1. Ch 2.3 & 2.4 Objective: To solve problems involving operations with integers.

2. Properties Commutative Property: a + b = b + a Two numbers can be added (or subtracted) in either order and the result is always the same. For example: 4 + 6 = 6 + 4 Associative: (a + b) + c = a + (b + c) Three numbers can be added (or subtracted) in any order and the result will always be the same. For example: (1 + 2) + 3 = 1 + (2 + 3) Identity Property: a + 0 = a The sum of 0 and a number will always be the number. For example: 11 + 0 = 11 Inverse Property: a + (-a) = 0 The sum of a number and its opposite will always result in 0.

3. Rules Same Sign (add) Find the sum of all of the numbers that have the same sign, then place their sign in front of the sum. For example: 4 + 5 = 9 -4 – 5 = - 9 (both the 4 and the 5 have a negative sign) Opposite Signs (subtract) Find the difference of the two numbers (the positive value and the negative value), then look for the Strongest/Heaviest number – that is the sign that you place in front of the result. For example: 4 – 9 = -5 (9 is the strongest number)

4. 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 Combining Integers +3 +4

5. 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 Combining Integers -4 -2

6. 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 +1 +5 Combining Integers

7. 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers -3 -5

8. Combining Integers 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers 5) + 5 + 9 = 6) - 3 - 13 = 7) 4 + 6 = 8) -6 + -7 = +14 -16 +10 -13

9. Combining Integers 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers 5) + 5 + 9 = 6) - 3 - 13 = 7) 4 + 6 = 8) -6 + -7 = +14 -16 +10 -13 1) - 3 + 4 = -3 +4 +1

10. Combining Integers 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers 5) + 5 + 9 = 6) - 3 - 13 = 7) 4 + 6 = 8) -6 + -7 = +14 -16 +10 -13 1) - 3 + 4 = +2 -4 +1 2) + 2 - 4 =-2

11. Combining Integers 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers 5) + 5 + 9 = 6) - 3 - 13 = 7) 4 + 6 = 8) -6 + -7 = +14 -16 +10 -13 1) - 3 + 4 = -5 +1 2) + 2 - 4 =-2 3) -5 + 1 =-4

12. Combining Integers 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers 5) + 5 + 9 = 6) - 3 - 13 = 7) 4 + 6 = 8) -6 + -7 = +14 -16 +10 -13 1) - 3 + 4 = +5 -12 +1 2) + 2 - 4 =-2 3) -5 + 1 =-4 4) +5 - 12 =-7 Difference of the numbers

13. 7 6 5 4 3 2 1 0 -2 -3 -4 -5 -6 -7 -8 Same Signs Different Signs 1) + 3 + 4 =+7 2) - 2 - 4 =-6 3) 1 + 5 =+6 4) -5 - 3 =-8 Sum of the numbers 5) + 5 + 9 = 6) - 3 - 13 = 7) 4 + 6 = 8) -6 + -7 = +14 -16 +10 -13 1) - 3 + 4 =+1 2) + 2 - 4 =-2 3) -5 + 1 =-4 4) +5 - 12 =-7 Difference of the numbers 5) - 8 + 3 = 6) +6 - 13 = 7) -9 + 15 = 8) +11 - 4 = -5 -7 +6 +7 Combining Integers

14. Combine the following integers. 1) -6 + 10 = 2) +4 - 9 = 3) -2 + -7 = 4) +6 + 8 = 5) -3 + 12 = 6) -4 - 13 = 7) +5 - 18 = 8) -2 + -6 = 9) +7 - 5 = 10) -9 + -8 = 11) -6 + 4 = 12) -2 + 19 = 13) -8 + -32 = 14) +15 - 4 = +4+4 -5 -9 +14 +9+9 -17 -13 -8-8 +2+2 -17 -2 +17 -40 +11

15. Combine the following integers. 1) -3 - - 10 = 2) +6 - - 9 = 3) -4 + -7 = 4) +6 - 8 = 5) -3 - - 12 = 6) -7 - 12 = 7) +5 - -18 = 8) -14 - - 6 = 9) +7 - - 5 = 10) -7 + -8 = 11) -6 - - 4 = 12) -5 + 16 = 13) -17 - - 12 = 14) +13 - - 4 = +7+7 +15 -11 -2-2 +9+9 -19 +23 -8-8 +12 -15 -2 +11 -5 +17

16. Commutative Properties Commutative Property of Addition a + b = b + a Commutative Property of Multiplication Example: 3 + 5 = 5 + 3 Example: Properties of Real Numbers

17. Are the following operations commutative? 1) Subtraction 2) Division a - b = b - a Counterexamples 8 - 5 = 5 - 8 3 = -3 Therefore, subtraction is not commutative. Counterexample - a single example that proves a statement false. Therefore, division is not commutative.

18. Associative Properties Associative Property of Addition ( a + b ) + c = a + ( b + c ) Associative Property of Multiplication Example: ( 4 + 11 ) + 6 = 4 + ( 11 + 6 )

19. 3 = 7 Are the following operations associative? 1) Subtraction 2) Division (a - b) - c = a - (b - c) (10 - 5) - 2 = 10 - (5 - 2) 5 - 2 = 10 - 3 Therefore, subtraction is not associative. Therefore, division is not associative.

20. Commutative vs. Associative Commutative( Flip-flop )Associative ( Re-group ) Commutative Situations 1) Drinking orange juice and then eating cereal. 2) Doing math homework and then science homework. Flip-flop Re-grouping

21. Identities Identity Property of Addition x + 0 = x Identity Property of Multiplication Properties of Zero Multiplication Property of Zero Division Property of Zero

22. Identify the property shown below. 1) (2 + 10) + 3 = (10 + 2) + 3 2) 3) (6 + 8) + 9 = 6 + (8 + 9) 4) 5) 6) 5 + 0 = 5 7) Comm. Prop. of Add. Mult. Prop. of Zero Assoc. Prop. of Add. Comm. Prop. of Mult. Identity Prop. of Add. Identity Prop. of Mult. Assoc. Prop. of Mult.

23. Use a property to simplify each expression below. Identify the property used. 1) 2) 7 + ( 43 + 29 ) Comm. Prop. of Mult. ( 7 + 43 ) + 29 ( 50 ) + 29 79 Assoc. Prop. of Add.