DICKINSON STATE UNIVERSITY
DICKINSON ND
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ELEMENTARY FRACTIONS
Some instructions and advice for teaching
Introduction
The materials presented below are not addressed to children, but to the persons
who want to teach children about fractions. In our opinion they present a bare
minimum of knowledge and understanding required for teaching this subiect.
Children are familiar with the idea of fractions even before they start school and
they may intuitively understand some simple fractions. After all they hear adults
use fractions such as 1/2 a pie, 1/3 cup of milk, 3/4 of a mile, 1/2 dozen
eggs and so on. However they do not understand fully the meaning of fractions.
Before children are taught about fractions they should be familiar at least with
natural numbers or even whole numbers and their addition, subtraction,
multiplication and to a certain extent division. For example they should realize
that dividing 10 by 4 gives 2 with a reminder of 2. More formally
10 : 4 = 2 x 4 + 2 .
It is also important for the teaching person to realize that the introduction of
whole number fractions automatically introduces the set of rational numbers and
this new set contains the set of whole numbers. All of it follows from the
definition of a rational number as the ratio m/n where m and
n are any two whole numbers with only one restriction. Namely, for intuitively
obvious reason, n must not be zero. Thus all rational numbers are fractions.
The upper portion of the fraction is called the numerator, whereas the lower part
is called the denominator. Every whole number m can be formally presented as
the fraction m/1. As a matter of fact we can make any number of different
fractions representing the same whole number. Not every fraction is a
rational number. For example  / 4 is a fraction,
but not a rational number.
This approach, however, will not work with children. It is much too abstract. We
have to explain them fractions by example using simple fraction interpretations.
Moreover, this should be limited to the rational fractions only.
What are simple interpretations of a rational fraction?
The square in Fig. 1 is divided into four equally sized parts, consequently each
of these parts is 1/4 of the square. If we treat the square as a unit or a
whole, then any of its parts is just 1/4. Then the fraction 3/4 tells
that we are considering three of these parts taken together, for example the three
parts colored in blue. Exactly the same scheme applies to the rectangle and circle
in Fig. 1. The rectangles in Fig. 2 illustrate the same kind
of interpretation. Considering the parts that are not yellow, we deal with 1/3
and 2/6 whereas for the yellow parts with 2/3 and 4/6
respectively. Both pairs represent equivalent fractions describing or symbolizing
the same rational number. Again, as for whole numbers, we can make any number of
fractional representations of a given rational fraction. For example 1/3
can be also represented as 3/9 or 4/12. These examples show that
simultaneous multiplication of the numerator and denominator of the fraction by
the same non-zero number changes the fraction representation, but it does not change
the rational number represented by the original fraction. This property is very
important if we want to compare, add or subtract fractions.
Fig. 1
Another possible interpretation of fractions can be illustrated with the twenty
small squares in Fig. 1. If we consider columns, this set is divided into four equal
groups of squares, five squares in each group (column). Thus, a single group constitutes
1/4 of the whole set, whereas all green squares constitute 3/4 of
the set. Using this interpretation we can operate only with a limited number of
fractions and the smallest fraction is one over the number of elements in the set.
Consequently it is less usefull than the formerly discussed interpretation.
Fig. 2
Comparing fractions
The easiest fractions to compare are fractions with the same denominator. Then the
fraction with the bigger numerator is obviously the bigger fraction. But
explaining it to children with a graphical representation like the one in Fig. 3
would not hurt.
Fig. 3
Comparing 5/8 with 6/10 presents a more complicated problem and even
adults have to think a moment to find the right answer. The graphical representation
of the problem shown in the upper part of Fig. 4 gives the answer instantly, but
it is not a practical method. The rational numbers 5/8 and 6/10
have many fractional representations. If we choose for both numbers their representations
with the the same denominator, formally called the common denominator, the problem will
be solved.
Fig. 4
The simplest, but not optimal, method to find such denominator is to multiply
the numerator and denominator of the first fraction by the denominator of the
second fraction and the numerator and denominator of the second fraction by the
denominator of the first fraction. In our example it replaces the original
fractions 5/8 and 6/10 by 50/80 and 48/80 respectively. The new
representation shows clearly that the first fraction is greater by 2/80.
This difference that can be represented as 1/40 is shown in the lower part
of Fig. 4.
Notice that the fraction 6/10 from our example can be reduced to its
simplest equivalent form 3/5. Repetition of the procedure from the previous
paragraph replaces the original fractions by 25/40 and 24/40.
Thus, the simplification of 6/10 simplified in turn the comparison
of both fractions showing right away that the difference between them is 1/40.
In this case we have achieved an optimal simplification of our analysis. But for
5/6 and 7/10 the simplification of the original fractions is not
possible and the simplest method of finding a common denominator is not good
enough for optimization. This method will replace the original fractions by
50/60 and 42/60 showing that the first fraction is larger by
8/60. Further simplification replaces this difference by the equivalent
fraction of 2/15.
The smallest common denominator for a set of fractions is called the least
common denominator. Representation of these fractions in the forms with the least
common denominator usually simplifies their comparison, addition and subtraction.
Before a detailed discussion about finding the least commom denominators we shall
talk about prime numbers. By definition every positive integer divisible
only by one and itself (without any reminder) is a prime number. Thus 1, 2, 3, 5,
7, 11, 13, 17, 19 are prime numbers, whereas 4, 6, 8, 9, 10, 12, 14, 15, 16,
18, 20 are not. Moreover, from the definition of prime numbers follows an
important property of non-prime positive integers. Namely each of them can be
factored and represented as the product of at least two prime numbers different
than 1. For example the non-prime numbers listed above can be presented as
2x2, 2x3, 2x2x2, 3x3, 2x7, 3x5,2x2x2x2, 2x3x3, 2x2x5 respectively.
Let us come back to our example with 5/6 and 7/10. The first
denominator 6 = 2x3 and the second 10 = 2x5. Both of them have
one common prime number 2, but the numbers 3 and 5 are not
common. Multiplying this common prime number by the prime numbers from both
denominators that are not common, we obtain 2x3x5 = 30 which is the
least common denominator for this case. To find a new numerator for this
denominator, the old numerator must be multiplied by all uncommon prime factors
of the other denominator. Applying this rule to the current example our
original fractions will be replaced by the equivalent fractions 25/30
and 21/30.
If instead of 10 the second
denominator is 20 = 2x2x5 then we have one common pair of twos and
another single 2 which is not common. Therefore multiplying the common
2 by the other noncommon prime factors we obtain the least common
denominator 2x2x3x5 = 60.
The method described above always produces the least common denominator.
First, the number created by the method is a common denominator because in the
process of its creation we take the first denominator multiplied by the noncommon
prime factors from the other denominator or vice-versa. Consequently we can
divide it without any reminder either by the first or/and by the second denominator.
To prove that it must be the least common denominator let us assume it is only
a common denominator but not the least common denominator. If this is true, then
we should be able to remove from it at least one prime factor and the
reminder would still be a common denominator. But if we do so, from the
construction of the original common denominator it follows that this remainder cannot
be divided by at least one of the denominators of the original fractions. Thus, our
assumption about the constructed common denominator as not the least common
denominator was false.
In concluding this section we may say that usually the best method to handle comparison,
addition or subtraction of fractions is to replace them, if possible, by their
simplest representations and find for these representations the least common
denominator. Replacement of a fraction by its simplest representation is typically
called the reduction to the lowest term. Here again we may use the prime numbers technique.
Consider the reduction of 18/30
18/30 = (2x3x3)/(2x3x5) = 3/5
Addition and subtraction of fractions
The most important methods useful for addition and subtraction were already discussed in
the preceding section. Here you will find a few new definitions and some examples.
If the fractional value is smaller than 1 but greater than -1,
the fraction is called a proper fraction. Otherwise the fraction is called
an improper fraction. Thus for any proper fraction made of positive integers its
denominator is always greater than the numerator. For the improper fraction of the
same sort, the numerator is either greater than or equal to the denominator. The
fractions 4/7, 7/9, 1/4, 2/4, -3/5 are proper fractions whereas 12/5, 5/5,
17/15, 10/5, -21/8 are improper fractions. Every improper fraction comprises
non-zero number of units and a proper fraction. For example 12/5 = 2 2/5 or
-21/8 = -2 5/8. The notation on the right sides of both examples is called
the mixed numbers notation because it mixes whole numbers and fractions. The fractions used
in this notation are usually proper fractions.
Using our knowledge about least common denominators let us try to add two proper
fractions 2/3 and 1/4 and visualize this process with the help of our
Fraction Addition and Subtraction Applet. After setting both fractions in the applet
we have to determine the least common denominator. In this case the first
denominator contains only one prime number 3 and the second contains
the prime number 2 twice. The least common denominator is 12. The result
of this choice is shown in Fig. 4 where the original fractions are visualized in the
narrow rectangles and their equivalent representations pertaining to the least
common denominator in the wide rectangles. The formal description of these rectangles
appears on the little blackboards on the right. The result after addition is shown in
Fig. 5. Formally this whole operation can be recodred as follows:
2/3 + 1/4 = 8/12 + 3/12 = (8 + 3)/12 = 11/12 .
Fig. 4
Fig. 5
The subtraction operation of these two fractions can be recorded as:
2/3 - 1/4 = 8/12 - 3/12 = (8 - 3)/12 = 5/12
and the visualization of the final result is shown in Fig. 6.
Fig. 6
In the former example, the two added proper fractions produced the proper fraction
too. But if we take 7/8 and 5/12 the result will be the improper
fraction. Indeed, after concluding that the least common denominator in this
case must be 24, we obtain
7/8 + 5/12 = 21/24 + 10/24 = (21 + 10)/24 = 31/24 = 1 7/24 .
This addition procedure is illustrated in Fig. 7 and Fig. 8.
Fig. 7
Fig. 8
Recall that a mixed number contains a whole number and a fraction. Also
recall that here we are only discussing here rational numbers. Thus it is
always possible to convert a mixed number with an improper fraction into
another mixed number with a proper fraction. In adding such mixed numbers
we may first add their whole parts and then add the fractional parts utilizing
the techniques described above. For example
3 7/4 + 2 17/5 = 4 3/4 + 5 2/5 = 4 + 5 + 3/4 + 2/5 =
9 + 15/20 + 8/20 = 9 + (15 + 8)/20 = 9 + 23/20 = 9 + 1 + 3/20 = 10 3/20
A similar technique may be used to subtract fractions. In the following example
we purpousely make the fractional part of the second mixed number greater than
the fractional part of the first mixed number:
4 2/7 - 2 3/4 = 4 - 2 + 2/7 - 3/4 = 2 + 8/28 - 21/28 = 2 - (21 - 8)/28 =
2 - 13/28 = 1 + 1 - 13/28 = 1 + 28/28 - 13/28 = 1 + (28 - 13)/28 = 1 + 15/28 = 1 15/28
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