REAL NUMBERS and Their Properties CHAPTER 1.1 REAL NUMBERS and Their Properties

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

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

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.

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

Two Kinds of Real Numbers Rational Numbers Irrational Numbers

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

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

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

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

Real Numbers Irrational numbers Rational numbers Integers Whole

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

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

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

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

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

Four Properties Distributive Commutative Associative Identity properties of one and zero

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

Algebra terms commute when they trade places

This is a statement of the commutative property for addition:

It also works for multiplication:

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

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

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

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

In algebra:

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

In algebra:

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

The Associative Property also works for multiplication:

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)

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

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

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

The identity property makes me think about my identity.

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

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

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

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

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

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

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

1)       26 +0 = 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)       9 + 8 = 8+ 9                        f) Associative Property of Add. 7)       If 32 + 4 = 36 and 36 = 62, then 32 + 4 = 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 5 + 6 = 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 + 0 = 54                                                             e) Transitive 16)    16 – 5 = 16 – 5                                                       f) Associative Property of Addition 17)    If 12 + 4 = 16 and 16 = 42, then 12 + 4 = 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)  6 +11 = 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) .1875                       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.

Example 2: Using the Commutative and Associate Properties Simplify each expression. Justify each step. 29 + 37 + 1 Commutative Property of Addition 29 + 37 + 1 = 29 + 1 + 37 Associative Property of Addition = (29 + 1) + 37 = 30 + 37 Add. = 67

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