# ## Lesson: Exponents and Powers of Ten: Developing the Concept

Now that students are familiar with exponents and powers of ten, they can use this knowledge to multiply and divide by powers of ten.

• Say: Now you will learn how to use patterns to help you multiply and divide by powers of 10.
Write the following problems on the board. • Ask: Who can tell us the products of these factors by using mental math?
Students should remember that multiplying whole numbers by 10, 100, or 1,000 adds 1, 2, or 3 zeros to the number: 8 x 10 = 80, 19 x 100 = 1,900, and 246 x 1,000 = 246,000.

Write this problem on the board. • Ask: Who can solve this problem, using any method?
Students should write the product 503.4 on the board after the equals sign. We will be creating a short table on the board.

On a different part of the chalkboard, have a student complete this table for a quick review for the class.

• 10³ = 1,000
• 10² =
• 101 =
• 100 =
• 10-1 = 0.1
• 10-2 =

This will help those students who need a review of the powers of 10. Place this multiplication problem just above the first problem. 5.034 x 10³ =

• Ask: What is the product of this number and this power of 10?
Students should solve with any method to find 5,034. Write this product after the equals sign. Do the same for 5.034 x 101 = 50.34, placing the problem under 5.034 x 10² = , forming a table.
• Ask: Can anyone see a pattern in the products?
Students should see that the pattern of the digits remains the same and the decimal point moves one place to the right each time the exponent increases by 1.
• Ask: Can anyone see how the exponent helps us find the product?
Students should see that the exponent matches the number of places the decimal point moves.
• Ask: When we multiply the numbers in these examples, why does the decimal point move to the right?
Students should answer that when we multiply by a whole number, the product is greater than the original number. Do not say that when multiplying by a power of 10, the product always increases. Complete the table with the class by putting these problems under the above problems.

5.034 x 100

5.034 x 10-1

5.034 x 10-2

• Ask: Can anyone use mental math to find the products?
Students who understand the patterns will know that the products are 5.034, 0.5034, and 0.0534.
• Ask: What happened to the placement of the decimal point?
Students should answer that multiplying by 100 or 1 did not move the decimal point. Multiplying by 10-1 and 10-2 moved the decimal point to the left.Write this sentence on the board. • Ask: What is an equivalent operation to multiplying by 10-1 (ten to the power of negative one)?
Students should respond that this is the same as dividing by 10 or multiplying by .
• Ask: What is another way of writing 5.034 x 10-2?
Students may respond with 5.034 x 0.01 or 5.034 x , but lead them to . Have a student solve this long division problem on the board to check the mental-math results. Make sure that students see that multiplying by 10-2 is the same as dividing by 10² or 100.
Ask students to come up with a rule about multiplying by powers of 10. Students should see this pattern: The decimal point moves to the right if the exponent is positive and to the left if the exponent is negative. The exponent also tells how many places to move the decimal point. If the exponent is zero, the decimal point does not move. Students learned earlier, that powers of 10 include negative exponents. It is not accurate to state that when we multiply by a power of 10 the decimal point always moves to the right. It is also not accurate to state that dividing by a power of 10 always moves the decimal point to the left. Make sure students understand that.

Have students complete the following table to deepen their understanding of the patterns involved and the mental math strategies they can use when multiplying and dividing decimals. ### Solutions to the Table

 Row 1 186 ÷ 10,000 186/10,000 .0186 Row 2 186 ÷ 1,000 186/1,000 .186 Row 3 186 ÷ 100 186/100 1.86 Row 5 186 ÷ 1 186/1 186 Row 7 186 ÷ 0.01 186/1 ÷ 1/100 186/1 × 100/1 = 18,600

Wrap-Up and Assessment Hints
Allow students to use the charts and tables they have created while practicing mental-math problems. Have students answer such questions as What is 2.67 times ten to the third power? (2,670) or What is four hundredths divided by ten squared? (0.04 ÷ 100 = 0.0004) This activity will strengthen their mental math skills, reinforce their place-value skills, and show the relationships between powers of ten and decimal multiplication and division.