Transcript Fractions

EMSE 3123
Math and Science in Education
Not Whole Numbers I:
Fractions
Presented by
Frank H. Osborne, Ph. D.
© 2015
1
Teaching the Meaning of Fractions
• So far, we have studied the teaching of whole
number concepts. Frequently we need to
express those that are not whole numbers.
• These are called rational numbers.
• One way to express rational numbers is via
fractions.
• Using manipulatives, we need first to have
something we can consider as a whole, or ‘1’.
• This can be divided in numerous ways.
2
Teaching the Meaning of Fractions
• Example: Take a piece of clay and find its
mass.
– Now, break off a piece amounting to ¾ of the
whole clay.
– How would you demonstrate that you have in
fact broken off ¾?
• To do this, you need to break the clay into 4
equal parts and then take 3 of these. We
express this in the following form.
3
Teaching the Meaning of Fractions
We write number of parts (numerator) over
the number of equal parts (denominator).
4
Teaching the Meaning of Fractions
A way to show this pictorially is to take a
square and identify it as the “whole”. The
square can then be divided into any
number of equal parts.
We use shading to indicate how much our
fraction has.
5
Teaching the Meaning of Fractions
We use shading to indicate how much our
fraction has.
6
Teaching the Meaning of Fractions
If more than one of the parts of is shaded, we
still use our definition to express the
number as
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Teaching the Meaning of Fractions
Formal instruction for associating a fraction
with a part of the whole should begin in
third grade asking the students to express
each not-whole-number as a fraction.
8
Teaching the Meaning of Fractions
Formal instruction for associating a fraction
with a part of the whole should begin in
third grade asking the students to express
each not-whole-number as a fraction.
9
Teaching the Meaning of Fractions
Children should also have experiences
working with objects, where a certain
number of objects represents a whole, and
each object is a fractional part of the whole.
For example, we have 8 pears.
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Teaching the Meaning of Fractions
Each pear is ?/8 of the whole?
Five pears are ?/8 of the whole?
?/8 of the pears are colored?
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Teaching the Meaning of Fractions
Each pear is ?/8 of the whole?
1/8
Five pears are ?/8 of the whole? 5/8
?/8 of the pears are colored?
4/8 = 1/2
12
Teaching the Meaning of Fractions
Circular or rectangular regions can also be
used as models to illustrate fractions.
Divide them into equal parts.
Some examples of rectangular regions are
shown in the activity on the next slides.
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Teaching the Meaning of Fractions
Write down different size relationships
among fraction from the strips below. Use
terminology of greater than, less than,
equals.
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Teaching the Meaning of Fractions
Just a few: 1/9 is less than 1/6
1/6 is greater than 1/9
1/4 is greater than 1/9.
1/8 is less than 1/3.
15
Teaching the Meaning of Fractions
• Children need to realize through use of
manipulatives that fractions with the same
numerator and denominator are all equal to 1.
We can show that 9/9=6/6=3/3=8/8=4/4=1.
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Teaching the Meaning of Fractions
• With the aid of manipulatives, children
should be able to order simple fractions,
first with the same numerator:
1/9, 1/8, 1/6, 1/4, 1/3,
• and then different numerators.
17
Teaching the Meaning of Fractions
• After mastering size relationships for
fractions less than 1, they can be
introduced to fractions greater than 1.
• For example, circular regions, like these
below, can be used to introduce fractions
greater than 1.
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Teaching the Meaning of Fractions
Rectangles, like those shown below, can
demonstrate that a fraction can be
expressed as a mixed number.
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Developing Concepts for Fractions
With Denominators of 10 and 100
• When working with fractions it is very
important to introduce fractions in which
the denominator is 10 or 100.
• This work prepares students for decimals
which are the next topic.
• Children must learn that there is another
way to write tenths. This permits early
introduction of decimals as another way of
expressing fractions with 100 or 100 as the
denominator.
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Developing Concepts for Fractions
With Denominators of 10 and 100
c. Use grids to answer each question:
1/10 = ?/100 2/10 = ?/100 10/10 = ?/100
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Developing Concepts for Fractions
With Denominators of 10 and 100
c. Use grids to answer each question:
1/10 = 10/100 2/10 = ?/100 10/10 = ?/100
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Developing Concepts for Fractions
With Denominators of 10 and 100
c. Use grids to answer each question:
1/10 = 10/100 2/10 = 20/100 10/10 = ?/100
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Developing Concepts for Fractions
With Denominators of 10 and 100
c. Use grids to answer each question:
1/10 = 10/100 2/10 = 20/100 10/10 = 100/100
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Developing Concepts for Fractions
With Denominators of 10 and 100
d. How would you demonstrate on the grids
which is bigger, 45/100 or 5/10?
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Developing Concepts for Fractions
With Denominators of 10 and 100
d. How would you demonstrate on the grids
which is bigger, 45/100 or 5/10?
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Developing Concepts for Fractions
With Denominators of 10 and 100
e. Demonstrate the answer to
50/100 + 12/100 = ?
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Developing Concepts for Fractions
With Denominators of 10 and 100
e. Demonstrate the answer to
50/100 + 12/100 = 62/100
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Developing Concepts for Fractions
With Denominators of 10 and 100
Once children understand the differences
between 1/10’s and 1/100’s, simple decimal
notation can be introduced as a shorter way of
writing fractions that have denominators of
tenths or hundredths, e.g.,
1/10 = .1
1/100 = .01
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Developing Concepts for Fractions
With Denominators of 10 and 100
Children should then be able to express any
number of tenths or hundredths in terms of
decimals e.g.,
2/10 = .2
2/100 = .02
3/10 = .3
3/100 = .03
.
.
.
.
9/10 = .9
9/100 = .09
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Addition and Subtraction of Fractions
Addition and subtraction of fractions using
manipulatives is a natural extension of the
activities we have already done.
Just specify a piece to represent the whole, then
represent each fraction in the addition or
subtraction problem with fraction pieces. Bring
the pieces together for addition, or take away for
subtraction, and express the answer in terms of the
whole.
We will start with ½ + ¾ as a demonstration.
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Addition and Subtraction of Fractions
Demonstration of ½ + ¾ .
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Addition and Subtraction of Fractions
When studying addition and subtraction it is
helpful to have all of the fraction pieces for a
particular set of problems lined up.
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Addition and Subtraction of Fractions
We can do many fraction manipulations with
the set of fraction pieces. For example, to add ½
and 1/3, we bring the corresponding pieces
together.
How much is this?
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Addition and Subtraction of Fractions
To find the answer
we compare it to the
whole by lining it up
with the complete set.
We see that the
answer is less than the
whole but more than
one-half of the whole.
The set also gives us
the exact answer.
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Addition and Subtraction of Fractions
For the exact
answer we use the
other fraction
pieces.
We see that our
answer is 10 of the
twelfths (10/12) or 5
of the sixths (5/6).
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Addition and Subtraction of Fractions
We could also express each of the fractions
in the problem in terms a smaller piece. Onehalf is the same lengths as 3 of the sixths
while 1/3 is the same as 2 of the sixths.
The longest rod that will fit into both pieces is
called the common denominator. This is a
concrete demonstration of how it is found.
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Addition and Subtraction of Fractions
Subtraction proceeds in a similar way. For
example, in order to subtract 1/6 from ½
(1/2 – 1/6)
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Addition and Subtraction of Fractions
To see how much the
answer is we line it up
with the rest of the
pieces.
We find that it
represents 2 of the
sixths, or 1 of the thirds.
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Addition and Subtraction of Fractions
Or we could have
also used our common
denominator method
seeing that ½ is the
same as 3 of the sixths.
Take 1 sixth away and
you have two sixths left
which is 1/3.
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Teaching Fractions with Cuisenaire Rods
Size Relationships
• We can assign any rod to represent a
whole or the number one.
• The other rods will have a corresponding
value in terms of the whole.
• Assume that the dark green (6) rod equals
one. What are the number values of all the
other rods?
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Line the other rods up
against the dark green one to
see the answer.
White =
Red =
Green =
Purple =
Yellow =
Dark Green =
Black =
Brown =
Blue =
Orange =
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Line the other rods up
against the dark green one to
see the answer.
White = 1/6 Dark Green = 1
Red =
1/3 Black = 7/6
Green = 1/2 Brown = 4/3
Purple = 2/3 Blue =
3/2
Yellow = 5/6 Orange = 5/3
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Using the rods to
answer, which is
larger?
a. 2/3 or ½?
b. 2/3 or 5/6?
c. 3/2 or 4/3?
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Using the rods to
answer, which is
larger?
a. 2/3 or ½?
b. 2/3 or 5/6?
c. 3/2 or 4/3?
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Now we assume that brown
(8) represents the number
one. What are the values of
all the rods?
White =
Red =
Green =
Purple =
Yellow =
Dark Green =
Black =
Brown =
Blue =
Orange =
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Now we assume that brown
(8) represents the number
one. What are the values of
all the rods?
White = 1/8 Dark Green = 3/4
Red = 1/4 Black = 7/8
Green = 3/8 Brown = 1
Purple = 1/2 Blue =
9/8
Yellow = 5/8 Orange = 5/4
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Using the rods to
answer, which is
larger?
a. ¾ or 5/8?
b. ½ or 3/8?
c. ¾ or 7/8?
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Teaching Fractions with Cuisenaire Rods
Size Relationships
Using the rods to
answer, which is
larger?
a. ¾ or 5/8?
b. ½ or 3/8?
c. ¾ or 7/8?
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Teaching Fractions with Cuisenaire Rods
Size Relationships
How many eighths make up 1 1/4?
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Teaching Fractions with Cuisenaire Rods
Size Relationships
How many eighths make up 1 1/4?
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Let the Orange (10) rod represent the
number one. What will represent
a. ½
b. 1/5
c. 1/10
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Let the Orange (10) rod represent the
number one. What will represent
a. ½
b. 1/5
c. 1/10
Note that ½ = 5/10 and 1/5 = 2/10. We need
to keep the common denominator in mind
when we add or subtract fractions.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Use your rods to verify that ½ + 1/5 = 7/10.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Use your rods to verify that ½ + 1/5 = 7/10.
In terms of the common denominator, we
have 5/10 + 2/10 = 7/10.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Use your rods to answer the following
a. ½ + 2/5
b. 3/5 + 1/10
c. 1/5 + 7/10
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Use your rods to answer the following
a. ½ + 2/5
b. 3/5 + 1/10
c. 1/5 + 7/10
Answers are: a. 5/10 + 2/10 = 9/10
b. 6/10 + 1/10 = 7/10
c. 2/10 + 7/10 = 9/10
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
Finding a common denominator is most
important. You cannot add fractions
unless the denominators are the same
size.
Example: We wish to add 1/3 + 1/2.
We see that the denominators are different. So
we need to find a common denominator.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
We take threes and twos and lay them end
to end until they line up evenly.
It takes 2 threes and 3 twos to make two
even rows. Therefore, the common
denominator is 6. Now we can deal with the
numerators.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
We used 2 threes, so we need to have 2 ones.
We used 3 twos, so we need 3 ones.
When we line them up, we get a total
numerator value of 5.
Answer: 1/3 + 1/2 = 2/6 + 3/6 = 5/6
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
As the children practice this procedure they
can move up to more complicated examples.
However, this is done the exact same way.
Example: We wish to add 2/7 + 3/5.
We see that the denominators are different. So
we need to find a common denominator.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
We take sevens and fives and lay them end
to end until they line up evenly.
It takes 5 sevens and 7 fives to make two
even rows. Therefore, the common
denominator is 35. Now we can deal with
the numerators.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
We used 5 sevens, so we need to have 5
twos. We used 7 fives, so we need 7 threes.
When we line them up, we get a total
numerator value of 31.
Answer: 2/7 + 3/5 = 10/35 + 21/35 = 31/35
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
The same concept applies to subtraction.
You can only subtract fractions
with equal size denominators.
Example: We wish to subtract 2/3 – 1/5.
We see that the denominators are different. So
we need to find a common denominator.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
We take threes and fives and lay them end
to end until they line up evenly.
It takes 5 threes and 3 fives to make two even
rows. Therefore, the common denominator is 15.
Now we can subtract the numerators.
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Teaching Fractions with Cuisenaire Rods
Addition and Subtraction of Fractions
We used 5 threes so we need 5 twos. We
used 3 fives so we need 3 ones which will be
subtracted from the twos.
Answer: 2/3 – 1/5 = 10/15 – 3/15 = 7/15
Remember that addition and subtraction of
fractions require a common denominator.
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Multiplication of Fractions
• Just as we did for whole number
multiplication, we want to teach the meaning
of multiplying by a whole number or another
fraction before teaching algorithms.
• This is done using concrete objects.
• We can use the repetitive addition model, or
the area model.
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Multiplication of Fractions
The addition model for multiplying fractions.
Using repetitive addition, multiplying a whole
number by a fraction such as 3 x ½ is the same as
3 x ½ = ½ + ½ + ½ = 3/2 = 1 ½
For another example,
2x3½=3½+3½=3+3+½+½=7
These can easily be demonstrated using fraction
pieces or Cuisenaire rods.
68
Multiplication of Fractions
How would we teach the meaning of
multiplying a fraction by a whole number such
as ½ x 3?
½x3=3x½=1½
A fraction multiplied by a whole number is the
same as taking the fractional part of the whole
number
½ x 3 = ½ of 3 = 1 ½
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Multiplication of Fractions
So, 3 x ½ is the same as taking half of the
number 3, which is 1 ½.
This means that if we take three wholes and
divide into two equal parts, each part is 1 ½.
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Multiplication of Fractions
As another example, multiplying the fraction
1/3 by 3
1/3 x 3 = 1/3 of 3 = 1
This means that if we have three wholes, and
divide them in to three equal parts, each is 1.
Multiplications such as 2/3 x 3 follow logically.
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Multiplication of Fractions
In words, the problem 2/3 x 3 means that we
find the fraction piece which is 1/3 of 3 and
take two of these pieces. As each third is 1,
then 2/3 is equal to 2.
1/3 of 3 = 1
2/3 of 3 = 2 (two 1/3 pieces)
3/3 of 3 = 3 (three 1/3 pieces)
72
Multiplication of Fractions
• Essentially, in multiplying a fraction by a whole
number, a certain number of objects represents
your whole, and you are taking a fractional part
of these objects as we demonstrated above.
• After students understand the meaning of
multiplying a whole number by a fraction they
can be introduced to the algorithm which is
cross-cancellation and multiplication.
73
Multiplication of Fractions
Cross-cancellation and multiplication.
Example: ¾ x 12
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Multiplication of Fractions
Multiplication of two fractions can follow the
same approach.
½ x 1/3 = ½ of 1/3
Which means to take a 1/3 fraction piece and
divide it into two equal parts (or, take ½ of this
1/3 piece).
Each part represents what part of the whole?
75
Multiplication of Fractions
½ x 1/3 = ½ of 1/3
Each part represents what part of the whole?
We see that if we take ½ of the 1/3 piece we get
a piece that corresponds to 1/6 of the whole.
76
Multiplication of Fractions
In Lab, we will work with exercises based on
this pattern.
77
Multiplication of Fractions
The area model for multiplying fractions.
We used the area approach in the
multiplication of whole numbers. We can use
the same method for multiplying fractions.
½ x 1/3 =
As shown above, the answer is 1/6.
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Multiplication of Fractions
As another example, we multiply
2/3 x 3/5 =
As shown, the answer is 6/15.
79
Multiplication of Fractions
• Division of fractions has the same
interpretation as division of whole numbers.
X ÷ Y means, “How many Ys fit into X?”
In the case of a whole number divided by a
fraction, such as
2 ÷ ½ means, “How many 1/2’s fit into 2?”
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Division of Fractions
2 ÷ ½ means, “How many 1/2’s fit into 2?”
One way to demonstrate the answer is by
repetitive subtraction:
2–½-½-½-½=0
We see that four ½’s fit into 2.
This can be illustrated by fraction pieces or
Cuisenaire rods. We will use the purple (4) to
represent the whole.
81
Division of Fractions
2 ÷ ½ means, “How many 1/2’s fit into 2?”
Or, “How many ½’s fit into 2 wholes?”
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Division of Fractions
Dividing two fractions together can be done
in a similar manner. ½ ÷ ¼ means how many
¼’s fit into ½. Using successive subtraction
½-¼-¼=0
Using Cuisenaire rods it is
done this way. We can see
that two ¼’s fit into ½.
83
Division of Fractions
For a set of exercises we will use orange 10
and red 2 together to make a whole.
What is the value of each color?
White =
Dark Green =
Red =
Black =
Green =
Brown =
Purple =
Blue =
Yellow =
Orange =
84
Division of Fractions
For a set of exercises we will use orange 10
and red 2 together to make a whole.
What is the value of each color?
White = 1/12 Dark Green = 1/2
Red = 1/6 Black = 7/12
Green = 1/4 Brown = 2/3
Purple = 1/3 Blue =
3/4
Yellow = 5/12 Orange = 5/6
We can use the rods to divide
1÷1/6. How many reds (1/6) make up one whole?
85
Division of Fractions
Simple activities can begin
as early as 2nd or 3rd grade.
Have the students learn the
meaning of mathematical
operations using
manipulatives before
teaching the algorithms.
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The End
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