Chapter. Fractions. Contents: A Representing fractions


 Phillip Lyons
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1 Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers
2 08 FRACTIONS (Chpter ) OPENING PROBLEM Things to think out: When Unle Pulo died he let ll his money to his sister s hildren, o whom re girls nd re oys. They re eh to get equl shres o the totl mount o E Wht prt o the inheritne does eh hild reeive? Wht prt do the girls reeive? How muh do the oys reeive in totl? The irle longside is divided into three equl portions. The one whole irle is divided into three, so the one portion tht is shded represents o the whole irle. We ommonly write this s the rtion. Two o the three portions re unshded, so this is or o the irle. On numer line, we hve divided the segment rom 0 to into three equl prts. We n then ple nd on the numer line. Qe_ We_ 0 In generl, n e written s the rtion. mens we divide whole into equl portions, nd then onsider o them. the numertor is the numer o portions onsidered the r indites division the denomintor is the numer o portions we divide whole into. The denomintor nnot e zero, s we nnot divide whole into zero piees. Other denomintors we desrie using dierent words: Denomintor Nme o portions hl third qurter ith sixth
3 FRACTIONS (Chpter ) 09 FRACTIONS ARE EVERYWHERE Qt_ open Qr_ remining \\Qw_ pple size 8\Qw_ \\Qw_ hnds Qe_ ull ACTIVITY Wht to do: FRACTIONS WE ALL KNOW Copy nd omplete the ollowing skethes to show: F E FUEL hl pst twelve petrol guge showing the tnk is lmost three qurters ull pizz with one ith o it missing.
4 0 FRACTIONS (Chpter ) A REPRESENTING FRACTIONS The rtion three eighths n e represented in numer o dierent wys: Words three eighths Digrm s shded region or s piees o pie Numer line 0 three eighths Symol 8 numertor r denomintor A rtion written in symoli orm with r is lled ommon rtion. PRINTABLE WORKSHEET EXERCISE A Copy nd omplete the ollowing tle: Symol Words Numertor Denomintor Mening Numer Line one hl three qurters One whole divided into two equl prts nd one is eing onsidered. One whole divided into our equl prts nd three re eing onsidered. 0 one hl 0 three qurters 0 two thirds two d 7 sevenths e One whole divided into nine equl prts nd seven re eing onsidered.
5 Rt_ q J_p_ Qt_ Qr_ FRACTIONS (Chpter ) Symol Words Numertor Denomintor Mening Numer Line 8 g 0 ACTIVITY Wht to do: ESTIMATING FRACTIONS d Mke your own rtion wheel s ollows: Use drwing ompss to drw two identil irles on two dierent oloured piees o rdord. Use your protrtor to mrk the rtions s shown on the seond irle. For exmple, 0 is (0o 0) = o, 8 is (0o 8) = o, 8 is (0o 8 then ) = o. Cut out oth piees. Mrk nd ut rdius on oth irles s shown. Interlok the irles. Ui_ 0 qa_p_ Qi_ Ty_ Qy_ ut Er_ ut We_ Qe_ Ti_ Et_ Qw_ Wt_ Ei_ Chllenge your prtner to guess the rtions you mke y looking t the reverse side whih hs no rtions written on it. Hve your prtner estimte the rtion whih dds to yours to mke one. Clik on the ion to lod gme or estimting rtions. Ply the gme until you eome good t reognising the size o dierent rtions. FRACTION ESTIMATOR
6 FRACTIONS (Chpter ) B FRACTIONS OF REGULAR SHAPES A good wy to lern out rtions is to divide regulr two dimensionl shpes. EXERCISE B DEMO Whih o the ollowing shded shpes does not show ive sixths? A B C D E F G H Copy the given shpe extly. Consider the lrge squre to e whole or. d e I eh retngle is hl o the one eore it, how muh o the shpe is unshded? Chek your nswer to y drwing grid within the lrge squre. Use the oundries o the shded squre s the dimensions o the smllest squres in your grid. How mny o the smllest squres it into your lrge squre? Wht rtion o the whole is the shded squre? Wht rtion o the whole is the unshded re? Qw_ Qr_ Using identil squre piees o pper, mke opies o this tngrm. Numer the piees on oth sheets. Cut one o the sheets into its seven piees. Use the piees to help you work out the ollowing: How mny tringles like piee would it into the lrgest squre? Wht rtion o the lrgest squre is piee? Wht rtion o piee is piee? d Wht rtion o the lrgest squre is eh tngrm piee? 7
7 FRACTIONS (Chpter ) C INVESTIGATION EQUAL FRACTIONS EQUAL FRACTIONS Wht to do: Use grid pper to onstrut identil squres with sides m long, or lik on the ion to otin templte. Use the grid lines on the pper to guide you. TEMPLATE Divide the irst squre into equl prts. Eh prt is one hl. One hl hs een shded. Divide the seond squre into qurters. Eh hl is now equivlent to two qurters or. Shde in the sme hl s you did in the irst squre. Divide the third squre into eighths. Shde in the one hl o the ig squre. Divide the ourth squre into sixteenths. Shde in the one hl o the ig squre. d In the ith squre show tht one hl equls. e Copy nd omplete: = = :::::: = :::::: = : Use protrtor to outline identil irles. From the entre o the irst irle, mesure nd rule lines, 0 o prt. Sine 0 o = 0 o, you hve divided the irle into thirds. Shde. In the seond irle drw lines 0 o prt. Sine 0 o = 0 o, you hve divided the irle into sixths. Shde. We_ Ry_ d In the third irle drw lines 0 o degrees prt. Shde the pproprite equl re. e Continue the pttern in the ourth irle. Copy nd omplete: = = ::: = :::::: : In the investigtion ove, you should hve ound tht: = = 8 = 8 = nd = = 8 = Notie how these numers re relted: = = 8 = 8 = = = 8 =
8 FRACTIONS (Chpter ) This suggests tht we n use multiples to ind rtions tht re equl. For exmple: = = = 8 = 0 = :::::: Multiplying or dividing oth the numertor nd the denomintor y the sme nonzero numer produes n equl rtion. For exmple: = = 0 nd = = 0 nd so = 0 = 0. 8 = 8 = 9 nd 9 = 9 = nd so 8 = 9 = : Exmple Express with denomintor 8: Sel Tutor = 7 9 s 9 =8g = 8 = s =8g = 8 EXERCISE C Express with denomintor 8: d Express with denomintor 0: d 0 e g h Express in sixteenths: 8 d 0 e 7 8 g 8 h Express in hundredths: d e 0 g h 7 0 Multiply to ind equl rtions: = = 0 d = 8 = 8 9 = 8 9 = e = = 0 0 DEMO 7 = 7 = 7 8 = 7 = 8
9 FRACTIONS (Chpter ) Divide to ind equl rtions: 8 = 8 = d 8 = 8 = 8 0 = 8 0 = e = = 0 = 0 = 8 0 = 0 = 9 7 Find i: = 7 = = 77 d = 7 e 7 = = g = 9 9 h 8 7 = 8 Find i: = = 0 = d = e 7 8 = = 7 9 g = h 8 = 8 D SIMPLIFYING FRACTIONS In Chpter we sw how there is proper order in whih the opertions in n expression should e perormed. We lled this order BEDMAS. The division line o rtions ehves like set o rkets. This mens tht the numertor nd denomintor must e ound eore doing the division. Exmple Sel Tutor Simpliy: = = = 0 8 = 0 = EXERCISE D. Simpliy: d +8 e +
10 FRACTIONS (Chpter ) g h + 7 i ( ) 8 LOWEST TERMS We n lso simpliy rtion y writing it s n equl rtion where the numertor nd denomintor re s smll s possile. 0 For exmple, 0 = in simplest orm. To write rtion in simplest or lowest terms, we need to remove the ommon tors rom the numertor nd denomintor. For exmple, nd 0 hve HCF =. So 0 = 0 =. Exmple Sel Tutor Simpliy to lowest terms: = is the HCF o nd 7g = 9 7 = 7 is the HCF o 7 nd g = 7 EXERCISE D. Simpliy to lowest terms: d e 77 g 8 8 h 0 i 00 j 000 Simpliy to lowest terms: d 9 e g 0 80 h 87 i 00 j 000 Simpliy: d 0 e 000 g 8 h 80 i 0 j 0 Whih o these rtions re in lowest terms? 0 d e 9 8 g h i 00 j k 00 l
11 FRACTIONS (Chpter ) 7 E FRACTIONS OF QUANTITIES In this setion we see how rtions re pplied to the rel world. They n desrie prt o quntity or group o ojets. When writing rtions tht involve mesurements it is importnt tht we use the sme units in the numertor nd the denomintor. Exmple Wht rtion o metre is 7 m? 7 m 7 m s rtion o metre = metre 7 m = 00 m = 7 00 Sel Tutor the sme unitsg metre = 00 m Exmple Mtthew ws given ox o hooltes. hd red wrppers, hd lue, hd gold nd hd green. Wht rtion o the hooltes hd red wrppers? Wht rtion o the hooltes did not hve gold wrppers? Sel Tutor Frtion with red wrppers = numer with red wrppers totl numer o hooltes = = hooltes did not hve gold wrppers. numer without gold wrppers Frtion without gold wrppers = totl numer o hooltes = EXERCISE E Wht rtion o eh o the ollowing dierent quntities hs een irled?
12 8 FRACTIONS (Chpter ) Use ull pk o plying rds to work out the ollowing questions. Clik on the link i you need to see wht ll o the rds look like. Clulte wht rtion o the ull pk re: red rds suh s spdes suh s DEMO es suh s d piture rds suh s e ll the odd numered rds ll the even numered lk rds In simplest orm, stte wht rtion o: metre is 0 m metres is 78 m kg is 00 g d week is dys e dy is hours Novemer is two dys g dede is one yer h dollrs is 7 ents kg = 000 g A dede is 0 yers. Wht rtion o one hour is: 0 minutes 0 minutes minutes d minutes? Wht rtion o one dy is: hour hours 0 minutes d minute? Gordon spent $ on drink nd $ on hooltes. Wht rtion o $0 did he spend? 7 Jenny sored 7 orret nswers in her test o 0 questions. Wht rtion o her nswers were inorret? 8 Lind hd g o 9 pples. She te nd she ed others to her horse. Wht rtion o her pples remin? 9 Jmes ws trvelling journey o 0 km. His r roke down ter 80 km. Wht rtion o his journey did he still hve to trvel? 0 Wht numer is: o 0 o o d o e o 0 o g o 0 h o 0 i o 0 j o k 8 o 8 l o 00? Trn strted his homework t 8: pm nd ompleted it t 9:08 pm. I he hd llowed one hour to do his homework, wht rtion o tht time did he use? Vijy hd 9 m o rope. He ut piees rom it, eh 0 m long. Wht rtion o the rope remined?
13 FRACTIONS (Chpter ) 9 Exmple Sel Tutor To ind On the irst dy o shool this yer, o the th grde lss were ged yers or older. I there were 7 students in the lss, how mny were yers or older? The ull numer is 7. So, 7 = 9students There were 9 students ged yers or older. Find: o people o 0 lollies d 0 o 0 g e o E8 o drinks o 0 minutes. Viktor only won one third o the gmes o tennis tht he plyed or his shool tem. I he plyed gmes, how mny did he win? One ith o the students t shool were sent euse o hiken pox. I there were students in the shool, how mny were sent? One sixth o the rs rom n ssemly line were pinted white. I rs me rom the ssemly line, how mny were pinted white? 7 Ling spent one third o her money on new dminton rket. I she hd 9 RMB eore she ought the rket, how muh did the rket ost? 8 While Evn ws on holidys, one eighth o the tomto plnts in his greenhouse died. I he hd 9 plnts live when he went wy, how mny were still live when he me home? 9 There re 0 o in ull revolution or turn. Find the numer o degrees in: i one qurter turn ii hl turn iii three qurters o turn Wht rtion o revolution is: i 0 o ii 0 o iii 0 o? o 7 we need to divide 7 into equl prts. Exmple 7 Sel Tutor o the irds in my viry re inhes. I there re irds in my viry, how mny inhes re there? o is = =8 So, o must e 8= There re inhes in my viry.
14 0 FRACTIONS (Chpter ) 0 One morning two iths o the pssengers on my us were shool hildren. I there were pssengers, how mny were shool hildren? Rihrd spent three qurters o his working dy instlling omputers, nd the reminder o the time trvelling etween jos. I his working dy ws 8 hours, how muh time did he spend trvelling? When Ssh plyed netll, she sored gol with seven eighths o her shots or gol. I she shot or gol times in mth, how mny gols did she sore? A usiness hired truk to trnsport oxes o equipment. The totl weight o the equipment ws 000 kg, ut the truk ould only rry 8 o the oxes in one lod. Wht weight did the truk rry in the irst lod? I there were 80 oxes, how mny did the truk rry in the irst lod? Answer the questions in the Opening Prolem on pge 08. F COMPARING FRACTION SIZES We oten wish to ompre the size o two rtions. For exmple, would you rther hve o lok o hoolte? or The sizes o two rtions re esily ompred when they hve the sme denomintor. For exmple, = = 9 nd = = 8 : Sine 9 > 8, 9 > 8 nd so >. We n show this on numer line. We_ =qk_w_ 0 Er_ =ql_w_ To ompre rtions we irst onvert them to equl rtions with ommon denomintor whih is the lowest ommon multiple o the originl denomintors. This denomintor is lled the lowest ommon denomintor or LCD.
15 FRACTIONS (Chpter ) Exmple 8 Sel Tutor Find the LCD o,, nd y irst inding the lowest ommon multiple o,, nd. The multiples o re The multiples o re The multiples o re ) the ommon multiples o, nd re:,, nd so on. ) the lowest ommon multiple is. ) the LCD o, nd is. LCD is the revition or Lowest Common Denomintor. Exmple 9 Sel Tutor Write the rtions,, nd with the lowest ommon denomintor or LCD. Hene write the originl rtions in sending order o size (smllest to lrgest). The lowest ommon multiple o, nd is 0. So, the LCD o,, nd is 0. = 0 0 = 0 0 = = 0 Now 0 < 0 0 < 0, so < <. = = 0 EXERCISE F Find the LCM o: 7,,, d, 8 e, 8, 9 0,, g,, h,, 9 Write eh set o rtions with the lowest ommon denomintor nd hene write the originl rtions in sending going up. order (smllest to lrgest): Desending Asending mens,,, mens 7 going down. d 8, 7 e 0, 7 9, g 8, h, 9 8 i, 7 0, By writing eh set o rtions with the lowest ommon denomintor, rrnge the rtions in desending order:,, 7 0, 8,, 7,
16 FRACTIONS (Chpter ) G IMPROPER FRACTIONS AND MIXED NUMBERS IMPROPER FRACTIONS All the rtions we hve looked t so r hve hd vlues etween zero nd one. This mens tht their numertors were less thn their denomintors. 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. represents : is n improper rtion. represents : To otin ive qurters or we tke two wholes, divide oth into qurters, then shde qurters. We n see rom the digrm tht is the sme s,or nd. MIXED NUMBERS When n improper rtion is written s whole numer nd rtion, it is lled mixed numer. For exmple, is mixed numer. It is oten neessry to hnge numer rom n improper rtion to mixed numer nd vie vers. 8 For exmple, =8 =wholes nd equl prts (thirds) let over. 8 So, =. 8 Another wy o doing this is: = + =+ = Exmple 0 Sel Tutor Write s whole numer or mixed numer: = = = 0 + =+ =
17 FRACTIONS (Chpter ) EXERCISE G Write s whole numer: 0 8 d 0 8 e 0 0 g 0 0 h 0 0 i 0 j 8 k l 7 Write s mixed numer: 7 8 d 9 e 7 g 7 h 8 i 7 j 9 k 09 l Exmple Sel Tutor Write rtion. s n improper =+ split the mixed numerg = 0 + write with ommon denomintorg = Write s n improper rtion: d e g h i 9 j 7 8 k 7 l Suppose we hve two die. We roll one to give the numertor o rtion nd the other to give the denomintor. Find: numertor is the upper e r denomintor is the lower e the smllest rtion it is possile to roll the lrgest proper rtion it is possile to roll the lrgest improper rtion whih is not whole numer tht it is possile to roll d the numer o dierent rtions it is possile to roll. e List the dierent omintions tht n e simpliied to whole numer. KEY WORDS USED IN THIS CHAPTER ² ommon rtion ² denomintor ² equivlent rtions ² improper rtion ² lowest ommon denomintor ² lowest ommon multiple ² lowest terms ² mixed numer ² numer line ² numertor ² proper rtion ² simplest orm
18 FRACTIONS (Chpter ) REVIEW SET A Wht rtion is represented y the ollowing? Express with denomintor : Find the lowest ommon multiple o: 8 nd, nd Write T or true nd F or lse: 9 = 0 7 = =9 d = 8 Find: o E o 00 g In lowest terms, stte wht rtion o: one week is dys one metre is m 7 Solve the ollowing prolems: Lex hd rton o eggs. He used o them to mke ke. Wht rtion o the eggs did he use? Dwn hs ts, dogs nd ish. Wht rtion o her pets re ts? Sr went on holidy or 0 dys. It rined on qurter o the dys. On how mny dys did it rin? 8 By writing eh set o rtions with ommon denomintor, rrnge the rtions in sending order (smllest to lrgest): 9, 8, 7 9 Whih is the greter, 7 or 9? 0 Write s whole numer or mixed numer: 7
19 FRACTIONS (Chpter ) REVIEW SET B Wht rtion is represented y the ollowing? Express with denomintor : 8 Find i: = 7 = Convert 9 8 to mixed numer. Wht rtion o $900 is $80? Wht rtion o 800 mis00 m? Express, nd 0 with lowest ommon denomintor. Hene write the originl rtions in desending order o size. Find o 8. Find the vlues o nd given tht = 0 = 7. 7 Write T or true nd F or lse: 7 = = = 8 7 = 8 Solve the ollowing prolems: A mn who weighed 90 kg went on diet nd lost 0 kg. Wht rtion o his originl weight did he lose? o lok o sheep numered 0. Find the size o the whole lok. 7 o the students o shool ttended ilm night. I there were 80 students in the shool, how mny ttended the ilm night? 9 By writing eh set o rtions with ommon denomintor, rrnge the rtions in desending order (lrgest to smllest): 7 0, 8, Write s n improper rtion: 7
20 FRACTIONS (Chpter ) ACTIVITY MAKING AN ORIENTEERING COURSE Wht to do: Otin mp o your shool nd its grounds, or lol prk or plyground. Divide your lss into smll groups nd design simple orienteering ourse leding rom one point or lndmrk to nother. The lndmrks might inlude distintive tree, the orner o uilding, or sign. Choose or lndmrks nd drw the ourse on your mp. Numer the lndmrks in the order you wnt them visited. Eh lndmrk must e lerly visile rom the previous one. Sout hll Drinking ountin N Sttue Strt Plque Sle :000 Cr prk Go to your strting lotion nd use ompss to mesure the ering o the irst lndmrk. Mesure the distnes etween lndmrks using trundle wheel, or i you do not hve one you n estimte them y ping them out. Use the sme person to pe out eh leg nd mesure the length o pe severl times to mke your estimte s urte s possile. Find the ering o eh lndmrk rom the previous one on your ourse, nd the distne etween them. Prepre tle o instrutions tht will enle others to ollow your ourse. Leg Lndmrk Compss ering Distne strt to to to to Swp your instrutions with nother group nd test out eh other s ourses. Did you orretly identiy eh other s lndmrks? How urte were your erings nd distnes?
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