• Observation during the Power of Ten group activity may be used to determine class comprehension level.
• Student presentations of Powers of Ten word problems may be used to evaluate individual skill level and understanding.
• The Lesson 2 Exit Ticket may be used to assess student mastery of lesson concepts.

As you begin the lesson, display the words product, quotient, and exponent on the board.

“Today we will be starting a new math lesson. There are some vocabulary words we should review before we start. Can somebody remind us what a product is?” (The answer to a multiplication question.)

“Who remembers what a quotient is?” (The answer to a division question.)

“I want you all to think about what an exponent is.”

Give students about a minute to think. Gauge student understanding of exponent by using the fist-to-five method.

“By a show of fingers I want you to let me know how certain you are that you know what an exponent is. If you are very sure you know and could even give me an example, raise your hand and show me all 5 fingers like this. If you do not have any idea what it is, raise your hand and show me zero fingers by closing your fist like this. If you have an idea but are not too sure, you can raise three or four fingers like this. You can use any number from 0 to 5. When I give the signal I want you to show me your fist-to-five response.”

If many students raise four or five fingers, ask a few to share the definition and/or an example of an exponent in their own words. If most students seem unsure, reassure them that they will learn it in today’s lesson. Use this as a guide to determine how much time to spend during the lesson on the basic definition and examples of exponents.

“In today’s lesson we will be working a lot with exponents and the number ten. We will be looking at powers of ten and patterns in products and quotients involving powers of ten. These patterns will help you make calculations with some very large and very small numbers quickly and easily.”

Display the following list or a similar list on the board:

• 724 × 10 =
• 724 × 100 =                                                                                                                                        
• 724 × 1,000 =                                                                                                                                       
• 724 ÷ 10 =                                                                                                                                               
• 724 ÷ 100 =                                                                                                                                           
• 724 ÷ 1,000 =

Either allow students to use calculators to find the products and quotients or use a calculator displayed for the class to find them. Ask students to write down three observations regarding the problems, products, and quotients on a piece of paper. Randomly select several students to share their observations. Call attention to student responses related to the patterns of the zeros and decimal point movement such as:

• “When we multiplied by 1,000 it was like our product was 724 with three zeros added on the end.” (Ask what happens when you multiply by 10. Emphasize that multiplying by 1,000 is like multiplying by 10 × 10 × 10.)
• “When we divided 724 by 100 the answer was still 7, then 2, then 4, but with a decimal between the 7 and 2 (or 2 places from the end of the number).” Ask students how many places the decimal is from the original placement of the decimal. They may say that there was not a decimal to begin with. Take this opportunity to discuss that a decimal can be placed at the end of any whole number, although we usually only show it if there will be a digit in the tenths place. Ask the class to describe a relationship or pattern that might explain the decimal moving left two places.

“You noticed many interesting patterns in this set of calculations. These are the types of number relationships and patterns we will be exploring today as we move through our lesson.”

Hand out base-ten blocks or the paper version from lesson 1 (see M-5-5-1_Paper Base-Ten Models in the Resources folder).

“In the problems we just looked at, you noticed several patterns involving zeros and the movement of the decimal point. We are going to use base-ten blocks to visualize these patterns. Once we understand how these patterns work, we can use them to efficiently answer many multiplication and division problems.

“Let’s start with 1 × 10 = 10. Use your base-ten blocks to represent 1 × 10.” Have two students show this on the board (look at student work to select a student to demonstrate each display shown below).

“We can see that 1 × 10 means one group of ten, or ten groups of one. All of these are the equivalent of one long. Ten ones becomes one group of ten or one long. Remember every group of ten can be regrouped to become one of the next larger place value. How could you adjust your blocks to represent 2 × 10? What about 5 × 10?”

“It looks like most of you made two groups of 10, which are equal to 20, and five groups of 10, which are 50. Without saying anything aloud, think about these examples and what do you notice about their answers.

“Next show me 10 × 10 with your blocks.”

“Using your block representation, please complete this sentence:

10 × 10 means _______________, or __________ flat(s).” (10 groups of 10; 1 flat)

“When we have 10 groups of 10, we need to regroup to one group in the hundreds place, leaving us zero tens and zero ones.

“Show me 20 × 10 and 12 × 10.”

“Let’s go a little bit higher. Use your blocks to represent 100 × 10.”

“Please complete this sentence: 100 × 10 means _________  or  __________, which is equivalent to _______.” (100 groups of 10 or 10 groups of 100; 1 cube or 1,000)

“In this case, we had so many hundreds that we could regroup them to one group of a thousand leaving a zero in the hundreds, tens, and ones places.” 

“Has anybody noticed a theme or pattern in what I have asked you to multiply by?” (always by 10, 100, or 1,000)

“How are all of the products related?” (They end in the same number of zeros in the power of ten we multiplied by.)

“These are what we call powers of 10. Exponents are used to tell us the power of a number. In other words, an exponent can be used to represent the number of times a number should be multiplied by itself. For example, think about 3². The base number we will multiply by is three. The exponent is two. This means that the number of times three will be multiplied together is two, or 3 × 3 = 9. How many of you got that right in your head?” A common error is to multiply 3 × 2 = 6 instead of 3 × 3 = 9. 

“What do 5³, 12², and 10³ represent? Raise your hand if you know at least one of the answers.” (5³ = 5 x 5 x 5 = 125; 12² = 12 x 12 = 144; 10 x 10 x 10 = 1,000)

“When we multiply by 100, it is like multiplying by 10 and 10 again. Another way to represent this is ‘multiply by 10².’ We read this as ‘ten to the second power’ or ‘ten squared’. Why ten squared? Think about your base-ten blocks. The block that is 10 ones by 10 ones, and contains 100 ones is a _________.” (flat, which is square)

“10² = 10 × 10 = 100.

“Fill in the blanks on this one: 10³  = ___ × __  × __  = _____.” (10 x 10 x 10 = 1,000)

“The expression 10³ can be read as ‘ten to the third power’ or ‘ten cubed.’ Which base-ten block is 10 ones by 10 ones by 10 ones?” (The cube.) “Yes! The cube that represents 1,000. The term cubed is used when labeling something to the third power or a measurement in three dimensions like a cube. For the remainder of our work with multiplying and dividing by powers of 10, we can use standard form numbers 10, 100, and 1,000, or the exponent form, such as 10² and 10³.”

Arrange students in groups of three for the next part of the lesson.

“We have mainly worked with multiplication problems. It is important to notice the patterns for division, too. Within your groups I would like each of you to divide two of the following problems using paper and pencil or base-ten blocks. Use any method you know. Each of you should do different problems. If you do not have enough people in your group to complete each problem, do more than two problems so you have all of the quotients to compare. When you are done, put your answers together in order.”

[Note: If it is too difficult or time consuming for some, or all, of your students to do long division, you may allow them to use a calculator. It may be more difficult to make some of the math connections with a calculator.]

• 6,800 ÷ 10 =                            45.68 ÷ 10 =               
• 6,800 ÷ 100 =                          45.68 ÷ 100 =
• 6,800 ÷ 1,000 =                       45.68 ÷ 1,000 =

Walk around the room helping students who are having trouble with the division process. When the groups have finished, continue to the next step.

“Let’s think about what you found.”

Hand out three sticky notes to each group. Give them 5–10 minutes to discuss the similarities and differences between the multiplication and division questions. Have each group use the sticky notes to write down one similarity, one difference, and one pattern or strategy it found that would help solve multiplication or division questions involving powers of 10. Ask each group to share its ideas. Make a place on the front board or poster paper to post each note. Place similarities together, differences together, and strategies together as groups continue to share. Be sure to emphasize correct thinking and strategies as they are presented. Use guiding questions to help students correct any misconceptions as they come up. Bring up any additional unique ideas or misconceptions you may have heard while observing, if they are not presented. Be sure to correct these misconceptions.

Summarize the patterns and strategies that students discovered. Observations should be similar to the following, but are not limited to the following:

• Multiplying by a power of 10 makes the value greater; dividing by a power of 10 makes the value smaller.
• Multiplying any whole or decimal number by 10 moves its decimal point right one place. For whole numbers this is like inserting a place holding zero at the end of the original number.
• Dividing any whole or decimal number by 10 moves the decimal point left one place.
• Multiplying any whole or decimal number by 100 (or 10²) moves its decimal point right two places. For whole numbers this is like inserting two place-holding zeros at the end of the original number.
• Dividing any whole or decimal number by 100 (or 10²) moves its decimal point left two places.
• Multiplying any whole or decimal number by 1,000 (or 10³) moves the decimal right three places. For whole numbers this is like inserting three place-holding zeros at the end of the original number.
• Dividing any whole or decimal number by 1,000 (or 10³) moves its decimal point three places to the left.
• Multiplying or dividing by 100 (or 10²) is like multiplying or dividing by 10 and 10 again.
• Multiplying or dividing by 1,000 (or 10²) is like multiplying or dividing by 10 three times in a row.
• Multiplying by any power of ten moves the decimal right (which may result in extra zeros being inserted at the end of the number), while dividing by any power of ten moves the decimal left (which may remove zeros from the end of the number).

Group Activity: Power of Ten Practice

For this activity students can work in the same group of three or switch to pairs. Hand out copies of the Power of Ten practice sheet (M-5-5-2_Power of Ten Practice and KEY.docx). Provide approximately 15 minutes for students to complete the questions and create a problem. Remind students that they will be presenting the problem they create (question 15) to the class at the end of the work time. Plan on approximately 15–25 minutes for students to present the problems and solutions they wrote.

Monitor students during work time. Within each pair or small group, ask the students to summarize their ideas and findings on the patterns of zeros and decimals from the lesson. Make suggestions or use clarifying questions to guide students who have misconceptions. During work time and presentations encourage students to correct or adjust any work that is inaccurate.

Each student should complete the exit ticket at the end of the lesson (M-5-5-2_Lesson 2 Exit Ticket and KEY.docx). Use the results to determine which additional optional instructional strategies may be helpful for individual students.

Extension:

• Routine: Throughout the school year, point out and discuss instances of powers of ten as they arise. Ask students to bring examples of powers of ten that they see when reading or making calculations in math or any other subjects. Have a special place in the room, such as a bulletin board, to display what students have found and shared with the class.
• Small Group: Use these or similar stations for students who are having difficulty with the concept of products and quotients using powers of ten. Students can work alone or in pairs as they move through stations 1 and 2.

Work Station 1:  Flash Cards

This activity will work best with a partner. Place one or more sets of the Power of Ten Flash Cards at the station (M-5-5-2_Flash Cards.docx). Students working in pairs will have one host and one player. The host will hold the cards up for the player to answer. The host will be able to see and check the answers on the back. Any question answered incorrectly should be placed back at the bottom of the pile by the host, to be attempted again at the end of the round. When the player is finished, students reverse roles.

Work Station 2:  Decimal Dash

Students will practice writing decimal numbers as a fraction with a denominator of 10, 100, or 1,000 and the reverse as quickly as they can. Place Decimal Dash sheets at the station (M-5-5-2_Decimal Dash and KEY.docx). Each student will need a copy. Also place a timer at the station. Students should try to complete the sheet in less than 1 minute, or any time that is posted at the station.

• Expansion: Make That Negative

This optional activity is appropriate for students who have shown proficiency with powers of ten in products and quotients and are looking for a challenge beyond the requirements of the standard. Students will multiply by negative powers of ten, and rewrite multiplication with negative powers of ten as division by positive powers of ten. Give each student or pair of students a Make That Negative sheet (M-5-5-2_Make That Negative and KEY.docx).