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Scientific Notation with Positive and Negative Powers of Ten

Benchmark

Use scientific notation with positive and negative powers of 10, with

appropriate treatment of significant digits, to solve real-world and math

problems.

Teaching Suggestions

Students have studied scientific notation with positive powers of 10 in the 7th

grade curriculum. The idea here is to review positive powers of 10 and extend to

negative powers of 10.

The reasons for scientific notation can be illustrated by writing a very long large

number on the board and a very long small number on the board. Scientists and

mathematicians use this notation to make calculations and communication moreefficient.

Review how to change the long large number to scientific notation with class

input. The two important parts are:

1.  A number between 1 and 10 (1≤ n < 10)

2.  A power of 10 that is multiplied by the number in Part 1 above (e.g. 105

)

 Example: The closest galaxy to our Milky Way Galaxy is the Andromeda Galaxy,

which contains

200,000,000,000 stars.

What would that be in scientific notation? (2 x 1011

)

Hand out the Powers of Ten for Scientific Notation Chart for reference and

discussion.

We know about millimeter, which are 0.001 of a meter. Some people use the thickness

of a CD as a benchmark for that length.

 But what is a nanometer? Our chart says it is 0.000000001 of a meter, or 1 trillionth

of a meter. It can also be written:

1 x 10-9

 NASA says that carbon nanotubes " could be used to create teeny tiny machines that 

can travel inside humans to attack cancers."

from NASAexplores: September 5, 2002

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Go over the two examples at the bottom of the chart. Both have more than one

number digit, which complicates the conversion. The power of 10 will indicate how

many places to move the decimal point, not how many zeroes or decimal places youwill end up with. The sign of the power indicates which direction to move the

decimal point.

Be sure that the students can convert both ways-from scientific notation to standard

notation and vice versa.

Assignment

Hand out Student Sheet: Scientific Notation

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Powers of Ten Chart

for Scientific Notation

Name Expanded Form Power of 10 for

Scientific

Notation

trillion 1,000,000,000,000 1012

billion 1,000,000,000 109

million 1,000,000 106

hundred thousand 100,000 105

ten thousand 10,000 104

thousand 1,000 103

hundred 100 102

ten 10 101

one 1 100

tenth 0.1 10-1

hundredth 0.01 10-2

thousandth 0.001 10-3

ten thousandth 0.0001 10-4

hundred thousandth 0.00001 10-5

millionth 0.000001 10-6

billionth 0.000000001 10-9

trillionth 0.000000000001 10-12

ExamplesThe mass of the sun is

1.989 x 1030

kg

which in expanded form is

1,989,000,000,000,000,000,000,000,000,000

The diameter of the HIV virus is 1.1 x 10-11

which in expanded form is

0.000000000011

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Student Sheet: Scientific Notation

1. Describe any patterns you see in the Powers of Ten for Scientific Notation Chartand examples.

Write using scientific notation.

2. 5 x 10 x 10 x 10 ________________ 3. 7,000,000,000 _________________

4. 0.00008 ________________ 5. 930,000 _________________

6. 0.00032 ________________ 7. 365000000 _________________

Write in expanded form.

8. 9 x 103

_________________________________________

9. 8.43 x 108

_________________________________________

10. 4 x 10-9

_________________________________________

11. 2.6 x 10-3

_________________________________________

Compare using >, < , or =.

12. 4 x 103

3 x 104

13. 2 x 10-6

2 x 10-7

14. 7 x 104

70,000 15. 9.21 x 103

921

16. 2.67 x 10-7

0.0000267

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Student Sheet: Scientific Notation -- Answer Key

1. Describe any patterns you see in the Powers of 10 for Scientific Notation Chart.Possible answers:

§  Positive powers of 10 indicate a number greater than 1; negative powers of 10 indicate a

number less than 1.§  The number of zeroes matches the exponent of 10 for the numbers equal to or greater than

1.

§  For numbers less than 1, the number of places to the right of the decimal matches the

absolute value of the exponent of 10.

§  When there is more than one digit other than zero in the number, the number of zeroes and 

the decimal places do not match the exponent of 10. In  al l cases, the exponent indicates the

number of places to move the decimal point when converting. The sign of the exponent 

indicates the direction.

Write using scientific notation.

2. 5 x 10 x 10 x 10 5 x 103

3. 7,000,000,000 7 x 109

4. 0.00008 8 x 10-5

5. 930,000 9.3 x 105

6. 0.00032 3.2 x 10-4

7. 365000000 3.65 x 108

Write in expanded form.

8. 9 x 103

9,000

9. 8.43 x 108

843,000,000

10. 4 x 10-9

0.000000004

11. 2.6 x 10-3

0.026

Compare using >, < , or =.

12. 4 x 103

3 x 104

13. 2 x 10-6

2 x 10-7

14. 7 x 104

70,000 15. 9.21 x 103

921

16. 2.67 x 10-7

0.0000267

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