Subtracting Mixed
Numbers
Objective To develop subtraction concepts related
tto mixed numbers.
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Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Find equivalent names for mixed numbers. [Number and Numeration Goal 5]
• Convert between fractions and
mixed numbers. [Number and Numeration Goal 5]
• Subtract mixed numbers. [Operations and Computation Goal 4]
• Use benchmarks to estimate differences. [Operations and Computation Goal 6]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Mixed-Number Spin
Math Journal 2, p. 255
Math Masters, pp. 488 and 489
per partnership: large paper clip
Students practice adding and
subtracting fractions and mixed
numbers with like and unlike
denominators.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 255. [Operations and Computation Goals 4 and 6]
Key Activities
Students subtract mixed numbers with like
denominators by renaming the minuend.
Students use benchmarks to estimate
differences.
Ongoing Assessment:
Informing Instruction See page 632.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Subtracting Mixed Numbers
Students use a counting-up algorithm to
explore mixed-number subtraction.
ENRICHMENT
Exploring a Pattern for Fraction Addition
and Subtraction
Math Masters, p. 225
Students explore patterns that result from
adding and subtracting unit fractions.
EXTRA PRACTICE
Math Boxes 8 3
5-Minute Math
Math Journal 2, p. 256
Geometry Template
Students practice and maintain skills
through Math Box problems.
5-Minute Math™, pp. 184 and 185
Students add mixed numbers.
Study Link 8 3
Materials
Math Journal 2, p. 254
Math Masters, p. 414
Study Link 82
slate
Math Masters, p. 224
Students practice and maintain skills
through Study Link activities.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 142, 143
630
Unit 8
Fractions and Ratios
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Getting Started
Mental Math and Reflexes
Math Message
Have students name a common denominator for
the following:
Solve Problems 1–3 at the top of journal page 254.
Use benchmarks to estimate the differences and to
check the reasonableness of your solutions.
halves and thirds sixths
fifths and thirds fifteenths
fourths and eighths eighths
fifths and eighths fortieths
halves, thirds, and fourths twelfths
halves, fifths, and sixths thirtieths
Study Link 8 2 Follow-Up
Have partners compare answers and resolve
differences. Ask volunteers how they know their
answers are in simplest form. A fraction is in simplest
form if the numerator and denominator have no
common factors except 1.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 254)
Ask students to share their estimating strategies for Problem 2
4 is close to 5 and
using benchmarks. Sample answer: I know that 4_
5
5 – 2 = 3. I estimated my answer to be about 3.
Ask volunteers to write their solutions on the board and explain
their strategies. Expect that most students will have subtracted
the whole-number and fraction parts separately and renamed the
differences in Problems 1 and 3 in simplest form.
▶ Subtracting Mixed Numbers
WHOLE-CLASS
ACTIVITY
with Renaming
Student Page
Date
Time
LESSON
Subtracting Mixed Numbers
8 3
Math Message
Subtract.
Write this problem on the board.
1
3_
-
1.
3
2.
1
4 _5
4
3.
–2
2 _2
2 _5
1
3
_
1 23
7 _6
2
– 2 _6
5
5 _2
4
1
Renaming and Subtracting Mixed Numbers
Fill in the missing numbers.
Ask: How does this problem differ from the Math Message
1 , is smaller than
problems? The fraction being subtracted from, _
3
2
_
the fraction that is being subtracted, 3 . Ask volunteers to suggest
1 to make the
solution strategies. Sample answer: Rename 3_
3
4
2
2
_
_
_
fraction part larger. 2 3 - 1 3 = 1 3
Remind students that when they add mixed numbers, some sums
might have to be renamed to write them in simplest form. For
7 could be renamed: 4_
7 =4+_
3 , or
4 +_
example, a sum of 4_
4
4
4
4
3 = 5_
3 . Emphasize that the two mixed numbers are
4+1+_
4
4
7 is another name for 5_
3.
equivalent; 4_
4
3 _4
–1 _4
4
4.
5
1
5 _4 = 4 _
4
5.
3
6=5_
3
6.
23
5
3 _6 = _
6
7.
8 _9 =
7
7
16
_
9
Subtract. Write your answers in simplest form. Show your work.
7 _3
2
8.
1
8 – _3 =
3 _2
3
5 – 2 _5 =
_3
1
10.
12.
3
1
7 _4 – 3 _4 =
2 _5
2
9.
11.
5
7
4 _8 – 3 _8 =
4
1
Isaac would like to practice the violin 7_
2 hours this week. So far, he has practiced
3
5_
4 hours. How many more hours does he need to practice this week?
1 _4 hours
3
Math Journal 2, p. 254
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Lesson 8 3
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1 to an equivalent mixed number with a
Have students rename 3_
3
7
4 Write their responses on the board
_
larger fraction part. 1 3 , or 2_
3
or a transparency, and illustrate the mixed numbers with pictures.
1
7
_
3_
3 = 13
1
4
_
3_
3 = 23
Ask students to use each of the responses to solve the problem.
Circulate and assist. When most students have finished, have
volunteers explain which mixed-number name was more efficient
7 you need
4 , because with 1_
for solving the subtraction problem. 2_
3
3
to rename the fraction again in order to express the difference as a
mixed number. Explain that deciding how to rename a fraction
depends on the problem. Sometimes it might be more efficient to
work with mixed numbers, other times it might be more efficient
to work with improper fractions. In this case, the more efficient
mixed-number name is a matter of personal preference.
Pose several problems. Encourage students to estimate each
difference using what they know about benchmarks with fractions.
They should use their estimates to check the reasonableness of
their solutions. In each case, ask students first how they would
rename the minuend and then how they would solve the problem.
Suggestions:
3 ; solution: 4_
2 8 = 7_
1
● 8 - 3_
3
Ongoing Assessment:
Informing Instruction
Watch for students who have difficulty
renaming mixed numbers for subtraction.
Have students practice the following steps:
1. Write the whole-number part of the
mixed number as an equivalent addition
expression that adds 1.
2
2
5_ = 4 + 1 + _
5
5
2. Use the denominator to rename the 1 as
a fraction.
5
2
2
5_ = 4 + _ + _
5
5
5
3. Combine the fraction parts.
7
2
5_ = 4 + _
5
5
632
Unit 8
3
3
●
3
1 6 = 5_
4 ; solution: 5_
6-_
●
3 - 1_
3 = 3_
8 ; solution: 2_
4 4_
4
4_
5
5 5
5
5
5
7
1
1
_
_
_
_
_
5 - 2 5 = 4 ; solution: 2 1
●
●
●
●
4
6
4
6
6
4
6
3
5 - 3_
5 = 5_
17 ; solution: 2_
11 6_
1
6_
12
12 12
12
2
Carlie purchased 10 feet of ribbon to make bows. She used
7 feet to make one bow. How much ribbon remains to make
2_
8
1 feet
additional bows? 7_
8
1
_
Sammy needs 3 4 cups of sugar to make sugar cookies. He only
3 cups of sugar. How much more sugar does he need for
has 1_
4
1 cups
the recipe? 1_
2
Fractions and Ratios
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Student Page
Write this problem on the board.
1
3_
-
Date
LESSON
83
3
_
1 12
Materials
Players
Time
Mixed-Number Spin
□
Math Masters, p. 488
□
large paper clip
2
Directions
Ask: How would you solve this problem? Expect that students will
1 is smaller than _
1 and might suggest renaming
likely recognize _
3
2
the fraction parts with common denominators. Ask volunteers to
model their strategies on the board. Sample answer: Use 6 as a
3
2 - 1_
common denominator. Rename the mixed numbers. 3_
6
6
8 . Subtract. 2_
8 - 1_
3 =_
5.
2 = 2_
Rename the minuend. 3_
6
6
6
6
6
Pose a few mixed-number subtraction number stories. Suggestions:
1 hours of viewing. So far,
● Daniel has a DVD set that provides 4_
2
5 of an hour. How many more hours of the DVD
he has watched _
6
4 , or 3_
2 hours
set does he still have to watch? 3_
6
●
3
2 feet of rope to make a jump rope. Her
Serena measured out 5_
3
1 feet long. How much
friends told her that it needed to be 6_
2
5 ft more
more rope did she need to measure out? _
6
1.
Each player writes his or her name in one of the boxes below.
2.
Take turns spinning. When it is your turn, write the fraction or mixed number
you spin in one of the blanks below your name.
3.
The first player to complete 10 true sentences is the winner.
Name
Name
+
<3
+
<3
+
>3
+
>3
-
<1
-
<1
-
1
<_
-
1
<_
2
+
>1
+
>1
+
<1
+
<1
+
<2
+
<2
-
=3
-
=3
-
>1
-
>1
+
1
>_
+
1
>_
+
<3
+
<3
+
>2
+
>2
2
2
2
Math Journal 2, p. 255
▶ Subtracting Mixed Numbers
PARTNER
ACTIVITY
248-287_EMCS_S_MJ2_U08_576434.indd 255
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(Math Journal 2, p. 254; Math Masters, p. 414)
Ask students to complete the journal page. Remind them to use
benchmarks to estimate the difference. Ask students to use an Exit
Slip (Math Masters, page 414) to explain their strategy for solving
Problem 10 on Math Journal 2, page 254.
2 Ongoing Learning & Practice
▶ Playing Mixed-Number Spin
PARTNER
ACTIVITY
(Math Journal 2, p. 255; Math Masters, pp. 488 and 489)
Algebraic Thinking Students practice estimating sums and
differences of mixed numbers with like and unlike denominators
by playing Mixed-Number Spin. Players use benchmarks to
estimate sums and differences as they record number expressions
that fit the parameters given on the Mixed-Number Spin record
sheet, Math Journal 2, page 255. Each partnership makes a
spinner using a large paper clip anchored by a pencil point to the
center of the spinner on Math Masters, page 488. They use the
numbers they spin to complete the number sentences on the
record sheet.
Links to the Future
Subtraction of mixed numbers with unlike
denominators continues in Sixth Grade
Everyday Mathematics.
Ongoing Assessment:
Recognizing Student
Achievement Journal
Page 255 Use the Mixed-Number Spin record sheet on
journal page 255 to assess students’ ability
to make estimates for adding and subtracting
mixed numbers. Students are making
adequate progress if they correctly complete
1
number sentences using the benchmarks _
2
and 1. Some students may be able to make
the estimates and comparisons using 2 and 3
as benchmarks.
[Operations and Computation Goals 4 and 6]
Ask students to save their spinners for future use. Copy Math
Masters, page 489 to provide additional record sheets.
Lesson 8 3
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Student Page
Date
Time
LESSON
1.
▶ Math Boxes 8 3
(Math Journal 2, p. 256)
Make a circle graph of the survey results.
Time Spent on Homework
Time Spent on Homework
Time
2.
title
Percent of Students
0–29 minutes
25%
30–59 minutes
48%
60–89 minutes
10%
90–119 minutes
12%
2 hours or more
5%
126 127
3.
423 thousand
b.
423,000
c.
18,000,000,000
d.
9,500,000
48%
25%
56 million
56,000,000
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 8-1. The skill in Problem 5
previews Unit 9 content.
10%
12%
5%
Write each numeral in
number-and-word notation.
a.
18 billion
95 hundredthousand
a.
All rectangles are
parallelograms.
T
b.
A kite is not
a parallelogram.
T
c.
Some parallelograms
are squares.
T
d.
Some trapezoids are
not quadrangles.
F
146
5.
Complete the “What’s My Rule?”
table and state the rule.
in
out
Rule
48
6
out = in ÷ 8
40
5
Find the area of the rectangle.
12 m
6m
8
0
0
16
2
Writing/Reasoning Have students write a response to the
following: Explain how you determined your answer to
Problem 3c. Sample answer: The answer is true because a
parallelogram has two pairs of parallel sides with opposite sides
congruent. When all sides of a parallelogram are the same length
and form right angles, the parallelogram is also a square.
Area = b ∗ h
1
_
1
Writing/Reasoning Have students write a response to the
following: Explain one advantage and one disadvantage to
using number and word notation in Problem 2. Sample
answer: One advantage is that the number is written the same
way that it is said aloud. One disadvantage is that you cannot
perform operations on them as easily.
True or False? Write T or F.
4
4.
INDEPENDENT
ACTIVITY
Math Boxes
8 3
72 m2
Area:
(unit)
231 232
189
Math Journal 2, p. 256
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▶ Study Link 8 3
INDEPENDENT
ACTIVITY
(Math Masters, p. 224)
Home Connection Students practice renaming and
subtracting mixed numbers.
3 Differentiation Options
Name
Date
STUDY LINK
▶ Subtracting Mixed Numbers
Time
Subtracting Mixed Numbers
83
71 72
11
= 2—
3_
8
8
3.
= 1—
2_
9
9
5.
= 3—
4_
5
5
3
10
1
8
3
2.
=
4_
6
3
11
_
4.
=
6_
7
5
10
_
6.
=
7_
3
6
—
5
3
2
6
7
5
3
Subtract. Write your answers in simplest form.
7.
5_
4
3
3_58
3
4 - _8 =
12.
3
=
5 - 2_
10
7
2_
10
- 1_
=
3_
5
5
1_45
3
Practice
16.
18.
9.
654 * 205 =
654 * 250 =
134,070
163,500
5
6 - _9 =
13.
3
=
7 - 4_
4
17.
19.
4
5
- 3_
=
4_
8
8
3
7
654 * 502 =
654 * 520 =
5
way to answer plus what?
2_15
11.
15.
5_
5
3
- 3_
5
2_13
10.
2
2
1
- 4_
3
2_12
14.
6_
3
8.
1
- 3_
4
15–30 Min
To explore mixed-number subtraction, have students use a
counting-up algorithm. Remind students that another way to
subtract is to count up. With whole numbers, they could solve
5 - 3 by thinking 3 plus what equals 5? Discuss how they would
2 - 3_
4 . Expect that students will
count up to solve this problem: 5_
5
5
4 plus what equals 5_
2 ?” Explain that counting up is one
suggest “3_
Fill in the missing numbers.
1.
SMALL-GROUP
ACTIVITY
READINESS
Study Link Master
4 to 5_
2.
Demonstrate how to count up from 3_
5
5
5_49
2_14
1. Count up, keeping track of the amounts used.
1 : 3_
4 +_
1 =4
1
_
Count up _
_1
2
5
328,308
Count up 1:
2:
Count up _
340,080
5
5
5
5
4+1=5
2 = 5_
2
5+_
5
5
1
2
_
5
3
1 +1+_
2 = 1_
2. Add. _
5
5
5
3.
2 and 3_
4 is 1_
So the difference between 5_
5
5
5
Math Masters, p. 224
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Unit 8
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Fractions and Ratios
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Teaching Master
Pose problems for students to solve by counting up. Suggestions:
7 2_
3
8 4_
2 - 1_
1
● 4_
● 7 - 2_
8
3
● 3_
5
-
8
_
1 45
8
_
1 45
10
●
5
Name
Date
LESSON
83
Addition and Subtraction Patterns
Add.
3 2_
1 - 2_
1
5_
4
4 2
_1
1. a. 1
+ _12 =
_1
_1
d. 4 + 5 =
PARTNER
ACTIVITY
ENRICHMENT
▶ Exploring a Pattern for Fraction
15–30 Min
Addition and Subtraction
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
2.
3.
Time
1_12 , or _32
9
_
20
_1
b. 2
+ _13 =
_1 _1
e. 5 + 6 =
_5
6
_1
c. 3
+ _14 =
7
_
12
11
_
30
Sample answer:
In each problem, the answer is the sum of the denominators
over the product of the denominators.
What pattern do you notice in Problems 1a–1e?
Use the pattern above to solve these problems.
_1
a. 6
1
+ _7 =
13
_
42
1
_
b. 10
1
+_
11 =
21
_
110
1
1
_
+_
100 =
c. 99
199
_
9,900
_1
_1
4. Do you think this pattern also works for problems like 8 + 3 ? Explain.
Sample answer: Yes. This pattern will work whenever two
unit fractions are added.
(Math Masters, p. 225)
Algebraic Thinking To apply students’ understanding of addition
and subtraction of fractions, have them find patterns in unit
fraction addition and subtraction problems. When most students
have completed the problems, discuss the pattern in Problem 1. In
each problem, the answer is the sum of the denominators over the
product of the denominators.
Problem 4 asks whether this pattern works for the sum of any two
1
1
1
1
_
_
_
unit fractions _
a + b . It does. Add a and b using the QCD method:
b+a
1∗b
1∗a
b
a
1
1
_
_
_
_
_
_
_
a + b = a∗b + b∗a = a∗b + b∗a = a∗b
5.
The plus signs in Problem 1 have been replaced with minus signs. Find each answer.
_1
a. 1
- _12 =
_1
d. 4
- _15 =
f.
_1
2
1
_
20
_1
b. 2
- _13 =
_1
e. 5
- _16 =
_1
6
_1
c. 3
- _14 =
1
_
12
1
_
30
Describe the pattern. Sample answer: The numerator is always 1.
The denominator of the result is the product of the
denominators of the subtracted fractions.
Math Masters, p. 225
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Have students verify that this pattern holds true for other unit
1 +_
1.
fractions, for example: _
3
5
The pattern for Problem 5 is similar. The denominator of the result
is the product of the denominators of the unit fractions being
subtracted. The numerator is always 1.
Problems 5a–5e involve the subtraction of two unit fractions
where the difference between the denominators is 1. The pattern
described in Problem 5f reflects that type of problem. You may
want to extend the Math Master activity by having students
find the results of the following subtraction problems (where
the difference between the denominators of the unit fractions is
greater than 1):
3
5
7
1 -_
1 _
1 -_
1 _
1 -_
1 _
_
_
_
2
5 10
3
8 24
5
12 60
Ask students to describe a pattern for subtracting any two unit
fractions. Sample answer: The numerator of the result is found
by subtracting the denominator of the first fraction from the
denominator of the second fraction. The denominator of the result
is the product of the denominators of the two unit fractions.
EXTRA PRACTICE
▶ 5-Minute Math
SMALL-GROUP
ACTIVITY
5–15 Min
To offer students more experience with adding mixed numbers,
see 5-Minute Math, pages 184 and 185.
Lesson 8 3
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