Math 6A ( Periods 2 & 5)

Dividing Integers 11-5

Essential Question: Is the quotient of two integers positive, negative or zero? How can you tell?

The rules for multiplying integers apply to dividing integers.

Look at your Fact Families

(2)(-3) = (-6)

so we know

-6 ÷ 2 = -3  and   -6 ÷ -3 = 2

The same life story applies as told about multiplying integers!  Hmmm… what was that story?

-15 ÷  3  ( integers with different signs) = - 5

12 ÷ 4 ( integers with same signs) = 3

12 ÷ (-3) ( integers with different signs) = -4

-12 ÷ (-4) ( integers with same signs) = +3

0 ÷  (-15) = 0

IMPORTANT:
What’s the general rule about dividing two integers with the same sign?
When dividing integers with the same sign, divide their absolute value and make the quotient positive.

What’s the general rule about dividing two integers with different sign?
When dividing integers with different sign, divide their absolute value and make the quotient negative.

-49/7 = -7

21/-3 = -7

Evaluating an Expression

Evaluate 10 - x² ÷ y  when x = 8 and y = -4
Substitute each value using {{HUGS}} 

10 – (8)² ÷ (-4)

10 -64÷ (-4)

10 –(-16)

10 + 16 = 26

Real Life Application

You measure the height of the tide using the support beams of a pier. Your measurements are  59 inches at 2PM and 8 inches at 8PM

What is the mean hourly change in the height?
How many hours are between  2 PM and 8 PM ?   6 hours
Use a model to help you solve this problem

The mean change in the height of the tide is -8.5 inches per hour

What is the mean of these integers:
3, -10, -2, 13, 11
First- add all the positives 3 + 13 + 11 = 27
Second add all the negative -10 + -2 = -12
Add them together +15
To get the mean divide by the number of integers you started with ( which was 5)
15/5 = 3
-8 -14 ÷ 2 + 5
Make sure you follow the Order of Operations ! So
-8 -7 + 5 
-15 + 5 
-10
Any number divided by ZERO is UNDEFINED  (You must know that term!)
4/0 = undefined  
You cannot divide by ZERO
However, notice 0/4 = 0
4 + 3÷0   since 3÷0  is undefined  --> the entire expression  is UNDEFINED! 

Math 6A (Periods 2 & 5)

Multiplying Integers 11-4


3 ⋅ -2 = -6
Its really repeated addition
or
-2 + -2 + -2 which we learned a few sections ago was equal to -6.

The product of a positive integer and a negative integer is a negative integer.

The product of ZERO and any integer is ALWAYS ZERO!!
a⋅0 = 0

Math imitates life...and Karma(?)
What was the story I told in class... it applies to
Multiplication & Division ...
+ ⋅ + = +
- ⋅ + = -
+ ⋅ - = -
- ⋅ - = +



The product of -1 and any integer equals the opposite of that integer.
(-1)(a) = -a

The product of two negative integers is a positive integer

For a product with NO ZERO factors:
-->if the number of NEGATIVE factors is odd, the product is negative
-->if the number of NEGATIVE factors is even, then the product is positive
Be a sign counter-- count your negatives!
Every integer and its opposite have equal squares!!

Remember-- if its all multiplication use the Associative & Commutative Properties of Multiplication to make your work EASIER!!

Using Exponents

Evaluate (-2)²
Write (-2)² as repeated multiplication

(-2)² = (-2)(-2) = 4

Evaluate
-5²
Notice that the exponent is ONLY touching the 5 so this is really
-(5∙5) = -25

Evaluate
(-4)³  =(-4)(-4)(-4) = 16(-4) = -64

Algebra:
Evaluate the expression when a = -2, b = 3, and c = -8
Always substitute your actual number in with {{Hugs}} that is  (  )
Example 1 ab would be ( -2)(3) = -6
Example 2 │a²c│= │(-2)²(-8)│ = │ (4)(-8)│=│-32│= 32
Example 3 -ab³ –ac  = - (-2)(3)² – (-2)(-8) = -(-2)(27) –(16) = -(-54) – 16 = +54 – 16 = 38
Did you get that one?
Look at each step to see what was done along the way!

Number Sense:
Find the next two numbers in the patterns:
7, -28, 112, -448, …
First what is happening to each?
It looks like each consecutive integer is 4 times the opposite of the previous number.
So the next two integers would be  1792,  -7168