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Making All Numbers Whole

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Making All Numbers

Whole

Making All Numbers

Whole

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Yes, Virginia

There Are No Fractions

Yes, Virginia

There Are No Fractions

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by

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

© 2009 Herbert I. Gross

Preface

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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.

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.

© 2009 Herbert I. Gross

Preface

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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.

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.

© 2009 Herbert I. Gross

Preface

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In this context, our approach is to show that once

students understand whole number arithmetic they also know the arithmetic of fractions.

How this is done is the subject

of Module 3.

© 2009 Herbert I. Gross

1 horse.

1 rabbit,

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

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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 numbers1.

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

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

Owner: Half and half;

© 2009 Herbert I. Gross

Definition

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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 notas numbers, but as quantities. By way of review …

A quantity is a phrase consisting of an adjective

and a noun.

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).

© 2009 Herbert I. Gross

1 rabbit,

1 horse.

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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.

In still other words,

1 “rabbit unit” is not the same as 1 “horse unit”.

© 2009 Herbert I. Gross

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Application to

The Language of Fractions

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

If this sounds a bit strange consider the following anecdote.

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.

© 2009 Herbert I. Gross

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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”.

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.

© 2009 Herbert I. Gross

The names for the unit fractions are,

halves, thirds, fourths, fifths, sixths, ...

and “nth’s”; where…

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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.

© 2009 Herbert I. Gross

The unit fractions “half”, “third”, etc.

are further symbolized as…

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1/2 is the symbol for a half.

1/3 is the symbolfor a third.

1/4 is the symbolfor a fourth.

1/5 is the symbolfor a fifth.

1/6 is the symbolfor a sixth.

1/n is the symbolfor an “nth”.

© 2009 Herbert I. Gross

Special Note

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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.

For example, if we’re

counting 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.

© 2009 Herbert I. Gross

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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.

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”.

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.

© 2009 Herbert I. Gross

Definition

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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 thedenominator(think of denomination which means size, a noun).

© 2009 Herbert I. Gross

Note

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The “tricky” part is that the greater the denominator, the smaller the size of each piece.

This can be remembered by the following joke…

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.

© 2009 Herbert I. Gross

Note

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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 the

whole into only 10 equally sized pieces.

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.

© 2009 Herbert I. Gross

Note

2If 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”

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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, thenoun is sevenths (not 7) where “sevenths” means 1 of what it takes 7 of to make the whole.2

© 2009 Herbert I. Gross

Note

3In 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”.

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

© 2009 Herbert I. Gross

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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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The Adjective/NounTheme

Concluding Remark

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.

© 2009 Herbert I. Gross