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Now in its second year, the Computer-Based Math Education Summit 2012. is fast becoming the hub of a major change in math education. A broad cross-section of leaders with a stake in STEM education from industry, technology, government, and education will attend to answer the question, "What are the steps to delivering computer-based math education worldwide?"
Computerbasedmath.org is a project to build a completely new math curriculum with computer-based computation at its heart - alongside a campaign to refocus math education away from historical hand-calculating techniques and toward relevant, real-world and conceptually interesting topics. | 677.169 | 1 |
Most people don?t think about numbers, or take them for granted. For the average person numbers are looked upon as cold, clinical, inanimate objects. Math ideas are viewed as something to get a job done or a problem solved. Get ready for a big surprise with Numbers and Other Math Ideas Come Alive . Pappas explores mathematical ideas by looking behind... more...
The present monograph is intended to provide a comprehensive and accessible introduction to the optimization of elliptic systems. This area of mathematical research, which has many important applications in science and technology. has experienced an impressive development during the past two decades. There are already many good textbooks dealing with... more...
This book provides an introduction to index numbers for statisticians, economists and numerate members of the public. It covers the essential basics, mixing theoretical aspects with practical techniques to give a balanced and accessible introduction to the subject. The concepts are illustrated by exploring the construction and use of the Consumer... more...
This is the eighth edition of the four-yearly review of mathematics education research in Australasia. Commissioned by the Mathematics Education Research Group of Australasia (MERGA), this review critiques the most current Australasian research in mathematics education in the four years from 2008-2011. The main objective of this review is to celebrate... more...
Lately there is an increasing interest in partial difference equations demonstrated by the enormous amount of research papers devoted to them. The initial reason for this increasing interest was the development of computers and the area of numerical analysis, where partial difference equations arise naturally when discretizing a partial differential... more... | 677.169 | 1 |
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Mathematics 300
MATH 300: Mathematical Foundations
Prerequisites: MATH 142 or ITEC 122 or permission of instructor, and MATH 152 and any MATH course numbered 200 or above
Credit Hours: (3)
A first course in the foundations of modern mathematics. The topics covered include propositional and predicate logic, set theory, the number system, induction and recursion, functions and relations, and computation. The methods of proof and problem solving needed for upper-division coursework and the axiomatic basis of modern mathematics are emphasized throughout the course. The level of the course is challenging but appropriate for students with a minimum of 3 semesters of college mathematics. Students who have earned credit for MATH 200 may not subsequently earn credit for MATH 300.
Detailed Description of Course
Course content includes:
The propositional calculus:
Propositional variables and logical connectives.
The use of truth tables to test for truth conditions.
Tautologies, and contradictions.
The predicate calculus:
Predicate functions, variables, and logical connectives.
The universal and existential quantifiers and their standard interpretations.
Validity and satisfiability.
Soundness and completeness.
Using the language of predicate calculus in mathematical proofs.
Naïve and formal set theory:
Standard set notation.
The set operations of union, intersection, symmetric difference, and power set.
The Zermelo/Frankel axioms and the axiom of choice.
Finite and transfinite sets, Cantor's theorem.
Functions and Relations:
Relations on sets, including transitive, symmetric, and reflexive relations.
Partial orders, equivalence relations, and partitions.
Functions on sets, including compound functions.
The Number System:
The sets of Natural Numbers and Integers; well-foundedness and proofs by induction, ordinality and cardinality, countability, the Peano axioms.
The Rational Numbers; rational number arithmetic and the field axioms.
The Real Numbers; irrationality, algebraic and transcendental numbers, Dedekind cuts, and the non-denumerability of the reals.
Other number systems; algebraic versus geometric closure of a field, extension by radicals (e.g., the Gaussian integers), transfinite ordinals and cardinals.
This is a traditional lecture course, but with a significant degree of classroom interaction encouraged and collaborative (group-learning) projects and assignments will be frequent. Students will use computers in and out of class to write their own computable functions and apply these programming techniques to solve problems in other topics in the course.
Student Goals and Objectives of the Course
The primary objective of the course is to prepare students for upper-division coursework in mathematics. Students will be able to
Comprehend and express themselves clearly in the language of modern mathematics, including first-order logic and formal set theory.
Employ the most common problem solving techniques and methods of proof needed in advanced coursework.
Understand the axiomatic foundations of the mathematics they have previously learned, and be able to approach the study of new topics such as modern algebra, number theory, and analysis using an axiomatic framework and the expository cycle of "definition-theorem-proof."
Assessment Measures
Graded tasks will include individual homework, quizzes, and written exams, including a cumulative final. Additional assessment measures may include collaborative projects or homework. | 677.169 | 1 |
Web Site Webmath.com This is a dynamic math website where students enter problems and where the site's math engine solves the problem. Students in most cases are given a step-by-... Curriculum: Mathematics Grades: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
8.
Web Site Prentice Hall Math Textbook Resources This site has middle school and high school lesson quizzes, vocabulary, chapter tests and projects for most chapters in each textbook. In some sections, ther... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12
9.
Web Site Dave's Short Trig Course Check out the short trigonometry course and learn the new way of learning trig. This short course breaks into sections and allows user to learn at his/her o... Curriculum: Mathematics Grades: 9, 10, 11, 12 | 677.169 | 1 |
Combining theory, methods and instructional activities in one convenient volume, Heddens, Speers and Brahier's Twelfth Edition of "Today's Mathematics" provides a valuable set of ideas and reference materials for actual classroom use. This combined coverage of content and methods creates a long-lasting resource, helping pre-service and in-service teachers see the relationship between what they teach and how they teach. Reflecting recent recommendations from the NCTM Standards, the text emphasizes how to introduce a concept at a given level to expand and reinforce it at successive levels.
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Rent Today's Mathematics, (Shrinkwrapped with CD inside envelop inside front cover of Text) 12th edition today, or search our site for other textbooks by James W. Heddens. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Wiley. | 677.169 | 1 |
BEGINNING ALGEBRA
9780131444447
ISBN:
0131444441
Edition: 4 Pub Date: 2004 Publisher: Prentice Hall
Summary: Clearly explained concepts, study skills help, and real-life applications will help the reader to succeed in learning algebra.
Martin-Gay, K. Elayn is the author of BEGINNING ALGEBRA, published 2004 under ISBN 9780131444447 and 0131444441. One hundred forty four BEGINNING ALGEBRA textbooks are available for sale on ValoreBooks.com, forty one used from the cheapest price of $0.65, or buy new starting at $13.9...90131444461-2-0-1 Orders ship the same or next business day. Expedite [more]
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Read More have been consistently praised for being at just the right level for precalculus students. The book also provides calculator examples, including specific keystrokes that show how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares readers for further courses in mathematics Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780534931902931902.
Customer Reviews
Algebra and Trigonometry with Analytic Geometry
by Earl William Swokowski
An ordinary text
This is a common secondary level text. Like most current texts of this type, it fails to give the reader any understanding of the breadth of this subject as developed over the last 150 years. There are a number of useful tricks of interest to engineers, computer graphics specialists, and other 3D modelers that this text does not even point to for additional reading. However, for its intended audience, it delivers adequately but still does not entice the imagination | 677.169 | 1 |
1. Classify, analyze and evaluate relations and functions. 2. Use the definition of radian measure to calculate arc length, radius, and angles on circles. 3. Measure angles, and calculate coterminal and reference angles, in both degrees and radians. 4. Calculate by hand, and using a calculator, the six basic trigonometric functions of angles, both in right triangles and on the unit circle. 5. Use basic trigonometric functions and their inverses to solve applications involving right triangles. 6. Graph basic trigonometric and inverse trigonometric functions. 7. Solve trigonometric equations, and prove and apply trigonometric identities. 8. Use sum and product of angle formulas and the laws of sines and cosines to solve applications involving triangles. 9. Cacluate sums and multiples of vectors, projections onto other vectors, dot products, and angles between vectors. 10. Graph and analyze parametric equations. 11. Graph and analyze polar equations. 12. Use technology to graph trigonometric functions and inverses, and to solve equations involving themMethods of presentation can include lectures, discussion, demonstration, experimentation, audiovisual aids, group work, and regularly assigned homework. Calculators / computers will be used when appropriate. Course may be taught as face-to-face, media-based, hybrid or online course.
VIII. Course Practices Required
(To be completed by instructor)
IX. Instructional Materials
Note: Current textbook information for each course and section is available on Oakton's Schedule of Classes.
Within the Schedule of Classes, textbooks can be found by clicking on an individual course section and looking for the words "View Book Information".
Textbooks can also be found at our Mathematics Textbooks page.
X. Methods of Evaluating Student Progress
(To be completed by instructor)
Evaluation methods can include assignments, quizzes, chapter or major tests, individual or group projects, computer assignments and/or a final examination | 677.169 | 1 |
Ithaca College, in New York, has developed and tested a projects-based first-year calculus course over the last 3 years which uses the graphs of functions and physical phenomena to illustrate and motivate the major concepts of calculus and to introduce students to mathematical modeling. The course curriculum is designed to: (1) emphasize on the unity of calculus; (2) focus on the effective teaching of the central concepts of calculus; (3) increase geometric understanding; (4) teach students to be good problem solvers; and (5) improve attitudes toward mathematics. The course centers on large, often open-ended, problems upon which students work both in and outside of class in groups, and individually, spending 2 to 3 weeks on each problem. Most of these projects are presented in such a way as to require a top-down analysis, in which the top level forces attention to a main idea, while the computations are required at the lowest levels. This approach enables students to recognize calculations as the "nuts and bolts" of a larger problem-solving process. Students' active participation, and clear written presentations of results are required. The course design is best represented by a spiral, which emphasizes the unity of calculus, while allowing for the continual review of the discipline's key skills and concepts, such as graphing, distance and velocity, multiple representations of functions, modeling, and top-down methodology. Three sample course problems are provided. (MAB) | 677.169 | 1 |
Prentice Hall Math Course 2 Study Guide and Practice Workbook 2004c 0131254561 WE HAVE NUMEROUS COPIES-PAPERBACK new in factory box.
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Very Good Unused workbook-no pages filled in; Unless specifically stated as present, assume no CD, DVD, access code or other support materials is available.
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About the Book
A math text creates a path for students - one that should be easy to navigate, with clearly marked signposts, built-in footholds, and places to stop and assess progress along the way. Research-based and updated for today's classroom, Prentice Hall Mathematics is that well-constructed path. An outstanding author team and unmatched continuity of content combine with timesaving support to help teachers guide students along the road to success. | 677.169 | 1 |
Details about Practical Problems in Mathematics for Masons:
Gain the math skills you need to succeed in the masonry with Practical Problems in Mathematics for Masons. Using a straightforward writing style and simple, step-by-step explanations this text is extremely reader-friendly. The book begins with basic arithmetic and then, once these basic topics have been mastered, progresses to algebra and then trigonometry. Practical Problems in Mathematics for Masons provides readers with realistic mathematical problems from the field providing a solid foundation for a career in masonry. This is the perfect resource for anyone entering the masonry industry, or simply looking to brush up on the necessary math.
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Rent Practical Problems in Mathematics for Masons 2nd edition today, or search our site for other textbooks by John Ball. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Delmar Cengage Learning. | 677.169 | 1 |
principle, Algebra 1 and some modest extensions are all the math background that is needed for this part of physics. I find however, that some students - even some with good grades in math - just get lost in a "forest" of algebra and cannot see the physics ideas that are the essence of the ma... | 677.169 | 1 |
A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video...
see more
A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video programs and coordinated books In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus In Simplest Terms to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Algebra In Simplest Terms
Select this link to open drop down to add material Algebra In Simplest Basics of probability and statistics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Basics of probability and statistics
Select this link to open drop down to add material Basics of probability and statistics Lecture Notes to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Calculus Lecture Notes
Select this link to open drop down to add material Calculus Lecture Notesers and Computational Statistics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Computers and Computational Statistics
Select this link to open drop down to add material Computers and Computational StatisticsNASA COLLABORATIVE 2007: Getting a Lift out of Calculus to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material CSUB-NASA COLLABORATIVE 2007: Getting a Lift out of Calculus
Select this link to open drop down to add material CSUB-NASA COLLABORATIVE 2007: Getting a Lift out of Calculus to your Bookmark Collection or Course ePortfolio
EnVision is a Web-based chat program that assists in the live communication of mathematical content. It is used by the author...
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EnVision is a Web-based chat program that assists in the live communication of mathematical content. It is used by the author to conduct online office hours in introductory courses. It allows students to log in anonymously and thus reduce anxieties that they may have about seeking assistanceVision: A tool for live web-based communication in the mathematical sciences to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material EnVision: A tool for live web-based communication in the mathematical sciences
Select this link to open drop down to add material EnVision: A tool for live web-based communication in the mathematical sciences to your Bookmark Collection or Course ePortfolio
Fraction Fundamentals is a Stand Alone Instructional Resource (StAIR) that teaches students basic concepts about fractions...
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Fraction Fundamentals is a Stand Alone Instructional Resource (StAIR) that teaches students basic concepts about fractions and tests their knowledge. This resource is suitable for 3rd grade students, but may be used in other grade levels if needed. By going through this PowerPoint, students learn what fractions are, where they see fractions, equivalent fractions, and comparing fractions. There are multiple choice and true/false questions throughout the presentation to assess student learning. Audio is avaliable on most slides to assist non-readers or visually impairedraction Fundamentals to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Fraction Fundamentals
Select this link to open drop down to add material Fraction in Art and Architecture to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Geometry in Art and Architecture
Select this link to open drop down to add material Geometry in Art and Architecture to your Bookmark Collection or Course ePortfolio
The author offers reflections on specific questions mathematicians and philosophers have asked about the infinite over the...
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The author offers reflections on specific questions mathematicians and philosophers have asked about the infinite over the centuries. He examines why explorers of the infinite, even in its strictly mathematical forms, often find it to be sublime Infinite Reflections to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Infinite Reflections
Select this link to open drop down to add material Infinite Reflections to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics. Convex Analysis introduces... more...
Clear, rigorous definitions of mathematical terms are crucial to good scientific and technical writing-and to understanding the writings of others. Scientists, engineers, mathematicians, economists, technical writers, computer programmers, along with teachers, professors, and students, all have the need for comprehensible, working definitions of mathematical... more...
"Krantz is a very prolific writer. He ? creates excellent examples and problem sets." ?Albert Boggess, Professor and Director of the School of Mathematics and Statistical Sciences, Arizona State University, Tempe, USA Designed for a one- or two-semester undergraduate course, Differential Equations: Theory, Technique and Practice, Second Edition... more...
This text provides a systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. Each chapter includes illustrative examples and a collection of exercises. more...
The subject of real analysis dates to the mid-nineteenth century - the days of Riemann and Cauchy and Weierstrass. Real analysis grew up as a way to make the calculus rigorous. Today the two subjects are intertwined in most people's minds. Yet calculus is only the first step of a long journey, and real analysis is one of the first great triumphs along... more...
You know mathematics. You know how to write mathematics. But do you know how to produce clean, clear, well-formatted manuscripts for publication? Do you speak the language of publishers, typesetters, graphics designers, and copy editors? Your page design-the style and format of theorems and equations, running heads and section headings, page breaks,... more...
This book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral,... more...
This book takes the reader on a journey around the globe and through centuries of history, exploring the transformations that mathematical proof has undergone from its inception at the time of Euclid and Pythagoras to its versatile, present-day use | 677.169 | 1 |
Cranston Physics. Discreet mathematics is the study of things that can only exist is defined states or at defined times. It is similar to the difference between integers (discreet) and all real numbers (continuous) with integers being only certain numbers and real numbers also including everything in between.
Fred B.
I strive to develop a student's Algebra skills from the basics of Middle School math and Pre - Algebra. I like to use models that help a student visualize the concepts of Algebra for greater true understanding (and less memorization). I have succeeded at helping Algebra 2 students by framing the su...
Ross L | 677.169 | 1 |
This popular Physical Chemistry text book is now available in electronic format. We have preserved much of the material of the former hard copy editions, making changes to improve understanding of the concepts in addition to including some of the recent discoveries in physical chemistry. Many chapters have new sections and the coverage of several chapters has been greatly expanded. The chapter on statistical mechanics, 15, has been completely rewritten.
The eBook has also been divided into smaller modules that are appropriate for specific courses in Physical Chemistry.
Easy to use Clebsch-Gordan coefficient solver for adding two angular momentums in Quantum Mechanics. This tool is created for my Quantum Mechanics II course offered by Dr. Thompson in Summer of 2007.
[Instruction]
Execute "GUI.m" script by invoking "GUI".
Inspired by a discussion with my father on how to solve sudokus, I decided to implement a GUI for MATLAB and play around with automatic solving. The result can be found here: You can use the GUI just for playing sudoku and having an online check or you may turn on the solving aids: Display tooltips showing all valid numbers so far, or have a semiautomatic or a automatic solver which evaluates the logical constraints. On top of that, a branching algorithm is implemented, which solves any arbitrary sudoku very fast.
Math Solver Free for Windows 8 is a handy tool for performing frequently used operations used for solving math problems. You can use this tool for solving quadratic equations or calculating the angles of a triangle.
The app also includes a unit converter and other useful tools for dealing with math problems by using your Windows 8 device.
Worksheet Generator for Chemistry is a handy and reliable software that helps you to easily and quickly create and customize your personal chemistry worksheets.
The application provides you with various exercise templates that allow you to adjust your worksheets. You are able to insert various chemistry exercises of different areas such as units and chemical formulae, thermochemistry, chemical kinetics, Redox reactions and organic chemistry | 677.169 | 1 |
Electromotive Force (EMF),
Active and Passive Elements,
Concept of Potential,
Potential Difference,
Ohm's Law,
Resistance,
Effect of Temprature on Resistance,
Temprature Coefficient of Resistance,
Power Dissipated in a Resistor,
Series connection of Resistors,
Parallel Connection of Resistors,
Short Circuits and Open Circuits,
and other topics.
Introduction to tracing curves,
Point of intersection with Axes,
Critical Points and Concavity,
Tracing a Parabola,
Transformations,
Symmetry,
Region of non-existence,
Tracing a Circle,
Tracing a Cubic Curve (point of Inflection),
and other topics.
Concept of Direction Cosines,
Direction Ratios,
Distance between 2 points and DRs of line Joining them,
Angle between Lines,
Equation of a Line,
Equation of a Plane,
Angle between Planes,
Prependicular distance from a point to a plane,
and other topics.
Concept of Equlibrium,
Eqilibrium of a 2-force System,
Eqilibrium of 3-Force System,
Free Body Diagram,
Reactions due to Supports and Connections,
Ex: Beam on a Hinge and Roller Support,
General Steps for solution of Equilibrium related Problems,
and other topics.
Introduction to Trusses,
Types of Trusses,
Method of Joints,
Ex: using Method of Joints,
Steps for Analysis by Method of Joints,
Method of Sections,
Ex: Using Method of Sections,
Steps for Analysis by Method of Sections,
Ex: Finding an unknown Axial Force in a member,
and other topics. | 677.169 | 1 |
The Smith-Minton family
of textbooks includes Calculus: Early Transcendentals
and Calculus: Late Transcendentals ,
now out in their fourth editions,
Calculus: Concepts and Connections and
translations into various languages. A Calculus Essentials
textbook has been completed.
Intended for students majoring in mathematics,
physics, chemistry, engineering and related fields, these best-selling texts follow traditional
table of contents yet also address many of the goals of calculus reform.
A strong emphasis on problem-solving provides plenty of opportunity for the use of technology,
exploratory techniques and real-world applications. Key features
that make the Smith & Minton series outstanding include: Appropriate use of technology (graphing calculators and computer
algebra systems)
A rich variety of interesting and unique applications from sports, medicine and other areas
Extensive use of graphics and numerical tables to introduce and develop concepts
Careful derivations and clear explanations with helpful warnings about common mistakes
Flexible exercise sets with writing problems and projects in each section
Candid discussions of advantages and disadvantages of techniques covered
The readability of these texts make them ideal for use with clickers, group projects and other pedagogical
techniques for increasing student involvement and understanding. Take a
guided tour of the second edition. | 677.169 | 1 |
I can't imagine how long it must have taken to put this together, as every lesson is packed full of not just lecture material, but explanations for each problem, hints, etc.
Boggles the mind.
The lesson begins with a lecture.
While the tutor is narrating, the words appear on the screen.
At any time during the lesson…
You can pause, fast forward, or rewind. All this time, the tutor is very clearly explaining the concept the student is seeing.
After the lessons, the problems begin. They start out pretty simple…
Then get more and more difficult as the lesson progresses.
The tutor actually narrates the problems, and many problems offer an optional hint the student can choose to listen to. The hint doesn't solve the problem for them but gives them tools they can use to think about how to solve the problem.
Here's the website. You can find placement tests for your students on the website, and the sets come with both the CD-ROMs, a large spiral bound workbook, and an answer key. (You can also just buy the CD's and forgo the workbook.)
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Comments
We've used Saxon Math for years and years, but we've looked at this for my youngest…just haven't made up our minds about changing. We'll have to revisit it again for next year.
Theresa in Albera
Income tax!!! Not only are they learning math but learning about real life!!
Bridget in Minnesota
I love the feel of real books, but I can see how everyone would like this.
Have a good day!
jennie w.
When I started homeschooling my daughter in 7th grade and she was working on algebra we tried several math methods. SInce helping her with math is sort of like the blind helping the blind it was incredibly difficult. We tried out Teaching Textbooks and it was wonderful! It's pretty much foolproof since there is so much back-up help available. We've been really happy with it.
Chris
Have you checked out spelling city? It's a great one for spelling and vocabulary. plus it's free.
Sandra
We use Teaching Textbooks!! We LOVE it!!
Michele
We are not a very "math minded" family. I loathe it, actually; so does my daughter. This program has helped us survive. We LOVE our Teaching Textbooks!
SarahinSC
We don't homeschool, but I can see that this program would be excellent even for a supplement/summer time. Looks like I could learn a thing or two too!
Vicky
I don't homeschool, but we purchased Teaching Textbooks for my son because he needs additional help with Math. We buy the one for the upcoming grade, and Joseph does the course during summer. He actually enjoys it, and we've seen his math grades jump from D's to A's as a result!
pip
That lesson on income taxes is BRILLIANT! I work in a tax office and it amazes me at how many adults truly do not know what income taxes are or how it's calculated. They think its a big lottery and if they get a refund they won. Every year I say "i wish the high schools would teach a class about income taxes"
Colleen
After several happy and successful years with Miquon and Singapore, we began Teaching Textbooks Pre-Algebra with my seventh grader. I'm super pleased. My son might even say grudgingly that he likes it, although he finds the guys voice annoying.
Michelle
is there a Canadian version of this? I am homeschooling and math is NOT my strong suit. NOt a big deal now as he is only little and the math is ok for me but as he (there are a lot of He(s) ) I will need more "extra" help!
Oh yes, we are Teaching Textbook converts here – couldn't imagine using anything else now. It's like having a math tutor who comes to your house (but you don't have to clean up for him). Love it!
Tasha
We've used TT for three years. Our kids LOVE it! For any parent who isn't gifted in the math department…it's a great help!
Perfect Dad
Do your children work through it by themselves, or are you there helping them? If it works well enough that the child can do it themselves then I can imagine that we should run out and buy it IMMEDIATELY!
sparky
Interesting that the jobs in the math problems are all low wage, service jobs.
ChristinaB
We started using TT this year. LOVE IT! My daughter has flown through 5th grade math and will be starting 6th grade math in a few days. I am totally sold on the program and will be using them FOREVER. The three little brothers will be using them too, as soon as they hit 3rd grade. Best thing, though — my 13 year-old autistic son can do these all by himself. :)
Oh, and I love the narrators' voices — so pleasant, calm, and not patronizing. I heart TT.
ChristinaB
My two kids that use it (ages 11 and 13) do it completely on their own. And one of them is autistic. I just look at their gradebooks when they are done, and it shows me exactly which problems they missed, and gives a score for each assignment. Yes, go to their website and buy it NOW.
jodi
This seems like a really cool program. I wish we had it for my classroom for computer time enrichment. The only problem is making sure they can do the problems by just reading them and not hearing them. I hate to be the standardized test goblin, but they will not be able to hear them out loud on the high stakes tests. Man it's testing season.
Lori
Love Spelling City!
Lori
Oh Sparky…..
Irene
This looks really cool! Is there anything similar out there for history?
Natalie V2
So… it gets more difficult as the lesson goes on, but gives the tools to learn how to solve the problems?
If it can walk me though filing our income taxes in a friendly lesson, then I think I know what I need to order asap!
Tricia
Ree,
We just started homeschooling this year and chose Teaching Textbooks to use for our math program and my girls LOVE it!! And better yet, I LOVE IT!!! We are using Math 4 and 7. THANK GOODNESS for these!!!
N S
I started using TT with my daughter this year and it has made ALL THE DIFFERENCE IN THE WORLD in our homeschool. She does two or more lessons a day just because it's fun and is a grade ahead. We love Teaching Textbooks! | 677.169 | 1 |
College Algebra with Smart CD (Windows Barnett, Ziegler, Byleen College Algebra/Precalculus series is designed to be user friendly and to maximize student comprehension. The goal of this series is to emphasize computational skills, ideas, and problem solving rather than mathematical theory. The large number of pedagogical devices employed in this text will guide a student through the course. Integrated throughout the text, the students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematically concepts. In each section, the worked examples are followed by matched problems that reinforce the concept being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A Smart CD is packaged with the seventh edition of the book. This CD tutorial reinforces important concepts, and provides students with extra practice problems.
1 Basic Algebraic Operations
1-1 Algebra and Real Numbers
1-2 Polynomials: Basic Operations
1-3 Polynomials: Factoring
1-4 Rational Expressions: Basic Operations
1-5 Integer Exponents
1-6 Rational Exponents
1-7 Radicals
Chapter 1 Group Activity: Rational Number Representations
Chapter 1 Review
2 Equations and Inequalities
2-1 Linear Equations and Applications
2-2 Systems of Linear Equations and Applications
2-3 Linear Inequalities
2-4 Absolute Value in Equations and Inequalities
2-5 Complex Numbers
2-6 Quadratic Equations and Applications
2-7 Equations Reducible to Quadratic Form
2-8 Polynomial and Rational Inequalities
Chapter 2 Group Activity: Rates of Change
Chapter 2 Review
1&2 Cumulative Review Exercises
3 Graphs and Functions
3-1 Basic Tools: Circles
3-2 Straight Lines
3-3 Functions
3-4 Graphing Functions
3-5 Combining Functions
3-6 Inverse Functions
Chapter 3 Group Activity: Mathematical Modeling in Business
Chapter 3 Review
4 Polynomials and Rational Functions
4-1 Polynomial Functions and Graphs
4-2 Finding Rational Zeros of Polynomials
4-3 Approximating Real Zeros of Polynomials
4-4 Rational Functions
4-5 Partial Functions
Chapter 4 Group Activity: Interpolating Polynomials
Chapter 4 Review
5 Exponential and Logarithmic Functions
5-1 Exponential Functions
5-2 The Exponential Function with Base e
5-3 Logarithmic Functions
5-4 Common and Natural Logarithmic Functions
5-5 Exponential and Logarithmic Equations
Chapter 5 Group Activity: Growth of Increasing Functions
Chapter 5 Review
3, 4, & 5 Cumulative Review Exercises
6 Systems of Equations and Inequalities
6-1 Systems of Linear Equations and Augmented Matrices
6-2 Gauss-Jordan Elimination
6-3 Systems Involving Second-Degree Equations
6-4 Systems of Linear Inequalities in Two Variables
6-5 Linear Programming
Chapter 6 Group Activity: Modeling with Systems of Equations
Chapter 6 Review
7 Matrices and Determinants
7-1 Matrices: Basic Operations
7-2 Inverse of a Square Matrix
7-3 Matrix Equations and Systems of Linear Equations
7-4 Determinants
7-5 Properties of Determinants
7-6 Cramer's Rule
Chapter 7 Group Activity: Using Matrices to Find Cost, Revenue, and Profit | 677.169 | 1 |
Calculus AB Syllabus Introduction Our study of calculus, the mathematics of motion and change, is divided into two major topics: differential and integral calculus.
Algebra I HSCEs - Alignment Worksheet 3 - Complete L1.2.4 Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data ...
Piecing Together PiecewiseFunctionsPiecewisefunctions, typically introduced in Algebra 2, help set the foundation for deeper explorations of the graphs of various ...
AP Calc Summer Assignments L AP Calculus AB Course Course Design and Philosophy It is my belief that students gain a deeper understanding of mathematics if they have ... | 677.169 | 1 |
Math Objectives Covered in this App:
8.NS1 KNOW THAT THERE ARE NUMBERS THAT ARE NOT RATIONAL, AND APPROXIMATE THEM BY RATIONAL NUMBERS.
8.EE1 WORK WITH RADICALS AND INTEGER EXPONENTS.
8.EE2 UNDERSTAND THE CONNECTIONS BETWEEN PROPORTIONAL RELATIONSHIPS, LINES, AND LINEAR EQUATIONS.
8.EE3 ANALYZE AND SOLVE LINEAR EQUATIONS AND PAIRS OF SIMULTANEOUS LINEAR EQUATIONS.
8.F1 DEFINE, EVALUATE, AND COMPARE FUNCTIONS.
8.F2 USE FUNCTIONS TO MODEL RELATIONSHIPS BETWEEN QUANTITIES.
8.G1 UNDERSTAND CONGRUENCE AND SIMILARITY USING PHYSICAL MODELS, TRANSPARENCIES, OR GEOMETRY SOFTWARE.
8.G2 UNDERSTAND AND APPLY THE PYTHAGOREAN THEOREM.
8.G3 SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING VOLUME OF CYLINDERS, CONES, AND SPHERES.
8.SP1 INVESTIGATE PATTERNS OF ASSOCIATION IN BIVARIATE DATA | 677.169 | 1 |
Warner Springs CalculusBosko C.
...Bosko had 95th percentile on his GRE score. Bosko scored 95th percentile on his GMAT Quantitative. Example for Probability of Events:
Q: What is the probability of drawing 2 red cards (one after another) out of regular 52-card deck?
Uzair Q.
...The course typically begins with Euler's method and moves on to topics such as Ordinary Differential Equations, Second Order Equation Systems, and Laplace Transforms. I enjoy teaching this subject because it has tremendous real-world application in areas such as circuit-design, physics, and population growth. I have founded and sold my own company.
Genevieve G | 677.169 | 1 |
Catalog Description
Prerequisites
TISP math completed and two years of high school algebra and one year of geometry
or MATH 1314: College Algebra
Course Curriculum
Basic Intellectual Compentencies in the Core Curriculum
Reading
Writing
Speaking
Listening
Critical thinking
Computer literacy
Perspectives in the Core Curriculum
Establish broad and multiple perspectives on the individual in relationship to the
larger society and world in which he/she lives, and to understand the responsibilities
of living in a culturally and ethnically diversified world.
Stimulate a capacity to discuss and reflect upon individual, political, economic,
and social aspects of life in order to understand ways in which to be a responsible
member of society.
Recognize the importance of maintaining health and wellness.
Develop a capacity to use knowledge of how technology and science affect their lives.
Develop personal values for ethical behavior.
Develop the ability to make aesthetic judgments.
Use logical reasoning in problem solving.
Integrate knowledge and understand the interrelationships of the scholarly disciplines.
To use appropriate technology to enhance mathematical thinking and understanding and
to solve mathematical problems and judge the reasonableness of the results.
To interpret mathematical models such as formulas, graphs, tables and schematics,
and draw inferences from them.
To recognize the limitations of mathematical and statistical models.
To develop the view that mathematics is an evolving discipline, interrelated with
human culture, and understand its connections to other disciplines.
Instructional Goals and Purposes
Panola College's instructional goals include 1) creating an academic atmosphere in
which students may develop their intellects and skills and 2) providing courses so
students may receive a certificate/an associate degree or transfer to a senior institution
that offers baccalaureate degrees.
General Description of Each Lecture or Discussion
After studying the material presented in the text(s), lecture, laboratory, computer
tutorials, and other resources, the student should be able to complete all behavioral/learning
objectives listed below with a minimum competency of 70%.
Sets, Linear Equations, and Functions
1. Give an example of and/or use in an applied situation the following symbols and
terms a. set builder (set specification) notation b. null or empty set c. element
d. universal set e. subset f. proper subset g. equality of sets h. total number of
possible subsets (and proper and nonempty) of a given set
9. Find the sum of a specified number of terms of a given geometric sequence.
10. Compute the amount (future value) of an ordinary annuity.
11. Compute the present value of an ordinary annuity.
12. Compute the regular payments required to amortize a debt.
13. Compute the amount that must be invested periodically in a sinking fund to discharge
a debt or other financial obligation at some specified time in the future.
Introduction to Probability (As Time Permits)Upon completion of this chapter, the student will be able to correctly
1. Apply the multiplication rule to find the number of ways an event can happen.
2. Determine the number of permutations ofnthings takenrat a time (both with and without repetition),nPr.CRS: V-A-1
3. Determine the number of permutations of n given objects when p of the n objects
are alike and of one kind, q of the objects are alike of a second kind, ..., up to
t others alike of still another kind.CRS: V-A-1
4. Determine the number of circular permutations of n distinct objects.CRS: V-A-1
5. Determine the number of combinations of n distinct objects taken r at a time,nCr.CRS: V-A-16.Optional:Use combinations to expand a binomial by the binomial theorem.
7.Optional:Find a specified term of a given binomial raised to a given power without expansion.
Upon completion of this section, the student will be able to correctly
Methods of Instruction/Course Format/Delivery
Methods employed will include Lecture/demonstration, discussion, problem solving,
analysis, and reading assignments.Homework will be assigned.Faculty may choose from, but are not limited to, the following methods of instruction:
(1) Lecture
(2) Discussion
(3) Internet
(4) Video
(5) Television
(6) Demonstrations
(7) Field trips
(8) Collaboration
(9) Readings
Assessment
Faculty may assign both in- and out-of-class activities to evaluate students' knowledge
and abilities. Faculty may choose from – but are not limited to - the following methods
Attendance
Book reviews
Class preparedness and participation
Collaborative learning projects
Compositions
Exams/tests/quizzes
Homework
Internet
Journals
Library assignments
Readings
Research papers
Scientific observations
Student-teacher conferences
Written assignments
Four (4)Major Examsat 15% each60%
Homework Notebook/Folder 10% Note: There will be no make-up exams. If you miss an exam your Final Exam percentage will be used as a substitute for
the missing grade. If youdo not missany exams, your one lowest Exam grade will be replaced by the Final Exam percentage
provided it (the Final Exam percentage) is higher. | 677.169 | 1 |
Subject: Mathematics (8 - 12) Title: Graphing Stations Description: This activity will be used to review all the forms of linear equations and to review graphing linear, their parallels and their perpendiculars. All the forms of linear equations will be reviewed for testing purposes. Students will participate in a round robin activity to help students master objectives before unit test.This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
Thinkfinity Lesson Plans
Subject: Mathematics Title: Graph ChartAdd Bookmark Description: This reproducible transparency, from an Illuminations lesson, contains the answers to the similarly named student activity in which students identify the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 Title: Movement with FunctionsAdd Bookmark Description: In this unit of 3 lessons from Illuminations, students use movement to reinforce the concepts of linear functions and systems of equations. Multiple representations are used throughout, along with tools such as motion detectors and remote-controlled cars. Students explore how position, speed, and varying motion are reflected in graphs, tables, and algebraic equations. Thinkfinity Partner: Illuminations Grade Span: 6,7,8,9,10,11,12 | 677.169 | 1 |
Reflecting the latest New York State curriculum change, this brand-new addition to Barron's Let's Review series covers all topics prescribed by the New York State Board of Regents for the new Integrated Algebra Regents exam, which replaces the Math A Regents exam. This book stresses rapid learning, using many step-by-step demonstration examples, helpful diagrams, enlightening "Math Fact" summaries, and graphing calculator approaches. Fourteen chapters review the following topics: sets, operations, and algebraic language; linear equations and formulas; problem solving and technology; ratios, rates, and proportions; polynomials and factoring; rational expressions and equations; radicals and right triangles; area and volume; linear equations and graphing; functions, graphs, and models; systems of linear equations and inequalities; quadratic and exponential functions; statistics and visual representations of data; and counting and probability of compound events. Exercise sections within each chapter feature a large sampling of Regents-type multiple-choice and extended response questions, with answers at the back of the book. Students will find this book helpful when they need additional explanation and practice on a troublesome topic, or when they want to review specific topics before taking a classroom test or the Regents exam. Teachers will value it as a lesson-planning aid, and as a source of classroom exercises, homework problems, and test questions.
The author is hampered by the fact that the Integrated Algebra is a brand new regents. There is no prior history. At the end of the volume, there is a good sample exam which emulates the actual ones to be administered in January and June of 2008. In short, don't throw out your old Math A book because it contains more pertinent regents practice questions.
A cursory review of the new volume disclosed that the content is similar to the Math A. The sample exam is slightly more complicated and theoretical.
Parts 2 and 3 of the new sample exam are significantly more complex than the comparable Math A. I suspect that the initial administrations of this new exam will produce a curve which reflects the learning transfer in the new course.
The examples in the new Integrated Algebra are pertinent. There are plenty of illustrations on how to utilize the statistical calculator to solve a plethora of problems-particularly in probability and statistics.
From a review of the sample exam, I believe that candidates should endeavor to maximize their score on Part I of the exam which is fairly comparable to the current Math A.
The new volume is replete with plenty of pertinent examples in polynomials, number theory, equations, elementary trigonometry and probability/statistical inference amongst many other topics. The acquisition would be a good value for the price charged. Students would benefit from this volume because it contains sufficient illustrations to amplify the concepts tested on the "live regents exam". Practice is helpful because the new regents criteria requires a minimum passing grade of 65 in all the regents subjects pertinent to obtaining a diploma in NYS.
The new Integrated Algebra matter will be tested again in one form or another on the Collegiate SAT.
This book is a complete solution for Regents and SAT. In fact, I believe it is a bit too difficult for Regents as standard of Regents is lower than SAT. The book has a lot of word problems, which is an excellent source for SAT students. I highly recommend it. | 677.169 | 1 |
2008 | ISBN-10: 1604564296 | 178 Pages | PDF | 3 MB This book describes in detail a series of new strategies to solve problems, mainly in mathematics. New techniques are presented which have been test...
ISBN: 9814725269 | 2015 | PDF | 276 Pages | 3 MB This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is writte...
English | Jan. 22, 2016 | ASIN: B01AZB6IBI | 1290 Pages | PDF | 10.43 MB The story of science is a fascinating one and, whatever the difficulties it needs to be told. Science and history continue to b...
English | 3 Dec. 2009 | ISBN: 0521588073 | 612 Pages | PDF | 20 MB This classic text is known to and used by thousands of mathematicians and students of mathematics throughout the world. It gives an i...
English | 2008 | ISBN-10: 3528031360 | PDF | 300 Pages | 2,6 MB This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homologi...
English | 1963 | ISBN: 1258259699 | PDF | 127 Pages | 6 mb Since the extensive use of tensors by Einstein, important applications in other fields, such as differential geometry, classical mechanics, a...
English | 2011-08-26 | ISBN: 0817682554 | PDF | 260 Pages | 1,5 MB Using an original mode of presentation, and emphasizing the computational nature of the subject, this book explores a number of the u... | 677.169 | 1 |
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DiveCd-ROM for SaxonAlgebra22nd & 3rdEdition
Teaches each of the 129 Lessons in the Saxon Math textbook. Each lecture is fifteen to twen- ty minutes long. Can be used with the Saxon Algebra 2 2nd or 3rd Edition Home Study Kit. A Solutions Manual is required. The only dif- ference between the 2nd...
From the publisher: WHAT'S ON THE DIVECD 120 LESSON VIDEO LECTURES 10 INVESTIGATION VIDEO LECTURES WEEKLY ASSIGNMENT CHART QUESTION AND ANSWER EMAIL SERVICE FREE CLEP PROFESSOR FOR COLLEGE ALGEBRA WHAT YOU NEED SAXONALGEBRA22ND OR 3RDEDITION HOME | 677.169 | 1 |
From modern-day challenges such as balancing a checkbook, following the stock market, buying a home, and figuring out credit card finance charges to appreciating historical developments by Pythagoras, Archimedes, Newton, and other mathematicians, this engaging resource addresses more than 1,000 questions related to mathematics. Organized into chapters that cluster similar topics in an easily accessible format, this reference provides clear and concise explanations about the fundamentals of algebra, calculus, geometry, trigonometry, and other branches of mathematics. It contains the latest mathematical discoveries, including newly uncovered historical documents and updates on how science continues to use math to make cutting-edge innovations in DNA sequencing, superstring theory, robotics, and computers. With fun math facts and illuminating figures, The Handy Math Answer Book explores the uses of math in everyday life and helps the mathematically challenged better understand and enjoy the magic of numbers"-- Provided by publisher. | 677.169 | 1 |
Chelmsford StatisticsRobert Z.
Introductory Algebra is the most important math course most secondary students will take. Not only does it lay the foundation for more advanced courses, it also teaches the student how to take real-life problems and translate them into the language of mathematics where they can be solved by some si... | 677.169 | 1 |
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This book provides an introduction to the theory of relativity and the mathematics used in its processes. Three elements of the book make it stand apart from previously published books on the theory of relativity. First, the book starts at a lower mathematical level than standard books with tensor calculus of sufficient maturity to make it possible to give detailed calculations of relativistic predictions of practical experiments. Self-contained introductions are given, for example vector calculus, differential calculus and integrations. Second, in-between calculations have been included, making it possible for the non-technical reader to follow step-by-step calculations. Thirdly, the conceptual development is gradual and rigorous in order to provide the inexperienced reader with a philosophically satisfying understanding of the theory. The goal of this book is to provide the reader with a sound conceptual understanding of both the special and general theories of relativity, and gain an insight into how the mathematics of the theory can be utilized to calculate relativistic effects147.99
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An Introduction to Analysis, Second Edition
9781577662327
ISBN:
1577662326
Edition: 2nd Pub Date: 2002 Publisher: Waveland Pr Inc
Summary: An Introduction to Analysis, Second Edition provides a mathematically rigorous introduction to analysis of real-valued functions of one variable. The text is written to ease the transition from primarily computational to primarily theoretical mathematics. Numerous examples and exercises help students to understand mathematical proofs in an abstract setting, as well as to be able to formulate and write them. The mater...ial is as clear and intuitive as possible while still maintaining mathematical integrity. The author presents abstract mathematics in a way that makes the subject both understandable and exciting to students.
James R. Kirkwood is the author of An Introduction to Analysis, Second Edition, published 2002 under ISBN 9781577662327 and 1577662326. One hundred six An Introduction to Analysis, Second Edition textbooks are available for sale on ValoreBooks.com, five used from the cheapest price of $53.87, or buy new starting at $92.33 | 677.169 | 1 |
Calculus eText (258 pgs)
2.
2
Copyright c 2014 Jim Fowler and Bart Snapp
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license,
visit or send a letter to Creative Commons, 543 Howard Street,
5th Floor, San Francisco, California, 94105, USA. If you distribute this work or a derivative, include the history of the document.
The source code is available at:
This text is based on David Guichard's open-source calculus text which in turn is a modification and expansion of notes written by
Neal Koblitz at the University of Washington. David Guichard's text is available at
calculus/ under a Creative Commons license.
The book includes some exercises and examples from Elementary Calculus: An Approach Using Infinitesimals, by H. Jerome Keisler,
available at under a Creative Commons license.
This book is typeset in the Kerkis font, Kerkis c Department of Mathematics, University of the Aegean.
We will be glad to receive corrections and suggestions for improvement at fowler@math.osu.edu or snapp@math.osu.edu.
7.
How to Read Mathematics
Reading mathematics is not the same as reading a novel. To read mathematics you
need:
(a) A pen.
(b) Plenty of blank paper.
(c) A willingness to write things down.
As you read mathematics, you must work alongside the text itself. You must write
down each expression, sketch each graph, and think about what you are doing.
You should work examples and fill in the details. This is not an easy task; it is in
fact hard work. However, mathematics is not a passive endeavor. You, the reader,
must become a doer of mathematics.
8.
0 Functions
0.1 For Each Input, Exactly One Output
Life is complex. Part of this complexity stems from the fact that there are many
relations between seemingly independent events. Armed with mathematics we seek
to understand the world, and hence we need tools for talking about these relations. Something as simple as a dictionary could be thought
of as a relation, as it connects words to definitions.
However, a dictionary is not a function, as there
are words with multiple definitions. On the other
hand, if each word only had a single definition, then
it would be a function.
A function is a relation between sets of objects that can be thought of as a
"mathematical machine." This means for each input, there is exactly one output.
Let's say this explicitly.
Definition A function is a relation between sets, where for each input, there
is exactly one output.
Moreover, whenever we talk about functions, we should try to explicitly state
what type of things the inputs are and what type of things the outputs are. In
calculus, functions often define a relation from (a subset of) the real numbers to (a
subset of) the real numbers.
While the name of the function is technically "f ," we
will abuse notation and call the function "f (x)" to
remind the reader that it is a function.
Example 0.1.1 Consider the function f that maps from the real numbers to
the real numbers by taking a number and mapping it to its cube:
1 → 1
−2 → −8
1.5 → 3.375
9.
calculus 9
and so on. This function can be described by the formula f (x) = x3
or by the
plot shown in Figure 1.
Warning A function is a relation (such that for each input, there is exactly one
output) between sets and should not be confused with either its formula or its
plot.
• A formula merely describes the mapping using algebra.
• A plot merely describes the mapping using pictures.
−2 −1 1 2
−5
5
x
y
Figure 1: A plot of f (x) = x3
. Here we can see that
for each input (a value on the x-axis), there is exactly
one output (a value on the y-axis).
Example 0.1.2 Consider the greatest integer function, denoted by
f (x) = ⌊x⌋.
This is the function that maps any real number x to the greatest integer less
than or equal to x. See Figure 2 for a plot of this function. Some might be
confused because here we have multiple inputs that give the same output.
However, this is not a problem. To be a function, we merely need to check that
for each input, there is exactly one output, and this is satisfied.
−2 −1 1 2 3 4
−2
−1
1
2
3
x
y
Figure 2: A plot of f (x) = ⌊x⌋. Here we can see that
for each input (a value on the x-axis), there is exactly
one output (a value on the y-axis).
Just to remind you, a function maps from one set to another. We call the set a
function is mapping from the domain or source and we call the set a function is
mapping to the range or target. In our previous examples the domain and range
have both been the real numbers, denoted by R. In our next examples we show that
this is not always the case.
Example 0.1.3 Consider the function that maps non-negative real numbers
to their positive square root. This function is denoted by
f (x) =
√
x.
Note, since this is a function, and its range consists of the non-negative real
numbers, we have that
√
x2 = |x|.
10.
10
See Figure 3 for a plot of f (x) =
√
x.
Finally, we will consider a function whose domain is all real numbers except for
a single point.
Example 0.1.4 Consider the function defined by
f (x) =
x2
− 3x + 2
x − 2
This function may seem innocent enough; however, it is undefined at x = 2.
See Figure 4 for a plot of this function. −8 −6 −4 −2 2 4 6 8
−4
−2
2
4
x
y
Figure 3: A plot of f (x) =
√
x. Here we can see that
for each input (a non-negative value on the x-axis),
there is exactly one output (a positive value on the
y-axis).
−2 −1 1 2 3 4
−3
−2
−1
1
2
3
x
y
Figure 4: A plot of f (x) =
x2
− 3x + 2
x − 2
. Here we
can see that for each input (any value on the x-axis
except for x = 2), there is exactly one output (a value
on the y-axis).
13.
calculus 13
0.2 Inverses of Functions
If a function maps every input to exactly one output, an inverse of that function
maps every "output" to exactly one "input." While this might sound somewhat
esoteric, let's see if we can ground this in some real-life contexts.
Example 0.2.1 Suppose that you are filling a swimming pool using a garden
hose—though because it rained last night, the pool starts with some water in
it. The volume of water in gallons after t hours of filling the pool is given by:
v(t) = 700t + 200
What does the inverse of this function tell you? What is the inverse of this
function?
Here we abuse notation slightly, allowing v and t to
simultaneously be names of variables and functions.
This is standard practice in calculus classes.
Solution While v(t) tells you how many gallons of water are in the pool after
a period of time, the inverse of v(t) tells you how much time must be spent to
obtain a given volume. To compute the inverse function, first set v = v(t) and
write
v = 700t + 200.
Now solve for t:
t = v/700 − 2/7
This is a function that maps volumes to times, and t(v) = v/700 − 2/7.
Now let's consider a different example.
Example 0.2.2 Suppose you are standing on a bridge that is 60 meters above
sea-level. You toss a ball up into the air with an initial velocity of 30 meters
per second. If t is the time (in seconds) after we toss the ball, then the height
at time t is approximately h(t) = −5t2
+ 30t + 60. What does the inverse of this
function tell you? What is the inverse of this function?
Solution While h(t) tells you how the height the ball is above sea-level at an
instant of time, the inverse of h(t) tells you what time it is when the ball is at a
given height. There is only one problem: There is no function that is the inverse
14.
14
of h(t). Consider Figure 7, we can see that for some heights—namely 60 meters,
there are two times.
While there is no inverse function for h(t), we can find one if we restrict the
domain of h(t). Take it as given that the maximum of h(t) is at 105 meters and
t = 3 seconds, later on in this course you'll know how to find points like this with
ease. In this case, we may find an inverse of h(t) on the interval [3, ∞). Write
h = −5t2
+ 30t + 60
0 = −5t2
+ 30t + (60 − h)
and solve for t using the quadratic formula
t =
−30 ± 302 − 4(−5)(60 − h)
2(−5)
=
−30 ± 302 + 20(60 − h)
−10
= 3 ∓ 32 + .2(60 − h)
= 3 ∓ 9 + .2(60 − h)
= 3 ∓
√
21 − .2h
Now we must think about what it means to restrict the domain of h(t) to values
of t in [3, ∞). Since h(t) has its maximum value of 105 when t = 3, the largest
h could be is 105. This means that 21 − .2h ≥ 0 and so
√
21 − .2h is a real
number. We know something else too, t > 3. This means that the "∓" that
we see above must be a "+." So the inverse of h(t) on the interval [3, ∞) is
t(h) = 3 +
√
21 − .2h. A similar argument will show that the inverse of h(t) on
the interval (−∞, 3] is t(h) = 3 −
√
21 − .2h.
2 4 6
20
40
60
80
100
t
h
Figure 7: A plot of h(t) = −5t2
+ 30t + 60. Here we
can see that for each input (a value on the t-axis),
there is exactly one output (a value on the h-axis).
However, for each value on the h axis, sometimes
there are two values on the t-axis. Hence there is no
function that is the inverse of h(t).
2 4 6
20
40
60
80
100
t
h
Figure 8: A plot of h(t) = −5t2
+ 30t + 60. While
this plot passes the vertical line test, and hence
represents h as a function of t, it does not pass the
horizontal line test, so the function is not one-to-one.
We see two different cases with our examples above. To clearly describe the
difference we need a definition.
Definition A function is one-to-one if for every value in the range, there is
exactly one value in the domain.
15.
calculus 15
You may recall that a plot gives y as a function of x if every vertical line crosses
the plot at most once, this is commonly known as the vertical line test. A function is
one-to-one if every horizontal line crosses the plot at most once, which is commonly
known as the horizontal line test, see Figure 8. We can only find an inverse to a
function when it is one-to-one, otherwise we must restrict the domain as we did in
Example 0.2.2.
Let's look at several examples.
Example 0.2.3 Consider the function
f (x) = x3 one-to-one and f −1
(x) = 3
√
x. See Figure 9.
−2 −1 1 2
−2
−1
1
2
f (x)
f −1
(x)
x
y
Figure 9: A plot of f (x) = x3
and f −1
(x) = 3
√
x. Note
f −1
(x) is the image of f (x) after being flipped over
the line y = x.
Example 0.2.4 Consider the function
f (x) = x2 not one-to-one. However, it is one-to-one on the
interval [0, ∞). Hence we can find an inverse of f (x) = x2
on this interval, and it
is our familiar function
√
x. See Figure 10.
−2 −1 1 2
−2
−1
1
2
f (x)
f −1
(x)
x
y
Figure 10: A plot of f (x) = x2
and f −1
(x) =
√
x.
While f (x) = x2
is not one-to-one on R, it is one-to-
one on [0, ∞).
0.2.1 A Word on Notation
Given a function f (x), we have a way of writing an inverse of f (x), assuming it exists
f −1
(x) = the inverse of f (x), if it exists.
On the other hand,
f (x)−1
=
1
f (x)
.
16.
16
Warning It is not usually the case that
f −1
(x) = f (x)−1
.
This confusing notation is often exacerbated by the fact that
sin2
(x) = (sin(x))2
but sin−1
(x) (sin(x))−1
.
In the case of trigonometric functions, this confusion can be avoided by using
the notation arcsin and so on for other trigonometric functions.
17.
calculus 17
Exercises for Section 0.2
(1) The length in centimeters of Rapunzel's hair after t months is given by
ℓ(t) =
8t
3
+ 8.
Give the inverse of ℓ(t). What does the inverse of ℓ(t) represent? ➠
(2) The value of someone's savings account in dollars is given by
m(t) = 900t + 300
where t is time in months. Give the inverse of m(t). What does the inverse of m(t)
represent? ➠
(3) At graduation the students all grabbed their caps and threw them into the air. The height
of their caps can be described by
h(t) = −5t2
+ 10t + 2
where h(t) is the height in meters and t is in seconds after letting go. Given that this
h(t) attains a maximum at (1, 7), give two different inverses on two different restricted
domains. What do these inverses represent? ➠
(4) The number n of bacteria in refrigerated food can be modeled by
n(t) = 17t2
− 20t + 700
where t is the temperature of the food in degrees Celsius. Give two different inverses on
two different restricted domains. What do these inverses represent? ➠
(5) The height in meters of a person off the ground as they ride a Ferris Wheel can be modeled
by
h(t) = 18 · sin(
π · t
7
) + 20
where t is time elapsed in seconds. If h is restricted to the domain [3.5, 10.5], find and
interpret the meaning of h−1
(20). ➠
(6) The value v of a car in dollars after t years of ownership can be modeled by
v(t) = 10000 · 0.8t
.
Find v−1
(4000) and explain in words what it represents. ➠
20.
20
a − δ a a + δ
L − ε
L
L + ε
x
y Figure 1.2: A geometric interpretation of the (ε, δ)-
criterion for limits. If 0 < |x − a| < δ, then we have
that a − δ < x < a + δ. In our diagram, we see that
for all such x we are sure to have L −ε < f (x) < L +ε,
and hence |f (x) − L| < ε.
Example 1.1.1 Let f (x) = ⌊x⌋. Explain why the limit
lim
x→2
f (x)
does not exist.
−2 −1 1 2 3 4
−2
−1
1
2
3
x
y
Figure 1.3: A plot of f (x) = ⌊x⌋. Note, no matter
which δ > 0 is chosen, we can only at best bound
f (x) in the interval [1, 2]. With the example of f (x) =
⌊x⌋, we see that taking limits is truly different from
evaluating functions.
Solution The function ⌊x⌋ is the function that returns the greatest integer less
than or equal to x. Since f (x) is defined for all real numbers, one might be
tempted to think that the limit above is simply f (2) = 2. However, this is not the
case. If x < 2, then f (x) = 1. Hence if ε = .5, we can always find a value for x
(just to the left of 2) such that
0 < |x − 2| < δ, where ε < |f (x) − 2|.
On the other hand, lim
x→2
f (x) 1, as in this case if ε = .5, we can always find a
value for x (just to the right of 2) such that
0 < |x − 2| < δ, where ε < |f (x) − 1|.
We've illustrated this in Figure 1.3. Moreover, no matter what value one chooses
21.
calculus 21
for lim
x→2
f (x), we will always have a similar issue.
Limits may not exist even if the formula for the function looks innocent.
Example 1.1.2 Let f (x) = sin
1
x
. Explain why the limit
lim
x→0
f (x)
does not exist.
Solution In this case f (x) oscillates "wildly" as x approaches 0, see Figure 1.4.
In fact, one can show that for any given δ, There is a value for x in the interval
0 − δ < x < 0 + δ
such that f (x) is any value in the interval [−1, 1]. Hence the limit does not exist.
−0.2 −0.1 0.1 0.2
x
y
Figure 1.4: A plot of f (x) = sin
1
x
.
Sometimes the limit of a function exists from one side or the other (or both)
even though the limit does not exist. Since it is useful to be able to talk about this
situation, we introduce the concept of a one-sided limit:
Definition We say that the limit of f (x) as x goes to a from the left is L,
lim
x→a−
f (x) = L
if for every ε > 0 there is a δ > 0 so that whenever x < a and
a − δ < x we have |f (x) − L| < ε.
We say that the limit of f (x) as x goes to a from the right is L,
lim
x→a+
f (x) = L
if for every ε > 0 there is a δ > 0 so that whenever x > a and
x < a + δ we have |f (x) − L| < ε.
Limits from the left, or from the right, are collectively
called one-sided limits.
24.
24
(10) In the theory of special relativity, a moving clock ticks slower than a stationary observer's
clock. If the stationary observer records that ts seconds have passed, then the clock
moving at velocity v has recorded that
tv = ts 1 − v2/c2
seconds have passed, where c is the speed of light. What happens as v → c from below?
➠
28.
28
However, consider
f (x) =
3 if x = 2,
4 if x 2.
and g(x) = 2. Now the conditions of Theorem 1.2.3 are not satisfied, and
lim
x→1
f (g(x)) = 3 but lim
x→2
f (x) = 4.
Many of the most familiar functions do satisfy the conditions of Theorem 1.2.3.
For example:
Theorem 1.2.4 (Limit Root Law) Suppose that n is a positive integer. Then
lim
x→a
n
√
x = n
√
a,
provided that a is positive if n is even.
This theorem is not too difficult to prove from the definition of limit.
41.
calculus 41
Example 2.2.4 Give the horizontal asymptotes of
f (x) =
6x − 9
x − 1
Solution From our previous work, we see that lim
x→∞
f (x) = 6, and upon further
inspection, we see that lim
x→−∞
f (x) = 6. Hence the horizontal asymptote of f (x) is
the line y = 6.
It is a common misconception that a function cannot cross an asymptote. As
the next example shows, a function can cross an asymptote, and in this case this
occurs an infinite number of times!
Example 2.2.5 Give a horizontal asymptote of
f (x) =
sin(7x)
x
+ 4.
Solution Again from previous work, we see that lim
x→∞
f (x) = 4. Hence y = 4 is
a horizontal asymptote of f (x).
We conclude with an infinite limit at infinity.
Example 2.2.6 Compute
lim
x→∞
ln(x)
5 10 15 20
−1
1
2
3
4
x
y
Figure 2.6: A plot of f (x) = ln(x).
Solution The function ln(x) grows very slowly, and seems like it may have a
horizontal asymptote, see Figure 2.6. However, if we consider the definition of
the natural log
ln(x) = y ⇔ ey
= x
Since we need to raise e to higher and higher values to obtain larger numbers,
we see that ln(x) is unbounded, and hence lim
x→∞
ln(x) = ∞.
43.
calculus 43
2.3 Continuity
Informally, a function is continuous if you can "draw it" without "lifting your pencil."
We need a formal definition.
Definition A function f is continuous at a point a if lim
x→a
f (x) = f (a).
2 4 6 8 10
1
2
3
4
5
x
y
Figure 2.7: A plot of a function with discontinuities
at x = 4 and x = 6.
Example 2.3.1 Find the discontinuities (the values for x where a function is
not continuous) for the function given in Figure 2.7.
Solution From Figure 2.7 we see that lim
x→4
f (x) does not exist as
lim
x→4−
f (x) = 1 and lim
x→4+
f (x) ≈ 3.5
Hence lim
x→4
f (x) f (4), and so f (x) is not continuous at x = 4.
We also see that lim
x→6
f (x) ≈ 3 while f (6) = 2. Hence lim
x→6
f (x) f (6), and so
f (x) is not continuous at x = 6.
Building from the definition of continuous at a point, we can now define what it
means for a function to be continuous on an interval.
Definition A function f is continuous on an interval if it is continuous at
every point in the interval.
In particular, we should note that if a function is not defined on an interval, then
it cannot be continuous on that interval.
−0.2 −0.1 0.1 0.2
x
y
Figure 2.8: A plot of.
Example 2.3.2 Consider the function,
see Figure 2.8. Is this function continuous?
44.
44
Solution Considering f (x), the only issue is when x = 0. We must show that
lim
x→0
f (x) = 0. Note
−| 5
√
x| ≤ f (x) ≤ | 5
√
x|.
Since
lim
x→0
−| 5
√
x| = 0 = lim
x→0
| 5
√
x|,
we see by the Squeeze Theorem, Theorem 1.3.5, that lim
x→0
f (x) = 0. Hence f (x) is
continuous.
Here we see how the informal definition of continuity being that you can
"draw it" without "lifting your pencil" differs from the formal definition.
We close with a useful theorem about continuous functions:
Theorem 2.3.3 (Intermediate Value Theorem) If f (x) is a continuous func-
tion for all x in the closed interval [a, b] and d is between f (a) and f (b), then
there is a number c in [a, b] such that f (c) = d.
The Intermediate Value Theorem is most frequently
used when d = 0.
For a nice proof of this theorem, see: Walk, Stephen
M. The intermediate value theorem is NOT obvious—
and I am going to prove it to you. College Math. J. 42
(2011), no. 4, 254–259.
In Figure 2.9, we see a geometric interpretation of this theorem.
a c b
f (a)
f (c) = d
f (b)
x
y
Figure 2.9: A geometric interpretation of the Inter-
mediate Value Theorem. The function f (x) is contin-
uous on the interval [a, b]. Since d is in the interval
[f (a), f (b)], there exists a value c in [a, b] such that
f (c) = d.
Example 2.3.4 Explain why the function f (x) = x3
+ 3x2
+ x − 2 has a root
between 0 and 1.
Solution By Theorem 1.3.1, lim
x→a
f (x) = f (a), for all real values of a, and hence
f is continuous. Since f (0) = −2 and f (1) = 3, and 0 is between −2 and 3, by
the Intermediate Value Theorem, Theorem 2.3.3, there is a c ∈ [0, 1] such that
f (c) = 0.
This example also points the way to a simple method for approximating roots.
Example 2.3.5 Approximate a root of f (x) = x3
+ 3x2
+ x − 2 to one decimal
place.
Solution If we compute f (0.1), f (0.2), and so on, we find that f (0.6) < 0
and f (0.7) > 0, so by the Intermediate Value Theorem, f has a root between
0.6 and 0.7. Repeating the process with f (0.61), f (0.62), and so on, we find
45.
calculus 45
that f (0.61) < 0 and f (0.62) > 0, so by the Intermediate Value Theorem,
Theorem 2.3.3, f (x) has a root between 0.61 and 0.62, and the root is 0.6
rounded to one decimal place.
47.
3 Basics of Derivatives
3.1 Slopes of Tangent Lines via Limits
Suppose that f (x) is a function. It is often useful to know how sensitive the value of
f (x) is to small changes in x. To give you a feeling why this is true, consider the
following:
• If p(t) represents the position of an object with respect to time, the rate of change
gives the velocity of the object.
• If v(t) represents the velocity of an object with respect to time, the rate of change
gives the acceleration of the object.
• The rate of change of a function can help us approximate a complicated function
with a simple function.
• The rate of change of a function can be used to help us solve equations that we
would not be able to solve via other methods.
The rate of change of a function is the slope of the tangent line. For now, consider
the following informal definition of a tangent line:
Given a function f (x), if one can "zoom in" on f (x) sufficiently so that f (x) seems to be
a straight line, then that line is the tangent line to f (x) at the point determined by x.
We illustrate this informal definition with Figure 3.1.
The derivative of a function f (x) at x, is the slope of the tangent line at x. To find
the slope of this line, we consider secant lines, lines that locally intersect the curve
48.
48
x
y
Figure 3.1: Given a function f (x), if one can "zoom
in" on f (x) sufficiently so that f (x) seems to be a
straight line, then that line is the tangent line to
f (x) at the point determined by x.
at two points. The slope of any secant line that passes through the points (x, f (x))
and (x + h, f (x + h)) is given by
∆y
∆x
=
f (x + h) − f (x)
(x + h) − x
=
f (x + h) − f (x)
h
,
see Figure 3.2. This leads to the limit definition of the derivative:
Definition of the Derivative The derivative of f (x) is the function
d
dx
f (x) = lim
h→0
f (x + h) − f (x)
h
.
If this limit does not exist for a given value of x, then f (x) is not differentiable
at x.
x x + h
f (x)
f (x + h)
x
y
Figure 3.2: Tangent lines can be found as the limit
of secant lines. The slope of the tangent line is given
by lim
h→0
f (x + h) − f (x)
h
.
54.
54
These exercises are computational in nature.
(7) Let f (x) = x2
− 4. Use the definition of the derivative to compute f ′
(−3) and find the
equation of the tangent line to the curve at x = −3. ➠
(8) Let f (x) =
1
x + 2
. Use the definition of the derivative to compute f ′
(1) and find the
equation of the tangent line to the curve at x = 1. ➠
(9) Let f (x) =
√
x − 3. Use the definition of the derivative to compute f ′
(5) and find the
equation of the tangent line to the curve at x = 5. ➠
(10) Let f (x) =
1
√
x
. Use the definition of the derivative to compute f ′
(4) and find the equation
of the tangent line to the curve at x = 4. ➠
55.
calculus 55
3.2 Basic Derivative Rules
It is tedious to compute a limit every time we need to know the derivative of a
function. Fortunately, we can develop a small collection of examples and rules that
allow us to compute the derivative of almost any function we are likely to encounter.
We will start simply and build-up to more complicated examples.
The Constant Rule
The simplest function is a constant function. Recall that derivatives measure the
rate of change of a function at a given point. Hence, the derivative of a constant
function is zero. For example:
• The constant function plots a horizontal line—so the slope of the tangent line is 0.
• If p(t) represents the position of an object with respect to time and p(t) is constant,
then the object is not moving, so its velocity is zero. Hence
d
dt
p(t) = 0.
• If v(t) represents the velocity of an object with respect to time and v(t) is constant,
then the object's acceleration is zero. Hence
d
dt
v(t) = 0.
The examples above lead us to our next theorem. To gain intuition, you should compute the derivative
of f (x) = 6 using the limit definition of the derivative.
Theorem 3.2.1 (The Constant Rule) Given a constant c,
d
dx
c = 0.
Proof From the limit definition of the derivative, write
d
dx
c = lim
h→0
c − c
h
= lim
h→0
0
h
= lim
h→0
0 = 0.
57.
calculus 57
Example 3.2.3 Compute
d
dx
x13
.
Solution Applying the power rule, we write
d
dx
x13
= 13x12
.
Sometimes, it is not as obvious that one should apply the power rule.
Example 3.2.4 Compute
d
dx
1
x4
.
Solution Applying the power rule, we write
d
dx
1
x4
=
d
dx
x−4
= −4x−5
.
The power rule also applies to radicals once we rewrite them as exponents.
Example 3.2.5 Compute
d
dx
5
√
x.
Solution Applying the power rule, we write
d
dx
5
√
x =
d
dx
x1/5
=
x−4/5
5
.
The Sum Rule
We want to be able to take derivatives of functions "one piece at a time." The sum
rule allows us to do this. The sum rule says that we can add the rates of change
of two functions to obtain the rate of change of the sum of both functions. For
example, viewing the derivative as the velocity of an object, the sum rule states that
the velocity of the person walking on a moving bus is the sum of the velocity of the
bus and the walking person.
64.
4 Curve Sketching
Whether we are interested in a function as a purely mathematical object or in
connection with some application to the real world, it is often useful to know what
the graph of the function looks like. We can obtain a good picture of the graph
using certain crucial information provided by derivatives of the function and certain
limits.
4.1 Extrema
Local extrema on a function are points on the graph where the y coordinate is larger
(or smaller) than all other y coordinates on the graph at points "close to" (x, y).
Definition
(a) A point (x, f (x)) is a local maximum if there is an interval a < x < b with
f (x) ≥ f (z) for every z in (a, b).
(b) A point (x, f (x)) is a local minimum if there is an interval a < x < b with
f (x) ≤ f (z) for every z in (a, b).
A local extremum is either a local maximum or a local minimum.
Local maximum and minimum points are quite distinctive on the graph of a
function, and are therefore useful in understanding the shape of the graph. In
many applied problems we want to find the largest or smallest value that a function
achieves (for example, we might want to find the minimum cost at which some task
65.
calculus 65
can be performed) and so identifying maximum and minimum points will be useful
for applied problems as well.
If (x, f (x)) is a point where f (x) reaches a local maximum or minimum, and if the
derivative of f exists at x, then the graph has a tangent line and the tangent line
must be horizontal. This is important enough to state as a theorem, though we will
not prove it.
Theorem 4.1.1 (Fermat's Theorem) If f (x) has a local extremum at x = a
and f (x) is differentiable at a, then f ′
(a) = 0.
−0.5 0.5 1 1.5 2 2.5 3
−2
2
f (x)
f ′
(x)
x
y
Figure 4.1: A plot of f (x) = x3
− 4x2
+ 3x and f ′
(x) =
3x2
− 8x + 3.
Thus, the only points at which a function can have a local maximum or minimum are
points at which the derivative is zero, see Figure 4.1, or the derivative is undefined,
as in Figure 4.2. This brings us to our next definition.
−3 −2 −1 1 2 3
−2
−1
1
2
f (x)
f ′
(x)
x
y
Figure 4.2: A plot of f (x) = x2/3
and f ′
(x) =
2
3x1/3
.
Definition Any value of x for which f ′
(x) is zero or undefined is called a
critical point for f (x).
Warning When looking for local maximum and minimum points, you are likely
to make two sorts of mistakes:
• You may forget that a maximum or minimum can occur where the deriva-
tive does not exist, and so forget to check whether the derivative exists
everywhere.
• You might assume that any place that the derivative is zero is a local
maximum or minimum point, but this is not true, see Figure 4.3.
Since the derivative is zero or undefined at both local maximum and local
minimum points, we need a way to determine which, if either, actually occurs. The
most elementary approach is to test directly whether the y coordinates near the
potential maximum or minimum are above or below the y coordinate at the point of
interest.
66.
66
It is not always easy to compute the value of a function at a particular point. The
task is made easier by the availability of calculators and computers, but they have
their own drawbacks—they do not always allow us to distinguish between values
that are very close together. Nevertheless, because this method is conceptually
simple and sometimes easy to perform, you should always consider it.
−1 −0.5 0.5 1
−2
2
f (x)f ′
(x)
x
y
Figure 4.3: A plot of f (x) = x3
and f ′
(x) = 3x2
. While
f ′
(0) = 0, there is neither a maximum nor minimum
at (0, f (0)).
Example 4.1.2 Find all local maximum and minimum points for the function
f (x) = x3
− x.
Solution Write
d
dx
f (x) = 3x2
− 1.
This is defined everywhere and is zero at x = ±
√
3/3. Looking first at x =
√
3/3,
we see that
f (
√
3/3) = −2
√
3/9.
Now we test two points on either side of x =
√
3/3, making sure that neither is
farther away than the nearest critical point; since
√
3 < 3,
√
3/3 < 1 and we
can use x = 0 and x = 1. Since
f (0) = 0 > −2
√
3/9 and f (1) = 0 > −2
√
3/9,
there must be a local minimum at x =
√
3/3.
For x = −
√
3/3, we see that f (−
√
3/3) = 2
√
3/9. This time we can use x = 0
and x = −1, and we find that f (−1) = f (0) = 0 < 2
√
3/9, so there must be a
local maximum at x = −
√
3/3, see Figure 4.4.
−1.5 −1 −0.5 0.5 1 1.5
−2
−1
1
2
f (x)f ′
(x)
x
y
Figure 4.4: A plot of f (x) = x3
−x and f ′
(x) = 3x2
−1.
68.
68
4.2 The First Derivative Test
The method of the previous section for deciding whether there is a local maximum
or minimum at a critical point by testing "near-by" points is not always convenient.
Instead, since we have already had to compute the derivative to find the critical
points, we can use information about the derivative to decide. Recall that
• If f ′
(x) > 0 on an interval, then f (x) is increasing on that interval.
• If f ′
(x) < 0 on an interval, then f (x) is decreasing on that interval.
So how exactly does the derivative tell us whether there is a maximum, minimum,
or neither at a point? Use the first derivative test.
Theorem 4.2.1 (First Derivative Test) Suppose that f (x) is continuous on
an interval, and that f ′
(a) = 0 for some value of a in that interval.
• If f ′
(x) > 0 to the left of a and f ′
(x) < 0 to the right of a, then f (a) is a
local maximum.
• If f ′
(x) < 0 to the left of a and f ′
(x) > 0 to the right of a, then f (a) is a
local minimum.
• If f ′
(x) has the same sign to the left and right of a, then f (a) is not a local
extremum.
Example 4.2.2 Consider the function
f (x) =
x4
4
+
x3
3
− x2
Find the intervals on which f (x) is increasing and decreasing and identify the
local extrema of f (x).
Solution Start by computing
d
dx
f (x) = x3
+ x2
− 2x.
71.
calculus 71
4.3 Concavity and Inflection Points
We know that the sign of the derivative tells us whether a function is increasing or
decreasing. Likewise, the sign of the second derivative f ′′
(x) tells us whether f ′
(x)
is increasing or decreasing. We summarize this in the table below:
f ′
(x) < 0 f ′
(x) > 0
f ′′
(x) > 0
Concave Up
Here f ′
(x) < 0 and f ′′
(x) > 0. This means
that f (x) slopes down and is getting less
steep. In this case the curve is concave
up.
Concave Up
Here f ′
(x) > 0 and f ′′
(x) > 0. This means
that f (x) slopes up and is getting steeper.
In this case the curve is concave up.
f ′′
(x) < 0
Concave Down
Here f ′
(x) < 0 and f ′′
(x) < 0. This
means that f (x) slopes down and is getting
steeper. In this case the curve is concave
down.
Concave Down
Here f ′
(x) > 0 and f ′′
(x) < 0. This means
that f (x) slopes up and is getting less
steep. In this case the curve is concave
down.
If we are trying to understand the shape of the graph of a function, knowing
where it is concave up and concave down helps us to get a more accurate picture. It
is worth summarizing what we have seen already in to a single theorem.
72.
72
Theorem 4.3.1 (Test for Concavity) Suppose that f ′′
(x) exists on an inter-
val.
(a) If f ′′
(x) > 0 on an interval, then f (x) is concave up on that interval.
(b) If f ′′
(x) < 0 on an interval, then f (x) is concave down on that interval.
Of particular interest are points at which the concavity changes from up to down
or down to up.
Definition If f (x) is continuous and its concavity changes either from up to
down or down to up at x = a, then f (x) has an inflection point at x = a.
It is instructive to see some examples and nonexamples of inflection pointsWe identify inflection points by first finding where f ′′
(x) is zero or undefined
and then checking to see whether f ′′
(x) does in fact go from positive to negative or
negative to positive at these points.
Warning Even if f ′′
(a) = 0, the point determined by x = a might not be an
inflection point.
Example 4.3.2 Describe the concavity of f (x) = x3
− x.
73.
calculus 73
Solution To start, compute the first and second derivative of f (x) with respect
to x,
f ′
(x) = 3x2
− 1 and f ′′
(x) = 6x.
Since f ′′
(0) = 0, there is potentially an inflection point at zero. Since f ′′
(x) > 0
when x > 0 and f ′′
(x) < 0 when x < 0 the concavity does change from down to
up at zero—there is an inflection point at x = 0. The curve is concave down for
all x < 0 and concave up for all x > 0, see Figure 4.6.
−1.5 −1 −0.5 0.5 1 1.5
−2
2
f (x)
f ′′
(x)
x
y
Figure 4.6: A plot of f (x) = x3
− x and f ′′
(x) = 6x.
We can see that the concavity change at x = 0.
Note that we need to compute and analyze the second derivative to understand
concavity, so we may as well try to use the second derivative test for maxima and
minima. If for some reason this fails we can then try one of the other tests.
75.
calculus 75
4.4 The Second Derivative Test
Recall the first derivative test, Theorem 4.2.1:
• If f ′
(x) > 0 to the left of a and f ′
(x) < 0 to the right of a, then f (a) is a local
maximum.
• If f ′
(x) < 0 to the left of a and f ′
(x) > 0 to the right of a, then f (a) is a local
minimum.
If f ′
(x) changes from positive to negative it is decreasing. In this case, f ′′
(x)
might be negative, and if in fact f ′′
(x) is negative then f ′
(x) is definitely decreasing,
so there is a local maximum at the point in question. On the other hand, if f ′
(x)
changes from negative to positive it is increasing. Again, this means that f ′′
(x)
might be positive, and if in fact f ′′
(x) is positive then f ′
(x) is definitely increasing,
so there is a local minimum at the point in question. We summarize this as the
second derivative test.
Theorem 4.4.1 (Second Derivative Test) Suppose that f ′′
(x) is continu-
ous on an open interval and that f ′
(a) = 0 for some value of a in that
interval.
• If f ′′
(a) < 0, then f (x) has a local maximum at a.
• If f ′′
(a) > 0, then f (x) has a local minimum at a.
• If f ′′
(a) = 0, then the test is inconclusive. In this case, f (x) may or may
not have a local extremum at x = a.
The second derivative test is often the easiest way to identify local maximum and
minimum points. Sometimes the test fails and sometimes the second derivative is
quite difficult to evaluate. In such cases we must fall back on one of the previous
tests.
78.
78
4.5 Sketching the Plot of a Function
In this section, we will give some general guidelines for sketching the plot of a
function.
Procedure for Sketching the Plots of Functions
• Find the y-intercept, this is the point (0, f (0)). Place this point on your
graph.
• Find candidates for vertical asymptotes, these are points where f (x) is
undefined.
• Compute f ′
(x) and f ′′
(x).
• Find the critical points, the points where f ′
(x) = 0 or f ′
(x) is undefined.
• Use the second derivative test to identify local extrema and/or find the
intervals where your function is increasing/decreasing.
• Find the candidates for inflection points, the points where f ′′
(x) = 0 or
f ′′
(x) is undefined.
• Identify inflection points and concavity.
• If possible find the x-intercepts, the points where f (x) = 0. Place these
points on your graph.
• Find horizontal asymptotes.
• Determine an interval that shows all relevant behavior.
At this point you should be able to sketch the plot of your function.
Let's see this procedure in action. We'll sketch the plot of 2x3
− 3x2
− 12x.
Following our guidelines above, we start by computing f (0) = 0. Hence we see that
the y-intercept is (0, 0). Place this point on your plot, see Figure 4.8.
−2 −1 1 2 3 4
−20
−10
10
20
x
y
Figure 4.8: We start by placing the point (0, 0).
90.
6 The Chain Rule
So far we have seen how to compute the derivative of a function built up from other
functions by addition, subtraction, multiplication and division. There is another
very important way that we combine functions: composition. The chain rule allows
us to deal with this case.
6.1 The Chain Rule
Consider
h(x) = (1 + 2x)5
.
While there are several different ways to differentiate this function, if we let
f (x) = x5
and g(x) = 1 + 2x, then we can express h(x) = f (g(x)). The question is,
can we compute the derivative of a composition of functions using the derivatives of
the constituents f (x) and g(x)? To do so, we need the chain rule.
f ′
(g(a))g′
(a)h
g′
(a)h
h
g(x)
y
x
a
Figure 6.1: A geometric interpretation of the chain
rule. Increasing a by a "small amount" h, increases
f (g(a)) by f ′
(g(a))g′
(a)h. Hence,
∆y
∆x
≈
f ′(g(a))g′(a)h
h
= f ′
(g(a))g′
(a).
Theorem 6.1.1 (Chain Rule) If f (x) and g(x) are differentiable, then
d
dx
f (g(x)) = f ′
(g(x))g′
(x).
Proof Let g0 be some x-value and consider the following:
f ′
(g0) = lim
h→0
f (g0 + h) − f (g0)
h
. | 677.169 | 1 |
Differential Equations with Linear Algebra
Overview that study systems of differential equations.
Because of its emphasis on linearity, the text opens with an introduction to essential ideas in linear algebra. Motivated by future problems in systems of differential equations, the material on linear algebra introduces such key ideas as systems of algebraic equations, linear combinations, the eigenvalue problem, and bases and dimension of vector spaces. These key concepts maintain a consistent presence throughout the text and provide students with a basic knowledge of linear algebra for use in the study of differential equations.
This text offers an example-driven approach, beginning each chapter with one or two motivating problems that are applied in nature. The authors then develop the mathematics necessary to solve these problems and explore related topics further. Even in more theoretical developments, an example-first style is used to build intuition and understanding before stating or providing general results. Each chapter closes with several substantial projects for further study, many of which are based in applications. Extensive use of figures provides visual demonstration of key ideas while the use of the computer algebra system Maple and Microsoft Excel are presented in detail throughout to provide further perspective and enhance students' use of technology in solving problems. Support for the use of other computer algebra systems is available online. | 677.169 | 1 |
1
00:00:00.3 --> 00:00:00
OK.
2
00:00:00 --> 00:00:03
This is lecture twelve.
3
00:00:03 --> 00:00:07
We've reached twelve lectures.
4
00:00:07 --> 00:00:14
And this one is more than the
others about applications of
5
00:00:14 --> 00:00:16
linear algebra.
6
00:00:16 --> 00:00:19
And I'll confess.
7
00:00:19 --> 00:00:26.46
When I'm giving you examples of
the null space and the row
8
00:00:26.46 --> 00:00:31.9
space, I create a little matrix.
9
00:00:31.9 --> 00:00:37
You probably see that I just
invent that matrix as I'm going.
10
00:00:37 --> 00:00:42
And I feel a little guilty
about it, because the truth is
11
00:00:42 --> 00:00:47
that real linear algebra uses
matrices that come from
12
00:00:47 --> 00:00:48.34
somewhere.
13
00:00:48.34 --> 00:00:52
They're not just,
like, randomly invented by the
14
00:00:52 --> 00:00:53
instructor.
15
00:00:53 --> 00:00:57.3
They come from applications.
16
00:00:57.3 --> 00:01:00
They have a definite structure.
17
00:01:00 --> 00:01:06
And anybody who works with them
gets, uses that structure.
18
00:01:06 --> 00:01:10.65
I'll just report,
like, this weekend I was at an
19
00:01:10.65 --> 00:01:13
event with chemistry professors.
20
00:01:13 --> 00:01:19
OK, those guys are row reducing
matrices, and what matrices are
21
00:01:19 --> 00:01:22
they working with?
22
00:01:22 --> 00:01:26
Well, their little matrices
tell them how much of each
23
00:01:26 --> 00:01:31
element goes into the -- or each
molecule, how many molecules of
24
00:01:31 --> 00:01:34
each go into a reaction and what
comes out.
25
00:01:34 --> 00:01:38
And by row reduction they get a
clearer picture of a complicated
26
00:01:38 --> 00:01:39
reaction.
27
00:01:39 --> 00:01:43
And this weekend I'm going to
-- to a sort of birthday party
28
00:01:43 --> 00:01:45.7
at Mathworks.
29
00:01:45.7 --> 00:01:49
So Mathworks is out Route 9 in
Natick.
30
00:01:49 --> 00:01:53
That's where Matlab is created.
31
00:01:53 --> 00:01:56
It's a very,
very successful,
32
00:01:56 --> 00:02:00
software, tremendously
successful.
33
00:02:00 --> 00:02:06
And the conference will be
about how linear algebra is
34
00:02:06 --> 00:02:07.9
used.
35
00:02:07.9 --> 00:02:13
And so I feel better today to
talk about what I think is the
36
00:02:13 --> 00:02:17
most important model in applied
math.
37
00:02:17 --> 00:02:20.95
And the discrete version is a
graph.
38
00:02:20.95 --> 00:02:23
So can I draw a graph?
39
00:02:23 --> 00:02:27
Write down the matrix that's
associated with it,
40
00:02:27 --> 00:02:31
and that's a great source of
matrices.
41
00:02:31 --> 00:02:33.9
You'll see.
42
00:02:33.9 --> 00:02:39
So a graph is just,
so a graph -- to repeat -- has
43
00:02:39 --> 00:02:40
nodes and edges.
44
00:02:40 --> 00:02:41
OK.
45
00:02:41 --> 00:02:46
And I'm going to write down the
graph, a graph,
46
00:02:46 --> 00:02:50
so I'm just creating a small
graph here.
47
00:02:50 --> 00:02:56
As I mentioned last time,
we would be very interested in
48
00:02:56 --> 00:03:00
the graph of all,
websites.
49
00:03:00 --> 00:03:03
Or the graph of all telephones.
50
00:03:03 --> 00:03:08
I mean -- or the graph of all
people in the world.
51
00:03:08 --> 00:03:14
Here let me take just,
maybe nodes one two three --
52
00:03:14 --> 00:03:20
well, I better put in an -- I'll
put in that edge and maybe an
53
00:03:20 --> 00:03:27.3
edge to, to a node four,
and another edge to node four.
54
00:03:27.3 --> 00:03:28
How's that?
55
00:03:28 --> 00:03:32
So there's a graph with four
nodes.
56
00:03:32 --> 00:03:37
So n will be four in my -- n
equal four nodes.
57
00:03:37 --> 00:03:43
And the matrix will have m
equal the number -- there'll be
58
00:03:43 --> 00:03:48
a row for every edge,
so I've got one two three four
59
00:03:48 --> 00:03:50
five edges.
60
00:03:50 --> 00:03:55
So that will be the number of
rows.
61
00:03:55 --> 00:03:59.17
And I have to to write down the
matrix that I want to,
62
00:03:59.17 --> 00:04:02.54
I want to study,
I need to give a direction to
63
00:04:02.54 --> 00:04:06
every edge, so I know a plus and
a minus direction.
64
00:04:06 --> 00:04:08
So I'll just do that with an
arrow.
65
00:04:08 --> 00:04:11
Say from one to two,
one to three,
66
00:04:11 --> 00:04:13
two to three,
one to four,
67
00:04:13 --> 00:04:15
three to four.
68
00:04:15 --> 00:04:19
That just tells me,
if I have current flowing on
69
00:04:19 --> 00:04:25
these edges then I know whether
it's -- to count it as positive
70
00:04:25 --> 00:04:31
or negative according as whether
it's with the arrow or against
71
00:04:31 --> 00:04:32
the arrow.
72
00:04:32 --> 00:04:36
But I just drew those arrows
arbitrarily.
73
00:04:36 --> 00:04:36
OK.
74
00:04:36 --> 00:04:41
Because I --
my example is going to come --
75
00:04:41 --> 00:04:47
the example I'll -- the words
that I will use will be words
76
00:04:47 --> 00:04:50.33
like potential,
potential difference,
77
00:04:50.33 --> 00:04:51
currents.
78
00:04:51 --> 00:04:55
In other words,
I'm thinking of an electrical
79
00:04:55 --> 00:04:55
network.
80
00:04:55 --> 00:04:59
But that's just one
possibility.
81
00:04:59 --> 00:05:03
My applied math class builds on
this example.
82
00:05:03 --> 00:05:09
It could be a hydraulic
network, so we could be doing,
83
00:05:09 --> 00:05:11
flow of water,
flow of oil.
84
00:05:11 --> 00:05:15
Other examples,
this could be a structure.
85
00:05:15 --> 00:05:20.39
Like the -- a design for a
bridge or a design for a
86
00:05:20.39 --> 00:05:23
Buckminster Fuller dome.
87
00:05:23 --> 00:05:27
Or many other possibilities,
so many.
88
00:05:27 --> 00:05:31
So l- but let's take potentials
and currents as,
89
00:05:31 --> 00:05:36
as a basic example,
and let me create the matrix
90
00:05:36 --> 00:05:40
that tells you exactly what the
graph tells you.
91
00:05:40 --> 00:05:44
So now I'll call it the
incidence matrix,
92
00:05:44 --> 00:05:47.1
incidence matrix.
93
00:05:47.1 --> 00:05:47
OK.
94
00:05:47 --> 00:05:51
So let me write it down,
and you'll see,
95
00:05:51 --> 00:05:54
what its properties are.
96
00:05:54 --> 00:05:58
So every row corresponds to an
edge.
97
00:05:58 --> 00:06:04
I have five rows from five
edges, and let me write down
98
00:06:04 --> 00:06:07
again what this graph looks
like.
99
00:06:07 --> 00:06:13
OK, the first edge,
edge one, goes from node one to
100
00:06:13 --> 00:06:15.25
two.
101
00:06:15.25 --> 00:06:21
So I'm going to put in a minus
one and a plus one in th- this
102
00:06:21 --> 00:06:25.51
corresponds to node one two
three and four,
103
00:06:25.51 --> 00:06:27.22
the four columns.
104
00:06:27.22 --> 00:06:33
The five rows correspond -- the
first row corresponds to edge
105
00:06:33 --> 00:06:33
one.
106
00:06:33 --> 00:06:38
Edge one leaves node one and
goes into node two,
107
00:06:38 --> 00:06:42
and that --
and it doesn't touch three and
108
00:06:42 --> 00:06:43
four.
109
00:06:43 --> 00:06:47
Edge two, edge two goes -- oh,
I haven't numbered these edges.
110
00:06:47 --> 00:06:50
I just figured that was
probably edge one,
111
00:06:50 --> 00:06:52
but I didn't say so.
112
00:06:52 --> 00:06:54
Let me take that to be edge
one.
113
00:06:54 --> 00:06:57
Let me take this to be edge
two.
114
00:06:57 --> 00:06:59
Let me take this to be edge
three.
115
00:06:59 --> 00:07:02.1
This is edge four.
116
00:07:02.1 --> 00:07:05
Ho, I'm discovering -- no,
wait a minute.
117
00:07:05 --> 00:07:06
Did I number that twice?
118
00:07:06 --> 00:07:08.12
Here's edge four.
119
00:07:08.12 --> 00:07:09.68
And here's edge five.
120
00:07:09.68 --> 00:07:09
OK?
121
00:07:09 --> 00:07:10
All right.
122
00:07:10 --> 00:07:13
So, so edge one,
as I said, goes from node one
123
00:07:13 --> 00:07:14
to two.
124
00:07:14 --> 00:07:18
Edge two goes from two to
three, node two to three,
125
00:07:18 --> 00:07:23
so minus one and one in the
second and third columns.
126
00:07:23 --> 00:07:27
Edge three goes from one to
three.
127
00:07:27 --> 00:07:34
I'm, I'm tempted to stop for a
moment with those three edges.
128
00:07:34 --> 00:07:39
Edges one two three,
those form what would we,
129
00:07:39 --> 00:07:44
what do you call the,
the little, the little,
130
00:07:44 --> 00:07:51
the subgraph formed by edges
one, two, and three?
131
00:07:51 --> 00:07:52
That's a loop.
132
00:07:52 --> 00:07:58.32
And the number of loops and the
position of the loops will be
133
00:07:58.32 --> 00:07:59
crucial.
134
00:07:59 --> 00:07:59
OK.
135
00:07:59 --> 00:08:03
Actually, here's a interesting
point about loops.
136
00:08:03 --> 00:08:08
If I look at those rows,
corresponding to edges one two
137
00:08:08 --> 00:08:12
three, and these guys made a
loop.
138
00:08:12 --> 00:08:18
You want to tell me -- if I
just looked at that much of the
139
00:08:18 --> 00:08:22
matrix it would be natural for
me to ask, are those rows
140
00:08:22 --> 00:08:23
independent?
141
00:08:23 --> 00:08:26
Are the rows independent?
142
00:08:26 --> 00:08:31
And can you tell from looking
at that if they are or are not
143
00:08:31 --> 00:08:32.49
independent?
144
00:08:32.49 --> 00:08:36
Do you see a,
a relation between those three
145
00:08:36 --> 00:08:37
rows?
146
00:08:37 --> 00:08:38
Yes.
147
00:08:38 --> 00:08:43
If I add that row to that row,
I get this row.
148
00:08:43 --> 00:08:49
So, so that's like a hint here
that loops correspond to
149
00:08:49 --> 00:08:56
dependent, linearly dependent
column -- linearly dependent
150
00:08:56 --> 00:08:56
rows.
151
00:08:56 --> 00:09:02
OK, let me complete the
incidence matrix.
152
00:09:02 --> 00:09:06
Number four,
edge four is going from node
153
00:09:06 --> 00:09:08
one to node four.
154
00:09:08 --> 00:09:14
And the fifth edge is going
from node three to node four.
155
00:09:14 --> 00:09:14
OK.
156
00:09:14 --> 00:09:16
There's my matrix.
157
00:09:16 --> 00:09:21
It came from the five edges and
the four nodes.
158
00:09:21 --> 00:09:27
And if I had a big graph,
I'd have a big matrix.
159
00:09:27 --> 00:09:31
And what questions do I ask
about matrices?
160
00:09:31 --> 00:09:35
Can I ask -- here's the review
now.
161
00:09:35 --> 00:09:40
There's a matrix that comes
from somewhere.
162
00:09:40 --> 00:09:45
If, if it was a big graph,
it would be a large matrix,
163
00:09:45 --> 00:09:49
but a lot of zeros,
right?
164
00:09:49 --> 00:09:52
Because every row only has two
non-zeros.
165
00:09:52 --> 00:09:56
So the number of -- it's a very
sparse matrix.
166
00:09:56 --> 00:10:00.15
The number of non-zeros is
exactly two times five,
167
00:10:00.15 --> 00:10:01
it's two m.
168
00:10:01 --> 00:10:03
Every row only has two
non-zeros.
169
00:10:03 --> 00:10:06
And that's with a lot of
structure.
170
00:10:06 --> 00:10:09
And -- that was the point I
wanted to begin with,
171
00:10:09 --> 00:10:14
that graphs,
that real graphs from real --
172
00:10:14 --> 00:10:19
real matrices from genuine
problems have structure.
173
00:10:19 --> 00:10:19
OK.
174
00:10:19 --> 00:10:25
We can ask, and because of the
structure, we can answer,
175
00:10:25 --> 00:10:29
the, the main questions about
matrices.
176
00:10:29 --> 00:10:35
So first question,
what about the null space?
177
00:10:35 --> 00:10:40
So what I asking if I ask you
for the null space of that
178
00:10:40 --> 00:10:40
matrix?
179
00:10:40 --> 00:10:46
I'm asking you if I'm looking
at the columns of the matrix,
180
00:10:46 --> 00:10:49
four columns,
and I'm asking you,
181
00:10:49 --> 00:10:51
are those columns independent?
182
00:10:51 --> 00:10:58.6
If the columns are independent,
then what's in the null space?
183
00:10:58.6 --> 00:11:01
Only the zero vector,
right?
184
00:11:01 --> 00:11:07
The null space contains --
tells us what combinations of
185
00:11:07 --> 00:11:13
the columns -- it tells us how
to combine columns to get zero.
186
00:11:13 --> 00:11:20
Can -- and is there anything in
the null space of this matrix
187
00:11:20 --> 00:11:24
other than just the zero vector?
188
00:11:24 --> 00:11:28
In other words,
are those four columns
189
00:11:28 --> 00:11:30.96
independent or dependent?
190
00:11:30.96 --> 00:11:31
OK.
191
00:11:31 --> 00:11:33
That's our question.
192
00:11:33 --> 00:11:37
Let me, I don't know if you see
the answer.
193
00:11:37 --> 00:11:40
Whether there's -- so let's
see.
194
00:11:40 --> 00:11:44.6
I guess we could do it
properly.
195
00:11:44.6 --> 00:11:47
We could solve Ax=0.
196
00:11:47 --> 00:11:53
So let me solve Ax=0 to find
the null space.
197
00:11:53 --> 00:11:53
OK.
198
00:11:53 --> 00:11:55
What's Ax?
199
00:11:55 --> 00:12:00
Can I put x in here in,
in little letters?
200
00:12:00 --> 00:12:05
x1, x2, x3, x4,
that's -- it's got four
201
00:12:05 --> 00:12:06
columns.
202
00:12:06 --> 00:12:12.6
Ax now is that matrix times x.
203
00:12:12.6 --> 00:12:15
And what do I get for Ax?
204
00:12:15 --> 00:12:21
If the camera can keep that
matrix multiplication there,
205
00:12:21 --> 00:12:23
I'll put the answer here.
206
00:12:23 --> 00:12:28.47
Ax equal -- what's the first
component of Ax?
207
00:12:28.47 --> 00:12:34
Can you take that first row,
minus one one zero zero,
208
00:12:34 --> 00:12:40
and multiply by the x,
and of course you get x2-x1.
209
00:12:40 --> 00:12:44
The second row,
I get x3-x2.
210
00:12:44 --> 00:12:49
From the third row,
I get x3-x1.
211
00:12:49 --> 00:12:54
From the fourth row,
I get x4-x1.
212
00:12:54 --> 00:13:00
And from the fifth row,
I get x4-x3.
213
00:13:00 --> 00:13:08
And I want to know when is the
thing zero.
214
00:13:08 --> 00:13:10
This is my equation,
Ax=0.
215
00:13:10 --> 00:13:16
Notice what that matrix A is
doing, what we've created a
216
00:13:16 --> 00:13:21
matrix that computes the
differences across every edge,
217
00:13:21 --> 00:13:24
the differences in potential.
218
00:13:24 --> 00:13:30
Let me even begin to give this
interpretation.
219
00:13:30 --> 00:13:38.12
I'm going to think of this
vector x, which is x1 x2 x3 x4,
220
00:13:38.12 --> 00:13:42
as the potentials at the nodes.
221
00:13:42 --> 00:13:49
So I'm introducing a word,
potentials at the nodes.
222
00:13:49 --> 00:13:57
And now if I multiply by A,
I get these -- I get these five
223
00:13:57 --> 00:14:01
components, x2-x1,
et cetera.
224
00:14:01 --> 00:14:05
And what are they?
225
00:14:05 --> 00:14:08
They're potential differences.
226
00:14:08 --> 00:14:11
That's what A computes.
227
00:14:11 --> 00:14:16
If I have potentials at the
nodes and I multiply by A,
228
00:14:16 --> 00:14:21
it gives me the potential
differences, the differences in
229
00:14:21 --> 00:14:24
potential, across the edges.
230
00:14:24 --> 00:14:24
OK.
231
00:14:24 --> 00:14:29.1
When are those differences all
zero?
232
00:14:29.1 --> 00:14:31
So I'm looking for the null
space.
233
00:14:31 --> 00:14:36
Of course, if all the (x)s are
zero then I get zero.
234
00:14:36 --> 00:14:40
That, that just tells me,
of course, the zero vector is
235
00:14:40 --> 00:14:42
in the null space.
236
00:14:42 --> 00:14:45
But w- there's more in the null
space.
237
00:14:45 --> 00:14:50
Those columns are -- of A are
dependent, right --
238
00:14:50 --> 00:14:54
because I can find solutions to
that equation.
239
00:14:54 --> 00:14:57
Tell me -- the null space.
240
00:14:57 --> 00:15:03.04
Tell me one vector in the null
space, so tell me an x,
241
00:15:03.04 --> 00:15:08
it's got four components,
and it makes that thing zero.
242
00:15:08 --> 00:15:11
So what's a good x to do that?
243
00:15:11 --> 00:15:16
One one one one,
constant potential.
244
00:15:16 --> 00:13:21
If the potentials are constant,
then all the potential
245
00:13:21 --> 00:11:41
differences are zero,
and that x is in the null
246
00:11:41 --> 00:11:28
space.
247
00:11:28 --> 00:10:21
What else is in the null space?
248
00:10:21 --> 00:08:52
If it -- yeah,
let me ask you just always,
249
00:08:52 --> 00:07:38
give me a basis for the null
space.
250
00:07:38 --> 00:05:37
A basis for the null space will
be just that.1
251
00:05:37 --> 00:06:21
That's --, that's it.
252
00:06:21 --> 00:07:31
That's a basis for the null
space.
253
00:07:31 --> 00:09:38
The null space is actually one
dimensional, and it's the line
254
00:09:38 --> 00:10:46
of all vectors through that one.
255
00:10:46 --> 00:12:36
So there's a basis for it,
and here is the whole null
256
00:12:36 --> 00:12:48
space.
257
00:12:48 --> 00:14:53
Any multiple of one one one
one, it's the whole line in four
258
00:14:53 --> 00:15:57
dimensional space.
259
00:15:57 --> 00:16:01
Do you see that that's the null
space?
260
00:16:01 --> 00:16:07
So the, so the dimension of the
null space of A is one.
261
00:16:07 --> 00:16:13
And there's a basis for it,
and there's everything that's
262
00:16:13 --> 00:16:14
in it.
263
00:16:14 --> 00:16:14
Good.
264
00:16:14 --> 00:16:19.6
And what does that mean
physically?
265
00:16:19.6 --> 00:16:24
I mean, what does that mean in
the application?
266
00:16:24 --> 00:16:27
That guy in the null space.
267
00:16:27 --> 00:16:34
It means that the potentials
can only be determined up to a
268
00:16:34 --> 00:16:35
constant.
269
00:16:35 --> 00:16:41
Potential differences are what
make current flow.
270
00:16:41 --> 00:16:45
That's what makes things
happen.
271
00:16:45 --> 00:16:48
It's these potential
differences that will make
272
00:16:48 --> 00:16:50
something move in the,
in our network,
273
00:16:50 --> 00:16:52
between x2- between node two
and node one.
274
00:16:52 --> 00:16:55
Nothing will move if all
potentials are the same.
275
00:16:55 --> 00:16:57
If all potentials are c,
c, c, and c,
276
00:16:57 --> 00:16:59.19
then nothing will move.
277
00:16:59.19 --> 00:17:02
So we're, we have this one
parameter, this arbitrary
278
00:17:02 --> 00:17:05
constant that raises or drops
all the potentials.
279
00:17:05 --> 00:17:09
It's like ranking football
teams, whatever.
280
00:17:09 --> 00:17:14
We have a, there's a,
there's a constant -- or
281
00:17:14 --> 00:17:19
looking at temperatures,
you know, there's a flow of
282
00:17:19 --> 00:17:24.9
heat from higher temperature to
lower temperature.
283
00:17:24.9 --> 00:17:28
If temperatures are equal
there's no flow,
284
00:17:28 --> 00:17:34
and therefore we can measure --
we can measure temperatures by,
285
00:17:34 --> 00:17:38
Celsius or we can start at
absolute zero.
286
00:17:38 --> 00:17:44
And that arbitrary -- it's the
same arbitrary constant that,
287
00:17:44 --> 00:17:47
that was there in calculus.
288
00:17:47 --> 00:17:50
In calculus,
right, when you took the
289
00:17:50 --> 00:17:55
integral, the indefinite
integral, there was a plus c,
290
00:17:55 --> 00:18:00
and you had to set a starting
point to know what that c was.
291
00:18:00 --> 00:18:06.18
So here what often happens is
we fix one of the potentials,
292
00:18:06.18 --> 00:18:08
like the last one.
293
00:18:08 --> 00:18:13
So a typical thing would be to
ground that node.
294
00:18:13 --> 00:18:15
To set its potential at zero.
295
00:18:15 --> 00:18:19.96
And if we do that,
if we fix that potential so
296
00:18:19.96 --> 00:18:25
it's not unknown anymore,
then that column disappears and
297
00:18:25 --> 00:18:29
we have three columns,
and those three columns are
298
00:18:29 --> 00:18:31
independent.
299
00:18:31 --> 00:18:36
So I'll leave the column in
there, but we'll remember that
300
00:18:36 --> 00:18:39
grounding a node is the way to
get it out.
301
00:18:39 --> 00:18:43
And grounding a node is the way
to -- setting a node -- setting
302
00:18:43 --> 00:18:47
a potential to zero tells us
the, the base for all
303
00:18:47 --> 00:18:48
potentials.
304
00:18:48 --> 00:18:50.46
Then we can compute the others.
305
00:18:50.46 --> 00:18:50
OK.
306
00:18:50 --> 00:18:55
But what's the --
now I've talked enough to ask
307
00:18:55 --> 00:18:58
what the rank of the matrix is?
308
00:18:58 --> 00:18:59
What's the rank then?
309
00:18:59 --> 00:19:01
The rank of the matrix.
310
00:19:01 --> 00:19:04
So we have a five by four
matrix.
311
00:19:04 --> 00:19:08
We've located its null space,
one dimensional.
312
00:19:08 --> 00:19:12
How many independent columns do
we have?
313
00:19:12 --> 00:19:14.4
What's the rank?
314
00:19:14.4 --> 00:19:15
It's three.
315
00:19:15 --> 00:19:21
And the first three columns,
or actually any three columns,
316
00:19:21 --> 00:19:23
will be independent.
317
00:19:23 --> 00:19:28
Any three potentials are
independent, good variables.
318
00:19:28 --> 00:19:33
The fourth potential is not,
we need to set,
319
00:19:33 --> 00:19:36
and typically we ground that
node.
320
00:19:36 --> 00:19:38
OK.
321
00:19:38 --> 00:19:39.32
Rank is three.
322
00:19:39.32 --> 00:19:40
Rank equals three.
323
00:19:40 --> 00:19:41.15
OK.
324
00:19:41.15 --> 00:19:45
Let's see, do I want to ask you
about the column space?
325
00:19:45 --> 00:19:50
The column space is all
combinations of those columns.
326
00:19:50 --> 00:19:54.4
I could say more about it and I
will.
327
00:19:54.4 --> 00:20:00
Let me go to the null space of
A transpose, because the
328
00:20:00 --> 00:20:06
equation A transpose y equals
zero is probably the most
329
00:20:06 --> 00:20:11
fundamental equation of applied
mathematics.
330
00:20:11 --> 00:20:15
All right, let's talk about
that.
331
00:20:15 --> 00:20:18.6
That deserves our attention.
332
00:20:18.6 --> 00:20:21
A transpose y equals zero.
333
00:20:21 --> 00:20:26.4
Let's -- let me put it on here.
334
00:20:26.4 --> 00:20:26
OK.
335
00:20:26 --> 00:20:29
So A transpose y equals zero.
336
00:20:29 --> 00:20:34
So now I'm finding the null
space of A transpose.
337
00:20:34 --> 00:20:39
Oh, and if I ask you its
dimension, you could tell me
338
00:20:39 --> 00:20:40
what it is.
339
00:20:40 --> 00:20:45
What's the dimension of the
null space of A transpose?
340
00:20:45 --> 00:20:51
We now know enough to answer
that question.
341
00:20:51 --> 00:20:57
What's the general formula for
the dimension of the null space
342
00:20:57 --> 00:20:58
of A transpose?
343
00:20:58 --> 00:21:02
A transpose,
let me even write out A
344
00:21:02 --> 00:21:03
transpose.
345
00:21:03 --> 00:21:07.26
This A transpose will be n by
m, right?
346
00:21:07.26 --> 00:21:07
n by m.
347
00:21:07 --> 00:21:11
In this case,
it'll be four by five.
348
00:21:11 --> 00:21:15.9
Those columns will become rows.
349
00:21:15.9 --> 00:21:23
Minus one zero minus one minus
one zero is now the first row.
350
00:21:23 --> 00:21:30
The second row of the matrix,
one minus one and three zeros.
351
00:21:30 --> 00:21:35
The third column now becomes
the third row,
352
00:21:35 --> 00:21:38.4
zero one one zero minus one.
353
00:21:38.4 --> 00:21:43
And the fourth column becomes
the fourth row.
354
00:21:43 --> 00:21:45.9
OK, good.
355
00:21:45.9 --> 00:21:48
There's A transpose.
356
00:21:48 --> 00:21:52
That multiplies y,
y1 y2 y3 y4 and y5.
357
00:21:52 --> 00:21:52
OK.
358
00:21:52 --> 00:21:57
Now you've had time to think
about this question.
359
00:21:57 --> 00:22:04
What's the dimension of the
null space, if I set all those
360
00:22:04 --> 00:22:04
-- wow.
361
00:22:04 --> 00:22:11
Usually -- sometime during this
semester, I'll drop one of these
362
00:22:11 --> 00:22:15
erasers behind there.
363
00:22:15 --> 00:22:18
That's a great moment.
364
00:22:18 --> 00:22:22
There's no recovery.
365
00:22:22 --> 00:22:29
There's -- centuries of erasers
are back there.
366
00:22:29 --> 00:22:30.38
OK.
367
00:22:30.38 --> 00:22:38.5
OK, what's the dimension of the
null space?
368
00:22:38.5 --> 00:22:45
Give me the general formula
first in terms of r and m and n.
369
00:22:45 --> 00:22:51
This is like crucial,
you -- we struggled to,
370
00:22:51 --> 00:22:59
to decide what dimension meant,
and then we figured out what it
371
00:22:59 --> 00:23:06
equaled for an m by n matrix of
rank r, and the answer was m-r,
372
00:23:06 --> 00:23:08.75
right?
373
00:23:08.75 --> 00:23:14
There are m=5 components,
m=5 columns of A transpose.
374
00:23:14 --> 00:23:19
And r of those columns are
pivot columns,
375
00:23:19 --> 00:23:22
because it'll have r pivots.
376
00:23:22 --> 00:23:24
It has rank r.
377
00:23:24 --> 00:23:29
And m-r are the free ones now
for A transpose,
378
00:23:29 --> 00:23:35.9
so that's five minus three,
so that's two.
379
00:23:35.9 --> 00:23:39
And I would like to find this
null space.
380
00:23:39 --> 00:23:41
I know its dimension.
381
00:23:41 --> 00:23:44
Now I want to find out a basis
for it.
382
00:23:44 --> 00:23:48
And I want to understand what
this equation is.
383
00:23:48 --> 00:23:52
So let me say what A transpose
y actually represents,
384
00:23:52 --> 00:23:56
why I'm interested in that
equation.
385
00:23:56 --> 00:24:02
I'll put it down with those old
erasers and continue this.
386
00:24:02 --> 00:24:07
Here's the great picture of
applied mathematics.
387
00:24:07 --> 00:24:09
So let me complete that.
388
00:24:09 --> 00:24:15
There's a matrix that I'll call
C that connects potential
389
00:24:15 --> 00:24:17
differences to currents.
390
00:24:17 --> 00:24:23.72
So I'll call these --
these are currents on the
391
00:24:23.72 --> 00:24:27.07
edges, y1 y2 y3 y4 and y5.
392
00:24:27.07 --> 00:24:31
Those are currents on the
edges.
393
00:24:31 --> 00:24:38
And this relation between
current and potential difference
394
00:24:38 --> 00:24:40
is Ohm's Law.
395
00:24:40 --> 00:24:43
This here is Ohm's Law.
396
00:24:43 --> 00:24:51
Ohm's Law says that the current
on an edge is some number times
397
00:24:51 --> 00:24:54.9
the potential drop.
398
00:24:54.9 --> 00:24:59
That's -- and that number is
the conductance of the edge,
399
00:24:59 --> 00:25:01
one over the resistance.
400
00:25:01 --> 00:25:06
This is the old current is,
is, the relation of current,
401
00:25:06 --> 00:25:09
resistance, and change in
potential.
402
00:25:09 --> 00:25:14
So it's a change in potential
that makes some current happen,
403
00:25:14 --> 00:25:19
and it's Ohm's Law that says
how much current happens.
404
00:25:19 --> 00:25:19
OK.
405
00:25:19 --> 00:25:28.06
And then the final step of this
framework is the equation A
406
00:25:28.06 --> 00:25:31
transpose y equals zero.
407
00:25:31 --> 00:25:36
And that's -- what is that
saying?
408
00:25:36 --> 00:25:39
It has a famous name.
409
00:25:39 --> 00:25:47
It's Kirchoff's Current Law,
KCL, Kirchoff's Current Law,
410
00:25:47 --> 00:25:52.2
A transpose y equals zero.
411
00:25:52.2 --> 00:25:56
So that when I'm solving,
and when I go back up with this
412
00:25:56 --> 00:26:00
blackboard and solve A transpose
y equals zero,
413
00:26:00 --> 00:26:04
it's this pattern of -- that I
want you to see.
414
00:26:04 --> 00:26:08
That we had rectangular
matrices, but -- and real
415
00:26:08 --> 00:26:12
applications,
but in those real applications
416
00:26:12 --> 00:26:15
comes A and A transpose.
417
00:26:15 --> 00:26:21
So our four subspaces are
exactly the right things to know
418
00:26:21 --> 00:26:22
about.
419
00:26:22 --> 00:26:23
All right.
420
00:26:23 --> 00:26:28
Let's know about that null
space of A transpose.
421
00:26:28 --> 00:26:31
Wait a minute,
where'd it go?
422
00:26:31 --> 00:26:32
There it is.
423
00:26:32 --> 00:26:33
OK.
424
00:26:33 --> 00:26:33
OK.
425
00:26:33 --> 00:26:37
Null space of A transpose.
426
00:26:37 --> 00:26:40
We know what its dimension
should be.
427
00:26:40 --> 00:26:44
Let's find out -- tell me a
vector in it.
428
00:26:44 --> 00:26:47
Tell me -- now,
so what I asking you?
429
00:26:47 --> 00:26:53
I'm asking you for five
currents that satisfy Kirchoff's
430
00:26:53 --> 00:26:54
Current Law.
431
00:26:54 --> 00:26:59
So we better understand what
that law says.
432
00:26:59 --> 00:27:03
That, that law,
A transpose y equals zero,
433
00:27:03 --> 00:27:08
what does that say,
say in the first row of A
434
00:27:08 --> 00:27:09
transpose?
435
00:27:09 --> 00:27:15
That says -- the so the first
row of A transpose says minus y1
436
00:27:15 --> 00:27:18
minus y3 minus y4 is zero.
437
00:27:18 --> 00:27:22
Where did that equation come
from?
438
00:27:22 --> 00:27:26.7
Let me -- I'll redraw the
graph.
439
00:27:26.7 --> 00:27:31
Can I redraw the graph here,
so that we -- maybe here,
440
00:27:31 --> 00:27:35
so that we see again -- there
was node one,
441
00:27:35 --> 00:27:40.02
node two, node three,
node four was off here.
442
00:27:40.02 --> 00:27:42
That was, that was our graph.
443
00:27:42 --> 00:27:45
We had currents on those.
444
00:27:45 --> 00:27:48
We had a current y1 going
there.
445
00:27:48 --> 00:27:53
We had a current y --
what were the other,
446
00:27:53 --> 00:27:58
what are those edge numbers?
y4 here and y3 here.
447
00:27:58 --> 00:28:00
And then a y2 and a y5.
448
00:28:00 --> 00:28:06
I'm, I'm just copying what was
on the other board so it's ea-
449
00:28:06 --> 00:28:08
convenient to see it.
450
00:28:08 --> 00:28:14
What is this equation telling
me, this first equation of
451
00:28:14 --> 00:28:17
Kirchoff's Current Law?
452
00:28:17 --> 00:28:20
What does that mean for that
graph?
453
00:28:20 --> 00:28:24
Well, I see y1,
y3, and y4 as the currents
454
00:28:24 --> 00:28:25.57
leaving node one.
455
00:28:25.57 --> 00:28:29
So sure enough,
the first equation refers to
456
00:28:29 --> 00:28:31
node one, and what does it say?
457
00:28:31 --> 00:28:34
It says that the net flow is
zero.
458
00:28:34 --> 00:28:39.42
That, that equation A transpose
y, Kirchoff's Current Law,
459
00:28:39.42 --> 00:28:44
is a balance equation,
a conservation law.
460
00:28:44 --> 00:28:48
Physicists, be overjoyed,
right, by this stuff.
461
00:28:48 --> 00:28:52
It, it says that in equals out.
462
00:28:52 --> 00:28:56
And in this case,
the three arrows are all going
463
00:28:56 --> 00:29:01
out, so it says y1,
y3, and y4 add to zero.
464
00:29:01 --> 00:29:03
Let's take the next one.
465
00:29:03 --> 00:29:09
The second row is y1-y2,
and that's all that's in that
466
00:29:09 --> 00:29:10.95
row.
467
00:29:10.95 --> 00:29:16
And that must have something to
do with node two.
468
00:29:16 --> 00:29:20
And sure enough,
it says y1=y2,
469
00:29:20 --> 00:29:24.31
current in equals current out.
470
00:29:24.31 --> 00:29:29
The third one,
y2 plus y3 minus y5 equals
471
00:29:29 --> 00:29:29
zero.
472
00:29:29 --> 00:29:37
That certainly will be what's
up at the third node.
473
00:29:37 --> 00:29:40
y2 coming in,
y3 coming in,
474
00:29:40 --> 00:29:43
y5 going out has to balance.
475
00:29:43 --> 00:29:47
And finally,
y4 plus y5 equals zero says
476
00:29:47 --> 00:29:53.08
that at this node,
y4 plus y5, the total flow,
477
00:29:53.08 --> 00:29:53
is zero.
478
00:29:53 --> 00:29:59
We don't -- you know,
charge doesn't accumulate at
479
00:29:59 --> 00:30:02
the nodes.
480
00:30:02 --> 00:30:04
It travels around.
481
00:30:04 --> 00:30:04
OK.
482
00:30:04 --> 00:30:11
Now give me -- I come back now
to the linear algebra question.
483
00:30:11 --> 00:30:16
What's a vector y that solves
these equations?
484
00:30:16 --> 00:30:22
Can I figure out what the null
space is for this matrix,
485
00:30:22 --> 00:30:27.7
A transpose,
by looking at the graph?
486
00:30:27.7 --> 00:30:31.42
I'm happy if I don't have to do
elimination.
487
00:30:31.42 --> 00:30:34
I can do elimination,
we know how to do,
488
00:30:34 --> 00:30:38
we know how to find the null
space basis.
489
00:30:38 --> 00:30:43
We can do elimination on this
matrix, and we'll get it into a
490
00:30:43 --> 00:30:48
good reduced row echelon form,
and the special solutions will
491
00:30:48 --> 00:30:50
pop right out.
492
00:30:50 --> 00:30:55
But I would like to -- even to
do it without that.
493
00:30:55 --> 00:31:00
Let me just ask you first,
if I did elimination on that,
494
00:31:00 --> 00:31:04
on that, matrix,
what would the last row become?
495
00:31:04 --> 00:31:10
What would the last row -- if I
do elimination on that matrix,
496
00:31:10 --> 00:31:15
the last row of R will be all
zeros, right?
497
00:31:15 --> 00:31:15
Why?
498
00:31:15 --> 00:31:18
Because the rank is three.
499
00:31:18 --> 00:31:20
We only going to have three
pivots.
500
00:31:20 --> 00:31:25
And the fourth row will be all
zeros when we eliminate.
501
00:31:25 --> 00:31:29
So elimination will tell us
what, what we spotted earlier,
502
00:31:29 --> 00:31:34
what's the null space -- all
the, all the information,
503
00:31:34 --> 00:31:37.4
what are the dependencies.
504
00:31:37.4 --> 00:31:42
We'll find those by
elimination, but here in a real
505
00:31:42 --> 00:31:46
example, we can find them by
thinking.
506
00:31:46 --> 00:31:46
OK.
507
00:31:46 --> 00:31:51
Again, my question is,
what is a solution y?
508
00:31:51 --> 00:31:57
How could current travel around
this network without collecting
509
00:31:57 --> 00:32:01.5
any charge at the nodes?
510
00:32:01.5 --> 00:32:02
Tell me a y.
511
00:32:02 --> 00:32:03
OK.
512
00:32:03 --> 00:32:08
So a basis for the null space
of A transpose.
513
00:32:08 --> 00:32:11
How many vectors I looking for?
514
00:32:11 --> 00:32:12
Two.
515
00:32:12 --> 00:32:15.48
It's a two dimensional space.
516
00:32:15.48 --> 00:32:19
My basis should have two
vectors in it.
517
00:32:19 --> 00:32:22
Give me one.
518
00:32:22 --> 00:32:24
One set of currents.
519
00:32:24 --> 00:32:27
Suppose, let me start it.
520
00:32:27 --> 00:32:31
Let me start with y1 as one.
521
00:32:31 --> 00:32:31
OK.
522
00:32:31 --> 00:32:38
So one unit of -- one amp
travels on edge one with the
523
00:32:38 --> 00:32:39
arrow.
524
00:32:39 --> 00:32:41
OK, then what?
525
00:32:41 --> 00:32:42
What is y2?
526
00:32:42 --> 00:32:46
It's one also,
right?
527
00:32:46 --> 00:32:51
And of course what you did was
solve Kirchoff's Current Law
528
00:32:51 --> 00:32:53
quickly in the second equation.
529
00:32:53 --> 00:32:54
OK.
530
00:32:54 --> 00:32:59
Now we've got one amp leaving
node one, coming around to node
531
00:32:59 --> 00:32:59
three.
532
00:32:59 --> 00:33:01
What shall we do now?
533
00:33:01 --> 00:33:06
Well, what shall I take for y3
in other words?
534
00:33:06 --> 00:33:10
Oh, I've got a choice,
but why not make it what you
535
00:33:10 --> 00:33:12
said, negative one.
536
00:33:12 --> 00:33:17
So I have just sent current,
one amp, around that loop.
537
00:33:17 --> 00:33:20
What shall y4 and y5 be in this
case?
538
00:33:20 --> 00:33:23
We could take them to be zero.
539
00:33:23 --> 00:33:26.68
This satisfies Kirchoff's
Current Law.
540
00:33:26.68 --> 00:33:31
We could check it patiently,
that minus y1 minus y3 gives
541
00:33:31 --> 00:33:33
zero.
542
00:33:33 --> 00:33:35
We know y1 is y2.
543
00:33:35 --> 00:33:39
The others, y4 plus y5 is
certainly zero.
544
00:33:39 --> 00:33:45
Any current around a loop
satisfies -- satisfies the
545
00:33:45 --> 00:33:46.39
Current Law.
546
00:33:46.39 --> 00:33:46
OK.
547
00:33:46 --> 00:33:50
Now you know how to get another
one.
548
00:33:50 --> 00:33:54.9
Take current around this loop.
549
00:33:54.9 --> 00:34:00
So now let y3 be one,
y5 be one, and y4 be minus one.
550
00:34:00 --> 00:34:06
And so, so we have the first
basis vector sent current around
551
00:34:06 --> 00:34:12
that loop, the second basis
vector sends current around that
552
00:34:12 --> 00:34:12
loop.
553
00:34:12 --> 00:34:18
And I've -- and those are
independent, and I've got two
554
00:34:18 --> 00:34:23
solutions --
two vectors in the null space
555
00:34:23 --> 00:34:27
of A transpose,
two solutions to Kirchoff's
556
00:34:27 --> 00:34:28
Current Law.
557
00:34:28 --> 00:34:34
Of course you would say what
about sending current around the
558
00:34:34 --> 00:34:35
big loop.
559
00:34:35 --> 00:34:37
What about that vector?
560
00:34:37 --> 00:34:41
One for y1, one for y2,
nothing f- on y3,
561
00:34:41 --> 00:34:46.3
one for y5, and minus one for
y4.
562
00:34:46.3 --> 00:34:47
What about that?
563
00:34:47 --> 00:34:52
Is that, is that in the null
space of A transpose?
564
00:34:52 --> 00:34:52
Sure.
565
00:34:52 --> 00:34:57
So why don't we now have a
third vector in the basis?
566
00:34:57 --> 00:35:01
Because it's not independent,
right?
567
00:35:01 --> 00:35:03
It's not independent.
568
00:35:03 --> 00:35:06
This vector is the sum of those
two.
569
00:35:06 --> 00:35:12.4
If I send current around that
and around that --
570
00:35:12.4 --> 00:35:17
then on this edge y3 it's going
to cancel out and I'll have
571
00:35:17 --> 00:35:22
altogether current around the
whole, the outside loop.
572
00:35:22 --> 00:35:27
That's what this one is,
but it's a combination of those
573
00:35:27 --> 00:35:28
two.
574
00:35:28 --> 00:35:33
Do you see that I've now,
I've identified the null space
575
00:35:33 --> 00:35:37
of A transpose --
but more than that,
576
00:35:37 --> 00:35:41
we've solved Kirchoff's Current
Law.
577
00:35:41 --> 00:35:45
And understood it in terms of
the network.
578
00:35:45 --> 00:35:45
OK.
579
00:35:45 --> 00:35:48
So that's the null space of A
transpose.
580
00:35:48 --> 00:35:55.5
I guess I -- there's always one
more space to ask you about.
581
00:35:55.5 --> 00:36:01
Let's see, I guess I need the
row space of A,
582
00:36:01 --> 00:36:05
the column space of A
transpose.
583
00:36:05 --> 00:36:10
So what's N,
what's its dimension?
584
00:36:10 --> 00:36:10.96
Yup?
585
00:36:10.96 --> 00:36:18
What's the dimension of the row
space of A?
586
00:36:18 --> 00:36:21
If I look at the original A,
it had five rows.
587
00:36:21 --> 00:36:23
How many were independent?
588
00:36:23 --> 00:36:27
Oh, I guess I'm asking you the
rank again, right?
589
00:36:27 --> 00:36:29
And the answer is three,
right?
590
00:36:29 --> 00:36:31.36
Three independent rows.
591
00:36:31.36 --> 00:36:34
When I transpose it,
there's three independent
592
00:36:34 --> 00:36:35
columns.
593
00:36:35 --> 00:36:39.75
Are those columns independent,
those three?
594
00:36:39.75 --> 00:36:45
The first three columns,
are they the pivot columns of
595
00:36:45 --> 00:36:46
the matrix?
596
00:36:46 --> 00:36:46
No.
597
00:36:46 --> 00:36:50
Those three columns are not
independent.
598
00:36:50 --> 00:36:54
There's a in fact,
this tells me a relation
599
00:36:54 --> 00:36:57.25
between them.
600
00:36:57.25 --> 00:37:01
There's a vector in the null
space that says the first column
601
00:37:01 --> 00:37:04
plus the second column equals
the third column.
602
00:37:04 --> 00:37:08.78
They're not independent because
they come from a loop.
603
00:37:08.78 --> 00:37:12
So the pivot columns,
the pivot columns of this
604
00:37:12 --> 00:37:15
matrix will be the first,
the second, not the third,
605
00:37:15 --> 00:37:17
but the fourth.
606
00:37:17 --> 00:37:22
One, columns one,
two, and four are OK.
607
00:37:22 --> 00:37:29
Where are they -- those are the
columns of A transpose,
608
00:37:29 --> 00:37:32
those correspond to edges.
609
00:37:32 --> 00:37:37
So there's edge one,
there's edge two,
610
00:37:37 --> 00:37:40
and there's edge four.
611
00:37:40 --> 00:37:45
So there's a --
that's like -- is a,
612
00:37:45 --> 00:37:46
smaller graph.
613
00:37:46 --> 00:37:51
If I just look at the part of
the graph that I've,
614
00:37:51 --> 00:37:57.66
that I've, thick -- used with
thick edges, it has the same
615
00:37:57.66 --> 00:37:58
four nodes.
616
00:37:58 --> 00:38:01
It only has three edges.
617
00:38:01 --> 00:38:07.8
And the, those edges correspond
to the independent guys.
618
00:38:07.8 --> 00:38:12
And in the graph there -- those
three edges have no loop,
619
00:38:12 --> 00:38:13
right?
620
00:38:13 --> 00:38:18
The independent ones are the
ones that don't have a loop.
621
00:38:18 --> 00:38:21
All the -- dependencies came
from loops.
622
00:38:21 --> 00:38:27.42
They were the things in the
null space of A transpose.
623
00:38:27.42 --> 00:38:31
If I take three pivot columns,
there are no dependencies among
624
00:38:31 --> 00:38:34
them, and they form a graph
without a loop,
625
00:38:34 --> 00:38:38
and I just want to ask you
what's the name for a graph
626
00:38:38 --> 00:38:39
without a loop?
627
00:38:39 --> 00:38:43
So a graph without a loop is --
has got not very many edges,
628
00:38:43 --> 00:38:44
right?
629
00:38:44 --> 00:38:47
I've got four nodes and it only
has three edges,
630
00:38:47 --> 00:38:51.9
and if I put another edge in,
I would have a loop.
631
00:38:51.9 --> 00:38:58
So it's this graph with no
loops, and it's the one where
632
00:38:58 --> 00:39:02
the rows of A are independent.
633
00:39:02 --> 00:39:08
And what's a graph called that
has no loops?
634
00:39:08 --> 00:39:10
It's called a tree.
635
00:39:10 --> 00:39:16.72
So a tree is the name for a
graph with no loops.
636
00:39:16.72 --> 00:39:21
And just to take one last step
here.
637
00:39:21 --> 00:39:26
Using our formula for
dimension.
638
00:39:26 --> 00:39:36
Using our formula for
dimension, let's look -- once at
639
00:39:36 --> 00:39:38
this formula.
640
00:39:38 --> 00:39:47
The dimension of the null space
of A transpose is m-r.
641
00:39:47 --> 00:39:48.52
OK.
642
00:39:48.52 --> 00:39:58
This is the number of loops,
number of independent loops.
643
00:39:58 --> 00:40:03
m is the number of edges.
644
00:40:03 --> 00:40:07.3
And what is r?
645
00:40:07.3 --> 00:40:11
What is r for our -- we'll have
to remember way back.
646
00:40:11 --> 00:40:17
The rank came -- from looking
at the columns of our matrix.
647
00:40:17 --> 00:40:18.88
So what's the rank?
648
00:40:18.88 --> 00:40:20
Let's just remember.
649
00:40:20 --> 00:40:25
Rank was -- you remember there
was one -- we had a one
650
00:40:25 --> 00:40:29
dimensional --
rank was n minus one,
651
00:40:29 --> 00:40:32
that's what I'm struggling to
say.
652
00:40:32 --> 00:40:38
Because there were n columns
coming from the n nodes,
653
00:40:38 --> 00:40:42
so it's minus,
the number of nodes minus one,
654
00:40:42 --> 00:40:47
because of that C,
that one one one one vector in
655
00:40:47 --> 00:40:49
the null space.
656
00:40:49 --> 00:40:53
The columns were not
independent.
657
00:40:53 --> 00:40:59
There was one dependency,
so we needed n minus one.
658
00:40:59 --> 00:41:02
This is a great formula.
659
00:41:02 --> 00:41:09
This is like the first shall I,
-- write it slightly
660
00:41:09 --> 00:41:10
differently?
661
00:41:10 --> 00:41:17.4
The number of edges -- let me
put things -- have I got it
662
00:41:17.4 --> 00:41:19.25
right?
663
00:41:19.25 --> 00:41:24.36
Number of edges is m,
the number -- r- is m-r,
664
00:41:24.36 --> 00:41:24.7
OK.
665
00:41:24.7 --> 00:41:31
So, so I'm getting -- let me
put the number of nodes on the
666
00:41:31 --> 00:41:32
other side.
667
00:41:32 --> 00:41:39
So I -- the number of nodes --
I'll move that to the other side
668
00:41:39 --> 00:41:46.39
-- minus the number of edges
plus the number of loops is -- I
669
00:41:46.39 --> 00:41:50
have minus, minus one is one.
670
00:41:50 --> 00:41:55
The number of nodes minus the
number of edges plus the number
671
00:41:55 --> 00:41:56
of loops is one.
672
00:41:56 --> 00:41:58
These are like zero dimensional
guys.
673
00:41:58 --> 00:42:00
They're the points on the
graph.
674
00:42:00 --> 00:42:03
The edges are like one
dimensional things,
675
00:42:03 --> 00:42:05.77
they're, they connect nodes.
676
00:42:05.77 --> 00:42:08.64
The loops are like two
dimensional things.
677
00:42:08.64 --> 00:42:11.3
They have, like,
an area.
678
00:42:11.3 --> 00:42:15
And this count works for every
graph.
679
00:42:15 --> 00:42:19
And it's known as Euler's
Formula.
680
00:42:19 --> 00:42:24
We see Euler again,
that guy never stopped.
681
00:42:24 --> 00:42:24
OK.
682
00:42:24 --> 00:42:29
And can we just check -- so
what I saying?
683
00:42:29 --> 00:42:36.5
I'm saying that linear algebra
proves Euler's Formula.
684
00:42:36.5 --> 00:42:41
Euler's Formula is this great
topology fact about any graph.
685
00:42:41 --> 00:42:46.73
I'll draw, let me draw another
graph, let me draw a graph with
686
00:42:46.73 --> 00:42:48.52
more edges and loops.
687
00:42:48.52 --> 00:42:50
Let me put in lots of -- OK.
688
00:42:50 --> 00:42:53
I just drew a graph there.
689
00:42:53 --> 00:42:57
So what are the,
what are the quantities in that
690
00:42:57 --> 00:42:58
formula?
691
00:42:58 --> 00:43:01
How many nodes have I got?
692
00:43:01 --> 00:43:02
Looks like five.
693
00:43:02 --> 00:43:05
How many edges have I got?
694
00:43:05 --> 00:43:08
One two three four five six
seven.
695
00:43:08 --> 00:43:10
How many loops have I got?
696
00:43:10 --> 00:43:12
One two three.
697
00:43:12 --> 00:43:15
And Euler's right,
I always get one.
698
00:43:15 --> 00:43:19
That, this formula,
is extremely useful in
699
00:43:19 --> 00:43:25
understanding the relation of
these quantities --
700
00:43:25 --> 00:43:30
the number of nodes,
the number of edges,
701
00:43:30 --> 00:43:34
and the number of loops.
702
00:43:34 --> 00:43:34
OK.
703
00:43:34 --> 00:43:42
Just complete this lecture by
completing this picture,
704
00:43:42 --> 00:43:44
this cycle.
705
00:43:44 --> 00:43:52
So let me come to the -- so
this expresses the equations of
706
00:43:52 --> 00:43:55.8
applied math.
707
00:43:55.8 --> 00:44:00
This, let me call these
potential differences,
708
00:44:00 --> 00:44:00
say, E.
709
00:44:00 --> 00:44:01
So E is A x.
710
00:44:01 --> 00:44:05
That's the equation for this
step.
711
00:44:05 --> 00:44:09.59
The currents come from the
potential differences.
712
00:44:09.59 --> 00:44:10
y is C E.
713
00:44:10 --> 00:44:17
The potential -- the currents
satisfy Kirchoff's Current Law.
714
00:44:17 --> 00:44:21
Those are the equations of --
with no source terms.
715
00:44:21 --> 00:44:26
Those are the equations of
electrical circuits of many --
716
00:44:26 --> 00:44:30
those are like the,
the most basic three equations.
717
00:44:30 --> 00:44:33
Applied math comes in this
structure.
718
00:44:33 --> 00:44:38
The only thing I haven't got
yet in the picture is an outside
719
00:44:38 --> 00:44:42
source to make something happen.
720
00:44:42 --> 00:44:46
I could add a current source
here, I could,
721
00:44:46 --> 00:44:52
I could add external currents
going in and out of nodes.
722
00:44:52 --> 00:44:55
I could add batteries in the
edges.
723
00:44:55 --> 00:44:57
Those are two ways.
724
00:44:57 --> 00:45:02
If I add batteries in the
edges, they, they come into
725
00:45:02 --> 00:45:03
here.
726
00:45:03 --> 00:45:07
Let me add current sources.
727
00:45:07 --> 00:45:11
If I add current sources,
those come in here.
728
00:45:11 --> 00:45:15
So there's a,
there's where current sources
729
00:45:15 --> 00:45:20
go, because the F is a like a
current coming from outside.
730
00:45:20 --> 00:45:24
So we have our edges,
we have our graph,
731
00:45:24 --> 00:45:29
and then I send one amp into
this node and out of this node
732
00:45:29 --> 00:45:32
--
and that gives me,
733
00:45:32 --> 00:45:35
a right-hand side in Kirchoff's
Current Law.
734
00:45:35 --> 00:45:40
And can I -- to complete the
lecture, I'm just going to put
735
00:45:40 --> 00:45:43
these three equations together.
736
00:45:43 --> 00:45:46
So I start with x,
my unknown.
737
00:45:46 --> 00:45:47
I multiply by A.
738
00:45:47 --> 00:45:51
That gives me the potential
differences.
739
00:45:51 --> 00:45:58
That was our matrix A that the
whole thing started with.
740
00:45:58 --> 00:45:59
I multiply by C.
741
00:45:59 --> 00:46:05
Those are the physical
constants in Ohm's Law.
742
00:46:05 --> 00:46:06
Now I have y.
743
00:46:06 --> 00:46:11
I multiply y by A transpose,
and now I have F.
744
00:46:11 --> 00:46:16
So there is the whole thing.
745
00:46:16 --> 00:46:19
There's the basic equation of
applied math.
746
00:46:19 --> 00:46:24
Coming from these three steps,
in which the last step is this
747
00:46:24 --> 00:46:25
balance equation.
748
00:46:25 --> 00:46:29
There's always a balance
equation to look for.
749
00:46:29 --> 00:46:33
These are the -- when I say the
most basic equations of applied
750
00:46:33 --> 00:46:36
mathematics --
I should say,
751
00:46:36 --> 00:46:37
in equilibrium.
752
00:46:37 --> 00:46:40.07
Time isn't in this problem.
753
00:46:40.07 --> 00:46:43
I'm not -- and Newton's Law
isn't acting here.
754
00:46:43 --> 00:46:47
I'm, I'm looking at the --
equations when everything has
755
00:46:47 --> 00:46:51
settled down,
how do the currents distribute
756
00:46:51 --> 00:46:53
in the network.
757
00:46:53 --> 00:46:58
And of course there are big
codes to solve the -- this is
758
00:46:58 --> 00:47:04
the basic problem of numerical
linear algebra for systems of
759
00:47:04 --> 00:47:08.85
equations, because that's how
they come.
760
00:47:08.85 --> 00:47:11
And my final question.
761
00:47:11 --> 00:47:17.5
What can you tell me about this
matrix A transpose C A?
762
00:47:17.5 --> 00:47:19
Or even A transpose A?
763
00:47:19 --> 00:47:23
I'll just close with that
question.
764
00:47:23 --> 00:47:28
What do you know about the
matrix A transpose A?
765
00:47:28 --> 00:47:31
It is always symmetric,
right.
766
00:47:31 --> 00:47:32
OK, thank.
767
00:47:32 --> 00:47:38
So I'll see you Wednesday for a
full review of these chapters,
768
00:47:38 --> 00:47:41
and Friday you get to tell me.
769
00:47:41 --> 00:47:44
Thanks. | 677.169 | 1 |
Cranston Statisticsue BAmy C.
Bernie L.
...It's also when the infamous "word problem" gets introduced. Algebra II extends what is covered in Algebra I, and typically includes more quadratic equations, and introduction to advanced factoring, and more simultaneous equations. It's fun too! | 677.169 | 1 |
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PRODUCT DESCRIPTION
Quadratics "Clue" Project for either Algebra 1 or Algebra 2. Fun way for your class to learn and practice quadratics by solving a murder mystery. Students will be required to graph and solve equations to determine who committed a murder, with what weapon, and the location of the crime.
This project can be done individually or in groups of up to 3. The project will last several days, probably up to a week depending on the ability of your students and whether or not you are on a block schedule.
Includes very detailed instructions on getting started as well as all the handouts you will need. I also made scans of my own answer keys. The answer keys are in my own hand writing, but look much better than the sample I scanned to the right, I promise!
Please check out the thumbnails!
My students really enjoy this project each year - I'm sure yours will too :)
=-=-=-=-=-=-=-=-=-=
UPDATE: A friend of mine recently kicked this off by using his computer and smartboard to introduce the victim and the "persons of interest" by showing their youtube videos to the class. Yes each character is a "real" youtube celebrity. If you have the means to stream the videos, I highly recommend it. It was quite entertaining for his kids and only took a few minutes.
1 word of warning though. when it comes to the honey badger, make certain you show the (clean) version and save the original one to watch with your teacher friends in the lounge lol. Even the (clean) version still contains one curse word near the end that starts with a "b", so use good judgement here please. Having fun is never worth helicopter parent drama :/
For this item, the cost for one user (you) is $4.99.
If you plan to share this product with other teachers in your school, please add the number of additional users licenses that you need to purchase.
Each additional license costs only $2.00. | 677.169 | 1 |
Expressions and Equations
Table of Contents
Terms
This chapter is a bridge between pre-algebra and algebra. It starts with a review of the material learned at the end of pre-algebra, and leads right into working with variables. It explains how to simplify expressions and solve equations; in addition, it teaches the formal properties which allow us to solve equations.
Much of the information covered in the first section can be found in the last chapter of the Pre-Algebra SparkNote. This section discusses variables and how they are used. It outlines the steps for translating word statements into algebraic expressions and algebraic equations.
The second section explains what a term is and discusses how to simplify expressions by combining like terms. Simplifying expressions makes expressions a lot easier to think about and work with, and a most algebra problems require this skill.
The next section is a first look at solving equations. It discusses the meaning of a solution and it explains how to find a solution set from a replacement set. Using the skills learned in the first two sections, the reader will begin to solve equations.
The fourth section details a different method of solving equations-- solving by inverse operations. The section also focuses on solving equations by reversing multiple operations.
In the fifth section, we learn the properties that allow us to perform inverse operations--the properties of equality and the properties of operations and identities. These properties, which all real numbers have, are important to know both for solving equations and for future proofs involving equations. They are interesting to learn because they reveal the nature of the numbers we work with in all of mathematics.
Using these properties, the final section shows how to solve an equation that contains variables on both sides. Many of the equations we will encounter in algebra are of this type, and it is essential to know how to solve them.
Overall, this chapter is the first in-depth look at what we will be doing in the rest of algebra. It is impossible to understand the material covered in future chapters without a thorough understanding of this chapter. In fact, this chapter is essential to understanding all future mathematics. Equations play a huge role in Geometry, Algebra II, and Calculus, as well as in higher math, so it is crucial to know how to work with them and solve them. | 677.169 | 1 |
This book comprehensively covers several hundred functions or function families. In chapters that progress by degree of complexity, it starts with simple, integer-valued functions then moves on to polynomials, Bessel, hypergeometric and hundreds moreBased on the authors? combined 35 years of experience in teaching, A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the way a typical mathematician works observing patterns, conducting experiments by means of looking at or creating examples, trying to understand... more...
This secondrevised andextendededition of the self-contained and unified approach to Bernstein functions and their subclassesbrings together old and establishs new connections. Applications of Bernstein functions in different fields of mathematics (such as probability theory, potential theory, operator theory, integral equations, functional... more...
Treated in this volume are selected topics in analytic &Ggr;-almost-periodic functions and their representations as &Ggr;-analytic functions in the big-plane; n -tuple Shilov boundaries of function spaces, minimal norm principle for vector-valued functions and their applications in the study of vector-valued functions and n -tuple polynomial and... more...
In 1821, Augustin-Louis Cauchy published a textbook to accompany his course in analysis at the Ecole Polytechnique. It is still regarded as one of the most influential mathematics book ever written, and this is the first English translation of that book. more...
Complex Analysis with Applications to Flows and Fields presents the theory of functions of a complex variable, from the complex plane to the calculus of residues to power series to conformal mapping. The book explores numerous physical and engineering applications concerning potential flows, the gravity field, electro- and magnetostatics, steady... more...
From the reviews: "The author, [...], has written a book which will be of service to all who are interested in this by now vast subject. [...] This is a book of many virtues: mathematical, historical, and pedagogical. Parts of it could be used for a graduate complex manifolds course. J.A. Carlson in Mathematical Reviews , 1987 "There are many... more... | 677.169 | 1 |
Graphing Calculator by Mathlab
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Graphing Calculator is an insanely complete app, thought for helping any need any algebra student may need. It also comes with a smooth interface and a supporting website and | 677.169 | 1 |
Details about Number Theory:
This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.
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Rent Number Theory 1st edition today, or search our site for other textbooks by Henri Cohen. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Springer. | 677.169 | 1 |
Introduction
As you learn more and more mathematical methods and skills, it is important to think about the purpose of mathematics and how it works as part of a bigger picture. Mathematics is used to solve problems which often arise from real-life situations. Mathematical modeling is a process by which we start with a real-life situation and arrive at a quantitative solution. Modeling involves creating a set of mathematical equations that describes a situation, solving those equations and using them to understand the real-life problem. Often the model needs to be adjusted because it does not describe the situation as well as we wish.
A mathematical model can be used to gain understanding of a real-life situation by learning how the system works, which variables are important in the system and how they are related to each other. Models can also be used to predict and forecast what a system will do in the future or for different values of a parameter. Lastly, a model can be used to estimate quantities that are difficult to evaluate exactly.
Mathematical models are like other types of models. The goal is not to produce an exact copy of the "real" object but rather to give a representation of some aspect of the real thing. The modeling process can be summarized as follows.
Notice that the modeling process is very similar to the problem solving format we have been using throughout this book. In this section, we will focus mostly on the assumptions we make and the validity of the model. Functions are an integral part of the modeling process because they are used to describe the mathematical relationship in a system. One of the most difficult parts of the modeling process is determining which function best describes a situation. We often find that the function we chose is not appropriate. Then, we must choose a different one, or we findthat a function model is good for one set of parameters but we need to use another function for a different set of parameters. Often, for certain parameters, more than one function describes the situation well and using the simplest function is most practical.
Here we present some mathematical models arising from real-world applications.
Example 1 Stretching springs beyond the "elastic limit"
A spring is stretched as you attach more weight at the bottom of the spring. The following table shows the length of the spring in inches for different weights in ounces.
a) Find the length of the spring as a function of the weight attached to it.
b) Find the length of the spring when you attach 5 ounces.
c) Find the length of the spring when you attach 19 ounces.
Solution
Step 1Understand the problem
Define weight in ounces on the spring
length in inches of the spring
Step 2 Devise a plan
Springs usually have a linear relationship between the weight on the spring and the stretched length of the spring. If we make a scatter plot, we notice that for lighter weights the points do seem to fit on a straight line (see graph). Assume that the function relating the length of the spring to the weight is linear.
Step 3 Solve
Find the equation of the line using points describing lighter weights:
(0, 2) and (4, 2.8).
The slope is
Using 4 Check
To check the validity of the solutions lets for small weights, the relationship between the length of the spring and the weight is a linear function.
For larger weights, the spring does not seem to stretch as much for each added ounces. We must change our assumption. There must be a non-linear relationship between the length and the weight.
Step 5 Solve with New Assumptions
Let's find the equation of the function by cubic regression with a graphing calculator. 6 Check
To check the validity of the solutions lets plot the answers to b) and c) on the scatter plot. We see that the answer to both b) and c) are close to the rest of the data.
We conclude that a cubic function represents the stretching of the spring more accurately than a linear function. However, for small weights the linear function is an equally good representation, and it is much easier to use in most cases. In fact, the linear approximation usually allows us to easily solve many problems that would be very difficult to solve by using the cubic function.
Example 2 Water flow
A thin cylinder is filled with water to a height of 50 centimeters. The cylinder has a hole at the bottom which is covered with a stopper. The stopper is released at time seconds and allowed to empty. The following data shows the height of the water in the cylinder at different times.
a) Find the height (in centimeters) of water in the cylinder as a function of time in seconds.
b) Find the height of the water when seconds.
c) Find the height of the water when seconds.
Solution:
Step 1 Understand the problem
Define the time in seconds
height of the water in centimeters
Step 2 Devise a plan
Let's make a scatter plot of our data with the time on the horizontal axis and the height of water on the vertical axis.
Notice that most of the points seem to fit on a straight line when the water level is high. Assume that a function relating the height of the water to the time is linear.
Step 3 Solve
Find the equation of the line using points describing lighter weights:
(0, 50) and (4, 35.7).
The slope is
Using
a) We obtain the function:
b) The height of the water when seconds is
c) The height of the water when seconds is
Step 4 Check
To check the validity of the solutions, let's when the water level is high, the relationship between the height of the water and the time is a linear function. When the water level is low, we must change our assumption. There must be a non-linear relationship between the height and the time.
Step 5 Solve with new assumptions
Let's assume the relationship is quadratic and let's find the equation of the function by quadratic regression with a graphing calculator.
a) We obtain the function
b) The height of the water when seconds is
c) The height of the water when seconds is
Step 6: Check
To check the validity of the solutions lets plot the answers to b) and c) on the scatterplot. We see that the answer to both b) and c) are close to the rest of the data.
We conclude that a quadratic function represents the situation more accurately than a linear function. However, for high water levels the linear function is an equally good representation.
Example 3 Projectile motion
A golf ball is hit down a straight fairway. The following table shows the height of the ball with respect to time. The ball is hit at an angle of 70 degrees with the horizontal with a speed of 40 meters/sec.
a) Find the height of the ball as a function of time.
b) Find the height of the ball when seconds.
c) Find the height of the ball when seconds.
Solution
Step 1Understand the problem
Define the time in seconds
height of the ball in meters
Step 2 Devise a plan
Let's make a scatter plot of our data with the time on the horizontal axis and the height of the ball on the vertical axis. We know that a projectile follows a parabolic path, so we assume that the function relating height to time is quadratic.
Step 3 Solve
Let's find the equation of the function by quadratic regression with a graphing calculator.
a) We obtain the function
b) The height of the ball when seconds is:
c) The height of the ball when seconds is:
Step 4 Check
To check the validity of the solutions lets plot the answers to b) and c) on the scatterplot. We see that the answer to both b) and c) follow the trend very closely. The quadratic function is a very good model for this problem
Example 4 Population growth
A scientist counts two thousand fish in a lake. The fish population increases at a rate of 1.5 fish per generation but the lake has space and food for only 2,000,000 fish. The following table gives the number of fish (in thousands) in each generation.
a) Find the number of fish as a function of generation.
b) Find the number of fish in generation 10.
c) Find the number of fish in generation 25.
Solution:
Step 1 Understand the problem
Define the generation number the number of fish in the lake
Step 2 Devise a plan
Let's make a scatterplot of our data with the generation number on the horizontal axis and the number of fish in the lake on the vertical axis. We know that a population can increase exponentially. So, we assume that we can use an exponential function to describe the relationship between the generation number and the number of fish.
Step 3 Solve
a) Since the population increases at a rate of 1.5 per generation, assume the function
b) The number of fish in generation 10 is: thousand fish
c) The number of fish in generation 25 is: thousand fish
Step 4 Check
To check the validity of the solutions, let's plot the answers to b) and c) on the scatter plot. We see that the answer to b) fits the data well but the answer to c) does not seem to follow the trend very closely. The result is not even on our graph!
When the population of fish is high, the fish compete for space and resources so they do not increase as fast. We must change our assumptions.
Step 5 Solve with new assumptions
When we try different regressions with the graphing calculator, we find that logistic regression fits the data the best.
a) We obtain the function
b) The number of fish in generation 10 is thousand fish
c) The number of fish in generation 25 is thousand fish
Step 6 Check
To check the validity of the solutions, let's plot the answers to b) and c) on the scatter plot. We see that the answer to both b) and c) are close to the rest of the data.
We conclude that a logistic function represents the situation more accurately than an exponential function. However, for small populations the exponential function is an equally good representation, and it is much easier to use in most cases.
Review Questions
In Example 1, evaluate the length of the spring for weight by
Using the linear function
Using the cubic function
Figuring out which function is best to use in this situation.
In Example 1, evaluate the length of the spring for weight by
Using the linear function
Using the cubic 3, evaluate the height of the ball when . Find when the ball is at its highest point.
In Example 4, evaluate the number of fish in generation 8 by
Using the exponential function
Using the logistic function
Figuring out which function is best to use in this situation.
In Example 4, evaluate the number of fish in generation 18 by
Using the exponential function
Using the logistic function
Figuring out which function is best to use in this situation.
Review Answers
2.6 inches
2.6 inches
Both functions give the same result. The linear function is best because it is easier to use.
5 inches
4.5 inches
The two functions give different answers. The cubic function is better because it gives a more accurate answer.
34.96 cm
35.07 cm
The results from both functions are almost the same. The linear function is best because it is easier to use.
-18.02 cm
3.5 cm
The two function give different results. The quadratic function is better because it gives a more accurate answer.
48.6 meters
3.7 seconds
51,000
55,000
The results from both functions are almost the same. The linear function is best because it is easier to use.
2,956,000
1,571,000
the two functions give different results; the logistic function is better because it gives a more accurate answer.
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See | 677.169 | 1 |
Essentials of College Algebra
Browse related Subjects ...
Read More specifically designed to provide a more compact and less expensive text for courses that do not include the more advanced topics covered in the longer text. Focused on helping students develop both the conceptual understanding and the analytical skills necessary to experience success in mathematics, the authors present each mathematical topic in this text using a carefully developed learning system to actively engage students in the learning process. The book addresses the diverse needs of today's students through a clear design, current figures and graphs, helpful features, careful explanations of topics, and a comprehensive package of available supplements and study aids.
Read Less
Hardcover. Instructor Edition: Same as student edition but has free copy markings. New Condition. SKU: 9780321912749Very good. Hardcover. Instructor Edition: Same as student edition with taped cover. Has minor wear and/or markings. SKU: 9780321912251-3-0-18 | 677.169 | 1 |
I'm not the original commenter, but I can say you do need the book for homework problems. You don't have to buy it if you don't mind going to the library and borrowing it from there for a few hours. I bought the binder version directly from the publisher for 60 dollars. You will still need to purchase access to Connect to do the online homework if you do that.
Agreeing with both repliers so far - you definitely need it for homework problems although you could borrow it from the library or easily share with a friend (make sure your schedules are compatible though). | 677.169 | 1 |
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The MATH 180 Course II curriculum transitions students to pre-algebra with an emphasis on building proportional reasoning with rates, ratios and linear relationships, and functions. Visual models bring coherence to instruction, making abstract concepts more concrete. Learn More
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SAM Data Dictionary for Enterprise Edition and Next Generation v2.4.x
The Data Dictionary is a resource created to assist administrators using Enterprise Edition and Next Generation v2.4.x with SAM roster imports and exports.
SAM Data Dictionary for Enterprise Edition and Next Generation v2.3.x
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Elementary Algebra / Edition 1
Suitable for either classroom use or self-paced study, Jacobs's popular text combines real-life examples, carefully structured exercises, and humor to help students learn and remember.
See more details below
Most Helpful Customer Reviews
I bought this book for my homeschooled 8th grader. We used Saxon Math so far, but she was always complaining how boring Math was. She still is not a Math enthusiast, but the "moaning and groaning" has diminished significantly - and she admitts that this book is "kind of" interesting. I think that is probably the best rating you can expect from a teenager! The accompanying teacher handbook is sufficient and helpful whenever we encounter a question. So far we have not had to call in the paternal back-up professor!
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More than 1 year ago
Harold R. Jacobs' clearly written text Elementary Algebra is an engaging introduction to the subject. Jacobs uses humor and references to real persons and events as he explains the concepts that underlie the algorithms he presents in the text. He introduces new topics incrementally. He first covers an idea in one context, then continually revisits it after the student has learned additional material. This reinforces the concept and contributes to the depth of the student's understanding, as do his well-chosen problem sets. His problems are not just drill. Different facets of each topic are explored in the problem sets, which helps the student gain a fuller understanding of the material. Jacobs begins by reviewing arithmetic, while incrementally adding algebraic concepts. He then introduces the reader to linear equations, systems of linear equations, and exponents before pausing for a midterm review. After the midterm review, Jacobs covers polynomial and rational expressions and equations, radicals, properties of the real numbers, and number sequences. The book concludes with a final review. Since each section contains a problem set for which the answers are provided in the back of the text, it is possible to use the book for self-study.
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More than 1 year ago
Because of its readability, I use this text with homeschooled kids who have to be on their own much of the week. Each lesson is accompanied by three or four sets of exercises. The first set is a couple of review problems. The second set has all the answers in the back of the book. The third set is like the second set, but with no answers in the book. The fourth set, if there is one, is a logic puzzle or brain teaser. The teacher guide for this book, with all the answers, is also available, but I don't see it at B&N. The book takes a different path through algebra than most, putting linear equations and simultaneous equations in front of work with polynomials of higher degree. For kids who cognitively might not be quite ready to handle the language of algebra, this gives them a bit gentler approach. For any student, having a guide who speaks the language is helpful. Many homeschooled kids drop out in math when they get to algebra for lack of a good guide. If you're the parent and can't be that person, hire a good tutor! The book does not include complex numbers. | 677.169 | 1 |
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Overview
Chapter 1 sets the stage by describing the games combinatorialists play. It introduces basic combinatorial structures and construction techniques. Chapter 2 discusses frames extensively and includes comprehensive lists of direct and recursive constructions. Chapter 3 provides known classes of RBIBD constructions. Chapter 4 deals with existence results and demonstrates the utility of the frame approach. Chapter 5 is a series of informative tables useful for researchers.
No other book tackles this demanding topic from these varied perspectives. Multi-faceted and written for easy access by different users, Frames and Resolvable Designs: Uses, Constructions and Existence is the right choice for students, as a text in advanced design theory; for researchers, as a resource complement to standard encyclopedic works; and for mathematicians and statisticians in this field, as a working handbook. | 677.169 | 1 |
Double Maths
Why choose Double Maths?
This means studying Maths A level and Further Maths A level together. It is a course for those with a strong interest in pursuing Maths beyond sixth form and demands a strong overall GCSE profile. Students often take it because they want to study Maths, physics or engineering at degree level and would find a stronger mathematical background helpful. It can be extremely rewarding for you if you enjoy Maths and are stimulated by the faster pace and greater depth. The course covers all available Pure Maths modules.
Further details of these courses can be found by opening the Open Evening Document link below.
How can you know whether you'll be suited to A level Maths?
You may be surprised to hear that this is by no means just about your GCSE grade. It is more important for you to reflect on the fact that you need to become fluent with algebra to become successful at AS and A level Maths, algebra being the language of the subject at this level. (It is possible to struggle with algebra at GCSE Maths yet obtain a high grade through proficiency in other topics such as arithmetic, geometry and statistics.
In preparation for your maths course we will be setting summer work, and there will be an early assessment of your algebraic skills to ensure that you are doing the most appropriate Maths course.
If you need help assessing your algebraic skills, you could seek advice from your teacher.
Features
Further Maths is essential for some university courses in Maths,
Further Maths is useful for some university courses in Economics, Computing, Physics and Engineering.
Students currently take five module exams in Year 12 and seven in Year 13.
No coursework.
Includes both mechanics and statistics modules, students are encouraged to use a graphical calculator. | 677.169 | 1 |
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PROFESSOR STRANG: OK, so this is the start. I won't be able to do it all in one day of what I think of as the number one model in applied math, in discrete implied math, I'll say. Let me review what our four examples are. Just so you see the big picture. So the first example was the springs and masses. That was beautiful. It's simple. The masses are all in a line, and the matrix K, the free-fixed and fixed-fixed and free-free come out closely related to our K t b matrices. So that was the natural place to start, and actually we also got a chance to do the most important equation in time. Ku''. Sorry Mu''+Ku=0. So that was a key example. Then least squares. Very important, I'm already getting questions from the class about problems that come up in your work, least square problems. Maybe I'll just mention that the professional numerical guys don't always go to A transpose A. If it's a badly conditioned problem, in that conditioning is a topic that was in 1.7 and we'll eventually come back to, if it's a badly conditioned problem matrix a then, a transpose a kind of makes it worse. So there's another way to orthogonalize in advance. And if you're working with orthogonal vectors, or orthonormal vectors, numerical calculations are as safe as they can be. Yeah. Wall Street is more like A transpose A. And the orthonormal is the safe way.
Alright, this is today's lecture. You'll see the matrix a for a graph, for a network. It's simple to construct, and it just shows up everywhere. Because networks are everywhere. And, just, looking ahead, trusses are there partly because they're the most fun. You'll enjoy trusses. I mean, it's kind of fun to figure out is the truss going to collapse or not. It's good. And actually, what's the linear algebra in there? The collapsing or not will depend on solutions to Au=0. Let me just recall the equation Au=0. If A is our key matrix in each example, it's different in each example. And we sort of hope that Au=0 doesn't have solutions, or that it has solutions we know. Because if Au=0 has solutions that's the case where a transpose a is not invertable and we have to do something. Very useful to review. What were the solutions to Au=0, in the case of springs? Well, there were some in the free-free case. The all ones vector was the solution u or all constant, was the solution u in the free-free case and that's why we couldn't invert it. But the fixed-free or the fixed-fixed, when we have one support or two supports, that removed the all ones solution. Good.
These squares, we assume there weren't any. We assumed because we wanted to work directly with a transpose a, the normal equations, so we assumed that the columns of a were independent. We assumed that there were no non-zero solutions to Au=0. Because if there were, that would have made a transpose a singular, and we would have had to do something different. Here, this'll be a lot like this one. Today, once you see A, you'll spot the solutions to Au=0. This is A for a network. And the solution is going to be that same guy, all ones. And that only tells us again that we have to ground a node, I may use an electrical term. Grounding a node is like fixing a displacement. Once you've fixed one of those, say at zero, whatever, but zero's the natural choice. Once you've said one of the potentials, one of the voltages is zero, then you know all the rest. You can find all the rest from our equations. So this is like this in having this all ones solution. And as you'll see with trusses, that could, depending on the truss, have more solutions. And if there are more solutions that's when the truss collapses. So the trusses need more than just a single support to hold up a whole truss. OK.
So that's the Au=0. Now we're ready for the lecture itself. Graphs and networks. OK, let me start with, what's a graph. A graph is a bunch of nodes and some or all of the edges between them. Let me take just a particular example of a graph. And this of course you spot in the book. Oh and everybody recognized that, and it's probably now corrected, that in the homework where it said 3.4 it meant 2.4, of course. And this is Section 2.4 now. Let me draw a different graph. Maybe it'll have four nodes, at those four edges, let me put in a fifth edge. OK, that's a graph. It's not a complete graph because I didn't include that extra edge. It's not a tree because there are some loops here. So complete graphs are one extreme where all the edges are in. A tree is the other extreme, where you have a minimum number of edges. It would only take probably three edges. So just while we're looking at it, there are a bunch of possible trees that would be sort of inside this graph. Sub-graphs of this graph, if I knock out those two edges I have a tree. Going out, or a tree could be like this. Or a tree could be like this. Anyway, five edges is in this graph, six in a complete graph, it would be three edges in a tree. OK, and the number of edges is always m. So five edges. And the number of nodes is always n, for nodes. So a will be five by four.
OK. And it's called, so we get a special name in this world, it's called the incidence matrix of the graph. The incidence matrix. Or, of course, these things come up so often they have other names, too. But incidence matrix is a pretty general name. OK, I have to number the nodes just so we can create the matrix, A one, two, three, four. And I have to number the edges. If I don't number them, I don't know which is which. So let me call this edge one, from one to two, and I'll draw an arrow on the edges. So from one to two, maybe this'll be edge two, from one to three. This'll be edge three. Oh no, let me put edge three there, would be a natural one, say from two to three. And how about edge four there, from two to four. And edge five going from three to four. OK, so now I have numbered, I've identified the nodes, and I've identified the edges. And there were five edges and four nodes. Usually m is bigger than n. We're in this, except for trees, m will be at least as large as n. And I've put arrows on, so you could say it's a directed graph. Because I've given a direction. You'll see that the directions, those arrow directions, which are just to tell me which way current should count as plus, if it's with the arrow, or which way it should count as minus if it's against the arrow. Of course, current could go either way. It's just, now I have a convention of which is plus and which is minus.
OK, so now let me tell you the incidence matrix. So everybody can get it right away, how do you create this incidence matrix? A five by four. So it's going to have five rows, one for every edge. So what's the row for edge one? And it's got four columns, one for every node. So these are the nodes. Nodes one, two, three, four. So there's a column for every node and a row for every edge. OK, edge one. This is just going to tell me everything about the graph. So exactly what's in that picture will be in this matrix. If I've erased one, I could reproduce it's by knowing the other one. OK, edge one goes from node one to node two. So it leaves node one, I'll put a minus one there. In the first column. And a plus one in the second column. Edge one doesn't touch nodes three and four. So there you go, that's edge one. Let me do edge two and then you'll be able to fill in the rest. So edge two goes from one to three, minus one, and a one. Edge three goes from two to three, I'll just keep going. Minus one and a one. Edge four goes from two to four. And edge five goes from three to four. OK. Simple, right? Got it. That matrix has got all the information that's in my picture, and the matrix, but the point about matrices is, they do something. They multiply vector u to produce something. They have a meaning beyond just a record of the picture. So a is a great thing. In fact, what does it do? Let's see.
So that's the matrix a that we work with. Oh, first tell me about Au=0. Because we brought up that subject already? Are those four columns independent? I've got four columns, they're sitting in five-dimensional space, there's plenty of room there for four independent vectors. Are these four columns independent vectors? No. No, they're not. Because what combination of them produces the zero vector? . If I take that column plus that, plus that, plus that, I'm multiplying by. So, A I'll just put that up here and then I won't have to write it again. A times , is five zeroes. So that u, that particular u, of all ones is, I would say, in the null space of the matrix? The null space is all the solutions at Au=0. In other words, so these four columns, tell me about the geometry again. These four columns, if I take all their combinations, yeah. Think about this. If I take all four combinations, all combination, any amount of this column, this column, this column, that fourth column, those are all vectors in five-dimensional space. Now, this isn't essential but it's good. Do you have an idea of what you'd get? What would you get if you took, so this, think of four vectors, pointing along, take all their combinations, that kind of fills in. Whatever fill in may mean. And what does it fill in? What do I get? What's your image? Frankly, I don't know. I can't visualize five-dimensional space. That well. But still, we can use words. What do you think?
You get a something subspace. You got a something, you get something flat. I don't know if you do. It's pretty flat, somehow. Like I'm just asking you to jump up from a case we know. Where we had columns in three-dimensional space and we took a combination and they gave us a plane. Right, when they were dependent? Now, how would you visualize the combinations in five-dimensional space? Just for the heck of it? It's some kind of a subspace, I would say. And what's its dimension, maybe that's what I want to ask you. What's the dimension? Do I get, like, a four-dimensional subspace of five-dimensional space when I take the combinations of these particular four guys? Yes or no? Do I get a four-dimensional subspace, whatever that may mean? No. Right answer, I don't. I don't. Somehow the dimension of that subspace, whatever I get, isn't four because this fourth guy is not contributing anything new. The fourth one is a combination of the first three. So I get a three-dimensional subspace. The rank of this matrix is three. If you allow me to introduce that key word, rank, is the number of independent columns. It tells you how big the matrix really is. You know, if the matrix, if I pile on a whole lot of zero columns, or a lot of zero rows, the matrix looks bigger. But of course it isn't truly bigger. The heart of the matrix, the core of the matrix is somehow just three. And actually, I tell you now and we'll see it happen, can I tell you the key result in the first half of linear algebra? It's this. That if I have three independent columns, and by the way any three are independent, it's just all four together are dependent. This has three independent columns, then the great fact is, it has three independent rows. That's kind of fantastic. Since it's such a beautiful and remarkable and basic fact, look at the rows. That what linear algebra is all about. Looking at a matrix by columns, and then by rows, and seeing what are the connections.
And the connection is, the key connection is, that these five rows, now what space are they in? What what space are these rows in? four-dimensional space. They only have four components. So I had four columns in 5-D, I have five rows in 4-D. But now, are those five rows independent? Let me just ask that question. Are those five independent rows, are they pointing in different directions, or could any combination give the zero vector in 4-D, looking at those five rows? What do you say, wait a minute. Five vectors, in four-dimensional space? Dependent, of course. Right. So they're dependent. There couldn't be five independent vectors in 4-D. But are there four in this particular case? And here's the great fact, no, there are three. If there are three independent columns and no more, then there are three independent rows and no more. And we'll get to see which rows are independent. And which are not. That's a question about A transpose, and we haven't got to A transpose yet. OK, are you OK with that incidence matrix? Because this is like the central matrix of our subject. We can figure out A transpose A, that's kind of fun. I do a transpose a then you'll see the core computations of this neat section. So if I do A transpose A, so I'm going to bring in a transpose and you know that I'm not just bringing it in from nowhere, that networks. The balance law is going to involve a transpose. So let's just anticipate.
What do you think a transpose a looks like? Now, how am I going to do this for you? May I write may I erase this for a moment, and try to squeeze in a transpose here? So that you can multiply it by site and see the answer, and then you'll see the pattern. That's the great thing about math. You do a few examples, and you hope that a pattern reveals itself. So let me show a transpose. So now I'm going to take that column and make it a row. I'm going to take that column and make it a row, it's going to be a little squeezed but we can do it. Take that column, . And the last column, . OK. So I just wrote a transpose here. And now could you help me with A transpose A. Which is the key matrix in the graph here. What size will it be? Everybody knows it's going to be square, it's going to be symmetric, and just tell me the size. Four by four. Right, we have a four by five times a five by four, we're expecting this to be four by four. And what's the first entry? Two. Right, take row one, dot it with column one. I get two ones and then a bunch of zeroes, so I just get a two. What's the next entry? Take row one against column two, can you do that in your head? Row one, column two, the top one is going to hit on a minus one, and I think that's all there is, right? Then this one hits a zero and those three zeroes, so. And then what about the next guy here? A minus one. And the last guy? A zero.
So that's row one of A transpose A. Can I just look at that for a moment before I fill in the rest? And then, when you fill in the rest it'll confirm the idea. Why do I have a zero there? Why did a zero appear in the 1, 4 position? If I look back at the graph, what is it about nodes one and four that told me ahead of time? You're going to get a zero in that A transpose A. Everybody see what nodes one and four are? Yeah, say it again. Not connected. No edge. Here there was an edge from node one to two. Here is an edge from node one to three. Those both produce the minus ones. And on the diagonal came the two to balance it. What does that two represent? That two represents the number of edges that do go into node one. See, that row is all about node one. So they're two edges into it, and then an edge out, and an edge out, and the edge out and the no edge. OK. So, now I know it's going to be a symmetric matrix, so I could speed up and fill those in. What's the next entry here? What's the guy on this diagonal? So that's row two against column two, so I have a one there, a one there, a one there, that makes a three. Why a three? Because there are, yeah, you got it. There are three edges into node number two. Three edges into node number two, and now I'm going to have some minus ones off the diagonal for those edges. So what are these entries going to be here? They're both minus ones. Edge two is connected to all three other nodes. So I'm going to see a minus one and a minus one there, and it's going to be symmetric. And I'm nearly there.
Of course, I'm describing a pattern that you're just seeing unfold, but I'm doing it that way so that you'll feel hey, I can write down a transpose a, or check it quite quickly, without doing this complete matrix multiplication. So what number goes there? That's to do with node three, and I see node three connected to all three other nodes, and so what do you expect there? Minus one there, and a minus one there, and what do you expect here? Two. And so now I have my matrix. The a transpose a matrix. And that's square and it's symmetric. Now I ask you, is it positive definite? Or is it only semi-definite? Right, we know that A transpose A is always positive definite in the best case. But only positive semi-definite if it's singular, if there's some vector in its null space, if a transpose a times some vector gives zero. If some combination of those columns gives me the zero column. Which is it? Have I got a singular matrix or an invertible matrix here? Singular. Why singular? Because a had some solutions to Au=0. So if Au equaled zero, then I could multiply both sides by a transpose, that same u, A transpose times zero will still be zero, it might be a different size zero, but it'll be zero. And what's the u, then? It's the all ones vector. What am I saying about the columns of A transpose A? They're dependent.. they add up because it's the vector that's guilty. Every row adds to zero. Every row adds to zero.
Let me just say for a moment, introduce two notation for the diagonal matrix. D, that's the diagonal matrix, two, three, three, two. And then I'll put in a minus sign, and this is and I'll call it W. So you can pick out what D and W are, but let me do it for sure. So D, the degree matrix, OK this is this is like fun because I'm not doing anything yet. I'm just giving names here. Two, three, three, two. The degree of a node, the degree means how many edges go from it. How many edges touch it. And W is also a great matrix, it's called the adjacency matrix. It's also beautiful. Now it'll have plus ones because I wanted minus W, so it has, these nodes are not adjacent to themselves but it's got this one and this one and this one this one and that one, and that's a zero. So there are five, the adjacency matrix tells me which nodes are connected to which other nodes. And of course the connections are going both ways. So I see five ones from five edges. And I see five more ones below the diagonal, because the edges are connecting both ways. Ones connected to three, and three is connected to one. One is not connected to four, and four is not connected to one. One is not connected to itself. By an edge. If we allowed, like, little self loops, then a one could appear on the diagram. But we don't. OK, so that's D and W.
Here are the key matrices. This is actually, I venture to say that any afternoon at MIT there's a seminar that involves these matrices. One name for this is the graph, Laplacian. From Laplace's equation and we'll see pretty soon where that name's coming from. But it's there. And should I think, I think I should, just about networks. Like where, does the networks come from? I think we've got networks all around us. Right? Electrical networks are the simplest, maybe in some ways the simplest to visualize. So that's the example, that's the language I'll use. Now, I get a network, I use the word network when there's a c_1, c_2, c_3, c_4, c_5. Those extra numbers. I've got the A, and now the network comes from the c part. That diagonal matrix, and if I'm talking electricity, these could be resistors. Status springs, they're resistors. So it's the conductance in those five resistors, are c_1, c_2, c_3, c_4, and c_5. So I'm ready for that. Ready for the C matrix, because we got the a matrix. And we've got A transpose A, but the the applications throw in a c matrix also. What are other applications, I was saying, like this one is the one, I'll use the word current, for flow in the edges, or I'll use the word flow. A network of oil, or natural gas, or water pipes would be just that, and then the electrical people study. Professor Vergasian in Course 6 studies the electric grid. The US electric grid, or the western, off in the western half of the US electric grid. So that's got a whole lot of things. Pumping stations. You see it? Actually, the world wide web, the internet, is a giant network that people would love to understand. And the phone company would love to understand those networks of phone calls. I mean, those are really, that's what, giant businesses are are dependent on understanding and maintaining networks.
OK, so I'm going to use resistors. Of course, I'm staying linear. And I'm staying steady state. So by staying linear there aren't any transistors in this net. By staying steady state, there aren't any capacitors or inductors. Those guys would be linear elements, but they would be coming in a time-dependent problem. A UTT problem. And I'm just staying now with Ku=f, I'm trying to create K. The stiffness matrix, which maybe here we might call the conductance matrix. OK, so ready for the picture now? That these come into? You know what the picture looks like, it's going to have the usual four, we'll start with these potentials u at the nodes, potentials at nodes, so those will be u_1, u_2, u_3, u_4. Voltages, if I'm really speaking, those units would be volts, and now comes the matrix A. And now I get, what do I get from A? What do I get from A? Key question. If I multiply A times u, and you know that's coming, right? If I multiply A times u, so I'll erase A transpose now, because we've got that. So there's A, and now I'll make space to multiply by u, alright? So now I want to look at Au. So A multiplies a bunch of potentials, a bunch of voltages. And let's just do this multiplication and see what it produces. This is the great thing about matrices, they produce something. OK, what's the first component of Au? Of course, Au is going to be five by five. It's going to be associated with edges. Right, u's associated with nodes, a u with edges. Just, the pattern is so nice. Alright, what's the first component? Just do that multiplication and what do you get? u_2-u_1. What do you get in the second component? Do that multiplication and you get u_3-u_1. The third one will be u_3-u_2. The fourth one would be u_4-u_2, and the fifth one will be u_4-u_3.
Just like our first difference matrices. But this one deals with, I mean, our first difference matrices were exactly like this when the graph was all in a line. The big step now is that the graph is not in a line, not even necessarily in a plane. Could be in, it's a bunch of points, and edges. Actually, the position of those points, we don't have to know are they in a plane. I think of them as nodes and edges. OK, what's the natural name for Au? I would call those potential differences, right? Voltage differences. So that's what we see here and those will be e. e_1, e_2, e_3, e_4, e_5 will be potential or voltage differences. Voltage drops, you might say. Potential differences, voltage drops. Oh well, now. When I say voltage drops, that's because, as we noted before, the current goes from a higher to a lower potential. It goes in the direction of the drop. And I think that what we need now is minus Au, for e. So I think we need a minus sign and it's quite common to have the minus sign. We saw it already with least squared.
And let me say also, so this is e. I'll abbreviate those always five e's I just wrote down, five of them. So you would remember there are five. We're talking about the currents. We're talking about, this is the e in e=IR. The voltage drop. That makes some current go. Now, also, just as with least squares, so it was great that we saw it before, there could be a source term here. So I'm completing the picture here, allowing the source term. And we'll come back to what does that mean, physically. But at that point could enter b, and b is really standing for batteries. I work hard to make the language match the initials. These letters. OK, now what? That step just involved A, nothing physical. Now comes the step that involves A, so w will be Ce. And these will be the currents on the edges. And that's the law, then, with a matrix C, of course C is our old friend c_1 to c_5. And tell me first, the name. Whose law is this? That the current is proportional to the voltage drop? Ohm. So this is Ohm's law. Instead of Hooke's law, it's Ohm's law. And I've written it with conductances, not resistances. So resistances are 1/R, the usual R in e=IR, would be, I'm more looking at it as i current equals Ce, instead of e=IR. So I'm flipping the the the resistance, or the impedance to give the conductance.
OK, and now finally can you tell me what the last step is going to be? If life is good, well you might wonder whether life is good, reading the papers, but it's still good here. OK, what matrix shows up there? Everybody knows it. A transpose. So the final equation, the balance equation, will be, let me write it so I don't catch it up here. Will be A transpose w equals whatever. Will be the balance equation. The current balance, it's the balance of currents, balance of charge, whatever you like to say. At each node, it's the balance at the nodes. Because when we're up on this line, we're in the node picture. We have four equations here, right? We're talking about at each node. Here we're talking about on each edge. There is so critical. These two variables. Which we're seeing physically as node variables and edge variables. That pair of variables just shows up everywhere. In displacements and stresses, it's fundamental in elasticity. And oh, there are just so many in optimization, it's everywhere. And a big part of this course is to see it everywhere. OK, why don't I, just so you see the main picture. We're going to have the A transpose C A matrix that I'm going to maybe call K again. And now of course there could be current sources. Just the way there could be forces that we had to balance. There could be, not always but there could be, current sources from outside. External current sources. So these are external voltage sources. These are external current sources. So in a way, we now have combined our first two examples, our springs and masses only had forces external. Our least squares problem had an external b. Measurement. This picture is the whole deal. It's gotta b and f, and actually I could put in even a little more.
Sources like, well, we already kind of caught on to the fact that we'd better ground the node or A transpose C A as it stands, A transpose C A as it stands will be singular. You know, it's the matrix, there's A transpose A and the C in the middle isn't going to help any. That's singular. If we wanted to be able to compute voltages, we've got to set one of them. It's like setting one temperature, it's like deciding where is absolute zero. Let's put absolute zero down here. u_4=0. Grounded the node. OK, so we've fixed a potential. So here's a boundary condition coming in u_4=0. That's another source term, another thing coming, you could say sort of from outside the A transpose C A. We could fix another voltage at, I mean, I'm thinking now about what's the picture. What's the whole problem? So the problem could have batteries, in the edges. It could have current sources into the nodes. It could fix u_1 at some voltage like ten. Our problem could fix - we must fix one of them. Otherwise our matrix isn't as singular. But once we've set up the matrix, and when we fix u_4=0 by the way, what happens to our matrix?
Let me take u_4=0, so this is a key step here. When I set u_4=0, I now know u_4. It's not an unknown any more. So I've removed u_4 from the problem. And then it'll be also removed from A transpose A. So this, is you could say, like a reduced A, or a grounded matrix A. It's now five by three. And A transpose A, what shape will the a transpose a matrix be? It'll be three by three, right? I now have five by three, three by five. Multiplying five by three gives me three by three. This column is gone, and that row is gone. Because the row came from A transpose and the column came from A, and we've just thrown them away. By grounding that node. Now give me the key fact about that A transpose A matrix? What what do you see there? Now, you see a reduced, a grounded A transpose A. What kind of a matrix have I got? Positive def. Good. Positive definite. It's now not singular any more, its determinant is some positive number. And everything is positive, its eigenvalues are all positive, everything's good about that matrix. OK, and I guess what I was starting to say here, if I wanted to fix, this would be a natural problem. Fix the top voltage at one, say. Fix u_1=1 and see how much current flows. That would be a natural question. What's the system resistance between the top node and the bottom, if I'm given, or the system conductance. If I'm given a c_1, a c_2, a c_3, a c_4, and a c_5, I could say I could fix that voltage at one, I could fix this at zero. Maybe one of the homework problems asks you for something like this. And then you find all the currents. And the voltages, you solve the problem. And you know what the currents are. You know the total current that leaves node one, enters node four when the voltages drop by one, between, right?
So current can flow down here, cross over here, down here whatever. Somehow all these five numbers are going to play a part in that system resistance. So that would be an interesting number to know. Out of those five numbers, somehow five c's, there's a system resistance between that node and that node. And we can find it by setting this to be one, this to be zero, having the reduced matrix - oh, well what will happen? How many unknowns well I have? Just do this mental experiment. Suppose I introduce u_1 to be one, for example. This is just one type of possible problem. If I take u_1 to be one, what happens to my matrix A? It loses its first column, too. u_1 is not unknown any more. u_1 will not be unknown. And that value one is somehow going to move to the right-hand side, right? People have asked me after class, well what happens if a boundary condition isn't zero? Suppose we have this fixed springs and we pull this spring down to make its displacement 12. Well, somehow that 12 is going to show up on the right side of the equation. It's a source, it's an external term. OK, so if we had u_1 equals whatever, this u_1 would disappear. I would only have a two by two problem. Because I would only have two, I now have only two unknown u's, right? So that's where sources can come. And can I just complete the picture of the source stuff? We could fix, we could. Look, here's what I'm going to say. External stuff. Sources can come into here. They can come into here. They can come into here, so of course everybody says why shouldn't they come in here? And the answer is we could send them here. So we could fix, we could fix some w's.
Of course, you understand we can't do everything. I mean, there's a limit to how much we can put on the system. We want to have some unknowns left. Some matrix still, but anyway. I like this picture now, it's more complete. That you now see the node variables and node equations, the edge variables, e and w. The currents. These guys are the big ones. w and u are what I think of as the crucial unknowns. e is sort of on the way. f is the source. But now we have the possibility of sources at all four positions. OK, let's see. If I wrote out, If I looked at A transpose C A, would you like to tell me, yeah. Have we got? No, we don't. I was going to say, what's a typical row of A transpose C A, can I just say it in words? It'll be too quick to really catch. So without the C, this is what we had. So what do you think that two becomes if there's an A transpose C A, if there's a C in the middle. Have you got the pattern yet? That two was there because of two edges. Edges one and two, it happened to be. So instead of the two, I'm going to see c_1+c_2. Right. When those were ones, I got the two. So this will be c_1+c_2, this'll be a minus c_1, and that'll be a minus c_1, when we do it out. And you could do it out for yourself. Just tell me what would show up there. In A transpose C A, so I'm talking now about A transpose C A. So instead of one plus one plus one, what do I have? What am I going to have, and you really want to multiply it out, because it's so nice to see it happen. What do I have? I'm looking at node two, I'm seeing three edges out of it. And instead of one, one, one, I'll have c_1+c_3+c_4. c_1+c_3+c_4 will be sitting here. And minus c_1 will be here, and minus c_3 will be here, and minus c_4 will be there. The pattern's just nice. So if you can read this part of the section, I'll have more to say Friday about the a transpose w, the balance. That critical point we didn't do yet. But the main thing, you've got it | 677.169 | 1 |
The Developmental Mathematics classes at Galveston College are designed to refresh and strengthen students' basic math and algebra skills. They help students brush up on – and sometimes to learn for the first time – the math that they will need to be successful in their first college level math class. A placement test score plus the date and grade from the student's last math class help determine the most appropriate course level for that student.
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In MATH0300, Basic Mathematics, students review techniques of simplifying, evaluating, adding, subtracting, multiplying, and dividing integers, fractions, and decimals and then practice using them in geometry formulas, ratios, rates, proportions, and percents. Additionally, they solve for the unknowns in equations.
In MATH0303, Introductory Algebra, students solve more complicated equations, as well as inequalities. They graph, find slopes of, and write linear equations. They review functions, polynomial operations, and exponent rules. They translate English words describing real world situations into the mathematical symbols required to solve the problems.
MATH0304, Intermediate Algebra, picks up right where MATH0303 finished. Students even can use the same access code for their MyMathLabs PLUS assignments. They cover factoring, rational expressions, rational exponents, set theory, and solving systems of equations with 2 or 3 variables, linear and quadratic equations, linear inequalities, and equations with absolute values.
CONTEXTUALIZED LEARNING: FUSING MATH INTO WELDING
In these hands-on contextualized courses, which were designed by Carolyn Harnsberry, students in the Welding Technology Program apply the mathematical knowledge and concepts learned in the math class to create and complete projects in the welding lab. The hands-on contextualized math classes for students enrolled in the Welding Technology Program are taught on site at Galveston College's Applied Technologies Center. Be sure to check the most current course schedule for current learning options. Check out the videos below for a look at how this program works.
After successfully completing MATH0304, students are ready for their first college level math course | 677.169 | 1 |
Introductory Algebra with MathZoneIntroductory Algebra prepares students for Intermediate Algebra by covering fundamental algebra concepts and key concepts needed for further study. Students of all backgrounds will be delighted to find a refreshing book that appeals to every learning style and reaches out to diverse demographics. Through down-to-earth explanations, patient skill-building, and exceptionally interesting and realistic applications, this worktext will empower students to learn and master algebra in the real world.
R PREALGEBRA REVIEW
R1 Fractions
R2 Operations with Fractions
R3 Decimals and Percents
1 REAL NUMBERS AND THEIR PROPERTIES
1.1 Introduction to Algebra
1.2 The Real Numbers
1.3 Adding and Subtracting Real Numbers
1.4 Multiplying and Dividing Real Numbers
1.5 Order of Operations
1.6 Properties of the Real Numbers
1.7 Simplifying Expressions
2 EQUATIONS, PROBLEM SOLVING, AND INEQUALITIES
2.1 The Addition and Subtraction Properties of Equality
2.2 The Multiplication and Division Properties of Equality: Applications with Percents | 677.169 | 1 |
1 definition
by
Matthew Garlick
Precalculus is a mathematics course typically taught in secondary schools and in colleges. Precalculus courses assume a prerequisite knowledge of concepts in intermediate algebra, and is generally designed to prepare students for the study of Calculus.
Topics that are typically covered in precalculus are exponential, logarithmic, polynomial, and trigonometric functions as well as complex numbers, polar coordinates, parametric equations, vectors. Topics in trigonometry include trigonometric graphs and identities as well as applications.
After completing Algebra 2 in 9th grade, my teacher determined that I was too smart for precalculus and therefore I enrolled in Advanced Algebra and Trigonometry. | 677.169 | 1 |
The history of the calculus is a fascinating story, inspired by the search for solutions to interesting problems. We do our students a disservice when we fail to share with them some of this exciting history. During 2005-07, with support from the National Science Foundation, I developed modules for teaching calculus concepts in a way that integrates the historical evolution of these concepts. This article contains three examples, which include several interactive applets produced using the software GeoGebra, a free, open-source software package. In addition, many of the exercises provided require GeoGebra and/or a computer algebra system (CAS) such as Maple, Mathematica, or a Texas Instruments (TI) Voyage 200 calculator. | 677.169 | 1 |
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Chapter 2: Real Numbers
Introduction
Given a number like 5, 0.75, or \begin{align*}\sqrt{2}\end{align*}, how would you classify it beyond just identifying it as a real number? In this chapter you'll learn three subsets of real numbers – rational numbers, irrational numbers, and integers – that you'll find in daily life. The number of students in your class – that's an integer. The interest rate on your car loan – that's a rational number. The ratio of a circle's circumference to its diameter – that's an irrational number. This chapter differentiates these various types of real numbers and explains important properties and rules that apply to them. You will learn how to perform operations on, and solve problems involving, rational numbers. The Concepts in this chapter will equip you with problem-solving strategies to solve problems involving rational numbers.
Chapter Summary
Summary
This chapter explains the difference between integers, rational numbers, and irrational numbers. It then focuses on adding, subtracting, multiplying, and dividing rational numbers. There are Concepts exploring number opposites, absolute values, the distributive property, and square roots. There are many real-world problems to encourage practice of guess and check and working backward strategies to solve them.
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Compare and graph integers, find absolute values, order and perform basic operations with rational numbers, and apply the commutative, associative, and distributive properties. | 677.169 | 1 |
Functions and Change A Modeling Approach to College AlgebraFunctions and Changeprovides an alternative to a traditional college algebra course for students who either will not take another math course or may go on to a business calculus course. The authors wrote this text for the many college algebra students who are poorly served by books that focus on preparing them for a course they will never take--traditional calculus. An informal approach offers clear explanations that leave out unnecessary technicalities so that students who are not comfortable with algebra will easily understand the concepts. Graphing calculator technology is integrated throughout the text as a descriptive and problem solving tool, and to help students understand abstract concepts. Functions are introduced by the "Rule of Four"--graphically, numerically, symbolically, and verbally. Diverse Modeling/Real Life Applications and Data reveal mathematics as an integral part of nature, science, and society. This real-world context allows students to check their answers against their own intuition and common sense while also making the topic more relevant and interesting to students. New!Chapter Review Exercises help students build their skills to remember key concepts from the chapter. Skill Building Exercises precede each exercise set in the text; in this edition these have expanded by approximately 5% per section. Key Idea Boxes highlight key ideas for every section. Workplace/life skills are covered by giving students plenty of opportunities to use Excel spreadsheets in the text and in the technology guide. This meets the revised AMATYC's 2006 Beyond Crossroads guidelines, which recommends teaching students workplace/life skills. Finance applications are now highlighted in Index of Applications AnotherLookwith enrichment exercises end each section in the first five chapters and is based on market feedback. These optional exercises provide a slight increase in algebraic presentation and symbol manipulation, along with general enrichment or alternative treatments of the material. While written in the same spirit as the rest of the text, theAnother Lookfeature contains more traditional coverage. Exercises include more practice setting up formulas, and a wider variety of non-science applications.
Note: Each chapter concludes with a Summary. Prologue: Calculator Arithmetic | 677.169 | 1 |
Details about Plane Trigonometry:
This revision of a best-selling plane trigonometry text for freshmen and sophomores maintains the trademarks of clear, concise exposition coupled with graded problems. The result is a fresh and modern version of a classic text. Major changes include an emphasis on the use of calculators and calculator-related exercises, the use of radian measures appearing after Chapter 5 to help prepare students for analytic geometry or calculus, the expansion of Chapter 9 to include more applications as well as the addition of polar, exponential functions, and a greater emphasis on graphing.
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Rent Plane Trigonometry 7th edition today, or search our site for other textbooks by E. Richard Heineman. Every textbook comes with a 21-day "Any Reason" guarantee. Published by McGraw-Hill Companies. | 677.169 | 1 |
ALEX Lesson Plans
Title: Discover the Roots of a Polynomial Function
Description:
In (9 - 12) Title: Discover the Roots of a Polynomial Function Description: In
Title: Investigating Parabolas in Standard Form
Description:
StudentsStandard(s): [MA2015] ALT (9-12) 34: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx),[F-BF3]
Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: Investigating Parabolas in Standard Form Description: Students
Title: Who am I? Find A Polynomial From Its Roots
Description:
StudentsStandard(s): [MA2015] AM1 (9-12) 11: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Understand the importance of using complex numbers in graphing functions on the Cartesian or complex plane. [N-CN9] (Alabama)
Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: Who am I? Find A Polynomial From Its Roots Description: Students
Thinkfinity Lesson Plans
Title: Building Connections
Description:
In Title: Building Connections Description: In Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 | 677.169 | 1 |
Multiple choice is fine for many areas of testing, but it isn't ideal for math. Find out why the University of Birmingham chose Maple T.A., and how they are now testing a student's true understanding of key concepts.
Disclaimer: This blog post has been contributed by Dr. Nicola Wilkin, Head of Teaching Innovation (Science), College of Engineering and Physical Sciences and Jonathan Watkins from the University of Birmingham Maple T.A. user group*. If you have arrived at this post you are likely to have a STEM ...
This module provides an introduction to the dynamics of rotation. The main concepts covered are the Moment of Inertia of a rotating body and Torque. Basic derivations are included to help understand how these physical quantities are related to Newton's laws of motion. Numerical examples and steps to create MapleSim simulations are included to enhance the learning experience.
This module covers the kinematics of pure rotation. The angular variables for rotational motion are described and the relation between these variables and the variables of linear motion are discussed. Numerical examples, plots, and steps to create a MapleSim simulation are included to enhance the learning experience
I began my experience with Maplesoft with the first academic year at the university, I loved it since the first application.
We have used this program in general in mathematical applications, and graph theory; and I obtained the best results in all tests related to the Maplesoft
Maplesoft language-level professional and very understandable so that each novice to use the program with some training and guidance.
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Mathematical language
Free Course
In our everyday lives we use we use language to develop ideas and to communicate them to other people. In this free course, Mathematical language, we examine ways in which language is adapted to express mathematical ideas.
By the end of this free course you should be able to:
Section 1: Sets
use set notation;
determine whether two given sets are equal and whether one given set is a subset of another;
find the union, intersection and difference of two given sets.
Section 2: Functions
determine the image of a given function;
determine whether a given function is one-one and/or onto;
find the inverse of a given one-one function;
find the composite of two given functions.
Section 3: The language of proof
understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
produce simple proofs of various types, including direct proof, proof by induction, proof by contradiction and proof by contrapositionMathematical language
Introduction
When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation, and this can lead to misunderstandings and confusion.
In mathematics we try to avoid these difficulties by expressing our thoughts in terms of well-defined mathematical objects. These objects can be anything from numbers and geometrical shapes to more complicated objects, usually constructed from numbers, points and functions. We discuss these objects using precise language which should be interpreted in the same way by everyone. In this unit we introduce the basic mathematical language needed to express a range of mathematical concepts.
Please note that this unit is presented through a series of downloadable PDF files.
This unit is an adapted extract from the Open University course Pure mathematics
(M208) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]
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Originally published: Tuesday, 28th June 2011
Last updated on: Monday, 25th July 2011 | 677.169 | 1 |
Please, inform me of books, websites, ways that helped you to get to the stage you are at now. Moreover, is there any good sources of information which help you to understand maths better or simply explain advanced maths in a basic way. | 677.169 | 1 |
11+ Maths Two by The Tutors
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This app is a set of 20 question papers, each paper consists of 50 questions in multiple-choice format; the questions cover all of the question types contained in the actual 11+ and independent school common entrance examinations and are in the format of the tests that the children will take.
With 1000 individual questions, this is the most comprehensive 11+ Mathematics App available for 11+ Grammar School selection tests and Independent school common entrance examinations. The app also contains 480 bonus question to help practice times tables instant recall.
The Tutors 11+ Mathematics Apps Volumes 1 & 2 contain different questions, if you purchase both apps you will have 1900 test questions and 980 bonus questions.
This app is ideal for:
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The Tutors 11+ Mathematics Volume 2 App gives you all of the above. We believe that you should get what you need, so we provide you with 1000 unique questions in 20 test papers that mirror the real tests in a single App, plus a bonus section with 480 times tables recall questions..
The Tutors have over 15 years experience preparing children for 11+ tests and have created one of the most comprehensive ranges of resources for 11+ Mathematics. We also publish 11+ verbal reasoning, English and vocabulary building resources; over 8 million questions have been purchased by schools, tuition centers and parents. The Tutors resources are recognized as being of the highest quality; when you use The Tutors resources you have the peace of mind that you are working with the best material available from a single author | 677.169 | 1 |
Find a SouthworthAlgebra skills include understanding how signed numbers work in equations, what graphs look like,and how to factor polynomials. Each of these key concepts are foundational to variations seen in Algebra I; they also provide the background for advancement into higher levels of Algebra and beyond. I ... | 677.169 | 1 |
Gilberts PrealgebraJoyce R.Andrew C.
...All You must also be able to analyz... | 677.169 | 1 |
Hi. This is the first lecture in MIT's course 18.06, linear algebra, and I'm Gilbert Strang. The text for the course is this book, Introduction to Linear Algebra.
And the course web page, which has got a lot of exercises from the past, MatLab codes, the syllabus for the course, is web.mit.edu/18.06.
And this is the first lecture, lecture one.
So, and later we'll give the web address for viewing these, videotapes. Okay, so what's in the first lecture? This is my plan.
The fundamental problem of linear algebra, which is to solve a system of linear equations.
So let's start with a case when we have some number of equations, say n equations and n unknowns.
So an equal number of equations and unknowns.
That's the normal, nice case.
And what I want to do is -- with examples, of course -- to describe, first, what I call the Row picture. That's the picture of one equation at a time. It's the picture you've seen before in two by two equations where lines meet.
So in a minute, you'll see lines meeting.
The second picture, I'll put a star beside that, because that's such an important one.
And maybe new to you is the picture -- a column at a time.
And those are the rows and columns of a matrix.
So the third -- the algebra way to look at the problem is the matrix form and using a matrix that I'll call A.
Okay, so can I do an example? The whole semester will be examples and then see what's going on with the example.
So, take an example. Two equations, two unknowns. So let me take 2x -y =0, let's say. And -x +2y=3.
Okay. let me -- I can even say right away -- what's the matrix, that is, what's the coefficient matrix? The matrix that involves these numbers -- a matrix is just a rectangular array of numbers. Here it's two rows and two columns, so 2 and -- minus 1 in the first row minus 1 and 2 in the second row, that's the matrix.
And the right-hand -- the, unknown -- well, we've got two unknowns. So we've got a vector, with two components, x and x, and we've got two right-hand sides that go into a vector 0 3.
I couldn't resist writing the matrix form, right -- even before the pictures. So I always will think of this as the matrix A, the matrix of coefficients, then there's a vector of unknowns.
Here we've only got two unknowns.
Later we'll have any number of unknowns.
And that vector of unknowns, well I'll often -- I'll make that x -- extra bold. A and the right-hand side is also a vector that I'll always call b.
So linear equations are A x equal b and the idea now is to solve this particular example and then step back to see the bigger picture. Okay, what's the picture for this example, the Row picture? Okay, so here comes the Row picture.
So that means I take one row at a time and I'm drawing here the xy plane and I'm going to plot all the points that satisfy that first equation. So I'm looking at all the points that satisfy 2x-y =0. It's often good to start with which point on the horizontal line -- on this horizontal line, y is zero.
The x axis has y as zero and that -- in this case, actually, then x is zero. So the point, the origin -- the point with coordinates (0,0) is on the line. It solves that equation.
Okay, tell me in -- well, I guess I have to tell you another point that solves this same equation.
Let me suppose x is one, so I'll take x to be one.
Then y should be two, right? So there's the point one two that also solves this equation.
And I could put in more points. But, but let me put in all the points at once, because they all lie on a straight line. This is a linear equation and that word linear got the letters for line in it.
That's the equation -- this is the line that ...
of solutions to 2x-y=0 my first row, first equation.
So typically, maybe, x equal a half, y equal one will work. And sure enough it does.
Okay, that's the first one. Now the second one is not going to go through the origin. It's always important.
Do we go through the origin or not? In this case, yes, because there's a zero over there. In this case we don't go through the origin, because if x and y are zero, we don't get three. So, let me again say suppose y is zero, what x do we actually get? If y is zero, then I get x is minus three.
So if y is zero, I go along minus three.
So there's one point on this second line.
Now let me say, well, suppose x is minus one -- just to take another x. If x is minus one, then this is a one and I think y should be a one, because if x is minus one, then I think y should be a one and we'll get that point. Is that right? If x is minus one, that's a one.
If y is a one, that's a two and the one and the two make three and that point's on the equation.
Okay. Now, I should just draw the line, right, connecting those two points at -- that will give me the whole line. And if I've done this reasonably well, I think it's going to happen to go through -- well, not happen -- it was arranged to go through that point. So I think that the second line is this one, and this is the all-important point that lies on both lines. Shall we just check that that point which is the point x equal one and y was two, right? That's the point there and that, I believe, solves both equations.
Let's just check this. If x is one, I have a minus one plus four equals three, okay. Apologies for drawing this picture that you've seen before. But this -- seeing the row picture -- first of all, for n equal 2, two equations and two unknowns, it's the right place to start. Okay.
So we've got the solution. The point that lies on both lines. Now can I come to the column picture? Pay attention, this is the key point. So the column picture.
I'm now going to look at the columns of the matrix.
I'm going to look at this part and this part.
I'm going to say that the x part is really x times -- you see, I'm putting the two -- I'm kind of getting the two equations at once -- that part and then I have a y and in the first equation it's multiplying a minus one and in the second equation a two, and on the right-hand side, zero and three. You see, the columns of the matrix, the columns of A are here and the right-hand side b is there. And now what is the equation asking for? It's asking us to find -- somehow to combine that vector and this one in the right amounts to get that one. It's asking us to find the right linear combination -- this is called a linear combination.
And it's the most fundamental operation in the whole course.
It's a linear combination of the columns.
That's what we're seeing on the left side.
Again, I don't want to write down a big definition.
You can see what it is. There's column one, there's column two. I multiply by some numbers and I add. That's a combination -- a linear combination and I want to make those numbers the right numbers to produce zero three. Okay.
Now I want to draw a picture that, represents what this -- this is algebra. What's the geometry, what's the picture that goes with it? Okay. So again, these vectors have two components, so I better draw a picture like that. So can I put down these columns? I'll draw these columns as they are, and then I'll do a combination of them.
So the first column is over two and down one, right? So there's the first column.
The first column. Column one.
It's the vector two minus one. The second column is -- minus one is the first component and up two.
It's here. There's column two.
So this, again, you see what its components are. Its components are minus one, two. Good.
That's this guy. Now I have to take a combination. What combination shall I take? Why not the right combination, what the hell? Okay. So the combination I'm going to take is the right one to produce zero three and then we'll see it happen in the picture. So the right combination is to take x as one of those and two of these.
It's because we already know that that's the right x and y, so why not take the correct combination here and see it happen? Okay, so how do I picture this linear combination? So I start with this vector that's already here -- so that's one of column one, that's one times column one, right there.
And now I want to add on -- so I'm going to hook the next vector onto the front of the arrow will start the next vector and it will go this way. So let's see, can I do it right? If I added on one of these vectors, it would go left one and up two, so we'd go left one and up two, so it would probably get us to there.
Maybe I'll do dotted line for that.
Okay? That's one of column two tucked onto the end, but I wanted to tuck on two of column two. So that -- the second one -- we'll go up left one and up two also.
It'll probably end there. And there's another one.
So what I've put in here is two of column two.
Added on. And where did I end up? What are the coordinates of this result? What do I get when I take one of this plus two of that? I do get that, of course.
There it is, x is zero, y is three, that's b. That's the answer we wanted.
And how do I do it? You see I do it just like the first component. I have a two and a minus two that produces a zero, and in the second component I have a minus one and a four, they combine to give the three.
But look at this picture. So here's our key picture.
I combine this column and this column to get this guy.
That was the b. That's the zero three.
Okay. So that idea of linear combination is crucial, and also -- do we want to think about this question? Sure, why not.
What are all the combinations? If I took -- can I go back to xs and ys? This is a question for really -- it's going to come up over and over, but why don't we see it once now? If I took all the xs and all the ys, all the combinations, what would be all the results? And, actually, the result would be that I could get any right-hand side at all.
The combinations of this and this would fill the whole plane.
You can tuck that away. We'll, explore it further.
But this idea of what linear combination gives b and what do all the linear combinations give, what are all the possible, achievable right-hand sides be -- that's going to be basic. Okay.
Can I move to three equations and three unknowns? Because it's easy to picture the two by two case.
Let me do a three by three example.
Okay, I'll sort of start it the same way, say maybe 2x-y and maybe I'll take no zs as a zero and maybe a -x+2y and maybe a -z as a -- oh, let me make that a minus one and, just for variety let me take, -3z, -3ys, I should keep the ys in that line, and 4zs is, say, 4. Okay.
That's three equations. I'm in three dimensions, x, y, z. And, I don't have a solution yet. So I want to understand the equations and then solve them. Okay.
So how do I you understand them? The row picture one way. The column picture is another very important way. Just let's remember the matrix form, here, because that's easy. The matrix form -- what's our matrix A? Our matrix A is this right-hand side, the two and the minus one and the zero from the first row, the minus one and the two and the minus one from the second row, the zero, the minus three and the four from the third row. So it's a three by three matrix. Three equations, three unknowns. And what's our right-hand side? Of course, it's the vector, zero minus one, four. Okay.
So that's the way, well, that's the short-hand to write out the three equations. But it's the picture that I'm looking for today. Okay, so the row picture.
All right, so I'm in three dimensions, x, y and z. And I want to take those equations one at a time and ask -- and make a picture of all the points that satisfy -- let's take equation number two.
If I make a picture of all the points that satisfy -- all the x, y, z points that solve this equation -- well, first of all, the origin is not one of them.
x, y, z -- it being 0, 0, 0 would not solve that equation. So what are some points that do solve the equation? Let's see, maybe if x is one, y and z could be zero. That would work, right? So there's one point.
I'm looking at this second equation, here, just, to start with. Let's see.
Also, I guess, if z could be one, x and y could be zero, so that would just go straight up that axis. And, probably I'd want a third point here. Let me take x to be zero, z to be zero, then y would be minus a half, right? So there's a third point, somewhere -- oh my -- okay. Let's see.
I want to put in all the points that satisfy that equation.
Do you know what that bunch of points will be? It's a plane. If we have a linear equation, then, fortunately, the graph of the thing, the plot of all the points that solve it are a plane.
These three points determine a plane, but your lecturer is not Rembrandt and the art is going to be the weak point here.
So I'm just going to draw a plane, right? There's a plane somewhere. That's my plane.
That plane is all the points that solves this guy.
Then, what about this one? Two x minus y plus zero z.
So z actually can be anything. Again, it's going to be another plane. Each row in a three by three problem gives us a plane in three dimensions.
So this one is going to be some other plane -- maybe I'll try to draw it like this. And those two planes meet in a line. So if I have two equations, just the first two equations in three dimensions, those give me a line. The line where those two planes meet. And now, the third guy is a third plane. And it goes somewhere.
Okay, those three things meet in a point.
Now I don't know where that point is, frankly.
But -- linear algebra will find it.
The main point is that the three planes, because they're not parallel, they're not special.
They do meet in one point and that's the solution.
But, maybe you can see that this row picture is getting a little hard to see. The row picture was a cinch when we looked at two lines meeting.
When we look at three planes meeting, it's not so clear and in four dimensions probably a little less clear.
So, can I quit on the row picture? Or quit on the row picture before I've successfully found the point where the three planes meet? All I really want to see is that the row picture consists of three planes and, if everything works right, three planes meet in one point and that's a solution.
Now, you can tell I prefer the column picture.
Okay, so let me take the column picture.
That's x times -- so there were two xs in the first equation minus one x is, and no xs in the third.
It's just the first column of that.
And how many ys are there? There's minus one in the first equations, two in the second and maybe minus three in the third.
Just the second column of my matrix.
And z times no zs minus one zs and four zs.
And it's those three columns, right, that I have to combine to produce the right-hand side, which is zero minus one four.
Okay. So what have we got on this left-hand side? A linear combination.
It's a linear combination now of three vectors, and they happen to be -- each one is a three dimensional vector, so we want to know what combination of those three vectors produces that one.
Shall I try to draw the column picture, then? So, since these vectors have three components -- so it's some multiple -- let me draw in the first column as before -- x is two and y is minus one. Maybe there is the first column. y -- the second column has maybe a minus one and a two and the y is a minus three, somewhere, there possibly, column two.
And the third column has -- no zero minus one four, so how shall I draw that? So this was the first component. The second component was a minus one. Maybe up here.
That's column three, that's the column zero minus one and four. This guy.
So, again, what's my problem? What this equation is asking me to do is to combine these three vectors with a right combination to produce this one. Well, you can see what the right combination is, because in this special problem, specially chosen by the lecturer, that right-hand side that I'm trying to get is actually one of these columns. So I know how to get that one.
So what's the solution? What combination will work? I just want one of these and none of these.
So x should be zero, y should be zero and z should be one. That's the combination.
One of those is obviously the right one.
Column three is actually the same as b in this particular problem. I made it work that way just so we would get an answer, (0,0,1), so somehow that's the point where those three planes met and I couldn't see it before. Of course, I won't always be able to see it from the column picture, either.
It's the next lecture, actually, which is about elimination, which is the systematic way that everybody -- every bit of software, too -- production, large-scale software would solve the equations.
So the lecture that's coming up.
If I was to add that to the syllabus, will be about how to find x, y, z in all cases. Can I just think again, though, about the big picture? By the big picture I mean let's keep this same matrix on the left but imagine that we have a different right-hand side. Oh, let me take a different right-hand side. So I'll change that right-hand side to something that actually is also pretty special.
Let me change it to -- if I add those first two columns, that would give me a one and a one and a minus three.
There's a very special right-hand side.
I just cooked it up by adding this one to this one.
Now, what's the solution with this new right-hand side? The solution with this new right-hand side is clear.
took one of these and none of those.
So actually, it just changed around to this when I took this new right-hand side.
Okay. So in the row picture, I have three different planes, three new planes meeting now at this point. In the column picture, I have the same three columns, but now I'm combining them to produce this guy, and it turned out that column one plus column two which would be somewhere -- there is the right column -- one of this and one of this would give me the new b. Okay.
So we squeezed in an extra example.
But now think about all bs, all right-hand sides.
Can I solve these equations for every right-hand side? Can I ask that question? So that's the algebra question.
Can I solve A x=b for every b? Let me write that down.
Can I solve A x =b for every right-hand side b? I mean, is there a solution? And then, if there is, elimination will give me a way to find it.
I really wanted to ask, is there a solution for every right-hand side? So now, can I put that in different words -- in this linear combination words? So in linear combination words, do the linear combinations of the columns fill three dimensional space? Every b means all the bs in three dimensional space.
Do you see that I'm just asking the same question in different words? Solving A x -- A x -- that's very important. A times x -- when I multiply a matrix by a vector, I get a combination of the columns. I'll write that down in a moment. But in my column picture, that's really what I'm doing. I'm taking linear combinations of these three columns and I'm trying to find b.
And, actually, the answer for this matrix will be yes. For this matrix A -- for these columns, the answer is yes. This matrix -- that I chose for an example is a good matrix. A non-singular matrix.
An invertible matrix. Those will be the matrices that we like best. There could be other -- and we will see other matrices where the answer becomes, no -- oh, actually, you can see when it would become no. What could go wrong? How could it go wrong that out of these -- out of three columns and all their combinations -- when would I not be able to produce some b off here? When could it go wrong? Do you see that the combinations -- let me say when it goes wrong. If these three columns all lie in the same plane, then their combinations will lie in that same plane. So then we're in trouble.
If the three columns of my matrix -- if those three vectors happen to lie in the same plane -- for example, if column three is just the sum of column one and column two, I would be in trouble. That would be a matrix A where the answer would be no, because the combinations -- if column three is in the same plane as column one and two, I don't get anything new from that.
All the combinations are in the plane and only right-hand sides b that I could get would be the ones in that plane.
So I could solve it for some right-hand sides, when b is in the plane, but most right-hand sides would be out of the plane and unreachable.
So that would be a singular case.
The matrix would be not invertible.
There would not be a solution for every b.
The answer would become no for that.
Okay. I don't know -- shall we take just a little shot at thinking about nine dimensions? Imagine that we have vectors with nine components.
Well, it's going to be hard to visualize those.
I don't pretend to do it. But somehow, pretend you do. Pretend we have -- if this was nine equations and nine unknowns, then we would have nine columns, and each one would be a vector in nine-dimensional space and we would be looking at their linear combinations. So we would be having the linear combinations of nine vectors in nine-dimensional space, and we would be trying to find the combination that hit the correct right-hand side b. And we might also ask the question can we always do it? Can we get every right-hand side b? And certainly it will depend on those nine columns. Sometimes the answer will be yes -- if I picked a random matrix, it would be yes, actually. If I used MatLab and just used the random command, picked out a nine by nine matrix, I guarantee it would be good.
It would be non-singular, it would be invertible, all beautiful. But if I choose those columns so that they're not independent, so that the ninth column is the same as the eighth column, then it contributes nothing new and there would be right-hand sides b that I couldn't get.
Can you sort of think about nine vectors in nine-dimensional space an take their combinations? That's really the central thought -- that you get kind of used to in linear algebra. Even though you can't really visualize it, you sort of think you can after a while. Those nine columns and all their combinations may very well fill out the whole nine-dimensional space. But if the ninth column happened to be the same as the eighth column and gave nothing new, then probably what it would fill out would be -- I hesitate even to say this -- it would be a sort of a plane -- an eight dimensional plane inside nine-dimensional space.
And it's those eight dimensional planes inside nine-dimensional space that we have to work with eventually.
For now, let's stay with a nice case where the matrices work, we can get every right-hand side b and here we see how to do it with columns. Okay.
There was one step which I realized I was saying in words that I now want to write in letters.
Because I'm coming back to the matrix form of the equation, so let me write it here. The matrix form of my equation, of my system is some matrix A times some vector x equals some right-hand side b. Okay.
So this is a multiplication. A times x.
Matrix times vector, and I just want to say how do you multiply a matrix by a vector? Okay, so I'm just going to create a matrix -- let me take two five one three -- and let me take a vector x to be, say, 1and 2. How do I multiply a matrix by a vector? But just think a little bit about matrix notation and how to do that in multiplication.
So let me say how I multiply a matrix by a vector.
Actually, there are two ways to do it.
Let me tell you my favorite way.
It's columns again. It's a column at a time.
For me, this matrix multiplication says I take one of that column and two of that column and add.
So this is the way I would think of it is one of the first column and two of the second column and let's just see what we get. So in the first component I'm getting a two and a ten. I'm getting a twelve there.
In the second component I'm getting a one and a six, I'm getting a seven. So that matrix times that vector is twelve seven. Now, you could do that another way. You could do it a row at a time. And you would get this twelve -- and actually I pretty much did it here -- this way.
Two -- I could take that row times my vector.
This is the idea of a dot product.
This vector times this vector, two times one plus five times two is the twelve. This vector times this vector -- one times one plus three times two is the seven.
So I can do it by rows, and in each row times my x is what I'll later call a dot product.
But I also like to see it by columns.
I see this as a linear combination of a column.
So here's my point. A times x is a combination of the columns of A. That's how I hope you will think of A times x when we need it.
Right now we've got -- with small ones, we can always do it in different ways, but later, think of it that way. Okay.
So that's the picture for a two by two system.
And if the right-hand side B happened to be twelve seven, then of course the correct solution would be one two.
Okay. So let me come back next time to a systematic way, using elimination, to find the solution, if there is one, to a system of any size and find out -- because if elimination fails, find out when there isn't a solution. Okay, thanks | 677.169 | 1 |
The book, Exploring Abstract Algebra with Mathematica, is intended for anyone trying to learn (or teach) abstract algebra (a difficult course for some students). Perhaps one of the reasons this course is often challenging is because of its formal and abstract nature. While some people are quite adept at thinking abstractly, many are helped by also thinking visually or geometrically. To this end, where possible, the Mathematica labs are designed to appeal to visualization of various algebraic ideas (as pioneered by Ladnor Geissinger in his software package Exploring Small Groups). Additionally, the nature of the Mathematica notebooks encourages an exploratory environment in which one can make and test conjectures. Viewing the notebooks as interactive texts allows an environment that can not be replicated by lecture alone. While many of the labs are designed to prepare the way for in-class discussion/lecture, they can also be used to extend examples seen in class.
There is no assumption about being able to program in Mathematica; users only need to know the basic concepts of using Mathematica, which are reviewed in Lab 0. Every lab starts with a set of goals as well as prerequisites, listing both mathematical assumptions as well as any assumptions about having used previous labs. Most labs are independent, though a few assume some experience with a previous lab. Although the labs are presented with the ring labs following the group labs, they can be just as easily used by those who prefer to do rings first (as one of us does). Questions are interspersed through the lab at the points where it is natural to ask. As with any text, one does not need to complete every question (although a few have dependency on previous questions). While the length of the labs vary from 40 minutes to 90 minutes in length, they typically require about 60 minutes. (Of course, this is a function of how many of the questions are assigned.) For adopters of the book, there are provisions for suggested minimal questions to be included, as well as which ones could be considered optional. Additionally, notebooks containing just the questions are available, as are partial solutions upon request by an instructor. Finally, a number of palettes for 3.x users are available to facilitate the use of the labs and the implementation of the packages.
The book comes with a CD containing all the labs, the packages, additional palettes and other related materials. It began shipping January, 1999. Note: the labs make no assumptions about what main text is being used; they should be suitable to accompany any text. See the Cross References subsection to see how a number of standard texts can be used with EAAM. | 677.169 | 1 |
This manual contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
See more details below
Student Solutions Manual for LaTorre/Kenelly/Reed/Carpenter/Harris/Biggers' Calculus Concepts: An Informal Approach to the Mathematics of Change, 5th / Edition 5 available in
Paperback
Overview
This manual contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. | 677.169 | 1 |
Westminster, CA Statistics SAndy L.
...There are many different ways you can describe a functions, the most popular are algebraic and graphical.
2. Know the basic function library. Knowing the basic properties of common will save you a lot of time in your calculus studies. | 677.169 | 1 |
Mathematical Tables of In ? (z) for Complex Argument is a compilation of tables of In ? (z), z = x + iy, calculated for steps in x and y of 0.01 and with an accuracy of one unit in the last (the sixth) decimal place. Interpolation is used to calculate In ? (z) for intermediate values and is carried out separately for the real and imaginary parts of... more...
Get ready for a trip around the world to find the many different shapes that surround you. You can find shapes in the places you go, games you play, and even the food you eat. In fact, shapes are everywhere! Can you find them? 32pp. more...
Circles, rectangles, triangles, and squares—these shapes are everywhere! You can even find them at school. Join the students in this book as they discover the many shapes both inside and outside their school. What shapes are in your classroom? 32pp. more...
This clearly written text is the first book on unitals embedded in finite projective planes. It provides a thorough survey of the research literature on embedded unitals. The book is well-structured with excellent diagrams and a comprehensive bibliography. more...
An important new perspective on AFFINE AND PROJECTIVE GEOMETRY This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view. Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and... more...
Containing numerous exercises, illustrations, hints and solutions, presented in a lucid and thought-provoking style, this text provides a wide range of skills required in competitions such as the Mathematical Olympiad. With more than fifty problems in Euclidean geometry, it is ideal for Mathematical Olympiad training and also serves as a supplementary... more...
The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed. The algorithms form a new calculus which... more... | 677.169 | 1 |
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Problem Solving
Course Number:
MTH-107
Minimum Credits:
3
This is a basic problem-solving course suitable for students in any major. It is a survey of a wide variety of problem-solving strategies and concepts. Students successfully completing this course will effectively communicate mathematically, utilize various strategies in analyzing problems, increase problem-solving persistence, and sharpen problem-solving skills. | 677.169 | 1 |
Screenshots
Details
Recently changed in this version
Version 1.3.5
Persistence
The app will now load all of you input, functions, and variables from your last run.
So now everytime you open the app it will look exactly like when you closed it.
Graph zooming improvements
The graph zooms and changes grids in a way that makes alot more sense now
The next version will be nearly an entire new subsystem, expect major speed improvements and no glitches.
Description
**Now Celebrating Over 5000 lines of code***
**Warning this is an alpha version, the app is incomplete and no number given by it should be considered correct until the app is actually release, you have been warned**
**This project has been orphaned. No one is currently developing it as the project was too large for one programmer to manage.**
Ever hated the fact that good graphing calculators are so expensive. When you have to have a good calculator but you don't want to pay $100, you can't do the work properly, and you can't get it done without hours of work, or at least an internet connection. Well now there is Mathulator.
It makes no sense that you have a phone or a tablet that is capable of doing everything that expensive calculator can do but still have to get it anyway. Not anymore. Mathulator aims to be the ultimate calculator, capable of doing anything those amazing graphing calculators can do, in color, with a touchscreen, and in your pocket wherever you are.
For those of you who are familiar with those fancy-shmancy it will still fit right in. The syntax and input is the same format as most new Texas Instruments calculators with only a few exceptions.
As of now there is a lot of Mathulator that doesn't work. Even more of it hasn't been written yet. There is very little left to do before the app is taken into beta. The last of it being an entire code cleanup that will fix hundreds of things that don't work quite properly yet. The reason why some things are being put out there broken is to see how issues come up most often to find the best ways. Other things are broken simply because the code that actually does the math is what need the overhaul the most so we will fix all the math code at once. So if there is a function that isn't working or an error where there shouldn't be during evaluation it will probably stay that way until beta. However if there is a problem with the interface we will try to fix it by the next update.
Tags:
mathulator | 677.169 | 1 |
Download NCERT Books for Class 7 – Free CBSE Books Class 7th
8 July 20105 Comments
NCERT textbooks are read by the students all over the world with great enjoyment & interest. They find them easy to read & learn. All the concepts are very well explained. All the NCERT books of Class VII are easily available free of cost here. One can download them & use them as desired.
There are seven subjects in class 7th. These are English, Hindi, Mathematics, Sanskrit, Science, Social Science & Urdu. NCERT Books of class VII are in English as well in Hindi languages. For English subject, NCERT book of Class 7 are An Alien Hand Supplementary Reader and Honeycomb. The Hindi books are Durva, Mahabharat & Vasant. For Maths, book is Mathematics for class VII. Sanskrit textbook is Ruchir. For S.st., the books are Our past II, Our Environment, Social & Political Life – II. There are 2 books for Urdu subject. These are Urdu Guldasta & Apni Zuban. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Highlighting the new aspects of MATLAB 7.10 and expanding on many existing features, MATLAB Primer, Eighth Edition shows you how to solve problems in science, engineering, and mathematics. Now in its eighth edition, this popular primer continues to offer a hands-on, step-by-step introduction to using the powerful tools of MATLAB.
New to the Eighth Edition
A new chapter on object-oriented programming
Discussion of the MATLAB File Exchange window, which provides direct access to over 10,000 submissions by MATLAB users
Major changes to the MATLAB Editor, such as code folding and the integration of the Code Analyzer (M-Lint) into the Editor
Explanation of more powerful Help tools, such as quick help popups for functions via the Function Browser
The new bsxfun function
A synopsis of each of the MATLAB Top 500 most frequently used functions, operators, and special characters
The addition of several useful features, including sets, logical indexing, isequal, repmat, reshape, varargin, and varargout
The book takes you through a series of simple examples that become progressively more complex. Starting with the core components of the MATLAB desktop, it demonstrates how to handle basic matrix operations and expressions in MATLAB. The text then introduces commonly used functions and explains how to write your own functions, before covering advanced features, such as object-oriented programming, calling other languages from MATLAB, and MATLAB graphics. It also presents an in-depth look at the Symbolic Toolbox, which solves problems analytically rather than numerically.
Synopsis
Featuring the use of MATLABA(R) 7.10, the eighth edition of this popular book provides a hands-on introduction to MATLAB that can be used at the undergraduate level. It covers object-oriented programming in MATLAB and the improvements that have been made to the MATLAB desktop and programming tools. A new chapter discusses MATLAB Central, a supporting website where users can exchange files, blogs, and links. In addition, the appendix of help topics has been updated to reflect only the most frequently used commands, making it quicker and easier to access needed information.
Synopsis
Highlighting the new aspects of MATLAB 7.10 and expanding on many existing features, this updated primer shows how to use the powerful tools of MATLAB to solve problems in science, engineering, and mathematics. | 677.169 | 1 |
enseignement secondaire mathématiques manuel Dictionnaires Étude et enseignement (secondaireexercices Manuels d'enseignement secondaire Premier cycle Termin | 677.169 | 1 |
Are you sure you're allowed to use a calculator on your test? I know I don't let my students use it.
As far as I'm aware, because the TI-84 does not have the CAS system like the TI-89/92/nspire series calculators, there aren't any programs that would simplify expressions. It looks like there's an app "PlySmlt2" that can factor quadratics. I imagine there's something similar for cubics.
The regular GRE or the GRE Math Subject Test? If the former, you'll be totally fine. If the latter, spend a month or so before the exam with the Princeton Review book and your old calculus book, doing several problems. It'll come back to you.
Working on writing a cover letter and polishing up my resume/CV to apply for a adjunct teaching position at a local community college. I'm hoping that the additional teaching load will be enough to help me pay for my wedding without bogging me down and keeping me from finishing my thesis.
Reasarch-wise, my adviser and I had a meeting today about these bisectors in complex hyperbolic space, so I'm playing around in Mathematica to explore these objects a little bit more.
With "just temperament," all of your notes' ratios align with the ratios of harmonic overtones, but the catch is that it's key-specific: if you're tuned to the key of C and try to play an Eb major chord (which is out of key, depending on what you know about music), it's going to sound pretty terrible. Our "equal temperament" uses the 12th root of 2 and gives us a really reasonable approximation of the harmonic overtone ratios in such a way that all chord types sound more-or-less the same, regardless of the key.
Not OP, but at my school the grad students grade for their own classes (if they TA/teach any) as well as some number of upper division classes. The 100-level and occasional 200-level classes are usually graded by undergrads.
Source: I'm a grad student at a large university, teaching a section of calculus and also grading for a 400-level math course.
What you've noticed is something that might be called an "embedding problem". The sphere is technically a 2-dimensional object (/u/math_inDaHood gives a way of visualizing why this is true), but we can only see the whole thing by "embedding" it into 3-dimensional space. In a similar vain, the cylinder and torus (a hollow donut) are both 2-dimensional objects but we require embedding them into 3-dimensional space to actually see them.
Another visual is to think about the sphere like a balloon and only draw lines that can be drawn without popping the balloon. Certainly we can't go through the center of the sphere, so we're stuck with lines that can be drawn on the surface. With this new constraint, what is the shortest "line" between two points?
It's "a guess" in the same way that a limit is just "a guess": we can get arbitrarily close to a value in a way that is consistent with our intuition for definite integrals of somewhat more tame functions.
As /u/private_feet said, Do Carmo's Differential Geometry of Curves and Surfaces is definitely one of the standards, and for good reason - it can be approached with only knowledge of proof structure and mathematics up to Calc III.
I also really liked Bröcker & Jänich's Introduction to Differential Topology. I think it covers very similar material to Do Carmo, but assumes a slightly more topological background and uses a bit more modern notation.
I've heard good things about Pressley's Elementary Differential Geometry and O'Neill's Elementary Differential Geometry, but I've never read either of them (However, I have read another of O'Neill's books, and I liked it).
Well, the square requirement ensures you get a matrix with the same dimensions (for example xTAy, for x and y vectors, gives you a scalar back). I'm not sure about the invertibility requirement, however - probably a condition to ensure singular and nonsingular don't end up being congruent.
I can't really argue with that; I don't find them particularly fascinating either. However, I'm assuming OP is teaching to a much younger crowd that (1) aren't interested in abstract mathematical concepts "just because" and (2) aren't mathematically mature enough to take a "bottom-up" approach that you would have in an undergraduate abstract algebra course.
As such, looking at naturally occurring group structures and then extrapolating the definition of a group is a much more intuitive (and historically accurate) approach. And the classification of finite groups could be introduced in terms of similar geometric objects, say by comparing all symmetries of a triangle and all possible rotations of a regular hexagon. With just these two examples of groups, you can then make the bold claim that every group of order 6 behaves like exactly one of these two groups, which by this point will hopefully perplex and intrigue some of the students. | 677.169 | 1 |
Description: This distinguished little book is a brisk introduction to a series of mathematical concepts, a history of their development, and a concise summary of how today's reader may use them. An expertly written book by a brilliant man, filled with valuable insights and impressive prose. | 677.169 | 1 |
It's an extremely elegant display of mathematics, and although the practical applications of the subject are rather limited it is used in many other scientific and engineering disciplines. I can provide customized help from exam prep to ongoing homework assistance. A well-developed vocabulary i... | 677.169 | 1 |
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Contents/Summary
Bibliography
Includes bibliographical references and index.
Publisher's Summary
This is a textbook for a course in Honors Analysis (for freshman/sophomore undergraduates) or Real Analysis (for junior/senior undergraduates) or Analysis-I (beginning graduates). It is intended for students who completed a course in "AP Calculus", possibly followed by a routine course in multivariable calculus and a computational course in linear algebra. There are three features that distinguish this book from many other books of a similar nature and which are important for the use of this book as a text. The first, and most important, feature is the collection of exercises. These are spread throughout the chapters and should be regarded as an essential component of the student's learning. Some of these exercises comprise a routine follow-up to the material, while others challenge the student's understanding more deeply. The second feature is the set of independent projects presented at the end of each chapter. These projects supplement the content studied in their respective chapters. They can be used to expand the student's knowledge and understanding or as an opportunity to conduct a seminar in Inquiry Based Learning in which the students present the material to their class. The third really important feature is a series of challenge problems that increase in impossibility as the chapters progress. (source: Nielsen Book Data) | 677.169 | 1 |
When students truly understand the mathematical concepts, it's magic. Students who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement of the mathematical world. That's why the new Ninth Edition of Musser, Burger, and Peterson's best-selling textbook focuses on one primary goal: helping students develop a true understanding of central concepts using solid mathematical content in an accessible and appealing format. The components in this complete learning program;from the textbook, to the eManipulative activities, to the online problem-solving tools and the resource-rich website;work in harmony to help achieve this goal.
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Rent Mathematics for Elementary Teachers: A Contemporary Approach, Student Hints and Solutions Manual 9th edition today, or search our site for other textbooks by Gary L. Musser. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Wiley. | 677.169 | 1 |
16+ Mathematics
Take a look at our resources for 16+ KS5 Maths including the NEW Maths in Context series, Edexcel AS and A Level Modular Mathematics, our Revise Edexcel AS Mathematics Revison Guide and Workbook as well as Advancing Maths for AQA and more. | 677.169 | 1 |
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 | 677.169 | 1 |
Education Announces Interactive E-books with ALEKS 360
BEST PRACTICES SERIES
McGraw-Hill Education has introduced ALEKS 360, a mathematics solution that combines an artificial intelligence and personalized learning program with a fully integrated, interactive e-book package. ALEKS 360 delivers assessments of students' math knowledge, guiding them in the selection of appropriate new study material, and recording their progress toward mastery of course goals. Through adaptive questioning, ALEKS accurately assesses a student's knowledge state and delivers targeted instruction on the exact topics a student is most ready to learn.
The e-books featured within ALEKS 360 are interactive versions of their physical counterparts, which offer virtual features such as highlighting and note-taking capabilities, as well as access to multimedia assets such as images, video, and homework exercises. E-books are accessible from ALEKS Student Accounts and the ALEKS Instructor Module for convenient, direct access.
The initial e-books to be offered in ALEKS 360 include: Introductory Algebra, Second Edition, by Julie Miller and Molly O'Neill; Intermediate Algebra, Second Edition, by Miller and O'Neill; College Precalculus, Second Edition, by John W. Coburn; and College Algebra, Second Edition, by Coburn. | 677.169 | 1 |
bought the book because I want my students to see what common core math questions look like. I only gave it 3 starts because it didn't include an answer sheet. If I want to use the assessments in the book I will need to create an answer sheet first for every test in the book. If I would have known that I might not have bought the book. I have enough papers to grade. I don't have time to take math tests.
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To be clear though, the answers are NOT included in this book. These are just the assessments themselves. The ANSWERS to these assessments are included in the Teacher Edition (Teacher Edition is set of manuals broken up by chapter/topic. It has the answers for the student edition, practice book, reteach book, enrich book, and assessment book). So just be aware that you will need to purchase the Teacher Edition (or create answer keys yourself) if you want the answers. | 677.169 | 1 |
Course Description: (Non-credit for mathematics major or minor) A continuation of MATH 2703. Special emphasis for teachers of grades P - 8. Logic; real numbers; basic and transformational geometry; measurement, including the metric system; problem solving; methods and material for teaching mathematics at the P - 8 level.
Topics: basic notions of geometry; geometry in three-dimensions, polygons, and curves; constructions; congruence; similarity; concepts of measurement in both the English and metric system; area; volume; mass; temperature; and motion geometry | 677.169 | 1 |
Unit 7
Unit Overview
In this optional unit, students begin by investigating the use models to solve practical problems. Students then look at the use of numbers in identifying objects, such as cars, books, or credit cards. Students complete the unit by applying the rules of networks to create shortest paths by distance, time, or money.
Instructional Flow
Instructional Flow (PDF) (In Development) Description of the typical order of textbook sections and topics taught in the unit. | 677.169 | 1 |
John Blackwood's new book makes mathematics so exciting I wish I could go back and study it again, this time with the life and energy Blackwood pours into it.
Here you will find lesson after lesson on the subjects covered in Grade 7 of the Waldorf curriculum: a tour de force review of geometric construction, from perpendicular angles to generalized spiroids; the harmonies and proportions of nature (spirals, Fibonnaci sequences, Golden angles, more); the nature of numbers themselves (numeric representation, binary numbers, units of measure, set theory, the magic of Pythagoras' Theorem and Bhaskara's demonstration).
What makes Blackwood's guide so outstanding is that in addition to really clear and easy-to-follow explanations of every single thing to be covered in these lesson blocks, practically every page has one or more main lesson page, colored drawing, or photograph that will inspire the master teacher and leave a clear, richly laden path for the new teacher. | 677.169 | 1 |
Trigonometry
Description: Often, trigonometry students leave class believing that they understand a concept but are unable to apply that understanding when they get home and attempt their homework problems. This mainstream yet innovative text is written by an experiencedOften, trigonometry students leave class believing that they understand a concept but are unable to apply that understanding when they get home and attempt their homework problems. This mainstream yet innovative text is written by an experienced professor who has identified this gap as one of the biggest challenges that trigonometry professors face. She uses a clear voice that speaks directly to students- similar to how instructors communicate to them in class. Students learning from this text will overcome common barriers to learning trigonometry and will build confidence in their ability to do mathematics | 677.169 | 1 |
Essential Mathematica for Students of Science
Tutorial Approach to Mastery of
Mathematica
Mathematica is a fully integrated
system for technical computing. Among the capabilities it offers are:
an interactive front end with notebook interface
numerical calculations with (practically) unlimited
precision
symbolic manipulation
special functions
graphics
typesetting
extensibility
Within a Mathematica notebook you can develop solutions
to complex problems that combine symbolic derivations, numerical calculations,
and graphical displays in an interactive document. The tools provided by Mathematica
will help you focus more on conceptual development and visualization than on
details of algebra or procedural programming. Your solution can be
documented and presented in a very appealing format using the typesetting tools
provided by the front end interface.
The materials provided here are based upon a
course, PHYS426,
that I have taught at the University of Maryland several times during the last
few years. The purpose of this course is to provide
mastery of the Mathematica tools and techniques most relevant to
scientists and engineers. No prior familiarity with Mathematica is
needed. The course consists of a collection of tutorials that present concepts and
practical techniques in a systematic fashion. Many people, including many
faculty, attempt to learn Mathematica in a very haphazard fashion,
learning just enough of the basic syntax to evaluate a few integrals, perform
simple numerical calculations, and make some basic graphs, without ever
understanding how the system works. Unfortunately, this approach often
leads to frustration because it provides no basis for interpreting errors or
unexpected output. It also tends to produce rather inefficient and clumsy
code. Effective use of Mathematica requires training the mind to
formulate problems in new ways, not just to apply new syntax to old
methods. If you take the time to work through the tutorials and exercises,
you will understand the system well enough to use it both efficiently and
creatively. You will be amazed at how much you can accomplish with so
little code, especially if you are a veteran programmer in traditional
languages. I now find Mathematica indispensable to my research
productivity and am confident that you soon will also.
The materials for this course consist of two types of notebooks. Language notebooks present the
basic concepts and techniques of the Mathematica system using relatively
simple exercises. Application notebooks present solutions to
more interesting problems at considerably greater depth. These notebooks
are intended to illustrate the use of Mathematica for research in the
form of polished documents which introduce a problem, outline the strategy for
its solution, implement that solution, discuss interesting examples, and draw
conclusions. The language notebooks are designed to present Mathematica
in a nearly linear fashion, building an edifice one block at a time, whereas
application notebooks use whatever tools are needed to solve a particular
problem or class of problems. Therefore, applications may use features of Mathematica
before they are covered in the course, but links to related course materials or
to on-line documentation are provided where appropriate. After all, in
research one does not choose problems according to the tools at hand, but rather
acquires the tools needed for a problem as those needs are recognized! You
are encouraged to jump back and forth between application and language
notebooks. The style of reading from page 1 to the end is passé.
The notebooks may be accessed using
the hyperlinks below. To view the files directly, you must configure your
web browser to launch either Mathematica or MathReader for files
with the nb extension. MathReader
is a free program that
will display or print a Mathematica notebook, but it cannot evaluate or
edit notebooks.
Alternatively, you may download files to your computer and launch the
application manually. To ensure that the hyperlinks between notebook files
work correctly, you must reproduce my file structure for the course.
Create a top directory for the course, such as c:\My Notebooks\tutorial course.
Save the file contentshere.
The entire directory tree can be downloaded from EssentialMathematicaArchive or
individual files can be obtained using the links tabulated below. After
downloading, the table of contents
provides the simplest means of accessing course materials. I recommend
creating a shortcut to this file on your desktop. If you open this file first
and then use its hyperlinks to open other notebooks, the hyperlinks between
notebooks should work properly. Although you can open notebooks directly,
hyperlinks between notebooks may not work properly that way. Nor will
these hyperlinks work if you open files from your browser without downloading
them. Nevertheless, you should be able to view and evaluate these
notebooks if your browser has been configured properly.
Most of the course materials originated in
1997, but undergo frequent revision as both Mathematica and the
course evolve. Dates for the last revision of each notebook are included
in the listings for your convenience. The most recent major revision
was in January 2006, updating the notebooks for Version 5.2.
Investigates the effects of magnetic interactions between the electron and
proton in hydrogen and their interactions with an external magnetic field,
known as the Zeeman effect for weak fields or Paschen-Back effect for strong
fields.
Fractals and deterministic chaos
A nonlinear discrete model for the population of species with
nonoverlapping generations that illustrates the phenomena of stable cycles,
chaos, and order within chaos. Introduces entropy and Lyapunov
exponents.
Data analysis
The following files should be saved in the subdirectory Data analysis
and constitute a minicourse in data analysis using Mathematica. The
MathLite notebook provides a very brief introduction to some of the
features needed to use Mathematica in an introductory physics laboratory
that can be covered in one lab period of about 3 hours. The DataAnalysis
notebook presents methods of linear and nonlinear least-squares and develops
some simple programs that should be useful for such a course. Improved
versions of these programs are supplied in the package Curfit.
Physics problems
This is a small collection of solutions to assorted physics problems that
illustrate a variety of numerical methods implemented within Mathematica.
A few more are under development but are not ready for posting.
Shooting methods are used to find bound-state energies and
wave functions for Schrödinger equation.
Jan., 2007
Advice for students
The notebooks which comprise this book are intended to teach Mathematica
by example and by practice. Ideally the course should be studied in a tutorial
format in which an instructor guides students through a progression of exercises
which develop skill and understanding through practice. However, if you are
studying Mathematica independently, please work the exercises — do not
just read the text and skip the exercises. Do not be discouraged if you have had
negative experiences in the past with an instructor in one of your courses who
expected you to learn Mathematica after just one or two lectures. It is
only through practice that one can train the mind to approach problem solving in
new ways and this takes time and patience. Mathematica provides many
tools which will help you to solve and visualize problems in new ways, and these
tools are well worth the effort needed to master them. The notebooks in this
book will guide you toward mastery of the most important tools in Mathematica
slowly, with exercises you can work at your own pace. By the time you complete
the two core notebooks, getting started and functions,
you will have acquired a solid foundation that will allow you to easily learn
and apply the particular tools most appropriate to your needs. Take the time
needed to understand the material presented in these notebooks well.
Advice for instructors
This book arose from a course that I teach at the
University of Maryland. Although I am a professor of physics, the course is
intended for advanced undergraduate and beginning graduate students in any field
of science, engineering, or mathematics. Thus, although many of the examples are
naturally from physics, many examples are also drawn from mathematics (iterated
functions and fractals), biology (population dynamics), and chemistry (chemical
oscillators). The prerequisites are calculus and introductory physics.
Too often students receive their first introduction to Mathematica
in just one or two class periods of a course which uses Mathematica as
one of the tools employed in the presentation of its primary subject. The
unfortunate student is then expected to master new concepts, syntax, and
programming techniques in a couple of hours. This impossible burden often
alienates the student so completely that he or she abandons Mathematica
once that course is finished. If any of your students have suffered through that
experience, please reassure them that the tools provided by Mathematica
are worth a considerable investment of effort and that your course will offer
sufficient time and practice to acquire mastery of them. Mastery of Mathematica
is like mastery of a second language — it requires time, patience, practice,
immersion, repetition, and more patience. Any second language becomes much
easier when a person begins to think in that language without having to
translate every thought. Languages cannot be learned by passively listening to
lectures, but require active immersion and practice.
My course was taught in a teaching theater using a
tutorial format. The teaching theater provides each student with a computer and
the instructor with equipment to project the instructor's monitor, transmit the
instructor's screen to each student monitor, and to view and/or project any
monitor in the room. Networking software permits communication or collaboration
between students and the instructor. Assignments were distributed and collected
electronically. All materials for the course were presented in the form of Mathematica
notebooks. Most classes began with a brief introduction to the lesson of
the day, with the instructor's monitor transmitted to the students. I would work
through and discuss the appropriate section of a notebook, and then the students
would work some of the exercises in that notebook as I circulated through the
room offering help and suggestions where needed. If several students were
experiencing similar problems, I might give hints to the entire class or might
return to my computer and show how a particular problem might be solved.
Occasionally, I would project the solution produced by a student and comment on
it or allow the student to present it. After students have completed the
exercises for one topic, a new topic would be introduced by the instructor and
then the students would perform the exercises for that section.
I found that it was very important to allow students to
work through the materials for themselves as much as possible. Students are very
anxious to place their hands on the keyboard and to dive right in; students tend
to have relatively little patience for dry lectures. Most students learn
relatively little from listening, but learn a lot from working the embedded
exercises and discussing strategies for the solution of specific problems. The
tutorial format requires more time than lecturing, but is much more effective.
Do not try to go too fast! It is more important for students to begin to think
in Mathematica style than to hear or read the syntax for thousands of
functions.
The essential core of an introductory course in Mathematica
is contained within the two notebooks getting started and
functions, each of which requires approximately three weeks to cover.
The student who has mastered these lessons is already quite proficient in the
use of Mathematica and should be able to identify and learn the
additional tools needed for his/her purposes with relatively little help from
the instructor. Therefore, the selection of topics for the remainder of a
standard one semester course permits flexibility to meet the needs of the class.
A sample curriculum is given below with estimated times.
The notebook on dimensional analysis, or
parts of it, should be assigned as homework soon after completion of getting
started. Within functions is a problem on Julia sets that
is easy and fun; I assigned it between two classes and then presented julia.nb
as an example of modular programming after the students turned in their
solutions. Depending upon the pace of the class, plotting can
either be covered in class or parts of it assigned as homework; I used the
diffraction problem as a homework assignment that reinforced techniques of
modular programming as well as covering plotting features. I chose to omit lists
to permit more time for differential equations. I also used frequency
analysis to shed additional insight upon limit cycles (van der Pol
oscillator) and the logistic map. A miniproject based upon Koch
was also a lot of fun.
Style sheet
If you would like to prepare notebooks with the
style used for these course materials, you will need the style sheet TutorialJJK.nb.
Solutions to exercises
Detailed solutions to all of the exercises in getting
started and functions and to many of the exercises in
other notebooks are available to qualified instructors upon request. These
solutions have not been made public so that instructors can assign homework
without the solutions being in general circulation.
Feedback
Please notify me of any errors you find. Comments
and suggestions are welcome. My e-mail address is jjkelly@umd.edu. | 677.169 | 1 |
Algebra 1 Final Exam
This 60 question test works great as a final exam, or practice for a state test. The test covers the following objectives:
-Solve multi-step equations and inequalities.
-Identify properties used while solving an equation or
Two versions of the same 6 page test are included, each with 56 questions. The questions on each version are nearly identical, with the exception of different numbers. Each version has TWO questions per standard. Sub-standards were not included
This Common Core aligned Algebra (order of operations and writing simple expressions) Assessment Pack is a complete formative and summative assessment package, ready to use, to assess your students understanding during your instruction of order of
Divisibility rules, or divisibility tests, have a wide range of applications in mathematics (finding factors, determining prime vs. composite, simplifying fractions, probability, etc.), but are often underemphasized in the classroom or not explored
This test contains 42 multiple choice questions on topics presented in my Algebra Guided Presentation Notes: Unit 1, which is available here on TPT. This is a PDF file.
This test assesses:
Using Variables
Order of Operations
Evaluating Algebraic
This is an excellent review for any student studying how to solve systems of linear equations. This is the 2 page practice test I always give my students the day before the real test, but you could also use it for a review before the midterm or
This is the exact test I give my students upon completing Unit 3 on Solving Inequalities. It goes perfectly with my Unit 3 Guided Presentation Notes available here on TpT.
This test has 30 questions and covers:
* Recognizing Solutions to an
This is a 23 question multiple choice test to assess students at the end of Unit 4 on Functions and Their Graphs. It is the test I give my students once we complete the guided notes, practice, and review for Unit 4.
It is on recognizing
This is a 26 item "Show your work" or practice test. It is made to go with my Unit 2 Guided Presentation Notes on Solving Equations and Word Problems available here on TPT. This can be used as the "real" test or as a review to get ready for the
This is the test I give at the end of Unit 6 on Solving Systems of Linear Equations and their Word Problems. It is four pages containing 23 questions based on similar question that have appeared on the state test.
I try hard to produce a test that
This is a 30 question multiple choice review of 7th grade common core curriculum. I used this as my Mid-Term exam, but it would also be a great review of the standards before your state test.
Topics Covered:
- Operating with rational numbers
-
13 tests, 4 versions of each. My students improved from 47% to 63% proficient or advanced on the California State Test the year that I added 6 of these tests as review and reinforcement in my Algebra 2 class (2010-11). They cover ESSENTIAL SKILLS
A twist on the traditional "matching" assessment, this "quiz" assesses students' understanding of several properties of exponents.
First, students work together to match specific expressions to equivalent, more simplified expressions. As needed,
This is an excellent review for any student learning to multiply and factor polynomials. This is the practice test I always give my students as a parter review a day or two before the real test, but you could also use it for a review before for the
This item is a handout consisting of 27 test questions. Most are multiple-choice and the rest are free response. It covers topics for "Slope, Intercepts, and Graphing Linear Equations" such as using slope formula, determining x & y-intercepts,
Download these "spot" quizzes to help you assess how students are doing on individual skills in Algebra I. This file contains 50 eight-question quizzes on topics covered in an Algebra I course. An answer key for each quiz is also provided.
This
This packet includes 18 study guides/informational sheets and assessments on a variety of Algebra skills. Included are: Slope, Graphing, Factoring polynomials, systems of equations, scatterplots, inequalities and graphing linear equations. Every
This is a "Gallon Man" worksheet that I created to help teach cups, pints, quarts, and gallons. All the finger and toes of the man are cups, the next largest limb segment is pints, the next largest next to the torso is quarts, and the actual body of
Multiplication and division test:
This set includes formative assessments for the Operations and Algebraic Thinking standards 1-4. Covered on these two assessments are the concepts of:
-1. Interpret a multiplication equation as a comparison, e.g.,
For fun ways to ensure your students do well on these quizzes, please check out my discounted Integer & Rational Numbers Combo Pack. It features 6 fun activities to make an otherwise dull topic pretty fun!
So, you have started teaching using
PLEASE READ THE ENTIRE DESCRIPTION before making your purchase.
I have incorporated several different strategies into my lessons over the years which have aided me in becoming a Georgia Certified Master teacher. I truly believe that the MOST
This is a 52 page packet that includes 18 quizzes, all with Answer Keys! There are a variety of middle school/algebra/pre-algebra topcs assessed. Each of these quizzes are sold separately, so please check out my product page. This bundle
This assessment includes 9 items that show understanding of Common Core State Standard 5.OA.3. Items on the assessment progress from identifying simple patterns to graphing ordered pairs and determining the relationships between corresponding terms
Algebra Exit Slips (Order of Operations and Simple Expressions) - 5th Grade - Formative Assessment at it's Best!
Don't wait for the big test to figure out who doesn't get it! With frequent math exit slips you can quickly assess your students and
In a world of multiple choice, standardized test we often lose sight on how much our students actually know. This assessment allows students to show their understanding of coordinates, the axes, and the quadrants.
A great open-ended assessment or
This item is a handout consisting of 23 test questions. About half are multiple-choice and the other half is free response. It covers topics for "Scatter plots, Correlation, and Line of Best fit" such as making predictions given an equation for a
"Trigonometry an Introduction" introduces the trig functions, sine, cosine and tangent. They are used to solve right triangles, oblique triangles, special triangles, and area of triangles. There are a total of 18 pages of problems and activities
In a world of multiple choice, standardized test we often lose sight on how much our students actually know. This assessment not only allows students to show their knowledge on Direct variation, it allows them to create graphs, tables, and
This is a power point warm-up or bell ringer activity or quick pop quiz for an Algebra 1 class. There are 10 pages of multiple choice questions. Each page gives an inequality and asks students to choose the correct number line for the inequality
I used this product along with several additional scenarios to help my students understand the flow between the following:
1. Identifying independent and dependent variables
2. Using those variables to create an equation
3. Using the equation to set of worksheets serves as a competency test for Basic Factoring. These questions all have a one as a coefficient in front of the squared term. There are five different forms of the test in this set.
An answer key for each of the forms is
This item is a handout consisting of 17 test questions. About half are multiple-choice and the other half is free response. It covers topics for "Real World Linear Equation/Table/Graph" such as writing and solving equations given word problems,
This quiz requires students to determine the slope and y-intercept of a linear function from a table or graph. (It does not ask them to find the slope or y-intercept from an equation.)
Part of the TEKS quiz series, available for all 7th and 8th
This quiz requires students to create an equation in slope-intercept form from a table, graph, situation, or two points.
Part of the TEKS quiz series, available for all 7th and 8th grade math TEKS. Each quiz contains ten questions of varying
There are 4 different versions (with keys) of an eight question quiz. It can be used as a pop quiz, a pre-assessment quiz, or a review quiz. I have taken the time to make sure that all the quizzes have the exact same type of questions, but
This item is a handout consisting of 22 test questions. About half are multiple-choice and the other half is free response. It covers topics for "Laws of Exponents" such as using one or multiple laws to simplify simple and complex expressions with
Here is the complete bundled unit for Algebra 2 for Unit 2. FUNCTIONS, EQUATIONS, & GRAPHS. The unit includes nine lessons which are presented to students with an 8-page Dinah Zike Bound-Book Style Foldable*, used with permission, that acts like
This test covers the content in common core standard MCC5.G1-2 & OA.3. The test includes a mix of multiple choice and open response questions to challenge students.
I created this test with the implementation of the common core standards in is a test that covers the concept of slope. Students are expected to be able to write the equation of a line given a slope and y intercept, calculate the slope given two points, calculate the slope of a line given the equation of a line, and
Math Operations Quiz
Are you a superstar?
I have complied a list of questions using identity, inverse, exponent, power, and cancellation properties. This should be a simple quiz for students beyond 8th grade math. I have yet to have even a
This item is a handout consisting of 22 test questions. Most are multiple-choice and the rest are free response. It covers topics for "Solving Systems Graphing, Substitution, Elimination " such as solving systems that produce infinite or no
3.OA.3 Use Multiplication within 100 to Solve Word Problems
This pack contains 25 multiplication problems designed, not only to test your students' multiplication skills, but also to see how well they understand the problem being presented to them. | 677.169 | 1 |
Provides a concise overview of the core undergraduate physics and applied mathematics curriculum for students and practitioners of science and engineering Fundamental Math and Physics for Scientists and Engineers summarizes college and university level physics together with the mathematics frequently encountered in engineering and physics calculations.... more...
There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics. The Great Mathematical Problems explains why these problems exist, why they matter, what drives mathematicians to incredible lengths to solve them and where... more...
This book reminds students in junior, senior and graduate level courses in physics, chemistry and engineering of the math they may have forgotten (or learned imperfectly) that is needed to succeed in science courses. The focus is on math actually used in physics, chemistry, and engineering, and the approach to mathematics begins with 12 examples of... more...
A follow-up to "Used to Know That", which helps you to brush up on your maths, science, history, geography and English language while enjoying a walk down memory lane - and remembering the stuff you really shouldn't have forgotten. more...
The ultimate collection of high-school-level mathematics problems. Former IMO participants have rescued these problems from old and scattered manuscripts, and produced the definitive source of IMO practice problems in one volume for the first time. more... | 677.169 | 1 |
multiplication: an interactive micro-course for beginners to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Matrix multiplication: an interactive micro-course for beginners
Select this link to open drop down to add material Matrix multiplication: an interactive micro-course for beginners to your Bookmark Collection or Course ePortfolio
Discussion for Matrix multiplication: an interactive micro-course for beginners
Claudia Ayerdis
(Student)
One of my favorite topics, matrix. This site is use full at giving the basics on how to multiply and the rules of multiplying matrix's. I like how it highlights the equation inputs with the outputs making it easier to remember the steps to be taken. This method was used in the class with my Finite math professor. He was able to explain and illustrate this method and make it easy for us to re-identify it on our examinations.
Technical Remarks:
This site is very short. Easy to get to and definitely easy to use.
Used in course
7 years ago
Amit Rajwani
(Student)
After browsing the Merlot website for at least 30 minutes, I came across to this website. It educates us about Matrix Multiplication that I have studied in my finite mathematics class. Multiplying matrices are really easy, but when u get the concept. It also can be tricky if you do not know how to solve it the right way. This website gives us more information on how to multiply matrices together. First, it tells us to multiply a row matrix with a column matrix of the same length. It also shows us an example on how to work through a matrices problem. I spent a few hours in understanding the material from this website and compared it to how I understood in my finite mathematics class. This is an excellent website because it has visuals, if you are not an English speaker you can still understand the concept by looking at the visuals. I think that Merlot has some great information that can educate us and help us understanding the material better. At last I would like to say that, this is a great resource for the students who have trouble understanding the concept of Matrix Multiplication.
Used in course
7 years ago
Elaine Aguilar
(Student)
I spent about 15 minutes reviewing and testing my knowledge on this webpage and thought this was one of the better sites I've discovered on MERLOT. The site is a brief introduction to multiplying matrices and the concepts were clear, easy to follow and the material is presented with simple explanations and excellent visuals. This site would have made my life a lot easier when I was learning matrix multiplication. This website would definitely be effective in enhancing a student's learning because it was interactive, making solving the application easy to follow and the visual highlights reinforce the material. First time users will find the software easy to use because its straightforward and very user-friendly. | 677.169 | 1 |
A full course in differential equations involves applications of derivatives to be studied after two or three semester courses in calculus. A derivative is the rate of change of one quantity with respect to another; for example, the rate...
The following is a complete listing of all the pages on this site. You can use this listing if the menus aren't working correctly. Note as well that as I work to update the site to ASP.NET and move some of the files around there may be times in whic...
The game of billiards is played on a table (known as a billiard table) upon which balls are placed. One ball (the "cue
ball") is then struck with the end of a "cue" stick, causing it to
bounce into other balls and off the sides
of the table. Rea... | 677.169 | 1 |
The younger son is beginning adventures in algebra, and I had a hard decision to make. He'd been using computer-based programs to learn math, but Mike and I decided we didn't want to go that route any longer. I had spent a lot of time looking into curriculum with the older son, so I already had a textbook available (Jacob's Elementary Algebra), and it's one that has received excellent reviews.
It's also 37 years old. Apparently there's a newer edition, but that's not the one I bought.
I had one concern with using this book. A lot of the standards surrounding math curriculum have changed and become standardized. There are a lot of texts available that have been evaluated and measure up to those standards. I was worried that by going with an older book, I was going to shortchange the younger son in his education. (I think that's something almost every homeschool parent worries about.) The problem with a lot of the modern curricula, though, is that I really don't like it. While I think the sciences generally benefit from taking a problem-solving approach, I'm not so sure that's the best way to do it with math. Sure, I think there are ways to teach it more effectively, especially in terms of using active learning strategies and hands-on learning. Reasoning is important, but so is process, and kids need to come out of the classroom very fluent in process and computation. I'm one of those old-fashioned types that thinks you're better off giving your kids a multiplication table than a calculator.
I had issues with one curriculum that was being used locally, for instance, because it taught division as repeated subtraction without teaching long division. It also taught matrix math and repeated sums without teaching the standard multiplication schemes. For those who are familiar with all the controversy over curricula and math standards, I'm sure this is old hat.
I was pleasantly surprised, then, to find that this 37 year old book assumes that the student knows long division and standard multiplication. However, in the first chapter (which is review), it introduced both matrix multiplication and repeated division as alternative methods. Repeated division was done side by side with long division as a way to show how long division works. However, it was not suggested as a good way to do division but to augment student understanding of long division. Matrix multiplication was proffered as a bonus problem, but I made sure younger son understood how to do it. I found with the older son that he was less likely to stumble on multiplication problems if he used the matrix method but would have a hard time keeping things straight with the standard method. It's a good tool to have in your toolbox, and I have even pulled it out when I had to do a fairly large problem by hand despite only having learned it about 10 years ago.
This left me feeling like this book was going to work just fine. In fact, I'm rather disappointed that I didn't get to use this book in high school. (It was already out of print, sadly.) Apparently, though, Amazon reviewers, internet philosophers, and other homeschooling parents really do know what they're talking about. Feynman may even have approved.
The younger son was very adamant that he wanted to take high school biology this year. He wasn't in my face about it, but whenever the question was put to him about whether he was sure he wanted to do that, he was pretty firm.
My approach to dealing with this, after seeing he was sure was, "What the hell?!" Worst case scenario is that he fails and has to retake it in four years with his age mates.
The first couple assignments were great. However, when he hit the second unit of the class, I started having second thoughts. It wasn't going well. And would failing a class leave a long term scar on his academic record?
He was worried, too, but he started asking me how he could improve things. I noted that he started saying he needed to "study harder," but when I asked him what he meant, he wasn't sure. I started giving him specific suggestions and pointers and told him that doing those things is what "study harder" meant.
I learned a few things from this experience. First, younger son didn't know how to study when he started this class. To anyone who has ever dealt with a bright kid, you'll identify this as a common problem. It's hard for kids to learn how to study when the subject matter they're tackling is relatively easy and doesn't require the type of effort that a seriously challenging class does…or any other life obstacle. I think we're all convinced this was a good experience in that regard. Second, I'm probably more worried about his grades than I thought, but I think I'm managing not to be a helicopter parent. There were some assignments he submitted that he didn't ask me to review. Some came back with really good grades and some didn't, but I really wanted this to be his own work. Honestly, it's a bit more stressful to be hands off than I thought. I keep reminding myself that I should be celebrating a good effort instead of relatively effortless higher grade (that probably indicates he wasn't seeing anything new).
To all of our surprise, he pulled his grade up to a B- for the first semester. This guarantees he won't be a straight A student in high school, but I personally think he got a lot more out of it now than if he'd taken it when he was supposed to.
Part of the fun of hanging out with my offspring for part of the day is the entertaining conversations we get to have. When he was younger, he had some awfully adorable misconceptions that resulted in a lot of fun. Now that he's older, his discussions have become more sophisticated.
I have been working at home, trying to finish up this PhD thing once and for all. Earlier this year, the place I worked was shut down and so I figured that if I had any desire to stay in academia (which I do), the PhD thing is kind of a necessary evil.
Because of the job situation, however, I also ended up with a new officemate: my younger son. It was actually a combination of factors: private school is expensive, middle school is a cesspool of derision and contempt (and therefore not the best place to develop social skills), and, finally, the younger son really wanted to take high school biology and no one would let him. Except me, being the overindulgent parent I am.
I have to admit that he's been a bit easier to deal with than his older sibling. It's amazing how much easier this education thing is when you're not dealing with ADHD. The younger son is amazingly self-sufficient and does a good job of keeping a schedule.
I have, however, discovered one major flaw in this plan. I had no idea how much middle schoolers talked. Mostly, he gets excited about the things he's learning in his class, which really tickles me. However, he wants to share everything with me. Every. Thing. I have learned more about genes and cell processes and reproduction in the past two months than I probably did during my own high school biology class. I have learned about social and mental and physical health. I am beginning to speak Spanish with a level of proficiency that has not been present since my teens. And mostly, I see him being happy and excited about learning again.
Unfortunately, he's not quite so receptive when I begin to talk about coding and arrays and debugging and compiler issues and, especially, writing. I have begun, as of late, to tell him that while I'm glad he's learning, I really need him to let me focus on my work, too. Someday, if he has to share an office with someone, this will be good real life practice for not making them insane. At least he's not asking to go out every ten minutes, like the dogs.
Like this:
This weekend, the Blue Angels were in town to perform at the Fargo AirSho. While we were watching them today, I made some comment about how amazing it is that they can keep such perfect formation despite the high speeds. The younger son asked how fast they fly, and I responded that they could go up to a few hundred miles per hour. He came back with:
I bet they're flying at a trillion nanometers per second.
I honestly had no idea since that required not only a conversion to more reasonable units for such a measurement as well as the fact that we'd have to hop between metric and English units.
I decided to check it out, and it turns out he wasn't far off. The Blue Angels use the F/A-18 Hornet, which wikipedia gives a top speed of Mach 1.8 or 1,190 miles per hour. The equivalent speed in nanometers/second is 531,977,600,000. In other words, it's half a trillion nanometers per second, so the younger son was only off by a factor of two when they're traveling at top speed (which they obviously weren't).
That's a wee bit faster than an unladen European Swallow, which has an airspeed velocity of about 11,176,000,000 nm/s (based on Wolfram Alpha's estimate of 25 mph). I'm sure you were just dying to know that.
Most people are familiar with the concept of microwaving a grape to make an arc. If not, the procedure is very simple: cut a grape in half but leave just a small bit of skin to connect to the two halves. Put the grape on a plate in the microwave, turn it on, and watch the sparks fly. (As a side note, I've been able to replicate this on a smaller scale when microwaving green beans.) This video explains it fairly clearly:
This week, we discovered another fun microwaving activity: soap. I can't be just any soap: it specifically has to be Ivory soap. Apparently it gets hot and the gas bubbles expand causing it to create a hot foam which grows fairly quickly. You can't do it with other soaps, however, because they're too hard and will explode.
We used a whole bar of soap with our experiment, but the younger son told us later that the demo he saw only used a smaller chunk. Be careful after you pull it out of the microwave: it's hot! Also, once it's cooled, you can use the soap, although it may be more useful to stick it into a soap sleeve than try to use it directly.
A couple days ago, @katiesci posted this opinion piece from Science by Eleftherios Diamandis on getting noticed. I was rather frustrated with the article because the way to get noticed was apparently to put in a lot of face time (which is probably decent advice) and to publish like crazy (also not bad advice), even if it means you have to work unrealistic schedules and foist all of your childcare duties onto your spouse.
It was this last part that got under my skin because it's so much a recapitulation of the status quo: you can't do anything else and be a scientist, forget balance if you want an academic career.
I have to admit I jumped to a pretty lousy conclusion when I read the following:
I worked 16 to 17 hours a day, not just to make progress on the technology but also to publish our results in high-impact journals. How did I manage it? My wife—also a Ph.D. scientist—worked far less than I did; she took on the bulk of the domestic responsibilities. Our children spent many Saturdays and some Sundays playing in the company lobby. We made lunch in the break room microwave.
I can't presume to know the dynamic between the author and his wife, and it may be that she was perfectly happy with this arrangement. Academic couples tend to understand better than others how frustrating this career path can be, and I know there were several occasions where either my husband or myself was bringing the other dinner/microwaving in the lobby or lunch room to help ease the stress of deadlines along with an empty stomach.
But what about the people for whom this is not an option? Most of the people I know get very upset if their spouse is putting in more than 60 hours per week. Are they just supposed to give up? What about people who are physically unable to work those types of hours? Even if you are physically capable, it's bad for you in the long run and turns out to be rather useless.
If anything, this just reinforced that to make it in science, you don't have to do good science, you just have to be willing to give up any semblance of a family life and turn into a squeaky wheel. I'm not sure what the author intended to convey, but reading this piece was rather disheartening.
Instead, I'd rather have heard about how the author's wife did it: how is it she was able to work less hours than him, raise their kids, and still manage to have an apparently successful career? At least, that's the implication at the end of the piece. To me, it sounds like she was able to handle a very unbalanced load successfully, and unless it's, "don't sleep," I would think she may have some advice worth sharing with the rest of us mere mortals. If you happen to be from Science magazine, could you please let her know?
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This is a pretty special week: Teradog's Gotcha Day was on Tuesday. Three years ago, we welcomed him into our family, thinking it was only going to be temporary. The truth is, we're foster failures. Despite Mike's insistence that he was just staying for a couple days, we ended up staying for a month before Mike asked about whether the rescue group had found him a new home. I said they hadn't been looking but I could contact them, if he wanted me to. By that point, he didn't want me to because that giant ball of fluff and love had steadfastly attached himself to Mike's hip.
We weren't sure how long he would be around, which was the really scary part. The vet couldn't figure out how old he was (his teeth were in bad shape) and said he could be anywhere from four to ten years old, his teeth indicating the high end of that range. We took the median, seven, which is getting old for a Newfoundland. He was also in very bad health. However, he's doing very well now (except for a bit of arthritis) and is happy, healthy, and generally content. We're hoping he will be around for a while longer.
Happy Newfie!
Today is another anniversary: I will have been writing at this blog for five years. While that's generally a happy thing, you may have noticed that things have been rather quiet the past couple months. That's because, after five years and not quite a month at my job, the research center I've been working for has turned into a support lab and all the research staff have either been terminated or will be let go as soon as funds on their respective projects are gone. Because of this, there hasn't been much to talk about. I'm spending a lot of time in front of the computer, working on my thesis, hanging out with my critters. While it lends itself to a lot of cute puppy and kitty pics (and often kitty AND puppy pics, probably snuggling), there hasn't been a whole lot of narrative material there unless you'd like me to get into the specifics of drooling and sleeping patterns of Newfoundlands. The only thing I am sure I could do on a fairly regular basis is complain about how certain programs are a pain to use, but I've already done that (probably ad nauseum).
All of this boils down to today being a good day to celebrate changes. Change is generally a stressful thing, but it's all in what you make of it. And there's certainly worse things to do than to hang around with domesticated bears. | 677.169 | 1 |
Prentice Hall Algebra 1 and Algebra 2 with Trigonometry / Edition 1
This textbook provides thorough coverage of all traditional Algebra 2 concepts and skills. At the beginning of the course, the lessons review and extend key Algebra 1 concepts and skills.
See more details below | 677.169 | 1 |
INTRODUCTION
The Ministry of Education's vision of "Thinking Schools, Learning Nation" gives impetus for the infusion of three initiatives; Thinking Skills, Information Technology (IT) and National Education into the curriculum. As we move towards a knowledge-based society which is powered by IT, the need to prepare our people for the challenges and opportunities of the future becomes obvious. Besides being proficient in the use of IT, pupils will need to be able to think creatively, learn independently and work successfully in teams. Above all, as Singapore's economy moves towards globalisation, they need to have a strong feeling for home and remain Singaporean in heart, mind and being. Against this background and with the Desired Outcomes of Education as the overarching aim, the mathematics syllabus was revised. This revised mathematics syllabus reflects the recent developments in mathematics education. The focus of the syllabus is mathematical problem solving. The emphasis is the development of concepts, skills and its underlying processes. This, together with the explication of thinking skills and the integration of IT in mathematics teaching and learning, will give leverage to the development of mathematical problem solving. This syllabus consists of two parts. Part A explains the philosophy of the syllabus and the spirit in which it should be implemented. It also spells out the aims and objectives of the mathematics programme. The framework of the mathematics programme summarises the essence of mathematics teaching and learning in schools. The learning of mathematics at all levels involves more than the basic acquisition of concepts and skills. It also involves an understanding of mathematical thinking, general problemsolving strategies, having positive attitudes to and an appreciation of mathematics as an important and powerful tool in everyday life. This framework forms the basis for mathematics teaching and learning in schools. The objectives of the mathematics programme for the foundation stage and the orientation stage are summarised to provide an overview of the concepts and skills introduced at each level. Part B gives the syllabus content for each level. Care has been taken to ensure that there is continuity from the primary to the secondary level. In the syllabus, the spiral approach is adopted to ensure that each topic is covered at appropriate levels in increasing depth. This enables pupils to consolidate the concepts and skills learnt and to develop further concepts and skills. The content for the EM3 stream repeats some of the important topics covered in the foundation stage. This is to ensure that pupils have a good understanding of basic mathematical concepts covered in the foundation stage before they proceed to other topics in the orientation stage. This syllabus is a guide for teachers to plan their mathematics programmes. Teachers need not be bound by the sequence of topics presented here but should ensure that hierarchy and linkages are maintained. Teachers should exercise flexibility and creativity when using the syllabus.
3
AIMS OF MATHEMATICS EDUCATION IN SCHOOLS
Mathematics education aims to enable pupils to • • • acquire and apply skills and knowledge relating to number, measure and space in mathematical situations that they will meet in life acquire mathematical concepts and skills necessary for a further study in Mathematics and other disciplines develop the ability to make logical deduction and induction as well as to explicate their mathematical thinking and reasoning skills through solving of mathematical problems use mathematical language to communicate mathematical ideas and arguments precisely, concisely and logically develop positive attitudes towards Mathematics including confidence, enjoyment and perseverance appreciate the power and structure of Mathematics, including patterns and relationships, and to enhance their intellectual curiosity
• • •
4
FRAMEWORK OF THE MATHEMATICS CURRICULUM
The conceptualisation of the mathematics curriculum is based on the following framework:
The primary aim of the mathematics curriculum is to enable pupils to develop their ability in mathematical problem solving. Mathematical problem solving includes using and applying mathematics in practical tasks, in real life problems and within mathematics itself. In this context, a problem covers a wide range of situations from routine mathematical problems to problems in unfamiliar contexts and open-ended investigations that make use of the relevant mathematics and thinking processes. The attainment of problem solving ability is dependent on five inter-related components - Concepts, Skills, Processes, Attitudes and Metacognition. 1 Concepts Concepts refer to the basic mathematical knowledge needed for solving mathematical problems. They cover the following: • • • • 2 Numerical concepts Geometrical concepts Algebraic concepts Statistical concepts
Skills Skills refer to the topic-related manipulative skills that pupils are expected to perform when solving problems. They include: • • • • estimation and approximation mental calculation communication use of mathematical tools
5
• • • 3
arithmetic manipulation algebraic manipulation handling data
Processes Processes refer to the thinking and heuristics involved in mathematical problem solving. Some thinking skills and heuristics which are applicable to problem solving at the primary level are listed below: Thinking skills: • Classifying • Comparing • Sequencing • Analysing Parts & Whole • Identifying Patterns & Relationships • Induction • Deduction • Spatial Visualisation Heuristics for problem solving: • Act it out • Use a diagram/model • Make a systematic list • Look for pattern(s) • Work backwards • Use before-after concept • Use guess and check • Make suppositions • Restate the problem in another way • Simplify the problem • Solve part of the problem (Refer to Appendix A for the definitions of the suggested thinking skills)
4
Attitudes Attitudes refer to the affective aspects of mathematics learning such as: • • • • enjoy doing mathematics appreciate the beauty and power of mathematics show confidence in using mathematics persevere in solving a problem
5
Metacognition Metacognition refers to the ability to monitor one's own thinking processes in problem solving. This includes: • • • constant and conscious monitoring of the strategies and thinking processes used in carrying out a task seeking alternative ways of performing a task checking the appropriateness and reasonableness of answers
This framework encompasses the whole mathematics curriculum from primary to secondary school.
WHOLE NUMBERS P3 1. Number notation and place values up to 10 000 2. Addition and subtraction of numbers up to 4 digits 3. Multiplication tables up to 10×10 4. Multiplication and division of numbers up to 3 digits by a 1-digit number 5. Odd and even numbers
1. Number notation and place values up to 100 000 2. Approximation and estimation 3. Factors and multiples 4. Multiplication of numbers • up to 4 digits by a 1-digit number • up to 3 digits by a 2-digit number 5. Division of numbers up to 4 digits by a 1-digit number and by 10
1. Multiplication and division of length, mass, volume and time (in compound units) 2. Multiplication and division of money (in compound units using decimal notation) 3. Units of measure of volume: cubic centimetre, cubic metre 4. Volume of • a cube and a cuboid • liquid 5. Area and perimeter of a square, a rectangle and their related figures
1. Addition and subtraction • like fractions • related fractions 2. Product of a proper fraction and a whole number 3. Mixed numbers and improper fractions
1. Number notation and place values up to 3 decimal places 2. Comparing and ordering 3. Addition and subtraction up to 2 decimal places 4. Multiplication and division up to 2 decimal places by 1digit whole number 5. Conversion between decimals and fractions 6. Approximation and estimation
10
PRIMARY 5 & PRIMARY 6 (EM1/EM2 STREAMS)
WHOLE NUMBERS P5 EM1 EM2 1. Number notation and place values up to 10 million 2. Approximation and estimation 3. Multiplication and division of numbers up to 4 digits by a 2digit whole number 4. Order of operations MONEY, MEASURES & MENSURATION 1. Conversion of units of measure involving decimals and fractions 2. Volume of a cube and a cuboid 3. Area of a triangle STATISTICS GEOMETRY FRACTIONS
1. Line graphs • reading and interpreting • solving problems
1. Angles • angles on a straight line • angles at a point • vertically opposite angles 2. 8-point compass 3. Properties of • a parallelogram • a rhombus • a trapezium • a triangle 4. Geometrical construction: Draw a square, a rectangle, a parallelogram, a rhombus and a triangle from given dimensions 5. Tessellation
1. Addition and subtraction of • mixed numbers • unlike fractions 2. Product of fractions 3. Concept of fraction as division 4. Division of a proper fraction by a whole number
P6 EM1 EM2
1. Area and circumference of a circle 2. Area and perimeter of a figure related to square, rectangle, triangle and circle 3. Volume of • a solid made up of cubes and cuboids • liquid
PRIMARY 5 & PRIMARY 6 (EM3 STREAM)
WHOLE NUMBERS P5 EM3 1. Number notation and place values up to 10 million 2. Addition and subtraction of numbers up to 4 digits 3. Multiplication of numbers • up to 4 digits by a 1-digit number • up to 3 digits by a 2-digit number 4. Division of • numbers up to 4 digits by a 1digit number • a 2 digit-number by a 2-digit number 5. Factors and multiples 6. Approximation and estimation P6 EM3 MONEY, MEASURES & MENSURATION 1. Money • decimal notation • 4 operations involving money in the decimal notation STATISTICS GEOMETRY 1. Perpendicular and parallel lines 2. Angles in degrees 3. Properties of a square and a rectangle 4. Symmetry
1. Number notation and place values up to 3 decimal places 2. Comparing and ordering 3. Addition and subtraction up to 2 decimal places 4. Multiplication up to 2 decimal places by a whole number up to 2 digits 5. Division up to 2 decimal places by a 1digit whole number 6. Multiplication and division up to 2 decimal places by tens, hundreds and thousands 7. Conversion between fractions and decimals 8. Approximation
P6 EM3
1. Average 2. Rate
1.
Direct proportion
1. Concept of percentage 2. Percentage of a quantity
14
PRIMARY 1
TOPICS/OUTCOMES
REMARKS
WHOLE NUMBERS
1 a) NUMBER NOTATION AND PLACE VALUES Pupils should be able to count to 100
a)
• •
Include completing sequences of consecutive numbers Include counting in tens and completing sequence
b) c) 2 d) e) f)
read and write numbers up to 100 in numerals and in words Recognise the place values of numbers (tens, ones) CARDINAL AND ORDINAL NUMBERS give a number to indicate the number of objects in a given set represent a given number by a set of objects use ordinal numbers such as first, second, up to tenth COMPARING AND ORDERING compare two or more sets in terms of the difference in number
d) e) f)
• • • •
Exclude the term 'cardinal number' Include visualising small sets up to 5 objects instead of counting one by one Include symbols, e.g. 1st, 2nd, 3rd, etc. Exclude the term 'ordinal number' Include the concept of one-to-one correspondence Include use of the phrases 'more than', 'less than' and 'fewer than' Include finding 'How many more/less?' Include use of the words: greater, greatest, smaller, smallest Exclude use of the symbols ' > ' and '<'
3 g)
g)
• • •
h)
compare numbers up to 100
h)
• •
i)
arrange numbers in increasing and decreasing order
15
PRIMARY 1
TOPICS/OUTCOMES
REMARKS
WHOLE NUMBERS
4 j) ADDITION AND SUBTRACTION illustrate the meaning of 'addition' and 'subtraction' j) • Include comparing two numbers within 20 and finding how much greater/smaller
k) l)
write mathematical statements for given situations involving addition and subtraction build up the addition bonds up to 9 + 9 and commit to memory
l)
• •
Include writing number stories for each number up to 10 Include sums such as the following: (i) +2 =7 (ii) 3 + = 12 Exclude box sums which are beyond addition bonds such as 9+ = 22
•
m) n)
o) p)
q) 5 r) s) t) u) 6 v)
recognise the relationship between addition and subtraction add and subtract numbers involving • 2-digit numbers and ones • 2-digit numbers and tens • 2-digit numbers and 2-digit numbers add 3 one-digit numbers carry out simple addition and subtraction mentally involving • 2-digit number and ones without renaming • 2-digit number and tens solve 1-step word problems on addition and subtraction MULTIPLICATION illustrate the meaning of multiplication as repeated addition write mathematical statements for given situations involving multiplication multiply numbers whose product is not greater than 40 solve 1-step word problems with pictorial illustrations on multiplication DIVISION divide a quantity not greater than 20 into equal sets: • given the number of objects in each set • given the number of sets
n)
• •
Exclude formal algorithm Include addition/subtraction with renaming
q)
•
Use numbers within 20
s) t,u)
• •
Use numbers with product not greater than 40 Exclude use of multiplication tables
v)
•
Exclude use of division symbol
16
PRIMARY 1
TOPICS/OUTCOMES
REMARKS
MONEY AND MEASURES
1 a) MEASUREMENT OF LENGTH AND MASS Pupils should be able to compare the lengths/masses of two or more objects in non-standard units
a)
• • •
Include use of simple approximation to measure lengths and masses Exclude finding the difference in length/mass Include the use of the following words: long, longer, longest short, shorter, shortest tall, taller, tallest high, higher, highest heavy, heavier, heaviest light, lighter, lightest
2 b) 3 c)
d) e)
f)
g)
TIME (12-HOUR CLOCK) tell time in terms of o'clock and half past MONEY tell the different denominations of • coins • notes match one coin/note of one denomination to an equivalent set of coins/notes of another denomination tell the amount of money • in cents (¢) up to $1 • in dollars ($) up to $100 add and subtract money • in dollars only • in cents only solve 1-step word problems on addition and subtraction of money • in cents only • in dollars only
b)
•
Exclude use of 24-hour clock
e)
• •
Include use of symbols '$' and '¢' Exclude combinations of dollars and cents
g)
•
Include finding 'How much more/less?'
17
PRIMARY 1
TOPICS/OUTCOMES REMARKS
STATISTICS
1 a) PICTURE GRAPHS Pupils should be able to make picture graphs of given data a) • • • b) read and interpret picture graphs b) • Include collecting and organising data Include both horizontal and vertical forms Include the use of symbolic representations, e.g. represents one child Exclude picture graphs with scales such as each represents 5 children
Exclude use of the words 'cube', 'cone', 'cylinder' in written or verbal form
18
PRIMARY 2
TOPICS/OUTCOMES REMARKS
WHOLE NUMBERS
1 a) b) c) d) 2 e) NUMBER NOTATION AND PLACE VALUES Pupils should be able to count to 1000 read and write numbers up to 1000 in numerals and in words recognise the place values of numbers (hundreds, tens, ones) compare and order numbers up to 1000 ADDITION AND SUBTRACTION add and subtract two numbers up to 3 digits
a) b) c) d) e)
• • • • • •
Include counting in tens and hundreds Include the use of zero as a place holder Include completing number sequences Include finding the difference Include formal algorithm Include use of abacus (refer to Appendix B)
f)
3 g) h) i) j) k)
carry out addition and subtraction mentally involving • 3-digit number and ones • 3-digit number and tens • 3-digit number and hundreds MULTIPLICATION AND DIVISION WITHIN THE 2, 3, 4, 5 AND 10 TIMES TABLES count in steps of 2, 3, 4, 5 and 10 build up the multiplication tables of 2, 3, 4, 5 and 10 and commit to memory multiply numbers within the multiplication tables write mathematical statements for given situations involving division divide numbers within the multiplication tables carry out multiplication and division within multiplication tables mentally WORD PROBLEMS solve 1-step word problems involving the four operations
g) h)
• •
Include completing number sequences leading to multiplication tables Include activities to help pupils see that multiplication is commutative
j) k)
• •
Include use of division symbol Exclude division with remainder
4 l)
19
PRIMARY 2
TOPICS/OUTCOMES REMARKS
MONEY AND MEASURES
1 a) MEASUREMENT OF LENGTH, MASS AND VOLUME Pupils should be able to estimate and measure • length in metres/centimetres • mass in kilograms/grams • volume in litres
a)
• • • • •
Include the use of appropriate instruments for measuring Include use of the appropriate measure and their abbreviations: cm, m, g, kg and l Exclude compound units Include concept of conservation of volume of liquid Exclude volume of solids
Exclude cases where the minute hand is between two numbers Include reading time, e.g. read '9.15' as 'nine fifteen'; '9.50' as 'nine fifty' Include use of 'a.m.' and 'p.m.' Include drawing hands on the clock face to show time Include use of abbreviations: h and min Include the concept of duration of time when reading time
4 e) f)
ADDITION AND SUBTRACTION OF MONEY read and write money using decimal notation add and subtract money in compound units
f)
• • •
Include making 'change' Include cases such as $2.50 + 60¢ and $5.75 − $3 Exclude cases such as $2.50 + $3.20 and $5.75 − $2.55
STATISTICS
1 a) b) c) PICTURE GRAPHS WITH SCALES Pupils should be able to make picture graphs using a scale representation read and interpret picture graphs with scales solve problems using information presented in picture graphs a) • Include both horizontal and vertical representations Include scales such as "Each represents 6 bags." Exclude cases involving the use of an incomplete symbolic representation such as
c)
• •
TOPICS/OUTCOMES
REMARKS
FRACTIONS
1 EQUAL PARTS OF A WHOLE Pupils should be able to a) a) 2 b) • • Exclude set of objects Include the use of symbols:
1 Recognise and name unit fractions up to 12
IDEA OF SIMPLE FRACTIONS recognise and name a fraction of a whole b)
COMPARING AND ORDERING FRACTIONS compare and order unit fractions and like fractions
21
PRIMARY 2
TOPICS/OUTCOMES
REMARKS
GEOMETRY
1 a) SHAPES AND PATTERNS Pupils should be able to identify and name • a semicircle (half circle) • a quarter circle identify the following shapes that make up a given figure: • square • rectangle • triangle • circle • semicircle • quarter circle complete patterns according to • shape • size • orientation • two of the above attributes LINES, CURVES AND SURFACES identify straight lines and curves draw a straight line of given length identify flat and curved faces of a 3-D object
b)
b)
•
Include tracing the outlines of figures formed and talking about the shapes used to form the figures
c)
c)
•
Include identifying the patterns and relationships
2 d) e) f)
d)
•
Include forming figures with straight lines and curves and describing how they form the figures
22
PRIMARY 3
TOPICS/OUTCOMES REMARKS
WHOLE NUMBERS
1 a) b) c) 2 d) NUMBER NOTATION AND PLACE VALUES Pupils should be able to read and write numbers up to 10 000 in numerals and in words recognise the place values of numbers (thousands, hundreds, tens, ones) compare and order numbers up to 10 000 ADDITION AND SUBTRACTION add and subtract numbers up to 4 digits d) • • e) 3 f) g) 4 h) carry out addition and subtraction mentally involving two 2-digit numbers MULTIPLICATION TABLES UP TO 10 × 10 count in steps of 2, 3, 4, 5, . . ., 10 build up the multiplication tables up to 10 × 10 and commit to memory MULTIPLICATION AND DIVISION BY A 1-DIGIT NUMBER multiply and divide numbers up to a 3digit number by a 1-digit number Include use of terms 'sum' and 'difference' Include use of abacus (refer to Appendix B)
f)
•
Include completing number sequences leading to multiplication tables
h)
• •
Include use of the terms 'product', 'quotient' and 'remainder' Exclude 2-step calculation such as: Find the product of 6 and the difference between 10 and 8
i) 5 j) 6 k)
carry out simple mental calculations ODD AND EVEN NUMBERS identify odd and even numbers WORD PROBLEMS solve up to 2-step word problems involving the four operations on whole numbers
k)
•
Include units of measure
23
PRIMARY 3
TOPICS/OUTCOMES REMARKS
MONEY AND MEASURES
1 a) UNITS OF MEASURE Pupils should be able to visualise the relative magnitudes of standard units • kilometre and metre • metre and centimetre • kilogram and gram • litre and millilitre • hour and minute • minute and second • year and month • month and day • year and day • week and day measure in compound units • length : kilometre, metre, centimetre • mass : kilogram, gram • time : hour, minute, second, day, week, month, year • area : square metre, square centimetre • volume : litre, millilitre carry out the following conversions, and vice versa: • kilometre to metre • metre to centimetre • kilogram to gram • litre to millilitre • hour to minute • minute to second • year to month • week to day ADDITION AND SUBTRACTION OF LENGTH, MASS, VOLUME AND TIME add and subtract in compound units • length • mass • volume • time a) • • Include use of the word 'capacity' Include use of abbreviations: km, m, cm, kg, g, l, ml, h, min, s
b)
b)
• •
Include estimating and measuring with different units Include use of the terms 'past' and 'to' such as '10 minutes past 5' and '15 minutes to 12'
c)
c)
• •
Include compound units Numbers involved should be within easy manipulation
2 d)
d)
• •
Exclude seconds Numbers involved should be within easy manipulation
24
PRIMARY 3
TOPICS/OUTCOMES REMARKS
MONEY AND MEASURES
3 e) 4 f) ADDITION AND SUBTRACTION OF MONEY add and subtract money in compound units using the decimal notation WORD PROBLEMS solve up to 2-step word problems involving money, length, mass, volume and time PERIMETER OF A RECTILINEAR FIGURE calculate the perimeter of a rectilinear figure • in centimetres • in metres
f)
• •
Include problems involving different units of measure Include problems involving concept of duration of time interval
5 g)
• •
Include estimating and measuring perimeter Exclude figures such as and
6 h) i) j) k)
AREA AND PERIMETER OF A SQUARE AND A RECTANGLE calculate the perimeter of square and rectangle compare the areas of shapes in nonstandard units estimate the area of a square and a rectangle in standard unit visualise the relative sizes of 1 square metre and 1 square centimetre
h)
•
Exclude use of formulae
k)
• •
l)
use formula to calculate the area of a square and a rectangle TOPICS/OUTCOMES
l)
•
Exclude conversion between cm2 and m2 Include estimating area in square metres and square centimetres Include use of abbreviations: cm2, m2
REMARKS
STATISTICS
1 a) b) c) BAR GRAPHS Pupils should be able to read scales on the axis read and interpret bar graphs solve problems using information given in bar graphs a) • Include both horizontal and vertical representations
25
PRIMARY 3
TOPICS/OUTCOMES
REMARKS
FRACTIONS
1 a) b) c) d) 2 e) EQUIVALENT FRACTIONS Pupils should be able to recognise and name equivalent fractions list the first 8 equivalent fractions of a given fraction with denominator not greater than 12 write the equivalent fraction of a fraction given the denominator/ numerator express a fraction in its simplest form COMPARING AND ORDERING compare and order related and unlike fractions with denominators up to 12 a) • Include the terms 'numerator' and 'denominator'
e)
• •
Include both increasing and decreasing order Number of fractions involved should not exceed 3 REMARKS
TOPICS/OUTCOMES
GEOMETRY
1 a) b) CONCEPT OF ANGLES Pupils should be able to associate an angle as a certain amount of turning identify right angles
b)
•
For identifying right angles in a figure, restrict to only right angles inside the figure Example: How many right angles are there inside the figure? (Answer: 5)
•
Exclude figures such as and
c) d)
tell whether a given angle is greater or smaller than a right angle identify angles in 2-D shapes
c)
•
Exclude use of the terms 'acute', 'obtuse' and 'reflex' angles
26
PRIMARY 4
TOPICS/OUTCOMES REMARKS
WHOLE NUMBERS
1 a) b) NUMBER NOTATION AND PLACE VALUES Pupils should be able to read and write numbers up to 100 000 in numerals and in words recognise the place values of numbers (ten thousands, thousands, hundreds, tens, ones) compare and order numbers up to 100 000 APPROXIMATION AND ESTIMATION round off numbers to the nearest 10 and 100 estimate the answers in calculations involving addition, subtraction and multiplication FACTORS AND MULTIPLES determine if a 1-digit number is a factor of a given whole number list all factors of a whole number up to 100
MULTIPLICATION BY A NUMBER UP TO 2 DIGITS multiply numbers up to • 4 digits by a 1-digit number • 3 digits by a 2-digit number DIVISION BY A 1-DIGIT NUMBER AND BY 10 divide numbers up to 4 digits by a 1-digit number and by 10 WORD PROBLEMS solve up to 3-step word problems involving whole numbers
l)
• •
Include units of measure Include checking reasonableness of answers
27
PRIMARY 4
TOPICS/OUTCOMES REMARKS
MONEY AND MEASURES
1 MULTIPLICATION AND DIVISION OF LENGTH, MASS, VOLUME AND TIME Pupils should be able to Multiply and divide in compound units • Length • Mass • Volume • Time MULTIPLICATION AND DIVISION OF MONEY multiply and divide money in compound units using decimal notation UNITS OF MEASURE OF VOLUME: CUBIC CENTIMETRE, CUBIC METRE build solids with unit cubes and state their volumes visualise the relative sizes of 1 cubic metre and 1 cubic centimetre VOLUME OF A CUBE/CUBOID AND LIQUID use formula to find the volume of a cuboid
a)
a)
• • •
Use whole numbers only Exclude seconds Numbers involved should be within easy manipulation
2 b) 3 c) d) 4 e)
•
Exclude conversion between m3 and cm3
e)
• • • • •
f) g) 5
use formula to find volume of liquid in a rectangular container recognise the equivalence of 1 litre/1000 ml and 1000 cm3 AREA AND PERIMETER OF A SQUARE, A RECTANGLE AND THEIR RELATED FIGURES find the area/perimeter of a figure made up of squares and/or rectangles find one dimension of a rectangle given the other dimension and • its perimeter • its area find the side of a square given • its perimeter • its area WORD PROBLEMS solve word problems involving • volume of solids/liquid • area and perimeter of squares and rectangles
f) g)
Include use of abbreviations: m3 and cm3 Include finding the volume of the solid made up of unit cubes of given dimension Exclude compound units Exclude compound units Include conversions between l, ml and cm3
h) i)
h)
•
Include finding the area of a figure by subtraction of areas
j)
j)
•
Exclude use of '
' sign
6 k)
k)
•
Exclude compound units
28
PRIMARY 4
TOPICS/OUTCOMES
REMARKS
STATISTICS
1 a) b) 2 c) 3 d) TABLES Pupils should be able to complete a table from given information read and interpret tables BAR GRAPHS complete a bar graph from given data WORD PROBLEMS solve problems using data presented in bar graphs and tables a) • Include collecting data and presenting the data in a table form
TOPICS/OUTCOMES
REMARKS
FRACTIONS
1 a) ADDITION AND SUBTRACTION Pupils should be able to add and subtract • like fractions • related fractions a) • • Denominators of given fractions should not exceed 12 Exclude sums involving more than 2 different denominators
2 b) c) 3 d) 4 e)
PRODUCT OF A PROPER FRACTION AND A WHOLE NUMBER recognise and name fractions as parts of a set of objects calculate the product of a proper fraction and a whole number MIXED NUMBERS AND IMPROPER FRACTIONS express an improper fraction as a mixed d) number, and vice versa WORD PROBLEMS solve up to 2-step word problems involving fractions
•
Include expressing an improper fraction/mixed number in its simplest form Include using unitary method to find the 'whole' given a fractional part Exclude question such as 'Express the number of girls as a fraction of the number of boys.' as it will be dealt with under the topic 'Ratio'
e)
• •
29
PRIMARY 4
TOPICS/OUTCOMES REM (i) 0.125 = 1 2 + + 10 100 1000 125
Include mental calculations involving addition and subtraction of 1-digit whole numbers/tenths and tenths Include division of whole number by whole number with decimal answers Include rounding off answers to 2 decimal places Include checking reasonableness of answers
d)
• • •
e) 4 f) 5 g)
h) 6 i)
carry out mental calculation within the multiplication tables CONVERSION BETWEEN DECIMALS AND FRACTIONS express a decimal as a fraction, and vice versa APPROXIMATION AND ESTIMATION round off decimals to • theInclude units of measure
h)
•
Include checking reasonableness of answers Include rounding off answers to a specified degree of accuracy Include checking reasonableness of answers
i)
• •
30
PRIMARY 4
TOPICS/OUTCOMES
REM estimate size of angles and measure angles in degrees using a protractor
a,b) •
Include use of the terms 'vertical' and 'horizontal'
c)
• • •
Include using notation such as ∠ABC and ∠x to name angles Exclude using variable such as x° to represent size of angle Exclude reflex angles
draw a given angle using a protractor SYMMETRY recognise symmetric figures
e) f)
• • •
Exclude drawing reflex angles Include identifying and visualising symmetry in the environment or in designs Exclude rotational symmetry Exclude finding the number of lines of symmetry of a symmetric figure
g) h) 4 i)
j)
determine whether a straight line is a line of symmetry of a figure complete a symmetric figure with respect to a given line of symmetry GEOMETRICAL FIGURES identify and name the following figures: • rectangle • square • parallelogram • rhombus • trapezium identify and name the following triangles: • isosceles • equilateral • right-angled
g)
•
j)
•
Exclude use of the terms 'scalene', 'acute' and 'obtuse'
31
PRIMARY 4
TOPICS/OUTCOMES
REMARKS
GEOMETRY
5 k) PROPERTIES OF A SQUARE AND A RECTANGLE state and use properties of • a square • a rectangle 2-D REPRESENTATION OF A 3-D SOLID visualise cubes and cuboids from drawings state the number of unit cubes that make up a solid visualise and identify the new solid formed by increasing/ decreasing the number of cubes of a given solid drawn on an isometric grid. k) • • Exclude the term 'diagonal' and its related properties Include finding angles and sides
6 l) m) n)
n)
•
Exclude asking pupils to draw the solid on an isometric grid
32
PRIMARY 5 (EM1/EM2)
TOPICS/OUTCOMES REMARKS
WHOLE NUMBERS
1 a) 2 b) c) 3 d) e) 4 f) 5 g) NUMBER NOTATION AND PLACE VALUES Pupils should be able to read and write numbers up to 10 million in numerals and in words MULTIPLICATION AND DIVISION multiply and divide numbers up to 4 digits by a 2-digit whole number multiply and divide numbers by tens, hundreds and thousands APPROXIMATION AND ESTIMATION round off numbers to the nearest 1000 estimate the answers in calculations involving addition, subtraction and multiplication ORDER OF OPERATIONS state the order of operations and carry out combined operations involving the 4 operations WORD PROBLEMS solve word problems involving whole numbers
d)
•
Include use of the approximation symbol ' ≈ '
f)
•
Include use of brackets
g)
•
Include rounding off answers to a specified degree of accuracy REMARKS
TOPICS/OUTCOMES
MENSURATION
1 CONVERSION OF MEASUREMENTS INVOLVING DECIMALS AND FRACTIONS Pupils should be able to convert measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa VOLUME OF A CUBE/CUBOID find one dimension of a cuboid given its volume and other dimensions find the edge of a cube given its volume solve up to 2-step word problems involving volume of a cube/cuboid and liquid
a)
a)
• •
Exclude measurements involving decimals for time Numbers involved should be within easy manipulation Exclude finding the area of a face given its volume and one dimension Exclude use of ' 3 ' sign Include problems involving the height of water level in rectangular tank Exclude finding volume of solid by the volume of liquid displaced Include identifying the base and its corresponding height Exclude finding the base or height of a triangle given its area
2 b) c) d)
b) c) d)
• • • •
3 e)
AREA OF A TRIANGLE use formula to find the area of a triangle
e)
• •
33
PRIMARY 5 (EM1/EM2)
TOPICS/OUTCOMES
REMARKS
STATISTICS
1 a) b) LINE GRAPHS Pupils should be able to read and interpret line graphs solve problems using information presented in line graphs a,b) • Exclude distance - time graphs
TOPICS/OUTCOMES
REMARKS
FRACTIONS
1 a) ADDITION AND SUBTRACTION Pupils should be able to add and subtract • mixed numbers • unlike fractions PRODUCT OF FRACTIONS calculate the product of 2 fractions CONCEPT OF FRACTION AS DIVISION associate a fraction with division DIVISION OF A PROPER FRACTION BY A WHOLE NUMBER divide a proper fraction by a whole number WORD PROBLEMS solve word problems involving fractions • • b) c) • • Include listing of equivalent fractions to identify fractions with common denominator Denominators of given fractions should not exceed 12 Exclude mixed numbers Include conversion between fractions and decimals
2 b) 3 c) 4 d) 5 e)
TOPICS/OUTCOMES
REMARKS
DECIMALS
1 a) b) MULTIPLICATION AND DIVISION Pupils should be able to multiply decimals up to 2 decimal places by a 2-digit whole number multiply and divide decimals up to 3 decimal places by tens, hundreds and thousands solve word problems involving decimals a) b) • • Include checking reasonableness of answers by estimation Exclude cases where the first nonzero digit in the answer is at the 4th decimal place such as 0.12 ÷ 1000 = 0.00012 Include rounding off answers to a specified degree of accuracy Include checking reasonableness of answers
c)
c)
• •
34
PRIMARY 5 (EM1/EM2)
TOPICS/OUTCOMES REMARKS
GEOMETRY
1 a) ANGLES Pupils should be able to identify and name • angles on a straight line • angles at a point • vertically opposite angles a) • • Exclude angles between parallel lines such as alternate angles, interior angles, corresponding angles Exclude use of the terms 'complementary' and 'supplementary'
b) c) d)
2 e) 3 f)
g) h)
i) 4 j)
5 k) l) m) n)
recognise that angles on a straight line add up to 180º and angles at a point add up to 360º recognise that vertically opposite angles are equal find unknown angles involving • angles on a straight line • angles at a point • vertically opposite angles 8-POINT COMPASS tell direction in relation to the 8-point compass PROPERTIES OF A PARALLELOGRAM, A RHOMBUS, A TRAPEZIUM AND A TRIANGLE state and find unknown angles involving the f) properties of • a parallelogram • a rhombus • a trapezium recognise and use the property that the angle g) sum of a triangle is 180º state and find unknown angles involving the properties of • an isosceles triangle • an equilateral triangle • a right-angled triangle recognise that the exterior angle of a triangle is equal to the sum of the interior opposite angles GEOMETRICAL CONSTRUCTION draw squares, rectangles, parallelograms, j) rhombuses and triangles from given dimensions TESSELLATION recognise shapes that can tessellate identify the shape used in a tessellation make different tessellations with a given shape draw a tessellation on dot paper
• • •
Exclude the term 'diagonal' and its related properties Exclude additional construction of lines Exclude problems where the skill of solving equations is required
• •
Use ruler, protractor and set squares Exclude cases where compasses are required
35
PRIMARY 5 (EM1/EM2)
TOPICS/OUTCOMES REMARKS
AVERAGE, RATE AND SPEEDTOPICS/OUTCOMES
REMARKS
RATIO AND PROPORTION
1 a) RATIO Pupils should be able to use ratio to show the relative sizes of • 2 quantities • 3 quantities interpret a given ratio • a:b • a:b:c recognise equivalent ratios reduce a given ratio to its lowest terms solve up to 2-step word problems involving ratio a) • Introduce the sign ' : '
b)
b,c) •
Include sums such as the following: (i) 1 : 2 (ii) 2 : = = : 8 1 : 4
c) d) e)
d)
•
Include reducing a : b : c to its lowest terms
TOPICS/OUTCOMES
REMARKS
PERCENTAGES
1 a) CONCEPT OF PERCENTAGE Pupils should be able to change fractions and decimals to percentages, and vice versa a) • • b) 2 c) 3 d) express a part of a whole as a percentage PERCENTAGE OF A QUANTITY calculate part of a whole given the percentage and the whole WORD PROBLEMS solve up to 2-step word problems involving percentage
Include use of the percentage notation '%' Include recognising the equivalence between percentage and fraction/decimal
d)
•
Exclude use of the terms 'profit' and 'loss'
36
PRIMARY 6 (EM1/EM2)
TOPICS/OUTCOMES REMARKS
MENSURATION
1 a) AREA AND CIRCUMFERENCE OF A CIRCLE Pupils should be able to identify and name the following parts of a circle: • centre • radius • diameter • circumference use the formula to find • circumference • area
b)
b)
•
Include finding area and perimeter of a figure made up of the following shapes: (i) a semicircle (ii) a quadrant (quarter circle) (Figure should be provided) Include use of π (to be
c)
solve word problems involving area and circumference of a circle
c)
• •
22 or 3.14) 7
Exclude finding the radius/diameter of a circle given its area
2
AREA AND PERIMETER OF A FIGURE RELATED TO SQUARE, RECTANGLE, TRIANGLE AND CIRCLE find the area and perimeter of a figure related to the following shapes: • rectangle • square • triangle • circle • semicircle • quadrant VOLUME OF A SOLID MADE UP OF CUBES/CUBOIDS AND VOLUME OF LIQUID solve word problems involving volume of • a solid made up of cubes and cuboids • liquid
d)
3 e)
37
PRIMARY 6 (EM1/EM2)
TOPICS/OUTCOMES REMTOPICS/OUTCOMES
REMARKS
GEOMETRY
1 a) ANGLES IN GEOMETRIC FIGURES Pupils should be able to find unknown angles in geometric figures using the properties of • angles on a straight line • angles at a point • vertically opposite angles • square, rectangle, parallelogram, rhombus, trapezium and triangle 2-D REPRESENTATION OF A 3-D SOLID visualise a prism and a pyramid from drawings NETS identify nets of • a cube • a cuboid • a prism • a pyramid identify the solid which can be formed by a net a) • Exclude additional construction of lines
2 b)
b)
• • •
Include cylinder Include the terms 'prism' and 'pyramid' Exclude net of cylinder
Include use of the formula Include activities for pupils to read, interpret and write speed in different units: km/h, m/min, m/s and cm/s Exclude conversion of units such as 10 km/h = ? m/min, and vice versa Exclude problems where rest time is involved in finding the average speed of a journey
TOPICS/OUTCOMES
REMARKS
RATIO AND PROPORTION
1 a) b) c) d) e) RATIO AND DIRECT PROPORTION Pupils should be able to express one value as a fraction of another given their ratio, and vice versa find how many times one value is as large as another given their ratio, and vice versa recognise that two quantities are in direct proportion solve direct proportion problems using unitary method solve word problems on ratio and direct proportion
TOPICS/OUTCOMES
REMARKS
PERCENTAGE
1 a) b) c) ONE QUANTITY AS A PERCENTAGE OF ANOTHER Pupils should be able to express one quantity as a percentage of another find the whole given a part and the percentage solve word problems
c)
•
Exclude finding percentage profit/loss 39
PRIMARY 6 (EM1/EM2)
TOPICS/OUTCOMES
REMARKS
ALGEBRA
1 a) ALGEBRAIC EXPRESSION IN ONE VARIABLE Pupils should be able to use a letter to represent an unknown number and write a simple algebraic expression in one variable for a given situation simplify algebraic expressions
WHOLE NUMBERS
1 a) b) 2 c) 3 d) NUMBER NOTATION AND PLACE VALUES Pupils should be able to read and write numbers up to 10 million in numerals and in words compare and order numbers up to 100 000 ADDITION AND SUBTRACTION add and subtract numbers up to 4 digits MULTIPLICATION AND DIVISION multiply numbers • up to 4 digits by a 1-digit number • up to 3 digits by a 2-digit number divide numbers • up to 4 digits by a 1-digit number • up to 2 digits by a 2-digit number multiply and divide numbers by tens, hundreds and thousands FACTORS AND MULTIPLES determine if a 1-digit number is a factor of a given whole number list all factors of a whole number up to 100
a)
•
Include completing number sequences
c)
• • •
Include the use of the terms 'sum' and 'difference' Include use of the term 'product'
d)
e)
e)
Include use of the terms 'quotient' and 'remainder'
f) 4 g) h)
h)
• •
Include finding common factor of 2 numbers Exclude finding highest common factor (H.C.F.)
i) j)
Determine if a whole number is a multiple of a given 1-digit whole number list the first 12 multiples of a given 1-digit whole number
j)
• •
Include finding common multiple of 2 numbers Exclude finding lowest common multiple (L.C.M.) Include use of the approximation symbol '≈' Include checking reasonableness of answers Include rounding off answers to a specified degree of accuracy
Include making 'change' Include visualising the relative magnitudes of standard units: - kilometre and metre - metre and centimetre - kilogram and gram Include use of abbreviations: km, m, cm, kg, g Numbers involved should be within easy manipulation Exclude compound units
• d) convert measurements of length and mass from a smaller unit of measure to a bigger unit, and vice versa e) perform the 4 operations with units of length and mass 3 WORD PROBLEMS f) solve up to 3-step word problems involving money, length and mass 4 AREA AND PERIMETER OF A SQUARE AND A RECTANGLE g) calculate the perimeter of a rectilinear figure d) e) • •
g)
• •
Include estimating and measuring perimeter Include finding the perimeter of a figure made up of unit squares Include use of abbreviations: cm2 and m2 Exclude conversion between cm2 and m2 Exclude compound units Include visualising the relative size of 1 square metre and 1 square centimetre
h) find the area of a figure drawn on square grid i) calculate the area and perimeter of a square and a rectangle
i)
• • • •
j)
find one dimension of a rectangle given the other dimension and • its perimeter • its area k) find the side of a square given • its perimeter • its area l) solve up to 3-step word problems involving area and perimeter of a square and a rectangle
k)
• •
Exclude use of '
' sign
l)
Exclude compound units
42
PRIMARY 5 (EM3)
TOPICS/OUTCOMES REMARKS
STATISTICS
1 a) b) 2 c) d) 3 e) 4 f) TABLES Pupils should be able to complete a table from given information read and interpret tables BAR GRAPHS read and interpret bar graphs complete a bar graph from given data LINE GRAPHS read and interpret line graphs PROBLEMS ON STATISTICS solve problems using information given in tables, bar graphs and line graphs a) • Include collecting data and presenting the data in a table form
c)
•
Include both horizontal and vertical representations
e)
•
Exclude distance - time graphs
TOPICS/OUTCOMES
REMARKS
FRACTIONS
1 a) b) c) 2 d) e) f) g) 3 h) 4 i) CONCEPTS OF FRACTIONS Pupils should be able to recognise and name a fraction of a whole identify and name fractions as parts of a set of objects associate a fraction with division EQUIVALENT FRACTIONS recognise and name equivalent fractions list the first 8 equivalent fractions of a given fraction with denominator not greater than 12 write the equivalent fraction of a fraction given the denominator/ numerator express a fraction in its simplest form IMPROPER FRACTIONS AND MIXED NUMBERS express an improper fraction as a mixed number, and vice versa COMPARING AND ORDERING compare and order unlike fractions with denominators up to 12
h)
•
Include expressing an improper fraction/mixed number in its simplest form Include both increasing and decreasing order Include use of listing of equivalent fractions Number of fractions involved should not exceed 3
i)
• • •
43
PRIMARY 5 (EM3)
TOPICS/OUTCOMES REMARKS
FRACTIONS
5 j) ADDITION AND SUBTRACTION add and subtract fractions • • • 6 k) l) 7 m) 8 n) PRODUCT OF FRACTIONS calculate the product of a proper fraction and a whole number calculate the product of 2 fractions DIVISION OF A PROPER FRACTION BY A WHOLE NUMBER divide a proper fraction by a whole number WORD PROBLEMS solve up to 2-step word problems involving fractions k,l) • Include unlike fractions and mixed numbers Include listing of equivalent fractions to identify fractions with common denominators Denominators of given fractions should not exceed 12 Exclude mixed numbers
TOPICS/OUTCOMES
REM: (i) 0.125 = (ii) 0.125 = 1 2 + + 1000 10 100 125
(iii) 21.203 = 21 + • Exclude 0.125 = b) compare and order decimals
1000
1 1 + + 200 10 50
44
PRIMARY 5 (EM3)
TOPICS/OUTCOMES REMARKS
DECIMALS
2 c) 3 d) e) ADDITION AND SUBTRACTION add and subtract decimals up to 2 decimal places MULTIPLICATION AND DIVISION multiply decimals up to 2 decimal places by a whole number up to 2 digits divide decimals up to 2 decimal places by a 1-digit whole number
d) e)
• • • •
Include checking reasonableness of answers Include division of whole number by whole number with decimal answers Include rounding off answers to 2 decimal places Include checking reasonableness of answers
f) g)
carry out mental calculation within the multiplication tables multiply and divide decimals up to 2 decimal places by tens, hundreds and thousands CONVERSION BETWEEN DECIMALS AND FRACTIONS express a decimal as a fraction, and vice versa APPROXIMATION AND ESTIMATION round off decimals to • TheExclude cases where the first nonzero digit in the answer is at the 4th decimal place such as 0.12 ÷ 1000 = 0.00012
4 h) 5 i)
h)
• •
Include comparing a decimal and a fraction Include units of measure e.g. round off the answer correct to the nearest m, kg, etc Include checking reasonableness of answers
i)
j) 6 k)
j)
•
k)
• •
Include rounding off answers to a specified degree of accuracy Include checking reasonableness of answers
45
PRIMARY 5 (EM3)
TOPICS/OUTCOMES REM measure and draw angles in degrees using a protractor
a,b) •
Include use of the terms 'vertical' and 'horizontal'
c)
• • • •
d)
identify and name • angles on a straight line • angles at a point • vertically opposite angles recognise that angles on a straight line add up to 180º and angles at a point add up to 360º recognise that vertically opposite angles are equal find unknown angles involving • angles on a straight line • angles at a point • vertically opposite angles PROPERTIES OF A SQUARE AND A RECTANGLE state and use properties of • rectangle • square SYMMETRY recognise symmetric figures
d)
• • •
Include using notation such as ∠ABC and ∠x to name angles Include the term 'right angle' Include estimating the size of angles Exclude use of the terms 'acute', 'obtuse' and 'reflex' Exclude drawing reflex angles Exclude angles between parallel lines such as alternate angles, interior angles, corresponding angles Exclude use of the terms 'complementary' and 'supplementary'
e) f) g)
3 h)
h)
• •
Exclude the term 'diagonal' and its related properties Include finding angles and sides Include identifying and visualising symmetry in the environment or in designs Exclude rotational symmetry Exclude finding the number of lines of symmetry of a symmetric figure
4 i)
i)
• • •
j) k)
determine whether a straight line is a line of symmetry of a figure complete a symmetric figure with respect to a given line of symmetry
j)
46
PRIMARY 6 (EM3)
TOPICS/OUTCOMES
REMARKS
MONEY, MEASURES AND MENSURATION
1 a) UNITS OF MEASURE Pupils should be able to use units of time and volume time : hour, minute, second, day, week, month, year volume of liquid : litre, millilitre a) • Include visualising the relative magnitudes of standard units - hour and minute - minute and second - year and month - year and day - month and day - week and day - litre and millilitre Include the use of abbreviations: l, ml, h, min, s
• b) c) d) e) find the duration of time interval convert measurements of time and volume of liquid from a smaller unit of measure to a larger unit, and vice versa perform the 4 operations with units of time and volume of liquid solve up to 3-step word problems involving time and volume of liquid AREA AND PERIMETER calculate the area and perimeter of • a triangle • figure related to square, rectangle and triangle solve up to 3-step word problems involving area and perimeter of squares, rectangles and triangles
d) e)
• •
Exclude compound units Include problems involving concept of duration of time interval, e.g. find duration of time, find starting time or arrival time Include identifying the base and its corresponding height Exclude compound units Exclude compound units Exclude finding the base or height of a triangle given its area
2 f)
f)
• •
g)
g)
• •
47
PRIMARY 6 (EM3)
TOPICS/OUTCOMES REMARKS
MONEY, MEASURES AND MENSURATION
3 h) i) VOLUME OF CUBE / CUBOID AND LIQUID find volume of solids made up of unit cubes calculate the volume of a cube, a cuboid and liquid in cubic centimetres/cubic metres
i)
• • •
• • • • j) k) l) find one dimension of a cuboid given its volume and the other two dimensions find the edge of a cube given its volume solve up to 3-step word problems involving volume j) k) l) • • • •
Include use of abbreviations: cm3, m3, l, ml Include finding volume of liquid in a rectangular container Include recognising the following equivalents: 1l = 1000 cm3 1 cm3 = 1 ml Include conversions between l, ml and cm3 Exclude conversion between m3 and cm3 Exclude compound units Include visualising the relative size of 1 cubic metre and 1 cubic centimetre Exclude use of the term 'area of a face' Exclude use of ' 3 ' sign Include problems involving the height of water level in rectangular tank Exclude finding volume of solid by the volume of liquid displaced
TOPICS/OUTCOMES
REM48
PRIMARY 6 (EM3)
TOPICS/OUTCOMES
REMARKS
GEOMETRY
1 a) PROPERTIES OF TRIANGLES Pupils should be able to identify and name the following triangles: • isosceles • equilateral • right-angled recognise and use the property that the angle sum of a triangle is 180º state and find unknown angles involving the properties of • an isosceles triangle • an equilateral triangle • an right-angled triangle ANGLES IN GEOMETRIC FIGURES find unknown angles in geometric figures using the properties of • angles on a straight line • angles at a point • vertically opposite angles • a square, a rectangle and a triangle GEOMETRICAL CONSTRUCTION draw a square, a rectangle and a triangle from given dimensions a) • Exclude use of the terms 'scalene', 'acute' and 'obtuse'
b) c)
b)
•
Exclude problems where the skill of solving equations is required
2 d)
3 e)
e)
• • •
Include providing a sketch of the figure to be constructed Use ruler, protractor and set squares Exclude cases where compasses are required
TOPICS/OUTCOMES
REMARKS
AVERAGE AND RATE49
PRIMARY 6 (EM3)
TOPICS/OUTCOMES REMARKS
DIRECT PROPORTION
1 a) b) c) DIRECT PROPORTION Pupils should be able to recognise that two quantities are in direct proportion solve direct proportion problems using the unitary method solve up to 3-step word problems on direct proportion
TOPICS/OUTCOMES
REMARKS
PERCENTAGE
1 a) CONCEPT OF PERCENTAGE Pupils should be able to change fractions and decimals to percentages, and vice versa a) • • b) 2 c) d) 3 e) express a part of a whole as a percentage PERCENTAGE OF A QUANTITY calculate part of a whole given the percentage and the whole calculate discount given its percentage WORD PROBLEMS solve up to 3-step word problems involving percentage Include use of the percentage notation '%' Include recognising the equivalence between percentage and fraction/decimal
d)
• • • • • •
Exclude finding percentage discount
e)
Include problems involving discount Exclude use of the terms 'profit' and 'loss' Exclude finding the whole given a part and the percentage Exclude finding percentage increase/decrease and percentage profit/loss Exclude expressing one quantity as a percentage of another
50
APPENDIX A
Definitions of Suggested Thinking Skills
• Classifying Using relevant attributes to sort, organise and group information • Comparing Using common attributes to identify commonalities and discrepancies across numerous sets of information • Sequencing Placing items in a hierarchical order according to a quantifiable value • Analysing Parts and Whole Recognising and articulating the parts that together constitute a whole • Identifying Patterns & Relationships Recognising the specific variations between two or more attributes in a relationship that yields a reliable or repeated scheme • Induction Drawing a general conclusion from clues gathered (from specific to general) • Deduction Inferring various specific situations or examples from given generalisations (from general to specific) • Spatial Visualisation Visualising a situation or an object and mentally manipulating various alternatives for solving a problem related to situation or object without benefit of concrete manipulatives
51
APPENDIX B
Use of Abacus for Addition and Subtraction
1. Structure of the Abacus (a) An abacus is a calculating instrument. It consists of a number of counting beads that slide up and down the column rods. (b) A horizontal bar divides the abacus nto two parts. The upper part consists of 5-unit beads and the lower part consists of 1-unit beads. Each 5-unit beads has a 'face value' of 'five' and each 1-unit bead has a 'face value' of 'one'. (c) Each column has a place value 10 times the place value of the column on its right. 2. General Rule for Manipulating Beads (a) Always use the thumb to push up the lower beads. This is the only time the thumb is used. It never goes above the bar. (b) Always use the forefinger to push down the lower beads. (c) Always use the forefingers to push up the upper beads. (d) When displaying numbers 6 to 9, slide the 1-unit bead(s) with the thumb and the 5-unit bead with the forefinger at the same time. 3. Use of Abacus For Addition And Subtraction When using the abacus to add or subtract numbers, we may have to subtract while adding or add while subtracting. Therefore, in learning to use the abacus for addition and subtraction, both addition and subtraction should be learnt at the same time. There are altogether five basic skills and one further skill of manipulating the beads for addition and subtraction. They are: 1. Skill A This is the most fundamental skill of using the 1-unit beads, 5-unit bead or both together for addition and subtraction. There are 35 cases each. Skill B In adding or subtracting 1, 2, 3 or 4, when there are not enough 1-unit beads to use on the column, you have to make use of the 5-unit bead. There are 10 cases each for addition and subtraction. Skill C In adding or subtracting 5, 6, 7, 8, or 9, when there are not enough 1-unit beads and 5-unit bead to use on the column, you have to make use of beads on the left column. There are 25 cases each for addition and subtraction. Skill D In adding or subtracting 1, 2, 3 or 4, when there are not enough 1-unit beads and 5-unit bead to use on the column, you have to make use of beads on the left column. There are 10 cases each for addition and subtraction.
2.
3.
4.
52
5.
Skill E This is a combination of Skill B and Skill C. There are 10 cases each for addition and subtraction. Further Skill Addition and subtraction involving multiple steps. | 677.169 | 1 |
The main goal of this class is to present a collection of mathematical tools for both understanding and solving problems
infieldsthatmanipulatemodelsoftherealworld, suchasrobotics, artificialintelligence, vision, engineering, orseveral
aspects of the biological sciences. Several classes at most universities each cover some of the topics presented in this
class, and do so in much greater detail. If you want to understand the full details of any one of the topics in the
syllabus below, you should take one or more of these other classes instead. If you want to understand how these tools
are implemented numerically, you should take one of the classes in the scientific computing program, which again
cover these issues in much better detail. Finally, if you want to understand robotics, vision, or other applied fields, you
should take classes in these subjects, since this course is not on applications.
On the other hand, if you do plan to study robotics, vision, or other applied subjects in the future, and you regard
yourself as a user of the mathematical techniques outlined in the syllabus below, then you may benefit from this course.
Of the proofs, we will only see those that add understanding. Of the implementation aspects of algorithms that are
available in, say, Matlab or LApack, we will only see the parts that we need to understand when we use the code.
In brief, we will be able to cover more topics than other classes because we will be often (but not always) un-
concerned with rigorous proof or implementation issues. The emphasis will be on intuition and on practicality of the
various algorithms. For instance, why are singular values important, and how do they relate to eigenvalues? What are
the dangers of Newton-style minimization? How does a Kalman filter work, and why do PDEs lead to sparse linear
systems? In this spirit, for instance, we discuss Singular Value Decomposition and Schur decomposition both because
they never fail and because they clarify the structure of an algebraic or a differential linear problem. | 677.169 | 1 |
This is an idiosyncratic introduction to complex variables, and represents the author's take on the subject after a lifetime working in the field. The book was originally published in German as Funktionentheorie in 1950, and almost immediately translated into English and published by Chelsea in 1954. The book was written as a text (although without exercises), but today it works better as a monograph, where you can pick and choose the interesting topics.
The first volume works along a leisurely path from the very simple and general to the very specific. It starts with properties of the complex numbers, then of the complex plane and the Riemann sphere. Then it starts working on some simple transformations of the sphere, specifically inversions and Möbius transforms, which are covered in detail, then there's a little bit of topology and contour integration, and we finally start looking at analytic functions. Then there's a good bit about Carathéodory's theory of continuous convergence (an alternative approach to uniform convergence). Finally, about 3/4 of the way through the first volume, we get to power series and detailed studies of several special functions.
The second volume is more specialized and was likely intended to be a second course after the basics were covered. The first half of the book covers a number of ideas in geometric function theory. The second half covers some triumphs of circa-1900 function theory, such as Picard's theorem on exceptional values of entire functions (a non-constant entire function omits taking on at most one finite value).
The book has a strong geometric slant, and the topics (although not the treatment) overlap a lot of Ahlfors's Complex Analysis. It is not very similar in topics or treatment to Needham's very modern geometric book Visual Complex Analysis, which really is an introductory textbook. | 677.169 | 1 |
Grapevine, TX ACTAmy S.
In Algebra I, we search for the unknown. It?s an abstract discipline ? a process of logically analyzing patterns ? that builds our problem-solving muscles and lays the foundation for all subsequent mathematics courses. While Algebra I presents many challenges to the student, it doesn?t have to be a miserable experience. | 677.169 | 1 |
Mathematics for Economists
9780393957334
ISBN:
0393957330
Pub Date: 1994 Publisher: Norton, W. W. & Company, Inc.
Summary: An abundance of applications to current economic analysis, illustrativediagrams, thought-provoking exercises, careful proofs, and a flexibleorganization-these are the advantages that Mathematics for Economists brings to today's classroom.
Simon, Carl P. is the author of Mathematics for Economists, published 1994 under ISBN 9780393957334 and 0393957330. Five hundred four Mathematics for Economists textbooks a...re available for sale on ValoreBooks.com, ninety six used from the cheapest price of $39.90, or buy new starting at $156.20 | 677.169 | 1 |
In this first lesson of a series of advanced algebra lessons, we will learn about graphing functions. By working through the materials step-by-step and using sample questions for each section, you will be able to better understand college-level algebra. | 677.169 | 1 |
In this reader, students find out what grows in a garden and about animal life in gardens. Students count to find out how many there are and then compare the quantities with other things found in the garden. Students learn about the comparing terms: more, less, and equal. more...
The highly acclaimed MEI series of text books, supporting OCR's MEI Structured Mathematics specification, has been updated to match the requirements of the new specifications, for first teaching in 2004. This series, well-known for accessibility and for a student friendly approach, has a wealth of features: worked examples, activities, investigation,...MASTER MATH: ALGEBRA 2 is a clear, comprehensive guide to the concepts and problem-solving techniques taught in the typical high school Intermediate Algebra or Algebra 2 class. As Algebra 2 becomes a graduation requirement in more states, it's more important than ever for students to prepare themselves to handle this intermediate-level course work.... more... | 677.169 | 1 |
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