# Common Core Standards: Math

### 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.

#### Drills

1. Write the number eight million as the product of a single digit and a power of 10.

8 × 106

First, we can write the number out longhand: eight million is 8,000,000. It's an 8 followed by six zeros, so 6 is the power of 10. Typically, the number of zeros after a number is the power we want for 10.

2. Write the number twenty thousand as the product of a single digit and a power of 10.

2 × 104

Twenty thousand is 20,000. It's a 2 followed by four zeros. So the power of 10 is 4. The other options are incorrect because 2 × 103 is two thousand, 2 × 105 is two hundred thousand, and 2 × 106 is two million.

3. Write the number seventy as the product of a single digit and a power of 10.

7 × 101

Seventy is a 7, followed by one 0. So the power of 10 in this notation is 1. We should automatically know that (A) is wrong because 100 = 1 and 7 × 1 = 7. We also know that 70 = 7 × 10, and 101 = 10. Either method gets us the same answer.

4. Find the product of (9 × 102)(1 × 105).

9 × 107

We multiply the two beginning terms together: 9 × 1 = 9. Next, we add the powers of 10 (because we're multiplying the tens together): 102 × 105 = 102 + 5 = 107. So our final answer is 9 × 107. Some overzealous eighth graders might be tempted to distribute, but that's unnecessary because everything is multiplied—even the terms in parentheses.

5. Find the product of (4 × 103)(2 × 106).

8 × 109

To find the product, multiply the single digits together and add the powers of 10. That means we have 4 × 2 = 8 and 3 + 6 = 9. Or 8 × 109.

6. Find the quotient.

4 × 106

To find the quotient, we divide 24 by 6 and subtract 8 by 2. Since 24 ÷ 6 = 4 and 8 – 2 = 6, our final answer should be 4 × 106. The other answers result from incorrectly combining the exponents of 10. It's so nice to have the numbers all lined up, isn't it?

7. Find the quotient.

4

This might be a half step past your comfort zone, but 1.2 × 105 = 12 × 104. If we take the quotient of 12 × 104 and 3 × 104, we end up with 12 ÷ 3 = 4 and 104 – 4 = 100. That means our final answer is 4 × 100 = 4 × 1 = 4. Not that bad, right?

8. One estimate says that the typical November attendance at Disneyland is approximately 6 × 105 visitors. December attendance is typically double that of November. About how many people visit Disneyland in December?

1.2 × 106

There are many ways to think about this problem, none of which are as fun as Space Mountain. If we want to double 6 × 105, we're multiplying it by 2, which is the same as 2 × 100. If we multiply 2 by 6, we get 12. Since our 10 has an exponent of 0, we get 12 × 105 as our answer. Even though (D) is almost right, notice the exponent! We really want (A), which just results from doing this: 12 × 105 = (1.2 × 10) × 105 = 1.2 × (10 × 105) = 1.2 × 101 + 5 = 1.2 × 106.

9. At the LotsaStuff Ice Cream Parlor, a banana split contains approximately 9 × 102 Calories. A bowl of vanilla ice cream contains just under 3 × 102 Calories. The banana split has how many times the Calories of the ice cream?

3

In order to find the ratio, we can divide 9 × 102 by 3 × 102. The 9 divided by 3 gives us 3. Since both the numerator and denominator have 102, they cancel each other out and we're left with 3. And for the record, you can claim banana splits are healthy—they do have bananas, after all.

10. The distance from Salt Lake City to New York City is approximately 2 × 103 miles. The distance from Salt Lake City to Denver is about 5 × 102 miles. The first distance is approximately how many times the second?

4