Readings for Session 11 –
Division of Whole Numbers,
Properties of Equality, and Solving Equations
How does the operation
of division used in the two following problems relate to sets?
An employer distributes
$18,300 equally in bonuses to fifteen employees. How much money
does each employee receive?
Each employee received
$720 when an employer distributed $15,840 equally in bonuses.
How many employees received bonuses?
Division is used to find the answer to both types of questions.
These two problems illustrate two different views or models for
division of whole numbers.
In the first problem, we know the number of sets (number
of employees) and need to find the number of elements for each
set (number of dollars each employee receives). The problem
“divides” the amount ($18,300) among 15
equivalent sets and asks the
question: “How many elements should each set receive?” This
concept of sharing objects equally among several sets is called
the partition model
(some call it the
partitive model) because a set of objects is partitioned
into several equal sized sets.
In the second problem, we know the number of elements in
each set (number of dollars each employee received) and need to
find how many sets there are (number of employees). The problem
“divides” the
amount ($15,840) into sets of $720 and asks the question: “How
many sets can be made?” This concept of finding the number of
set that can receive the same number of objects is called the
repeated-subtraction
model (sometimes called the
measurement model)
because we may find the solution by finding the number of times
the amount can be subtracted from the original.
For the first problem if each of the fifteen employees
received $1,220 all the $18,300 would be distributed. Stating
the result this way, we note that 15
∙ 1220 = 18300.
In the second problem, after subtracting $720 twenty-two times,
all the $15,840 would be distributed. Again, stating the result
this way, we note that 22
∙ 720 = 15,840;
the inverse of repeated addition. This relationship from both
models leads to division as the inverse operation of
multiplication. That is,
a
÷
b
=
c only when
b
∙
c =
a.
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