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1 Introduction 1. Whol numbrs Hom pag: Can you bat this? Numbr systms Laugh Zon 5 1. Whol numbr problms 6 Invstigation: Palindroms Magic squars Numbr pyramids and cross-numbr totals Estimating and rounding Ordr of oprations 16 Activity: Th four 4s puzzl; Puzzling yar; What numbr am I? 18 Invstigation: Th multiplication targt gam Mntal maths stratgis Numbr sntncs Invstigation: Odds and vns 4 Maths in Action: Calculating th Grat Wall 5 Chaptr 1 rviw: Prsonal Larning Activity 1 7 Rviw qustions 1 7. Displaying data Hom pag: Counting th catch 31.1 Survys and frquncy tabls 3. Avrags 35 Laugh Zon 38 Invstigation: Using avrags 39 Computr invstigation: Collcting statistics 41.3 Bar graphs 4.4 Lin graphs 45 Maths in Action: Th mystry of th Incas 49.5 Dividd bar and sctor graphs 5.6 Th bst statistical graph 53 Invstigation: Survys 54 Chaptr rviw: Prsonal Larning Activity 55 Rviw qustions Numbr pattrns Hom pag: Th truth is out thr in th numbrs Multipls Divisibility 6 Activity: What numbr am I? Factors 63 Invstigation: Th siv of Eratosthns Prim and composit numbrs 66 Invstigation: Goldbach s conjctur Squar and cub numbrs 69 Laugh Zon 71 Maths in Action: Kping it scrt Othr spcial numbrs 74 Computr invstigation: Fibonacci and othr numbr pattrns 77 Chaptr 3 rviw: Prsonal Larning Activity 3 78 Rviw qustions Fractions Hom pag: Fractions giv you rhythm Equivalnt fractions 8 Invstigation: Fraction wall Impropr fractions and mixd numbrs Ky prcntags Comparing fractions Probability Adding and subtracting fractions 91 Activity: Unit fractions 94 Maths in Action: Egyptian fractions Multiplying fractions 97 Invstigation: Idal fractions Dividing fractions 101 Activity: What fractions ar w? 10 Laugh Zon 103 Chaptr 4 rviw: Prsonal Larning Activity Rviw qustions iii

2 5. Dcimals Hom pag: Richtr scal a bit shaky Plac valu 108 Activity: Donna s dcimal Comparing dcimals Rounding off Convrting dcimals to fractions Addition of dcimals Subtraction of dcimals 117 Activity: Tn dollars and ighty-nin cnts 118 Laugh Zon Multiplication of dcimals by whol numbrs Multiplication of dcimals by multipls of Multiplication of dcimals by othr dcimals Division of dcimals by whol numbrs 15 Invstigation: Diving scors Division of dcimals by multipls of Division of dcimals by othr dcimals 130 Maths in Action: Dcimal drinks 13 Chaptr 5 rviw: Prsonal Larning Activity Rviw qustions Masurmnt Hom pag: Spaking volums about Archimds Mtric units 138 Activity: Mtric masurs Primtr 144 Laugh Zon Ara 147 Activity: Rmoving matchsticks 148 Invstigation: How many squars on a chssboard? Ara of a rctangl Ara of a triangl 15 Maths in Action: From fingrs to ft to mtrs Volum Volum of rctangular prisms Tim 160 Chaptr 6 rviw: Prsonal Larning Activity 6 16 Rviw qustions Rlationships Hom pag: Bubbl algbra bursts onto th scn Algbra ruls Finding a formula Pronumrals 169 Laugh Zon Dscribing pattrns algbraically 17 Invstigation: Cups and countrs 175 Activity: Cutting string 176 Activity: Th handshak problm Grid rfrncs 178 Invstigation: Chss pic tours 180 Maths in Action: Whr in th world? Th Cartsian plan 185 Computr invstigation: Scattrplots Latitud and longitud 189 Chaptr 7 rviw Prsonal Larning Activity Rviw qustions Angls Hom pag: In th stps of th dinosaur Masuring angls 196 Activity: Count th angls Drawing angls 00 Invstigation: Dot papr angls 0 Maths In Action: Billiard ball bouncs Dscribing angls 05 Invstigation: Lin dsigns in angls Complmntary and supplmntary angls 08 Laugh Zon 10 Activity: Ella s angls Angls in a rvolution Vrtically opposit angls 14 Chaptr 8 rviw Prsonal Larning Activity 8 16 Rviw qustions 8 16 iv HEINEMANN MATHS ZONE 7

3 9. Equations Hom pag: Soaking up th gravy quation Numbr sntncs 0 9. Using pronumrals in quations 9.3 Using a flowchart 4 Activity: Th mindradr gam Building xprssions with flowcharts Solving quations using backtracking 7 Maths in Action: Pdal-powrd flight 31 Invstigation: Guss, chck and improv 33 Computr invstigation: Solving quations using substitution Solving problms with quations 35 Activity: Algbraic puzzls 38 Laugh Zon 39 Chaptr 9 rviw: Prsonal Larning Activity 9 40 Rviw qustions Shaps Hom pag: Bringing shaps to lif Triangls 44 Invstigation: Polyiamonds 46 Invstigation: Angl sum in a triangl Angl sum in a triangl Quadrilatrals 50 Activity: Quadrilatral quandaris Angl sum in a quadrilatral 53 Laugh Zon Polygons 57 Invstigation: Angl sum in a polygon Compass constructions 6 Maths@Work: Graphic dsignr Plan shaps with curvs Transformations 69 Computr invstigation: Using Microworlds or LOGO Solids 7 Chaptr 10 rviw: Prsonal Larning Activity Rviw qustions Rich tasks 75 Workbook and txtbook answrs 80 Glossary and indx 34 v

4 T h Tasmanian Aquacultur and Fishris Institut is undrtaking a statisticallybasd rsarch projct to hlp prdict th numbr of abalon at any particular tim. Until now th catch limit has bn calculatd using imprcis mthods, basd on th rports by divrs as to how many abalon ar in an ara. Whil divrs rports will still b valuabl information, th nw projct will us a captur sampling procss to gt som ral data rlatd to th abalon population. With this industry worth mor than $?5 million pr yar to th Australian conomy, it is asy to s why it nds to b managd wll. hi.com.au w Word sarch 31

5 In this topic w talk a lot about data. Data is simply information that has bn collctd in som way. On of th asist ways to collct data is to conduct a survy. In a survy only a fraction of th total population is qustiond or obsrvd as opposd to a cnsus which involvs all of th population. Having collctd th data w must thn prsnt it in som way. A frquncy tabl is oftn usd for this. If w survyd a class of Yar 7 studnts about th numbr of brothrs and sistrs thy hav, th rsults might look lik this:, 3, 5, 0, 1, 1,, 0, 3, 1, 4, 0, 7, 1, 3,,,, 1, 1, 0, 1, 1,, 4. To summaris th data, w first draw up a tabl with thr columns. Th first column shows what is bing survyd. Th scond column is th tally column, whr w count th numbr of tims ach catgory occurs. Notic that is usd to rprsnt th numbr 5. Th third column is th frquncy column, whr w ntr th tallis as numbrs. W can add up th frquncy column to chck that w did not miss any numbrs. Frquncy tabl: Numbr of brothrs and sistrs Tally Frquncy Somtims th data w collct is so sprad out that w nd to group th rsults so that w only hav btwn fiv and tn rows in our frquncy tabl. In th following data th valus rang from 40 to 84: 73, 84, 68, 45, 5, 44, 45, 5, 66, 4, 43, 40, 53, 47, 8, 76, 4, 57, 65, 81, 80, 40, 56, 7, 74, 83, 41, 66, 76, 75, 68, 81, 8, 79, 58, 81, 78, 80, 78, 76. W could us class intrvals of 40 49, 50 59, 60 69, and 80 89, or smallr ons of 40-44, 45 49, 50 54, 55 59, tc. up to HEINEMANN MATHS ZONE

6 xrcis.1 w Ex.1 Q1 6 Worksht C.1 Survys and frquncy tabls hi.com.au Qustions 7 10 can b don by th class as a whol with rsults bing rcordd on th blackboard. 7 Find out how many pts ach of th studnts in your class owns by survying your class. Draw up a frquncy tabl of your rsults. (a) How many mmbrs of your class hav no pts? (b) How many mmbrs of your class hav on pt? (c) What numbr of pts do th majority of th mmbrs of your class hav? 8 Find out how many hours of tlvision ach studnt in your class watchs on a normal wk night by survying your class. Draw up a frquncy tabl of your rsults. Round your answrs to th narst half hour. (a) How many of your class mmbrs watch no tlvision on a normal wk night? (b) How many of your class mmbrs watch lss than 1 hour of tlvision on a normal wk night? (c) How many of your class mmbrs watch 3 or mor hours of tlvision on a normal wk night? (d) How much tlvision do most of your classmats watch on a wk night? () Do you watch mor or lss tlvision than th majority of th mmbrs of your class? Or do you watch about th sam amount as th majority of th mmbrs? 9 Find out th favourit school subjct of ach of th studnts in your class by survying your class. Draw up a frquncy tabl of your rsults. (a) What is th favourit subjct of th mmbrs of your class? (b) What is th scond favourit subjct of th mmbrs of your class? (c) How many studnts had no favourit subjct? 10 Find out th hair colour of ach of th studnts in your class by survying your class. Draw up a frquncy tabl of your rsults. (a) What ar your catgoris? (b) Why was this hardr to do than th othr survys? (c) What was th most common hair colour? (d) What was th scond most common hair colour? displaying DATA 33

7 11 Draw up a frquncy tabl that has a total of 0 pics of data shard btwn th possibl outcoms 1,, 3, 4, 5, and 6. No individual frquncy is to b gratr than 4. 1 Look at th following data st: (a) Draw up a frquncy tabl using class intrvals of 0 9, 30 39, tc. (b) Dscrib what you find in your frquncy tabl. 13 Sixty 1-yar-old studnts wr tstd to find thir puls rat whn rsting. Th following figurs wr obtaind (bats pr minut): Draw up a frquncy tabl using class intrvals of 51 60, 61 70, tc. 14 Th following is a list of birth wights (in grams) of 30 babis: Draw a frquncy tabl for this data using th class intrvals , , tc. 34 HEINEMANN MATHS ZONE

8 On of th most important things that statistics hlps us to do is to work out avrags. Th us of th word avrag in statistics is a bit diffrnt from in vryday lif. In mathmatics, w hav to b mor prcis about th maning of words. Usually whn popl spak about avrags thy man on valu that is typical or rprsntativ of a whol group of valus. Oftn w hav a vagu ida that th avrag is somwhr in th middl of th group of valus. Look at th following maths tst rsults (out of 10) of a group of svn studnts: 9, 4, 5, 7, 8, 7, Just by looking at th rsults (don t do any On avrag I m warm. calculations), what do you think th avrag rsult would b? Thr ar actually thr diffrnt typs of avrags in statistics, and thy ar calculatd in diffrnt ways. Man This is th typ of avrag you hav possibly com across bfor. To find th man, w add up all th valus and divid th sum by th numbr of valus. Mdian Th mdian is th middl valu whn th data is placd in ascnding or dscnding ordr. If thr is an vn numbr of valus, w us th man of th middl two valus; that is, w add thm togthr and thn divid by. Mod Th mod is th valu that occurs most oftn. For th data 9, 4, 5, 7, 8, 7, : man mdian mod First find th sum: = 4 Thn count th valus: 7 sum of valus man = numbr of valus 4 = = 6 First plac th valus in ordr, thn find th middl on: mdian = 7 Thr ar mor 7s than any othr valu so: mod = 7 Tutorial Tutorial Tutorial displaying DATA 35

9 Somtims a st of rsults can hav mor than on rsult which occurs most frquntly. If thr ar two valus which occur most frquntly w say th rsults ar bimodal. If thr ar mor than two valus which occur most frquntly w usually say th rsults hav no mod. xrcis. Avrags w Ex. Q1 3 Worksht C. Worksht C.3 Tstr Qustions Qustions 4 (a) Writ down fiv tst rsults that hav thir man, mdian and mod all qualling 6. Don t hav th rsults all th sam. (b) Writ down fiv tst rsults whr th man and th mdian ar 7 and th mod is 9. (c) Writ down fiv tst rsults whr th man is 7 and th mdian and mod ar 6. 5 Th Kora Japan 00 World Cup in soccr was playd in mor stadiums than vr bfor. Th following tabl shows th city and capacity of th diffrnt stadiums. Intractiv Qustions 36 HEINEMANN MATHS ZONE

10 Kora Capacity Japan Capacity Soul Inchon Suwon Dajon Dagu Jonju Gwangju Ulsan Busan Sogwipo Sapporo Miyagi Niigata Ibaraki Saitama Shizuoka Kob Osaka Oita Yokohama (a) Find th (i) man and (ii) mdian capacity for th stadiums in Kora. (b) Which stadium is closst to th man capacity in Kora? (c) Find th (i) man and (ii) mdian capacity for th stadiums in Japan. (d) Which stadium is closst to th man capacity in Japan? () How do th sizs of th stadiums in th two countris compar? (f) Find th (i) man and (ii) mdian capacity for all twnty stadiums. (g) Which stadium is closst to th man capacity for all twnty stadiums? 6 A matchbox indicats that th box contains 50 matchs. Th matchs in ach of 30 boxs wr countd and th rsults obtaind appar in th following tabl. Numbr of matchs Numbr of boxs (a) What is th modal numbr of matchs pr box? (b) Writ th 30 valus out in ordr from smallst to largst, i.. 48, 48, 48, 49, 5, and hnc find th mdian numbr of matchs pr box. (c) What is th total numbr of matchs in th 30 boxs? (d) Find th man numbr of matchs pr box. 7 A small company has a managr, an assistant managr, two offic workrs and tn factory workrs. Th managr is paid $ pr yar, th assistant managr $55 000, th offic workrs $ ach, and th factory workrs $3 500 ach. (a) Find th man, mdian and mod for th annual incom of all th popl in th company. (b) Suggst a us for ach of th thr valus obtaind. (c) Which figur do you think dos th bst job of dscribing th avrag incom at th factory? Explain your answr. Homwork.1 displaying DATA 37

11 Th data displays blow hav many numbrs missing. Find th numbr that ach lttr rprsnts and thn arrang th lttrs in th ordr givn by th corrsponding answrs to find th cartoon caption. Show all working in your book. 51,, 65, 4, 6, 3,, 54, 7, 39, 4, 17, 4, For th data 9, 9, 13, 0, 4 8, 37, 18, 36, 46, 1,, 36, 39, 35, 8, 39, 47, 17, 4, 5, 53, 1, 57,, 59, 6 man = E mdian = S mod = B Us th numbrs abov to complt th following frquncy tabl. For th data 7, 7, 7, 39, 41, 100, 187, 00 Scor Frquncy 0 9 K R 19 W L A C U O T Y Total 35 man = D mdian = H mod = M For th data 1, 1, 5, 10, 60, 69, 70, 7 man = G mdian = I mod = N , 1 3, ? , , HEINEMANN MATHS ZONE

12 Using avrags You will nd: A calculator. Round off your answrs to dcimal placs. 1 Brookval High s mixd basktball tam has had th following rsults in thir matchs for Brookval HS 35 Stat Hill SC 1 th sason so far. Th nxt gam is a crucial on, and it is Brookval HS 17 Kingston HS 40 against th traditional rival East Brookval Brookval HS 33 Wakfild HS 30 HS. Suppos you ar th tam statistician, and th coach has askd you to calculat som avrags. Brookval HS 35 Fullrton SC 6 Whn popl talk about avrags to do with sport, thy ar narly always talking about th man. (a) What is th avrag Brookval scor? (b) What is th avrag scor of Brookval s opponnts? (c) What is th rang of Brookval s scors? Th rang is th diffrnc btwn th largst valu and th smallst valu. (d) What is th rang of Brookval s opponnts scors? () Th coach wants you to mak a prdiction, on th basis of your calculations, about th nxt gam. What do you think th final scor against East Brookval will b? (f) What ar som problms with making ths sorts of prdictions? Your coach has a thory that your tam only wins if thy play a tam which is at last 5 cm shortr on avrag. H asks you to tst his thory. Th Brookval High basktball playrs, togthr with thir hights, ar shown opposit. Tim Su Rajina Jff (a) What is th man hight? (b) What is th mdian hight? 163 cm 156 cm 156 cm 160 cm (c) What is th mod hight? Lann Sam Chuck Magic (d) Which on of th thr typs of avrags is probably th wrong on to us? Why? () What is th rang of th hights? 16 cm 159 cm 160 cm 156 cm displaying DATA 39

13 3 You obtain th statistics of th tams you hav playd against so far. Thy ar givn in th tabl blow. Tam Stat Hill SC Kingston HS Wakfild HS Fullrton SC Hights (cm) 148, 151, 15, 160, 148, 148, 149, , 158, 156, 155, 157, 160, 16, , 146, 159, 15, 144, 158, 157, , 154, 148, 151, 153, 15, 156, 148 (a) Calculat th mans of th hights of ach of ths tams. (b) Compar ths with th man you workd out in Qustion (a). Do your rsults back up your coach s thory? 4 Your coach wants you to prdict whthr you will win your nxt gam if his thory is tru. You hav managd to obtain som information on East Brookval s tam. Th East Brookval playrs, togthr with thir hights, ar shown in th nxt diagram. (a) What is th man hight? (b) What is th mdian hight? (c) What is th mod hight? (d) Why is thr such a big diffrnc btwn th man and th mdian? () What is th rang of th hights? How dos th rang compar to Brookval s rang? (f) Compar th mans of Brookval s and East Brookval s playrs. According to your coach s thory, will Brookval win? (g) You hav to dcid which avrag to us. Why is th avrag you choos crucial in this cas? Which on do you dcid on? Why? What dos this dpnd on? (h) Suppos you find out th day bfor that Tiny is injurd and won t b playing. H is going to b rplacd with a playr who is 151 cm tall. What do you prdict will happn? Grald Flur Andrs Con Slin John Christin Tiny 149 cm 154 cm 15 cm 151 cm 154 cm 153 cm 164 cm 179 cm 40 HEINEMANN MATHS ZONE

14 Collcting statistics Lt s collct som data basd on a survy to th qustion In which month of th yar wr you born? 1 Survy your classmats (or anothr group) and ntr th rsults in column A of a spradsht. To mak things simplr, ntr only th numbr of th month, i.. 9 for Sptmbr. W ll assum you hav no mor than 40 rsponss. Add th following hadings to th sht. In E4 ntr th formula =AVERAGE(A1:A40). In E5 ntr =MODE(A1:A40) and ntr similar formula into E6, E7 and E8. In E9 ntr =COUNT(A1:A40), which will count how much data you hav collctd. If you wr to chang th qustion and/or th data, ths statistics would b producd for th nw data. 3 To complt th frquncy distribution tabl rquirs a nw formula, COUNTIF. In H6 ntr th formula =COUNTIF($A$1:$A$40,G6). This formula looks in clls A1 to A40 and COUNTs thm IF thy ar lik G6. That is, it will count how many 1s it finds. Go back to H6 and mov th mous to th bottom right-hand cornr of th cll. Th cursor should chang into th black cross calld th fill handl. Drag it down to H17 and th spradsht should complt th tabl. Why us th $ signs? Thy ar to fix whr th data is. 4 Chang th survy qustion to collct som othr numrical data and ntr it. For xampl, which dat in th month wr you born? (Answr could b 7, for instanc.) displaying DATA 41

15 w DIY Summary Worksht C.8 Prsonal Larning Activity 1 Writ a list of th diffrnt typs of graphs in this chaptr. Think of at last on xampl of whn you could us ach on, and giv a rason for your choic. Your frind wants to know what man, mdian and mod ar, how to find thm and why thy ar all calld avrag. Explain th answrs to thir qustions. You could us numbrs from an xrcis in this chaptr or mak up your own numbrs. Rviw qustions w Rviw Q1 6 7 A numbr of yar 7 studnts wr survyd about thir sho siz. Th rsults wr: , 5, 5 --, 5, 4 --, 4, --, --, 3 --, 6,, 5 --, 1 --, 1, 3 --, , 3, 4, 5, 5 --, --,, 3 --, 4 --, 3, 1, 3 --, 1 --, 4, 3 (a) Draw up a frquncy tabl to show this information. (b) Which sho siz occurrd most frquntly? (c) Which sho siz occurrd last frquntly? 8 A numbr of familis wr survyd as to th numbr of TV sts in thir hous. Th rsults wr as follows. Numbr of TV sts Numbr of familis (a) How many familis wr survyd? (b) Find th mod of TV sts pr family. (c) Writ out th data as a list (that is, 0, 0, 0, 1, 4) and hnc find th mdian numbr of TV sts pr family. (d) Find th man numbr of TV sts pr family. 9 Twnty-fiv Yar 7 boys and twnty-fiv yar 7 girls had thir hight masurd to th narst cntimtr. Th rsults ar as follows. Boys: ,. displaying DATA 55

16 Girls: (a) Draw up an ungroupd frquncy tabl for th boys. (b) Find th (i) mod (ii) mdian and (iii) man hight for th boys. (c) Draw up an ungroupd frquncy tabl for th girls. (d) Find th (i) mod (ii) mdian and (iii) man hight for th girls. () Dscrib th similaritis and diffrncs btwn th hights of th boys and th girls. (f) Draw a combind ungroupd frquncy tabl for boys and girls. (g) Find th (i) mod (ii) mdian and (iii) man hight for th studnts. (h) Dscrib th hight of th studnts. 10 Look at th following graph, which shows th avrag amount of rainfall (cm) in th city of Jakarta ovr on yar. (a) What typ of graph is this? (b) What is unusual about th position of th vrtical scal? (c) What is Jakarta s scond drist month? (d) What is th highst avrag monthly rainfall? () Which four months form th rainy sason? Avrag amount of rainfall in Jakarta cm J FMAM J J A S OND 0 11 Look at th following graph, which shows th prcntag of popl who wr out of work during th priod from 1981 to 000. % Unmployd Unmploymnt rat in Australia Yar 56 HEINEMANN MATHS ZONE

17 (a) What typ of graph is it? (b) What dos th jaggd lin on th vrtical axis man? (c) Whn did unmploymnt pak in th priod from 1981 to 000? (d) In what yars was unmploymnt at its lowst? () What was th lowst rat of unmploymnt during th tim shown on th graph? (f) Approximatly what prcntag of popl wr unmployd in lat 1996 and arly 1997? 1 Look at th following information..3,.5,.6 How much Australians spnd on pts ach yar ($ million) Expnditur typ Dogs Cats Othr Total Food Vt chargs and prscriptions Pt car products/quipmnt Pt srvics Othr xpnss Total Sourc: BIS Shrapnl (a) Draw a bar chart showing th various xpnditur typs for dogs. (b) Draw a dividd bar chart to show th total xpnditur for th thr catgoris of pts. (c) In a sctor graph that shows th various xpnditur typs for cats, what sctor would b biggr than half th pi? (d) Explain why a lin graph could not b usd for any of ths sts of data. w Rplay Assignmnt displaying DATA 57