Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language © 2006 Herbert I. Gross An Innovative Way to Better

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?

© 2006 Herbert I. Grossnextnext

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.

© 2006 Herbert I. Grossnextnextnext

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

nextnext

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


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


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


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

nextnext

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

© 2006 Herbert I. Grossnext

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.


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

person”.


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.


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


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


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




A

A

C

B

B

C

D D E F FE

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


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


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.


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

© 2006 Herbert I. Grossnextnextnextnextnextnextnextnext

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


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.


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


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


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


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 ?


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


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.


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


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

© 2006 Herbert I. Grossnextnextnextnext

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.

© 2006 Herbert I. Grossnext

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Mathematics as a Second Language Mathematics as a Second Language Mathematics as a Second Language © 2006 Herbert I. Gross An Innovative Way to Better