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# Equivalent exponential situations

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A class activity to deepen understanding of the nature of exponential relations.

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### Equivalent exponential situations

1. 1. A bacteria gets *16 every hour hours bacteria 0 10 1 2 3
2. 2. A bacteria gets *16 every hour hours bacteria 0 10 1 160 2 2560 3 40960
3. 3. A bacteria gets *4 every half-hour hours bacteria 0 10 .5 1 1.5 2 2.5 3
4. 4. A bacteria gets *4 every half-hour hours bacteria 0 10 .5 40 1 160 1.5 640 2 2560 2.5 10240 3 40960
5. 5. Coincidence? hours bacteria hours bacteria 0 10 0 10 .5 40 1 160 1.5 640 2 2560 2.5 10240 3 40960 1 2 3 160 2560 40960
6. 6. A bacteria gets doubled every ¼ hour hours bacteria 0 10 .25 .5 .75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
7. 7. A bacteria gets doubled every ¼ hour hours bacteria 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960
8. 8. Coincidence? hours bacteria hours bacteria 0 10 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960 .5 1 1.5 2 2.5 3 40 160 640 2560 10240 40960
9. 9. 2 coincidences?!?!? hours bacteria hours bacteria hours bacteria 0 10 0 10 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960 .5 1 160 1 1.5 2 2560 2 2.5 3 40960 3 40 160 640 2560 10240 40960
10. 10. Are these 3 situations identical? hours bacteria hours bacteria hours bacteria 0 10 0 10 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960 .5 1 160 1 1.5 2 2560 2 2.5 3 40960 3 40 160 640 2560 10240 40960
11. 11. Are these 3 situations identical? hours bacteria 0 10 hours bacteria hours bacteria 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 160 10 .75 1 0 10 0 According to these .25 tables, yes they are! .5 .5 40 20480 3 40960 1 1.5 2 2560 2 2.5 3 40960 3 160 640 2560 10240 40960 20 40
12. 12. Let’s see what the algebra has to say: x 0 10 1 160 2 2560 3 40960
13. 13. Rule using x = #hrs, y = #bacteria x y = 10(2)x 0 10 *16 every hour 1 160 2 2560 3 40960
14. 14. Rule using x = #hrs, y = #bacteria x y = 10(16)x 0 10 *16 every hour 1 160 2 2560 3 40960
15. 15. Rule using x = #hrs, y = #bacteria x y = 10(16)x x 0 10 0 10 .5 40 = 10(4)1 1 160 = 10(4)2 1.5 640 = 10(4)3 2 2560 = 10(4)4 2.5 10240 = 10(4)5 3 40960 = 10(4)6 1 2 3 160 2560 40960
16. 16. Rule using x = #hrs, y = #bacteria x y = 10(16)x x y = 10(1.41)2x 0 10 0 10 .5 40 = 10(4)1 quadrupled every half hour  After each hour, it’s happened twice 3 2560 40960 160 = 10(4)2 640 = 10(4)3 2 2560 = 10(4)4 2.5 2 160 1 1.5 1 10240 = 10(4)5 3 40960 = 10(4)6
17. 17. Rule using x = #hrs, y = #bacteria x y = 10(16)x x y = 10(4)2x 0 10 0 10 .5 40 = 10(4)1 quadrupled every half hour  After each hour, it’s happened twice 3 2560 40960 160 = 10(4)2 640 = 10(4)3 2 2560 = 10(4)4 2.5 2 160 1 1.5 1 10240 = 10(4)5 3 40960 = 10(4)6
18. 18. Rule using x = #hrs, y = #bacteria x y = 10(16)x x y = 10(4)2x x 0 10 0 10 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960 .5 1 160 1 1.5 2 2560 2 2.5 3 40960 3 40 160 640 2560 10240 40960
19. 19. doubled every quarter hour  After each hour, it’s happened 4 times Rule using x = #hrs, y = #bacteria x y = 10(16)x x y = 10(4)2x x y = 10(1.189)4x 0 10 0 10 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960 .5 1 160 1 1.5 2 2560 2 2.5 3 40960 3 40 160 640 2560 10240 40960
20. 20. doubled every quarter hour  After each hour, it’s happened 4 times Rule using x = #hrs, y = #bacteria x y = 10(16)x x y = 10(4)2x x y = 10(2)4x 0 10 0 10 0 10 .25 20 .5 40 .75 80 1 160 1.25 320 1.5 640 1.75 1280 2 2560 2.25 5120 2.5 10240 2.75 20480 3 40960 .5 1 20 1 1.5 2 40 2 2.5 3 80 3 40 160 640 2560 10240 40960
21. 21. A bacteria gets doubled every hour hours bacteria 0 10 1 2 3
22. 22. A bacteria gets doubled every hour hours bacteria 0 10 1 20 2 40 3 80
23. 23. A bacteria gets * 1.4142135’d every half-hour hours bacteria 0 10 .5 1 1.5 2 2.5 3
24. 24. A bacteria gets * 1.4142135’d every half-hour hours bacteria 0 10 .5 14.14 1 20 1.5 28.28 2 40 2.5 56.56 3 80
25. 25. A bacteria gets * 1.4142135’d every half-hour hours bacteria hours bacteria 0 10 0 10 .5 14.14 1 20 1.5 28.28 2 40 2.5 56.56 3 80 1 2 3 20 40 80
26. 26. A bacteria gets * 1.18920711’d every ¼ hour hours bacteria 0 10 .25 .5 .75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
27. 27. A bacteria gets * 1.18920711’d every ¼ hour hours bacteria 0 10 .25 11.89 .5 14.14 .75 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 67.26 3 80
28. 28. A bacteria gets * 1.18920711’d every ¼ hour hours bacteria hours bacteria 0 10 0 10 .25 11.89 .5 14.14 .75 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 67.26 3 80 .5 1 1.5 2 2.5 3 14.14 20 28.28 40 56.56 80
29. 29. A bacteria gets * 1.18920711’d every ¼ hour hours bacteria hours bacteria hours bacteria 0 10 0 10 0 10 .25 11.89 .5 14.14 .75 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 67.26 3 80 .5 1 20 1 1.5 2 40 2 2.5 3 80 3 14.14 20 28.28 40 56.56 80
30. 30. Are these 3 situations identical? hours bacteria hours bacteria hours bacteria 0 10 0 10 0 10 .25 11.89 .5 14.14 .75 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 67.26 3 80 .5 1 20 1 1.5 2 40 2 2.5 3 80 3 14.14 20 28.28 40 56.56 80
31. 31. Are these 3 situations identical? hours bacteria 0 10 hours bacteria hours bacteria 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 20 11.89 .75 1 0 10 0 According to these .25 tables, yes 14.14 are! .5 they .5 67.26 3 80 1 1.5 2 40 2 2.5 3 80 3 20 28.28 40 56.56 80 10 14.14
32. 32. Let’s see what the algebra has to say: x 0 10 1 20 2 40 3 80
33. 33. Rule using x = #hrs, y = #bacteria x y = 10(2)x 0 10 Doubled every hour 1 20 2 40 3 80
34. 34. Rule using x = #hrs, y = #bacteria x y = 10(2)x 0 10 Doubled every hour 1 20 2 40 3 80
35. 35. Rule using x = #hrs, y = #bacteria x y = 10(2)x x 0 10 0 10 0.5 14.14 = 10(1.4142135)1 1 20 = 10(1.4142135)2 1.5 28.28 = 10(1.4142135)3 2 40 = 10(1.4142135)4 2.5 56.56 = 10(1.4142135)5 3 80 = 10(1.4142135)6 1 2 3 20 40 80
36. 36. Rule using x = #hrs, y = #bacteria x y = 10(2)x x y = 10(1.41)2x 0 10 0 10 14.14 1.4142135’d every half = 10(1.4142135)1 hour 1 20  After each hour, it’s happened twice = 10(1.4142135)2 1.5 28.28 = 10(1.4142135)3 2 40 = 10(1.4142135)4 2.5 56.56 = 10(1.4142135)5 3 80 = 10(1.4142135)6 0.5 1 2 3 20 40 80
37. 37. Rule using x = #hrs, y = #bacteria x y = 10(2)x x y = 10(1.41)2x 0 10 0 10 14.14 1.4142135’d every half = 10(1.4142135)1 hour 1 20  After each hour, it’s happened twice = 10(1.4142135)2 1.5 28.28 = 10(1.4142135)3 2 40 = 10(1.4142135)4 2.5 56.56 = 10(1.4142135)5 3 80 = 10(1.4142135)6 0.5 1 2 3 20 40 80
38. 38. Rule using x = #hrs, y = #bacteria x y = 10(2)x x y = 10(2½ )2x 0 10 0 10 0.5 1 20 14.14 (2½ )’d every half = 10(1.4142135)1 hour  After each hour, it’s happened twice = 10(1.4142135)2 3 80 28.28 = 10(1.4142135)3 2 40 = 10(1.4142135)4 2.5 40 20 1.5 2 1 56.56 = 10(1.4142135)5 3 80 = 10(1.4142135)6
39. 39. Rule using x = #hrs, y = #bacteria x y = 10(2)x x y = 10(2½ )2x x 0 10 0 10 0 10 .25 11.89 .5 14.14 .75 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 67.26 3 80 0.5 1 20 1 1.5 2 40 2 2.5 3 80 3 14.14 20 28.28 40 56.56 80
40. 40. Rule using x = #hrs, y = ji#bacteria = 10(1.189207115) 1 x y = 10(2)x 0 10 x =10 (1.189207115) =10 (1.189207115) 12ji 3 80 1.25 23.78 28.28 1.75 33.63 2 40 47.56 2.5 40 20 2.25 2 16.82 1.5 4ji 14.14 1 = 10(1.189207115) 3ji 11.89 .75 20 10 .5 1 0 .25 = 10(1.189207115) 2ji 56.56 2.75 67.26 3 80
41. 41. 1.18920711’d every quarter hour  After each hour, it’s happened 4 times 1ji = 10(1.189207115) y = 10(2½ )2x x y = 10(1.189)4x Rule using x = #hrs, y = #bacteria x y = 10(2)x 0 10 x 0 10(1.189207115) 2ji 10 = 1.5 2 40 28.28 =10 (1.189207115) 2 40 2.5 56.56 =10 (1.189207115) 12ji 3 80 3 80 4ji 1 20 1.25 23.78 1.5 1 20 = 10(1.189207115) 3ji 16.82 28.28 1.75 33.63 2 40 2.25 20 14.14 .75 1 11.89 .5 14.14 10 .25 0.5 0 47.56 2.5 56.56 2.75 67.26 3 80
42. 42. 1.18920711’d every quarter hour  After each hour, it’s happened 4 times 1ji = 10(1.189207115) y = 10(2½ )2x x y = 10(1.189)4x Rule using x = #hrs, y = #bacteria x y = 10(2)x 0 10 x 0 10(1.189207115) 2ji 10 = 1.5 2 40 28.28 =10 (1.189207115) 2 40 2.5 56.56 =10 (1.189207115) 12ji 3 80 3 80 4ji 1 20 1.25 23.78 1.5 1 20 = 10(1.189207115) 3ji 16.82 28.28 1.75 33.63 2 40 2.25 20 14.14 .75 1 11.89 .5 14.14 10 .25 0.5 0 47.56 2.5 56.56 2.75 67.26 3 80
43. 43. (21/4)’d every quarter hour  After each hour, it’s happened 4 times Rule using x = #hrs, y = #bacteria x y = 10(2)x x y = 10(2½ )2x x y = 10(21/4)4x 0 10 0 10 0 10 .25 11.89 .5 14.14 .75 16.82 1 20 1.25 23.78 1.5 28.28 1.75 33.63 2 40 2.25 47.56 2.5 56.56 2.75 67.26 3 80 0.5 1 20 1 1.5 2 40 2 2.5 3 80 3 14.14 20 28.28 40 56.56 80