1. Lesson 1.3 Properties of Real Numbers
Objective - To identify properties of real numbers and use them to solve problems.
Commutative Properties
Commutative Property of Addition
a + b = b + a
Example:
3 + 5 = 5 + 3
Commutative Property of Multiplication
Example:
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series
Algebra 1 by James Wenk © 2003 published by TEACHINGpoint

2. Are the following operations commutative?
1) Subtraction
Counterexamples
a - b = b - a
8 - 5 = 5 - 8
3 = -3
Therefore, subtraction is
not commutative.
2) Division
Therefore, division is
not commutative.
Counterexample - a single example that proves
a statement false.
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

3. ( a + b ) + c = a + ( b + c ) ( 4 + 11 ) + 6 = 4 + ( 11 + 6 )
Associative Properties
Associative Property of Addition
( a + b ) + c = a + ( b + c )
Example:
( ) + 6 = 4 + ( )
Associative Property of Multiplication
Example:
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

4. Are the following operations associative?
1) Subtraction
(a - b) - c = a - (b - c)
(10 - 5) - 2 = 10 - (5 - 2)
Therefore, subtraction is
not associative. =
3 = 7
2) Division
Therefore, division is
not associative.
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

5. Commutative vs. Associative
( 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.
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

6. x + 0 = x Identities Identity Property of Addition
Identity Property of Multiplication
Properties of Zero
Multiplication Property of Zero
Division Property of Zero
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

7. 8 0 = ? 8 ounce glass Division by Zero?
= ?
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

8. 8 0 = ? 8 8 = 1 time 8 ounce glass Division by Zero? 8 oz.
= ?
= 1 time
8 oz.
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

9. 8 0 = ? 8 8 = 1 time 8 4 = 8 ounce glass Division by Zero? 4 oz.
= ?
= 1 time =
4 oz.
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

10. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 ounce glass Division by Zero?
= ?
= 1 time
= 2 times
4 oz.
4 oz.
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

11. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 8 ounce glass
Division by Zero?
= ?
= 1 time
= 2 times =
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

12. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 8 ounce glass
Division by Zero?
= ?
= 1 time
= 2 times =
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

13. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 8 ounce glass
Division by Zero?
= ?
= 1 time
= 2 times =
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

14. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 ounce glass
Division by Zero?
= ?
= 1 time
= 2 times
= 4 times
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

15. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 1 = 8 ounce glass
Division by Zero?
= ?
= 1 time
= 2 times
= 4 times =
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

16. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 1 = 8 times
Division by Zero?
= ?
= 1 time
= 2 times
8 oz.
= 4 times
= 8 times
8 ounce glass
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

17. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 1 = 8 times 8 =
Division by Zero?
= ?
= 1 time
= 2 times
= 4 times
= 8 times
8 ounce glass =
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

18. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 1 = 8 times
Division by Zero?
= ?
= 1 time
= 2 times
8 oz.
= 4 times
= 8 times
8 ounce glass
= 16 times
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

19. 8 0 = ? 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 1 = 8 times
Division by Zero?
= ?
= 1 time
= 2 times
= 4 times
= 8 times
8 ounce glass
= 16 times
= ?
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

20. 8 0 = Undefined 8 8 = 1 time 8 4 = 2 times 8 2 = 4 times 8 1 = 8 times
Division by Zero?
= Undefined
= 1 time
= 2 times
= 4 times
= 8 times
8 ounce glass
= 16 times
= ?
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

21. Review
0 8 =
8 0 =
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

22. Review
0 8 =
8 0 =
undefined
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

23. x = 0 x = -5 x = 0 or x = 0 x = -9 x = 1 or x = 3 x = -7
In the expression below the variable x can
represent any real number except what? 1)
x = 0 4)
x = -5
x = 0 or 2)
x = 0 5)
x = -9
x = 1 or 3) 6)
x = 3
x = -7
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

24. Identify the property shown below.
1) 7 + ( ) = ( ) + 5
Assoc. Prop. of Add. 2)
Mult. Prop. of Zero 3)
Comm. Prop. of Mult. 4)
Comm. Prop. of Mult.
5) = 8 + 7
Comm. Prop. of Add.
6) = 12
Identity Prop. of Add. 7)
Identity Prop. of Mult.
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

25. 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
Identity Prop. of Add. 7)
Identity Prop. of Mult.
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

26. Use a property to simplify each expression below.
Identify the property used. 1)
Comm. Prop. of Mult.
2) 7 + ( )
Assoc. Prop. of Add.
( ) + 29
( 50 ) + 29 79
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

27. Use a property to simplify each expression below.
Identify the property used. 3)
Mult. Prop. of Zero 4)
Assoc. Prop. of Mult.
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

28. Closure Property A set of numbers is said to be ‘closed’ if the
numbers produced under a given operation are
also elements of the set.
Tell whether the whole numbers are closed under
the given operation. If not, give a counterexample.
1) Addition
Closed
3) Multiplication
5 + 7 = 12
Closed
6 + 0 = 6
2) Subtraction
Not Closed
4) Division
Not Closed
5 - 7 = -2
2 8 = 0.25
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series

29. Closure Property A set of numbers is said to be ‘closed’ if the
numbers produced under a given operation are
also elements of the set.
Tell whether the integers are closed under the
given operation. If not, give a counterexample.
1) Addition
Closed
3) Multiplication
= -2
Closed
= 4
2) Subtraction
Closed
4) Division
Not Closed
= 12
2 8 = 0.25
Algebra I by James Wenk © 2003 published by TEACHINGpoint as part of the Expert Systems for Teachers Series