Simplifying Fractions
Douglas Ruby
Math for Middle School Teachers
04.534/201
Dr. Regina Panasuk
Page 1
Abstract
Working fluently with fractions requires the ability to compare equivalent and
different fractions. This process requires substantial number sense and numeric
manipulative skill. Key to understanding the magnitude of a fraction is to be able to
simplify that fraction. Subordinate skills for simplifying fractions include knowledge of
divisibility rules, ability to perform prime factorization, ability to recognize that two
numbers are relatively prime, understanding of commutative and associative properties of
multiplication, and a thorough understanding and ability to use the multiplicative identity.
Development of these subordinate skills is non-trivial, yet is important to long-term
mathematical success.
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Introduction
Simplify! A “simple” word that seems to raise anxiety levels in students, be they
middle school students simplifying fractions or college students simplifying the results of
complex, multiple step, calculus problems. Beginning with simplification of numeric
fractions, mathematics stresses the notion that we need to reduce the results of any
mathematical process to its simplest form. When studying fractions, simplification is key
to the development of number sense. Simplification is the process that allows a student to
understand the magnitude of a fraction as compared with another fractions, an integer, a
decimal, or a percent. Further, as one proceeds on into Algebra, Trigonometry, and
Calculus, the ability to simplify polynomial fractions and other kinds of functional
expressions is important to understanding the behavior of the underlying mathematical
function. Yet even in more advanced mathematics, the processes used to simplify rational
expressions are dependent on the basic skills and mathematical knowledge developed
during middle school.
The place and significance of simplifying fractions in the middle school curriculum
According to the National Council of Teachers of Mathematics, initial work with
fractions begins in Grades 3-5. However, in a normal academic progression, mastery of
fractions should be achieved during Grades 6-8. In its Number and Operations standard
for Grades 6-8, the 2000 Principles and Standards of the NCTM expects that students
will be able to “compare and order fractions, decimals, and percents efficiently and find
their appropriate locations on the number line”. Yet in the 1988 National Assessment of
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Education Progress (NAEP), less than 1/3 of the 13-year-old students tested could
correctly order the fractions ¾, 9/16, 5/8, and 2/3. (NCTM 2000, p214-216)
A typical sequence for a sixth or seventh grade math sequence (or even high
school pre-Algebra and adult developmental math) is:
1. Addition and Subtraction of whole (natural) numbers
2. Multiplication and Division of whole numbers
3. Number Theory including prime numbers, divisibility rules, factoring including
prime factorization, and greatest common factor (GCF) and least common
multiple (LCM)
4. Fractions
a. Fractional Equivalence
b. Multiplication and Division of Fractions
c. Least Common Denominator
d. Addition and Subtraction of Fractions
e. Mixed numbers and improper fractions
5. Decimals
a. Conversion between Fractions and Decimals including repeating decimals
b. Addition and subtraction of decimals
c. Multiplication and Division of decimals
d. Rounding and estimation
6. Percents
a. Basic understanding of percent problems.
b. Conversion between Percent and Fractions and Decimals
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7. Problems of Rate and Ratio and Proportion
8. Integers including negative numbers and properties of zero.
9. Geometric Figures including properties of triangles, squares, circles, cubes and
other volumetric figures.
In the outline above, the treatment of number theory, fractions, decimals, and
percent is expanded, since the problem of reducing fractional results and dealing with
fractional equivalence tends to be spread across these four topic areas. The basic prior
knowledge of prime numbers, divisibility rules, factorization, and prime factorization are
taught in number theory along with Greatest Common Factor (GCF) and Least Common
Multiple (LCM). Typically, the notions of equivalent fractions, simplifications, and Least
Common Denominator (LCD) are woven into the treatment of the arithmetic operations
to fractions including multiplication/division and addition/subtraction.
It seems that in some texts, simplification becomes an after thought in the
manipulation of fractions. One older seventh grade textbook reviewed for this paper,
while seemingly exemplary in its use of visual manipulatives and concrete examples,
covers the whole process of simplification of fractions under the title “Simplest Name” in
just two pages out of 350 with just one additional reminder later in its text (Wells,
Abbott, Yacono, and Spence, 1978). In searching for lesson materials, one online lesson
plan used by a 6th grade teacher in Wisconsin stated that learning how to simplify
fractions was important “because in the future, all of your answers must be in simplest
form” with no notion of why this is so! (Horn 2001)
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Fractions, a brief historical overview
In order to understand the relation of fractions with other aspects of our number
system such as decimal notation, a brief history of fractions is in order.
The notion of fractions dates to earliest recorded history. Ancient Babylonians
used a sexigesimal (base 60) number system that had implied fractions. The Babylonian
system was similar to a place value system, except it contained neither the zero digit nor
an explicit “decimal point” (or its equivalent). For example, the Babylonian number:
was found on a Babylonian tablet from the Yale collection. This number is an
approximation for the square root of two using the Babylonian symbols for 1, 24, 51, and
10. In modern fractions, this number is 1
since 2
24
60
51
60 2
10
= 1.414212963. Note that
60 3
1.414213562, the Babylonians had clearly developed sophisticated
mathematical methods for manipulating fractions, since this approximation is correct to 5
decimal places. (Ulearn Today, 2003) Of course, the notion of simplifying a fraction was
not necessary, since like today’s decimal system, a fraction be written in only one way in
sexigesimal notation.
In at least two other ancient cultures, mechanisms for dealing with fractions seem
to evolve from the problems of division. In ancient China, numeric calculations were
done using counting boards. Fractional results, when two numbers cannot be evenly
divided were treated as what we now call common fractions. While the modern concept
of decimal notation did not exist, the Chinese did use the device of the common
denominator in order to make addition and subtraction of fractions possible. (Katz 1998)
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Since common denominators were used, some form of “simplification” of results of
addition, subtraction, multiplication or division would be necessary.
Egyptian culture had the notion of numbers that were not evenly divisible, hence
had a fractional portion. Egyptians used unit fractions in the hieroglyphic form of 1/n
where n=2, 3, etc. It is interesting to note that Egyptian hieratic form did include a
fraction for 2/3, clearly not a unit fraction. When the result of a division did not result in a
unit fraction, then the result would be written as a sum of fractions. This appears in the
Rhind Papyrus with the problem of dividing 6 loaves of bread amongst 10 men. The
result (3/5 in modern fractions) is expressed as ½ + 1/10 loaves. (Katz 1998). In Eqyptian
notation, simplification should not be a significant factor, since numerators are always
“1” and denominators resulting from addition can be expressed as a larger or smaller unit
fraction.
While we trace much of our modern mathematical thought to the great Greek
mathematicians, our modern concept of fraction is not one of those. Euclid had a very
well developed notion of ratio (ergo rational numbers) when comparing either continuous
magnitudes or discreet numbers. Euclid’s Book V stipulates the notion of equal ratio, or
the notion that a:b = c:d. Today, we would also express this as a/b = c/d, however this
fractional notion was not part of Greek mathematical rigor, since the Greeks used the
Egyptian notion of unit fractions, or “parts” of a whole. By the 5th century BC, Greek
mathematicians had found that the ratio of a diagonal of a square to its one of its sides
(the “square root” of 2) was a quantity that was incommensurable, or that it could not be
expressed as a number using their number systems. This discovery is the first time that
the number system was divided into “rational” and “irrational”, though the use of these
Page 7
two terms came much later. Again, if we look at Greek mathematics, the current concept
of simplifying fractions would not have been part of the calculus.
Our common form of ordinary fractions seems to begin in early Hindu
mathematical thought. Both Brahmagupta (c. 628 AD) and Bhaskara (c. 1150 AD) wrote
fractions much as we do today, yet without the fraction bar. The use of the fraction bar
comes to us from Arabic mathematicians. The use of the fraction bar itself is attributed to
al-Hassar. Its use in European texts comes to us from Fibonacci’s Liber abaci (1202). By
the late middle ages, arithmetic using fractions (essentially similar to the way we do
today) was common. Arabic mathematicians wrote the fraction in front of the rest of the
number rather than behind it as we do today (Ulearn Today 2003).
Of more interest than the notational form, however, is the actual computational
use of “decimal fractions”, or the result of taking a common fraction like 1/3 and
expressing it as a power of 10. The first complete use of decimal fractions is attributed to
al-Samaw in his Treatise on Arithmetic written in 1172 AD. While the basics of a place
value and a decimal fraction system (we call this decimal today) were in place by the
1200’s from Arabic sources, it is not until the late 1500’s that the great French
mathematician Francois Viete’ introduced the full decimal fraction system that we now
use for most calculation. It is at this point that calculating using pure decimal notation
took over from calculations using a combination of integers and fractions (Katz 1998).
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Main ideas and concepts of the fractional simplification
Working fluently with fractions requires the ability to compare equivalent and
different fractions. This process requires substantial number sense and numeric
manipulative skill. Key to understanding the magnitude of a fraction is to be able to
simplify that fraction. Subordinate skills for simplifying fractions include knowledge of
divisibility rules, ability to perform prime factorization, ability to recognize that two
numbers are relatively prime, understanding of commutative and associative properties of
multiplication, and a thorough understanding of and ability to use the multiplicative
identity. Development of these subordinate skills is not trivial and is important to longterm mathematical development. . It is the personal experience of this writer that many
students are deficient in the basic ability to recognize that a fraction needs to be
simplified.
While a purely algorithmic process for learning how to simplify fractions is NOT
recommended, we will use just such as process to analyze the component skills required
to perform simplification. To break the process of fractional simplification down into its
component parts, lets look at it as a three-step process.
Step 1 – Factoring
The first step in simplifying a fraction is to factor it. Please note, however, that the
process of factoring a number may one of the more difficult concepts since in order to
factor a number a student must know:
1. Divisibility rules. Typically, the rules for divisibility by two, three, four, five, and
nine are required at minimum. Divisibility by six and eight can be developed from the
rules of divisibility by two and three.
Page 9
2. Prime numbers. Generally, students need to become facile with knowledge of the
primes up to 100.
3. Prime factorization. The ability to reduce a number to its prime factors is generally a
requirement for simplifying the numerator and denominator of fractions.
So for example, if the fraction is
24
we will need to factor both the numerator and
36
denominator into their component primes. The numerator is 24 = 2
denominator is 36 = 2
2
3
2
2
3 while the
3.
Step 2 – Determine the Greatest Common Factor.
A strategy for determining the greatest common factor between two numbers is
required. Once we factor 24 and 36 in our fraction
2
24
, we need to recognize that 12 = 2
36
3 is the greatest common factor between the two numbers.
Step 3 – Use Multiplicative Identity to simplify the fraction.
Having determined that the Greatest Common Factor of 24 and 36 is 12, we now
divide both the numerator and the denominator by 12:
24 12
36 12
2
. Since both 2 and 3
3
are prime, we recognize that the process of simplification is now complete. We show a
“one” in dotted lines around the division of numerator and denominator by 12 to illustrate
the multiplicative (or “divisive”?) identity. This highlights the notion that we are not
actually changing the value of the result since we are effectively just multiplying by one.
A student must also recognize that the result of simplification may not be to reduce the
fraction to a ratio of primes. Fractional simplification is also complete when the
numerator and denominator are “relatively prime”, that is they have no common factors.
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Misconceptions
This purely algorithmic approach to simplifying fractions is fraught with pitfalls and
misconceptions. Possible ways to introduce error include:
1. Incomplete factoring in Step 1. A very common problem is that a student might use
rules of divisibility to factor a number but fail to complete the process. So the 24 in
the numerator may be seen as 2 x 2 x 2 x 3 but the 36 in the denominator as 2 x 2 x 9.
This could lead the student to then incorrectly determine that 4 is the greatest
common factor instead of 12.
2. Improper assessment of Greatest Common Factor. In addition to incorrect
determination of greatest common factor due to incomplete factoring, the process of
determining the correct power of a prime to use in the greatest common factor is not
obvious to all students. In this writer’s experience, students might see that there is a
23 in 24 and a 32 in 36, therefore assume that 23
32 is the greatest common factor.
This, of course, is incorrect. Further, when using words to describe the process of
coming up with the Greatest Common Factor, to say “use the lowest power of the
common factors” is not terribly helpful to math students who are unclear on the
concepts to begin with.
3.
Misunderstanding of “Relatively Prime”. If a fraction were reduced to 11
by 5
7
22
35
2 divided
11 2
, then there is no common factor. Even though 22 and 35 are
7 5
not themselves prime, they are “relatively prime”. In this case, the fraction cannot be
further reduced. However, it is this writer’s experience that, nonetheless, students
may choose to write the fraction as 2/7 or 11/35 feeling that they must somehow
“reduce” the fraction, or else.
Page 11
Some examples of problems, activities, or puzzles involving simplifying fractions
Since divisibility rules have been covered quite nicely in the pedagogy of the Math
for Middle School Teachers course, I would like to address the issue of recognizing
whether a number is prime or not. A tool that can be used to develop prime numbers is a
10 table as follows:
prime number sieve. Lets create a 10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
First we place a circle around the two (2) and cross out all multiples of two. Next we
place a circle around the three and cross out all multiples of three. Note that the even
multiples of three will already be crossed out. Next we place a circle around the next
number that is not crossed out which should be 5. We then cross out all multiples of five.
We continue this process of circling the next lowest number available and crossing out all
of its multiples. This process will result in circling all of the prime numbers between 1
Page 12
and 100 (not including 1 which is not considered prime). An exercise like this that allows
the student to “discover” primes without even having to necessarily know their definition
could be used to help understanding and recognition. Further, some additional inspection
will show that by the time one reaches the number 11, we have crossed out all primes
which are multiples of the numbers between 1 and 10, therefore, we already know all of
the prime numbers between one (12) and one hundred (102).
In order to address another common misconception about simplifying fractions,
we need to help students understand “equivalent” fractions completely. Fractions are not
equivalent just because we can simplify them to the same relatively prime numerator and
denominator, but they are equivalent because the produce the same ratio (the Greek
concept) or have the same magnitude. Using geometric diagrams can show magnitude.
Perhaps one of the best ways to do this is using fraction bars. In the example below, we
show four equivalent fractions, 3/4, 6/8, 9/12, and 18/24.
3/4
6/8
9/12
18/24
0
¼
½
¾
1
This diagram can be used to help understand not only fractional equivalence, but also
factoring and greatest common factors. One can see by looking at the dotted lines and
comparing with the fully reduced fraction (3/4) that the greatest common factor of 6 and
Page 13
8 is two, of 9 and 12 is three, and of 18 and 24 is six. Further, one can observe that
reducing 18/24 to 6/8 is an incomplete simplification since it can be further reduced to ¾.
Conclusion
The modern concepts of decimal numbers, fractions, and percent are relatively
recent developments in the history of mathematical development. As we saw, early
Babylonian, Egyptian, and Greek math largely dealt with the problems of fractional
equivalence and simplification by avoiding the issue. Current systems of common
fractions had their antecedents in Hindu, Arabic, and Medieval thought. The ability to
manipulate and perform fractional and mixed arithmetic was a defining characteristic of
mathematical literacy during the medieval period. Once the decimal system came into
wide use in the mid 1500’s, calculation became easier, but the standard for magnitude
and comparison became more difficult, as we now had to become fluent in fractional and
decimal equivalence.
Key to understanding fractional equivalence and to making fraction to decimal
conversion more straightforward is the ability to reduce or simplify fractions. This
process is often taught in a mechanistic, algorithmic way as a small part of the middle
school curriculum. The process of understanding the concept of simplification and its
subordinate concepts of prime numbers, prime factorization, and Greatest Common
Factors can be made more concrete with visual explanations. As students continue to
develop their mathematical skills, simplification plays an increasingly critical role in
understanding the function of the mathematical equations of Algebra, Trigonometry, and
Calculus.
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References
Eves, H. (1990) An introduction to the history of mathematics, Saunders College
Publishing.
Horn, J. (2001) Simplifying Fractions, Chilton, WI, retrieved on April 5, 2003 from
http://www.chilton.k12.wi.us/staff/hornj/review/6-5/index.htm
Katz, V. (1998) A History of Mathematics: An Introduction, 2nd ed., Addison-Wesley
Educational Publishers, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for
school mathematics, Reston, Va.; National Council of Teachers of Mathematics.
O’Connor, J. and Robertson, E. (2002), The MacTutor History of Mathematics archive,
School of Mathematics and Statistics, University of St. Andrews, Scotland,
retrieved from http://www-gap.dcs.st-and.ac.uk/~history/
Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000) Linking fractions, decimals
and percent using an area model; Implementing Standards Based Mathematics
Instruction; Teachers College Press, Chap. 4,
Ulearn Today, (2003) The history of fractions, Ulearn today library, retrieved on March
31, 2003 from
http://www.ulearntoday.com/magazine/physics_article1.jsp?FILE=fractions
Wells, D., Abbott, J., Yacono, M. & Spence, R. (1978) Growth in mathematics, Harcourt,
Brace, Jovanovich
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