Areas of Rectangles

Transcription

Areas of Rectangles

Areas of Rectangles
Objective To reinforce students’ understanding of area
concepts and units of area.
c
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Teaching the Lesson
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Fraction Division Review
• Multiply fractions and mixed numbers
to find the area of a rectangle. Math Journal 2, p. 306
Student Reference Book, pp. 80–80B
Students use visual models and
number stories to solve fraction
problems.
[Operations and Computation Goal 5]
• Use a formula to calculate the areas
of rectangles. [Measurement and Reference Frames Goal 2]
• Compare inch and centimeter measures
for length and area. [Measurement and Reference Frames Goal 3]
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Comparing Perimeter and Area
Math Boxes 9 4
Math Masters, p. 266
per partnership: 2 six-sided dice,
36 centimeter cubes
Students use centimeter grids to compare
the perimeters and areas of rectangles.
Math Journal 2, p. 307
Students practice and maintain skills
through Math Box problems.
for Irregular Figures
Key Activities
Study Link 9 4
Students review area concepts and the
names and notations for common area
units. They find areas of rectangles by
counting and by applying an area formula.
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Informing Instruction See page 725.
Ongoing Assessment:
Recognizing Student Achievement
Use an Exit Slip (Math Masters,
page 414). [Measurement and Reference
Frames Goal 2]
Key Vocabulary
area square units base height formula variable
ENRICHMENT
Math Masters, pp. 267 and 436
3 different-colored pencils or markers scissors
Students compare the perimeters and
areas of irregular polygons.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, p. 212
Students calculate the areas of rectangles.
EXTRA PRACTICE
Area: Tiling and Using a Formula
Math Masters, pp. 293A and 436
Students find the areas of rectangles with
fractional units by tiling and using a formula.
ELL SUPPORT
Materials
Math Journal 2, pp. 304 and 305
Student Reference Book, p. 188
Study Link 93 Math Masters, p. 414
transparency of Math Masters, p. 436 Class
Data Pad inch ruler slate roll of paper
towels, wax paper, or aluminum foil (optional)
Building a Math Word Bank
Differentiation Handbook, p. 142
Students define and illustrate the terms
length, height, base, and width.
Advance Preparation
For Part 1, display a set of unit squares. (See Planning Ahead, Lesson 9-2.) Use a roll of paper or foil
to demonstrate carpet rolls for Math Journal 2, page 305, Problem 2.
Teacher’s Reference Manual, Grades 4–6 pp. 220–222, 233, 234, 236
722
Unit 9
Coordinates, Area, Volume, and Capacity
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Getting Started
Mental Math and Reflexes
Math Message
Have students write fractions as equivalent decimals
and percents. Suggestions:
Read page 188 of the Student Reference Book,
and write two important facts about area.
2
2
_
⎯⎯; 66 _
0.66
%
3
3
4
_
0.8; 80%
5
8
_
0.32; 32%
25
19
_ 0.95; 95%
20
24
_
0.48; 48%
50
3
_
0.375; 37.5%
8
Study Link 9 3 Follow-Up
Have partners compare answers and resolve any
differences.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 188)
Ask volunteers to share what they wrote about area. Use their
responses and the display of common units of area to review basic
area concepts. Emphasize the following points:
Area is a measure of the surface, or region, inside a closed
boundary. It is the number of whole and partial unit squares
needed to cover the region without gaps or overlaps.
NOTE It is more precise to talk about the area of a rectangular region, the
area of a triangular region, and so on. However, it is customary to refer to area
in terms of the figure that is the boundary: the area of a rectangle, the area of a
triangle, and so on.
Area is measured in square units. There are many units to
choose from, and some choices make more sense than others.
Call students’ attention to the classroom display of unit squares
and to alternative ways of writing the units: square inch, sq in., or
in2; square meter, sq m, or m2; and so on.
Ask students to share the relationships they observe among the
units—for example, a square meter is larger than a square yard.
There are 9 square feet in a square yard and 144 square inches in
a square foot. A square inch is larger than a square centimeter.
Student Page
Measurement
Area
Area is a measure of the amount of surface inside a closed
boundary. You can find the area by counting the number of
squares of a certain size that cover the region inside the
boundary. The squares must cover the entire region. They must
not overlap, have any gaps, or extend outside the boundary.
1 cm
1 cm
1 square centimeter
(actual size)
Sometimes a region cannot be covered by an exact number of
squares. In that case, count the number of whole squares and
fractions of squares that cover the region.
1 in.
1 in.
Area is reported in square units. Units of area for small regions
are square inches (in.2), square feet (ft2), square yards (yd2),
square centimeters (cm2), and square meters (m2). For large
regions, square miles (mi2) are used in the United States, while
square kilometers (km2) are used in other countries.
1 square inch
(actual size)
You may report area using any of the square units. But you
should choose a square unit that makes sense for the region
being measured.
Examples
The area of a field-hockey field is reported below in three different ways.
Area of the field is
6,000 square yards.
54,000 square feet.
7,776,000 square inches.
Area ⫽ 6,000 yd
Area ⫽ 54,000 ft
Area ⫽ 7,776,000 in.2
2
2
100 yd
60 yd
300 ft
180 ft
3,600 in.
2,160 in.
Although each of the measurements above is correct, reporting
the area in square inches really doesn’t give a good idea about
the size of the field. It is hard to imagine 7,776,000 of anything!
The International Space
Station (ISS) orbits the
Earth at an altitude of
250 miles. It is 356 feet
wide and 290 feet long,
and has an area of over
100,000 square feet.
Student Reference Book, p. 188
Lesson 9 4
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Student Page
Date
Time
LESSON
▶ Finding the Area of a Rectangle
Areas of Rectangles
1 cm2
1 cm
1 cm
D
Ask volunteers to define the terms base and height. The term
base is often used to mean both a side of a figure and the length of
that side. The height of a rectangle is the length of a side adjacent
to the base.
base
(or length)
E
B
A
Fill in the table. Draw rectangles D, E, and F on the grid.
Rectangle
Base (length)
2
4
2.5
A
B
C
2.
Ask students to decide upon the phrasing of a common definition
for these vocabulary terms. Record the student definitions on the
Class Data Pad.
F
C
1.
PARTNER
ACTIVITY
(Math Journal 2, p. 304)
height
(or width)
94
D
6 cm
E
3.5 cm
F
3 cm
Height (width)
cm
cm
cm
5
4
2.5
2
4
3.5
cm
cm
cm
cm
Ask a volunteer to draw a rectangle on the board and label the
base and height.
Area
10
16
6.25
cm2
cm2
cm2
12 cm2
cm
14 cm2
cm
10.5 cm2
height
Write a formula for finding the area of a rectangle.
Area =
base ∗ height (b ∗ h), or length ∗ width (l ∗ w)
base
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In Fourth Grade Everyday Mathematics, students found the area
of a rectangle by counting unit squares. Then they developed a
formula for finding the area of a rectangle. Expect that students
might use either method—formula or counting squares—to find
the areas of the rectangles on journal page 304.
With the counting method, some rectangles enclose partial
grid squares, and students must count and add the full and
partial squares to find areas. For example, rectangle C encloses
1 = 2 cm2), and
4 full squares (4 cm2), 4 half-squares (4 ∗ _
2
1 cm2). Its total area is 4 + 2 + _
1 = 6_
1 cm2.
1 quarter-square (_
4
4
4
4 full squares
4 cm2
4 half-squares
2 cm2
+1 quarter-square
1 cm2
_
total area
1 cm2
6_
4
4
Each half-square has an area
1 cm2.
of _
2
1 cm
The quarter-square is _
2
1 cm wide.
long and _
2
1 cm2
1 cm
1 cm
C
Assign journal page 304, Problem 1. Circulate and assist.
724
Unit 9
3/23/11 2:47 PM
▶ Discussing Formulas for
WHOLE-CLASS
DISCUSSION
the Area of a Rectangle
(Math Journal 2, p. 304; Math Masters, p. 436)
Algebraic Thinking Ask volunteers to give the dimensions of
rectangles A–F as other volunteers draw the rectangles on a
transparency of Math Masters, page 436.
Ask: What do you notice about the relationship between the base
and height and the actual area of each figure? The base multiplied
by the height is equal to the area.
Reinforce this rule:
If the length of the base and the height of a rectangle are
known, the area can be found by multiplying the length of the
base by the height.
Such a rule is called a formula. The formula can be written in
abbreviated form as:
NOTE An alternative formula for the area of
a rectangle is A = l ∗ w, where l stands for
the length and w stands for the width of the
rectangle. Students are familiar with both
versions of the formula from Fourth Grade
Everyday Mathematics.
A = b ∗ h,
where A stands for the area, b stands for the length of the base,
and h stands for the height. Ask students to complete Problem 2
on journal page 304. Remind students that letters used in this
way are called variables. Add the abbreviated formula to the
definitions on the Class Data Pad, and have students write the
abbreviated formula after their answers for Problem 2 on journal
page 304.
Refer students to the rectangles drawn on the transparency. Have
students apply the formula for the rectangles in Problem 1. Ask
volunteers to record a number model for the area of each rectangle
on the transparency. For example, 2 cm ∗ 5 cm = 10 cm2. Have
students check their total count of the squares with the product
from the number model. For rectangles C and E, ask students to
1 cm ∗ 2_
1 cm = 6_
1 cm2,
think about the decimals as fractions (2_
2
2
4
1 cm = 14 cm2).
and 4 cm ∗ 3_
2
Ask partners to estimate, in inches, the length of the sides of the
rectangles on journal page 304. Then have students measure the
sides of rectangle C using their inch rulers. (Each side of rectangle
C is about 1 inch long.) Ask students what the area of rectangle C
is when the unit is inches. 1 square inch Point out that there are
1 centimeters in 1 inch, and about 6.25, or 6_
1
about 2.5, or 2_
2
4
square centimeters in 1 square inch.
Student Page
Date
Time
LESSON
Area Problems
9 4
1.
A bedroom floor is 12 feet by 15 feet
(4 yards by 5 yards).
Floor area =
Floor area =
2.
3.
12 ft
(4 yd)
square yards
6 ft
(2 yd)
30 ft, or 10 yd
4.
Calculate the areas for the
figures below.
9 yd
Fill in the missing lengths for
the figures below.
a.
12 ft
6 yd
PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
N
G
Have students complete journal page 305. Circulate and assist.
12 yd
INDEPENDENT
ACTIVITY
6 yd
square feet
Imagine that you want to buy carpet for
the bedroom in Problem 1. The carpet
comes on a roll that is 6 feet (2 yards)
wide. The carpet salesperson unrolls the
carpet to the length you want and cuts
off your piece. What length of carpet
will you need to cover the bedroom floor?
a.
▶ Applying the Area Formulas
180
20
15 ft (5 yd)
6 yd
30
ft
30
360 ft2
ft
3 yd
Area =
72
b.
yd2
b.
12 12 ft
12
ft
15
yd
4 ft
8 ft
4 12 ft
2 ft
Area =
76
ft2
25
yd
375 yd2
15
25 yd
yd
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Lesson 9 4
725
4/7/11 9:30 AM
Student Page
Date
Time
LESSON
2.
9
Liz, Juan, and Michael equally share _
12 of a pizza.
a.
To show how the 9 pieces can be distributed,
write the student’s initial on each piece that he
or she is getting.
b.
Each student will get
c.
Write a number model to show what fraction
of the whole pizza each student gets.
3
J M
L
L
pieces.
M J
9
_
0
12
1
FEET
2
3
18
4
5
18
1
players. So, 6 ÷ _
3 =
6
.
1
_
Write an open number model to show how much of
Mr. Shower’s prize went to each department.
2
1
_
b.
5.
3
1
_
÷3=_
12 , or 4
When Mr. Showers won a large amount of money, he donated half to a local
college. The funds were divided equally among 5 departments.
a.
4.
Watch for students who hesitate with Problem 2. Ask: Which unit makes more
sense to work with, feet or yards? yards Refer students to Problem 1 and ask
how many yards of carpet are needed. 20 square yards Illustrate Problem 2
using a roll of paper towels, waxed paper, or aluminum foil to demonstrate how a
piece of carpet is cut from a roll. As the “carpet” unrolls, the area increases. For
example, in Problem 2, each foot unrolled adds an additional 6 square feet or 2
square yards. As you unroll the model, have students shade the grid in Problem
1 to show how much floor is covered. Prompt students to tell you where to cut
the model.
J
M
L
A football team orders a 6-foot-long submarine sandwich. Each player
1
will eat _
3 of a foot of sandwich. Draw tick marks on the ruler to help you
find how many players the sandwich will serve.
The sandwich will serve
3.
Ongoing Assessment: Informing Instruction
Fraction Division Review
9 4
1.
10
Each department received
When most students have completed the journal page, ask
volunteers to share their solution strategies. Ask questions like
the following:
÷5=p
of the prize.
The stone is a unit of weight used in the United Kingdom
1
and Ireland. One pound equals _
14 stone. Anna’s pen pal
in Ireland weighs 6 stones. How much is this in pounds?
84
pounds
●
1
Write a number story that can be solved by dividing 3 by _
4 . Solve your problem.
Answers vary.
292-332_EMCS_S_MJ2_G5_U09_576434.indd 306
In Problem 2, how will you cut the carpet to make it fit the
bedroom? If students determine that they will need a piece of
carpet 30 feet or 10 yards long, they will need to cut it in half
to fit the bedroom. Each of the two pieces will be 2 yards by
5 yards.
4/6/11 11:01 AM
2 yd
10 yd
2 yd
4 yd
2 yd
5 yd
●
Student Page
Date
Time
LESSON
Math Boxes
9 4
In Problem 4, how did you find the missing lengths? Encourage
students to use multiplication/division relationships and
open number sentences. For example, 360 ÷ 12 = h, or
25 ∗ b = 375.
3
_
1. A rope 4 meter long is cut into 6 equal pieces. 2. What is volume of the prism?
Choose the best answer.
3
4
0
240 units3
meters
a.
b.
30 units3
Write a number model to describe
the problem.
4
3.
90 units2
Draw lines on the rope to show how
long each piece will be.
3
_
÷ 6 = _18
90 units3
>
8 * 105
4.
b.
12.4 million
c.
7,000,000
d.
82
e.
5.4 * 102
<
=
>
429
* 15
6,435
12,400,000
7 * 105
c.
28
<
Solve.
a.
80,000
b.
134
* 82
10,988
706
*189
[Measurement and Reference Frames Goal 2]
133,434
5,400
220 221
5.
false
true
86,076 is divisible by 9. true
908,321 is divisible by 2. false
5,278 is divisible by 3.
b.
79,002 is divisible by 6.
c.
d.
19 20
6.
Write true or false.
a.
Exit Slip
Use an Exit Slip (Math Masters, page 414) to assess students’ ability to
calculate area. Have students write a response to the following: Explain how you
found the area for the figure in Problem 3b on journal page 305. Students are
making adequate progress if they multiply the side lengths to find the areas of the
rectangles in order to find the total area.
197
80A 80B
Compare. Use <, >, or =.
a.
Ongoing Assessment:
Recognizing Student Achievement
1
11
Complete the “What’s My Rule?” table,
and state the rule.
Rule:
∗2+1
in
out
4
9
7
15
11
23
9
19
6
13
231 232
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Unit 9
4/6/11 11:01 AM
4/7/11 9:30 AM
Study Link Master
Name
2 Ongoing Learning & Practice
▶ Fraction Division Review
PARTNER
ACTIVITY
Date
STUDY LINK
Time
More Area Problems
9 4
1.
Rashid can paint 2 square feet of fence in 10 minutes. Fill in the
missing parts to tell how long it will take him to paint a fence
that is 6 feet high by 25 feet long. Rashid will be able to paint
150 sq ft of fence in . 12 hr 30 min
2.
Regina wants to cover one wall of her room with wallpaper. The wall is
9 feet high and 15 feet wide. There is a doorway in the wall that is 3 feet
wide and 7 feet tall. How many square feet of wallpaper will she need to buy?
(hours/minutes)
(area)
(Math Journal 2, p. 306; Student Reference Book, pp. 80–80B)
104 105
189
114 square feet
Have students review information about fraction division problems
on pages 80–80B of the Student Reference Book. Discuss the use
of visual models and number stories. Then have students complete
the journal page.
Calculate the areas for the figures below.
3.
4.
4 yd
1 ft
2 ft
12 yd
4 yd
9 ft
4 ft
4 yd
10 yd
2 ft
▶ Math Boxes 9 4
INDEPENDENT
ACTIVITY
5.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 9-2. The skill in Problem 6
previews Unit 10 content.
Writing/Reasoning Have students write a response to the
following: Explain how you could determine the volume of
the rectangular prism in Problem 2 by counting the unit
cubes. Then write a number model for the formula you could use to
find the volume. Sample answers: I could count the number of
unit cubes in the bottom layer first. The width is 6 and the length
of the base is 5. So there are 30 unit cubes in the bottom layer.
The height is 3, so there are three layers of 30. The volume is
90 units3. The formula is B ∗ h = V. The number model is
(5 ∗ 6) ∗ 3 = 90 units3.
80
yd2
Area =
33
Fill in the missing lengths for the figures below.
▶ Study Link 9 4
Area =
6m
60 cm
3,000 cm
2
ft 2
33 m
6.
50 cm
5 ft
198 m 2
60 cm
6m
33 m
50 cm
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INDEPENDENT
ACTIVITY
(Math Masters, p. 265)
Teaching Master
Home Connection Students solve area problems.
Name
LESSON
9 4
Date
Roll 2 six-sided dice. The numbers on top
are the lengths of 2 sides of a rectangle.
3 Differentiation Options
READINESS
▶ Comparing Perimeter and Area
PARTNER
ACTIVITY
Time
Rectangle
Draw the rectangle in the grid below.
A
Record the perimeter and the area
of the rectangle in the table.
B
Use centimeter cubes to find other
rectangles that have the same area,
but different perimeters. Draw the
rectangles and record their perimeters
and areas in the table.
Perimeter
Area
Answers vary.
C
D
E
F
Repeat until you have filled the table. You
might need to roll the dice several times.
5–15 Min
(Math Masters, p. 266)
To support students’ understanding of perimeter and area, have
partners roll 2 six-sided dice to determine the dimensions of a
rectangle. They draw a rectangle with those dimensions and find
the perimeter and area of the rectangle. Partners then find other
rectangles with the same area, but different perimeters. They
repeat this process until the table on page 266 is completed.
Discuss what conclusions can be drawn from the table. Sample
answers: Different perimeters can have the same area; the
dimensions are factor pairs for the area.
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Lesson 9 4
727
4/7/11 9:30 AM
Teaching Master
Name
Date
LESSON
Time
9 4
▶ Comparing Perimeter and
Cut 6 rectangles that are 6 columns by 7 rows from the centimeter grid paper.
Record the area and the perimeter of one of these rectangles in Problem 1.
Divide each rectangle by using 3 different colored pencils to shade three connected parts
with the same number of boxes. The parts must follow the grid, and the squares must be
connected by sides.
Area for Irregular Figures
Divide each rectangle in a different way.
(Math Masters, pp. 267 and 436)
1.
42 cm
2
Perimeter =
26 cm
To apply students’ understanding of perimeter and area to
irregular figures, have partners divide rectangles and compare the
relationship between perimeter and area. When students have
finished, have them share the parts they cut from their rectangles
and discuss questions such as the following:
Record the perimeters for the divisions of the 6 rectangles in the table.
Perimeters
Rectangle
Part 1
1
Part 2
Part 3
Answers vary.
2
3
4
5
6
14 square centimeters
What is the range of the perimeters for each of the parts? Answers vary.
●
Can you use the area of a figure to predict the perimeter of that
figure? No
●
Can you use the perimeter of a figure to predict the area of that
figure? No
4.
5. a.
py g
p
What is the area for each of the parts?
g
3.
15–30 Min
For a rectangle that is 6 cm by 7 cm:
Area =
2.
PARTNER
ACTIVITY
ENRICHMENT
Perimeter and Area of Irregular Figures
b.
Describe one relationship between perimeter and area.
Sample answer: Shapes with the same
area can have different perimeters.
Is the relationship the same for rectangles and irregular figures? Explain.
Guide students to conclude that perimeter and area are
independent measures.
Answers vary.
254-293_497_EMCS_B_MM_G5_U09_576973.indd 267
EXTRA PRACTICE
3/23/11 1:09 PM
▶ 5-Minute Math
SMALL-GROUP
ACTIVITY
5–15 Min
To offer students more experience with calculating the area of
rectangles, see 5-Minute Math, page 212.
EXTRA PRACTICE
▶ Area: Tiling and Using
PARTNER
ACTIVITY
15–30 Min
a Formula
(Math Masters, pp. 293A and 436)
Teaching Master
Name
Date
LESSON
94
To find the area, students use grid paper to tile rectangles that
have fractional side lengths. They also use the formula for the area
of a rectangle to show that both methods produce the same result.
Time
For each rectangle below, cut out a rectangle from the centimeter grid paper (Math Masters,
page 436) that has the same dimensions. Follow the directions for each problem.
1.
The length of the base of the rectangle
1
is 6 cm and the height is 2_2 cm.
ELL SUPPORT
2.
a.
Tape the centimeter grid over the rectangle, and then
use the counting method to find the area of the rectangle.
b.
Use the formula to write an open number
model that can be used to find the area.
c.
Area =
15
15 cm
_
6 ∗ 2 12 = A
▶ Building a Math Word Bank
2
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 142)
cm2
The length of the base of the rectangle below is 12_2 cm and the height is 2_2 cm.
1
1
a.
b.
Use the formula to find the area.
b.
Use the formula to find the area of the rectangle below.
4
cm2
cm2
p
1
1_2 cm
g
cm2
cm2
py g
3. a.
31_14
31_14
15_34
15_3
To provide language support for area, have students use the Word
Bank Template found on Differentiation Handbook, page 142. Ask
students to write the terms length, height, base, and width; draw
pictures relating to each term; and write other related words. See
the Differentiation Handbook for more information.
10 _2 cm
1
c.
Explain why the formula and the counting method produce the same area.
Sample answer: Each row has the same number of squares. So,
multiplying the base by the height gives you the number of squares
there are in all. Counting the squares gives you the same area.
Math Masters, p. 293A
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Unit 9
3/22/11 9:31 AM
3/23/11 2:47 PM
Name
Date
LESSON
94
Time
For each rectangle below, cut out a rectangle from the centimeter grid paper (Math Masters,
page 436) that has the same dimensions. Follow the directions for each problem.
1.
The length of the base of the rectangle
1
is 6 cm and the height is 2_2 cm.
a.
2.
b.
Use the formula to write an open number
model that can be used to find the area.
c.
Area =
cm2
cm2
1
1
The length of the base of the rectangle below is 12_2 cm and the height is 2_2 cm.
a.
b.
b.
cm2
Use the formula to find the area.
cm2
Use the formula to find the area of the rectangle below.
cm2
cm2
1_2 cm
1
10 _2 cm
1
c.
Copyright © Wright Group/McGraw-Hill
3. a.
Explain why the formula and the counting method produce the same area.
293A
293A_EMCS_B_MM_G5_U09_576973 .indd 293A
3/22/11 9:31 AM

Areas of Rectangles

Transcription

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