Download EQUIVALENT FRACTIONS: AN INQUIRY ACTIVITY

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EQUIVALENT FRACTIONS: AN INQUIRY ACTIVITY
Goal: The fifth grade student will be able to recognize and generate equivalent forms of
fractions, using fractional forms of the number one. (NCTM Standards of Numbers and
Operations from the Principles and Standards for School Mathematics, 2000, Grades 3-5,
p. 392, and Academic Content Standards: K-12 Mathematics for Ohio (Number, Number
Sense and Operations Standard, Grade Four, Standard #1, 2001, p. 137 and Grade 5, p.
63.) The students will already have learned how to compare fractions for relative size.
This lesson can also be used for remedial work in Integrated Math I (9th Grade).
Pre-Activity: Fractions of Pizzas
Since many people like to eat pizza, we are going to use drawings of pizzas throughout
this activity.
Let’s compare some pizza drawings:
Pizza A is a whole pizza, Pizza B was cut into
not cut into slices.
two pieces of the same size;
one piece was eaten, and only
one piece remains.
Pizza C was cut into
eight identical pieces;
only four are left.
1. Compare the amount of pizza that remains of Pizza B as compared to that of Pizza C.
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2. What fraction best describes the amount of Pizza B that remains?_______________
3. What fraction of Pizza C remains?_______________________________________
4. Based on your answer for Problem 1, compare the fractions that you found in
Problems 2 and 3. What can you say about the value of these fractions? (Hint: Are some
of them larger than the others?)
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ACTIVITY I: Multiplying Numerator and Denominator of
Fractions by the Same Nonzero Number
Let’s look at some fractions in table form:
2
2
1
2
(Y)
3
3
4
4
5
5
6
6
2
4
(Y)
1
3
1
4
1
5
2
3
(G)
2
5
(O)
4
4
(G)
(O)
NOTE: The letters in the colored cells are labeled Y for yellow, G for green and O for
orange, for people who do not distinguish colors.
Fill out the table above, multiplying the fractions found in the first column by the
fractions at the top of the other columns. The first square has been completed as an
example
1 2
2
(  =
).
2 2
4
3
What is the integer value for each of the following fractions? ________
2
3
4
=______; = _______;
= _______ ;
2
3
4
What can you conclude about the integer value of all of the fractions in the first row?
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Do you think that multiplying the fractions in the first column by numbers such as
2
and
2
3
changed the value of the fractions listed in the first column? Explain your answer.
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So, in general, what should happen to the value of a fraction when you multiply by
2 3
4
fractions such as , and ?
2 3
4
Further Investigation:
Look at the yellow squares (Y). The fractions in them should look familiar; they are the
same ones that represented the leftover pieces of pizzas B and C.
Based on what you discovered earlier about the fractions of pizzas B and C, what do you
conclude about the value of the fractions in the yellow squares? How big are they in
relation to each other?
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Look at the fractions in the green squares in your chart (G). The fraction
2
is shown
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below:
Shade in the number of squares in the rectangle below that you think best represents the
fraction you wrote in the second green square:
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What can you conclude about the value of the fractions in the green squares?
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Now look at the fractions in the orange squares in your chart (O). The fraction
2
is
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shown below:
Shade in the number of squares in the rectangle below that you think best represents the
fraction you wrote in the second orange square.
What can you conclude about the fractions in the orange squares?
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DEFINITION: The fractions that you wrote in your chart, obtained by multiplying
2 3
n
or dividing each fraction by forms of one, such as , , or, in general, , where n is
2 3
n
any nonzero integer, are called EQUIVALENT FRACTIONS.
EXTENSION: Look at your chart again. Based on what you have just learned, do you
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think that
is equivalent to ? Why or why not?
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4
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If you did not have a chart, how could you find the answer to the questions above?
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Based on what you have just discovered, what can you conclude about the fractions in
any single row of your chart?
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Since you multiplied the numerator and denominator of the fractions in the first column
of your chart to get the rest of the fractions in that row, summarize what you have
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discovered about multiplying the numerator and denominator of a fraction by the same
nonzero integer n, such as 2 or 3:
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When you multiply both numerator and denominator of a fraction by the same
n
number, you are really multiplying the original fraction by another fraction, ,
n
where n is a nonzero integer.
How can we find the value of the fraction
n
?
n
a
can also be called a division expression, a divided
b
by b, as long as b does not equal zero.)
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(Hint: Remember that any fraction
Fill out the third column of the chart below:
Nonzero integer n
Fraction,
1
2
3
12
n
n
n n = ?
1
1
2
2
3
3
12
12
Look at what you wrote down in the third column of the chart, and then answer the
following questions:
Compare the answers that you wrote in the third column of the chart. How big are they in
relation to each other? Is one value bigger than the others? Explain your answer.
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What can you conclude about the value of
n
, as long as n is a nonzero integer?
n
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Based on what you have discovered in this activity, explain how to obtain equivalent
fractions using multiplication:
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Definition: Fractions of the form
n
, where n  0, are called a form of one.
n
n
(where n  0)
n
hypothesize what you think will happen when you divide fractions by a form of one:
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Based on what you have just discovered about multiplying fractions by
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ACTIVITY II: Dividing Numerator and Denominator of a
Fraction by the Same Nonzero Number
Fill out the table below, dividing the fractions found in the first column by the fractions at
the top of the other columns. The first square has been completed as an example
2
2
6
12
(R)
6
18
(B)
6
24
(G)
12
18
(Y)
3
3
6
6
3
6
(R)
(B)
(G)
(Y)
NOTE: The letters in the colored cells are labeled R for red, B for blue, G for green, and
Y for yellow, for people who do not distinguish colors.
What is the decimal value for each fraction in a red square (R)?
The fraction in the first square,
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= 6 divided by 12 = _____ (decimal)
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The other fraction in a red square is ____ = ________ (decimal)
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What can you conclude about the decimal value of all of the fractions in the row with the
red squares?
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What is the decimal value for each fraction in a blue square (B)?
6
= 6 divided by 18 = _____ (decimal)
18
The other fraction in a blue square is ____ = _______ (decimal)
What can you conclude about the decimal value of all of the fractions in the row with the
blue squares?
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What is the decimal value for each fraction in a green square (G)?
6
= 6 divided by 24 = _____ (decimal)
24
The other fraction in a green square is ____ = ________ (decimal)
What can you conclude about the decimal value of all of the fractions in the row with the
green squares?
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What is the decimal value for each fraction in a yellow square (Y)?
12
= 12 divided by 18 = _____ (decimal)
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The other fraction in a yellow square is ____ = _______ (decimal)
What can you conclude about the decimal value of all of the fractions in the row with the
yellow squares?
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Based on what you have just discovered, what can you conclude about the fractions in
any single row of this chart?
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Summarize what you have learned about dividing fractions by a form of one, such as
2
,
2
3
n
or, in general, , where n is any nonzero integer:
3
n
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What do you conclude about multiplying or dividing any number (including
n
fractions) by , where n is any nonzero integer?
n
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APPLICATION:
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A. Given the fraction , find four equivalent fractions with numerators larger than
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three: ______, ______, ______. ______.
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, find two equivalent fractions with denominators smaller
16
than 16: ______, ______.
B. Given the fraction
EXTENSION:
Fill in Column Two of the table below (under Fraction Two), with your answers from
Question A above, and then complete the chart:
Fraction
One
3
5
Fraction
Two
Numerator of Fraction One
Times Denominator of
Fraction Two
Numerator of Fraction Two
Times Denominator of
Fraction One
10
3
5
3
5
3
5
What do you notice about the numbers in columns three and four?
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Does this give you a way to see if two fractions are equivalent? Explain your answer:
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4
1
is equivalent to :
12
4
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Using the strategy you just discovered, prove or disprove that
GENERALIZE:
A. Name two ways to find equivalent fractions:
1. __________________________________________________________________
2. __________________________________________________________________
B. Describe two ways to see if two fractions are equivalent:
1. __________________________________________________________________
2. __________________________________________________________________
EXTENSION: Why do you think that n must be nonzero, when using n to find
equivalent fractions? Let’s explore this idea a little deeper:
What happens when you multiply a number by zero?
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Would multiplying two fractions by zero give you equivalent fractions? Explain your
answer:
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Why can’t you divide both fractions by zero?
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Based on your answers above, why must n have to be a nonzero number?
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FURTHER EXTENSION: Can you think of any practical uses for equivalent
fractions? Describe them below:
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