Interacting with mathematics in Key Stage 3. Year 9 proportional reasoning: mini-pack

Transcription

1 Intercting with mthemtics in Key Stge Yer 9 proportionl resoning: mini-pck

2 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

3 Contents Yer 9 proportionl resoning: smple unit Introduction Ojectives Differentition Resources 6 Key mthemticl terms nd nottion 6 Unit pln 8 Supplementry mterils Prompts for orl nd mentl strters: phse 1 10 Prompts for orl nd mentl strters: phse 1 Prompts for orl nd mentl strters: phse 1 Prompts for min ctivities in phse 1 1 Prompts for min ctivities in phse 1 Prompts for min ctivities in phse Prompts for finl plenry in phse 8 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

4 Yer 9 proportionl resoning: smple unit Introduction This unit is sequel to the Yer 8 multiplictive reltionships unit. It provides n opportunity to revise, consolidte nd extend ides introduced in Yer 8 nd to mke links to other mthemticl strnds, prticulrly shpe nd spce. Mking such links, especilly with visul contexts, cn help pupils to understnd proportion. There is perhps no other single concept in secondry mthemtics which is so importnt nd so pervsive. Drwing ttention to multiplictive reltionships s they rise in other topics provides opportunities to ssess pupils understnding of the key ides nd their ility to use nd pply these. For some pupils this unit will represent second opportunity to come to terms with the ig ides involved in multiplictive thinking. Other pupils my hve een using nd pplying strtegies developed from the Yer 8 unit nd will e redy to refocus on proportionl resoning nd develop more sophisticted methods for solving prolems. This Yer 9 unit hs een developed through flexile use of the Smple medium-term plns for mthemtics. In plnning the unit severl decisions were mde tht ffect the medium-term plns for mthemtics. The ojectives for the unit re minly drwn from Numer 1, with some from Numer nd Shpe, spce nd mesures. It is tught during the utumn term of Yer 9 nd replces Numer 1. Some ojectives concerning use of the clcultor re covered here nd will need less time in Numer. Ojectives on clcultion from the existing Numer 1 will need to e covered in Numer. Proportion prolems re sometimes solved quite esily using n informl pproch, prticulrly when the numers involved re simple multiples of ech other, s in recipe conversion. But pupils lso need to e le to del with the generl cse; this requires higher level of understnding nd prolem-solving skills, using the opertions of multipliction nd division. A key feture of the orl nd mentl strters in this unit is the utomting of some of the processing tsks. Throughout the unit, clcultors remove the urden of rithmetic, which helps pupils to focus on the reltionships involved. This unit hs een structured into three phses for teching. Phse 1 (out three lessons) Phse 1 uilds on the ide of scling numers, introduced in the Yer 8 mini-pck, to consider successive sclings nd their inverses. It re-exmines the use of tulr rrys to set out proportionl reltionships nd find unknowns (followed through in phses nd ). Phse (out three lessons) Phse links numericl work to geometry the invrint properties of enlrgement nd the concept of similrity. It considers reltionships within nd etween similr shpes. Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

5 Phse (out three lessons) Phse considers prolems from vriety of contexts (numericl, grphicl, sptil, sttisticl), to identify the underlying reltionships of proportionlity. It revises nd develops methods for solving proportion prolems. The phse plces prticulr emphsis on prolems involving percentge increses nd decreses. Ojectives A Understnd the effects of multiplying nd dividing y numers etween 0 nd 1; use the lws of rithmetic nd inverse opertions; recognise nd use reciprocls. B C D E F G H Enter numers into clcultor nd interpret the disply in context (negtive numers, frctions, decimls, percentges, money, metric mesures, time). Recognise when frctions or percentges re needed to compre proportions; solve prolems involving percentge chnges. Use proportionl resoning to solve prolem, choosing the correct numers to tke s 100%, or s whole; understnd nd use proportionlity nd clculte the result of ny proportionl chnge using multiplictive methods; understnd the implictions of enlrgement for re nd volume; compre two rtios; interpret nd use rtio in rnge of contexts, including solving word prolems. Enlrge -D shpes; recognise the similrity of the resulting shpes; identify the scle fctor of n enlrgement s the rtio of the lengths of ny two corresponding line segments; recognise tht enlrgements preserve ngle ut not length. Solve incresingly demnding prolems nd evlute solutions; explore connections in mthemtics cross rnge of contexts: numer, lger, shpe, spce nd mesures nd hndling dt; generte fuller solutions. Represent prolems nd synthesise informtion in lgeric, geometric or grphicl form; move from one form to nother to gin different perspective on the prolem. Suggest extensions to prolems, conjecture nd generlise; identify exceptionl cses or counter-exmples, explining why; justify generlistions, rguments or solutions; pose extr constrints nd investigte whether prticulr cses cn e generlised further. Differentition Ides for progression in orl nd mentl strters re provided in set of prompts. Support in the first phse could e provided y devoting more time to revising ides developed in the Yer 8 multiplictive reltionships: mini-pck. The unit refers to the Frmework supplement of exmples, prticulrly in phse of the unit. In the cse of ech ojective the pitch of the work is ccessile through Yers to 9 (nd Yer 9 extension) nd exmples cn e chosen ppropritely. Further ides for vrying the level of chllenge re detiled in the prompts for min ctivities. For some pupils, it would e pproprite to revisit these concepts through lessons, nd 1 in the Yer 9 ooster kit: mthemtics. Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

6 Resources Clcultors, scissors Pper: two lrge squre sheets, one smller squre sheet (clss demonstrtion) two sheets ech of A nd trimmed A pper for folding nd cutting (per pir of pupils) Selected sets of proportion prolems, drwn from the Yer 8 prolem nk nd other suitle sources (see Prompts for min ctivities in phse pge ) Resource sheets to prepre s OHTs nd/or hndouts (included in the school file nd on the CD-ROM): Scling line segments ( sheets) Ct fces Photogrphic enlrgements Shdows ( sheets) Four prolems: mking the links Supplementry notes (pges 10 8 of the mini-pck): Prompts for orl nd mentl strters: phse 1 Prompts for orl nd mentl strters: phse Prompts for orl nd mentl strters: phse Prompts for min ctivities in phse 1 Prompts for min ctivities in phse Prompts for min ctivities in phse Prompts for finl plenry in phse Key mthemticl terms nd nottion scling, scle fctor, multiplier inverse opertion, inverse multiplier reciprocl rtio: expressed using rtio nottion ( : ) or in equivlent frction ( ), deciml or percentge forms proportion: s frction or prt of whole ( proportion of ) s equlity of rtios, i.e. direct proportion ( in proportion, proportionl to ) rte, per, for every, in every unitry method enlrge, enlrgement, centre of enlrgement similr, similrity 6 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

7 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

8 Unit pln Orl nd mentl strter Min teching Notes Plenry Ojectives A, B Phse 1 (three lessons) Ojectives A, D, H Higher/lower Give frction multipliers. Pupils hve to sy whether the result will e higher or lower. Introduce the term reciprocl nd include unitry frctions. Aout Give multipliers (frctions, then decimls) to pply to specified numer. Pupils hve to pproximte the size of the result. Clcultor quick Give clcultion (e.g. ). Pupils use clcultors nd discuss key sequence. Repeted scling Introduce repeted scling (e.g. ) using multipliction only, representing numers y line segments. Reclling work from the Yer 8 multiplictive reltionships unit, identify scle fctors ( nd ), the equivlent single scle fctor ( ) nd inverses. Give reminder of the terms scle fctor, multiplier, inverse multiplier. Check tht deciml scle fctors give consistent nswers. Discuss other exmples, working towrds the generl result =. Proportionl sets Referring to the plenry of the previous lesson, remind pupils of the terms rtio nd in proportion. Give tle with three columns nd up to ten rows of numers in proportion. Leve some entries lnk. The tsk is to find these entries using reltionships etween rows or columns. Pupils hve to select pproprite rrys to clculte ech unknown, setting out tulr extrcts nd equtions. Discuss lterntive pproches nd mix exmples suitle for mentl nd clcultor methods. Set A Set B Set C x x x =. or x =. y.. z 8 t... c c Support: See Yer 8 multiplictive reltionships unit. Sclings of line segments cn e expnsions or reductions. Mthemticlly, these re ll enlrgements, sometimes with scle fctors less thn 1. At suitle point, discuss links etween scle fctors using frction nd rtio nottion, e.g. if : : c = :: the scle fctors etween them re,,, etc. Extension: Consider how to plce dt so tht missing entries cn e found (lst pge of prompt sheet). Generte three proportionl sets, given specified rtios, e.g. : = : nd : c = 10 :. Strt y choosing entries for one set. (Could e set s homework.) For given unknown, pupils offer severl different tulr extrcts from which it cn e clculted. Discuss efficient choices. Ojectives C, D Phse (three lessons) Ojectives D, E, F, G, H (Frmework pp. 19, 1 1) Equivlent lists Give rtio of two quntities. Pupils give scle fctor s frction nd deciml, nd the percentge increse or decrese. Sme for inverse nd other rtios. Wes For exmple, if = 8 (:8), give some more equivlents. Set out s we digrm nd include entries for 1 nd 1. Folding pper Techer demonstrtion: Strt with squre, fold long shorter line of symmetry nd cut in hlf. Repet four more times. Arrnge rectngles nd discuss, noting two sets with the sme proportions (1 : 1 nd : 1). Clss exercise (in pirs): repet for rectngles, hlf the clss strting with metric sheet (e.g. A), hlf with non-metric sheet. Arrnge rectngles, tulte lengths nd widths, find scle fctors/rtios. Ct fces (OHT nd resource sheet) Which fces re similr nd how do you know? Consider: rtios of fetures within fce tht re preserved in similr fce; multipliers (s frction, deciml nd percentge) tht scle etween similr fces. Emphsise tultion of dt nd mke links with phse 1 of the unit. Define mthemticl similrity: dimensions in proportion sme shpe, different size. Note tht ll squres re similr. Extension: For squre folding use visulistion nd discuss. Include re scle fctors. Discuss results from metric nd non-metric rectngles, noting tht, for the former, ll rectngles hve the sme proportions. Choose plenry depending on progress with min ctivities. For exmple: As well s squres nd circles, wht other shpes re lwys similr? Rise suitle thinking point relting to photogrphic enlrgements (prompt sheet). 8 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

9 Orl nd mentl strter Min teching Notes Plenry Splits For exmple, divide 6 in different rtios (e.g. 1 :, :, 1::, ). Set out s we digrm. Photogrphic enlrgements (OHT or resource sheet) Given three similr photogrphs, discuss wht vlues need to e given to determine six dimensions (lengths nd widths), six sclings etween photogrphs, nd six internl rtios etween lengths nd widths. Support: Guide pupils, reveling more informtion s needed. Extension: Concentrte on generl principles out wht vlues might e given. Extended plenry OHT Shdows 1: Given lengths of shdows nd height of child, find heights of trees. OHT Shdows : Given length of child s shdow lter in the dy, find lengths of shdows of trees. Ojectives A, B, C, D Phse (three lessons) Ojectives B, C, D, F, G, H Working towrds fluency (Extend to up to 1 minutes) Using clcultors with wkwrd numers Rpid conversion etween rtio, frction, deciml nd percentge forms Numers nd quntities, using rtes (clerly stted per ) Cover these clcultions: Expressing proportions: Finding proportions of : of Compring proportions: less thn, equl to, or greter thn Compring quntities: of less thn, equl to, or greter thn of c d Using nd pplying rtes: e.g. mchines need 1 hours mintennce; how long for mchines? ( hours per 1 mchine, multiplier ) c d Strtegies for solving prolems involving multipliction, division, rtio nd proportion Collect suitle nk of prolems, rnging from level to level /8. Use Y8 mini-pck prolem nk (pp. 8), Frmework (pp.,, 1,, 81, 1, 16, 1, 9,, 69), suitle textooks. Drw from numericl (including percentge increses nd decreses), grphicl, geometricl (enlrgement, mps nd plns), sttisticl contexts. Activities (my include mini-plenries): Choose one prolem: discuss lterntive strtegies for solving; chnge numers (e.g. mke them more difficult) nd consider how methods cn e dpted; sk different or supplementry questions from sme context. (See Y8 mini-pck prompts, pp ) Choose smll set of prolems: concentrte on extrcting nd orgnising dt (e.g. putting into tulr form) efore deciding on possile methods of solution, rther thn working prolems through to n nswer. (See Y8 minipck prompts, pp. 1.) Ask pupils to mke up similr prolems for prtner to solve. Give prt solutions nd sk pupils to continue nd complete solution, or give complete solution nd sk pupils to evlute efficiency of strtegy chosen nd/or identify errors. Prolem-solving strtegies to e revised nd developed: Trnslte prolem into form which will help with the solution: e.g. extrct pproprite dt nd put in tulr form. Estimte the nswer: sk Will it e igger or smller?, Will it e greter or less thn 1?, etc.; use knowledge of the effect of multiplying or dividing y numers greter thn or less thn 1. Consider scling methods y finding multiplier. When using unitry method involving division, clrify rtes expressed prt wy to solution, e.g. Is it euros per pound or pounds per euro? Phse of the Yer 8 multiplictive reltionships unit outlined generl pproch to developing prolemsolving strtegies. Set the expecttions for Yer 9 group y collecting suitle prolem nk. Give prticulr ttention to: prolems involving rtes nd chnge of units; prolems involving percentges; the common structure (reltionship of proportionlity) pplying to different contexts. (See Yer 9 phse prompts, pp. ). Support: Choose esier exmples from the collection. Extension: Include prolems where the reltionships involved re not proportions. Mini-plenries nd short finl plenries should not introduce new prolems, ut reflect on: successful strtegies nd their trnsferility; efficiency of strtegies; emerging misconceptions; developing coherent written rguments. Use the plenry to ssess progress nd decide the focus for the next lesson. 9 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

10 Prompts for orl nd mentl strters: phse 1 The following ctivities im to utomte some of the processing tsks which re essentil to speedy solution of rtio nd proportion prolems. In this sense they re similr to the kinds of ctivities primry techers use with Key Stge pupils in order to secure rpid responses to prolems involving multipliction nd division. This does not exclude understnding ut seeks to dd speed to pupils use of sic knowledge. It is importnt to incorporte oth pce nd discussion within these strters. Generlly few firly speedy responses should e sought efore results re compred. Pupils re then in position to exchnge strtegies nd improve on responses, redy for nother quick tch of questions. Higher/lower This is quick-pced ctivity requiring mentl pproch. Pupils could respond y showing one of two crds higher or lower. Give scle fctor (perhps point to one on the ord). Ask whether the scling will give result tht is higher or lower thn the originl vlue. Prt wy through, sk pupils to explin how the vlue of the scle fctor helps to indicte the size of the result. Introduce the term reciprocl nd write reciprocls of the vlues used so fr. Ask the sme set of questions for the reciprocls nd note the difference in the direction of the scling etween the reciprocl pirs. Suggested scle fctors re given here ut this list should e vried. Scle fctor An lterntive is to specify higher or lower nd hve pupils suggest scle fctors using whiteord or equivlent. 10 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

11 Aout This is quick-pced ctivity requiring mentl pproch. Pupils could respond y using whiteords or equivlent. Give vlue nd scle fctor (perhps point to these on the ord; see the tle on the right). Initilly use only frctions, then only decimls, then mixture. Ask Aout how ig will the resulting vlue e? Prt wy through sk some pupils to explin their strtegies in generl for prticulr scle fctors, e.g. How do we find the pproximte 8 effect of scling nything y fctor of? Suggestions re given here ut this list should e vried. Alterntively specify the pproximtion nd sk for the scle fctor. Which scle fctors would give result which is: out hlf of the originl? out the sme s the originl? out twice the size of the originl? out ten times the originl? Vlues Scle Scle to fctor fctor scle Clcultor quick This ctivity, which requires the use of clcultors, needs mesured pce. Pupils could respond using whiteords or equivlent. An OHP clcultor is useful for the follow-up discussion. Give clcultion, perhps orlly or reveled on the OHP. See the suggestions opposite. Pupils use their clcultors to perform quick clcultion nd write their solutions on whiteords or equivlent. Ask pupil to scrie selection of the responses on the ord. Encourge pupils to use rguments of pproximtion to confirm the correct nswer. Discuss the key sequence for ech clcultion (perhps demonstrted on the OHP clcultor). Prticulr ttention could e pid to key sequences which give incorrect solutions if these re common to numer of pupils. Clcultion / If the preceding points hve not een lengthy, sk wht other form the clcultion could tke in order to prompt the sme key sequence. 11 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

12 Prompts for orl nd mentl strters: phse Equivlent lists In this quick-pced ctivity, mentl pproches should e used, ut clcultors my e needed for frction-to-deciml conversion. Pupils could respond using whiteords or equivlent. An OHT with copies of the scling line segments resource sheet my help in the discussion of why percentge increses nd decreses re different. Strt the ctivity y giving n initil rtio nd sk for full list of equivlent wys of stting this reltionship. For exmple: The rtio p : q is : Wht is the scling from p to q: s frction? s deciml? 0. Is this n increse or decrese? Wht percentge decrese does this represent? decrese % of p Inverse The rtio q : p is : Wht is the scling from q to p: s frction? s deciml? 1. Is this n increse or decrese? Wht percentge increse does this represent? increse 1 % of q Wes In this quick-pced ctivity, mentl pproch or clcultor methods could e used, ut pupils should e told which is expected. Pupils could respond on lrge sheet of pper showing their we or contriute to clss we. Give n equivlence (or pproximte equivlence) etween two vlues. See the suggestions on the right. Ask pupils to generte more equivlences from the given fct. Link these equivlences in we digrm. Ask pupils to comment riefly on how prticulr entries were clculted. Strting points = 8 (or current equivlence) gllons = 9 litres miles = 8 km kg = 11 l Include entries for unit vlue on ech side of the equivlence nd highlight these s rtes. Be explicit in the lnguge to descrie rte: per. Drw out the importnce of certin key entries in enling whole string of other entries to e generted nd linked in the we. 1 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

13 Splits Mentl pproches should e used in this quick-pced ctivity. Pupils could respond on lrge sheet of pper showing their we or contriute to clss we. Give quntity including units. Exmples could include: 6, kg, 100 cm,. ml Ask pupils to split the quntity into given rtios. Record the splits using we digrm (see elow). Ask pupils to comment riefly on how prticulr entries were clculted. Drw out the equivlence of some splits nd the links etween others. Show these equivlences nd links on the we. When pproprite drw out generl method for dividing in given rtio. : 16: 0 6 :: 8: 8: 0 : 1: 1: 1 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

14 Prompts for orl nd mentl strters: phse Where pproprite: expect rpid conversion etween frctions (lowest terms), deciml frctions, percentges; expect rtes, which should e clerly stted s per ; expect the use of clcultor where numers re wkwrd. Where proportion is sked for this could e frction, deciml frction or percentge. Expressing proportions Numers, first smller thn second: Wht is s frction of 8? Wht is s percentge of 8? Wht is s proportion of 8? Numers, first lrger thn second: Wht is 1 s frction/percentge/proportion of 8? Quntities, first smller thn second (nswer is not in units, e.g. ): Wht is s frction/percentge/proportion of 1? Quntities, first lrger thn second (nswer is not in units, e.g. kg): Wht is kg s frction/percentge/proportion of 8 kg? Quntities (mixed units), first smller thn second (nswer is not in units, e.g. or p): Wht is 60p s frction/percentge/proportion of? Quntities (mixed units), first lrger thn second (nswer is not in units, e.g. m or cm): Wht is m s frction/percentge/proportion of 0 cm? Expressing proportions nd their inverses As ove, del with vrious forms: frctions/deciml frctions/percentges; numers/quntities; sme units/mixed units. Exmples: Wht is 6 s proportion of? Wht is s proportion of 6? Expressing rtes These should e written s this per tht. Exmples: 6 miles trvelled t constnt speed for hours 8 split eqully etween 6 children 1 hours of work to e covered y 9 clerks Inverting rtes Using the previous exmples: Wht is the mening of tht per this? When is it useful? hours per mile children per pound clerks per hour 1 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

15 Finding proportions of Numers, proportions less thn 1, some terminting decimls nd some not: Wht is three fifths of 8? Wht is 0.6 of 8? Wht is 60% of 8? Numers, proportions greter thn 1, some terminting decimls nd some not: Wht is ten sixths of? Wht is 1.6 of? Wht is 16% of? Quntities, proportions less thn 1, some terminting decimls nd some not (nswer is in units, e.g. ): Wht is three sevenths of 1? Quntities, proportions greter thn 1, some terminting decimls nd some not (nswer is in units, e.g. litres): Wht is four thirds of 1 litres? Compring proportions Numers, first smller thn second: Wht is greter: s proportion of 8, or s proportion of 90? Numers, first lrger thn second: Wht is greter: 6 s proportion of 8, or s proportion of 9? Quntities, first smller thn second (nswer is not in units, e.g. minutes): Wht is greter: minutes s proportion of 18 minutes, or minutes s proportion of 10 minutes? Quntities, first lrger thn second (nswer is not in units, e.g. cm): Wht is greter: 18 cm s proportion of cm, or 16 cm s proportion of 6 cm? Quntities (mixed units), first smller thn second (nswer is not in units, e.g. or p): Wht is greter: 60p s proportion of, or 90p s proportion of.? Quntities (mixed units), first lrger thn second (nswer is not in units, e.g. m or cm): Wht is greter: m s proportion of 0 cm, or. m s proportion of cm? Inverting proportions nd compring As ove, del with vrious forms: frctions/deciml frctions/percentges; numers/quntities; sme units/mixed units. Exmple: Wht is greter: 9 s proportion of 8, or 1 s proportion of 0? Given the previous fct, wht is greter: 8 s proportion of 9, or 0 s proportion of 1? Compring rtes Wht is the greter rte: 6 miles trvelled t constnt speed for hours, or 60 miles trvelled t constnt speed for hours? 1 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

16 Compring quntities Numers, proportions less thn 1, some terminting decimls nd some not: Which is the greter: one fifth of 8 or one seventh of 90? Numers, proportions greter thn 1, some terminting decimls nd some not: Which is the greter: five thirds of or seven qurters of? Quntities, proportions less thn 1, some terminting decimls nd some not (nswer is in units, e.g. litres): Which is the greter: 16% of or % of 16? Quntities, proportions greter thn 1, some terminting decimls nd some not (nswer is in units, e.g. litres): Which is the greter: ten thirds of 1 litres or ten sevenths of 0 litres? Using nd pplying rtes mchines need 1 hours of mintennce. How mny mchines cn e serviced in hours? Wht is the rte we use in this clcultion? (Not expecting numericl nswer ut rte, e.g. mchines per hour ) Wht is the clcultion? (Not expecting numericl nswer ut clcultion, e.g. ) 1 Wht is the nswer? mchines need 1 hours of mintennce. How mny hours re needed to service mchines? Wht is the rte we use in this clcultion? (Not expecting numericl nswer ut rte, e.g. hours per mchine ) Wht is the clcultion? (Not expecting numericl nswer ut clcultion, e.g. ) 1 Wht is the nswer? 16 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

17 Prompts for min ctivities in phse 1 Repeted scling In the Yer 8 unit, pupils used the imge of prllel numer lines to explore multiplictive reltionships. This imge is continued, ut simplified to the use of line segments. Note tht: The digrms cn e rough sketches. It helps to think in terms of expnsion nd reduction of the line segment. Depending on the clss, you might introduce the mthemticl term enlrgement, for exmple: expnsion y fctor of is equivlent to enlrgement y scle fctor of ; 1 reduction y fctor of is equivlent to enlrgement y scle fctor of. This will help pupils to mke link with enlrgement in phse of the unit. It will of course e necessry to discuss the pprent contrdiction in using the term enlrgement for reduction! 1 1 Quickly recp the principle of scling etween ny two numers in either direction using only multipliction. For most pupils the interim steps of scling down to 1 will e dropped fter the initil recp. For exmple, sk: Wht single numer could you multiply y to scle it to 1? Wht out the inverse from 1 to? Wht single numer could you multiply 1 y to scle it to? Wht out the inverse from to 1? Wht numer could you multiply y to scle it to in single step? Wht is the single multiplier coming ck from to? Move on to consider successive sclings. 1 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

18 Use the prompts elow for vriety of strt, middle nd end points including non-integer vlues. For exmple, sk: Wht single numer could you multiply y to scle it to? Wht single numer could you multiply y to scle it to? Wht numer could you multiply y to scle it to in single step? How is this relted to the two steps we hve descried from to nd from to? Cn you see tht multiplying y the single scle fctor ( ) is equivlent to multiplying in two steps y the two seprte scle fctors ( nd )? Use your clcultor to check ech step of the scling nd to confirm this equivlence. (Point out the rounding error which cn occur here.) Cn you see tht the single multiplier (1.) is the product of the multipliers used in the two-step scling (1. 0.1)? Repet the ove for the inverse sclings, from to to noting tht eginning with the frctionl form of the scle fctors is useful here. Link to multipliction nd division of frctions Although not ddressed in this unit, these points re worth noting: Successive sclings nd cn e thought of s expnding y fctor of nd reducing y fctor of, then expnding y fctor of c nd reducing y fctor of. Expnsion nd reduction y fctors of cn e oserved to undo one nother. c c This my help to explin the result = in terms more meningful thn cncelling the s. c The inverse of is. However, the inverse cn lso e expressed s the multiplier. So nd re equivlent opertors, which explins the infmous (fmilir ut frequently not understood) rule for dividing y frction: turn the frction upside down nd multiply. 18 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

19 Proportionl sets Tles 1 nd show proportionl sets of numers. The min ctivity is to give pupils some of the entries nd sk them to clculte the others. A suggestion is given in the incomplete tle on the right clerly enough entries re positioned to mke the tsk possile ut lso chllenging. Tle 1 Set A Set B Set C Set A Set B Set C Tle Set A Set B Set C Set A Set B Set C The presenttion of this tsk is importnt. This is not routine completion of entire rows or columns. The im is to use the tle s n opportunity to drw out the structures which help to develop proportionl understnding. You could model the following pproch. Finding missing entry To find missing entry, three relted entries re required. A good strtegy is to set out the entries in -y- rry nd then to identify the multiplier. (The multiplier my e etween ny pir of numers.) x. Encourge pupils to estimte the size of the missing entry. (Refer ck to the orl nd mentl strters Higher/lower nd Aout.) Write the clcultion s n eqution which is then solved mentlly or using clcultor: x =. 19 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

20 Drwing together solutions nd compring strtegies Consider lterntive rrys nd clcultions nd discuss their merits. At suitle stge consider the scle fctor etween the columns nd its inverse. This cn e seen s sfety net y which ll missing entries cn e clculted if simpler strtegy is not pprent. For exmple: Scling from A to B involves multiplying y rtio = (Note lso the inverse rtio = ) Scling from B to C involves multiplying y rtio = (Note lso the inverse rtio = ) Scling from A to C involves multiplying y rtio = (The inverse is = ) This is equivlent to the product of the two sclings = = Pupils will sometimes use division to complete the clcultion, for exmple: c c c c c 11? Remind pupils tht dividing y 10 cn lso e thought of s multiplying y or 0.1. Extending the tsk y posing further questions For exmple: Which entries must e plced in the tle in order to e le to complete ll missing entries? Wht is the mximum numer of items which could e entered nd the tsk remin impossile? Wht is the minimum numer of entries needed nd wht is importnt out their loction? For ny one empty cell Wht is the est strting point? How mny different strting points re there? 1 10 Simplifying the tsk Pre-select the rry to e used to find the missing entry. Use the scle fctor etween columns systemticlly. 0 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

21 Prompts for min ctivities in phse Folding pper Two pper-folding ctivities follow. The first, sed on squre, cn e used s prcticl demonstrtion or visulistion exercise. Notes for oth re given elow. The second, sed on rectngle, is prcticl ctivity. Folding squre sheet This is prcticl demonstrtion for the mjority of pupils. Resources: Two identicl lrge squre sheets of pper. A smller squre sheet to check for similrity (not produced y folding the lrger ones) nd rectngle (sides in rtio of out : 1). Strt y showing pupils the identicl squre sheets. Stick one sheet on the ord nd fold the other sheet to mke rectngle nd cut into two hlves. Stick one hlf on the ord nd repet with the other hlf (folding prllel to pir of sides, long the shorter line of symmetry). Do this three more times. Ask volunteers to rrnge the six shpes on the ord in different wys, e.g. lign to show centre of enlrgement. Below re some useful points to drw out: There re three squres, with sides in the rtio 1 : 1, ech successive one eing enlrgement of the previous squre. There re three rectngles, with sides in the rtio : 1, ech successive one eing enlrgement of the previous rectngle. 1 Shpes whose proportions re the sme, one eing n enlrgement of the other, re descried s mthemticlly similr. Show pupils the smller squre sheet nd sk whether it elongs to the set of similr shpes. Check y mesuring (to check rtio) nd y ligning (to check centre of enlrgement). Conclude tht ll squres re similr. Show pupils the other rectngle (e.g. rtio : 1) nd sk whether it elongs to the set of similr shpes. Check y mesuring (to check rtio) nd y ligning (to check centre of enlrgement). Conclude tht only certin rectngles re similr nd tht this depends on the rtio of length to width. 1 Folding squre sheet: visulistion This lterntive visulistion exercise is suitle for higher-ttining pupils. For generl guidnce on conducting visulistion exercises, see the Prompts for orl nd mentl strters in phse 1 in the Yer 9 geometricl resoning unit. 1 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

22 Visulistion script Imgine lrge squre sheet of pper. Ask yourself: Wht is the rtio of length to width? (Check tht pupils understnd tht it is 1 : 1.) Imgine folding it in hlf long line of symmetry. Wht shpe do you get? (Tke nswers from the group two possiilities re rectngle nd right-ngled isosceles tringle. Ask the clss to focus on folding prllel to side to get rectngle. There should e no further discussion until the end of the visulistion.) Imgine the rectngle you get fter folding squre sheet in hlf. Ask yourself: Wht is the rtio of length to width? (Extension: How does the re compre with the re of the squre?) Now fold the rectngle in hlf long the shorter line of symmetry. Wht shpe do you get? Wht is the rtio of length to width? (Extension: How does the re compre with the re of the originl squre?) Fold gin long line of symmetry (not digonl). Wht shpe do you get? Wht is the rtio of length to width? Fold gin long the shorter line of symmetry. Wht shpe do you get? Wht is the rtio of length to width? Fold one more time long line of symmetry (not digonl). Wht shpe do you get? Wht is the rtio of length to width? At the end of the visulistion, sk pupils to descrie the sequence of six shpes they sw nd the reltionships etween them. You my wish to hve set of shpes prepred to demonstrte nd check pupils oservtions. Drw out the sme points s for the prcticl ctivity. Folding rectngulr sheet Ask pupils to work in pirs nd split the clss into two hlves. Provide one hlf of the clss with two sheets of A pper nd the other with two identicl sheets of non-metric pper (e.g. trimmed A). Ask them to set one sheet side nd to fold the other sheet long the shorter line of symmetry nd cut it into two hlves. Set one hlf side nd repet with the other hlf. Do this three more times. Now sk pupils to: Arrnge the sheets in different wys nd consider whether ny of the rectngles re mthemticlly similr (e.g. lign to show centre of enlrgement). Mesure nd tulte lengths nd widths of rectngles which they think re similr nd clculte scle fctors within the tle. On the ord compre the results from the two hlves of the clss, mking sure tht you disply ccurte dt. Allow ll pirs few minutes to consider the resons for the difference (they my wish to form fours to discuss the two sets of results). In the plenry rise these questions for discussion: Considering the non-metric sheet, re ny rectngles mthemticlly similr? Wht hppens with A pper? Why do you think tht stndrd metric pper sizes re designed in this wy? (You hve een looking t A, A nd A6, nd smller sizes which re not nmed. They re derived from A0 pper, which hs n re of 1 m.) Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

23 Ct fces Distriute copies of the Ct fces resource sheet to pirs of pupils. If you use selection of sizes of sheet (e.g. A, A, A), dditionl questions on the preservtion of rtios cross the enlrged sheets will e possile when pupils re redy to shre their results. Descrie the following tsks: Find the fces which re similr y clculting the rtio of two dimensions within fce nd confirming tht this is preserved. Record the fctor which scles from one similr fce to nother (s frction, deciml nd percentge). Record the inverse of ech of these. Check some dditionl mesurements on the similr fces nd confirm tht they lso hve incresed y the scling identified. Consider the sclings etween the three circulr fces A, B nd C. Confirm tht the scling from A to C cn e clculted y multiplying the scling from A to B nd the scling from B to C. Photogrphic enlrgements Show the set of similr rectngles given on the resource sheet Photogrphic enlrgements (OHT, poster size or A sheets). Explin tht: This is set of photogrphs which re enlrgements of ech other (i.e. it is set of similr rectngles). There re 18 vlues missing from the digrm (on poster you might like to hve these vlues hidden y smll sticky notes lift the flp ), ut they cnnot e found y mesuring s the rectngles re not drwn ccurtely. Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

24 Of the missing vlues: six re dimensions the width nd length of ech rectngle; six re sclings etween the three rectngles in oth directions; six re internl rtios written s sclings from the length to the width nd vice vers. Pose the chllenge: I will give you some of the vlues. You hve to sk for the minimum you need to work out ll the other vlues. Wht vlues do you wnt me to give? (Which flps would you choose to lift?) The difficulty of this tsk is lrgely determined y the decision mking required s to which numers re reveled. More guidnce on this choice could e given to support some groups of pupils. Follow up: Pupils should explore lterntive sets of given vlues nd consider wht is the lest numer they need. They might tulte dimensions nd the sclings etween them s follows: length width photo 1 photo photo Extension: Wht is the lrgest numer of vlues you could e given nd still e unle to find ll 18 vlues? Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

25 Prompts for min ctivities in phse Phse of the Yer 8 multiplictive reltionships unit outlined generl pproch to developing prolem-solving strtegies. The development of these strtegies is continued here. You need to collte nk of ppropritely pitched Yer 9 prolems. Importnt sources for this will e questions from pst Key Stge test ppers, textooks nd the Frmework supplement (pges,, 1,, 81, 1, 16, 1, 9,, 69). The prolem nk included in the Yer 8 mterils (pges 8 in the multiplictive reltionships mini-pck) is still likely to e useful resource for this unit. Use the ides elow for further guidnce pproprite to the Yer 9 context. Also refer to the Yer 8 Prompts for min ctivities in phse on pges 19 of the multiplictive reltionships mini-pck. Rtes nd chnge of units Exmple 1: Speed 1 1 A cr verges 6 mph. How fr does it trvel in hrs? To the nerest hr, how long does it tke to trvel miles? This exmple is typicl of mny prolems where rte is given. Mny pupils will pproch these prolems successfully without tultion recognising tht multipliction or division is required. However, discussion of the question in its tulr form cn e of vlue to ll pupils in clrifying the reltionships in the prolem. Extrct the dt into tulr form, noting tht one of the entries will e 1. Support y simple questions such s those rehersed in the orl nd mentl strters. Distnce Time The first nswer is distnce, so the units will e miles. (miles) (hours) It will e igger thn 0; it will e out 10 miles. 6 1 The second nswer is time. The units will e hours.?. The nswer will e igger thn..? It will e out 6. Exmple : Density The density of pint is 1. g/cm. Wht is the mss of litres of pint? In this exmple, there is chnge of units. To highlight the structure of the question the dt cn gin e tulted. This my initilly e solved s two-stge clcultion. It might e pproprite to comine the two scle fctors into one if the clcultion is to e repeted for series of quntities or for tultion in spredsheet. Mss Volume Volume The first row shows rte given in the question. This (g) (cm ) (litres) will chnge for different pint The second row shows conversion rte. This will lwys e the sme.?? The finl nswer is mss. The unit will e grms The nswer will e igger thn 1.; it will e etween 000 nd 00 grms. 100 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

26 Percentge chnges Exmple : Svings Wht is the finl mount fter % interest is dded to my svings of 8? My friend s svings mount to 90 fter % interest hs een dded. Wht ws the originl mount of her svings efore interest ws dded? Terms like increse (or decrese ) my trigger dditive thinking, ut it is lwys possile nd sometimes necessry to relte the originl nd finl vlues multiplictively. Prt : Originl Finl This tle gives concise overview of the prolem. The first row shows which mount to consider s 100%. All? entries 8? re mounts of money; the units will e. Ech entry in the? 90 first column will e smller thn the corresponding entry in in the second column finl mount finl mount = 8 = From ny originl mount ( s) the finl mount ( t) cn e clculted: t = s 1.0 Prt : The second prt of this inverse clcultion is mde possile ecuse of the pproch through considering scling methods nd finding multiplier. Discuss the fct tht the reduction here is not % ut out 6.%: originl mount 90 originl mount = 90 = From ny finl mount ( t) the originl mount ( s) cn e clculted: s = t Ask pupils to mke up similr prolems for their neighour to solve using different finl mounts nd different interest rtes. Exmple : Prices If n item originlly selling for 0 is reduced y 0% nd then y further 0%, do I get it for nothing? If the price is incresed y 0% nd then decresed y 0%, does it return to its originl vlue? Misconceptions relting to successive percentge chnges cn e exposed here if pupils nswer yes to either of these questions. This usully results from dditive rther thn multiplictive thinking. The tles opposite give concise overview of the prolem. The first two rows show which mount to consider s 100%. 6 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

27 Prt First Second Third Note tht single scling of = 0. revels tht the third price is % of the first price ?? Prt third price = =.0 First Second Third Note tht single scling of = 0. revels tht the third price is % of the first price ?? third price = =.0 Links etween different strnds of mthemtics List the spects of mthemtics with which pupils re fmilir (it is proly etter not to include using nd pplying mthemtics t this point): numer lger shpe, spce nd mesures hndling dt For ech spect sk the clss: Wht is it out? Wht sorts of prolems do you solve in this spect of mths? Wht types of tle, digrm or grph re used? Use the resource sheet Four prolems: mking the links s hndout, OHT or poster. For ech prolem sk pupils to: discuss from which strnd of mthemtics the question could e drwn; extrct the dt nd put it into tulr form do not solve the prolem t this stge; discuss wys of constructing tle (with hedings); compre ny lterntives nd decide which is est. Considering the collection of tles for ll four prolems, sk pupils to: compre tles for different prolems, noting the similrity in structure, ech eing proportion in which the nswer cn e found y finding nd pplying scle fctor; continue the prolems through to solutions. Finlly point out tht these exmples illustrte how n importnt mthemticl ide (proportion) cn underpin pprently very different situtions. Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00

28 Prompts for finl plenry in phse Shdows This is suggested s n extended plenry to conclude phse of the unit. Resources: The four resource sheets Shdows will need to e copied on to cette. You will find these sheets in the school file. Show Shdows 1 which is sketch of child, some trees nd their shdows. Ask pupils wht they notice out the figures nd their shdows nd how they cn tell tht the shdows re ll cst t the sme time of dy. Discuss the generl question of using the height of the child nd length of the child s shdow to estlish something out the reltionship etween the height of ech tree nd the length of its shdow. Use Shdows 1 overly to illustrte the mesurements which re known (lengths of the shdows nd height of the child) nd those which need to e clculted (heights of the trees). For simplicity Shdows 1 cn e removed nd the overly left in plce to minimise distrctions in the disply. Use the tsk s n opportunity to reinforce the strtegies which hve een developed throughout the unit, nmely: Write the three known vlues nd the one unknown s -y- rry nd decide which scling will help find the unknown (using ides from phse 1). Use the multiplier to decide whether the clcultion will result in higher or lower result nd lso estlish out how ig the result will e (using ides from the orl nd mentl strters). Confirm this y reference to the context of the question. Use clcultors to quickly find the height of ech tree nd reinforce efficient clcultor use (ides from orl nd mentl strters). Show Shdows, sketch of the sme trees nd the sme child, nd sk wht is noticele out the shdows now. Are ny of the reltionships from the first sitution trnsferle to this new context? Ask how much informtion is needed here to clculte the lengths of the shdows. Use Shdows overly nd the key points descried ove to clculte the length of the shdow of ech tree. Note tht prt of the picture hs deliertely fllen off the sheet. At this stge, reltionships re well enough estlished tht it is still possile to find solution y visulising the complete tringle. 8 Intercting with mthemtics Yer 9 proportionl resoning: mini-pck Crown copyright 00