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

A solid knowledge of powers of ten and exponents will help students remember the place-value names.

Prerequisite Skills and Concepts: Students need to be familiar with exponents and the place-value chart.

Preparation: Display the place-value chart below on the chalkboard or overhead projector. • Say: When we multiply a number by itself several times, we can write this by using exponents.
Write the example 10 x 10 x 10 = 1,000 = 10³ on the board. Write 10³ large for demonstration purposes.
• Say: The 10 is called the base, and the 3 is called the exponent. The exponent indicates the number of times the base is used as a factor.
• Ask: Who can show us how to use exponential notation for 10 as a factor four times? Students should respond by writing 104 = 10 x 10 x 10 x 10 = 10,000.
• Say: The product 10,000 is called a power of 10. Another name for ten thousand is 104, which is read “ten to the fourth power.”
• Ask: What is the product of 10 x 10? Who can show us how to write this power of 10 by using exponents?
Students should respond with 10 x 10 = 100 = 10². Point out that 100 and 10² name the same number.
• Say: This can be read as “ten to the second power,” or “ten squared.”
• Ask: Who can tell us what digit is in the hundreds place on the place-value chart?
Students should indicate the 7 is in the hundreds place.
• Ask: What would be another way to indicate the hundreds place, using exponents?
Students should respond with 10². Some may want to use 10 x 10, but point out that the exponential notation will be easier to write when we use larger numbers. In the appropriate space on the chart, under the 7 in the hundreds place, have a student write the power of 10 using exponents. (10²)
• Ask: Using exponential notation, what power of 10 can represent the thousands place?
Students should reply with 10³. Have a student write this in the appropriate place under the 4.
• Ask: Who can complete the powers of 10 for the whole-number places by using exponents?
Students should see the pattern and fill in 106, 105, 104, 101, and 100. If some students need to review exponents or cannot see the pattern, remind them that 101 = 10 and 100 = 1.
• Ask: What patterns can you see in the powers of 10 on the chart?
Students should see that the exponents are positive numbers in sequence. Some will notice that 1,000 has 3 zeros and its power of 10 has an exponent of 3.
• Ask: Who can predict the powers of 10 for the decimal places?
Students should see the pattern of exponents decreasing from 106 to 100. Continuing this pattern for the decimal places gives 10-1, 10-2, 10-3, and 10-4. Have students write these powers of 10 on the chart.
• Ask: Can this chart be extended to show even greater or smaller numbers? Will the pattern continue?
Students should answer that the pattern will continue in both directions.
• Ask: Can you see other patterns?
Some students may see the relationship between the exponent and the number of zeros in the standard form.

10³ = 1,000        104 = 10,000

Have students create their own place-value charts modeled on the chart that appears in Lesson 1 of their textbook. This chart includes millionths through hundred billions. Have students include a row under the chart to list the powers of 10 in exponential notation. This will help to reinforce the relationship between powers of 10 and place-value positions. Give printed blank charts with all 18 places to each student. This will help students create neat, orderly charts that they can keep in their notebooks.