Fundamental Tactics for Solving Problems

Slides:



Advertisements
Similar presentations
Chapter 2 Functions and Graphs.
Advertisements

Vectors 5: The Vector Equation of a Plane
Digital Lesson on Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables.
Matrix Algebra Matrix algebra is a means of expressing large numbers of calculations made upon ordered sets of numbers. Often referred to as Linear Algebra.
CMPT 354, Simon Fraser University, Fall 2008, Martin Ester 52 Database Systems I Relational Algebra.
The Terms that You Have to Know! Basis, Linear independent, Orthogonal Column space, Row space, Rank Linear combination Linear transformation Inner product.
Chapter 1 Systems of Linear Equations
ECE 667 Synthesis and Verification of Digital Systems
ENGG2013 Unit 5 Linear Combination & Linear Independence Jan, 2011.
LIAL HORNSBY SCHNEIDER
Fall 2002CMSC Discrete Structures1 Yes, No, Maybe... Boolean Algebra.
Mathematical problem solving strategies used by pre-service primary school teachers Zsoldos-Marchis Iuliana Babes-Bolyai University This research was.
1 New York State Mathematics Core Curriculum 2005.
Logarithmic Functions  In this section, another type of function will be studied called the logarithmic function. There is a close connection between.
Introduction Information in science, business, and mathematics is often organized into rows and columns to form rectangular arrays called “matrices” (plural.
2IV60 Computer Graphics Basic Math for CG
The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard.
1 February 24 Matrices 3.2 Matrices; Row reduction Standard form of a set of linear equations: Chapter 3 Linear Algebra Matrix of coefficients: Augmented.
ECON 1150 Matrix Operations Special Matrices
Chapter 3: Linear Algebra I. Solving sets of linear equations ex: solve for x, y, z. 3x + 5y + 2z = -4 2x + 9z = 12 4y + 2z = 3 (can solve longhand) (can.
Matrices And Linear Systems
Systems of Linear Equation and Matrices
MATH 250 Linear Equations and Matrices
Brandon Graham Putting The Practices Into Action March 20th.
Chapter 1 Section 1.1 Introduction to Matrices and Systems of Linear Equations.
Review of Matrices Or A Fast Introduction.
Multivariate Statistics Matrix Algebra II W. M. van der Veld University of Amsterdam.
A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore:
4.2 An Introduction to Matrices Algebra 2. Learning Targets I can create a matrix and name it using its dimensions I can perform scalar multiplication.
University of Memphis Mathematical Sciences Numerical Analysis “Algebra” (from the Arabic al-jabr, (the mending of) “broken bones”) literally refers to.
Quantum One: Lecture Representation Independent Properties of Linear Operators 3.
Multivariate Statistics Matrix Algebra I W. M. van der Veld University of Amsterdam.
2.6 Formulas A literal equation – an equation involving two or more variables. Formulas are special types of literal equations. To transform a literal.
Deep Dive into the Math Shifts Understanding Focus and Coherence in the Common Core State Standards for Mathematics.
Section 2.3 Properties of Solution Sets
Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson.
Weikang Qian. Outline Intersection Pattern and the Problem Motivation Solution 2.
Chapter 1 Linear Algebra S 2 Systems of Linear Equations.
College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson.
Chapter 1 Systems of Linear Equations Linear Algebra.
Boolean Algebra Computer Architecture. Digital Representation Digital is an abstraction of analog voltage –Voltage is a continuous, physical unit Typically.
Arab Open University Faculty of Computer Studies M132: Linear Algebra
Multi-linear Systems and Invariant Theory
3.2 Solving Systems Algebraically When you try to solve a system of equations by graphing, the coordinates of the point of intersection may not be obvious.
Grade 7 & 8 Mathematics Reporter : Richard M. Oco Ph. D. Ed.Mgt-Student.
Digital Lesson Graphs of Equations. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The graph of an equation in two variables x and.
5 5.1 © 2016 Pearson Education, Ltd. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.
#1 Make sense of problems and persevere in solving them How would you describe the problem in your own words? How would you describe what you are trying.
Introduction to Symmetry Analysis Brian Cantwell Department of Aeronautics and Astronautics Stanford University Chapter 1 - Introduction to Symmetry.
Boolean Algebra.
Lecture from Quantum Mechanics. Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 6.
Lesson 4 – 1.8 – Solving Absolute Value Equations & Inequalities Math 2 Honors - Santowski 7/6/20161 Math 2 Honors.
Chapter 7 Algebraic Structures
5 Systems of Linear Equations and Matrices
… and now for the Final Topic:
Linear Algebra Lecture 4.
Propositional Calculus: Boolean Algebra and Simplification
CSE 541 – Numerical Methods
Chapter 3 Linear Algebra
Linear Algebra Lecture 3.
6 Systems of Linear Equations and Matrices
Chapter 1: Linear Equations in Linear Algebra
MCS680: Foundations Of Computer Science
Chapter 5: Exponential and Logarithmic Functions
Sec 3.5 Inverses of Matrices
Matrix Operations Ms. Olifer.
Some thoughts from the conference
3.5 Perform Basic Matrix Operations Algebra II.
The Graph of an Equation Objective: 1. Sketch graphs of equations 2. Find x- and y-intercepts of graphs of equations 3. Find equations of and sketch graphs.
Presentation transcript:

Fundamental Tactics for Solving Problems Prepared by Almira Cattleya L. Cariño

Fundamental Tactics for Solving Problems What is symmetry? Symmetry in mathematics

Fundamental Tactics for Solving Problems TACTICS are broadly applicable mathematical methods that often simplify problems. STRATEGY alone rarely solves problems; we need the more focused power of tactics (and often highly specialized tools as well) to finish the job. Most of the strategic ideas are plain common sense. In tactical ideas in this easy t use, are less “natural” as few people would think of them. Lets return to our mountaineering analogy for a moment. An important climbing tactics is the rather non- obvious idea ( meant to be taken literally ).

What is Symmetry? Symmetry is a model topic for study in school. It is embedded in reality, it is conceptually simple for younger pupils, and concrete examples around. symmetry also provides various opportunities for students to enjoy learning mathematics. It helps students visualize different geometry concepts and connect learning with their real-life experience. Symmetry involves finding or imposing order in a concrete way for examples by reflections.

Symmetry in mathematics Symmetric functions In algebra In geometry

Symmetric functions In the case of symmetric functions, the value of the output is invariant under permutations of variables. From the form of an equation one may observe that certain permutations of the unknowns result in an equivalent equation. In that case the set of solutions is invariant under any permutation of the unknowns in the group generated by the aforementioned permutations. For example (a − b)(b − c)(c − a) = 10, for any solution (a,b,c), permutations (a b c) and (a c b) can be applied giving additional solutions (b, c, a) and (c, a, b). a2c + 3ab + b2c remains unchanged under interchanging of a and b.

In algebra A symmetric matrix, seen as a symmetric function of the row- and column number, is an example. The second order partial derivatives of a suitably smooth function, seen as a function of the two indexes, is another example. See also symmetry of second derivatives. A relation is symmetric if and only if the corresponding boolean-valued function is a symmetric function. A binary operation is commutative if the operator, as function of two variables, is a symmetric function. Symmetric operators on sets include the union, intersection, and symmetric difference.

In geometry By considering the coordinate space we can consider the symmetry in geometric terms. In the case of three variables we can use e.g. Schoenflies notation for symmetries in 3D. In the example the solution set is geometrically in coordinate space at least of symmetry type C3. If all permutations were allowed this would be C3v. If only two unknowns could be interchanged this would be Cs. In fact, prior to the 20th century, groups were synonymous with transformation groups (i.e. group actions). It's only during the early 20th century that the current abstract definition of a group without any reference to group actions was used instead.