PROGRAMA

PENTRU DISCIPLINA MATEMATICA

EVALUAREA NATIONALA

PENTRU ELEVII CLASEI A VIII – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbOaqaaaaaaaaaWdbiaa=nbiaaa@37C2@ A

2011

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PROGRAMA PENTRU DISCIPLINA MATEMATICA

I. STATUTUL DISCIPLINEI

Pentru anul scolar 2010/2011, in cadrul Evaluarii Nationale pentru elevii clasei a VIII-a, matematica are statut de disciplina obligatorie.

Testul la matematica este o proba scrisa cu durata de 2 ore.

II. COMPETENTE DE EVALUAT

1. Utilizarea notiunii de numar real si a relatiilor dintre multimile de numere studiate

2. Identificarea proprietatilor operatiilor cu numere reale

3. Aplicarea operatiilor cu numere reale in calcule variate

4. Analizarea unor situatii practice cu ajutorul rapoartelor, procentelor, proportiilor

5. Identificarea unor probleme care se rezolva cu ajutorul ecuatiilor, inecuatiilor sau a sistemelor de ecuatii, rezolvarea acestora si interpretarea rezultatului obtinut

6. Aplicarea in rezolvarea problemelor a elementelor de logica si de teoria multimilor

7. Utilizarea elementelor de calcul algebric

8. Alegerea metodei adecvate de rezolvare a problemelor in care intervin dependente functionale sau calculul probabilitatilor

9. Aplicarea teoriei specifice functiei de forma f:ℝ→ℝ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFGI8FfYJH8YrFfeuY=Hhbbe9s8qspq0xc9fs0xc9q8qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaGG6aGaeSyhHeQaeyOKH4QaeSyhHekaaa@3C90@ , f( x )=ax+b, a,b∈ℝ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqarVepK0de9LqFHe9Lqpepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaamyyaiaadIhacqGHRaWkcaWGIbGaaiilaiaaykW7caaMc8UaamyyaiaacYcacaWGIbGaeyicI4SaeSyhHekaaa@47E6@

10. Utilizarea proprietatilor figurilor geometrice si a corpurilor geometrice in probleme de demonstratie si de calcul

11. Reprezentarea, prin desen, a unor figuri geometrice si a unor corpuri geometrice utilizând instrumente geometrice

12. Transpunerea in limbaj matematic a enuntului unei situatii-problema

13. Analizarea si interpretarea rezultatelor obtinute prin rezolvarea unei probleme practice cu referire la figurile geometrice si la unitatele de masura

14. Investigarea valorii de adevar a unor enunturi si construirea unor generalizari

15. Redactarea coerenta si completa a solutiei unei probleme

II. CONTINUTURI ARITMETICA SI ALGEBRA

Multimi

Multimi: relatii (apartenenta, egalitate, incluziune); submultime; operatii cu multimi (reuniunea, intersectia, diferenta, produsul cartezian). Multimi finite, multimi infinite.

Multimile: ℕ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0dg9q8qqaqFn0dXdir=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqWIvesPaaa@3B19@ , ℤ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiablssiIcaa@3658@ , ℚ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiablQriKcaa@3650@ , ℝ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabl2riHcaa@3650@ , ℝ\ℚ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabl2riHkaacYfacqWIAecPaaa@38A0@ , ℕ⊂ℤ⊂ℚ⊂ℝ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiablwriLkabgkOimlablssiIkabgkOimlablQriKkabgkOimlabl2riHcaa@4098@ .

Scrierea numerelor naturale in baza zece.

Propozitii adevarate si propozitii false.

Impartirea cu rest a numerelor naturale. Divizibilitatea in ℕ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiablwriLcaa@364C@ : definitie, divizor, multiplu; proprietati ale relatiei de divizibilitate; criteriile de divizibilitate cu 10, 2, 5, 3; numere prime si numere compuse; numere pare si numere impare; numere prime intre ele; descompunerea unui numar natural in produs de puteri de numere prime; cel mai mare divizor comun si cel mai mic multiplu comun.

Divizibilitatea in ℤ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiablssiIcaa@3658@ : definitie, divizor, multiplu.

Fractii subunitare, echiunitare, supraunitare; reprezentari echivalente ale fractiilor; fractii ireductibile.

Scrierea unui numar rational sub forma de fractie ordinara sau fractie zecimala.

Reprezentarea pe axa a numerelor reale. Compararea si ordonarea numerelor reale.

Valoarea absoluta (modulul), partea intreaga si partea fractionara a unui numar real. Opusul si inversul unui numar real. Rotunjirea si aproximarea unui numar real.

Intervale in ℝ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabl2riHcaa@3650@ : definitie, reprezentare pe axa.

Radacina patrata a unui numar natural patrat perfect; algoritmul de extragere a radacinii patrate dintr-un numar natural; scrierea unui numar real pozitiv ca radical din patratul sau.

Reguli de calcul cu radicali. Introducerea factorilor sub radical. Scoaterea factorilor de sub radical. Rationalizarea numitorului de forma a b , a± b MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaakaaabaGaamOyaaWcbeaakiaacYcacaaMc8UaaGPaVlaadggacqGHXcqSdaGcaaqaaiaadkgaaSqabaaaaa@3F81@ cu a∈ ℤ ∗ , b∈ℕ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolablssiIoaaCaaaleqabaGaey4fIOcaaOGaaiilaiaaykW7caaMc8UaamOyaiabgIGiolablwriLcaa@425A@ . Operatii cu numere reale: adunarea, scaderea, inmultirea, impartirea, ridicarea la putere cu exponent numar intreg. Ordinea efectuarii operatiilor si folosirea parantezelor. Factorul comun.

Media aritmetica a unor numere rationale pozitive. Media geometrica a doua numere reale pozitive.

Rapoarte si proportii: raport; proprietatea fundamentala a proportiilor; proportii derivate; aflarea unui termen necunoscut dintr-o proportie; marimi direct proportionale si marimi invers proportionale; regula de trei simpla.

Procente: p% dintr-un numar real; aflarea unui numar rational când cunoastem p% din el; aflarea raportului procentual. Rezolvarea problemelor in care intervin procente.

Calculul probabilitatii de realizare a unui eveniment.

Calcul algebric

Calcul cu numere reprezentate prin litere: adunare, scadere, inmultire, impartire, ridicarea la putere cu exponent numar intreg.

Formulele de calcul prescurtat: (a±b) 2 = a 2 ±2ab+ b 2 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacqGHXcqScaWGIbGaaiykamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadggadaahaaWcbeqaaiaaikdaaaGccqGHXcqScaaIYaGaamyyaiaadkgacqGHRaWkcaWGIbWaaWbaaSqabeaacaaIYaaaaaaa@4574@

(a+b)(a−b)= a 2 − b 2 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacqGHRaWkcaWGIbGaaiykaiaacIcacaWGHbGaeyOeI0IaamOyaiaacMcacqGH9aqpcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOyamaaCaaaleqabaGaaGOmaaaaaaa@436C@

(a+b+c) 2 = a 2 + b 2 + c 2 +2ab+2bc+2ac MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacqGHRaWkcaWGIbGaey4kaSIaam4yaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadogadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamyyaiaadkgacqGHRaWkcaaIYaGaamOyaiaadogacqGHRaWkcaaIYaGaamyyaiaadogaaaa@4EF8@

Descompunerea in factori: metoda factorului comun; utilizarea formulelor de calcul prescurtat; gruparea termenilor si metode combinate.

Rapoarte de numere reale reprezentate prin litere. Simplificare. Operatii cu rapoarte (adunare, scadere, inmultire, impartire, ridicare la putere cu exponent numar intreg).

Functii

Notiunea de functie.

Functii definite pe multimi finite exprimate cu ajutorul unor diagrame, tabele, formule; graficul unei functii, reprezentarea geometrica a graficului.

Functii de tipul f:A→ℝ, MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGbbGaeyOKH4QaeSyhHeQaaiilaaaa@3BC4@ f( x )=ax+b, MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadggacaWG4bGaey4kaSIaamOyaiaacYcacaaMc8oaaa@3FA6@ a,b∈ℝ, MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9siVeYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVlaadggacaGGSaGaamOyaiabgIGiolabl2riHkaacYcaaaa@3CF4@ unde A=ℝ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2da9iabl2riHcaa@3920@ sau o multime finita; reprezentarea geometrica a graficului functiei f ; interpretare geometrica.

Ecuatii, inecuatii si sisteme de ecuatii

Rezolvarea in ℝ×ℝ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHeQaey41aqRaeSyhHekaaa@3AAC@ a sistemelor de ecuatii de forma:

{ a 1 x+ b 1 y= c 1 a 2 x+ b 2 y= c 2 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafaqabeGabaaabaGaamyyamaaBaaaleaacaaIXaaabeaakiaadIhacqGHRaWkcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaamyEaiabg2da9iaadogadaWgaaWcbaGaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaamiEaiabgUcaRiaadkgadaWgaaWcbaGaaGOmaaqabaGccaWG5bGaeyypa0Jaam4yamaaBaaaleaacaaIYaaabeaaaaaakiaawUhaaaaa@49F3@ , a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ∈ℝ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaaMc8UaamyyamaaBaaaleaacaaIYaaabeaakiaacYcacaaMc8UaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaaMc8UaamOyamaaBaaaleaacaaIYaaabeaakiaacYcacaaMc8Uaam4yamaaBaaaleaacaaIXaaabeaakiaacYcacaaMc8Uaam4yamaaBaaaleaacaaIYaaabeaakiabgIGiolabl2riHcaa@4EE3@ .

Rezolvarea in ℝ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3725@ a inecuatiilor de forma ax+b≤0 ( <, ≥, > ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadIhacqGHRaWkcaWGIbGaeyizImQaaGimaiaaykW7caaMc8+aaeWaaeaacqGH8aapcaGGSaGaaGPaVlaaykW7caaMc8UaeyyzImRaaiilaiaaykW7caaMc8UaaGPaVlaaykW7cqGH+aGpaiaawIcacaGLPaaaaaa@506E@ , a∈ℝ*, b∈ℝ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFC0xe9vqGqpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolabl2riHkaacQcacaGGSaGaaGPaVlaaykW7caWGIbGaeyicI4SaeSyhHekaaa@41DE@ .

Probleme cu caracter aplicativ care se rezolva cu ajutorul ecuatiilor, inecuatiilor si al sistemelor de ecuatii. Utilizarea metodelor aritmetica sau algebrica pentru rezolvarea unor probleme.

GEOMETRIE

Masurare si masuri

Figuri si corpuri geometrice:

1. Punctul, dreapta, planul, semiplanul, semidreapta, segmentul de dreapta, unghiul

- pozitii relative, clasificare; conventii de desen si de notatii

- paralelism si perpendicularitate in plan si in spatiu; axioma paralelelor; unghiuri cu laturile respectiv paralele; unghiul a doua drepte in spatiu; drepte perpendiculare; dreapta perpendiculara pe un plan; distanta de la un punct la un plan; plane paralele; distanta dintre doua plane paralele;

- teorema celor trei perpendiculare; distanta de la un punct la o dreapta;

- proiectia ortogonala a unui punct, segment sau a unei drepte pe un plan;

- unghiul unei drepte cu un plan; lungimea proiectiei unui segment;

- unghiul diedru; unghiul plan corespunzator unui unghi diedru; masura unghiului a doua plane; plane perpendiculare;

- simetria fata de un punct in plan; simetria fata de o dreapta in plan.

- calculul unor distante si masuri de unghiuri pe fetele sau in interiorul corpurilor studiate.

2. Triunghiul

- perimetrul si aria;

- suma masurilor unghiurilor unui triunghi;

- unghi exterior unui triunghi;

- linii importante in triunghi si concurenta lor;

- linia mijlocie in triunghi;

- triunghiul isoscel si triunghiul echilateral – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@ proprietati;

- criteriile de congruenta a triunghiurilor;

- triunghiul dreptunghic – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@ teorema inaltimii; teorema catetei; teorema lui Pitagora si reciproca ei;

sinusul, cosinusul, tangenta, cotangenta; rezolvarea triunghiului dreptunghic;

- teorema lui Thales si reciproca ei;

- teorema fundamentala a asemanarii;

- triunghiuri asemenea – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@ criteriile de asemanare a triunghiurilor.

3. Patrulaterul convex

- perimetrul si aria (paralelogramul, dreptunghiul, rombul, patratul, trapezul);

- suma masurilor unghiurilor unui patrulater convex;

- paralelogramul – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@ proprietati referitoare la laturi, unghiuri, diagonale;

- paralelograme particulare (dreptunghi, romb, patrat) – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@ proprietati;

- trapezul; linia mijlocie in trapez;

- trapeze particulare (isoscel si dreptunghic) – MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@ proprietati.

4. Cercul

- centru, raza, diametru, disc;

- unghi la centru;

- coarde si arce in cerc (la arce congruente corespund coarde congruente si reciproc; proprietatea diametrului perpendicular pe o coarda; proprietatea arcelor cuprinse intre doua coarde paralele; proprietatea coardelor egal departate de centru);

- unghi inscris in cerc; masura unghiului inscris in cerc;

- lungimea cercului; aria discului;

- calculul elementelor(latura, apotema, perimetru, arie) in poligoane regulate: triunghi echilateral, patrat.

5. Corpuri geometrice

Paralelipipedul dreptunghic, cubul; prisma dreapta cu baza triunghi echilateral, patrat sau dreptunghi;

piramida triunghiulara regulata, tetraedrul regulat, piramida patrulatera regulata:

- reprezentarea lor prin desen; conventii de desen si de notatii;

- descrierea elementelor lor (vârfuri, muchii, fete laterale, baze, diagonale, inaltimi);

- desfasurari;

- aria laterala, aria totala, volumul.

NOTA: Programa de examen este realizata in conformitate cu prevederile programelor scolare in vigoare. Subiectele pentru examenul de evaluare nationala 2011 se elaboreaza in baza prevederilor prezentei programe si nu vizeaza continutul unui manual anume.