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Division
Division of
whole numbers may be thought of as repeated
subtraction. In general, division
problems partition a whole into subsets.
For instance, the expression 6 ¸
3 = 2 may be thought of as asking two types of questions, called partitive and measurement division. The
meaning of each type of division is illustrated in Figure 3.3. Note the
different interpretations assigned to the terms divisor and quotient in
these models.
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Partitive
Division: 6 ¸
3 = 2
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Measurement
Division: 6 ¸
3 = 2
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A
whole is partitioned into a known number of subsets. The quotient answers the question, “How
many elements are in each subset?”
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A
whole is partitioned into an unknown number of subsets, each having the
same number of elements. The
quotient answers the question, “How many subsets are there?”
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Maria divides 6 candies equally among 3 friends. How many candies does each friend receive?
Set Model
·
Dividend: There are 6
candies
·
Divisor: Three friends
receive candy
·
Quotient: Each friends
receives 2 candies
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Maria distributes 6 candies 3 at a time to her friends. How many friends receive candies?
Set Model
·
Dividend: There are 6
candies
·
Divisor: Each friend
receives 3 candies
·
Quotient: Two friends
receive candy
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Maria’s candy bar has 6 parts.
If she divides the candy equally among 3 friends. How many parts does each friend receive?
Area Model
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Maria’s candy bar has 6 rectangular parts. If she distributes the candy 3 at a time
to her friends, how many friends receive candy?
Area Model
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Figure 3.3: Partitive vs.
Measurement Division
In
some divisions, a portion, or remainder,
of the original set is not assigned to any of the subsets. Figure 3.4 illustrates such a division
using both partitive and measurement concept models. Note that the meaning associated with the
term remainder is the same in both
models.
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Bill divides 7 candies equally among 3 friends. How many candies does each friend
receive?
Partitive Interpretation for 7 ¸ 3 = 2 R 1
·
Dividend: There are 7
candies
·
Divisor: Three friends
receive candy
·
Quotient: Each friends
receives 2 candies
·
Remainder: One candy
is left over
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Bill distributes 7 candies three at a time to his friends. How many friends receive candy?
Measurement
Interpretation for 7 ¸ 3 = 2 R 1
·
Dividend: There are 7
candies
·
Divisor: Each friend
receives 3 candies
·
Quotient: Two friends
receive candy
·
Remainder: One candy
is left over
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Figure 3.4: Remainders
Example 3.4
Create
a concept model for the word problem, “Jose gives 12 baseball cards to his best
friends, Jimmy, Hector, and Peter. If
each friend receives the same number of cards, how many cards does each
friend receive?” Is this an example of
partitive or measurement division?
Solution 3.4
This
is an example of partitive division.
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