1.5: Kuzidisha na Gawanya Integers

Last updated
Save as PDF

Page ID: 177924

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

Kuzidisha integers
Gawanya integers
Kurahisisha maneno na integers
Tathmini maneno ya kutofautiana na integers
Tafsiri misemo ya Kiingereza kwa maneno ya algebraic
Tumia integers katika programu

Utangulizi wa kina zaidi wa mada yaliyofunikwa katika sehemu hii inaweza kupatikana katika sura ya Prealgebra, integers.

Kuzidisha integers

Kwa kuwa kuzidisha ni shorthand ya hisabati kwa kuongeza mara kwa mara, mfano wetu unaweza kutumika kwa urahisi ili kuonyesha kuzidisha kwa integers. Hebu tuangalie mfano huu halisi ili kuona ni mwelekeo gani tunaoona. Tutatumia mifano sawa ambayo tulitumia kwa kuongeza na kuondoa. Hapa, tutatumia mfano tu kutusaidia kugundua mfano.

Tunakumbuka kwamba$a\cdot b$ ina maana kuongeza$a,\, b$ nyakati. Hapa, sisi ni kutumia mfano tu kutusaidia kugundua mfano.

Picha mbili zinaonyeshwa kwa upande mmoja. Picha upande wa kushoto ina equation mara tano tatu juu. Chini ya hii inasoma “kuongeza 5, mara 3.” Chini ya hii inaonyesha safu tatu za counters bluu, na counters tano katika kila mstari. Chini ya hili, inasema “chanya 15.” Chini ya hii ni equation “mara 5 3 sawa na 15.” Picha upande wa kulia inasoma “hasi mara 5 tatu. Ya tatu ni katika mabano. Chini ya hii inasoma, “ongeza hasi tano, mara tatu.” Chini ya hii ni counters kumi na tano nyekundu katika safu tatu za tano. Chini ya hii inasoma” “15 hasi”. Chini hii ni equation hasi mara tano 3 sawa hasi 15.” — Kielelezo$\PageIndex{1}$

Mifano miwili ijayo ni ya kuvutia zaidi.

Ina maana gani kuzidisha$5$ na$−3$? Ina maana ya kuondoa$5, 3$ nyakati. Kuangalia uondoaji kama “kuchukua mbali,” inamaanisha kuchukua$5, 3$ nyakati. Lakini hakuna kitu cha kuchukua, kwa hiyo tunaanza kwa kuongeza jozi zisizo na upande kwenye nafasi ya kazi. Kisha tunachukua mara$5$ tatu.

Takwimu hii ina nguzo mbili. Katika mstari wa juu, safu ya kushoto ina maneno mara 5 hasi 3. Hii ina maana kuchukua 5, mara tatu. Chini ya hili, kuna makundi matatu ya counters tano nyekundu hasi, na chini ya kila kundi la counters nyekundu ni kundi linalofanana la counters tano za bluu nzuri. Nini kilichoachwa ni hasi kumi na tano, kilichowakilishwa na counters 15 nyekundu. Chini ya counters ni equation 5 mara hasi 3 sawa hasi 15. Katika mstari wa juu, safu ya kulia ina maneno hasi mara 5 hasi 3. Hii ina maana kuchukua hasi 5, mara tatu. Chini ya hili, kuna makundi matatu ya counters tano za bluu nzuri, na chini ya kila kikundi cha counters bluu ni kundi linalofanana la counters tano nyekundu hasi. Nini kilichoachwa ni chanya kumi na tano, kilichowakilishwa na counters 15 za bluu. Chini ya counters bluu ni equation hasi 5 mara hasi 3 sawa 15. — Kielelezo$\PageIndex{2}$

Kwa muhtasari:

\[\begin{array} {ll} {5 \cdot 3 = 15} &{-5(3) = -15} \\ {5(-3) = -15} &{(-5)(-3) = 15} \end{array}\]

Kumbuka kwamba kwa kuzidisha namba mbili saini, wakati:

ishara ni sawa, bidhaa ni chanya.
ishara ni tofauti, bidhaa ni hasi.

Tutaweka haya yote pamoja katika chati hapa chini.

KUZIDISHA NAMBA ZILIZOSAINIWA

Kwa kuzidisha namba mbili zilizosainiwa:

Jedwali$\PageIndex{1}$
Ishara sawa	Bidhaa	Mfano
Chanya mbili	Chanya	$7\cdot 4 = 28$
Mbili hasi	Chanya	$-8(-6) = 48$

Jedwali$\PageIndex{2}$
Ishara tofauti	Bidhaa	Mfano
Chanya$\cdot$ hasi	Hasi	$7(-9) = -63$
$\cdot$Chanya hasi	Hasi	$-5\cdot 10= -50$

Zoezi$\PageIndex{1}$

Kuzidisha:

$-9\cdot 3$
$-2(-5)$
$4(-8)$
$7\cdot 6$

Jibu

\[\begin{array} {ll} {} &{-9\cdot 3} \\ {\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \end{array}\]
\[\begin{array} {ll} {} &{-2(-5)} \\ {\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \end{array}\]
\[\begin{array} {ll} {} &{4(-8)} \\ {\text{Multiply, with different signs.}} &{-32} \end{array}\]
\[\begin{array} {ll} {} &{7\cdot 6} \\ {\text{Multiply, with different signs.}} &{42} \end{array}\]

Zoezi$\PageIndex{2}$

Kuzidisha:

$-6\cdot 8$
$-4(-7)$
$9(-7)$
$5\cdot 12$

Jibu

$-48$
$28$
$-63$
$60$

Zoezi$\PageIndex{3}$

Kuzidisha:

$-8\cdot 7$
$-6(-9)$
$7(-4)$
$3\cdot 13$

Jibu

$-56$
$54$
$-28$
$39$

Wakati sisi kuzidisha idadi na$1$, matokeo ni idadi sawa. Nini kinatokea wakati sisi kuzidisha idadi na$−1$? Hebu kuzidisha idadi chanya na kisha idadi hasi$−1$ na kuona nini sisi kupata.

\[\begin{array} {lll} {} &{-1\cdot 4} &{-1(-3)}\\ {\text{Multiply.}} &{-4} &{3} \\ {} &{-4\text{ is the opposite of 4.}} &{3\text{ is the opposite of } -3} \end{array}\]
Kila wakati sisi kuzidisha idadi na$−1$, sisi kupata kinyume chake!

KUZIDISHA KWA -1

\[−1a=−a\]

Kuzidisha idadi kwa$−1$ anatoa kinyume chake.

Zoezi$\PageIndex{4}$

Kuzidisha:

$-1 \cdot 7$
$-1(-11)$

Jibu

\[\begin{array} {ll} {} &{-1\cdot 7} \\ {\text{Multiply, noting that the signs are different}} &{-7} \\ {\text{so the product is negative.}} &{-7\text{ is the opposite of 7.}} \end{array}\]
\[\begin{array} {ll} {} &{-1(-11)} \\ {\text{Multiply, noting that the signs are different}} &{11} \\ {\text{so the product is positive.}} &{11\text{ is the opposite of -11.}} \end{array}\]

Zoezi$\PageIndex{5}$

Kuzidisha:

$-1\cdot 9$
$-1\cdot(-17)$

Jibu

$-9$
$17$

Zoezi$\PageIndex{6}$

Kuzidisha:

$-1\cdot 8$
$-1\cdot(-16)$

Jibu

$-8$
$16$

Kugawanya Integers

Nini kuhusu mgawanyiko? Idara ni operesheni inverse ya kuzidisha. Hivyo,$15\div 3=5$ kwa sababu$5 \cdot 3 = 15$. Kwa maneno, maneno haya anasema kwamba$15$ inaweza kugawanywa katika makundi matatu ya tano kila kwa sababu kuongeza tano mara tatu anatoa$15$. Angalia baadhi ya mifano ya kuzidisha integers, kufikiri sheria za kugawa integers.

\[\begin{array} {ll} {5\cdot 3 = 15\text{ so }15\div 3 = 5} &{-5(3) = -15\text{ so }-15\div 3 = -5} \\ {(-5)(-3) = 15\text{ so }15\div (-3) = -5} &{5(-3) = -15\text{ so }-15\div (-3) = 5} \end{array}\]

Idara ifuatavyo sheria sawa na kuzidisha!

Kwa mgawanyo wa namba mbili saini, wakati:

ishara ni sawa, quotient ni chanya.
ishara ni tofauti, quotient ni hasi.

Na kumbuka kwamba tunaweza daima kuangalia jibu la tatizo mgawanyiko kwa kuzidisha.

KUZIDISHA NA MGAWANYIKO WA IDADI ZILIZOSA

Kwa kuzidisha na mgawanyiko wa namba mbili zilizosainiwa:

Ikiwa ishara ni sawa, matokeo ni chanya.
Ikiwa ishara ni tofauti, matokeo ni hasi.

Jedwali$\PageIndex{3}$
Ishara sawa	Matokeo
Chanya mbili	Chanya
Mbili hasi	Chanya
Ikiwa ishara ni sawa, matokeo ni chanya.

Jedwali$\PageIndex{4}$
Ishara tofauti	Matokeo
Chanya na hasi	Hasi
Hasi na chanya	Hasi
Ikiwa ishara ni tofauti, matokeo ni hasi.

Zoezi$\PageIndex{7}$

$-27\div 3$
$-100\div (-4)$

Jibu

\[\begin{array} {ll} {} &{-27 \div 3} \\ {\text{Divide, with different signs, the quotient is}} &{-9} \\ {\text{negative.}} &{} \end{array}\]
\[\begin{array} {ll} {} &{-100 \div (-4)} \\ {\text{Divide, with signs that are the same the}} &{25} \\ {\text{ quotient is negative.}} &{} \end{array}\]

Zoezi$\PageIndex{8}$

Gawanya:

$-42\div 6$
$-117\div (-3)$

Jibu

$-7$
$39$

Zoezi$\PageIndex{9}$

Gawanya:

$-63\div 7$
$-115\div (-5)$

Jibu

$-9$
$23$

Kurahisisha Maneno na Integers

Nini kinatokea wakati kuna idadi zaidi ya mbili katika kujieleza? Utaratibu wa shughuli bado unatumika wakati hasi zinajumuishwa. Kumbuka Mpendwa wangu Shangazi Sally?

Hebu jaribu mifano fulani. Tutaweza kurahisisha maneno ambayo hutumia shughuli zote nne na integers-Aidha, kutoa, kuzidisha, na mgawanyiko. Kumbuka kufuata utaratibu wa shughuli.

Zoezi$\PageIndex{10}$

Kurahisisha:

$7(-2)+4(-7)-6$

Jibu: \[\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\ {\text{Multiply first.}} &{-14+(-28)-6} \\ {\text{Add.}} &{-42-6} \\{\text{Subtract}} &{-48} \end{array}\]

Zoezi$\PageIndex{11}$

Kurahisisha:

$8(-3)+5(-7)-4$

Jibu: $-63$

Zoezi$\PageIndex{12}$

Kurahisisha:

$9(-3)+7(-8)-1$

Jibu: $-84$

Zoezi$\PageIndex{13}$

Kurahisisha:

$(-2)^{4}$
$-2^{4}$

Jibu

\[\begin{array} {ll} {} &{(-2)^{4}} \\ {\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\ {\text{Multiply}} &{4(-2)(-2)} \\{\text{Multiply}} &{-8(-2)} \\{\text{Multiply}} &{16} \end{array}\]
\[\begin{array} {ll} {} &{-2^{4}} \\ {\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\cdot 2\cdot 2 \cdot 2)} \\ {\text{Multiply}} &{-(4\cdot 2\cdot 2)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Multiply}} &{-16} \end{array}\]

Angalia tofauti katika sehemu (1) na (2). Katika sehemu (1), exponent ina maana ya kuongeza kile ni katika mabano,$(−2)$ kwa$4^{th}$ nguvu. Katika sehemu (2), exponent ina maana ya kuongeza tu$2$ kwa$4^{th}$ nguvu na kisha kuchukua kinyume.

Zoezi$\PageIndex{14}$

Kurahisisha:

$(-3)^{4}$
$-3^{4}$

Jibu

$81$
$-81$

Zoezi$\PageIndex{15}$

Kurahisisha:

$(-7)^{2}$
$-7^{2}$

Jibu

$49$
$-49$

Mfano unaofuata unatukumbusha kurahisisha ndani ya mabano kwanza.

Zoezi$\PageIndex{16}$

Kurahisisha:

$12-3(9 - 12)$

Jibu: \[\begin{array} {llll} {} &{12-3(9 - 12)} \\ {\text{Subtract parentheses first}} &{12-3(-3)} \\ {\text{Multiply.}} &{12-(-9)} \\{\text{Multiply}} &{-(8\cdot 2)} \\{\text{Subtract}} &{21} \end{array}\]

Zoezi$\PageIndex{17}$

Kurahisisha:

$17 - 4(8 - 11)$

Jibu: $29$

Zoezi$\PageIndex{18}$

Kurahisisha:

$16 - 6(7 - 13)$

Jibu: $52$

Zoezi$\PageIndex{19}$

Kurahisisha:

$8(-9)\div (-2)^{3}$

Jibu: \[\begin{array} {ll} {} &{8(-9)\div(-2)^{3}} \\ {\text{Exponents first}} &{8(-9)\div(-8)} \\ {\text{Multiply.}} &{-72\div (-8)} \\{\text{Divide}} &{9} \end{array}\]

Zoezi$\PageIndex{20}$

Kurahisisha:

$12(-9)\div (-3)^{3}$

Jibu: $4$

Zoezi$\PageIndex{21}$

Kurahisisha:

$18(-4)\div (-2)^{3}$

Jibu: $9$

Zoezi$\PageIndex{22}$

Kurahisisha:

$-30\div 2 + (-3)(-7)$

Jibu: \[\begin{array} {ll} {} &{-30\div 2 + (-3)(-7)} \\ {\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\ {\text{Multiply.}} &{-15+ 21} \\{\text{Add}} &{6} \end{array}\]

Zoezi$\PageIndex{23}$

Kurahisisha:

$-27\div 3 + (-5)(-6)$

Jibu: $21$

Zoezi$\PageIndex{24}$

Kurahisisha:

$-32\div 4 + (-2)(-7)$

Jibu: $6$

Tathmini Maneno ya kutofautiana na Integers

Kumbuka kwamba kutathmini maneno ina maana ya kubadilisha idadi kwa variable katika kujieleza. Sasa tunaweza kutumia namba hasi pamoja na idadi nzuri.

Zoezi$\PageIndex{25}$

Wakati$n=−5$, tathmini:

$n+1$
$−n+1$.

Jibu

\[\begin{array} {ll} {} &{n+ 1} \\ {\text{Substitute}-5\text{ for } n} &{-5+1} \\ {\text{Simplify.}} &{-4} \end{array}\]
\[\begin{array} {ll} {} &{-n+ 1} \\ {\text{Substitute}-5\text{ for } n} &{-(-5)+1} \\ {\text{Simplify.}} &{-4} \\{\text{Add.}} &{6} \end{array}\]

Zoezi$\PageIndex{26}$

Wakati$n=−8$, tathmini:

$n+2$
$−n+2$.

Jibu

$-6$
$10$

Zoezi$\PageIndex{27}$

Wakati$y=−9$, tathmini:

$y+8$
$−y+8$.

Jibu

$-1$
$17$

Zoezi$\PageIndex{28}$

Tathmini$(x+y)^{2}$ wakati$x = -18$ na$y = 24$.

Jibu: \[\begin{array} {ll} {} &{(x+y)^{2}} \\ {\text{Substitute }-18\text{ for }x \text{ and } 24 \text{ for } y} &{(-18 + 24)^{2}} \\ {\text{Add inside parentheses}} &{(6)^{2}} \\{\text{Simplify.}} &{36} \end{array}\]

Zoezi$\PageIndex{29}$

Tathmini$(x+y)^{2}$ wakati$x = -15$ na$y = 29$.

Jibu: $196$

Zoezi$\PageIndex{30}$

Tathmini$(x+y)^{3}$ wakati$x = -8$ na$y = 10$.

Jibu: $8$

Zoezi$\PageIndex{31}$

Tathmini$20 -z $ wakati

$z = 12$
$z = -12$

Jibu

\[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }12\text{ for }z.} &{20 - 12} \\ {\text{Subtract}} &{8} \end{array}\]
\[\begin{array} {ll} {} &{20 - z} \\ {\text{Substitute }-12\text{ for }z.} &{20 - (-12)} \\ {\text{Subtract}} &{32} \end{array}\]

Zoezi$\PageIndex{32}$

Tathmini$17 - k$ wakati

$k = 19$
$k = -19$

Jibu

$-2$
$36$

Zoezi$\PageIndex{33}$

Tathmini$-5 - b$ wakati

$b = 14$
$b = -14$

Jibu

$-19$
$9$

Zoezi$\PageIndex{34}$

Tathmini:

$2x^{2} + 3x + 8$lini$x = 4$.

Jibu

Mbadala$4$ kwa ajili ya$x$. Tumia mabano ili kuonyesha kuzidisha.

\[\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\ {\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\ {\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\ {\text{Multiply.}} &{32 + 12 + 8} \\{\text{Add.}} &{52} \end{array}\]

Zoezi$\PageIndex{35}$

Tathmini:

$3x^{2} - 2x + 6$lini$x =-3$.

Jibu: $39$

Zoezi$\PageIndex{36}$

Tathmini:

$4x^{2} - x - 5$lini$x = -2$.

Jibu: $13$

Tafsiri Maneno kwa Maneno na Integers

Kazi yetu ya awali kutafsiri Kiingereza hadi algebra pia inatumika kwa misemo ambayo ni pamoja na idadi nzuri na hasi.

Zoezi$\PageIndex{37}$

Tafsiri na kurahisisha: jumla ya$8$ na$−12$, iliongezeka kwa$3$.

Jibu: \[\begin{array} {ll} {} &{\text{the } \textbf{sum} \text{of 8 and -12, increased by 3}} \\ {\text{Translate.}} &{[8 + (-12)] + 3} \\ {\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\{\text{brackets with an absolute value sign.}} \\{\text{Add.}} &{-1} \end{array}\]

Zoezi$\PageIndex{38}$

Tafsiri na kurahisisha: jumla ya$9$ na$−16$, iliongezeka kwa$4$.

Jibu: $(9 + (-16)) + 4 - 3$

Zoezi$\PageIndex{39}$

Tafsiri na kurahisisha: jumla ya$-8$ na$−12$, iliongezeka kwa$7$.

Jibu: $(-8 + (-12)) + 7 - 13$

Tulipoanzisha kwanza alama za uendeshaji, tuliona kwamba maneno yanaweza kusomwa kwa njia kadhaa. Wao ni waliotajwa katika chati hapa chini.

Jedwali$\PageIndex{5}$
$a−b$
\ (a-b\)” data-valign="top">$a$ minus$b$ tofauti ya$a$ na$b$ $b$ kuondolewa kutoka$a$ $b$ chini ya$a$

Kuwa makini kupata na b katika utaratibu sahihi!

Zoezi$\PageIndex{40}$

Tafsiri na kisha kurahisisha

tofauti ya$13$ na$−21$
Ondoa$24$ kutoka$−19$.

Jibu

\[\begin{array} {ll} {} &{\text{the } \textbf{difference } \text{of 13 and -21}} \\ {\text{Translate.}} &{13 - (-21)} \\ {\text{Simplify.}} &{34} \end{array}\]
\[\begin{array} {ll} {} &\textbf{subtract }24 \textbf{ from }-19 \\ {\text{Translate.}} &{-19 - 24} \\ {\text{Remember, subtract b from a means }a - b} &{} \\{\text{Simplify.}} &{-43} \end{array}\]

Zoezi$\PageIndex{41}$

Tafsiri na kurahisisha

tofauti ya$14$ na$−23$
Ondoa$21$ kutoka$−17$.

Jibu

$14 - (-23); 37$
$-17 - 21; -38$

Zoezi$\PageIndex{42}$

Tafsiri na kurahisisha

tofauti ya$11$ na$−19$
Ondoa$18$ kutoka$−11$.

Jibu

$11 - (-19); 30$
$-11 - 18; -29$

Mara nyingine tena, kazi yetu ya awali ya kutafsiri Kiingereza hadi algebra huhamisha maneno ambayo yanajumuisha kuzidisha na kugawa integers. Kumbuka kwamba neno muhimu la kuzidisha ni “bidhaa” na kwa mgawanyiko ni “quotient.”

Zoezi$\PageIndex{43}$

Tafsiri kwa kujieleza kwa algebraic na kurahisisha ikiwa inawezekana: bidhaa ya$−2$ na$14$.

Jibu: \[\begin{array} {ll} {} &{\text{the product of }-2 \text{ and } 14} \\ {\text{Translate.}} &{(-2)(14)} \\{\text{Simplify.}} &{-28} \end{array}\]

Zoezi$\PageIndex{44}$

Tafsiri kwa kujieleza kwa algebraic na kurahisisha ikiwa inawezekana: bidhaa ya$−5$ na$12$.

Jibu: $-5(12); -60$

Zoezi$\PageIndex{45}$

Tafsiri kwa kujieleza kwa algebraic na kurahisisha ikiwa inawezekana: bidhaa ya$8$ na$-13$.

Jibu: $-8(13); -104$

Zoezi$\PageIndex{46}$

Tafsiri kwa kujieleza algebraic na kurahisisha kama inawezekana: quotient ya$−56$ na$−7$.

Jibu: \[\begin{array} {ll} {} &{\text{the quotient of }-56 \text{ and } -7} \\ {\text{Translate.}} &{-56\div(-7)} \\{\text{Simplify.}} &{8} \end{array}\]

Zoezi$\PageIndex{47}$

Tafsiri kwa kujieleza algebraic na kurahisisha kama inawezekana: quotient ya$−63$ na$−9$.

Jibu: $-63\div (-9); 7$

Zoezi$\PageIndex{48}$

Tafsiri kwa kujieleza algebraic na kurahisisha kama inawezekana: quotient ya$−72$ na$−9$.

Jibu: $-72\div (-9); 8$

Tumia Integers katika Maombi

Tutaelezea mpango wa kutatua programu. Ni vigumu kupata kitu kama hatujui nini sisi ni kuangalia kwa au nini kuiita! Kwa hiyo tunapotatua programu, sisi kwanza tunahitaji kuamua ni shida gani inatuuliza kupata. Kisha tutaandika maneno ambayo inatoa taarifa ili kuipata. Tutaweza kutafsiri maneno katika kujieleza na kisha kurahisisha kujieleza ili kupata jibu. Hatimaye, sisi muhtasari jibu katika sentensi ili kuhakikisha ni mantiki.

Jinsi ya kutumia Mkakati wa Kutatua Maombi na Integers

Zoezi$\PageIndex{49}$

Joto la Urbana, Illinois asubuhi moja lilikuwa$11$ digrii. Katikati ya mchana, joto lilikuwa limeshuka hadi$−9$ digrii. Ilikuwa tofauti gani ya joto la asubuhi na alasiri?

Jibu

Hatua ya 1. Soma tatizo. Hakikisha maneno yote na mawazo yanaeleweka.
Hatua ya 2. Tambua kile tunachoulizwa kupata.	tofauti ya joto la asubuhi na alasiri
Hatua ya 3. Andika maneno ambayo inatoa taarifa ili kuipata.	tofauti ya$11$ na$-9$
Hatua ya 4. Tafsiri maneno kwa kujieleza.	$11 - (-9)$
Hatua ya 5. Kurahisisha maneno.	$20$
Hatua ya 6. Andika sentensi kamili inayojibu swali.	Tofauti katika joto ilikuwa digrii 20.

Zoezi$\PageIndex{50}$

Joto la Anchorage, Alaska asubuhi moja lilikuwa$15$ digrii. Katikati ya mchana joto lilikuwa limeshuka hadi$30$ digrii chini ya sifuri. Ilikuwa tofauti gani katika joto la asubuhi na alasiri?

Jibu: Tofauti katika joto ilikuwa$45$ digrii.

Zoezi$\PageIndex{51}$

Joto la Denver lilikuwa$−6$ digrii wakati wa chakula cha mchana. By sunset joto alikuwa imeshuka kwa$−15$ digrii. Ilikuwa tofauti gani katika joto la chakula cha mchana na jua?

Jibu: Tofauti katika joto ilikuwa$9$ digrii.

TUMIA MKAKATI WA KUTATUA MAOMBI NA INTEGERS.

Soma tatizo. Hakikisha maneno yote na mawazo yanaeleweka
Tambua kile tunachoulizwa kupata.
Andika maneno ambayo inatoa taarifa ili kuipata.
Tafsiri maneno kwa kujieleza.
Kurahisisha maneno.
Jibu swali kwa sentensi kamili.

Zoezi$\PageIndex{52}$

Timu ya soka ya Mustangs ilipata adhabu tatu katika robo ya tatu. Kila adhabu iliwapa hasara ya yadi kumi na tano. Idadi ya yadi imepotea nini?

Jibu

Hatua ya 1. Soma tatizo. Hakikisha maneno yote na mawazo yanaeleweka.
Hatua ya 2. Tambua kile tunachoulizwa kupata.	idadi ya yadi waliopotea
Hatua ya 3. Andika maneno ambayo inatoa taarifa ili kuipata.	mara tatu adhabu ya$15$ yadi
Hatua ya 4. Tafsiri maneno kwa kujieleza.	$3(-15)$
Hatua ya 5. Kurahisisha maneno.	$-45$
Hatua ya 6. Andika sentensi kamili inayojibu swali.	Timu ilipoteza$45$ yadi.

Zoezi$\PageIndex{53}$

Bears alicheza vibaya na alikuwa na adhabu saba katika mchezo. Kila adhabu ilisababisha hasara ya$15$ yadi. Idadi ya yadi iliyopotea kutokana na adhabu ni nini?

Jibu: Bears waliopotea$105$ yadi.

Zoezi$\PageIndex{54}$

Bill inatumia ATM juu ya chuo kwa sababu ni rahisi. Hata hivyo, kila wakati anatumia anashtakiwa ada ya $2. Mwezi uliopita alitumia ATM mara nane. Kiasi gani ilikuwa jumla ya ada yake kwa ajili ya kutumia ATM?

Jibu: Ada ya $16 ilitolewa kutoka akaunti yake ya kuangalia.

Dhana muhimu

Kuzidisha na Idara ya Nambari mbili zilizosainiwa
- Same Ishara-bidhaa ni chanya
- Ishara tofauti-bidhaa ni hasi
Mkakati wa Maombi
1. Tambua kile unachoulizwa kupata.
2. Andika maneno ambayo inatoa taarifa ili kuipata.
3. Tafsiri maneno kwa kujieleza.
4. Kurahisisha maneno.
5. Jibu swali kwa sentensi kamili.

Ishara sawa	Bidhaa	Mfano
Chanya mbili	Chanya	\(7\cdot 4 = 28\)
Mbili hasi	Chanya	\(-8(-6) = 48\)

Ishara tofauti	Bidhaa	Mfano
Chanya\(\cdot\) hasi	Hasi	\(7(-9) = -63\)
\(\cdot\)Chanya hasi	Hasi	\(-5\cdot 10= -50\)

Malengo ya kujifunza

Kuzidisha integers

KUZIDISHA NAMBA ZILIZOSAINIWA

Zoezi\(\PageIndex{1}\)

Zoezi\(\PageIndex{2}\)

Zoezi\(\PageIndex{3}\)

Zoezi\(\PageIndex{4}\)

Zoezi\(\PageIndex{5}\)

Zoezi\(\PageIndex{6}\)

Kugawanya Integers

KUZIDISHA NA MGAWANYIKO WA IDADI ZILIZOSA

Zoezi\(\PageIndex{7}\)

Zoezi\(\PageIndex{8}\)

Zoezi\(\PageIndex{9}\)

Kurahisisha Maneno na Integers

Zoezi\(\PageIndex{10}\)

Zoezi\(\PageIndex{11}\)

Zoezi\(\PageIndex{12}\)

Zoezi\(\PageIndex{13}\)

Zoezi\(\PageIndex{14}\)

Zoezi\(\PageIndex{15}\)

Zoezi\(\PageIndex{16}\)

Zoezi\(\PageIndex{17}\)

Zoezi\(\PageIndex{18}\)

Zoezi\(\PageIndex{19}\)

Zoezi\(\PageIndex{20}\)

Zoezi\(\PageIndex{21}\)

Zoezi\(\PageIndex{22}\)

Zoezi\(\PageIndex{23}\)

Zoezi\(\PageIndex{24}\)

Tathmini Maneno ya kutofautiana na Integers

Zoezi\(\PageIndex{25}\)

Zoezi\(\PageIndex{26}\)

Zoezi\(\PageIndex{27}\)

Zoezi\(\PageIndex{28}\)

Zoezi\(\PageIndex{29}\)

Zoezi\(\PageIndex{30}\)

Zoezi\(\PageIndex{31}\)

Zoezi\(\PageIndex{32}\)

Zoezi\(\PageIndex{33}\)

Zoezi\(\PageIndex{34}\)

Zoezi\(\PageIndex{35}\)

Zoezi\(\PageIndex{36}\)

Tafsiri Maneno kwa Maneno na Integers

Zoezi\(\PageIndex{37}\)

Zoezi\(\PageIndex{38}\)

Zoezi\(\PageIndex{39}\)

Zoezi\(\PageIndex{40}\)

Zoezi\(\PageIndex{41}\)

Zoezi\(\PageIndex{42}\)

Zoezi\(\PageIndex{43}\)

Zoezi\(\PageIndex{44}\)

Zoezi\(\PageIndex{45}\)

Zoezi\(\PageIndex{46}\)

Zoezi\(\PageIndex{47}\)

Zoezi\(\PageIndex{48}\)

Tumia Integers katika Maombi

Zoezi\(\PageIndex{49}\)

Zoezi\(\PageIndex{50}\)

Zoezi\(\PageIndex{51}\)

TUMIA MKAKATI WA KUTATUA MAOMBI NA INTEGERS.

Zoezi\(\PageIndex{52}\)

Zoezi\(\PageIndex{53}\)

Zoezi\(\PageIndex{54}\)

Dhana muhimu