Lesson
5 Regrouping With Mixed Numbers
Monitoring Progress:
Quiz 1
Regrouping With Mixed Numbers
When adding mixed numbers, when do
we regroup?
We have added and subtracted mixed numbers, but we haven’t had to
regroup yet. Sometimes we have to regroup the fractional part when
we add or subtract mixed numbers. This happens in addition when the
two fractions add up to more than 1. We do this in the “simplify” step
of LAPS.
L—LOOK at the problem carefully.
• Make sure the numbers are lined up correctly.
• Decide if addition or subtraction is supposed to be
performed.
22
3
+ 12
3
In this problem, the fractions and whole numbers are lined up
correctly, and we need to add.
A—ALTER the problem if necessary.
In this problem, the denominators are the same, so we do not
have to alter the fractions.
Unit 3 • Lesson 5 197
Lesson 5
P—PERFORM the operation.
Now we add. We start by adding the fractional parts of the two
numbers.
22
3
+ 12
3
4
3
Next, we add the whole numbers.
22
3
+ 12
3
34
3
S—SIMPLIFY the answer.
Our answer is not in its simplest form. In this problem, the answer
includes an improper fraction. An improper fraction is a fraction that’s
equal to or bigger than 1. We have to regroup so that we have a whole
number and a fraction that is less than 1.
22
3
+ 12
3
34
3
198 Unit 3 • Lesson 5
This fraction is bigger
than 1. We need to
regroup.
Lesson 5
2
+
1
2
3
2
3
3 1
3+3
We break the fraction 4
3 into two fractions.
One should be equal to 1. In this case it’s 3
3.
The other fraction should be less than 1.
Here it’s 1
3 . After we break up the fraction,
we regroup.
3
3 or 1
2
+
1
2
3
2
3
3 1
3+3
We leave the fractional part, or 1
3 , in the
“fractions place.” We regroup by moving
the 3
3 , or 1, to the ones place.
1
1
2
3
2
3
4
1
3
2
+
We finish the problem by adding the whole
numbers together. Here, we add the whole
number parts 2 + 1 = 3. Then we add the
number we regrouped: 3 + 1 = 4. We get 4.
Our answer is 4 13 .
Unit 3 • Lesson 5 199
Lesson 5
When subtracting mixed numbers, when do
we regroup?
When we add mixed numbers, the regrouping takes place in the
“Simplify” step of LAPS. Subtraction is different. When we regroup in
subtraction, it’s because the fractional part of the top number is smaller
than the fractional part of the bottom number. We have to change
that in the “Alter” step of LAPS. Let’s review the steps for subtracting
whole numbers.
34
18
We cannot work the problem the way it is. The top number in the ones
place is smaller than the bottom number in the ones place. We have
to regroup.
Let’s look at the regrouping in expanded form.
30 4
− 10 8
20 + 10
−
10
We can’t subtract 8 from 4. We have to
regroup.
4
8
20 10 + 4
− 10
8
20 14
− 10 8
10 6
Answer: 16
200 Unit 3 • Lesson 5
We rewrite the 30 as 20 + 10. Then we
borrow the 10 and move it into the ones
place.
We already have 4 in the ones place, so we
add the 10 to the 4.
Now we have 14. We subtract 8 from 14
and get 6. We subtract 20 − 10 in the tens
place. We get 10. We combine 10 + 6 = 16.
The answer is 16.
Lesson 5
The process for regrouping mixed numbers in subtraction is similar
to the process for regrouping whole numbers. When we subtract, we
need to regroup if the top fractional part is smaller than the bottom
fractional part.
Example 1
Regroup with mixed numbers when subtracting.
L—LOOK at the problem carefully.
• Make sure the numbers are lined up correctly.
• Decide if addition or subtraction is supposed to be performed.
In this problem, the fractions and whole numbers are lined up correctly,
and we need to subtract.
32
5
− 14
5
A—ALTER the problem if necessary.
The fractions do have a common denominator, so we do not have to
change the denominators. But we cannot subtract the fractions the way
they are written. The bottom fraction is larger than the top fraction. We
need to regroup before we can subtract.
3 2
5
− 1 4
5
2+ 5
5
3 2
5
−
1 4
5
The top fraction, 2
5 , is smaller than the bottom
4
fraction, 5 , so we need to regroup.
We break the whole number on the top into a
whole number plus a fraction equal to 1. For
this problem, we rewrite 3 as 2 + 1 or, using
fifths, 2 + 5
5.
Unit 3 • Lesson 5 201
Lesson 5
5
2 2
5+ 5
− 1 4
5
2 7
5
− 1 4
5
We need to move the 5
5 over and combine
2
it with the fraction, 5 . This will give us a
bigger fraction on top.
2
7
We add 5
5 to 5 to get 5 . Now we
can subtract.
P—PERFORM the operation.
Now we subtract.
2 7
5
− 1 4
5
We start by subtracting the
fractional parts.
3
5
2 7
5
Next, we subtract the whole numbers.
− 1 4
5
1 3
5
S—SIMPLIFY the answer.
We have the answer we want because it’s a mixed number in
its simplest form. So we don’t need to do anything in this step.
Answer: 1 3
5
202 In subtraction, the regrouping
takes place in the “Alter” step.
We have to adjust the numbers
first before we can perform the
operation.
Apply Skills
Monitoring Progress
Reinforce Understanding
Turn to Interactive Text,
page 102.
Quiz 1
Use the mBook Study Guide
to review lesson concepts.
Unit 3 • Lesson 5
Lesson 5
Homework
Activity 1
Convert the improper fractions to mixed numbers.
1.
4
3
3.
12
5
1
3
2
5
2.
27
4
4.
57
8
3
4
1
8
Activity 2
Use LAPS to add and subtract. Be sure to regroup if necessary.
1.
3 43
2.
+ 1 13
5 12
3.
+ 2 32
7 15
4.
− 4 35
8 14
− 6 24
Activity 3
Solve the word problems. Simplify your answers.
Molding is a decorative strip placed around windows or walls to make a room
look better. The Scatter Plots are fixing the molding on the windows in the
kitchen. That means the group members will have to cut pieces of molding so
they fit.
1
1. The molding they need to put on the bottom of the window is 4 3 feet
long. They have a piece of molding that is 4 14 feet long. Is the piece long
enough? What is the difference between the piece they have and what
they need?
2. The molding along each side of the window needs to be 4 38 feet long. They
need two pieces. What is the total length of the two pieces?
1
12
3
4
Activity 4 • Distributed Practice
Solve.
1.
3,000
+ 8,000
11,000
2.
4,002
 1,987
2,015
3.
67
98
4.
90q7,200
6,566
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Unit 3 • Lesson 5 203