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About the Book: This text has been carefully designed for flexible use for First Semester M.C.A. course of Uttar Pradesh Technical University (U.P.T.U.), and it contains the following features: Precise mathematical language is used without excessive formalism and abstraction. Over 900 exercises (problem sets) in the text with many different types... more...
Presents methods for solving counting problems and other types of problems that involve discrete structures. This work illustrates the relationship of these structures to algebra, geometry, number theory and combinatorics. It addresses topics such as information and game theories. more...
Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools... more...
This book provides algorithms and ideas for computationalists. Subjects treated include low-level algorithms, bit wizardry, combinatorial generation, fast transforms like the Fourier transform, and fast arithmetic for both real numbers and finite fields. Various optimization techniques are described and the actual performance of many given implementations... more...
This book presents the latest findings on one of the most intensely investigated subjects in computational mathematics--the traveling salesman problem. It sounds simple enough: given a set of cities and the cost of travel between each pair of them, the problem challenges you to find the cheapest route by which to visit all the cities and return home... more... more... | 677.169 | 1 |
The notion of geometric combinatorics is quickly getting a much broader meaning. At present it covers not only a structure of polytopes and simplicial complexes but many further topics and interesting connections to other fields of mathematics. It is worth looking at the contents of this book, which contains written versions of the lecture series presented at a three-week program organised at the IAS/Park City Mathematics Institute in 2004. Counting of lattice points in polyhedra and connections to computational complexity is discussed in lectures by A. Barvinok. Root systems, generalised associahedra and combinatorics of clusters form the topic of the lectures by S. Fomin and N. Reading. Combinatorial problems inspired by topics from differential topology (Morse theory) and differential geometry (the Hopf conjecture) are studied by R. Forman. M. Haiman and A. Woo treat topics around Catalan numbers and Macdonald polynomials (the positivity conjecture). D. N. Kozlov discusses in his lectures chromatic numbers, morphism complexes and Stiefel-Whitney characteristic classes. Lectures by R. MacPherson cover topics such as equivariant homology, intersection homology, moment graphs and linear graphs and their cohomology. R. P. Stanley discusses topics connected with hyperplane arrangements and M. L. Wachs treats poset topology. The book ends with a contribution by G. M. Ziegler on convex polytopes. The book contains an enormous amount of interesting material (including a substantial numbers of exercises).
The aim of this book is to provide a rigorous foundation of the real number system. The first step is a treatment of natural numbers and their properties, which are stated as axioms (only the last chapter outlines the possibility of a construction of natural numbers on a set-theoretical basis). The real numbers are then defined as infinite sequences of decimal digits. At this point, it is possible to introduce the ordering of real numbers and prove the supremum property. The operations of addition and multiplication are first defined for numbers with finite decimal expansion (by shifting the decimal point, the problem is reduced to addition or multiplication of natural numbers); by means of a limit process, they are extended to all real numbers. The book also discusses additional interesting topics, in particular the definition of powers with real exponents, exponential and logarithm functions, Egyptian fractions, computer implementation of arithmetic operations and the uniqueness of real numbers up to isomorphism. The text is elementary and contains numerous remarks on the history of the subject.
The core of the book consists of contributions presented at the Abel bicentennial conference held at the University of Oslo, June 3-8, 2002, commemorating the 200th anniversary of Niels Henrik Abel's birth. The volume does not contain all the contributions of invited speakers at the conference and not all of the contributors attended the conference. The book contains the opening address of King Harald V and 25 papers devoted to various aspects of Abel's work. However, the reader can also find here papers treating topics that can be considered as mathematics of the next generation. This includes Manin's paper on applications of non-commutative geometry in Abelian class field theory for real quadratic fields, Fulton's paper on quantum cohomology of homogeneous varieties, Kassel's contribution on Hopf-Galois extensions to non-commutative algebras from the point of view of topology, van den Bergh's paper on non-commutative crepant resolutions of singularities and Chas and Sullivan's joint paper on closed string operators in topology leading to Lie bialgebras and higher string algebras. The other contributions describe many aspects of Abel's work as well as parts of number theory, analysis, algebra and geometry having roots in it. The book contains a huge amount of information of a historical and mathematical nature. It can be recommended not only to those working in fields having roots in one of Abel's versatile contributions to mathematics, but also to anybody who likes to read how ideas can influence future development.
This book consists of 33 essays trying to show to a nonmathematical community what mathematics and its applications really are, why they are so important and how they influence our day to day life. The essays may be read independently. Thanks to a long experience with mathematics as a researcher and teacher, the author provides many creative discussions and examples, varying from simple to more abstract structures of mathematics. He tries to provide a leitmotif to illustrate the relationship between mathematics and common sense. He writes about more than sixty major topics in mathematics, many of which have significant connections to other branches of knowledge (e.g. cosmology, physics, teaching, logic, philosophy, languages). The reader can find discussions on the nature of logic, numbers, counting and discounting, mathematical thinking, deductions, intuition and creativity, problem solving, conceptions of space, mathematical operations, structures, objects, paradoxes, theorems and proofs, as well as meditations on the influence of the media and wars on the development of mathematics and its position in the society. The author states and answers many interesting questions from many points of view. At the end of each essay the references to material that is both popular and professional are given. The book can be recommended to all who are interested in mathematics and its nature, beauty and role in modern society and science.
This small booklet brings the reader to a strange place called Numberland, where all the (integer) numbers live in a big hotel. An experienced reader soon recognizes that the hotel presented here is nothing other than a version of the famous Hilbert hotel, which was constructed as a tool to illustrate the problems in connection with countability. The small size of the book, the style of writing, a (large) number of illustrations and especially the "fairy-tale-like" language all indicates that the booklet is meant for children, probably around or under ten years of age.
And at this point the reviewer was beginning to get a little unsure as to whether this rather difficult piece of mathematics should be presented and explained to children of that age. Maybe, the children should first become reliably accustomed to the notions of "more than", "larger", and even "finitely and infinitely many", and only after that, at a proper age, should they be faced with facts like "there are as many odd integers as there are integers" or "there are as many integer fractions as integers themselves", which are the main "results" of the book. Also, since the concept of uncountability is not at all addressed (which is correct), the child reader can possibly be driven to the misleading realization that all infinite sets are "equally large". So there remains a small question if the topic is suitable for children of a "fairy-tale" age. However, if the answer to this question is "yes", then nothing can stop the reviewer from claiming that the booklet is written in a very nice way, presenting all the ideas clearly (at least for the adult reader) and in a concise yet comprehensive form. | 677.169 | 1 |
Triumphs--Foundations for Algebra 1
Math Triumphs is an intensive intervention resource for students who are two or more years below grade level. The series accompanies Glencoe "Algebra ...Show synopsisMath Triumphs is an intensive intervention resource for students who are two or more years below grade level. The series accompanies Glencoe "Algebra 1," "Geometry," and "Algebra 2" and provides step-by-step intervention, vocabulary support, and data-driven decision making to help students succeed in high school mathematics 9780078908460-4-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780078908460.
Description:In very good unmarked condition. Cover has small crease. Your...In very good unmarked condition. Cover has small crease. Your purchase benefits world-wide relief efforts of Mennonite Central Committee Multiple Copies. MI. 1C. | 677.169 | 1 |
He explains calculations effectively and includes detailed glossaries throughout the book. It is a practical manual for carpenters and other trade professionals to have on their book shelves. It is a great book to give to a young person enrolled in either high school or college carpentry programs or to share with your local building trades instructor. Mathematics instructors would do well to give it a read too because it may help them engage young people in their classrooms who are interested in construction. It is a good reference book to have in your office if you struggled with math and work in construction trades, the building supply industry or are a design professional.
The book could be improved with some photographs illustrating some of the math applications and we would encourage the author to think about adding some in subsequent printings of the book. We would encourage Mr. Williams to add more text about the value of these calculations. We bet he has some great stories to share about how he and others learned the importance of some of the mathematical concepts. Having said this, we believe the first edition of this book is valuable to read, share and use and we applaud his efforts to help young people master construction math.
Reviews:
5.0 out of 5 stars Very Useful, October 30, 2011
By
E. Stole "Me" (LV usa)
(REAL NAME)
This review is from: Applying Mathematics to Construction: Carpentry Mathematics & Estimating (Paperback)
I've read many construction books and math text books that do a pretty good job at enhancing my understanding of carpentry mathematics, but I must say this book tops them all. The methods used in the mathematical sense will blow your mind away. Lot's of times I found myself saying, "Wow I didn't even know it could be done this way" , or "That made it so much easier".
This book will definitely help you in many areas, even if you think you know it all, you don't. It takes a gifted mind to be able to come up with these formulas that no one else really uses. I recommend this book to anyone seeking to better themselves in estimating, framing, exterior AND interior finish and trim work. instead of beating yourself up trying to figure out the math part of a project, take a moment and go through this book and you will be amazed at how much easier things can be done.
"
About Kenneth
Hi my name is Kenneth Williams sr. I am a native New Orleanian who grew up in the Fisher Housing projects. I am the author of Applying Mathematics to Construction. I am a master instructor at Delgado Community College where I teach carpenrty... | 677.169 | 1 |
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Mathematical Proofs : A Transition to Advanced Mathematics
Mathematical Proofs : A Transition to Advanced Mathematics
Summary
Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise providing solid introductions to relations, functions, and cardinalities of sets. | 677.169 | 1 |
0387950605
9780387950600
Understanding Analysis (Undergraduate Texts in Mathematics):This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.
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Rent Understanding Analysis (Undergraduate Texts in Mathematics) 1st edition today, or search our site for Stephen textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. | 677.169 | 1 |
Photoshop
Adobe Photoshop is a graphics-editing software program, part of the Adobe Creative Suite. A variety of tools with multiple image-editing functions are the key elements of this software. These tools typically fall under the categories of drawing; painting; measuring and navigation; selection; typing; and retouching. Photoshop is a powerful tool which can be used to do much more than retouch photographs. The user has the ability to manipulate graphic images and type. Files can be saved in a wide variety of formats.
Algebra 1Algebra 2
Algebra 2 expands and builds on the concepts learned in Algebra 1. Algebra 1 is the foundation for all mathematics that comes after it. It is essential for students to master concepts, properties and rules, but more important for them to learn to think mathematically and think critically. I take a problem solving approach, teaching students how to ask questions. Learning how to ask the right questions is one of the best paths to becoming a successful student of mathematics.
Elementary Math
Elementary mathematics consists of mathematics topics frequently taught at the primary school level. The most basic topics in elementary mathematics are number sense, arithmetic and geometry. Students learn basic operations and are expected to memorize math facts such as common sums or products. Long division is learned during the primary years and an introduction to basic problem solving is also included. Elementary mathematics is used in everyday life in activities such as making change, cooking, and buying in a store. It is also an essential step on the path to understanding science. In elementary mathematics, the student begins to learn how to think mathematically and correctly interpret data.
English
Mastering the basics of using the Englishlanguage is essential for any student, or any speaker of another language who hopes to be gainfully employed.
Geometry
Geometry is the branch of mathematics that deals with shapes such as rectangles or circles (plane geometry) or 3-D solids in space (solid geometry) and their properties. It also includes learning deductive proof – which requires the student to be able to think linearly and employ supporting statements when justifying an argument. Example: demonstrating that two triangles are congruent (or similar) using given information, plus supporting postulates or theorems.
Grammar
Grammar is the structure and system of a language. In English, there are eight parts of speech (nouns, verbs, adjectives, adverbs, pronouns, prepositions, conjunctions and interjections). Grammar also includes what is called syntax, which is a set of rules related to the way words and phrases are arranged in a sentence, in order for it to make sense.
Macintosh
I am very experienced in use of a Macintoshcomputer, having used one for over 20 years. I am an expert in understanding and use of the operating system, OS X. I can effectively guide any new users, or users switching from Windows. I am also qualified in supporting users in the use of all built-in Apple applications (Mail, Contacts, Preview, Calendar, Dictionary, iPhoto, iTunes, System Preferences, etc.).
Microsoft Word
Microsoft Word is the most commonly used "word processor" software in the world. Word These include things like tables, columns, and inserting graphics. You can even use Word to do what is called a "Mail Merge" – personalizing form letters with each recipient's name, address and other information.
PrealgebraSAT Math
The SAT Math Level 1 test is a one-hour multiple choice test given on algebra, geometry, basic trigonometry, algebraic functions, elementary statistics and a few miscellaneous topics. A student chooses whether to take the test depending upon college entrance requirements for the schools in which the student is planning to apply. The SATMath Level 2 test covers more advanced content.
Study Skills
To be an effective student, it is important to have a complete set of tools in your student "toolbox." These tools include keeping your materials organized and always appearing in class with all the necessary requirements. Tools also include an organized system of study. This can differ greatly with the subject matter being studied. In all subjects, knowing key terms is essential. Paraphrasing key ideas is also a must. Studying should begin with getting a quick overview of the section at hand, then going back and reading the material section by section. Use of a highlighter or taking notes is helpful 09/10 | 677.169 | 1 |
About This Book
This book is
intended for all the mathematicians, engineers, and physicists who have
to know, or who want to know, more about the modern theory of
quaternions. Primarily, as the title page suggests, it is an exposition
of the quaternion and its primary application in a rotation operator.
In a parallel fashion, however, the conventional or more familiar matrix
rotation operator is also presented. This parallel presentation affords
the reader the opportunity for making comparative judgments about which
approach is preferable, for very specific applications.
This book readily divides into three major areas of concern:
The first three or four chapters present introductory material which
establishes terminology and notation to be used later on.
The mathematical properties of quaternions are then presented, including
quaternion algebra and geometry. This is followed by more advanced special
topics in spherical trigonometry, along with an introduction to quaternion
calculus and perturbation theory, required in certain situations involving
dynamics and kinematics.
Lastly, state-of-the-art applications are discussed. A six degree-of-freedom
electromagnetic position and orientation transducer is presented. With
this we end with a discussion of computer graphics, necessary for the
development of applications in Virtual Reality.
The writing of this book was early-on supported by the United States Air
Force, whose objective was to provide a primer on quaternions, suitable for
self-study. Our primary concern was that the book be written at a level such
that much of the subject matter would be accessible to those with a modest
background in mathematics. With this in mind, the quaternion is defined and
its algebra is introduced and developed. Several applications of the
quaternion, the quaternion rotation operator, and quaternion rotation
sequences are presented. A preview of the Table of Contents will provide the
reader with a measure of the intent and scope of this book. | 677.169 | 1 |
Wavelet Theory. An Elementary Approach with Applications
A self-contained, elementary introduction to wavelet theory and applications
Exploring the growing relevance of wavelets in the field of mathematics, Wavelet Theory: An Elementary Approach with Applications provides an introduction to the topic, detailing the fundamental concepts and presenting its major impacts in the world beyond academia. Drawing on concepts from calculus and linear algebra, this book helps readers sharpen their mathematical proof writing and reading skills through interesting, real-world applications.
The book begins with a brief introduction to the fundamentals of complex numbers and the space of square-integrable functions. Next, Fourier series and the Fourier transform are presented as tools for understanding wavelet analysis and the study of wavelets in the transform domain. Subsequent chapters provide a comprehensive treatment of various types of wavelets and their related concepts, such as Haar spaces, multiresolution analysis, Daubechies wavelets, and biorthogonal wavelets. In addition, the authors include two chapters that carefully detail the transition from wavelet theory to the discrete wavelet transformations. To illustrate the relevance
of wavelet theory in the digital age, the book includes two in-depth sections on current applications: the FBI Wavelet Scalar Quantization Standard and image segmentation.
In order to facilitate mastery of the content, the book features more than 400 exercises that range from theoretical to computational in nature and are structured in a multi-part format in order to assist readers with the correct proof or solution. These problems provide an opportunity for readers to further investigate various applications of wavelets. All problems are compatible with software packages and computer labs that are available on the book's related Web site, allowing readers to perform various imaging/audio tasks, explore computer wavelet transformations and their inverses, and visualize the applications discussed throughout the book.
Requiring only a prerequisite knowledge of linear algebra and calculus, Wavelet Theory is an excellent book for courses in mathematics, engineering, and physics at the upper-undergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level.
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READ MORE >
Preface.
Acknowledgments.
1 The Complex Plane and the Space L2(R).
1.1 Complex Numbers and Basic Operations.
Problems.
1.2 The Space L2(R).
Problems.
1.3 Inner Products.
Problems.
1.4 Bases and Projections.
Problems.
2 Fourier Series and Fourier Transformations.
2.1 Euler's Formula and the Complex Exponential Function.
Problems.
2.2 Fourier Series.
Problems.
2.3 The Fourier Transform.
Problems.
2.4 Convolution and B-Splines.
Problems.
3 Haar Spaces.
3.1 The Haar Space V0.
Problems.
3.2 The General Haar Space Vj.
Problems.
3.3 The Haar Wavelet Space W0.
Problems.
3.4 The General Haar Wavelet Space Wj.
Problems.
3.5 Decomposition and Reconstruction.
Problems.
3.6 Summary.
4 The Discrete Haar Wavelet Transform and Applications.
4.1 The One-Dimensional Transformation.
Problems.
4.2 The Two-Dimensional Transformation.
Problems.
4.3 Edge Detection and Naive Image Compression.
5 Multiresolution Analysis.
5.1 Multiresolution Analysis.
Problems.
5.2 The View from the Transform Domain.
Problems.
5.3 Examples of Multiresolution Analyses.
Problems.
5.4 Summary.
6 Daubechies Scaling Functions and Wavelets.
6.1 Constructing the Daubechies Scaling Functions.
Problems.
6.2 The Cascade Algorithm.
Problems.
6.3 Orthogonal Translates, Coding and Projections.
Problems.
7 The Discrete Daubechies Transformation and Applications.
7.1 The Discrete Daubechies Wavelet Transform.
Problems.
7.2 Projections and Signal and Image Compression.
Problems.
7.3 Naive Image Segmentation.
Problems.
8 Biorthogonal Scaling Functions and Wavelets.
8.1 A Biorthogonal Example and Duality.
Problems.
8.2 Biorthogonality Conditions for Symbols and Wavelet Spaces.
Problems.
8.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair.
Problems.
8.4 Decomposition and Reconstruction.
Problems.
8.5 The Discrete Biorthogonal Wavelet Transformation.
Problems.
8.6 Riesz Basis Theory.
Problems.
9 Wavelet Packets.
9.1 Constructing Wavelet Packet Functions.
Problems.
9.2 Wavelet Packet Spaces.
Problems.
9.3 The Discrete Packet Transform and Best Basis Algorithm.
Problems.
9.4 The FBI Fingerprint Compression Standard.
Appendix A: Huffman Coding.
Problems.
References.
Topic Index.
Author Index.
"The book, putting emphasize on an analytic facet of wavelets, can be seen as complementary. to the previous Patrick J. Van Fleet's book, DiscreteWavelet Transformations: An Elementary. Approach with Applications, focused on their algebraic properties." (Zentralblatt MATH, 2011)
"Requiring only a prerequisite knowledge of calculus and linear algebra, Wavelet theory is an excellent book for courses in mathematics, engineering, and physics at the upper-undergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level." (Mathematical Reviews, 2011) | 677.169 | 1 |
New to This Edition
Math and Drug Calculations Online for Nursing and Health Professions is a complete drug calculations online course that provides students an opportunity for application and practice. Since it is not tied to a specific text, this online course is suitable for all drug calculations classes. It incorporates the ratio and proportion, fractional equation, formula, and dimensional analysis methods and presents a step-by-step approach to the calculation and administration of drug dosages. Animations, voice-overs, and interactive self-assessment activities are used to provide an engaging and interactive course platform for students. This online course consists of three comprehensive modules; Module 1: Math Introduction and Overview, Module 2: Medication Administration, and Module 3: Medication Administration in Specialty Areas. Each module includes practice problems to promote active learning and quizzes that instructors can use to evaluate students' understanding of content presented in the course. A comprehensive test bank of approximately 300 questions is also provided for instructor's to build quizzes and test.
Key Features
Includes the four drug calculation methods (ratio and proportion, fractional equation, formula, and dimensional analysis) to expose the user to key calculation methods so they can apply the method which works best for them.
Modules are organized by topic sections that include an overview, objectives, what you need to know, example problems, practice problems, and one or more quizzes.
Follows abbreviation and dose designation recommendations from the Joint Commission on Accreditation of Healthcare Organizations (JCAHO) and the Institute for Safe Medication Practices (ISMP).
Many of the math practice problems include a tutorial button for each of the four drug calculation methods that provides a step-by-step tutorial to solving the problem in the chosen method.
Animations demonstrate various concepts related to drug calculation and administration, with some animations requiring user participation.
Interactive self-assessment activities are incorporated throughout the course to allow users to apply their knowledge in context.
Voice-overs enhance the step-by-step explanation of medication administration procedures and the drug calculation methods demonstrated throughout the course.
One or more Quizzes are included within each module to evaluate understanding of all the major topics covered in that particular module.
Provides the latest drug administration techniques and devices with detailed explanations of the various ways to administering drugs, including oral, intravenous, intramuscular, subcutaneous and other routes.
Includes the most up-to-date, commonly used drugs so users have exposure to what is being used in the "real world" of clinical practice.
Presents information on infusion pumps (enteral, single, multi-channel, PCA, and insulin) to help users understand their increased use in drug administration.
Related Links
Math and Drug Calculations Online for Nursing and Health Professions is a completely new drug calculations online course | 677.169 | 1 |
Pages
Saturday, August 31, 2013
Algebra 2 Interactive Notebook Pages for Unit 1
This year, I have resolved to do a much better job at the interactive notebook in Algebra 2 than last year. Last year, we had 12 students in my entire school who were enrolled in Algebra 2. This year, that number is just under 40. This is both exciting and kinda terrifying. First of all, it means that more students are prepared and willing to take Algebra 2. At the same time, my Algebra 2 students this year are greater in number and much more varied in level. This presents many challenges. But, these are challenges I am excited to attempt to meet.
Our first unit in Algebra 2 is an introduction to functions, function notation, domain and range, intercepts, maximums and minimums, intervals of increasing and decreasing, finding solutions, and transformations. My goal is to create a foundation which I can build off of once we start linear functions. I am also working hard to prove to my students that they are capable of doing Algebra 2 level work. Many of my students have extremely low confidence. We are also learning how to use the graphing calculator. This is the first experience any of my students have had with a graphing calculator, and I am working hard to make it a positive one.
So far, my Algebra 2 students are loving our interactive notebooks. They thank me on an (almost) daily basis for making Algebra 2 visual, fun, and easy. I have some students who are complaining right now that Algebra 2 is too easy. I told them that they just had to wait. Before they knew it, we would be exploring logarithms, exponentials, conic sections, and all kinds of other exciting mathematical relations. After going on and on about how excited I was about everything we were going to be learning and studying this year, one student asked, "Do you like math?" I was a bit taken aback by this question. Are there math teachers who don't like math? Of course, I like math. I love math! I eat, breath, and sleep math.
As usual, I have embedded the files for these foldables at the end of the post. If you have trouble viewing them, please make sure that you have Flash/Shockwave installed. If that does not correct the problem, please send me an e-mail and let me know what documents you are needing. I will be happy to send them to you!
My Algebra 2 Interactive Notebook
Algebra 2 Unit 1 Table of Contents (Thus Far)
I have already blogged about the NAGS foldable I had my Algebra 2 students create here.
NAGS Foldable - Outside
NAGS Foldable - Inside
NAGS Chart
I still haven't found a better way to practice differentiating between function/not a function than this card sort. I blogged about this last year.
Function / Not a Function Card Sort
We also created a Frayer Model for the word "function."
Function / Vertical Line Test Frayer Model
I stole this coordinate plane foldable from Ms. Haley and her wonder Journal Wizard blog! I think this is a big improvement over the coordinate plane foldable I did with my students last year. I created a template for this foldable which I have embedded below.
Parts of the Coordinate Plane Foldable
Parts of the Coordinate Plane Foldable
Parts of the Coordinate Plane Foldable
Our notes over independent and dependent variables were less than exciting. Maybe next year I will come up with a card sort or something. Hmm...
Independent and Dependent Variable Notes
Last year, my students had a TERRIBLE time remembering the difference between domain and range. This summer, at the amazing Common Core Training I received from the Oklahoma Geometry and Algebra Project (OGAP), I was introduced to an amazing resource--Shmoop. They have amazing commentary for each and every high school common core math standard! I learned about the DIXROY acronym from their commentary on F-IF.1.
I was able to re-use the domain/range notation foldable that I created last year for my Algebra 2 students. My students were VERY confused by the different notations. I haven't yet figured out a way to introduce these notations without overwhelming my students. They recovered, eventually.
Domain and Range Notation Foldable - Outside
Domain and Range Notation Foldable - Inside
I downloaded the domain and range cards from this blog post. There are 32 cards which give my students 32 opportunities to practice finding the domain and range!
Domain and Range Foldable
We made a tiny envelope to hold our 32 cards. Let me just say - having the students cut out all 32 cards took WAY too much time. I was about ready to pull my hair out. I think we might of spent half of a fifty minute class period just cutting these cards out. But, we used them a lot, so I think it was worth it.
I LOVED the envelope template that Kathryn (iisanumber.blogspot.com) posted earlier this summer. I downsized her template to the exact size needed to fit the domain and range cards I linked to earlier.
As you can see, this foldable perfectly holds the domain and range practice cards from our handy-dandy envelope!
The foldable is made to perfectly hold our domain and range practice cards that are housed in the envelope.
Students fold over the domain tabs to help them determine the left-most and right-most points on the graph. If the graph goes approaches negative or positive infinity, the students leave the flap open where it reads positive or negative infinity. I wanted my treatment of domain and range to be much more hands-on this year, and I think this foldable does the trick!
After doing many, many cards together, I had students find the domain and range of all 32 cards as homework. They had to write the domain and range in both interval and algebraic notation. (And, the discrete graphs had to have their domain written in set notation.) The next day, I gave them an answer key to use the check their work.
The Domain and Range Foldable in Action
One of the main thing my students need to be able to do on their Algebra 2 EOI is to describe graphs. This foldable is an attempt to introduce my students to the concepts of x-intercepts, y-intercepts, relative maximums, relative minimums, increasing intervals, decreasing intervals, roots, solutions, and zeros.
Describing Characteristics of Graphs Foldable - Outside
Because there is so much information on this one foldable, this was a perfect opportunity to use COLOR WITH A PURPOSE. Each term was marked with a different color. And the corresponding part of the graph was marked with the same color. This is one of my favorite foldables that we have done this year!
I've posted some close-ups of the flaps if you'd like to see what I had my students write.
Close-up of Right Flaps
Close-Up of Left Flaps
Last year, my Algebra 2 students really struggled with the concept of an inverse. So, this year, I decided to start talking about inverses very early in the school year. This will allow us to revisit the concept over and over as we explore different types of function in a much more in depth manner. By the time the EOI rolls around, my students should no longer be scared when they see the word inverse!
This foldable was inspired by @druinok's post from February.
Inverse of a Function Foldable - Outside
I want my students to be able to find the inverse if they are given a set of points, a graph, or an equation. Since we have only just started exploring functions in general, the examples we went through were quite simplistic. We will explore much more complicated inverses as the year progresses!
A lot of my students were terrified when I told them that we would be learning about inverses. By the end of the lesson, they were amazed that inverses were actually quite easy.
Inverse of a Function Foldable - Inside
Inverse of a Function - Important Fact!
I still have to figure out how I want to introduce transformations to my Algebra 2 students. That topic should end our first unit. Hmm...
Wow, wow, wow! That's an excellent collection of foldables. I love what you did with the domain and range foldable. I've had my kids use post-its before to mark off the lowest and highest values, but I love how the flaps show +/- infinity as well. My other favorite is the foldable that goes over all the characteristics of a graph. By the way, thanks for the shoutout on the envelope. Glad it could help!
I just have to write to tell you how amazing you are for sharing all this, and to thank you from the bottom of my heart. You are an excellent resource, and I almost feel as though I'm "stealing" these ideas from you, but the fact of the matter is that I'm teaching five preps (Geometry, Algebra II, PreCalc, AP Calc, and Business Math), and I just don't have enough time to devote to planning each lesson that I would like. I've been dying to try interactive notebooks with my Alg II students, and thanks to your insight, I'll be able to do it, at least part-time, this year. Thank you again and again!
To answer your question-- yes, there are math teacher who don't like math. I am one of them, LOL. I am a middle school special education teacher who is only teaching math this year, which is quite possibly my least favorite subject-- though my kids can relate to the fact that I struggled with math during middle school. :P While many of these foldables are beyond what I will cover with my students, you have given me a lot of great ideas. I will definitely be consulting your blog as a resource!. :)
I love, love, love your blog! I am new to Alg 2 and to interactive notebooks and you have such great ideas. I wish I would have remembered to come to your blog last week as it would have saved me some headaches. Out of curiosity, are you teaching from the Common Core State Standards?
Thank you! This year, I am not teaching directly to the Common Core State Standards. We have one year left of testing over our old standards, so I am teaching primarily to them. However, I attended an amazing Common Core training this summer, and I am trying to match my teaching to the CCSS whenever possible.
I absolutely love your blog! I have implemented an Interactive Notebook in all of my math classes this year and I love it so much. Your ideas, as well as many other bloggers, are so fantastic! I figured you might get a chuckle on some of the following student commentary this week:
-Ugh this class is like kindergarten. -But kindergarten was fun! -Stop complaining or Miss Nelson will take away the markers and crayons, guys! -Miss Nelson, I'm glad you don't make math hard. -Do we need markers AGAIN? -I don't want to take a boring math class next trimester. Can you teach {insert other math class here} next time? -I like coloring. -I don't like coloring. -When are we gonna be done with these foldy things? -Why do we have to glue something EVERY day, Miss Nelson? -Someone stole my notebook!!! -Wait... I found it. -This is weird. -Can you cut this for me, Miss Nelson? -Who wrote my name on my notebook?
Oh, the joys of interactive notebooks and math class. Thank you for providing me with an excellent resource for helping remedial students who are preparing to retake their state Algebra ECA in 2 months! Keep up the amazing work!
Thanks! I LOVE your students' comments. Your students sound a lot like mine! This week, I asked my students to draw a table in their interactive notebooks. I was looking for an x/y chart like the one I had already drawn on the Smart Board. As we started filling out the chart, one of my students became really confused. When I had asked my students to draw a table, she had taken this to mean a literal table... There is definitely never a dull moment when working with high school students!
I love your blog! There are so many great ideas. I am curious about how you go about making the foldables and visuals while lecturing. Do you lecture as you make them or make them first and talk about them second...or finally (haha) lecture then make the visuals to reinforce the concepts?
Hi Jill! I know this isn't exactly the answer you are looking for, but I do a combination of all the different ways you mentioned. Usually, I lecture as my students make them, but I've also done them both of the other ways. Sometimes, I even let my students vote on which way they would prefer me to do it.
If it's a topic that I believe my students might already be familiar with, I will probably lecture as we make the foldables. If it is a completely foreign topic to my students, we will usually make the foldable, talk about the material, and then reference the foldable as we work practice problems. This past week, I introduced my students to transformations of functions. We spent an entire class period exploring transformations with as little lecture as possible. Then, on the second day we created a foldable to summarize what we had discovered on the previous day.
I just want to say THANK YOU for your blog. It's been a few years since I took Algebra I & II in high school and now that I'm in college algebra it seems that I've forgotten much of what I learned. Your simple foldables (which I am a huge fan of) have put a lot of the ideas into an easy to understand format that just might help me get through this math class. THANK YOU again! | 677.169 | 1 |
ELCT 57
Tech Math Elec I
Course info & reviews
This course is designed to provide a basis for a clear mathematical understanding of the principles of DC electricity and electronics, and their analysis. Covered are algebra, equations, power of 10, units and dimensions, special products and factoring, algebraic fractions, fractional equations, graphs, simultaneous equations, determin... | 677.169 | 1 |
This course is designed to strengthen basic math skills. Topics include properties, rounding, estimating, comparing, converting, and computing whole numbers, fractions, and decimals. Upon completion, students should be able to perform basic computations and solve relevant mathematical problems. There is a $7.50 lab fee for this course.
MAT 060 ESSENTIAL MATHEMATICS 3 lecture 2 lab 4 credit
Prerequisite: MAT 050 or appropriate placement test score
Corequisite: None
This course is a comprehensive study of mathematical skills which should provide a strong mathematical foundation to pursue further study. Topics include principles and applications of decimals, fractions, percents, ratio and proportion, order of operations, geometry, measurement, and elements of algebra and statistics. Upon completion, students should be able to perform basic computations and solve relevant, multi-step mathematical problems using technology where appropriate. There is a $7.50 lab fee for this course.
MAT 070 INTRODUCTORY ALGEBRA 3 lecture 2 lab 4 credit
Prerequisite: MAT 060 or appropriate placement test score
Corequisite: ENG 085 or RED 080
This course establishes a foundation in algebraic concepts and problem solving. Topics include signed numbers, exponents, order of operations, simplifying expressions, solving linear equations and inequalities, graphing, formulas, polynomials, factoring, and elements of geometry. Upon completion, students should be able to apply the above concepts in problem solving using appropriate technology. This course is also available through the Virtual Learning Community (VLC). There is a $7.50 lab fee for this course.
MAT 080 INTERMEDIATE ALGEBRA 3 lecture 2 lab 4 credit
Prerequisite: MAT 070 or appropriate placement test score
Corequisite: ENG 085 or RED 080
This course continues the study of algebraic concepts with emphasis on applications. Topics include factoring; rational expressions; rational exponents; rational, radical, and quadratic equations; systems of equations; inequalities; graphing; functions; variations; complex numbers; and elements of geometry. Upon completion, students should be able to apply the above concepts in problem solving using appropriate technology. There is a $7.50 lab fee for this course.
MAT 090 ACCELERATED ALGEBRA 3 lecture 2 lab 4 credit
Prerequisite: MAT 060 or appropriate placement test score
Corequisite: ENG 085 or RED 080
This course covers algebraic concepts with emphasis on applications. Topics include those covered in MAT 070 and MAT 080. Upon completion, students should be able to apply algebraic concepts in problem solving using appropriate technology. There is a $7.50 lab fee for this course.
These courses are offered through the Math and Physics Department.
MAT 095 ALGEBRAIC CONCEPTS 3 lecture 3 credit
Prerequisites: None
Corequisites: None
This course covers algebraic concepts with an emphasis on applications. Topics include linear, quadratic, absolute value, rational and radical equations, sets, real and complex numbers, exponents, graphing, formulas, polynomials, systems of equations, inequalities, and functions. Upon completion, students should be able to apply the above topics in problem solving using appropriate technology. This course is designed for students attending East Carolina University and is only offered on the ECU campus.
This course is a comprehensive review of arithmetic with basic algebra designed to meet the needs of certificate and diploma programs. Topics include arithmetic and geometric skills used in measurement, ratio and proportion, exponents and roots, applications of percent, linear equations, formulas, and statistics. Upon completion, students should be able to solve practical problems in their specific areas of study. This course is only offered for diploma-level students. There is a $7.50 lab fee for this course.
This course provides an activity-based approach to utilizing, interpreting, and communicating data in a variety of measurement systems. Topics include accuracy, precision, conversion, and estimation within metric, apothecary, and avoirdupois systems; ratio and proportion; measures of central tendency and dispersion; and charting of data. Upon completion, students should be able to apply proper techniques to gathering, recording, manipulating, analyzing, and communicating data. There is a $7.50 lab fee for this course.
This course develops the ability to utilize mathematical skills and technology to solve problems at a level found in non-mathematics-intensive programs. Topics include applications to percent, ratio and proportion, formulas, statistics, function notation, linear functions, probability, sampling techniques, scatter plots, and modeling. Upon completion, students should be able to solve practical problems, reason and communicate with mathematics, and work confidently, collaboratively, and independently. This course is also available through the Virtual Learning Community (VLC). There is a $7.50 lab fee for this course.
This course introduces the concepts of plane trigonometry and geometry with emphasis on applications to problem solving. Topics include the basic definitions and properties of plane and solid geometry, area and volume, right triangle trigonometry, and oblique triangles. Upon completion, students should be able to solve applied problems both independently and collaboratively using technology. There is a $7.50 lab fee for this course.
This course provides an integrated approach to technology and the skills required to manipulate, display, and interpret mathematical functions and formulas used in problem solving. Topics include simplification, evaluation, and solving of algebraic and radical functions; complex numbers; right triangle trigonometry; systems of equations; and the use of technology. Upon completion, students should be able to demonstrate an understanding of the use of mathematics and technology to solve problems and analyze and communicate results. There is a $7.50 lab fee for this course.
MAT 122 ALGEBRA/TRIGONOMETRY II 2 lecture 2 lab 3 credit
Prerequisite: MAT 121, MAT 161, MAT 171, or MAT 175
Corequisite: None
This course extends the concepts covered in MAT 121 to include additional topics in algebra, function analysis, and trigonometry. Topics include exponential and logarithmic functions, translation and scaling of functions, Sine Law, Cosine Law, vectors, and statistics. Upon completion, students should be able to demonstrate an understanding of the use of technology to solve problems and to analyze and communicate results. There is a $7.50 lab fee for this course.
This course provides an introduction in a non-technical setting to selected topics in mathematics. Topics may include, but are not limited to, sets, logic, probability, statistics, matrices, mathematical systems, geometry, topology, mathematics of finance, and modeling. Upon completion, students should be able to understand a variety of mathematical applications, think logically, and be able to work collaboratively and independently. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
This course is an introduction to descriptive and inferential statistics. Topics include sampling, distributions, plotting data, central tendency, dispersion, Central Limits Theorem, confidence intervals, hypothesis testing, correlations, regressions, and multinomial experiments. Upon completion, students should be able to describe data and test inferences about populations using sample data. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
Enrollment in MAT 161 more than three times requires the written permission of the Math & Physics Department chair.
This course provides an integrated technological approach to algebraic topics used in problem solving. Emphasis is placed on equations and inequalities; polynomial, rational, exponential and logarithmic functions; and graphing and data analysis/modeling. Upon completion, students should be able to choose an appropriate model to fit a data set and use the model for analysis and prediction. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural science/mathematics.
MAT 162 COLLEGE TRIGONOMETRY 3 lecture 3 credit
Prerequisite: MAT 161
Corequisite: None
This course provides an integrated technological approach to trigonometry and its applications. Topics include trigonometric ratios, right triangles, oblique triangles, trigonometric functions, graphing, vectors, and complex numbers. Upon completion, students should be able to apply the above principles of trigonometry to problem solving and communication. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural science/mathematics.
MAT 175 PRECALCULUS 4 lecture 4 credit
Prereq: MAT 161
Corequisite: MAT 175A
This course provides an intense study of the topics which are fundamental to the study of calculus. Emphasis is placed on functions and their graphs with special attention to polynomial, rational, exponential, and logarithmic and trigonometric functions, and analytic trigonometry. Upon completion, students should be able to solve practical problems and use appropriate models for analysis and prediction. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
MAT 175A PRECALCULUS LAB 2 Lab 1 credit
Prerequisites: MAT 161
Corequisites: MAT 175
This course is a laboratory for MAT 175. Emphasis is placed on experiences that enhance the materials presented in the class. Upon completion, students should be able to solve problems, apply critical thinking, work in teams, and communicate effectively. This course has been approved to satisfy the Comprehensive Articulation Agreement pre-major and/or elective course requirement. There is a $7.50 lab fee for this course.
MAT 223 APPLIED CALCULUS 2 lecture 2 lab 3 credit
Prerequisite: MAT 122
Corequisite: None
This course provides an introduction to the calculus concepts of differentiation and integration by way of application and is designed for engineering technology students. Topics include limits, slope, derivatives, related rates, areas, integrals, and applications. Upon completion, students should be able to demonstrate an understanding of the use of calculus and technology to solve problems and to analyze and communicate results. There is a $7.50 lab fee for this course.
MAT 263 BRIEF CALCULUS 3 lecture 3 credit
Prerequisite: MAT 161, MAT 171, or MAT 175
Corequisite: None
This course introduces concepts of differentiation and integration and their applications to solving problems; the course is designed for students needing one semester of calculus. Topics include functions, graphing, differentiation, and integration with emphasis on applications drawn from business, economics, and biological and behavioral sciences. Upon completion, students should be able to demonstrate an understanding of the use of basic calculus and technology to solve problems and to analyze and communicate results. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics.
MAT 271 CALCULUS I 3 lecture 2 lab 4 credit
Prerequisite MAT 172 or MAT 175 with a grade of C or better
Corequisites: None
This course covers in depth the differential calculus portion of a three-course calculus sequence. Topics include limits, continuity, derivatives, and integrals of algebraic and transcendental functions of one variable, with applications. Upon completion, students should be able to apply differentiation and integration techniques to algebraic and transcendental functions. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics. There is a $7.50 lab fee for this course.
MAT 272 CALCULUS II 3 lecture 2 lab 4 credit
Prerequisites: MAT 271
Corequisites: None
This course provides a rigorous treatment of integration and is the second calculus course in a three-course sequence. Topics include applications of definite integrals, techniques of integration, indeterminate forms, improper integrals, infinite series, conic sections, parametric equations, polar coordinates, and differential equations. Upon completion, students should be able to use integration and approximation techniques to solve application problems. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics. There is a $7.50 lab fee for this course.
MAT 273 CALCULUS III 3 lecture 2 lab 4 credit
Prerequisites: MAT 272
Corequisites: None
This course covers the calculus of several variables and is third calculus course in a three-course sequence. Topics include functions of several variables, partial derivatives, multiple integrals, solid analytical geometry, vector-valued functions, and line and surface integrals. Upon completion, students should be able to solve problems involving vectors and functions of several variables. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics. There is a $7.50 lab fee for this course.
MAT 280 LINEAR ALGEBRA 3 lecture 3 credit
Prerequisites: MAT 271
Corequisites: None
This course provides a study of linear algebra topics with emphasis on the development of both abstract concepts and applications. Topics include vectors, systems of equations, matrices, determinants, vector spaces, linear transformations in two or three dimensions, eigenvectors, eigenvalues, diagonalization and orthogonality. Upon completion, students should be able to demonstrate both an understanding of the theoretical concepts and appropriate use of linear algebra models to solve application problems. This course has been approved to satisfy the Comprehensive Articulation Agreement pre-major and/or elective course requirement.
MAT 285 DIFFERENTIAL EQUATIONS 3 lecture 3 credit
Prerequisites: MAT 272
Corequisites: None
This course provides an introduction to ordinary differential equations with an emphasis on applications. Topics include first-order, linear higher-order, and systems of differential equations; numerical methods; series solutions; eigen values and eigen vectors; Laplace transforms; and Fourier series. Upon completion, students should be able to use differential equations to model physical phenomena, solve the equations, and use the solutions to analyze the phenomena. This course has been approved to satisfy the Comprehensive Articulation Agreement pre-major and/or elective course requirement. | 677.169 | 1 |
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Problems in Geometry at Putnam County High School
by
Debra Newsome
In response to calls for reform, in both the teaching and the learning
of school mathematics, the National Council of Teachers of Mathematics has
taken a major stand on the content and emphasis of the mathematics curriculum.
In producing its Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989) and other documents, professional educators have assumed leadership
roles in two critical areas: (1) the creation of a vision of mathematics
in an increasingly technological society with a diverse variety of needs
and requirements and (2) the design of a set of standards to guide curriculum
revision within this vision. These goals as well as the broader goals of
getting students to value math, gain confidence in their own ability, and
become comfortable as mathematical thinkers place the teacher in the role
as a coach and catalyst for knowledge acquisition.
The state of Georgia has taken a close look at the NCTM Standards in producing
its Quality Core Curriculum (QCC). Indeed, the Geometry or Informal Geometry
guide states:
Geometry provides students with a way to link their
perceptions of the real world with the mathematics that
allows them to solve a variety of problems they will
encounter not only in other disciplines but also in their
lives....Geometry should provide students with visual and
concrete representations that help them gain insight into
important areas of mathematics and their applications to
the real world...High school geometry must extend beyond
the traditional treatment of geometry as a deductive
system and provide students with a broad view of
geometry and its applications...(p. 1).
This small sample of statements from the introduction seeks to link a Georgia
high school course in geometry directly to Standards 1, 2, 3, 7, 8, and
14, and indirectly to the remaining Standards. Despite these affirmations,
the Putnam County High School geometry courses for college-bound students
and the geometry topics for general or vocational students have continued
to be taught predominantly through textbooks and strongly teacher-centered
instruction, especially for lower-achieving students. Manipulatives have
been limited to 3-dimensional models of geometric solids to be held up in
the front of the classroom. Tools have been limited to compass and straight
edge and then used exclusively to complete paper-and-pencil constructions
for college-bound students. Instructors outside the field have rarely been
sought and interdisciplinary lessons have been severely limited by the knowledge
base of individual teachers.
The role of the teacher as a facilitator rather than as a dictator is perhaps
the greatest distinction between the "traditional" mathematics
classrooms at PCHS and those envisioned by the writers of the standards.
Too many secondary teachers are uncomfortable granting students permission
to move around the room, not to mention allowing interaction with their
classmates! Students are complacently resigned to assume the passive role
of being "talked at" by the teacher. Many adults still view mathematics
classrooms as being the same as they were during personal educational experiences.
Students must now accept a far greater and active role in their own education
to receive far greater rewards. Through group and individual projects, assignments,
and the like, the student is able to explore and see the integration of
mathematical topics that emphasize the body of mathematics as a whole "greater
than the sum of its parts."
The current role of the teacher in many classrooms at PCHS is that of an
information-giver, despite personal desires to create active rather than
passive learners. Teachers need to accept new roles as leaders and inventors
and be facilitated in these roles, but time is lacking to share with and
support other teachers. The mathematics and science classrooms at PCHS are
located across the hall from each other and opportunities to collaborate
have been initiated during the past school year and should continue to be
actively sought. Interdisciplinary teams are nonexistent in most high schools
although the practice of common planning is now part of the middle school
concept. Many primary teachers enhance learning of specific content with
instructional units, integrating a variety of subjects, as opposed to the
isolated content areas in the high school curriculum. Students should be
accustomed to the integration of topics based on experiences at the primary
and middle school levels.
Although publishers have sought to meet the varied mathematical needs of
teachers and students, textbooks can not continue as the main source of
geometry instruction. It is not in the spirit of the Standards nor the interest
of the students to approach applications as side bars or "extra"
exercises. Emphasis on pencil and paper exercises reinforced by correct
answers will not facilitate full appreciation of a subject so visual in
content and scope, yet this has been the nature of the resources used by
mathematics teachers at Putnam County High School. Despite the adoption
of new textbooks which include a variety of supplemental materials, the
focus remains on the textbook as a primary source of instruction. The complaint
raised continues to be lack of time to explore a variety of presentation
methods and materials prior to classroom implementation. Students must be
shown that there is more to geometry than those topics which appear in a
textbook but this engagement can occur only when teachers accept the responsibility
to facilitate rather than dictate learning.
One reason for the existence of a narrow approach to the teaching of high
school geometry is tradition. Teachers tend to adopt teaching styles and
methods most like those to which they themselves were exposed. Many people
hold the childhood views of school as authoritarian institutions in which
somebody smart stands in front of a room and tries to pass information on
to large groups of students. The content of textbooks, too, is often a reproduction
of those used by previous generations despite the current penchant for visual
displays. Emphases in content topics may change, but the typical classroom
activities carry on as they always have. Society must realize that not only
can students (and teachers) learn from each other, but also from manipulatives,
other tools and technologies, a spectrum of qualified personnel, and a wide
variety of situations.
Tradition breeds the idea that schools should operate on the assumptions
of the past and therefore schools are having a difficult time adapting to
the ever-changing needs of the present. Even with the large volume of, the
retrieval speed of, and the variety of sources of information available
today, our school finds it difficult to stay abreast of current trends.
Rural schools, like Putnam County High, are especially at risk for lagging
behind, since they are often further removed by geography from major business
and industry influences. Information as to the current and future needs
of employers and post-secondary institutions may be disseminated by informal
or other means which may be unavailable or inaccessible to rural districts.
Smaller population centers lack the financial and human resources that are
taken for granted in urban and surrounding suburban centers. The low population
density in rural areas limits the attention given to education by the media
and hence even fewer resources may be identified or accessed (Bracey, 1992).
Availability of financial resources continues to be a major player in the
determination of curriculum and its implementation. The diversity in size
and resources impinges negatively on rural communities in the applications
of geometry to which students are exposed. Smaller systems or those less
financially able, have had to be content with less technology, specifically
computers, software, and multimedia applications. Rural schools face strong
competition for resources from suburban schools whose populations include
a larger tax base to provide money for materials, experimental and/or innovative
programs, and instructors of the highest caliber. Furthermore, a wider variety
of business and industry personnel in non rural areas provides resourceful
contacts that can increase motivation for students to learn geometry, especially
if knowledge acquired can be applied in a local job or is directly related
to a field chosen for further study.
The world apart from school depends on the successful conception, implementation,
and completion of projects that involve the cooperation of many individuals.
Students are too often left to forage through the same old curriculum, in
the same old manner, with the same old results, namely poor student achievement,
motivation, and inspiration. An education system suffused with individual
and group projects, particularly apprenticeships and hands-on experiences
can fill the void in genuine student understanding. Assessment, although
often looked upon with disfavor by students, must become part of the learning
process rather than the objective. Alternative assessment advocates maintain
that the proof of a person's capacity is found in their ability to perform
or produce, not in their ability to answer on cue. Students value the opportunity
to discover for themselves what they have mastered, without the need for
teacher approval. Mathematics teachers at PCHS are seeking new ways to assess
students, but these processes require further study since no department
member is experienced in alternate assessment methods.
Geometry offers a wonderful opportunity for students to expand their knowledge,
gain confidence, and explore new interests. Projects completed independently
of the classroom or cooperatively within the classroom have allowed students
at PCHS to explore new areas of geometric applications while pursuing their
own interests. Art, architecture, construction, design, drafting, forestry,
history, map-making, photography, research, teaching, and many other areas
all provide opportunities for students to use geometry in a field of interest.
My students have explored a variety of topics including tessellations, scale
models, art, blueprints, fractals, geometry in the workplace, and bridges.
These projects, when presented to classmates, offered other opportunities
for personal growth and inspiration for peers. As their teacher, it has
been a joy to experience the interest of this aspect of my students' geometric
experience.
Teachers, too, can benefit from the experiences of their peers and expand
their knowledge base to provide interesting activities for their students.
Multi-disciplinary committees can be formed to plan for units that would
reinforce concepts in all areas. Math conference participants have shared
a variety of projects and plans for implementing programs designed to provide
innovative opportunities for students to connect and integrate concepts
and activities within the disciplines of mathematics, science, and technology
education. If business personnel can be persuaded to provide input, perhaps
offering real problems occurring on the job, students may experience the
power of geometry through sources outside of the classroom. Study for an
advanced degree has offered opportunities to extend and explore standard
topics within the framework of technological innovation. The door that has
been opened by this knowledge will benefit teachers and students at PCHS
for years to come.
With graphing calculators and computer software, students can now experiment
with parameter changes that previously had to be time-consumingly drawn
by hand and visualize how the graphs are effected without the tedium of
re-drawing. Many maintain that these visualizations allow students to make
abstract connections, yet students are only now beginning to experience
technology at Putnam County High School. Graphing and computer technologies
which have been available at other schools throughout the state have only
now been made available to students at PCHS, and presently only on a severely
limited basis. As recently as the past school year, students have been graphing
exclusively using paper-and-pencil, which has had obvious limitations to
the depth of mathematical experiences. Technology is quite expensive and
financial considerations have often been given as justification for lack
of spending in Putnam and other rural counties. Extra effort must be made,
however, to supplement direct expenditures on technology with alternate
sources of materials acquisition such as aid-in-kind. Putnam County leaders
must overcome their aversion to seeking financial sources outside of the
local budget. Grant money may be sought, business leaders may donate or
lend materials and/or speakers, or appeals to the community may result in
volunteers who can share their real-world uses of geometry with students.
Despite statements in curriculum guides or mandates from administration
as to the content to be taught in geometry, teachers make the ultimate decisions
as to the depth and scope of these objectives and how they will be carried
out in the classroom. It is evident that student-centered activities must
come from teacher leadership positions. Moves to broaden the approach to
teaching geometry will not occur without a direct effort by teachers in
the classroom to refocus the direction and methods of instruction. Most
teachers in Putnam County have not been trained in the use nor the application
of technology toward teaching methodology in the classroom. This training
process, though slow to be implemented, will surely benefit all parties.
Many sources offer suggestions on how to make mathematics more interesting.
The current high school student seems to be under the influences of "entertainment".
Leisure and fun are pervasive in their lives in the form of fast-paced computer
and video technology and students seem to expect education to come in the
form of entertainment. Many books activities have been written to make mathematics
fun. The "hook" of fun is designed to increase success and encourage
the further study of mathematics. Activities can be designed that are fun
but also require application of concepts learned in the classroom. For example,
rope and chalk can be used outside to build a hopscotch or a basketball
court after a unit in geometry on compass and straight edge constructions.
Origami activities produce objects of beauty and geometric significance.
Tessellation exploration and creation can be as practical as quilting or
as creative as design opportunities.
The use of manipulatives is recommended by professional mathematics teachers
and their associations (NCTM, 1989). Teachers are inundated with catalogs
offering manipulative merchandise for sale, but these accouterments cost
money. Creative teachers and students, however, can generate many items
with available scrap materials, donations, and redesign of projects. For
example, a hypsometer can be used to measure the angle of elevation or depression
but a crude representation can be constructed of tag board or adapted from
a protractor and a paper clip. Low budget activities such as paper folding,
geoboard lessons, tangrams, etc. are often readily shared among teachers
at professional conferences or work sites.
Practice lab experiences, although normally confined to science classrooms,
can provide opportunities for students to brainstorm problem solving techniques
and explore results. For example, Hunt (1978) describes three methods that
are commonly used to determine the height of an object. Working cooperatively,
students are likely to determine these and other methods through discovery
or research. The same problem can be solved in geometry after a unit on
basic trigonometric functions, employing a hypsometer or its equivalent.
Labs can be involved and require additional resources or as basic as paper
folding explorations. Explorations using The Geometer's Sketchpad await
the geometry students at PCHS for this school year. NCTM yearbooks and addenda
offer other resources for exploration. Materials included with adopted textbooks
also offer suggestions for lab activities and these may be fully accessible
to student groups.
Opportunities for changing the traditional narrow approach to high school
geometry exist. Students, teachers, business and community leaders, and
parents can exert a positive influence over the materials and methods used
in the classroom to provide innovative and interesting explorations in geometry.
Every successful innovation has the power to spark the imagination of a
mind.
Bibliography
Bracey, G. W. The second Bracey report on the condition of public education.
Phi Delta Kappan, 74, 104-117,1992. | 677.169 | 1 |
Introduction to Algebra Author(s): No creator setIntroduction to Fractions Lesson on fractions. Fractions are introduced and the basics are taught in this video. Examples problems are shown, solved, and explained with picture representations. More lesson can be found at | 677.169 | 1 |
Professional Commentary: This three-part activity illustrates the use of iteration, recursion, and algebra to model and analyze the changing amount of medicine in an athlete's body. The activity is adapted from High School Mathematics at Work, a publication from the National Research Council....
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Professional Commentary: Graphing calculators are sophisticated devices that can run small computer programs and draw the graph represented by complex equations in an instant. In the last few years, they have become mandatory in many high school mathematics classes and can be used on the SAT and advanced placement exams and other standardized tests....
Professional Commentary: How does one go about finding the volume of an irregular shape? Students get a preview of integral calculus as they compute volume of a solid as the sum of volumes of slices of the solid....
Professional Commentary: The purpose of this activity is to discover relationships among the volumes of simple solids such as cones, cylinders, pyramids, cubes, and spheres. A discussion of the underlying mathematical ideas is included.... | 677.169 | 1 |
Outlines and Highlights for Excursions in Modern Mathematics by Peter Tannenbaum, Isbn : 9780321568038
Student Resource Guide To Accompany Excursions In Modern Math
Videos on DVD with Optional Subtitles for Excursions in Modern Mathematics
Summary
Student Resource Guidecontains full worked out solutions to odd-numbered exercises from the text, "selected hints" that point the reader in#xA0;one of many#xA0;directions leading to a solution and keys to student success including lists of skills#xA0;that will#xA0;help prepare for chapter exams. | 677.169 | 1 |
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9780618627196
ISBN:
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Edition: 1 Pub Date: 2006 Publisher: Houghton Mifflin College Div
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Mathematics and Quantitative Literacy (3 Crs.)
Students will have the "ability to understand numerical data and use mathematical methods for analysis and problem solving" (PASSHE BOG Policy 1993-01). Mathematics is the science of numbers and their operations, interrelations, combinations, generalizations and abstractions and of space configurations and their structure, measurement, transformations and generalizations. | 677.169 | 1 |
5069657 / ISBN-13: 9780205069651
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This text provides a simple account of classical number theory, as well as some of the historical background in which the subject evolved. It is ...Show synopsisThis text provides a simple account of classical number theory, as well as some of the historical background in which the subject evolved. It is intended for use in a one-semester, undergraduate number theory course taken primarily by mathematics majors and students preparing to be secondary school teachers. Although the text was written with this readership in mind, very few formal prerequisites are required. Much of the text can be read by students with a sound background in high school mathematics | 677.169 | 1 |
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Summary: The only book that introduces differential geometry through a combination of an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach. Starting with basic geometric ideas, Differential Geometry uses basic intuitive geometry as a starting point to make the material more accessible and the formalism more meaningful. The book presents topics through problems to provide re...show moreaders with a deeper understanding. And, it introduces hyperbolic geometry in the first chapter rather than in a closing chapter as in other books. An important reference and resource book for any reader who needs to understand the foundations of differential geometry3675 +$3.99 s/h
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By Dan Kalman
Uncommon Mathematical Excursions is for anyone who knows and appreciates the basics of algebra, geometry, and calculus, and would like to learn more. It offers teachers an opportunity to deepen their knowledge of mathematics, not by exploring the far reaches of the subject, but by returning to the core. More generally, for the old hand and new devotee alike, this book will surprise, intrigue, and delight readers with unexpected aspects of old and familiar subjects. The book is particularly recommended for professional development and continuing education of secondary and college mathematics teachers.
Table of Contents
Preface
Acknowledgements
I. The Province of Polynomia
Horners Form
Polynomial Potpourri
Polynomial Roots and Coefficients
Solving Polynomial Equations
II. Maxiministan
Leveling with Lagrange: Constrained Maxima and Minima with Lagrangian Functions
A Maxmini Miscellany
Envelopes and the Ladder Problem
Deflection on an Ellipse
III. The Calculusian Republic
A Generalized Logarithm for Exponential-Linear Equations
Envelopes and Asymptotes
Derivatives Without Limits
Two Calculusian Miracles
References
Index
About the Author
About the Author
Dan Kalman has been writing about and teaching mathematics for 30 years. A graduate of Harvey Mudd College (BS, 1974) and the University of Wisconsin (PhD, 1980) he is a Professor of Mathematics at American University, Washington, DC. He previously held faculty positions at the University of Wisconsin, Green Bay, and Augustana College, Sioux Falls, among other institutions, and worked for several years as an applied mathematician at the Aerospace Corporation. He also served for one year as an Associate Executive Director of the MAA. Kalman's mathematical writing has been recognized with multiple MAA awards: Allendoerfer Awards in 1998 and 2002, Pólya Awards in 1994 and 2002, and an Evans Award in 1997. He is the author of one previous book, Elementary Mathematical Models, published by the MAA in 1997. Kalman has served on the Editorial Boards for several MAA publications, including Mathematics Magazine, MAA FOCUS, Math Horizons, and the Spectrum and Classroom Resource Materials book series. | 677.169 | 1 |
InterpolIn the mathematical subfield of numerical analysis, interpolation is a procedure that assists in "reading between the lines" in a set of tables by constructing new data points from existing points. This rigorous presentation includes such topics as displacement symbols and differences, divided differences, formulas of interpolation, much more. 1950 edition. | 677.169 | 1 |
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066947 to Go: A Mathematics Handbook (Math Handbooks)
Algebra To Go is a unique new handbook designed to help demystify algebra for students. Modeled after the Math On Call handbook, Algebra To Go is a student-friendly resource that covers key and often complex math topics in a way that's clear and easily understandable for students - from numeration and number theory to estimation, linear and non-linear equations, geometry, and data | 677.169 | 1 |
Acceleration is both a buzz-word and a set of solutions in developmental mathematics. In a basic way, the New Life model is based on acceleration to college mathematics for most of our students. The courses in the New Life model — Mathematical Literacy and Algebraic Literacy — are being well received; dozens of colleges have implemented one or both courses.
However, we are resisting a simple change that promises significant improvement at little risk — eliminating any college course prior to the level of beginning algebra or mathematical literacy. I'm talking about courses called pre-algebra, basic math, and or arithmetic. I believe that these courses have insignificant benefits while presenting risks to students.
The vast majority of these courses focus on procedural skills in a few content domains (decimals, fractions, percents, very basic geometry, and perhaps extremely limited algebraic skills). Historically, these courses are a relatively recent development from a remedial point of view:
The myth that we must fill all student deficiencies before they can take a college-level math course.
We all have deficiencies; human beings have a capacity to function in spite of them. We tend to accept without question the surface logic that says a student needs to master arithmetic before they can master algebra. [The New Life courses do not de-emphasize algebra; our focus is on diverse mathematics and understanding, including algebra.] A course like beginning algebra or Math Lit continues to be one of the key gatekeepers to college success.
At the global level, I have never seen any study reporting a large correlation between pre-algebra (or arithmetic) skills and success in beginning algebra; sure, there are a few studies (including my own) that show a significant correlation … due primarily to large sample sizes. Significance does not show a meaningful relationship in all cases. A correlation of 0.2 to 0.3 is only connected with 5% to 10% of the variation in outcomes; other student factors (like high school GPA) have larger correlations.
At the micro level, we often justify a pre-algebra course by justifying the components. Fractions are needed before algebra, because the algebra course covers rational expressions. Other content areas have similar rationales. This justification has two major problems:
The need in the target course is artificially imposed in many cases ('needed for calculus, so we do this in beginning algebra'). [This is a pre-calculus course has the responsibility for this need.]
The pre-algebra content is almost always a procedurally bound, right answer obsessed quick tour with no known transfer to an algebraic setting.
When the New Life model was developed, we did not assume any particular content connections. We looked at the content of Mathematical Literacy, and determined that nature of the knowledge needed before students would have a reasonable chance of success. The list of prerequisites to Math Lit is quite short:
Understand various meanings for basic operations, including relating each to diverse contextual situations
Use arithmetic operations to solve stated problems (with and without the aid of technology)
Order real numbers across types (decimal, fractional, and percent), including correct placement on a number line
Use number sense and estimation to determine the reasonableness of an answer
Apply understandings of signed-numbers (integers in particular)
For the vast majority of students, any gaps in these areas can be handled by just-in-time remediation. This list certainly does not justify a prerequisite course. A similar analysis from a beginning algebra reference would yield a similar list, I believe.
In spite of what we know, we continue to offer courses before beginning algebra or Math Lit, and continue to require students to pass them before progressing in the sequence.
This has been a long-debated topic in AMATYC — why does an arithmetic-based course need to be a prerequisite to algebra? Essentially, I think this is our problem — these courses are security blankets for us. We feel like we are doing the safe thing and helping our students by giving them this 'chance to be successful'; we believe that these courses offer real benefits for students, even though the data is pretty clear that they do not (in general).
It is uncomfortable, perhaps even scary, for us to consider the possibility that all students be placed into beginning algebra or Mathematical Literacy. We worry about the risk. We seem unconvinced that another math course in a sequence is creating known risks and problems for our students.
We can easily see the problem by a simulation. Let's assume that 70% of the students pass pre-algebra, that 80% of those continue to beginning algebra (or Math Lit), and 60% of these pass.
Enter pre-algebra, pass beginning algebra … about 34%
Compare this to these same students starting out in beginning algebra. There is no sequence; the percent who pass beginning algebra is simply the pass rate for a group with somewhat higher risk.
Skip pre-algebra, pass beginning algebra … about 40% to 50%
The real world is not as rosy as the first scenario. At my college, less than 50% of our pre-algebra passers complete beginning algebra (and a fourth of these barely pass, having little chance at the next level).
We should be very upset by the situation. Few researchers talk about this, but we know. Pre-algebra (and arithmetic) courses tend to have a higher (sometimes much higher) proportion of minority students, as well as people with employment and economic problems. Community colleges are supposed to be about upward mobility; instead, we've created a system which has been shown to keep certain groups from advancing.
Let go of that security blanket called pre-algebra (or arithmetic). Take the very small risk of helping a lot more students get though their mathematics and their program. Completion leads to economic opportunity. Let's get out of the way, as much as possible!
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The AMATYC "New Life Project" has a curricular vision that includes two new courses that can replace the traditional developmental mathematics courses. The first of the new courses is Mathematical Literacy for College Students (often shortened to just Mathematical Literacy, or even MLCS); this course shares content with the Carnegie Foundation Pathways (Quantway™ and Statway™) and with the Dana Center (Foundations of Mathematical Reasoning).
With the New Life Project, our work is based on faculty making choices and working with publishers to develop materials. Currently, two commercial texts are available for MLCS (either published or soon-to-be published); some faculty have also custom published their own materials, or adapted existing materials. The initial MLCS pilots started about two years ago.
As of fall 2013, here is a summary of the known MLCS course implementations (number of sections):
State
count
AL
1
AZ
2
CA
12
CO
29
GA
1
IA
19
IL
31
MA
7
MI
7
NY
26
TX
18
The actual total is definitely higher than this (153 sections), as we know of other colleges using one of the new Math Lit textbooks. A few more colleges are implementing MLCS in Spring 2014, and several more colleges are implementing Fall 2014.
I think it is worth noting that all of this progress is being made without special grants; no mandates are involved, and we have no 'staff' in the New Life project. What we do have — dozens of dedicated faculty, willing colleges, and publishers willing to work with us.
The New Life Project is a voluntary effort (AMATYC Developmental Mathematics Committee) with considerable collaboration with the Carnegie Foundation and the Dana Center — especially in providing curricular expertise to those organizations. We can be proud of this progress and our work together.
The use of Mathematical Literacy will continue to grow. Our work will increase the emphasis on the second course — Algebraic Literacy; for information on Algebraic Literacy, you can see a presentation from this year's Summit on Developmental Mathematics at
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Like humans in general, our students develop expectations based on experience. Habits form, often without awareness or conscious effort. Behaviors exhibit, which are used to measure knowledge. In assessments, we often confuse correct behavior with correct knowledge.
Symbolic work can be difficult for novices. We (experts) see large amounts of information in short symbolic statements. For a novice, symbols are like a map to a city never visited — yes, we can remember how to get from point A to point B on the map … but without any understanding of what these points mean in the city.
On a recent test in my beginning algebra classes, two mistakes were made by at least 20% of the students (one or both):
-3² + 5² = 2^4 = 16
8^6 divided by 8^2 = 1^6
The first error is a coincidental 'right answer' for a very wrong method. The second one, not at all. Both involve over-generalizations of 'same number' rules. Obviously, there is a very high probability that the students making one or both of these errors have low study skills or habits (like not doing any practice outside of class).
My concern is not these particular students, nor these particular errors. My concern is our overall approach to mathematics. We tend to take one of these approaches to symbolism in mathematics:
Emphasize context and reasoning, and measure understanding by correctly completing related problems with differences in details.
Some reform models take approach #2 to the extreme — very few symbolic procedures are introduced, and most of what is done is arithmetic; algebraic models are used but carried out with technology more than symbolic procedures. We need to learn how to balance the 'symbols' and 'reasoning' aspects of mathematics — and be willing to embrace both as critical in all mathematics courses.
Clearly, there is much (perhaps a majority) of our traditional algebra curriculum that involves symbolic work without a purpose now or in the student's future. I seriously doubt that solving a radical equation by squaring each side twice will ever be a survival skill in a student's future.
Just as clearly (to me, at least), many of our students will need good understanding of various symbolic structures in mathematics, in future science courses (hard science and soft science). Terms, exponents, coefficients, subscripts, groupings, equations, inequalities … are involved in stating properties in sciences and in using predictive models.
When we assess the mastery of symbolism, we need to deal with much more than 'correct answers'. In the ideal situation, assessments would be done in a one-on-one verbal interview so the expert can probe into the novice's understanding based on the individual learner. Lacking that luxury, we will need to use diverse assessment tools that deal with process and connections, as well as answers.
Sadly, I had integrated some of this assessment into the beginning algebra class about two weeks ago — dealing with the adding terms error (first error above). On a worksheet, students were faced with adding like terms (10x^4 + 6x^4) before we had dealt with them formally in class. Something like half the students added the exponents as well as adding the 'terms' (coefficients). About 40% of these students apparently maintained this erroneous method up until the test.
Correct answers are only correlated with correct knowledge; students are always seeking the simplest rules for achieving correct answers — which can lead to totally wrong rules. Mathematical symbolism can be a window into the houses where students keep their math knowledge. Too often, however, symbolism is confused with the knowledge and correct answers stop the assessment process.
We need to slow down our courses. Learning mathematics is not a fast or spontaneous activity. Learning mathematics is hard work for both us and our students.
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This semester, we have more than 150 sections of Mathematical Literacy offered at colleges across the country … and these are outside of the grant-related work (such as Quantway™). In other words, the New Life Mathematical Literacy course is now the most implemented reform math course in the United States.
Getting to this point is the result of the incredible effort of dozens of math faculty, many of whom have been members of the New Life wiki at Our work has not involved large grants from foundations; rather, collaboration and local initiative have allowed us to create significant change.
However, change is not the same as progress. Progress involves sustained efforts which achieve explicit goals. We have achieved more than other efforts … but we "are not there yet".
Where are we headed? How will we know when we have arrived? These are not questions for which we create singular responses and data-based conclusions; these are questions for a profession to use as standards for our work.
In the world of process and product design, one set of strategies involves having people seeing themselves in the situation that they are trying to create. For example, we might ask 100 math faculty to imagine that the mathematics curriculum works like it is supposed to. What does this look like? What does it sound like? What does it smell like?
For our work, here are some answers I would give to those questions about what we are trying to achieve:
Students text each other about the latest exciting math problem.
Students pass every math class unless something unexpected comes up.
Over 10% of students major in a STEM field and over 10% of degrees are awarded in STEM fields.
Students learn diverse mathematics, with understanding, in both pre-college and college math courses.
Fewer students are in college-prep math classes than are in college level math classes.
Half of the students who start in college-prep math classes change their goals to be more STEM-like.
Math faculty are the happiest faculty on campus.
Part of our difficulty has been that we have not had a goal in mind — beyond having higher pass rates. Higher pass rates is not a design standard; it's a production standard (and a poor one, at that).
Progress would exist if we would judge that we are substantially closer to achieving our goals. If we don't articulate our goals (like the 7 statements above), we can never have progress … because we are not directing our efforts towards anything. Change is cheap; progress is where the power is.
I started off this post thinking a next step, like getting the Algebraic Literacy course on the radar — and I still think that is very important. Or, thinking about salvaging the college algebra and pre-calculus curriculum, which is very important. I hope that you will be involved with one or both of those reform efforts. Overall, however, I am concluding that we need to have more conversations about our goals. What does progress look like? How do we know when we are there, as opposed to where we are now? | 677.169 | 1 |
ALEX Lesson Plans
Title: Systems of Linear Inequalities Project
Description:
The
Subject: Mathematics (9 - 12) Title: Systems of Linear Inequalities Project Description: The
Title: Systems of Equations: What Method Do You Prefer?
Description:
TheStandard(s): [MA2010] (8) 10: Analyze and solve pairs of simultaneous linear equations. [8-EE8] [MA2010] AL1 (9-12) 19: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. [A-REI5 ALC (9-12) 2: Solve application-based problems by developing and solving systems of linear equations and inequalities. (Alabama) (8 - 12) Title: Systems of Equations: What Method Do You Prefer? Description: The
Thinkfinity Lesson Plans
Title: Shedding the Light
Description:
In
Standard(s):S1] PHY (9-12) 7: Describe properties of reflection, refraction, and diffraction. DM1 (9-12) 3: Use the recursive process and difference equations to create fractals, population growth models, sequences, series, and compound interest models. (Alabama)
Subject: Mathematics,Science Title: Shedding the Light Description: In Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 absolute | 677.169 | 1 |
TrigonometryALCULATE THIS: TRIGONOMETRY JUST GOT A LOT EASIER TO LEARN!Now anyone with an interest in basic, practical trigonometry can master it -- without formal training, unlimited time, or a genius IQ. In "Trigonometry Demystified, best-selling author Stan Gibilisco provides a fun, effective, and totally painless way to learn the fundamentals and general concepts of trigonometry. With "Trigonometry Demystified you master the subject one simple step at a time -- at your own speed. Unlike most books on trigonometry, this book uses prose and illustration... MOREs to describe the concepts where others leave you pondering abstract symbology. This unique self-teaching guide offers questions at the end of each chapter and section to pinpoint weaknesses, and a 100-question final exam to reinforce the entire book. Simple enough for beginners but challenging enough for professional enrichment, "Trigonometry Demystified is your direct route to learning or brushing up on trigonometry. Learn all aspects of trigonometry: * How angles are expressed * The relationships between angles and distances * Calculating distances based on parallax * Coordinate systems and navigation * And much more! Text provides a totally painless way to learn the fundamentals and general concepts of trigonometry. Uses prose and illustrations to describe the concepts, offers questions at the end of each chapter and section, and includes a 100-question self-test. Softcover.
Stan Gibilisco is one of McGraw-Hillís most diverse and best-selling authors. His clear, friendly, easy-to-read writing style makes his electronics titles accessible to a wide audience and his background in mathematics and research make him an ideal handbook editor. He is the author of The TAB Encyclopedia of Electronics for Technicians and Hobbyists Teach Yourself Electricity and Electronics, and The Illustrated Dictionary of Electronics. Booklist named his book, The McGraw-Hill Encyclopedia of Personal Computing, one of the Best References of 1996. | 677.169 | 1 |
Creating interactive models in Mathematica allows students to explore hard-to-understand concepts, test theories, and quickly gain a deeper understanding of the materials being taught firsthand. This screencast shows you how get started creating interactive models in Mathematica. Includes Chinese audio.
When working in Mathematica, you will often find it useful to view groups of functions that relate to a specific subject area or set of tasks. The Documentation Center includes guide pages and the function navigator for this purpose. Learn more in this "How to" screencast. Includes Japanese audio.
Mathematica provides several convenient ways to find information about functions. In addition to searching the documentation or navigating the guide pages, you can access documentation on functions directly from within your notebook. Learn more in this "How to" screencast. Includes Japanese audio.
Mathematica can run its calculations on other computers that have Mathematica installed. Passing computations to other, potentially more powerful, machines can increase the efficiency of your work. Learn more in this "How to" screencast. Includes Japanese audio.
Mathematica offers great flexibility for adding text to graphics; you can add text interactively using the Drawing Tools palette or programmatically using various graphics primitives. Learn more in this "How to" screencast. Includes Japanese audio.
Palettes give you immediate access to many features built into Mathematica, from creating syntactically complete expressions and inserting special characters to building up charts and slide shows, all through a convenient point-and-click interface. Learn more in this "How to" screencast. Includes Japanese audio.
Mathematica allows Greek letters to be integrated into symbol names, strings, graphics, and text. You can input Greek letters by using palettes or keyboard shortcuts. Learn more in this "How to" screencast. Includes Japanese audio.
You may want to export data from Mathematica to a spreadsheet. Excel is one example of a common spreadsheet format that Mathematica supports. Learn more in this "How to" screencast. Includes Japanese audio.
Geophysics professor Frank Scherbaum walks through an example of how he used Mathematica to develop an integrated system for students, teachers, and researchers to use in their probabilistic seismic hazard analysis work. Includes Spanish audio.
Geophysics professor Frank Scherbaum walks through an example of how he used Mathematica to develop an integrated system for students, teachers, and researchers to use in their probabilistic seismic hazard analysis work. Includes Japanese audio.
This screencast helps you get started using Mathematica by introducing some of the most basic concepts, including entering input, understanding the anatomy of functions, working with data and matrix operations, and finding functions. Includes Spanish audio.
Mathematica gives students the power to manipulate interactive graphics and develop complex data models. High-school teacher Abby Brown shares the success she experiences by using Mathematica in her classroom. Includes Spanish audio.
William Meyer, the vice president of technology at Scattering Solutions, LLC, describes an example of using Mathematica's data-analysis capabilities to save time and money on drug screening. Includes Japanese audio.
This video features John Kiehl, co-owner of Soundtrack Recording Studio, who shares an example of Mathematica using powerful set theory and pattern-matching capabilities to make and produce music. Includes Japanese audio. | 677.169 | 1 |
OCAD 0800
ELEMENTARY ALGEBRA (4 cr.)
This course is appropriate for you if you have a sound background in basic arithmetic but have not been exposed to algebra or if you need to strengthen your basic algebra skills. Topics may include properties of real numbers, order of operations, linear and quadratic equations, exponents, polynomials, graphing and systems of linear equations. Successful completion of OCAD 0800 ensures placement into MATH 1050, MATH 1070, MATH 1080, MATH 2500, ECON 2200, ECON 2250, or PSYC 2050. You are referred to this course based on your results on the mathematics/statistics placement assessment. See Summary of Financial Procedures for pricing information. Offered under the S/U grading option only. (Note: OCAD credits are not applied to graduation credit requirements.) | 677.169 | 1 |
, algebra 2 and calculus algebra 1, algebra 2 and calculus | 677.169 | 1 |
Calculus Revisited: Single Variable Calculus
Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course. The series was first released in 1970 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time.
Is our political ideology simply the result of a genetic coin toss? Mounting evidence suggests that biology may be a factor. In this video, Academic Earth explores some of the key research into the biology of politics. | 677.169 | 1 |
Course Content and Outcome Guide for MCH 120
Date:
22-AUG-2011
Posted by:
Curriculum Office
Course Number:
MCH 120
Course Title:
Machine Shop Math
Credit Hours:
2
Lecture hours:
0
Lecture/Lab hours:
20
Lab hours:
0
Special Fee:
$12
Course Description
Covers instruction and practice in working with whole numbers, fractions, decimals, formulas, inch and metric systems, formulas, calculating simple and direct indexing. Introduces how to apply the use of the inch/metric systems, dividing/index head and formulas as they pertain to thread calculations, gear calculations, speed and feed calculations, and taper calculations. Prerequisite: MCH 100. Audit available.
Addendum to Course Description
Applying Shop Math - Math skills are very important to the machinist in his/her daily work. The machinist must be able to calculate accurately and with reasonable speed. This module will provide instruction and practice in working with whole numbers, fractions, and decimals conversions.
Shop Math - Inch & Metric -In the Machine Shop, accurate workmanship depends on accurate measurements. The metric system of measurement is being adopted by many industries in an effort to be competitive in foreign markets. This module will introduce the student to the principles of the inch and metric systems of measurement.
Shop Math/Formulas - The machinists frequently makes calculations to solve for the unknown value needed to produce a part. A formula tells what values using symbols needs; what computations are necessary to combine those values by using operation signs; and what order to combine them by using grouping signs. In this module the student will learn how to apply the use of formulas as they pertain to Thread Calculations and Taper Calculations.
Percent, Charts, Graphs & Angles - There are times when the print given to the technician does not specifically provide all of the required information to allow the machinist to complete the work piece. At these times, the technician may have to use math procedures to calculate the missing information.To produce a superior product in the manufacturing process, machinists need tools to evaluate quality. Statistical Process Control (SPC) includes Percent, Graphs and Charts which are those tools that help the technician interpret whether the process is in or out of control. This module will introduce the student to the tools of SPC. This module will help the student learn these procedures and calculations.
Intended Outcomes for the course
Upon successful completion of this course students will be able to:
Calculate decimal equivalents of fractions noted on blue prints.
Convert inch to metric and metric to inch from dimensions on blue prints.
Apply mathematical formulas as appropriate to thread and taper calculations on shop drawings.
Course Activities and Design
MCH 120
Through direct instruction and practice students use formulas to determine tapers, thread pitch, and depth from shop drawings and blue prints. Students perform calculations in metric and English that include conversion of fractions to decimal equivalents, and conversion form metric to English units. | 677.169 | 1 |
MAPLE BASICS
The Maple interface has a long list of palettes and even handwriting recognition for finding symbols and one can edit mathematical expressions to make them look like real mathematics without being limited by the executable Maple syntax. Context sensitive right-click menus allow one to do most elementary mathematical operations without knowing any syntax.
This "Clickable Calculus" makes Maple easier for elementary users. It also has an easily accessible Math Dictionary.
It has a tools menu with Assistants (dialog windows to import and analyze data), Tutors (Java applet windows that guide you through many calculus and algebra routines), and Tasks (that uses Help to guide you in accomplishing a list of mathematical tasks).
In document mode, one can freely create a mathematical report without the constraint of input, output and text regions, much like a MathCAD worksheet.
Worksheet mode allows one to see the input region syntax popped in by right-click menu choices, giving more information about what Maple is doing and allowing easy editing of command parameters.
Math (2d) versus Maple (1d) notation in the input region
When the cursor is in an input region, the"text" and "math" entries in the tool bar will toggle between Standard Math Notation and Maple Notation. Standard Math is preferred for using the (floating, more limited) palettes to input expressions. Maple notation shows you the Maple syntax of the input expression. You can also use the Format Menu and choose convert to change from 2D Math input (the old Standard Math Notation) to 1D Math input = Maple input, if you want to learn the Maple command syntax for a WYSIWYG math expression.
An important tip for 2D Math input is that you must use the right arrow key to continue inputting an expression after raising to an exponent or dividing by a denominator (using the forward slash for division, asterisk for multiplication), in fact you can use all 4 arrows to move around an expression to edit its various pieces, while when entering from the palette, the tab key moves you through the characters to be replaced.
Calculus packages:
Student[Calculus1] is very useful for Calc 1 and Calc 2.
Student[MultivariateCalculus] is helpful for Calc 3 for tasks involving a scalar function of 2 or more independent variables.
Student[VectorCalculus] is helpful for Calc3 for doing vector calculus tasks, i.e., those tasks involving vector functions of one or more independent variables.
VecCalcis an external package to accompany Stewart Multivariable Calculus CalcLabs, available on citrixweb.
Linear algebra packages:
LinearAlgebra utilizes the new way of doing vectors and matrices in Maple, available from the palette insertion.
Student[LinearAlgebra] uses the same structures but aimed at teaching linear algebra with additional Tutor commands. In both cases commands are named by joining together capitalized key words, like "ReducedRowEchelonForm". Here it is useful to use command auto-completion so one can just type the first few characters and then Control, Space Bar to bring up a list of all commands starting with those letters to select from with the mouse (only one starts with "Red"). This feature is case sensitive. | 677.169 | 1 |
Also Available As:
Triangular Arrays with Applications
Thomas Koshy
Description
Triangular arrays are a unifying thread throughout various areas of discrete mathematics such as number theory and combinatorics. They can be used to sharpen a variety of mathematical skills and tools, such as pattern recognition, conjecturing, proof-techniques, and problem-solving techniques.
While a good deal of research exists concerning triangular arrays and their applications, the information is scattered in various journals and is inaccessible to many mathematicians. This is the first text that will collect and organize the information and present it in a clear and comprehensive introduction to the topic. An invaluable resource book, it gives a historical introduction to Pascal's triangle and covers application topics such as binomial
coefficients, figurate numbers, Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers, graph theory, Fibonomial and tribinomial coefficients and Fibonacci and Lucas polynomials, amongst others. The book also features the historical development of triangular arrays, including short biographies of prominent mathematicians, along with the name and affiliation of every discoverer and year of discovery. The book is intended for mathematicians as well as computer scientists, math and science teachers, advanced high school students, and those with mathematical curiosity and maturity. | 677.169 | 1 |
JOHN WAS RIGHT! - SOME THINGS HAVEN'T CHANGED EVEN AFTER MORE THAN A HUNDRED YEARS!
Homeschool educators are constantly faced with the dilemma of deciding whether or not their son or daughter needs to take a separate high school geometry course because some academic institution wants to see geometry on the high school transcript. Or, because the publishers offer it as a separate math textbook in their curriculum - implying it is to be taken as a separate course. Remembering, of course, that selling three different math books brings in thirty-three percent more revenue than selling just two.
John Saxon's unique methodology of combining algebra in the geometric plane and geometry in the algebraic plane all in the same math textbook had solved that dilemma facing home school educators for these past twenty-five years. However, unknown to John, this same problem had been addressed over a hundred years earlier at the University of Chicago.
Knowledge of this information came to me by way of a gift from my wife and her sisters. After their mother's death in 2003, my wife and her sisters spent the next six years going through some sixty years of papers and books accumulated by their parents and stored in the attic and basement of the house they all grew up in. When asked by friends why it was taking them so long, one of the daughters replied "Mom and Dad took more than a half century to fill the house with their memories. It won't hurt to take a couple more years to go through them."
Among some of the treasures they found in the basement were letters to their great-grandfather written by a fellow soldier while both were on active duty serving in the Union Army. One of these letters was written to their great-grandfather while his friend was assigned to "Picket Duty" on the "Picket Line." His friend was describing to their great-grandfather the dreary rainy day he was experiencing. He wrote that he thought it was much more dangerous being on "Picket Duty" than being on the front lines, as the "Rebels" were always sneaking up and shooting at them from out of nowhere.
The treasure they found for me was an old math book that their father had used while a sophomore in high school in 1917. The book is titled "Geometric Exercises for Algebraic Solution - Second Year Mathematics for Secondary Schools." It was published by the University of Chicago Press in October of 1907.
The authors of the book were professors of mathematics and astronomy at the University of Chicago, and they addressed the problem facing high school students in their era. Students who had just barely grasped the concepts of the algebra 1 text, only to be thrown into a non-algebraic geometry textbook and then, a year or more later being asked to grasp the more complicated concepts of an algebra 2 textbook. The book they had written contained algebraic concepts combined with geometry. It was designed as a supplement to a geometry textbook so the students would continue to use algebraic concepts and not forget them.
John never mentioned these authors - or the book - so I can only assume that he never knew it existed. For if he had, I feel certain that it would have been one more shining light for him to shine in the faces of the high-minded academicians that he - as did these authors - thought were wreaking havoc with mathematics in the secondary schools. In the preface of their textbook, the professors wrote:
"The reasons against the plan in common vogue in secondary schools
of breaking the continuity of algebra by dropping it for a whole year
after barely starting it, are numerous and strong ... With no other
subject of the curriculum does a loss of continuity and connectiveness
work so great a havoc as with mathematics ... To attain high educational
results from any body of mathematical truths, once grasped, it is profoundly
important that subsequent work be so planned and executed as to lead the
learner to see their value and to feel their power through manifold uses."
So, should you blame the publishers for publishing a separate geometry textbook? Or is it the fault of misguided high-minded academicians who - after more than a hundred years - still demand a separate geometry text from the publishers? I am not sure, but thankfully, this decision need not yet face the home school educators using John Saxon's math books for the original home school third editions of John Saxon's algebra one and algebra two textbooks still contain geometry as well as algebra - as does the advanced mathematics textbook.
Any home school student using John Saxon's math textbooks who successfully completes algebra one, (2nd or 3rd editions), algebra 2, (2nd or 3rd editions), and at least the first half of the advanced mathematics (2nd edition) textbook, has covered the same material found in any high school algebra one, algebra two and geometry math textbook - including two-column formal proofs. Their high school transcripts - as I point out in my book - can accurately reflect completion of an algebra one, algebra two, and a separate geometry course.
When home school educators tell me they are confused because the school website offers different materials than what is offered to them on the Saxon Homeschool website, I remind them that - unless they want to purchase a hardback version of their soft back textbook - they do not need anything being offered on the Saxon School website. In fact, they are getting a better curriculum by staying on the Homeschool website. You can still purchase the original versions of John Saxon's math textbooks that he intended be used to develop "mastery" as recommended by the University of Chicago mathematics professors over a hundred years ago.
Because many of you do not have a copy of my book, I have reproduced that list from page 15 of the book so you can see what editions of John Saxon's original math books are still good whether acquired used or new. These editions will remain excellent math textbooks for several more decades.
Math 54, 2nd Ed (Hardcover) - or - new 3rd Ed (Softcover).
Math 65, 2nd Ed (Hardcover) - or - new 3rd Ed (Softcover).
Math 76, 3rd Ed (Hardcover) - or - new 4th Ed (Softcover).
Math 87, 2nd Ed (Hardcover) - or - new 3rd Ed (Softcover).
Algebra 1/2, 3rd Ed (Hardcover).
Algebra 1, 3rd Ed (Hardcover).
Algebra 2, 2nd or 3rd Ed (Hardcover) - content is identical.
Advanced Mathematics, 2nd Ed (Hardcover)..
Calculus, 1st or 2nd Ed (Hardcover).
Physics, 1st Ed (Hardcover) - there is no second edition of this book.
"May you have a very Blessed and Merry Christmas."
November 2011
SHOULD YOU GRADE THE DAILY SAXON MATH ASSIGNMENTS?
I continue to see comments on familiar blogs about correcting - or grading - the daily work of Saxon math students. That is a process contrary to what John Saxon intended when he developed his math books. Unlike any other math book on the market today, John's math books were designed to test the student's knowledge every week. Why would you want to have students suffer the pains of getting 100 on their daily work when the weekly test will easily tell you if they are doing well?
I always tell homeschool educators that grading the daily work, when there is a test every Friday, amounts to a form of academic harassment to the student. Like everything else in life, we tend to apply our best when it is absolutely necessary. Students will accept minor mistakes and errors when performing their daily "practice" of math problems. They know when they make a mistake and rather than redo the entire problem, they recognize the correction necessary to fix the error and move on without correcting it. They have a sense when they know or do not know how to do a certain math problem; however, when they encounter that all important test every Friday, - as I like to describe it - they put on their "Test Hat" to do their very best to make sure they do not repeat the same error!
In sports, daily practice ensures the individual will perform well at the weekly game, for without the practice, the game would end in disaster. The same concept applies to daily piano practice. While the young concert pianist does not set out to make mistakes during the daily practice for the upcoming piano recital, he quickly learns from his mistakes. Built into John Saxon's methodology are weekly tests (every four lessons from Algebra 1/2 through Calculus) to ensure that classroom as well as homeschool educators can quickly identify and correct these mistakes before too much time has elapsed.
In other words, the homeschool educator as well as the classroom teacher is only four days away from finding out what the student has or has not mastered during the past week's daily work. I know of no other math textbook that allows the homeschool educator or the classroom teacher this repetitive check and balance to enable swift and certain correction of the mistakes to ensure they do not continue. Yes, you can check daily work to see if your students are still having trouble with a particular concept, particularly one they missed on their last weekly test, which can be correlated to their latest daily assignment. However, as one home school educator stated recently on one of the blogs "Yes, they must get 100 percent on every paper or they do not move on." While this may be necessary in other math curriculums that do not have 30 or more weekly tests, it is a bit restrictive and punitive in a Saxon environment.
John Saxon realized that not all students would master every new math concept on the day it is introduced, which accounts for the delay allowing more than a full week's practice of the new concepts before being tested on them. He also realized that some students might need still another week of practice for some concepts which accounts for his using a test score of eighty percent as reflecting mastery. Generally, when a student receives a score of eighty on a weekly test, it results from the student not yet having mastered one or two of the new concepts as well as perhaps having skipped a review of an old concept that appeared in the assignment several days before the test. When the students see the old concept in the daily work, they think they can skip that "golden oldie" because they already know how to do it! The reason they get it wrong on the test is that the test problem had the same unusual twist to it that the problem had that the student skipped while doing the daily assignment.
In all the years I taught John Saxon's math at the high school, I never graded a single homework paper. I did monitor the daily work to ensure it was done and I would speak with students whose test grades were falling below the acceptable minimum of eighty percent. I can assure you that having the student do every problem over that he failed to do on his daily assignments does not have anywhere near the benefit of going over the problems missed on the weekly tests because the weekly tests reveal mastery - or lack thereof - while the daily homework only reveals their daily memory!
October 2011
SAXON MATH - WHEN USED CORRECTLY - IS STILL THE BEST MATH CURRICULUM!
Over the last twenty-some years, I have heard just about every story told by public and private school classroom teachers, school administrators, and home school educators about how difficult John Saxon's math books are. I believe that home school educators who speak poorly about John Saxon's math books are like the classroom teachers and administrators I encountered who always blamed the math book for students' poor showing. They never became aware that misusing John Saxon's math books was a major contributing factor to the student's difficulty. They never received or asked for any special training on how to correctly use John Saxon's math books. Why should they? After all, isnt one math textbook just like another?
More than a decade ago, while briefing the superintendent of a large school district in Colorado I related to him - from my observations - that the vast majority of the districts math teachers were not properly using John Saxon's math books. His district had been using (rather misusing) John's books for more than five years and the students math scores were getting worse, not better! I told him that it was not the books, but rather this misuse of the books that accounted for the district's failing math scores.
In the middle of my briefing, he stopped me and said, "What I hear you saying Art, is that we bought a new car, and since we already knew how to drive, we saw no reason to read the owner's manual wouldn't you agree?" To which I replied "It's worse than that, sir! You all thought you had purchased a car with an automatic transmission, but Saxon is a stick shift! It is critical that certain procedures be followed, just as well as some be dropped, or you will strip the gears!" I went on to tell him that while I was not the owner of the company, if John Saxon were alive today, standing up here addressing them under these circumstances, he would tell them to either use his books correctly or get rid of them, and blame someone else's textbooks for the failing grades.
On another occasion while briefing administrators at a school district in the state of Missouri whose math teachers were guilty of the same misuse, I promised their superintendent the same results I had promised the superintendent in Colorado that if they would make some adjustments and use the books correctly, they would develop a successful math program and their students math scores would go up. The district decided to implement the changes I had recommended. About eighteen months later, I received a letter from the superintendent. She wrote that their middle school students had scored the highest of any middle school in the state on their end-of-year math exams.
During these past several decades of advising and assisting homeschool parents about curriculum choices for their children, I noticed many of their calls and email to me were the result of having received inaccurate or inadequate - sometimes downright erroneous - advice. This erroneous advice came from other home school parents, discussion groups, well-meaning but uninformed or inexperienced publishing company employees, or from well meaning, but inexperienced employees of homeschool textbook distributors.
With all the new math books and supplemental math products on the market today - and textbook publishers promising every home school parent that if you use their books, your sons and daughters would score well on the ACT or SAT tests - I thought it appropriate to take this opportunity to defend John's math books for the benefit of the home school educators and their students.
I do not sell John Saxon's math books - I never have! But I firmly believe that John's math books remain the best math textbooks on the market today. I also believe that some of these new book fads, advertising how their product makes math fun, will ultimately leave your child short of mastering the requisite fundamentals of mathematics necessary to succeed in the collegiate realm of engineering, architecture, science, medicine, et al.
Students fail calculus - not because they do not understand the calculus - but because they never mastered the fundamentals of algebra, and they will fail a basic algebra course if they have not mastered the concepts of decimals, fractions and percents. John Saxon's math books were designed to create mastery of mathematics at all levels, and the infusion of repetition over time (referred to by Dr. Benjamin Bloom as "automaticity") creates this mastery at every level for every student who uses John Saxon's math books properly.
While home school students have a great deal more academic flexibility than the public or private classroom students do, they can just as easily fall prey to the same difficulties in mathematics as the public classroom students if they are using one of John Saxon's math books incorrectly. Parents of home school students who have displayed poor progress while using John Saxon's math books, generally have unknowingly contributed to the student's poor performance by taking shortcuts and preventing the student from receiving the full benefit of John Saxon's methods.
Earlier this year, in the January and February news articles, I went over the essential Do's and Don't's together with my comments and recommendations on how to correct them and have your child enjoy mathematics and realize mastery of the material using the best mathematics curriculum on the market today - John Saxon's math books!
If you need to discuss a special situation concerning your son or daughter's math progression or difficulty, please do not hesitate to either email or call me at:
Email:[email protected] Telephone: 580-234-0064 (CST)
"Do not worry about your difficulties in mathematics; I can assure you that mine are far greater"
Albert Einstein
September 2011
WHAT ARE FORMULA CARDS? WHAT ARE THEY USED FOR?
WHERE CAN I GET THEM?
Having been repeatedly threatened by my high school math teachers that I would be doomed to fail their tests if I did not memorize all those math formulas, I was somewhat surprised later in a college calculus course when the professor handed out "formula cards" containing over ninety geometry, trigonometry identities, and calculus formulas. He explained that they could be used on his tests. He did not bat an eye as he handed them out and reminded us that selecting the correct formulas and knowing how and when to use them was far more important than trying to memorize them or write them on the desk top.
So, when I started teaching at the high school, I announced to the students that they could make "formula cards" by using 5 x 8 inch cards, lined on one side and plain on the other. It never failed. Immediately, one of the students would ask why I did not have them printed off and handed out, saving them a considerable amount of time and money creating their own.
I told my students that whenever they encountered a formula in their textbooks, writing it down would strengthen the connection more than if they just read it and tried to recall it later while working a problem. Reading the formula in the textbook was their first encounter and there would not yet be a strong connection between what they were reading and what they tried to remember. However, when they took the time to create a formula card for that particular formula, they would be strengthening that connection. As they used the card when doing their daily assignments, they would continue the process and eventually place the formula in their long term memory.
So, how can you get formula cards? Simple! Each student makes his own. I allowed my students to use them starting with Math 87 or Algebra 1/2. One young lady in my Algebra 2 class used blue cards for geometry formulas and white cards for the algebra formulas to save her time looking through the cards. The cards should be destroyed after completion of the course, requiring the next student to make his own. Then how do you make formula cards?
Have the students use 5 x 8 cards - and write or print clearly and big. On the plain side of the card they print the title of the formula such as the formula for the area of a sector found on page 16 of the third edition of the Algebra 2 textbook.
So, on the front of the card (the plain side) in the center of the card the student would print::
AREA OF A SECTOR
When you turn the card over, in the upper right hand corner is the page number of the formula to enable the student to immediately go to that page should he need more information (in this case p 16). Recording the page number saves flipping through the book looking for the information and wasting time, especially when the student encounters a difficult problem some twenty lessons later.
After writing down the appropriate page number, they neatly record the formula: (double checking to make sure they have recorded it correctly.)
Area of Sector = Pc/360 times [(pi)(r)]^2 (where the piece (Pc) equals the part of the sector given.)
NOTE: If diameter is given remember to divide by two before squaring the value.
Remember, students may also use the formula cards on tests - and if you watch them - the dog eared cards seldom get looked at after awhile.
For those of you concerned about students taking the ACT or SAT, unless they have changed their policy, students are given a sheet of formulas for the math portion of the test. Again, this requires the student to know which formula to select and what to do with it - rather than remembering all those formulas!
August 2011
IS THERE ANY VALUE TO USING A SEPARATE GEOMETRY TEXTBOOK?
Have you ever seen an automobile mechanic's tool chest? Unless things have changed, auto mechanics do not have three or four separate tool chests. They have one tool chest that contains numerous file drawers separating the tools necessary to accomplish their daily repair work. But the key is that all of these tools are in a single tool chest.
What if the auto mechanic purchased several tool chests thinking to simplify things by neatly separating the specific types of tools from each other into separate tool chests rather than in separate drawers in the one tool chest? Each separate tool chest would then contain a series of complete but distinctly different tools. If mechanics did this, there would now exist the possibility that they would find themselves trying to remember which tool chest contained which tools - and - the extra tool chests would cost them more!
It is somewhat like that in mathematics. Each division of mathematics has its strengths and weaknesses and like the auto mechanic who selects the best tool for a specific job, so the physicist, engineer, or mathematician selects the best math procedure to meet the needs of what they are doing.
But how do we address the argument that geometry provides a distinct and essential thought process unlike that used in algebra? The advent of computers has provided educators with an alternative course titled Computer Programming. A computer programming course teaches students the same methodology or thought process that the two-column proofs of geometry do. Basically, it teaches the student that he cannot go through a door until he has opened it - meaning - the student must use valid statements that are logically and correctly placed to reach a valid conclusion and to prove that conclusion valid by having the computer program work correctly.
Before computers, educators in the United States felt that providing the separate geometry course would benefit those students interested in literature and the arts, who enjoyed the challenge of geometry without the burden of algebra, while still allowing students entering the fields of science and engineering, who had to take more math, to take the course also. When I was in high school, most geometry teachers taught only geometry, they never taught an algebra course - as I soon learned!
I recall encountering that little known fact when I took high school geometry from one such teacher. I was sharply rebuked early in the school year when I kept using the term "equal" to describe two triangles that had identical measures of sides and angles. The first time I said the two triangles that contained identical angles and sides were "equal," she told me I was wrong. She then proceeded to tell me and the class that the only correct term to describe two identical triangles was the term "congruent." She did not say my answer was technically correct, but that in the geometry class, we used the term "congruent" rather than "equal" - she specifically pointed out that I was "wrong."
The next day in geometry class, I really got in trouble when I stood and read Webster's definition of the word congruent.
Just before I was told to go to the office and tell the principal that I was being rude, I asked her why the two triangles could not also be said to be equal since they had identical angles and sides and were equal in size. Then I drove the final nail in my coffin when I proceeded to read Webster's definition of "equal."
Equal - adj. 1) Being the same in quantity or size...
Half a century later, when I taught both the algebra and geometry concepts simultaneously while using John Saxon's math books at a rural high school, I made it clear to the students that while they should become familiar with the terminology of the subject, they were free to interchange terms as long as they were correctly applied. I also made it clear to them that the object of learning geometry and algebra was to challenge and expand their thought processes and for them to understand the strengths and weaknesses of each and apply whichever math tool best served the problem being considered.
While many tout the separate geometry textbook as necessary to enable a child to concentrate on a single subject rather than attempting to process both geometry and algebra simultaneously, I would ask them how a young geometry student can solve for an unknown side in a particular triangle without some basic knowledge of algebraic equations. In other words, if you are going to use a separate geometry textbook, it cannot be used by a student who has not yet learned how to manipulate algebraic equations. This means that a separate geometry book is best introduced after the student has successfully completed an algebra 1 course.
For most students this means placing the separate geometry course between the Algebra 1 and Algebra 2 courses creating a gap of some fifteen months between them. (Two summers off, plus the nine month geometry course). As was true in my high school days, this situation creates a problem for the vast majority of high school students who enter the Algebra 2 course having forgotten much of what they had learned in the Algebra 1 course fifteen months earlier.
So what am I getting at? Must we have a separate geometry textbook for students who cannot handle the geometry and algebra concurrently? Well, let me ask you, if students can successfully study a foreign language while also taking an English course or successfully study a computer programming course while also taking an algebra course, why can't they study algebra and geometry at the same time, as John Saxon designed it?
Successful completion of John Saxon's Algebra 2, (2nd or 3rd editions) not only gives the student a full years' credit for the Algebra 2 course, but it also incorporates the equivalent of the first semester of a regular high school geometry course. I said "Successful Completion" for several reasons. FIRST: The student has to pass the course and SECOND: The student has to complete all 129 lessons.
Whenever I hear home school educators make the comment that "John Saxon's Algebra 2 book does not have any two-column proofs," I immediately know that they stopped before reaching lesson 124 of the book which is where two-column proofs are introduced. The last six lessons of the Algebra 2 textbook (2nd or 3rd editions) contain thirty-one problems dealing with two-column proofs. The following year, in the first half of the Advanced Mathematics textbook, they not only encounter some heavy duty algebra concepts, but they also complete the equivalent of the second semester of a regular high school geometry course. The first thirty of these sixty lessons contain more than forty problems dealing with two-column proofs.
So why then did John Saxon not want to publish a separate geometry textbook? As I mentioned in one of my newsletters several years ago, a group of professors who taught mathematics and science at the University of Chicago bemoaned the fact that educators continued to place a geometry course between basic algebra (Algebra 1) one and the advanced algebra course (Algebra 2) to the detriment of the student. AND THEY WROTE THIS 104 YEARS AGO!
In the preface to their book titled "Geometric Exercises for Algebraic Solution," published in 1907, the professors explained that it is this lengthy "void" that prevents students from retaining the necessary basic algebra concepts learned in basic algebra to be successful when encountering the rigors of advanced algebra.
We remain one of the only - if not the only - industrialized nations that have separate math textbooks for each individual math subject. When foreign exchange students arrive at our high schools, they come with a single mathematics book that contains geometry, algebra, trigonometry, and when appropriate, calculus as well. Is it any wonder why we are falling towards the bottom of the list in math and science?
When students take a separate geometry course without having gradually been introduced to its unique terminology and concepts, they encounter more difficulty than do students using John Saxon's math books The beauty of using John's math books, from Math 76 through Algebra 1 is that students receive a gradual introduction to the geometry terminology and concepts.
If you are going to use John Saxon's math books through Advanced Mathematics or Calculus you do not need a separate geometry book. This means you must use the third editions of John Saxon's Algebra 1 and Algebra 2 books because the current owners of Saxon Publishers (HMHCO) have stripped all of the geometry from the new fourth editions of their versions of Algebra 1 and Algebra 2. And you do not want a student to go from the fourth edition of Algebra 2 to the Saxon Advanced Mathematics textbook.
NOTE: Please Click Here to watch a short video on how to receive credits for Geometry, Trigonometry and Pre-calculus using John Saxon's Advanced Mathematics textbook - and how to record the course titles on the student's transcript.
July 2011
HOW MUCH TIME SHOULD STUDENTS SPEND ON MATH EACH DAY?
One of several arguments advanced by home school educators regarding the efficiency of the Saxon math curriculum is that from Math 54 through Advanced Mathematics the courses require too much time to complete the daily assignment of thirty problems. Their solution to this often takes one of two approaches. Either they allow the student to take shortcuts to reduce the time spent on daily assignments, or they find another math curriculum that takes less time – you know – you've heard them say, "We found another math curriculum that is more fun, easier, and it does not require so much time."
In this year's, January and February newsletters, I addressed some of the ramifications of taking these shortcuts when using John Saxon's math books. In these two articles, I described in detail the effects upon students who used some or all of them, so I will not go over them again here. I would ask you to read those two newsletters if you have not already done so. What I want to discuss here is what may be causing the excessive amount of time taken by the students and also, what constitutes excessive time to an educator who taught in a public classroom using Saxon math books for over a decade.
While I was teaching high school mathematics in a rural Oklahoma high school, I would often go and watch my students who were on the the high school track, basketball, or football teams during their practice sessions after school. I was able to chat with the mothers and fathers who were also watching these practices. This one-on-one conversation often gave me an insight into their priorities regarding their children's education.
While they sometimes complained about the rigors of my math classes, they never once complained about the length of time their sons and daughters were out on the field in the heat or cold - or on the basketball court – practicing – after just spending six academic hours in the classroom. In fact, when coaches were forced to cancel a practice for one reason or another, some of the parents would vocally complain that the practices should continue. They expressed concern that skipping practice would take the "edge" off their son or daughter's playing ability and inhibit their athletic "sharpness" for the next game.
Strange then that some parents would complain the 45 – 60 minutes spent each day on mathematics "practice" would be excessive - and more unusual - that they would seek an easier course of action. They never discussed the ramifications that doing so might take the "edge" off their child's math "sharpness" for the next math course or the state mandated math test. I never heard the high school parents complain about watching the tough daily drills and practices run by the coaches. I never heard a parent complain about the hour spent each day by the students diligently practicing their piano lessons, or having to come in before school early each day to spend 45 minutes in the weight room.
At least several times each week I receive email from home school parents who express concern that their son or daughter was taking an inordinate amount of time on their daily math assignment in one of the books from Math 54 through Advanced Mathematics. The solution to the excessive time spent by students using the Advanced Mathematics textbook is easy to resolve. The solution to that unique situation is explained in a short video clip (Click Here to view that video).
I have interacted with several thousand parents and students in the twelve years that I taught mathematics at that rural high school. I have also advised thousands more home school educators and home school students in the succeeding decade after my retirement while serving as one of the Homeschool Curriculum Advisors (for Math 76 through Calculus and Physics) for Saxon Publishers and later for Harcourt-Achieve who bought the company from John's children. And while every child and home school situation is different, my experiences have shown me that there exist several situations that contribute to excessive time spent on daily work by students, whether home schooled or attending a public or private classroom.
These situations are:
The Student is in The Wrong Level Math Course: If after lesson thirty in any Saxon math book, students continue to receive 80% on the weekly tests of twenty questions, within a maximum of fifty minutes with no partial credit (all right or all wrong) and no calculator (until Algebra 1), then they are in the correct level Saxon math book. If the test scores are constantly below that or if they fall below an 80 on their first five or so tests, then that is a good indication they are in the wrong level Saxon math book. This situation can result from any one or more of the following conditions:
They did not finish the previous Saxon math book.
They took shortcuts in the preceding math book.
Their previous math book was not a Saxon math book.
They did not take the weekly tests in the previous math book, using the daily grade as an indication of their level of proficiency.
Their last five tests in the preceding course were well below 80% (minimal mastery).
The Student is Required to Re-do Math Problems from Yesterday's Lesson: Why do we want students to get 100% on their daily practice for the weekly test? When we grade their daily work and have them go over the ones they missed on the previous day's assignment, nothing is accomplished except to "academically harass" the students. The daily work reflects nothing but the status of the students' temporary learning curve. It is the weekly tests and not the daily work that reveal what the student has mastered from the previous weeks and months of work. Not every student masters every concept the day it is introduced, which is why there is a four to five day delay from when the concept is introduced to when it is tested. In the twelve years that I taught John Saxon's math books in high school, I did not grade one homework paper – but I did grade the weekly tests which reflected what the students had mastered as opposed to their daily work which did not. Remember, John Saxon's math books are the only books I am aware of that use weekly tests to evaluate a student's progress. There are a minimum of thirty weekly tests in every one of John's math books from Math 54 on.
Too Much Time is Spent on The Warm Up Box: From Math 54 through Math 87, there is what used to be called a "Warm Up" box at the top of the first page of every lesson. I recall watching a sixth grade teacher waste almost thirty minutes of class time while three boys took turns giving different opinions as to how the "Problem of The Day" was to be solved – and arguing as to which had the better approach. After class, I reminded the teacher that the original purpose of the box was to get the students settled down and "focused" on math right after the second bell rang. I said to her, "Why not immediately review a couple of the problems from yesterday's lesson at the start of class for the few who perhaps did not grasp the concept yesterday? Then move immediately to the new lesson." This process would take about 10 to 20 minutes and would leave students with about 40 minutes of remaining class time to work on their new homework assignment.
NOTE: In any of John Saxon's math books from Math 54 through Algebra 2, the "A" and "B" students will get their 30 problems done in less than 40-50 minutes. The "C" students will require more than an hour.
The Student is Required to Do All of The Daily Practice Problems: The daily practice problems were created for teachers to use on the blackboard when teaching the lesson's concept so they did not have to create their own or use the homework problems for demonstrating that concept. Many of the lessons from Math 54 through Math 87 have as many as six or more such problems and if the student understands the concept, they are not necessary. If the student has not yet grasped the concept, having the student do six or more additional practice problems of the same concept will only further frustrate him. Remember, not every student grasps every concept on the day it is introduced. The five minutes spent on review each day is essential to many students.
The Student is a Dawdler or a Dreamer: There is nothing wrong with being a "Dreamer," but some students just look for something to keep them from doing what they should be doing. I call these students "Dawdlers." I recall the first year I taught. I had to constantly tell some students in every class to stop gazing out the window at the cattle grazing in the field outside our classroom – and get on their homework. That summer, I replaced the clear glass window and frame with a frosted glass block window - and in the following eleven years I had absolutely no problem with my "Dawdlers."
The Student is Slowed by Distractions: Is the student working on the daily assignment in a room filled with activity and younger siblings who are creating all sorts of distractions? Even the strongest math student will be distracted by excessive noise or by constantly being interrupted by younger siblings seeking attention. Did you leave the student alone in his room only to find he was on his cell phone talking or texting with friends or listening to the radio? Or worse, does he have a television or computer in his room and does he use the computer to search the internet for a solution to his math problems or engage in something equally less distracting by watching the television?
Please do not misinterpret what I have discussed here. If you desire to do all of the above and the student takes two hours to complete a daily assignment, and both you and the student are satisfied, then that is acceptable. But if you are using this excessive time as an excuse for your child's frustration and as an argument against John Saxon's textbooks, I would remind you of what John once told a school district that did everything John had asked them not to do and they were now blaming John's books for their district's low math test scores. John told them, "If you want to continue your current practices, get rid of my books and buy someone else's book to blame."
June 2011
JUST WHAT IS THE DIFFERENCE BETWEEN MATH 87 AND ALGEBRA 1/2 ?
There appear to be varying explanations regarding whether a student should use Math 87 or Algebra 1/2 after completing the third or fourth edition of John Saxon's Math 76 textbook. Let me see if I can shed some light on the best way to determine which one, or when both, should be used.
First, whenever I discuss the Math 76 textbook in this article, I am talking about the third or fourth editions of that book. I am not talking about the old first or second editions of John's Math 76 books. These two older editions have been out of print for almost fifteen years now and their content, while acceptable at the time, would not now enable a math student to proceed successfully through either the second or third editions of Math 87 or the third edition of Algebra 1/2.
Second, whenever I discuss using the Math 87 textbook, I am talking about either the second or third editions of that book and not the older first edition which has also been out of print for more than a decade.
Let me assure you that, except for the new soft cover, the addition of a solutions manual, and the varied numbering of the pages, there is absolutely no difference between the 120 lessons and the 12 Investigations of the hard cover second edition of Math 87 and the new soft cover third edition of Math 87. Oh yes, the new third edition has added a TOPIC A at the end of the book (after Investigation 12) dealing with Roman Numerals and Base 2. And even though the marketing folks at the publishers have added the word "Prealgebra" under the cover title of the soft cover third edition textbook, these two additional topics, while interesting and nice to know, are not pre-algebra material and will not create any shortfall for the student in Algebra 1 or even later in Algebra 2.
Both the Math 87 and Algebra 1/2 textbooks prepare the student for any Algebra 1 course. The main difference between using the Math 87 and the Algebra 1/2 books depends upon the student's success in the Math 76 textbook. The Math 87 book starts out a little slower than the Algebra 1/2 book does because it assumes the student needs the additional review resulting from the student encountering difficulty in the latter half of the Math 76, textbook. Also, If you were to open to any lesson in the Algebra 1/2 textbook, you would immediately notice that the "Warm-Up' box common in all Saxon math books from Math 54 through Math 87, is noticeably absent.
If there exists a math savvy student of John Saxon's Math 76 textbook, who received test grades of 85 or better on the last five tests in Math 76 (50 - 55 minute test period, no calculator, and no partial credit), then that student would be more challenged and, from my teaching experience, much better off in the Algebra 1/2 book. However, if his last five test scores are below 85, then from my experiences, that indicates that student should proceed to Math 87, and upon completion of that book, if his or her last five tests are now 80 or better (minimal mastery), then that student can easily skip Algebra 1/2 and go on to success in Algebra 1.
If however, a student encounters difficulty going through both Math 76 and Math 87, then proceeding through the Algebra 1/2 textbook before attempting Algebra 1 will allow the student to regain his confidence. Doing so will further ensure the student has mastered a solid conceptual base necessary for success in any Algebra 1 course.
Students will fail an Algebra 1 course if they have not mastered fractions, decimals and percents which are emphasized before the student reaches that course. I realize that not every child fits neatly into a specific mold, but John Saxon's Math 76, Math 87, and Algebra 1/2 textbooks allow the Saxon home school educator sufficient flexibility to satisfy every student's needs and to ensure the students' success in any Algebra 1 course.
May 2011
ARE JOHN SAXON'S ORIGINAL MATH BOOKS GOING THE WAY OF THE DINOSAUR?
In the past several weeks I have been asked by some home school educators whether or not I will create my teaching DVD "videos" for the new fourth editions of Algebra 1 and Algebra 2, and the resulting new first edition of Geometry now being sold on the Saxon Homeschool website by the new owners of Saxon Publishers.
The answer is no, I will not do so. My creation of the current DVD video series for John's math books, based upon rock solid editions created by John Saxon, was not to make money. Using my Saxon classroom teaching experiences, I wanted to create a classroom environment for home school students who wanted to master high school mathematics using John's unique math books. However, publishing math textbooks redesigned to be like all the other math textbooks on the market is not what John intended when he created his unique style of math books.
John Saxon would not have sanctioned gutting his Algebra 1 and Algebra 2 textbooks of their geometry to create a separate geometry textbook. He believed that using a separate geometry textbook was not conducive to mastering high school mathematics. More importantly, each of John's math books had an author - an experienced classroom mathematician - behind them. These three new editions, created under his Saxon title, do not.
When Harcourt-Achieve bought John Saxon's dream - Saxon Publishers - from his children, I made the comment that the new owners were certainly smart enough to recognize the uniqueness of John's books. I predicted that they would not change the content of John's books. Certainly, I commented. "They would never take their prize winning bull and grind it up into hamburger" – or so I thought!
Well the new owners of Saxon Publishers appear to have done just that, and the time has come for me to apologize because they are now selling the hamburger on the Homeschool website. I have previously cautioned home school Saxon users not to use the new fourth editions of Algebra 1 and Algebra 2 then offered only on the school website because the company had gutted all geometry from them to enable them to publish a separate geometry textbook desired by the public school system. But they are now selling them on the Homeschool site as well.
Having been affiliated with one of the larger publishing companies - after Saxon Publishers was sold - I observed that the driving force in the company was not so much the education of the children, but the quarterly profit statement. And that is okay, but being around their VP's and upper level executives showed me that to them "a book is a book is a book." I still believe they have not the foggiest idea of just how unique and powerful John's math books are when used correctly. However, I may be wrong, because they may have already observed that it is this "uniqueness" that requires special handling and that requires special training, and that costs money – reducing quarterly profits.
If you are serious about using John Saxon's original math series through high school, I recommend you not buy these new fourth editions of Algebra 1 and Algebra 2. I strongly recommend you immediately acquire the home school editions of John's math books that I discussed in my February 2010 news article - which include the third editions of Algebra 1 and Algebra 2. The news article not only explains the correct editions that will still be good for several more decades, but it explains the correct sequencing of the books as well.
I do not believe the publishing company will long suffer the expense of publishing both the third and fourth editions of Algebra 1 and Algebra 2. It is my opinion they may well stop printing and selling the third editions of Algebra 1 and Algebra 2 when current stocks run out. This will then require that home school educators using Saxon math books buy the separate geometry book also. After all, "Don't you make more money from selling three books than you do from just selling two?"
Maybe I am wrong, and the publishers of John Saxon's math books will not stop printing the third editions of Algebra 1 and Algebra 2 as I am predicting - but then again I could be wrong - again!
April 2011
WHY IS THERE A "LOVE – HATE" RELATIONSHIP WITH SAXON MATH BOOKS?
Over the past twenty-five years, I have noticed that parents, students, and educators I have spoken to, either strongly like or, just as strongly, dislike John Saxon's math books. During my workshops at home school conventions, I am often asked the question about why this paradigm exists. Or, as one home school educator put it, "Why is there this Love – Hate relationship with Saxon math books?" It is easy to understand why educators like John's math books. They offer continuous review while presenting challenging concepts in increments rather than overwhelming the student with the entire process in a single lesson. They allow for mastery of the fundamentals of mathematics.
In an interview with William F. Buckley on the FIRING LINE in 1983, John Saxon responded to educators who were labeling his books as "blind, mindless drill." He accused them of misusing the word "drill." John reminded the listeners that:
"Van Cliburn does not go to the piano to do piano drill. He practices - and - Reggie Jackson does not take batting drill, he takes batting practice."
John went on to explain that
"Algebra is a skill like playing the piano, and practice is required for learning to play the piano. You do not teach a child to play the piano by teaching him music theory. You do not teach a child algebra by teaching him advanced algebraic concepts that had best be reserved till his junior year in college when he has mastered the fundamentals."
As John would say, "Doing precedes Understanding - Understanding does not precede doing."
It is my belief that, "John Saxon's math books remain the best math books on the market today for mastery of math concepts!" Successful Saxon math students cannot stop telling people how they almost aced their ACT or SAT math test, or CLEP'd out of their freshman college algebra course. And those who misuse John Saxon's math books, and ultimately leave Saxon math for some other "catchy – friendly" math curriculum, rarely tell you that their son or daughter had to take a no-credit algebra course when they entered the university because they failed the entry level math test. Yes, they had learned about the math, but they did not master or retain it.
Just what is it that creates this strong dislike of John Saxon's math books? During these past twenty-five years I have observed there are several general reasons that explain most of this strong dislike. Any one of these – or a combination of several – will create a situation that discourages or frustrates the student and eventually turns both the parent and the student against the Saxon math books.
Here are several of those reasons:
ENTERING SAXON MATH AT THE WRONG LEVEL: Not a day goes by that I do not receive an email or telephone call from frustrated parents who cannot understand why their child is failing Saxon Algebra 1 when they just left another publisher's pre-algebra book receiving A's and B's on their tests in that curriculum. I explain that the math curriculum they just left is a good curriculum, but it is teaching the test, and while the student is learning, retention of the concepts is only temporary because no system of constant review is in place to enable mastery of the learned concepts.
Every time I have encountered this situation, I have students take the on-line Saxon Algebra 1 placement test - and without exception, these students have failed that test. That failure tends to confuse the parents when I tell them the test the student just failed was the last test in the Saxon Pre-Algebra textbook. Does this tell you something? This same entry level problem can occur when switching to Saxon at any level in the Saxon math series from Math 54 through the upper level algebra courses; however, the curriculum shock is less dynamic and discouraging when the switch is made after moving from a fifth grade math curriculum into the Saxon sixth grade Math 76 book.
MIXING OUTDATED EDITIONS WITH NEWER ONES: There is nothing wrong with using the older out-of-print editions of John Saxon's original math books so long as you use all of them – from Math 54 to Math 87. However, for the student to be successful in the new third edition of Algebra 1, the student has to go from the older first edition of Math 87 to the second or third edition of Algebra ½ before attempting the third edition of the Saxon Algebra 1 course.
But when you start with a first edition of the Math 54 book in the fourth grade and then move to a second or third edition of Math 65 for the fifth grade; or you move from a first or second edition of the sixth grade Math 76 book to a second or third edition of the seventh grade Math 87 book, you are subjecting the students to a frustrating challenge which in some cases does not allow them to make up the gap they encounter when they move from an academically weaker text to an academically stronger one.
The new second or third editions of the fifth grade Saxon Math 65 are stronger in academic content then the older first edition of the sixth grade Math 76 book. Moving from the former to the latter is like skipping a book and going from a fifth grade to a seventh grade textbook. Again, using the entire selection of John's original first edition math books is okay so long as you do not attempt to go from one of the old editions to a newer edition. If you must do this, please email or call me for assistance before you make the change.
SKIPPING LESSONS OR PROBLEMS: How many times have I heard someone say, "But the lesson was easy and I wanted to finish the book early, so I skipped the easy lesson. That shouldn't make any difference." Or, "There are two of each type of problem, so why do all thirty problems? Just doing the odd numbered ones is okay because the answers for them are in the back of the book." Well, let's apply that logic to your music lessons.
We will just play every other musical note when there are two of the same notes in a row. After all, when we practice, we already know the notes we're skipping. Besides, it makes the piano practice go faster. Or an even better idea. When you have to play a piece of music, why not skip the middle two sheets of music because you already know how they sound and the audience has heard them before anyway.
My standard reply to these questions is "Must students always do something they do not know how to do? Can they not do something they already know how to do so that they can get better at it? The word used to describe that particular phenomenon is "Mastery!"
USING A DAILY ASSIGNMENT GRADE INSTEAD OF USING THE WEEKLY TEST GRADES: Why would John Saxon add thirty tests to each level math book if he thought they were not important and did not want you to use them? Grading the daily assignments is misleading because it only reflects students' short term memory, not their mastery. Besides, unless you stand over students every day and watch how they get their answers, you have no idea what created the daily answers you just graded.
Doing daily work is like taking an open book test with unlimited time. The daily assignment grades reflect short term memory. However, answering twenty test questions - which came from among the 120 – 150 daily problems the students worked on in the past four or five days - reflects what students have mastered and placed in long term memory. John Saxon's math books are the only curriculum on the market today that I am aware of that require a test every four or five lessons. Grading the homework and skipping the tests negates the system of mastery, for the student is then no longer held accountable for mastering the concepts.
MISUSE OF THE SAXON PLACEMENT TESTS: When students finish one Saxon level math book, you should never administer the Saxon placement test to see what their next book should be. The placement tests were designed to see at what level your child would enter the Saxon series based upon what they had mastered from their previous math experiences. They were not designed to evaluate Saxon math students on their progress. The only valid way to determine which the next book to use would be is by evaluating the student's last four or five test scores in their current book. If those test scores are eighty or better, in a fifty minute test – using no partial credit – then they are prepared for the next level Saxon math book.
In March of 1993, in the preface to his first edition Physics textbook, John wrote about "The Coming Disaster in Science Education in America." He explained that this was a result of actions by the National Council of Teachers of Mathematics (NCTM). He went on to explain that the NCTM had decided, with no advance testing whatsoever, to replace testing for calculus, physics, chemistry and engineering with a watered-down mathematics curriculum that would emphasize the teaching of probability and statistics and would replace the development of paper-and-pencil skills with drills on calculators and computers. John Saxon believed that this shift in emphasis ". . . would leave the American student bereft of the detailed knowledge of the parts that permit comprehension of the whole."
If you use the books as John Saxon intended them to be used, you will join the multitude of other successful Saxon users who value his math books. I realize that every child is different. And while the above situations apply to about 99% of all students, there are always exceptions that justify the rule. If your particular situation does not fit neatly into the above descriptions, please feel free to email me at [email protected] or call me at (580) 234-0064 (CST). If you email me, please include your telephone number and I will call you at my expense.
March 2011
ARE THE NEW SAXON MATH BOOKS BETTER THAN THE OLD EDITIONS?
Some of you may remember that in the summer of 2004, the Saxon family sold Saxon Publishers to Harcourt Achieve. Just to put everything in perspective, Harcourt Achieve, Inc. was then owned by the Harcourt Corporation which in turn was later acquired by the multi-billion dollar conglomerate Reed-Elsevier who then sold Harcourt, Inc. to Houghton Mifflin creating the current company (that owns Saxon Publishers) which is now the Houghton-Mifflin Harcourt Company also known as HMHCO. It all reminds me of when the Savings and Loan Companies got the nickname "Velcro banks" because they changed names so often before they disappeared the way of the dinosaurs.
When I published my June 2007 news article, I told readers "Not to worry!" As I had said earlier when Harcourt acquired John Saxon's publishing company in 2004, the new sale should not affect the quality of John's books. I asked the obvious question, "Why would anyone buy someone's prize-winning 'Blue Ribbon Bull' to make hamburger with?" I did not believe that this new sale would change John's books much either, and I told the readers that if these changes became more than just cosmetics, I would certainly keep them informed.
Well, it is time to mention that some of the changes are no longer cosmetic. Some of the new editions are not what John Saxon had intended for his books. These new editions are vastly different, and both home school educators as well as classroom teachers must be aware of these changes and be selective about what editions and titles they should and should not use if they desire to continue with John Saxon's methods and standards.
Initially, these revised new editions were offered only to the public and private schools and not to the home school community. However, introduction of their new geometry textbook to the home school web site tells me that it may not be long before the new fourth editions of Algebra 1 and Algebra 2 replace the current third editions now offered on the website.
I could be wrong. Perhaps they added the geometry textbook to the home school website because some home school parents were unaware that a full year of high school geometry was already offered within the Algebra 2 textbook (first semester of geometry) and the first sixty lessons of the Advanced Mathematics textbook (second semester of geometry). Additionally, placing the geometry course in between the Saxon Algebra 1 and Saxon Algebra 2 textbooks is a sure formula for student frustration in Saxon Algebra 2 since the new geometry book does not contain algebra content. The only reference to "Geometry" in the new fourth edition of Saxon Algebra 1 is a reference in the index to "Geometric Sequences" found in lesson 105. That term is not related to geometry. It is the title given to an algebraic formula dealing with a sequence of numbers that have a common ratio between the consecutive terms.
It would be my hope that the senior executives at HMHCO would recognize the uniqueness and value of the current editions of John Saxon's math books that continue his methods and standards. However, to ensure you have the correct editions of John Saxon's math books, as he published them, you can go to my February 2010 news article where I list the correct editions to use from Math 54 through Calculus. The editions I referenced in that article will be good for many more decades.
February 2011
What are some of the main causes for student frustration or failure when using John Saxon's math books? (Part 2)
Last month we discussed the ESSENTIAL DO'S when using John Saxon's math books.
This month we will go over the ESSENTIAL DONT'S:
Don't Skip the First 30 – 35 Lessons in the Book. Many home school parents still believe that because the first thirty or so lessons in every Saxon math book appear to be a review of material in the last part of the previous textbook, they can skip them. Let's review the two elements of automaticity. The two critical elements are: repetition - over time!
Yes, some of the early problems in the textbook appear similar to the problems found in the last part of the previous textbook. They have, however, been changed from the previous textbook to ensure that the student has mastered the concept. Remember, part of the concept of mastery involves leaving the material for a period of time and then returning to it. Students are supposed to have sixty to ninety days off in the summer to rest their thought processes. They need this review to reinstate that thought process! Additionally, while the first lessons in the books do contain some review, they also contain new material as well.
I would add what I have asked thousands of home school parents these past nine years. "Must students always have to do something they do not know how to do? Why can't they do something they already know how to do? What is wrong with building or reinforcing their confidence in mathematics through review?"
Don't Skip Textbooks. Skipping a book in Saxon is like tearing out the middle pages of your piano sheet music and then attempting to play the entire piece while still providing a meaningful musical presentation. In my book, under the specific textbook descriptions, I discuss any legitimate textbook elimination based upon specific abilities of the individual students. However, these recommendations vary from student to student depending upon their background and ability.
Don't Skip Problems in the Daily Assignments. When students complain that the daily workload of thirty problems is too much, it is generally the result of one of the following conditions:
Students are so involved in a multitude of activities that they cannot spend the thirty minutes to an hour each day required for Saxon mathematics.
Students are at a level above their capabilities and unable to adequately process the required concepts in the allotted time because of this difficulty.
The student is either a dawdler or just lazy!
Doing just the odd or just the even numbered problems in a Saxon math book is not the solution to those difficulties. As I mention in one of the early chapters in my book, there are two of each type of problem for several reasons - and doing just the odd or even is not one of those reasons!
Don't Skip Lessons. Incremental Development literally means introducing complicated math concepts to the students in small increments, rather than having them tackle the entire concept all at once. It is essential that students do a lesson a day and take a test every four to five lessons, depending on what book they are using. So what happens when you skip an easy lesson or two?
Very simply, the student cannot process the new material satisfactorily without having had a chance to read about it, and to understand its characteristics. Some students attempt to fix this shortcoming by then working on several lessons in a single day, to catch up to where they should be in the book. This technique is also not recommended. As I have told my classroom students on numerous occasions, "The only way to eat an elephant is one bite at a time."
While my book goes into more detail, I believe these few simple rules about what TO DO and what NOT TO DO to ensure success when using John's math books will benefit home school educators who use, or are contemplating using, Saxon math books.
So long as you use the books and editions I referenced in my book, and later re-iterated in my February 2010 news article, you will find that Saxon math books remain the best math books on the market today - if used correctly! Those referenced books and editions will be good for your child's math education - from fourth grade through their senior year in high school - for several more decades – or longer!
January 2011
What are some of the main causes for student frustration or failure when using John Saxon's math books? (Part 1)
The unique incremental development process used in John Saxon's math textbooks,; coupled with the cumulative nature of the daily work make them excellent textbooks for use in either a classroom or home school environment. If the textbooks are not used correctly, however, they will eventually present problems for the students.
The uniqueness of John Saxon's method of incremental development, coupled with the cumulative nature of the daily work in every Saxon math textbook, requires specific rules be followed to ensure success – and ultimately mastery!
In the next several news articles, we will discuss the ESSENTIAL DO'S and DON'T'S when using John Saxon's math books.
This month we will discuss the ESSENTIAL DO'S.
Do Place the Student in the Correct Level Math Book. Probably the vast majority of families who dislike John Saxon's math books do so because the student is using a math book above his or her capability. Since all of John's math books were written at the appropriate reading level of the student (or a grade level below), the problem is not one of students not being able to read the material presented to them, but their not being able to comprehend the math concepts being presented to them. This frustration is then interpreted as being created by the book and not by incorrect placement of the student.
Do Always Use the Correct Edition. Using the wrong edition of a Saxon math book can quickly lead to insurmountable problems. For example, moving from the first or second edition of Math 76 to the second or third edition of Math 87, or the third edition of Algebra ½ would be like moving from Math 65 to Algebra ½ in the current editions. For more information on which editions of John's books are still valid, read the earlier published February 2010 Newsletter, or read pages 15 – 18 in my book.
Do Finish The Entire Book. Finishing the entire textbook is critical to success in the next level book. I know, parents and teachers often ask me, "Why finish the last twenty or so lessons when much of that same material is presented in the first thirty or so lessons of the next level textbook?" While the first twenty or so lessons of the next level Saxon book may appear to cover the same concepts as the last thirty or so lessons in the previous book, the new textbook presents the review concepts in different and more challenging ways. Additionally, there are new concepts mixed in with them. The review is used to enable a review of necessary concepts while building the student's confidence back up after a few months off during the summer. Then comes the argument from some home school educators, "But we do not take any break between books – we go year round, so the review is not necessary."
My only reply to that is "Why must students always do something they do not know how to do? Can't they sometimes just review to build their confidence by doing something they already know how to do? If they are continuing year round, and already know how to do some of the early concepts in the next textbook, then it won't take them long to do their daily assignment. I once had a public school superintendant ask me "Which is more important, mastery or completing the book?" To which I replied, "They are synonymous."
Do All of the Problems - Every Day. There is a reason the problems come in pairs, and it is not so the student can do just the odd or even problems. The two problems are different from each other to keep the student from memorizing the procedure, as opposed to mastering the concept. Students who cannot complete the thirty problems each day in about an hour are either dawdling, or are at a level of mathematics above their capabilities, based upon their previous math experiences.
Do Follow the Order of the Lessons. I am often asked by parents at workshops and in email "Why study both lessons seventeen and eighteen?After all, they both cover the same concept?"Why not just skip lesson eighteen and go straight to lesson nineteen?" Why do both lessons? Well, because the author took an extremely difficult math concept and separated it into two different lessons. This allowed the student to more readily comprehend the entire concept, a concept which will be presented again in a more challenging way later in lesson twenty-seven of that book!
Do Give All of the Scheduled Tests – On Time. In every test booklet, in front of the printed Test 1 is a schedule for the required tests. Not testing is not an option! I have often heard home school parents say, "He does so well on his daily work; why test him?" To which I reply, "The results of the daily work reflect memory – the results of the weekly tests reflect mastery!" The results of the last five tests in every book give an indication of whether or not the student is prepared for the next level math book. Scores of eighty or better on any test reflect minimal mastery achieved. Scores of eighty or better on the last five tests in the series tell you the student is prepared to advance to the next level.
In the February newsletter, we will discuss the ESSENTIAL DON'T'S when using John's books. | 677.169 | 1 |
books.google.com - ".... Mathematics from an Advanced Standpoint
". His three-part treatment begins with topics associated with arithmetic, including calculating with natural numbers, the first extension of the notion of number, special properties of integers, and complex numbers. Algebra-related subjects constitute the second part, which examines real equations with real unknowns and equations in the field of complex quantities. The final part explores elements of analysis, with discussions of logarithmic and exponential functions, the goniometric functions, and infinitesimal calculus. 1932 ed. 125 figures.
Very good book, advanced explanations for many things either taken for granted or new perspectives to look at things. Interesting things like impossibility to inscribe heptagon using compass and ...Read full review | 677.169 | 1 |
Galois Theory (Universitext)
Book summary
A clear, efficient exposition of this topic with complete proofs and exercises, covering cubic and quartic formulas; fundamental theory of Galois theory; insolvability of the quintic; Galoiss Great Theorem; and computation of Galois groups of cubics and quartics. Suitable for first-year graduate students, either as a text for a course or for study outside the classroom, this new edition has been completely rewritten in an attempt to make proofs clearer by providing more details. It now begins with a short section on symmetry groups of polygons in the plane, for there is an analogy between polygons and their symmetry groups and polynomials and their Galois groups - an analogy which serves to help readers organise the various field theoretic definitions and constructions. The text is rounded off by appendices on group theory, ruler-compass constructions, and the early history of Galois Theory. The exposition has been redesigned so that the discussion of solvability by radicals now appears later and several new theorems not found in the first edition are included. [via] | 677.169 | 1 |
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Geometry: With Geometry Explorer
Publisher:
McGraw-Hill
Number of Pages:
463
Price:
103.44
ISBN:
0-07-312990-9
Given the number of high quality geometry texts published within the past ten years or so, one could be forgiven for assuming that the subject has undergone some sort of renaissance within mathematics education. In reality, however, geometry still has no coherent presence within school mathematics and it continues to play only a minor role within undergraduate mathematics.
So here is a book whose overall aim is to 'develop an intuitive feel for geometry whilst also establishing sound understanding of underlying proofs and abstractions'. This, of course, is the common challenge faced by authors of university geometry texts, who necessarily have the two-fold task of laying foundations that should have been provided at school, whilst simultaneously transporting students into the realm of advanced geometric topics. But having taught geometry to undergraduate students, I long ago ceased to be amazed by their lack of basic geometric knowledge, and I have sometimes spent more time going backwards than forwards.
In addressing this dilemma, Michael Hvidsten has invoked the use of new technology in the form of Geometry Explorer (provided with the book). This, of course, is not the only way in which geometric intuition can be strengthened, and it can't be expected to totally compensate for many previous years of geometric neglect. Nonetheless, although employed in a supplementary fashion, I very much like the use to which it is put. For example, in the very first exercise project, students use it to explore many properties of the Golden Rectangle. Much later on, it is used to investigate the tiling of the hyperbolic plane, and there are many other 'projects' that also involve the application of this software. In addition, students may be inclined to experiment with it of their own volition — and the more the better.
Anyway, the book begins with a stimulating chapter on axiomatic methods, with historical commentary running throughout. It considers the strengths and weaknesses of various sets of axioms including those of Euclid, Hilbert and Birkhoff. These are considered in the context of consistency, independence and completeness, followed by some explanation of Gödel's Incompleteness theorem. Chapters 2 to 6 are respectively devoted to Euclidean Geometry, Analytic Geometry, Constructions, Transformation Geometry and Symmetry.
Chapters 7 and 8 provide a cogent introduction to Non-Euclidean Geometry and Non-Euclidean Transformations and I particularly liked the author's treatment of all this material. Initial motivation comes from a re-examination of Euclid's axioms and the history of the parallel postulate. Also, Project 2 (in the very first chapter) requires the use of Geometry Explorer to test the validity of Euclidean properties in hyperbolic geometry, thereby setting the scene for later major investigation of the Poincaré model. The Klein model is also introduced and shown to be structurally equivalent to that of Poincaré. Marvellous stuff!
The last chapter provides an appealing introduction to Fractal Geometry together with discussion of many relevant ideas such as contraction mappings, algorithmic geometry and space-filling curves. And Geometry Explorer is again brought into action via 'Projects' on Snowflake Curves, Complex Branching Systems and IFS Ferns.
Intended for use with mathematics students at 'junior or senior collegiatelevel', the book requires a background in geometry provided by elementary high school and some expertise with matrix algebra and groups is also recommended. Generally, I very much like this book but, for the following reasons, I have considerable reservations regarding its use with such a target group.
Treatment of much of the material, although soundly formulated, could be very demanding for those with weak geometric backgrounds (the majority of undergraduate students, of course).
There is a rather sudden transition from the synthetic/axiomatic approach of Euclid to the transformational approach of Klein. And, although isometries are applied to analysis of symmetry and tiling patterns etc, this algebraic machinery is not employed to reveal a range of geometric results akin to the theorems of Euclid. At a very basic level, for example, if two triangles have equal sides, one can find an isometry that maps one onto the other, thus linking the ideas of Euclidean and Kleinian congruence.
The exercises are predominantly of an investigative nature and many of them require reports to be written by students. This is all very well, except that students need considerable preparation to meet the demands of such a research-based approach to learning.
Of course, no review is complete without mention of one or two minor quibbles. Firstly, it was not the Arabs who introduced symbols such as x2, and I still shudder on seeing expressions like 1/0 = ∞. Finally, no history of vector analysis is complete without mention of Willard Gibbs!
However, if only for the chapters on axiomatics, non-Euclidean and fractal geometry, this book should be regarded as a very valuable addition to the existing literature. Moreover, it is suitable for use with a much wider readership than specified by the author, but it wouldn't be my first choice as a self-tuition manual for those with weak geometric backgrounds.
Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible. | 677.169 | 1 |
Geometry Lessons in the Waldorf School – Volume 2
Freehand Form Drawing and Basic Geometric Construction in Grades 4 and 5
Ernst Schuberth
Softbound
Includes CD-ROM with additional exercises and color plates
$18.00
This book is the second volume* of mathematician and Waldorf teacher Ernst Schuberth's Geometry Lessons in the Waldorf School. With an abundance of black and white drawings and a clear, descriptive text, Schubreth covers the free form drawing and basic constructions taught in the Waldorf school fourth and fifth grades.
A real bonus is the accompanying CD Rom that has offers additional exercises and color plates for the teacher to study. | 677.169 | 1 |
& You: The Power & Use of Mathematics
As the world around us changes and information comes at warp speed, it is more important than ever to be quantitatively literate. Yet most U.S. ...Show synopsisAs the world around us changes and information comes at warp speed, it is more important than ever to be quantitatively literate. Yet most U.S. students leave high school with quantitative skills far below what they need and what employers are seeking, and virtually every college finds that many students need remedial mathematics. Based on the latest educational research, "Math & YOU" helps students develop the quantitative skills needed to be successful in school and the workplace, using real data, problems based on everyday situations, and activities built around topics that are recognizable and relevant. With this approach, students become comfortable with quantitative ideas and proficient in applying them. In addition, to support the printed text, "Math & YOU" provides an online eBook accompanied by additional teaching aids, all part of a robust companion Web site. Hardcover edition available upon request. Ask your local W.H. Freeman representative. Math & YOU HallmarksConfidence with Mathematics. One of the goals of the ""Math & YOU"" program is to help students become comfortable with quantitative ideas and proficient in applying them. Students routinely quantify, interpret, and check information such as comparing the total compensation of two job offers, or comparing and analyzing a budget Cultural Appreciation. "Math & YOU" provides examples and exercises that help student to understand the nature of mathematics and its importance for comprehending issues in the public realm. Logical Thinking. "The Math & YOU" program develops habits of inquiry, prepares students to look for appropriate information, and exposes them to arguments so that they can analyze and reason to get at the real issues. Making Decisions. One of the main threads of the "Math & YOU" program is to help students develop the habit of using mathematics to make decisions in everyday life. One of the goals of the text is for students to see that mathematics is a powerful tool for living. Mathematics in Context. The "Math & YOU" program helps students to learn to use mathematical tools in specific settings where the context provides meaning. Number Sense. "The Math & YOU" program begins with a chapter that reviews the meaning of numbers, estimation and measuring. Throughout the rest of the program students develop intuition, confidence, and common sense for employing numbers. Practice Skills. Throughout the "Math & YOU" program students encounter quantitative problems that they are likely to encounter at home or work. This helps students become adept at using elementary mathematics in a wide variety of common situations 1608406024 Purchased like new but not a guarantee that...Fine.
Description:New. 1608406024 Purchased like new but not a guarantee that...New | 677.169 | 1 |
Offering
10+ subjects
including logic logic logic | 677.169 | 1 |
Bob Miller's Basic Math and Pre-Algebra for the Clueless
Bob Miller's fail-safe methodology helps students grasp basic math and pre-algebra. All of the courses in the junior high, high school, and college mathematics curriculum require a thorough grounding in the fundamentals, principles, and techniques of basic math and pre-algebra, yet many students have difficulty grasping the necessary concepts. Utilizing the author's acclaimed and patented fail-safe methodology for making mathematics easy to understand, Bob Miller's Basic Math and Pre-Algebra for the Clueless enhances students' facility in these techniques and in understanding the basics. This valuable new addition to Bob Miller's Clueless series provides students with the reassuring help they need to master these fundamental techniques, gives them a solid understanding of how basic mathematics works, and prepares them to perform well in any further mathematics courses they take.
Tcl/Tk is enjoying a resurgence of popularity and interest in the computing community due to the fact that it is relatively easy to learn, powerful, fast, permits rapid development, and runs on all ... | 677.169 | 1 |
Guide students through the Further Maths certificate with this handy practice book, featuring short topic explanations, worked examples and loads of graded practice exercises that will stretch and challenge. | 677.169 | 1 |
Numbers
This unit will help you understand more about real numbers and their properties. It...
This
By the end of this unit you should be able to:
explain the relationship between rational numbers and recurring decimals;
explain the term irrational number and describe how such a number can be represented on a number line;
find a rational and an irrational number between any two distinct real numbers;
solve inequalities by rearranging them into simpler equivalent forms;
solve inequalities involving modulus signs;
state and use the Triangle Inequality;
use the Binomial Theorem and mathematical induction to prove inequalities which involve an integer n;
explain the terms bounded above, bounded below and bounded;
use the strategies for determining least upper bounds and greatest lower bounds;
state the Least Upper Bound Property and the Greatest Lower Bound Property;
explain how the Least Upper Bound Property is used to define arithmetical operations with real numbers;
Hi,
I've spoken to the course team and this unit, along with several others in Maths, is currently being looked at. Due to a system change some of the file formats are not compatible and so the course teams are deciding how to deal with the problem.
I have asked to be informed as soon as they have made their decision and I will get back to you when I hear from them.
Many apologies for the problems on this unit.
Best wishes
OpenLearn Moderator
Copyright & revisions
Publication details
Originally published: Wednesday, 29 | 677.169 | 1 |
Cliffs Quick Review for Geometry - 01 edition
Summary: When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts -- from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry -- and get the best possible grade.
At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented wri...show moreters who know how to cut to the chase -- and zero in on the essential information you need to succeed. ...show less
Ed Kohn, MS is an outstanding educator and author with over 33 years experience teaching mathematics. Currently, he is the testing coordinator and math department chairman at Sherman Oaks Center for Enriched Studies condition. Interior is tight and bright. Paperback cover has moderate scuffing and corner bumps from shelf and reader wear. FREE shipping upgrade - If you live in the USA this book will arrive in...show more 4 to 6 business days when you choose the lowest cost shipping method when checking out. 100% Satisfaction Guaranteed. We ship promptly and worldwide | 677.169 | 1 |
Deep Dive into Mathematica's Numerics: Applications and Tips
Andrew Moylan
In this course from the Wolfram Mathematica Virtual Conference 2011, you'll learn how to best use Mathematica's numerics functions in advanced settings. Topics include techniques and best practices for using multiple numerics functions together, advanced numeric features, and understanding precision and accuracy. | 677.169 | 1 |
Linear Algebra
A college level study of linear algebra which includes systems of equations, matrices, vector spaces, linear transformations, bases, dimension, eigenvalues, eigenvectors, and orthogonality. This course emphasizes computational techniques, geometry and theoretical structure, creative problem solving, and proofs. Upon successful completion of the course, the student will receive 3 credit hours from the Central Virginia Community College.
Course Materials:
Textbook: David C. Lay, Linear Algebra and its Applications, 2nd edition. We will also use graphing calculators.
Typical Hours for Students Session I (7:30-10:10)Session II (8:25-11:05)
Period 1: 7:30-8:20 Period 2: 8:25-9:15
Period 2: 8:25-9:15 Period 3: 9:20-10:10
Period 3: 9:20-10:10 Period 4: 10:15-11:05 | 677.169 | 1 |
More About
This Textbook
Overview
The new 3rd edition of Cynthia Young's Algebra & Trigonometry continues to bridge the gap between in-class work and homework by helping readers overcome common learning barriers and build confidence in their ability to do mathematics. The text features truly unique, strong pedagogy and is written in a clear, single voice that speaks directly to students and mirrors how instructors communicate in lectures. In this revision, Young enables readers to become independent, successful learners by including hundreds of additional exercises, more opportunities to use technology, and a new themed modeling project that empowers them to apply what they have learned in the classroom to the world outside the classroom. The seamlessly integrated digital and print resources to accompany Algebra & Trigonometry 3e offer additional tools to help users experience success.
Meet the Author
Cynthia Young received her BA in Math Education from UNC in 1990, has two masters, one in Mathematical Sciences from UCF in 1993 and a second in Electrical Engineering from the University of Washington in 1997. Finally, she received a PhD in Applied Mathematics from the University of Washington in 1996. She is already a tenured professor at UCF and is very actively involved in the supervision of UCF's graduate and undergraduate research assistants. Before becoming an award-winning Associate Professor at UCF, Cynthia taught High School. Cynthia received numerous grants and was named the principal investigator on six military and academic research projects. She has been an administrator/advisor to the Florida Space Institute at the Kennedy Space Center since 1998. Cynthia is a veteran presenter at conferences and conventions and has published over a dozen journal articles. In addition, she has been a contributor to several texts, including a College Algebra workbook for McGraw-Hill. Lastly, she edited the Marcel Decker's Optical Engineering Encyclopedia | 677.169 | 1 |
course provides Mechanical Engineering students with an awareness of various responses exhibited by solid engineering...
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This these properties characterize material response; quantitative skills to deal with materials-limiting problems in engineering design; and a basis for materials selection in mechanical design.
This is a free online course offered by the Saylor Foundation.'The courses included in this program are designed for the high...
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This is a free online course offered by the Saylor Foundation.'The courses included in this program are designed for the high school student preparing for college or the adult learner who needs a refresher course or two in mathematics.Each of the courses in this series includes instructional videos and practice problems from Khan Academy™ (Khan Academy™ is a library of over 3,000 videos covering a range of topics, including math and physics) that will help you master the foundational knowledge necessary for success in College Algebra (MA001: Beginning Algebra) and beyond.These courses focus on the ways in which math relates to common "real world" situations, transactions, and phenomena, such as personal finance, business, and the sciences. This "real world" focus will help you grasp the importance of the mathematical concepts you encounter in these courses and understand why you need quantitative and algebraic skills in order to be successful both in college and in your day-to-day-life.' lab series designed to familiarize students with using computer models to answer biochemical questions. ...
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Undergraduate lab series designed to familiarize students with using computer models to answer biochemical questions. Ideally, this lab would be taught as a supplement to a concurrent lecture course. Students are assumed to have completed one year of undergraduate calculus.Topics include acid-base chemistry, Gibbs free energy, Michaelis-Menten kinetics, enzyme inhibition, hemoglobin, and the Bohr effect. Math skills used include graphing (2-D and 3-D), algebra, logarithms, and numerical solutions to systems of equations | 677.169 | 1 |
This fast, effective tutorial helps you master core algebraic concepts -- from monomials, inequalities, and analytic geometry to functions and variations, roots and radicals, and word problems -- and get the best possible grade.
This fast, effective tutorial helps you master core physics concepts -- from classical mechanics, thermodynamics, and electricity to magnetism, light, and nuclear physics -- and get the best possible grade. | 677.169 | 1 |
books.google.com - Designed for students preparing to engage in their first struggles to understand and write proofs and to read mathematics independently, this is well suited as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology. The book teaches in detail how... Accompaniment to Higher Mathematics | 677.169 | 1 |
Based on a series of lectures for adult students, this lively and entertaining book proves that, far from being a dusty, dull subject, geometry is in fact full of beauty and fascination. The author's infectious enthusiasm is put to use in explaining many of the key concepts in the field, starting with the Golden Number and taking the reader on... more...
Learn geometry at your own pace What are congruent circles? How do you find the hypotenuse of a triangle? What is the sum of the angles in a decagon? How can you apply geometric equations to your daily life? With the unbeatable study companion Geometry: A Self-Teaching Guide, you'll discover the answers to these questions and many more. This thorough... more...
You, too, can understand geometry---- just ask Dr. Math ? ! Are,...Meyer's Geometry and Its Applications, Second Edition , combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text... more...
Deformable objects are ubiquitous in the world, on various levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications, ranging from medicine to security. This book provides an overview of the state of science in analysis and synthesis of non-rigid shapes. more...
Like other areas of mathematics, geometry is a continually growing and evolving field. Computers, technology, and the sciences drive many new discoveries in mathematics. For geometry, the areas of quantum computers, computer graphics, nanotechnology, crystallography, and theoretical physics have been particularly relevant in the past few years. There... more... | 677.169 | 1 |
Graphing Calculator ti-84? What is the difference between the ti-84 and the ti-84 plus silver edition?
(See answer below)
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There isn't a huge difference between the TI-84 Plus and the TI-84 Plus Silver Edition. Aside from a few minor aspects, they could even be considered the same calculator. Here are some of the ...
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Complete conversation
User: Graphing Calculator ti-84? What is the difference between the ti-84 and the ti-84 plus silver edition?
Weegy: There isn't a huge difference between the TI-84 Plus and the TI-84 Plus Silver Edition. Aside from a few minor aspects, they could even be considered the same calculator. Here are some of the advantages the Silver Edition has over the regular edition:
- More Flash ROM (SE has 1.5 MB user accessible Flash memory while regular only has 480 KB)
- Comes with more preloaded apps (though you could simply transfer them onto a regular from online)
- Comes in different colors
- Supports interchangeable faceplates
Aside from those, the two are pretty much the same thing. Auto Answered|Score 1
Weegy: What is the difference between a Texas Instruments a TI 84 calculator ... What is the difference between a ti-89 titanium graphing calculator and a ti-84 silver plus graphing ... Auto Answered|Score .7617
Weegy: I have a TI-84 Plus calculator and when I try to go play the game I put on it I can't because there is a ... What is the file extension 8XU for the TI-84 plus silver edition? ... Auto Answered|Score .6248
Weegy: What is the file extension 8XU for the TI-84 plus silver edition? ... What is the file extension 8XU for the TI-84 plus silver edition? How do I download games on a TI-84 Plus ... Auto Answered|Score .6046 | 677.169 | 1 |
Subject: Mathematics (8 - 12) Title: Systems of Equations: What Method Do You Prefer? Description: The purpose of this lesson is to help students apply math concepts of solving systems of equations to real life situations. The students will use the three methods of graphing, substitution, and elimination to solve the system of equations Pick's Theorem as a System of EquationsAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students gather three examples from a geoboard or other representation to generate a system of equations. The solution provides the coefficients for Pick s Theorem. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Road Trip!Add Bookmark Description: In this Illuminations lesson, students investigate the famous Traveling Salesman Problem by considering the shortest route between five northeastern cities. Three different algorithms for finding the shortest route are explored, and students are encouraged to look for others. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics,Science Title: Northwestern CrowsAdd BookmarkSubject: Mathematics,Science Title: Whelk-Come to MathematicsAdd BookmarkSubject: Mathematics Title: Isosceles Triangle Investigation Add Bookmark Description: This student interactive, from an Illuminations lesson, allows students to investigate the relationship between the area of the triangle and the length of its base. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Least Squares RegressionAdd Bookmark | 677.169 | 1 |
A Transition to Advancedfully addressing the frustration many students feel as they make the transition from beginning calculus to a more rigorous level of mathematics, A Transition to Advanced Mathematics provides a firm foundation in the major ideas needed for continued work in the discipline. The authors guide students to think and to express themselves mathematically--to analyze a situation, extract pertinent facts, and draw appropriate conclusions. With their proven approach, Smith, Eggen, and St. Andre introduce students to rigorous thinking about sets, r... MOREelations, functions and cardinality. The text also includes introductions to modern algebra and analysis with sufficient depth to capture some of their spirit and characteristics. Bridge | 677.169 | 1 |
books.google.com - The... standard mathematical tables and formulae
CRC standard mathematical tables and formulae, Volume 30
The handbook for modern mathematics, filled with tables, formulae, equations, and descriptions.
From inside the book
Review: CRC Standard Mathematical Tables and Formulae
User Review - Amarqu
Review: CRC Standard Mathematical Tables and Formulae
User Review - Adam Marqu | 677.169 | 1 |
Stewart, the author of the best-selling calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this book to address a problem they frequently saw in their calculus courses. Many students were not prepared to "think mathematically" but attempted to memorize facts and mimic examples. This trigonometry text has been designed specifically to help students learn to think mathematically and to develop true problem-solving skills. Patient, clear, and accurate, this text consistently illustrates how useful and applicable trigonometry is to real life. | 677.169 | 1 |
I've always wanted to excel in how babylonians solved specified system of equations, it seems like there's a lot that can be done with it that I can't do otherwise. I've browsed the internet for some good learning resources, and consulted the local library for some books, but all the data seems to be targeted at people who already understand the subject. Is there any resource that can help new students as well?
Well of course there is. If you are confident about learning how babylonians solved specified system of equations, then Algebrator can be of great benefit to you. It is made in such a manner that almost anyone can use it. You don't need to be a computer expert in order to use the program.
I agree, a good software can do miracles . I used a few but Algebrator is the greatest. It doesn't make a difference what class you are in, I myself used it in Intermediate algebra and Algebra 2 too, so you don't have to worry that it's not on your level. If you never had a software before I can tell you it's not hard, you don't need to know anything about the computer to use it. You just have to type in the keywords of the exercise, and then the software solves it step by step, so you get more than just the answer.
A extraordinary piece of algebra software is Algebrator. Even I faced similar problems while solving angle complements, factoring and perpendicular lines. Just by typing in the problem workbookand clicking on Solve – and step by step solution to my math homework would be ready. I have used it through several algebra classes - Algebra 2, Algebra 2 and College Algebra. I highly recommend the program. | 677.169 | 1 |
This book presents the fundamentals of multiple regression, linear modelling, multivariate analysis, and other statistical methods for the elucidation of complicated data. The author uses the basic terms of matrix algebra to provide a clear and accessible guide for biologists, engineers, students of statistics, and others concerned with data analysis. Numerical methods for matrices are described and the book contains a set of algorithms to make such methods generally available.
Introducing Matrices
Determinants
Inverse Matrices
Linear Dependence and Rank
Simultaneous Equations and Generalized Inverses
Linear Spaces
Quadratic Forms and Eigensystems
Matrix Algorithms | 677.169 | 1 |
Missouri Western State University
Developmental Math Program (DMP)
COURSE NAME: Foundations for
University Mathematics
COURSE OBJECTIVES:
The objective of this course is to
provide the student with the fundamentals of arithmetic and algebra, and the
other mathematical skills necessary to succeed in College Algebra, Finite
Mathematics or Contemporary Problem Solving. Assessment of a student's knowledge
and skill level in the following will determine the specific topics that a
student will study.
Arithmetic operations and
properties of real numbers.
Subtract, multiply and divide
algebraic expressions.
Properties of exponents of
algebraic expressions.
Linear equations and
inequalities in a single variable.
Factoring algebraic expressions.
Solving quadratic equations by
factoring.
Graphing and manipulation of
linear equations and inequalities in two variables.
Graph linear equations and
inequalities in two variables, and graph quadratic equations. | 677.169 | 1 |
A Fresh Start for
Collegiate Mathematics:
Rethinking the
Courses below Calculus
[email protected]
[email protected]
College Algebra and Precalculus
Each year, more than 1,000,000 students
take college algebra, precalculus, and
related courses.
The Focus in these Courses
Most college algebra courses and certainly all
precalculus courses were originally intended
and designed to prepare students for calculus.
Most of them are still offered in that spirit.
But only a small percentage of the students have
any intention of going on to calculus!
Enrollment Flows
Based on several studies of enrollment flows into
calculus:
• Less than 5% of the students who start college
algebra courses ever start Calculus I
• Virtually none of the students who pass college
algebra courses ever start Calculus III
• Perhaps 30-40% of the students who pass
precalculus courses ever start Calculus I
• Only about 10% of students in college algebra
are in majors that require calculus.
Why Students Take These Courses
• Required by other departments
• Satisfy general education requirements
• To prepare for calculus
• For the love of mathematics
What the Majority of Students Need
• Conceptual Understanding, not rote
manipulation
• Realistic applications and mathematical
modeling that reflect the way mathematics
is used in other disciplines
• Fitting functions to data
• Recursion and difference equations – the
mathematical language of spreadsheets
The Link to Calculus
Calculus and Related Enrollments
In 2000, about 676,000 students took Calculus,
Differential Equations, Linear Algebra, and
Discrete Mathematics
(This is up 6% from 1995)
Over the same time period, however, calculus
enrollment in college has been steady, at best.
Calculus and Related Enrollments
In comparison, in 2000, 171,400 students took one
of the two AP Calculus exams – either AB or BC.
(This is up 40% from 1995)
In 2004, 225,000 students took AP Calculus exams
In 2005, about 240,000 took AP Calculus exams
Reportedly, about twice as many students take
calculus in high school, but do not take an AP
exam.
Some Implications
Today more students take calculus in high school
than in college
And, as ever more students take more
mathematics, especially calculus, in high school,
we should expect:
• Fewer students taking these courses in college
• The overall quality of the students who take
these courses in college will decrease.
Another Conclusion
We should anticipate the day, in the
not too distant future, when
college calculus, like college algebra,
becomes a semi-remedial course.
(Several elite colleges already have stopped
giving credit for Calculus I.)
Who Are the Students?
Based on the enrollment figures, the students
who take college algebra and related courses are
not going to become mathematics majors.
They are not going to be majors in any of the
mathematics intensive disciplines.
Associates Degrees in Mathematics
In 2000,
• There were 564,933 associate degrees
• Of these, 675 were in mathematics
This is one-tenth of one percent!
Bachelor's Degrees in Mathematics
In 2000,
• There were 457,056 bachelor's degrees
• Of these, 3,412 were in mathematics
This is seven-tenths of one percent!
Some Conclusions
Few, if any, math departments can exist
based solely on offerings for math and
related majors. Whether we like it or not,
mathematics is a service department at
almost all institutions.
And college algebra and related courses
exist almost exclusively to serve the needs of
other disciplines.
Some Conclusions
If we fail to offer courses that meet the
needs of the students in the other
disciplines, those departments will
increasingly drop the requirements for
math courses. This is already starting to
happen in engineering.
Math departments may well end up offering
little beyond developmental algebra courses
that serve little purpose.
Responding to the
Challenges
Four Special Invited Conferences
• Rethinking the Preparation for Calculus,
• Forum on Quantitative Literacy,
• CRAFTY Curriculum Foundations Project,
• Reforming College Algebra,
Common Recommendations
• ―College Algebra‖ courses should stress
conceptual understanding, not rote manipulation.
• "College Algebra‖ courses should be real-world
problem based:
Every topic should be introduced through a
real-world problem and then the mathematics
necessary to solve the problem is developed.
Common Recommendations
• ―College Algebra‖ courses should focus on
mathematical modeling—that is,
– transforming a real-world problem into
mathematics using linear, exponential and
power functions, systems of equations,
graphing, or difference equations.
– using the model to answer problems in
context.
– interpreting the results and changing the
model if needed.
Common Recommendations
• "College Algebra‖ courses should emphasize
communication skills: reading, writing,
presenting, and listening.
These skills are needed on the job and for
effective citizenship as well as in academia.
• "College Algebra‖ courses should make
appropriate use of technology to enhance
conceptual understanding, visualization, inquiry,
as well as for computation.
Common Recommendations
• ―College Algebra‖ courses should feature
student-centered rather than instructor-centered
pedagogy.
- They should include hands-on activities
rather than be all lecture.
- They should emphasize small group projects
involving inquiry and inference.
Important Volumes
• AMATYC Crossroads Standards.
• NCTM, Principles and Standards for School
Mathematics.
• CUPM Curriculum Guide: Undergraduate
Programs and Courses in the Mathematical
Sciences, MAA Reports.
• Ganter, Susan and Bill Barker, Eds., A
Collective Vision: Voices of the Partner
Disciplines, MAA Reports.
Important Volumes
• Madison, Bernie and Lynn Steen, Eds.,
Quantitative Literacy: Why Numeracy Matters for
Schools and Colleges, National Council on
Education and the Disciplines, Princeton.
• Baxter Hastings, Nancy, Flo Gordon, Shelly
Gordon, and Jack Narayan, Eds., A Fresh Start
for Collegiate Mathematics: Rethinking the
Courses below Calculus, MAA Notes.
AMATYC Crossroads Standards
• In general, emphasis on the meaning and use of
mathematical ideas must increase, and attention to
rote manipulation must decrease.
• Faculty should include fewer topics but cover
them in greater depth, with greater understanding,
and with more flexibility. Such an approach will
enable students to adapt to new situations.
• Areas that should receive increased attention
include the conceptual understanding of
mathematical ideas.
CUPM Curriculum Guide
• All students, those for whom the (introductory
mathematics) course is terminal and those for
whom it serves as a springboard, need to learn
to think effectively, quantitatively and logically.
• Students must learn with understanding,
focusing on relatively few concepts but treating
them in depth. Treating ideas in depth includes
presenting each concept from multiple points of
view and in progressively more sophisticated
contexts.
CUPM Curriculum Guide
• A study of these (disciplinary) reports and the
textbooks and curricula of courses in other
disciplines shows that the algorithmic skills that
are the focus of computational college algebra
courses are much less important than
understanding the underlying concepts.
• Students who are preparing to study calculus
need to develop conceptual understanding as well
as computational skills.
Voices of the Partner
Disciplines
CRAFTY's Curriculum
Foundations Project
Curriculum Foundations Project
A series of 11 workshops with leading
educators from 17 quantitative disciplines to
inform the mathematics community of the
current mathematical needs of each
discipline.
The results are summarized in the MAA
Reports volume: A Collective Vision: Voices
from the Partner Disciplines, edited by Susan
Ganter and Bill Barker.
What the Physicists Said
• Students need conceptual understanding first,
and some comfort in using basic skills; then a
deeper approach and more sophisticated skills
become meaningful. Computational skill
without theoretical understanding is shallow.
What the Physicists Said
• Students should be able to focus a situation
into a problem, translate the problem into a
mathematical representation, plan a solution,
and then execute the plan. Finally, students
should be trained to check a solution for
reasonableness.
What the Physicists Said
• The learning of physics depends less directly
than one might think on previous learning in
mathematics. We just want students who can
think. The ability to actively think is the most
important thing students need to get from
mathematics education.
What Business Faculty Said
• Courses should stress conceptual
understanding (motivating the math with the
―why's‖ – not just the ―how's‖).
• Students should be comfortable taking a
problem and casting it in mathematical terms.
• Courses should use industry standard
technology (spreadsheets).
Common Themes from All Disciplines
• Strong emphasis on problem solving
• Strong emphasis on mathematical modeling
• Conceptual understanding is more
important than skill development
• Development of critical thinking and
reasoning skills is essential
Common Themes from All Disciplines
• Use of technology, especially spreadsheets
• Development of communication skills
(written and oral)
• Greater emphasis on probability and
statistics
• Greater cooperation between mathematics
and the other disciplines
A Fresh Start for Collegiate
Mathematics:
Rethinking the Courses below
Calculus
Nancy Baxter Hastings
Florence Gordon
Sheldon Gordon
Jack Narayan
MAA Notes, January 2006
A Fresh Start to Collegiate Math
Introduction
Nancy Baxter Hastings Overview of the Volume
Jack Narayan &
Darren Narayan The Conference: Rethinking the Preparation for Calculus
Lynn Steen Twenty Questions about Precalculus
Background
Mercedes McGowen Who are the Students Who Take Precalculus?
Steve Dunbar Enrollment Flow to and from Courses below Calculus
Deborah Hughes What Have We Learned from Calculus Reform? The Road to
Hallett Conceptual Understanding
Calculus and Introductory College Mathematics: Current Trends and
Susan Ganter Future Directions
A Fresh Start to Collegiate Math
New Visions for Introductory Collegiate Mathematics
Shelly Gordon Preparing Students for Calculus in the Twenty-First Century
Bernie Madison Preparing for Calculus and Preparing for Life
Don Small College Algebra: A Course in Crisis
Scott Herriott Changes in College Algebra
One Approach to Quantitative Literacy: Mathematics in Public
Janet Andersen Discourse
The Transition from High School to College
Zal Usiskin High School Overview and the Transition to College
Dan Teague Precalculus Reform: A High School Perspective
Eric Robinson & The Influence of Current Efforts to Improve School
John Maceli Mathematics on Preparation for Calculus
A Fresh Start to Collegiate Math
The Needs of Other Disciplines
Bill Barker and Fundamental Mathematics: Voices of the Partner
Susan Ganter Disciplines
Rich West Skills versus Concepts
Allan Rossman Integrating Data Analysis into Precalculus Courses
Student Learning and Research
Assessing What Students Learn: Reform versus
Florence Gordon Traditional Precalculus and Follow-up Calculus
Student Voices and the Transition from Standards-Based
Rebecca Walker Curriculum to College
A Fresh Start to Collegiate Math
Implementation
Robert Megginson Some Political and Practical Issues in Implementing Reform
Implementing Curricular Change in Precalculus: A Dean's
Judy Ackerman Perspective
Alternatives to the One-Size-Fits-All Precalculus/College
Bonnie Gold Algebra Course
Preparing for Calculus and Beyond: Some Curriculum Design
Al Cuoco Issues
Lang Moore and
David Smith Changing Technology Implies Changing Pedagogy
Sheldon Gordon The Need to Rethink Placement in Mathematics
Influencing the Mathematics Community
Launching a Precalculus Reform Movement: Influencing the
Bernie Madison Mathematics Community
Bonnie Saunders Mathematics Programs for the "Rest of Us"
Sheldon Gordon Where Do We Go from Here: Forging a National Initiative
A Fresh Start to Collegiate Math
Ideas and Projects that Work , Part 1
Doris An Alternate Approach: Integrating Precalculus into
Schattschneider Calculus
College Algebra Reform through Interdisciplinary
Bill Fox Applications
Elementary Math Models: College Algebra Topics and a
Dan Kalman Liberal Arts Approach
Brigette Lahme,
Jerry Morris and
Elias Toubassi The Case for Labs in Precalculus
Ideas and Projects that Work , Part 2
Gary Simundza The Fifth Rule: Experiential Mathematics
Darrell Abney and
James Reform Intermediate Algebra in Kentucky Community
Hougland Colleges
Marsha Davis Precalculus: Concepts in Context
A Fresh Start to Collegiate Math
Benny Evans Rethinking College Algebra
Sol Garfunkel From the Bottom Up
Florence Gordon &
Shelly Gordon Functioning in the Real World
Deborah Hughes
Hallett Importance of a Story Line Functions as a Model
Nancy Baxter Using a Guided-Inquiry Approach to Enhance Student Learning
Hastings in Precalculus
Allan Jacobs Maricopa Mathematics
Linda Kime Quantitative Reasoning
Developmental Algebra: The First Course for Many College
Mercedes McGowan Students
Allan Rossman Workshop Precalculus: Functions, Data and Models
Chris Schaufele &
Nancy Zumoff The Earth Math Projects
Don Small Contemporary College Algebra
A Fresh Start to Collegiate Math
Ernie Danforth,
Brian Gray,
Arlene Kleinstein,
Rick Patrick and Mathematics in Action: Empowering Students with
Sylvia Svitak Introductory and Intermediate College Mathematics
Todd Swanson Precalculus: A Study of Functions and Their Applications
David Wells
Lynn Tilson Successes and Failures of a Precalculus Reform Project
Distribution Plan
With support from the NSF, the MAA
has developed a distribution plan to
provide one free copy to any department
that requests one.
Announcements will be sent to all
department chairs informing them of the
details in February.
Common Themes
Common Themes
• Conceptual Understanding is more important
than rote manipulation
• The Rule of Four: Graphical, Numerical,
Algebraic and Verbal Representations
• Realistic Applications via Math Modeling
• Non-routine problems and assignments
• Algebra in Context – Not Just Drill
Common Themes
• Families of Functions – Linear, Exponential,
Power, Logarithmic, Polynomial, and
Sinusoidal
• The significance of the parameters in the
different families of functions
• Limitations of the models developed –
the practical significance of the domain
and range
Common Themes
• Data Analysis
• Connections to Other Disciplines
• Writing and Communication
• More Active Classroom Environment –
Group Work, Collaborative Learning,
Exploratory Approach to Mathematics
• Use of Technology in Teaching and Learning
Conceptual Understanding
• What does conceptual understanding mean?
• How do you recognize its presence or absence?
• How do you encourage its development?
• How do you assess whether students have
developed conceptual understanding?
What Does the Slope Mean?
Comparison of student response to a problem on the final
exams in Traditional vs. Reform College Algebra/Trig
Brookville College enrolled 2546 students in 1996 and 2702 students
in 1998. Assume that enrollment follows a linear growth pattern.
a. Write a linear equation giving the enrollment in terms of the year t.
b. If the trend continues, what will the enrollment be in the year 2016?
c. What is the slope of the line you found in part (a)?
d. Explain, using an English sentence, the meaning of the
slope.
e. If the trend continues, when will there be 3500 students?
Responses in Traditional Class
1. The meaning of the slope is the amount that is gained in years
and students in a given amount of time.
2. The ratio of students to the number of years.
3. Difference of the y's over the x's.
4. Since it is positive it increases.
5. On a graph, for every point you move to the right on the x-
axis. You move up 78 points on the y-axis.
6. The slope in this equation means the students enrolled in 1996.
Y = MX + B .
7. The amount of students that enroll within a period of time.
8. Every year the enrollment increases by 78 students.
9. The slope here is 78 which means for each unit of time, (1
year) there are 78 more students enrolled.
Responses in Traditional Class
10. No response
11. No response
12. No response
13. No response
14. The change in the x-coordinates over the change in the y-
coordinates.
15. This is the rise in the number of students.
16. The slope is the average amount of years it takes to get 156
more students enrolled in the school.
17. Its how many times a year it increases.
18. The slope is the increase of students per year.
Responses in Reform Class
1. This means that for every year the number of students
increases by 78.
2. The slope means that for every additional year the number of
students increase by 78.
3. For every year that passes, the student number enrolled
increases 78 on the previous year.
4. As each year goes by, the # of enrolled students goes up by 78.
5. This means that every year the number of enrolled students
goes up by 78 students.
6. The slope means that the number of students enrolled in
Brookville college increases by 78.
7. Every year after 1996, 78 more students will enroll at
Brookville college.
8. Number of students enrolled increases by 78 each year.
Responses in Reform Class
9. This means that for every year, the amount of enrolled
students increase by 78.
10. Student enrollment increases by an average of 78 per year.
11. For every year that goes by, enrollment raises by 78
students.
12. That means every year the # of students enrolled increases
by 2,780 students.
13. For every year that passes there will be 78 more students
enrolled at Brookville college.
14. The slope means that every year, the enrollment of students
increases by 78 people.
15. Brookville college enrolled students increasing by 0.06127.
16. Every two years that passes the number of students which is
increasing the enrollment into Brookville College is 156.
Responses in Reform Class
17. This means that the college will enroll .0128 more students
each year.
18. By every two year increase the amount of students goes up
by 78 students.
19. The number of students enrolled increases by 78 every 2
years.
Understanding Slope
Both groups had comparable ability to calculate the slope of a
line. (In both groups, several students used x/y.)
It is far more important that our students understand what
the slope means in context, whether that context arises in a
math course, or in courses in other disciplines, or eventually
on the job.
Unless explicit attention is devoted to emphasizing the
conceptual understanding of what the slope means, the
majority of students are not able to create viable
interpretations on their own. And, without that understanding,
they are likely not able to apply the mathematics to realistic
situations.
Further Implications
If students can't make their own connections with a
concept as simple as slope, they won't be able to create
meaningful interpretations on their own for more
sophisticated concepts. For instance,
• What is the significance of the base (growth or decay
factor) in an exponential function?
• What is the meaning of the power in a power function?
• What do the parameters in a realistic sinusoidal model
tell about the phenomenon being modeled?
• What is the significance of the factors of a polynomial?
• What is the significance of the derivative?
• What is the significance of a definite integral?
Further Implications
If we focus only on manipulative skills
without developing
conceptual understanding,
we produce nothing more than students
who are only
Imperfect Organic Clones
of a TI-89
Results of the Study
The study involved 10 common questions on the
final exam in college algebra/trigonometry, most
of which were basically computational in nature.
The students in the reform sections outscored
those in the traditional, algebraic-oriented,
sections, on 7 of the 10 questions.
Follow-Up Results in Calculus
The students involved in the precalculus study
were then followed in Calculus I the next term.
The calculus course was a reform course with
emphasis also on conceptual understanding, not
just manipulation.
Follow-Up Results in Calculus
On every weekly quiz, on every class test, and on
the final exam, the students from the reform
sections of precalculus consistently scored higher
than the students from the traditional sections.
On an attitudinal survey, the students from the
reform section had significantly better attitudes
toward mathematics, its usefulness, and the
importance of technology for problem solving.
Follow-Up Results in Calculus
77% of the students who had been in a reform
section of precalculus ended up receiving a
passing grade in Calculus I.
41% of those who had been in a traditional
section of precalculus received a passing
grade in Calculus I.
Developing Conceptual
Understanding
Conceptual understanding cannot be just an add-on.
It must permeate every course and be a major focus
of the course.
Conceptual understanding must be accompanied by
realistic problems in the sense of mathematical
modeling.
Conceptual problems must appear in all sets of
examples, on all homework assignments, on all project
assignments, and most importantly, on all tests.
Otherwise, students will not see them as important.
Conclusions
We cannot simply concentrate on teaching the mathematical
techniques that the students need. It is as least as important
to stress conceptual understanding and the meaning of the
mathematics.
We can accomplish this by using a combination of realistic
and conceptual examples, homework problems, and test
problems that force students to think and explain, not just
manipulate symbols.
If we fail to do this, we are not adequately preparing our
students for successive mathematics courses, for courses in
other disciplines, and for using mathematics on the job and
throughout their lives.
Some Illustrative Examples
of Problems
to Develop or Test for
Conceptual Understanding
Identify each of the following functions (a) - (n) as linear, exponential,
logarithmic, or power. In each case, explain your reasoning.
(g) y = 1.05x (h) y = x1.05 (m) (n)
x y x y
0
(i) y = (0.7)t (j) y = v0.7 0 3 5
1
(k) z = L(-½) (l) 3U – 5V = 14 1 5.1 7
2
2 7.2 9.8
3
3 9.3 13.7
For the polynomial shown,
(a) What is the minimum degree? Give two different
reasons for your answer.
(b) What is the sign of the leading term? Explain.
(c) What are the real roots?
(d) What are the linear factors?
(e) How many complex roots does the polynomial have?
The following table shows world-wide wind power
generating capacity, in megawatts, in various
years.
Year 1980 1985 1988 1990 1992 1995 1997 1999
Wind
power 10 1020 1580 1930 2510 4820 7640 13840
15000
10000
5000
0
1980 1985 1990 1995 2000
(a) Which variable is the independent variable and which
is the dependent variable?
(b) Explain why an exponential function is the best model
to use for this data.
(c) Find the exponential function that models the relation-
ship between power P generated by wind and the year t.
(d) What are some reasonable values that you can use for
the domain and range of this function?
(e) What is the practical significance of the base in the
exponential function you created in part (c)?
(f) What is the doubling time for this exponential function?
Explain what it means.
(g) According to your model, what do you predict for the
total wind power generating capacity in 2010?
Biologists have long observed that the larger the area of
a region, the more species live there. The relationship is
best modeled by a power function. Puerto Rico has 40
species of amphibians and reptiles on 3459 square miles
and Hispaniola (Haiti and the Dominican Republic) has
84 species on 29,418 square miles.
(a) Determine a power function that relates the number
of species of reptiles and amphibians on a Caribbean
island to its area.
(b) Use the relationship to predict the number of
species of reptiles and amphibians on Cuba, which
measures 44218 square miles.
The accompanying table and associated scatterplot give
some data on the area (in square miles) of various
Caribbean islands and estimates on the number species
of amphibians and reptiles living on each.
Number o f Speci es
Island Area N 100
Redonda 1 3 80
60
Saba 4 5
40
Montserrat 40 9
20
Puerto Rico 3459 40 0
Jamaica 4411 39 0 15000 30000 45000
Area (square miles)
Hispaniola 29418 84
Cuba 44218 76
(a) Which variable is the independent variable and which
is the dependent variable?
(b) The overall pattern in the data suggests either a power
function with a positive power p < 1 or a logarithmic
function, both of which are increasing and concave down.
Explain why a power function is the better model to use
for this data.
(c) Find the power function that models the relationship
between the number of species, N, living on one of these
islands and the area, A, of the island and find the
correlation coefficient.
(d) What are some reasonable values that you can use for
the domain and range of this function?
(e) The area of Barbados is 166 square miles. Estimate the
number of species of amphibians and reptiles living there.
Write a possible formula for each of the following
trigonometric functions:
The average daytime high temperature in New York as a
function of the day of the year varies between 32F and
94F. Assume the coldest day occurs on the 30th day and
the hottest day on the 214th.
(a) Sketch the graph of the temperature as a function of
time over a three year time span.
(b) Write a formula for a sinusoidal function that models
the temperature over the course of a year.
(c) What are the domain and range for this function?
(d) What are the amplitude, vertical shift, period,
frequency, and phase shift of this function?
(e) Predict the high temperature on March 15.
(f) What are all the dates on which the high temperature is
most likely 80?
Some Other Issues
Regarding the Need
to Refocus the Courses
below Calculus
The Need to Rethink
Placement
in Mathematics
Rethinking Placement Tests
Four scenarios:
1. Students come from traditional curriculum
into traditional curriculum.
2. Students from Standards-based curriculum
into traditional curriculum.
3. Students from traditional curriculum into
reform curriculum.
4. Students from Standards-based curriculum
into reform curriculum.
A National Placement Test
1. Square a binomial.
2. Determine a quadratic function arising from a
verbal description (e.g., area of a rectangle
whose sides are both linear expressions in x).
3. Simplify a rational expression.
4. Confirm solutions to a quadratic function in
factored form.
5. Completely factor a polynomial.
6. Solve a literal equation for a given unknown.
A National Placement Test
7. Solve a verbal problem involving percent.
8. Simplify and combine like radicals.
9. Simplify a complex fraction.
10. Confirm the solution to two simultaneous linear
equations.
11. Traditional verbal problem (e.g., age problem).
12. Graphs of linear inequalities.
A Modern High School Problem
Given the complete 32-year set of monthly CO2
emission levels (a portion is shown below), create
a mathematical model to fit the data.
Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Avg
1968 322 323 324 325 325 325 324 322 320 320 320 322 323
1969 324 324 325 326 327 326 325 323 322 321 322 324 324
A Modern High School Problem
1. Students first do a vertical shift of about 300 ppm and then fit
an exponential function to the transformed data to get:
F (t ) 1.656e 0.03923t 299.5
2. They then create a sinusoidal model to fit the monthly
oscillatory behavior about the exponential curve
1
S t 3.5sin 2 t 0.5
24
3. They then combine the two components to get
1
C t F t S t 1.656e 0.03923 t
3.5sin 2 t 299
24
4. They finally give interpretations of the various parameters
and what each says about the increase in concentration and use
the model to predict future or past concentration levels.
Placement, Revisited
Picture an entering freshman who has taken
high school courses with a focus on problems
like the preceding one and who has developed
an appreciation for the power of mathematics
based on understanding the concepts and
applying them to realistic situations.
What happens when that student sits down to
take a traditional placement test? Is it
surprising that many such students end up
being placed into developmental courses?
What a High School Teacher Said
―If you try to teach my students with the
mistaken belief that they know the mathematics
I knew at their age, you will miss a great
opportunity. My students know more
mathematics than I did, but it is not the same
mathematics; and I believe they know it
differently. They have a different vision of
mathematics that would be helpful in learning
calculus if it were tapped.‖
Dan Teague
The Need to Rethink
Course Content
What Can Be Removed?
How many of you remember that there used to
be something called the Law of Tangents?
What happened to this universal law?
Did triangles stop obeying it?
Does anyone miss it?
What Can Be Removed?
• Descartes' rule of signs
• The rational root theorem
• Synthetic division
• The Cotangent, Secant, and Cosecant
were needed for computational purposes.
Just learn and teach a new identity:
1 tan x cos2 x
2 1
How Important Are Rational Functions?
• In DE: To find closed-form solutions for several differential
equations, (usually done with CAS today, if at all)
• In Calculus II: Integration using partial fractions–often all four
exhaustive (and exhausting) cases
• In Calculus I: Differentiating rational functions
• In Precalculus: Emphasis on the behavior of all kinds of
rational functions and even partial fraction decompositions
• In College Algebra: Addition, subtraction, multiplication,
division and especially reduction of complex fractional
expressions
In each course, it is the topic that separates the ―adults‖ from the
―children‖! But, can you name any realistic applications that
involve rational functions? Why do we need them in excess?
Challenges
to Be Faced
The Challenges Ahead
• Convincing the math community
1. Conducting a series of extensive tracking
studies to determine how many (or how few)
students who take these courses actually go on
to calculus.
2. Identifying and highlighting ―best practices‖
in programs that reflect the goals of this
initiative.
The Challenges Ahead
• Convincing college administrators to
support (both academically and
financially) efforts to refocus the courses
below calculus.
What Can Administrators Do?
When the University of Michigan wanted
to change to calculus reform, including
going from large lectures of 800 students
to small classes of 20 taught by full-time
faculty, the department argued to the dean
that by saving only 2% of the students who
fail out because of calculus, the savings to
the university would exceed the $1,000,000
annual additional instructional cost. The
dean immediately said ―Go for it.‖
The Challenges Ahead
Convincing academic bodies outside of
mathematics to allow alternatives to
traditional college algebra courses to fulfill
general education requirements.
An Example: Georgia
The state education department in Georgia
had a mandate for general education that
every student must take college algebra. A
group of faculty from various two and four
year colleges across the state lobbied for
years until they finally convinced the state
authorities to allow a course in
mathematical modeling at the college
algebra level to serve as an alternative for
satisfying the Gen Ed math requirement.
The Challenges Ahead
Convincing the testing industry to begin
development of a new generation of
placement and related tests that reflect the
NCTM Standards-based curricula in the
schools and the kinds of refocused courses
below calculus in the colleges that we hope
to being about.
The Challenges Ahead
Gaining the active support of
representatives of a wide variety of other
disciplines in the effort to refocus the
courses below calculus.
• CRAFTY and MAD (Math Across the
Disciplines) committee have launched a
second round of Curriculum Foundations
workshops to address this issue.
The Challenges Ahead
Gaining the active support of
representatives of business, industry, and
government in this initiative.
• Discussions are underway about
revisiting some of the participants in the
Forum on Quantitative Literacy.
The Challenges Ahead
Developing a faculty development
program to assist faculty, especially part
time faculty and graduate TA's, to teach
the new versions of these courses.
• NSF has funded a demonstration project
through CRAFTY involving 11 schools.
• 200 other departments wanted to be part
of this project.
The Challenges Ahead
Influencing teacher preparation programs
to rethink the courses they offer to prepare
the next generation of teachers in the spirit
of this initiative.
The Challenges Ahead
Influencing funding agencies such as the
NSF to develop new programs that are
specifically designed to promote both the
development of new approaches to the
courses below calculus and the widespread
implementation of existing ―reform‖
versions of these | 677.169 | 1 |
Mathematics: The Next Generation
Subject:
Overview
Mathematics is important to us all. So it is important to enable young mathematicians, clear-thinking and passionate about their subject, to contribute at the highest level. Peter Cameron will talk about his experience designing and presenting a course for first-semester university students aiming to produce mathematicians. | 677.169 | 1 |
Why study mathematics? Many students are attracted to mathematics at school by the clear unequivocal nature of the answers to the questions. Mathematics is a discipline in which, at university level too, precise propositions can lead, through elegant arguments, to far-reaching consequences, including surprising applications. The clear-cut nature of the subject means that a higher proportion of mathematics students obtain first-class degrees than in most other subjects.
Why study mathematics at Essex?
Students in a big city university often feel that the staff do not know them at all. This is not the case at Essex where students and staff have regular contact with each other. The transition from school/college to university is therefore not as daunting and you will soon get to know the other people in the Department. We maintain an 'open door' policy which means that whenever you are stuck on a problem, you are likely to find a lecturer available to give help.
Our Department is home to one of the area co-ordinators for the East of England Further Mathematics Support
Programme, which promotes and aids the teaching of A-level Further Mathematics, meaning we are at the forefront of what is happening in schools and colleges.
Flexibility to change
courses We recognise that the journey from school/college to university is a trip into the unknown. This is particularly the case if you choose a joint honours course and have not studied your chosen second subject before coming to university. We allow you the flexibility to experiment with new subjects: most students can change from their initial choice of course to single-honours courses up to the start of the second year, and many can change from joint honours to single honours up to the start of the final year.
Flexibility to change modules
Our courses are tailored with particular outcomes in mind, so in most cases there are rather few options in the first two years of a course. However, in all courses there are options in the third year. In any term, when you have options, you are encouraged to sit in on several modules before making a final choice, and staff are happy to give advice.
Departmental scholarships and
bursaries Major scholarships (worth £2,000 over two years for single-honours students) are awarded to those who obtain AAA (including A in Mathematics) in three full A-levels. Minor scholarships (worth £1,000 over two years for single-honours students) are awarded to those who obtain ABB or AAC (including A in Mathematics) in three full A-levels. If you are taking a joint honours course you will also receive these scholarships, but at half the full rate. Renewal of scholarships is subject to good performance in the first-year examinations.
To qualify for one of these awards you must have placed Essex as your firm UCAS choice. Additionally, a one-off payment of £250 will be awarded to you if you achieve grade A or B in A-level Further Mathematics. Single-honours students who do well in the second-year exams are awarded a final-year bursary of £500. For full details, please visit: prospective_students/undergraduates/ scholarships.aspx for full details. | 677.169 | 1 |
Jobs
Mathematics GCSE
Introduction to course
1year, assessed by exams
GCSE Maths is essential for entry onto many university level courses - and employers look for it, too, in most business areas. If you obtained a grade D from school, then during your time with us our friendly teachers will help you to improve your grade with this very popular resit course.
Course Details
We follow a modular Edexcel course and you may sit a combination of higher and foundation modules, depending on how you progress with the course.
Entry Requirements
This GCSE resit course is suitable for students who have previously obtained a grade D in GCSE Maths at school and wish to improve it to a grade C.
Where the course lead
GCSE Maths grade C is an essential requirement for nearly all university degree courses.
This course is not suitable with
GCSE re-sit English (you will have to sit English first and do Maths the following year).
Course Assessment
You will be given the opportunity to sit this exam in November or you will sit 2 modules in March and 2 modules in May.
Sir George Monoux College tries to ensure the accuracy of the information contained in this web site. However, such accuracy cannot be guaranteed. The College reserves the right to make changes in regulations, the offering and structure of courses and programmes without notice. | 677.169 | 1 |
Product Details
Combinatorics: A Guided Tour by David Mazur
Combinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Pólya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques. The text emphasizes the brands of thinking that are characteristic of combinatorics: bijective and combinatorial proofs, recursive analysis, and counting problem classification. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses. | 677.169 | 1 |
book's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students. The primary ...Show synopsisThis book's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students. The primary focus of the pedagogy, presentation and other elements is to ease the transition into algebra; for example, emphasis is placed on basic arithmetic operations within algebraic contexts. The Second Edition includes a greater integration of NCTM and AMATYC standards, including more emphasis on visualization, problem solving and data analysis.Hide synopsis
Description:Paperback. Instructor Edition: Same as student edition with...Paperback. Instructor Edition: Same as student edition with additional notes or answers. New Condition. SKU: 978032162919728862. arithme... | 677.169 | 1 |
UA preparatory math goes virtual
Apr 26, 2011 By La Monica Everett-Haynes
Is this what flashes across your mental screen when you think about math? The UA's mathematics department is piloting a new course, Math 100, which is designed to help students who struggle with university-level math. The course provides personalized instruction with a heavy emphasis on tutoring, peer support and the use of technology.
The University of Arizona's math department is experimenting with a novel approach to early math instruction – one with a heavy emphasis on technology and peer-to-peer tutoring.
Arguably, few other required college-level courses elicit the same frustration or the intimidation factor as mathematics.
Some commonly talk about holding a hatred for math, believe they are no good at it or think up strategies to avoid it all together.
But one University of Arizona team is working to unravel the enigmatic nature of math for the very students who struggle the most with it – those who do not test into college-level math.
Math 100, now in the second semester of its pilot phase, has a heavy emphasis on both self-paced progress and peer-to-peer support while being offered through Elluminate, a web-conferencing system.
"Students are so used to being online. We thought that if we put the course online we could interact more," said Michelle Woodward, who coordinates the pilot course being offered by the UA mathematics department.
The number of section offerings will be expanded during the fall to accommodate more UA students who do not test into algebra-level mathematics.
Woodward said the course is being emphasized and expanded because it is especially important for new students to grasp college math, especially algebra – a curricular core – early.
Algebraic skills have long been associated with giving students the ability to think in more complex ways. A student's ability to comprehend algebra has long been upheld as an indication of college-readiness, particularly for study in science and engineering-related disciplines.
"It's the foundational material they need to be prepared for college algebra," Woodward said.
"My whole goal in this is to make an online environment that is as close to what students would do in person. I want the environment to be as interactive as possible," Woodward said, adding that another program, the ALEKS Learning Module, provides both structure and flexibility while also offering the course content.
"I have done a lot of work with students who needed individualized plans. ALEKS does that for me," she said. "I could not do that for 300 students, it doesn't replace me – it frees me up to work with students individually, the kind of work I didn't have time to do before."
Over the course of the semester, the 300 students currently enrolled in one dozen Math 100 sections meet three hours weekly, receiving self-paced instruction mediated by Elluminate. Students complete assignments, learning to master algebraic expressions and graphing techniques and, all the while, ALEKS tracks their progress.
"We are able to personalize the lessons much better than we have. It's been wonderful," said Cheryl Ekstrom, a mathematics lecturer who initiated the idea to incorporate Elluminate. "You aren't stuck listening to a lecture on things you already know or breezing by things you don't understand."
This is in direct contrast to more established and traditional ways of teaching math.
"In a traditional class, it doesn't matter if it's hard for you," said Shailendra Simkhada, an electrical engineering senior also studying math.
"Each day in a regular class, you might get a new chapter or deadline to meet but, here, they can work at their own pace," he said.
"It's not that they do less work, but if you don't understand something you get more information and one-on-one help so that they stay on track," he added.
If fact, students designate their goals at the start of the class, deciding what sections they want to master and what math class they hope to test into at the end of the term.
Students also engage in weekly virtual classroom meetings, sharing their computer screens and conversing online with student leads and support staff – UA students who are advanced in math and receive more than 15 hours of training.
Kirandeed Banga, a UA sophomore studying biology, is a member of the student lead and support staff.
Each week, Banga joins the other leads and support staff members in a classroom in the Math Building where they each log online to tutor and monitor student work.
"With it being completely online, it's hard to get their trust. But we try to talk to them as much as possible," said Banga who, like others on the team, also offer office hours.
"And we put them into virtual groups, so they are also able to help one another," she added. "They obviously are used to the technology, so they can adapt to it."
Also built into the design of the course is extensive support to the UA students facilitating the class.
Ivvette Rios, a UA math and French major, observes the virtual sessions and conducts weekly meetings with all of the students offering tutoring and support. Her role is to ensure that the leads and support staff have everything they need to appropriately help the hundreds of students enrolled.
Rios said the time for self-evaluation and self-reflection is critical for those involved, and helps to ensure that the structure is working well for all involved.
"We are always thinking of ways we can do this better; to make it more and more like our everyday experience," Rios said. "It's work out way better than we thought it would."
Leo Shmuylovich knows a lot about how tutoring can take a student from confused to confident. The Washington University graduate student has worked as a tutor for several test preparatory companies over the years, helping ...
(PhysOrg.com) -- New research from the University of Notre Dame suggests that even though adults tend to think in more advanced ways than children do, those advanced ways of thinking don't always override old, incorrectConsidering how many fools can calculate, it is surprising that it should be thought either a difficult or tedious task for any other fool to learn how to master the same tricks. Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the text-books of advanced mathematics-and they are mostly clever fools-seldom take the trouble to show you how easy the calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way." Calculus Made Easy, Silvanus P. Thompson, Prologue, 1910 population | 677.169 | 1 |
Calculus : Early Transcend (Cloth) - 2nd edition
ISBN13:978-1429208383 ISBN10: 1429208384 This edition has also been released as: ISBN13: 978-1429260091 ISBN10: 1429260092
Summary:
What's the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching-supported by Rogawski's Calculus Second Edition-the most successful new calculus text in 25 years! Widely adopted in its first edition, Rogawski's Calculus worked for instructors and students by balancing formal precision with a guiding conceptual...show more focus. Rogawski engages students while reinforcing the relevance of calculus to their lives and future studies. Precise mathematics, vivid examples, colorful graphics, intuitive explanations, and extraordinary problem sets all work together to help students grasp a deeper understanding of calculus. Now Rogawski's Calculus success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning95 +$3.99 s/h
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The following computer-generated description may contain errors and does not represent the quality of the book: These examples are intended to provide a complete course of elementary algebra for classes in which the bookwork is supplied by the teacher. In the choice of the subjects included, and in their arrangement, I have throughout followed the recommendations of the Committed on the Teaching of Mathematics appointed by the Mathematical Association. Among new features may be mentioned: The postponement until after easy simultaneous equations of the long rules for multiplication and division. The postponement until after quadratics of complicated fractions, H.C.F., L.C, M., Square root, and Literal Equations. The early introduction and extensive use of Graphs. The inclusion of some of the applications to Geometry which form such a prominent feature in modem continental text-books. The treatment of fractional indices from a numerical point of view, so as to lead up to the use of four-figure logarithm tables. The stress laid oo numerical checks of all kinds. The large selection of problems, including very easy ones. | 677.169 | 1 |
Synopses & Reviews
Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.
Publisher Comments:
Need to learn MATHEMATICA? Problem SOLVED
Take full advantage of all the powerful capabilities of Mathematica with help from this hands-on guide. Filled with examples and step-by-step explanations, Mathematica Demystified takes you from your very first calculation all the way to plotting complex fractals.
Using an intuitive format, this book explains the fundamentals of Mathematica up front. Learn how to define functions, create 2-D graphs of functions, write basic programs, and use modules. You'll move on to 3-D graphics, calculus, polynomial, linear, and differential equations, dynamical systems, and fractals. Hundreds of examples with concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.
This self-teaching guide offers: A quick way to get up and running on MathematicaCoverage of Mathematica 6 and 7Tips for avoiding and correcting syntax errorsDetails on creating slideshow presentations of your workNo unnecessary technical jargonA time-saving approach to performing better on an exam or at work
Simple enough for a beginner, but challenging enough for an advanced user, Mathematica Demystified is your shortcut to mastering this fully integrated technical computing software.
Synopsis:
The fast and easy way to learn MathematicaAbout the Author
Jim Hoste, Ph.D. (Claremont, CA) is a Professor at Pitzer College, and a member of the Mathematical Sciences Research Institute. Hes been an Associate Editor of the Journal of Knot Theory and Its Ramifications since 1991, and has authored dozens of research publications. | 677.169 | 1 |
This is a short, focused introduction to MATLAB, a comprehensive software system for mathematical and technical computing. It contains concise explanations of essential MATLAB commands, as well easily understood instructions for using MATLAB's programming features, graphical capabilities, simulation models, and rich desktop interface. Written for MATLAB 7 it can also be used with earlier (and later) versions of MATLAB.This book teaches how to graph functions, solve equations, manipulate images, and much more. It contains explicit instructions for using MATLAB's companion software, Simulink, which allows graphical models to be built for dynamical systems. MATLAB's new "publish" feature is discussed, which allows mathematical computations to be combined with text and graphics, to produce polished, integrated, interactive documents.
MATLAB is an incredibly rich and complicated software system; this book focuses first on the essentials, then develops finer points through numerous examples
Reviews & endorsements
"Major highlights of the book are completely transparent examples of classical yet always intriguing mathematical, statistical, engineering, economics, and physics problems. In addition, the book explains a seamless use with Microsoft Word for integrating MATLAB outputs with documents, reports, presentations, or other on-line processes. Advanced topics with examples include: Monte Carlo simulation, population dynamics, and Linear Programming. Overall, it is an outstanding textbook, and, likewise, should be an integral part of the technical reference shelf for most IT professionals. It is a great resource for wherever MATLAB is available!"
ACM UbiquityBrian R. Hunt, University of Maryland, College Park Brian R. Hunt is an Associate Professor of Mathematics at the University of Maryland.
Ronald L. Lipsman, University of Maryland, College Park Ronald L. Lipsman is a Professor of Mathematics and Associate Dean of the College of Computer, Mathematical and Physical Sciences at the University of Maryland.
Jonathan M. Rosenberg, University of Maryland, College Park Jonathon M. Rosenberg is a Professor of Mathematics at the University of Maryland.
Kevin R. Coombes, University of Texas, M. D. Anderson Cancer Center Kevin R. Coombes is an Associate Professor of Biostatistics and Biomathematics at the M.D. Anderson Cancer Center, University of Texas.
John E. Osborn, University of Maryland, College Park John E. Osborn is a Professor of Mathematics at the University of Maryland.
Garrett J. Stuck, University of Maryland, College Park Garrett J. Stuck is a Professor of Mathematics at the University of Maryland | 677.169 | 1 |
Euless Algebra 2Bob These more advanced courses are primarily taught by math and science experts at the University. My background in linear algebra is significant due to my advanced science related to my research endeavors in advanced science systems. Linear Algebra also used differential equations | 677.169 | 1 |
ISBN13:978-0201347302 ISBN10: 020134730X This edition has also been released as: ISBN13: 978-0201384086 ISBN10: 0201384086
Summary: This...show more. Students using this text will receive solid preparation in mathematics, develop confidence in their math skills and benefit from teaching and learning techniques that really work. For mathematics teachers. ...show less
(Each chapter begins with a ''Preliminary Problem'' and concludes with a ''Hint for Solution to the Preliminary Problem,'' ''Questions from the Classroom,'' ''Chapter Outline,'' ''Chapter Review,'' and a ''Selected Biography.'') (*indicates optional section.)
Integers and the Operations of Addition and Subtraction. Multiplication and Division of Integers. Divisibility. Prime and Composite Numbers. Greatest Common Divisor and Least Common Multiple. *Clock and Modular Arithmetic.
5. Rational Numbers as Fractions.
The Set of Rational Numbers. Addition and Subtraction of Rational Numbers. Multiplication and Division of Rational Numbers. Proportional Reasoning.
Instructor's EditionBook Daddy Fort Wayne, IN
Hardcover Very Good 020134730X Purchase Protected By Our Satisfaction Guarantee | 677.169 | 1 |
Precalculus: Mathematics for Calculus
9780840068071
ISBN:
0840068077
Edition: 6 Pub Date: 2011 Publisher: Brooks Cole
Summary: Designed to give students a background in mathematics theory and introduce them to mathematics concepts this textbook is comprehensive without being daunting. Students are introduced to modelling and problem solving and they are given a rigorous workout on what they have learned as they work through the book. It has many graphs that chart mathematical ideas that students can assimilate with ease. It is written in a c...lear and readable style that will aid comprehension and enjoyment. This is just one of the many cheap math textbooks we have available for students to acquire in great condition.
Stewart, James is the author of Precalculus: Mathematics for Calculus, published 2011 under ISBN 9780840068071 and 0840068077. Nine hundred seventy three Precalculus: Mathematics for Calculus textbooks are available for sale on ValoreBooks.com, two hundred nine used from the cheapest price of $89.30, or buy new starting at $155.18 Free Edition: Same as student edition but has free copy markings. Almost new condition. SKU:97808... [more] | 677.169 | 1 |
Mathematics helps us make sense of the world and reveals methods for using known information to find unknown information. We regularly study the most efficient methods for reaching solutions, but also realize that examining different solution methods helps develop more flexible problem solving skills.
Our department is focused on instilling students with enduring understandings in mathematics. We seek to help students become efficient users of algorithms who can articulate their thinking and understand how to apply mathematics in different contexts. For example, if you are taking a prescription medication and miss a dose, why do the instructions warn against simply taking a double dose? Or how does a country with over 300 million people in 50 states fairly apportion a mere 435 seats in the House of Representatives?
Mathematics is the language of the universe. We strive to help students become fluent users of this language.
Quote
"You understand how to think mathematically when you are resourceful, flexible, and efficient in your ability to deal with new problems in mathematics." | 677.169 | 1 |
Mathematical Equation Evaluator is a mobile based student guiding application which is specially designed for local ordinary level students and advanced level students who are working in the different contexts related to complex mathematical formulas. The present process of solving complex mathematical problems is, writing down the mathematical formulae and simplifying them for solving the problem easily with a lesser number of variables. Assigning the given values to the final formula and calculating the final answer or the value by using a scientific calculator in step by step are the last two steps involved. The developed mobile application Mathematical Equation Evaluator system will eliminate the complexity and delays in the manual solving of the complex mathematical equations. The system users can enter the mathematical equation to be solved (or to be simplified) via the camera of the mobile phone in to the system. First the system will identify the equation string and detect the variables and the constants. Then the users can enter the known values for the variables in the equation. Finally, the system will generate the answer for the equation, with the step-by-step explanation displaying how the system has calculated the answer. The application is to be used by students after they have worked out the problem for answers. The application will show the answer and the steps of reaching the answer for the student to verify the result of the calculation attempt. In addition to that, the teachers can use this application to buildup answer scripts for mathematics papers. The most important feature of this project is the capability of inputting the equations via the mobile phone camera. Then, the students can scan the equation via the camera attached to the phone and evaluate the equation easily and quickly. Optical Character Recognition technology is used in implementing the above. Complex mathematical equations are solved by using predefined mathematical equations (axioms) and by simplifying the given equations in to smaller terms or simplifying the given equation to have lesser number of unknown variables. In this stage, it is required to follow a large set of steps to find the correct answer. Previously, a paper based system and calculators were used in solving the equation. But it's not efficient and can lead to errors, because, the equation to solve and values are on a paper and the calculations are processing in the calculator. There is more possibility to occur mistakes in data entering to the calculator from the paper and vice-versa. In addition to that, it will consume more time in doing the calculations. Another aspect that leads to difficulties in mathematical equation solving is that, for some mathematical equations, it's not provided a correct answer with it. Therefore the students and lectures may get confused with different potential answers. Then, some of the complex mathematical equations need using recursive methods in solving the final equation. So the person who is doing the calculation will be bored with the calculation task, and such long and recursive calculations are having a higher risk of getting incorrect evaluations during the intermediate steps. The developed solution, Mathematical equation evaluator is a mobile application, targeted for mobile devices and tablets (with Android operating system). This application aims to provide solutions to above mentioned problems that are faced by students in learning mathematics. The application summery is as follows. There are two types of information gathering techniques using by the application for receiving input data in to the system. First technique is capturing the equation via mobile device's camera and system automatically detecting the equation by using the OCR technology and the second technique is manually entering the equation details in to the system.
1. Capturing the mathematical equation via mobile device's camera and automatically detecting the equation by using the OCR technology 2. Support for manually entering the equation details in to the system via the keypad. 3. Final solution is displayed with steps of solving. 4. Variable to find value should be the only literal in left the hand side of the equal mark and it should be marked as ? (question mark) when asking for known values. | 677.169 | 1 |
The Mathematics Curriculum: Counting and Configurations what they could look like. There is then a detailed look at probability, graph theory as a network and also some topics for classroom | 677.169 | 1 |
Welcome to your online math class! The following steps should help you get started. This is intended to be a very short overview of what is expected of you the first week of class. All details are found on documents posted on Blackboard.
Log on to Blackboard. Download each document under "Course Information" and print them out for future reference.. Doing this will help you decide if you are in the correct course. It will also provide you with your homework assignment, due dates, test schedule, minimum computer requirements, etc.
Complete the online orientation found on Blackboard. This is required!
Purchase your materials.
Log in to MyMathLab. To do this you will need your ACC email address set up. You will also need my courseID provided on the handout you printed from my Blackboard documents. Lastly, you will need an access code that will come with the MyMathLab materials you purchased.
Begin your lessons in MyMathLab. This is where the majority of your learning will take place.
Log in to Blackboard weekly to read any announcements from me. This is also where you will receive your homework and test grades. You should log on to Blackboard at least once a week. | 677.169 | 1 |
A convenient single source for vital mathematical concepts, written by
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Description:
Structures are defined by laws of composition, rules of generation,
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
APSC 171
WORKBOOK
Interactive Class Notes
prepared by
Leo Jonker
with the assistance of
Cesar Aguilar
Queen's University
Fall 2008
Foreword
These are so-called "incomplete class notes", or "interactive class notes",
prepared for your use in APSC 171. The notes reflect the way in which the
instructors will present the course material in the lectures. Gaps have been
left in the notes, some as blanks in formulas or text, others as problems or
unanswered questions. These blanks should be filled in by you as you study
the material in this course.
What's our reason for using interactive notes? They work especially
well if the instructor uses transparencies instead of the blackboard to present
the material. Transparencies are a good presentation medium. The organi-
zation of material is often clearer on transparencies than on blackboard; it is
easier to bring back examples, definitions and other refresher material; visual
cues (colour, size, point-form presentation) can help to high-light what is im-
portant; diagrams better because they have been prepared beforehand; the
instructor can face the class and stay aware of student reaction. The main
purposes of the interactive notes, however, is to transfer some of these same
advantages to you, the student. In fact, even in a class taught on the black-
board, the use of interactive notes should enhance the learning experience.
Interactive notes eliminate the need for tedious copying from the board or
overhead when you should be thinking. They also eliminate the frustration
of trying to, in a few minutes, copy a diagram that took the teacher a long
time or a lot of practice to make. The second purpose of interactive notes
is to encourage student involvement beyond passive listening and reading.
By reading the notes before class and trying to fill in the gaps and doing
the problems you will become committed to the material in a way that will
enhance the lectures.
i
ii
How should you use these notes? They are intended to complement
the textbook and to highlight what is important. By their organization into
weeks they also provide a guideline indicating approximately what we can
expect to cover each day. Read ahead in the text - even browsing will
be useful. Then look at the notes, and try to answer some of the
questions before you come to class. Bring the notes to class - you will
need them. Treat the notes as a workbook. At the end of the course, with
the blanks filled in and perhaps with the addition of some material of your
own, these interactive notes will constitute your class notes for the course.
Use the extra blank pages or the margins to write down your own questions.
Ask these questions in class. You can be sure that if a question seems
important to you it will seem so to at least ten others in the class. Additional
"Notes" pages are included at the end of each week for you to write things
that come up in class but are not part of the notes (such as additional
examples used by your instructor). You may also wish to bring to class some
coloured pens or pencils to highlight important ideas. Finally, keep a list
of your comments about the interactive notes themselves and let us know
what they are, so that suggestions for improvement can be incorporated for
the benefit of future classes.
The textbook: The textbook for the course this year is Calculus Early
Transcendentals, Edition 6e, by James Stewart. This is a new edition. Last
year we used Edition 5e. All references in these notes will be to the new
edition, followed by the corresponding reference to the old edition given in
square brackets.
Contents
Material to review on your own 1
Week 1 3
Course Unit: How do we describe a moving object? . . . . . . . . 4
1.1 Four Ways to Represent a Function . . . . . . . . . . . . . . . 5
10.1 & 13.1 Other Types of Functions . . . . . . . . . . . . . . . . 9
13.2 Velocity and Parametric Curves . . . . . . . . . . . . . . . . . 14
Week 2 27
3.3 [3.4] Derivatives of Trigonometric Functions . . . . . . . . . . . 28
13.2 & 13.4 Velocity and Acceleration . . . . . . . . . . . . . . . . 37
Week 3 49
Course Unit: More About Derivatives . . . . . . . . . . . . . . . . . 50
1.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 54
3.5 [3.6] Implicit Differentiation . . . . . . . . . . . . . . . . . . . 62
3.6 [3.8] Logarithmic Differentiation . . . . . . . . . . . . . . . . . 67
Week 4 69
iii
iv CONTENTS
3.7 [3.3] Rates of Change in Science . . . . . . . . . . . . . . . . . 70
3.10 [3.11] Differentials and Linear Approximations . . . . . . . . 76
4.4 Indeterminate Forms and l'Hopital's Rule . . . . . . . . . . . . 85
Week 5 93
Course Unit: Integration . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 & 5.2 What is an Integral? . . . . . . . . . . . . . . . . . . . . 96
Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Area Under a Graph . . . . . . . . . . . . . . . . . . . . . . . 101
Week 6 117
5.3 The Fund. Theorem of Calculus, Part 2 . . . . . . . . . . . . . 118
5.2 Properties of Integrals . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . 131
Week 7 141
5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1 Areas between Curves . . . . . . . . . . . . . . . . . . . . . . . 150
Week 8 159
6.2 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3 Volumes by Cylindrical Shells . . . . . . . . . . . . . . . . . . 167
6.4 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Week 9 185
7.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . 186
CONTENTS v
7.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Week 10 209
7.7 Approximate Integration . . . . . . . . . . . . . . . . . . . . . 210
7.8 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 216
Week 11 229
Course Unit: Differential Equations . . . . . . . . . . . . . . . . . . 230
9.1 Modeling with Differential Equations . . . . . . . . . . . . . . 231
Exponential Growth (see also 3.8 [9.4]) . . . . . . . . . . . . . 231
Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . 236
Week 12 249
9.3 Separable Differential Equations . . . . . . . . . . . . . . . . . 250
9.4 [9.5] The Logistic Differential Equation . . . . . . . . . . . . . 259
How to study in this course
Some of you, perhaps most of you, come in with good study habits. All
of you are excellent mathematics students - you must be, to get the marks
needed for admission to Queen's. However, being good at mathematics in
high school can be a real trap, for it can make you complacent about study
habits, and give you the false impression that you can rely on your facility for
calculations. You will find that in your university mathematics courses, the
volume and the pace is such that you cannot afford to do things in a hurry.
The only way to maintain the pace is to really understand the fundamental
ideas very well. Here are some do's and dont's that will help you achieve
this:
• Do not study the material by doing the homework questions, and then
checking the text only when you get stuck. Read the week's section of
the text first. Then read the interactive notes, and try the classroom
questions before you get to class. Go to the tutorials. Only after that
should you attempt the homework questions, as a final check on your
understanding.
• Read definitions and statements of theorems carefully. The powerful
ideas in mathematics are often subtle and depend on the precise use of
terms.
• Do not imagine that the course is about memorizing a bunch of for-
mulas. If you do not understand the concepts behind the formulas you
have gained very little.
• Do not do lots and lots of problems on a topic. It is much better to do
two or three and then to reflect on them.
Material to review on your own
Study Sections: 1.2, 1.3, 1.5, 1.6, 2.1-2.3 (to page 103
[108]), 2.7-2.9, 3.1, 3.2, 3.4 [3.5], and 3.6 [3.8] in the textbook
• Basic Functions and their Graphs (Polynomial, Power,
Rational, Trigonometric, Exponential, Logarithmic)
• Inverse Functions
• Derivatives and Rates of Change
• Derivatives of Polynomials, Exponential and Logarithmic
Functions
• Differentiation Rules (Product, Quotient, and Chain Rules)
The topics listed above represent what we expect you to know from the
start of this course. All of the topics are covered in the Ontario high
school curriculum, so in principle you should not need to review any of
this material. If you have a weak background in differential calculus, then
you should have reviewed the Quick and Easy Differential Calculus
Notes this summer, available on the APSC 171 website. These notes
introduce differential calculus in a highly intuitive manner, which fits well
with the tenor of the rest of the course. They also review some other
important topics, such as inverse functions and logarithmic functions, of
which we judged you were most likely to be unsure.
1
2 MATERIAL TO REVIEW ON YOUR OWN
Week 1
Study Sections: 1.1, 10.1 and 13.1 in the textbook
• Course Unit: Describing Moving Objects
• Different Ways of Representing Functions
• Vector-valued Functions, or Parametric Curves
3
4 WEEK 1
Course Unit: How do we describe a moving
object?
The first unit of the course will concern the description and study of moving
objects. These objects will move, not only along a line, but also in two-
dimensional and three-dimensional space. We will try to relate formulas to
geometric descriptions, and to problems of velocity, force and acceleration.
While this is happening, you will find that the skills you are learning in the
calculus course complement the mechanics you are studying in the first few
weeks of your physics course. By having one subject reinforce the other, we
hope that both will benefit.
To start this unit off, we begin with a problem, which we will come back to
at the close of the unit. Here is that problem.
Problem to be solved at the end of week 2, at the end of the first
unit of the course1 FOUR WAYS TO REPRESENT A FUNCTION 5
1.1 Four Ways to Represent a Function
A function is a rule or process that assigns to each
input a corresponding output.
A function can come in a variety of forms, as the following example will
illustrate.
Example 1. Suppose we have a wheel of radius 1, with a red dot at the top
of the wheel's rim and a green dot at the extreme right. If we turn the wheel
by an angle θ, the heights f (θ) and g(θ) of the two dots above (or below) a
horizontal line through the centre of the wheel vary with the angle. Discuss
the functions f (θ) and g(θ).
R
1 G
6 WEEK 1
1.1 FOUR WAYS TO REPRESENT A FUNCTION 7
8 WEEK 1
This example shows that there are (at least) four ways to represent a
function:
• Verbal description
• Table
• Graph
• Formula
A function is not necessarily given by a formula.
In every one of the descriptions provided, the function involves two varying
quantities, with one of them, the input, giving rise to, or "causing", the
other one, which is then called the output.
For this reason, a function may be thought of as a machine with input and
output.
x f (x)
input
f output
The input variable is also referred to as the "independent variable" and the
output as the "dependent variable".
In the above example the term "machine" can be taken quite literally: You
can imagine turning a crank that makes the wheel go around and stop at
the angle (the input) you choose, and then reading off the y-coordinate of
the green (or red) dot. This y-coordinate is the output. In other cases the
machine picture may have to be understood in a more figurative sense.
10.1 & 13.1 OTHER TYPES OF FUNCTIONS 9
10.1 & 13.1 Other Types of Functions
The functions familiar to us from our high school courses are all functions
for which both the input and the output are single numbers. We do not have
to restrict ourselves to functions of that type, however.
Functions of several variables
For one thing, there is no reason why a function might not require more
than one input. For example, if we want to think of the temperature in
Ontario at noon today, the answer cannot be specified unless we are given a
precise location. That is, we need both a longitude (an x-coordinate) and a
latitude (a y-coordinate) before we can say what the temperature is at the
point (x, y). In this setting, the temperature is the output of a function of
two variables:
T = f (x, y) .
We can still think of this function as a machine, but this machine works only
if the input you give it is a pair of numbers:
(x, y) f (x, y) = T
input
f output
2 in R
in R
We will be studying such functions extensively in the second term.
The domain of a function is the set of inputs the function will accept (the
coordinates corresponding to locations in Ontario in this case). The range
is the set of the outputs it can produce (that is, the range of temperatures
in Ontario at noon today).
10 WEEK 1
Vector-valued functions
We can also vary the type of function under discussion by taking a single
number as an input, but admitting a pair of numbers as the output:
t r(t) = (x(t), y(t))
input
r output
in R in R2
This is precisely what we get if we think of the moving green dot in the
problem we have been discussing:
Example 2. How do the coordinates of the green dot vary as θ varies?
10.1 & 13.1 OTHER TYPES OF FUNCTIONS 11
Suppose the dot is rotating at an even rate, say at one revolution per 2π
seconds. Then at time t the location of the dot is given by coordinates
(cos t, sin t), for at one revolution per 2π seconds, the angle θ (measured in
radians) is always precisely equal to the elapsed time. We can think of the
moving location G of the green dot as a vector-valued function
r(t) = (cos t, sin t) .
The input is time (t) and the output is a location (x, y) = (cos t, sin t), or
equivalently, the position vector cos t, sin t of that location. Another term
for such a function is (parametric) plane curve. This is a natural way to
describe it if we think of the green dot as a moving particle whose position
depends on the parameter time. Notice that we can also, if we want, describe
this vector-valued function by means of the two (ordinary) functions giving
the coordinates separately:
x(t) = cos t
y(t) = sin t
12 WEEK 1
Concept Question 1. If we wanted to write down the displacement of the
green dot between times t = 0.5 and t = 1.6, which of the following would be
the right expression?
A. cos(1.6 − 0.5), sin(1.6 − 0.5) ;
B. sin(0.5), cos(0.5) ;
C. cos(1.6) − cos(0.5), sin(1.6) − sin(0.5) ;
D. cos(1.6), sin(1.6) − cos(0.5), sin(0.5) .
10.1 & 13.1 OTHER TYPES OF FUNCTIONS 13
Example 3. Sketch and identify the curve defined by the parametric equa-
tions, x(t) = t2 − 2t and y(t) = t + 1. What is the displacement of a particle
moving along this curve from t = 0 to t = 3? Sketch this vector on your
diagram.
14 WEEK 1
13.2 Velocity and Parametric Curves
Just as we learned how to calculate the derivative of a function whose output
is a single real number, we also want to discuss the derivative of a function
whose output is a point in the plane (or, equivalently, the position vector for
that point). To do this we consider what we mean by velocity. We define
velocity as the rate of change of position. That is, it is obtained by dividing
a displacement by the length of time needed to produce that displacement.
For example, using again the function
r(t) = cos t, sin t ,
if we divide the displacement between times 0.5 and 1.6 by the length of that
time period, we get
1
cos(1.6) − cos(0.5), sin(1.6) − sin(0.5) .
1.6 − 0.5
This can be thought of as the average velocity of the dot between times
0.5 and 1.6 because if we traveled with this constant velocity for a time
period of 1.1 (i.e. from t = 0.5 to t = 1.6), we would produce exactly the
desired displacement. Notice that this is a vector in the same direction as
the displacement vector, but multiplied by the scalar 1/1.1. In other words,
this vector is a bit shorter than the displacement vector. Of course the actual
velocity of the dot changes continuously as it moves along the circle. We will
discuss this motion next.
13.2 VELOCITY AND PARAMETRIC CURVES 15
Concept Question 2. Suppose we replaced 1.6 by 0.5 + h and calculated
the average velocity. It would give us
1
cos(0.5 + h) − cos(0.5), sin(0.5 + h) − sin(0.5) .
h
If now we let h get smaller and smaller (h → 0) then this vector indicating
average velocity will
A. Shrink to a point, because the displacement vector gets smaller and smaller;
B. Get longer and longer because we are dividing the displacement by a
smaller and smaller number h;
C. Stay more or less the same length.
16 WEEK 1
We would like to study this process more closely. It is key to understanding
the concept of derivative for a vector-valued function (that is, for a plane
curve). Writing a for 0.5, for convenience, the average velocity becomes
1
[r(a + h) − r(a)]
h
which can also be written as
r(a + h) − r(a)
.
h
Our question about what this becomes as h → 0 is a question about
r(a + h) − r(a)
lim .
h→0 h
This is the definition of the derivative at a of the vector-valued function
r(t).
Notice that it looks the same as the definition of the derivative of an "ordi-
nary" function. The difference is that the numerator is a vector and not a
scalar quantity. If you imagine successive instances of average velocity as h
gets smaller and smaller, you might get something like this diagram:
In the diagram you see two vectors representing average velocities between
times 0.5 and 1.6 (h = 1.1) and between 0.5 and 0.8 (h = 0.3), as well as the
limit (the dashed arrow) to which these average velocity vectors converge as
13.2 VELOCITY AND PARAMETRIC CURVES 17
h → 0. This limit vector is called the derivative of the vector-valued func-
tion r(t) at t = a. This derivative represents the instantaneous velocity
of r(t) at t = a, and we write it as r′ (a):
1
r′ (a) = lim [r(a + h) − r(a)]
h→0 h
Example 4. Suppose a vector-valued function is given by the formula r(t) =
t, t2 . What is its derivative at time t = a?
18 WEEK 1
The length of the velocity vector is the speed.
The example on page 17 demonstrates a general principle:
When a vector-valued function is given by the formula
r(t) = x(t), y(t)
then its velocity at a can be calculated by taking derivatives of the
components:
r′ (a) = x′ (a), y ′ (a)
When a vector-valued function arises in a problem as the path of a moving
particle, it is sometimes helpful to draw that path. To do so is, in effect,
to draw the range of the function. One way to do that is to plot the points
(x(t), y(t)) for a set of values of t and then to connect these with a smooth
curve. Another method that is often useful is the method of eliminating
the parameter. To do this for the vector-valued function r(t) = t, t2 ,
begin with the two formulas
x=t
and
y = t2 ,
and use algebra to eliminate the parameter t. You may have used the same
idea already when you solved Example 3.
13.2 VELOCITY AND PARAMETRIC CURVES 19
Example 5. Use elimination of the parameter to describe the path r(t) =
t, t2 .
You can generate a picture of this curve (without the velocity vectors) very
easily, using the Maple command
> plot([t,tˆ2, t=−3.5..3.5]);
20 WEEK 1
Concept Question 3. A particle moves so that its location at time t is
1
given by r(t) = ( t3 − t, t), where the first coordinate measures distance
3
along a horizontal axis and the second coordinate measures vertical distance.
At what value(s) of t is the particle moving in a perfectly vertical direction?
A. At t = 1
B. At t = 0
√
C. At t = 0 and t = ± 3
D. At t = ±1
Parametric curves (or space curves) in three dimensions
A vector-valued function r(t) can also have a triple of numbers as its output,
as in r(t) = (x(t), y(t), z(t)). We can think of this triple as a set of coor-
dinates in three-dimensional space, so (x(t), y(t), z(t)) can be thought of as
describing the path of a moving point in three-space. For this reason, the
curve consisting of all outputs (x(t), y(t), z(t)) is also called a space curve.
Of course we can also think of r(t) as the position vector of that point. In
that case we would probably write r(t) = x(t), y(t), z(t) .
Paths in three-space are handled in the same way as paths in two-dimensional
space, especially when it comes to calculating the velocity.
13.2 VELOCITY AND PARAMETRIC CURVES 21
Example 6. A particle moves in such a way that its position at time t is
1 t − 1 2t
given by r(t) = 2 , , e . What is its velocity at time t = 1? Also,
t +1 t−5
what is the domain of this vector-valued function?
22 WEEK 1
Example 7. What does the particle do when t gets close to 5?
13.2 VELOCITY AND PARAMETRIC CURVES 23
For the preceding two questions it was important to understand interval
notation; let's review it here. (See also Appendix A in the textbook.)
(2, 7) means: { x | 2 < x < 7 };
[−5, −3) means: { x | − 5 ≤ x < −3 };
(−7, ∞) means: { x | − 7 < x < ∞ }.
Never write (−7, ∞], for ∞ is not a number, so you cannot include it at
the end of an interval.
Basic set notation is also discussed in Appendix A in the textbook: Consider
the sets S1 and S2 , let S3 be the set consisting of the elements of S1 together
with those of S2 , or S3 = S1 ∪ S2 ; S1 ∪ S2 is called the union of S1 and S2 .
S1 S2
We can also have a set with its elements consisting of those that belong to
both S1 and S2 . This is called the intersection of S1 and S2 , and can be
written as S4 = S1 ∩ S2 .
S1 S2
x ∈ S means "x is a member of the set S ".
24 WEEK 1
Example 8. When I sketch the curve
r(t) = cos(t) sin(t3 ), cos(t) cos(t3 ), sin(t)
using the Maple commands,
> with(plots):
> spacecurve([cos(t)∗cos(tˆ3),cos(t)∗sin(tˆ3), sin(t)], t=-4..4, num-
points=1000);
I get the following picture, which makes it look as if that curve stays on a
sphere centered at the origin. Prove that this is the case.
13.2 VELOCITY AND PARAMETRIC CURVES 25
Notes
26 WEEK 1
Week 2
Study Sections: 3.3 [3.4], 13.2 (first three pages) and
13.4 (first 4 pages) in the textbook
• Derivatives of Trigonometric Functions
• Acceleration and Parametric Curves
• Application to Motion Problems
27
28 WEEK 2
3.3 [3.4] Derivatives of Trigonometric Func-
tions
We would like to calculate the velocity of the moving green dot
r(t) = (cos t, sin t)
in Example 2. In order to do that we have to know how to calculate the
derivative of the functions sin and cos.
To begin our discussion of derivatives of these trigonometric functions, we
want to do three things first:
1. Define the "secondary trigonometric ratios" secant, cosecant and cotan-
gent,
2. Introduce the formulas for the sine and cosine of the sum of two angles,
YOU MUST MEMORIZE THESE
sin t
3. Investigate what happens to the expression as t gets closer and
t
closer to 0 (that is, as t → 0).
Introducing the secondary trigonometric ratios is best done using a pic-
ture. We will use x instead of t to denote the input variable.
c b
x
a
cos x = a/c sec x = c/a = 1/ cos x
sin x = b/c csc x = c/b = 1/ sin x
tan x = b/a = sin x/ cos x cot x = a/b = cos x/ sin x = 1/ tan x
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 29
We also need a formula that expresses the sine of the sum of two angles in
terms of the sines and cosines of those angle. Here is that formula:
Theorem W2.1 (see text pg. A29)
sin(x + y) = sin x cos y + cos x sin y
To see why this is so, consider a circle of radius 1, with angles x, y, w and z
drawn in as shown:
P
z
O x 1 R
y
w
Q
Note that P Q is perpendicular to OR.
Example 9. Finish the following calculations to prove the identity:
It is easy to see that
PR sin x
PR = = tan x =
1 cos x
QR
QR = = =
1
PQ =
OQ =
sin z = cos ( )
30 WEEK 2
Now use the Sine Law to calculate
sin(x + y)
=
tan x + tan y
Next multiply both sides by tan x + tan y and get
sin x sin y
sin(x + y) = cos x cos y + =
cos x cos y
Theorem W2.2 (see text pg. A29)
cos(x + y) = cos x cos y − sin x sin y
This identity follows from the one we just proved if we keep in mind that
if z = π − x as in the preceding discussion, then cos z = sin x. That is,
2
cos(π/2−x) = sin x and cos x = sin(π/2−x) The rest is just some calculation:
cos(x + y) = sin(π/2 − (x + y)) =
These theorems have the following simple corollary:
Corollary W1.1 (see text pg. A29)
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2 (x) − sin2 (x)
MEMORIZE THESE FORMULAS
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 31
Theorem W2.3 (see text pg. 190 [212])
sin θ
lim =1
θ→0 θ
A rigorous proof of this limit is given in the textbook on pages 190-191 [212].
Here we will find this limit experimentally:
sin x
Consider the graph of the function f (x) = . Apart from the little circle
x
on the y-axis, this is what we get if we plot it in Maple, using the command:
> plot(sin(x)/x, x=-1..1);
1
−1.0 −0.5 0 0.5 1.0
Notice that the function f (x) = sin x is undefined at x = 0, because with
x
x = 0 we cannot do the calculation. This is indicated by the little circle,
which was drawn in by hand - Maple does not draw it for you. Whether
or not 0 is in the domain of f (i.e. whether or not "f is defined at 0") has
nothing to do with the limit. The graph certainly suggests that
sin x
lim = 1.
x→0 x
32 WEEK 2
The following analogy may help explain what we mean by the limit: If the
graph in the diagram represents a road, and the vertical line through 0 a
canal, and if the little circle indicates that there is no bridge at that point,
then the limit is the y-value where you would fall into the canal if you came
along the road.
Another (also not totally reliable) way to show that this limit is equal to 1
is to use a calculator to test the function at values close to zero. (See text,
page 91 [93].)
x sin x/x
± 0.5 0.95885108
± 0.1 0.99833417
± 0.001 0.99999983
Maple will even check the limit directly for you: Simply enter
> Limit(sin(x)/x, x=0)=limit(sin(x)/x, x=0);
and press "return". (Here limit(...) does the calculation, while the "inert"
form of the command, Limit(...), gets Maple to write out the expression
sin x
lim ) Of course, none of these three methods is completely conclusive.
x→0 x
For a conclusive argument you have to go to the proof given in the text on
pages 190-191 [212].
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 33
Concept Question 4. What do you think the limit is likely to be in the
cos θ − 1
Corollary on the next page? That is, what do you think lim is equal
θ→0 θ
to, and why?
A. 0 because the numerator goes to zero as θ → 0
B. 1 because the limit in the theorem we just did was 1
C. ∞ because the denominator goes to zero as θ → 0
D. We cannot tell without doing some work.
A degree of confirmation of the answer can be obtained using Maple, either
by graphing the expression, using the command
> plot((cos(x)-1)/x, x= −2..2);
or by asking Maple to take the limit, using the command
> Limit((cos(x)-1)/x, x=0)=limit((cos(x)-1)/x, x=0);
Try it yourselves!
34 WEEK 2
Corollary W4.1 (see text pg. 192 [213])
cos θ − 1
lim =
θ→0 θ
Example 10. Prove the corollary using the theorem
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 35
d
Example 11. Prove the following theorem: dx sin x = cos x. Note this
is where you will need the trigonometric identity sin(x + y) = sin x cos y +
cos x sin y.
36 WEEK 2
Table of Derivatives of Trigonometric Functions
Once you have the derivatives of the sine and cosine functions you can use
differentiation rules to deduce the others. Try a few to make sure you can
do it:
d
1. dx
sin x = cos x ;
d
2. dx
cos x = − sin x ;
d
3. dx
tan x = sec2 x ;
d
4. dx
csc x = − csc x cot x ;
d
5. dx
sec x = sec x tan x ;
d
6. dx
cot x = − csc2 x.
MEMORIZE THESE!
13.2 & 13.4 VELOCITY AND ACCELERATION 37
13.2 & 13.4 Velocity and Acceleration
We will now return to the green dot of Example 2. Its path was given, as a
function of time, by the formula
r(t) = (cos t, sin t) .
We are now in a position to calculate its velocity:
r′ (t) = − sin t, cos t .
Here are some of those velocity vectors:
1
0.5
0
-1 -0.5 0 0.5 1
-0.5
-1
Now what would it mean to take a second derivative of a vector-valued func-
tion? In physics we associate the second derivative of position with acceler-
ation. In calculus, in high school, you will have learned to relate the second
derivative to the concavity of the graph of a function. We will see that both
points of view continue to make sense when we deal with vector-valued func-
tions: the second derivative of the path of a moving particle measures how
38 WEEK 2
fast and in which direction a particle is accelerating; and, when the acceler-
ation vector points in a direction different from that of the velocity, it also
indicates the "amount" by which the path curves. Think of the green dot as
a vehicle with you as a passenger. As you zip around the circle, you will find
yourself pushed against the side of the vehicle. This indicates that you are
accelerating in the direction in which the vehicle pushes against your body.
A greater speed or a tighter circle will produce a greater acceleration
Consider the picture, drawn above, of the moving green dot and its velocities
at t = 0, t = π/4, and t = 3. The second derivative is always the derivative
of the first derivative, so we should think about the rate at which the veloc-
ity changes. To do this we could begin with an average rate of change
of velocity over a short time period. So consider the velocities at t = 0
and t = π/4. We have to calculate the difference between these two vectors
(subtract one from the other) and then divide that by the time increase. To
think of this geometrically is a little difficult, for to do it we first have to
place the two velocity vectors at the same point, as in the following diagram:
1
0.5
0
-1 -0.5 0 0.5 1
-0.5
-1
13.2 & 13.4 VELOCITY AND ACCELERATION 39
The vector on the right of the second figure is the velocity at t = 0:√ ′ (0) √
r =
0, 1 ; the vector on the left is the velocity at t = π/4: r′ (π/4) = −1/ 2, 1/ 2 .
The vector at the top of the second figure is their difference, the vector
r′ (π/4) − r′ (0). Its coordinates are easily calculated:
√ √
r′ (π/4) − r′ (0) = −1/ 2, 1/ 2 − 1 = −0.707, −0.293 .
This vector represents the change from the velocity at t = 0 to the velocity
at t = π/4. The (average) rate of change between these times is found by
dividing this change in velocity (a vector) by the change in time (a scalar):
1
−0.707, −0.293 = −0.9, −.373 .
π/4
By this estimate, the acceleration of the green dot at time 0 is given ap-
proximately by the vector −0.9, −.373 . To get instantaneous, rather than
average, acceleration we should turn this into a limit:
1 ′
r′′ (0) = lim (r (h) − r′ (0))
h→0 h
On page 18 we saw that this instantaneous rate is calculated by taking deriva-
tives of the components of the vector-valued function r′ (t) = − sin t, cos t .
Example 12. Calculate the acceleration at t = 0.
40 WEEK 2
If
r = (x(t), y(t))
then
r′ = x′ (t), y ′(t)
and
r′′ = x′′ (t), y ′′ (t)
Notice that in the case of the green dot describing the circle at a constant
rate of rotation, we have r′′ (0) = −r(0). The acceleration is towards the
center! It is easy to see that (for uniform circular motion) this is true at any
value of t, not just t = 0:
d2 d2
r′′ (t) = cos t, 2 sin t = − cos t, − sin t = −r(t) .
dt2 dt
Example 13. Suppose you are a passenger in the back seat of a taxi that is
speeding along so that its location at time t (seconds) is r(t) = (100t2 , 10t),
measured in meters. What does the path of the taxi look like; what is your
acceleration at time t = 0; and given that your mass is 70 kilograms, and
assuming that you forgot to put on your seat belt, with what force is the car
door pressing against you at that instant?
13.2 & 13.4 VELOCITY AND ACCELERATION 41
Concept Question 5. As t increases from 0 to 10, which of the following
alternatives best describes what happens?
A. The force of the door against your body stays the same
B. The force of the door against your body decreases, but the force of the
back of your seat against your body increases
C. The force of the door against your side remains the same, but you find
yourself sliding forward towards the back of the seat in front of you
The calculation of acceleration in three-dimensional space is exactly the same
as in two dimensions, for exactly the same reasons. To find the acceleration
all you have to do is differentiate each of the (three) components twice:
If the position of a moving particle is given by
r(t) = (x(t), y(t), z(t)) ,
then its acceleration is given by
r′′ (t) = x′′ (t), y ′′(t), z ′′ (t) .
42 WEEK 2
Example 14. The acceleration due to gravity is 9.8 meters per second per
second. Suppose that between times t = −10 and t = 10, measured in seconds,
a stunt plane follows the path
r(t) = (200t, 5t3, 800 − 5t2 )
without going upside down or banking significantly. Will the pilot ever be
lifted off his seat? Notice that in this problem, and in all cases un-
less indicated otherwise, the third component (the z-component)
is assumed to measure vertical distance
13.2 & 13.4 VELOCITY AND ACCELERATION 43
Example 15. Describe the parametric curve
r(t) = (2 cos t, sin t) .
44 WEEK 2
Concept Question 6. Given the answer to the preceding question, describe
the following parametric curve:
r(t) = (2 cos t, sin t, t) .
A. It is an ellipse tilted in the direction of the x-axis
B. It is an ellipse tilted in the direction of the y-axis
C. It is a helix
D. It is a curve in the plane that spirals outward
13.2 & 13.4 VELOCITY AND ACCELERATION 45
Solution of the problem stated at the start of this unit
We now complete the first unit in the course by revisiting the problem posed
at its beginning, on page 4 What does that tell us about the way this object moves?
If we ignore the last coordinate, and focus on the first two, we are effectively
looking at the projection of the path of the particle on the (x, y)-plane. In
other, words, if we imagine the sun shining from the "end" of the z-axis
(i.e. infinitely far away in that direction), then the first two coordinates
describe the movement of the object's shadow on the (x, y)-plane. But we
saw in earlier examples that (cos t, sin t) rotates around a circle of radius 1
centered at the origin at a constant rate of one revolution per unit time. The
z-coordinate ln(t + 1) simply tells us how far the object is above (or below)
the (x, y)-plane. Sketch the graph of ln t below:
3
2
1
0
-3 -2 -1 0 1 2 3
-1
-2
-3
46 WEEK 2
What is the effect of changing ln t to ln(t + 1)? Sketch the graph of ln(t + 1).
3
2
1
0
-3 -2 -1 0 1 2 3
-1
-2
-3
NOTE: You can plot both of these graphs in Maple, with the commands
> plot(ln(t), t=0..5, scaling=constrained);
> plot(ln(t+1), t=0..5,scaling=constrained);
A question that springs to mind is "if this function indicates the height of
the object at time t, what is the greatest height reached by it? " This is really
a question about the range of the function ln t. Does it have a horizontal
asymptote as t → ∞ or is every height eventually exceeded by the object?
From the origins of ln t as the inverse of the exponential function ex we know
that the range of ln t is the same as the domain of ex which, we know, is
the entire real line. Thus this object keeps rising as it circles, and eventually
reaches any height whatsoever, even though it rises more an more slowly, and
thus takes longer and longer to achieve a given increase in height.
2. What is its speed at, say, time t = 0?
13.2 & 13.4 VELOCITY AND ACCELERATION 47
3. When will you experience the greatest acceleration?
4. If you were to fall off at time t = 1, what would happen to you, especially
in the case of zero gravity?
48 WEEK 2
Notes
Week 3
Study Sections: 1.6, 3.5 [3.6], 3.6 [3.8] in the textbook
• Course Unit: More Derivative Techniques and Applications
• Inverse Trigonometric Functions
• Implicit Differentiation
• Derivatives of Inverse Trigonometric Functions
• Logarithmic Differentiation
49
50 WEEK 3
Course Unit: More About Derivatives
In the next two weeks, we return to functions of a single variable and ex-
pand our knowledge of differential calculus. This is a continuation of the
material that you should already know from high school, and which is cov-
ered in the Quick and Easy Differential Calculus Notes. Here we will learn
more derivative techniques and see how derivatives are used to model many
physical processes. We will also learn how derivatives can be used to approx-
imate functions; such approximations can be used to estimate experimental
measurement errors.
Let's begin with a question to motivate a discussion of inverse trigonometric
functions, the first topic in this section:
Example 16. A projectile is fired from the origin, with angle of elevation α
(radians) and speed 60 meters per second.
α
Assuming that air resistance is negligible and that the only force on the pro-
jectile is gravity (producing a downward acceleration of g = 9.8 meters per
second per second), answer the following questions:
1. Find a formula for the position of the projectile at time t.
2. How far down range will the projectile land?
3. At what angle α should the projectile be launched so that it will land
300 meters down range?
COURSE UNIT: MORE ABOUT DERIVATIVES 51
52 WEEK 3
COURSE UNIT: MORE ABOUT DERIVATIVES 53
54 WEEK 3
1.6 Inverse Trigonometric Functions
We will now apply our understanding of inverse functions to discuss inverse
trigonometric functions more carefully. We would like to be able to talk
about the inverse of the sine function, for example, as we did when we solved
Example 16. As you will have learned in high school, and as reviewed in
the Quick and Easy Calculus notes available through the course website, a
function must be one-to-one in order for it to have an inverse function. In
the case of the function f (x) = sin x, one can clearly observe by graphing
the function that it does not possess this property (does not pass horizontal
line test) and that it therefore does not have an inverse. How then can we
find an inverse for f (x) = sin x? What did the calculator think it was doing
when we used the sin−1 button? We get around this problem by analyzing
f (x) = sin x only from − π to π . The graph of the sin function from − π to
2 2 2
π
2
is shown in the following graph:
f (x) = Sin x
1
−π
2
π
2
−1
For convenience we could call this new function Sin (x), where
Sin (x) = sin(x)
provided − π ≤ x ≤ π .
2 2
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 55
Sin (x) satisfies the horizontal line test and therefore has an inverse function
which we call the inverse sine function and denote it as
sin−1 (x).
Note that
1
sin−1 (x) = .
sin(x)
In fact the practice of writing the inverse of the sine function as sin−1 is a
bad habit that we have to get used to. It is too well-established to change; in
particular, that is how the function is indicated on calculators. As mentioned
earlier, the inverse of sin is also often written as
arcsin(x) ,
and its graph is the following one:
π
2
f (x) = arcsin x
−1 1
−π
2
The domain of arcsin is: [−1, 1] .
The range of arcsin is: − π , π .
2 2
56 WEEK 3
Note that, as always, the graph can be produced using Maple. Simply enter
the command > plot(arcsin(x), x=-2..2, scaling=constrained);
Since arcsin undoes what sin does, and vice-versa, the following equations
are true, but only for the specified values of x:
sin−1 (sin x) = x, for − π ≤ x ≤ π
2 2
sin(sin−1 x) = x, for −1 ≤ x ≤ 1.
Example 17. What is the value of arcsin(0.5)?
Example 18. sin(−7π/5) = 0.951, so what is the value of sin−1 (0.951)?
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 57
The inverse of cosine is obtained by a calculation similar to the way the
inverse of sine was determined. We analyze cosine from 0 to π; this is shown
in the following graph:
1 f (x) = Cos x
0
π
−1
For convenience, we could call this new function Cos (x) where
Cos (x) = cos(x)
provided 0 ≤ x ≤ π.
Cos (x) satisfies the horizontal line test and therefore has an inverse function
which we call the inverse cosine function and denote it as
cos−1 (x),
noting that
1
cos−1 x = .
cos x
The inverse cosine function is also often written as
arccos x,
and it is graphed on the following page.
58 WEEK 3
f (x) = arccos x
π
−1 1
The domain of arccos is: [−1, 1].
The range of arccos is: [0, π].
Concept Question 7. When you enter cos−1 2 on your calculator, it ob-
jects. Why is that?
A. The numbers involved are too large for the calculator to handle
B. The calculator does not understand this business of taking the inverse
using only part of the cosine function
C. The cosine function does not really have an inverse
D. The number 2 is outside the domain of the function arccos
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 59
Concept Question 8. What is the value of arccos(cos(7π/6))?
A. 7π/6
B. 210
C. 5π/6
60 WEEK 3
The inverse of tangent is determined in the same way, only analyzing it from
− π to π , this is shown in the following graph:
2 2
f (x) = T an x
−π
2
π
2
As done before, we name this portion of the tan function T an (x), where
T an(x) = tan x
provided − π < x < π .
2 2
T an (x) satisfies the horizontal line test and therefore has an inverse, which
we call the inverse tangent function and denote it as
tan−1 x,
once again noting that
1
tan−1 x = .
tan x
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 61
The inverse tangent function is also often written as
arctan x,
and its graph is the following one:
π
f (x) = tan−1 x
2
−π
2
This graph can be produced in Maple, using the command
> plot(arctan(x), x=-5..5, scaling=constrained);
If you want the horizontal asymptotes as well, use the command
> plot([arctan(x), Pi/2, -Pi/2], x=-5..5, scaling=constrained);
The domain of arctan is: (−∞, ∞).
The range of arctan is: (− π , π ).
2 2
Example 19. What is the value of arctan(1)?
62 WEEK 3
3.5 [3.6] Implicit Differentiation
The relation between an "input" x and "output" y may be given explicitly.
Here is an example: √
y = 1 − x2 .
This same relation can be given implicitly:
x2 + y 2 = 1.
When the relation is given implicitly, there are often several y's for a given x,
unless you add some information, e.g. y ≥ 0. Sometimes an implicit relation
can be turned into an explicit relation by algebra, as in the above example,
although usually this cannot be done! For example, consider the relation:
cos(x − y) = y sin x.
dy
You might think that to calculate dx you must first write y explicitly in
terms of x. In this section we learn that this is not necessary. Hence the
term "Implicit Differentiation".
Consider the following example:
√
x 1 + y + y x − 1 = x + 5.
In this relation, think of y as somehow dependent on x. That is y = y(x); in
other words y is a function of x. To help you think this way, you can write
the expression in this way:
√
x 1 + y(x) + y(x) x − 1 = x + 5.
Now differentiate both sides:
dy
1 dx dy √ 1 1
1 + y(x) + x · · + · x − 1 + y(x) · · √ = 1.
2 1 + y(x) dx 2 x−1
dy
Next, collect terms involving dx
and go back to writing y instead of y(x):
x √ dy y
√ + x−1 =1− 1+y− √ .
2 1+y dx 2 x−1
3.5 [3.6] IMPLICIT DIFFERENTIATION 63
dy
Solving for dx
gives:
√
dy 1 − 1 + y − 2√y
x−1
= √ .
dx √x + x−1
2 1+y
As in the example just demonstrated, it is typical of implicit differentiation
that the answer involves both x and y.
You can think of √
x 1+y +y x −1 = x +5;
or, to express it more completely in set notation,
√
{(x, y) ∈ R2 | x 1 + y + y x − 1 = x + 5} ,
as some sort of curve in the (x, y)-plane.
(2,3)
Of course, this curve is not presented to us as a parametric curve (as a
vector valued function). It may be possible to find a way to parametrize this
particular set of points, but it is probably difficult. Fortunately we do not
need to. Note that (x, y) = (2, 3) satisfies the equation, therefore (2, 3) lies
on the curve. We can calculate the slope at the point (2, 3) by substituting
dy
for x = 2 and y = 3 into the formula obtained earlier for dx . This works out
dy
to dx = − 5 . This tells us that the slope of the curve (that is, the tangent
3
line to the curve) at (2,3) is equal to − 5 . Note that we are not speaking of
3
a velocity here, for to have a velocity you first need a parametrization. If
we did have a parametrization for this curve - that is, if we had a particle
moving along this curve - then its velocity as it passes through the point
(2, 3) would lie along the tangent line whose slope we just calculated.
64 WEEK 3
Example 20. Find the points where the ellipse x2 − xy + y 2 = 3, crosses the
x-axis and show that the tangent lines at these points are parallel.
3.5 [3.6] IMPLICIT DIFFERENTIATION 65
Example 21. Let y = sin−1 x, and differentiate the equation sin(sin−1 x) =
d
x, or sin(y) = x, to get a formula for sin−1 (x).
dx
Notice that Maple calculates derivatives for you. In this case you can do it
in two ways: You can work with the function arcsin directly and enter
> D(arcsin);
or you can start with the expression arcsin(x) (that is; put the variable
in), and enter
> diff(arcsin(x), x);
66 WEEK 3
Here is a list of the derivatives of the inverse trigonometric functions. Mem-
orize the first three of these. They will be used a lot in the course.
d 1
sin−1 x = √ ;
dx 1 − x2
d 1
tan−1 x = ;
dx 1 + x2
d 1
cos−1 x = −√ ;
dx 1 − x2
d 1
sec−1 x = √ .
dx x x2 − 1
Example 22. Differentiate y = tan−1 (sin x).
3.6 [3.8] LOGARITHMIC DIFFERENTIATION 67
3.6 [3.8] Logarithmic Differentiation
dy
Example 23. Find when y = (x2 + 1)x .
dx
Maple is not fazed by logarithmic derivatives. Try
> diff((x∧2+1)∧x, x);
68 WEEK 3
Notes
Week 4
Study Sections: 3.7 [3.3], 3.10 [3.11], 4.4 in the text-
book
• Rates of Change in Science
• Differentials and Linear Approximations
• l'Hopital's Rule
69
70 WEEK 4
3.7 [3.3] Rates of Change in Science
A large part of the difficulty in applying mathematical methods to applied
problems is the matter of translating the English-language formulation of a
problem or of a scientific law into a mathematical sentence. It is especially
important to recognize a derivative when it appears in a discussion and to
know how to express it mathematically.
Example 24. "When water drains out of a tank, the rate of flow is propor-
tional to the square root of the volume of water left". Translate this into a
mathematical sentence.
3.7 [3.3] RATES OF CHANGE IN SCIENCE 71
Concept Question 9. Newton's Law of Cooling states that the rate at
which an object cools off (or heats up) is proportional to the difference between
the temperature of the object and the temperature of its surroundings. We
want to translate this into a mathematical sentence for a cup of coffee placed
in a room at 20 o C. What variable quantities must be given names so that
we can do this translation?
A. The temperature T of the object, and the time t.
B. The temperature T , the time t, and the rate R at which the object cools
off.
C. The temperature T , the time t, and the difference s between the tem-
perature of the surroundings and the temperature of the object
D. The temperature T , the time t, the difference s between the temperature
of the surroundings and the temperature of the object, and the rate R
at which the object cools.
72 WEEK 4
Example 25. Isothermal compressibility measures the compressibility of a
gas kept at constant temperature. It is given by the formula:
1 dV
β=− ,
V dP
(see page 200 in the text book), where V is volume and P is pressure.
• Why is there a derivative?
• Why a minus sign?
• Why the factor 1/V ?
3.7 [3.3] RATES OF CHANGE IN SCIENCE 73
Example 26. In a certain chemical reaction a substance A is transformed
into a substance B. Because the reaction is autocatalytic, the product B is
produced at a rate that is proportional to the product of the concentrations of
A and B. Assuming that the concentrations of A and B always add up to
K, answer the following questions:
• Translate this into a mathematical sentence.
• Sketch a rough graph of the concentration of B over time.
74 WEEK 4
Many laws of nature, when modeled by mathematics, take the form of a
differential equation. A differential equation is an equation that in-
volves an unknown function and some of its derivatives. To solve a
differential equation is to find a formula for the unknown function. There
will be a unit on differential equations at the end of this term.
Example 27. Under certain circumstances (abundant resources) the rate of
growth of a population is proportional to the size of the population. Translate
this into a differential equation.
3.7 [3.3] RATES OF CHANGE IN SCIENCE 75
Example 28. (Hooke's Law) Suppose a wooden block is attached to a hor-
izontal spring on a frictionless table. One end of the spring is fixed to the
table. Then the acceleration of the block is proportional to the distance by
which the spring is extended from its equilibrium position. Translate this into
a differential equation.
0
76 WEEK 4
3.10 [3.11] Differentials and Linear Approxi-
mations
When you look at a very small part of a (differentiable) function, it looks
linear.
• Zoom into a graph and it will look like a line.
• Zoom into a table (i.e. check nearby values) and the values go up by
(roughly) equal amounts.
Consider the function y = f (x). What do we mean by saying that f ′ (x0 )
is the slope of f at x0 ? It means that if we change x from x0 to x0 + ∆x,
resulting in a change of y from y0 = f (x0 ) to y0 + ∆y, then the ratio between
the changes is (approximately) f ′ (x0 ). In other words,
∆y ≈ f ′ (x0 )∆x .
The idea behind derivatives is that this approximate relationship becomes
precise in the limit as the changes in the variables go to zero. In other words,
as ∆x and ∆y get smaller and smaller, the ratio between them becomes more
and more precisely equal to f ′ (x0 ). Leibniz (see page 170 in the textbook),
who along with Newton invented calculus, liked to think of the quantities
as "infinitesimals" - infinitely small changes whose sizes can nevertheless be
compared to each other. To indicate their status as infinitely small limiting
quantities, we use the symbols dx and dy in place of ∆x and ∆y. They
are then referred to as differentials. Because they describe the limiting
situation, the relationship between them becomes exact:
dy = f ′ (x0 )dx .
You can also think of the relationship between ∆x, ∆y, dx and dy in terms
of the relationship between the graph of f and its tangent line at x0 : The
tangent line is what the graph becomes as you zoom in. Therefore, if we let
∆x = dx then we can think of ∆y as the corresponding change in the value
given by f , and dy as the corresponding increase given by the tangent line.
When we zoom in (that is when ∆x is very small) then
∆y ≈ dy.
The following diagram illustrates this relationship between the small changes:
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 77
y = f (x)
slope = f ′ (x0 )
dy
∆y
x0 x0 + ∆x
dx = ∆x
In summary, we have the equation relating the differentials
dy = f ′ (x0)dx.
and the equation relating small changes in the independent and dependent
variables:
∆y ≈ f ′ (x0)∆x.
when ∆x is small. Notice that in this formula we are merely repeating
something we have known from the time we first learned to differentiate:
∆y
∆x
≈ f ′ (x0).
78 WEEK 4
√
Example 29. Use differentials to get an approximate value for 4.03.
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 79
√
The method of using differentials to approximate 4.03 can be summarized
as follows:
√
• We have a function f (x) (in this case x) and a point a (in this case
4) where we know the value f (a).
• We have a nearby point x (in this case 4.03).
• We want an approximate value for f (x).
To solve this problem we let
∆x = x − a,
∆y = f (x) − f (a).
We know that
∆y ≈ f ′ (a) · ∆x.
Thus
f (x) − f (a) ≈ f ′ (a)(x − a).
That is
f (x) ≈ f (a) + f ′ (a)(x − a).
This is the linear approximation or the tangent line approximation of
f at a.
The right hand side,
L(x) = f (a) + f ′ (a)(x − a),
80 WEEK 4
is called the linearization of f at a. Think of this formula as a recipe for
a linear function L(x) which is very nearly equal to f (x) as long as x is close
to a. The ingredients that you have to supply are written in boldface letters.
Example 30. The function y = L(x) is a linear equation, so it represents a
line. What line is it and why?
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 81
Concept Question 10. If e.5 is approximated by using the tangent line
to the graph of f (x) = ex at (0, 1), and we know that f ′ (0) = 1, then the
approximation is1
A. 0.5
B. 1 + e.5
C. 1 + .5
1
We thank the Good Question project at for
this problem.
82 WEEK 4
Example 31. The edge of a cube was found to be 30 cm with a possible
error in measurement of 0.1 cm. Use differentials to estimate the maximum
possible error in computing the surface area of the cube.
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 83
Concept Question 11. A function f and its derivative have the following
values at 0 and 10:
f (0) = 2 f (10) = 5
f ′ (0) = −1 f ′ (10) = 1
What is a good estimate for f (0.2)?
A. (0.02) × (5 − 2) + 2 = 2.06
B. 2
C. (0.2) × (−1) = −0.2
D. (−1) × (0.2) + 2 = 1.8
84 WEEK 4
Example 32. Find the linearization of f (x) = 1/x at a = 4.
Example 33. Find the equation of the tangent line to the hyperbola y = 1/x
at x = 4.
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 85
4.4 Indeterminate Forms and l'Hopital's Rule
l'Hopital's Rule is a convenient method for calculating limits. It is also a nice
application of linear approximations. Here is an example to demonstrate how
it works: Suppose we want to find the following limit:
sin x
lim .
x→0 x
Both sin x and x have limit 0. Therefore we would end up with the expression
0
.
0
This expression is called an indeterminate form. When an indetermi-
nate form of this type arises in calculating limits, l'Hopital's Rule says
that you should differentiate the numerator and denominator separately.
(Caution: Do not confuse this with the quotient rule!)
If we were to apply l'Hopital's Rule to our example, that is if we were to
differentiate the numerator and denominator, we would end up with
cos x
.
1
If we then take the limit of this new expression we get
cos x
lim = 1.
x→0 1
l'Hopital's Rule says that this limit is equal to the limit we originally began
with, namely
sin x
lim = 1.
x→0 x
Theorem W4.1 L'Hopital's Rule (see text pg. 299 [308])
If lim f (x) = 0 and lim g(x) = 0. Then
x→a x→a
f (x) f ′ (x)
lim = lim ′
x→a g(x) x→a g (x)
provided the latter exists.
86 WEEK 4
ex − 1
Example 34. Find lim .
x→0 sin x
Maple can handle limits like this quite well: Enter
> limit((exp(x)-1)/sin(x), x=0);
2
ex
Example 35. Find lim 2 .
x→0 x − 2
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 87
Why does this method work?
Consider the following diagrams, which show generic functions f and g whose
limits as x → a are both 0 (in fact, f (a) = g(a) = 0):
f (x)
g lim =?
x→a g(x)
a
f
Now if we zoom in near a, the graphs of the functions will begin to resemble
their tangent lines at a:
g
x
a
f
That is, for x close to a, we have these linearizations, represented by the two
straight lines in the picture:
f (x) ≈ f (a) + f ′ (a)(x − a) = f ′ (a)(x − a) ,
g(x) ≈ g(a) + g ′(a)(x − a) = g ′(a)(x − a) .
88 WEEK 4
Notice that we used the fact that f (a) = g(a) = 0. Therefore we have
f (x) f ′ (a)(x − a) f ′ (a)
≈ ′ = ′ .
g(x) g (a)(x − a) g (a)
Since all of these approximations get better and better as x → a, we conclude
that
f (a) f ′ (a)
lim = ′ .
x→a g(a) g (a)
If this were a more rigorous mathematics course, we would find that there
are several points at which this proof is not totally clear, but it captures the
essence of why l'Hopital's Rule works the way it does. For a complete treat-
ment covering the subtleties that have been glossed over in our discussion,
check Appendix F in the textbook.
l'Hospital's Rule can also be used when we get an indeterminate of the form
±∞
,
±∞
0
instead of , and when a limit is taken for x → ∞.
0
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 89
ex
Example 36. Calculate lim 3 .
x→∞ x
Maple does this limit for you if you enter
> limit(exp(x)/x∧3, x=infinity);
90 WEEK 4
Notes
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 91
92 WEEK 4
Week 5
Study Sections: 5.1 and 5.2 in the textbook
• What is an Integral?
• The integral as a way to calculate an accumulated total
• Riemann Sums
• Connection to Areas
93
94 WEEK 5
Course Unit: Integration
If we had to summarize the first four weeks of the course, we would have
to say that the focus was on differentiation. We learned derivatives of some
new functions (at least for most of us) and we learned to differentiate and
interpret vector-valued functions.
All differentiation problems ask the same basic question: At what rate does
a process change, and how does that rate of change relate to other charac-
teristics of the process? This entire discussion of rates of change depended on
one key idea: No matter how complicated a time-dependant process is, when
you examine change over a sufficiently small interval, the behaviour seems
linear; that is, over a very small time interval, the rate seems constant. The
analysis was always directly or indirectly related to this simple ideaIn the remainder of the course we will study integration. Again, the analysis
is made possible by the observation that on a very small scale all processes
look linear. This time, though, we will use this fact to see how regarding
a process as an accumulation of infinitely many small linear steps allows us
to calculate the accumulated total even when the rate of accumulation is
far from linear. Integration is always in some way about finding the
total at the end of a process of accumulation.
COURSE UNIT: INTEGRATION 95
Suppose we want to find the volume of the hat pictured above. The (solid)
shape pictured is the shape you get if you take the region between the graph
of the function 1/(x2 + 3x + 2) and the x-axis, bounded by the y-axis and the
line x = 1, and rotate it around the vertical axis. Initially it may seem that
this question has nothing do to with a process of accumulation. However,
there are several ways to turn this into an accumulation.
One option is to think of this figure as a stack of infinitely many infinitely
thin disks of decreasing radius piled on top of each other. This would allow us
to think of its volume as the accumulated total of the volumes of these disks.
If each disk is thin enough (and infinitely thin is very thin), and because they
have a circular cross-section, we can think of these disks as (infinitely thin)
cylinders. Since we know how to calculate the volume of a circular cylinder,
we may be able to use this analysis to arrive at a calculation for the total
volume. This method is described more fully at the end of week 7.
A second option is to think of the solid figure as a nested set of cylindri-
cal shells. Imagine the solid separated into infinitely many infinitely thin
cylindrical shells. Think of growth rings in the trunk of a tree; or imagine
a very fine set of concentric circular cookie cutters of various radii. Each
concentric cylindrical shell has a height determined by its distance from the
centre, and radius equal to that distance. This will allow us to determine
the volume (area multiplied by infinitely small thickness) of a typical shell.
We can think of the volume of the solid figure as the accumulated total of
these small volumes. This method of cylindrical shells is the one that turns
out to work best for this solid. The method of cylindrical shells is discussed
in Week 8. For a complete solution of this problem you should go to Week
9.
On the way, we will examine a lot of other problems that relate more obvi-
ously to a process of accumulation, and use integration to solve them.
96 WEEK 5
5.1 & 5.2 What is an Integral?
Concept Question 12. Suppose you are travelling by car on a long stretch
of isolated road. Your car is old. The speedometer works, but the odometer
is broken. You make the observations indicated on the following table:
time t (hours) 0 0.25 0.5 1 1.25 1.5
speed v (km/h) 55 60 65 65 60 50
You want to use the information in the table to estimate how far you travelled
in 1.5 hours. Which of the following calculations makes the most sense to
that end?
1
A. (55 + 60 + 65 + 65 + 60 + 55) × 1.5
6
B. 0.25 × 55 + 0.25 × 60 + 0.5 × 65 + 0.25 × 65 + 0.25 × 60
C. 0.25 × (55 + 60 + 65 + 65 + 60 + 50)
D. 55 ÷ 0.25 + 60 ÷ 0.25 + 65 ÷ 0.5 + 65 ÷ 0.25 + 60 ÷ 0.25
5.1 & 5.2 WHAT IS AN INTEGRAL? 97
98 WEEK 5
Here is a principle that we will see again and again as we learn about inte-
gration:
To say that we are studying the accumulation of a quantity s that accu-
mulates at a varying rate v(t) is to say
• first of all that
ds
= v(t) ;
dt
and secondly that
• an estimate of the total accumulated as t varies from a to b can
be obtained by sampling v in each of a number of subintervals
of the total interval [a, b] and adding the products of rate times
subinterval-length (v × ∆t) for each of these intervals.
We would be able to estimate total distance more accurately if we sampled
the speed of the car more frequently.
Example 37. For example, suppose we had sampled the speed every 6 min-
utes, and that we measured the speeds
v(0), v(0.1), v(0.2), · · · v(1.4), v(1.5)
How could we combine these numbers to obtain a better estimate of the dis-
tance travelled from time 0 to time 1.5?
5.1 & 5.2 WHAT IS AN INTEGRAL? 99
100 WEEK 5
Riemann Sums
Both the left sum and the right sum are examples of Riemann sums. A
Riemann sum is any sum obtained in the process of trying to calculate the
total of an accumulation that takes place at rate v(t) by partitioning an
interval a ≤ t ≤ b into n subintervals:
a = t0 < t1 < t2 < · · · < tn−1 < tn = b ,
and then choosing a representative point t∗ in (or on the edge of) the ith
i
interval for each i. Each term of the Riemann sum has the form
rate × subinterval-length
where the rate (for the ith subinterval) is taken as v(t∗ ) and the subinterval
i
length is the length of time represented by that subinterval, namely ∆ti =
ti − ti−1 . In effect, you are pretending that the rate of accumulation is v(t∗ )
i
not just at that one point of time, but throughout the ith subinterval. Thus
the Riemann sum representing the accumulated total is
v(t∗ ) × ∆t1 + v(t∗ ) × ∆t2 + v(t∗ ) × ∆t3 + · · · + v(t∗ ) × ∆tn ,
1 2 3 n
or n
v(t∗ ) × ∆ti
i
i=1
5.1 & 5.2 WHAT IS AN INTEGRAL? 101
Area Under a Graph
Another problem that gives rise to a Riemann Sum occurs when calculating
the area under a graph. Suppose we want to calculate the area under the
1
graph of f (x) = 1+x2 between 0 and 1.
1
f (x) =
1 1 + x2
−1 1
If the function were constant, the answer would be obvious:
Area = (function value) × (length of interval).
In general we get an approximate value by dividing the interval into (n) sub-
intervals and sampling the function once in each sub-interval: f (x∗ ), where
i
x∗ is a sample point in the the interval [xi−1 , xi ]:
i
0 = x0 < x1 < x2 < . . . < xn = 1.
If each interval [ xi−1 , xi ] is very small, then the function f (x) is "practically"
constant on [ xi−1 , xi ]. Therefore the area under the graph of f (x) between
xi−1 and xi is approximately
f (x∗ ) · (xi − xi−1 ) .
i
height ∆xi =width
We could then add these approximate values of the area under f for each
sub-interval, producing another "Riemann Sum":
n
f (x∗ ) · ∆xi .
i
i=1
102 WEEK 5
We can represent this process in terms of a picture:
1 �
∗
f (x1 )�
x0 = 0 x1 x2 x3 1
x∗
1 x∗
2 x∗
3 x∗
4
1
e.g. if x1 = 4 , x2 = 1 ,
2
and x3 = 3 ,
4
we could choose x∗ = 1 ,
1 8
x∗ = 3 ,
2 8
x∗ =
3
5
8
and x∗ = 7 .
4 8
Therefore,
1 64
f (x∗ ) = f
1 = ,
8 65
3 64
f (x∗ ) = f
2 = ,
8 73
5 64
f (x∗ ) = f
3 = ,
8 89
7 64
f (x∗ ) = f
4 = .
8 113
So the approximate area under f (x) between 0 and 1 is
64
Area ≈ 65
·1+
4
64
73
· 1
4
+ 64
89
·1+
4
64
113
· 1
4
Area ≈ 0.7867.
5.1 & 5.2 WHAT IS AN INTEGRAL? 103
We have calculated a Riemann Sum for a partition of [ 0, 1 ] into 4 equal
sub-intervals, choosing each "sample point" in the center of the sub-interval.
This is known as the "midpoint sum", or "midpoint rule".
We could have chosen the left end of each sub-interval instead (x∗ = xi−1 )
1
and we would get the "left sum". Similarly we could choose the right end of
each interval and calculate the "right sum". In the following picture the left
sum is indicated by solid lines and the right sum by dashed lines.
1
1 f (x) =
1 + x2
1
Because this function f (x) is decreasing on [ 0, 1 ], it follows that the
Right sum ≤ Area ≤ Left sum.
Summary
In certain problems that involve the accumulation of some quantity over an
x-interval [ a, b ] (e.g. gas used, area), an approximate answer may be found
by partitioning the interval and pretending that on each subinterval the
rate f (x) at which the accumulation takes place is constant. The obvious
calculation then produces a Riemann Sum:
n
f (x∗ )∆xi ,
i
i=1
where f (x∗ )
i represents the "rate" of accumulation on the interval, and ∆xi
the interval length.
104 WEEK 5
The true value of the accumulated total can be found in principle by doing
the calculation over and over again with finer partitions (i.e. ∆xi → 0 or
b−a
equivalently, as n → ∞ since n = ). In other words the Riemann Sums
∆xi
n
f (x∗ )∆xi ,
i
i=1
converge to the true value of the accumulated quantity. The true value is
known as the "Integral", and is denoted by the symbol
b
f (x) dx.
a
This is read as the "Integral of f (x) from a to b". It is the limit of the
approximating Riemann Sums. f (x) is called the integrand and a and b are
known as the limits of integration.
Notice how the symbol for the integral resembles the expression for the Rie-
mann sums that approximate it: The integral sign resembles an elon-
gated S, for "sum"; The index i and its limits 1 and n are replaced by a and
b; f (x∗ ) becomes f (x); and ∆xi becomes dx. In terms of developing your
i
intuition, it is not unreasonable to think of the integral as an infinite
(Riemann) sum of infinitely many small increments.
Thus, in the examples we have discussed so far,
1.5
The total distance traveled in Example 37 is v(t) dt
0
1
1
The total area discussed on page 101 is dx
0 1 + x2
In each case, the Riemann sums we computed are approximations to these
integrals. Notice that in the first case, the integrand is the rate of accu-
mulation of the underlying process. We will see in a moment that the
second case can also be interpreted that way.
5.1 & 5.2 WHAT IS AN INTEGRAL? 105
Concept Question 13. Water is pouring into a reservoir at a varying rate
of f (t) cubic meters per hour. The total volume (measured in cubic meters)
of water in the reservoir at time t (hours) is g(t). Then which of the following
statements are necessarily true?
5
1. The amount of water in the reservoir at time t = 5 is equal to f (t) dt
0
2. The increase in the amount of water in the reservoir between times 0
5
and 5 is f (t) dt
0
3. At time t = 3, water is flowing into the reservoir at a rate equal to
g ′ (3)
4. The increase in the amount of water in the reservoir between times 0
and 5 is g(5) − g(0)
A. 1 and 4 only
B. 2 and 4 only
C. 3 and 4 only
D. 2, 3 and 4 only
E. all
106 WEEK 5
Here is a more formal definition of the integral concept: Let P be a partition
of [ a, b ],
P : a = x0 ≤ x1 ≤ . . . xn−1 ≤ xn = b.
Choose x∗ in [xi−1 , xi ] and let ||P || = max {∆xi }. Then the integral of f
i
from a to b is defined to be
b n
f (x) dx = lim f (x∗ ) · ∆xi
i
a ||P ||→0
i=1
if the limit exists.
The function f is called integrable on [ a, b ] if this limit exists; that is, if
successive approximations get closer and closer to a fixed amount (which we
then decree to be the integral), and don't fluctuate erratically or go off to
infinity. You might think that this would always be the case. You would be
nearly correct in supposing this, but there are cases when a function does not
have an integral. For example suppose the function has a vertical asymptote
inside the interval [a, b]. In that case you would not expect to be able to
talk about the area under its graph. Or what about a really wild function
that is highly discontinuous (lots of sudden increases and decreases in value -
an extreme example is the function that takes the value 1 at rational inputs
and 0 at irrational inputs)? Fortunately, many (essentially all 'reasonable')
functions are integrable:
Theorem W5.1 (see text pg. 368 [380])
If f is continuous on [a, b], then f is integrable on [a, b].
The role of Riemann sums
1. They are needed to say what we mean by an integral.
2. They enable us to decide which integral is appropriate in a word
problem.
3. They can also be used to give an approximate value of the integral,
as seen in the next example.
5.1 & 5.2 WHAT IS AN INTEGRAL? 107
2
Example 38. Use the midpoint rule with n = 4 to approximate x2 dx.
0
108 WEEK 5
You can get Maple to draw the diagram associated with the midpoint rule
for this integral by entering
> with(student): (this enables the extra procedures you need)
> middlebox(x∧2, x=0..2, 4);
You can then produce this Riemann sum by typing
> middlesum(x∧2, x=0..2, 4);
and find the value of this sum by typing
> evalf(middlesum(x∧2, x=0..2, 4));
Concept Question 14. The answer we found in the preceding question is
A. An underestimate because the first rectangle is very small
B. An overestimate because this is an increasing function.
C. An underestimate because the graph of f is concave upward
D. An overestimate because we used the midpoints
5.1 & 5.2 WHAT IS AN INTEGRAL? 109
The Relationship Between the Integral and the Area
Under a Graph
We started our discussion of integration by stressing the fact that an integral
problem is always at heart a problem in which something accumulates. We
were able to interpret the distance and speed problem that way, and were
were able to present the gas consumption problem that way. In each case
the analysis resulted in a limit of Riemman sums, in which each term was
the product of the rate of accumulation and a time interval. So far we have
not discussed the problem of the area between the input axis and the graph
of a function that way. We want to show next that an area problem can also
be regarded as an accumulation problem in the same way; and conversely,
that any accumulation problem can be thought of as a problem of finding
the area between the input axis and the graph of the function that gives the
rate of accumulation.
It helps if we think of the function f as being positive for the moment. To
b
examine the process of accumulation implied in an area calculation a f (x) dx
we should identify an intermediate stage in the process that gives rise to
the total area. We do this by considering the area from the left end a of
the interval to an intermediate point u, which will be considered variable.
Suppose the area under the graph and above the interval [a, u] is thought of
as a total, called G(u) that accumulates as we let u increase gradually from
a to b. That is,
u
G(u) = f (x) dx,
a
where a ≤ u ≤ b. If this is the total that accumulates, what represents the
rate of accumulation? In other words, what is G′ (u)?
110 WEEK 5
Example 39. From first principles determine what G′ (u) is.
5.1 & 5.2 WHAT IS AN INTEGRAL? 111
In effect we have proved one of the most important theorems in the theory
of integration. A more rigorous calculus course would pay closer attention
to subtleties we have glossed over in our discussion, but our presentation
contains the central idea in the proof.
Theorem W5.2 Fundamental Theorem of Calculus, Part1 (see text
pg. 381 [396])
u
If f (x) is continuous on [ a, b ], then G(u) = a f (x) dx
is continuous as well as differentiable, and G′ (u) = f (u).
Another way of saying that f is the derivative of G is that G is an anti-
derivative of f . This theorem confirms what we saw in several examples: f
is the rate at which G accumulates. In the case of the distance problem and
the problem of gas consumption, the integrand was the rate of accumulation
already by virtue of the physics of the situation. In the case of an area
problem we just proved it assuming f is a positive function. The proof
works equally well if the function is not positive everywhere.
However, the relationship between the integral and area is a bit more com-
plicated when the function f is negative at some places on the interval [a, b].
The Riemann sum for a partition a = x0 < x1 < x2 < · · · < xn−1 < xn = b
has the form n
f (x∗ ) ∆xi .
i
i=1
Here is a picture representing this sum:
112 WEEK 5
f (x∗ ) ∆xi > 0
i
x
a b
f (x∗ ) ∆xi < 0
i
Each term in the Riemann sum represents the area of a corresponding rect-
angle as f (x∗ ) ∆xi . That makes sense, except that it means that the integral
i
does not exactly calculate the area between the graph and the x-axis, for
whenever the graph dips below the x-axis, the corresponding terms in the
Riemann sum are negative. That is, areas of little rectangles get subtracted.
b
The conclusion is that the integral f (x) dx is equal to A − B + C in the
a
example in the diagram.
A C
x
a b
B
Integral = A − B + C
When a function dips below the axis, the integral and the area between
the graph an the axis are no longer the same thing.
5.1 & 5.2 WHAT IS AN INTEGRAL? 113
There is a general principle that when you do an integral to calculate a total
quantity that accumulates at a varying rate, then the integral (between x = a
and x = b say) can be thought of as the area under the graph (or better:
between the horizontal axis and the graph) of the function expressing the
rate of accumulation. This principle is repeated more explicitly in the "Net
Change Theorem" discussed on page 131.
You may have already encountered some quantities in high school physics
that were represented as areas under graphs, without knowing exactly why
these quantities could be thought of as areas under graphs and thus, as
integrals. Such examples may include:
• displacement as the area under a velocity-time graph
Here displacement, s = v(t) dt, where velocity v(t) represents the
rate at which displacement is accumulated, i.e. v(t) = s′ (t).
• (change in) velocity as the area under an acceleration-time graph
Here velocity, v = a(t) dt, where acceleration a(t) represents the rate
at which velocity is accumulated, i.e. a(t) = v ′ (t).
• work as the area under a power-time graph
Here work, W = P (t) dt, where power P (t) represents the rate at
which work is accumulated, i.e. P (t) = W ′ (t).
In each of these cases, you were likely asked to compute a certain quantity
(such as displacement or work) by finding the area under a simple curve.
Justification for doing so, beyond the fact that the units worked out, was
likely limited or missing. Only with an understanding of accumulation (and
thus, integration) does it make sense to connect these quantities with areas.
In the next few weeks, we will see further examples where basic physical defi-
nitions, such as displacement = velocity × time, or work = force × distance,
applicable to simple situations of constant velocity or force, when general-
ized to situations of varying velocity or force, take the form of integrals. You
should see those discussions as explanations of these statements that connect
displacement, velocity, and work to areas under graphs.
114 WEEK 5
Notes
5.1 & 5.2 WHAT IS AN INTEGRAL? 115
116 WEEK 5
Week 6
Study Sections: 5.2-5.4 in the textbook
• The Second Part of the Fundamental Theorem of Calculus
• Integral Calculations using Antiderivatives
• Properties of Integrals
• The Indefinite Integral
117
118 WEEK 6
5.3 The Fund. Theorem of Calculus, Part 2
When we are looking for the value of an integral, calculating Riemann sums
is a lot of work; and when you have calculated one, it is not necessarily
obvious how much your answer differs from the true value (that is, from the
integral).
Fortunately, the Fundamental Theorem of Calculus provides a very simple
and clever way to do many integral calculations directly. The Second part
of the Fundamental Theorem of Calculus (see the next page) shows how it
does that.
Consider the example of the area under the graph of the function f (x) = x2
between x = 0 and x = 2. We calculated a Riemann sum for that when
we did Example 38. The Fundamental Theorem of Calculus, Part 1, tells us
that if we let u
G(u) = x2 dx ,
0
then
G′ (u) = u2 ;
That is, G(u) is an anti-derivative of u2 .
Example 40. Does this information help us get a formula for G(u)?
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 119
Once we have a formula for G, we can calculate the integral immediately, for
2
G(2) = x2 dx .
0
Example 41. Calculate G(2)
Theorem W6.1 Fundamental Theorem of Calculus, Part 2 (see
text pg. 384 [398])
If f is continuous and F is an anti-derivative of f , then
b
f (x) dx = F (b) − F (a).
a
The idea is that if somehow we can come up with the formula F for a function
whose derivative is the integrand f , then to get the integral of f over the
interval [a, b], all we have to do is plug in and subtract F (b) − F (a). This
idea is important enough that we should do the proof.
120 WEEK 6
The proof hinges on the fact that if you have a formula for one anti-derivative
F (x) of a function f (x) then you can immediately write down what the other
anti-derivatives look like, for they will all have the form F (x) + C where C
is an unspecified constant.
Example 42. Prove that if F (x) and G(x) both have the same derivative f ,
then they differ by a constant.
Example 43. Prove the Fundamental Theorem of Calculus, Part 2
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 121
Using the Fundamental Theorem of Calculus, Part 2
To use Part 2 of the Fundamental Theorem of Calculus, we need to know
formulas for the anti-derivatives of functions. We already know quite a few.
Example 44. Complete the following table of basic anti-derivatives by asking
yourself the question, "f (x) is the derivative of what function, F (x)?".
Function f (x) Anti-derivative F (x)
xn (n = 1)
1
x
ex
cos x
sin x
sec2 x
1
√
1 − x2
1
1 + x2
122 WEEK 6
Example 45. Find the derivative of ln(|x|) by considering x > 0 and x < 0
separately.
x3 − 3x + 5
Example 46. Find the most general anti-derivative of f (x) = .
x
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 123
Concept Question 15. Suppose we want to calculate the area under one
section of the graph of sin x, the part from 0 to π.
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3
Then we should calculate
A. cos(π)
B. cos(0) − cos(π)
C. cos(π) − cos(0)
D. sin(π) − sin(0)
124 WEEK 6
4 √
Example 47. Calculate x dx.
0
Notation
The expression F (b) −F (a) comes up so often that there is a special notation
for it. It is written as b
F (x) or [F (x)]b
a
a
Applying this notation to the previous example gives
4
4√
2 3/2 2 2 16
x dx = x = (4)3/2 − (0)3/2 = .
0 3 3 3 3
0
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 125
π/3
Example 48. Calculate 3 sec2 θ dθ.
π/4
Maple does integrals very easily. You can refer to the Maple notes for Week
6 to see some of the ways it can do integration. For this integral, all you
have to do is to enter
> int(3*(sec(x))∧2, x=Pi/4..Pi/3);
Note that Pi has to be capitalized!
126 WEEK 6
5.2 Properties of Integrals
Before we go on to refine our skill at calculating integrals, we should first
reflect on some basic properties of integrals that derive from their origins as
limits of more and more accurate Riemann sums.
b a
1. If a > b then f (x) dx = − f (x) dx (by convention).
a b
b
2. If a = b then f (x) dx = 0.
a
b
3. c dx = c (b − a).
a
b b b
4. f (x) ± g(x) dx = f (x) dx ± g(x) dx.
a a a
b b
5. c f (x) dx = c f (x) dx.
a a
b c b
6. f (x) dx = f (x) dx + f (x) dx.
a a c
Properties 3-6 need proofs; they depend on the analogous properties for
Riemann Sums. See pages 373-375 [387-390] in the textbook for careful
proofs of all of these properties, as well as for properties 7-10 appearing
later.
5.2 PROPERTIES OF INTEGRALS 127
Example 49. Prove Property 4.
128 WEEK 6
2 4
Example 50. Given that f (x) dx = 3 , f (x) dx = 2 and
0 1
4 2
f (x) dx = 4, what is the value of f (x) dx ?
0 1
5.2 PROPERTIES OF INTEGRALS 129
Other Properties of Integrals
b
7. If f (x) ≥ 0 for a ≤ x ≤ b then f (x) dx ≥ 0.
a
b b
8. If f (x) ≥ g(x) for a ≤ x ≤ b then f (x) dx ≥ g(x) dx.
a a
b
9. If m ≤ f (x) ≤ M for a ≤ b then m (b − a) ≤ f (x) dx ≤ M (b − a).
a
b b
10. f (x) dx ≤ |f (x)| dx.
a a
The following diagrams illustrate the ideas of Properties 8 and 9:
f
g
a b
M
f
m
a b
In the second diagram, it can be observed that the area under f (x) in [ a, b ]
is greater than the area under m in [ a, b ] but less than the area under M.
130 WEEK 6
Example 51. Let f (x) be as shown
A C E
a b
B D
Sketch |f (x)| and so verify Property 10.
5.4 THE INDEFINITE INTEGRAL 131
5.4 The Indefinite Integral
The second part of the Fundamental Theorem of Calculus,
b
f (x) dx = F (b) − F (a),
a
reduces the problem of doing an integral to that of finding the anti-derivative
F . For that reason the anti-derivative is often called the indefinite integral
and written as
f (x) dx = F (x) .
The actual integral is often referred to as the definite integral to distinguish
it from the indefinite integral. Notice that these terms are confusing. The
definite integral is a number (the outcome of an integration) while
the indefinite integral is a function (the antiderivative needed to
calculate a definite integral). Note also that f (x) = F ′ (x), so f gives
the rate of change of F . Notice that this observation was made much earlier,
when we started our discussion of integration. For we saw then that when
an integral is associated with a process of accumulation then the rate of
accumulation is always precisely the integrand. We can express it this way:
b
F ′ (x) dx = F (b) − F (a).
a
"The integral of a rate of change is the total change". The textbook calls
this the Net Change Theorem. (See page 394 [408].)
132 WEEK 6
u
Concept Question 16. If G(u) = F ′ (x) dx, then
a
A. F is the derivative of G
B. G is the derivative of F
C. F and G are equal
D. F and G differ by a constant
Example 52. Suppose water is flowing into a tank at a rate given by r(t) =
200 − 4t L/min. How much water is added to the tank during the first 10
minutes of filling?
5.4 THE INDEFINITE INTEGRAL 133
√ 3
Example 53. Find t− √
t
dt.
Note: When you are asked to find an indefinite integral, as in the previous
question, it is important to add the "+C" to indicate that there is more than
one anti-derivative, and that all of them differ from each other by a constant.
Finding antiderivatives is surprisingly difficult. You would think that if we
know how to find derivatives then we should know how to find antiderivatives.
In fact, however, the latter is much more difficult, as illustrated next.
1
Example 54. Calculate dx
x3 +1
134 WEEK 6
Guess and Check
Next week we will study the "Substitution Rule", or "Method of Substitu-
tion", the first of a list of techniques for finding anti-derivatives. To prepare
us for the Method of Substitution, let me explain an informal method I call
"guess and check".
Example 55. Find cos(5x) dx.
5.4 THE INDEFINITE INTEGRAL 135
Example 56. Find cos(x2 ) dx.
What if we were given the problem x cos(x2 ) dx?
136 WEEK 6
So why does "guess and check" work in one case but not the other ?
In the case of cos(5x) dx, we were successful essentially because we were
able to rewrite it as
1
5 5 cos (5x) dx.
derivative of "outer function"
"inner function" 5x
In other words, we were able to interpret the integrand as the result of a
chain-rule differentiation:
1 du
5 dx
cos ( u ) dx.
5 5x
The same does not work for
cos(x2 ) dx.
You cannot turn this integral into
2x cos( x2 ) dx.
du u
dx
5.4 THE INDEFINITE INTEGRAL 137
Concept Question 17. Which of the following anti-differentiations can be
done by the guess-and-check method, do you think?
3 2
1. x2 ex dx 2. x2 ex dx
2 2
3. xex dx 4. ex dx
5. x2 ex dx
A. 1
B. 5
C. 2, 3, and 4
D. 1 and 3
E. 4
138 WEEK 6
Example 57. Calculate cos(x) esin(x) dx.
Next week we will turn these ideas into a formal procedure called
"Substitution".
5.4 THE INDEFINITE INTEGRAL 139
Notes
140 WEEK 6
Week 7
Study Sections: 5.5, and 6.1 in the textbook
• The Method of Substitution
• Areas
141
142 WEEK 7
5.5 Substitution Rule
We want to formalize the "guess and check" method we introduced at the
end of last week. That will give us the Method of Substitution. Consider
the previous example:
esin(x) cos(x) dx.
Here is the procedure:
1. Let u = sin x. (a guess)
2. Differentiate u: du = cos x dx.
3. Rewrite the integral entirely in terms of u and du:
eu du.
If this step cannot be carried out, it means that you should go
back and try a different choice of u!
4. Solve this anti-derivative:
eu du = eu + C.
Again, if this step cannot be carried out, it means that you
should go back and try a different choice of u!
5. Rewrite u in terms of x:
eu = esin x .
Therefore
esin(x) cos(x) = esin x + C .
5.5 SUBSTITUTION RULE 143
3x
Example 58. Calculate dx .
1 + x2
144 WEEK 7
5
Example 59. Calculate dx.
1 + 9x2
The method of substitution is really all about the chain rule. We will learn
many methods of integration, but the method of substitution is the first
one you should think of first when you encounter an integral question. You
should always determine what the "inner function" is and then try to see if
the derivative of this "inner function" is in the integrand. If it is, chances are
the method of substitution will easily solve the problem. Once you do many
examples using this method, you will become so familiar with the outlined
procedure that you will not need to follow it step by step. It will become so
automatic that your answers will be one-liners.
5.5 SUBSTITUTION RULE 145
Concept Question 18. What substitution(s) could you choose for the in-
definite integral
x cos(1 + x2 ) dx ?
(1.) u = cos(1 + x2 ) (2.) u = x2 (3.) u = sin(1 + x2 )
(4.) u = 1 + x2 (5.) u = x
A. Option 1
B. Options 1 and 3
C. Options 2 and 3
D. Options 2, 3, and 4
E. All options
146 WEEK 7
4 1 1
Example 60. Calculate 1 x2
1+ x
dx.
5.5 SUBSTITUTION RULE 147
Example 61. Calculate tan x dx.
148 WEEK 7
π
Example 62. Find 0
4
tan x dx.
Special procedure for using substitution on a definite integral:
u
In the last example, we evaluated F (cos x) at x = π and x = 0 and sub-
4
tracted. But when x = π then u = cos( π ) = √2 and when x = 0 then
4 4
1
u = cos(0) = 1. These are the u-values that correspond to x = 0 and
1
x = π . So the solution can also be expressed as F √2 − F (1), instead of
4
F cos π − F (cos(0) . In other words
4
x= π u= √1
4 2 1
tan x dx = − du,
x=0 u=1 u
√
= − ln |1/ 2| + ln |1|,
√
= − ln(1/ 2).
That is, when we use the method of substitution on a definite integral, we
can omit Step 5 provided we change the limits of integration when we change
variables.
5.5 SUBSTITUTION RULE 149
π 2 /4
√
cos( t)
Example 63. √ dt
π 2 /9 t
150 WEEK 7
6.1 Areas between Curves
We are now ready to use integration in some simple applications.
Example 64. Find the area between y = x and y = x2 .
6.1 AREAS BETWEEN CURVES 151
Here we do this problem a second time, using a more intuitive and informal
method called Short-cut Riemann sums:
152 WEEK 7
Example 65. Find the area between x = 1 − y 2 and x = y 2 − 1.
6.1 AREAS BETWEEN CURVES 153
Example 66. Find the area between y = x3 − 2x and y = −x2 .
154 WEEK 7
6.1 AREAS BETWEEN CURVES 155
Concept Question 19. Suppose we want to find the area between the curves
x = y − y 3 and x = y 2 − y. Using the diagram shown below, determine which
of the integrals or integral combinations will give the correct answer.
1.5
1 (0,1)
y
0.5
x
0 1 2 3 4 5 6
0
-0.5
-1
-1.5
(6,-2)
-2
1
A. ((y 2 − y) − (y − y 3)) dy
−2
1
B. ((y − y 3 ) − (y 2 − y)) dy
−2
0 1
C. (y 2 − y) dy + (y − y 3) dy
−2 0
0 1
D. ((y − y 3 ) − (y 2 − y)) dy + ((y 2 − y) − (y − y 3)) dy
−2 0
0 1
E. ((y 2 − y) − (y − y 3)) dy + ((y − y 3) − (y 2 − y)) dy
−2 0
156 WEEK 7
6.1 AREAS BETWEEN CURVES 157
Notes
158 WEEK 7
Week 8
Study Sections: 6.2, 6.3 and 6.4 in the textbook
• Volumes by slicing
• Volumes by Cylindrical Shells
• Work
159
160 WEEK 8
6.2 Volumes
In the integration problems considered in this section the accumulated total
is a volume. In each of the examples, this total is obtained by regarding
the solid figure as a stack of infinitely thin slices. For this reason it can be
referred to as the "method of slices". If the slices have a simple shape (say
if they are circular so that they can be thought of as infinitely thin cylinders)
so that we can obtain a simple formula for the volume of a typical slice, then
we can "add" (that is, integrate) the volumes of these slices to get a total
volume for the figure.
Example 67. Find the volume of the solid obtained by rotating the triangle
bounded by x = 0, y = 0 and x + y = 1 about the x-axis.
6.2 VOLUMES 161
162 WEEK 8
Example 68. Find the volume of the solid obtained by rotating the region
bounded by x = 0 and x = y − y 2 about the y-axis.
6.2 VOLUMES 163
Concept Question 20. Suppose a vase is such that when you fill it with
water up to depth h (measured in cm) the surface area of the water in the
vase is A(h) square centimeters. Then if you fill the vase up to 30 cm, the
integral that is equal to the volume of water in the vase is
30
A. hA(h) dh
0
30
B. 2πA(h) dh
0
30
C. A′ (h) dh
0
30
D. A(h) dh
0
164 WEEK 8
Example 69. Find the volume of the solid obtained by rotating the region
bounded by y = x3 and x = y 2 about the x-axis.
6.2 VOLUMES 165
Example 70. Find the volume of a ball of radius r.
166 WEEK 8
6.3 VOLUMES BY CYLINDRICAL SHELLS 167
6.3 Volumes by Cylindrical Shells
In this kind of application of integration, the accumulated total is a volume
consisting of infinitely many infinitely thin concentric cylindrical shells. The
goal is to find an expression for the volume of a typical shell, and then to
add (that is, integrate) these volumes to get the total. (Note that Example
69 done using washers can also be done using cylindrical shells.)
Example 71. Find the volume of the solid produced by rotating the region
bounded by y = x and y = 4x(1 − x) about the y-axis.
168 WEEK 8
6.3 VOLUMES BY CYLINDRICAL SHELLS 169
Concept Question 21. Suppose we have a function f (x) that is positive
on the interval [1, 2] and suppose that D is the region between the graph of
f , the lines x = 1, x = 2, and the x-axis.
D
1 2
If we rotate the region D about the y-axis to form a solid S, which of the
following integrals represents the volume of S?
2 2
A. f (x) dx B. 2πf (x) dx
1 1
2 2
C. 2πxf (x) dx D. πx2 f (x) dx
1 1
1
E. πx2 f (x) dx
0
170 WEEK 8
Example 72. Find the volume generated by rotating about the line x = −1
the region that lies in the first quadrant and is bounded by y = x2 , y = 4
and x = 0 .
6.3 VOLUMES BY CYLINDRICAL SHELLS 171
172 WEEK 8
Concept Question 22. We learned earlier, on page 165 that the volume of
4
a solid ball of radius r is π r 3 . Suppose we have a formula for the area of
3
the surface of a sphere of radius r. Imagine it is A(r) square units. Then
we can think of a ball of radius R as an accumulation of concentric spherical
shells each of thickness dr, starting at the center of the ball and going out
towards its outer shell. Which of the following are then true?
r R
4 4
1. π r3 = A(u) du 2. π R3 = A(r) dr
3 0 3 0
r
d 4 d 4
3. A(r) = π r 3 dr 4. A(r) = π R3
dr 0 3 dr 3
d 4
5. A(r) = π r3
dr 3
A. All of these
B. None of them
C. 2
D. 1, 2, and 5
E. 5
6.4 WORK 173
6.4 Work
The basic formula for work is the product of force times distance. If force
is measured in Newtons and distance in meters, the answer is in Joules. In
the case of a problem that requires an integral for its solution, this means
that we have to find a way to divide the process into infinitely small steps so
that on each step the amount of work may be computed as a simple product
of force and distance.
Example 73. When a particle is x meters from the origin, a force measuring
cos πx N acts on it. How much work is done by moving the particle from
3
x = 1 to x = 2 ?
174 WEEK 8
Example 74. Hooke's Law tells us that when a spring is extended, the force
(measured in Newtons) with which the spring pulls back is equal to product
of the distance by which the spring has been extended x (beyond its relaxed
length) and a constant k characterizing the stiffness of the spring, where k is
called the "spring constant". Calculate the work done in extending a spring
from x = 1 cm to x = 3 cm. We will assume that the units of k are N/cm so
that the final answer has units of N cm. What is the general formula for the
work done in extending a spring x units from its relaxed position?
6.4 WORK 175
Example 75. An aquarium 2 m long, 1 m wide and 1 m deep is full of water.
Find the minimum amount of work needed to pump half of the water out of
the aquarium.
176 WEEK 8
6.4 WORK 177
Comments on the aquarium problem:
1. Strictly speaking, the assumptions behind the problem are highly ide-
alized. In any real situation the water would come out of the hose with
some amount of kinetic energy, and this extra energy adds to the work
done. In addition to that a certain amount of work is needed to over-
come internal friction in the water, friction in the pump, and so on.
To calculate the minimum amount of work required is to ignore these
effects. Even if in real life the amount of work required is always some-
what more than what this calculation tells, it is nevertheless helpful to
know that it gives the absolute minimum that could be reached.
2. The method used really hinges on the conservation of energy: energy
gained = work done. We calculated this work by calculating the in-
crease in (potential) energy in the horizontal slabs of water.
3. To minimize the work needed, we imagine the pumping done "slowly"
so no kinetic energy is created.
4. In principle, if we knew what happened to each particle of water, we
could do a more detailed and "realistic" analysis. It would require
knowing where the hose is placed (on the bottom of the aquarium or
higher) and a calculation of the work done on or by each individual
water particle as it is pushed down the tube and then up again, or
(in other cases) as it sinks closer to the bottom of the aquarium. In
practice this picture becomes far too complicated to use. The power
of the principle of energy conservation is in its ability to simplify the
problem.
But then we would have to take into account the downward pressure
of the water, which "helps" the pump. The final answer would be the
same.
178 WEEK 8
Concept Question 23. A large cylindrical tank is filled with water. There
is a drain in the center of the bottom of the tank, two meters above the surface
of a lake. A hose is attached to the drain, and the tank is allowed to empty
through the hose onto the surface of the lake. We want to calculate the loss
of potential energy of the water as it runs from the tank to the surface of the
lake. We realize that the problem requires an integral - we have to think of
the water as a parametrized family of "pieces" so that for each such piece the
energy loss can be calculated. How should we choose those "pieces" of water
and why?
A. Horizontal slabs because that's what we did in the preceding question
B. Horizontal slabs because all the points in a horizontal slab are the same
distance above the surface of the lake
C. Horizontal slabs because when it is at rest water surface is always hor-
izontal
D. Cylindrical shells because the tank is cylindrical
E. Cylindrical shells because each such shell is at a constant radius from
the center, where the drain is located
6.4 WORK 179
Example 76. The parabola y = x2 is rotated about the y-axis, and filled
with water to the level y = 3. How much work is required to pump the water
out through a hole located at y = 4? (Assume all scales are in meters.)
180 WEEK 8
Concept Question 24. The angular momentum of a point mass rotating
about an axis at ω radians per second is defined as the product of the mass,
the square of the distance from the centre, and the angular velocity ω (that
is, mr 2 ω). Suppose we have a large metal cylinder of uniform density rotat-
ing about its axis. If we want to use an integral to calculate the total angular
momentum of the cylinder we have to think of the cylinder in terms of a para-
metric family of "pieces" chosen in such a way that the angular momentum
of each piece is easily calculated. How should we do that, and why?
A. Horizontal slices of vertical thickness dy, because the volumes of these
slices are given by π r 2 dy
B. Horizontal slices of vertical thickness dy, because the cylinder is rota-
tionally symmetric
C. Cylindrical shells because the object is a cylinder
D. Cylindrical shells because on each such shell the radius to the axis is the
same at every point, and thus the angular momentum of such a shell is
easily calculated.
6.4 WORK 181
182 WEEK 8
Notes
6.4 WORK 183
184 WEEK 8
Week 9
Study Sections: 7.1 and 7.4 in the textbook
• Integration by Parts
• Partial Fractions
185
186 WEEK 9
7.1 Integration by Parts
In Week 6 we learned that the key to integration is the ability to find
anti-derivatives. In Week 7 we learned the first technique for finding anti-
derivatives: The Method of Substitution. To allow us to apply integration
to a variety of problems, we need additional techniques for finding anti-
derivatives. Integration by Parts is the first of these.
The main idea of Integration by Parts is to use the Product Rule in reverse.
d
f (x) g(x) = f ′ (x) g(x) + f (x) g ′(x).
dx
If we integrate the expression on the right:
f ′ (x) g(x) dx + f (x) g ′(x) dx = f (x) g(x),
which is what we started with. If we let u = f (x) and v = g(x) then
du = f ′ (x) dx,
dv = g ′ (x) dx.
Therefore the equation can be written in the form
v du + u dv = uv,
u dv = uv − v du.
given simpler
The main goal of Integration by Parts is to end up with a simpler integral,
v du, than the given integral, u dv.
7.1 INTEGRATION BY PARTS 187
For example, consider the integral xex dx. Let
u=x dv = ex dx (this is our guess)
∴ du = dx ∴ v = ex
dif f erentiated integrated
Thus
xex dx = xex − ex dx = xex − ex .
original simpler
After the first step, if the new integral is much more complicated than the
original one, go back and set u and dv equal to something else so that you end
up with an easier integral to solve. You can avoid this by mentally checking
what v du will be before actually choosing what u and dv are.
Example 77. Calculate x sin(4x) dx.
188 WEEK 9
Analysis of our solution to Example 77:
Could we have foreseen that our choice of u and dv would work? Let's review
what we did:
x sin(4x) dx
differentiate integrate
1 cos(4x)
leaving out sign
and constants
So the new integral would be something like
cos(4x) dx.
What would of happened if we would have chosen u and dv to equal something
else?
dv
x sin(4x) dx
u
integrate differentiate
x2 cos(4x)
ignore constants for
this "rough work"
The new integral would be something like
x2 cos(4x) dx.
It is worse! This is obviously a lot more complicated to solve. Integration
by Parts is very easy once you learn to make smart choices for u and dv.
Practice a lot.
7.1 INTEGRATION BY PARTS 189
2
Example 78. Calculate x3 ex dx.
190 WEEK 9
Example 79. Calculate ln x dx. Use your answer to find the area under
the graph of ln x between x = 1 and x = 2.
7.1 INTEGRATION BY PARTS 191
1
Example 80. Calculate 2
0
sin−1 (x) dx.
192 WEEK 9
7
x5
Concept Question 25. If you want to do the integral √ dx
0 1 + x3
using integration by parts then you should choose
1
A. u = x5 and dv = √ dx
1 + x3
x
B. u = x4 and dv = √ dx
1 + x3
x2
C. u = x3 and dv = √ dx
1 + x3
x3
D. u = x2 and dv = √ dx
1 + x3
x2
E. u = √ and dv = x3 dx
1+x 3
7.4 PARTIAL FRACTIONS 193
7.4 Partial Fractions
This is the last of the techniques of integration (that is, techniques for anti-
differentiation) covered in this course. If you look in the textbook you will see
that there are many other methods we do not include. With the availability
of computer packages such as Maple, much of the hard work in finding anti-
derivatives has been automated. The techniques we have covered are included
because a certain amount of facility with a few basic methods is needed to
know how to get problems to the point where computer packages can take
over.
The method of Partial Fractions is purely algebraic. It consists of a series
of algorithmic steps that simplify the integrand so that an anti-derivative can
be easily found. For example, consider the integral
x+5
dx.
x2 + x − 2
At first glance this integral seems difficult to solve, but by using the method
of Partial Fractions, we can break the integrand into
2 1 x+5
− = 2 .
x−1 x+2 x +x−2
Check for yourself that this equation is true. We can then write the integral
as
2 1
dx − dx.
x−1 x+2
This integral is now very easy to solve.
Our goal then is to find this kind of Partial Fraction Decomposition.
This method is used only for expressions of the form
P (x)
(rational function),
Q(x)
where P and Q are polynomials.
A proper rational function is one for which the degree of P is strictly less
than the degree of Q. To be able to use the method of Partial Fractions, we
194 WEEK 9
must first make sure that the integrand is a proper rational function; this
is our first step:
Step 1.
Turn P (x) into an expression involving a proper rational function and a
Q(x)
polynomial:
P (x) R(x)
= S(x) + .
Q(x) Q(x)
The method we use to turn P (x) into a proper rational function is Long
Q(x)
Division. For example, to solve
x2 − 4x + 3
dx,
x−5
we do the following:
x+1 S(x)
x−5 x2 − 4x + 3 P (x)
Q(x) x2 − 5x
x+3
x−5
8 R(x)
so,
P (x) = S(x) · Q(x) + R(x)
P (x) R(x)
∴ = S(x) + .
Q(x) Q(x)
Thus with our example we get
x2 − 4x + 3 8
dx = (x + 1) + dx
x−5 x−5
x2
= + x + 8 ln |x − 5| + C.
2
7.4 PARTIAL FRACTIONS 195
x2 + 1
Example 81. Find dx.
x2 − 1
196 WEEK 9
The next next step (after we have turned the integral into one involving a
proper rational) consists of four cases, distinguished by what happens when
you factor the denominator.
Step 2. CASE I:
The denominator is the product of distinct linear factors. Say
Q(x) = (a1 x + b1 )(a2 x + b2 ) . . . (ak x + bk ).
The goal is to look for numbers A1 , . . . , Ak so that
R(x) A1 A2 Ak
= + + ...+ .
Q(x) a1 x + b1 a2 x + b2 ak x + bk
"Partial Fractions"
x
Example 82. Find dx.
x2 + 3x + 2
7.4 PARTIAL FRACTIONS 197
198 WEEK 9
3x2 − 2
Example 83. Find dx.
(x − 1)(x − 2)(x + 1)
7.4 PARTIAL FRACTIONS 199
We are now at the point where we can complete the problem we presented
on page 96 to introduce integration. Before going on to the second case of
the method of partial fractions, we will solve that problem.
Example 84. That is, find the volume of the "hat-shaped" solid you get
when you take the region between the graph of the function 1/(x2 + 3x + 2)
and the x-axis, bounded by the y-axis and the line x = 1, and rotate it around
the vertical axis200 WEEK 9
7.4 PARTIAL FRACTIONS 201
Step 2. CASE II:
The polynomial Q(x) is a product of linear factors, some of which are re-
peated. Rule: If (ax + b) occurs to the power r, then instead of one term,
put down the following r terms:
A1 A2 Ar
+ 2
+ ...+ .
ax + b (ax + b) (ax + b)r
dx
Example 85. Find .
x2 (x− 1)2
202 WEEK 9
7.4 PARTIAL FRACTIONS 203
Why do we have to learn these rules for setting these problems up?
It is natural at this point to wonder why we have to set up the question
precisely as we did. Why could we not reduce the number of variables by
letting
1 B C D
= 2+ + ?
x2 (x − 1)2 x x − 1 (x − 1)2
If we did this we would get the equations
C =0
B−C +D =0
2B =0
B =1
You can see immediately that this system of equations does not have a so-
lution. That is, there do not exist numbers B, C, and D that satisfy all four
equations at once.
1
Concept Question 26. Suppose we want to integrate dx.
(x − 1)(x + 2)2
How should we set up the partial fractions?
A B A B C
(A.) + (B.) + +
x−1 x+2 x−1 x+2 x+2
A B A B C
(C.) + (D.) + +
x − 1 (x + 2)2 x − 1 x + 2 (x + 2)2
A B C D
(E.) + 2
+ +
x − 1 (x − 1) x + 2 (x + 2)2
204 WEEK 9
1
Example 86. Find dx.
(x − 1)(x + 2)2
7.4 PARTIAL FRACTIONS 205
Step 2. CASES III AND IV:
The polynomial Q(x) contains quadratic factors. We will omit these
cases.
206 WEEK 9
Notes
7.4 PARTIAL FRACTIONS 207
208 WEEK 9
Week 10
Study Sections: 7.7 and 7.8 in the textbook
• Approximate Integration
• Improper Integrals
209
210 WEEK 10
7.7 Approximate Integration
Approximate Integration is very important when we cannot find a formula
for the anti-derivative of a given function. For example, the functions in the
integrals
2 √
e−x dx and 1 + x3 dx
do not have a formula for their anti-derivatives.
We have already learned and used a method for calculating an approximate
value for an integral: Riemann Sum.
1 x
Suppose we wanted to know how close we could get to calculating 0 x2 +1
dx
by using a simple Riemann Sum. By integrating, we know that
1
x 1
dx = ln 2 = 0.34657 . . . .
0 x2 +1 2
x
Here are some values for f (x) = between 0 and 1:
x2 +1
x 0 0.25 0.5 0.75 1
f (x) 0 0.23529 . . . 0.4 0.48 0.5
In the next few examples we will see that, even using the same set of numbers,
some ways of combining them gives more accurate results than others.
7.7 APPROXIMATE INTEGRATION 211
Example 87. Calculate the R.S. using two intervals and their midpoints.
(This is the "Midpoint Rule".)
Example 88. Calculate the R.S. using four intervals and their left-end
points. (This is the "left end-point approximation".)
212 WEEK 10
Example 89. Calculate the R.S. using four intervals and their right-end
points. (This is the "right end-point approximation".)
Example 90. Calculate the average of the left- and right-end point approx-
imations.
7.7 APPROXIMATE INTEGRATION 213
The general formula for the left end-point approximation is
∆x(f (x0 ) + f (x1 ) + . . . + f (xn−1 )),
and the general formula for the right end-point approximation is
∆x(f (x1 ) + f (x2 ) + . . . + f (xn )).
This average of the left end-point and right end-point approximations is
known as the "Trapezoidal Rule". The Trapezoidal Rule is best under-
stood with a picture:
f (x)
xi−1 xi
From the picture, we can observe that the area of the trapezoid in the ith
subinterval is
f (xi−1 ) + f (xi )
Area = ( xi − xi−1 ) · .
2
This area is equal to the average of the areas of the rectangles with base
( xi − xi−1 ) and heights f (xi−1 ) and f (xi ). We can write the Trapezoidal
Rule in terms of the formula:
∆x
f (x0 ) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn−1 ) + f (xn ) .
2
You should memorize this formula.
214 WEEK 10
b
In practice, when you have to integrate a f (x) dx, you would like to know
beforehand how many intervals to divide [ a, b ] into to get the desired ac-
curacy. In other words, we would like to know the "error" made in approx-
b
imating a f (x) dx using the Trapezoidal Rule. The error is the difference
b
between the value of a f (x) dx and the approximated value obtained using
the Trapezoidal Rule. There are theorems for this, but we will not use them
in this course. The accuracy of the Trapezoidal Rule depends not only on the
number of intervals but also on the second derivative. The following theorem
gives a formula for calculating the error made when using the Trapezoidal
Rule.
Theorem W9.1 Error Bounds (see text pg. 499 [515])
If |f ′′(x)| ≤ K on [ a, b ], then the error ET made
b
in approximating a f (x) dx using the Trapezoidal
Rule with n partitions is
K(b − a)3
|ET | ≤
12n2
Why does this Error Bound theorem involve the second derivative?
You can 'sort of' see why the second derivative is important when estimating
the error. If you go back to the figure on page 213, you will see that the
error, the difference between the sum of the areas of the trapezoids and the
true area under the curve is just the sum of the areas of curved segments
such as the one above the trapezoid in the figure. If the second derivative is
zero, the graph is a straight line, and the graph will coincide with the tops
of the trapezoids, allowing no error at all. When the second derivative is
not equal to zero, it measures the amount by which the graph is curved (the
rate at which the slope changes). Therefore, if the second derivative is very
large, it is curved a lot, and each segment will have a rather large area, thus
producing a rather large error.
7.7 APPROXIMATE INTEGRATION 215
Simpson's Rule is an even better method. It is based on approximation by
pieces of parabola:
y
a b x
The formula for Simpson's Rule is: (n must be even)
∆x
f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + 2f (x4 ) + · · ·
3
· · · + 2f (xn−2 ) + 4f (xn−1 ) + f (xn ) .
You should MEMORIZE this formula.
Example 91. Apply Simpson's Rule to the previous example.
216 WEEK 10
7.8 Improper Integrals
So far in our study of integration, we have dealt with functions that were
always continuous on the interval that we were integrating on. When we
integrated f (x) over [ a, b ], the function f (x) did not have a discontinuity in
[ a, b ], and a and b were always finite. In this section we will introduce the
concepts of
1. integrating f (x) over an infinite interval, and
2. integrating f (x) over an interval [ a, b ] where the f (x) contains a dis-
continuity in [ a, b ].
Hence the term "Improper Integrals".
Improper Integrals of Type 1: Infinite Intervals
These integrals contain ∞ and/or −∞ in their limits of integration:
∞ b
f (x) dx or f (x) dx.
a −∞
For example, consider the integral
∞
1
dx.
1 x2
By calculating this integral, we will be finding the area under f (x) = x12
between x = 1 to x = ∞ and we will be able to show that its value is finite.
1
f (x) =
x2
...
1
7.8 IMPROPER INTEGRALS 217
∞
1
Example 92. Using the rule stated below, calculate dx, and so justify
1 x2
the statement made on the previous page.
∞ t
Rule : f (x) dx = lim f (x) dx .
a t→∞ a
Example 93. Is the area under the graph of f (x) = e−3x , extending from
x = 0 to infinity, finite? If so, what is its value?
218 WEEK 10
A nice way to explore improper integrals of Type I is to ask how the family
of functions 1/xp , for various positive values of p, relate to each other, and
what happens to them when we integrate them from 1 to ∞. Here are some
examples of such functions, generated by the Maple command
> plot([1/sqrt(x), xˆ(-1), xˆ(-2)], x= 0.. 6, y= 0.. 4);
4
3
y 2
1
0
0 1 2 3 4 5 6
x
Notice what goes wrong when you leave out the y= 0.. 4 part from the Maple
command. As you can see, all these functions cross at the point (1,1).
Concept Question 27. Which of the following expectations seems reason-
able to you?
∞
A. 1/xp dx is more likely to be finite when p is small than when p is
1
large, for then the anti-derivative will be smaller.
∞
B. 1/xp dx is more likely to be finite when p is small than when p is
1
large, for then the graph of 1/xp is closer to the horizontal axis.
∞
C. 1/xp dx is more likely to be finite when p is large than when p is
1
small, for then the graph of 1/xp is closer to the horizontal axis.
∞
D. 1/xp dx is more likely to be finite when p is large than when p is
1
small, for then the graph of 1/xp is farther from the vertical axis.
7.8 IMPROPER INTEGRALS 219
∞
Example 94. Calculate 1/xp dx
1
220 WEEK 10
∞
3
Example 95. Find x2 e−x dx . (# 14 in the text)
−∞
7.8 IMPROPER INTEGRALS 221
Improper Integrals of Type 2: Discontinuous Integrand
b
These are integrals, a f (x) dx, where f has a vertical asymptote somewhere
in [ a, b ]. For example, consider the integral
1
1
dx.
0 x2
1
Even though the function f (x) = x2 has a discontinuity in the specified
interval (at x = 0), we will learn techniques that will show that this integral
diverges to infinity. That is, the area under f (x) = x12 between x = 0 and
x = 1 is infinite.
.
.
.
1
f (x) =
x2
1
222 WEEK 10
Like Type 1 problems, the solution technique for Type 2 problems involves
taking a limit. In this case, since the function is undefined at the x location
of its vertical asymptote, say at x = a, we replace a with t and then take the
limit as t approaches a.
1
1
Example 96. Calculate dx and so verify that the area under the graph
0 x2
1
of 2 from x = 0 to x = 1 is infinite.
x
7.8 IMPROPER INTEGRALS 223
4
1
Example 97. Calculate dx.
0 x2 +x−6
You can easily miss that this is an improper integral!.
224 WEEK 10
7.8 IMPROPER INTEGRALS 225
Example 98. Newton's Law of Gravitation states that two bodies with masses
m1 and m2 attract each other with a force of magnitude
m1 m2
F =G
r2
where r is the distance between the two bodies and G = 6.67×10−11N ·m2 /kg2
is the gravitational constant. Assuming that the Earth's mass is 5.98 × 1024
kg and that its radius is 6.37 × 106 meters, calculate the work required to
propel a 1000-kg satellite out of the Earth's gravitational field. In problems
like this it is correct to act as if the mass of the Earth were concentrated at
its center.
226 WEEK 10
Notes
7.8 IMPROPER INTEGRALS 227
228 WEEK 10
Week 11
Study Sections: 9.1, 3.8 [9.4] in the textbook
• Course Unit: Differential Equations
• Modeling with Differential Equations
• Exponential Growth
• Harmonic Motion
229
230 WEEK 11
Course Unit: Differential Equations
As pointed out on several other occasions in the course, most laws of nature
and science, once they are translated into mathematics, take the form of a
differential equation.
A differential equation is quite unlike the other equations we have studied.
For one thing, it involves derivatives, often even second or higher derivatives.
But this is not the most important difference between a differential equation
and the kinds of equations we have been used to and continue to study
in our courses. In those more familiar problems involving equations, the
goal is to find the number x or perhaps the pair of numbers, x and y, that
constitutes the solution of the equation or set of equations. By contrast, in
a differential equation the unknown is the function f itself, and
not the variable! A related characteristic of a differential equation is that
when we write down a differential equation involving a variable x (and there
is always at least one variable), the equation is understood to hold for
all values of the variable. Thus we are looking for a particular f that
satisfies the equation for all values of x.
Concept Question 28. Which of the following functions is a solution to
the differential equation x′′ (t) = −36x(t)?
A. x(t) = −6t3
B. x(t) = cos(6t + 2)
C. x(t) = e−6t
D. x(t) = −e−6t
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 231
9.1 Modeling with Differential Equations
This week we will focus on two particular differential equations and find
their solutions. The first of these is the differential equation that gives rise
to exponential growth.
Exponential Growth (see also 3.8 [9.4])
Suppose we are in a situation where, for a certain species, the population of
size y grows in such a way that its rate of growth is always 3 times as big as
the size of the population at that moment, we would translate that as
dy
= 3y
dt
Here y is (the output of) a function of the variable t. We are looking for
a formula for y in terms of t, which satisfies the differential equation for all
values of t.
Example 99. Find a function y = f (t) that satisfies this differential equa-
tion. HINT: If you cannot think of it immediately, try to do it first for the
dy
differential equation = y.
dt
232 WEEK 11
What we have discovered in this problem is the origin of exponential growth!
When population size is given by the formula y = ket we have a population
that grows exponentially, as most of you will have learned in high school.
Exponential growth is not something that falls out of the sky as a peculiar
preference exhibited by populations! There is a clear and simple explanation
for exponential growth, and it centers on this differential equation: The rate
at which young are born into a population is in direct proportion to the size
of the population. The ratio k between the number of offspring per unit
time and the size of the population is called "fecundity", and depends on the
organism. It is proverbially higher for rabbits than for humans. In any case
when we translate "the rate at which young are born into a population is in
direct proportion to the size of the population" into mathematics, we get
dy
= ky .
dt
The fact that the solutions for this differential equation all have the form y =
Aekt is the reason why, in the presence of abundant resources, populations
grow exponentially.
Are there other functions that satisfy this differential equation?
We promised to show that functions of the form y = ekt are the only solutions
to the differential equation
dy
= ky .
dt
The next exercise will do that for us:
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 233
Example 100. Suppose y(t) is a solution of the differential equation
dy
= ky.
dt
d y(t)
(a) Evaluate .
dt ekt
(b) What does your calculation in part (a) tell you about the most general
form a solution of the differential equation can take ?
234 WEEK 11
Theorem: If a function y(t) satisfies the differential equation
dy
= ky(t)
dt
then y(t) necessarily has the form
y(t) = Aekt
for some constant A. The constant A is the value taken by y at t = 0.
Sometimes, a differential equation can be turned into the equation for expo-
nential growth by means of an appropriate substitution. Newton's Law of
Cooling is an example;
Example 101. Newton's Law of Cooling claims that if a hot cup of coffee
is left on the table, then the rate of cooling is always proportional to the
difference between the temperature of the coffee and ambient temperature in
the room. Write this down as a differential equation, assuming the room
temperature at 20◦ .
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 235
Example 102. By concentrating on the temperature difference rather than
the temperature, turn this into an exponential growth question, and then solve
this problem: If it takes 5 minutes for the temperature of the coffee to go from
100◦ to 80◦ , how hot will the coffee be after 10 minutes?
236 WEEK 11
Harmonic Motion
A very different differential equation arises when we study the motion of an
object at the end of a spring, as in the next diagram:
0 x
In this diagram, the grey object on the left is attached to the table, while
the block on the right is allowed to slide without friction. 0 indicates where
the right end of the spring is when the system is at rest. Thus x indicates
the distance by which the spring is extended. Hooke's law tells us that the
force exerted by the spring on the block has magnitude proportional to the
extension x, and is of course in the opposite direction:
F = −kx .
Here k is the "spring constant", a positive constant that describes the stiffness
of the spring. If the spring is very stiff, k is large, for then a small extension
is enough to produce a large force. Using Newton's second law, this leads to
the equation
d2 x
m 2 = −kx ,
dt
where m is the mass of the block and t is time. Notice that this, too, is a
differential equation. It involves a quantity, x, that is a function of time,
t, and it gives a relation between the function and one of its derivatives.
This differential equation is inherently more complicated than the one for
exponential growth, because it involves the second derivative of the function
x = x(t). Remember that, when we study a differential equation, the goal is
to find a formula for the unknown function. In this case we want an expression
for x in terms of t. Of course you expect k to occur in this formula as well
and, as in the case of the solution to the differential equation for exponential
growth, we should not expect the formula to be uniquely determined by the
differential equation, but to depend on one (or more) arbitrary constants.
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 237
To find a formula for x(t), we will first consider a special case of the differ-
ential equation by assuming that k = 1 and m = 1. Then we have
d2 x
= −x .
dt2
Example 103. This differential equation invites us to find a function x(t)
whose second derivative is its own negative. Try to find such a function.
d2 x
We have found a set of solutions for the differential equation = −x. Does
dt2
d2 x
this help us discover solutions for the differential equation m 2 = −kx? To
dt
help us find solutions to this, we could begin by putting the two positive
constants m and k together, and then giving the name ω to the square root
of their ratio:
d2 x k
2
= − x = −ω 2 x .
dt m
Example 104. Find a solution to this differential equation.
238 WEEK 11
Are there other solutions? You may have noticed that if A and φ are
constants, then x(t) = A cos(ωt+ φ) is a solution and yet has a different form
from the one we discovered. You can check easily that this function satisfies
the differential equation. Does this mean that this is a new type of solution?
In fact it does not, for we will now show that these two only look like different
functions, and that, using appropriate trigonometric identities, we can show
that each B cos(ωt) + C sin(ωt) can be transformed into A cos(ωt + φ) for
suitable A and φ, and conversely.
First of all, you should note that allowing A to be negative in the expression
x(t) = A cos(ωt + φ) is redundant, for if we add π to φ we change the sign of
the expression:
A cos(ωt + φ + π) = −A cos(ωt + φ)
The reason for this is that cos(x + π) = − cos(x) for any x. Thus, we may
as well assume that A is chosen to be positive.
Now recall the trigonometric identity proved in Week 2 on page 30:
cos(a + b) = cos a cos b − sin a sin b .
If we apply this with a = ωt and with b = φ, we obtain
A cos(ωt + φ) = A [cos(ωt) cos(φ) − sin(ωt) sin(φ)]
= [A cos(φ)] cos(ωt) + [−A sin(φ)] sin(ωt) .
Notice that t is the only variable in this expression; the other letters represent
constants. The question before us is: Can B and C be chosen so that this
is exactly the same as B cos(ωt) + C sin(ωt) for all t? Clearly we can, by
simply letting
B = A cos(φ) = A cos(−φ)
C = −A sin(φ) = A sin(−φ)
Conversely, we could ask whether we can always solve for A and φ in terms
of B and C. We are not interested in knowing an actual formula so much
as knowing that it is possible, so that we can say with confidence that the
two families of solutions we found are really one and the same - that for
each expression in one family there is a corresponding one (defining the same
function) in the other. To see that this can always be done, suppose a number
φ and a (positive) number A are given. The we can imagine a line segment
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 239
of length A emanating from the origin and making the angle −φ with the
horizontal axis:
−φ
A
Example 105. What are the coordinates of the endpoint of the line segment?
This shows that for any A and any φ the associated numbers B and C are just
the coordinates of the endpoint of the line segment we drew; and, conversely,
if B and C are given, then we let A be the length of the segment from the
origin to the point (B, C) and φ the angle it makes with the horizontal axis.
Thus we have shown geometrically that we can always solve for B and C if
we know A and φ, and conversely.
240 WEEK 11
The upshot of this discussion is that
The family of functions
B sin(ωt) + C cos(ωt) ,
where B and C are arbitrary constants, and the family of functions
A cos(ωt + φ) ,
where φ is an arbitrary constant and A an arbitrary positive constant,
are really the same family. Each member of this family of functions is a
solution to the differential equation
x′′ (t) = −ω 2 x(t)
arising from Hooke's Law.
By another, rather long, argument it can be shown that there are no solutions
arising from Hooke's Law, other than the ones in this family of functions.
We will simply assume this.
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 241
What do these solutions look like? We should now describe the func-
tions in this family of solution, for they tell us how the block will move under
the influence of the spring.
To do this it is easiest to work with the expression
A cos(ωt + φ) .
We begin by observing that these functions are all periodic:
Concept Question 29. x(t) = A cos(ωt + φ) is periodic with period
A. 2π
B. ω
C. φ/ω
D. 2π/ω
242 WEEK 11
We can study it in terms of transformations applied to the function cos(t).
We know what the graph of the cosine function looks like:
cos(t)
1
−2π 0 2π
As a first step in transforming this to the function A cos(ωt + φ), we now add
φ to the variable t. We learned in high school that this will have the effect
of moving the graph to the left by the amount φ:
cos(t + φ)
1
−2π −φ 0 2π
As a second step, we now multiply the variable t by the constant ω. You
will remember that the effect of this is to compress the graph towards the
vertical axis (or to expand it away from the vertical axis if ω < 1):
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 243
cos(ωt + φ)
1
−2π φ
−ω 2π
As a final step, we multiply the function by the constant A. This will have
the effect of stretching the graph vertically by that factor:
A cos(ωt + φ
A
1
−2π φ
−ω 2π
The upshot of this discussion is that A cos(ωt + φ) is a"sinusoidal" function
(that is, it is shaped like a sine or cosine function) with amplitude A.
The function is periodic with period 2π/ω, for if we increase t by that
much, the input into the cosine function is increased by exactly 2π. In
terms of the motion of the block, it means that it will vibrate back and
forth, between extreme positions −A and A, and that takes 2π/ω time units
between successive instants at position A. The motion described by the
function A cos(ωt + φ) is known as simple harmonic motion.
244 WEEK 11
Concept Question 30. Recall that in an application to spring motion ω =
k
, where k is the stiffness of the spring and m is the mass of the sliding
m
block. Suppose the motion of the block is started by releasing the block (at
zero velocity) from a non-equilibrium position. Which of the following will
change the time it takes for the block to go through one cycle of its periodic
motion?
A. The block is given a small push in the direction away from the equilib-
rium position
B. The block is released closer to the equilibrium position
C. A weight is attached to the top of the block
Example 106. Suppose that for a block attached to a spring (with spring
constant k) on a frictionless table the potential energy is 0 when the block is
in the equilibrium position, what is the potential energy when the extension
of the spring is x?
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 245
Example 107. If an object undergoes harmonic motion, so that its position
function is a solution of the differential equation
k
x′′ (t) = − x(t)
m
, prove from the differential equation that
m(x′ (t))2 + k(x(t))2 = a constant
What does this equation tell you physically?
246 WEEK 11
Notes
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 247
248 WEEK 11
Week 12
Study Sections: 9.3, 9.4 [9.5] in the textbook
• Separable Differential Equations
• The Logistic Equation
249
250 WEEK 12
9.3 Separable Differential Equations
Last week we looked at two very particular differential equations, and found
the solutions for both of them. Our methods were ad hoc: There is no
indication how the techniques we used to find these solutions might be used
to solve other differential equations. Eventually we want to learn techniques
that can be applied to a wide variety of differential equations. Such a study
of differential equations is not taken up seriously until second year. This
week we will conclude the course by demonstrating a method for finding
solutions for one relatively simple class of differential equations that includes
exponential growth and Newton's Law of Cooling as special cases.
A differential equation is said to be of first order when it involves first
derivatives only. The simplest kind of first order equation is one in which the
first derivative can be isolated on one side of the equation. This will produce
an equation of the form
dy
= f (x, y)
dx
where f (x, y) indicates an expression involving x and y but no derivatives.
This type of equation has a very nice visualization. Think of the unknown
function y as measured along the dependent axis, and x as independent
variable in the usual way. We are looking for a solution function y(x). The
differential equation tells you that when x has a given value and y has a
particular value, then the slope of the function you are trying to find is equal
to the value of f (x, y). For example, suppose the differential equation is
dy
= xy
dx
and that a solution y(x) of this equation has the value 3 when x = 2, then
the slope of the graph of y(x) at x = 2 is 6. This observation allows us to
think of finding a solution to this kind of differential equation in terms of a
picture.
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 251
Example 108. On the following grid, draw the slopes for this differential
equation at a number of points. The result is know as the slope field of the
differential equation.
4
3
2
1
0
-1
-2
-3
-4
-4 -3 -2 -1 0 1 2 3 4
Maple is particularly well suited to draw a slope field. The Maple Notes
discuss this more fully in the section for Week 11. Here are the commands
for producing a slope field for the differential equation y ′ = x2 /100:
with(DEtools):
and then
DEplot(diff(y(x),x)=x∧2/100,y, x=-10..10, y=-10..10,
arrows=LINE);
252 WEEK 12
10
y(x)
5
K
10 K 5 0 5 10
x
K 5
K
10
You can use this slope field to sketch the graphs of some solutions of the
differential equation.
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 253
A separable differential equation is a differential equation in which the
expression for the derivative can be isolated and written as a product of
two functions, one involving only the independent variable and the other
involving only the dependent variable. That is, it is a separable differential
equation if it can be written in the form
dy
= g(x)f (y) .
dx
Concept Question 31. Which of the following are separable?
du
1. dt
= u cos(t)
2. x′ (t) = kt
dy
3. 4 dx = xy
4. T ′ (t) = T (t) − 20
A. All of them
B. Number 3. only
C. Numbers 1. and 3.
254 WEEK 12
Example 109. Suppose the function f (y) has a root at y0 ; that is, suppose
f (y0) = 0. Use this information to find a solution of the differential equation
dy
= g(x)f (y) that takes the value y0 when x = 0.
dx
This example allows us to focus on the values of y for which f (y) = 0. For
such values of y it is convenient to define h(y) = 1/f (y). The differential
equation then becomes
dy g(x)
= .
dx h(y)
The method for solving separable equations hinges on our ability to separate
the variables as follows:
h(y)dy = g(x)dx .
Writing it this way, invites integrating both sides:
h(y)dy = g(x)dx .
Before we go on, we should justify this formal pushing around of symbols. Do
we have any justification for believing that doing this will lead to a solution
for the differential equation? An indefinite integral is an anti-derivative (and
always involves an arbitrary constant), so the last equation can be thought
of as follows: Suppose we have found anti-derivatives H(y) = h(y)dy and
G(x) = g(x) dx (each hiding an arbitrary constant of course). Now put
these equal to each other, H(y) = G(x), and solve this for y in terms of x.
Do we know that y = y(x) will then be a solution to the differential equation?
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 255
Example 110. Use implicit differentiation on H(y) = G(x) to show that y
is a solution to the differential equation
dy g(x)
= .
dx h(y)
Now that we have provided this justification, the method for substitution of
variables can be summarized as follows:
Separable differential equations: Separate the variables to produce
an equality of differential expressions
h(y)dy = g(x)dx ,
Find anti-derivatives of both sides (thus introducing an arbitrary con-
stant) giving
H(y) = G(x)
and solve for y in terms of x.
256 WEEK 12
Example 111. Solve the differential equation
(x2 + 1)y ′ = xy .
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 257
Example 112. In your second term physics course you will learn how to
analyze electric circuits. In particular, you will learn that the electric current
I in the circuit shown below
R
E L
switch
will satisfy the differential equation
dI
L + RI = E(t)
dt
Find an expression for the current (measured in Amperes) when the resistance
R = 12Ω (Ohms), the inductance L = 4 henries (H), and the power source E
gives a constant voltage of 60 volts (V), and the switch is turned on at time
t = 0.
258 WEEK 12
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 259
9.4 [9.5] The Logistic Differential Equation
Last week we discussed the exponential growth model, which is valid when
there is no limit on resources. Under those circumstances, a population is
likely to grow at a rate proportional to its size. The logistic differential
equation provides a more versatile model for population growth, one that
includes the effect of limited resources. To model this situation, we imagine
that the environment has a certain carrying capacity K. If the population
should ever reach the level K, it will not be able to grow any bigger. The
Logistic differential equation is simplest model that has this feature:
dP P
= kP 1− .
dt K
Notice that this differential equation resembles the differential equation for
exponential growth. The constant k can still be thought of as "fecundity",
and if we let the carrying capacity K become infinite, we get exponential
growth.
Concept Question 32. Assuming that K and k are positive constants,
when will the function P (t) decrease?
A. When P is greater than K.
B. When P is greater than K/2
C. When t is negative
D. When t < P/K
260 WEEK 12
Example 113. At what value of P does P increase most rapidly?
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 261
Example 114. (Taken from Stewart) The Pacific halibut fishery has been
modeled by the differential equation
dy y
= ky 1 −
dt K
where y(t) is the biomass (the total mass of the members of the population)
in kilograms at time t (measured in years), the carrying capacity is estimated
to be K = 8 × 107 kg, and k = 0.71 per year.
(a) If y(0) = 2 × 107 kg, find the biomass a year later.
(b) How long will it take for the biomass to reach 4 × 107 kg?
262 WEEK 12
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 263
264 WEEK 12
Notes
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 265 | 677.169 | 1 |
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Math
Calculus
Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists)
Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).
The Calculus BC AP exam is a super set of the AB exam. It covers everything in AB as well as some of the more advanced topics in integration, sequences and function approximation. This tutorial is great practice for anyone looking to test their calculus mettle! | 677.169 | 1 |
Elements of Number Theory (Dover Phoenix Editions)
Synopses & Reviews
Publisher Comments:
"A very welcome addition to books on number theory."—Bulletin, American Mathematical Society
Clear and detailed in its exposition, this text can be understood by readers with no background in advanced mathematics; only a small part requires a working knowledge of calculus. One of the most valuable characteristics of this book is its stress on learning number theory by means of demonstrations and problems. More than 200 problems and full solutions appear in the text, plus 100 numerical exercises. Some of these exercises deal with estimation of trigonometric sums and are especially valuable as introductions to more advanced studies. Translation of 1949 Russian edition.
Synopsis:
Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1949 edition.
Synopsis:Synopsis:
Related Subjects
Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1949 edition.
"Synopsis"
by Hold All,"Synopsis"
by Ingram, | 677.169 | 1 |
Elementary Numerical Analysis
9780471433378
ISBN:
0471433373
Edition: 3 Pub Date: 2003 Publisher: Wiley
Summary: Offering a clear, precise, and accessible presentation, complete with MATLAB programs, this new Third Edition of Elementary Numerical Analysis gives students the support they need to master basic numerical analysis and scientific computing. Now updated and revised, this significant revision features reorganized and rewritten content, as well as some new additional examples and problems. The text introduces core areas... of numerical analysis and scientific computing along with basic themes of numerical analysis such as the approximation of problems by simpler methods, the construction of algorithms, iteration methods, error analysis, stability, asymptotic error formulas, and the effects of machine arithmetic.
Kendall Atkinson is the author of Elementary Numerical Analysis, published 2003 under ISBN 9780471433378 and 0471433373. Six hundred sixty eight Elementary Numerical Analysis textbooks are available for sale on ValoreBooks.com, one hundred ten used from the cheapest price of $58.68, or buy new starting at $68 | 677.169 | 1 |
Eighth Grade Mathematics Curriculum
This course is aligned with the Common Core Standards for 8th grade mathematics and focuses on three critical areas.
First, the students will learn to formulate expressions and equations, show the association of data with a linear equation, and to solve linear equations. The students will become comfortable using the linear equation, y = mx + b. They will understand that m represents the rate of change of the two variables. In addition they will understand the graphs are lines through the origin. The students will become comfortable using a linear equation to describe the relationship between two values in bivariate data. They will also be able to solve problems with one linear equation and systems with two linear equations.
Second, the students will learn to understand functions and to use a function to describe quantitative relationships. They will understand that functions help to describe situations where one quantity is dependent on another. Third, the students will learn to assess two- and three-dimensional shapes using distance, angle, and similarity using ideas about distance and angles and how they behave. The students will understand the Pythagorean Theorem and be able to explain why it is true. They will learn to use the theorem to find distances between points on the coordinate plane, to find lengths, and to analyze triangles. The students will complete their study of volume by learning to solve for the volume of cones, cylinders, and spheres. | 677.169 | 1 |
Esta investigação visa analisar o modo como a resolução de tarefas de natureza exploratória e investigativa, envolvendo o uso da calculadora da gráfica, contribui para a compreensão e aprendizagem das funções quadráticas dos alunos. A metodologia insere-se no paradigma interpretativo e segue uma abordagem qualitativa, baseada em estudos de caso. Foram seleccionados, de uma escola secundária, uma turma do 10.º ano de escolaridade (científico-humanístico) e dois alunos desta turma. A recolha de dados recorreu a duas entrevistas clínicas realizadas individualmente a dois alunos, uma antes outra depois da unidade de ensino "Funções quadráticas", complementada por observação de aulas, registos áudio, resoluções de tarefas de investigação e relatórios escritos produzidos pelos alunos. Os resultados obtidos mostram que os alunos revelam diversas dificuldades na compreensão do conceito de função em diferentes representações e essas dificuldades não foram superadas após a realização da unidade de ensino. Também permitem concluir que os alunos sabem identificar as propriedades da função nas representações gráfica e algébrica revelando, portanto, que reificaram algumas propriedades da função afim e da função quadrática. A realização de tarefas de investigação, por parte dos alunos, possibilitou a utilização de vários processos característicos da actividade matemática. No entanto, na resolução de problemas, alguns utilizaram principalmente processos algébricos e usaram processos gráficos apenas quando a natureza da tarefa proporciona. Outros usaram também processos gráficos com a ajuda da calculadora. Os processos matemáticos utilizados durante o trabalho investigativo foram influenciados pela natureza da tarefa, conhecimento adquirido, experiência prévia e competência em usar a calculadora gráfica. This study aims at analyzing how the resolution of exploratory and research tasks using the graphic calculator contributes to the understanding and learning of quadratic functions by students. The methodology is based on the interpretative paradigm and follows a qualitative approach on case studies. A grade-ten class (science and humanities course) from a secondary school and two students from this class were selected. The collecting of data involved two clinical interviews to two students separately: one before the teaching unit "Quadratic functions", the other afterwards. Class observation, audio recordings, resolution of research tasks and written reports by the students also contributed to the data collection. The results show that the students revealed several difficulties in what regards understanding the concept of function in different representations. Those difficulties were not overcome after the study unit was taught. They also indicate that students were able to identify the properties of the function in the graphic and algebraic representations and thus understood some of the properties of the linear and quadratic functions. When students carried out research tasks, they used several distinctive mathematical processes. However, when solving problems, some used mainly algebraic processes and only when the nature of the task allowed it, did they use graphic processes. Others also used graphic processes with the help of the graphic calculator. The mathematical processes used during research were influenced by the nature of the task, the previous practice and ability to use the graphic calculator. | 677.169 | 1 |
Science Books
Practical Algebra: A Self-Teaching Guide, 2nd Edition
Practical Algebra If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you're a student who needs an introduction to the subject, here's the perfect book for you.
Practical Algebra is an easy and fun-to-use workout program that quickly puts you in command of all the basic concepts and tools of algebra.
With the aid of practical, real-life examples and applications, you'll learn:The basic approach and application of algebra to problem solvingThe number system (in a much broader way than you have known it from arithmetic)Monomials and polynomials; factoring algebraic expressions; how to handle algebraic fractions; exponents, roots, and radicals; linear and fractional equationsFunctions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and moreAuthors Peter Selby and Steve Slavin emphasize practical algebra throughout by providing you with techniques for solving problems in a wide range of disciplines—from engineering, biology, chemistry, and the physical sciences, to psychology and even sociology and business administration.
Step by step, Practical Algebra shows you how to solve algebraic problems in each of these areas, then allows you to tackle similar problems on your own, at your own pace.
Self-tests are provided at the end of each chapter so you can measure your mastery..
For more information about the title Practical Algebra: A Self-Teaching Guide, 2nd Edition, read the full description at Amazon.com, or see the following related books:
Algebra for Dummies — One of the most commonly asked questions in a mathematics classroom is, "Will I ever use this stuff in real life?" Some teachers can give a good, ... > read more
Physics — Improving the Game When it comes to teaching and learning physics, most pedagogical innovations were pioneered in Cutnell and Johnson | 677.169 | 1 |
Core-Plus Mathematics
is a five-year project funded by the National Science Foundation to
develop student and teacher materials for a complete three-year high
school mathematics curriculum for all students, plus a fourth-year
course continuing the preparation of students for college mathematics.
The Contemporary Mathematics in Context curriculum builds upon the
theme of mathematics as sense-making. Throughout it acknowledges,
values, and extends the informal knowledge of data, shape, change,
and chance that students bring to situations and problems.
Each year Contemporary Mathematics in Context features strands
of algebra and functions, geometry and trigonometry, statistics
and probability, and discrete mathematics. These strands are connected
within and across units by fundamental ideas such as symmetry, function,
matrices, data analysis and curve-fitting. The strands also are
connected across units by mathematical habits of mind such as visual
thinking, recursive thinking, searching for and describing patterns,
making and checking conjectures, reasoning with multiple representations,
inventing mathematics, and providing convincing arguments. The strands
are linked further by the fundamental themes of data, representation,
shape, and change. Important mathematical ideas are continually
revisited through these connections so that students can develop
a robust understanding of mathematics. Numerical, graphics, and
programming/link capabilities of graphics calculators are being
capitalized on to enable students to develop versatile ways of dealing
with realistic situations. The curriculum is designed to make more
mathematics accessible to more students, while at the same time
challenging the most able students.
Differences in student performance and interest can be accommodated
by the depth and level of abstraction to which topics are pursued,
by the nature and degree of difficulty of applications, and by providing
opportunities for student choice of homework tasks and projects.
Instructional practices promote mathematical thinking through the
use of rich problem situations that involve students, both in collaborative
groups and individually, in investigating, conjecturing, verifying,
applying, evaluating, and communicating mathematical ideas. Comprehensive
assessment of student understanding and progress through both curriculum-embedded
assessment opportunities and supplementary assessment tasks enables
monitoring and evaluation of each student's performance in terms
of mathematical processes, content, and dispositions.
For more information about the program
and its implementation, contact: | 677.169 | 1 |
_________________
it took 2minutes to me to choose IMO B; here is my reasoning: A. this is evident from the last line of argument "even though the deepest mathematical......" C. we can find it paraphrasing "The graphical illustrations mathematics teachers use enable students to learn geometry more easily...." d. again same reasoning as in C. e. whole argument support this option. B. i dint find any support in argument for this option.
OA please, please let me know if my reasoning is correct and also please tell the difficulty level of this problem, and is 2mins OK? i found this question little bit difficult.
Last edited by gmatstar10 on 23 Apr 2010, 21:35, edited 1 time in total.Your reasoning is in reversed order. For strengthen questions, you need to doubt the argument and take the stated answer choices as CORRECT to support the conclusion not the other way round.
IMO B.
poohv005 wrote....graphical illustrationsenable students to learngeometry more easily by providing them with an intuitive understanding of geometric concepts, which makes it easier to acquire the ability to manipulate symbols
so having preliminary understanding of math concepts makes it easier to acquire the ability to manipulate symbols .
in B we have controversial reasoning, stating that: thouse who are good in manipulating symbols do not necessarily have any mathematical understanding. - vice versa reasoning. | 677.169 | 1 |
Related Products
Summary
Precalculus: A Problems-Oriented Approach offers a fairly rigorous lead-in to calculus using the right triangle approach to trigonometry. A graphical perspective gives students a visual understanding of concepts. The text may be used with any graphing utility, or with none at all, with equal ease. Modeling provides students with real-world connections to the problems. The author is know for his clear writing style and numerous quality exercises and applications.
Table of Contents
Algebra Background for Precalculus
1
(34)
Sets of Real Numbers
1
(5)
Absolute Value
6
(5)
Polynomials and Factoring
11
(9)
Quadratic Equations
20
(15)
Coordinates, Graphs, and Inequalities
35
(66)
Rectangular Coordinates
35
(12)
Graphs and Equations, A Second Look
47
(10)
Equations of Lines
57
(12)
Symmetry and Graphs
69
(8)
Inequalities
77
(8)
More on Inequalities
85
(16)
Functions
101
(74)
The Definition of a Function
101
(12)
The Graph of a Function
113
(17)
Techniques in Graphing
130
(12)
Methods of Combining Functions. Iteration
142
(12)
Inverse Functions
154
(21)
Polynomial and Rational Functions. Applications to Iteration and Optimization
175
(95)
Linear Functions
175
(14)
Quadratic Functions
189
(9)
More on Iteration. Quadratics and Population Growth
198
(17)
Applied Functions: Setting up Equations
215
(12)
Maximum and Minimum Problems
227
(11)
Polynomial Functions
238
(13)
Rational Functions
251
(19)
Exponential and Logarithmic Functions
270
(87)
Exponential Functions
271
(9)
The Exponential Function y = ex
280
(9)
Logarithmic Functions
289
(12)
Properties of Logarithms
301
(9)
Equations and Inequalities with Logs and Exponents
310
(11)
Compound Interest
321
(11)
Exponential Growth and Decay
332
(25)
Trigonometric Functions of Angles
357
(51)
Trigonometric Functions of Acute Angles
357
(10)
Algebra and the Trigonometric Functions
367
(7)
Right-Triangle Applications
374
(9)
Trigonometric Functions of Angles
383
(12)
Trigonometric Identities
395
(13)
Trigonometric Functions of Real Numbers
408
(75)
Radian Measure
408
(9)
Radian Measure and Geometry
417
(10)
Trigonometric Functions of Real Numbers
427
(11)
Graphs of the Sine and the Cosine Functions
438
(16)
Graphs of y = A sin(Bx - C) and y = A cos(Bx - C)
454
(8)
Simple Harmonic Motion
462
(5)
Graphs of the Tangent and the Reciprocal Functions
467
(16)
Analytical Trigonometry
483
(55)
The Addition Formulas
483
(9)
The Double-Angle Formulas
492
(10)
The Product-to-Sum and Sum-to-Product Formulas
502
(5)
Trigonometric Equations
507
(11)
The Inverse Trigonometric Functions
518
(20)
Additional Topics in Trigonometry
538
(62)
The Law of Sines and the Law of Cosines
538
(15)
Vectors in the Plane, A Geometric Approach
553
(7)
Vectors in the Plane, An Algebraic Approach
560
(8)
Parametric Equations
568
(8)
Introduction to Polar Coordinates
576
(9)
Curves in Polar Coordinates
585
(15)
Systems of Equations
600
(72)
Systems of Two Linear Equations in Two Unknowns
600
(11)
Gaussian Elimination
611
(8)
Matrices
619
(12)
The Inverse of a Square Matrix
631
(7)
Determinants and Cramer's Rule
638
(13)
Nonlinear Systems of Equations
651
(7)
Systems of Inequalities
658
(14)
Analytic Geometry
672
(80)
The Basic Equations
672
(9)
The Parabola
681
(11)
Tangents to Parabolas (Optional)
692
(3)
The Ellipse
695
(15)
The Hyperbola
710
(11)
The Focus-Directrix Property of Conics
721
(9)
The Conics in Polar Coordinates
730
(6)
Rotation of Axes
736
(16)
Roots of Polynomial Equations
752
(70)
The Complex Number System
752
(8)
Division of Polynomials
760
(7)
Roots of Polynomial Equations: The Remainder Theorem and the Factor Theorem | 677.169 | 1 |
At Earlham, we strive to teach our students mathematical fundamentals and problem-solving skills that they can apply in a variety of disciplines or in further study of mathematics. Mathematics students may participate in weekly "mathophiles" seminars and informal lunches, attend regional meetings of professional mathematicians, and participate in mathematically related off-campus programs during the academic year or the summer.
The Association for Women in Mathematics (Careers That Count) describes mathematics as "… a powerful tool for solving practical problems and a highly creative field of study, combining logic and precision with intuition and imagination. The basic goal of mathematics is to reveal and explain patterns — whether the pattern appears as electrical impulses in an animal's nervous system, as fluctuations in stock market prices, or as fine detail of an abstract geometric figure."
Earlham College, an independent, residential college, aspires to provide the highest-quality undergraduate education in the liberal arts, including the sciences, shaped by the distinctive perspectives of the Religious Society of Friends (Quakers). | 677.169 | 1 |
The opinions found on these pages are my own. They are not the opinions of the school or district where I work.
Sunday, May 15, 2011
SIP Day introduction to New Algebra curriculum (textbook)
I went to the Algebra-Geometry professional development at the high school. The district has decided to go with a traditional textbook from Pearson, "Algebra Foundations" and Geometry Foundations", that has been updated with a lot of 21st century razzle dazzle.
Karla and a few of our math teachers have taken the time to go through the entire series with a fine toothed comb. They have created a binder and web site to help teachers quickly identify the necessary content that meets current and future standards. They also created alternative Solve it activities to better introduce some of the sections.
The textbook is designed using the principals of Understanding by Design, Grant Wiggins was a consultant. Each chapter has Big Ideas and Essential Questions.
At the beginning of each chapter are seven 21st century additions.
Video introduction
Real students introduce the chapter and explain how it is applicable to the real world.
Math vocabulary
The vocabulary is recorded so students can hear the words.
Solve it
A launch problem designed to introduce each section - we may have substituted our own.
Dynamic Activities
Interactive graphs and such so student can connect Algebra to graphs.
Download
The examples are solved online or on the DVD with narration.
Online Homework
Each student can be given their assignments online.
Extra Practice
Self explanatory.
The entire textbook is online. (All teachers can and should be able to get access if you don't ask me or Karla, or another teacher for the code)
If you use the stock examples the student can replay it at home though not with your words.
Teachers can create separate classes with individual students.
Assignments can be created (well they are already created) and assigned to your virtual class.
There seem to be three standard types of assignments games - worksheets - and tutorials.
I watched one tutorial it might be a good review or supplementary lesson, the one I watched did not include the objective or a summary.
If you prefer every student in all of your classes can access the online textbook under the same user name and password.
The online textbook doesn't seem to track time spent on the assignment or give a grade when the student is finished. The student can click a button that notifies the teacher when they have finished an assignment.
The Entire textbook is on DVD. Same as above, but when your student tries to claim s/he doesn't have Internet access you can give them the DVD. (We have rights to make copies as needed) | 677.169 | 1 |
Math
This program prepares students for college mathematics, and to that end, it offers six year-long courses. These courses include: Pre-algebra, Algebra 1, Geometry, Algebra 2, Pre-Calculus and Calculus. Students are required to take two years of math in the Middle School and three years of math in the Upper School. The math department determines placement of each student. | 677.169 | 1 |
An on-line resource for instructors and student users of the Lial/Hornsby/Miller
Paperback Series. Teachers can find materials to enhance their courses, while students can strengthen their understanding through interactive tutorials and study aids or explore concepts through real-world applications and Web links for further research. (To access the InterAct tutorials over the Web, you will need to download the InterAct plug-in, for Windows only.) Topics cover basic college mathematics, introductory algebra, intermediate algebra with early functions and graphing, introductory and intermediate algebra (a combined approach), and pre-algebra, with a Spanish glossary of English-to-Spanish translations for key mathematical terms used. | 677.169 | 1 |
Basic Mathematics - 7th edition
Summary: Patient and clear in his explanations and problems, Pat McKeague helps students develop a thorough understanding of the concepts essential to their success in mathematics. Each chapter opens with a real-world application. McKeague builds from the chapter-opening applications, such as the average amount of caffeine in different beverages, and uses the application as a common thread to introduce new concepts, making the material more accessible and engaging for student...show mores. Diagrams, charts, and graphs are emphasized to help students understand the material covered in visual form. McKeague's unique and successful EPAS system of Example, Practice, Answer, and Solution actively involves students with the material and thoroughly prepares them for working the Problem Sets. The Sixth Edition of BASIC MATHEMATICS also features a robust suite of online course management, testing, and tutorial resources for instructors and students. This includes iLrn Testing and Tutorial, vMentor live online tutoring, the Digital Video Companion CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE), powered by iLrn. TLE provides a complete courseware package, featuring a diagnostic tool that gives instructors the capability to create individualized study plans. With TLE, a cohesive, focused study plan can be put together to help each student succeed in math Title. We're a Power Distributor; Your satisfaction is our guarantee!
$257.95 +$3.99 s/h
New
Textbook Barn Woodland Hills, CA
PAPERBACK New 0495559741 Premium Books are Brand New books direct from the publisher sometimes at a discount. These books are NOT available for expedited shipping and may take up to 14 business day...show mores to receive. ...show less
$276.97 +$3.99 s/h
New
PROFESSIONAL & ACADEMIC BOOKSTORE Dundee, MI
0495559741 | 677.169 | 1 |
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