## 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 10^{4}= 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 10*^{4}, 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 10^{6}, 10^{5}, 10^{4}, 10^{1}, and 10^{0}. If some students need to review exponents or cannot see the pattern, remind them that 10^{1}= 10 and 10^{0}= 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 10^{6}to 10^{0}. 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 10

^{4}= 10,000Have 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.