Objective To solve problems involving operations with integers'

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