Naural Mahs raeges Bk 1 Wre by A Baker BPhl, DpRdg ad Jhy Baker BcHs, PhD Cpyrgh 2006 A & Jhy Baker ad Blake Educa Frs publshed 2006 Blake Educa Py Ld ABN 50 074 266 023 108 Ma Rd Clay uh VC 3168 Ph: (03) 9558 4433 Fax: (03) 9558 5433 www.blake.cm.au Publsher: Ly Dcks eres edr: ae D Erre Desger: Dma Desg llusras: hlh Grd ad Oscar Brw Cver desg: Dma Desg Typeseer: Ps Pre-press Grup Pred by Thumbprs Ud, Malaysa. Ths publca s cpyrghed. N par f hs bk may be reprduced by ay meas whu wre permss frm he publsher. COPYNG OF TH BOOK BY EDUCATONAL NTTUTON A purchasg educaal su may ly phcpy pages wh hs bk accrdace wh The Ausrala Cpyrgh Ac 1968 (he Ac) ad prvded he educaal su (r bdy ha admsers ) has gve a remuera ce he Cpyrgh Agecy Lmed (CAL) uder he Ac. Fr deals f he CAL lcece fr educaal sus, cac: Cpyrgh Agecy Lmed Level 19, 157 Lverpl ydey, NW. 2000 COPYNG BY NDVDUAL OR NON-EDUCATONAL NTTUTON Excep as permed uder he Ac (fr example fr far dealg fr he purpses f sudy, research, crcsm r revew) par f hs bk may be reprduced, sred a rereval sysem, r rasmed ay frm by ay meas, whu he prr wre apprval f he publsher. All equres shuld be made he publsher. Naal Lbrary f Ausrala BN: 1-921143-38-X BN: 978-1-921143-38-0 1.Mahemacs udy ad eachg (Prmary).. Tle 372.7
Ces ad Naural Mahs Plaer CONTENT rduc U 1: Early Number ese 2 U 2: Early Number raeges 14 U 3: Exedg Number raeges 26 U 4: Paers Number, pace ad Measureme 38 U 5: Begg Chace ad Daa 50 U 6: Number Cex 62 U 7: hape ad Ps 74 U 8: Explrg Number uas 86 U 9: frmal Measureme 98 U 10: hape ad he Evrme 110 U 11: Usg Paers 122 U 12: Pug all Tgeher 134 Acvy hees 147 Task Cards 174 Mahs Plaer The Bk 1 Mahs Plaer s he sfware package prvded a he fr f hs bk whch supprs he maeace f plaed us f wrk cver he syllabus. The Mahs Plaer has facles fr: crprag he acves frm hs bk ad her schemes a cmplee mahemacs prgress maag assessme recrds cjuc wh aaed wrk samples summarsg class ad dvdual prgress. Each u cas w examples f he ype f assessme acvy ha wll allw sudes shw her udersadg f a parcular pc. Examples f sude respses hese ad he prblemased suas are gve he asscaed publcas Naural Mahs raeges Assessme Gude fr Bk 1 wh wrk samples. Ths bk shws hw wrk samples ca be aaed creae a prfl f he sude s acheveme. The bk als gves a summary f key vcabulary, sraeges, represeas ad udersadgs ha sudes a hs level mgh be expeced demsrae. v C e s 2006 Blake Educa Naural Mahs raeges Bk 1
Bg deas Mahs Bg deas Mahs The mahemacal ce f hs seres s rgased arud 12 bg deas ha are releva eachg mahs a hs level. Ths char gves a vervew f he bg deas ad her lks he 12 us f wrk. B g N1.1 Number sese N1.2 N1.3 Add ad subrac Mulplca ad dvs U 1 U 2 U 3 U 4 U 5 U 6 Early Number ese Early Number raeges Exedg Number raeges Paers Begg Chace Number, pace ad Daa ad Measureme Number Cex d e a s M a h s N1.4 Mey M1.1 Measureme M1.2 Tme ` 1.1 hape 1.2 Ps space CD1.1 CD1.2 PA1.1 PA1.2 Chace Daa Paer Equvalece A full cvers char ha relaes hese bg deas ae ad Naal Gudeles s suppled separaely. v 2006 Blake Educa Naural Mahs raeges Bk 1
Ces ad Naural Mahs Plaer Bg deas Mahs U 7 U 8 U 9 U 10 U 11 U 12 hape ad Ps Explrg Number uas frmal Measureme hape ad he Evrme Usg Paers Pug all Tgeher Represeg, cmparg ad rderg umbers 20 usg effecve cug sraeges. Usg meal sraeges ad frmal represeas f add ad subrac. Usg frmal sraeges ad represeas f mulplca ad dvs. Recgsg cs ad her uses rasacs. Measurg by drec ad drec cmpars usg a rage f -sadard us. Usg he laguage f me ad sequecg he days f he week. Namg, descrbg ad csrucg wh 2-D ad 3-D shapes. B g d e a s Usg psal laguage ad frmal spaal plas. Usg he laguage f chace descrbe everyday suas. Cllecg, erpreg ad represeg daa. Creag ad descrbg paers based smple rules. Represeg umber a varey f ways ad develpg he ccep f equvalece. M a h s 2006 Blake Educa Naural Mahs raeges Bk 1 v
rduc rduc r d u c The acves hs bk prvde sarg ps fr hree-par lesss ha fcus he bg deas fr eachg mahs 5 7-year-lds. s eded ha he acves als be used cjuc wh he Bk 1 Mahs Plaer CD-ROM, whch eables he eacher maa cmplee class recrds f prgress. Accrdgly, hs bk s rgased 12 us, wh each u cag meal rues, prblemased suas, a vesga, games ad assessme acves mach he bg deas mahemacs. The us are eded as sarg ps fr eachers buld su he rage f learers ha hey are wrkg wh. The mahs currcula are dvded fve srads: Number (cludg Mey) pace Measureme Chace ad Daa Paers ad Algebra Wh each f he srads here are a umber f bg deas, r cceps, whch fcus he syllabus. Ths bk s rgased us f wrk based he curre research he develpmeal sequece whch sudes geerally acqure hse cceps. Gudg he chce f acves has bee he research ha suggess ha chldre are capable f graspg key mahemacal cceps a a earler age ha prevusly hugh. Researchers als urge eachers help chldre hk mahemacally raher ha merely memrse algrhms ad he her cmpuaal sklls (see Checkley, K. 1999). Three-par lesss Bh he Mahs Plaer ad hs bk are much mre ha a facly help eachers mee her syllabus requremes. The acves prvde he ype f resurce eeded mpleme a hree-par less prcess. ule, a hree-par less cludes: a meal rue develp he sude s self-cfdece ad reperre mahemacal hkg a prblemased sua where he sude apples her w hkg a sua ha hey ca egage wh a me fr reflec whch sraeges ad slus are shared, cmpared ad frmalsed, hrugh whch: 1. we beg frm where he sudes are 2. we buld her udersadg hrugh he sharg f deas 3. frmal mahemacal mehds are subrdae mehds ha he sudes ve ad whch wrk effecvely fr hem 4. sudes lear value each her s deas, wrkg as a cmmuy f learers raher ha as dvduals. Ths apprach he eachg ad learg f mahemacs has s rs research fdgs, ad brgs hese fdgs lfe hrugh acves ha have bee fud fully egage sudes mahemacal dscvery, dscuss ad udersadg. v 2006 Blake Educa Naural Mahs raeges Bk 1
rduc Par 1: Meal Rues Par 2: Prblemased uas Number sese, fluecy wh mahemacal ad sraegc hkg ad esma sklls are he fudaal buldg blcks f all laer mahemacs. Wrkshees ad meal arhmec ess are aahema rsk-akg, reflecve hkg ad seekg u effce sraeges ha wll develp aumacy umber facs based deep udersadg. Three years ag we bega ques he relevace f paper ad pecl, wrkshees ad drll ad kll mehds he develpme f fudaal basc umber facs ad udersadgs. We bega by esg a few meal acves ha vlved he whle class smulaeusly fu ad releva acves. As we dd s, we bserved ha whe sudes are egaged meal acves, cera cds eed be prese fr hem ba maxmum beefs. These are: Ccree maerals eed be prvded fr sudes use as ls by sudes. Feedback s mmedae, ad vlves sharg ad dscussg sraeges as well as shwg equal respec fr all sudes. Errrs are see as learg pprues fr all. Quess prvde success fr all as well as challeges fr sme. All sudes eed be egaged a her w level durg he prcess. Par 3: Reflec udes see hemselves as a cmmuy f learers where everye has a rle play he develpme f hkg ad learg. was wh hese crera md ha we bega explre he peal f meal rues as we have chse call hem. The purpse f meal rues s develp useful sraeges ha wll lead masery ad a sld fuda basc mahs cceps. Meal sraeges as far as pssble shuld relae he mehds ha sudes develp uvely ad wh her w culure. They shuld als relae as far as pssble he ways whch hse sraeges are appled he real wrld. Ths meas ha mahemacs sruc mus use cexs ad pedagges ha allw sudes use her w culural, ehc, ad geder prefereces ad appraches. Lads-Bllgs, 1994 Whe we refer he cds ha eed be prese fr he effecve develpme f meal sraeges, we see ha hs vew s clearly refleced. The meal rues make a excelle less sarer as hey aruse ehusasm ad ecurage he sudes feel par f a learg cmmuy. They eed las mre ha e mues, bu ha me every sude has bee egaged ad challeged ake rsks wh her curre udersadg. r d u c P a r 1 2006 Blake Educa Naural Mahs raeges Bk 1 v
rduc Classrm maageme Fr each meal rue, prvde a lamaed m-whebard f he ask resurce card fr each sude ad suable wrg maerals. We call hese m-whebards, as fel hghlgher pes ca be used rg r mark deas he lamaed cards. r d u c P a r 1 The eacher begs by psg smple, clsed quess ha eable everye be successful., he quess chage a mre pe ype, where mre ha e aswer ca be fud. Ths eables sudes beg wrk a her w level. Fally, he prcess s flpped, ad he sudes ask he quess, ryg deerme a slu he prblem ha he eacher has psed. Targe sraeges gve a fcus he less. Clsed quess eable he eacher see wh has g ad whch sraeges are beg used. Ope quess shw he sudes ha here s fe mre ha e mehd ad mre ha e rgh aswer a ques. Flp quess gve he sudes he ppruy pracse he laguage f mahs. Targe sraeges Cug ad cug ubsa Number recg Esma Clsed quess YOGHURT TUB Fr hs rue gve each sude he lamaed margare ub card ad ask hem recrd her aswers her w way he card. Fr example hey mgh draw he cuers beg drpped r hey mgh use umbers. 178 2006 Blake Educa Naural Mahs raeges Bk 1 Lse as drp he cuers he margare ub. Hw may cuers are he ub? There are fur cuers he ub s wre he umber r draw fur cuers he ub. f drp w mre, hw may wll here be he? There are w cuers he ub. Lse ad cu as drp sme mre he ub. Hw may are he ub w? Whe he sudes are gd a hs, sar wh a lw eve umber ad drp he cuers by ws ad see hw hey cu r recrd he. Ope quess Lse as shake he ub. Esmae frm he sud whe rale he ub, hw may are he ub? There were hree cuers he ub. drpped a few mre. Lse as rale he ub. Esmae hw may mgh be he ub w? pu cuers he ub w a a me. There are less ha e cuers my ub. Hw may cuers mgh have my ub? Flp quess ask card 3 Card masers are prvded fr he m-whebards. Task Card 3 (4 cards) have sx cuers my ub. Yur jb s ask me quess fd u hw may were my ub beg wh. Ecurage he sudes use cuers r marks, ad fd a way f keepg rack f wha hey have fud u frm he quess. Yu may eed mdel hs a few mes. As he sudes ge gd a hs, ecurage hem ask rcker quess r crease he umber f cuers beg used he ub. Ne: Each lamaed m-whebard ca be reused fr up w weeks by chagg he level f he ques ce as he sudes vcabulary, sklls ad sraeges mprve. v 2006 Blake Educa Naural Mahs raeges Bk 1
rduc Par 1: Meal Rues Par 2: Prblemased uas Par 3: Reflec We use he erm prblemased sua descrbe he ype f acves ha wll allw sudes egage wh realsc ( hem) suas as descrbed he research frm he Freudehal sue. The suas prvde he kds f challeges ha ecurage sudes csruc her w deas, sraeges ad mahemacal udersadgs as hey grapple wh hem. The sudes, as descrbed earler, are develpg her w mahemacal ls, whch ca be frmalsed by he eachers whe apprprae. The prblemased suas prvded have mulple ery ps ad may mehds f slu. f he umbers are hard, hey ca be reduced; f hey are easy, hey ca be creased. me sudes wll draw pcures r ac u he slu wh bjecs, whereas hers may use a mre symblc apprach usg umbers r alles. me wll prese slus a rgased fash, whereas hers wll be mre muddled. s he sharg ad reflecg he rage f sraeges ha wll brade he pssbles fr he sudes ad allw hem eer mahemacal hkg frm her very frs expereces. The fcus he prmary classrm s shfg wards a emphass mahemacal reasg ad prblem slvg a rue sese. Ths ew fcus helps sudes lear descrbe, cmpare ad dscuss her mulple appraches slvg real prblems. he classrms where we have bee wrkg, we have ed ha all he sudes egage well wh he prblems ad have shw creased eres mahs alg wh a really frm ccepual udersadg. s dffcul each he sudes algrhms ad prcedures whe hey are ready fr hem ad have frm fudas place. The reflec, as descrbed furher ad cluded each f he preseed prblemased suas, s ceral hs apprach. Par f he prepara fr he reflec s he prcess f bservg he sraeges ha he sudes use ad f lseg her explaas. Frm he frma gahered, s pssble exed, csldae ad frmalse learg durg he reflec prcess. Whe he sudes are wrkg, s pssble gaher frma abu wha hey d kw ad wha hey ca d. Fr sace whe a sude s uch cug all he hgs draw r se u, a smple ques such as D yu have cu hem all r s here smehg else yu ca d? may ac as a prmp frm whch he sudes ca demsrae ha hey ca cu r cu by 2s. Aag he wrk samples makes pssble recrd hs frma s ha decss abu fuure plag ca be made. A rage f wrk samples wll eveually gve a clear pcure f a sude s prgress wards udersadg he bg deas. r d u c P a r 2 2006 Blake Educa Naural Mahs raeges Bk 1 x
rduc r d u c P a r 2 Classrm maageme The bdy f a hree-par less s fe ake up wh a prblemased sua whch he acvy s rduced wh as lle scaffldg as pssble. The acvy ca be srucured eable he sudes wrk depedely, small grups r cllabravely larger grups, eher as hey wsh r su he eacher s assessme purpses. The prblemased suas requre he sudes wrk mahemacally, draw her w expereces ad fe ve her w mehds f recrdg ad fdg a slu. We gve a specfca f he prblem ha ca be dsplayed a prme place fr all refer. The resurces ls s a sugges ly eeds M be alred yur class s eeds. The acvy gude makes suggess fr rug he acvy. Afer he vesga, each sua leads he fal par f he less he reflec sage. HOW MANY THNG DD THE CATERPLLAR EAT ALTOGETHER? Hw may hgs dd he Very Hugry Caerpllar ea algeher? Resurces The sudes pages fr The Very Hugry Caerpllar, cug maerals, Kd Px. Acvy gude Use fve f he sudes pages (Mday Frday) ad ask hem fd a way f wrkg u hw may hgs he caerpllar ae algeher. Tell he sudes ha hey ca use cug bjecs f hey wa bu ha yu wa hem draw her mehd f wrkg u s ha yu ca fllw her hkg. Kd Px ca als be used fr hs acvy. Yu may wa chage hs acvy su he dffere ery levels f yur sudes, s fr sme sudes Mday Wedesday wll be eugh, whereas fr hers he whle week ca be used. Fr sme sudes shwg her meal hkg s sll mpra whchever cmba s chse. Reflec Use hree r fur f he sudes mehds as he bass f he reflec, s ha he sudes see a rage f dffere sraeges. D be afrad use a example wh a errr because fe he sudes lear frm her msakes ad fx up as hey expla her mehds. Afer hearg abu he dffere mehds, ask he sudes: Whch mehd f recrdg made cleares yu hw he aswer was fud? N1.1 Represeg, cmparg ad rderg umbers 20 usg effecve cug sraeges. f yu dd hs acvy aga mrrw, wuld yu use a dffere sraegy r he same e? Why? A lk he ms releva bg dea helps prvde a fcus fr bserva f he acvy. As he sudes are wrkg he prblem, he eacher has pprues bserve mehds f recrdg, sraeges used, prblems ecuered ad fx-up sraeges used. Ths s mpra prepara fr he fal reflec sage. x 2006 Blake Educa Naural Mahs raeges Bk 1
rduc Par 1: Meal Rues Par 2: Prblemased uas he busy classrm he ed f he less appraches all quckly ad as a resul he reflec s fe egleced. Ye he reflec s he ms mpra par f he less. s he me whe he sudes use mahemacal laguage expla wha hey have de ad hey see ha here are may sraeges fr slvg prblems, ad ha sme are mre effecve ha hers. s als he me whe he eacher ca frmalse a parcular dea, ccep r prcess ad scaffld he sudes he ex level. fac here are sme wh g s far as say ha f yu dd d a reflec he he sudes wll prbably rea hg. The develpme f a cmmuy f learers wh share, lse ad lear frm each her s a he hear f hs apprach mahemacs. The reflec me ses up he mahemacal culure f he classrm wh s gh-k cmmuy f learers. allws fr mahemacal md jureys ad adds he exceme f learg mahemacs. The prcples f rgrus reflec are: he defca f a rage f sraeges share ad dscuss he use f e r mre errrs shw he value f checkg resuls ad f develpg a fx-up sraegy Par 3: Reflec celebrag rsk-akg, veveess, mahemacal reasg ad learg frm msakes buldg, exedg ad preseg mre frmal mehds f recrdg as sudes demsrae readess fr hem psve, csrucve feedback wh a fcus feed frward wha yu wll d ex me. Thrugh he dalgues ad parcpa f all sudes he class, he reflec sage becmes crucal he develpme f a cmmuy f learers, hrugh whch acve vlveme learg mahemacs s successfully fsered. r d u c P a r 3 2006 Blake Educa Naural Mahs raeges Bk 1 x
rduc r d u c Ccree maerals OTHER CONDERATON may seem lke a cradc say ha hads- maerals culd be used as par f meal rues bu le s explre hs dea a lle furher. s ur belef ha meal mahs ca vlve he use f ccree maerals ad des have be ally absrac. Fr sace whe sudes are frs becmg famlar wh dubles ad ear dubles (dubles plus r mus e), he use f maerals such as Ufx cubes ca becme a meal rue. The sudes ca be asked clsed quess such as hw me a ear duble ha makes seve. The sudes may hld up a sack f 3 ad a sack f 4 shw duble 3 plus 1 r hey may hld up w sacks f 4 ad he sap ff 1 Ufx cube shw duble 4 ake 1. The beauy f hs s ha he sudes ca acually see he cubes ad shw hw hey mach he sraeges. Ths meas ha laguage ad vsual magery are cmbed chuk he frma a meagful whle. here are w ps keep md as we dscuss he uses f ls: Frs, meag s here he l; sudes csruc meag fr. ecd, meag develped fr ls ad meag develped wh ls bh resul frm acvely usg ls. Teachers d eed prvde lg demsras befre allwg sudes use ls; eachers jus eed be aware ha whe sudes are usg ls, hey are wrkg w frs smulaeusly: wha he l meas ad hw ca be used effecvely udersad smehg else. Heber, J.e al. 1997 Ths use f veed ls s equally mpra whe he sudes are wrkg he prblemased suas descrbed belw. Yu wll ce ha we have prvded hads- resurces fr each meal rue. These ca be ph-elarged, reduced r cped as apprprae. We lamae urs because we kw ha hey wll be used me ad me aga, ad we wa sudes erac wh hem. They are used repeaedly ad have uses usde hse ally preseed. The sudes ejy usg waer-sluble hghlgher pes ad a ssue clea hem. The use f darker clured pes meas ha we ca see wha he sudes recrd ad als wach her hkg as hey fd her aswers. The sudes ca hld her cards up fr everye see ad hs meas ha hey see a brad rage f pssble aswers durg he pe quess. The resurces are als used develp adapve reasg durg he flp quesg. Feedback Feedback shuld be mmedae ad useful, ad shuld creae a w-w sua fr all raher ha he cmpeve w-lse sua ha s may sudes are famlar wh. By hs we mea ha here s place fr he ver-learg f umber facs r fr he sressful learg ad esg pracces ha fe ypfy meal arhmec. The e s replace hs wh a sua where sudes share her slus ad sraeges, where hey csder he beefs f dffere appraches smehg as smple as 4 plus 7 ad, s dg, receve valuable feedback fr makg cmmes such as he fllwg: dd my rabw fac 10 ad he added 1. k 1 frm he 7ad pu wh he 4 ad kew had duble 5 plus 1. dd a urarud ad cued frm 7 ul g 11. jus kew ha fac. hugh was 10 because k 1 frm 7 ad gave he 4, bu he frg he exra 1 ha was lef ver. Teacher feedback s barely ecessary, s? The las example shws ha he sude, whle lseg he hers, realsed ha a errr had bee made ad wdered hw ad where. x 2006 Blake Educa Naural Mahs raeges Bk 1
rduc There was embarrassme abu he errr, jus a wllgess share wh hers a cmm errr such mehds. The fal quess frm he eacher ehace hs feedback by requrg he sudes reflec wha was sad as hey aswer quess such as: Whch sraegy dd yu hk wrked bes hs example ad why? Whch sraegy wuld yu lke explre a lle furher? Whe we d aher e, wll yu sck wh he sraegy yu jus used r ry a dffere e? Thrugh hs prcess he sudes have bee egaged he feedback prcess ad have bee asked self-reflec ad use ha reflec as a plag l ready fr ex me. Egageme The e r s mues se asde fr meal rues are fas ad pacey. They may vlve ccree maerals, umber cards, 100 squares, dce, ble ps,... The sudes egage wh he acves because hey are dffere he res f he less. Whe we frs bega explrg he meal acves ha we sugges here, we had dea hw much fu ad f curse hw much learg wuld flw frm hem. We s realsed ha we dd eed make up a ew meal acvy every day because he aure f he asks ad he sudes eres hem mea ha hey culd be used ad easly adaped ver several days, hece he erm meal rues. We w use he rues fr several csecuve days, all he me wachg see he level f egageme ad, f curse, we swch a ew rue f we hk he eres s dwdlg. As we rduce each rue, we use he mea-laguage f he sraeges r prcess ha g wh. A frs was ur e smply mmerse he sudes he mea-laguage bu hey were s capvaed by wrds, such as subse ad use, ha hey s waed use hem. Wachg he sudes egage wh he acves has bee rewardg fr us ad fr hem. Whe he sudes are havg fu ad are egaged, hey seem be hugry fr mre. We have see eve he swched-ff learers re-egage hrugh he meal rues. Learg pprues appears ha here s lle value parcpag meal acves whch are easy r whch are already well develped. Acves, he, eed be jus he edge f he sudes cmfr zes, scaffldg hem he ex level. f hs s he case he bvusly errrs cmpua are gg ccur frm me me. We fud a frs whe we wrked hs way, sudes wuld be dersve ad scker a errrs. We als fud ha sme sudes wuld have a g fr fear f falure. was eresg hwever see hw quckly hs mdse ured arud. The sudes seemed embrace he dea f usg errrs as learg pprues ad were fe heard sayg, Oh gd, a learg ppruy. Very fe as a chld s explag her sraegy, hey ce her w errrs ad are kee fx hem up he sp. Oher mes, hugh, he errr s ced. Fr sace, recely a 6-year-ld respdg a readg f The Very Hugry Caerpllar shwed hw she had draw w rws f fds ha he caerpllar had eae durg he week. he cued 2s bu ly uched e hg a a me. he was uperurbed by her errr; she had cued by 1s earler ad wre he umber 26 her paper. Oe f he her sudes was very mpressed by her ably cu s fas ad s far by 2s, bu aher asked her why she ly uched e hg a a me. he sd perplexed lkg a her page ad asked he her sude cme up ad shw wha she mea. he laughed u lud whe she saw wha she shuld have de ad prceeded repea her cug crrecly hs me. r d u c 2006 Blake Educa Naural Mahs raeges Bk 1 x
rduc We recall whe chldre have waed share msakes; afer preseg her slus, hey expla her errrs. The chldre als becme very supprve f e aher ad udersad ha errrs are a aural par f dg mahemacs. Errrs fe ca lead ew udersadgs abu he ccep. Traf, P. & Thesse, D. 1999 r d u c Cmmuy f learers T ga he ms frm hese acves, he sudes eed becme a cmmuy f learers. They eed really lse he deas f hers, gve psve feedback, ask quess, make suggess ad cmparss, ad fally evaluae he sraeges preseed by hers. They eed feel safe ake a rsk, prese her deas ad cmme he deas f hers. They eed lear jusfy her vewps ad sck wh hem. Fr sace, f afer hearg hw ear dubles ca be used fr a add a chld sll prefers a cu- sraegy he hey shuld be able expla why hey prefer. Ad a he ed f he day f he respse s, Because kw always gves me a crrec aswer, he ha jusfca has be see by all as vald fr ha sude a ha me, ad as such shuld be respeced. Learg be a member f a mahemacal cmmuy meas akg wershp f he gals ad accepg he rms f scal erac. Why s mpra ha classrms becme mahemacal cmmues ad ha all sudes parcpae? Because such cmmues prvde rch evrmes fr develpg deep udersadgs f mahemacs. Heber, J. e al. 1997 xv 2006 Blake Educa Naural Mahs raeges Bk 1
rduc REFERENCE The refereces gve belw are he key surces fr ur explaa f he develpmeal sequece asscaed wh pcs a hs level. 1. Checkley, K. (1999) Mah he early grades: Layg a fuda fr laer learg. Currculum Magaze. 2. Cpley, J.V. (2000) The Yug ude ad Mahemacs. Naal Assca fr he Educa f Yug Chldre, Washg DC. 3. Heber, J. e al. (1997) Makg ese: Teachg ad Learg Mahemacs wh Udersadg. Heema, NH. 4. Hu, J. (1999) Mahs he Early Grades, Currculum Jural. ACD. 5. Kam, C., (1989) Yug Chldre Cue Reve Arhmec: 2d grade. mplcas f Page s Thery. Teacher s Cllege Press, N.Y. 6. Lads-Bllgs, G. (1994) The Dreamkeepers: uccessful Teachers f Afrca Amerca udes. Jssey-Bass, a Fracsc. 7. Naal Research Cucl (2001) Addg Up: Helpg Chldre Lear Mahemacs. The Naal Academes Press, Washg DC. 8. Russell,.J. (2001) Chagg he Elemeary Mahemacs Currculum: Obsacles ad Challeges, D. Zhag, T. awada & J.P. Becker (eds) Prceedgs f he Cha Japa U.. emar Mahemacs Educa. 9. Traf, P.R. & Thesse, D. (1999) Learg Thrugh Prblems: Number ese ad Cmpuaal raeges. Heema, NH. 10. va Hele, P.M. (1999) Develpg gemerc hkg hrugh acves ha beg wh play, Teachg Chldre Mahemacs, 5 (6), pp. 310 16. r d u c 2006 Blake Educa Naural Mahs raeges Bk 1
U 11 Usg Paers Fcus Paers ccur everywhere aural phemea ad mahemacs. They frm he bass f much f ur everyday lves where we rgase urselves accrdg aurally ccurrg paers me ad aural cycles such as he seass ad eves fr celebra. Paers are used musc ad dace, ar ad desg, games ad everyday uses f mahemacs. The ably sp paers, creae paers ad make predcs based hem s fudameal mahemacal ad adapve reasg. Chage paers als frms par f hs mahemacal reasg. Fr sace kwg ha yu ca cu frwards fd a al bu ha yu ca cu back he same way ad arrve a he rgal sarg p s a ceral par f udersadg relashps ad bewee umbers. a smlar way kwg ha dffere pahways ca beg ad fsh a he same place s mpra spaal kwledge. The acves hs u are eded allw sudes explre ad egage wh paers ad chage ways ha wll buld fudas fr mahemacal reasg ad vsual memry. Cex The cex chse fr hs u s ha f playg Paer deecves. Ms sudes kw ha deecves bserve suas ad lk fr clues. They kw ha deecves he use ha frma sr u muddles r puzzlg suas. As paer deecves hey wll d he same. The sudes wll bserve places where paers are fud, es he paers ad wrk paer puzzles. As deecves hey wll repr ad prese her fdgs. 122 2006 Blake Educa Naural Mahs raeges Bk 1
Develpmeal sequece he early sages f wrkg wh paers, sudes beg : 1. defy he elemes wh a smple paer 2. defy he elemes wh mre cmplex paers 3. creae smple paers based e prpery r characersc (red r blue, square r rud) 4. creae mre cmplex paers usg w r mre prperes r characerscs (square ad red) 5. cue paers begu by hers 6. raslae paers frm e frm aher (seeg he relashp bewee a AB AB AB paer made wh shapes ad he same paer made wh claps ad clcks) 7. see he relashps bewee paers erms f umber, paer sequece, ps, ad s 8. predc he effecs f chages a paer. 2006 Blake Educa Naural Mahs raeges Bk 1 123
UNT 11 Usg Paers WHAT N A GRD? M e a l Targe sraeges Laguage f ps (ex, besde, abve, belw, afer, befre) Laguage f drec (bewee, arud, ver, uder, up, dw, lef, rgh) Laguage f mveme (frwards, backwards, sdeways) Explrg pahways Clsed quess There s smee he space abve me. Wh am? Wha umber s he space bewee he pgle ad he sheep? was ar ad he mved hree spaces up he grd ad fur spaces he rgh. Wh am? Task Card 20 ask card 20 Where a grd? 21 22 23 24 25 20 16 17 19 18 19 17 16 20 11 12 13 14 15 10 6 79 8 97 10 6 1 2 3 4 5 ar 2006 Blake Educa Naural Mahs raeges Bk 1 195 r u e s Ope quess am he mddle rw. Wh mgh be? There s smee hree spaces my lef/rgh. Wh mgh be? There are e/w/hree amals he spaces arud me. Wh mgh be? have made a pah frm ar Fsh jus usg empy spaces. Wha umbers dd mee? Flp quess am hkg f a amal. Yu ca ask me quess abu ps, drec r mveme fd u wh am. Wha quess culd yu ask? Yu may eed mdel hs usg smple ps quess ad shwg he sudes hw elmae amals frm her bards. Each me yu play, yu wll be able rduce mre cmplex laguage ad pahways. 124 2006 Blake Educa Naural Mahs raeges Bk 1
Usg Paers 11 UNT PATTERN TO 20 Targe sraeges kp cug 20 Explrg equal-szed jumps Lkg fr relashps paers Clsed quess bega ar ad made hree jumps f 3. Wha umber dd fsh? sared 2 ad made fur jumps f 2. Wha umber dd fsh? bega ar ad fshed he umber 9. made hree jumps. Wha sze were my jumps? Ope quess bega ar ad laded 10. Wha sze jumps mgh have made? made hree equal-szed jumps. Where mgh have laded? jumped by 4s. Where mgh have laded? Flp quess Task Card 21 (5 umber les) ask card 21 We are gg play a game called Guess my Jump Paer. Yu eed ask quess fd u where sared my paer frm ad where fshed. Als ask abu he sze f he jump ad he umber f jumps made. Yu may eed mdel he quess ad he elma prcess fr he sudes frs. Make sure he sudes have had expereces buldg up cug paers 20 by placg clured cuers he grds befre rducg he ex vers f he flp. made equal-szed jumps ge 20. Yu ca ask me quess fd u hw may jumps made ad hw bg each jump was. The ws s ha yu ca ly ask me quess abu he umbers ha laded. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ar 196 2006 Blake Educa Naural Mahs raeges Bk 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ar M e a l r u e s 2006 Blake Educa Naural Mahs raeges Bk 1 125
UNT 11 Usg Paers P R O B L E M A T E D T U A T O N Resurces PATTERN N PATH The ler has w ypes f les, recagles ad squares. He was le a pah lke hs pahway bu he ca decde whch paer d r hw may f each le he wll eed. Wha wuld yu sugges? Phcpy he les ad pahways frm bh ask cards dffere clured card ad cu u he square ad recagular les; glue. Acvy gude Ths acvy requres he sudes vesgae paers ha perae mre ha e drec. The paer mus repea acrss he pah ad alg. Ths wll be a ew challege fr may f he sudes. 198 Alhugh hey wll be cverg he area, sme les 2006 Blake Educa Naural Mahs raeges Bk 1 may verlap he edges as wuld happe real lfe ad wll eed be cu ff r crssed ff. Ths wll 2006 Blake Educa Naural Mahs raeges Bk 1 197 allw he ppruy alk abu par ad whle relashps ad he use f fracs. me sudes wll aurally clude hese deas her preseas, fr sace by reprg ha her pah k 12 les ad 5 par r half les. hw he sudes hw her pahs ca be made permae by glug he les ps. As he sudes make her paers ad cu he les, selec a varey fr sharg. Lk fr sme cmplex paers as well as sme eresg mehds f wrkg u hw may were used. Reflec Based he wrk seleced, ask quess such as he fllwg: Whch paers are smlar sme way? Whch paer s rckes sp ad why? Hw dd yu wrk u hw may f each le was used? Pahway les ask card 23 Pahways Dd all he pahways use he same umber f les? Why/Why? Task Cards 22, 23 ask card 22 PA1.1 Creag ad descrbg paers based smple rules. 126 2006 Blake Educa Naural Mahs raeges Bk 1
Usg Paers 11 UNT COUNTNG PATTERN Whch umbers he umber le are laded ms fe whe yu vesgae cug paers? Whch umbers d yu lad leas fe? Resurces Lamaed cpy f he umber le 1 20 (card masers), sme clured pes r cuers, a calcular fr each grup. Acvy gude Fr hs acvy he sudes wll be vesgag he effecs f dffere umber paers. Beg cug 2s, 3s... ad he ecurage hse wh eed mre challege vesgae wha happes whe hey ry mxed paers such as cu- 4, cu-back 1 ype f paers. Expla he sudes ha hey eed fd a away f keepg a recrd f whch umbers are laded each paer ad ha hey ca use clured cuers ad/r pes recrd ha. hw he sudes he key sequece: fr vesgag he cug paers ad sugges ha e pers he grup ake respsbly fr creag ad r checkg he paer sequeces. As he sudes are wrkg, ask hem predc ad he fd u wha happes whe hey cu back he same paer as hey used fr cug frward, sarg frm he las cued umber f curse. Observe he sudes as hey wrk ad selec dffere appraches ad paers fr he reflec. Reflec 2 + = = Beg he reflec by askg he sudes whch umbers were laded ms frequely ad whch were laded leas frequely. Ask hem ffer reass as why hs mgh be he case. elec w r hree dffere mehds f recrdg. Ask he sudes whch mehds shwed he dffere paers ms clearly as well as he umber f mes each umber was laded ad why. PA1.1 Creag ad descrbg paers based smple rules. P R O B L E M A T E D T U A T O N 2006 Blake Educa Naural Mahs raeges Bk 1 127
UNT 11 Usg Paers P R O B L E M A T E D T U A T O N Resurces GRD PATTERN Whch dffere pahway paers hs grd wll ake yu frm a sarg p a fshg p, bh chse by yu? Whch pahway was he lges/shres? Lamaed cpy f he 12 x 16 grd, cuers, clured pes. Acvy gude Fr hs acvy he sudes wll vesgae dffere pahway paers, such as frward 2, rgh 3, as hey ry make a cuus pah frm a seleced sar a seleced fsh ps. Allw me fr he sudes explre paer mves befre preseg he prblem hem. Fsh Whe yu rduce he prblem, demsrae a pahway 2006 Blake Educa Naural Mahs raeges Bk 1 199 paer hem ad he ask hem predc wha wuld happe f yu backracked by fllwg he paer seps reverse. Tell he sudes ha whe hey have fud a pahway paer ha js he w pss, hey ca wre he pahway paer dw, mark he sar p bu he fsh p her grd, ad he ask a parer fllw her pahway. As a exra challege he sudes ca ry markg bxes ad ry fd a paer ha wll pass hrugh her marked bxes ad sll mee he fshg p. Reflec elec a varey f paer pahways use fr he reflec cludg e ha des j sar fsh crrecly, ad ask he sudes expla: he repeag paer whch paers are smlar sme way ad hw w paers dffer wha happes whe a paer s reversed 12 x 16 grd PA1.1 Creag ad descrbg paers based smple rules. wha prblems hey had whe hey were ryg dffere pahways ad wha sraeges hey used fx hem up. Task Card 24 The shw he sudes he pahway ha des wrk ad ask fr sme fx-up sraeges. ar ask card 24 128 2006 Blake Educa Naural Mahs raeges Bk 1
CLOED PATHWAY Usg Paers Wha dffere paer pahways he grd make a clsed shape? Wha s he larges/smalles clsed pahway ha yu ca fd? Resurces Lamaed cpy f he 12 x 16 grd, clured pes. Acvy gude Ths acvy asks he sudes vesgae pahways ha make clsed shapes. As he sudes vesgae pahways, ecurage hem csder he pssbly f hem ever clsg up. Whle hey may be able make hese predcs a frs, ecuragg hem hk ad pla ahead wll mprve her vsualsa. Ask he sudes recrd he es hey d fd, gvg each e a dffere clur. Challege he sudes vesgae he ypes f mves ha wll resul larger/smaller shapes ha he es already fud. Reflec As he sudes repr ad shw her fdgs, remd hem ha hey are paer deecves ad ha hey shuld have draw sme ccluss frm her bservas. Fr sace, ask hem wha hey have ccluded abu: pahs ha wll ever clse ad why hw make a really wggly pah ha clses hw may dffere ypes f mves ca be cluded a paer ha sll clses hw make he smalles pssble clsed shape wh a pah hw make he lges clsed pah. 12 x 16 grd ar Task Card 24 2006 Blake Educa Naural Mahs raeges Bk 1 ask card 24 Fsh PA1.1 Creag ad descrbg paers based smple rules. 199 11 UNT P R O B L E M A T E D T U A T O N 2006 Blake Educa Naural Mahs raeges Bk 1 129
UNT 11 Usg Paers PAPER FOLDNG v e s g a Whch shapes ca yu make whe yu fld a pece f paper hree r eve fur mes? Clur shapes ha have he same umber f sdes he same clur, eve f hey are smaller r larger. Fd a way f shwg hw may f each shape yu have made. Hw may shapes algeher? Resurces A5 paper wrks bes fr hs vesga. The vesga Demsrae he prcess f cug as yu fld a pece f paper hree mes, explag ha flds ca be made ay drec acrss he paper. Ope u yur paper ad demsrae hw draw pecl alg he fld les make he shapes clear. Beg cug hw may f each ype f shape yu have ad csruc a smple able r graph shw cmparss bewee he umbers f each shape. Allw me fr he sudes fld her paper ad clur he shapes befre hey fd her w way f recrdg hw may f each shape. Reflec ve e r w sudes draw e r w f her shapes he bard, makg sure ha he umber f sdes each e s wre udereah. Ask sudes f ay shapes have bee mssed ad, f s, whch es. udes ca cmpare hw may shapes hey made. They may be able gve reass as why here are dffere umbers f shapes as well as a rage f dffere shapes. The sudes mgh lke call u hw may f a parcular shape hey fud ad, wh he help f a calcular, ca creae a class recrd fd u whch was he ms cmm shape. 130 2006 Blake Educa Naural Mahs raeges Bk 1
Usg Paers 11 UNT A game fr 2 r mre players PATTERN FUN Resurces Paer ad ac dce made frm he lamaed ask card. Task Card 25 ask card 25 Paer ad ac dce Jump Clap Hw play (cmpevely) Players ake urs hrw he paer de, whch ells hem he paer sequece. The ac de s he hrw w r hree mes decde he paer acs. The player he has creae he paer sequece acs, ad repea fve mes whu a errr r whu laughg s much ha hey ca cue. The her players are paer deecves wh wach check ha he paer acs ad s repeas are played crrecly. A crrec cmpleed paer scres e p. A he ed f hree games, he player wh he ms ps ws. Hw play (cperavely) Make hs a cperave game by sag ha a eam ly ws f all s members have full marks, whch case he paer deecves call u he paer esure ha s crrecly perfrmed. Varas The sudes ca make her w dce mach her w deas ad eress. Clck kp Wggle hake ABBA AABB ABAB ABC ABBC 200 2006 Blake Educa Naural Mahs raeges Bk 1 AABCC g a m e s 2006 Blake Educa Naural Mahs raeges Bk 1 131
UNT 11 Usg Paers NUMBER ON PEG A E M E N T A C T V T E Resurces Pecls, scssrs, umbered cuers, glue sck. Prr expereces The sudes wll be ready fr hs acvy f hey have had expereces wh: explrg cug paers sequecg ad rderg umbers 1 10 defyg mssg umbers sequeces cug sequeces already begu Observer s gude Acvy hee 23 Acvy hee 23 Numbers pegs These chldre are pug umbers pegs make umber sequeces. Cu u hese umbers ad pu hem he pegs cmplee he sequeces. The acvy preseed he shee requres he sudes defy each f he cug sequeces preseed ad cmplee hem usg he umbers a he bm f he shee. There are dffere ways whch hs ca be de, fr example: The sudes ca cu u he umbers a he bm f he page ad glue hem ps. The sudes ca place umbered cuers he umbers a he bm f he page ad he mve he cuers ps he umber sequeces. The sudes ca crss ff he umbers a he bm f he page as hey draw he umbers eeded cmplee he sequeces. By prvdg pecls, scssrs ad umbered cuers, he sudes make her w chce as he apprach hey wa ake, ad dg s wll demsrae her level f develpme. Name 170 2006 Blake Educa Naural Mahs raeges Bk 1 PA1.1 Creag ad descrbg paers based smple rules. 132 2006 Blake Educa Naural Mahs raeges Bk 1
Usg Paers 11 UNT PYTHON PATTERN Prr expereces The sudes wll be ready fr hs page f hey have had expereces wh: creag paers ad cug paers labellg paers wh umbers cmparg leghs Observer s gude The acvy preseed he shee requres he sudes defy he paer u preseed ad he wre he umber labels fr he paer. The prcess s he reversed wh he sudes makg a paer mach he gve umber labels. Fally he sudes ca brg her w level f cmplexy he ask as hey creae ad label a paer f her w. Ths acvy ecurages he sudes : mve fluely bewee pcures ad symbls udersad ha a paer ca be shw a umber f ways. Whe creag her w paers, sme sudes wll frs use he cards make he paer ad he label he paer, whereas, hers wll wre he umber labels frs ad he creae he paer. Bh appraches are perfecly accepable. The secd apprach hwever mgh dcae a hgher degree f cfdece wh umbers ad wh he develpme f absrac hkg. Name Pyh paers Acvy hee 24 2006 Blake Educa Naural Mahs raeges Bk 1 Acvy hee 24 Jm ad Ta made hs pyh paer. Cpy her pyh ad make lger. Wre he umber paer fr yur pyh. Make a pyh paer ha maches hs umber Make yur w pyh paer ad wre s umber paer. PA1.1 Creag ad descrbg paers based smple rules. 171 A E M E N T A C T V T E 2006 Blake Educa Naural Mahs raeges Bk 1 133
Acvy hee 23 Name Numbers pegs These chldre are pug umbers pegs make umber sequeces. Cu u hese umbers ad pu hem he pegs cmplee he sequeces. 170 2006 Blake Educa Naural Mahs raeges Bk 1
Name Acvy hee 24 Pyh paers Jm ad Ta made hs pyh paer. Cpy her pyh ad make lger. Wre he umber paer fr yur pyh. Make a pyh paer ha maches hs umber. Make yur w pyh paer ad wre s umber paer. 2006 Blake Educa Naural Mahs raeges Bk 1 171
ask card 20 ask card 20 Where a grd? 194 2006 Blake Educa Naural Mahs raeges Bk 1
ask card 21 ask card 21 ar ar ar ar ar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2006 Blake Educa Naural Mahs raeges Bk 1 195
ask card 22 ask card 22 Pahway les 196 2006 Blake Educa Naural Mahs raeges Bk 1
ask card 23 ask card 23 Pahways 2006 Blake Educa Naural Mahs raeges Bk 1 197
ask card 24 ask card 24 12 x 16 grd 198 2006 Blake Educa Naural Mahs raeges Bk 1
ask card 25 ask card 25 Paer ad ac dce Jump Clap Clck kp Wggle AABB hake ABAB ABC AABCC ABBA ABBC 2006 Blake Educa Naural Mahs raeges Bk 1 199
Based wrld s bes pracce! Ths seres prvdes he cre kwledge ad udersadg f he bg deas r cceps sudes requre becme cfde ad ehusasc mahs users. Ths bk s rgased welve us f wrk based he curre research he develpmeal sequece whch sudes geerally acqure hse cceps. Each u s dvded fve secs: Meal rues 10-mue less sarers wh suggesed clsed ad pe quess desged egage sudes ad aruse her ehusasm Prblemased suas challeges ha ecurage sudes wrk mahemacally wh pe-eded real-lfe suas ad csruc her w deas. These lesss clude a reflec sess where mahemacal laguage s used descrbe successful sraeges ad mre frmal mehds are rduced ad demsraed. Games fu acves desged refrce he sraeges develped each u. vesgas pe-eded vesgas ecurage sudes es ad exed her sklls. Assessme acves cslda acves ha sudes shuld readly accmplsh a he ed f each u. The seres ecurages he use f readly avalable ccree maerals ad s suppred by ver 50 phcpable acvy shees ad ask cards. The CD-ROM cluded wh hs bk s desged help eachers pla ad persalse her mahs prgram ad recrd dvdual sude s prgress. The Naural Mahs raeges seres s a cmplee schl prgram, whch als ecurages he use f supplemeal resurces esure a varey f mahs eachg ad learg expereces.