INTRODUCTION TO
FRACTIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
Introduction to Fractions
• A fraction represents the number of equal parts
of a whole
• Fraction = numerator (up North)
denominator (Down south)
= numerator/denominator
• Numerator = # of equal parts
• Denominator = # of equal parts that make up a
whole
• Example: My husband and I ordered a
large Papa John’s pizza. The large pizza
is cut into 8 (equal) slices. If my husband
ate 3 slices, then he ate
• 3/8 of the pizza
Types of Fractional Numbers
A proper fraction is a fraction whose value
is less than 1 (numerator < denominator)
An improper fraction is a fraction whose
value is greater than or equal to 1
(numerator > denominator)
A mixed number is a number whose value
is greater than 1 made up of a whole part
and a fraction part
Converting Between Fraction
Types
• Any integer can be written as an improper
fraction
• Any improper fraction can be written as a
mixed number
• Any mixed number can be written as an
improper fraction
Integer  Improper Fraction
•
•
•
•
The fraction bar also represents division
The denominator is the divisor
The numerator is the dividend
The original integer (number) is the
quotient
• To write an integer as a division problem,
what do we divide a number by to get the
number?
One . . . n = n/1
Ex: Write 17 as an improper fraction
• 17 = 17 / ?
• 17 divided by what is 17?
1
• Therefore, 17 = 17 / 1
Improper Fraction  Mixed Number
• Denominator: tells us how many parts make up
a whole
• Numerator: tells us how many parts we have
• How many wholes can we make out of the parts
we have?
 Divide the numerator by the denominator  the
quotient is the whole part
• How many parts do we have remaining?
 The remainder (over the denominator) makes up
the fraction part
Ex: Write 11/8 as a mixed number.
How many parts make up a whole?
8
Draw a whole with 8 parts:
How many parts do we have?
11
To represent 11/8 we must shade 11 parts . . .
But we only have 8 parts. Therefore, draw
another whole with 8 parts . . .
Keep shading . . . 9 10 11
This is what 11/8 looks like.
Given the representation of 11/8, how many
wholes are there? 1
Dividing 11 parts by 8 will tell us how many wholes
we can make: 11/8 = 1 R ?
The remainder tells us how much of another
whole we have left: 1 R 3
Since 8 parts make a whole, we have 3/8 left.
Therefore, 11/8 = 1 3/8.
Mixed Number  Improper Fraction
• Denominator: tells us how many parts make up a whole.
Chop each whole into that many parts. How many parts
do we get?
 Multiply the whole number by the denominator.
• Numerator: tells us how many parts we already have.
How many parts do we now have in total?
 Add the number of parts we get from chopping the
wholes to the number of parts we already have
 Form the improper fraction:
# of parts
# of parts that make a whole
Ex: Write 2 5/8 as an improper fraction.
Draw the mixed number
Looking at the fraction, how many parts make
up a whole? 8
Chop each whole into 8 pieces.
How many parts do we now have? 8 + 8 + 5
= 8 * 2 + 5 = 21
= parts from whole + original parts
Therefore 2 5/8 = 21/8
Finding Equivalent Fractions
• Equal fractions with different denominators
are called equivalent fractions.
• Ex: 6/8 and 3/4 are equivalent.
The Magic One
• We can find equivalent fractions by using the
Multiplication Property of 1:
for any number a, a * 1 = 1 * a = a (magic one)
• We will just disguise the form of the magic one
• Do you agree that 2/2 = 1?
• How about 3/3 = 1?
• 4/4 = 1?
• 25/25 = 1? 17643/17643 = 1?
• 1 has many different forms . . .
• 1 = n/n for any n not 0
Ex: Find another fraction equivalent to 1/3
1/3 = 1/3 * 1
= 1/3 * 2/2
= 2/6
or
1/3 = 1/3 * 1
= 1/3 * 3/3
= 3/9
We can write 1/3 many
ways just be using the
Magic One
Ex: Find a fraction equivalent to ½ but with
a denominator of 8
1/2 = 1/2 * 1
= 1/2 * 4/4
= 4/8
Notice:
We can write 1/2 many
ways just be using the
Magic One. We want a
particular denominator –
8. What can we multiply
2 by to get 8?
4
so choose the
form of the
Magic One
Ex: Find a fraction equivalent to 2/3 but with
a denominator 12
2/3 = 2/3 * 1
= 2/3 * 4/4
= 8/12
We can write 2/3 many
ways just be using the
Magic One. We want a
particular denominator –
12. What can we
multiply 3 by to get 12?
4
so choose the
form of the
Magic One
Simplest Form of a Fraction
• A fraction is in simplest form when there
are no common factors in the numerator
and the denominator.
Ex: Simplest Form
Ex: 6/8 and 3/4 are equivalent
The fraction 6/8 is written in simplest form as 3/4
=
=
=1x
Magic one
Ex: Write 12/42 in simplest form
• First prime factor the numerator and the
denominator:
• 12 = 2 x 2 x 3 and 42 = 2 x 3 x 7
• Look for Magic Ones
• Simplify
=
=
=1x1x
=
Notice: 2 x 3 = 6 = GCF(12, 42)
 factoring (dividing) out the GCF will simplify the fraction
Ex: Write 7/28 in simplest form
• What is the GCF(7, 28)?
=7
– Hint: prime factor 7 = 7
–
prime factor 28 = 2 x 2 x 7
=
=
= 1x
=
Dividing out the GCF from the numerator and denominator
simplifies the fraction.
Ex: Write 27/56 in simplest form
• What is the GCF(27, 56)? = 1
– Hint: prime factor 27 = 3 x 3 x 3
–
prime factor 56 = 2 x 2 x 2 x 7
There is no common factor to the numerator and
denominator (other than 1)
Therefore, 27/56 is in simplest form.