Develop Skills and Strategies
Lesson 15
Part 1: Introduction
CCSS
Numerical Expressions with Exponents
6.EE.A.1
In previous lessons, you learned about powers of 10. Take a look at this problem.
A sports store orders a box of table tennis balls. The dimensions of the box are
10 inches by 10 inches by 10 inches. Inside the box, there are 6 layers of table
tennis balls. Each layer is made up of 6 rows and 6 columns of table tennis balls.
How many table tennis balls are in the box?
Explore It
Use the math you know to solve the problem.
Write a multiplication expression to represent the volume of the box. Write the expression you wrote for the volume of the box using exponents. Write a multiplication expression to represent the number of table tennis balls in one
layer of the box. Write a multiplication expression to represent the number of table tennis balls in all
6 layers of the box. Write an expression using exponents to represent the total number of table tennis
balls in the box. Explain how you could find the number of table tennis balls in the box.
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L15: Numerical Expressions with Exponents
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Part 1: Introduction
Lesson 15
Find Out More
You already know that multiplication is a shorter way to write repeated addition. You use
exponents to write repeated multiplication in a shorter way.
Problem
Shorter way to write it
How to read it
Repeated Addition
5151515
Repeated Multiplication
5•5•5•5
5•4
​5​4​
5 times 4
5 raised to the fourth power
Numbers raised to the second or third power are often read in specific ways.
​52​ ​is read “five squared.”
5
​ ​3​is read “five cubed.”
Expressions written with exponents are called exponential expressions.
The number being multiplied by itself is called the base.
The exponent shows how many times you multiply the base by itself.
5 is the base.
3 is the exponent.
​53​ ​
The base of an exponential expression can be any kind of number.
​72​ ​ 5 7 • 7
5
1
1​​ ​ ··13 ​  2​​ ​ 5 ··​ 3 ​ • ··​ 13 ​ • ··​ 13 ​• ··​ 13 ​ • ··​ 13 ​
1​​ 0.2 23​​ ​
5 0.2 • 0.2 • 0.2
When you multiply measurements with units, the units are also multiplied.
In the example on the previous page, to find the volume of the box, you multiply
10 inches • 10 inches • 10 inches.
10 inches • 10 inches • 10 inches 5 ​10​3​​inches​3​
5 10 • 10 • 10 ​inches​3​
5 1,000 ​in.​3​
This is why area is measured in square units and why volume is measured in cubic units.
Reflect
1 Is ​2​7​equal to 2 ? 7? Explain.
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Part 2: Modeled Instruction
Lesson 15
Read the problem below. Then explore different ways to understand how to write
and evaluate expressions with exponents.
Julie’s brother says that instead of paying her the $40 he owes her, he will give her
$2 today and double the amount she has each day for 6 days. Should Julie accept
her brother’s offer? Why or why not?
Model It
Use multiplication to represent the problem.
Find the amount of money Julie has each day and then double it to find the amount she
has the next day.
Day 1
2
Day 2
2•254
Day 3
4•258
Day 4
8 • 2 5 16
Day 5
16 • 2 5 32
Day 6
32 • 2 5 64
Model It
Represent the problem with repeated multiplication.
Each day, Julie gets two times the amount of money she got the previous day.
Day
1
2
3
4
5
6
146
Amount of Money Julie
Has from Her Brother
2
2•2
2•2•2
2•2•2•2
2•2•2•2•2
2•2•2•2•2•2
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Part 2: Guided Instruction
Lesson 15
Connect It
Now you will solve the problem from the previous page by writing a power.
2 Look at the first Model It on the previous page. Why do you double 8 to find the
amount of money Julie will have on Day 4? 3 Look at the second Model It. How can you tell that the amount from the previous day
is being doubled? 4 In the second Model It, which expression represents the amount of money Julie will
have on Day 6? 5 How can you write an expression for the amount of money Julie has on Day 6 using
exponents? 6 How much money will Julie have on Day 6? Should she take her brother’s offer? Explain.
7 Matt thinks that if Julie starts with $2, she will have $12 after 6 days. Explain what
Matt is doing wrong.
Try It
Use what you just learned to solve these problems. Show your work on a separate
sheet of paper.
8 Maria is buying new carpet for her bedroom. Her bedroom is in the shape of a square
and the length of each side is 12 feet. Write and simplify an exponential expression to
find how much carpet she needs. 9 Write and simplify a multiplication expression to represent 5
​ ​4.​ L15: Numerical Expressions with Exponents
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147
Part 3: Modeled Instruction
Lesson 15
Read the problem below. Then explore different ways to understand evaluating
expressions with exponents.
Michael, Morgan, and Keegan are in the same class. The teacher asks the students
to write and simplify the expression “6 plus 4 cubed.” Here are the expressions
each person wrote.
Michael
(6 1 4)3
Morgan
63 1 43
Keegan
6 1 43
Which students will get the correct answer? Which, if any, of the expressions are
equivalent? Explain.
Model It
Use the order of operations to simplify Michael’s expression.
First add 6 and 4. Then raise 10 to the third power.
6 1 4 5 10
103 5 10 • 10 • 10
Model It
Use the order of operations to simplify Morgan’s expression.
First find 63 and 43.63 5 6 • 6 • 6
43 5 4 • 4 • 4
5 216
5 64
Then add 63 and 43.216 1 64
Model It
Use the order of operations to simplify Keegan’s expression.
First find 43.43 5 4 • 4 • 4
Then add 6.
148
5 64
64 + 6
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Part 3: Guided Instruction
Lesson 15
Connect It
Now you will solve the problem from the previous page using the order of
operations with exponents.
10 Look at Michael’s expression. Why do you add 6 + 4 first?
11 Look at Morgan’s expression. Why do you simplify each exponential expression first?
12 The teacher says the expression equals 70. Which student wrote the expression
correctly? 13 How might the teacher have said the expression if she wanted students to write
Michael’s expression? 14 Corwin tries to evaluate ​42​ ​ • 5 1 6, as shown below.
​4​2​ • 5 1 6
5 ​42​ ​ • 11 5 16 • 11
5 176
Explain why his answer is incorrect. Try It
Use what you just learned to solve these problems.
2
​ ​  ​ ​ 15 Evaluate: ​ 6
3
··
16 Evaluate: 3 1 ​52​ ​ • 4 L15: Numerical Expressions with Exponents
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149
Part 4: Guided Practice
Lesson 15
Study the student model below. Then solve problems 17–19.
Student Model
The volume of 4 cubes
is 4 times the volume of
1 cube.
Write and evaluate an expression using exponents to find the
total volume of the 4 cubes shown below.
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
Look at how you could show your work using labeled equations.
Volume of 1 cube: 5 lwh
5 5 cm • 5 cm • 5 cm
5 ​53​ ​ ​cm​3​
Pair/Share
Is (4 • 5​)3​ ​equal to 4 • 53?
Why or why not?
What is the relationship
between the number of
the space and the power
that shows the number of
grains on that space?
Volume of 4 cubes: V 5 4 • 5 • 5 • 5 cm3
V 5 4 • 125 ​cm​3​
V 5 500 ​cm​3​
3
3
Solution: V 5 4 • ​5​ ;​ 500 c​m​ ​
17 In an old story, a man puts 2 grains of rice on the first space of a chess
board. He puts 4 grains on the second space, 8 grains on the third
space, and so on. Write and evaluate an expression with exponents
to find how many grains of rice the man puts on the tenth space.
Show your work.
Pair/Share
What do you think is
meant by the phrase
“exponential growth”?
150
Solution: L15: Numerical Expressions with Exponents
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Part 4: Guided Practice
18 Moira does one-half of a homework assignment on Monday.
On Tuesday, Wednesday, and Thursday, she does one-half of the
homework she has left over from the day before. Write and evaluate
an expression using exponents to find the fraction of the original
assignment Moira will have to do on Friday in order to completely
finish her homework.
Lesson 15
Try using a picture to
model the assignment.
Show your work.
Pair/Share
Solution: 3
19 Evaluate: ​ (6 1 2​)​   ​ 2 12 
​ 
5
··········
A​ 2 ​
5
··
B 90 ​  2 ​
5
··
If Moira kept on
doing one-half of the
remaining assignment
instead of finishing it on
Friday, would she ever
finish her homework?
How does the order of
operations apply to
evaluating this
expression?
C100
D 509 ​  3 ​
5
··
Jerry chose A as the correct answer. How did he get that answer?
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Pair/Share
Why does everyone
need to follow the order
of operations when
evaluating expressions?
151
Part 5: Common Core Practice
Lesson 15
Solve the problems.
1
2
 ​? 
Which best describes the first step in evaluating the expression 4 • }}}}
​  53 1 13 
(8 2 5​)​ ​
A simplify ​5​3​
B
multiply 4 • 53
C
subtract 8 2 5
D
add 53 1 1
In a science-fiction story, a spaceship travels 3 times faster each minute than it traveled
during the minute before. If the ship travels at 3 km/hr during its first minute of flight,
which expression shows the ship’s speed during the 15th minute?
A
3 • 15
B​15​3​
C​3​15​
D
3 152
3•3
From the choices on the left, write inside the box each expression equivalent to 34 3 32.
32 3 34
36
12 3 6
33 3 33
38
81 3 9
Expressions equivalent to 34 3 32.
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Part 5: Common Core Practice
4
Lesson 15
Look at the cube below.
5 in.
5 in. 5 in.
Which statement correctly calculates the volume of the cube? Choose True or False for each
statement.
5
A
Simplify the expression 5 1 5 1 5.
True
False
B
Simplify the expression 5 3 5 3 5.
True
False
C
Simplify the expression 3 3 5.
True
False
D
Simplify the expression 5 3 25.
True
False
E
Simplify the expression 53.
True
False
Write each of the numbers 1, 4, 9, 16, and 25 as a base raised to the second power. Explain
why these numbers sometimes are called “perfect squares.”
6 Trey knows that 3 1 4 5 4 1 3 and 3 • 4 5 4 • 3. He says that ​34​ ​ 5 ​43​ ​. Is Trey correct? Explain
your answer.
Self Check Go back and see what you can check off on the Self Check on page 143.
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153
Develop Skills and Strategies
Lesson 15
(Student Book pages 144–153)
Numerical Expressions with Exponents
Lesson Objectives
The Learning Progression
•Write numerical expressions involving
whole-number exponents.
In keeping with the Common Core goal of developing
deeper student understanding of expressions and
equations, this lesson extends students’ understanding
of writing and evaluating numeric expressions to
include powers with whole-number exponents.
Students will write numeric expressions. They will also
evaluate powers with whole-number exponents to solve
real-world problems.
•Evaluate numerical expressions involving
whole-number exponents.
Prerequisite Skills
•Use the order of operations to evaluate expressions.
•Fluently multiply with decimals using a standard
algorithm.
Teacher Toolbox
•Fluently multiply with fractions using a standard
algorithm.
Teacher-Toolbox.com
Prerequisite
Skills
•Evaluate powers of 10.
Ready Lessons
•Use whole-number exponents to denote powers
of 10.
Tools for Instruction
Interactive Tutorials
✓
✓
6.EE.A.1
✓
✓
✓
Vocabulary
base: the number that is multiplied by itself when it is
raised to a certain power
exponent: a number that shows how many times a
base is multiplied by itself
exponential expression: expressions written with
exponents
CCSS Focus
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–8 (see page A9 for full text)
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157
Part 1: Introduction
Lesson 15
At a Glance
Develop skills and strategies
Students read a word problem and answer questions
that connect repeated multiplication and powers.
Lesson 15
Part 1: introduction
ccss
6.ee.a.1
numerical expressions with exponents
Step By Step
in previous lessons, you learned about powers of 10. take a look at this problem.
•Tell students this page models how to use powers to
write numerical expressions.
A sports store orders a box of table tennis balls. The dimensions of the box are
10 inches by 10 inches by 10 inches. Inside the box, there are 6 layers of table
tennis balls. Each layer is made up of 6 rows and 6 columns of table tennis balls.
How many table tennis balls are in the box?
•Have students read the problem at the top of the page.
explore it
•Work through Explore It as a class.
use the math you know to solve the problem.
•Emphasize the value of 10 as 10 ? 10 ? 10 5 1,000
and not 10 ? 3 or 30.
Write a multiplication expression to represent the volume of the box.
•Point out to students that 10, the base, is being
multiplied by itself 3 times.
layer of the box.
3
10 • 10 • 10
Write the expression you wrote for the volume of the box using exponents.
103
Write a multiplication expression to represent the number of table tennis balls in one
6•6
Write a multiplication expression to represent the number of table tennis balls in all
6 layers of the box.
3
•Ask, What is the base number for 6 ? [6] Following the
pattern used to find the value of 103, what expression do
you predict to be equivalent to 63? [6 ? 6 ? 6]
6•6•6
Write an expression using exponents to represent the total number of table tennis
balls in the box.
63
Explain how you could find the number of table tennis balls in the box.
63 means multiply 6 by itself 3 times, so 63 5 6 • 6 • 6
•Ask student pairs or groups to explain how to
calculate the number of table tennis balls in the box.
ELL Support
3
144
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3
Display 10 and 6 for students numerically and in
words. Point to and label the parts of the power as
the vocabulary is used.
SMP Tip: Recognizing the structure of 10 as
10 ? 10 ? 10 prepares students to make use of the
same structure in determining the value of 63
(SMP 7).
3
Visual Model
Model powers with unit cubes.
Mathematical Discourse
•Why is it important to know the value of the base
and the exponent?
The base tells you which number is being
multiplied by itself. The exponent tells you
how many times to multiply the base by itself.
•Explain why 32 and 23 are not equivalent.
32 is 3 ? 3 5 9, and 23 is 2 ? 2 ? 2 5 8. Even
though the same two numbers are used, they
are not representing the same expression.
Materials: unit cubes
•Model 2 ? 2, 3 ? 3, and 4 ? 4 with unit cubes.
(2 rows of 2, etc).
•Ask, Which powers are being represented?
[22, 32, 42]
•Represent 2 ? 2 ? 2 ? 2 (two sets of 22). Stack the
sets. Model 33 and 43.
•Discuss what observations students have made.
158
L15: Numerical Expressions with Exponents
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Part 1: Introduction
Lesson 15
At a Glance
Students write numerical expressions using powers
with whole numbers. Students will explore a visual
model to understand exponents.
Step by Step
Part 1: introduction
Lesson 15
Find out More
You already know that multiplication is a shorter way to write repeated addition. You use
exponents to write repeated multiplication in a shorter way.
Repeated addition
5151515
Problem
shorter way to write it
how to read it
•Read Find Out More as a class.
Repeated Multiplication
5•5•5•5
5•4
54
5 times 4
5 raised to the fourth power
Numbers raised to the second or third power are often read in specific ways.
•Clarify the vocabulary. The power is the entire
expression, which includes the base and the
exponent. The base is the number which is being
multiplied by itself. The exponent represents the
number of times the base is multiplied by itself.
52 is read “five squared.”
Base has multiple meanings. Show the base of a
figure. Discuss how it is the support of the figure.
A good example is to show a cylinder and describe
how the circle is repeated to create the cylinder.
53
5 is the base.
3 is the exponent.
The base of an exponential expression can be any kind of number.
1 ··13 2
5
72 5 7 • 7
ELL Support
53 is read “five cubed.”
Expressions written with exponents are called exponential expressions.
The number being multiplied by itself is called the base.
The exponent shows how many times you multiply the base by itself.
1
5 ··3 • 1 • 1 • 1 • 1
3 ··
3 ··
3 ··
3
··
1 0.2 23
5 0.2 • 0.2 • 0.2
When you multiply measurements with units, the units are also multiplied.
In the example on the previous page, to find the volume of the box, you multiply
10 inches • 10 inches • 10 inches.
10 inches • 10 inches • 10 inches 5 103 inches3
5 10 • 10 • 10 inches3
5 1,000 in.3
This is why area is measured in square units and why volume is measured in cubic units.
Reflect
1 Is 27 equal to 2 ? 7? Explain.
no; 27 is 2 multiplied by itself 7 times, or 2 • 2 • 2 • 2 • 2 • 2 • 2 5 128,
Concept Extension
but 2 • 7 5 14.
Model exponents.
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145
Materials: counters
•Tell students you will create a model to show 24.
•Use chips to show 22 or 2 ? 2. Ask, What happens
to the number of counters when I show the
next power of 2 or 23? [The number of counters is
multiplied by 2.] Show this new number of
counters. [8 counters]
4
•Repeat this question and process for 2 . Show this
new number of counters. [16 counters]
Real-World Connection
Encourage students to think about everyday
situations where people may need to write and
evaluate expressions with powers. Have volunteers
share their ideas.
Examples: sewing (material for a square quilt),
volume of a cubic container, epidemiologist (to study
the growth and spread of disease)
•Ask students to predict the number of counters
for 21. Have a volunteer show the model to the
class and explain their reasoning. [The number of
counters is 2. Following the model backwards, the
number of counters is divided by two, so 4
divided by 2 is 2 counters, for 21.]
•Ask students to predict the number of counters
for 20. Have a volunteer show this model and
explain. [The number of counters is one.
Following the model backwards, the number of
counters is divided by two, so 2 divided by 2
would be 1 counter for 20.]
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159
Part 2: Modeled Instruction
Lesson 15
At a Glance
Part 2: Modeled instruction
Students write and use powers to solve a real-world
problem. Students will explore visual models to
understand exponents.
Read the problem below. then explore different ways to understand how to write
and evaluate expressions with exponents.
Julie’s brother says that instead of paying her the $40 he owes her, he will give her
$2 today and double the amount she has each day for 6 days. Should Julie accept
her brother’s offer? Why or why not?
Step By Step
•Read the problem at the top of the page as a class.
Model it
use multiplication to represent the problem.
•Read the first Model It. Ask, What do you notice about
the amount of money Julie has each day? [The amount
doubles each day.]
Find the amount of money Julie has each day and then double it to find the amount she
has the next day.
Day 1
2
•Direct students to the second Model It. Ask, Do you
see a pattern in the number of 2s as we go down the
rows of the table? If so, describe it. [Each row adds one
more 2.]
Day 2
2•254
Day 3
4•258
Day 4
8 • 2 5 16
Day 5
16 • 2 5 32
Day 6
32 • 2 5 64
Model it
Represent the problem with repeated multiplication.
Each day, Julie gets two times the amount of money she got the previous day.
Day
1
2
3
4
5
6
146
160
Lesson 15
amount of Money julie
has from her brother
2
2•2
2•2•2
2•2•2•2
2•2•2•2•2
2•2•2•2•2•2
L15: Numerical Expressions with Exponents
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L15: Numerical Expressions with Exponents
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Part 2: Guided Instruction
Lesson 15
At a Glance
Students revisit the problem on page 146 to write and
evaluate numerical expressions using whole numbers in
a real-world application.
Part 2: guided instruction
Lesson 15
connect it
now you will solve the problem from the previous page by writing a power.
2 Look at the first Model It on the previous page. Why do you double 8 to find the
Step By Step
•Read Connect It as a class. Point out that the
questions refer to the problem from page 146.
•Review the meaning of base (the number being
multiplied by itself), exponent (the number of times
the base is being multiplied by itself), and power
(the expression which includes the base and the
exponent).
•Have students work the Try It problems by
themselves or in small groups.
SMP Tip: Mathematical vocabulary allows
students to share ideas clearly and precisely
(SMP 6) in discussions and explanations of their
reasoning. Use the words base, exponent, and power
regularly and share your expectation that students
use them in class.
amount of money Julie will have on Day 4? Possible answer: the total amount of
money julie had on Day 3 was $8. so you double this amount to get $16, the
amount of money she has on Day 4.
3 Look at the second Model It. How can you tell that the amount from the previous day
is being doubled? When you double an amount, you multiply the amount by 2.
the first 2 in each expression is being multiplied by the amount of money
julie has from the day before.
4 In the second Model It, which expression represents the amount of money Julie will
have on Day 6? 2 • 2 • 2 • 2 • 2 • 2
5 How can you write an expression for the amount of money Julie has on Day 6 using
exponents?
$64; yes, julie should take her brother up on his offer, because she will get
her $40 back and an additional $24.
7 Matt thinks that if Julie starts with $2, she will have $12 after 6 days. Explain what
Matt is doing wrong.
Possible response: Matt multiplied 2 • 6 to get 12 instead of finding 26.
try it
use what you just learned to solve these problems. show your work on a separate
sheet of paper.
8 Maria is buying new carpet for her bedroom. Her bedroom is in the shape of a square
and the length of each side is 12 feet. Write and simplify an exponential expression to
2
2
find how much carpet she needs. (12 feet) 5 12 feet • 12 feet 5 144 feet ;
Maria will need 144 ft2 of carpet.
9 Write and simplify a multiplication expression to represent 54.
Use a table to evaluate exponents.
•Create a 3-column table. Write 52 in the left
column. Ask a volunteer to write an equivalent
expression in the middle column and another
volunteer to record the whole-number value of the
expression in the right column.
5 • 5 • 5 • 5 5 625
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Concept Extension
26
6 How much money will Julie have on Day 6? Should she take her brother’s offer? Explain.
Copying is not permitted.
147
Try It Solutions
8Solution: 122 5 12 3 12 5 144 square feet
9Solution: 5 3 5 3 5 3 5 5 625
ERROR ALERT: Students who write 5 3 4 5 20
misunderstand the meaning of base and exponent.
•Repeat this process for 53, 54, and 55.
•Ask students to describe patterns they may see in
the table. [As the exponent increases, the number
of 5s in the expression increases; values for each
expression are multiplied by 5 as the exponent
increases; values are divided by 5 as the exponent
decreases.]
•Repeat with other powers. Discuss if the patterns
appear to hold true.
•Ask students to compare their tables. Ask, What
mathematical conjectures can you make? [The
powers with a large base increase in value at a
faster rate than those with a smaller base.]
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Part 3: Modeled Instruction
Lesson 15
At a Glance
Part 3: Modeled instruction
Students evaluate expressions which include whole
numbers as exponents by applying the order of
operations. Students will explore visual models to
understand exponents.
Lesson 15
Read the problem below. then explore different ways to understand evaluating
expressions with exponents.
Michael, Morgan, and Keegan are in the same class. The teacher asks the students
to write and simplify the expression “6 plus 4 cubed.” Here are the expressions
each person wrote.
Step By Step
Michael
(6 1 4)3
•Read the problem at the top of the page as a class.
Morgan
63 1 43
keegan
6 1 43
Which students will get the correct answer? Which, if any, of the expressions are
equivalent? Explain.
•Read the first Model It. Have a volunteer read the
example aloud or describe the order of operations
involved in simplifying Michael’s expression.
Model it
use the order of operations to simplify Michael’s expression.
First add 6 and 4.
Then raise 10 to the third power.
•Read the second Model It. Have a volunteer read the
example aloud or describe the order of operations
involved in simplifying Morgan’s expression.
6 1 4 5 10
103 5 10 • 10 • 10
Model it
use the order of operations to simplify Morgan’s expression.
First find 63 and 43.
63 5 6 • 6 • 6
43 5 4 • 4 • 4
5 216
•Read the third Model It. Have a volunteer read the
example aloud or describe the order of operations
involved in simplifying Keegan’s expression.
Then add 63 and 43.
5 64
216 1 64
Model it
use the order of operations to simplify keegan’s expression.
First find 43.
43 5 4 • 4 • 4
5 64
Then add 6.
148
Hands-On Activity
Use cubes to understand exponents and
model volume.
Materials: unit cubes
• Distribute unit cubes to groups of 2–4 students.
•Have students create models for 13, 23, 33, and 43.
Then ask volunteers to share their models.
•Lead a discussion with students about how their
models connect the concept of exponents with
that of volume.
•Guiding questions for students may be: What is
the smallest cube? [The smallest square is one cube
itself.] What is the largest cube? [This will depend
on the number of cubes available.] How can the
volume of each cube be determined? [By counting
the individual cubic units or writing a power.]
How can the volume of each cube be written as a
power? [Possible powers include 13, 23, 33, 43, etc.]
162
64 + 6
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Mathematical Discourse
•Use the numbers 2, 3, and 6 to write mathematical
expressions that include the exponents as whole
numbers. You may use any mathematical operations
with which you are familiar.
Display student expressions for all to see.
•Can you predict which expressions will have a
greater value without performing the mathematical
operations? Explain how you know.
Predictions will vary. Expressions with large
exponents will likely have greater values than
those with small exponents or no exponents.
•What expression can you create to get the smallest
value using these 3 numbers? What expression can
you create to get the largest value using these
3 numbers?
Expressions will vary. Encourage students to
offer new expressions during the discussion as
their understanding of exponents change.
L15: Numerical Expressions with Exponents
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Part 3: Guided Instruction
Lesson 15
At a Glance
Students revisit the problem on page 148.
Step By Step
•Read Connect It as a class. Be sure to point out that
the questions refer to the problem from page 148.
•Emphasize the importance of the order of operations.
Review the correct order of operations with students:
parentheses, exponents, multiplication and division
(left to right rule), and finally addition and
subtraction (left to right rule).
•For problem 14, discuss as a class Corwin’s error and
evaluate correctly. Have students show each step and
explain why each step was completed in that order.
2
Part 3: guided instruction
Lesson 15
connect it
now you will solve the problem from the previous page using the order of
operations with exponents.
10 Look at Michael’s expression. Why do you add 6 + 4 first?
use the order of operations and do what’s inside the parentheses first.
11 Look at Morgan’s expression. Why do you simplify each exponential expression first?
use the order of operations and simplify exponents before adding.
12 The teacher says the expression equals 70. Which student wrote the expression
correctly? keegan
13 How might the teacher have said the expression if she wanted students to write
Michael’s expression? Possible answer: the quantity 6 1 4 cubed.
14 Corwin tries to evaluate 42 • 5 1 6, as shown below.
42 • 5 1 6
5 42 • 11
5 16 • 11
5 176
Explain why his answer is incorrect. he added 5 and 6 first instead of simplifying
the exponent first. the correct answer is 86.
4 ?516
try it
16 ? 5 1 6 — (exponents come first)
use what you just learned to solve these problems.
2
12
15 Evaluate: 6
3
··
80 1 6 — (multiplication comes next)
86 — (addition is the only operation left)
ELL Support
Provide students with a graphic organizer to
remember the order of operations.
16 Evaluate: 3 1 52 • 4
103
149
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Try It Solutions
15 Solution: 12; 6 ? 6 5 36. 36 divided by 3 is 12.
Students may write ​ 36  ​ . Remind them to simplify.
3
··
16 Solution: 103; Apply the order of operations.
52 5 25. 25 ? 4 is 100. 3 1 100 5 103.
ERROR ALERT: Students who wrote 112 added
before multiplying, thus violating the order of
operations.
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163
Part 4: Guided Practice
Lesson 15
Part 4: guided Practice
Lesson 15
study the student model below. then solve problems 17–19.
Student Model
The volume of 4 cubes
is 4 times the volume of
1 cube.
Write and evaluate an expression using exponents to find the
total volume of the 4 cubes shown below.
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
Lesson 15
18 Moira does one-half of a homework assignment on Monday.
On Tuesday, Wednesday, and Thursday, she does one-half of the
homework she has left over from the day before. Write and evaluate
an expression using exponents to find the fraction of the original
assignment Moira will have to do on Friday in order to completely
finish her homework.
5 cm
5 cm
1 ··12 2
4
51•1•1•15 1
2 ··
2 ··
2 ··
2
··
16
···
Look at how you could show your work using labeled equations.
Pair/share
volume of 1 cube: 5 lwh
5 5 cm • 5 cm • 5 cm
5 53 cm3
Pair/share
Is (4 • 5)3 equal to 4 • 53?
Why or why not?
What is the relationship
between the number of
the space and the power
that shows the number of
grains on that space?
volume of 4 cubes: V 5 4 • 5 • 5 • 5 cm3
V 5 4 • 125 cm3
V 5 500 cm3
3
Try using a picture to
model the assignment.
Show your work.
5 cm
5 cm
Part 4: guided Practice
Solution:
3
Solution: V 5 4 • 5 ; 500 cm
17 In an old story, a man puts 2 grains of rice on the first space of a chess
Moira will need to do 1 of her original assignment.
16
···
3
19 Evaluate: (6 1 2) 2 12
5
··········
2
a
5
··
b 90 2
5
··
How does the order of
operations apply to
evaluating this
expression?
board. He puts 4 grains on the second space, 8 grains on the third
space, and so on. Write and evaluate an expression with exponents
to find how many grains of rice the man puts on the tenth space.
c
Show your work.
Jerry chose a as the correct answer. How did he get that answer?
10
2 5 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 5 1,024
If Moira kept on
doing one-half of the
remaining assignment
instead of finishing it on
Friday, would she ever
finish her homework?
100
D 509 3
5
··
he did not follow the order of operations. he cubed 2 before
adding 6 and 2.
Pair/share
Why does everyone
need to follow the order
of operations when
evaluating expressions?
Pair/share
What do you think is
meant by the phrase
“exponential growth”?
150
Solution:
210 5 1,024; 1,024 grains of rice
L15: Numerical Expressions with Exponents
151
L15: Numerical Expressions with Exponents
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At a Glance
Solutions
Students write and evaluate numerical expressions
involving whole-number exponents.
Ex The example shows how to write and evaluate a
numerical expression involving whole-number
exponents.
Step by Step
•Ask students to solve the problems individually and
interpret their answers in the context of the
problems.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
17 Solution: 1,024 grains of rice; Students could solve
the problem by evaluating the following:
210 5 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2. (DOK 2)
18 Solution: Moira will need to do ​ 1  ​ of her original
16
··
homework; Students could solve the problem by
evaluating ​1 ​  1 ​  24​ 5 ​ 1 ​ ? ​ 1 ​ ? ​ 1 ​ ? ​ 1 ​ . (DOK 2)
2
··
2
··
2
··
2
··
2
··
19 Solution: C; Jerry did not follow the order of
operations, instead cubing 2 before adding 6 and 2.
Explain to students why the other two answer
choices are not correct:
  ​  does not simplify to
B is not correct because ​ 500
5
····
90 ​ 2 ​.
5
··
D is not correct because the entire numerator
should be divided by 5. (DOK 3)
164
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Part 5: Common Core Practice
Part 5: common core Practice
Lesson 15
2
Part 5: common core Practice
4
Solve the problems.
1
Lesson 15
Look at the cube below.
5 in.
53 1 1 ?
Which best describes the first step in evaluating the expression 4 • }}}}
(8 2 5)3
A simplify 53
B
multiply 4 • 53
C
subtract 8 2 5
D
add 53 1 1
5 in. 5 in.
Which statement correctly calculates the volume of the cube? Choose True or False for each
statement.
In a science-fiction story, a spaceship travels 3 times faster each minute than it traveled
during the minute before. If the ship travels at 3 km/hr during its first minute of flight,
which expression shows the ship’s speed during the 15th minute?
A
3 • 15
B
153
C
315
D
3•3
Lesson 15
5
A
Simplify the expression 5 1 5 1 5.
B
Simplify the expression 5 3 5 3 5.
C
Simplify the expression 3 3 5.
D
Simplify the expression 5 3 25.
E
Simplify the expression 53.
True
3 False
True
3 False
3 True
3 True
3 True
False
False
False
Write each of the numbers 1, 4, 9, 16, and 25 as a base raised to the second power. Explain
why these numbers sometimes are called “perfect squares.”
Possible answer: 12, 22, 32, 42, 52; each number can be written as a whole-number
squared.
3
From the choices on the left, write inside the box each expression equivalent to 34 3 32.
32 3 34
36
12 3 6
33 3 33
38
81 3 9
Expressions equivalent to 34 3 32.
6
32 3 34
36
33 3 33
81 3 9
Trey knows that 3 1 4 5 4 1 3 and 3 • 4 5 4 • 3. He says that 34 5 43. Is Trey correct? Explain
your answer.
Possible answer: no; 34 is 3 multiplied by itself 4 times, 3 • 3 • 3 • 3 5 81. in contrast,
43 is 4 multiplied by itself 3 times, 4 • 4 • 4 5 64.
self check Go back and see what you can check off on the Self Check on page 143.
152
L15: Numerical Expressions with Exponents
L15: Numerical Expressions with Exponents
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At a Glance
Students write and evaluate numeric expressions which
include powers with whole-number exponents to solve
problems that might appear on a mathematics test.
Solutions
1Solution: C: Apply the order of operations to
evaluate the parentheses first, (8 2 5). (DOK 2)
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153
4Solution: A False; B True; C False; D True; E True
(DOK 2)
5Solution: 12, 22, 32, 42, 52; Each number can be
written as a whole number squared. (DOK 3)
6Solution: No; 34 is 3 multiplied by itself 4 times,
3 ? 3 ? 3 ? 3 5 81. In contrast, 43 is 4 multiplied by
itself 3 times, 4 ? 4 ? 4 5 64. (DOK 3)
2Solution: C: The spaceship is going “3 times” faster
each time (the base). This happens for 15 minutes
(the exponent); 315 (DOK 2)
3Solution: 32 3 34, 36, 33 3 33, 81 3 9; Evaluate each
expression to find equivalent expressions. (DOK 1)
L15: Numerical Expressions with Exponents
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165
Differentiated Instruction
Lesson 15
Assessment and Remediation
•Ask students to evaluate 5 1 72 ? (8 2 6). [5 1 49 ? 2 5 5 1 98 5 103]
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Ask students to evaluate 28 2 (5 2 2)3.
[28 2 (3)3 5 28 2 27 5 1]
•If a student is still having difficulty, review this lesson, and consult Ready Instruction, Level 5, Lesson 19.
If the error is . . .
Students may . . .
To remediate . . .
108
not have followed the
order of operations.
Review the order of operations; parentheses, exponents,
multiplication and division (follow the left to right rule), and finally
addition and subtraction (follow the left to right rule). Explain that
8 2 6 should be evaluated first. The difference, 2, should then be
multiplied by 49 (from 72). Finally, add 5.
33
have evaluated the
power incorrectly.
Reteach the meaning of powers; 72 is equivalent to 7 ? 7, not 7 1 7 or
7 ? 2. Model exponential growth to see how the multiplication is
growing by using counters or an area model.
38
not have followed the
order of operations AND
additionally evaluated
the power incorrectly.
Provide a graphic organizer as a reference for the order of
operations. Ask students to create a model to represent the power
being evaluated.
have made a simple
calculator error.
Consider the use of a multiplication table or basic calculator.
Any other
answer
Hands-On Activity
Challenge Activity
•Revisit page 150, problem 17 with students.
•Revisit page 150, problem 17 with students.
•Ask, How would the problem change if the man put
3 grains of rice on the first space, 9 grains on the
second space, 27 grains on the third space, and so on?
How many grains of rice would be on space 11?
•Tell students, A friend of the man wants to follow a
similar pattern by putting 4 grains of rice on the first
space, 16 grains of rice on the second space, 64 grains
of rice on the third space, and so on.
•Ask students to use a table, numeric expressions
involving exponents, and/or a model to support
their solution. [The model should support 311 as
their resulting solution.]
•Now say, Compare and contrast the two friends by
creating a table, using numerical expressions
involving exponents, and modeling the growth.
Explain to the friends the observations you have
made. [Student observations should indicate the
friend used powers of 4, while the man used
powers of 2. Observations should also indicate
that the friend’s pattern grows at a much faster
rate.]
•Note: Students will question the effectiveness of
the model as it grows rapidly. Ask students to
provide a model to show initial growth.
•Note: The model will grow rapidly. Ask students
to provide a model to show initial growth.
166
L15: Numerical Expressions with Exponents
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