Scientific Notation – A Different Approach

Some students struggle with converting numbers to and from scientific notation. I believe that this is mostly due to memorizing/recalling only part of the rule. Today I’ll share a technique that I have found to be quite successful.

Confusion for Some Students

As an example of how this usually goes wrong, if I ask my students to write 93,000,000 in scientific notation, one of them will tell me it’s $9.3 \times 10^7$. When I ask why the power of 10 is a positive 7, they reply that it’s 7 because they had to move the decimal point 7 places, and it’s positive because they had to move it to the left.

So, what about converting $8.5 \times 10^4$? Some students see the positive power of 4, and based on the previous example think that they need to move the decimal point 4 places to the left, because in that problem a move to the left is associated with a positive power of 10. They fail to see that they need to do the opposite in this case.

My Approach

I build a table of powers of 10 for exponents running from 1 to 7 and -1 to -7.

$10^1=10 \\ 10^2=100 \\ . \\ . \\ . \\ 10^7=10,000,000$

$10^{-1}=0.1 \\ 10^{-2}=0.01 \\ . \\ . \\ . \\ 10^{-7}=0.0000001$

I then point out that all of the “big” numbers are associated with powers of 10 that are positive, and all of the “small” numbers are associated with powers of 10 that are negative. (We define a “big” number to be 10 or larger, and a “small” number to be less than 1.)

So, when we convert 93,000,000 to scientific notation, my students know that the exponent is positive because the original number was a “big” number. When I ask them to convert $8.5 \times 10^4$ to a decimal number, they know that the exponent of positive 4 tells them to move the decimal point by 4 places in such a way that will make 8.5 into a bigger number, and that will be by moving the decimal point to the right.

$0.000029=2.9 \times 10^{-5}$ : The exponent is a negative 5, not a positive 5, because the original number was a “small” number.

To convert $7.3 \times 10^{-11}$ to a decimal number, we move the decimal point 11 places to the left because a negative power of 10 is associated with a “small” number and to make 7.3 smaller the decimal has to move to the left.

Summary

I hope that you find this helpful. If you have any questions, or if you have any other topics you are interested in, you can reach me through the contact page at my web site – georgewoodbury.com. Please comment if you have a different way of approaching scientific notation.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

• 1. jedward706  |  October 13, 2010 at 1:16 pm

I focus on place value by reminding students that we generally work in a base 10 (decimal) system, associating this with the number of digits used [in part, because I will often teach how to work with other bases when I present scientific notation].
__ __ __ __ . __ __ __ __

Filling this in with the place values, with emphasis on the pattern in the exponents seems to help students understand the meaning and use of scientific notation. I make sure they associate the digit just to the left of the decimal with the place value. When they can get the place value picture
$… 10^4 10^3 10^2 10^1 10^0 . 10^{-1} 10^{-2} 10^{-3} …$ conceptually tied to scientific notation, they never have to remember what “direction” to move the decimal point!

Don’t know that I was clear…submitting this in a rush… I would be happy to explain further.

• 2. jdsfunsdijfn  |  February 17, 2011 at 7:48 am

i askedd how is scientific notation different/similar from other topics ?