2 Fractions and ratios

Transcription

1 Frtions nd rtios Number nd lgebr Setion Disussion prompts Diret students to exmine the opening photo for this unit on pges nd of their Student Resoure Book. Ask students how the rrngement of blls on the pool tble ould represent whole. When beginning gme of pool, the blls re rrnged by pling them into tringulr rk so tht they re in position. When the blls re inside the rk, they n be onsidered to be one whole. Ask students to ount the number of pool blls in the imge nd write one whole s frtion. There re blls in the imge; this n be represented by the frtion. You my like to pose some questions tht demonstrte whether students understnd the onept of frtion. For exmple: How mny blls re yellow? () Wht frtion of the blls is yellow? ( ) How mny blls re blk? () Wht frtion of the blls is blk? ( ) How mny blls re green or blue? ( + ) Wht frtion of the blls re green or blue? ( ) Consolidte the mening of numertor nd denomintor. As onrete id, you n provide eh group of students with ounters to represent the pool blls. Consider simple ddition nd subtrtion of frtions with the sme denomintor. For exmple, of the blls re yellow nd of the blls re blk. Demonstrte tht the frtion of blls tht re yellow or blk is +. Frtions A Understnding length nd rtios Number nd lgebr Wht does frtion desribe nd how re frtions used eh dy? Consider simple multiplition by Ask students to ome up with their own Students ould be sked to bring in n identify proper frtions, improper use digrm to find frtion of frtion. For exmple, wht is onethird or frtion questions. item or two eh from home to further frtions nd mixed numbers whole-number mount of the blls? () How n the ounters be used to help work this out? Brinstorm other everydy ples where demonstrte tht frtion represents find the highest ommon ftor of use digrm to divide whole numbers prt of whole. (divide the ounters into three groups) items n be divided into prt of whole. two numbers by frtion For more ble students, disuss whether Answers will vry: some suggestions Setion Prerequisite identify nd write equivlent frtions lulte simple powers nd squre roots you n find,, et. of the blls nd ould be tubs of yoghurt in six pk; knowledge list frtions with the sme denomintor of whole numbers if the result is whole number. multipk of Smrtie boxes, then the Students need to be ble to: in sending or desending order understnd how to write the omprison smller pkets of Smrties; blok Disuss equivlent frtions by looking t write frtion with the orret dd nd subtrt frtions with the of two quntities of the sme type nd of hoolte. of the ounters nd reognising tht this is the sme s numertor nd denomintor sme denomintor reognise tht order is importnt.. find the lowest ommon multiple of two or three numbers A B C D E F G H Understnding frtions Equivlent frtions Adding nd subtrting frtions Multiplying frtions Dividing frtions Powers nd squre roots of frtions Understnding rtios Working with rtios Before ontinuing this unit, omplete the Preview on pges of your Student Setion Curriulum links The profiieny strnds Understnding, Flueny, Problem solving nd Resoning, re fully integrted into the ontent of the units. The following ontent desriptions re onsidered in this unit. Compre frtions using equivlene. Lote nd represent positive (nd negtive) frtions nd mixed numbers on number line (ACMNA). Note: negtive frtions nd mixed numbers re onsidered in Unit. Solve problems involving ddition nd subtrtion of frtions, inluding those with unrelted denomintors (ACMNA). Multiply nd divide frtions nd deimls using effiient written strtegies nd digitl tehnologies (ACMNA). Express one quntity s frtion of nother, with nd without the use of digitl tehnologies (ACMNA). Reognise nd solve problems involving simple rtios (ACMNA). Notes Professionl Support

2 Frtions nd Rtios Student Progress Book preview Deep Lerning Kit A rd Writing frtions Intervention tsk B rd Finding the highest ommon ftor Intervention tsk C rd Adding nd subtrting frtions Intervention tsk C rd Finding the lowest ommon multiple Intervention tsk D rd Frtions of n mount Intervention tsk E rd Dividing frtions Intervention tsk F rd Powers nd squre roots of whole numbers Intervention tsk G rd Compring numbers of objets Intervention tsk Professionl support online File.0: Frtion wll Reproduible mster File.0: Frtion wll strips Reproduible mster Diret students to omplete Preview on pges of their Student preview Frtions nd rtios Wht does frtion desribe nd how re frtions used eh dy? To nswer eh question, shde one bubble or write your nswer in the box provided. A Wht frtion of the shpe hs been shded in eh of the following? b A Shde of the hoolte br shown below. A Stte whether eh of the following is proper frtion, improper frtion or mixed number. b Wht is the highest ommon ftor of nd? Whih frtion is equivlent to? Whih frtion is equivlent to? Write these frtions in sending order (smllest to lrgest).,,,, Desribe the number of red jellybens ompred to G proper frtion d the number of blue jellybens. e b Write these frtions in desending order (lrgest to smllest).,,,,, Clulte eh of these. b d + + D 0 Use the digrm below to help nswer the following questions. Wht is of equl to? How mny lots of re in the retngle below? b How mny lots of re in the two retngles below? How mny lots of re in the three retngles below? F Clulte eh of these. b d Questions nd refer to the digrm below. to to to to Kit. Students n write this informtion in the pproprite setion of My lerning nd then tik them off s they re ompleted. Those students who require intervention n omplete ll the nominted rds before ommening the first topi. However, it my be more benefiil to omplete eh rd just prior to ommening the mthing topi in the unit. Fous: To reognise tht the number of equl prts of eh shpe represents the denomintor of frtion nd tht the shded prts represent the numertor; nd to distinguish between proper frtion, n improper frtion nd mixed number proper frtion mixed number proper frtion improper frtion You my need to undertke some expliit tehing so students understnd how frtion is formed. They need to reognise tht the denomintor represents the number of equl prts the whole is divided into nd the numertor represents the required number of prts. or B B B B C,,,,,,,,, D Wht is the lowest ommon multiple of, nd? Ask students to ut out the strips nd fold the first strip into two equl prts. How mny prts re there? This beomes the denomintor s the strip is folded into hlves. The frtion will be. Shde in one-hlf nd omplete the frtion representing the shded portion of the strip. Fold the seond strip into three equl prts. How mny prts re there? This E Eh question reltes to prerequisite G Desribe the number of green jellybens ompred skill for this unit. If students experiene f mixed number b Wht is of equl to? to the number of yellow nd pink jellybens. diffiulty with ny of the questions, to to to to the denomintor. These lists n be used reord the mthing rd to to find the HCF. For exmple for 0 be ompleted (see below). You my like, the ftors of re,, nd nd to dvise students individully of the the ftors of 0 re,, nd 0. The rds they will need to omplete. A Understnding frtions Diret students to omplete A To reinfore this onept File.0 beomes the denomintor s the strip highest number ommon to both lists Alterntively, students n refer to the preview Q rd : Writing frtions if they hd Frtion wll strips n be provided for is folded into thirds. The frtion will be is, so divide both the numertor nd Preview referene rd in Deep Lerning diffiulty with these questions or require n tivity.. Shde in one-third nd then two- the denomintor by to obtin _. more prtie t this skill. thirds, ompleting the frtions representing the shded portions. Also sk students to write frtion to represent the unshded portion of eh strip, so tht they re wre tht they re not lwys looking for the shded omponent nd tht they should red the question refully. Using the orret lnguge, suh s hlves, thirds, qurters nd so on, Frtions nd rtios for eh type of frtion will enble students to understnd tht frtions re different to whole numbers. A rd Writing frtions Fous: To write frtions with the orret numertor nd denomintor Resoures: ruler, oloured penils Students identify the numertor nd the denomintor of frtion. For given digrms, students write frtion to represent the mount of irle or retngle tht is shded. They re guided to first ount the totl number of equl prts in the digrm to identify the denomintor nd then ount the number of required (shded) prts to identify the numertor. Students lso drw their own digrms to represent given frtions. B Equivlent frtions preview Q Fous: To find the HCF for two numbers nd identify equivlent frtions Diret students to omplete B rd : Finding the highest ommon ftor if they hd diffiulty with Q or require more prtie t this skill. You my need to undertke some expliit tehing so students understnd tht for equivlent frtions they re looking for either multiplition or division ftors. When finding n equivlent frtion by dividing, students n mke list of ll of the ftors of both the numertor nd B rd Finding the highest ommon ftor Fous: To find the HCF of two numbers Resoures: oloured penils or highlighters Students find ftor pirs for speifi numbers nd ompre speifi lists to initilly find ommon ftors. They then use this omprison to find the HCF. Professionl Support

3 Frtions nd Rtios Student Progress Book preview C Adding nd subtrting frtions preview Q nd Q Fous: To dd nd subtrt frtions with the sme denomintor, nd to use knowledge of multiples to find the LCM for given group of numbers Diret students to omplete C rd : Adding nd subtrting frtions if they hd diffiulty with Q or require more prtie t this skill. You my need to undertke some expliit tehing so students understnd tht frtion is single entity. Use lnguge suh s hlves, qurters, thirds nd eighths, rther thn sying over two, et. When sking students to dd over nd over, they n be onfused s to wht is required. If you sk students to dd three-qurters nd two-fifths, most students will reognise tht they re two frtions nd be ble to write them ordingly. Diret students to omplete C rd : Finding the lowest ommon multiple if they hve diffiulty with Q or require more prtie t this skill. You my need to undertke some expliit tehing so students understnd wht multiples re nd how to ompre lists of multiples to identify the lowest ommon multiple for two or more numbers. C rd Adding nd subtrting frtions Fous: To dd nd subtrt frtions with the sme denomintor Resoures: oloured penils Students re guided through ddition nd subtrtion of frtions with the sme denomintor, by shding digrms. They develop shortut nd then onsolidte their understnding by performing number of ddition nd subtrtion problems. C rd Finding the lowest ommon multiple Fous: To find the LCM of two or three numbers Resoures: oloured penils or highlighters Students write lists of multiples for speifi numbers nd ompre these lists to initilly find ommon multiples. They then use this omprison to find the LCM. D Multiplying frtions preview Q0 Fous: To find frtion of whole number using visul id Diret students to omplete D rd : Frtions of n mount if they hd diffiulty with this question or require more prtie t this skill. You my need to undertke some expliit tehing so students understnd how to mke use of the visul model. D rd Frtions of n mount Fous: To find the frtion of wholenumber mount Students re guided through problems where they find hlf of whole number nd then qurter of whole number, relting the denomintor of the frtion to finding tht number of groups of equl size. E Dividing frtions preview Q Fous: To divide whole into prts using visul id Diret students to omplete E rd : Dividing frtions if they hd diffiulty with this question or require more prtie t this skill. You my need to undertke some expliit tehing so students understnd wht hppens when dividing whole numbers by frtions. It my be useful for students to hve onrete model to experiment with. Students ould be given opies of File.0: Frtion wll strips. Cut out one strip nd then ut the strip in hlf to model. Red the problem s how mny hlves in? Hve students ount the number of hlves to obtin the result of. Repet with two strips for, three strips for nd four strips for to obtin the results of, nd. Red eh problem s how mny hlves in? Now model,, nd by utting strips into thirds. Agin, red eh problem s how mny thirds in? Further modelling ould be done to show how to divide by qurters, fifths et. Continue until students n identify the link between the whole number nd the denomintor of the frtion. An lterntive strtegy is to ount the number of prts in one whole nd then multiply this nswer by the whole number being divided. Students n refer to File.0: Frtion wll. For exmple, for, students n see 0 tht there re 0 lots of in one. 0 There would be 0 lots of in. 0 So 0 0. E rd Dividing frtions Fous: To find how mny lots of frtion divide evenly into whole numbers using digrms Resoures: ruler Students re guided through problems where they use digrm to divide whole numbers by simple frtions. They omplete questions with visul support nd then questions written in mthemtil sentenes. F Powers nd squre roots of frtions preview Q Fous: To demonstrte n understnding tht the power indites the repeted multiplitions required nd the squre root is relted to squring Diret students to omplete F rd : Powers nd squre roots of whole numbers if they hd diffiulty with this question or require more prtie t this skill. You my need to undertke some expliit tehing so students understnd wht the power or index represents. Refer students to F Powers nd squre roots for dditionl support. F rd Powers nd squre roots of whole numbers Fous: To work with powers of, nd nd find squre roots of perfet squres Students prtise repeted multiplition problems. They lso use ptterns relted to squring number to write the squre root of number. G Understnding rtios preview Q nd Q Fous: To reognise tht quntities of similr items n be ompred when written s rtio, using the word to, nd tht the order is importnt Diret students to omplete G rd : Compring numbers of objets if they hd diffiulty with these questions or require more prtie t this skill. You my need to undertke some expliit tehing using onrete model so students understnd this onept. Ask students to empty their penil ses on the tble. Questions n be posed, suh s: How mny oloured penils? How mny highlighters? How n you ompre oloured penils to highlighters? Highlighters to oloured penils? And so on. Emphsise tht order is importnt. G rd Compring numbers of objets Fous: To desribe the number of given objets, ompred to nother lot of given objets, with order being importnt Resoures: oloured penils Students ount the number of shpes in digrm nd how to write these vlues s omprison, using the term to. The importne of writing the order orretly for the omprison is emphsised. There is no preview question for H Working with rtios. Professionl Support

4 A Understnding frtions Unit Frtions nd rtios Deep Lerning Kit A rd Writing frtions Intervention tsk A rd Mixed numbers nd improper frtions Additionl skill prtie Professionl support online File.0: Retngles Reproduible mster File.0: Cirles Reproduible mster Student online support A etutor A Guided exmple A Test yourself tsk Fous: To write frtion bsed on visul model, nd to distinguish between proper frtions, improper frtions nd mixed numbers Students need to understnd tht when breking whole into prts, the number of prts beomes the denomintor of the frtion. They re guided to mke different visul representtions of the sme mount. This reinfores tht the number of equl prts defines the denomintor. The orienttion of the prts my be different but still equl. Students write frtions to represent the shre of pizz eten by different people. They desribe the differene between proper frtions, improper frtions nd mixed numbers. A Understnding frtions Consider how four people ould hve equl shres of this pizz. Drw retngle to represent the pizz nd show one person s shre. Drw two other possible retngles representing one person s shre of the pizz. Wht frtion of the pizz is displyed in questions nd? A frtion n be desribed s proper frtion or n improper frtion. Whih nme is given to the type of frtion you listed in question? proper frtion Imgine there re now three pizzs nd eh one is ut into eight equl slies, s shown below. The pizzs will be shred between Mry nd her two brothers, Xvier nd Dvid. Mry nd Dvid et slies of pizz between them. Write this number of slies s frtion of one whole pizz. Wht nme (proper or improper) is given to this type of frtion? improper frtion Explin the mjor differene between the type of frtion obtined in question nd the type obtined in question. The numertor in proper frtion is less thn the denomintor nd the frtion is less thn one. The numertor in n improper frtion is greter thn the denomintor nd the frtion is greter thn one. Write the frtion obtined in question in terms of the number of whole pizzs nd the number of slies. Wht nme is given to this type of frtion? mixed number Both re lrger thn one. 0 Wht does n improper frtion nd mixed number hve in ommon? Improper frtions nd mixed numbers re greter How re improper frtions nd mixed numbers different from proper frtions? thn one nd proper frtions re less thn one. Xvier ets eight slies of pizz. Write the number tht represents his shre of the pizzs in two different wys., In your own words, write definition for eh of the following nd provide n exmple tht hs not lredy been used bove. proper frtion b improper frtion mixed number exmple Convert the mixed number to n improper frtion. think write Write the mixed number. Express the mixed number s the sum of whole number nd frtion. + _ Tke note of the denomintor (in this se it is ) nd write eh whole number s seven sevenths. + _ + _ Add ll of the sevenths nd write the nswer. e e exmple Convert the improper frtion to mixed number. think write Write the improper frtion. As eighths mke whole, group eighths into three lots of eighths nd eighth left over. + _ + _ + Rewrite eh lot of eighths s whole number _ Write the totl s whole number nd frtion. _ key ides A frtion is prt or portion of whole. It n be written s one whole number (numertor) over nother (denomintor). The two numbers re seprted by horizontl line (vinulum). Proper frtions hve numeril vlue tht is less thn. The numertor in proper frtion is smller thn the denomintor. the numertor shows the number Some exmples re:,,. of equl prts out of the whole vinulum Improper frtions hve numeril vlue tht the denomintor shows the totl is greter thn. The numertor in n number of prts in the whole improper frtion is lrger thn the denomintor. Some exmples re:,,. Mixed numbers lso hve numeril vlue tht is greter thn. They ontin whole number nd proper frtion omponent. Some exmples re:,,. Whole numbers n be written s frtions. For exmple,,,,. Wht frtion of the digrm is shded in eh of the following? b d Wht frtion of eh digrm in question is not shded? Rule up tble with three olumns. Title the olumns Proper frtions, Improper frtions nd Mixed numbers. Ple eh of these frtions into the pproprite olumn:,,,,,,,,. proper frtions:,, Convert eh of these mixed numbers to improper frtions. improper frtions:,, mixed number:,, b d e 0 f Convert eh of these improper frtions to mixed numbers. b d 0 e f 0 A Understnding frtions As tsk to be undertken either in lss or t home, diret students to omplete the etutor, Guided exmple nd Test yourself. POTENTIAL DIFFICULTY Some students my not reognise tht there is link between the number of equl prts nd the denomintor of the frtion. Deep Lerning Kit A rd Writing frtions Fous: To write frtions with the orret numertor nd denomintor Resoures: ruler, oloured penils Students identify the numertor nd the denomintor of frtion. For given digrms, students write frtion to represent the mount of irle or retngle tht is shded. They re guided to first ount the totl number of equl prts in the digrm to identify the denomintor nd then ount the number of required (shded) prts to identify the numertor. Students lso drw their own digrms to represent given frtions. A rd Mixed numbers nd improper frtions Fous: To onvert between mixed numbers nd improper frtions Students re guided to use visul model to find mixed number. They identify the denomintor nd the numertor of the frtion by ounting prts. Additionl prtie questions similr to the Now try these questions re lso provided. Some possible questions to sk: Convert _ to n improper frtion.( whole lss Ensure students understnd tht Students my find the following If students require dditionl support, nd the other demonstrting this ) Convert _ the number of prts of the whole is reproduible msters helpful when diret them to A rd. frtion on retngle or irle. Inlude to mixed number. (0 Fous: To onsolidte student ) the denomintor of the frtion nd ompleting onversions: File.0: improper frtions nd mixed numbers Wht frtion hs this shpe been understnding of proper frtions, Q llows students to lssify different the number of required prts is the Retngles nd File.0: Cirles. for whih more thn one retngle or divided into? (fifths) improper frtions nd mixed numbers, types of frtions. numertor. irle needs to be used. Ensure students nd to onvert between improper Diret students to the key ides. You Q nd Q llow students to onvert strt by using the denomintor to frtions nd mixed numbers Ensure students understnd the my like them to opy this summry. between improper ftions nd differene between proper frtions, identify the number of equl prts tht mixed numbers. Wht frtion of the shpe hs been After students omplete the Disover improper frtions nd mixed numbers. whole retngle or irle needs to be shded? ( _ divided into. You my like to disuss Fous: To write nd lssify ) tsk, onsolidte their understnding. Demonstrte A etutor, or diret Wht frtion of the shpe hs not been Ensure students re ble to orretly frtions nd to onvert between the onept tht s the number of students to do this independently. shded? ( equl prts inreses, the size of eh _ improper frtions nd mixed numbers identify the numertor nd the ) Whih is lrger, or? ( _ denomintor of frtion. Reinfore prt dereses. ) tht the vinulum is written horizontlly nd not s slsh. numertor denomintor vinulum Diret students to the exmples. Exmple shows how to onvert mixed number to n improper frtion. Exmple shows how to onvert n improper frtion to mixed number. In Q nd Q, students identify the numertor nd the denomintor of frtions. For those students experiening diffiulty with these questions, intervention through smll group tehing of the onept my be neessry. Provide students with opies of File.0: Retngles nd File.0: Cirles. Hve students work in pirs, with one person suggesting frtion b d For dditionl prtie, diret students to A rd. 0 whole lss Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

5 A Understnding frtions Unit Frtions nd rtios Deep Lerning Kit A rd Apple pies Problem-solving tsk A rd Sndwih frtions Problem-solving tsk A rd Cho Blok design Problem-solving tsk Professionl support online File.0: Retngles Reproduible mster File.0: Cirles Reproduible mster File.0: Number line Reproduible mster whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: Write proper frtion. Lbel the numertor, denomintor nd the vinulum. numertor (Possible nswer: _ vinulum) denomintor Wht type of frtion is? Explin. (improper frtion s the numeril vlue is more thn ; the numertor is lrger thn the denomintor) Cn be hnged into nother type of frtion? If so, wht type? (yes, mixed number) Wht will hnge into when onverted? ( ) Wht type of frtion is _? Explin. (mixed number; whole number nd proper frtion omponent) Estimte wht frtion of this nt s whole body length is mde up of its hed. The Austrlin stte nd territory flgs re shown t right. Use this mp of Austrli to estimte the required frtions in the Stte wht frtion of the flgs inlude the Southern Cross in following questions. their design. Austrlin Cpitl Territory New South Wles Wht frtion of Austrli is the stte of Vitori? 0 b Stte wht frtion of the flgs inlude flower in their design. b Wht frtion of Austrli is the stte of Tsmni? 00 Stte wht frtion of the flgs inlude blue in their olours. Wht frtion of Austrli is the stte of Queenslnd? d Stte wht frtion of the flgs do not inlude nimls in their design. Whih digrm orretly displys tht d Wht frtion of Austrli is the stte of New South Wles? 0 of the shpe hs been shded? Northern Territory Queenslnd e Wht frtion of Austrli is the stte of Western Austrli? Provide reson for your seletion. f Wht frtion of Western Austrli is the stte of A B C New South Wles? g Wht frtion of South Austrli is the stte of Tsmni? B. The retngle is divided into three equl setions nd one of the setions is South Austrli Tsmni shded. The other two retngles re not divided into three equl setions. Write the frtion tht is desribed by these sttements. orret nswers on quiz ontining questions b the number of months in yer onsisting of dys Copy this digrm of jug into your workbook. The sle indites mesurements in litres. $ hnge from $0 note Vitori Western Austrli 0 Indite the wter level for mesurement of litres on your jug. There re 0 minutes in one hour. Wht frtion of n hour is: (Hint: first onvert the improper frtion to mixed number.) minutes? b minutes? 0 0 minutes (s n improper frtion)? d minutes (s mixed number)? 0 Repet the method used in question to indite the wter level on 0 0 jug for eh of these mesurements. Georgin, Roisin nd Vness hve orgnised lunh for their lss nd deide to order lrge pizzs. Eh pizz will be ut into eight equl slies. The girls estimte tht eh person will et two slies nd there will be litres b litres litres mximum of people. Convert both frtions to improper frtions nd use the symbol < (is less thn) or > (is greter thn) to mke true sttement. Write the number of pizzs required s: i n improper frtion ii mixed number. b d > < > < b Will eh person reeive the estimted two slies of pizz if five lrge pizzs hve been ordered? The number line below hs been divided into four equl intervls from 0 to. Eh intervl hs been further No, beuse there will only be 0 slies. subdivided into three equl prts. The girls budget does not llow them to order extr pizzs. Wht is the minimum number of equl slies eh pizz n be ut into, to ensure everyone reeives two slies? nine slies d Wht frtion of the pizz will be left over if the pizzs re ut in the wy suggested in prt? 0 0 e On the dy of the lunh, three people were wy nd of the pizzs were eten. If eh of the pizzs were Wht frtion of n intervl is eh prt? ut into eight slies, how mny slies were eten? slies b Lbel eh mrk on the sle with the vlue it represents. 0 Consider the imge of the Colosseum shown here. Drw number line from 0 to nd mrk the position of eh of these frtions. By estimtion, how mny levels high is the b d e tllest point of the Colosseum? Write down the vlues shown for d on the number line. b Wht frtion of the height of the Colosseum is one level? b d or or or 0 or Wht frtion of the levels ontin rhes? Gemm nd Cindy disgree on whih frtion is lrger, or. Desribe how number lines n be used to ompre the frtions nd deide whih is lrger. reflet Explin how you n ompre the size of frtions. A Understnding frtions students with File.0: Number line. It shows mjor intervl from 0 to (but the mjor mrks n be hnged to ny whole numbers). For eh of the two frtions being ompred, hve students use ruler to divide the line segment into the required number of minor intervls. For exmple, for, divide the line segment into four equl intervls, nd for, divide nother number line segment into five equl intervls. Mrk the required frtion on eh number line nd ompre. If students re still experiening diffiulty, it my be esier to demonstrte whih frtion is lrger using File.0: Retngles. Hve students shde eh required frtion on seprte retngle nd then ompre the two retngles. Repet with different pirs of frtions. Deep Lerning Kit A rd Apple pies Fous: To determine nd write frtions ssoited with shring pple pies Students the different possibilities of shring three pple pies eqully mong six, twelve, eighteen nd nine people. They need to onsider wht frtion to ut the pies into nd wht frtionl quntities eh person ets. A rd Sndwih frtions Fous: To the reltionship between proper nd improper frtions Students different lof designs nd sndwih uts using frtions. As n extension they design lof of bred. A rd Cho Blok design 0 Fous: To investigte the reltionship between frtions, ftors nd division Cn _ be hnged into nother Q nd Q involve writing frtions to Q involve estimting frtion. inequlity sign with the open side should first identify the size of the Students identify how mny equl-sized type of frtion? If so, wht type? desribe different situtions. Students Ressure students tht similr nswer towrds the lrger frtion. minor intervl (one-seventh). smller prts of blok of hoolte n be (yes, improper frtion) will need to understnd tht eh flg to tht given in the Answers setion is In Q, students lbel given number Q requires students to use number shred mong different groups of people s 0 Wht will _ hnge into when onverted? ( ) questions Fous: To pply understnding of frtions to different ontexts For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. in Q represents one out of eight flgs. In Q, students will need to understnd tht frtions re written to represent equl prts of visul representtion. Q nd Q require writing frtion from worded sttement. Students need to orretly identify eh numertor nd denomintor. Q involves the division of pizz into slies. File.0: Cirles my be useful in visulising the problem. eptble. Q nd Q0 involve onverting between mixed numbers nd improper frtions to mesure quntity for ooking. In Q, students ompre mixed numbers nd improper frtions. They my need to be reminded to onvert the frtions to improper frtions before ompring their size. A helpful hint for students n be to insert the 0 line by onsidering the size of the minor intervls. Eh minor intervl represents one-third. In Q, students represent rnge of frtions on number line similr to tht shown in Q. They my need to be reminded tht sine ll of the frtions hve denomintor of, eh minor intervl should represent one-fifth. Q requires students to identify the frtionl vlues represented. They lines to ompre two frtions. For students experiening diffiulty with this onept, strtegies re disussed in the smll group setion tht follows. smll group Fous: To ompre frtions nd to represent frtions on number line Students who experiene diffiulty in ompring frtions nd representing frtions on number line will need to be expliitly shown how to do this. Provide frtion of the originl blok. They onsider the benefits of ertin blok designs, nd design their own blok of hoolte. whole lss: reflet Possible nswer: Frtions n be ompred using number lines. Alterntively, they n use visul model for eh frtion nd ompring them side by side. Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

6 B Equivlent frtions Unit Frtions nd rtios Deep Lerning Kit B rd Finding the highest ommon ftor Intervention tsk B rd Simplifying frtions Additionl skill prtie Professionl support online File.0: Frtion wll Reproduible mster Student online support B etutor B Guided exmple B Test yourself tsk Fous: To build on student understnding tht different frtions n represent the sme mount nd tht these types of frtions re lled equivlent frtions Students need to reognise tht the kes re ll the sme size; the differene is the number of equl prts tht the kes hve been divided into. They re guided to write frtions tht represent the ied portions of eh individul ke. Through this representtion they _ 0 0. Students mke the onnetion by looking t visul models tht equivlent frtions re identil in vlue they re different representtions. whole lss Fous: To onsolidte student Some possible questions to sk: understnding tht different frtions wrpping; there re hlves. Students n equivlent frtion, by multiplying In Q, students find equivlent Some students my experiene diffiulty Do equivlent frtions hve the sme n represent the sme mount nd to lerly see tht _ the numertor nd the denomintor frtions, by seleting multiplition in reognising ftors tht hve been or different vlues? (sme) exmine how equivlent frtions n. Brek the groups of six into hlf gin to produe qurters. by the sme ftor. Exmple shows ftor. used. A sound knowledge of the times How n n equivlent frtion be be found _. And so on. After students omplete the Disover tsk, onsolidte their understnding. Ensure tht ll students hve rrived t the understnding tht different frtions n represent the sme mount. Conrete exmples re exellent for exploring equivlent frtions. For exmple, pk of yoghurt or pudding in whih there re two bloks of six wrpped in rdbord shows tht one whole is. Open the rdbord 0 B Equivlent frtions Look t eh of the kes below. Iing hs been spred over some of the top of eh ke nd the ke hs been ut into equl piees. Let s look t the mount of eh ke tht hs been ied. ke A ke B ke C ke D Wht do you notie bout the mount of ke Cke Totl Number of Frtion of tht hs been ied in eh se? number of piees tht the ke tht The sme mount of ke is ied. equl piees hve been ied hs been ied b Copy nd omplete this tble for eh ke. Without drwing digrm, explin how the A Totl number of next ke in the series would look in terms of B 0 equl piees:, 0 the totl number of equl piees, the number of C number of piees tht hve been piees tht hve been ied nd the frtion of the ke tht hs been ied. D 0 ied: 0, frtion 0 of the ke tht hs been ied: 0 The frtions representing the mount of eh ke tht hs been ied re lled equivlent frtions. They represent the sme vlue. d How ould you show tht two frtions re equivlent? Express them with the sme denomintor nd then ompre the numertors. Consider the retngle to the right., divide the numertor One possible nswer Wht frtion of the retngle is shded? nd the denomintor is ; multiply the b Write the frtion in simpler wy nd explin how this ws done. by. numertor nd the denomintor by. Write n equivlent frtion for this frtion using lrger denomintor nd explin how this ws done. Now onsider this next retngle. No, sine the numertor nd the denomintor hve no ommon ftors (other Wht frtion of the retngle is shded? thn ) nd hene the frtion nnot b Cn this frtion be written in simpler wy? Explin your nswer. be simplified. Crete two equivlent frtions to represent the mount of shding in this retngle. Some possible nswers re: 0, 0 Does the numeril vlue of frtion hnge one n equivlent frtion is reted? Explin. No; equivlent frtions hve the sme vlue. exmple think write Fill in the Write the two frtions nd determine the 0 gp to mke reltionship between the two numertors. Sine equivlent 0, the numertor of the seond frtion frtions for is obtined by multiplying the numertor of the 0 0 first frtion by.. To find the equivlent frtion, use the sme 0 reltionship (multiply by ) to obtin the denomintor of the seond frtion. 0 Multipks of mini erels re lso good to use s there re eight prts in the whole. Brs of hoolte nd bgs of lollies, suh s Smrties, n lso reinfore the onept of equlsized groups being mde up of smller omponents. Demonstrte B etutor or diret students to do this independently. Diret students to the exmples. Exmple shows how to rete n how to simplify frtions, by dividing numertor nd denomintor by the sme ftor. Diret students to the key ides. You my like them to opy this summry. Fous: To find equivlent frtions Q nd Q involve writing frtions represented by visul model. e e exmple think write Write eh frtion in its simplest form. b Write the frtion nd determine the highest ommon ftor (HCF) of the numertor nd the denomintor. HCF. Divide both the numertor nd the denomintor by, the HCF. This proess is lso known s nelling. Write the nswer. b Write the mixed number. Determine the HCF of the numertor nd b the denomintor of the frtion omponent. HCF. Leve the whole number s it is nd divide both the numertor nd the denomintor by, the HCF. Write the nswer. key ides Equivlent frtions hve the sme numeril vlue. An equivlent frtion n be formed by multiplying both the numertor nd the denomintor of frtion by the sme vlue (see Exmple A). Dividing both the numertor nd the denomintor of frtion by the sme vlue lso produes n equivlent frtion (see Exmple B). This proess is lled simplifying or nelling. A frtion is in its simplest form one the numertor nd the denomintor hve been divided by the highest ommon ftor (HCF). Mixed numbers n be simplified by keeping the whole number the sme nd simplifying the frtion omponent. Stte wht frtion of eh retngle is shded. b Stte wht frtion of eh retngle in question is not shded. b Crete two equivlent frtions to represent eh of the frtions found in: i question ii question. i,, b,, 0 0,, ii 0,, b,,, 0 0, Fill in the gps to mke equivlent frtions for eh of the following. (Hint: hnge mixed numbers to improper frtions first.) b d 0 0 e f g h 0 0 i Write eh frtion in its simplest form. 0 b d 0 e f or or g 0 h 0 i j k l Q llows students to find the multiplition ftor nd then use this ftor to find the missing omponent of n equivlent frtion. Q llows students to find equivlent frtions by dividing both the numertor nd the denomintor by the sme ftor. For those students experiening diffiulty with these questions, intervention through smll group tehing of the onept my be neessry. Exmple A Multiplying by is the sme s multiplying by, so the vlue of the equivlent frtion is the sme. Exmple B tbles is helpful. Students without this knowledge might find it esiest to use lultor to find the ftor required. B Equivlent frtions When nelling to simplify, students n list ll of the ftors of both the numertor nd the denomintor nd use these lists to find the HCF. Reinfore the onept tht wht is done to the top of frtion is done to the bottom of frtion to keep the vlue the sme (or equivlent). Diret students to B rd for further support. File.0: Frtion wll n be used s referene when finding equivlent frtions. For dditionl prtie, diret students to B rd. As tsk to be undertken either in lss or t home, diret students to omplete the etutor, Guided exmple nd Test yourself. POTENTIAL DIFFICULTY Some students my hve diffi ulty identifying equivlent frtions. Emphsise tht they will be looking for multiplition or division ftors, not ddition nd subtrtion ptterns. Deep Lerning Kit B rd Finding the highest ommon ftor Fous: To find the HCF of two numbers Resoures: oloured penils or highlighters Students find ftor pirs for speifi numbers nd ompre lists to find the HCF. B rd Simplifying frtions Fous: To write frtions in their simplest form Resoures: oloured penils Students use visul model to link equivlent frtions. They use the HCF to simplify frtions, nd rete equivlent frtions. Additionl prtie questions similr to the Now try these questions re lso provided. whole lss formed? (multiply or divide the top nd the bottom of frtion by the sme ftor) Do frtions only hve one or two equivlent frtions? (no, there re mny possibilities for eh frtion) Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

7 B Equivlent frtions Unit Frtions nd rtios Deep Lerning Kit B rd Quilt design Investigtive tsk B rd In step with time Problem-solving tsk B rd Clss distintion Problem-solving tsk Professionl support online File.0: Frtion wll Reproduible mster File.0: Frtion wll Reproduible mster File.0: Quilt design Reproduible mster whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: Wht n be sid bout the numeril vlue of equivlent frtions? (sme) Wht mthemtil opertions do you use when forming equivlent frtions? (multiplition or division) Wht is possible equivlent frtion for _? ( _,,, 0 0, ) questions Fous: To pply understnding of equivlent frtions to ompre different quntities For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. smll group Fous: To ompre frtions by finding equivlent frtions with the sme denomintor Students who experiene diffiulty in progressing independently to ompring frtions my need to be expliitly shown how to do this. It is esier to ompre frtions when they hve the sme denomintor. To identify the denomintor, ensure students n find the lowest ommon multiple of the individul denomintors (provide C rd : Finding the lowest ommon multiple). As visul model, frtion wll n be used to find n equivlent frtion to eh, with this denomintor. See File.0: Frtion wll or File.0: Frtion wll. Try this with _ nd _ Deep Lerning Kit or _ nd _. B rd Quilt design Fous: To investigte simplifying to equivlent frtions in the rel-life ontext of the tesselltions in quilt design Resoures: oloured pens nd penils, File.0: Quilt design Students identify nd simplify frtions of ertin olours for tessellted quilt design. They ompre different quilt designs nd design their own pttern subjet to frtionl onstrints. B rd In step with time Fous: To identify equivlent frtions within the ontext of frtions of n hour using n nlogue lok Students re guided through series of Q nd Q involve writing frtions sttements tht require explortion of wht to desribe situtions. Students need Q requires students to onvert suh s of frtions nd then find equivlent Q nd Q involve finding possible frtion of n hour ertin time period to tke re to use the orret vlues frtions into mixed numbers in order lest-ommon-multiple-tool.html to frtions for eh so tht the list of denomintors of frtions with given represents. for the numertor nd the denomintor. to ompre them. ssist them. Multiplying the two given frtions n be ordered. In Q, ommon denomintor. Knowledge of B rd Clss distintion These frtions re then simplified using the HCF. Q require students to find equivlent proper nd improper frtions. In Q, students need to onvert between mixed numbers nd improper frtions. There re different nswers possible, depending on the multiplition ftor used. Q 0 involve students representing different frtions on number line so they n be ompred Convert these frtions to equivlent frtions using the LCD nd then rrnge them in sending order.,,,,,,,,,,,, Lrni sored of her tem s gols. Wht frtion of the tem s b Drw number line nd mrk the position of the equivlent frtions. Wht length is represented by the gols did Lrni sore? Express your nswer in simplest form. intervl between eh mrk on the sle? 0 0 Ahmed spent minutes of 0-minute footbll gme on the field. Write the originl frtion below its equivlent frtion on the number line. Wht frtion of time, in simplest form, did he spend off the field? List eh set of frtions in sending order (smllest to lrgest). Write four equivlent frtions for.,,, 0,,,,, b,,,,,,,,,, d,,, 0, 0, 0 Crete three proper frtions tht re equivlent to. 0,, 0,,,,,,,,,,,,,,, 0,, 0,, 0, List eh set of frtions in desending order (lrgest to smllest). Crete three improper frtions tht re equivlent to.,,,,, 0,, b,,,,, 0,,,,, d,,,,, Crete three mixed numbers tht re equivlent to.,,,, 0,,,,,,,,, 0,,,,,,,,, Write three frtions tht re smller thn. b Write three frtions tht re lrger thn Write frtion tht is smller thn.,,.,, Write three frtions between nd. d Write three frtions between nd b Write frtion tht is greter thn with the sme denomintor s tht in prt.,,.,, 0 e Write three frtions between 0 0 Ple these frtions on number line nd omment on their position with respet to. nd. f Write three frtions between nd., 0, 0,, When looking t the number line, frtions smller thn will be positioned to the left of. Frtions lrger thn will Drw number line nd mrk the position of eh of these frtions. be positioned to the right of. Philip ws ompring two of his test results. He sored on the first test nd 0 on the seond. On whih test did Philip perform better?, 0,,,,,, seond test b Using the number line, list the frtions in sending order (smllest to lrgest). Wht do you notie? 0 Zh nd Melnie re ompring their results fter n fternoon of rhery. Explin how to order frtions tht hve the sme denomintor.,, 0,,,,, ( ), The numertors re in From ttempts, Zh sored 0 bullseyes. Melnie sored bullseyes sending order. Compre the numertors nd ple them in sending order (from smllest to lrgest). from 0 ttempts. Who hit the bullseye on the rhery trget with more 0 Copy the number line on the right nd divide eh intervl ury? Justify your nswer. Melnie; into three equl prts. Lbel eh of these prts. ompred to 0 The ommon denomintor of three different frtions is. Wht ould the b Convert eh of these frtions to mixed number, simplifying where neessry, nd then mrk its position denomintors of the frtions be?,, on the number line. i 0 The ommon denomintor of three different frtions is. Wht ould the ii iii iv v vi denomintors of the frtions be?,, Comment on the position of the lrgest frtion ompred to the other frtions. Lrgest frtion is furthest right. d Comment on the position of the smllest frtion ompred to the other frtions. This digrm displys setion of frtion wll. Frtion wlls n be used to obtin equivlent frtions or Smllest frtion is furthest left. to ompre frtions. Convert both frtions to mixed numbers nd use the < (is less thn) or > (is greter thn) symbol to mke Wht do you notie bout eh true sttement. horizontl setion of the wll? > b < > d < Eh horizontl strip is equl to one unit. 0 b Wht frtion is shown by eh segment e < in the seond setion down? f > g < h > Write eh of these frtions in simplest form nd list those tht re equivlent frtions. Wht frtion is shown by eh segment in the third setion down?, 0,,,, 0,,, 0 0,,,,, 0 0, The lowest ommon d Wht do you notie bout eh segment Determine the lowest ommon denomintor (LCD) of the frtion of prtiulr horizontl setion? pir denomintor is the lowest nd. (Hint: find the lowest ommon multiple of nd.) Eh segment of prtiulr horizontl strip is equl. ommon multiple of e Use the frtion wll provided to list the b Convert eh of the originl frtions to equivlent frtions using the numbers in the set of equivlent frtions for. the LCD s the denomintor., 0,, 0, denomintors. f Use the frtion wll provided to list Stte whih frtion of the given pir is lrger. four frtions tht re lrger thn.,,, Using the tehnique outlined in question, stte whih frtion of eh given pir is lrger. g Use grph or grid pper nd follow the teher s instrutions to rete your own frtion wll. nd b 0 nd nd d nd h Work in pirs nd use the frtion wll to design reflet e nd 0 f 0 nd g nd h Wht lultions ould be 0 0 nd three questions you ould sk your lssmtes. performed using the frtion wll? Q sks students to simplify frtions so tht they n identify those frtions tht re equivlent. In Q, students find the lowest ommon denomintor for two given frtions. They my need to be reminded to the multiples for eh of the denomintors to identify the lowest ommon multiple. Knowledge of multiplition fts (times tbles) will be helpful. Alterntively, students my like to use n Internet lultor denomintors n provide ommon denomintor, but this my not be the lowest ommon denomintor. Q llows students to find equivlent frtions with ommon denomintors to ompre given frtions. Strtegies used in Q will be helpful. For students still experiening diffiulty with this onept, strtegies re disussed in the smll group setion tht follows. In Q, students identify the lowest ommon denomintor for given group students lso ple these vlues on number line. Q is n open, thinking tsk for whih students use their understnding of equivlent frtions to write frtions tht re lrger or smller thn given frtion or re between two given frtions. Q nd Q0 re pplition questions where students ompre results hieved. Equivlent frtions with the sme denomintor must be found first. ftors will be helpful. A website suh s lultors/mth/ommonftors.php my be useful for some students. Q s frtion wll. File.0: Frtion wll n be provided to students for this question. Students re guided to understnd wht eh segment in the wll represents nd how it n be used to find equivlent frtions. They use the frtion wll to design three questions for lssmtes. B Equivlent frtions Fous: To identify nd drw ertin physil hrteristis of lss of students ording to given numeril onstrints Resoures: oloured penils Students drw ertin physil hrteristis of lss of students ording to given informtion. Equivlent nd simplified frtions re expeted s nswers. overmtter Professionl Support

8 C Adding nd subtrting frtions Unit Frtions nd rtios Deep Lerning Kit C rd Adding nd subtrting frtions Intervention tsk C rd Finding the lowest ommon multiple Intervention tsk C rd Adding nd subtrting frtions Additionl skill prtie Professionl support online File.0: Retngles Reproduible mster File.0: Cirles Reproduible mster File.0: Frtion wll Reproduible mster File.0: Frtion Wll Reproduible mster Student online support C etutor C Guided exmple C Test yourself tsk Fous: To dd or subtrt frtions Students re guided through ddition nd subtrtion of two frtions with the sme denomintor using visul model. They re then guided through the ddition nd subtrtion of frtions with different denomintors. Students n use File.0: Retngles nd File.0: Cirles to rete their own visul models. whole lss Diret students to the exmples. Exmple If students find these questions diffiult, intervention through smll group For dditionl prtie, diret students ) Fous: To onsolidte student shows how to subtrt mixed numbers diret them to C rd. tehing of the onept my be to C rd. Is it possible to simply subtrt from? understnding of the ddition nd with the sme denomintor. Exmple subtrtion of frtions In Q, the denomintors remin the neessry. Why? (no, ommon denomintor needs As tsk to be undertken either in shows how to dd frtions with different to be found) sme, but students re dding nd lss or t home, diret students to denomintors. subtrting mixed numbers. omplete the etutor, Guided exmple Complete this subtrtion, showing ll Diret students to the key ides. You nd Test yourself. working. ( _ _ ) my like them to opy this summry. After students omplete the Disover tsk, onsolidte their understnding. Ensure tht students understood tht the denomintors of the frtions to be dded or subtrted need to be the sme. Demonstrte C etutor or diret students to do this independently. In this tsk, we will onsider how to dd nd subtrt frtions using shded retngles to represent the frtions. First let s develop rule for dding two frtions. C Adding nd subtrting frtions + The denomintors re the sme. The sum of the numertors of the frtions tht re being dded equls the numertor of the nswer. Wht do you notie bout the denomintor of eh frtion in the digrm? Wht do you notie bout the numertors of the frtions being dded nd the numertor of the nswer? Copy nd omplete this digrm to represent. Wht do you notie when subtrting frtions tht hve the sme denomintor? Keep the denomintor nd subtrt the numertors. This digrm shows the ddition of two mixed numbers. Wht do you notie? Convert them to improper frtions. Chek tht they re written with ommon + denomintor nd dd the numertors. + Fous: To dd nd subtrt frtions Q involves dding or subtrting frtions with the sme denomintors. + or Copy nd omplete this digrm to represent. Write sentene to explin how to dd or subtrt frtions tht hve the sme denomintor. When dding nd subtrting frtions with the sme denomintor, keep the denomintor nd then dd or subtrt the numertors. Prior to dding Write sentene to explin how to dd or subtrt mixed numbers tht hve the sme denomintor. nd subtrting mixed numbers, onvert them to improper frtions. Chek tht they re written with ommon denomintor nd then dd or subtrt the numertors. Consider this digrm. + + Comment on the denomintor of eh frtion in the digrm. + + The denomintors re different. However, in this exmple, one is multiple of the other. 0 Use the digrm in question to explin wht must be done to dd frtions with different denomintors. Convert the frtions so tht they hve ommon denomintor. Drw digrm similr to the one in question to represent. Write sentene to explin how to dd or subtrt frtions tht hve different denomintors. When dding nd subtrting frtions with different denomintors, first find the equivlent frtions with the lowest ommon denomintor. Then dd or subtrt the numertors. exmple think write Clulte. Write the frtion problem. Convert eh mixed number to n improper frtion. Chek if the denomintors re the sme. They re both. Subtrt the numertors. The denomintor stys the sme. Convert the improper frtion to mixed number. Q involve finding ommon denomintor before ompleting the ddition or subtrtion. For those students struggling to find the lowest ommon multiple to identify the LCD, diret them to C rd. For those students experiening diffiulty with these questions, + exmple think write Clulte +. Write the frtion problem. + Find the LCD. Tht is, determine the lowest ommon multiple of the denomintors (). Write eh frtion s n equivlent + frtion with denomintor of. 0 + Add the numertors together. The denomintor stys the sme. Convert the improper frtion to mixed number. key ides To dd (or subtrt) frtions tht hve the sme denomintor, keep the denomintor the sme nd dd (or subtrt) the numertors. To dd (or subtrt) frtions tht hve different denomintors, first find the equivlent frtions with the lowest ommon denomintor (LCD) nd then dd (or subtrt) the numertors. Before dding or subtrting mixed numbers, onvert them to improper frtions nd hek tht they re written with ommon denomintor. When performing lultions, if the nswer is n improper frtion it should be onverted to mixed number. Clulte eh of these. + b + + d + + or e f g h 0 or e Clulte eh of these. + b + d e + f g + h 0 0 i + + j k + l + e Clulte eh of these. + or b + d 0 or e + f g + h 0 i j + 0 k + l 0 0 m n + o + 0 p or 0 For eh pir of frtions to be dded or subtrted, write the LCD tht n be used to perform the lultion. + b + d Perform the lultions shown in question. or b d It my be benefiil for some students to be shown how to use visul model suh s frtion wll to dd or subtrt two frtions. Refer to File.0: Frtion wll or File.0: Frtion wll File.0: Retngles templte nd File.0: Cirle templte n be provided. or or C Adding nd subtrting frtions Deep Lerning Kit C rd Adding nd subtrting frtions Fous: To dd nd subtrt frtions with the sme denomintor Resoures: oloured penils Students re guided through ddition nd subtrtion of frtions with the sme denomintor by shding digrms. They develop shortut nd then onsolidte their understnding by performing number of ddition nd subtrtion problems. C rd Finding the lowest ommon multiple Fous: To find the LCM of two or three numbers Resoures: oloured penils or highlighters Students write lists of multiples for speifi numbers nd then ompre lists to find the LCM. C rd Adding nd subtrting frtions Fous: To dd nd subtrt two frtions with different denomintors Resoures: oloured penils Students find the lowest ommon denomintor (by finding the LCM of the denomintor) for two frtions with different denomintors. They then write eh frtion s n equivlent frtion with the sme denomintor nd dd or subtrt. Additionl prtie questions similr to the Now try these questions re lso provided. whole lss Some possible questions to sk: Is it possible to simply dd nd? Why? (yes, sme denomintors) Wht will the nswer be? Is there nother wy to write this frtion? ( _ Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

9 C Adding nd subtrting frtions Unit Frtions nd rtios Deep Lerning Kit C rd It just ll dds up Ativity C rd Puzzling pumpkins Problem-solving tsk C rd Conotion ups Problem-solving tsk C rd Egyptin frtions Investigtive tsk Professionl support online File.0: It just ll dds up Reproduible mster whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: Desribe the proess if I wnt to dd two frtions with the sme denomintor. (keep the denomintor the sme nd dd the numertors) Desribe the proess if I wnt to subtrt two frtions with different denomintors. (find equivlent frtions with the LCD nd subtrt the numertors) questions Fous: To pply understnding of ddition nd subtrtion of frtions A group of friends orgnise to th up for lunh nd deide to order pizz. Eh person estimtes wht frtion of pizz he or she n et. Luis n et, Deln, Nis, Toms nd Christopher of pizz. Den ompleted 0 km run in of n hour while Mrk ompleted How mny full pizzs will need to be ordered? b Wht frtion of pizz will be left over? the run in of n hour. Lil is plnning to trvel overses t the end of the yer nd needs to sve money so Whih runner hd the fster time? Den she devises budget. She hs deided to llote of her weekly py to living expenses, b Wht ws the differene between the two times? of n hour to rent, to reretion nd the reminder to svings. Clulte wht frtion of Lil s py will go towrds living expenses, m Clulte eh of these. 0 rent nd reretion. + + b b Wht frtion of her py will go towrds svings? Explin your m d nswer. + e + 0 f will go towrds svings. 0 To dd or subtrt mixed numbers tht hve different 0 The totl length round the outside of the tringle t right is m. denomintors, we first onvert eh mixed number to n The length mesurements of two of the sides re known. improper frtion nd then ontinue s we hve for erlier Wht is the length of the unknown side? 0 m ddition or subtrtion problems with different denomintors. Json heked the petrol guge on his motorbike t the beginning of Consider +. the dy nd it indited the tnk ws 0 full. At the end of the dy, Write the ddition problem nd then onvert eh mixed the petrol guge indited the tnk ws / full. Wht frtion of number to n improper frtion. + petrol ws used? b Find the LCD to write the frtions s equivlent frtions While on mp, students were required to omplete km bushwlk / nd then ontinue s before to find the result. 0 over three dys. They overed + km on the first dy nd km on the Clulte eh of these. seond dy. Wht distne will they need to wlk on the third dy? + b + d 0 km 0 e b How muh further did they wlk on the first dy thn on the seond? + f + g 0 h 0 km i Clulte eh of the following nd omment on wht you observe. + j + k l The frtion is multiplied by i ii iii iv eh time nd the nswer Clulte eh of these. is the finl frtion of the problem. b Predit the nswer for the next two lultions in the pttern. Test your predition. + + b d + nd 0 Complete eh of the following by filling in the gps. Explin the pttern you observe. e + f 0 + g + + h +,,,,,, b 0,,,,,, A ftory worker is employed to mke t-shirts. On verge, eh t-shirt tkes (or ) 0 ; inreses by eh time ; inreses by eh time minutes for utting,,,, 0,,, d 0, 0,,,, minutes for sewing nd minutes for finishing. Wht is the totl time required to mke one t-shirt? ; dereses by (or ) eh time (or ); dereses by eh time 0 Rmon hs prt-time job in fé. This tble displys the hours she worked in one week of the minutes Find the frtion tht is hlfwy between eh of these pirs of frtions. shool holidys. 0 nd 0 b nd or nd d nd 0 0 Copy the tble into your book. Dy Number of Number Number e nd f nd g nd h 0 nd hours nd of minutes of hours b Complete the seond nd third olumns of 0 minutes worked worked worked the tble by onverting the hours nd minutes Find pir of frtions with different denomintors tht: Mondy h min worked into minutes nd vie vers. dd to give one b dd to give hve differene of Tuesdy h 0 min. Express the time worked eh dy in hours in 0 nd 0 nd nd 0 simplest frtionl form nd omplete the lst The sum of three frtions is. Given tht one of the frtions is 0, find the other two frtions. Wednesdy h 0 min 0 olumn of the tble. (Hint: express the (Note: there is more thn one nswer.) nd minutes s frtion of n hour.) Thursdy h 0 Use single pir of brkets to mke the following d How mny hours (in simplest frtionl form) Fridy 0 h 0 min 0 0 equtions true. reflet Explin why you nnot dd hs Rmon spent working t the fé in this Sturdy h 0 min 0 0 b + or subtrt the numertors of frtions week? tht hve different denomintors. hours ( ) ( + ) C Adding nd subtrting frtions Deep Lerning Kit C rd It just ll dds up Fous: To reognise the reltionship between frtions tht dd up to whole Resoures: oloured penils, File.0: It just ll dds up Students brek retngles into ertin setions nd shde the required prts. They write n ssoited ddition sttement involving frtions. C rd Puzzling pumpkins Fous: To use the ddition of frtions in rel life ontext Students the problem of purhsing ertin quntities of pumpkin. The greengroer hs ut the whole pumpkin into smller frtionl mounts. C rd Conotion ups Fous: To lulte frtionl quntities using one, one-hlf, one-third nd one-qurter Students different quntities tht n be obtined in rel-life ontext using only the ddition nd subtrtion of the frtions one whole, one-hlf, one-third nd one-qurter. C rd Egyptin frtions Fous: To represent frtions using the nient Egyptin numertion system of symbols nd hieroglyphis Students interpret the nient Egyptin For some students, it my be pproprite numertion system nd write their own not to omplete ll the questions numbers nd frtions using these symbols. but to work on one or more or Some ddition of frtions is required in the rds to develop nd deepen lter prt of this tivity. As n extension, their understnding. students develop their own numertion Q involves ompring two frtions Q involves onverting mesurements Q involves finding the differene restritions. There re mny possible Students who experiene diffiulty in system nd desribe how to form frtions with different denomintors. They of time into frtions. Some students between frtions in given list to nswers for these. solving worded problems involving in this system. need to understnd tht finding the my need to be reminded of ommon identify pttern. Students use this Q is hllenging question in whih frtions n be stepped through n differene mens to subtrt the smller whole lss: reflet time onversions: hour 0 minutes, frtion from the lrger frtion. hour 0 minutes, hour minutes. Q involve rnge of more diffiult ddition nd subtrtion questions. Q is n pplition question in whih students extrt the frtionl vlues from the worded question before dding. For students experiening diffiulty with this onept, strtegies re disussed in the smll group setion tht follows. Q re pplition questions in whih students use ddition nd subtrtion of frtions to solve problems. In Q, students omplete sequene of subtrtion lultions nd observe pttern tht ours in the results. They then predit the nswers for the next two lultions in the sequene. ommon differene to lulte the next four vlues in the list. Students who hve diffiulty with this n use lultor to find the differene. In Q, students find the frtion tht is hlfwy between two given frtions. Mny students my benefit from lss disussion of how to tkle problems of this type. Q nd Q re open, thinking questions in whih students identify frtions tht stisfy speifi students re to inlude single pir of brkets to mke n eqution true; tht is, to ensure tht the LS nd RS re equl. Remind students of the order of opertions rules (BIDMAS). Clultor use my be lso pproprite. smll group Fous: To pply understnding of ddition nd subtrtion of frtions to problem-solving tsks exmple of how to extrt the required informtion from written question. Some students my lso find ddition nd subtrtion of frtions very hllenging. Disuss the option of using lultor nd perhps step them through more omplited ddition nd subtrtion questions using the orret funtions on their lultor so tht they experiene degree of suess. Possible nswer: You nnot dd or subtrt the numertors of frtions with different denomintors s they do not represent the sme-sized piees of whole. Diret students to omplete the pproprite setion of My lerning in their Student Progress Book Professionl Support

10 D Multiplying frtions Unit Frtions nd rtios Deep Lerning Kit D rd Frtions of n mount Intervention tsk D rd Multiplying frtions Additionl skill prtie Student online support D etutor D Guided exmple D Test yourself tsk Fous: To build on student knowledge of multiplying frtions Students re guided to see wy of multiplying two frtions, using n re model. The two frtions represent the length nd width of retngle drwn on m m squre tht hs been divided into equl-sized prts. Sine the re of the retngle is found by multiplying the length by the width, the re represents the result of multiplying the two frtions. Students ompre the frtion obtined for the re with the originl two frtions to observe reltionship between the numertors nd the denomintors. They verify this reltionship by repeting the tsk for other pirs of frtions. whole lss Fous: To onsolidte student understnding of multiplying frtions D Multiplying frtions A smll retngulr vegetble grden hs the dimensions shown. We n lulte the re of this grden by multiplying the length of the retngle by the width. Tht is: Are length width m m But how n we lulte? In this tsk, we will use our knowledge of finding the re of retngle to develop method for multiplying frtions. Copy the squre shown. If the side length is m, use the rule m Are length width to lulte the re of the squre in m. m m m m The squre hs been divided into equl-sized retngles. How mny smll retngles re there? Wht frtion of the squre does eh of the smll retngles represent? Use your nswers to questions nd to write the re of the squre s frtion. One side of the squre hs been divided into five equl intervls. If this side is m or m long, mrk length of m long this side of the squre. The other side of the squre hs been divided into three equl intervls. If this side is m or m long, mrk length of m long this side of the squre. Shde the prt of the squre tht represents the retngulr vegetble grden. m How mny smll retngles hve been shded? m Use your nswers to questions nd to write the re of the vegetble grden in m s frtion. Eh smll retngle represents n re of m. There re eight smll retngles so the re is _ m Why is the nswer to question (the re of the vegetble grden) the sme s the nswer to? 0 Compre the numertor of the frtion you obtined in question with the numertors of the two frtions you re multiplying. Cn you see reltionship? Compre the denomintor of the frtion you obtined in question with the denomintors of the two frtions you re multiplying. Cn you see reltionship? Repet this tsk by drwing digrm to represent. Clulte the re by ounting the smll retngles. Does the reltionship you found in questions 0 nd still exist? Try t lest two more exmples. The reltionship is the sme. to hek whether the reltionship you hve found works in ll ses. The reltionship is the sme in ll ses. Write sentene to explin how to multiply frtions. To multiply frtions, multiply the numertors together nd multiply the Think bout how you dd or subtrt frtions denomintors nd how you multiply frtions. Compre the together. methods used nd omment on ny differenes or similrities you observe. exmple think write Clulte. Write the multiplition problem. Look for ny ommon ftors between numertors nd denomintors (there re none). Multiply the numertors together ( ) nd then multiply the denomintors together ( ). _ m e e exmple think write Clulte. Write the multiplition problem. Look for ny ommon ftors between numertors nd denomintors. The numbers nd hve HCF of. The numbers nd hve HCF of. Cnel nd by dividing both numbers by (shown in green). Then nel nd by dividing both numbers by (shown in pink). Write the result obtined fter nelling. Multiply the numertors together ( 0) nd then multiply the denomintors together ( ). 0 Convert the improper frtion to mixed number. key ides To multiply frtions, follow these steps. Note: in exmple, we ould hve First look for ny ftors ommon to the numertor nd first multiplied the numertors denomintor. together nd the denomintors Cnel number in the numertor nd number in the denomintor by dividing both numbers by the HCF. This will simplify the lultion. together nd then nelled by dividing by ny ommon ftors. However, it is esier to nel Cnelling n our vertilly; for exmple, 0 before multiplying so tht we re or digonlly; for exmple, not deling with lrge numbers. Then multiply the numertors together nd multiply the denomintors together. The denomintors do not need to be the sme when multiplying frtions. Mixed numbers must be onverted to improper frtions before multiplying. b Whole numbers n be expressed s frtions with denomintor of. For exmple, Use digrms nd the re method to lulte eh of these. b d Chek your nswers to question by performing the multiplition s shown in exmple. b 0 Clulte eh of these. b d e 0 f 0 g 0 h 0 Clulte eh of these. Where pproprite, nswers should be written s mixed numbers. b 0 d e f 0 g h 0 i j k 0 l m n o 0 0 p 0 0 Clulte eh of these. Remember tht whole number n be expressed s frtion with denomintor of. b d 0 After students omplete the Disover tsk, onsolidte their understnding. Diret students to the exmples. denomintors. Some students my For those students experiening Ensure students n see tht, when Exmple provides worked exmple hve diffiulty with nelling nd diffiulty with these questions, For _ _, n I first nel before Fous: To multiply two frtions 0 multiplying frtions, the numertors of how to multiply frtions without n be enourged to multiply first intervention through smll group multiplying? nd then simplify their nswer. (no, remind students tht it is not re multiplied together nd the denomintors re multiplied together. They should reognise tht the method used is different to dding nd subtrting frtions. For further support, diret students to D rd. Demonstrte D etutor or diret students to do this independently. m m m m m. needing to simplify. Exmple provides worked exmple of how to nel before multiplying. Diret students to omplete this multiplition without first nelling. Ask them to simplify their nswer by dividing both the numertor nd the denomintor by the HCF. Disuss the result. Diret students to the key ides. You my like them to opy this summry. The re n be lulted in two wys: by ounting the smll retngles in the digrm s in question nd by using the rule Are length width; tht is, by multiplying the length of the retngle (in this se, m) by the width (in this se, m). The vlue for the re should be the sme in both ses. So re of the vegetble grden is m or m m. This mens. Q direts students to use the re model to multiply two frtions. The results re heked in Q by using the method of multiplying numertors together nd denomintors together. Q involves multiplying frtions without the need to nel ftors or simplify. In Q, nelling n be performed before multiplying numertors nd Q requires students to multiply whole number by frtion. It is useful to write the whole number s frtion; tht is, write the whole number over. Students n be enourged to use the pproprite steps on their lultors to hek their nswers.. tehing of the onept my be neessry. D Multiplying frtions For dditionl prtie, diret students to D rd. As tsk to be undertken either in lss or t home, diret students to omplete the etutor, Guided exmple nd Test yourself. d 0 or 0 0 or 0 d POTENTIAL DIFFICULTY Some students my experiene diffi ulty when nelling. Remind them of the following: Numertor nd denomintor of the sme frtion n be nelled or simplifi ed. Cnelling n be performed between numertor nd denomintor of different frtions. Cnelling nnot be performed numertor to numertor, or denomintor to denomintor. Deep Lerning Kit D rd Frtions of n mount Fous: To find the frtion of wholenumber mount Students re guided through problems in whih they relte the denomintor of the frtion to the number of groups of equl size. They omplete questions with visul support nd then questions written in mthemtil sentenes. D rd Multiplying frtions Fous: To multiply frtions fter first nelling the ommon ftors to numertor nd denomintor Students re guided through the nelling proess before multiplying numertors together nd denomintors together. They multiply proper frtions, improper frtions nd mixed numbers. Additionl prtie questions similr to the Now try these questions re lso provided. whole lss Some possible questions to sk: Why is it helpful to nel before multiplying? (working with smller numbers) possible to nel denomintor to denomintor) Wht should be done first when lulting _ 0? (hnge mixed number to improper frtion nd write whole number s frtion) Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

11 D Multiplying frtions Unit Frtions nd rtios Deep Lerning Kit D rd Frtions musi to the er Investigtive tsk D rd Pizz perfet Problem-solving tsk D rd All things equl Investigtive tsk whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: How is multiplying frtions different from dding or subtrting frtions? (no need for ommon denomintor) Why do we try to nel frtions first before multiplying? (to work with smller numbers) How n frtions in multiplition lultion be nelled? (divide number in ny numertor nd number in ny denomintor by ommon ftor, preferbly the HCF) Wht should you do with mixed number in multiplition lultion? (write it s n improper frtion) questions Fous: To pply understnding of multiplying frtions For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. Q involve mixed numbers nd improper frtions. Students will need to onvert mixed numbers to improper frtions before multiplying. sign hs been repled with the word students my lulte the vlue of the trining. It my be helpful to remind opertions to find the seond frtion smller nd esier to mnge. Q involves the multiplition of three of. Previous onsidertions, suh s disount but need to be reminded to Diret students to omplete the pproprite frtions. Students will need to hnge nelling, hnging mixed numbers subtrt this vlue from the originl setion of My lerning in their Student mixed numbers to improper frtions. nd writing whole numbers s frtions prie to find the disounted prie. Cnelling will ssist in this question, re still required. s numbers will be lrger. Q sks students to find errors mde. Q involve the multiplition of frtion by whole number. Hilire works t the lol swimming pool s lifegurd. Her shift begins t. pm nd ends t. pm. Q involve further multiplition questions in whih the multiplition Q fouses on rel-life senrio where students identify different frtionl mounts represented in one lrge bth of upkes. They re guided to use visul model to represent the upkes. Q involves lulting the new selling prie for disounted TV. Some Alterntively, students n lulte _ of the originl prie. You my like to disuss why this option is lso orret. In Q, students lulte the distne jogged before stopping. Remind students to onvert the mixed number to n improper frtion before ompleting the lultion. Copy this grid into your workbook. b Shde of the grid. Using different olour, shde of. d Wht frtion of the grid does of represent? Write the hours Hilire works in one shift s: e Clulte 0 i mixed number. hours ii n improper frtion. 0 f Compre your nswers to prts d nd e. Wht do you notie? Sme nswer b If Hilire hs the sme shift five nights week, lulte the g Repet prts f for of. is obtined. totl number of hours she works s frtion. hours Besides multiplying proper nd improper frtions, we n lso multiply mixed numbers. Before multiplying, h Copy nd omplete this sentene. onvert eh mixed number to n improper frtion. Consider In mthemtis, the word of n be repled by. 0. multiplition symbol Write the multiplition problem nd then onvert eh mixed number to n improper frtion. 0 Clulte eh of these. 0 b Continue s before to find the result. Remember to lwys look for ny ommon ftors. of b of 0 of Clulte eh of these. Where pproprite, nswers should be written s mixed numbers. d of e of f of b 0 d 0 Find the following frtion quntities s speified by the units shown in the brkets. e f g h The first one hs been ompleted for you s n exmple. Clulte eh of these. of hours (minutes) minutes b of yer (months) months Exmple of $0 (dollrs) d 0 b d 0 $ of the yer (weeks) weeks e of minutes (seonds) f of hours (minutes) e 0 f g h 0 seonds of 00 litres (litres) litres 0 minutes Whilst orreting his students homework, teher found three different nswers to the question g of weeks (dys) dys h of $00 (dollrs) $0 minutes Find i in simplest form. The nswers given were:, nd. of of km (km) Whih nswer is orret? hours is inorret; the whole numbers were multiplied rther thn onverting the mixed numbers to improper frtions. km is inorret; the denomintors were dded. k of 0 yers (months) 00 months l of $.0 (ents) 0 ents b Comment on why the other two nswers re inorret. Explin the error mde in eh se. Clulte eh of the following. Amy bked bth of upkes nd rrnged them on try in six by six pttern. Drw digrm of the rrngement nd shde of the digrm. How mny b Remember to write eh 0 upkes does this represent? d e b Amy deides to tke f whole number s frtion 0 with denomintor of. of the upkes to shool nd shre them with her Refer to your nswers to question. Copy nd omplete the following lssmtes. How mny upkes will she tke to shool? using the words lrger or smller. Upon rriving t shool, Amy relised tht of the upkes were When multiplying whole number by proper frtion the nswer is lwys smller thn the whole number. squshed in her bg. How mny of the upkes were squshed? When multiplying whole number by n improper frtion or mixed number the nswer is lwys lrger A deprtment store tkes off the retil prie of its television sets during the thn the whole number. stok-tke sles. Wht will n LCD television set tht is normlly sold for Look t the different-oloured Vesp motorsooters shown. Count the number of red $0 now ost? $ Your friend mnges to omplete How ould you use this digrm to illustrte the sttement Vesps nd divide by the of the km run before stopping. How fr of the Vesps re red? totl number of Vesps. did she jog prior to stopping? 0 Boris spent of n hour trining for his upoming tennis mth. Roger spent times longer thn Boris in prepring for his mth. Boris: minutes, Roger: minutes (or hours minutes) How long did eh boy spend trining? b How muh longer thn Boris did Roger spend trining? hours When two frtions re multiplied, their produt (or nswer) is. Wht possible vlues ould the two frtions b How mny Vesps re there? How mny Vesps re red? be? List three pirs of possible frtions. d Wht is the reltionship between One possible nswer is: _ number of red Vesps nd the nswers obtined in prts b nd? Chllenge! Clulte eh of these. reflet e Using your nswers to prts b nd, opy nd omplete this sttement: totl number of Vesps nd 0. Explin to friend why it of equls. b f Rewrite the sttement in prt e s mthemtil eqution. Wht opertion hs repled the word of? + ( + is useful to fi rst simplify frtions before multiplying them., multiplition Q involves ompring the mount of time two different tennis plyers spend students tht there re 0 minutes in hour, s working with minutes my be esier. Agin, remind students to onvert the mixed number to n improper frtion before ompleting the lultion. Q is n open, thinking question in whih students find three different possible sets of frtions tht n be multiplied to hieve given result. D Multiplying frtions More ble students my use the ft tht multiplition nd division re inverse in the pir. Q is hllenging question for whih students perform frtion lultions tht ontin more thn one opertion. Remind students to use the order of opertions rules (BIDMAS). They n hek their nswers using lultor or other digitl tehnology. smll group Fous: To pply multiplition skills to worded problems nd hllenge tsks Students who experiene diffiulty in pplition questions my need to be expliitly shown how to extrt the relevnt informtion from the text of worded problems. One students hve deiphered the question, they re generlly ble to demonstrte the required mthemtis. Deep Lerning Kit D rd Frtions musi to the er Fous: To frtions using musil notes nd their ssoited time bets Students re guided through series of musil explntions nd questions tht involve whole numbers nd frtions used to find the number of bets for ertin notes nd brs. D rd Pizz perfet Fous: To use problem-solving strtegies nd the ddition, subtrtion nd multiplition of frtions in rel-life ontext Students re given ertin informtion bout pizzs eten by fmily nd re then required to determine how the pizzs re ut nd wht pizz eh fmily member ets. D rd All things equl Fous: To determine frtion of frtions in rel-life ontext Students investigte the possibilities of shring fudge ndy tht hd lredy been ut into frtionl quntities. whole lss: reflet Possible nswer: It is useful to simplify frtions before multiplying s the numbers tht you will be working with will be Professionl Support

12 E Dividing frtions Unit Frtions nd rtios Deep Lerning Kit E rd Dividing frtions Intervention tsk E rd Dividing frtions Additionl skill prtie Professionl support online File.0: Frtion wll strips Reproduible mster File.0: Retngles Reproduible mster Student online support E etutor E Guided exmple E Test yourself tsk Fous: To how to divide quntity by frtion nd to build link between multiplition nd division Students use visul models to how to divide quntity by frtion. They first use irles divided into equl setions to model shring two pizzs mong three boys. Students re then guided to onsider dividing retngles into qurters to look t how mny lots of _ re in one whole, two wholes nd three wholes. They repet this tsk with fifths nd sevenths. File.0: Retngles n be provided. E Dividing frtions After trining session, three boys look forwrd to eting their fvourite food. They order two lrge pizzs nd deide to shre them. Consider whether it is possible to evenly divide two pizzs mongst three boys. Drw two irles nd divide eh of them into three equl setions. b Wht do the three equl setions in eh irle represent? Eh boy s shre of pizz How mny equl setions re there ltogether? six d Use three different olours to illustrte how the two irles n be divided into three equl shres. e From your findings in prt d, stte wht frtion of whole pizz eh boy will reeive. Copy nd omplete the following sentenes. Look t these retngles. How mny lots of Dividing number by is the sme s re in multiplying tht number by. one whole? b How mny lots of re b Dividing number by is the sme s multiplying tht number by. in two wholes? Dividing number by How mny is the sme s lots of multiplying tht number by. re in three wholes? The retngle t right hs been divided into tenths. Wht do you notie bout your nswers to Eh nswer is obtined by multiplying the whole question? The shded setion of the retngle represents. number by the denomintor,. Use the digrm to determine wht Repet questions nd but this time drw 0 is equl to. Tht is, how mny lots of retngles tht re divided into fifths. Consider how 0 re there in? How mny lots of does this ompre to the onlusions you mde in re in one whole, two wholes nd question? Explin your nswer.. Sme method is used; three wholes., 0,. Eh nswer is obtined by multiplying tht is, dividing by the whole number by the denomintor,. is the sme s multiplying Chllenge! Drw digrm to represent Repet questions nd but this time drw. by 0. 0 Tht is, how mny lots of retngles tht re divided into sevenths. re there in two wholes? Consider how mny lots of (Hint: drw two retngles nd divide eh into re in one whole, two thirds. Shde eh lot of wholes nd three wholes. in different olour.,,. Eh nswer is obtined by multiplying the whole number by the When sking how mny lots of denomintor,. How mny lots of do you hve?) re in one whole, we re sking for the result when is divided by. 0 Clulte. (Remember tht n be written This n be written s. s.) Compre your nswer to the result obtined in question. ; sme result Use the digrms nd your results for questions, nd to nswer the following. Cn you see shortut to use when dividing b by frtion? Write sentene to explin the d e f reltionship between multiplying nd dividing 0 frtions. How n you use this reltionship to g h i mke dividing by frtions esier? The result obtined when dividing by frtion is the sme s tht obtined when multiplying by the frtion fter it hs been turned upside down. This reltionship mkes it esier to nswer division problems without needing to drw digrms. A division problem n be hnged to multiplition problem, whih is esier to work out. exmple think write Clulte. Write the division problem. Chnge it from division problem to multiplition problem. Reple with nd turn the frtion tht follows the division sign upside down. Note: the proess of turning the frtion upside down is lled tking the reiprol. Multiply the numertors together nd then multiply the denomintors together. key ides To divide frtions, follow these steps. Chnge from division to multiplition problem by repling the Exmple A division sign with multiplition sign nd turning the frtion tht follows upside down. Proeed s for multiplition problem by first nelling, if possible, then multiplying the numertors together nd the denomintors Exmple B together (see Exmple A). The reiprol of is. The proess of turning frtion upside down is lled tking the The reiprol of is reiprol (see Exmple B). (sine n be expressed s ). When dividing mixed numbers, first onvert them to improper frtions. The reiprol of Where pproprite, onvert the finl nswer to mixed number. (sine when expressed s n improper frtion is ). Rewrite eh of the following division problems in the form How mny lots of re there in? How mny lots of b How mny lots of How mny lots of d How mny lots of re there in? re there in? re there in? re there in Drw digrm to help you find the nswer to eh of the division problems in question. Write eh division problem in question s multiplition problem. Chek your nswers to question by performing the lultions to question. Remember tht whole numbers n be written with denomintor of. Write the reiprol of eh of these. b d b d e f e Clulte eh of these. Where pproprite, nswers should be written s mixed numbers. b 0 d e 0 f g h i j 0 k 0 l m n o 0 p Clulte eh of these. Where pproprite, nswers should be written s mixed numbers. b d 0 e f g 0 h Students tht there is link between dividing by frtion nd multiplying by its reiprol nd see tht this is shortut tht n be used to b d mke dividing by frtion esier to perform. Demonstrte E etutor or diret For those students experiening diffiulty Cut out five strips to model _. In Q, students write the division whole lss whole lss students to do this independently. with these questions, intervention Cut eh strip into qurters. How problems in Q s multiplition Some possible questions to sk: Fous: To onsolidte student Diret students to the exmple. It shows through smll group tehing of the mny re there? (0) problems. Q requires students to understnding of dividing frtions how to hnge division problem to onept my be neessry. Students n Cut out three strips to model _ multiplition problem. be given opies of File.0: Frtion. How mny lots of _ re there in one omplete the lultion by multiplying whole? () Cut eh strip into eighths. How by the reiprol. This llows them to How mny lots of Diret students to the key ides. You wll strips nd enourged mny re there? () hek tht they obtin the sme results _ re there in? () my like them to opy this summry. to experiment. s in Q. After students omplete the Disover tsk, onsolidte their understnding. Ensure tht ll students hve rrived t the understnding tht dividing number by frtion is the sme s multiplying the number by the reiprol of the frtion. Diret students to E rd for further support. Fous: To divide frtions Q nd Q enourge students to use visul model to onsolidte their understnding of dividing by frtion. Cut out one strip to model _. Cut the strip into thirds. How mny re there? () Cut out one strip to model _. Cut eh strip into fifths. How mny re there? () This model n be used to represent number of different division questions. After number of trils, sk the students if they n see link between the whole number nd the denomintor. When the students offer ny reltionships s suggestions, hve them model the suggestion, to see if it works. E Dividing frtions Q llows students to prtise writing the reiprol of frtion. In Q, students omplete division problems using the reiprol to hnge the division to multiplition. Q involves first writing eh whole number s frtion with denomintor of. For dditionl prtie, diret students to E rd. As tsk to be undertken either in lss or t home, diret students to omplete the etutor, Guided exmple nd Test yourself. POTENTIAL DIFFICULTY Some students my mke errors by tking the reiprol of the fi rst number rther thn the seond when hnging division problem to multiplition problem. One wy of tkling this is to emphsise the order of the steps they follow. Write the fi rst frtion. Chnge to. Turn the seond frtion upside down. Deep Lerning Kit E rd Dividing frtions Fous: To find how mny lots of frtion divide evenly into whole numbers using digrms Resoures: ruler Students re guided through problems where they use digrm to divide whole numbers by simple frtions. They omplete questions with visul support nd then questions written in mthemtil sentenes. E rd Dividing frtions Fous: To divide frtions by onverting to multiplition problem Students re stepped through the proess of hnging division problem into multiplition problem. They divide proper frtions, improper frtions nd mixed numbers. Additionl prtie questions similr to the Now try these questions re lso provided. How n this be written s division problem? ( _ ) How n this be written s multiplition problem? ( ) Explin how we n lulte _ _. (write s multiplition problem, _ _ ) Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

13 E Dividing frtions Unit Frtions nd rtios Deep Lerning Kit E rd Very deliious frtions Problem-solving tsk E rd Pper uts Investigtive tsk whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: How mny lots of _ re there in one whole? () How mny lots of _ re there in? () Cn you omplete the following without working? () b () () d (0) Wht strtegy hve you used? ( whole number by denomintor of frtion) Wht is the reiprol of _? ( _ ) Rewrite s multiplition problem. Clulte the nswer. ( ) questions Fous: To pply understnding of dividing frtions to different ontexts For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. Q is reltively esy pplition question relted to reipe. Some students my find it useful to disuss wht is required in eh prt with lssmte. A new housing estte is to be developed on hetres of lnd. If eh blok mesures hetres, how mny bloks will there be? The ingredients to mke stiky dte pudding While mrking test ppers, teher obtined three different nswers to the question Divide into for four people re shown below.. The nswers obtined were, nd _ 0. up of hopped pitted dtes Only one of these nswers is orret. Whih one is it? Misred question nd lulted to obtin. Converted mixed numbers to improper frtions, repled the tespoon of birbonte of sod b Explin how the two inorret nswers might hve been obtined. division sign with multiplition sign but did not turn the 0 grms of butter seond frtion upside down to obtin 0 Use wht you hve lernt from question to help you nswer the following. 0. up of sugr eggs Divide into. b Divide into. Divide into. up of self-rising flour d Divide into. e Divide into. f Divide into. A triple mixture of stiky dte pudding is needed to feed people. How muh of eh ingredient would be required? ups of hopped pitted dtes, tespoons of birbonte of sod, 0 grms of butter, ups of sugr, eggs, Mdush nd Cynthi hve been trining for their shool s ups of self-rising flour b To feed six people, one-nd--hlf mixture of stiky dte pudding is needed. How muh of eh upoming wlkthon. It tkes them 0 of n hour to ingredient is required? ups of hopped pitted dtes, tespoons of birbonte of sod, 0 grms of butter, up of sugr, omplete one lp. eggs, How muh of eh ingredient is ups required of self-rising for the flour reipe to feed two people? up of hopped pitted dtes, tespoon of If the time llowed for the wlkthon is hours, birbonte of sod, 0 grms of butter, up of sugr, egg, wht is the mximum number of lps the girls ould d How muh of eh ingredient is required up of self-rising for the reipe flour to feed three people? up of hopped pitted dtes, tespoon of birbonte of sod, grms of butter, up of sugr, eggs, up of self-rising flour omplete? Besides dividing proper nd improper frtions, we n lso divide mixed numbers. The first step is to onvert b Comment on whether you think the girls ould No, not ble eh mixed number to n improper frtion. We then ontinue s we hve for erlier division problems. omplete the mximum number of lps. to mintin sme pe Consider finding the result to. Mth eh step shown below with its orresponding line of nd will need to inlude Clulte eh of these. rest/mel breks. working A G in the lultion. Write the lines of working in the orret order in your workbook. b 0 d Steps B, D, F, C, G, A, E Lines of working 0 Write the division problem. A e 0 f g h 0 Chnge eh mixed number to n improper frtion. 0 B Chnge to multiplition problem by repling with nd turning the Ginlu must pk kg of pples into plsti bgs tht n hold C frtion tht follows upside down. 0 mximum of kg. Cnel nd 0 by dividing eh by the HCF of. D How mny 0 bgs n he fill? bgs Write the result fter nelling. E b How mny kilogrms of pples will be left over? Multiply the numertors together nd then multiply the denomintors together. F kg 0 When dividing mixed number by proper frtion, Chnge the improper frtion to mixed number. G do you obtin smller or lrger number? Explin. Lrger number, sine we re dividing by vlue tht is less thn. Clulte eh of these. Where pproprite, nswers should be written s mixed numbers. After dividing two proper frtions, mixed number is b d obtined. Write three possible division problems tht show this. e f g 0 h Some possible nswers re:,, After dividing two proper frtions, whole number is obtined. i Write three possible division problems tht show this. j k 0 0 l Some possible nswers re:,, A bth of hoolte pudding reipe requires After dividing two mixed numbers, proper frtion is obtined. Write three possible division problems tht kg of sugr. How mny bthes of hoolte pudding n be mde from this pket of sugr? show this. Some possible nswers re:,,. After one frtion is divided by nother, the nswer is Over the shool holidys Brinn worked totl of hours stking shelves t. Write three possible division problems tht show this. the supermrket. If she worked Some possible nswers re: hour shifts, how mny shifts did she work? Chllenge! Clulte eh of, these., Remember. to think refully bout whih opertions re performed first. Students in tehnology nd design lss re mking ostumes for the shool + prodution. A totl of metres of fbri hs been purhsed. If eh grment b ( requires ) metres of fbri, how mny grments n be mde? 0 Rhys shres + 0 reflet Wht would you tell your pizzs eqully between himself nd eight friends. Wht frtion of d ( ) ( + pizz will eh person reeive? ) ( ) friend it s importnt to remember when dividing frtions? whole number by mixed number. experiening diffiulty with this Q onsiders rel-life pplition tht involve more thn one opertion. Q guides students through the division In Q, they divide mixed number onept, strtegies re disussed in the where students lulte the number of They my need to be reminded to whole lss: reflet of frtions involving mixed numbers. by whole number. You my need to smll group setion tht follows. bgs of pples tht n be mde from follow the order of opertions rules Q llows students to prtise questions involving mixed numbers. In Q, students need to refer to the photo of bg of sugr nd understnd tht it holds kg. They then lulte how mny lots of _ kg re in kg or _. Q re rel-life pplition questions in whih students identify the different vlues required for eh division problem. In Q nd Q, students divide remind them to write the whole number s frtion with denomintor of. Q involves dividing mixed number by proper frtion. Q nd Q0 involve n lterntive wy of expressing division problem. In Q, students onsider how eh of three nswers to the one question is obtined nd then deide whih is orret. Q0 llows them to perform more of this type of problem. For students Q involves division lultion but then requires students to think bout how relisti the nswer is. In Q, students omplete division problems in whih nelling n be used to simplify the multiplition lultion. You my need to emphsise tht nelling should not be ttempted until the division problem is written s multiplition problem. kg. Sine the result of the division problem is mixed number, they need to think bout the pproprite nswer to give. Q re open, thinking questions in whih students demonstrte their understnding of dividing frtions. They my like to disuss nd verify their nswers with lssmte. Q is hllenging question for whih students omplete frtion problems E Dividing frtions (BIDMAS). You my like them to hek their nswers with lultor or other digitl tehnology. smll group Fous: To understnd how to write division lultions when expressed in different forms Students who experiene diffiulty with Q nd Q0 my need ssistne in reognising the divisor (nd dividend) in the worded division problems. Disuss different wys division problem n be posed: Divide by., How mny lots of _ re in?, How mny times does go into _?, Divide into. Show tht ll these expressions re equivlent to _ _. Students my find it esier to first onsider different division expressions involving whole numbers only nd then extend the onept to frtions. Disussing the errors mde in Q n be vluble lerning tool. Deep Lerning Kit E rd Very deliious frtions Fous: To lulte the quntities for reipe tht involves frtionl mounts for the ingredients Resoures: lultor (optionl) Students lulte the quntities of the ingredients to mke hzelnut bisotti using reipe tht mkes quntity of 0 bisotti. As n extension, they re given list of ingredients nd must determine the mximum number of bisotti tht n be mde. E rd Pper uts Fous: To determine frtionl sizes of pper using multiplition nd division of frtions Students the nming of pper sizes nd their reltionship to the A0 size pper. They nswer series of questions to guide them, informlly exploring frtionl quntities to bse two. As n extension, students design their own system of pper sizes from given informtion. Possible nswer: Division questions n be rewritten s multiplition questions. The division sign is hnged to multiplition sign nd the reiprol of the seond frtion is used. Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

14 F Powers nd squre roots of frtions Unit Frtions nd rtios Deep Lerning Kit F rd Powers nd squre roots of whole numbers Intervention tsk F rd Squres nd squre roots of frtions Additionl skill prtie Student online support F etutor F Guided exmple F Test yourself tsk Fous: To squres nd squre roots of frtions Students build on their understnding tht the power, or index, of number indites the number of times the bse is written in repeted multiplition. They first squre whole numbers, explining the proess used, nd then pply this to squring frtions. Students tht there re two methods of squring frtion: multiplying the frtion by itself, nd squring the numertor nd squring the denomintor. Students then onsider the squre root of whole numbers nd think bout the method used. They use their nswers obtined fter squring frtions (Q) to lulte the squre root of mthing frtions. Guidne is provided to omplete further squre root lultions. F Powers nd squre roots of frtions Finding the squre, squre root or other powers of frtions is very similr to finding squres, squre roots or other powers of whole numbers. In this tsk, we will strt with whole numbers nd then develop method for finding squres nd squre roots of frtions. ( ) ( ) ( ) First onsider finding the squre of whole number. This is the sme s rising the number to the power of. Clulte eh of these. b d 0 00 Explin how you obtined your nswers to question. Multiplied eh number by itself. 0 Now onsider finding the squre root of whole number. Clulte eh of these. Clulte eh of these. b d _ 00 0 b d Explin how you obtined your nswers to question 0. To lulte squre root, find the number tht when squred Explin how you obtined your nswers to question gives the number under the squre root sign.. Multiplied the numertors together nd multiplied the Use your nswers to question to help you denomintors together. lulte eh of these. Write wht you expet the nswer to be for eh of these. b d 00 0 ( ) b ( ) ( ) d ( 0 ) 00 Copy nd omplete the following to show how to Explin how question reltes to question. lulte the squre root of eh frtion. Writing the squre of frtion is the sme s writing the frtion Copy nd omplete eh of the following to show b multiplied by itself. _ two methods of squring frtion. ( ) or ( ) _ d b ( ) or ( ) 0 Explin how you obtined your nswers to question. Found the squre root of the numertor nd the squre root of the denomintor seprtely. ( ) or ( ) Copy nd omplete eh of the following to lulte the squre root of eh frtion. d ( 0 ) 0 0 or ( _ 0 ) b Explin the two methods tht n be used to lulte the squre of frtion. d 00 Multiply the frtion by itself or squre the numertor nd 00 squre the denomintor. Use both of the methods you stted in question to 0 obtin nswers to these. Explin the differene between finding the squre of ( ) b ( ) frtion nd finding the squre root of frtion. or ( ) Finding the squre of frtion nd finding the squre root of frtion or ( ) re opposite or inverse opertions. For exmple, ( ) nd. exmple think write Clulte ( ). Write the frtion problem. ( Convert the mixed number to n improper frtion. ( ) or ( ) Squre the numertor nd squre the denomintor. (Alterntively, write s the multiplition of two frtions.) Clulte the numertor nd then the denomintor. Convert the improper frtion to mixed number. exmple think write Clulte 00. Write the squre root problem. 00 _ Express s the squre root of the numertor divided by 00 the squre root of the denomintor. _ Clulte the squre root of the numertor nd the 0 squre root of the denomintor. Simplify by nelling. Divide the numertor nd the denomintor by the HCF of. 0 Write the nswer. key ides A frtion n be squred in two wys. Multiply the frtion by itself. For exmple, ( ). Squre the numertor nd then squre the denomintor. For exmple, ( ). To obtin the squre root of frtion, find the squre root of the numertor nd then find the squre root of the denomintor. For exmple, _. A mixed number must first be onverted to n improper frtion before squring or finding the squre root. Where possible, the finl nswer n then be expressed s mixed number (see exmple). Frtions n lso be rised to higher powers. For exmple, ( nd ( ) ). Clulte eh of these. ( ) b ( ) ( ) d ( 0 ) e ( 00 ) f ( ) e Clulte eh of these. ( ) b ( ) ( ) d ( ) e ( ) f ( ) e Clulte eh of these. b d e f or Clulte eh of these. b d e f They fous on the link between finding rising frtion to power, or when Similrly, when writing the squre Diret students to the exmples. whether to multiply the frtion by itself, Some possible questions to sk: squres nd squre roots of numbers. lulting the squre root of frtion, root of frtions, enourge students Exmple shows how to squre mixed or squre the numertor nd then squre In the frtion ( _ ) identify the bse nd whole lss both the numertor nd the denomintor to write the squre root over the whole number. Exmple shows how to the denomintor. You my like them to the power. (Bse _, power ) of the frtion re ffeted by the power or frtion so tht they n see tht both lulte the squre root of frtion. try both methods so they see tht the Cn you identify the error in the Fous: To onsolidte student the squre root symbol. the numertor nd the denomintor re Diret students to the key ides. You sme result is obtined. working shown? ( _ understnding of squres nd squre When writing squres, students ffeted by the squre root symbol. my like them to opy this summry. In Q, students squre mixed numbers. ) (only numertor hs been squred) roots of frtions should be enourged to put brkets Some students my need further They need to remember to hnge eh Wht is round the frtion so tht they n support working with powers nd mixed number to n improper frtion ( _ )? ( _ ) After students omplete the Disover tsk, see tht both the numertor nd the squre roots of whole numbers before Fous: To lulte squres nd first. Cn you identify the error in the onsolidte their understnding. Ensure denomintor re ffeted by the index. they tkle frtions. Diret these squre roots of frtions Q involves finding the squre root of working shown? tht students n reognise tht when students to F rd. proper frtions. All vlues used for (squre root of not lulted) Demonstrte F etutor or diret Q involves squring proper nd the numertors nd denomintors re Wht is? ( _ ) students to do this independently. improper frtions. Students n hoose perfet squres. Diret students to omplete the pproprite setion of My lerning in their Student ) Exmple _ or F Powers nd squre roots of frtions In Q, students find the squre root of improper frtions nd then mixed numbers. For dditionl prtie, diret students to F rd. As tsk to be undertken either in lss or t home, diret students to omplete the etutor, Guided exmple nd Test yourself. POTENTIAL DIFFICULTY If students don t use brkets to enlose the whole frtion, they my only squre the numertor. For exmple, if lulting the squre of without the use of brkets, students my write. When squring mixed numbers, some students my squre the whole number nd the frtion omponent seprtely. The mixed number should be hnged to n improper frtion nd then squred. Deep Lerning Kit F rd Powers nd squre roots of whole numbers Fous: To work with powers of, nd nd find squre roots of perfet squres Students prtise repeted multiplition problems. They lso use ptterns relted to squring number to write the squre root of number. F rd Squres nd squre roots of frtions Fous: To find squres nd squre roots of frtions Students lulte squres nd squre roots of frtions. Additionl prtie questions similr to the Now try these questions re lso provided. whole lss Professionl Support

15 F Powers nd squre roots of frtions Unit Frtions nd rtios Deep Lerning Kit F rd Oigys, Shnens nd Doblovihs Problem-solving tsk F rd Drinking duos Problem-solving tsk F rd Pik the pttern Problem-solving tsk whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: How do you write one-hlf squred orretly? [( _ ) ] How is this lulted? ( _ ) Wht steps would be needed to lulte ( _ )? [( _ ) ( ) 0 ] Wht steps would be needed to lulte? [ _ questions _ ] _ Fous: To pply understnding of the lultion of squres, squre roots nd powers of frtions to different ontexts For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. Q is reltively simple pplition tht should be essible to most students. Some students my require some ssistne in extrting the required informtion from the question. Three lots of the whole number or frtion re multiplied together. Alterntively, rise the numertor of frtion to the power of nd rise the denomintor of the frtion to the power of. Work with prtner to investigte how to lulte eh of these where number is rised to power of. Write sentene to explin the method you use. Your grndfther hs number of vinyl LP reords in b d ( ) e ( ) f ( ) his musi olletion. The front over of one of these Desribe how to lulte the nswer when frtion is rised to ny power. is in the shpe of squre. For the front over shown, work out the re in m by lulting the squre of 0 Clulte eh of these. the length or (length length). ( ) b ( ) ( ) d ( ) e ( 00 m ) f ( or 0 or ) In eh digrm, squre hs been shded. Use the Clulte eh of these. length shown to nswer the following. i ( ) ii ( ( ) b ( ) ( ) d ( ) e ( ) f ( ) Clulte the vlue of (length length) or (length) ) 0 or Find the nswer to eh of these. to find the re of the shded region. iii ( ) iv ( ) ( ) b ( 0 ) ( ) d ( ) e ( ) f ( b Write the frtion of the lrge squre tht is shded. i ii ) 0 Crefully study eh of these frtions:, ( ), ( ), ( _ 0 Compre your nswers to prts nd b. Wht do iii iv ). 0 you notie? m The nswers re the sme. Arrnge them in sending order. ( ), ( ), ( ), The frtion rised to the highest power is the smllest frtion i ii iii iv b Comment on your findings. Is this wht you expeted? beuse the denomintor is lrger thn the numertor. Crefully study eh of these frtions:, ( ), ( ), ( ). Proper frtions rised to powers of Arrnge them in sending order. or more will be smller thn the, ( ), ( ), ( ) originl frtion. Improper frtions b Comment on your findings. The frtion rised to the highest power is the lrgest frtion rised to powers of or more will be beuse the numertor is lrger thn the denomintor. lrger thn the originl frtion. Using your results from questions nd, write short sttement explining wht hppens when proper frtions nd improper frtions re rised to powers of or more. Anlyse eh pir of frtions nd determine whether they will produe the sme nswer. Explin your A rel estte gent sttes the size of squre blok of lnd resoning. No, sine both the No, sine both Yes, sine both is squre kilometres. ( numertor nd ) nd b ( ) nd the numertor ( the numertor ) nd d ( + ) nd ( ) + ( Find the squre root of the re. denomintor nd denomintor nd denomintor ) Explin how the length of the blok of lnd n be worked must be squred. must be squred. re squred. Two students produed two different nswers when sked to lulte ( out if you know tht the re of squre is length length. ). Exmine eh student s No, sine the sum within the pir of lultion proess nd explin whih is orret. brkets must first be b Clulte the length of the blok of lnd for sle. km lulted nd then Olivi s lultion Kte s lultion How long is the fene tht runs long the boundry of the Kte is orret squred. Remember beuse the mixed to use the orret lnd? number must first ( d If the lnd is for sle t $00 000, lulte the ost per ) + ( ) ( ) ( ) order of opertions. km be onverted to n improper frtion squre kilometre. + $ 000 nd then squred. Clulte eh of these ( ) b + ( ) ( or ) or d + ( ) e ( ) + f + First simplify eh frtion nd then lulte the result. or 0 or 0 Clulte eh of these. Remember to think refully bout whih opertions re performed first. b 0 d 0 e f 0 ( ) b ( ) or Explin why it ws importnt to first simplify eh of the frtions in question before finding its squre root. d + e or f ( ) Jk produed these workings nd nswer when sked to lulte 0 or 0. Comment on whether his nswer nd the proess he used re orret Chllenge! Clulte eh of these. Simplifying the frtion ( ) b Both the proess for finding the solution nd the ( ) + 0 first mde it possible nswer re inorret beuse the mixed number must to find the squre roots 0 + or d (( ) first be onverted to n improper frtion nd then the ) ( ) of the numertors nd squre root found. denomintors. e _ or 0 f reflet How do you know in whih or order to perform opertions on frtions? 0 or or 0 In Q, students re gin sked to onsider whether the working out 0 m onept, strtegies re disussed in the smll group setion tht follows. The power indites how mny lots of the frtion to multiply together. If the power is, then we multiply four lots of the frtion together. If the power is, then we multiply five lots of the frtion together nd so on. Alterntively, we n rise the numertor to the given power nd then rise the denomintor to tht power. Q requires students to write n explntory sttement of their findings Q re hllenging questions in whih students perform lultions Students who experiene diffiulty in extending their knowledge of the squring of frtions to powers of,, nd my need to review the onept of repeted multiplition overed extensively in F Powers nd squre roots before ttempting Q. Students who re struggling to omplete the repeted multiplitions ould be enourged to use their lultors. Deep Lerning Kit F rd Oigys, Shnens nd Doblovihs Fous: To determine the ptterns for different groups of frtions using frtions rised to ertin powers Students re guided to determine ptterns tht exist with ertin groups of frtions nd then form more frtions for these groups given ertin onstrints. F rd Drinking duos Fous: To ompre results when frtions re multiplied s ompred to when they re rised to power Students ompre the wter drinking ptterns of two trining prtners under ertin onditions. They multiply, rise to power nd subtrt frtions. F rd Pik the pttern Fous: To investigte ptterns for sequene of frtions nd find the sum of the series of frtions This is more hllenging tivity. Students investigte the pttern for sequene of frtions nd lulte the sum of the onvergent series of frtions. whole lss: reflet Possible nswer: The order of opertions used for frtions is the sme s the order of opertions for whole numbers. 0 Q uses the re model to onsolidte shown is orret. Explining possible In Q, students onsider proper to Q nd Q. This my be good involving more thn one opertion. B Brkets student understnding of squring errors llows students to deepen their frtion tht is rised to different opportunity for lss disussion You my need to remind students of the I Indies frtions. understnding of the onept. powers nd ompre the size of the so students n ommunite their order of opertions rules (BIDMAS). Q nd Q require students to nlyse Q involves rising whole numbers nd resulting frtions. They my need to be understnding. Clultor or other digitl tehnology DM Division nd Multiplition possible nswers to identify ommon then frtions to the power of three nd guided to the onlusion tht the lrger Q involves finding the length of use my be pproprite. (left to right) errors mde. explining the method used. Students the power, the smller the resulting squre blok of lnd by lulting the AS Addition nd Subtrtion smll group Q requires tht students simplify the pply this to Q0 nd then extend the frtion. squre root of its re. Students pply (left to right) frtion before finding the squre root. powers to nd in Q nd Q. Similrly, Q involves rising n ddition (or multiplition) skills to Diret students to omplete the pproprite Fous: To understnd rising Q sks students to nlyse why You my need to remind them tht improper frtion to different powers. lulte the perimeter nd division setion of My lerning in their Student frtions to powers greter thn two simplifition ws required before mixed numbers must be onverted to skills to lulte the ost per squre lulting the squre root. improper frtions first. For students kilometre. experiening diffiulty with this F Powers nd squre roots of frtions Professionl Support

16 G Understnding rtios Unit Frtions nd rtios Deep Lerning Kit G rd Compring numbers of objets Intervention tsk G rd Writing rtios Additionl skill prtie Student online support G etutor G Guided exmple G Test yourself tsk Fous: To how to write omprisons s rtio Students look t three different senrios to ompre quntities. They re introdued to the term rtio nd re guided to write eh omprison s rtio. Students re lso guided to reognise tht order is importnt nd tht rtios ompre quntities of the sme kind. whole lss Fous: To onsolidte student understnding of rtios After students omplete the Disover tsk, onsolidte their understnding. Ensure tht students understnd the onept tht rtios n be used to ompre items tht re similr. They must lso reognise tht the order of rtio is importnt. Diret students to G rd for further support. Demonstrte G etutor or diret students to do this independently. Q involves writing rtios when senrios nd sk speifi questions relted to This will reinfore the ide tht the POTENTIAL DIFFICULTY Diret students to the key ides. You re given in number formt only. some omprisons. A strting point for quntities hve to be of similr type. Students must red eh question my like them to opy this summry. Students should note tht the units in questions ould be: For dditionl prtie, diret students the omprison need to be the sme. refully to ensure tht the order of their How mny blue ounters re there? to G rd. In Q, students write rtios for worded rtio is orret. The lrger number or the How mny red ounters re there? As tsk to be undertken either in omprisons. smller number my pper fi rst. The Fous: To write rtios to How n you ompre blue ounters lss or t home, diret students to numbers in the rtio must orrespond represent different senrios to red ounters? omplete the etutor, Guided exmple with the order in the question. nd Test yourself. In Q nd Q, students write rtios ompring different mounts of similr items shown in digrms. Some possible nswers inlude: the number of people wering ties ompred to the number of people not wering ties, the number of people with drk-oloured hir ompred to the number of people with lightoloured hir. 0 Drker shde of pink ompred to the shde produed in question. G Understnding rtios To wht rtio is ll bout, we re going to look t three different senrios. Senrio First, let s look t the photo of group of people. We n mke mny different omprisons. s. For exmple, one ould be to ompre the number of hildren to the number of dults. Cn you see other omprisons? Copy nd omplete the following sentenes. In the photo, there re mles ompred to 0 femles. b In the photo, there re dults ompred to hildren. The totl number of people is the sme in eh prt of question. Wht do you notie bout the totl in eh of the omprisons mde in question? Write four more omprisons using the photo. A omprison of two or more quntities is lled rtio. In the photo, there re hildren ompred to dults, so the rtio n be written s :. Notie tht we use numbers seprted by olon. Rewrite the omprisons from question s rtios using numbers nd olon. :0 Rewrite the omprisons from question s rtios using numbers nd olon. : Senrio Our seond senrio involves mixing pint. To mke prtiulr shde of pink, one prt of red pint nd four prts of white pint re mixed together. Write the bove informtion s rtio. : Wht would mixture with rtio of : represent? Two prts of red pint to three prts of white pint. Desribe the olour tht would be produed with the rtio stted in question. Comment on whether mixture with rtio of : would produe the sme shde s mixture with rtio of :. Clerly explin your resoning. : would produe drker shde s there is more red pint. : would produe slightly lighter shde s there is less red pint. For those students experiening diffiulty with these questions, intervention through smll group tehing of the onept my be neessry. A onrete model n be useful to ssist students with this onept. Give students rnge of oloured ounters 0 Explin why the order of eh mount (prt) listed in the rtio is importnt. The order of eh mount (prt) of pint is importnt beuse the first number reltes to the red pint Senrio nd the seond reltes to the white, nd you need the right proportion of eh to produe the orret shde. Our lst senrio involves two students who re exited bout their next siene lss, s they re going to mke slime. The ingredients required to mke slime re: ups of wter up of ornstrh tespoon of food olouring. After the experiment, the students re required to write report nd inlude omprison of ingredients using rtios. Write the rtio of ornstrh to wter. : b Write the rtio of wter to ornstrh. : Are the two rtios found in question the sme? Explin. No; they represent different omprisons. The order is importnt. The students disuss whether rtio ould be written for the mount of ornstrh used ompred to the mount of food olouring used. They both onluded tht rtio ould not be written. Comment on whether you gree with the students. The quntities of ornstrh nd food olouring nnot be ompred s they re mesured in different units; tht is, ups nd tespoons. Conlusion Let s onsider ll three senrios. Desribe wht quntities or units of mesurement we were ompring in the lss photo, ns of pint nd slime ingredients questions. From your observtions mde in question, opy nd omplete the following sentene. Rtios ompre quntities of the sme kind. Senrio (lss photo) ompres the number of people. Senrio (pint) ompres prts of pint. Senrio (slime ingredients) ompres ups of ingredients. How would you ompre red ounters to blue ounters? This strtegy n be extended to sk students whether you n ompre ounters to something different. key ides A rtio is omprison of two or more quntities of the sme kind. Before writing rtio, the numbers must be in the sme unit of mesurement. Rtios do not require units. Tht is, they re written s whole numbers with no units shown. A rtio must be written in the order of the worded desription given. For exmple, one prt ordil to five prts wter is written s :. Rtios do not ontin frtions or deimls. They only ontin whole numbers. Write these omprisons s rtios. the number of red jelly bens ompred to the number of yellow jelly bens : b the number of ornge jelly bens ompred to the totl number of jelly bens : the number of pink jelly bens ompred to the number of yellow jelly bens ompred to the number of blue jelly bens :: d the number of red nd blue jelly bens ompred to the number of pink nd blk jelly bens : Express eh of these s rtio. i the number of shded prts to the number of non-shded prts ii the number of shded prts to the totl number of prts b i : i : ii : Write eh of these s rtio in the given order. m to 0 m :0 b kg to kg : min to min : d $ to $ : e weeks to weeks : f litres to litres : Write eh of these s rtio in the given order. Collingwood sored points ompred to St Kild s sore of points. : b Kyl spent hours on Febook nd hour doing homework. : The fstest reorded tennis serve for mle is km/h ompred to 0 km/h for femle. :0 d Austrli won three gold, four silver nd nine bronze medls in lst yer s World Swimming Chmpionships. :: e Twenty-eight students tried out for the volleybll tem while students tried out for the netbll tem. : f To mke ordil, pour one prt ordil to four prts wter. : ii : i : ii : G Understnding rtios Deep Lerning Kit G rd Compring numbers of objets Fous: To desribe the number of given objets ompred to nother lot of given objets, with order being importnt Resoures: oloured penils Students ount the number of shpes in digrm nd how to write these vlues s omprison, using the term to. The importne of writing the order orretly for the omprison is emphsised. G rd Writing rtios Fous: To write rtios to represent the omprison of quntities Students ount the number of symbols in wether forest nd represent different omprisons in rtio form, using the symbol : insted of the word to. Additionl prtie questions similr to the Now try these questions re lso provided. whole lss Some possible questions to sk: Is rtio restrited to only hving two numbers? (no) Wht words ould be used to reple the rtio symbol in worded questions? (to, ompred to) Cn m to km be represented by :? (no, units need to be the sme) How n we write the rtio orretly? (:000) Cn frtions nd deimls be inluded in rtios? (no, only whole numbers) Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

17 G Understnding rtios Unit Frtions nd rtios Resoures Deep Lerning Kit G rd Reipes in rtio Problem-solving tsk G rd Bowling strike rtes Problem-solving tsk whole lss Remind students of wht ws lernt in the Disover setion. Some prompts re: Cn the number of stff rs be ompred to the number of mie in the siene lb? (no, the items being ompred should be of similr type) Ask students to stte rtios for the items on their tble. (notebooks to textbooks, pens to penils, lultors to ersers) questions Fous: To pply understnding of rtios to different senrios For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. Q, Q nd Q involve using given informtion in rel-life ontext to write rtios. Q is n open question tht requires students to think of their own exmples to illustrte rtio. Q guides students to reognise tht two quntities of the sme type n be ompred but not written s rtio unless they re in the sme unit. Q involves deiding whether given sttement n be written s rtio. smll group Fous: To write omprisons of quntities in different units s rtios Students struggling with Q my need expliit instrution in onverting units. Disover rds provided for Units nd my provide further support. Refer to C rd : Time units, C rd : Converting units of time, G rd : Converting units of mss, A rd : Converting units of length. Alterntively, these onversion reltionships re pproprite for Q nd n be provided s referene: hour 0 minutes, minute 0 seonds, week dys, yer months, $ ents, tonne 000 kg, L 000 ml, dozen, km 000 m, m 00 m. Deep Lerning Kit G rd Reipes in rtio Fous: To ompre nd determine the quntities of ingredients in ommon reipes using rtios Students use ommon reipes for pnkes, muffins, quihe, sones, frittt nd repes to ompre the mounts of bsi ingredients using rtios. They determine the ingredient quntities in given senrios. G rd Bowling strike rtes Fous: To lulte nd ompre strike rtes for different bowlers using rtios Students lulte bowling strike rtes using the rtio of bowled blls to wiket blls for group of riket plyers nd mke omprisons nd ntiipte likely senrios. diffiulty with this question, strtegies Agin, students my need guidne of one quntity to nother is written s Q relies on students understnding whole lss: reflet re disussed in the smll group setion s to how to find missing vlue by rtio wheres omprison of prt to the soring in Austrlin Rules footbll. This llows students to demonstrte Possible nswer: Comprisons of similr tht follows. performing lultion first. whole is generlly written s frtion. They need to know tht gol sores things n be written s rtios. their understnding tht rtios re ompring quntities of the sme type. For quntities of the sme type but in different units, students should reognise tht rtio n still be written one eh quntity is onverted to the sme unit. In Q, students onvert units so tht rtio n be written for eh omprison. For students experiening Convert the quntities to the sme unit nd then write the rtio for eh of these omprisons. 0 dys to weeks 0: b minutes to hours :0 m to. m :0 Write the number of footbll plyers shown d ents to $. : e kg to. tonne :00 f ml to. L :00 to the number of umpires s rtio. : : g months to yers :0 h bnns to dozen bnns i ml to L :000 Write eh of these quntities s rtio. j hours to minutes 00: k. km to m 00: l seonds to. hours :00 A reipe requires five ups of flour to one up of fruit nd hopped nuts. : In prtiulr survey, out of 00 people preferred Dzzle toothpste to their regulr brnd. The reminder preferred their regulr brnd. Write rtio for eh of these. b A mixture ontins kg of snd, kg of ement nd kg of grvel. :: the number of people tht prefer Dzzle toothpste to the totl number of people surveyed :00 In group of 00 people, re left-hnded, re both left- nd :: b the number of people tht prefer their regulr brnd of toothpste to the totl number of people surveyed :00 the number of people tht prefer Dzzle toothpste to the number of people who prefer their regulr right-hnded nd the rest re right-hnded. brnd : d Brittny nd Luren ontribute eqully to Mx obtined sore of 00 (or out of 00) for his Geogrphy test. friend s present. : Write rtio for the mrks reeived ompred to the totl number of mrks. :00 e There re pproximtely three mle pssengers to every five b Write rtio for the mrks lost ompred to the totl number of mrks. :00 femle pssengers on the trin in the morning. : DAY FORECAST Write rtio for the mrks reeived ompred to the mrks lost. : List n exmple to illustrte eh of these rtios. You my like to use digrm to show your exmple. tody mx showers 0 Look t the pentgon shown. : b : : d : tomorrow mx showers Wht is the rtio of the shded setions to the non-shded setions? : Mondy mx windy b Wht is the rtio of the non-shded setions to the totl number of setions? : From the given seven-dy forest, write rtios for the following. Tuesdy mx showers Wht is the rtio of the shded setions to the totl number of setions? : the number of windy dys to the number of sunny dys : Wednesdy mx sunny d Comment on whether ny of the rtios in prts, b nd n be expressed s frtions. b the number of showery dys to the totl number of dys : Explin your resoning. Prts b ( ) nd ( ) n be written s frtions s they re expressed s prt of the whole. Thursdy mx sunny the number of dys with mximum temperture of C to e Comment on whether ny of the rtios in prts, b nd nnot be expressed s frtions. Explin your the number of dys with mximum temperture of C to the Fridy mx sunny resoning. Prt nnot be written s frtion s it is ompring shded to non-shded setions. number of dys with mximum temperture of C :: f Copy nd omplete the following sentene. Consider this elephnt nd its hndler. The mss of eh Rtios n be expressed s frtions when ompring the rtio of prt to its whole. Convert one of the is shown. Msses re in different units. vlues so tht the Geoff mixes litres of blue pint with litres of yellow pint to rete shde of green for his studio. Explin why the rtio of the elephnt hndler s mss to mesurements re in Write the rtio of the mount of blue pint to the mount of green pint. : the elephnt s mss nnot be expressed s :. the sme unit. b Suggest wht ould be done to one of the mesurements b Wht frtion of the new shde is blue pint? The bigger unit (tonnes) ws so tht rtio ompring their msses n be written. tonnes Wht frtion of the new shde is yellow pint? onverted to the Disuss whih mesurement ws seleted nd why. Elio nd Getno work t the lol supermrket nd re lled in fter shool to over some shifts. They both smller unit (kilogrms). d Write the rtio of the elephnt hndler s mss to the strt t pm. Elio is ble to work till pm while Getno must leve t pm. elephnt s mss. (Hint: 000 kg tonne.) 000: kg Wht is the rtio of the number of hours worked by Elio ompred to those worked by Getno? : Stte whether eh of these n be written s rtios. Give b Wht is the totl mount of time worked by both boys? hours reson for your nswer. Wht frtion of the totl mount of time did Elio work? The rtio is :. Gols n be onverted to points Yes; both quntities re of ( gol points) so tht the sores n be ompred. m ompred to m the sme kind (distne). Brry ould not understnd why his response to n Aussie rules question ws mrked inorret. The question Yes; both b ml of juie ompred to kg of hiken No; the quntities sked whether 0 gols nd behinds ompred to points ould be written s rtio, to whih Brry quntities re of the sme kind $ ompred to 0 ents re not of the sme replied No, sine the quntities involved different units. Explin to Brry why his response ws inorret. (mount of kind. The first is d km ompred to hours money) lthough pity in ml Mke up four omprisons nd sk your friends whether they re shown in nd the other is No; the quntities re not of the sme kind. The first rtio n be written for eh, stting resons for their reflet Wht type of omprisons different units. mss in kg. is distne in km nd the other is n mount of nswers. n be written s rtio? time in hours. Q involves interpreting informtion from worded question to write the required rtios. They my need to be reminded tht order is importnt nd tht lultions suh s subtrtion my need to be performed to obtin the required vlues. In Q, students the notion tht test result n be written s rtio. Q0 involves the nlysis of shded prts nd non-shded prts of given shpe. Students the reltionship between rtios nd frtions. In Q, students onsider the rtio of blue pint to yellow pint but lso need to lulte the totl mount of green pint mde so frtion n be formed. They should be wre tht omprison Similrly, Q involves writing rtio when ompring the number of hours worked by two employees t lol supermrket but frtion when ompring the time worked by one employee to the totl mount of time worked. points nd behind sores point. Converting both mounts to points enbles the omprison to be written s rtio. Q is n open, thinking question in whih students write omprisons of their own nd then swp them with lssmte to deide if writing rtio is pproprite. G Understnding rtios Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

18 H Working with rtios Unit Frtions nd rtios Deep Lerning Kit H rd Simplifying rtios Additionl skill prtie Professionl support online File.0: Rtio retngles Reproduible mster Student online support H etutor H Guided exmple H Test yourself tsk Fous: To equivlent rtios nd how to obtin them Students build on their understnding of how rtio is written nd tht order is importnt. Students re guided to equivlent rtios, whih re different wys of representing the sme mount. As with equivlent frtions, students n write equivlent rtios if they perform the sme opertion (multiplition or division) on ll numbers in the originl rtio. whole lss Fous: To onsolidte student understnding of equivlent rtios After students omplete the Disover tsk, onsolidte their understnding. Ensure students reognise tht equivlent rtios n be reted by multiplying or dividing the originl rtio by the sme ftor. They lso need to reognise tht rtios n be simplified in muh the sme wy s frtions, by dividing by the HCF. found using multiplition pttern not n ddition pttern. For dditionl prtie, diret students to H rd. As tsk to be undertken either in lss or t home, diret students to omplete the etutor, Guided exmple nd Test yourself. POTENTIAL DIFFICULTY Some students my struggle to fi nd the HCF of the numbers in rtio. Ressure them tht they n still fi nd the simplifi ed rtio by performing more thn one division. As long s they n identify ommon ftor, they n repet the proess of dividing until they reh the simplest form of the rtio. Deep Lerning Kit H rd Simplifying rtios Fous: To write rtios in simplest form Students identify nd then divide by the HCF of the rtio. They prtise simplifying number of rtios in this wy. Students re lso given omprisons where the units re different. They onvert the quntities to the sme unit before writing s rtio nd simplifying. Additionl prtie questions similr to the Now try these questions re lso provided. whole lss Some possible questions to sk: Wht opertions n be used to rete equivlent rtios? (multiplition nd division) Wht do we divide by when simplifying rtios? (HCF) Find the HCF for these rtios nd simplify. : (, :) were not hnged to m. Remind Q involves expressing rtios in simplest through smll group tehing of the the other three retngles ( of 0 prts, students tht rtios nnot be written form. Students need to identify the HCF. onept my be neessry. of prts nd of 0 prts). Write b : (, :) Demonstrte H etutor, or diret with frtions or deimls. This exmple In Q, students re sked to find Students need to reognise tht rtios for the number of shded prts students to do this independently. lso demonstrtes tht rtios should be 00:0 (0, :) written in simplest form. d 0: (, :) Diret students to the exmples. Exmple demonstrtes how to simplify the rtio :, by dividing both numbers in the rtio by the HCF of. Exmple shows how the quntities in the omprison first need to be expressed in the sme unit (mm) so tht rtio n be written. It would be useful to disuss why the quntities try A:, try B: 0, try C:, try D: 0 Diret students to the key ides. You my like them to opy this summry H Working with rtios We hve seen previously tht when we ompre the rtio of prt to its whole or totl, the rtio n be expressed s frtion. Now we will whether there re other similrities between frtions nd rtios. For fmily lunh, four trys of lsgne were bked. The lsgne in the first try ws ut into equl retngulr slies, the seond into 0 equl retngulr slies, the third into equl retngulr slies nd the fourth into 0 equl retngulr slies. After the lunh, it ws notied tht the sme mount of lsgne hd been eten from eh try. Copy nd omplete the following tble for eh try of lsgne. Try Number of slies eten Originl number of slies A B C D For eh try of lsgne, write s rtio the number of slies eten ompred to the originl number of slies. try A: :, try B: :0, try C: :, try D: :0 Look for pttern in your nswers to question. Write the next three rtios you would expet. 0:, :0, : Use your nswers to question to write eh rtio s frtion. Eh frtion represents the frtion of Explin wht eh frtion represents. lsgn eten from the whole try. Tht is, the number of slies eten out of the totl number of slies in the try. Compre the frtions you hve obtined. Wht is speil bout them? They re equivlent frtions. D Write the three rtios you listed for question s frtions. Are these 0 frtions equivlent to eh other? How n you tell?, 0,. These frtions re equivlent s eh n be simplified to The rtios listed in questions nd re lled equivlent rtios.. Like frtions, rtios n be written in their simplest form by dividing eh number How ould you show tht two rtios re equivlent? of the rtio by the sme vlue. Two rtios re equivlent if you obtin the sme rtio fter simplifying eh. Consider this retngle, whih hs been divided into equl prts. Write s rtio the number of shded prts ompred to the totl number of prts. 0:0 0 Write two equivlent rtios tht ontin lrger numbers thn those obtined in question. Some possible nswers re: 0:0, 0:0. Eh number in the originl b Explin how you reted these equivlent rtios. rtio is multiplied by the sme vlue. For exmple, the equivlent rtio 0:0 is obtined by multiplying 0 nd 0 by. Write two equivlent rtios tht ontin smller numbers thn those obtined in question. :0, : Eh number in the originl rtio b How were these equivlent rtios reted? is divided by the sme vlue. From your observtions mde in questions 0 nd, opy nd omplete the following sentene. Numbers in rtio n be multiplied or divided by the sme vlue to rete n equivlent rtio. Fous: To simplify rtios by finding equivlent rtios 0 0 equivlent rtios. They identify the vlue tht hs been multiplied or divided to the rtio nd then pply the sme vlue to omplete the rtios. Q llows students to prtise onverting one or more omponents of the rtio so tht the units re the sme before simplifying. A B C For those students experiening diffiulty with these questions, intervention exmple Express the rtio : in simplest form. think equivlent rtios represent the sme mount. Digrms similr to those in the Disover tsk n be used to demonstrte this onept. File.0: Rtio retngles ontins retngles tht re the sme size but divided into different number of prts (, 0, nd 0). Guide students to shde two-fifths of the first retngle ( of prts) nd then shde the equivlent mount in eh of write Write the rtio nd determine the HCF of eh number : in the rtio. HCF. : Divide eh number in the rtio by the : HCF of. : exmple Write the omprison m to mm s rtio in simplest form. think write Write the quntities in the order given. m to mm Convert the quntities to the sme unit. Tht is, express them 0 mm to mm in the smller unit (mm). Remember tht m 0 mm. Write the omprison s rtio now tht the quntities re 0: in the sme unit. Divide eh number in the rtio by the HCF. HCF. 0: (0 0 nd.) key ides Numbers in rtio n be multiplied or divided by the Note: rtios n be simplified in more thn sme vlue to rete n equivlent rtio. one step if it is diffiult to work out the HCF. Rtios re expressed with whole numbers only. In exmple, we ould hve first divided both numbers by the ommon ftor of A rtio my be expressed in simplest form by dividing the nd then divided the resulting numbers by numbers in the rtio by the HCF. the ommon ftor of to obtin the rtio in simplest form. Try it for yourself! e Express eh of these rtios in simplest form. : : b : : : : d : : e : : f 0: : g : h : i 0:0:0 j :: k :: l ::0 Fill in the gps to omplete the equivlent rtios. : : : 0 b : 0 : : 0 0 :0 0:0 : d : : : e : 0: 0 : f : : : e Write eh of these omprisons s rtios in simplest form. Remember to first onvert to the sme unit in eh se. 00 g to. kg : b ml to. L : to $. : d m to 0. km :0 e h min to min : f weeks to dys : g weeks to yers h.0 million to i seonds to minutes : : :: :: :: :: : : : ompred to the number of unshded prts. Compre the rtios to see tht they represent the sme omprisons but re expressed differently. Repet this tsk but this time strt by shding onefifth or four-fifths. (Note: this file n lso be used to repet the Disover tsk.) By using the digrms, it n be emphsised tht equivlent rtios re H Working with rtios Diret students to omplete the pproprite setion of My lerning in their Student Professionl Support

19 H Working with rtios Unit Frtions nd rtios Deep Lerning Kit H rd Fuelling motorbikes nd go-krts Problem-solving tsk H rd Quilt ptterns Problem-solving tsk whole lss Remind students of wht ws lernt in the Disover lesson. Some prompts re: Find the HCF for these rtios nd simplify. 000:00:00 (00, 0::) ::0 (, ::0) :: (, ::) questions Fous: To pply understnding of rtios, equivlent rtios nd simplifying rtios to different types of questions For some students, it my be pproprite not to omplete ll the questions but to work on one or more or rds to develop nd deepen their understnding. Q involve worded questions for whih students gther the informtion tht they need either from the written question, pitures or tble. The rtios need to be written in simplest form. No unit onversions re required. In Q, students use mesurements in the photos to write rtios in simplest form. They need to ensure tht the quntities re whole numbers in the sme units. Q is n open, thinking question in whih students write three different rtios tht n be simplified to :. A fruit bowl ontins pples, ornges nd bnns. Write the rtio of the number of pples to the number of ornges in simplest form. : b Write the rtio of the number of pples to the number of bnns in simplest form. : Write the rtio of the number of ornges to the number of bnns to the number of pples in simplest form. :: In n upset win in n AFL round of footbll, Crlton bet Geelong with sore of points to points. Write the rtio of the sores in simplest form. : In lss of students, re girls. How mny students re boys? b Write the rtio of girls to boys in simplest form. : A brekfst erel mix is mde using kg of ots, kg of dried fruit nd kg of whet flkes. Write the rtio of ots to dried fruit to whet flkes in simplest form. :: b Express eh ingredient s frtion of the whole mix.,, Lst week, Shnnen ompleted extr hores round home nd erned $. This week, she erned $ less. : Write s rtio, in simplest form, the mount of lst week s poket money to this week s poket money. The tble shows the ttendne t eh performne of the lol shool prodution. Performne Number of people ttending Thursdy evening Fridy evening Sturdy mtinee 0 Use the photos to write eh of the following rtios in simplest form. The rtio of the height of the tller person to the height of the shorter person. : b The rtio of the mount of liquid in the ornge juie rton to the mount of liquid in the milk rton. :0 Sturdy evening 0 The rtio of the length of netbll ourt to its width is :. Wht is the width of the ourt? m Find the totl ttendne to the shool prodution over the three dys. The rtio of the number of supporters for the visiting tem to the b Write s rtio in simplest form: number of supporters for the home tem t Sturdy s gme ws i the number ttending Thursdy s performne to the number ttending Fridy s performne : 0:. If there were 00 supporters for the visiting tem, how 0 m ii the number ttending Sturdy s mtinee performne to the number ttending Sturdy s evening mny supporters for the home tem were there? 0 performne. : Provide three different rel-world exmples of rtios tht If the venue n set mximum of 0 people, write s rtio in simplest form the totl ttendne to the re equivlent to :. You my like to use digrms to reflet How n equivlent rtios totl possible ttendne over the three dys. :0 illustrte your rtios. be used to solve problems?. m m Some possible nswers re: :, 0:0, Write three rtios tht simplify to :. 0:. Some possible nswers Write three rtios tht re equivlent to :0. re: :, 0:00, :. 0 Brett is unsure if the rtios 0: nd : re equivlent. Explin to Brett how he n determine whether the pir of rtios re equivlent. Write eh rtio in its simplest form. They both simplify to :. Determine whether the following pirs of rtios re equivlent, providing reson for your nswer. Yes, sine the rtios n Yes, sine the rtios n No, sine the rtios : nd : be expressed in the sme b 0: nd : be expressed in the sme : nd : nnot be expressed simplest form. simplest form. in the sme simplest Crete equivlent rtios by finding the missing numbers. form. : : 0 b 0:0 : 00:00 : d : 000:0 000 e : : f : 0:0 The rtio of ts to dogs t the niml shelter is :. If there re ts, how mny dogs re t the niml shelter? The rtio of boys to girls in shool orhestr is :, nd there re girls in it. How mny boys re in the orhestr? 0 b How mny students re in the shool orhestr? The rtio of pple juie to sod wter for fruit punh reipe is :. If the reipe requires 0 ml of pple juie, how muh sod wter is required? 0 ml H Working with rtios equivlent rtio nd the pproprite number in the originl rtio to be multiplied by this ftor. Deep Lerning Kit H rd Fuelling motorbikes nd go-krts Fous: To lulte the quntities of petrol nd engine oil to mke -stroke fuel using the onepts of rtio Students re to determine the quntities of petrol nd -stroke engine oil to mke fuel for motorbikes nd go-krts using speified rtios for eh vehile. As n extension, they onsider how 0: fuel mix n be hnged to : mix, nd vie vers. H rd Quilt ptterns Fous: To determine the quntities of shpes nd olour required to mke given quilt design using the onepts of rtio Resoures: Internet ess (optionl) Students re guided to reognise geometri ptterns nd rtio quntities in repeted pttern quilt design. As n extension, they n design their own quilt. You might wish to mke vilble some other interesting quilt designs bsed on repeting rtio pttern. whole lss: reflet Possible nswer: Equivlent rtios n be used to solve problems when missing mount needs to be identified. You need to first work out the multiplition ftor used to produe the equivlent rtio. Diret students to omplete the pproprite setion of My lerning in their Student In Q, students lulte the missing experiening diffiulty with this smll group quntity desription in the orret order Similrly, Q involves writing three vlue to omplete n equivlent rtio. onept, strtegies re disussed in the bove the rtios. equivlent rtios to :0. This The problems provided re more smll group setion tht follows. Fous: To use equivlent rtios to For exmple, in Q, the setting out llows students to demonstrte their diffiult thn in erlier questions. solve problems ould be: understnding of equivlent rtios. Remind students to relte first ple to ts to dogs first ple nd seond ple to seond ple. : :? Q0 is more hllenging question in whih students onsider whether two rtios re equivlent. You my like to disuss the strtegy of writing eh rtio in its simplest form. This onept is further d in Q. Q involve solving worded problems. In eh se, students write the given rtio nd the inomplete equivlent rtio nd then proeed s they did in Q. For students Q is n open, thinking question in whih students write rtios involving rel-life senrios tht result in simplified rtio of :. You my like them to work with prtner or in smll groups. They re enourged to use digrms to represent their rtios visully. Students who experiene diffiulty with solving worded problems my struggle with interpreting the given informtion. Enourge them to set out the informtion in logil wy by writing the rtio nd the inomplete equivlent rtio under one nother. It is lso benefiil to write the This setting out emphsises the orret order nd highlights the unknown vlue. It mkes it esier for students to identify both the ftor needed to produe the Professionl Support

20 Frtions nd rtios revise hoose Answer: D. prts out of, A: ounted number of blue prts to obtin. B: ounted totl number of prts s 0 insted of. C: ounted number of prts tht re not blue. Answer: B.. obtined by multiplying numertor nd denomintor by. Answer: D. _ + _ + A: dded numertors together nd denomintors together. B: multiplied numertors together nd denomintors together. C: subtrted numertors nd subtrted denomintors. Answer: A. of _ _ B: dded numertors together nd denomintors together. C: subtrted numertors nd subtrted denomintors. D: ross-multiplied to obtin. Answer: B. 0 0 _ A: divided the numertors ( ) nd divided the denomintors (0 ) to obtin. C: subtrted the numertors ( ) nd subtrted the denomintors (0 ) to obtin. D: dded the numertors ( + ) nd dded the denomintors (0 + ) to obtin. Answer: B. ( ) _ _ A: both numertor nd denomintor hve been multiplied by. Frtions nd rtios revise E Clulte eh of these. b A survey of 00 fmilies found tht 0 hve one summrise pet nd 00 hve two pets. d Crete summry of the unit using the key terms below. You my like to write prgrph, rete onept mp Wht frtion of fmilies hve one pet? Write or use tehnology to present your work. E Clulte eh of these. this in simplest form. frtion improper frtion highest ommon ftor (HCF) squre root b b Wht frtion of fmilies hve two pets? Write this in simplest form. numertor proper frtion lowest ommon denomintor (LCD) rtio d denomintor equivlent frtions reiprol equivlent rtios Write rtio for the number of fmilies with 0 Clulte eh of these. one pet ompred to those with two pets. : vinulum simplifying power F ( ) b ( ) ( 0 ) d ( ) d The survey lso found tht of fmilies hve mixed number nelling squre 000 three or more pets. How mny fmilies hve F Clulte eh of these. three or more pets? b d hoose nswer e Wht frtion of fmilies do not hve pets? How mny fmilies is this?, fmilies A The frtion of the shpe A Copy nd omplete this tble. G Use the photo to write eh of the following Corret bout f A news reporter sid tht in every 0 people omprisons s rtios. one pet, tht is oloured blue is: hve one pet, nd in every 0 people hve two A B Mixed number Improper frtion beuse pets. Are they orret? Use your knowledge of 0 0, but not equivlent frtions to support your nswer. C D b bout two pets, beuse 0. Brendn wnts to buy new mp plyer worth $0. B An equivlent frtion to is: He reeives $0 poket money every week. A If he sves of his poket money eh week, B C 0 D d 0 how muh is this? + $ C is equl to: B Simplify eh of these frtions. b How mny weeks would it tke to sve $0? A B C D 0 b 0 00 d How muh would he hve sved fter weeks? $ d Express your nswer to prt s frtion of his Wht is of? B Ple these frtions in sending order. D weekly poket money in the form of: A B 0 C D,,,,, 0,, 0,,, i n improper frtion C Clulte eh of these. the number of pink Smrties to the number ii mixed number. 0 E is equl to: of yellow Smrties e If, fter weeks, Brendn spends of his A B C D + b : b the number of yellow Smrties to the svings, how muh does he hve left? d 0 + $ 0 number of green Smrties f Brendn hs sved of his weekly poket The vlue of ( : F ) is: C Clulte eh of these. the number of ornge Smrties to the totl money. How muh is this nd how long would it A B C D + b number of Smrties : tke t the rte mentioned in prt? $, weeks G Written s rtio, 0 m to m is: H Write the following s rtios in their simplest g If he insted sved of his poket money d + form. eh week, how long would it tke to hieve A 0: B 0:0 D Clulte eh of these. the mount of svings lulted in prt f? : (Hint: divide by b :0.) C 0:00 D 0:000 b 0 : weeks : h Write the initil mount sved eh week A mixture of 00 ml ordil to. litres of 0 0 d H to $.0 : (prt ) ompred to the seond mount sved wter expressed s rtio in its simplest form d weeks to dys eh week (prt g) s rtio in simplest form. is equl to: D Clulte eh of these. : : H The rtio of snkes to lizrds in reptile zoo A 00:. B 00:00 b is :. If there re 0 lizrds, how mny snkes ) _ ) _ 0 ( _ _ b ( _ _ ( 0) _ 000 ) _ d ( _ _ _ _ _ b _ _ _ d _ pink to yellow : b yellow to green : ornge to totl : :, HCF, : b :0, HCF, : :$.0, :0, HCF, : d weeks: dys, : HCF, : snkes to lizrds :?:0 Ftor 0 so. There re snkes in the zoo. nlyse 0 00 _ b _ 0:00 : d e + _ of fmilies hve no pets. Number of fmilies with no pets 00 f in 0 0 _ ; this mthes the survey results for one pet. in 0 _ ; in the survey, of people 0 hd two pets. These frtions re not equivlent. 0 C: hs been dded to both the C : D d re there in the zoo? numertor nd the denomintor. He sves $ eh week. b 0 D: the numertor hs been ubed but the It will tke weeks to sve $0. denomintor hs only been squred. After weeks he will hve sved Answer: C. 0 m to m 0 m to B: rtio orret but not simplified. HCF, _ d _ _ + _ + + _ 0 _ b HCF, _ $. 00 m 0:00 D: the vlues of the two quntities hve HCF 0, 0 b _ _ 00 _ d HCF, _ d i 0 ii _ 0 Two mixed numbers: A: m not onverted to m before writing been dded together. nd b e _ _ _ _ _ _ _ rtio. Remining frtions to hve ommon nswer denomintor of 0. d _ He hs spent $ nd hs $ left. _ + + _ + 0 B: mistkenly used m 0 m rther f Mixed number Improper frtion d _ 0 0 _ _ 0, _ 0 0, 0 0, _ thn m 00 m. 0 He hs sved $. Asending order is smllest to lrgest. b 0 _ _ _ _ D: mistkenly used m 000 m rther This will tke weeks., _, 0, _, _, _ thn m 00 m. b _ 0 0 g _ _ _ weeks _ + b Answer: C. 00 ml to. L 00 ml to d b _ or _ 0 0 _ _ $ per week _ 00 ml 00:00 : _ _ d _ $ sved in weeks. d A: L not onverted to ml before writing 0 + _ b _ d _ _ _ h : : rtio. _ _ nlyse A Understnding Frtions nd frtions rtios Professionl Support

21 Frtions nd rtios onnet Deep Lerning Kit onnet rd The big ke stll Problem-solving tsk onnet rd Designing stirse sulpture Problem-solving tsk Professionl support online File.0: Ctering for birthdy prty Assessment rubri File.0: The big ke stll Assessment rubri File.: Designing stirse sulpture Assessment rubri onnet tsk Fous: To use fmilir ontext to onnet the key ides of frtions nd rtios Students re to pply their frtion nd rtio skills to prepre food nd drink for prty. This is n open tsk in whih students will need to dpt reipes designed for smll number of prty guests to ter for mny more prty guests. The tsk requirements re expressed using everydy lnguge so tht students n use the urrent quntities s the bsis for their new reipes. Students work with quntities mesured in kilogrms, grms, ups, tespoons nd tblespoons. 0 Frtions nd rtios onnet Ctering for birthdy prty Food, musi nd gret ompny mke for suessful prty. When you re plnning prty it is importnt to know how mny people re ttending nd to estimte how muh food nd drink eh person my onsume. How re frtions nd rtios relted to prepring food for birthdy prty? Complete the Connet setion on pges of your Student Progress s Book to show ll your working nd nswers to this tsk. In totl, people (inluding the hosts) re expeted to be t the prty. Of these people, re mle nd 0 re femle. The menu will inlude: pizz, hiken, sld, fruit, drinks nd ke. Ingredients required for some of the reipes re given. Smoky brbeue hiken kebbs (serves ) kg skinless nd boneless hiken thighs tblespoons olive oil pit bred mixed sld leves Sue: brown onions up golden syrup up ider vinegr up wter ups tomto puree up Worestershire sue tblespoons hot English mustrd up olive oil tespoons smoked ground pprik tespoon hot hilli powder up brown sugr se slt nd rked blk pepper Choolte brownie nd rspberry trifles (serves ) 00 g drk hoolte 0 g unslted butter ups brown sugr eggs up oo powder up plin flour tespoon bking powder 0 g punnets rspberries up rème de ssis ups rem up iing sugr Your tsk is to: lulte the number of pizzs required for the prty determine the new quntities of ingredients required to ter for people, given tht eh reipe serves eight people determine the new quntities of ingredients required to ter for ny given number of guests obtin reipe for fruit punh nd djust it to ter for the number of guests lulte the rtio of mles to femles t the prty determine whih quntities n nd nnot be ompred s rtio. Besides reording ll working nd nswers to the tsk in your Student Progress Book, you my like to present your findings s report. Your report ould be in the form of: poster booklet PowerPoint presenttion video other (hek with your teher). Choolte rmel slie (serves ) up plin flour up desited oonut up brown sugr 0 g butter up golden syrup 00 g ns sweetened ondensed milk g drk hoolte tespoons vegetble oil As dditionl preprtion for the unit test, omplete the Review on pges 0 of the Student You my like to hve students disuss the tsk requirements in smll groups to identify strtegies for: problem-solving proess. Students n An ssessment rubri n be Deep Lerning Kit Students onsider the: estimting the number of pizzs use this s foundtion to presenting downloded from Professionl Support required onnet rd number of possible ustomers their findings in report. online (see file.0). The big ke stll lulting the new quntities needed for eh reipe to ter for people tering for ny number of guests finding reipe for fruit punh. Diret students to omplete the pproprite setion of Connet in their Student Progress Book (pges ). This setion provides sffolding to the tsk to guide students through the Enourge students to be retive in presenting their report but stress tht they should show orret lultions with pproprite resoning. They need to justify their findings nd inlude ny ssumptions they hve mde. Smple nswers re provided to the onnet tsk (see pges ). Students n undertke n lterntive or dditionl onnet tsk using the investigtions provided in the Deep Lerning Kit. Diret them to onnet rd : The big ke stll nd/or onnet rd : Designing stirse sulpture. Fous: To onnet the key ides of frtion nd rtio to prepre for lss ke stll Students re initilly guided to ompre reipe quntities using frtions nd rtios. They then prepre detiled report determining the quntities of ingredients required nd how mny items will be for sle. number of bthes of eh tret to be mde number of slies eh ke is to be ut into list of required ingredients mount of wsted ingredients tht re left over from reipe. As n extension, students n ost ll spets of this tivity. Frtions nd rtios An ssessment rubri n be downloded from the Professionl Support online (see file :0). onnet rd Designing stirse sulpture Fous: To onnet the key ides of frtions nd rtio to design stirse sulpture Resoures: ruler, lultor, rdbord (optionl), stiky tpe (optionl) This is hllenging, retive, open-ended tivity. Students re to show ll their lultions nd digrms to support their design. Students onsider the: length, width nd height of the first step of their design frtion to use to determine the mesurements for the seond step length, width nd height of the seond, third, fourth, et, steps type of objet to ple on the steps rtio of the size of objets height of eh objet. As n extension, they my like to mke sle model of their sulpture design. An ssessment rubri n be downloded from the Professionl Support online (see file..) Professionl Support

22 Frtions nd rtios Student Progress Book onnet If piees of pizz re left over t the end of the night, how mny of the pizzs were eten? Number of piees eten _ There were whole pizzs eten nd three piees from nother pizz. If only three-qurters of up of flour is vilble, find the quntities required to mke three-qurters of the hoolte brownie nd rspberry trifle reipe. Ensure tht the quntities of ingredients for the fruit punh will ter for the number of guests. Adjust them if neessry. Explin the resoning used s to how muh beverge you hve llowed for eh guest. Ctering for birthdy prty How re frtions nd rtios relted to prepring food for birthdy prty? In totl, people (inluding the hosts) re expeted to be t the prty. Of these people, re mle nd 0 re femle. The menu will inlude pizz, hiken, sld, fruit, beverges nd ke. From pst experiene, mles tend to et more food thn femles. So when ordering pizz, it is estimted tht the mles will eh hve four piees while the femles will eh hve three piees of pizz. If eh pizz is divided into eight piees, how mny pizzs will be required? mles: piees femles: 0 0 piees totl number of piees + 0 number of pizzs 0 pizzs will be required. How mny piees of pizz will remin? Wht frtion of the pizz does this represent? Number of piees in pizzs There will be two piees remining. Frtion remining from one pizz _ or _ Frtion remining from totl mount of pizz _ or _ The lists of ingredients for smoky brbeue hiken kebbs nd hoolte brownie nd rspberry trifles re given on pges 0 of your Student Book. Eh reipe serves eight. Find the quntities of eh ingredient required to ter for people. Multiply eh mount by. Smoky brbeue hiken kebbs (serves ) kg hiken thighs, tblespoons olive oil, pit bred, mixed sld leves, brown onions, _ ups golden syrup, ups ider vinegr, ups wter, _ ups tomto puree, up Worestershire sue, tblespoons hot English mustrd, _ up olive oil, tespoons smoked ground pprik, _ tespoons hot hilli powder, up brown sugr, slt nd pepper Choolte brownie nd rspberry trifles (serves ) 00 g drk hoolte, 0 g unslted butter, _ ups brown sugr, eggs, up oo powder, ups plin flour, _ tespoon bking powder, 0 g punnets rspberries, ups rème de ssis, ups rem, _ up iing sugr Find the quntities of eh ingredient required if you were tering for people. Hlve the mounts lulted in Q (or multiply the originl mounts by _ ). Multiply eh quntity by _. Choolte brownie nd rspberry trifles 0 g drk hoolte, _ g unslted butter, _ ups brown sugr, eggs, _ up oo powder, _ up plin flour, _ tespoon bking powder, 0 g punnets rspberries, _ ups rème de ssis, _ ups rem, _ up iing sugr Smoky brbeue hiken kebbs (serves ) kg hiken thighs, tblespoons olive oil, pit bred, mixed sld leves, brown onions, _ up golden syrup, _ ups ider vinegr, up wter, _ ups tomto puree, _ up Worestershire sue, tblespoons hot English mustrd, _ up olive oil, tespoons smoked ground pprik, _ tespoon hot hilli powder, _ up brown sugr, slt nd pepper The originl reipe for hoolte brownie nd rspberry trifles serves eight. How mny will the new reipe serve? Choolte brownie nd rspberry trifles (serves ) 00 g drk hoolte, g unslted butter, _ ups brown sugr, eggs, _ up oo powder, _ ups plin flour, _ tespoon bking powder, 0 g punnets rspberries, _ ups rème de ssis, ups rem, _ up iing sugr _ Serves six people Using ookbook, the Internet or by other mens, obtin reipe for non-loholi fruit punh nd list the ingredients. Answers will vry. One possible nswer is: Wtermelon fruit punh ( drinks) kg wtermelon peeled, hopped litre pinepple juie _ up hopped fresh mint leves litre sprkling minerl wter Answers will vry. It is estimted mles will eh onsume drinks nd femles drinks. Totl drinks + 0 drinks. Need to multiply by to ter for drinks. Wtermelon fruit punh ( drinks) kg wtermelon peeled, hopped litres pinepple juie ups hopped fresh mint leves litres sprkling minerl wter 0 Clulte the rtio of expeted number of mles to femles t the prty. :0 or : The list of ingredients for hoolte rmel slie is given on pge of your Student Book. Explin whih ingredients n be ompred s rtio nd if there re ny ingredients tht nnot. Cup mesures n be ompred s rtio (plin flour, oonut, brown sugr, golden syrup). Mss in grms n be ompred s rtio (butter, sweetened ondensed milk, drk hoolte). Vegetble oil (in tespoons) nnot be ompred to ny of the other quntities s rtio. If ll the invited guests s well s four unexpeted guests (two mles nd two femles) ttended the prty, lulte the new rtio of number of mles to femles. : or : Comment on how the rtio in question ompres with tht obtined in question 0. Even though n equl number of mles nd femles rrived s unexpeted guests, the rtios re not equivlent. The proportion of mles to femles t the prty is now slightly lower. Originl rtio of mles to femles: : : New rtio of mles to femles: : 0: Professionl Support