Thank you for participating
in Teach It First!
This Teach It First Kit contains a Common Core Support Coach, Foundational
Mathematics teacher lesson followed by the corresponding student lesson. We are
confident that using this lesson will help you achieve your assessment preparation
goals for your entire class.
Common Core Support Coach was built from the ground-up to provide students
with scaffolded support for content required by the Common Core State Standards
for Mathematics (CCSS-M). The program features 20 lessons which focus on
20 critical skills essential to mastery of Common Core State Standards. These
skills were chosen because they are critical components of the mathematical
progressions that connect earlier standards to this grade level and from this grade
level to the next. While each lesson connects to prior foundational skills and
concepts, it can be viewed as an independent unit of instruction. In this way, the 20
lessons allow teachers to differentiate instruction according to the requirements of
each student.
Key to the philosophy behind Common Core Support Coach is the recognition that
math skills and concepts are part of a progression that begins early in students’
lives and continues beyond their current grade level with increased complexity and
depth. For students, achieving true understanding of CCSS-M at any grade level
means mastery of prior content that connects to this grade and mastery of content
that connects within the grade. Often, students who cannot cope with a specific
part of their grade’s curriculum are missing one or more understandings that would
allow mastery. Common Core Support Coach supplies the missing pieces.
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LESSON
12
S
uppor
tCoa
c
h,
Ma
t
he
ma
t
i
c
s
,
T
e
a
c
he
rE
di
t
i
on,
Gr
a
de4
Adding and Subtracting Fractions
and Mixed Numbers
Plug In Adding and Subtracting Fractions
FOUNDATIONAL
UNDERSTANDING
Plug In
Adding and
Subtracting
Fractions
Student Edition .
pp. 114–115
power up
Decomposing
Fractions
ON-LEVEL
TARGET
ready to go
Adding and
Subtracting
Fractions and
Mixed Numbers
MATERIALS
• Math Tool: Labeled
Fraction Strips, p. A25
(Student Edition p. 257)
OBJECTIVES
• Add and subtract
fractions with like
denominators.
CONCEPTS AND SKILLS
VOCABULARY
Add fractions by joining parts of a
whole, and subtract fractions by
separating parts of a whole.
CCSS: 4.NF.3.a
• Understand how to show Show how to decompose fractions
a fraction as the sum of
and mixed numbers using the
its parts.
sum of smaller fractions and visual
models.
CCSS: 4.NF.3.b
• Add and subtract mixed
numbers, and solve
word problems with
mixed numbers.
• like
denominators
• sum
• difference
• unit fraction
• mixed number
Rewrite mixed numbers as improper • improper
fractions to add and subtract.
fraction
Solve word problems involving
addition and subtraction of mixed
numbers with like denominators.
CCSS: 4.NF.3.c, 4.NF.3.d
Build Background
Talk to students about reasons to add or subtract fractions in real life. 3
1
__
For example, you have ​ __
4 ​ quart of milk. You drink ​ 4  ​ quart of the milk. Explain that subtracting the fractions can tell you how much milk is left.
MP4 Have students discuss additional examples of real situations that
involve adding and subtracting fractions.
Tell students they will work on understanding how to use fraction strips
and equations to add and subtract fractions.
english language
learners
MP4 Use Math Tool: Labeled
Fraction Strips to demonstrate
a whole divided into thirds,
fourths, fifths, and sixths, to
help ELL students understand
parts and whole. Have
students label each part, say
the name of each part, then
complete the sentence frame:
This whole fraction strip shows
____ (three thirds, four fourths,
five fifths, or six sixths).
90 Le ss o n 1 2
CC13_Mth_G4_TM_PDF.indd 90
Introduce and Model
Introduce Concepts and Vocabulary Guide students through the
information about adding and subtracting fractions. Emphasize the like
denominators. Use Words to Know to clarify students’ understanding
of vocabulary. MP7 Have students give examples of other fraction pairs
with like denominators. MP2 Ask students to demonstrate to a partner
their understanding of sum and difference.
Support Discussion MP1 Have partners discuss briefly before group
discussion. As needed, have students name each denominator.
© 2014 Triumph Learning, LLC
20/05/13 4:01 PM
LESSON
Plug In
DO
Adding and Subtracting Fractions
1
2
Youcanaddorsubtractfractionswithlike denominatorsbyaddingorsubtracting
thenumerators .Thedenominatorstaysthesame .
Addingfractionsmeansjoiningparts .
1
1
111 __
2
__
____
53 __
3 13 5 3 Subtractingfractionsmeanstaking
awayparts .
1
4
2
Thesumis__
3 .
DIS
1
4
sum
theanswerinanaddition
problem
difference
theanswerina
subtractionproblem
114 Writethedifference .
1
__
Add__
4 14 .
1
2
1
__
Usetwo__
4piecestomodel4 .
2
__
Useone__
4piecetomodel4 .
1
1
5
3
5
1
Thereare two __
5piecesleft .
3
1
__
__
5 255
2
5
1
4
__
Add__
616 .
1
Checkthatthefractionshave
likedenominators .
Bothfractionshaveadenominatorof
2
Addthenumerators .
3
Writethesum .
1 1 4
5
1
4
__
65_________
5
61__
6
6
.
6
PRACTICE
1
4
1
3
Combineandcountthenumber
offractionpieces .
4
Writethesum .
1
4
1
1
4
4piecesinall .
Thereare three __
2
1
__
__
4 145
3
4
1
1
8
1
8
3
8
1
8
1
8
1
8
3
8
2
1
8
6
1
6
1
6
5
6
1
6
2
6 8
Add or subtract.
5
7
3 __
102__
105
2
10
2
1
4 __
51__
55
3
1
6
1
6
3
6
Labeled
Fraction Strips
can be found on p. 257.
5
115
Lesson 1 2
Prompt: How do you add (or subtract) fractions with
like denominators?
Sentence Starter: To add (or subtract) fractions with
like denominators…
Model Application
DO
1
5
Use fraction strips to add or subtract the fractions.
Writeanadditionorsubtractionsentenceof2fractionswith
likedenominators .Describethesentence .
3
2
1
Possible response: __
6 1 __
6 5 __
6 . I wrote two fractions with the same denominator, 6.
3
The sum is __
6 .
A Youcanusefractionstripstoaddfractions .
2
4
DO
1
After I take away two __
4 pieces,
1
there is one __
4 piece left.
CU S S
DO
Countthenumberoffraction
piecesleft .
1
5
C Youcanaddandsubtractfractionsusingpaperandpencil .
1
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
like denominators Know
twoormore
denominatorsthatare
thesame
1
4
Thedifferenceis__
4 .
I see! There are
two __31 pieces.
Words
to
3
3
322
2
1
__
4 2__
4 5_____
4 5__
4 1
3
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
1
3
I am taking away
3
parts from __
5.
B Youcanuseafractionstriptosubtractfractions .
3 __
1
Subtract__
5 25 .
3
1
__
Usethree__
piecestomodel
5 .
5
1
1
__
Crossoutone__
piecetoshow
5 .
5
Lesson 12
12
12 Adding and Subtracting Fractions and Mixed Numbers
Adding and Subtracting Fractions
and Mixed Numbers
A Guide students through using fraction strips to add fractions. Remind them that they are joining parts of the same whole.
DO B Monitor to make sure students count the remaining parts
correctly. Remind them that they are separating parts of the same whole.
DO C Remind students that the fractions have the same denominators. Monitor to make sure students add only the numerators.
spotlight on
MATHEMATICAL LANGUAGE
MP6 Support students in
using mathematical language
as they work:
How can you tell if the
fractions have like
denominators?
Which number is the sum?
Which number is the
difference?
Practice and Assess
Ask students to complete practice items 1–4 on page 115 independently
or in pairs. Monitor ongoing work.
Observe whether students are correctly adding and subtracting
fractions. Use the chart below, as needed, to address any difficulties. Observation
Action
Students add or
subtract denominators
as well as numerators
for fractions with like
denominators.
Have students circle the numerators before
adding or subtracting. Remind students that the
denominators remain the same and that they
should only add or subtract the numerators.
© 2014 Triumph Learning, LLC
CC13_Mth_G4_TM_PDF.indd 91
Adding and S ubtracting F ractions and Mixed Numbers 91
20/05/13 4:01 PM
power up Decomposing Fractions
FOUNDATIONAL
UNDERSTANDING
Plug In
Adding and
Subtracting
Fractions
power up
Decomposing
Fractions
Student Edition .
pp. 116–117
ON-LEVEL
TARGET
ready to go
Adding and
Subtracting
Fractions and
Mixed Numbers
MATERIALS
• Math Tool: Labeled
Fraction Strips, p. A27
(Student Edition p. 261)
OBJECTIVES
• Add and subtract
fractions with like
denominators.
CONCEPTS AND SKILLS
Add fractions by joining parts of a
whole, and subtract fractions by
separating parts of a whole.
CCSS: 4.NF.3.a
• Understand how to show Show how to decompose fractions
a fraction as the sum of
and mixed numbers using the
its parts.
sum of smaller fractions and visual
models.
CCSS: 4.NF.3.b
• Add and subtract mixed
numbers, and solve
word problems with
mixed numbers.
VOCABULARY
• like
denominators
• sum
• difference
• unit fraction
• mixed number
Rewrite mixed numbers as improper • improper
fractions to add and subtract.
fraction
Solve word problems involving
addition and subtraction of mixed
numbers with like denominators.
CCSS: 4.NF.3.c, 4.NF.3.d
Build Background
Talk to students about reasons to decompose fractions in real life. 3
For example, if you need ​ __
4 ​ teaspoon of sugar in a recipe and you have
3
1
__
only a ​ 4  ​ teaspoon to measure, you can decompose ​ __
4 ​ into unit fractions to
3
1
__
__
find how many ​ 4  ​ teaspoons to use to get ​ 4 ​ teaspoon.
MP4 Have students discuss additional examples of real situations in
which they might need to decompose fractions.
Tell students they will learn how to show fractions and mixed numbers as
the sum of smaller fractions.
english language
learners
MP6 ELL students may need
extra practice in reading
the names of fractions and
mixed numbers, particularly
for halves and thirds. Write
several fractions and mixed
numbers on the board, and
have students read them
aloud. Then dictate various
fractions and mixed numbers,
and have students write them. Go over any discrepancies
with students until they are
confident about reading and
writing fractions and mixed
numbers.
92 Le ss o n 1 2
CC13_Mth_G4_TM_PDF.indd 92
Introduce and Model
Introduce Concepts and Vocabulary Guide students through the
information about decomposing fractions. Emphasize that the smaller
fractions have the same denominator as the original fraction. Use Words
to Know to clarify students’ understanding of vocabulary. MP6 Have
students use their own words to explain the difference between a unit
fraction and a mixed number. Support Discussion MP3 Have partners discuss briefly before group
discussion. As needed, have students review the unit fractions that were
used in the Introduction box.
5
Prompt: How can the denominator in the fraction __
​ 12  ​help you
answer this question?
5
Sentence Starter: T he denominator in __
​ 12  ​means I should
use…
© 2014 Triumph Learning, LLC
20/05/13 4:01 PM
12 Adding and Subtracting Fractions and Mixed Numbers
Decomposing Fractions
3
Youcanalsoshow__
4
anotherway .
Youcanseparate,or
decompose,afraction
intosmallerfractions .
3
1
4
Youcanshow__
4 asasum
ofunit fractions .
1
3
4
1
4
1
1
1
1
4
1
4
2
CU S S
1
4
1
4
1
4
1
4
4
1
Findthecorrectunitfraction .
2
Findthenumberofunit
4
fractionsthatequal__
6 .
3
116 4
Show__
6 asthesumoftwofractions .
Combinetheunitfractions
tomaketwofractions .
1
1
1
1
4 ____ ____ ____ ____
1 6 1 6 1 6 __
6 5 6 4
__
6 5
3
6
1
1__
6
Writethewhole-numberpart
asafraction .
3
Writethefractionpartasa
sumofunitfractions .
4
Writethemixednumberas
asumofthefractions .
CU S S
1
2
7
1
1
7
2 ____
__
____
15____
7 7 5 7 1 7 7
1
1
2
1__
75____
7 1____
7 1____
7 6
5 3
1
3 2
4
ways to separate them into two groups: __
7 1 __
7 ; __
7 1 __
7 ; __
7 1 __
7 .
PRACTICE
Write the fraction as the sum of two fractions.
5
1 __
9
1
9
5
__
95
1
9
1
9
1
9
1
9
3
2 1
4
Possible answers: __
1 __
9 ; __
9 1 __
9 9
1
Write the mixed number as the sum of fractions.
2
2 2__
35
3
3
1
2
3 1__
4 5
3
3
1
1
3
1
1
4
3
4
1
1
4
1
1
4
117
Lesson 1 2
Model Application
DO
1
Couldyouwrite__
7 asthesumoftwofractionsinmorethanoneway?
Explain .
1
Possible response: Yes, because there are six __
7 parts, there are many different
mixed number
anumberthathasawhole-numberpart
andafractionpart
Towrite__
12asthesumofunitfractions,whatunitfractionwouldyouuse?
Explain .
1
Possible response: I would use __
12 , because it has the same denominator, with 1 as the numerator.
A Youcanwriteafractionasthesumoffractions .
DO
DIS
I know! 1 whole
is the same as
4 fourths.
5
2
2
1__
7 5
1
24 4 4 4
unit fraction
afractionwith1inthenumerator
DIS
1
4
1
4
1
4
because it has 1 in
the numerator.
Know
1
4
1
4
1
4 4 4
1
__
4 is a unit fraction
Words
to
1
4
2__
451111__
4 4
Write1__
7 asthesumoffractions .
Writethemixednumberas
thesumofthewhole-number
partandthefractionpart .
1
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
3
4
1
4
1
4
2
DO
Youcanshowamixed numberasasumof
smallerfractions .
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
1
4
I remember! A
fraction with the
same numerator
and denominator
is equal to 1.
B Youcanwriteamixednumberasasumoffractions .
Lesson 12
POWER uP
A Guide students through writing the sum of fractions. Ask: What is
4
another way to write ​ __
6 ​as the sum of fractions?
DO B Monitor students as they write the mixed number as a sum of
fractions. Ask: What fraction can you write to show the whole number 1?
Support Discussion MP7 Have partners discuss briefly before group
discussion. As needed, have students list different addition facts for the
sum of 6.
Prompt: How will all of the fractions you add be the same?
How will they be different?
Sentence Starter: They will all have…
COMMON ERRORS
Students often make mistakes
writing whole numbers as
fractions. Use Math Tool:
Labeled Fraction Strips for 1
5
6
3 __
4 __
2 __
__
whole, ​ __
2 ​, ​ 3 ​, ​ 4 ​, ​ 5 ​, and ​ 6 ​. Have
students use the strips to help
them write the numerator for
each of the following fractions
so that it equals 1 whole, then
read each fraction aloud:
____
____ ____ ____
​  2   ​,  ​ ____
3   ​,  ​  4   ​,  ​  5   ​,  ​  6   ​. 
Practice and Assess
Ask students to complete practice items 1–3 on page 117 independently
or in pairs. Monitor ongoing work.
Observe whether students correctly write the sum of smaller fractions. Use the chart below, as needed, to address any difficulties.
Observation
Action
Students incorrectly
write the fraction
as a sum of smaller
fractions.
Have students answer these questions to find their
mistake and correct it: Are the denominators the
same as the denominator in the original fraction?
Is the sum of the numerators equal to the numerator
of the original fraction?
© 2014 Triumph Learning, LLC
CC13_Mth_G4_TM_PDF.indd 93
Adding and S ubtracting F ractions and Mixed Numbers 93
20/05/13 4:01 PM
and Subtracting Fractions
ready to go Adding
and Mixed Numbers
FOUNDATIONAL
UNDERSTANDING
Plug In
Adding and
Subtracting
Fractions
power up
Decomposing
Fractions
OBJECTIVES
• Add and subtract
fractions with like
denominators.
ON-LEVEL
TARGET
MATERIALS
• Lesson 12 Quiz,
Assessment Manual .
pp. 26–27
• Lesson 12 Quiz Answer
Key, Assessment Manual
• Math Tool: Labeled
Fraction Strips, p. A27
(Student Edition p. 261)
• Index cards (suggested)
Add fractions by joining parts of a
whole, and subtract fractions by
separating parts of a whole.
CCSS: 4.NF.3.a
• Understand how to show Show how to decompose fractions
a fraction as the sum of
and mixed numbers using the
its parts.
sum of smaller fractions and visual
models.
CCSS: 4.NF.3.b
ready to go • Add and subtract mixed
Adding and
Subtracting
Fractions and
Mixed Numbers
Student Edition .
pp. 118–123
CONCEPTS AND SKILLS
numbers, and solve
word problems with
mixed numbers.
VOCABULARY
• like
denominators
• sum
• difference
• unit fraction
• mixed number
Rewrite mixed numbers as improper • improper
fractions to add and subtract.
fraction
Solve word problems involving
addition and subtraction of mixed
numbers with like denominators.
CCSS: 4.NF.3.c, 4.NF.3.d
Build Background
Talk to students about reasons to add and subtract mixed numbers in real
2
1
__
life. For example, if you have 5​ __
3 ​ pounds of apples and use 2​ 3  ​ pounds to
make an apple pie, will you have enough apples for another apple pie?
Explain that subtracting mixed numbers can help you find the answer.
MP4 Have students discuss additional examples of real situations that
involve adding and subtracting mixed numbers.
Tell students they will add and subtract mixed numbers by writing
equivalent improper fractions.
Introduce and Model
english language
learners
MP6 To support ELL students’
understanding of different
kinds of fractions, have
them write examples of
the following terms: mixed
number, improper fraction,
and unit fraction. Then write
examples of each, and have
students label them and read
the labels aloud.
94 Le ss o n 1 2
CC13_Mth_G4_TM_PDF.indd 94
Introduce Concepts and Vocabulary Guide students through the
information about mixed numbers. Point out that students first rewrite
the mixed numbers as improper fractions before adding. Emphasize
to students to add only the numerators. Use Words to Know to clarify
students’ understanding of vocabulary. MP6 Ask: What is the value of
any fraction whose numerator and denominator are the same? What is
the name for this kind of fraction?
Support Discussion MP1 Have partners discuss briefly before group
discussion. If needed, have students look back at the Introduction box.
Prompt: How would you write the whole-number part of the
mixed number as the sum of fractions?
Sentence Starter: I would write the 2 as…
© 2014 Triumph Learning, LLC
20/05/13 4:01 PM
12 Adding and Subtracting Fractions and Mixed Numbers
WORk TOgEThER
Youcanwriteequivalent
improperfractionstoadd
andsubtractmixednumbers .
Toaddandsubtractmixednumbers,rewriteeachmixednumberasan
improper fraction .Add2__
111__
1 .
3
3
1
Write2__
3asasumof
fractions .
1
1
Write1__
3asasumof
fractions .
1
1__
3511__
3
5__
31__
3 1__
3
1
1
5__
31__
3
31311
5________
3
7
__
53
311
5____
3 4
__
53
__
2__
3511113
3
3
The equivalent
improper fraction
for 2__31 is __73 .
1
6
2
4 .
•Write1__
4as__
7
2
DO
1
7
7
1
2__
75__
7 1__
7 1__
7
I get it! I can add and
subtract mixed numbers
by finding equivalent
improper fractions. Then,
I can add and subtract the
improper fractions.
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
3
1
4
__
71__
7 5__
7
A mixed number can be written as the sum of fractions with the same denominator.
gO!
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
To add or subtract fractions with like denominators, add or subtract the numerators and keep the denominator.
POWER uP
3
Writeeachmixednumberas
animproperfraction .
2
Addtheimproperfractions .
3
Writetheanswerasamixed
number .
3
1__
55
1
2__
55
5
5
5
5
1
1
3
1
1__
5 12__
55
19
5
3
5
5
5
3
5
5
5
8
5
1
8
5
1
1
5
5
5
1
5
11
5
1
5
5
5
5
11
5
19
5
1
5 1 1 1 1 1 1
4
3 __
5 1
4
5
4
__
5 1__
512__
55
DIS
CU S S
Howisaddingmixednumbersdifferentthanaddingfractions?
Possible response: It is different because before you add, you may have to change the mixed numbers to improper fractions. 119
lesson link
Connect to Foundational Understanding Skills learned in the Plug In
and Power Up are referenced in the Lesson Link. Explain that adding
and subtracting fractions and writing mixed numbers as the sum of
fractions can help students use improper fractions to add and subtract
mixed numbers.
Work Together Explain that students will use improper fractions to
subtract mixed numbers. Point out that they should rewrite each mixed
number as an improper fraction before they subtract.
DO A Monitor students as they add. Remind students that the
denominator of the improper fraction is the same as the denominator of
the whole number.
Support Discussion MP1 MP6 Have partners discuss briefly before
group discussion. As needed, have students first explain the difference
between mixed numbers and fractions.
Prompt: What should you do to mixed numbers before you
can add or subtract them?
Sentence Starter: Before adding or subtracting mixed
numbers, I should…
CC13_Mth_G4_TM_PDF.indd 95
3
1
__
Add1__
5 125 .
1
Lesson 1 2
© 2014 Triumph Learning, LLC
7
3
7
4
__
4 5__
41__
4
5
1
6
13
A Youcanwritemixednumbersasimproperfractionstoadd .
numbers as __
2 1 __
2 1 __
2 5 __
2 . Plug In
118 2
6
2
2
3
1
2
4
__
511__
4 514
2
53__
3
Howwouldyouwrite2__
2asanimproperfraction?
1
1
Possible response: First, I would write 2 __
2 as 1 1 1 1 __
. Then write the whole 2
lInk
lEssOn
13
1
5__
4 1__
45__
4
1
2
improper fraction
Know
afractionwithanumeratorthatisgreaterthanorequaltothedenominator
CU S S
4
__ __ __ __
3__
4214 5424 54
531__
3
Words
to
DIS
7
3
3
1
2
__
__
3__
4214 514
5111111__
3
The equivalent
improper fraction
4
for 1__31 is __
3.
4
2
2
1__
4511__
4
•Write__
4 as1__
4 .
3
3
3
11
2
__
3 1__
31__
31__
3
3 5__
1
4
__ __ __ __
5__
4 14 14 1454
•Thedifferenceis__
4 .
Rewritethesumas
amixednumber .
1
2
__
3__
451111114
13
•Write3__
4as__
4 .
7
4
11
__
3 5__
3
3 1__
1
3
Addtheimproper
fractions .
1
I can rewrite
each whole as
4 fourths.
__
Subtract3__
4214 .
1
Lesson 12
READY TO gO
Adding and Subtracting Fractions
and Mixed Numbers
COMMON ERRORS
After students change the
whole-number part of the
mixed number to fractions,
they may forget to include
the fractional part in their
calculations. Have students
circle the fraction in the mixed
number and in the calculations
to help them remember to
include the fraction. SPOTLIGHT ON
MATHEMATICAL PRACTICES
MP6 Attend to Precision
Use Math Tool: Labeled
Fraction Strips to model the
Work Together example. Have students use fraction
strips for fourths to show the
subtraction. Have partners
explain to each other how the
fraction strips show the same
results as the Work Together
example.
Adding and S ubtracting F ractions and Mixed Numbers 95
20/05/13 4:01 PM
12 Adding and Subtracting Fractions and Mixed Numbers
READY TO gO
PRACTICE
Add. Make a model to check your answer. 1
3 __
3 2
2
1 1__
3 11__
35
2
__
1 1 3 5
__
2
2
1__
3 11__
35 3 1
3
10
__
__
3 5 3 1
2
1__
3 5
1
1
Add or subtract.
1
1
1
120 1
1
1
__
3
5
thicknessoftheboards?
1
1
7
__
8 17
__
8
1
__
8 1
5
8
__
8 15
__
8
2
5
1
8
__
8 7
__
8
2
__
8 5
1
5
8
__
8 15
__
8 1
__
8 1
5
17
6
__
6
6
__
6
1 1 1
6
1
__
__
1 6 5 6
16
7
__
6 __ 1 6 5
5
1
1
1
7
__
__
1 6 5 6 23
___
6 5
__
6
1
__
6
1
__
6
5
1
1
1
1
1
6
6
6
5
__
1 6 5
5
__
3 __
1 6 5 6
4
__
6 5
16
__
6 I have to rewrite the
mixed numbers as
improper fractions.
2
3 __
4 cm
1
3
Afterafewhours,thedepthdecreasedby1__
3
5 inches . 1 __
5 in.
Whatisthenewdepthofthewaterinthepuddle?
HINT
Sometimes the answer will not be an improper fraction.
DIS
CU S S
Follow the Steps 3
4
4
__
6
8
__
4 or 2
2 4 cm
1
7 Thewaterinapuddlewas3__
5inchesdeep .
__
8
5
3 __
6
2
Add__
51__
512__
5 .
Writethemixednumberasanimproperfraction:
2
2__
55
1
1
1
1
2
__
5 5
5
__
5 5
__
5 1
1
2
__
5 5
12
__
5 Addthefractions:
4
3
__
__
5 15 1
12
__
5 5
19
__
5 I separate the
improper fraction
into as many
wholes as I can,
with parts of a
whole left over.
Writetheanswerasamixednumber:
19
__
5 4
3
5
5
5
4
5
__
5 1
__
5 1
__
5 1
__
5 5
1
1
1
1
1
1
__
5 2
__
__
__
5 15 125 5
4
4
3 __
5 121
Less o n 1 2
ADDITIONAL PRACTICE
Provide students with
additional practice to add and
subtract mixed numbers and
fractions:
1
1
6 Acarpenterwillnailaboardthatis1__
4cmthickto
1
aboardthatis2__
4cmthick .Whatisthecombined
1
3 __
3
1 4 cm
1
7
4
1
2__
6 5
1
1
1__
5
6
4
1
2__
6 11__
65
23
___
6
1
1
2
1
__
2__
8218 5
4
1
3 2__
6 11__
65
1
3
1
5 __
411__
45
Solve.
1
__
3 Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
1
7
1__
8 5
1
3
__
3
__
8
1
7
2 2__
821__
8 5
2__
85
3
__
3 3
__
7 6
2
4 1__
7 2__
75
5
__
3 Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
5
2
3
__ __ 5 3 1 3 5
10
5
__
__
3 5
3
Add or subtract. 5
3
__
2​ __
6 ​ 2 1​ 6 ​ 
2
5​ __
5 ​ 2 4
__
1​ __
8 ​  1 ​ 8 ​
2 1 3​ __
3 ​
1
2
Support Independent Practice
1–3 Remind students to read the HINT. If needed, ask: What improper
fractions will you add? What improper fractions will you subtract?
4–5 What steps will you follow to solve these problems?
6–7 How do you know if you should add or subtract?
Support Discussion MP1 Have partners discuss briefly before group
discussion. As needed, have students read aloud each step.
Prompt: What should you do before adding these numbers?
Sentence Starter: The first thing I should do is…
english language
learners
MP6 To help ELL students
understand the meaning of
sum or difference, provide
examples of addition and
subtraction problems. Write
the words sum and difference
on index cards. Have students
label the correct card to show
the answer in each type of
problem. As students become
more familiar with the terms,
connect the words add
with sum and subtract with
difference. 96 Le ss o n 1 2
CC13_Mth_G4_TM_PDF.indd 96
Problem Solving
Model the Four-Step Method MP1 MP8 Guide students through the
four-step method using think-aloud strategies. Point out that students
can use fraction strips to check that their answers are correct.
1
Think Aloud When I check the answer, I first make a model to show 4​ __
4 ​. 
3
2
__
__
Then I cross out 3​ 4 ​to find that ​ 4 ​is left.
Support Problem-Solving Practice Have students use the Checklist as
they complete each step.
Prompt: What should you do before adding mixed numbers?
Prompt: What should you do to an answer that is an
improper fraction?
Prompt: How can you check your answer?
© 2014 Triumph Learning, LLC
20/05/13 4:01 PM
12 Adding and Subtracting Fractions and Mixed Numbers
READY TO gO
Use the problem-solving steps to help you. rise and shine!
READ
PlAn
1
1 Luisrodehisbike3__
3milesonSaturdayand
2
2__
3 milesonSunday .Howmanymilesdidhe
Sultanaismakingpancakesforbreakfast .
1
Therecipecallsfor4__
4cupsofflour .She
2
hasalreadymeasured3__
4 cups .Howmuch
moreflourdoesSultananeedtomeasure?
rideinall?
Youneedtosubtract
SoLvE
CHECK
4
2 Agneshashiked2__
5 kilometersofatrailthatis
1
kilometerslong .Howmuchfartherdoesshe
4__
5
havetogotohiketheentirelengthofthetrail?
2
3 __
4 from
1
4 __
4 .
CHECKLIST
READ
PLAN
SoLvE
2
1 __
5 kilometers
CHECK
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
17
1
____
4__
45 4 14
2
3__
4 5____
4 17
14
3
____ ____
Subtracttheimproperfractions:____
4 2 4 5 4 Makeamodel .
1
2
Shade4__
4 .Thencrossout3__
4 .
3
Sultananeedstomeasure
__
4 cupmoreflour .
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
1
2
__
Subtract4__
4234 .
Rewritethemixednumbersasimproperfractions .
122 READ
•Whatistheproblemaskingyoutofind?
flour
Theamountof
stillneededfortherecipe
•Howcanyousolvetheproblem?
ChECk
CHECKLIST
PLAN
6 miles
•Whatdoyouneedtoknowtosolvetheproblem?
1
4 __
Therecipecallsfor 4 cupsofflour .
2
3 __ Sultanahasmeasured 4 cupsofflour .
sOlvE
I can make a model to
check my answer!
PRACTICE
Lesson 12
PROblEm sOlvIng
7
3 Peetamixes__
8 cupoffrozenstrawberries,
5
2
__
8cupoffrozenblackberries,and__
8 cupoffrozen
blueberriestomakeasmoothie .Howmanycups
offruitareinthesmoothie?Writeyouransweras
animproperfractionandasamixednumber .
6
14
__
cups; 1 __
8 cups
8
CHECKLIST
READ
PLAN
SoLvE
CHECK
123
Lesson 1 2
Explore Student Thinking MP3 Invite students to explain their
reasoning for deciding on addition or subtraction to solve each problem. Have them provide a similar word problem that would be solved in the
same way.
spotlight on
mathematical practices
MP1 Solving Problems
After completing the word .
problems, have partners .
work together to check .
their answers by modeling .
the problem.
Assess
Use the table below to observe whether students accurately add and
subtract mixed numbers and to address any difficulties, as needed,
before the quiz.
When students are ready, assign the Lesson 12 Quiz.
1
Observation
Errors in adding or subtracting mixed
numbers are frequent; general confusion
about rewriting mixed numbers as
improper fractions.
Action
Have students use Math Tool: Labeled Fraction Strips for
1
2
__
fourths to model 2​ __
4  ​ 1 1​ 4 ​. Have them write the sum as an
improper fraction and a mixed number. Repeat with other
examples.
2
Observation
Makes occasional errors when adding
and subtracting mixed numbers; some
understanding of rewriting the numbers.
Action
Provide additional practice problems for adding and
subtracting mixed numbers. Guide students through each
step until they master the skill.
3
Observation
Adds and subtracts mixed numbers
accurately.
Action
Assign the Lesson 12 Quiz.
© 2014 Triumph Learning, LLC
CC13_Mth_G4_TM_PDF.indd 97
Adding and S ubtracting F ractions and Mixed Numbers 97
20/05/13 4:01 PM
LESSON
S
uppor
tCoa
c
h,
Ma
t
he
ma
t
i
c
s
,
S
t
ude
ntE
di
t
i
on,
Gr
a
de4
12
Adding and Subtracting Fractions
and Mixed Numbers
Plug In
Adding and Subtracting Fractions
You can add or subtract fractions with like denominators by adding or subtracting
the numerators. The denominator stays the same.
Adding fractions means joining parts.
1
111
2
1
3
3
3 2 2 __
2 _____
1
__
​ 4 ​ 2 ​ __
​  5 ​ 4 ​
4 ​ 5 ​  4   
1
3
1
4
2
The sum is ​ __
3 ​.
like denominators
two or more
denominators that are
the same
Know
DIS
CU S S
1
4
1
The difference is ​ __
4 ​. 
I see! There are
two ​ __31  ​pieces.
Words
to
1
4
After I take away two  __
​41  ​pieces,
there is one __
 ​41  ​piece left.
sum
the answer in an addition
problem
difference
the answer in a
subtraction problem
Write an addition or subtraction sentence of 2 fractions with like denominators. Describe the sentence. A You can use fraction strips to add fractions.
DO
114 2
1
__
Add ​ __
4 ​ 1 ​ 4 ​. 
1
2
1
__
Use two ​ __
4 ​  pieces to model ​ 4 ​.
2
__
Use one ​ __
4 ​  piece to model ​ 4 ​. 
1
1
4
1
3
Combine and count the number
of fraction pieces.
4
Write the sum.
There are
2
1
__
​ __
4 ​ 1 ​ 4 ​  5
1
4
1
1
4
​ __
4 ​  pieces in all.
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
1
__ ____​  5 ​ __ ​
​ __
3 ​ 1 ​ 3 ​ 5 ​  3   
3
Subtracting fractions means taking
away parts.
L e ss o n 1 2
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09/05/13 5:47 PM
12 Adding and Subtracting Fractions and Mixed Numbers
I am taking away
parts from ​__
 35 ​.
B You can use a fraction strip to subtract fractions.
3
DO
1
2
3
4
1
__
Subtract ​ __
5 ​ 2 ​ 5 ​. 
1
5
3
1
__
Use three ​ __
5 ​  pieces to model ​ 5 ​. 1
1
__
Cross out one ​ __
5 ​  piece to show ​ 5 ​. 
Count the number of fraction
pieces left. Write the difference.
1
5
1
5
3
5
1
​ __
5 ​  pieces left.
There are
3
1
__
​ __
5 ​ 2 ​ 5 ​  5
C You can add and subtract fractions using paper and pencil.
1
DO
4
__
Add ​ __
6 ​  1 ​ 6 ​.
1
Check that the fractions have
like denominators.
both fractions have a denominator of
2
Add the numerators.
3
Write the sum. 1
1
4 _________
__
 
​ 5
​ 
 
​  5
​ 6 ​  1 ​ __
6
6 
.
PRACTICE
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
Use fraction strips to add or subtract the fractions.
1
1
8
1
8
3
8
1
8
1
8
1
8
3
8
2
1
8
1
6
1
6
5
6
Add or subtract.
5
7
3 __
​ 10  ​ 2 ​ __
10  ​ 5
2
1
4 __
​ 5 ​ 1 ​ __
5 ​  5
1
6
1
6
1
6
2
6 Labeled
Fraction Strips
can be found on p. 257.
115
CC13_Mth_G4_SE_PDF.indd 115
09/05/13 5:47 PM
POWER UP
Decomposing Fractions
3
You can also show ​ __
4 ​ another way.
3
1
4
You can show ​ __
4 ​ as a sum
of unit fractions.
1
4
3
4
1
4
1
3
4
1
4
1
1
4
1
4
You can show a mixed
number as a sum of
smaller fractions.
2
4
1
4 4 4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
5
To write ​ __
12  ​  as the sum of unit fractions, what unit fraction would you use?
Explain.
116 4
Show ​ __
6 ​ as the sum of two fractions.
1
Find the correct unit fraction.
2
Find the number of unit
4
fractions that equal ​ __
6 ​.
3
1
mixed number
a number that has a whole-number part
and a fraction part
A You can write a fraction as the sum of fractions.
DO
4
I know! 1 whole
is the same as
4 fourths.
unit fraction
a fraction with 1 in the numerator
CU S S
DIS
1
4
24 4 4 4
because it has 1 in
the numerator.
Know
1
__
2​ __
4 ​  5 1 1 1 1 ​ 4 ​ 
1
4
__
​ 41  ​is a unit fraction
Words
to
1
1
4
Combine the unit fractions
to make two fractions.
1
4 ____ ____ ____ ____
​  1 ​  6   ​  1 ​  6    ​ 1 ​  6   ​ 
​ __
6 ​ 5 ​  6   
4
​ __
6 ​ 5
1
1 ​ __
6 ​  
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
You can separate, or
decompose, a fraction
into smaller fractions.
L e ss o n 1 2
CC13_Mth_G4_SE_PDF.indd 116
09/05/13 5:47 PM
12 Adding and Subtracting Fractions and Mixed Numbers
I remember! A
fraction with the
same numerator
and denominator
is equal to 1.
B You can write a mixed number as a sum of fractions.
2
DO
Write 1​ __
7 ​ as the sum of fractions.
Write the mixed number as
the sum of the whole-number
part and the fraction part.
1
Write the whole-number part
as a fraction.
2
Write the fraction part as a
sum of unit fractions.
3
CU S S
DIS
1
1
2
__ ​ 5 ​ ____
____
1 5 ​ ____
 
7   ​ ​ 
7
7   ​  1 ​  7   ​ 
2
____ ____ ____
1​ __
7 ​ 5 ​  7   ​  1 ​  7   ​  1 ​  7   ​ 
Write the mixed number as
a sum of the fractions.
4
2
1​ __
7 ​ 5
6
Could you write ​ __
7 ​ as the sum of two fractions in more than one way?
Explain.
PRACTICE
Write the fraction as the sum of two fractions.
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
5
1 __
​ 9 ​
1
9
5
__
​ 9 ​ 5
1
9
1
9
1
9
1
9
1
Write the mixed number as the sum of fractions.
2
2 2​ __
3 ​ 5
2
3 1​ __
4 ​ 5
3
3
1
1
1
1
1
117
CC13_Mth_G4_SE_PDF.indd 117
09/05/13 5:47 PM
Adding and Subtracting Fractions
and Mixed Numbers
READY TO GO
To add and subtract mixed numbers, rewrite each mixed number as an improper fraction. Add 2__
​ 1 ​ 1 1__
​ 1 ​. 
3
1
Write 2​ __
3 ​  as a sum of
fractions.
1
3
1
Write 1​ __
3 ​  as a sum of
fractions.
1
__
1​ __
3 ​  5 1 1 ​ 3 ​  
1
1
__
5 ​ __
3 ​ 1 ​ 3 ​ 
__
2​ __
3 ​  5 1 1 1 1 ​ 3 ​  
3
3
__ __
5 ​ __
3 ​ 1 ​ 3 ​ 1 ​ 3 ​ 
31311
5 ​ ________
 
​  
3 
311
7
4
5 ​ __
3 ​
The equivalent
improper fraction
for 2​__
 31  ​is ​__
 73 ​.
improper fraction
a fraction with a numerator that is greater than or equal to the denominator
CU S S
DIS
1
How would you write 2​ __
2 ​  as an improper fraction?
lesSon link
Plug In
To add or subtract
fractions with like
denominators, add or subtract the numerators and keep the denominator.
3 __
1
4
__
​ 7 ​  1 ​ __
7 ​ 5 ​ 7 ​
118 The equivalent
improper fraction
for 1​__
 31  ​is ​__
 43 ​.
3 __
3 __
3 __
11
2
__
 
​ 1
​ 
 
​ 1
​ 
 
​ 1
​ 
​ 3  ​ 5 ​ __
3
3
3
3 ​
2
5 1 1 1 1 1 1 ​ __
3 ​
2
5 3 1 ​ __
3 ​ 2
5 3​ __
3 ​
POWER UP
A mixed number can be
written as the sum of
fractions with the same
denominator.
1
7
7
1
__ __ __
2​ __
7 ​  5 ​ 7 ​ 1 ​ 7 ​ 1 ​ 7 ​ 
GO!
I get it! I can add and
subtract mixed numbers
by finding equivalent
improper fractions. Then,
I can add and subtract the
improper fractions.
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
Know
Rewrite the sum as a mixed number.
1
5 ​ ____
​  
3   
5 ​ __
3 ​
Words
to
7
4 __
11
__
​ 3 ​ 1 ​ __
3 ​ 5 ​ 3  ​
1
3
Add the improper
fractions.
L e ss o n 1 2
CC13_Mth_G4_SE_PDF.indd 118
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12 Adding and Subtracting Fractions and Mixed Numbers
work together
You can write equivalent improper fractions to add and subtract mixed numbers.
1
13
2
6
1
1
4
2
4
4
13
1
__ __ __ __
5 ​ __
4 ​ 1 ​ 4 ​ 1 ​ 4 ​ 1 ​ 4 ​  5 ​ 4  ​
2
2
__
1​ __
4 ​ 5 1 1 ​ 4 ​
7
4
6
2
__ __
5 ​ __
4 ​ 1 ​ 4 ​ 5 ​ 4 ​
3
__
•Write ​ __
4 ​ as 1​ 4 ​.
1
1
__
3​ __
4 ​  5 1 1 1 1 1 1 ​ 4 ​ 
•The difference is ​ __
4 ​.
7
2
__
Subtract 3​ __
4 ​  2 1​ 4 ​.
__
•Write 3​ __
4 ​  as ​ 4  ​.
__
•Write 1​ __
4 ​ as ​ 4 ​.
I can rewrite
each whole as
4 fourths.
1
2
6
13
7
__ __ __ __
3​ __
4 ​  2 1​ 4 ​ 5 ​ 4  ​ 2 ​ 4 ​ 5 ​ 4 ​
3
3
7
4 __
__
​ 4 ​ 5 ​ __
4 ​ 1 ​ 4 ​ 3
3
__
5 1 1 ​ __
4 ​ 5 1​ 4 ​
__
__
3​ __
4 ​  2 1​ 4 ​ 5 1​ 4 ​
A You can write mixed numbers as improper fractions to add.
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
DO
3
1
__
Add 1​ __
5 ​ 1 2​ 5 ​. 
1
Write each mixed number as
an improper fraction.
2
Add the improper fractions.
3
Write the answer as a mixed
number.
3
1​ __
5 ​ 5
1
2​ __
5 ​  5
3
1
5
1
1
1
__
1​ __
5 ​ 1 2​ 5 ​  5
3
5
1
5
5
1
1
1
5
1
1
1
1
__
1​ __
5 ​ 1 2​ 5 ​  5
CU S S
DIS
How is adding mixed numbers different than adding fractions? 119
CC13_Mth_G4_SE_PDF.indd 119
09/05/13 5:47 PM
Ready to Go
practice
Add. Make a model to check your answer.
2
2
__
1 1​ __
 
​ 1
1​ 
3
3 ​ 5
2
1​ __
3 ​ 5
2
1
2
__
1​ __
3 ​ 1 1​ 3 ​5
5
1
1
5
5
5
1
1
1
5
1
1
1
5
Add or subtract.
1
7
__
2 2​ __
8 ​  2 1​ 8 ​ 5
1
2​ __
8 ​  5
7
1​ __
8 ​ 5
1
1
1
5
1
1
5
1
5
7
__
2​ __
8  ​ 2 1​ 8 ​ 5
2
1
5
HINT
Sometimes the
answer will not
be an improper
fraction.
5
4
2​ __
6 ​ 5
1
1​ __
6 ​  5
4
1
__ 5
2​ __
6 ​ 1 1​ 6 ​ 
120 1
1
5
1
1
5
1
5
1
1
5
5
1
1
1
5
1
1
1
5
5
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4
1
__
3 2​ __
6 ​ 1 1​ 6 ​  5
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12 Adding and Subtracting Fractions and Mixed Numbers
Add or subtract.
6
2 __
4 1​ __
7 ​ 2 ​ 7 ​ 5
3
1
5 __
​ 4 ​ 1 1​ __
4 ​  5
Solve.
I have to rewrite the
mixed numbers as
improper fractions.
1
6 A carpenter will nail a board that is 1​ __
4 ​  cm thick to
1
a board that is 2​ __
4 ​  cm thick. What is the combined
thickness of the boards?
1
1
1 4 cm
2 4 cm
1
7 The water in a puddle was 3​ __
5 ​  inches deep. 3
After a few hours, the depth decreased by 1​ __
5 ​ inches. What is the new depth of the water in the puddle?
DIS
CU S S
Follow the Steps
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
3
4 __
2
__
Add ​ __
5 ​ 1 ​ 5 ​ 1 2​ 5 ​.
Write the mixed number as an improper fraction:
2
2​ __
5 ​ 5
1
1
5
1
1
5
Add the fractions:
3
4 __
​ __
 
​ 1
​ 
5
5 ​ 1
5
Write the answer as a mixed number:
4
5
1
1
1
5
1
1
1
3
I separate the
improper fraction
into as many
wholes as I can,
with parts of a
whole left over.
2
__
__
​ __
5 ​ 1 ​ 5 ​ 1 2​ 5 ​ 5
121
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Ready to Go
Problem solving
Rise and Shine!
READ
Sultana is making pancakes for breakfast. 1
The recipe calls for 4​ __
4 ​  cups of flour. She
2
has already measured 3​ __
4 ​ cups. How much
more flour does Sultana need to measure?
PLAN
•What is the problem asking you to find?
the amount of
still needed for the recipe
•What do you need to know to solve the problem?
The recipe calls for
cups of flour.
Sultana has measured
cups of flour.
•How can you solve the problem?
You need to subtract
1
.
2
__
Subtract 4​ __
4 ​  2 3​ 4 ​.
Rewrite the mixed numbers as improper fractions. 1
____
4​ __
4 ​  5 ​  4   ​ 
2
____
3​ __
4 ​ 5 ​  4   ​ 
____ ____
Subtract the improper fractions: ​ ____
4    ​ 2 ​  4   ​  5 ​  4   ​ 
CHECK
Make a model. 1
2
  Then cross out 3​ __
Shade 4​ __
4 ​. 4 ​. Sultana needs to measure
122 cup more flour.
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SOLVE
from
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12 Adding and Subtracting Fractions and Mixed Numbers
practice
I can make a model to
check my answer!
Use the problem-solving steps to help you.
1
1 Luis rode his bike 3​ __
3 ​  miles on Saturday and
2
2​ __
3 ​ miles on Sunday. How many miles did he
ride in all?
CHECKLIST
Read
Plan
Solve
Check
4
2 Agnes has hiked 2​ __
5 ​ kilometers of a trail that is
1
4​ __
5 ​  kilometers long. How much farther does she
have to go to hike the entire length of the trail?
CHECKLIST
Read
Plan
Solve
Duplicating any part of this book is prohibited by law. © 2014 Triumph Learning, LLC
Check
7
3 Peeta mixes ​ __
8 ​ cup of frozen strawberries,
5
2
__
​ 8 ​ cup of frozen blackberries, and ​ __
8 ​ cup of frozen
blueberries to make a smoothie. How many cups
of fruit are in the smoothie? Write your answer as
an improper fraction and as a mixed number.
CHECKLIST
Read
Plan
Solve
Check
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