Power of 10
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Understanding Place Value
Power of 10
Institute for Mathematics Learning: http://ime.math.arizona.edu/progressions/
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
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 Tasks for 5.NBT.1 and 5.NBT.2 are listed at
the end of this document.
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Building Powers of Ten
Common Core Standard:
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.
Additional/Supporting Standards:
5.NBT.1, 5.NBT.3 Understand the place value system.
5.NBT.5 Perform 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:
1. “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.
2. 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?”
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
3. 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 write the 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:
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
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
Students think of the exponents as factors.
For example:
103 = 10 x 3
101 = 10 x 1
Suggestions
• Students should build square arrays and record
them as exponents. For example:
3 x 3 = 32
• 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
Exponential
Words and/or Pictures
Expression
Notation
10 x 10 x 10
ten cubed
10
Standard
Notation
103
1,000
101
10
10 x 10 x 10 x 10
ten to the fourth power
104
10,000
10 x 10 x 10 x 10 x 10
ten to the fifth power
105
100,000
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Building Powers of 10
Multiplication
Expression
Words and/or Pictures
10 x 10
ten squared
Exponential
Notation
Standard
Notation
102
100
10 x10 x 10
101
104
105
I Discovered That:
101 = _______
102 = _______
103 = _______
104 = _______
105 = _______
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Value of Bills
Common Core Standard:
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.
Additional/Supporting Standards:
5.NBT.1, 5.NBT.3 Understand 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 20
40 x 20
400 x 20
4,000 x 20
40,000 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?
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
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
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
Suggestions
• 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
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Value of Bills
What if you had 8 of the same bill in your pocket?
Number of
Bills
Value of the Each Bill
8
$10
8
$100
8
$1,000
8
$10,000
8
$100,000
Total Value
Expression with an
Exponent
$80
8 x 101
What if you 8 of the same bill in your pocket?
Number of
Bills
Value of the Each Bill
8
$10
8
$100
8
$1,000
8
$10,000
8
$100,000
NC DEPARTMENT OF PUBLIC INSTRUCTION
Total Value
Expression with an
Exponent
FIFTH GRADE
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
1
2
Original
Value
$0.25
$0.25
Expression when
Multiplying by 10
$0.25 x 10 = $2.50
$0.25 x 10 x 10 = $25.00
Expression with an
Exponent
$0.25 x 101
$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?
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Mass of Supplies
Common Core Standard:
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.
Additional/Supporting Standards:
5.NBT.1, 5.NBT.3 Understand 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 103
6 x 105
5.2 x 103
3 x 104 + 4 x 103
4 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?
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Possible Misconceptions/Suggestions:
Possible Misconceptions
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
Suggestions
• 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:
Scotch Tape
1 bin: 16.5 x 101, 165 grams
1 crate: 16.5 x 103, 16,500 grams
1 basket: 16.5 x 102, 1,650 grams
1 pallet: 16.5 x 104, 165,000 grams
Paper Clips
1 bin: 23.8 x 101, 238 grams
1 crate: 23.8 x 103, 23,800 grams
1 basket: 23.8 x 102, 2,380 grams
1 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 g
3 x 23.8 x 102 = 3 x 2,380 = 7,140 g
14,850 + 7,140 = 21,990
21,990 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 g
47,600 + 495 = 48,095 g
3 x 16.5 x 101 = 3 x 165 = 495g
48,095 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 g
4 x 23.8 x 103 = 4 x 23,800 g = 95,200 g
330,000 + 95,200 g = 425,200 g
425,200 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 g
6,410,000 x 0.002 = $12,820.00
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
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
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
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 Mass
to Find 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
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Between the Stars
Common Core Standard:
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.
Additional/Supporting Standards:
5.NBT.1, 5.NBT.3 Understand 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 102
7 x 104 ÷ 102
7 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?
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE

Why does this relationship make sense?
Have students complete the second page of the activity sheet. Pose the same questions.
Possible Misconceptions/Suggestions:
Possible Misconceptions
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
Suggestions
• 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:
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 A
Star B
Star C
Star D
Star E
3.2 x 102 km from the sun
103 km farther from the sun than Star A
104 km closer to the sun than Star B
101 km farther from the sun than Star B
102 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 F
Star G
Star H
Star I
Star J
4.9 x 103 km from the sun
102 km farther from the sun than Star F
102 km closer to the sun than Star F
102 km farther from the sun than Star G
103 km closer to the sun than Star G
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
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 A
Star B
Star C
Star D
Star E
3.2 x 102 km from the sun
103 km farther from the sun than Star A
104 km closer to the sun than Star B
101 km farther from the sun than Star B
102 km farther to the sun than Star B
Based on the measurements listed above find the actual distance from the sun of each
star. Make sure you write an equation and show your work.
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Between the Stars (Part 2)
There are five stars that are seen in a straight line through a telescope. The stars are
the following distance from the sun:
Star F
Star G
Star H
Star I
Star J
4.9 x 103 km from the sun
102 km farther from the sun than Star F
102 km closer to the sun than Star F
102 km farther from the sun than Star G
103 km closer to the sun than Star G
Based on the measurements listed above find the actual distance from the sun of each
star. Make sure you write an equation and show your work.
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE
Formative Instructional and Assessments Tasks
on Number and Operations in Base Ten 1 & 2






5.NBT.1
Task 1: Value of a Digit
Task 2: Danny & Delilah
Task 3: Value of a Digit
Task 4: Comparing Digits
5.NBT.2
Task 1: Veronica’s Statement
Task 2: Distance from the Sun
These tasks can be found on the
5th Grade Assessment Page
located on the Mathematics Wiki:
http://3-5cctask.ncdpi.wikispaces.net/home
NC DEPARTMENT OF PUBLIC INSTRUCTION
FIFTH GRADE