Download Clock Fractions and Common Denominators

Clock Fractions and
Common Denominators
Objective To provide additional references for fraction concepts.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Find common denominators. [Number and Numeration Goal 5]
• Use clock models and pencil-and-paper
algorithms to add and subtract fractions. [Operations and Computation Goal 4]
• Use benchmarks to estimate sums and
differences. [Operations and Computation Goal 6]
Key Activities
Students use a clock face to find equivalent
fractions and to model addition and
subtraction of fractions. They use the
multiplication rule, a multiplication table, and
reference lists to find common denominators.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Fraction Capture
Math Journal 1, p. 198
Math Masters, p. 460
per partnership: 2 six-sided dice
Students practice comparing fractions
and finding equivalent fractions.
Math Boxes 6 9
Math Journal 1, p. 199
Students practice and maintain skills
through Math Box problems.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Modeling Fractions with a Military Clock
Math Masters, p. 179
Students apply the clock model to a different
unit by using military time.
EXTRA PRACTICE
Writing Elapsed Time Number Stories
Math Masters, p. 180
Students write a number story using fractions
to represent elapsed time.
Study Link 6 9
Math Masters, p. 178
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 194. [Operations and Computation Goal 4]
Key Vocabulary
common denominator unlike denominators
Materials
Math Journal 1, pp. 194–197
Student Reference Book, p. 401
Study Link 68
Math Masters, p. 177
Multiplication Table Poster demonstration
clock
Advance Preparation
For Part 1, display the Multiplication Table Poster on the board or elsewhere for student use. When you
use the Everyday Mathematics posters with English language learners, display either the English version
only or both English and Spanish versions simultaneously; do not display the Spanish version only.
Teacher’s Reference Manual, Grades 4–6 pp. 141, 142, 242, 243
Lesson 6 9
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Getting Started
Mental Math and Reflexes
Math Message
Fraction addition and subtraction: Write the problems on the board, and have
students estimate using benchmarks and then solve. Students use the estimate
to assess the reasonableness of the answers.
Complete Part 1 on journal
page 194.
7
1
1
_
+ 2_ = ? 3_
+ 2_
+ 1_
=?6
2_
4
2
4
- 4 + 2_
= ? 4_
6_
2
4
4
5
3
7
3_ - 1_ = ? 1_
5
3
1
7
1_ + 1_ + 3_ = ? 6_
3
3
1
1
4_
+ 2 - 1_
+ 1_
= ? 6_
8
4
2
8
8
4
8
1
8
4
1
8
8
1
16
4
1
16
1
3
Study Link 6 8
Follow-Up
Have partners compare answers and
resolve differences.
NOTE Working with elapsed time can
provide students with more practice using
a clock face to add and subtract fractions.
Remind students that elapsed time is the time
that passes between a given starting and
ending time. Give them various problems that
involve finding elapsed time and then adding
or subtracting the amounts. Ask students to
give the elapsed time in minutes and then in
fractions of an hour. They can then use the
clock face to add or subtract.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 1, p. 194)
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
N
G
Review the answers to Part 1 with the class. You might want to
pose a few additional easy problems that have mixed numbers or
fractions greater than 1. Suggestions:
1 hours? 150 min
● How many minutes are in 2_
2
Student Page
Date
䉬
Part 1: Math Message
Whole
Rule
The numbers on a clock face divide one hour into
1
twelfths. Each 1
2 of an hour is 5 minutes.
11
10
hour
1.
1
4. 3
5
hr min
20
hr min
2.
5
12
5.
1
4
hr 25
10
10.
13.
3
12
12
8
8. 12
hr
hr 56 hr
1
12 hr 6
4
hr
hr 11.
3
5
14. 3
hr 4
3 hours is equivalent to how many minutes? 90 min
_
4
2
4
7
min
5
6
3.
1
2
hr 30
min
6.
1
6
hr 10
min
2
3
1
9. 3
hr
hr 19
hr
2
20
12
hr
2
hr 6
12. 2
12
hr 4
15. 12
hr 1
6
1
3
hr
hr
Make sure students understand that they may use the clock model
to help them answer the problems on the journal page. At times,
students might want to “think minutes,” as in the example for
Part 3. At other times, students might want to look at the clock
face divided into twelfths. Using a demonstration clock, work
several of the problems in Part 2 with the class before students
work in partnerships. Assign the remainder of the journal page.
hr
Part 3
夹
Ongoing Assessment:
Recognizing Student Achievement
Use clock fractions, if helpful, to solve these problems. Write each answer as a fraction.
3
1
Example: 4 3 ?
Think: 45 minutes 20 minutes 25 minutes
3
1
5
So 4 3 1
2
5
16. 12
19.
8
,
3
12
1
2 2
1 3 1
22. 4
1
3 or
1
3
7
12
2
3
17.
3
4
5
,
2
4 4
20.
5
4
2
4 23.
1
3
PARTNER
ACTIVITY
3
Using the clock face, fill in the missing numbers. The first one has been done for you.
hr ●
2
2
Part 2
1
7. 4
5 hours? 75 min
In _
(Math Journal 1, p. 194)
1
8
min
15
hr 12
9
How many minutes does each of the following fractions and mixed
numbers represent? The first one has been done for you.
1
12
●
Subtract Fractions
Clock Fractions
6 9
5 hours? 150 min
In _
▶ Using a Clock to Add and
Time
LESSON
●
1
4 or 114
8
,
3
12
1
2 18.
11
12
3
4
21.
2
3
1
6 1
12
24.
5
6
3
4 or
5
6
2
3
Journal
Page 194
Problems 16–24
Use journal page 194, Problems 16–24 to assess students’ ability to use a
visual model to add and subtract fractions with unlike denominators. Students
are making adequate progress if they correctly solve Problems 16–24.
1
12
[Operations and Computation Goal 4]
Math Journal 1, p. 194
424
Unit 6
Using Data; Addition and Subtraction of Fractions
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Student Page
▶ Discussing Strategies for
WHOLE-CLASS
DISCUSSION
Adding and Subtracting Fractions
Date
LESSON
6 9
䉬
Time
Number Strip Fractions
Name the strips that you used for the numerator and denominator. Then list the
fractions formed by the two strips. Sample answers:
Problem 1
(Math Journal 1, p. 194)
Strip
Name
Numerator:
Discuss students’ solutions to Part 3. Expect that some students
converted most fractions to minutes, did the operation, and then
converted the answer in minutes back to a fraction. Others may
have converted all fractions to twelfths and found the answer
without any reference to the clock or time.
Denominator:
7
9
Fractions List
7 14 21 28 35 42 49 56 63 70
ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ
9 , 18 , 27 , 36 , 45 , 54 , 63 , 72 , 81 , 90
Problem 2
Strip
Name
Numerator:
Denominator:
10
4
Fractions List
10 20 30 40 50 60 70 80 90 100
ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ
4 , 8 , 12 , 16 , 20 , 24 , 28 , 32 , 36 , 40
Problem 3
Strip
Name
Adjusting the Activity
Numerator:
Denominator:
Pose fraction problems with denominators of 30 and 60.
Suggestions follow:
3
5 _
50
5
• _
+_
, or _
4
60 60
6
A U D I T O R Y
18
28
1 _
14
• _
+_
, or _
30
3 30
15
K I N E S T H E T I C
T A C T I L E
9
27
Explain how you can use a multiplication table to find equivalent fractions for ᎏᎏ.
V I S U A L
NOTE A clock face is a convenient model for fraction operations involving
halves, thirds, fourths, fifths, sixths, twelfths, and even thirtieths and sixtieths. The
link between fractions and their equivalents in minutes allows students to add and
subtract fractions with unlike denominators without rewriting the fractions with a
common denominator.
▶ Using a Multiplication Table
Fractions List
3 6 9 12 15 18 21 24 27 30
ᎏᎏ ᎏᎏ ᎏᎏ ᎏ
ᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10
Problem 4
5
25 _
50
5
• _
+_
, or _
12
60 60
6
3
1
I can look for a column that has both 9 and 27. Then I can find
the rows for those numbers. The 1 row has 9 for the numerator,
and the 3 row has 27 for the denominator. I can then list the
other fractions made by the numbers in the 1 row that correspond to
the numbers in the 3 row.
Math Journal 1, p. 195
WHOLE-CLASS
ACTIVITY
to Explore Equivalent Fractions
(Math Journal 1, p. 195; Math Masters, p. 177)
Any two rows of a multiplication table can be used to form
equivalent fractions. Display the Multiplication Table Poster on
the board. Ask partners to cut the strips from Math Masters,
page 177 and place them in the middle of their workspace.
Each student takes one strip. Ask students to make true
statements about the numbers on their strip. The numbers are
multiples of the first number on the strip; the numbers have a
common factor. Tell them that a strip can be named by its
smallest number, for example, the “4 strip.”
One partner is “numerators” and the other is “denominators.”
Partners then match their strips, laying the numerator strip
above the denominator strip. Tell students that the columns
form fractions.
For Problem 1, on journal page 195, ask students to write
down the strip names and then list all of the fractions formed
by the matches.
Partners then take two different strips and repeat this process
for Problems 2 and 3. Each strip should be used only once.
Circulate and assist.
Ask students what they notice about their lists of fractions.
The numerators and denominators are multiples of the numbers
in the first column; the fractions are equivalent.
Lesson 6 9
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Student Page
Date
Time
LESSON
Using a Common Denominator
6 9
䉬
Study the examples. Then work the problems below in the same way.
Example 1:
2
1
?
3
6
Unlike
Denominators
2
3
2
3
Unlike
Denominators
4
6
4
6
1
6
1
6
5
6
2
1. 3
1
3. 3
5
6
3
4
2
3
2
9
2
3
Common
Denominators
6
9
1
3
2
5
Common
Denominators
10
12
9
12
3
4 ?
13
16
3
4
3
4
Common
Denominators
13
16
12
16
1126
1
16
4
9 ?
Unlike
Denominators
5
15
165
11
15
5
15
6
15
Unlike
Denominators
5
4. 6
Unlike
Denominators
Common
Denominators
10
12
9
12
1
12
6
9
29
8
9
2
5 ?
1
3
2
5
5
6
3
4
13
2. 16
2
9 ?
Unlike
Denominators
5
3
?
6
4
Example 2:
Common
Denominators
5
6
4
9
5
6
4
9
Common
Denominators
15
18
8
18
15
18
188
Ask volunteers to match two of their remaining strips and write
the fraction from the first column on the board. Use these
fractions to demonstrate the multiplication rule.
Example:
8
4 = _
4∗2 =_
_
9
9∗2
18
Ask students to use the appropriate strips to give another
4 . Then ask a volunteer to write the
equivalent fraction for _
9
number model for this change using the multiplication rule.
32 = _
4∗8
Sample response: _
72
9∗8
Refer students to the Multiplication Table Poster. Explain that
for any two rows, the equivalent fractions are the result of
multiplying the fraction in the first column by another name
8 , depending on the column. So the second
2 or _
for 1, such as _
2
8
2 , the third column is the
column is the result of multiplying by _
2
3
_
result of multiplying by 3 , and so on.
7
18
▶ Using a Common Denominator
(Math Journal 1, pp. 196 and 197; Student Reference
Book, p. 401)
Math Journal 1, p. 196
Adjusting the Activity
Refer students to the Equivalent
Fractions, Decimals, and Percents table on
Student Reference Book, page 401. Ask
students to describe the similarities and
differences between the structure of this
table and their work with multiplication table
number strips. 1 row of the table is similar
to two number strips; the table shows the
equivalent decimals and percents for the
fractions.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
PARTNER
ACTIVITY
ELL
Algebraic Thinking Introduce the next activity by discussing the
following points:
●
It is easy to add or subtract fractions if they have the same
denominator, usually called a common denominator. To
support English language learners, discuss the
meaning of common in this mathematical context.
●
One way to add or subtract fractions with different
denominators, usually called unlike denominators, is to
rewrite the fractions with a common denominator.
●
One way to find common denominators is to use the
multiplication rule (or the division rule) for finding equivalent
fractions. Ask volunteers to express the rules with variables.
a÷n
a = _
a ∗ n; _
a = _
_
b
b∗n b
b÷n
Have students look at Example 1 and Example 2 at the top of
Math Journal 1, page 196. Ask: How could you use benchmarks to
estimate the solution to each problem? Sample answer: For Problem
2 is greater than _
1 and _
1 is less than _
1 , so the
1, I know that _
3
2
6
2
5 and _
3 are
answer will be close to 1. For Problem 2, I know that _
6
4
both close to 1, so my answer will be close to zero.
NOTE Some students may realize that _16 is less than _13 and conclude that the
answer will be less than 1.
Then work through the examples to illustrate the use of the
multiplication rule to find common denominators. Pose one or two
similar problems as needed.
426
Unit 6
Using Data; Addition and Subtraction of Fractions
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Student Page
In addition to using the multiplication rule to find equivalent
fractions, students can also refer to the Table of Equivalent
Fractions, Decimals, and Percents on page 401 of the Student
Reference Book.
Assign journal pages 196 and 197. Remind students of the
importance of using benchmarks to estimate the solution and then
assess the reasonableness of their answers. Students may choose
to solve Problems 7 and 8 by finding a common denominator.
NOTE In Problems 1, 2, 5, 6, 7, and 8, on journal pages 196 and 197, the
common denominator is the same as one of the original denominators. In
Problems 3 and 4, the common denominator is different from both of the
original denominators.
Date
Time
LESSON
Using a Common Denominator
6 9
䉬
12
5. 4
3
2 ?
6.
Unlike
Denominators
12
4
3
2
3
2
Common
Denominators
12
4
6
4
64
1
8
4 , or
Unlike
Denominators
1
1
11
116
6
3
8
3
8
11
16
1
1
5
516
in.
Write a number sentence to show how you solved the problem.
7186 2136 5156
Three boards are glued together. The diagram below shows the
thickness of each board. What is the total thickness of the three boards?
8.
1"
2
5"
38
2 Ongoing Learning & Practice
▶ Playing Fraction Capture
Common
Denominators
17
16
166
17
16
6
16
42
A piece of ribbon is 72 in. long. If a piece
3
2 1
6 in. long is cut off, how long is the remaining piece?
7.
continued
1
3
11
6 8 ?
7
68
in.
3"
24
Write a number sentence to show how you solved the problem.
358 48 268 678
PARTNER
ACTIVITY
Math Journal 1, p. 197
(Math Journal 1, p. 198; Math Masters, p. 460)
Players roll dice, form fractions, and claim corresponding sections
of squares. The rules are on Math Journal 1, page 198, and the
gameboard is on Math Masters, page 460. Remind students of the
1 when playing
importance of using the benchmark fraction of _
2
this game.
▶ Math Boxes 6 9
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 199)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-6. The skill in Problem 6 previews
Unit 7 content.
Student Page
Date
Writing/Reasoning Have students write a response to
the following: Explain your answer to the question in
Problem 3 and how you chose the values for the data set.
Because the average cannot be greater than the maximum in the
data set, 53 inches cannot be Esther’s average since 50 is the
maximum number. I chose 5 numbers for the data set that could
be added together and divided by 5 so that the average would
equal 53.
Time
LESSON
6 9
䉬
1.
Math Boxes
In the figure below, write the correct
fraction in each of the smaller regions.
Check to see that the fractional parts add
up to 1.
2.
1
16
1
4
Answers vary.
3
16
Estimate the measure of ⬔M:
1
2
The measure of ⬔M is about
75
3.
Esther did 5 standing jumps. Her longest
jump was 50 in. Could her average jump
be 53 in.?
4.
▶ Study Link 6 9
Answers vary.
a.
51, 52, 53, 54, 55
c.
(Math Masters, p. 178)
pinkie finger
b.
pencil
length: ______ cm
length: ______ cm
width: ______ cm
width: ______ cm
notebook
length: ______ cm
width: ______ cm
121
Home Connection Students solve problems similar to
those on journal pages 196 and 197. This page reinforces
the idea that a common denominator can be determined
by finding fractions equivalent to the given fractions.
.
Measure the length and width of each of the
following objects to the nearest centimeter.
Create a data set for Esther’s jumps that
could have this average.
INDEPENDENT
ACTIVITY
37°
204
No
M
5.
Measure each line segment to the nearest
1
in.
6.
8
a.
118
in.
b.
59
a. 5
88
b. 11
120
c. 7
228, or 214 in.
94
d. 4
183
183
Rename each fraction as a mixed number
or a whole number.
102
e. 6
4
115
8
1717
23 24, or 23 12
17
62 63
Math Journal 1, p. 199
Lesson 6 9
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Study Link Master
Name
Date
Time
Adding and Subtracting Fractions
STUDY LINK
69
䉬
Multiplication Rule
65
68–71
To find a fraction equivalent to a given fraction,
multiply the numerator and the denominator
of the fraction by the same number.
4
9
2
6
3
9
4
12
5
15
5
ᎏᎏ
8
2
ᎏᎏ
5
6
18
⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ...
1
3
4
9
3
9
1
9
⫺ ᎏᎏ ⫽ ᎏᎏ ⫺ ᎏᎏ ⫽ ᎏᎏ
10
16
4
10
15
24
6
15
20
32
8
20
25
40
10
25
30
48
12
30
⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ...
14
35
16
40
18
45
⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ ⫽ ...
Both fractions can be rewritten with the
common denominator 40.
5
ᎏᎏ
8
2
5
25
40
16
40
41
40
7
⫹
4
ᎏᎏ
5
1ᎏ15ᎏ
⫽
2
10
ᎏᎏ ⫽ ᎏᎏ
3
15
4
12
ᎏ ᎏ ⫽ ᎏᎏ
5
15
10
12
ᎏᎏ ⫹ ᎏᎏ
15
15
or 1ᎏ175ᎏ
8
2. ᎏᎏ
9
⫺
5
ᎏᎏ
6
⫽ ᎏ1168ᎏ
5
15
ᎏᎏ ⫽ ᎏᎏ
6
18
16
15
ᎏᎏ ⫺ ᎏᎏ
18
18
1
3
3. ᎏᎏ
4
22
ᎏᎏ,
15
⫹
1
1ᎏᎏ
2
2 ᎏ4ᎏ
⫽
1ᎏ12ᎏ ⫽ ᎏ32ᎏ ⫽ ᎏ64ᎏ
3
6
9
ᎏᎏ ⫹ ᎏᎏ ⫽ ᎏᎏ, or
4
4
4
8
ᎏᎏ
9
⫽
⫽
2ᎏ41ᎏ
1
ᎏᎏ
18
1
Lisa was 4 feet 10 ᎏᎏ inches tall at the end of fifth grade. During the year, she
2
3
had grown 2 ᎏᎏ inches. How tall was Lisa at the start of fifth grade?
4.
4
4
7ᎏ34ᎏ
feet
15–30 Min
(Math Masters, p. 179)
1
ᎏᎏ
18
⫽
▶ Modeling Fractions with a
Military Clock
1
40
⫹ ᎏᎏ ⫽ ᎏᎏ ⫹ ᎏᎏ ⫽ ᎏᎏ, or 1ᎏᎏ
Find a common denominator. Then add or subtract.
2
1. ᎏᎏ
3
PARTNER
ACTIVITY
ENRICHMENT
2
5
Example 2: ᎏᎏ ⫹ ᎏᎏ ⫽ ?
9 is a common denominator.
4
ᎏᎏ
9
aºn
bºn
⫽ ᎏᎏ
5
8
1
3
Example 1: ᎏᎏ ⫺ ᎏᎏ ⫽ ?
1
ᎏᎏ
3
a
ᎏᎏ
b
3 Differentiation Options
in.
To apply students’ understanding of the fractional units
on a 12-hour clock face, have students use a 24-hour
military clock face model to add, subtract, and find
equivalent fractions. When they have finished the page, have
students describe similarities and differences between using the
12-hour clock and the 24-hour clock.
1
Bill was baking two different kinds of bread. One recipe called for 3ᎏᎏ cups of
2
1
flour. The other called for 2ᎏᎏ cups of flour. How much flour did Bill need in all?
5.
3
5ᎏ56ᎏ
cups
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
▶ Writing Elapsed Time
Math Masters, p. 178
15–30 Min
Number Stories
(Math Masters, p. 180)
Students write a number story using fractions to
represent elapsed time. Ask students to exchange and
solve each other’s problems and then share their
solution strategies.
Teaching Master
Teaching Master
Name
Date
Time
Fractions in Military Time
LESSON
69
䉬
Name
Date
LESSON
69
䉬
Time
Writing Elapsed-Time Number Stories
1
Rule
Whole
0
55
50
21
22
23
00 01 02
5
03
04
20
19
day
10
11
10
05
45 18
14
13
12 11
10
1
On a military clock, the whole is 1 day or 24 hours. ᎏᎏ is one hour. The time
24
shown on this clock face is 08:14:42 (8 hours, 14 minutes, and 42 seconds).
Using the clock face, write the fractions as days, hours, and minutes. The first
one has been done for you.
18
2. ᎏᎏ
24
⫽
10
3. ᎏᎏ
24
1
4. ᎏᎏ
2
⫽
Rule
Whole
hour
5
Use fractions to represent amounts of elapsed time and write a number story for a
partner to solve.
25
30
⫽
3
6
09
35
2
1. ᎏᎏ
24
2
4
7
08 20
15
1
8
07
16
12
9
06 15
17
40
The numbers on a clock face divide one hour into twelfths. Each ᎏᎏ of an hour is
12
5 minutes.
1
12
3
4
5
12
hour ⫽
of a day ⫽ 2 hours ⫽
of a day ⫽
18
of a day ⫽
10
1
48
120
minutes
Example:
8
Maria started her piano practice at 3:15. She practiced for ᎏᎏ of an hour. At what time did
12
she finish practicing?
1
12
8
12
Think: ᎏᎏ hour ⫽ 5 minutes; ᎏᎏ hour is 8 ⴱ 5, or 40 minutes; 40 minutes more than 3:15 is 3:55.
Maria finished practicing at 3:55.
Your Elapsed-Time Number Story:
hours ⫽
1,080
minutes
hours ⫽
600
minutes
Answers vary.
of a day
Your Partner’s Solution:
5.
Explain how you found your answer for Problem 4.
1
ᎏᎏ
2
Sample answer: hour is equal to 30 minutes.
In one day, there are 1,440 minutes, so
30
3
1
ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ.
1,440
144
48
Math Masters, p. 179
428
Unit 6
Answers vary.
Explain your answer.
Answers vary.
Math Masters, p. 180
Using Data; Addition and Subtraction of Fractions
EM3cuG5TLG1_424-428_U06L09.indd 428
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