Decimal Fractions Decimal Fractions Curriculum Ready www.mathletics.com Decimalfractionsallowustobemoreaccuratewithourcalculationsandmeasurements. Becausemostofushavetenfingers,itisthoughtthatthisisthereasonthedecimalfractionsystem isbasedaroundthenumber10! Sowecanthinkofdecimalfractionsasbeingfractionswithpowersof10 inthedenominator. WriteinthisspaceEVERYTHINGyoualreadyknowaboutdecimalfractions. s a go! Give thi Q Tomakedark-greencolouredpaint,youcanmixyellowandbluetogether,usingexactly0.5(half)asmuch yellowasyoudoblue. Howmuchdark-greenpaintwillyoumakeifyouuseallofthe12.5 mLofbluepaintyouhave? Work through the book for a great way to do this Decimal Fractions H 6 1 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Decimal Fractions Place value of decimal fractions Decimalfractionsrepresentpartsofawholenumberorobject. thousandths thousands of • thousandths Millionths Tens ThousandsHundredsTens Ones TenthsHundredthsThousandthsTen HundredMillionthsTen W H O L E D E C I M A L 1 10 10 # 000 000 000 000 000 100 100 # ' 1000 1000 # ' # Decimalpoint 10 10 ' 000 000 # 100 ' 1 ' 10 ' ' st 1 1 decimalplace:'10 ==# onetenth 10 nd 1 2 decimalplace:'10 ==# onehundredth 100 Add‘th’tothe namefordecimal rd 1 3 decimalplace:'10 ==# onethousandth placevalues 1000 th 1 4 decimalplace:'10 ==# onetenthousandth etc... 10 000 Writetheplacevalueofeachdigitinthenumber465.2703 4 6 5 . 2 7 0 3 Multiplybymultiplesof 10 Dividebymultiplesof 10 Expanded forms Place values 4......41# 00 = 400 = 4hundred 6......61# 0 = 60 = 6tens(orsixty) Integerparts 5......51# = 5 = 5ones(orfive) 1 2 2......21' 02# = = 2 tenths 1stdecimalplace `or 10 j 10 1 7 7......71' 00or 7 # = = 7 hundredths 2nddecimalplace ` 100 j 100 1 0 00...... '1000or 0 # = = 0 thousandths 3rddecimalplace ` 1000 j 1000 1 3 33...... '10 000or 3 # = = 3 tenthousandths 4thdecimalplace ` 10 000 j 10 000 2 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions Place value of decimal fractions 1 Writethedecimalfractionthatrepresentsthese: a 2hundredths b 9tenths c 1tenthousandth 0.02 Alwaysputazeroinfront (calledaleading zero)when therearenowholenumbers d 3thousandths e 6hundredthousandths f 8millionths 2 Write thefractionthatrepresentsthese: a 3tenths b 7thousandths c 1hundredth d 9tenthousandths e 51hundredths f 11tenthousandths 3 Writetheplacevalueofthedigitwritteninsquarebracketsforeachofthesedecimalfractions: a 31.325 b 10.231 c 451.046 6 @ 6 @ 6 @ d 50.05043 e 60.79264 f 08.56309 6 @ 6 @ 6 @ 4 Circlethedigitfoundintheplacevaluegiveninsquarebrackets: a [tenths] b [thousandths] c [hundredthousandths] 8.171615 4.321230 100.1001001 d [hundredths] e [tenthousandths] f [millionths] 9.12421 16.123210 3.120619 Decimal Fractions H 6 3 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions Place value of decimal fractions Each digitismultipliedbytheplacevalueandthenaddedtogetherwhenwritinganumberinexpandedform. Writethedecimalfraction23.401inexpandedform 1 1 1 23.4012=+##10 31++4 ##0 +1 # Multiplyeachdigitbyitsplacevalue 10 100 1000 1 1 =+21##0314++##1 Zerodigitscanberemovedtosimplify 10 1000 5 Writethesedecimalfractionsinexpandedform: a 41. 9 = b 29. 281 = c 40. 2685 = d 37. 4932 = e 02. 306 = f 0. 0085 = ALUE CE V OF LA DE P CI M A S L N 6 Simplifythesenumberswritteninexpandedform: O F I R T A C C T A I R O 1 1 F a ## # N L 11++4 6 = S A 10 100 M I ..../...../20... P C L E A D C E F O V A E L 1 1 U b 41##09++10##+=7 10 100 1 1 1 c 51##00 ++2100##12++1 ##+=8 10 100 1000 1 1 1 1 1 d 61##++8 5 ##++0 2 ##+=9 10 100 1000 10 000 100 000 Psst:Remembertoincludealeading zerofortheseones. 1 1 1 e 2 ##++0 3 # = 10 100 1000 1 1 1 1 f 6 ##++7 0 # +=1 # 100 1000 10 000 100 000 1 1 1 1 1 g 3 ##++4 1 ##++0 8 # = 10 100 1000 10 000 100 000 4 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Decimal Fractions Approximations through rounding numbers Lookatthesetwostatementsmadeaboutateamofsnowboarders: • Theyhaveattempted4937trickssincestarting= Accurate statement • Theyhaveattemptednearly5000trickssincestarting= Rounded offapproximation Roundingoffvaluesisusedwhenagreatdealofaccuracyisnotneeded. Thenextdigitfollowingtheplacevaluewhereanumberisbeingroundedofftoistheimportantpart. Nextdigit 0 1 2 3 4 5 6 7 8 9 Closertolowervalue,soround down Closertohighervalue,soround up Leavetheplacevalueunchanged Add1totheplacevalue Herearesomeexamplestoseehowweroundoffnumbers. Roundthesenumbers (i) 2462tothenearesthundred 2462 Thedigit‘4’isinthehundredsposition 2462 The nextdigitisa6,soround upbyadding1to4 2500 Changetheothersmallerplacevaluedigitsto0’s ` 2462. 2500 roundedo t thenearesthundred (ii) 0.3145toonedecimalplace(ortothenearesttenth) 0.3145 Thedigit‘3’isinthefirstdecimalplace 0.3145 Thenextdigitisa1,soround down 0.3 Write decimalfractionwithonedecimalplaceonly ` 03..1450. 3 roundedtoonedecimalplace (iii) 26.35819 tofourdecimalplaces(ortothenearesttenthousandth) 26.35819 Thedigit‘1’isinthefourthdecimalplace 26.35819 Thenextdigitisa9,soroundupbyadding1to1 26.3582 Writedecimalfractionwithfourdecimalplacesonly ` 26.3 58192. 63. 582 roundedtofourdecimalplaces Decimal Fractions H 6 5 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Your Turn Decimal Fractions APPROXIM S AT R . I E 5 O B 4 N Approximations through rounding numbers M 4 U T N H R G O N 4 U I G D H N 3 U R 1 Roundthesewholenumberstotheplacevaluegiveninsquarebrackets. O ..../...../20... a [nearestten] b [nearesthundred] c [nearestthousand] (i) 536 . (i) 14 302 . (i) 98 542 . (ii) 8514 . (ii) 4764 . (ii) 18 401 . (iii) 93025 . (iii) 80 048 . (iii) 120510 . 2 Roundthesedecimalfractionstothedecimalplacesgiveninthesquarebrackets. a [nearesttenth] b [nearesthundredth] c [nearestthousandth] (i) 07. 3 . (i) 24. 06 . (i) 10.4762 . (ii) 34. 7 . (ii) .0007 . (ii) 03. 856 . (iii) 11.85 . (iii) 1.003 . (iii) 00. 48640 . 3 Approximatethefollowingdistancemeasurements: a Agroupofpeopleforman8.82 mlonglinewhentheystandtogether. (i) Howlongisthislinetothenearest10cm(i.e.1decimalplace)? . (ii)Whatistheapproximatelengthofthislinetothenearest10metres? . b Underamicroscopethelengthofadustmitewas0.000194 m (i) Approximatethelengthofthisdustmitetothenearesttenthousandth . ofametre. (ii) Approximatethelengthofthisdustmitetothenearesthundredthofametre. c IfLichenCityis3 458 532 mawayfromMossCity: (i) Whatisthisdistanceapproximatedtothenearestkm? . (i.e.nearestthousand) (ii)Whatistheapproximatedistancebetweenthecitiestothenearest100 km? . (iii)Arethedigits2, 3oreven5importanttoincludewhendescribingthetotal distancebetweenthetwocities?Brieflyexplainherewhy/whynot. 6 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions Approximations through rounding numbers Roundingupcanaffectmorethanonedigitwhenthenumber9isinvolved. Round0.95toonedecimalplace 0.95 Thedigit‘9’isinthetenthsposition 0.95 The nextdigitisa5,soround upbyadding1to9 9roundsupto10,so the 9becomes0and1 1.0 Changetheothersmallerplacevaluedigitsto0s isaddedtothedigitinfront. ` 09..51. 0 roundedtoonedecimalplace 4 Roundoffthesenumbersaccordingtothesquarebrackets. a [onedecimalplace] b [nearestten] c [twodecimalplaces] 19. 8 . 398 . 11.899 . d [nearestones] e [threedecimalplaces] f [threedecimalplaces] .799 . 01. 398 . 21. 995 . g [nearestthousand] h [nearestones] i [fourdecimalplaces] 49798 . 1999. .989999 . 5 Approximatethesevalues: a Acallcentrereceivesanaverageof2495.9callseachdayduringonemonth. (i) Approximatethenumberofcallsreceivedtothenearesthundreds. (ii) Approximatelyhowmanythousandsofcallsdidtheyreceive? . (iii) Estimatethenumberofcallsreceiveddailythroughoutthemonth. b Aswimmingpoolhadaslowleak,causingittoempty9599.5896 Linoneweek. (i) Howmuchwaterwaslosttothenearest10litres? . (ii) HowmuchwaterwaslosttothenearestmLif1mL = 1 L? . 1000 (iii) Isthedigit6importantwhenapproximatingtothenearestwholelitre? Brieflyexplainherewhy/whynot. Decimal Fractions H 6 7 Mathletics Passport © 3P Learning SERIES TOPIC How does it work? Decimal Fractions Decimal fractions on the number line Thesmallestplacevalueinadecimalfractionisusedtopositionpointsaccuratelyonanumberline. • Decimalfractionsarebasedonthenumber10,sotherearealwaystendivisionsbetweenvalues Eg:Hereisthevalue3.6onanumberline: 6 Sixtenthsof thewayfrom 3.0 3.6 4.0 3.0to4.0 • Themajorintervalsonthenumberlinearemarkedaccordingtothesecond lastdecimalplacevalue 8 1.240 1.248 1.250 Soitseightthousandthsof thewayfrom1.240to1.250 Herearesomemoreexamplesinvolvingnumberlines: (i) Whatvaluedotheplottedpointsrepresentonthenumberlinesbelow? 4 a) 0.1 0.2 Pointisfourstepsfrom0.1towards0.2,sotheplottedpointis:0.14 9 b) 10.06 10.07 Pointisninestepsfrom10.06towards10.07,sotheplottedpointis:10.069 (ii) Roundthevalueoftheplottedpointsbelowtothenearesthundredth. 3 a) 2.14 2.15 Pointisthreestepsfrom2.14towards2.15,sotheplottedpointis2.143 ` thevalueoftheplottedpointtothenearesthundredthis:2.14 5 b) 8.79 8.80 Pointisfivestepsfrom8.79towards8.80,sotheplottedpointis8.795 ` thevalueoftheplottedpointtothenearesthundredthis:8.80 8 H 6 Decimal Fractions SERIES TOPIC Mathletics Passport © 3P Learning How does it work? Your Turn Decimal Fractions L FRACT MA I I 4 O C N S E Decimal fractions on the number line D O N T E H N E I 1 L Displaythesedecimalfractionsonthenumberlinesbelow: N U R M E ..../...../20...B a 0.7 b 2.1 0.0 1.0 2.0 3.0 c 0.13 d 9.15 0.1 0.2 9.1 9.2 e 2.34 f 5.212 2.3 2.4 5.21 5.22 2 Labelthesenumberlinesandthendisplaythegivendecimalfractiononthem: