ADDING MIXED NUMBERS
Introduction: Student’s will be able to use bar models (fraction bars) and decomposition
to add mixed numbers with regrouping. Students will also see the use of improper
fractions when adding mixed numbers. (5NS 2.3)
Materials: Fraction bars
Key Concepts: Equivalent forms of one, decomposition, equivalent fractions, least
common multiple/denominator, regrouping
“We have learned what happens when we add fractions and we have a sum greater than
one whole. Today we will examine what to do when we add a mixed number and our
fractional part is greater than one whole. Let’s look at an example!”
7 2
Example #1 (I do): 1 +
8 8
Build It:
1
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Ask:
• Turn to your partner and tell them what you notice about my fractional parts?
(Greater than one, more eights than one whole, etc.) Have students share out their
observations.
• Class, how many eighths do I have in all? Let’s count them together. (Have
students chorally count out the nine eighths pieces)
• What do we call a number like nine eighths? (Improper Fraction)
• Is there a way for me to regroup and rewrite nine eighths? (As a mixed number)
1
• How would I regroup and rewrite nine eighths as a mixed number? ( 1 )
8
Use fraction bars to build and model the two mixed numbers. Collect like pieces and
9
1
show that =1+ .
8
8
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MCC@WCCUSD 02/28/13
Bar Model
(Fraction Bars)
Decomposition
7 2
1 +
8 8
7 2
1 +
8 8
2+7
8
8 1
=1 + +
8 8
1
=1+1+
8
1
=2 +
8
1
=2
8
7 2
=1+ +
8 8
7+ 2
=1 +
8
9
=1 +
8
8 1
=1 + +
8 8
1
=1 + 1+
8
1
=2 +
8
1
=2
8
=1+
Improper Fractions
7 2
1 +
8 8
15 2
= +
8 8
15 + 2
=
8
17
=
8
8 + 8 +1
=
8
8 8 1
= + +
8 8 8
1
= 1+1+
8
1
= 2+
8
1
=2
8
Ask:
• What happened to the sum of the fractional part of our number? (It was an
improper fraction, it was greater than one whole)
• What did we need to do to our fractional part? (Decompose, break into one whole
and a fractional unit, simplify)
2
3
Example #2 (We do): 3 + 2
3
4
Build It/ Draw It:
1
1
1
1
1
3
1
1
4
Page 2 of 9
1
4
1
3
1
4
MCC@WCCUSD 02/28/13
Bar Model
(Fraction Bars)
2
3
3 +2
3
4
Decomposition
Improper Fractions
2
3
3 +2
3
4
2 3
= 3+ 2 + +
3 4
2 ⎛ 4 ⎞ 3 ⎛ 3⎞
=5+ ⎜ ⎟ + ⎜ ⎟
3 ⎝ 4 ⎠ 4 ⎝ 3⎠
8 9
= 5+ +
12 12
8+9
= 5+
12
12 5
=5+ +
12 12
5
= 5 +1+
12
5
=6+
12
5
=6
12
2
3
3 +2
3
4
2
3
=3+ + 2 +
3
4
2 3
=3+ 2 + +
3 4
2⎛ 4⎞ 3⎛3⎞
=5+ ⎜ ⎟+ ⎜ ⎟
3⎝ 4⎠ 4⎝3⎠
8 9
=5+ +
12 12
8 +9
=5+
12
12 5
=5+ +
12 12
5
= 5 +1+
12
5
=6
12
11 11
+
3 4
11 ⎛ 4 ⎞ 11 ⎛ 3 ⎞
= ⎜ ⎟+ ⎜ ⎟
3 ⎝ 4⎠ 4 ⎝3⎠
44 33
= +
12 12
44 + 33
=
12
77
=
12
12 12 12 12 12 12 5
= + + + + + +
12 12 12 12 12 12 12
5
=1 + 1 + 1 + 1 + 1 + 1 +
12
5
=6+
12
5
=6
12
=
“With your partner, find the sum of the following mixed numbers using your fraction
bars.”
2 1
2 +3
3 2
Partner You Try #1 (You Do a Think, Pair, Share):
Build It/ Draw It:
1
1
1
1
1
3
1
3
1
Page 3 of 9
1
2
MCC@WCCUSD 02/28/13
Bar Model
(Fraction Bars)
2 1
2 +3
3 2
2⎛ 2⎞ 1 ⎛3⎞
= 2 + 3+ ⎜ ⎟ + ⎜ ⎟
3⎝ 2⎠ 2⎝3⎠
4 3
=5+ +
6 6
4+3
=5+
6
6 1
=5+ +
6 6
1
= 5 +1+
6
1
=6+
6
1
=6
6
1
4
Example #3 (We do): 4 + 3
2
5
Build It/ Draw It:
1
1
1
1
Decomposition
Improper Fractions
2 1
2 +3
3 2
2
1
= 2 + + 3+
3
2
2 1
= 2 + 3+ +
3 2
2⎛ 2⎞ 1 ⎛3⎞
=5+ ⎜ ⎟+ ⎜ ⎟
3⎝ 2⎠ 2⎝3⎠
4 3
=5+ +
6 6
4+3
=5+
6
6 1
=5+ +
6 6
1
= 5 +1+
6
1
=6+
6
1
=6
6
1
1
Page 4 of 9
2 1
2 +3
3 2
8 7
= +
3 2
8⎛ 2⎞ 7 ⎛ 3⎞
= ⎜ ⎟+ ⎜ ⎟
3⎝ 2 ⎠ 2 ⎝ 3⎠
16 21
= +
6 6
16 + 21
=
6
37
=
6
6 6 6 6 6 6 1
= + + + + + +
6 6 6 6 6 6 6
1
=1+1+1+1+1+ 1+
6
1
=6+
6
1
=6
6
1
1
2
1 1 1 1
5 5 5 5
MCC@WCCUSD 02/28/13
Bar Model
(Fraction Bars)
1
4
4 +3
2 5
1 4
= 4 + 3+ +
2 5
1⎛5⎞ 4⎛ 2⎞
=7+ ⎜ ⎟+ ⎜ ⎟
2⎝5⎠ 5⎝ 2⎠
5 8
=7+ +
10 10
5+8
=7+
10
10 3
=7+ +
10 10
3
= 7 +1+
10
3
= 8+
10
3
=8
10
Decomposition
Improper Fractions
1
4
4 +3
2 5
1
4
4 +3
2 5
1
4
= 4 + + 3+
2
5
1 4
= 4 + 3+ +
2 5
1⎛5⎞ 4⎛ 2⎞
=7+ ⎜ ⎟+ ⎜ ⎟
2⎝5⎠ 5⎝ 2⎠
5 8
=7+ +
10 10
5+8
=7+
10
10 3
=7+ +
10 10
3
= 7 +1+
10
3
= 8+
10
3
=8
10
9 19
= +
2 5
9 ⎛ 5 ⎞ 19 ⎛ 2 ⎞
= ⎜ ⎟+ ⎜ ⎟
2⎝5⎠ 5 ⎝ 2⎠
45 38
= +
10 10
45 + 38
=
10
83
=
10
10 10 10 10 10 10 10 10 3
= + + + + + + + +
10 10 10 10 10 10 10 10 10
3
=1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
10
3
= 8+
10
3
=8
10
3
7
1 +2
4
8
Build It/ Draw It:
You Try # 2:
1
1
1
4
1
4
1
4
1
1 1 1 1 1 1 1
8 8 8 8 8 8 8
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MCC@WCCUSD 02/28/13
Bar Model (Fraction
Bars)
3
7
1 +2
4
8
3 7
=1 + 2 + +
4 8
3⎛2⎞ 7
=3+ ⎜ ⎟ +
4⎝2⎠ 8
6 7
=3+ +
8 8
6+7
=3+
8
8 5
=3+ +
8 8
5
= 3 +1+
8
5
=4+
8
5
=4
8
€
Decomposition
Improper Fractions
3
7
1 +2
4
8
3
7
=1 + + 2 +
4
8
3 7
=1 + 2 + +
4 8
3 " 2$ 7
= 3+
+
4 # 2% 8
6 7
= 3+ +
8 8
6+7
=3+
8
8 5
=3+ +
8 8
5
= 3 +1+
8
5
=4 +
8
5
=4
8
3
7
1 +2
4
8
7 23
= +
4 8
7 ⎛ 2 ⎞ 23
= ⎜ ⎟+
4⎝2⎠ 8
14 23
= +
8 8
14 + 23
=
8
37
=
8
8 8 8 8 5
= + + + +
8 8 8 8 8
5
=1+1+1+1 +
8
5
=4+
8
5
=4
8
“Now that we have found the sum of two mixed numbers let’s take a look and see if we
can apply the same strategies with three mixed numbers.”
Example # 4 (I do/We do): In one week during P.E., Mary ran 4
3
miles, Lauren ran
5
2
3
miles, and Dante ran 3 miles. In one week, how many miles total did the three
5
10
students run?
1
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MCC@WCCUSD 02/28/13
3
3
2
4 +1 + 3
5 10 5
Build It/ Draw It:
1
1
1
Bar Model
(Fraction Bars)
3
3
2
4 +1 + 3
5 10 5
5 3
= 4 +1+ 3 + +
5 10
3
= 4 +1 + 3 +1 +
10
3
=9 +
10
3
= 9 miles
10
1
1
1
1 1 1
5 5 5
1 1 1
10 10 10
1
1
Decomposition
1
5
1
5
Improper Fractions
3
3
2
4 +1 + 3
5 10 5
3 3 2
= 4 +1+ 3 + + +
5 10 5
3+ 2 3
=8 +
+
5
10
5 3
=8+ +
5 10
3
= 8 +1+
10
3
= 9+
10
3
= 9 miles
10
Page 7 of 9
3
3
2
4 +1 + 3
5 10 5
23 13 17
= + +
5 10 5
23 ⎛ 2 ⎞ 13 17 ⎛ 2 ⎞
= ⎜ ⎟+ + ⎜ ⎟
5 ⎝ 2 ⎠ 10 5 ⎝ 2 ⎠
46 13 34
= + +
10 10 10
46 +13 + 34
=
10
93
=
10
10 10 10 10 10 10 10 10 10 3
= + + + + + + + + +
10 10 10 10 10 10 10 10 10 10
3
=1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
10
3
= 9+
10
3
= 9 miles
10
MCC@WCCUSD 02/28/13
You Try # 3: Carla bought candy at the Candy Shop for the party. She purchased 2
2
3
2
1
pounds of red vines, and 1 pounds of gummy bears.
5
3
How many pounds of candy did Carla buy for the party?
2
2 1
2 + 3 +1
3
5 3
Build It/ Draw It:
pounds of chocolate almonds, 3
1
1
1
1
1
1
3
1
3
1
1
5
1
5
1
3
Bar Model
(Fraction Bars)
2 2 1
2 + 3 +1
3 5 3
1 2 2
=6+ + +
3 3 5
3 2
=6+ +
3 5
2
= 6 +1 +
5
2
=7+
5
2
= 7 pounds of candy
5
Decomposition
2 2 1
2 + 3 +1
3 5 3
2 2 1
= 2 + 3 +1+ + +
3 5 3
2+1 2
=6+
+
3
5
3 2
=6+ +
3 5
2
= 6 +1+
5
2
=7+
5
2
= 7 pounds of candy
5
Page 8 of 9
Improper Fractions
2 2 1
2 + 3 +1
3 5 3
8 17 4
= + +
3 5 3
8 ⎛ 5 ⎞ 17 ⎛ 3 ⎞ 4 ⎛ 5 ⎞
= ⎜ ⎟+ ⎜ ⎟+ ⎜ ⎟
3⎝ 5⎠ 5 ⎝ 3⎠ 3⎝ 5⎠
40 51 20
= + +
15 15 15
40 + 51 + 20
=
15
111
=
15
15 15 15 15 15 15 15 6
= + + + + + + +
15 15 15 15 15 15 15 15
6
=1 + 1 + 1 + 1 + 1 + 1 + 1 +
15
2• 3
=7+
3• 5
2
=7+
5
2
= 7 pounds of candy
5
MCC@WCCUSD 02/28/13
Do: Have students present out on the overhead/ board their answers. Look for different
ways that students approached the problem.
Debrief: Ask students to discuss what they discovered today when building the sum of
mixed numbers and when using decomposition. Ask what strategies they used today
when adding mixed numbers.
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MCC@WCCUSD 02/28/13