1.8 What Comes Next? What Comes Later?

Transcription

1 What Comes Next? What Comes Later? A Practice Uderstadig Task For each of the followig tables, CC BY Hiroaki Maeda describe how to fid the ext term i the sequece, write a recursive rule for the fuctio, describe how the features idetified i the recursive rule ca be used to write a explicit rule for the fuctio, ad write a explicit rule for the fuctio. idetify if the fuctio is arithmetic, geometric or either Example: ?? To fid the ext term: add 3 to the previous term Recursive rule: 0 = 5, = To fid the th term: start with 5 ad add 3 times Explicit rule: = Arithmetic, geometric, or either? Arithmetic Fuctio A 1. How to fid the ext term: 2. Recursive rule: 3. To fid the th term: 4. Explicit rule: ?? 5. Arithmetic, geometric, or either?

2 36 SECONDARY MATH 1 // MODULE 1 Fuctio B 6. How to fid the ext term: 7. Recursive rule: 8. To fid the th term: 9. Explicit rule: 10. Arithmetic, geometric, or either? Fuctio C 11. To fid the ext term: 12. Recursive rule: 13. To fid the th term: 14. Explicit rule: 15. Arithmetic, geometric, or either? Fuctio D 16. To fid the ext term: 17. Recursive rule: 18. To fid the th term: 19. Explicit rule: 20. Arithmetic, geometric, or either? ??

3 37 Fuctio E 21. To fid the ext term: 22. Recursive rule: 23. To fid the th term: 24. Explicit rule: 25. Arithmetic, geometric, or either? Fuctio F 26. To fid the ext term: 27. Recursive rule: 28. To fid the th term: 29. Explicit rule: 30. Arithmetic, geometric, or either? ?? Fuctio G 31. To fid the ext term: 32. Recursive rule: 33. To fid the th term: 34. Explicit rule: 35. Arithmetic, geometric, or either?

4 38 SECONDARY MATH 1 // MODULE 1 Fuctio H 36. To fid the ext term: 37. Recursive rule: 38. To fid the th term: 39. Explicit rule: 40. Arithmetic, geometric, or either?

5 1.8 What Comes Next? What Comes Later? Teacher Notes A Practice Uderstadig Task Purpose: The purpose of this task is to practice writig recursive ad explicit formulas for arithmetic ad geometric sequeces from a table. This task also provides practice i usig tables to idetify whe a sequece is arithmetic, geometric, or either. The task exteds studets experieces with sequeces to iclude geometric sequeces with alteratig sigs, ad more work with fractios ad decimal umbers i the sequeces. Core Stadards: F.BF.1: Write a fuctio that describes a relatioship betwee two quatities.* a. Determie a explicit expressio, a recursive process, or steps for calculatio from a cotext. F.LE.1. Distiguish betwee situatios that ca be modeled with liear fuctios ad with expoetial fuctios. F.LE.2. Costruct liear ad expoetial fuctios, icludig arithmetic ad geometric sequeces, give a graph, a descriptio of a relatioship, or two iput-output pairs (iclude readig these from a table). Stadards for Mathematical Practice of Focus i the Task: SMP7 Look for ad make use of structure. SMP8 Look for ad express regularity i repeated reasoig. The Teachig Cycle: Lauch (Whole Class): Remid studets of the work doe i previous tasks i this module. Ask, What strategies are you workig o for writig formulas for arithmetic ad geometric sequeces?

6 Studets may offer some ideas that will be helpful i this task. Had out the task ad work through the example with the class. It will be importat to clarify the two prompts that ask studets to describe what they are seeig i the table. Writig a verbal descriptio before writig the formula ca promote deeper uderstadig. Explore (Small Group): Moitor studet work o the task. Fuctios A ad B should be familiar ad accessible for most studets. Fuctio F is more challegig; although studets may recogize a patter, it is either arithmetic or geometric. It is ot ecessary for studets to write the formulas, sice this is a ufamiliar fuctio type. If you fid that a group is etirely stuck o the problem, you might ask them to move o to some of the others ad go back to fiish Fuctio F. Watch their work o Fuctio H to be sure that they cosider the alteratig sig i their formula. If they have t icluded it i the formula, ask them how their formulas will produce the egative sig o some of the terms. Allow them to work to fid a way to iclude the egative sig i their formula. Watch for miscoceptios that might make for a productive discussio ad liste for studets geeralizig strategies for writig equatios. Idetify groups to preset their work for Fuctios B ad E that have show the first differece o their tables ad ca articulate how they have used the commo differece or commo ratio to write the equatios. Discuss (Whole Class): Start the discussio with the studet presetatio of their work o Fuctio D. Their work will probably look somethig like: Differece > -9 > -9 > -9 Fuctio B To fid the ext term: Add -9 to the previous term Recursive rule: (1) = 8, () = ( 1) 9 To fid the th term: Start with -8 ad add -9-1 times Explicit rule: () = 8 9( 1) Arithmetic, geometric, or either? Arithmetic

7 Ask studets how they were able to use the commo differece of -9 i the formulas. Ask how the first term, -8, shows up i each formula. Next, ask a group to show their work with Fuctio C. It should look somethig like this: Differece > 4 > 12 > 36 Commo ratio 3 Fuctio C To fid the ext term: Multiply the previous term by 3 Recursive rule: (1) = 2, () = 3( 1) To fid the th term: start with 2 ad multiply by 3 (-1) times Explicit rule: () = 2(3) Arithmetic, geometric, or either? Geometric Ask studets how they were able to use the commo ratio of 3 i the formulas. The, ask how the first term, 2, appears i the formulas. If time allows, you may wish to have some of the others preseted. If ot, move to askig studets to geeralize their thikig about writig formulas for arithmetic ad geometric fuctios with the followig chart: How do you use: Arithmetic Sequece Recursive Formula Arithmetic Sequece Explicit Formula Geometric Sequece Recursive Formula Geometric Sequece Explicit Formula Commo ratio or differece? First term?

8 Ask if there are ay other strategies that they have foud for writig formulas ad record them beeath the chart. Leave the chart i view for studets to use i upcomig tasks. Aliged Ready, Set, Go Homework: Sequeces 1.8

9 39 SECONDARY MATH I // MODULE READY, SET, GO %%%%%%Name% %%%%%%Period%%%%%%%%%%%%%%%%%%%%%%%Date% READY Topic:CommoRatios Fid%the%commo%ratio%for%each%geometric%sequece.% % 1.2,4,8,16 2.,1,2,4,8 3.85,10,820, ,5,2.5,1.25 SET Topic:Recursiveadexplicitequatios Fillitheblaksforeachtable;thewritetherecursiveadexplicitequatioforeachsequece. 5.%%Table%1% x" y" Recursive: Explicit: 6.%%Table%2% % % % 7.%%Table%3% % % % 8.%%Table%4% % % % 1 3 % % % Recursive: Recursive: Recursive: Explicit: Explicit Explicit:

10 40 SECONDARY MATH I // MODULE GO Topic:Writigequatiosofliesgiveagraph. Write%each%equatio%of%the%lie%i% = " + %form.%%name%the%value%of%m%ad%b.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Recall%that%m%is%the%slope%or%rate%of%chage%ad%bis%the%yGitercept.% % 9. = =Equatio: 11. = =Equatio: 10. = =Equatio: 12. = =Equatio: