MATHEMATICS AND STATISTICS 1.2

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1 MATHEMATICS AND STATISTICS. Apply lgeric procedures in solving prolems Eternlly ssessed 4 credits AS 907 Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndrd. Simplifying lgeric epressions In lger, vrile is letter such s, used to stnd for numer. Algeric terms nd epressions re formed y pplying opertions (such s +,,, ) to vriles. Emple. The numer which is doule y is y which is written y. The numer multiplying y,, is clled the coefficient of y, nd y is clled the product of nd y.. The numer which is 5 more thn hlf is written + 5 or + 5. An lgeric epression is mde up of terms dded (or sutrcted) together. Like terms (terms with the sme vriles) cn e simplified y ddition or sutrction. Emple. 7y + y = 0y. 8 = = 6 Eercise A: Algeric epressions. Write down s lgeric epressions: d. The squre of less thn e. The squre root of more thn y f. p is multiple of 6. Wht is the net multiple of 6 fter p? g. A dog costs $d per month to feed. A ct costs $7 less per month to feed. Wht is the cost of feeding dog nd ct for month? Simplify your nswer. h. Mum is yers older thn her son, Mrk. Mrk is 4 yers younger thn his sister, Disy. Dd is two yers older thn Mum. If Mum is m yers old wht is the sum of ll four ges? Simplify your nswer. i. Hn uys kg of pples t $ per kilogrm nd kg of pers t $p per kilogrm. i. How much does this cost ltogether? Ans. p. 97. The product of 7 nd. The sum of y nd 4 ii. How much chnge does Hn get if she pys $d for her fruit? c. The numer which is more thn five times the numer w iii. Wht restrictions re there on the vlue of d? ESA Pulictions (NZ) Ltd, Freephone , ISBN

2 Achievement Stndrd 907 (Mthemtics nd Sttistics.) AS 907. Simplify where possile the following c. s t s + t Multiplying nd dividing terms Any lgeric terms cn e multiplied together. Indices re used to simplify repeted multipliction, e.g. y y is written y Emple. 4 = [ 4 =, = ]. y.6y = yy [dot mens ] = y [using indices] d. p + 8q + 9p 4q e. 7f f 0f f f Division of lgeric terms is est written in frction form. Simplify using =. Note tht if ll fctors in the numertor cncel, then fctor of will remin in the numertor. For emple, 9 =. Emple g =. 0y 5 = y [dividing top nd ottom y 5] [simplifying nd cncelling ] h. 4w 5 w i j. c c c + c. Find the missing term in ech of the following simplifictions = 5 If powers of vriles occur, these cn e written s repeted multiplictions efore cncelling s efore. Emple Q. Simplify y 4 y A. Using repeted multiplictions: y 4 y = y y y y = y y y y = y [cncelling (twice) nd y y ] Ans. p pq qp + r = pq r c. m + n m + mn = n mn Eercise B: Multiplying nd dividing terms. Write ech of the following in simplest form.. 7..y d. + = c. 5 w d. 6 e. 7d + = d d e.. f. 5c g. y y y h. 5. ESA Pulictions (NZ) Ltd, Freephone , ISBN

3 MATHEMATICS AND STATISTICS. Eternlly ssessed 4 credits Investigte reltionships etween tles, equtions nd grphs Sequences of numers A sequence is n ordered list of numers (these numers re clled terms of the sequence). For emple, for the sequence, 4, 9, 6, 5, the st term is, the nd term is 4, the rd term is 9, the 4th term is 6 nd so on. By oserving ptterns in sequence, rule for the vlues of the terms my e found. This rule is often epressed in terms of the position of the term in the sequence. Emple For the sequence, 4, 9, 6, 5, The differences etween terms re, 5, 7, 9, [4 =, 9 4 = 5, ] These differences increse y ech time. So, the net difference is 9 + = nd the 5th term is 5 + = 6 Similrly, the 6th term is 6 + = 49, nd so on. Liner sequences A liner sequence increses (or decreses) stedily y the sme mount (the first difference) from one term to the net. Emple. The liner sequence, 6, 9,, 5, is formed y dding to ny term to get the net term. A generl reltionship etween the position of the term in the sequence, nd its vlue cn e found using tle. Position of term, Vlue of term, y Clerly, the vlue of ech term is equl to the position of the term. This cn e written in the form y =. By dding to ech term of the sequence, 6, 9,, this liner sequence is formed: 4, 7, 0,,. The generl rule for this sequence is y = + The generl rule for liner sequence is lwys of the form: y = m + c Where is the position of the term in the sequence, m is the stedy increse, nd c is constnt. Emple The rule for the sequence Position of term, Vlue of term, y is y = 4 + c [since the terms increse stedily y 4] By sustituting = into the rule to get 6 = 4 + c, it cn esily e seen tht c = The rule for the sequence is therefore y = 4 + Liner sequences rise in mny prcticl situtions which my involve sptil ptterns of shpes. Letters relting to the ojects (e.g. M for the numer of mtchsticks) my lso e used in the rules. Emple Joined hegons re mde with mtchsticks. The tle summrises the informtion. Numer of hegons, n... Numer of mtchsticks, M The rule is M = 5n + c [M increses stedily y 5] Sustituting n = gives 6 = 5 + c, so c = The rule is M = 5n + AS 908 ESA Pulictions (NZ) Ltd, Freephone , ISBN

4 44 Achievement Stndrd 908 (Mthemtics nd Sttistics.) Eercise A: Liner sequences Ans. p Ptterns re mde from lrge nd smll squres.. Find the generl rule for the following sequences. AS 908. n T. n T T = T = c. n T d. n T Find rule which gives the totl numer of squres, s, in the nth pttern. 6. A sequence of tiling ptterns is shown. T = T =. A sequence of joined squres is mde using mtches. Find the rule for the numer of mtches, m, needed for joined sequence of n squres.. Find rule which gives the numer of green tiles, G, in the nth pttern.. The digrms show rectngulr tles nd people seted round them. tle tles tles. Find rule which gives the numer of tiles, L, long the length of the nth pttern. 6 people 0 people 4 people Find the rule for the numer of tles, n, nd the numer of people, p, who cn sit round them. c. Find rule which gives the numer of tiles, W, long the width of the nth pttern. 4. Te towels re hung on line. By shring peg, it only tkes pegs to hng out tetowels, 4 pegs to hng out tetowels, nd so on. Write down rule for the numer of pegs, p, needed to hng out n te towels. d. Find rule which gives the numer of tiles, P, round the perimeter of the nth tile. ESA Pulictions (NZ) Ltd, Freephone , ISBN

5 MATHEMATICS AND STATISTICS.6 Eternlly ssessed 4 credits Apply geometric resoning in solving prolems Finding unknown ngles When finding the size of unknown ngles in figure, t lest two steps of resoning will e required. The following rules for finding unknown ngles re lredy fmilir. Rule Digrm Emple Adjcent ngles on stright line dd to 80 ( s on line) Angles t point dd to 60 ( s t point) Verticlly opposite ngles re equl (vert opp s) + = 80 c + + c = 60 = 50 = 40 ( s on line) 00 0 = 40 ( s t point) 80 = 80 (vert opp s) Solution = 80 ( s on line) + 70 = 80 [simplifying] = 0 [sutrcting 70 ] = 55 y = 0 (vert opp s) = 55 0 y = 5 Eercise A: Finding unknown ngles. Find the vlue of in the digrm elow. Give geometricl reson for your nswer AS 90 Ans. p. 5 An revited form of ech rule is cceptle when giving geometric resons. Emple Find the vlue of nd y in the digrm elow. Give geometricl resons for your nswers. y 0. Find the vlue of in the digrm elow. Give geometricl reson for your nswer ESA Pulictions (NZ) Ltd, Freephone , ISBN

6 00 Achievement Stndrd 90 (Mthemtics nd Sttistics.6). Light rys re reflected from mirror s shown in the figure. 70 Find the size of ngle. Give resons nd show ll working.. Find the vlue of c. Find the vlue of y (give reson for your nswer) AS Tom cuts cke into two pieces so he gets twice s much s his little sister. He mkes sketch of the figure to work out the ngle t the centre of his slice of cke. Form n eqution nd solve it to find this ngle. Give resons. 7. Find the vlue of in the digrm elow. Give geometricl reson for your nswer. 5. The figure shows lptop with its lid open. Angles nd prllel lines A trnsversl is line which cuts through given pir of lines to form vrious pirs of ngles. trnsversl corresponding ngles lternte ngles Find the ngle through which the lid hs een opened. Form n eqution nd solve it to find this ngle. Give resons. 6.. Give geometric reson for the reltionship + = 80 y co-interior ngles When the trnsversl cuts pir of prllel lines, the following rules result (mtching rrows show prllel sides). Rule Digrm Emple Corresponding ngles on prllel lines re equl. (corr s // lines) = 50 = 50 (corr s // lines) ESA Pulictions (NZ) Ltd, Freephone , ISBN

7 MATHEMATICS AND STATISTICS. Eternlly ssessed 4 credits Demonstrte understnding of chnce nd dt The sttisticl enquiry cycle (PPDAC) The sttisticl enquiry cycle summrises the steps involved in sttisticl investigtion. Conclusion Anlysis Prolem Sttisticl enquiry cycle Dt Pln Sttisticl litercy involves the ility to understnd, interpret nd evlute the results of sttisticl investigtions undertken y others. Dt collection It is often imprcticl to crry out census (n investigtion involving every memer of popultion of interest). Insted, portion of the popultion, known s smple, is investigted. The smple should e rndom (ech memer of the popultion hs the sme chnce of eing chosen) so tht the chrcteristics of the smple re similr to those of the popultion. If the smple does not ccurtely reflect the chrcteristics of the whole popultion then the smple is sid to e ised. Sttisticl nlysis of rndom smple llows inferences (conclusions) to e mde out the popultion s whole. In sttisticl investigtion, dt my e collected y the investigtors themselves. Alterntively, dt set my e otined from secondry source it is importnt tht this dt source is listed. Selecting rndom smple Suppose you re given list of 500 students, nd you hve to select rndom smple of 0 nmes from the list. Drwing nmes from ht is good relile method tht hs no is. A quicker method is to use the rndom numers on your clcultor, s follows: give ech student numer from to 500 set your scientific clcultor to produce rndom numers from to 500 (Press: nd tke the whole numer prt, ignoring repetitions) otin the first thirty rndom numers identify the smple of 0 students y mtching the numers to the nmes. Emple Select rndom smple of 8 nmes from the following list of 5 nmes: John, Will, Anne, Helen, Henry, Tom, Liz, Luke, Nthn, Jco, Angus, Amy, Jck, Slly, Pt Solution Enter the nmes in tle nd give numer to ech nme. Numer Nme Numer Nme John 9 Nthn Will 0 Jco Anne Angus 4 Helen Amy 5 Henry Jck 6 Tom 4 Slly 7 Liz 5 Pt 8 Luke Using your clcultor, otin 8 numers in the rnge 5 (Press: nd tke the whole numer prt only, ignoring repetitions) A typicl result might e:,, 4, 8, discrd (no repets), 7, 4,, 9 Highlight or tick the nmes, s shown. AS 907 ESA Pulictions (NZ) Ltd, Freephone , ISBN

8 48 Achievement Stndrd 907 (Mthemtics nd Sttistics.) Ans. p. AS 907 In sttisticl investigtion it is importnt tht the dt collection method is verified s rndom, so tht the smple is representtive of the popultion (shres its fetures). This llows inferences to e mde out the popultion. Eercise A: Selecting rndom smple Use the rndom numers on your clcultor to nswer the questions in Eercise A.. Select 8 nmes from the list of nmes in the tle. Numer Nme Srh Millie Phiz Boz Liz John Bill Ann Jmes Brry Amy Petr List the eight numers, nd in the tle highlight your choices. Rndom numers used:. Allocte numer to ech country in the tle. Select 0 countries from the list of 5. Highlight your selection nd list the numers. Numer Country Brzil Peru Bolivi Sudn Liy Kore Angol Zmi USA NZ UK Chin Niger Polnd Kuwit Rndom numers used: Dt collection methods There re three min methods of gthering smple dt. Oservtion wtching nd ccurtely recording the informtion, e.g. counting crs ( discrete dt), mesuring plnt heights ( continuous dt), etc. Orl interviews n investigtor sks questions nd records responses, e.g. opinions out products, or ttitudes to vrious issues ( qulittive dt). Written questionnires lso involve questions nd responses, ut in written form. Answers re recorded in vrious wys in words, choosing from multichoice options, or using scle (e.g. rting from 0 to 0). The initil dt gthered is clled rw dt. Dt orgnistion After collection, the rw dt re often orgnised into tles. This llows fetures of the dt set to e seen more clerly. Frequency distriution tles An effective wy to orgnise rw dt is the frequency distriution tle. If there is smll numer of dt vlues, then scores re listed seprtely.. At Mountin High School there is oys soccer tem nd girls netll tem. The plyers in ech squd re listed in the tle. Select smple of 7 plyers from the soccer tem nd 5 plyers from the netll tem. Soccer tem Numer Nme Luke Peter John In Crl Andy Sen Krl Mike Kurt Amos Sng Mrk Joel Aln Rndom numers used: Numer Nme Netll tem Grce Meg Emm Ell Lorn Srh Prue Emily Indi Rose Lur Binc Rndom numers used: Emple The times tken in seconds for smple of 5-yerold children to complete simple puzzle re shown elow: 6, 5, 5, 7, 7, 6, 5, 9, 7, 6, 6, 6, 8, 6, 5, 9, 5, 6, 9, 5, 6, 5, 7, 8, 6 The dt re orgnised, using tllies, into frequency tle. Time (seconds) Tlly Frequency Totl 5 9 children took 6 seconds. 5 children took more thn 7 seconds. ESA Pulictions (NZ) Ltd, Freephone , ISBN

9 ANSWERS Achievement Stndrd 907 Mthemtics nd Sttistics. Eercise A: Algeric epressions (pge ).. 7. y + 4 c. 5w + d. ( ) e. y + f. p + 6 g. d 7 h. 4m 58 i. i. + p ii. d p iii. d + p... c. s d. p 6q e. f f f. 0 + g. h. w 5 i. 4 j r c. mn; m d. ; e. 8d; d Eercise B: Multiplying nd dividing terms (pge ) y c. 5w d. e. 6 f. 5c g. 6y h d. p g. pq 4st.. d. mnp c. e. 4 f. h. y. c. wz c. 4 d. pq 5 Eercise C: Order of opertions (pge )... c. 6y d. 8 e. 4 f. g. h... ( + ) = 6. 8 (4 ) + = c. ( ) = d. (y + y) y y = y e. (p + 4p) (p p) = f. w (w w) = w g. (6 4) + = h. (p + q) (q q) = 0 Eercise D: Multiplying nd dividing with indices (pge 4) c. 6 d. e. f. 8 g. 5 h c. d. 8 e. 0 6 f. 6 5 y g. 6 c h p. 5 c. 9 d. g. e. f. pq h y 4y Eercise E: Indices with powers nd roots (pge 5) c. 8 8 d. e. 8w f. 4 6 g. 64 h. 9 7y 6 8y 6 ANSWERS ESA Pulictions (NZ) Ltd, Freephone , ISBN

10 ANSWERS 98 Answers.. y 8. 0 c. 8 d. 0y e. g. h p d. e. 9 g. 5 h. 6 8 f. p t 5 c. 4 f. 4 4 Eercise F: Bsic lgeric frctions.. y (pge 6) d. e. 6 7 g... d h. 7 8c g c.. 0y 7 e. y h. 4 4 d. 6 e. 5 y g. 8 y f. 5c c. f c. 6y h. q. d. 4 e. 0 c g. 8 5 d. 5p 8 h. y. 4y Eercise G: Epnding rckets (pge 8) c. 0y 5 d. y + 8 f. c c. c d f. 4 y c. 5 e f g. y + y y c. 6 d. + 6 e c. + d. + 6 e f. 8 Eercise H: Epnding pirs of rckets (pge 9) c. + 8 d. + 5 e f g y 6y + 9 c. 6 d. 4y + 0y + 5 e. 9y 4 f p + 6p c d e. 8 Eercise I: Fctorising using the distriutive lw (pge 0).. 7( + ). ( + ) c. y( y) d. 5( + ) e. p(p 4) f. (c ) g. ( + c) h. (4 + y).. ( ). y ( + y ) c. y 4 ( + y) d. 4 ( ) e. p (p p + ) f. (4 ) g. 8 ( ) h. 8 ( + 8 ) Eercise J: Fctorising qudrtics (pge ).. ( + 5)( + ). ( + 6)( + 7) c. ( 8)( + 6) d. ( + 8)( 5) e. ( + 4)( 4) f. (5y + )(5y ) g. ( 8)( ) h. ( 7)( + ).. ( + 5)( 4). 5( + )( ) c. ( + 6)( ) d. 5( 4)( + ) e. ( 7)( 4) f. 0( + )( + ) g. 8( + )( ) h. ( 8)( 4).. ( + )( + ). ( + )( + ) c. (5 + )( ) d. (7 )( + ) e. (4 + )( + ) f. ( )( + ) g. ( )( ) h. ( + 5)( ) 4.. ( + 4)( 4). 4( + ) ESA Pulictions (NZ) Ltd, Freephone , ISBN

11 INDEX dd (frctions) 6 djcent (side) lgeric epression lgeric frctions lternte ngles 00 nlysis (dt) 7 ngle of depression 4 ngle of elevtion 4 ngle sum of tringle 04 rc 8 rrowhed 07 symptote 9 verge 5 is of symmetry 7 ck-to-ck stem-nd-lef plot 57 r grph 60 se ngles 05 erings 9 BEDMAS BEMA ised 47 o-nd-whisker grph 55 census 47 certin (event) 77 chnging suject of formul chord circle 8 circumference 8 clinometer 4 clusters 55 coefficient co-interior ngles 00 composite r grph 60 conclusion 7 concyclic points 4 conditionl proilities 90 continuous dt 48, 49 corresponding ngles 00 cosine cyclic qudrilterl 4 dt 7 dt source 47 decy curve 9 difference of two squres 9 discrete dt 48, 49 distriution 55 distriutive lw 8, 0 divide (frctions) 6 dot plots 55 elimintion method eqully likely outcomes 8 eqution of line 5 equilterl tringle 04 eperimentl proility 77 eponent 4 eponentil equtions 4 eponentil sequences 88 eterior ngle 04 fctored form (prol) 77 fctorised 0 formul 0 frequency distriution tle 48 generl form (line) 57 generl rule (sequence) 64 grdient 55 grdient-intercept form 59 growth curve 90 hegon 04 horizontl lines 55 hypotenuse 7, impossile (event) 77 inde, 4 indices, 4 inference 47 intercepts method 57 interqurtile rnge 5 interviews (sttisticl) 48 inverse trig functions 5 isosceles trpezium 07 isosceles tringle 04, kite 07 like terms liner equtions 4 liner grphs 6 liner inequtions 7 liner sequence 4 long-run eperimentl proility 77 long-term trend 66 lower qurtile 5 men 5 medin 5 mode 5 multiply (frctions) 6 multivrite sttisticl dt 84 oservtion (sttisticl) 48 opertions opposite side optiml solution 9 ordered stem-nd-lef plot 57 outcomes 8 outliers 55 prol 7 prllel lines 00 prllelogrm 07 perfect squres 9 ESA Pulictions (NZ) Ltd, Freephone , ISBN