Download 6 +

7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
+4
Same Signs
1) + 3 + 4 = +7
+3
Different Signs
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs
1) + 3 + 4 = +7
2) - 2 - 4 = -6
-2
-4
Different Signs
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs
+5
+1
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
Different Signs
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs Sum of the numbers
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
-5
-3
Different Signs
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs Sum of the numbers
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
Different Signs
5) + 5 + 9 = +14
6) - 3 - 13 = -16
7) 4 + 6 = +10
8) -6 + -7 = -13
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs Sum of the numbers
-3 +4
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
Different Signs
1) - 3 + 4 = +1
5) + 5 + 9 = +14
6) - 3 - 13 = -16
7) 4 + 6 = +10
8) -6 + -7 = -13
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs Sum of the numbers
+2
-4
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
Different Signs
1) - 3 + 4 = +1
2) + 2 - 4 = -2
5) + 5 + 9 = +14
6) - 3 - 13 = -16
7) 4 + 6 = +10
8) -6 + -7 = -13
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To solve problems involving
operations with integers.
Combining Integers
Same Signs Sum of the numbers
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
-5
Different Signs
+1
1) - 3 + 4 = +1
2) + 2 - 4 = -2
3) -5 + 1 = -4
5) + 5 + 9 = +14
6) - 3 - 13 = -16
7) 4 + 6 = +10
8) -6 + -7 = -13
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To add integers using
properties of real numbers.
Combining Integers
Same Signs Sum of the numbers
+5
-12
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
5) + 5 + 9 = +14
6) - 3 - 13 = -16
7) 4 + 6 = +10
8) -6 + -7 = -13
Different Signs Difference of the numbers
1) - 3 + 4 = +1
2) + 2 - 4 = -2
3) -5 + 1 = -4
4) +5 - 12 = -7
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
Objective- To add integers using
properties of real numbers.
Combining Integers
Same Signs Sum of the numbers
1) + 3 + 4 = +7
2) - 2 - 4 = -6
3) 1 + 5 = +6
4) -5 - 3 = -8
5) + 5 + 9 = +14
6) - 3 - 13 = -16
7) 4 + 6 = +10
8) -6 + -7 = -13
Different Signs Difference of the numbers
1) - 3 + 4 = +1
2) + 2 - 4 = -2
5) - 8 + 3 = -5
6) +6 - 13 = -7
3) -5 + 1 = -4
4) +5 - 12 = -7
7) -9 + 15 = +6
8) +11 - 4 = +7
Combine the following integers.
1) -6 + 10 = +4
8) -2 + -6 = -8
2) +4 - 9 = -5
9) +7 - 5 = +2
3) -2 + -7 = -9
10) -9 + -8 = -17
4) +6 + 8 = +14
11) -6 + 4 = -2
5) -3 + 12 = +9
12) -2 + 19 = +17
6) -4 - 13 = -17
13) -8 + -32 = -40
7) +5 - 18 = -13
14) +15 - 4 = +11
Properties of Real Numbers
Commutative Properties
Commutative Property of Addition
a+b=b+a
Example:
3+5=5+3
Commutative Property of Multiplication
a  b  b  a
Example:
4  7  7  4
Are the following operations commutative?
1) Subtraction
a-b=b-a
2) Division
a  b  b  a
Counterexamples
8-5=5-8
3 = -3
Therefore, subtraction is
not commutative.
8  4  4  8
1
2 
2
Therefore, division is
not commutative.
Counterexample - a single example that proves
a statement false.
Associative Properties
Associative Property of Addition
(a+b)+c = a+(b+c)
Example:
( 4 + 11 ) + 6 = 4 + ( 11 + 6 )
Associative Property of Multiplication
(a  b)  c  a  (b  c)
Example:
( 2  5)  4  2  (5  4)
Are the following operations associative?
1) Subtraction
(a - b) - c = a - (b - c)
(10 - 5) - 2 = 10 - (5 - 2)
5 - 2 = 10 - 3
3 = 7
Therefore, subtraction is
not associative.
2) Division
( a  b )  c  a  ( b  c)
(80  4)  2  80  (4  2)
(20)  2  80  (2)
10  40
Therefore, division is
not associative.
Commutative vs. Associative
Commutative ( Flip-flop )
Associative ( Re-group )
(5  3)  7  7  (5  3)
(5  3)  7  5  (3  7)
Flip-flop
(5  3)  7  (3  5)  7
Flip-flop
Commutative Situations
1) Drinking orange juice
and then eating cereal.
2) Doing math homework
and then science homework.
Re-grouping
Identities
Identity Property of Addition
x+0=x
Identity Property of Multiplication
x 1  x
Properties of Zero
Multiplication Property of Zero
x0  0
Division Property of Zero
x  0  undefined
Identify the property shown below.
1) (2 + 10) + 3 = (10 + 2) + 3 Comm. Prop. of Add.
2) 5  (7  4)  (7  4)  5 Comm. Prop. of Mult.
3) (6 + 8) + 9 = 6 + (8 + 9) Assoc. Prop. of Add.
4) (10  4)  3  10  (4  3) Assoc. Prop. of Mult.
5) 5  0  0 Mult. Prop. of Zero
6) 5 + 0 = 5 Identity Prop. of Add.
7) 5  1  5 Identity Prop. of Mult.
Use a property to simplify each expression below.
Identify the property used.
1) 5  19  2
Comm. Prop. of Mult.
5  2  19
10  19
190
2) 7 + ( 43 + 29 )
( 7 + 43 ) + 29
( 50 ) + 29
79
Assoc. Prop. of Add.