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Chapter 6: Rational Number Operations and Properties
6.3 Multiplying Rational Numbers
6.3.1. Modeling rational number multiplication
6.3.1.1. Repeated addition can be used when we have a whole number
4 4 4 4 12
2
times a rational 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
1 1
number times a rational number: 2   ? see figure 6.14, p. 307
2 2
6.3.1.3. Area model can also be used for multiplying a mixed number times
1 1
a rational number: 2   ? see figure 6.15, p. 308
2 2
6.3.1.4. Additionally the area model can be used to show multiplication of a
rational number times a rational number.
6.3.2. Multiplying rational numbers in fraction form
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
1
by unit 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. Class demonstration using paper folding to show multiplication of
rational numbers:
1 1 1
6.3.2.4.1.   half OF a third
2 3 6
2 3 1
6.3.2.4.2.   two-thirds OF three-fourths
3 4 2
1 2 1
6.3.2.4.3.   half OF two-thirds
2 3 3
3 4 3
6.3.2.4.4.   three-fourths OF four-fifths
4 5 5
6.3.2.5. Integer rod steps (always use least number of rods possible)
6.3.2.5.1. Represent the factors of the original problem
6.3.2.5.2. Run a race to a tie
6.3.2.5.3. Represent the 2nd factor only using the new base and find the
part of the numerator indicated by the original 1st factor
6.3.2.5.4. The top and bottom rods now form the answer
6.3.2.6. Class demonstration using integer rods to show multiplication of
rational numbers:
6.3.2.6.1. http://arcytech.org/java/
2 3 1
6.3.2.6.2.  
3 4 2
1 2 1
6.3.2.6.3.  
2 3 3
3 4 3
6.3.2.6.4.  
4 5 5
3 4
6.3.2.6.5.   1
4 3
3
4 49
4 3
4

6.3.2.6.6. 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. Basic properties for multiplication of rational numbers
6.3.3.2.1. Closure property: For rational numbers ba and cd , ba  cd is a
unique rational number
6.3.3.2.2. Identity property: A unique rational number, 1, exists such that
1 ba  ba  1  ba ; 1 is the multiplicative identity element
6.3.3.2.3. Zero property: For each rational number
a
b
, 0  ba  ba  0  0
6.3.3.2.4. Commutative property: For rational numbers
a
b
a
b
and
c
d
,
 cd  cd  ba
6.3.3.2.5. Associative property: For rational numbers
a
b
,
c
d
, and
e
f
,
6.3.3.2.6. Distributive property: For rational numbers
a
b
,
c
d
, and
e
f
,
ba  cd  ef  ba  cd  ef 
a
b
 cd  ef   ba  cd   ba  ef 
6.3.3.2.7. Multiplicative inverse: For every nonzero rational number
a
b
,a
unique rational number, ba , exists such that ba  ba  ba  ba  1
6.3.3.3. Property for multiplying an integer by a unit fraction: For any
integer a and any unit fraction b1 , a  b1  ba
6.3.3.4. Using the properties to verify (prove) the procedure for
multiplication of rational numbers: see p. 313 (Yes, these will be back
on the quizzes and test )
6.3.4. Multiplying rational numbers in decimal form
6.3.4.1. Procedure for multiplying rational numbers in decimal form (besides
just grab my calculator ):
6.3.4.1.1. Multiply the factors as if they are whole numbers, then place the
decimal point into the appropriate place in the product
6.3.4.1.2. The product of a factor with m places to the right of the decimal
point and a factor with n places to the right of the decimal point will
have m + n decimal places to the right of the decimal point in the
product (including zeros that are part of the product.
6.3.4.2. Example 1
6.3.4.2.1. 4000 x 25 = 100000
6.3.4.2.2. 400 x 25 = 10000
6.3.4.2.3. 40 x 25 = 1000
6.3.4.2.4. 4 x 25 = 100
6.3.4.2.5. 0.4 x 25 = 10.0
6.3.4.2.6. 0.04 x 25 = 1.00
6.3.4.2.7. 0.004 x 25 = 0.100
6.3.4.2.8. 0.0004 x 25 = 0.0100
6.3.4.3. Example 2
6.3.4.3.1. 4000 x 0.25 = 1000.00
6.3.4.3.2. 400 x 0.025 = 10.000
6.3.4.3.3. 40 x 0.0025 = 0.1000
6.3.4.3.4. 4 x 0.00025 = 0.00100
6.3.4.3.5. 0.4 x 0.25 = 0.100
6.3.4.3.6. 0.04 x 0.025 = 0.00100
6.3.4.3.7. 0.004 x 0.0025 = 0.0000100
6.3.4.3.8. 0.0004 x 2.5 = 0.00100
6.3.5. Estimation strategies
6.3.5.1. rounding
6.3.5.2. front-end estimation
6.3.5.3. substituting compatible numbers
6.3.5.4. Where does the decimal point go?
6.3.5.4.1. 6.25 x 0.89 = 55625
6.3.5.4.2. 4.3 x 0.49 = 2107
6.3.5.4.3. 5.75 x 1.39 = 79925
6.3.6. Problems and Exercises p. 316
6.3.6.1. Home work: 1, 6, 7, 8ac, 9, 10, 14, 15abcf, 16abdef, 17ab, 19,
20ad, 22, 26acd, 29