̂ÓÔ ¯·ÚÙ›, ı· apple·›ÍÔ˘Ì ̤۷ ÛÙËÓ Ù¿ÍË ÙÔapple·È¯Ó›‰È Ì ÙÔ ı¤·ÙÚÔ. ÙÔ ÂapplefiÌÂÓÔ ∫ÂÊ¿Ï·ÈÔı· ·Û¯ÔÏËıԇ̠̠appleÚÔ‚Ï‹Ì·Ù·.fiÛÔÈ Â›Ó·È ÔÈ Ì·ıËÙ¤˜ ÛÙËÓ Ù¿ÍË ÌÔ˘123451Ë2Ë3Ë4Ë5Ë1 2 31 2 31 2 31 2 31 2 3


33OÚÁ¿ÓˆÛË Û˘ÏÏÔÁÒÓ –∞ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50O Ù·Ì›·˜∞ÓÙ·ÏÏ¿ÛÛÔ˘Ì ‰¤Î· appleÚ¿ÛÈÓ˜ Ì¿ÚΘ Ì ̛· ÎfiÎÎÈÓË.1◗ fiÛ˘ ·Í›·˜ Ì¿ÚΘ ¤¯ÂÈ Ô ˘ı·ÁfiÚ·˜; ......◗ fiÛ˘ ·Í›·˜ Ì¿ÚΘ ¤¯ÂÈ Ë Àapple·Ù›·; ......◗ ÔÈo˜ ¤¯ÂÈ appleÂÚÈÛÛfiÙÂÚ˘ ·Í›·˜ Ì¿ÚΘ; ......ªÂÙÚÔ‡ÌÂ Î·È ·ÓÙ·ÏÏ¿ÛÛÔ˘Ì ÙȘ Ì¿ÚΘ.∞ÓÙ·ÏÏ¿ÛÛÔ˘Ì ‰¤Î· appleÚ¿ÛÈÓ˜ Ì¿ÚΘ Ì ̛· ÎfiÎÎÈÓË.AappleÔÙ¤ÏÂÛÌ·2 5AappleÔÙ¤ÏÂÛÌ·AappleÔÙ¤ÏÂÛÌ·12OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ·ÓÙÈηٿÛÙ·ÛË ‰¤Î· ·ÓÙÈÎÂÈÌ¤ÓˆÓ Ì ¤Ó· ·ÓÙÈΛÌÂÓÔ ›Û˘ ·Í›·˜.


ÂÓfiÙËÙ·5Ë10ªÂÙÚÒ Ì¤¯ÚÈ ÙÔ 50.2°Ú¿Êˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜, fiappleˆ˜ Ê·›ÓÂÙ·È ÛÙÔ apple·Ú¿‰ÂÈÁÌ·.31520›ÎÔÛÈ˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜.›ÎÔÛÈ ÙÚ›·ÙÚÈ¿ÓÙ·4ÙÚÈ¿ÓÙ· apple¤ÓÙÂÛ·Ú¿ÓÙ·Û·Ú¿ÓÙ· Ù¤ÛÛÂÚ·appleÂÓ‹ÓÙ·appleÂÓ‹ÓÙ· apple¤ÓÙÂappleÂÓ‹ÓÙ· ÂÓÓ¤·10 + 3 = ...ÀappleÔÏÔÁ›˙ˆ Ù· ·ıÚÔ›ÛÌ·Ù·.10 + 8 = ...10 + 10 + 7 = ... 10 + 10 + 10 + 5 = ...52. OÈ Ì·ıËÙ¤˜ ÛÙËÓ ·Ú¯‹ ·ÚÈıÌÔ‡Ó appleÚÔÊÔÚÈο ·Ó¿ 1 ̤¯ÚÈ ÙÔ 50.∑ËÙԇ̠Âapple›Û˘ ·applefi Ù· apple·È‰È¿ Ó· ·ÚÈıÌ‹ÛÔ˘Ó ·Ó¿ 10 ̤¯ÚÈ ÙÔ 50Î·È ÛÙË Û˘Ó¤¯ÂÈ· Ó· ÁÚ¿„Ô˘Ó Ì¤Û· ÛÙ· ΢ÎÏÈο appleÏ·›ÛÈ· ÙÔ˘˜ apple¤ÓÙ ·ÚÈıÌÔ‡˜.13


34ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (I)OÈ ¿‚·Î˜1∫ÈÓ¤˙ÈÎÔ˜ ¿‚·Î·˜.O «¿‚·Í Ù˘ ·Ï·Ì›ÓÔ˜» ¯ÚÔÓÔÏÔÁÂ›Ù·È ·applefi ÙÔÓ 5Ô ‹ ÙÔÓ6Ô ·ÈÒÓ· apple.Ã. ∂›Ó·È Ô apple·Ï·ÈfiÙÂÚÔ˜ ¿‚·Î·˜ Î·È ÊÈÏÔÍÂÓ›ٷÈÛÙÔ ∂ıÓÈÎfi ∞Ú¯·ÈÔÏÔÁÈÎfi ªÔ˘ÛÂ›Ô ÙˆÓ ∞ıËÓÒÓ.Ì·ı·›ÓˆÂ ¤Ó·Ó ·ÚÈıÌfi Ì ‰‡Ô „ËÊ›· – ÁÈ· apple·Ú¿‰ÂÈÁÌ· ÙÔ 26 – ÙÔ„ËÊ›Ô ·applefi Ù· ‰ÂÍÈ¿ (2 6) ‰Â›¯ÓÂÈ ÙȘ ÌÔÓ¿‰Â˜ Î·È ÙÔ „ËʛԷapplefi Ù· ·ÚÈÛÙÂÚ¿ (2 6) ÙȘ ‰Âο‰Â˜.¢M2 6¢Âο‰Â˜MÔÓ¿‰Â˜2 6Œ¯Ô˘Ì 26 Ì¿ÚΘ. AÓÙ·ÏÏ¿ÛÛÔ˘Ì 10 Ì ̛·.2‰Âο‰Â˜6ÌÔÓ¿‰Â˜14∂ÈÛ¿ÁÔ˘ÌÂ Î·È ·ÛÎԇ̠ÙÔ˘˜ Ì·ıËÙ¤˜ ÛÙȘ ¤ÓÓÔȘ ÙˆÓ ‰Âο‰ˆÓ Î·È ÙˆÓ ÌÔÓ¿‰ˆÓ.


ÂÓfiÙËÙ·5Ë∏ ΢ڷ-ª·ÚÈÒ, Ë appleÔÓËÚ‹ ·ÏÂappleÔ‡, ¯ÚˆÛÙ¿ 14 ·‚Á¿ÛÙËÓ Î˘Ú·-apple¿appleÈ· Î·È ı¤ÏÂÈ Ó· Ù· ÂappleÈÛÙÚ¤„ÂÈ.2£· ÛÔ˘ ‰ÒÛˆfiÛ· ϤÂÈ Ô ·ÚÈıÌfi˜ 14. £· ÛÔ˘ ‰ÒÛˆ‰ËÏ·‰‹ 1 Î·È 4.∏ ·ÏÂappleÔ‡ ¤‰ˆÛ ÛÙËÓ Î˘Ú·-apple¿appleÈ· fiÛ··‚Á¿ Ù˘ ¯ÚˆÛÙÔ‡Û ‹ ÙËÓ ÍÂÁ¤Ï·ÛÂ;µÚ›ÛΈ ÙȘ ‰Âο‰Â˜ Î·È ÙȘ ÌÔÓ¿‰Â˜. ˘ÌappleÏËÚÒÓˆ ÙȘ ÈÛfiÙËÙ˜.3ñ ∆Ô 27 ¤¯ÂÈ ... ‰Âο‰Â˜ Î·È ... ÌÔÓ¿‰Â˜.27 = 10 + 10 + 7ñ∆Ô 14 ¤¯ÂÈ ............................ 14 = .....................ñ∆Ô 22 ¤¯ÂÈ ............................ 22 =.....................ñ∆Ô 36 ¤¯ÂÈ ............................ 36 =.....................ñ∆Ô 44 ¤¯ÂÈ ............................ 44 =.....................15


35∞ıÚÔ›ÛÌ·Ù· Ì appleÔÏÏÔ‡˜ fiÚÔ˘˜O ·ÚÈıÌfi˜-ÛÙfi¯Ô˜°È· Ó· ÎÂÚ‰›ÛÂÈ Î¿appleÔÈÔ˜, appleÚ¤appleÂÈ Ó· Û¯ËÌ·Ù›ÛÂÈÙÔÓ ·ÚÈıÌfi 10 ‰È·Ï¤ÁÔÓÙ·˜ ÙÚÂȘ οÚÙ˜.10 1 2 34 5 61ÔÈÔ˜ ΤډÈÛÂ;ŒÏÏË Mapple¿Ìapple˘ ÿÏÓÙ·5 2 14 2 4 6 2 15 + 2 + 1 = ... 4 + ... + ... = ... ... + ... + ... = ...+521...4+ 24...6+ 21...16OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙÔÓ ˘appleÔÏÔÁÈÛÌfi ·ıÚÔÈÛÌ¿ÙˆÓ Ì appleÂÚÈÛÛfiÙÂÚÔ˘˜ ·applefi ‰‡Ô fiÚÔ˘˜.


ÂÓfiÙËÙ·5ËÀappleÔÏÔÁ›˙ˆ ·ıÚÔ›ÛÌ·Ù· Ì ÙÚÂȘ appleÚÔÛıÂÙ¤Ô˘˜.2 ·˘Ùfi ÙÔ apple·È¯Ó›‰È ΤډÈÛ·Ó Î·È Ù· ÙÚ›· apple·È‰È¿.µÚ›ÛΈ appleÔÈ· ‹Ù·Ó Ë ÙÚ›ÙË Î¿ÚÙ· Î·È Û˘ÌappleÏËÚÒÓˆ ÙȘ ÈÛfiÙËÙ˜.ŒÏÏË Mapple¿Ìapple˘ ÿÏÓÙ·35 2 3 2 1 55 + 2 + ... = 10 3 + 2 + ... = ... ... + ... + ... = ...˘ÌappleÏËÚÒÓˆ ÙȘ ÙÚÂȘ οÚÙ˜, ÁÈ· Ó· ¤¯ˆ ¿ıÚÔÈÛÌ· ›ÛÔ Ì 9.493 22 4542. ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· ̤¯ÚÈ ÙÔ 10 Ì ÙÚÂȘ appleÚÔÛıÂÙ¤Ô˘˜, ·applefi ÙÔ˘˜ ÔappleÔ›Ô˘˜ Ô appleÚÒÙÔ˜ appleÚÔÛıÂÙ¤Ô˜Â›Ó·È ÌÂÁ¿ÏÔ˜ ·ÚÈıÌfi˜ Î·È ÔÈ ‰‡Ô ˘applefiÏÔÈappleÔÈ Â›Ó·È ÙÔ 1 ‹ ÙÔ 2 (apple.¯. 5 + 1 + 1 Î.Ïapple.).17


36K›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ›ÙÔ ı¤·ÙÚÔ˘˙ËÙԇ̠ÁÈ· ÙÔÓ ÙÚfiappleÔ Ì ÙÔÓ ÔappleÔ›Ô Î·ıfiÌ·ÛÙ ÛÙÔ ı¤·ÙÚÔ.11. Xڈ̷ٛ˙ˆ ÙȘ ı¤ÛÂȘappleÔ˘ ‰Â›¯ÓÂÈ ÙÔ ÂÈÛÈÙ‹ÚÈÔ.EIITHPIOÕÙÔÌ·: 2ÂÈÚ¿: 3Ë£¤ÛÂȘ: 1, 22. EappleÈϤÁˆ ÌÈ· ı¤ÛË ÛÙÔ ‰ÈappleÏ·Ófi Û¯¤‰ÈÔÎ·È Áڿʈ ÙÔ˘˜ ·ÓÙ›ÛÙÔÈ¯Ô˘˜ ·ÚÈıÌÔ‡˜ÛÙÔ ÂÈÛÈÙ‹ÚÈÔ.EIITHPIOÕÙÔÌ·: 2ÂÈÚ¿: ---------£¤ÛÂȘ: ---------KHNH1Ë 1 2 32Ë 1 2 33Ë 1 2 34Ë 1 2 35Ë 1 2 318OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ·applefiÎÙËÛË Ù˘ ÈηÓfiÙËÙ·˜ Ó· appleÚÔÛ‰ÈÔÚ›˙Ô˘ÓÙȘ ı¤ÛÂȘ Î·È Ó· ÎÈÓÔ‡ÓÙ·È Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ›.


ÂÓfiÙËÙ·5Ë12345EÓÙÔapple›˙ˆ Î·È appleÂÚÈÁڿʈ ÙȘ ı¤ÛÂȘ ÙˆÓ ˙ÒˆÓ.12345¯Â‰È¿˙ˆ ‰ÂÍÈ¿ Ù· ›‰È· ·ÓÙÈΛÌÂÓ· Î·È ÛÙȘ ›‰È˜ ı¤ÛÂȘ.12345BÚ›ÛΈ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔÓ ·ÚÈıÌfi ÙˆÓ ‚ËÌ¿ÙˆÓ.2342 appleÚÔ˜ Ù· Âapple¿ÓˆappleÚÔ˜ Ù· ‰ÂÍÈ¿appleÚÔ˜ Ù· Âapple¿ÓˆappleÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿appleÚÔ˜ Ù· οو19


37appleÚÔ‚Ï‹Ì·Ù· appleÚÔ‚Ï‹Ì·Ù·123 x 4567+896538-ÚÔ‚Ï‹Ì·Ù·∫¿Óˆ ÂÚˆÙ‹ÛÂȘ˘Ó‰¤ˆ ÙȘ ÂÚˆÙ‹ÛÂȘ Ì ÙȘ ÂÈÎfiÓ˜.1ÔÈÔ˜ ¤¯ÂÈ ÙȘ appleÂÚÈÛÛfiÙÂÚ˜ Ìapple›ÏȘ;ÔÈ· ̤ڷ ‰Â›¯ÓÂÈ ÙÔ ËÌÂÚÔÏfiÁÈÔ;fiÛ· ‚È‚Ï›· ÌÂٷʤÚÂÈ Ô µ·ÁÁ¤Ï˘;N›ÎÔ˜B·ÁÁ¤Ï˘∞apple·ÓÙÒ ÛÙȘ ÂÚˆÙ‹ÛÂȘ.°È¿ÓÓ˘◗ fiÛ· ‚È‚Ï›· ÌÂٷʤÚÂÈ Ô µ·ÁÁ¤Ï˘; ..........................................................◗ ÔÈ· ̤ڷ ‰Â›¯ÓÂÈ ÙÔ ËÌÂÚÔÏfiÁÈÔ; ..............................................................◗ ÔÈÔ˜ ¤¯ÂÈ ÙȘ appleÂÚÈÛÛfiÙÂÚ˜Ìapple›ÏȘ; ........................................................................................................ª·˙› Ì ÙÔ˘˜ Û˘ÌÌ·ıËÙ¤˜ ÌÔ˘ ‚Ú›ÛΈ Î·È ¿ÏϘ ÂÚˆÙ‹ÛÂȘ.20


72x15ÂÓfiÙËÙ·5£¤ÏÔ˘Ì ӷ ‚¿ÏÔ˘Ì ̤۷ ÛÙ·Î˘ÎÏÈο appleÏ·›ÛÈ· ÙÔ˘˜ ·ÚÈıÌÔ‡˜ 1, 2, 3Î·È 4.  οı ÌÈÎÚfi ·ÎÏÔ ‚¿˙Ô˘ÌÂ¤Ó·Ó ‰È·ÊÔÚÂÙÈÎfi ·ÚÈıÌfi.∆ÔappleÔıÂÙԇ̠ÙÔ˘˜ ·ÚÈıÌÔ‡˜ ÌÂÙ¤ÙÔÈÔÓ ÙÚfiappleÔ ÒÛÙÂ, fiÙ·ÓappleÚÔÛı¤ÙÔ˘Ì οıÂÙ·, ÔÚÈ˙fiÓÙÈ· ηÈ΢ÎÏÈο, Ó· ‚Ú›ÛÎÔ˘Ì ÙÔ ¿ıÚÔÈÛÌ· 10.5Ë2ŒÏÏË Mapple¿Ìapple˘ M·Ú›·3∏ ŒÏÏË ¤¯ÂÈ ...... ÏÂappleÙ¿.O ªapple¿Ìapple˘ ¤¯ÂÈ ...... ÏÂappleÙ¿.∏ ª·Ú›· ¤¯ÂÈ ...... ÏÂappleÙ¿.7 ÏÂappleÙ¿◗ ÔÈÔ˜ ÌappleÔÚ› Ó· ·ÁÔÚ¿ÛÂÈ ÙÔ Ìapple·ÏfiÓÈ; ......................................................◗ fiÛ· ÏÂappleÙ¿ ¤¯Ô˘Ó Ô Mapple¿Ìapple˘ Î·È Ë ŒÏÏË Ì·˙›; ........................................◗ fiÛ· Ú¤ÛÙ· ı· apple¿ÚÂÈ Ë M·Ú›·; ..................................................................◗ fiÛ· ÏÂappleÙ¿ ¯ÚÂÈ¿˙ÂÙ·È ·ÎfiÌË Ë ŒÏÏË,ÁÈ· Ó· ·ÁÔÚ¿ÛÂÈ ÙÔ Ìapple·ÏfiÓÈ; .......................................................................21


38Âapple·Ó·ÏËappleÙÈÎfiÌ¿ıËÌ·123 x 4567+896538-Âapple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·∫ÈÓÔ‡Ì·È Ì appleÚÔÛÔ¯‹ ÁÈ· Ó· ÌË Ì ʿÂÈ Ô ¿ÎÌ·Ó.1appleÚÔ˜ Ù· Âapple¿ÓˆappleÚÔ˜ Ù· ‰ÂÍÈ¿appleÚÔ˜ Ù· Âapple¿ÓˆT¤ÏÔ˜appleÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿appleÚÔ˜ Ù· οوAÚ¯‹◗ ∞appleoʇÁˆ ÙÔÓ ¿ÎÌ·Ó; ˘ÌappleÏËÚÒÓˆ ÙË ‰È·‰ÚÔÌ‹.◗ ∫ÂÚ‰›˙ˆ ‹ ¯¿Óˆ;....................................................................................µ¿˙ˆ Û ·ÎÏÔ Ù· ÓÔÌ›ÛÌ·Ù· appleÔ˘ ¯ÚÂÈ¿˙ÔÓÙ·ÈÁÈ· Ó· ·ÁÔÚ¿Ûˆ ÙÔ apple·È¯Ó›‰È.29 ú22


72x15ÂÓfiÙËÙ·5˵ڛÛΈ Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi.3°Ú¿Êˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ì ϤÍÂȘ Î·È Û˘ÌappleÏËÚÒÓˆ Ù· ·ıÚÔ›ÛÌ·Ù·.23 ............................................ ›ÎÔÛÈ ÙÚ›·.....................................................20 + 3=12 .............................................. = .....................................................46 ............................................ = .....................................................11 .............................................. = .....................................................50 ............................................ = .....................................................¯ËÌ·Ù›˙ˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ ÛÙÔ˘˜ ¿‚·Î˜.4510 + 10 + 4 30 + 3 10 + 10 + 10 50 + 33. ∏ ‰·ÛοϷ ϤÂÈ appleÚÔÊÔÚÈο ÛÙÔ˘˜ Ì·ıËÙ¤˜ ÙȘ ‰Âο‰Â˜ Î·È ÙȘ ÌÔÓ¿‰Â˜ ÂÓfi˜ ·ÚÈıÌÔ‡Î·È ÔÈ Ì·ıËÙ¤˜ ‚Ú›ÛÎÔ˘Ó appleÔÈÔ˜ Â›Ó·È Ô ·ÚÈıÌfi˜ Î·È ÙÔÓ ÁÚ¿ÊÔ˘Ó ÛÙÔ appleÏ·›ÛÈÔ.23


123 x 4567+89


EÓfiÙËÙ· 6Ë:MONA¢E KAI ¢EKA¢E –°EøMETPIKA XHMATA – XPONO6538-72x15842394041424344∫ÂÊ¿Ï·ÈÔ 39Ô:ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (ππ)∫ÂÊ¿Ï·ÈÔ 40fi:°ÂˆÌÂÙÚÈο Û¯‹Ì·Ù·∫ÂÊ¿Ï·ÈÔ 41Ô:√ ¯ÚfiÓÔ˜∫ÂÊ¿Ï·ÈÔ 42Ô:ÚÔÛı¤ÛÂȘ Ì ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜∫ÂÊ¿Ï·ÈÔ 43Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·∫ÂÊ¿Ï·ÈÔ 44Ô:2Ô ∫ÚÈÙ‹ÚÈÔ AÍÈÔÏfiÁËÛ˘EÓfiÙËÙ· 6˪¤Û· ·applefi ÙÔ apple·È¯Ó›‰È «O Ù·Ì›·˜» ηÈÙȘ ·ÓÙ·ÏÏ·Á¤˜ ÓÔÌÈÛÌ¿ÙˆÓ ÛÙÔ ∫ÂÊ¿-Ï·ÈÔ 39 ı· ÂÍÂÙ¿ÛÔ˘ÌÂ Î·È apple¿ÏÈÙȘ ÌÔÓ¿‰Â˜ Î·È ÙȘ ‰Âο‰Â˜.ÙÔ ∫ÂÊ¿Ï·ÈÔ 40 ı· ÂÍÂÙ¿ÛÔ˘Ì ٷۯ‹Ì·Ù· Î·È ı· Û˘Óı¤ÛÔ˘Ì ÂÈÎfiÓ˜appleÚÔÛÒappleˆÓ appleÔ˘ Â›Ó·È Û¯Â‰È·Ṳ̂Ó˜ ÌÂÁˆÌÂÙÚÈο Û¯‹Ì·Ù·.¯ÂÙÈο Ì ÙËÓ ¤ÓÓÔÈ· ÙÔ˘ ¯ÚfiÓÔ˘ ÛÙÔ∫ÂÊ¿Ï·ÈÔ 41 ı· ‚¿ÏÔ˘Ì Û ¯ÚÔÓÔ-ÏÔÁÈ΋ ÛÂÈÚ¿ ʈÙÔÁڷʛ˜ appleÔ˘ apple·-ÚÔ˘ÛÈ¿˙Ô˘Ó ÁÂÁÔÓfiÙ· ·applefi ÙËÓ Î·ıË-ÌÂÚÈÓ‹ ˙ˆ‹, ı· ‰È·‚¿ÛÔ˘Ì ËÌÂÚÔÏfi-ÁÈ· Î·È ı· Ì¿ıÔ˘Ì ӷ ϤÌ ÛÙË ÛÂÈÚ¿ÙȘ Ë̤Ú˜ Ù˘ ‚‰ÔÌ¿‰·˜. ∆¤ÏÔ˜, ÛÙÔ∫ÂÊ¿Ï·ÈÔ 42 ı· ·ÛÎËıԇ̠۠appleÚÔ-Ûı¤ÛÂȘ Ì ÙË Ì¤ıÔ‰Ô Ù˘ ˘apple¤Ú‚·Û˘Ù˘ ‰Âο‰·˜.AÓÙ·ÏÏ¿ÛÛÔ˘Ì ‰¤Î·appleÚ¿ÛÈÓ˜ Ì¿ÚΘ Ì ̛· ÎfiÎÎÈÓË252‰Âο‰Â˜5ÌÔÓ¿‰Â˜


39ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (II)O Ù·Ì›·˜∞ÓÙ·ÏÏ¿ÛÛˆ Ù· ÓÔÌ›ÛÌ·Ù· ÙÔ˘ 1 ∂ÀƒøÌ ÓÔÌ›ÛÌ·Ù· ›Û˘ ·Í›·˜ Î·È fiÛÔ ÙÔ ‰˘Ó·ÙfiÓ ÌÂÁ·Ï‡ÙÂÚ˘ ·Í›·˜.1µ¿˙ˆ Û ·ÎÏÔ Ù· ÓÔÌ›ÛÌ·Ù· appleÔ˘¯ÚÂÈ¿˙ÔÓÙ·È ÁÈ· Ó· Û¯ËÌ·ÙÈÛÙ› ÙÔ›‰ÈÔ appleÔÛfi Ì ٷ ‰ÈappleÏ·Ó¿ ÓÔÌ›ÛÌ·Ù·ÙÔ˘ 1 ∂Àƒø.fiÛ· Â›Ó·È fiÏ· Ù· ∂Àƒø;fiÛ· Â›Ó·È fiÏ· Ù· ∂Àƒø;26˘Ó¯›˙Ô˘Ì ÙËÓ ÂÍ¿ÛÎËÛË ÙˆÓ Ì·ıËÙÒÓ ÛÙÔ ‰Âη‰ÈÎfi Û‡ÛÙËÌ· ·Ú›ıÌËÛ˘ Ì ٷ ÓÔÌ›ÛÌ·Ù· Î·È ÙÔ˘˜ ¿‚·Î˜.


ÂÓfiÙËÙ·6ËÀappleÔÏÔÁ›˙ˆ ÙÔ ¿ıÚÔÈÛÌ· ÙˆÓ ‰Âο‰ˆÓ Î·È ÙˆÓ ÌÔÓ¿‰ˆÓ ÂÓfi˜ ·ÚÈıÌÔ‡.2¯ËÌ·Ù›˙ˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ ÛÙÔ˘˜ ¿‚·Î˜.25 33 42 50316 ÏÂappleÙ¿∫¿ı apple·È‰› ı¤ÏÂÈ Ó· ·ÁÔÚ¿ÛÂÈ ÙÔ Ìapple·ÏfiÓÈ.µ¿˙ˆ Û ·ÎÏÔ Ù· ¯Ú‹Ì·Ù· appleÔ˘ appleÚ¤appleÂÈÓ· ‰ÒÛÂÈ Î¿ı apple·È‰› ÁÈ· Ó· ·ÁÔÚ¿ÛÂÈÙÔ Ìapple·ÏfiÓÈ.42. ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· Ù˘ ÌÔÚÊ‹˜ 10+Ó, 20+Ó Î.Ïapple., fiappleÔ˘ Ó ¤Ó·˜ ÌÔÓÔ„‹ÊÈÔ˜ ·ÚÈıÌfi˜.27


40°ÂˆÌÂÙÚÈο Û¯‹Ì·Ù·∞ÛÙ›· appleÚfiÛˆapple··Ú·ÙËÚÒ ÙȘ ÂÈÎfiÓ˜ Î·È ÙȘ Û˘Ó‰¤ˆ Ì ÙÔ ·ÓÙ›ÛÙÔÈ¯Ô Û¯‹Ì·.1·ÎÏÔ˜ ÙÚ›ÁˆÓÔ ÔÚıÔÁÒÓÈÔ ÙÂÙÚ¿ÁˆÓÔ∫fi‚ˆ Û¯‹Ì·Ù· Î·È Ù· ÎÔÏÏÒ ÁÈ· Ó· Û¯ËÌ·Ù›ÛˆÙÔ ‰ÈÎfi ÌÔ˘ ·ÓıÚˆapple¿ÎÈ.28OÈ Ì·ıËÙ¤˜ ·Ó·ÁÓˆÚ›˙Ô˘Ó ÙȘ ÌÔÚʤ˜ Î·È ÔÓÔÌ¿˙Ô˘Ó Ù· Û¯‹Ì·Ù·.


ÂÓfiÙËÙ·6Ë·Ú·ÙËÚÒ Ù· ·ÓÙÈΛÌÂÓ· Î·È Ù· Û˘Ó‰¤ˆÌ ٷ ·ÓÙ›ÛÙÔȯ· ÁˆÌÂÙÚÈο Û¯‹Ì·Ù·.2·‚Ԙ·ÏÈÓ‰ÚÔ˜ÛÊ·›Ú·ÛÙÂÚÂfi ÔÚıÔÁÒÓÈԯ‰ȿ˙ˆ ÙÔ ‰¤ÓÙÚÔ Ì ηٿÏÏËÏ· ÁˆÌÂÙÚÈο Û¯‹Ì·Ù·.3¯Â‰È¿˙ˆ ‰›appleÏ· Ù· ›‰È· Û¯‹Ì·Ù· ¯ÚËÛÈÌÔappleÔÈÒÓÙ·˜ ÙÔ ¯¿Ú·Î·.429


41O ¯ÚfiÓÔ˜∆Ô ËÌÂÚÔÏfiÁÈÔ·Ú·ÙËÚÒ Î·È Û˘ÌappleÏËÚÒÓˆ ÙȘ ÂÏÏÈapple›˜Î·Ú٤Ϙ ÙÔ˘ ËÌÂÚÔÏÔÁ›Ô˘.1¯ı˜ Û‹ÌÂÚ· ·‡ÚÈÔ........................................................................ºÂ‚ÚÔ˘·Ú›Ô˘ºÂ‚ÚÔ˘·Ú›Ô˘ºÂ‚ÚÔ˘·Ú›Ô˘·Ú·ÙËÚԇ̠ÙȘ ÂÈÎfiÓ˜ Î·È Û˘˙ËÙ¿ÌÂ.30OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ¤ÓÓÔÈ· ÙÔ˘ ¯ÚfiÓÔ˘.


ÂÓfiÙËÙ·6ËOÈ ÂappleÔ¯¤˜2ÕÓÔÈÍË K·ÏÔη›ÚÈ ºıÈÓfiappleˆÚÔ XÂÈÌÒÓ·˜◗ ÔÈ· ÂappleÔ¯‹ Â›Ó·È Ù· ÁÂÓ¤ıÏÈ¿ ÛÔ˘; ..............................................................◗ ÔÈ· ÂappleÔ¯‹ Â›Ó·È Ù· ÃÚÈÛÙÔ‡ÁÂÓÓ·; ............................................................◗ ÔÈ· ÂappleÔ¯‹ Â›Ó·È Ë ÁÈÔÚÙ‹ Ù˘ ÚˆÙÔÌ·ÁÈ¿˜; ............................................◗ ÔÈ· ÂappleÔ¯‹ ÎÏ›ÓÔ˘Ó Ù· Û¯ÔÏ›·; ................................................................OÈ Ë̤Ú˜ Ù˘ ‚‰ÔÌ¿‰·˜3¢5T6T78910K11◗ ÙȘ 6 ÙÔ˘ Ì‹Ó· Â›Ó·È Ë̤ڷ .......................................................................◗ ÙȘ 10 ÙÔ˘ Ì‹Ó· Â›Ó·È Ë̤ڷ .....................................................................◗ ÙȘ 8 ÙÔ˘ Ì‹Ó· Â›Ó·È Ë̤ڷ .......................................................................◗ ÙȘ 11 ÙÔ˘ Ì‹Ó· Â›Ó·È Ë̤ڷ .....................................................................31


42ÚÔÛı¤ÛÂȘ Ì ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ª¤Û· ÛÙÔ ÎÔ˘Ù› ˘apple¿Ú¯Ô˘Ó 9 ηڷ̤Ϙ Î·È ¤Íˆ·applefi ÙÔ ÎÔ˘Ù› ¿ÏϘ 4. fiÛ˜ Â›Ó·È fiϘ ÔÈ Î·Ú·Ì¤Ï˜;1O ˘ı·ÁfiÚ·˜ ÌÂÙÚ¿ ÁÈ·Ó· ‚ÚÂÈ ÙÔ ¿ıÚÔÈÛÌ· 9 + 4.∞applefi ÙÔ 9 ·Ó‚·›Óˆ4 ·ÚÈıÌÔ‡˜(9, 10, 11, 12, 13)Î·È ‚Ú›ÛΈ ÙÔ 13.∏ Àapple·Ù›· ˘appleÔÏÔÁ›˙ÂÈ ÙÔ ¿ıÚÔÈÛÌ·9 + 4.∞Ó appleÚÔÛı¤Ûˆ ÛÙÔ9 ÙÔ 1, ı· ¤¯ˆ 10.10 Î·È 3 = ......4 = 1 + 39 + 4 = ...9 + 4 = 9 + 1 + 3 = ...9+4+110+3ÀappleÔÏÔÁ›˙Ô˘Ì Ì ÙÔ ·ÚÈıÌËÙ‹ÚÈÔÀappleÔÏÔÁ›˙ˆ Ì ÙÔ ·ÚÈıÌËÙ‹ÚÈÔ, fiappleˆ˜ Ë Àapple·Ù›·, ÙÔ ¿ıÚÔÈÛÌ· 9 + 4.Œ¯Ô˘Ì 9. ÚÔÛı¤Ùˆ ·ÎfiÌË 1ÁÈ· Ó· Á›ÓÔ˘Ó 10.Ù· 10 appleÔ˘ ¤¯ˆ appleÚÔÛı¤Ùˆ ¿ÏÏ· 3.32OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ÂÎÙ¤ÏÂÛË ÙˆÓ appleÚÔÛı¤ÛÂˆÓ Ì ÙË Ì¤ıÔ‰Ô Ù˘ ˘apple¤Ú‚·Û˘ Ù˘ ‰Âο‰·˜.


ÂÓfiÙËÙ·6ËÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ Û˘Ìappleϋڈ̷ ÙÔ˘ ·ÚÈıÌÔ‡ 10.2§‡ÓÔ˘Ì ٷ appleÚÔ‚Ï‹Ì·Ù· Î·È Û˘˙ËÙ¿ÌÂ∏ ÔÊ›· ¤¯ÂÈ Ì¤Û· ÛÙÔ ÎÔ˘Ù› 7 ÎÔ‡ÎϘ.£¤ÏÂÈ Ó· ‚¿ÏÂÈ ¿ÏϘ 4.fiÛ˜ ı· Â›Ó·È ÔÈ ÎÔ‡ÎϘ̤۷ ÛÙÔ ÎÔ˘Ù›;ª¤Û· ÛÙÔ ÎÔ˘Ù› ˘apple¿Ú¯Ô˘Ó 8 Ìapple¿Ï˜.∞Ó ‚¿ÏÔ˘Ì ̤۷ ÛÙÔ ÎÔ˘Ù›¿ÏϘ 6 Ìapple¿Ï˜,applefiÛ˜ ı· Á›ÓÔ˘Ó fiϘ Ì·˙›;37 87 + ... = 10 8 + ... = 1010 + ... = ... 6 = ... + ...7 + 4 = ... 8 + 6 = ...ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.6 + ... = 10 8 + ... = 10 9 + ... = 107 + 3 + 4 = ... 9 + 1 + 6 = ... 7 = 3 + ...6 + 4 + 2 = ... 8 + 2 + 5 = ... 9 = 4 + ...42. ¢›ÓÔ˘Ì ÛÙÔ˘˜ Ì·ıËÙ¤˜ ·ÚÈıÌÔ‡˜ ÌÂÁ·Ï‡ÙÂÚÔ˘˜ ÙÔ˘ 5 Î·È ÙÔ˘˜ ηÏԇ̠ӷ ‚ÚÔ˘Ó ÙÔ Û˘Ìappleϋڈ̿ ÙÔ˘˜,ÒÛÙ ӷ ¤¯Ô˘Ì ¿ıÚÔÈÛÌ· 10 (apple.¯. 7 Î·È applefiÛÔ Î¿ÓÂÈ 10;).33


43Âapple·Ó·ÏËappleÙÈÎfiÌ¿ıËÌ·123 x 4567+896538-Âapple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·ªÂÙÚÒ Ù· Û¯‹Ì·Ù· appleÔ˘ ÌÔÈ¿˙Ô˘Ó ÌÂٷ͇ ÙÔ˘˜Î·È Áڿʈ ÙÔ˘˜ ·ÓÙ›ÛÙÔÈ¯Ô˘˜ ·ÚÈıÌÔ‡˜.1ÙÂÙÚ¿ÁˆÓ· ÙÚ›ÁˆÓ· ·ÎÏÔÈ ÔÚıÔÁÒÓÈ·ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.27 + ... = 10 4 + ... = 10 2 + ... = 106 + 4 + 3 = ... 9 + 1 + 4 = ... 8 = 2 + ...5 + 5 + 7 = ... 7 + 3 + 6 = ... 6 = 4 + ...34


72x15ÂÓfiÙËÙ·∂ÓÒÓˆ ÙȘ ϤÍÂȘ Ì ÙȘ ·ÓÙ›ÛÙÔȯ˜ ÂÈÎfiÓ˜.XÂÈÌÒÓ·˜ ÕÓÔÈÍË ºıÈÓfiappleˆÚÔ K·ÏÔη›ÚÈ6Ë3ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.9 + 4 =8 + 6 =49+48+6+1 +310109 + 7 = 7 + 4 =+7 +49 710 1035


123 x 4567+89


EÓfiÙËÙ· 7Ë:XAPA•EI, AZ§ – PO£EHKAI AºAIPEH – H YEPBAHTH ¢EKA¢A6538-72x1584245464748495051∫ÂÊ¿Ï·ÈÔ 45Ô:÷ڿÍÂȘ, apple·˙Ï Î·È ÌˆÛ·˚ο∫ÂÊ¿Ï·ÈÔ 46Ô:ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ‰È„‹ÊÈˆÓ Î·È ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ∫ÂÊ¿Ï·ÈÔ 47Ô:∏ appleÚfiÛıÂÛË Î·È Ë ·Ê·›ÚÂÛˈ˜ ·ÓÙ›ÛÙÚÔʘ appleÚ¿ÍÂȘ –∏ ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜∫ÂÊ¿Ï·ÈÔ 48Ô:ÀappleÔÏÔÁÈÛÌÔ› –∂appleÈÛÙÚÔÊ‹ ÛÙËÓ appleÂÓÙ¿‰·∫ÂÊ¿Ï·ÈÔ 49Ô:ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË –¢È„‹ÊÈÔÈ Î·È ÌÔÓÔ„‹ÊÈÔÈ ·ÚÈıÌÔ›∫ÂÊ¿Ï·ÈÔ 50fi:ÚÔ‚Ï‹Ì·Ù·∫ÂÊ¿Ï·ÈÔ 51Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·EÓfiÙËÙ· 7Ë∞Ú¯Èο, ÛÙË ÁˆÌÂÙÚ›· ı··Û¯ÔÏËıԇ̠̠ÙË Û‡ÓıÂÛË apple·˙Ï‚·ÛÈ˙fiÌÂÓˆÓ ÛÙÔ apple·È¯Ó›‰È Ù¿ÁÎÚ·Ì,appleÚÔÎÂÈ̤ÓÔ˘ Ó· ·ÛÎËıԇ̠ÛÙËÓ·Ó¿Ï˘ÛË Î·È ÙË Û‡ÓıÂÛË ÙˆÓÛ¯ËÌ¿ÙˆÓ.ÙÔ ∫ÂÊ¿Ï·ÈÔ 46 ÛÙËÓ Ù¿ÍË ı·apple·›ÍÔ˘Ì ÙÔ apple·È¯Ó›‰È «ºÈ‰¿ÎÈ», ÁÈ· Ó··ÛÎËıԇ̠ÛÙËÓ appleÚfiÛıÂÛË ‰È„‹ÊÈÔ˘Ì ÌÔÓÔ„‹ÊÈÔ ·ÚÈıÌfi Î·È ÛÙËÓ·Ê·›ÚÂÛË ÌÔÓÔ„‹ÊÈÔ˘ ·applefi ‰È„‹ÊÈÔ·ÚÈıÌfi.Ù· ÂapplefiÌÂÓ· ÎÂÊ¿Ï·È· ÙfiÛÔ Ì ÙË‚Ô‹ıÂÈ· ÂappleÔappleÙÈÎÔ‡ ˘ÏÈÎÔ‡(·ÚÈıÌËÙ‹ÚÈÔ Î·È ‚¿ÛÂȘ) fiÛÔ Î·È ¯ˆÚ›˜·˘Ùfi ı· ÂÎÙÂϤÛÔ˘Ì appleÔÏϤ˜appleÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘÂÊ·ÚÌfi˙ÔÓÙ·˜ ÙË Ì¤ıÔ‰Ô Ù˘˘apple¤Ú‚·Û˘ Ù˘ ‰Âο‰·˜, ÙË Ì¤ıÔ‰ÔÙ˘ ÂappleÈÛÙÚÔÊ‹˜ ÛÙËÓ appleÂÓÙ¿‰· ηıÒ˜Î·È ¿ÏϘ ÌÂıfi‰Ô˘˜.ª¤Û· ÛÙË ÊˆÏÈ¿ ˘apple¿Ú¯Ô˘Ó 9 Ì˘ÚÌ‹ÁÎÈ·.∞Ó ÌappleÔ˘Ó ·ÎfiÌ· 6, applefiÛ· ı· Â›Ó·È fiÏ·;∫·Ù·Û΢¿˙ˆ ÙÔ Ù¿ÁÎÚ·Ì.9+6+110+5


45÷ڿÍÂȘ, apple·˙Ï Î·È ÌˆÛ·˚ο∆Ô Ù¿ÁÎÚ·Ì∫·Ù·Û΢¿˙ˆ ÙÔ Ù¿ÁÎÚ·Ì.1ªÂ Ù· ÎÔÌÌ¿ÙÈ· ÙÔ˘ Ù¿ÁÎÚ·Ì Û˘Óı¤Ùˆ Ù· apple·Ú·Î¿Ùˆ Û¯‹Ì·Ù·.38OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙȘ ¯·Ú¿ÍÂȘ ηıÒ˜ Âapple›Û˘ ÛÙËÓ ·Ó¿Ï˘ÛËÎ·È ÙË Û‡ÓıÂÛË ÙˆÓ Û¯ËÌ¿ÙˆÓ Ì apple·˙Ï Î·È ÌˆÛ·˚ο.


ÂÓfiÙËÙ·7ËOÈ ÁÚ·Ì̤˜ Ù˘ appleÚfiÛıÂÛ˘.2∂ÓÒÓˆ Ì ÙÔ ¯¿Ú·Î· ¤Ó·Ó ·ÚÈıÌfi ·applefi ÙËÓ appleÚÒÙË ÁÚ·ÌÌ‹Ì ¤Ó·Ó ¿ÏÏÔ ·applefi ÙËÓ ÙÚ›ÙË.∆È apple·Ú·ÙËÚÒ;˘Ó¯›˙ˆ ÙÔ ¯ÚˆÌ·ÙÈÛÌfi Ì ÙÔÓ ›‰ÈÔ ÙÚfiappleÔ.·Ú·ÙËÚÒ Ù· Û¯‹Ì·Ù· appleÔ˘ ÂÌÊ·Ó›˙ÔÓÙ·È ÛÙÔ appleÏ·ÎfiÛÙÚˆÙÔ.339


46ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ‰È„‹ÊȈÓÎ·È ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ·›˙Ô˘Ì ÙÔ «ºÈ‰¿ÎÈ»∫·ÓfiÓ˜ ÙÔ˘ apple·È¯ÓȉÈÔ‡ÃÚÂÈ¿˙ÂÙ·È ¤Ó· ˙¿ÚÈ ÛÙÔÔappleÔ›Ô ÔÈ appleÏ¢ڤ˜ Ì ÙÔ˘˜·ÚÈıÌÔ‡˜ 4, 5 Î·È 6 ı· ¤¯Ô˘ÓÎfiÎÎÈÓÔ ¯ÚÒÌ·, ÂÓÒ ÔÈappleÏ¢ڤ˜ Ì ÙÔ˘˜ ·ÚÈıÌÔ‡˜ 1,2 Î·È 3 ı· ¤¯Ô˘Ó ÌappleÏ ¯ÚÒÌ·.O ·ÚÈıÌfi˜ ·applefi ÙÔÓ ÔappleÔ›ÔÍÂÎÈÓ¿ÌÂ Â›Ó·È ÙÔ 12. ∫¿ıÂapple·›ÎÙ˘ Ì ÙË ÛÂÈÚ¿ Ú›¯ÓÂÈ ÙÔ˙¿ÚÈ. ŸÙ·Ó ÙÔ ˙¿ÚÈ ‰Â›¯ÓÂÈ ÌÈ·ÎfiÎÎÈÓË appleÏÂ˘Ú¿, appleÚÔ¯ˆÚÔ‡ÌÂÌappleÚÔÛÙ¿ ÙfiÛ˜ ı¤ÛÂȘ fiÛ˜‰Â›¯ÓÂÈ ÙÔ ˙¿ÚÈ. ŸÙ·Ó ÙÔ ˙¿Úȉ›¯ÓÂÈ ÌÈ· ÌappleÏ appleÏÂ˘Ú¿,appleËÁ·›ÓÔ˘Ì apple›Ûˆ ÙfiÛ˜ı¤ÛÂȘ fiÛ˜ ‰Â›¯ÓÂÈ ÙÔ ˙¿ÚÈ.¡ÈÎËÙ‹˜ ı· Â›Ó·È ·˘Ùfi˜ appleÔ˘ı· ÊÙ¿ÛÂÈ appleÚÒÙÔ˜ ÛÙÔ 30.1˘ÌappleÏËÚÒÓˆ Ù· ÛÙÔȯ›· appleÔ˘ Ï›appleÔ˘Ó ÛÙÔ˘˜ apple·Ú·Î¿Ùˆ apple›Ó·Î˜.AÚÈıÌfi˜ ·applefi ÙÔÓÔappleÔ›Ô ÍÂÎÈÓ¿ÌÂZ¿ÚÈM·Ú›· 12 4N›ÎÔ˜ 12 3X¿Ú˘ 12 6AÚÈıÌfi˜ ·applefi ÙÔÓÔappleÔ›Ô ÍÂÎÈÓ¿ÌÂZ¿ÚÈM·Ú›· 22 5N›ÎÔ˜ 18 3X¿Ú˘ 14 6AÚÈıÌfi˜ ÛÙÔÓÔappleÔ›Ô ÊÙ¿ÓÔ˘ÌÂAÚÈıÌfi˜ ÛÙÔÓÔappleÔ›Ô ÊÙ¿ÓÔ˘ÌÂ40OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ÂÎÙ¤ÏÂÛË appleÚfiÛıÂÛ˘ ‰È„‹ÊÈÔ˘ Ì ÌÔÓÔ„‹ÊÈÔ·ÚÈıÌfi Î·È ·Ê·›ÚÂÛ˘ ÌÔÓÔ„‹ÊÈÔ˘ ·applefi ‰È„‹ÊÈÔ ·ÚÈıÌfi ¯ˆÚ›˜ ÎÚ·ÙÔ‡ÌÂÓÔ.


ÂÓfiÙËÙ·7ËÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ Ù· ·ıÚÔ›ÛÌ·Ù· Î·È ÙȘ ‰È·ÊÔÚ¤˜.214§‡ÓÔ˘Ì ٷ appleÚÔ‚Ï‹Ì·Ù· Î·È Û˘˙ËÙ¿ÌÂ.ª¤Û· ÛÙÔ Î·Ï¿ıÈ ˘apple¿Ú¯Ô˘Ó 14 Ì‹Ï·.∞Ó ‚¿Ïˆ ·ÎfiÌË 3 Ì‹Ï·, applefiÛ· ı· Â›Ó·È Ù· Ì‹Ï· ̤۷ ÛÙÔ Î·Ï¿ıÈ;3°Ú¿Êˆ ÙËÓ appleÚ¿ÍË Î·È ÙÔ ·appleÔÙ¤ÏÂÛÌ·.O °ÈÒÚÁÔ˜ ›¯Â 17 ·˘ÙÔÎÈÓËÙ¿ÎÈ·. ÿÚÈÛ 3 ·˘ÙÔÎÈÓËÙ¿ÎÈ·ÛÙÔ˘˜ Ê›ÏÔ˘˜ ÙÔ˘. fiÛ· ·˘ÙÔÎÈÓËÙ¿ÎÈ· ¤¯ÂÈ ÙÒÚ·;°Ú¿Êˆ ÙËÓ appleÚ¿ÍË Î·È ÙÔ ·appleÔÙ¤ÏÂÛÌ·.ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ ·appleÔÙ¤ÏÂÛÌ·.5 + 2 = ... 2 + 2 = ... 6 + 3 = ...15 + 2 = ... 12 + 2 = ... 16 + 3 = ...4 – 2 = ... 8 – 4 = ... 9 – 5 = ...14 – 2 = ... 18 – 4 = ... 19 – 5 = ...2 + 6 = ... 4 + 5 = ... 2 + 7 = ...12 + 6 = ... 14 + 5 = ... 12 + 7 = ...42. ∏ ‰·ÛοϷ appleÚÔÙ›ÓÂÈ appleÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ, ÙÔ ·appleÔÙ¤ÏÂÛÌ· ÙˆÓ ÔappleÔ›ˆÓ ‰ÂÓÍÂappleÂÚÓ¿ ÙÔ 10 (apple.¯. 4 + 3, 7 – 2 Î.Ïapple.). OÈ Ì·ıËÙ¤˜ ˘appleÔÏÔÁ›˙Ô˘Ó Î·È ÁÚ¿ÊÔ˘Ó ÙËÓ appleÚ¿ÍË Ì¤Û· ÛÙÔ appleÏ·›ÛÈÔ.41


47H appleÚfiÛıÂÛË Î·È Ë ·Ê·›ÚÂÛË ˆ˜ ·ÓÙ›ÛÙÚÔʘappleÚ¿ÍÂȘ – ∏ ˘apple¤Ú‚·ÛË Ù˘ ‰Âο‰·˜∆· Ì˘ÚÌ‹ÁÎÈ·ª¤Û· ÛÙË ÊˆÏÈ¿ ˘apple¿Ú¯Ô˘Ó 9 Ì˘ÚÌ‹ÁÎÈ·.∞Ó ÌappleÔ˘Ó ·ÎfiÌ· 6, applefiÛ· ı· Â›Ó·È fiÏ·;19 + ... = 1010 + ... = ...9 + 6 = ...9+610∞Ó Ù· 6 Ì˘ÚÌ‹ÁÎÈ· appleÔ˘ Ìapple‹Î·Ó ̤۷ ÛÙË ÊˆÏÈ¿ ‚ÁÔ˘ÓÍ·Ó¿ ¤Íˆ, applefiÛ· ı· Ì›ÓÔ˘Ó Ì¤Û· ÛÙË ÊˆÏÈ¿;Aapplefi Ù· 15, ·Ó‚Á¿Ïˆ Ù· 5, ı·ÌÔ˘ Ì›ÓÔ˘Ó 10.Œ‚Á·Ï· Ù· 5.°È· Ó· ‚Á¿Ïˆ 6‚Á¿˙ˆ ¿ÏÏÔ 1.15 – 5 = 1010 – 1 = ...15 – 6 = ...15–61042OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ÂÎÙ¤ÏÂÛË appleÚÔÛı¤ÛÂˆÓ Î·È ·Ê·ÈÚ¤ÛÂˆÓ Ì ÙË Ì¤ıÔ‰Ô Ù˘ ˘apple¤Ú‚·Û˘ Ù˘‰Âο‰·˜ ηıÒ˜ Âapple›Û˘ ÛÙËÓ ·ÓÙÈÌÂÙÒappleÈÛË Ù˘ appleÚfiÛıÂÛ˘ Î·È Ù˘ ·Ê·›ÚÂÛ˘ ˆ˜ ·ÓÙ›ÛÙÚÔÊˆÓ appleڿ͈Ó.


ÂÓfiÙËÙ·7ËÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙȘ appleÚÔÛı¤ÛÂȘ Î·È ÙȘ ·Ê·ÈÚ¤ÛÂȘ.2O ∞Ó¤ÛÙ˘ ›¯Â ̤۷ ÛÙÔ ÎÔ˘Ù› 14 ÌappleÈÛÎfiÙ·.ŒÊ·Á ٷ 5. fiÛ· ÌappleÈÛÎfiÙ· ÙÔ‡ ¤ÌÂÈÓ·Ó;31414 – ... = 1010 – ... = ...14–514 – 5 = ...ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.107 + 4 = 9 + 3 =4+ 4 + 37 910 1011 – 4 = 12– 3 =– 4 – 311 1210 102. ÚÔÙ›ÓÔ˘Ì ÛÙÔ˘˜ Ì·ıËÙ¤˜ appleÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ. OÈ appleÚÔÛı¤ÛÂȘ Â›Ó·È Ù˘ ÌÔÚÊ‹˜ 10+Ó(apple.¯. 10 + 3, 10 + 7 Î.Ïapple.) Î·È ·ÓÙÈÛÙÔ›¯ˆ˜ ÔÈ ·Ê·ÈÚ¤ÛÂȘ Ù˘ ÌÔÚÊ‹˜ 1Ó–Ó (apple.¯. 14 – 4, 16 – 6 Î.Ïapple.).43


48ÀappleÔÏÔÁÈÛÌÔ› –∂appleÈÛÙÚÔÊ‹ ÛÙËÓ appleÂÓÙ¿‰·¯ËÌ·Ù›˙ˆ ·ÚÈıÌÔ‡˜ Ì ٷ ‰¿¯Ù˘Ï·fiÛ· Â›Ó·È fiÏ· Ù· ‰¿¯Ù˘Ï· οı ÊÔÚ¿;1Î·È Î·È Î·È5 + ... = ... ... + ... = ... ... + ... = ...¯ËÌ·Ù›˙ˆ ·ÚÈıÌÔ‡˜ Ì ÙÔ ·ÚÈıÌËÙ‹ÚÈÔfiÛ˜ Â›Ó·È ÔÈ ¯¿ÓÙÚ˜ Û οı ·ÚÈıÌËÙ‹ÚÈÔ;6 = 5 + ...... = 5 + ... ... = 5 + ...6 = 5 + 17 = 5 + 26 + 7 = ...8 = 5 + ...6 = ... + ...8 + 6 = ...7 = ... + ...9 = ... + ...7 + 9 = ...44OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙË ‰È·‰Èηۛ· ˘appleÔÏÔÁÈÛÌÔ‡ ÙÔ˘ ·ıÚÔ›ÛÌ·ÙԘ̠ÙË ÛÙÚ·ÙËÁÈ΋ Ù˘ «ÂappleÈÛÙÚÔÊ‹˜ ÛÙËÓ appleÂÓÙ¿‰·».


ÂÓfiÙËÙ·7ËÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ ¿ıÚÔÈÛÌ·.2∏ ª·Ú›· ¤¯ÂÈ 6 ηڷ̤Ϙ. ∏ ÌËÙ¤Ú· Ù˘ Ù˘ ‰›ÓÂÈ ·ÎfiÌ· 5.fiÛ˜ Â›Ó·È fiϘ Ì·˙› ÔÈ Î·Ú·Ì¤Ï˜ Ù˘ ª·Ú›·˜;3O ˘ı·ÁfiÚ·˜ ÌÂÙÚ¿ ÁÈ·Ó· ‚ÚÂÈ ÙÔ ¿ıÚÔÈÛÌ· 6 + 5.∏ Àapple·Ù›· ÙÔappleÔıÂÙ› ÙȘ ηڷ̤Ϙ̠ÙË ‰È¿Ù·ÍË appleÔ˘ ¤¯Ô˘Ó ÔÈ ÎÔ˘ÎΛ‰Â˜ÛÙÔ ˙¿ÚÈ Î·È ˘appleÔÏÔÁ›˙ÂÈ.6 = 5 + 1∞applefi ÙÔ 6 ·Ó‚·›Óˆ5 ·ÚÈıÌÔ‡˜(7, 8, 9, 10, 11)Î·È ‚Ú›ÛΈ ÙÔ 11.6 + 5 = ...6 + 5 = 5 + 1 + 56 + 5 = ...ÀappleÔÏÔÁ›˙ˆ fiappleˆ˜ Ë Àapple·Ù›· Ù· apple·Ú·Î¿Ùˆ ·ıÚÔ›ÛÌ·Ù·.7 + 5 7 + 87 = 5 + ...7 = 5 + ...7 = 5 + ...8 = 5 + ...7 + 8 = ...2. ∏ ‰·ÛοϷ appleÚÔÙ›ÓÂÈ ·ıÚÔ›ÛÌ·Ù· Ù˘ ÌÔÚÊ‹˜ 5+Ó ‹ Ó+5 Ì ÙÈÌ‹ ÌÈÎÚfiÙÂÚË ‹ ›ÛË ÙÔ˘ 10.OÈ Ì·ıËÙ¤˜ ‚Ú›ÛÎÔ˘Ó Ù· ·ıÚÔ›ÛÌ·Ù· Î·È Ù· ÁÚ¿ÊÔ˘Ó Ì ۇ̂ÔÏ· ̤۷ ÛÙ· appleÏ·›ÛÈ·.45


49ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË –¢È„‹ÊÈÔÈ Î·È ÌÔÓÔ„‹ÊÈÔÈ ·ÚÈıÌÔ›ª·ÓÙ‡ˆ ÙÔÓ ·ÚÈıÌfi1ΤÊÙÔÌ·È ¤Ó·Ó‰È„‹ÊÈÔ ·ÚÈıÌfi.OÈ ÌÔÓ¿‰Â˜ ÙÔ˘ ·ÚÈıÌÔ‡Â›Ó·È 4. ∞Ó ·Ê·ÈÚ¤Ûˆ ÙȘÌÔÓ¿‰Â˜ ·applefi ÙÔÓ ·ÚÈıÌfi,‚Ú›ÛΈ ÙÔ 10.ÔÈÔ˜ Â›Ó·È Ô ·ÚÈıÌfi˜appleÔ˘ ÛΤÊÙËη;°Ú¿Êˆ ÙÔÓ ·ÚÈıÌfi Î·È ÙËÓ appleÚ¿ÍË.... – ... = ...µÚ›ÛΈ apple¿ÓÙ· ÙÔÓ ·ÚÈıÌfi 82ΤÊÙÔÌ·È ¤Ó·Ó ‰È„‹ÊÈÔ·ÚÈıÌfi. O ·ÚÈıÌfi˜ ·˘Ùfi˜Â›Ó·È ÌÈÎÚfiÙÂÚÔ˜ ·applefi ÙÔ 20.∞applefi ÙÔÓ ·ÚÈıÌfi appleÔ˘ÛΤÊÙËη ·Ê·ÈÚÒ ÙȘ ÌÔÓ¿‰Â˜.∞applefi ·˘Ùfi appleÔ˘ ‚Ú›ÛΈ ·Ê·ÈÚÒÙÔ 2 Î·È ‚Ú›ÛΈ ÙÔ 8.°Ú¿Êˆ ÙÔÓ ·ÚÈıÌfi Î·È ÙȘ appleÚ¿ÍÂȘ.... – ... = ... ... – 2 = 846OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ÂÎÙ¤ÏÂÛË appleÚÔÛı¤ÛÂˆÓ Î·È ·Ê·ÈÚ¤ÛÂˆÓ ÓÔÂÚ¿Ì ÙË ÛÙÚ·ÙËÁÈ΋ Ù˘ «ÂappleÈÛÙÚÔÊ‹˜ ÛÙËÓ appleÂÓÙ¿‰·».


ÂÓfiÙËÙ·7˵ڛÛΈ Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi.3∞Ê·›ÚÂÛË Ì appleÚfiÛıÂÛË appleÚÔ˜ Ù· Âapple¿Óˆ.4ª¤Û· Û ÌÈ· ηÛÂÙ›Ó· ¤¯ˆ 13 ÎËÚÔÌappleÔÁȤ˜.∞Ó ‚Á¿Ïˆ ÙȘ 9, applefiÛ˜ ÎËÚÔÌappleÔÁȤ˜ ı· Ì›ÓÔ˘Ó̤۷ ÛÙËÓ Î·ÛÂÙ›Ó·;13ÀappleÔÏÔÁ›˙ˆ ÙËÓ ·Ê·›ÚÂÛË 13 – 9.9 Î·È 1 Ì·˜ οÓÔ˘Ó 10.10 Î·È 3 Ì·˜ οÓÔ˘Ó 13.∞Ó¤‚Ëη 4.9 + 1 = 1010 + 3 = 13ÕÚ· 13 – 9 = ...ÀappleÔÏÔÁ›˙ˆ ÙȘ appleÚÔÛı¤ÛÂȘ Î·È ÙȘ ·Ê·ÈÚ¤ÛÂȘ Ì ٷ ‰ÈappleÏ¿.56 + 6 = ... 7 + 7 = ... 8 + 8 = ... 9 + 9 = ...12 – 6 = ... 14 – 7 = ... 16 – 8 = ... 18 – 9 = ...3. ÚÔÙ›ÓÔ˘Ì ¤Ó·Ó ‰È„‹ÊÈÔ ·ÚÈıÌfi ·Ó·Ê¤ÚÔÓÙ·˜ ÙÔÓ ·ÚÈıÌfi ÙˆÓ ÌÔÓ¿‰ˆÓ Î·È ÙˆÓ ‰Âο‰ˆÓ ÙÔ˘ (apple.¯.«appleÔÈÔ˜ Â›Ó·È Ô ·ÚÈıÌfi˜ appleÔ˘ ¤¯ÂÈ 6 ÌÔÓ¿‰Â˜ Î·È 2 ‰Âο‰Â˜»). OÈ Ì·ıËÙ¤˜ ‚Ú›ÛÎÔ˘Ó Î·È ÁÚ¿ÊÔ˘Ó ÙÔÓ ·ÚÈıÌfi.47


50appleÚÔ‚Ï‹Ì·Ù· appleÚÔ‚Ï‹Ì·Ù·123 x 4567+896538-4 9 23 5 78 1 6ÚÔ‚Ï‹Ì·Ù·∆Ô Ì·ÁÈÎfi ÙÂÙÚ¿ÁˆÓÔÙÔ ÙÂÙÚ¿ÁˆÓÔ ·˘Ùfi˘apple¿Ú¯ÂÈ Î¿ÙÈ ÙÔ Ì·ÁÈÎfi.ÚÔÛı¤Ùˆ ÙÔ˘˜ ÙÚÂȘ·ÚÈıÌÔ‡˜ ÔÚÈ˙ÔÓÙ›ˆ˜,ηı¤Ùˆ˜ Î·È ‰È·ÁˆÓ›ˆ˜.∆È apple·Ú·ÙËÚÒ;1O ¯ÒÚÔ˜ ÛÙ¿ıÌ¢Û˘ÙÔ ¯ÒÚÔ ÛÙ¿ıÌ¢Û˘ ÌÈ·˜ appleÔÏ˘Î·ÙÔÈΛ·˜ ˘apple‹Ú¯·Ó 5 ·˘ÙÔΛÓËÙ·.∆Ô ÌÂÛË̤ÚÈ ‹Úı·Ó Î·È ÛÙ¿ıÌÂ˘Û·Ó ¿ÏÏ· 6.∆Ô ‚Ú¿‰˘ ¤Ê˘Á·Ó Ù· 3.fiÛ· ·˘ÙÔΛÓËÙ· ¤ÌÂÈÓ·Ó ÛÙÔ ¯ÒÚÔ ÛÙ¿ıÌ¢Û˘;248


72x15ÂÓfiÙËÙ·Œ¯ˆ ÛÙËÓ ÙÛ¤appleË ÌÔ˘ 15 ÏÂappleÙ¿. ∞Ó ÙÔ ¤Ó· ÓfiÌÈÛÌ· Â›Ó·È ÙˆÓ 10 ÏÂappleÙÒÓ,Ù· ˘applefiÏÔÈapple· ÓÔÌ›ÛÌ·Ù· ÙÈ ÌappleÔÚ› Ó· ›ӷÈ;µÚ›ÛΈ ‰È¿ÊÔÚÔ˘˜ Û˘Ó‰˘·ÛÌÔ‡˜ Ì ٷ ÓÔÌ›ÛÌ·Ù·.∑ˆÁÚ·Ê›˙ˆ Ù· ÓÔÌ›ÛÌ·Ù·.7Ë310ÏÂappleÙ¿·Ú·ÙËÚÒ ÙȘ ÂÈÎfiÓ˜.¢È·Ù˘appleÒÓˆ ‰Èο ÌÔ˘ appleÚÔ‚Ï‹Ì·Ù·.4£·Ï‹˜NÂʤÏËMapplefiÓ·6 ¯ÚfiÓˆÓ10 ¯ÚfiÓˆÓ13 ¯ÚfiÓˆÓ49


51Âapple·Ó·ÏËappleÙÈÎfiÌ¿ıËÌ·123 x 4567+896538-Âapple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·∫¿appleÔȘ ηڷ̤Ϙ Â›Ó·È Ì¤Û· Î·È Î¿appleÔȘ ¤Íˆ ·applefi ÙÔ ÎÔ˘Ù›.18 710fiÛ˜ Â›Ó·È fiϘ ÔÈ ÎfiÎÎÈÓ˜ ηڷ̤Ϙ;+ =fiÛ˜ Â›Ó·È fiϘ ÔÈ appleÚ¿ÛÈÓ˜ ηڷ̤Ϙ;+ =fiÛ˜ Â›Ó·È fiϘ ÔÈ ÎfiÎÎÈÓ˜ Î·È ÔÈ appleÚ¿ÛÈÓ˜ηڷ̤Ϙ appleÔ˘ ‚Ú›ÛÎÔÓÙ·È Ì¤Û· ÛÙ· ÎÔ˘ÙÈ¿;+ =∂¿Ó Ê¿ˆ 6 ÌÔ‚ ηڷ̤Ϙ, applefiÛ˜ ı· Ì›ÓÔ˘Ó;– =∂¿Ó Ê¿ˆ 5 ÎfiÎÎÈÓ˜ ηڷ̤Ϙ, applefiÛ˜ ı· Ì›ÓÔ˘Ó;– =50


72x15ÂÓfiÙËÙ·ÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ ·appleÔÙ¤ÏÂÛÌ· ÙˆÓ appleÚÔÛı¤ÛˆÓ.1Ë2ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.10 42ŸÏ· Ì·˙› Â›Ó·È 20.5 55ŸÏ· Ì·˙› Â›Ó·È 18.10 106ŸÏ· Ì·˙› Â›Ó·È 30.5 54ŸÏ· Ì·˙› Â›Ó·È 17.¯Â‰È¿˙ˆ ‰›appleÏ· ÙËÓ ›‰È· ÂÈÎfiÓ·.34˘Ó¯›˙ˆ ÙÔ ¯ÚˆÌ¿ÙÈÛÌ· Ì ÙÔÓ ›‰ÈÔ ÙÚfiappleÔ.52. ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· Ù˘ ÌÔÚÊ‹˜ Ó+5 ‹ 5+Ó, fiappleÔ˘ Ó ÔÈ ·ÚÈıÌÔ› 6, 7 ‹ 8.ÚÔÙ›ÓÔ˘Ì Âapple›Û˘ ·ıÚÔ›ÛÌ·Ù· appleÔ˘ Â›Ó·È Ù· ÌÂÁ¿Ï· ‰ÈappleÏ¿ (‰ËÏ·‰‹ 6+6, 7+7 Î.Ïapple.).51


123 x 4567+89


EÓfiÙËÙ· 8Ë:API£MOI MEXPI TO 70 – PA•EI –METPHH – YMMETPIA6538-72x15842525354555657∫ÂÊ¿Ï·ÈÔ 52Ô:√È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 70∫ÂÊ¿Ï·ÈÔ 53Ô:∂ÈÛ·ÁˆÁ‹ ÛÙÔÓ appleÔÏÏ·appleÏ·ÛÈ·ÛÌfi∫ÂÊ¿Ï·ÈÔ 54Ô:ª¤ÙÚËÛË ÌÂÁÂıÒÓ∫ÂÊ¿Ï·ÈÔ 55Ô:ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË ‰È„‹ÊȈӷÚÈıÌÒÓ∫ÂÊ¿Ï·ÈÔ 56Ô:∂ÈÛ·ÁˆÁ‹ ÛÙË Û˘ÌÌÂÙÚ›·∫ÂÊ¿Ï·ÈÔ 57Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·EÓfiÙËÙ· 8ËÙÔ ∫ÂÊ¿Ï·ÈÔ 52, appleÚÔÎÂÈ̤ÓÔ˘ Ó·ÂÍÔÈÎÂȈıԇ̠̠ÙÔ˘˜ ·ÚÈıÌÔ‡˜ ̤¯ÚÈ ÙÔ70, ı· apple·›ÍÔ˘Ì ¤Ó· apple·È¯Ó›‰È appleÔ˘ ϤÁÂÙ·È«ÌÂÙÚËÙ‹˜ ÙˆÓ ¯ÈÏÈÔ̤ÙÚˆÓ». ÙÔ ÂapplefiÌÂÓÔÎÂÊ¿Ï·ÈÔ ı· ‰Ô‡Ì appleÚÔ‚Ï‹Ì·Ù· ·applefi ÙËÓηıËÌÂÚÈÓ‹ ˙ˆ‹, ÛÙ· ÔappleÔ›·apple·ÚÔ˘ÛÈ¿˙ÔÓÙ·È ÔÚÁ·ÓˆÌ¤Ó˜ ÔÌ¿‰Â˜fiappleˆ˜ Â›Ó·È Ù· ̤ÏË ÙÔ˘ ÛÒÌ·ÙÔ˜, Î·È ı·apple·ÚÔ˘ÛÈ¿ÛÔ˘Ì ÌÈ· Ó¤· appleÚ¿ÍË appleÔ˘Ï¤ÁÂÙ·È «appleÔÏÏ·appleÏ·ÛÈ·ÛÌfi˜».ÙÔ ∫ÂÊ¿Ï·ÈÔ 54 ı· ·ÛÎËıԇ̠ÛÙȘÌÂÙÚ‹ÛÂȘ. ÙË Û˘Ó¤¯ÂÈ· ÛÙÔ ∫ÂÊ¿Ï·ÈÔ 55ı· ·Û¯ÔÏËıԇ̠̠appleÚÔ‚Ï‹Ì·Ù·appleÚfiÛıÂÛ˘ Î·È ·Ê·›ÚÂÛ˘ Ì ‰Âο‰Â˜¯ÚËÛÈÌÔappleÔÈÒÓÙ·˜ Ù· ÓÔÌ›ÛÌ·Ù· Î·È Ù·¯·ÚÙÔÓÔÌ›ÛÌ·Ù· ÙÔ˘ ∂Àƒø. ∫·ÙfiappleÈÓ ÛÙÔ∫ÂÊ¿Ï·ÈÔ 56 οÓÔÓÙ·˜ ÛÙ¿Ìapple˜ ̉ÈappleÏÒÛÂȘ Î·È Ì ÙÔÓ Î·ıÚ¤ÊÙË ı·ÂÈÛ¿ÁÔ˘Ì ÌÈ· ηÈÓÔ‡ÚÈ· ¤ÓÓÔÈ·: ÙËÛ˘ÌÌÂÙÚ›·.TÔ Ì¤ÙÚÔ5 5 5 5 20


52OÈ ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 70O ÌÂÙÚËÙ‹˜ ÙˆÓ ¯ÈÏÈÔ̤ÙÚˆÓ1O ÌÂÙÚËÙ‹˜ Á˘Ú›˙ÂÈ ·Ó¿ ¤Ó· ¯ÈÏÈfiÌÂÙÚÔ. µÚ›ÛΈ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜.ÔÈ· ΢ÎÏ¿ÎÈ· ı· Á˘Ú›Ûˆ;4 8°˘Ú›˙ˆ ÌfiÓÔ ÙÔ ·ÚÈÛÙÂÚfi ΢ÎÏ¿ÎÈ, ÒÛÙ ӷ ·Ó‚·›ÓÂÈ ·Ó¿ ¤Ó·Ó ·ÚÈıÌfi.µÚ›ÛΈ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ ı· Û¯ËÌ·ÙÈÛÙÔ‡Ó.4 0ÂÓ‹ÓÙ· EYPøÂÓ‹ÓÙ· ÏÂappleÙ¿2E›ÎÔÛÈ EYPøE›ÎÔÛÈ ÏÂappleÙ¿54∞ÛÎԇ̠ÙÔ˘˜ Ì·ıËÙ¤˜ ÛÙË ¯Ú‹ÛË ÙˆÓ ·ÚÈıÌÒÓ Ì¤¯ÚÈ ÙÔ 70.


ÂÓfiÙËÙ·µÚ›ÛΈ ÙÔÓ appleÚÔËÁÔ‡ÌÂÓÔ Î·È ÙÔÓ ÂapplefiÌÂÓÔ ·ÚÈıÌfi.65 66 678Ë3°Ú¿Êˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ì ϤÍÂȘ.56 6660 5670 61˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.453. ∏ ‰·ÛοϷ ϤÂÈ ¤Ó·Ó ·ÚÈıÌfi ·applefi ÙÔ 50 ̤¯ÚÈ ÙÔ 70Î·È ÔÈ Ì·ıËÙ¤˜ ‚Ú›ÛÎÔ˘Ó ÙÔÓ appleÚÔËÁÔ‡ÌÂÓÔ Î·È ÙÔÓ ÂapplefiÌÂÓÔ ·ÚÈıÌfi.55


53∂ÈÛ·ÁˆÁ‹ ÛÙÔÓ appleÔÏÏ·appleÏ·ÛÈ·ÛÌfiªÂÙÚ¿Ì ̿ÙÈ·, ·˘ÙÈ¿ Î·È ‰¿¯Ù˘Ï·ªÂÙÚ¿Ì ٷ ‰¿¯Ù˘Ï·1ÀappleÔÏÔÁ›˙ˆ Ì appleÂÓÙ¿‰Â˜ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜.1 ¯¤ÚÈ1 ÊÔÚ¿ ÙÔ 5 =2 ¯¤ÚÈ·+ =2 ÊÔÚ¤˜ ÙÔ 5 =3 ¯¤ÚÈ·+ + =3 ÊÔÚ¤˜ ÙÔ 5 =4 ¯¤ÚÈ·+ + + =4 ÊÔÚ¤˜ ÙÔ 5 =ÀappleÔÏÔÁ›˙ˆ Ì ‰Âο‰Â˜.3 ÊÔÚ¤˜ ÙÔ 10: 10 + 10 + 10 = 302 ÊÔÚ¤˜ ÙÔ 10: 10 + ......................4 ÊÔÚ¤˜ ÙÔ 10: .............................5 ÊÔÚ¤˜ ÙÔ 10: .............................56 ∂ÈÛ¿ÁÔ˘Ì ÙËÓ ¤ÓÓÔÈ· ÙÔ˘ appleÔÏÏ·appleÏ·ÛÈ·ÛÌÔ‡ ˆ˜ Âapple·Ó·Ï·Ì‚·ÓfiÌÂÓË appleÚfiÛıÂÛË.


ÂÓfiÙËÙ·8ËMÂÙÚÒ ·Ó¿ 10 ̤¯ÚÈ ÙÔ 100 Î·È ·Ó¿ 5 ̤¯ÚÈ ÙÔ 50.210 20 ... 100ªÂÙÚÒ Ù· ÎÂÚ¿ÛÈ·.3ŸÏ· Ù· ÎÂÚ¿ÛÈ· Â›Ó·È ...ªÂÙÚÒ ·Ó¿ 2.+2 +2 +2 +2 +2+240 2 4 6 8 10 122 ÊÔÚ¤˜ ÙÔ 2: 2 + 2 = 43 ÊÔÚ¤˜ ÙÔ 2: 2 + 2 + 2 = ...................4 ÊÔÚ¤˜ ÙÔ 2: ........................................5 ÊÔÚ¤˜ ÙÔ 2: ........................................6 ÊÔÚ¤˜ ÙÔ 2: ........................................2. OÈ Ì·ıËÙ¤˜ ÌÂÙÚ¿Ó appleÚÔÊÔÚÈο ·Ó¿ 10 ̤¯ÚÈ ÙÔ 100 Î·È ·Ó¿ 5 ̤¯ÚÈ ÙÔ 50.57


54ª¤ÙÚËÛË ÌÂÁÂıÒÓªÂÙÚÒ ÙËÓ appleÈı·Ì‹ ÌÔ˘1H appleÈı·Ì‹ ÌÔ˘H appleÈı·Ì‹ ÌÔ˘ Â›Ó·È ...H appleÈı·Ì‹ ÌÔ˘ Â›Ó·È ...H appleÈı·Ì‹ ÌÔ˘ Â›Ó·È ...◗ ¯Â‰È¿˙ˆ ÙËÓ appleÈı·Ì‹ ÌÔ˘ Î·È ÙË ÌÂÙÚÒ ÌÂ Û˘Ó‰ÂÙ‹Ú˜,͇ÛÙÚ˜ Î·È Û‚ËÛÙ‹Ú˜.◗ ˘ÌappleÏËÚÒÓˆ ÙÔÓ apple›Ó·Î· Ì ‚¿ÛË ÙȘ ÌÂÙÚ‹ÛÂȘ appleÔ˘ οӈ.◗ ÔÈÔ˜ Â›Ó·È Ô ÌÂÁ·Ï‡ÙÂÚÔ˜ Î·È appleÔÈÔ˜ Ô ÌÈÎÚfiÙÂÚÔ˜ ·ÚÈıÌfi˜:ÙˆÓ Û˘Ó‰ÂÙ‹ÚˆÓ, ÙˆÓ Û‚ËÛÙ‹ÚˆÓ ‹ ÙˆÓ Í˘ÛÙÚÒÓ;¢ÈηÈÔÏÔÁ‹ÛÙ ÙËÓ ·apple¿ÓÙËÛ‹ Û·˜.TÔ Ì¤ÙÚÔªÂÙÚÒ Ì ÙÔ Ì¤ÙÚÔ◗ ∏ Ù¿ÍË ÌÔ˘ ¤¯ÂÈ Ì‹ÎÔ˜ ۯ‰fiÓ... ̤ÙÚ·.◗ ∏ Ù¿ÍË ÌÔ˘ ¤¯ÂÈ appleÏ¿ÙÔ˜ ۯ‰fiÓ ... ̤ÙÚ·.58OÈ Ì·ıËÙ¤˜ ÌÂÙÚÔ‡Ó Ì‹ÎË, ÂappleÈÊ¿ÓÂȘ Î·È ¯ˆÚËÙÈÎfiÙËÙ· Ì ÌË Û˘Ì‚·ÙÈΤ˜Î·È Û˘Ì‚·ÙÈΤ˜ ÌÔÓ¿‰Â˜ ̤ÙÚËÛ˘.


ÂÓfiÙËÙ·8˪ÂÙÚÒ ÙË ¯ˆÚËÙÈÎfiÙËÙ·2ŒÓ· Ï›ÙÚÔ ÓÂÚÔ‡ ÁÂÌ›˙ÂÈ Û¯Â‰fiÓ 4 appleÔÙ‹ÚÈ·.fiÛ· appleÔÙ‹ÚÈ· ı· ÁÂÌ›ÛÂÈÙÔ ÌÈÛfi Ï›ÙÚÔ ÓÂÚÔ‡;Ãڈ̷ٛ˙ˆ Ù· appleÔÙ‹ÚÈ·appleÔ˘ ı· ÁÂÌ›ÛÂÈ.fiÛ· appleÔÙ‹ÚÈ· ı· ÁÂÌ›ÛÔ˘ÓÙ· ‰‡Ô Ï›ÙÚ· ÓÂÚÔ‡;Ãڈ̷ٛ˙ˆ Ù· appleÔÙ‹ÚÈ· appleÔ˘ ı· ÁÂÌ›ÛÔ˘Ó.ÔÈÔ ‚·ÁfiÓÈ ÎÔ˘‚·Ï¿ Ù· appleÂÚÈÛÛfiÙÂÚ· ÎÈ‚ÒÙÈ·;°Ú¿Êˆ ÛÙ· ÙÂÙÚ¿ÁˆÓ· ÙÔÓ ·ÚÈıÌfi ÙˆÓ ÎÈ‚ˆÙ›ˆÓ.3A B °59


55ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË ‰È„‹ÊȈӷÚÈıÌÒÓ (π)∆Ô appleÔ‰‹Ï·ÙÔ165 ú£·Ï‹˜ ÿÏÓÙ· Mapple¿Ìapple˘O £·Ï‹˜ ¤¯ÂÈ ... ú. ∏ ÿÏÓÙ· ¤¯ÂÈ ... ú. O ªapple¿Ìapple˘ ¤¯ÂÈ ... ú.◗ ÔÈÔ apple·È‰› ÌappleÔÚ› Ó· ·ÁÔÚ¿ÛÂÈ ÙÔ appleÔ‰‹Ï·ÙÔ; ...........................◗ fiÛ· ¯Ú‹Ì·Ù· ¯ÚÂÈ¿˙ÂÙ·È ·ÎfiÌË Ô £·Ï‹˜ ÁÈ· Ó· ·ÁÔÚ¿ÛÂÈÙÔ appleÔ‰‹Ï·ÙÔ; ...........................◗ fiÛ· ¯Ú‹Ì·Ù· ¯ÚÂÈ¿˙ÂÙ·È ·ÎfiÌË Ô ªapple¿Ìapple˘ ÁÈ· Ó· ·ÁÔÚ¿ÛÂÈÙÔ appleÔ‰‹Ï·ÙÔ; ...........................60OÈ Ì·ıËÙ¤˜ ·ÛÎÔ‡ÓÙ·È Û appleÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ‰È„‹ÊÈˆÓ ·ÚÈıÌÒÓ,·applefi ÙÔ˘˜ ÔappleÔ›Ô˘˜ ÙÔ˘Ï¿¯ÈÛÙÔÓ Ô ¤Ó·˜ Â›Ó·È ‰Âο‰·.


ÂÓfiÙËÙ·8ËÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ Ù· ·appleÔÙÂϤÛÌ·Ù· ÙˆÓ appleڿ͈Ó.2∑ˆÁÚ·Ê›˙ˆ Ù· ÏÈÁfiÙÂÚ· ÓÔÌ›ÛÌ·Ù· appleÔ˘ ı·¯ÚÂÈ·ÛÙÒ ÁÈ· Ó· ·ÁÔÚ¿Ûˆ Ù· ·ÓÙÈΛÌÂÓ·.324 ÏÂappleÙ¿45 ÏÂappleÙ¿62 ÏÂappleÙ¿ 73 ÏÂappleÙ¿ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔÓ ·ÚÈıÌfi appleÔ˘ Ï›appleÂÈ.60 + 10 = ... 43 + 20 = ... 67 – 20 = ...40 + 20 = ... 52 – 30 = ... 40 + ... = 7060 – 30 = ... 35 + 40 = ... 30 + ... = 6070 – 40 = ... 58 – 30 = ... 20 + ... = 7042. ÚÔÙ›ÓÔ˘Ì ÛÙÔ˘˜ Ì·ıËÙ¤˜ appleÚÔÛı¤ÛÂȘ Ì ·ÚÈıÌÔ‡˜ appleÔ˘ appleÂÚȤ¯Ô˘Ó ÌfiÓÔ ‰Âο‰Â˜(apple.¯. 20 + 20, 30 + 20 Î.Ïapple.).61


56EÈÛ·ÁˆÁ‹ ÛÙË Û˘ÌÌÂÙÚ›·OÈ ÛÙ¿Ìapple˜ Î·È Ô Î·ıÚ¤ÊÙ˘1BOY¢AETHºÙÈ¿¯Óˆ ÛÙ¿Ìapple˜.∆È apple·Ú·ÙËÚÒ;˘ÌappleÏËÚÒÓˆ Ù· ¯ÚÒÌ·Ù· ÛÙȘ ÛÙ¿Ìapple˜.62OÈ Ì·ıËÙ¤˜ ÂÈÛ¿ÁÔÓÙ·È Î·È ·ÛÎÔ‡ÓÙ·È ÛÙËÓ ¤ÓÓÔÈ· Ù˘ ·ÍÔÓÈ΋˜ Û˘ÌÌÂÙÚ›·˜.


ÂÓfiÙËÙ·8ËÀappleÔÏÔÁ›˙ˆ Î·È Áڿʈ Ù· ·appleÔÙÂϤÛÌ·Ù· ÙˆÓ ·Ê·ÈÚ¤ÛˆÓ.2·Ú·ÙËÚÒ ÙȘ ÂÈÎfiÓ˜ ηȂ¿˙ˆ Û ·ÎÏÔ ÙȘ Û˘ÌÌÂÙÚÈΤ˜.3·Ú·ÙËÚÒ ·˘Ùfi ÙÔ¯ÚÈÛÙÔ˘ÁÂÓÓÈ¿ÙÈÎÔ ‰¤ÓÙÚÔ.OÈ Ìapple¿Ï˜ Â›Ó·È ¯ÚˆÌ·ÙÈṲ̂ÓÂ˜Û˘ÌÌÂÙÚÈο;Ãڈ̷ٛ˙ˆ ÙȘ Ìapple¿Ï˜ Û˘ÌÌÂÙÚÈο.42. £¤ÙÔ˘Ì ÛÙÔ˘˜ Ì·ıËÙ¤˜ ·Ê·ÈÚ¤ÛÂȘ ÛÙȘ Ôappleԛ˜ Ô ·Ê·ÈÚÂÙ¤Ô˜ Â›Ó·È ÌÂÁ¿ÏÔ˜ ·ÚÈıÌfi˜, ÂÓÒ apple·Ú¿ÏÏËÏ·appleÚÔÛʤÚÔÓÙ·È ÁÈ· Âapple›Ï˘ÛË Ì appleÚfiÛıÂÛË appleÚÔ˜ Ù· Âapple¿Óˆ (apple.¯. 15 – 9, 13 – 8 Î.Ïapple.).63


57Âapple·Ó·ÏËappleÙÈÎfiÌ¿ıËÌ·123 x 4567+896538-Âapple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·∏ appleÂÙ·ÏÔ‡‰·1∞ÓÙÈÁڿʈ ÙÔ Û¯‹Ì· Û ¤Ó· ʇÏÏÔ ¯·ÚÙÈÔ‡.∆Ô ‰ÈappleÏÒÓˆ Î·È Îfi‚ˆ.∞ÓÔ›Áˆ.ÀappleÔÏÔÁ›˙ˆ Î·È Û˘ÌappleÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ appleÔ˘ Ï›appleÔ˘Ó.15 + 4 = ... 12 + 5 = ...6 + ... = 26 17 – 2 = ...12+ 619– 5216 – 2 = ... 21 + ... = 2964


72x15ÂÓfiÙËÙ·µÚ›ÛΈ Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi.8Ë3·Ú·ÙËÚÒ Ù· apple·Ú¿ı˘Ú· Î·È ÙËÓ applefiÚÙ·. ÔÈÔ Â›Ó·È ÙÔ appleÈÔ ÌÂÁ¿ÏÔ;4Ãڈ̷ٛ˙ˆ Ù· apple·Ú¿ı˘Ú· Î·È ÙËÓ applefiÚÙ· Ì ٷ ¯ÚÒÌ·Ù·appleÔ˘ appleÚ¤appleÂÈ, ÍÂÎÈÓÒÓÙ·˜ ·applefi ÙÔ ÌÈÎÚfiÙÂÚÔÎ·È appleÚÔ¯ˆÚÒÓÙ·˜ appleÚÔ˜ ÙÔ ÌÂÁ·Ï‡ÙÂÚÔ.1Ô 2Ô 3Ԍӷ ÎÔ˘Ù› ¤¯ÂÈ Ì¤Û· 5 ÎËÚÔÌappleÔÁȤ˜. fiÛ˜ÎËÚÔÌappleÔÁȤ˜ ı· ˘apple¿Ú¯Ô˘Ó ̤۷ Û 4 ÎÔ˘ÙÈ¿;5ª¤Û· Û 4 ÎÔ˘ÙÈ¿ ı· ˘apple¿Ú¯Ô˘ÓÎËÚÔÌappleÔÁȤ˜.3. ÚÔÙ›ÓÔ˘Ì οı ÊÔÚ¿ ¤Ó·Ó ‰È„‹ÊÈÔ ·ÚÈıÌfi ·applefi ÙÔ 50 ̤¯ÚÈ ÙÔ 70, ·Ó·Ê¤ÚÔÓÙ·˜ ÙÔÓ ·ÚÈıÌfiÙˆÓ ÌÔÓ¿‰ˆÓ Î·È ÙˆÓ ‰Âο‰ˆÓ ÙÔ˘ (apple.¯. «appleÔÈÔ˜ Â›Ó·È Ô ·ÚÈıÌfi˜ appleÔ˘ ¤¯ÂÈ 4 ÌÔÓ¿‰Â˜ Î·È 6 ‰Âο‰Â˜;»).65


123 x 4567+89


EÓfiÙËÙ· 9Ë:API£MOI MEXPI TO 100 – PA•EI –BAPO – °EøMETPIKA XHMATA6538-72x1584258596061626364∫ÂÊ¿Ï·ÈÔ 58Ô:√È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 100 – ÃÚ‹Ì·∫ÂÊ¿Ï·ÈÔ 59Ô:∂ÌappleÂÈÚÈÎfi˜ appleÔÏÏ·appleÏ·ÛÈ·ÛÌfi˜Î·È ‰È·›ÚÂÛË∫ÂÊ¿Ï·ÈÔ 60fi:µ¿ÚÔ˜ – §ÂÈÙÔ˘ÚÁ›· ˙˘Á·ÚÈ¿˜∫ÂÊ¿Ï·ÈÔ 61Ô:÷ڿÍÂȘ Û¯ËÌ¿ÙˆÓ– ·˙Ï Î·È appleÏ·ÎfiÛÙÚˆÙÔ∫ÂÊ¿Ï·ÈÔ 62Ô:ÚÔ‚Ï‹Ì·Ù·∫ÂÊ¿Ï·ÈÔ 63Ô:∂apple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·∫ÂÊ¿Ï·ÈÔ 64Ô:3Ô ∫ÚÈÙ‹ÚÈÔ AÍÈÔÏfiÁËÛ˘EÓfiÙËÙ· 9ËÙÔ ∫ÂÊ¿Ï·ÈÔ 58 apple·›˙ÔÓÙ·˜ ÙÔ apple·È-¯Ó›‰È «ºÈ‰¿ÎÈ» ı· Ì¿ıÔ˘Ì ÙÔ˘˜·ÚÈıÌÔ‡˜ ̤¯ÚÈ ÙÔ 100. ÙÔ Âapplefi-ÌÂÓÔ ÎÂÊ¿Ï·ÈÔ ı· ·ÎÔ‡ÛÔ˘Ì ¤Ó·apple·Ú·Ì‡ıÈ Ì ٷ ÙÚ›· ÁÔ˘ÚÔ˘Ó¿ÎÈ·Î·È Ì¤Û· ·applefi ÙËÓ Âapple›Ï˘ÛË Û¯ÂÙÈ-ÎÒÓ appleÚÔ‚ÏËÌ¿ÙˆÓ ı· ·ÛÎËıÔ‡ÌÂÛÙÔÓ appleÔÏÏ·appleÏ·ÛÈ·ÛÌfi Î·È ÙȘ ÌÔÈ-Ú·ÛȤ˜.ÙÔ ∫ÂÊ¿Ï·ÈÔ 60 ı· appleÂÈÚ·Ì·ÙÈ-ÛÙԇ̠̠ÙË Ì¤ÙÚËÛË ÙÔ˘ ‚¿ÚÔ˘˜Î·È ı· Ì¿ıÔ˘Ì ÁÈ· Ù· ‰È¿ÊÔÚ· ›-‰Ë ˙˘Á·ÚÈÒÓ. ∫·ÙfiappleÈÓ ÛÙÔ ∫ÂÊ¿-Ï·ÈÔ 61 ı· ·Û¯ÔÏËıÔ‡ÌÂ Î·È apple¿ÏÈÌ ¯·Ú¿ÍÂȘ, apple·˙Ï Î·È appleÏ·Îfi-ÛÙÚˆÙ·, ÂÓÒ ÛÙÔ ÂapplefiÌÂÓÔ ÎÂÊ¿-Ï·ÈÔ ı· χÛÔ˘Ì appleÚÔ‚Ï‹Ì·Ù·.T· 100 EYPø3 3 3 3 12


58OÈ ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 100 – ÃÚ‹Ì·TÔ Êȉ¿ÎÈ∫·ÓfiÓ˜ ÙÔ˘ apple·È¯ÓȉÈÔ‡ÃÚÂÈ¿˙ÂÙ·È ¤Ó· ˙¿ÚÈ ÛÙÔ ÔappleÔ›Ô ÔÈappleÏ¢ڤ˜ Ì ÙÔ˘˜ ·ÚÈıÌÔ‡˜ 4, 5Î·È 6 ı· ¤¯Ô˘Ó ÎfiÎÎÈÓÔ ¯ÚÒÌ·,ÂÓÒ ÔÈ appleÏ¢ڤ˜ Ì ÙÔ˘˜·ÚÈıÌÔ‡˜ 1, 2 Î·È 3 ı· ¤¯Ô˘ÓÌappleÏ ¯ÚÒÌ·.•ÂÎÈÓ¿Ì ·applefi ÙÔ 0. ∫¿ı apple·›ÎÙ˘Ì ÙË ÛÂÈÚ¿ Ú›¯ÓÂÈ ÙÔ ˙¿ÚÈ.ŸÙ·Ó ÙÔ ˙¿ÚÈ ‰Â›¯ÓÂÈ ÌÈ· ÎfiÎÎÈÓËappleÏÂ˘Ú¿, appleÚÔ¯ˆÚԇ̠ÌappleÚÔÛÙ¿ Ùfi-Û˜ ı¤ÛÂȘ fiÛ˜ ‰Â›¯ÓÂÈ ÙÔ ˙¿ÚÈ.ŸÙ·Ó ÙÔ ˙¿ÚÈ ‰Â›¯ÓÂÈ ÌÈ· ÌappleÏÂappleÏÂ˘Ú¿, appleËÁ·›ÓÔ˘Ì apple›Ûˆ ÙfiÛ˜ı¤ÛÂȘ fiÛ˜ ‰Â›¯ÓÂÈ ÙÔ ˙¿ÚÈ.¡ÈÎËÙ‹˜ ı· Â›Ó·È ·˘Ùfi˜ appleÔ˘ ı·ÊÙ¿ÛÂÈ appleÚÒÙÔ˜ ÛÙÔ 100.1˘ÌappleÏËÚÒÓˆ Ù· ÛÙÔȯ›· appleÔ˘ Ï›appleÔ˘Ó ÛÙÔÓ apple·Ú·Î¿Ùˆ apple›Ó·Î·.AÚÈıÌfi˜ ·applefi ÙÔÓÔappleÔ›Ô ÍÂΛÓËÛ·ÓZ¿ÚÈ£¿ÏÂÈ· 30 4¢ËÌ‹ÙÚ˘ 80 3KÔÚ›Ó· 40 6AÚÈıÌfi˜ ÛÙÔÓÔappleÔ›Ô ¤ÊÙ·Û·ÓÌ·ı·›ÓˆT· 100 úT· 100 ÏÂappleÙ¿ ·Í›˙Ô˘Ó fiÛÔ 1ú10 ÊÔÚ¤˜ ÙÔ 10 =68∞ÛÎԇ̠ÙÔ˘˜ Ì·ıËÙ¤˜ ÛÙË ¯Ú‹ÛË ÙˆÓ ·ÚÈıÌÒÓ Ì¤¯ÚÈ ÙÔ 100.


ÂÓfiÙËÙ·ªÂÙÚÒ ·Ó¿ 10 ̤¯ÚÈ ÙÔ 100 Î·È ·ÓÙ›ÛÙÚÔÊ·.9Ë210 20 ... 10080ÔÁ‰fiÓÙ·ÔÁ‰fiÓÙ· ÂÓÓ¤·°Ú¿Êˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ì ϤÍÂȘ.90ÂÓÂÓ‹ÓÙ·10089 97 7268 86 99ÂηÙfi3¢È·‚¿˙ˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Î·È Û˘ÌappleÏËÚÒÓˆ Ù· ÎÂÓ¿.∆ M6 868 = ÂÍ‹ÓÙ· ÔÎÙÒ 68 = 60 + 8∆ M76 = ....................................... 76 = .....................4∆ M88 = ....................................... 88 = .....................∆ M90 = ....................................... 90 = .....................∆ M99 = ....................................... 99 = .....................2. OÈ Ì·ıËÙ¤˜ ÌÂÙÚÔ‡Ó ·Ó¿ 10 ̤¯ÚÈ ÙÔ 100. ∂apple›Û˘ ·Ó‚·›ÓÔ˘Ó ·Ó¿ 10 ÍÂÎÈÓÒÓÙ·˜ ·applefi ÌÈ· ÔappleÔÈ·‰‹appleÔÙ‰Âο‰·. ∆¤ÏÔ˜, ηÙ‚·›ÓÔ˘Ó ·Ó¿ 10 ·applefi ÙÔ 100.69


59ÔÏÏ·appleÏ·ÛÈ·ÛÌfi˜ Î·È ‰È·›ÚÂÛË∆· ÙÚ›· ÁÔ˘ÚÔ˘Ó¿ÎÈ·1∆· ÙÚ›· ÁÔ˘ÚÔ˘Ó¿ÎÈ· ‚Á‹Î·Ó ÛÙÔ ‰¿ÛÔ˜ ÁÈ· Ó· Ê¿ÓÂ.∫¿ı ÁÔ˘ÚÔ˘Ó¿ÎÈ ¤Ê·ÁÂ:‰‡Ô Ì·ÓÈÙ¿ÚÈ·◗ fiÛ· Ì·ÓÈÙ¿ÚÈ· ¤Ê·Á·Ó Î·È Ù· ÙÚ›· ÁÔ˘ÚÔ˘Ó¿ÎÈ·Ì·˙›;ŸÏ· Ì·˙› ¤Ê·Á·Ó ... Ì·ÓÈÙ¿ÚÈ·.Ù¤ÛÛÂÚ· ‚ÂÏ·Ó›‰È·◗ fiÛ· ‚ÂÏ·Ó›‰È· ¤Ê·Á·Ó Î·È Ù· ÙÚ›· ÁÔ˘ÚÔ˘Ó¿ÎÈ·Ì·˙›;ŸÏ· Ì·˙› ¤Ê·Á·Ó ... ‚ÂÏ·Ó›‰È·.T· ÙÚ›· ÁÔ˘ÚÔ˘Ó¿ÎÈ· ‚Ú‹Î·Ó ÛÙÔ ‰¿ÛÔ˜ 9 ηڇ‰È· ÂÓÓ¤· ηڇ‰È·Î·È ·appleÔÊ¿ÛÈÛ·Ó Ó· Ù· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ÌÂٷ͇ ÙÔ˘˜.fiÛ· ηڇ‰È· ı· apple¿ÚÂÈ Î¿ı ÁÔ˘ÚÔ˘Ó¿ÎÈ;∫¿ı ÁÔ˘ÚÔ˘Ó¿ÎÈ ı· apple¿ÚÂÈ ... ηڇ‰È·.70∂Í·ÛÎԇ̠ÙÔ˘˜ Ì·ıËÙ¤˜ Û ÂÌappleÂÈÚÈΤ˜ ηٷÛÙ¿ÛÂȘ appleÔÏÏ·appleÏ·ÛÈ·ÛÌÔ‡ Ì ÙË ÌÔÚÊ‹ Ù˘Âapple·Ó·Ï·Ì‚·ÓfiÌÂÓ˘ appleÚfiÛıÂÛ˘ Î·È ‰È·›ÚÂÛ˘ Ì ÙË ÌÔÚÊ‹ Ù˘ ÌÔÈÚ·ÛÈ¿˜.


ªÂÙÚÒ ·Ó¿ 10 ̤¯ÚÈ ÙÔ 100 Î·È ·Ó¿ 5 ̤¯ÚÈ ÙÔ 50.10 20 ... 100ÂÓfiÙËÙ·9Ë21 ÏÔ˘ÏÔ‡‰ÈÀappleÔÏÔÁ›˙ˆ Ù· apple¤Ù·Ï·.apple¤Ù·Ï· 1 ÊÔÚ¿ ÙÔ 3 = 32 ÏÔ˘ÏÔ‡‰È·+ =3 ÏÔ˘ÏÔ‡‰È·apple¤Ù·Ï· 2 ÊÔÚ¤˜ ÙÔ 3 =+ + =apple¤Ù·Ï· 3 ÊÔÚ¤˜ ÙÔ 3 =4 ÏÔ˘ÏÔ‡‰È·+ + + =4 ÊÔÚ¤˜ ÙÔ 3 =1 ·˘ÙÔΛÓËÙÔÀappleÔÏÔÁ›˙ˆ ÙȘ Úfi‰Â˜.Úfi‰Â˜ 1 ÊÔÚ¿ ÙÔ 4 = 42 ·˘ÙÔΛÓËÙ·+ =3 ·˘ÙÔΛÓËÙ·+ + =4 ·˘ÙÔΛÓËÙ·+ + + =Úfi‰Â˜ 2 ÊÔÚ¤˜ ÙÔ 4 =5 ·˘ÙÔΛÓËÙ·Úfi‰Â˜ 3 ÊÔÚ¤˜ ÙÔ 4 =2. OÈ Ì·ıËÙ¤˜ ÌÂÙÚÔ‡Ó appleÚÔÊÔÚÈο ·Ó¿ 10 ̤¯ÚÈ ÙÔ 100 Î·È ·Ó¿ 5 ̤¯ÚÈ ÙÔ 50.Úfi‰Â˜ 4 ÊÔÚ¤˜ ÙÔ 4 =+ + + + =5 ÊÔÚ¤˜ ÙÔ 4 =3471


60µ¿ÚÔ˜ – §ÂÈÙÔ˘ÚÁ›· ˙˘Á·ÚÈ¿˜¢È¿ÊÔÚ˜ ˙˘Á·ÚȤ˜.1ÔÈÔ Â›Ó·È ‚·Ú‡ÙÂÚÔ;∑˘Á›˙ˆ Ù· ‰È¿ÊÔÚ· ·ÓÙÈΛÌÂÓ· Î·È ‚Ú›ÛΈ ÙÔ appleÈÔ ‚·Ú‡.∆Ú›· ›‰È· ÎÔ˘ÙÈ¿ Â›Ó·È ÁÂÌ¿Ù· Ì ¿ÌÌÔ, ηʤ Î·È ‚·Ì‚¿ÎÈ.∞ÚÈıÌÒ Ù· ÎÔ˘ÙÈ¿ Ì ÙÔ˘˜ ·ÚÈıÌÔ‡˜ 1, 2 Î·È 3ÍÂÎÈÓÒÓÙ·˜ ·applefi ÙÔ ‚·Ú‡ÙÂÚÔ Î·È appleÚÔ¯ˆÚÒÓÙ·˜ appleÚÔ˜ ÙÔ ÂÏ·ÊÚ‡ÙÂÚÔ.‚·Ì‚¿ÎÈ ¿ÌÌÔ˜ ηʤ˜72OÈ Ì·ıËÙ¤˜ ¤Ú¯ÔÓÙ·È Û appleÚÒÙË Âapple·Ê‹ Ì ÙËÓ ¤ÓÓÔÈ· ÙÔ˘ ‚¿ÚÔ˘˜ Î·È ÙË Ì¤ÙÚËÛ‹ ÙÔ˘.


ÂÓfiÙËÙ·9Ë·Ú·ÙËÚԇ̠ÙȘ ÂÈÎfiÓ˜ Î·È Û˘˙ËÙ¿ÌÂ.2◗ ∏ ·ÚÎÔ‡‰· Â›Ó·È ‚·Ú‡ÙÂÚË ·applefi Û¤Ó·;◗ ∆Ô ·ÚÎÔ˘‰¿ÎÈ ˙˘Á›˙ÂÈ fiÛÔ Î·È Ë ÌÂÁ¿ÏË ·ÚÎÔ‡‰·;◗ O ÂϤʷÓÙ·˜ Â›Ó·È ÂÏ·ÊÚ‡ÙÂÚÔ˜ ·applefi ÙËÓ ·ÚÎÔ‡‰·;◗ ∆Ô ÂÏÂÊ·ÓÙ¿ÎÈ ˙˘Á›˙ÂÈ fiÛÔ Ô ÌÂÁ¿ÏÔ˜ ÂϤʷÓÙ·˜;◗ ÔÈ· ¿ÏÏ· ˙Ò· Ì ÌÂÁ¿ÏÔ ‚¿ÚÔ˜ ÁÓˆÚ›˙ÂȘ;µ¿˙ˆ Û ·ÎÏÔ Ù· appleÚ¿ÁÌ·Ù· appleÔ˘ ÓÔÌ›˙ˆfiÙÈ Â›Ó·È appleÈÔ ÂÏ·ÊÚ¿ ·applefi ¤Ó· ‚È‚Ï›Ô.3----------------------------------------ÎÔ˘Ù¿ÏÈ ÎÔ˘Ó¤ÏÈ Ìapple·ÏfiÓÈ Û·ÏÔ˜ ÏÔ˘ÏÔ‡‰È73


61÷ڿÍÂȘ Û¯ËÌ¿ÙˆÓ – ·˙Ï –Ï·ÎfiÛÙÚˆÙÔ∏ ‰È·‰ÚÔÌ‹ ÙÔ˘ ÎÔ˘ÓÂÏÈÔ‡¢È·‚¿˙ˆ ÙȘ ÈÛÙÔڛ˜ Î·È ÙÚ·‚Ò ÁÚ·Ì̤˜.1◗ ŒÓ· ÎÔ˘ÓÂÏ¿ÎÈ ÍÂΛÓËÛ ·applefi ÙË ÊˆÏÈ¿ ÙÔ˘Î·È ¤ÊÙ·Û ̤¯ÚÈ ÙÔ Ì¤ÚÔ˜ fiappleÔ˘ ˘apple¿Ú¯Ô˘Ó Ù· ηÚfiÙ·.∞applefi ÂΛ apple‹Á ÛÙÔ Î›ÙÚÈÓÔ ÏÔ˘ÏÔ‡‰È.•ÂÎÔ˘Ú¿ÛÙËΠÏÈÁ¿ÎÈ Î·È Âapple¤ÛÙÚ„ ÛÙË ÊˆÏÈ¿ ÙÔ˘.∆È Û¯‹Ì· ¤¯ÂÈ Ë ‰È·‰ÚÔÌ‹ ÙÔ˘;∞apple¿ÓÙËÛË: .....................................................................................◗ ∆Ô ÎÔ˘ÓÂÏ¿ÎÈ ÍÂΛÓËÛ ·applefi ÙÔ appleÚ¿ÛÈÓÔ Ï¿¯·ÓÔÎ·È apple‹Á ÛÙÔ ÎfiÎÎÈÓÔ ÏÔ˘ÏÔ‡‰È. ∞applefi ÙÔ ÎfiÎÎÈÓÔÏÔ˘ÏÔ‡‰È apple‹Á ÛÙÔ Î›ÙÚÈÓÔ Ï¿¯·ÓÔ Î·È ·applefi ÂΛÛÙÔ ÌÔ‚ ÏÔ˘ÏÔ‡‰È. ∞applefi ÙÔ ÌÔ‚ ÏÔ˘ÏÔ‡‰ÈÂapple¤ÛÙÚ„ ÛÙÔ appleÚ¿ÛÈÓÔ Ï¿¯·ÓÔ.∆È Û¯‹Ì· ¤¯ÂÈ Ë ‰È·‰ÚÔÌ‹ ÙÔ˘;∞apple¿ÓÙËÛË: .....................................................................................74OÈ Ì·ıËÙ¤˜ ı· Û˘Ó¯›ÛÔ˘Ó ÙËÓ ÂÍ¿ÛÎËÛ‹ ÙÔ˘˜ ÛÙȘ ¯·Ú¿ÍÂȘ ÙˆÓ Û¯ËÌ¿ÙˆÓ,ÛÙË Û‡ÓıÂÛË apple·˙Ï Î·È ÛÙËÓ Î·Ù·Û΢‹ appleÏ·ÎfiÛÙÚˆÙˆÓ.


ÂÓfiÙËÙ·9ˢÌappleÏËÚÒÓˆ ÙȘ ÁÚ·Ì̤˜ Ì ÙÔ¯¿Ú·Î· Î·È Û˘Ó¯›˙ˆ ÙÔ ¯ÚˆÌ·ÙÈÛÌfi Ì ÙÔÓ ›‰ÈÔ ÙÚfiappleÔ.2Ãڈ̷ٛ˙ˆ οı ÎÔÌÌ¿ÙÈ Ì ÙÔ ›‰ÈÔ ¯ÚÒÌ·.375


62appleÚÔ‚Ï‹Ì·Ù· appleÚÔ‚Ï‹Ì·Ù·123 x 4567+896538-ÚÔ‚Ï‹Ì·Ù·OÈ Î¿ÚÙ˜ Ì ٷ ·ÁÁ›·1∏ ¡·Ù·Ï›· ·ÁfiÚ·Û οÚÙ˜ ·applefi ÙÔ ÌÔ˘Û›Ô.ÿÚÈÛ ÛÙÔ Ê›ÏÔ Ù˘ ÙÔÓ ÃÚ‹ÛÙÔ 8 οÚÙÂ˜Î·È Ù˘ ¤ÌÂÈÓ·Ó ·˘Ù¤˜ appleÔ˘ ‚ϤappleÂȘ.fiÛ˜ ·ÁfiÚ·ÛÂ Û˘ÓÔÏÈο;∞apple¿ÓÙËÛË: ∏ ¡·Ù·Ï›· ·ÁfiÚ·Û ... οÚÙ˜ ·applefi ÙÔ ÌÔ˘Û›Ô.˘˙ËÙ¿Ì ÛÙËÓ Ù¿ÍË ÙÈ ‰Â›¯ÓÔ˘Ó ·˘Ù¤˜ ÔÈ Î¿ÚÙ˜.76


72x15ÂÓfiÙËÙ·∏ Ù¿ÍË Ù˘ ÃÚÈÛÙ›Ó·˜ ¤¯ÂÈ 18 apple·È‰È¿.∂appleÙ¿ ·applefi ·˘Ù¿ ÊÔÚÔ‡Ó Á˘·ÏÈ¿.fiÛ· apple·È‰È¿ ‰Â ÊÔÚÔ‡Ó Á˘·ÏÈ¿;9Ë2∞apple¿ÓÙËÛË: ... apple·È‰È¿ ‰Â ÊÔÚÔ‡Ó Á˘·ÏÈ¿.ºÙÈ¿¯ÓÔ˘Ì ¤Ó· apple·ÚfiÌÔÈÔ appleÚfi‚ÏËÌ· ÁÈ· ÙË ‰È΋ Ì·˜ Ù¿ÍË Î·È ÙÔ Ï‡ÓÔ˘ÌÂ.∏ Ù¿ÍË Ì·˜ ¤¯ÂÈ ......... apple·È‰È¿............ ·applefi ·˘Ù¿ ......................................................fiÛ· apple·È‰È¿ ............................................................ ;∞apple¿ÓÙËÛË: .............................................................. οı ʈÏÈ¿ ˘apple¿Ú¯Ô˘Ó 3 appleÔ˘Ï¿ÎÈ·.ÙÔ ‰¤ÓÙÚÔ ˘apple¿Ú¯Ô˘Ó 4 ʈÏȤ˜.fiÛ· Â›Ó·È fiÏ· Ù· appleÔ˘Ï¿ÎÈ· ÛÙÔ ‰¤ÓÙÚÔ;377


63Âapple·Ó·ÏËappleÙÈÎfiÌ¿ıËÌ·123 x 4567+896538-Âapple·Ó·ÏËappleÙÈÎfi Ì¿ıËÌ·Ãڈ̷ٛ˙ˆ Ì ÙÔ ›‰ÈÔ ¯ÚÒÌ·Ù· Û¯‹Ì·Ù· appleÔ˘ Â›Ó·È ›‰È·.12ªÔ˘ ·Ú¤ÛÂÈ appleÔχ Ó· ÙÚ҈ηÚfiÙ·.∞Ó Î¿ı ̤ڷ ÙÚÒˆ3 ηÚfiÙ·, applefiÛ· ηÚfiÙ·ı· Ê¿ˆ Û ÌÈ· ‚‰ÔÌ¿‰·;78


72x15ÂÓfiÙËÙ·µÚ›ÛΈ Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi.9Ë3˘ÌappleÏËÚÒÓˆ Ì ÙÔ ¯¿Ú·Î· ÙÔ ‰Â‡ÙÂÚÔ Û¯‹Ì·ÒÛÙ ӷ Á›ÓÂÈ ›‰ÈÔ Ì ÙÔ appleÚÒÙÔ.4µ¿˙ˆ Û ·ÎÏÔ ÙÔ ÂÏ·ÊÚ‡ÙÂÚÔ.5appleÔÚÙÔοÏÈ ÏÂÌfiÓÈ OÈ Ìapple·Ó¿Ó˜ ˙˘Á›˙Ô˘Ó ................3. ÚÔÙ›ÓÔ˘Ì οı ÊÔÚ¿ ¤Ó·Ó ‰È„‹ÊÈÔ ·ÚÈıÌfi ·applefi ÙÔ 70 ̤¯ÚÈ ÙÔ 100, ·Ó·Ê¤ÚÔÓÙ·˜ ÙÔÓ ·ÚÈıÌfi ÙˆÓÌÔÓ¿‰ˆÓ Î·È ÙˆÓ ‰Âο‰ˆÓ ÙÔ˘ (apple.¯. «appleÔÈÔ˜ Â›Ó·È Ô ·ÚÈıÌfi˜ appleÔ˘ ¤¯ÂÈ 3 ÌÔÓ¿‰Â˜ Î·È 9 ‰Âο‰Â˜;»).79


ªÂ ·applefiÊ·ÛË Ù˘ ∂ÏÏËÓÈ΋˜ ∫˘‚¤ÚÓËÛ˘ Ù· ‰È‰·ÎÙÈο ‚È‚Ï›· ÙÔ˘¢ËÌÔÙÈÎÔ‡, ÙÔ˘ °˘ÌÓ·Û›Ô˘ Î·È ÙÔ˘ §˘Î›Ԣ Ù˘appleÒÓÔÓÙ·È ·applefi ÙÔÓ√ÚÁ·ÓÈÛÌfi ∂ΉfiÛˆ˜ ¢È‰·ÎÙÈÎÒÓ µÈ‚Ï›ˆÓ Î·È ‰È·Ó¤ÌÔÓÙ·È ‰ˆÚÂ-¿Ó ÛÙ· ¢ËÌfiÛÈ· ¯ÔÏ›·. ∆· ‚È‚Ï›· ÌappleÔÚ› Ó· ‰È·Ù›ıÂÓÙ·È appleÚÔ˜appleÒÏËÛË, fiÙ·Ó Ê¤ÚÔ˘Ó ‚È‚ÏÈfiÛËÌÔ appleÚÔ˜ ·applefi‰ÂÈÍË Ù˘ ÁÓËÛÈfiÙË-Ù¿˜ ÙÔ˘˜. ∫¿ı ·ÓÙ›Ù˘appleÔ appleÔ˘ ‰È·Ù›ıÂÙ·È appleÚÔ˜ appleÒÏËÛË Î·È ‰Â ʤÚÂÈ‚È‚ÏÈfiÛËÌÔ, ıˆÚÂ›Ù·È ÎÏ„›Ù˘appleÔ Î·È Ô apple·Ú·‚¿Ù˘ ‰ÈÒÎÂÙ·È Û‡Ì-ʈӷ Ì ÙȘ ‰È·Ù¿ÍÂȘ ÙÔ˘ ¿ÚıÚÔ˘ 7, ÙÔ˘ ¡fiÌÔ˘ 1129 Ù˘ 15/21ª·ÚÙ›Ô˘ 1946 (ºEK 1946, 108, Aã).BIB§IOHMO∞apple·ÁÔÚ‡ÂÙ·È Ë ·Ó·apple·Ú·ÁˆÁ‹ ÔappleÔÈÔ˘‰‹appleÔÙ ÙÌ‹Ì·ÙÔ˜ ·˘ÙÔ‡ ÙÔ˘ ‚È‚Ï›Ô˘, appleÔ˘Î·Ï‡appleÙÂÙ·È ·applefi ‰ÈηÈÒÌ·Ù· (copyright), ‹ Ë ¯Ú‹ÛË ÙÔ˘ Û ÔappleÔÈ·‰‹appleÔÙ ÌÔÚÊ‹, ¯ˆÚ›˜ÙË ÁÚ·appleÙ‹ ¿‰ÂÈ· ÙÔ˘ ·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘.


123 x 4567+896538-72x1581


123 x 4567+896538-72x1583


KÂÊ¿Ï·ÈÔ 40KÂÊ¿Ï·ÈÔ 57

More magazines by this user
Similar magazines