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Objective
To add and subtract mixed numbers with
like denominators.
1
materials
Teaching the Lesson
Key Activities
Students practice adding and subtracting mixed numbers that have fractions with
like denominators.
Math Journal 1, pp. 132 and 133
Student Reference Book,
pp. 84–86
Study Link 4 3
Key Concepts and Skills
• Convert between fractions and mixed numbers. [Number and Numeration Goal 5]
• Use multiplication and division facts to find equivalent fractions and to simplify fractions.
[Operations and Computation Goal 2]
• Add and subtract mixed numbers with like denominators. [Operations and Computation Goal 3]
Teaching Master (Math Masters,
p. 117)
scissors
See Advance Preparation
Key Vocabulary
mixed number • proper fraction • improper fraction • simplest form
Ongoing Assessment: Recognizing Student Achievement Use journal page 133.
[Operations and Computation Goal 3]
2
materials
Ongoing Learning & Practice
Students practice estimating sums of fractions by playing Fraction Action, Fraction Friction.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
materials
Differentiation Options
READINESS
Students use a calculator
to practice counting by
fractions and converting
between improper fractions
and mixed numbers.
ENRICHMENT
Students read a poem
about fractions.
EXTRA PRACTICE
Students use bills and
coins to model and simplify
mixed numbers.
Additional Information
Advance Preparation For the Math Message in Part 1, make one copy of
Math Masters, page 117 for every two students.
272
Unit 4 Rational Number Uses and Operations
Math Journal 1, p. 134
Student Reference Book, p. 317
Study Link Master (Math Masters,
p. 118)
Game Master (Math Masters,
p. 446)
Geometry Template; calculator
Teaching Master (Math Masters,
p. 119)
Math Talk: Mathematical Ideas in
Poems for Two Voices
calculator; coins and bills of various
denominations
Technology
Assessment Management System
Journal page 133, Problems 9, 12, 14,
and 15
See the iTLG.
Getting Started
Mental Math and Reflexes
Math Message
Students rename improper fractions as mixed
numbers. Suggestions:
Complete a copy of the Math Message problem.
3
1
1
2
2
5
2
1
3
3
75
3
18
4
4
5
1
1
4
4
13 1
2
6
6
108
3
21
5
5
Study Link 4 3 Follow-Up
Go over the answers.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
NOTE If students are fairly skilled at finding
(Math Masters, p. 117)
sums and differences of fractions, this lesson
may take less than one day.
Ask a volunteer to demonstrate and explain how to use the paper
ruler to measure the line segment. Sample answer: Line up the
right end of A
苶B
苶 with the mark for 4 in. on the ruler. The left end
5
of 苶
AB
苶 is aligned with the mark for 16 in. The total length of A
苶B
苶 is
5
5
4 in. 16 in., or 4 16 in. This ruler shows the two parts of a mixed
number: the whole number and the fraction. A mixed number
can be viewed as the sum of a whole number and a fraction.
Discuss why fractions greater than 1 are easier to interpret when
3
written as mixed numbers. For example, 24 clearly represents a
number greater than 2 but less than 3. This is not as obvious
3
11
when 24 is written as the improper fraction 4.
Writing Mixed Numbers
Teaching Master
Name
WHOLE-CLASS
DISCUSSION
in Simplest Form
Date
LESSON
Time
Math Message
4 4
Cut out the ruler below. Use it to measure line segment AB to the
1
nearest inch.
16
A
Review the meanings of proper fraction, improper fraction, and
simplest form with the class.
B
—
length of AB A fraction in which the numerator is less than the
denominator is called a proper fraction. A proper fraction
names a number that is less than 1.
3
9
5
16
4 in.
0
1
2
3
4
5
inches
0
Examples: 8, 10, 4
A fraction in which the numerator is equal to or greater than
the denominator is called an improper fraction. An
improper fraction names a number that is greater than or
equal to 1.
5 7 9
Examples: 5, 2, 3
Name
Date
LESSON
Time
Math Message
4 4
Cut out the ruler below. Use it to measure line segment AB to the
1
nearest inch.
16
A
B
—
length of AB 0
1
2
3
4
5
inches
Math Masters, p. 117
Lesson 4 4
273
Student Page
Date
Time
LESSON
A mixed number is in simplest form if the fraction part is a
proper fraction in simplest form.
Adding and Subtracting Mixed Numbers
4 4
2
5
4
5
Example 1: 1 2 ?
84–86
Step 1
Add the fractions. Then add the
whole numbers.
6
3 65 3 5
2
15
5
1
3 5 5
4
2 5
1
3 1 5
1
4 5
6
3 5
Add. Write your answers in simplest form.
1
4 5
1.
2
14
2.
2
3 5
3
1 4
414
7
735
5
4 8
Example 2:
1
2 8
1
5 4
3.
3
2 4
2
Step 2
If necessary, rename the difference.
5
4 8
5
4 8
1
2 8
1
2 8
4
4
Example 3:
2
13
1
2 8 2 2
2 8
1
5 3
9
be written using proper fractions, the fraction part is not required to be in
simplest form. Students should know how to name fractions in simplest form
for standardized tests. However, they may often find it helpful to work with
fractions or mixed numbers that are not in simplest form when they are
computing with fractions.
2
18
3
4
?
Step 1
Subtract the fractions.
Then subtract the whole numbers.
8
NOTE While Everyday Mathematics requests that mixed-number answers
4
18
4.
4
Examples: 27 is in simplest form. 45 and 39 are not in
simplest form because they contain an improper fraction.
4
4
38 is not in simplest form because 8 is not in simplest form.
Step 2
If necessary, rename the sum.
5
Write 34 on the board and ask a volunteer to rename it in
simplest form. Use pictures similar to those below to show
the procedure.
?
Notice that the fraction in the first mixed number is less than the fraction in the second
2
1
1
mixed number. Because you can’t subtract 3 from 3, you need to rename 5 3.
Step 1
Rename the first mixed number.
1
1
5 3 4 1 3
3
Step 2
Subtract the fractions.
Then subtract the whole numbers.
1
4 3 3
4
5 3
1
4 3
2
13
1
4
4
1
2
13
4
4 3 4 3
1
1
1
4
1
4
1
4
1
4
2
3 3
132
Math Journal 1, p. 132
5
34 1
1
1
1
Time
4 4
5
7 88
7 8
5
3 8
6.
1
4 5
2 4
3
2 5
1
2
4
5
5
7.
2
10.
1
2
1
3 5
5
11.
1 5
3
14
3
1
3
3
5
1
4
3
1
3
7
6 8
1
5
42 52
2
9
23 33
10
A U D I T O R Y
1
4
7 12 inches
14. Mr. Ventrelli is making bread. He adds
cups
1
of white flour and 14 cups of wheat flour. The
recipe calls for the same number of cups of
water as cups of flour. How much water
should he add?
2 12 cups of water
15. Evelyn’s house is between Robert’s
and Elizabeth’s. How far is Robert’s
house from Elizabeth’s?
1
2
3
4
mi
3
1 4 mi
2 miles
Robert’s
Evelyn’s
1
16 22
ELL
3
3 8
1
14
Elizabeth’s
133
Math Journal 1, p. 133
274
1
44
Have students model mixed numbers using bills and quarters. To
5
model 34, use $3 and 5 quarters. Five quarters is equal to $1 and 1 quarter.
Therefore, $3 and 5 quarters $4 and 1 quarter.
1
3 4
3
4
12.
2 6
6
7
8.
3
13. Joe has a board that is 8 4 inches long. He cuts
1
off 14 inches. How long is the remaining piece?
1
4
Adjusting the Activity
2
2 3
1
3
1
1
3
Add or subtract.
4 6
1
4
5
4 8
9.
1
4
8
8
3 8
8
7 8
3
75 8
Step 2
Subtract the fractions. Then subtract
the whole numbers.
871
1
84–86
Step 1
Rename the whole number.
3 4
1
1
4
Write several such mixed numbers on the board. Have students
rename them in simplest form. Suggestions:
5
Example 4: 8 3 8 ?
5.
4
4
1
4
354 can be renamed as 414.
Adding and Subtracting Mixed Numbers cont.
1
4
1
4
Student Page
LESSON
1
4
Date
1
4
1
3
1
5
4
3
Unit 4 Rational Number Uses and Operations
K I N E S T H E T I C
T A C T I L E
V I S U A L
Student Page
Adding Mixed Numbers
INDEPENDENT
ACTIVITY
with Like Denominators
Fractions
Addition of Mixed Numbers
One way to add mixed numbers is to add the fractions and the
whole numbers separately. This may require renaming the sum.
(Math Journal 1, p. 132; Student Reference Book, p. 84)
5
Example
7
Find 48 28.
Step 1: Add the fractions.
Step 2: Add the whole
Step 3: Rename the sum.
numbers.
2
4
Write the following problem on the board: 15 25 ?
Ask students to solve and then share their strategies. Go over
the steps on journal page 132 before having students complete
Problems 1–4 on their own. Bring the class together to share
solutions. If necessary, provide more practice, particularly with
problems that require renaming. Refer students to page 84 in the
Student Reference Book.
5
4
5
8
4
2
7
8
12
8
2
7
5
8
6
12
8
7
8
12
6 8
6
8
8
61
4
8
1
2
7
7
1
48 28 72
4
8
4
8
If the fractions do not have the same denominator, first rename
the fractions so they have a common denominator.
3
Example
2
Find 34 53.
Step 1: Rename and add
5
3
3
4
2
3
3
5
2
Step 3: Rename the sum.
Step 2: Add the whole
numbers.
the fractions.
3
9
12
8
12
17
12
3
5
8
9
12
8
12
17
12
8
17
12
8 12
12
8 1 9
5
12
5
12
5
12
5
34 53 912
Subtracting Mixed Numbers
PARTNER
ACTIVITY
Some calculators have special keys for entering mixed numbers.
3
Example
2
Solve 34 53 on a calculator.
On Calculator A: Key in 3
with Like Denominators
Unit
On Calculator B: Key in 3
(Math Journal 1, pp. 132 and 133; Student Reference Book, p. 85)
3
3
4
n
d
4
5
+
5
Unit
2
2
n
3
Enter
=
d
3
Check Your Understanding
Solve Problems 1–3 without a calculator. Solve Problem 4 with a calculator.
1
Go over the three subtraction examples on journal pages 132
and 133 (Examples 2–4). Ask students to solve Problem 5. If
successful, they should continue and complete the page. You may
need to provide additional practice before continuing or refer
students to page 85 in the Student Reference Book.
Ongoing Assessment:
Recognizing Student Achievement
7
4
1. 28 78
1
2
2. 35 22
3
4
3. 63 34
Ch
k
6
4. 149 87
417
Student Reference Book, p. 84
Journal
Page 133
Problems
9, 12, 14, and 15
Use journal page 133, Problems 9, 12, 14, and 15 to assess
students’ ability to add mixed numbers with like denominators. Students are making
adequate progress if they can calculate the sums in Problems 9, 12, 14, and 15.
Some students may be able to calculate the differences in Problems 5–8, 10, 11,
and 13.
[Operations and Computation Goal 3]
Student Page
Fractions
Subtraction of Mixed Numbers
If the fractions do not have the same denominator, first rename
them as fractions with a common denominator.
7
3
Find 38 14.
Example
Step 1: Rename the
fractions.
3 7
8
1 3
4
7
8
1 6
8
3
Step 2: Subtract the
Step 3: Subtract the whole
fractions.
3 7
1
38 14 28
numbers.
3 8
7
3 7
1 6
1 6
8
8
1
8
8
2 1
8
To subtract a mixed number from a whole number, first rename
the whole number as the sum of a whole number and a fraction
that is equivalent to 1.
2
Find 5 23.
Example
Step 1: Rename the whole
number.
2 2
3
2
3
2 2
3
Step 3: Subtract the whole
fractions.
4 3
5
Step 2: Subtract the
1
5 23 23
numbers.
4 3
4 3
3
2 2
3
1
3
3
2 2
3
1
3
2 When subtracting mixed numbers, rename the larger mixed
number if it contains a fraction that is less than the fraction in
the smaller mixed number.
Example
1
3
Find 75 35.
Step 1: Rename the larger
mixed number.
7 1
5
3 3
5
1
3
Step 2: Subtract the
Step 3: Subtract the whole
fractions.
numbers.
6 6
5
6 6
5
6 6
3 3
3 3
3 3
5
3
75 35 35
5
5
3
5
5
3 3
5
Student Reference Book, p. 85
Lesson 4 4
275
Game Master
Name
Date
Time
Fraction Action, Fraction Friction Card Deck
1
2
1
3
2
3
1 2
4 3
2 Ongoing Learning & Practice
1
4
Playing Fraction Action,
PARTNER
ACTIVITY
Fraction Friction
3
4
1
6
1
6
5
6
1
12
1
12
5
12
5
12
7
12
7
12
11
12
11
12
(Student Reference Book, p. 317; Math Masters, p. 446)
Distribute one set of 16 Fraction Action, Fraction Friction cards
and one or more calculators to each group of two or three players.
Review game directions on page 317 of the Student Reference
Book. Play a few practice rounds with the class.
Math Boxes 4 4
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 134)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 4-2. The skill in Problem 5
previews Unit 5 content. Students will need the Percent
Circle on the Geometry Template to complete Problem 5.
Math Masters, p. 446
Study Link 4 4
INDEPENDENT
ACTIVITY
(Math Masters, p. 118)
Home Connection Students practice addition and
subtraction of mixed numbers.
Study Link Master
Student Page
Date
Time
LESSON
Name
4 4
18
a. 45 2
5
26
b. 39
2
3
1
3
a. 10
5
7
10
56
c. 80
7
10
25
d. 625
1
25
5
1
b. 12
3
3
4
7
4
c. 9
9
1
3
6
d. 8
0
9
4
Write 2 fractions equivalent to .
e.
27
12
18
8
3
4
In a national test, eighth-grade students answered the problem shown in the
top of the table at the right. Also shown are the 5 possible answers they
were given and the percent of students who chose each answer.
Sample answers:
a.
4. Thomas Jefferson was born in 1743.
George Washington was born m years
earlier. In what year was Washington born?
Choose the best answer.
following set of numbers.
1.5, 2.8, 3.4, 4.5, 2.2, 8.4
3.1
b. mean
3.8
m 1743
The mean would double.
1743 m
240
5. The table below shows the results of a
survey in which people were asked
which winter Olympic sport they most
enjoyed watching. Use a Percent Circle
to make a circle graph of the results.
Favorite
Sport
Percent of
People Surveyed
Luge
35%
Ice hockey
15%
Figure skating
40%
Other
10%
Winter Olympic Sports Preferences
28%
D. 21
27%
14%
1
4
1 inches
(unit)
3
Add or subtract. Write your answers as mixed numbers in simplest form. Show your work on
the back of the page. Use number sense to check whether each answer is reasonable.
1
3
1
1
1
1
2
2
2
4
3
4. 3 1 5. 4 2 6. 1 4
4
4
1
4
3
3
2
3
Circle the numbers that are equivalent to 24.
7
6
4
3
7
11
4
Practice
Solve mentally.
8.
5 º 18 90
9.
6 º 41 246
145 146
Unit 4 Rational Number Uses and Operations
C. 19
Tim is making papier-mâché. The recipe calls for 14 cups of paste. Using only
1
1
1
-cup, -cup, and -cup measures, how can he measure the correct amount?
2
4
3
1
Sample answer: He can use three 2-cup measures and
1
one 4-cup measure.
134
276
24%
3
14
Math Journal 1, p. 134
7%
B. 2
3.
Luge
Other
Figure
skating
A. 1
A board is 68 inches long. Verna wants to cut enough
1
so that it will be 58 inches long. How much should she cut?
7.
Ice
hockey
Explain why B is the best estimate.
2.
1743 m
136 137
7
Percent Who Chose
This Answer
E. I don’t know.
m 1743
Suppose you multiplied each data value
by 2. What would happen to the mean?
Possible
Answers
Both fractions are close to
1, so their sum should be
close to 2.
83
74
12
Estimate the answer to 13 8.
You will not have enough time to solve
the problem using paper and pencil.
What mistake do you think the
students who chose C made?
They may have added
only the numerators.
b.
3. Find the median and mean for the
a. median
1.
2. Add or subtract. Then simplify.
Time
ⴙ, ⴚ Fractions and Mixed Numbers
4 4
1. Write each fraction in simplest form.
Sample answers:
Date
STUDY LINK
Math Boxes
Math Masters, p. 118
10.
9 º 48 432
11.
7 º 45 315
Teaching Master
Name
3 Differentiation Options
Date
LESSON
Time
Fraction Counts and Conversions
4 4
Most calculators have a function that lets you repeat an operation, such
1
as adding to a number. This is called the constant function. To use
4
1
the constant function of your calculator to count by s, follow one of the
4
key sequences below, depending on the calculator you are using.
PARTNER
ACTIVITY
READINESS
Using a Calculator for
Calculator A
Calculator B
Op1 +
Press:
1
n
4
d
Op1
1
14
Display: 5
Fraction Counts
Display:
1.
1
a.
b.
2.
1
4
6
4
1
4
1
1?
2
6
6
How many counts of are needed to display ?
How many counts of
are needed to display
Use a calculator to convert mixed numbers to improper
fractions or whole numbers.
11
11
3
7
4
4
a. 2 b. 1 4
3.
5
4
K
Using a calculator, start at 0 and count by 4s to answer the
following questions.
c.
Reading about Fractions
0
Display: 14
Display: PARTNER
ACTIVITY
+ +
1
5
4
(Math Masters, p. 119)
ENRICHMENT
4
Press:
Press:
To provide experience with fractions and mixed numbers, have
students use the constant function on a calculator to define and
generate fraction counts. They also study and apply patterns
involving unit fractions, improper fractions, and mixed numbers.
1
Press:
Op1 Op1 Op1 Op1 Op1
5–15 Min
4
3
4
4
2 d.
6
12
4
3 1
4
How many s are between the following numbers?
3
a. 4
c.
and 2
3
4
1 and 4
5
9
6
b. 4
d.
3
4
and 2 1
2
3 and 4 5
6
5–15 Min
in Poetry
Math Masters, p. 119
Literature Link To further explore fractions, have students
read the poem “Proper Fractions” in Math Talk:
Mathematical Ideas in Poems for Two Voices. Suggest that
students recite the poem in their spare time and present it to
the class.
PARTNER
ACTIVITY
EXTRA PRACTICE
Modeling Mixed Numbers
5–15 Min
with Bills and Coins
To provide extra practice simplifying mixed numbers, have
students use bills and coins to model renaming procedures.
Suggestions:
4
13
24 $2 and 4 quarters; 3
9
3
21
0 $2 and 13 dimes; 3 10
1
14 $1 and 9 quarters; 34
31
11
32
0 $3 and 31 nickels; 4 20
Lesson 4 4
277