INTEGERS
What is an Integer?
An integer is a positive or negative whole number, including 0. …-3, -2, -1, 0, 1, 2, 3…
There are “4” Integer Operations
4 Integer Operations Addition + Subtraction - Multiplication x Division ÷
Rule #1 for Adding Integers (+) The sum of two positive integers is always positive. 5 + 1 = 6
Rule #2 for Adding Integers (+) The sum of two negative integers is always negative. -5 + (-1) = -6
Rule #3 for Adding Integers (+) The sum of a positive and a negative integer could be positive, negative, or zero.
Rule #3 for Adding Integers Continued When you add a positive and negative integer, you are really subtracting. Then, you give the answer the sign of the greater absolute value. 5 + (-1) = +4 -5 + 1 = -4 -5 + (+5) = 0
Let’s Practice “Addition” 1) 5 + 6 = -3 + (-2) = -6 + 5 = 8 + (-7) = -9 + 9 =
Let’s Check 1) 5 + 6 = 11 -3 + (-2) = -5 -6 + 5 = -1 8 + (-7) = 1 1) 5 + 6 = 11 -3 + (-2) = -5 -6 + 5 = -1 8 + (-7) = 1 -9 + 9 = 0
Subtracting Integers
Rules for Subtracting Integers (-) To subtract an integer, add its opposite. You will need to correctly change all subtraction problems into addition problems!
How do you change a subtraction problem into an addition problem?
We ADD the OPPOSITE Eg: 14 - (-3)
OR… there are three steps: 1. Keep the first integer the same. (Keep) 2. Flip the subtraction sign into an addition sign. (Flip) 3. Take the opposite of the number that immediately follows the newly placed addition sign. (Change)
Think … Keep, Flip, Change Examples: 5 – (-2) = 5 + 2 = 7 5 – (-2) = 5 + 2 = 7 -5 – 2 = -5 + (-2) = -7
Let’s Practice “Subtraction” 1) 5 – 2 = 2) -3 – 4 = 3) -1 – (-2) = 4) -5 – (-3) = 5) 7 – (-6) =
Let’s Check 1) 5 – 2 = 5 + (-2) = 3 2) -3 – 4 = -3 + (-4) = -7 1) 5 – 2 = 5 + (-2) = 3 2) -3 – 4 = -3 + (-4) = -7 3) -1 – (-2) = -1 + 2 = 1 4) -5 – (-3) = -5 + 3 = -2 5) 7 – (-6) = 7+ 6 = 13
Multi-Step Questions: -8 + 5 – (-2)
2) -4 + (-3) + (-30) + (-2)
3) -1 + (-1) + (-3) – (-3) – (-4)
4) 12 – (-3) + (-12) -5
5) -4 – 5 – 10 – (-2)
Multiplying & Dividing Integers
Did you know that the rules for multiplication and division are the same?
Guess what…. They are!
Rules for Multiplying or Dividing Integers The product (or quotient) of two integers with the same signs is POSITIVE. The product (or quotient) of two integers with different signs is NEGATIVE.
Rules Summary for Multiplication (& Division) + x + = + - x - = + + x - = - - x + = -
When there are more than 2 integers… The product of an even number of negative integers is positive. Ex: (-1)(-2)(-2)(-4) = ______ The product of an odd number of negative integers is negative. Ex: (-1)(-4)(-3) = ______
Let’s Practice “Multiplication” 1) 6 x (-3) = 2) 3 x 3 x -3 = 3) -4 x 5 x (-2) = 4) -6 x (-2) = 5) -7 x (-8) = Will the answer be positive or negative? (-2) x (-3) x (-4) x (-5) x (-6)
Let’s Check 1) 6 x (-3) = -18 2) 3 x 3 x -3 = -27 3) -4 x 5 x -2 = +40
Let’s Practice “Division” 1) 18 ÷ (-2) = 2) -48 ÷ (-6) = 3) -27 ÷ 9 = 4) 64 ÷ 8 = 5) 30 ÷ (-5) =
Let’s Check 1) 18 ÷ (-2) = -9 2) -48 ÷ (-6) = 8 3) -27 ÷ 9 = -3 1) 18 ÷ (-2) = -9 2) -48 ÷ (-6) = 8 3) -27 ÷ 9 = -3 4) 64 ÷ 8 = 8 5) 30 ÷ (-5) = -6
The rules for division are exactly the same as those for multiplication. If we were to take the rules for multiplication and change the multiplication signs to division signs, we would have an accurate set of rules for division.
Rules for Dividing Integers (÷) The quotient of two integers with the same signs is POSITIVE. The quotient of two integers with different signs is NEGATIVE.
Rules Summary for Division Positive ÷ Positive = Positive Negative ÷ Negative = Positive Positive ÷ Negative= Negative Negative ÷ Positive = Negative
Let’s Practice “Division” 1) 18 ÷ (-2) = 2) -48 ÷ (-6) = 3) -27 ÷ 9 = 4) 64 ÷ 8 = 5) 30 ÷ (-5) =
Let’s Check 1) 18 ÷ (-2) = -9 2) -48 ÷ (-6) = 8 3) -27 ÷ 9 = -3 1) 18 ÷ (-2) = -9 2) -48 ÷ (-6) = 8 3) -27 ÷ 9 = -3 4) 64 ÷ 8 = 8 5) 30 ÷ (-5) = -6
Order of Operations Brackets Exponents Division Multiplication Addition Subtraction
Examples: 1) 2 + 42
2) (24-3) ÷ 7
3) 4 + 3 x (-10)
4) (12 x 3) ÷ (1 + 3)
5)(16 – 7 + 3) ÷ [ 4(-2) + 22]
6)
7)
Let’s Review…
What is an integer?
ANSWER An integer is a positive or negative whole number, including 0.
Can you give an example of an integer?
ANSWER …-3, -2, -1, 0, 1, 2, 3…
What are the four operations?
ANSWER The four operations are: addition, subtraction, multiplication, and division.
How do you add integers?
ANSWER The sum of two positive integers is always positive. The sum of two negative integers is always negative. When you add a positive and negative integer, you are really subtracting. Then, you give the answer the sign of the greater absolute value.
How do you subtract integers?
ANSWER To subtract an integer, add its opposite. Same, Change, Change
How do you multiply integers?
ANSWER If the signs are the same, your answer is always positive. If the signs are different, your answer is always negative.
How do you divide integers?
ANSWER If the signs are the same, your answer is always positive. If the signs are different, your answer is always negative. *Multiplication and Division Rules are the same!
15 10 6 The End -7 -9 3 -8