Transcript next That is… - Adjective Noun Math

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Mathematics
as a
Second Language
Arithmetic Revisited
© 2010 Herb I. Gross
Developed by
Herb I. Gross and Richard A. Medeiros
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Lesson 2 Part 4
Whole Number
Arithmetic
Division or
“Unmultiplication”
© 2010 Herb I. Gross
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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.
In essence, a formula is a recipe that tells
us how to find the value of one quantity
once we know the value of one or more
other quantities.
© 2010 Herb I. Gross
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For example, we all know that there are
12 inches in a foot. Thus, to convert feet to
inches, we simply multiply the number of
feet by 12.
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.
© 2010 Herb I. Gross
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In terms of a more verbal description,
the “recipe”
(which we usually refer to as a formula)
may be stated as follows…
Step 1:
Start with the number of feet.
Step 2:
Multiply by 12.
Step 3:
The product is the number of
inches.
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Thus, if we start with 5 feet, the recipe
tells us that…
Step 1:
Start with 5 feet.
Step 2:
Multiply by 12 (inches per foot).
Step 3:
The product is 60 (inches).
© 2010 Herb I. Gross
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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 by 12
to get the number of inches.
On the other hand, if we had started with 60
inches and wanted to know the number of
feet with which we started, we would have
had to realize that we obtained 60 inches
after we multiplied the number of feet by 12.
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That is, in terms of “fill-in-the-blank”,
the computation is…
60 inches = 12 × _____ feet
Since we obtained 60 after we multiplied
by 12, to find out what number we
had before we multiplied by 12, we have
to “unmultiply” by 12 to determine
the number of feet.
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In more familiar terms, to convert 60 inches
into feet, we divide 60 by 12.
60 inches ÷ 12 = _____ feet
In this context, “divide” is the more
formal term for “unmultiply”.1
note
1 Our
own experience seems to indicate that students learn to internalize division
better once they connect it with “unmultiplication. For example, they seem to better
understand the meaning of 2,600 ÷ 13 = ___, if we paraphrase the problem into the
equivalent form 13 × ____ = 2,600.
© 2010 Herb I. Gross
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An Algebraic “Digression”
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 (because
it suggests inches) represent the number of
inches. The equal sign replaces the word
“is”.
© 2010 Herb I. Gross
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An Algebraic “Digression”
That is…
English
Step 1:
Start with the number
of feet.
Step 2:
Multiply by 12.
Step 3:
The product is the
number of inches.
© 2010 Herb I. Gross
Algebra
F
12 × F (or F × 12)
12 × F = I
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An Algebraic “Digression”
12 × F = I
Because the “times sign” ( × ) and the
letter x are easy to confuse (especially when
we’re writing by hand), we do not use the
“times sign” in algebra. That is, 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 much more
natural it is to say such things as “I have 12
dollars” as opposed to “I have dollars 12”).
© 2010 Herb I. Gross
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An Algebraic “Digression”
12F = I
It is also traditional to write the letter that
stands by itself (namely, the “answer”), on
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 recipe.
© 2010 Herb I. Gross
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Revisiting the Division Algorithm
for Whole Numbers
Again, we assume that you know the
standard algorithm for performing long
division. What is not always as clear to
students is that division is really a form of
repeated, rapid subtraction.
Let’s look at a typical kind of problem that
one uses to illustrate a division problem.
© 2010 Herb I. Gross
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The Problem
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?
© 2010 Herb I. Gross
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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…
1 3
–
2 1 7
2 8 2 1
–
2
–
9
0
© 2010 Herb I. Gross
1 3
× 2
2 6
1 3
× 1
1 3
1 3
× 7
9 1
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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!
More specifically, the “fill in the blank”
problem we are being asked to solve is…
13 × _____ = 2821
© 2010 Herb I. Gross
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Since 13 × 2 = 26, our adjective/noun
theme tells us that…
13 × 2 hundred = 26 hundred
… or in the language of place value…
13 × 200 = 2600
Hence, after we have packed 200 cartons,
we have packed 2,600 books.2
note
since 13 × 300 = 3,900 and there are only 2,821 books, we know that we will
need fewer than 300 cartons. Thus, we already know that we will need more than
200 but less than 300 cartons.
2And
© 2010 Herb I. Gross
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Since we started with 2,821 books and
2,600 books have now been packed, there
are now 2,821 – 2,600 or 221 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.
© 2010 Herb I. Gross
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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.
© 2010 Herb I. Gross
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We can now write the result in
tabulated form as shown below.
2 8 2
– 2 6 0
2 2
– 1 3
9
– 9
1 books
0 books
1 books left
0 books
1 books left
1 books
0 books left
© 2010 Herb I. Gross
200 cartons
10 cartons
+ 7 cartons
217 cartons
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In our opinion, repeated, rapid
subtraction is easier to understand.
However, the “traditional” recipe appears
to be more familiar.
2
1 3
2 8
–2 6
2
– 1
1 7
2 1
2
3
9 1
– 9 1
0
© 2010 Herb I. Gross
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However, if we simply insert the “missing
digits” into the above format
(which are emphasized in color),
2 8 2
– 2 6 0
2 2
– 1 3
9
– 9
7
1
0
1
0
1
1
0
0
we see that the numbers in the emended
diagrams look exactly the same.
© 2010 Herb I. Gross
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0
1
0
1
1
2
1 3
2 8
–2 6
2
– 1
1
2
0
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9
– 9
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Note
In terms of
having students
see the above
transition more
gradually, we
might want to
introduce the
intermediate
representation…
© 2010 Herb I. Gross
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1 0
2 0 0
1 3
2 8 2 1
– 2 6 0 0
2 2 1
– 1 3 0
9 1
– 9 1
0
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In checking a division problem,
Note 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.
More specifically…
2 1 7
× 1 3
6 5 1
2 1 7
2 8 2 1
© 2010 Herb I. Gross
Notice, however, that when we
Note perform the multiplication this
way, the rows in the above
multiplication (namely, 651 and 2,170) have
no relationship to the three rows in the
traditional division algorithm (namely,
2,600, 130 and 91).
2 1 7
2 1 7
1 3
2 8 2 1
× 1 3
–2 6 0 0
1
2
2
6 5 1
– 1 3 0
2 1 7 0
9 1
– 9 1
2 8 2 1
0
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© 2010 Herb I. Gross
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The reason is that the division
Note
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 above check by
multiplication showed that if
there were 217 books in a carton,
then 13 cartons would have to be used.3
note
3In
other words, the multiplication problem we did was 13 × 217. However, to check
the division problem, the multiplication problem should have been 217 × 13.
© 2010 Herb I. Gross
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However, a more enlightening
Note
way to perform the check is to
compute the product in the less
traditional but more accurate form 217 × 13.
Indeed, when we do the problem that way
we see an “amazing” connection
between multiplication
1 3
and division.
× 2 1 7
Namely…
© 2010 Herb I. Gross
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1 3
2 6 0
2 8 2
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0
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Notice that when written in this form we
see that the same three numbers we are
adding to get the product (namely 91, 130
and 2,600 are the same three numbers
(in the reverse order) we subtracted to find
the quotient.
2 1 7
1 3
1 3
2 8 2 1
× 2 1 7
–2 6 0 0
2 2 1
9 1
– 1 3 0
1 3 0
9 1
2 6 0 0
– 9 1
2 8 2 1
0
© 2010 Herb I. Gross
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×
÷
This illustrates quite vividly the
connection between division and
multiplication as inverse processes.4
note
4The
reason for using the traditional check is that the fewer digits in the bottom
number, the fewer rows we will need to compute the product, thus saving paper. Yet
this format, inspired by economic frugality, obscures the connection that illustrates
why we often refer to division as being the inverse of multiplication.
© 2010 Herb I. Gross
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Beware of the Missing Zero
When students memorize the traditional
long division recipe, they often have trouble
with what we might call the missing 0.
For example, suppose instead of 2,821
books, there were 2,652 books. In this
case, the number of cartons would have
been 2,652 ÷ 13. However, in performing
the division the following problem arises.
© 2010 Herb I. Gross
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Namely, the division proceeds normally
until they get to…
The tendency is then to
observe that since 13
doesn’t “go into” 5,
the student brings
down the next digit (2)
and then finishes the
problem as follows…
© 2010 Herb I. Gross
2 4
1 3
2 6 5 2
–2 6
– 5 2
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If students understood the concept of
“unmultiplying”, they would see that it is
impossible that 2652 ÷ 13 = 24
(or equivalently that it is impossible that
13 × 24 = 2652).
Namely, 100 thirteen’s is 1300;
therefore, 24 thirteen’s is less than 1300.
More symbolically,
13 × 24 < 1300 < 2652
© 2010 Herb I. Gross
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The “trick” is that
every time we “bring
down” a digit from the
dividend, we have to
place a digit above it in
the quotient.
Hence, the correct
computation would be…
2 0 4
1 3
2 6 5 2
–2 6
– 0
5
– 5 2
0
Since 200 × 13 = 2600, we see that
204 is a “reasonable” answer.
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This problem wouldn’t occur if
students used our adjective/noun theme.
More specifically…
2 0 4
0
1 3
© 2010 Herb I. Gross
2 6 5
–2 6 0
5
– 5
2
0
2
2
0
2 0 0 thirteen’s
+ 4 thirteen’s
2 0 4 thirteen’s
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In any event, this concludes our
discussion of the role of place value
in whole number arithmetic.
Our next endeavor will be to show
that a knowledge of whole number
arithmetic, together with our
adjective/noun theme, is all we need
in order to do the arithmetic
associated with fractions.
© 2010 Herb I. Gross