equivalent exponential situations
DESCRIPTION
A class activity to deepen understanding of the nature of exponential relations.TRANSCRIPT
![Page 1: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/1.jpg)
A bacteria gets *16 every hour hours bacteria
0 10
1
2
3
![Page 2: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/2.jpg)
A bacteria gets *16 every hour hours bacteria
0 10
1 160
2 2560
3 40960
![Page 3: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/3.jpg)
A bacteria gets *4 every half-hour hours bacteria
0 10
.5
1
1.5
2
2.5
3
![Page 4: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/4.jpg)
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
![Page 5: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/5.jpg)
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
![Page 6: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/6.jpg)
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
![Page 7: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/7.jpg)
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
![Page 8: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/8.jpg)
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
![Page 9: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/9.jpg)
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
![Page 10: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/10.jpg)
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
![Page 11: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/11.jpg)
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!
![Page 12: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/12.jpg)
Let’s see what the algebra has to say: x
0 10
1 160
2 2560
3 40960
![Page 13: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/13.jpg)
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 160
2 2560
3 40960
*16 every hour
![Page 14: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/14.jpg)
Rule using x = #hrs, y = #bacteria x y = 10(16)x
0 10
1 160
2 2560
3 40960
*16 every hour
![Page 15: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/15.jpg)
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
![Page 16: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/16.jpg)
= 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
![Page 17: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/17.jpg)
= 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
![Page 18: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/18.jpg)
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
![Page 19: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/19.jpg)
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
![Page 20: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/20.jpg)
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
![Page 21: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/21.jpg)
A bacteria gets doubled every hour hours bacteria
0 10
1
2
3
![Page 22: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/22.jpg)
A bacteria gets doubled every hour hours bacteria
0 10
1 20
2 40
3 80
![Page 23: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/23.jpg)
A bacteria gets *1.4142135’d every half-hour hours bacteria
0 10
.5
1
1.5
2
2.5
3
![Page 24: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/24.jpg)
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
![Page 25: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/25.jpg)
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
![Page 26: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/26.jpg)
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
![Page 27: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/27.jpg)
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
![Page 28: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/28.jpg)
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
![Page 29: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/29.jpg)
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
![Page 30: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/30.jpg)
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
![Page 31: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/31.jpg)
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!
![Page 32: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/32.jpg)
Let’s see what the algebra has to say: x
0 10
1 20
2 40
3 80
![Page 33: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/33.jpg)
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
Doubled every hour
![Page 34: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/34.jpg)
Rule using x = #hrs, y = #bacteria x y = 10(2)x
0 10
1 20
2 40
3 80
Doubled every hour
![Page 35: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/35.jpg)
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
![Page 36: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/36.jpg)
= 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
![Page 37: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/37.jpg)
= 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
![Page 38: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/38.jpg)
= 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
![Page 39: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/39.jpg)
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
![Page 40: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/40.jpg)
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
![Page 41: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/41.jpg)
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
![Page 42: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/42.jpg)
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
![Page 43: Equivalent exponential situations](https://reader033.vdocuments.site/reader033/viewer/2022051210/54bd92544a7959bb098b45d7/html5/thumbnails/43.jpg)
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