Workbook 3rd Issue

M·ıËÌ·ÙÈο Aã¢ËÌÔÙÈÎÔ‡ M·ıËÌ·ÙÈο Ù˘ º‡Û˘ Î·È Ù˘ Zˆ‹˜

TÂÙÚ¿‰ÈÔ EÚÁ·ÛÈÒÓ TPITO TEYXO™

™Y°°PAºEI™

KPITE™-A•IO§O°HTE™

EIKONO°PAºH™H ºI§O§O°IKH E¶IME§EIA Y¶EY£YNO™ TOY MA£HMATO™ KATA TH ™Y°°PAºH Y¶EY£YNH TOY Y¶OEP°OY

E•øºY§§O ¶POEKTY¶øTIKE™ EP°A™IE™

÷ڿϷÌÔ˜ §ÂÌÔÓ›‰Ë˜, ∫·ıËÁËÙ‹˜ ÙÔ˘ ¶·ÓÂÈÛÙËÌ›Ô˘ ¢˘ÙÈ΋˜ ª·Î‰ÔÓ›·˜ Aı·Ó¿ÛÈÔ˜ £ÂÔ‰ÒÚÔ˘, ∂Î·È‰Â˘ÙÈÎfi˜ A¯ÈÏϤ·˜ K·„¿Ï˘, K·ıËÁËÙ‹˜ ÙÔ˘ ¶·ÓÂÈÛÙËÌ›Ô˘ M·Î‰ÔÓ›·˜ ¢ËÌ‹ÙÚÈÔ˜ ¶ÓÂ˘Ì·ÙÈÎfi˜, §¤ÎÙÔÚ·˜ ÙÔ˘ ¶·ÓÂÈÛÙËÌ›Ô˘ ¢˘ÙÈ΋˜ M·Î‰ÔÓ›·˜ £ÂÔ‰fiÛÈÔ˜ Z·¯·ÚÈ¿‰Ë˜, AÓ·ÏËÚˆÙ‹˜ K·ıËÁËÙ‹˜ ÙÔ˘ ¶·ÓÂÈÛÙËÌ›Ô˘ AıËÓÒÓ M·Ú›· KÔÙÛ·ÎÒÛÙ·, ™¯ÔÏÈ΋ ™‡Ì‚Ô˘ÏÔ˜ £ÂfiÊÈÏÔ˜ T˙ÒÚÙ˙˘, ∂Î·È‰Â˘ÙÈÎfi˜

∫ˆÓÛÙ·ÓÙ›ÓÔ˜ ∞ÚÒÓ˘, ™ÎÈÙÛÔÁÚ¿ÊÔ˜-∂ÈÎÔÓÔÁÚ¿ÊÔ˜ ºÚfiÛˆ •ÈÍ‹, ºÈÏfiÏÔÁÔ˜

°ÂÒÚÁÈÔ˜ ∆‡·˜, ªfiÓÈÌÔ˜ ¶¿Ú‰ÚÔ˜ ÙÔ˘ ¶·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘ M·Ú›· XÈÔÓ›‰Ô˘-MÔÛÎÔÊfiÁÏÔ˘, E›ÎÔ˘ÚÔ˜ K·ıËÁ‹ÙÚÈ· ÙÔ˘ ¶·ÓÂÈÛÙËÌ›Ô˘ AÈÁ·›Ô˘ AÓ‰Ú¤·˜ °ÎÔÏÊÈÓfiÔ˘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 ÚÔÁÚ·ÌÌ¿ÙˆÓ ÛÔ˘‰ÒÓ Î·È Û˘ÁÁÚ·Ê‹ Ó¤ˆÓ ÂÎ·È‰Â˘ÙÈÎÒÓ ·Î¤ÙˆÓ» ¶∞π¢∞°ø°π∫√ π¡™∆π∆√À∆√ ªÈ¯¿Ï˘ ∞Á. ¶··‰fiÔ˘ÏÔ˜ √ÌfiÙÈÌÔ˜ ∫·ıËÁËÙ‹˜ ÙÔ˘ ∞.¶.£. ¶Úfi‰ÚÔ˜ ÙÔ˘ ¶·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘ ¶Ú¿ÍË Ì ٛÙÏÔ:

«™˘ÁÁÚ·Ê‹ Ó¤ˆÓ ‚È‚Ï›ˆÓ Î·È ·Ú·ÁˆÁ‹ ˘ÔÛÙËÚÈÎÙÈÎÔ‡ ÂÎ·È‰Â˘ÙÈÎÔ‡ ˘ÏÈÎÔ‡ Ì ‚¿ÛË ÙÔ ¢∂¶¶™ Î·È Ù· ∞¶™ ÁÈ· ÙÔ ¢ËÌÔÙÈÎfi Î·È ÙÔ NËÈ·ÁˆÁ›Ի ∂ÈÛÙËÌÔÓÈÎfi˜ À‡ı˘ÓÔ˜ ŒÚÁÔ˘ °ÂÒÚÁÈÔ˜ ∆‡·˜ MfiÓÈÌÔ˜ ¶¿Ú‰ÚÔ˜ ÙÔ˘ ¶·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘ ∞Ó·ÏËÚˆÙ‹˜ ∂ÈÛÙËÌÔÓÈÎfi˜ À‡ı˘ÓÔ˜ ŒÚÁÔ˘ °ÂÒÚÁÈÔ˜ √ÈÎÔÓfiÌÔ˘ MfiÓÈÌÔ˜ ¶¿Ú‰ÚÔ˜ ÙÔ˘ ¶·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘

ŒÚÁÔ Û˘Á¯ÚËÌ·ÙÔ‰ÔÙÔ‡ÌÂÓÔ 75% ·fi ÙÔ ∂˘Úˆ·˚Îfi ∫ÔÈÓˆÓÈÎfi ∆·ÌÂ›Ô Î·È 25% ·fi ÂıÓÈÎÔ‡˜ fiÚÔ˘˜.

À¶√Àƒ°∂π√ ∂£¡π∫∏™ ¶∞π¢∂π∞™ ∫∞𠣃∏™∫∂Àª∞∆ø¡ ¶∞π¢∞°ø°π∫√ π¡™∆π∆√À∆√

÷ڿϷÌÔ˜ §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‹˜

TÂÙÚ¿‰ÈÔ EÚÁ·ÛÈÒÓ TPITO TEYXO™

OP°ANI™MO™ EK¢O™Eø™ ¢I¢AKTIKøN BIB§IøN A£HNA

¢ÔÌ‹ ÙÔ˘ ‚È‚Ï›Ô˘ AÚÈıÌfi˜ ÎÂÊ·Ï·›Ô˘

Xڈ̷ÙÈο ۇ̂ÔÏ· K¿ı ÎÂÊ¿Ï·ÈÔ, ·Ó¿ÏÔÁ· Ì ÙË ıÂÌ·ÙÈ΋ ÂÚÈÔ¯‹ ÛÙËÓ ÔÔ›· ·Ó·Ê¤ÚÂÙ·È, ¤¯ÂÈ ¤Ó· ¯ÚÒÌ·. OÈ ÂÚÈÔ¯¤˜ Â›Ó·È ÔÈ ÂÍ‹˜: AÚÈıÌÔ›

T›ÙÏÔ˜ ÎÂÊ·Ï·›Ô˘

™‡Ì‚ÔÏÔ - ÎÏÂȉ› ÁÈ· ÙÔ Â›‰Ô˜ Ù˘ ÂÚÁ·Û›·˜ Ô˘ ·ÎÔÏÔ˘ı› *

¶Ú¿ÍÂȘ °ÂˆÌÂÙÚ›· MÂÙÚ‹ÛÂȘ ¶ÚÔ‚Ï‹Ì·Ù· E·Ó¿ÏË„Ë

EÈÎÔÓ›‰È·

(ۇ̂ÔÏ· ÎÏÂȉȿ)

™ÙËÓ Â¿Óˆ ·ÚÈÛÙÂÚ‹ ÁˆÓ›· οı ‰Ú·ÛÙËÚÈfiÙËÙ·˜ ˘¿Ú¯ÂÈ ¤Ó· ·fi Ù· ·ÎfiÏÔ˘ı· ۇ̂ÔÏ·: O ¶˘ı·ÁfiÚ·˜ Ô˘ ÛΤÊÙÂÙ·È - ™‡Ì‚ÔÏÔ ÛΤ„˘:: EÌÊ·Ó›˙ÂÙ·È Û ‰Ú·ÛÙËÚÈfiÙËÙ˜ ÓÔÂÚÒÓ ˘ÔÏÔÁÈÛÌÒÓ. H ̤ÏÈÛÛ· - ™‡Ì‚ÔÏÔ ÂÚÁ·ÙÈÎfiÙËÙ·˜: EÌÊ·Ó›˙ÂÙ·È Û ‰Ú·ÛÙËÚÈfiÙËÙ˜ ÂÊ·ÚÌÔÁ‹˜ Î·È ÂÌ¤‰ˆÛ˘. O Û·ÏÔ˜ ȯÓËÏ¿Ù˘ - ™‡Ì‚ÔÏÔ ·Ó·Î¿Ï˘„˘ : EÌÊ·Ó›˙ÂÙ·È ÛÙȘ ‰Ú·ÛÙËÚÈfiÙËÙ˜ Ô˘ ÂÈÛ¿ÁÔ˘Ó ÙÔ˘˜ Ì·ıËÙ¤˜ ÛÙË Ó¤· ÁÓÒÛË. O ÂϤʷÓÙ·˜ - ™‡Ì‚ÔÏÔ ÌÓ‹Ì˘: EÌÊ·Ó›˙ÂÙ·È ÛÙȘ ‰Ú·ÛÙËÚÈfiÙËÙ˜ Â·Ó¿Ï˄˘. OÌ¿‰· Ì·ıËÙÒÓ - ™‡Ì‚ÔÏÔ ÔÌ·‰ÈÎfiÙËÙ·˜: EÌÊ·Ó›˙ÂÙ·È Û ‰Ú·ÛÙËÚÈfiÙËÙ˜ Ô˘ Â›Ó·È ‰˘Ó·Ùfi Ó· Á›ÓÔ˘Ó Û ÔÌ¿‰Â˜.

4

AÚÈıÌfi˜ ÛÂÏ›‰·˜

¢

˘

˘ Ô › Ï ‚ È ¢ÔÌ‹ ÙÔ˘ ‚ AÚÈıÌfi˜ ‰Ú·ÛÙËÚÈfiÙËÙ·˜

¢È‰·ÎÙÈÎÔ› ÛÙfi¯ÔÈ ÙÔ˘ ÎÂÊ·Ï·›Ô˘

™ËÌ›ˆÛË ÁÈ· ÙÔ ‰¿ÛηÏÔ ÛÙÔ˘˜ ÓÔÂÚÔ‡˜ ˘ÔÏÔÁÈÛÌÔ‡˜ 5

OÈ ‹ÚˆÂ˜ ÙÔ˘ ‚È‚ Ï›Ô˘ ¶˘ı·ÁfiÚ·˜ Ô ™¿ÌÈÔ˜

(ÂÚ›Ô˘ 600 .Ã.)

O ¶˘ı·ÁfiÚ·˜ ‹Ù·Ó ¤Ó·˜ ÛÔ˘‰·›Ô˜ Ì·ıËÌ·ÙÈÎfi˜ Ù˘ ·Ú¯·ÈfiÙËÙ·˜ Ô˘ ÁÂÓÓ‹ıËΠÛÙË ™¿ÌÔ. ÿ‰Ú˘Û ÌÈ· Û¯ÔÏ‹, ÙË Û¯ÔÏ‹ ÙˆÓ ¶˘ı·ÁÔÚ›ˆÓ, ÔÈ ÔÔ›ÔÈ ÌÂÏÂÙÔ‡Û·Ó ÙË ÊÈÏÔÛÔÊ›·, Ù· Ì·ıËÌ·ÙÈο Î·È ÙȘ ÂÈÛً̘. ∂›¯Â ‰·ÛοÏÔ˘˜ ÌÂÁ¿ÏÔ˘˜ ÛÔÊÔ‡˜ Ù˘ ·Ú¯·ÈfiÙËÙ·˜ Î·È Ù·Í›‰Â„ ÛÙËÓ ∞Û›· Î·È ÙËÓ ∞›Á˘ÙÔ fiÔ˘ ÌÂϤÙËÛ ÙËÓ ·ÈÁ˘Ùȷ΋ ÊÈÏÔÛÔÊ›·, Ù· Ì·ıËÌ·ÙÈο, ÙËÓ ·ÛÙÚÔÓÔÌ›· Î·È ÙËÓ È·ÙÚÈ΋. O ¶˘ı·ÁfiÚ·˜ ¤ÌÂÈÓ ÁÓˆÛÙfi˜ ˆ˜ Ô ¿ÓıÚˆÔ˜ Ô˘ ¤‚ÏÂ ·ÓÙÔ‡ ·ÚÈıÌÔ‡˜.

O ¶˘ı·ÁfiÚ·˜

H KÔÚ›Ó·

6

O

˘

‚ È Ï ‚ › Ô ˘ ˘ Ô Ù OÈ ‹ÚˆÂ˜ À·Ù›· Ë ∞ÏÂÍ·Ó‰ÚÈÓ‹

(370-415 Ì.Ã.)

∏ À·Ù›· ‹Ù·Ó Ë ÚÒÙË Á˘Ó·›Î· Ì·ıËÌ·ÙÈÎfi˜ ÛÙËÓ πÛÙÔÚ›·. °ÂÓÓ‹ıËΠÛÙËÓ ∞ÏÂÍ¿Ó‰ÚÂÈ·. ◊Ù·Ó ÎfiÚË ÙÔ˘ ÊÈÏfiÛÔÊÔ˘ £¤ˆÓ·, ‰È¢ı˘ÓÙ‹ ÙÔ˘ ¶·ÓÂÈÛÙËÌ›Ô˘ Ù˘ ∞ÏÂÍ¿Ó‰ÚÂÈ·˜. °È· ÙÔ ÏfiÁÔ ·˘Ùfi ›¯Â ÙËÓ Ù‡¯Ë Ó· ·ÔÎÙ‹ÛÂÈ Û¿ÓÈ· ÌfiÚʈÛË, Û ÌÈ· ÂÔ¯‹ Ô˘ Ë ı¤ÛË Ù˘ Á˘Ó·›Î·˜ ÛÙËÓ ÎÔÈÓˆÓ›· ‹Ù·Ó Ôχ ‰È·ÊÔÚÂÙÈ΋ ·fi fi,ÙÈ Û‹ÌÂÚ·. ™˘Ó¤¯ÈÛ ÙȘ ÛÔ˘‰¤˜ Ù˘ ÛÙËÓ ∞ı‹Ó· Î·È ÙË ƒÒÌË ÂÓÙ˘ˆÛÈ¿˙ÔÓÙ·˜ fiÏÔ˘˜ fiÛÔÈ ÙËÓ Û˘Ó·Ó·ÛÙÚ¤ÊÔÓÙ·Ó Ì ÙÔ Ó‡̷, ÙË ÛÂÌÓfiÙËÙ·, ÙËÓ ÔÌÔÚÊÈ¿ Î·È ÙËÓ Â˘ÁψÙÙ›· Ù˘. ∂ÈÛÙÚ¤ÊÔÓÙ·˜ ÛÙËÓ ∞ÏÂÍ¿Ó‰ÚÂÈ· Ôχ Û‡ÓÙÔÌ· ·Ó·‰Â›¯ıËΠ۠ÌÂÁ¿ÏË ‰·ÛοϷ Ù˘ ÊÈÏÔÛÔÊ›·˜ Î·È ÙˆÓ Ì·ıËÌ·ÙÈÎÒÓ.

H Y·Ù›·

O MÂϤÙ˘

H ÿÏÓÙ· H B¿Ûˆ

7

¶ÂÚȯfiÌÂÓ·

Xڈ̷ÙÈο ۇ̂ÔÏ· AÚÈıÌÔ› ¶Ú¿ÍÂȘ E·Ó¿ÏË„Ë

°ÂˆÌÂÙÚ›· MÂÙÚ‹ÛÂȘ ¶ÚÔ‚Ï‹Ì·Ù·

8

¢ÔÌ‹ ÙÔ˘ ‚È‚Ï›Ô˘

4-5

OÈ ‹ÚˆÂ˜ ÙÔ˘ ‚È‚Ï›Ô˘

6-7

¶ÂÚȯfiÌÂÓ·

8-9

· Ó Â Ì ¶ÂÚȯ fi

Ó·

EÓfiÙËÙ· 5Ë: √π ∞ƒπ£ª√π ª∂Ãπ ∆√ 50, ª√¡∞¢∂™ ∫∞π ¢∂∫∞¢∂™ – ∆∂∆ƒ∞°ø¡π™ª∂¡√ Ã∞ƒ∆π

33 34 35 36 37 38

°ã ¶ÂÚ›Ô‰Ô˜ ∞ÚÈıÌÔ›: ¶Ú¿ÍÂȘ:

∫ÂÊ¿Ï·ÈÔ 33Ô: √ÚÁ¿ÓˆÛË Û˘ÏÏÔÁÒÓ – √È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50 . . . 10-11 ∫ÂÊ¿Ï·ÈÔ 34Ô: ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (π) . . . . . . . . . . . . . . . . . . . . 12-13

°ÂˆÌÂÙÚ›·:

∫ÂÊ¿Ï·ÈÔ 35Ô: ∞ıÚÔ›ÛÌ·Ù· Ì ÔÏÏÔ‡˜ fiÚÔ˘˜ . . . . . . . . . . . . . . . 14-15

ªÂÙÚ‹ÛÂȘ:

∫ÂÊ¿Ï·ÈÔ 36Ô: ∫›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ› . . . . . . . . . . . . . . 16-17 ∫ÂÊ¿Ï·ÈÔ 37Ô: ¶ÚÔ‚Ï‹Ì·Ù· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-19 ∫ÂÊ¿Ï·ÈÔ 38Ô: ∂·Ó·ÏËÙÈÎfi Ì¿ıËÌ· . . . . . . . . . . . . . . . . . . . . . . 20-21

Bã ¶ÂÚ›Ô‰Ô˜ ∞ÚÈıÌÔ›: ¶Ú¿ÍÂȘ:

°ÂˆÌÂÙÚ›·: ªÂÙÚ‹ÛÂȘ:

√È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50 – ™‡ÛÙËÌ· ·Ú›ıÌËÛ˘, ÌÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜. ∞Ê·ÈÚ¤ÛÂȘ Ì ·ÚÈıÌÔ‡˜ ̤¯ÚÈ ÙÔ 10 – ∞ıÚÔ›ÛÌ·Ù· Ì ÔÏÏÔ‡˜ fiÚÔ˘˜ – ¶ÚÔÛı¤ÛÂȘ Ì ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜. ÿڷÍË ÁÚ·ÌÌÒÓ – ∫›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ› – °ÂˆÌÂÙÚÈο Û¯‹Ì·Ù·. ªÔÙ›‚· – √ ¯ÚfiÓÔ˜.

EÓfiÙËÙ· 6Ë: ª√¡∞¢∂™ ∫∞π ¢∂∫∞¢∂™ – °∂øª∂∆ƒπ∫∞ ™Ã∏ª∞∆∞ – Ã√¡√™

39 40 41 42 43

∫ÂÊ¿Ï·ÈÔ 39Ô: ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (ππ) . . . . . . . . . . . . . . . . . . . 22-23

√È ·ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 100. ¶ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ‰È„‹ÊÈˆÓ Î·È ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ – ¶ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ Ì ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ – ¶ÔÏÏ·Ï·ÛÈ·ÛÌfi˜. ÷ڿÍÂȘ, ·˙Ï, Ï·ÎfiÛÙÚˆÙÔ Î·È ÌˆÛ·˚ο – °ÂˆÌÂÙÚÈο Û¯‹Ì·Ù· – ™˘ÌÌÂÙÚ›·. ª¤ÙÚËÛË Û˘Ó¯ÒÓ ÌÂÁÂıÒÓ – µ¿ÚÔ˜ – ¡ÔÌ›ÛÌ·Ù·.

EÓfiÙËÙ· 7Ë: Ã∞ƒ∞•∂π™, ¶∞∑§ – ¶ƒ√™£∂™∏ ∫∞π ∞º∞πƒ∂™∏ – ∏ À¶∂ƒµ∞™∏ ∆∏™ ¢∂∫∞¢∞™

45 46

∫ÂÊ¿Ï·ÈÔ 45Ô: ÷ڿÍÂȘ, ·˙Ï Î·È ÌˆÛ·˚ο . . . . . . . . . . . . . . . . . 32-33 ∫ÂÊ¿Ï·ÈÔ 46Ô: ¶ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ‰È„‹ÊÈˆÓ Î·È ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ . . . . . . . . . . . 34-35

47

∫ÂÊ¿Ï·ÈÔ 47Ô: ∏ ÚfiÛıÂÛË Î·È Ë ·Ê·›ÚÂÛË ˆ˜ ·ÓÙ›ÛÙÚÔʘ Ú¿ÍÂȘ – ∏ ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ . . 36-37

48 49

∫ÂÊ¿Ï·ÈÔ 48Ô: ÀÔÏÔÁÈÛÌÔ› – ∂ÈÛÙÚÔÊ‹ ÛÙËÓ ÂÓÙ¿‰· . . . . . . . . 38-39

50 51

∫ÂÊ¿Ï·ÈÔ 50fi: ¶ÚÔ‚Ï‹Ì·Ù· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42-43

∫ÂÊ¿Ï·ÈÔ 49Ô: ¶ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË – ¢È„‹ÊÈÔÈ Î·È ÌÔÓÔ„‹ÊÈÔÈ ·ÚÈıÌÔ›. . . . . . . . . . . . . . 40-41

∫ÂÊ¿Ï·ÈÔ 51Ô: ∂·Ó·ÏËÙÈÎfi Ì¿ıËÌ· . . . . . . . . . . . . . . . . . . . . . . 44-45

∫ÂÊ¿Ï·ÈÔ 40fi: °ÂˆÌÂÙÚÈο Û¯‹Ì·Ù· . . . . . . . . . . . . . . . . . . . . . . . 24-25 ∫ÂÊ¿Ï·ÈÔ 41Ô: √ ¯ÚfiÓÔ˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-27 ∫ÂÊ¿Ï·ÈÔ 42Ô: ¶ÚÔÛı¤ÛÂȘ Ì ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜ . . . . . . . . . 28-29 ∫ÂÊ¿Ï·ÈÔ 43Ô: ∂·Ó·ÏËÙÈÎfi Ì¿ıËÌ· . . . . . . . . . . . . . . . . . . . . . . 30-31

9

33

OÚÁ¿ÓˆÛË Û˘ÏÏÔÁÒÓ – ∞ÚÈıÌÔ› ̤¯ÚÈ ÙÔ 50

1 ÀÔÏÔÁ›˙ˆ ·ıÚÔ›ÛÌ·Ù· Ì ‰Âο‰Â˜.

2 ªÂÙÚÒ Ù· ΢‚¿ÎÈ· Î·È ˘ÔÏÔÁ›˙ˆ Ùo ¿ıÚoÈÛÌ¿ ÙÔ˘˜.

10+ 3

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

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

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

10

ŸÏ· ›ӷÈ

ŸÏ· ›ӷÈ

ŸÏ· ›ӷÈ

ŸÏ· ›ӷÈ

1. ¶ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· ̤¯ÚÈ ÙÔ 50 ÛÙ· ÔÔ›· Â·Ó·Ï·Ì‚¿ÓÂÙ·È ÙÔ 10 (.¯. 10 + 10 + 10 + 7). ¶ÚÔÙ›ÓÔ˘Ì Â›Û˘ ·ıÚÔ›ÛÌ·Ù· ÛÙ· ÔÔ›· Û ¤Ó·Ó ·ÚÈıÌfi ÛÙÚÔÁÁ˘ÏÒÓ ‰Âο‰ˆÓ (10, 20, 30, 40) ÚÔÛı¤ÙÔ˘Ì ÙÔ 10 (.¯. 30 + 10).

ÂÓfiÙËÙ· 5 Ë

3 ¶fiÛ· ÏÂÙ¿ ¤¯ÂÈ Î¿ı ·È‰›;

™˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

10+ 10 + ... = ...

.......................... = ...

.......................... = ... 4 µÚ›ÛΈ Ù· ·ıÚÔ›ÛÌ·Ù·.

10+ 4 = ...

10+ 9 = ...

10+ 10+ 3 = ...

10+ 10+ 1= ...

10+10+10+ 6 = ... 10+10+10+ 7 = ... 11

34

ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (I)

1 ¶fiÛ· Â›Ó·È Î¿ı ÊÔÚ¿ Ù· ÌÔχ‚È· Î·È Ù· ÙÂÙÚ·ÁˆÓ¿ÎÈ·; µ¿˙ˆ Û ·ÎÏÔ ÙÔ ÛˆÛÙfi ·ÚÈıÌfi.

23

32

14

41

35

53

45

54 2

™˘ÌÏËÚÒÓˆ ÙÔÓ ÂӉȿÌÂÛÔ ·ÚÈıÌfi.

21, ..., 23

19, ..., 21

39, ..., 41

29, ..., 31

36, ..., 38

49, ..., 51 3

°Ú¿Êˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ì ϤÍÂȘ.

11 22 36 12

12 29 44

18 33 50

ÂÓfiÙËÙ· 5 Ë

4 ™¯ËÌ·Ù›˙ˆ Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi.

5 °Ú¿Êˆ ÙÔ ·ÔÙ¤ÏÂÛÌ· ÙˆÓ ڿ͈Ó. ¶·ÚÔ˘ÛÈ¿˙ˆ ÙȘ Ú¿ÍÂȘ Î·È Ì ϤÍÂȘ. ∞Ó ¯ÚÂÈ·ÛÙ›, ¯ÚËÛÈÌÔÔÈÒ ÙÔ ·ÚÈıÌËÙ‹ÚÈÔ.

10+ 7 = 17 ‰¤Î· Î·È ÂÙ¿ = ‰ÂηÂÙ¿

10+ 9 = ... ........................ = ........................ 30 + 5 = ... ........................ = ........................ 14 – 4 = 10 ‰Âη٤ÛÛÂÚ· ‚Á¿˙ˆ Ù¤ÛÛÂÚ· = ‰¤Î·

16 – 6 = ... ........................ = ........................ 28 – 8 = ... ........................ = ........................ 4. ∏ ‰·ÛοϷ ϤÂÈ ÚÔÊÔÚÈο ÛÙÔ˘˜ Ì·ıËÙ¤˜ ÙȘ ‰Âο‰Â˜ Î·È ÙȘ ÌÔÓ¿‰Â˜ ÂÓfi˜ ·ÚÈıÌÔ‡ Î·È ÔÈ Ì·ıËÙ¤˜ ‚Ú›ÛÎÔ˘Ó ÔÈÔ˜ Â›Ó·È Ô ·ÚÈıÌfi˜ Î·È ÙÔÓ ÁÚ¿ÊÔ˘Ó ÛÙÔ Ï·›ÛÈÔ. 13

35

∞ıÚÔ›ÛÌ·Ù· Ì ÔÏÏÔ‡˜ fiÚÔ˘˜

1 ™˘ÌÏËÚÒÓˆ ÙȘ ÙÚÂȘ οÚÙ˜ ÁÈ· Ó· ¤¯ˆ ¿ıÚÔÈÛÌ· ›ÛÔ Ì 8.

8 2

1

4

3

2

3 2

™˘ÌÏËÚÒÓˆ ÙȘ ÙÚÂȘ οÚÙ˜ ÁÈ· Ó· ¤¯ˆ ¿ıÚÔÈÛÌ· ›ÛÔ Ì 10.

10 7

1

3

4

2

2 3

ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

2 + 4 + 3 = ... 4 + 1+ 5 = ... 3 + 2 + ... = 10 6 + 1+ ... = 9 14

7 +2 1 ...

2 +3 2 ...

5 +1 ... 10

ÂÓfiÙËÙ· 5 Ë

4 ÀÔÏÔÁ›˙ˆ ·ıÚÔ›ÛÌ·Ù· Ì ÙÚÂȘ ÚÔÛıÂÙ¤Ô˘˜.

5 OÈ ÓÈÊ¿‰Â˜ ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

9

3

1

1 7

1

2 2

10

8

4

3

6 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

4 +2 + 1 + 2 = ...

8 +... = 10

5 +... = 10

6 + 1 + 2 + 1 = ...

6 +... = 9

5 +2 + 3 + 1 = ...

2 +2 + 2 + 2 = ...

6 +4 + 2 = ...

4. ¶ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· ̤¯ÚÈ ÙÔ 10 Ì ÙÚÂȘ ÚÔÛıÂÙ¤Ô˘˜, ÂÎ ÙˆÓ ÔÔ›ˆÓ ÔÈ ‰‡Ô Ú¤ÂÈ Ó· Â›Ó·È fiÌÔÈÔÈ, ‰ËÏ·‰‹ Ù· ‰ÈÏ¿ ·ıÚÔ›ÛÌ·Ù· (Ó+Ó), Î·È Ô ÙÚ›ÙÔ˜ ÙÔ 1 ‹ ÙÔ 2 (.¯. 2 + 2 + 1, 4 + 4 + 1 Î.Ï.). 15

36

K›ÓËÛË Û ÙÂÙÚ·ÁˆÓÈṲ̂ÓÔ ¯·ÚÙ›

™¯Â‰È¿˙ˆ ÛÙÔ ‰ÂÍÈfi Û¯¤‰ÈÔ fi,ÙÈ ·ÎÚÈ‚Ò˜ ‚Ϥˆ ÛÙÔ ·ÚÈÛÙÂÚfi.

™¯Â‰È¿˙ˆ Ì ÙÔ ¯¿Ú·Î· ÙÔ ÛÈÙ¿ÎÈ ÛÙÔ ‰ÈÏ·Ófi Û¯¤‰ÈÔ.

™ÙÔ ‰ÈÏ·Ófi Û¯¤‰ÈÔ Ì·˘Ú›˙ˆ ÙÂÙÚ¿ÁˆÓ· ÛÙȘ ›‰È˜ ı¤ÛÂȘ Ì ·˘Ù¤˜ Ô˘ ‚Ϥˆ ·ÚÈÛÙÂÚ¿.

16

1

2

3

ÂÓfiÙËÙ· 5 Ë

4 °Ú¿Êˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜.

5 ™¯Â‰È¿˙ˆ ÙË ‰È·‰ÚÔÌ‹ Û‡Ìʈӷ Ì ÙȘ Ô‰ËÁ›Â˜ ÁÈ· Ó· ‚ÚÂÈ ÙÔ ·È‰› ÙÔ ·ÁˆÙfi.

1

ÚÔ˜ Ù· Â¿Óˆ

2

ÚÔ˜ Ù· ‰ÂÍÈ¿

T¤ÏÔ˜

3

ÚÔ˜ Ù· Â¿Óˆ

4

ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿

AÚ¯‹

6 EÓÒÓˆ Ì ÙË ÛÂÈÚ¿ ÙÔ˘˜ ·ÚÈıÌÔ‡˜.

4. H ‰·ÛοϷ ÚÔÙ›ÓÂÈ ‰È„‹ÊÈÔ˘˜ ·ÚÈıÌÔ‡˜ ̤¯ÚÈ ÙÔ 20, ÙÔ˘˜ ÔÔ›Ô˘˜ ÔÈ Ì·ıËÙ¤˜ ÁÚ¿ÊÔ˘Ó ÛÙ· Ï·›ÛÈ·. 17

37

¶ÚÔ‚Ï‹Ì·Ù·

1 ∏ ΢ڛ· ª·Ú›· ¤ÊÙÈ·Í 10 ÁÏ˘Î¿. ™Â οı ·È‰› ı· ‰ÒÛÂÈ ·fi 2 ÁÏ˘Î¿.

¶fiÛ· ÁÏ˘Î¿ ı· ‰ÒÛÂÈ Û fiÏ· Ù· ·È‰È¿ Ì·˙›; ........ ¶fiÛ· ÁÏ˘Î¿ ı· Ù˘ Ì›ÓÔ˘Ó; ........

2 ∆Ô ÓËÛ› ÙÔ˘ 10

™ÙÔ ÓËÛ› ÙÔ˘ 10 ÙÔ ‚·ÙÚ·¯¿ÎÈ ÊÙ¿ÓÂÈ ¿ÓÙ· ˉÒÓÙ·˜ 10 ‚ÔÙÛ·Ï¿ÎÈ·. ■ ∞Ó Â›Ó·È ÛÙÔ ‚ÔÙÛ·Ï¿ÎÈ 5, Ú¤ÂÈ Ó· ˉ‹ÍÂÈ

3

‚ÔÙÛ·Ï¿ÎÈ·.

■ ∞Ó Â›Ó·È ÛÙÔ ‚ÔÙÛ·Ï¿ÎÈ 2, Ú¤ÂÈ Ó· ˉ‹ÍÂÈ

‚ÔÙÛ·Ï¿ÎÈ·.

2 1

■ ∞Ó Â›Ó·È ÛÙÔ ‚ÔÙÛ·Ï¿ÎÈ 7, Ú¤ÂÈ Ó· ˉ‹ÍÂÈ

18

‚ÔÙÛ·Ï¿ÎÈ·.

ÂÓfiÙËÙ· 5 Ë

3 ¢È·Ù˘ÒÓˆ ‰Èο ÌÔ˘ ÚÔ‚Ï‹Ì·Ù· Û‡Ìʈӷ Ì ÙȘ ÂÈÎfiÓ˜.

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

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

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

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

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

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

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

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

4

8 ηڷ̤Ϙ

O ªÂϤÙ˘ Î·È Ë ∫ÔÚ›Ó· ÌÔÈÚ¿˙ÔÓÙ·È ÙÔ ·Î¤ÙÔ Ì ÙȘ ηڷ̤Ϙ. O ηı¤Ó·˜ ı· ¿ÚÂÈ ÙfiÛ˜ ηڷ̤Ϙ fiÛ˜ Î·È Ô ¿ÏÏÔ˜.

O ηı¤Ó·˜ ı· ¿ÚÂÈ ..... ηڷ̤Ϙ.

19

38

E·Ó·ÏËÙÈÎfi Ì¿ıËÌ·

1 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ Ù· ·ıÚÔ›ÛÌ·Ù·.

2 ™˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Î·È ÙËÓ Ú¿ÍË. À¿Ú¯Ô˘Ó Î·È Î·È °Ú¿Êˆ ÙËÓ Ú¿ÍË:

ŸÏ· Ì·˙› ›ӷÈ

3 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

5 +2 1 2 + 4 + 2 + 1 = ... 2 + ... = 10 ... 7 + 2 + 1 = ... 5 + ... = 9 4 +1 ... 3 + 3 + ... = 10 3 + ... = 10 10 3 +2 + 3 + 1 = ... 3 +... = 6

20

4 +3 2 ... 4 +4 ... ...

1. ¶ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· ̤¯ÚÈ ÙÔ 10 Ì ÙÚÂȘ ÚÔÛıÂÙ¤Ô˘˜.

ÂÓfiÙËÙ· 5 Ë

4 ÀÔÏÔÁ›˙ˆ ÙÔ ¿ıÚÔÈÛÌ· ÙˆÓ ‰Âο‰ˆÓ Î·È ÙˆÓ ÌÔÓ¿‰ˆÓ ÂÓfi˜ ·ÚÈıÌÔ‡.

5 °Ú¿Êˆ Î·È Ï¤ˆ fiÛ˜ Â›Ó·È ÔÈ ‰Âο‰Â˜ Î·È ÔÈ ÌÔÓ¿‰Â˜. OÓÔÌ¿˙ˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ ÚÔ·ÙÔ˘Ó.

MÔÓ¿‰Â˜

MÔÓ¿‰Â˜

MÔÓ¿‰Â˜

¢Âο‰Â˜

¢Âο‰Â˜

¢Âο‰Â˜

6 ªÂÙÚÒ Î·È ‚Ú›ÛΈ ÙÔ ¿ıÚÔÈÛÌ·.

10+ –– + –– + –– =

10+ –– + –– + –– + –– =

4. ¶ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· Ù˘ ÌÔÚÊ‹˜ 10+Ó, 20+Ó Î.Ï., fiÔ˘ Ó ÌÔÓÔ„‹ÊÈÔ˜ ·ÚÈıÌfi˜.

21

39

ªÔÓ¿‰Â˜ Î·È ‰Âο‰Â˜ (II)

1 ∞Ê·ÈÚÒ ÙȘ ÌÔÓ¿‰Â˜ ·fi ‰È„‹ÊÈÔ ·ÚÈıÌfi.

2 µ¿˙ˆ Û ·ÎÏÔ Ù· ÓÔÌ›ÛÌ·Ù· Ô˘ ¯ÚÂÈ¿˙ÔÓÙ·È.

13 ÏÂÙ¿

34 ÏÂÙ¿

48 ÏÂÙ¿

3 ™˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó. °Ú¿Êˆ ÙȘ ‰Âο‰Â˜ Ì ÎfiÎÎÈÓÔ ¯ÚÒÌ·.

0 1 2 3 10 12 21 23

5 7 15 16 27 34

40

42 51

22

29 38

46 54

9

49 58 59

1. ¶ÚÔÙ›ÓÔ˘Ì ·Ê·ÈÚ¤ÛÂȘ Ù˘ ÌÔÚÊ‹˜ 1Ó–Ó, 2Ó–Ó Î.Ï., fiÔ˘ Ó Ô ·ÚÈıÌfi˜ Ô˘ Â›Ó·È ›ÛÔ˜ Ì ÙÔ „ËÊ›Ô ÙˆÓ ÌÔÓ¿‰ˆÓ ÙÔ˘ ‰È„‹ÊÈÔ˘ ·ÚÈıÌÔ‡.

ÂÓfiÙËÙ· 6 Ë

4 µÚ›ÛΈ ÙȘ ‰Âο‰Â˜ Î·È ÙȘ ÌÔÓ¿‰Â˜. ™˘ÌÏËÚÒÓˆ ÙȘ ÈÛfiÙËÙ˜.

∆Ô 12 ¤¯ÂÈ ….. ‰Âο‰Â˜ Î·È ….. ÌÔÓ¿‰Â˜.

10+ 2 12 = ...................…..

∆Ô 29 ¤¯ÂÈ ….........................................

29 = ........................

∆Ô 33 ¤¯ÂÈ ….........................................

33 = ........................

5 µÚ›ÛΈ ÙÔ ·ÔÙ¤ÏÂÛÌ· ÙˆÓ ڿ͈Ó. ¶·ÚÔ˘ÛÈ¿˙ˆ ÙȘ Ú¿ÍÂȘ Î·È Ì ϤÍÂȘ.

10+ 2 = ... ‰¤Î· Î·È ........................ = ........................ 20 + 6 = ... ........................ = ........................ 28 – 8 = ... ........................ = ........................ 34 – 4 = ... ........................ = ........................ 6 ™˘ÌÏËÚÒÓˆ ÙȘ ÈÛfiÙËÙ˜.

10 + 9 = ... 17 – 7 = ... 10 + 10 + 4 = ... 10 + ... = 16 12 – 2 = ... 25 = ... + ... + ... 8 + 8 = ...

7 + 5 = ... 13 – ... = 10

7 + 7 = ... 18 – ... = 10 37 = ... + ... + ... + ... 23

40

°ÂˆÌÂÙÚÈο Û¯‹Ì·Ù·

1

µ¿˙ˆ Û ·ÎÏÔ Ù· Û¯‹Ì·Ù· Ô˘ ÌÔÈ¿˙Ô˘Ó. ªÂÙÚÒ fiÛ· Â›Ó·È Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi.

3 ÛÊ·›Ú˜

·‚ÔÈ

·ÏÈÓ‰ÚÔÈ

ÛÙÂÚ¿ ÔÚıÔÁÒÓÈ· 24

ÂÓfiÙËÙ· 6 Ë

2 Ãڈ̷ٛ˙ˆ Ì ÙÔ ›‰ÈÔ ¯ÚÒÌ· ·˘Ù¿ Ô˘ ÌÔÈ¿˙Ô˘Ó.

3 ªÂÙÚÒ Ù· Û¯‹Ì·Ù· Ô˘ ÌÔÈ¿˙Ô˘Ó ÌÂٷ͇ ÙÔ˘˜ Î·È Áڿʈ ÙÔ˘˜ ·ÓÙ›ÛÙÔÈ¯Ô˘˜ ·ÚÈıÌÔ‡˜. Y¿Ú¯Ô˘Ó

Y¿Ú¯Ô˘Ó Y¿Ú¯Ô˘Ó

Y¿Ú¯Ô˘Ó

Y¿Ú¯Ô˘Ó

Y¿Ú¯Ô˘Ó 25

41

O ¯ÚfiÓÔ˜

1 µ¿˙ˆ ·ÚÈıÌÔ‡˜ ÛÙȘ ÂÈÎfiÓ˜ ÁÈ· Ó· ‰Â›Íˆ ÙË ¯ÚÔÓÈ΋ ÛÂÈÚ¿ ÙÔ˘˜.

1 2 µ¿˙ˆ ·ÚÈıÌÔ‡˜ ÛÙȘ Ë̤Ú˜ Ù˘ ‚‰ÔÌ¿‰·˜ Û‡Ìʈӷ Ì ÙË ¯ÚÔÓÈ΋ ÛÂÈÚ¿ ÙÔ˘˜.

TÚ›ÙË ¶·Ú·Û΢‹

1

K˘Úȷ΋

¶¤ÌÙË

¢Â˘Ù¤Ú·

TÂÙ¿ÚÙË

™¿‚‚·ÙÔ

3 ∑ˆÁÚ·Ê›˙ˆ ÎÂÚ¿ÎÈ· ÛÙËÓ ÙÔ‡ÚÙ· ÁÈ· Ó· ‰Â›Íˆ ÙËÓ ËÏÈΛ· ÌÔ˘.

26

ÂÓfiÙËÙ· 6 Ë

4

µ¿˙ˆ ·ÚÈıÌÔ‡˜ ÛÙȘ ÂÈÎfiÓ˜ οı ÈÛÙÔÚ›·˜ ÁÈ· Ó· ‰Â›Íˆ ÙË ¯ÚÔÓÈ΋ ÛÂÈÚ¿ ÙÔ˘˜.

1

2

5 ¶·Ú·ÙËÚԇ̠ÙËÓ ÂÈÎfiÓ· Î·È Û˘˙ËÙ¿ÌÂ.

∞ÁÒÓ·˜ ‰ÚfiÌÔ˘ ÕÓÓ·

ŒÏÏË

T›Ó·

¶ÔÈÔ ÎÔÚ›ÙÛÈ ¯ÚÂÈ¿˙ÂÙ·È ÏÈÁfiÙÂÚÔ ¯ÚfiÓÔ ÁÈ· Ó· ÊÙ¿ÛÂÈ ÛÙÔ Ù¤ÚÌ·; 27

42

¶ÚÔÛı¤ÛÂȘ Ì ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜

1 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ Û˘Ìϋڈ̷ ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ.

2 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

9+5=

7+6=

+5

+6

9

7 +1

+4 10

10

8+7=

9+8=

+7 8

9

10

28

+8

10

1. H ‰·ÛοϷ ÚÔÙ›ÓÂÈ ÛÙÔ˘˜ Ì·ıËÙ¤˜ ·Û΋ÛÂȘ Û˘ÌÏËÚÒÌ·ÙÔ˜ ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ (.¯. «3 Î·È fiÛÔ Ì·˜ οÓÂÈ 7;»).

ÂÓfiÙËÙ· 6 Ë

3 ÀÔÏÔÁ›˙ˆ ·ıÚÔ›ÛÌ·Ù· Ì ÙËÓ ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜.

4 ™˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

10

10 6

4 2

ŸÏ· Ì·˙› Â›Ó·È 20.

5 5

ŸÏ· Ì·˙› Â›Ó·È 30.

5

5

ŸÏ· Ì·˙› Â›Ó·È 18.

10

5

ŸÏ· Ì·˙› Â›Ó·È 17.

5 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ ·ÔÙ¤ÏÂÛÌ·.

8 + 3 = ...

7 + 4 = ...

9 + 7 = ...

9 + 6 = ...

8 + 4 = ...

3 + 9 = ...

2 + 9 = ...

8 + 5 = ...

8 + 8 = ...

3. ∏ ‰·ÛοϷ ÚÔÙ›ÓÂÈ Î¿ı ÊÔÚ¿ ¤Ó· ¿ıÚÔÈÛÌ· (.¯. 9 + 6, 8 + 4, Î.Ï.). OÈ Ì·ıËÙ¤˜ ˘ÔÏÔÁ›˙Ô˘Ó ÓÔÂÚ¿ Î·È ·ÚÔ˘ÛÈ¿˙Ô˘Ó ÚÔÊÔÚÈο ÙÔÓ ÙÚfiÔ Ì ÙÔÓ ÔÔ›Ô ˘ÔÏfiÁÈÛ·Ó. 29

43

E·Ó·ÏËÙÈÎfi Ì¿ıËÌ·

1 ÀÔÏÔÁ›˙ˆ ·ıÚÔ›ÛÌ·Ù· Ì ÙËÓ ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜.

2 ™˘ÌÏËÚÒÓˆ Ù· ËÌÂÚÔÏfiÁÈ·.

¯ı˜

Û‹ÌÂÚ·

·‡ÚÈÔ

TÂÙ¿ÚÙË

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

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

7

...

... 3

O Ù·¯˘‰ÚfiÌÔ˜ ª¤Û· ÛÙÔ Û¿ÎÔ ÌÔ˘ ¤¯ˆ 9 ÌÈÎÚÔ‡˜ ʷΤÏÔ˘˜ Î·È 7 ÌÂÁ¿ÏÔ˘˜ ʷΤÏÔ˘˜. ¶fiÛÔÈ Â›Ó·È fiÏÔÈ ÔÈ Ê¿ÎÂÏÔÈ Ì·˙›;

30

1. ∏ ‰·ÛοϷ ÚÔÙ›ÓÂÈ Î¿ı ÊÔÚ¿ ¤Ó· ¿ıÚÔÈÛÌ· (.¯. 9 + 4, 8 + 3 Î.Ï.). OÈ Ì·ıËÙ¤˜ ˘ÔÏÔÁ›˙Ô˘Ó ÓÔÂÚ¿ Î·È ·ÚÔ˘ÛÈ¿˙Ô˘Ó ÚÔÊÔÚÈο ÙÔÓ ÙÚfiÔ Ì ÙÔÓ ÔÔ›Ô ˘ÔÏfiÁÈÛ·Ó.

ÂÓfiÙËÙ· 6 Ë

4 ¶fiÛ· ¯Ú‹Ì·Ù· ›ӷÈ; ™˘Ó‰¤ˆ Ì ̛· ÁÚ·ÌÌ‹ Ù· ›Û· ÔÛ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

∂›Ó·È ... ÏÂÙ¿.

5 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

6 + 6 = ...

7 + 6 = ...

5 + 7 = ...

20+ ... = 25

6 + 5 = ...

20+ 6 = ...

8 + 8 = ...

9 + 9 = ...

37 – 7 = ... 8 + 5 = ...

43 – ... = 40

30 + ... = 38 10 + 10 + 10 + 10 = ... 7 + 7 = ...

10 + 10 + 10 + 9 = ... 31

45

÷ڿÍÂȘ, ·˙Ï Î·È ÌˆÛ·˚ο

1 ªÂÙÚÒ Î·È Áڿʈ ÙÔÓ ·ÚÈıÌfi ÙˆÓ Û¯ËÌ¿ÙˆÓ Ù˘ ·Ú·Î¿Ùˆ ÂÈÎfiÓ·˜.

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

2 ∂ÓÒÓˆ Ì ÙÔ ¯¿Ú·Î· Ù· ÛËÌ›· Ô˘ ¤¯Ô˘Ó ÙÔ ›‰ÈÔ ¯ÚÒÌ·. OÓÔÌ¿˙ˆ Ù· Û¯‹Ì·Ù· Ô˘ ‰ËÌÈÔ˘ÚÁÔ‡ÓÙ·È Ì ·˘ÙfiÓ ÙÔÓ ÙÚfiÔ Î·È Ù· ¯ÚˆÌ·Ù›˙ˆ Ì ÙÔ ·ÓÙ›ÛÙÔÈ¯Ô ¯ÚÒÌ· ÙˆÓ ÛËÌ›ˆÓ.

32

ÂÓfiÙËÙ·

7 Ë

3 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ·ıÚÔ›ÛÌ·Ù· Ì ÙÚÂȘ ÚÔÛıÂÙ¤Ô˘˜.

4 ∂ÓÒÓˆ Ì ÌÈ· ÁÚ·ÌÌ‹ Ù· ÎÔÌÌ¿ÙÈ· Ô˘, ·Ó Û˘Ó‰ÂıÔ‡Ó, Û¯ËÌ·Ù›˙Ô˘Ó ¤Ó· ÙÂÙÚ¿ÁˆÓÔ, ÙÚ›ÁˆÓÔ ‹ ÔÚıÔÁÒÓÈÔ.

3. ¶ÚÔÙ›ÓÔ˘Ì ÚÔÛı¤ÛÂȘ ̤¯ÚÈ ÙÔ 10 Ì ÙÚÂȘ ÚÔÛıÂÙ¤Ô˘˜, ÛÙȘ Ôԛ˜ ÔÈ ‰‡Ô ÚÔÛıÂÙ¤ÔÈ Â›Ó·È fiÌÔÈÔÈ (.¯. 2 + 2 + 3, 3 + 3 + 2 Î.Ï.).

33

46

¶ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ‰È„‹ÊÈˆÓ Î·È ÌÔÓÔ„‹ÊÈˆÓ ·ÚÈıÌÒÓ

1 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ Ù· ·ıÚÔ›ÛÌ·Ù· Î·È ÙȘ ‰È·ÊÔÚ¤˜.

2 ∆· ÁÂÓ¤ıÏÈ· Ù˘ MfiÓ· ∏ MfiÓ· ÁÈÔÚÙ¿˙ÂÈ Ù· ÁÂÓ¤ıÏÈ¿ Ù˘. ¶fiÛˆÓ ¯ÚfiÓˆÓ Â›Ó·È; ŒÓ· ÌÂÁ¿ÏÔ ÎÂÚ› ÈÛÔ‰˘Ó·Ì› Ì 10 ÌÈÎÚ¿ ÎÂÚ¿ÎÈ·.

∏ MfiÓ· Â›Ó·È ........ ¯ÚfiÓˆÓ. ¶fiÛˆÓ ¯ÚfiÓˆÓ ı· Â›Ó·È ¤ÂÈÙ· ·fi 3 ¯ÚfiÓÈ·;

¶fiÛˆÓ ¯ÚfiÓˆÓ ‹Ù·Ó ÚÈÓ ·fi 2 ¯ÚfiÓÈ·;

ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ.

ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ.

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

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

3 YÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ ·ÔÙ¤ÏÂÛÌ·.

14+ 1 = ... 15 + 3 = ... 15 – 3 = ... 18 – 4 = ... 15 + 2 = ... 13 + 3 = ... 14 – 2 = ... 17 – 2 = ... 14 + 4 = ... 12 + 2 = ... 16 – 3 = ... 16 – 4 = ... 34

1. ¶ÚÔÙ›ÓÔ˘Ì ÛÙÔ˘˜ Ì·ıËÙ¤˜ ÚÔÛı¤ÛÂȘ ‰È„‹ÊÈÔ˘ Ì ÌÔÓÔ„‹ÊÈÔ ·ÚÈıÌfi Î·È ·Ê·ÈÚ¤ÛÂȘ ÌÔÓÔ„‹ÊÈÔ˘ ·fi ‰È„‹ÊÈÔ ·ÚÈıÌfi (.¯. 12 + 2, 16 – 3 Î.Ï.).

ÂÓfiÙËÙ·

7 Ë

4 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ Ù· ·ıÚÔ›ÛÌ·Ù· Î·È ÙȘ ‰È·ÊÔÚ¤˜.

5 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

2 + ... = 4 12 + ... = 14 6 – ... = 3 16 – ... = 13

5 + ... = 9 15 + ... = 19 8 – ... = 7 18 – ... = 17

2 + ... = 7 12 + ... = 17 9 – ... = 2 19 – ... = 12 6

£¤Ïˆ Ó· ‚¿Ïˆ 9 ÌÈÛÎfiÙ· Û 3 Û·ÎÔ˘Ï¿ÎÈ·. µÚ›ÛΈ Î·È Áڿʈ ÙÚÂȘ ‰È·ÊÔÚÂÙÈÎÔ‡˜ ÙÚfiÔ˘˜.

... + ... + ... = ... ... + ... + ... = ... ... + ... + ... = ... 4. ∏ ‰·ÛοϷ ÚÔÙ›ÓÂÈ ÛÙÔ˘˜ Ì·ıËÙ¤˜ ÚÔÛı¤ÛÂȘ ‰È„‹ÊÈÔ˘ Ì ÌÔÓÔ„‹ÊÈÔ ·ÚÈıÌfi Î·È ·Ê·ÈÚ¤ÛÂȘ ÌÔÓÔ„‹ÊÈÔ˘ ·fi ‰È„‹ÊÈÔ ·ÚÈıÌfi (.¯. 15 + 4, 19 – 3 Î.Ï.). 35

47

H ÚfiÛıÂÛË Î·È Ë ·Ê·›ÚÂÛË ˆ˜ ·ÓÙ›ÛÙÚÔʘ Ú¿ÍÂȘ – ∏ ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜

1 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜.

14 – 6 =

16 – 7 =

–6

–7

14

16 –4

–2 10

10

13 – 5 =

17 – 9 =

–5

–9

13

17 10

10

12 – 5 =

13 – 7 =

–5 12

13 10

36

–7

10

ÂÓfiÙËÙ·

7 Ë

2 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙȘ ÚÔÛı¤ÛÂȘ.

3 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ ·ÔÙ¤ÏÂÛÌ·.

9 + 8 = ...

7 + 4 = ...

4 + 8 = ...

17 – 8 = ...

11 – 4 = ...

12 – 4 = ...

18 + 4 = ...

17 + 5 = ...

19 + 4 = ...

22 – 4 = ...

22 – 5 = ...

23 – 4 = ... 4

Ãڈ̷ٛ˙ˆ Ù· Ì·ÏfiÓÈ· Ì ÙÔ ¯ÚÒÌ· Ô˘ Ú¤ÂÈ.

6+6 5+6

7+5

4+8

3+9

9+6

7+7

8+6

8+2

14 12 2. ¶ÚÔÙ›ÓÔ˘Ì ·ıÚÔ›ÛÌ·Ù· Ô˘ Â›Ó·È ‰˘Ó·Ùfi Ó· ˘ÔÏÔÁÈÛÙÔ‡Ó Ì ÙË Ì¤ıÔ‰Ô Ù˘ ˘¤Ú‚·Û˘ Ù˘ ‰Âο‰·˜ (.¯. 8 + 5, 9 + 7, 3 + 8 Î.Ï.).

37

48

ÀÔÏÔÁÈÛÌÔ› – ∂ÈÛÙÚÔÊ‹ ÛÙËÓ ÂÓÙ¿‰·

1 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙȘ ·Ê·ÈÚ¤ÛÂȘ.

2 ¶fiÛÔ Î¿ÓÔ˘Ó 8 + 6; ªÂÙÚÒ

ÀÔÏÔÁ›˙ˆ

8 + 6 = ...

8 = 5 + ... 6 = ... + ... 8 + 6 = ...

ÀÔÏÔÁ›˙ˆ Ù· ·Ú·Î¿Ùˆ ·ıÚÔ›ÛÌ·Ù·.

7+6

8+5

7 = ... + ... 6 = ... + ... 7 + 6 = ...

8 = ... + ... 8 + 5 = ... 3

ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ ·ÔÙ¤ÏÂÛÌ·.

38

9 + 5 = ...

9 + 7 = ...

8 – 5 = ...

9 + 6 = ...

10+ 6 = ...

10– 5 = ...

10+ 5 = ...

7 – 5 = ...

9 – 5 = ...

1. ∏ ‰·ÛοϷ ÚÔÙ›ÓÂÈ ÛÙ· ·È‰È¿ ·Ê·ÈÚ¤ÛÂȘ Ù˘ ÌÔÚÊ‹˜ Ó – 5, fiÔ˘ Ó ¤Ó·˜ ·ÚÈıÌfi˜ ·fi ÙÔ 6 ̤¯ÚÈ ÙÔ 10 (.¯. 7 – 5, 8 – 5 Î.Ï.).

ÂÓfiÙËÙ·

7 Ë

4 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ ¿ıÚÔÈÛÌ·.

5 ÀÔÏÔÁ›˙ˆ Ù· ‰ÈÏ¿ ·ıÚÔ›ÛÌ·Ù·.

6+6

7+7

6 = 5 + ... 6 + 6 = ...

7 = ... + ... 7 + 7 = ...

8+8

9+9

8 = ... + ... 8 + 8 = ...

9 = ... + ... 9 + 9 = ... 6

¶fiÛ· fi‰È· ¤¯Ô˘Ó Ù· ‰‡Ô ¯Ù·fi‰È·;

T· ‰‡Ô ¯Ù·fi‰È· ¤¯Ô˘Ó ... fi‰È·.

¶fiÛ· Â›Ó·È fiÏ· Ù· ¯Ú‹Ì·Ù·;

ŸÏ· Ù· ¯Ú‹Ì·Ù· Â›Ó·È ... ú.

4. ∏ ‰·ÛοϷ ÚÔÙ›ÓÂÈ ·ıÚÔ›ÛÌ·Ù· Ì ÚÔÛıÂÙ¤Ô˘˜ ·fi ÙÔ 5 ̤¯ÚÈ ÙÔ 10 (.¯. 10 + 5, 7 + 5 Î.Ï.). 39

49

¶ÚfiÛıÂÛË Î·È ·Ê·›ÚÂÛË – ¢È„‹ÊÈÔÈ Î·È ÌÔÓÔ„‹ÊÈÔÈ ·ÚÈıÌÔ›

1 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ ·ÔÙ¤ÏÂÛÌ· ÙˆÓ ·Ê·ÈÚ¤ÛˆÓ.

2 ÀÔÏÔÁ›˙ˆ ÙȘ ·Ê·ÈÚ¤ÛÂȘ Ì ÚfiÛıÂÛË ÚÔ˜ Ù· Â¿Óˆ.

12 – 8 = ... 8 + ... = 10 10 + ... = 12

13 – 7 = ...

14 – 9 = ...

15 – 12 = ...

3 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

9 + 6 = ... 10 + ... = 17 8 + 4 = ... 15 – ... = 10 40

13 - 8 = ... 10 + 12 = ... 12 – 7 = ... 8 + 9 = ...

7 + 4 = ... 16 – 7 = ... 23 – 3 = ... 16 – 8 = ...

1. ¶ÚÔÙ›ÓÔ˘Ì ·Ú¯Èο ·Ê·ÈÚ¤ÛÂȘ ÌÔÓÔ„‹ÊÈÔ˘ ·fi ‰È„‹ÊÈÔ ·ÚÈıÌfi Ù˘ ÌÔÚÊ‹˜ 1Ó – Ó, (.¯. 15 – 5, 17 – 7 Î.Ï.). ™ÙË Û˘Ó¤¯ÂÈ· ÚÔÙ›ÓÔ˘Ì ·Ê·ÈÚ¤ÛÂȘ, fiˆ˜ 12 – 4, 17 – 8 Î.Ï., ÔÈ Ôԛ˜ ÂÎÙÂÏÔ‡ÓÙ·È Î·È Ì ÙËÓ ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜.

ÂÓfiÙËÙ·

7 Ë

4 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙÔ ·ÔÙ¤ÏÂÛÌ· ÙˆÓ ·Ê·ÈÚ¤ÛˆÓ.

5 ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

+3

+6

9

7 10

10

13

8

11

6

10

10

–6

–7

13

9

15

10

12

10

8

10

11

7

10

4. ¶ÚÔÙ›ÓÔ˘Ì ·Ê·ÈÚ¤ÛÂȘ fiˆ˜ 12 – 6, 15 – 7, 16 – 8 Î.Ï. OÈ Ì·ıËÙ¤˜ ˘ÔÏÔÁ›˙Ô˘Ó ÓÔÂÚ¿ Î·È ÂÍËÁÔ‡Ó ÙÔÓ ÙÚfiÔ ˘ÔÏÔÁÈÛÌÔ‡.

41

50

¶ÚÔ‚Ï‹Ì·Ù· ™ÙËÓ ·˘Ï‹ ÙÔ˘ Û¯ÔÏ›Ԣ

1

™ÙËÓ ·˘Ï‹ ÙÔ˘ Û¯ÔÏ›Ԣ ‚Á‹Î·Ó 5 ÎÔÚ›ÙÛÈ·. ™Â Ï›ÁÔ ‚Á‹Î·Ó Î·È ·ÁfiÚÈ·. ª¤ÙÚËÛ· Ù· ·ÁfiÚÈ· Î·È ‚ڋη fiÙÈ ‹Ù·Ó 3 ·Ú·¿Óˆ ·fi Ù· ÎÔÚ›ÙÛÈ·. ¶fiÛ· ‹Ù·Ó Ù· ·ÁfiÚÈ·;

¶fiÛ· ‹Ù·Ó fiÏ· Ì·˙› Ù· ·È‰È¿;

∆· ·ÁfiÚÈ· ‹Ù·Ó ...

ŸÏ· Ù· ·È‰È¿ Ì·˙› ‹Ù·Ó ...

2 O ¶˘ı·ÁfiÚ·˜ ›¯Â 4 ÁÚ·ÌÌ·ÙfiÛËÌ·. ∏ ª·Ú›Ó· ÙÔ˘ ¤‰ˆÛ ¿ÏÏ· 2. ∏ ∫·ÙÂÚ›Ó· ÙÔ˘ ¤‰ˆÛ 2 ·ÎfiÌË Î·È Ô ∞fiÛÙÔÏÔ˜ ÙÔ˘ ¤‰ˆÛ ÌÂÚÈο ·ÎfiÌË. ∆ÒÚ· Ô ¶˘ı·ÁfiÚ·˜ ¤¯ÂÈ 10 ÁÚ·ÌÌ·ÙfiÛËÌ·. O ∞fiÛÙÔÏÔ˜ ÙÔ˘ ¤‰ˆÛ ... ÁÚ·ÌÌ·ÙfiÛËÌ·. 42

ÂÓfiÙËÙ·

7 Ë

3 ªÔÚ›˜ Ó· ‚ÚÂȘ ¤Ó·Ó ÙÚfiÔ ÁÈ· Ó· ¤¯ÂȘ 12 ÏÂÙ¿ Ì ٤ÛÛÂÚ· ÓÔÌ›ÛÌ·Ù·; ∑ˆÁÚ¿ÊÈÛ ٷ ÓÔÌ›ÛÌ·Ù·.

4 ■ ™ÙÔ Ì¿ıËÌ· Ù˘ Á˘ÌÓ·ÛÙÈ΋˜ 8 ·È‰È¿ ·›˙Ô˘Ó Ô‰fiÛÊ·ÈÚÔ, ÂÓÒ Ù· ˘fiÏÔÈ· 6 ·›˙Ô˘Ó Î·Ï·ıÔÛÊ·›ÚÈÛË. ¶fiÛ· Â›Ó·È fiÏ· Ù· ·È‰È¿ ÛÙÔ Ì¿ıËÌ· Ù˘ Á˘ÌÓ·ÛÙÈ΋˜;

A¿ÓÙËÛË

¢È·Ù˘ÒÓˆ ¤Ó· ‰ÈÎfi ÌÔ˘ Úfi‚ÏËÌ· ■ ™Î¤„Ô˘ ¤Ó· ·ÚfiÌÔÈÔ Ì ÙÔ ÚÔËÁÔ‡ÌÂÓÔ Úfi‚ÏËÌ·, ¯ÚËÛÈÌÔÔÈÒÓÙ·˜ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ 9 Î·È 7.

¢ÒÛ ÙÔ Úfi‚ÏËÌ· Ó· ÙÔ Ï‡ÛÂÈ Ô ‰ÈÏ·Ófi˜ ÛÔ˘. 43

51

E·Ó·ÏËÙÈÎfi Ì¿ıËÌ·

1 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙȘ Ú¿ÍÂȘ.

2 ™˘ÁÎÚ›Óˆ ÙÔ˘˜ ‚fiÏÔ˘˜.

°ÈÒÚÁÔ˜

N›ÎÔ˜

O °ÈÒÚÁÔ˜ ¤¯ÂÈ 12 ‚fiÏÔ˘˜ Î·È Ô ¡›ÎÔ˜ 8 ‚fiÏÔ˘˜. ¶fiÛÔ˘˜ ÂÚÈÛÛfiÙÂÚÔ˘˜ ‚fiÏÔ˘˜ ¤¯ÂÈ Ô °ÈÒÚÁÔ˜; ªÂ ÙÔÓ ›‰ÈÔ ÙÚfiÔ ˘ÔÏfiÁÈÛ ÙȘ ·Ú·Î¿Ùˆ Ú¿ÍÂȘ:

12 ‚fiÏÔ˘˜

8 ‚fiÏÔ˘˜

13 – 7 = ...

11 – 9 = ...

13 – 8 = ...

11 – 7 = ... 3

ÀÔÏÔÁ›˙ˆ Î·È Û˘ÌÏËÚÒÓˆ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ Ï›Ô˘Ó.

3 + ... = 5 13 + ... = 15 4 + ... = 8 14 + ... = 18 5 + ... = 9 15 + ... = 19 44

4 + ... = 7 14 + ... = 17 6 – ... = 3 16 – ... = 13 8 – ... = 6 18 – ... = 16

9 – ... = 7 19 – ... = 17 8 – ... = 2 18 – ... = 12 7 – ... = 6 17 – ... = 16

1. ¶ÚÔÙ›ÓÔ˘Ì ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ÔÈ Ôԛ˜ ÚÔÛʤÚÔÓÙ·È ÁÈ· Â›Ï˘ÛË Ì ÙËÓ ˘¤Ú‚·ÛË Ù˘ ‰Âο‰·˜, Ì ٷ ‰ÈÏ¿ ·ıÚÔ›ÛÌ·Ù· ‹ Ì ¿ÏÏÔ˘˜ ÙÚfiÔ˘˜ (.¯. 7 + 8, 2 + 9, 18 – 9, 13 – 5 Î.Ï.).

ÂÓfiÙËÙ·

7 Ë

4 ÀÔÏÔÁ›˙ˆ Î·È Áڿʈ ÙȘ Ú¿ÍÂȘ.

5 Ãڈ̷ٛ˙ˆ ÙȘ Ì¿Ï˜ Ì ÙÔ ·ÓÙ›ÛÙÔÈ¯Ô ¯ÚÒÌ·.

11 8 7 9 4 + 7 = ...

14 – 6 = ...

13 – 4 = ... 13 – 6 = ...

17 – 8 = ... 2 + 9 = ... 15 – 7 = ...

15– 8 = ...

4. ¶ÚÔÙ›ÓÔ˘Ì ÛÙÔ˘˜ Ì·ıËÙ¤˜ ÚÔÛı¤ÛÂȘ ‰È„‹ÊÈÔ˘ Ì ÌÔÓÔ„‹ÊÈÔ ·ÚÈıÌfi Î·È ·Ê·ÈÚ¤ÛÂȘ ÌÔÓÔ„‹ÊÈÔ˘ ·fi ‰È„‹ÊÈÔ ·ÚÈıÌfi (.¯. 12 + 2, 16 – 3 Î.Ï.). 45

ªÂ ·fiÊ·ÛË Ù˘ ∂ÏÏËÓÈ΋˜ ∫˘‚¤ÚÓËÛ˘ Ù· ‰È‰·ÎÙÈο ‚È‚Ï›· ÙÔ˘ ¢ËÌÔÙÈÎÔ‡, ÙÔ˘ °˘ÌÓ·Û›Ô˘ Î·È ÙÔ˘ §˘Î›Ԣ Ù˘ÒÓÔÓÙ·È ·fi ÙÔÓ √ÚÁ·ÓÈÛÌfi ∂ΉfiÛˆ˜ ¢È‰·ÎÙÈÎÒÓ µÈ‚Ï›ˆÓ Î·È ‰È·Ó¤ÌÔÓÙ·È ‰ˆÚÂ¿Ó ÛÙ· ¢ËÌfiÛÈ· ™¯ÔÏ›·. ∆· ‚È‚Ï›· ÌÔÚ› Ó· ‰È·Ù›ıÂÓÙ·È ÚÔ˜ ÒÏËÛË, fiÙ·Ó Ê¤ÚÔ˘Ó ‚È‚ÏÈfiÛËÌÔ ÚÔ˜ ·fi‰ÂÈÍË Ù˘ ÁÓËÛÈfiÙËÙ¿˜ ÙÔ˘˜. ∫¿ı ·ÓÙ›Ù˘Ô Ô˘ ‰È·Ù›ıÂÙ·È ÚÔ˜ ÒÏËÛË Î·È ‰Â ʤÚÂÈ ‚È‚ÏÈfiÛËÌÔ, ıˆÚÂ›Ù·È ÎÏ„›Ù˘Ô Î·È Ô ·Ú·‚¿Ù˘ ‰ÈÒÎÂÙ·È Û‡Ìʈӷ Ì ÙȘ ‰È·Ù¿ÍÂȘ ÙÔ˘ ¿ÚıÚÔ˘ 7, ÙÔ˘ ¡fiÌÔ˘ 1129 Ù˘ 15/21 ª·ÚÙ›Ô˘ 1946 (ºEK 1946, 108, Aã).

BIB§IO™HMO

∞·ÁÔÚ‡ÂÙ·È Ë ·Ó··Ú·ÁˆÁ‹ ÔÔÈÔ˘‰‹ÔÙ ÙÌ‹Ì·ÙÔ˜ ·˘ÙÔ‡ ÙÔ˘ ‚È‚Ï›Ô˘, Ô˘ ηχÙÂÙ·È ·fi ‰ÈηÈÒÌ·Ù· (copyright), ‹ Ë ¯Ú‹ÛË ÙÔ˘ Û ÔÔÈ·‰‹ÔÙ ÌÔÚÊ‹, ¯ˆÚ›˜ ÙË ÁÚ·Ù‹ ¿‰ÂÈ· ÙÔ˘ ¶·È‰·ÁˆÁÈÎÔ‡ πÓÛÙÈÙÔ‡ÙÔ˘.