INTRODUCTION TO FRACTIONS

MSJC ~ San Jacinto CampusMath 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 =

The remainder tells us how much of another whole we have left:

1 R ?

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 We can write 1/3 many ways just be using the Magic One

= 1/3 * 2/2

= 2/6

1/3 = 1/3 * 1

= 1/3 * 3/3

= 3/9

or

Ex: Find a fraction equivalent to ½ but with a denominator of 8

1/2 = 1/2 * 1 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?

= 1/2 * 4/4

= 4/8

4

so choose the form of the Magic One

Notice:

Ex: Find a fraction equivalent to 2/3 but with a denominator 12

2/3 = 2/3 * 1 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?

= 2/3 * 4/4

= 8/12

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

= = = 1 x

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

= = = 1 x 1 x =

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)?– Hint: prime factor 7 = 7– prime factor 28 = 2 x 2 x 7

= = = 1 x =

Dividing out the GCF from the numerator and denominator simplifies the fraction.

= 7

Ex: Write 27/56 in simplest form

• What is the GCF(27, 56)?– 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)

= 1

Therefore, 27/56 is in simplest form.