Chpter Frtions Wht you will lern - Nming frtions - Equivlent frtions - Compring frtions - Aing n sutrting frtions - Multiplying y frtion Breking the reor - Frtion of quntity - Diviing y frtion - Converting frtions n eimls In the moern worl there is n inresing nee for greter ury in mesuring length, time, weight n other things. This inresing nee for ury hs ment tht we often nee to use smller prts of the unit of mesurement eing use. These prts re the frtions tht we will stuy in this hpter. For exmple, worl reors n now e roken y n improvement of mere one hunreth of seon. 0 Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
New Zeln Curriulum Level Numer strtegies n knowlege Unerstn ition n sutrtion of frtions, eimls n integers Fin frtions, eimls n perentges of mounts expresse s whole numers, simple frtions n eimls Apply simple liner proportions, inluing orering frtions Level Numer strtegies n knowlege Unerstn opertions on frtions, eimls, perentges n integers Know ommonly use frtion, eiml n perentge onversions 0 Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Do now List the prime ftors of the following numers. 0 Fin the highest ommon ftor of: n 0 n, 0 n Whih of the following represents? A C Whih of the following represents? A C Prior knowlege B B D Wht frtion of the shpe is represente y the yellow region? numer line sening orer ivisiility ommon ftors esening orer < less thn multiples prime numers > greter thn Wht frtion of these shpes is she re? Work out: + + lots of Arrnge in orer from smllest to lrgest:,.,, 0.,.0, 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Nming frtions Three stuents look t this igrm, ut give three ifferent nmes for wht they see. Sue sys I n see she lue. This is her igrm. Ryn sys I n see she lue. He rws igrm to explin this. Jo sys I think it shows she lue. She rws igrm to explin this. Drw the igrms tht Ryn n Jo use for their explntions. Key ies A quntity tht is prt of whole is frtion. Nming prts of frtion: numertor numer of prts out of the whole line enomintor numer of equl prts in the whole Three eighths is written s. There re prts in one whole n there re three of them. In proper frtion the numertor is less thn (<) the enomintor; e.g. in, < In n improper frtion the numertor is greter thn (>) the enomintor; e.g. in, > A mixe numer onsists of whole numer n proper frtion; e.g. EXAMPLE + + Eh igrm represents one whole ivie into numer of prts. i ii Write the frtion of the igrm tht is she yellow: using wors using numers; i.e. s proper frtions. Chpter Frtions 0 Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Solution Explntion i two sevenths ii seven twelfths i ii EXAMPLE EXAMPLE i There re two yellow fes, out of seven. ii There re seven yellow loks, out of. i The numertor is the numer of yellow fes. The enomintor is the totl numer of fes. ii The numertor is the numer of yellow loks. The enomintor is the totl numer of loks. Write the improper frtion s mixe numer. Gin hs ut pples into hlves for her lss. There re hlves left fter morning te. How mny pples is this? Solution Explntion = The enomintor is so there re prts in every whole four fourths (qurters) or. Arrnge in groups of this gives wholes n three qurters reminer: = = + + + Rewrite the mixe numer Eh piee is hlf or. The enomintor is so there re prts (hlves) in eh whole. There will e wholes n one hlf. s n improper frtion. Jeremih hs one n three qurter pies left over from his prty. Write this s n improper frtion. Solution = One n three qurter pies = = Explntion One whole ontins three thirs. groups of three thirs + two thirs = eight thirs + + = The whole pie hs four qurters, n there re more qurters. + = 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exerise Exmple Exmple Exmple Exmple Eh shpe shown on the next pge represents one whole. Prt of the shpe is she. i How mny prts re in eh shpe? ii How mny prts re she? iii In wors, wht frtion is she? iv Use numers to write the frtion tht is she. v In wors, wht frtion is not she? vi Write the frtion tht is not she, s proper frtion. e f g i h Write eh s proper frtion. one hlf three qurters three fifths five twelfths e four sevenths f six tenths g four ninths h two thirs i five eighths Write these improper frtions s mixe numers. You my like to sketh igrms to illustrte your nswer. g 0 0 e h Illustrte these mixe frtions using retngles n she the frtion given. f 0 g e h f Chpter Frtions 0 Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple Look t your nswers for Questions n. Explore other strtegies tht you oul use to rry out these hnges in formt for the frtions. Rewrite eh mixe numer s n improper frtion. e f Stte whether eh is mixe numer, proper frtion, n improper frtion or whole numer. g 0 0 e h Use igrm to show how lrge kes oul e shre eqully mong eight people. Diego s Pizzs ut their pizzs into slies to sell in their lunh r. They ut their stnr pizzs into sixths n their grne pizzs into eighths. There re stnr slies left fter lunh. How mny pizzs re left? There re grne slies left fter lunh. How mny pizzs re left? 0 Eh iruit of the shool fiel is qurter of kilometre. How mny kilometres i Json run if he rn seven iruits? ENRICHMENT: Egyptin frtions The nient Egyptins only wrote frtions s unit frtions. (A unit frtion must hve one s the numertor; e.g. one fifth, one seventh, one everything.) So to represent they wrote + + or in shortest form s +. Think of the fmily of twelfths. They woul hve written + or s. How woul they hve written,,,? s f 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Equivlent frtions A frtion wll like this one n e use to fin equivlent frtions. Use this frtion wll to fin: ll the equivlent frtions for n the simplest forms of, n. 0 How oul you simplify frtions without using the wll? Disuss with prtner. Key ies Equivlent frtions re equl they inite the sme shre of whole. 0 = An equivlent frtion n e rete y: 0 = multiplying the numertor n the enomintor y the sme numer: is the sme s 0 = 0 0 = 0 iviing the numertor n the enomintor y the sme numer. This is lle simplifying euse we nel ommon ftors in the frtion: e.g. = 0 0 0 0 0 0 or. 0 is the ommon ftor so it n e simplifie. This leves, so is equivlent to. A frtion tht nnot e simplifie is lle simple frtion. 0 Chpter Frtions 0 Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
EXAMPLE Fill in the missing numer to mke n equivlent frtion. = = 0 Solution = = 0 EXAMPLE Explntion Fin three frtions tht re equivlent to. Solution = = = = = = From the frtion wll, is equivlent to. So the missing numer is. From the frtion wll, is equivlent to. So we nee to multiply y. 0 NOTE: we o not multiply the whole numer. Or we oul use nother igrm: 0 0 0 0 0 0 Explntion Using the frtion wll, the frtions equivlent to re:,, Or we oul show this using igrm: 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple Exmple Exmple Write the frtion in its simplest form. Write the frtion 0 in its simplest form. Solution = = or = 0 = 0 = Exerise Explntion Hlve n, then hlve gin. is ommon to n. This mens tht the numertor n the enomintor n e ivie y. is ommon to oth 0 n. This mens tht the numertor n enomintor n e ivie y. Or we n use igrm: 0 Copy n fill in the missing numers to mke n equivlent frtion. 0 = = = e = f = g = i = j = k 0 = Write four frtions tht re equivlent to eh of the following. e f 0 g h l h = = = i j k l Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple Exmple Exmple Fill in the missing numers to omplete eh set of equivlent frtions. e = = 0 = = = 000 = = = 0 = = 00 = 0 = = 0 = 0 = 00 Write eh frtion in its simplest form. e 0 0 i f j 0 g k = = = = 0 = 0 = 0 = = 000 = = 00 Look t the frtion wll on pge 0 n write own ll the sets of equivlent frtions for: Drw frtion wll tht ivies one whole into five fifths. A lyer of tenths n nother of twentieths. Use your frtion wll to show the sets of equivlent frtions for: Keep your frtion wlls for future referene. Look t the pttern in this tle. If we form eh pir into frtion we get the following series of equivlent frtions:,,, Count y s Count y s Using your frtion wll, or otherwise, explin why this works. Copy n omplete these tles. i ii 0 0 0 0 0 Count y s Count y s Count y? Count y? h l 0 00 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
ENRICHMENT: Fir shre Show how eh shpe n e ivie into the numer of equl res require. four equl res of the sme shpe three equl res of the sme shpe two equl res tht re ifferent shpes i ii iii A wy of utting irulr irthy ke into equl slies is to ivie the irumferene into equl lengths n ut to the mile. Here is n exmple of ke ivie into sevenths for the irthy girl n her six guests. Use this metho of eqully iviing the length roun the irumferene to try to ut eh of these kes into six equl slies. Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Compring frtions Tm, Hine n Petr unerstn equivlent frtions, ut sometimes hve iffering opinions out the size of frtion. Petr sys Twenty eight of the forty texts I sent yestery were piture messges. Tm sys Tht mens tht more thn hlf of them were pitures messges. Hine sys Tht mens tht less thn three qurters of them were piture messges. Who is orret? Who is the more urte? Wht woul e more urte sttement? Key ies If frtions hve the sme enomintor, the frtion with the lrger numertor is the lrger frtion; e.g. > If frtions hve the sme numertor, the frtion with the lrger enomintor is the smller frtion; e.g. > If neither the numertor nor enomintor re the sme, we n: onvert to equivlent frtions to ompre them enhmrk on numer line y ompring to known frtions suh s hlf, qurter. EXAMPLE Insert the symol < or > etween eh pir of frtions., Solution > Explntion, The enomintors re the sme so the size of eh piee is the sme, n is lrger thn so there re more piees in. So > Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Solution Explntion > EXAMPLE Whih is the lrger of eh pir of frtions? or Solution 0 is lrger thn, so the ninths re smller thn the eighths. The numertors re the sme, so there is the sme numer of piees. So > 0 or Explntion is more thn hlf n is less thn hlf. 0 0 0 Or we oul onvert them to equivlent frtions with the sme enomintor. = 0 0 > >, = 0 is more thn n is less thn. 0 0 Or we oul onvert them to equivlent frtions with the sme enomintor. =, = 0 0 0 > 0 0 > 0 0 Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
EXAMPLE Write these frtions in orer from smllest to lrgest., n Solution,,,,, 0,,,,, 0 Explntion Hlf of = (enhmrk ). is smller thn, n so fits to the left of. 0 0 is out Chnge = 0 to mke the enomintor even, n enhmrk. Hlf of =. Beuse 0 is lrger thn, it fits to the right of 0 0 0 is out will fit etween wholes n wholes on numer line. Hlf of = (Benhmrk.) Beuse is smller thn, fits to the left of. is out 0 The unit ftions in orer re,,. Both. = so > 0 0 0 0 0 0 n. re lrger thn Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exerise Exmple Exmple Exmple Exmple Exmple Exmple Orer eh pir of frtions, using either < or > symols. e n n f n n Orer eh pir of frtions, using either < or > symols. n n e n n g n n 0 Drw numer line for eh set of frtions, s inite. i Ple eh frtion of the set on tht line. ii Use your numer line to help you write eh set of frtions in size orer, from smllest to lrgest. Line from 0 to with ivisions, to show e,,,, f Line from 0 to with ivisions, to show e,,,, Line from 0 to with 0 ivisions, to show e,,, 0 e f 0 f Line from 0 to with ivisions, to show e,,,, f Line from 0 to with 0 ivisions, to show e,,, f i Drw numer line n ple these frtions on it. ii Write the frtions in orer from smllest to lrgest.,,,, 0,, 0 0,,, Arrnge eh set of frtions in orer from smllest to lrgest.,,,, Put the following frtions in orer from smllest to lrgest.,,,, 0 0,,,, e, 0,,,,,,,,, 0 f f h n 0 n n n,,,,,,,, 0,,,,,, A group of friens orere pizz for inner. Ge te of pizz, Fergus te n Anton te. Who te the most pizz? Who te the lest pizz?, Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Angie is ompleting survey on how she spens her y. These re her t: Ativity Eting Sleeping Trvelling Working Relxing Frtion of y Put her tivities in orer of time spent on them. ENRICHMENT: Squeezing frtions Fin three frtions etween Fin five frtions etween n. Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Aing n sutrting frtions Mere ets segments of n ornge whih hs segments. Tom ets setions of -setion lok of hoolte. Mere sys: I hve eten three ninths of my ornge n hve six ninths left over. If I these two frtions together, I hve one whole. She writes + =. Tom sys: I h one whole lok of hoolte n fter eting two sevenths, I still hve five sevenths to et. If I sutrt two sevenths from one whole, I hve five sevenths left. He writes =. Investigte the orretness of these sttements. Key ies Frtions with the sme enomintor n e e or sutrte esily, euse they re the sme size piees. The numertors re e or sutrte. e.g. + = If frtions o not hve the sme enomintor, they must e rewritten efore ing or sutrting. Chnge frtions to equivlent frtions with the sme enomintor. A or sutrt the numertors. + = Write the nswer in its simplest form or s mixe numer. EXAMPLE 0 Work out the solution n write it in its simplest form or s mixe numer. + + Solution Explntion + = + = + = + or + = + + = + = Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
EXAMPLE Work out the following. Write the solution in its simplest form or s mixe numer. Solution Explntion = = = or = EXAMPLE Evlute: + Solution + = + 0 0 = 0 = = = Explntion = Use equivlent frtions with the sme enomintor: LCD = = 0 = 0 + = = 0 Use equivlent frtions with sme enomintor: LCD = = = = = = + = 0 0 0 = 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple 0, Exmple 0, EXAMPLE Work out the nswers to the following. George n Jmes re eting pizz for inner. George ets pizzs n Jmes ets pizzs. How muh pizz i they et ltogether? Solution + = + = + = = = = 0 = Exerise Explntion The mthemtil opertion we nee to o is ition s the prolem sks for ltogether. + Write s improper frtions. Convert to equivlent frtions. LCD = = Convert k to mixe numer. + = Write s improper frtions. Convert to equivlent frtions. Lowest ommon enomintor = = + = Work out n write the solution in its simplest form or s mixe numer. e i + + f j + + g k + 0 0 0 Work out n write the solution in its simplest form or s mixe numer. + + h l + + e + f Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple Exmple Evlute: e + Evlute: + f 0 + e + g + + f 0 h + During Top Tem ompetition, four memers of rugy, hokey n soer tem ompete in oonut-eting ompetition. The frtion of oonut eh person te is shown in this tle. Hokey Rugy Soer Plyer A Plyer B Plyer C Plyer D Rnk the tems on their oonut-eting ility. Work out the ifferenes in the mount of oonut eten y Plyer B n Plyer D in eh oe. George n Peter re tol they my eh hve of hoolte r. Sione fins the hoolte r efore they hve hne to et the hoolte n tkes of the r. 0 Wht frtion of the r is left for the two friens? Is there enough of the hoolte r left for George n Peter to hve their shre of the r? Justify your nswer. At DVD shop, of ll its DVDs re ol Westerns, n re other ol movies. The rest re reent 0 releses. Wht frtion of its DVDs re reent releses? Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Mrs Tmhere hs just h new house uilt. She wnts to know wht frtion of the ost of the house ws the uiler s profit. He gve her opy of this tle whih shows the ost of eh prt of the uiling proess s frtion of the totl ost of the house. Wht frtion of the ost of the house ws his profit? Costs of uiling house Tsk Frtion Tsk Frtion Tsk Frtion Conrete & loks 0 Roof Eletril 0 Pinting 0 Timer Plstering 0 Pluming 0 Lour 0 Joinery 0 Su ivers re wre of the time spent on ive. Fin the totl time spent if Pet spens minutes serhing for orl n minutes olleting strfish. 0 This qurilterl hs perimeter of metres. The longest sie is metres n is twie the length of the opposite prllel sie. How long is eh of the other sies if they re equl? m A plnt nursery sells mulh y the trtor uket lo. On one Stury the nursery h 00 uket los t the strt of the y. They sol the following equivlent numer of uket los:,,. They then reeive lo of ukets k from n erlier orer. Wht o they hve in stok t the en of the y? ENRICHMENT: Mgi frtions Complete these mgi squres. Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Multiplying y frtion Lin spent three fifths of her wges on expenses. Of the money she spent on expenses, qurter ws spent on foo. Wht frtion of her wges i Lin spen on foo? Lin expline her metho of working this prolem out to her frien: I rew these igrms: = = I she the igrm of in yellow, n I she the igrm of in re. Then I put the two retngles on top of eh other. There re squres out of 0 tht hve oule shing. So the nswer is tht Lin spent = Use Lin s metho to work out, n. 0 of her wges on foo. Disuss with your prtner how you woul work out the nswers to these multiplitions of frtions without rwing the igrm. Key ies When multiplying frtion y whole numer, multiply the numertor y the whole numer. If the frtion is mixe numer, then turn the frtion into n improper frtion first. Multiplying frtion y nother frtion less thn gives n even smller frtion. Numertors re multiplie together n enomintors re multiplie together: e.g. = Mixe frtions re hnge to improper frtions efore eing multiplie: e.g. eomes Sometimes it is possile to simplify first using ommon ftors: e.g. = Both n n e ivie y their ommon ftor. Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
EXAMPLE Four fifths of the pupils t Hui shool re y pupils. Two thirs of the y pupils trvel to shool y us. Wht frtion of the shool s pupils trvel to shool y us? Solution Two thirs of four fifths = So of = = = of the shool s pupils trvel y us. EXAMPLE Fin: three n three fifths of four Solution Three n three fifths of four = = = = Explntion Explntion This is frtion of frtion prolem. In mthemtis of mens to multiply. So we wnt = She n she of retngle. Ple the yellow she retngle over the top of the lue she retngle. There re now smll ivisions (ells) in the retngle: ells re oth yellow n lue (green). This is eight fifteenths ( ). We n sy tht two thirs of four fifths equls eight fifteenths: of = OR Multiply numertors n multiply enomintors. two n one thir of four n three sevenths. Write s mthemtil sttement. Chnge to n improper frtion. Multiply the numertors ross the top of frtion n multiply the enomintors ross the ottom of frtion. Chnge to mixe frtion. Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple Exmple Exmple Solution Explntion Two n one thir of four n three sevenths = = = = = 0 Exerise e Write s mthemtil sttement. Mixe frtions written s improper frtions. Multiply the numertors ross the top of frtion n multiply the enomintors ross the ottom of frtion. Or we oul simplify efore multiplying. = Simplify frtion to mixe numer. = 0 Sketh igrms to show how the result is otine in these lultions. Evlute: e of is of f of Work out (you my wish to use simplifying): e f of of of is g g of is 0 Work out the following, giving your nswers s mixe numers where pproprite. of e f h h of of 0 g h of Smnth is mking skirts for herself n her four friens in ne group. Eh skirt nees m of mteril. How muh mteril oes she nee? Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
A phone outlet si tht two thirs of their moile phone ustomers re uner 0 yers of ge. Three qurters of these ought smrt phones. Wht frtion of the outlet s ustomers re uner 0 yers of ge n ought smrt phone? Mr Snfor is mking six new shelves for the lirry. Eh shelf is to e m long. How muh timer oes he nee? Hlf of shool s Sports Aemy re oys. Three fifths of the oys list rugy s their mjor sport. One thir of the girls list netll s their mjor sport. Wht frtion of the emy hve rugy s their mjor sport? Wht frtion of the emy hve netll s their mjor sport? One lp of yle re is km. Sm yles lps. How fr hs he yle? 0 Christopher nees ns of pint for his eroom. The re to e pinte in Mtthew s room is times the re of Christopher s room. How mny ns of pint will Mtthew require? This igrm shows the frtion of flow in the hnnels of river. The frtion of the flow t the strt of eh hnnel is shown. Wht frtion of the full river flow woul pss A, B, C, D, E, F, G n H? B ENRICHMENT: Ptterns in frtions A E F Mison is prtising multiplying frtions. Mison sys she hs foun pttern in the enomintors n writes this: =, =, = 0, n 0 re ll in times tle. Wht pttern woul you expet to see if multiplying y? Wht pttern woul you expet to see if multiplying y? D C G H Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Frtion of quntity Ashley hs m of rion. She uses of it to tie up present. Disuss with prtner how to work out of the length of the rion in entimetres. How oul you use the nswer to work out of the length of the rion? Key ies The wor of n e hnge to multiplition sign; e.g. of = Doule numer lines re useful wy of lulting frtions of quntity; e.g. of 0 pssengers n e foun y using these numer lines. EXAMPLE 0 0 0 0 0 0 pssengers Solve the following. Slly sves two thirs of her weekly $0 poket money for the en-of-yer mp. How muh oes she sve eh week? Crpet is on sle for $0 per metre. Anru hs mesure his room n nees metres to rpet it. How muh woul the rpet ost him? Solution of 0 mens 0 = $0 Explntion 0 = 0 Simplifying = 0 or simplify first 00 = 0 Or we oul use oule numer lines; the quntity of $0 is mthe ginst. then mthes ginst $0. 0 0 $0 $0 $0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Solution Explntion Exmple Exmple 0 = 0 Exerise = 00 f Clulte these mounts. 0 = 0 = 00 = $00 or simplify first 00 = $00 Or we oul use oule numer lines. The quntity of $0 is mthe ginst. then mthes ginst $00. 0 0 $0 $0 $00 $0 of of 00 of of 0 e of f of g of ents h of ents i of $0 j of $0 k of $0 l of $0 0 Pul gives two fifths of g of sweets to his frien Gry. How mny sweets oes Gry reeive? How mny sweets re left for Pul? Stuents re llowe to keep one twentieth of every ox of hoolte rs they sell. There re 0 rs in every ox. Hone sells oxes. How mny hoolte rs o they keep from eh ox? How mny rs oes Hone get to keep? For seven weeks Rj gives four ninths of his weekly poket money to his sister. He reeives $ eh week, how muh hs he given his sister fter weeks? Sushil ought kg of tril mix t the ulk ins. It osts $.00 per kilo. How muh i she py for the tril mix? Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
A photogrpher hire for weing tkes 0 photogrphs in the hours she works. of the photogrphs re tken t the hurh. How mny photogrphs is this? of the photogrpher s time is spent t the hurh. How mny minutes is this? Of ll the photogrphs, inlue the rie. In how mny photogrphs is the rie inlue? How mny photogrphs o not inlue the rie? e If the reeption tkes up of the 0 photogrpher s time, how muh of her time is spent t the reeption? (Give your nswer in minutes.) Tim ought kg of mixe nuts of whih were penuts. How mny kilogrms of penuts i Tim uy? The foo tehnology lss is mking muffins for funriser. They nee to lulte wht quntities of ingreients they nee for 0 lueerry muffins. Complete this tle for them. Reipe for muffins ups fresh or frozen lueerries eggs 00 g flour 0 g sugr up of milk up utter tespoons king power ENRICHMENT: Movie tikets Quntity for 0 muffins Two fifths of the tikets sol t the Bunnyville Movie Thetre re sol to hilren. During July the thetre sells 0 tikets. Amission for hil is $.0. If three fifths of ll tiket sles is pi s inome, running osts n tx, how muh profit oes Bunnyville Thetre mke from the sle of tikets to hilren uring July? 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Diviing y frtion Two friens, Try n Jill hve isussion out iviing y frtion. Whih girl is orret? is the sme s euse it s just like multiplition. It n e written either wy! Drw igrms tht Jill oul use to support her sttement. Key ies NO! You re wrong Try, euse: = n = Diviing y frtion is the sme s fining how mny of those frtions there re in tht numer. When iviing whole numer y frtion less thn, the result is lrger thn the originl numer; e.g. four ivie y one thir mens: How mny thirs re there in four wholes? 0 0 There re thirs in four wholes. Division y frtion n lso e one y multiplying y the frtion flippe upsie own. = is the sme s writing = Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Diviing frtion y whole numer gives smller frtion, euse we re sking how mny whole numers re in tht frtion; e.g. mens how mny fours re there in? 0 there is n eighth of in OR = is the sme s writing =. The ivision n e hnge to multiplition y flipping the seon frtion. EXAMPLE Solve: The retker nees ui metres of grvel for new pth. His triler n hol of ui metre. How mny triler los will he nee? Three qurters of pizz is to e shre eqully etween two oys. How muh pizz o they eh reeive? Solution = = = = = Explntion Rewrite s mthemtil sttement: we nee to know how mny s there re in. Drw igrm to show tht there re lots of in. Or use numer line to show there re skips of in. 0 Or ivision prolem is written s multiplition flip over the seon frtion. Rewrite s mthemtil sttement: shre evenly mens use ivision. Drw igrm to show ivie into two prts =. Or use numer line to fin how ig eh skip is if there re skips in. 0 OR Rewrite s multiplition. = Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
EXAMPLE Exmple Exmple Exmple Work out: Exerise g Solve: e Solve: e f g Solve: e Solution = = = 0 0 = = 0 = Explntion f f 0 There re pizzs left over fter te. If six people re to shre them for lunh the next y, wht frtion of pizz will eh person get? Rewrite s multiplition flip over the seon frtion. A frm of hetres is to e split into poks, eh mesuring hetres. How mny poks n e me? Mixe numers nee to e hnge to improper frtions efore hnging ivision to multiplition n flipping over the seon frtion. Improper frtion is hnge k to mixe numer. g g h h Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Ie rem is soope out t out of litre per soop. If I hve litres left, how mny soops will I e le to serve? A groer hs kilogrms of mnrins to pk into kilogrm gs. How mny gs n he fill n how mny kilogrms of mnrins will e left? Trys of met t the uther s shop weigh kg. If Ky ought six trys for her resturnt, n eh mel she prepres for her ustomers requires kg of met, how mny mels n Ky prepre? ENRICHMENT: Mesuring time In ys of ol, one wy of mesuring time ws to mrk urning nle, s nles urn t firly onstnt rte. 0 m If this type of nle tkes hours 0 minutes to e use ompletely, where will it hve urne own to y noon if it ws lit t o lok in the morning? Mke n urte sle rwing of this nle n mrk every 0 minutes, then mrk where it will hve urne own to y miy. Reserh the history of time on the internet. Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
- Converting frtions n eimls John sys, All eimls re reurring. Sue sys, Wht out = 0.? John replies, It is 0.000000000+ whih is reurring. Do you gree with John? Sue sys Wht out π? Does π written s eiml reur? Investigte. Key ies To hnge eiml to frtion rewrite it with enomintor of 0 or 00 n then simplify if possile. To hnge frtion to eiml, rewrite s n equivlent frtion with enomintor of 0 or 00 or, ivie the numertor y the enomintor. or use oule numer line: e.g. = 0. In hnging frtion to eiml we often rete reurring eimls. A reurring eiml is eiml tht repets. We show the repeting pttern using ots or rs over the numers. 0... = 0. or 0. # n 0. = 0. # # or 0. Here re some frtion eiml onversions tht you shoul know. Frtion Deiml 0. 0. 0. 0. 0. 0. 0. 0. 0. EXAMPLE Convert the following to eimls. Solution Explntion 0. Rewrite s n equivlent frtion with enomintor of 0. = 0 = 0. OR use oule numer line. 0 0 0. 0 0 0. Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Solution Explntion Exmple Exmple =. Look t the frtionl prt of the numer. Rewrite s equivlent frtion with enomintor of 00. = 00 = 0. Or use oule numer line. = 0. = 0. or 0.# Divie y = 0. = 0.# # or 0. EXAMPLE 0 Divie y = 0. Convert the following to frtions. 0. 0.. Solution 0. = 0 = 0. =. = 00 = 000 Exerise h Explntion Convert these to eimls. e i 0 00 Convert these to eimls. e i 00 represents tenths, whih n e simplifie to. represents hunreths, whih n e simplifie to. represents thousnths, whih nnot e simplifie. f j f j 00 00 0 0 0. g g 000 h h 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Exmple Exmple 0 Exmple 0 We know tht = 0. n = 0.. Use this informtion to onvert the following to eimls. e = 0., so how mny fifths re there in: f 0.? 0.? Convert these frtions to eimls. 0 e i 00 f j 00 Convert these mixe numerls to eimls. e i f j 00 00 g 0 g k 0 h h l Convert eh set of frtions to eimls n then write in orer from smllest to lrgest.,, e,, f,, g If we know tht = 0.#, wht is the eiml vlue of:,, 0,???? Convert these eimls to frtions. 0. 0. 0. 0. h 0, 0,, 0 e 0.0 f 0.0 g 0.0 h 0.0 i 0. j 0. k 0. l 0. m 0.00 n 0. o. p. 0 Convert these eimls to frtions. 0.00 0.0 0.0 0. e. f.0 g. h.0 i 0. j. k. l 0.00 Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
In Netll the shooters performnes re etermine y the gols sore ivie y the totl numer of gols ttempte. In the first hlf, Ann ttempte 0 gols n sore. In the seon hlf, she sore gols out of 0 ttempts. Whih ws her etter hlf? In the first hlf, Jen sore out of ttempts. In the seon hlf, she sore 0 out of ttempts. Whih ws her etter hlf? Who h the etter soring performne? Joseph n Alii plye hess on their omputers. Alii si, I hve plye gmes n eten the omputer times. Joseph si, I hve plye gmes n eten it times so I m etter plyer thn you. Ws Joseph orret in sying this? ENRICHMENT: Reurring eimls Express, n s eimls n use the pttern to preit the eimls for, n. Write the eimls for,, n then preit the eimls for,,. Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Thinking mthemtilly Vnishing shpes Visul effets re feture of rt n esign. The use of olour n shpe n e omine to proue imges tht pper s though they re three-imensionl, even though they re rwn on pge! In this pplition we will omine mthemtil onstrution n shing to proue three-imensionl effet. Unerstning the onstrution On sheet of grph pper, rule squre of sie m. Step : Divie the squre into two hlves s shown. Step : Divie the hlf-squre into two equl hlves to mke two qurters. Step : Now ivie the qurter-squre into two equl hlves. Step : Continue the proess until you onstrut this shpe. of = of = Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Creting optil illusions If you refully pply shes of hosen olour, you n rete remrkle optil illusions. This tivity n e omplete on omputer where wie rnge of gruting shes of olour re ville to rete this effet. Strting with squre of sie m, onstrut eh of these frture pitures. Inlue the frtion tht eh re represents of the originl squre. Use olour to rete optil illusions like tht shown ove. of Extening the ie Strting with squre of sie m, evelop your own vnishing squre. You oul strt with ifferent frtion or reuse one of the frtions ove, ut experiment with other wys of iviing the squre. For exmple, ivision gin se on is shown elow. Wll poster Mke poster of your oloure onstrution. Show the frtion of the originl squre for eh re. 0 Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Chpter summry Nming frtions A quntity tht is prt of whole is frtion. Nming prts of frtion: n u m e r t o r e n o m i n t o r numer of prts out of the whole line numer of equl prts in the whole Three eighths is written s. There re prts in one whole n there re three of them. In proper frtion the numertor is less thn (<) the enomintor. In n improper frtion the numertor is greter thn (>) the enomintor. Equivlent frtions Equivlent frtions re equl they inite the sme shre of whole. An equivlent frtion n e rete y: multiplying the numertor n the enomintor y the sme numer. iviing the numertor n the enomintor y the sme numer. This is lle simplifying euse we nel ommon ftors in the frtion. A frtion tht nnot e simplifie is lle simple frtion. Compring frtions If frtions hve the sme enomintor, the frtion with the lrger numertor is the lrger frtion; e.g. > If frtions hve the sme numertor, the frtion with the lrger enomintor is the smller frtion; e.g. > If neither the numertor nor enomintor re the sme, we n: onvert to equivlent frtions to ompre them enhmrk on numer line y ompring to known frtions suh s hlf, qurter. Aing n sutrting frtions Frtions with the sme enomintor n e e or sutrte esily, euse they re the sme size piees. The numertors re e or sutrte. If frtions o not hve the sme enomintor, they must e rewritten efore ing or sutrting. Chnge frtions to equivlent frtions with the sme enomintor. A or sutrt the numertors. Write the nswer in its simplest form or s mixe numer. Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Multiply y frtion When multiplying frtion y whole numer, multiply the numertor y the whole numer. If the frtion is mixe numer, turn the frtion into n improper frtion first. Multiplying frtion y nother frtion less thn gives n even smller frtion. Numertors re multiplie together n enomintors re multiplie together: Mixe frtions re hnge to improper frtions efore eing multiplie. Sometimes it is possile to simplify using ommon ftors. The wor of n e hnge to multiplition sign: e.g. of = Doule numer lines re useful wy of lulting frtions of quntity. Diviing y frtion Diviing y frtion is the sme s fining how mny of those frtions there re in tht numer. When iviing whole numer y frtion less thn, the result is lrger thn the originl numer. Division y frtion n lso e one y multiplying y the frtion flippe upsie own. Diviing frtion y whole numer gives smller frtion, euse we re iviing tht frtion prt into smller piees. Converting frtions n eimls To hnge eiml to frtion rewrite it with enomintor of 0 or 00 n then simplify if possile. To hnge frtion to eiml: rewrite s n equivlent frtion with enomintor of 0 or 00 Or, ivie the numertor y the enomintor. Or, use oule numer line In hnging frtion to eiml we often rete reurring eimls. A reurring eiml is eiml tht repets. We show the repeting pttern using ots or rs over the numers. Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Review questions Give three frtions tht re equivlent to. Wht is the sum of n? Wht is the LCD of, n? Benhmrk. Is it loser to or whole? Explin. Fin of. Rewrite: (Do not solve.) Evlute: Evlute: Evlute: 0 Write these frtions in orer from smllest to lrgest.,,,,,, 0 Fin the nswers to the following. + + Evlute: + e f g i h 0 j Write these frtions s eimls. Write these eimls s frtions in their simplest form. 0.... Of 0 ie rems sol y fst-foo outlet t the riket, were vnill n strwerry. The reminer were hoolte. Wht frtion of the sles ws hoolte? Of the $0 weekly ernings from lwn-mowing usiness, ws pi in tx, in wges, in foo n the rest nke. Wht frtion ws nke? How muh ws nke? Chpter Frtions Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
A totl of people ttene shopping spree. Of the people, eie to stop shopping n et lunh while the rest ontinue to visit the stores. Wht frtion of people ontinue shopping? Wht frtion of the group eie to eh lunh? Wht frtion shoppe t some time uring the y? An re of hetres is to e suivie into equl hetre lots n the reminer use for prkln. How mny lots n e me? How muh ln will remin for prk? Wht frtion of the overll evelopment is prkln? Mthemtis & Sttistis for the New Zeln Curriulum: Yer Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00
Unorrete smple pges, pps Cmrige University Press Fgn, Gooey, Lwrene et l 0 ISBN --0-0- Ph 0 00