Research—Best Practices
Putting Research
into Practice
From Our Curriculum Research Project
UNIT 4 | Overview | 293O
Resea rch
This reasoning applies to decimal factors as well as to whole number
factors, so students now have a general method. They can look at
specific examples to understand and use the rule that the product
will have the number of decimal places that is the sum of the places
in the factors. Often this rule is only a rote rule about moving
the decimal point, and it can easily be forgotten or confused. Our
students had a foundation of reasoning about shifting the multiplied
number to get larger or smaller, so they could use this understanding
to check answers and remember the rule.
UNIT 4
Dr. Karen C. Fuson,
Math Expressions Author
The most difficult aspect of multiplying by decimals is that
multiplying by a decimal less than 1 gives a result that is less than the
multiplied number. This is new for students because up to now their
experience with whole number multiplication leads them to expect
a larger result. Students need strong visual and conceptual supports
to understand this new result. We found that it was powerful to use
what students know about multiplying by 10 and 100 to find out
what happens when they multiply by 0.1 and 0.01. By thinking about
numbers place by place, they see that multiplying by 10 shifts each
place to the left: it gets one place bigger. But multiplying by onetenth (0.1) shifts each place to the right: it gets one place smaller as
they find one-tenth of each place. Multiplying by 100 shifts a number
two places to the left (two places larger), and multiplying by 0.01
shifts a number two places to the right (two places smaller).
Contents
Planning
Research & Math Background
From Current Research
Multidigit Multiplication
Researchers have reported a preliminary learning progression of
multidigit methods for third- to fifth-grade classrooms in which
teachers fostered children’s invention of algorithms (Baek 1998). These
methods moved from (a) direct modeling with objects or drawings
(such as by ones and by tens and ones), to (b) written methods
involving repeatedly adding—sometimes by repeated doubling, a
surprisingly effective method used historically, to (c) partitioning
methods. The partitioning methods ranged from partitioning using
numbers other than 10, partitioning one number into tens and ones,
and partitioning both numbers into tens and ones.
Fuson, Karen C. “Toward Computational Fluency in Multidigit Multiplication and
Division.” Teaching Children Mathematics, 9.6, (Feb. 2003): 300–305.
Deepening Understanding
Ultimately, the meanings of decimal numbers and multiplication, as
well as the concepts developed in mental computation, are used as
tools by the students to discover generalizations, communicate their
ideas about decimal multiplication, and justify why rules work. By
engaging in these processes, students have opportunities to build
connections to other mathematical topics (such as proportional
reasoning) and to other content areas (science and social studies). In
forming these links, students learn mathematical content with deeper
understanding.
Rathouz, Margaret M. “Making Sense of Decimal Multiplication.” Mathematics
Teaching in the Middle School, 16.7, (March 2011): 430–437.
Other Useful References
Ashlock, Robert B. Error Patterns in
Computation: Using Error Patterns
to Help Each Student Learn, 10th ed.
Boston: Allyn & Bacon, 2010.
Schielack, Jane. Focus in Grade 6:
Teaching with Curriculum Focal
Points. Reston, VA: NCTM, 2010.
293P | UNIT 4 | Overview
Van de Walle, John A. Elementary and
Middle School Mathematics: Teaching
Developmentally, 7th ed. Boston:
Allyn & Bacon, 2010.
Getting Ready to Teach Unit 4
Using the Common Core Standards for Mathematical
Practice
The Common Core State Standards for Mathematical Content indicate
what concepts, skills, and problem solving students should learn. The
Common Core State Standards for Mathematical Practice indicate how
students should demonstrate understanding. These Mathematical
Practices are embedded directly into the Student and Teacher Editions
for each unit in Math Expressions. As you use the teaching suggestions,
you will automatically implement a teaching style that encourages
students to demonstrate a thorough understanding of concepts, skills,
and problems. In this program, Math Talk suggestions are a vehicle
used to encourage discussion that supports all eight Mathematical
Practices. See examples in Mathematical Practice 6.
UNIT 4
Mathematical Practice 1
Make sense of problems and persevere in solving them.
Teacher Edition: Examples from Unit 4
MP.1 Make Sense of Problems Act It
Out Ask three student volunteers to go to
the board and enact the shift patterns that
will occur when $325 is multiplied by 10.
Begin by first writing 6 large line segments
for the place values, and then put a comma
in the middle. Write 3, 2, and 5 in the
appropriate places. Assign one student to
be the 3 hundreds, another to be the 2
tens, and the other to be the 5 ones.
Ask the students at the board to shift
positions to show how $325 changes when
it is multiplied by 10 and to explain their
shift as they do it. Write the numbers in
their new places above the old numbers.
Lesson  1
MP.1 Make Sense of Problems
Reasonable Answers Encourage Small
Groups of students to discuss how they
can decide whether or not their answers
are reasonable. Possible methods might
include:
→Problem
21 Students can recognize that
their answer should be between 4 × 25
= 100 and 5 × 25 = 125.
→Problem
22 A reasonable answer would
be close to the product of 3 × 3.
→Problem
23 Rounding the factors to 2
and 70 would predict an answer that
was close to 140 pounds.
Lesson  11
ACTIVITY 2
ACTIVITY 1
Mathematical Practice 1 is integrated into Unit 4 in the following ways:
Act it Out
Check Answers
Look for a Pattern
Reasonable Answers
UNIT 4 | Overview | 293Q
MATH BACKGRO UN D
Students analyze and make conjectures about how to solve a problem.
They plan, monitor, and check their solutions. They determine if their
answers are reasonable and can justify their reasoning.
Contents
Planning
Research & Math Background
Mathematical Practice 2
Reason abstractly and quantitatively.
Students make sense of quantities and their relationships in problem
situations. They can connect diagrams and equations for a given
situation. Quantitative reasoning entails attending to the meaning
of quantities. In this unit, this involves reasoning about shift patterns
to determine how multiplying by tens or multiplying by decimals will
affect the product.
Teacher Edition: Examples from Unit 4
MP.2 Reason Abstractly and
Quantitatively Connect Symbols and
Words As Student Pairs focus on Student
Book page 107, they should discuss the
explanation of exponential form. One
partner should then answer the odd
numbered exercises and the other partner
should answer the even numbered
exercises. They can then check each
other’s work and discuss any discrepancies.
Lesson  1
ACTIVITY 3
MP.2 Reason Abstractly and
Quantitatively Remind students that
they explored shift patterns using
exponents in Lesson 1. Review with them
that the exponent is equal to the number
of zeros to the right of the 1 (101 = 10,
102 = 100, 103 = 1,000).
Write the following problems on the
board and have students identify which
way the digits shift. Then have them
write the exponential form as repeated
multiplication and solve.
14 × 101 left one space, 14 × 10 = 140
14 × 102 left two spaces, 14 × 10 × 10 =
1,400
14 × 103 left three spaces, 14 × 10 × 10 ×
10 = 14,000
Lesson  9
Mathematical Practice 2 is integrated into Unit 4 in the following ways:
Connect Diagrams and Equations
Connect Symbols and Words
293R | UNIT 4 | Overview
ACTIVITY 2
Mathematical Practice 3
Construct viable arguments and critique the reasoning of others.
Students use stated assumptions, definitions, and previously established
results in constructing arguments. They are able to analyze situations and
can recognize and use counterexamples. They justify their conclusions,
communicate them to others, and respond to the arguments of others.
Students are also able to distinguish correct logic or reasoning from
that which is flawed, and—if there is a flaw in an argument—explain
what it is. Students can listen to or read the arguments of others,
decide whether they make sense, and ask useful questions to clarify or
improve the arguments.
Math Talk is a conversation tool by which students formulate ideas
and analyze responses and engage in discourse. See also Mathematical
Practice 6: Attend to Precision.
Elicit from the class that Place Value
Sections and Expanded Notation are set
up so that each place is multiplied by each
place, while Place Value Rows and the
Short Cut method are set up so that the
entire top number is multiplied by each
place in the multiplier.
Lesson  4
ACTIVITY 2
What’s the Error? WHOLE CLASS
MP.3, MP.6 Construct Viable
Arguments/Critique Reasoning of
Others Puzzled Penguin On the board,
write the three dollar amounts and the
way Puzzled Penguin rounded them:
$1.35 ≈ $1.4
$0.07 ≈ $0.1
$12.39 ≈ $12.4
•Is Puzzled Penguin correct?
•What did Puzzled Penguin do wrong?
Puzzled Penguin did not write the cents
to the hundredths place.
•How can we round to the nearest
tenth and still write the dollar amount
correctly?
Elicit from students that to write the
dollar amount for ten cents, they should
write $0.10. Students should identify that
0.1 and 0.10 are two possible ways to
write one tenth.
Lesson  10
ACTIVITY 2
Mathematical Practice 3 is integrated into Unit 4 in the following ways:
Compare Methods
Puzzled Penguin
UNIT 4 | Overview | 293S
MATH BACKGRO UN D
MP.3 Construct Viable Arguments
Compare Methods On page 115 of the
Student Book, Small Groups of students
can compare all of the multiplication
methods. Use Exercises 7 and 8 to discuss
the similarities and differences among the
methods.
UNIT 4
Teacher Edition: Examples from Unit 4
Contents
Research & Math Background
Planning
Mathematical Practice 4
Model with mathematics.
Students can apply the mathematics they know to solve problems that
arise in everyday life. This might be as simple as writing an equation
to solve a problem. Students might draw diagrams to lead them to
a solution for a problem. Students apply what they know and are
comfortable making assumptions and approximations to simplify a
complicated situation. They are able to identify important quantities in
a practical situation and represent their relationships using such tools
as diagrams, tables, graphs, and formulas.
Teacher Edition: Examples from Unit 4
MP.4 Model with Mathematics Ask the
class to solve Problem 2 on Student Book
page 111. Draw a few large rectangles
on the board and invite several student
volunteers to work there. One model is
shown below.
39 × 54
30
+
9
+
50
4
30 × 4
= 120
30 × 50 = 1,500
9×4
= 36
9 × 50 = 450
+
50
30
+
9
4
1 1
30 × 50 = 1,500
30 × 4 =
120
9 × 50 =
450
9 × 4 = + 36
2,106
Lesson  3
MP.4 Model with Mathematics
Concrete Objects Read the first problem
on Student Book page 119 together. As
volunteers explain the calculations done
in each line of the chart, have them
act out the multiplication by using play
money. Students can see that modeling
3 groups of 9 cents is the same as writing
3 × $0.09. Discuss the answers to Problems
1 and 2.
The problem about Mia is more complex,
but can be solved by multiplying each
place value by each place value, just as
with whole numbers. These basic kinds
of multiplications are similar to the first
problem, except now the results are
combined. Expanded Notation allows
students to see each partial product for
each place value. When multiplying by
20, students use the shift pattern for
multiplying by 10 discussed in Lesson 1.
Lesson  6
ACTIVITY 1
Mathematical Practice 4 is integrated into Unit 4 in the following ways:
Class MathBoard
Concrete Objects
Write an Equation
293T | UNIT 4 | Overview
ACTIVITY 1
Mathematical Practice 5
Use appropriate tools strategically.
Students consider the available tools and models when solving
mathematical problems. Students make sound decisions about when each
of these tools might be helpful. These tools might include paper and
pencil, a straightedge, a ruler, or the MathBoard. They recognize both
the insight to be gained from using the tool and the tool’s limitations.
When making mathematical models, they are able to identify quantities
in a practical situation and represent relationships using modeling tools
such as diagrams, grid paper, tables, graphs, and equations.
Modeling numbers in problems and in computations is a central focus
in Math Expressions lessons. Students learn and develop models to
solve numerical problems and to model problem situations. Students
continually use both kinds of modeling throughout the program.
The first Student Leader shows a 1-dollar bill.
The second Student Leader says and shows
1 dollar multiplied by 10. (ten 1-dollar bills)
Student Leader 1 writes $10 on the board.
Then, Student Leader 1 shows a 10-dollar
bill. Student Leader 2 says and shows $10
multiplied by 10. (ten 10-dollar bills) Student
Leader 1 writes $100 on the board.
Discuss how the digits shift to the left every
time they are multiplied by 10. The 1 keeps
shifting another place to the left. ($1, $10,
$100)
Lesson  1
ACTIVITY 1
MP.4, MP.5 Model with Mathematics/
Use Appropriate Tools Class
MathBoard Exercise 5 on Student Book
page 142, provides an opportunity
to model the connection shared by
multiplication and repeated addition.
Invite one student to sketch six confused
cloudwings on the Class MathBoard while
the other students sketch at their desks.
After each butterfly’s wingspan is labeled
in inches (1.26), have students write and
complete the repeated addition and the
multiplication that can be used to find the
combined length of all six wingspans.
1.26 + 1.26 + 1.26 + 1.26 + 1.26 +
1.26 = 7.56
1.26 × 6 = 7.56
Point out that multiplication is a fast
and efficient way to perform repeated
addition.
Lesson  12
ACTIVITY 1
Mathematical Practice 5 is integrated into Unit 4 in the following ways:
Concrete Objects
Class MathBoard
UNIT 4 | Overview | 293U
MATH BACKGRO UN D
MP.4, MP.5 Model with Mathematics/Use
Appropriate Tools Concrete Objects Ask
two student volunteers to be Student
Leaders who use play money to act out these
multiplication problems. The rest of the class
can form Small Groups and follow along,
drawing the money on their MathBoards.
UNIT 4
Teacher Edition: Examples from Unit 4
Contents
Planning
Research & Math Background
Mathematical Practice 6
Attend to precision.
Students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They
state the meaning of the symbols they choose. They are careful about
specifying units of measure to clarify the correspondence with quantities
in a problem. They calculate accurately and efficiently, expressing
numerical answers with a degree of precision appropriate for the problem
context. Students give carefully formulated explanations to each other.
Teacher Edition: Examples from Unit 4
MP.6 Attend to Precision Explain a
Method Ask a volunteer to explain the
Expanded Notation method. Each factor is
separated into its tens and ones and then
all 4 pair combinations formed by the tens
and ones are multiplied to produce the 4
partial products.
Ask another volunteer to compare
variations A and B. Variation A shows the
4 multiplication equations that give the
partial products; variation B shows only
the partial products.
Lesson  3
ACTIVITY 2
Put the following problem on
the board: 4 × (3 + 6) = 4 × 3 + 4 × 6. Ask
students to identify which property this
represents. Distributive Property Ask students
to summarize what the Distributive Property
does. Students’ summaries should include the
idea that the product of a number and a sum
is equal to the sum of the products of the
number and each of the addends.
M AT H TA L K
Lesson  3
MATH TALK
in ACTION
ACTIVITY 1
S tudents discuss how they know if an
answer is reasonable.
Write the following problem on the board:
0.73 × 4.8 = 35.04
Does this answer seem reasonable?
Tony: No, it doesn’t seem reasonable.
MP.6 Attend to Precision Explain
Solutions Have a few students solve the
problems at the board and explain the
steps they used. Encourage other students
to ask questions and provide assistance if
necessary.
Lesson  11
ACTIVITY 2
Maria: Since the first factor is a number less than
one, the product should be less than the second
factor. Because 35.04 is greater than 4.8, the answer
can’t be right.
Tony: I would expect the product to be greater than
0.73 and less than 4.8.
Maria: Besides, there are 3 decimal places in both
factors so I would expect 3 decimal places in the
product.
Good thinking everyone.
Lesson  8
ACTIVITY 2
Mathematical Practice 6 is integrated into Unit 4 in the following ways:
Describe a Method
Explain a Method
293V | UNIT 4 | Overview
Explain a Solution
Explain Solutions
Identify Relationships
Puzzled Penguin
Mathematical Practice 7
Look for and make use of structure.
Students analyze problems to discern a pattern or structure. They
draw conclusions about the structure of the relationships they have
identified.
Teacher Edition: Examples from Unit 4
5 × 4 = 20
5 × 5 = 25
5 × 6 = 30
5 × 7 = 35
5 × 8 = 40
Lesson  2
Lesson  7
ACTIVITY 1
ACTIVITY 1
Mathematical Practice 7 is integrated into Unit 4 in the following ways:
Identify Relationships
UNIT 4 | Overview | 293W
MATH BACKGRO UN D
5 × 3 = 15
MP.7 Look for Structure Identify
Relationships Encourage students to work
in pairs to solve problems that include
a whole number and a decimal. Have
Student Pairs use pencil and paper to
multiply 294 by 0.1 and by 0.01. Provide
3 more whole numbers to multiply by
0.1 and by 0.01. Discuss the pattern that
students notice. Students should be able
to see that the digits in the whole number
are moved over 1 and 2 places to the
right when they multiply by 0.1 and by
0.01, respectively.
UNIT 4
MP.7 Look for Structure Identify
Relationships You may want to point
to the 5-row on the Class Multiplication
Table Poster to review the five-pattern
there. Or write these problems on the
board and have students give the answers
quickly. They will be reminded that every
other answer ends in zero because it is a
multiple of 10. (Even numbers of 5s make
multiples of 10.)
5 × 2 = 10
5 × 1 = 5
Contents
Research & Math Background
Planning
Mathematical Practice 8
Look for and express regularity in repeated reasoning.
Students use repeated reasoning as they analyze patterns, relationships,
and calculations to generalize methods, rules, and shortcuts. As
they work to solve a problem, students maintain oversight of the
process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
Teacher Edition: Examples from Unit 4
MP.8 Use Repeated Reasoning
Generalize Discuss the key idea by asking
students to determine the pattern in the
number of decimal places in the factors
and in the product. When you multiply a
decimal number by a whole number, the
product has the same number of decimal
places as the decimal factor.
Ask students to consider how the key idea
can assist them as they multiply by decimals.
Lesson  6
MP.8 Use Repeated Reasoning Draw
Conclusions Focus students’ attention
on Student Book page 136. The
multiplications in Exercises 29–31 and
33–35 might appear to be exceptions to
the generalization. However, as students
perform the multiplications, they will see
that they are not exceptions. Review the
idea about where zeros do and do not
change the value of a decimal number.
Lesson  9
ACTIVITY 1
ACTIVITY 3
Mathematical Practice 8 is integrated into Unit 4 in the following ways:
Conclude
Draw Conclusions
Generalize
Student EDITION: Lesson 12, pages 141–142
Name
4-12
Date
The table below shows typical
wingspans, in centimeters and inches,
of interesting flying insects.
Insects are sometimes called bugs, and are the
most numerous living things on Earth.
Buckeye Butterfly
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Stuart Hogton/Alamy Images; (b) ©pasmal/amanaimages/Corbis
There are nearly one million different kinds
of known insects, and most scientists believe
more than that number of insects have never
been discovered and given names.
confused cloudywing butterfly
Atlas moth
The table below shows typical lengths, in
centimeters and inches, of interesting insects.
monarch butterfly
sleepy orange butterfly
Emperor Dragonfly
damselfly
Length
Insect
crazy ant
green stinkbug
3.5
4.5
1.8
19.1
7.5
parasitic wasp
0.15
0.06
chickweed geometer moth
2.25
0.9
0.25
0.1
buckeye butterfly
5.7
2.24
Atlas Moth
0.6
3.8
1.5
spotless ladybug
0.5
0.2
emperor dragonfly
7.8
3.1
praying mantis
10
4
giant water bug
11.6
4.6
1.9
0.75
Solve. Use the table above.
Show your work.
Solve. Use the table above.
10
megachile pluto bee
4. In centimeters, which wingspan is twice as long as the
wingspan of a chickweed geometer moth?
sleepy orange butterfly
5. In inches, the total wingspan of six confused cloudywing
butterflies is about the same as the wingspan of which insect?
damselfly
Walkingstick Insect
Show your work.
1. Eight green stinkbugs, marching one behind the other,
are about the same length in inches as which insect?
6. How does the total wingspan in centimeters of 60 parasitic
wasps compare to the wingspan of 2 sleepy orange butterflies?
The wingspans are equal; 0.15 × 60 = 2 × 4.5
giant water bug
7. An Atlas moth has a wingspan of 11 inches. Suppose the
moth was placed on the cover of your math book. What is
the difference between the length of the wingspan and your book?
2. In centimeters, which insect is about forty times as long
as a crazy ant?
praying mantis
Answers will vary; the wingspan will typically be greater than every
dimension of the book.
3. The total length in inches of fifty spotless ladybugs is
about the same length as which insect?
5_MNLESE824482_U04L12.indd 141
1.26
11.0
9
in.
25
firefly
3.2
28
cm
1.5
walkingstick
wingspan
cm
in.
Insect
8. Write and solve a problem of your own using data from the table.
Possible answer: The wingspan in centimeters of a monarch butterfly is
walkingstick
293X | UNIT 4 | Overview
Date
Class Activity
► Math and Insects
UNIT 4 LESSON 12
Name
4-12
Class Activity
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Robert Marien/Corbis; (b) ©Bert Hoferichter/Alamy Images
Focus on Mathematical Practices
Unit 4 includes a special lesson that
involves solving real world problems
and incorporates all 8 Mathematical
Practices. In this lesson, students use
what they know about operating
with decimals to compare the
wingspans of various insects.
about the same as the wingspans of four geometer moths.
Focus on Mathematical Practices
141
21/02/12 1:46 PM
142
UNIT 4 LESSON 12
5_MNLESE824482_U04L12.indd 142
Focus on Mathematical Practices
21/02/12 1:45 PM
Getting Ready to Teach Unit 4
Learning Path in the Common Core Standards
In this unit, students multiply whole numbers and decimals. Their
work involves finding the products of two-digit whole numbers and
decimals both less than 1 and greater than 1, and estimating products
by rounding factors.
Visual models and real world situations are used throughout the unit
to illustrate important number and operation concepts.
Math Expressions
VOCABULARY
As you teach this unit, emphasize
understanding of these terms:
•shift patterns
•Place Value Rows
•Place Value Sections
•New Groups Above
•New Groups Below
•Short Cut Method
See the Teacher Glossary
Help Students Avoid Common Errors
Math Expressions gives students opportunities to analyze and correct
errors, explaining why the reasoning was flawed.
UNIT 4
In this unit we use Puzzled Penguin to show typical errors that
students make. Students enjoy teaching Puzzled Penguin the correct
way, and explaining why this way is correct and why the error is
wrong. The following common errors are presented to students as
letters from Puzzled Penguin and as problems in the Teacher Edition
that were solved incorrectly by Puzzled Penguin:
MATH BACKGRO UN D
→ Lesson 2: Assuming the number of zeros in a product is
always the same as the number of zeros in the factors
→ Lesson 4: Regrouping incorrectly when finding a
product
→ Lesson 7: Shifting the digits in the wrong direction
→ Lesson 8: Unable to apply properties to simplify
finding products
→ Lesson 9: Not using zero as a placeholder and using an
exponent as a factor
→ Lesson 10: Rounding cents incorrectly
In addition to Puzzled Penguin, there are other suggestions
listed in the Teacher Edition to help you watch for situations that
may lead to common errors. As part of the Unit Test Teacher Edition
pages, you will find a common error and prescription listed for each
test item.
UNIT 4 | Overview | 293Y
Contents
Research & Math Background
Planning
Lessons
Shift Patterns in
Multiplication
1
2
Multiplying Whole Numbers by Powers of Ten Real world scenarios
involving money are used to introduce students to the concept of
shifting digits. For whole numbers, Jordan's salary is given as $243 per
week, and students consider the amount earned in 1, 10, 100, and
1,000 weeks.
2
,
×1
4
3
,
compare, and how numbers round
for decimals to thousandths.
New at Grade 5 is the use of whole
$
2
4
,
number exponents to denote
powers of 10. Students understand
3
0
10 × $243 = $2,430
or decimal that many places to
,
the left. For example, multiplying
,
by 104 is multiplying by 10 four
times. Multiplying by 10 once shifts
,
every digit of the multiplicand one
place to the left in the product
After 100 Weeks
2
4
,
3
0
0
10 × 10 × $243
= $24,300
the value of each place is 10 times
,
the value of the place to its right.
,
So multiplying by 10 four times
shifts every digit 4 places to the
,
left. Patterns in the number of 0s
After 1,000 Weeks
4
3
,
,
,
0
in products of whole numbers and
0
0
a power of 10, and the location of
the decimal point in products of
× 1000
, 10 × 10 × 10 × $243
= $243,000
,
Jordan's earnings show students that the result of multiplying by
1, 10, 100 and 1,000 shifts the digits in the multiplicand to the left.
Multiplying by 10 shifts the digits one place to the left, by 100 shifts
the digits two places, and by 1,000 shifts the digits three places.
293Z | UNIT 4 | Overview
(the product is ten times as large)
because in the base-ten system
,
× 100
why multiplying by a power of 10
shifts the digits of a whole number
,
× 10
2
system to the relationship between
adjacent places, how numbers
After 10 Weeks
$
understanding of the base-ten
,
1 × $243 = $243
$
Place Value and Shift Patterns Students extend their
Jordan‘s Weekly Earnings
$
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
decimals with powers of 10, can be
explained in terms of place value.
Multiplying Decimals by Powers of Ten For decimals, the same
concept is presented by giving the manufacturing cost of an item as
$0.412, and having students consider the cost for the production of
1, 10, 100, and 1,000 items.
Cost of a Red Phantom Marble
$
4
.
1
2
.
×1
1 × $0.412 = $0.412
.
.
×
.
10 Red Phantom Marbles
.
4
1
2
UNIT 4
$
.
× 10
.
.
MATH BACKGRO UN D
10 × $0.412 = $4.12
.
100 Red Phantom Marbles
$
4
1
.
2
0
.
× 100
.
.
100 × $0.412 = $41.20
.
1,000 Red Phantom Marbles
$
4
1
2
.
0
.
0
× 1,000
.
. 1,000 × $0.412 = $412.00
.
Again students see that multiplying by 10, 100, and 1,000 shifts the
digits in the multiplicand to the left. Multiplying by 10 shifts the
digits one place to the left, by 100 shifts the digits two places, and by
1,000 shifts the digits three places.
UNIT 4 | Overview | 293AA
Contents
Research & Math Background
Planning
Patterns in Multiplying with Zeros Exploring the Shift Pattern of
zeros in the table below enables students to generalize that shifting
digits any number of places to the left shifts the decimal point the
same number of places to the right. It also gives students a way to
predict the number of zeros in a product.
×
3
a.
30
b.
2×3
=6
2
300
2 × 30
= 2 × 3 × 10
= 6 × 10
= 60
c.
2 × 300
= 2 × 3 × 100
= 6 × 100
= 600
3,000
d.
2 × 3,000
= 2 × 3 × 1,000
= 6 × 1,000
= 6,000
Powers of 10 Students work with expanded and exponential forms
f.
g.
h.
of powerse.of 10. This enables
students to connect
factors of 10,
100,
20 × 3
20 × 30
20 × 300
20 × 3,000
and 1,000 to repeated multiplication and exponents.
20
= 2 × 10 × 3
××
101
= 610
10 =
= 60
100 = 10 × 10
= 2 × 10 × 3
= 2 × 10 × 3
= 2 × 10 × 3
× 10 exponential
× 100
× 1,000
1
form: 10
= 6 × 100
= 6 × 1,000
= 6 × 10,000
2
exponential
= 600
= 6,000form: 10 = 60,000
1,000 = 10 × 10 × 10
i.
j.
exponential form: 103
k.
This concept is200
extended
to multiplying
of 10.
×3
200 × 30 with powers
200 × 300
l.
200 × 3,000
= 2 × 100 × 3
= 2 × 100 × 3
= 2 × 100 × 3
= 2 × 100 × 3
200
2
3 =
×
×
=
×
×
×
×
×
=
6
×
100
×
10
×
100
5 10 10 5 10 3 10
3 10 10 10 × 1,000
= 600
= 6 × 1,000
= 6 × 10,000
= 6 × 100,000
= 6,000
= 60,000
= 600,000
Patterns With Fives and Zeros In Lesson 2, students discover that
using a pattern
of zeros to n.
predict the number
m.
o. of zeros in a product
p.
requires special
attention
for some
combinations
of
factors.
For
2,000
×3
2,000 × 30
2,000 × 300
2,000 × 3,000
= product
2 × 1,0002 × 50 =
2 × 1,000one
× 3zero
= 2in×the
1,000
×3 =
2 ×two
1,000 × 3
example, the
contains
factors
but
2,000
×
3
×
10
×
100
×
1,000
zeros in the product.
= 6 × 1,000
= 6 × 10,000
= 6 × 100,000
= 100
=
6,000
the
product of 2 =×60,000
5 is 10
2 ×=50
600,000
= 6 × 1,000,000
= 6,000,000
Lesson 2 makes students aware that using a pattern of zeros
(e.g., counting zeros in the factors) must be done with care when
predicting the reasonableness of a product.
Generalizations The real world examples in Lessons 1 and 2 lead
students to the following place value generalizations.
→Multiplying
by 10 produces a number 10 times as great and shifts
each digit one place to the left.
→Multiplying
by 100 produces a number 100 times as great and shifts
each digit two places to the left.
→Multiplying
by 1,000 produces a number 1,000 times as great and
shifts each digit three places to the left.
293BB | UNIT 4 | Overview
Lessons
Multidigit
Multiplication
3
4
5
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
In these lessons students discuss, analyze, and draw models for
contexts involving multiplication of two-digit numbers, and work
with methods that can be used to find the products.
Place Value Sections One way to model the multiplication of
two-digit numbers, such as 43 × 67, is to use a Place Value Sections
model, which shows the relationship the Distributive Property
shares with multiplication—the areas of the four smaller rectangles
represent the four partial products of the multiplication.
Place Value Sections
43 × 67
+
3
multiplication is understanding
the role played by the Distributive
Property. This allows numbers
+
= 60  
+
​     67
  
  
7 ​
     
43 = 40 + 3
___
40 × 60 = 2,400
40 × 7 = 280
3 × 60 = 180
3 × 7 = 21
1
2,881
7
40 × 7 40
= 280
3× 7
= 21
+ 7
to be decomposed into base-ten
+
3
units, products of the units to be
computed, then combined. By
decomposing the factors into like
base-ten units and applying the
Distributive Property, multiplication
computations are reduced to
single-digit multiplications and
products of numbers with multiples
of 10, 100, and 1,000. Students can
Short Cut
Place Value Rows
43 × 67
New Groups
Above
2 2
67
2
40
+
3
67
× 40
2,680
2
67
×3
201
2,680
+201
2,881
​   
67 
​ 
 
× 43
__
​ 
​   
201 
2,680
__
2,881
Distributive Property Using expanded notation to find
the product of two-digit numbers involves place value
because the Distributive Property decomposes the
factors into base ten units, and produces the four factor
pairs (40 × 60, 3 × 60, 40 × 7, and (3 × 7) that represent
the partial products of the multiplication.
New Groups
Below
​ 
​   
67 
 
× 43
__
1 2
​
​   
81 
 
2 2
480
__
1
2,881
connect diagrams of areas or arrays
to numerical work to develop
understanding of general base-ten
multiplication methods.
43 × 67 = (40 + 3) × (60 + 7)
= (40 + 3) × 60 + (40 + 3) × 7
= 40 × 60 + 3 × 60 + 40 × 7 + 3 × 7
These lessons demonstrate that although a rectangular area model
can be used to represent all multiplication situations, the partial
products of those situations can be recorded in different ways.
UNIT 4 | Overview | 293CC
MATH BACKGRO UN D
60
base-ten methods for multi-digit
Expanded Notation
40 × 60 = 2,400
3 × 60 = 180
part of understanding general
UNIT 4
40
60
Multi-digit Multiplication Another
Contents
Multiplication With
Decimal Numbers
Research & Math Background
Planning
Lessons
6
7
8
9
Decimal Shift Patterns The activities presented in Lesson 6 enable
students to compute decimal products symbolically and explore shift
patterns in those products. The first pattern they explore shows
that the product of a whole number and a decimal has the same
number of decimal places as the decimal factor. Since multiplication is
repeated addition, addition is used to verify this relationship.
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
Decimal Products and
Factors General methods used
for computing products of whole
numbers extend to products of
decimals. Because the expectations
2 × 0.3 ton = 0.6 ton
because 0.3 + 0.3 = 0.6
for decimals are limited to
thousandths and expectations for
factors are limited to hundredths
at this grade level, students will
Students also explore the effect of multiplying by 0.1, 0.01, and 0.001,
and again see the pattern that the product of a whole number and a
decimal has the same number of decimal places as the decimal factor.
multiply tenths with tenths and
tenths with hundredths, but they
need not multiply hundredths
with hundredths. Before students
consider decimal multiplication
x
0.3
0.03
0.003
more generally, they can study the
2
2 × 0.3
= 2 × 3 × 0.1
= 6 × 0.1
= 0.6
2 × 0.03
= 2 × 3 × 0.01
= 6 × 0.01
= 0.06
2 × 0.003
= 2 × 3 × 0.001
= 6 × 0.001
= 0.006
effect of multiplying by 0.1 and by
0.01 to explain why the product
is ten or a hundred times as small
as the multiplicand (moves one
or two places to the right). They
can then extend their reasoning
The patterns shown in the table above demonstrate how the product
of a whole number and a decimal factor can be computed using ones,
tens, and hundreds, or using tenths, hundredths, or thousandths.
They give students an opportunity to recognize that the shift pattern
for multiplying whole numbers is the opposite of the shift pattern for
multiplying decimals. In whole number multiplication, the digits in
the multiplicand move to the left. When the multiplication consists of
one or more decimal factors, the digits shift to the right. Both shifts
often require students to use zeros as placeholders.
293DD | UNIT 4 | Overview
to multipliers that are single-digit
multiples of 0.1 and 0.01 (e.g., 0.2
and 0.02, etc.).
Shift Patterns with Decimals The activities in Lesson 7 provide
students with additional opportunities to explore shift patterns of
decimals. Students first work with a real world example of saving a
portion of a $213 monthly income.
Leon‘s Earnings
$
2
1
3
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
Using Reasoning Students can
.
also reason that when they carry
out the multiplication without the
×1
decimal point, they have multiplied
each decimal factor by 10 or 100,
1 × $213 = $213
so they will need to divide by
Students first explore the shift using 0.1 as a factor.
2
.
1
3
the correct answer. Also, students
can use reasoning about the sizes
Save 0.1 Each Month
$
those numbers in the end to get
0
of numbers to determine the
For example, 3.2 × 8.5 should be
× 0.1
close to 3 × 9, so 27.2 is a more
reasonable product for 3.2 × 8.5
0.1 × $213 = $21.30
than 2.72 or 272. This estimation-
Students then explore the shift using 0.01 as a factor.
2
.
1
cases students will encounter in
later grades. For example, it is not
Save 0.01 Each Month
$
all cases, however, especially in
3
easy to decide where to place the
decimal point in 0.023 × 0.0045
based on estimation. Students
× 0.01
0.01 × $213 = $2.13
These activities involve one decimal factor, and lead students to the
following generalizations:
→Multiplying
by 0.1 produces a shift of one place to the right
and produces a product 10 times as small. Multiplying by 0.1 is
1
__
equivalent to multiplying by ​ 10
 ​ or
 
dividing by 10.
can summarize the results of their
reasoning such as those above as
specific numerical patterns and
then as one general overall pattern
such as "the number of decimal
places in the product is the sum of
the number of decimal places in
each factor."
→Multiplying
by 0.01 produces a shift of two places to the right
and produces a product 100 times as small. Multiplying by 0.01 is
1
___
equivalent to multiplying by ​ 100
  ​ or
  dividing by 100.
UNIT 4 | Overview | 293EE
MATH BACKGRO UN D
based method is not reliable in
UNIT 4
placement of the decimal point.
Contents
Planning
Research & Math Background
Two Decimal Factors Also in Lesson 7, students explore shift patterns that occur
when both factors are decimals. The symbolic exercises and real world problems
they work with lead them to infer the following:
→The
number of decimal places in a product is the same as the total number of
decimal places in the factors.
Connect Decimals to Whole Numbers and Fractions The algorithms used to
multiply decimals are the same as those used for whole numbers. Whether
the factors are decimals or whole numbers, the digits in the product are the
same. Reminding students of this connection will allow them to focus on the
relationships between multiplying whole numbers and multiplying decimals and
will promote understanding.
The rules about placing a decimal point in the product connects multiplying
decimals to fractions. When multiplying fractions, we multiply the numerators as
if they are whole numbers, and then multiply the denominators, for example
__
__
___
​ 1 ​ × 10
​ 1 ​ = 100
​  1  ​.  The number of zeros in the product of the denominator corresponds
10
to the number of decimal places, which we use to place the decimal point.
Application The activities in Lesson 8 give students an opportunity to practice
and apply the decimal shift patterns they learned in Lesson 7. Their work involves
computing products symbolically and solving problems in real world contexts.
Properties In Lesson 8, students also complete equations using the Commutative
Property of Multiplication, the Associative Property of Multiplication, and the
Distributive Property.
Compare and Contrast Shift Patterns In Lesson 9, students compare and contrast
shift patterns of whole number multipliers (10 and 100) to those of decimal
multipliers (0.1 and 0.01).
Whole Number Multipliers
1. When you multiply by 10, the number
10 times as big. The digits
gets
1
left .
place(s) to the
shift
Decimal Number Multipliers
2. When you multiply by 0.1, the number
___
​ 101  ​  as big. The digits shift
gets
1
place(s) to the right .
3. When you multiply by 100, the
. When you multiply by 0.01, the
4
1
____
​ 100
   ​ 
100 times as big. The
as big. The
number gets
number gets
2
2
place(s) to
place(s) to
digits shift
digits shift
right
left
.
.
the
the
Students conclude from these relationships that:
→multiplying
by 10 or 0.1 results in a shift of one place.
→multiplying
by 100 or 0.01 results in a shift of two places.
→the
difference is the direction of the shift.
293FF | UNIT 4 | Overview
Powers of 10 Students also explore shift patterns using powers of
10, and infer that the shift pattern of the digits is described by the
number of times 10 is used as a factor (i.e. the exponent).
101 = 10
102 = 10 × 10 = 100
103 = 10 × 10 × 10 = 1,000
4
0.4 × 10 = 0.4 × 101 =
0.4 × 100 = 0.4 × 10 × 10 = 0.4 × 102 =
0.4 × 1,000 = 0.4 × 10 × 10 × 10 = 0.4 × 103 =
40
400
Lessons
10 11
MATH BACKGRO UN D
Reasonable
Answers
UNIT 4
Generalization In Lesson 9, students also extend the Big Idea from
Lesson 7—the number of decimal places in a product is the same as
the total number of decimal places in the factors—to include 5-pattern
products, or products such as 0.8 × 0.5. Although such products are
typically written using only one decimal place (0.8 × 0.5 = 0.4), they
extend the Big Idea because there are two decimal places in the factors
and two in the product (0.8 × 0.5 = 0.40) before the product is written
in simplest form.
Check Products The activities and exercises in Lesson 10 remind
students that they know a variety of strategies that they can use for
checking exact products for reasonableness. One strategy is to use
rounding to create an estimate that is compared to an exact answer,
and is implemented by students on computations such as these.
Estimated Answer
23.24 × 39 ≈
26.12.3 × 3.7 ≈
20
12
Exact Answer
40
×
×
800 ≈
4
≈
48
24 × 39 =
936
12.3 × 3.7 = 45.51
When making estimates, students are encouraged to use patterns and,
whenever possible, mental math. For example, a pattern of basic facts
and zeros to create an estimate for Exercise 23 above could be 2 × 4
= 8 and 2 × 40 = 80. So, 20 × 40 = 800. The basic fact 2 × 4 = 8, and
the generalization the number of zeros in the factors is equal to the
number of zeros in the product, enables many students to complete
the estimate using only mental math.
UNIT 4 | Overview | 293GG
Contents
Planning
Research & Math Background
Check for Reasonableness Also in Lesson 10, students use estimation
to check the reasonableness of given products, such as:
24.5 × 4 = 98 0.56 × 30 = 1.68 15.2 × 2.03 = 30.856
Although strategies for checking these products may vary, many
students will use rounding to decide. For example:
→24.5 ×
4 = 98 is reasonable because 24.5 is about 25, and
25 × 4 = 100.
→0.56 ×
1
30 = 1.68 is not reasonable because 0.56 is about ​2_1​ ,
  and ​_​ of
 
2
30 is 15.
→15.2 ×
2.03 = 30.856 is reasonable because 15.2 is close to 15, 2.03
is close to 2, and 15 × 2 = 30.
Practice The activities in Lesson 11 involve practice, giving students
an opportunity to apply the concepts they have learned in Unit 4. The activities range from performing symbolic multiplication
computations to solving real world multiplication word problems, and
involve either one or two decimal factors.
Focus on
Mathematical Practices
Lesson
12
The Standards for Mathematical Practice are included in every lesson
of this unit. However, there is an additional lesson that focuses on all
eight Mathematical Practices. In this lesson, students use what they
know about multiplying whole numbers and decimals to complete
computations related to measurements of insects.
293HH | UNIT 4 | Overview