chapter 1.1 real numbers and their properties standard: af 1.3 apply algebraic order of operations...
TRANSCRIPT
![Page 1: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/1.jpg)
CHAPTER 1.1
REAL NUMBERS and Their Properties
![Page 2: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/2.jpg)
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
![Page 3: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/3.jpg)
Set
Set Notation
Natural numbers
Whole Numbers
Rational Number
Integers
Irrational Number
Real Numbers All numbers associated with the number line.
Vocab
ula
ry
![Page 4: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/4.jpg)
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
![Page 5: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/5.jpg)
Essential Questions:
• How do you know if a number is a rational number?
• What are the properties used to evaluate rational numbers?
![Page 6: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/6.jpg)
Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers
![Page 7: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/7.jpg)
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
![Page 8: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/8.jpg)
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
![Page 9: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/9.jpg)
Venn Diagram: Naturals, Wholes, Integers, Rationals
Naturals1, 2, 3...
Wholes0
Integers11 5
Rationals
6.7
59
0.8
327
Real Numbers
![Page 10: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/10.jpg)
Venn Diagram: Naturals, Wholes, Integers, Rationals
Naturals1, 2, 3...
Wholes0
Integers11 5
Rationals6.7
59
0.8
327
Real Numbers
![Page 11: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/11.jpg)
Irrational numbersRational numbers
Real Numbers
Integers
Wholenumbers
![Page 12: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/12.jpg)
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
![Page 13: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/13.jpg)
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
![Page 14: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/14.jpg)
Venn Diagram: Naturals, Wholes, Integers, Rationals
Naturals1, 2, 3...
Wholes0
Integers11 5
Rationals
6.7
59
0.8
327
Real Numbers
![Page 15: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/15.jpg)
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
![Page 16: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/16.jpg)
Properties
A property is something that is true for all situations.
![Page 17: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/17.jpg)
Four Properties
1. Distributive
2. Commutative
3. Associative
4. Identity properties of one and zero
![Page 18: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/18.jpg)
We commutewhen we go back and forth
from work to home.
![Page 19: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/19.jpg)
Algebra terms commute
when they trade placesx y
y x
![Page 20: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/20.jpg)
This is a statement of thecommutative property
for addition:
x y y x
![Page 21: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/21.jpg)
It also works for multiplication:
xy yx
![Page 22: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/22.jpg)
![Page 23: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/23.jpg)
Distributive Property
A(B + C) = AB + BC
4(3 + 5) = 4x3 + 4x5
![Page 24: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/24.jpg)
Commutative Propertyof addition and multiplication
Order doesn’t matter
A x B = B x A
A + B = B + A
![Page 25: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/25.jpg)
To associate with someone means that we like to
be with them.
![Page 26: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/26.jpg)
The tiger and the pantherare associating with eachother.
They are leaving thelion out.
( )
![Page 27: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/27.jpg)
In algebra:
( )x y z
![Page 28: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/28.jpg)
The panther has decided tobefriend the lion.
The tiger is left out.
( )
![Page 29: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/29.jpg)
In algebra:
( )x y z
![Page 30: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/30.jpg)
This is a statement of theAssociative Property:
( ) ( )x y z x y z
The variables do not change their order.
![Page 31: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/31.jpg)
The Associative Propertyalso works for multiplication:
( ) ( )xy z x yz
![Page 32: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/32.jpg)
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)
![Page 33: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/33.jpg)
The distributive property onlyhas one form.
Not one foraddition . . .and one for
multiplication
. . .because both operations areused in one property.
![Page 34: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/34.jpg)
4(2x+3)=8x+12
This is an exampleof the distributive
property.
8x 124
2x +3
![Page 35: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/35.jpg)
Here is the distributiveproperty using variables:
( )x y z xy xz
xy xz
y +z
x
![Page 36: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/36.jpg)
The identity
property makes
me thinkabout
myidentity.
![Page 37: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/37.jpg)
The identity property for addition asks,
“What can I add to myselfto get myself back again?
_x x0
![Page 38: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/38.jpg)
The above is the identity propertyfor addition.
_x x0
is the identity elementfor addition.0
![Page 39: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/39.jpg)
The identity property for multiplication
asks, “What can I multiply to myself
to get myself back again?
(_ )x x1
![Page 40: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/40.jpg)
The above is the identity propertyfor multiplication.
1
is the identity elementfor multiplication.1
(_ )x x
![Page 41: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/41.jpg)
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
![Page 42: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/42.jpg)
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.
![Page 43: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/43.jpg)
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
![Page 44: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/44.jpg)
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.
![Page 45: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/45.jpg)
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.
![Page 46: CHAPTER 1.1 REAL NUMBERS and Their Properties STANDARD: AF 1.3 Apply algebraic order of operations and the commutative, associative, and distributive](https://reader034.vdocuments.us/reader034/viewer/2022052603/56649d755503460f94a557d9/html5/thumbnails/46.jpg)
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