PRESENTATION ON THE PROPERTIES OF THE
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The document discusses the six properties of operations on integers: 1. The closure property states that adding or multiplying integers results in an integer. 2. The commutative property does not change the result of addition or multiplication when the order of numbers is changed. 3. The associative property does not change the result when grouping numbers that are added or multiplied is changed. 4. The distributive property states that when two numbers are added/subtracted and multiplied by a factor, the result is the same as multiplying each number by the factor and then adding/subtracting. 5. The identity properties are: addition identity of 0, and multiplication identity of 1. 6. The inverse properties are: additive
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PROPERTIES OF THE OPERATIONS ON INTEGERS
The document discusses the six properties of operations on integers:
1) Closure property - the result of any integer operation is also an integer
2) Commutative property of addition - the order of addends does not change the sum
3) Associative property - the grouping of numbers being added does not change the value
4) Distributive property - multiplying the sum/difference of two numbers by a factor equals multiplying each addend and adding/subtracting the products
5) Identity properties of addition and multiplication - adding/multiplying a number by 0/1 respectively equals the original number
6) Inverse properties of addition and multiplication - adding/multiplying a number by its inverse equals 0/1 respectively
Translating Expressions
The document provides examples and explanations for translating word problems and phrases into algebraic expressions. It gives examples such as "18 less than a number" being translated to "x - 18" and "the product of a number and 5" being "5n". It also provides word problems like writing an expression for the total cost of admission plus rides at a county fair. The document teaches learners how to identify keywords that indicate mathematical operations when translating word phrases into algebraic notation.
Multiplying and dividing fractions
1) This document provides instructions on multiplying and dividing fractions. It explains how to multiply and divide fractions by multiplying or dividing their numerators and denominators.
2) Visual representations are used to demonstrate multiplying fractions, such as fractions multiplied by whole numbers or other fractions. Mixed numbers are also covered.
3) Cancelling terms before and after calculations is discussed as a way to simplify fractions. Dividing fractions is explained as turning the second fraction upside down and multiplying instead of dividing.
Dividing polynomials
This document provides instructions on dividing polynomials using long division and synthetic division. It explains that long division can be used to divide polynomials just as it is used to divide numbers. An example of long division of polynomials is shown. Synthetic division is introduced as an alternative method that is used when the coefficient of the highest term is 1. Examples of both long division and synthetic division of polynomials are provided and step-by-step workings are shown. Some notes are included that synthetic division should be used when the coefficient is 1 and that testing for zero remainder determines if a binomial is a factor.
Multiplying polynomials
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
Polynomials and factoring
The document discusses polynomials and factoring polynomials. It defines polynomials as expressions with terms added or subtracted, where terms are products of numbers and variables with exponents. It provides examples of monomials, binomials, trinomials, and polynomials based on the number of terms. It also discusses finding the greatest common factor of a polynomial to factor out a monomial.
Lesson 1: Special Products
This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.
Subtracting integers
The document discusses subtracting integers using number lines, algebra tiles, and the rule for subtracting integers. It provides examples of subtracting integers with each method and asks students to practice subtracting integers using the different approaches. The document also includes a section connecting subtracting integers to concepts of poverty, hunger, and righteousness from the Beatitudes in the Gospel of Matthew.
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PROPERTIES OF THE OPERATIONS ON INTEGERS
The document discusses the six properties of operations on integers:
1) Closure property - the result of any integer operation is also an integer
2) Commutative property of addition - the order of addends does not change the sum
3) Associative property - the grouping of numbers being added does not change the value
4) Distributive property - multiplying the sum/difference of two numbers by a factor equals multiplying each addend and adding/subtracting the products
5) Identity properties of addition and multiplication - adding/multiplying a number by 0/1 respectively equals the original number
6) Inverse properties of addition and multiplication - adding/multiplying a number by its inverse equals 0/1 respectively
Translating Expressions
The document provides examples and explanations for translating word problems and phrases into algebraic expressions. It gives examples such as "18 less than a number" being translated to "x - 18" and "the product of a number and 5" being "5n". It also provides word problems like writing an expression for the total cost of admission plus rides at a county fair. The document teaches learners how to identify keywords that indicate mathematical operations when translating word phrases into algebraic notation.
Multiplying and dividing fractions
1) This document provides instructions on multiplying and dividing fractions. It explains how to multiply and divide fractions by multiplying or dividing their numerators and denominators.
2) Visual representations are used to demonstrate multiplying fractions, such as fractions multiplied by whole numbers or other fractions. Mixed numbers are also covered.
3) Cancelling terms before and after calculations is discussed as a way to simplify fractions. Dividing fractions is explained as turning the second fraction upside down and multiplying instead of dividing.
Dividing polynomials
This document provides instructions on dividing polynomials using long division and synthetic division. It explains that long division can be used to divide polynomials just as it is used to divide numbers. An example of long division of polynomials is shown. Synthetic division is introduced as an alternative method that is used when the coefficient of the highest term is 1. Examples of both long division and synthetic division of polynomials are provided and step-by-step workings are shown. Some notes are included that synthetic division should be used when the coefficient is 1 and that testing for zero remainder determines if a binomial is a factor.
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Lesson 1: Special Products
This document provides examples and rules for working with exponents and polynomials. It begins by showing the step-by-step working of three multiplication problems involving exponents. It then states the product rule for exponents and the rule for raising a power to another power. The document encourages positive thinking and hard work. It ends by having the reader say a phrase aloud together to reinforce a growth mindset towards math.
Subtracting integers
The document discusses subtracting integers using number lines, algebra tiles, and the rule for subtracting integers. It provides examples of subtracting integers with each method and asks students to practice subtracting integers using the different approaches. The document also includes a section connecting subtracting integers to concepts of poverty, hunger, and righteousness from the Beatitudes in the Gospel of Matthew.
Subtracting integers
The document discusses the rule for subtracting integers. The rule states that to subtract, you add the opposite. So for example, 5 - (-3) means 5 + 3, since the opposite of -3 is 3. It provides examples of subtracting integers like -5 - (-3) and -8 - (-4) using the rule. It also has practice problems for students to evaluate expressions with variables like finding a - b - (-6) when a = 8 and b = -15.
Adding and subtracting polynomials
Adding and subtracting polynomials involves combining like terms. Like terms are terms that have the same variables and the same exponents. To add polynomials, terms are grouped by their like terms and the coefficients are combined by adding them. To subtract polynomials, the process is the same as adding the opposite of the second polynomial. The opposite of each term is found by changing its sign. Then the like terms are combined in the same way as when adding.
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Study about integers. For complete integers lectures (with captions in many different languages), visit my course link on udemy . Link is
https://www.udemy.com/course/basics-of-math/?referralCode=6FF51E8FD0CE20F99D35
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PRESENTATION ON THE PROPERTIES OF THE
- 1. PROPERTIES OF THE OPERATIONS IN THE SET OF INTEGERS
- 2. OBJECTIVES . Identify the Six Properties in the Operation of Integers . Demonstrate and apply the Six Properties in the Operation of Integers
- 3. DRILL Look for the missing numbers 15 + 20 = 20 + _____ 5 X 2 = 2 X _____ 125 = 5 X _____
- 4. REVIEW Answer the following operations 25 + 5 =_____ -21 + (-13) =_____ 14 + (-7) =_____
- 5. Kuwentong Pangkabuhayan Mang Arturo have five carabaos, 4 ducklings and 6 chickens in his backyard. Counting the number of legs, Who among of you can give the mathematical sentence using only plus (+) and equal (=) symbols of the problem?
- 6. Exactly! Fill the blanks with the correct number that will make it exact. (8 + ___ ) + 1 = ___ + ( 5 + 1 ) What are the numbers in the blanks 5 8 What property is illustrated?
- 7. THE SIX PROPERTIES OF THE OPERATIONS ON THE SET OF INTEGERS
- 8. 1. CLOSURE PROPERTY Two integers that are added and multiplied remain as integers. The set of integers is closed under addition and multiplication. The Closure Property for real numbers states that if a and b are real numbers, then a + b is a unique real number. Example 1: Adding two real numbers produces another real number. 15 + 16_ 21 The number "21" is a real number
- 9. : Multiplying two real numbers produces another real number The number "312" is a real number.
- 10. 2. COMMUTATIVE PROPERTY Changing the order of two numbers that are either being added or multiplied does not change the value. a+b = b+a ab = ba Examples: 2 + 3 = 3 + 2, since 2 + 3 = 5 and also 3 + 2 = 5. (-16) +( -5) = (-5) + (-16) 100 + 99 = 99 + 100 (2) (3) = (3) (2), since (2)(3) = 6 and also (3)(2) = 6. (-4) (-15) = (-15) (-4) Note: Subtraction and Division are not commutative.
- 11. 3. ASSOCIATIVE PROPERTY Changing the grouping of numbers that are either being added or multiplied does not change its value. (a + b) + c = a + (b + c) (ab) c = a (bc) Examples: (10 + 25) + 5 = 10 + (25 + 5) 2(6 X 4) = (2 X 6)4
- 12. 4. DISTRIBUTIVE PROPERTY When two numbers have been added/subtracted and then multiplied by a factor, the result will be the same when each number is multiplied by the factor and the products are then added / subtracted. a ( b + c) = ab + ac a ( b – c) = ab – ac Examples: 2( 8 + 9 ) = 2X8 + 2X9 Checking 2 X 17 = 16 + 18 34 = 34 2( 8 – 9) = 2X8 - 2X9 2X (-1) = 16 – 18 -2 = -2
- 13. 5. IDENTITY PROPERTY 5a. Additive Identity - states that the sum of any number and 0 is the given number. Zero is the additive identity. a + 0 = a Examples: 4 + 0 = 4 -10 + 0 = -10 99 + 0 = 99
- 14. 5. IDENTITY PROPERTY 5b. Multiplicative Identity - states that the product of any number and 1 is the given number, a • 1 = a. One is the multiplicative identity. a X 1 = a Examples: 12 x 1 = 12 -32 x 1 = -32 99 x 1 = 99
- 15. 6. INVERSE PROPERTY 6a. Additive Inverse - states that the sum of any number and its additive inverse is zero. The additive inverse of a positive number is the negative of that number, that is a + ( -a) = 0 Example: 10 + (-10) = 0 -19 + 19 = 0
- 16. 6. INVERSE PROPERTY 6b. Multiplicative Inverse Property states that the product of any number and its multiplicative inverse or reciprocal is 1. The multiplicative inverse of the number a is Examples:
- 17. ABSTRACTION We already finished discussing the six properties on the operation of integers; 1. What is Closure Property? 2. What is Commutative Property? 3. What is Associative Property? 4. What is a Distributive Property? 5. What is Identity? a. Identity Property of Addition? b. Identity Property of Multiplication? 6. What is Inverse Property? a. Inverse Property of Addition? b. Inverse Property of Multiplication?
- 18. Kuwentong Pangkabuhayan Mang Arturo have five carabaos, 4 ducklings and 6 chickens in his backyard. Counting the number of legs, Who among of you can give the mathematical sentence using only plus (+) and equal (=) symbols of the problem?