Transcript 8. Place Value Division

Taking the Fear
out of Math
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#5
Unultiplying
Whole Numbers
Division
81 ÷ 9
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Formulas as a Way to “Unite”
Multiplication and Division
A rather easy way to see the relationship
between multiplication and division is in
terms of simple formulas.
For example, we all know that there are 12
inches in a foot. Thus, to convert the
number of feet to the number of inches, we
simply multiply the number of feet by 12.
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As a specific example, to convert 5 feet
into an equivalent number of inches, we
first notice that 5 feet may be viewed as
5 × 1 foot and we may then replace 1 foot
by 12 inches to obtain that
5 feet = 5 × 12 inches or 60 inches.
In terms of a more
verbal description,
the “recipe” (which
we usually refer to
as a formula) may be
stated as follows…
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Step 1: Start with the
number of feet.
Step 2: Multiply by 12.
Step 3: Finally,
replace feet by the
number of inches.
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The verbal description above is easy
to understand but cumbersome to write
(and it gets even more cumbersome as the
“recipe” contains more and more steps).
The algebraic shortcut is to let a
“suggestive” letter of the alphabet
represent the number of feet (perhaps
the letter “F” because it suggests feet,
and similarly let a letter such as I to
represent the number of inches
(because it suggests inches).
Use the equal sign to represent the word
“is”.
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The verbal recipe and algebraic formula
are shown below…
English - “Recipe”
Step 1: Start with a
quantity, the number
of feet.
Step 2: Multiply by 12
Step 3: Replace feet
by inches and the
product is I.
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Algebra
F
F × 12 (or 12 ×
F)
F × 12 = I
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Algebra Notes
Because × and x are easy to confuse
(especially when we’re writing by hand), we
do not use the “times sign” in algebra.
Rather than write 12 × F we would write 12F
(even if it were in the form F × 12, we would
write 12F probably because of how natural it
is to say such things as “I have 12 dollars”
as opposed to “I have dollars 12”).
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Algebra Notes
It is also traditional to write the letter
that stands by itself (namely, the
“answer”) to the left of the equal sign
but this is not really vital. In any case
when we write I = 12F it is
“shorthand” for the verbal situation
described in Steps 1, 2, and 3.
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In studying algebra, many students are
stymied by the symbolism. For whatever
reason, there seems to be an anxiety among
many beginning students when letters are
used to represent numbers. Therefore, it is
a nice transition (and it also highlights an
aspect of learning a language) to have
students learn to translate “English” into
“Algebra” and also from “Algebra” into
English. Skill in being able to maneuver
easily between “English” and “Algebra” can
go a long way in helping a student better
understand algebra.
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In any case, the verbal process gives us
an easy way to see the relationship between
multiplication and division. Namely, when
we started with 5 feet, we multiplied 5 by 12
to get the number of inches.
On the other hand, if we started with
60 inches and wanted to know the number
of feet with which we started, we would
have to realize that we obtained 60 inches
after we multiplied by 12.
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We obtained 60 inches from 5 feet after we
multiplied by 12. To determine the number
of feet we started with, we must unmultiply
by 12. Unmultiplying is actually division,
so 5 feet = 60 inches ÷ 12.
In more familiar terms, to convert
60 inches into feet, we divide 60 by 12.
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The Division Algorithm for
Whole Numbers
Again, we assume that the students
know the standard algorithm for
performing long division.
What is not always clear to students is
that division is really a form of rapid
subtraction.
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Let’s look at a typical kind of problem
that one uses to illustrate a division problem.
For example, consider the following
question…
A certain book company ships
its books in cartons, each of
which contains 13 books. If a
customer makes an order for
2,821 books, how many cartons
will it take to ship the books?
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To answer this question we divide 2,821
by 13 to obtain 217 as the quotient.
The usual algorithm is illustrated below.
13
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217
2,821
2 6
22
13
91
91
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However, even if it is successfully
memorized by students, this algorithm is
rarely understood by them. However, with a
little help from our “adjective/noun” theme
the “mysticism” is quickly evaporated!
The “fill in the blank” problem we are being
asked to solve is…
13 × ___ = 2,821
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13 × ___ = 2,821
Since 13 × 2 = 26, our adjective/noun
theme tells us that 13 × 200 = 2,6001
Hence, after we have packed 200 cartons,
we have packed 2,600 books.
note
1
Since we also know that 13 × 300 = 3,900, we know that there aren't enough books
left to fill an additional hundred cartons. In other words, at this stage we know
that we need more than 200 cartons but less than 300 cartons in order to pack
the books.
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Since we started with 2,821 books,
there are now 221 (that is, 2,821 – 2,600)
books still left to pack.
Since 13 × 1 = 13, we know that 13 × 10 =
130.
Hence, after we pack an additional
10 cartons there are still 91
(that is, 221 – 130) books still left to pack).
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Finally, since 13 × 7 = 91, we see that it
takes 7 more cartons to pack the
remaining books.
Thus, in all, we had to use 217
(that is, 200 + 10 + 7) cartons.
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We can now write the above result in
tabulated form as shown below.
2 , 8 2 1 books
– 2 6 0 0 books
2 2 1 books left
– 1 3 0 books
9 1 books left
– 9 1 books
0 books left
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2 0 0 cartons filled
1 0 more cartons
7 more cartons
217 cartons filled
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The previous format is less familiar
than the traditional algorithm, but it
seems to better exhibit the “rapid,
repeated subtraction” principle.
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However, if we simply
insert the “missing”
digits” into the
13
traditional algorithm
(they are emphasized
in red type), we see
that the numbers in the
emended version look
exactly the same as the
numbers in the
previous format.
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7 cartons
1 0 cartons
2 0 0 cartons
2 , 8 2 1 books
2 6 0 0 books
2 2 1 books
1 3 0 books
9 1 books
9 1 books
0 books
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Notes
In checking a division problem,
we multiply the quotient by the
divisor, and if we divided
correctly, the product we
obtain should be equal to the
dividend. In terms of the
present illustration, the usual
check is to show that
217 × 13 = 2,821, and we
usually perform the check as
shown.
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217
× 13
651
2170
2821
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Notes
The rows in the above multiplication
(namely, 651 and 2170) have no relationship
to the three rows in the division shown in
the above computation (2600, 130 and 91).
The reason is that the division problem
showed us that if there were 13 books in
each carton, then 217 cartons would have to
be used. On the other hand, the check by
multiplication showed that if there were 217
books in a carton, then 13 cartons would
have to be used.
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Notes
However, a more
enlightening way to
perform the check is to
compute the product in the
less traditional but more
accurate form 217 × 13.
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13
× 217
91
130
2600
2821
Indeed, when we do the problem this
way we see an “amazing” connection
between multiplication and division.
217
13
13
2,821
× 217
2 600
91
221
130
130
2600
2
91
2821
note
91
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We put in the 0’s for emphasis. That is, because the 6 in 26 was placed directly under
the 8, and since the 8 was already holding the hundreds place, we did not need the
two 0’s as place holders. We are not advocating that the above format replace the
usual long division algorithm, but rather that by using the above format a “few”
times, the usual algorithm will seem more natural (or logical) to use.
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Notice that when written in this form
we see that the same three numbers we
add to get the product (namely 91, 130
and 2,600 are the same three numbers
(in the reverse order) we subtract to find
the quotient.
This illustrates quite vividly the
connection between division and
multiplication as inverse processes.
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Closing Notes on the
Adjective/Noun Theme
Students often tend to be careless when it
comes to keeping track of
place holders when doing division.
For example, given a
problem such as
2,613 ÷ 13 they will
often write.
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13
21
2,613
2 6
13
13
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With a little number sense they should
see that this answer is not even
plausible because 13 ×100 is only 1,300
and 13 × 21 is less than 13 × 100.
Hence 13 × 21 is much too small to be
equal to 2,613.
13
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21
2,613
2 6
13
13
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However, using our adjective/noun
theme, we see that…
13
1
20 0
2,613
2 600
13
13
200 thirteen’s
1 thirteen
201 thirteen’s
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division
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In our next
presentation, we
introduce the concept
of factors and
multiples.