Transcript 7.NS.2 Multiply rational numbers power point

Multiplying
Rational Numbers
© William James Calhoun
To multiply rational numbers.
The rules for multiplying numbers are different from adding and
subtracting numbers.
You need to keep the rules for adding and subtracting numbers in your
head.
Keep those rules separate from the multiplying numbers rules we are
about to discuss.
Remember multiplying numbers is actually a quick way of adding
numbers by grouping them.
© William James Calhoun
2.6.1 MULTIPLYING TWO RATIONAL NUMBERS
The product of two numbers having the same sign is positive.
The product of two numbers having different signs in negative.
The short-and-sweet is that multiplying rational numbers is just the same
as all the multiplying you have done before.
The only new additions to the rules-of-old are the following:
1) A positive times a positive is a positive.
2) A positive times a negative is a negative.
3) A negative times a negative is a positive.
Commit these three new additions to memory.
© William James Calhoun
EX1β
EXAMPLE 1α: Find each product.
2
a. (-9.8)4
b.   3 
  
 4  3 
Negative times positive yields
negative.
All that is left is to multiply 9.8 by 4
and put a negative sign on the result.
Negative times negative yields a positive.
Multiply the numbers.
Reduce.
(9.8)4 = 39.2
 3  2  6 1
    
 4  3  12 2
-39.2
EXAMPLE 1β: Find each product.
 4
a. 20   
 5
b. (-1.4)7
© William James Calhoun
EX2β
2
5
EXAMPLE 2α: Evaluate a   if a = 2.
6
2
Plug 2 in for a.
 5   2  5  5   2  25   2  25   50 25

2 
 




Exponent.
 36  1  36  36 18
6 6
6
Multiply.
Multiply again. Reduce.
3
2
EXAMPLE 2β: Evaluate a   if a = 3.
 3
© William James Calhoun
EX3β
EXAMPLE 3α: Simplify each expression.
a. (2b)(-3a)
b. 3x(-3y) + (-6x)(-2y)
Positive times negative yields a negative.
Multiplying with letters, so letter
configuration will change.
Multiply the numbers.
2*3=6
a’s and b’s form new letter configuration:
ab
So, the answer - keeping the sign in mind
- is:
-6ab
First term will be negative; second
positive.
Multiply the numbers in both terms.
3*3=9
and
6 * 2 = 12
Now handle the changes in letter
configurations.
x * y = xy
and
x * y = xy
Bring it all together.
Combine like terms.
-9xy + 12xy = 3xy
EXAMPLE 3β: Simplify each expression.
a. (-2a)(3b) + (4a)(-6b)
b. (5x)(-3y) + (-7x)(4y)
© William James Calhoun
Question: What is -1 times 5?
Answer: -5
Question: What is -1 times -14?
Answer: 14
Question: So what does multiplying by -1 do to any number?
Answer: Multiplying by -1 changes only the sign of a number.
2.6.2 MULTIPLICATIVE PROPERTY OF -1
The product of any number and -1 is its additive inverse.
-1(a) = -a and a(-1) = -a
© William James Calhoun
EX4β
3  1  2 

EXAMPLE 4α: Find    4  3   4  1 .
 4  3  5 
3  1   3  13  39 13

Handle the first pair.    4        

 4  3   4  3  12 4
 13  2 
We now have:
  3   4  1
 4  5 
 13   2   13   17  221
Handle the first pair. 
No reduction is needed.
  3      
 4   5   4   5  20
 221 
We now have:

  4  1
 20 
884 221
 221 
Handle the first pair. 
No reduction is needed.
4


 
20
5
 20 
221
 221 
We now have:
Finally the answer!

  1  
5
 5 
© William James Calhoun
EXAMPLE 4β: Multiply.
 2  1  9  1 
a.        
 3  6  5  2 
 5
 1  11 
b.     4   7  
 6
 5  12 
© William James Calhoun
17-31 odd
35-39 odd, 43-51 odd
© William James Calhoun