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

1. A bacteria gets *16 every hour
hours
bacteria
0
10
1
2
3

2. A bacteria gets *16 every hour
hours
bacteria
0
10
1
160
2
2560
3
40960

3. A bacteria gets *4 every half-hour
hours
bacteria
0
10
.5
1
1.5
2
2.5
3

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. 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. 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. 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. 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. 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. 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. 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. Let’s see what the algebra has to say:
x
0
10
1
160
2
2560
3
40960

13. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
*16 every hour
1
160
2
2560
3
40960

14. Rule using x = #hrs, y = #bacteria
x
y = 10(16)x
0
10
*16 every hour
1
160
2
2560
3
40960

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. 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. 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. 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. 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. 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. A bacteria gets doubled every hour
hours
bacteria
0
10
1
2
3

22. A bacteria gets doubled every hour
hours
bacteria
0
10
1
20
2
40
3
80

23. A bacteria gets
*
1.4142135’d
every half-hour
hours
bacteria
0
10
.5
1
1.5
2
2.5
3

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. 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. 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. 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. 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. 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. 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. 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. Let’s see what the algebra has to say:
x
0
10
1
20
2
40
3
80

33. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
Doubled every hour
1
20
2
40
3
80

34. Rule using x = #hrs, y = #bacteria
x
y = 10(2)x
0
10
Doubled every hour
1
20
2
40
3
80

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. 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. 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. 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. 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. 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. 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. 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. (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