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Lesson 6.1
Reteach
Name
Addition with Unlike
Denominators
Karen is stringing a necklace with beads. She puts green beads on
1
3
_ of the string and purple beads on __
2
10 of the string. How much of
the string does Karen cover with beads?
You can use fraction strips to help you add fractions with
unlike denominators. Trade fraction strips of fractions with unlike
denominators for equivalent strips of fractions with like denominators.
Use fraction strips to find the sum. Write your answer in
simplest form.
1
3
_ 1 __
2 10
Step 1
Use a 1_2 strip and three
1
__
10 strips to model fractions
with unlike denominators.
Step 2
Trade the 1_2 strip for
1
five __
10 strips.
3 5 __
5 1 __
3
1 1 __
_
Step 3
sum
2 10 10 10
Add the fractions with
like denominators.
3 5 __
8
5 1 __
__
Step 4
10 10 10
Write the answer in simplest form.
4
8 5_
__
10 5
_ of the string with beads.
So, Karen covers 4
5
Use fraction strips to find the sum. Write your answer in
simplest form.
3
1 __
8 4
3
1. __
Reteach
© Houghton Mifflin Harcourt Publishing Company
1
1 __
3 4
7
1 ___
6 12
2
2. __
5
3. __
R50
Grade 5
Lesson 6.2
Reteach
Name
Subtraction with Unlike
Denominators
You can use fraction strips to help you subtract fractions with unlike
denominators. Trade fraction strips of fractions with unlike denominators
for equivalent strips of fractions with like denominators.
Use fraction strips to find the difference. Write your answer in
simplest form.
1 2 __
1
_
2 10
Step 1
Use a 1_2 fraction strip to
model the first fraction.
Step 2
Trade the 1_2 strip for
1
five __
10 strips.
5 2 __
1 5 __
1
1 2 __
_
2
Step 3
10
10
10
Subtract by taking
1
away __
10 .
5 2 __
1 5 __
4
__
10 10 10
1 5 __
4 . Written in simplest form, __
4 5_
2.
_ 2 __
So, 1
2 10 10
10 5
Use fraction strips to find the difference. Write your answer in
simplest form.
7
1. __
__
21
8 2
2
2. __
__
__ 2 1
4. 1
9
5. ___
2
3
Reteach
© Houghton Mifflin Harcourt Publishing Company
5
3. __
__
21
3 4
10
1
2 __
6 3
4
2 __
5
2
6. __
3
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5
2 ___
12
Grade 5
Lesson 6.3
Reteach
Name
Estimate Fraction Sums
and Differences
You can round fractions to 0, to 1_2 , or to 1 to estimate sums
and differences.
41_
1
Estimate the sum. _
6 9
0
6
Find 4_6 on the number line.
Is it closest to 0, 1_2 , or 1?
The fraction 4_6 is closest to 1_2 .
Step 1
0
9
3
6
4
6
5
6
6
6
1
1
2
1
9
2
9
3
9
4
9
0
To estimate the sum 4_6 1 1_9 ,
add the two rounded numbers.
Step 3
2
6
0
Find 1_9 on the number line.
Is it closest to 0, 1_2 , or 1?
The fraction 1_9 is closest to 0.
Step 2
1
6
5
9
6
9
7
9
8
9
9
9
1
1
2
1
1
_105_
2
2
1 is about 1
41_
_.
So, _
6 9
2
Estimate the sum or difference.
0
6
0
1
6
2
6
3
6
1
2
4
6
5
6
6
6
0
8
1
0
5
8
6
8
7
8
8
8
1
1
2
3
2 __
6 8
1
6. __
7
5. __
Reteach
4
8
__
25
8 6
3
1 __
6 8
© Houghton Mifflin Harcourt Publishing Company
3
8
5
3. __
2
2. __
4
4. __
2
8
__
17
6 8
1
1 __
6 8
4
1. __
1
8
7
1 __
6 8
R52
Grade 5
Lesson 6.4
Reteach
Name
Factors
The factors of a number are the numbers that divide evenly into it.
Prime factors are the factors of a given number that are prime.
A prime number has exactly two factors, 1 and itself. A composite
number has more than two factors.
You can use division to find the factors of a number.
Find the factors of 45.
Step 1
You know that 45 is an odd number so
it cannot be divided by 2. Try dividing by 3.
45 ÷ 3 = 15
So, 3 × 15 = 45.
3 and 15 are factors of 45.
Step 2
Identify the factors as prime or composite numbers.
3 is a prime number. Its factors are 1 and itself.
15 is a composite number. It’s factors are: 1, 3, 5, and itself.
Step 3
You can divide 15 further because it is not a prime number.
15 ÷ 3 = 5
3 and 5 are prime factors.
So, the prime factors of 45 are: 3, 3, and 5
Step 4
You can write 45 as a product of its prime factors.
Write them in order from least to greatest.
3 × 3 × 5 = 45
Write the number as the product of prime factors.
1.
8
Reteach
© Houghton Mifflin Harcourt Publishing Company
2.
3.
15
R53
30
4.
50
Grade 5
Lesson 6.5
Reteach
Name
Common Denominators and
Equivalent Fractions
You can find a common denominator of two fractions.
A common denominator of two fractions is a common multiple
of their denominators.
7
Find a common denominator of 1_6 and __
10 . Rewrite the pair of
fractions using a common denominator.
Step 1
Identify the denominators.
The denominators are 6 and 10.
Step 2
List the multiples of the greater denominator, 10.
Multiples of 10: 10, 20, 30, 40, 50, 60, ...
Step 3
Check if any of the multiples of the greater denominator
are evenly divisible by the other denominator.
Both 30 and 60 are evenly divisible by 6.
7
Common denominators of 1_6 and __
10 are 30 and 60.
Step 4
Rewrite the fractions with a denominator of 30.
Multiply the numerator and the denominator of each fraction
by the same number so that the denominator results in 30.
3 5 5 __
5
151
_
_____
6
635
30
7 3 3 5 21
7 5 ______
__
__
10
10 3 3
30
Use a common denominator to write an equivalent fraction for each
fraction.
5 2
1. ___, __
3 5
2. __, __
__, 1
__
3. 2
9
__, ___
4. 3
8 6
common denominator:
12 9
common denominator:
9 6
common denominator:
Reteach
© Houghton Mifflin Harcourt Publishing Company
4 10
common denominator:
R54
Grade 5
Lesson 6.6
Reteach
Name
Add and Subtract Fractions
To add or subtract fractions with unlike denominators, you need to
rename them as fractions with like denominators. You can do this
by making a list of equivalent fractions.
5 1_
1
Add. __
12 8
Step 1
Step 2
Step 3
10, __
15, 20
5 , __
__
__
5
Write equivalent fractions for __
12 .
12 24 36 48
2 , __
3
1
_, __
Write equivalent fractions for 1_8 .
8 16 24
Rewrite the problem using the equivalent fractions.
Then add.
5 1_
1 becomes __
10 1 __
3 5 __
13.
__
12 8
24 24 24
9 21
_
Subtract. __
10 2
Step 1
Step 2
Step 3
Stop when you
find two fractions
with the same
denominator.
18, __
27, 36
9 , __
__
__
9
Write equivalent fractions for __
10 .
10 20 30 40
4, __
5
1
_, 2
_, 3
_, _
Write equivalent fractions for 1_2 .
2 4 6 8 10
Rewrite the problem using the equivalent fractions.
Then subtract.
9 2_
1 becomes ___
9 2 __
5 5 __
4 . Written in simplest form, __
4 5_
2.
__
10 2
10 10 10
10 5
Find the sum or difference. Write your answer in simplest form.
2
1. __
__
11
3
1
2. __
1
2 __
8 4
3
6. __
9
7
5. __
Reteach
© Houghton Mifflin Harcourt Publishing Company
2
2
1 __
5
1
3. __
4
__
11
6
9
7. ___
2
2 __
4 3
__
24
10 5
R55
1
4. __
5
3
1 __
4
5
2 __
9 6
8
8. __
Grade 5
Lesson 6.7
Reteach
Name
Add and Subtract Mixed Numbers
When you add or subtract mixed numbers, you may need to
rename the fractions as fractions with a common denominator.
Find the sum. Write the answer in simplest form. 5 3_4 1 2 1_3
Step 1
Model 5 3_4 and 2 1_3 .
Step 2
A common denominator for 3_4 and 1_3 is 12,
9
1
4
_
__
so rename 5 3_4 as 5 __
12 and 2 3 as 2 12 .
Step 3
Add the fractions.
4 5 13
9 1 __
__
__
12
Step 4
12
12
Add the whole numbers
51257
Add the sums. Write the answer in simplest form.
1
13, or 8__
13 1 7 5 7__
__
12
12
12
1.
1 5 8__
_ 1 2_
So, 53
4
3
12
Find the sum or difference. Write your answer in simplest form.
1.
1
2 1 4__
2__
9
6
Reteach
© Houghton Mifflin Harcourt Publishing Company
2.
3
__ 1 5__
105
6
4
3.
R56
5
__ 2 9__
117
8
6
4.
1
__ 2 14__
183
5
2
Grade 5
Lesson 6.8
Reteach
Name
Subtraction with Renaming
You can use a common denominator to find the difference of two mixed numbers.
3
1 2 2_
Estimate. 9_
6
4
Step 1
Estimate by using 0, 1_2 , and 1 as benchmarks.
3
_ 2 2_
91
92356
6
4
So, the difference should be close to 6.
Step 2
Identify a common denominator.
3 A common denominator of 6 and 4 is 12.
1 2 2_
9_
6
4
Step 3
Write equivalent fractions using the common denominator.
1 5 9 1 _____
2
1 3 2 5 9__
9_
632
12
6
3 5 2 1 _____
9
3 3 3 5 2__
2_
433
12
4
Step 4
Rename if needed. Then subtract.
9
2 , __
2 as 814
__.
Since __
, rename 9__
12
12
12
12
9
5
14
__
__
__
Subtract. 8 2 2 5 6
12
12
12
5.
1 2 23
_ 5 6__
So, 9_
6
4
12
5
Since the difference of 6 __
12 is close to 6, the answer is reasonable.
Estimate. Then find the difference and write it in simplest form.
1.
2.
Estimate:
__ 2 35
__
51
3
6
3.
5
__ 2 2___
71
4
12
4.
Estimate:
6.
Estimate:
5
3 2 1__
7___
16
8
Reteach
© Houghton Mifflin Harcourt Publishing Company
Estimate:
__ 2 33
__
92
5
4
__ 2 27
__
82
3
9
5.
Estimate:
Estimate:
11
__ 2 1__
24
18
9
R57
Grade 5
Lesson 6.9
Reteach
Name
Algebra • Patterns with
Fractions
You can find an unknown term in a sequence by finding a rule
for the sequence.
Find the unknown term in the sequence.
7 , 2, _, 23
_, 1__
_
12
5 10
5
Step 1
Find equivalent fractions with a common denominator
for all of the terms.
The denominators are 5 and 10. A common denominator is 10.
6
4 and 23
_ 5 1__
_ 5 2__
12
5
5
10
10
Step 2
Write the terms in the sequence using the common denominator.
6
4 , 1__
7 , 2, _, 2__
1__
10 10
10
Step 3
Write a rule that describes the pattern.
The sequence increases. To find the difference between terms,
subtract at least two pairs of consecutive terms.
3
7 2 1__
4 5 __
1__
10
10 10
3.
So, a rule is to add __
10
Step 4
3
7 5 __
2 2 1__
10 10
Use the rule to find the unknown term.
3 to the third term to find the unknown term.
Add __
10
3
3 5 2__
2 1 __
10
10
Write a rule for the sequence. Then, find the unknown term.
1.
1, _, 5__
1, 6
__, 3__
22
3 2
6
2.
Rule:
Reteach
© Houghton Mifflin Harcourt Publishing Company
1, _, 2
__, 37
__, 3__
41
2 8 4
Rule:
R58
Grade 5
Lesson 6.10
Reteach
Name
Problem Solving • Practice Addition
and Subtraction
Makayla walks for exercise. She wants to walk a total of 6 miles.
On Monday, she walked 2 5_6 miles. On Tuesday, she walked 1 1_3 miles.
How many more miles does Makayla need to walk to reach her goal?
Read the Problem
Solve the Problem
What do I need to find?
I need to find
• Start with the equation.
6
5 2 56 1 1 13 1 x
________________
_
the distance that
Makayla needs to walk.
• Use subtraction to work backward and
rewrite the equation.
5
1
6
2
2
2
1
5x
6
3
________________
_
the distance she
wants to walk
and
the distance
she has already walked.
Subtraction is the inverse operation of
addition.
.
What information do I need to use?
I need to use
.
.
_
3111256
I can work backward to
________
solve the problem.
1.
_
_
15
2 76
_
3_
56
6
6
5
5
1 5 21 2
_
_
22 5 22
21_
6
6
6
3
3 16
1 56
Estimate to show that your answer is
reasonable.
6 5
_
I can write an equation
6 5 2 5_6 1 1 1_3 1 x
Then
_
• Subtract to find the value of x.
How will I use the information?
First
_
.
Ben has 5 3_4 cups of sugar. He uses
2
_ cup of sugar to make cookies. Then
3
he uses 2 1_2 cups of sugar to make fresh
lemonade. How many cups of sugar
does Ben have left?
_
1 56 more miles
So, Makayla has to walk __
to reach her goal.
2.
Cheryl has 5 ft of ribbon. She cuts a
3 3_4 -ft strip to make a hair bow. Then
she cuts a 5_6 -ft strip for a border on a
scrapbook page. Is there enough ribbon
for Cheryl to cut two 1_3 -ft pieces to put
on a picture frame? Explain.
________
________
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© Houghton Mifflin Harcourt Publishing Company
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________
Grade 5
Lesson 6.11
Reteach
Name
Algebra • Use Properties of Addition
You can use the properties of addition to help you add fractions with
unlike denominators.
Use the Commutative Property and the Associative Property.
Add.
7 1 21
_
_32_5 1 1__
5
15+
7 1 21
7 1 32
1
_ 5 1__
_ 1 2_
_32_5 1 1__
5 _ 15
5+
5
15+
_
Use the Commutative Property to
order fractions with like denominators.
5+
7 1 32
_ 1 21
_
5 1__
Use the Associative Property to group
fractions with like denominators.
7 1 53
_
5 1__
Use mental math to add the fractions
with like denominators.
9
7 1 5__
5 1__
Write equivalent fractions with like
denominators. Then add.
15
15
5
5
15
15
1
__ 5 7__
5 616
15
15
Rename and simplify.
Use the properties and mental math to solve. Write your answer in
simplest form.
1.
3 1 __
4
_5__7 1 ___
14+ 7
2.
_2__5 1 5__9+ 1 7__9
3.
3 1 __
3
7 1 5__
_3___
10
4+ 4
4.
5 1 4__
7
2 1 3___
2___
12
3
12
5.
1
__ 1 21
__ 1 5__
33
5
8
8
6.
_4__73 1 21__6+ 1 35__7
_
+
Reteach
© Houghton Mifflin Harcourt Publishing Company
R60
_
+
Grade 5