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Arithmetic
Operations with Decimals – Multiplication and Division
Division
Division of decimal numbers may be modeled as repeated
subtraction or measurement. For instance,
the operation 0.75 ¸ 0.20 may be thought of asking the question, “How many
times may the quantity 0.20 be subtracted from the quantity 0.75?” In the context of a word problem, one might
ask, “A cookie recipe calls for 0.20 of a cup of sugar. If Alice has 0.75 cups of sugar, how
many batches of cookies can she make?”
The expanded algorithm and explanation presented in Figure 6.5
approach the operation  directly. Doing so retains the original numbers and
preserves the meaning of the context presented in the word problem.
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Concept Model
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Discussion
Assuming that the large square represents a one, or unit, each column
represents a tenth and each
small square represents a hundredth. The dividend 0.75 is modeled using 75
shaded small squares and the divisor 0.20 is modeled using shaded pairs
of columns. The quantity 0.20 may
be subtracted 3 times (gray, red, green), leaving a remainder of 15 (blue)
small squares. Written as a
fraction of the divisor, this equals
 .
So, 0.20 may
be subtracted 3.75 times from 0.75.
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Expanded
Algorithm
Using an expanded algorithm, the same result is
obtained as follows:
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Figure 6.5:
Division as Repeated Subtraction
In addition to modeling the operation , Figure 6.5 provides a context in which to explore the
mathematical basis for another familiar rule: When dividing a decimal number
into a whole number or decimal number,
1. Move
the decimal point in the divisor to the right until a whole number is
obtained. Call the number of places
moved m.
2. Move
the decimal point in the dividend to the right m places.
3. Use
the standard division algorithm for dividing a whole number into a decimal
number.
Applying this rule to the concept model in Figure 6.5, one
may move the decimal points in the divisor and dividend …
§ One
place to the right, resulting in the operation  , or
§ Two
places to the right, resulting in the operation  .
The “rule” asserts that the quotients associated with
these operations are identical to that of the original operation, . Of these
alternative representations, the second is more easily related to the concept
model in Figure 6.5. Instead of
viewing the small squares as having area 0.01, think of them as having area 1
(See Figure 6.6). The shaded regions
may then be thought of as representing the operation . Using the same
argument used to explain why = 3.75, we may assert that = 3.75 also.
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Concept Model
    
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Discussion
Assuming that the small squares each represent a one, or unit, each column
represents a ten and the large
square represents a hundred. The dividend 75 is modeled using 75
shaded small squares and the divisor 20 is modeled using shaded pairs of columns. By inspection, it is clear that the
quantity 20 may be subtracted 3 times (gray, red, green), leaving a
remainder of 15 (blue) small squares.
Represented as a fraction of the divisor, this remainder may be
written as 15/20 or 0.75. So, 20
may be subtracted 3.75 times from 75.
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Expanded
Algorithm
Using an expanded algorithm, the same result is
obtained as follows:
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Figure 6.6: An
Equivalent Division
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