Expressions and Equations 8.EE.A.3
3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.
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Sounds impressive, doesn't it? Well, don't be fooled. With the right words, even socks can sound extraordinary. Whether your students know it or not, numbers can be the same way.
One one-thousandth might sound like a really impressive number with that many syllables, but it's actually very tiny. A billion, on the other hand, doesn't sound very big—certainly not as big as 1,000,000,000. Since words can be so deceiving, mathematicians usually prefer to use numbers. (That's probably why they're mathematicians.)
But even numbers can be a bit misleading since we can write very big or very small numbers as single digits times a power of 10. We do this because it's easier to write 3 × 109 instead of 3,000,000,000. It's also why most of us prefer "sock" to "Podiatric Thermal Insulator 3000X." Unless we're telling the in-laws about our next big idea.
It's important for students to understand why we multiply these single digits by ten. Essentially, our whole number system is based on powers of ten. Change the power of ten, and you move over one digit. That's why we have the tenths and hundredths places past the decimal point.
We can also use this notation to divide and multiply big numbers faster and easier. For instance, if we wanted to divide three billion by a thousand, we could write:
Since the rules of exponents still apply, we can divide the 3 by 1 (which is 3) and subtract the exponents on the 10s (which will give us 106). So our final answer is 3 × 106, or 3,000,000.
Students should be able to apply this same concept to really small numbers (meaning the exponent on the 10 will be negative), and convert to and from this notation. They should also be able to compare these numbers and perform operations with them.
As long as you stress that both forms of the number are identical and that the exponent rules still apply, students shouldn't have too much trouble. In no time, they'll be ready to knock the Podiatric Thermal Insulators off these ultra-big numbers.