Lesson
10-5A
Modeling Addition and
Subtraction of Mixed Numbers
fraction strips
Common
Core
How can you model addition of
mixed numbers?
5.NF.2 Solve word problems
involving addition and
subtraction of fractions
referring to the same
whole… by using visual
fraction models or equations
to represent the problem.
Also 5.NF.1
Bill has 2 boards he will use to make picture
frames. What is the total length of the
boards Bill has to make picture frames?
11 ​feet
1​ ___
12
5   ​feet
2​ ___
12
Choose an Operation Add to find the total length.
Another
Example How can you model subtraction of mixed numbers?
5  ​2 1​ ___
11 ​.
Find 2​ ___
12
12
Step 1
Model the number
you are subtracting
5  ​.
from, 2​ ___
12
Step 2
1
1
1 1 1 1 1
12 12 12 12 12
1
If the fraction you will be subtracting is
greater than the fraction of the number you
model, rename 1 whole.
Since ___
​ 11
 ​. ___
​  5  ​,
12 12
rename 1 whole
12 ​.
as ​ ___
12
Use your renamed model to cross out the
11 ​.
number that you are subtracting, 1​ ___
12
1
1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 1 1
12 12 12 12 12
1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 1 1
12 12 12 12 12
6  ​left.
There are ___
​ 12
5  ​2 1​ ___
11 ​5 ___
So, 2​ ___
​  6  ​.
12
12 12
6  ​5 __
Simplify: ​ ___
​ 1 ​
12 2
Explain It
5  ​renamed as 1​ __
17 ​?
1. In the example above, why is 2​ ___
12
12
Guided Practice
Do you know HOW?
Use fraction strips to find each sum or
difference. Simplify, if possible.
2 ​1 2​ __
4 ​
1. 1​ __
5
5
3 ​
1 ​2 3​ __
2. 4​ __
4
4
3 ​2 2​ __
5 ​
3. 5​ __
1 ​1 2​ __
2 ​
4. 3​ __
6
30
6
3
3
MATHEMATICAL
PRACTICES
Do you UNDERSTAND?
5. How is renaming to add mixed
numbers different from regrouping to
add whole numbers?
6. Construct Arguments When adding
two mixed numbers is it always
necessary to rename the fractional
sum? Explain.
Step 2
Step 1
Model the addends and add the fractional parts.
1
1
5  ​
2​ ___
12
11
1 1​ ___ ​
12
16 ​
​ ___
12
Now add the whole numbers, including
the renamed fraction.
1
1
1
1
1 1 1 1 1
12 12 12 12 12
1
1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12
1 1 1 1
12 12 12 12
2111154
4  ​.
Rename ___
​ 16
 ​as 1​ ___
12
12
5  ​1 1​ ___
11 ​5 4​ ___
4  ​.
So, 2​ ___
12
12
12
4  ​5 4​ __
1 ​
Simplify: 4​ ___
12
3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
1
1 ​feet.
The total length of the boards is 4​ __
3
1 1 1 1
12 12 12 12
Independent Practice
In 7 and 8, use each model to find the sum or difference.
Simplify if possible.
2 ​cups of walnuts and
7. Model Charles used 1​ __
3
2 ​cups of cranberries to make breakfast
2​ __
3
bread. How many cups of walnuts and
cranberries did he use in all?
1
1
3
1
3
1
1
1
3
8. Terrell ran to his friend’s apartment in
5 ​minutes. It took Terrell 4​ __
1 ​minutes to
2​ __
6
6
go back home. How much more time
did Terrell take to get home?
1
3
1
1
1
1
6
1
6
1
6
1
6
1
6
1
6
1
6
In 9 through 16, use fraction strips to find each sum or difference.
Simplify if possible.
3 ​1 1​ __
3 ​
9. 2​ __
5
5
5  ​1 1​ ___
7  ​
10. 4​ ___
12
12
9  ​1 3​ ___
7  ​
11. 4​ ___
10
10
3 ​1 2​ __
3 ​
12. 5​ __
4
4
5 ​
1 ​2 4​ __
13. 9​ __
6
6
3 ​2 9​ __
5 ​
14. 12​ __
8
8
1 ​2 7​ __
2 ​
15. 8​ __
3
3
8 ​
7 ​2 10​ __
16. 13​ __
9
9
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Lesson 10-5A
31
MATHEMATICAL
Problem Solving
PRACTICES
For 17 and 18, use fraction strips to solve. Simplify, if possible.
17. Kit said, “On summer vacation,
4 ​weeks with my grandma
I spent 2​ __
7
6 ​weeks with my aunt.”
and 1​ __
7
2 ​gallons of white paint
18. Hannah used 1​ __
3
2 ​gallons of green
for the ceiling and 3​ __
3
paint for the walls of her kitchen.
a How many weeks is that in all?
a How much paint did Hannah use in all?
b How many weeks longer did Kit
spend with her grandmother than
with her aunt?
b How much more green paint did
Hannah use than white paint?
For 19 and 20, use the map at the right.
19. Ben left the museum and walked 4 blocks
to his next destination. What was it?
20. Think About the Structure Ben
walked from the restaurant to the
bus stop. Then he took the bus to the
stadium. If he took the shortest route,
how many blocks did Ben travel?
y
10
9
8
7
6
5
4
3
2
1
0
Stadium
Museum
Bus Stop
Restaurant
1 2 3 4 5 6 7 8 9 10 x
21. Be Precise What is the value of the underlined digit in 6.275?
A 7 ones
B 7 tenths
C 7 hundredths
For 22 and 23, use the table at the right.
22. How many inches longer is a Hercules beetle
than a ladybug?
23. What is the difference between the largest
and the smallest stag beetles?
24. Raoul’s heart beats about 72 times each minute.
How many times does his heart beat in 15 minutes?
D 7 thousandths
Beetles by Length
Beetle
Length in inches
Hercules beetle
3 ​
6​ __
1 ​
​ __
Ladybug
Stag beetle
25. Nicole, Tasha, Maria, and Joan ran a relay race. Nicole ran the first leg
13 ​minutes. Tasha ran the second leg in 2​ ___
1  ​minutes.
of the race in 1​ ___
15
15
7  ​minutes. Joan ran the last leg in
Maria ran the third leg in 1​ ___
15
2  ​ minutes to finish the race.
2​ ___
15
a Reason How can you find how much faster Maria ran than Joan?
b The team wanted to run the race in less than six minutes. Did they
meet their goal? Explain.
32
4
4
1 ​to 2​ __
4 ​
1​ __
8
8
The human body is very complex. Two systems of the human body are
the muscular and circulatory systems.
1. An average ten-year-old weighs about
1 ​pounds and has muscles that weigh
86​ __
2
1 ​pounds. Find the difference
about 34​ __
5
between these two weights.
2. Use the information given in Exercise 1.
About how much do the muscles of two
average ten-year-olds weigh? Write your
answer as a mixed number.
3. Your body temperature can change
from hour to hour. Tony’s temperature
was 98.6°F at 2 p.m. His temperature
increased 0.6°F during the first hour,
0.2°F the second hour, and 0.1°F during
the third hour. It decreased 0.8°F during
the fourth hour. What was Tony’s
temperature at 6 p.m.?
4. The circulatory system includes the
heart, the blood, and the blood vessels.
A person is born with about 0.25 liter
of blood. An adult has about 5 liters
of blood. What is the difference
between the amount of blood an
adult has and the amount of blood a
newborn baby has?
Blood flow in
In 5 and 6, use the following information.
The left ventricle of an average person can pump
about 315 liters of blood in 1 hour.
Aorta
5. Reason Estimate to find about how many
liters of blood the left ventricle can pump
in 1 minute. Hint: There are 60 minutes in
an hour.
Blood flow out
6. Reason The left ventricle pumps blood
away from the heart into your body’s largest
artery, the aorta. About how much blood
is pumped into the aorta in 10 minutes?
A 5 liters
C 50 liters
B 10 liters
D 500 liters
Left ventricle
Right ventricle
Lesson 10-5A
33