Taking the Fear
out of Math
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#6T
Comparing
Fractions
7
15
2
3
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Preface
Once we have internalized the meaning of
a common fraction, there are times when
we can tell, simply by looking, which of
two fractions is greater.
For example, students who have not
internalized the definition often believe
that 7/15 is greater than 2/3 because 7 is
greater than 2, and 15 is greater than 3.
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However, if we have internalized what a
common fraction means we see at once
that 7/15 is less than half while 2/3 is more
than half.
Classroom Note
In terms of manipulatives that are easy for
even the youngest students to understand,
imagine that there are 15 pencils on the
table and they take 7 of them (that is, they
have taken 7 pencils).
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They will be able to see that there are
more pencils left (8) than the number of
pieces they have taken (7). So they have
taken less than half of the pencils
7 pencils
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8 pencils
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And if there are 3 pencils on the table
and they take 2 of them (that is, they have
taken 2/3 of the pencils), they have taken
more than they have left on the table and
hence they have taken more than half of
the pencils.
2 pencils
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1 pencil
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However, there are times when it is not as
easy to determine by sight which of the
two fractions names the greater amount.
For example, while it is relatively easy to
see that both 2/5 and 3/8 are less than 1/2,
it is not as easy to see which of the two
fractions is the greater one. So we will
soon develop a technique that works
all the time.
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Rational Numbers
The study of common fractions begins
with what happens when we divide
two whole numbers.
In such a situation, the quotient might be a
whole number, but it doesn’t have to be.
So to extend the whole number system,
we agree to define the quotient of two
whole numbers (provided the divisor
is not 0) to be a rational number.
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Rational Numbers
Thus, every whole number is a
rational number (for example, 6 is also
6 ÷ 1) but not every rational number
is a whole number.
Although 5 ÷ 3 is not a whole number, it is
a rational number which we may denote by
the common fraction 5/3.
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In the previous lesson, we introduced
the concept of unit fractions.
From a non-mathematical
point of view, notice that the
two statements “John is
taller than Bill” and “Bill is
shorter than John” do not
sound the same.
Yet, both say the same thing
from a different point of
emphasis.
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The same relationship exists between
non zero whole numbers and a special set
of common fractions called unit fractions.
We “complement” the set of numbers
1, 2, 3, 4, 5, 6 etc., with a new set of
numbers, 1/1, 1/2, 1/3, 1/4, 1/5, 1/6 etc. To see
the connection between these two sets of
numbers visually, consider the “corn
bread” (i.e., rectangle) that appears below.
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Looking at the rectangle, it is impossible to
tell whether we started with the smaller
rectangle and marked it off 3 times to
obtain the larger rectangle, or whether we
started with the larger rectangle and divided
it into 3 smaller pieces of equal size.
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In terms of our corn bread model,
if we put in the units, we do not confuse
1 piece with 1 corn bread.
1 Corn Bread
1 piece
However, once the units are omitted it is
impossible to know without being
told what the 1 is modifying.
1
1
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For example, if the adjective 1 is
modifying “piece”, it means that the
adjective 3 is modifying
the (whole) “corn bread”
1 piece 1 piece 1 piece
Corn Bread
3 pieces
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However, if we started with the
corn bread as our unit and used the
adjective 1 to modify it, then each piece of
the corn bread is modified by the adjective
1/ because each piece is 1 of what it takes 3
3
of to equal the whole corn bread.
1/ Corn1/Bread1/
3
3
3
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Notes
What we’re calling our “corn breads” can
represent any amount.
Thus, if our “corn bread” represented $24,
1/ of the “corn bread” would represent $8
3
because $24 ÷ 3 = $8.
The fact that 3 × 8 = 24 allows us to say
that 24 is 3 times 8 and equivalently
that 8 is 1/3 of 24.
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Notes
Thus, in much the same way that “John is
taller than Bill”, and “Bill is shorter than
John” are two different ways of transmitting
the same information, “28 ÷ 4 = 7” and
“1/4 of 28 = 7 ” are also two different
ways of transmitting the same information.1
note
We should be cautious here. 4 pens at $7 each is not the same “event”
as 7 pens at $4 each even though the cost is the same in both cases.
In this context, 1/4 of 28 = 7 is an answer to 28 ÷ 4 = ___.
And 1/7 of 28 = 4 is an answer to 28 ÷ 7 = ___. Hence, 1/4 of 28 = 7 and
1/ of 28 = 4 are both correct ways of saying that the product of 4 and 7 is 28.
7
1
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An ordinary ruler, marked off in inches,
uses other unit fractions as well.
For example, if the ruler is marked off in
eighths of an inch. We can use 1/8 as a unit
fraction, meaning that any measurement
on the ruler can be given in terms of the
number of inches and also in terms of the
number of 1/8 of an inch.
1/
8
2/
8
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3/
8
41
5/
/8inch
8
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8/
8
=1
6/
8
7/
8
8/
8
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That is, just as we can count 1, 2, 3, 4, 5
etc., we can also count, 1 eighth, 2 eighths,
3 eighths, 4 eighths, 5 eights, etc., or in
more mathematical terms, 1 × 1/8 , 2 × 1/8 ,
3 × 1/8 , 4 × 1/8 , 5 × 1/8 , etc.
In this context, as shown in the figure below,
the shaded region which we call 5/8 is the
same size as 5 × 1/8. In this form, it is easy
to see why 5 is the called the numerator (it
counts the number of pieces) and why the
denominator is referred to as “eighths”.
1/
8
1/
8
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1/
8
1/
8
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1/
8
1/
8
1/
8
1/
8
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Notes
There is a tendency to define 5/8 by saying
we divide the unit into 8 pieces and then
take 5 of the pieces.
This leads to a conceptual problem when
we define 9/8. Namely, if there are only 8
pieces, then you can’t take 9 of them.
However, you can take 9 of what it would
take 8 of to make the whole.
1/
8
2/
8
3/
8
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41
5/
/8inch
8
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6/
8
7/
8
8/
8
9/
8
10/
8
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Historical Geometrical Note
The ancient Greek mathematicians knew
how to divide a line segment into any
number of pieces of equal length2.
Viewing the original length as representing
1 unit, and assuming that the segment was
divided into n pieces of equal length, each
of the smaller pieces was named by the
1/ .
unit
fraction
n
note
2 The concept of dividing a line segment into any number of pieces of equal length
gave rise to the concept of the number line. In this context, we may think of
the corn breads we’ve drawn as being “thick” number lines. Thus when we
talk about dividing the corn bread into 5 pieces of equal size, it corresponds to
the ancient Greeks dividing a line segment into 5 pieces of equal length.
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Historical Geometrical Note
The ancient Egyptians were enamored with
unit fractions, and they were particularly
interested in expressing all common
fractions as sums of unit fractions.
For example, they were intrigued by such
facts as 5/6 could be written as the sum of
the two unit fractions, 1/2 and 1/3 (1/2 + 1/3 =
5/ ) and in that context they viewed unit
6
fractions as the “building blocks” for
obtaining all common fractions.
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Anecdotal Aside
This is an interesting way of illustrating
that 1 + 1 can equal 5 if the numbers are
not modifying the same noun.
More specifically…
1 half + 1 third = 5 sixths
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Comparing Fractions
1
8
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1
5
In our next presentation,
we will discuss
equivalent fractions.
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