Transcript Slide 1

Mathematics
as a
Second Language
An Innovative Way
to
Better Understand Arithmetic
by
Herbert I. Gross & Richard A. Medeiros
© 2006 Herbert I. Gross
next
3/4
Part 2
Fractions are
numbers, too
next
© 2006 Herbert I. Gross
Division
Rates
Common Fractions
next
© 2006 Herbert I. Gross
Two corn breads are to be
divided equally among 3 people.
How many corn breads does each
person get?
next
© 2006 Herbert I. Gross
Key Point
2÷3=?
Is by definition
another way of
saying
3×?=2
3×0=0
Therefore ? must be
greater than zero.
3×1=3
Therefore ? must be
less than one.
next
© 2006 Herbert I. Gross
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.
next
© 2006 Herbert I. Gross
Definition
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.
next
© 2006 Herbert I. Gross
Key Point
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.
next
© 2006 Herbert I. Gross
A rate usually appears as a phrase
that consists of two nouns
separated by the word “per”.
Example
6 apples ÷ 3 children = 2 apples per child
next
© 2006 Herbert I. Gross
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
next
next
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
next
© 2006 Herbert I. Gross
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”.
next
© 2006 Herbert I. Gross
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.
next
© 2006 Herbert I. Gross
While 2/3 still means 2 of what it takes 3 of, it
also answers the division problem 2 ÷ 3.
Special Note
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.
next
© 2006 Herbert I. Gross
As a check, notice that 3 × 2/3 =
3 × 2 thirds = 6 thirds = 2.
2 of 3
2 of 3
2 of 3
(where each color represents 2/3 of a corn
bread; that is 2 of what it takes 3 of to make a
corn bread)
next
© 2006 Herbert I. Gross
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 A
bread
corn B
bread
corn A
bread
corn B
bread
corn C
bread
corn C
bread
2 corn breads divided by 3 persons = 2/3 corn
breads per person.
D corn D
bread E
E corn bread
F
F
next
© 2006 Herbert I. Gross
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
next
© 2006 Herbert I. Gross
Geometric Version
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.
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.
next
© 2006 Herbert I. Gross
Pictorially
If the 7 people are named A, B, C, D, E, F, G,
we see that...
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
And since the pieces all have the same size, the
result may be rewritten as...
A A A A B B B
B C C C C D D
D D E E E E F
F F F G G G G
next
© 2006 Herbert I. Gross
A Note about Improper Fractions
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.
next
© 2006 Herbert I. Gross
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.
next
© 2006 Herbert I. Gross
Each corn bread is sliced into 4 equally sized
pieces, and each person receives
1 piece from each of the 5 corn breads.
A
A
A
A
A
Thus if one person is labeled A, A receives 5 of
what it takes 4 of to make a whole corn bread.
next
© 2006 Herbert I. Gross
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.
next
© 2006 Herbert I. Gross
We often prefer to write improper fractions
as mixed numbers.
A A A A
A
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.)
next
© 2006 Herbert I. Gross
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.
Problem ?
If it cost $3 to buy 5 pens, and the pens are
equally priced, how much did each pen cost?
next
© 2006 Herbert I. Gross
Solution
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.
$1
$1
$1
next
© 2006 Herbert I. Gross
Note
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.
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.
next
© 2006 Herbert I. Gross
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
next
© 2006 Herbert I. Gross
The corn bread represents $1, another name
which is 100 cents.
1/5
1/5 1 dollar
3/5
1/5
1/5
1/5
20cents
20cents
20cents
20cents
1/5 60
1/5
cents
1/5
1/5 20cents
1/5
100
cents
If the corn bread is sliced into 5 equally sized
pieces, each piece is 1/5 of the corn bread.
1/5 of 100 cents is 20 cents. Therefore, 3/5 of
the corn bread is 3 × 20 cents or 60 cents.
next
© 2006 Herbert I. Gross
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.
next
© 2006 Herbert I. Gross