Opening problem. 94 FRACTIONS (Chapter 5) The students in Amelia s class have all been given a week to do a project. So far, Amelia has done 3 8,

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1 Chpter Frtions Contents: A Frtions B Frtions s division C Proper nd improper rtions D Frtions o quntities E Frtions on number line F Equl rtions G Compring rtions H Adding nd subtrting rtions

2 FRACTIONS (Chpter ) Opening problem The students in Ameli s lss hve ll been given week to do projet. So r, Ameli hs done, Chrlie hs done, nd Mtild hs done. Things to think bout: Hs Ameli ompleted more o the projet thn Chrlie? b Hs Ameli ompleted more o the projet thn Mtild? Whih o the problems or b ws esier to solve? Wht mde the other one hrder? d The next night, Ameli ompleted nother ompleted now? o her projet. Wht totl rtion hs she Every dy we see quntities whih n be expressed s rtions. It is thereore importnt tht we n understnd, ompre, dd, nd subtrt rtions. Qt_open Qr remining \\Qw_pple size _Qw _Qw hnds Qe ull

3 FRACTIONS (Chpter ) A FRACTIONS A rtion is prt o ny quntity. For exmple, hoolte br is divided into equl prts. I George tkes o the prts, we sy tht George hs tken o the hoolte br. In words, is two iths. is rtion whih shows tht we hd whole, we divided it into equl prts, nd we re looking t o them. numertor br denomintor {z } tken by George The numertor shows how mny prts we re looking t. The denomintor shows how mny equl prts there re ltogether. Ativity Frtion wlls The rtion wll below shows some o the dierent wys o dividing whole into equl prts. whole Qw Qw hlves Qe Qe Qe thirds Qr Qr Qr Qr qurters Qt Qt Qt Qt Qt iths Qy Qy Qy Qy Qy Qy sixths Qu Qu Qu Qu Qu Qu Qu sevenths Qi Qi Qi Qi Qi Qi Qi Qi eighths Qo Qo Qo Qo Qo Qo Qo Qo Qo ninths Aq_p Aq_p Aq_p Aq_p Aq_p Aq_p Aq_p Aq_p Aq_p Aq_p tenths

4 FRACTIONS (Chpter ) Wht to do: Use the rtion wll to omplete this tble: Number o equl prts One prt s rtion Frtion in words one whole one hl All prts orm the rtion PRINTABLE TABLE b d one qurter one ith e g one eighth h i 0 EXERCISE A Write rtion to show: three qurters b two thirds two iths d our iths e three eighths ive eighths g two sevenths h three tenths i one hundredth Write these rtions in words: b d e i 0 j g k 00 h 0 l 00 For eh o the ollowing rtions, stte the numertor: b For eh o the ollowing rtions, stte the denomintor: b d d

5 FRACTIONS (Chpter ) Wht rtion o the squre is shded? b d e g h i j k l Copy the irles nd shde the rtions given: b d Drw digrm to represent the ollowing rtions: b d Is o this tringle shded? Explin your nswer. Wht rtion o the dots re red? b 0 Wht rtion o the ts re: blk b white?

6 FRACTIONS (Chpter ) Wht rtion o the lowers re: i in the vse ii lying on the tble? b Wht rtion o the lowers re: i tulips ii disies? b Wht rtion o the hildren re: i wering hts ii not wering hts? Wht rtion o the hildren re: i boys ii girls? Give the rtion shown in eh digrm: b BUTTER kg MILK MILK L BUTTER kg? L? Copy nd omplete the ollowing skethes to show: b F? E FUEL d glss whih is hl ull o wter e petrol guge showing the tnk is three qurters ull pizz with one third missing PRINTABLE DIAGRAMS the ontiner is ull ive ninths o the blls re blue.

7 FRACTIONS (Chpter ) B FRACTIONS AS DIVISION When we write rtion suh s br, the br indites division. So, n be written s, nd n be written s. We n see this by dividing pizz into our equl portions. Eh person will get one qurter o the pizz. Bill Jne pizz people = o pizz eh Emm Tony Exmple Write the ollowing divisions s rtions: b = b = EXERCISE B Write the ollowing divisions s rtions: b 0 d e Suppose pizzs re shred eqully between people. Kim Look t wht Kim gets. Wht rtion o pizz is this? b Chek tht the other two people eh get the sme mount s Kim. Copy nd omplete:... pizzs... people =... o pizz eh. Jnie Teegn Exmple Write the ollowing rtions s divisions: b = b =

8 00 FRACTIONS (Chpter ) Write the ollowing rtions s divisions: d Exmple b e 0 Write the ollowing rtions s divisions, nd hene s whole numbers: b = = b = = Write the ollowing rtions s divisions, nd hene s whole numbers: 0 e b g 0 d h 0 C PROPER AND IMPROPER FRACTIONS A rtion whih hs numertor less thn its denomintor is lled proper rtion. A rtion whih hs numertor greter thn its denomintor is lled n improper rtion. For exmple: is proper rtion. is n improper rtion. = + =+ To see how improper rtions our, suppose you nd riend shre three ookies. Eh person reeives three hlves o ookie, whih is ookies. We n lso see tht = + =+. So eh person reeives one nd hl ookies. We n write this s ookies. you riend

9 FRACTIONS (Chpter ) 0 When n improper rtion is written s whole number nd proper rtion, it is lled mixed number. For exmple, is mixed number. wholes nd one hl. It mens two We n write mixed numbers s improper rtions, nd vie vers. For exmple, t lss pini there were pple pies, eh ut into qurters. Sm te one qurter o pie. We see there re now pies remining. 0 Eh whole pie hs qurters, nd we hve qurters o the third pie, so we hve lots o plus qurters = qurters. So, =. Exmple Write s n improper rtion. is wholes nd one third. Eh whole hs thirds, so there re +=thirds in totl. ) = EXERCISE C Determine whether eh o the ollowing is proper rtion, n improper rtion, or mixed number: e b g Wht mixed number do these digrms show? b 0 m m L d h 0 TAST TASTY CHEESE kg L TASTY CHEESE kg TASTY CHEESE kg TASTY CHEESE kg

10 0 FRACTIONS (Chpter ) This digrm shows pizzs. How mny hlves re there in pizzs? b Copy nd omplete: = :::: Wht mixed number is represented by this digrm? b How mny qurters re shded? Copy nd omplete:... = :::: Write s n improper rtion: b d e g h i j 0 Exmple Write s mixed number. is thirds. This is whole, nd thirds let over. So, = Ater the shool pini there were qurter sndwihes let over. How mny whole sndwihes n be ormed rom the qurters? b One the whole sndwihes hve been ormed, how mny qurters re let over? Copy nd omplete: = :::::: Write s mixed number: b d e g h i 0 j

11 FRACTIONS (Chpter ) 0 rrots re shred eqully between horses. How mny rrots does eh horse reeive? Give your nswer s mixed number. D FRACTIONS OF QUANTITIES Chris hs gol blls. He gives one third o them to his brother Josh. How mny gol blls does Josh reeive? To ind out, we n divide the gol blls into equl groups. We see tht o blls is blls. We lso notie tht =. Eh group ontins o the gol blls. So, to ind o number, we divide the number by. Exmple Find o. o = = EXERCISE D Find: o 0 b o o d e o 0 o g o 0 h o o 0 i o 0 j 0 o 0 k o l o 00

12 0 FRACTIONS (Chpter ) Find: o 0 people b o 0 lollies d 0 o 0 g e o $ o drinks o 0 minutes Exmple On the irst dy o shool this yer, there were Yer students in lss. o the students were ged yers or older. How mny students were yers or older? To ind o, we need to divide into equl prts. The ull lss is students. So, o is =students. There were students ged yers or older. Viktor plyed gmes o tennis or his shool tem. He won one third o them. How mny gmes did Viktor win? O the 0 students t shool, one ith were bsent with hiken pox. How mny students were bsent? One sixth o the rs rom n ssembly line were pinted white. I 0 rs me rom the ssembly line, how mny were pinted white? Ling hd $00 in her bnk ount. She spent one ith o her money on new bdminton rquet. How muh did the rquet ost? While Evn ws on holidys, one eighth o the tomto plnts in his greenhouse died. I he hd plnts live when he went wy, how mny plnts: died b were still live? Exmple Find o 0. o 0 is 0 = ) o 0 is =

13 Find: o b o d o e o g 0 o 0 h o i 0 FRACTIONS (Chpter ) 0 o o o 00 pssengers were on the bus one morning. Two iths o the pssengers were shool hildren. How mny shool hildren were on the bus? 0 Rihrd spent three qurters o his working dy instlling omputers, nd the reminder o the time trvelling between jobs. I his working dy ws hours, how muh time did Rihrd spend instlling omputers? Ssh shot or gol times during netbll mth. She sored gol with seven eighths o her shots. How mny gols did Ssh sore? A business hired truk to trnsport boxes o equipment. The totl weight o the equipment ws 000 kg, but the truk ould only rry o the boxes in one lod. Wht weight did the truk rry in the irst lod? b I there were 0 boxes o equl weight, how mny did the truk rry in the irst lod? E FRACTIONS ON A NUMBER LINE In Chpter, we pled nturl numbers on number line. 0 0 We n do the sme thing with rtions. For exmple, to ple the rtion on number line, we divide the intervl between 0 nd into equl prts. Eh o the smll intervls hs length. Er 0 We ount intervls rom 0, nd mrk with dot.

14 0 FRACTIONS (Chpter ) Exmple Ple the rtions nd on number line. Sine these rtions both involve sevenths, we divide the number line into intervls o length. Tu Eu 0 EXERCISE E Ple the ollowing rtions on number line: nd d nd g 0, 0, nd 0 b nd e nd h,, nd Identiy the vlue indited by the red dot: b i nd nd,, nd 0 0 d 0 0 e 0 0 g h Ple the rtion on number line. b On the sme number line, ple the rtion. Whih vlue is lrger, or? Explin your nswer. Ple the rtions nd on the sme number line. b Wht do you notie bout these rtions?

15 FRACTIONS (Chpter ) 0 F EQUAL FRACTIONS Two rtions re equl i they desribe the sme mount. They lie t the sme ple on the number line. For exmple, we n represent the rtions nd by shding digrms. We see the sme mount is shded in eh se, so =. is shded is shded Investigtion DEMO Equl rtions Wht to do: Use grid pper to onstrut three identil squres with sides m long, or lik on the ion to obtin templte. b Divide the irst squre into two equl hlves. Shde one o the hlves, so tht o the squre is shded. TEMPLATE Divide the seond squre into our equl qurters. Shde two o the qurters, so tht o the squre is shded. d e Do you think tht the rtions nd re equl? Chek your nswer by pling them on number line. Divide the third squre into six equl sixths. How mny sixths do you need to shde, to mke rtion equl to nd? Drw two irles with rdius m, or print them using the templte. b From the entre o the irst irle, mesure nd rule lines, 0 ± prt. Sine 0 ± = 0 ±, you hve divided the irle into thirds. Shde two o the setors, whih is o the irle. We From the entre o the seond irle, mesure nd rule lines, 0 ± prt. Sine 0 ± = 0 ±, you hve divided the irle into ninths. Shde six o the setors, whih is o the irle. Yo d Do you think tht the rtions nd them on number line. re equl? Chek your nswer by pling

16 0 FRACTIONS (Chpter ) In the Investigtion, you should hve ound tht =, nd =. Notie how these numbers re relted: * / * / = or = = or = * / * / This suggests tht: Multiplying or dividing both the numertor nd the denomintor by the sme non-zero number produes n equl rtion. This rule llows us to write given rtion with dierent numertor or with dierent denomintor, without hnging the rtion s vlue. Exmple 0 Express with denomintor : b b = =g = =g = =g = = = EXERCISE F. Write with denomintor: b d 0 Write with denomintor: 0 0 b 0 0 d Express with denomintor : b d e 0 Express with denomintor 0: b d 0 e g h i 0 j 0

17 FRACTIONS (Chpter ) 0 Express in hundredths: b d 0 e 0 g h 0 i 00 j 00 Whih two o the ollowing rtions re equl? A 0 B C D E F G H 0 0 Gme Equl rtions Clik on the ion to ply gme where you must ind equl rtions. GAME SIMPLEST FORM We sy tht rtion is written in simplest orm i it is written with the smllest possible whole number numertor nd denomintor. For exmple, the rtion beuse we n write s. is not written in simplest orm, / = / To write rtion in simplest orm, we must ind the lrgest number tht is tor o both the numertor nd the denomintor. We then divide the numertor nd denomintor by this vlue. Exmple Write in simplest orm: 0 0 = 0 = is tor o both nd 0g b b = = is the lrgest tor o both nd g A rtion is in simplest orm when the numertor nd denomintor do not hve ny tors in ommon, exept.

18 0 FRACTIONS (Chpter ) EXERCISE F. Write in simplest orm: b d 0 e g 0 h 0 0 i j Whih o these rtions is written in simplest orm? A B C 0 D 0 E G We oten wish to ompre the size o two rtions. COMPARING FRACTIONS For exmple, i you were oered or 0 o blok o hoolte, whih would you hoose? I two rtions re written with the sme denomintor, we n simply ompre the sizes o the numertors. Exmple Whih is lrger: or b or? Convert mixed numbers to improper rtions beore ompring them. is lrger thn, so is lrger thn. b s n improper rtion is. is lrger thn, so is lrger thn. I two rtions do not hve the sme denomintor, we write one o them s n equl rtion whih hs the sme denomintor s the rtion we re ompring with. We n then ompre the numertors. Exmple Whih is lrger: or 0? We multiply the numertor nd denomintor o rtions hve denomintor 0. = = 0 by, so tht both 0 is greter thn 0,so 0 is greter thn.

19 FRACTIONS (Chpter ) EXERCISE G Whih is lrger: d or or b or e or Keith nd Croline te sushi or dinner. Keith hd piees o sushi. Croline ut her sushi piees into thirds, nd te o the thirds. Who hd more sushi or dinner? or or? Whih is lrger: or b or or d or e or or 0 g or h or i or? Arnold spends o his inome on rent, nd more on rent or on groeries? Trent nd Meredith eh own ge o birds. o his inome on groeries. Does he spend Trent s ge Meredith s ge b Wht rtion o Trent s birds re yellow? Wht rtion o Meredith s birds re yellow? In whih ge is there greter rtion o yellow birds? Disussion Are improper rtions lwys lrger in size thn proper rtions?

20 FRACTIONS (Chpter ) H ADDING AND SUBTRACTING FRACTIONS A pizz is divided into equl piees. Sm tkes piees, nd Pm tkes piees. This mens tht together they hve tken totl o piees. Sm Notie tht Sm hs tken o the pizz, Pm hs tken, nd together they hve tken. Pm So, + = + = Sm ets o his piees o pizz, so he hs piees remining. We n lso sy tht Sm took o the pizz, he te o the pizz, nd he hs o the pizz remining. So, = = To dd or subtrt rtions with the sme denomintor, we dd or subtrt the numertors. The denomintor stys the sme. Exmple Find: b + = b + = + = = =

21 FRACTIONS (Chpter ) EXERCISE H. Find: + b d + e + g h i + Find: + b d + e Exmple Find: =+ + =+ 0 =+ = Find: + + d + + b e Exmple Find +, giving your nswer in simplest orm. + = + = = is tor o both nd g =

22 FRACTIONS (Chpter ) Find, giving your nswer in simplest orm: b + d + + e Exmple Find +. + To dd or subtrt mixed numbers, we irst onvert them to improper rtions. = + = + = = Find: + d b e Simon nd Shne went hiking. On the irst dy they wlked o the totl distne. They hd steep limb on the seond dy nd only wlked o the totl distne. Wht rtion o the totl distne ws ompleted ter dys? Leh wrote pges o story beore te, nd nother hd she ompleted? pges ter te. How mny pges Spiros hd 0 b o bg o ertiliser. He used 0 Wht rtion o bg o ertiliser ws let? o bg or his tomtoes. Suppose ull bg ontins 0 kg o ertiliser. How mny kilogrms o ertiliser does Spiros hve let?

23 FRACTIONS (Chpter ) Srh nd Jne went pple piking. Srh piked bgs nd Jne piked bgs. b How mny bgs o pples did they pik ltogether? How mny more bgs did Jne pik thn Srh? ADDING AND SUBTRACTING FRACTIONS WITH UNEQUAL DENOMINATORS Sometimes the rtions we wnt to dd or subtrt do not hve the sme denomintor. For exmple, suppose Anit drinks o n o sot drink, nd Meliss drinks drink? o the n. Wht rtion o the n did they Qw Ei In the sme wy tht it is esier to ompre two rtions i they hve the sme denomintor, it is lso esier to dd or subtrt rtions i they hve the sme denomintor. In the sitution bove, we need to ind +. We n write with denomintor by multiplying the numertor nd denomintor by. So, we hve + = + = + Multiplying the numertor nd denomintor by the sme number produes n equl rtion. = So, Anit nd Meliss drnk o the n o sot drink.

24 FRACTIONS (Chpter ) Exmple Find: b + 0 = onverting to thsg b + 0 = + 0 onverting to 0thsg = = = = 0 = 0 EXERCISE H. Find: b + d + e Find: + 0 b d + 00 Exmple Find: + b + b = + onverting to improper rtionsg = onverting to improper rtionsg = + onverting to qurtersg = onverting to thirdsg = + 0 = = = = =

25 FRACTIONS (Chpter ) Find: + b + 0 d e +0 Find: b d Joshu bked ke to shre with riends. Lis te, nd Rebe te rtion o the ke did the girls et between them? o the ke. Wht Every dy, Angus eeds his hikens o lrge tub o eed. I Angus tub is ull t the strt 0 o the dy, how muh is let ter he hs ed his hikens? Smnth is n rtist. She spends hours on Sturdy pinting portrit, nd urther hours inishing it o on Sundy. How long did it tke her to pint the portrit? tonnes o erth needs to be removed to level housing blok. A truk moves tonnes in the irst lod. How muh erth still needs to be moved? KEY WORDS USED IN THIS CHAPTER ² denomintor ² equl rtions ² rtion ² improper rtion ² mixed number ² number line ² numertor ² proper rtion ² simplest orm Review set Wht rtion is represented by the ollowing? b

26 FRACTIONS (Chpter ) Wht rtion o the rs in this r prk re blue? Write the ollowing divisions s rtions: b Write s mixed number: b Express with denomintor : b 0 Find: o $00 b o 00 g Ple the ollowing rtions on number line: nd b nd An thlete runs o re in the irst hour nd 0 in the seond hour. Wht rtion o the re hs he ompleted? o m nd Whih is lrger: 0 or 0 b or or d or? 0 Find: b + + d Prtie test A Clik on the ion to obtin this printble test. Multiple Choie PRINTABLE TEST

27 FRACTIONS (Chpter ) Prtie test B Short response Copy this irle nd shde o it. Wht mixed number is represented by this digrm? b Write the mixed number s n improper rtion. Write these rtions s divisions, nd hene s whole numbers: 0 b Write s n improper rtion: b Write in simplest orm: b Write with denomintor: b 0 Srh went on holidy or 0 dys. It rined on one qurter o the dys. On how mny dys did it rin? Whih is greter, or 0? Find: + + b A ylist ompletes o her trining ride in the irst hour nd Wht rtion o her ride hs she ompleted? in the seond hour.

28 0 FRACTIONS (Chpter ) Prtie test C Extended response Answer the Opening Problem on pge. At brbeue, Adm te susges. susges, nd Jill te b Write eh o these vlues s n improper rtion. How mny susges did they et in totl? How mny more susges did Adm et thn Jill? Ple the rtion on number line. b On the sme number line, ple the rtion. d Whih o these rtions do you think is lrger? Chek your nswer to by writing the rtions with the sme denomintor. Judy hs to write 0 Christms rds to send to her riends nd mily. She writes o them on Mondy. How mny rds did Judy write on Mondy? b How mny rds does she still need to write? Judy writes o the remining rds on Tuesdy. i How mny rds did she write on Tuesdy? ii How mny rds does she still need to write? Cleb hd mthemtis test nd spelling test on the sme dy. The results o eh test re shown below. Mthemtis test Spelling test X X X X X X X X X X 0 X X 0 X X X X X X X 0 X b Wht rtion o the mthemtis questions did Cleb nswer orretly? Wht rtion o the spelling questions did Cleb nswer orretly? In whih test did Cleb nswer greter rtion o questions orretly?