Multiplying and Dividing Fractions
6.3 Multiplying and Dividing Fractions  
6.3.1. Modeling multiplication of fractions  
6.3.1.1. Repeated
addition can be used when we have a whole number times a rational number: 

6.3.1.2. Joining of
equalsized groups can be used when we have a mixed number times a rational number :see figure 6.12, p. 326 

6.3.1.3. Area model can
also be used for multiplying a mixed number times a rational number:see figure 6.13, p. 326 

6.3.1.4. Additionally
the area model can be used to show multiplication of a rational number times a rational number. 

6.3.1.5. Your turn p. 327: Do the practice and reflect  
6.3.2. Multiplying fractions  
6.3.2.1. Fraction with a numerator of one is called a unit fraction  
6.3.2.2. Generalization
about multiplying rational numbers represented by unit fractions: For rational numbers 

6.3.2.3. Procedure for
multiplying rational numbers in fraction form: For rational numbers 

6.3.2.4. Your turn p. 329: Do the practice and reflect  
6.3.2.5. Class
demonstration using paper folding to show multiplication of rational numbers: 

6.3.2.5.1.half OF a third  
6.3.2.5.2.twothirds OF threefourths  
6.3.2.5.3.half OF twothirds  
6.3.2.5.4.threefourths OF fourfifths  
6.3.2.6. Integer rod steps (always use least number of rods possible)  
6.3.2.7.
Class demonstration using integer rods to show multiplication of
rational numbers: 

6.3.2.7.1.  
6.3.2.7.2.  
6.3.2.7.3.  
6.3.2.7.4.  
6.3.3. Properties of rational number multiplication  
6.3.3.1. Basic properties of rational numbers  
6.3.3.1.1. Multiplicative inverse ( reciprocal ) analogous to additive inverse property  
6.3.3.2. Your turn p. 330: Do the practice and reflect  
6.3.3.3. Basic properties for multiplication of rational numbers  
• Closure property: For
rational numbers is a unique
rational number 

• Identity property: A
unique rational number, 1, exists such that ; 1 is the multiplicative identity element 

• Zero property : For each rational number  
• Commutative property: For rational numbers  
• Associative property: For rational numbers  
• Distributive property: For rational numbers  
• Multiplicative
inverse: For every nonzero rational number b/a , a unique rational number, a/b , exists such that 

6.3.3.4.
Property for multiplying an integer by a unit fraction: For any integer
a and any unit fraction 

6.3.3.5.
Using the properties to verify (prove) the procedure for multiplication
of rational numbers: see p. 331 

6.3.4. Modeling Division of fractions  
6.3.4.1. used to separate a quantity into groups of the same size  
6.3.4.2. no remainders in division of rational numbers  
6.3.4.3. Partition model – fig. 6.16 p. 332  
6.3.4.4. Measurement model – fig.6.17 p. 332  
6.3.4.5. Integer rod steps (always use least number of rods possible)  
6.3.4.6. Class demonstration using integer rods to show division of rational numbers:  
6.3.4.6.1.  
6.3.4.6.2.  
6.3.4.6.3.  
6.3.4.6.4. 3/2  
6.3.4.6.5.  
6.3.4.6.6.  
6.3.4.6.7. 3/10  
6.3.5. Definition and properties of rational number division  
6.3.5.1.
Definition of rational number division in terms of multiplication : for
rational numbersif and only if e/f is a unique rational number such that 

6.3.5.2.
Closure property of division for nonzero rational numbers: For nonzero rational numbers is a unique nonzero rational number 

6.3.6. Dividing fractions  
6.3.6.1.
Procedure for dividing fractions – multiplying by the reciprocal method :
for rational numbers a/b and c/d , where c, b, and d ≠ 0, 

6.3.6.2.
Procedure for dividing fractions – common denominator method : for
rational numbers a/b and c/d , where c ≠ 0, 

6.3.6.3.
Procedure for dividing fractions – complex fraction method: for rational numbers a/b and c/d , where c ≠ 0, 

6.3.6.4.
Procedure for dividing fractions – missing factor method : for rational numbers a/b and c/d , where c, b, and d ≠ 0,, where To find f,


6.3.7. Estimation strategies  
6.3.7.1. rounding  
6.3.7.2. frontend estimation  
6.3.7.3. substituting compatible numbers  
6.3.7.4. Where does the decimal point go?  
6.3.7.4.1. 6.25 x 0.89 = 55625  
6.3.7.4.2. 4.3 x 0.49 = 2107  
6.3.7.4.3. 5.75 x 1.39 = 79925  
6.3.8. Problems and Exercises p. 340  
6.3.8.1. Home work: 1, 6, 7, 8, 9ac, 10, 14, 15, 16, 17, 18 
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