Fraction and Mixed-Number
Addition and Subtraction
Objective To guide students in the use of pattern blocks to
add and subtract fractions and mixed numbers.
a
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Teaching the Lesson
Key Concepts and Skills
• Identify the whole or the ONE. [Number and Numeration Goal 2]
• Represent fractions and mixed numbers
with pattern blocks. [Number and Numeration Goal 2]
• Identify equivalent fractions. [Number and Numeration Goal 5]
• Model fraction and mixed-number addition
and subtraction with pattern blocks. [Operations and Computation Goal 5]
Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Ongoing Learning & Practice
1 2
4 3
Playing Angle Tangle
Student Reference Book, p. 230
Math Masters, p. 457
protractor straightedge
Students practice measuring and
estimating the measures of angles.
Math Boxes 7 5
Math Journal 2, p. 199
Students practice and maintain skills
through Math Box problems.
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Exploring Fractions that Sum to One
Math Masters, p. 217
Students divide cakes into thirds, fourths,
sixths, and eighths and write number models.
EXTRA PRACTICE
Solving Frames-and-Arrows Problems
Math Masters, p. 393
Students practice adding and subtracting
fractions.
Study Link 7 5
• Identify a triangle, hexagon, trapezoid,
and rhombus. [Geometry Goal 2]
Math Masters, p. 216
Students practice and maintain skills
through Study Link activities.
Key Activities
Students model fraction and mixed-number
sums and differences with pattern blocks.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 198. [Operations and Computation Goal 5]
Materials
Math Journal 2, pp. 198–198B
Study Link 7 4
pattern blocks calculator slate overhead pattern blocks (optional) colored
chalk (optional)
Advance Preparation
For the optional Readiness activity in Part 3, obtain a copy of Full House: An Invitation to Fractions
by Dayle Ann Dodds (Candlewick Press, 2009).
Teacher’s Reference Manual, Grades 4–6 pp. 29–35, 142, 149, 150
592
Unit 7
Fractions and Their Uses; Chance and Probability
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Getting Started
Mental Math and Reflexes
Write fractions and mixed numbers on the board for students to decompose on their slates in at least two different ways.
Suggestions: Sample answers are given.
5 _
5
3 _
5
2
1
4
_
_
_
_
_
11 11 = 11 + 11 ; 11 = 11 + 11
_7 _7 = _1 + _6 ; _7 = _3 + _3 + _1
8 8
_3 _3
5 5
=
8
_1
5
+
8 8
8
_1 + _1 ; _3
5
5 5
=
8
_2
5
+
8 _
8
5
3 _
8
4
4
_
_
_
_
_
15 15 = 15 + 15 ; 15 = 15 + 15
5
5
3
9
12 _
12
2 12
_
= _ + _ + _; _ = _ + _
8
_1
5
25 25
7 _
7
_
10 10
=
15
5
_
10
+
15
15 15
2 _
7
1
_
_
10 ; 10 = 10
+
15
2
_
10
+
3
6
3
2 2
2
2
1
1
3_3 3_3 = 2 + _3 + _3 ; 3_3 = _3 + _3 + _3 + _3
5
5
6
5
5
3
2
1_6 1_6 = _6 + _6 ; 1_6 = 1 + _6 + _6
9
9
9
3
4
4
4
4
1
2_ 2_ = _ + _ + _; 2_ = 1 + _ + _ + _
15
4
_
10
9
9
9
9
9
9
9
Math Message
Study Link 7 4 Follow-Up
If the hexagon pattern block is the whole, what
fractions are represented by the trapezoid, the
rhombus, and the triangle?
Ask students to share their solution strategies for
Problem 3.
9
9
Sample answer: _23 of 15 quarters = 10 quarters, or $2.50;
3
_
5 of 10 nickels = 6 nickels, or 30 cents. Therefore, Rosa spent
$2.50 + $0.30 = $2.80. Each can costs 35 cents, so she was
able to buy 8 cans with her money.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Review answers. The trapezoid represents _12 , the rhombus
represents _13 , and the triangle represents _16 .
Ask students to use a trapezoid, a rhombus, and a triangle to form
a hexagon. Ask what number model describes what they just did.
1
_
+ _13 + _16 = 1
2
Tell students that in this lesson they will use pattern blocks
to model fraction and mixed-number addition and subtraction
problems. For all of the activities, the hexagon pattern block
represents the whole, or the ONE.
Modeling Fraction and
WHOLE-CLASS
ACTIVITY
Mixed-Number Sums
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
E
M
SO
S
SOLVING
OL
O
LV
VIIN
VIN
NG
G
Use pattern blocks on the overhead to work through fraction and
mixed-number addition problems with the class. Suggestions:
●
2
_
6
+ _16 = _36 , or _12
●
2_13 + _13 = 2_23
●
1
_
6
+ _36 = _46 , or _23
●
1_12 + _12 = 1_22 , or 2
●
1
_
3
+ _13 = _23
●
2_13 + 3_13 = 5_23
●
2
_
3
+ _16 = _56
●
2_12 + 1_36 = 3_22 , or 4
●
1
_
3
+ _12 = _56
●
1_13 + 1_16 = 2_36 , or 2_12
●
5
_
6
+ _13 = _76 , or 1_16
●
2_46 + 1_23 = 4_13
Adjusting
the Activity
ELL
Use colored chalk that matches the pattern
block colors to write the number model on
the board: _12 + _13 + _16 = 1.
AUDITORY
KINESTHETIC
TACTILE
Links to the Future
These activities provide exposure to adding
and subtracting fractions using manipulatives.
Paper-and-pencil addition and subtraction of
fractions is a Grade 5 Goal.
Lesson 7 5
593-597_EMCS_T_TLG2_G4_U07_L05_576906.indd 593
VISUAL
593
3/3/11 3:12 PM
Modeling Fraction and Mixed-Number Addition
with Like Denominators
Have students use pattern blocks to find the sum of fractions and
mixed numbers with like denominators.
Example 1: _13 + _13 = ?
Have students take 2 rhombuses and combine them to form _23 .
2
3
Example 2: 1_16 + _36 = ?
Have students take 1 hexagon and 1 triangle to represent 1_16 , and
3 triangles to represent _36 . Then have students combine them to
form 1_46 .
4
16
Modeling Fraction and Mixed-Number Addition with
Unlike Denominators
You can either let students devise their own methods or
demonstrate the following step-by-step procedure. This process
involves an extra step of trading in blocks for one kind of block.
Example 1: _23 + _16 = ?
Step 1: Model the fractions to be added.
2
3
1
6
Step 2: Combine the blocks to show the sum.
2
3
⫹ 16
Step 3: Trade for one kind of block.
2
3
⫹ 16
Step 4: Name the fraction for the sum.
2
3
594
⫹ 16 ⫽ 56
Unit 7 Fractions and Their Uses; Chance and Probability
593-597_EMCS_T_TLG1_U07_L05_576906.indd 594
1/27/11 3:37 PM
Example 2: 1_23 + 1_16 = ?
Step 1: Model the numbers to be added.
2
1
13
16
Step 2: Combine the blocks to show the sum.
1
+
1
2
1
2
1
+ 3+6
Step 3: Trade for one kind of block.
1
+
1
+ 3+6
Step 4: Name the mixed number for the sum.
1
+
1
2
1
5
+ 3+6 = 26
Modeling Fraction and
Mixed-Number Differences
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
Modeling Fraction and Mixed-Number Subtraction
with Like Denominators
Now have students use pattern blocks to find differences of
fractions and mixed numbers with like denominators.
Example 1: _23 - _13 = ?
Have students take 2 rhombuses and then remove or cover
1 rhombus to show the difference is _13 .
Example 2: 2_36 - 1_16 = ?
Have students take 2 hexagons 3 triangles and then remove or
cover 1 hexagon and 1 triangle to show the difference is 1_26 .
Modeling Fraction and Mixed-Number Subtraction
with Unlike Denominators
Ask students how they could use pattern blocks to solve _56 – _23 .
After a few minutes, ask students to share their approaches.
Subtraction can be harder to model than addition, so students’
methods may be awkward. Below are two approaches.
Cover-Up Method
Model both fractions with pattern blocks. Then put the blocks
representing the smaller fraction on top of the blocks representing
the larger fraction. The part of the larger fraction that remains
uncovered is the difference.
Lesson 7 5
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594A
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Example 1: _56 - _23 = ?
Step 1: Model the fractions with pattern blocks.
5
6
2
3
Step 2: Cover up the larger fraction with the smaller fraction.
5
6
⫺ 23
Step 3: The uncovered part of the larger fraction is the difference.
The uncovered part is a triangle representing _16 . So,
5
_
- _23 = _16 .
6
Example 2: 2_56 - 1_13 = ?
Step 1: Model the mixed numbers with pattern blocks.
5
1
13
26
Step 2: Cover up the larger mixed number with the smaller mixed
number.
5
1
2 6 – 13
Step 3: The uncovered part of the larger mixed number is
the difference. The uncovered part is 1 hexagon and
3 triangles representing 1_36 . So, 2_56 - 1_13 = 1_36 , or 1_12 .
Take-Away Method
Model the larger fraction with pattern blocks. Then take away
blocks representing the smaller fraction, trading for blocks of
the proper size if necessary. The remaining blocks represent
the difference.
Example 1: _56 - _23 = ?
Step 1: Model the larger fraction with pattern blocks.
Trade 4 triangles
for 2 rhombuses.
Take away
2 rhombuses.
5
6
Step 2: Remove blocks representing the smaller fraction. The
smaller number, _23 , can be represented by 2 rhombuses. Since
there are no rhombuses, trade 4 triangles for 2 rhombuses.
Then remove 2 rhombuses. (See margin.)
594B Unit 7 Fractions and Their Uses; Chance and Probability
593-597_EMCS_T_TLG1_U07_L05_576906.indd 594B
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Student Page
Step 3: The block(s) that are left represent the difference.
Date
Time
LESSON
Fraction and Mixed-Number Sums & Differences
75
1.
5
_
6
2 = _
1
-_
3
Use pattern blocks to find fractions that add up to 1 whole.
Draw lines to show the blocks you used. Write a number
model to show that the sum of your fractions is 1.
Whole
55
hexagon
Sample answers:
6
_1 _1 _1 _1 _1 _1
1
1
1
1
_
+_
+_
+_
=1 6 + 6 + 6 + 6 + 6 + 6 =
3
3
6
6
Example 2: 1_12 - _16 = ?
1 1_6 + 1_3 + 1_2 = 1
Step 1: Model the larger mixed number with pattern blocks.
1
_ + 1_
2
2
2.
1
=
1
1
_ + 1_ + 1_ =
3
3
3
1
3
_ + 1_ =
6
2
1
2
Use pattern blocks to find fractions that add up to _
3 . Draw lines to show the blocks
2
you used. Write a number model to show that the sum of your fractions is _
3.
12
Step 2: Remove blocks representing the smaller number. The
smaller number, _16 , can be represented by a triangle. Since
there are no triangles, trade 1 trapezoid for 3 triangles.
Then remove 1 triangle.
1
_ + 1_ = _2
3
3
3
1
_ + 1_ = _2
2
6
3
1
_ + _2 = _2
3
6
3
Solve. You may use pattern blocks or any other method.
3
_
3. 6
2
-_
6 =
2
1
_
_
5. 3 - 6 =
_1
6
_3
6 , or
_1
2
2
_
4. 2
1
-_
2 =
5
1
_
_
6. 6 - 2 =
_1
2
_2
_1
6 , or 3
Math Journal 2, p. 198
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1/26/11 5:03 PM
Trade 1 trapezoid
for 3 triangles.
Step 3: The block(s) that are left represent the difference.
1
12 –
1
6
2
1
= 1 6 , or 1 3
Pose fraction and mixed number subtraction problems for students
to solve with pattern blocks. Suggestions:
●
5
_
6
- _26 = _36 , or _12
●
1_56 - _46 = 1_16
Student Page
Date
Time
LESSON
2
_
3
- _13 = _13
●
●
2
_
3
- _12 = _16
●
1_12 - _22 = _12
●
1
_
2
- _13 = _16
●
1_12 - _23 = _56
●
2
_
3
- _16 = _36 , or _12
●
1_13 - _56 = _36 , or _12
●
2_23 - 2_13 = _13
75
7.
Fraction and Mixed-Number Sums & Differences
Use pattern blocks to find three different pairs of mixed numbers
4
that add up to 2_
6 . Use your Geometry Template to illustrate the
mixed numbers. Write a number model to show that the sum of
4
each pair of mixed numbers is 2_
6 . Sample answers:
cont.
Whole
hexagon
a.
Number model:
1_13 + 1_13 = 2_46
Number model:
1_16 + 1_36 = 2_46
Number model:
2_12 + _16 = 2_46
b.
c.
Solve. You may use pattern blocks or any other method.
8.
10.
1
1
_
2_
2 – 12 =
1
1
_
1_
6 – 3 =
1
_5
6
9.
11.
2
1
_
1_
3 – 13 =
5
1
_
1_
2 – 6 =
_1
3
_4
6,
or _23
Math Journal 2, p. 198A
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Lesson 7 5
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595
3/3/11 3:12 PM
Student Page
Date
Solving Fraction Number Stories
75
1.
Solving Fraction and Mixed-
Time
LESSON
3
1
_
Rithik ate _
6 of a cheese pizza. He then ate 6 of a veggie pizza.
_4
of a pizza, or _23 of a pizza
What fraction of a pizza did he eat in all? 6
a.
_3
6
Number model:
b.
+ _16 = _46
Sample number
models are given.
less
Did he eat more or less than a whole pizza?
How do you know?
PARTNER
ACTIVITY
Number Addition and Subtraction Problems
(Math Journal 2, pp. 198 and 198A)
Sample answer: _66 would be 1 whole pizza, and _46 < _66.
2.
_3
a.
4
How far did she walk after school?
_2
mile
1
3
+ _4 = _4
_4
mile,
or 1 mile
How far did she walk in all? 4
_1
_2
_1
_4
4 + 4 + 4 = 4
Number model:
4
Number model:
b.
3.
Students use pattern blocks to solve fraction and mixed-number
addition problems. They use pattern blocks or any other method to
solve fraction and mixed-number subtraction problems.
1
Karina walked _
4 of a mile to school. After school, she walked
2
1
_
_
4 of a mile to the store, and then 4 of a mile back to her home.
Stephano is making pancakes and waffles for his guests.
_2
3
Number model:
_2
_1
13 - 13 =
Number model:
3,
4
2
+ _3 = _3
2
_
b. Stephano has 1 3 cups of milk. Will he have
any left over? If so, how much milk will be left?
Yes.
Ongoing Assessment:
Recognizing Student Achievement
1
or 1_3 cups
_4
2
2
_
He needs _
3 cup of milk for the pancakes and 3 cup of
milk for the waffles. How much milk does he need in all?
a.
_1
3
cup
3
1
2
_
Kumba has one dollar. He spent _
2 of the dollar on a pencil and 10 of the dollar on an eraser.
7
__
a.
What fraction of the dollar did he spend?
_1
b.
+
2
Number model:
2
__
10
=
10
7
__
10
1-
Number model:
10
=
[Operations and Computation Goal 5]
3
__
10
What fraction of the dollar does he have left?
7
__
Use journal page 198, Problem 1 to assess students’ ability to use pattern
blocks to solve fraction addition problems. Students are making adequate
progress if they are able to show at least one combination of pattern blocks that
adds up to 1 and write an appropriate number model. Some students might be
able to show multiple solutions to the problem.
_1
Try This
4.
Journal
page 198
Problem 1
3
__
10
Math Journal 2, p. 198B
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3/3/11 12:39 PM
Using a Calculator to Add
WHOLE-CLASS
ACTIVITY
and Subtract Fractions
Have students practice entering fraction addition and subtraction
problems on their calculators.
Example 1: _23 + _16 = _56
On a TI-15: 2 n 3
d
On a Casio fx-55: 2
+1 n 6
3+1
d
6
Example 2: _12 - _13 = _16
On a TI-15: 1 n 2
Student Page
Date
On a Casio fx-55: 1
Time
LESSON
Math Boxes
75
1
2
_
_
1. Circle 5 of all the triangles. Mark Xs on 3
of all the triangles.
2.
Sample answer:
b.
c.
d.
(
)
( )
9.1 =(28.4 - 1.1) ÷ 3
9 ∗(2.5 + 3.5)= 54
8.2 - 5.2 + 2.5 = 0.5
A (5,0)
5
B (3,5)
4
C (1,4)
2
C
3
0
D
0
T
E
1
D (1,1)
1
3
4
Y
acute
R
A
2
This angle is an
(acute or obtuse) angle.
5
E (2,4)
144
5.
red blocks,
blue blocks,
green blocks, and
orange blocks.
You put your hand in the bag and, without
looking, pull out a block. About what
fraction of the time would you expect to
get a blue block?
4
__
20 ,
or
_1
5
93 143
6.
A bag contains
6
4
7
3
If 1 centimeter on a map represents
10 kilometers, then
a.
6 cm represent
b.
19.5 cm represent
c.
d.
e.
3
5.5
0.5
45
185-218_EMCS_S_MJ2_G4_U07_576426.indd 199
2 Ongoing Learning & Practice
Playing Angle Tangle
60 km.
195 km.
cm represent 30 km.
PARTNER
ACTIVITY
(Student Reference Book, p. 230; Math Masters, p. 457)
cm represent 55 km.
cm represent 5 km.
145
Students play Angle Tangle to practice measuring and estimating
the measures of angles. See Lesson 6-6 for additional information.
Math Journal 2, p. 199
596
PARTNER
ACTIVITY
Have students solve number stories involving fractions and mixed
numbers. They can use pattern blocks or any other method to solve.
Draw and label a 45° angle.
Sample
answer:
B
3
(Math Journal 2, p. 198B)
150
4.
Plot and label each point on the
coordinate grid.
2-1
d
Number Stories
13.6 - 5 + 8 = 0.6
59
3.
-1 n 3
Solving Fraction
Insert parentheses to make these
number sentences true.
a.
d
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Unit 7 Fractions and Their Uses; Chance and Probability
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Math Boxes 7 5
Study Link Master
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 199)
Name
Date
STUDY LINK
Fractions
75
1.
2.
Study Link 7 5
less
Sample answer: 0.75 + 0.10 = 0.85
1
Jillian draws a line segment 2 _4 inches long. Then she makes the
2
line segment 1 _4 inches longer. How long is the line segment now?
1
4.
INDEPENDENT
ACTIVITY
3 _4
3
inches
2
2 4 in.
3.
55 57
3
1
Jake has _4 of a dollar. Maxwell has _
10 of a dollar.
Do they have more or less than $1.00 in all?
Number model:
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 7-7. The skill in Problem 6
previews Unit 8 content.
Time
1 4 in.
A pizza was cut into 6 slices. Benjamin ate
1
1
_
_
3 of the pizza and Dana ate 2 . What fraction
of the pizza was left?
1
_
6
7
Rafael drew a line segment
7
2 _8 inches long. Then he erased
4
_
8 inch. How long is the line
segment now?
2 8 in.
3
_
28
inches
4
8
?
(Math Masters, p. 216)
5.
Home Connection Students solve fraction addition and
subtraction problems.
in.
Two hexagons together are one whole. Draw line segments to divide each
whole into trapezoids, rhombuses, and triangles. Write a number model
to show how the parts add up to the whole.
Sample answers:
1
_
4
1
_
+ _14 + _14 + _14 = 1
4
+
3
_
12
+
3
_
6
2
_
=1
4
+
3
_
6
=1
Practice
3 Differentiation Options
1
_
6. 4
8
of 32 =
45 = _ of 50
9
10
7.
8.
7
_
8 of 56 =
49
22 = _ of 24
11
12
9.
Math Masters, p. 216
READINESS
Exploring Fractions
SMALL-GROUP
ACTIVITY
203-246_EMCS_B_MM_G4_U07_576965_NEW_PAGES.indd 216
1/26/11 2:58 PM
5–15 Min
that Sum to One
(Math Masters, p. 217)
Literature Link To explore the addition of fractions with
like denominators, read Full House: An Invitation to
Fractions by Dayle Ann Dodds (Candlewick Press, 2009). Record
the number model to show how the cake was divided among the
people. Have students divide the cakes and write number models
to show how one cake can be evenly divided among different
numbers of people.
EXTRA PRACTICE
Solving Frames-and-Arrows
INDEPENDENT
ACTIVITY
5–15 Min
Problems
Teaching Master
Name
Date
LESSON
Time
Dividing Cakes
75
Use a straightedge to divide the cakes below among different numbers of people.
Make sure each person gets an equal share. Write a number model to show
what you did.
1.
2.
Six people
Number model:
_1
(Math Masters, p. 393)
6
_1
+
6
_1
+
6
_1
+
6
_1
+
6
6
Three people
Number model:
_1
+
_1
3
=1
44
_1
+
3
_1
+
3
=1
To provide practice adding and subtracting fractions, have
students solve Frames-and-Arrows problems. Use Math Masters,
page 393 to create problems to meet the needs of individual
students or have students create and solve their own problems.
3.
Four people
Number model:
Planning Ahead
_1 + _1 + _1 + _1 = 1
4
4
4
4
4.
Eight people
_1 + _1 + _1 + _1 +
_1 + _1 + _1 8+ _1 8= 18
8
8
8
8
Number model: 8
In preparation for Lesson 7-6, direct students to remove Activity
Sheets 5 and 6 from the back of the journal and cut the Fraction
Cards apart. Have students write their initials on the cards
for identification.
Math Masters, p. 217
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Lesson 7 5
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597
1/27/11 3:38 PM
Name
STUDY LINK
75
1.
Date
Time
Fractions
55 57
3
1
Jake has _4 of a dollar. Maxwell has _
10 of a dollar.
Do they have more or less than $1.00 in all?
Number model:
2.
1
Jillian draws a line segment 2 _4 inches long. Then she makes the
2
line segment 1 _4 inches longer. How long is the line segment now?
1
2
2 4 in.
3.
inches
1 4 in.
A pizza was cut into 6 slices. Benjamin ate
1
1
_
of the pizza and Dana ate _ . What fraction
3
2
of the pizza was left?
4.
Rafael drew a line segment
7
2 _8 inches long. Then he erased
4
_
8 inch. How long is the line
segment now?
7
2 8 in.
inches
4
8
?
5.
in.
Two hexagons together are one whole. Draw line segments to divide each
whole into trapezoids, rhombuses, and triangles. Write a number model
to show how the parts add up to the whole.
Copyright © Wright Group/McGraw-Hill
Practice
1
_
6. 4
of 32 =
7.
9
=_
10 of 50
8.
7
_
8 of 56 =
9.
11
=_
12 of 24
216
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