Fractions and Decimals
Objectives To provide experience with renaming fractions
as
a decimals and decimals as fractions; and to develop an
understanding of the relationship between fractions and division.
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Ongoing Learning & Practice
Key Concepts and Skills
Math Boxes 7 8
• Read and write decimals through
hundredths. Math Journal 2, p. 204
Students practice and maintain skills
through Math Box problems.
[Number and Numeration Goal 1]
• Represent a shaded region as a fraction
and a decimal. [Number and Numeration Goal 2]
• Rename fractions with 10 and 100
in the denominator as decimals. Study Link 7 8
Math Masters, p. 226
Students practice and maintain skills
through Study Link activities.
[Number and Numeration Goal 5]
• Use equal sharing to solve division
problems. [Operations and Computation Goal 4]
Key Activities
Students rename fractions as decimals and
decimals as fractions. They also explore the
relationship between fractions and division.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 203. [Number and Numeration Goal 5]
Materials
Math Journal 2, pp. 203, 342, and 343
Student Reference Book, p. 46
Study Link 77
Math Masters, p. 426 (optional)
transparency of Math Masters, p. 426 base-10 blocks calculator slate overhead base-10 blocks (optional)
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Creating Base-10 Block Designs
Math Masters, p. 442
base-10 blocks
Students make a design with base-10 blocks,
copy the design on a grid, and write a
decimal and a fraction to describe what
part of the grid is covered by the blocks.
ENRICHMENT
Finding Fractions, Decimals,
and Percents on Grids
Math Masters, p. 227
Students shade a 10-by-10 grid to represent
fractions and find the percent and decimal
equivalencies.
ENRICHMENT
Designing a Baseball Cap Rack
Math Masters, pp. 227A and 227B
Students use fractions with denominators
of 10; 100; or 1,000 to design a baseball
cap rack.
EXTRA PRACTICE
Taking a 50-Facts Test
Math Masters, pp. 412 and 414; p. 416
(optional)
pen or colored pencil
Students take a 50-facts test. They use a
line graph to record individual and optional
class scores.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 62, 63, 153, 154
Lesson 7 8
609
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6
Content Standards
Getting Started
4.NF.1, 4.NF.5, 4.NF.6
Mental Math and Reflexes
Math Message
Write a fraction on the board. Students write an
equivalent fraction on their slates. Suggestions:
Write the following fractions as decimals:
1
_
10
Sample answers:
3
1 _
2 _
_
,
2
1
_
4
1
_
3
4
2
_
,
8
2
_
,
6
3
1 _
2 _
_
,
6 _
2 _
12
_
,
5 10 15
50 _
5 _
25
_
,
100 10 50
3 _
6 _
9
_
,
4 8 12
6
3
_
12
3
_
9
9 3 18
5 _
10 _
50
_
,
8 16 80
3 _
6 _
9
_
,
5 10 15
7
_
10
32
_
100
9
_
100
Study Link 7 7 Follow-Up
Have students compare answers and share the
name-collection boxes they created.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Masters, p. 426)
Display a transparency of Math Masters, page 426 as you discuss
the answers. Remind students that the square is the “whole.” You
can color the grid sections to show fractional parts or cover them
with base -10 blocks.
Color or cover one column of the bottom grid.
●
1
What fractional part of the square is this? _
10
Teaching Aid Master
Name
Date
Time
Base-10 Grids
1
,
10
●
or 0.1
1
How would you write _
as a decimal? 0.1
10
3
7
Repeat with other fractions in tenths, including _
and _
.
10
10
Adjusting the Activity
ELL
Provide students with base-10 blocks and a copy of Math Masters,
page 426, so they can model the decimal numbers at their desks.
A U D I T O R Y
Math Masters, p. 426
426-429_458_467_490_EMCS_B_MM_G4_PROJ_576965.indd 426
610
3/5/11 2:31 PM
Unit 7 Fractions and Their Uses; Chance and Probability
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
Date
Next, color (or cover) one small square of the top grid on
the transparency.
●
Time
LESSON
Fractions and Decimals
78
䉬
Whole
1
What fractional part of the square is this? _
100
large square
2.
2
ᎏᎏ
10
夹
2
ᎏᎏ
10
or 0.1
2
1
,
100
1
ᎏᎏ
5
or 0.01
How would you write
1
_
100
as a decimal? 0.01
is shaded.
⫽
2
10
⫽ 0.
9
_
.
100
Repeat with other fractions in hundredths, including
and
Also, give students practice converting decimals into fractions; for
example, 0.3 and 0.25.
Tell students that in this lesson they will use a base -10 grid as
a tool to help them rename fractions as decimals.
⫽
5
3
5. ᎏ5ᎏ
⫽
6
4
6. ᎏ5ᎏ
⫽
2
1
ᎏᎏ
2
2
ᎏᎏ
5
2
2
is shaded.
2
ᎏᎏ
5
⫽
夹
4
⫽ 0.
1
ᎏᎏ ,
100
or 0.01
5
⫽ 0.
5
⫽ 0.
6
⫽ 0.
8
10
10
8
10
4
How many tenths?
4
10
7.
32
_
100
is shaded.
How many tenths?
4.
How many tenths?
1
ᎏᎏ
5
●
⫽ 0.
夹
3.
1
ᎏᎏ
2
of the square is shaded.
How many tenths?
1
ᎏᎏ,
10
61
夹
夹
1.
夹
8.
1
ᎏᎏ
4
is shaded.
1
ᎏᎏ
4
⫽
25 ⫽ 0. 25
100
3
ᎏᎏ
4
is shaded.
3
ᎏᎏ
4
⫽
75 ⫽ 0. 75
100
Math Journal 2, p. 203
Links to the Future
Do not be concerned with reducing fractions to simplest form when converting
between decimals and fractions. At this stage, it is enough for students simply
to make the conversions. Naming fractions in simplest form is a Grade 5 Goal.
Renaming Fractions as Decimals
INDEPENDENT
ACTIVITY
and Decimals as Fractions
(Math Journal 2, pp. 203, 342, and 343; Math Masters, p. 426)
Students complete journal page 203. Discuss answers, using a
transparency of Math Masters, page 426. For each problem,
ask by which number the numerator and denominator were
multiplied to obtain the second fraction.
Ask students to record the decimals in the Equivalent Names for
Fractions table on journal pages 342 and 343.
Ongoing Assessment:
Recognizing Student Achievement
Journal page 203
Problems
1–4, 7, and 8
Use journal page 203, Problems 1–4, 7, and 8 to assess students’ ability to
rename tenths and hundredths as decimals with the assistance of a visual
model. Students are making adequate progress if they are able to name the
number of tenths or hundredths shaded on the grid as a fraction and rename the
fraction as a decimal. Some students may be able to solve Problems 5 and 6 on
journal page 203, which do not include a visual prompt.
[Number and Numeration Goal 5]
Lesson 7 8
611
Student Page
Date
Math Boxes
78
Complete the name-collection box.
1.
4
_
5
+
_3
5
2.
Sample
answers:
_1
5
10
10
-
1
__
10
15
51
Use pattern blocks to help you solve
these problems.
1
_
1
_
3 + 3 =
b.
2
2
_
_
6 + 3 =
c.
5
1
_
_
6 - 6 =
d.
4
1
_
_
6 - 2 =
4.
_2
3
_6 _3
,
6 3 , or 1
_4
_2
6 , or 3
_1
acute
For the TI-15:
(acute or
A
R
T
6
The measure of ∠ART is
40
There are 252 pages in the book Ming is
reading for his book report. He has two
weeks to read the book. About how many
pages should he read each day?
18
Use Problem 7 on journal page 203 to model renaming fractions
as decimals on a calculator.
45
∠ART is an
obtuse) angle.
55–57
5.
(Math Journal 2, p. 203)
2
__
80%
a.
blue blocks,
red blocks,
green block, and
orange blocks.
You put your hand in the bag and, without
looking, pull out a block. About what
fraction of the time would you expect to
get a red block?
8
__
9
__
6.
Enter the fraction _14 (press 1 n 4 d d ).
Then press
1 yard
the width of your journal 1 foot
the length of
your largest toe
c.
1 inch
the length of your shoe 1 foot
d.
0.25
Enter the fraction _14 (press 1
Then press
4).
. 0.25
Use Problem 8 to model renaming a decimal as a fraction.
For the TI-15:
130
Math Journal 2, p. 204
185-218_EMCS_S_MJ2_G4_U07_576426.indd 204
F D.
For the Casio fx-55:
93 142
143
.
the height of the door
b.
pages
°
Tell if each of these is closest to 1 inch,
1 foot, or 1 yard.
a.
WHOLE-CLASS
ACTIVITY
Decimals with a Calculator
A bag contains
8
2
1
4
0.8
3.
Renaming Fractions as
Time
LESSON
1/27/11 10:51 AM
Enter the decimal 0.75, then press
75
F D. _
100
For the Casio fx-55:
Enter the decimal 0.75, then press
Discussing Fractions and
. _34
WHOLE-CLASS
DISCUSSION
Division
(Student Reference Book, p. 46)
Read and discuss “Fractions and Division” on page 46 of the
Student Reference Book. Have students apply their understanding
of division to equal-sharing division problems. For example:
Study Link Master
Name
Date
STUDY LINK
Nina and her mother baked 4 dozen cookies for the book club
meeting. The club has 8 members. How many cookies are there
for each member?
Time
Fractions and Decimals
78
Write 3 equivalent fractions for each decimal.
Sample answers:
61
Four dozen equals 4 ∗ 12, or 48. The number models 48/8 = 6,
48
48 ÷ 8 = 6, and _
= 6 fit this problem. The first and second
8
number models suggest “dealing out” the 48 cookies to the 8 club
members. Each member would get 6 cookies. The third number
48
model, _
= 6, suggests dividing each cookie into eighths and
8
1
_
giving 8 of every cookie to each person. Each person would end up
with 48 eighths. If the 48 eighths were reassembled, they would
be equivalent to 6 cookies.
Example:
8
_
4
_
80
_
2
_
_1
20
_
1.
0.20
2.
0.6
6
_
10
5
_
10
_6
_3
5
_1
2
_3
60
_
100
50
_
100
75
_
3.
0.50
4.
0.75
8
4
100
5
10
0.8
100
5
10
100
Write an equivalent decimal for each fraction.
9.
53
Shade more than _
100 of the square and less than
8
_
10 of the square. Write the value of the shaded part
as a decimal and a fraction.
10.
0.3
Decimal:
0.70
70
_
Fraction:
100
7
_
7. 10
0.7
3
_
10
63
_
6. 100
0.63
5.
_2
8. 5
0.4
Sample answer:
NOTE
is called an improper fraction because the numerator is greater than
the denominator. Improper fractions have numerators that are greater than or
equal to their denominators.
11
Shade more than _
100 of the square and less than
_1 of the square. Write
the value of the shaded part
4
as a decimal and a fraction.
Decimal:
Fraction:
0.2
2
_
10
Sample answer:
Also discuss problems in which the divisor is greater than the
dividend. For example:
Practice
11.
702
= 78 ∗ 9
12.
461 ∗ 7 =
3,227
13.
975
Adam ordered 3 pizzas for a party. There will be 5 people at
the party. How much pizza is there for each person?
= 39 ∗ 25
Math Masters, p. 226
203-246_EMCS_B_MM_G4_U07_576965.indd 226
612
48
_
8
1/25/11 9:58 AM
Unit 7 Fractions and Their Uses; Chance and Probability
Teaching Master
Name
Point out that this problem and the cookie problem are both about
sharing. The main difference is that in this problem, each share is
less than one whole pizza. Draw 3 pizzas on the board or on the
overhead transparency, and divide each one into fifths for the
5 people. If Adam’s guests are named Bob, Charles, Darryl, and
Ed, the pizzas could be shared in the following way:
A B
E
D
C
A B
E
D
C
D
Time
Designing a Baseball Cap Rack
78
Karen plans to design and construct two identical horizontal racks to display her
baseball cap collection. She has 12 different caps to hang on pegs. Karen’s
sister suggested that she add extra pegs for caps she may get in the future.
Karen measured the width of some caps and decided that the pegs need to be
2
2 decimeters (_
10 meter) apart. Also, in order to fit on her wall, each rack cannot
be more than 160 centimeters long.
Help Karen design one of the identical racks. Use metric units. Fill in the blanks
below as you create the design. Sample answers are given.
A B
E
Date
LESSON
C
Each of Karen’s racks will have
2.
The total length of the rack will be
3.
The first peg will be
4.
In the space below, draw a rough sketch of the rack. Include the measurements
in your sketch.
10
10 cm
Help students see how the number model 3 / 5 = _35 fits this
problem. The left side, 3 / 5, suggests dividing 3 pizzas among
5 people. The right side, _35 , tells how much each person would get.
8
1.
pegs for hats.
160
centimeters.
centimeters from the edge of the rack.
2 dm
160 cm
5.
Write a fraction addition number sentence to show the total length of the rack.
10
2
2
2
2
2
2
2
_
_
_
_
_
_
_
Sample answer: _
100 + 10 + 10 + 10 + 10 + 10 + 10 + 10 +
10
160
_
_
100 = 100 m, or 160 cm
Explain that in high school and beyond, the symbol ÷ is almost
never used for division. Division is usually shown with a slash (/)
or a fraction bar (—).
6. Could there be 9 pegs on the rack? Explain your answer.
No. Nine hats would require about 200 cm, and the rack can
only be 160 cm long.
Math Masters, p. 227A
2 Ongoing Learning & Practice
Math Boxes 7 8
227A-227B_EMCS_B_MM_G4_U07_576965.indd 227A
3/3/11 10:44 AM
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 204)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 7-6. The skill in Problem 6
previews Unit 8 content.
Writing/Reasoning Have students write a response to the
following: Explain how you solved Problem 2. Sample answer:
There are 15 blocks in the bag, and 2 of them are red. So the
2
chance of getting a red block is _
.
15
Teaching Master
Name
Study Link 7 8
LESSON
78
INDEPENDENT
ACTIVITY
(Math Masters, p. 226)
Home Connection Students rename decimals as fractions
and fractions as decimals. They color fractional parts of
a base-10 grid and write the value of the shaded part as
a decimal and a fraction.
Date
Time
Designing a Baseball Cap Rack
continued
Use your answers to Problems 1 and 2 on Math Masters, page 227A to fill in
the blanks in the sentence below. Sample answers:
Each rack is
160
centimeters long and has
8
pegs.
At the lumberyard, Karen discovered she could spend less if she was willing to
glue leftover pieces of wood together instead of using one long piece. She
measured several boards and wrote down the lengths:
7
_
10 meter
7.
85
_
100 meter
35
_
100 meter
3
_
10 meter
20
_
100 meter
9
_
10 meter
8
_
10 meter
55
_
100 meter
75
_
100 meter
15
_
100 meter
Sample answers are given.
Can Karen use these pieces to create two racks of the length you planned?
Explain why or why not. Show your work.
9
9
16
7
7
_
_
_
_
Yes. Rack one: _
10 meter and 10 meter; 10 + 10 = 10 m,
or 160 cm
75
55
3
Rack two: _
meter, _
meter, and _
meter;
100
100
10
75
55
3
160
_
_
_
_
100 + 100 + 10 = 100 m, or 160 cm
Pegs come in two different packages:
5-pack for $3.79 or 2-pack for $1.99
8.
Explain how Karen can purchase the pegs for her racks, spending as little
money as possible.
Karen needs 16 pegs. She should buy three 5-packs and
one 2-pack for $13.36 because this is cheaper than
buying two 5-packs and three 2-packs for $13.55.
Math Masters, p. 227B
227A-227B_EMCS_B_MM_G4_U07_576965.indd 227B
3/23/11 12:43 PM
Lesson 7 8
613
3 Differentiation Options
READINESS
Creating Base -10 Block Designs
INDEPENDENT
ACTIVITY
30+ Min
(Math Masters, p. 442)
Decimal:
0.24
24
Fraction: 100
To explore representing fractions and decimals on a base-10 grid,
have students make a design on a base-10 block flat with cubes
and then copy the design onto one of the grids shown on Math
Masters, page 442. Students determine how much of the flat is
covered by their design and express this number as a decimal
and a fraction. (See margin.) Students may choose to exchange as
many cubes as possible for longs, which would result in a certain
number of longs (tenths) and cubes (hundredths).
ENRICHMENT
Finding Fractions, Decimals,
PARTNER
ACTIVITY
15–30 Min
and Percents on Grids
(Math Masters, p. 227)
To further investigate fraction, decimal, and percent
equivalencies, have students shade a base-10 grid to show
1 _
_
, 1 , _1 , and _46 . Encourage students to discuss patterns they
8 3 6
see and strategies they used. Ask: How could you have found
the percent equivalent for _46 without shading the grid? Sample
answer: Use the answer for _16 and multiply by 4.
ENRICHMENT
Designing a Baseball Cap Rack
Teaching Master
Name
Date
LESSON
Time
Sample answers:
61 62
1.
2.
1
Fraction: _8 =
Decimal:
Percent:
12_12
1
Fraction: _3 =
100
0.125
12.5 %
3.
15–30 Min
(Math Masters, pp. 227A and 227B)
Fraction, Decimal, and Percent Grids
78
Fill in the missing numbers. Shade the grids.
PARTNER
ACTIVITY
Decimal:
Percent:
33_13
To further investigate the relationships among fractions with
denominators of 10; 100; or 1,000, have students design two
identical racks to display a baseball cap collection. Encourage
students to work with a partner to complete the activity.
EXTRA PRACTICE
Taking a 50-Facts Test
100
0.333
33_13 %
SMALL-GROUP
ACTIVITY
5–15 Min
(Math Masters, pp. 412, 414, and 416)
4.
See Lesson 3-4 for details regarding the administration of the
50-facts test and the recording and graphing of individual and
optional class results.
1
Fraction: _6 =
Decimal:
Percent:
16 _46
100
0.166
16_46 %
4
Fraction: _6 =
Decimal:
Percent:
Math Masters, p. 227
203-246_EMCS_B_MM_G4_U07_576965.indd 227
614
66 _46
100
0.666
66_46 %
227
1/25/11 9:58 AM
Unit 7 Fractions and Their Uses; Chance and Probability
Name
LESSON
78
Date
Time
Designing a Baseball Cap Rack
Karen plans to design and construct two identical horizontal racks to display her
baseball cap collection. She has 12 different caps to hang on pegs. Karen’s
sister suggested that she add extra pegs for caps she may get in the future.
Karen measured the width of some caps and decided that the pegs need to be
2
2 decimeters (_
10 meter) apart. Also, in order to fit on her wall, each rack cannot
be more than 160 centimeters long.
Help Karen design one of the identical racks. Use metric units. Fill in the blanks
below as you create the design.
1.
Each of Karen’s racks will have
2.
The total length of the rack will be
3.
The first peg will be
4.
In the space below, draw a rough sketch of the rack. Include the measurements
in your sketch.
5.
Write a fraction addition number sentence to show the total length of the rack.
pegs for hats.
centimeters.
centimeters from the edge of the rack.
227A
Copyright © Wright Group/McGraw-Hill
6. Could there be 9 pegs on the rack? Explain your answer.
Name
LESSON
78
Date
Time
Designing a Baseball Cap Rack continued
Use your answers to Problems 1 and 2 on Math Masters, page 227A to fill in
the blanks in the sentence below.
Each rack is
centimeters long and has
pegs.
At the lumberyard, Karen discovered she could spend less if she was willing to
glue leftover pieces of wood together instead of using one long piece. She
measured several boards and wrote down the lengths:
Copyright © Wright Group/McGraw-Hill
7.
7
_
10 meter
85
_
100 meter
35
_
100 meter
3
_
10 meter
20
_
100 meter
9
_
10 meter
8
_
10 meter
55
_
100 meter
75
_
100 meter
15
_
100 meter
Can Karen use these pieces to create two racks of the length you planned?
Explain why or why not. Show your work.
Pegs come in two different packages:
5-pack for $3.79 or 2-pack for $1.99
8.
Explain how Karen can purchase the pegs for her racks, spending as little
money as possible.
227B