The document outlines a 11-day unit on number theory including lessons on prime and composite numbers, factors and prime factorization, greatest common factor, properties of operations, the distributive property, combining like terms, and solving equations using the distributive property. It provides learning standards, vocabulary words, and example problems for each topic as well as a unit review worksheet covering all the lessons.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
umerical algorithm for solving second order nonlinear fuzzy initial value pro...IJECEIAES
The purpose of this analysis would be to provide a computational technique for the numerical solution of second-order nonlinear fuzzy initial value (FIVPs). The idea is based on the reformulation of the fifth order Runge Kutta with six stages (RK56) from crisp domain to the fuzzy domain by using the definitions and properties of fuzzy set theory to be suitable to solve second order nonlinear FIVP numerically. It is shown that the second order nonlinear FIVP can be solved by RK56 by reducing the original nonlinear equation intoa system of couple first order nonlinear FIVP. The findings indicate that the technique is very efficient and simple to implement and satisfy the Fuzzy solution properties. The method’s potential is demonstrated by solving nonlinear second-order FIVP.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
umerical algorithm for solving second order nonlinear fuzzy initial value pro...IJECEIAES
The purpose of this analysis would be to provide a computational technique for the numerical solution of second-order nonlinear fuzzy initial value (FIVPs). The idea is based on the reformulation of the fifth order Runge Kutta with six stages (RK56) from crisp domain to the fuzzy domain by using the definitions and properties of fuzzy set theory to be suitable to solve second order nonlinear FIVP numerically. It is shown that the second order nonlinear FIVP can be solved by RK56 by reducing the original nonlinear equation intoa system of couple first order nonlinear FIVP. The findings indicate that the technique is very efficient and simple to implement and satisfy the Fuzzy solution properties. The method’s potential is demonstrated by solving nonlinear second-order FIVP.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
The matrix game theory gives a mathematical background for dealing with competitive or antagonistic situations arise in many parts of real life. Matrix games have been extensively studied and successfully applied to many fields such as economics, business, management and e-commerce as well as advertising. This paper deals with two-person matrix games whose elements of pay-off matrix are fuzzy numbers. Then the corresponding matrix game has been converted into crisp game using defuzzification techniques. The value of the matrix game for each player is obtained by solving corresponding crisp game problems using the existing method. Finally, to illustrate the proposed methodology, a practical and realistic numerical example has been applied for different defuzzification methods and the obtained results have been compared
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
The matrix game theory gives a mathematical background for dealing with competitive or antagonistic situations arise in many parts of real life. Matrix games have been extensively studied and successfully applied to many fields such as economics, business, management and e-commerce as well as advertising. This paper deals with two-person matrix games whose elements of pay-off matrix are fuzzy numbers. Then the corresponding matrix game has been converted into crisp game using defuzzification techniques. The value of the matrix game for each player is obtained by solving corresponding crisp game problems using the existing method. Finally, to illustrate the proposed methodology, a practical and realistic numerical example has been applied for different defuzzification methods and the obtained results have been compared
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
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This learner's module discusses or talks about the topic Radical Expressions. It also teaches how to recognize basic radical notation. It also teaches the multiplication and division of Radical Expressions.
1. Unit 4: Number Theory Approximate Pacing: 11 days
NYS Common Core Textbook
Topic On Core Vocabulary
Learning Standard Resource
4.OA.4 Prime & Composite / Divisibility Rules 4-1 - divisible, prime,
composite
4.OA.4 Factors & Prime Factorization 4-2 - factor, prime factorization
6.NS.4 Greatest Common Factor 4-3 1-6 greatest common factor
6.NS.4 Properties p. 669 & 1-5 - Commutative Property
~Commutative Property (addition & multiplication) Associative Property
~Associative Property (addition & multiplication) Identity Property of Zero
~Identity Property of Zero (addition) Identity Property of One
~Identity Property of One (multiplication) Property of Zero
~Property of Zero (multiplication)
6.NS.4 Distributive Property p. 669 3-5 Distributive Property
6.NS.4 Distributive Property with GCF - 3-5
6.EE.3 Two Equivalent Expressions / - 3-5 like terms, equivalent
6.EE.4 Distributive Property with Variables expression
Common task:
Secret Number
Unit 4 Lesson 1 Prime & Composite / Divisibility Rules
Unit 4 Lesson 2 Factors & Prime Factorization
Unit 4 Lesson 3 Greatest Common Factor
Unit 4 Lesson 4 Properties
Unit 4 Lesson 5 Distributive Property
Unit 4 Lesson 6 Distributive Property with GCF
Unit 4 Lesson 7 Two Equivalent Expressions (Combining Like Terms)
Unit 4 Lesson 8 Distributive Property with Variables (Combining Like Terms)
Unit 4 Lesson 8 Day 2 Solving Distributive Property with Variables (Combining Like Terms)
2. Name ____________________________________ Date _________________________
Mrs. Labuski / Mrs. Portsmore Period _________ Unit 4 Review
Unit 4 Lesson 1 Prime & Composite / Divisibility Rules
Directions: Tell whether each number is divisible by 2, 3, 4, 5, 6, 9 or 10.
1) 118 _______________________________________________
2) 342 _______________________________________________
3) 170 ______________________________________________
4) 90 _______________________________________________
5) Which numbers is 284 NOT divisible by: 2, 3, 4, 5, 6, 9, and 10
Directions: Tell whether each number is PRIME or COMPOSITE.
6) 121 _______ 7) 61 _______
8) 97 _______ 9) 77 _______
10) 118 _______ 11) Identify the only EVEN number that is PRIME _______
Unit 4 Lesson 2 Factors & Prime Factorization
Directions: List all of the factors of each number. Use “List Method”
12) 60 13) 72
14) 85 15) 56
Directions: Write the Prime Factorization of each number in exponential form.
16) 65 17) 99
18) 76 19) 46
3. Unit 4 Lesson 3 Greatest Common Factor
Directions – Use Factor Trees to find the GCF.
20) 36 and 60
21) 50, 75 and 125
Unit 4 Lesson 4 Properties
Directions – Identify the PROPERTY shown and the missing variable.
Menu of Properties
- Commutative Property of Addition/Multiplication
- Associative Property of Addition/Multiplication
- Identity Property of ZERO With Addition
- Identity Property of ONE With Multiplication
- Property of Zero With Multiplication
22) 3 x (2 x 6) = (3 x 2) x n 23) 64 + 18 = x + 64
n = _______ x = ______
Property = _________________ Property = _________________
24) 99 x n = 99 25) 34 x (32 X 64) = (y x 32) x 64
n = _______ y = _______
Property = _________________ Property = _________________
26) n + 79 = 79 27) 10098 x 0 = r
n = _______ n = _______
Property = _________________ Property = _________________
4. Unit 4 Lesson 5 Distributive Property
Directions – Solve using the Distributive Property.
28) 12(x + 1) 29) 7(a + 13)
30) 23x(x + y) 31) 12x(8x – 5y)
Unit 4 Lesson 6 Distributive Property with GCF
Write each of the following sums as two factors of their GCF and a sum.
32) 24 + 16 33) 30 + 60 34) 49 + 63
______________ _______________ ________________
35) 35a + 40a 36) 9cd + 12c 37) 27x + 72x
______________ _______________ ________________
38) 49x2 + 63x 39) 6a + 15a2 40) 12x2y + 16xy
______________ _______________ ________________
Unit 4 Lesson 7 Two Equivalent Expressions (Combining Like Terms)
Simplify.
41) 8k2 + 4k - 3k2 + 32 - k + 5 42) 10x3 + 5y2 + 2xy - 4y2 + 4xy - x3
__________________________ __________________________
43) 3a + 2b2 + 6c + a - 2c + b2 + c 44) 12x4 + 6x2 + 5x3 - x2 + 2xy - 8x4
__________________________ __________________________
45) 9p6 + q2 + 6p + 5q2 + 5p - 5q2 46) h2 + 4h + 4h2 - h + 4 + h2 + 7h
5. __________________________ __________________________
47. Find the perimeter of the rectangle. 48. Find the perimeter of the triangle.
Unit 4 Lesson 8 Distributive Property with Variables (Combining Like Terms)
Distribute. Simplify.
49. 6(7 + 2m) + 3(5m – 4) 50. 6x(4x + 7) + x
Unit 4 Lesson 8 Day 2 Solving Distributive Property with Variables
Distribute. Simplify. Plug. Solve. (In that order!)
51) 6(4 + m) + 3(6m – 5); x = 2 52) 2m(4 + m) + 3(2m + 8); x = 3
Simplify. Solve.
53) 5y + 2y = 21 54) 9x – 3x = 42
Write as an algebraic expression.
55) 6 plus m multiplied by 6 ___________
56) 54 divided by the sum of 9 and a number_______
Write as an equivalent expression.
6. 57) x + x = _____________________ 58) x • x = __________________________