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

Rational Number

Integers

Irrational Number

Real Numbers All numbers associated with the number line.

Vocab

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Set A collection of objects.

Set Notation { }

Natural numbers

Counting numbers {1,2,3, …}

Whole Numbers

Natural numbers and 0.{0,1,2,3, …}

Rational Number

Integers Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …}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.

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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 NUMBERS161/23.56-81.3333…-3/4

Irrational Numbers

• 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

Venn Diagram: Naturals, Wholes, Integers, Rationals

Naturals1, 2, 3...

Wholes0

Integers11 5

Rationals

6.7

59

0.8

327

Real Numbers

Venn Diagram: Naturals, Wholes, Integers, Rationals

Naturals1, 2, 3...

Wholes0

Integers11 5

Rationals6.7

59

0.8

327

Real Numbers

Irrational numbersRational numbers

Real Numbers

Integers

Wholenumbers

Whole numbers and their opposites.

Natural Numbers - Natural counting numbers.

1, 2, 3, 4 …

Whole Numbers - Natural counting numbers and zero.

0, 1, 2, 3 …

Integers -… -3, -2, -1, 0, 1, 2, 3 …

Integers, fractions, and decimals.Rational Numbers -

Ex: -0.76, -6/13, 0.08, 2/3

Rational Numbers

AnimalReptile

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!

LizardGecko

Making Connections

Venn Diagram: Naturals, Wholes, Integers, Rationals

Naturals1, 2, 3...

Wholes0

Integers11 5

Rationals

6.7

59

0.8

327

Real Numbers

ReminderReminder

• 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

• IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever.

• Examples: π

3

2

Properties

A property is something that is true for all situations.

Four Properties

1. Distributive

2. Commutative

3. Associative

4. Identity properties of one and zero

We commutewhen we go back and forth

from work to home.

Algebra terms commute

when they trade placesx y

y x

This is a statement of thecommutative property

for addition:

x y y x

It also works for multiplication:

xy yx

Distributive Property

A(B + C) = AB + BC

4(3 + 5) = 4x3 + 4x5

Commutative Propertyof 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 pantherare associating with eachother.

They are leaving thelion out.

( )

In algebra:

( )x y z

The panther has decided tobefriend the lion.

The tiger is left out.

( )

In algebra:

( )x y z

This is a statement of theAssociative Property:

( ) ( )x y z x y z

The variables do not change their order.

The Associative Propertyalso works for multiplication:

( ) ( )xy z x yz

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 onlyhas one form.

Not one foraddition . . .and one for

multiplication

. . .because both operations areused in one property.

4(2x+3)=8x+12

This is an exampleof the distributive

property.

8x 124

2x +3

Here is the distributiveproperty using variables:

( )x y z xy xz

xy xz

y +z

x

The identity

property makes

me thinkabout

myidentity.

The identity property for addition asks,

“What can I add to myselfto get myself back again?

_x x0

The above is the identity propertyfor addition.

_x x0

is the identity elementfor addition.0

The identity property for multiplication

asks, “What can I multiply to myself

to get myself back again?

(_ )x x1

The above is the identity propertyfor multiplication.

1

is the identity elementfor multiplication.1

(_ )x x

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

Associative Property of Addition

The factors are grouped differently.

Solving Equations; 5 Properties of Equality

Reflexive For any real number a, a=a

SymmetricProperty

For all real numbers a and b, if a=b, then b=a

TransitiveProperty

For all reals, a, b, and c, if a=b and b=c, then a=c

1)       26 +0 = 26                             a) Reflexive2)       22 · 0 = 0                                                     b) Additive Identity             3)       3(9 + 2) = 3(9) + 3(2)                                                 c) Multiplicative identity4)       If 32 = 64 ¸2, then 64 ¸2 = 32                            d) Associative Property of Mult.5)       32 · 1 = 32                                                     e) Transitive6)       9 + 8 = 8+ 9                        f) Associative Property of Add.7)       If 32 + 4 = 36 and 36 = 62, then 32 + 4 = 62            g) Symmetric8)       16 + (13 + 8) = (16 +13) + 8                                  h) Commutative Property of Mult.9)       6 · (2 · 12) = (6 · 2) · 12                                         i) Multiplicative property of zero10)  6 ∙ 9 = 6 ∙ 9                                j) Distributive•Complete the Matching Column (put the corresponding letter next to the number)•Complete the Matching Column (put the corresponding letter next to the number)11)    If 5 + 6 = 11, then 11 = 5 + 6                                a) Reflexive12)    22 · 0 = 0                                                                b) Additive Identity             13) 3(9 – 2) = 3(9) – 3(2)                                               c) Multiplicative identity14)    6 + (3 + 8) = (6 +3) + 8                                          d) Associative Property of Mult.15)    54 + 0 = 54                                                             e) Transitive16)    16 – 5 = 16 – 5                                                       f) Associative Property of Addition17)    If 12 + 4 = 16 and 16 = 42, then 12 + 4 = 42             g) Symmetric18)    3 · (22 · 2) = (3 · 22) · 2                                      h) Commutative Property of Addition19)    29 · 1 = 29                                                              i) Multiplicative property of zero20)  6 +11 = 11+ 6                                                           j) DistributiveC.21) Which number is a whole number but not a natural number?a) – 2               b) 3                  c) ½                 d) 022) Which number is an integer but not a whole number?a) – 5               b) ¼                 c) 3                  d) 2.523) Which number is irrational?a)                   b) 4                  c) .1875                       d) .3324) 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

29 + 37 + 1 = 29 + 1 + 37 Commutative Property of Addition

= (29 + 1) + 37

= 30 + 37

Associative Property of Addition

= 67

Add.

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

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