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Algebra I 0011 0010 1010 1101 0001 0100 1011

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1-1 Operations with Fractions Mrs. Adams 6th Grade October 5, 2009

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Students will be learning how to… 0011 0010 1010 1101 0001 0100 1011

•

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Understand improper fractions

(difference between numerator

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Identify prime numbers

and denominator)

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Multiply fractions

Properly identify fractions

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Properly factor whole numbers

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Use reciprocals

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Understand proper fractions

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Divide fractions and simplify

Adding/Subtracting Fractions vs. Multiplying/Dividing Fractions 0011 0010 1010 1101 0001 0100 1011 •

What is the difference between adding/subtracting fractions and multiplying/dividing fractions?

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• • When adding or subtracting fractions the denominators must be set equal to each other. When multiplying or dividing fractions the denominators do not have to be set equal to each other (finding the LCD).     

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See link: Adding Fractions with Different Denominators

See link: Multiplying and Dividing Fractions with Different Denominators

Fractions 0011 0010 1010 1101 0001 0100 1011 The numbers used most often are whole numbers,  0, 1, 2, 3, 4, and so on, For counting and fractions such as  ⅓ ,⅔, ⅜, and so on. 

 

 

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In a fraction the top number is called the numerator and the bottom number is called the denominator. numerator 9 denominator 10

There are two types of fractions:  1. Proper fractions where the numerator is less than the denominator; for example, 9  2. Improper fractions where the numerator is greater than or equal to the denominator; for 10 example, 



10 10 or 10 9

Prime Numbers and Factorization

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Any whole number can be stated as a product of two or more whole numbers, called factors of the number. For example, Product Factors 12=2 x 6 12=1 x 12 12=4 x 3 12=2 x 2 x 3



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2 and 6 are factors 1 and 12 are factors 4 and 3 are factors 2, 2, and 3 are factors To factor a whole number is to write the number as a product of factors. It will be necessary to factor whole numbers so that the factors are prime numbers.  1. Prime Number- any whole number greater than 1 whose only factors are the number itself and 1. The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.    

 

Write each number as a product of prime factors.

       

Thus 36=2 x 2 x 3 x 3

36 2 x 18 2x2x9 2x2x3x3

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Divide 36 by 2 = 18 Divide 18 by 2 = 9 Divide 9 by 3 = 3

Reducing fractions to lowest terms

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A fraction is reduced to lowest terms when the only factor common to the numerator and the denominator is 1.  To reduce a fraction to lowest terms:  1. Write the numerator and the denominator as a product of prime factors.  2. Divide the numerator and the denominator by all common factors. 1. 14 2 ⋅ 7  Write as a product of prime factors = 21 3 ⋅ 7  Divide numerator and denominator by common factor 7  2 =  3 2. Now reduce to lowest terms. 

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Products and quotients of fractions

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To multiply two or more fractions, we use the following procedure.  To multiply fractions:  1. Write the numerator and the denominator as an indicated product (do not multiply).  2. Reduce the resulting fraction to lowest terms. 



Multiply the following fractions and reduce to lowest terms. 1. Multiply numerators 

2 5 2 ⋅5 ⋅ = 3 7 3⋅ 7



Multiply denominators



=



5 3 5⋅3 ⋅ = 6 4 6⋅4 5⋅3 2 ⋅ 3⋅ 2 ⋅ 2

=

5 2⋅2⋅2

=

5 8

Multiply denominators

=

Factor

2.

Fraction cannot be factored any further

Divide numerator and denominator by 3



10 21

Multiply in the denominator



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Products and quotients of fractions continued…pt.2

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Suppose we multiply the fractions

5 6 5⋅6 ⋅ = 6 5 6⋅5 30 = 30

=1



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When the product of two numbers is 1, we call each number the reciprocal of the other number.  Thus, 5 6 

    



and are reciprocals, 6 5 2 7 are reciprocals, and 7 2 14 13 are reciprocals, and 13 14

We can see that the reciprocal of any fraction is obtained by interchanging the numerator and the denominator. The reciprocal of a fraction is used to divide fractions.

Products and quotients of fractions continued…pt.3

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To divide two fractions:  1. Multiply the first fraction by the reciprocal of the second fraction.  2. Reduce the resulting product to lowest terms. 



Divide the following fractions and reduce to lowest terms. 1. 

7 6 7 7 ÷ = ⋅ 8 7 8 6

  

=

7⋅7 8⋅6

=

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  

Multiply by the reciprocal of

6 7

Multiply numerators Multiply denominators

2.



4 3 4 7 ÷ = ⋅ 5 7 5 3

Multiply by the reciprocal of

=

4⋅7 5⋅3

Multiply numerators Multiply denominators

=

28 13 or1 15 15

Perform indicated operations

3 7

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Practice Problems 1.1 0011 0010 1010 1101 0001 0100 1011 Write each number as a product of prime factors. (Factor) 1. 64 2. 42 3. 24 Multiply the following fractions and reduce to lowest terms. 1. 2 4 

⋅ = 6 5

 

2.

  

3.

3 1 ⋅ = 7 2 1 5 ⋅ = 2 8

Divide the following fractions and reduce to lowest terms. 1. 

3 1 ÷ = 2 2. 7 3 1 ÷ = 4 4 

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Resources 0011 0010 1010 1101 0001 0100 1011

• Factoring • • Multiplying Fractions • • Dividing Fractions

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