1. REAL NUMBERS and Their Properties
CHAPTER 1.1
REAL NUMBERS and Their Properties

2. STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions: and justify each step in the process.
Student Objective:
Students will apply order of operations to solve problems with rational numbers and apply their properties, by performing the correct operations, using math facts skills, writing reflective summaries, and scoring 80% proficiency

3. Set
Set Notation
Natural
numbers
Whole
Numbers
Integers
Vocabulary
Rational
Number
Irrational
Number
Real Numbers
All numbers associated with the number line.

4. Set
A collection of objects.
Set Notation
{ }
Natural
numbers
Counting numbers {1,2,3, …}
Whole
Numbers
Natural numbers and 0.
{0,1,2,3, …}
Integers
Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …}
Vocabulary
Rational
Number
A real number that can be expressed as a ratio of integers (fraction)
Irrational
Number
Any real number that is not rational.
Real Numbers
All numbers associated with the number line.

5. Essential Questions: How do you know if a number is a rational number?
What are the properties used to evaluate rational numbers?

6. Two Kinds of Real Numbers
Rational Numbers
Irrational Numbers

7. Rational Numbers
A rational number is a real number that can be written as a ratio of two integers.
A rational number written in decimal form is terminating or repeating.
EXAMPLES OF RATIONAL NUMBERS 16
1/2
3.56 -8
1.3333…
-3/4

8. Irrational Numbers Square roots of non-perfect “squares” Pi- īī
An irrational number is a number that cannot be written as a ratio of two integers.
Irrational numbers written as decimals are non-terminating and non-repeating.
Square roots of non-perfect “squares”
Pi- īī 17

9. Real Numbers Venn Diagram: Naturals, Wholes, Integers, Rationals

10. Real Numbers Venn Diagram: Naturals, Wholes, Integers, Rationals

11. Real Numbers Irrational numbers Rational numbers Integers Whole

12. Rational Numbers Natural Numbers - Natural counting numbers.
1, 2, 3, 4 …
Whole Numbers -
Natural counting numbers and zero.
0, 1, 2, 3 …
Integers -
Whole numbers and their opposites.
… -3, -2, -1, 0, 1, 2, 3 …
Rational Numbers -
Integers, fractions, and decimals.
Ex:
-0.76, -6/13, 0.08, 2/3

13. Making Connections
Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. Rational Numbers are classified this way as well!
Animal
Reptile
Lizard
Gecko

14. Real Numbers Venn Diagram: Naturals, Wholes, Integers, Rationals

15. Reminder
Real numbers are all the positive, negative, fraction, and decimal numbers you have heard of.
They are also called Rational Numbers.
IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever.
Examples: π

16. Properties
A property is something that is true for all situations.

17. Four Properties Distributive Commutative Associative
Identity properties of one and zero

18. We
commute
when we
go back
and forth
from work
to home.

19. Algebra terms commute
when they trade places

20. This is a statement of the
commutative property
for addition:

21. It also works for
multiplication:

23. Distributive Property
A(B + C) = AB + BC
4(3 + 5) = 4x3 + 4x5

24. Commutative Property of addition and multiplication
Order doesn’t matter
A x B = B x A
A + B = B + A

25. To associate with someone
means that we like to
be with them.

26. ( ) The tiger and the panther are associating with each other.
They are leaving the
lion out.
( )

27. In algebra:

28. ( ) The panther has decided to befriend the lion.
The tiger is left out.
( )

29. In algebra:

30. This is a statement of the Associative Property:
The variables do not change
their order.

31. The Associative Property also works for multiplication:

32. Associative Property of multiplication and Addition
Associative Property  (a · b) · c = a · (b · c)
Example: (6 · 4) · 3 = 6 · (4 · 3)
Associative Property  (a + b) + c = a + (b + c)
Example: (6 + 4) + 3 = 6 + (4 + 3)

33. The distributive property only has one form.
Not one for
addition
. . .and one for
multiplication
. . .because both operations are
used in one property.

34. 4(2x+3) =8x +12 This is an example of the distributive property. 2x +3

35. Here is the distributive property using variables:
y z x xy xz

36. The
identity
property
makes me
think
about my
identity.

37. The identity property for addition asks, “What can I add to myself
to get myself back again?

38. The above is the identity property
for addition.
is the identity element
for addition.

39. The identity property for multiplication asks,
“What can I multiply to myself
to get myself back again?

40. The above is the identity property
for multiplication.
is the identity element
for multiplication.

41. Identity Properties
If you add 0 to any number, the number stays the same.
A + 0 = A or = 5
If you multiply any number times 1, the number stays the same.
A x 1 = A or 5 x 1 = 5

42. Example 1: Identifying Properties of Addition and Multiplication
Name the property that is illustrated in each equation.
A. (–4)  9 = 9  (–4) B.
(–4)  9 = 9  (–4)
The order of the numbers changed.
Commutative Property of Multiplication
The factors are grouped differently.
Associative Property of Addition

43. Solving Equations; 5 Properties of Equality
Reflexive
For any real number a, a=a
Symmetric Property
For all real numbers a and b, if a=b, then b=a
Transitive Property
For all reals, a, b, and c, if a=b and b=c, then a=c

44. 1)       = 26                             a) Reflexive
2)       22 · 0 = 0                                                     b) Additive Identity            
3)       3(9 + 2) = 3(9) + 3(2)                                                 c) Multiplicative identity
4)       If 32 = 64 ¸2, then 64 ¸2 = 32                            d) Associative Property of Mult.
5)       32 · 1 = 32                                                     e) Transitive
6)       = 8+ 9                        f) Associative Property of Add.
7)       If = 36 and 36 = 62, then = 62            g) Symmetric
8)       16 + (13 + 8) = (16 +13) + 8                                  h) Commutative Property of Mult.
9)       6 · (2 · 12) = (6 · 2) · 12                                         i) Multiplicative property of zero
10)  6 ∙ 9 = 6 ∙ 9                                j) Distributive
Complete the Matching Column (put the corresponding letter next to the number)
11)    If = 11, then 11 = 5 + 6                                a) Reflexive
12)    22 · 0 = 0                                                                b) Additive Identity            
13) 3(9 – 2) = 3(9) – 3(2)                                               c) Multiplicative identity
14)    6 + (3 + 8) = (6 +3) + 8                                          d) Associative Property of Mult.
15)    = 54                                                             e) Transitive
16)    16 – 5 = 16 – 5                                                       f) Associative Property of Addition
17)    If = 16 and 16 = 42, then = 42             g) Symmetric
18)    3 · (22 · 2) = (3 · 22) · 2                                      h) Commutative Property of Addition
19)    29 · 1 = 29                                                              i) Multiplicative property of zero
20)  = 11+ 6                                                           j) Distributive C.
21) Which number is a whole number but not a natural number?
a) – 2               b) 3                  c) ½                 d) 0
22) Which number is an integer but not a whole number?
a) – 5               b) ¼                 c) 3                  d) 2.5
23) Which number is irrational?
a)                   b) 4                  c)                       d) .33
24) Give an example of a number that is rational, but not an integer.              
25) Give an example of a number that is an integer, but not a whole number.
26) Give an example of a number that is a whole number, but not a natural number. 
27) Give an example of a number that is a natural number, but not an integer.

45. Example 2: Using the Commutative and Associate Properties
Simplify each expression. Justify each step.
Commutative Property of Addition =
Associative Property of Addition
= (29 + 1) + 37 =
Add.
= 67

46. Exit Slip! Name the property that is illustrated in each equation.
1. (–3 + 1) + 2 = –3 + (1 + 2)
2. 6  y  7 = 6 ● 7 ● y
Simplify the expression. Justify each step. 3.
Write each product using the Distributive Property. Then simplify
4. 4(98)
5. 7(32)
Associative Property of Add.
Commutative Property of Multiplication 22
392
224