Transcript Fractions Are Numbers Too: Part 5

Mathematics
as a
Second Language
An Innovative Way
to
Better Understand Arithmetic
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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Part 5
Fractions are
numbers, too
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© 2007 Herbert I. Gross
Multiplying
X
Common Fractions
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© 2007 Herbert I. Gross
There is a close connection between how we
multiply fractions and how we take a fractional
part of a fractional part.
Based on the explanation that is given in
many text books, an expression such as
2/5 x 3/8 would be read as “ 2/5 of 3/8”.
One drawback of using the number line as a
model for multiplication is that it makes it difficult
to see the connection between “of” and “times”.
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© 2007 Herbert I. Gross
However, using the 2-dimensional property of
the corn bread, we may begin slicing the corn
bread vertically into 8 pieces of equal size (so
we can take 3/8 of it).
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
corn bread
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© 2007 Herbert I. Gross
And to anticipate taking 2/5 of 3/8 we might
also want the same corn bread to be sliced
horizontally into 5 equally sized pieces.
1/5
1/5
corn1/5bread
1/5
1/5
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© 2007 Herbert I. Gross
The first three columns is 3/8 of the corn bread.
The first two rows is 2/5 of the corn bread.
3/8
2/5
The shaded region shows 2/5 of 3/8.
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© 2007 Herbert I. Gross
However, this figure gives us a clue as to why
“times” also has the same meaning as it does
when we multiply whole numbers. Namely we
may view the whole corn bread in the figure as
a rectangle whose dimensions are 8 eighths
by 5 fifths.
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© 2007 Herbert I. Gross
Thus the shaded region is a rectangle whose
length is 3 eighths and whose width is 2 fifths.
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© 2007 Herbert I. Gross
In other words, the area of the shaded region,
on one hand is…
2 fifths × 3 eighths… or…
2/5 × 3/8
while on the other hand it is…
2 /5 of 3/8
Hence:
2/5 × 3/8 = 2 /5 of 3/8
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© 2007 Herbert I. Gross
We can see that the shaded region in the figure
consists of 6 of the 40 “little squares” into which
the corn bread has been divided.
1
2
3
4
5
6
Thus we also say that…
2/5 × 3/8 = 2 /5 of 3/8 = 6/40
© 2007 Herbert I. Gross
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If we look at the shaded rectangle below and
omit the nouns, the rectangle is 3 units by 2
units. Thus the adjective part of the area is 6
3
2
1
2
3
4
5
6
However, what the 6 modifies depends on the
units that are being used in the measurements.
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© 2007 Herbert I. Gross
Examples
3 inches × 2 inches = 6 “inch inches”
= 6 inches² = 6 square inches
3 inches × 2 feet = 6 “inch feet”
(An “inch foot” may seem unfamiliar, but it simply
represents the area of a rectangle whose dimensions
are 1 inch by 1 foot.)
3 kilowatts × 2 hours = 6 kilowatt hours
3 eighths × 2 fifths = 6 “eighth fifths”
3 eighths × 2 fifths = 6 fortieths
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© 2007 Herbert I. Gross
To visualize why 1 “eighth fifths” = 1 fortieth,
look at the diagram below. The shaded region
below illustrates why 1 eighth × 1 fifth
= 1 fortieth.
eighths
1
fifths
40
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© 2007 Herbert I. Gross
Conceptually 3/8 of 2/5 and 2/5 of 3/8 appear to
be different. However, notice that the shaded
region can be viewed as being either 2/5 of 3/8
of the corn bread or as 3/8 of 2/5 of the corn
bread.
Key Point
In terms of area, the area of a rectangle
remains unchanged if we interchange its base
and height.
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© 2007 Herbert I. Gross
In other words, the area of the shaded region
can be represented by…
3/8
2 /5
1
2
3
4
5
6
3/8 × 2/5
© 2007 Herbert I. Gross
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2 /5
or…
3/8
1
2
3
4
5
6
2/5 × 3/8.
© 2007 Herbert I. Gross
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Key Point
In terms of our adjective / noun theme, the
numerators are the adjectives, and the
denominators are the nouns. Hence to
find the product of two common fractions,
we multiply the two numerators to obtain
the numerator of the product, and we
multiply the two denominators to find the
denominator of the product.
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© 2007 Herbert I. Gross
Important Note
The above discussion might seem like a round
about way to describe what could have been
described in just one sentence, namely…
To find the product of two common fractions,
we multiply the two numerators to obtain the
numerator of the product, and we multiply the
two denominators to find the denominator of
the product.
Example
3/8 × 2/5 = (3 × 2) / (8× 5) = 6/40
© 2007 Herbert I. Gross
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However, the “mechanical rule” while easy
to use, gives no insight as to what the
product means. Moreover, if one thinks
it’s logical to “multiply numerators and
multiply denominators”, it is then not too
surprising that one also thinks it is logical
to add fractions by “adding the numerators
and adding the denominators”; a
procedure that is easy to do, but which
yields an incorrect answer (unless we are
trying to compute a weighted average).
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© 2007 Herbert I. Gross
Using
the Corn Bread and
Number Line
While the corn bread, makes it easier to see
the connection between “of” and “times”, the
number line is also convenient to use if we
want to find a fractional part of a fractional part.
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© 2007 Herbert I. Gross
Illustrative Example
How much is 3/7 of 4/5?
Answer
12/35
If all we want is the correct answer, we
need only use the facts that…
3/7 of 4/5 = 3/7 x 4/5 = (3 × 4)/(7 × 5) = 12/35
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© 2007 Herbert I. Gross
Suppose we weren’t aware of the above
“recipe”. We can visualize our corn bread as
being in one piece.
To take 4/5 of a number tells us that it would be
helpful if the number was divisible by 5; and to
take 3/7 of a number tells us that it would be
helpful if the number was divisible by 7.
5, 10, 15, 20, 25, 30, 35,
35 40
7, 14, 21, 28, 35
35, 42, 49, 56
35 would be a common multiple.
© 2007 Herbert I. Gross
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Since 35 is a common multiple of 7 and 5, we
can imagine that the corn bread is pre-sliced
into 35 pieces of equal size.
35
corn bread
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© 2007 Herbert I. Gross
12
0
1/7
28 35
2/7
3/7
4/7
5/7
6/7
7/7
corn bread
0
1/5
2/5
3/5
4/5
3/7 of 4/5 of the corn bread =
3/7 of 4/5 of 35 pieces =
3/7 of (4/5 of 35 pieces) =
3/7 of 28 pieces =
12 pieces =
12/35 of the corn bread
© 2007 Herbert I. Gross
That is, 3/7 of 4/5 =12/35
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Note
Notice that as used in the preceding
slides, the corn bread is simply a
“thick” number line. That is, if we
think of a line segment that is 1 unit
long, we can subdivide it into 35
equally- sized segments and proceed
word for word as we did previously.
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© 2007 Herbert I. Gross
12
0
1/7
0
28 35
2/7
1/5
3/7
4/7
2/5
5/7
6/7
3/5
7/7
4/5
1
3/7 of 4/5 of the number line =
3/7 of 4/5 of 35 pieces =
3/7 of (4/5 of 35 pieces) =
3/7 of 28 pieces =
12 pieces =
12/35 of the number line
© 2007 Herbert I. Gross
That is, 3/7 of 4/5 =12/35
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Summary
It is rather cumbersome to divide the number
line into 35 equally sized segments. For that
reason it might be easier to visualize the 35
pieces if we use the corn bread instead of the
number line.
Namely, we can divide
the corn bread into 5
equally sized pieces
vertically…
corn bread
and we can divide the
corn bread into 7
equally sized pieces
horizontally.
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© 2007 Herbert I. Gross
If we shade the first 4 columns of the corn
bread, we obtain the area of 4/5 of the
corn bread.
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The first 3 rows of the shaded region
represent 3/7 of the previously shaded
region...
that is 3/7 of 4/5 of the corn bread.
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© 2007 Herbert I. Gross
The diagram illustrates graphically why
the answer is 12/35, and why we multiply
the numerators to get 12 and multiply the
denominators to get 35.
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© 2007 Herbert I. Gross