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Conference Proceedings Detail Page
This paper is part of an ongoing investigation of how students use and understand mathematics in introductory physics. Our previous research [1] revealed that differences in score as large as 50% can be observed between numeric and symbolic versions of the same question. We have expanded our study of numeric and symbolic differences to include 10 pairs of questions on a calculus based introductory physics final exam. We find that not all physics problems exhibit such large differences and that in the cases where a large difference is observed that the largest difference occurs for the poorest students. With these 10 questions we have been able to develop phenomenological categories to characterize the properties of each of the questions. We will discuss what question properties are necessary to observe differences in score on the numeric and symbolic versions. We will also discuss what insights these categories give us about how students think about and use symbols in physics.
Physics Education Research Conference 2007
Part of the PER Conference series
Greensboro, NC:
August 1-2, 2007
Volume 951, Pages 200-203
Disclaimer: ComPADRE offers citation styles as a guide only. We cannot offer interpretations about citations as this is an automated procedure. Please refer to the style manuals in the Citation Source Information area for clarifications.
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Mathematica!
One area of mathematics that has its roots deep in philosophy is the study of logic. Logic is the study of formal reasoning based upon statements or propositions. (Price, Rath, Leschensky, 1992) Logic evolved out of a need to fully understand the details associated with the study of mathematics.
Logic
Business Calculus
spring 2009 course descriptionspring 2009 syllabusinstructions for the TI-83/84course reviewstudy tips Calculus is usually a major change for math students. This is appropriate, because calculus is the study of change: slope, velocity, growth rate, and other ways that we describe how one quantity changes with respect to another. Calculus is also perceived as difficult, and historically for the scientific community it was. It took about 300 years of concentrated effort to develop calculus as a usable and well-founded discipline!
Peter Alfeld, --- Department of Mathematics, --- College of Science --- University of Utah a study guide by Peter Alfeld. I wrote this page for students at the University of Utah. You may find it useful whoever you are, and you are welcome to use it, but I'm going to assume that you are such a student (probably an undergraduate), and I'll sometimes pretend I'm talking to you while you are taking a class from me. Let's start by me asking you some questions. If you are interested in some suggestions, comments, and elaborations, click on the Comments.
Understanding Mathematics
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Odyssey Algebra
04/01/05
CompassLearning ( has expanded its entire suite of Odyssey products, including Odyssey Algebra for middle schools and secondary education. The browser-based curriculum will help teachers offer a comprehensive approach to math education, while providing a platform that supports a variety of instructional strategies and learning styles. Odyssey Algebra has 13 chapters and 131 objectives to cover in an entire school year. The curriculum's online features include interactive tutorials that are woven throughout the program and aids such as online calculators, graph paper, number lines, protractors, spreadsheets and rulers. The program also provides additional offline materials for students that are designed to extend learning beyond the classroom
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From Lie theory to matrices, this site covers most of the topics included in linear algebra. Thorough notation and diagrams enhance the review pages and clarify the theorem explanations. Topics include linear transformation, systems of equations, and a complete overview of terms and definitions. Eigenvalues and eigenvectors, as well as many relevant theorems, can be found under the matrices tab. As with all the MathWorld sites, the information is abstracted and intended for the very advanced student. This resource is part of the Teaching Quantitative Skills in the Geosciences collection.
Intended for grade levels:
College (13-14)
College (15-16)
Graduate / Professional
Type of resource:
Text:
Ref. material
Subject:
Mathematics
Technical requirements:
No specific technical requirements, just a browser required
Cost / Copyright:
No cost
These pages may not be copied, mirrored, or reproduced in bulk without permission of the author. Reproduction for commercial purposes is not permitted, nor is use of robots to create archival copies. Copyright 1999-2002 Wolfram Research, Inc.
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Explained: Matrices
Dec 06, 2013 by Larry Hardesty
Among the most common tools in electrical engineering and computer science are rectangular grids of numbers known as matrices. The numbers in a matrix can represent data, and they can also represent mathematical equations. In many time-sensitive engineering applications, multiplying matrices can give quick but good approximations of much more complicated calculations.
Matrices arose originally as a way to describe systems of linear equations, a type of problem familiar to anyone who took grade-school algebra. "Linear" just means that the variables in the equations don't have any exponents, so their graphs will always be straight lines.
The equation x - 2y = 0, for instance, has an infinite number of solutions for both x and y, which can be depicted as a straight line that passes through the points (0,0), (2,1), (4,2), and so on. But if you combine it with the equation x - y = 1, then there's only one solution: x = 2 and y = 1. The point (2,1) is also where the graphs of the two equations intersect.
The matrix that depicts those two equations would be a two-by-two grid of numbers: The top row would be [1 -2], and the bottom row would be [1 -1], to correspond to the coefficients of the variables in the two equations.
In a range of applications from image processing to genetic analysis, computers are often called upon to solve systems of linear equations—usually with many more than two variables. Even more frequently, they're called upon to multiply matrices.
Matrix multiplication can be thought of as solving linear equations for particular variables. Suppose, for instance, that the expressions t + 2p + 3h; 4t + 5p + 6h; and 7t + 8p + 9h describe three different mathematical operations involving temperature, pressure, and humidity measurements. They could be represented as a matrix with three rows: [1 2 3], [4 5 6], and [7 8 9].
Now suppose that, at two different times, you take temperature, pressure, and humidity readings outside your home. Those readings could be represented as a matrix as well, with the first set of readings in one column and the second in the other. Multiplying these matrices together means matching up rows from the first matrix—the one describing the equations—and columns from the second—the one representing the measurements—multiplying the corresponding terms, adding them all up, and entering the results in a new matrix. The numbers in the final matrix might, for instance, predict the trajectory of a low-pressure system.
Of course, reducing the complex dynamics of weather-system models to a system of linear equations is itself a difficult task. But that points to one of the reasons that matrices are so common in computer science: They allow computers to, in effect, do a lot of the computational heavy lifting in advance. Creating a matrix that yields useful computational results may be difficult, but performing matrix multiplication generally isn't.
One of the areas of computer science in which matrix multiplication is particularly useful is graphics, since a digital image is basically a matrix to begin with: The rows and columns of the matrix correspond to rows and columns of pixels, and the numerical entries correspond to the pixels' color values. Decoding digital video, for instance, requires matrix multiplication; earlier this year, MIT researchers were able to build one of the first chips to implement the new high-efficiency video-coding standard for ultrahigh-definition TVs, in part because of patterns they discerned in the matrices it employs.
In the same way that matrix multiplication can help process digital video, it can help process digital sound. A digital audio signal is basically a sequence of numbers, representing the variation over time of the air pressure of an acoustic audio signal. Many techniques for filtering or compressing digital audio signals, such as the Fourier transform, rely on matrix multiplication.
Another reason that matrices are so useful in computer science is that graphs are. In this context, a graph is a mathematical construct consisting of nodes, usually depicted as circles, and edges, usually depicted as lines between them. Network diagrams and family trees are familiar examples of graphs, but in computer science they're used to represent everything from operations performed during the execution of a computer program to the relationships characteristic of logistics problems.
Every graph can be represented as a matrix, however, where each column and each row represents a node, and the value at their intersection represents the strength of the connection between them (which might frequently be zero). Often, the most efficient way to analyze graphs is to convert them to matrices first, and the solutions to problems involving graphs are frequently solutions to systems of linear equations.
In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians—the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, ...
The maximum-flow problem, or max flow, is one of the most basic problems in computer science: First solved during preparations for the Berlin airlift, it's a component of many logistical problems and a staple ...
It took only a few years for high-definition televisions to make the transition from high-priced novelty to ubiquitous commodity—and they now seem to be heading for obsolescence just as quickly. At the ...
(PhysOrg.com) -- A new analysis of number randomness in Sudoku matrices could lead to the development of more difficult and multi-dimensional Sudoku puzzles. In a recent study, mathematicians have found that
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Video tutorial & screenshots
Editorial review
This review applies to version 2.0.
The latest version of this software will soon be reviewed by our
informers.
Graphmatica is a small program designed for students and teachers of Mathematics. With it, you can graph your own equation just typing it over a text field and pressing enter.
Graphmatica is very easy-to-use and have a lot of interesting options. You can easily switch the graph paper (ex. to show polar coordinates instead of the standard x-y graph). Moreover you can change any graph color: the background, grids, and function line colors can be altered to easily match your needs.
Looking at the math features, after you draw a line over the graph you can easily (I mean 1-click) find derivatives for such equation. The same occurs with numerical integrations: just select the area and the program makes the rest.
Personally, I found this program very interesting. The only drawback is that you could use it just for 30 days, after that you have to pay for it. Another issue to mention is that you have no help at all when typing the equations. If you misspelled something, then you get just an error message. It will be crucial to read carefully the help file before to start using it seriously.
Pros
Very fast drawing.
Complete set of graphs.
Cons
The errors on equations are not very descriptive.
This software was checked for viruses
and was found to be clean.
Click here
to see antivirus report.
What's new in version 2.2 beta
Corrected domain specifications in one more demo file and a help page.
XP-style common controls are now loaded on 64-bit versions of Windows.
The installer is now digitally signed to verify that you have downloaded an authentic version of the program.
What's new in version 2.0
1. Fixed infinite loop parsing comma operator in domains and 2-variable functions when decimal separator is also set to ",".
2. Fixed crash graphing equations with free variables that have no on-screen solutions.
3. Added independently-settable font for bottom labels.
4. Graphs are now recalculated automatically upon changing the Theta Range.
5. The fraction characters ¼, ½, and ¾ are now accepted in place of normal decimals (before they could cause an infinite loop in the equation parser).
6. The custom increment for point table spacing now works for polar graphs as well as cartesian and parametric.
7. Switched from WinHelp to HTML help format, which is supported on Vista.
Publisher's description
raphmatica version 2.0 is designed especially for Windows 95/98/ME/2000/XP/Vista and Windows NT 3.51/4
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Prealgebra and Introductory Algebra - 3rd edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Prealgebra& Introductory Algebra, Third Edition was written to help students effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for3rd$9.18 +$3.99 s/h
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Sorry, CD MISSING. May have minimal notes/highlighting, minimal wear/tear. Please contact us if you have any Questions.
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0321644905 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!!
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mathematics program offers students the essential practice they need to demonstrate math proficiency. Organized by the key strands in math, these eight books help students develop a better understanding of math concepts, prepare for exams, and build skills that translate into greater confidence.Interest Level: 6-12Reading Level: 3-4
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More About
This Textbook
Overview
MATH FOR THE AUTOMOTIVE TRADE, 5E is an up-to-date, highly practical book that helps readers develop a real-world understanding of math concepts and applications in the modern automotive repair trade. Written at a beginner's level, this book is a comprehensive instructional workbook that shows readers how to solve the types of math problems faced regularly by automotive technicians. Unique to MATH FOR THE AUTOMOTIVE TRADE, 5E are realistic practice exercises that allow readers to determine if their answers fall within manufacturers' specifications and repair orders that are completed by finding the appropriate information in the professional literature and reference material, included in the book's valuable appendices.
Editorial Reviews
From the Publisher
"It is one of the best books on this topic I have seen. With the few items I have recommended I feel it would make a great addition to our automotive program."Anthony Allegro, Lincoln Tech (Union New Jersey Campus)
Related Subjects
Meet the AuthorWilliam J. de Kryger is a retired Professor of Automotive Technology in the Department of Engineering and Technology at Central Michigan University. He has been a certified ASE Master Technician for over thirty five years and is a member of the Society of Automotive Engineers. During his career as a professional technician, he was employed by both automotive dealerships and independent garages in Michigan, California, and Hawaii. Other professional services have included: expert witness for automotive cases, automotive arbitration for the National Center for Dispute Settlement, annual study abroad leader to Vladimir State University, Vladimir, Russia, and annual Visiting Professor of Automotive Technology to Instituta Technologico Y de Superiores de Monterrey, Campus Toluca, Mexico. William J. de Kryger holds an Associate's Degree in Automotive Services and a Bachelor of Science in Technical Education from Ferris State University, and a Master of Arts in Higher Education from Central
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Starting at $62 second book of a three-part series, An Introduction to Algebraic, Graphical, and Numerical Problem Solving, Fourth Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and the accompanying practice exercises. Along with the activities and the exercises within the text, MathXL®and MyMathLab®have been enhanced to create a better overall learning experience for the reader. Technology integrated throughout the text helps readers interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops readers' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines.
Author Biography
The Consortium for Foundation Mathematics is a team of fourteen co-authors, primarily from the State University of New York and the City University of New York systems. Using the AMATYC Crossroads standards, the team developed an activity-based approach to mathematics in an effort to reach the large population of college students who, for whatever reason, have not yet succeeded in learning mathematics.
Table of Contents
Chapter 1. Number Sense
Cluster 1. Introduction to Problem Solving
Activity 1.1 The Bookstore
Objectives:
1. Practice communication skills.
2. Organize information.
3. Write a solution in sentences.
4. Develop problem-solving skills.
Activity 1.2 The Classroom
Objectives:
1. Organize information.
2. Develop problem-solving strategies.
• Draw a picture.
• Recognize a pattern.
• So a simpler problem.
3. Communicate problem-solving ideas.
Activity 1.3 Properties of Arithmetic
Objectives:
1. Identify and use the commutative property in calculations.
2. Use the distributive property to evaluate arithmetic expressions.
3. Use the order of operations convention to evaluate arithmetic expressions.
4. Identify and use the properties of exponents in calculations
5. Covert numbers to and from scientific notation.
6. Identify, understand, and use formulas.
7. Use the order of operations convention in formulas involving whole numbers.
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Find a HaledonPrealgebra
Discussion of the subject will start from the real number system. Addition, subtraction, multiplication and division will follow, using signed numbers including integers. Fractions, decimals, exponents, graphs,and first order variable expression/equation will be considered.
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The TI-84 Plus graphing calculator offers three times the memory, more than twice the speed and a higher contrast screen than the TI-83 Plus model. It's keystroke-for-keystroke compatible, too-84 Plus graphing calculator offers three times the memory, more than twice the speed and a higher contrast screen than the TI-83 Plus model. Its keystroke-for-keystroke compatible, too.
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97806184701Math Excursions With Cd Plus Smarthinking
A new text for the liberal arts math course by a seasoned author team, Mathematical Excursions, is uniquely designed to help students see math at work in the contemporary world. Using the proven Aufmann Interactive Method, students learn to master problem-solving in meaningful contexts. In addition, multi-part Excursion exercises emphasize collaborative learning. The text's extensive topical coverage offers instructors flexibility in designing a course that meets their students' needs and curriculum requirements.
The Excursions activity and corresponding Excursion Exercises, denoted by an icon, conclude each section, providing opportunities for in-class cooperative work, hands-on learning, and development of critical-thinking skills. These activities are also ideal for projects or extra credit assignments. The Excursions are designed to reinforce the material that has just been covered in the section in a fun and engaging manner that will enhance a student's journey and discovery of mathematics.
The proven Aufmann Interactive Method ensures that students try concepts and manipulate real-life data as they progress through the material. Every objective contains at least one set of matched-pair examples. The method begins with a worked-out example with a solution in numerical and verbal formats to address different learning styles. The matched problem, called Check Your Progress, is left for the student to try. Each problem includes a reference to a fully worked out solution in an appendix to which the student can refer for immediate feedback, concept reinforcement, identification of problem areas, and prevention of frustration.
Each Chapter Opener begins with a short introduction to a real data application, which is then highlighted again in one Excursions activity and in the corresponding Excursion Exercises at the end of a section. This specific Excursion will be denoted by an icon. A section-by-section table of contents is accompanied by a brief summary of the topics that will be covered in the chapter.
A section called Problem-Solving Strategies in Chapter 1 introduces students to the inductive and deductive reasoning strategies they will use throughout the text.
An Instructor's Annotated Edition features icons denoting tables and art that appear in PowerPoint and Word files on the Instructor CD-ROM and web site; worked-out solutions to all Check Your Progress exercises and answers to all exercises; and a time-saving listing, Suggested Assignments.
A supportive Question/Answer feature at key points throughout the text encourages students to pause and reflect on the concept being discussed and to answer the question. The answer is located in a footnote on the same page.
Extension exercises placed near the end of each exercise set present a combination of Critical Thinking, Cooperative Learning, and Exploration exercises to provide further challenge and concept extension.
Take Note boxes in the margins alert students to a point requiring special attention or amplify a concept being developed.
Math Matters essay boxes throughout the text help motivate students by demonstrating how and why math is applicable to contemporary, real-life situations. Accompanying graphs and figures help students visually interpret the material.
Point of Interest notes provide relevant, contemporary information that helps motivate learning by giving context to concepts being presented. Historical Notes offer additional context by highlighting important mathematical developments or famous individuals who have made major advancements in their fields
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Preparing for the Placement Exams
Math Exam
Workshops
These workshops provide an opportunity to review pre-college mathematics that you may have forgotten since you last took a math course. Plan to attend a workshop in advance of taking or retaking the math placement exam. You will need time to practice the skills you have reviewed in the workshop in order to be better prepared to take the placement exam. The time spent reviewing and studying now will help ensure you start college math at the appropriate level, and with your math skills refreshed and ready for new challenges.
The topics reviewed in each workshop are included below. Students are free to attend either workshop or a combination of the two.
Reservations are required and can be made by contacting Kyle Schwieterman, ACE/Math Lecturer, at kschwiet@iusb.edu or by calling (574) 520-4665. Kyle can answer any questions you have about the workshops.
Workshop Schedule - Spring 2014
Day
Date
Time
Topics
Location
Facilitator
Monday
May 5
5:00p - 7:00p
Part A
Northside 376
Kyle Schwieterman
Wednesday
May 7
5:00p - 7:00p
Part B
Northside 376
Kyle Schwieterman
There is no cost to attend a workshop,
* List of topics covered in the math workshops
Part A: Advanced Arithmetic
Meaning of fractions and decimals
Operations on fractions and decimal numbers
Meaning and use of percents and proportions
Operations and signed numbers
Square roots and cube roots
The Pythagorean theorem
Area and perimeter of rectangles and triangles
Part B: Beginning Algebra
Evaluation of expressions and equations
Solution of linear equations
Meaning of integer exponents
Application of rules of integer exponents
Operations on polynomials
Factoring quadratic polynomials
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AP Calculus
Ap Calculus is an advanced math class designed for students seeking to further their knowledge in mathematics. The course spans over three terms and covers Calculus I and parts of calculus II.
AP Calculus Syllabus
The AP Calculus AB course is designed to provide students with an AP experience equivalent to that of a college course in single variable calculus. This course will develop a students understanding of the concepts of calculus and provide experience with its methods and applications. The course includes limits, differentiation and integration with problem solving and applications. Problems will be expressed graphically, numerically, analytically, and verbally. This three term course is intended for students who have a thorough knowledge of college preparatory mathematics including algebra, geometry, and pre-calculus.
Instructional Procedures & Support
Students are expected to bring all supplies to class every day including the textbook, a notebook, pencil and graphing calculator. Written assignments will be given daily and should be completed before the next class period. Homework will be checked for completeness at the start of each day. Late work will not be accepted unless it is due to an excused absence. Quizzes will be given daily following a short discussion of the previous day's assignment. Tests will be given approximately once per week. Students will be allowed to retake one test per term with the condition that all assignments leading up to that test have been completed. Test retakes must be completed before or after school prior to the next test (approximately one week).
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Students
preparing to take an introductory course in physics in college may want to review some, or
all, of the math topics listed below. The student with a good high school background in
math should be familiar with most of these topics.
Students planning on
enrolling in the College Physics sequence [111 & 112] should review the topics
highlighted in red. College Physics is the non-calculus based introductory sequence.
Potential General Physics [125 & 126] students may wish to review all topics. These
are calculus based introductory courses.
[Note: These materials were
created in Microsoft WORD and are saved in this form. Ifyour
computer's word processor will not reproduce these, copies are obtainable from your
instructor]
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97898123826 Sixth Conference on Function Spaces, held in Wroclaw, Poland, in September 2001. It discusses the following topics: convex analysis; operator theory; interpolation theory; theory of real functions; theory of analytic functions; bifurcation theory; Fourier analysis; functional analysis; measure theory; geometry of Banach spaces; and the history of mathematics
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Arithmetic Mean
Modern Calculator is a beautiful advanced replacement for the standard Windows calculator with a completely customizable skin-supported interface and some useful instruments that will allow you to make specific types of calculations.
ASP Did You Mean - The search query spell checker ASP Did You Mean is a COM ActiveX Server Component for providing accurate spelling suggestions for search strings. Its works the "Did You Mean" spelling corrections feature seen on Google ,
NumberMate is an arithmetic and scientific calculator, adding machine, and time calculator that documents calculations on multiple tapes which you can edit. It can make and record complex sets of interrelated calculations, print and save tapes.
This bilingual problem-solving mathematics software allows you to work through 36319 arithmetic and pre-algebra problems with guided solutions, and encourages to learn through in-depth understanding of each solution step and repetition rather than th
| 677.169 | 1 |
This course is one of the quickStep series offered by Siemens in AC Drives. These are FREE on-line industrial knowledge building tutorials. quickSTEPs are a great start for industry novices moving into technical jobs or...
This course is one of the quickStep series offered by Siemens in AC Motors. These are FREE on-line industrial knowledge building tutorials. quickSTEPs are a great start for industry novices moving into technical jobs or...
This is a video about Niko Henderson, an engineer for Easton Sports. He uses science, mathematics, engineering and innovative testing to help produce some of the fastest bikes on the road. Treat your students to a rare...
An idiosyncratic and personal selection of particularly important or particularly intriguing mathematical equations, not all of them complicated. Mathematical Constants; The definition of Pi; The definition of e; A...
This Public Broadcasting Service NOVA website contains information on shock waves and the speed of sound. It has a useful illustration of that shows how a sonic boom occurs at the intersection of the cone of the...
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Summary: ELE. The new edition features increased emphasis on the computing technologies commonly used in such coureses. New to This Edition
&q...show moreuot;Technology Step by Step" sections show how to solve basic problems using Minitab software, the TI-83 graphing calculator, or Excel.
More examples and exercises based on actual data.
Features
Statistics Today problems open every chapter. These real-life problems, accompanied by a photo or graphic and sometimes a news item, show students the relevance of the chapter's topic. The answer is provided at chapter end.
Procedure Tables embody the book's step by step approach. These boxes summarize methods for solving various types of common problems. Worked examples include EVERY step.
Critical Thinking Challenges at the end of each chapter extend chapter concepts into new areas, inviting students to think about and apply what they have learned.
Allan G. Bluman is Professor of Mathematics at Community College of Allegheny County, near Pittsburgh. For the McKeesport and New Kensington Campuses of Pennsylvania State University, he has taught teacher-certification and graduate education statistics courses. Prior to his college teaching, he taught mathematics at a junior high school.
Professor Bluman received his B.S. from California State College in California, Penn.; his M.Ed. from the University of Pittsburgh; and, in 1971, his Ed.D., also from the University of Pittsburgh. His major field of study was mathematics education.
In addition to Elementary Statistics: A Step by Step Approach, Third Edition, and Elementary Statistics: A Brief Version, the author has published several professional articles and the Modern Math Fun Book (Cuisenaire Publishing Company). He has spoken and presided at national and local mathematics conferences and has served as newsletter editor for the Pennsylvania State Mathematics Association of Two-Year Colleges. He is a member of the American Statistical Association, the National Council of Teachers of Mathematics, and the Mathematics Council of Western Pennsylvania.
Al Bluman is married and has two children. His hobbies include writing, bicycling, and swimming.
Book has appearance of only minimal use. All pages are undamaged with no significant creases or tears.With pride from Motor City. All books guaranteed. Best Service, Best Prices.
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AlphaBookWorks Alpharetta, GA
007231694224.59 +$3.99 s/h
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walker_bookstore tempe, AZ
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Books Revisited Chatham, NJ
Very good.
$79
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Summary: This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Books a la Carte also offer a great value-this format costs significantly less than a new textbook.
The first book of theMathematics in Actionseries,Prealgebra Problem Solving,Fourth Editionillustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and accompa...show morenying practice exercises.
This unique approach helps students increase their knowledge of mathematics, sharpen their problem-solving skills, and raise their overall confidence in their ability to learn. Technology integrated throughout the text helps students interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops students' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines
| 677.169 | 1 |
FYI, there is a book called Perspective Made Easy with illustrated masters
that can be reproduced found in the Sax VAR catalog. It includes lessons,
drawings and assignments. Maybe this is what would help you. Product #572
929J $17.95 Judy Nagel BTW we also carry the eyewitness on Perspective
book and the Masters of Illusion video if you're interested.
Check the eyewitness series on Perspective. It's useful enough to buy for
the classroom library. There are mathmatical layouts I believe and lots of
diagrams that could be photographed in slide form for a lecture accompanying
your class.
I am developing a lesson on linear perspective (one point) for an 8th grade
math class which I will be team teaching with the math teacher. He is not
an artist and I am no mathematician. Does anyone know of math formulas that
I can apply to the lesson, preferably dealing with the calculation of
distances between objects (or transversals on a 1 point persp. grid) as they
recede into the distance. I have found many books which talk about the
ideas of Alberti, Leonardo, and Brunelleschi, but none that show their
calculations.
Oh, and if anyone is looking for a video that deals with this topic, a
really good one is called Masters of Illusion from the National Gallery of
Art. I got my copy from the local public library.
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Algebra I Training [Mega] (Leído 48 veces)
Algebra I is one of the most critical courses that students take in high school. Not only does it introduce them to a powerful reasoning tool with applications in many different careers, but algebra is the gateway to higher education. Students who do well in algebra are better prepared for college entrance exams and for college in general, since algebra teaches them how to solve problems and think abstractly-skills that pay off no matter what major they pursue.
For other news, visit my profile every day! To Unzip the files use 7zip or WinRar. I recommend to download as fast as you can or you will lose file you need ( Links dead because of Copyright Infringement )
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More About
This Textbook
Overview
Designed for a one-semester beginning or introductory algebra course, this successful worktext is appropriate for lecture, learning center, laboratory, or self-paced courses.Maintaining its hallmark features of carefully detailed explanations and accessible pedagogy, this edition of Beginning Algebra also addresses the AMATYC and NCTM Standards. In addition to the changes incorporated into the text, a new integrated video series and multimedia tutorial program are also available
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Mathletics is a remarkably entertaining book that shows readers how to use simple mathematics to analyze a range of statistical and probability-related questions in professional baseball, basketball, and football, and in sports gambling. How does professional baseball evaluate hitters? Is a singles hitter like Wade Boggs more valuable than a power hitter like David Ortiz? Should NFL teams pass or run more often on first downs? Could professional basketball have used statistics to expose the crooked referee Tim Donaghy? Does money buy performance in professional sports? In Mathletics, Wayne Winston describes the mathematical methods that top coaches and managers use to evaluate players and improve team performance, and gives math enthusiasts the practical tools they need to enhance their understanding and enjoyment of their favorite sports--and maybe even gain the outside edge to winning bets. Mathletics blends fun math problems with sports stories of actual games, teams, and players, along with personal anecdotes from Winston's work as a sports consultant. Winston uses easy-to-read tables and illustrations to illuminate the techniques and ideas he presents, and all the necessary math concepts--such as arithmetic, basic statistics and probability, and Monte Carlo simulations--are fully explained in the examples. After reading Mathletics, you will understand why baseball teams should almost never bunt, why football overtime systems are unfair, why points, rebounds, and assists aren't enough to determine who's the NBA's best player--and much, much more. In a new epilogue, Winston discusses the stats and numerical analysis behind some recent sporting events, such as how the Dallas Mavericks used analytics to become the 2011 NBA champions.
Master the business modeling and analysis techniques that help you transform data into bottom-line results. For more than a decade, Wayne Winston has been teaching corporate clients and MBA students the most effective ways to use Excel to solve business problems and make better decisions. Now this award-winning educator shares the best of his expertise in this hands-on, scenario-focused guide--fully updated for Excel 2010!
Use Excel to solve real business problems--and sharpen your edge!
Model investment risks and returns
Analyze your sales team's effectiveness
Create best, worst, and most-likely case scenarios
Compare lease vs. buy, and calculate loan terms
See how price, advertising, and seasonality affect sales
Manage inventory with precision
Quantify the value of customer loyalty
Calculate your break-even number and ROI
Maximize scheduling efficiency
Express "home-field advantage" in real numbers
Project company growth, predict election results, and more!
Plus--introduce yourself to PowerPivot for Excel
Your companion web content includes:
Downloadable eBook
Hundreds of scenario-based practice problems
All the book's sample files--plus customizable templates
Customer Service Note: We are sorry for the inconvenience, but the code for accessing the eBook version of this title was accidentally left out of the first printing. However, the code is available for buyers of the first printing by contacting O'Reilly Media, the official distributor for Microsoft Press books, at mspbooksupport@oreilly.com or 800-889-8969. All subsequent printings include the access code and instructions inside the book.
Master business modeling and analysis techniques with Microsoft Excel 2013, and transform data into bottom-line results. Written by award-winning educator Wayne Winston, this hands-on, scenario-focused guide shows you how to use the latest Excel tools to integrate data from multiple tables--and how to effectively build a relational data source inside an Excel workbook.
Solve real business problems with Excel--and sharpen your edge
Summarize data with PivotTables and Descriptive Statistics
Explore new trends in predictive and prescriptive analytics
Use Excel Trend Curves, multiple regression, and exponential smoothing
Master advanced Excel functions such as OFFSET and INDIRECT
Delve into key financial, statistical, and time functions
Make your charts more effective with the Power View tool
Tame complex optimization problems with Excel Solver
Run Monte Carlo simulations on stock prices and bidding models
Apply important modeling tools such as the Inquire add-in this award-winning educator shares the best of his classroom experience in this practical, business-focused guide. Each chapter advances your data analysis and modeling expertise using real-world examples and learn-by-doing exercises. You also get all the book's problem-and-solution files on CD--for all the practice you need to solve complex problems and work smarter with Excel.
Learn how to solve real business problems with Excel!
Create best, worst, and most-likely scenarios for sales
Calculate how long it would take to recoup a project's startup costs
Plan personal finances, such as computing loan terms or saving for retirement
Estimate a product's demand curve
Simulate stock performance over a year
Determine which product mix will yield the greatest profits
Interpret the effects of price and advertising on sales
Assign a dollar value to customer loyalty
Manage inventory and order quantities with precision
Create customer service queues with short wait times
Estimate the probabilities of equipment failure
Model business uncertainties
Get new perspectives on data with PivotTable dynamic views
Help predict quarterly revenue, outcomes of sporting events, presidential elections, and more!
On the CD:
Practice files for all the book's exercises
Solutions for problem setsMaster the analysis and business modeling techniques that help you transform your data into bottom-line results. Award-winning business professor and corporate consultant Wayne Winston shares the best of his real-world experience in this practical guide--now updated for Excel 2007. Use Wayne's proven practices and hands-on exercises to help you work smarter, make better decisions, and gain the competitive edge.
Solve real-world business problems with Excel 2007!
Maximize profits--determine NPV, optimize your product mix, calculate ROI
Create best, worst, and most-likely case scenarios for sales
Analyze investment performance and help minimize risk
Track your personal finances, calculate loan terms, and plan for retirement
Use trend and seasonality to forecast revenue
Estimate a product's demand curve and manage inventory with precision Interpret the effects of price and advertising on sales
Assign a dollar value to customer loyalty
Solve work scheduling problems and shorten wait times for customer service queues
PLUS--predict outcomes of sporting events, presidential elections, and more!
CD Includes:
Sample files from each chapter
Problem-and-solution sets--build your expertise!
Fully searchable eBook
A Note Regarding the CD or DVD
The print version of this book ships with a CD or DVD. For those customers purchasing one of the digital formats in which this book is available, we are pleased to
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This course is primarily for liberal arts and education majors, and emphasizes mathematical systems and reasoning. Course content includes sets, symbolic logic, and elementary probability and such optional topics as basic statistics, game theory, or linear programming.
VII. Required Course Content and Direction
Learning Goals:
Course Specific
The student will be able to:
analyze, represent, and solve elementary problems in logic, set theory and probability.
recognize and apply the characteristics of a mathematical structure.
analyze and apply the concepts and principles of mathematics in varying situations.
Category III: Critical Thinking and Problem Solving: The student will be able to reason from what they know to form new knowledge, draw conclusions, solve problems, explain, decide, and/or predict (Inductive and/or Deductive Reasoning Skills).
The instructor may elect to cover logic, set theory, or probability in more depth or discuss some optional topics, such as game theory, statistics, or linear programming.
Assessment Methods for Core Learning Goals:
All Core Critical Thinking and Problem Solving, College Level Mathematics or Science, and Discipline-Specific Course Objectives will be assessed as follows:
The student will apply mathematical concepts and principles to identify and solve problems presented through informal assessment, such as oral communication among students and between teacher and students and, for the core, formal written assessment using open-ended questions reflecting theoretical and applied situations.
Reference, Resource, or Learning Materials to be used by Students:
Departmentally selected textbook. Details provided by the instructor of each course section. See Course Format.
VIII. Teaching Methods Employed
Use of lecture, recitation, problem solving, and class discussion as appropriate.
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MATH CLUB @ WMU
Mathematics Organizations
Mathematical Associations For Students
The diversity and vitality of the mathematical community is expressed by the wide variety
of mathematics organizations. It's never too early to get involved!
Below are a few pages aimed at the needs of students, especially undergraduates:
Founded in 1888 to further the interests of mathematical research and scholarship,
the AMS serves the national and international community through its publications,
meetings, advocacy and other programs.
A public voice of mathematics education supporting teachers to ensure equitable mathematics
learning of the highest quality for all students through vision, leadership,
professional development and research.
SIAM fosters the development of applied mathematical and computational methodologies
needed in these various application areas. Through publications, research, and community,
the mission of SIAM is to build cooperation between mathematics and the worlds of science and technology.
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... more...
This book gives a systematical presentation of stochastic approximation methods for models of American-type options with general pay-off functions for discrete time Markov price processes. It is the first volume of the comprehensive two volumes monograph. more...
This photocopy master book, which has proven extremely popular over the years, provides a range of 30+ problem solving activities using strategies such as: Developing logical thinking; Using number concepts to develop logical thinking; Logical reasoning; Developing visual imagery; and Pattern perception using number. more...
Photocopy Master book. Includes problem solving strategies such as Guess and Check, Act It Out, Make A Model, Look for a Pattern, Construct a Table and so on. These strategies are applied to a range of interesting problem situations. Children will enjoy the variety of characters that provide an amusing element to the serious business of solving mathematical... more...
Photocopy Master book. Students are required to utilise a range of problem
solving strategies in their approach to reaching solutions for these
interesting problems. Cool cartoon characters add a highly motivating element
to the process of working through the problems.
more...
New look versions of Pythagoras, Galileo and Archimedes are some of the
characters presented in cartoon form in this photocopy master book, lending a stimulating
element to problem solving. A variety of brain teasers is also included
for copying onto cards to make class sets.
more...
Sequential blackline master activities
in the area of geometry and spatial mathematics. Covers the major learning areas
such as identifying different types of angles, using a protractor to measure
angles, using known rules to calculate the size of angles and construction of
angles using either a compass or a protractor. more...
Learn how to easily do quick mental math calculations Speed Math for Kids is your guide to becoming a math genius--even if you have struggled with math in the past. Believe it or not, you have the ability to perform lightning quick calculations that will astonish your friends, family, and teachers. You'll be able to master your multiplication tables... more...
Bob Miller's fail-safe methodology helps students grasp basic math and pre-algebra All of the courses in the junior high, high school, and college mathematics curriculum require a thorough grounding in the fundamentals, principles, and techniques of basic math and pre-algebra, yet many students have difficulty grasping the necessary concepts.... more...
Everything you need to know to ace the math sections of the NEW SAT!... more...
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2Strong algebra skills are crucial to success in applied calculus. This text is designed to bolster these skills while students study applied calculus. As students make their way through the calculus course, this supplemental text shows them the relevant algebra topics and points out potential problem spots. the table of contents is organized so that the algebra topics are arranged in the order in which they are needed for applied calculus.
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Description new Precalculus eText by Eric Schulz, William Briggs, and Lyle Cochran strengthens the connection between precalculus and calculus in a way that's never been done before. The eText seamlessly integrates narrative text, interactive figures, and instructive author videos to immerse students in a rich learning environment.
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Journal of Online Mathematics and its Applications offers articles, learning modules, "mathlets" (single-purpose learning tools), reviews of online resources, and a developers' area. Search contents of the journal by type of resource (e.g., article), by subject (e.g., number concepts, data presentation, plane geometry), or both. The journal makes extensive use of graphics, animations, video clips, and other media. Articles and other materials are peer reviewed.
Classroom Capsules is on-line version of part of The College Mathematics Journal. The purpose of the capsules is to make easily available, by web search, short mathematical articles, which can give a student new mathematical insights, and instructors effective teaching strategies. The articles might represent a nice application, an unusual point of view or example, or they might give the instructor a non-standard way of presenting the topic in class.
MathDL Mathematical Communication is a collection of instructional strategies, materials, and references for having students write and speak about mathematics, whether for the purpose of learning mathematics or learning to communicate as mathematicians.
Course Communities in Undergraduate Mathematics (or Course Communities for short) contains a new collection of online resource recommendations. So far we have listed over 340 resources: PDF files, applets, videos, and other formats.
The following types of materials will appear in Loci:
Articles
Modules: innovative, class-tested, web-based learning materials that can be used for guided explorations of particular topics in mathematics
Should math educations be separated into the two tracks? Math education is either practical or abstract. One is useful in everyday life and the other is essential for science, engineering or math investigation.
Discussion Question(s)
1.
How can we communicate about math to reduce math anxiety?
2.
How do we learn to communicate math effectively, in writing or orally?
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Jun 27, 2008
The National Institute of Standards and Technology (NIST) has released a five-chapter preview of the much-anticipated online Digital Library of Mathematical Functions (DLMF). In development for over a decade, the DLMF is designed to be a modern successor to the 1964 "Handbook of Mathematical Functions," a reference work that is the most widely distributed NIST publication (with over a million copies in print) and one of the most cited works in the mathematical literature (still receiving over 1,600 yearly citations in the research literature). The preview of the new DLMF is a fully functional beta-level release of five of the 36 chapters.
The DLMF is designed to be the definitive reference work on the special functions of applied mathematics. Special functions are "special" because they occur very frequently in mathematical modeling of physical phenomena, from atomic physics to optics and water waves. These functions have also found applications in many other areas; for example, cryptography and signal analysis. The DLMF provides basic information needed to use these functions in practice, such as their precise definitions, alternate ways to represent them mathematically, illustrations of how the functions behave with extreme values and relationships between functions.
The DLMF provides various visual aids to provide qualitative information on the behavior of mathematical functions, including interactive Web-based tools for rotating and zooming in on three-dimensional representations. These 3-D visualizations can be explored with free browsers and plugins designed to work in virtual reality markup language (VRML). Mouse over any mathematical function, and the DLMF provides a description of what it is; click on it, and the DLMF goes to an entire page on the function. The DLMF adheres to a high standard for handbooks by providing references to or hints for the proofs of all mathematical statements. It also provides advice on methods for computing mathematical functions, as well as pointers to available software.
The complete DLMF, with 31 additional chapters providing information on mathematical functions from Airy to Zeta, is expected to be released in early 2009. With over 9,000 equations and more than 500 figures, it will have about twice the amount of technical material of the 1964 Handbook. An approximately 1,000-page print edition that covers all of the mathematical information available online also will be published. The DLMF, which is being compiled and extensively edited at NIST, received initial seed money from the National Science Foundation and resulted from contributions of more than 50 subject-area experts worldwide. The NIST editors for the DLMF are Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark.
Related Stories
The National Institute of Standards and Technology has released the Digital Library of Mathematical Functions (DLMF) and its printed companion, the NIST Handbook of Mathematical Functions, the much-anticipated successors
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0471849022
9780471849025
Discrete Mathematics With Algorithms:This first-year course in discrete mathematics requires no calculus or computer programming experience. The approach stresses finding efficient algorithms, rather than existential results. Provides an introduction to constructing proofs (especially by induction), and an introduction to algorithmic problem-solving. All algorithms are presented in English, in a format compatible with the Pascal programming language. Contains many exercises, with answers at the back of the book (detailed solutions being supplied for difficult problems).
Back to top
Rent Discrete Mathematics With Algorithms 1st edition today, or search our site for Michael O. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Wiley.
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Geometry, Perspective Drawing, and Mechanisms
Description: The aim of this book is to examine the geometry of our world and, by blending theory with a variety of every-day examples, to stimulate the imagination of the readers and develop their geometric intuition. It tries to recapture the excitement thatMore...
The aim of this book is to examine the geometry of our world and, by blending theory with a variety of every-day examples, to stimulate the imagination of the readers and develop their geometric intuition. It tries to recapture the excitement that surrounded geometry during the Renaissance as the development of perspective drawing gathered pace, or more recently as engineers sought to show that all the world was a machine. The same excitement is here still, as enquiring minds today puzzle over a random-dot stereogram or the interpretation of an image painstakingly transmitted from Jupiter.The book will give a solid foundation for a variety of undergraduate courses, to provide a basis for a geometric component of graduate teacher training, and to provide background for those who work in computer graphics and scene analysis. It begins with a self-contained development of the geometry of extended Euclidean space. This framework is then used to systematically clarify and develop the art of perspective drawing and its converse discipline of scene analysis and to analyze the behavior of bar-and-joint mechanisms and hinged-panel mechanisms. Spherical polyhedra are introduced and scene analysis is applied to drawings of these and associated objects. The book concludes by showing how a natural relaxation of the axioms developed in the early chapters leads to the concept of a matroid and briefly examines some of the attractive properties of these natural structures
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Real World Algebra
Book Description: Algebra is often taught abstractly with little or no emphasis on what algebra is or how it can be used to solve real problems. Just as English can be translated into other languages, word problems can be "translated" into the math language of algebra and easily solved. Real World Algebra explains this process in an easy to understand format using cartoons and drawings. This makes self-learning easy for both the student and any teacher who never did quite understand algebra. Includes chapters on algebra and money, algebra and geometry, algebra and physics, algebra and levers and many more. Designed for children in grades 4-9 with higher math ability and interest but could be used by older students and adults as well. Contains 22 chapters with instruction and problems at three levels of difficulty
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Linear Systems and Optimization | Convex Optimization II
Course Content:
Resources
This page contains links to various interesting and useful sites that relate in
some way to convex optimization. It goes without saying that you'll
be periodically checking things using google and wikipedia. The wikipedia
entry on convex optimization (and related topics) could be improved
or extended.
1. Stephen
Boyd's research page. There's a lot of material there, and you
don't have to know every detail in every paper, but you should certainly
take an hour or more to browse through these papers.
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Watch 3,500 videos FREE for 1 week. No Credit Card required.
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About the Course & AP Exam Strategies
Transcript
[0:00:00] My name is John Postovit and this is advanced placement calculus AB. In this course we're going to cover a ton of stuff. We're going to do the problems involving the basics of calculus. And we're also going to do problems involving those strange theoretical things that you'll definitely see on AP test. You can use this of course in a couple of different ways. You could use it all year long as a supplement to your regular calculus course. I have a lot of episodes where I do lots of problems involving things like the chain rule, things like finding areas between curves, finding volumes, all sorts of the essentials that you need. Or, maybe using this course at the very end of the year in that last month or two while you were leaving for the AP test. Either way, I first recommend that you go to the bonus materials, down with the follow alongs.
The reason I want you to do that is because, I often do a lot of writing and you have more coordinated, organised notes if you're writing them down and can look at the tips and the hints and the problem itself, without having to madly scribble that down. You can just concentrate on taking good notes for yourself. Of course you can always stop the video to give yourself a second to catch up.
After you watch every episode, I'd also say that you ought to try the extra practise problems. I've written up a bunch of practise problems along the lines of what I show you during this course, so that you can try two or three of them on your own.
[0:02:00] These practise problems also involve the solution including every step, it's probably first it'll do that sometimes that the calculus text book will say, "Well okay, here's a problem. Do this problem and there is one example, and by the way we left out a bunch of steps because you're just too smart to need all the steps." Well I'm assuming that you're like me and when you were taking calculus, sometimes I didn't see those steps that were left off. And I had to think how did they get from there to there. In my extra practices, I've tried to show you everything. Well settle in, relax a little bit, and I hope you enjoy the course.
Before you walk into the room on that fateful day where you're going to be a total success on the calculus test, you need to know how it's organised, so that you're prepared and ready to go. Sections, questions and time. Tests are broken up into two large sections. The first large section is multiple choice. The second one is free response. Each of these two sections is weighted the same. So you get an equal amount of credit for multiple choice and free response on the whole.
When they score it, they'll score this part, they'll score this part, add up the results and then compare that to their standards for what's considered a one, two, three, four or five on the test. Depending what college you go to, you'll need a three or a four, or sometimes even a five in order to get credit for moving on to the next level of calculus. The whole test takes 3 hour and 15 minutes and that doesn't include the fiddling around time for passing out the test, reading the rules, collecting the tests and all of that.
Now within each part of multiple choice and free response, there is two subsections. So when you get the test, the first thing they'll tell you is, "Put your calculator away." And they give you part A. They say, "Do this part A and when your done stop. Put our head on down and just sit soundly."
[0:04:00] Then on part B they'll allow you to get your calculator out to do just part B. Well, look at the timing on the number of questions. Part A, 28 questions in 55 minutes, that's about two minutes a question. Part B, 17 questions, 50 minutes, that's about three minutes a question. That should tell you something right there. They're allowing you more time because they think you'll need it.
Part B is more likely to have some unusual theoretical problems that might have to pause and think about just to organise your thoughts. And part A will tend to have the more straight forward ones that don't require a lot of calculations, or they're fairly direct derivative, chain rules, that kind of thing.
A little bit of strategy, you might have felt, go through the test first and decide which problems to do. I wouldn't recommend that because of the amount it takes. But I'd recommend something very similar. Look at question number one, and if you immediately think you know, "I think that's more than a two minute question." Circle it and then go on to the next one. Then you can go back later to those harder questions that you circled. The reason I recommend that is because hard and easy, on multiple choice they're all worth the same. So you don't want to spend a time on a hard question when you could be knocking out some of those thirty second questions.
Go back later to finish the harder ones, same thing for part B of the multiple choice.
Free response, you're allowed a total of 90 minutes and that's broken up into two sections. The first part takes a calculator. So after you finish this they'll let you keep your calculator. The second part doesn't and there is only three questions for each part. What that tells you is that, each of those three questions has a lot of subsections. So there may be four or five things you need to find for each of those questions. But still, I would recommend the strategy of looking it over first.
[0:06:00] Now this, since they are only three questions probably is worth reading each question and deciding which one to do first. You want to do the one first that you immediately know how to do. Then say you're going through one of those questions and you get to the second or last part and you think, "I don't know how to do that." Think about it for just a little while, but not too long. And if you can't do the last part, that's okay. It's free response and you get credit for each sub-section that you do of each question. And actually within each sub section there is different things they are looking for. They'll be looking for having the integral written up properly, looking for some of the details, the solution etcetera. So remember you can get partial credit, don't panic and give up. If you can write something down that you think can makes sense, do it. 15 minutes a question so that's a lot of steps.
Now after you've done this 45 minutes they'll say, "Stop," to have you put away your calculator. Then for the next 45 minutes, you'll have part B. Three questions again, tend to be questions that don't allow a calculator. Which means that they tend to be more theoretical questions where you'll have charts to work from or with the calculations are very simple. Whereas part A, with the calculator, will often have integrals that you can't do by hand, and where you'll have to approximate the solutions by using some of the functions on your graphing calculator.
If you finish the three questions on part B and still have some time left, they're going to let you go back to part A, on the free response section only. They won't let you have your calculator back because no one else in the room will have one. But at least you can go in and maybe fill in some more of the theoretical parts and almost finish some of the problems that you're asked to do in the calculator section.
Now that I've shown you how the test is organised, I'm going go to show you some general strategies for preparing for the test and for actually taking the test.
[0:08:0] I'm going to begin by covering some basic strategies that can really help you earn high score on AP test. First strategy, go back and review. That's kind of vague isn't it? So I want to give you some specific hints on how you could do that.
First thing I would recommend doing, is going and finding all your old tests. Find the problems that you got wrong the first time, that's what you should concentrate on. Look at the problems you got wrong in the first time, and then stop and think, "Do I know how to do this now?" If you do, then don't spend a lot of time on it. Just enough time to remember how it works. Go back and review.
The reason I'm saying this is because in order to score well on the AP test, you really have to know the calculus. You have to be able to do the basic operations like differentiation and integration, and even things like chain rule, which is fairly basic. You have to be able to them instinctively. So you don't have to concentrate on how you do that when you're trying to figure out what the problem is asking for. There are a lot of problems where it's not entirely clear what they want. There are a lot of problems where it'll say do such and such, find the rate of change of an integral. It won't just always say, "Here is something integrated." Sure you have some of those problems where you just do an operation, but knowing how to do those isn't enough.
Next, do the official AP practise test. Of course before you do that, this course is designed to get you through that stuff. What I've done is, I've gone through the officially released course description which includes, multiple choice questions and free response questions. And I've modelled a lot of what I'm teaching you on what they are asking you there. And sometimes I've taken it a step beyond.
[0:10:00] Once you've done that, then you'll want to do the practise problems that I give you, so that you get these down a little bit better. And then go do the official test, which is going to have a lot of similarities. Take that official test, try to figure out which technique you are using. You might even want to go back and say, "Hey! What episode was this on?" And go through it. It's really important to do the AP practise test.
Next thing you need to do, is be really strategic on the free response section. First off, the problems are weird. They are seriously weird problems because, the AP people want to find out if you really know calculus, or whether not you're just parroting a bunch of operations. So they've put in questions where you have to know the theory behind calculus, and you know how to be calm and collected. To read a problem and realise, "Well are they asking me to integrate it, or to eliminate some or do the derivative. Just what are they asking for?" You have to be strategic on it.
Also, while you're doing the free response section, you have to make sure that the scorer can read what you're doing. That's actually humans that grade these. The way it's set up is, the AP people hire calculus teachers to come in during the summer and grade your test. Now, they are given really strict guidelines on how it should be graded, how many points you get for each part. Do you notice I said each part? yeah. Because you get points for everything that you do right. But there is a lot of particularities that they look for.
For example, if you're not using correct notation, you're probably going to loose some points. If you skip a detail at the end of an integral like putting the constant integration, that's right.
[0:12:00] One of the points that you can get is actually for just writing plus C at the end of the integral. Make your writing clear. Again these are humans that are grading this. If they can't read what you did, guess how many points you get? Zero. That's right.
Also since you're under time constraints, you want yo page through the entire section, that entire section that you're allowed to do at that point and see which problems you can do first, most quickly. Otherwise there's chance that you might just sit there looking at a problem and then your head gets filled up and starts buzzing and you start thinking, "How do I do this one?" You panic, and that actually makes it harder for you to do the easier problems.
Last thing, if you're doing the test and you suddenly realised, "You know what I made a big mistake." Don't spend time erasing. Cross it off, it's faster to cross it off, and the AP scores are told if something is crossed off, they just ignore it. So be sure as you're going through the course, to follow along with how I model the ways to write out problems.
Guess, guessing is evil. I think so but also, one of your goals of course is to pass the AP test, and there's no way you're going to get every question right. There is no way. Some of the multiple choice questions will have often 70-80-90% of the students who take the test will get those right. There are other questions where only 20-30% of the students taking the test that year, get that one right. And that doesn't mean that they qualified the question, not at all. It's still included.
So be prepared, there is going to be some that you just can't do, and that's okay.
[0:14:00] Because they don't really compare you against you know, "Did you get 90% on the test? Did you get 90% right and you can't pass unless you get 70% right." That's not how it's done at all. It's a normed test. What they do is research on students actually taking the test, and they look to see how an overall very large group of students does on the test. And then they figure out what the passing score is and 70% probably isn't a passing score, it's probably lower than that. That being said, guessing in the multiple choice, you don't want just to randomly guess. Sorry they see it coming on that one.
If they didn't correct for guessing, you would score a little bit higher than you should. Because there is five possible choices and if you guessed on every one, you would probably get a 20% on the test and they don't want that to happen. So what they do is they give you one point for every question you get right on multiple choice. They give you nothing for something you leave blank, notice I said leave blank. It's okay to leave questions blank. If you put an answer down and it's wrong, they take away a quarter of a point.
On the average by doing that, the statistics show that if you take a quarter point off for everything that is wrong, it balances out the results of guessing. But still it doesn't mean that you can't guess sometimes. If you can narrow down a little bit and you know that say, two of the possible choices are obviously bogus, and you know it's one of three. Make an educated guess on those one or three, because the odds are a little bit in you're favour at that point.
If you're an old school guy like me, you've got to be told this one, use a calculator when possible. I grew up in the dark ages of slide rules. So I don't instinctively think of calculators.
[0:16:00] Or maybe you're a mathematical tough guy, you can do algebra all day long and you're right every time and yo just hate using that calculator. I'd still recommend you do it, when you're allowed to do it. The AP test has basically four sections. There is multiple choice with the calculator, multiple choice without. There is free response with the calculator, free response without.
Especially on the free response part, that involves the calculator, if you don't have a calculator you can't do it because you will be given crazy integrals that you can't integrate by hand. And you don't even want to do that many steps on it by hand, you want to do as little as possible by hand.
For example, later on the course you'll see some problems where I set up an integral and then I don't even try to integrate the thing. What I do is I type it in undistributed, I type it into the calculator and let the calculator find the area underneath that curve. Notice I said undistributed. I know it's instinct because you've done it for years, you always simplify. But you don't want to on this, because the calculator doesn't care if you simplified, as long as you use the correct number of parentheses to make the calculator understand the order of operations. Simplifying just increases the chances that you make some kind of an error.
Finally, when you are using the calculator on the free response section, you have to be careful on how you write down your answer. Part of the required answer is going to be set up for the problem. In other words a correctly written derivative or a correctly written integral or a graph, something like that. You have to write that down, even though you get final answer with your calculator, you have to. If you just write down, "I pressed this button on the calculator and then I pressed graph then I pressed second calculate integrate and got this answer."
[0:18:00] Sorry, they're not going to give you any credit for that. They don't care what brand of calculator you use. There is lots of good brands are out there. What they care about is that, you know how to write it out in technical form, write out the calculus with all the correct symbols and then get the answer using whatever calculator you have, and they don't care about those steps.
The functions you should really know on your calculator are finding zeros, usually by using a graph. Finding approximations for intersections between lines, that's a very important one, especially for finding limits of integrals. And finding the area underneath curves. Usually as you use that one, you have to find an integral but find the area underneath a curve for some formula that you can't integrate.
So what you've seen as some strategies for both reviewing for the exam and actually taking the AP exam.
Our introductory episode began with giving you some strategies that you can use to get ready for the test, to really solidify your knowledge for the calculus. Along with some hints that you can use during the test to increase your score, to format your answers in a way that you're going to get maximum credit on it.
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Matrices as a rectangular array of real
numbers, equality of matrices, addition,
multiplication by a scalar and product of
matrices, transpose of a matrix, determinant of
a square matrix of order up to three, inverse of
a square matrix of order up to three, properties
of these matrix operations, diagonal, symmetric
and skew-symmetric matrices and their
properties, solutions of simultaneous linear
equations in two or three variables.
Addition and multiplication rules of probability,
conditional probability, Bayes Theorem,
independence of events, computation of
probability of events using permutations and
combinations.
Trigonometry: Trigonometric functions, their
periodicity and graphs, addition and subtraction
formulae, formulae involving multiple and submultiple angles, general solution of
trigonometric equations.
Relations between sides and angles of a
triangle, sine rule, cosine rule, half-angle
formula and the area of a triangle, inverse
trigonometric functions (principal value only).
Analytical geometry (2 dimensions):
Cartesian coordinates, distance between two
points, section formulae, shift of origin.
Equation of a straight line in various forms,
angle between two lines, distance of a point
from a line; Lines through the point of
intersection of two given lines, equation of the
bisector of the angle between two lines,
concurrency of lines; Centroid, orthocentre,
incentre and circumcentre of a triangle.
Equation of a circle in various forms, equations
of tangent, normal and chord.
Parametric equations of a circle, intersection of
a circle with a straight line or a circle, equation
of a circle through the points of intersection of
two circles and those of a circle and a straight
line.
Equations of a parabola, ellipse and hyperbola
in standard form, their foci, directrices and
eccentricity, parametric equations, equations of
tangent and normal.
Locus Problems.
Analytical geometry (3 dimensions):
Direction cosines and direction ratios, equation
of a straight line in space, equation of a plane,
distance of a point from a plane.
Limit and continuity of a function, limit and
continuity of the sum, difference, product and
quotient of two functions, L'Hospital rule of
evaluation of limits of functions.
Even and odd functions, inverse of a function,
continuity of composite functions, intermediate
value property of continuous functions.
Derivative of a function, derivative of the sum,
difference, product and quotient of two
functions, chain rule, derivatives of polynomial,
rational, trigonometric, inverse trigonometric,
exponential and logarithmic functions.
Derivatives of implicit functions, derivatives up
to order two, geometrical interpretation of the
derivative, tangents and normals, increasing
and decreasing functions, maximum and
minimum values of a function, Rolle's Theorem
and Lagrange's Mean Value Theorem
Using Logarithmic Tables
Authors
Express the given number "n" in the form of m * 10pwhere 1≤m<10 and p is an integer(positive or negative whole number).
For example number 2 is expressed as 2*100
log n become equal to p + log m
log 2 becomes equal to 0 + log 2
p is called the characeristic and log m is called the mantissa. Mantissa is read from the logarithmic tables.
Logarithmic tables are show three sets of columns
i) the first set of column on the extreme left contains numbers from 10 to 99.
ii) in the seocnd set there 10 columns headed by 0,1,2,...,9
iii) after this, in the third set there 9 more columns headed by 1,2,3...9. These are known as mean differences.
As 1≤m<10, the mantissa is for a number between 1 and 10. Hence the interpretation of the first set of column in the table is 1.0 to 9.9, If you add the digit in the second set one more digit is added to the number. Which mean 1.0 becomes 1.01. If we add a digit in the third column on more digit is added to the number. Which means 1.01 becomes 1.011.
Hence log 2 = 0 + 0.3010 = 0.3010
How to see its antilogarithm.
Antilogaritm tables are written from .00. If mantissa of a logarithm is .00, then antilogarithm is 1.000
Antilogarithm of .3010 is equal to 2.000
As the characteristic of the number is 0 the number is 2.0*100. Which is equal to 2.
Suppose the problem is to find 2^(1/6). It is 2 to the power (1/6).
When we take logarithms, it becomes (1/6)* log 2 which is equal to (1/6)*(0.3010) = 0.0617 (rounded)
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Today at Central Library Cooperative of Knowledge
Upon entering high school students should have completed all the necessary math courses available at the junior high level. Students should know basic math such as addition, subtraction, multiplications and division. They should also have an understanding of algebra, geometry, percentages, decimals, data analysis and probability, along with other junior...
An online Bible study course can be useful in helping an individual to learn and understand the teachings of the Bible, and how to apply them practically in daily life. This is particularly ideal for those who have no time to spare because of other commitments. Online Bible study courses...
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Political science, the study of government and politics, public issues and the processes through which conflict over such issues are resolved, is fascinating, surprising, and affects everyone, every day, in many ways. And an education in political science involves much more than merely learning about politics and history.
As the world...
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Find a San Anselmo ACTAlgebra 2 introduces independent and dependent variables and how their solution can be determined by for linear relationships for two or three variables. Algebra 2 also gives an overview of more complex mathematical functions like basic trigonometric functions, power functions, logarithms, and
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Synopses & Reviews
Publisher Comments:
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Book News Annotation:
A brief introduction for undergraduate students and teachers, presenting detailed proofs and explanations of the elementary components of classical algebraic number theory. Begins with divisibility and Gaussian primes, and proceeds to ideal classes and class numbers and the Fermat conjecture. A slightly revised and totally unexpurgated edition of the 1975 publication by the Mathematical Association of America. Annotation c. by Book News, Inc., Portland, OR (booknews@booknews.com)
Synopsis:
Excellent intro to basics of algebraic number theory. Gausian primes; polynomials over a field; algebraic number fields; algebraic integers and integral bases; uses of arithmetic in algebraic number fields; more. 1975
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Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
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More About
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Overview
This book is a short, focused introduction to Mathematica, the comprehensive software system for doing mathematics. Written for the novice, this engaging book contains an explanation of essential Mathematica commands, as well as the rich Mathematica interface for preparing polished technical documents. Mathematica can be used to graph functions, solve equations, perform statistics tests, and much more. In addition, it incorporates word processing and desktop publishing features for combining mathematical computations with text and graphics, and producing polished, integrated, interactive documents. You can even use it to create documents and graphics for the Web. This book explains everything you need to know to begin using Mathematica to do all these things and more. Written for Mathematica version 3, this book can also be used with earlier versions of the software. Intermediate and advanced users may even find useful information here, especially if they are making the switch to version 3 from an earlier version.
Editorial Reviews
From the Publisher
"Coombes and colleagues offer a 'short, focused introduction to Mathematica,' designed to be useful to beginners as well as more experienced Mathematica users. The text is well written; graphics are used effectively." Choice
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This is the eBook version of the printed book. If the book includes supplemental materials, the supplemental materials are not included within the eBook version. The Holt McDougal Larson Secondary Mathematics series is designed for students studying pre-algebra, algebra, geometry, and advanced algebra. It supports students working at the regular level, and also provides significant challenge and support for advanced and Pre-AP students.Larson Mathematics is Built for the Common Core State StandardsThe new tables of contents for the Holt McDougal Larson Secondary Mathematics series reflect comprehensive alignment to the new Common Core State Standards. Correlations are placed at the front of the texts, and on every lesson. Larson Pre-Algebra addresses both accelerated 7th grade standards and all 8th grade standards, and is appropriate for students preparing to study algebra. Algebra 1, Geometry, and Algebra 2 cover the course-level standards, including the plus standards.Larson Mathematics Takes Students to the Next Level of ComprehensionThe program is designed to emphasize conceptual understanding, to focus on critical thinking and reasoning, and to develop strong skills through mathematical modeling and problem solving. The Activity Generator provides full Investigations for all Standards. The investigations are leveled for universal access and include a section of "Drawing Conclusions" to foster discourse and develop reasoning. Lessons open with "Essential Questions" in the Teacher's Editions to foster discourse about the topic. Teachers also have "Key Questions" for Examples as they are working through lessons. Performance Tasks are available for formative and summative assessments.Larson Mathematics Paves the Way to Higher MathematicsThe Holt McDougal Larson Secondary Mathematics series provides a superior foundation for the study of advanced level mathematics. Lead author, Dr. Ron Larson, is nationally recognized for his award-winning precalculus and calculus level textbooks, and he has written this series so that students will be "college and career ready".Larson Mathematics Utilizes Technology to Enhances LearningWith the interactive Online Editions, students have access to their book, Lesson Tutorial Videos, additional practice with immediate feedback, and intervention resources. eText Student Editions come to life on a variety of devices. The Teacher One Stop CD-ROM with Test and Practice Generator supplies teachers with everything they need to plan and manage lessons. The Activity Generator provides additional tasks, investigations and activities correlated to the Common Core State Standards. Interactive Whiteboard Lessons provide an interactive pathway to present the Explorations Lessons. The Interactive Answers and Solutions CD-ROM allows teachers to select questions, as well as customize answer keys with stepped-out solutions. Assessment resources provide diagnosis tools as well as remediation opportunities.Strengths of the Holt McDougal Larson Secondary Mathematics SeriesLaser Focus on Developing the Standards for Mathematical PracticeThe Standards for Mathematical Practice are embedded in every lesson through the exercise sets, and correlated in the Teacher's Editions. The Explorations in Core Math component presents lessons through guided-discovery activities that encourage development of the Practices. The Activity Generator includes a bank of activities written specifically to develop the Practices.Attention to Problem Solving, Modeling, and Performance TasksMath modeling and the integration of relevant applications is paramount to the design of the Holt McDougal Larson Secondary Mathematics series. Every lesson begins with "Before, Now, and Why?" that connects to a real-world application problem that incorporates the math of the lesson. Real-World Situations are used in Examples as the lessons progress. Every homework set focuses on relevant, real-world Problem Solving. The Assessment Book includes authentic assessments for use in performance-based tasks. The Activity Generator includes a bank of Performance Tasks and authentic assessment tools.Comprehensive Interactive Whiteboard LessonsHolt McDougal Larson Secondary Mathematics includes full, comprehensive standards-based lessons for use on interactive whiteboards. Every lesson begins with an Essential Question, followed by Explore activities and Examples, and then closes with Standardized Test Prep a lesson Summary. These lessons integrate the use of interactive whiteboard tools, digital manipulatives, and student response systems.Opportunities for Challenging Students on the Pre-AP TrackThe Larson Secondary Mathematics series is known to provide rigorous coursework, and includes instruction on the "Plus" Standards. There are "C-level" exercises in both the Practice and Problem Solving exercise sets. There are "C-level" and "Challenge" practice pages in the Chapter Resources for every lesson. Several Extension Lessons are placed throughout each course so that teachers may extend content at appropriate points during instruction. The Pre-AP Resources book provides an alternate pacing and assignment guide to use the texts with advanced students. It also includes best practices and teaching strategies for extending content of the lessons appropriately. The Activity Generator provides C-level investigations for lessons, and there are additional Problem Solving Applications for lessons and extended Projects to culminate work through the chapters.Best Common Core Assessment PreparationIn addition to the course-specific test generator provided on the Teacher One Stop, the On Core 6-12 Deluxe ExamView Assessment Suite provides over 11,000 items correlated to the Common Core State Standards and written to DOK Levels 1-4 with which to prepare additional standards-based practice and assessment
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MATH IIIA: PRECALCULUS
A primer course for calculus exploring linear, polynomial, rational, exponential, logarithmic and trigonometric functions. The graphing calculator is introduced and used throughout the course to operate with real-life data and applications. Students are taught to examine a situation from numerical, graphical and analytical perspectives. The last trimester deals with analytic trigonometry: fundamental identities, solving trigonometric equations, sum and difference formulas, Law of Sines and Law of Cosines. The course ends with sequences, series, sigma notation, and multivariable systems of equations and inequalities.
MATH III: INTERMEDIATE ALGEBRA
Intermediate Algebra provides a straightforward program of study with continual reinforcement and progressive confidence building. The topics covered include a chapter on real numbers, linear equations, functions and graphs, systems of linear equations and inequalities, polynomials, factoring, simplifying expressions, radical equations and complex rational expressions. Practice of sample SAT's are offered to keep the students comfortable with this particular type of examination format. The graphing calculator is introduced and used frequently in this course. The course ends with sequences, series, sigma notation, and multivariable systems of equations and inequalities.
Projective Geometry
Projective Geometry covers a historical view of the mathematical breakthroughs symbolizing new realms of thought. A comprehensive and hands-on experience of projective transformations and in particular, the geometrical experience of infinity, gives students a real experience of synthesizing these new ideas.
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Thanks for visiting ARIS or MathZone. We have retired ARIS and MathZone, but no worries! We've replaced them with Connect and ConnectPlus, our new generation of digital learning products with improved user experience and enhanced functionality.
Elementary & Intermediate Algebra, 3/e content in context. The authors use a three-pronged approach (I. Communication, II. Pattern Recognition, and III. Problem Solving) to present the material and stimulate critical thinking skills. Items such as Math Anxiety boxes, Check Yourself exercises, and Activities
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When you are able to conquer Algebra, you can conquer the world! Algebra 2 can't be properly understood without a thorough understanding of Algebra 1. It makes sense, but in actuality the concepts and functions of Algebra 2 are built upon Algebra 1
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Mathematics as a Constructive Activity [NOOK Book]NOOK Study
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Table of Contents
Contents: Preface. Introduction in Exemplification in Mathematics. Learner-Generated Examples in Classrooms. From Examples to Example Spaces. The Development of Learners' Example Spaces. Pedagogical Tools for Developing Example Spaces. Strategies for Prompting and Using Learner-Generated Examples. Mathematics as a Constructive Activity. Append
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Math: Factoring
Factoring polynomials is one of the most important skills you will learn and use now and in your future math classes. Factoring is an essential skill that can make or break you and your grade. Join us for a "factoring summary," and learn four straight-forward things you can do to make it in factoring and your math class.
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This is a guide to some of the ways in which the Worldwide Web can be used as a resource for teachers and learners of mathematics. Browsing the Internet can be very time-consuming, so this is just one person's selection of tutorials, interactive materials and mathematical tools. The selection given here, along with the accompanying notes, was collated by Pam Bishop, Assistant Director of the LTSN Mathematics, Statistics and Operational Research (OR) Network (see for the original version).
Recently the Mathematics, Statistics and OR Network has added a section to its Web site, 'Accessing Maths and Stats By Other Means' ( which includes tools for students with disability. Help with finding mathematical sites on the Internet is available via the online tutorial 'The Internet Mathematician' (
Some of these sites use special browsers or plug-ins which have to be installed as part of Netscape or Internet Explorer. If you want to take advantage of such facilities on machines in your own institution, you (or a network manager) may have to download and install the correct version for your machine or network.
Many other sites can be searched for within the Network's resource database at or the 'Internet Guide to Mathematics' at
Given the dynamic nature of the Web, URLs and site content are liable to change but the information here was correct at the time of publication
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These video lectures of Professor Gilbert Strang teaching 18.06 were recorded in Fall 1999 and do not correspond precisely to the current edition of the textbook. However, this book is still the best reference for more information on the topics covered in each lecture.
Related Resources
Okay. This is lecture five in linear algebra. And, it will complete this chapter of the book. So the last section of this chapter is two point seven that talks about permutations, which finished the previous lecture, and transposes, which also came in the previous lecture.
There's a little more to do with those guys, permutations and transposes. But then the heart of the lecture will be the beginning of what you could say is the beginning of linear algebra, the beginning of real linear algebra which is seeing a bigger picture with vector spaces -- not just vectors, but spaces of vectors and sub-spaces of those spaces. So we're a little ahead of the syllabus, which is good, because we're coming to the place where, there's a lot to do.
Okay. So, to begin with permutations.
Can I just -- so these permutations, those are matrices P and they execute row exchanges.
And we may need them. We may have a perfectly good matrix, a perfect matrix A that's invertible that we can solve A x=b, but to do it -- I've got to allow myself that extra freedom that if a zero shows up in the pivot position I move it away. I get a non-zero.
I get a proper pivot there by exchanging from a row below.
And you've seen that already, and I just want to collect the ideas together. And principle, I could even have to do that two times, or more times.
So I have to allow -- to complete the -- the theory, the possibility that I take my matrix A, I start elimination, I find out that I need row exchanges and I do it and continue and I finish. Okay.
Then all I want to do is say -- and I won't make a big project out of this -- what happens to A equal L U? So A equal L U -- this was a matrix L with ones on the diagonal and zeroes above and multipliers below, and this U we know, with zeroes down here.
That's only possible. That description of elimination assumes that we don't have a P, that we don't have any row exchanges. And now I just want to say, okay, how do I account for row exchanges? Because that doesn't. The P in this factorization is the identity matrix. The rows were in a good order, we left them there. Maybe I'll just add a little moment of reality, too, about how Matlab actually does elimination. Matlab not only checks whether that pivot is not zero, as every human would do.
It checks for is that pivot big enough, because it doesn't like very, very small pivots. Pivots close to zero are numerically bad. So actually if we ask Matlab to solve a system, it will do some elimination some row exchanges, which we don't think are necessary. Algebra doesn't say they're necessary, but accuracy -- numerical accuracy says they are. Well, we're doing algebra, so here we will say, well, what do row exchanges do, but we won't do them unless we have to.
But we may have to. And then, the result is -- it's hiding here. It's the main fact.
This is the description of elimination with row exchanges.
So A equal L U becomes P A equal L U.
So this P is the matrix that does the row exchanges, and actually it does them -- it gets the rows into the right order, into the good order where pivots will not -- where zeroes won't appear in the pivot position, where L and U will come out right as up here.
So, that's the point. Actually, I don't want to labor that point, that a permutation matrix -- and you remember what those were. I'll remind you from last time of what the main points about permutation matrices were -- and then just leave this factorization as the general case. This is -- any invertible A we get this. For almost every one, we don't need a P. But there's that handful that do need row exchanges, and if we do need them, there they are. Okay, finally, just to remember what P was. So permutations, P is the identity matrix with reordered rows.
I include in reordering the possibility that you just leave them the same. So the identity matrix is -- okay. That's, like, your basic permutation matrix -- your do-nothing permutation matrix is the identity. And then there are the ones that exchange two rows and then the ones that exchange three rows and then then ones that exchange four -- well, it gets a little -- it gets more interesting algebraically if you've got four rows, you might exchange them all in one big cycle. One to two, two to three, three to four, four to one.
Or you might have -- exchange one and two and three and four. Lots of possibilities there. In fact, how many possibilities? The answer was (n)factorial. This is n(n-1)(n-2)... (3)(2)(1).
That's the number of -- this counts the reorderings, the possible reorderings. So it counts all the n by n permutations. And all those matrices have these -- have this nice property that they're all invertible, because we can bring those rows back into the normal order. And the matrix that does that is just P -- is just the same as the transpose.
You might take a permutation matrix, multiply by its transpose and you will see how -- that the ones hit the ones and give the ones in the identity matrix. So this is a -- we'll be highly interested in matrices that have nice properties.
And one property that -- maybe I could rewrite that as P transpose P is the identity. That tells me in other words that this is the inverse of that.
Okay. We'll be interested in matrices that have P transpose P equal the identity.
There are more of them than just permutations, but my point right now is that permutations are like a little group in the middle -- in the center of these special matrices. Okay.
So now we know how many there are.
Twenty four in the case of -- there are twenty four four by four permutations, there are five factorial which is a hundred and twenty, five times twenty four would bump us up to a hundred and twenty -- so listing all the five by five permutations would be not so much fun. Okay.
So that's permutations. Now also in section two seven is some discussion of transposes.
And can I just complete that discussion.
First of all, I haven't even transposed a matrix on the board here, have I? So I'd better do it. So suppose I take a matrix like (1 2 4; 3 3 1). It's a rectangular matrix, three by two. And I want to transpose it.
So what's -- I'll use a T, also Matlab would use a prime.
And the result will be -- I'll right it here, because this was three rows and two columns, this was a three by two matrix. The transpose will be two rows and three columns, two by three.
So it's short and wider. And, of course, that row -- that column becomes a row -- that column becomes the other row.
And at the same time, that row became a column.
This row became a column. Oh, what's the general formula for the transpose? So the transpose -- you see it in numbers. What I'm going to write is the same thing in symbols. The numbers are the clearest, of course. But in symbols, if I take A transpose and I ask what number is in row I and column J of A transpose? Well, it came out of A.
It came out A by this flip across the main diagonal.
And, actually, it was the number in A which was in row J, column I.
So the row and column -- the row and column numbers just get reversed. The row number becomes the column number, the column number becomes the row number. No problem.
Okay. Now, a special -- the best matrices, we could say. In a lot of applications, symmetric matrices show up. So can I just call attention to symmetric matrices? What does that mean? What does that word symmetric mean? It means that this transposing doesn't change the matrix.
A transpose equals A. And an example.
So, let's take a matrix that's symmetric, so whatever is sitting on the diagonal -- but now what's above the diagonal, like a one, had better be there, a seven had better be here, a nine had better be there.
There's a symmetric matrix. I happened to use all positive numbers as its entries. That's not the point.
The point is that if I transpose that matrix, I get it back again. So symmetric matrices have this property A transpose equals A. I guess at this point -- I'm just asking you to notice this family of matrices that are unchanged by transposing. And they're easy to identify, of course. You know, it's not maybe so easy before we had a case where the transpose gave the inverse.
That's highly important, but not so simple to see.
This is the case where the transpose gives the same matrix back again. That's totally simple to see.
Okay. Could actually -- maybe I could even say when would we get such a matrix? For example, this -- that matrix is absolutely far from symmetric, right? The transpose isn't even the same shape -- because it's rectangular, it turns the -- lies down on its side. But let me tell you a way to get a symmetric matrix out of this.
Multiply those together. If I multiply this rectangular, shall I call it R for rectangular? So let that be R for rectangular matrix and let that be R transpose, which it is.
Then I think that if I multiply those together, I get a symmetric matrix. Can I just do it with the numbers and then ask you why, how did I know it would be symmetric? So my point is that R transpose R is always symmetric. Okay? And I'm going to do it for that particular R transpose R which was -- let's see, the column was one two four three three one. I called that one R transpose, didn't I, and I called this guy one two four three three one.
I called that R. Shall we just do that multiplication? Okay, so up here I'm getting a ten. Next to it I'm getting two, a nine, I'm getting an eleven. Next to that I'm getting four and three, a seven. Now what do I get there? This eleven came from one three times two three, right? Row one, column two.
What goes here? Row two, column one.
But no difference. One three two three or two three one three, same thing.
It's going to be an eleven. That's the symmetry.
I can continue to fill it out. What -- oh, let's get that seven. That seven will show up down here, too, and then four more numbers.
That seven will show up here because one three times four one gave the seven, but also four one times one three will give that seven. Do you see that it works? Actually, do you want to see it work also in matrix language? I mean, that's quite convincing, right? That seven is no accident. The eleven is no accident.
But just tell me how do I know if I transpose this guy -- How do I know it's symmetric? Well, I'm going to transpose it. And when I transpose it, I'm hoping I get the matrix back again.
So can I transpose R transpose R? So just -- so, why? Well, my suggestion is take the transpose.
That's the only way to show it's symmetric.
Take the transpose and see that it didn't change.
Okay, so I take the transpose of R transpose R.
Okay. How do I do that? This is our little practice on the rules for transposes.
So the rule for transposes is the order gets reversed.
Just like inverses, which we did prove, same rule for transposes and -- which we'll now use.
So the order gets reversed. It's the transpose of that that comes first, and the transpose of this that comes -- no.
Is that -- yeah. That's what I have to write, right? This is a product of two matrices and I want its transpose.
So I put the matrices in the opposite order and I transpose them. But what have I got here? What is R transpose transpose? Well, don't all speak at once.
R transpose transpose, I flipped over the diagonal, I flipped over the diagonal again, so I've got R.
And that's just my point, that if I started with this matrix, I transposed it, I got it back again. So that's the check, without using numbers, but with -- it checked in two lines that I always get symmetric matrices this way.
And actually, that's where they come from in so many practical applications. Okay.
So now I've said something today about permutations and about transposes and about symmetry and I'm ready for chapter three. Can we take a breath -- the tape won't take a breath, but the lecturer will, because to tell you about vector spaces is -- we really have to start now and think, okay, listen up.
What are vector spaces? And what are sub-spaces? Okay. So, the point is, The main operations that we do -- what do we do with vectors? We add them. We know how to add two vectors.
We multiply them by numbers, usually called scalers.
If we have a vector, we know what three V is.
If we have a vector V and W, we know what V plus W is.
Those are the two operations that we've got to be able to do.
To legitimately talk about a space of vectors, the requirement is that we should be able to add the things and multiply by numbers and that there should be some decent rules satisfied. Okay.
So let me start with examples. So I'm talking now about vector spaces. And I'm going to start with examples. Let me say again what this word space is meaning. When I say that word space, that means to me that I've got a bunch of vectors, a space of vectors. But not just any bunch of vectors. It has to be a space of vectors -- has to allow me to do the operations that vectors are for.
I have to be able to add vectors and multiply by numbers.
I have to be able to take linear combinations.
Well, where did we meet linear combinations? We met them back in, say in R^2.
So there's a vector space. What's that vector space? So R two is telling me I'm talking about real numbers and I'm talking about two real numbers.
So this is all two dimensional vectors -- real, such as -- well, I'm not going to be able to list them all. But let me put a few down.
|3; 2|, |0;0|, |pi; e|.
So on. And it's natural -- okay.
Let's see, I guess I should do algebra first. Algebra means what can I do to these vectors? I can add them. I can add that to that.
And how do I do it? A component at a time, of course. Three two added to zero zero gives me, three two. Sorry about that.
Three two added to pi e gives me three plus pi, two plus e. Oh, you know what it does.
And you know the picture that goes with it.
There's the vector three two. And often, the picture has an arrow. The vector zero zero, which is a highly important vector -- it's got, like, the most important here -- is there.
And of course there's not much of an arrow. Pi -- I'll have to remember -- pi is about three and a little more, e is about two and a little more.
So maybe there's pi e. I never drew pi e before.
It's just natural to -- this is the first component on the horizontal and this is the second component, going up the vertical. Okay.
And the whole plane is R two. So R two is, we could say, the plane.
The xy plane. That's what everybody thinks.
But the point is it's a vector space because all those vectors are in there. If I removed one of them -- Suppose I removed zero zero. Suppose I tried to take the -- considered the X Y plane with a puncture, with a point removed.
Like the origin. That would be, like, awful to take the origin away.
Why is that? Why do I need the origin there? Because I have to be allowed -- if I had these other vectors, I have to be allowed to multiply three two -- this was three two -- by anything, by any scaler, including zero. I've got to be allowed to multiply by zero and the result's got to be there.
I can't do without that point. And I have to be able to add three two to the opposite guy, minus three minus two.
And if I add those I'm back to the origin again.
No way I can do without the origin.
Every vector space has got that zero vector in it.
Okay, that's an easy vector space, because we have a natural picture of it. Okay.
Similarly easy is R^3. This would be all -- let me go up a little here. This would be -- R three would be all three dimensional vectors -- or shall I say vectors with three real components.
Okay. Let me just to be sure we're together, let me take the vector three two zero.
Is that a vector in R^2 or R^3? Definitely it's in R^3.
It's got three components. One of them happens to be zero, but that's a perfectly okay number.
So that's a vector in R^3. We don't want to mix up the -- I mean, keep these vectors straight and keep R^n straight. So what's R^n? R^n.
So this is our big example, is all vectors with n components. And I'm making these darn things column vectors. Can I try to follow that convention, that they'll be column vectors, and their components should be real numbers.
Later we'll need complex numbers and complex vectors, but much later. Okay.
So that's a vector space. Now, let's see. What do I have to tell you about vector spaces? I said the most important thing, which is that we can add any two of these and we -- still in R^2.
We can multiply by any number and we're still in R^2.
We can take any combination and we're still in R^2.
And same goes for R^n. It's -- honesty requires me to mention that these operations of adding and multiplying have to obey a few rules. Like, we can't just arbitrarily say, okay, the sum of three two and pi e is zero zero.
It's not. The sum of three two and minus three two is zero zero. So -- oh, I'm not going to -- the book, actually, lists the eight rules that the addition and multiplication have to satisfy, but they do.
They certainly satisfy it in R^n and usually it's not those eight rules that are in doubt. What's -- the question is, can we do those additions and do we stay in the space? Let me show you a case where you can't.
So suppose this is going to be not a vector space.
Suppose I take the xy plane -- so there's R^2.
That is a vector space. Now suppose I just take part of it. Just this.
Just this one -- this is one quarter of the vector space.
All the vectors with positive or at least not negative components. Can I add those safely? Yes. If I add a vector with, like, two -- three two to another vector like five six, I'm still up in this quarter, no problem with adding.
But there's a heck of a problem with multiplying by scalers, because there's a lot of scalers that will take me out of this quarter plane, like negative ones.
If I took three two and I multiplied by minus five, I'm way down here. So that's not a vector space, because it's not -- closed is the right word. It's not closed under multiplication by all real numbers. So a vector space has to be closed under multiplication and addition of vectors.
In other words, linear combinations.
It -- so, it means that if I give you a few vectors -- yeah look, here's an important -- here -- now we're getting to some really important vector spaces.
Well, R^n -- like, they are the most important.
But we will be interested in so- in vector spaces that are inside R^n. Vector spaces that follow the rules, but they -- we don't need all of -- see, there we started with R^2 here, and took part of it and messed it up. What we got was not a vector space. Now tell me a vector space that is part of R^2 and is still safely -- we can multiply, we can add and we stay in this smaller vector space. So it's going to be called a subspace. So I'm going to change this bad example to a good one. Okay.
So I'm going to start again with R^2, but I'm going to take an example -- it is a vector space, so it'll be a vector space inside R^2. And we'll call that a subspace of R^2.
Okay. What can I do? It's got something in it. Suppose it's got this vector in it. Okay.
If that vector's in my little subspace and it's a true subspace, then there's got to be some more in it, right? I have to be able to multiply that by two, and that double vector has to be included.
Have to be able to multiply by zero, that vector, or by half, or by three quarters.
All these vectors. Or by minus a half, or by minus one. I have to be able to multiply by any number. So that is going to say that I have to have that whole line. Do you see that? Once I get a vector in there -- I've got the whole line of all multiples of that vector. I can't have a vector space without extending to get those multiples in there.
Now I still have to check addition.
But that comes out okay. This line is going to work, because I could add something on the line to something else on the line and I'm still on the line.
So, example. So this is all examples of a subspace -- our example is a line in R^2 actually -- not just any line. If I took this line, would that -- so all the vectors on that line.
So that vector and that vector and this vector and this vector -- in lighter type, I'm drawing something that doesn't work. It's not a subspace.
The line in R^2 -- to be a subspace, the line in R^2 must go through the zero vector. Because -- why is this line no good? Let me do a dashed line.
Because if I multiplied that vector on the dashed line by zero, then I'm down here, I'm not on the dashed line.
Z- zero's got to be. Every subspace has got to contain zero -- because I must be allowed to multiply by zero and that will always give me the zero vector.
Okay. Now, I was going to make -- create some subspaces. Oh, while I'm in R^2, why don't we think of all the possibilities.
R two, there can't be that many.
So what are the possible subspaces of R^2? Let me list them. So I'm listing now the subspaces of R^2. And one possibility that we always allow is all of R two, the whole thing, the whole space. That counts as a subspace of itself. You always want to allow that.
Then the others are lines -- any line, meaning infinitely far in both directions through the zero.
So that's like the whole space -- that's like whole two D space. This is like one dimension.
Is this line the same as R^1 ? No. You could say it looks a lot like R^1. R^1 was just a line and this is a line.
But this is a line inside R^2. The vectors here have two components. So that's not the same as R^1, because there the vectors only have one component.
Very close, you could say, but not the same.
Okay. And now there's a third possibility. There's a third subspace that's -- of R^2 that's not the whole thing, and it's not a line.
It's even less. It's just the zero vector alone. The zero vector alone, only. I'll often call this subspace Z, just for zero. Here's a line, L. Here's a plane, all of R^2. So, do you see that the zero vector's okay? You would just -- to understand subspaces, we have to know the rules -- and knowing the rules means that we have to see that yes, the zero vector by itself, just this guy alone satisfies the rules. Why's that? Oh, it's too dumb to tell you. If I took that and added it to itself, I'm still there. If I took that and multiplied by seventeen, I'm still there.
So I've done the operations, adding and multiplying by numbers, that are required, and I didn't go outside this one point space. So that's always -- that's the littlest subspace. And the largest subspace is the whole thing and in-between come all -- whatever's in between.
Okay. So for example, what's in between for R^3? So if I'm in ordinary three dimensions, the subspace is R, all of R^3 at one extreme, the zero vector at the bottom. And then a plane, a plane through the origin. Or a line, a line through the origin. So with R^3, the subspaces were R^3, plane through the origin, line through the origin and a zero vector by itself, zero zero zero, just that single vector.
Okay, you've got the idea. But, now comes -- the reality is -- what are these -- where do these subspaces come -- how do they come out of matrices? And I want to take this matrix -- oh, let me take that matrix. So I want to create some subspaces out of that matrix. Well, one subspace is from the columns. Okay.
So this is the important subspace, the first important subspace that comes from that matrix -- I'm going to -- let me call it A again. Back to -- okay.
I'm looking at the columns of A.
Those are vectors in R^3. So the columns are in R^3.
The columns are in R^3. So I want those columns to be in my subspace. Now I can't just put two columns in my subspace and call it a subspace.
What do I have to throw in -- if I'm going to put those two columns in, what else has got to be there to have a subspace? I must be able to add those things.
So the sum of those columns -- so these columns are in R^3, and I have to be able -- I'm, you know, I want that to be in my subspace, I want that to be in my subspace, but therefore I have to be able to multiply them by anything.
Zero zero zero has got to be in my subspace.
I have to be able to add them so that four five five is in the subspace. I've got to be able to add one of these plus three of these. That'll give me some other vector. I have to be able to take all the linear combinations. So these are columns in R^3 and all there linear combinations form a subspace. What do I mean by linear combinations? I mean multiply that by something, multiply that by something and add. The two operations of linear algebra, multiplying by numbers and adding vectors.
And, if I include all the results, then I'm guaranteed to have a subspace. I've done the job.
And we'll give it a name -- the column space.
Column space. And maybe I'll call it C of A.
C for column space. There's an idea there that -- Like, the central idea for today's lecture is -- got a few vectors. Not satisfied with a few vectors, we want a space of vectors. The vectors, they're in -- these vectors in -- are in R^3 , so our space of vectors will be vectors in R^3. The key idea's -- we have to be able to take their combinations.
So tell me, geometrically, if I drew all these things -- like if I drew one two four, that would be somewhere maybe there. If I drew three three one, who knows, might be -- I don't know, I'll say there.
There's column one, there's column two.
What else -- what's in the whole column space? How do I draw the whole column space now? I take all combinations of those two vectors.
Do I get -- well, I guess I actually listed the possibilities. Do I get the whole space? Do I get a plane? I get more than a line, that's for sure. And I certainly get more than the zero vector, but I do get the zero vector included. What do I get if I combine -- take all the combinations of two vectors in R^3 ? So I've got all this stuff on -- that whole line gets filled out, that whole line gets filled out, but all in-between gets filled out -- between the two lines because I -- I allowed to add something from one line, something from the other. You see what's coming? I'm getting a plane. That's my -- and it's through the origin. Those two vectors, namely one two four and three three one, when I take all their combinations, I fill out a whole plane. Please think about that. That's the picture you have to see. You sure have to see it in R^3 , because we're going to do it in R^10, and we may take a combination of five vectors in R^10, and what will we have? God knows. It's some subspace.
We'll have five vectors. They'll all have ten components. We take their combinations. We don't have R^5 , because our vectors have ten components. And we possibly have, like, some five dimensional flat thing going through the origin for sure. Well, of course, if those five vectors were all on the line, then we would only get that line. So, you see, there are, like, other possibilities here.
It depends what -- it depends on those five vectors. Just like if our two columns had been on the same line, then the column space would have been only a line. Here it was a plane.
Okay. I'm going to stop at that point. That's the central idea of -- the great example of how to create a subspace from a matrix.
Take its columns, take their combinations, all their linear combinations and you get the column space.
And that's the central sort of -- we're looking at linear algebra at a higher level. When I look at A -- now, I want to look at Ax=b. That'll be the first thing in the next lecture. How do I understand Ax=b in this language -- in this new language of vector spaces and column spaces. And what are other subspaces? So the column space is a big one, there are others to come
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Synopses & Reviews
Publisher Comments:
This new title in Barron's E-Z Series contains everything students need to prepare themselves for an algebra class. Separate chapters focus on fractions, integers, ratios, proportions, expressions, equations, inequalities, graphing, statistics and probability basics, word problems, and more. Review questions and chapter reviews all have answers. The fast-growing E-Z Series presents new, updated, and improved versions of Barron's longtime popular Easy Way books. New cover designs, new interior layouts, and more graphic material than ever make these books ideal as self-teaching manuals. Teachers have discovered that E-Z titles also make excellent supplements to classroom textbooks. Skill levels range between senior high school and college-101 standards. All titles present detailed reviews of the target subject plus short quizzes and longer tests to help students assess their learning progress. All exercises and tests come with answers.
Synopsis:
(back cover)
Pre-Algebra makes sense when you approach it the E-Z way! Open this book for a clear, concise, step-by-step review of:
Fractions
Integers
Ratios
Proportions
Equations
Inequalities
Statistics and Probability Basics . . . and more
Measure your progress as you learn Pre-Algebra:
Take time to review and understand each important concept
Solve each chapter's problems and check your answers And discover that learning Pre-Algebra can be E-Z
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Many mathematical statistics texts are heavily oriented toward a rigorous mathematical development of probability and statistics, without much attention paid to how statistics is actually used.. In contrast, Modern Mathematical Statistics with Applications, Second Edition strikes a balance between mathematical foundations and statistical practice.
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2013-2014 Mathematics Courses
Mathematics has been an undeniably effective tool in humanity's ongoing effort to understand the nature of the world around us, yet the mantra of high-school students is all too familiar: What is math good for anyway?When am I ever going to use this stuff? What serves to explain the puzzling incongruity between the indisputable success story of mathematics and students' sense of the subject's worthlessness? Part of the explanation resides in the observation that all too many mathematics courses are taught in a manner that entirely removes the subject matter from its proper historical, social, and cultural context—naturally leaving students with the distinct impression that mathematics is a dead subject, one utterly devoid of meaningfulness and beauty. In reality, mathematics is one of the oldest intellectual pursuits, its history a fascinating story filled with great drama, extraordinary individuals, and astounding achievements. This seminar focuses on the role played by mathematics in the emergence of civilization and follows their joint evolution over nearly 5,000 years to the 21st century. We will explore some of the great achievements of mathematics and examine the full story behind those glorious achievements. The ever-evolving role of mathematics in society and the ever-intertwined threads of mathematics, philosophy, religion, and culture provide the leitmotif of the course. Specific topics to be explored include the early history of mathematics, logic and the notion of proof, the production and consumption of data, the analysis of conflict and strategy, and the concept of infinity. Readings will be drawn from a wide variety of sources (textbooks, essays, articles, plays, and fictional writings), connecting us to the thoughts and philosophies of a diverse set of scholars; a partial list includes Pythagoras, Euclid, Galileo, René Descartes, Isaac Newton, Immanuel Kant, Lewis Carroll, John Von Neumann, John Nash, Kurt Gödel, Bertrand Russell, Jorge Luis Borges, Kenneth Arrow, and Tom Stoppard.
An Introduction to Statistical Methods and Analysis
An introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental ideas of statistical analysis used to gain insight into diverse areas of human interest. The use, abuse, and misuse of statistics will be the central focus of the course. Topics of exploration will include the core statistical topics in the areas of experimental study design, sampling theory, data analysis, and statistical inference. Applications will be drawn from current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical software will be introduced and used extensively in this course, but no prior experience with the software is assumed. This seminar is an invaluable course for anybody planning to pursue graduate work and/or research in the natural sciences or social sciences. No college-level mathematical knowledge is required.
Calculus II: Modeling With Differential Equations
An infectious disease spreads through a community: What is the most effective action to stop an epidemic? Populations of fish swell and decline periodically: Should we change the level of fishing allowed this year to have a better fish population next year? Foxes snack on rabbits: In the long term, will we end up with too many foxes or too many rabbits? Calculus can help us answer these questions. We can make a mathematical model of each situation, composed of equations involving derivatives (called differential equations). These models can tell us what happens to a system over time which, in turn, gives us predictive power. Additionally, we can alter models to reflect different scenarios (e.g., instituting a quarantine, changing hunting quotas) and then see how these scenarios play out. The topics of study in Calculus II include power series, integration, and numerical approximation, all of which can be applied to solve differential equations. Our work will be done both by hand and by computer. Conveniently, learning the basics of constructing and solving differential equations (our first topic of the semester) includes a review of Calculus I concepts Prerequisite: Calculus I (differential calculus in either a high-school or college setting).
Calculus I: The Study of Motion and Change
Calculus is the study of rates of change of functions (the derivative), accumulated areas under curves (the integral), and how these two ideas are (surprisingly!) related. The concepts and techniques involved apply to medicine, economics, engineering, physics, chemistry, biology, ecology, geology, and many other fields. Such applications appear throughout the course, but we will focus on understanding concepts deeply and approach functions from graphical, numeric, symbolic, and descriptive points of view Facility with high-school algebra and basic geometry are prerequisites for this course. Prior exposure to trigonometry and/or precalculus is highly recommended. No previous calculus experience is necessary or desired.
Discrete Mathematics: A Gateway to Advanced Mathematics
This seminar is an introduction to the world of elegant mathematics, beyond that encountered in high school, under the guise of an introductory survey course in discrete mathematics. We will touch on the tips of many icebergs! The subject of discrete mathematics houses the intersection of mathematics and computer science; it is an active area of research that includes combinatorics, graph theory, geometry, and optimization. The topics in this course are selected to give an idea of the types of thinking used in a variety of discrete mathematics research areas. Learning the facts and techniques of discrete mathematics is inextricably intertwined with reasoning and communicating about discrete mathematics. Thus, at the same time as surveying discrete mathematics, this course is an introduction to rigorous reasoning and to writing convincing arguments. These skills are necessary in all of mathematics and computer science and very applicable to law and philosophy. Conference work will explore additional mathematical topics. The seminar is essential for students planning advanced study in mathematics and highly recommended for students studying computer science, law, or philosophy or who seek to enhance their logical thinking and problem-solving skills. Prerequisite: Prior study of Calculus or equivalent preparation.
Topics in Multivariable and Vector Calculus
Compared to the familiar single-variable territory of Calculus I and II, multivariable calculus is a foreign land. Imagine, if you will, that instead of a function taking a single input and producing a single output, we either use one input and get multiple outputs (vector functions) or use several inputs and get one output (multivariable functions). And yes, there are even functions that have several inputs and multiple outputs! In this new realm, we will investigate lines and planes, curves and surfaces, and multidimensional generalizations of these objects, with a focus on those functions that can be visualized in three dimensions. For both vector and multivariable functions, we will address the basic questions of calculus: How do we measure rates of change? How do we find areas and volumes? How can we interpret derivatives and integrals both geometrically and for practical purposes? Fascinatingly, each of these questions has more than one answer. We will examine gradients and directional derivatives, maxima and minima and saddle points, double and triple integrals, integrals taken along curves, and more—as time permits. This seminar is essential for students intending to pursue engineering, physics, mathematics, graduate study in economics, or rocket science and is recommended for students pursuing chemistry or computer science. Prerequisites: Calculus I and Calculus II.
Topology: The Nature of Shape and Space
Topology, a modernized version of geometry, is the study of the fundamental, underlying properties of shapes and spaces. In geometry, we ask: How big is it? How long is it? But in topology, we ask: Is it connected? Is it compact? Does it have holes? To a topologist there is no difference between a square and a circle and no difference between a coffee cup and a donut because, in each case, one can be transformed smoothly into the other without breaking or tearing the mathematical essence of the object. This course will serve as an introduction to this fascinating and important branch of mathematics. Conference work will be allocated to clarifying course ideas and exploring additional mathematical topics. Successful completion of a yearlong study of Calculus is a prerequisite and completion of an intermediate-level course (e.g., Discrete Mathematics, Linear Algebra, Multivariable Calculus, or Number Theory) is strongly recommended.
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Develop Algebraic Thinking 3-5 - MAT-925Use manipulatives, real-world problem-solving, and captivating activities to engage your students in developmentally appropriate algebraic thinking. This ONLINE course will provide you with numerous opportunities to reflect on current research and pedagogy related to algebraic thinking in the intermediate grades. Make direct application to your own classroom through the design and development of lessons that explore growth patterns, tables, variables, and coordinate graphs. All of the readings and activities are built upon the Common Core standards. Teachers may complete this course with or without students
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2014-04-16T13:10:28ZNumerical solution of Hodgkin-Huxley's partial differential system for nerve conduction
Numerical solution of Hodgkin-Huxley's partial differential system for nerve conduction
Morton, John Baird
A numerical solution to Hodgkin and Huxley's partial differential
system for the propagated action potential is presented. In
addition a three dimensional demonstration of the absolute refractory
period is given. Lastly, theoretical evidence supporting
Rushton's hypothesis is presented.
Graduation date: 1967
1967-05-04T00:00:00ZA FORTRAN to ALGOL translator
A FORTRAN to ALGOL translator
Hill, Edward Burlingame
FORTRAN is readily feasible to translation into ALGOL since
they share many common features. Most of the features that are
unique to FORTRAN can be translated by restricting them somewhat.
The translator will handle explicit declarations of each item
in a block, compensate for the differences in various operators,
compensate for the different storage techniques and provide a
simple input /output scheme.
Running test cases through the FORTRAN programs and
through the translated ALGOL programs indicated that the ALGOL
programs take longer to execute than their FORTRAN counterparts.
Graduation date: 1969
1968-10-16T00:00:00ZThe extension problem for functions invariant under a group
The extension problem for functions invariant under a group
Chang, Bai-Ching for extending f invariantly to
the whole space, and thus generalizing the classical Tietze extension
theorem.
Graduation date: 1967
1967-05-05T00:00:00ZA study of unique factorization in quadratic integral domains
A study of unique factorization in quadratic integral domains
Van Enkevort, Ronald Lee
This thesis studies the question of unique factorization in
quadratic integral domains. In the first chapter many general
theorems and definitions from algebraic number theory are introduced.
The second chapter considers an integral domain in which
unique factorization holds. The necessary theorems to prove
unique factorization are developed. The third chapter concerns an
integral domain in which unique factorization fails. That it fails
is proved and then ideals are introduced to indicate how unique
factorization would be restored in terms of ideals.
Graduation date: 1967
1966-07-18T00:00:00Z
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5th edition of Model Building in Mathematical Programming discusses the general principles of model building in mathematical programming and demonstrates how they can be applied by using several simplified but practical problems from widely different contexts. Suggested formulations and solutions are given together with some computational experience to give the reader a feel for the computational difficulty of solving that particular type of model. Furthermore, this book illustrates the scope and limitations of mathematical programming, and shows how it can be applied to real situations. By emphasizing the importance of the building and interpreting of models rather than the solution process, the author attempts to fill a gap left by the many works which concentrate on the algorithmic side of the subject.
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Starting at $54Solidly grounded in up-to-date research, theory and technology, Teaching Secondary Mathematicsis a practical, student-friendly, and popular text for secondary mathematics methods courses. It provides clear and useful approaches for mathematics teachers, and shows how concepts typically found in a secondary mathematics curriculum can be taught in a positive and encouraging way. The thoroughly revised fourth edition combines this pragmatic approach with truly innovative and integrated technology content throughout. Synthesized content between the book and comprehensive companion website offers expanded discussion of chapter topics, additional examples and technological tips. An online graphic calculator invites students to interact with the math content of the book, while video clips provide chapter by chapter illustrations of classroom instructional strategies. Each chapter features tried and tested pedagogical techniques, problem solving challenges, discussion points, activities, mathematical challenges, and student-life based applications that will encourage students to think and do. New in this edition: A companion website replete with chapter-by-chapter video lessons, teacher tools, problem solving Q&As, helpful links and resources, and embedded graphing calculators. Updated icons, including NCTM core standards and math history, that connect to key strands that readers will find indispensible as they build their professional knowledge and skills. Problem solving challenges and sticky questions featured in each chapter encourage students to think through everyday issues and possible solutions. Fresh interior design to better highlight pedagogical elements and key features, all to better engage students.
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Beginning and Intermediate Algebra: The Language & Symbolism ofIntended for schools that want a single text covering the standard topics from Beginning and Intermediate Algebra. Topics are organized by using the principles of the AMATYC standards as a guide, giving strong support to teachers using the text. The book's organization and pedagogy are designed to work for students with a variety of learning styles and for teachers with varied experiences and backgrounds. The inclusion of multiple perspectives -- verbal, numerical, algebraic, and graphical -- has proven popular with a broad cross section of students.
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Precalculus: Finish vectors. Law of sines and law of cosines quiz due. Vector test and project next week. Register at for on line quizzes. Algebra II: Lessons 6.4 and 6.5. Bellwork quiz on 6.4 is Thursday.
Bonus next test. All Classes. Write your answer next to your name on your test paper.
____________________ lived around the year 250 BC and among other things, developed a method for determining the volume of objects with irregular shapes
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BASIC MATH HANDBOOK ... The first part contains a formula sheet of commonly referred to formulas in the water industry. ... By analyzing units (called dimensional analysis) in a formula or mathematical calculation you can determine if the problem is set up correctly.
BasicFormulas and Mathematical Results Used in the Analysis 219 Let us call r∗ the expected rate of return on contributions that an individual who enters the system at time 1 and retires at time R would
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Entering a basic formula A formula is a worksheet entry that performs a calculation. You can use the following mathematical operators in formulas. Operator Description + Addition - Subtraction * Multiplication / Division You ...
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Basic Water and Wastewater Formulas Summary Operators obtaining or maintaining their certification must be able to calculate complex formulas and conversion factors. This Tech Briefprovides basic examples of these formulas and conversion factors.
Australian Journal of Basic and Applied Sciences, 5 ... mathematical symbols and mathematicalformulas. The MDS can help distance learners understand mathematical knowledge, with the freedom and flexibility offered by the online environment.
SAY, "Today we will discuss basic spreadsheet formulas using mathematical operators with which you're already familiar (such as addition, subtraction, division , and multiplication). I will show you how you could use each formula to approachThis proficiency is basic to safely administering medications and intravenous fluids. Enclosed is a booklet to guide you in mastering the mathematical competencies ... There are multiple mathematicalformulas that may be used to calculate dosages
you may need to brush up on some basicmathematical concepts to appreciate fully the understanding my texts are trying to convey. ... Because statistical formulas frequently contain fractions, and changing the form of the fraction can make the formula look very different, it is
Simple Addition Formula in an Excel 2007 Document Formulas allow calculations to be performed on data and entered into the spreadsheet. Excel 2007 offers many formulas for basic number crunching, such as addition or subtraction,
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Elementary Technical Mathematics - 9th edition
ISBN13:978-0495012740 ISBN10: 0495012742 This edition has also been released as: ISBN13: 978-0495113492 ISBN10: 0495113492
Summary: Ewen/Nelson's ELEMENTARY TECHNICAL MATHEMATICS, Ninth Edition is a well-respected, extremely user-friendly text. It emphasizes essential math skills and consistently relates math to practical applications so students can see how learning math will help them on the job. The applications are drawn from a wide array of technical fields, making the text useful to a broad range of students. Annotated examples and visual images are used to engage students and assist with prob...show morelem solving. Comprehensive and well-organized, this text engages students, providing them with a solid foundation in mathematical principles that will help them to succeed in the current course and beyond.
Features
This book features a large number of applications from a wide variety of technical areas noted by marginal icons, including auto-diesel mechanic, industrial and construction trades, electronics, agriculture, allied health, CAD/drafting, HVAC, welding, and aviation. Marginal icons are used to call attention to the applications, making it easier for instructors and students to benefit from them.
A laminated reference card of useful, frequently referenced information--such as metric system prefixes, English weights and measures, metric and English conversion, and formulas from geometry--is provided, perforated for easy removal and use.
Chapter 1 reviews basic concepts in such a way that students or the entire class can easily study only those sections they need to review.
The use of a scientific calculator has been integrated in an easy to use format throughout the text to reflect its nearly universal use in technical classes and on the job.
A comprehensive introduction to basic algebra is presented for those students who need it as a prerequisite to more advanced algebra courses. However, this book is written to allow the omission of selected sections or chapters without the loss of continuity, to meet the needs of specific students.
GROUP PROJECTS entitled ACCENT ON TEAMWORK are included at the end of each chapter.
A chapter review and a chapter test appear at the end of each chapter, as an aid for students in preparing for quizzes and exams. Cumulative reviews are provided at the end of every three chapters to help students review for comprehensive exams
New to this Edition
Coverage of descriptive statistics has been expanded in response to user feedback.
A new Section 2.7 on Engineering Notation makes the text even more practical for your students.
The use of author annotations in the examples has been substantially increased, providing the student with more guidance on the proper use of mathematical operations.
The interior design was enhanced to make key material more accessible and to make the text easier to use as a study tool.
The number of manufacturing applications has been increased, and many of them have been updated.
A new Ch. 16 on binary and hexadecimal systems has been added, for those who need to cover this material in their course.
Excessive wear to corners and fore edge. A readable copy. All pages are intact, and the cover is intact (the dust cover may be missing). Pages can include considerable notes--in pen or highlighter--bu...show moret the notes cannot obscure the text. Your item will be shipped within 2 business days. Thank You ...show less
$2.9011114
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Course Detail
Registration
Mathematics for Educators: Math as a Second Language
MAED 205 Z1 (CRN: 61105)
3 Credit Hours—Seats Available!
Jump Navigation
MAED 205 Z1 is a contract program.
To register for a contract course, you must first have permission of the organization sponsoring the course.
About MAED 205 Z1
This course lays the groundwork for all the Vermont Mathematics Initiative courses that follow. A major theme is understanding algebra and arithmetic through language. The objective is to provide a solid conceptual understanding of the operations of arithmetic, as well as the interrelationships among arithmetic, algebra, and geometry. Topics include arithmetic vs. algebra; solving equations; place value and the history of counting; inverse processes; the geometry of multiplication; the many faces of division; rational vs. irrational numbers and the one-dimensional geometry of numbers. All of the topics in this course are taught in the context of the mathematics curriculum in grades K-8.
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Web Resources
Lesson Plans
Title: Graphs From the Unit Circle
Description:Standard(s): [MA2013] PRE (9-12) 26: Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. (Alabama)
Graphs From the Unit Circle
Assessments
Title: Prerequsites for Calculus Quiz
Description:
Students will take this self-assessment to make sure they are prepared to take Calculus.
Standard(s): [MA2013] PRE (9-12) 13: (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal's Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) [A-APR5] [MA2013] PRE (9-12) 33: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. [F-TF8] (Alabama) [MA2013] PRE (9-12) 3: (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. [N-CN6] [MA2013] PRE (9-12) 14: (+) Represent a system of linear equations as a single matrix equation in a vector variable. [A-REI8] [MA2013] PRE (9-12) 20: Determine the inverse of a function and a relation. (Alabama) [MA2013] PRE (9-12) 22: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. [F-BF4c] [MA2013] PRE (9-12) 24: (+) Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. [F-BF5] [MA2013] PRE (9-12) 25: Compare effects of parameter changes on graphs of transcendental functions. (Alabama) [MA2013] PRE (9-12) 26: Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. (Alabama) [MA2013] PRE (9-12) 27: Use the sum, difference, and half-angle identities to find the exact value of a trigonometric function. (Alabama) [MA2013] PRE (9-12) 34: (+) Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems. [F-TF9] [MA2013] PRE (9-12) 12: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* (Extend to infinite geometric series.) [A-SSE4] (Alabama)
Interactives/Games
Title: Sine Function (with Unit Circle)
Description:
Standard(s): [MA2013] PRE (9-12) 26: Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. (Alabama)
Sine Function (with Unit Circle)
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The authors define general vector spaces and linear mappings at the outset and base all subsequent developments on these ...
More About
This Book
The authors define general vector spaces and linear mappings at the outset and base all subsequent developments on these concepts. This approach provides a ready-made context, motivation, and geometric interpretation for each new computational technique. Proofs and abstract problem-solving are introduced from the start, offering students an immediate opportunity to practice applying what they've learned. Each chapter contains an introduction, summary, and supplementary exercises. The text concludes with a pair of helpful appendixes and solutions to
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Linear Algebra
9780135367971
ISBN:
0135367972
Edition: 2 Pub Date: 1971 Publisher: Prentice Hall
Summary: This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems. Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms For all readers interested in linear algebra. ...> Hoffman, Kenneth is the author of Linear Algebra, published 1971 under ISBN 9780135367971 and 0135367972. Five hundred sixty eight Linear Algebra textbooks are available for sale on ValoreBooks.com, nine used from the cheapest price of $76.42, or buy new starting at $177.94 [more
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Proofs and Fundamentals : A First Course in Abstract Mathematics - 00 edition
Summary: Proofs and Fundamentals: A First Course in Abstract Mathematics is designed as a 'transition' course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra, and real analysis.
This 3-pa...show morert work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material including sets, functions, and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics, and the Peano Postulates. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and writing are never compromised. The material is presented in the way that mathematicians actually use it; good mathematical taste is preferred to overly clever pedagogy. There is a key section devoted to the proper writing of proofs. The text has over 400 exercises, ranging from straightforward examples to very challenging proofs.
The excellent exposition, organization and choice of topics will make this text valuable for classroom use as well as for the general reader who wants to gain a deeper understanding of how modern mathematics is currently practiced by mathematicians. ...show less
2. Strategies for Proofs 2.1 Mathematical Proofs-What They are and Why We Need Them 2.2 Direct Proofs 2.3 Proofs by Contrapositive and Contradiction 2.4 Cases, and If and Only If 2.5 Quantifiers in Theorems 2.6 Writing Mathematics
8. Number Systems 8.1 Back to the Beginning 8.2 The Natural Numbers 8.3 Further Properties of the Natural Numbers 8.4 The Integers 8.5 The Rational Numbers 8.6 The Real Numbers and the Complex Numbers 8.7 Appendix: Proof of Theorem 8.2.1113.90
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This course covers the maths behind how your computer stores and manipulates data. You'll learn how to read binary and hexadecimal, how both integers and floating point numbers are stored and the limitations of using them. Advice on best practices and how to work effectively with boolean values and bitwise operators [Read more...]
Progress in the theory of economic equilibria and in game theory has proceeded hand in hand with that of the mathematical tools used in the field, namely nonlinear analysis and, in particular, convex analysis. Jean-Pierre Aubin, one of the leading specialists in nonlinear analysis and its application to economics, has written a rigorous and concise – yet still elementary and self-contained – textbook providing the mathematical tools needed to study optima and equilibria, as solutions to problems, arising in economics, management sciences, operations research, cooperative and non-cooperative games, fuzzy games etc. It begins with the foundations of optimization theory, and mathematical programming, and in particular convex and nonsmooth analysis. Nonlinear analysis is then presented, first game-theoretically, then in the framework of set valued analysis. These results are then applied to the main classes of economic equilibria. The book contains numerous exercises and problems: the latter allow the reader to venture into areas of nonlinear analysis that lie beyond the scope of the book and of most graduate courses. [Read more...]
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MTHS 644
Foundations of Analysis
Reid,James D.
01/28/2008 - 05/10/2008
Tuesday 06:00 PM - 08:30 PM
Science Tower 109
A classical trichotomy splits mathematics into three parts -- Algebra, Analysis, and Geometry -- and this classical view still has a certain validity. Broadly speaking most people might view Algebra as the art of symbolic manipulations and Geometry as the study of spatial shapes and relationships. That leaves Analysis.
For purposes of discussion, we view Analysis as having to do with the study of infinitesimal processes as codified in the concept of a limit. The gateway to Analysis, the basic ideas, form the content of what is called Calculus. This course will study the foundations of Calculus, the motivation for its existence, and the structure of the resulting theory.
The basic idea is that of "limits." This appears notably in discussing continuity of functions, in the idea of the derivative, and in the definition of the integral. We will make a careful study of this fundamental concept and examine some of its most important applications. Participants in the course should come away with a solid understanding of the fundamental ideas of Calculus, and through many examples, a deep appreciation of its power.
Weekly problem sets will be assigned to help in the assimilation of ideas. Grades will be based on these assignments together with participation in class.
As prerequisites we cite familiarity with the idea of notation of functions and the representation of functional relationships by means of graphs, though these will be reviewed and every effort will be made to make the course accessible to those in attendance. Most important is the willingness to think hard and the patience to let things sink in.
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Certify your skills to increase your chance of being awarded!
to solve second order differential equation - open to bidding
CLOSED
Bids
18the project is about second order differential equation solving the algorithms of earth's magnetic filed and Finite-difference equations will be formulated and solved that describe current flow induced by time-varying electromagnetic fields in the Earth. Students will write their own code in a programming language of their choice to solve the associated second-order differential equation. An introduction to the background science and explanation of the various algorithms will be given in class. In addition, students will be expected to identify and describe a scientific problem of relevance to them for which numerical computations could provide an answer or deeper understanding, and to write code to carry out the appropriate computations
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This book is very clear and easy to use. The author is clearly well informed in the area, and his years in lecturing reflect his ability to communicate effectively. I used this book as a companion to my lectures at uni and was very useful in providing alternative definitions and extra examples. A must have for an undergrad student in maths.
The book is intended for extensive use of undergradurate Maths students. It is recommended by University Professors as the Core texts and introduction of anything that is related to Abstract Algebra courses. Good use of examples, with fairly clearly laid out definitions. !The material used in this book will be useful for any undergrad students throughout the whole three year course! (This is the recommended course text in my Uni, you can't go wrong with it!) Anyway,(enough said!) a MUST buy not only because it covers a good understanding of the course, but by far the most important reason - it is a one of the very few grumpy Maths books that a student can afford, cheap enough to get it without the hassle of getting it out from the library! PS. Put it that way, I am forced to buy it NOT because I am a nerd. I want to pass my exams too, but when the Professor was too intelligent for ordinary humanoids like myself and the rest of my classmates it would be mission impossible. So don't come emailing me with Maths questions unless you are studying at the primary level!!
This book is really useful for the module that I have been doing. Not only describes abstract algebra, but goes into a lot of detail, giving another angle to look at things. There is also alot of interesting information on the players and the history of abstract algebra. All in all very helpful to anyone studying this.
I used this book as a supplement to a course in algebra at master level. Allenby uses a simple step by step approach to the subject supplied with simple examples and some exercises (not that simple). What I realy like with this book is the historical tour of algebra and the brief presentation of many of the worlds famoust mathematicians. I think the simple approach and the historical elements is main reason for still using this book when I am plying with the math.
5 of 5 people found the following review helpful
4.0 out of 5 starsGood for self-study22 Sep 2009
By Brian Jo - Published on Amazon.com exercise up to Chapter 5, I've found about 5-6 typos in the print, which is decent for a math text. Brief solutions for most problems are in the back, which makes this book ideal for self-study.
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This textbook will prepare students for future Algebra courses and also consolidate the core ideas of previous mathematics courses, so that it will serve students of varied mathematics backgrounds. The contents of this course are many of the fundamental ideas and procedures necessary to be successful in subsequent mathematics courses and many careers
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Introductory Algebra - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief.Introductory Algebra, Fourth Editionwas written to provide students with a solid foundation in algebra and to help stuents make the transition to intermediate algebra. The new edition offers new resources like theStudent Organizerand now includesStudent Resourcesin the back of the book to help students on their que...show morest for
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Differential Equations in Action
Making Math Matter
Intermediate
Join 31,611 Students
We're still building the full experience
Approx.
2
months
(work at your own pace)
Class Summary
In this course you will examine real world problems -- rescue the Apollo 13 astronauts, stop the spread of epidemics, and fight forest fires -- involving differential equations and figure out how to solve them using numerical methods.
What Will I Learn?
By the end of this course, you'll develop an intuition for the use of differential equations in the applied sciences. You'll also learn how to build mathematical models for systems of differential equations. Along the way, you'll learn how to translate mathematical expressions into Python code, and solve some really cool problems!
What Should I Know?
You'll need a basic knowledge of programming for this course, around the level of CS 101 or equivalent. You'll also need to understand trigonometry at the high school level, as well as basic vector algebra. This class will primarily involve solving equations numerically rather than analytically, but some exposure to calculus and physics at the level of PH 100 wouldn't hurt.
Syllabus
Lesson 1: Rescuing Apollo 13, Part 1
Introduction to the Forward Euler Method
Lesson 2: Rescuing Apollo 13, Part 2
Comparing solvers, Heun's Method and Symplectic Euler Method
Lesson 3: Analyzing the Spread of Diseases
Implicit methods and stiffness
Lesson 4: There's Plenty of Fish in the Sea?
Stability, sensitivity, and optimization
Lesson 5: Antilock Brake Systems
Friction, equilibria, and control theory
Lesson 6: The Path of a Forest Fire
Partial differential equations and heat conduction
Lesson 7: Advanced Applications of Numerical Methods
Chaos, software, and predictive capabilityJörn Loviscach
Instructor
Jörn Loviscach is a professor of technical mathematics and computer engineering. In a former life, he was a professor of computer graphics, animation, and simulation, worked as editor and senior editor for several computer magazines. For three years he also served as deputy editor-in-chief of c't Magazin für Computertechnik, a renowned German biweekly. His research integrates human-computer interaction, visual computing, and audio and music computing. Both as a researcher and as a practitioner he is interested in applying digital media to education. He has published over 1800 video lectures on his YouTube channel on mathematics and computer science, and his channel is one of the most popular of its kind in Germany.
Miriam Swords Kalk
Course Developer
Miriam feels inspired by the inherent beauty of math and loves helping others develop confidence and excitement as they learn. Having the opportunity to do this with thousands of students all over the world through her work at Udacity is a dream come true. Outside of work, she enjoys singing, baking, teaching yoga, and tutoring high schoolers. Miriam attended Stanford University, where she earned a B.S. in Physics, a B.A. in Philosophy and Religious Studies, and an M.A. in Religious Studies.
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Find an El Mirage Prealgebra solutions are very carefully detailed, and important concepts are particularly emphasized for attention. The student is urged to ask questions in discussing those problems, and, in turn, I ask peripheral questions to ensure good basic comprehension. Reading involves developing a comprehension of what the writer means
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Numerical Mathematics and Computing
9780495114758
ISBN:
0495114758
Pub Date: 2007 Publisher: Thomson Learning
Summary: Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. The text also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors. A more t...heoretical text with a different menu of topics is the authors' highly regarded NUMERICAL ANALYSIS: MATHEMATICS OF SCIENTIFIC COMPUTING, THIRD EDITION.
Cheney, Ward is the author of Numerical Mathematics and Computing, published 2007 under ISBN 9780495114758 and 0495114758. Three hundred four Numerical Mathematics and Computing textbooks are available for sale on ValoreBooks.com, one hundred seventeen used from the cheapest price of $12.21, or buy new starting at $66.88.[read more
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0055720 / ISBN-13: 9780870055720
Mathematics for Retail Buying
The top-selling introductory retailing textbook/workbook, through four highly successful editions, has been revised. This popular classic text meets ...Show synopsisThe top-selling introductory retailing textbook/workbook, through four highly successful editions, has been revised. This popular classic text meets the needs of students who will be directly or indirectly involved in the activities of merchandising and buying at the retail level. Tepper continues to delineate the essential concepts, practices, procedures, calculations, and interpretations of figures that relate to the many factors that produce profit. She also describes the place of computers in retailing, updates MOR figures, and adds case studies. Clear instructions, multiple practice problems, and current terminology and data make this the best retail buying math book available. New to This edition: -- A chapter describing the impact of computer technology on the growth of retailing, the cardinal principles of successful merchandising, the role of computers in retailing, the functions of buying, the circular nature of the merchandising process, and the responsibilities of the buyer -- Provides a "road map" of the mathematical factors in the text -- Numerous work and practice problems test the understanding of fundamental principles -- Three-hole punched text pages allow students to interweave review problem answers with the text statement of the review problems -- Additional case studies utilizing practical retailing situations and difficulties in real-life merchandising -- Selected answers at the back of the book
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This is a free mathematical calculator, which is able to add, subtract, multiply and divide numbers in binary format (used in computers).
Any number can be represented by any sequence of bits (binary digits), they are commonly written using the symbols 0 and 1.
Best math tool for school and college! If you are a student, it will helps you to learn arithmetics for computer science and electrical engineeringMay 19 2013 internet connection issue solved!!!!
IMPORTANT There is no virus on this app, what some programs are detecting as virus is just ads. You can confirm this using Kaspersky antivirus.
BinaryCC is a great simple app that allow you to convert from and to BIN, HEX, DEC, OCT. It's also a simple calculator.
Note: -The internet and network permission are for the Ads -The Ads supports the developers to keep this app free for you.
Features: -Instantly convert to BIN,HEX, DEC and OCT with just one input. -Automatically block the buttons acording to the base you are using (only in touchscreen phones). -Simple calculator (+, -, *, /) for all the basesThis free app is a math calculator, which calculates the values for the most important trigonometric functions, like: Sine, Cosine, Tangent, Arcsine, Arccosine, Arctangent.
Best math tool for school and college! If you are a student, it will helps you to learn geometry!
Note: Trigonometric functions are used for computing unknown lengths and angles in triangles (in navigation, engineering and physics). A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillatorsThis free math app is a statistical calculator with various functions:
- Statistics: You are able to calculate the mean, median, variance, maximum and minimum for a set of numbers.
- Statistical Distributions: You are able to calculate the values of different Statistical Distributions. The following distributions are available: Binomial Distribution, Normal Distribution, Students t-Distribution, F-Distribution, Exponential Distribution, Poisson Distribution, Chi Squared Distribution
- Frequency Table: You are able to create a frequency table for a list of numbers. Just enter the numbers, separated by comma.
Best mathematical tool for school and college! If you are a student, it will helps you to learn statistics and probability theory.
Note: Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events.
This free app is an Electric Circuit Calculator. You are able to calculate: - the current in parallel circuits (both total and partial) - the voltage in series circuits (both total and partial) - the resistance in parallel and series circuits
The best tool for school and college! If you are a student it will helps you to learn electricity and electrical engineering which
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Author(s): LarsonCOLLEGE ALGEBRA: REAL MATHEMATICS, REAL PEOPLE, 6th Edition, is an ideal student and instructor resource for courses that require the use of a graphing calculator. The quality and quantity of the exercises, combined with interesting applications and innovative resources, make teaching easier and help students succeed. Retaining the series' emphasis on student support, selected examples throughout the text include notations directing students to previous sections to review concepts and skills needed to master the material at hand. The book also achieves accessibility through careful writing and design--including examples with detailed solutions that begin and end on the same page, which maximizes readability. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles. Reflecting its new subtitle, this significant revision focuses more than ever on showing students the relevance of mathematics in their lives and future careers.
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There are only three types of solutions: no solution, one solution, or infinitely MANY solutions. Mr. Defining Wizard will explain how to analyze and determine which type of solution you have for your problem (graphically and algebraically). Don't forget to check your answersgebra Geometry Formulae is an ideal free app for all students above 12th grade, college graduates, engineering graduates and students preparing for various exams. We have compiled all the algebra, geometry and statistics related formulas to cover all the Math's formulas. The maths topics covered in this free app are: ALGEBRA *Basic Properties and Facts *Factoring and Solving Formulas *Factoring and solving Methods (completing the squares methods etc...) *Functions and Graphs *Common Algebraic ErrorsA straight forward algebraic technique to solvingHow many times have you been told to "reduce or simplify fractions"? Calculating the GCF is a common routine to begin to reduce your fractions completely. Master this skill and you will not be afraid of fractions any more!
Prime Factorization is a technique used to find out what all numbers are made up of. This insight is a preliminary step for many math problems such as: add, subtract, reduce, or simplify fractions. Mastering this skill will definitely prepare you for more advanced problems. Give it a try, and have fun!
A nifty algebraic technique, so good it deserves two names. The Algebraic or Elimination method hints at both the process and result of the method. In this app, Mr. Defining Wizard explains how to take advantage of this method. Don't forget to check your answers. Solutions should be the same if you use the graphing, substitution, or matrices method.
When solving logarithm equations, we can only rely on our knowledge of the logarithm properties. Understanding what the properties mean will help you in the long run. Review with Mr. Defining Wizard all properties so you are prepared for any problem you face.
What values can x be? What values can y be? Practice with Mr. Defining Wizard and develop a good eye for determining domain and range for any equation which includes: fractions, square roots, and linear equations.
Simply put, math problems containing <, >, <=, and >= are called inequalities and math problems containing = is called an equation. Solutions to both inequalities and equalities are numbers that make the statements true. Practice these type of problems with Mr. Defining Wizard for additional tips
Did you know that all inverse equations are symmetric across the line y=x? Why is this? Mr. Defining Wizard explains the definition of inverse and reviews a simple problem for you to refer back to for future referencing. Horizontal Line Test and Vertical Line Test are not reviewed in this app.
Dealing with a system that involves 3 or more variables and/or 3 or more equations are best solved using matrices. Learn the rules to solving a system of equations using matrices as an alternate method to the substitution and elimination methods.
A graphical representation of a system may lead to your solution. In this app, Mr. Defining Wizard explains how to take advantage of this method. Don't forget to check your answers. Solutions should be the same if you use the substitution, addition, or matrices method.
Mr. Defining Wizard keeps it real brief and simple in this app. When solving a system of equations, what method should you use? This app will help you become familiar with your options when solving a system of equations: graphing, substitution, addition, and matrices method
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Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
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algebra from the viewpoint of computergraphics. It covers most vector and matrix topics needed for college-level computer graphicstext books. Most graphics texts cover these subjects in an appendix, but it is often too short.This tutorial covers the same material at greater length, and with many examples.
of their appealing blood-and-gore covers, mass trade books on game programming require thesame understanding of vectors and matrices as more staid text books (and usually defer thesetopics to the same skimpy mathematical appendix).This tutorial is useful for more than computer graphics. Vectors and matrices are used in allscientific and engineering fields, and any other field that uses computers (are there any thatdon't?) In many fields, the vocabulary used for vectors and matrices does not match that usedin computer graphics. But the ideas are the same, and reading these notes will take only aslight mental adjustment.These notes assume that you have studied plane geometry and trigonometry sometime in thepast. Notions such as "point", "line", "plane", and "angle" should be familiar to you. Othernotions such as "sine", "cosine", "determinant", "real number", and the common trig identitiesshould at least be a distant memory.These pages were designed at 800 by 600 resolution with "web safe" colors. They have been(somewhat) tested with not-too-old versions of Netscape Navigator and Internet Explorer, using"Times Roman" font (the usual browser default font). Many pages require Javascript, and somepages require Java. If you lack these (or are behind a firewall that does not allow these inside)you will be able to read most pages, but the interactive features will be lost.Some sections are more than three years old and have been used in several lecture sections(and hence are "classroom tested" and likely to be technically correct and readable). Othersections have just been written and might fall short of both goals.This tutorial may be freely downloaded and used as long as copyright and authorshipinformation is not removed. (They are contained in HTML comments on each page.) Peoplewho wish to reward this effort may do so by going to their local public library and checking outany long neglected, lonely book from the stacks.
This chapter discusses the objects of computer graphics---vectors and points---and how they arerepresented in a computer---as column matrices. A column matrix is a mathematical object that hasmany uses besides its use in computer graphics. These notes discuss only the aspects that are used incomputer graphics.
Chapter Topics:
q
Computer graphics as modeling and viewing.
q
Geometrical points and vectors.
q
Column and row matrices.
q
Calculating displacements.
q
Equality of column matrices.
q
Names for column matrices.
q
Representing points with column matrices.
Computer graphics books use one of two ways to represent points and vectors. Some books use rowmatrices; other books use column matrices. The two methods are exactly equivalent (although someformulae have to be adjusted). These notes use column matrices.Some graphics books use the term "column vector" for the object that these notes call a "columnmatrix." This is just a variation in terminology and does not affect the concepts or formulae presentedhere.
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Table of Contents Exponential and Logarithmic Series Complex Quantities. De Moivre-s Theorem Expansions of sin n, and cos n.Series for sin , and cos in Powers of Expansions of Sines and Cosines of Multiple Angles and of Powers of Sines and Cosines Exponential Series for Complex Quantities Circular Functions for... more
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Check your work-and your understanding-with this manual, which provides solutions for all of the odd-numbered exercises in the text. You will also find strategies for solving additional exercises and many helpful hints and warnings. The classic in the series of highly respected Swokowski/Cole mathematics texts retains the elements that have... more
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This text uses the graphing utility to enhance the study of mathematics. Technology is used as a tool to solve problems, motivate concepts, and explore mathematical ideas. Sullivan's Series "Enhanced with Graphing Utilities" provides clear and focused coverage. Many of the problems are solved using both algebra and a graphing... more
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The Dugopolski Precalculus series for 1999 is technology optional. With this approach, teachers will be able to choose to offer either a strong technology-oriented course, or a course that does not make use of technology. For departments requiring both options, this text provides the advantage of flexibility. College Algebra and... more
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Primarily designed as a textbook, Trigonometry is a unique treatise on vectors, aiming at providing a fairly complete account of the basic concepts required to build a strong foundation for a student endeavouring to study this subject. The analytical approach to the major theories Inverse Circular Functions, De-Moivre s Theorem,... more
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23Kau mathematics courses in an easy-to-read format. TheTable of Contents
Basic Concepts And Properties
Sets, Real Numbers, and Numerical Expressions
Operations with Real Numbers
Properties of Real Numbers and the Use of Exponents
Algebraic Expressions
Summary
Review Problem Set
Test
Equations, Inequalities, And Problem Solving
Solving First-Degree Equations
Equations Involving Fractional Forms
Equations Involving Decimals and Problem Solving
Formulas
Inequalities
More on Inequalities and Problem Solving
Equations and Inequalities Involving Absolute Value
Summary
Review Problem Set
Test
Polynomials
Polynomials: Sums and Differences
Products and Quotients of Monomials
Multiplying Polynomials
Factoring: Use of the Distributive Property
Factoring: Difference of Two Squares and Sum or Difference of Two Cubes
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In:
TI-85 Graphing Calculator
The Texas Instruments TI-85 is a graphing calculator which can be used
in Algebra, Geometry, Trigonometry, Statistics, Calculus, Business/Accounting,
and Physics courses. The calculator has on-keypad functions such as SOLVER
(to solve equations of one variable), SIMULT (to solve equations of more
than one variable where specific values are inserted for those variables),
POLY (to solve polynomial equations of various degree), as well as menus
for matrices, vectors, and complex numbers. The calculator graphs functions
of one variable.
NOTE: This calculator is for LIBRARY USE ONLY and cannot be taken
out of the building.
The TI-85 comes with four AAA batteries.
The loan period for the TI-85 is two hours and cannot be renewed.
It can be picked up from and returned to the Circulation Desk. There is
one TI-85 available on a first come, first serve basis.
| 677.169 | 1 |
Algebra I – Unit 2
Ascension Parish Comprehensive Curriculum
Concept Correlation
Unit 2: Measurement
Time Frame: Regular – 12 days
Block – 6 days
Big Picture: (Taken from Unit Description and Student Understanding)
This unit is an advanced study of measurement.
It includes the topics of precision and accuracy and investigates the relationship and difference between the two.
The investigation of absolute and relative error and how they each relate to measurement is included.
Students should understand significant digits. Significant digits need to be studied along with the computations that can be performed using
them.
Students should be able to find the precision of an instrument and determine the accuracy of a given measurement.
Students should see error as the uncertainty approximated by an interval around the true measurement.
Activities Focus GLEs
Guiding Questions GLEs
17 - Distinguish between precision and accuracy (M-1-H)
Concept 1: 12 – What Does it mean to be
(Comprehension)
Measurement Accurate? (GQ 7) 4, 17
6. Can students 18 - Demonstrate and explain how the scale of a measuring
instrument determines the precision of that instrument (M-1-
determine the 13 – How Precise is Your
4, 17, H) (Application)
precision of a given Measurement Tool? (GQ 6)
18 19 - Use significant digits in computational problems (M-1-H) (N-
measurement
2-H) (Application)
instrument? 14 – Temperature – How
7. Can students 4, 17, 20 - Demonstrate and explain how relative measurement error is
Precise Can You Be? (GQ 6)
determine the 18 compounded when determining absolute error (M-1-H) (M-2-
H) (M-3-H) (Analysis)
accuracy of a 15 – Repeatability and
measurement? Precision (GQ 8, 9) 17 21 - Determine appropriate units and scales to use when solving
8. Can students measurement problems (M-2-H) (M-3-H) (M-1-H)
(Application)
differentiate between 16 –Precision vs. Accuracy
what it means to be 17
precise and what it 17 – Absolute Error (GQ 9) 18, 20
Algebra I – Unit 2 - Measurement
Algebra I – Unit 2
means to be
accurate? 18– Relative Error (GQ 9 ) 4, 5,
9. Can students discuss 20
the nature of 19 – What's the Cost of Those
precision and 4, 17,
Bananas? (GQ 6, 9) Reflections
accuracy in 18
measurement and 20 – What are Significant
note the differences Digits? (GQ 10) 4, 19
in final measurement
values that may
21 – Calculating with
result from error?
Significant Digits (GQ 10) 4, 19
10. Can students
perform basic
mathematical 22 – Measuring the Utilities
operations using You Use(GQ 10)
19
significant digits?
11. Can students
determine the most
appropriate units and
scales to use when 23 – Which Unit of
solving measurement 5, 21
Measurement? (GQ 11)
problems?
Algebra I – Unit 2 - Measurement
Algebra I – Unit 2
Unit 2 – Measurement (LCC Unit 6)
GLEs
*Bolded GLEs are assessed in this unit
4 Distinguish between an exact and an approximate answer, and recognize errors
introduced by the use of approximate numbers with technology (N-3-H) (N-4-H) (N-
7-H) (Application)
5 Demonstrate computational fluency with all rational numbers (e.g., estimation,
mental math, technology, paper/pencil) (N-5-H) (Application)
17 Distinguish between precision and accuracy (M-1-H) (Comprehension)
18 Demonstrate and explain how the scale of a measuring instrument determines
the precision of that instrument (M-1-H) (Application)
19 Use significant digits in computational problems (M-1-H) (N-2-H) (Application)
20 Demonstrate and explain how relative measurement error is compounded when
determining absolute error (M-1-H) (M-2-H) (M-3-H) (Analysis)
21 Determine appropriate units and scales to use when solving measurement
problems (M-2-H) (M-3-H) (M-1-H) (Application)
Purpose/Guiding Questions: Vocabulary:
Determine the precision of a given Accurate
measurement instrument Precision
Determine the accuracy of a Absolute Error
measurement Relative Error
Differentiate between what it Significant Digits
means to be precise and what it Computations with Significant Digits
means to be accurate Appropriate Unit of Measurement
Discuss the nature of precision
and accuracy in measurement and
note the differences in final
measurement values that many
result from error
Calculate using significant digits
Key Concepts (Math Across High School):
demonstrate an understanding of precision, accuracy, and significant digits
Assessment Ideas:
Portfolio Assessment: The student will create a portfolio divided into the following
sections:
1. Accuracy
2. Precision
3. Precision vs. Accuracy
4. Absolute error
Algebra I – Unit 2 - Measurement 13
Algebra I – Unit 2
5. Relative error
6. Significant digits
In each section of the portfolio, the student will include an explanation of each,
examples of each, artifacts that were used during the activity, sample questions given
during class, etc. The portfolio will be used as an opportunity for students to
demonstrate a true conceptual understanding of each concept.
The student will complete learning logs using such topics as:
o Darla measured the length of a book to be 11 1 inches with her ruler and 11 1
4 2
inches with her teacher's ruler. Can Darla tell which measurement is more
accurate? Why or why not? (She cannot tell unless she knows which ruler is
closer to the actual standard measure)
o What does it mean to be precise? Give examples to support your explanation.
o What is the difference between being precise and being accurate? Explain your
answer.
o Explain the following statement: The more significant digits there are in a
measurement, the more precise the measurement is.
o When would it be important to measure something to three or more significant
digits? Explain your answer.
Resources:
Video Clip – Drew Brees and Sports Science
Glencoe Geometry textbook - Ch. 1
Graphic Organizers:
organizers/printable/6293.html and
god3TaxVg
ABC Passing the GEE
ILEAP Practice workbook
Create your own organizers using:
and
Plato – Refer to end of Concept 1
Algebra I – Unit 2 - Measurement 14
Algebra I – Unit 2
Instructional Activities
Activity 12: What Does it Mean to be Accurate? (LCC Unit 6)
(GLEs: 4, 17)
Materials List: paper, pencil, three or more different types of scales from science
department, three or more different bathroom scales, student's watches, Internet access,
What Does It Mean To Be Accurate? BLM, sticky notes
This unit on measurement will have many new terms to which students have not
yet been exposed. Have students maintain a vocabulary self-awareness chart
(view literacy strategy descriptions) for this unit. Vocabulary self-awareness is
valuable because it highlights students' understanding of what they know, as well
as what they still need to learn, in order to fully comprehend the concept.
Students indicate their understanding of a term/concept, but then adjust or change
the marking to reflect their change in understanding. The objective is to have all
terms marked with a + at the end of the unit. A sample chart is shown below.
Word + - Example Definition
accuracy
precision
Relative error
Absolute error
Significant
digits
Be sure to allow students to revisit their self-awareness charts often to monitor
their developing knowledge about important concepts. Sample terms to use
include accuracy, precision, significant digits, absolute error, and relative error.
Have students use the What Does It Mean To Be Accurate? BLM to complete this
activity.
Talk with students about the meaning of ―accuracy‖ in measurement. During the
discussion, the following video may be used to supplement and encourage further
dialog regarding the meaning of accuracy in real life contexts.
( Accuracy indicates how
close a measurement is to the accepted ―true‖ value. For example, a scale is
expected to read 100 grams if a standard 100 gram weight is placed on it. If the
scale does not read 100 grams, then the scale is said to be inaccurate.
Algebra I – Unit 2 - Measurement 15
Algebra I – Unit 2
If possible, obtain a standard weight from one of the science teachers along with
several scales. With students, determine which scale is closest to the known value
and use this information to determine which scale is most accurate. Next, ask
students if they have ever weighed themselves on different scales—if possible,
provide different scales for students to weigh themselves. Depending on the scale
used, the weight measured for a person might vary according to the accuracy of
the instruments being used. Unless ―true‖ weight is known, it cannot be
determined which scale is most accurate (unless there is a known standard to
judge each scale).
Generally, when a scale or any other measuring device is used, the readout is
automatically accepted without really thinking about its validity. People do this
without knowing if the tool is giving an accurate measurement. Also, modern
digital instruments convey such an aura of accuracy and reliability (due to all the
digits it might display) that this basic rule is forgotten—there is no such thing as a
perfect measurement. Digital equipment does not guarantee 100% accuracy. Note:
If some students object to being weighed, students might weigh their book bags or
other fairly heavy items. Adjust the BLM if this is done.
Have all of the students who have watches to record the time (to the nearest
second) at the same moment and hand in their results. Post the results on the
board or overhead—there should be a wide range of answers. Ask students,
Which watch is the most accurate? Students should see that in order to make this
determination, the true time must be known. Official time in the United States is
kept by NIST and the United States Naval Observatory, which averages readings
from the 60 atomic clocks it owns. Both organizations also contribute to UTC, the
world universal time. The website has the official U.S. time,
but even its time is ―accurate to within .7 seconds.‖ Cite this time at the same time
the students are determining the time from their watches to see who has the most
accurate time.
Lead students in a discussion as to why their watches have different times (set to
home, work, and so on) and how their skill at taking a reading on command might
produce different readings on watches that may be set to the same time.
Ultimately, students need to understand that accuracy is really a measure of how
close a measurement is to the ―true‖ value. Unless the true value is known, the
accuracy of a measurement cannot be determined.
Activity-Specific Assessment
The student will write a paragraph explaining in his/her own words what it means to be
accurate. He/she will give an example of a real-life situation in which a measurement taken
may not be accurate.
Algebra I – Unit 2 - Measurement 16
Algebra I – Unit 2
Activity 13: How Precise is Your Measurement Tool? (LCC Unit 6)
(GLE: 4, 17, 18)
Materials List: paper, pencil, rulers with different subdivisions, four-sided meter sticks,
toothpicks, What is Precision? BLM, wall chart , blue masking tape
Discuss the term ―precision‖ with the class. Precision is generally referred to in one
of two ways. It can refer to the degree to which repeated readings on the same
quantity agree with each other. We will study this definition in Activity 15.
Have students use the What is Precision? BLM for this activity.
Precision can also be referred to in terms of the unit used to measure an object.
Precision depends on the refinement of the measuring tool. Help students to
understand that no measurement is perfect. When making a measurement, scientists
give their best estimate of the true value of a measurement, along with its uncertainty.
The precision of an instrument reflects the number of digits in a reading taken from
it—the degree of refinement of a measurement. Discuss with students the degree of
precision with which a measurement can be made using a particular measurement
tool. For example, have on hand different types of rulers (some measuring to the
1
nearest inch, nearest 1 inch, nearest 1 inch, nearest 1 inch, nearest 16 inch, nearest
2 4 8
centimeter, and nearest millimeter) and discuss with students which tool would give
the most precise measurement for the length of a particular item (such as the length of
a toothpick). Have students record measurements they obtain with each type of ruler
and discuss their findings.
Divide students into groups. Supply each group with a four-sided meter stick. (This
meter stick is prism-shaped with different divisions of a meter on each side. The
meter stick can be purchased at NASCO, and other suppliers.)
Cover the side of the meter stick that has no subdivisions with two strips of masking
tape and label it as side 1. (You need two layers of masking tape so the markings on
the meter stick will not show through the tape. The blue tape works better as the
darker color prevents markings from showing through better.) Repeat this with the
other sides of the stick such that side 2 has decimeter markings, side 3 has centimeter
markings, and side 4 has millimeter markings. Have students remove the tape from
side 1 and measure the length of a sheet of paper with that side and record their
answers. Repeat with the other sides of the meter stick in numerical order. Post a wall
chart similar to the one below and have each group record their measurements:
Algebra I – Unit 2 - Measurement 17
Algebra I – Unit 2
Length of Paper
Side 1 Side 2 Side 3 Side 4
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Average
Have students calculate the averages of each column. Lead students to discover
that the measurements become closer to the average with the increase in divisions
of the meter stick.
Help students understand that the ruler with the greatest number of subdivisions
per unit will provide the most precise measure.
Have students complete the following RAFT writing assignment (view literacy
strategy descriptions) in order to give students a creative format for demonstrating
their understanding of precise measurement.
Role- millimeter ruler
Audience-decimeter ruler
Format-advertisement
Topic-Buy my subdivisions
Once RAFT writing is completed, have students share with a partner, in small
groups, or with the whole class. Students should listen for accurate information
and sound logic in the RAFTs.
Activity-Specific Assessment
The student will keep a log of the various measurements that are taken at different
measurement stations. The student will record each measurement of each item and then
decide which measurement would be more precise. The student will be required to justify
each answer with a written explanation.
Activity 14: Temperature—How Precise Can You Be? (LCC Unit 6)
(GLEs: 4, 18, 17)
Algebra I – Unit 2 - Measurement 18
Algebra I – Unit 2
Materials List: paper, pencil, thermometers
Have students get in groups of three. Provide each team with a thermometer that is
calibrated in both Celsius and Fahrenheit. Have each team record the room
temperature in both oC and oF. Have students note the measurement increments of the
thermometer (whether it measures whole degrees, tenths of a degree, etc.) on both
scales. Make a class table of the temperatures read by each team. Ask students if it is
possible to have an answer in tenths of a degree using their thermometers and why or
why not?
It is important that students understand that the precision of the instrument depends
on the smallest division of a unit on a scale. If the thermometer only has whole
degree marks, then it can only be precise to one degree. If the thermometer has each
degree separated into tenths of a degree then the measurement is precise to the nearest
tenth of a degree. Regardless of the measurement tool being used, this idea of the
precision of the instrument holds true.
Activity 15: Repeatability and Precision (GLE: 17) (LCC Unit 6)
Materials List: paper, pencil
As stated in Activity 2, precision can also refer to the degree to which repeated readings
on the same quantity agree with each other.
Present students with the following situations:
Jamaal made five different measurements of the solubility of nickel (II)
chloride in grams per deciliter of water and obtained values of 35.11,
35.05, 34.98, 35.13, and 35.09 g/dL.
Juanita made five different measurements of the solubility of nickel (II)
chloride in grams per deciliter of water and obtained values of 34.89,
35.01, 35.20, 35.11, and 35.13 g/dL.
Have students work with a partner to discuss ways to determine which set of
measurements is more precise.
Have students come up with a method for determining which set of measurements is the
most precise. Lead students to the determination that the set that has the smallest range is
a more precise set of measurements.
Provide students with additional measurement situations so that they have the opportunity
to practice determining the more precise set of measurements when given a group of
measurements.
Algebra I – Unit 2 - Measurement 19
Algebra I – Unit 2
Activity 16: Precision vs. Accuracy (LCC Unit 6)
(GLE: 17)
Materials List: paper, pencil, Target BLM transparency, Precision vs. Accuracy BLM,
sticky notes
Student Questions for Purposeful Learning or SQPL (view literacy strategy
descriptions) is a strategy designed to gain and hold students' interest in the
material by having them ask and answer their own questions. Before beginning
the activity, place the following statement on the board:
Accuracy is telling the truth. Precision is telling the same story
over and over again.
Have students pair up and, based on the statement, generate two or three questions
they would like answered. Ask someone from each team to share questions with
the whole class and write those questions on the board. As the content is covered
in the activity, stop periodically and have students discuss with their partners
which questions could be answered, and have them share answers with the class.
Have them record the information in their notebooks.
Create a transparency of the Target BLM which includes the target examples
shown below and have students determine if the patterns are examples of
precision, accuracy, neither or both. Cover boxed descriptions with sticky notes
and remove as the lesson progresses. After the lesson provide students with
Target BLM to include in their notes.
If you were trying to hit a bull's eye (the center of the target) with each of five
darts, you might get results such as in the models below. Determine if the results
are precise, accurate, neither or both.
Neither Precise Nor Accurate
This is a random-like
pattern, neither precise
nor accurate. The darts
are not clustered together
and are not near the
bull's eye.
Algebra I – Unit 2 - Measurement 20
Algebra I – Unit 2
Precise, Not Accurate
This is a precise
pattern, but not
accurate. The darts are
clustered together but
did not hit the intended
mark.
Algebra I – Unit 2 - Measurement 21
Algebra I – Unit 2
Accurate, Not Precise
This is an accurate
pattern, but not precise.
The darts are not
clustered, but their
average position is the
center of the bull's eye.
Precise and Accurate
This pattern is both
precise and accurate.
The darts are tightly
clustered, and their
average position is the
center of the bull's eye.
Lead a class discussion reviewing the definitions of precision and accuracy and revisit the class-
generated questions.
Use the Precision vs. Accuracy BLM and present the examples to students. Lead a class
discussion using the questions on the BLM.
Provide students with more opportunities for practice in determining the precision and/or
accuracy of data sets.
Algebra I-Unit 2-Measurement 21
Algebra I – Unit 2
Activity-Specific Assessment
The student will be quizzed on the difference between being precise and being
accurate. Given examples similar to the ones in the activity, the student will answer
questions about the measurements.
Activity 17: Absolute Error (GLEs: 18, 20) (LCC Unit 6)
Materials List: paper, pencil, Absolute Error BLM, three different scales, 2 different beakers,
measuring cup, meter stick, 2 different rulers, calculator, cell phone, wrist watch
In any lab experiment, there will be a certain amount of error associated with the calculations.
For example, a student may conduct an experiment to find the specific heat capacity of a certain
metal. The difference between the experimental result and the actual (known) value of the
specific heat capacity is called absolute error. The formula for calculating absolute error is as
follows:
Absolute Error = Observed Value - Actual Value
Review absolute value with students and explain to them that since the absolute value of the
difference is taken, the order of the subtraction will not matter.
Present the following problems to students for a class discussion:
Luis measures his pencil and he gets a measurement of 12.8 cm but the actual measurement is
12.5 cm. What is the absolute error of his measurement?
( Absolute Error = 12.8 - 12.5 .3 .3 cm )
A student experimentally determines the specific heat of copper to be 0.3897 oC. Calculate the
student's absolute error if the accepted value for the specific heat of copper is 0.38452 oC.
( Absolute Error = .3897-.38452 0.00518 0.00518 )
Place students in groups and have them rotate through measurement stations. Have students use
the Absolute Error BLM to record the data. After students have completed collecting the
measurements, present them with information about the actual value of the measurement. Have
students calculate the absolute error of each of their measurements.
Examples of stations:
Station Measurement Instruments Actual Value
1 Mass 3 different scales 100 gram weight
2 Volume 2 different sized Teacher measured
beakers and a volume of water
measuring cup
Algebra I-Unit 2-Measurement 22
Algebra I – Unit 2
3 Length Meter stick, rulers Sheet of paper
with 2 different
intervals
4 Time Wrist watch,
calculator, cell
phone
Activity 18: Relative Error (GLEs: 4, 5, 20) (LCC Unit 6)
Materials List: paper, pencil
Although absolute error is a useful calculation to demonstrate the accuracy of a measurement,
another indication is called relative error. In some cases, a very tiny absolute error can be very
significant, while in others, a large absolute error can be relatively insignificant. It is often more
useful to report accuracy in terms of relative error. Relative error is a comparative measure. The
formula for relative error is as follows:
Absolute Error
Relative Error = 100
Actual value
To begin a discussion of absolute error, present the following problem to students:
Jeremy ordered a truckload of dirt to fill in some holes in his yard. The company
told him that one load of dirt is 5 tons. The company actually delivered 4.955
tons.
Chanelle wants to fill in a flowerbed in her yard. She buys a 50-lb bag of soil at a
gardening store. When she gets home she finds the contents of the bag actually
weigh 49.955 lbs.
Which error is bigger?
The relative error for Jeremy is 0.9%. The relative error for Chanelle is 0.09%.
This tells you that measurement error is more significant for Jeremy's purchase.
Use these examples to discuss with students the calculation of relative error and how it relates to
the absolute error and the actual value of measurement. Explain to students that the relative error
of a measurement increases depending on the absolute error and the actual value of the
measurement.
Provide students with an additional example:
In an experiment to measure the acceleration due to gravity, Ronald's group
calculated it to be 9.96 m/s2. The accepted value for the acceleration due to
gravity is 9.81 m/s2. Find the absolute error and the relative error of the group's
calculation. (Absolute error is .15 m/s2, relative error is 1.529%.)
Algebra I-Unit 2-Measurement 23
Algebra I – Unit 2
Provide students with more opportunity for practice with calculating absolute and relative error.
Activity-Specific Assessment
The student will solve sample test questions, such as:
Raoul measured the length of a wooden board that he wants to use to build a ramp. He
measured the length to be 4.2 m. The absolute error of his measurement is .1 m. His
friend, Cassandra, measured a piece of molding to decorate the ramp. Her measurement
was .25 m with an absolute error of .1 m. Find the relative error of each of their
measurements. Whose measurement was better? Explain your answer. (Raoul – 2%,
Cassandra – 40%, Raoul because his percentage of relative error was smaller.)
Activity 19: What's the Cost of Those Bananas? (LCC Unit 6)
(GLEs: 4, 17, 18)
Materials List: paper, pencil, pan scale, electronic scale, fruits or vegetables to weigh
The following activity can be completed as described below if the activity seems reasonable
for the students involved. If not, the same activity can be done if there is access to a pan scale
and an electronic balance. If done in the classroom, provide items for students to measure—
bunch of bananas, two or three potatoes, or other items that will not deteriorate too fast.
Have the students go to the local supermarket and select one item from the produce
department that is paid for by weight. Have them calculate the cost of the object using the
hanging pan scale present in the department. Record their data. At the checkout counter, have
the students record the weight given on the electronic balance used by the checker. Have
students record the cost of the item. How do the two measurements and costs compare? Have
students explain the significance of the number of digits (precision) of the scales.
Activity 20: What are Significant Digits? (LCC Unit 6)
(GLEs: 4, 19)
Materials List: paper, pencil
Discuss with students what significant digits are and how they are used in measurement.
Significant digits are defined as all the digits in a measurement one is certain of plus the first
uncertain digit. Significant digits are used because all instruments have limits, and there is a
limit to the number of digits with which results are reported. Demonstrate and discuss the
process of measuring using significant digits.
After students have an understanding of the definition of significant digits, discuss and
demonstrate the process of determining the number of significant digits in a number. Explain
Algebra I-Unit 2-Measurement 24
Algebra I – Unit 2
to students that it is necessary to know how to determine the significant digits so that when
performing calculations with numbers they will understand how to state the answer in the
correct number of significant digits.
Rules For Significant Digits
1. Digits from 1-9 are always significant.
2. Zeros between two other significant digits are always significant
3. One or more additional zeros to the right of both the decimal place and another
significant digit are significant.
4. Zeros used solely for spacing the decimal point (placeholders) are not significant.
Using a chemistry textbook as a resource, provide problems for students to practice in
determining the number of significant digits in a measurement.
In their math learning logs (view literacy strategy descriptions) have students respond to the
following prompt:
Explain the following statement:
The more significant digits there are in a measurement, the more precise the
measurement is.
Allow students to share their entries with the entire class. Have the class discuss the entries to
determine if the information given is correct.
Activity 21: Calculating with Significant Digits (GLEs: 4, 19) (LCC Unit 6)
Materials List: paper, pencil,
Discuss with students how to use significant digits when making calculations. There are different
rules for how to round calculations in measurement depending on whether the operations involve
addition/subtraction or multiplication/division. When adding, such as in finding the perimeter,
the answer can be no more PRECISE than the least precise measurement (i.e., the perimeter must
be rounded to the same decimal place as the least precise measurement). If one of the measures is
15 ft and another is 12.8 ft, then the perimeter of a rectangle (55.6 ft) would need to be rounded to
the nearest whole number (56 ft). We cannot assume that the 15 foot measure was also made to
the nearest tenth based on the information we have. The same rule applies should the difference
between the two measures be needed.
When multiplying, such as in finding the area of the rectangle, the answer must have the same
number of significant digits as the measurement with the fewest number of significant digits.
There are two significant digits in 15 so the area of 192 square feet, would be given as 190 square
feet. The same rule applies for division.
Algebra I-Unit 2-Measurement 25
Algebra I – Unit 2
Have students find the area and perimeter for another rectangle whose sides measure 9.7 cm and
4.2 cm. The calculated area is (9.7cm)(4.2cm) = 40.74 sq. cm, but should be rounded to 41 sq cm
(two significant digits). The perimeter of 27.8 cm would not need to be rounded because both
lengths are to the same precision (tenth of a cm).
After fully discussing calculating with significant figures, have students work computational
problems (finding area, perimeter, circumference of 2-D figures) dealing with the topic of
calculating with significant digits. A chemistry textbook is an excellent source for finding
problems of calculations using significant digits.
Activity 22: CC Activity 9: Measuring the Utilities You Use (LCC Unit 6)
(GLE: 19)
Materials List: paper, pencil, utility meters around students' households, utility bills
Have students find the various utility meters (water, electricity) for their households. Have
them to record the units and the number of places found on each meter. Have the class get a
copy of their family's last utility bill for each meter they checked. Have students answer the
following questions: What units and number of significant digits are shown on the bill? Are
they the same? Why or why not? Does your family pay the actual ―true value‖ of the utility
used or an estimate?
If students do not have access to such information, produce sample drawings of meters used
in the community and samples of utility bills so that the remainder of the activity can be
completed.
Activity 23: Which Unit of Measurement? (LCC Unit 6)
(GLEs: 5, 21)
Materials List: paper, pencil, centimeter ruler, meter stick, ounce scale, bathroom scale, quarter,
cup, gallon jug, bucket, water
Divide students into groups. Provide students with a centimeter ruler and have them measure
the classroom and calculate the area of the room in centimeters. Then provide them with a
meter stick and have them calculate the area of the room in meters. Discuss with students
which unit of measure was most appropriate to use in their calculations. Ask students if they
were asked to find the area of the school parking lot, which unit would they definitely want to
use. What about their entire town? In that case, kilometers would probably be better to use.
Provide opportunities for discussion and/or examples of measurements of weight (weigh a
quarter on a bathroom scale or a food scale) and mass (fill a large bucket with water using a
cup or a gallon jug) similar to the linear example of the area of the room. Use concrete
examples for students to visually explore the most appropriate units and scales to use when
solving measurement problems.
Algebra I-Unit 2-Measurement 26
Algebra I – Unit 2
Activity-Specific Assessment
The student will be able to determine the most appropriate unit and/or instrument to use
in both English and Metric units when given examples such as:
How much water a pan holds
Weight of a crate of apples
Distance from New Orleans to Baton Rouge
How long it takes to run a mile
Length of a room
Weight of a Boeing 727
Weight of a t-bone steak
Thickness of a pencil
Weight of a slice of bread
Algebra I-Unit 2-Measurement 27
Algebra I – Unit 2
PLATO Instructional Resources
GLE 17: Plato
o Applied Math – Estimating
o Chemistry I –Meas. & Calc.: Uncertainty…
GLE 18: Plato
o Applied Math – Using Linear Measurement Tools
o Chemistry I –Meas. & Calc.: Uncertainty..
GLE 19: Plato
o Chemistry I – Meas. & Calc.: Uncertainty…
GLE 20: Plato
o Chemistry I – Meas. & Calc.: Uncertainty…
GLE 21: Plato
o Applied Math-Using Linear Measurement Tools
o Chemistry I –Meas. & Calc.: Uncertainty
Algebra I-Unit 2-Measurement 28
Algebra I – Unit 2 changes
Number
* If you suggest an activity substitution, please attach a copy of the activity narrative formatted
like the activities in the APCC (i.e. GLEs, guiding questions, etc.).
Algebra I-Unit 2-Measurement
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Summary: From the Core-Plus Mathematics ProjectMathematics That Makes Sense to More StudentsThis innovative program engages students in investigation-based, multi-day lessons organized around big ideas. Important mathematical concepts are developed in relevant contexts by students in ways that make sense to them. Students in ''Contemporary Mathematics in Context work collaboratively, often using graphing calculators, so more students than ever before are able to learn important and broadly us...show moreeful mathematics. Courses 1, 2, and 3 comprise a core curriculum that will upgrade the mathematics experience for all your students. Course 4 is designed for all college-bound students.Research-Based and Classroom-TestedDeveloped with funding from the National Science Foundation, each course in ''Contemporary Mathematics in Context is the product of a four-year research, development, and evaluation process involving thousands of students in schools across the country. The result is a program rich in modern content organized to make active student learning a daily occurrence in your classroomGood
Borgasorus Books, Inc. MO Wentzville, MO
Hardcover Good 0078275458
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Trigonometry Workbook: Teacher Guide [NOOK Book] ...
More About
This Book starters, students will actually like using them. Here's why: • Math is explained in simple language, in an easy-to-follow style • The workbooks allow students to learn at their own pace and master the subject • 15 lessons break down the material into the basics • Each lesson is fully devoted to a key math concept and includes many step-by-step examples • Paced instruction with drills and quizzes reinforces learning • The innovative "Math Flash" feature offers helpful tips and strategies in each lesson—including advice on common mistakes to avoid • Skill scorecard measures the student's progress and success • Every answer to every question, in every test, is explained in full detail • A final exam is included so students can test what they've learned When students apply the skills they've mastered in our workbooks, they can do better in class, raise their grades, and score higher on the all-important end-of-course, graduation, and exit exams. Some of the math topics covered in the Ready, Set, Go! Trigonometry Workbook include: • Trigonometric Ratios • Graphing Points and Angles • Special Angles • Inverse Trigonometric Values • Solving Triangles • Solving Four-Sided Figures • Solving Trigonometric Equations and more! Whether used in a classroom, for home or self study, or with a tutor, this workbook gets students ready for important math tests and exams, set to take on new challenges, and helps them go forward in their studies 4, 2014
Good book keep writeing
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Anonymous
Posted January 29, 2014
The story deserves it
I really loved the phone like thing idea but now i must read more! You finsh one part and are looking at all results for the next one you must find out that clif hanger!
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Anonymous
Posted January 29, 2014
Chapter 5- Light
The brightness of even a dim bulb would have been enough by now to blind Ana. So twenty or so super-high-watt bulbs was like pure agony after the last fifteen minutes or so. <br> Ana groaned. She wondered if this was some kind of defense mechanism. Leave the place ridiculously dark, then have extra-bright lighting blind th intruders. However, once she couod get both eyes to stay open without feeling like she was trying to look straight at the sun, she realized this place was all but abandoned. <br>The room she was in was not unlike her science classroom at school: lots of odd-looking bottles of who-knows-what, lab supplies strewn across a table in one end of the room. All it needed was her teacher, Mr. Speck. Had he been there, she would have easily believed herself to be in class. Waking up from a weird dream brought on by a longer than average lecture. Except it obviously wasn't a dream. Was it? <br> She decided to test the theory. If she was dreaming, she should be able to do whatever, right? So she should be able to teleport out. Ana closed her eyes, imagining herself appearing in her room. She stayed that way for about thirty seconds, concentrating. Then she opened her eyes again. <br> She was still in the mysterious room that resembled a science lab, standing there with her sock rope tied around her waist. Of course not. Either she was having a very strange dream, or this was real. Best she keep going with the latter assumption for now. A dream might not have much consequences, but if this was real...<br> She staggered over to the table to get a closer look at the contents. A bunch of vials of stuff she couldn't identify, a couple of unused petri dishes, and...a small grey box. <br> The box was about the size of her fist, with the thickness of about half an inch, and had a metallic sheen to it. She picked it up and turned it over in her good hand. A mark on the bottom caught her eye. <br> The mark looked familiar somehow...it took her a second to recognize it. It looked like the 'call' button on a cellphone. She wondered if she should press it or not. Then a loud buzz took her by surprise. It was coming from the box! She dropped it, startled, then saw how it had landed. <br>Button down. Meaning the button had just been pressed. Wonder what this one would trigger? 'Hopefully not another elevator...' she thought. <br>It wasn't an elevator, of course. It was something much stranger.
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Survival Skills for
Quantitative Courses
These units will help you teach yourself the skills you need to
succeed in introductory science courses. Some of them will be very easy, some
you may never have seen before. Every person's preparation for these
courses is different and these units are designed to bring everyone to
the same level. Each unit has a series of associated problems.
Complete these problems to test your mastery of the material.
| 677.169 | 1 |
Arithmetic and pre-algebra
The first math topic. Start here if you want to learn the basics (or just want to make sure you know these topics). After this, you should be ready for algebra. This topic includes videos from the former developmental math playlists.
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Mu Alpha Theta (Math)
About Us
Mu Alpha Theta is a national honor society in mathematics for the junior college student. Its purpose is to stimulate interest in mathematics by providing public recognition of superior mathematical scholarship and by promoting various mathematical activities. Initiation is held in the spring.
The goal of the club is to promote interest in math and engineering among Blinn students.
| 677.169 | 1 |
Atlanta ACT Math study cardiac dynamics and other complex dynamical systems, where differential equations play a key roll. The
| 677.169 | 1 |
Quantitative Literacy
Drake students will learn to reason with the symbols and components of mathematical languages as well as effectively use the principles that underlie these operations. Courses that satisfy this requirement will have mathematical reasoning as their principal focus. They may also address questions that engage learners with the world around them and help them to analyze quantitative claims that arise from the study of civic, political, scientific, or social issues. Quantitative literacy courses may be focused on the mathematical needs of a specific discipline or on a specific interdisciplinary issue or problem. these courses will engage students to achieve these student learning outcomes. Students will be able to:
Analyze and present solutions to problems using symbols and components from mathematical languages and their underlying principles.
Identify and execute appropriate mathematical operations for a given question
| 677.169 | 1 |
The treatise on conic sections by the Hellenistic mathematician Apollonius from Perga is regarded as a supreme achievement of Greek mathematics and maintained its authority right up to the 18th century. This new edition is the first to consider all Greek and Arabic sources, with the Arabic texts being presented in the first ever critical edition. Both... more...
Traditional methods for handling spatial data are encumbered by the assumption of separate origins for horizontal and vertical measurements. Modern measurement systems operate in a 3-D spatial environment. The 3-D Global Spatial Data Model: Foundation of the Spatial Data Infrastructure offers a new model for handling digital spatial data, the global... more...
This book is designed for grades K–2 instruction and provides step-by-step mathematics lessons that incorporate the use of the TI-10 calculator throughout the learning process. The 30 lessons included present mathematics in a real-world context and cover each of the five strands: number and operations, geometry, algebra, measurement, and data... more...
This book is designed for grades 3–5 instruction and provides step-by-step mathematics lessons that incorporate the use of the TI-15 calculator throughout the learning process. The 30 lessons included present mathematics in a real-world context and cover each of the five strands: number and operations, geometry, algebra, measurement, and data... more...
Maths is enjoying a resurgence in popularity. So how can you avoid being the only dinner guest who has no idea who Fermat was or what he proved, and what Fibonacci?s sequence or Pascal?s triangle are? The more you know about Maths, the less of a science it becomes. 30 Second Maths takes the top 50 most engaging mathematical theories, and explains... more...
| 677.169 | 1 |
Area Formulas - MAT-955 your students develop area formulas for parallelograms, triangles, and trapezoids. Using hands-on activities, video demonstrations, animations, and comics, your students will build a strong understanding of area formulas and how to use them in problem solving. This course is built around core propositions from the National Board for Professional Teaching Standards as well as national content standards.
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"The information that I have gathered while participating in various courses at Fresno Pacific University has impacted my classroom tremendously. The students are able to learn with a more hands-on approach which has improved their understanding of the material being presented."
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A Best Answer: An answer that clearly demonstrates how to solve the problem the student is facing. Remember, we're here to help students learn the material. Craft a solution as if you were sitting side-by-side with the student, explaining each step and your reason for doing so. This often means that you go back to the fundamental concepts behind the question in order to solve it, that you explain in words as well as mathematical symbols, and that you write clearly so someone at any level could understand how to solve the problem. Students come to Chegg with various levels of experience and expertise, so it's important that we offer them the support they need.
Some Key Components of a Winning Answer, in order of our experts' ranking of their importance:
Answer more questions to build up your reputation with the community. The higher your rating, the more reputable your answers appear.
Explain each step in words and equations. Write clearly, using proper grammar, punctuation, and spelling.
Upload a supporting diagram. This can be done using software and then saving as a .jpg or .png or just by simply drawing an illustrative diagram and then taking a photo with your camera phone.
Break big chunks of text up into paragraphs when you move on to a new idea.
Start the answer with a clear introduction to the problem-solving method you'll be using. This will help start things off on a clear foot.
Upload your profile with a photo of yourself and a headline. This will make your answers seem more human and easy to relate to.
Post as yourself, not anonymously. This communicates authenticity.
Software Suggestions for Creating Graphs and Diagrams:
Microsoft Excel for business, accounting, basic math, and tons of subjects in between. Creating graphs and tables is so easy in Excel that we always recommend you start here first. Then you can save them and import as jpegs. This is clearer than a hand-drawn table or diagram and looks super professional.
DrawPlus. This is a free drawing program you can download on the Serif website. It will help you create clear diagrams and figures for your answer.
Specialized software for certain subjects (this could include Avogadro for chemists, MATLAB for mathematics, PSPICE for engineering, etc.).
Pencil and paper. Take a photo of a graph or diagram you've drawn on paper (as long as it's super clear). Please note you can't currently rotate the image, so make sure you rotate it properly beforehand.
Here's an example:
Question: (x-2)^2 = 144
Solution:
This question requires familiarity with square roots and exponents.
Step 1) You need to remove the exponent from the left side of the equation to solve for x, so take the square root of both sides.
√[(x-2)^2)] = √144
Step 2) Solve for the square root. One the left side, the square root cancels out an equation that is squared. On the right side, the square root of 144 is 12 (i.e., 12^2 = 144).
(x-2) = 144
Step 3) Add 2 to each side of the equation to so that x is the only variable on the left side of the equation.
[...] I always include the required formulas and diagrams needed in the question. It is important for me to help students understand the concepts behind the problems, and including these is the best way to do this. I usually spend 2-3 minutes answering an average problem and maybe more than 15 minutes if the problem is complex. I make sure I answer each and every question with the utmost concentration, since my errors will have an impact on the student who asked the question. [...]
The probability that a customer would purchase a washing machine is 0.2. The probability that the customer would purchase a washing machine or a dryer is 0.30. The probability that the customer would purchase a dryer is 0.25. The probability that a customer would purchase a washing machine but not a dryer is
| 677.169 | 1 |
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Starting at $0Intended for developmental math courses in intermediate immediate feedback, reinforcing the concept, identifying problem areas, and, overall, promoting student success. New!Interactive Exercisesappear at the beginning of an objective's exercise set (when appropriate), and provide students with guided practice on some of the objective's underlying principles. New!Think About ItEx. New!Important Pointshave been highlighted to capture students' attention. With these signposts, students are able to recognize what is most important and to study more efficiently. New!Coverage of evaluating functions, graphing functions, and the vertical line test have been added to Section 3.2. New!An explanation that the x-coordinate of an x-intercept is a zero of a function can now be found in Section 3.3. New!Improved Introductions to exponential and logarithmic functions in Chapter 10 will lead to greater student understanding of and interest in these topics. New Media!Two key components have been added to the technology package: HM Testing (powered by Diploma) and, as part of the Eduspace course management tool, HM Assess, an online diagnostic assessment tool.
Table of Contents
Review of Real Numbers
Introduction to Real Numbers
Operations on Rational Numbers
Variable Expressions
Verbal Expressions and Variable Expressions
First-Degree Equations and Inequalities
Equations in One Variable
Coin, Stamp, and Integer Problems
Value Mixture and Motion Problems
Applications: Problems Involving Percent
Inequalities in One Variable
Absolute Value Equations and Inequalities
Linear Functions and Inequalities in Two Variables
The Rectangular Coordinate System
Introduction to Functions
Linear Functions
Slope of a Straight Line
Finding Equations of Lines
Parallel and Perpendicular Lines
Inequalities in Two Variables and by Using Matrices
Application Problems
Solving Systems of Linear Inequalities
Polynomials and Exponents
Exponential Expressions
Introduction to Polynomials
Multiplication of Polynomials
Division of Polynomials
Factoring Polynomials
Special Factoring
Solving Equations by Factoring
Rational Expressions
Introduction to Rational Functions
Operations on Rational Expressions
Complex Fractions
Rational Equations
Proportions and Variation
Literal Equations
Rational Exponents and Radicals
Rational Exponents and Radical Expressions
Operations on Radical Expressions
Radical Functions
Solving Equations Containing Radical Expressions
Complex Numbers
Quadratic Equations and Inequalities
Solving Quadratic Equations by Factoring or by Taking Square Roots
Solving Quadratic Equations by Completing the Square and by Using the Quadratic Formula
| 677.169 | 1 |
Algebra Fundamentals for Ultrasound Techs: A Student's Guide
Description: Anyone who wants to become an ultrasound technologist knows that understanding algebra and how it serves as a foundation for physics is a top priority. In this compact guidebook, a longtime teacher who has helped many students provides clearMore...
Anyone who wants to become an ultrasound technologist knows that understanding algebra and how it serves as a foundation for physics is a top priority. In this compact guidebook, a longtime teacher who has helped many students provides clear explanations and analysis to help you land your dream job. Even if you struggle with learning what everyone else seems to understand about math, don't despair. Take proactive steps to understand algebra fundamentals by referring to this guide, which offers answers to numerous questions and specific guidance, such as how exponents make multiplication easy;how to calculate in mathematics using scientific notation;how equations, relationships and graphics can help you;how fractions, decimals, and percentages work; andhow variables in equations can be solved. The guidebook includes practice problems, easy-to-follow explanations, answer keys, and a glossary defining key terms. Stop living in fear, and start seeking good employment. It begins with unraveling the mysteries of algebra
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Teaching and Learning Guide 6: Non- Linear Equations
Section 2: The Concept of Quadratic Functions
1. The concept of quadratic functions
The difficulties in moving the focus away from linearity are perhaps two fold. First, as teachers we
have to undo the previous conception of a market or particular function and say that the real world
is very unlikely to be linear and thus we must alter how to try to model and represent real world
situations. Yet we need to do this in a way that does not lead the students to suddenly 'lose' all of
the economic understanding they have built up from this simplifying assumption. This is not easy
but is perhaps more straightforward than the second problem which is getting students to tackle
non-linearity in a mathematical setting when they have only just grasped the rudiments of linear
equations.
Clearly, the lecturer needs to build here on the assumption that linear equations have been taught
and the students have learned how to use them, interpret them in an economically sensible
manner and that they are comfortable when faced with problems that require solutions to linear
equation models. If you are uncertain about the students grasp of linearity, then please refer to
Teaching Guides 2 and 3 on this subject before reading any further.
In many respects, the difficulty with non-linearity is that students rarely encounter it in the early
years of their undergraduate degrees (if at all) and thus meaningful application is not always
forthcoming in parallel subject teaching. Again this reflects the more difficult application of non-
linear models within undergraduate teaching although that does not mean it is never done and nor
does it mean it has no value. This does not mean they do not realise that the assumption of
linearity is likely to be invalid, but that we do not do enough to show why it is useful, how
conclusions change and perhaps as importantly why assuming linearity is not always helpful.
2. Presenting the concept of quadratic functions
One approach is use a simple excel worksheet to demonstrate visually what happens when each
of the parameters of a quadratic equation are changed. Lecturers could plot the values of x and y
for a given quadratic equation (y = a + bx +cx2), perhaps one chosen by the students in a lecture
or small group and then illustrate this function. Values of a, b and c in the equation can be altered
and the equation re-plotted against the original graph.
Page 4 of 23
Teaching and Learning Guide 6: Non- Linear Equations
It is helpful to begin by switching between positive and negative versions of the same number
before altering the absolute value of the number. This demonstrates very quickly to students that
changing the value of a shifts the line up and down, it changes the value at which the line cuts the
y axis (x=0). Changing the value of b shifts the line left and right.
Finally and most importantly switching the value of c from positive or negative gives the line its 'u'
or 'n' shape, while increasing or decreasing the absolute value of c makes the slops steeper or
shallower. In the sample below we show the effect of a change in the c parameter from 1 to -1.
These worksheets could be mounted on the module website to allow students to work with in their
own time.
2
Function: y = a + bX + cX
Values a b c
old values 3 2 1
new values 3 2 -1
15000
X Old Series New Series
-100 9803 -10197
-99 9606 -9996 10000
-98 9411 -9797
-97 9218 -9600
-96 9027 -9405 5000
-95 8838 -9212
-94 8651 -9021 Old Series
-93 8466 -8832 0
New Series
-100
-84
-68
-52
-36
-20
-4
12
28
44
60
76
92
-92 8283 -8645
-91 8102 -8460
-90 7923 -8277
-5000
-89 7746 -8096
-88 7571 -7917 -10000
-87 7398 -7740
-86 7227 -7565
-85 7058 -7392 -15000
-84 6891 -7221
-83 6726 -7052
-82 6563 -6885
3. Delivering the concept of quadratic functions to small or larger groups
Students could benefit from a kinaesthetic and visual method of delivery. Lecturers could create a
range of laminated cards which cover the values 0 -1- inclusive, the variables x and y and the
power '2'. Students select cards and create their own quadratic in front of them and in pairs they
try to sketch what their quadratic might look like. Students could look at the quadratics and
sketches from other pairs.
The lecturer could select some or all of the quadratics and using Excel project what the functions
actually look like. This could be simple, effective and 'instant' way for students to develop a feeling
Page 5 of 23
Teaching and Learning Guide 6: Non- Linear Equations
for what a quadratic looks like and the function is constructed and behaves according to its
constituent coefficients.
Links with the online question bank
Questions on quadratic functions can be found on the METAL website at:
Lecturers might find it useful to refer to the questions on adding polynomials at
as a
precursor to the subsequent material or cubic and polynomial functions. These questions could
also be used to further differentiate teaching and learning.
Video clips
Although there are no clips which deal specifically with quadratics, lecturers might want to look at
the clip 1.09 at
which examines powers and indices in the context of gold mining and oil exploration.
4. Discussion Questions
Ask students to think of any relationships which are u-shaped. Could students who follow
geography courses make links e.g. mathematically modelling u-shaped valleys or river beds?
Obvious links to the natural world abound e.g. the wear on a step or the trajectory of a thrown
tennis ball. Students could see how many quadratic relationships they could identify and describe
within a week. Higher ability students might want to reflect on why the relationship is quadratic as
opposed to say a linear function.
5. Activities
Learning Objectives
LO1: Students learn to calculate values of y for different values of x using quadratic
functions
LO2: Students learn how to tabulate and plot quadratic functions
LO3: Students learn that many economic relationships are rarely linear and some can be
expressed using a quadratic expression.
Page 6 of 23
Teaching and Learning Guide 6: Non- Linear Equations
Task One
To begin with, let us assume that we are in a small group setting, such as a tutorial or problems
class, and we will show later how the activity can be adjusted for the large group (lecture theatre)
setting.
The first stage is to split the group into pairs or threes, which can be done with the people sitting
next to each other – there is no need to enforce randomness on this procedure as in fact that
might be threatening for some students and again create barriers to learning. The second stage is
to provide each group with a piece of graph paper with a relatively simple non-linear function
written onto it (e.g. y = 10 + 4 x 2 ).
Each group could have the same function or you could vary them so that there is exposure to a
number of equations. Another possibility is for the students to be given a series of equations and
they chose one of them to examine, which puts the element of control into their hands, again
helping to create a more positive atmosphere around the learning.
The exercise they are then set is to calculate the change in the dependent variable ( y ) for the
same (or possibly different) changes in the independent variable ( x ) . These can be written down
in tabular form on the sheet and then they can be plotted. Thus for example, for y = 10 + 4 x 2 it
might look like this:
(x ) (y)
0 10
1 14
2 26
3 46
Page 7 of 23
Teaching and Learning Guide 6: Non- Linear Equations
X=0
40
30
20
10
Y=0
0 1 2 3 4
The outcome of this exercise is that it should be apparent that for a given move from one value in
(x ) the change in ( y ) varies and thus the previously simple relationship between ( x ) and ( y ) found
in linear equations no longer holds. Crucially too the impact of changes in ( x ) on ( y ) depends on
where you start with ( x ) – a low value or a high value for instance.
Another issue here of course is how the lines are plotted. At first it is best to make explicit that you
simply want to plot the points as straight lines between two pairs of co-ordinates. Once this has
been done then, discussion of what happens to all the values in between the whole numbers we
have chosen and that can lead to plotting a smooth curve. This reinforces the staging process
from linear models to non-linear as the shape becomes exacerbated.
In essence, the exercise builds on prior learning as it requires manipulation of two variables to find
solutions to an equation and then plotting these on a graph, both of which they will have done
previously. In a large group setting the sheets can be handed out at the start of the session and
students can work on their own. Values for x are provided and they have to find y. They then offer
their answers to the lecturer who has a plot on the whiteboard or computer at the front of the
lecture theatre. As answers are plotted the non-linear nature of the relationship should become
Page 8 of 23
Teaching and Learning Guide 6: Non- Linear Equations
apparent quite readily. This can be done using Excel and plotting from the data therein which
could be pre-loaded with more data points than they provide in responses but which allow for the
curvature of equations to be emphasised.
An Example of an Excel chart with Data:
y= 10 + 4x2
x y
0 10
1 14
2 26
3 46
4 74
5 110
Task Two
Given the demand function QD = 75 – (1/4)P
a. Find the total revenue function written in terms of Q [TR = P(Q)]
b. Calculate the point at which total revenue is equal to zero
c. Calculate the point at which total revenue is maximised.
Task Three
A manufacturer faces two types of costs in its production process, fixed costs which are equal to
£1000 and variable costs which are equal to £2 for each tyre produced.
a. State the total cost function for this firm
b. Calculate average costs
c. Calculate total costs if 700 tyres are produced.
Page 9 of 23
Teaching and Learning Guide 6: Non- Linear Equations
Task Four
Using the total revenue function calculated from question 1 and the total cost function calculated
in question 2.
a. Derive the profit function for the firm
b. What is the value of profits when production equals 125 units?
Task Five
A company discovers the following economic information about its costs and demand function:
Demand Data Cost Data
1
Q = 33 − p Fixed costs are £200
2
Variable costs are £8 per unit
i) Derive the profits function for the firm
ii) What is the breakeven quantity?
iii) At what output level would profits be maximised?
ANSWERS
Task One
See Task
Task Two
1 2
a. TR = 75 p − P
4
b. TR=0, p=300
c. TR is maximised when P=150 (see diagram below)
Page 10 of 23
Teaching and Learning Guide 6: Non- Linear Equations
6000 TR
5000
4000
Total Revenue
3000
2000
1000
0
0 50 100 150 200 250 300
Price
Task Three
Let t denote the number of tyres produced
a. TC = 1000 + 2t
1000
b. AC = TC/t = +2
t
c. TC=£2400 when t=700
Task Four
1
a. Profit = TR-TC = 300Q − Q 2 − 1000 − 2Q
4
b. Profit = -£26,250 i.e. a loss
Page 11 of 23
Teaching and Learning Guide 6: Non- Linear Equations
Task Five
i)
1
Q = 33 − p
2
⇒ p = 66 − 2Q
TR = p × Q = 66Q − 2Q 2
π = TR − TC = 66Q − 2Q 2 − 200 − 8Q
π = −2Q 2 + 58Q − 200
ii) Breakeven occurs when π=0
π = −2Q 2 + 58Q − 200 =0
Factorising we get:
(−2Q + 50)(Q − 4) = 0
Q=4 or Q=25
So, the firm breaks even when output is 4 units and also when output is 25 units
∂π
iii) Profits maximised =0
∂Q
π = −2Q 2 + 58Q − 200
∂π
= −4Q + 58
∂Q
∂π
=0
∂Q
− 4Q + 58 = 0
58
Q= = 14.5units
4
Page 12 of 23
Teaching and Learning Guide 6: Non- Linear Equations
6. Top Tips
One of the best ways of keeping students engaged with non-linear ideas is to use as often as
possible graphical representations to illustrate points especially when dealing with more
complex equations. A visual cue often helps a more intuitive understanding of the ideas.
It often helps to approach quadratic equations in a structured way such as the following:
1. Determine the basic shape using the coefficient on the squared term.
2. Find the y intercept by substituting in x=0.
3. Find the x intercepts (if possible) using the formula.
7. Conclusion
Create opportunities for students to use both graphical and algebraic representations to help them
really acquire a solid sense of what a quadratic function looks like and how it is composed. This
would provide a good foundation for the next section.
Section 3: Cubic and other polynomial functions
1. The concept of cubic and other polynomial functions
Higher order polynomial functions, such as cubic, share some of the problems of quadratic
equations in that they are difficult to visualise and the equations look "messy". They do not
however, typically create problems and issues to the student that are distinct. It is also generally
the case that we actually offer very little in the way of mathematical tools to deal with them. We
rarely go beyond simply plotting them for different values of x. The mathematical tools applied to
them, such as differentiation, are covered in a later section of this guide.
Compared to linear and quadratic functions higher order functions have few properties that are
particularly useful in describing economic concepts, perhaps explaining why we often spend so
little time with them. This guide does not therefore spend a long time explaining why a cubic cost
function is of particular economic relevance for example.
Page 13
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