introducing integers copyright 2015 scott storla

Copyright 2015 Scott Storla

Introducing Integers


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.


Whole numbers

Definition – Integers The natural numbers their negatives and 0.

... 3, 2, 1,0,1, 2, 3...

The Integers


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


The Integers

3

3 3


Introducing Integers


Absolute Value


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


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.


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


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.


4 3 9

12 9

3

3

Count the number of operators, discuss theorder of the operations and then simplify.


13 1 10

14 10

14 10

4



18 16 2 3

18 16 6

18 10

18

28

10



14 11 15 13

14 11 2

14 11 2

1

3 2



Absolute Value


Multiplying and Dividing Integers


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.


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.


4 3 5

12 5

60


7 4 3 4

28 12

16


8 6

2 3

2

8

4


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


9 9 9

3 3 3

( 3)( 3)( 3)

9( 3)

27


8 5 2 15

5 2

40 30

5 2

10

5 2

10

10

1



Multiplying and Dividing Integers


Adding Integers With Similar Signs


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


2 3 2 3

5

11

6



6 3 5 2

63

9 7



3 3 3

3 3

9 3

6

12

6

2


3 3 3

3 3

9 3

6

12

6

2


4 2 6 6 1

8 6 6

14 6

20

8 6 6 1



Adding Integers With Similar Signs


Adding Integers With Different Signs


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.


10 4

Simplify

4 12

3 4

14 12

10 12

6

1

8

2

2


6 3 1

3 1

4


6 3 1

3 1

4


10 5 2 4 12

5 2 4 12

7 4 12

11 12

1



8 12 11 4 6

4 11 4 6

15 4 6

11 6

5



4 15 20 8 2

4 5 8 2

20 16

4

20 8 2



112 2 6 2

12 2 6 2

12 2 4

12 8

4



Adding Integers With Different Signs


Subtracting Integers


Introducing the Procedure


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.


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.


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


10 4

Simplify

10 4

14

14

10 4

10 4

10 4

6

10 4


8 18

Simplify

8 18

10

26

8 18

8 18

8 18

26

8 18


Introducing the Procedure


Continuing With the Procedure


12 2 6

12 2 6

10 6

16



18 2 15 8 5

18 2 15 8 5

20 15 8 5

5 8 5

3 5

2



11 18 14 3 12

11 18 14 3 12

7 14 3 12

7 3 12

4 12

8


11 18 14 3 12

11 18 14 3 12

7 14 3 12

7 3 12

4 12

8


7 2 4 3( )( ) ( )

14 12

26

7 2 4 3( )( ) ( )



15 5 4 12( )( )

15 5 4 12( )( )

20 8( )( )

160



3 4 5 10( )

3 4 5 10( )

3 9 10( )

27 10

37



6 14 6 4 2 8( )

6 14 6 4 2 8( )

6 8 4 6( )

48 24

48 24

72



Continuing With the Procedure


Problems With a Factor of -1


6 4 7( )

6 4 71( )

1

6 1 3( )

6 3

9

Simplify


2 1 5 4 7

2 1 5 1 4 7

2 6 1 11

12 11

1

Simplify


1 5 1 5

5 1 5

5 5

10


1 5 1 5

5 1 5

5 5

10


8 16 6 2( )

8 16 1 6 2( )

8 16 1 4( )

8 16 4

8 4 4

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

( )( ) ( ( ))

( )( ) ( ( ))

( )( ) ( )

( )


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

( )( ) ( ( ))

( )( ) ( ( ))

( )( ) ( )

( )


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

{ [ ( )]}

{ [ ( )]}

{ [ ]}

{ [ ]}

{ }

{ }


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

{ [ ( )]}

{ [ ( )]}

{ [ ]}

{ [ ]}

{ }

{ }


Problems With a Factor of -1


See introductory algebra resources


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



Show 5 7 is equivalent to2 .

5 7

Rewrote 7 as terms.

5 (5 2)

_____________________.

( 5 5) 2

_____________________.

0 2

_____________________.

2



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



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


Integer

Automaticity

Quiz


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


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


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



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



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



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



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



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



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



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


2 7 3

2 1 7 1 3 1

9 1 3 1

12 1

12

1 12

Simplify

introducing integers copyright 2015 scott storla

Documents

scott storla copyright

copyright scott storla

scott storlacount

scott storla32 copyright

scott storla25 copyright

scott storla4 copyright

scott storla17 copyright

scott storla19 copyright