Scientific Notation

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

In this lesson we go over scientific notation. First the slides dedicate time to explaining how our base 10 number system allows us to simply move decimals right or left when multiplying or dividing by 10. The lesson then builds on this idea to demonstrate how very large and very small numbers can be represented more easily in scientific notation. Students will see how positive exponents move decimals right for large numbers and negative exponents move decimals left for very small numbers.

So without further ado, read through the slides below to get a feel for how to convert numbers into scientific notation!

Sci 1Sci 2Sci 3Sci 4Sci 5Sci 6Sci 7Sci 8Sci 9Sci 10Sci 11Sci 12Sci 13

Sci 14Sci 15Sci 16

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Operations in Scientific Notation

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • Scientific Notation
  • Converting Numbers
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Multiplying and dividing by 10 mentally
  • Converting numbers in standard form into scientific notation
  • Converting numbers in scientific notation back into standard form
  • 10’s and decimals
  • The real wonder in this section is the power of our base 10 number system.
  • Odds are you’ve heard someone say something about that before but probably never really understood what they were talking about.
  • Well, today’s the day! It all has to do with how we count.
  • We count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
  • Ten is just one with a zero behind it. One ten.
  • Then 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.
  • Twenty is two and then zero, or two tens.
  • The pattern continues all the way to ten tens:
  • 10’s and Decimals
  • Ten hundreds is then 1000, ten thousands is 10,000.
  • And so on and so forth…..forever.
  • Just as impressive is what happens behind a decimal point!
  • Remember our place values?
  • Our decimal places follow a pattern of tens as well, it seems.
  • So then, how can we exploit this pattern to fuel our lazy mathematics?
  • Multiplying 10’s
  • Remember this? (Nod slowly and cautiously.)
  • We multiply our first digit from the right in the bottom number by everything on the top. In this case that means 0 x 2 and 0 x 5.
  • Then the second digit by everything on the top. But this time we have a place holder zero since we are in the second place.
  • Add em up:
  • Hmm, wasn’t really a very difficult addition was it?
  • Multiplying 10’s
  • This first line of multiplication really didn’t affect anything, did it?
  • In fact, all we did was slap on a zero to our original number.
  • This pattern can follow for any multiple of 10 in fact.
  • 26 x 100 = 2600
  • 84 x 1000 = 84000
  • You simply add the zeroes onto the end of the number you are multiplying.
  • But there is another kind of number that will be even more useful to understand.
  • 10’s and decimals
  • So what happens with a number like this?
  • Let’s multiply our decimal by 10 again with the same process:
  • Remember, with decimals we count the number of decimal places in our original number:
  • Perfect! So we have two places, and then we count that many places from the right on our answer:
  • Thus, our final product will be the decimal number:
  • 10’s and Decimals
  • As always, as lazy mathematicians, there must be some quicker way to do this, right?
  • Remember that multiplying by ten just means adding a zero to our number.
  • But then with decimals we have to count our decimal places.
  • Hmmm quicker, but how are these two numbers really different?
  • Our decimal just moved once to the right.
  • And ten has just one zero in it!
  • So to multiply decimals by ten, we just move the decimal once to the right!
  • 10’s and Dividing
  • You can hopefully see that multiplying by larger multiples of 10 (100, 1000, 10000) will just move our decimal more places, however many zeroes our multiple has.
  • Let’s think critically now about division.
  • Before we saw that division and multiplication are inverses.
  • This means opposites, but what is the opposite of moving our decimal to the right?
  • Moving our decimal to the left!
  • Therefore when multiplying or dividing by 10, 100, 1000…etc simply move the decimal the same number of places as zeroes you see!
  • Multiplying to the right, and dividing to the left.
  • Scientific Notation
  • Ok, ok. So what is all of this new information for? 9 slides and we haven’t even mentioned the title yet…c’mon.
  • I hear you! But we needed all that information to understand this:
  • Well that certainly clears it all up…right?
  • Maybe looking at this animal in pieces will help a bit.
  • We start with a decimal number.
  • Then we multiply it by a multiple of ten (because now we know that only moves the decimal.
  • But what about this exponent business?
  • Scientific Notation
  • Let’s get a reminder of our handy 10’s chart.
  • 10 raised to a power has the same number of zeroes as the exponent.
  • Scientific Notation
  • So multiplying in a form like this would mean to just move the decimal the same number of places as the exponent.
  • In this example:
  • We moved 4 places to the right, but we only had 3 places to the right of the decimal.
  • To fix this, we simply add a zero to the end to make up for the missing place (there are infinite zeroes behind a decimal because there are infinite place values).
  • Thus, our number written in standard form would be:
  • Scientific Notation
  • Alright, so then what was all that effort about dividing then?
  • I’m so glad you asked.
  • Now we have a negative exponent.
  • One of our exponent properties said that negative exponents are just flipped to the bottom, like dividing, right?
  • And what did we say about dividing by 10’s?
  • That’s right, decimal goes to the left!
  • Yet again, we are going to have to put some zeroes in those blank spaces.
  • Scientific Notation
  • So our number in standard form would be:
  • Let’s take note here that the negative exponent made a smaller number, very small in fact.
  • Multiplying due to a positive exponent made a larger number like so:
  • I like to remember it as “Multiply=Big, Divide=Small” to check.
  • Some of you may prefer to remember that multiplying moves the decimal right and dividing moves the decimal left, however.
  • Whichever makes more sense to you, think of it that way!
  • Really Big…And Really Small
  • I know what you’re thinking. “That seems like a lot of work just to make a number look different…Why?”
  • As always, mathematicians are lazy.
  • Would you rather write this:
  • Sixty-four quadrillion, seven hundred thirty trillion
  • Or this:
  • Scientific notation saves a lot of space and time.
  • This is especially true in scientific fields such as physics where really big or really small numbers pop up a lot, some much bigger than even that number!
  • Take Note!
  • As a final thought, let’s really look at the form of a number in scientific notation.
  • The decimal number will always come first.
  • And it will always have EXACTLY one digit to the left of the decimal in the ones place.
  • Every number in scientific notation will always have a “x10” to some power which indicates how many places and which direction to move the decimal.
  • Here are two examples that are NOT in scientific notation:
  • conclusion
  • Multiplying any whole number by a multiple of 10 will add that many zeroes to the number.
  • Multiplying a decimal number by a multiple of ten will move the decimal to the right the same number of places as zeroes.
  • Dividing a decimal number by a multiple of ten will move the decimal to the left the same number of places as zeroes.
  • Scientific notation can be used to represent very large and very small numbers.
  • Scientific notation is always a decimal number with exactly one digit to the left of the decimal followed by “x10” raised to a power.
  • Positive exponents represent how many places to move the decimal right.
  • Negative exponents represent how many places to move the decimal left.