Download Multiply 3-Digit and 4-Digit Numbers with Regrouping

LESSON
2.11
Multiply 3-Digit and 4-Digit
Numbers with Regrouping
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit
whole number, and multiply two two-digit numbers, using strategies
based on place value and the properties of operations. Illustrate and explain the
calculation by using equations, rectangular arrays, and/or area models.
MATHEMATICAL PRACTICES
MP4 Model with mathematics. MP8 Look for and express regularity in repeated reasoning.
F C R Coherence:
Standards Across the Grades
Before
Grade 4 After
3.OA.A.3 4.NBT.B.5 5.NBT.B.5
F C R Rigor:
Level 1: Understand Concepts....................Share and Show (
Checked Items)
Level 2: Procedural Skills and Fluency.......On Your Own
Level 3: Applications..................................Think Smarter and Go Deeper
Learning Objective
Use regrouping to multiply a multidigit number
by a 1-digit number.
Language Objective
Student pairs demonstrate to a small group how
you can use regrouping to multiply.
Materials
MathBoard
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 61J.
About the Math
Professional Development
Teaching for Depth
For most computations, students start with an estimate
to establish what a reasonable answer should be. With an
estimated answer, the estimate is the answer. Students
seldom check an estimated answer.
• Students should utilize other methods for determining
if their estimated answer is reasonable. For example, if
students have rounded to determine the answer, they
might determine a range of estimates (round up and
round down) to check their estimates.
Interactive Student Edition
Personal Math Trainer
Math on the Spot
iTools: Base-Ten Blocks
HMH Mega Math
Professional Development Videos
125A
Chapter 2
Daily
Daily Routines
Routines
Common Core
Common Core
Problem of the Day 2.11
Write an expression that shows how to
multiply 6 × 435 using place value and
expanded form. Possible answer: (6 × 400) +
(6 × 30) + (6 × 5)
1 ENGAGE
with the Interactive Student Edition
Essential Question
How can you use regrouping to multiply?
Making Connections
Invite students to tell you what they know about different methods
for multiplication.
What does it mean to use repeated addition to multiply 356 by 4?
Add 4 groups of 356 or 356 groups of 4. What does it mean to use
expanded form to multiply 356 by 4? Write the multidigit number in
expanded form and then perform multiplication. (4 × 300) + (4 × 50) +
(4 × 6)
Vocabulary
• Interactive Student Edition
• Multimedia eGlossary
Learning Activity
Fluency Builder
Common Core Fluency
Standard 4.NBT.B.4
Adding Whole Numbers Write the following
problems on the board. Have students
complete the problems and check their
answers with a partner.
327 + 4,618 = 4,945
26,806 + 7,495 = 34,301
583 + 6,279 = 6,862
Remind students who rewrite addition
problems in vertical form to align the
numbers by place value. Point out that
regrouping may or may not be needed in
every place of a problem.
Connect the story to the problem. Ask the following questions.
• How many loaves of bread does the diner order each week? 217
• How many weeks of deliveries will there be? 4
• What multiplication problem will let you determine the number
of loaves ordered in four weeks? 4 × 217
Literacy and Mathematics
Choose one or more of the following activities.
• Have students work in partners to generate a list of additional
questions that could be asked in relation to the scenario described
in the problem.
• Have students explain which details in the problem lead them to
conclude that multiplication should be used to solve it.
How can you use
regrouping to multiply ?
Lesson 2.11 125B
LESSON
2.11
2 EXPLORE
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers,
DO
NOT
EDIT--Changes
made
through
"File info"
using
strategies
based onmust
placebevalue
and
the properties
of operations. Illustrate and explain the calculation by using equations,
CorrectionKey=B
rectangular arrays, and/or area models.
ELL Strategy:
Multiply 3-Digit and 4-Digit Numbers
with Regrouping
Frontload
MATHEMATICAL PRACTICES
• Why do you estimate before you multiply?
I estimate so I will know what is a reasonable answer.
MP1 Make sense of problems and
persevere in solving them.
• Explain how you know if the estimate will
be less than or greater than the actual
product. Since one factor is rounded up and the
other stayed the same, the estimate will be greater than
the exact product.
Unlock
Unlock the
the Problem
Problem
Alley Spring, in Missouri, produces an
average of 567 million gallons of water
per week. How many million gallons of
water do the springs produce in 3 weeks?
Multiply. 3 × 567
Possible estimate:
1,800
600
= __
Estimate. 3 × __
THINK
RECORD
STEP 1
Multiply the ones.
2
56 7
× 3
_
1
21 ones
3 × 7 ones = _
Regroup the 21 ones.
Regroup the 21 ones
as 2 tens and 1 one.
STEP 2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Ron and Patty Thomas/Getty Images
Unlock the Problem
Number and Operations in Base
Ten—4.NBT.B.5
MATHEMATICAL PRACTICES
MP1, MP3, MP6
Essential Question How can you use regrouping to multiply?
Help students by anticipating and reducing
difficulties students might have knowing how
to respond to questions beginning with how
much and how many.
• Write on the board:
How many boys are in our class?
How many girls are there?
How much time do you take to eat your
lunch?
How much time do we spend doing math
problems each day?
• Pair students. Have students ask each other
these questions, pointing out which are
exact and which are estimates.
Lesson 2.11
Name
Multiply the tens.
22
567
× 3
_
01
18 tens
3 × 6 tens = _
Add the regrouped tens.
18 tens + 2 tens = 20 tens
Regroup the 20 tens.
Regroup 20 tens
as 2 hundreds 0 tens.
STEP 3
Multiply the hundreds.
22
567
15 hundreds
3 × 5 hundreds = _
Add the regrouped hundreds.
15 hundreds + 2 hundreds = 17 hundreds
× 3
__
1,701
17 hundreds is the same as
1 thousand 7 hundreds.
1,701 million gallons
So, Alley Spring produces _
of water in 3 weeks.
DEEPER
To extend their thinking, ask students about a
strategy for finding a closer estimate.
MP2 Reason abstractly and
quantitatively.
• How can you find an estimate that is closer
to the exact answer? I can round the greater
factor to the nearest ten.
• What is the new estimate? 3 × 570 = 1,710
MP3 Construct viable arguments and
critique the reasoning of others.
• How does the method for multiplying
a 3-digit number by a 1-digit number
compare to multiplying a 2-digit number
by a 1-digit number? Possible answer: In both
cases, each digit in the multi-digit number is multiplied
by the 1-digit number starting with the ones digit and
moving left, regrouping as needed. To multiply a 3-digit
number, there is also a hundreds digit to multiply.
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Lesson 2.11
Reteach
Name Lesson 2.11
Enrich
Name
Multiply 3-Digit and 4-Digit 
Numbers with Regrouping
Multiplication Mystery
There’s something mysterious in the water off the coast of Florida.
To discover what it is, find the products and use the decoder below.
The first letter has been done for you.
When​you​multiply​3-digit​and​4-digit​numbers,​you​may​need​to​regroup.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Estimate. Then find the product.
​​​​
​ $1,324​
​​​
3 ​​​​​   ​7
__
​
Step 1​ Estimate​the​product.​​​​​​​​​​​​​​​​$1,324​rounds​to​$1,000;​$1,000​3​7​5​$7,000.
1. Letter
2
Step 2  Multiply​the​4​ones​by​7.
​​ 
​​​​
​$1,324 
3   ​​​​​​7
Regroup​the​28​ones​as​2​tens​8​ones.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​
__
​
8
1​2
Step 3​ Multiply​the​2​tens​by​7.
​​ 
​​​​
​$1,324​
Add​the​regrouped​tens.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​
3   ​​​​​​7
__
Regroup​the​16​tens​as​1​hundred​6​tens.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​
68​
2​​1​2
Step 4​ Multiply​the​3​hundreds​by​7.
​$1,324​
​​​​
​​ 
Add​the​regrouped​hundred.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​
3   ​​​​​​7
__
Regroup​the​22​hundreds​as​2​thousands​
​
2​hundreds.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​ 268
2​​1​2
1: 2 3 6,532 2. Letter 2: 5 3 245 3. Letter 3: 3 3 4,893 4. Letter 4: 7 3 198
Answer: 13,064
Answer: 1,2 2 5
Answer: 1 4,6 7 9
Code: Use the ten
thousands digit and
the thousands digit.
13 Letter: M
Code: Use the
thousands digit.
Code: Use the ten
Code: Use the
thousands digit and thousands digit.
the thousands digit.
14 Letter: N
1 Letter: A
5. Letter
Step 5​ Multiply​the​1​thousand​by​7.
​$1,324​
​​​​
​​ 
Add​the​regrouped​thousands.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​
3   ​​​​​​7
__
$9,268​
Answer: 5 1 6
Answer: 5,8 40
Code: Use the ten
thousands digit and
the thousands digit.
20 Letter: T
Code: Use the
hundreds digit.
Code: Use the
thousands digit.
5 Letter: E
5 Letter: E
IT’S A M A N A T E E !
8.
Possible estimates are given.
$49,000
15,000
6,000 2. Estimate:$3,200
3. Estimate:
4. Estimate:
Estimate. Then find the product.
 
6,368
​ 2,637​​
​​​​
​​​
 3​​  ​5​
__
​
 
$3,312
 
13,185
​​​​
​​​
​ $6,900​​
3   ​​​​​​7​
__
​
 
$48,300
The product of 5 and another number has the code for E in its ones place.
What digit could be in the ones place of the other number? Explain.
Since the code for E is 5, the product of 5 and the digit in the ones
1. Estimate:
​​​​
​ $828​​​​​
3   ​​​4​
__
​
1 Letter: A
Answer:1,3 8 6
5: 6 3 3,411 6. Letter 6: 4 3 129 7. Letter 7: 8 3 730
Answer: 2 0,4 6 6
So,​7​3​$1,324​5​$9,268.
Since​$9,268​is​close​to​the​estimate​of​$7,000,​the​answer​is​reasonable.
​​​​
​ 3,184​​
​​​
 3​​  ​2​
__
​
Differentiated
Instruction
place must end in 5. So, the digit is 1, 3, 5, 7, or 9.
9.
Is the product of a 4-digit number and a 1-digit number
always a 5-digit number? Explain.
Possible answer: No. For example, in 4,000 3 1 5 4,000, the
product is not a 5-digit number.
125 Chapter 2
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
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Chapter Resources
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Example
The table shows the prices of three
vacation packages. Jake, his parents,
and his sister want to choose a package.
These are multistep problems. Before solving
these problems, give students an opportunity
to share their answers to the Math Talk
questions.
• What word in Example A tells you that
you do not need an exact answer? the word
A About how much would Package C cost
Jake’s family?
• How do the words in Example B tell you
that you need an exact answer? Possible
Example
Use an estimate or an exact answer.
“about”
STEP 1
STEP 2
STEP 3
Estimate the cost for
2 adults.
Estimate the cost for
2 children.
Add to estimate the
total cost.
$1,400
2 × $699
2 × $484
↓
Example A
• Why couldn’t you just multiply 4 ∙ $700
to estimate the total for Package C? Possible
+ $1,000
______
↓
$1,400
2 × $700 = __
answer: The problem asks “how much more,” not
“about how much more.”
$1,000
2 × $500 = __
Math
Talk
So, Package C would cost Jake’s family about $2,400.
$2,400
MATHEMATICAL PRACTICES 1
Analyze How did you use the
information to know that you
needed an estimate?
• Why don’t you need to regroup when you
multiply 2 ∙ 700? Possible answer: 14 hundreds
B Jake’s family wants to compare the total costs of
Packages A and C. Which plan costs more? How much
more does it cost? Possible explanation: the question uses the word “about,”
which tells you that an estimate is needed.
Adults
$1,299
×
2
__
$2,598
Children
$619
×
2
__
$1,238
Subtract to compare the
total costs of the packages.
is the same as 1 thousand 4 hundreds. So, I can just
write the 14 in the answer.
Package C
Total Cost
Adults
$2,598
Children
$699
×
2
__
+ $1,238
__
$484
×
2
__
$1,398
$3,836
$968
Total Cost
Math
Talk
MATHEMATICAL PRACTICES 1
Make Sense of Problems How
did you use the information to
know that you needed an exact
answer?
A would cost _
$1,470 more
So, Package _
C .
than Package _
­answer with a dollar sign.
+ $ 968
__
$2,366
$3,836 Possible explanation: the question asks
how much more, which is an exact answer.
− $2,366
__
$1,470
• How do you label the answer? I label the
$1,398
• How is estimating with dollar amounts
­different from estimating with whole
­numbers? Possible answer: I need to write a dollar
© Houghton Mifflin Harcourt Publishing Company
Package A
answer: the estimated total would be much greater
because 2 of the 4 people are children and the price for
children is much less than it is for adults.
sign in front of the estimate.
Example B
• Why is the $2,400 estimate for Package C
greater than the $2,366 actual price for
Package C? I rounded $699 and $484 up to the next
hundred.
126
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COMMON ERRORS
Error Students may write the regrouped
number in the wrong location.
Advanced
Advanced
Learners
Learners
Visual
Individual
Materials 1-Centimeter Grid Paper (see eTeacher Resources)
•Challenge students to fill in unknown digits to
complete multiplication sentences.
•Give students the problems below. Have them
use grid paper and the given digits to complete
each problem.
1. 2, 3, and 5
3 × 3, 5 2 7 = 10,581
2. 2, 4, and 6
5 × 6, 4 7 2 = 32,360
COMMON
ERRORS
1
Example
​      
619   
​
∙ 2 _
8 Springboard to Learning Emphasize that a
regrouped number is written over the place
immediately to the left of the place that is
being multiplied. Have students draw lines to
help them keep the places aligned.
1
​ 6 1 9 
    
​  
∙ 2
__
8
Lesson 2.11 126
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Name
3 EXPLAIN
Share
Share and
and Show
Show
MATH
BOARD
1. Tell what is happening in Step 1
of the problem. Possible answer:
Share and Show
MATH
Hands
On
BOARD
multiply 6 × 4 ones = 24 ones, regroup 24
Estimate. Then find the product.
2
1
2,400
2. Estimate: __
Then
603
Rt I
On Your Own
If students complete the checked exercises
correctly, they may continue with the
remaining exercises.
DEEPER
MP2 Reason abstractly and
quantitatively. Encourage students to
develop different ­strategies for multiplying.
• In Exercise 9, how could ­multiplying 2 ∙ 440
help you find the product of 4 ∙ 440? Possible
answer: 2 × 440 = 880; 880 + 880 = 1,760
127 Chapter 2
STEP 4
1 42
1,2 74
1,2 74
×
6
__
644
×
6
__
7,644
$40,000
4. Estimate: __
1,935
×
7
___
13,545
Estimate. Then find the product.
$24,000
5. Estimate: __
$8,326
×
5
___
$41,630
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how you can use
estimation to find how many
digits the product 4 × 1,861 will
have.
Possible estimates are given.
$21,000
6. Estimate: __
$3,316
×
8
___
$26,528
© Houghton Mifflin Harcourt Publishing Company
Possible answer: this could help you see that you may
have made a regrouping error. If you made a regrouping
error, you may have more digits in the product than
there should be.
1, 2 74
×
6
__
44
On
On Your
Your Own
Own
8.
Use Math Talk to check students’
understanding of the usefulness of
estimating a product.
• How would it be helpful to know the
number of digits a product will have?
1,2 74
×
6
__
4
4 digits; possible explanation: the product will be
between 4,000 and 8,000, so it will have 4 digits.
Differentiate Instruction with
• Reteach 2.11
• Personal Math Trainer 4.NBT.B.5
• RtI Tier 1 Activity (online)
Math
Talk
1 42
14,000
3. Estimate: __
× 4
__
a student misses the checked
exercises
If
STEP 3
42
Possible estimates are given.
2,412
Quick Check
STEP 2
2
ones as 2 tens 4 ones.
The first problem connects to the learning
model. Have students use the MathBoard to
explain their thinking. Have students describe
what is happening in Step 3.
Use the checked exercises for QuickCheck.
3
STEP 1
$2,900
×
7
___
$20,300
$24,000
7. Estimate: __
$4,123
×
6
___
$24,738
DEEPER
Mr. Jackson has $5,400 to buy supplies for the
school computer lab. He buys 8 boxes of printer ink that cost
$149 each and 3 printers that cost $1,017 each. How much
money will Mr. Jackson have left after he buys the printer
ink and printers?
$1,157
Practice: Copy and Solve Compare. Write <, >, or = .
●
< 8 × 658
12. 740 × 7 ●
9. 5 × 352 = 4 × 440
●
> 5 × 2,834
13. 4 × 3,645 ●
10. 6 × 8,167 < 9,834 × 5
●
> 4,365 × 3
14. 6,573 × 2 ●
11. 3,956 × 4 < 5 × 7,692
Chapter 2 • Lesson 11
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MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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15.
DEEPER
Airplane tickets to Fairbanks,
Alaska, will cost $958 each. Airplane tickets to
Vancouver, Canada, will cost $734. How much
can the four members of the Harrison family
save on airfare by vacationing in Vancouver?
WRITE
4 ELABORATE
Math t Show Your Work
Problem Solving • Applications
MATHEMATICAL PRACTICES
$896
16.
Have students discuss how to decide
when to estimate an answer or find an
exact answer.
SMARTER
Philadelphia, Pennsylvania,
is 2,147 miles from Salt Lake City, Utah, and
2,868 miles from Portland, Oregon. What is the
difference in the round-trip distances between
Philadelphia and each of the other two cities?
Explain whether you need an estimate or an
exact answer.
SMARTER
In Exercise 16, students should recognize
that the one-way distances between cities is
given, but they need to compare the
round-trip distances. Therefore, this is a
multistep problem involving multiplication
and subtraction.
1,442 mi; possible explanation: exact answer
since the question asks for a difference
17.
MATHEMATICAL
3 Verify the Reasoning of Others
PRACTICE
Joe says that the product of a 4-digit number
and a 1-digit number is always a 4-digit number.
Does Joe’s statement make sense? Explain.
Math on the Spot
Video Tutor
No. Possible explanation: when there are
regrouped thousands, the product of a
Use this video to help students model and
solve this type of Think Smarter problem.
4-digit number and a 1-digit number can
have 5 digits.
SMARTER
What number is
150 more than the product of 5 and
4,892? Explain how you found the
answer.
© Houghton Mifflin Harcourt Publishing Company
18.
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
24,610; Possible explanation: first find the
product 5 × 4,892 = 24,460. Then add 150
to the product; 24,460 + 150 = 24,610.
128
DIFFERENTIATED INSTRUCTION
D
INDEPENDENT ACTIVITIES
MP3 Construct viable arguments and
critique the reasoning of others.
Exercise 17 requires students to use higher
order thinking skills to decide if Joe’s
conclusion is correct.
SMARTER
Students should recognize that Exercise 18 is
a multistep problem involving multiplication
then addition. If students set up the problem
correctly but still have the wrong answer,
check for computational and regrouping
errors.
5 EVALUATE Formative
Assessment
Differentiated Centers Kit
Activities
Product Power
Literature
Multiplying a
Good Deed
Essential Question
Games
Multiplication
Marathon
(BNFT
Students complete
purple Activity Card
5 by multiplying
multi-digit numbers
by single-digit
numbers.
Students read about
how Ramon uses
multiplication to
find how many cans
can be collected for
a food drive.
Students take turns
using number cards
to make and solve
2-digit by 1-digit
multiplication
problems.
Using the Language Objective
Reflect Have student pairs demonstrate to a
small group to answer the Essential Question.
How can you use regrouping to multiply?
Possible answer: I start at the right. First, I multiply the
ones, then the tens, then the hundreds, and finally the
thousands. I regroup when the product in any place
is 10 or more. I record the regrouped number in the
column above the next greater place.
Math Journal
WRITE
Math
Explain how finding 4 × 384 can help you
find 4 × 5,384. Then find both products.
Lesson 2.11 128
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Practice and Homework
Name
Lesson 2.11
Multiply 3-Digit and 4-Digit Numbers
with Regrouping
COMMON CORE STANDARD—4.NBT.B.5
Use place value understanding and properties
of operations to perform multi-digit arithmetic.
Possible estimates are given.
Practice and Homework
Estimate. Then find the product.
4,000
1. Estimate: _
Use the Practice and Homework pages to
provide students with more practice of the
concepts and skills presented in this lesson.
Students master their understanding as they
complete practice items and then challenge
their critical thinking skills with Problem
Solving. Use the Write Math section to
determine student’s understanding of content
for this lesson. Encourage students to use their
Math Journals to record their answers.
30,000
2. Estimate: _
$7,200
3. Estimate: _
15,000
4. Estimate: _
5,339
×
6
__
32,034
$879
×
8
__
$7,032
3,182
×
5
__
15,910
$27,000
6. Estimate: _
16,000
7. Estimate: _
6,300
8. Estimate: _
1 2 2
1,467
×
4
__
5,868
15,000
5. Estimate: _
4,616
×
3
__
13,848
$2,854
×
9
__
$25,686
7,500
×
2
__
15,000
948
×
7
__
6,636
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. Lafayette County has a population of 7,022
11.
10. A seafood company sold 9,125 pounds of fish
people. Columbia County’s population
is 8 times as great as Lafayette County’s
population. What is the population of
Columbia County?
last month. If 6 seafood companies sold the
same amount of fish, how much fish did the
6 companies sell last month in all?
56,176 _
people
___
_
_
_
pounds
__54,750
_
_
_
_
_
Math Explain how finding 4 × 384 can help you
WRITE
find 4 × 5,384. Then find both products.
Check students’ work.
Chapter 2
11/6/14 7:19 AM
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4_MNLESE342194_C02P11.indd 129
129
129 Chapter 2
Lesson Check (4.NBT.B.5)
1. By recycling 1 ton of paper,
6,953 gallons of water are saved.
How many gallons of water are saved
by recycling 4 tons of paper?
27,812 gallons
2. Esteban counted the number of steps it
took him to walk to school. He counted
1,138 steps. How many steps does he take
walking to and from school each day?
Continue concepts and skills practice with
Lesson Check. Use Spiral Review to engage
students in previously taught concepts and
to promote content retention. Common Core
standards are correlated to each section.
2,276 steps
Spiral Review (4.NBT.A.2, 4.NBT.A.3, 4.NBT.B.4, 4.NBT.B.5)
What is the word form of this number?
thirteen thousand, four hundred six
5. Connor scored 14,370 points in a game.
Amy scored 1,089 fewer points than
Connor. How many points did Amy score?
13,281 points
130
4. In one year, the McAlister family drove
their car 15,680 miles. To the nearest
thousand, how many miles did they drive
their car that year?
16,000 miles
6. Lea buys 6 model cars that each cost $15.
She also buys 4 bottles of paint that each
cost $11. How much does Lea spend on
model cars and paint?
$134
© Houghton Mifflin Harcourt Publishing Company
3. A website has 13,406 people registered.
FOR MORE PRACTICE
GO TO THE
Personal Math Trainer
Lesson 2.11 130