USE REAL-LIFE DATA TO MOTIVATE YOUR STUDENTS 1

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1 USE REAL-LIFE DATA TO MOTIVATE YOUR STUDENTS 1 Rober E. Kowalczk and Adam O. Hausknech Universi of Massachuses Darmouh Mahemaics Deparmen, 285 Old Wespor Road, N. Darmouh, MA rkowalczk@umassd.edu and ahausknech@umassd.edu Real-life daa provides an eremel rich environmen for developing, learning, and appling mahemaics. Wheher ou are inroducing a new opic, epanding on a curren opic, or demonsraing how mahemaics can be applied, presen echnolog makes using daa a cinch. Graphing calculaors, compuer algebra ssems, and man oher mahemaics sofware packages all have some capabili for handling daa. In his presenaion, we will use he mahemaics eploraion sofware package TEMATH o ineracivel demonsrae how daa can be an inegral par of he eaching of mahemaics. We will presen eamples of how modeling daa can be used o moivae sudens in college algebra, precalculus, and calculus. In-Class Daa Gahering A he beginning of he semeser when we discuss linear funcions wih our college algebra sudens, we have hem perform he wave eperimen. We have he firs four sudens in he firs row of he classroom perform a wave b having each suden sand up and hen si down in sequence. Usuall some suden has a sop wach and we assign ha suden o be he imekeeper. Firs, we measure he ime i akes for he firs four sudens o perform he wave. Ne we add four more sudens o he wave and measure he ime i akes he eigh sudens o perform he wave. We coninue in his wa unil he enire class does he wave. The following wave daa was generaed b Prof. Kowalczk's class his semeser: s (number of sudens in he wave) (ime in sec o complee he wave) We use TEMATH o plo he daa and hen ask our sudens wha mahemaical model (funcion) would be useful o describe he wave daa. We hen find he leas squares line fi o he daa as is shown in Figure 1. Mos of our sudens know how o calculae he slope of a line, bu, he have considerable difficul in inerpreing he meaning of he slope in erms of he unis in he problem. We promp our sudens o correcl inerpre he slope of he wave model.28 sec/suden as he average ime i akes each suden o perform he wave. We also have hem inerpre he meaning of he -inercep.96 sec as he reacion ime of he firs suden o sar he wave afer he word GO is shoued. Figure 2 shows a comparison of he wave daa from Prof. Hausknech's class and Prof. Kowalczk's class. This comparison provides a rich environmen in which sudens can compare linear funcions and inerpre he slopes and -inerceps. For eample, iniial reacion imes are he same bu sudens in Prof. Hausknech's class ake wice as long o perform heir par of he wave. Technolog in Collegiae Mahemaics", Addison-Wesle Publishing Co., 1997, p

2 1 1 (s) = s (s) = s (s) = s 35 s 35 s Figure 1 Wave Daa Figure 2 Comparing Waves Linear Trend Analsis I is imporan o use daa ha is relevan o he sudens and he socie he live in. In his da and age of violence, we can use mahemaics o sud he rend of violen crime in he Unied Saes. The 1995 World Almanac conains he raes for violen crime (per 1, inhabians) for he ears 1973 o Afer seing he base ear 1973 o =, ploing he scaer plo for he violen crime raes, and finding he leas squares rend line, we see from Figure 3 ha over his wen ear period, he violen crime rae is increasing on average b 15.5 more crimes per 1, inhabians per ear. Noe ha in his problem, we are no ineresed in finding a funcion ha fis all he daa, bu a funcion ha models he rend of hese varing crime raes. 8 = Figure 3 Violen Crime Raes in he US ( ) Performing an Eperimen and Developing Mahemaical Models o Fi he Daa In he ransiion beween linear funcions and quadraic funcions, we have our sudens perform he following eperimen a home or in heir dorm: Place a flashligh (wih a round face) one inch from a wall and measure he diameer of he circular area of ligh. Repea his for disances of 2, 3,..., 8 inches. Plo he daa for he diameer of he circle of ligh as a funcion of he disance of he flashligh from he wall. Find he leas squares line ha bes fis he daa and inerpre he meaning of he slope and -inercep. Technolog in Collegiae Mahemaics", Addison-Wesle Publishing Co., 1997, p

3 Plo he daa for he area of he circle of ligh on he wall as a funcion of he disance of he flashligh from he wall. Find he leas squares quadraic fi for he daa. Discuss he relaionship beween ligh area and disance of he flashligh from he wall. How does he brighness of he ligh change as he area increases? How would ou calculae he inensi of he ligh for he differen size circles? In performing his eperimen, our sudens develop an undersanding for he need of funcions oher han linear. A pical suden's daa and models are shown in Figure 4. D 15 A 16 D() = A() = 3^ Figure 4 Fiing Linear and Quadraic Daa Fiing Daa Obained from Reference Sources To coninue he heme of using funcions o model daa, we ne inroduce our sudens o daa ha can no be modeled wih a polnomial. Since mos of our college algebra L sudens are business majors, we use business growh daa o moivae he sud of 15 eponenial funcions and eponenial growh. As one eample, he 1993 World Almanac conains daa for he amoun of L() = Ep(.115) life insurance purchased (millions of dollars) for he ears Seing he base ear 194 o =, our sudens hen find he leas squares eponenial fi o he daa and he inerpre he consans a and r in he model f () = ae r. Figure 5 presens he eponenial fi for he insurance daa. Observe ha he amoun of 5 life insurance purchased has increased a a earl rae of roughl 1% over his 5 ear period. Figure 5 Ownership of Ordinar Life Insurance in he US Technolog in Collegiae Mahemaics", Addison-Wesle Publishing Co., 1997, p

4 Comparing Eponenial Growh o Polnomial Growh When inroducing eponenial funcions, we ofen sress he fac ha eponenial funcions grow a lo faser han polnomial funcions. This noion becomes imporan when modeling daa ha has an imporan impac on socie. For eample, in his da of budge cus, Medicare is in he forefron. The 1994 Saisical Absrac of he US conains he earl Medicare medical pamens (billions of dollars) for he ears 197 o Afer seing he base ear 197 o =, ploing he scaer plo, finding he leas squares eponenial fi and he quadraic polnomial fi, we observe ha Medicare pamens are rising sharpl, bu, he good news is ha he are rising a a quadraic rae and no an eponenial rae. See Figure p() =.2^ f() = 8.2 ep(.14) Figure 6 Medicare Medical Pamens (billion dollars) 21 Tesing he Accurac of Eising Well-known Models Anoher wa we use daa in he classroom is o check he accurac of well-known models presened in he e book. For eample, we bring conainers of ho waer and hermomeers o class and have he sudens record he emperaure of he waer over ime o es he accurac of Newon's Law of Cooling. In preparing his eperimen for our sudens, we discovered ha Newon's Law of Cooling is valid onl when he difference beween he objec emperaure and he ambien emperaure is no oo large. We sared wih 9 C waer and le i cool o room emperaure (23 C) over a four hour (241 minue) period. The recorded daa and he model for Newon's Law of Cooling are ploed in Figure 7 below. Noice he poor fi. In all he eperimens we performed, waer alwas cooled faser han wha was prediced b Newon's Law of Cooling. When he emperaure of he objec is closer o ambien emperaure (wihin 27 C in his case), Newon's Law of Cooling models he daa well (see Figure 8 below). C 7 C Figure 7 Large Temperaure Difference Figure 8 Small Temperaure Difference Technolog in Collegiae Mahemaics", Addison-Wesle Publishing Co., 1997, p

5 Finding Formulas and Making Conjecures We also have our sudens generae daa mahemaicall for he purpose of recognizing paerns and making conjecures. For eample, we ell our sudens ha he sum L+ n can be represened b a quadraic funcion wih raional coefficiens. I is suggesed ha he evaluae his sum for hree differen values of n and hen find he quadraic inerpolaing polnomial ha passes hrough he hree poins. Sudens are ne asked o find a formula for L + n 2. Using he sraeg he developed o find his formula, he are hen asked o find a formula for 1 k + 2 k + 3 k +L + n k, for k = 3,4,5,6,7,8 and o wrie all formulas as polnomials wih raional coefficiens. Demonsraing he Imporance of Parameric Equaions Wih he power of oda's echnolog, parameric equaions are becoming a popular opic in mahemaics. There are an endless number of moivaing eamples for eploring he concep of parameric equaions. For eample, one migh ask, How do modern-da priners produce smooh leers and graphics? As an aemp o answer o his quesion, we have our sudens draw leers of he alphabe b using cubic polnomials o paramericall inerpolae a se of daa poins or b using a se of conrol poins and Bezier curves. Figure 9 shows an eample of drawing he leer C using Bezier curves Figure 9 Bezier Curves Bibliograph [1] TEMATH - Tools for Eploring Mahemaics Version 1.5 b Rober Kowalczk and Adam Hausknech, Brooks/Cole, [2] A Guide o TEMATH - Tools for Eploring Mahemaics b Rober Kowalczk and Adam Hausknech, Brooks/Cole, [3] Algebra Eperimens Eploring Linear Funcions, Mar Jean Winer and Ronald J. Carlson, Addison-Wesle Publishing Co., [4] The World Almanac and Book of Facs, Pharos Books, [5] 1994 Saisical Absrac of he Unied Saes. [6] The 1995 World Almanac. Technolog in Collegiae Mahemaics", Addison-Wesle Publishing Co., 1997, p