Fractions Module Three: Equivalent Fractions, Part One
Module three
E Q U I V A L E N T F R A C T I O N S , p art one
Great job! You have done a lot of work to really understand the concept of fractions. Now it’s
time to move on to another big topic with fractions, equivalent fractions.
The first step is to get out your fraction pieces.
I really urge you to use the fraction pieces. The fraction pieces give you the powerful
effects of physical modeling, which enhances conceptual understanding. So—please trust
me on this, and use your fraction pieces. It will take an extra five to ten minutes, not
much time for the positive effects.
Note that for a fraction to be equivalent, the new
fraction must fit exactly over the original fraction
pieces. Close matches don’t count.
__
2
__
1
As an example, 4 is equivalent to 2 .
Put them on top of each other, and see
that they exactly cover each other.
__
1 __
1
4
4
__
1
2
Three looks at equivalent frac tions
Look 1 at equivalent fractions: Use your fraction pieces
Equivalent fractions are a very, very important concept in fractions. Like all concepts, we begin with concrete objects. We will use fraction pieces for the concrete representation.
The word equivalent has the same sound as equal in it.
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Fractions Module Three: Equivalent Fractions, Part One
My name is Bernice German. Some people
call me the Math Whisperer, some call me
Bernice, some call me Mrs. German, my
children call me Mom. They are all names
for the same person—me. The names are
different, but I’m the same person. Math
Whisperer is equivalent to Mrs German.
the Math Whisperer
Bernice
Mrs. German
Mom
What other names do you have?
Here is another example. Say you have a dollar bill. This is equivalent to four quarters.
is equivalent to
__
1
2
pieces as well as the __
1
and __
1
4
8 pieces.
__
2 have the same amount of surface. That is why
1 and __
Area is the amount of surface. 2
4
they are equivalent. Two fourths has the same area as one half—there are just more pieces.
Take out your This is a picture of the situation:
1 - for reference
30
__
1
2
One-half
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is equivalent to
__
1 __
1
4
4
Two-fourths
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Fractions Module Three: Equivalent Fractions, Part One
Remember the denominator tells us the number of pieces the one is cut into.
The numerator tells us the number of pieces we have.
__
1
__
1
1 __
2
4
4
The one is cut into 2 pieces,
so the denominator is 2.
The numerator is 1.
The one is cut into 4 pieces,
so the denominator is 4.
The numerator is 2.
Now it’s your turn. In the space below, use your fraction pieces to trace.
Trace a
__
1
2
fraction piece. Then trace over it four
Based on your tracing,
So ,
__
1
2
__
1
is equivalent to
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2
__
1
covers the same area as
?
__
8
fraction pieces.
8
?
__
8
.
.
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Fractions Module Three: Equivalent Fractions, Part One
Look 2 at equivalent fractions: Pizza
Joe gets four fourths.
Jo gets two halves.
Who gets more pizza, Jo or Joe?
The correct answer is that they get the same amount of pizza. Jo gets one giant slice that is
half of the pizza. Joe gets two big slices, each of which is one fourth of the pizza, for a total of
two fourths of the pizza. They get the same amount of yummy pizza.
If you want, you can tell your parent that the Math Whisperer says your homework is to check
this out for yourself by ordering two pizzas. Cut one in half, and cut the other into fourths.
Eat half of one, and two fourths of the other. Do you notice a difference?
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Fractions Module Three: Equivalent Fractions, Part One
Look 3 at equivalent fractions: Algebra
In picture form:
The one has been divided into two
1 .
equal parts. Each is __
2
Now I divide each of the one halves
into four equal parts. Each of these
1 .
parts is __
__
1
8
8
__
1
8
There is an algebraic way to go from
__
1
2
to
__
4
.
8
_______
1*4= 4
2*4= 8
Each of the halves was divided into four equal pieces.
(The pink half and the white half were each divided into four equal pieces.)
And we used a factor of 4 to go from
__
1
2
to
__
4
8
.
_______
1*4= 4
2*4= 8
Factor of 4
__
1
Is this a coincidence?! The original fraction piece (
) divided into
2
four equal pieces, and a factor of 4?! Your mission, should you choose
to accept it, is to find out!
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33
Fractions Module Three: Equivalent Fractions, Part One
Ac tivit y 1: T wo looks at equivalent frac tions
Equivalent
Fractions
__
1
=
__
2
__
1
=
__
3
__
2
=
__
4
__
1
=
__
2
__
3
=
__
6
__
2
=
__
4
3
3
3
4
4
5
34
6
Picture
How many equal
parts did you
make from each
original part?
Each of the one
thirds is divided
into 2 equal pieces
Algebraic
form:
___
1*2
3*2
=
__
2
6
9
6
8
8
10
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Fractions Module Three: Equivalent Fractions, Part One
Equivalent
Fractions
__
2
=
__
6
__
1
=
__
2
__
1
=
__
3
__
1
=
__
2
__
3
=
__
6
__
4
=
__
8
5
5
5
7
7
4
Picture
How many equal
parts did you
make from each
original part?
Algebraic
form:
15
10
15
14
14
8
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35
Fractions Module Three: Equivalent Fractions, Part One
Is this a coincidence?! Look at some of your examples in the activity you just did.
_______
1 3 = 3 Here each of the two halves was divided into 3 pieces. And so on.
*
2*3= 6
Try to make a general case here. Do your best!
_______
1 n = means each of the halves is divided into ________(how many) pieces?
*
2*n=
_______
1 * n = means each of the thirds is divided into ________(how many) pieces?
3*n=
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Fractions Module Three: Equivalent Fractions, Part One
Ac tivit y 2: more work with equivalent frac tions
Picture
Symbol
__
1
___
1*6
=
__
6
__
6
____
6÷6
=
__
1
__
1
___
1*
=
__
2
__
2
____
2÷
=
__
1
2
12
4
8
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Relationship
2*6
12 ÷ 6
4*
8÷
12
Explanation
__
1
2 is equivalent to
__
6
they
12 because
cover the same
area
2
8
__
1
4
is equivalent to
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Fractions Module Three: Equivalent Fractions, Part One
Picture
Symbol
Relationship
Explanation
__
3
8
__
9
24
__
1
4
__
6
24
___
1*8
=
__
8
____
8 ÷8
=
__
1
3*8
24 ÷ 8
24
3
__
2
3
__
16
24
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is equivalent to
because they
cover the same
area
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Fractions Module Three: Equivalent Fractions, Part One
Ac tivit y 3:Using pic tures to find equivalent
frac tions—the area model
Original
Fraction
__
1
2
Picture
Final
Fraction
__
4
8
__
1
__
3
__
1
__
5
__
1
__
3
__
1
__
2
__
1
__
2
__
2
__
4
__
1
__
3
2
2
4
4
3
5
3
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Picture
Factor for
Numerator and
Denominator
For every 1 piece
1 , there are 4
of __
2
4
pieces for __
8
6
10
12
8
6
10
9
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Fractions Module Three: Equivalent Fractions, Part One
Ac tivit y 4: Finding equivalent frac tions
with your frac tion pieces
Use your fraction pieces for this activity.
__
1
2
__
1
2
__
2
4
__
3
6
__
5
10
__
1
3
__
1
5
__
2
3
__
2
5
40
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Fractions Module Three: Equivalent Fractions, Part One
__
5
10
__
3
12
__
1
4
Thinking about it:
__
and __
5
?
__
and __
1
?
a. What is the same about the list of equivalent fractions for 1
2
b. What is the same about the list of equivalent fractions for 3
12
10
4
c. How can you tell without fraction pieces that two fractions are equivalent?
(more about this later)
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Fractions Module Three: Equivalent Fractions, Part One
Ac tivit y 5: More equivalent frac tion prac tice
Circle the fractions that are equivalent to the given fraction.
__
1
2
__
1
3
__
1
4
__
2
5
__
2
3
42
__
4
__
50
25
8
__
5
__
2
__
120
240
12
6
3
__
4
__
120
300
150
__
3
__
120
480
20
__
200
500
12
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6
__
100
300
1
__
11
__
5
1
55
__
8
__
80
320
33
__
80
120
2
700
__
50
__
300
900
__
16
__
6
__
3
6
25
__
8
125
5
__
5
__
4
8
75
__
9
6
100
__
4
__
100
400
14
__
20
__
20
100
5
__
12
__
12
50
30
__
3
2
46
__
25
2
__
50
__
14
42
__
10
__
50
__
16
36
150
150
18
__
350
__
4
2
__
12
13
__
75
__
50
20
__
22
6
__
14
2
__
22
__
4
__
4
10
42
__
5
10
__
11
__
30
__
4
12
__
4
1
33
__
4
12
164
__
3
12
__
75
__
11
__
82
20
__
9
__
2
1
__
3
__
30
__
9
12
20
__
250
__
16
24
350
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Fractions Module Three: Equivalent Fractions, Part One
Ac tivit y 6: Equivalent frac tion dominoes
Dominoes have been played for over 300 years. This version is a great equivalent fraction
exercise, too.
In case you don’t know how to play, here is an explanation for playing dominoes:
This is a single domino:
If the value on the right of one domino matches the value on the left of the second domino,
they can be connected. Whether they are connected horizontally or vertically depends on
the second value on the second domino. (Generally up and down on the domino are irrelevant, unless it bothers the students to read a number upside down.)
You can work in pairs on this, taking turns to add one domino each and check your partner’s
work. Or you can play by yourself. Either way, you practice and learn!
Matches can be either
numbers or pictures
__
5
10
__
1
2
__
2
4
__
1
3
__
2
6
__
1
4
__
2
8
__
1
2
So print pages 44 and 45 to play. You can play by yourself or with a partner.
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Fractions Module Three: Equivalent Fractions, Part One
44
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Fractions Module Three: Equivalent Fractions, Part One
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Fractions Module Three: Equivalent Fractions, Part One
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