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Chapter 6: Rational Number Operations and Properties
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
4 4 4 4 12
2
number: 3     
or 2
5 5 5 5 5
5
6.3.1.2. Joining of equal-sized groups can be used when we have a mixed number times
1 1
a rational number: 2   ? see figure 6.12, p. 305
2 2
6.3.1.3. Area model can also be used for multiplying a mixed number times a rational
1 1
number: 2   ? see figure 6.13, p. 305
2 2
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. 306: 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
1
fractions: For rational numbers a1 and b1 , a1  b1  ab
6.3.2.3. Procedure for multiplying rational numbers in fraction form: For rational
numbers ba and cd , ba  cd  ac
bd
6.3.2.4. Your turn p. 308: Do the practice and reflect
6.3.2.5. Class demonstration using paper folding to show multiplication of rational
numbers:
1 1 1
6.3.2.5.1.   half OF a third
2 3 6
2 3 1
6.3.2.5.2.   two-thirds OF three-fourths
3 4 2
1 2 1
6.3.2.5.3.   half OF two-thirds
2 3 3
3 4 3
6.3.2.5.4.   three-fourths OF four-fifths
4 5 5
6.3.2.6. Integer rod steps (always use least number of rods possible)
6.3.2.6.1. Represent the factors of the original problem
6.3.2.6.2. Run a race to a tie
6.3.2.6.3. Represent the 2nd factor only using the new base and find the part of the
numerator indicated by the original 1st factor
6.3.2.6.4. The top and bottom rods now form the answer
6.3.2.7. Class demonstration using integer rods to show multiplication of rational
numbers:
6.3.2.7.1. http://arcytech.org/java/
2 3 1
6.3.2.7.2.  
3 4 2
1 2 1
6.3.2.7.3.  
2 3 3
3 4 3
6.3.2.7.4.  
4 5 5
3 4
6.3.2.7.5.   1
4 3
3
4 49
4 3
4

6.3.2.7.6. 1  2 
think of as 1 2     2 
5 4
5
4
5 10

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. 309: Do the practice and reflect
6.3.3.3. Basic properties for multiplication of rational numbers
 Closure property: For rational numbers ba and cd , ba  cd is a unique rational
number
 Identity property: A unique rational number, 1, exists such that
1 ba  ba  1  ba ; 1 is the multiplicative identity element
, 0  ba  ba  0  0

Zero property: For each rational number

Commutative property: For rational numbers

Associative property: For rational numbers
a
b
,
c
d
, and
e
f
,

Distributive property: For rational numbers
a
b
,
c
d
, and
e
f
,
ba  cd  ef  ba  cd  ef 
a
b

a
b
 cd  ef   ba  cd   ba  ef 
a
b
and
c
d
a
b
,
 cd  cd  ba
Multiplicative inverse: For every nonzero rational number
a
b
, a unique
 ba  ba  ba  1
6.3.3.4. Property for multiplying an integer by a unit fraction: For any integer a and
any unit fraction b1 , a  b1  ba
6.3.3.5. Using the properties to verify (prove) the procedure for multiplication of rational
numbers: see p. 311 (Yes, these will be back on the quizzes and test )
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. 312
6.3.4.4. measurement model – fig.6.17 p. 312
6.3.4.5. Integer rod steps (always use least number of rods possible)
6.3.4.5.1. Represent the factors of the original problem
6.3.4.5.2. Run a race to a tie
6.3.4.5.3. Represent the 2nd factor 1st and then represent the other factor using the new
base
6.3.4.5.4. Redraw the new representation of the 2nd factor and then place the numerator
of the new representation of the 1st factor on top
6.3.4.5.5. the top two rows are the simplification and give the answer to the division
rational number,
b
a
, exists such that
a
b
6.3.4.6. Class demonstration using integer rods to show division of rational numbers:
6.3.4.6.1. http://arcytech.org/java/
6.3.4.7. Your turn p. 316: Do the practice and reflect
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
a
c
a c e
e
numbers
and , c  0,   if and only if
is a unique rational number such
b
d
b d f
f
e c a
that  
f d b
6.3.5.2. Closure property of division for nonzero rational numbers: For nonzero
a
c a c
rational numbers
and ,  is a unique nonzero rational number
b
d b d
6.3.6. Dividing fractions
6.3.6.1. Procedure for dividing fractions – multiplying by the reciprocal method: for
a
c
a c a d
rational numbers
and , where c, b, and d  0,   
b
d
b d b c
6.3.6.2. Procedure for dividing fractions – common denominator method: for rational
a
c
a c ad bc ad


numbers
and , where c  0,  
b
d
b d bd bd bc
6.3.6.3. Procedure for dividing fractions – complex fraction method: for rational
a a d a d


a
c
a c b b c b c ad
numbers
and , where c  0,   


b
d
b d c cd
1
bc
d d c
6.3.6.4. Procedure for dividing fractions – missing factor method: for rational
a
c
a c
a c
numbers
and , where c, b, and d  0,   f , where   f . To find f,
b
d
b d
b d
a d d c
a c a d ad
    f  f , so    
b c c d
b d b c bc
6.3.6.5. Your turn p. 319: Do the practice and reflect
6.3.7. Estimation strategies
6.3.7.1. rounding
6.3.7.2. front-end 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
6.3.7.5. Your turn p. 320: Do the practice and reflect
6.3.8. Problems and Exercises p. 320
6.3.8.1. Home work: 1, 6, 7, 8, 9ac, 10, 14, 15, 16, 17, 18