Transcript Adjectives that modify Nouns

Taking the Fear
out of Math
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#7
Multiplying
Mixed
Numbers
3 13
1
×2 2
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Let’s again begin by using
a “real world” example…
How much will you have to
pay for candy in order to buy
21/2 pounds, if the candy
costs $4.50 per pound?
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Stated in its present form, the problem
is a “simple” arithmetic problem. Namely,
at $4.50 per pound, 2 pounds would cost $9
and a half pound would cost half of $4.50 or
$2.25. Therefore, the total cost is $11.25.
Notice that in the language of mixed
numbers, we have found the answer to…
21/2 pounds × 41/2 dollars per pound.
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Notice that because the plus sign is
“missing”, it is easy to overlook the fact
that when we multiply 21/2 by 41/2 , we must
use the distributive property.
For example, there is a tendency by students
to multiply the two whole numbers to get 8
and the two fractions to get 1/4.1
However, since 21/2 is greater than 2, we
know that 21/2 × 4 1/2 is greater than
2 × 41/2, and since 2 × 41/2 = 9, we know
1/ × 41/ is greater than 9.
that
2
2
2
note
1 Learning by rote presents a tendency to confuse how we multiply two mixed
numbers with how we add two mixed numbers.
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Notes
We have multiplied 21/2 pounds by 41/2
dollars per pound and obtained 111/4
dollars as the product.
► This validates our observation that the
product of 21/2 and 41/2 is greater than 9.
► In addition to the fact that our result
validates that the product is greater than 9,
we have learned to use the distributive
property to get the exact answer
(41/2 × 21/2 = 111/4).
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Notes
► Notice that if you were the store owner
and believed that 21/2 × 41/2 = $81/4, you
would have “short changed” yourself
by $3 on this transaction.
► So even if it does seem “natural” or
“logical” to multiply the whole numbers and
multiply the fractions; it just doesn’t work
in the real world!
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Whether it is more difficult to multiply
the “correct” way is not the issue.
The issue is that if we want to multiply
mixed numbers, we have to pay attention
to the distributive property.
(21/2 × 41/2) = (2 × 4) + (2 × 1/2) + (1/2 × 4) + (1/2 × 1/2)
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=
8
=
111/4
+
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1
+
2
+
1/
4
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In schematic form, we may represent
the distributive property as follows…
2
+
1/
2
×
×
×
4
+
1/
2
2
1
1/
4
8
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Notice that the area model may be used
to help visualize the distributive property
(just as we did in our discussion of
whole number multiplication).
1/
2
4
2
1/
2
2×4=8
4×
1/
2
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1/
=2
10
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9
2 × 1 /2 = 1
+
2
×
1/
2
=
11/4
1/
4
+ 21/4
= 111/4
next
1/
2
4
2
2×4=8
1/
2
4 × 1 /2 = 2
2 × 1 /2 = 1
1/
2
× 1 /2 = 1 / 4
The “big” rectangle has dimensions
41/2 by 21/2.
Its area is the sum of the areas of the
four smaller rectangles inside. That is, the
total area is 8 + 1 + 2 + 1/4 = 111/4.
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1/
2
4
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2
2×4=8
1/
2
4 × 1 /2 = 2
2 × 1 /2 = 1
1/
2
× 1 /2 = 1 / 4
The shaded rectangles show the region
that’s represented by (4 × 2) + (1/2 × 1/2);
which is the region that is represented by
when we say “multiply the whole numbers
and multiply the two fractions” (the regions
in white represent the error in computing
(4 × 1/2) + (2× 1/2) in this way).
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If we prefer not to use the distributive
property, we may convert both mixed
numbers to improper fractions and solve
the problem that way. In other words…
21/2 × 41/2 = 5/2 × 9/2
=
=
(5×9)/
(2×2)
45/
4
= 111/4
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To generalize the above result, the recipe
for computing the product of two mixed
numbers by using improper fractions is…
Rewrite the mixed number problem as an
equivalent improper fraction problem.
Solve the resulting improper fraction problem.
Convert the answer from an improper fraction
into a mixed number.
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Notes
It is a good idea to have students
estimate the answer even before they do
the actual computation.
In this case…
4 < 41/2 < 5
×
2 < 21/2 < 3
8 < ?
< 15
Thus, any answer that is 8 or less or is 15
or greater must be incorrect.
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Adjective/Noun
Remember that the actual arithmetic
involves only the adjectives. The noun that
the answer modifies has to be determined
by the actual problem.
That is, there are many “real world”
problems that can be solved by knowing
that 41/2 × 21/2 = 111/4,
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Adjective/Noun
For example…
If the problem was to find the area of
a rectangle whose length is 41/2 inches
and whose width is 21/2 inches,
the answer would be 111/4 square inches.
4 1/2 inches
2 1/2 inches
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111/4 square inches
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Our example then would be…
41/2 inches × 21/2 inches
= (41/2 × 21/2) inch-inches
= 111/4 inches2
= 111/4 square inches
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Applying the same adjective/noun
theme to another example…
If an object moved at a constant speed of
41/2 miles per hour for 21/2 hours, it would
travel 111/4 miles during this time.
4 1/2 miles/hour
2 1/2 hours
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111/4 miles
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62/3 ÷ 33/4 = ?
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In our next section we
will discuss the process
of dividing one mixed
number by another
mixed number.
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