Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Transcription

1 Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction - Left to Right.. Follow the correct order of opertions to evlute epressions. Evlute: Rememer to use the correct Order of Opertions Evlute for =, =, c=, d=0. c d. c d. d c Solve the following using the correct order of opertions: ( 8. (. 9.. ( ( 0.

2 Nme Period Order of Opertions Prctice Alger. When evluting epressions, work using the correct order of opertions: P (Prenthesis Do ll grouped opertions first. E (Eponents Do ll opertions involving eponents. M D (Mult./Div. Do ll multipliction nd division from left to right. A S (Add./Su. Do ll ddition nd sutrction lst - from left to right. Solve:.. ( 9. ( 6 (. [ (9 ]. 6 (9 6. ( ( (

3 Nme Period Order of Opertions Prctice Alger. Evlute for =, =, c=6. c (.. ( c. [ c ( ] (. c ( 6. 6( c 7. c 8. c 9. ( c c 0. c

4 Integer Addition Alger. notes: Integers re positive nd negtive Whole Numers like Integers re NOT decimls or frctions. Adding nd sutrcting integers cn seem unnecessril complicted. Tr the following prctice prolems first: Prctice:.... (. 6. ( ( If ou got ll of these right, ou lred hve proven method for dding nd sutrcting integers. Close our ers, sit quietl, nd continue using our own method. If ou missed even one, p close ttention nd tke notes. notes: Adding Integers: Sme Sign Sum When dding integers with the sme sign, find the sum nd keep the sign of oth numers.. (.. (. Different Sign Difference When dding integers with different signs, find the difference nd keep the sign of the igger numer... (. (. Mied Review Add:. (. ( 8. 7 ( 8. ( 6. ( ( 8. (

5 Sutrction Alger. Sutrcting Integers: SMATO Sutrction Mens Add The Opposite Sutrcting Integers is more complicted thn dding integers. To sutrct integers, chnge sutrction to ddition nd switch the sign of the second numer. Then, follow the two rules we hve lerned for dding integers. Emples: SMATO Chnge to ddition (. 0 ( Prctice: Chnge to ddition, then solve.. 9 ( ( ( 7 Adding nd Sutrcting Rtionls: Use the sme rules for frctions nd decimls s ou would for integers: Sme Sign Sum, Different Sign Difference, SMATO. Emples: (.7. Prctice: ( (.0.

6 Alger Mtrices: A mtri is rectngulr tle of numers. Horizontl lines re clled rows. Verticl lines re clled columns. 9 A 7 0 B 7 0 C Mtri A nd B re mtrices. C is mtri. Q: In mtri C, which numer is in the second row, third column? Mtri ddition/sutrction. To dd A+B, simpl dd the corresponding prts. You cn onl dd or sutrct mtrices with the sme dimensions. Sutrction is esier if ou simpl dd the opposite. 9 A 7 0 B 8 0 B A 6 7 B A Prctice: Solve the following using the given mtrices: A 0 9 B 7 0 C. A+B. A-C. B-C. A+C Mtrices.+

7 Alger Mtri multipliction: Multipling mtri sclr ( numer shown outside the mtri involves multipling ech term the sclr. 9 A A Complete the following opertions involving mtrices: A 6 B 0 C. A. B. B A. C B Solving Mtri Equtions: E: 9 7 One-step. Solve for B: Two-steps. Solve for A: 0 B A Prctice: Solve for the missing mtri in ech prolem elow:. 7 9 A A Mtrices.+

8 Alger Complete the following prolems using the given mtrices: Write impossile if prolem cnnot e solved. 9 A 7 0 B 7 0 C. B A. A. C C. A C. B A 6. Solve for mtri X: 6 7 X Mtri Opertions Nme Period.+

9 Comining Like Terms Alger. Comining Like Terms: When dding or sutrcting numers nd vriles, ou cn onl comine like terms. L i k e t e r m s contin the sme vriles, with the sme eponents in single product. Here re some sets of like terms: 9 6m m m n n n c c c Prctice: Mtch ech pir or set of like terms elow: 7 9 You cnnot dd or sutrct unlike terms. It is like tring to dd pples nd ornges. Prctice: Simplif the following: z z. 6. m m m m. 6. 9c 7 You cnnot simplif numer si ecuse there re NO LIKE TERMS.

10 Comining Like Terms Alger. Prctice: Simplif c c c Prctice: Simplif c 7c Prctice: Simplif

11 Like Terms Retech: Nme Period Alger Comining like terms is just like dding nd sutrcting integers: Simplif You cn onl comine terms with the sme vriles nd eponents: Simplif. Write SIMPLIFIED if there re no terms which cn e comined. Circle like terms s ou comine them m 7 n n m. 6. w w w w w

12 Like Terms Retech: Frction review: Solve. Nme Period Alger Simplif ech comining like terms m mn mn m n 7mn m n mn m

13 Nme Period Like Terms Prctice Alger. Simplif ech. Write simplified if no terms cn e comined... m 7n n m... w w w w z z m m m m 7ce m ce m

14 Eponents Alger 8. Bse: The repeted fctor in power. In the epression n³, n is the se. Eponent: Represents the numer of times fctor is eing multiplied. In the epression n³, the ³ is the eponent. The epression The epression The epression mens tht ou multipl mens tht ou multipl ( mens tht ou multipl ( ( ( ( Prctice: Write-out without using eponents:... (. ( Prctice: Write using eponents... rssr Prctice: Evlute. (solve. 7...

15 Eponents Alger 8. One of the esiest ws to multipl epressions using eponents is to write them out in fctored form, nd then recomine terms using eponents: E. n(n nnn nnn 6n Prctice: Simplif.. (. (. 7 ( Rules: When multipling vriles with eponents, simpl dd eponents: E. n n n n 8 or 7 ( Prctice: Simplif. 0. (7. ( ( m n m n The sme rules ppl for positive nd negtive eponents. Prctice: Simplif... ( Chllenge: Find the Perimeter AND Are of ech shded figure elow: note: ll ngles re right ngles. 7

16 Eponents nd Division Alger 8. Review: Multipl.. (. (. ( Dividing Monomils: You cn write-out vriles nd eponents, or simpl sutrct eponents: Emples: Prctice: Divide/ Simplif. Answers should hve positive eponents Wht is negtive eponent? Look t the following pttern in our own numer sstem: 876. The 8 is in the plce 0. The 7 is in the plce 0. The 6 is in the plce 0 0. The is in the plce 0. The is in the plce 0. The is in the plce 0.

17 Eponents nd Division Alger 8. Negtive Eponents: A negtive eponent cn e epressed s positive eponent in the denomintor: Emples: Notice tht negtive eponent in the denomintor cn lso e epressed s positive in the numertor. Prctice: Rewrite with positive eponents: ( 6. Negtive Eponents: The esiest w to simplif epressions with negtive eponents is to egin rewriting them: Emples: Prctice: Rewrite with positive eponents, then simplif: You cn lso use the sutrction method, ut it ecomes much more confusing.

18 Nme Period Division With Eponents Alger. Simplif ech. Your nswers should e written with positive eponents ( 7. 0( ( (6.

19 Division With Eponents Nme Period Alger More chllenging prolems: Simplif ech. Your nswers should e written with positive eponents ( (

20 Eponents Alger 8. Rising power to power: Prctice: Simplif ech using wht ou know out eponents.. (. (. 8 ( Emples: Rising power to power.. (. ( 6. ( Prctice: Rising power to power.. (. ( 6. (.. ( 6. (

21 Alger Eponents nd Division 8. Rising frction to power: When rising frction to power, ppl the eponent to the numertor nd the denomintor: Emples: Prctice: Tr these es ones: Write-out if necessr.... Prctice: Tr these more difficult prolems. Like most of the mth we hve done, there re mn ws to get the right nswer. Answers should hve positive eponents You cn simplif wht is in prenthesis efore or fter ppling the eponent.

22 Nme Period Eponents Retech: Multipling Alger Write ech epression out without using eponents (write smll!:.. (. ( Rewrite ech epression using eponents:.. 6. Simplif ech using the rules for eponents. 7. ( 8. ( 9. 7 ( 0. ( 8. (. 0 (. ( 6. (. ( m n ( n (7 8. [( ]

23 Nme Period Eponents Retech: Dividing Alger Write ech epression out without using eponents (write smll!: e ( 0.. ( 0 Simplif ech, then rewrite ech epression using eponents:. 0.. Rewrite ech with positive eponents: DO NOT SIMPLIFY, just rewrite using ll positive eponents: Rewrite ech then simplif: Tke our time nd complete severl steps ( ( mn m n

24 Quick Review Alger.7 Cut-out the following nd give ech tle set (8 sets. Mtch the letters to the proper numers to find the clue. ns: red it ckwrds (red ckwrds. ( s.. ( d. (. r.. ( w k ( c. 6

25 8. 9 (. Alger 8 9 c 9 c t. 6. i d. 8. ( e r. 7

26 Prctice Quiz: Chpter ( Alger Solve for =, =, c= Nme Period. c.. c(.. ( c (. Simplif:

27 Prctice Quiz: Chpter ( Alger Simplif: All nswers should e written with positive eponents. Nme Period (.. ( (

28 Prctice Quiz: Chpter (/7 Alger Solve for =-, =, c= Nme Period. c.. c(.. ( c ( c. Simplif:

29 Prctice Quiz: Chpter (/7 Alger Simplif: All nswers should e written with positive eponents. Nme Period (.. 8 (. (

30 The Distriutive Propert Alger.7 The Distriutive Propert sttes: For n numers,, nd c: ( c c Emples: distriute the distriute the ( ( 6 ( r s r s ( Multipl the term outside the prenthesis oth terms inside. Prctice: Rewrite using the Distriutive Propert.. 9(. (. 7(. (. ( 6. 8 ( 7 Distriuting the negtive: E: 8 ( ( distriute the - distriute the - distriute the negtive (-. 7 ( Prctice: Rewrite using the Distriutive Propert.. (. (. 8 (. 6 (. ( 6. ( ( 7

31 The Distriutive Propert Alger.7 Prctice: Rewrite the Following Using the Distriutive Propert:. c (c. (. (. ( 8 7. ( ( Prctice: Fill-in the lnks. The GCF hs een fctored out for ou.. 8 (. ( (. 6 Prctice: Fctor the Following (Reverse the Distriutive Propert

32 Distriuting Division Alger.7+ You cn use the Distriutive Propert with division. Emple:. 6 Divide 6 6 nd Prctice: 6 Therefore Prctice: Answers will include frctions Chllenge Set: Some nswers will include frctions

33 Nme Period Distriutive Propert Alger.7+ Rewrite nd Simplif using the Distriutive Propert:. 7 (. (. 8(. (6. ( 7 6. ( 7. 6 ( 8. ( 9. 6 ( 6 0. (9. ( ( 7. c ( c

34 Nme Period Distriutive Propert Alger.7+ Simplif: Distriuting division ( unn ers : c Fctor (Rewrite using the Distriutive Propert in reverse:. 6 8.

35 Nme Period Distriutive Retech: Multipling Alger Rewrite ech prolem elow using the Distriutive Propert. Multipl ech term outside of the prenthesis ll terms inside the prenthesis. Creful with our signs nd rememer our rules for multipling with eponents.. (. (. (. (. ( 6. ( 7. ( 8. ( 9. ( Now tr distriuting some negtives. Rememer our integer rules. 0. (. (. ( For # ove: DISTRIBUTE THE - NOT THE. Answer: -+6. Tr the following similr prolems nd comine like terms wherever possile to simplif our nswer.. (. c c( 7. ( 6. ( 7 7. (

36 Nme Period Distriutive Retech: Dividing Alger Rewrite ech prolem elow using the Distriutive Propert. Bunn-Er ech term in the numertor with the term in the denomintor. Creful with our signs! The finl three nswers involve frctions. The frctions should e simplified nd left s coefficients

37 Nme Period Review Alger.7+ Rewrite nd Simplif using the Distriutive Propert:. (. (. (. ( 6. (. ( 7. ( 8. (7 9. ( 0. c c ( c c

38 Nme Period Distriutive Propert Alger.7+ Simplif: Distriuting division ( unn ers : mn mn 8m 6c 6c 8c 7. 8c 8. Fctor Ech: Reverse Distriution:

39 Quiz Review Distriutive Propert: Alger ( 00. ( ( 00. ( 00. ( ( Distriuting Division: Fctoring: m n mn mn

40 Quiz Review Order of Opertions: 00. ( ( (6 00. ( ( Alger ( 00. ( 6 ( ( ( Comining Like Terms ( ( 600. ( ( 00. Eponents: ( ( ( (

41 Prctice Quiz Alger.7+ Solve for =-, =, c= Nme Period. c ( c c. ( c ( c.. c Simplif: (.. 0 (.

42 Prctice Quiz Simplif: Nme Period Alger Rewrite Using the Distriutive Propert nd Simplif where possile: 7. ( ( ( Pledge: write-out nd sign.

43 Fctoring Alger.7+ Reversing the Distriutive Propert is clled Fctoring. Emple: Rewrite the Following Using the Distriutive Propert:. (. (. ( Answers should e:... 6 To fctor n epression:. Look for the GCF of ll terms, including the vriles.. Plce the GCF outside of the prenthesis. c. Divide ech originl term the GCF to get the terms inside the prenthesis. Emples: Prctice: Fill-in the lnks.. 9 (. 7 (. (. 6 8 ( Prctice: Fctor the following Prctice: Fctor the following c. 8 7

44 Nme Period Fctoring Alger.7+ Rewrite ech fctoring (using the Distriutive Propert in reverse: c c

45 Writing Epressions/ Equtions Alger. Words to indicte: Addition Sutrction Multipliction Division Rewrite s n epression:. The sum of seven nd.. The quotient of nd.. Five times the sum of c squred nd nine.. Nine less thn w.. Twice incresed nineteen. 6. Two times the sum of nd nineteen. 7. Hlf the product of nd decresed the quntit plus. In ech cse elow, replce numer with the vrile n :. The sum of seven nd numer.. The quotient of numer nd three.. Five times the sum of numer squred nd nine.. Nine less thn numer.. Three times numer incresed ten. 6. Two times the sum of numer nd four. 7. The product of numer nd incresed the numer squred. In the following prolems, tr to use vriles tht represent wht is eing given in the prolem (For emple, could e used to represent the numer of pples. c could represent the cost. etc.. The numer of pples incresed si.. Hlf the cost.. Twice the numer of cs incresed three times the numer of uses.. Nine less thn the numer of ds.. Three times s mn computers incresed ten. 6. One third of the totl numer of os nd girls. 7. The cost incresed 0%.

46 Writing Epressions/ Equtions Alger. In word prolems, the word IS usull mens equls. Terms with no equls sign re clled epressions. If there is n equls sign, it is clled n eqution. The following should e written s equtions using vriles.. The sum of seven nd numer is 6.. Twent is three times numer incresed ten.. One fifth of the totl numer of os nd girls is nine more thn the numer of girls.. Four less thn the numer of pinepples is twice the numer of pers.. Tom is three ers ounger thn his sister Ktie. The word WHAT usull mens USE A VARIABLE, often we use. E. Wht is the sum of nd numer: = + n. Wht is the totl numer of crs nd trucks?. Wht is 0% of the totl cost?. Wht numer is three times the sum of itself nd seven? Defining vrile: To solve mn word prolems, ou must use vrile to represent n unknown quntit (or quntities. Red the following emple: Mrgret hs sket of pples nd pers. The numer of pples equls three more thn twice the numer of pers. If there re pieces of fruit ltogether, how mn pples nd pers re there? Using p for pers nd for pples, write two equtions tht could help ou solve this prolem. Write three equtions: Am is five inches tller thn Jmes. Jmes is twice s tll s Pmel. Pmel is inches shorter thn Am.

47 Prctice: Writing Epressions Nme Period Alger Write n epression for ech sttement elow. If ou need help, there is list of nswers on the ck of the sheet to choose from. Write the epression/eqution AND the letter tht goes with it. There will not e word/phrse spelled.. Together, Alice nd Bets hve $6.. Nine less thn numer.. The product of nine nd numer is four.. Si times the difference of nd is 6.. Four less thn the product of nine nd numer is the numer itself. 6. Nine less thn the product of four nd numer. 7. Nine dded to the quotient of numer nd four. 8. Nine decresed numer. 9. Four less thn the product of nine nd numer is nine more thn the product of four nd the sme numer. 0. Brend hs thirt-si less thn Am.. Four less thn the product of nine nd numer.. Nine more thn numer.. Nine times the sum of numer nd four equls the sme numer.. Four times the sum of numer nd nine.. Three more thn the quotient of two numers. 6. The product of two numers is One-fourth the sum of numer nd nine. 8. Nine is four more thn Nnc s ge. 9. Four more thn the product of numer nd nine. 0. The product of nine nd the sum of numer nd four

48 Prctice: Writing Epressions Alger Answer list for the front of the sheet: Ech nswer elow ppers once, one nswer is unused. (note: There will not e word or phrse spelled with the nswers on the front. n 9. n 9. n 9 r. 9 n d. Nme Period w. ( 9 n. n 9 s. 6 h. 9n i. 9n n 9 u. k. 9( n n m. 6 n. 6 p. 9 n 6 q. 6( 6 n t. 9n o. 9n 9 n e. c. 9(. 9 n f. 9n n Simplif ech epression elow nd find the nswer ove. The letters will crete phrse. 8 mn 8mn mn. 7n 7 n.. n. n. ( ( n 7 6n n 6. n n n 8. n n 9. n n 8. ( n ( n.. ( 9. 6 n nswer:!

49 Quick Review Alger.7 This set of equtions cn e grouped into sets of letters which cn e rerrnged into words tht form question. To group the letters, find epressions or equtions tht re equl. Rerrnge the letters in ech set to form words nd rerrnge the words to form question. Rise our hnd when ou know the nswer to the question. W. ( H. A. T. ( I. 6 S. 8 0 T. ( H. E. Twice numer decresed ten. C. U. B. ( E. Twice the cue of numer decresed.

50 Alger R. 8 7 O. 6 O. 6 6 T. 6 9 ( 6 7 O. ( F. 6 E. 9 I. ( G. H. 0 ( 7 8 T.?. The sum of numer squred nd two.

51 Test Review Like Terms: Alger ( ( Distriution: ( ( ( Fctoring:

52 Test Review Eponents: Alger ( ( ( [( ] Equtions nd Epressions: Write. 00. Twice the sum of numer nd seven. 00. The quotient of four nd numer is incresed. 00. Mr is five ers older thn seven times her dog Peches ge. 00. The cost of c ride if ou p $.0 per mile, nd tip the driver $ is $ A rectngle is twice s wide s it is long. The re of the sme rectngle is inches squred. Write two equtions to descrie this rectngle In right tringle, the shortest side is five inches less thn hlf s long s the longest side. The middle side is four inches longer thn twice the shortest side. The longest side is three times the shortest side. Drw the tringle nd lel ll three sides, using for the shortest side.

53 Alger. Simplif. Simplif. Distriute. Simplif Distriute ( ( ( 6. Simplif 0 7. Greg, Hnk, Iris, Josh, nd Kell ech collect silver dollrs. Kell hs three more thn twice s mn s Josh, who hs three more thn twice s mn s Iris, who hs three more thn twice s mn s Hnk, who hs three more thn twice s mn s Greg, who hs three. How mn does Kell hve? 8. Comine like terms: Fctor the nswer ove. Nme Period Trick Review Prolems

54 Trick Review Prolems Nme Period Alger 0. Fctor. Which is greter, or?. Solve for : Find the numertor: z z z Chllenge: When ou solve the following, how mn zeroes re in the nswer? 9 0

55 Prctice Test (,7 Simplif:. 7 Nme Period Alger (9.. (7. 6. ( (

56 Prctice Test (,7 Write n epression or eqution for ech: Nme Period Alger.9 0. Five times the difference of numer nd eleven. 0.. Cludi is eleven ers older thn her rother Jmes... The quotient of nd five decresed twelve is fifteen.. Rewrite ech using the Distriutive Propert:. ( (.. ( Fctor Completel (reverse distriution:

57 Prctice Test (th Alger.7+ Solve for =, =-, c= Nme Period. c ( c c. ( c ( c.... Simplif: c c c 9c c ( 9c ( (

58 Prctice Test (th Alger.7+ Rewrite Using the Distriutive Propert nd Simplif where possile: Nme Period. (.. (.. ( c Write ech sentence s n lgeric epression or eqution. DO NOT TRY TO SOLVE OR SIMPLIFY. 8. Meredith is three ers older thn her cousin Nin Three less thn twice the squre of numer Si more thn the numer of crs. 0.. Four times the sum of numer nd two is eight less thn the sme numer... The quotient of nd is three less thn the product of nd..

59 Prctice Test (th Solve for =, =-, c= Nme Period Alger... ( c. ( c c Simplif: c ( c ( (

60 Prctice Test (th Alger Rewrite Using the Distriutive Propert nd Simplif where possile: Nme Period. (.. (.. ( Write ech sentence s n lgeric epression or eqution. DO NOT TRY TO SOLVE OR SIMPLIFY. 8. Tim hs 9 dollrs more thn Rchel The sum of numer squred nd ten is twent-si Five less thn is divided three. 0.. Four more thn the quotient of numer nd two... The difference of nd is nine more thn cued..