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A class activity to deepen understanding of the nature of exponential relations.
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A bacteria gets *16 every hour hours bacteria
0 10
1
2
3
A bacteria gets *16 every hour hours bacteria
0 10
1 160
2 2560
3 40960
A bacteria gets *4 every half-hour hours bacteria
0 10
.5
1
1.5
2
2.5
3
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
Coincidence? hours bacteria
0 10
1 160
2 2560
3 40960
hours bacteria
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
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
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
Coincidence? hours bacteria
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
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
2 coincidences?!?!? hours bacteria
0 10
1 160
2 2560
3 40960
hours bacteria
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
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
Are these 3 situations identical? hours bacteria
0 10
1 160
2 2560
3 40960
hours bacteria
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
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
Are these 3 situations identical? hours bacteria
0 10
1 160
2 2560
3 40960
hours bacteria
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
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
According to these tables, yes they are!
Let’s see what the algebra has to say: x
0 10
1 160
2 2560
3 40960
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 160
2 2560
3 40960
*16 every hour
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
*16 every hour
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
x
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
= 10(4)1
= 10(4)2
= 10(4)3
= 10(4)4
= 10(4)5
= 10(4)6
= 10(4)1
= 10(4)2
= 10(4)3
= 10(4)4
= 10(4)5
= 10(4)6
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
x y = 10(1.41)2x
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
quadrupled every half hour
After each hour, it’s happened twice
= 10(4)1
= 10(4)2
= 10(4)3
= 10(4)4
= 10(4)5
= 10(4)6
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
x y = 10(4)2x
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
quadrupled every half hour
After each hour, it’s happened twice
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
x y = 10(4)2x
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
x
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
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
x y = 10(4)2x
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
x y = 10(1.189)4x
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
doubled every quarter hour After each hour, it’s happened 4
times
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 20
2 40
3 80
x y = 10(4)2x
0 10
.5 40
1 160
1.5 640
2 2560
2.5 10240
3 40960
x y = 10(2)4x
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
doubled every quarter hour After each hour, it’s happened 4
times
A bacteria gets doubled every hour hours bacteria
0 10
1
2
3
A bacteria gets doubled every hour hours bacteria
0 10
1 20
2 40
3 80
A bacteria gets *1.4142135’d every half-hour hours bacteria
0 10
.5
1
1.5
2
2.5
3
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
A bacteria gets *1.4142135’d every half-hour hours bacteria
0 10
1 20
2 40
3 80
hours bacteria
0 10
.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
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
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
A bacteria gets *1.18920711’d every ¼ hour hours bacteria
0 10
.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
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
A bacteria gets *1.18920711’d every ¼ hour hours bacteria
0 10
1 20
2 40
3 80
hours bacteria
0 10
.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
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
Are these 3 situations identical? hours bacteria
0 10
1 20
2 40
3 80
hours bacteria
0 10
.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
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
Are these 3 situations identical? hours bacteria
0 10
1 20
2 40
3 80
hours bacteria
0 10
.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
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
According to these tables, yes they are!
Let’s see what the algebra has to say: x
0 10
1 20
2 40
3 80
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
Doubled every hour
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
Doubled every hour
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
= 10(1.4142135)1
= 10(1.4142135)2
= 10(1.4142135)3
= 10(1.4142135)4
= 10(1.4142135)5
= 10(1.4142135)6
= 10(1.4142135)1
= 10(1.4142135)2
= 10(1.4142135)3
= 10(1.4142135)4
= 10(1.4142135)5
= 10(1.4142135)6
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(1.41)2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
1.4142135’d every half hour
After each hour, it’s happened twice
= 10(1.4142135)1
= 10(1.4142135)2
= 10(1.4142135)3
= 10(1.4142135)4
= 10(1.4142135)5
= 10(1.4142135)6
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(1.41)2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
1.4142135’d every half hour
After each hour, it’s happened twice
= 10(1.4142135)1
= 10(1.4142135)2
= 10(1.4142135)3
= 10(1.4142135)4
= 10(1.4142135)5
= 10(1.4142135)6
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(2½ )2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
(2½ )’d every half hour
After each hour, it’s happened twice
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(2½ )2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
x
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
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x
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
= 10(1.189207115) 1ji
= 10(1.189207115) 2ji
= 10(1.189207115) 3ji
=10 (1.189207115) 4ji
=10 (1.189207115) 12ji
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(2½ )2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
x y = 10(1.189)4x
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
1.18920711’d every quarter hour After each hour, it’s happened 4
times = 10(1.189207115) 1ji
= 10(1.189207115) 2ji
= 10(1.189207115) 3ji
=10 (1.189207115) 4ji
=10 (1.189207115) 12ji
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(2½ )2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
x y = 10(1.189)4x
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
1.18920711’d every quarter hour After each hour, it’s happened 4
times = 10(1.189207115) 1ji
= 10(1.189207115) 2ji
= 10(1.189207115) 3ji
=10 (1.189207115) 4ji
=10 (1.189207115) 12ji
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
x y = 10(2½ )2x
0 10
0.5 14.14
1 20
1.5 28.28
2 40
2.5 56.56
3 80
x y = 10(21/4)4x
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
(21/4)’d every quarter hour After each hour, it’s happened 4
times
