introducing integers copyright 2015 scott storla
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Copyright 2015 Scott Storla
Introducing Integers
Copyright 2015 Scott Storla
If all we have are the whole numbers,
{0,1,2,3…},
we can find this difference 5 – 2 = 3
but we can’t find this difference 2 – 5 = ?
To find the difference 2 – 5 we need to
expand our set of numbers.
Copyright 2015 Scott Storla
Whole numbers
Definition – Integers The natural numbers their negatives and 0.
... 3, 2, 1,0,1, 2, 3...
The Integers
Copyright 2015 Scott Storla
Positive
Money we have
Yards gained
Degrees above 0
Floors above ground
Above sea level
Negative
Money we owe
Yards lost
Degrees below 0
Floors below ground
Below sea level
Copyright 2015 Scott Storla
The Integers
3
3 3
Copyright 2015 Scott Storla
Introducing Integers
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Absolute Value
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Positive
Money we have
Yards gained
Degrees above 0
Floors above ground
Above sea level
Negative
Money we owe
Yards lost
Degrees below 0
Floors below ground
Below sea level
Copyright 2015 Scott Storla
Integers have two attributes,
a “size” (how much, how many) and
a sign (positive or negative).
For instance $2 has a size of 2 and it’s
positive, we have $2.
– $2 also has a size of 2 and it’s negative, we
owe $2.
To discuss the size of a number we use the
absolute value operator.
Copyright 2015 Scott Storla
3 3
3 3 3
Absolute value bars, return
the size of a number.
Absolute value is both an operator and an implicit grouping symbol.
Absolute Value
Copyright 2015 Scott Storla
Operations and Operators
Operation Operator(s)
Addition +
Subtraction
Multiplication
Division
Absolute value
Root
Power 2
Logarithm log ln
Exponential 10 e
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping Operations; in the numerator or denominators of fractions. inside absolute value bars. in radicands or exponents.
1. Start to the left and work right simplifying each operation beyond the basic four as you come to them.
2. Start again to the left and work right simplifying each multiplication or division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each addition or subtraction as you come to them.
Copyright 2015 Scott Storla
4 3 9
12 9
3
3
Count the number of operators, discuss theorder of the operations and then simplify.
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13 1 10
14 10
14 10
4
Count the number of operators, discuss theorder of the operations and then simplify.
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18 16 2 3
18 16 6
18 10
18
28
10
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
14 11 15 13
14 11 2
14 11 2
1
3 2
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
Absolute Value
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Multiplying and Dividing Integers
Copyright 2015 Scott Storla
Solver/Writer Pairs
1. The solver doesn’t hold a writing instrument. The writer does.
2. The solver tells the writer how to transform the problem. a) Count the number of operators. b) Discuss the order for the operations. c) Carry out the order describing one transformation per
line.
3. The writer only includes justified work that is explained well.
4. One solver will be called on to finish the problem using the recorded process.
Copyright 2015 Scott Storla
Procedure – Multiplying or Dividing Two Integers
1. Multiply or divide the absolute values of the two integers.
2. If originally both integers had the same sign, then the result is positive. If they originally had different signs, then the result is negative.
Copyright 2015 Scott Storla
4 3 5
12 5
60
Count the number of operators, discuss theorder of the operations and then simplify.
7 4 3 4
28 12
16
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8 6
2 3
2
8
4
Count the number of operators, discuss theorder of the operations and then simplify.
40 8 5
1
5 5
1
25
1
25
Copyright Scott Storla 2015
9 9 9
3 3 3
( 3)( 3)( 3)
9( 3)
27
Count the number of operators, discuss theorder of the operations and then simplify.
9 9 9
3 3 3
( 3)( 3)( 3)
9( 3)
27
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8 5 2 15
5 2
40 30
5 2
10
5 2
10
10
1
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
Multiplying and Dividing Integers
Copyright 2015 Scott Storla
Adding Integers With Similar Signs
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Procedure – Adding Two Integers with Similar Sign
1. When adding two integers with a common sign add their absolute values and use the common sign.
2 9 4
11 4
15
Simplify
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2 3 2 3
5
11
6
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
6 3 5 2
63
9 7
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright Scott Storla 2015
3 3 3
3 3
9 3
6
12
6
2
Count the number of operators, discuss theorder of the operations and then simplify.
3 3 3
3 3
9 3
6
12
6
2
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4 2 6 6 1
8 6 6
14 6
20
8 6 6 1
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
Adding Integers With Similar Signs
Copyright 2015 Scott Storla
Adding Integers With Different Signs
Copyright 2015 Scott Storla
Procedure – Adding Two Integers with Different Signs
1. Subtract the smaller absolute value from the larger.
2. Attach the original sign of the number that had the larger absolute value.
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10 4
Simplify
4 12
3 4
14 12
10 12
6
1
8
2
2
Copyright Scott Storla 2015
6 3 1
3 1
4
Count the number of operators, discuss theorder of the operations and then simplify.
6 3 1
3 1
4
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10 5 2 4 12
5 2 4 12
7 4 12
11 12
1
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
8 12 11 4 6
4 11 4 6
15 4 6
11 6
5
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
4 15 20 8 2
4 5 8 2
20 16
4
20 8 2
Count the number of operators, discuss theorder of the operations and then simplify.
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112 2 6 2
12 2 6 2
12 2 4
12 8
4
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
Adding Integers With Different Signs
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Subtracting Integers
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Introducing the Procedure
Copyright 2015 Scott Storla
Careful! Meanings for the symbol
Depending on the situation you may find it easier to think of the dash symbol, , as
subtraction, as the negative of, as an opposite or as a factor of 1 . Sometimes it’s
helpful to change your point of view within the same problem. With practice you will
develop ways of understanding which meaning is the most appropriate.
Copyright 2015 Scott Storla
Procedure – Subtracting Two Integers
1. Change the subtraction to addition.
2. Change the number that follows the subtraction to its opposite.
3. Follow the procedure for adding two integers.
Copyright 2015 Scott Storla
Procedure – Subtracting Two Integers
1. Change the subtraction to addition.
2. Change the number that follows the subtraction to its opposite.
3. Follow the procedure for adding two integers.
4 7 8 2
4 7
11 8 2
Simplify
3 2
5
28
Copyright 2015 Scott Storla
10 4
Simplify
10 4
14
14
10 4
10 4
10 4
6
10 4
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8 18
Simplify
8 18
10
26
8 18
8 18
8 18
26
8 18
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Introducing the Procedure
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Continuing With the Procedure
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12 2 6
12 2 6
10 6
16
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
18 2 15 8 5
18 2 15 8 5
20 15 8 5
5 8 5
3 5
2
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
11 18 14 3 12
11 18 14 3 12
7 14 3 12
7 3 12
4 12
8
Count the number of operators, discuss theorder of the operations and then simplify.
11 18 14 3 12
11 18 14 3 12
7 14 3 12
7 3 12
4 12
8
Copyright 2015 Scott Storla
7 2 4 3( )( ) ( )
14 12
26
7 2 4 3( )( ) ( )
Count the number of operators, discuss theorder of the operations and then simplify.
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15 5 4 12( )( )
15 5 4 12( )( )
20 8( )( )
160
Count the number of operators, discuss theorder of the operations and then simplify.
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3 4 5 10( )
3 4 5 10( )
3 9 10( )
27 10
37
Count the number of operators, discuss theorder of the operations and then simplify.
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6 14 6 4 2 8( )
6 14 6 4 2 8( )
6 8 4 6( )
48 24
48 24
72
Count the number of operators, discuss theorder of the operations and then simplify.
Copyright 2015 Scott Storla
Continuing With the Procedure
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Problems With a Factor of -1
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6 4 7( )
6 4 71( )
1
6 1 3( )
6 3
9
Simplify
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2 1 5 4 7
2 1 5 1 4 7
2 6 1 11
12 11
1
Simplify
Copyright Scott Storla 2015
1 5 1 5
5 1 5
5 5
10
Count the number of operators, discuss theorder of the operations and then simplify.
1 5 1 5
5 1 5
5 5
10
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8 16 6 2( )
8 16 1 6 2( )
8 16 1 4( )
8 16 4
8 4 4
Simplify
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14 3 6 2 12 3 3
14 1 3 6 2 1 12 3 3
14 1 9 2 1 12 9
14 18 1 21
14 18 21
32 21
11
( )( ) ( ( ))
( )( ) ( ( ))
( )( ) ( )
( )
Count the number of operators, discuss theorder of the operations and then simplify.
14 3 6 2 12 3 3
14 1 3 6 2 1 12 3 3
14 1 9 2 1 12 9
14 18 1 21
14 18 21
32 21
11
( )( ) ( ( ))
( )( ) ( ( ))
( )( ) ( )
( )
Copyright 2015 Scott Storla
2 2 2 2 2
2 1 2 1 2 2 2
2 1 2 1 2 4
2 1 2 1 6
2 1 2 6
2 1 4
2 4
6
{ [ ( )]}
{ [ ( )]}
{ [ ]}
{ [ ]}
{ }
{ }
Count the number of operators, discuss theorder of the operations and then simplify.
2 2 2 2 2
2 1 2 1 2 2 2
2 1 2 1 2 4
2 1 2 1 6
2 1 2 6
2 1 4
2 4
6
{ [ ( )]}
{ [ ( )]}
{ [ ]}
{ [ ]}
{ }
{ }
Copyright 2015 Scott Storla
Problems With a Factor of -1
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See introductory algebra resources
Copyright 2015 Scott Storla
Fill in each blank with the appropriate property
Show 3 2 is equivalent to1.
3 2
Rewrote subtraction as adding an opposite.
3 2
Rewrote 3 as terms.
(1 2) 2
_____________________.
1 (2 2)
_____________________.
1 0
_____________________.
1
Copyright 2015 Scott Storla
Fill in each blank with the appropriate property
Show 5 7 is equivalent to2 .
5 7
Rewrote 7 as terms.
5 (5 2)
_____________________.
( 5 5) 2
_____________________.
0 2
_____________________.
2
Copyright 2015 Scott Storla
Fill in each blank with the appropriate property
Show 5 7 is equivalent to 12 .
5 7
Rewrote 5 and 7 as products.
1 5 1 7
_____________________.
(5 7)( 1)
_____________________.
(12)( 1)
_____________________.
12
Copyright 2015 Scott Storla
Fill in each blank with the appropriate property
Given ( 1)( 1) 1 Show 3 5 is equivalent to 15 .
3 5
Rewrote as factors.
( 1 3)( 1 5)
_____________________.
( 1 3 1) 5
_____________________.
( 1 1 3) 5
_____________________.
( 1 1) (3 5)
Given and multiplied 3 and 5.
1 15
_____________________.
15
Copyright 2015 Scott Storla
Integer
Automaticity
Quiz
Copyright 2015 Scott Storla
Simplify
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
( 2)( 5) 5 3
2 5 2 8
2 5 14
2
2 5 142
5 2 8 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
10
7
7
3
7
15
6
7
7
10
Answers
Copyright 2015 Scott Storla
Simplify
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
3
3
16
13
13
2
2
3
3
21
Answers
1.
2.
3.
4.
5.
5 8
5 8
8 2
4 9
4 9
6.
7.
8.
9.
10.
63
63
9 12
9 12
9 12
6.
7.
8.
9.
10.
63
63
9 12
9 12
9 12
1.
2.
3.
4.
5.
5 8
5 8
8 2
4 9
4 9
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
( 12)( 15) 17.09 0.4
45 1825 72
12 8
5 12 7
44
1432
3 10511 72
4 21
5 2
17 40 9.511 18
Copyright 2015 Scott Storla
What’s the sign of the result?
1. ( 12)( 15) 6. 17 40
2. 45 1825 72
7. 12 8
3. 5 12 7
8. 44
1432
4. 3 10511 72
9. 4 1
5 8
5. 17 40 10. 9 18
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
12 5 1 4
7 1813
25 72 4 1
5 8
5 12 3
1 42 3
8 12 4 15 8
8 12 1
94
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 12 5 6. 1 4
2. 7 18
1325 72
7. 4 15 8
3. 5 12 3 8.
1 42 3
4. 8 12 9. 4 15 8
5. 8 12 10. 1
94
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
75 2
9 8 10
44 12 63 4 24 7 3
1 13
2 4 4 8 2
3 105 311 72 8
4 66 2
4 40 12 41 17 5
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 7
5 29
6. 8 10
2. 44 12 6 7. 3 4 24 7 3
3. 1 1
32 4
8. 4 8 2
4. 3 105 311 72 8
9. 4 66 2
5. 4 40 12 10. 41 17 5
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 12 15 6. 1 4
2. 458 887 7. 54 97
3. 31 22 8. 32 9
4. 82 12 9. 37 12
5. 8 12 10. 9 1
Copyright 2015 Scott Storla
What’s the sign of the result?
1. 5 17 2 6. 8 10
2. 44 12 6 7. 4 7 23
3. 12 4 30 8. 4 8 2
4. 11 3 55 9. 4 66 2
5. 4 40 12 10. 4 1 35
Copyright 2015 Scott Storla
2 7 3
2 1 7 1 3 1
9 1 3 1
12 1
12
1 12
Simplify
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