lesson 15 ccls numerical expressions with...

©Curriculum Associates, LLC Copying is not permitted.L15: Numerical Expressions with Exponents144

Lesson 15 Part 1: Introduction

Numerical Expressions with Exponents

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.

CCLS6.EE.1

©Curriculum Associates, LLC Copying is not permitted.145L15: Numerical Expressions with Exponents

Lesson 15Part 1: Introduction

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 Repeated MultiplicationProblem 5 1 5 1 5 1 5 5 ? 5 ? 5 ? 5

Shorter way to write it 5 ? 4 5 4

How to read it 5 times 4 5 raised to the fourth power

Numbers raised to the second or third power are often read in specific ways.

5 2 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.

5 3

3 is the exponent.

The base of an exponential expression can be any kind of number.

7 2 5 7 ? 7 1  1 ·· 3 2 5 5 1 ·· 3 ? 1 ·· 3 ? 1 ·· 3 ? 1 ·· 3 ? 1 ·· 3 1 0.2 2 3 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.

Lesson 15

©Curriculum Associates, LLC Copying is not permitted.

L15: Numerical Expressions with Exponents146

Part 2: Modeled Instruction

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 Day 2 Day 3 Day 4 Day 5 Day 62 2 ? 2 5 4 4 ? 2 5 8 8 ? 2 5 16 16 ? 2 5 32 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.

DayAmount of Money Julie

Has from Her Brother1 22 2 ? 23 2 ? 2 ? 24 2 ? 2 ? 2 ? 25 2 ? 2 ? 2 ? 2 ? 26 2 ? 2 ? 2 ? 2 ? 2 ? 2

Lesson 15


Part 2: Guided Instruction

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 .

Lesson 15



Part 3: Modeled Instruction

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 Morgan Keegan(6 1 4)3 63 1 43 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. 6 1 4 5 10

Then raise 10 to the third power. 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

5 64

Then add 6. 64 + 6

Lesson 15


Part 3: Guided Instruction

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 4 2 ? 5 1 6, as shown below.

4 2 ? 5 1 65 4 2 ? 11 5 16 ? 115 176

Explain why his answer is incorrect.

Try It

Use what you just learned to solve these problems.

15 Evaluate: 6 2 ·· 3

16 Evaluate: 3 1 5 2 ? 4

Student Model

Lesson 15



Part 4: Guided Practice

What do you think is meant by the phrase “exponential growth”?

Pair/Share

Is (4 ? 5 ) 3 equal to 4 ? 53? Why or why not?

Pair/Share

Study the student model below. Then solve problems 17–19.

Write and evaluate an expression using exponents to find the

total volume of the 4 cubes shown below.

5 cm

5 cm5 cm

5 cm

5 cm5 cm

5 cm

5 cm5 cm

5 cm

5 cm5 cm

Look at how you could show your work using labeled equations.

Solution:

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.

Solution:

Volume of 1 cube: 5 lwh 5 5 cm ? 5 cm ? 5 cm 5 5 3 cm 3

Volume of 4 cubes: V 5 4 ? 5 3 cm 3 V 5 4 ? 125 cm 3 V 5 500 cm 3

V 5 4 ? 5 3 ; 500 c m 3

The volume of 4 cubes is 4 times the volume of 1 cube.

What is the relationship between the number of the space and the power that shows the number of grains on that space?

Lesson 15


Part 4: Guided Practice

If Moira kept on doing one-half of the remaining assignment instead of finishing it on Friday, would she ever finish her homework?

Pair/Share

Why does everyone need to follow the order of operations when evaluating expressions?

Pair/Share

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.

Show your work.

Solution:

19 Evaluate: (6 1 2 ) 3 2 12 ·········· 5

A 2 ·· 5

B 90 2 ·· 5

C 100

D 509 3 ·· 5

Jerry chose A as the correct answer. How did he get that answer?

Try using a picture to model the assignment.

How does the order of operations apply to evaluating this expression?

Lesson 15



Part 5: Common Core Practice

Solve the problems. Mark your answers to problems 1–4 on the Answer Form to the right. Be sure to show your work.

1 Evaluate: 3 ? 1  1 } 3 2

2

A 1 } 3

B 1 } 2

C 1

D 2

2 Ruby paints 3 walls of her shed. Each wall measures 8 feet by 8 feet. Then she paints the door, which is 3 feet by 7 feet. What is the total area Ruby paints?

A 65 ft2

B 213 ft2

C 213 ft

D 65 ft

3 Which best describes the first step in evaluating the expression 4 ? 53 1 1 }}}} (8 2 5 ) 3 ?

A simplify 5 3

B multiply 4 ? 53

C subtract 8 2 5

D add 53 1 1

1 B C D

2 B C D

3 B C D

4 B C D

Answer Form

Number Correct 4

Lesson 15


Part 5: Common Core Practice

Self Check Go back and see what you can check off on the Self Check on page 143.

4 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 ? 3

5 Write the numbers 1, 4, 9, 16, and 25 as a base raised to an exponent. Explain why these numbers sometimes are called “perfect squares.”

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6 Trey knows that 3 1 4 5 4 1 3 and 3 ? 4 5 4 ? 3. He says that 3 4 5 4 3 . Is Trey correct? Explain your answer.

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numerical expressions with exponentsLesson 15

L15: Numerical Expressions with Exponents 147©Curriculum Associates, LLC Copying is not permitted.

(Student Book pages 144–153)

Lesson objectives

• Write numerical expressions involving whole-number exponents.

• 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.

• Fluently multiply with fractions using a standard algorithm.

• Evaluate powers of 10.

• Use whole-number exponents to denote powers of 10.

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

the LeaRning PRogRession

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.

ccLs Focus

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

stanDaRDs FoR MatheMaticaL PRactice: SMP 1–8 (see page A9 for full text)

Toolbox Teacher-Toolbox.com

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✓

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Prerequisite Skills 6.EE.1

Ready Lessons

Tools for Instruction

Interactive Tutorials ✓

L15: Numerical Expressions with Exponents148©Curriculum Associates, LLC Copying is not permitted.

Lesson 15Part 1: introduction

at a gLance

Students read a word problem and answer questions that connect repeated multiplication and powers.

steP by steP

• Tell students this page models how to use powers to write numerical expressions.

• Have students read the problem at the top of the page.

• Work through Explore It as a class.

• Emphasize the value of 103 as 10 ? 10 ? 10 5 1,000 and not 10 ? 3 or 30.

• Point out to students that 10, the base, is being multiplied by itself 3 times.

• Ask, What is the base number for 63? [6] Following the pattern used to find the value of 103, what expression do you predict to be equivalent to 63? [6 ? 6 ? 6]

• Ask student pairs or groups to explain how to calculate the number of table tennis balls in the box.

Display 103 and 63 for students numerically and in words. Point to and label the parts of the power as the vocabulary is used.

eLL support

sMP tip: Recognizing the structure of 103 as 10 ? 10 ? 10 prepares students to make use of the same structure in determining the value of 63 (SMP 7).

Model powers with unit cubes.

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.

visual Model

• 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.

Mathematical Discourse

©Curriculum Associates, LLC Copying is not permitted.L15: Numerical Expressions with Exponents144

Lesson 15 Part 1: introduction

numerical expressions with exponents

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.

ccLs6.ee.1

10 • 10 • 10

10 3

6 3

6 • 6

6 • 6 • 6

6 3 means multiply 6 by itself 3 times, so 6 3 5 6 • 6 • 6



at a gLance

Students write numerical expressions using powers with whole numbers. Students will explore a visual model to understand exponents.

steP by steP

• Read Find Out More as a class.

• 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.

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.

eLL support

Model exponents.

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]

• Repeat this question and process for 24. Show this new number of counters. [16 counters]

• 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.]

concept extension

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)

Real-World connection



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 Repeated Multiplication

Problem 5 1 5 1 5 1 5 5 ? 5 ? 5 ? 5

shorter way to write it 5 ? 4 5 4

how to read it 5 times 4 5 raised to the fourth power

Numbers raised to the second or third power are often read in specific ways.

5 2 is read “fi ve squared.” 5 3 is read “fi ve 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.

5 3

3 is the exponent.

The base of an exponential expression can be any kind of number.

7 2 5 7 ? 7 1  1 ·· 3 2 5 5 1 ·· 3 ? 1 ·· 3 ? 1 ·· 3 ? 1 ·· 3 ? 1 ·· 3 1 0.2 2 3 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.

no; 2 7 is 2 multiplied by itself 7 times, or 2 • 2 • 2 • 2 • 2 • 2 • 2 5 128,

but 2 • 7 5 14.

150 L15: Numerical Expressions with Exponents


Lesson 15Part 2: Modeled instruction

at a gLance

Students write and use powers to solve a real-world problem. Students will explore visual models to understand exponents.

steP by steP

• Read the problem at the top of the page as a class.

• Read the first Model It. Ask, What do you notice about the amount of money Julie has each day? [The amount doubles each day.]

• 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.]

Lesson 15



Part 2: Modeled instruction

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 Day 2 Day 3 Day 4 Day 5 Day 6

2 2 ? 2 5 4 4 ? 2 5 8 8 ? 2 5 16 16 ? 2 5 32 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.

Dayamount of Money julie

has from her brother

1 2

2 2 ? 2

3 2 ? 2 ? 2

4 2 ? 2 ? 2 ? 2

5 2 ? 2 ? 2 ? 2 ? 2

6 2 ? 2 ? 2 ? 2 ? 2 ? 2


Lesson 15Part 2: guided instruction

at a gLance

Students revisit the problem on page 146 to write and evaluate numerical expressions using whole numbers in a real-world application.

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.

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.

• 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.]

concept extension

Lesson 15


Part 2: guided instruction

connect it

now you will solve the problem from the previous page by writing a power.

2 Look at the fi rst Model It on the previous page. Why do you double 8 to fi nd 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 off er? 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 fi nd how much carpet she needs.

9 Write and simplify a multiplication expression to represent 5 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.

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.

$64; yes, julie should take her brother up on his offer, because she will get

her $40 back and an additional $24.

2 • 2 • 2 • 2 • 2 • 2

2 6

Possible response: Matt multiplied 2 ? 6 to get 12 instead of finding 2 6 .

5 ? 5 ? 5 ? 5 5 625

(12 feet)2 5 12 feet ? 12 feet 5 144 feet2;

Maria will need 144 ft2 of carpet.

tRy it soLutions

8 Solution: 122 5 12 3 12 5 144 square feet

9 Solution: 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.



Lesson 15Part 3: Modeled instruction

at a gLance

Students evaluate expressions which include whole numbers as exponents by applying the order of operations. Students will explore visual models to understand exponents.

steP by steP

• Read the problem at the top of the page as a class.

• Read the first Model It. Have a volunteer read the example aloud or describe the order of operations involved in simplifying Michael’s expression.

• Read the second Model It. Have a volunteer read the example aloud or describe the order of operations involved in simplifying Morgan’s expression.

• Read the third Model It. Have a volunteer read the example aloud or describe the order of operations involved in simplifying Keegan’s expression.

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.]

hands-on activity• 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.

Mathematical Discourse

Lesson 15



Part 3: Modeled instruction

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 Morgan keegan

(6 1 4)3 63 1 43 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. 6 1 4 5 10

Then raise 10 to the third power. 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

5 64

Then add 6. 64 + 6


Lesson 15Part 3: guided instruction

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.

42 ? 5 1 6

16 ? 5 1 6 — (exponents come first)

  80 1 6 — (multiplication comes next)

  86 — (addition is the only operation left)

tRy it soLutions

15 Solution: 12; 6 ? 6 5 36. 36 divided by 3 is 12.

Students may write 36 ··

 3 . Remind them to simplify.

16 Solution: 103; Apply the order of operations. 52 5 25. 25 ? 4 is 100. 3 1 100 5 103.

Provide students with a graphic organizer to remember the order of operations.

eLL support

ERROR ALERT: Students who wrote 112 added before multiplying, thus violating the order of operations.

Lesson 15


Part 3: guided instruction

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 fi rst?

11 Look at Morgan’s expression. Why do you simplify each exponential expression fi rst?

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 4 2 ? 5 1 6, as shown below.

4 2 ? 5 1 6

5 4 2 ? 11

5 16 ? 11

5 176

Explain why his answer is incorrect.

try it

use what you just learned to solve these problems.

15 Evaluate: 6 2 ·· 3

16 Evaluate: 3 1 5 2 ? 4

use the order of operations and do what’s inside the parentheses first.

use the order of operations and simplify exponents before adding.

keegan

Possible answer: the quantity 6 1 4 cubed.

he added 5 and 6 first instead of simplifying

the exponent first. the correct answer is 86.

12

103



Lesson 15Part 4: guided Practice

at a gLance

Students write and evaluate numerical expressions 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.

soLutions

Ex The example shows how to write and evaluate a numerical expression involving whole-number exponents.

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.

18 Solution: Moira will need to do 1 ··

 16

of her original

homework; Students could solve the problem by

evaluating 1  1 ··

 2 2 4 5 1

·· 

2 ? 1

·· 

2 ? 1

·· 

2 ? 1

·· 

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:

B is not correct because 500 ····

 5 does not simplify to

90 2 ··

 5 .

D is not correct because the entire numerator should be divided by 5.

Lesson 15


Part 4: guided Practice

If Moira kept on doing one-half of the remaining assignment instead of finishing it on Friday, would she ever finish her homework?

Pair/share

Why does everyone need to follow the order of operations when evaluating expressions?

Pair/share

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 fi nd the fraction of the original assignment Moira will have to do on Friday in order to completely fi nish her homework.

Show your work.

Solution:

19 Evaluate: (6 1 2 ) 3 2 12 ·········· 5

a 2 ·· 5

b 90 2 ·· 5

c 100

D 509 3 ·· 5

Jerry chose a as the correct answer. How did he get that answer?

Try using a picture to model the assignment.

How does the order of operations apply to evaluating this expression?

1  1 ·· 2 2 4 5 1 ·· 2 • 1 ·· 2 • 1 ·· 2 • 1 ·· 2 5 1 ··· 16

Moira will need to do 1 ··· 16 of her original assignment.

he did not follow the order of operations. he cubed 2 before

adding 6 and 2.

Student Model

Lesson 15



Part 4: guided Practice

What do you think is meant by the phrase “exponential growth”?

Pair/share

Is (4 ? 5 ) 3 equal to 4 ? 53? Why or why not?

Pair/share

study the student model below. then solve problems 17–19.

Write and evaluate an expression using exponents to find the

total volume of the 4 cubes shown below.

5 cm

5 cm5 cm

5 cm

5 cm5 cm

5 cm

5 cm5 cm

5 cm

5 cm5 cm

Look at how you could show your work using labeled equations.

Solution:

17 In an old story, a man puts 2 grains of rice on the fi rst 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 fi nd how many grains of rice the man puts on the tenth space.

Show your work.

Solution:

volume of 1 cube: 5 lwh

5 5 cm ? 5 cm ? 5 cm

5 5 3 cm 3

volume of 4 cubes: V 5 4 ? 5 3 cm 3

V 5 4 ? 125 cm 3

V 5 500 cm 3

V 5 4 ? 5 3 ; 500 c m 3

The volume of 4 cubes is 4 times the volume of 1 cube.

What is the relationship between the number of the space and the power that shows the number of grains on that space?

2 10 5 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 5 1,024

2 10 5 1,024; 1,024 grains of rice


Lesson 15Part 5: common core Practice

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.

steP by steP

• First, tell students that they will write and evaluate powers with whole-number exponents. Then have students read the directions and answer the questions independently. Remind students to fill in the correct answer choices on the Answer Form.

• After students have completed the Common Core Practice problems, review and discuss correct answers. Have students record the number of correct answers in the box provided.

soLutions

1 Solution: A; apply the order of operations and

simplify to 1 ··

 3 .

2 Solution: B; The area of each wall is 82. There are 3 walls; 3(82). The area of the door is (3 ? 7). So, 3(82) 1 (3 ? 7) 5 213 ft2.

3 Solution: C: Apply the order of operations to evaluate the parentheses first, (8 2 5).

4 Solution: C: The ship is going “3 times” faster each time (the base). This happens for 15 minutes (the exponent); 315

5 Solution: 12, 22, 32, 42, 52; Each number can be written as a whole number squared.

6 Solution: 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.

Lesson 15



Part 5: common core Practice

Solve the problems. Mark your answers to problems 1–4 on the Answer Form to the right. Be sure to show your work.

1 Evaluate: 3 ? 1  1 } 3 2

2

A 1 } 3

B 1 } 2

C 1

D 2

2 Ruby paints 3 walls of her shed. Each wall measures 8 feet by 8 feet. Then she paints the door, which is 3 feet by 7 feet. What is the total area Ruby paints?

A 65 ft2

B 213 ft2

C 213 ft

D 65 ft

3 Which best describes the fi rst step in evaluating the expression 4 ? 53 1 1 }}}} (8 2 5 ) 3 ?

A simplify 5 3

B multiply 4 ? 53

C subtract 8 2 5

D add 53 1 1

1 A B C D

2 A B C D

3 A B C D

4 A B C D

answer Form

numbercorrect 4

Lesson 15


Part 5: common core Practice

self check Go back and see what you can check off on the Self Check on page 143.

4 In a science-fi ction 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 fi rst minute of fl ight, which expression shows the ship’s speed during the 15th minute?

A 3 ? 15

B 15 3

C 3 15

D 3 ? 3

5 Write the numbers 1, 4, 9, 16, and 25 as a base raised to an exponent. 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 3 4 5 4 3 . Is Trey correct? Explain your answer.

Possible answer: 1 2 , 2 2 , 3 2 , 4 2 , 5 2 ; each number can be written as a whole-number

squared.

Possible answer: no; 3 4 is 3 multiplied by itself 4 times, 3 • 3 • 3 • 3 5 81. in contrast,

4 3 is 4 multiplied by itself 3 times, 4 • 4 • 4 5 64.

Lesson 15

assessment and Remediation

L15: Numerical Expressions with Exponents156©Curriculum Associates, LLC Copying is not permitted.

challenge activityhands-on activity• Revisit page 150, problem 17 with students.

• 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. 

• 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: The model will grow rapidly. Ask students to provide a model to show initial growth.

• 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?

• 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.]

• Note: Students will question the effectiveness of the model as it grows rapidly. Ask students to provide a model to show initial growth.

• 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.

Any other answer

have made a simple calculator error.

Consider the use of a multiplication table or basic calculator.

Differentiated instruction

lesson 15 ccls numerical expressions with...

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