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Clasa a-V-a

COALA GENERAL PLATCA

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Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: Scrierea i citirea numerelor naturale. irul numerelor naturale. Reprezentarea

numerelor naturale pe o ax.DATA:

Def:nelee! "r#n numr naturalo no#une !a$e!a$#%& folo'#$& la nu!&ra$ (# %al%ula$.

Nu!erele na$urale 'e e)"r#!& %u a*u$orul %#frelor: +,-,,/,0,1,2,3,4,5 6%are 'un$ n#($e '#!7olur#8 9 diferena dintre un numr i o cifr?

Def:Nu!erele na$urale '%r#'e a'$fel: +,-,,/-+,--,,0+ for!ea;& irul numerelor naturale6un (#r#nf#n#$8.

9 a%e'$ (#r 'e "oa$e re"re;en$a (# raf#%:,C,D %are au %a (# %oordona$e nu!erele na$urale -,,/,0 ? dea'e!enea un#$a$ea de !&'ura e'$e 'e!en$ul A>.

C#$#rea unu# nu!&r na$ural: 24.15 %#frele un#$lor, %#fra ;e%#lor, %#fra 'u$elor, %#fra !##lor, ;e%#lor de !##,

e$%. 6al$e e)e!"le8

Exerciii 2: S& 'e %#$ea'%& 6de "e $a7l&8: --.345, 0./12, --.012

Exerciii 3: Da# e)e!"lu de nu!&r %are are %#fra ;e%#lor /. Da# e)e!"lu de nu!&r de / %#fre %are are %#fra'u$elor . Da# e)e!"lu de nu!&r de 0 %#fre %u %#fra ;e%#lor (# %#fra un#$lor /. e$%.

Exerciii : Da# $oa$e nu!erele na$urale de doua %#fre %are 'e "o$ '%r#e %u %#frele - (# .Dar $oa$e nu!erele na$urale de $re# %#fre %are 'e "o$ '%r#e %u %#frele -, , +@ Dar %u %#frele /,2,3 @

Te!a:

+ / 0 - -0 -1 -2 -3

A > C D N P B

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-. Da$# $re# e)e!"le de nu!ere na$urale de / %#fre %are au %#fra 'u$elor 0.. Enu!era$# $oa$e nu!erele na$urale de doua %#fre %are au %#fra ;e%#lor ./. Enu!era$# $oa$e nu!erele na$urale de $re# %#fre %are 'e "o$ '%r#e folo'#nd %#frele ,1,+.

Te!a:-. Da$# $re# e)e!"le de nu!ere na$urale de / %#fre %are au %#fra 'u$elor 0.. Enu!era$# $oa$e nu!erele na$urale de doua %#fre %are au %#fra ;e%#lor .

/. Enu!era$# $oa$e nu!erele na$urale de $re# %#fre %are 'e "o$ '%r#e folo'#nd %#frele ,1,+.

Te!a:-. Da$# $re# e)e!"le de nu!ere na$urale de / %#fre %are au %#fra 'u$elor 0.. Enu!era$# $oa$e nu!erele na$urale de doua %#fre %are au %#fra ;e%#lor ./. Enu!era$# $oa$e nu!erele na$urale de $re# %#fre %are 'e "o$ '%r#e folo'#nd %#frele ,1,+.

Te!a:-. Da$# $re# e)e!"le de nu!ere na$urale de / %#fre %are au %#fra 'u$elor 0.

. Enu!era$# $oa$e nu!erele na$urale de doua %#fre %are au %#fra ;e%#lor ./. Enu!era$# $oa$e nu!erele na$urale de $re# %#fre %are 'e "o$ '%r#e folo'#nd %#frele ,1,+.


Te!a:

-. Da$# $re# e)e!"le de nu!ere na$urale de / %#fre %are au %#fra 'u$elor 0.. Enu!era$# $oa$e nu!erele na$urale de doua %#fre %are au %#fra ;e%#lor ./. Enu!era$# $oa$e nu!erele na$urale de $re# %#fre %are 'e "o$ '%r#e folo'#nd %#frele ,1,+.




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012 413+/

1 -0-

01 --/

10/42/

Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: !"unarea numerelor naturale.DATA:

Def:Lu&! nu!erele na$urale (# / (# o! def#n# #umaa%e'$or dou& nu!ere na$urale %a f##nd un nu!&r na$uraleal %u /. A%e'$ nu!&r na$ural e'$e 1.

= '%r#e! o"era#a efe%$ua$&: / F 1, unde 1 e'$e #umanu!erelor (# / , #ar ,/ 'e nu!e'% termenii'u!e#.

Lu&! al$e dou& 'u!e: / 0 F 3 (# 0 / F 3. Se o7'er& %&:or#%are ar f# dou& nu!ere na$urale a,b ae! %& a + b = b + a $comutati%itate&

Lu&! 'u!ele / 60 28 (# 6/ 08 2 de unde o7'er&! %&:or#%are ar f# dou& nu!ere na$urale a,b ae! %&: a + (b + c) = (a + b) + c $a#ociati%itate&

Cal%ul&! / + F / (# + / F / (# o7'er&! %&:or#%are ar f# a nu!&r na$ural ae! %&: a + 0 = 0 + a = a $element neutru pentru a"unarea numerelor

na$urale8

'oncluzie: adunarea nu!erelor na$urale e'$e %o!u$a$#&, a'o%#a$#& (# are ele!en$ neu$ru ;ero.

Di#cuie:"u$e! "r## or#%e nu!&r na$ural %a f##nd o 'u!& de nu!ere na$urale. 6e). F - -, 1 F /8 9 A'$fel da%& adun&! nu!ere !ar#: 0-1 /012 , no# ae! 60+++ -++ + 18 6/+++ 0++ 1+ 28 (#adun&! "e rnd o7#nnd 3+++ 1++ 3+ -- F 31++ 4- F 314-

= 'e %orelea;& %u '%r#erea nu!erelor unul 'u7 al$ul: 0-1 /012

FFFFF 314-

Exerciii 1: S& 'e %al%ule;e: a8 /01 20-? 78 012 210? %8 -/0 ++ /0? e$%.

Exerciii 2: S& 'e efe%$ue;e: a8 - 65 8 64 / 38 0 2 1 ? 78 -/ 62 008 3? %8 - 5 - 55 - 555 ? d8 6453 /8 6-5 4-8?

Exerciii 3: a8 S& 'e '%r#e nu!&rul 0/3 %a 'u!& de $re# nu!ere na$urale? 78 S& 'e '%r#e /- %a 'u!& de $re#nu!ere na$urale f#e%are %u $re# %#fre? %8 S& 'e '%r#e 0 %a 'u!& de / nu!ere na$urale, de o %#fr&.

Exerciii :Re%on'$ru##:

TE1. Efe%$ua# adun&r#le: a8. /14 ? 78. -1-0/? %8. -/012345-+----/-0-1-2 2. S%r#e# nu!&rul - a8. %a 'u!& de do# $er!en#? 78. %a 'u!& de $re# $er!en#? %8. %a 'u!& de $re# nr. "are. 3. S%r#e# nu!&rul -00 a8. %a 'u!& de $re# nu!ere nenule? 78. %a 'u!& de $re# nu!ere %on'e%u$#e.

. Cal%ula# :-555555-+--- (. Re%on'$#$u##:


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/3 =54-5

3210 = 3+/4

04 = 12/

Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: Sc"erea numerelor naturale.DATA:

Def:Lu&! %a e)e!"lu nu!erele 1 (# /, "iferenaa%e'$or dou& nu!ere na$urale e'$e un nu!&r na$ural %are a f#no$a$ %u 1 =/ , ad#%& .

= d#n 1 H / F %#$#! un "e#czut 618, un #cztor6/8 (# o "iferen68.= d#ferena nu!erele 1 (# / '=a o7#nu$ "r#n o"era#a de #c"ere.

)*#er%aii:er#f#%&! n#($e "ro"r#e$ ale '%&der##:= %al%ul&! 1 =/ F ? d#ferena / = 1 nu e'$e un nu!&r na$ural de%# '%&derea nu!erelor na$urale nu e'$e

%o!u$a$#&?= %al%ul&! 3 H 60 = /8 F 3 H - F 2 (# 63 = 08 H / F +, de%# '%&derea nu!erelor na$urale nu e'$e a'o%#a$#&.= %al%ulnd 3 H + F 3, o7'er&! %& d#ferena d#n$re or#%e nu!&r na$ural (# ;ero e'$e de'%&;u$ul 6nu (# #ner'8.

)*#er%aii:S%&derea e'$e o"era#a #ner'& adun&r##, efe%$ele lor 'e anulea;& re%#"ro%.= e): /0=--F/, /--F/0

= n %on'e%#n& '%&derea 'e "oa$e "ro7a: da%& 14=-F02 a$un%# $re7u#e %a 02-F14

D#ferene !a# %o!"l#%a$e: 0-1 = /012 e'$e de fa"$ 0+++ -++ + 1 H /+++ H 0++ H 1+ H 2 , 'e %orelea;&%u !e$oda "r#n '%r#erea nu!erelor unul 'u7 al$ul:

0-1 =/012

FFFFF F225

Exerciii 1: Cal%ula#: a80/ H /0 ? 78 -++ H /0? %8 /0 H /0 H 3? d8 012 H 6-0/= 38 H / F

e8 6/0 H -+38 H 0/ ?

Exerciii 2: Cal%ula#: a8-+=-/? 78. -01+=-/5? %8 6+/=-++8=1-? d8. -1=-++=6-+=08J?e8. 12=K14=1+=6/1-/8J f8 01 H K6/0-8=-3J H -/

Exerciii 3:Re%on'$ru## e)er%###le:

Exerciii :Su!a a dou& nu!ere e'$e 0+. S& 'e afle nu!erele ($##nd %& unul e'$e %u - !a# !are de%$ %el&lal$.

Exerciii (:Cal%ula#: a8 531/--+420 ++++ 78 55...1/--++...20 ++++

TEA1. Cal%ula#: a8. -=-3? 78. -++=155? %8. -21-+=-1/2? d8. -++++++-=-+/++? e8 6/+/=++8=3-? f8 -11=--+=6-+=048J? f8. 15=K2-=1/=6/4-/8J.TEA

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1. Cal%ula#: a8. -=-3? 78. -++=155? %8. -21-+=-1/2? d8. -++++++-=-+/++? e8 6/+/=++8=3-? f8 -11=--+=6-+=048J? f8. 15=K2-=1/=6/4-/8J.

TEA1. Cal%ula#: a8. -=-3? 78. -++=155? %8. -21-+=-1/2? d8. -++++++-=-+/++? e8 6/+/=++8=3-? f8 -11=--+=6-+=048J? f8. 15=K2-=1/=6/4-/8J.






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Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: 'ompararea i or"onarea numerelor naturale.DATA:

Di#cuie:d#n 'u!a 2 F 0 , 'e "o$ dedu%e ur!&$oarele:= 2 e'$e %u !a# !are de%$ 0, de%# 2 !a# !are de%$ "a$ru, %are 'e '%r#e: + , ? n a%ela(# $#!" 0 e'$e

!a# !#% de%$ 2, %eea %e 'e '%r#e +.

Def:

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Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: nmulirea numerelor naturale. /roprieti. )r"inea efecturii operaiilor.DATA:

Def:

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e8 -/08-4/06/ ++ f8 328-4-61- + 8 854--+6-2840-+6/0 + 8 -//+/+/0 +

TEA-.Cal%ula#: a8 5-+-+3---+ ++ 78 1-13-1-14 + %8 ./-321 d8 1/+

e8 -/08-4/06/ ++ f8 328-4-61- + 8 854--+6-2840-+6/0 +

8 -//+/+/0 +

TEA-.Cal%ula#: a8 5-+-+3---+ ++ 78 1-13-1-14 + %8 /-321 d8 1/+

e8 -/08-4/06/ ++ f8 328-4-61- + 8 854--+6-2840-+6/0 + 8 -//+/+/0 +

COALA GENERAL PLATCAProf. Poenaru Radu

CLASA: V-a

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TITLUL LECIEI: Di#tri*uti%itatea 0nmulirii faa "e a"unare i #c"ere. actorul comun.DATA:

Di#tri*uti%itatea 0nmulirii fa "e a"unare i #c"ere 6"r#n e)e!"lu8:= or#%are ar f# nu!erele na$urale cba ,, , ae!: acabcba +=+ 86= or#%are ar f# nu!erele na$urale cba ,, , ae!: acabcba = 86

Exerciii:Cal%ula# n dou& !odur#: a8 826/ + 78 ( )/1-+ %8 ( ) ( )--/1/ +++

d8 ( )[ ]31./-+ ++/ro*lem:Un !un%#$or e)e%u$& 0 de "#e'e de a%ela(# $#" "e or&. C$e "#e'e e)e%u$& !un%#$orul lu%rnd - ;#le %$e 4 ore "e ;#@

Scoaterea factorului comun: D#'%u#e du"& d#'$r#7u$##$a$e (# a"o# "r#n e)e!"lu: ( ) 1/1/ +=+ , de%# (# ( )1/1/ +=+ .

Exerciii:Efe%$ua# %al%ulele "r#n '%oa$erea fa%$orulu# %o!un: a8 /2/4 78 1/113 + %8 0./440 + d8 1+-++-+++ e8 -++0+ +

Mn %on$#nuare, e)er%### d#er'e:

Exerciii: Cal%ula#:a8 -+-+ + 78 -00 %8 03103++ + d8 ( ) ( )21102450/2-210 + e8 ( ) 2--/+ + f8 ( )[ ] -+3-2/100/30 +

/ro*lem:C$e ;erour# are nu!&rul na$ural eal %u "rodu'ul nu!erelor na$urale %on'e%u$#e de la - la -+#n%lu'#@ Dar al nu!erelor na$urale %on'de%u$#e de la 0 la -1@

/ro*lem:De %$e or# 7a$e #n#!a unu# %o"#l n$r=un an, da%& n$r=un !#nu$ 7a$e, n !ed#e, de 31 de or#@

Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12 e8 ( )[ ]/-/34 f8 /-4545123 +++

Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12 e8 ( )[ ]/-/34 f8 /-4545123 +++

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Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12 e8 ( )[ ]/-/34 f8 /-4545123 +++

Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12 e8 ( )[ ]/-/34 f8 /-4545123 +++

Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12

e8 ( )[ ]/-/34 f8 /-4545123 +++

Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12 e8 ( )[ ]/-/34 f8 /-4545123 +++

Te!&: Cal%ula#: a8 ( )-1/01 ++ 78 ( )-//- %8 -133/ + d8 -0/12 e8 ( )[ ]/-/34 f8 /-4545123 +++


CLASA: V-aTITLUL LECIEI: )r"inea efecturii operaiilor. Exerciii i pro*leme.DATA:

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Se $re% n re#'$& 6oral8 o"era##le '$ud#a$e !"reun& %u "ro"r#e$le lor.

Exerciii: Cal%ula#: a8 /0.50231 + 78 /1052140 %8 0/54354 + d8 5413/0 ++ e8 54/-10+ ++

Exerciii:Cal%ula#: a8 /-2543-45 ++++ 78 20-0/221045 +++(comutatiitate) 78 525354550/.- +++++++

Exerciii:Cal%ula#: a8 -/012/43 + 78 ( ) -11//-/ ++ %8 ( )[ ] 0/-0/20 + d8 ( )[ ] -0:01+3/1+ +

Exerciii:Efe%$ua# "r#n '%oa$erea fa%$orulu# %o!un: a8 1//-3 78 0414-24 + %8 2+-.+ d8 -++++/++ +

/ro*lem:Un au$o!o7#l %on'u!& /l "e 0+! de dru!. C$ %on'u! a aea du"& 2 ;#le "e d#'$ana ;#ln#%& de-2+!@

Te!&:-. Cal%ula#: a8 0/0/543 78 0/4/=/23/ %8 ( ) ---30 d8 ( )[ ]-0/4//12 +

. Co!"ara# nu!erele a(# bunde: aF 012/0/ (# bF ( ) -2-01/

Te!&:-. Cal%ula#: a8 0/0/543 78 0/4/=/23/ %8 ( ) ---30 d8 ( )[ ]-0/4//12 +. Co!"ara# nu!erele a(# bunde: aF 012/0/ (# bF ( ) -2-01/

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Te!&:-. Cal%ula#: a8 0/0/543 78 0/4/=/23/ %8 ( ) ---30 d8 ( )[ ]-0/4//12 +

. Co!"ara# nu!erele a(# bunde: aF 012/0/ (# bF ( ) -2-01/



CLASA: V-aTITLUL LECIEI: e#t "e %erificareDATA:

-. Cal%ula#: a8 =+ /254/ 78 215- /10/ F

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%8 =45-30 d8 =050/023-

e8 = 10/- f8 ( )1/012 +

8 ( )[ ] =+ 4/1.0/01

. Ordona# %re'%&$or nu!erele: a8 /,+,/,120,213,-+ 78 243,423,342,-43

/. Cal%ula# "r#n '%oa$erea fa%$orulu# %o!un: a8 0/10 78 1+4+-++ +

0. S& 'e afle dou& nu!ere %are aduna$e dau 01 ($##nd %& unul e'$e %u -+ !a# !are de%$ %el&lal$.

1. Care e'$e %el !a# !#% nu!&r %are 'e "oa$e for!a rearan*nd %#ferele nu!&rulu#: 51+/43@

Pun%$a*:-: -" "en$ru a8,%8,e8 (# +,1" "en$ru 78,d8,f8,8 F 1"

: +,1" "en$ru a8,78 F -" /: +,1" "en$ru a8,78 F -" 0: -"

1: -"

To$al 5" -" 6of#%#u8 F -+"


Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: )r"inea efecturii operaiilor. actor comun. Exerciii i pro*leme.DATA:

Exerciii:Cal%ula#: a8 /-2543-45 ++++ 78 20-0/221045 +++(comutatiitate) 78 525354550/.- +++++++

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Exerciii:Cal%ula#: a8 -/012/43 + 78 ( ) -11//-/ ++ %8 ( )[ ] 0/-0/20 + d8 ( )[ ] -001+3/1+ +

Exerciii:Cal%ula#: a8 = 001+01/0 78 ( ) = --55341-.55342-. %8 = //031+/031//0

/ro*lem:Un au$o!o7#l %on'u!& /l "e 0+! de dru!. C$ %on'u! a aea du"& 2 ;#le "e d#'$ana ;#ln#%& de-2+!@

/ro*lem:Afla# $re# nu!ere %on'e%u$#e %are aduna$e dau 2/.

/ro*lem:Tre# nu!ere aduna$e dau 21. $#! %& "r#!ul e'$e de dou& or# !a# !are de%$ al do#lea (# %& al $re#leae'$e %u 1 !a# !are de%$ al do#lea. S& 'e afle nu!erele.

Te!& 6$e!a (# %a$ea e)er%### d#n %ar$e8?


CLASA: V-aTITLUL LECIEI: mprirea. eorema 0mpririi cu re#t.DATA:

Lu&! o !"&r#re 1+ : 4 de%#: 1+ 4 . Mn a%ea'$& !"&r#re ae! de!"&r#$, !"&r#$or, %$ (# re'$.04 2

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Teore!a !"&r#r## %u re'$: or#%are ar f# nu!erele na$urale d(# i, +i e)#'$& nu!erele na$urale c(# r, un#%e,a'$fel n%$ rcid += , unde ir< . = rela#a "oa$e f# e)"l#%a$& a'$fel: do7#e%$e "o$ f# a(e;a$e n irndur# de %$e c!a# "u#n %ele ro7#e%$e %are

ar for!a un rnd #n%o!"le$, de%# rQi!= $eore!a !"&r#r## %u re'$ %on'$#$u#e "ro7a !"&r#r## %u re'$: da%& ae! d: i %are d& %$ul c(# re'$ul r,

a$un%# ae! (# %& rcid += .= e)e!"lu: 4:/F %$ul (# re'$ul , de%# 4 F / + .= e)e!"le: "en$ru !"&r#$or# %u (# / %#fre.

)*#:da%& re'$ul o7#nu$ e'$e ;ero a$un%# '"une! %&0mprirea e#te exact. o! aea cid = , deoare%e re'$ule'$e ;ero.

Exerciii:De$er!#na# %$ul (# re'$ul !"&r#r#lor: a8 01 : 2 78 12 : 4 %8 0/1 : -+ d8 -+ : 0 e8 0/ : --

Te!&: De$er!#na# %$ul (# re'$ul !"&r#r#lor: a8 /- : / 78 12 : 1 %8 0/ : -- d8 /- : -2







CLASA: V-aTITLUL LECIEI: mprirea. /roprieti. )*#er%aiiDATA:

D#'%u#e: "ro7a !"&r#r## %u re'$, fara re'$.

)*#:!"&r#rea (# n!ul#rea 'un$ o"era## #ner'e, ad#%& una anulea;& "e %ealal$&.

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)*#:da%& a,b'e !"ar$ e)a%$ la nu!&rul ca$un%# (# !"&r#rea lu# a+bla nu!&rul ce'$e e)a%$&.= e)e!"le: 0 (# -+ 'e !"ar$ e)a%$ la , -+ 0 'e !"ar$e e)a%$ la .

)*#:= !"&r#rea la ;ero nu e)#'$&?= !"&r#rea unu# nu!&r da$ la - ne d& %$ul a%el nu!&r (# re'$ul ;ero.

Exerciii:Cal%ula# (# a"o# er#f#%a# re;ul$a$ele o7#nu$e: a8 -24+:12 78 -24+ : /+ %8 -24 : /?

d8 -45+5 : -4 e8 /01+ : -5 f8 5-0+ : /1 8 /13++ : /++

Exerciii:6d#n !anual8


CLASA: V-aTITLUL LECIEI: /uterea unui numr naturalDATA:

Con'#der&! n!ul#r#le: = // , a%ea'$a o "u$e! no$a "re'%ur$a$ ./ %are 'e %#$e($e / la "u$erea a doua.= .... a%ea'$a 'e !a# no$ea;& 0. (# 'e %#$e($e do# la "u$erea a "a$ra.= "r#n --a o! nelee aaa ... de -- or#.

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Def:

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Te!&: Cal%ula#: a8 .-/ 78 /1 %8 ..+ d8 .-++ e8 1/ f8 3.

Te!&: Cal%ula#: a8 .-/ 78 /1 %8 ..+ d8 .-++ e8 1/ f8 3.

Te!&: Cal%ula#: a8 .-/ 78 /1 %8 ..+ d8 .-++ e8 1/ f8 3.

Te!&: Cal%ula#: a8 .-/ 78 /1 %8 ..+ d8 .-++ e8 1/ f8 3.

Te!&: Cal%ula#: a8 .-/ 78 /1 %8 ..+ d8 .-++ e8 1/ f8 3.


CLASA: V-aTITLUL LECIEI: /trat perfect. 'u*ul unui numr natural. $ora 2&DATA:

A!#n$#!: un nu!&r o7#nu$ "r#n r#d#%area la "u$erea a doua a unu# nu!&r na$ural 'e nu!e($e "&$ra$ "erfe%$.= e)e!"le: -- F .-- , 0++ F ..+ , /5= .

Def:un nu!&r o7#nu$ "r#n r#d#%area la "u$erea a $re#a a unu# nu!&r na$ural 'e nu!e($e %u7 "erfe%$.= e)e!"le: = //3= de%# 3 e'$e %u7ul lu# /

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= /020= de%# 20 e'$e %u7ul lu# 0 = /4= de%# 4 e'$e %u7ul lu#

Exerciii:Cal%ula# "&$ra$ele (# %u7ur#le nu!erelor: 1, 4, 5, -+, --, -, -++.

)*#: n$r=un (#r de %al%ule n %are ae! adun&r#, '%&der#, n!ul#r#, !"&r#r# (# "u$er# o! efe%$ua na#n$e%al%ulul "u$er#lor, a"o# n!ul#r#le (# !"&r#r#le, a"o# adun&r#le (# '%&der#le.

Exerciii:Cal%ula#: a8 0/ + 78 -// %8 ( ) /3:3// d8 /.. ./0 + e8 ( 0:55 / f8 ( 0: /0 8 ..:.... 10/. +++

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ../ ../. + d8 ( 5:52 e8 ( )/. 1/:01 f8 -+/.1.-+ ... + 8 ( )[ ] -+:-+../-0 0

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ( ) ../ ../. + d8 ( ) 5:52 e8 ( )/. 1/:01 f8 ( )-+/.1.-+ ... + 8 ( -+:-+../-0 0

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ( ) ../ ../. + d8 ( ) 5:52 e8 ( )/. 1/:01 f8 ( )-+/.1.-+ ... + 8 ( -+:-+../-0 0

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Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ../ ../. + d8 ( 5:52 e8 ( )/. 1/:01 f8 -+/.1.-+ ... + 8 ( )[ ] -+:-+../-0 0

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ../ ../. + d8 ( 5:52 e8 ( )/. 1/:01 f8 -+/.1.-+ ... + 8 ( )[ ] -+:-+../-0 0

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ../ ../. + d8 ( 5:52 e8 ( )/. 1/:01 f8 -+/.1.-+ ... + 8 ( )[ ] -+:-+../-0 0

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ( ) ../ ../. + d8 ( ) 5:52 e8 ( )/. 1/:01 f8 -+/.1.-+ ... + 8 ( -+:-+../-0 0

Te!a: Cal%ula#: a8 0/ .. + 78 -0:-2 2 %8 ( ) ../ ../. + d8 ( ) 5:52 e8 ( )

/. 1/:01 f8 ( )-+/.1.-+ ... + 8 ( -+:-+../-0 0


CLASA: V-aTITLUL LECIEI: Reuli "e calcul cu puteriDATA:

Cal%ul&!: ( ) ( ) ./1./

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

+

====

Pr#n al$e e)e!"le (# eneral#;are "u$e! dedu%e for!ula: nmnm aaa += .

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Pr#n al$e e)e!"le o7'er&! %& nmnm aaa =: .

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Te!&:-. Cal%ula#: a8 -./ + 78 ( ) -21/-0 + %8 -/-0 d8 /:01+/:/+01 . Cal%ula#: a8 ( )/1/11 0.:. 78 -3:33 /1-1+ %8 -20/0:/0/0 310 d8 ( ) -4. /.






CLASA: V-aTITLUL LECIEI:

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%8 ( )= 10/11 /

d8 ( = -+-/ /2:

e8 =++ ... /-12/5540//

/. De$er!#na# %$ul (# re'$ul !"&r#r#lor: a8 /-4 : 23

78 0/0 : /0

%8 --/5-/4 : -0/

0. Cal%ula#: a8 ( ) = .0242 0-0:-0

78 ( =45/+ :/:2

%8 ( ) =3-3+54 :1:-+

d8 ( ) =.33./1 3:33


Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI:

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d8 55 54 H /1 F

e8 60 238 H 4+ F

/. Cal%ula#: a8 =1/0

78 = 0-2

%8 =4-/1

d8 =-10/

0. Cal%ula#: a8 =1:-01

78 =:4+

%8 =:-02

Te'$ de ealuareDATA: NUE:

-. Cal%ula#: 6-"8 a8 =.-/ 6-"8 78 =+ ./ 0.

6-"8 %8 = 10/ /:// 6-"8 d8 ( )=+ :-0/1

. Co!"ara# nu!erele:6+,1"8 a8 /0. (# .4.

6-"8 78 -11 (# 1 3

6+,1"8 %8

2+

. (#

01

/

/. Cal%ula#:6-"8 a8 ( ) = 1.:. -031+/

6-"8 78 ( = 152+23.+1+ 1:11:11 6-"8 %8 ( ) = 15/+000/01 :/::2

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Te'$ de ealuareDATA: NUE:

-. Cal%ula#: 6-"8 a8 =./ 6-"8 78 =+ ./ 0.

6-,1"8 %8 = 01 6-,1"8 d8 =-4-3

6-"8 e8 0/3 0/ F6-"8 f8 11/ : / F

. Co!"ara# nu!erele:6-"8 a8 /0. (# .4.

6-"8 78 -11 (# 1 3

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/. De$er!#na#: a8 $oa$e nu!erele de for!a +1a d##;#7#le %u 1?

78 $oa$e nu!erele de for!a b/+ d##;#7#le %u ?

%8 $oa$e nu!erele de for!a ba+2 d##;#7#le %u -+?

d8 $oa$e nu!erele de for!a aa+1 d##;#7#le %u ?

Lu%rare 'e!e'$r#alaNu!e:

-. Cal%ula#: a8 =+/1

/. 78 =

1.

10 %8 =+

2.

2-- d8

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Prof. Poenaru RaduCLASA: V-aTITLUL LECIEI: ul$#!#: no$a##, ele!en$, a"ar$enen&. Su7!ul#!#, rela#a de #n%lu;#une.DATA:

/artea 1: mulimi

E)e!"le de !ul#!#: -8 !ul#!ea ele#lor %la'e# a 1=a no$a$& A F KPaul, Iul#a = !ul#!e %u 3 ele!en$e

= 'e "oa$e '%r#e (# %a: Kele eleul a"ar#ne %la'e# a 1=a

8 !ul#!ea for!a$& d#n nu!erele +,,0,2,4: K+,,0,2,4 = "oa$e f# '%r#'& 'u7 for!a: Kn nu!&r na$ural n "ar (# n Q -+

/8 !ul#!ea nu!erelor na$urale: N F K+,-,,/,,n,.

Mn e)e!"lul -8 o! '"une %&: = Paul e'$e un element al !ul#!## A (# o! '%r#e %& PaulA? = Andre# nu e'$e un ele!en$ al !ul#!## A de%#: Andre#A?/artea 2: #u*mulimi

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A!#n$#!: da%& A (# > 'un$ !ul#!# (# A are $oa$e ele!en$ele !ul#!## > a$un%#, '"une! %& > e'$e o'u7!ul#!e a !ul#!## A.

e): ae! !ul#!#le A F K-,,0,2,3 (# > F K,2. Toa$e ele!en$ele lu# > 'un$ (# ele!en$e ale lu# A de%# > e'$e o'u7!ul#!e a lu# A.

- " nu este o submulime a lui #

- $ = %&,', nu este o submulime a lui "*

A!#n$#!: = n e)e!"lul an$er#or ae! %& > e'$e 'u7!ul#!e a lu# A, (# o! e)"r#!a a%ea'$a "r#n: >A , %are'e %#$e($e !ul#!ea > e'$e #n%lu'& n !ul#!ea A.

= deoare%e A nu e'$e #n%lu'& n >, o! '%r#e: A >.

A!#n$#!: !ul#!ea %are nu are n#%# un ele!en$, nu!#$& mulimea %i", (# %are 'e no$ea;& %u .

)*#:deoare%e !ul#!ea #d& nu are n#%# un ele!en$, ea e'$e #n%lu'& n or#%e !ul#!e %u %el "u#n un ele!en$.

Ex 1: Care d#n ur!&$oarele "ro"o;### 'un$ ade&ra$e (# %are 'un$ fal'e:

a8 - N 78 1

K+,,/,1,3 %8 + (# 'un$ ele!en$e ale !ul#!## A F K-,,/,0,1

d8 K/,0K/,0,1,2,3 e8 L2,1K1 f8 L+K 8 > F K0,1 e'$e o 'u7!ul#!e a lu# A F K+,-,,/,0,2,3

Ex 2: F K)N ) Q 1 (# C F K)N ) Q -+ (# ) . Care d#n ur!&$oarele"ro"o;### 'un$ ade&ra$e: a8 A> 78 $# %8 $" d8 #/ e8 ".,- f8 "#

Te!&: -8 Care 'un$ ele!en$ele !ul#!#lor: A F K )N ) Q 0 , > F KVN V Q + (# V/ ,C F K;N ; W0 (# ; Q 5@

8 Cal%ula#: a8 = 8-/6/ +/ 78 ( ) =3/+ : /8 Care d#n ur!&$oarele nu!ere 'un$ d##;#7#le %u : 01, 2+, /++, -/, 44+0, 521 @

Te!&:-8 Care 'un$ ele!en$ele !ul#!#lor: A F K )N ) Q 0 , > F KVN V Q + (# V/ ,

C F K;N ; W0 (# ; Q 5@ 8 Cal%ula#: a8 [ ]= 8-/6/ +/ 78 ( ) =3/+ : /8 Care d#n ur!&$oarele nu!ere 'un$ d##;#7#le %u : 01, 2+, /++, -/, 44+0, 521 @


C F K;N ; W0 (# ; Q 5@ 8 Cal%ula#: a8 = 8-/6/ +/ 78 ( ) =3/+ : /8 Care d#n ur!&$oarele nu!ere 'un$ d##;#7#le %u : 01, 2+, /++, -/, 44+0, 521 @

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C F K;N ; W0 (# ; Q 5@ 8 Cal%ula#: a8 [ ]= 8-/6/ +/ 78 ( ) =3/+ : /8 Care d#n ur!&$oarele nu!ere 'un$ d##;#7#le %u : 01, 2+, /++, -/, 44+0, 521 @

Te!&: -. Cal%ula#: a8 /40 -23 F 78 01 H -4 F %8 -00 : - F d8 =/ e8 =./ . er#f#%a# %are d#n ur!&$oarele "ro"o;### 'un$ ade&ra$e:

a8 /K-,/,1 78 - (# K+,,/,0,1 %8 + K+,-,0,1


a8 /K-,/,1 78 - (# K+,,/,0,1 %8 + K+,-,0,1

Te!&: -. Cal%ula#: a8 /40 -23 F 78 01 H -4 F %8 -00 : - F d8 =/

e8 =

.

/ . er#f#%a# %are d#n ur!&$oarele "ro"o;### 'un$ ade&ra$e:a8 /K-,/,1 78 - (# K+,,/,0,1 %8 + K+,-,0,1


a8 /K-,/,1 78 - (# K+,,/,0,1 %8 + K+,-,0,1

Te'$ de ealuare

DATA: NUE:

6"8 -. Se dau !ul#!#le: A F K+,,/, > F K+,-,0, C F K+,-,,/.S& 'e de$er!#ne: A> , AC, C X A (# A X C.

6/"8 . Se dau !ul#!#le ,N,P d#n d#ara!ele al&$ura$e.Se %er:

a8 de$er!#na# !ul#!#le ,N,P? 78 de$er!#na#: N , P N, X P (# X N.

6"8 /. Se dau !ul#!#le: A F K+,,0, > F K,1,2, C F K/,1,3.De$er!#na#:

N

P

-

+

/

1

02

3

4

5

-+

--

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a8 A X 6A>8 F78 A 6> X C8 F%8 C X > X 6 A X >8 J Fd8 6A>8 6>C8 F

6"8 0. S& 'e de$er!#ne ele!en$ele !ul#!#lor:a8 A F { },1U ++,+ F { }.,2U >

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)*#:a re;ola o e%ua#e n'ea!n& a de$er!#na !ul#!ea 'olu##lor e%ua#e#, ad#%& a afla nu!erele,a,c"en$ru%are "ro"o;###le !a$e!a$#%e 'un$ ade&ra$e.

Ex:a8 0/=++ , '%&de! n a!7# !e!7r# "e /, "en$ru a afla "e: =+=+ /0///U0/ ++ -=+ . A'$fel 'olu#a e%ua#e# e'$e !ul#!ea K-.

78 0 =a , !"&r#! n a!7# !e!7r# %u "en$ru a afla "e: LK:0: === /aa .

Exerciiu:Re;ola# e%ua##le:a8 -+1=++ 78 2/ =+ %8 -+1 =+ d8 -+=++ e8 -1-/=++

em1:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++ . Cal%ula#: a8 ( ) =22/0 0:. 78 ( ) =--25:-/ 1+. %8 = 1+/ -...---

d8 ( ) ( )[ ] = 1504/05/+ ----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++em1:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++ . Cal%ula#: a8 ( ) =22/0 0:. 78 ( ) =--25:-/ 1+. %8 = 1+/ -...---

d8 ( ) ( )[ ] = 1504/05/+ ----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++


d8 ( ) ( )[ ] = 1504/05/+ ----1:/

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em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++


d8 ( ) ( )[ ] = 1504/05/+ ----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++


d8 ( ) ( )[ ] = 1504/05/+ ----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++ COALA GENERAL PLATCA

Prof. Poenaru RaduCLASA: V-a

TITLUL LECIEI: E%ua$## #n !ul$#!ea nr. na$uraleDATA:

Relu&!:

E%ua#e: o "ro"o;##e !a$e!a$#%& %u o ne%uno'%u$& + .

Re;olarea une# e%ua##: aflarea alor## lu#"en$ru %are "ro"o;##a e'$e ade&ra$&, ad#%& aflarea 'olu#e#.

e$ode de re;olare a une# e%ua##: = folo'#! "ro"r#e$le eal#$lor? = folo'#! "ro7a adun&r##Y'%&der## 'au "ro7a n!ul#r##Y!"&r#r##?

E)e!"le: a8 0 =+ .= n$re7area e'$e !ul#$ %u %#ne ne d& 0, (# r&'"un'ul l "u$e! o7#ne d#n "ro7a n!ul#r##: da%&

n!ul#$ %ue'$e 0 , a$un%# 0 : F.= 0 =+ 'e "oa$e re;ola folo'#nd "ro"r#e$le eal#$lor? "u$e! a'$fel !"&r# n a!7## !e!7r#

%u a%ela(# nu!&r, ur!&r#nd '& o7#ne! o eal#$a$e de enulF ? a'$fel, o! !"&r# %u na!7# !e!7r# (# o! o7#ne :0 ==+

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78 31=++ = 'e "oa$e re;ola %u "ro7a adun&r## 'au "r#n "ro"r#e$le eal#$lor? "r#n "ro"r#e$le

eal#$lor, "u$e! '%&dea un 1 n a!7# !e!7r#, (# a*une! la .1311 ==+ ++ .

%8 1/. =+ + = dor#! '& a*une! d#n nou la o eal#$a$e de enul F ? a'$fel, o! el#!#na na#n$e "e /,

'%&;nd n a!7## !e!7r# "e /: ../1//. ==+ ++ . A'$fel a*une! la o e%ua#e deenul %ele#a d#n e).a8 (# o! !"&r# n a!7# !e!7r# %u , o7#nnd: -.:..:. == ++ .

Exerciii:Re;ola# e%ua##le:-8. /=+ 8. 3/=++ /8. 4. =+ 08. -11 =+ 18. 5/ =++ 28. 5 =+ ++ 38. / =+ 48. .0. 0:./ =+

Exerciii:Re;ola# e%ua##le:-8. / =++ 8. 33=++ /8. 20/ =+ 08. -11 =+ 18. /20/8234/6 0 =++ +

em1:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++ . Cal%ula#: a8 ( ) =22/0. 0:. 78 ( ) =--25:-/ 1+. %8 = 1+/ -...---

d8( ) ( )[ ] =

1504/05/+

----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++

em1:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++ . Cal%ula#: a8 ( ) =22/0. 0:. 78 ( ) =--25:-/ 1+. %8 = 1+/ -...---

d8 ( ) ( )[ ] = 1504/05/+ ----1:/

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em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++

em1:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++

. Cal%ula#: a8 ( ) =22/0 0:. 78 ( ) =--25:-/ 1+. %8 = 1+/ -...---

d8 ( ) ( )[ ] = 1504/05/+ ----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++


d8 ( ) ( )[ ] = 1504/05/+ ----1:/

em2:-. Re;ola# e%ua##le: a8 3+-0 =+ 78 22 =+ %8 +0 =+ d8 -21=++CLASA: V-aTITLUL LECIEI:

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. Cal%ula#: a8 ( = -+41 : 78 ( ) =30 /:/ %8 ( ) =/0 1:1 d8 =323 /:1:-1 e8 ( = 340 0:0 f8 ( ) ( ) =02+2+0 /:/ 8 ( ) = 442301/0 :/:/

/. Da# $re# d##;or# a# nu!erelor: 0+, -21, 01, -4, 5+0. Re;ola# e%ua##le: a8 -420 =++ 78 -/1/ =+ %8 -1- =+ + d8 000 =++

CLASA: V-aTITLUL LECIEI:

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+. Ser#u (# #ne n dou& dou& %u$## %ele 4+ de %re#oane ale 'ale. Da%& d#n "r#!a %u$#e '=ar lua 0- de %re#oane (#'=ar "une n a doua %u$#e, a$un%# n "r#!a ar f# de $re# or# !a# !ul$e %re#oane de%$ n a doua. C$e %re#oane areSer#u n f#e%are %u$#e@

777. Re;ola# e%ua##le:

1. 0/=+ 2. 0/=++ 3. 04 =+ . 3/ =++ (. -+/ += ++ +. 431 +=+ ++

7V. 6RE\OLAREA PRO>LEELOR CU AZUTORUL ECUAIILOR8

1. S& 'e afle un nu!&r, ($##nd %& adunndu=l %u 0 o7#ne! 02.2. Ce nu!&r $re7u#e 'a '%&de! d#n -04 "en$ru a o7#ne nu!&rul 2@3. S& 'e afle un nu!&r, ($##nd %a n!ul#ndu=l %u 0 o7#ne! a%ela(# re;ul$a$ %a a$un%# %nd l adun&! %u -+.. lad are de dou& or# !a# !ul# 7an# de%$ Paul. Da%& lad ar da lu# Paul - le# a$un%# e# ar aea a%eea(#'u!&. C# le# are lad (# %# are Paul@(. Mn$r=o $a7&r& 'un$ de $re# or# !a# !ul# 7e# de%$ fe$e. Da%& n $a7&r& ar !a# en# 54 de fe$e, nu!&rul7e#lor ar f# eal %u nu!&rul fe$elor. C# 7e# (# %$e fe$e 'un$ n $a7&r&@

COALA INTERNAIONAL H CLUZ

=A /R)45E6E

7. 6ETODA REDUCERII LA UNITATE8

1. 1 de !ere %o'$& + le#. C&$ %o'$& 3 de !ere@ Dar 0+ de !ere@2. a&. - ! de '$of& %o'$& 0+ de le#. C&$ %o'$& 4 ! d#n a%e#a(# '$of&@

*&. 3 ! de "n;& %o'$& 33 de le#. C&$ %o'$& un !e$ru d#n a%e#a(# "n;&@

77. 6ETODA

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. Pe lad l=a %o'$a$ a%#;##onarea a dou& %&r# 'u!a de -1+ de le#. C$ a %o'$a$ f#e%are %ar$e da%& d#ferena de"re d#n$re ele e'$e de + de le#@(. eda are n$r=o %u$#e nu!a# 7#le de $re# %ulor#: ro(##, al7ene (# nere. Nu!a# 3 d#n ele nu 'un$ nere (#nu!a# /5 nu 'un$ ro(##. Nu!&rul 7#lelor ro(## e'$e de dou& or# !a# !#% de%$ nu!&rul 7#lelor nere. C$e 7#le def#e%are %uloare 'un$ #n %u$#a ede#@+. N#na (# #ne n dou& dou& %u$## %ele 4+ de %re#oane ale 'ale. Da%& d#n "r#!a %u$#e '=ar lua 0- de %re#oane (#'=ar "une n a doua %u$#e, a$un%# n "r#!a ar f# de $re# or# !a# !ul$e %re#oane de%$ n a doua. C$e %re#oane areN#na n f#e%are %u$#e@

777. Re;ola# e%ua##le:

1. 0/=+ 2. 0/=++ 3. 04 =+ . 3/ =++ (. -+/ += ++ +. 431 +=+ ++7V. 6RE\OLAREA PRO>LEELOR CU AZUTORUL ECUAIILOR8

1. S& 'e afle un nu!&r, ($##nd %& adunndu=l %u 0 o7#ne! 02.2. Ce nu!&r $re7u#e 'a '%&de! d#n -04 "en$ru a o7#ne nu!&rul 2@3. S& 'e afle un nu!&r, ($##nd %a n!ul#ndu=l %u 0 o7#ne! a%ela(# re;ul$a$ %a a$un%# %nd l adun&! %u -+.. Ralu%a are de dou& or# !a# !ul# 7an# de%$ Ana. Da%& Ralu%a #=ar da Ane# - le# a$un%# ele ar aea a%eea(#

'u!&. C# le# are Ralu%a (# %# are Ana@(. Mn$r=o $a7&r& 'un$ de $re# or# !a# !ul# 7e# de%$ fe$e. Da%& n $a7&r& ar !a# en# 54 de fe$e, nu!&rul7e#lor ar f# eal %u nu!&rul fe$elor. C# 7e# (# %$e fe$e 'un$ n $a7&r&@

Te'$ de ealuare

6"8 -. Re;ola$# e%ua$##le: a8 0/=++ 78 +1 =+ %8 51. =+ + d8 -1-- =+ +

6"8 . Sa 'e afle doua nu!ere %are aduna$e dau 32 '$##nd %a unul e'$e de $re# or# !a# !are de%a$ %elalal$.

6"8 /. Sa 'e afle nu!arul na$ural n'$##nd %a du7lul 'au e'$e %u 01 !a# !#% de%a$ 3/.

6"8 0. Cal%ula$#: a8 =+ / / 78 ( ) =-20-25:-/ /5.+. %8 ( =45/+ :/:2 d8 ( ) =1+5-4 -++:-+

6-"8 1. Sa 'e afle $re# nu!ere na$urale a,b,cda%a: ae'$e %u !a# !are de%a$ b, be'$e %u 1 !a# !are de%a$ c'#'u!a lor e'$e 01.

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Te'$ de ealuare

6"8 -. Re;ola$# e%ua$##le: a8 0/=++ 78 +1 =+ %8 51. =+ + d8 -1-- =+ +

60"8 . Cal%ula$#: a8 /01 / F 78 /0 H -4 F %8 /0 -2 F d8 =//0 e8 =/-0 f8 =-/0 8 =+ /01/ 8 /0- H -/ F


CLASA: V-a

TITLUL LECIEI: 8umere rationale. ractii or"inare: fractii ec9iunitare#u*unitare#upraunitare.DATA:

Def: 'e nu!e'$e fractie, una 'au !a# !ul$e "ar$# eale #n %are a fo'$ #!"ar$#$ un #n$re.

8otatie: da%a #!"ar$#! #n$reul a#n b"ar$# eale, o! no$a ba: ad#%ab

a.

= ex: #!"ar$#! !e$r# de "an;a la 0 oa!en#, #n !od eal:= f#e%are o! "r#!e'$e %a$e o 7u%a$a d#n %e# 0!,

or#%are d#n a%e'$e 7u%a$# e'$e o "a$r#!e d#n %e#

0! '# o o! no$a0

-.

= "r##! "e f#ura fra%$##le -00,

0.,

0/ =

)*#: a8 deoare%e fra%$#ab

are"re;#n$a #!"ar$#rea lu# a#n b"ar$# eale, o! '$## %a ba

b

a:= de%# l#n#a de fra%$#e

re"re;#n$a o #!"ar$#re. 78 ele!en$ele fra%$#e#: nu!#$or, nu!ara$or.

Exercitii: Ce fra%## 'un$ re"re;en$a$e n ur!&$oarele de'ene @

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1& 2& 3&

& (& +&

;& &

11& 12& 13& 1& 1(&

1+& 1;& 1&

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Exercitii: [a(ura# n %on$#nuare a'$fel n%$ de'enul '& #lu'$re;e fra%#a al&$ura$&.

1&0. 2&

.- 3&

00

&1

/ (&

1

0 +&

1

1 ;&

1

0

& /-

11& /.

12& //

13&.

/ 1&

.

- 1(& -

1+&1

.

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1;&/

-

Te!a: -8 Cal%ula$#: a8 =++ ../ .1/. 78 =0:/01/2 %8 =/0:0 031+ d8 ( ) =13/+/ :

8 Sa 'e re"re;#n$e "e un de'en fra%$##le:--

-+,

5

.,

3

-,

1

/

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Te!a: -8 Cal%ula$#: a8 =++ ../ .1/. 78 =0:/01/2 %8 =/0:0 031+ d8 ( ) =13/+/ :


-+,

5

.,

3

-,

1

/

Te!a: -8 Cal%ula$#: a8 ( )=++

../ .1/.

78 =0:/01/2 %8 =/0:0 031+

d8 ( ) =13/+/

: 8 Sa 'e re"re;#n$e "e un de'en fra%$##le:

--

-+,

5

.,

3

-,

1

/

Te!a: -8 Cal%ula$#: a8 ( ) =++ ../ .1/. 78 =0:/01/2 %8 =/0:0 031+ d8 ( ) =13/+/ :


-+,

5

.,

3

-,

1

/

Te!a: -8 Cal%ula$#: a8 =++ ../ .1/. 78 =0:/01/2 %8 =/0:0 031+ d8 ( ) =13/+/ :


-+,

5

.,

3

-,

1

/

Te!a: -8 Cal%ula$#: a8 ( ) =++ ../ .1/. 78=

0:/01/2 %8=

/0:0 031+ d8( ) =

13/+/ :


-+,

5

.,

3

-,

1

/


CLASA: V-aTITLUL LECIEI: 8umere rationale. ractii or"inare: fractii ec9iunitare#u*unitare#upraunitare.

DATA:

Pe de'en ur!ar#! fra%$##le5

5,

3

3,

1

1,

.

.'# o7'era! %a ele for!ea;a #n$reul.

Con%lu;#e: a$un%# %and nu!#$orul '# nu!ara$orul une# fra%$## 'un$ eale, fra%$#a for!ea;a un #n$re, a%ea'$afra%$#e 'e nu!e'$e ec9iunitara.

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Ur!ar#! "e de'en fra%$##le:--

/,

5

1,

3

-,

1

/%are ne dau alor# !a# !#%# de%a$ #n$reul '# 'e or nu!# fra%$##

#u*unitare.

Lua! a"o# -- "ar$# d#n$r=un #n$re #!"ar$#$ la 1 '# o7'era! 6"e de'en8 %a a%e'$ea ne dau o aloare !a# !arede%a$ #n$reul.

Con%lu;#e: a$un%# %and nu!ara$orul une# fra%$## e'$e !a# !are de%a$ nu!#$orul, ae! o fra%$#e #upraunitara.

Exercitiu: Care d#n ur!a$oarele fra%$## 'un$ 'u7un#$areYe%#un#$areY'u"raun#$are:

--

-+,

5

-.,

3

-,

2

0,

4

-+,

3

3,

0

/,

5

-/,

1

/

Te!a: -8 Care d#n ur!a$oarele fra%$## 'un$ e%#un#$areY'u7un#$areY'u"raun#$are:-2

-2,

--

-+,

3

4,

/

/,

5

.,

0

1,

3

.-,

-1

-/.

8 Care d#n ur!a$oarele nu!ere 'un$ d##;#7#le %u ,/,1,-+: /+, 5/0, -+, -2, 0/, 52

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Te'$ de ealuareCLASA: V-aDATA:

6/"8 -. Cal%ula$#: a8 =+ /. .. 78 = 05/+.+ .:.. %8 ( ) =--5/0+ :

d8 ( = 23+/+ :0 e8 = ./1./ ... f8 ( ) =/2+/+ :0

6"8 . Se dau !ul$#!#le AF{ }/,.,- , >F { }0,.,+ '# CF { }0 +,+ .Se %er: a8 =#"

78 =$#

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%8 " X =# d8 ( )= $#"

6"8 /. Re;ola$# e%ua$##le: a8 2+/ =+ 78 -01/ =++ %8 -3- =+ d8 -41=++

6-"8 0. Cal%ula$#: ( =+ -5:./1/./ ..

6-"8 1. Cal%ula$# 6een$ual "r#n '%oa$erea fa%$orulu# %o!un8:= 0/010

/0/0/0

5ucrare #eme#triala

Nu!e:

-8 Cal%ula#:

a8 =+

1

- 78 =

4

1

4

5 %8 =

+

/

/

-

/

0 d8 =

-+

/

1

4

e8 =+ -0-32 f8 =+ -++/-+-1/

8 Cal%ula#: a8 =+ /1,+0.,+ 78 =+ ./,--,+ %8 = /,01,2 d8 =+ 4,++2,-+.,/ e8 =+ 22,--,0/,. f8 ( )= /0,+..,/1,-

/8 Re;ola# e%ua##le: a8 5/ =+ 78 5-2 =++ ++ %8 ( ) ( ) -+-/- =+++ ++

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08 Cal%ula#: a8 =+-+

././,+ 78 =++

-+++

54

-+

1+/,2 %8 =

+

.

-1

-1,-

100947571-clasa-v (1)

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