Mathematics as a

Second Language

Mathematics as a

Second Language

Mathematics as a

Second Language

© 2006 Herbert I. Gross

An Innovative Way to

Better Understand Arithmeticby

Herbert I. Gross & Richard A. Medeiros

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1/23/45/6

7/8

9/10Fractions arenumbers, too

Part 2Part 2

next© 2006 Herbert I. Gross

DivisionDivision

RatesRates

Common FractionsCommon Fractionsnext

© 2006 Herbert I. Gross

Two corn breads are to be divided equally among 3 people.

How many corn breads does each person get?

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2 ÷ 3 = ?

Key Point

Is by definition another way of

saying3 × ? = 2

3 × 0 = 0 Therefore ? must be greater than zero.

3 × 1 = 3 Therefore ? must be less than one.

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Key Point

There are no whole numbers greater than 0 but less than 1. Yet it is just as logical to want to divide 2 corn

breads among 3 persons as it is to divide 6 corn breads among 3

persons.Hence to answer our question,

common fractions had to be invented.© 2006 Herbert I. Gross

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When one quantity is divided by another, the quotient (answer) is called a rate.

The words “rate” and “ratio” have the same origin. In this context a rational number is any number that can be obtained as the

quotient of two whole numbers. So while the quotient 2 ÷ 3 is not a whole number, it

is a rational number.

Definition

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Every whole number is a rational number (for example 6 = 6 ÷ 1,

12 ÷ 2, etc.), but not every rational number is a whole number.

In the language of sets, the whole numbers are a subset of the rational

numbers.

Key Point

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A rate usually appears as a phrase that consists of two nouns

separated by the word “per”.

6 apples ÷ 3 children = 2 apples per child

Example

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6 dollars ÷ 3 tickets = 2 dollars per ticket

6 miles ÷ 3 minutes = 2 miles per minute

6 students ÷ 3 teachers = 2 students per teacher

Note

In terms of the adjectives 6 ÷ 3 is always equal to 2. However, what noun the 2 modifies depends on what nouns the 6

and 3 are modifying.© 2006 Herbert I. Gross

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Now look at the connection between, say 2 ÷ 3 and 2/3. In

terms of the adjective/noun theme and “corn breads”, suppose there

are 2 corn breads to be shared equally among 3 persons.

corn bread corn bread

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Each of the corn breads can be sliced into 3 equally sized pieces, and thus paraphrasing

the problem into sharing 6 pieces of corn bread among 3 persons.

corn bread corn bread

In this case, 6 is divided by 3 to obtain 2 as the adjective and the noun is now “pieces per

person”.

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Therefore, each of the 3 persons receives 2 pieces of the corn bread. Since there are 3

pieces per corn bread each person receives 2 of what it takes 3 of to make the whole corn

bread.

This is the same 2/3 that was discussed in the previous presentation.

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While 2/3 still means 2 of what it takes 3 of, it also answers the division problem 2 ÷ 3.

While 2/3 means the same in both cases, there is a conceptual difference between dividing 1 corn bread into 3 equally sized pieces and taking 2 of these pieces; and dividing 2 corn breads equally among 3

people.

Special Note

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As a check, notice that 3 × 2/3 = 3 × 2 thirds = 6 thirds = 2.

(where each color represents 2/3 of a corn bread; that is 2 of what it takes 3 of to make a

corn bread)

2 of 3 2 of 3

2 of 3

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Just as 6 ÷ 3 = 2 is a relationship between 3 numbers, so also is 2 ÷ 3 = 2/3. And just as 6 corn breads divided by 3 persons = 2 corn

breads per person… corn bread corn bread

corn bread corn bread

corn bread corn bread

corn bread corn bread

A

A

C

B

B

C

D D E F FE

2 corn breads divided by 3 persons = 2/3 corn breads per person.

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This helps to explain why mathematicians use common fractions to represent division

problems.

For example, rather than write 4 ÷ 7, they will often write 4/7. Namely 4 ÷ 7 means the

number which when multiplied by 7 yields 4 as the product. That is…

7 × 4/7 = 7 × 4 sevenths = 28 sevenths (of a unit) = 28 of what it takes 7 of to make a unit

= 4 units

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In terms of the corn bread model, the numerator represents the number of corn breads, and the denominator represents the number of people who are sharing the corn breads. Thus 4/7 (4 ÷ 7) may be interpreted as sharing 4 corn

breads among 7 persons.

Geometric Version

In this case the corn bread is sliced into 7 equally sized pieces, and each person is

given one piece from each of the four corn breads.

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Pictorially

And since the pieces all have the same size, the result may be rewritten as...

A B C D E F G A B C D E F G

A B C D E F G A B C D E F G

If the 7 people are named A, B, C, D, E, F, G, we see that...

A A

AA

A A A A

B

B

B

B

B B B B

C

C

C

C

C C C C

D

D

D

D

D D

D D

E

E

E

E

E E E E

F

F

F

F

F F F F

G

G

G

G

G G G G

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A common fraction is called improper if the numerator is equal to or greater than the

denominator.

For example, 5/4 is called an improper fraction (as opposed to a proper fraction

which is a fraction in which the numerator is less than the denominator). It is the answer

to the division problem 5 ÷ 4.

A Note about Improper Fractions

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In terms of the corn bread model, improper fractions occur when we have more corn breads than persons to share these corn breads. In particular 5/4 is the amount of

corn breads each person receives if 5 corn breads are shared equally among 4 persons.

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Each corn bread is sliced into 4 equally sized pieces, and each person receives

1 piece from each of the 5 corn breads.

Thus if one person is labeled A, A receives 5 of what it takes 4 of to make a whole corn

bread.

A A A A A

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And since all 20 pieces have the same size…

A A A A A

the above figure may be rewritten in the form.

5 of what it takes four of to make the whole corn bread.

A A A AA

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We often prefer to write improper fractions as mixed numbers.

A mixed number is the sum of a whole number plus a proper fraction. As illustrated

in the diagram above, each person would receive 1 whole corn bread plus 1 piece from the remaining corn bread. (Mixed numbers will be discussed in a later presentation.)

A A A AA

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Let’s close this section with a typical example that shows in terms of division and

our adjective/noun theme that common fractions are just names for numbers.

If it cost $3 to buy 5 pens, and the pens are equally priced, how much did each pen cost?

Problem ?

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To ask the question a slightly different way, we are asked to find the rate “dollars per pen”.

That is “How much is 3 dollars ÷ 5 pens?”.

Based on the previous discussion, the answer is 3/5 dollars per pen.

Solution

$1 $1 $1

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If fractions had never been invented, it would be tempting to answer the question in terms of the rate “cents per pen”. In this case, 3 dollars

would have been rewritten as 300 cents; and the answer would have been

300 cents ÷ 5 pens or 60 cents per pen.

Note

60 cents and 3/5 of a dollar are equivalent ways of expressing the same amount. That is,

if we prefer to change the noun “cents” to “dollars”, 60 cents becomes 3/5 of a dollar.

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60 cents and 3/5 of a dollar are equivalent ways of expressing the same amount. That is,

if we prefer to change the noun “dollars” to “cents”, 3/5 of a dollar equals 3/5 of 100 cents

which in turns becomes 3 x (100 ÷ 5) or 60 cents.

This can be illustrated in terms of the corn bread model :

corn bread

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The corn bread represents $1, another name which is 100 cents.

1 dollar

100 cents

1/5 1/5 1/5 1/5 1/5

If the corn bread is sliced into 5 equally sized pieces, each piece is 1/5 of the corn bread.

1/5 1/5 1/5 1/5 1/5

1/5 of 100 cents is 20 cents. Therefore, 3/5 of the corn bread is 3 × 20 cents or 60 cents.

20cents 20cents 20cents 20cents 20cents

3/5

60 cents

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Key Point

If you are comfortable with the quantity “60 cents” but uncomfortable with the quantity “3/5 of a dollar”, it is

probably more of a language (vocabulary) problem than a

mathematics problem.

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