Lecture1

Definitions:

One: A single unit.
1 ≡ One

Addition: Combine like units.
+ ≡ Addition

Equal: Same quantity of units.
= ≡ Equal

Associated: Grouped together

( ) ≡ Associated.

1 ≡ ( 1 )

2 ≡ ( 1 + 1 )

3 ≡ ( 2 + 1 )

4 ≡ ( 3 + 1 )

5 ≡ ( 4 + 1 )

6 ≡ ( 5 + 1 )

and so on…

Examples:

Show: 2 + 2 = 4

2 + 2 = 2 + ( 1 + 1 ) by Definition of 2

= (2 + 1) +( 1 ) by Associative Property of Addition

= ( 3 ) +( 1 ) by Definition of 3

= ( 3 + 1 ) by Associative Property of Addition

= 4 by Definition of 4

Hence, 2 + 2 = 4.

Exercises:

1. Indicate the justification for each step:

Show: 3 + 2 = 5

3 + 2 = 3 + ( 1 + 1 ) by Definition of ____________

= (3 + 1) +( 1 ) by ____________ Property of Addition

= ( 4 ) +( 1 ) by ____________of 4

= ( 4 + 1 ) by Associative Property of ____________

= 5 by ____________ of 5. Hence, 3 + 2 = 5.

2. Indicate the justification for each step:

Show: 4 + 2 = 6

4 + 2 = 4 + ( 1 + 1 ) by Definition of ____________

= (4 + 1) +( 1 ) by ____________ Property of Addition

= ( 5 ) +( 1 ) by ____________of 5

= ( 5 + 1 ) by Associative Property of ____________

= 6 by ____________ of 6

Hence, 4 + 2 = 6.

3. Indicate the justification for each step:

Show: 5 + 2 = 7

5 + 2 = 5 + ( 1 + 1 ) by Definition of ____________

= (5 + 1) +( 1 ) by ____________ Property of Addition

= ( 6 ) +( 1 ) by ____________of 6

= ( 6 + 1 ) by Associative Property of ____________

= by ____________ of 7

Hence, 5 + 2 = 7.

4. Indicate the justification for each step:

Show: 6 + 2 = 8

6 + 2 = 6 + ( 1 + 1 ) ________________________________

= (6 + 1) +( 1 ) ________________________________

= ( 7 ) +( 1 ) ________________________________

= ( 7 + 1 ) ________________________________

= ________________________________

Hence, 6 + 2 = 8

5. Indicate the proper values for each step:

Show: 7 + 2 = 9

7 + 2 = 7 + ( _____ +_____ ) by Definition of 2

= (_____+______) + ( 1 ) by Associative Property of Addition

= (________) +( 1 ) by Definition of 8

= (______+______) by Associative Property of Addition

= ________ by Definition of 9

Hence, 7 + 2 = 9.

6. Indicate the proper values and justification for each step:

Show: 8 + 2 = 10

8 + 2 = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) +( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, 8 + 2 = 10.

7. Indicate the proper values and justification for each step:

Show: 9 + 2 = 11

9 + 2 = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) +( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, 9 + 2 = 10.

8. Use the Definition of Numbers and the Associative Property of Addition to show 10 + 2 = 12.

10 + 2 =

=

=

=

=

Hence, 10 + 2 = 12.

9. Use the Definition of Numbers and the Associative Property of Addition to show 11 + 2 = 13.

11 + 2 =

=

=

=

=

Hence, 11 + 2 = 13.

Solutions:

1. Indicate the justification for each step:

Show: 3 + 2 = 5

3 + 2 = 3 + ( 1 + 1 ) by Definition of 2

= (3 + 1) +( 1 ) by ASSOCIATIVE Property of Addition

= ( 4 ) +( 1 ) by DEFINITION of 3

= ( 4 + 1 ) by Associative Property of ADDITION

= 4 by DEFINITION of 4

Hence, 3 + 2 = 5.

2. Indicate the justification for each step:

Show: 4 + 2 = 6

4 + 2 = 4 + ( 1 + 1 ) by Definition of 2

= (4 + 1) +( 1 ) by ASSOCIATIVE Property of Addition

= ( 5 ) +( 1 ) by DEFINITION of 5

= ( 5 + 1 ) by Associative Property of ADDITION

= 6 by DEFINITION of 6

Hence, 4 + 2 = 6

3. Indicate the justification for each step:

Show: 5 + 2 = 7

5 + 2 = 5 + ( 1 + 1 ) by Definition of 2

= (5 + 1) +( 1 ) by ASSOCIATIVE Property of Addition

= ( 6 ) +( 1 ) by DEFINITION of 6

= ( 6 + 1 ) by Associative Property of ADDITION

= by DEFINITION of 7

Hence, 5 + 2 = 7

4. Indicate the Justification for each step:

Show: 6 + 2 = 8

6 + 2 = 6 + ( 1 + 1 ) by Definition of 2

= (6 + 1) +( 1 ) by Associative Property of Addition

= ( 7 ) +( 1 ) by Definition of 7

= ( 7 + 1 ) by Associative Property of Addition

= 8 by Definition of 8

Hence, 6 + 2 = 8.

5. Indicate the values for each step:

Show: 7 + 2 = 9

7 + 2 = 7 + ( 1 + 1 ) by Definition of 2

= ( 7 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 8 ) +( 1 ) by Definition of 8

= ( 8 + 1 ) by Associative Property of Addition

= 9 by Definition of 9

Hence, 7 + 2 = 9.

6. Indicate the proper values and justification for each step:

Show: 8 + 2 = 10

8 + 2 = 8 + ( 1 + 1 ) by Definition of 2

= ( 8 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 9 ) +( 1 ) by Definition of 9

= ( 9 + 1) by Associative Property of Addition

= 10 by Definition of 10

Hence, 8 + 2 = 10.

7. Indicate the proper values and justification for each step:

Show: 9 + 2 = 11

9 + 2 = 9 + ( 1 + 1 ) by Definition of 2

= ( 9 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 10 ) +( 1 ) by Definition of 10

= ( 10 + 1) by Associative Property of Addition

= 11 by Definition of 11

Hence, 9 + 2 = 11.

8. Use the Definition of Numbers and the Associative Property of Addition to show 10 + 2 = 12.

Show: 10 + 2 = 11

10 + 2 = 10 + ( 1 + 1 ) by Definition of 2

= ( 10 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 11 ) +( 1 ) by Definition of 11

= ( 11 + 1) by Associative Property of Addition

= 12 by Definition of 12

Hence, 10 + 2 = 12.

9. Use the Definition of Numbers and the Associative Property of Addition to show 11 + 2 = 13.

Show: 11 + 2 = 13

11 + 2 = 11 + ( 1 + 1 ) by Definition of 2

= ( 11 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 12 ) +( 1 ) by Definition of 12

= ( 12 + 1) by Associative Property of Addition

= 13 by Definition of 13

Hence, 11 + 2 = 13.