Examine each chart, what connections are there between the ratio!p!n and your findings in Task 2.1.1? Explain your reasoning.

Transcription

1 Algebra II: Strad. Liear Fuctios; Topic. Liear Patters; Task.. TASK..: FINITE DIFFERENCES. Refer to your charts from Task.. ad add two colums to each chart: a. A colum o the left of the chart labeled Chage i the umber of cars: b. A colum o the right side of the chart labeled Chage i : Examie each chart, what coectios are there betwee the ratio ad your fidigs i Task..? Explai your reasoig. Chages i # of cars Positio i Sequece Number of cars i trai (each square represets trai car) of Trai Chage of Frot + back + car with (top + bottom) + () Frot + back + cars with (top + bottom) + () Frot + back + cars with (top + bottom) + () + () 8 0 Frot + back + cars with (top + bottom) + () Stays the same + (# of cars i trai)(rate) + () = + + December 0, 00. Esurig Teacher Quality: Algebra II, produced by the Charles A. Daa Ceter at The Uiversity of Texas at Austi for the Texas Higher Educatio Coordiatig Board.

2 Algebra II: Strad. Liear Fuctios; Topic. Liear Patters; Task.. Chage i # of cars Positio i Sequece Number of cars i trai of Trai Chage i P Frot + back + horizotal + Frot + back + horizotal + Frot + back + horizotal + Stays the same + (what chages) + ()... + () Chages i # of cars Positio Sequece Number of cars i Trai of Trai Chages i P (Frot + back) + (bottom with legth + top with legth ) car + () (Frot + back) + ( bottom ad top segmets) cars + () +() +() 8 +() 7 (Frot + back) + ( bottom ad top segmets) cars + + December 0, 00. Esurig Teacher Quality: Algebra II, produced by the Charles A. Daa Ceter at The Uiversity of Texas at Austi for the Texas Higher Educatio Coordiatig Board.

3 Algebra II: Strad. Liear Fuctios; Topic. Liear Patters; Task.. Chage i # of cars Positio Sequece Number of cars i trai of trai Chage i (Frot + back) + ( top + bottom) car + () (Frot + back) + ( top + bottom) cars + () (Frot + back) + ( top + bottom) cars + () (Frot + back) + ( top + bottom) cars + () 0 8 Stays the same + rate of chage (what chages) (Frot + back) + ( top + bottom) cars + () + +. Why is it importat to cosider the ratio rather tha simply comparig (the rate) for each chage i whe determiig whether or ot there is a liear patter? May times fiite differece examples are misleadig i that they typically illustrate fiite differeces over itervals of equal legth (i most cases legth= uit). Thus, participats may icorrectly assume that if they were to calculate fiite differeces for, say, =,,, 7, 9, for a liear fuctio the chage i p would ot be costat that because the fiite differeces were ot costat, the fuctio is ot liear. Here we hope to see aswers that illustrate that although the chage i p is ot costat they would have to check the ratio before makig ay coclusios. Math otes Ivestigatig fiite differeces is a typical task for middle school studets. The focus of this task is i uderscorig the importace of lookig at the ratio of the differeces rather tha oly the resultig differece i the fuctio values without regard to the chage i the idepedet variable. Teachig otes This task may be assiged as homework after the istructor illustrates how to expad the charts (addig the extra colums) ad demostrates fidig the differeces for a few rows. A class discussio should take place at the begiig of the ext sessio where December 0, 00. Esurig Teacher Quality: Algebra II, produced by the Charles A. Daa Ceter at The Uiversity of Texas at Austi for the Texas Higher Educatio Coordiatig Board.

4 Algebra II: Strad. Liear Fuctios; Topic. Liear Patters; Task.. participats share their results i groups. Esure that the idea that the graph of ay fuctio that produces a costat ratio of fiite differeces must be a lie or that the fuctio must be liear. December 0, 00. Esurig Teacher Quality: Algebra II, produced by the Charles A. Daa Ceter at The Uiversity of Texas at Austi for the Texas Higher Educatio Coordiatig Board.

5 Algebra II: Strad. Liear Fuctios; Topic. Liear Patters; Task.. TASK..: FINITE DIFFERENCES TASK..: FINITE DIFFERENCES. Refer to your charts from Task.. ad add two colums to each chart: a. A colum o the left of the chart labeled Chage i the umber of cars: b. A colum o the right side of the chart labeled Chage i : Examie each chart. What coectios are there betwee the ratio fidigs i Task..? Explai your reasoig. ad your. Why is it importat to cosider the ratio rather tha simply comparig (the rate) for each chage i whe determiig whether or ot there is a liear patter? Explai ad give a example that clarifies your explaatio. December 0, 00. Esurig Teacher Quality: Algebra II, produced by the Charles A. Daa Ceter at The Uiversity of Texas at Austi for the Texas Higher Educatio Coordiatig Board.