Contents
Planning
Research & Math Background
Research—Best Practices
Putting Research
into Practice
From our Curriculum Research Project:
Multidigit Addition and Subtraction Methods
Dr. Karen C. Fuson,
Math Expressions Author
We show three methods for multidigit addition: the common
algorithm (New Groups Above), plus two methods found to be
effective during the research project, New Groups Below and Show
All Totals. These methods are introduced to help students see and
discuss core mathematical ideas about addition and subtraction.
New Groups Below Method
• Students record a regrouped digit on the line below the addition
exercise, instead of above the addition exercise.
• Students see the tens and ones, or hundreds and tens, more closely
together than in the New Groups Above method.
Show All Totals Method
• Students add in each place, record the total for each place, then
add these totals to find the sum.
Multidigit Subtraction Methods
To subtract multi-digit numbers, we teach students to ungroup all the
places before they subtract. This approach reduces errors and helps
develop conceptual understanding of multidigit subtraction. Some
students make the common error of consistently subtracting the
smaller digit in a place-value column from the larger digit, even if the
smaller digit is on top.
To help students remember to ungroup in subtraction, they
• Draw a “magnifying glass” around the top number to prepare for
ungrouping.
• “Look inside” the magnifying glass to see which places need to be
ungrouped.
413P | UNIT 4 | Overview
From Current Research: Accessible Methods
for Multidigit Addition
UNIT 4
Method B [New Groups Below] is taught in China and has been
invented by students in the United States. . . . [T]his method [where]
the new 1 or regrouped 10 (or new hundred) is recorded on the
line separating the problem from the answer . . . requires that
children understand what to do when they get 10 or more in a given
column. . . . Method C [Show All Totals], reflecting more closely many
students’ invented procedures, reduces the problem [of carrying] by
writing the total for each kind of unit on a new line. The carryingregrouping-trading is done as part of the adding of each kind of unit.
Also, Method C can be done in either direction.
National Research Council. “Developing Proficiency with Whole Numbers.” Adding
It Up: Helping Children Learn Mathematics. Washington, D.C.: National Academy
Press, 2001. p. 203.
Research
Other Useful References: Addition and Subtraction
Fuson, Karen C. "Developing
Mathematical Power in Whole
Number Operations" A Research
Companion to Principles and
Standards for School Mathematics
NCTM. Reston, VA. 2003, Chapter 6:
pp. 68–75.
Van de Walle, John A., Karp, Karen S., and
Bay-Williams, Jennifer M., Elementary
and Middle School Mathematics:
Teaching Developmentally. 7th ed.
Allyn & Bacon, 2009.
Number and Operations Standard for
Grades 3–5. Principles and Standards
for School Mathematics. Reston,
VA: National Council of Teachers of
Mathematics, 2000. pp. 148–155.
UNIT 4 | Overview | 413Q
Contents
Planning
Research & Math Background
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.
Mathematical Practice 1
Make sense of problems and persevere in solving them.
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.
Teacher Edition: Examples from Unit 4
MP.1, MP.4 Make Sense of Problems/
Model with Mathematics Draw
a Diagram Ask students to solve
Problem 11. Allow students to use any
method they choose. If students have
difficulty, suggest they try making place
value drawings.
•Draw a ten stick for each box of books.
Make a hundred box for each group of
10 ten sticks. Count to find the total:
100, 200, 300, 310, 320, 330, 340, 350.
There are 350 books in all.
Lesson  4
MP.1, MP.4 Make Sense of Problems/
Model with Mathematics Draw a
Diagram Read aloud Problem 1 on
Student Book page 231. Invite three to
six students to go to the classroom board
and solve the problem, relating each
step of a proof drawing to each step of a
numerical method. Other students work
on MathBoards at their seats.
ACTIVITY 2
Mathematical Practice 1 is integrated into Unit 4 in the following ways:
Make Sense of Problems
Analyze the Problem
413R | UNIT 4 | Overview
Lesson  8
ACTIVITY 1
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 place value, rounding numbers,
assessing reasonableness of answers, and addition and subtraction of
multidigit numbers.
Teacher Edition: Examples from Unit 4
Make a Thousand to Subtract Ask
another volunteer to find the answer to
1,000 - 800 by making a thousand.
With a Drawing
Mentally
•Think: 800 + ? = 1,000.
Lesson  5
ACTIVITY 1
•Start with 800. Add 200 to get to one
thousand.
•You have added a total of 200, so
800 + 200 = 1,000, or equivalently,
1,000 - 800 = 200.
Lesson  8
ACTIVITY 3
Mathematical Practice 2 is integrated into Unit 4 in the following ways:
Reason Abstractly and Reason Quantitatively
Reason Quantitatively
UNIT 4 | Overview | 413S
MATH BACKG ROUND
•We are going to round 368 to the
hundreds place. What digit is in the
hundreds place? 3 Let’s underline
the 3 so we remember the place we
are rounding to.
MP.2 Reason Quantitatively Continue
to have students use strategies to count
on to subtract.
UNIT 4
MP.2 Reason Abstractly and
Quantitatively Write 368 on the left side
of the board, leaving room above and
below the number. Ask students to do the
same on their MathBoards.
Contents
Planning
Research & Math Background
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.
is a conversation tool by which students formulate ideas
and analyze responses and engage in discourse. See also MP.6 Attend
to Precision.
MATH TALK
Teacher Edition: Examples from Unit 4
MP.3 Construct Viable Arguments
Compare Methods Ask students to
compare the two methods, explaining
what is different and what is the same.
Then, have students complete the
subtraction.
Lesson  13
ACTIVITY 1
What’s the Error? WHOLE CLASS
MP.3, MP.6 Construct Viable
Arguments/Critique Reasoning of
Others Puzzled Penguin Give students
time to read the letter from Puzzled
Penguin on Student Book page 228. Then
ask for volunteers to tell how they would
respond. Ask a volunteer to go to the
board and explain why Puzzled Penguin
didn’t round correctly.
Lesson  6
Mathematical Practice 3 is integrated into Unit 4 in the following ways:
Construct a Viable Argument
Critique the Reasoning of Others
Puzzled Penguin
413T | UNIT 4 | Overview
Compare Methods
Justify Conclusions
ACTIVITY 2
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
ACTIVITY 1
sum
224
138
addend
86
addend
Explain to students that they can write
an addition equation that shows that the
two addends in the Math Mountain add
to make the total.
Lesson  14
ACTIVITY 1
Mathematical Practice 4 is integrated into Unit 4 in the following ways:
Model with Mathematics
Write an Equation
Draw a Diagram
UNIT 4 | Overview | 413U
MATH BACKG ROUND
Lesson  17
MP.4 Model with Mathematics Draw
a Diagram Make a Math Mountain for
Problem 1. You might tell students that
they can visualize the total at the top
breaking into two pieces, one of which
rolls down one side of the mountain and
one of which rolls down the other side.
UNIT 4
MP.1, MP.4 Make Sense of Problems/
Model with Mathematics Write an
Equation The multistep word problems
in this lesson are complex. Encourage
students to organize and keep track of
their work by taking notes and labeling
their drawings and equations.
Contents
Planning
Research & Math Background
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.
Teacher Edition: Examples from Unit 4
MP.5 Use Appropriate Tools Secret Code
Cards Ask students which Secret Code
Cards are needed to build 368. Choose a
volunteer to build the number.
400
368
300
300
60
8
6 0
3 0
8
Tell students that to figure out how
to round to the hundreds place, they
should “open up” the Secret Code Cards,
separating 3 hundreds from the rest of
the number.
Lesson  5
ACTIVITY 1
MP.5 Use Appropriate Tools Addition
Table Direct students’ attention to the
addition table. Make sure students can
find the sum of two numbers by locating
the first addend in the left column and
the second addend in the top row. Then
move to the right and down until the
column and row meet.
Point out that patterns in the addition
table can be found in each row, column,
and diagonal. Ask students to describe
patterns they see and to explain why the
patterns work this way.
+ 0 1
2
3
4
5
6
7
8
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10 11
2
2
3
4
5
6
7
8
9
10 11 12
3
3
4
5
6
7
8
9
10 11 12 13
4
4
5
6
7
8
9
10 11 12 13 14
5
5
6
7
8
9
10 11 12 13 14 15
6
6
7
8
9
10 11 12 13 14 15 16
7
7
8
9
10 11 12 13 14 15 16 17
8
8
9
10 11 12 13 14 15 16 17 18
9
9
10 11 12 13 14 15 16 17 18 19
9 10
10
10 10 11 12 13 14 15 16 17 18 19 20
Lesson  17
Mathematical Practice 5 is integrated into Unit 4 in the following ways:
Use Appropriate Tools
MathBoard
413V | UNIT 4 | Overview
Secret Code Cards
Addition Table
ACTIVITY 1
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, express 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
ACTIVITY 1
Select two or three students to show their
place value drawings and explain their
thinking. Encourage other students to listen
carefully and ask questions. Make sure
students give the label, or unit, for their
answer. If they forget, ask questions like the
following:
• You said the answer is 71. 71 what? 71 rolls
Lesson  3
MP.6 Attend to Precision Explain
Solutions Ask a volunteer to answer
Questions 5 and 6. Student explanations
should cover the following points:
MATH TALK
in ACTION
ACTIVITY 2
S tudents discuss the order of steps when
solving problems with more than one
step.
•There are not enough tens to subtract
9 tens. So, students need to ungroup a
hundred to make 10 more tens.
Can you change the order of the steps to solve
Problem 4?
•There are not enough ones to subtract
5 ones. So, students need to ungroup a
ten to make 10 more ones.
Why not?
Lesson  15
ACTIVITY 1
Matilda: No
Matilda: Because finding the number of pizzas
Liz delivered depends on knowing the number of
pizzas Finn delivered.
Lesson  17
ACTIVITY 1
Mathematical Practice 6 is integrated into Unit 4 in the following ways:
Attend to Precision
Describe Methods
Explain an Example
Explain Solutions
Explain a Representation
Puzzled Penguin
UNIT 4 | Overview | 413W
MATH BACKG ROUND
Lesson  11
Use the Solve and Discuss
structure for Problem 21. Invite a few
students to work at the board, while the
other students work on their MathBoards. Ask
students to solve using a place value drawing.
M AT H TA L K
UNIT 4
MP.6 Attend to Precision Describe
Methods Have student volunteers present
their different solution methods. Students
who used Math Expressions in a previous
grade may use the Expanded method
or the Ungroup First method. Other
students are likely to use the Common
U.S. method. These methods are shown
for your reference. Please do not “teach”
any of these methods at this time. Allow
students to show and explain the methods
they are already using.
Contents
Planning
Research & Math Background
Mathematical Practice 7
Look for 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
MP.7 Use Structure Have students build
the numbers, one at a time, with their
Secret Code Cards. For each number, ask
questions about place value, in mixed
order. For example:
•What digit is in the tens place?
•What is the value of the digit in the tens
place?
•What digit is in the thousands place?
•What is the value of the digit in the
thousands place?
•What digit is in the ones place?
MP.7 Look for Structure Identify
Relationships Tell students that, even
though they know the answer to
Problem 2, they should work through the
subtraction and make a proof drawing
to show how grouping in addition and
ungrouping in subtraction are related.
Ask students what subtraction you should
write. Then, ask them how to start the
proof drawing.
224
– 86
•What is the value of the digit in the
ones place?
•What digit is in the hundreds place?
•What is the value of the digit in the
hundreds place?
Lesson  3
ACTIVITY 1
Mathematical Practice 7 is integrated into Unit 4 in the following ways:
Look for Structure
Identify Relationships
Use Structure
413X | UNIT 4 | Overview
Lesson  14
ACTIVITY 1
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 This is an example of the
Associative Property of Addition, which
states that grouping the addends in
different ways does not change the sum.
Mat h Backgro und
Lesson  5
UNIT 4
MP.8 Use Repeated Reasoning
Generalize Some may already know the
rounding rule. Explain that because 450 is
exactly halfway between 400 and 500, it
is impossible to figure out how to round
it. For this reason, people have agreed to
round up whenever a number is exactly
halfway between 2 hundred numbers. So
we round 450 up to 500.
Problem 11 on Student Book page 223
asks students to explain the rule for
rounding to the nearest hundred.
•Do you think these properties work for
subtraction, too? Explain. No, because
6 - 2 = 4, but 2 - 6 does not equal 4;
5 - (2 - 1) = 4 but (5 - 2) - 1 = 2.
Lesson  17
ACTIVITY 1
ACTIVITY 1
Mathematical Practice 8 is integrated into Unit 4 in the following ways:
Student EDITION: Lesson 18, pages 253–254
4-18
Name
Date
4-18
Class Activity
Name
Date
Class Activity
► Math and Maps
► Use a Table
The Pony Express was a mail service from St. Joseph,
Missouri, to Sacramento, California. The Pony Express
service carried mail by horseback riders in relays.
It took the Pony Express 10 days to
deliver letters between Sacramento
and St. Joseph. Today we send
emails that are delivered within a
few minutes. The chart below
shows the number of emails sent in
a month by different students.
Number of Emails Sent last Month
Name
Number
Robbie
Samantha
Ellen
Bryce
Callie
528
462
942
388
489
Use the information in the table for Problems 4–6.
Write an equation and solve the problem.
Use the information on the map for Problems 1–3.
Write an equation and solve the problem.
4. How many more emails did Robbie send than Callie?
39 more; 528 - 489 = n, n = 39
1. How many miles did the Pony Express riders travel
on a trip from Sacramento to Salt Lake City?
5. How many more emails did Ellen send than Bryce
and Samantha combined?
700 miles; 167 + 533 = n, n = 700
92 more; 942 - (388 + 462) = n, n = 92
2. The total distance from St. Joseph to Fort Laramie
is 616 miles. How many miles is it from Julesburg
to Fort Laramie?
6. Tamara said that Robbie and Bryce together sent
806 emails. Is her answer reasonable? Explain.
Then find the actual answer to see if you are correct.
168 miles; 616 - 178 - 270 = n or 616 = 178 + 270 + n, n = 168
No, her answer is not reasonable. 528 rounds to
3. Write and solve a problem that can be answered
using the map.
500 and 388 rounds to 400; together that is 900
emails. So, 806 is too low. 528 + 388 = 916
Problems may vary.
UNIT 4 LESSON 18
3_MNLESE824536_U04L18.indd 253
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (laptop) ©Stockbyte/Getty Images; (tablet) ©Tetra Images/Alamy Images
Focus on Mathematical Practices
Unit 4 includes a special lesson
that involves solving real world
problems and incorporates all eight
Mathematical Practices. In this lesson,
students use what they know about
reading a map and a table to solve
word problems involving addition
and subtraction.
Draw Conclusions
Identify Patterns
© Houghton Mifflin Harcourt Publishing Company
Use Repeated Reasoning
Generalize
Focus on Mathematical Practices
253
08/03/12 12:42 AM
254
UNIT 4 LESSON 18
3_MNLESE824536_U04L18.indd 254
Focus on Mathematical Practices
13/03/12 5:20 PM
Contents
Planning
Research & Math Background
Getting Ready to Teach Unit 4
Learning Path in the Common Core Standards
The lessons in this unit develop multidigit addition and subtraction
methods that are meaningful and easily used by students. Placevalue activities build understanding of the base ten numeration
system and provide the foundation to understand the grouping and
ungrouping concepts that students use to add and subtract. Students
use drawings to show grouping and ungrouping, and then describe
and discuss the process.
The activities in this unit help students gain practical understanding
of addition and subtraction and the relationship between the two
operations. They begin to see addition and subtraction as inverse
operations and apply their knowledge of these concepts and skills
to problem solving. Estimation provides students with methods to
validate their answers.
Help Students Avoid Common Errors
Math Expressions gives students opportunities to analyze and correct
errors, explaining why the reasoning was flawed.
In this unit, we use Puzzled Penguin to show typical errors that
students make. Students enjoy teaching Puzzled Penguin the correct
way, why this way is correct, and why Puzzled Penguin made the
error. Common errors are presented in Puzzled Penguin features in
the following lessons:
→ Lesson 6: When rounding to the hundreds, rounding
down although the number is closer to the greater
hundred
→ Lesson 8: Misaligning place values when adding
multidigit numbers
→ Lesson 11: When subtracting multidigit numbers,
subtracting the lesser digit from the greater digit in
each place rather than ungrouping
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 a part of
the Unit Test Teacher Edition pages, you will find a common
error and prescription listed for each test item.
413Z | UNIT 4 | Overview
Math Expressions
VOCABULARY
As you teach the unit, emphasize
understanding of these terms.
•ten stick
•hundred box
•thousand bar
•Secret Code Cards
•Counting On strategy
•Make a Ten strategy
•place value drawings
•proof drawing
•Show All Totals method
•New Groups Below
method
•New Groups Above
method
•Make a Thousand
­strategy
•ungrouping
•Math Mountain
See the Teacher Glossary.
Place Value
Concepts
Lessons
1
2
3
4
Place value drawings In Math Expressions, students use place value
drawings to help them conceptualize numbers and understand
the relative sizes of place values. Students begin by making these
drawings on the dot-grid side of their MathBoards. They show ones
by circling individual dots, tens by drawing lines through groups of
ten dots, and hundreds by drawing squares around groups of 100
dots. The terms ones, ten sticks, and hundred boxes are used to
describe the three representations. The drawing below represents the
number 247. It shows:
•2 hundred boxes (2 squares that each contain 100 dots) = 200
•4 ten sticks (4 line segments that each connect 10 dots) = 40
•7 ones (7 circles that each contain 1 dot) = 7
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
Base ten Units The power of the
base ten system is in repeated
bundling by ten: 10 tens make a
unit called a hundred. Repeating
this process of creating new units
by bundling in groups of ten
creates units called thousand, ten
thousand, hundred thousand . . . .
UNIT 4
MATH BACKG ROUND
The place value drawing is a beneficial model because it helps
students visualize the magnitude of numbers. For example, in this
model, students can see that the 2 in the hundreds place represents
200 dots, and they can develop a sense of the relative size of 200 dots.
Once students have a conceptual understanding of the number of
ones contained inside each place, they move to drawings without
dots. For example, the drawing below shows 176. Because these types
of drawings do not need to be perfectly scaled, students can make
them quickly. Grouping ten sticks and ones in subgroups of five helps
to avoid errors and to make the drawings easier to read.
1 hundred
7 tens
6 ones
Students broaden their understanding of place value as they extend
their models to include thousands. Students represent one thousand
using a thousand bar. Students then apply these understandings to
sketch models for numbers in the thousands. The model shows 2
thousands, 3 hundreds, 6 tens, and 8 ones, or 2,368.
UNIT 4 | Overview | 413AA
Contents
Research & Math Background
Planning
3 0 0 7 0 2
Secret Code Cards Students explore place value by assembling Secret
Code Cards to form multidigit numbers. The cards show place values.
On the front of each card, the value of a number in a certain place
appears, such as 300. The back of each Secret Code Card has a place
value drawing representation of the number shown on the front.
This drawing helps students to further understand the value of
each number by showing a pictorial representation of the base ten
form. To show 372, for example, students select cards representing
3 hundreds, 7 tens, and 2 ones. They can then show the number
pictorially, in base ten form, or in standard form:
300
Hundreds Card
Tens Card
Ones Card
2
70
2
7 0
3 0
2
70
70
3 0 0 7 0 2
300
300
2
70
7 0
3 0
2
3 0 0 7 0 2
2
300
2
70
2 of the number in the upper left corner.
Each card has a300small70version
So even after the number 372 is assembled, students can see that the
3 represents 300, the 7 represents 70, and the 2 represents 2. Students
continue to extend their understanding of place value as they use the
cards to model numbers in the thousands.
300
1000
300
200
300
7 0
3 0
2
30
7
3 0
3 0
3 2
1
7
Using the cards is beneficial for students because the cards emphasize
how the position of the digit in the number determines the value of
the digit. For example, with the cards students can more easily see
that a 2 on the hundreds card is 200, while a 2 on the tens card is 20.
413BB | UNIT 4 | Overview
Comparing, Rounding,
and Estimation
Lessons
1
5
6
Comparing An understanding that, in the base ten system, one of
a greater unit is always greater than nine or fewer of a lesser unit
provides students with the foundation necessary to compare numbers
of increasing value. To help students connect their understanding of
place value to comparing, they are encouraged to make place value
drawings. They can see from the drawings below that there are more
hundreds in 312 than 176, so 312 > 176.
Comparing Comparing magnitudes
of two-digit numbers draws on
the understanding that 1 ten
is greater than any amount of
ones represented by a one-digit
number. Comparing magnitudes
of three-digit numbers draws on
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
the understanding that 1 hundred
312
(the smallest three-digit number)
176
is greater than any amount of
tens and ones represented by a
two-digit number.
Rounding Students use their place
2
3 0 0 7 0 2
In this example, students use secret code cards to show that since 68
is closer to 100 than to 0, 368 rounds to 400.
6 0
3 0
8
when moving to the right across
400
368
300
7 0
3 0
2
8
2
60
numbers to the nearest 10 or 100.
They need to understand that
300
the places in a number (e.g., 456),
60
8
3 0 0 6 8
0
70
300
70
400
368
300
value understanding to round
300
Students use these models to extend their understanding of rounding
to rounding 4-digit numbers to the nearest hundred and 2- and
3-digit numbers to the nearest ten.
the digits represent smaller units.
When rounding to the nearest 10 or
100, the goal is to approximate the
number by the closest number with
no ones or no tens and ones (e.g.,
so 456 to the nearest ten is 460; and
to the nearest hundred is 500).
300
Estimation Throughout this unit, students apply their understanding
of rounding to estimate answers and to determine if an answer is
reasonable. They learn that they can find an approximate answer to a
problem involving addition or subtraction by rounding the numbers in
the problem and either adding or subtracting. They can use that answer
to estimate the solution or check a given solution to the problem.
UNIT 4 | Overview | 413CC
MATH BACKG ROUND
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
UNIT 4
Rounding In this unit, students learn how to use rounding to
estimate. They model rounding numbers using both place value
drawings and secret code cards. The model below helps students
understand that since there are more than 5 sticks, 368 rounds up to 400.
Contents
Addition of
Whole Numbers
Research & Math Background
Planning
Lessons
7
8
9
10
Computations The uniformity of the base ten system facilitates
understanding of place value concepts, but it also provides the
foundation for successfully completing standard algorithms for
computation within the base ten system. Once students understand
that numbers are composed of ones, tens, hundreds, and so on,
they can use this understanding to decompose and compose units in
computations.
Modeling Addition Before numeric methods are presented, Lesson 7
encourages students to use place value drawings to add. The
following place value drawing shows how students add 586 and 349.
The drawing allows students to visualize the regrouping of 10 ones as
1 ten and 10 tens as 1 hundred.
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
Computations Standard algorithms
for base ten computations with
the four operations rely on
decomposing numbers written in
base ten notation into base ten
units. The properties of operations
then allow any multidigit
computation to be reduced
to a collection of single-digit
computations. These single-digit
computations sometimes require
the composition or decomposition
586
of a base ten unit.
349
new hundred
new ten
586
349
9 hundreds
413DD | UNIT 4 | Overview
3 tens
5 ones
Numeric Addition Methods After students explore addition using
place value drawings, three methods for adding numbers are
presented. The variety of algorithms is beneficial for students because
it allows them to choose the algorithm that best suits their learning
style and the one that feels the most natural to them. The algorithms
themselves emphasize grouping and ungrouping to address common
errors that are learning obstacles for students. The examples below
show how to use these three numeric methods to add 249 and 386.
Students learn that the order in which the steps of the method are
recorded does not change the value of the answer.
Show All Totals
Method
New Groups
Below Method
Students add in each
place, record the total
for each place, then
add these totals to
find the sum. This can
be done from the left
or right.
Students record a
regrouped digit on
the line below the
next left column
beginning from the
right.
1 1
​
​   
249 
 
 
+
386
__
635
adding and subtracting within
1000. They achieve fluency with
strategies and algorithms that are
based on place value, properties of
operations, and/or the relationship
between addition and subtraction.
The regrouped digit
is recorded above the
next left column.
MATH BACKG ROUND
​   
249 
​
 
 
+
386
__
1 1
635
Algorithms Students continue
UNIT 4
​  249 
​
 
 
  
+ 386
__
500
​
 
 
​  120 
  
+ 15
__
635
New Groups
Above Method
from THE PROGRESSIONS FOR
THE COMMON CORE STATE
STANDARDS ON NUMBER AND
OPERATIONS IN BASE TEN
Proof Drawings Proof drawings are used to visually illustrate
the grouping process in addition and the ungrouping process in
subtraction. At first, students link each step of a proof drawing to
each step of a numerical method. Students then begin to do only the
numerical method, but they can think of a drawing to self-correct.
Occasionally, it is helpful for students to make a proof drawing to
explain their numerical method to someone else and to keep the
meanings attached to the numerical method. Students use a proof
drawing, such as the one below, to show that 249 + 386 = 635
6 hundreds
3 tens
5 ones
UNIT 4 | Overview | 413EE
Contents
Subtraction of
Whole Numbers
Planning
Research & Math Background
12
14
Lessons
11
13
Modeling Subtraction As with addition, before numeric subtraction methods
are presented, students are encouraged to use place value drawings to subtract.
The following place value drawings show two ways that students might model
134 – 58. Notice that, although students’ models look different, in each model,
they ungroup a hundred and a ten and cross out to show taking away 58.
Student 1
Student 2
Numeric Subtraction Methods Lesson 11 presents numeric methods for subtracting
numbers in the hundreds. The subtraction algorithms parallel the addition algorithms
students have learned. This parallel presentation benefits students because they can
use their understanding of place value and addition to subtract.
In numeric addition methods, students use composition to group units of lesser
values to units of greater values, for example, grouping 10 ones as 1 ten, or 10 tens as 1 hundred. With subtraction, students use the opposite action. They
decompose numbers to ungroup units of greater value into units of lesser value.
For example, they ungroup 1 ten to 10 ones or 1 hundred to 10 tens. The following
examples show how students use three numeric methods to subtract 58 from 134.
Expanded Method
Ungroup First Method
Students write each number in Students determine the
expanded form. They ungroup necessary ungrouping. Then
they subtract in any direction.
as needed to subtract. They
subtract in each place. Then
they add the differences.
413FF | UNIT 4 | Overview
Common U.S. Method
Students ungroup tens,
subtract ones, ungroup
hundreds, subtract tens, and
so on.
Common Errors To address students’ common errors in problems
where it is necessary to ungroup multiple times, students are
encouraged to do all the ungrouping before subtracting. Students
can ungroup either from left to right or from right to left. Once
everything is ungrouped, students can subtract the places in any
order. To facilitate students’ decisions about when to ungroup,
students draw a “magnifying glass” around the top number.
This enables them to focus in on the digits in the first number to
determine if there are enough in each place value to subtract.
UNIT 4
As with addition, students are asked to make proof drawings to show
ungrouping and to verify that their answers are correct.
MATH BACKG ROUND
Subtracting Across Zeros In Lesson 12, students subtract across zeros.
The work that students have done connecting numeric methods
to conceptual understanding facilitates understanding of how to
subtract across zeros. The following example shows how students
can use the Ungroup First method to subtract 138 from 500. Asking
students to use a proof drawing along with the numeric method
encourages them to connect the conceptual underpinnings of
subtraction with each algorithmic step.
9
4 1010
500
– 138
362
UNIT 4 | Overview | 413GG
Contents
Relating Addition
and Subtraction
Research & Math Background
Planning
Lessons
14 16
Math Mountains In Lesson 14, a Math Mountain is introduced to
help students conceptualize the relationship between addition and
subtraction. Students visualize the total at the top of the mountain
breaking into two pieces, one of which rolls down one side and one
of which rolls down the other side. The following shows a Math
Mountain for 224, 138, and 86.
sum
224
138
addend
86
addend
Students can use the Math Mountain to write related addition and
subtraction equations, such as:
138
addend
+
86 =
224
addend sum
and
224
sum
-
=
86
138
addend addend
By thinking about addition and subtraction in terms of a total and
two parts, students can understand the relationship between addition
and subtraction as well as use the Math Mountain to represent the
relationships in word problems.
Grouping and Ungrouping Students analyze the proof drawings for
related addition and subtraction problems to connect the grouping
in addition with the ungrouping in subtraction. As
they develop this skill, they begin to see that the
138
before grouping model for addition correlates to
+ 86
11
the after ungrouping model for subtraction. The
224
after grouping model for addition correlates to the
before ungrouping model for subtraction. Notice
11
that when adding in the example below, students
1 12 14
group 1 hundred, 11 tens, and 14 ones to get
224
2 hundreds, 2 tens, and 4 ones. When subtracting,
-86
138
students ungroup 2 hundreds, 2 tens, and 4 ones as
1 hundred, 11 tens, and 14 ones.
413HH | UNIT 4 | Overview
Before grouping: 1h 11t 14o
After grouping: 2h 2t 4o
Before ungrouping: 2h 2t 4o
After ungrouping: 1h 11t 14o
Lessons
5
Problem
Solving
6
9
10
15 16
17
Problem Solving Plan In Math Expressions a research-based problem
solving approach that focuses on problem types is used.
•Interpret the problem
•Represent the situation
•Solve the problem
•Check that the answer makes sense
Focus on
Mathematical Practices
672 addend
228
addend
672 + 228 = 900
900 - 228 = 672
Lesson
18
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 adding and subtracting whole numbers to solve
problems involving maps and email.
UNIT 4 | Overview | 413II
MATH BACKG ROUND
Multistep Problems The equations that students write for problems
with more than one step may include more than one operation.
Sometimes the order in which these operations are completed is not
important. Students are encouraged to consider a variety of ways to
solve different problems.
sum
900
UNIT 4
Choosing the Operation The unit presents, in the same lesson, word
problems that can be solved using addition and subtraction. This
gives students the opportunity to decide which operation to use.
Students learn to attend closely to the structure of the problem and
use a model to help them to determine what operation is indicated.
Students are asked to show their work by writing an equation to
represent the problem.
Contents
Planning
Research & Math Background
NOTES:
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
Building a Math Talk Community
M AT H TA L K
Frequent opportunities for students to explain
their mathematical thinking strengthen the learning
community of your classroom. As students actively question,
listen, and express ideas, they increase their mathematical
knowledge and take on more responsibility for learning. Use
the following types of questions as you build a Math Talk
community in your classroom.
Elicit student thinking
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
• So, what is this problem about?
• Tell us what you see.
• Tell us your thinking.
Support student thinking
• What did you mean when you said _______?
• What were you thinking when you decided to _______?
• Show us on your drawing what you mean.
• Use wait time: Take your time…. We’ll wait….
_________________________________________________
Extend student thinking
_________________________________________________
• Restate: So you’re saying that
_________________________________________________
• Now that you have solved the problem in that way, can you
think of another way to work on this problem?
_________________________________________________
_________________________________________________
• How is your way of solving like _______’s way?
• How is your way of solving different from _________’s way?
Increase participation of other students in the conversation
_________________________________________________
• Prompt students for further participation: Would someone
like to add on?
_________________________________________________
• Ask students to restate someone else’s reasoning: Can you
repeat what _______ just said in your own words?
_________________________________________________
• Ask students to apply their own reasoning to someone
else’s reasoning:
_________________________________________________
• Do you agree or disagree, and why?
• Did anyone think of this problem in a different way?
_________________________________________________
_________________________________________________
_________________________________________________
• Does anyone have the same answer, but got it in a
different way?
• Does anyone have a different answer? Will you explain
your solution to us?
Probe specific math topics:
_________________________________________________
_________________________________________________
_________________________________________________
413JJ | UNIT 4 | Overview
• What would happen if _______?
• How can we check to be sure that this is a correct answer?
• Is that true for all cases?
• What pattern do you see here?