1. 1-4 Properties of
Real Numbers
I CAN:
Identify and use properties of real numbers

2. Get Ready!
•

3. Focus Question
• Why are the properties of real numbers, such as commutative
and associative properties, useful?

4. BIG Ideas
• Relationships that are always true for real numbers are called
properties, which are rules used to rewrite and compare
expressions.
• Important properties include commutative, associative and
identity properties, the zero property of multiplication and the
multiplication property of -1.

5. Vocabulary to Know
• Equivalent Expressions
• Algebraic expressions that have the same value for all values of
the variables(s).

6. Vocabulary to Know
• Property
• Rules used to rewrite and compare expressions

7. Properties of Real Numbers
• Commutative Properties of Addition and Multiplication
• Changing the order of the addends does not change the sum.
• Changing the order of the factors does not change the product.

8. Properties of Real Numbers
• Commutative Properties of Addition and Multiplication
• Changing the order of the addends does not change the sum.
• Changing the order of the factors does not change the product.
Algebra Example
Addition a+b=b+a 18 + 54 = 54 + 18
Multiplication a·b=b·a

9. Properties of Real Numbers
• Associative Properties of Addition and Multiplication
• Changing the grouping of the addends does not change the sum.
• Changing the grouping of the factors does not change the
product.

10. Properties of Real Numbers
• Associative Properties of Addition and Multiplication
• Changing the grouping of the addends does not change the sum.
• Changing the grouping of the factors does not change the
product.
Algebra Example
Addition (a + b) + c = a + (b + c) (23 + 9) + 4 = 23 + (9 + 4)
Multiplication (a · b) · c = a · (b · c) (7 · 9) · 10 = 7 · (9 · 10)

11. Properties of Real Numbers
• Identity Properties of Addition and Multiplication
• The sum of any real number and 0 is the original number.
• The product of any real number and 1 is the original number.

12. Properties of Real Numbers
• Identity Properties of Addition and Multiplication
• The sum of any real number and 0 is the original number.
• The product of any real number and 1 is the original number.
Algebra Example
Addition a+0=1
Multiplication a·1=a 67 · 1 = 67

13. Properties of Real Numbers
• Zero Property of Multiplication
• The product of a and 0 is 0. a · 0 = 0 18 · 0 = 18

14. Properties of Real Numbers
• Zero Property of Multiplication
• The product of a and 0 is 0. a · 0 = 0 18 · 0 = 18
• Multiplication Property of -1
• The product of -1 and a is –a. -1 ·a = - a -1 · 9 = -9

15. Identifying Properties
•

16. Identifying Properties
•

17. Properties & Mental Math
• You can use properties to help you solve some problems using
mental math.
• A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs
$1.25. What is the total cost for a ticket, a drink, and
popcorn? Use mental math.

18. Properties & Mental Math
• You can use properties to help you solve some problems using
mental math.
• A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs
$1.25. What is the total cost for a ticket, a drink, and
popcorn? Use mental math.
• Use the Commutative Property of Addition
• (7.75 + 2.40) + 1.25 = (2.40 + 7.75) + 1.25

19. Properties & Mental Math
• You can use properties to help you solve some problems using
mental math.
• A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs
$1.25. What is the total cost for a ticket, a drink, and
popcorn? Use mental math.
• Use the Commutative Property of Addition
• (7.75 + 2.40) + 1.25 = (2.40 + 7.75) + 1.25
• Use the Associative Property of Addition
• 2.40 + (7.75 + 1.25)

20. Properties & Mental Math
• You can use properties to help you solve some problems using
mental math.
• A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs
$1.25. What is the total cost for a ticket, a drink, and
popcorn? Use mental math.
• Use the Commutative Property of Addition
• (7.75 + 2.40) + 1.25 = (2.40 + 7.75) + 1.25
• Use the Associative Property of Addition
• 2.40 + (7.75 + 1.25)
• Simplify inside parentheses
• 2.40 + 9

21. Properties & Mental Math
• You can use properties to help you solve some problems using
mental math.
• A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs
$1.25. What is the total cost for a ticket, a drink, and
popcorn? Use mental math.
• Use the Commutative Property of Addition
• (7.75 + 2.40) + 1.25 = (2.40 + 7.75) + 1.25
• Use the Associative Property of Addition
• 2.40 + (7.75 + 1.25)
• Simplify inside parentheses
• 2.40 + 9
• Add
• 11.40
• State your solution: The total cost is $11.40.

22. Properties & Mental Meth
• A can holds 3 tennis balls. A box holds 4 cans. A can holds 6
boxes. How many tennis balls are in 10 cases? Use mental
math!

23. Writing Equivalent
Expressions
• 5(3n)

24. Writing Equivalent
Expressions
• 5(3n)
• Put numbers together…
• (5 · 3)n

25. Writing Equivalent
Expressions
• 5(3n)
• Put numbers together…
• (5 · 3)n
• Simplify 15n

26. Writing Equivalent
Expressions
• (4 + 7b) + 8

27. Writing Equivalent
Expressions
• (4 + 7b) + 8
• Use the Commutative Property of Addition
• (7b + 4) + 8

28. Writing Equivalent
Expressions
• (4 + 7b) + 8
• Use the Commutative Property of Addition
• (7b + 4) + 8
• Use the Associative Property of Addition
• 7b + (4 + 8)

29. Writing Equivalent
Expressions
• (4 + 7b) + 8
• Use the Commutative Property of Addition
• (7b + 4) + 8
• Use the Associative Property of Addition
• 7b + (4 + 8)
• Simplify
• 7b + 12

30. Writing Equivalent
Expressions
•

31. Writing Equivalent
Expressions
•

32. Writing Equivalent
Expressions
•

33. Writing Equivalent
Expressions
•

34. •
Writing Equivalent
Expressions

35. Simplify Each Expression
•

36. Vocabulary to Know
• Deductive Reasoning
• the process of reasoning logically from given facts to a
conclusion.

37. Vocabulary to Know
• Counterexample
• To show that a statement is not true, find an example for which
the statement is not true. A example showing that a statement is
false is a counterexample. You need only one counterexample to
prove that a statement is false.

38. Using Deductive Reasoning and
Counterexamples
• Is the statement true or false? If false, give a counterexample.
• a·b=b+a

39. Using Deductive Reasoning and
Counterexamples
• Is the statement true or false? If false, give a counterexample.
• a·b=b+a
• False
• 5·3≠3+5

40. Using Deductive Reasoning and
Counterexamples
• (a + b) + c = b + (a + c)

41. Using Deductive Reasoning and
Counterexamples
• (a + b) + c = b + (a + c)
• True. You can use the properties of real numbers to prove this
is true.

42. Using Deductive Reasoning and
Counterexamples
• (a + b) + c = b + (a + c)
• True. You can use the properties of real numbers to prove this
is true.
• Use the Associative Property of Addition
• (a + b) + c = (b + a) + c

43. Using Deductive Reasoning and
Counterexamples
• (a + b) + c = b + (a + c)
• True. You can use the properties of real numbers to prove this
is true.
• Use the Associative Property of Addition
• (a + b) + c = (b + a) + c
• Use the Commutative Property of Addition
• (a + b) + c = (a + b) + c

44. Got It?
• Is each statement true or false? If it is false, give a
counterexample. If it is true, use the properties of real
numbers that the expressions are equivalent.
• a. For all real numbers j and k, j · k = (k + 0) · j
• b. For all real numbers m and n, m(n + 1) = mn + 1

45. Got It?
• Is each statement true or false? If it is false, give a
counterexample. If it is true, use the properties of real
numbers that the expressions are equivalent.
• a. For all real numbers j and k, j · k = (k + 0) · j
• b. For all real numbers m and n, m(n + 1) = mn + 1
• c. Is the statement in part (A) false for every pair of real
numbers a and b?

46. Focus Question Answer
• Why are the properties of real numbers, such as the
commutative and associative properties, useful?

47. BIG Ideas
• Relationships that are always true for real numbers are called
properties, which are rules used to rewrite and compare
expressions.
• Important properties include commutative, associative and
identity properties, the zero property of multiplication and the
multiplication property of -1.

48. Assignment
• Pages 29-31
• 1-4
• 7-19 odd
• 20-34 even
• 35-45
• 47-56