Integers

SOL Objective 7.3 The student will - a. model addition, subtraction, multiplication and division of integers; and b. add, subtract, multiply and divide integers.

Integers -4 -3 -2 -1 0 1 2 3 4 The set of integers: {…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…} Negative integers are integers that are less than zero. Positive integers are integers that are greater than zero. Zero is neither positive or negative. Negative Integers Positive Integers -4 -3 -2 -1 0 1 2 3 4

Absolute Value -4 -3 -2 -1 0 1 2 3 4 |3| “the absolute value of 3” Two numbers are opposites if they are the same distance from 0 on a number line and are on opposite sides of 0. 3 and -3 are opposites -4 -3 -2 -1 0 1 2 3 4 Absolute Value |3| “the absolute value of 3” “How far away am I from zero?”

How would you use an integer to write… 1. a bank deposit of $250 2. 350 feet below sea level 3. a gain of 12 yards 4. a bank withdrawal of $50 5. a loss of 3 yards 6. an altitude of 4,750 feet 7. 60 seconds before launch

Write these numbers in order, from least to greatest 1. -2, |-2|, 3, |-4| 2. |4|, |-3|, 2, 1 -4 -3 -2 -1 1 2 3 4

Adding Integers Use number chips Use a number line Use addition rules Link -4 -3 -2 -1 1 2 3 4

Adding Integers Addition Rules Same Signs: the sum of two positive integers is positive the sum of two negative integers is negative Add the absolute values and use the same sign for the answer 2 + 4 = 6 |-2| + |-6| 2 + 6 = 8 -2 + (-6) = -8

Adding Integers Addition Rules Different Signs: find the difference of the absolute values give the answer the same sign as the greater absolute value 5 + (-6) |5| |-6| 5 6 6 – 5 = 1 = -1 Subtract the smaller number from the larger number Give the answer the same sign as the greater absolute value

Adding Integers Addition Rules Different Signs: drop the signs subtract the smaller number from the bigger number steal the sign from the bigger number 5 + (-6) 5 6 6 – 5 = = - 1 1

Adding Integers Addition Rules Opposites: any number added to its opposite equals zero 5 + (-5) = 0 -173 + 173 = 0

WE DON’T! Keep Change Opposite Subtracting Integers How do we subtract integers like: -2 – 5? WE DON’T! We turn them into addition problems… we add its opposite by using KCO Keep Change Opposite

Subtracting Integers K C O -2 - 5 = -7 -2 + -5 -2 + (-5) = -7

Subtracting Integers 4 – 7 = -4 – 7 = 4 – (-7) = -4 – (-7) = 4 + (-7) = -3 KCO -4 + (-7) = -11 KCO 4 + 7 = 11 KCO -4 + 7 = 3 KCO

Multiplying Integers When you multiply… the answer is always… two positive integers positive two negative integers positive a positive and a negative integer negative a negative and a positive integer negative 4 • 5 = 20 -4 • -5 = 20 4 • -5 = -20 -4 • 5 = -20 If the signs are the same, the answer’s positive If the signs are different, the answer’s negative

Multiplying Integers Evaluate xyz when x = 2, y = -4,and z = 6 Substitute & Simplify = -8(6) = -48 Evaluate a2 when a = -3 a2 = (-3)2 = -3 • (-3) = 9

Multiplying Integers -xyz - ( )( )( ) -2 -4 6 - (8) (6) - (48) = -48 Evaluate -xyz when x = -2, y = -4,and z = 6 -xyz - ( )( )( ) -2 -4 6 - (8) (6) - (48) = -48

(-2)(-2)(3)(-2)(-1)(-1)(-1)(-1)(-2)(2) (-2)(2)(5)(-1)(-1)(-1) (-2)(-2)(3)(-2)(-1)(-1)(-1)(-1)(-2)(2)

Dividing Integers Use the same rules for division that you use for multiplication 20 ÷ 5 = 4 -20 ÷ -5 = 4 If the signs are the same, the answer’s positive 20 ÷ -5 = -4 -20 ÷ 5 = -4 If the signs are different, the answer’s negative