Download - CHAPTER 1.1 REAL RATIONAL NUMBERS (as opposed to fake numbers?) and Properties Part 1 (introduction)
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CHAPTER 1.1REAL RATIONAL NUMBERS
(as opposed to fake numbers?)
and PropertiesPart 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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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.
,2
Vocab
ula
ry
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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 NUMBERS161/23.56-81.3333…-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- īī
17
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Irrational numbersRational numbers
Real Numbers
Integers
Wholenumbers
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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
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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
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Venn Diagram: Naturals, Wholes, Integers, Rationals
Naturals1, 2, 3...
Wholes0
Integers11 5
Rationals
6.7
59
0.8
327
Real Numbers
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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
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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 commutewhen we go back and forth
from work to home.
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Algebra terms commute
when they trade placesx y
y x
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This is a statement of thecommutative property
for addition:
x y y x
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It also works for multiplication:
xy yx
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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 doesn’t 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 pantherare associating with eachother.
They are leaving thelion out.
( )
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In algebra:
( )x y z
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The panther has decided tobefriend the lion.
The tiger is left out.
( )
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In algebra:
( )x y z
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This is a statement of theAssociative Property:
( ) ( )x y z x y z
The variables do not change their order.
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The Associative Propertyalso works for multiplication:
( ) ( )xy z x yz
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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 onlyhas one form.
Not one foraddition . . .and one for
multiplication
. . .because both operations areused in one property.
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4(2x+3)=8x+12
This is an exampleof the distributive
property.
8x 124
2x +3
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Here is the distributiveproperty using variables:
( )x y z xy xz
xy xz
y +z
x
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The identity
property makes
me thinkabout
myidentity.
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The identity property for addition asks,
“What can I add to myselfto get myself back again?
_x x0
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The above is the identity propertyfor addition.
_x x0
is the identity elementfor addition.0
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The identity property for multiplication
asks, “What can I multiply to myself
to get myself back again?
(_ )x x1
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The above is the identity propertyfor multiplication.
1
is the identity elementfor multiplication.1
(_ )x x
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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
Associative Property of Addition
The factors are grouped differently.
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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
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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.
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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.
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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
22
392
224