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Struggling with higher level maths Struggling with higher level maths I am a second year math major here. I took calc III, linear algebra, and differential equations so far, and I realized that I don't have very good study habits. I went to a public high school and I was smart enough to get by without developing good studying skills. During the break, I tried to study ahead for this numerical methods class I'll be taking this semester, but I wasn't able to learn as much as I had wished. If there are any successful (grad) students, post-docs, or educators out there, what are some study habits or advice that you can give to an aspiring mathematician? Practice tutor someone in a math subject that you know and notice how they ask questions or where they get hung up on a problem. Then begin tutoring yourself, ask yourself questions and write them down and then try to find good answers for them. Studying is noticing the techniques used in solving problems, being able to reconstruct the solution from your knowledge looking for the best way to solve problems. Struggling with higher level maths If you don't spend much time studying, analyze how you do spend your time. Maybe that will tell you that you really want to do something besides become a mathematician. Drill and understanding are two different things. You can understand a topic but not be able to do work its problems rapidly. If you had an easy time in high school, perhaps you never had to drill yourself.
Breadcrumbs Algebra Touch 1.0 for OSX – See Why People Enjoy Doing Math Last Updated on Tuesday, 3 July 2012 04:00Written by iUseMacTuesday, 3 July 2012 04:00 Seattle, Washington – Regular Berry Software is pleased to announce Algebra Touch 1.0 for OS X. Algebra Touch is an educational app for OS X and iOS for learning and practicing algebra. The app incorporates an equation editor with iCloud sync so that users may create their own sets of problems and access those problems across all of their devices automatically. Algebra Touch also features 21 interactive lesson topics and supporting practice problems. Algebra Touch enables the excitement of learning through simplified, interactive instruction, with styling and functionality usually reserved for electronic gaming. Algebra Touch for iOS is used in classrooms to supplement lectures and add a tangible method of practicing and exploring math principles. The principles and problem solving capabilities of algebra are, in themselves, fascinating. By simply facilitating the process of discovery, Algebra Touch makes math fun. Feature Highlights: * Appropriate for learning or reviewing of algebra * For students of any age or gender * Enjoy the wonderful conceptual leaps of algebra, without the tedium of traditional methods * Drag to rearrange, click to simplify, and draw lines to eliminate identical terms * Distribute by clicking and sliding, Factor Out by dropping terms on one another * Easily switch between lessons and randomly generated practice problems * Users may create their own sets of problems * Topics include: Simplification, Like Terms, Commutativity, and Order of Operations * Additional topics: Factorization, Prime Numbers, Elimination, and Isolation * Advanced topics: Variables, Solving Equations, Distribution, Factoring Out, and Substitution * Includes support for iCloud sync, which will work with the iOS version of this app as well Located in Seattle, Washington, Regular Berry Software is an educational app software company with the goal to reveal to students that math problems are just puzzles, and can be fun if you know the rules. Copyright (C) 2012 Regular Berry Software LLC. All Rights Reserved. Apple, the Apple logo, iPhone, iPod and iPad are registered trademarks of Apple Inc. in the U.S. and/or other countries. This entry was posted on Tuesday, July 3rd, 2012 at 16:00 and is filed under Mac News. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.
Useful for advanced algebra, trigonometry, computer science, chemistry, and statistics. This durable calculator is a great combination of advanced scientific functions and solar power.Operates in low light; never needs batteries.
Sections of the MAA At present, the MAA has 29 sections, which are defined by ZIP or postal code. Sections are a vital component of the MAA, and a significant part of the Association's activity is centered on them. Each section holds at least one professional meeting per year, usually in the Spring. Section meetings include, but not limited to: Invited lectures Contributed papers Panel discussions Other activities designed to promote and improve collegiate level mathematics Programs of upcoming section meetings, if not available via the links below, may be obtained from the appropriate section secretary. Many sections also conduct activities that involve both high school and college students. These include: Sponsoring mathematics contests Advising state departments of education on teacher certification in the mathematical sciences Working with high schools and colleges on course content and curricula Providing lecturers to colleges and high schools Active MAA members who reside in the United States and Canada have the benefit of being affiliated with one of 29 sections. Find My Section Enter Zip Code or Canadian Postal Code Borders for each section are roughly illustrated in the map below. Click map to see the full-sized version.
How do structural engineers predict how much the Bay Bridge will sway during an earthquake? How does the Federal Reserve analyze stock market fluctuations to help determine when to cut interest rates? What model do bio logists use to describe the rate of stem cell growth in human embryos? The answer to all of these questions involves mathematical objects called functions. The study of functions is where calculus begins! We will mostly look at functions which will be important to us in calculus namely, polynomial, logarithmic and exponential functions. One of the great accomplishments in modern mathematics is to apply these functions to study real world situations. Calculus teaches you how to use functions to study velocities and accele rations of moving bodies, find the firing angle that gives a cannon its greatest range, or calculate the area of irregular regions in the plane. Calculus is a really fun subject because you will learn to use powerful ideas that took centuries to develop. It is also a really challenging subject because it requires solid algebra skills and has thought provoking concepts. Homework: Problems will be as signed at the end of each class. It is important for your success in the course that you attempt to do those problems before the following class meeting. The struggle to solve them prepares you for the following class. You may discuss the homework by forming a group and studying with your peers. If you need help please come to my office or go to Sichel 105, the Academic Support and Achievement Program, and ask for a tutor. Act fast and do not fall behind. A completed homework assignment should be folded lengthwise in half. On the outside front half, print your name, the assignment number , the due date of the assignment, and the time you spent doing the assignment. Please staple your homework. Attendance: Attendance is required and roll will be taken at the beginning of each hour. If you are not in your seat when roll is taken, you may be considered absent, so be on time. You are allowed to miss three classes without affecting your grade. After your grade is dropped one step (A- to B+, C+ to C, etc.). Thereafter, each two successive absences your grade is dropped one step further. If there is a major illness or incapacitation, speak to your instructor. SMC athletes are excused to attend team commitments but are responsible for notifying me ahead of time (see below). Exams: There will be three midterm examinations and a final exam. Suppose a student receives the following grades. First Midterm Second Midterm Third Midterm Final Exam Homework Grade Then the lowest of the Midterm/Final Exam grades above is dropped. If you miss a midterm exam that is the grade you drop. The final grade for the course is the average of the remaining five grades, in this example a B-. The Homework Grade cannot be dropped. Honor code: Students are expected to abide by the SMC honor code when taking exams and doing course work.
The 2013-2014 Loudoun County Public Schools Program of Studies can be found here: Here is the information about Math Courses available for 2013-2014 (please pay particular attention to the teacher-recommended level for your student when considering a selection): Algebra I 540000 Grades: 9-12 Credit: 1 Prerequisite:Mathematics 8 Algebra I incorporates all of the concepts and skills necessary for students to pursue the study of rigorous advanced mathematics.The arithmetic properties of numbers are extended to include the development of the real number system.The fundamental concepts of equality, functions, multiple representations, probability, and data analysis guide the activities that allow students to enhance critical thinking skills.Computers are used as tools to enhance the problem solving process and provide students with visual models that augment the learning of algebraic concepts.Graphing calculators are utilized to enhance the understanding of functions and provide a powerful tool for solving and verifying solutions to equations and inequalities. Emerging technologies are incorporated into the curriculum as they become available. Geometry 550000 Grades: 9-12 Credit: 1 Prerequisite: Algebra I Geometry is the unified study of plane, solid, and coordinate geometric concepts which provides students with the skills requisite for the study of advanced mathematics. Investigations of lines, planes, congruence, similarity, areas, volumes, circles, and three-dimensional shapes are incorporated to provide a complete course of study. Formal and informal deductive reasoning skills are developed and applied to the construction of formal proofs. Opportunities for inquiry-based learning through hands-on activities and experiences that allow for utilizing computer software to explore major concepts and develop critical thinking skills are provided.An emphasis on reasoning, critical thinking, and proof permeates the course and includes two-column proofs, paragraph proofs, and coordinate proofs.Graphing calculators are utilized to enhance the understanding of functions and provide a powerful tool for solving and verifying solutions to equations and inequalities. Emerging technologies are incorporated into the curriculum as they become available.Mathematical communication, reasoning, are emphasized throughout the course. Functions, Algebra, and Data Analysis 565000 Grades: 9-12 Credit: 1 Prerequisite: Algebra I Designing experiments and building mathematical models to describe the experimental results allow students to strengthen conceptual understandings of linear, quadratic, exponential, and logarithmic functions.Within the context of mathematical modeling and data analysis, students will study functions and their behaviors, systems of inequalities, probability, experimental design and implementation, and analysis of data. Data is generated by practical applications arising from science, business, and finance. Students solve problems that require the formulation of linear, quadratic, exponential, or logarithmic equations or a system of equations.Through the investigation of mathematical models and interpretation/analysis of data from real life situations, students strengthen conceptual understandings in mathematics and further develop connections between algebra and statistics.Graphing calculators and other emerging technologies are incorporated into instruction to enhance teaching and learning.Mathematical communication, reasoning, problem solving, critical thinking, and multiple representations are emphasized throughout the course. Algebra II 560000 Grades: 9-12 Credit: 1 Prerequisite: Algebra I and Geometry Algebra IIGraphing calculators and other emerging technologies are incorporated into instruction to enhance teaching and learning.Mathematical communication, reasoning, problem solving, critical thinking, and multiple representations are emphasized throughout the course. Algebra II/Trigonometry, weighted 0.5 571000 Grades: 9-12 Credit: 1 Prerequisite: Algebra I and Geometry Algebra II/TrigonometryThe study of trigonometry includes trigonometric definitions, applications, equations and inequalities. The connections between right triangle ratios, trigonometric functions, and circular functions, are emphasized.Graphing calculators and other emerging technologies are incorporated into instruction to enhance teaching and learning.Mathematical communication, reasoning, problem solving, critical thinking, and multiple representations are emphasized throughout the course. Statistics and Probability 597700 Grades: 10-12 Credit: 0.5 Prerequisite: Algebra II Elementary probability and statistics are studied with an emphasis on collecting data and interpreting data through numerical methods. Specific topics include the binomial and normal distributions, probability, linear correlation and regression, and other statistical methods. Students are expected to understand the design of statistical experiments.They are encouraged to study a problem, design and conduct an experiment or survey, and interpret and communicate the outcomes.Through meaningful activities and simulations, students are provided with experiences that models the means by which data are collected, used, and analyzed.This course enables students to be wise users of statistical methods and more critical consumers of statistical materials.The use of computers and calculators should enhance the learning process and provide students with experiences working with emerging technologies. Discrete Mathematics 599700 Grades: 10-12 Credit: 0.5 Prerequisite: Algebra II Discrete Mathematics involves applications using discrete variables rather than continuous variables.Modeling and understanding finite systems is central to the development of the economy, the natural and physical sciences, and mathematics itself. Discrete Mathematics introduces the topics of social choice as a mathematical application, matrices and their uses, graph theory and its applications, and counting and finite probability, as well as the processes of optimization, existence, and algorithm construction. Emerging technologies are incorporated into thecurriculum as they become available. Advanced Functions and Modeling 572000 Grades: 10-12 Credit: 1 Prerequisite: Algebra II Advanced Functions and Modeling provides opportunities for students to deepen understanding and knowledge of functions based mathematics. Problem solving and critical thinking provide the structure in which functions (polynomial, exponential, logarithmic, transcendental, and rational) are studied. Experimental design provides the foundation for data gathering, curve sketching, and curve fitting in order to provide a graphical interpretation of real world situations.Graphing calculators and other emerging technologies along with the precepts of transformational graphing are incorporated into instruction to enhance teaching and learning.Mathematical communication, reasoning, problem solving, critical thinking, and multiple representations are emphasized throughout the course. Advanced Algebra/Precalculus 585000 Grades: 10-12 Credit: 1 Prerequisite: Algebra II Advanced Algebra/Precalculus emphasizes polynomial, exponential, logarithmic, and rational functions, theory of equations, sequences and series, conic sections, limits, mathematical induction, and the Binomial Theorem . Trigonometry topics include triangular and circular definitions of the trigonometric functions, establishing identities, special angle formulas, Law of Sines, Law of Cosines, and solutions of trigonometric equations.Constructing, interpreting, and using graphs of the various function families are stressed throughout the course of study. Students are encouraged to explore fundamental applications of the topics studied with the use of graphing calculators. Emerging technologies are incorporated into the curriculum as they become available. Mathematical Analysis, weighted 0.5 586000 Grades: 10-12 Credit: 1 Prerequisite: Algebra II/Trigonometry or Advanced Algebra/Precalculus Mathematical Analysis introduces mathematical induction, matrix algebra, vectors, and the Binomial Theorem. A detailed treatment of function concepts provides opportunities to explore mathematics topics deeply and to develop an understanding of algebraic and transcendental functions, parametric and polar equations, sequences and series, conic sections, and vectors. Mathematical Analysis also includes precalculus topics such as limits and continuity, the derivative of functions of a single variable and curve sketching.The course of study is enhanced by making connections of the concepts presented to other disciplines. Students routinely use graphing calculators as tools for exploratory activities and for solving rich application problems. Emerging technologies are incorporated into the curriculum as they become available. Computer Mathematics 593000 Grades: 10-12 Credit: 1 Co-requisite: Algebra II Computer Mathematics provides students with experiences in workplace computer applications, personal finance, essential algebra skills necessary for college mathematics, and computer programming techniques and skills. Students solve problems that can be set up as mathematical models. Students develop and refine skills in logic, organization, and precise expression, thereby enhancing learning in other disciplines. Programming should be introduced in the context of mathematical concepts and problem solving. Students define a problem; develop, refine, and implement a plan; and test and revise the solution. For students entering the 9th grade for the first time in 2010-2011 or after, Computer Mathematics may not count as one of the four mathematics courses required for an Advanced Studies diploma. Computer Science A Advanced Placement, weighted 1.0 with AP exam 595100 Grades: 11-12 Credit: 1 Prerequisite: Computer Mathematics Advanced Placement Computer Science A is taught according to the syllabus for Computer Science A available through the College Entrance Examination Board.Major topics in AP Computer Science A include programming methodology, algorithms, and data structures.Topics are extended to include constructs,data types, functions, testing, debugging, algorithms, and data structures.The JAVA programming language is used to implement computer based solutions to meaningful problems.Treatments of computer systems and the social implications of computing are integrated into the course.College AB Advanced Placement, weighted 1.0 with AP exam 585100 Grades: 11-12 Credit: 1 Prerequisite:Mathematical Analysis or Advanced Algebra/Precalculus Advanced Placement Calculus AB explores the topics of limits/continuity, derivatives, and integrals.These ideas are examined using a multi-layered approach including the verbal, numerical, analytical, and graphical analysis of polynomial, rational, trigonometric, exponential, and logarithmic functions and their inverses.The student is expected to relate the connections among these approaches. Students are also required to synthesize knowledge of the topics of the course to solve applications that model physical, social, and/or economic situations.These applications emphasize derivatives as rates of change, local linear approximations, optimizations and curve analysis, and integrals as Reimann sums, area of regions, volume of solids with known cross sections, average value of functions, and rectilinear motions BC Advanced Placement, weighted 1.0 with AP exam 586100 Grades: 11-12 Credit: 1 Prerequisite:Mathematical Analysis or Calculus AB—Advanced Placement Advanced Placement Calculus BC is intended for students who have a thorough knowledge of analytic geometry and elementary functions in addition to college preparatory algebra, geometry, and trigonometry. Although all of the elements of the Advanced Placement Calculus AB course are included, it provides a more rigorous treatment of these introductory calculus topics. The course also includes the development of the additional topics required by the College Entrance Examination Board in its syllabus for Advanced Placement Calculus BC. Among these are parametric, polar, and vector functions; the rigorous definition of limit; advanced integration techniques; Simpson's Rule; length of curves; improper integrals; Hooke's Law; and the study of sequences and series.The use of the graphing calculator is fully integrated into instruction and students are called upon to confirm and interpret results of problem situations that are solved using available technologyStatistics Advanced Placement, weighted 1.0 with AP exam 598100 Grades: 10-12 Credit: 1 Prerequisite: Algebra II The Advanced Placement Statistics course explores the concepts and skills according to the syllabus available through the College Entrance Examination Board.These topics includecollecting and interpreting data through numerical methods, binomial and normal distribution, probability, linear correlation and regression, analysis of variance, and other descriptive statistical methods. Students should be able to transform data to aid in data interpretation and prediction and test hypotheses using appropriate statisticsMultivariable Calculus (only offered through NVCC--see Counselor) Dual Enrollment, weighted 0.5 583000 Grades: 11-12 Credit: 1 Prerequisite: Calculus BC—Advanced Placement Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus in several variables.Topics may include Euclidean 3-space, vector functions, derivatives and curvature and torsion, Rn space, surface normals, the Taylor polynomial, power and Taylor series, multivariable integration, vector function integration, and theorems by Gauss, Green, and Stokes.
Instructor Class Description Beginning Scientific Computing Introduction to the use of computers to solve problems arising in the physical, biological, and engineering sciences. Application of mathematical judgment, programming architecture, and flow control in solving scientific problems. Introduction to MATLAB routines for numerical programming, computation, and visualization. Prerequisite: either MATH 125, Q SCI 292, MATH 128, or MATH 135. Offered: AWSpS. Class description This course is intended to provide an introduction to the use of computers to solve scientific and engineering problems. Various computational approaches to solve mathematical problems, such as solution of a set of linear equations, curve fitting, solution of differential equations and more (see syllabus) will be presented. The approaches will be covered along with a discussion of their limitations, eventually providing a mathematical judgment in selecting tools to solve scientific problems. MATLAB will be used as the primary environment for numerical computation. Overview of MATLAB's syntax, code structure and algorithms will be given. Although the subject matter of Scientific Computing has many aspects that can be made rather difficult, the material in this course is an introduction to the field and will be presented in a simple as possible way. Theoretical aspects will be mentioned throught the course, but more complicated issues such as proofs of relevant theorems/schemes will not be presented. Applications will be emphasized. Student learning goals MATLAB programming language Set of computational tools to solve basic mathematical problems Limitations of the computational approach General method of instruction MWF - lectures with the help of the computer and on the board T - programming days Eli Shlizerman Date: 09/30/2009 Office of the Registrar For problems and questions about this web page contact icd@u.washington.edu, otherwise contact the instructor or department directly. Modified:May 17, 2013
Factoring and Solving Polynomials - Trinomials this presentation, students are first presented with a challenge problem that I expect them to be unable to solve (but they can solve it after the lesson, which builds their confidence). The lesson focuses on solving 2nd degree trinomials by figuring out the factors of the 3rd term and using those to determine what the coefficient of the middle term should be. I tried to use animations to visualize the thought process behind the method. The last slide includes my homework assignment for this day where I forgot which page it was on! heh Presentation (Powerpoint) File Be sure that you have an application to open this file type before downloading and/or purchasing. 286
Tag Archives: high school math From teachhighschoolmath.blogspot.com Mr. Sladkey's top 10 tips for Math Class 1. Do the example problems from the book. Most math text books have examples that are completely worked out with the solution given. Take a piece of paper and hide … Continue reading → AUTHOR: Neil Priddy Grade Level(s): 10, 11, 12 Subject(s): Mathematics/Measurement OVERVIEW: Parallax is not only used by astronomers to determine the distance of many stars, and other heavenly objects, but is also one of the ways humans use to determine … Continue reading →
A curve fitting program: Lorentzian, Sine, Exponential & Power series are available models to match your data. A Lorentzian series is recommended for real data especially for multiple peaked data. Another improved productivity example. Free Digital Challenge is a set of interactive activities for use in teaching basic digital concepts. The activities give students immediate feedback to reinforce correct responses. All student responses are corrected and graded by the program. Voltmeter Challenge is designed to help you teach students to analyze wiring and troubleshoot circuits using digital voltmeters. Troubleshooting circuits consists of wiring, resistors, lamps, relays, coils, diodes, switches and PC boards. Trigonometry Challenge is designed to help students learn to do calculations related to right triangles and sine waves. Solutions to oblique triangles using the Law of Sines and the Law of Cosines are includes. Solid State Challenge consists of twelve activities to help you teach and learn basic solid-state circuit concepts and troubleshooting. Each use of an activity has new component values and parameters assigned. This program provides realistic troubleshooting activities of almost unlimited variety on DC Power Supplies. Students practice troubleshooting half-wave, full-wave, and bridge type power supply circuits using a voltmeter or oscilloscope.
Programs in Mathematics Mathematics is a more general program that focuses on the analysis of quantities, magnitudes, forms, and their relationships, using symbolic logic and language. It includes instruction in algebra, calculus, functional analysis, geometry, number theory, logic, topology and other mathematical specializations. Mathematics Internships Summer intern position performing test engineering tasks in the Test and Evaluation (T&E) organization in support of various commercial airplane, aerospace and defense product test programs. Participate in various laboratory and/or flight...> Job Description: Full-time, year-round instructor for a unique curriculum that focuses on logical reasoning using advanced mathematical concepts in a fun, exciting atmosphere of games and other challenges. This is NOT a tutoring company;...> Leading in-home tutoring service now seeking qualified tutors for summer months! Experienced tutors for all grades and all subject areas. Local work and flexible hours. New college graduates and retirees welcome. Tutors are needed for all...> What we do: Tri-Ed Tutoring provides private in-home tutoring services for students in grades K-College. Current High Need Areas: Currently looking for tutors in a variety of subjects. However, the most needed areas include upper...> Shmoop ( is a digital curriculum company that makes learning, teaching, and test prep materials that are—dare we say it—smart and fun. We''re looking for an math and science image production intern who''s interested in...> Westminster College at Mesa, a four-year national liberal arts college, invites applications for a part-time, non-tenure track position in the area of Math with teaching responsibilities in in College Algebra, Statistics, and/or Calculus.> Mathematics or Statistics Quantitative Analyst Our Stamford CT. based Consulting Firm specializes in mathematical modeling of risk for financial institutions. This position will be full-time (40 hours per week) during the summer and...> Job Descriptions: Project Description:The summer internship program provides opportunities for educators to learn the dynamic world of large-scale assessment. During the 8-week internship program, interns will work closely with experienced...> Job Descriptions: Project Description:The Assessment Development Division ofEducational Testing Serviceis seeking mathematics teachers at the high school and college level AND graduate students to work with test development staff
The curriculum that will be covered this year is as follows: Discussion of Irrational numbers; work with radicals and integer exponents; understand connections between proportional relationships, lines and linear equations; analyze and solve linear equations and pairs of simultaneous linear equations; functions; use of functions to model problems; understand congruence and similarity in geometry; use the Pythagorean Theorem; solve problems using volume of cylinders, cones and spheres; investigate patterns of association in data(scatter plots, outliers, clustering, etc) ALGEBRA Text: Holt McDougal Algebra I The curriculum to be covered this year include the following: Perform arithmetic operations on polynomials ; understand the relationship between zeros and factors of polynomials; use polynomial identities to solve problems; rewrite rational expressions; create equations that describe numbers or relationships; solve equations and inequalities with one variable; solve systems of equations; represent and solve equations and inequalities graphically Skills taught: basic map skills: oragainzation of information, note-taking and note revision, writing as a fundamental tool of learning; writing various reports and summations, disscussion as a mode of learning (Aristotilian method); using/preparing tables and charts, timelines, using the internet to monitor current events, and using the internet as a resource for a variety of reports (written, poster and group reports), using research as preparation for debates and written statements of opinions on critical issues.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. MATH 3283W. Sequences, Series, and Foundations: Writing Intensive. Spring 2009Homework 1. Problems and Solutions I. Writing Intensive Part 1 (5 points). Check whether or not each of the following statements can be true for some values ("true" or "false") These notes by Mikhail Safonov serve as a supplementary material to the textbook by Weyne Richter "Sequences, Series and Foundations. Math 2283 and 3283W"Sequences, Series and FoundationsChapter 1. Truth, Falsity and Mathematical Induction1 Truth Table Math 5615H. Name (Print)October 5, 2011.Midterm Exam.60 points are distributed between 5 problems. You have 50 minutes (2:30 pm 3:20 pm) to work on these problems. No books, no notes. Calculators are permitted, however, for full credit, you need to sho Math 5615H: Introduction to Analysis I.Fall 2011Homework #2 (due on Wednesday, September 21). 50 points are divided between 5 problems, 10 points each. #1. Let F be a field. Show that there exist not more that two different solutions solutions of the eq Math 5615H: Introduction to Analysis I.Fall 2011Homework #4 (due on Wednesday, October 5). 50 points are divided between 5 problems, 10 points each. #1. Let f be a mapping of A to B. Show that for each B1 B and B2 B, their inverse images satisfy the pro Math 5615H: Introduction to Analysis I.Fall 2011Homework #6 (due on Wednesday, October 19). 50 points are divided between 5 problems, 10 points each. #1. Show that for an arbitrary set E in a metric space (X, d), the set E of its limit point is closed. Math 5615H: Introduction to Analysis I.Fall 2011Homework #7 (due on Wednesday, October 26). 50 points are divided between 4 problems. You can use the following Theorem which was proved in class. Theorem. A subset K of a metric space (X, d) is compact in Math 5615H: Introduction to Analysis I.Fall 2011Homework #1. Problems and short Solutions. #1. Prove that 6 and 2 + 3 are NOT rational. Proof. If p := 6 Q, then p2 = 6, and we get a contradiction in the same way as in Example 1.1 in the textbook. If q : Math 5615H: Introduction to Analysis I. Homework #2. Problems and Solutions.Fall 2011#1. Let F be a field. Show that there exist not more that two different solutions solutions of the equation x x = 1. Is it possible that there is only one solution to t Math 5615H: Introduction to Analysis I. Homework #6. Problems and Solutions.Fall 2011#1. Show that for an arbitrary set E in a metric space (X, d), the set E of its limit point is closed. Proof. Let p be a limit point of E . Then r > 0, the set Gr := Nr Math 5615H: Introduction to Analysis I. Homework #7. Problems and Solutions.Fall 2011You can use the following Theorems which were discussed in class. Theorem 1. A subset K of a metric space (X, d) is compact in X if and only if every infinite subset E Math 5615H: Introduction to Analysis I. Homework #13. Problems and Solutions.Fall 2011#1. If f is a continuous mapping of a metric space X into a metric space Y , prove that f (E) f (E) for every set E X. (E denotes the closure of E). Show, by an exampl Math 8601: REAL ANALYSIS. Fall 2010 Problems for Final Exam on Saturday, December 18, 4pm6pm, VinH 1. This Final Exam will be based on the material from the textbook, in the following Sections: 0.50.6, 1.11.5 ,2.12.5, 2.6 (Theorems 2.40, 2.41), 3.13.2, 3. Math 8601: REAL ANALYSIS. Fall 2010 Some problems for Midterm Exam #1 on Wednesday, October 6. You will have 50 minutes (10:10 am11:00 am) to work on 5 problems, 2 of which will be selected from the following list. It is recommended to prepare solutions o Math 8601: REAL ANALYSIS. Fall 2010 Problems for Midterm Exam #2 on Wednesday, November 17. This Midterm will be based on the material for the textbook up to (including) Section 2.3. You will have 50 minutes (10:10 am11:00 am) to work on 5 problems, 2 of Math 8601: REAL ANALYSIS.Fall 2010Homework #1 (due on W, September 15). Updated on Sat, September 11. 50 points are divided between 5 problems, 10 points each. #1. Let F be a compact subset of Rn . Show that there are point x0 , y0 F , such that diamF : Math 8601: REAL ANALYSIS.Fall 2010Homework #5. Problems and Solutions. #1. Let E be a Lebesgue measurable set in R1 with Lebesgue measure m(E) > 0. Show that for any < 1, there is an open interval I = (a, b) such that m(E I) > m(I). Proof. Suppose that Math 8601. December 18, 2010. Final Exam. Problems and Solutions. #1. Let A be an arbitrary set, and for each A, let an open ball B Rn be defined. Show that there is a finite or countable subset A0 A, such that B =A A0B .Proof. The open set :=AB =j Ed Tell Angela (Anqi) Liu #6 Response: Purpose, Audience, Design (Reading 12)Sec. 002 20415897 A public speaker must be audience-oriented (99). To be more specific, the speaker should take the audience into account when selecting, a topic, preparing, Ed Tell Angela (Anqi) Liu #3 Response: Purpose and ResponsibilitySec. 002 20415897 Try as we may to reveal ourselves as much as possible when bonding with people, we are no longer free to disregard the context in which the communication is taking plac
Definite Integral This section becomes easier to move on to the concept once they have fully grasped the idea of indefinite integrals and had opportunities to practice solving them. In some ways, indefinite integrals can be seen as a way to introduce the 'theory' withdefinite integralsoffering more scope for lecturers and students to apply this knowledge to real world problems. Many studentsgrasp integrationbut make silly errors when completing the two part calculation. Simplepeer assessmentcould help students to think about setting each step of their work minimizing these small but significant slips. This work could be made more engaging by inviting students to mix and match integration problems withgraphical representations. This section needs a lot of practice to get better understanding. Broadly, this section makes the root of everyscientific organization and studies.
Naive Lie Theory (Undergraduate Texts in Mathematics) Book Description: In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra. This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history. John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994). Buyback (Sell directly to one of these merchants and get cash immediately) Currently there are no buyers interested in purchasing this book. While the book has no cash or trade value, you may consider donating it
Composite solids. - Mathematics objective:On completion of the lesson the student will be able to: dissect composite solids into simpler shapes so that the volume can be calculated, calculate the volume of a variety of composite solids, and use formulae appropriately
Mathcad Prime It's the essential tool for engineers, scientists, technicians - in fact anyone who works with numbers and equations. Mathcad is a free-form desktop environment where you can solve equations automatically, plot graphs instantly, combine them with diagrams, tables, annotations and text and produce documents without wasting time.
Site Navigation Outreach Programs F.L. Griffin Mathfest Since 1987, the Reed College mathematics department has sponsored a Saturday workshop in the spring for area high school students and teachers interested in mathematics enrichment. The workshop gives students and teachers a hands-on opportunity to work on interesting problems in a mathematical topic complementary to the high school curriculum. Recent topics have included "Turing Machines," "Trigonometry by Example," "To Capture Infinity," "The Mathematics of Perspective Drawing," and "The Theory of the Rainbow." There is no cost to participants. Lunch and campus tour provided. Registration is limited to 25 students, plus teachers. Program schedule: Saturday, February 9, 2013 9:30 am to 3:00 pm The Mathematics of Ornamental Art Prerequisites: • Junior or senior standing in high school • Familiarity with the basic concepts of Euclidean geometry • Serious interest in the topic • Pre-registration required
Purchasing Options Features Provides students with computational skills and the ability to read and write proofs, helping them succeed in further mathematical studies Begins with the calculus of one variable and then proceeds to multivariable calculus Presents all of the theorems and definitions in a careful and complete manner Weaves historical facts into the main presentation, making the text interesting and more motivating for students Emphasizes the reasons why certain topics are being studied Solutions manual available with qualifying course adoption Summary This textbook strengthens students' computational skills, teaches them how to prove theorems, and helps them understand proofs. Designed for junior and senior undergraduates, the book includes historical background for many concepts and theorems. It begins with the calculus of one variable and progresses to multivariable calculus in the second half of the book. An introductory section in each chapter reviews the material covered in earlier calculus courses, easing students into the more serious material. A solutions manual is available with qualifying course adoption. Table of Contents Sequences and Their Limits. Real Numbers. Continuity. The Derivative. The Indefinite Integral. The Definite Integral. Infinite Series. Sequences and Series of Functions. Functions of Several Variables. Derivatives. Implicit Functions and Optimization. Integrals Depending on a Parameter. Integration in Rn.
Sure, you could use it, but not as a full blown resource. You're in college, and this seems like it was written for middle/high school. Second of all, there's a whole chapter on quadratic equations, which is not linear. No. I would suggest you buy some typing paper, printer paper, and start writing. Write big and be neat enough for someone to follow what you do. Here's what to do, using the examples in the book copy them down, write them out on the typing paper. As you write think to yourself, where did this come from and or what is my next step. But continue to copy the problem, example, down until finished. Now, put that paper to the side and do the same example again, think as you go, but, if you get stuck look back and keep going and thinking, "what do I do next". Ok, do it again and again until you no longer need to look back. At this point, you're in good shape try a problem, look back if necessary, but don't panic, addition, subtraction, multiplication, division is all that's required. You are just learning new ways of doing things you already know how to do, solve an equation or group of equations. If you don't use all 500 or so sheets of paper in learning one section, you're doing something wrong. :-) You're learning the steps and procedure for the method being taught. For this method, you do this, this way, so to speak, write.
a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. The first half forms an introductory graduate course on vorticity and incompressible flow. The second half comprises a modern applied mathematics graduate course on the weak solution theory for incompressible flow.
Supplies needed for this class are a 3 ring binder, loose paper, and dividers. This will be used exclusively for General Physics. A scientific calculator is necessary, a graphing calculator is not recquired. This is a high school level introductory course in Physics appropriate for any student interested in science. It will emphasize concepts as well as mathematical models as they apply to our studies. Supplies needed for this class are a three ring binder, loose paper and dividers. This will be used exclusively for AP Physics. A scientific calculator is necessary, a graphing calculator is not recquired. This is an algebra based, introductory course in Physics equivalent to a college course as outlined by the AP College Board.
MATH 112 Introductory Survey For each of the topics listed below, rate your understanding of the topic on a scale of 1 to 10. Here a 1 corresponds to "I don't understand this topic" and a 10 correspondes to "I would feel confident teaching this topic". You may put your name on this paper if you wish, but you are not required to. When you have finished, please use the remainder of the page to write down what you would like to learn in MATH112.
Why study Mathematics? If you enjoy Maths, particularly the logical processes of algebra and problem solving, then A-level Maths is the course for you. You will be able to develop your skills further and begin to see how Maths can relate to the real world. What do I study? The course is divided into three probability distributions and discrete random variables The A2 is very similar in that you study a further two core units together with one statistics unit. Where does Mathematics take me? A wide area of opportunity is open to students with an A-level in Mathematics. They can apply for degrees in areas such as Accounts, Business, Computing, Engineering, Geography, Law, Management, any Science course, Psychology, Technology, to name but a few. All Mathematics qualifications at this level are much sought after and highly valued by employers and universities. What do I need? A minimum grade B in GCSE Mathematics. There is no coursework in this A-level. DOUBLE MATHS – MATHS & FURTHER MATHS Why study Double Maths? As the title suggests, this course will lead to two A-levels in Mathematics. It is the qualification to gain if you are good at and enjoy Mathematics. This qualification is highly regarded by all Higher Education institutions and employers and can be studied alongside any other A-levels. What do I study? The course is divided into four the normal distribution and discrete random variables • mechanics – covering areas such as forces, vectors and the motion of particles In the first year you will study three core units, one mechanics unit and two decision units. In the second year you will study a further three core units, two statistics units and one more mechanics unit. Where does Double Maths take me? This course is almost essential for a Maths degree. The two A-levels will also be highly valued for any degree with a Maths element, e.g. Business, Computing, Engineering, Geography, Law, Management, Insurance, any Science subject, Technology etc. They are also useful when applying for highly competitive areas such as law and psychology. It goes without saying that any employer would value two A-levels in Mathematics extremely highly. What do I need? A minimum grade A in GCSE Mathematics. There is no coursework in these A levels. USE OF MATHEMATICS Why study Use of Mathematics? This course is aimed at students who wish to develop their knowledge of Mathematics. It covers mathematical and statistical techniques used in A-level subjects such as Biology, Business Studies, Economics, Geography and Psychology. If you enjoy Maths, this course would provide useful support for these subjects as well as being a qualification in its own right. What do I study? The first year course is divided into three modules: • algebra • data analysis • decision maths In the second year you would study Calculus and Mathematical comprehension; you would also complete a piece of coursework in year two. Where does Use of Mathematics take me? Use of Mathematics is intended to support students who need Maths or Statistics in their degree course. These skills are embedded in a large number of degree courses such as Business, Geography, Economics, Law, Management and Psychology. All Mathematics qualifications at this level are much sought after and highly valued by employers and universities. What do I need? The entry requirement is Grade C from the higher tier of GCSE Mathematics. You will also need a Graphic Display Calculator – the Casio fx-9750G would be ideal. MEDIA STUDIES Why study Media Studies? Media Studies is a rigorous academic subject both visually interesting and dynamic in its content. It is vocationally relevant to a range of professions from Journalist to Magazine Editor, Film Scriptwriter to TV Producer to Advertising Copywriter. It is also one of the fastest expanding courses at university. Studying the media gives you a chance of understanding one of the most powerful and influential forms of mass communication available. What do I study? Year 1 (AS) Unit 1: Media Representations and Responses A diverse and exciting examination of the content of media forms, e.g. Advertising, Television, New Digital Media, Film, Radio and Newspapers analysing how they communicate with their target audience. Students will also examine how individuals and social groups are represented in a wide range of media. The unit carries an external assessment of a 2½ hour written exam.
Hi Friends. Ever since I have started variables and expressions worksheet for 5th graders at college I never seem to be able to learn it well. I am a topper at all the other branches, but this particular chapter seems to be my weak point. Can some one help me in learning it well? Thanks for the tip. Algebrator is quite a life-saving algebra software. I was able to get answers to difficulties I had about radical inequalities, cramer's rule and multiplying matrices. You only need to type in a problem, click on Solve and you get the all the solutions you need. You can use it for so many, like Algebra 1, Algebra 1 and Algebra 2. I would highly recommend Algebrator. Is the software really that helpful? I'm just concerned because the software might not really help because it only solves the problem per ?e. I like to understand how a problem is answered and not only find out the answer. Nevertheless, could you give me a link for this software? Algebrator is a user friendly product and is certainly worth a try. You will also find lot of exciting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more enjoyable.
Signature Math Core Courses for Algebra Readiness and Algebra I Solving algebra. Sounds simple when you say it, but getting students to solve algebra is a real challenge today in education. Statistics support the fact that students continue to struggle in overcoming this critical milestone in obtaining a high school diploma. Signature Math solves the algebra problem by combining 181 computer-based lessons and 35 hands-on, small-group activities to build mastery of pre-algebra and Algebra I concepts. Signature Math is a blended learning model of one-to-one computing and teacher-led instruction that combines 35 small-group, hands-on learning activities and 181 computer-based lessons. Students build mastery of critical math concepts and experience real-world learning applications that make Algebra meaningful and relevant. Students begin by taking an assessment to determine their level of knowledge of pre-algebra and Algebra I concepts and are then assigned Individualized Prescriptive Lessons™ (IPLs). As students master each concept, they advance to the next lesson. Learning is reinforced and the concepts are applied in hands-on, Culminating Group Activities (CGAs), which are teacher-led in a whole-class learning environment.
Math 132: Precalculus II PLEASE NOTE Beginning Summer 2009, this course will be known as Math 142; only the course number will change. Course Description Math 132 is the second course in a two-quarter precalculus sequence that also includes Math 131. Topics include: polynomial, rational, trigonometric, and inverse trigonometric functions; and applications involving these functions and functions from Math 131. Who should take this course? Generally, students seeking to take the 151–152–153 calculus sequence take the 131–132 precalculus sequence first. Some students in programs like business take this course (in place of Math 140) and then take Math 150 instead of Math 132. You should consult the planning sheet for your program and consult an advisor to determine if this sequence is appropriate for you. Who is eligible to take this course? The prerequisite for this course is Math 131 with a grade of 2.0 or higher. Is this course transferable? This course transfers to the University of Washington as UW Math 120 if both Math 131 and Math 132 are taken; consult an advisor or see the Transfer Center to determine transferability to other institutions. What textbook is used for this course? The first edition of Precalculus Concepts and Functions: A Unit Circle Approach by Michael Sullivan and Michael Sullivan III; a lower-priced custom version comprising Chapters 1–7 and 9 is available through the EdCC Bookstore. What else is required for this course? Students are required to have a graphing calculator; the TI-83 Plus or TI-84 Plus is recommended.
LAMBDA^2 - Math is everywhere "An approach to make math easily accessible to every visually impaired pupil, student and employed person." or "How to combine the advantages of LaTeX, LiTeX, LAMBDA and others in an all-in-one solution." Math should not be accessible only to sighted and special trained visually impaired persons. With LAMBDA^2 a new European research project will start with the perspective to provide an all-in-one solution that provides all mathematical needs for school and university use and beyond. Therefore several additional developments are planed, like a connection to LaTeX, different office products like MS Word, Excel, etc. and a wide range of mathematical tools (Maple, Mathematica, MatLab, etc.). Another important aspect is the usability of the program and the display and presentation of the formulas and graphical output for visually impaired as well for sighted people (e.g. teachers). To fulfill the needs of the future users and to define the future improvements, we offer a workshop to experience the advantages and disadvantages of the known mathematical languages (LaTeX, LiTeX, LAMBDA) as basis for a discussion round to define the functions of the upcoming all-in-one solution LAMBDA^2.
Math 105/Fall 2012 This course surveys contemporary thinking about complex systems, coexistence and strategy, through a mathematical lens. We will explore system theories, cooperation, networks, and related ideas. We will use them as a framework to develop relevant math concepts, such as sets, algebra and statistics, and simultaneously explore their social context and think critically about ways to use and question them. Students will gain broadly applicable math skills and resources to develop them further, and a survey of culturally relevant discourses. The instructor will work with students to develop class projects relevant to their interests. Students will be evaluated on a mix of coursework and projects. Understanding parentheses. Understanding multiplying without a times symbol as in 2⁢x⁢(1-x) (or with a little dot, as in 3⋅4). Precedence. The commutative property. The associative property. The special roles of 1 and 0. The distributive property. Multiply through a set of parentheses. I think though that a lot of this basic arithmetic has to be considered prerequisite for this class. I could spend a whole semester teaching it, and that wouldn't serve the other students. I can tutor students, or give them time to bone up using the videos. They can catch up on some arithmetic skills while we go ahead with iterating maps and stuff. But also, we can do review over a few weeks while moving into the dynamics stuff. That was one thought, but also the class was announced without prerequisites. I don't think it's fair to grade students down for not having had algebra or prealgebra in high school. I am going to try to see how far I can go without requiring the background. This means if we need fractions and decimals, say, which we do, I have to give them what they need. I guess a short lecture plus some resources to use might be okay. (But the one I did on the first full day wasn't enough.) ideas lecture the idea of a "system" Find Adam Smith quote about "imaginary machines". Maybe think of the whole class as being about imaginary machines. A system is a thing that has parts, and it moves or changes, and how it moves or changes depends on the state of its parts. Its moving and changing is called dynamics. "A system is an imaginary machine invented to connect together in the fancy those different movements and effects which are already in reality performed. The machines that are first invented to perform any particular movement are always the most complex, and succeeding artists generally discover that, with fewer wheels, with fewer principles of motion, than had originally been employed, the same effects may be more easily produced." Mathematics and the sciences Physics's triumphs Models, equations, "physics envy" math lecture OK, let's get into dynamics. Iterating a discrete-time map (with as little notation as possible) At first: no notation. Call them "multiply by two", etc. Later: x→something. A variable. The rule, the right hand side. The right hand side is a little bit of algebra. No plotting, no equals sign, no trying to solve what happens, just look at what happens, and do some thinking about it in plain English. postmortem I am so unhappy! Last year I started off by throwing too much at people all at once, assuming the functions, variables, and stuff would make sense to them. I tried to fix that this year and it didn't work! What good is it to do a 15-minute crash review of the associative law and all that? Even if it was effective, I ended up losing people by suddenly dropping the rule for simplifying fractions on them without any review. The anxiety level in the room went up and up, and I sent them home with a homework assignment that I'm afraid they're going to cry over. Maybe I should have made two versions of the homework, one with the logistic question and one without. Also, maybe I should just insist that the class has algebra as a prerequisite. Update: I told them not to do the math homework, I'll give them homework for the coming week. Lecture notes I think I can teach some fundamental concepts about dynamics without math background. Because I'm a practitioner, you get my perspective on the craft - including things that aren't in any textbook. I'm trying something new with this class, and I might make some missteps. It's very important to me to be reasonable and not ask you to do things that you can't be expected to do. We'll have to review some things along the way, like fractions, negative numbers, variables. I'll also introduce some fundamental concepts that you probably never had, like unbounded growth. I threw in a lot of things together last week, and it would have been better to take on a small number of new things at a time. It might have made it hard to follow, and hard to know what's expected of you. It might have given you reason to worry about whether I'm going to put you in a difficult position by asking you to do things you aren't in a position to do, and it might have given you reason to worry about whether I can be trusted to treat you fairly. These things are important to me, and I'm committed to checking myself when I need to and being fair. Revisit the doubling model, and the halving model. First, just the doubling. We start with the behavior of the parts. Each individual creates one more each day. Every one individual becomes two. We ask a question about the number of pads. We could ask other questions, like how the sun changes their color, but we are asking this one. It seems natural to consider the time period of one day. Never mind what time of day they do it. This is a choice we make, how to look at the system, which parts to leave out. The system is the population. Its behavior is driven by what the parts do. The individuals are the parts. How to get from behavior of the parts to behavior of the system? At least two steps. First rule for the parts behavior to rule for behavior of the system. Then from system-level rule to outcome. Last week we really just looked at getting from rule to outcome. How do we get the rule? Here are two ways. First: At the beginning of the day, there are so many lily pads. They are all there at the end of the day, and each of them has produced a new one. So there are so many parents, and the same number of children. The original number, and that many more. Add the number to itself. Draw a rectangle: row of parents above, row of children below. What is multiplication? Multiplying a number by two means adding that number to itself - adding two of that number together. So adding the number of parents to the number of children means multiplying the number of either by two. Second: Here are these lily pads at the beginning of the day. There are so many of them. Then each of them becomes two. Now there are that many pairs. The new number of pairs is two times the number of parents. Two plus two plus ... This is adding together all the columns of the rectangles instead of the rows. This may seem like beating a dead horse, because you already know about adding and multiplying, but it actually does something profound for us: it turns what the parts do into what the whole does. Sometimes it's not easy to do that. The reason we're doing it with these lily pads is because we can actually do this step with them, using only addition and multiplication. This gives us the rule for the system: Multiply by two. Now we try to understand what that rule does to the system, and what outcomes it produces. Some terminology! We created a system by choosing to investigate the number of lily pads. In this case they are imaginary, but they could be real. Anyway, we investigate the system by creating a model. First a mental model then a mathematical one. The system is different from the actual lily pads, and the model is different from the system. We omit various things: for instance, an industrial accident or disease could decimate the pond, but it'll never happen in our model. Even more so: someone pointed out last week, they will fill the pond and stop growing. Above a certain number, we shouldn't trust the model. In real situations, sometimes all we have is a model (the Attenborough climate video, for example). So we have to do two things: try to understand what the model does, and try to understand when you can and can't take the model's behavior as prediction. The rule "multiply by 2" is the update rule for the system's dynamics. We follow the model's dynamics through time by iterating the rule. When we iterate the rule, we're looking at a dynamical system. A dynamical system has a state and an update rule. In this one, the state is the number of pads. The next state is determined by the current state, according to the update rule. The state is all you need to know. The subject of discreteness came up, but without the words. Discrete means separate, well-defined objects. In this model there are always discrete lily pads - we can count them. This means the state is always a counting number: 1, 2, 3, .... Time is also discrete in this model: we consider one day, then the next day, then the next day. We don't try to describe what happens each second, each fraction of a second. This is how we can talk about the "next state". Not discrete is continuous. We won't do continuous time because it requires more advanced math tools. We will talk about continuous state, when the time comes. So the behavior of the multiply-by-two system. We've boiled it down to this rule, multiply by two, so the only thing remaining to get us from rule to outcome is to ask what happens when you iterate the rule - what happens when you multiply by two repeatedly? We did that some last week. 1, 2, 4, 8, 16. The sizes get bigger. Actually they get a lot bigger pretty quickly. The expansion speeds up. The bigger it gets, the faster it grows. Does it ever stop? We had some discussion about this last week. It's worth discussing. First of all, we have to distinguish between the model and the actual system. Of course the pond will stop filling up. There are probably several ways that can happen. But what does the model do? This connects to a fairly profound question: is there a biggest number? And it lets us look at how we answer the question. Suppose there was a biggest value for the system. Then what would happen next? We would multiply it by two! The next value would be even bigger! So that value wouldn't actually be the biggest. This is contradictory, so there can't be a biggest value. This is a proof. Proof is how mathematicians figure out what's true about things we can't investigate with our eyes and hands, like very big numbers. Note: this is probably a distraction that I should skip. The state of the model grows without bound. A bound would be a "ceiling value" that it never crosses. Also, it grows exponentially, as someone pointed out. Another term is that it grows geometrically. That's the term for the same thing when we use discrete time steps. Why? If we want to look at declining population without getting into complications immediately, we shouldn't look at lily pads, or at least not small numbers of them. Consider pond scum - algae on the pond - that's cheerful, because more lily pads and less algae makes people happy. So for simplicity, let's say that they don't give birth, and each day half of them die. Algae are microscopic creatures, so there are really a lot of them. It's easier to measure them in pounds, or kilograms, than to try to count them. Why do half of them die? Maybe because of weed killer in the water. If you are a tiny creature floating around, every day is like flipping a coin whether you encounter a lethal dose. So to really make all the steps from what each cell does to what the whole does we would have to deal with probability and statistics... Like the growth model, lots of things behave like this model too: a bouncing ball, the temperature of a bottle of vodka in the freezer, the amount of radiation in our milk from the Fukushima meltdown Let's not do a bouncing ball, because the bounces get smaller but they also come sooner and sooner Bottle of vodka: especially simple if we use Celsius temperature because 0 is freezing. So if the temperature of the freezer is 0, the temperature of the bottle will keep getting closer to zero. It'll be half as much about every 30 minutes or something. To completely work this one from the beginning we would have to know some things about molecules and heat. Unfortunately, the amount of radiation from Fukushima takes longer to cut in half (but fortunately there isn't very much of it here in North America). To do this one we would have to know some things about atoms and radiation. So, good things and bad things. Anyway, let's stick with pond scum, and we can make an approximation by saying that exactly half of them die every day. It's very reasonable. So the total amount of scum, in pounds, cuts in half each day. If we start with 1000 pounds, next day we have 500 pounds. If we have 2 pounds, next day we'll have one pound. And if we have one pound, next day we'll have one half pound. We leave the realm of counting numbers. We are measuring now, and fractions of pounds are completely sensible for this. So one half is less than one. Where is it on the number line? Q: if we have one half pound, what do we have the next day? A: one quarter pound. Enough to make an awful hamburger. Where is that on the number line? Stop and talk about (positive) fractions and positive numbers. Use the number line, or a yardstick for height. We are dealing with amounts of things rather than counting numbers. Distances, weights, fractional parts of a whole. Slicing a pie, for example. Or dividing up a gram or an ounce of something. The amount can get smaller and smaller; zero is at the bottom. These measurements are not discrete, they're continuous, although our time measurements - day after day - are still discrete. A discrete-time, continuous-state model. This is what happens when our population keeps cutting in half. It keeps getting smaller, and closer to zero. Zero is the floor. In fact, each day we go half of the remaining way to zero. So our steps also get smaller and smaller - we are slowing down. So it's getting smaller, and it's going to keep doing that for a good while. Eventually, in the real-life system, we'll get down to things that don't divide, just like we would with lily pads, but it's a long way off. In the model that will never happen, we just keep splitting in half forever. So some qualititative behavior: the amount of pond scum always gets smaller; it shrinks more and more slowly; it never gets to zero, and it never goes beyond zero. So zero is a lower bound. Is that enough? If not: If we start the decay with different amounts, we get the same kind of behavior. One of Eno's favorite quotes, from the managerial-cybernetics theorist Stafford Beer, would become a fundamental guiding principle for his work: Instead of trying to specify it in full detail," Beer wrote in his book The Brain of the Firm, "you specify it only somewhat. You then ride on the dynamics of the system in the direction you want to go." Eno also derived inspiration from Stafford Beer's related definition of a "heuristic." "To use Beer's example: If you wish to tell someone how to reach the top of a mountain that is shrouded in mist, the heuristic 'keep going up' will get him there," Eno wrote. Eno connected Beer's concept of a "heuristic" to music. Week 5: Sept. 26 I think this week we don't necessarily get to logistic population growth yet. I think the sequence is plots with different independent axes; variables; functions and cobwebbing; then changing the linear growth function to a density-dependent one. I should also set them up with equilibrium example. Make logistic-growth applet, show it but wait to develop the model? Also I wanted to give them an overview. I am teaching them a streamlined introduction to dynamical systems. This will include things like state space, parameters, equilibria, attractors, basins of attraction, bifurcation, chaos. Later in the semester we'll also talk about self-organization, complex systems, evolution, networks, swarms. We'll get to these things in time. I am going to use the terms I gave you last week in this week's lecture, however. Look what we can do now that we have graphs. We can compare lots of these things at a glance. Sometimes up, sometimes down. What do you think makes the difference? A. Above or below 1! Also, what does "Multiply by 1" do? Two classes of outcomes and a borderline case. 1 is a special parameter value. We'll see more of this kind of thing when we do bifurcations. We can do that now: plot ultimate value vs. growth rate. Terminology: plot something against something else. We can get nimble, plot all kinds of things against other things, ask various questions. let's do this one starting at, say, 1 starting at a whole lot of different initial values. We could also plot ultimate value vs. initial value. In this system, it wouldn't be much of a plot except when growth rate is 1. No - this is silly to do here. oops, I should have taken it out of the notes. I did it in class. an especially useful one is next value vs. current value. Let's do that one for growth rate of 2. Start with 1, get 2. review coordinates, introduce the ordered pair. Given 2, we get 4. Given 4, we get 8. An odd connection between the values on the two axis. We're doing a mental transportation from one axis to the other. We would not do that with latitude and longitude! But on this plane, there is this connection between the two different directions. The next value becomes the current value. But we don't have to follow the sequence. We can just draw the whole rule. Given 3, we get 6. etc. Given 1/2, we get 1; given 0, we get 0. Given 3 1/2, can we tell what we get by looking, without calculating? When ready: let's do it for "Multiply by 1/2". OK, it looks similar but different. Any thoughts? Now, what about for "Multiply by 1"? This is a special model in which nothing changes! Also, a special place on the plane where the values are equal. Known as the diagonal. The diagonal wouldn't be meaningful in longitude and latitude. Places where the numbers match aren't special places the way the North Pole is. Likewise, if we're plotting a time series, the diagonal wouldn't be meaningful, because who cares if the number of lily pads equals the number of days - we could be measuring in hours and then it wouldn't be equal, but everything that matters would be the same. But here it is meaningful. Well, so far it's just the location of this especially useless model, which is about the same as not having any dynamics at all. But look what it's useful for... See, we start at 1 and go up to find that the next value is 2. So we had a question and an answer. The question is "Starting at current size 1, what do we get next?" That's a location on the horizontal axis, or a vertical line. The answer is this point - or really it's the vertical location, or a horizontal line. We asked a question by providing one coordinate, and the answer is the other one. Next thing, now that we're at 2, is to ask what comes next. That means we put 2 on the bottom axis. We take it off the left axis and put it on the bottom axis. We had this horizontal line, and now we use this vertical line. Look where they meet! You can use the diagonal to shift your "now" from tomorrow to today. I.e. to make "next value" into "current value". So what? It's a trick, but you can do that just by looking at one axis and then the other. But look: we can make it into a shortcut. Here we are at (1,2). That is a question-answer pair: 1 today, 2 tomorrow. To ask the next question - 2 today, what tomorrow - we slide horizontally to (2,2) on the diagonal, and up from there. Skip the axes completely. This is the cobweb diagram. This is how you can read the outcome off the picture of the rule. Do cobweb diagram for "Multiply by 2" with various starting points. Do cobweb diagram for "Multiply by 1/2". And for "Multiply by 1". The diagonal is where the next value is the current value - no motion. What does it look like when the value is getting larger? What does it look like when the value is getting smaller? Maybe this week, maybe not: A little preview: Could I draw this loopy squiggle on the plane and use it to do a cobweb diagram? Well, if I am here, where do I go next? There are three or five places on that vertical line. In order to know how to do it, we need a particular shape: only one place above each location on the horizontal axis. When you have this question-answer relationship, it's a function. Something that given one thing, gives you another thing. There are lots of functions in life, though we may not call them that. Every person has a height and weight. Given a person, a quantity. Names have length. And something like "Add 1" or "Divide in half" is a function: given a number, it gives you the next number. So is the population size time series: for any day, a size. When we have an independent and dependent axis, we have a function, because we're talking about the same thing: we use the horizontal location as a question and get the vertical location as an answer. Another preview: the granovetter threshold model There's a few people demonstrating in Tahrir Square. Maybe there are a lot of people who would join if the crowd was big enough - but how big? What if we could ask them? So for a given size, there are so many people who would consider that big enough. If we have 100 people, how many people say 100 is enough to get them to participate? More than 100, we hope. So if there are so many people that will do it even if nobody else does - that might be how we got started - and so many people that will do it if 10 people do, which is at least as many as for zero - and so on up, how do we find out what happens? Cobweb. And maybe the tipping example: this curve is just below the diagonal most of the way; but then if just a few people change from naysayers to enthusiasts, they raise the curve over there on the left at say, 50. But if those people are satisfied if there's 50, they're also satisfied if there's 100, and 200, ... They raise the curve all the way to the right (to the point where they came from). And look, that's enough to set off all these other people, and make the whole thing catch fire! That can happen, I think. (But how often are we right on the edge like that, we don't know...) A third preview, sometime soon we're going to consider the logistic map, which looks like this on the cobweb diagram. What? Well, not going into the details right now, but the relationship between current size and next size of the population goes like this. Over here, it's above the diagonal, which means it goes up; over here, it's below the diagonal, which means it goes down. Where it crosses, it doesn't go anywhere. So maybe that's where it ends up? They are the qualitative behavior of positive and negative feedback processes in general. Neither is a good model forever. After a long decline, you arrive at the end, and after an exponential growth phase you tend to hit a limit and get a different kind of system behavior. Note that it's not trivial that there are the general behaviors of self-amplifying and self-damping things - recall the "add 1" and "subtract 1" models, which are different. They do arise, but they're more exceptional. (note to self: I wonder if I'll get to present the reason why, which is the linearization of a system. I guess not, but I might want to look for a way to sneak it in from some sideways angle.) We're getting to models with "limits to growth", but need to lay groundwork first. We looked at these two models in terms of rules and outcomes - how to start with something that follows certain rules, and figure out what the outcome of that process is. We identified two different kinds of rules: the rule that the parts follow (produce 1 new offspring each day) and the rule that the whole follows (multiply the population size by 2) and how to get from the parts rule to the whole rule (drawing a rectangle, which stands for how we get from addition to multiplication). The update rule is powerful because you can predict the future of the state just from knowing the current state, without needing to know a bunch of other stuff. and then how to get from the whole rule to the whole outcome, by iterating. Plotting and graphing (I'm using the two words interchangeably) Similar to locating things on a map. We do the same thing with more abstract things: by identifying events with pairs of numbers, we can locate them on a plane and look at patterns of events in a form that lets our visual abilities work on them. There's room for a lot of creativity using the two axes, and other ways of representing things on a page/blackboard/screen, as the tools we have to work with. And some standard ways. Using the horizontal axis as the independent quantity. The question-answer relationship; the vertical-line test for functions. Time series, very useful for our systems with numerical state and update rules. We also looked briefly at the growth rate as a parameter, plotting "ultimate outcome" vs. growth rate. There's a general pattern here, though we haven't done enough examples yet to see that it's general: qualitatively different outcomes for different ranges of parameters, with borderline cases separating them. Like different regions on a world map. Threshold values. Knowing where they are can mean knowing how a system could be radically different, and maybe even knowing what to do about it. We could have also plotted ultimate outcome vs. starting pop size. But instead we got into plotting the rule rather than outcome: we visualize the rule by plotting next state vs. current state. In this form, the doubling system looks like a straight line, and the halving system looks like a different straight line. It's the relationship with the diagonal that makes the difference between the qualitatively different outcomes, and explains the borderline case in between. And we used cobwebbing to read off long-terms outcomes from the diagram of the rule, without needing to know the symbolic ("multiply by two") version of the rule. A preview of things to come (maybe even today): Using symbols to describe rules and outcomes, and learn things about them. Things that happen in a population model that includes limits to growth. fractions I promised to talk about how decimals and fractions relate (why 1/2 is sometimes called 0.5) but then I didn't. So I need to do that. Needs to present them as proportions, in the way I'll use for the logistic map. Mini-lecture on fractions and decimals! Fractions and decimals are two ways of talking about the same things. A fraction describes a portion of something. A simple example: part of a pie. I might cut the pie into quarters. Quarters is Latin for fourths, which we also call them. I might eat one fourth and leave the other three. We have developed special notation for this: we write 14. This amount of pie is one fourth of a pie. So 14 means of four equal parts, one of them. The two parts are the denominator on the bottom: of how many equal parts in total, and the numerator on the top: how many of those. This is a generalization of the idea of a portion: it could be a fourth of a pie, or of a bottle of water, or of the world's ocean water, or of all the dentists in a survey... Sometimes there's more than one way to write a fraction: if I took 24 of the pie, it's the same amount as if I just cut the pie in two pieces and took one of them: 12. That's also the same as if I cut it into six and took three of them. One way to look at this relationship is: here's a pie cut in half. Here's the half I'll eat. If I slice up the part I eat and the part I don't eat, I'm still eating the same amount: now there are 3 times as many parts here and 3 times as many parts there, and 3 times as many parts all together: now we would call that 36, but it's the same amount of the pie as before. We had 12 of the pie - one slice of two - and we tripled the whole number of slices : 6 - and we tripled the number I'm eating - 36 - and it's the same amount. This is why we can multiply the numerator and denominator by the same amount and the fraction we get is equal to the one before. We can also do that in reverse. If the pizza is cut into 8 slices and I eat two of them, it's simple enough that we can visualize: we could have just cut it into 4 pieces - we can consider pairs of little slices - and I would have eaten one of them. If I slice the pie into 8 pieces and eat 2, and then I eat 1 more, clearly I've eaten 3 out of 8, that's right, 28+18=38. I just introduced the equal sign. If I eat 3 of those slices, and then I eat the other 5, I've eaten all 8, and it still works: 38+58=88 and 88 is the same as one whole pie. That's right, 88=1. Where there's adding, there's subtracting: if I eat 3 of the slices, how many are left: clearly 5. That's because we can take 38 away from 1 just like what we just did, only backwards: 1 (the whole pie) is 8 slices, 88, and taking away 3 of those 8 leaves 5: 1-38=88-38=58. It's fine to chain equal signs. All those things are equal to each other. And by the way, when we were first looking at the "divide by 2" system where half of the algae cells die off each day, and we said that it always gets smaller but never reaches zero, maybe it wasn't fully clear why that is. I think it's pretty clear when we use this pie analogy for situation. Here's the whole pie, half the pie, then when we divide again we cut it into twice as many pieces and have one of them... the piece is always half as big as the one before (which is not a surprise), but we can see that it is always smaller and there's always still some amount of pie, not zero. decimals There have been times and places where people used only fractions to handle parts of things. for instance, Egyptian notation: instead of 38, one had to write 14+18. But the decimal system makes us a lot more powerful. We can handle arbitrary parts of things and easily tell at a glance how big they are, and which are bigger than others, round them up and down to various amounts of precision, and it's much simpler to multiply and divide them. We use it for whole numbers already. We reuse the same numbers to represent larger or smaller amounts, depending on position. Example: 656 - 6 100's plus 5 10's plus 6 ones. each position is "worth" 10 times as much as the one to its right. If we return to the convention of drawing numbers as spatial locations, that is, use a "number line", look how a decimal number like that is a really efficient way of finding a precise location. It's like transportation: first you fly to Boston, then you take a train to Providence, then take a local bus from the train station, then walk from the bus stop. Get into the general area, and then navigate the finer and finer details. The same: here are these big regions, 100 numbers at a time, we choose one of these, and then it's divided into these regions, 10 at a time, like city blocks, choose one of these, and then get off the bus and walk to the exact number. In both cases, we might need even finer detail. Even after you get in the house, you might want to go into the living room, and then go and sit on the couch. With the numbers, we can keep going by cutting these intervals into tenths. Take this one, it has ten subdivisions, take the right one of those. The way we write it is, we just keep going to the right. We didn't need a decimal point before, we just understood that the right edge of the number is the ones. Now we add the decimal point so we can tell which position is which, because now we have some numbers to the left of the ones and some numbers to the right as well. so I wanted to address the question, "why is 12 also written as 0.5", because that is what came up in class a couple weeks ago, when I said I would talk about decimals and fractions. So this picture is a start towards answering that. every time we home in on the parts of a region, we have it cut into ten equal slots. Five is right in the middle. We know that half of ten is 5, and half of 100 is 50, and so forth. Half means half the distance to zero. Do you see why? let's go back to the pie for that. If we compare a certain amount of the pie to a minute hand that can go around the pie, if it's at half-past we get half the pie and if it's right on the hour, no pie. However much we have, if we shift to having half as much we're going half of the way down to zero. In the same way, dividing in half on the number line is going halfway between there and zero. I think these shifts of reference can be confusing. Slices of things, amounts of rotation, distances on a meter stick, fractions with one number over another, decimal numbers, horizontal and vertical distances... take time with them and don't lose people. Once we buy that, we can see that dividing one in half takes us to 0.5 because it's the same division into ten slices with five in the middle. In each case, going from 10 to 5, with the 5 one position to the right of where the 1 was. I forgot to pre-teach mean and standard deviation! I have to write better notes and use them better! show the rest of the video also in class Hand out new homework (due in 2 weeks) Talk about next week's midterm Hand out sample test as a study guide and a promise about what to expect Reading for 2 weeks away. notes Stuff I mostly haven't gotten to yet, so I'm carrying it along from week to week until I do: Spaces the state space parameter space number of dimensions, maybe have fun with Especially: Negative feedback and self-regulation. "The system goes to an equilibrium." Like a thermostat. Or homestasis of the human body. Or Gaia. It's going to do something according to its nature. Should we "go with the flow"? Trust it? Try to anticipate it? Or just find out? Like Adam Smith's invisible hand. Joanna Macy video, maybe. In preparation for going on to the Curtis video. Variables and functions I introduced the "function", while drawing plots, and mentioned the vertical-line test. I haven't said anything about using letters for variables or anything that uses that since the first day (except to say we'll get to it later). I think it's time. notes that I might use sometime, but not right now: Multiple kinds of plots: trajectory of state variable, time series of state variable vs. time, map of next state vs. current state Simplifying expressions. Applying operations precisely vs. using intuition. Add to/Subtract from both sides. Divide from both sides. Dividing by a negative? Multiplying by a negative? The issue of dividing by zero. Cancelling something out when it's not zero. Fractions and decimals. Fractions that include variables, parentheses, etc. But it's a good time to note that we're starting to have a lot of quantities, and it's time to fix up some names. "When a twelfth-century youth fell in love he did not take three paces backward, gaze into her eyes, and tell her she was too beautiful to live. He said he would step outside and see about it. And if, when he got out, he met a man and broke his head - the other man's head, I mean - then that proved that his - the first fellow's - girl was a pretty girl. But if the other fellow broke his head - not his own, you know, but the other fellow's - the other fellow to the second fellow, that is, because of course the other fellow would only be the other fellow to him not the first fellow who - well if he broke his head, then his girl - not the other fellow's but the fellow who was the - Look here, if A broke B's head, then A's girl was a pretty girl, but if B broke A's head, then A's girl wasn't a pretty girl but B's girl was." Jerome K. Jerome, Idle Thoughts of an Idle Fellow, 1890 (D.M. Campbell, The Whole Craft of Number, 140) Week 7: Oct. 10 Week 8: Oct. 17 Overview of where we're going with the population models: the logistic model Writing dynamics symbolically, using variables Review what we've seen cobweb diagrams do: equilibria, attractors Preview the simplest logistic model Talk about projects More on rules/outcomes in games, art reading: maybe chapter from Gleick, Chaos? Or something else about chaos. Sci. Am. 1986 article? Can I get a copy from the library? math lecture do an overview of things to come dynamical systems can do a lot more than the multiply and divide models we've been looking at. In order to see more interesting behaviors we need a bit more complexity in our models. In order to do that, we need better tools for working with them. I'm going to develop the logistic population growth model. In symbolic form (x→r⁢x⁢(1-x)), with variables, parameters, and equations. We'll work up to doing some things with variables and these symbolic forms. It's like learning a language. In graphical form via the cobweb diagram It looks like this, and we'll do some building up to this too. Via simulation, looking at its time series. It looks like this, on the computer. direct damping, overshoot, chaos The logistic map in historical context. (fill in story of May, chaos, Laplace etc.) We're going to need notation. First the use of a letter. We can introduce a letter to some things we've been doing and it doesn't make much difference. Redraw population vs. time, with letter labels on the axes. Same as using English-language labels, except that you need to be told what the letters mean. Rewrite our update rules. "Multiply by 2" ⇒ "x→2⁢x". It's more compact, but basically the same as before. Here's what we had, and then here's what we had multiplied by 2. But let's pause and make sure we understand it. A variable name is like a pronoun. Sometimes it can refer to a particular person and sometimes not. For instance, if I say, "whoever has the attendance sheet, could they please pass it to me?" - I don't actually know who I'm referring to. I'm just saying whoever it is, this is what I have to say to you. And I might say that every week, and it might be different people at different times, even though I say the same words. I could write it on my notes and plan to say it every week. It's the same no matter who the people are. This is like that. This is quantitative, so instead of "whoever", it's "however much" or "however many". "However many plants there are, there will be two times that many tomorrow". We can use it every day, even though there will be different amounts: x will mean different things at different times. So it's very similar to things we do in regular speech, but it's a different language that might take some practice. So that's "x" -- we should also talk about "→": this is notation for an update rule, and it means "whatever x is, it gets updated to this other thing." So x→2⁢x is "replace x by 2⁢x", as we all know, but it could be other things. We can talk about x→3⁢x, x→4⁢x, x→12⁢x, which we could already, but also x→2⁢x+1, x→(x-1)⁢x, lots of other things. Are they meaningful? Sometimes. Sometimes it's useful to look at these things even when they're not directly meaningful. We have several different ways to work with the dynamics of a system now. Ideally you'll become able to bounce between them and carry several in your mind at the same time. You can use them to check each other. You might need some practice to get there. Using words and thinking about the actual system. This is when we say "the population size doubles every day." If it's not too unwieldy to put into words, this might be all we need to understand the long-term outcome. Using a math operation on the numerical that represents the system's state. This is "multiply by 2". It's similar to using English, but we could do it without knowing what the number actually stands for, and we could do it mechanically without having an intuition for whether the answer we get is reasonable or not. Also, we could program a calculator or computer to do it. Doing it graphically on a cobweb diagram. This is a way you can get a qualitative picture of what's going on without needing to know exact numbers. If you work well with visuals, this might work well for you. But it's a peculiar, abstract visual that can take some getting used to, and it's best when you combine it with thinking about the update rule another way as well. Doing it symbolically. This is similar to using a math operation verbally, but a lot more powerful. You get best results when you use more than one mode at once. The last question on the midterm was like that, to let you choose which way works best for you. I drew a curve for cobwebbing, and I also wrote a description of what the model does, so you could use the description to ask whether you have it doing something that makes sense or not, or even just use it to get the answer and skip the cobweb. Anyway, we'll build on that. I don't expect you to become a hotshot with variables and symbolic math expressions just like that. Here's an example of the kind of question you've got on the homework this week: If the update rule for our system is x→x-10, and the current state of the system is 30, what is the next state? Things you'll need for the HW: Fractions and division. we talked about parts of things like 35, but we didn't talk about things like x5 or 205. but if you wanted to, you could maybe work it out from what we did talk about. But I'm not going to do that to you. 35 means you are counting fifths of a pie, and you count three of them. It's that much of a thing. 35 of the people in the room, or the population size is 35 of what it used to be, or whatever. We are counting fifths, and we count 20 of them. It takes more than one pie to get to that many slices. (draw sliced pies) It takes 4 pies. Why is this? Because in each pie, you get 5 of these slices, so with 4 pies you get 4×5=20 slices. This is the reason that 205 is 4. But does this seem familiar? This is division. We did the same thing as if we asked, what is 20 divided by 5. In fact, fractions and division are the same thing. If you take 20 divided by 5, you get a regular whole number 4, and if you take 1 divided by 5 you get this other thing, 15, which is not a whole number, it's a fraction. This notation, with an upper and lower number and a bar, means division. So if you see 102, you don't need to do anything very special, you just say "what's 10 divided by 2" and that's what it is, 5. Nonlinear systems can do lots of other things, like, for instance, everything that anything in the universe does. (At least, if you are okay with the materialist faith underpinning the sciences.) FYI it's called linear because in the "next value vs. current value" plot, it's a straight line. We don't need to get caught up in the difference between linear and nonlinear systems. But since I started giving you cobweb diagrams to work with, we've seen some of the other things systems can do. remember the threshold model? Should I review the threshold model? When I introduced the threshold model, I drew a diagram that stops at 700 people demonstrating. That 700 is the place where the curve for the update rule meets the diagonal. Since the curve draws the next value against the current value, and the diagonal is the place where the two values are the same, the place where the curve meets the diagonal is the place where the next value is the same as the current value. That's a fixed point. If your system is there, it stays there. It's also an attractor, because the system is attracted to it. In the homework, I gave you another threshold model, that looks more like this. from here, you go up to the upper crossing, and from here, you go down to the lower crossing. Here in the middle is a repellor - the system moves away from it. It's like the continental divide. The number line is divided into two regions - the values that end up here, and the values that end up there. These are basins of attraction of the attractors. These fixed points are the simplest kind of attractor. There are other possibilities, for instance, it can oscillate up and down forever. other notes, things that might not go anywhere Computer models some examples. maybe bring them in earlier. Agent-based models and other simulations. some examples of complex models that produce the same behavior as simple models? some that don't? Week 9: Oct. 24 (class cancelled) Week 10: Oct. 31 project presentations. Week 11: Nov. 7 pop dyn chaos, bifurcations class outline: scheduling - presentations last week, final the week before? mistake on the hw logistic map lecture complex systems lecture Attractors. cycles, maybe in cobweb diagram, maybe in computer examples. initial conditions, basin of attraction. Deterministic vs. random. mention random noise, stochastic dynamics Bifurcation, bifurcation diagrams. a bit about more complex attractors, chaos logistic map lecture Now we can talk about the logistic model We were ready for it in a way, in the first couple weeks, when we were looking at the unbounded growth model, and Probably the best way to motivate the logistic map: birth requires both a parent and an empty space. Do it with proportions from the start. How much of the lawn has dandelions (a common weed) growing on it? half? Somewhere from none (0) to all (1). Draw. By the way, why am I doing this? Because we've been talking about exponential growth and limits to growth, and I want to give you an inside perspective on sustainability and overshoot? Yes. But also because this system will seem boring for a certain amount of time and then it will suddenly erupt into chaos. Watch for it. How many seeds does a lawnful of dandelion produce? A lot. We don't actually care how many, so much as we care how many new plants it'll lead to. Say each plant's seeds is expected to produce 1.2 new plants, if all the seeds land on congenial soil. So far so good, looks like a "multiply by 1.2" system, except with this extra clause. Congenial soil meaning unoccupied. How much of the lawn is UNoccupied? The remainder. If 34 of the lawn is occupied, 14 is unoccupied, which is 1-34. "Multiply by 1.2 and then multiply by one minus the population size" It's time to improve our language. It's time to call the population size x. This might be unfamiliar, or uncomfortable, or seem mysterious, but stay with me and see what it lets us do. So if the population size is x, what did we have so far? It produces a lot of seeds, enough to make 1.2⁢x on the next week. If that was it, we would have "multiply by 1.2" but we could also call it "replace x by 1.2⁢x." But what about the land-use issue? Because x is a proportion, how much of the lawn is occupied, the inverse quantity, how much is unoccupied, is also a proportion, the rest of the lawn. 1-x. So the lawn produces enough fluff to seed 1.2⁢x of the lawn. But only (1-x) proportion of that is actually able to set seed. I should draw pictures. Maybe a cloud of fluff, and the places it lands, and a fraction of it that is viable. Now that we're using variable names, let's run with it, since we want to consider various rates of reproduction, not just 1.2: call that r for reproductive rate, and our rule for this model is, instead of "Multiply by 2", "replace x by r⁢x⁢(1-x)". So we've gotten from describing how the parts update (they put out seeds and the seeds might succeed, depending on free space) to how the whole updates (according to this symbolic expression). How to get from here to an outcome? Where do we begin with trying to understand this rule? well, before we even do that, maybe we should go back to the old model(s) in the new language of r and x. We considered "Multiply by 2", "Multiply by 1/2", even "Multiply by 1", we were varying that number up and down (that parameter, to be precise), but we haven't considered this other object, "Multiply by r", so before we ask about a more complicated rule with r and x maybe we should do that. So there's some number, called r, and we don't know what number it is, because it can be various numbers. It's a variable (it's also a parameter, but let's wait on that). Again, a variable name is like a pronoun. Sometimes it can refer to a particular person and sometimes not. For instance, if I say, "whoever has the attendance sheet, could they please bring it up here?" - I don't actually know who I'm referring to. I'm just saying whoever it is, this is what I have to say to you. And I might say that every week, and it might be different people at different times, even though I say the same words. The update rule is like that. Without knowing what x is, we say "x→2⁢x", and we can keep using it even though x is different things at different times. This is like that. We might spend some time on this "multiply by r", and then say, well, let's say r is 2 and ask certain questions, and then later say, now let's say r is 12. And we can say some things about it regardless of what r is, just like I can say no matter who the person is, this is my message to them. Like what? Well, for one thing, we know that multiplying by one leaves the number you multiply unchanged. So when the population size is 1, "multiply by r" gives you r. For another thing, anything times zero is zero. So when the population size is zero, "multiply by r" always gives you some more zero. (A reasonable biology model will almost always give you that, because life doesn't spontaneously appear.) What about x? As long as we have to deal with it, we might as well get some use out of it, no? Instead of saying "when the population size is 1, 'multiply by r' gives you r", I could just be saying "when x is 1, r⁢x is r". That might take some getting used to, but you have to admit it's more to the point. So the familiar old "Multiply by 2" could also be called "replace x by 2⁢x", again, and the general-purpose "multiply by r" would be "replace x by r⁢x". Or "x→r⁢x". It gets shorter and shorter. Brevity is useful, by the way. The less it takes to do something, the more of it you can do... What else do we know about x→r⁢x? We have 0→0 and 1→r. Not much else, except I guess that when x is really large, then r⁢x is as well. This is like saying that when we drew "Multiply by 2" and the others as curves, on the "current size vs. next size" plot, they all slope upward as we go to the right. So we know some things about it. But as we know from last time, in some of these multiply models the population grows and in some of them it shrinks. So without knowing what r is, we don't know enough to say what kind of outcome we get. This is true with the more complicated system too. The dandelion system, or logistic population growth model, has the form: 'At the end of an essay on cybernetics that Grey wrote in Colin Ward's Anarchy #25, 1963, (one shilling and sixpence), and which lies here open in front of me, he concluded: "we find no boss in the brain, no oligarchic ganglion, or glandular Big Brother...If we must identify biological and political systems our own brains would seem to illustrate the capacity and limitations of an anarcho-syndicalist community."' Networks and graphs Networks have been the symbol of complex systems since the beginning of system theory in the 1940s or before. Now that we have easy access to computers we can study them effectively. As I have mentioned before, a curious thing about mathematical models is that when a model become popularly known, the main take-away message is often not the conclusions of the model but its assumptions. (And this has clear implications in understanding how math can be used in service of propaganda.) For instance, the most influential message of classical economics is not that supply and demand balance each other, but that people are self-interested; the most influential message of game theoretical models of cooperation is not that defection is the prediction unless there is reciprocity, kin selection or various other things, but that cooperation is hard because of temptation to defect - that's an assumption of the models; and with networks it's not that degree distribution matters, or that there are special positions in a network, but that it's possible to be organized in a way that doesn't have a central position of "power over". Though, of course, a centralized, hierarchical form is one kind of network structure - it's just not the only kind. Classifying structure intro 3 categories: centralized, decentralized and distributed. These are relative terms - how you describe a particular system depends on what you're comparing it with. One network or organization is more decentralized or more distributed than another. So we often hear "network" used to talk about something that is decentralized or distributed, like a coalition or other leaderless organization. Or used to refer to something that uses the Internet, which is the most conspicuous network in our lives presently. Social networks have always existed, but now online social networks are an important part of our daily lives, and there's a lot of debate about whether they're changing the landscape of possibilities for social and political change. There's also the horizontal structure of many social movements, i.e. leaderless. Horizontal vs. vertical. Again, these are relative terms, and there are degrees of horizontal and vertical. (The idea of the tyranny of structurelessness relates to this.) It seems like horizontal structures are becoming more popular, even expected. In social movements there is sometimes a lot of tension between horizontalists and verticalists - for example: At the time I was only vaguely aware of the background: that a month before, the Canadian magazine Adbusters had put out the call to "Occupy Wall Street", but had really just floated the idea on the internet, along with some very compelling graphics, to see if it would take hold; that a local anti-budget cut coalition top-heavy with NGOs, unions, and socialist groups had tried to take possession of the process and called for a "General Assembly" at Bowling Green. The title proved extremely misleading. When I arrived, I found the event had been effectively taken over by a veteran protest group called the Worker's World Party, most famous for having patched together ANSWER one of the two great anti-war coalitions, back in 2003. They had already set up their banners, megaphones, and were making speeches—after which, someone explained, they were planning on leading the 80-odd assembled people in a march past the Stock Exchange itself. The usual reaction to this sort of thing is a kind of cynical, bitter resignation. "I wish they at least wouldn't advertise a 'General Assembly' if they're not actually going to hold one." Actually, I think I actually said that, or something slightly less polite, to one of the organizers, a disturbingly large man, who immediately remarked, "well, fine. Why don't you leave?" But as I paced about the Green, I noticed something. To adopt activist parlance: this wasn't really a crowds of verticals—that is, the sort of people whose idea of political action is to march around with signs under the control of one or another top-down protest movement. They were mostly pretty obviously horizontals: people more sympathetic with anarchist principles of organization, non-hierarchical forms of direct democracy, and direct action. I quickly spotted at least one Wobbly, a young Korean activist I remembered from some Food Not Bomb event, some college students wearing Zapatista paraphernalia, a Spanish couple who'd been involved with the indignados in Madrid… I found my Greek friends, an American I knew from street battles in Quebec during the Summit of the Americas in 2001, now turned labor organizer in Manhattan, a Japanese activist intellectual I'd known for years… My Greek friend looked at me and I looked at her and we both instantly realized the other was thinking the same thing: "Why are we so complacent? Why is it that every time we see something like this happening, we just mutter things and go home?" – though I think the way we put it was more like, "You know something? Fuck this shit. They advertised a general assembly. Let's hold one." So we gathered up a few obvious horizontals and formed a circle, and tried to get everyone else to join us. Almost immediately people appeared from the main rally to disrupt it, calling us back with promises that a real democratic forum would soon break out on the podium. We complied. It didn't happen. My Greek friend made an impassioned speech and was effectively shooed off the stage. There were insults and vituperations. After about an hour of drama, we formed the circle again, and this time, almost everyone abandoned the rally and come over to our side. One fundamental result of living experiments like the Occupy movement is that it does seem possible for leaderless organizational structures to work. 1968, groupuscules, wallerstein again... foucault, deleuze + guattari all that said, let's look at results from network science as well. Graph theory. Is built on top of set theory. A network is a graph, which is a set of "vertices" together with a set of "edges" connecting them, either directed (arrows) or undirected (arcs). Example: social network websites. On Facebook friend relationships are undirected (if I'm your friend, it also means you're my friend). On Google+, Twitter, Diaspora* and some others, they use "follower" relationships which are directed - I can follow you without you following me. A network can be connected or not. It may contain cycles or it may be a tree. Populist researchers work to map out the connections among the "1%", while government agencies and corporations - Google, Facebook, the AT&T/US government partnership - record and map the connections among regular people - because knowledge of how people are organized confers power to intervene. I talked about distributions a bit, let's spell out what a degree distribution looks like. This is statistics. Make a bar chart of how many of the vertices have degree 0, how many have degree 1, and on up. What is a logarithm? What is a log-log plot? Why does a straight line mean there are important hubs in the network that have a lot of spokes? Or: do it myself. If you just wire up an E-R random graph, what you get depends on the density of links - there is a threshold, and above it the graph is connected (has a giant cluster) Difference between those graphs and scalefree graphs Three major ideas from network research The small world phenomenon. The Milgram experiment (the small world one, not the fascism one), "six degrees of separation", Kevin Bacon - as in the Barabasi video a small-world network has clustering (friends of friends know each other) and short paths between most points. Two main ways to get one: a localized graph with bridges added, or a scale-free graph. The "strength of weak ties". The less-conspicuous relationships, the ones that aren't active most of the time, and are easily forgotten, can be the most important ones in getting a job, or otherwise making things happen. May be related to the bridge links that make a localized, provincial network into a small-world network. These links are a kind of social capital. If they are related to bridge links in small-world networks, then creating and maintaining them isn't only self-serving, it's also for the common good, because they make it easier for something good to spread across the network and increase potentially fruitful or transformative encounters between disjoint communities. Power-law graphs. Airline network vs. highway network. When there are hubs, you need to deal with the network differently. Hubs are influential: use them to get something to happen; they are weak points if one wants to interrupt something that is happening in a network (disrupt an opponent's organization; stop the spread of a disease). Structural holes If you are in the special position of connecting two disjoint communities, you are in a privileged position and can use it to your advantage. You have insider information from one that the other doesn't yet have (in both directions), and can broker that information or use it yourself. This is a form of structural power. Boston Commons; "They hang the man and flog the woman, That steal the goose from off the common, But let the greater villain loose, That steals the common from the goose" "commoning" and a third form of property There does seem to be something to it sometimes - consider dishes in the sink in a group house. Compare to the public goods problem taxes, infrastructure, public radio, etc. Intimately connected to the theory of government. Very similar - people want the results of cooperation but they are tempted to be selfish. In the commons scenario, selfishness is actively degrading the common resources, in the public goods problem it's passively declining to contribute to the shared good. Other solutions: communication, tags, spatial proximity, relatedness, application of heuristics developed in other settings, group selection, reputation, a norm of conformity, punishment, incentives (see for list with citations). mention my escape from prisoner's dilemma and related results, Turner and Chao for example. Save sequential selection for the Darwin stuff, when we will talk about Gaia. Play byproduct cooperation game with your neighbor, for cookies. Use the cards. Ways to think critically about claims about cooperation and temptation and altruism. (Numbered items are from Peter Taylor's paper and his book.) Question the structure of the game. Some famous "prisoner's dilemma" scenarios are actually snowdrift, chicken, or even byproduct cooperation games, if they're two-person games at all, and the same caveats and more go for N-person games and "tragedies". Question the assumption of "rational self interest". Sometimes people have other-regarding preferences. Question the limitation to two choices. There are usually others. Positive social change often is often created by discovering or inventing new options when the ones that are given aren't sufficient. 1. Interpret systemness as problematic. What's been declared extraneous to the system? possibility of changing the system's dynamics. For example, changing the game to a different one. connections not acknowledged, for instance social relationships between "game players" boundaries of the system are seen as permeable Inequality among individuals within the system colors their options, including response to developments "outside" the system 2. Interpret the rhetorical effects of models "simpling": "Like sampling, 'simpling' is a technique for reducing the complexity of reality to manageable size. Unlike sampling, 'simpling' does not keep in view the relation between its own scope and the scope of the reality with which it deals ...It then secures a sense of progress by progressively readmitting what it has first denied. 'Simpling' ... is unfortunately easily confused with genuine simplification by valid generalization." (Hymes 1974, See Taylor for citation) privileging certain interests. For instance use of the T.O.C. model strengthens the political position of players with disproportionate power, since it makes the effects of inequality invisible. Who benefits? Evolution and economics "Self interest", cooperation ecological tragedies and market failures. Darwin's history, the basic ideas Natural selection, descent with modification How it works Variation: some individuals in a population are different from others. Heredity: offspring resemble their parents more often than they resemble unrelated inviduals. Natural selection: different variants leave different numbers of offspring We might expect the traits we observe to be adaptations, i.e. the result of optimizing; but they could be an exaptation as Gould argued, and could also be the result of an evolution process that does not optimize -- evolution can even pessimize, even in a one-species system, when the population affects the environment: you can have a tragedy of the commons outcome. Mention the Gaia controversy, but we'll discuss it later evolution is a kind of adaptation process evolution of things other than organisms culture, institutions, technologies. universes? other adaptation processes including learning The blurring between biology and society in metaphors of interest, politics, economics "Diversifying your portfolio" to manage risk, whether you are a pension fund or a milkweed cast seeds far and wide so if there's a drought or landslide, some seeds will be elsewhere put some in the "seed bank", i.e. have them lie dormant for some years so if there are bad years they will be around afterward. "Biological markets", "Bionomics" Is there Capitalism, Marxism in biology? We tend to project features of our current society into nature - biological markets, for instance - but if we were less resistant to considering alternative social possibilities we could also consider things we see in nature as things we could do socially... byproduct mutualism for instance when we say it does we are reifying the politics of Eurocentric domination, faith in science and technology, etc. hence Gould's argument to the contrary. Humans are not "higher" than bacteria, just newer. does complexity increase? It's not clear. Meanwhile, it's not like the bacteria are going away. but Kauffman's adjacent possible is a nice frame. "Putting an optimization program into practice requires a general theory of optimality, which evolutionists have taken directly from the economics of capitalism. It is assumed that organisms are struggling for resources that are in short supply, a postulate introduced by Darwin after he read Malhus's Essay on the Principle of Population. The organism must invest time and energy to acquire these resources, and it reinvests the return from this investment partly in acquiring fresh supplies of resource and partly in reproducing. ... The optimizing theory of allocation assumes that time allocation will be close to optimal for maximizing total investment in reproduction, or growth of the firm. In such theories the criterion of optimality is efficiency, whether of time or invested energy, yet the moralistic and ideological overtones of "efficiency," "waste," "maximum return on investment," and "best use of time" seem never to have come to the consciousness of evolutionists, who adhere to these social norms unquestioningly." The Dialectical Biologist, pp. 25-26Notes critiques of Dawkins. critiques of gene ideology. Levins and Lewontin.Week 14: Nov. 28 Notes for a project presentation of my own, on the student debt situation People have to borrow. Stafford loans and Pell grants aren't enough, so people need private loans. Predatory, deceptive practices, high and adjustable interest rates. Effectively indentured for life Means you have to take whatever work you can get and can't do work that's meaningful, important, constructive, take risks. Student debt exempted from bankruptcy The universities are using students' debt to keep afloat and profit from investments secured by them. (See the bunker video: "The banks lend to the students. The students pay us huge tuition... We spend money that is not ours. Then the students pay the banks."). Is there a student debt bubble? How bubbles work: Dutch tulip craze example. People believe they are a good investment, therefore they are a good investment. You can buy high and sell even higher to another person just like you. A sort of leaderless pyramid scheme. Recognizable as a positive feedback process. When the mystique fails, people get stuck with bad investments. Lose their fortunes, homes. Debt bubble: a certain kind of debt is seen as a good investment by investors. Student debt for instance, because people keep borrowing and they can be forced to pay back no matter how much it hurts them. They sell packages of students' debts to each other and use them as collateral on other investments. But more and more students are defaulting, because the job market sucks so hard and they have medical debt, foreclosure issues, etc. If all those packages of debt lose their value, investors will be in trouble and the trouble will ripple out through the economy, as it did during the housing meltdown. What will happen to students? I don't know. In the case of mortgages, people lost their homes. Student debtors don't have such tangible collateral. But they could end up in debtor's prison or something because of the bankruptcy exception, or just indentured for life as many already are. Loans will become less available, and more people will have to do without college education. It'll become more a privilege of the rich. Tuition keeps universities afloat, whether public or private - they use it directly and to secure other investments. So they will become poorer and continue to cut all "inessentials" such as arts and humanities, becoming more like business schools, to function as a career investment. More of the current trend toward undermining tenure and academic freedom, cutting benefits and commitments by phasing out tenured positions for contingent adjunct and student teaching. With housing costs, a student could owe $46000 at end of first year. 99% of applicants are accepted, 46% of those who come graduate. 82% of students receive financial aid. So many will probably end up with serious debt and no degree. Graeber's point about the ethics of debt - why should it be more important to pay a monetary debt than to make sure someone has health care or housing, or keep babies from dying?
Smartboard This module explains the use of Smartboards in classrooms and demonstrates multiple applications that can be used for social studies, science, language arts, and math integration. Author(s): Nesli Monroe License information Related content Rights not set No related items provided in this feed Advanced Algebra II: Conceptual Explanations This is the Conceptual Explanations part of Kenny Felder's course in Advanced Algebra II. It is intended for students to read on their own to refresh or clarify what they learned in class. This text is designed for use with the "Advanced Algebra II: Homework and Activities" and the "Advanced Algebra II: Teacher's Guide" collections (coming soon) to make up the entire course. Author(s): No creator set License information Related content No related items provided in this feed Applied Finite Mathematics This module contains all 10 chapters of the Applied Finite Mathematics open textbook by Rupinder Sekhon. NOTE: This book is a work in progress and has not yet been marked up in CNXML. You can download individual chapter files from their respective modules. Author(s): Rupinder Sekhon License information Related content Rights not set No related items provided in this feed Elementary Algebra Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques. Author(s): No creator set
This book is designed to form the basis of a one-year course in discrete mathematics for first-year computer scientists or software engineers. The materials presented cover much of undergraduate algebra with a particular bias toward the computing applications. Topics covered include mathematical logic, set theory, finite and infinite relations and mapping, graphs, graphical algorithms and axiom systems. It concludes with implementations of many of the algorithms in Modula-2 to illustrate how the mathematics may be turned into concrete calculations. Numerous examples and exercises are included with selected solutions to the problems appearing in the appendix.
Category Archives: Algebra II Josten's will be here on Monday and Tuesday, April 15 and 16 to give out cap and gown and announcement orders. If anyone owes a balance please bring money Monday or Tuesday to receive your order. They are only taking … Continue reading → This nine weeks we will discuss the following: logarithmic and exponential functions, sequence and series, z-scores, normal distributions, permutations and combinations. Additionally, you will prepare for your Algebra 2 SOL exam which will be administered on May 15th. You will … Continue reading → We will discuss direct, inverse, and joint variations on the next class period. Please review the following file before class. VariationNotes. You may want to print the file and attempt the examples as well. Pay close attention to the formula … Continue reading → factoring practice.WS Remember that when simplifying rational expressions you will need to factor using the GCF method or the try and error method. I am attaching a handout for you to use to practice factoring. There are also videos on … Continue reading →
The text comprises explanations and examples of basic arithmetic operations applied to whole numbers and fractions, a lengthy section on commercial arithmetic, and a brief account of square and cube roots at the end.
Mathematical Modelling in One Dimension: An Introduction Via Difference and Differential EquatiUses a wide variety of applications to demonstrate the universality of mathematical techniques in describing and analysing natural phenomena. Difference and Differential Equations in Mathematical Modelling demonstrates the universality of mathematical techniques through a wide variety of applications. Learn how the same mathematical idea governs loan repayments, drug accumulation in tissues or growth of a population, or how the same argument can be used to find the trajectory of a dog pursuing a hare, the trajector... MOREy of a self-guided missile or the shape of a satellite dish. The author places equal importance on difference and differential equations, showing how they complement and intertwine in describing natural phenomena. The universality of mathematical techniques is demonstrated through a wide variety of applications and a description of basic methods for their analysis. The author places equal importance on difference and differential equations, showing how they complement and intertwine in describing natural phenomena.
In Algebra 1, we'll begin with review of the fundamentals of Order of Operation, integers, fractions, and percents. With good grounding, we'll then explore the various algebraic tools of understanding the art of problem solving. We learn the language and working tools of the math tribe.
(Adopted from Chapter Zero Instructor Resource Manual by Carol Schumacher with a nod to Dr. Dana C. Ernst) Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principal ambitions is to make you the student independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much "classroom knowledge" is fairly superficial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are "getting it" comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills. Expectations This course will be different than most math classes that you have taken. You are used to being asked to do things like: "solve for ," "take the derivative of this function," "integrate that function," etc. Accomplishing tasks like these usually amounts to mimicking examples that you have seen in class or in your textbook. The steps you take to "solve" problems like these are always justified by mathematical facts (theorems), but rarely are you paying explicit attention to when you are actually using these facts. Furthermore, justifying (i.e., proving) the mathematical facts you use may have been omitted by the instructor. And, even if the instructor did prove a given theorem, you may not have taken the time or have been able to digest the content of the proof. This course is all about "proof." Mathematicians are in the business of proving theorems and this is exactly our endeavor. You will be exposed to what "doing" mathematics is really all about. In a typical course, math or otherwise, you sit and listen to a lecture. (Hopefully) These lectures are polished and well-delivered. You may have often been lured into believing that the instructor has opened up your head and is pouring knowledge into it. I absolutely love lecturing and I do believe there is value in it, but I also believe that the reality is that most students do not learn by simply listening. You must be active in the learning you are doing. I'm sure each of you have said to yourselves, "Hmmm, I understood this concept when the professor was going over it, but now that I am alone, I am lost." In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called the Moore method (after R.L. Moore, a former professor of mathematics at the University of Texas, Austin). Modifications of the Moore method are also referred to as inquiry-based learning (IBL) or discovery-based learning. Much of the course will be devoted to students proving theorems on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the work "produce" because I believe that the best way to learn mathematics is by doing mathematics. I learned to ride a bike by getting on and then falling off, and in a similar fashion, you will learn mathematics in this course by attempting it and sometimes falling off. In this course, everyone will be required to read and interact with course notes on your own; write up quality proofs to assigned problems; present proofs on the board to the rest of the class; participate in discussions centered around a student's presented proof; call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar on first glance. As the semester progresses, it should become clear to you what the expectations are. Course Notes We will not be using a textbook this semester, but rather we will be using a theorem-sequence adopted for inquiry-based learning and the Moore method for teaching mathematics. The theorem-sequence that we are using is an adaptation of the notes by Ron Taylor by The Journal of Inquiry Based Learning in Mathematics. The published original version of the notes can be found here Attendance Regular attendance is expected and is vital to success in this course. If you miss more than 6 classes you cannot pass this class. Proofs More or less all of the work you will be assessed on in this course involves writing or presenting proofs. You will be assigned proofs for practice, proofs to read, proofs to present, and the exams will involve doing proofs. It will be a semester long exercise in learning proofs by doing proofs. Traditionally in a course like this students are discouraged from working togather but, unlike a traditional Moore method course, you are allowed and encouraged to work together. You can use the online forum at or you can meet up and work togather. You should however be careful that you aknowledge any help you recieve. I have written some Proof guidelines to give you a sense of what I will look for when grading your proofs. Class Presentations Most days there will be proofs presented by students. These will be written up in sets (several at a time). Then each proof will be presented by its author. To steamline this process I will ask that you claim proofs in advance (in the online forum) to present. This way you can see what proofs are still open for presentation. You will notice that the grade calculation includes a class participation component. This gives you incentive to pay attention to the presentations. You will get graded on how you interact with the people presenting. Also, you should keep a notebook with all of the proofs presented in class. To make this easier I will ask that each proof presented be written up in the online forum. You will recieve some participation credit for this. Exams There will be a midterm exam and a cumulative final exam. All exams will may consist of an in-class part and a take-home part. Each exam will be worth roughly 25 percent of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time. Rules of the Game You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other, the course notes, me, and your own intuition. Basis for Evaluation Your final grade will be determined by the scores of your presentations, class participation, and exams. grade calculation Additional Information Getting Help There are many resources available to get help. First, I recommend that you work on homework in groups as much as possible. You should come see me whenever you can. Also, you are strongly encouraged to ask questions in the course forum, as I will post comments there for all to benefit from. Closing Remarks (Adopted from pages 202-203 of The Moore Method: A Pathway to Learner-Centered Instruction by C.A Coppin, W.T. Mahavier, E.L. May, and G.E. Parker) There are two ways to approach this class. The first is to jump right in and start wrestling with the material. The second is to say, "I'll wait and see how this works and then see if I like it and put some problems on the board later in the semester after I catch on." The second approach isn't such a good idea. If you try every night to do the problems, then either you will get a problem (Shazaam!) and be able to put it on the board with pride or you will struggle with the problem, learn a lot in your struggle, and then watch someone else put it on the board. When this person puts it up you will be able to ask questions that help you and the others understand it, as you say to yourself, "Ahhh, now I see where I went wrong and now I can do this one and a few more for the next class." If you do not try problems each night, then you will watch the student put the problem on the board, but perhaps will not quite catch all the details and then when you study for the exams or try the next problems you will have only a loose idea of how to tackle such problems. And then the anxiety will build and build and build. So, take a guess what I recommend that you do. If you are struggling too much, then there are resources available for you. Work together and help each other learn. Use the course forum! I am always happy to help you. If my office hours don't work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don't wait until it is too late if you need help. Ask questions! NC Policy The NC policy has changed beginning with this semester. For a 100% refund the date is August 23 and for a 50% refund the date is September 1. The last day to obtain an NC is Friday Oct. 23. This is a hard deadline and will be enforced as such. The department will not approve any late NC requests. Students must request an NC through MetroConnect; faculty approval is no longer required. Holidays: Observance of religious holidays follows College policy, which is posted on the web at in the Academic and Campus Policies for Students section. Each student is responsible for understanding and abiding by the policy. Accommodations for Students with Disabilities The Metropolitan State College of Denver is committed to making reasonable accommodations to assist individuals with disabilities in reaching their academic potential. If you have a disability that may impact your performance, attendance, or grades in this class and are requesting accommodations, then you must first register with the Access Center, located in the Auraria Library, Suite 116, 303-556-8387. The Access Center is the designated department responsible for coordinating accommodations and services for students with disabilities. Accommodations will not be granted prior to my receipt of your faculty notification letter from the Access Center. Please note that accommodations are never provided retroactively (i.e., prior to the receipt of your faculty notification letter.) Once I am in receipt of your official Access Center Faculty Notification Letter, I would be happy to meet with you to discuss your accommodations. All discussions will remain confidential. Further information is available by visiting the Access Center website
Maths everywhere This Instruments TI-83 calculator and the book Tapping into Mathematics With the TI-83 Graphics Calculator (ISBN 0201175479). After studying this unit, you should: be able to describe your view of what mathematics is; have begun to recognise different types of written mathematics and developed your skill at reading it; be able to tackle mathematical problems using a calculator and with understanding for basic arithmetic, percentages, square roots, reciprocals and powers; be able to express and interpret numbers in scientific notation, both in writing and on your calculator; be able to give some examples of common mathematical 'doing–undoing' pairs of operations; be more attuned to noticing mathematical questions arising from the world around you. Maths everywhere Introduction This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize In order to complete this unit you will need to have obtained a Texas Instruments TI-83 calculator and the book Tapping into Mathematics With the TI-83 Graphics Calculator by Barrie Galpin and Alan Graham (eds), Addison Wesley, 1997 (ISBN 0201175479). This unit is from our archive and is an adapted extract from Open mathematics (MU120
This course aims to develop an understanding of elementary linear algebra and differential calculus. It provides students with the skills of solving scientific and engineering problems formulated as linear systems, one dimensional optimization problems and nonlinear algebraic equations. Course Intended Learning Outcomes (CILOs) Upon successful completion of this course, students should be able to: apply mathematical and computational methods to a range of scientific and engineering applications involving linear algebra, calculus and complex numbers. 2 6. the combination of CILOs 1--6 32 hours in total Learning through tutorials is primarily based on interactive problem solving allowing instant feedback. 2 2 hours 3, 4 2 hours 1 1 hour 5 2 hours Learning through take-home assignments helps students understand basic concepts and techniques of basic linear algebra, single variable calculus, complex numbers, and some applications in engineering science. 1--5 after-class Learning through online examples for applications helps students apply mathematical and computational methods to some problems in engineering applications. 5 after-class Learning activities in Math Help Centre provides students extra help. 1--5 statistics to see how well the students have learned the basic concepts, and techniques of basic linear algebra and single variable calculus as well as some applications. Hand-in assignments 1--5 0-15% These are skills based assessment to see whether the students are familiar with the basic concepts, techniques of linear algebra, elementary calculus, complex numbers and their related applications in engineering. Examination 6 70% Examination questions are designed to see how far students have achieved their intended learning outcomes. Questions will primarily be skills and understanding based to assess the student's versatility in elementary linear algebra and calculus, and complex numbers Part III Keyword Syllabus Differentiation and Applications. Integration and Applications. Complex Numbers. Vectors, Matrices, Determinants and System of Linear Equations. Coordinate Geometry in Space, Vector Equations of Lines and Planes.
Free Maths Textbook " The Free High School Science Textbook (FHSST) project is our contribution towards furthering Science Education in South Africa. As young South Africans who believe in building up our country, we want to use our skills as scientists to help our next generation by providing free science and mathematics textbooks for Grades 10-12 to all South African learners. Science education is about more than Physics, Chemistry and Mathematics... It's about learning to think and to solve problems which are valuable skills that can be applied through all spheres of life. Teaching these skills to our next generation will help them when it is their turn to make a difference to our country." - FHSST website
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. Math 250-CMATLAB Assignment #11 Revised 5/25/00LAB 1: Introduction to MATLABIn this lab you will learn how to use MATLAB to create and operate on matrices and vectors. The commands needed to do this are short and easy to remember, because MATL Math 250-CMATLAB Assignment #31 Revised 5/24/00LAB 3: Complete Solution to Ax = bIn this lab you will use MATLAB to study the complete solution of the linear equation Ax = b, where A is a given m n matrix, b is a given m 1 column vector and Math 250-CMATLAB Assignment #41 Revised 5/23/00LAB 4: Vector Spaces and Approximate Solutions to Ax = bIn this lab you will use MATLAB to study four topics: 1) The subspace spanned by a set of vectors, and the important concepts of independenc Math 250-CMATLAB Assignment #51 Revised 5/23/00LAB 5: A = QR Factorization, Determinants, and Eigenvalues/EigenvectorsIn this lab you will use MATLAB to study three topics: 1) How to transform a given set of basis vectors into an orthonormal b Math 250-CMATLAB Assignment #61 Revised 5/23/00LAB 6: Symmetric and Positive-Definite Matrices, Singular Value Decomposition, and Linear TransformationsIn this lab you will use MATLAB to transform a matrix into a diagonal matrix by finding its Intro. to Linear Algebra 250-C Extra Credit Project 1 - Graphs and Matrices Please write all answers on separate sheets of paper. Your answers should be numbered and in the same order in which the problems appear. Your project should be stapled and y Intro. to Linear Algebra 250-C Extra Credit Project 2 - Graphs and Markov Processes Introduction: In this project we will learn about the connection between linear algebra and market distribution. Suppose that, every day, there are three types of ent Math 250-CMATLAB Assignment #11 Revised 9/15/02LAB 1: Matrix and Vector Computations in MATLABIn this lab you will use MATLAB to study the following topics: How to create matrices and vectors in MATLAB. The commands to do this are short and e Math 250-CMATLAB Assignment #21 Revised 8/20/02LAB 2: Linear Equations and Matrix AlgebraIn this lab you will use MATLAB to study the following topics: Solving a system of linear equations by using the row reduced echelon form of the augmente Math 250-CMATLAB Assignment #31 Revised 10/15/02LAB 3: LU Decomposition and DeterminantsIn this lab you will use MATLAB to study the following topics: The LU decomposition of an invertible square matrix A. How to use the LU decomposition to Math 250-CMATLAB Assignment #41 Revised 10/25/02LAB 4: General Solution to Ax = bIn this lab you will use MATLAB to study the following topics: The column space Col(A) of a matrix A The null space Null(A) of a matrix A. Particular solutions Math 250-CMATLAB Assignment #51 Revised 11/13/02LAB 5: Eigenvalues and EigenvectorsIn this lab you will use MATLAB to study these topics: The geometric meaning of eigenvalues and eigenvectors of a matrix Determination of eigenvalues and eige Math 250-CMATLAB Assignment #61 Revised 11/11/02LAB 6: Orthonormal Bases, Orthogonal Projections, and Least SquaresIn this lab you will use MATLAB to study the following topics: Geometric aspects of vectors: the norm of a vector, the dot prod Math 250-CMATLAB Assignment #11 Revised 9/12/01LAB 1: Matrix Computations in MATLABIn this lab you will learn how to use MATLAB to create and operate on matrices and vectors. The commands needed to do this are short and easy to remember, becau Math 250-CMATLAB Assignment #31 Revised 12/12/01LAB 3: DeterminantsIn this lab you will use MATLAB to study the key aspects of the determinant of a square matrix: how it changes under row operations and matrix multiplication how it can be ca Math 250-CMATLAB Assignment #41 Revised 12/11/01LAB 4: Vector Spaces and General Solution to Ax = bIn this lab you will use MATLAB to study the following topics: The subspace spanned by a set of vectors, and the fundamental concepts of indepe Math 250-CMATLAB Assignment #51 Revised 12/12/01LAB 5: Orthogonal Subspaces, QR Factorization, and Inconsistent Linear SystemsIn this lab you will use MATLAB to study four topics: The Gram-Schmidt Algorithm that transforms a given set of basi Math 250-CMATLAB Assignment #61 Revised 12/12/01LAB 6: Eigenvalues and EigenvectorsIn this lab you will use MATLAB to study three topics: The geometric meaning of eigenvalues and eigenvectors of a matrix Determination of eigenvalues and eige
This edition retains all the features of the second edition, including a handy thesaurus, and adds to them by way of: approximately 1000 new entries, primarily drawn from key areas of secondary study - technology, science, media, popular culture additional 'Common Error' and 'Origin' b... The Maths Tracks New South Wales Homework Books are designed to support the content of class activities and lessons. Each book consists of 40 homework pages, each with a student self-assessment task, a Homework Record and space for parent-teacher comments. A removable answer section is also provided
Oxford International Maths for Cambridge Secondary 1 Thorough preparation for Cambridge Checkpoint and a flying start for Cambridge IGCSE Help your students excel in the Cambridge Checkpoint test and lay the best possible foundations for the Cambridge IGCSE. With a huge focus on extension and challenge, this course will rigorously prepare students for strong achievement at Checkpoint level and beyond. We are working with Cambridge International Examinations towards endorsement of these titles. Features Comprehensive - all aspects of the curriculum are thoroughly covered so you can trust in complete preparation Excel at Checkpoint - questions are designed to match the Checkpoint style, building confidence Extension and challenge - a huge focus on extension material to ensure a flying start at Cambridge IGCSE Practice - over 400 pages of rigorous practice to make sure students truly understand all the material Eliminate confusion - detailed worked examples help students understand every step in complex problems Customisable activities - digital exercises in PDF and Word format so you can tailor lessons exactly to your class Internationally focused - with examples from all over the world, so material is genuinely relevant to your students
Mathematics Meets Technology - Brian Bolt - Paperback 9780521376921 ISBN: 0521376920 Publisher: Cambridge University Press Summary: A resource book which looks at the design of mechanisms, for example gears and linkages, through the eyes of a mathematician. Readers are encouraged to make models throughout and to look for further examples in everyday life. Suitable for GCSE, A level, and mathematics/technology/engineering courses in Further Education.
Search Journal of Online Mathematics and its Applications: Journal of Online Mathematics and its Applications Tool Building: Web-based Linear Algebra Modules by David E. Meel and Thomas A. Hern Eigenizer Tool and Sample Activity Working with Eigenizer, similar to Transformer2D, involves coordinated actions between defining the column vectors of the matrix of transformation. The yellow box controls the column vectors defining the matrix of transformation. In particular, the green vector controls the first column vector and the blue vector controls the second column vector. By grabbing the ends of these two vectors, you can construct any 2x2 matrix. Below the yellow box is a box that controls the vector x. As you move x (the red vector) about the domain of the transformation, you can watch both x and the image T(x) (the magenta vector) change in the large area to the right of the screen, depicting the codomain of the transformation. The movement of the vector T(x) depends on the nature of the matrix of transformation. The large codomain region also displays information concerning the length of the vector x, the length of the vector T(x), the radian measure of the angle between these two vectors, and a lambda approximator. Two buttons at the bottom of this region control the display of the Eigen Equations in a red box above the codomain box. Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page. Sample Exploratory Activity: This sample activity provides a guided exploration of eigenvalues and eigenvectors of a particular matrix and includes a set of questions that can be asked for any other matrix, as well as some general questions about the tool and observations made from interacting with the tool. Using the vectors contained in the yellow box, move (by click dragging) the green vector to (3,-1) and the blue vector to (-2,2). This should construct the following matrix of transformation: . Move the red vector x (in the domain portion of the tool) and observe the movement of the magenta vector, T(x). If possible, move the vector x so that the vectors T(x) and x are collinear. Note: This can be aided by examining the angle measure at the bottom of the y-axis in the range portion of the tool. Click on the "Show lambda 1 equation" button, and observe whether the equation is true, i.e., do the vectors displayed on the right and left sides of the equation match each other? If they do not match, click on the "Show lambda 2 equation" button, and check if truth is found. Move the red vector x (in the domain portion of the tool) so that it and the T(x) vector are collinear in a different location. Redo step 3 for this new location. Given your exploration (and perhaps some additional ones), answer the following questions: What are the eigenvalues for this matrix? What are corresponding eigenvectors for these eigenvalues? What does the lambda approximator do? If an eigenvalue is positive, what does this mean concerning the form of collinearity between the vector x and the vector T(x)? If an eigenvalue is negative, what does this mean concerning the form of collinearlity between the vector x and the vector T(x)? Can you have a matrix with two positive eigenvalues or two negative eigenvalues? Explain why or why not For a given eigenvalue, is there a unique eigenvector or a set of corresponding eigenvectors? If unique, explain why it's unique and if there is a set, explain how to describe the set. Define a matrix that does not have any real eigenvalues? In general, what would be the nature of the column vectors of such a matrix? Is it possible to define a matrix that has eigenvalues of -3 and 2? If it is possible, state how many such matrices could be constructed, and provide a specific example of at least one. Is it possible to define a matrix that has only a single eigenvalue, say -2? Explain why or why not. The tool allows students to explore specific matrices as well as hypothesize about possible matrices with particular properties. It is with the latter type of explorations that the worlds of geometry and computation can fuse. Students need to think beyond computations that MatLab might be able to perform and ponder the possibilities of "what if?".
book is designed to help bridge the gap between GCSE and AS Level Maths. It's full of clear notes and helpful practice to recap the most difficult topics from GCSE Maths that students need when going on to study AS Level Maths. Everything you need to know for all the exam boards is explained clearly and simply, in CGP's chatty straightforward style
This homeschool teacher's guide accompanies the Saxon Math 2 StudentWorkbooks. Scripted lessons are included for each chapter, with dialogue, chalkboard sketches and more. Reduced student pages are also included for easy tracking and communication between student and parent. Lesson preparation lets teachers know the materials they'll need and any beforehand preparation. Answers are lightly overlaid on the reduced student pages. 735 pages, softcover,spiral-bound Math 2, Home Study Teacher's Edition Review 1 for Math 2, Home Study Teacher's Edition Overall Rating: 4out of5 Date:April 26, 2011 momof4 Gender:female Quality: 5out of5 Value: 4out of5 Meets Expectations: 4out of5 I used Saxon I and liked it a lot so I decided to continue my kids math with Saxon II.
Math--that four-letter word! Abstract Math anxiety is extremely prevalent in the general population, but tends to have gender and age as key aspects to occurrence. A factor in this may be that many elementary teachers tend to be math phobicpho·bic adj. Of, relating to, arising from, or having a phobia. n. One who has a phobia. . This paper looks at two returning adult college students who participated in a Math Anxiety Workshop as well as follow-up mathematics tutoring and their battles with math anxiety. Introduction As an instructor of preservice teachers, I am always faced with instances of math anxiety in my students. I teach classes in mathematics methods of instruction to both prospective secondary and elementary teachers. Interestingly, even the students who will be secondary teachers have phobiasPhobiasDefinition A phobia is an intense but unrealistic fear that can interfere with the ability to socialize, work, or go about everyday life, brought on by an object, event or situation. that need to be addressed, these tending most often to be in the areas of problem solvingproblem solving Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. , probability, or geometry. However, it is with elementary educationelementary education or primary education Traditionally, the first stage of formal education, beginning at age 5–7 and ending at age 11–13. majors that I run into the most severe cases of math anxiety. It seems to me, after many years of working with these students, that about two-thirds of them come in with feelings of inadequacy about their math ability, fears of having to teach math, and tendencies toward avoidance of math. My present elementary education major students have been required to pass three university math courses; one college algebraalgebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as and two focused on elementary mathematicsElementary mathematics consists of mathematics topics frequently taught at the primary and secondary school levels. The most basic are arithmetic and geometry. The next level is probability and statistics, then algebra, then (usually) trigonometry and pre-calculus. , and still have these reservations. Given the amount of information that can be found on the internet regarding math anxiety (a Google searchGoogle is owned by Google, Inc. whose mission statement is to "organize the world's information and make it universally accessible and useful". The largest search engine on the web, Google receives several hundred million queries each day through its various services. turned up 1,510,000 listings on this topic), I know this is common across the nation. Specifically, two recent articles in the National Council of Teachers of Mathematics' (NCTM NCTM National Council of Teachers of Mathematics NCTM Nationally Certified Teacher of Music NCTM North Carolina Transportation Museum NCTM National Capital Trolley Museum NCTM Nationally Certified in Therapeutic Massage ) journal Teaching Children Mathematics (Wolodko, Willson, and Johnson, 2003; Guillaume and Kirtman, 2005) share studies of preservice elementary teachers' backgrounds in mathematics and how to deal with negative attitudes, a further indication that this problem is wide spread. Additionally, it appears that starting in adolescence, math anxiety is far more common in females than in males (Hyde, Fennema, and Lamon, 1990). Since the vast majority of elementary teachers are female, this highlights how common math anxiety is and how easy it may be to pass on the same problems to generations of students. However, it is not just teachers who are burdened with this phobiaphobia: see neurosis. phobia Extreme and irrational fear of a particular object, class of objects, or situation. A phobia is classified as a type of anxiety disorder (a neurosis), since anxiety is its chief symptom. . In fact many people may choose not to pursue a career as a teacher because of their fear of taking and teaching mathematics courses. Walker and Karp (2005) indicate that math phobia is so widespread it "profoundly affects the country's policies, teaching practices, and ultimately students' performance" (p.39). Having math anxiety may not, of course, mean a person cannot understand or do certain mathematical tasks, but it can inhibit the ability to learn or successfully demonstrate that learning. While a distinction must be made between math anxiety and test anxiety, which means that students may not be able to convince others of what they know mathematically (Kazelskis, Reeves and Kirsh, 2000), the two are tightly connected. Since testing is the most common means of determining acquisition of math knowledge and skills for instructors, many students tend to exhibit test anxiety primarily in mathematics rather than in other subjects. So while a math anxiety that involves avoidance or negative self-talk may cause a test anxiety, it is also possible that a test anxiety may generate a conditioned math anxiety (Hembree, 1990). In either case, the two appear to be linked for most students with math anxiety. A workshop on math anxiety During a time I was working in student support servicessupport services Psychology Non-health care-related ancillary services–eg, transportation, financial aid, support groups, homemaker services, respite services, and other services setting, I coordinated the math tutoring services for the college. This involved one-on-one tutoring, training of peer tutors A peer tutor is anyone who is of a similar status as the person being tutored. In an undergraduate institution this would usually be other undergraduates, as distinct from the graduate students who may be teaching the writing classes. , running review sessions, and student advising. A large number of students I worked with were adults returning to college, many of them extremely fearful of the math they would need to do after many years out of school. I decided that it might be advantageous for these students to offer a workshop on how to deal with their math anxiety. I examined several of the seminal self-help books in this area (Fear of Math: How to Get Over It and Get On with Your Life, Zaslavsky, 1994; Math Anxiety Reduction, Hackworth, 1985; Mind over Math, Kogelman and Warren, 1979; Overcoming Math Anxiety, Tobias, 1978) to design activities and discussion questions. At the first meeting eight participants, only one of which was of traditional college age, showed up. Six, all women, finished the six-week program, with two of the participants attending only the first session, and two continuing to meet with me after the workshop for advising and tutoring appointments. We began the sessions describing that anxiety affects us in three ways: physically, cognitively (or intellectually), and affectivelyaf·fec·tive adj.Psychology 1. Influenced by or resulting from the emotions. 2. Concerned with or arousing feelings or emotions; emotional. (emotionally). Since the participants were fully aware that they were suffering from and with math anxiety, they had a cognitive recognition of the problem and were taking a reasoned step to address it. However, dealing with the other two aspects would be key to overcoming their fears. We generated this listing of characteristics of the three domains present during an anxiety period: The first assignment for the workshop was to write a math autobiography. Students were instructed to think over their experiences in and outside of school that may have influenced their attitudes toward math. They were asked to try to pinpoint when they might have begun to feel uncomfortable or negative about their abilities or their experiences in math. From this we were able to develop a spirited discussion about the root causes of their fears. The ones discussed appear to be commonly reported by others (Perry, 2004) and fall into the three categories of poor or insensitive teachers, breaks or gaps in learning, and attitudes of others. When it came to teachers, the workshop participants said that many of their teachers did relatively little to explain the reasoning behind what they were learning but usually only presented processes. Most admitted what was so bothersome to them was that it appeared the other students in the class always understood what was going on. They felt they were the only ones who were struggling with the material and so did not blame the teacher for poor teaching, just for not being able to help them learn. However, there were also some horrifying stories of teachers bullying and embarrassing students when they didn't know the answer. In particular, several students recounted having to stand at the board trying to finish a computation, for which they had no clue where to begin, while the rest of the class laughed. A student mentioned she much preferred when instructors used a white board or PowerPoint so that they wouldn't need to see the blackboard(1) See Blackboard Learning System. (2) The traditional classroom presentation board that is written on with chalk and erased with a felt pad. Although originally black, "white" boards and colored chalks are also used. . Gaps in learning could be explained in several ways. Though several people mentioned they got lost on a particular topic or level (usually fractions, long division or algebra) and just never felt confident or competent after that, the reasons for the gap were often not the result of the student at all. In one case, a woman recounted moving and changing schools. At the new school, the class was well ahead of where she was in the old school and she never got the opportunity to "fill in the missing pieces." Similarly, a woman reported she had missed about two months of school due to a serious illness when she was in fourth grade. Though she could make up the content from the other subjects she never quite got back up to speed in math. Finally, one woman reported that her mother died when she was young and she was in math class when the principal came to her class to tell her. The negative connection was never broken for her. Finally, the attitudes of others appeared to be influential in participants' attitudes. Several of the women reported school counselors A school counselor is a counselor and educator who works in schools, and have historically been referred to as "guidance counselors" or "educational counselors," although "Professional School Counselor" is now the preferred term. who told them it was fine to do poorly in math because women do need it as much as men. Almost all of them had mothers who said that they had done poorly in math as Mathematics courses named Math A, Maths A, and similar are found in: Mathematics education in New York: Math A, Math A/B, Math B Mathematics education in Australia: Maths A, Maths B, Maths C well so it was probably genetic. However, there were again some horror storieshorror story Story intended to elicit a strong feeling of fear. Such tales are of ancient origin and form a substantial part of folk literature. They may feature supernatural elements such as ghosts, witches, or vampires or address more realistic psychological fears. of parents, siblingssiblingsnpl (formal) → frères et sœurs mpl (de mêmes parents), or teachers telling them they were stupid and that they would never be able to amount to something. Once we examined the root causes, we analyzed the "truth" of the situations. By examining the events as "history," participants were able to intellectually see the irrationality of the fear. For another activity, we spent some time doing an exercise that required them to look at various math symbols and expressions and told to indicate their immediate response as comfortable, uncomfortable, or panic. This revealed two things to the participants. First, that they were comfortable or only mildly uncomfortable in seeing many of the symbols and only a few really bothered them. This gave them a sense that the problem wasn't nearly as overwhelming as they thought. Second, it clearly pinpointed the areas of "gaps" in knowledge. It indicated for them a place to go back to for relearningre·learn·ing n. The process of regaining a skill or ability that has been partially or entirely lost. re·learn v.. We even went through a few "math lessons" on fractions and equations and they were quickly and easily they could comprehend the material. A reminder that a forty-year-old has a much better background to understand the math information than a ten-year old did, finally was clear to them. We were then able to concentrate on the ways we ourselves in our learning. The class was able to discuss elements for improving study habits like setting short-term goals, rewarding themselves for accomplishments, being more assertiveas·ser·tive adj. Inclined to bold or confident assertion; aggressively self-assured. as·sertive·ly adv. in seeking out help, and generating more positive talk. In role-playing a tutoring session, participants were encouraged to say aloud everything they were thinking as they worked. Most often they were thinking things like, "I'll never get this" or "This is so stupid" or even "I'm so stupid!" We worked on changing that talk to "What am I sure I know about this problem?" and "What do I do first?" Once the talking was focused on the topic rather than on the feelings, participants were much better able to concentrate on the task. Though dealing with the emotional aspect of the fear is certainly the hardest thing to overcome, building a cognitive awareness and minimizing the physical symptoms is necessary reduce the feelings of inadequacy. To deal with the physical aspects, we practiced common therapeutic techniques of breathing exercises, imaging, and desensitizationdesensitization or hyposensitization Treatment to eliminate allergic reactions (see allergy) by injecting increasing strengths of purified extracts of the substance that causes the reaction. . Using relaxation messages to reduce tenseness and slow the heart rate, one is less likely to be overcome by the anxiety. The imaging we did helped them develop a routine to control breathing and other physical responses and we were gradually able to spend more time during the imaging in the "math place." However, acknowledging that students were currently in classes and under time constraints In law, time constraints are placed on certain actions and filings in the interest of speedy justice, and additionally to prevent the evasion of the ends of justice by waiting until a matter is moot. , we identified things we could do to help in the process. Participants were encouraged to discuss the issue of their math anxiety with their math instructors, and I sometimes advocated on their part. They found that often their instructors were willing to help in several ways such as letting them take tests in the instructor's office or in the tutoring lab, extend the time limit for the test, meet with the instructor in his/her office to get questions answered instead of in front of the whole class, and bring tape recorderstape recorder, device for recording information on strips of plastic tape (usually polyester) that are coated with fine particles of a magnetic substance, usually an oxide of iron, cobalt, or chromium. The coating is normally held on the tape with a special binder. into class so they could double check their notes. A closer look at two students As mentioned previously, of the six participants, only two kept close contact with me after the workshop. These two had the most dramatic stories and had reached the highest level of anxiety. They also had much in common. Both were in their late 30s, studying for business degrees, had extremely negative educational backgrounds, had significant people in their life telling them they wouldn't be able to finish, had jobs that were integral to the financial welfare of their families, and reported similar anxiety reactions to math and tests. Most importantlyAdv.1.most importantly - above and beyond all other consideration; "above all, you must be independent" above all, most especially , both were enrolled in calculuscalculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value. the following semesterse·mes·ter n. One of two divisions of 15 to 18 weeks each of an academic year. [German, from Latin (cursus) s. They had been routinely using the tutoring facilities at the college, which had been instrumental in their successful completion of math classes to that point. When "Emily" and "Rose" started their calculus class, both continued to come to the tutoring lab for help. After taking her first test, Emily came into my office in tears. She had done horribly, she "blanked out" on many of the answers, and she was sure she had failed. We discussed whether she was able to do some breathing and during the test and she said she did, but then she worried she would run out of time, so didn't continue (Emily did not have a math instructor who was willing to make any adaptations for her anxiety). I said we would go over the test when she got it back. Rose was not as upset as Emily about her test, but did admit that she also struggled with "blanking out" during the test. Emily returned with her test, on which she received a D, again in tears. We spent quite a while as I focused her attention on each item and went over what she had done correctly and how far she had gotten on each item, and that she had attempted each item in spite of her fear of not finishing in time. Rose would not show me her test, just said she didn't do well and we focused on the new work. This was pretty much the pattern established for the rest of the semester, with Emily bringing in her tests and going over them with me. I would point out each improved grade and how much more she accomplished each time. She became less upset and more analytical the about process each time, knowing this is what we would do. Rose continued to come to the tutoring center for help with homework, but did not share any information about her grades. Although Emily ended up in the hospital at one point towards the end of the semester, and missed a test, she was able to complete the final and earn a B in the class. Rose did not finish the semester. What appears to be key here that differentiates the two women's experiences is Emily's ability to deal with the cognitive aspect of her anxiety. Ho et al (2000) point out that there are two different dimensions to test anxiety, and cognitive, similar to those domains mentioned previously in relation to math anxiety. According toaccording to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. Ho et al, "the test anxiety measures may have tapped more into cognitive anxiety, which is related to interference of task completion" (p. 375). In other wordsAdv.1.in other words - otherwise stated; "in other words, we are broke" put differently , Emily was better able to control the problems she was having both in learning math and in test-taking, by not only attending to the physical and emotional aspects, but by learning to focus her thinking. Rose, getting the same emotional support and continued reminders for handling her stress, never cognitively addressed her progress by identifying her successes and well as her errors. Conclusion My experiences with the Math Anxiety Workshop and individual students appear to be consistent with the literature on math and test anxiety identified in this paper. However, both experience and research highlight the cognitive nature of the phobia that must be as carefully and consistently attended to as the physical and emotional domains. So in summary, and in addition to making efforts not to create the anxiety, instructors should: 1. Acknowledge that math anxiety is real, is pervasive, and is debilitatingde·bil·i·tat·ing adj. Causing a loss of strength or energy. Debilitating Weakening, or reducing the strength of. Mentioned in: Stress Reduction . 2. Provide accommodations, especially for test taking, to help alleviate the conditions that aggravate the anxiety. 3) Be assertive in working with instructors to get needed information and accommodations, as well as getting questions answered. 4) Practice relaxation and breathing techniques in less stressful times to get efficient in their use. 5) Go back to the math where the gaps exist, relearnVerb1.relearn - learn something again, as after having forgotten or neglected it; "After the accident, he could not walk for months and had to relearn how to walk down stairs" those concepts and skills before trying to move forward. Hackworth, R.D. Math anxiety reduction. . H & H Publishing, 1985. Hembree, R. The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33-46, 1990. Ho, H.Z.; Senturk, D, Lam, A,G., Zimmer, J.M., Hong, S, Okamoto, Y, Chiu, S.Y., Nakazawa, Y, and Wang, C.P. The affective and cognitive dimensionsCognitive dimensions are design principles for notations & programming language design, described by researcher Thomas R.G. Green. The dimensions can be used to evaluate the usability of an existing interface, or as heuristics to guide the design of a new one. of math anxiety; a cross-national study. Journal for Research in MathematicsEducation, 31, 362-379, 2000. Kogelman, S. and Warren, J. Mind over math McGraw-Hill Publishing, 1979. Zaslavsky, C. Fear of math: How to get over it and get on with your life. , 1994. Melissa Freiberg, University of Wisconsin-Whitewater The University of Wisconsin–Whitewater (also known as UW-Whitewater) is part of the University of Wisconsin System, located in Whitewater, Wisconsin. It became Wisconsin's second public college on April 21, 1868 when it opened its doors to 39 students taught by nine Melissa Freiberg, Ph.D. is an Associate Professor of Curriculum and Instruction specializing in mathematics education. 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It's mainly intuitive. Once you know the techniques, you will be amazed at how powerful they can be. Topics in Algebra 1 include handling linear equations and inequalities, graphing, word problems, exponents and radicals, quadratic equations and factoring.
College Algebra Credits: 3Catalog #20804212 College Algebra includes fundamental topics covered in Intermediate Algebra with a more careful look at the mathematical details and a greater emphasis on the concept of function. It covers quadratic, polynomial, rational, exponential and logarithmic functions, equations and inequalities; the use of matrices and determinants in solving linear systems of equations, solving non-linear systems; sequences and series. Enrollment Requirements: Prereq: Intermediate Algebra 20-804-201 or 20-804-203 with a grade of "C" or better OR COMPASS: Algebra 66-99 or College Algebra 1-45 Course Offerings last updated: 09:01:45Students will receive a welcome letter one week before the class starts that includes instructions for accessing the online course. Students are expected to complete basic introductory activities before the first day of class. Coursework for this class is due weekly. Assessments for this class include proctored exams
1439046964 9781439046968 1111788103 9781111788100 Ninth Edition features a new design, enhancing the Aufmann Interactive Method and the organization of the text around objectives, making the pages easier for both students and instructors to follow. «Show less... Show more» Preface xiii AIM for Success xxiii Whole Numbers Prep Test Introduction to Whole Numbers To IdentifyStatistics and Probability Prep Test Pictographs and Circle Graphs To read a pictograph To read a circle graph Bar Graphs and Broken-Line Graphs To read a bar graph To read a broken-line graph Histograms and Frequency Polygons To read a histogram To read a frequency polygon Statistical Measures To find the mean, median, and mode of a distribution To draw a box-and-whiskers plot Introduction to Probability To calculate the probability of simple events Focus on Problem Solving: Inductive Reasoning Projects and Group Activities: Collecting, Organizing, Displaying, and Analyzing Data Summary Concept Review Review Exercises Test Cumulative Review Exercises U.S. Customary Units of Measurement Prep Test Length To convert measurements of length in the U.S. Customary System To perform arithmetic operations with measurements of length To solve application problems Weight To convert measurements of weight in the U.S. Customary System To perform arithmetic operations with measurements of weight To solve application problems Capacity To convert measurements of capacity in the U.S. Customary System To perform arithmetic operations with measurements of capacity To solve application problems Time To convert units of time Energy and Power To use units of energy in the U.S. Customary System To use units of power in the U.S. Customary System Focus on Problem Solving: Applying Solutions to Other Problems Projects and Group Activities: Nomographs Averages Summary Concept Review Review Exercises Test Cumulative Review Exercises The Metric System of Measurement Prep Test Length To convert units of length in the metric system of measurement To solve application problems Mass To convert units of mass in the metric system of measurement To solve application problems Capacity To convert units of capacity in the metric system of measurement To solve application problems Energy To use units of energy in the metric system of measurement Conversion Between the U.S. Customary and the Metric Systems of Measurement To convert U.S. Customary units to metric units To convert metric units to U.S. Customary units Focus on Problem Solving: Working Backward Projects and Group Activities: Name That Metric Unit Metric Measurements for Computers Summary Concept Review Review Exercises Test Cumulative Review Exercises Rational Numbers Prep Test Introduction to Integers To identify the order relation between two integers To evaluate expressions that contain the absolute value symbol Addition and Subtraction of Integers To add integers To subtract integers To solve application problems Multiplication and Division of Integers To multiply integers To divide integers To solve application problems Operations with Rational Numbers To add or subtract rational numbers To multiply or divide rational numbers To solve application problems Scientific Notation and the Order of Operations Agreement To write a number in scientific notation To use the Order of Operations Agreement to simplify expressions Focus on Problem Solving: Drawing Diagrams Projects and Group Activities: Deductive Reasoning Summary Concept Review Review Exercises Test Cumulative Review Exercises Introduction to Algebra Prep Test Variable Expressions To evaluate variable expressions To simplify variable expressions containing no parentheses To simplify variable expressions containing parentheses Introduction to Equations To determine whether a given number is a solution of an equation To solve an equation of the form x + a = b To solve an equation of the form ax = b To solve application problems using formulas General Equations: Part I To solve an equation of the form ax + b = c To solve application problems using formulas General Equations: Part II To solve an equation of the form ax + b = cx + d To solve an equation containing parentheses Translating Verbal Expressions into Mathematical Expressions To translate a verbal expression into a mathematical expression given the variable To translate a verbal expression into a mathematical expression by assigning the variable Translating Sentences into Equations and Solving To translate a sentence into an equation and solve To solve application problems Focus on Problem Solving: From Concrete to Abstract Projects and Group Activities: Averages Summary Concept Review Review Exercises Test Cumulative Review Exercises Geometry Prep Test Angles, Lines, and Geometric Figures To define and describe lines and angles To define and describe geometric figures To solve problems involving the angles formed by intersecting lines Plane Geometric Figures To find the perimeter of plane geometric figures To find the perimeter of composite geometric figures To solve application problems Area To find the area of geometric figures To find the area of composite geometric figures To solve application problems Volume To find the volume of geometric solids To find the volume of composite geometric solids To solve application problems The Pythagorean Theorem To find the square root of a number To find the unknown side of a right triangle using the Pythagorean Theorem To solve application problems Similar and Congruent Triangles To solve similar and congruent triangles To solve application problems Focus on Problem Solving: Trial and Error Projects and Group Activities: Investigating Perimeter Symmetry Summary Concept Review Review Exercises Test Cumulative Review Exercises FINAL EXAM APPENDIX 587 Table of Geometric Formulas Compound Interest Table Monthly Payment Table Table of Measurements Table of Properties 592 Solutions to you Try Its Answers to the Selected Exercises Glossary Index Index of Applications 8
Algebra II: Factoring Introduction and Summary Algebra I dealt with some factoring--we leaned how to factor equations of the form a2 + bx + c , as well as perfect square trinomials and the difference of squares. This chapter explains how to factor other polynomials. Section one explains how to factor trinomials of degree 2 with a leading coefficient--that is, trinomials of the form ax2 + bx + c , where a , b , and c are integers. This section outlines the steps for factoring these trinomials. The process for factoring ax2 + bx + c is a generalization of the process for factoring x2 + bx + c , which we learned in Algebra I. The second section explains how to factor some polynomials of degree 3. First, it deals with polynomials which are the difference of cubes, then with polynomials which are the sum of cubes. Finally, the second section explains how to factor equations of the form ax3 + bx2 + cx + d where = . The next section focuses on fourth degree polynomials. It explains how to factor a difference of fourth powers, as well as some fourth-degree trinomials. Finally, in the fourth section, we learn one of the most important uses of factoring--finding roots. The roots of a function are the solutions to f (x) = 0 ; i.e. the points at which y = f (x) crosses the x -axis. Learning how to find roots will help when graphing polynomial equations. Learning how to find the number of roots will also allow us to approximate the shape of a graph without plugging in points. Finding the roots of an equation becomes especially important in the study of polynomials in Algebra II and higher mathematics. Thus, it is crucial to understand how to factor an equation. Factoring takes practice; it is more useful to try several problems and get a feel for factoring than it is to memorize a set of steps for factoring. This chapter does provide a set of steps--they are meant to be used as a framework or skeleton until the reader becomes more familiar with factoring. The reader is encouraged to practice factoring, as it will come up a lot in Algebra II.
1 CLEP ® College Mathematics: At a Glance 1 Description of the Examination The CLEP® College Mathematics examination covers material generally taught in ... The examination places little emphasis on arithmetic calculations, and it does not contain any questions that require the use of a calculator. If your algebra skills are strong enough, you will be branched into the college-level mathematics test for course placement. If your algebra skills are weak, you will be branched down into the arithmetic section of the test.
Paperback Click on the Google Preview image above to read some pages of this book! Nelson QMaths for the Australian Curriculum 7 ï½ï½ï½ 10 is a brand new series that has been developed to support teachers implementing the Australian Mathematics Curriculum for Years 7 ï½ï½ï½ 10 students in Queensland. A comprehensive range of resources are available in printed form and in digital form on NelsonNet to support the Nelson QMaths series. This is the Teacher's Edition of the Year 9 student textbook. It contains the same content as the student book with additional page-by-page wraparound information to assist teachers with lesson planning and instruction. It includes suggests for integrating key aspects of the curriculum (capabilities, proficiencies, technology, and cross-curriculum priorities) into the teaching of each topic.
97807637149 Analysis, Revised Edition (Jones and Bartlett Books in Mathematics) The Way Of Analysis Gives A Thorough Account Of Real Analysis In One Or Several Variables, From The Construction Of The Real Number System To An Introduction Of The Lebesgue Integral. The Text Provides Proofs Of All Main Results, As Well As Motivations, Examples, Applications, Exercises, And Formal Chapter Summaries. Additionally, There Are Three Chapters On Application Of Analysis, Ordinary Differential Equations, Fourier Series, And Curves And Surfaces To Show How The Techniques Of Analysis Are Used In Concrete Settings
Math 2413 Calculus I Information LSC-CyFair Math Department Course Description Functions, limits, continuity, differentiation and integration of algebraic and trigonometric functions, applications of differentiation and an introduction to applications of the definite integral. Course Learning Outcomes The student will: • Develop solutions for tangent and area problems using the concepts of limits, derivatives, and integrals. • Draw graphs of algebraic and transcendental functions considering limits, continuity, and differentiability at a point. • Determine whether a function is continuous and/or differentiable at a point using limits. • Use differentiation rules to differentiate algebraic and transcendental functions. • Identify appropriate calculus concepts and techniques to provide mathematical models of real-world situations and determine solutions to applied problems. • Evaluate definite integrals using the Fundamental Theorem of Calculus. • Articulate the relationship between derivatives and integrals using the Fundamental Theorem of Calculus. • Use implicit differentiation to solve related rates problems. Getting Started Materials Review Exercises in MyMathLab that can be copied into your own course. Copy "getting started" assignments from Chapter 1 of jezek34081 in MyMathLab.
Math Mechanixs is a FREE and easy to use scientific and engineering math software program. (FREE registration is required after 30 days of usage). Math Mechanixs has an integrated Scientific Calculator and Math Editor that allows the you to compute complex functions and expressions while keeping detailed notes on your work. You can save your worksheet and share it with others. Math Mechanixs also includes a comprehensive and extendable function library with over 280 predefined functions. There are 91 advanced statistical functions including a complete set of random number generators and continuous distribution functions following a variety of statistical distributions. The function library is also extendable allowing you to create your own functions and categories. The function library is even more powerful when combined with our unique function solver, which provides a quick and easy way to solve real and complex roots of polynomials. Math Mechanixs has a calculus utility for performing single, double and triple integration and differentiation plus a curve fitting utility for data modeling using an nth order polynomial. A matrix utility allows you to perform matrix mathematics plus solve sets of linear algebraic equations. There is also an integrated context sensitive help system with numerous tutorials in .wmv file format which will significantly reduce the time it takes to learn Math MechanixsMarketBuddy Windows 3.0 - Market any product on the Internet. MarketBuddy can reduce the human effort required to market any product or service using the power and worldwide reach of the Internet.
What covers in a math workshop? •Review past/current contents each week. •Improve study and time management skills as well as reduce math anxiety. •Develop academic support network linked to specific classes with other students. •Work collaboratively to critically analyze course contents to improve understanding of complex material with workshop facilitators. •Provide opportunity to become actively involved in the course material. •Discover study and test preparation strategies. Do I need to sign up? No. Students just show up during the time scheduled. Does it count for extra credit? It depends on each instructor. Does it count for the DLA? It depends on each instructor. Some instructors might allow Math 80/81 workshop to be counted as DLAs. Welcome to Santa Ana College Math Center! The Math Center is a resource center that provides individual and group assistance in mathematics. The Math Center also facilitates Directed Learning Activities. Faculty instructors, instructional assistants, and Student tutors are available to assist students with challenging topics, answer questions, encourage understanding, and provide support for all math students. Students also have access to textbooks, graphing calculators, instructional videos, and computer programs. Math Center's Goals To help some students further develop basic skills in mathematics and keep them coming to school. To assist other students to further sharpen their pre-existing math skills and advance through math courses. To guide all students toward success in math and encourage them to excel through their scholastic endeavors and beyond.
This course introduces the basic concepts and techniques of linear algebra and calculus which are appropriate to building science and technology. It is aimed at students without a pass at A-Level in Pure Mathematics (or its equivalent). concepts from basic linear algebra and single variable calculus. appropriately apply mathematical methods to a range of application in building science and technology. 2 5. the combination of CILOs 1--5 20 hours in total Learning through tutorials is primarily based on interactive problem solving allowing instant feedback. 2 2 hours 3 2 hours 1 1 hour 4 2 hours Learning through take-home assignments helps students understand basic concepts and techniques of linear algebra and single variable calculus, and their applications. 1--4 after-class Learning through online examples for applications helps students apply mathematical methods to some problems in building science applications. 4 after-class Learning activities in Math Help Centre provides students extra help. 120%Coursework 802 15-30% Questions are designed for the first part of the course to see how well the students have learned concepts and techniques of elementary calculus and their applications. Hand-in assignments 1--4 0-15% These are skills based assessment to help students demonstrate advanced concepts and techniques of basic linear algebra and single variable calculus, as well as some applications in building science and technology. Examination 5 70% Examination questions are designed to see how far students have achieved their intended learning outcomes. Questions will primarily be skills and understanding based to assess the student's versatility in linear algebra and univariate calculus
Summary: This chapter gives a detailed analysis of how teaching with variation is helpful for students' learning of algebraic equations by using typical teaching episodes in grade seven in China. Also, it provides a demonstration showing how variation is used as an effective way of teaching through the discussion after the analysis.
A fascinating tour through parts of geometry students are unlikely to see in the rest of their studies while, at the same time, anchoring their excursions to the well known parallel postulate of Euclid. The author shows how alternatives to Euclids fifth postulate lead to interesting and different patterns and symmetries, and, in the process of examining geometric objects, the author incorporates the algebra of complex and hypercomplex numbers, some graph theory, and some topology. Interesting problems are scattered throughout the text. Nevertheless, the book merely assumes a course in Euclidean geometry at high school level. While many concepts introduced are advanced, the mathematical techniques are not. Singers lively exposition and off-beat approach will greatly appeal both to students and mathematicians, and the contents of the book can be covered in a one-semester course, perhaps as a sequel to a Euclidean geometry course.
Required Materials: There is no text for this course, however you will need the following items: A large collection of square paper – you can buy official origami paper, you can cut regular paper into squares, or you can look for memo cubes with square paper inside. I recommend you buy at least some official origami paper – many designs are nicer with the two colored origami paper. You'll want a variety of sizes as well. You must bring your own origami paper to class everyday. Pens of multiple colors – we'll be marking the crease lines of our origami patterns. This will be much easier to follow if you have a variety of different colors to use. A ruler – this is quite frequently useful for making crease lines. Any origami book: your group will from time to time present models of your choice to the class. There are also websites with models available. An especially good book for beginners is Origami, Plain and Simple by Robert Neale and Thomas Hull. Course Goals: In this course we will be making origami models and studying the underlying mathematics of these models. The design of this course is discovery based and open ended . That is, the in-class activities and assignments for this course will consist of open ended problems, so you will have a great deal of input into what topics we will cover in this course depending on what you discover and think about when solving this problems. This will most likely feel very different from other math courses that youíve had in the past. Hopefully, it will be more fun and exciting this way; however, it may be slightly more frustrating until you get the hang of it. There will also be a strong group work component. Course Content: We will study connections between paper folding and topics in number theory, combinatorics, and geometry. In particular, this course will cover selected topics from: Basic Geometry: How can we use geometry to analyze our models? How do the dimensions of our models relate to the size of paper that we use? How can we form a 30 degree angle? Can we trisect angles? How do we divide a piece of paper into perfect thirds? Topics in geometry include the Pythagorean Theorem, similar triangles, angles, and properties of parallel lines. Polygons and Polyhedra: How can we use origami to construct polygons and polyhedra of a given number of sides.This will introduce us to modular origami where we use multiple pieces of paper to form interesting shapes. Mathematical topics include Eulerís formula, coloring theorems, Hamilton cycles,and Buckyball classification and edge coloring. Flat Folding: How can we determine from the crease pattern alone if an origami pattern will fold flat? Partial answers include Maekawa's Theorem and Kawasaki's Theorem. Course Structure: There will be significant out of class assignments for this course – they will consist of smaller daily individual or group assignments and two larger group assignments. Grading: Your grade will be based on the following categories: attendance/participation (20%), daily homework (20%),daily reflection prompt (20%)and group projects (40%).I am also happy to do Johnston contracts and evaluations. Attendance and Participation: We will cover a huge amount of material in each 3-hour class session. Thus attendance is required and will be a part of your grade. If you are sick or have some other dire emergency you may excuse up to two absences by calling or emailing me before 2 pm on the day you miss class. You are still responsible for any material that is covered during your absence and must get and complete the homework assignment for the next class. You must also actively participate in class to receive full credit in this category. Required participation includes discussing homework at the beginning of class, working on in-class activities, and generally behaving in a way that maintains and supports a good learning environment. Daily Reflection Prompt: This is a new course and new material for me and we are using activities and materials from a colleague of mine who is working on writing a text for a course like this. As such your feedback on how class is going is extremely important. Every day I will ask you to reflect on class and email your responses to me. Full credit will be given to all thoughtfully completed assignments. Daily Homework: There will be daily assignments. These will include both folding of origami models and analysis of the crease design. We will start class everyday with a discussion of problems. Most days, I will check to make sure that everyone has completed their assignment, but not collect and grade them. In this case, full credit will be given to all thoughtfully completed assignments. However, occasionally I will collect and grade these assignments. You are encourage to work in your groups, but must complete each task individually as well, unless otherwise stated. That is, you may discuss the homework in groups but everyone should individually fold each model and write up the analysis in his or her own words. Occasionally, your group will teach the class how to fold an object of your choice. Group Projects: You will work in groups everyday and there will be two large group projects. The groups you sit in on the second day of class will be your group assignment for the entire month. These group projects will consist of a presentation to the class and a paper. The first group project will be due on Friday May 13 and will be on some aspect of the culture or history of origami. The second group project will be the final for the class and will be due on Thursday May 26. In this project you will make two models of your choice, one out of a single sheet of paper and one using modular origami (multiple sheets of paper) and perform a mathematical analysis of the crease patterns and model. More details on these below. First Group Project Info By Friday May 6, you should have decided upon a group for your final project. These groups should have 3-4 people and ideally would be the people at your table that you usually work with. However, you will need to meet outside of class for this, and I know lots of you work various hours, so you should make sure your schedules allow time for your group to meet before deciding on a group. Then we'll switch tables around as necessary. You will have the same group for both projects unless some problem arises with schedules. Mini-Project: Teach the class a fun origami fold. Starting next week each group will pick any origami object they like and present it to the class as our back from break warm-up fold. We will randomly decide which group goes on each of the following dates: Tuesday May 10, Thursday May 12, Monday May 16, and Thursday May 19, and Friday May 20. Project 1: Pick any area involving the culture and history of origami, write a paper and make a presentation to the class. You should also pick an origami object that relates to your topic (maybe rather loosely) and analyze the crease pattern of this object. This could include any aspect of the history of origami or any or the modern day culture of origami. For example, you could talk about how origami evolved out of Japanese culture or pick a modern day person that designs or studies origami. Each group should pick a different topic. TOPIC DUE BY MONDAY MAY 9. Presentation: On Friday May 13, each group will present their project to the class. Your presentation should include a summary of your research area and include teaching the class how to fold your origami object. Your presentation should last about 20-30 minutes. You may include some of your analysis of your crease pattern. Turn-In: You should turn in a 3-5 page paper describing your topic in the history or culture of origami, a completely folded version of your origami model, and an unfolded version of your origami model with all your crease lines marked. You should also include some analysis of your crease patterns, in addition to the 3-5 pages of history/culture. For example, find some geometry in your crease pattern where you can find dimensions or angles. Final Group Project Info Part 1: The 2D model: You should pick an origami model that folds flat and analyze it from a geometrical, flat foldable, and 2-colorable perspective. You should turn in: one crease pattern with all crease lines marked and labeling all vertices, edges, angles, and mountain valley assignments. A second crease pattern that only uses the crease lines in the final folded product and this should be 2 colored. A third folded model that is 2 colored (using only the crease lines that are used in the final folded object.) The write-up should contain: For the geometric analysis, you should find all the angles and dimensions of your model. Your write-up of this should include detailed information about how you know all the dimensions and angles based on how you folded the paper. That is, you should analyze each step of the folding and discuss what it tells you about the geometry of the crease lines. For the flat-foldable analysis: For this analysis you should discuss how each of our theorems about flat foldability are satisfied in your model. In your write-up you should examine each vertex, compute its degree and the number of mountains and valleys that it has. You should check all your angle calculations satisfy the other flat foldable theorems. For the 2- colorable analysis: Use the theorem about 2-colorability and your previous calculations to explain why you know your model must be 2 colorable (that is, how you know before you color it.) Part 2: The 3D model: Option 1: Find a modular unit that we have not used in class that can construct multiple polyhedral models, analyze the unit, make several smaller models and analyze the models you have made. Option 2: Use any modular unit (even ones we have used in class), analyze the unit, make a single very large model, and analyze the model you have made. In particular, PhiZZ units can make a very nice torus (doughnut) or large buckyball. Your analysis of the unit should discuss the dimensions and angles of the unit and how this effects how it can be put together with other units. That is, you should describe what type of faces it can form, how many faces can meet at a vertex, and thus, what type of polyhedra you can make with it. For each polyhedra you make, you should find the number of faces, vertices, and edges. You should verify that Euler's formula holds and determine if there are any other relationships that must hold between faces and vertices or edges and vertices or edges and faces and explain how you know this based on how the object is constructed.
As in previous editions, the focus in INTRODUCTORY ALGEBRA remains on the Aufmann Interactive Method (AIM). Students are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. Student engagement is crucial to success. Presenting students with worked examples, and then providing them with the opportunity to immediately solve similar problems, helps them build their confidence and eventually master the concepts. Simplicity is key in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully constructed hierarchy of objectives. Each exercise mirrors a preceding objective, which helps to reinforce key concepts and promote skill building. This clear, objective-based approach allows students to organize their thoughts around the content, and supports instructors as they work to design syllabi, lesson plans, and other administrative documents. New features like Focus on Success, Apply the Concept, and Concept Check add an increased emphasis on study skills and conceptual understanding to strengthen the foundation of student success. The Ninth Edition also features a new design, enhancing the Aufmann Interactive Method and making the pages easier for both students and instructors to follow. show more show lessList price: $239.95 Edition: 9th 2013 Publisher: Brooks/Cole Binding: Trade Paper Pages: 688 Size: 8.25" wide x 10.75" long x 0.75" tall Weight: 2
Mathematics The Mathematics Grade 9 and 10 courses are based upon the 1999 Ministry of Education and Training publication, Mathematics: Ontario Secondary School Curriculum, Grades 9 and 10. These courses provide the student with a solid foundation in mathematics prior to pursuing further courses in the various streams beyond grade 10. The Grade 11 and 12 Mathematics courses build on the knowledge of concepts and skills developed in earlier grades, extending students' knowledge in new areas and requiring them to solve more complex problems. Students in Grade 11 and 12 Mathematics will continue to develop key skills and make connections through the exploration of applications. The understanding of abstract mathematics is central to these courses. Skill acquisition is also an important part of the programme.
Our Editors also recommend: A Geometric Introduction to Topology by C. T. C. Wall First course in algebraic topology for advanced undergraduates. Homotopy theory, the duality theorem, relation of topological ideas to other branches of pure mathematics. Exercises and problems. 1972 edition. Counterexamples in Topology by Lynn Arthur Steen, J. Arthur Seebach, Jr. Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Numerous problems and exercises correlated with examples. 1978 edition. Bibliography. Topology for Analysis by Albert Wilansky Three levels of examples and problems make this volume appropriate for students and professionals. Abundant exercises, ordered and numbered by degree of difficulty, illustrate important topological concepts. 1970 edition. Intuitive Concepts in Elementary Topology by B.H. Arnold Classroom-tested and much-cited, this concise text is designed for undergraduates. It offers a valuable and instructive introduction to the basic concepts of topology, taking an intuitive rather than an axiomatic viewpoint. 1962 edition. Undergraduate Topology by Robert H. Kasriel This introductory treatment is essentially self-contained and features explanations and proofs that relate to every practical aspect of point set topology. Hundreds of exercises appear throughout the text. 1971 edition. Point Set Topology by Steven A. Gaal Suitable for a complete course in topology, this text also functions as a self-contained treatment for independent study. Additional enrichment materials make it equally valuable as a reference. 1964 edition. Introduction to Knot Theory by Richard H. Crowell, Ralph H. Fox Appropriate for advanced undergraduates and graduate students, this text by two renowned mathematicians was hailed by the Bulletin of the American Mathematical Society as "a very welcome addition to the mathematical literature." 1963 edition. General Topology by Stephen Willard Among the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Includes historical notes and over 340 detailed exercises. 1970 edition. Includes 27 figures. Introduction to Topology: Third Edition by Bert Mendelson Concise undergraduate introduction to fundamentals of topology — clearly and engagingly written, and filled with stimulating, imaginative exercises. Topics include set theory, metric and topological spaces, connectedness, and compactness
Mammoth, AZ ACT Math we must now deal with division. Division causes more difficulties because we cannot divide by zero. Algebra 2 mainly involves ratios of some sort: with rational functions there could be a zero in the denominator; radical functions could have a negative exponent and hence a zero in the denominator is important to understand the basic concepts of algebra before continuing to Algebra II. Students will learn to solve equations and inequalities. They will become proficient in factoring and simplifying algebraic fractions
This program offers all the algebra content students need to master in an accessible, informal format. Features Student Edition now contains more prerequisite skills practice, a new English/Spanish Glossary, sections on Standardized Test Practice and Mixed Problem-Solving, and references to the TI-84 Plus graphing calculator Resources are now conveniently arranged by chapter, saving you time and effort in preparing lessons Technology New! ExamView® Pro Testmaker CD-ROM allows you to create customized tests and study guides in minutes. Add or edit existing questions, integrate graphics, and more! Built-in state and national correlations. New! Online Learning Center gives you access to many valuable resources connected to your Glencoe textbook. The Online Learning Center is organized into two parts—the Student Center and the Teacher Center—with links to a wide variety of appropriate online resources. New! What's Math Got to Do With It? Real-Life Video video series engages students with relevant problem-based examples that show how math is used in real life.
MATH Essential Math Essential Math focuses on mastering the skills needed to be successful in life and to get ready for Algebra. Topics for Essential Math include problem solving, estimation, decimals, percents, fractions, proportions, money skills and basic geometry-students will learn to handle the basics with a deeper understanding and a greater degree of accuracy. The class is co-taught by a special education teacher and a mathematics teacher. The length of the course is one or more semesters, depending on the needs of the individual student. In Essential Math, students use the Holt McDougal Mathematics series within the learning management setting to view videos of topics and complete practice assignments. The student will be scheduled to meet in a webinar with one of the teachers at least weekly, either individually or in a small group. Class activities include the following for each mathematics topic: Weekly meetings where students and instructor can discuss math topics, course progress, any questions, and just to get to know one another better Topic Preview to review skills and include re-learning activities as needed Short videos that present the topic and show how to solve example problems Topic practice with feedback Vocabulary practice Problem solving and estimation practice Throughout the course, students will actively participate in their own learning and will also benefit from personal feedback to help them continue to improve their mathematic skills. Course materials: Holt McDougal Mathematics series Standards met: This course does not meet high school standards. It may be counted as an elective unless listed as a math course in a student's IEP In Math Skills and Topics, which is a one-quarter class, students review and learn about various math topics and how to overcome math or test anxiety. Throughout the course, students will practice to help prepare for standardized math tests, such as the MCA Math test or the Math GRAD test. Class activities include the following: attend weekly class meetings to work on solving practice math test questions together Course materials: : the Holt Mathematics series will be used as resources. Other readings and websites will be assigned and provided in the course. Software: Students need to have Microsoft Word and Excel or Open Office Standards met: This course is designed to help students overcome math anxiety, to help prepare for a standardized math test or retest, or for students who want to explore and review math topics, including topics from algebra 1, geometry, statistics, and probability. It does not meet high school math standards, but can be counted as an elective. Credit: 0.25 for one quarter, counts as a general elective Honors Opportunity: No Beginning Algebra focuses on the concepts of algebra-writing and simplifying expressions, solving equations, graphing on the coordinate system and looking at the rules of algebra and the properties of numbers. It is a one-semester class designed to get a student ready to succeed in Algebra 1A In Beginning Algebra, students use the Holt McDougal Mathematics Course 3 online text. Students are able to view videos of topics, see step by step solutions to problems, and access supplemental materials through this text. Class activities include the following: Periodic meetings where students and instructor can discuss math topics, course progress, any questions, and just to get to know one another better Chapter preview (Are You Ready?) to review skills and include re-learning activities as needed Presentations that explain the lesson and show how to solve example problems Exercises for each section Vocabulary practice Chapter projects, quizzes and tests Zany Brainy extra credit problems each week Throughout the course, students will actively participate in their own learning and will also benefit from personal feedback to help them improve their mathematic skills. Standards met: This course is designed to get the student ready for Algebra 1 and does not meet high school standards. It may be counted as an elective unless listed as a math course in the student's IEP. Mastering the concepts of algebra is very important to continued success in other mathematics courses. In Algebra 1 A, students will learn the why and the how of proportional reasoning and variation, linear equations including recursive sequences, graphing linear equations, solving equations and inequalities, and data analysis including graphs and statistics. Algebra 1 B looks at solving systems of equations and inequalities, using exponents, functions, and classifying numbers. Both courses use an online text that focuses on developing an understanding of the topic, investigating the exciting worlds of algebra, and gaining skills that can evolve and be adapted to new situations. Neat, online interactive tools are available to help algebra really come alive. Class activities will be varied to include the following: Investigations that explore the concepts of algebra Reading guide worksheets to help students focus on key concepts from the text Daily practice with reflection Discussion topics designed to let students learn from and about their classmates Engaging projects in which students gather their own data from the world around them to examine different aspects of algebra and to learn how algebra is used in various careers Quizzes and tests to evaluate students' progress and understanding Phone check in assignments where students and instructor can discuss course progress, any questions, and just to get to know one another better Throughout the courses, students will actively participate in their own learning and will also benefit from personal feedback to help them continue to improve their mathematic skills. In this one-semester course, students develop their problem solving skills in order to work with graphs, statistics, and introductory probability concepts. The focus is on mathematical literacy and practical problem solving; examples are drawn from music, sports, economics, public health, and other areas of life. Class activities include discussions, experiments, projects, and using software to make sense of statistics. Course materials: Excel or Open Office; Fathom software; online readings from a variety of sources Standards met: This course meets all standards and benchmarks in the Data Analysis & Probability strand of the Minnesota 2007 Math Standards for Grades 9-11. Credit: 0.5 Honors Opportunity: Yes Prerequisites: None—this course is appropriate for students at all levels who need to meet the Data Analysis & Probability standards or for those who need a one-semester math elective. Students use The Geometer's Sketchpad software to explore geometric concepts and make discoveries. Topics covered include angles and angle relationships; parallel and perpendicular lines; transformations, symmetry and tessellations; coordinate geometry; triangle relationships; quadrilaterals; polygons and polyhedra; Pythagorean Theorem and special right triangles; circles; perimeter, area, and volume; congruent triangles and proofs; and similarity and trigonometry. Course materials: The Geometer's Sketchpad software; online readings from a variety of sources; teacher created notes, videos and tutorials. Standards met: This course meets all standards and benchmarks in the Geometry & Measurement strand of the Minnesota 2007 Math Standards for Grades 9-11. Credit: 0.5 Honors Opportunity: No Prerequisites: None—this course is particularly appropriate for students who do not have the time in their schedule for a full-year sequence. It is important, however, for students to consult with their counselor regarding number of math credits as the year-long Sketchpad Geometry A and B course may be a better option. In this course, students will regularly use The Geometer's Sketchpad software to explore geometric concepts and make discoveries. Additional practice with geometric properties and theorems will be provided through our online textbook, worksheets, and online games. You will apply the geometry skills you learn to a variety of situations and problem types. Topics covered in Geometry A include reasoning and proof, parallel and perpendicular lines, triangle relationships, and quadrilaterals. In Geometry B, topics include area, volume, circles, and right triangle trigonometry. Course materials include video clip examples, interactive applets for inquiry-based learning, and real world application problems. Students also explore geometric relationships using The Geometer's Sketchpad, a powerful mathematical modeling tool. Course activities include discussions, labs, and problem sets—offering each student a mathematics learning environment where they can understand and excel. The Algebra 2 course will prepare students for a college Algebra class. Topics covered in Algebra 2 A include a review of algebraic properties, linear functions and graphs, linear systems, matrices, quadratic equations and functions, polynomials and polynomial functions. Topics covered in Algebra 2 B include radical functions and rational exponents, exponential and logarithmic functions, rational functions, quadratic relations, sequences, series, and probability and statistics. Course materials include video clip examplesThe Precalculus courses will prepare students for college Calculus. Topics covered in Precalculus A include algebraic and periodic functions, trigonometric properties, applications of trigonometric functions, and circular and parametric functions. In Precalculus B, topics include fitting functions to data, probability, polar equations, complex numbers, sequence, series, limits and derivatives. Course materials include examples from diverse online sourcesCalculus will tie together the mathematics you have learned in previous courses. The courses are designed to for the college-bound student, and when taken together will cover much of the same content as a first semester college calculus class. Calculus A topics covered include: Properties of limits, intermediate value theorem, derivatives, chain rule, derivatives of products and quotients, related rates, integrals, Riemann sums, the fundamental theorem of calculus. Calculus B Topics covered include: calculus of exponential and logarithmic functions, l'Hospital's Rule, critical points and points of inflection, integration by parts. Passing the Advanced Placement Exam in May will allow students to earn college credit. Students who choose to not take the AP Exam will be able to work on the Calculus B material through the last day of the academic year in June. In addition to an online text, students will access both teacher created and Web based resources. Learning will take place through a variety of methods, including the following: auto-graded assignments designed to give you immediate feedback, Sketchpad activities to help you discover and explore topics, and semester projects. Chapter tests and quizzes will also be given. In Discrete Math students will explore unique real world problems that cannot be directly solved through writing an equation or applying a common formula. The course does not require learning a large number of definitions, formulas, and theorems; instead a creative mind, problem solving skills, and visualization will be helpful! Discrete Math will cover a variety of topics to help us answer some real world questions: Euler circuits (What is the best route for the mailman to take?) Voting methods (Will we get a different winner if we hold a different type of election?) Map coloring (How many colors are needed so that no countries that are touching are the same color?) Matrices and tournaments (How can we determine a winner if all individuals have not played each other?) Fair division (How many seats should Minnesota have in Congress?) Course materials: The Geometer's Sketchpad software; online readings from a variety of sources SCIENCE Physical Science: Matter and Energy In this one-semester 9th grade course, students use journals, discussions, at-home and online labs, and structured web quests to explore introductory physical science concepts which will prepare them for high school chemistry and/or physics. Course materials: Online edition of Physical Science: Concepts in Action (Prentice Hall, 2006 Physical Science. This one-semester course focuses on how the Earth has changed over time and how it continues to change. Topics include: interactions of Earth systems; human impact on Earth systems; geology and plate tectonics; climate and climate change; and a descriptive history of the universe and solar system. Course activities include weekly journal and discussion assignments; webquests; online and at-home labs—all with an emphasis on critical thinking and reasoning from evidence. Students are asked to apply the concepts they are learning to their home community—through assignments such as a geology tour of your area and a state-of-your-watershed report. Course materials: Online edition of Earth Science (Prentice Hall, 2006); other selected web sites. Students are asked to identify native geological features and building stone in their communities, so some flexible transportation should be considered. Other labs are conducted online and supplies are not required Earth and Space Science. Biology deals with living systems. In each course, students consider basic concepts of biology, and how different biologists use their studies of living systems to try and answer questions. Students also look at how scientists describe the biological world; practice some of the thinking, observing, and communication skills that scientists use; and apply biological ideas to the world around them. Each course gives students the opportunity to participate in online discussions, conduct some biological investigations (labs and fieldwork) away from the computer, and complete unique assignments to help them develop the building blocks for further biology studies. Throughout both courses, assignments are designed to give students some freedom and creativity in the assignments that they complete, while engaging with important content. For example, in Biology A, students to write a newsletter on an ecosystem for possible publication; in Biology B, students to write a letter to Charles Darwin. Units covered in Biology A include: Scientific Process and Basic Chemistry, Ecology, Cells, Genetics, and Biotechnology and Bioethics. Topics covered in Biology B include the following: Darwin's Theory of Evolution, Evolution of Populations, The History of Life, Classification, Bacteria and Viruses, Protists, Fungi, Plants, Sponges and Cnidarians, Worms and Mollusks, Arthropods and Echinoderms, Nonvertebrate Chordates, Fishes and Amphibians, Reptiles and Birds, Mammals, and Human Systems. Course materials: Online edition of Prentice Hall Biology (the Miller/Levine "dragonfly book", 2006); other selected web sites. Each semester, students will need to provide some common household supplies including colored paper, yarn, markers, and several grocery supplies that are easily found—the list is available on the MNOHS web site and will be updated one week before the start of each semester. Click here for details. Before a student can enroll in a MNOHS science course, MNOHS must receive a permission form signed by a parent or guardian (if the student is under 18). Standards met: These courses meet all standards and benchmarks in the following strands of the 2009nMinnesota Science Standards for Grades 9-12: The Nature of Science and Engineering; Life Science. This one-semester course offers an overview of essential biology content including: cells, diversity of organisms, interdependence of life, heredity, biological populations change over time, flow of matter and energy, and the human organism. This course will blend online demonstrations, scientific research, home laboratory activities, and course discussions to help reinforce concepts. Course materials: Online edition of Prentice Hall Biology (the Miller/Levine "dragonfly book", 2006); other selected web sites. Students will need to provide some common household supplies including colored paper, yarn, markers, and several grocery supplies that are easily found one-semester course meets all standards and benchmarks in the following strands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering; Life Science. This quarter-length course meets all benchmarks of the Evolution strand of the Minnesota Academic Standards in Life Science while focusing on companion animals in a social context. Topics include: the genetics, breeding, and evolution of companion animals; the small animal industry; small animal safety; responsible pet ownership; animal rights: and animal welfare. Assignments vary from individual research, to creative writing about companion animals. For example, students write a letter from the perspective of an animal to inform his or her owner that they are sick. Students also have the opportunity to engage in discussions about current events that deal directly with companion animals, genetics, and evolution. Course materials: Online readings from a variety of sources; other selected web sites. Students will need to provide some common household supplies including colored paper, markers, and several grocery supplies that are easily found. Students will also be asked to visit several local businesses as a part of this course, so some flexible transportation should be considered Anatomy and Physiology to help students meet the biology graduation requirement. It meets some of the standards and benchmarks in the following strands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering; Life Science. Chemistry helps us to make sense of the world we live in—from why soap cleans greasy plates to whether or not biofuels are a beneficial energy path. First semester topics include lab techniques and safety; scientific methods; measurement; chemical and physical change; kinetic theory and states of matter; atomic structure; periodic table and trends; electron configuration; chemical bonding; and an intro to nanotechnologies. Second semester topics include laboratory safety; problem solving and dimensional analysis; chemical quantities; the gas laws; chemical reactions; balancing equations; solution chemistry; reaction rates and equilibrium; conservation of mass and energy; acids, bases, salts; carbon-based chemistry; and nuclear chemistry. Students use online graphing techniques, at-home labs and virtual labs to investigate chemistry concepts. Authentic scenarios are presented each week for analysis and discussion; these allow students to construct their own meaning of the concepts presented. Course materials: Online edition of Chemistry (Prentice Hall, 2005); selected web sites. Students are asked to purchase an inexpensive set of science equipment and common household materials Chemistry A and B will have surpassed the following strands and substrands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering (all); Physical Science (Substrands 1, 4, and part of 3); Chemistry (all). Credit: 1.0 (Semester A = 0.5 credit, Semester B = 0.5 credit.) Honors Opportunity: Yes Prerequisites: None for Chemistry A. Algebra 1 and Chemistry A, or the equivalent, for Chemistry B. Physics A addresses the concepts of one and two dimensional motion, Newton's Laws of motion, vectors, forces, and momentum. Physics B addresses the concepts of work, gravity, planetary motion, waves, light, sound, and Einstein's Theory of Relativity. Course materials: Conceptual Physics (Prentice Hall, 2009 Physics A and B will have surpassed the following strands and substrands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering (all); Physical Science (Substrands 2 and 3); Physics (all). Credit: 1.0 (Semester A = 0.5 credit, Semester B = 0.5 credit.) Honors Opportunity: Yes Prerequisites: Physics A and B may be taken independently of one another. Algebra 1 is required for both courses. This quarter-length course meets all benchmarks of the Human Interactions with Living Systems substrand of the Minnesota Academic Standards in Life Science, as well as some benchmarks of the Structure and Function substrand, while focusing on human anatomy and physiology. Topics include: the nervous system, nutrition and digestion, circulation and the respiratory system, how the body responds to disease and infection, internal regulation of the body and reproduction and development. Assignments will vary from simulations to connect body systems to labs requiring at home participation Biology of Companion Animals, to help students meet the biology graduation requirement. It meets some of the standards and benchmarks in the following strands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering; Life Science. This quarter-length course meets all benchmarks of the Interdependence Among Living Systems substrand of the Minnesota Academic Standards in Life Science while focusing on Minnesota forests. Topics include: tree anatomy and physiology, tree identification, forest ecology, silviculture, forest protection and management. During this course, students will completing a variety of assignments ranging from completing research on forests in Minnesota to exploring trees and forest ecosystems in their local communities. Students also have the opportunity to engage in discussions about current events that deal directly with forestry and ecology Food Science, to help students meet the biology graduation requirement. It meets some of the standards and benchmarks in the following strands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering; Life Science. This quarter-length course meets all benchmarks of the Structure and Function substrand of the Minnesota Academic Standards in Life Science while focusing on food science. Topics include: microbiology, cell structure and function, food safety, food preservation, nutrition, food development and packaging. Students will complete both laboratory and computer based assignments. Students also have the opportunity to engage in discussions about current events that deal directly with food science topics Forestry, to help students meet the biology graduation requirement. It meets some of the standards and benchmarks in the following strands of the 2009 Minnesota Science Standards for Grades 9-12: The Nature of Science and Engineering; Life Science. LANGUAGE ARTS Myths and Legends A & B For centuries, people have sought to explain the world around them, and this quest has led to a rich variety of legends and myths. In Myths and Legends, students will explore myths, legends and folklore from diverse world cultures. Reading and comprehension strategies are emphasized each week, and students will review writing skills and formal essay formats as well. The vocabulary units in both semesters center around Greek and Latin prefixes, bases, and suffixes, which will give students skills to decode and define new words they encounter. Semester A will focus on myths from various cultures that work to explain creation; nature and the elements; and life cycles. Students will write and present a research essay on an ancient culture to review research skills and the MLA format. Semester B will focus on the heroic cycle, and students will read a version of The Odyssey and John Steinbeck's novella The Pearl. Students will also read and discuss local folk heroes from Minnesota. As a final project, students will choose a modern novel or film to analyze based on the heroic cycle. Course materials: All required works are available online or provided within the course. Standards met: All strands of the Minnesota Language Arts standards are addressed. The American Literature Survey courses explore a variety of writings that reflect the rich history, diverse cultures, and of the United States. The first semester focuses on short stories and poetry from a variety of authors and time periods and the novel To Kill a Mockingbird by Harper Lee. Students will review plot structure and literary techniques. The second semester focuses on American folklore, famous essays and speeches. We will also be reading the play Inherit the Wind by Jerome Lawrence and Robert Lee. Students will continue to refine writing skills and focus on critical thinking skills. Course materials: Most required works are available online or provided within the course. The required novel for American Literature A (Harper Lee's To Kill a Mockingbird) and play for American Literature B (Inherit the Wind by Jerome Lawrence and Robert Lee) may be borrowed from a library, purchased, or provided by the school. Standards met: All strands of the Minnesota Language Arts standards are addressed. The English Survey courses focus on reading and writing skills through the use of high-interest literature and current event articles. Grammar and vocabulary refreshers are also featured in each week's assignments. The first semester focuses on the theme of technology and how its rapid changes are affecting our society. Students read short stories by Ray Bradbury, a novella by Ayn Rand (Anthem), and others. They review and practice several different essay forms. The second (B) semester's theme is the changing perceptions of society throughout each decade. Students read samples of literature and non-fiction articles from several historic periods, and continue to practice and refine writing skills. The novel for this semester is Stanley Gordon West's Until They Bring the Streetcars Back. Course materials: Most required works are available online or provided within the course. The two required novels, Ayn Rand's Anthem, and Stanley Gordon West's Until They Bring the Streetcars Back, may be borrowed from a library, purchased, or provided by the school. Standards met: All strands of the Minnesota Language Arts standards are addressed. In these courses, students explore short stories, articles, drama, poetry, and non-fiction. Through guided reading experiences, online discussion, writing assignments and responding to reflection questions, students examine the components and structure of each genre. In addition, students conduct web quests, perform research, and develop creative writing projects. Students will improve their expository writing, reading comprehension, and comprehension and synthesis skills. Students practice all stages of the writing process including pre-writing, rough draft and final draft. Our topics will be serious, satirical, imaginative, dramatic and thoughtful: something for everyone! This year-long sequence surveys the central themes of American literature. Semester A begins with Native American stories and the colonial period, and concludes with a close look at impact of slavery on American intellectual thought. Semester B covers the Civil War years to the present. Students read a wide variety of stories, plays, essays, poems, journals and historical accounts from a variety of authors with diverse perspectives. To explore these writings, students engage in all-class discussions, conduct guided as well as research-oriented web quests, and answer questions about the readings. During first semester, students read one novel chosen from a recommended reading list, and develop a critical essay. Several times during each semester, students pull back and write a reflective essay on the themes covered. Creative writing opportunities ask students to write free verse poetry and to imagine a fictional character's journal. This semester-long course explores literary voices through time and many cultures in an attempt to discover the ideas and ideals that make people similar, or that open doors to new ways of seeing and being. We begin by reading creation stories and mythologies from diverse cultures. Other works include Modernist poetry and fiction ("The Metamorphosis" by Kafka); ancient Greek tragedy (Oedipus the King by Sophocles); wisdom literature of ancient China and Japan; and modern African fiction (Things Fall Apart by Chinua Achebe). Short stories, essays, travelogues, biographies and memoirs from around the world round out the reading experiences. Students make connections to their own lives and times in reflective reader response journals, participate in threaded class discussions, use the writing process to produce fiction, poetry, creative non-fiction, and analytical writing, make extensive use of internet resources to conduct author studies, and actively work on vocabulary development. Students step back in time to explore our literary roots with a semester-long survey of British Literature. The course covers The Anglo-Saxon Period through the Elizabethan, Romantic, and Victorian periods. As they study such classic works as Beowulf, Macbeth, and "The Rime of the Ancient Mariner," students hone their skills as close readers, listeners, viewers, and critical thinkers. They practice both analytical and creative writing. Other activities include virtual tours of England, creating a "Shakespearian" style sonnet, engaging with Jonathan Swift in satirical social commentary, exploring the tenets of nature, spontaneity, and self-expression that inspired Romantic poets like Wordsworth and Keats, and appreciating the emerging voices of Romantic and Victorian women novelists. Vocabulary study, working with literary terms, class discussions, guided practice with study and reading strategies, writing mini-lessons, and reader response journals are an integral part of each week's course work. Research Learning brings the "information age" to life by helping students to design an independent learning project proposal and write a 5-8 page research-based persuasive paper formatted in MLA style. The course starts at the beginning – with inquiry. What does the student want to learn more about? Inquiry questions drive the research learning process. Students then learn to use library databases, the internet, and local experts to find and evaluate a variety of information resources, conduct original research, compile notes and data, cite their sources, prepare an annotated Bibliography, take a point of view, create a thesis, write to and for a specific audience, and document their learning growth—all skills that promote high school success and college readiness. The media has become one of the most powerful institutions in the world as access to newspapers, blogs, data, videos, Facebook and Twitter has exploded across the Internet. This semester-long introduction to journalism focuses on the role of journalism in a democracy as well as writing a variety of articles. This class is for students who want to learn to be savvy consumers of the news and try their hand at writing the news. In the first 8 weeks, students will evaluate, explore and identify various news sources and aspects of the news. The second 8 weeks will focus on writing articles, from editorials to feature articles. Throughout the course, students will be reading, interacting and sharing news experiences, from the web, to the TV and around the world. In this one-semester composition course, students develop written communication skills. To achieve that goal, students practice description, word choice, sentence variety, imagery and many other techniques as they are used in sketches, essays, stories, speeches and poems. Writing assignments vary from paragraph descriptions to a full research paper on a student-selected topic. Many types of writing are practiced, including film reviews, poetry, character sketches and incident essays. By building their writing skills, students are better able to express their ideas for school assignments, in the workplace, and in personal messages. Who knows, some students may even write the first chapters of a novel! Course materials: No required text Standards met: All benchmarks in the Writing Strand of the Minnesota Language Arts standards are addressed. SOCIAL STUDIES Civics This course will examine all elements of our nation's government including how our society formed, the reasons we chose to have a democratic government, and the problems our nation faces. We will look at these elements by using our textbook, researching the Internet, reading case studies, and through group discussions. As a member of this course you will become more informed about our nation and learn new methods of understanding and researching. Course materials: Civics (Prentice HallIn this course we will examine the land, culture, environment, and the impact of humans around the world. We will explore many regions including the United States, Canada, Europe, Russia, China, Southeast Asia, the Middle East, and Africa. We will learn how-to read maps, what different cultures eat, and how life differs from region-to-region. Course materials: World Geography (McDougal Littell, 2003). We will also use many research articles and Internet sites; links and attachments will be provided in class. Standards met: This course meets all standards and benchmarks in the Geography strand of the Minnesota Academic Standards in History and Social Studies for Grades 9-12. This year-long course surveys the geographical, intellectual, political, economic and cultural development of the American people and places. Semester A focuses on colonization to the beginning of the 20th Century. Semester B covers the history of the United States during the 20th century. Student will read and listen to a wide variety of historical events and personal stories. The course will allow each student to pursue individual historical interests alongside the standard curriculum. Through weekly assignments, course discussions, and research projects, students will learn the critical aspects of American history and the details that textbooks cannot cover. Course materials: The Americans (McDougal Littell, 2003) Standards met: All strands of the Minnesota United States History standards are addressed. This year-long sequence surveys the evolution of world societies. Semester A focuses on ancient times through 1500. Semester B examines 1500 through the present times. In addition to the text, students read and listen to a wide variety of historical events and personal stories. These courses will help students to become familiar with the world's societies and cultures, as well as with developments in politics, religious thought, philosophy, economics and literature. The courses include historical, multicultural, geographical, economic, technological, social, political and current event strands which are taught both independently and integrated with one another throughout. Both courses allow students to pursue individual historical interests alongside the standard curriculum. Through weekly assignments, course discussions, research projects, and exams, students will learn the critical aspects of World History and the details that textbooks cannot cover. Standards met: These courses meet all standards and benchmarks in the World History and Historical Skills strands of the Minnesota Academic Standards in History and Social Studies for Grades 9-12. Credit: 1.0 (Semester A = 0.5 credit, Semester B = 0.5 credit.) This course examines the basic principles and structure used in economic decision making; topics include the analysis of economic institutions; social issues; and the basic objectives of efficiency, equity, stability, and growth of economic activity. In class we will take a hands-on approach and deal with real life decisions, personal finance, and economic choices and outcomes in the long and short-run. Course materials: Economics (Prentice Hall, 2005). We will also use many research articles and Internet sites; links and attachments will be provided in class. Standards met: This course meets all standards and benchmarks in the Economics strand of the Minnesota Academic Standards in History and Social Studies for Grades 9-12. This course will take an in-depth look at the U.S. Government by examining each of the three branches (Legislative, Executive, and Judicial) including: their roles and responsibilities, outside influences, and the current leaders in each branch. We will also research government programs, the foundation of democracy, election policies, past leaders—and we will write legislation. Course materials: Magruder's American Government (Prentice Hall, 2003This semester-long course will introduce students to the scientific study of human behavior. Course topics will include research methods, historic and modern approaches to the study of psychology, sensation, consciousness, learning, memory, intelligence, heredity and environment, development of the individual, motivation, emotion, perception, personality, abnormal behavior and therapy. Course Materials: Psychology: Principles in Practice (Holt) Standards met: This course addresses all strands of the American Psychology Association National Standards. Electives Visual Arts Explore visual arts! Using a variety of media (such as pencil, paint, collage and sculpture), students create projects that range from the political to the personal and whimsical—for example, a collage that makes a powerful visual statement about an important issue or a Picasso-like sculpture splashed with color and pattern. A key focus is the language of art, known as the Elements and Principles of Design. Some key art movements are studied as well as the larger question: "What is Art?" The course utilizes a wealth of internet art resources. For example, a favorite project is the Independent Artist Study, where students "circle the virtual globe" as they examine the life and work of a favorite artist and create a piece of art in the same style. Feedback and reflection are other important parts of the learning process, facilitated by online class discussion boards and the students' interactive personal Art Journal. This class will open students' eyes to new ideas about art and creativity. Course Materials: Students are asked to purchase an inexpensive set of art materials—the list is available on the MNOHS web site and will be updated one week before the start of each semester. Standards met: Students come to understand the Elements and Principles of Design and can apply them in art creation and analysis. Study the world through the arts. Explore all types of visual art. Learn to think about art as it relates to you and the world around. See the connection between art and history. What does it all really mean? This class will open new doors to you and encourage you to see the arts in new ways. Students will have the opportunity to create several art projects of their own—such as an art gallery brochure and a collage (cut and paste, or digital). Students are also encouraged to share art they discover or create themselves. Course Materials: Internet Resources Standards: Students come to understand the Elements and Principles of Design and can apply them in art creation and analysis. This course is designed for students with an interest in gaining introductory experience in a variety of media art forms. The goal is to produce media arts projects that will teach students the steps, in all aspects of photo editing, video production, and animation. Students will learn key concepts related to successful photographic composition and manipulation, creation of multiple genres of video production, as well as gain experience creating stop-motion animation. These projects are designed to give students the opportunity to develop skills that may proceed to independent work beyond this course. Course Materials: Adobe Photoshop Elements and Premiere software will be provided by MNOHS. Students need access to a digital camera. Standards met: Most strands of the Minnesota Media Arts standards are addressed. In this semester length course, students will explore fundamental musical concepts while engaging with questions such as "What is Music?" and "Can Music Tell a Story?". The course teaches the elements of music including staves, clefs, notes, meter and rhythm, keys, scales, basic harmonization, and form. Students work with music theory software and music notation software to learn basic fundamentals; this allows them to create their own compositions. Through guided listening assignments, students will explore particular aspects of music—for example, rhythm and meter, musical style, or featured instruments. One of the more important features of this class is developing and sharing each student's tastes in music by introducing favorite bands, singers, and composers to one another. Honors Opportunity: Yes Prerequisite: The course is designed for students at various levels of musical experience who want to do different things in music. Students should have an interest in working with traditional music notation. This first-year Spanish sequence will introduce students to the basics of Spanish and give them confidence to begin to express themselves in a new language. While "traveling" around the Spanish-speaking world, students communicate about their family and friends, likes and dislikes, school, food and healthy lifestyle choices. Lessons 1 (Holt) Standards met: These courses meet national standards, developed with input from the World Languages Quality Teaching Network. This second-year Spanish sequence will build on the basics covered in Spanish 1 in order to increase students' confidence to express themselves in a second language. While "traveling" around the Spanish-speaking world, students will be communicating about their family and friends, their neighborhoods and cities, discussing daily life, vacations, childhood experiences 2 (Holt) Standards met: These courses meet national standards, developed with input from the World Languages Quality Teaching Network. Credit: 1.0 (Semester A = 0.5 credit, Semester B = 0.5 credit.) Honors Opportunity: No Prerequisites: Spanish 1 A and B, or the equivalent, for Spanish 2A. Spanish 2A, or the equivalent, for Spanish 2B. This third-year Spanish course will build on the material covered in Spanish 2 in order to increase students' confidence to express themselves more completely in a second language. While "traveling" around the Spanish-speaking world, students will be communicating about shopping, vacations, pastimes and sports, As in previous levels, there Levels 2 and 3 (Holt)br> Standards met: These courses meet national standards, developed with input from the World Languages Quality Teaching Network. In the first semester of this fourth-year Spanish course, students will build on the material covered in previous levels to gain confidence in all four skills areas—reading, writing, listening and speaking. While "traveling" around the Spanish-speaking world, students will be communicating about family, the fine arts, and the media. Lessons will focus on specific strategies to help students improve their skills. During the second semester, students will be reading a variety of short stories, poetry and other writings from Hispanic authors from around the world. Students will continue to review grammar in context of the writings and have ample opportunity to listen to and speak the language by listening to the audio samples and recording spoken exercises using Voice Thread. Students will communicate frequently with the instructor through writings and oral activities and discussions. The demands of our changing world make career planning an essential part of any high school curriculum. Career trends are dynamic, change as fast as our technology and evolve with global economic realities. This course invites students through a purposeful exploration to help identify strengths of their personalities, interests, intelligence, learning styles, and values. Then they investigate various career and education options that are well suited to their strengths. Students learn and practice essential workforce skills (SCANS) including how to write dynamite resumes and cover letters and how to ace a job interview. In addition, students are introduced to relevant tools and processes that they will use to examine post-secondary education options. As they have identified career possibilities in the first part of this course, they also learn about the education and training requirements of each career field they may enter. Students will learn about various funding options available to pay for school, they will explore types of school/training choices and gain important understanding of the criteria they must meet to enter this post-secondary phase of their career. Students who complete this course will have the skills and knowledge needed to continue with effective career processes today and in the future. Course Materials: Students link to some of the most important and up to date career and education resources available on the internet. In addition to ISEEK and other well-known career sites, we provide access to an array of sites that comprise a cutting edge career seeking process.) In a new 'global economy' it is becoming clear that competition for jobs will be fierce. Career trends are dynamic, change as fast as our technology and evolve with global economic realities. Workers need to acquire new job-content skills for a modern economy and to develop those soft-skills or transferable skills that employers have always valued. Soft skills are universally demanded by employers and include knowing how to communicate effectively, make decisions, handle conflict, show initiative on the job and become dependable and responsible employees and business owners. This course will also introduce concepts related to consumer economics, and other life-long learning applications. In the course students will identify and further develop essential work skills, and will apply them to the work setting. The job site will be a laboratory. Students will be asked to recognize and practice skill sets on the job and to report/discuss these concepts with other students in the course. The real life experiences will help to bring the course concepts to life for the student workers. Students are required to be employed before enrolling in the course. Course Materials: Students link to some of the most important and up to date career and education resources available on the internet. In addition to iSEEK and other well-known career sites, we provide access to an array of sites that comprise a cutting edge skill development process. Students will also be required to seek cooperation from supervisors in the student's work place. Teens in the 21st century face many choices when it comes to their health, and health information seems to change daily. There are always new theories, discoveries, and treatments. This course enables students to gain the skills necessary to make healthy and informed decisions; critical thinking skills are emphasized. Students consider what they will need to know to live a healthy lifestyle and how they will keep these concepts close throughout their life. Students will explore a broad range of topics that are determined by the Minnesota Academic Health Areas. A limited number of behaviors contribute markedly to today's major killers—heart disease, cancer, and injuries. These behaviors, often established during youth, place young people at significantly increased risk for serious health problems, both now and in the future. They include: Tobacco use Unhealthy dietary behaviors Inadequate physical activity Alcohol and other drug use Sexual behaviors that can result in HIV infection, other sexually transmitted infections, and unintended pregnancies Behaviors that may result in intentional injuries (violence and suicide) and unintentional injuries (motor vehicle crashes) By using interactive Internet sites and the most current sources and information available, students will focus on health promotion and risk reduction. Course materials: Online readings from a variety of sources. Standards met: This course meets local standards, developed with input from the Minnesota Health and Physical Education Quality Teaching Network. The best way to live a healthy life is to prevent health problems before they occur. This course creates opportunities for students to apply new skills and knowledge to experiencing the benefits of a physically active lifestyle. Students will be required to complete a mixture of physical activities, online assignments, tests, and a research paper—and to record in their workout log cardiovascular, flexibility, strength and endurance activities. They will learn about proper weight, good diet, and managing stress. By the end of the course, they will have gained the knowledge needed to begin developing healthy habits that will last a lifetime. Course materials: Online readings from a variety of sources. Standards: This course meets local standards, developed with input from the Minnesota Health and Physical Education Quality Teaching Network. Credit: 0.5 Honors Opportunity: No Prerequisite: Before a student can enroll in the PE course, MNOHS must receive a permission form signed by a parent or guardian (if the student is under 18). ELL Essential Skills focuses on English reading, writing, speaking and listening while developing skills you need to succeed in other courses. The learning activities are designed for you as an individual English Language Learner (ELL). This course is taught by an ESL teacher. It is one or more semesters long – depending on the needs of the individual student. Course Materials: No textbook used; Materials, including various online websites and resources, provided by the teacher on a weekly basis For general information
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Mathematics Senior School Mathematics The school provides all our students with the opportunity to progress in Mathematics. All our students are taught basic numeracy that is directed at allowing them to use mathematical knowledge required by our society. Our college also teaches the courses that build the mathematical skills and knowledge necessary to enter University courses in Mathematics and the Sciences. ' Another of our goals is to develop an awareness of patterns and the beauty inherent in Mathematics. It can create a passion for this enriching subject. We are progressively adopting the National Curriculum and it will be taught in grade 7 to 10 in 2012. In grade 7 and 8 students do a common course as they mature their mathematical understanding. Student's interest and demonstrated achievement are used to place students in more suitable levels by grade 9. However students are given many opportunities to change to a different level mathematics course as the need arises. There are courses that suit all our students from grades 7 to 12. Mathematical extra curricula activities are encouraged with the annual competition in the Math's Relay and National Mathematics competitions. Teachers are always willing to provide tutorial assistance and we also have available on line tutorial assistance. Below are examples of topics covered in Grade 11 and 12 Mathematics at St Mary's College. Specialised Mathematics This subject gives that extra mathematical knowledge useful for University student with their Mathematical based subjects Complex numbers Matrices Integral calculus to 3D Sequences Mathematics Methods This subject is required subject for doing Mathematics at University. Differential and Integral Calculus Functions Statisical Distributions Trigonometric Relations Applied Mathematics This subject gives most students the background needed for many University subjects from accounting to psychology etc. Algebraic modeling Calculus Applied Geometry Data Analysis Finance Mathematics Applied Foundation This subject is a preparation for the pretertiary Maths Applied for students who need to improve their pass in Grade 10 Advanced Maths. It covers a lot of the topics of the pretertiary course but covers the basics underpinning them and allows students more confidently attempt Maths applied the following year. finance space algebra probability Workplace Maths This subject gives most numeric skills in real life and in particular simulated workplace based contexts. It is about developing self confidence using mathematics and real world applications. core maths skills measurement consumer maths technology For information about enrolling in our Senior School, please contact the Enrolments Officer by email, telephone (03) 6234 3381, or view or download the Senior School brochure.
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Quantitative Problem Solving in Applied Science, Natural Sciences Mathematics and Commerce: "Hands on, Heads up" Learning Quantitative Problem Solving What is "Problem Solving"? This is a form of learning based on discovery: to solve the problem, you must both think and compute systematically. It is different from both "exercise solving", in which past routines are applied to solve similar problems, and a "trial and error" approach is used to match correct formula for the problem. A central idea in problem solving is the use of "concepts", which are the fundamental general ideas on which other notions may be built. In any subject, there are usually only a few basic concepts (sometimes expressed as formula), which are applied in a variety of ways or situations. For example, basic concepts include limit of function in math, and t- test in statistics, Newton's 2nd Law in physics, mole in chemistry, and liability in accounting. Identifying and deeply understanding key concepts, and developing an organizational structure to allow you to recall how they relate to each other are essential elements in expert problem solving The "spiral of learning" occurs when basic concepts are used repeatedly to solve a variety of problems. The central concept is the core of the spiral, and various applications spin out from, and loop back to, that concept. Frequently re-visiting those basic concepts allows you to firmly fix them in your long-term memory, where they can be quickly recalled and applied. People learn in different ways, and have different preferred styles of relating to their world, seeking sensory input, making information meaningful, and patterns of learning. It is very helpful to understand your own preferred learning style, and use methods that both mesh with and challenge your style. See the free "Index of Learning Styles" by Felder and Silverman. and refer to the "Working with Your Preferred Learning Style" resource on the Learning Strategies web site Self-Reflection Questions Do you: 1. understand your own approach: strengths and weaknesses? 2. focus on concepts to increase understanding, and as an organizational framework? 3. learn material sequentially? 4. look for the "spiral of learning": repetition and expansion of basic concepts? 5. develop a systematic, methodical approach, to talk yourself through each step? 6. compute accurately, and eventually… quickly 7. persist? 8. get help when needed? What is YOUR Approach to Quantitative Problem Solving? Awareness of your own attitudes and habits is a good starting point to see your strengths and areas to change. Click on the "Evidence Based Components" questionnaire to assess your approach. Characteristics of Expert Problem Solvers 1. Attitude Characteristics * Optimistic: you believe "I can do it" * Confident: the problem really does have a reasonable, but perhaps difficult, solution * Willing to persevere: you aim for a complete and well reasoned solution, not an immediate or superficial one * Concern for accuracy in reading: you concentrate, re-read and paraphrase to increase understanding, and translate unfamiliar words or terms * Concern for accuracy in thinking: you work at a moderate to slow pace initially, perform operations carefully, check answers periodically, and draw conclusions at the end not part way through. 2. Skill Characteristics * Systematic approach: you have a plan to follow, which i. reduces the panic ii. allows you to monitor your thought processes iii. helps isolate errors in logic or computation * Sound knowledge of basic concepts, which you mentally organize so you can recall and apply them * Computational skill, at a good speed * Habit of vocalizing or "thinking aloud": you talk yourself through all thoughts i. how to start the problem ii. steps to break problems into parts iii. decisions iv. analyses v. conclusions * Awareness of your own thought processes: What did I do or learn? How did I do or learn this? How effective was my process? Typical Characteristics of Novice Problem Solvers 1. You don't believe that persistent analysis is essential, therefore your effort and motivation to persist is weak. 2. You are careless in their reasoning. 3. You don't break problem into component parts and go step-by-step, therefore there are errors in logic and computation. 4. You focus on individual details, and don't see how details relate to concepts. Therefore, every problem feels new…how overwhelming! 5. Formula-memorizing is the main strategy. 6. You get behind in your learning, and then sequential learning is hampered. 7. You lose confidence in your ability to solve problems, due to lack of success. Strategies to Improve Problem Solving Skills 1. Use Time and Resources Effectively * Work on courses regularly: keep up so you can build on past knowledge (sequential learning), and get help quickly for difficulties. * Do all the questions assigned, rather than dividing questions among group members, as you will get more practice with the concepts your Professor expects you to know. Aim for accuracy, then speed. Start assignments at least a week ahead of the due date, so you have time for help if needed. * Use study groups to compare completed solutions to assigned problems. Teaching someone is a very effective learning and study technique. * Choose problems wisely: learn to apply a specific concept to solve a variety of related problems. Start with simpler ones, and work up. Identify the relevant concept and practice until you know when and how to apply it, i.e. you may not need to do all questions. * Set a time limit: attempt a new problem every @ 15-20 minutes. If you can't complete a problem, check your "thinking strategies" and change to a new problem. Get help with the problems you couldn't complete, at tutorial, etc. * Do some uncalculated solutions: If you are confident in your calculations-set up the solution but don't finish the calculation. * Learn the necessary background and skills: find out from professor, course outline, etc. what the course involves and upgrade before the course begins if you don't feel confident about the prerequisites. * Find and use help resources: use tutors, professors, TAs, friends, text, internet. For example: in accounting, economics, and finance texts, it is common to find examples that are quite similar to the problems at the end of the chapter. Work through the logic of the examples to develop a better understanding of how best to start the homework problems, if you run into trouble. 2. Develop Strategies to Organize Your Thinking * Quantitative Concept Summary Strategy Concepts are general organizing ideas, are there are often very few of them taught in a course, along with their many applications (ie. the spiral of learning). Key concepts may be identified by: * reading the learning objectives on the course outline or the course description, * referring to the lecture outline to identify recurring themes, * thinking about the common aspects of problems you are solving. Learn and understand the small amount of information essential to each concept. If in doubt, ask the professor what is important for you to "get". For more information, click here for the Quantitative Concept Summary Strategy description, Concept Summary form, and an example of a Concept Summary for Ordinary simple Annuities. View the video at (click Online Resources, scroll to "Math", select desired topic and format) * General Problem Solving Method Use a methodical, thorough approach to solve problems logically from first principles. Refer to the self-assessment questionnaire by Woods et al. (2000) in this guide to remind yourself of target activities you need to focus on. Steps involve: * Engage with the problem * Define and understand the problem- what is being asked? Express your thinking in several ways, such as verbally, graphically or pictorially, and finally mathematically * Explore links between the current problem and related ones you have previously solved. * Plan how you will solve the problem * Do it ? * Evaluate your method and result, and revise as needed Click here for more information on the General Problem Solving Strategy. * Decision Steps Strategy This strategy is a specific application of the General Problem Solving Strategy described above, and is suitable for use in statistics, accounting and other applied problem solving situations. During the lecture or when reading course notes, focus on the process of solving the problem, instead of on the computation. When your professor is lecturing, listen to their comments on how steps are inked from one to another. This helps you identify the "decision steps" that lead to correct application of a concept. Ask yourself "Why did I move from this step to this step?" Click here for more information on the Decisions Steps Strategy, and examples of Decision Steps in Calculus and Decision Steps for Rational Expressions. View the video at (click Online Resources, scroll to "Math", select topic and format) * Problem Solving Homework Strategy Use homework as a learning tool. Effective learning of the concepts and general methods will reduce the number of problems you may need to solve to feel confident in your knowledge and computations. Click here for details on the Problem Solving Homework Strategy. * Range of Problems Strategy Exams will challenge you to apply your knowledge to new situations, so prepare by creating questions or problems that are slightly different in some variable from your homework problems. Actively think about the range of problems that are associated with a concept. Think in terms of both i. level of difficulty of the problems and ii. common kinds of difficult problems. Use this to anticipate different kinds of difficult problems for exam preparation, and solve some practice problems to test yourself. This is an excellent activity for a study group. Click here for common examples of difficult problems using the Range of Problems Strategy. View the video at (click Online Resources, scroll to "Math", select topic and format) . Some Evidence-Based Components of Expert Problem-Solving1a Observe yourself as you solve problems, and rate how frequently you DO any of the following. Progress toward internalizing these targets, aiming for doing these activities 80-100% of the time. Targets for expert problem-solving 20% 40% 60% 80% 100% 1. I describe my thoughts aloud as I solve the problem. 2. I occasionally pause and reflect about the process and what I have done. 3. I don't expect my methods for solving problems to work equally well for others.b 4. I write things down to help overcome the storage limitations of short-term memory (where problem-solving takes place). 5. I focus on accuracy and not on speed. 6. I interact with others. b 7. I spend time reading the problem.c 8. I spend up to half the available time defining the problem.d 9. When defining problems, I patiently build up a clear picture in my mind of the different parts of the problem and the significance of each part.e 10. I use different tactics when solving exercises and problems.f 11. I use an evidence-based systematic strategy (such as read, define the stated problem, explore to identify the real problem, plan, do it, and look back). I am flexible in my application of the strategy. 12. I monitor my thought processes about once per minute while solving problems. Endnotes a Problem-solving contrasts with exercise-solving. In exercise-solving, the solution methods are quickly apparent because similar problems have been solved in the past. bAn important target for team problem solving. c Successful problem solvers may spend up to three times longer than unsuccessful ones in reading problem statements. d Most mistakes are made in the definition stage! e The problem that is solved is not the problem written in the textbook. Instead, it is your mental interpretation of that problem. f Some tactics that are ineffective in solving problems include: 1. trying to find an equation that includes precisely all the variables given in the problem statement, instead of trying to understand the fundaments needed to solve the problem; 2. trying to use solutions from past problems even when they don't apply; 3. trial and error Quantitative Concept Summary Strategy Taken from: Fleet, J., Goodchild, F. and Zajchowski, R., "Learning for Success", 2006 See for several completed examples. Click on SFU Q Conference, used with permission. Purpose: to provide a structure for organizing fundamental, general ideas. The mental work involved in constructing the summary helps clarify the basic ideas and shift the information from working memory to long-term memory. This is an excellent study tool, for quick review. Method: The organizational elements are i. Concept Title You can identify key ideas by referring to the course outline, chapter headings in the text, lecture outline. Sometimes concepts are thought of individually, other times they are meaningfully grouped for better recall. Eg. Depreciation, Capital Cost Allowance, and Half-Year Rule; acid, base and PH.. ii. Use general categories to organize material, and then add specific details as appropriate. Sample general categories may include: * Allowable key formula- check summary page of text or ask professor * Definitions- define every term, unit and symbol * Additional important information- sign conventions, reference values, meaning of zero values, situations in which formula do not work, etc * Simple examples or explanations- use your own words, diagrams, or analogies to deepen your thinking and check your understanding * List of relevant knowns and unknowns- to help you know which concepts are associated with which problems, use crucial knowns to help distinguish among problems. QUANTITATIVE CONCEPT SUMMARY Concept Title: Allowable Key Formula: Definitions of each symbol, and its units; Additional important information: (eg. sign conventions, special characteristics, when concept doesn't work, special cases, etc) Simple examples, explanations, cases: Relevant knowns, and unknowns: (and words or phrases from word problems that signal these) By permission from website of R. Zajchowki <> Concept Summary for Ordinary simple Annuities Used with permission General Problem Solving Strategy based on D.R. Woods, "Problem–based Learning", 1994 A systematic approach to problem solving helps the learner gain confidence, and is used consistently as a "blue print" by expert problem solvers as a way to be methodical, thorough and self-monitoring. This model is used in life generally, as well as in the sciences. The steps are not linear, and multiple processes are happening in your brain simultaneously, but the basic template hinges on effective questioning as you carry out various steps 1. Engage * Invest in the problem through reading about it and listening to the explanation of what is to be resolved. Your goal is to learn as much as you can about the problem before you begin to actually solve it, and to develop your curiosity (which is very motivating). Successful problem solvers spend two to three times longer doing this than unsuccessful problem solvers. Say "I want to solve this, and I can". 2. Define the stated problem…a challenging and time consuming task * Understand the problem as it is given you, ie. "What am I asked to do?" * Ask "What are the givens? the situation? the context? the inputs? the knowns? etc. * Determine the constraints on the inputs, the solution and the process you can use. For example, "you have until the end of class to hand this solution in" is a time constraint. * Represent your thinking conceptually first, by reading the problem, drawing a pictorial or graphic representation or mind map (see example attached), and then a relational representation. * Then represent your thinking computationally, using a mathematical statement 3. Explore and search for important links between what you have just defined as a problem, and your past experience with similar problems. You will create a personal mental image, trying to discover the "real" problem. Ultimately, you solve your "best mental representation" of the problem. * Guestimate an answer or solution, and share your ideas of the problem with others for added perspective. * Self-monitoring questions include: What is the simplest view? Have I included the pertinent issues? What am I trying to accomplish? Is there more I need to know for an appropriate understanding? 4. Plan in an organized and systematic way * Map the sub-problems * List the data to be collected * Note the hypotheses to be tested * Self-monitoring questions include: What is the overall plan? Is it well structured? Why have I chosen those steps? Is there anything I don't understand? How can I tell if I'm on the right track? 5. Do it * Self-monitoring questions include: Am I following my plan, or jumping to conclusions? Is this making sense? 6. Look back and revise the plan as needed. Significant learning can occur in this stage, by identifying other problems that use the same concepts (remember the spiral of learning?) and by evaluating your own thinking processes. This builds confidence in your problem solving abilities. * Self-monitoring questions include: Is the solution reasonable? Is it accurate? (you will need to check your work to know this!) Does the solution answer the problem? How might I do this differently next time? How would I explain this to someone else? What other kinds of problems can I solve now, because of my success? If I was unsuccessful, what did I learn? Where did I go off track? Decision Step Strategy: Applying the General Method to a Specific Problem Taken from: J. Fleet, F. Goodchild, R. Zajchowski, "Learning for Success", 2006 See for a completed example. Click on SFU Q Conference Purpose: to help learners focus on the process of solving problems, rather than on the mechanics of formula and calculations. The focus is on correct application of concepts to specific situations. This strategy helps you to increase your awareness of the mental steps you make in problem solving, by "forcing" you to articulate your inner dialogue regarding procedure. Method: Identify the key decisions that determine what calculations to perform. In lecture, try to record the decision steps the professor uses but may not write down or post. i. Analyze solved examples, using brief statements focusing on steps you find difficult: * What was done in this step? * How was it done; what formula or guideline was followed? * Why was it done? * Any spots or traps to watch out for? ii. Test run the decision steps on a similar problem, and revise until the steps are complete and accurate. Decision Steps in Calculus Used with permission Decision Steps for Rational Expressions Used with permission Problem Solving Homework Strategy This strategy encourages a deep understanding of concepts and procedures in calculation. The time you spend on this will reduce the amount of time you may spend in "plug and chug" attempts to do the homework, and reduce the amount of time you will need for studying later on. 1. Prepare for the homework questions.. * review class notes and understand the concepts in the examples. This might take 30 - 45 minutes. * write the first line of a sample problem, close the book, and work as far as you can without looking. * refer back to notes, and then again attempt sample * repeat over again until you can solve the sample problem both accurately and quickly. You will have memorized the rules in the process. This might take 1 hour. 2. Start the homework questions. Interrogate your problem solutions: ask questions about the problem and your method of solving it. E.g. 1. What are the givens? Can the givens be classified as Assets, Liabilities, Owner's Equity, Income, Expenses, etc? Is there any Depreciation? 2. What is required? 3. Can I diagram this? 4. What concepts are referred to? Theorems? Operations? 5. Is the problem similar to others I solved/How? 6. What more do I need to understand this? 7. Are there any "tricks" to the question? If so, how do I deal with them? 3. Keep track of problems you have trouble solving, isolate the particular difficulty, and get help to figure it out. Drill these problems until you are both accurate and fast in solving them. Range of Problems Strategy: Common Types of Difficult Problems Taken from: J. Fleet, F. Goodchild, R. Zajchowski, Learning for Success, 2006 Expand your thinking in preparation for exams, where problems are not exactly the same as you have previously solved. Work from an existing problem, and make it more challenging by adding or changing: Hidden knowns: needed information is hidden in a phrase or diagram Eg. "at rest" means initial v = 0 in physics. Multipart-same concept: a problem may comprise 2 or more sub-problems, each involving the same concept. This type of problem can be solved only by identifying the given information in light of these sub-problems Mulitpart-different concepts: same idea as above, but the sub-problems involve the use of different concepts Multipart-simultaneous equations: same idea as above, but no single sub-problem can be solved by itself. You may have 2 unknowns and 2 equations or 3 unknowns and 3 equations, and you will need to solve them simultaneously, eg. using substitution, comparison, addition and subtraction, matrices, etc. Work backwards: some problems look different because to solve them you have to work in reverse order from problems you have previously solved Letters only: when known quantities are expressed in letters, problems can look different. If you follow the decision steps, they are not usually as difficult. "Dummy variables": sometimes a quantity that you think should be a known is not specified because it is not really needed- that is, it cancels out. Eg. mass in work-energy problems, temperature in gas-law problems. Red herrings, unnecessary information: a problem may give you more information than is needed, which is confusing if you think you should use everything provided. Resources Online: last accessed May 2010. Use this free inventory, the Index of Learning Styles, to assess preferred learning styles, and get additional information on interpretation of your profile , last accessed May 2010. click "Online Resources", scroll to "Math", select topic and format There are 3 videos on Problem Solving illustrating general ideas (Problem Solver I), differences in applying concepts vs. formula chasing (Problem Solver II), and applying the Decision Steps strategy (Problem Solver III). , last accessed May 2010. Click on SFU Q Conference. The personal web site for Richard Zajchowski, with examples of completed Concept Summaries, Decision Steps and other strategies Books: Fleet, J, Goodchild, F, Zajchowski, R Learning for Success: Effective strategies for students, Thomson Nelson, 4th ed, 2006 Whimbey, A, Lockhead, J, Problem Solving & Comprehension, New Jersey: Lawrence Erlaum Associates, 5th ed., 1991 Woods, DR, Problem-based Learning: How to gain the most from PBL, Waterdown, ON: DR Woods, 1994 Mar. 2009 1 Woods, D.R., Felder, R.M., Rugarcia, A., Stice, J.E. (2000). The Future of Engineering Education III: Developing Critical Skills. Chemical Engineering Education, 34 (2), 108-117. --------------- ------------------------------------------------------------ --------------- ------------------------------------------------------------ Learning Strategies Development Queen's University
501 CHALLENGING LOGIC AND REASONING PROBLEMS 501 CHALLENGING LOGIC AND REASONING PROBLEMS 2nd Edition (R) N E W Y O R K Copyright (c) 2005 LearningExpress, LLC. All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by LearningExpress, LLC, New York. Library of Congress Cataloging-in-Publication Data: 501 challenging logic & reasoning problems. p. cm.--(LearningExpress skill builders practice) Includes bibliographical references. ISBN 1-57685-534-1 1. Logic--Problems, exercises, etc. 2. Reasoning--Problems, exercises, etc. 3. Critical thinking--Problems, exercises, etc. I. LearningExpress (Organization) II. Title: 501 challenging logic and reasoning problems. III. Series. BC108.A15 2006 160'.76--dc22 2005057953 Printed in the United States of America 9 8 7 6 5 4 3 2 1 Second Edition ISBN 1-57685-534-1 For information or to place an order, contact LearningExpress at: 55 Broadway 8th Floor New York, NY 10006 Or visit us at: Contents INTRODUCTION vii QUESTIONS 1 ANSWERS 99 v Introduction This book--which can be used alone,with other logic and reasoning texts ofyour choice,or in com- bination with LearningExpress's Reasoning Skills Success in 20 Minutes a Day--will give you practice dealing with the types of multiple-choice questions that appear on standardized tests assessing logic, reasoning, judgment, and critical thinking. It is designed to be used by individuals working on their own and by teachers or tutors helping students learn, review, or practice basic logic and reasoning skills. Practice on 501 logic and reasoning questions will go a long way in alleviating test anxiety, too! Maybe you're one of the millions of people who, as students in elementary or high school, never understood the necessity of having to read opinion essays and draw conclusions from the writer's argument. Or maybe you never understood why you had to work through all those verbal analogies or number series questions. Maybe you were one of those people who could never see a "plan of attack" when working through logic games or critical think- ing puzzles. Or perhaps you could never see a connection between everyday life and analyzing evidence from a series of tedious reading passages. If you fit into one of these groups, this book is for you. First, know you are not alone. It is true that some people relate more easily than do others to number series questions, verbal analogies, logic games, and reading passages that present an argument. And that's okay; we all have unique talents. Still, it's a fact that for most jobs today, critical thinking skills--including analytical and log- ical reasoning--are essential. The good news is that these skills can be developed with practice. Learn by doing. It's an old lesson, tried and true. And it's the tool this book is designed to give you. The 501 logic and reasoning questions that follow will provide you with lots of practice. As you work through each set of questions, you'll be gaining a solid understanding of basic analytical and logical reasoning skills--all without mem- orizing! The purpose of this book is to help you improve your critical thinking through encouragement, no frustration. v i i -INTRODUCTION- A n O v e r v i e w Working on Your Own If you are working alone to improve your logic skills or 501 Challenging Logic and Reasoning Problems is prepare for a test in connection with a job or school, divided into 37 sets of questions: you will probably want to use this book in combination with its companion text, Reasoning Skills Success in 20 Sets 1-4: Number Series Minutes a Day, 2nd Edition, or with some other basic Sets 5-6: Letter and Symbol Series reasoning skills text. If you're fairly sure of your basic Sets 7-8: Verbal Classification logic and reasoning abilities, however, you can use 501 Sets 9-11: Essential Part Challenging Logic and Reasoning Problems by itself. Sets 12-17: Analogies Use the answer key at the end of the book not Sets 18-19: Artificial Language only to find out if you got the right answer, but also to Set 20: Matching Definitions learn how to tackle similar kinds of questions next Set 21: Making Judgments time. Every answer is explained. Make sure you under- Set 22: Verbal Reasoning stand the explanations--usually by going back to the Sets 23-27: Logic Problems questions--before moving on to the next set. Sets 28-31: Logic Games Sets 32-37: Analyzing Arguments Tutoring Others This book will work well in combination with almost Each set contains between 5-20 questions, any analytical reasoning or logic text. You will proba- depending on their length and difficulty. The book is bly find it most helpful to give students a brief lesson specifically organized to help you build confidence as in the particular operation they'll be learning-- you further develop your logic and reasoning skills. number series, verbal classification, artificial language, 501 Challenging Logic and Reasoning Problems begins logic problems, analyzing arguments--and then have with basic number and letter series questions, and then them spend the remainder of the session actually moves on to verbal classification, artificial language, answering the questions in the sets. You will want to and matching definition items. The last sets contain stress the importance of learning by doing and of logic problems, logic games, and logical reasoning checking their answers and reading the explanations questions. By the time you reach the last question, carefully. Make sure they understand a particular set of you'll feel confident that you've improved your critical questions before you assign the next one. thinking and logical reasoning abilities. A d d i t i o n a l R e s o u r c e s H o w t o U s e T h i s B o o k Answering the 501 logic and reasoning questions in this Whether you're working alone or helping someone book will give you lots of practice. Another way to brush up his or her critical thinking and reasoning improve your reasoning ability is to read and study on skills, this book will give you the opportunity to prac- your own and devise your own unique methods of tice, practice, practice! attacking logic problems. Following is a list of logic and reasoning books you may want to buy or take out of the library: v i i i -INTRODUCTION- REASONING CRITICAL THINKING Reasoning Skills Success in 20 Minutes a Day Critical Thinking by Alec Fisher (Cambridge (2nd Edition) by LearningExpress University Press) Critical Reasoning: A Practical Introduction by Brainplay: Challenging Puzzles & Thinking Anne Thomson (Routledge) Games by Tom Werneck (Sterling) Attacking Faulty Reasoning: A Practical Guide to Challenging Critical Thinking Puzzles by Fallacy-Free Arguments by T. Edward Damer Michael A. Dispezio and Myron Miller (Wadsworth) (Sterling) Thinking Critically: Techniques for Logical Rea- Becoming a Critical Thinker: A User-Friendly soning by James H. Kiersky and Nicholas J. Manual by Sherry Diestler (Prentice Hall) Caste (Wadsworth) ANALOGIES LOGIC 501 Word Analogy Questions by Learning- Essential Logic: Basic Reasoning Skills for the Express Twenty-First Century by Ronald C. Pine Analogies for Beginners by Lynne Chatham (Oxford University Press) (Dandy Lion Publications) Increase Your Puzzle IQ: Tips and Tricks for Cracking the MAT (3rd Edition) by Marcia Building Your Logic Power by Marcel Danesi Lerner (Princeton Review) (Wiley) Amazing Logic Puzzles by Norman D. Willis (Sterling) Challenging Logic Puzzles by Barry R. Clarke (Sterling) i x
This item is printed on demand. If you've ever taken a graduate statistics course and discovered that you've forgotten how to divide a fraction or turn a fraction into a perc [more] This item is printed on demand. If you've ever taken a graduate statistics course and discovered that you've forgotten how to divide a fraction or turn a fraction into a percentage, then this handy guide to mathematics is for you. Each topic is provided.[less]
workshop Math 807T - Using Math to Understand Our World Workshop 2009 Introduction In this class we're going to look at some real life problems that people really used math to solve- problems that are important to business, science, medicine, and other sectors of society. You've probably already done a bit of this in your other classes. Can you give some examples? What questions did you answer? In this class we will face more real-world questions such as "How can you tell the time of death of a dead body?" and "How can you keep an outbreak of an infectious disease from turning into an epidemic?" or "How can you identify children who are at risk of developing adulthood obesity?" We'll start with questions of this type and then we will think about how we might use mathematics to answer the questions. This is a little bit different from what you usually do in a math class, where you are given problems that have already been formulated as math questions, but it is very much like what is done in the real world. These types of questions are the type that many applied mathematicians study. An applied mathematician is someone who describes a real-world situation using the language of mathematics and then uses that mathematics to answer important questions- basically applying mathematics to the real world. A pure mathematician is someone who proves theorems about mathematics. Often applied mathematicians use pure mathematics, for example, a lot of pure mathematics went into developing RSA cryptography. Remember all those theorems! This semester we will be applied mathematicians. 1 One note: Applied mathematics does not mean everyday mathematics. Everyday math- ematics is the type we use to answer questions like: "If carpeting cost $7.50 per square foot, how much will it cost to carpet a 9 × 16 ft room?" or "If Limited Too is having a 40% off sale, how much is that $56 dress that I've been wanting?". Everyday mathematics is important, but it is not quite the same as applied mathematics. In everyday mathematics, it is clear what the math question is. In applied mathematics, while it might be clear what the question is, it is not always clear what the math question is (for example, you know that you want to know if a teacher is cheating on a standardized test, but you may not know how exactly to use mathematics to determine that). Applied mathematicians must think first about how to put the problem into mathematical language. Even before that, they must understand the problem well, be it in biology, finance, medicine, business or whatever. And then the mathematical analysis used to answer the question is usually more complex, with more steps, than in everyday mathematics. Many of the problems we will address in this workshop will be closer to everyday mathematics than applied mathematics because it takes more than a day to solve an applied mathematics problem. But the problems in this workshop have been chosen to develop your ability to deal with more complex applied mathematics problems which you will face in the projects. Coming up with a mathematical statement of a real-life problem is called modeling. Sometimes a model can be just a simple equation. Here's a popular example. It turns out that crickets chirp faster when it's warm outside and slower when it's colder. In fact, you can estimate the temperature by counting the number of times a cricket chirps in a minute. If C is the number of chirps in a minute, then the temperature T is T = C/4 + 37. The equation T = C/4 + 37 is a model of how the temperature depends on the number of chirps (or vice versa). It's not exact- if you count the number of chirps, divide by 4, and add 37, most of the time you will not get exactly the current temperature. But it's close. It's a good model of what happens in real life. How do you think biologists came up with this model? Solving problems in applied mathematics can be an arduous endeavor. As applied mathe- maticians, we will have to read a lot of background information about our real-life problems. We will need to know what factors affect our real-world system and how these factors in- teract. We may have to try out several models before coming up with one that seems to work. Or we may use an already-existing model, but we may have to read a lot in order to understand how to use it. Solving a real applied mathematics problem is so involved that 2 we will only solve 5 or 6 problems this semester, once we complete the workshop. You've probably got the idea by now, that an important part of applied mathematics is coming up with the models. Often models are written in terms of functions that show how one quantity depends on another quantity (for example, temperature and chirps). For this reason, we will first spend some time reviewing what you already know about functions and learning about some new functions. The most common types of functions used in modeling are: linear, polynomial (usually quadratic or cubic), exponential and trigonometric. Today we will discuss linear functions, polynomials, and exponential functions, but mostly exponential functions. 1 Linear Functions What do you know about linear functions? What makes a function linear? What would the graph of a linear function look like? A table? A formula? Is the cricket function linear or not linear? How do you know? 3 0 0 .1 .01 Is this linear? How can you tell? .2 .04 1 1 If you're given some data from an experiment, how can you tell whether or not it is linear? Here is some data about the temperature of a cooling cup of coffee. Is it linear? Time (in mins) Temperature (◦ F) 0 200 10 181 20 163 30 146 One thing to keep in mind is that real life data is usually not as pretty as made-up data. It doesn't always fit a linear, polynomial or exponential pattern exactly and you are stuck deciding which sort of model would be best for your problem. This data looks like it might be fairly closely approximated by a linear model, but we will see later that an exponential model is better. Here are some exercises to limber up your linear skills, and maybe even get you into the modeling mood. Exercises 1. Graph the two lines y = 4 − 2x and 4y = 12x + 7 (preferably without a calculator). Where do they intersect? Be precise. 4 2. Acme Car Rental offers cars at $40 a day and 15 cents a mile. Zoomy Car Rental offers cars at $50 a day and 10 cents a mile. For each rental company, express the rental cost mathematically if you are going to rent the car for three days. Which company offers the better deal? 3. Consider the problem of search and rescue teams trying to find lost hikers in remote areas of the West. To search for an individual, members of the search team separate and walk parallel to one another through the area to be searched. If the search team members are close together they will be more likely to be successful than if they are far apart. Let d be the distance between searchers (and suppose they are all the same distance apart). In a study called An Experimental Analysis of Grid Sweep Searching, a lot of data about searcher distances and success rates was recorded. The following table comes from that report Separation distance Percent found 20 90 40 80 60 70 80 60 100 50 Write a function P (d) to model the success rate. If d = 0, what is P ? Does this make sense? If P = 0, what is d? Does this make sense? 4. Consider the models you used in the previous questions. In which cases do you think your model was exact and in which cases do you think it was a good approximation? Support your answers. 5. You can often use linear functions (and other types of functions) to represent a trade-off between two things. For example, the function in 3 might be thought of as representing a trade-off between searcher distance and success rate. Here's another trade-off that one of your students might face. Emma goes into the candy store to buy some tootsie rolls and some York peppermint patties. Tootsie roles are 2 cents each and peppermint patties are 5 cents each. Emma has $1 and she's going to spend it all. She needs to figure out how many of each to buy. What are her options? Write a linear function to represent her options and draw a graph that shows all the options. Label your axes. If Emma buys 14 peppermint patties, how many Tootsie rolls can she buy? What if she buys only 6 peppermint patties? (Remark: it's ok that your graph includes fractional values of candies- we'll just ignore those values for this application). 5 6. Governments must make trade-offs similar to the one that Emma had to make in Exercise 5. For example, there is an ongoing battle in the government as to how much money to spend on defense programs and how much to spend on social programs. This is the famous "guns and butter" problem. There is only so much money and the more you spend on defense, the less you have for butter and vice versa. Suppose the government has $12,000,000 to spend on guns and butter and guns cost $400 each and butter costs $2000 a ton. Write a linear function that represents the governments options and draw a graph. Label your axes. 7. Here is a more complicated trade-off problem. For now, just read this problem. It will be our fourth project later this semester. One day, a few years ago (in the days of VCR's), Wendy Hines received the following somewhat desperate email from her friend Tom: Dear Wendy, VCR tape will record 120 minutes in LP mode: it will record (3X) 360 minutes in EP mode. We frequently record tapes for later viewing. If a movie is e.g. 137 minutes, it obviously will not fit on the tape in LP mode ( 17 minutes short). If I want to record most of the movie in the best quality mode (i.e. LP=60 min.of recording per hour)then I must record some portion in the slower EP mode(120 min per hour) in order to get most of the movie recorded in the better quality LP (60 min per hr) mode. With my inadequate and antiquated memory of math, I often guess at the necessary mix of recording speeds.. Unfortunately, I sometimes estimate wrongly-resulting in missing the last few minutes of a movie....very frustrating!!! Intuitively, I know there must be an algebraic formula to indicate how much EP and LP recording time must be allocated....but I cannot come up with a successful formula........Your challenge: is there such a formula that can be relied upon that is better than my "guessing" and prevent "short" taping incidences. If there is such a formula..it could save my marriage. Thanks for listening, Tom Here's where the work of an applied mathematician really starts. Tom, who is a psychologist, has not stated his problem in a way that is very easy to understand, and he has stated it in English, not math. Our main challenge when we get to this project will be to figure out exactly what Tom is saying and figure out exactly what his trade-off is. 6 2 How Does Your Function Grow? Linear functions grow at a constant rate- i.e. they grow (or decrease) by the same amount from step to step. But in real life there are a lot of things whose growth rate is always increasing or decreasing. Imagine, for example, a glass of cold lemonade warming up in a hot room. Does the lemonade change temperature by the same amount every ten minutes? How do you think a graph of the temperature might look? Suppose a population of bacteria doubles every 6 hours (which it is likely to do). Does it increase by the same amount every 6 hours? To know what functions to use for a model, we have to have some understanding of how different functions grow or decrease. In this section we'll explore this some. Task 1: Use your calculator to graph the functions f (x) = x, f (x) = x2 , f (x) = x3 , f (x) = x4 and f (x) = x5 for x between 0 and 10 and draw graphs on the same axis here. How do these functions compare? 7 These, of course, are examples of polynomial functions, but a polynomial function can have many terms. For example 3x3 + 2x2 − 6x is a polynomial. If the highest power in your polynomial is 2, then we might call the polynomial a quadratic function (though it's still a polynomial, too). If the highest power is 3, we call the polynomial a cubic function. What do we call it if the highest power is 1? People who do modeling as a profession need to know a lot about how different types of functions behave, how to make a function have certain values at certain places, and grow or decrease in the right way at other places. It takes a lot of training to become fluent at modeling. Task 2: Graph the polynomial function f (x) = 3x3 + 2x2 − 6x. How low does it go? How high does it go? Where is the function zero? Usually an applied mathematician has to work backwards. She or he has to take a graph (probably made up of data points) and find a function that nearly fits it. In the next few tasks we will learn something interesting about polynomial functions. Task 3: Make a table that shows the values of the function f (x) = 2x for integer values of x (use x = 0, 1, 2, 3, 4, 5, 6). Add an extra column onto your table and in that column write the differences between consecutive entries in the second column. Now make another table for f (x) = x2 , only this time add on two extra columns and in each of those columns write the difference between consecutive entries in the previous column. Do the same thing for f (x) = x3 and f (x) = x4 . What do you notice? 8 9 Task 4: Try the same thing for f (x) = x2 − x and the function in Task 2. What happens? 10 Task 5: How might what you just learned be useful? Suppose a laboratory scientist is studying how much a metal rod will stretch when you pull on the ends. He does some experiments and takes some data which is in the table below. What sort of function should you use to model the amount of stretching as a function of the applied force (you don't have to come up with the specific function, just figure out what sort of function you would want to look for)? Applied force Amount of stretching 1N .09 mm 2N .76 mm 3N 2.61 mm 4N 6.24 mm 5N 12.25 mm 6N 21.24 mm 7N 33.81 mm In real life, data is never this nice. The differences never work out exactly. You might get the following data and then when you work out the differences, you would have to decide if the 3rd differences were close enough to being constant. Once you decided to go with a cubic polynomial for your model, you'd have to then figure out exactly which cubic polynomial works best. There's a lot of work, and math, involved in coming up with polynomial models. But once you finally have your model, you can use it to predict how much the rod will stretch under even greater forces, or just under forces that you haven't tried yet (like 4.5 N). This ability to predict is crucial in science and industry. Applied force Amount of stretching 1N .08 mm 2N .76 mm 3N 2.6 mm 4N 6.25 mm 5N 12.27 mm 6N 21.24 mm 7N 33.8 mm By the way, the cubic polynomial that fits the data (exactly!) in the first table is s(f ) = .1s3 − .01s2 . The data in the second table comes pretty close to matching this. Task 6: Bacteria often reproduce by simply splitting in two, and then each half grows to the size of the original one. Imagine the following scenario: a single bacterium is sitting in a Petri dish filled with agar (yummy stuff that bacteria like to eat). The bacterium splits into two. Each of those grow and split into two more, so now there are four. Each of those 11 four split into two more, etc. Count the bacteria after each division. Make a table that shows on one side the number of times you have counted the bacteria so far, and on the other side the number of bacteria you counted each time (let your first entry be 1 and 1). Look at the differences. What happens? Can you think up a function for the number of bacteria of the form B = f (n) where n is the number of times you have counted and B is the number of bacteria? Even though this function for B matches the story exactly, it is really an approximation. In real-life, the number of bacteria is probably not exactly B, as a few bacteria might die, or a bacteria might occasionally split into three and not two, or not split at all. A function of the form of B is called an exponential function (can you guess why?). Exponential functions grow faster than any power function (and hence any polynomial). Here are some graphs that show the functions f (x) = x2 , f (x) = x5 and f (x) = 2x . The power function may be bigger at first, but the exponential function always beats it out in the end. 12 Exercises (These will be handed in) 1. Below is some data showing the stopping distance of an Alpha Romeo sports car for different speeds. Speed (in mph) Stopping distance (in feet) 70 177 40 57.8 130 610.5 100 361 140 708 160 925 Find a model (i.e., an equation) for the stopping distance and use it to predict the stopping distance if the car were going 200 mph. (hint: assume your model has the form y = kxn where k is a fixed number and n is a power). 2. Notice that in the data above, the speeds are given every 30 mph, and the differences work out almost exactly. In real life, data is seldom so nice. Here is some more realistic stopping distance data for Toyota's new Escargo. Speed (in mph) Stopping distance (in feet) 35 74 50 149 90 490.5 100 590 Can you find a reasonable model for this data? This is pretty challenging, but remem- ber that from the above exercise you have some idea of what form your model might take. Once you've got a model you're happy with, use it to figure out the stopping distance at interstate speeds (assuming you're going the speed limit). 3. Gulliver, in his travels, discovered that the Lilliputions were increasing 2.6% in popu- lation each year. Here is a table of the population for the ten years that Gulliver spent in Lilliputia (the population is measured in thousands). 13 Year Population (in thousands) 1780 255.97 1781 262.63 1782 269.46 1783 276.46 1784 283.65 1785 291.03 1786 298.59 1787 306.36 1788 314.32 1789 322.49 If Gulliver goes back to Lilliputia in ten years (i.e., in 1799) how many Lilliputions will there be? What will the increase have been between 1798 and 1799, both in raw numbers and in percentages? What if he goes back in 40 years? In 50? What's going to happen to the Lillipution population in the long run? How does this compare to the bacteria population we talked about earlier? 4. Graph the functions f (x) = x20 and f (x) = 1.5x . Which function is bigger? Explain. 5. In which problems did you use exponential models and in which did you use power models? 14 3 More About Exponential Functions Modeling with polynomial functions is a very interesting and deep topic that we have only scratched the surface of. Unfortunately we don't have time to do more than that. But at least this gives you a little flavor for the topic. We will spend the rest of the Workshop talking about exponential functions. They are very useful in modeling- they come up all the time, more often than polynomial functions -and they are not too hard to work with. Here is an everyday example where exponential functions describe what's happening. Task 1: Suppose you deposit $100 into a savings account and the savings account earns 8% interest. Normally interest is compounded (i.e., added on) several times a year. Let's suppose that in this case interest is compounded quarterly (i.e., every three months). That means that 8/4% is added on each quarter (you divide the interest rate by the number of times it is compounded each year). So after the first three months the bank adds $2 to your account. How much money will you have after 2 quarters if you always put the interest back into your savings account? 3 quarters? a year? Can you find a function, M (q), that tells you how much money you will have after q quarters? Remark: Normally when a bank lists interest rates for savings accounts, they list two numbers- the regular interest rate (or nominal rate) and the APY. APY stands for annual percentage yield and is the percentage increase of the principal in a year's time. Normally, unless the interest is compounded only annually, the APY is a bit larger than the nominal rate. Task 2: What is the APY for the 8% savings account above? When Wendy Hines wrote these notes, she checked out Wells Fargo and found out that they give a whopping 2.23% interest rate for accounts between $10,000 and $25,000, compounded quarterly. What is the APY? 15 Task 3: We've used exponential functions to model the value of a bank account accruing interest and the size of a growing population. How are these two problems similar? Before, we said that exponential functions grow faster than almost every other function, but it may seem that at a 2.23% interest rate, or even at an 8% interest rate, your money doesn't grow too quickly. Sometimes you have to wait a bit for the exponential function to really take off. Here's an old story about a forgotten savings account- it goes something like this. One day, in 1996, a man by the name of Samuel Johnson was in his attic going through a trunk that had belonged to his grandfather. In the trunk he found an old bank passbook that had apparently been in his family for a long time. The last transaction in the passbook was dated July 31, 1790. The balance at that time was $244.82. On top of the page was printed the current interest rate: 4.5% compounded quarterly. Task 4: How much money was the account worth in 1996? Samuel took the passbook to the bank, which still existed, but they no longer had any record of the account and they weaseled out of paying Samuel his money. Exponential functions may grow slowly at first, but as you see, at some point they will really take off. Any time you put some money in the bank and don't touch it and let it accrue interest (i.e., let the interest compound), the value of your savings will be given by an exponential function. Compound interest has a huge impact on how debts and savings grow. If you google "compound interest" you will get literally millions of hits, most of them from financial companies trying to explain to you how compound interest makes you a lot of money, and why, to get the full effect of compound interest, you need to start saving money early. 16 Task 5: For example, suppose at age 40, you invested $5000 in a money market fund that makes 6% compounded annually. How much will this investment be worth when you retire? What if you had made the investment when you were 25? What if instead your parents had invested it for you when you were born? What if they had been able to find a fund that paid 9% annually? If you added $100 a month to the fund, your money would grow even faster. You'd be amazed! It takes some more math to calculate savings when you're also making a monthly or yearly contribution, so we'll save that for a project later in the semester. In that project, we will also see how credit card debt grows exponentially, if you don't pay it all off each month. One way that people like to think about exponential functions is to talk about doubling times. Task 6: If you invest $100 at an interest rate of 8% compounded quarterly, how long does it take for your investment to double (assuming that all interest is put back into the account and that you don't add or take out anything from the account)? How long does it take for it to double again? And again? 17 Each exponential function has its own fixed doubling time. Exponential functions grow so quickly because more and more is doubled each time. Task 7: What is the doubling time for the $5000 investment at 6% compounded annually? When will it double again? How much money will you have after the first doubling time? After it doubles again? And again? What is the doubling time for the function f (x) = 1.5x ? Task 8: There is a famous story about the meeting between a Chinese emperor and the inventor of the game of chess. The emperor was so delighted by the new game that he offered the inventor anything he wanted in the kingdom. The inventor said that all he wanted was some grains of rice. "I would like one grain of rice on the first square of the chessboard, two grains on the second, four grains on the third, and so on. I would like all of the grains of rice that are put on the chessboard in this way." Thinking this would amount to no more than a bushel of rice, the emperor readily agreed. Let's try this. What do you think will happen? How many grains of rice will be on the last square? (Extra credit if you can figure out how many were on the entire chessboard). Do you think this is more or less than a bushel? Notice that the amount in the last square is more than the amount on all the previous squares put together. In fact, this is true for every square: the amount on any square is more than the amount on all the previous squares put together. This shows how doubling can make things grow amazingly quickly. 18 Task 9: Discuss: which would you rather (1) I give you $10000 a year for 64 years or (2) I give you $.01 the first year, $.02 the second year, $.04 the third year and so on, doubling the previous years amount each year, for 64 years? The enormity of exponential growth has very important real-world implications. With these examples in mind, think about what it means to say something like "world oil con- sumption is growing at a rate of 2.3% per year". In 2000, world oil consumption was about 27,740,000,000 barrels. The total amount of oil believed to remain in the earth is about 1027 billion barrels. If we do the math and add up the total amount used for the next several years (finding out how much is used in 2020 is not hard, but adding up the total amount used between now and 2020 is a little harder), we would discover that we will run out of oil in about 27 years, unless consumption is drastically reduced. Even still, there is a finite amount of oil in the earth, and we will run out sooner or later. Task 10: If oil consumption continues to grow at a rate of 2.3% per year, how long until consumption doubles? How many barrels will be used per year then? How many barrels will be used per year after it doubles a second time? When we do our project about exponential growth of credit card debt, we will be able 19 to apply the math we learn there to predict growth of the national debt, which is currently a little over $8 trillion. Scary, huh? And we will be able to show that the world will run out of oil in 27 years. Every exponential function has the form f (x) = Cbx where C and b are fixed numbers. C is called the initial value because it is the value of f (x) = Cbx when x = 0, for example in the interest problems, C is the initial principal. What is C in the rice problem? The number b is called the base. It can be any positive number. What is b in the interest problems? in the bacteria problem? in the rice problem? Exercises (These will be handed in) 1. Review the uses of exponential functions we have talked about so far. In which exam- ples are exponential models exact and in which are they approximations? 2. Using your results from Exercise 3 in Section 2, what is the doubling time for the population of Lilliputia? The world's population currently has a doubling time of about 38 years. How big of a problem do you think this is? 3. What are C and b for the Lilliputian population model in Exercise 3 of Section 2? 4. If a 5% interest rate is compounded monthly, what is the APY? 5. Suppose Wendy Hines initially invests $1000 in the account from Exercise 4 when her daughter is born. Write a function that shows how much the account is worth after n years. Sketch a graph of this function that goes up to 90 years. How much will the account be worth when she goes to college? When she retires? 20 4 The Exponential Number e The most common base for exponential functions is the number e. The value of e is about 2.718. Does your calculator have an e button? If so, you can type e1 and see several digits of e. To write down the digits of e exactly would require an infinite number of digits after the decimal place, and so it is easier to just write "e". You might wonder why such a weird looking number is so popular. Unfortunately that's pretty hard to explain. One early appearance of e actually came out of the work of a mathematician named Jacob Bernoulli, in the late 1600's (there was a whole family, including three generations, of Bernoulli mathematicians). Bernoulli wanted to understand how compound interest worked to cause investments and debts to grow. He came up with the formula P (1 + r/n)n to describe the value of an investment with principal P invested at a rate r compounded n times per year. Task 1: When we say "value" what do we mean? the value when? Is this formula the same as the one you found for compound interest? Bernoulli wondered, "What happens if I compound more often? Will that have a big effect on how fast the value of an investment increases?" So he compared, for example, the growth of investments with quarterly compounding, monthly compounding and daily compounding. Task 2: Does it make a big difference how often interest is compounded? 21 Bernoulli, being a mathematician, wondered what would happen if compounding was continuous. What does it even mean to "compound continuously"? Well it's more often than every hour. Task 3a: What would n be if you compounded every hour? It's more often than every minute. Task 3b: What would n be if you compounded every minute? It's more often than every second. Task 3c: What would n be if you compounded every second? Suppose, for the moment, that the interest rate is r = 1 (in percents that's 100%- a good deal!) and that your initial investment is $1. Task 4a: What would the balance be after a year if you compound hourly? every minute? every second? What happens to the expression (1 + 1/n)n as n gets larger and larger? We say that n 1 e = lim 1+ . n→∞ n It turns out that r n er = lim 1+ . n→∞ n 22 Task 4b: Use your calculator and different values of r to convince yourself of this. Record the computations that you did here. If we compound continuously then after a year, with a principal of P and an interest rate of r, we have r n P lim 1 + = P er n→∞ n dollars. Task 5: Give a formula for the value of the investment after ten years. After t years. The number e is not a very good base for investment formulas that don't use continuous compounding, nor is it very good for the bacteria division or rice problem, but it turns out to be very convenient for many scientific applications. Exercises 1. Graph ex and e2x . How do they compare? Now graph ex · ex and e2x . Make an observation. Do you remember any laws of exponents that could account for this observation? 2. Suppose you are lucky enough to find a bank with an 8% interest rate compounded continuously. Suppose you deposit $100. Write a formula for the amount of money in the account after t years. What is the APY? 23 5 The Undoing of the Exponential Function Task 1: On your calculator, compute e2 . There is another curious button on your calculator called ln. Compute ln of the number you just got. What happened? Now compute e10 and then compute ln of that number. Compute ln(e3.4 ). Compute ln(ex ) for a few other values of x, or even graph ln(ex ). What happens? There's a word for this. We say ln x is the of the exponential function ex . VERY IMPORTANT FACT: ln(ex ) = x Task 2: Suppose we invest $100 at 8% compounded continuously. Can you use the VERY IMPORTANT FACT, instead of trial and error, to figure out how long it takes the in- vestment to double? In order to find out how long it takes the value of the investment to double, do you really need to know the amount of the principal? Why or why not? How long would it take the investment to double if the investment rate were 7%? Task 3: Solve the following: 4e3x = 24, 400e.01t = 1000 24 Every exponential function has it own associated logarithm function. The function f (x) = 2x has the associated logarithm function g(y) = log2 y. The function f (x) = 10x has the associated logarithm function g(y) = log10 y. The function f (x) = bx has the associated logarithm function g(y) = logb y. The number b is called the base just as it is for exponential functions. The natural logarithm is really just loge . It's called the natural logarithm just because it's used so often. Task 4: What is the base of log2 y? loge y? ln y? Task 5: Write a VERY IMPORTANT FACT for log2 . What is log2 8? log2 32? In this class, we'll only use the natural logarithm. The algebra of exponentials and logarithms is very useful in applied mathematics and is a topic in Algebra II. Unfortunately, we don't have time do study this topic in any depth. We will just use logarithms to help us solve problems. Exercises 1. Suppose we deposit $5000 into a bank account that gives 5% interest compounded continuously. Use ln to determine when the balance in the account would be $1,000,000. 2. Repeat Task 5 for log10 y. What is the base of log10 y? What is log10 10? log10 10000? log10 3? log10 30? 3. What happens if you take the logb of a negative number (for any base b)? 4. Graph ln x. Explain why the graph looks the way it does (the next problem might help you with this one). 5. Graph both ln x and ex on the same graph. Do you notice any graphical relationship between these two functions? Can you think of a reason why they might be related in this way? You could also compare log10 x and 10x if you want another example to look at. 25 6 Review Is your head spinning? Let's review. We talked a little bit about linear functions. We talked even less about polynomials. We saw a few situations in the exercises that we could model using linear functions and we saw an example where we could use a quadratic model (the stopping distance exercise). We talked a lot about exponential functions. Linear functions have a steady increase, polynomials grow faster, but exponentials grow the fastest. Many real-life things grow exponentially. We talked a lot about how investments grow exponentially if you leave them sit and always add the interest back in. We also saw that bacteria division and rice-grain doubling are exponential. We talked about examples of exponential population growth. If something increases by a fixed percent over each time period, then it is growing exponentially. We talked about the special exponential base e, which we will use quite often in the weeks to come. We also saw how logarithms "undo" exponentials and how to use ln to solve equations with e. We learned how to find the doubling time of exponential functions. These skills will soon be very useful as we try to solve some real-world problems. Review Exercises (These are to be handed in) Here are some problems to try that will help you to consolidate what we have learned today and help you move your focus towards exponential functions with base e. 1. Describe some things you might do to determine whether some given data is linear, polynomial or exponential. You may refer to exercises or tasks. 2. Populations are often thought to grow exponentially until they begin to run out of resources, at which time they begin to level off. Suppose we have a population of rabbits in a park and on May 1 we did a rabbit census (don't laugh- people really do this) and found that there were 426 rabbits in the park. If t stands for the number of days since May 1, then we might model this population as R(t) = 426ert where r is the per capita growth rate per day (that is, the number of new rabbits each existing rabbit makes, on average, per day). Presumably (hopefully!) r will be substantially less than one. How do we measure r in practice and how do we use this model to predict things about the growth of the population? Well suppose our park workers do a second census on June 1 and find that there are now 600 rabbits. How could you use that information to find r? Find r and use that to determine the doubling time for the rabbit population. What will the population be on Sept. 1? Why is this information useful? Notice that with this model, we could have fractions of rabbits. That's ok, it's just an approximation. 26 3. Brazil experienced exponential inflation during the last half of the 20th century (before its currency was revalued). Suppose the price of a loaf of bread was given by b(t) = .35e.34t where t is years since 1950. What was the price of bread in 1950? (the Brazilian unit of currency is the real- pronounced "hay-yal")? What was the price in 1995? You can probably imagine why they got rid of lower denominational notes. By 1995, the smallest note was the million real. What was the yearly inflation rate during this period? By what year was the price of bread one real? 4. Solve (a) 3e2x = 20 (b) 170e.7x = 420 (c) ln(ex ) = 1 (this one is really quick if you spot the trick) 5. Fill in the blanks. If oil consumption increases by a fixed percentage each year (i.e., the same percentage every year) then we say that consumption is growing . If a savings account grows with a fixed interest rate year after year (and you don't take any money out or put any money in, except for the interest) then your savings grows . If a population doubles every 10 years then it is growing . 6. If we deposit $1000 and are lucky enough to find a bank that compounds continuously, write a function that describes the value of our savings after t years if the interest rate is 7%. Assume that we don't withdraw or deposit any money into the account after the initial deposit, except that we always return the interest back to the account. 7. Suppose some quantity can be modeled as q(t) = q0 ert . There is a rule called The Rule of 70 which says that the doubling time is approximately 70/r. Where does this come from? How accurate is it? 27 7 Exponential Decay In most of our previous examples, the exponent was always positive, but it can be negative, too. Task 1: Graph the function f (x) = e−x . What is f (0)? f (1)? f (3.5)? We say that such a function decays exponentially. Here is another example of exponential decay. Consider a full glass of water in a straight up and down glass. Now pour half of the water out and note the new height of the water. Now pour half of that water out and note the height again. Continue doing this. Task 2: Write a function h(n) for the height of the water after the nth pouring (so h(0) = 1). Notice that here the base is less than 1. How is this like having a negative exponent? Here is some space to draw the function f (x) = (1/2)x (in the example above, n could only be an integer, but x can be anything). 28 There are many real-life examples of exponential decay. Perhaps the most famous one is radioactive decay. All atoms are made up of protons, neutrons and electrons. Uranium atoms have lots and lots of protons, neutrons and electrons and, because uranium atoms are very unstable, sometimes these particles go zinging off to other places (we say the uranium "decays" or "breaks down"). This is called radioactivity. The particles that zing off are called alpha particles and beta particles and they can do damage to living cells that they zing into. Once alpha particles and beta particles have zung off, the remaining atom is no longer a uranium atom. The new atom is also radioactive, however, and more alpha and beta particles will zing off. Eventually, as alpha and beta particles zing off, the uranium will be transformed into something no longer radioactive, but this takes a long time. Uranium is a well-known radioactive element, but because it decays into other radioactive elements, it can be a little confusing to talk about the decay of uranium. Instead, lets talk about the decay of radioactive iodine isotopes (an isotope is a version of an element that has a different number of neutrons than the regular version does- isotopes tend to be radioactive). Iodine-129 and iodine-131 are both radioactive, but when they undergo radioactive decay, they turn into nonradioactive elements. As time goes on, more and more of the iodine isotope decays until eventually there is no more radioactive iodine left. The remaining substance is safe. It turns out that if I0 is the amount of iodine isotope initially in a lump of stuff (measured in milligrams perhaps), then I(t) = I0 e−rt is the amount of iodine isotope after time t where the decay rate is r. This is really an approximation; it won't be exact, but will be pretty close. Decay rates for most radioactive substances have been determined in the lab. Remark: $1 for anyone who can find the contradiction in the above paragraph. What chemists usually measure, rather than the decay rate itself, is what's called the half-life. This is analogous to the doubling time in exponential growth. The half-life is the time it takes for half of the radioactive substance to decay. Task 3: What would r be if the half-life of a radioactive substance is 20 years? (be careful with the signs) As mentioned, when uranium undergoes radioactive decay, other radioactive elements result. The half-life of uranium is about 760 million years, but in nuclear reactors uranium 29 is broken down much more quickly than that. Two of the byproducts of the decay of uranium are radioactive iodine-129 and iodine-131. These are part of what we call "nuclear waste". The half-life of iodine-129 is 15.7 million years and the half-life of iodine-131 is 8 days. Here are two problems about iodine-129 and iodine-131. Exercises (These are to be handed in) 1. The Snake River Plain aquifer is the most important underground water resource in the northwest U.S. It is the sole source of drinking water for 200,000 people. It is the main source of irrigation water for crops and fisheries in Idaho. Over 75% of trout eaten in the U.S. comes from Idaho fisheries (and where do your potatoes come from?). In 2001, a PhD student named Michelle Boyd at the Idaho National Engineering and Environmental Laboratory (INEEL) measured the amounts of various nuclear contaminants in the aquifer. She knew that there were problems because INEEL used to be a big producer of nuclear weapons from 1950 until the end of the Cold War, and they took little care to dispose of their nuclear waste carefully. Basically they just dumped it into the aquifer. Nuclear waste is no longer being dumped into the aquifer, but of course what's already been dumped is still there. She found out that there are areas in the aquifer where the amount of iodine-129 is 3.82 picocuries per liter of water (a curie is a standard unit of radiation- it would be hard to explain exactly what it means; a picocurie is a trillionth of a curie). The highest amount that is considered safe by the FDA is 1 picocurie per liter. How long will it be until water in the aquifer is safe? 2. I-131 is sometimes used in medical imaging. It is injected into the blood and will collect on certain kinds of tumors. It is also used to treat hyperthyroid (overactive thyroid). When administered to a patient, I-131 (because it's iodine) accumulates in the thyroid where it decays. As it decays, the particles that zing off kill part of the gland, which is good if your thyroid is overactive. I-131 has a half life of 8 days. Suppose it takes 72 hours to ship I-131 from the producer to the hospital. What percentage of the original amount shipped actually arrives at the hospital? Suppose it is stored at the hospital for anther 48 hours before it is used. What percentage of the original amount is left when it it used? How long will it be before the I-131 is completely gone? 30
comprehensive math textbook for Grade 11. It can be downloaded, read on-line on a mobile phone, computer or iPad. Every chapter has links to on-line video lessons and explanations. Summary presentations at the end of each chapter offer an overview of the content covered, with key points highlighted for easy revision. Topics covered are: language of mathematics, exponents, surds, error margins, quadratic sequences, finance, quadratic equations, quadratic inequalities, simultaneous equations, mathematical models, quadratic functions and graphs, hyperbolic functions and graphs, exponential functions and graphs, gradient at point, linear programming, geometry, trigonometry, statistics, independent variables, dependent events. This book is based upon the original Free High School Science Text series.
Houston ACT algebraic operations, elementary equations, laws of integral exponents, factoring and radical notation, rational expressions and an introduction to the Cartesian coordinate system. Topics include basic arithmetical operations on integers and rational numbers, order of operat... ...Disc type o...With the proper train of thought and a bit of practice, the formulas can be deducted and learned easily. Microsoft Word is not only about typing letters and doing homework. It can also be used to create trifolds and even mail merge, in conjunction with Excel.
Number Theory This text provides a simple account of classical number theory, as well as some of the historical background in which the subject evolved. It is ...Show synopsisThis text provides a simple account of classical number theory, as well as some of the historical background in which the subject evolved. It is intended for use in a one-semester, undergraduate number theory course taken primarily by mathematics majors and students preparing to be secondary school teachers. Although the text was written with this readership in mind, very few formal prerequisites are required. Much of the text can be read by students with a sound background in high school mathematics pages have writing or highlighting on them205048145 May show one or several of the following...Fair. 0205048145 May show one or several of the following characteristics: Heavy wear, small or large amounts of highlighting/writing, or missing dustcover
Linear algebra developed from two concrete requirements: on the one hand the release from linear sets of equations, on the other hand the computational description of geometrical objects, analytic geometry in such a way specified. (Therefore some authors call linear algebra linear geometry.) One states that the function <math> A< /math> special characteristics has, it is a linear illustration: Is <math> x< /math> a solution for the set of equations <math> A (x) =b< /math>, and <math> y< /math>a solution of the set of equations <math> A (y) =c< /math>, then is analytic geometry the other origin of linear algebra is in the computational description 2 - and three-dimensional (Euclidean) area, also "opinion area" mentioned. Points in the area know assistance of a coordinate system by Tripel <math> (x_1, x_2, x_3)< /math> by numbers to be described. ThatType of illustration of the shift leads to the term of the vector, the direction and amount of the shift indicates. Many physical dimension, for example forces, have always this aspect of direction. There one also vectors by Zahlentripel <math> (a_1, a_2, a_3)< /math> to describe, blurs the separation between vectors can andPoints: one point <math> P< /math> its radius vector corresponds, to that from the origin after <math> P< /math> shows. Many of the illustration types, for example turns around axles by the origin or reflections on levels by the origin, regarded in classical geometry, belong to that to the class linear illustrations, which was mentioned already above. related terms in certain way is the term of the vector spacealready too generally. One can assign a dimension to each vector space, for example has the level dimension 2 and the area dimension 3. There are however vector spaces, whose dimension is not finite, and many of the well-known characteristics are lost. It has itselfbut when very successfully proved to equip infinite-dimensional vector spaces with an additional topological structure; the investigation of topological vector spaces is the subject of the Funktionalanalysis. The remainder of this article concerns itself with the case of finite dimensions. In the literature becomeVectors differently of other sizes differentiated: Small letters, fat-printed small letters, underlined small letters or small letters with an arrow over it are used. This article uses small letters. A matrix is indicated by a "raster" by numbers. Here is a matrix with 4 linesand 3 columns: Individual elements of a vector becomeColumn vectors usually by an index indicated: 2. Element of the vector A indicated above would be then A 2 =7. Sometimes in line vectors an exponent is used, whereby one must watch out whether a vector indexing or an exponent is present: Withthe above example b one has for instance b 4 =7. Array elements are indicated by two indices. The elements are represented by small letters: m 2.3 =2 is the element of the 2. Line in the 3. Column. Endomorphismen and square stencils during the representation of onelinear illustration - described how under matrix - there is the special case of a linear illustration <math> f< /math> a finite-dimensional vector space in itself (a so-called. Endomorphismus). One knows then the same basis <math> v< /math> for Urbild and picture coordinates and receives a square usesMatrix <math> A = {} _vf_v< /math>, so that the application of the linear illustration of the link multiplication with <math> A< /math> corresponds. The twice Hintereinanderausführung of this illustration corresponds then to the multiplication with <math> A^2< /math> etc., and one knows all polynomialen expressions with <math> A< /math> (Sums of multiples ofPowers of <math> A< /math>) as linear illustrations of the vector space understand. determinants a determinantis a special function, which assigns a number to a square matrix. This number gives information over some characteristics to the matrix. For example it shows itself by it whether a matrix is invertable. A further important application is the computation of thecharacteristic polynomial and thus the eigenvalues of the matrix. There are closed formulas for the computation of the determinants, like the Laplace' expansion theorem or the Leibniz formula. These formulas are however rather from theoretical importance, since their expenditure rises with larger stencils strongly.In practice one can compute determinants at the easiest, by bringing the matrix with the help of the Gauss algorithm in upper or lower triangle form, the determinant is then simply the product of the main diagonal elements. in that <math> the n< /math> - width unit power of a matrix <math> A< /math> occurs. The behavior of such a matrix with exponentiation is not easy to recognize; however math <the n> /math< becomes> - width unit power of a diagonal matrixsimply by exponentiation of each individual diagonal entry computes. If it now a invertable matrix <math> T< /math> gives, so that <math> T^ {- 1} A T< /math> Diagonal form has, leaves itself the exponentiation of <math> A< /math> to the exponentiation of a diagonal matrix attribute in accordance with the equation <math> (T^ {- 1} A T) ^n =T^ {- 1} A^n T< /math> (the left side of this equation is then <math> the n< /math> - width unit power of a diagonal matrix). Its behavior (with exponentiation, in addition, with other operations) shows itself general more easily by Diagonalisierung of a matrix. <math> \ Phi< /math> is howeveralso at the same time eigenvalue of the original matrix <math> A< /math> (with self-vector <math> do< /math>), the eigenvalues remain unchanged with transformation of the matrix. The diagonal form of the matrix <math> A< /math> arises thus from their eigenvalues, and around the eigenvalues of <math> A< /math> to find, one must examine, for which numbers <math> x< /math> the linear set of equations <math> outer ones = xu< /math> a solution different of zero <math> u< /math> has (or, differently expressed, the matrix <math> xE-A< /math>is not invertable). Diagonalisierbarkeit whether a matrix is diagonalisierbar, depends on the used counting range. <math> A< /math>is z. B. over the rational numbers not diagonalisierbar, because the eigenvalues <math> \ Phi< /math> and <math> 1 \ Phi< /math> surds are. In addition, the Diagonalisierbarkeit can fail independently of the counting range, if not "sufficient" eigenvalues are present; thus for instance the Jordan form matrix has < math> \ begin {pmatrix} 1&1 \ \ 0&1 \ ends {pmatrix}< to /math> onlythe eigenvalue 1 (as solution of the quadratic equation <math> (x-1) ^2 = 0< /math>) and is not diagonalisierbar. With sufficient large counting range (z. B. over the complex numbers) however each matrix can be diagonalisieren or transformed into Jordan standard format. There the transformation, means this last statement corresponds to a matrix the basis change of a linear illustration that one can always select a basis, which is illustrated "in a simple manner" to a linear illustration with sufficient large counting range: In the case of the Diagonalisierbarkeit each basis vector becomeson a multiple of itself shown (is thus a self-vector); in the case of the Jordan form on a multiple of itself plus possibly. the previous basis vector. This theory of the linear illustration can be generalized on bodies, which are not "sufficient large"; inthem other standard formats must be regarded beside the Jordan form (z. B. the Frobenius standard format).
Advances in linear and integer programming Linear and integer programming are mathematical techniques that are concerned with optimization, that is with finding the best possible answer to a problem. They are often associated with the wider field of operations research. They have been studied and researched since the late 1940s and elements of them are now taught in undergraduate and graduate programmes in mathematics/operations research worldwide. In recent years, after the advent of interior point methods, there has been an explosion of research into linear programming, as well as further steady advances in integer programming. This research has been reported, as one would expect, piece by piece in the research literature, i.e. at conferences and in journals. The reason for assembling this book was to bring together in a single text an accessible exposition of these advances. With contributions from acknowledged experts in their field this book deals with: Whilst some may read this book whole it is likely that the majority of readers will be most interested in particular chapters. For this reason each chapter has been written so that it can be read and studied separately.
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Precalculus: Functions and Graphs (3rd Edition) Making the transition to calculus means being prepared to grasp bigger and more complex mathematical concepts. Precalculus: Functions and Graphs is designed to make this transition seamless, by focusing now on all the skills that you will need in the future. The foundation for success begins with preparation and Precalculus: Functions and Graphs will help you succeed in this course and beyond. Customer Reviews: great! By Bad - January 15, 2013 This is a great book for anyone who is interested in learning more about this topic. Very clear to read, fast shipping. good looking book By Dwayne Peyton - September 18, 2011 The book looked brand new!!! This book is great but i dont need it anymore since i dropped the class GREAT By John G. Stien - September 29, 2011 Product arrived in a prompt and timely manner. Also as a bonus i received a student solutions manual with the package that has proven to be useful on more than one occasion. 5 big large stars! yeaaa!
Search Course Communities: Course Communities Mixing Problems Maplet Course Topic(s): Ordinary Differential Equations \| Modeling, ODE This maplet combines visualization, mixing word problems, and then fill in the blank values/functions for the maplet. The maplet provides good hints and shows whether the inputs are correct or not correct.
The Complete Book of Algebra and Geometry offers children in grades 5-6 easy-to-understand lessons in higher math concepts, skills, and strategies. This best-selling, 352 page workbook teaches children how to understand algebraic and geometric languages and operations. Children complete a variety of activities that help them develop skills and then complete lessons that apply these skills and concepts to everyday situations. Including a complete answer key this workbook features a user friendly format perfect for browsing, research, and review. Basic Skills Include: •Order of Operations •Numbers •Variables •Expressions •Integers •Powers •Exponents •Points •Lines •Rays •Angles •Area Over 4 million in print! The best-selling Complete Book series offers a full complement of instruction, activities, and information about a single topic or subject area. Containing over 30 titles and encompassing preschool to grade 8 this series helps children succeed in every subject area
Jan 3 (Thu): introduction and review of solving linear equation: write system of equations in matrix notation; Gauss elimination; basics of solving system; a few M/O examples ... key differences in octave and matlab: see here for example Read notes 1.1.1- 1.1.4. Jan 8 (Tue): Login info given to students. Norms of vector and matrices, + examples. An interesting link is here where you can learn application of linear algebra for fun. Notice: I'd like to move my office hours to every Thursday 3:30pm-5:00pm from next week on. Read notes 1.1.5 - 1.1.6. Jan 10 (Thu): More examples on calculate matrix norms, condition number and relative error. Some information about Rotation matrix can be found here (take a look at the 2D examples; the idea for reflection matrix would be similar; the matrix used in class is just one example). Read notes 1.1.7. (1.1.8 for summary of M/O commands) Homework hw1 due Jan 17 Solution1hw1_solution (for TA information, see main page of this course). Jan 15 (Tue): Lagrangian interpolation + examples in M/O; here is another way to view Lagrange interpolation polynomial, which I didn't have time to cover in class. You can think about why that is equivalent to the p(x) we learned in class (hint: what is the solution of a_i s formally?). Read notes 1.2.1-1.2.2. Jan 17 (Thu) Cubic spline interpolation and its linear equations (construction procedure). Read notes 1.2.3-1.2.5 Homework hw2 due Jan 24 Solution 2 (please see problems 1,3,4,5,7 in the attached file): solutions1.2 m-file you may use for the homework:, plotspline,splinemat Jan 22 (Tue) Extra example for Cubic spline interpolation; Finite difference approximation; solving boundary value problem (BVP) via system linear equations. Read notes 1.2.5-1.2.6; 1.3.1-1.3.2 Feb 5 (Tue) Dr. Sarah will cover a class for me since I have to travel to US for a conference. Orthonormal basis, orthogonal matrices and unitary matrices Read notes 3.3.1-3.3.2 (Notice that 3.2.* is not lectured; please read that part by yourselves.) Feb 7 (Thu) Quick review on complex numbers and orthogonal and unitary matrices; introduced Fourier series. Read notes 3.4.1-3.4.4 Homework: problem 2 of problems3.2, problem 1(skip the Parseval's formula part),3,4 of problems3.4due Feb 14 (note: try to calculate c_n's in problem; a_n and b_n's are for bonus. Although I didn't introduce the real form of Fourier series, you can still try to derive them. Idea: split e^{ix} into cos(x) and sin(x).) Feb 14 (Thu) More on eigenvalue problem; briefly included (sub)space, linear independence, basis ... Homework: calculate a_n, b_n's in problem 1,3,4 in previous problem set (see problems_3.4 in Feb 7) and answer the question about Parseval's formula in question 1 there; problem 1, 2 (a,c,e) of problems2.1 problem 1 (in part c, we mentioned algebraic multiplicity in class, please find the definition of geometric multiplicity in lecturenotes so that you can do that question), 3, 5 ofproblems4.1 due Feb 28 solutions solutions3.4solutions3.2 (other solutions will be released in a later time solutions4.1) Read notes 4.1.7-4.1.12 Mar 5 (Tue) Chemical system and reaction matrix. FYI: sample exams from previous years (notice they may have different topics) can be found in the following here (search old tests and exams e.g.). Mar 7 Midterm test (MCLD 228 one hour basically) No cell phone, no calculators, ... as usual exams. * Solutions to midterm exam will be released at a later time. You are expected to work on those on your own first. Homework: problem 3,4,5,6,7 of problem set 2.1 (see the one in Feb 14) due Mar 14solutions2.1 Mar 12 (Tue) Resistor network Mart 14 (Thu) Hermitian matrices and real sysmmetric matrices Homework problem 1, 2 of problems2.3 and problem 1,2 ofproblems4.2; problem 3 and 4 in problem set 4.2 are for bonus (you probably need to do a little bit more reading on that section). Due Mar 21 solutions2.3solutions4.2 Read notes 4.2.1-4.2.5 Mar 19 (Tue) Power method Here the midterm problems are attached, please redo prob 2 and 3 if your score for that question less than or equal to 6 (due Mar 28), no extra credit. The solutions will be released in a later time. Midterm307202_2013 solutions (as reference, the way to solve the problem may not be unique) to the midterm solns Read notes 4.3
Problem Solving 9780759342644 0759342644 Summary: Problem Solving provides students with a general approach and strategies to solve problems in real life. The text is easy to read and geared mainly for students who dislike math. Problems throughout the book range from easy to difficult, and require minimal mathematical experience. While possessing knowledge is one important requirement to solving problems, there are many others. Problem Solving focuses on providing ...strategies to help students become proactive, successful, and confident problem solversProblem Solving provides students with a general approach and strategies to solve problems in real life. The text is easy to read and geared mainly for students who dislike m [more] Problem Solving provides students with a general approach and strategies to solve problems in real life. The text is easy to read and geared mainly for students who dislike math. Problems throughout the book range from easy to difficult, and require
What can you do to help students learn the advanced math that is required in so much of today's industries and technologies? What helpful insights come from cognitive science, comparative anthropology, and educational psychology?
Start your semester off on the right foot: participate in these FREE reviews designed to prepare you for specific UB courses in which certain mathematical skills are required. You can participate in person or online through our virtual classroom. RSVP through UB's Web calendar to ensure that you reserve a seat. All reviews are taught by ALC Math Coordinator Yoosef Khadem. If you missed a review or wish to watch the session again, click the links below to watch the recordings and download the handouts. Algebra Review for MATH 111 If you're registering for UB's College Algebra (MATH 111) or Introductory Statistics (MATH 115), brush up on your algebra skills with this free workshop designed just for your class. Pre-Algebra Review for DVMA 93 If you're registering for UB's Introductory Algebra (DVMA 93), prepare to do your best by attending a special pre-algebra review. Research on student course performance indicates this review may be particularly important if you scored 50 or below on the Accuplacer Elementary Algebra Test. We also recommend attending if Accuplacer generated an arithmetic score below 79 for you. Introductory Algebra Review for DVMA 95 and MATH 115 If you're registering for UB's Intermediate Algebra (DVMA 95), prepare to do your best by attending a special introductory algebra review. This review is open to anyone who placed into DVMA 95 but will be especially useful for those who scored between 83 and 95 on the Accuplacer Elementary Algebra Test. Statistics Review for HSMG 632 Algebra Review for OPRE 315 If you're registering for UB's Business Applications of Decision Science (OPRE 315), take advantage of this free review. Topics will include solving and graphing linear equations, solving and graphing linear inequalities, solving two equations with two unknowns and formulating basic word problems. Statistics Review for OPRE 202 and 504 If you're registering for UB's Statistical Data Analysis (OPRE 202) or Data Analysis and Decisions (OPRE 504), take advantage of this free review. Topics will include basic probability, calculating measures of location and variation, working with normal distribution and calculating probabilities using the normal curve.
As part of ongoing research into cognitive processes and student thought, we have investigated conflicts between physics and mathematics intuitions in advanced mechanics students. Students compared various damped and undamped harmonic motions using both differential equations and verbal descriptions of physical systems. We present evidence from a reformed sophomore-level mechanics class which contains both both tutorial1 and lecture components. Preliminary data suggest that mathematics and physics intuitions, even in advanced students, are poorly linked and occasionally lead to conflicting predictions.
Written by an expert in mathematical education, this well-organized book reflects the author's forty-plus years of teaching and writing experience. This book constitutes a complete and multi-faceted exercise in critical thinking. The author focuses on explanations that are based on only high school basic algebra and geometry (both of which are reviewed in context) in an effort to appeal to a broad range of readers who are attracted to mathematics and its ideas. Well-conceived, illuminating, and entertaining diagrams and other figures are found throughout the book. Teachers, at both the high school and college level, will especially enjoy this book since it provides insight to answer student questions such as "Why is this important?" The history of mathematics is discussed when appropriate and provides a foundation for further learning.
MA 201 College Algebra Instructor: Ann Ostberg Semester: Spring Course Description This course is designed to explore the concepts of college algebra. It will include a study of linear, quadratic, rational, polynomial, and radical equations; relations and functions; rectangular coordinate system and graphs; systems of equations and inequalities; exponential and logarithmic functions; and matrices. Course Schedule Unit Topic Learning Objectives 1 Welcome ·This section reviews the different sets of numbers such as natural numbers, whole numbers, integers, and rational numbers and their properties.It continues with graphing on a number line and associated topics such as intervals, inequality symbols, absolute value, and distance. ·This section reviews exponents and the rules of exponents along with order of operations, evaluating expressions, and scientific notation. 2 rational exponents, radicals, and polynomials ·This section reviews rational exponents.Rational exponent is another name for fractional exponents.Radical expressions such as square roots, cube roots, etc. are presented along with the methods to simplify and combine (addition and multiplication).Remember rationalizing the denominator?We will also rationalize the numerator in this section. ·This section covers polynomials.We will define, add, subtract, multiply, and divide. Remember the term FOIL?A new term might be conjugate binomials. 3 factoring of the polynomials and algebraic fractions ·This section lays the foundation of a key process in algebra:factoring polynomials.Many factoring techniques will be presented.Practice as much as possible to become proficient! ·This section reviews the basic concepts of fractions; however, it extends it to algebraic fractions.The techniques to simplifying, multiplying, dividing, adding, and subtracting algebraic fractions are based on simple fractions that you learned in grade school.To make things more interesting, we will also work with complex fractions. 4 equations and their applications ·In this section, you'll learn about properties of equality, linear equations, rational equations, and formulas.All of these concepts are vital to your ability to work with equations. ·This section extends the concepts of linear equations to the practical application of linear equations.You may be familiar to these as 'story problems.'Pay close attention to the Strategy for Modeling with Equations in your text. 5 quadratic equations and their applications and complex numbers ·This section introduces quadratic equations. These are second-degree equations (they have an exponent of 2 on the variable).The key topics are solving the quadratic equation, i.e., finding the values that make the equation a true statement.In order to solve, the methods of zero-product, completing the square, square root property, and the quadratic formula will be used. ·This section looks at some applications of quadratic equations.We will place emphasis on the geometric problems, uniform motion problems, and flying object problems. ·This section investigates complex numbers.Complex numbers developed as solutions to certain quadratic equations.There are two types of complex numbers: real and imaginary.We will be looking at the definition and simplification of imaginary numbers and complex numbers.The techniques of FOIL will prove useful as you do arithmetic with complex numbers. 6 polynomial and radical equations and inequalities ·This section teaches solving polynomial equations by factoring and the methods used in solving radical equations.So your factoring skills will be put to the test in this section.Radical equations aren't trying to protest anything but they are equations that involve square roots and cube roots. ·This section looks at the properties of inequalities.You will utilize those properties to solve linear inequalities and compound inequalities.Of particular emphasis will be solving quadratic and rational inequalities. Of these types, you must make a separate chart to actually find the solution. 7 absolute value ·This section on absolute value concludes the chapter.Emphasis will be placed on the definition of absolute value and solving equations and inequalities involving absolute value.We will look at equations with two absolute values – these are actually easier than they sound. 8 rectangular coordinate system ·In this section, you will be able to graph linear equations on the Rectangular Coordinate Plane.On this plane, one may also find the distance and/or the midpoint between two points.Some basic applications will be presented.Be sure to understand these formulas and the terminology associated with this topic. ·This section focuses on the slope of lines.This will include horizontal, vertical, perpendicular, and parallel lines.An emphasis will be placed on the nonvertical lines (simply all lines that are not vertical).You will want to memorize the formula for slope. ·This section focuses on the methods of taking graphical information (points, slope, etc) and translating that into an algebraic equation.This was actually a very revolutionary development in the history of mathematics!Key formulas are the point-slope form and the slope-intercept form. 9 graphs of equations and ends with proportions and variations ·This section covers the graphing of other types of equations.Emphasis will be placed on the symmetries of graphs, miscellaneous graphs such as absolute value, square root, quadratic (parabolas), and circles.The formula for a circle will be presented. ·This section presents the topics of proportion and variation.You have undoubtedly worked proportion problems in earlier math classes.You will use the process of 'cross-multiplication.'Variation problems might be a bit new, however, they are practical problems.Direct variation implies that as one element (x) increases so does the other (y).Think of as you eat more calories, you will gain more weight.Indirect or inverse variation implies that as one element increases (x) the other (y) decreases. 10 functions and function notation and quadratic functions ·This section begins the discussion on functions.The variables, x and y, will be given new ideas:independent variable and dependent variable for x and y, respectively.The x values will also be referred to as the domain. The y values will be considered the range of the function.Understand the notation that is used to denote a function.We will conclude with drawing the graphs of functions:actually you have already done this.Using the 'vertical line test' will allow you to tell whether a graph is the graph of a function. ·This section looks specifically at the quadratic function.You are already familiar with this function as its graph is a parabola.The quadratic equation is very important as it has many practical applications. Besides knowing how to graph the parabola, finding the vertex is very useful.The vertex will tell the maximum or minimum value of a situation. 11 polynomials and other functions. Translating graphs and rational functions ·This section focuses on graphing polynomial functions.Did you know that there were 'even' and 'odd' functions? We will also look at increasing and decreasing function.Think of going up a hill versus going down a hill.Finally, we will graph piecewise-defined functions.These functions often frustrate students until they realize that it is just like taking a piece from two or more pies (graphs), say apple and cherry, and placing them on your plate (coordinate plane), side by side. ·This section will demonstrate how one can move graphs of equations to the right, left, up, down, and flip!These are called translations.The 'flip' is called a reflection about the x- or y-axes.For all of these translations, they will start with the basic graph of the equation. ·This section will provide a brief introduction to rational functions. The definitions of asymptotes, rational function, and their related domain will be discussed. We will not focus on the graphing of rational equations. 12 functions and inverse functions ·This section focuses on special functions called inverse functions.Only one–to-one functions can have inverse functions.Visually, this can be determined from the graph of the function using The Horizontal Line Test. You will also learn to write the 'inverse' of an equation. 13 exponential functions and their graphs, applications of exponential functions, and logarithmic functions ·This section will look at a special type of function, called an exponential function.You will always be able to recognize an exponential function as the variable (x) is the exponent.For example, , where b is a constant.The graphs of all exponential functions are similar in shape and go through the point (0, 1).A practical application of exponential functions is finding the value of compound interest.If the interest is compounded continuously, a special number, e, is used. The value of e is 2.7182818.... ·This section discusses some of the applications of exponential functions.We will focus only on radioactive decay and Malthusian population growth. ·This section covers logarithmic functions and their graphs.A logarithmic function is simply the inverse of an exponential function.The inverse is more commonly written as.If the base (b) is 10, it is called a common logarithm. If the base (b) is e, it is called a natural logarithm.The shape of the graph is similar to the exponential function, except it passes through the point (1, 0) instead of (0, 1).These graphs can also be written so that they have horizontal and vertical movement. 14 logarithmic functions, properties of logarithms, and exponential and logarithms equations ·This section focuses on the applications of logarithmic functions.We will only look at those applications from electrical engineering, geology, and population growth. ·This section investigates the properties of logarithms.The key thing to remember is that a logarithm IS an exponent!Remember the properties associated with exponents and you will see how they become applicable to the properties of logarithms.The Change-of-Base Formula allows one to calculate the value of any logarithm. ·This section shows how one may solve exponential and logarithmic equations.The steps are straightforward. Be sure to be able to write equations in exponential form to logarithmic form and vice versa.Carbon-14 dating is an important application that involves logarithmic equations. 15 linear equations, determinants, and graphs of linear inequalities ·This section covers the various methods that one can use to solve systems of linear equations in two variables:the graphing method, the substitution method, and the addition method.We will also look at the characteristics of a system with either infinitely many solutions or no solutions (inconsistent).Finally, the techniques will be utilized to solve a system of linear equations in three variables.Systems of linear equations are crucial in solving linear programming problems. ·This section investigates another method of solving a system of linear equations.It involves the use of matrices.Through the use of determinants (which are found from a matrix), one can find the solution to systems.Techniques involving determinants can also be used to find the equation of a line and the area of a triangle. ·The final section of this semester will discuss graphing linear inequalities.You will utilize the techniques of graphing equations and then determining which sections of the graph represent the solution area.Graphing a system of linear inequalities is also used in linear programming.
Math Middle school math is the base building period where the concepts of mathematics are taught keeping in mind the transition to high school math. Middle school is an important pathway from the elementary school to the high school. Our expertise at Educational Services Aug knows that and therefore we aim to provide a balanced approach to learning mathematics at the middle school level. The concepts of the all important algebra make an appearance at the dying part of the course. Our online curriculum for middle school maths is designed to make the students interested by presenting the subject in a new light. Through interactive associations help the students deepen their basic understanding of mathematical concepts along with the development of the mathematical thinking ability that are needed to be applied while solving problems. The critical high school period is essential in the advancements of students in the right track so that they can develop that all important problem solving skill later on. For that reason having rigorous lessons that are supported by the multiple learning modes is crucial. By the correct implementation of the EducationalServices Aug middle school math curriculum students can be excited into the learning of mathematics in the optimal manner. Students and learners must understand each problem and develop the tenacity to stick to the problem until he is able to chalk out a solution procedure. The EducationalServices Aug program encourages students to think on their own and question the procedure so that the solution technique comes as a part of their thinking process. The ability to reason the validity of every step is of vital importance in the problem solving arguments. The basic aim is to prepare the students in the use of the mathematical tools and also give special attention to the precision of the results obtained. The educators also play an important role in the shaping of the curriculum according to the capabilities of the learners. Assessments conducted at EducationalServices Aug helps these educators gain an insight into the gradual progress of the students and by the formulation of the course plan based on the results of these formulation the curriculum is adapted keeping in mind the needs of individual learners. The extensive courses on algebra, geometry and foundational mathematics are imparted along with special courses like math expeditions and straight curve mathematics. The EducationalServices Aug program allows students to learn at their pace and develop in a personalized way.
Math - Algebra I Written by Administrator \| 17 November 2009 Course Description: This class is a study of the language, concepts, and techniques of Algebra that will prepare students to approach and solve problems following a logical succession of steps. Skills taught in the course lay groundwork for upper level math and science courses and have practical uses. This course is primarily offered to grades 9-12, however, 8th graders who have demonstrated exceptional abilities in mathematics may request to take this course. The following topics are covered in this course:
Algebra Readiness Made Easy: Grade 6: An Essential Part of Every Math Curriculum (Best Practices in Action) This Algebra Readiness series makes meeting this mandate easy and fun, even in the younger grades. Each book features dozens of reproducibles that give students practice in different problem-solving strategies and algebraic concepts. Students learn to identify variables, solve for the values of unknowns, identify and continue patterns, use logical reasoning, and so much more. Includes 10 full-color transparencies with problem-solving steps and word problems, perfect for whole-class learning.
DESCRIPTION:This is a university-level course in Differential and Integral Calculus, equivalent to one semester of Calculus at most universities. The AP Calculus course is designed to develop the student's understanding of the concepts of Calculus and to provide experience with its methods and applications. The course emphasizes a multi-representational approach to Calculus with concepts, results, and problems expressed geometrically, algebraically, numerically, analytically, and verbally. Successful completion of the AP Calculus Course also provides the student with an MCB4UI credit. Math is a subject that builds upon itself; attending class, being on time, and participating in class are not only essential for good progress, but are sometimes essential for survival of the class. Unexcused absences result in no make up for the work. EVALUATION:70% is based on tests, quizzes, assignments. 30% is based on the summative assessment, which includes the June exam.
Offering a helpful introduction to basic algebraic concepts, this guide is useful for any student in pre-algebra and beyond as a reference tool at any level of algebra. The 6-page guide covers all the major topics included in a pre-algebra class. This guide is laminated and is three-hole punched for easy use.
Essentials Of Basic College Mathematics - With Cd - 2nd edition Summary: TheTobey/Slater seriesbuilds essential skills one at a time by breaking the mathematics down into manageable pieces. This practical ''building block'' organization makes it easy for readers to understand each topic and gain confidence as they move through each section. The authors provide a ''How am I Doing?'' guide to give readers constant reinforcement and to ensure that they understand each concept before moving on to the next. With Tobey/Slater, readers have a tutor and study com...show morepanion with them every step of the way. Whole Numbers, Fractions, Decimals, Ratio and Proportion, Percent. For all readers interested in basic college mathematics. ...show less All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU:9780321570659-5-0 $19.8721

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