INTRODUCTION TO FRACTIONS

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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)

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### 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 . . .

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.

8

+ 8

+ 5

How many parts do we now have?

= 8 * 2 + 5 = 21

= parts from whole + original parts

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

or

1/3 = 1/3 * 1

= 1/3 * 3/3

= 3/9

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

Notice:

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

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

= 7

=

=

=

1 x

=

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)?
• Hint: prime factor 27 = 3 x 3 x 3
• prime factor 56 = 2 x 2 x 2 x 7

= 1

There is no common factor to the numerator and denominator (other than 1)

Therefore, 27/56 is in simplest form.