CHAPTER 1.1 REAL RATIONAL NUMBERS

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CHAPTER 1.1REAL RATIONAL NUMBERS

• (as opposed to fake numbers?)

and Properties Part 1 (introduction)
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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

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A collection of objects.








Set
Set Notation

Natural numbers
Counting numbers 1,2,3,
Whole Numbers
Natural numbers and 0. 0,1,2,3,
Positive and negative natural numbers and zero
-2, -1, 0, 1, 2, 3,
Integers
Vocabulary
Rational Number
A real number that can be expressed as a ratio of
integers (fraction)
Any real number that is not rational.
Irrational Number
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?

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Two Kinds of Real Numbers

• Rational Numbers
• Irrational Numbers

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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

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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- ii

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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.
-0.76, -6/13, 0.08, 2/3
Ex
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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
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Venn Diagram Naturals, Wholes, Integers,
Rationals
Real Numbers
Rationals
Integers
Wholes
Naturals
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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 p

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

1. Distributive
2. Commutative
3. Associative
4. Identity properties of one and zero

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We commute when we go back and forth from work
to home.
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Algebra terms commute when they trade places
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This is a statement of the commutative
property for addition
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It also works for multiplication
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Distributive Property
A(B C) AB BC
4(3 5) 4x3 4x5
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Commutative Propertyof addition and
multiplication
Order doesnt matter
A x B B x A A B B A
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To associate with someone means that we like to
be with them.
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The tiger and the panther are associating with
each other.
They are leaving the lion out.
( )
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In algebra
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The panther has decided to befriend the lion.
The tiger is left out.
( )
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In algebra
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This is a statement of the Associative Property
The variables do not change their order.
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The Associative Property also works for
multiplication
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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)
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The distributive property only has one form.
Not one for addition
. . .and one for multiplication
. . .because both operations are used in one
property.
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This is an example of the distributive property.
4(2x3)
8x
12
2x 3
8x 12
4
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Here is the distributive property using variables
y z
x
xy xz
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The identity property makes me
think about my identity.
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The identity property for addition asks, What
can I add to myself to get myself back again?
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The above is the identity property for addition.
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The identity property for multiplication asks,
What can I multiply to myself to get myself
back again?
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The above is the identity property for
multiplication.
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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
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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
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Solving Equations 5 Properties of Equality
Reflexive For any real number a, aa
SymmetricProperty For all real numbers a and b, if ab, then ba
TransitiveProperty For all reals, a, b, and c, if ab and bc, then ac
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• 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)
• 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

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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.
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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
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