2.2 Introduction to Fractions 1 A fraction is a number that can
express values that are not necessarily whole numbers. They are
used to represent values that come between the whole numbers. By
using a fraction, the portion can be described exactly as to its
size in comparison to the whole. For example: 4/5 implies that if
the whole were divided into five equal parts that 4/5 would
comprise of four of the five parts. The top number of the fraction
is called the called the numerator and the bottom number is called
the denominator. Numerator & Denominator Next Slide Therefore
the top number of the fraction is representative of the portion to
be described or expressed and the bottom number is representative
of the number of divisions, sections, or portions in which the
whole has been divided. For example:
Slide 2
2.2 Introduction to Fractions 2 Proper Fractions A proper
fraction is a fraction whose numerator is smaller than its
denominator. All proper fractions are less than 1. Improper
Fractions An improper fraction is a fraction whose numerator is
larger than or equal its denominator. Mixed Numbers A mixed number
is a number written as a whole number and a fraction. It represents
the sum of the whole number and the fraction. All mixed numbers are
greater than 1. Next Slide
Slide 3
2.2 Introduction to Fractions 3 Fractions Equaling "1" Any
fraction with a non-zero denominator that is equal to its numerator
is equal to one. If portion is the same as the number of divisions,
then the amount described is the whole. For example: Fractions
Equaling "0" Any fraction with a numerator of zero and a non-zero
denominator is equal to zero For example: Undefined Fractions Since
division by zero is undefined, any fraction with a denominator of
zero is undefined. For example: Next Slide
Slide 4
2.2 Introduction to Fractions 4 Converting an Improper Fraction
to a Mixed Number To change an improper fraction to a mixed number:
Divide the denominator into the numerator. The quotient is the
whole number. The fraction part of the mixed number as the
remainder in the numerator and the original denominator as the
denominator. Example 1. Convert 9/4 to a mixed number: whole number
denominator numerator Your Turn Problem #1 Convert 17/5 to a mixed
number.
Slide 5
2.2 Introduction to Fractions 5 Converting a Mixed Number to an
Improper Fraction To change a mixed number into an improper
fraction, multiply the denominator by the whole number and then add
that product to the numerator. This sum becomes the numerator of
the improper fraction and the denominator of the improper fraction
is the same as the denominator from the mixed number. 1.Multiply
whole number and denominator. 2.Add numerator. 3.Write result over
original denominator. Your Turn Problem #2
Slide 6
2.2 Introduction to Fractions 6 Equivalent Fractions Recall the
Multiplicative Identity (also called the multiplication property of
1): a 1 = a. When we multiply a number by 1, we get the same
number. Example: 5 1 = 5 Also, any fraction with the same numerator
and denominator is equal to 1.,, If we take a fraction, such as ,
and multiply it by 1, we get the same number. Next Slide
Slide 7
2.2 Introduction to Fractions 7 A fraction written in lowest
terms is when the fraction is expressed using the lowest numerator
and denominator possible. Reducing Fractions (Simplifying)
represents that fraction written in lowest terms because it is
impossible to find an equivalent fraction that represents that
value using a smaller numerator or denominator. Reducing is the
process of converting a fraction to an equivalent fraction in
lowest terms. It is done by dividing the numerator and denominator
by the largest number that divides evenly into both numbers. The
largest number that divides evenly into a set of numbers is called
the gcf, (greatest common factor). Actually, it is not absolutely
necessary to use the gcf. It will just take more steps if we dont
use the gcf. Next Slide
Slide 8
2.2 Introduction to Fractions 8 Step 1. Find the gcf for both
numerator and denominator. Step 2. Divide both numerator and
denominator by the gcf. So 6 is the largest number that divides
evenly into both numerator and denominator. Ideally, we want to
divide both numerator and denominator by the gcf, but any common
factor will work. The question you are asking yourself is what
number divides evenly (goes into) both numbers? Some may think of
6. Others may think of 2 since the numbers are both even. Or some
may think of 3. Any will work. However, if we use a number that is
not the gcf, we need to continue the process until the numerator
and denominator have no factors in common. Alternative method: Next
Slide
Slide 9
2.2 Introduction to Fractions 9 Your Turn Problem #3 Simplify
the following: Step 1. Find the gcf for both numerator and
denominator. Step 2. Divide both numerator and denominator by the
gcf. So 15 is the largest number that divides evenly into both
numerator and denominator. Alternative Method: What number divides
evenly into both 30 and 75? Since the numbers end in a 0 and a 5,
they are divisible by 5. Next, 6 and 15 are divisible by what
number? Answer: 3 Answer
Slide 10
2.2 Introduction to Fractions 10 Prime Factorization Method of
Reducing This method can be more efficient when the fraction
contain large numbers. This method requires writing the prime
factorization of both numerator and denominator. Since any number
divided by itself equals one, we will line out any like factors
that are contained in both numerator and denominator. Your Turn
Problem #4 Step 1. Find the prime factorization for both numerator
and denominator. Step 2. Rewrite the fraction using the prime
factorizations. Procedure for Reducing using Prime Factorization
Step 1: Find the prime factorization of the numerator and
denominator. Step 2: Rewrite the fraction using the prime
factorizations. Step 3: Line out, one for one, like factors that
are contained both in the numerator and denominator. Step 4:
Multiply the remaining not lined-out factors together. Step 3. Line
out like factors. Step 4. Multiply the remaining factors. Answer:
The End B.R. 5-24-08