Malangizo othetsera Phunziro §1. Malingaliro, Mutu I. Malingaliro. Zosonkhanitsira, buku la Algebra 10. Zam’kati mwa phunziro 1 2 3 4 5 6 7 tsamba 9 10 buku lophunzirira Basic Algebra 10 limaphatikizapo kaphatikizidwe ka mafomula, malingaliro, njira zothetsera zochitika za algebra zomwe zikuphatikizidwa mu Mabuku othandizira ophunzira kuchita bwino masamu a giredi 10.

Mukuwona: Kuthetsa Maphunziro 10 a Algebra 1 tsamba 9

Chiphunzitso

I. Malingaliro. Ndime yokhala ndi zosintha

1. Ndime

Ndime iliyonse iyenera kukhala yowona kapena yabodza.

Mawu sangakhale oona ndi abodza.

Chitsimikizo chowona chimatchedwa mawu olondola. Mawu abodza amatchedwa mawu olakwika.

Mwachitsanzo:

Nambala 2 ndi nambala yayikulu ndi mawu owona.

5 yogawidwa ndi 3 ndi yabodza.

2. Ndime yomwe ili ndi kusintha

Mwachitsanzo: Ganizirani mawu awa:

(a): “7 + x = 3”

(b): “n ndi nambala yoyamba”

Pezani zikhalidwe ziwiri za x, n kuti (a), (b) mupeze chiganizo chimodzi chowona ndi chimodzi zabodza.

Chiganizo (a) ndi (b) ndi zitsanzo za mawu okhala ndi variable.

II. Kukana lingaliro

Chidziwitso cha chiganizo chotsutsa cha P ndi \ (\ overline P \), tili ndi:

\(\ overline P \) ndizowona pamene P ndi zabodza.

\(\ overline P \) ndi zabodza pamene P ndi zoona.

Mwachitsanzo:

Malingaliro P: “\(\pi \) ndi nambala yomveka”. Tili ndi: \(\ overline P :\) “\(\pi \) si nambala yomveka”.

Q: “Kuchuluka kwa mbali ziwiri za makona atatu ndikokulirapo kuposa mbali yachitatu”. Tili ndi: \(\ overline Q :\) “Kuchuluka kwa mbali ziwiri za makona atatu sikuli kwakukulu kuposa mbali yachitatu”.

III. Ndime yotsatirayi

Mawu oti “Ngati P ndiye Q” amatchedwa chiganizo chotsatira ndipo amatanthauzidwa ndi \(P \Rightarrow Q\).

Mawu akuti \(P \Rightarrow Q\) ndi zabodza pokhapokha ngati P ali woona ndipo Q ndi wabodza.

Zigawo za masamu nthawi zambiri zimakhala za \(P \Rightarrow Q\)

P ndiye lingaliro, Q ndiye mapeto a chiphunzitsocho.

P ndi gawo lokwanira la Q, kapena Q ndi chofunikira kwa P.

Mwachitsanzo:

Poganizira lingaliro: “Ngati makona atatu ABC ali ndi ngodya ziwiri zofanana ndi 600, ndiye kuti ABC ndi makona atatu ofanana”.

GT: Triangle ABC ili ndi ngodya ziwiri zofanana ndi 600.

KL: ABC ndi makona atatu ofanana.

IV. Chigamulo chosiyana – ndime ziwiri zofanana

Chigamulo \ ( Q \ Rightarrow P \ ) chimatchedwa kusokoneza kwa ndime \ (P \ Rightarrow Q\).

Ngati zonse \(P \Rightarrow Q\) ndi \(Q \Rightarrow P\) zili zoona, ndiye kuti P ndi Q akunenedwa kuti ndi ofanana. Kenako timatanthauzira \(P \Leftrightarrow Q\) ndikuwerenga kuti P ndi ofanana ndi Q, kapena P ndi chofunikira komanso chokwanira cha Q, kapena P ngati ndipo ngati Q.

V. Zizindikiro \(\forall \) ndi \(\ zilipo \).

Mwachitsanzo: Kutengera ndime zotsatirazi:

P: “Nambala yachilengedwe iliyonse ndi yayikulu kuposa yosiyana nayo”.

Q: “Pali nambala yomveka yochepera kuposa momwe imakhalira”.

Tchulani kukana mawu omwe ali pamwambawa. Ganizirani zowona ndi zabodza za mawu P, Q, \(\ overline P \), \(\ overline Q \).

Tili ndi:

+ \(\ overline P :\) “Pali nambala yachilengedwe yochepera kapena yofanana ndi mnzake”.

+ \(\ overline Q :\) “Nambala iliyonse yomveka ndi yayikulu kuposa kapena yofanana ndi yosiyana”.

+ P ndi zabodza, \(\ overline P \) ndizowona chifukwa 0 alibe zotsutsana.

+ Q ndi yolondola, \(\ overline Q \) ndiyolakwika, mwachitsanzo \(\frac{1}{2} Nayi Buku loyankha mafunso ndi zolimbitsa thupi mu gawo la zochitika za ophunzira sgk Algebra 10 .

Funso

1. Yankhani mafunso 1 tsamba 4 buku la Algebra 10

Poyang’ana zithunzi ziwiri pamwambapa, werengani ndi kuyerekezera ziganizo zomwe zili kumanzere ndi kumanja.

Yankhani:

Ziganizo kumanzere ndi ziganizo zotsimikizira, zowerengera zoona ndi zabodza.

Ziganizo za kumanja sizinganene kuti ndi zoona kapena zabodza.

2. Yankhani mafunso 2 masamba 4 buku la Algebra 10

Perekani zitsanzo za ziganizo zomwe ziri ziganizo ndi ziganizo zomwe siziri.

Yankhani:

Zitsanzo za ziganizo zomwe ndi ziganizo:

5 ndi nambala yoyamba.

Chitsulo ndi chitsulo.

Zitsanzo za ziganizo zomwe sizili ziganizo:

Kodi lero ndi lachingati?

Kumwamba ndi kokongola kwambiri!

3. Yankhani mafunso 3 masamba 5 buku la Algebra 10

Ganizirani chiganizo $“x> 3”$. Pezani zikhalidwe ziwiri zenizeni za x kuti kuchokera pachiganizo chomwe mwapatsidwa, pezani chowonadi chimodzi ndi chimodzi zabodza.

Yankhani:

Kwa $x = 5$, mawu omwe adalandira ndi oona.

Kwa $x = 1$, mawu omwe adalandira ndi abodza.

4. Yankhani mafunso 4 masamba 6 buku la Algebra 10

Samalani mawu otsatirawa:

$P: $“ π ndi nambala yomveka”;

$Q: $“Kuchuluka kwa mbali ziwiri za makona atatu ndikokulirapo kuposa mbali yachitatu”.

Ganizirani zowona ndi zabodza za mawu omwe ali pamwambawa ndi zonena zawo zoipa.

Yankhani:

$P$ clause: ndi zabodza

Ndime yolakwika $P$: “ π si nambala yomveka”;

Mawu akuti $Q$: ndi mawu owona

Malingaliro olakwika $Q$: “Kuchuluka kwa mbali ziwiri za makona atatu sikuli kwakukulu kuposa mbali yachitatu”.

5. Yankhani mafunso 5 masamba 6 buku la Algebra 10

Kuchokera ku ziganizo:

$P$: “Mvula ya Kumpoto Kum’mawa Ikubwera”

$Q$: “Kuyamba kuzizira”

Nenani mfundoyo $P ⇒ Q$

Yankhani:

$P ⇒ Q$: “Ngati mphepo yamkuntho ya kumpoto chakum’mawa ibwera, kudzakhala kozizira.”

6. Yankhani mafunso 6 masamba 7 buku la Algebra 10

Apatsidwa makona atatu $ABC$. Kuchokera ku zigawo

$P$: “Katatu $ABC$ ili ndi ngodya ziwiri zofanana ndi 60o”

$Q$: “$ABC$ ndi makona atatu ofanana”

Tchulani mfundo yakuti $P ⇒ Q$. Nenani zongopeka, zomaliza ndi kubwerezanso chiphunzitsochi malinga ndi zofunikira komanso zokwanira.

Yankhani:

$P ⇒ Q$: “Ngati makona atatu $ABC$ ali ndi ngodya ziwiri zofanana ndi 60o, ndiye $ABC$ ndi makona atatu ofanana”

Lingaliro: “Katatu $ABC$ ili ndi ngodya ziwiri zofanana ndi 60o”

Pomaliza: “$ABC$ ndi makona atatu ofanana”

Bwezeraninso chiphunzitsochi ngati chofunikira: “$ABC$ ndi makona atatu ofanana ndi chinthu chofunikira kuti makona atatu $ABC$ akhale ndi ngodya ziwiri zofanana ndi madigiri 60”

Bwezeraninso mfundoyi ngati yokwanira: “Makona atatu $ABC$ okhala ndi makona awiri ofanana ndi madigiri 60 ndi malo okwanira $ABC$ kukhala makona atatu ofanana”

7. Yankhani mafunso 7 masamba 7 buku la Algebra 10

Kupatsidwa makona atatu $ABC$. Ganizirani mawu otsatirawa a fomu $P Q$

a) Ngati $ABC$ ndi makona atatu ofanana ndiye kuti $ABC$ ndi makona atatu a isosceles.

b) Ngati $ABC$ ndi makona atatu ofanana ndiye kuti $ABC$ ndi makona atatu a isosceles okhala ndi ngodya yofanana ndi 60o.

Tchulani ziganizozo $Q ⇒ P$ motsatira ndipo ganizirani zowona ndi zabodza.

Yankhani:

a) Ngati $ABC$ ndi makona atatu a isosceles ndiye kuti $ABC$ ndi makona atatu ofanana.

 Izi ndi zabodza

b) Ngati ABC ndi makona atatu a isosceles ndipo ili ndi ngodya ya madigiri 60, ndiye kuti ABC ndi makona atatu ofanana.

 Awa ndi mawu olondola.

8. Yankhani mafunso 8 masamba 8 buku la Algebra 10

Tchulani malingaliro otsatirawa:

$∀n ∈ Z : n + 1 > n$

Kodi mawuwa ndi oona kapena abodza?

Yankhani:

Pa $n$ iliyonse mumagulu onse, $n + 1$ ndi yaikulu kuposa $n$.

 Mawuwa ndi olondola.

9. Yankhani mafunso 9 masamba 8 buku la Algebra 10

Tchulani malingaliro otsatirawa:

$∃ x ∈ Z $: x2 $= x$

Kodi mawuwa ndi oona kapena abodza?

Yankhani:

Pali nambala x mugulu la manambala onse kuti x masikweya ndi ofanana ndi $x$.

Mawuwa ndi oona chifukwa $0 ∈ Z$; 02 $= 0$

10. Yankhani mafunso 10 masamba 8 buku la Algebra 10

Tchulani kukana mawu otsatirawa:

$P$: “Zinyama zonse zimatha kusuntha”.

Yankhani:

“Kukhalapo kwa nyama zosayenda”

11. Yankhani mafunso 11 masamba 9 buku la Algebra 10

Tchulani kukana mfundo zotsatirazi

$P$: “Pali wophunzira m’kalasi amene sakonda kuphunzira Masamu”.

Yankhani:

“Ophunzira onse m’kalasi amakonda kuphunzira Masamu”

Nawa Malangizo othetsera phunziro 1 2 3 4 5 6 7 tsamba 9 10 Algebra 10 zoyambira. Chonde werengani mutuwo mosamala musanawuthetse!

Masewera olimbitsa thupi

91neg.com imakupatsirani njira yonse yothetsera zochitika za algebra 10 ndi mayankho atsatanetsatane a phunziro 1 2 3 4 5 6 7 masamba 9 10 buku la Basic 10 Algebra of Lesson §1. Malingaliro mu Mutu I. Malingaliro. Zasonkhanitsidwa kuti mufotokozere. Zambiri za phunziro lililonse zitha kuwonedwa pansipa:

Konzani phunziro 1 2 3 4 5 6 7 tsamba 9 10 buku la Algebra 10

1. Konzani vuto 1 patsamba 9 buku la Algebra 10

Ndi chiganizo chiti mwa chiganizo chomwe chili ndi chiganizo ndipo ndi chiani chosinthika?

a) \(3 + 2 = 7\);

b) \(4 + x = 3\);

c) \(x + y > 1\);

d) \(2 – \sqrt{5 } 1\), zoona pamene \(\kumanzere\{\kuyamba{matrix} x=1\\ y=2 \kumapeto{matrix}\kumanja.\), zabodza pamene \ (\kumanzere\{\kuyamba{matrix} x=0\\y=0 \kumapeto{matrix}\kumanja.\) . Kotero chiganizo ichi si lingaliro.

 Ichi ndi chiganizo chomwe chili ndi zosintha.

d) Chiganizo \(2 – \sqrt{5}

2. Konzani vuto 2, tsamba 9, buku la Algebra 10

Onani ngati ziganizo zotsatirazi ndi zoona kapena zabodza ndipo tchulani mawu ake oyipa.

a) $1794$ imagawidwa ndi $3$;

b) \(\sqrt{2}\) ndi nambala yomveka:

c) \(\pi0\).

Kukana mfundoyi: \(k=\) ndi chiganizo: \(\ overline{k}=”\ left | -1.25 \ right |> 0″\).

3. Konzani vuto 3 patsamba 9 buku la Algebra 10

Kwa ziganizo zotsatirazi:

Ngati $a$ ndi $b$ agawidwa ndi $c$ ndiye $a + b$ amagawidwa ndi $c$ ($a, b, c$ ndi manambala).

Nambala zomaliza mu $0$ zimagawidwa ndi $5$.

Makona atatu a isosceles ali ndi mizere iwiri yofanana.

Makona atatu a congruent ali ndi madera ofanana.

a) Tchulani chosiyana ndi chilichonse mwa zomwe zili pamwambazi.

b) Tchulani mawu onse omwe ali pamwambawa, pogwiritsa ntchito lingaliro la “mikhalidwe yokwanira”.

c) Tchulani mawu onse omwe ali pamwambawa, pogwiritsa ntchito lingaliro la “zofunikira”.

Yankho:

a) Island clause

Kusiyanitsa koyamba ndi: “Ngati $a + b$ igawika ndi $c$, ndiye $a$ ndi $b$ amagawidwa ndi $c$”. Ndime iyi ndiyolakwika.

Chotsalira chachiwiri ndi: “Nambala zogawikana ndi $5$ zonse zimatha $0”. Ndime iyi ndi yolakwika.

Kutsutsana kwa lingaliro lachitatu ndi: “Ngati makona atatu ali ndi mizere iwiri yofanana, ndiye katatu ndi isosceles”. Mawu awa ndi olondola.

Kutsutsana kwa lingaliro lachinayi ndi: “Makona atatu okhala ndi madera ofanana ndi ofanana”. Ndime iyi ndiyolakwika.

b) Pogwiritsa ntchito lingaliro la “zikhalidwe zokwanira”, ndiye:

Mawu oyamba akuti: “Kuti $a + b$ igawidwe ndi $c$, chokwanira ndi chakuti $a$ ndi $b$ amagawidwa ndi $c$”

Lingaliro lachiwiri likuti: “Kuti chiwerengero chigawike ndi $ 5 $, chikhalidwe chokwanira ndi chakuti chiwerengero chomaliza cha chiwerengerocho ndi $ 0”.

Lingaliro lachitatu likuti: “Kuti makona atatu okhala ndi apakati awiri akhale ogwirizana, chokwanira ndi chakuti makona atatu ndi isosceles”.

Lingaliro lachinayi limati: “Kuti makona atatu akhale ndi malo ofanana, chikhalidwe chokwanira ndi chakuti makona atatuwa ndi ofanana”.

c) Pogwiritsa ntchito mfundo yofunikira ndiye:

Mawu achiwiri akuti: “Kuti $ a $ ndi $ b $ zigawike ndi $ c $, chofunikira ndi chakuti chiwerengerocho chigawidwe ndi $ 5 $”.

Lingaliro lachiwiri likuti: “Kuti chiwerengero chithere mu $ 0, chikhalidwe chiyenera kukhala kuti chiwerengerocho chigawidwa ndi $ 5”.

Lingaliro lachitatu likuti: “Kuti makona atatu akhale isosceles, m’pofunika kuti makona atatu akhale ndi apakati awiri ofanana”.

Lingaliro lachinayi likuti: “Kuti makona atatu akhale ofanana, chofunikira ndi chakuti ali ndi malo ofanana”.

4. Konzani buku la Algebra 10 patsamba 4

Tchulani mawu otsatirawa, pogwiritsa ntchito lingaliro la “zofunikira komanso zokwanira”.

a) Nambala yomwe kuchuluka kwake kwa manambala kumagawika ndi $9 kumagawika ndi $9 ndi mosemphanitsa.

b) Paralelogalamu yokhala ndi ma diagonal a perpendicular ndi rhombus ndi mosemphanitsa.

c) A quadratic equation ali ndi njira ziwiri zosiyana ngati kusankha kwake kuli kolimbikitsa.

Yankho:

a) Chofunikira komanso chokwanira kuti nambala igawike ndi $ 9 ndikuti kuchuluka kwa manambala ake kumagawika ndi $9.

b) Mkhalidwe wofunikira komanso wokwanira kuti quadrilateral ikhale rhombus ndi quadrilateral yomwe ili parallelogram yokhala ndi ma diagonal awiri a perpendicular.

c) Mkhalidwe wofunikira komanso wokwanira kuti quadratic equation ikhale ndi mayankho awiri osiyana ndikuti tsankho lake ndilabwino.

5. Kuthetsa mavuto 5 masamba 10 buku Algebra 10

Gwiritsani ntchito notation \(\forall , \exists\) kuti mulembe ziganizo zotsatirazi

a) Nambala iliyonse yochulukitsidwa ndi 1 ndiyofanana nayo yokha;

b) Pali nambala yomwe imadziwonjezera ku ziro;

c) Nambala kuphatikizira kubweza kwake ndi 0.

Yankho:

a) Wochulukitsa aliyense ndi $1$ ndi wofanana nayeyekha

KH: \(\forall x \mu \mathbb{R}: x.1=x\);

b) Kukhala ndi nambala yodziphatikiza yokha ndikofanana ndi 0

KH: \(\ alipo \ mu \ mathbb{R}: a+a=0\);

c) Nambala kuphatikizira kubweza kwake ikufanana ndi 0 .

KH: \(\forall x \mu \mathbb{R}: x+(-x)=0\).

6. Konzani phunziro 6, tsamba 10, buku la Algebra 10

Nenani ziganizo zotsatirazi mokweza ndipo onani ngati zili zoona kapena zabodza

a) \(\forall x \mu R: x^2>0\);

b) \(\ alipo n \mu N: n^2=n\);

c) \(\forall n \mu N: n \leq 2n\);

d) \(\ alipo x \ mu R: x2 > 0″ ndi zabodza.

b) Pali nambala yachilengedwe imodzi yofanana ndi masikweya ake.

Izi ndi zowona, mwachitsanzo: 12=1.

c) Nambala yachilengedwe iliyonse ndi yocheperapo kapena yofanana ndi kawiri.

Izi ndi zoona chifukwa kusalingana: \(2n>n\Leftrightarrow n>0\) ndizoona pa manambala achilengedwe onse n.

d) Pali nambala imodzi yeniyeni yomwe ili yocheperapo poyerekeza ndi yake.

Awa ndi mawu olondola. Mwachitsanzo: \(\frac{1}{3}

7. Kuthetsa mavuto 7 masamba 10 buku la Algebra 10

Pangani mawu olakwika paziganizo zotsatirazi ndikuziwona ngati zoona kapena zabodza

a) \(\forall n \mu N: n\) amagawidwa ndi n;

b) \(\ ilipo x \mu Q: x^2=2\);

c) \(\ onse x \ mu R: xc) \(\ ilipo x \mu \mathbb{R} :x\geq x+1\)

Izi ndi zabodza, chifukwa kusalingana: \(x\geq x+1\Leftrightarrow 0\geq 1\) kulibe yankho.

Onaninso: Algebra Yabwino Kwambiri, Masamu a Geometry Pokha Pafoni

d) \(\onse x \mu \mathbb{R} :3x \neq x^2+1\)

Izi ndi zabodza, mwachitsanzo: \(x=\frac{3+\sqrt{5}}{2}\) ndiye \(3\left (\frac{3+\sqrt{5}}{2 }) \kumanja )=\kumanzere (\frac{3+\sqrt{5}}{2} \kumanja )^2+1\) ndizowona.

Chotsatira chotsatira:

Zabwino zonse ndi homuweki yanu ndikuthetsa masamu a giredi 10 ndi mayankho 1 2 3 4 5 6 7 tsamba 9 10 buku la Algebra 10!