1. Practical Estimation Using Scientific Notation

2. Created by Dick Heckathorn

3. Can you quickly approximate an answer to the following problem?
How many seconds are there in one year?

4. Can you quickly determine which is larger?
3684 x x ? or
63.72 x 273 x ?

5. Can you quickly approximate an answer to the following problem?

6. A review of place holders

7. 57,246.9318 5 ten-thousands 9 tenths 7 thousands 3 hundredths
2 hundreds 1 thousandths
4 tens 8 ten-thousandths
6 ones

8. A review of what happens to a number when it is multiplied by ten (10).

9. 2
2 x 10 = 20
2 x 10 x10 = 200
2 x 10 x 10 x 10 = 2000
2 / 10 = 0.2
2 / 10 / 10 = 0.02
2 / 10 / 10 / 10 = 0.002

10. A review of numbers that are different by a power of 10.

11. Number How number is formed
1 = 1
10 = 1 x 10
100 = 1 x 10 x 10
1000 = 1 x 10 x 10 x 10
10000 = 1 x 10 x 10 x 10 x 10
= 1 x 10 x 10 x 10 x 10 x 10
= 1 x 10 x 10 x 10 x 10 x 10 x 10

12. A review of how numbers are changed to power of ten format.

13. Power
Number How number is formed of 10
1 1 x no tens 100
10 1 x
100 1 x 10 x
1,000 1 x 10 x 10 x
10,000 1 x 10 x 10 x 10 x
100,000 1 x 10 x 10 x 10 x 10 x
1,000,000 1 x 10 x 10 x 10 x 10 x 10 x

14. A review of how numbers are changed to power of ten format.

15. Power
of 10 Number How number is formed
/ 10
/ 10 / 10
/ 10 / 10 / 10
/ 10 / 10 / 10 / 10
/ 10 / 10 / 10 / 10 / 10
/ 10 / 10 / 10 / 10 / 10 x 10

16. Scientific Notation
Is a system in which the numbers are expressed as the product of the
COEFFICIENT
which is a number that is equal to or greater than one but less than ten
and the appropriate
POWER OF TEN

17. Here is an example.

18. The Number
63847

19. Same number in scientific notation
63847
Same number in scientific notation
x 104
Coefficient Power of Ten

20. Let us now examine the steps to change a number to scientific notation format.

21. Number to be Changed
76348

22. 76348
First move the decimal point so that one non-zero digit is to the left of the decimal point.
7.6348

23. Next determine what you did with the decimal point.

24. It was moved 4 places to the left.
76348
7.6348
It was moved 4 places to the left.

25. How does this number compare to the original number?

26. The number is 10,000 (104) times smaller.
76348
7.6348
The number is 10,000 (104) times smaller.

27. 76348
7.6348
To make the number in scientific notation the same as the original number, what must we do to the powers of ten?

28. We must multiply the number by ten to the fourth power (4).
76348
7.6348
We must multiply the number by ten to the fourth power (4).
x 104

29. Number to be Changed

30.
First move the decimal point so that one non-zero digit is to the left of the decimal point.
3.857

31. Next determine what you did with the decimal point.

32. It was moved 4 places to the right.
3.857
It was moved 4 places to the right.

33. How does this number compare to the original number?

34. The number is 10,000 (104) times larger.
3.857
The number is 10,000 (104) times larger.

35.
3.857
To make the number in scientific notation the same as the original number, what must we do to the powers of ten?

36.
3.857
We thus multiply by 10 to the minus four power (10-4).
3.857 x 10-4

37. Check your answer on the next slide when finished.
Change the following number to scientific notation format.
724000
Check your answer on the next slide when finished.

38. 724000
The answer is
7.24 x 104

39. Check your answer on the next slide when finished.
Change the following number to scientific notation format.
Check your answer on the next slide when finished.

40.
The answer is
5.17 x 10-5

41. Let us now solve a problem where the numbers are written in scientific notation.

42. The Problem
(3.1 x 102) x (2.0 x 104)

43. First separate the coefficients & powers of ten.
(3.1 x 102) x (2.0 x 104)
First separate the coefficients & powers of ten.
(3.1 x 2.0) x (102 x 104)

44. (3.1 x 102) x (2.0 x 104)
(3.1 x 2.0) x (102 x 104)
Next show how the powers of ten are combined.
(3.1 x 2.0) x 10(2 + 4)

45. (3.1 x 102) x (2.0 x 104)
(3.1 x 2.0) x (102 x 104)
(3.1 x 2.0) x 10(2 + 4)
Finally finish the calculation.
6.2 x 106

46. Let us solve another problem where the numbers are written in scientific notation.

47. The Problem

48. First separate the coefficients & powers of ten.

49. Next show how the powers of ten are combined.

50. Finally finish the calculation.

52. Let us now solve a problem by first changing the number to scientific notation format and then determine the answer.

53. The Problem
6749 x 263

54. First write the number in
6749 x 263
First write the number in
scientific notation.
(6.749 x 103) x (2.63 x 102)

55. Next combine the coefficients and powers of ten.
(6.749 x 103) x (2.63 x 102)
Next combine the coefficients and powers of ten.
(6.749 x 2.63 ) x

56. Next combine the coefficients and the powers of ten.
(6.749 x 103) x (2.63 x 102)
(6.749 x 2.63 ) x
Next combine the coefficients and the powers of ten.
17.75 x 105

57. Finally write in proper scientific notation.
6749 x 263
(6.749 x 2.63 ) x
17.75 x 105
Finally write in proper scientific notation.
1.775 x 106

58. Set up the following problem on a piece of paper showing each step.
x 6734 x 27
When finished check your answer on the next slide.

59. The answer is:
8.42 x 102

60. Fermi Questions By
Dick Heckahtorn

61. To solve Fermi questions such as how may blades of grass are there inside the fence that surrounds the soccer field, one needs to learn how to round a number to its nearest order of magnitude.

62. In the desired format is:
For example:
6.35 x 105
In the desired format is:
106

63. In the desired format is:
For example:
2.35 x 105
In the desired format is:
105

64. But what about x 105?

65. 4.35 x 105?
The answer is:
106
How many of you said 105 ?
How can this be?

66. Since we are dealing only with orders of magnitude, we need to determine the half-way point between one order of magnitude and the next.
For example between
101 and 102.

67. Would you agree that the half way point between
101 and 102 is ?

68. I hope you agree because 101.5 is the midpoint between
101 and 102 ?

69. Our task then is quite simple.
What is the value of ?

70. If you did, you have responded with an answer that most students give.
Did you say 50?
If you did, you have responded with an answer that most students give.
But 50 is an incorrect answer.

71. You see, we are not dealing with an ordinary number scale.
We are dealing with an exponential scale.
This is true because each order of magnitude is 10 times larger than the previous.

72. Without going into a lot of mathematical explanation, perform the following steps on your calculator.

73. 1. Press the ‘2nd’ key
2. Press the ‘LOG’ key
3. Enter 1.5
4. Press the ‘ENTER’ key
The answer is ……..

74. Did you get ?
You should if you have the calculator set on float. Otherwise you will get a rounded value.

75. An Example

76. Examine the following:
order of magnitude = 0
order of magnitude = 1
order of magnitude = 5
order of magnitude = -1
order of magnitude = -3
order of magnitude = -5