making all numbers whole making all numbers whole or yes, virginia there are no fractions yes,...

Making All Numbers Whole

Making All Numbers Whole

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Yes, VirginiaThere Are No Fractions

Yes, VirginiaThere Are No Fractions

byHerbert I. Gross, Judith Bender, & Richard A. Medeiros

© 2009 Herbert I. Gross

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Of all the topics that cause students (and, often, teachers as well) anxiety, perhaps the

understanding of fractions heads the list. Once the “fear” of fractions sets in, it casts a negative

pall on the rest of the students’ mathematical experiences.

Preface

On the other hand, at least at the computational level, most people do not have the same

problem when internalizing the arithmetic of whole numbers.


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Thus the cornerstone of our approach, which we call “Mathematics as a Second Language”, is that

by viewing numbers as adjectives that modify nouns, every problem that involves common

fractions can be transformed into an equivalent problem that involves only whole numbers.

Preface


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The goal of our approach is to help students perceive mathematics as a unified whole

whereby one topic flows in a seamless way from the previous ones.

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In this context, our approach is to show that oncestudents understand whole number arithmetic

they also know the arithmetic of fractions.

Preface


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How this is done is the subject of Module 3.

In particular, our approach converts any problem whose solution requires a

knowledge of fractions to an equivalent problem whose solution requires only a knowledge of

whole numbers 1.


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To see our approach from a non-threatening point of view, simply consider the anecdote below.

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Customer: How much horse meat do you use Customer: How much horse meat do you use when you make rabbit stew?when you make rabbit stew?

Owner: Half and half; Owner: Half and half;

1By way of review the whole numbers are 0, 1, 2, 3, etc…note

1 rabbit, 1 rabbit, 1 horse.1 horse.

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This joke emphasizes how we usually think of numbers as being adjectives that modify

nouns. For example, in talking about distance, we will say 3 inches,

3 feet, 3 meters, 3 miles etc, but never just “3” by itself. In this context, we see amounts not as

numbers, but as quantities. By way of review …


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A quantity is a phrase consisting of an adjective and a noun.

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Definition

The adjective is a number (in the above example, 3), and the noun is the unit (in the above example, inches, feet, meters, or miles).


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In the rabbit stew joke, with respect to the nouns “rabbit” and “horse”, the number of each (the adjective, 1) is the same, but the

quantities (of meat) are very different.

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In still other words, 1 “rabbit unit” is not the same as 1 “horse unit”.

1 horse.1 horse.1 rabbit, 1 rabbit,

A novel way to introduce fractions is by asking “What’s my noun?”

Application toApplication to


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The Language of FractionsThe Language of Fractions

Clerk: Do you want the pizza sliced into 6 pieces or 8 pieces?

Customer: Please cut it into 6 pieces because I can’t eat 8 pieces.

If this sounds a bit strange consider the following anecdote.

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To be more precise, the word “piece” as it is used in “6 pieces” means “1 of the 6 equally

sized pieces into which the pizza is cut”; while as it is used in “8 pieces”, it means “1 of the 8 equally sized pieces into which the (same) pizza is cut”.


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Of course, such phrases as “1 of the 6 equally sized pieces into which the

pizza is cut” and “1 of the 8 equally sized pieces into which the (same) pizza is cut” are

cumbersome to write. Hence, we use an abbreviation which we call

a unit fraction.


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A half means 1 of the 2 parts of equal size that equal the whole unit.

A third means 1 of the 3 parts of equal size that equal the whole unit.

A fourth means 1 of the 4 parts of equal size that equal the whole unit.

A fifth means 1 of the 5 parts of equal size that equal the whole unit.

A sixth means 1 of the 6 parts of equal size that equal the whole unit.

...An “nth” means 1 of the n parts of equal size that equal

the whole unit.

The names for the unit fractions are, halves, thirds, fourths, fifths, sixths, ...

and “nth’s”; where…

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1/2 is the symbol for a half.

1/3 is the symbol for a third.

1/4 is the symbol for a fourth.

1/5 is the symbol for a fifth.

1/6 is the symbol for a sixth.

1/n is the symbol for an “nth”.

The unit fractions “half”, “third”, etc. are further symbolized as…

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When we count in the usual way; that is, 1, 2, 3, 4, 5..., we are assuming that we know the

noun that these adjectives are modifying.


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For example, if we’recounting doughnuts, we do not say,1 doughnut, 2 doughnuts, 3 doughnuts... because we know

from the context that the noun is doughnuts. In a similar way we can count by halves, thirds,

fourths, fifths, etc.

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Special Note


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For example, if the whole (be it a pizza or anything else) is divided into 7 pieces

of equal size and we take 3 of these pieces, we represent this quantity by saying “3 sevenths of the pizza” and writing it as “3/7 of the pizza”.

Thus, for example, we might count 1 seventh, 2 sevenths, 3 sevenths, 4 sevenths, etc.

We may think of 3 sevenths as 3 × 1 seventh, and we abbreviate this as 3/7.

In this context, 3/7 is an adjective modifying “of the pizza”; and with respect to 3/7, 3 is the

adjective and sevenths is the noun.


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A number such as 3/7, which we read as 3 sevenths, is called a common fraction.

In terms of unit fractions, it is an abbreviation for 3 × 1/7.

The top number (in this case, 3) is called the numerator, and it tells us how many “pieces” we

are taking (think of the word “enumerate”which means to count; to count asks the question

“how many?” and “how many” is an adjective).

The bottom number (in this case, 7 but read as sevenths) tells us the size of each piece relative

to the whole. For that reason it is called the denominator (think of denomination which means

size, a noun).

The “tricky” part is that the greater the denominator, the smaller the size of each piece. This can be remembered by the following joke…


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Note

A man was so grateful to God for surviving a serious operation that he increased his donation to the church from one 10th of his salary to one

20th of his salary.

In other words, if we divide the whole into 20 equally sized pieces, each piece is smaller than it

would have been if we had divided thewhole into only 10 equally sized pieces.


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Note

In still other words, as the number of people who get a

piece of the same pie increases, the

smaller the size of each piece becomes.

In summary, a common fraction is itself a quantity in which the numerator (“top” number) is the

adjective and the denominator (“bottom” number) is the noun. More specifically, if the denominator

is 7, the noun is sevenths (not 7) where “sevenths” means 1 of what it takes 7 of to make

the whole.2


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Note

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2 If we only think of the word “numerator” as being another name for the word “top”, it would have been wiser to use the word “top” because most people already know what “top” means. A similar argument applies to “denominator” versus “bottom”

Although 3/7 is itself a quantity, it is most often used as the adjective part of another quantity 3.


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Note

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3 In a sense this is equivalent to what is called an adjective phrase in English grammar. For example, in the sentence

“She wore a dark red dress”, “dark red” is an adjective phrase in which the adjective “dark” is modifying the adjective “red” and together they form an adjective phrase that modifies the

noun “dress”.

For example, if a carton of books contains 35 books, saying 3/7 of a carton is another way of saying 3 sevenths of

35 books. In this way the carton of books plays the role of the “pizza”, and a book plays the role of a “piece of the pizza”. If we divide the carton of books

into 7 pieces of equal size, then each piece (that is, 1 seventh of the carton) represents 5 books; and therefore 3 sevenths of the carton represents

3 × 5 books, or 15 books.© 2009 Herbert I. Gross

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Note

Hopefully, this discussion has highlighted the importance of being able to internalize the arithmetic of whole numbers. With this in

mind, we conclude this part of our dialogue and will next turn our attention to showing how the

adjective/noun theme gives us a unifying thread for understanding all of whole number arithmetic.


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TheThe Adjective/ Adjective/NounNoun ThemeTheme

Concluding RemarkConcluding Remark

making all numbers whole making all numbers whole or yes, virginia there are no fractions yes,...

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