Institute for Mathematics Learning

Power of 10

Understanding Place Value

Power of 10

Institute for Mathematics Learning:

Power of 10 Lessons

Understand the place value system.

5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Building Powers of Ten

Standard: 5.NBT.2 | Additional /Supporting Standard(s): 5.NBT.1, 5.NBT.3, 5.NBT.5

Mathematical Practice: 2, 3, 4, 7

Student Outcomes: I can make models of several powers of ten. I can demonstrate the meaning of exponential form. I can make conjectures about patterns in powers of ten

Value of Bills

Standard: 5.NBT.2 | Additional /Supporting Standard(s): 5.NBT.1, 5.NBT.3Mathematical Practice: 2, 4, 7

Student Outcomes:I can write an expression with exponents to represent powers of ten. I can make connections between multiplying by 10 and the exponent in powers of ten.

Mass of Supplies

Standard: 5.NBT.2 | Additional /Supporting Standard(s): 5.NBT.1, 5.NBT.3

Mathematical Practice: 2, 4, 7

Student Outcomes:I can write an expression with exponents to represent powers of ten. I can make connections between multiplying by 10 and the exponent in powers of ten.

Between the Stars

Standard: 5.NBT.2 | Additional /Supporting Standard(s): 5.NBT.1, 5.NBT.3

Mathematical Practice: 2, 4, 7

Student Outcomes: I can write an expression with exponents to represent powers of ten. I can make connections between multiplying by 10 and the exponent in powers of ten.

List of Formative Instructional and Assessments Tasksfor 5.NBT.1 and 5.NBT.2 are listed at the end of this document.

Building Powers of Ten

Common Core Standard:

Understand the place value system.

5.NBT.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Additional/Supporting Standards:

5.NBT.1, 5.NBT.3Understand the place value system.

5.NBT.5Perform operations with multi-digit whole numbers and with decimals to hundredths.

Standards for Mathematical Practice:

2.Reason abstractly and quantitatively

3.Construct viable arguments and critique the reasoning of others

4.Model with mathematics

7.Look for and make use of structure

Student Outcomes:

I can make models of several powers of ten
I can demonstrate the meaning of exponential form
I can make conjectures about patterns in powers of ten

Materials:

Base-ten blocks
Powers of Ten Sheet (one per student)

Advance Preparation:

Make copies of the Powers of Ten Sheet
Gather base-ten blocks
Consider how you will pair students
Students should be familiar with base-ten blocks and have a strong understanding of place value to 100,000
Students should have experience multiplying with tens
Students should have experience building square arrays such as 32
Students should be familiar with square numbers

Directions:

“How could you represent 10 x 10?” Allow many student suggestions. If no one suggests a base-ten block, ask how these blocks could be used to show 10 x 10.
Show students the flat base-ten block (sometimes called a “hundreds block”). “What shape is this block?” “Why does it make sense that it is a square?”

Distribute base-ten blocks to each student. Have the students use their blocks to prove that 10 x 10 = 100 and that this array forms a square. Students make need to share materials and work together.

4.On the board writethe following:10 x 10 = 102

Tell students that 102 is read “ten squared.” Ask the class to explain how it might have been given that name (the array for 10 x 10 is a square).

5. Distribute the Powers of Ten chart to each student. Point out that they have just worked on the first row.

6. Ask students to use their blocks to build 10 x 10 x 10. Have students share their strategies. Did they recognize that this could be rewritten as 10 x 100 and could be built as ten hundreds blocks stacked into a cube?

7. On the board write the following:10 x 10 x 10 = 103

Tell the class that 103 is read “ten cubed.” Ask the class to explain how it might have have been that name (the base-ten blocks form a cube of 10 flats).

8. Continue the chart by working with 101. Allow students to debate with one another how this might be written and built. Help the students make connections between the number of tens being multiplied and the number used in the exponent.

10 x 10 multiplies two tens, so the exponent is a 2

10 x 10 x 10 multiplies three tens so the exponent is a 3

101 has an exponent of one, so it must have only one 10

9.Explain to the students that 101 tells us that we are working with one ten. On the board write the following:10 = 101

Students can use a tens block (rod) to show 101

10.Challenge students to try the next row in their chart with a neighbor. Allow the students ample time to struggle with the work. As you circulate, listen to see if students are making connections to the exponents that they have already explored.

11.When most have finished invite students to share their thinking about 104. The students are likely to have a variety of answers. Engage a lively debate by asking students to justify their thinking and explain their reasoning. Students may question one another. They may use blocks, observations, patterns, and logic to prove their answer.

12.At a reasonable stopping point, help students see that 104 is written as 10 x 10 x 10 x 10

This might be written in words as “ten to the fourth power.” A picture is not necessary here.

Write on the board:10 x 10 x 10 x 10 = 104

13. For the final row of the chart, have students work with their partner. Then engage the class in another discussion of reasoning.

14. To summarize, ask students to complete the “I Discovered That…” section of the recording sheet. Ask students, “What pattern do you notice?” Students should see that each the number of zeros is changing. Ask students to explain why this pattern makes sense.

Questions to Pose:

As Students Work

How did you figure out that…?
How could you represent that with blocks?
How could you represent that with multiplication?

In Class Discussion

Why might some people think that 104 = 40? Where is the error in their reasoning?
What patterns do you notice in the standard notations?
Why does this pattern make sense?

Possible Misconceptions/Suggestions:

Possible Misconceptions / Suggestions
Students think of the exponents as factors.
For example:
103 = 10 x 3
101 = 10 x 1 / •Students should build square arrays and record them as exponents. For example:

•Return to 10 x 10 = 100. Use the flat base-ten block to show that 10 x 10 creates a square array. Explain that mathematicians shortened 10 x 10 to 102. Observe that 102 is not the same as 10 x 2, but that it represents 10 x 10. Make note of the location and size of the exponent.Connect these observations with other exponents.

Special Notes:

Before this lesson, students should conduct an investigation of square numbers using arrays. See the suggestions in above table.
A follow up lesson should support students as they apply their understanding of powers of ten to create scientific notations such as 4 x 103 = 4,000

Solutions:Note – Students may also use the phrase “ten to the power of ____”

Multiplication Expression / Words and/or Pictures / Exponential Notation / Standard Notation
10 x 10 x 10 / ten cubed / 103 / 1,000
10 / / 101 / 10
10 x 10 x 10 x 10 / ten to the fourth power / 104 / 10,000
10x10x10x10x10 / ten to the fifth power / 105 / 100,000

Building Powers of 10

Multiplication Expression / Words and/or Pictures / Exponential Notation / Standard Notation
10 x 10 / / 102 / 100
10 x10 x 10
101
104
105

I Discovered That:

101 = ______

102 = ______

103 = ______

104 = ______

105 = ______

Value of Bills

Common Core Standard:

Understand the place value system.

5.NBT.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Additional/Supporting Standards:

5.NBT.1, 5.NBT.3Understand the place value system.

Standards for Mathematical Practice:

2.Reason abstractly and quantitatively

4.Model with mathematics

7.Look for and make use of structure

Student Outcomes:

I can write an expression with exponents to represent powers of ten
I can make connections between multiplying by 10 and the exponent in powers of ten

Materials:

Value of Bills Activity Sheet

Advance Preparation:

Make copies of the Value of Bills Activity Sheet

Lesson:

Activity 1: Display the following equations and have students mentally compute the following products:

4 x 2040 x 20400 x 204,000 x 2040,000 x 20

Have students share their products one at a time. Record them.

Ask students, “Is there a strategy that you used to find the product of each expression?” and

“How do the products relate to one another?”

Have students create equivalent equations for each so that each expression equals 8 x __.

Example: 400 x 20 = 8 x ___

Ask students if they notice a pattern.

Activity 2: Distribute the Value of Bills activity sheet.

Remind students or tell them that exponents are used as a way to represent repeated multiplication, such as 10 x 10 x 10 can be written as 103.

Have them work on the front page of the activity sheet.

Questions to Pose as students work:

What strategies are you using to solve this task?
What relationship do you notice between the exponents and the context of your task?

Discussion after most students have finished the front side of the sheet

What relationship did you notice between the exponents and the context of your task?
Why does this relationship make sense?

Possible Misconceptions/Suggestions:

Possible Misconceptions / Suggestions
Students think of the exponents as a factor instead of the number of times they multiply by 10.
For example:
103 = 10 x 3
101 = 10 x 1 / •Return to 10 x 10 = 100. Use the flat base-ten block to show that 10 x 10 creates a square array. Explain that mathematicians shortened 10 x 10 to 102. Observe that 102 is not the same as 10 x 2, but that it represents 10 x 10. Make note of the location and size of the exponent. Connect these observations with other exponents.

Solutions:

4 bills

4 $10 bills: 4 x $10 = $40 = 4 x 101 4 $100 bills: 4 x $100 = $400 = 4 x 102

4 $1,000 bills: 4 x $1,000 = $4,000 = 4 x 103 4 $10,000 bills: 4 x $10,000 = $40,000 = 4 x 104

4 $100,000 bills: 4 x $100,000 = $400,000 = 4 x 105

8 bills

8 $10 bills: 8 x $10 = $80 = 8 x 101 8 $100 bills: 8 x $100 = $800 = 8 x 102

8 $1,000 bills: 8 x $1,000 = $8,000 = 8 x 103 8 $10,000 bills: 8 x $10,000 = $80,000 = 8 x 104

8 $100,000 bills: 8 x $100,000 = $800,000 = 8 x 105

Investing $0.25 at the bank

After Month 1: $0.25 x 10, $2.50, $0.25 x 101 After Month 2: $0.25 x 10 x 10, $25.00, $0.25 x 102

After Month 3: $0.25 x 10 x 10 x10, $250.00, $0.25 x 103

After Month 4: $0.25 x 10 x 10 x 10 x 10, $2,500.00, $0.25 x 104

After Month 5: $0.25 x 10 x 10 x 10 x 10 x 10, $25,000.00, $0.25 x 105

After Month 6: $0.25 x 10 x 10 x 10 x 10 x 10 x 10, $250,000.00, $0.25 x 106

Investigating $1.37 at the bank

After Month 1: $1.37 x 10, $13.70, $1.37 x 101

After Month 2: $1.37 x 10 x 10, $137.00, $1.37 x 102

After Month 3: $1.37 x 10 x 10 x10, $1,370.00, $1.37 x 103

After Month 4: $1.37 x 10 x 10 x 10 x 10, $13,700.00, $1.37 x 104

After Month 5: $1.37 x 10 x 10 x 10 x 10 x 10, $137,000.00, $1.37 x 105

After Month 6: $1.37 x 10 x 10 x 10 x 10 x 10 x 10, $1,370,000.00, $1.37 x 106

Value of Bills

What if you had 8 of the same bill in your pocket?

Number of Bills / Value of the Each Bill / Total Value / Expression with an Exponent
8 / $10 / $80 / 8 x 101
8 / $100
8 / $1,000
8 / $10,000
8 / $100,000

What if you 8 of the same bill in your pocket?

Number of Bills / Value of the Each Bill / Total Value / Expression with an Exponent
8 / $10
8 / $100
8 / $1,000
8 / $10,000
8 / $100,000

Value of Bills (page 2)

A bank promises that if you invest money the value be multiplied by 10 each month. If you start with $0.25 complete the table.

Month / Original Value / Expression when Multiplying by 10 / Expression with an Exponent
1 / $0.25 / $0.25 x 10 = $2.50 / $0.25 x 101
2 / $0.25 / $0.25 x 10 x 10 = $25.00 / $0.25 x 102
3
4
5
6

What if you invested with the bank a dollar, a quarter, a dime and two pennies?

Month / Original Value / Expression when Multiplying by 10 / Expression with an Exponent
1
2
3
4
5
6

What relationship do you notice between the exponent and your expression when multiplying by 10?

Mass of Supplies

Common Core Standard:

Understand the place value system.

5.NBT.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Additional/Supporting Standards:

5.NBT.1, 5.NBT.3Understand the place value system.

Standards for Mathematical Practice:

2.Reason abstractly and quantitatively

4.Model with mathematics

7.Look for and make use of structure

Student Outcomes:

I can write an expression with exponents to represent powers of ten
I can make connections between multiplying by 10 and the exponent in powers of ten

Materials:

Mass of Supplies Activity Sheet

Advance Preparation:

Make copies of the Mass of Supplies Activity Sheet

Lesson:

Activity 1: Display the following equations and have students compute the following products:

4 x 1036 x 1055.2 x 1033 x 104 + 4 x 1034 x 102 x 102

Have students share their products one at a time. Record them.

Ask students, “What strategy did you use to find the value of each expression?”

Activity 2: Distribute the Mass of Supplies activity sheet.

Remind students or tell them that exponents are used as a way to represent repeated multiplication, such as 10 x 10 x 10 can be written as 103.

Have them work on the front page of the activity sheet.

Questions to Pose as students work:

What strategies are you using?
What relationship do you notice between the exponents and the context of your task?

Discussion after most students have finished the front side of the sheet

What relationship do you notice between the exponents and the context of your task?
Why does this relationship make sense?

Possible Misconceptions/Suggestions:

Solutions:

Scotch Tape

1 bin: 16.5 x 101, 165 grams1 basket: 16.5 x 102, 1,650 grams

1 crate: 16.5 x 103, 16,500 grams1 pallet: 16.5 x 104, 165,000 grams

Paper Clips

1 bin: 23.8 x 101, 238 grams1 basket: 23.8 x 102, 2,380 grams

1 crate: 23.8 x 103, 23,800 grams1 pallet: 23.8 x 104, 238,000 grams

6 crates of tape: 6 x 16.5 x 103 = 6 x 16,500 = 99,000 g

99,000 x 0.002 = $198.000

4 pallets of paper clip: 4 x 23.8 x 104 = 4 x 238,000 = 952,000 g

952,000 x 0.002 = $1,904.00

9 baskets of tape and 3 baskets of paper clips

9 x 16.5 x 102 = 9 x 1,650 = 14,850 g3 x 23.8 x 102 = 3 x 2,380 = 7,140 g

14,850 + 7,140 = 21,99021,990 x 0.002 = $43.98

2 crates of paper clips and 3 bins of tape

2 x 23.8 x 103 = 2 x 23,800 = 47,600 g3 x 16.5 x 101 = 3 x 165 = 495g

47,600 + 495 = 48,095 g48,095 x 0.002= $96.19

2 pallets of tape and 4 crates of paper clips

2 x 16.5 x 104 = 2 x 165,000 = 330,000 g4 x 23.8 x 103 = 4 x 23,800 g = 95,200 g

330,000 + 95,200 g = 425,200 g425,200 x 0.002 = $850.40

10 pallets of tape and 20 pallets of paper clips

10 x 16.5 x 104 = 10 x 165,000 = 1,650,000 g 20 x 23.8 x 104 = 20 x 238,000 = 4,760,000 g

1,650,000 + 4,760,000 = 6,410,000 g6,410,000 x 0.002 = $12,820.00

Mass of Supplies

Salisbury Supply Company packages their supplies in various containers. The smallest container is a box. Ten boxes can be combined into a bin. Ten bins can be combined into a basket. Ten baskets can be combined into a crate. Ten crates can be combined into a pallet. Write an expression with exponents and find the mass for each sized container.

A box of scotch tape has a mass of 16.5 grams

Mass of 1 bin / Mass of 1 basket
Mass of 1 crate / Mass of 1 pallet

A box of paper clips weighs 23.8 grams

Mass of 1 bin / Mass of 1 basket
Mass of 1 crate / Mass of 1 pallet

Troubleshooting with the Supply Company

The cost of shipping supplies depends on the mass of their shipment. The cost is determined by multiplying the mass by 0.002.

Use the table below to determine the cost of each shipment.

Remember 1 box of tape weighs 16.5 g and 1 box of paper clips weighs 23.8 g.

Shipment / Expression with Exponents to Find Mass / Mass / Cost
6 crates of tape
4 pallets of paper clips
9 baskets of tape and
3 baskets of paper clips
2 crates of paper clips and 3 bins of tape
2 pallets of tape and 4 crates of paper clips
10 crates of tape and 20 crates of paper clips

Between the Stars

Common Core Standard:

Understand the place value system.

5.NBT.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Additional/Supporting Standards:

5.NBT.1, 5.NBT.3Understand the place value system.

Standards for Mathematical Practice:

2.Reason abstractly and quantitatively

4.Model with mathematics

7.Look for and make use of structure

Student Outcomes:

I can write an expression with exponents to represent powers of ten
I can make connections between multiplying by 10 and the exponent in powers of ten

Materials:

Mass of Supplies Activity Sheet

Advance Preparation:

Make copies of the Mass of Supplies Activity Sheet

Lesson:

Activity 1: Display the following equations and have students compute the following products:

7 x 102 x 1027 x 104 ÷ 1027 x 105÷103

Have students share their products one at a time. Record them.

Have students write equivalent equations that are equal to the total and also equal to 7 x 10

Example: 7 x 105÷103 = 700 = 7 x 102

Ask students, “What strategy did you use to find the value of each expression?”

Activity 2: Distribute the Between the Stars activity sheet.

Remind students or tell them that exponents are used as a way to represent repeated multiplication, such as 10 x 10 x 10 can be written as 103.

Have them work on the front page of the activity sheet.

Questions to Pose as students work:

What strategies are you using?
What relationship do you notice between the exponents and your final answer?

Discussion after most students have finished the front side of the sheet

What relationship do you notice between the exponents and your final answer?
Why does this relationship make sense?

Have students complete the second page of the activity sheet. Pose the same questions.

Possible Misconceptions/Suggestions:

Solutions:

Star B: 3.2 x 102 x 103 = 3.2 x 105 = 320,000

Star C: 3.2 x 102 x 103 ÷ 104 = 3.2 x 101 = 32

Star D: 3.2 x 102 x 103 ÷ 101 = 3.2 x 104 = 32,000

Star E: 3.2 x 102 x 103 x 102 = 3.2 x 107 = 32,000,000

Star A3.2 x 102 km from the sun

Star B103 km farther from the sun than Star A

Star C104 km closer to the sun than Star B

Star D101 km farther from the sun than Star B

Star E102 km farther to the sun than Star B

Part 2

Star G: 4.9 x 103 x 102 = 4.9 x 105 = 490,000

Star H: 4.9 x 103 ÷ 102 = 4.9 x 10 = 49

Star I: 4.9 x 103 x 102 ÷ 102 = 4.9 x 103 = 4,900

Star J: 4.9 x 103 x 102 ÷ 103 = 4.9 x 102 = 490

Star F4.9 x 103 km from the sun

Star G102 km farther from the sun than Star F

Star H102 km closer to the sun than Star F

Star I102 km farther from the sun than Star G

Star J103 km closer to the sun than Star G

Between the Stars

There are five stars that are seen in a straight line through a telescope. The stars are the following distance from the sun:

Star A3.2 x 102 km from the sun