Multiplying Mixed Numbers © Math As A Second Language All Rights Reserved next #7 Taking the Fear out of Math 1313 3 1212 2 ×

MultiplyingMixed

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MultiplyingMixed

Numbers© Math As A Second Language All Rights Reserved

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#7

Taking the Fearout of Math

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Let’s again begin by using a “real world” example…

© Math As A Second Language All Rights Reserved

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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 $9and 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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next 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, weknow that 21/2 × 4 1/2 is greater than

2 × 41/2, and since 2 × 41/2 = 9, we know that 21/2 × 41/2 is greater than 9.

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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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► This validates our observation that the product of 21/2 and 41/2 is greater than 9.


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We have multiplied 21/2 pounds by 41/2 dollars per pound and obtained 111/4

dollars as the product.

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► 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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► 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.


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► 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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next 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)

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= (2 × 4) + (2 × 1/2) + (1/2 × 4) + (1/2 × 1/2)

= 8 + 1 + 2 + 1/4

= 111/4

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next In schematic form, we may represent the distributive property as follows…

2

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1/2

1/4

+

×

4

8

×

1/2+

×

12

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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).

2

1/2

4 1/2

2 × 4 = 8 2 × 1/2 = 1

4 × 1/2 = 2 1/2 × 1/2 = 1/4

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9

+ 21/4

10 + 11/4 111/4=

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The “big” rectangle has dimensions 41/2 by 21/2.

2

1/2

4 1/2

2 × 4 = 8 2 × 1/2 = 1

4 × 1/2 = 2 1/2 × 1/2 = 1/4

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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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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).

2

1/2

4 1/2

2 × 4 = 8 2 × 1/2 = 1

4 × 1/2 = 2 1/2 × 1/2 = 1/4

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next 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

= 111/4

= (5×9)/(2×2)

= 45/4

To generalize the above result, the recipe for computing the product of two mixed

numbers by using improper fractions is…


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Convert the answer from an improper fraction into a mixed number.

Solve the resulting improper fraction problem.

Rewrite the mixed number problem as an equivalent improper fraction problem.

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In this case…


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It is a good idea to have studentsestimate the answer even before they do

the actual computation.

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4 < 41/2 < 5

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2 < 21/2 < 3×

8 < ? < 15Thus, any answer that is 8 or less or is 15

or greater must be incorrect.

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Remember that the actual arithmetic involves only the adjectives. The noun that the answer modifies has to be determined

by the actual problem.


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That is, there are many “real world” problems that can be solved by knowing

that 41/2 × 21/2 = 111/4,

Adjective/Noun


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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.

Adjective/Noun

4 1/2 inches

111/4 square inches2 1/2 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

Applying the same adjective/noun theme to another example…


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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

111/4 miles2 1/2 hours

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In our next section we will discuss the process

of dividing one mixed number by another

mixed number.


62/3 ÷ 33/4 = ?

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Multiplying Mixed Numbers © Math As A Second Language All Rights Reserved next #7 Taking the Fear out of Math 1313 3 1212 2 ×