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Differential Equations
Description
Differential equations provide one of the most powerful mathematical tools for understanding the natural world. Since rates of change are commonly expressed using derivatives, differential equations arise whenever some continuously varying quantities and their rates of change in space or time are known or postulated. Whether seeking to understand biological processes, behaviours of solids or liquids, ecological systems, or mechanical systems, differential equations provide essential insights. The modelling range of differential equations extends well into the world of human endeavour, for example, they are important for understanding financial markets, or even traffic flows.
This course introduces students to fundamental problems in differential equations. It will introduce students to mathematical modelling, exploring a wide breadth of application areas, and will investigate solution techniques, including methods for numerical computation of solutions.
Availability
Callaghan
Semester 2 - 2015
Learning Outcomes
1. Have the skills to build effective differential equations models and appreciate their implications for answering questions across the natural and human worlds.
2. Be able to classify the different classes of differential equation models, how they arise and what characterises them.
3. Be aware of solution and analytic approaches to important classes of differential equations arising from the mathematical modelling of physical, chemical and biological systems.
4. Be equipped to solve important classes of differential equations analytically and numerically. | 677.169 | 1 |
02011543The goal of Intermediate Algebra: Concepts and Applications, 7e is to help today s students learn and retain mathematical concepts by preparing them for the transition from skills-oriented intermediate algebra courses to more concept-oriented college-level mathematics courses, as well as to make the transition from skill to application. This edition continues to bring your students a best-selling text that incorporates the five-step problem-solving process, real-world applications, proven pedagogy, and an accessible writing style. The Bittinger/Ellenbogen series has consistently provided teachers and students with the tools needed to succeed in developmental mathematics. This edition has an even stronger focus on vocabulary and conceptual understanding as well as making the mathematics more accessible to students. Among the features added are new Concept Reinforcement exercises, Student Notes that help students avoid common mistakes, and Study Summaries that highlight the most important concepts and terminology from | 677.169 | 1 |
MAP
Objectives: The objective of this course is to present the foundations of many basic
computer related concepts and provide a coherent development to the students for the
courses like Fundamentals of Computer Organization, RDBMS, Data Structures,
Analysis of Algorithms, Theory of Computation ,Cryptography, Artificial Intelligence
and others. This course will enhance the student's ability to think logically and
mathematically. Prerequisites: Knowledge of basic concepts on Sets, different operations on sets,
binary operations, functions.
2. From Book # 2 Chapter – 1, articles 1.2 to 1.5
Chapter-3 article 3.3 Accomplishment of the student after completing the course : The student will be able to apply concepts to RDBMS, perform minimization of Boolean
functions, shall learn the fundamentals representations methods of graphs and trees. They
shall be able to use different logical reasoning to prove theorems. | 677.169 | 1 |
This report includes the original plenary papers from the joint ICMI-ICAM meeting, highlights collaborative research on the educational links between mathematics and industry, and features papers on the topic from a host of international contributors. more...
Here are a range of articles, from introductory texts to state-of-the-art research in biomathematics, with topics ranging from population genetics, population dynamics, speciation, adaptive dynamics, game theory, kin selection and stochastic processes. more...
This book offers ideas, blue-prints and actions that can help raise public awareness of the importance of mathematical sciences in our contemporary society. It covers national experiences, exhibitions, mathematical museums, and other popularization activities. more...
Newsletter_0609.pdf ""[This book] starts with some historical facts on digital publishing, together with a survey of the effort to convert existing classical sources to a digital form. It also contains examples of drawbacks of electronic publishing and comments on the influence of big specialised companies on the publishing of mathematical journals.... more...
Initial Considerations
Topics of Elementary Statistics
Introductory Notions
General Ideas
Variables
Populations and Samples
Importance of the Form of the Population
First Ideas of Interference on a Normal Population
Parameters and Estimates
Notions on Testing Hypotheses
Inference of the Mean of a Normal Population
Inference... more...
Parallel Computations focuses on parallel computation, with emphasis on algorithms used in a variety of numerical and physical applications and for many different types of parallel computers. Topics covered range from vectorization of fast Fourier transforms (FFTs) and of the incomplete Cholesky conjugate gradient (ICCG) algorithm on the Cray-1 to... more... | 677.169 | 1 |
This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.Practice makes perfect! Get perfect with a thousand and one practice problems! 1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more... more...
A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing shows how to use a collection of mathematical techniques to solve important problems in applied mathematics and computer science areas. The book discusses fundamental tools in analytical geometry and linear algebra. It covers a wide range of topics,... more... | 677.169 | 1 |
Undergraduate Algebra, 2nd Edition
Synopses & Reviews
Publisher Comments:
Undergraduate Algebra is a text for the standard undergraduate algebra course. It concentrates on the basic structures and results of algebra, discussing groups, rings, modules, fields, polynomials, finite fields, Galois Theory, and other topics. The author has also included a chapter on groups of matrices which is unique in a book at this level. Throughout the book, the author strikes a balance between abstraction and concrete results, which enhance each other. Illustrative examples accompany the general theory. Numerous exercises range from the computational to the theoretical, complementing results from the main text.For the third edition, the author has included new material on product structure for matrices (e.g. the Iwasawa and polar decompositions), as well as a description of the conjugation representation of the diagonal group. He has also added material on polynomials, culminating in Noah Snyders proof of the Mason-Stothers polynomial abc theorem. About the First Edition:The exposition is down-to-earth and at the same time very smooth. The book can be covered easily in a one-year course and can be also used in a one-term course...the flavor of modern mathematics is sprinkled here and there.- Hideyuki Matsumura, Zentralblatt
Book News Annotation:
Splendid undergraduate text, intended to function as a companion to the distinguished author's Linear algebra and to provide young mathematicians with a secure command of the fundamentals of groups, rings, fields, and related structures. Ten chapters, many excellent problems, written with exemplary clarity and with exceptional sensitivity to what young readers might on first encounter consider to be "scary". Departs from the previous edition (1987) by the inclusion of some new material and exercises. The author has been very well served by the production people at Springer, who have produced a physically beautiful book at a reasonable price. (NW) | 677.169 | 1 |
Product Description
The Matrix Algebra Tutor: Learning by Example DVD Series teaches students about matrices and explains why they're useful in mathematics. This episode teaches students about inconsistent and dependent systems in matrix math. Sometimes, a system of equations does not have a well-defined solution. For example, if we have two equations that represent two lines and these lines are exactly parallel and never intersect, then these equations do not have a solution. In this program, these ideas are explored. Grades 9-College. 54 | 677.169 | 1 |
Search Results
This algebra lesson helps students make the connection between functions and their graphs. The model of the level of water in a bathtub is used. Students will watch the graph and a chart of the depth of the water at...
Students in need of experience constructing and interpreting statistical graphs will find this exercise useful. The lesson uses data from past presidential elections; students will construct a variety of graphs (bar...
This lesson uses data from the Center for Disease Control to give students experience with the Chi-Square Statistic. Students will use spreadsheets to organize data, which they will then perform statistical analysis on....
This lesson plan has students create a confidence interval based on the historic snowfall records of a town in Ohio, the data for which is available online. The class will calculate the mean and standard deviation,...
This lesson helps students understand financial topics (interest rates, FICO scores and loan payments) in a mathematical context. Students will calculate monthly payments for a car or home based on the best interest... | 677.169 | 1 |
BEGINNING ALGEBRA
9780131444447
ISBN:
0131444441
Edition: 4 Pub Date: 2004 Publisher: Prentice Hall
Summary: Clearly explained concepts, study skills help, and real-life applications will help the reader to succeed in learning algebra.
Martin-Gay, K. Elayn is the author of BEGINNING ALGEBRA, published 2004 under ISBN 9780131444447 and 0131444441. One hundred twenty five BEGINNING ALGEBRA textbooks are available for sale on ValoreBooks.com, fourteen used from the cheapest price of $1.00, or buy new starting at $13.9...9 | 677.169 | 1 |
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 |
Mathematics: Mathematics
About this page
This page provides tips for choosing the best resources to assist with your mathematics studies.
New Library Resource
Everyday Math by Stan Gibilisco
Call Number: 510 GIBILIS 2013
ISBN: 9780071790130
Publication Date: 2012-08-14
If you cannot tell the difference between your Roman and Arabic numerals, or if when someone asks 'what is pi' you say "delicious," you need EverydayMath DeMYSTiFieD. Detailed examples make it easy to understand the material, and end-of- chapter quizzes and a final exam help reinforce key ideas. | 677.169 | 1 |
Tag InfoThe study of algorithms is the branch of mathematics that specializes in finding efficient (mostly in the terms of time and space complexity) for various computational problems. It often involves combinatorics, number theory and geometry.
The field also includes data structures, computational models and proofs of lower bounds for certain algorithms. | 677.169 | 1 |
Introduction Surficial Features Spatial Data General Organization of the Book
Mathematical Morphology: An Introduction Birth of Mathematical Morphology Elements of Set Theory and Logical Operations Grid Utilized for Morphological Transformations Theory of Structuring Elements Four Basic Principles of the Theory of Mathematical Morphology... more... | 677.169 | 1 |
books.google.com - The... Through Problems
Problem-Solving Through Problems
The and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate. | 677.169 | 1 |
This book has been specifically written for the new two-tier Edexcel linear GCSE specification for first examination in 2008. The book is targeted at the B to A* grade range in the Foundation tier GCSE, and it comprises units organised clearly into homeworks designed to support the use of the Higher Plus Students' Book in the same series. Each unit offers:
BLA review test focusing on prior topics for continual reinforcement BLTwo sets of questions that relate directly to individual lessons in the unit, providing ample practice BLA synoptic homework that covers the whole unit, so students consolidate the key techniques BLFull answers in the accompanying teacher book
It forms part of a suite of four homework books at GCSE, in which the other three books cater for grade ranges G to E, E to C and D to B.
Book Description OUP Oxford, 20063353735
Book Description OUP Oxford347740 | 677.169 | 1 |
College Algebra
9780077221287
ISBN:
0077221281
Edition: 3 Pub Date: 2008 Publisher: McGraw-Hill Companies, The
Summary: The Barnett Graphs & Modelsseries in college algebra and precalculus maximizes student comprehension by emphasizing computational skills, real-world data analysis and modeling, and problem solving rather than mathematical theory. Many examples feature side-by-side algebraic and graphical solutions, and each is followed by a matched problem for the student to work. This active involvement in the learning process helps... students develop a more thorough understanding of concepts and processes. A hallmark of the Barnett series, the function concept serves as a unifying theme. A major objective of this book is to develop a library of elementary functions, including their important properties and uses. Employing this library as a basic working tool, students will be able to proceed through this course with greater confidence and understanding as they first learn to recognize the graph of a function and then learn to analyze the graph and use it to solve the problem. Applications included throughout the text give the student substantial experience in solving and modeling real world problems in an effort to convince even the most skeptical student that mathematics is really useful.
Barnett, Raymond A. is the author of College Algebra, published 2008 under ISBN 9780077221287 and 0077221281. Two hundred forty College Algebra textbooks are available for sale on ValoreBooks.com, seventy two used from the cheapest price of $51.09, or buy new starting at $100 | 677.169 | 1 |
Geometry and Algebra (Basher Series)
Hardcover
Item is available through our marketplace sellers.
Overview secrets of their world and how they like to throw their numbers about. Bringing his charming manga-style artwork and tongue-and-cheek approach to explaining the basics, Basher brings a whole new spin to the world of higher math.
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Editorial Reviews
School Library Journal
Gr 4–6—This creative team introduces the components of algebra and geometry as cartoon-style characters. The book begins with a brief introduction to the subject of mathematics and Pythagoras. Then, the concepts are grouped together in eight chapters, including "Counting Crew" (Roman numerals, base 10, etc.), "Graph Gang" (vector, line, etc.), "Shape Sisters" (perimeter, area, etc.), "Trig-Athletes" (sine, cosine, etc.), and "In the Round" (circle, pi, etc.). Each chapter begins with an introduction and then the concepts are presented on a spread. One page features a drawing of the concept's character, while the opposing page provides a brief introduction to its characteristics and personality. The author describes a Mobius strip in detail, but never calls it by name. There is some crossover between this title and Green's Math: A Book You Can Count On (Kingfisher, 2010). The information is presented in a chatty tone. For example, Negative Number is introduced as living in, "…chilly, subzero zone. It's brrr, no doubt!" and is portrayed as an ice cube holding a thermometer. Along with the narrative, which is written in the first person from the concept's point of view, basic information and key facts are provided in bullet points. This book is certain to appeal to number lovers and would be an effective supplement to a mathematics curriculum.—Maren Ostergard, King County Library System, Issaquah, WA | 677.169 | 1 |
Windows software for examining and verifying or rejecting concepts, principles, definitions, assumptions and theorems, for middle school, high school, and college. With fully dynamic quantitative scales, interactive dynamic tools, and data for demonstrating mathematical concepts and developing understanding through experience and exploration. Use in class and for preparing class materials and homework (including equations, graphs, geometric shapes (Pythagoras), and activities) with applications such as MS Word and Powerpoint. Also available at | 677.169 | 1 |
English
Ideal for instructors who teach a precalculus level course and wish to include a comprehensive review of algebra at the beginning, this text introduces trigonometry first with a right triangle approach and then with the unit circle. As the best-selling text in the field, Algebra and Trigonometry provides unparalleled exercises, motivating real-life applications, a supportive pedagogical design, and innovative ancillaries and resources, making it a complete solution for both students and instructors.
New!Model It real-life applications in nearly every text section are multi-part exercises that require students to generate and analyze mathematical models. First referenced in the Why You Should Learn It at the beginning of each section, these interesting applications illustrate why it is important to learn the concepts in each section.
New! Enlarged printable graphs in many exercise sets contain problems asking students to draw on the graph provided.For added convenience, in the Sixth Edition these proofs have been moved from an appendix to the end of relevant chapters.
New!P.S. Problem Solving, at the conclusion of each chapter, features a collection of thought-provoking and challenging exercises that further explore and expand upon the concepts of the chapter. These exercises have unique characteristics that set them apart from traditional algebra and trigonometry exercises.
New! A wealth of student success tools includes How to Study This Chapter, a chapter-opening study guide that includes What you should learn (section-opening objectives), Important Vocabulary, a list of Study Tools, and a list of Additional Resources to help the student prepare for the chapter; Why you should learn it, a section-opening, real-life application or a reference to other branches of mathematics, illustrating the relevance of the section's content; and What did you learn?, a concise chapter summary organized by section. These objectives are correlated to the chapter Review Exercises to help students prepare for exams.
Abundant, up-to-date Real-Life Applications are integrated throughout the examples and exercises and identified by a globe icon to reinforce the relevance of the concepts being learned.
A wide variety of Exercises, including computational, conceptual, and applied problems are carefully graded in difficulty to allow students to gain confidence as they progress. Each exercise set includes Synthesis Exercises that promote further exploration of mathematical concepts, critical-thinking skills, and writing about mathematics, and Review Exercises that reinforce previously learned skills and concepts.
Special Algebra of Calculus examples and exercises highlight the algebraic techniques used in calculus to show students how the mathematics they are learning now will be used in future courses.
Optional graphing technology support is provided in marginal point-of-use instructions that encourage the use of graphing technology as a tool to visualize mathematical concepts, to verify other solution methods, and to facilitate computation. In addition, the section An Introduction to Graphing Utilities helps the student become familiar with the basic functionality of a graphing utility. The use of technology is optional in this text; all exercises that require the use of a graphing utility are clearly identified by an icon.
Explorations preceding the introduction of selected topics provide the opportunity to engage students in active discovery of mathematical concepts and relationships, often through the power of technology. Explorations strengthen students' critical- thinking skills and help develop an intuitive understanding of theoretical concepts.
All Examples have been carefully chosen to illustrate a particular mathematical concept or problem-solving skill. Every example contains step-by-step solutions, most with line-by-line explanations that lead students through the solution process, making it easy for students to understand the concepts being explained.
Designed for the three-semester course for math and science majors, the Larson/Hostetler/Edwards series continues its tradition of success by being the first to offer both an Early Transcendental version as well as a new Calculus with Precalculus text. This was also the first calculus text to use computer-generated graphics (Third Edition), to include exercises involving the use of computers and graphing calculators (Fourth Edition), to be available in an interactive CD-ROM format (Fifth Edition), and to be offered as a complete, online calculus course (Sixth Edition). Every edition of the book has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. The Seventh Edition also expands its support package with an all-new set of text-specific videos.
P.S. Problem-Solving Sections, an additional set of thought-provoking exercises added to the end of each chapter, require students to use a variety of problem-solving skills and provide a challenging arena for students to work with calculus concepts.
Getting at the Concept Exercises added to each section exercise set check students' understanding of the basic concepts. Located midway through the exercise set, they are both boxed and titled for easy reference.
Review Exercises at the end of each chapter have been reorganized to provide students with a more effective study tool. The exercises are now grouped and correlated by text section, enabling students to target concepts requiring review.
The icon "IC" in the text identifies examples that appear in the Interactive Calculus 3.0 CD-ROM and Internet Calculus 2.0 web site with enhanced opportunities for exploration and visualization using the program itself and/or a Computer Algebra System.
Think About It conceptual exercises require students to use their critical-thinking skills and help them develop an intuitive understanding of the underlying theory of the calculus.
Modeling Data multi-part questions ask students to find and interpret mathematical models to fit real-life data, often through the use of a graphing utility.
Section Projects, extended applications that appear at the end of selected exercise sets. may be used for individual, collaborative, or peer-assisted assignments.
True or False? Exercises, included toward the end of many exercises sets, help students understand the logical structure of calculus and highlight concepts, common errors, and the correct statements of definitions and theorems.
Motivating the Chapter sections opening each chapter present data-driven applications that explore the concepts to be covered in the context of a real-world setting.
The transition to upper-level math courses is often difficult because of the shift in emphasis from computation (in calculus) to abstraction and proof (in junior/senior courses). This book provides guidance with the reading and writing of short proofs, and incorporates a gradual increase in abstraction as the chapters progress. This helps students prepare to meet the challenges of future courses such as abstract algebra and elementary analysis.
* Clearly explains principles and guides students through the effective transition to higher-level math * Includes a wide variety of applications, technology tips, and exercises, including new true/false exercises in every section * Provides an early introduction to eigenvalues/eigenvectors * Accompanying Instructor's Manual and Student Solutions Manual (ISBN: 0-12-058622-3)
This interactive, text-specific software is a comprehensive learning source with a multitude of features to promote student success. The wealth of multimedia features combined with the complete text on CD-ROM make it an excellent tool for use at home, in classroom presentations, as a tutorial, or in math laboratories first with a unit circle approach and then with the right triangle.For a complete listing of features, see Larson/Hostetler, College Algebra, 5/e | 677.169 | 1 |
Mathematics
Shows students the relevance of statistics in real-world settings. Video series for college and high school classrooms and adult learners. Shows students the relevance of statistics in real-world settings. Video series for college and high school classrooms and adult learners.
A step-by-step look at algebra concepts. Video instructional series for college and high school classrooms and adult learners. A step-by-step look at algebra concepts. Video instructional series for college and high school classrooms and adult learners.
A multidisciplinary resource for middle and high school teachers to use photographs and photographic ephemera to convey content and teach visual analysis skills. A multidisciplinary resource for middle and high school teachers to use photographs and photographic ephemera to convey content and teach visual analysis skills.
This video workshop for middle and high school teachers presents strategies for improving how typical Algebra 1 topics are taught. This video workshop for middle and high school teachers presents strategies for improving how typical Algebra 1 topics are taught.
Practical examples highlight the basic concepts of data analysis and statistics in this video- and Web-based course for K-8 math teachers. Practical examples highlight the basic concepts of data analysis and statistics in this video- and Web-based course for K-8 math teachers.
Discover how geometric reasoning can be used as a method of problem solving in this video- and Web-based course for K-8 teachers. Discover how geometric reasoning can be used as a method of problem solving in this video- and Web-based course for K-8 teachers.
Examine some of the major ideas in measurement and their practical applications in this video- and Web-based course for K-8 teachers. Examine some of the major ideas in measurement and their practical applications in this video- and Web-based course for K-8 teachers.
This video- and Web-based course for K-8 teachers examines three main categories in the Number and Operations strand of the math standards. This video- and Web-based course for K-8 teachers examines three main categories in the Number and Operations strand of the math standards.
Explore the big ideas in algebra - patterns, functions, and linearity - in this video- and Web-based course for K-8 teachers. Explore the big ideas in algebra - patterns, functions, and linearity - in this video- and Web-based course for K-8 teachers.
This video workshop for K-12 teachers and administrators features seven leading educators sharing their ideas on how children really learn. This video workshop for K-12 teachers and administrators features seven leading educators sharing their ideas on how children really learn.
This video workshop for K-12 teachers and administrators encourages educators to analyze existing theories about how children learn. This video workshop for K-12 teachers and administrators encourages educators to analyze existing theories about how children learn.
This video library for K-12 educators looks at the many types of mathematics assessments and what they reveal about student learning. This video library for K-12 educators looks at the many types of mathematics assessments and what they reveal about student learning.
This course for high school teachers and college level instruction explores mathematical concepts as tools for understanding real-world phenomena. This course for high school teachers and college level instruction explores mathematical concepts as tools for understanding real-world phenomena.
This video workshop for middle school math teachers covers essential topics missed in most U.S. math curricula. This video workshop for middle school math teachers covers essential topics missed in most U.S. math curricula.
This video workshop for K-12 educators investigates how math teaching can be structured to resonate with children's own often complex ideas. This video workshop for K-12 educators investigates how math teaching can be structured to resonate with children's own often complex ideas.
Explore the brain's inborn abilities for learning math and tools to nurture it in this documentary for K-8 teachers and administrators. Explore the brain's inborn abilities for learning math and tools to nurture it in this documentary for K-8 teachers and administrators.
This video library for high school teachers of advanced math courses shows real classrooms where the NCTM standards guide the lessons. This video library for high school teachers of advanced math courses shows real classrooms where the NCTM standards guide the lessons. | 677.169 | 1 |
Love and Math: The Heart of Hidden Reality
by Edward Frenkel Publisher Comments
A New York Times Science Bestseller What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, werent even told they existed? Alas, this is how math is... (read more)
Visions of Infinity: The Great Mathematical Problems
by Ian Stewart Publisher Comments
It is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanize them. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. Such challenges offer a... (read more)
The Best Writing on Mathematics (Best Writing on Mathematics)
by Mircea Pitici Publisher Comments
This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2012 makes available to a wide audience... (read more)
Oxford Illustrated Math Dictionary
by Oxford University Press Publisher Comments
The Oxford Illustrated Math Dictionary explains academic vocabulary at a level appropriate for high-beginning and intermediate students, which accelerates their mastery of content and allows them to be successful in content-area classes and Content... (read more)
Ballparking: Practical Math for Impractical Sports Questions
by Aaron Santos Publisher Comments
How fat would you need to be to completely block a hockey goal? How much weight could you lift if you were ant-sized? How hard would you have to hit a baseball to hit the Goodyear blimp? In this amusing and enlightening book of offbeat sports... (read more)
The Joy of X: A Guided Tour of Math, from One to Infinity
by Steven Strogatz Publisher Comments
Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In The Joy of x, Steven Strogatz expands on his hit New York Times series to explain the big ideas... (read more)
Math Word Problems for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
Covers percentages, probability, proportions, and more Get a grip on all types of word problems by applying them to real life Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a... (read more)
Thinking in Numbers: On Life, Love, Meaning, and Math
by Daniel Tammet Publisher Comments
Thinking in Numbers is the book that Daniel Tammet, bestselling author and mathematical savant, was born to write. In Tammet's world, numbers are beautiful and mathematics illuminates our lives and minds. Using anecdotes, everyday examples, and... (read more)
The Mathematics of Life
by Ian Stewart Publisher Comments
Biologists have long dismissed mathematics as being unable to meaningfully contribute to our understanding of living beings. Within the past ten years, however, mathematicians have proven that they hold the key to unlocking the mysteries of our world... (read more)
Finite Markov Processes and Their Applications
by Marius Iosifescu Publisher Comments
A self-contained treatment of finite Markov chains and processes, this text covers both theory and applications. Author Marius Iosifescu, vice president of the Romanian Academy and director of its Center for Mathematical Statistics, begins with a review... (read more)
Mathematical Modeling: Case Studies from Industry
by Ellis Cumberbatch Book News Annotation
This volume presents actual industrial problems in order to expose
students of high-level modeling courses to the modeling process, as
well as to impress them with the diverse nature of applied math in
technology transfer. These 13 contributions are... (read more)
Mathematical Programming
by Steven Vajda Publisher Comments
Written by a trailblazer in the field, this classic of mathematical programming and operational research first appeared nearly 50 years ago. It remains as relevant today as at the time of its initial publication, offering advanced undergraduates | 677.169 | 1 |
01303196mediate Algebra for College Students (3rd Edition)
The goal of this series is to provide readers with a strong foundation in Algebra. Each book is designed to develop readers' critical thinking and problem-solving capabilities and prepare readers for subsequent Algebra courses as well as service math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. Algebra and Problem Solving. Functions, Linear Functions, and Inequalities. Systems of Linear Equations and Inequalities. Polynomials, Polynomial Functions, and Factoring. Rational Expressions, Functions, and Equations. Radicals, Radical Functions, and Rational Exponents. Quadratic Equations and Functions. Exponential and Logarithmic Functions. Conic Sections and Nonlinear Systems of Equations. Polynomial and Rational Functions. Sequences, Probability, and Mathematical Induction. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra. | 677.169 | 1 |
books.google.com - The book covers various topics of computer algebra methods, algorithms and software applied to scientific computing. One of the important topics of the book is the application of computer algebra methods for the development of new efficient analytic and numerical solvers, both for ordinary and partial... Algebra in Scientific Computing | 677.169 | 1 |
The canvas is set up at 1275x1875 pixels (4x6 inches with .25 inch bleed area). Contains two PSD files that are well organized and highly detailed for easy use. The files contains groups that are color coded and properly labelled.
Gary Rockswold teaches algebra in context, answering the question, "Why am I learning this?" By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswold focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses.
This set of six CDs offers high school students basic and useful practice in algebra 1, algebra 2, statistics, geometry, trigonometry and calculus. In the Algebra 1 CD, for example, students choose from 12 chapters and pick a sub topic. After a written (and pictorial) demonstration of the concept (e.g. ratio and proportion), students answer 10 questions to show comprehension. Hint and Solution icons lead to useful and straightforward help, and general progress is described after each section. All things considered, although dry in format and somewhat lacking in depth, this handy set of CDs offers understandable explanations of difficult subjects. | 677.169 | 1 |
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 |
Hi, I'm a teacher but not a math teacher. I have a son who is struggling
with math and I want to find the general math textbook that I used when I
was a student. It explained concepts very well and I liked it a lot.
It had a grass-green cover, it was a little bigger than a trade paperback
book, it had a multi-colored stripe across the front under the title, I
think the title was just "Mathematics". It was similar in size to the
white English Composition textbook and it might have been by the same
publisher but I don't know who that publisher was. It would have been
geared towards middle or high school aged students.
Any help would be greatly appreciated. | 677.169 | 1 |
is a physics course designed for the home school student with a relatively good command of trigonometry. The student should be at least halfway through trigonometry before starting this course. It contains thorough examples, experiments to be done in the home, and detailed solutions to all student exercises. 602 pages, hardback | 677.169 | 1 |
The master theme of the Ninth Grade will be to further develop in students all of their higher cognitive and affective skills, conforming them to the more complex concepts of the secondary school, and to the more difficult task of applying principles, concepts, and ideals in a much wider field of study than they have had in the elementary grades. Now, students will be obliged to thoroughly develop good study habits to help in the intellectual digestion of more difficult subject matter. Students will continue to be prepared to enter the world as productive members of their community, ready and able to steadfastly defend their Faith and to apply traditional Catholic principles to their everyday lives.
Theology: The Augustinian method for teaching religion continues this year with the second book of the Our Quest for Happiness series, Through Christ Our Lord Text. This year, the student will study the Gospels and their accounts of the life of Christ. Students will examine the deeds of love of the Divine Son in redeeming us and restoring hope. Further, this year students will delve into a study of the Mass and the Eucharist. An overview of morality, as well as an examination of the third and fourth commandments, is also covered this year. Through Christ our Lord Answer Key is keyed to the text and available for the regular review questions after each section, as well as for the end-of-chapter review questions. Other books used: The Rosary Novena Booklet, The Way of the CrossBooklet, and TheDouay-Rheims New Testament.
Math:Saxon Algebra I Text is the suggested text for this year's math course, although students coming to OLVS without having successfully completed Saxon's Algebra ½, are required to complete that introductory course before tackling Algebra I. There are some important concepts and skills that must be mastered in order to more thoroughly understand the more difficult mathematical processes that will follow in the next couple of years. Saxon Algebra I covers arithmetic of (evaluation of) expressions involving signed numbers, exponents and roots, properties of real numbers, absolute value and equations. Integrated into the text is an extensive amount of Geometric concepts as well. The three-book Saxon Algebra I Set includes: Saxon Algebra I Textbook; Saxon Algebra I Test Forms; Saxon Algebra I Answer Key. Saxon Algebra 1 Solutions Manual is also available, and strongly recommended.
Grammar: This grammar and compositon course, part of a series originally published (like Voyages in English) by Loyola University Press, was at one time the standard in Catholic high schools of the United States. The textbook, Correct Writing Text, is used in conjunction with the reference manual, Writing Handbook. Both of these books must be used together so that the students can acquire a thorough understanding of the necessary grammatical rules and concepts. Using the Traditional Catholic Speller 9, the students will also learn the spelling of hundreds of words and their meanings, including Catholic terms. Other book used:Correct Writing Answer Key.
Literature: This year the students will be concentrating on perfecting their skills at analyzing literature by identifying (and commenting upon) the literary elements of character, setting, plot and theme. Students will read, The Scarlet Pimpernel, by Baroness Orczy – "The year is 1792 during the French Revolution.Daily, tumbrels roll over the cobbled streets of Paris bearing new victims to the insatiable Madame guillotine…Thus the stage is set for one of the most enthralling novels of historical adventure ever written.The mysterious figure known as the Scarlet Pimpernel, sworn to rescue helpless men, women and children from their doom; his foe, the French agent Chauvelin; and the lovely Lady Marguerite Blakeney, a French exile married to an English lord – all play their parts in a suspenseful tale."Another author who was able to do so, while writing in the poetic English of the mid-19th century, was Cardinal Wiseman. His Fabiola, a thrilling story about the persecutions of the early Christians of Rome, has a number of sub-plots with suspenseful twists. Cardinal Wiseman artfully meshes the true history of the early Church, the persecutions endured by the first Christians, and the glorious army of martyrs of the catacombs, into an historical novel, which keeps readers spellbound. Come Rack, Come Rope, by Robert Hugh Benson takes place in England during the 16th century under the rule of Elizabeth I. Families, like the FitzHerberts, heard Mass and received the Sacraments in secret. Concealed places were built especially for the priests so they could hide in an emergency.Families were divided; men were thrown into jail; and neighbor turned against neighbor.A traitor in the midst meant capture, torture and certain death for the undercover priests.There exists also a charming and holy love between a young man and lady, but God comes first!Will he become a priest for God?This real life drama provides vivid descriptions and profound insights into the persecutions of Catholics during the age of the Protestant Reformation. Kidnapped, by Robert Louis Stevenson, assigned to the boys, isset in the aftermath of the Jacobite Rebellion of 1745 and sustains a gripping narrative. David Balfour, a young lowlander who is tricked by his miserly uncle, survives attempted murder, kidnap and shipwreck. In the company of Alan Breck, a Jacobite, he escapes through the Highlands. TheStory of the Other Wise Man by Henry van Dyke, is a touching tale of sacrifice and perseverance of the fourthpilgrim (the other wise man), whose great desire to see the Messiah was denied, yet accomplished in the denial. He wanders for years in search of and eventually finds, the one Whom he sought.
History: This year's World History is centered on the history of the Catholic Faith as embodied in The Story of the Church Text. For, what is history if not the telling of the story of Christ and His Church? This survey of the world stresses that the roots of the One, True Faith lie in the beginning of the world, the Fall of Man, and his subsequent Redemption. As the publishers state, "To know and to understand the Church is a sacred duty for every Catholic. . . This book has been written for boys and girls with the hope that as they learn the story it tells, their love of Christ in His Church will increase and that a lifelong interest in Church history will be enkindled. The Story of the Church is told in three parts: first in the days of the Roman Empire, then through the Ages of Faith, and finally in Modern Times, Christ in His Church advances down the ages unto the fulfillment of His mission." Other books used: Story of the ChurchTest Booklet, Story of the Church Test Answer Key.
Science: This course is based on the book, Physical Science Text, by Dr. Jay Wile. It was written specifically for the home-schooled student; the writing and presentation style Dr. Wile uses gives the reader the impression that the author is sitting across the kitchen table tutoring one-on-one. The experiments use common, readily available materials, and the author walks the students through each of the labs in a logical easy-to-follow manner. The text is intended to stimulate new questions that will cause the students to expand his knowledge. Topics covered include: the atmosphere and the weather; the structure of the earth; and an introduction to physics, touching on motion, Newton's Laws, gravity, and astrophysics. This course will help students understand how the entire universe, from atoms to galaxies, is governed and maintained by the infinite power of our God. Physical Science Solutions and Tests contains: (1) answers to the Study Guide found at the end of each module in the text; (2) tests for each module; and (3) answers to each of the modular tests. | 677.169 | 1 |
Shin Takahashi, Iroha Inoue, and Trend Co., Ltd
How can one make first-semester linear algebra material more appealing (or at least more palatable) to leery students? Why not try dressing it up in a story of young love? For those unfamiliar with the term, manga refers to a type of Japanese comics. The Manga Guide to Linear Algebra is just one in a series of manga presentations of scientific material such as statistics, molecular biology, and, perhaps most ambitiously, the universe. This particular guide seems aimed at high school and college students, though it could also appeal to general readers looking for a gentle refresher course on the topic.
The book's main characters are mathematically-talented college student Reiji, his fetching tutee Misa, and Tetsuo, Misa's older brother. The storyline is simple: Tetsuo, the head of a karate club, allows Reiji to join the club on the condition that Reiji tutors Misa in linear algebra. The book details the tutoring sessions, while providing occasional interludes which either further the book's story or provide supplemental mathematical material.
The literary quality of the work is about what you'd expect of a book that is primarily a mathematical text: the plot is superficial and predictable, though I enjoyed the author's successfully drawn parallel between Misa's efforts to learn linear algebra and Reiji's efforts to learn karate. I was also mildly disappointed in the book's stereotyped gender roles: while Misa is intelligent, she experiences no personal growth in the book (in contrast to Reiji, and even Tetsuo). As a woman in the field of mathematics, I would have preferred it if Misa's main talents hadn't been making lunch for Reiji and looking pretty.
The book's mathematical discussions range from elegant to confusing. Its author clarifies that the book is intended as a "complement to more comprehensive literature, not a substitute," and as such it is moderately successful. Many introductory linear algebra topics, as well as preliminary material, are reviewed in the book, and the author takes advantage of the manga format to provide some deft visual demonstrations of techniques and concepts. I also like that the book discusses some more abstract areas of linear algebra, such as linear independence and subspaces of Rn.
That being said, the book should certainly not be used as a solo text by someone who hasn't already studied the material. Many algorithms or concepts that students find difficult are rushed through. For instance, when Reiji covers Gaussian elimination, he provides only two examples, and both involve matrices whose non-augmented versions reduce to the identity matrix. The book's scope is limited — it focuses primarily on square, invertible matrices, and contains little to no discussion of inconsistent systems of linear equations or of those systems that have more than one solution. The book's method of computing determinants for general square matrices seems unnecessarily convoluted, requiring the explicit use of permutations.
At least once the text states something that is patently false: Reiji tells Misa that you "can generally never find more than n different eigenvalues and eigenvectors for any n ×n matrix," which is true about eigenvalues but false about eigenvectors: every eigenspace of every matrix contains infinitely many elements. While this may be merely the result of a mistranslation, this is the type of error that can seriously confuse someone who isn't already comfortable with linear algebra.
Thus, while I applaud the unique format of the book, and while portions of it provide solid summaries of linear algebra techniques, the book is less than stellar in its execution. It could work well as a study guide for someone who just needs a review of linear algebraic concepts, but I hesitate to recommend it to someone, like Misa, who is struggling with the subject. That said, I encourage authors and publishers to continue publishing creative texts such as this.
Jessica K. Sklar is an associate professor of Mathematics at Pacific Lutheran University. Trained as an algebraist, she considers herself more of a generalist these days, with particular interest in recreational mathematics, and in the communication and writing of math. She recently coedited (with Elizabeth S. Sklar) the collection of essays Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media(McFarland & Co., 2012), and in 2011, she and coauthor Gene Abrams (University of Colorado at Colorado Springs) received an MAA Carl B. Allendoerfer Award for their article "The Graph Menagerie: Abstract Algebra and the Mad Veterinarian" (Mathematics Magazine, June 2010). Her homepage is and you may contact her at sklarjk@plu.edu.
Comments
Karate, Math, and Romance! In this Manga Guide, Reiji seeks to learn karate. The captain of the karate team, Tetsuo, agrees to let Reiji join if he tutors his sister, Misa, in linear algebra, which is where the math comes in. As it turns out, Reiji has a crush on Misa, and that's where the romance comes in.
Linear algebra is presented here by way of the linear transformation. Topics covered are: matrices, vectors, linear independence and spanning, subspaces and dimension, linear transformations, and eigenvalues and eigenvectors. The text is heavy on computation and light on abstract theory. Most of the examples are done in two and three dimensions, allowing pictures to be used to show the geometry of linear transformations. The only drawback is that the book is light on applications. But if you're a visual learner and you want to learn linear algebra, this just might be the book for you! | 677.169 | 1 |
free Calculus eBooks
This is a substantially updated, extended and reorganized third edition of an introductory text on the use of integral transforms. Chapter I is largely new, covering introductory aspects of complex variable theory. Emphasis is on the development of techniques and the connection between properties of transforms and the kind of problems for which they provide tools. Around 400 problems are accompanied in the text. It will be useful for graduate students... more...
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This monograph presents recent contributions to the topics of almost periodicity and almost automorphy. Several new methods, including the methods of invariant subspaces and uniform spectrum, as well as various classical methods, such as fixed point theorems, are used to obtain almost periodic and almost automorphic solutions to some linear and non-linear evolution equations and dynamical systems. Almost periodicity and almost automorphy are also... more...
The theme of this unique work, the logarithmic integral, lies athwart much of twentieth century analysis. It is a thread connecting many apparently separate parts of the subject, and is a natural point at which to begin a serious study of real and complex analysis. Professor Koosis' aim is to show how, from simple ideas, one can build up an investigation that explains and clarifies many different, seemingly unrelated problems; to show, in effect, how... more...
University Calculus, Early Transcendentals, Second Edition helps readers successfully generalize and apply the key ideas of calculus through clear and precise explanations, clean design, thoughtfully chosen examples, and superior exercise sets. This text offers the right mix of basic, conceptual, and challenging exercises, along with meaningful applications. This significant revision features more examples, more mid-level exercises, more figures,... more... | 677.169 | 1 |
IMPACT Mathematics: Algebra and More, Course 3 is part of an exciting 3-course program developed in cooperation with Education Development Center, Inc. It makes mathematics accessible to more of your students. They spend less time reviewing topics from previous grades and more time progressing carefully and successfully toward the completion of Algebra 1 by the end of grade 8. Informal-to-formal concept development ensures that students build necessary skills and develop conceptual understanding. | 677.169 | 1 |
third edition illustrates the mathematical concepts that a game developer needs to develop 3D computer graphics and game engines at the professional level. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D programming such as illumination and visibility determination. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series.
The third edition expands upon topics that include projections, shadows, physics, cloth simulation, and numerical methods. All of the illustrations have been updated, and the shader code has been updated to the latest high-level shading language specifications.
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As a word of warning, do not purchase this book expecting it to teach you math fundamentals. If you do not have a background of at least algebra and trigonometry (and preferably a bit of calculus), you owe it to yourself to pick up another book and brush up on these fundamentals. While there are a few appendices covering a handful of topics, they are less about explaining the topic and more of reference pages.
Mathematics for 3D Game Programming and Computer Graphics is an excellent reference book for anyone doing 3D work. The topics are very to the point and few pages are wasted explaining basic math principles (hence the warning about having a decent math background). The book probably won't teach anyone who doesn't know they underlying principles but will be your go-to reference for any algorithm you implement.
The book starts with the reviews of the requisite vector, matrix, transformation (including rotations by quaternions) and basic geometry for a view frustum, but quickly dives into more advanced topics. Ray tracing is covered for all areas of use, from light maps to reflections. The lighting chapter covers texturing using several map types as well as lighting models with a very enjoyable discussion of specular reflection models.
Solid chapters on culling using bounding volumes and portal systems, shadowing and curve algorithms round out the first half of the book. The second half is devoted to the mathematics of physics, with chapters on basic collision detection, linear and rotational physics. The simulation of fluids and cloth (one of the more difficult physical models to accurately compute in a game) gets it's own chapter and it's a highlight for anyone implementing character clothing animation or a realistic water volume.Read more ›
Finally, no more searching through all my college math textbooks for the reference I need for real-time 3D software development. The basics of vectors and matrices are of course included, but in much more depth than you got in school, more than likely - and with emphasis on how they are useful in 3D game programming. So many game developers lack an intuitive feel for such basics as transformation matrices, dot products, and cross products and are hobbled by this; just read up to chapter three and the lights will go on, so to speak. The chapter on lighting is particularly, well, enlightening - not only are the various lighting models explained in detail (including some I was unfamiliar with before), but the author provides means for accomplishing them in real-time using texture and vertex shaders. The notation used in the book is modern and consistent, and the code samples clearly written. I believe this is the first volume to combine complete mathematical explanations of essential 3D computer graphics operations with practical advice on how to implement the sometimes complex math efficiently in real-time systems. The chapters on picking and collision detection are also complete and include practical advice on implementation in addition to the theory behind it. This is not a book for most high school math students - the author assumes you've at least been through some higher level math and can talk the basic language of mathematics. However, it does not presuppose that you are familiar with anything but basic calculus, and more importantly, it doesn't assume that you're familiar with some quirky notational system specific to the author. I haven't been in a math class for ten years, but I had no trouble understanding any concepts introduced in this book upon the first read. I don't forsee this volume leaving my desk anytime soon!
To be honest, while I find this book to be a decent reference, I find it to be pretty inaccessible in terms of sitting down and reading through it in an attempt to learn the concepts. As a non-math major (I'm actually an engineer and software developer) these math concepts are by no means beyond me. But rather than simply being presented with equation after equation, proof after proof, what I find a lot more valuable is more discussion on the usage of these equations. Specifically I'd like to see examples, diagrams, and code, and there is precious little of any of that in this book.
In other words, this book is very much like what you expect to find in a very dry upper devision college math text for the consumption of math majors who are used to such things. But for a non math major just trying to make use of these concepts in order to get the job done and make games? eh, not so much.
Still, I do think this book is useful as a reference when I want to look up an equation as there are a ton of them crammed into this book, but for me, I just don't find this book to be very good as a learning tool.
This book is just what I have been looking for: something that presents and cogently explains the math that is most useful for implementing 2d and 3d computer graphics. If the Kindle edition did not have the problems it has, I would give it 5 stars. However, it gets a poor rating for two reasons. One, the diagrams are too small! Other Kindle documents allow the reader to scale images, but not this one. Two, and this is just INEXCUSABLE: The Kindle edition, but not the print edition, has errors that make the equations and proofs worthless. I can't quote examples exactly because special characters don't show up properly, but here's a description of three examples:
Yes, this is being addressed now. We (the Publisher) will be reloading new files to Amazon's print replica program so the layout, including diagrams, notations and other pedagogy will mirror the print version. This new version should be live within a few days. Our apologies for any inconvenience this has caused. | 677.169 | 1 |
Calculus
Topics span techniques of differential and integral calculus, infinite sequences and series, the calculus of inverse functions, vectors, equations of planes and lines in space, and the differential and integral calculus of multivariable functions.
Assistance is available on a one-on-one or small group basis. Please bring all class notes and handouts with you when asking for assistance. Textbooks, calculators and solutions manuals are available in the Learning Commons.
Textbook Information
Hardcopies of this textbook are available for use at the Calculus tables in the Learning Commons.
MAC 2311, 2312, 2313
Calculus: Early Transcendentals, 7e, Stewart
Handouts
The Learning Commons offers handouts to help students understand important concepts in calculus.
Functions and Models Nonlinear Models Introduction to the Derivative Techniques of Differentiation: The Product and Quotient Rules, The Chain Rule, Derivatives of Logarithmic and Exponential Functions, Derivatives of Trigonometric Functions, Implicit Differentiation Applications of the Derivative The Integral Further Techniques and Applications of the Integral Functions of Several Variables Trigonometric Functions and Calculus Calculus Applied to Probability and Statistics
SOS Math calculus page. An advertising-supported site that has tutorials, worked examples, and practice problems with detailed solutions.
Sequences Series Limits and Continuity Differentiation Integration Techniques of Integration Local Behavior of Functions Power Series Fourier Series | 677.169 | 1 |
Wikipedia Mathematics
The free encyclopedia's entries on mathematics. A wiki is a collection of interlinked web pages, any of which can be visited and edited by anyone at any time. Many pages also available in a range of foreign languages.
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2 Pi Productions - Jake Scott
Instructional math rap includes the original songs "Triangle Experts," "Special Right Triangles," "Graphing Trig Functions," "Quadratic Formula," and "The Best Pythagorean Rap Ever." Lyrics include "The cosine is sine with a phase shift of pi/If you think
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Addison-Wesley Higher Education - Pearson Education
Publisher of programs and materials for primary and secondary school students and teachers; higher education textbooks and multimedia in major academic subjects for students and professors; books and multimedia for business professionals, practicing engineers,
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AIMMS - Paragon Decision Technology
This optimization software for advanced modeling concepts features solvers for major mathematical programming types and a full graphical user interface both for developers and end-users. Read about features and licensing options; see industrial applications,
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American Educational Research Association (AERA)
An Association concerned with improving the educational process by encouraging scholarly inquiry related to education and by promoting the dissemination and practical application of research results. Twelve divisions: Administration, Curriculum Studies,
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Avances de Investigación en Educación Matemática
The official publication of the Spanish Society for Research in Mathematics Education (SEIEM, Sociedad Española de Investigación Matemática) welcomes contributions in either Spanish or Portuguese. Freely download PDFs of past articles,
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Brainers - PedagoNet
More than 60 word problems and puzzles designed to develop logic and problem-solving skills. Submit your answers and suggestions to become part of the Wall of Fame. Site is in English, French, and Spanish.
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C.a.R. Geometry Program - Rene Grothmann
A dynamic geometry program that simulates constructions with a ruler and compass (Compasses and Ruler). C.a.R. runs on Java as a local application, with Java Web Start, or as an applet in browsers. The free and open source program includes documentation
...more>> | 677.169 | 1 |
Elementary Algebra For College Students Early Graphing
9780136134169
ISBN:
0136134165
Edition: 3 Pub Date: 2007 Publisher: Prentice Hall
Summary: Angel's texts are a proven favorite among students and instructors alike. The Angel texts consistently receive praise for their readability - short, clear sentences are used to ensure the text is readable even for those with weak reading skills- and for the abundance of detailed, worked-out examples... more than any other text! In this revised, 3rd edition of Elementary Algebra Early Graphing for College Students, An...gel continues to focus on the needs of the students taking this class and the instructors teaching them.
Angel, Allen R. is the author of Elementary Algebra For College Students Early Graphing, published 2007 under ISBN 9780136134169 and 0136134165. One hundred thirty four Elementary Algebra For College Students Early Graphing textbooks are available for sale on ValoreBooks.com, eight used from the cheapest price of $41.94, or buy new starting at $201.74134169-4-1-3 Orders ship the same or next business day. Expedited [more]
Missing components. May include moderately worn cover, writing, markings or slight discoloration. SKU:97801361341690136134176013613417636134176-2-0-1 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.[less] | 677.169 | 1 |
Beginning Algebra
Beginning Algebra
This is a free online course offered by the Saylor Foundation.'...
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This is a free online course offered by the Saylor Foundation.
' exposed to. This course is also intended to provide you with a strong foundation for intermediate algebra and beyond.
This course will begin with a review of some math concepts formed in pre-algebra, such as order of operations and simplifying simple algebraic expressions to get your feet wet. You will then build on these concepts by learning more about functions, graphing of functions, evaluation of functions, and factorization. You will spend time on the rules of exponents and their applications in distribution of multiplication over addition/subtraction Algebra to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Beginning Algebra
Select this link to open drop down to add material Beginning Algebra to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Algebra, Trigonometry, and StatisticsA Tour of the Calculus
Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these things possible and in doing so finds a correspondence between real numbers and the real world is the subject of this dazzling book by a writer of extraordinary clarity and stylistic brioMathematics: A Very Short Introduction
The aim of this audiobook is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and listeners of this audiobook will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought Mathematics Is Power
William Goldbloom Bloch is a respected professor of mathematics at Wheaton College. This intriguing lecture series, Mathematics Is Power, delves into both the history of mathematics and its impact on people's everyday lives from a non-mathematician's perspective. Bloch first examines the history of mathematics and age-old questions pertaining to logic, truth, and paradoxes. Moving on to a discussion of how mathematics impacts the modern world, Bloch also explores abstract permutations such as game theory, cryptography, and voting theory.
The King of Infinite Space: Euclid and His Elements
Geometry defines the world around us, helping us make sense of everything from architecture to military science to fashion. And for over 2,000 years, geometry has been equated with Euclid's Elements, arguably the most influential book in the history of mathematics. In The King of Infinite Space, renowned mathematics writer David Berlinski provides a concise homage to this elusive mathematician and his staggering achievements.
Math is one of those subjects that, if not explained correctly from an early age, can cause anxiety and even boredom for many of its learners. Without the right mindset about its usefulness, a positive attitude and a willingness to try, students quickly feel disconnected from this important field of study. This guide intends to provide any math learner the mental tools they need to successfully tackle any mathematical challenge Build a Computer (For Beginners): Eighth Edition
The audiobook, How to Build a Computer (For Beginners), outlines step-by-step instructions on how to build a computer from the bare bones to the finished product. The book includes a list of choices for tools and supplies necessary to successfully build a computer on your own; using the author's educated knowledge and background on computer software and hardwareZombies and Calculus
Zombies and Calculus is the account of Craig Williams, a math professor at a small liberal arts college in New England, who, in the middle of a calculus class, finds himself suddenly confronted by a late-arriving student whose hunger is not for knowledge. As the zombie virus spreads and civilization crumbles, Williams uses calculus to help his small band of survivors defeat the hordes of the undead.
Out of the Labyrinth: Setting Mathematics Free
Who hasn't feared the math Minotaur in its labyrinth of abstractions? Now, in Out of the Labyrinth, Robert and Ellen Kaplan - the founders of The Math Circle, the popular learning program begun at Harvard in 1994 - reveal the secrets behind their highly successful approach, leading listeners out of the labyrinth and into the joyous embrace of mathematics.
Publisher's SummaryFor the price, it is great. It might should be longer or more in depth but then it would cost more.
How would you have changed the story to make it more enjoyable?
It is a little dry but there are not many math books on here as of now.
What three words best describe James Powers's voice?
Straight-forward, easy, fast.
Do you think Algebra, Trigonometry, and Statistics needs a follow-up book? Why or why not?
Sure, hopefully as cheap again.
Any additional comments?
It seems to be the perfect book to throw on when I cannot decide which book to do next. I do not really care if I absorb it through each time; repeated usage will do the trick. Quick and painless but overall helpful.
to not talk through problems like one should already know or understand it. If one did already understand, to talk through each step. A narrator with more energy, less boring voice. Sounds like he is dying | 677.169 | 1 |
...
Show More aspects of the modern theory to the point where they will be equipped to read advanced treatises and research papers. This book includes many more exercises than the first edition, offers a new chapter on pseudodifferential operators, and contains additional material throughout.The first five chapters of the book deal with classical theory: first-order equations, local existence theorems, and an extensive discussion of the fundamental differential equations of mathematical physics. The techniques of modern analysis, such as distributions and Hilbert spaces, are used wherever appropriate to illuminate these long-studied topics. The last three chapters introduce the modern theory: Sobolev spaces, elliptic boundary value problems, and pseudodifferential operators | 677.169 | 1 |
Access
Available online
More options
Contents/Summary
Bibliography
Includes bibliographical references (page 87).
Publisher's Summary
Write On! Math is a programme that offers specific strategies and projects designed to keep your students engaged during maths class, strengthen their mathematics, and teach them technical writing skills. Write On! Math is a programme that will teach students systematically how to take better notes in maths class. Total concentration is a prerequisite to learning how to take better notes. Therefore, a by-product of taking better notes is staying focused in class. Possibly, as a teacher, you at one time remarked to a colleague how you understood something better (or even for the first time!) when you had to teach it. There is no better way to ensure you know something well than to have to teach it to somebody else. The Write On! Math programme requires students to do exactly that--that is why it improves their mathematics as well as teaches them a valuable writing technique not taught in English class. Write On! Math will improve the way you present material to your students in class and on your handouts. (source: Nielsen Book Data) | 677.169 | 1 |
Best math tool for school and college! If you are a student, it will helps you to learn! Note: In linear algebra, the determinant is a value associated with a square matrix. The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space | 677.169 | 1 |
This eBook introduces the student to number patterns and sequences, including odd and even numbers, square numbers, square roots, cube numbers, cube roots, factors, prime numbers, multiples, linear sequences, square number and cube number sequences, Fibonacci number sequences, triangular number sequences and sequences of the powers of 2, 3, 4, 5 and 10.
This eBook introduces the subject of algebra to the student encompassing, inverse operators, equations, the order of precedence, algebraic conventions, BODMAS, expressions, formulae, factorising, rearranging and solving linear, quadratic and simultaneous equations as well as inequalities.
This eBook introduces the significant scientific notation of the very large, the intermediate and the very small in terms of numbers and algebra through an exploration of indices, the rules of indices and standard index form.
Decimals considers the significance of the position of the decimal point, compares & converts 'fractions-decimals-percentages', considers 'multiplying & dividing' decimals by 10 and 100, rounds decimals to the nearest whole number, tenth, or hundredth, considers 'less than', 'greater than' and '=' signs in arithmetic and walks the student through the addition and subtraction methods with decimalsThis eBook introduces the subject of measures and measurement, and looks at both metric and imperial units of measurement, the process and accuracy of reading scales, limits on the accuracy of measurements and compound measurements.
This eBook introduces the related subjects of Pythagoras' theorem, trigonometry and similarity, as Pythagoras' theorem relates to all right-angles triangles, trigonometry as it relates to angles and ratios of sine, cosine and tangent in right-angled triangles, angles of elevation and depression as well as similarity and congruence.
This eBook introduces the subject of transformation as it relates to translations, reflections, rotations and enlargements either as individual operations or composite operations. In this eBook we illustrate each of these translations using right-angled triangles, but the principles developed extend to all 2D shapes as well as to 3D shapes using extensions.
This eBook introduces the subjects of angles, bearings and scale drawings. To angles as it relates to angular turn about a point, angles in polygons, angle facts and inter-relational aspects with parallel as well as crossing lines, to bearings as they relate to navigation and scale drawings as an aspect of technical drawing.
Loci, Constructions and 3D Co-ordinates is an introductory text on loci and their characteristics, constructing triangles, the bisector of a line and the perpendicular bisector of a line as well as using 3D co-ordinates
This eBook introduces 2D planar and 3D solids (shapes) and their corners (vertices), faces, edges, lines and planes of symmetry and order of rotational symmetry. It introduces the student to regular and irregular polygons, quadrilaterals, triangles, circles, nets of solids, cuboids, prisms and cylinders including formulating the algebra that describes their various areas, perimeters and volumes.
Mathematics Principles V10 – now comes with an interactive Tablet and Smartphone App.
See Additional Notes at the back of the book for instructions to download the accompanying interactive App which brings the 250+ topics to life by allowing you to insert your own values. Visually on a phone or tablet it looks almost identical to the eBook.
EasyA Maths Edition: Higher Level is a complete summary of the maths higher level Junior Certificate exam. It contains all the information that students need to do well in their Junior Certificate exam. The information is condensed and easier to understand and it saves students time as they dont have to search a maths books for the information they want to know.This book is for you if you would like to be involved in your school-going kid's math education and need to get your own basics right, or if you decided to extend your education, may be involving some computer programming, or statistics and want to be up to speed in junior high school math before taking the next step. This is not on teaching techniques.
The Practically Cheating Statistics Handbook TI-83 Companion Guide picks up where the the Practically Cheating Statistics Handbook left off, by giving students simple, step-by-step instructions for solving the most common statistics problems with the TI-83 calculator. The guide walks you through each problem type, telling you exactly what buttons to press without leaving out any details!
Math is a special and important learning in education. Even though Math is hard to some people, it is not hard to learn if you follow a good guide. This book is a good guide that will help high/middle school students learn basic and advanced skills with important concepts and skills carefully designed into questions and solution for students to master. This book will escort you to your success. | 677.169 | 1 |
Summary
This intermediate algebra text, based on standards in the AMATYC Crossroads document, motivates college math students to develop mathematical literacy and a solid foundation for future study in mathematics and other disciplines. This third book of a three-book series presents mathematical concepts and skills through relevant activities derived from real-life situations. These activities are meaningful to students because they illustrate how mathematics arises naturally from real-world situations and problems. The Mathematics in Action series is based on the assumption that students learn mathematics best by doing mathematics in a meaningful context. Therefore, the text takes a collaborative approach to learning. Students take an active role in their own learning by working in groups, thereby developing communication skills, a sense of independence, and a can-do attitude about mathematics. Technology is integrated throughout the book so that students learn to interpret real-life data numerically, symbolically, and graphically. Regardless of their level of preparation for the course, students can use this text to increase their knowledge of mathematics, their problem-solving skills, and their overall confidence in their ability to learn.
Table of Contents
Preface
xiii
Function Sense
1
(180)
Modeling with Functions
1
(42)
Parking Problems
1
(12)
Distinguish between input and output
Define a function
Represent a function numerically and graphically
Write a function using function notation
Fill 'er Up
13
(10)
Objectives: Determine the equation (symbolic representation) that defines a function
Write the equation to define a function
Determine the domain and range of a function
Identify the independent and the dependent variables of a function
Stopping Short
23
(9)
Use a function as a mathematical model
Determine when a function is increasing, decreasing, or constant
Use the vertical line test to determine if a graph represents a function
Graphs Tell Stories
32
(4)
Describe in words what a graph tells you about a given situation
Sketch a graph that best represents the situation described in words
What Have I Learned?
36
(1)
How Can I Practice?
37
(6)
Linear Functions
43
(53)
Walking for Fitness
43
(8)
Determine the average rate of change
Depreciation
51
(12)
Interpret slope as an average rate of change
Use the formula to determine slope
Discover the practical meaning of vertical and horizontal intercepts
Develop the slope/intercept form of an equation of a line
Use the slope/intercept formula to determine vertical and horizontal intercepts
A Visit to the Architect
63
(10)
Determine the slope-intercept form of a line given the slope and vertical intercept
Determine the slope-intercept form of a line given two points on the line
Compare the slopes of parallel lines
Skateboard Heaven
73
(9)
Write an equation of a line in general form Ax + By = C
Write the slope-intercept form of a linear equation given the general form
College Tuition
82
(6)
Determine a line of best fit with a straightedge
Determine the equation of a regression line using a graphing calculator | 677.169 | 1 |
What is Math Bridge Academy?
Mathematics Bridge Academy is a three week program designed to help junior high and high school students fill in gaps in math knowledge and accelerate to new math skills.
What should I expect from the academy?
Pre-testing to discover gaps and target areas for acceleration
Online software access with practice tailored to each student's needs Exit-testing to assess progress Instructors with teaching experience at BOTH high school level and at Brazosport College Tutors available to help students one-on-one Activities to build critical thinking skills needed for college and career success Campus tours and information on opportunities for early college credit
Career presentation from industry partners and faculty and staff
Who can participate?
In the summer of 2015, Math Bridge Academy will be open to any student entering, exiting, or needing additional practice in Algebra I. All sections cover the same math so students simply choose the dates and times that work best for them, regardless of current grade or math level.
What does it cost?
In the summer of 2015, Math Bridge Academy will have a $41 fee that students pay at registration. This fee covers all supplies and 16 weeks of access to online software that students will use during the academy and for further practice at home.
All participants will be required to turn in a signed copy of the Code of Conduct by the end of the first week of their academy. You may choose to print from the link below, or receive a copy on the first day | 677.169 | 1 |
● ArizMATYC Sessions Summary
Attanucci, Frank J. A note on the proportional partitioning of line segments, triangles and tetrahedra[View Presentation]
In the first part of this paper I solve the following problem: Where can one place a point G inside or on a triangle so that line segments from G to each of the vertices divide the triangle into three sub triangles whose areas A1, A2 and A3, respectively, satisfy the proportion: A1:A2:A3 = w1:w2:w3, where the wi's are non-negative constants with positive sum? I then state and prove an analogous result for tetrahedra. I finish with a theorem concerning the centroid of n! points. Along the way, everything is made more intriguing by allowing the wi's to be parameterized functions.
Beaudrie, Brian Improving the mathematical readiness of middle-achieving, college-bound students[View Presentation]
Based on more than six years of research, this presentation will discuss a practical two-tiered strategy designed to help college-bound students be better prepared for credit-bearing college mathematics courses before beginning their post-secondary education.
Dudley, Anne & Watkins, Laura Active math
The presenters will share a variety of activities to engage students in active math. These activities will require students to get up and move to help them retain the math. The presenters will share activities from many levels of mathematics and invite attendees to share their favorite activities as well.
Dumitrascu, Gabriela Generalization in mathematics from elementary school to college
The program will be divided into two parts. First, I will use a power point presentation format to describe a way to define the practice of generalization in mathematics. The presentation will include a theoretical definition and examples from textbooks how this definition may be used to organize mathematics instruction from elementary school level to high school and college levels. In the second part of the program, the participants will discuss how the practice of generalization is reflected in the Common Core State Standards for Mathematics.
Hughes-Hallett, Deborah & Lozano, Guadalupe Mathematics and sustainability
The emphasis in the Common Core Curriculum Standards on mathematical structure and modeling, and the worldwide concern with sustainability, can be combined into engaging materials for use in pre-calculus and calculus, as well as in quantitative reasoning and college algebra.
Knapp, Jessica Teaching foundations: new courses to prepare future teachers
A statewide Project NEXT grant has developed a set of course materials to improve the mathematical content knowledge for pre-service elementary school teachers in Arizona. We will discuss the rubric of higher order thinking developed by the Teaching Foundations faculty to serve as a guide in the course preparation as well as present some of the courses and course materials. We will explain the courses which have been developed and discuss some of the early results from the piloters. We will also give examples of the course materials being used and compare them to recommendations made by other mathematics education research projects.
Mayo, Tim Lucky Larry and lines of verse
A sampling of "Lucky Larry" problems will be worked out live, with humorous anecdotes and poetry lines used by the instructor in his classes. The pedagogical merits of these problems will be discussed. Attendees will have the opportunity to present "Lucky Larry" problems they have encountered.
McCallum, William The Illustrative Mathematics Project
The Illustrative Mathematics Project is building a community of mathematicians, educators, and teachers dedicated to designing high quality tasks to illustrate the Common Core State Standards in Mathematics. The core of the project is an interactive website where tasks are submitted, reviewed, edited and published by the community.
Mendel, Marilou WeBWorK demonstration[View Presentation]
This session will provide attendees with an overview of WeBWorK, an open-source online homework system. WeBWorK is hosted by the MAA and is accompanied by a National Problem Library that consists of more than 20,000 homework problems. Students are provided with individualized problems, immediate feedback on the correctness of their responses, and an opportunity to make repeated attempts to obtain the correct answers. Instructors are provided with automatic grading of assignments and detailed statistical information about the performance of the students. We will examine the student and instructor interfaces, explore the National Problem Library, and discuss how to obtain WeBWorK.
Schettler, Jordan & Charles Collingwood Occupy Calculus! The math behind the 99%[View Presentation]
We will explore Lorenz curves for income and explain how tools from integral and differential calculus can be used to analyze trends in the unequal distribution of wealth. We will discuss a powerful student project which helps build analytical skills and social consciousness.
Welch, Eric Unlocking the secrets of counting problems
Participants will solve problems suitable for 4th through 8th grade students that clarify the overarching concepts and elicit the habits of mind used to solve systematic listing and counting problems. The concepts range from the addition principle of counting to the meaning behind the formulas for permutations and combinations. Bring your willingness to change hats, approaching problems like a middle grades student while thinking about instructional moves like a middle school teacher. | 677.169 | 1 |
Mathematics: Applied eBooks
For every student who has asked their math teacher, "When will I ever use this?" applied mathematics is the answer ... as long as they choose a career that involves it. If you're looking for a mathematics eBook that can be used in real-world situations, this is the place.
There are over 200 eBooks in the category Mathematics: Applied. Use our eBook search to find a specific book or author. | 677.169 | 1 |
from 100 providers and counting...
Sequences and Series will challenge us to think very carefully about "infinity." What does it mean to add up an unending list of numbers? How can an infinite task result in a finite answer? These questions lead us to some very deep concepts—but also to some powerful computational tools which are used not only in math but in many quantitative disciplines.
This is a follow-on for the Coursera class "Principles of Functional Programming in Scala", which so far had more than 100'000 inscriptions over two iterations of the course, with some of the highest completion rates of any massive open online course worldwide.
In this course, you will learn all
of the major principles of microeconomics normally taught in a quarter or
semester course to college undergraduates or MBA students.
Perhaps more importantly, you
will also learn how to apply these principles to a wide variety of real
world situations in both your personal and professional lives. In this way, the Power of Microeconomics will
help you prosper in an increasingly competitive environment. | 677.169 | 1 |
WolframAlpha
If you haven't tried WolframAlpha, you really need to head over there immediately after reading this post.
The great thing about this site is its ease of finding computational data. Do you have an algebraic equation you need solved? Type it in the search box and watch the magic happen. Ask for the population of France and get both the number and a chart of population growth for the last 30 years. Just for fun, type in the following formula:
Taylor series of sin^3(x)
No, I don't know what it means either, but Wolfram Alpha does!
Here is the question. Would your math students benefit from using this site? Maybe not if you have a typical math class. If you expect your kids to sit down and do their homework in the confines of their room armed only with a calculator and their wits, you aren't paying attention. Kids today sit down with their laptop, mobile phone, calculator, iPod, and television all at the ready. They are connected. They are social.
So maybe to use this site you might have to rethink what you want from your students for their homework. Maybe you want them to find the answer here and then explain in class how the answer was solved. Yes, they will get step-by-step instructions for algebra problems. Here is an example of a linear equation I just made up:
If you assign a problem and then have the students discuss how it is solved so that they can teach others in the class, then Wolfram Alpha would be a great resource. It would ensure that the student is getting immediate, positive, correct feedback on how to solve problems.
Of course, there are a ton of other ways to use Wolfram Alpha. You just need to check them out for yourself. OK, you don't have to go all the way over to the Wolfram Alpha site. You can search right here and see for yourself.
One Comment;
I agree with you wholeheartedly… WolframAlpha is an AMAZING tool. I am currently working on it to submit science content (on an elementary level)… I would love some feedback as to what questions and information you would like to see on the site/project! You can also email Robert with information too!! (you must love mathforum too??!!! Am I right?? | 677.169 | 1 |
Introduction to the Geometry of Complex Numbers
by Roland Deaux Publisher Comments
Geared toward readers unfamiliar with complex numbers, this text explains how to solve the kinds of problems that frequently arise in the applied sciences, especially electrical studies. To assure an easy and complete understanding, it develops topics... (read more)
Modern Geometry / With CD (02 Edition)
by David A. Thomas Book News Annotation
A textbook for graduate and undergraduate mathematics education
students who will teach junior and senior high school mathematics. It
introduces both Euclidean and non-Euclidean geometry in a manner
consistent with the recommendations of the National... (read more)
Topos Theory (Dover Books on Mathematics)
by P. T. Johnstone. Publisher Comments
One of the best books on a relatively new branch of mathematics, this text is the work of a leading authority in the field of topos theory. Suitable for advanced undergraduates and graduate students of mathematics, the treatment focuses on how topos... (read more)
Convex Polyhedra (Springer Monographs in Mathematics)
by A. D. Alexandrov Publisher Comments
This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems... (read more)
Analytic Geometry (7TH 92 Edition)
by Gordon B. Fuller Publisher Comments
Tailored for a first course in the study of analytic geometry, the text emphasizes the essential elements of the subject and stresses the concepts needed in calculus. This new edition was revised to present the subject in a modern, updated manner. Color... (read more)
Analytical Conics
by Barry Spain Publisher Comments
This concise text introduces students to analytical geometry, covering basic ideas and methods. Readily intelligible to any student with a sound mathematical background, it is designed both for undergraduates and for math majors. It will prove... (read more)
Elementary Geometry for College Students (5TH 11 - Old Edition)
by Daniel C. Alexander Publisher Comments... (read more)
Lectures on Analytic and Projective Geom. (11 Edition)
by Dirk J. Struik Publisher Comments
Based on a historic approach taken by instructors at MIT, this text is geared toward junior and senior undergraduate courses in analytic and projective geometry. Starting with concepts concerning points on a line and lines through a point, it proceeds to... (read more)
Geometry
by David A Brannan Publisher Comments
This richly illustrated and clearly written undergraduate textbook captures the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of the space. The... (read more)
Research Problems in Discrete Geometry
by Peter Brass Publisher Comments
Although discrete geometry has a rich history extending more than 150 years, it abounds in open problems that even a high-school student can understand and appreciate. Some of these problems are notoriously difficult and are intimately related to deep | 677.169 | 1 |
The ANL is a discussion group. You can read and
post messages concerning homework problems anytime.
In addition a TA will be available at scheduled hours.
Hours will be posted shortly
What is the ANL:
The ANL is a discussion group and is another way
for you to get help with homework problems. It mainly offers you an opportunity to
help fellow students.
How to use the ANL:
Click the "discuss" button on your
homework set. This will take you to ANL. Its a good idea to always check the ANL,
especially if you solved a problem and might be able to help other students.
What can be posted and what
not:
In principle you are free to post any information you think
might be helpful to fellow students for solving their problems. We will try to
be as tolerant as possible, but will delete inappropriate messages or wrong
postings frequently to make the ALN as useful as possible. In particular we may
delete some messages that just give away the answer (this pertains to certain
types of problems) and reduce the learning effect to zero. As in every group you
should help others learn, not do the work for them. | 677.169 | 1 |
Polynomial Addition and Subtraction
iOS iPadEducation
With the Polynomial Addition and Subtraction app you can practice adding and subtracting polynomials with a step-by-step method. And you can get additional help at each step.
The app saves your most recent problems and allows you to play back previously solved problems. You can play back problems step by step or with an animation. You can stop the playback at any time and continue to solve the problem yourself.
The app has several settings that allow you to set the difficulty to suit your needs. You can either pick answers for each step from multiple choices or you can type in the answers yourself. The former method is a great way to start using the app and then later you can start to type in your answers. Eventually you can hide the current subtraction or addition step and solve the problems without any help.
You can have whole number and fraction coefficients and you can control how hard random problems the app generates. And you have complete freedom to set your own problems with up to forth-degree polynomials.
When you type in your answers you can use a custom keyboard that enables you to enter variables and exponents with just one tap. Each keyboard in the app also contains a built-in calculator.
For several types of steps the app contains help balloons that break the steps into even smaller sub steps that help you solve the steps. Just tap the "Help me" button to open a balloon that will guide you to enter the correct answer.
This app is part of the iDevBooks math app series. Other algebra apps in the series include Polynomial Long Division and Polynomial Multiplication.
Other iDevBooks math apps
iDevBooks math apps have been reviewed and endorsed by Wired.com, IEAR.org, Edudemic.com, Teachers with Apps, and other respected sites and organizations.
New ideas to make this app better are welcome. Please visit idevbooks.com to give feedback.
Privacy
This app has no ads or in-app purchases and it does not transmit any data during the operation of the app. This app also does not contain any links to other apps or the web.
What's New
- When selecting the method for solving each step, the "Type in answers" option did not always work. - Sometimes the app did not detect that the iPad was rotated into landscape orientation. - Interface updates for iOS 7.
If you have any feedback please visit idevbooks.com. There are now 24 iDevBooks math apps. | 677.169 | 1 |
few introductory comments are in order: (1) This is *not* intended to be a first look at the subject of linear algebra, at least from the "computational side". (2) This is an undergraduate level text, though typically students will not encounter this material before their junior or senior years. (3) There is some overlap with a graduate level course in linear algebra, though this book is not comprehensive enough for a course at that level.
Ok, now that we've gotten that out of the way...
We used this as the primary textbook as a cross-listed advanced undergraduate/beginning graduate course I took in linear algebra. I had to supplement this book with outside reading/assignments to fulfill the balance of the course requirements. Contrary to what you might expect, you do not need an "introductory linear algebra course" (read that as "linear algebra for engineers") to successfully navigate this book. Actually, much (not all) of the material covered in this book should be discussed in any decent undergraduate course in ordinary differential equations (Boyce & DiPrima's ODE text makes a decent reference).
Here, you'll find that the emphasis is on learning the theoretical side of linear algebra. While there is a chapter (Chapter 3) on basic matrix algebra (wholly unnecessary in my opinion), the main use of matrices here is to express linear operators in a form more suited for computations, e.g., the determination of eigenvalues and eigenvectors. Right away, in Chapter 1, vector spaces are introduced and many familiar (some unfamiliar) examples are given. Just as in an abstract algebra course, you define a list of axioms for vector spaces (later, inner product spaces) and see what you can do with them...quite a lot, as it turns out!Read more ›
For reference, I have done only a few of the problems and read no other books on Linear Algebra.
That aside, I can still attest that this is a good book. The proofs throughout are short, straightforward, and remarkably free of even trivial errors. The organization is sensible (at least from a theoretical perspective), and any definitions are generally introduced when the motivation has been established. The book is best geared for a math major, but I think the clarity is good enough to make it suitable for physics and engineering majors as well. To keep the book lively there are some well-developed examples in linear diffeq, economics, and einstein's relativity among others. These extra sections can ofcourse be skipped without loss of continuity. As per the problems, they are mostly of a trivial nature (dealing with concrete numbers) with a couple of intermediate proofs towards the end of each section.
My only gripe is that the authors take little initiative to ascribe geometric interpretations to results whenever possible; especially in the chapter on inner products. Frankly, it's easier to remember pictures then verbose thereoms.
If you do plan to read the book, I would recommend two semesters of calculus and possibly a preliminary course in abstract mathematics (sets and proofs).
This book blew me away! it's just great, it really explains everything (also an axiomatic approach of determinants which is very important to gain a thorough understanding of what determinants actually mean and which will help you when you are going to study multilinear algebra, exterior algebras in abstract algebra etc), the book gives you a good insight in the stucture of linear operators on a finite-dimensional vector space and provides lots of examples and useful applications (e.g. in economics and physics). There was a comment of one customer which criticized the fact that the theory doesn't offer an explanation of quotient spaces, this is true but in fact this is not important in the area of linear algebra, quotient structures should be studied in abstract algebra, when more algebraic structure has been developed so that one can really understand what quotients of algebraic structures are about. So the book is great, but I would recommend some knowledge about polynomials, fields, algebraic closure, vector space before starting to read this one.
I used the 3rd Ed. in UC Berkeley's MATH 110: Linear Algebra, then used the 4th while grading homework for the same class the next year. I think the book is fairly comprehensive (though by itself not enough to prepare one for grad school), and very well-written. The exercises at the end of each section span a wide range of difficulty. The book is self-contained, except for a few basic results from the calculus (one has to know the linearity properties of derivatives and definite integrals, i.e. derivative of linear combination is linear combination of derivatives and similarly for integrals), yet does sort of assume prior knowledge of linear algebra. At UC Berkeley students have already taken MATH 54: Linear Algebra & Differential Equations, which includes a brief treatment of vector spaces, linear transformations, eigenvalues, etc. I wouldn't say this book is "not for the faint of heart," as some reviewers put it. I think it's ideally suited--essential, in fact--for entering juniors majoring in the any of the mathematical sciences. If this book is your first exposure to linear algebra, then I highly, HIGHLY recommend chapters 12 and 13 of Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition), and chapters 1-5 of Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications. | 677.169 | 1 |
books.google.com - Int.... Algebra for College Students
Introductory Algebra for College Students
Int.
About the author (1997)
Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from the City College of New York. His unusual combination of academic interests led him toward a Master of Arts in mathematics from the University of Miami and a doctorate in behavioral sciences from Nova University. Bob is most energized by teaching mathematics and has taught a variety of mathematics courses at Miami-Dade College for nearly 30 years. He has received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College, and was among the first group of recipients at Miami-Dade College for an endowed chair based on excellence in the classroom. Bob has written "Intermediate Algebra for College Students, Introductory Algebra for College Students, Essentials of Intermediate Algebra for College Students, Introductory and Intermediate Algebra for College Students, Essentials of Introductory and Intermediate Algebra for College Students, Algebra for College Students, Thinking Mathematically, College Algebra, Algebra and Trigonometry," and "Precalculus," all published by Pearson Prentice Hall. | 677.169 | 1 |
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About the book:
A comprehensive handbook for scientists, engineers, and advanced students, this book presents--in a lucid and accessible form--classical areas of mathematics like algebra, geometry, and analysis and also areas of current interest. The handbook includes over 450 graphs, figures and illustrations. There is an extensive, thoroughly cross-referenced index which lists over 1,400 terms. 450 illustrations.
Hardcover, ISBN 081763858X Publisher: Birkhäuser Boston, 1995 Brand New, Unread Copy in Perfect Condition. A+ Customer Service!
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Better World Books via United States
Hardcover, ISBN 081763858X Publisher: Birkhäuser Boston1763858X Publisher: Birkhäuser Boston1763858X Publisher: Birkhäuser Boston1763858X Publisher: Birkhäuser Boston Birkhäuser Boston1763858X Publisher: Birkhäuser Boston, 1995 Used - Good, Usually ships in 1-2 business days, Good clean copy with no missing pages might be an ex library copy; may contain some notes and or highlighting, 1995 Used - Very Good. Former Library book. Great condition for a used book! Minimal wear. Shipped to over one million happy customers. Your purchase benefits world literacy!
Hardcover, ISBN 081763858X Publisher: Birkhäuser Boston, 1995 Used. This Book is in Good Condition. Clean Copy With Light Amount of Wear. 100% Guaranteed. | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
'In writing Elementary Algebra, John Redden had simple but important aims:Lay a solid foundation in mathematics through...
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'In writing Elementary Algebra, John Redden had simple but important aims:Lay a solid foundation in mathematics through algebra, the basis of all mathematical modeling used in a variety of disciplinesGuide students from the basics to more advanced techniques in mathematics PEDAGOGICAL FEATURES:REAL WORLD APPLICATIONS:With diverse exercise sets, students have the opportunity to solve plenty of practice problems. Elementary Algebra has applications incorporated into each and every exercise set. John makes use of the classic "translating English sentences into mathematical statements" subsections in chapter 1 and as the text introduces new key terms.VIDEO EXAMPLES AND ACTIVITIES:Embedded video examples are present, while the importance of practice with pencil and paper is still stressed. This text respects the traditional approaches to algebra instruction while enhancing it with today's technology.OPEN AND MODULAR FORMAT:While algebra is one of the most diversely applied subjects, students find it to be a difficult hurdle in their education. With this in mind, John Redden wrote Elementary Algebra in a format that allows instructors to modify it and leverage their individual expertise and maximize student experience and success.MORE VALUE AT AN AFFORDABLE PRICE:Using the online text in conjunction with a printed version of the text could promote greater understanding (at a lower cost than most algebra texts). to your Bookmark Collection or Course ePortfolio
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'College Algebra is an introductory text for a college algebra survey course. The material is presented at a level intended...
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'College Algebra is an introductory text for a college algebra survey course. The material is presented at a level intended to prepare students for Calculus while also giving them relevant mathematical skills that can be used in other classes. The authors describe their approach as "Functions First," believing introducing functions first will help students understand new concepts more completely.Each section includes homework exercises, and the answers to most computational questions are included in the text (discussion questions are open-ended). Graphing calculators are used sparingly and only as a tool to enhance the Mathematics, not to replace it Algebra to your Bookmark Collection or Course ePortfolio
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'Beginning and Intermediate Algebra was designed to reduce textbook costs to students while not reducing the quality of...
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'Beginning and Intermediate Algebra was designed to reduce textbook costs to students while not reducing the quality of materials. This text includes many detailed examples for each section along with several problems for students to practice and master concepts. Complete answers are included for students to check work and receive immediate feedback on their progress. Topics covered include: pre-algebra review, solving linear equations, graphing linear equations, inequalities, systems of linear equations, polynomials, factoring, rational expressions and equations, radicals, quadratics, and functions including exponential, logarithmic and trigonometric. Each lesson also includes a World View Note which describes how the lesson fits into math history and into the world, including China, Russia, Central America, Persia, Ancient Babylon (present day Iraq) and more and Elementary Algebra to your Bookmark Collection or Course ePortfolio
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'Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems...
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'Algebra is one of the main branches in mathematics. The book series of elementary algebra exercises includes useful problems in most topics in basic algebra. The problems have a wide variation in difficulty, which is indicated by the number of stars Exercise Book I to your Bookmark Collection or Course ePortfolio
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A text suitable for a Basic Algebra developmental course based on the representation of real world situations and on an...
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A text suitable for a Basic Algebra developmental course based on the representation of real world situations and on an appeal to reason as opposed to the recall of memorized "mathematical facts" and "skills״.Reasonable Basic Algebra, moreover, is a standalone version of part of a three semester course of study to start with Arithmetic and to culminate with Differential Calculusasonable Basic Algebra to your Bookmark Collection or Course ePortfolio
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The text is suitable for high-school Algebra 1, as a refresher for college students who need help preparing for college-level...
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The text is suitable for high-school Algebra 1, as a refresher for college students who need help preparing for college-level mathematics, or for anyone who wants to learn introductory algebra such as homeschoolers. Available on web or download as PDF or Kindle version. It is free to use online or download for a single printing. It is not generally openly licensed Algebra to your Bookmark Collection or Course ePortfolio
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Technology in the Upper-Level Curriculum - Geometry
Always ready to try technology, I decided to teach DePauw's junior-level geometry course with a variety of tools: Geometer's Sketchpad (Jackiw, 1992-95 ), the LénártSphere (Lénárt, undated), and PoincaréDraw (The Gap Group, 2000 -- written by Robert Foote and Nathan of Wabash College to demonstrate the Poincaré disk model of hyperbolic geometry). I continually felt challenged and stimulated by these students -- I am pleased that atmosphere has been repeated during several subsequent semesters. The question I have asked myself about this course was, "Why did we all feel so energized by this course? Why did the software do so much for the students?"
My students had all studied Euclidean geometry thoroughly in high school. They had been taught how to write proofs for geometry, and it was easy for me to show them how to transform their 2-column proofs into paragraph form. I then sent them to the computer lab to see what Geometer's Sketchpad would do. What was truly exciting for them was the enhanced capacity for visualization that came from the dynamic software. This became clear when I insisted that they work out illustrations for one problem set by hand, with a straightedge and compass. They struggled with the hand sketches and invariably missed the "special cases" that made refinements of the conjectures necessary. Later, I had them work with the Lénárt Sphere, and they fairly quickly recreated some of the crisis of 19th century geometry. Finally, they learned about hyperbolic geometry and appreciated how much PoincaréDraw helped in visualizing a "different" geometry. In the lab "Inscribed and Circumscribed Circles" students work through the same constructions in all three geometries and compare the results. [For the original Scientific Notebook file, click on the icon at the right.]
Our modern students do not have much experience with visualization in earlier courses -- in either two or three dimensions -- and this limits their abilities to think about geometry. The dynamic software packages (Geometer's Sketchpad and PoincaréDraw) enhanced their visualization and allowed them to "think gometrically." They had already understood the basic concepts and had been taught to formalize these concepts. But the new capacity to visualize, provided by the software, opened up creative avenues for them. As one student said at the end of the semester during a taped discussion, the course showed them the "wild side of math." More than in abstract algebra or real analysis, the students understood and were stimulated by the potential for mathematical exploration. | 677.169 | 1 |
National 5 Mathematics
This Course is valid from August 2013.
The National 5 Mathematics Course enables learners to select and apply mathematical techniques in a variety of mathematical and real-life situations. Learners interpret, communicate and manage information in mathematical form.
Subject updates
Notification of changes
01-JUN-2014
National 4 Mathematics Unit assessment support packs
Selected National 4 Mathematics Unit assessment support packs, including the National 4 Mathematics Added Value Unit Test, have been revised. Most of the revisions have been to clarify marking guidance, but some questions have been amended. The Added Value Test has been amended, in light of centre feedback, to make it less predictable. These assessments are now available on the secure website.
Unit Assessment Support
February 2014
As a result of feedback, the Mathematics Added Value Unit Test was amended in February 2014 to include the option for assessors to apply an overall threshold of 26 out of 43 when making a judgement on candidate performance. The existing guidance on judging operational and reasoning skills separately is also still a valid approach. The National 4 Lifeskills Mathematics Added Value Test is unchanged.
Additional support materials have been published in February 2014 for Outcome 2. A set of questions for the assessment/reassessment of Outcome 2 (reasoning) in National 4 Expressions and Formulae and RelationshipsIn response to feedback from teachers and lecturers, we made some additions, updates and minor amendments to the Unit assessment support packs for National 4 and National 5 Mathematics in October 2013.
These updates are to provide clarification and guidance for teachers/lecturers; particularly across the 'Judging evidence' tables, however the Outcomes and Assessment Standards remain unchanged. The Unit assessment support packs are to assist teachers/lecturers in using their own professional judgement when planning and recording their assessments and while we would encourage use of these latest packs, the previous versions will be accepted for verification purposes.
The updated Unit assessment support packs are now available and teachers/lecturers can arrange access to them through their SQA Co-ordinator.
This guidance document provides an introduction to new recording tables in the Unit assessment support packs.
Unit assessment overview
01-JUN-2014
This document provides general advice on assessment, judging evidence and re-assessment and outlines the approach taken in SQA-produced Unit assessment support packs.
Assessors also have freedom and flexibility to produce their own assessments, or use or adapt SQA-produced Unit assessment support packs. In all cases, Unit assessments have to demonstrate competency across all Assessment Standards.
Conditions of assessment
01-JUN-2014
Assessments for Mathematics must be carried out under supervised closed-book conditions. While this is not a change to the conditions of assessment, the UAS packs will be updated to include the term 'closed-book' in order to avoid any uncertainty in the future.
Notification of changes
01-JUL-2014
Selected Unit assessment support packs have been revised. Most of the revisions have been to clarify marking guidance, but some questions have been amended. These packs are now available on the secure website.
Assessments that were completed by candidates during 2013-14 session will be valid for verification in 2014-15.
Unit Assessment Support
February 2014
As a result of feedback, additional support materials have been published in February 2014 for Outcome 2. A set of questions for the assessment/reassessment of Outcome 2 (reasoning) in National 5 Expressions and Formulae, Relationships and ApplicationsThis guidance document provides an introduction to new recording tables in the Unit assessment support packs.
National Parent Forum Scotland have also produced their Revision in a Nutshell series to help learners to prepare for new National 5 examinations.
Unit Assessment Support
These documents contain details of Unit assessment task(s), show approaches to gathering evidence and how the evidence can be judged against the Outcomes and Assessment Standards. Teachers/lecturers can arrange access to these confidential documents through their SQA Co-ordinator. | 677.169 | 1 |
is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations. | 677.169 | 1 |
9144037279 a modern programming environment such as MATLAB it is possible, through simple commands, to perform advanced calculations on a personal computer, To use this powerful tool, students do not need to understand algorithmic details, but they do need a familiarity with numerical methods and the properties of algorithms. This text, for an introductory course in scientific computation at an advanced undergraduate level, is a revision and translation of a Swedish work by Eld n and Wittmeyer-Koch. It offers an introduction to basic ideas in numerical analysis including classical algorithms for solutions of nonlinear equations and linear systems of algebraic equations as well as ordinary differential equations plus methods for error analysis, interpolation, integration and approximation. Moreover, the text covers important applications in science and engineering such as floating point computer arithmetic and standard functions, splines, finite elements and discrete cosine transform. The authors focus on simplicity and readability in order to help students prepare for more complex mathematical software; they also offer useful materials on their personal Web site homepages. Useful end of chapter questions on theory and computer exercises make this a valuable tool for students as well as professionals | 677.169 | 1 |
60All NRICH live resources 2012 the UtilitiesCan you find a way to connect each house to the utilities without any pipes crossing? 01 Dec 1999 00:00:01 +0000Can You Traverse It?How can you decide if a graph is traversable? 01 Dec 1999 00:00:01 +0000Picture This! 01 Nov 2012 00:00:01 +0000Advanced Mathematics on Dotty GridsA dotty grid is a very simple mathematical structure that offers potential for very deep thought... 01 Mar 2014 00:00:01 +0000Can You Find... Trigonometric Edition Part 2Can you find trig graphs to satisfy a variety of conditions? 01 Jun 2015 00:00:01 +0100Curvy CubicsUse some calculus clues to pin down an equation of a cubic graph. 01 Jun 2015 00:00:01 +0100Can You Find... Cubic CurvesCan you find equations for cubic curves that have specific features? 01 Jun 2015 00:00:01 +0100Equation or Identity (2)Here are some more triangle equations. Which are always true? 01 Apr 2015 00:00:01 +0100Nested SurdsCan you find values that make these surd statements true? 01 May 2015 00:00:01 +0100Nine EigenExplore how matrices can fix vectors and vector directions. 01 Feb 2011 00:00:01 +0000Factorial FragmentsHere you have an expression containing logs and factorials! What can you do with it? 01 Dec 2014 00:00:01 +0000Two-way FunctionsThis gives you an opportunity to explore roots and asymptotes of functions, both by identifying properties that functions have in common and also by trying to find functions that have particular properties. You may like to use the list of functions in the Hint, which includes enough functions to complete the table plus some extras.You might like to work on this problem in a pair or small group,
or to compare your table to someone else's to see where you have used the same functions and where not. 01 Nov 2014 00:00:01 +0000Picture the Process IHow does the temperature of a cup of tea behave over time? What is the radius of a spherical balloon as it is inflated? What is the distance fallen by a parachutist after jumping out of a plane? After sketching graphs for these and other real-world processes, you are offered a selection of equations to match to these graphs and processes. 01 Nov 2014 00:00:01 +0000Giants Poster 01 Sep 2014 00:00:01 +0100Which Quadratic?In this activity you will need to work in a group to connect different representations of quadratics. 01 Sep 2014 00:00:01 +0100 | 677.169 | 1 |
Details about Precalculus with Limits:
Offering more algebra review than other texts, Precalculus with Limits encourages students to actively participate in math and focus on the link between concepts and applications. The proven Aufmann Interactive Method helps students learn the process of working out problems by providing a step-by-step example with annotations accompanied by a You-Try-It exercise. Students can then pinpoint mistakes by consulting the complete solutions in the appendix.
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Rent Precalculus with Limits 1st edition today, or search our site for Richard N. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning. | 677.169 | 1 |
An Introduction to the Theory of Numbers
9780471625469
ISBN:
0471625469
Edition: 5 Publisher: Wiley
Summary: The Fifth Edition of one of the standard works on number theory, written by internationally-recognized mathematicians. Chapters are relatively self-contained for greater flexibility. New features include expanded treatment of the binomial theorem, techniques of numerical calculation and a section on public key cryptography. Contains an outstanding set of problems.
Ivan Niven is the author of An Introduction ...to the Theory of Numbers, published under ISBN 9780471625469 and 0471625469. Seven hundred eighty four An Introduction to the Theory of Numbers textbooks are available for sale on ValoreBooks.com, eighty used from the cheapest price of $29.78, or buy new starting at $112.91 | 677.169 | 1 |
Overview
Main description
An easy-to-use guide that takes the fear out of
geometry
Bob Miller's Geometry for the Clueless tackles a subject
more than three million students face every year.
Miller acts as a private tutor, painstakingly covering
the high school curriculum as well as post secondary
courses in geometry.
Author comments
Bob Miller has been a lecturer in
Mathematics at City College of New York for more than
28 years.
Back cover copy
Student tested and APPROVED!
If you suffer from math anxiety,
Then sign up for private tutoring with Bob Miller!
Do theorems, figures, and angles leave your head spinning?
If so, you are like hundreds of thousands of other students who face math-especially, geometry-with fear.
Luckily, there's a cure: Bob Miller's Clueless series!
Like the teacher you always wished you had (but never thought existed), Bob Miller brings a combination of knowledge, empathy, and fun to the often-troubling subject of geometry. He breaks down the learning process in an easy, nontechnical way and builds it up again using his own unique methods. "Basically, the Clueless books are my notes. Exactly the way I teach: friendly, clear . . . with some humor and plenty of emotion!!!
Meant to bridge the gulf between you and geometry, the second edition of Geometry for the Clueless is packed with all of the information you need to conquer geometry. The new edition of this intensive guide gives you:
Completely updated to cover new methods and curriculum approaches in geometry
PLUS! All new chapters cover: Improvement of Reasoning; Indirect Proofs; Constructions; and sets.
"I am always delighted when a student tells me that he or she hated math... but taking a class with me has made math understandable... even enjoyable." Now it's your turn. Sharpen your #2 pencils and let Bob Miller show you how never to be clueless again! | 677.169 | 1 |
Snohomish PrecalculusMary Ann L.Steve V.
...Descriptive and Inferential statistics can be understood by most students, with the right teacher and the right motivation. Are you motivated to learn? Discrete mathematics is a blend of many different elements of logic, combinatorics and graph theory | 677.169 | 1 |
Get the skills you need to solve problems and equations and be ready for algebra class. Whether you′re a student preparing to take algebra or a parent who wants to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you′ll build necessary skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations. Look inside and discover topics such as: Understanding fractions, decimals, and percents Unraveling algebra word problems Grasping prime numbers, factors, and multiples Working with graphs and measures Solving single and multiple variable equations Want more? Let Basic Math & Pre–Algebra Workbook For Dummies help you out even further. You′ll find 280+ pages with hundreds of practice problems featuring ample workspace to work out the problems. Each problem includes a step–by–step answer set to identify where you went wrong (or right). This helpful workbook will get you up to speed with basic math and pre–algebra before you know it!
Most Helpful Customer Reviews
A pair of books which complement each other well. The chapters in each book mesh into each other well and its easy to plot a path whereby you tackle the chapter in the main book and then complete the relevant exercises in the workbook. I also found on a number of occasions that the explanation given in the workbook helped to further illustrate a point made in the main book i.e. reading an explanation of a concept for the second time in a slightly different set of words helped.
I think the approach taken is better suited for self-learning adults rather than children below 16 years.
As a bundle together this would qualify as a 5 star but for one problem which is the small print used to represent symbols such as the equals and division symbols or fractions. It is very small. My eyes are OK but I often found myself misinterpreting an equals symbol for a division symbol and vice-versa.
I did wonder if the two books could have just as easily been merged into one but at the available price its very good value.
Just bought these two books as a combined sale. Had a quick look thru them today and they both look totally superb. And the authour has done an excellent job in making something which can seem formidable to many people be easily within their grasp.
Most Helpful Customer Reviews on Amazon.com (beta)
Excellent Resource for Students and Parents to Gain Basic Concepts25 Sept. 2009
By
Andrea Polk
- Published on Amazon.com
Format: Paperback
Verified Purchase
This is a great package deal: 2 books - "Basic Math & Pre-Algebra for Dummies" and "Basic Math & Pre-Algebra Workbook for Dummies". Purchased this way you save a bit of money over purchasing them separately, and one reinforces the concepts laid out in the other. A great purchase for the beginning Pre-Algebra student or anyone who wants to understand the building block basics of what Pre-Algebra is and how it works.
First, a bit of background on me. The only 'D' I got in college was in Algebra. All A's and B's otherwise, and I think the 'D' was given to me out of pity so I could get my Bachelor's and move on! Little did I know that years later I'd have a daughter, who in 6th grade would then be taking Pre-Algebra and need my help. "Ask your Father..." was my first inclination, and then I spotted these books.
Now I am reading these books and using the knowledge I'm gaining to help her gain a basic understanding of the subject. Concepts I missed way back when, are clearer now and I find that I'm able to explain concepts with ease after reading about each subject. (Maybe that's because I'm no longer getting graded, but I do think the books help too!)
The workbook and book have tear-out cheat sheets, with basic explanations about the Order of Operations, Inequalities, Place Values, Negative and Absolute Values, Basic Algebra Conventions, and Solving Algebraic Equations. Practice exercises in the workbook are coordinated with the chapters in the book, for easy reference. Comments throughout both the book and workbook are peppered with humor. The simple organization of each book makes the ideas being covered seem far less daunting that I remembered too.
The books also cover 'how to use the book' and explain what the icons mean throughout the book. Definitions are clear and illustrations are often accented by a 'tip' that helps you know how to 'think' about attacking certain problems. I liked that feature!
The book touches on Basic Geometry and alternative numbers (Roman numerals, Mayan numerals, Binary numbers, Hexadecimal numbers, Prime-based numbers and 10 Curious types of numbers). Preparing the reader for what comes next in their mathematics education - and probably sells more books!
Overall, this is an excellent bundled (book/workbook) supplement to any-age student of Pre-Algebra, or for any parent who wants to assist their child in understanding the basic concepts of this subject.
29 of 31 people found the following review helpful
Extremely helpful if you ignore a few typos14 Jan. 2010
By
D. Garcia
- Published on Amazon.com
Format: Paperback
Verified Purchase
I plan on going back to school soon and needed a study guide so I would't have to end up in remedial classes. Since I have to pay my tuition all out of pocket, it would just be a waste of money, so I figured I should learn and relearn everything that I either copied off of someones' paper in high school or just winged without knowing a thing. This book is very helpful and it does every chapter in a way that builds up from the one before so you really get it. I believe I got stumped in school when we began doing multiplication and negatives. I have a BAD, and I mean BAD memory (kinda like Dory a la Finding Nemo), and seeing the huge times table really threw me for a loop that I had to remember all of that. This book cuts it down to half which of course is a no brainer as 3x8 is 8x3, but you couldnt tell me that when I was 10, that thing had me so discombobulated (Yay, big word! Im ready for colige now!). Flashcards may seem kiddish and might embarrass you if, say, you leave one at your work desk accidentally and a coworker sees it(ehem). They really do work though, even for me, who can't remember a thing. I had to keep THREE flashcards on my desk for a WEEK and look at them sporadically or think of them when I was away from my desk to remember just THREE! It really, really helps to remember those pesky X tables as much as I hate to admit it. Negatives I never really could wrap my head around until now. I suppose its because once you're an adult and have a bank account, you know a negative, or an overdraft = an overdraft fee, a bad, bad thing, lol. So it makes it easy to say -$3(overdraft)- $8(the overdraft fee your bank charges to your account, i wish!)= -$11 since it would give you MORE of a negative bank balance so its easy to remember that the answer would still have a negative sign, instead of memorizing all the rules rules rules. After that I was on a roll. The rest of it is rather methodical and easy if you memorized your x tables and negatives. The workbook is a must if you want it to sink in, so the bundle is your best option if you're really serious about studying. Its also a great refresher course if you happen to be one of those lucky people that did learn it all but forgot after so many years of cobwebs in the brain(the mnemonic PEMDAS, remember that?). The only downside is the book does have a few typos, but its nothing you can't decipher for yourself. The moral of the story here children, is really learn your stuff early on becuase you will have to waste your time doing it all over again someday anyway if you want to get anywhere in life.
27 of 29 people found the following review helpful
Good for brushing up on math skills16 July 2010
By
shernandez
- Published on Amazon.com
Format: Paperback
Verified Purchase
I'm going back to school after 25 years so this has helped me brush up on my math skills that I had forgotten or haven't used in awhile. Don't get too discouraged if you are having troubles getting some of your answers to match the answer key. Some of the answers are WRONG!!! I spent a lot of time on some problems trying to figure out HOW they could have come to their answers and why I came to mine...I came across wrong answers in the key at least once or twice through each section.
5 of 5 people found the following review helpful
Great for Arithmetic Review, Not so great for Pre-Algegra10 Jun. 2010
By
Amazon Customer
- Published on Amazon.com
Format: Paperback
Verified Purchase
The book gives very good review of basic mathematics. When it comes to the last couple of chapters, where they go over beginning albegra, the examples were very good for VERY basic algebra. Then they give you more complicated problems without detailed examples of how the problems are solved. SO, 5 stars for basic math and 2 stars for the pre-algebra = an average score of 3.5.
4 of 4 people found the following review helpful
Useful but let down by untidy approach to presenting information7 Jan. 2014
By
SRK
- Published on Amazon.com
Format: Paperback
First of all let me say this book and its companion are worth purchasing. Some the the tips and approaches to covering things I hated in school including a streamlined version of the dreaded times table are excellent.
The downside however is the lack of practice material as you work through the main textbook. This is instead offered in a workbook. The problem here is twofold. Firstly you have to oscillate between different books disrupting learning and secondly there is no uniformity in the way the two books present information or cross reference (page numbers to refer to for practice in the workbook) making it hard to match up the chapters in one book with the other.
The end result is using these books in tandem can feel untidy and at times chaotic. I hope in future one combined book is created to mitigate the problem or cross referencing instituted. | 677.169 | 1 |
What are the math subjects that a person with high-school math background needs to learn to reach the point of learning and understanding different techniques of mathematical optimization? It would be helpful if the subjects are listed in the order that is most convenient to follow.
4 Answers
4
I don't think there is a one size fits all answer, it depends on what subfield of optimisation you might want to learn about, but, based on my personal experience, here is somewhere that I think you could start.
"The only background required of the reader is a good knowledge of advanced
calculus and linear algebra. If the reader has seen basic mathematical analysis (e.g., norms, convergence, elementary topology), and basic probability theory, he or she should be able to follow every argument and discussion in the book."
In other words, pretty much what Glougloubarbaki says, a bit of real analysis, linear algebra and differential calculus. Most of the material is covered in this course and this one. What is missing is some of multivariable calculus (I couldn't think of a free, on-line, complete with exercises/solutions course for this - but there exists a myriad of textbooks on the subject). Also whenever you get to optimisation you might want to check out this course and this other one - they're great.
It depends whether you want to study optimization (academically, as a student or as a researcher), or use optimization techniques to solve practical problems. Most of the answers above assume the first option. I'll address the second one, since it's more common.
It's mostly a matter of knowing what tools exist and which problems they can be applied to successfully. To do this, you need some mathematical background, but not a great deal. The specifications of the tools should tell you when and where they can be used. In typical cases, you will be using commercial software packages whose internals are entirely hidden from you, so you can't even see what they're doing, still less understand them.
Using an optimization package as a "black box" with no understanding is somewhat risky, and your results might be complete rubbish if you're unlucky. But to exercise caution and apply common-sense sanity checks, you don't need to be an expert in optimization theory. Most users of optimization are certainly not experts in the underlying theory.
note that those topics take typically a year to cover for a student majoring in math
–
GlougloubarbakiMar 28 '13 at 10:46
2
I believe this answer is very incomplete. Graphs are an important part of optimization and so is Mathematical Programming (which as a whole is much, much more than mathematical analysis).
–
user69867Mar 28 '13 at 10:53 | 677.169 | 1 |
...Mathematics particularly is a subject where knowledge gaps in elementary arithmetic and algebra can cause cascading problems. A student who can easily factor a quadratic equation with multiple techniques may misunderstand "simpler" ideas, such as the meaning of negative exponents. A student who | 677.169 | 1 |
Purpose
This textbook and Internet resource provides introductory information, concept or skill development in Mathematics for grade 9, 10, 11, and 12 students who are at grade level in a single student situation.
Brief Description
CK-12's Geometry delivers a full course of study in the mathematics of shape and space for the high school student, relating the ancient logic and modern applications of measurement and description to its essential elements, processes of reasoning and proof, parallel and perpendicular lines, congruence and similarity, relationships within triangles and among quadrilaterals, trigonometry of right triangles, circles, perimeter, area, surface area, volume, and geometric transformations.
This digital textbook was reviewed for its alignment with the content standards only; California's Social Content Review criteria were not applied. Districts, schools, and individuals planning to take advantage of this free textbook are reminded to conduct their own review to determine whether this resource meets their instructional needs. | 677.169 | 1 |
Working in tandem with the Harvard-Smithsonian Center for Astrophysics, the team of experts at the Annenberg Media Foundation has created this excellent instructional series. In this eight-part program, educators can...
This site is intended for undergraduate students in physics and mathematics who need a helping hand with those late-night study sessions. Particularly enjoyable sections found here are "What dX Actually Means" and...
These lesson ideas from the New York Times offer suggestions for ways to draw on real world issues and statistics to develop lessons in mathematics. For example, in one lesson "students convert statistics about gun...
Project Mathematics, funded mostly by grants from the National Science Foundation, \"produces videotape-and-workbook modules that explore basic topics in high school mathematics in ways that cannot be done at the...
The Math Archives of the University of Tennessee-Knoxville specialize in teaching resources in Mathematics, but are also an excellent repository for mathematics software, bibliographies, and preprints. The Archives... | 677.169 | 1 |
Summary
Designed for first-year developmental math students who need support in beginning algebra, Elementary Algebra, 4/e, retains the hallmark features for which the Larson team is known: abundant, high-quality applications; the use of real data; the integration of visualization (figures and graphs) throughout; and extensive opportunities for self-assessment (mid-chapter quizzes, review exercises, tests, and cumulative tests). In developing supportive new features for the Fourth Edition, the authors' goal is for students to come away from the class with a firm understanding of algebra and how it functions as a modern modeling language.
Table of Contents
Note: Each chapter is preceded by Motivating the Chapter, includes a Mid-Chapter Quiz, and concludes with What Did You Learn? (Chapter Summary), Review Exercises, and a Chapter Test | 677.169 | 1 |
Louisiana Comprehensive Curriculum, Revised 2008
Algebra II
Unit 5: Quadratic and Higher Order Polynomial Functions
Time Frame: Approximately six weeks
Unit Description
This unit covers solving quadratic equations and inequalities by graphing, factoring, using
the Quadratic Formula, and modeling quadratic equations in real-world situations. Graphs of
quadratic functions are explored with and without technology, using symbolic equations as
well as using data plots.
Student Understandings
Students will understand the progression of their learning in Algebra II. They studied first-
degree polynomials (lines) in Unit 1, and factored to find rational roots of higher order
polynomials in Units 2, and were introduced to irrational and imaginary roots in Unit 4. Now
they can solve real-world application problems that are best modeled with quadratic
equations and higher order polynomials, alternating from equation to graph and graph to
equation. They will understand the relevance of the zeroes, domain, range, and
maximum/minimum values of the graph as it relates to the real-world situation they are
analyzing. Students will distinguish between root of an equation and zero of a function, and
they will learn why it is important to find the zeroes of an equation using the most
appropriate method. They will also understand how imaginary and irrational roots affect the
graphs of polynomial functions.
Guiding Questions
1. Can students graph a quadratic equation and find the zeroes, vertex, global
characteristics, domain, and range with technology?
2. Can students graph a quadratic function in standard form without technology?
3. Can students complete the square to solve a quadratic equation?
4. Can students solve a quadratic equation by factoring and using the Quadratic
Formula?
5. Can students determine the number and nature of roots using the discriminant?
6. Can students explain the difference in a root of an equation and zero of the
function?
7. Can students look at the graph of a quadratic equation and determine the nature
and type of roots?
8. Can students determine if a table of data is best modeled by a linear, quadratic, or
higher order polynomial function and find the equation?
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9. Can students draw scatter plots using real-world data and create the quadratic
regression equations using calculators?
10. Can students solve quadratic inequalities using a sign chart and a graph?
11. Can students use synthetic division to evaluate a polynomial for a given value and
show that a given binomial is a factor of a given polynomial?
12. Can students determine the possible rational roots of a polynomial and use these
and synthetic division to find the irrational roots?
13. Can students graph a higher order polynomial with real zeroes?
Unit 52. Predict the effect of operations on real numbers (N-3Algebra
Grade 9
14. Graph and interpret linear inequalities in one or two variables and systems of
linear inequalities (A-2-H) (A-4-H)
15. Translate among tabular, graphical, and algebraic representations of functions and
real life situations (A-3-H) (P-1-H) (P-2-H)
Grade 10
5. Write the equation of a line of best fit for a set of 2-variable real-life data presented
in table or scatter plot form, with or without technology (A-2-H) (D-2-H)
Grade 11/12
4. Translate and show the relationships among non-linear graphs, related tables of
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GLE # GLE Text and Benchmarks) (P516. Represent translations, reflections, rotations, and dilations of plane figures using
sketches, coordinates, vectors, and matrices (G-3-H)Algebra IIUnit 5Quadratic and Higher Order Polynomial Functions 90
Louisiana Comprehensive Curriculum, Revised 2008
GLE # GLE Text and Benchmarks mathematical 5Quadratic & Higher Order Polynomial Functions
5.1 Quadratic Function – give examples in standard form and demonstrate how to find
the vertex and axis of symmetry.
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5.2 Translations and Shifts of Quadratic Functions discuss the effects of the symbol
before the leading coefficient, the effect of the magnitude of the leading coefficient,
the vertical shift of equation y = x2 c, the horizontal shift of y = (x c)2.
5.3 Three ways to Solve a Quadratic Equation – write one quadratic equation and show
how to solve it by factoring, completing the square, and using the quadratic formula.
5.4 Discriminant – give the definition and indicate how it is used to determine the nature
of the roots and the information that it provides about the graph of a quadratic
equation.
5.5 Factors, x-intercept, y-intercept, roots, zeroes – write definitions and explain the
difference between a root and a zero.
5.6 Comparing Linear functions to Quadratic Functions – give examples to compare and
contrast y = mx + b, y = x(mx + b), and y = x2 + mx + b, explain how to determine if
data generates a linear or quadratic graph.
5.7 How Varying the Coefficients in y = ax2 + bx + c Affects the Graph discuss and
give examples.
5.8 Quadratic Form – Define, explain, and give several examples.
5.9 Solving Quadratic Inequalities – show an example using a graph and a sign chart.
5.10 Polynomial Function – define polynomial function, degree of a polynomial, leading
coefficient, and descending order.
5.11 Synthetic Division – identify the steps for using synthetic division to divide a
polynomial by a binomial.
5.12 Remainder Theorem, Factor Theorem – state each theorem and give an explanation
and example of each, explain how and why each is used, state their relationships to
synthetic division and depressed equations.
5.13 Fundamental Theorem of Algebra, Number of Roots Theorem – give an example of
each theorem.
5.14 Intermediate Value Theorem state theorem and explain with a picture.
5.15 Rational Root Theorem – state the theorem and give an example.
5.16 General Observations of Graphing a Polynomial – explain the effects of even/odd
degrees on graphs, explain the effect of the use of leading coefficient on even and
odd degree polynomials, identify the number of zeroes, explain and show an example
of double root.
5.17 Steps for Solving a Polynomial of 4th degree – work all parts of a problem to find all
roots and graph.
Activity 1: Why Are Zeroes of a Quadratic Function Important? (GLEs: Grade 9: 36;
Grade 11/12: 2, 4, 5, 6, 7, 8, 9, 10, 16, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Math Log Bellringer BLM, Zeroes of a
Quadratic Function BLM
In this activity, the students will plot data that creates a quadratic function and will determine
the relevance of the zeroes and the maximum and minimum of values of the graph. They will
also examine the sign and magnitude of the leading coefficient in order to make an educated
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guess about the regression equation for some data. By looking at real-world data first, the
symbolic manipulations necessary to solve quadratic equations have significance.
Math Log Bellringer:
One side (s) of a rectangle is four inches less than the other side. Draw a rectangle
with these sides and find an equation for the area A(s) of the rectangle.
Solution: A(s)= s(s - 4) = s2 – 4s
s4
s
Activity:
Overview of the Math Log Bellringers:
As in previous units, each in-class activity in Unit 5™ document or PowerPoint™ slide, and project using a TV or digital
projector, or print and display using a document or overhead projector. A sample
enlarged Math Log Bellringer Word™ Use the Bellringer to relate second-degree polynomials to the name "quadratic" equations
(area of a quadrilateral). Discuss the fact that this is a function and have students
identify this shape as a parabola.
Zeroes of a Quadratic Function BLM:
Distribute the Zeroes of a Quadratic Function BLM. This is a teacher/student
interactive worksheet. Stop after each section to clarify, summarize, and stress
important concepts.
Zeroes: Review the definition of zeroes from Unit 2 as the x-value for which the
yvalue is zero, thus indicating an x-intercept. In addition to the answers to the
questions, review with the students how to locate zeroes and minimum values of a
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function on the calculator. (TI83 and 84 calculator: GRAPH CALC (2nd TRACE)
2: zero or 3: minimum)
Local and Global Characteristics of a Parabola: In Activity 2, the students will
develop the formulas for finding the vertex and the equation of the axis of
symmetry. In this activity, students are simply defining, identifying, and reviewing
domain and range.
Reviewing 2nd Degree Polynomial Graphs: Review the concepts of end-behavior,
zeroes and leading coefficients.
Application: Allow students to work this problem in groups to come to a consensus.
Have the students put their equations on the board or enter them into the overhead
calculator. Discuss their differences, the relevancy of the zeroes and vertex, and the
various methods used to solve the problem. Discuss how to set up the equation from
the truck problem to solve it analytically. Have the students expand, isolate zero,
and find integral coefficients to lead to a quadratic equation in the form y = ax2 + bx
+ c. Graph this equation and find the zeroes on the calculator. This leads to the
discussion of the reason for solving for zeroes of quadratic equations.
Activity 2: The Vertex and Axis of Symmetry (GLEs: 4, 5, 6, 7, 8, 9, 10, 16, 27, 28, 29)
Materials List: paper, pencil, graphing calculator
In this activity, the student will graph a variety of parabolas, discovering the changes that
shift the graph vertically, horizontally, and obliquely, and will determine the value of the
vertex and axis of symmetry.
Math Log Bellringer:
(1) Graph y1 = x2, y2 = x2 + 4, and y3 = x2 – 9 on your calculator, find the zeroes and
vertices, and write a rule for the type of shift f(x) + k.
(2) Graph y1 = (x – 4)2, y2 = (x + 2)2 on your calculator, find the zeroes and vertices,
and write a rule for the type of shift f(x + k).
(3) Graph y1 = x2 – 6x and y2 = 2x2 12x on your calculator. Find the zeroes and
vertices on the calculator. Find the equations of the axes of symmetry. What is the
relationship between the vertex and the zeroes? What is the relationship between
the vertex and the coefficients of the equation?
Solutions:
(1) Zeroes: y1: {0}, y2: none, y3: {±3}. Shift up if k >0 and down if
k<0
(2) Zeroes: y1: {4}, y2: {2}. Vertices: y1: (4, 0), y2: (2, 0). Shift
right if k < 0, shift left if k > 0
(3) Zeroes: y1: {0, 6}, y2: {0, 6}. vertices: y1: (3, 9), y2: (3, –18),
axes of symmetry x = 3. The xvalue of the vertex is the
midpoint between the xvalues of the zeroes. A leading
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coefficient changes the yvalue of the vertex.
Activity:
Use the Bellringer to begin the development of the formula for finding the vertex of a
quadratic function in the form f(x) = ax2 + bx
b
Have the students set ax2 + bx equal to 0 to find the zeroes, 0 and .
a
b
Have the students find the midpoint between the zeroes at to find the xvalue or
2a
abscissa of the vertex and the axis of symmetry.
Have the students substitute the abscissa into the equation f(x) = ax2 + bx to find the
b
ordinate of the vertex f .
2a
Assign problems from the textbook for students to apply the formula for the vertex
b b 2
, f to practice graphing functions in the form f(x) = ax + bx + c.
2a 2a
Have the students develop a set of steps to graph a factorable quadratic function in the
form f(x) = ax2 + bx + c without a calculator:
1. Find the zeroes by factoring the equation and applying the Zero Property of
Equations.
b b
2. Find the vertex by letting x and y f .
2a 2a
3. Graph and make sure that the graph is consistent with the endbehavior property that
says, if a > 0 the graph opens up and if a < 0 it opens down
Application:
The revenue, R, generated by selling games with a particular price is given by R(p) = –
15p2 + 300 p + 1200. Graph the revenue function without a calculator and find the price that
will yield the maximum revenue. What is the maximum revenue? Explain
in real world terms why this graph is parabolic.
Solution: price = $10, maximum revenue = $2700. A larger price
will generate more revenue until the price is so high that no one
will buy the games and the revenue declines.
Activity 3: Completing the Square (GLEs: Grade 9: 6; Grade 10: 1; Grade 11/12: 1, 2,
4, 5, 9, 24, 29)
Materials List: paper, pencil
In this activity, students will review solving quadratic equations by factoring and will learn to
solve quadratic equations by completing the square.
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Math Log Bellringer:
Solve the following for x:
(1) x2 – 8x + 7 = 0
(2) x2 – 9 = 0
(3) x2 = 16
(4) x2 = –16
(5) (x – 4)2 = 25
(6) (x – 2)2 = –4
(7) Discuss the difference in the way you solved # 1 and #3
Solutions:
(1) x = 7, 1, (2) x = 3, –3, (3) x = 4, –4, (4) x = 4i, –4i, (5) x = 9, –1, (6) x = 2i + 2, –
2i+2, (7) To solve #1, I factored and used the Zero Property of Equations. To solve
#3, I took the square root of both sides to get ±.
Activity:
Use the Bellringer to:
(1) Review the rules for factoring and the Zero Property of Equations for problems #1
and #2.
(2) Review the rules for taking the square root of both sides in problems #3 and 4 with
real and complex answers, reiterating the difference between the answer for 16 and
the solution to the equation x2 = 16. (The solution to 16 4 is only the positive root,
but the solutions to x2 = 16 are ±4.)
(3) Discuss the two methods that can be used to solve problem #5: (1) expand, isolate
zero, and factor or (2) take the square root of both sides and isolate the variable.
(4) Discuss whether both of these methods can be used to solve problem #6.
Have students factor the expressions x2 + 6x + 9 and x2 –10x + 25 to determine what
properties of the middle term make these the square of a binomial (i. e. (x ± c)2). (Rule: If
the leading coefficient is 1, and the middle coefficient is double the ±square root of the
constant term, then it is a perfect square of a binomial (i.e. 6 2 9 and 10 2 25 ).
Have students check their conclusions by expanding (x + d)2 = x2 + 2dx + d2 and (x d)2 =
x2 2dx + d2. These are called perfect square trinomials.
Have students find c so the expressions x2 + 8x + c and x2 – 18x + c will be squares of
binomials or perfect square trinomials. Name this process "completing the square" and
have the students develop a set of steps to solve by this process.
(1) Move all constants to the right side.
(2) If the leading coefficient is not 1, factor out the leading coefficient and divide both
sides by the leading coefficient.
(3) Take ½ the middle coefficient of x and square it to find the constant, adding the same
quantity to the both sides of the equation.
(4) Write the perfect square trinomial as a binomial squared.
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(5) Take the square root of both sides making sure to get ±.
(6) Isolate x for the two solutions.
Guided Practice: Solve 3x2 + 18x 9 = 15 by completing the square showing all the steps.
Solution: Steps:
1. 3x2 + 18x = 24
2. x2 + 6x = 8
3. x2 + 6x + 9 = 8 + 9
4. (x + 3)2 = 17
5. x + 3 = 17
6. x 3 17 or x 3 17
Assign problems from the textbook to practice solving quadratic equations by completing
the square whose solutions are both real and complex.
Application:
Put students in pairs to solve the following application problem:
(1) A farmer has 120 feet of fencing to fence in a dog yard next to the barn. He will use
part of the barn wall as one side and wants the yard to have an area of 1000 square
feet. What dimensions will the three sides of the yard be? (Draw a picture of the
problem. Set up an equation to solve the problem by completing the square showing
all the steps.)
(2) Suppose the farmer wants to enclose four sides with 120 feet of fencing. What are the
dimensions to have an area of 1000 square feet? (Draw a picture of the problem. Set
up an equation to solve the problem. Find the solution by completing the square
showing all the steps.)
(3) Approximately how much fencing would be needed to enclose 1000 ft2 on four sides?
Discuss how you determined the answer.
Solutions:
(1) Perimeter: w + w + length = 120 length = 120 2w
Area: (120 2w)w = 1000
120w 2w2 = 1000 BARN
2(w 60w) = 1000
2
w2 60w = 500 w
w 60w + 900 = 500 + 900
2
1202w
(w 30)2 = 400
w 30 = ±20
w = 50 or w = 10, so there are two possible scenarios: (1) the three sides of
the yard could be (1) 10, 10 and 100 ft.or (2) 50, 50 and 20 feet
(2) Perimeter: 2w + 2 lengths = 120 length = 60 2w
Area: w(60 w) = 1000
BARN
60w w2 = 1000
w 60w = 1000
2
(w2 60w + 900) = 1000 + 900 w
(w 30)2 =10 60 w
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There is not enough fencing to enclose 1000 ft2.
(4) I need to get a positive number when I complete the square so considering the
equation w2 bw + c = 1000 + c, c must be > 1000 therefore ½b >
1
b 1000 31.623 b 63.245. Since 2b = perimeter, you will need
2
approximately 126.491 ft of fencing.
Activity 4: The Quadratic Formula (GLEs: Grade 9: 4, 6; Grade 10: 1; Grade 11/12: 1,
2, 4, 5, 9, 10, 29)
Materials List: paper, pencil, graphing calculator
Students will develop the quadratic formula and use it to solve quadratic equations.
Math Log Bellringer:
Solve the following quadratic equations using any method:
(1) x2 – 25 = 0
(2) x2 + 7 = 0
(3) x2 + 4x =12
(4) x2 + 4x = 11
(5) Discuss the methods you used and why you chose that method.
Solutions: (1) x = 5, –5, (2) x i 7 , (3) x = –6, 2, (4) x 2 15 ,
(5) Answers will vary: factoring, isolating x2 and taking the square root of
both sides, and completing the square.
Activity:
Use the Bellringer to check for understanding of solving quadratic equations by all
methods. Emphasize that Bellringer problem #4 must be solved by completing the square
because it does not factor into rational numbers.
Use the following process of completing the square to develop the quadratic formula.
ax2 + bx + c = 0
ax2 + bx = c
b c
x2 x
a a
2 2
b b b c
x x
2
a 2 a 2a a
2
b b2 4ac
x 2 2
2a 4a 4a
b b 2 4ac
2
x
2a 4a 2
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b 2 4ac
2
b
x
2a 4a 2
b b 2 4ac
x
2a 2a
b b 2 4ac
x
2a
Use the quadratic formula to solve all four Bellringer problems.
Use the math textbook for additional problems.
Relating quadratic formula answers to graphing calculator zeroes: Have the students put
y = x2 + 4x – 7 in their calculators, find the zeroes, and then use the quadratic formula to
find the zeroes. Use the calculator to find the decimal representation for the quadratic
formula answers and compare the results. Discuss difference in exact and decimal
approximation.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 5: Using the Discriminant and the Graph to Determine the Nature of the Roots
(GLEs: Grade 9: 6; Grade 10: 1; Grade 11/12: 1, 2, 4, 5, 6, 7, 9, 10, 27, 28, 29)
Materials List: paper, pencil, graphing calculator
In this activity, students will examine the graphs of shifted quadratic functions, determine the
types of roots and zeroes from the graph and from the discriminant, and describe the
difference in a root and zero of a function.
Math Log Bellringer:
Find the roots of the following functions analytically.
(1) f(x) = x2 + 4x –5
(2) f(x) = x2 – 5
(3) f(x) = x2 4x + 4
(4) f(x) = x2 3x + 7
(5) Graph the above functions on your calculator and describe the differences in the
graphs, zeroes, and roots.
3 i 19
Solutions: (1) x = –5, 1, (2) x 5 , (3) x = 2, (4) x
2
(5) #1 has two zeroes and two real rational roots,
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#2 has two zeroes and two real irrational roots,
#3 has one zero and one real rational double root,
#4 has no zeroes and two complex (imaginary) roots
Activity:
Use the Bellringer to check understanding of finding zeroes and relating them to the
graph. Review the definition of double root from Unit 2 and what it looks like on a graph.
Have students set up the Quadratic Formula for each of the equations in the Bellringer.
4 36 0 20 4 0 3 19
o Solutions: (1) , (2) , (3) , (4)
2 2 2 2
o Have students determine from the set ups above what part of the formula determines
if the roots are real or imaginary, rational or irrational, one, two or no roots.
o Define b2 4ac as the discriminant and have the students develop the rules
concerning the nature of the solutions of the quadratic equation.
1. If b2 4ac = 0 one zero and one real double root
2. If b2 4ac > 0 two zeroes and two real roots
3. If b2 4ac < 0 no zeroes and two imaginary roots
o Emphasize the difference in the word root, which can be real or imaginary, and the
word zero, which refers to an x-intercept of a graph.
Assign problems from the textbook to practice predicting solutions using the
discriminant.
Application:
Put students in pairs to determine if the following application problem has a solution
using a discriminant: The length of the rectangle is twice the length of the side of the
square and the width of the rectangle is 5 less than the side of the square. The area of a
square is 40 more than the area of a rectangle. Find the length of the side of the square.
(1) Draw pictures with the dimensions and set up the equation to compare areas. Use a
discriminant to determine if this scenario is possible. Explain why your solution is
possible or not.
(2) Find a scenario that would make the solution possible, discuss, and solve.
Solution:
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(1) s2= 2s(s 5) + 40 0 = s2 10s + 40. The 2s
s
discriminant = 60 therefore a solution is not s5
possible,
(2) Answers will vary, but one scenario is an
area of a square that is < 25 more than the
area of the rectangle.
Activity 6: Linear Functions versus Quadratic Functions (GLEs: Grade 9: 6, 15, 29;
Grade 10: 1, 5, 20, 27; Grade 11/12: 4, 6, 7, 8, 9, 10, 16, 19, 22, 27, 28)
Materials List: paper, pencil, graphing calculator
In this activity, the students will discover the similarities and differences in linear and
quadratic functions and data.
Math Log Bellringer:
Graph without a calculator: y = 4x – 8 and y = x(4x – 8). Find the x- and y- intercepts
of both and the vertex of the parabola.
Solutions:
(1) xintercept: (2, 0), yintercept: (0, -8)
(2) xintercept: (2, 0) and (0, 0), yintercept: (0, 0), vertex: (1, 4)
Activity:
Using the Bellringer for discussion, have the students check other pairs of equations in
the form y = mx + b and y = x(mx + b) to make conjectures.
Have students graph the Bellringer equations on their calculators
and adjust the window to x: [1, 3] and y: [–1, 1]. Have them discuss
that both graphs look like a line with the same x-intercept.
Give the students the following data and ask them which one is a line and why, while
reviewing the method of finite differences used in Activity 8 in Unit 2.
x 2 3 4 5 6 7 8 10
y1 –2 0 2 4 6 8 10 12
y2 –4 0 8 20 36 56 80 108
y
Solution: y1 is a line because the slope, , is always = 2.
x
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y
Have students find the twice on y2 data to prove it is a quadratic function. Make a
x
scatter plot of the data and find the equation of the best fit line in the form y = mx + b
and the equation of the best fit parabola in the form y = x(mx+b). Zoom out to determine
if the data fits the equations. (These are also referred to as regression equations and
prediction equations.) Solutions: y1 = 2x 6, y2 = x(2x 6)
Have students work several more examples such as the one below. Use the method finite
differences to determine what type of polynomial should represent the function, plot the
data on the calculator, and use the regression feature of the calculator to find the best fit
equation.
x 2 1 0 1 2 3 4 5
y1 28 14 10 10 8 2 26 70
Solution: The polynomial is cubic. f(x) = x3 + 2x2 x + 10
Activity 7: How Varying the Coefficients in y = ax2 + bx + c Affects the Graphs
(GLEs: 2, 4, 5, 6, 7, 8, 9, 10, 16, 19, 27, 28)
Materials List: paper, pencil, graphing calculator, Graphing Parabolas Anticipation Guide
BLM, The Changing Parabola Discovery Worksheet BLM
In this activity, students will discover how changes in the equation for the quadratic function
can affect the graph in order to create a best-fit parabola.
Math Log Bellringer:
Graph y1 = –4x + 6 and y2 = x(–4x + 6) without a calculator, discuss similarities, then
describe the method you used to graph the equations.
Solution:
Students should say that the graphs both have the same zero at x = 3/2.
Answers to the discussion may vary. They could have graphed y1 by finding
the yintercept and using the slope to graph, or they could have plotted
points. Students could have found the zeroes in y2 at x = 0 and 3/2 by using
the Zero Property of Equations or the quadratic formula, and they could have
found the vertex by finding the midpoint between the zeroes or by using
b b
, f .
2a 2a
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Louisiana Comprehensive Curriculum, Revised 2008
Activity:
Use the Bellringer to check for understanding of the relationship between y = mx + b and
y = x(mx+b) before going on to other changes.
Distribute the Graphing Parabolas Anticipation Guide BLM.
An anticipation guide is a modified form of opinionnaire (view literacy strategy
descriptions) which promotes deep and meaningful understandings of content area
topics by activating and building relevant prior knowledge, and by building interest in
and motivation to learn more about particular topics. Anticipation guides are
developed by generating statements about a topic that force students to take positions
and defend them. The emphasis is on students' points of view and not the
"correctness" of their opinions.
In the Graphing Parabolas Anticipation Guide BLM, the students will use their prior
knowledge of translating graphs to predict how changes in a, b, and c in the equation
y = ax2 + bx + c will affect the graph.
This should take approximately five minutes after which the students will discover
exactly what happens to the graph using The Changing Parabola Discovery
Worksheet BLM. There is no Graphing Parabolas Anticipation Guide with Answers
BLM because the answers may vary based on the students' opinions.
The Changing Parabola Discovery Worksheet:
On this worksheet the students will use their graphing calculators to graph the
parabola y = ax2 + bx + c with various changes in the constants to determine how
these changes affect the graph, and they will compare their answers to the predictions
in the anticipation guide.
Teach the following graphing technique before distributing the worksheet. Instead of
graphing every equation individually, students can easily change the constants in one
of three ways on the TI83 and TI84 graphing calculator. Practice with the following
example: y = x2 + a for a = {2, 0, 4}
(1) Type three related equations: y1 = x2, y2 = y1 2, y3 = y1
+ 4. (y1 is found under VARS, YVARS, 1:Function…,
1:Y1)
(2) Use a list: y1 = x2 + {2, 0, 4} (brackets are found above
the parentheses.)
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Louisiana Comprehensive Curriculum, Revised 2008
(3) Use the Transformation APPS:
Turn on the application by pressing APPS Transfrm
ENTER ENTER
Enter the equation y1 = x2 + A (Use the letters A, B,
C, or D for constant that will be changed.)
Set the window by pressing WINDOW cursor to
SETTINGS, set where A will start, in this example
A = 2, and adjust the step for A to Step = 1.
GRAPH and use the cursor to change the
values of A.
When finished, uninstall the transformation APP by
pressing APPS Transfrm, 1:Uninstall
For more information see the TI 83/TI84 Transformation App Guidebook at
Distribute The Changing Parabola Discovery Worksheet BLM and arrange the
students in pairs to complete it. Circulate to make sure they are graphing correctly.
The answers to "why the patterns occur" will vary. When the students finish the
worksheet, list the answers from the students on the board reviewing all the
information they have learned in previous units, such as finding the vertex from
b b and the axis of symmetry from x b , as well as, using the
, f
2a 2a 2a
discriminant b 4ac to determine when there are real roots.
2
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 8: Parabolic Graph Lab (GLEs: Grade 10: 20; Grade 11/12: 4, 6, 9, 10, 19, 20,
22, 24, 28, 29)
Materials List: paper, pencil, graphing calculator, Drive the Parabola Lab BLM, Drive the
Parabola Collection and Analysis BLM, the following for each lab group CBR motion
detector with cable to connect to graphing calculator, large truck or ball, ramp or board set on
books, Drive the Parabola Lab Teachers Information BLM
Algebra IIUnit 5Quadratic and Higher Order Polynomial Functions 104
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Students will collect data with a motion detector to determine a quadratic equation for the
position of a moving object and use the equation to answer questions.
Math Log Bellringer:
The position of a falling object with initial velocity of 50 ft/sec thrown up from a
height of 100 feet is given by f(t) = –16t2 + 50t + 100.
(1) Graph the equation on your calculator adjusting the window to see the intercepts
and vertex.
(2) Find the maximum height of the object and the time it gets to this height.
(3) Find the time the object hits the ground.
Solutions:
(1)
(2) The maximum height is 139.063 ft. at 1.563 seconds
(3) It hits the ground in 4.5106 sec.
Activity:
Use the Bellringer to check for understanding of the meaning of the vertex and the
zeroes.
Drive the Parabola Lab:
Divide the students into groups and distribute the Drive the Parabola Lab BLM and
the Drive the Parabola Lab Collection and Analysis BLM
Each group will collect data from the motion detector and use the data to answer
questions.
Groups can share the motion detectors, because once the data is collected, the
analysis can be finished with the calculator. After the students collect the data, they
should link to a partner's calculator to transfer data so everyone in the group can do
the analysis. Students can finish for homework if there isn't enough time in class.
Teacher Note: If motion detectors are unavailable, use the following data and the Drive the
Parabola Data Collection and Analysis BLM.
time (sec) 0 0.2 .4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
distance (m) 0.273 0.123 0 .095 .1632 0.204 .2176 .2038 .1628 .0946 .001 .1238 .2739 .4514 .656
Activity 9: Solving Equations in Quadratic Form (GLEs: Grade 9: 6; Grade 10: 1;
Grade 11/12: 1, 2, 5, 9, 27)
Materials List: paper, pencil
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The students will examine equations that are not truly quadratic but in which they can use the
same strategies to solve.
Math Log Bellringer:
Solve the following for t: 2t2 – 4t + 1 = 0 and discuss which method you used and
why.
2 2 2 2
Solution: t and t , I used the quadratic formula because I could not
2 2
factor the equation.
Activity:
Use the Bellringer to review the quadratic formula making sure to have students use the
b b 2 4ac
variable t in the quadratic formula t and in the answer, then write the
2a
two answers separately. Substitute (s - 3) for t in the equation and ask them how to solve.
Remind students to check the answers to prove that they are solutions and not extraneous
roots.
Solution:
2(s 3)2 4(s 3) + 1 = 0
2 2 2 2
s 3 or s 3
2 2
2 2 2 2
s 3 or s 3
2 2
8 2 8 2
Finding a common denominator: s and s
2 2
Define quadratic form as any equation that can be written in the form at2 + bt + c where t
is any expression of a variable. Have students identify the expression that would be t in
the following to make the equation quadratic form:
(1) x4 +7x2 + 6 = 0
(2) 2(y +4)2 + (y + 4) + 6= 0
(3) x 3 x 4 0
(4) s4 + 2s2 = 0
Solutions: (1) t = x2, (2) t x , (3) t = y + 4, (4) t = s2
Have students work in pairs to solve the problems above making sure to check answers
for extraneous roots.
5
Solution: (1) x = i, i 6 , (2) y = , 6 , (3) x =16, (4) s = {0, 3i }
2
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Application:
In a certain electrical circuit, the resistance of any R, greater than 6 ohms, is found by
solving the quadratic equation (R – 6)2 = 4(R – 6) + 5. Show all of your work.
(1) Find R by solving the equation using quadratic form.
(2) Find R by first expanding the binomials and factoring.
(3) Find R by expanding the binomials then quadratic formula.
(4) Find R by graphing f(R)= (R – 6)2 – 4(R – 6) – 5 and finding the zeroes.
(5) Discuss which of the above methods you like the best and why both solutions for R
are not used.
Solution: R = 11 ohms, 5 ohms is not valid for the initial conditions
Activity 10: Solving Quadratic Inequalities (GLEs: Grade 9: 6, 14; Grade 10: 1; Grade
11/12: 2, 4, 5, 6, 8, 9, 10, 24, 27, 29)
Materials List: paper, pencil, Solving Quadratic Inequalities by Graphing BLM
In this activity, students will solve quadratic inequalities by using a sign chart and by
interpreting a graph. This concept was first introduced in Activity 9 in Unit 2 and is
expanded here to include problems with nonreal roots.
Math Log Bellringer:
Solve the following without a calculator:
(1) 8 – 2x > 0
(2) (x – 4)(x + 3) > 0
(3) x2 9 < 0
(3) Discuss how you found the solution to #2 and #3 and why.
Solutions: (1) x > 6, (2) x < –3 or x > 4, (3)3 < x < 3, (4) I found where both
factors were positive or where both factors were negative.
Activity:
Use the Bellringer to check for students' understanding of the Zero Property for
Inequalities:
(1) If ab > 0, then either a and b are both positive or a and b are both negative.
(2) If ab < 0 then either a or b is negative but not both.
o Students will usually forget that there are two scenarios for each situation, forget
to factor, or try to take the square root of both sides of an inequality without using
absolute value. (i.e., x 2 x )
o Revisit the number line method used in Unit 2. Have students draw a number line
and locate the zeroes for Bellringer #2 and 3, then test values in each interval and
write + and signs above that interval on the number line. Discuss the use of and
or or, intersection or union, and how to express the answers in interval notation or
set notation.
#2
++++++ ) ( ++ +++
3 4 +
Algebra IIUnit 5Quadratic+ + +Higher OrderPolynomial + + + + +
+ and + + + Functions 107
#3 [ ]
3 3
Louisiana Comprehensive Curriculum, Revised 2008
Revisit how the graphs of y = 8 – 2x, y = (x – 4)(x + 3), and y = x2 9 can assist the
students in solving the inequalities in the Bellringer. What global characteristics of the
graphs are important? (Solution: the zeroes and end-behavior) Have students graph the
Bellringers to verify their answers.
Solutions: (1) , (2) , (3)
Solving Quadratic Inequalities by Graphing:
In this Solving Quadratic Inequalities by Graphing BLM, the students will first
complete a SPAWN writing (view literacy strategy descriptions) prompt. SPAWN is
an acronym that stands for five categories of writing prompts: Special Powers,
Problem Solving, Alternative Viewpoints, What If, and Next. In the first section of
this BLM the students will answer a "What If" writing prompt concerning using
graphs to help solve inequalities if the zeroes are not real.
Distribute the Solving Quadratic Inequalities by Graphing BLM and give students a
few minutes to complete the SPAWN writing prompt individually. Ask several
students to share their answers.
The students should then continue the worksheet in which they will find the zeroes
and roots and graph the related equations to solve the inequalities. Stress that it is not
important to find the vertices of the parabolas, just the zeroes and end-behavior.
When students have finished the worksheet, revisit the SPAWN prompt and refine the
procedure for finding the roots and end-behavior in order to determine if there are any
solutions or not.
To check for understanding, assign the following problems to be solved individually:
Write the related "y =" equation, find the roots, zeroes and graph without a
calculator, then write the solution to the inequality in interval notation.
(1) x2 + 3x > 0
(2) x2 2x < 2
(3) x2 2x + 2 > 0
(4) x2 8 < 0
Solutions:
(1) y = x2 + 3x, zeroes: x = 3, 1, roots: x = 3, 0
Solution to inequality: (, 3) (0, )
(2) y = x2 2x 2, zeroes, x 1 2 ,
roots: x 1 2 ,
Solution to inequality: 1 2, 1 2
Algebra IIUnit 5Quadratic and Higher Order Polynomial Functions 108
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(3) y = x2 2x + 2; zeroes: none, roots, x = 1 ± i
Solution to inequality: (, )
Algebra IIUnit 5Quadratic and Higher Order Polynomial Functions 109
Louisiana Comprehensive Curriculum, Revised 2008
(4) y = x2 8, zeroes: none, roots 2i 2 ,
Solution to inequality:
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 11: Synthetic Division (GLEs: Grade 9: 5; Grade 10: 2; Grade 11/12: 2, 5)
Materials List: paper, pencil
In this activity, students will use synthetic division to divide a polynomial by a first-degree
binomial.
Math Log Bellringer:
Divide by hand to simplify the following quotients:
(1) 7 1342 (2) x 2 x 3 4 x 2 7 x 14
(3) Discuss the difference in writing the answer to 7/5 in these two ways:
2
7 5 1 Remainder 2 or 7 5 1
5
5 4
Solutions: (1) 191 (2) x 2 6 x 5 , (3) See Activity for discussion
7 x2
Activity:
remainder
quotient
divisor
Use Bellringer #1 to review elementary school terminology: divisor dividend .
Rewrite this rule in Algebra II form: dividend remainder and relate to Bellringer
quotient
divisor divisor
problem 2.
Review the definition of polynomial and the steps for long division, stressing descending
2 x 3 3x 100
powers and missing powers. Have students divide .
x4
2 x 2 8x 35
Solution: x 4 2 x 0 x2 3x 100 with a remainder of 40 2 x 2 8x 35 40
3
x4
Introduce synthetic division illustrating that in the long division problems, the variable is
not necessary, and if we had divided by the opposite of 4, we could have used addition
instead of subtraction. Rework the problems using synthetic division.
Solution: 4| 2 0 3 100
8 32 140
2 8 35 40
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Have students develop the steps for synthetic division:
(1) Set up the coefficients in descending order of exponents.
(2) If a term is missing in the dividend, write a zero in its place.
(3) When dividing by the binomial x – c, use c as the divisor (c is the value of x that
makes the factor x – c = 0).
c c
(4) When dividing by the binomial ax – c, use as the divisor. ( is the value of x that
a a
makes the factor ax – c = 0.)
Have students practice the use of synthetic division to simplify the following and write
dividend remainder
the answers in equation form as quotient .
divisor divisor
(1) (2x3 + 5x2 –7x –12) (x + 3)
(2) (x4 –5x2 – 10x – 12) (x + 2)
Solutions:
2x 3 + 5x 2 -7x -12 0
(1) (2x2 – x – 4) +
x+3 x3
4 2
x -5x - 10x - 12 4
(2) x3 2 x2 x 8
x+2 x2
Use the math textbook for additional problems.
Activity 12: Remainder and Factor Theorems (GLEs: Grade 9: 6; Grade 10: 1; Grade
11/12: 1, 2, 5, 6, 9, 25)
Materials List: paper, pencil, graphing calculator, Factor Theorem Discovery Worksheet
BLM
In this activity, the students will evaluate a polynomial for a given value of the variable using
synthetic division, and they will determine if a given binomial is a factor of a given
polynomial.
Math Log Bellringer:
Use long division and synthetic division to simplify the following problem.
(2x3 + 3x2 8) (x 4)
80
Solution: 2x2 + 11x + 44 +
x4
Activity:
Factor Theorem Discovery Worksheet:
In this worksheet, the students will use synthetic division to find a relationship
between the remainder when dividing a polynomial by (x c) and the value of the
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polynomial at f(c) developing the Remainder Theorem. They will use this information
to determine when (x c) is a factor of a polynomial thus developing the Factor
Theorem.
Distribute the Factor Theorem Discovery Worksheet BLM. This worksheet should be
used as guided discovery. Allow students to work in pairs or groups stopping after
each section to ascertain understanding.
After questions #1 and #2 under the Synthetic Division section, have a student write
the answers on the board for others to check.
After questions #3 and #4, ask students to complete the Remainder Theorem. It states:
If P(x) is a polynomial and c is a number, and if P(x) is divided by x – c, then the
remainder equals P(c).
In the beginning of the Factor Theorem section, have students verbalize the definition
of factor two or more numbers or polynomials that are multiplied together to get a
third number or polynomial. Allow the students to complete the problems in this
section to develop the Factor Theorem: If P(x) is a polynomial, then x – c is a factor
of P(x) if and only if P(c) = 0. Have students define a depressed polynomial. Make
sure students understand that the goal of this process is to develop a quadratic
depressed equation that can be solved by quadratic function methods, such as the
quadratic formula or simple factoring.
When the theorems have been developed, have students practice the concepts using
the Factor Theorem Practice section of the BLM.
Assign additional problems from the math textbook if necessary.
Activity 13: The Calculator and Exact Roots of Polynomial Equations (GLEs: Grade 9:
4, 6; Grade 10: 1; Grade 11/12: 1, 2, 5, 6, 7, 9, 10, 25)
Materials List: paper, pencil, graphing calculator, Exactly Zero BLM
In this activity, the students will use the calculator and a synthetic division program to help
find the exact roots of polynomial equations.
Math Log Bellringer:
Graph f(x) = x3 + 5x2 18 on your graphing calculator and find all zeroes. Discuss
how you know how many roots and zeroes exist.
Teacher Note: Students must ZOOM IN around –3 to find both negative zeroes.
Solution: zeroes: {–3, –3.646, 1.646}, The
degree of the polynomial tells how many
roots there are; but some roots may be
imaginary and some may be double roots,
so there are at most three different roots and at most three different zeroes.
Algebra IIUnit 5Quadratic and Higher Order Polynomial Functions 112
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Activity:
Use the Bellringer to review the following concepts from Unit 2:
(1) finding zeroes of a polynomial on a graphing calculator
(2) determining the maximum number of roots for a polynomial equation
(3) remembering what a double zero looks like on a graph
(4) approximate values vs exact values
Have the students decide how to use the integer root they found from the graphs and from
synthetic divisions to find the exact answers of the Bellringer problems.
Solutions: Use the integer root x = 3 and synthetic division to find the depressed
equation which is a quadratic equation. Then use the quadratic formula to find the
exact roots 3, 1 7, 1 7
The problem with using the Factor Theorem is finding one or more of the rational roots to
use in synthetic division to create a depressed quadratic equation. The students can find
the integer or rational roots found on the calculator and synthetic division to find the
irrational or imaginary roots.
o Have students find the exact roots and factors for the following equation explaining
their reasoning: x4 6x3 + 13x2 24x + 36 = 0.
Solutions: From the graph, it is obvious that there is a double root at
x = 3, so 3 would be used twice once in synthetic division in the original equation
and then in the depressed equation to get to a quadratic equation that can be solved.
3 | 1 6 13 24 36 3 | 1 3 4 12
3 9 12 -36 3 0 12
1 3 4 12 0 1 0 4 0
Depressed quadratic equation: x + 4 = 0 x = ±2i
2
Roots: {3, 3, ±2i}, factors: (x 3)2(x 2i)(x + 2i)
If the students are going to use the calculator to find the rational roots, then it is logical
that they could use the calculator to run a synthetic division program that will generate
that depressed equation. This program is available for the TI 83 and 84 at the following
website.
Exactly Zero BLM:
On the Exactly Zero BLM, the students will practice finding the exact zeroes by first
graphing the function on the calculator to find one or more rational roots and then
using these roots in synthetic division (either by hand or using the program).
Repeated use of synthetic division will generate a depressed quadratic equation which
can then be solved by one of the methods for solving quadratic equations.
Distribute the Exactly Zero BLM and allow the students to work in pairs.
Algebra IIUnit 5Quadratic and Higher Order Polynomial Functions 113
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When students complete the worksheet, check their answers and assign the following
problem to be worked individually.
Find the roots and factors of the following equation:
x4 6x3 2x2 6x +5 = 0
Solution: Roots: {1, 1, 2 + i, 2 i}, factors: (x 1)2( x2 i))(x 2 + i)
Activity 14: The Rational Root Theorem and Solving Polynomial Equations (GLEs:
Grade 9: 4, 6; Grade 10: 1; Grade 11/12: 1, 2, 5, 6, 7, 9, 25, 27)
Materials List: paper, pencil, graphing calculator, Rational Roots of Polynomials BLM,
Exactly Zero BLM from Activity 13
In this activity, the students will use the Rational Root Theorem and synthetic division to
solve polynomial equations.
Math Log Bellringer: Distribute the Rational Roots of Polynomials BLM. Have students
complete the vocabulary self awareness (view literacy strategy descriptions) chart. They
should rate their personal understanding of each number system with either a "+"
(understands well), a "" (limited understanding or unsure), or a "" (don't know). They
should then look back at the Exactly Zero BLM completed in Activity 13 and list all the roots
found and place them in the correct category in the chart. Have students refer back to the
chart later in the unit to determine if their personal understanding has improved. For terms in
which students continue to have checks and minuses, additional teaching and review may be
necessary.
Complex Number System Terms + Root from Exact Zero BLM
1 integer
2 rational number
3 irrational number
4 real number
5 imaginary number
6 complex number
Activity:
Use the Bellringer to make sure students can classify types of numbers, a skill begun in
Unit 4.
Rational Roots of Polynomials:
The remainder of the Rational Roots of Polynomials BLM should be a teacher guided
interactive worksheet.
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Louisiana Comprehensive Curriculum, Revised 2008
Have students define rational number. Possible student answers: (1) a repeating or
terminating decimal, (2) a fraction, (3) p where p and q are integers and q ≠ 0.
q
Have students list the rational roots in each of the Exactly Zero BLM problems from
Activity 13.
o What is alike about all the polynomials that have integer rational roots? Solution:
leading coefficient of 1.
o What is alike about all the polynomials that have fraction rational roots? Solution:
The leading coefficient is the denominator.
State the Rational Root Theorem: If a polynomial has integral coefficients, then any
rational roots will be in the form p where p is a factor of the constant and q is a
q
factor of the leading coefficient.
Discuss the following theorems and how they apply to the problems above:
o Fundamental Theorem of Algebra: Every polynomial function with complex
coefficients has at least one root in the set of complex numbers
o Number of Roots Theorem: Every polynomial function of degree n has exactly n
complex roots. (Some may have multiplicity.)
o Complex Conjugate Root Theorem: If a complex number a + bi is a solution of a
polynomial equation with real coefficients, then the conjugate a – bi is also a
solution of the equation.
Have students decide how to choose which of the many rational roots to use to begin
synthetic division. Relate back to finding the zeroes on a calculator by entering a
lower bound and upper bound.
Discuss continuity of polynomials. Develop the Intermediate Value Theorem for
Polynomials: (as applied to locating zeroes). If f(x) defines a polynomial function
with real coefficients, and if for real numbers a and b the values of f(a) and f(b) are
opposite signs, then there exists at least one real zero between a and b.
Have students apply the Rational Root Theorem to solve the last polynomial.
Assign additional problems in the math textbook for practice.
Activity 15: Graphing Polynomial Functions (GLEs: Grade 9: 4, 6; Grade 10: 1; Grade
11/12: 1, 2, 4, 5, 6, 7, 8, 9, 10, 16, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Solving the Polynomial Mystery BLM
In this activity, the students will tie together all the properties of polynomial graphs learned
in Unit 2 and in the above activities to draw a sketch of a polynomial function with accurate
zeroes and end-behavior.
Math Log Bellringer: Graph on your graphing calculator. Adjust WINDOW to see
maximum and minimum y values and intercepts. Find exact zeroes and exact roots.
(1) f(x) = x3 – 3x2 – 5x + 12
(2) f(x)= x4 – 1
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(3) f(x)= –x4 + 8x2 + 9
(4) f(x)= –x3 – 3x
(5) Discuss the difference in zeroes and roots
Solutions:
(1) zeroes {–2, 1.5, 3.5}, roots {–2, 1.5, 3.5}, (2) zeroes: {1, –1}, roots: {1, –1, i, –i}
(3) zeroes: 3, 3 , roots: 3, 3, i, i , (4) zeroes: {0}, roots: i 3
(5) Zeroes are the xintercepts on a graph where y = 0. Roots are solutions to a
one variable equation and can be real or imaginary.
Activity:
Use the Bellringer to review the following:
(1) Unit 2 concepts (endbehavior of odd and even degree polynomials, how end-
behavior changes for positive or negative leading coefficients).
(2) Unit 5 concepts (the Number of Roots Theorem, Rational Root Theorem, and
synthetic division to find exact roots).
(3) What an imaginary root looks like on a graph (i.e. imaginary roots cannot be located
on a graph because the graph is the real coordinate system.) (Students in Algebra II
will be able to sketch the general graph with the correct zeroes and end-behavior, but
the particular shape will be left to Calculus.)
Before assigning the problem of graphing a polynomial with all of its properties, ask the
students to write a GIST (view literacy strategy descriptions).
GISTing is an excellent strategy for helping students paraphrase and summarize
essential information. Students are required to limit the GIST of a concept to a set
number of words. Begin by reminding students of the fundamental characteristics of a
summary or GIST by placing these on the board or overhead:
(1) Shorter than the original text
(2) A paraphrase of the author's words and descriptions
(3) Focused on the main points or events
Assign the following GIST: When you read a mystery, you look for clues to solve the
case. Think of solving for the roots of a polynomial equation as a mystery. Discuss all
the clues you would look for to find the roots of the equation. Your discussion should
be bulleted, concise statements, not full sentences, and cover about ½ sheet of paper.
When students have finished their GISTs, create a list on the board of characteristics
that should be examined in graphing a polynomial.
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Solving the Polynomial Mystery:
In the Solving the Polynomial Mystery BLM, the students will combine all the
concepts developed in this unit that help to find the roots of a higher degree
polynomial and will check to see if their GISTing was complete.
Distribute the Solving the Polynomial Mystery BLM. This is a noncalculator
worksheet. Allow students to work in pairs circulating to make sure they are applying
all the theorems correctly.
When students have completed the graph have them check it on their graphing
calculators finding both the graph and the decimal approximations of the roots. Make
sure all the elements in the worksheet intercepts, roots, end-behavior, and ordered
pairs in the chart are located on the graph. (They will not be able to find the
maximum and minimum points by hand until Calculus.)
Have students return to their GISTs and add any concepts they had forgotten.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.) y = x2, y = –x2, y = x2 + 4, y = (x + 4)2
(2) solving quadratic equations by using the quadratic formula
(3) solving quadratic equations by completing the square
Administer three comprehensive assessments:
(1) graphing quadratic functions
(2) solving quadratic equations and inequalities and application problems
(3) using synthetic division and the Factor Theorem to graph polynomials
Activity-Specific Assessments
Activities 4, 7, 10 and 15: Evaluate the Critical Thinking Writing using the following
rubric:
Grading Rubric for Critical Thinking Writing ActivitiesAlgebra IIUnit 5Quadratic and Higher Order Polynomial Functions 117
Louisiana Comprehensive Curriculum, Revised 2008
3 pts./solution - correct equations, showing work, correct answer
3 pts./discussion - correct conclusion
Activity 4: Critical Thinking Writing
John increased the area of his garden by 120 ft2. The original garden was 12 ft. by 16
ft., and he increased the length and the width by the same amount. Find the exact
dimensions of the new garden and approximate the dimensions in feet and inches.
Discuss which method you used to solve the problem and why you chose this method.
Solution:
x 14 2 79 , dimensions = 2 2 79 X 2 2 79 15ft. 9 in. X 19 ft. 9 in
Activity 7: Critical Thinking Writing
Answer the following questions using the conclusions from The Changing Parabola
Discovery Worksheet BLM.
(1) Discuss what happens to the zeroes of the equation y = x2 + 8x + c, why if c=0,
and when c .
(2) Discuss what happens to the zeroes of the equation y = x2 + bx – 5, why if b=0,
and when b .
(3) Discuss what happens to the zeroes of this equation y = ax2 + x – 5, why when
a>0, and what happens to the positive zero when a 0.
Solutions:
(1) When c = 0, the zeroes are {0, 8}. As c , the graph of y = x2 + 8x + c
moves up with the yintercept moving up. When the discriminant b2 4ac =
64 4c > 0 or c > 16, there are no real zeroes and two imaginary roots.
When c = 16, there is one real zero at x = 4 and a double real root.
(2) When b = 0 there are two real roots and two zeroes at x 5 with a
yintercept of 5. There will always be zeroes or real roots because b2 4ac
= b2 + 20 is always >0. As b , the yintercept remains at y = 5 and the
b
axis of symmetry which is x moves left. As b becomes larger and
2a
larger, the constant becomes less significant. If the constant is ignored, the
equation becomes y = x2+bx or y= x(x+b) which has the zeroes 0 and –b.
(3) When a > 0, the graph is a parabola opening up, and as a 0, the zeroes
become wider and wider apart. As a 0, the equation starts looking like the
equation y = x 5 which is a line with a zero at x = 5, so the positive zero
approaches 5.
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Activity 10: Critical Thinking Writing
A truck going through the parabolic tunnel over a two-lane highway has the following
features: the tunnel is 30 feet wide at the base and 15 feet high in the center.
(1) Sketch your tunnel so that the base is on the x-axis and the x intercepts are ±15.
(2) Find the equation of the parabola. What do the variables x and y represent?
(3) The truck is 10 feet high. Determine the range of distances the truck can drive
from the center of the tunnel and not hit the top of the tunnel.
(a) Find the inequality you will be solving.
(b) Find the zeroes and sketch of the related equation.
(c) Express your exact answer to the range of distances in feet and inches.
(4) Discuss how you set up the equation for the parabola and how you solved the
problem.
Solutions: (1)
1
(2) y x 2 15 , y = the height of the tunnel a distance of x
15
from the center of the tunnel
1
(3a) x 2 15 10
15
1
(b) related equation y x 2 5 , zeroes: x 5 3
15
(c) Distance from center of the tunnel < 5 3 ft 8 ft 8"
Activity 15: Critical Thinking Writing
One of your rational roots in The Polynomial Mystery BLM is a fraction. Discuss the
difference in the graph if you use the factor (x – ½ ) or the factor (2x – 1). Which one
is correct for this problem and why?
Solution: 2x – 1 is correct for this problem. Both equations have the same zeroes, but
one has higher and lower minimum points. Since f(x) has a leading coefficient of 4,
my factors must expand to 4x4 + …
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Algebra II
Unit 6: Exponential and Logarithmic Functions
Time Frame: Approximately four weeks
Unit Description
In this unit, students explore exponential and logarithmic functions, their graphs, and
applications.
Student Understandings
Students solve exponential and logarithmic equations and graph exponential and logarithmic
functions by hand and by using technology. They will compare the speed at which the
exponential function increases to that of linear or polynomial functions and determine which
type of function best models data. They will comprehend the meaning of a logarithm of a
number and know when to use logarithms to solve exponential functions.
Guiding Questions
1. Can students solve exponential equations with variables in the exponents and
having a common base?
2. Can students solve exponential equations not having the same base by using
logarithms with and without technology?
3. Can students graph and transform exponential functions?
4. Can students graph and transform logarithmic functions?
5. Can student write exponential functions in logarithmic form and vice versa?
6. Can students use the properties of logarithms to solve equations that contain
logarithms?
7. Can students find natural logarithms and anti-natural logarithms?
8. Can students use logarithms to solve problems involving exponential growth and
decay?
9. Can students look at a table of data and determine what type of function best
models that data and create the regression equation?
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Unit 62. Evaluate and write numerical expressions involving integer exponents (N-2-H)e.g., estimation,
mental math, technology, paper/pencil)3. Describe the relationship between exponential and logarithmic equations (N-2-H)
Algebra
Grade 9
8. Use order of operations to simplify or rewrite variable expressions (A-1-H) (A-2-H)
10. Identify independent and dependent variables in real-life relationships (A-1-H)
15. Translate among tabular, graphical, and algebraic representations of functions and
real-life situations (A-3-H) (P-1-H) (P-2-H)
Grade 11/12)(P-5-H)
10. Model and solve problems involving quadratic, polynomial, exponential,
logarithmic, step function, rational, and absolute value equations using technology
(A-4-H)
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GLE # GLE Text and Benchmarks17. Discuss the differences between samples and populations (D-1-H)35. Determine if a relation is a function and use appropriate function notation (P-1-H)Algebra IIUnit 6Exponential and Logarithmic Functions 122
Louisiana Comprehensive Curriculum, Revised 2008 6Exponential and Logarithmic Functions
6.1 Laws of Exponents – write rules for adding, subtracting, multiplying and dividing
values with exponents, raising an exponent to a power, and using negative and
fractional exponents.
6.2 Solving Exponential Equations – write the rules for solving two types of exponential
equations: same base and different bases (e.g., solve 2x = 8x – 1 without calculator; solve
2x = 3x – 1 with and without calculator).
6.3 Exponential Function with Base a – write the definition, give examples of graphs
with a > 1 and 0 < a < 1, and locate three ordered pairs, give the domains, ranges,
intercepts, and asymptotes for each.
6.4 Exponential Regression Equation give a set of data and explain how to use the
method of finite differences to determine if it is best modeled with an exponential
equation, and explain how to find the regression equation.
6.5 Exponential Function Base e – define e, graph y = ex and then locate 3 ordered pairs,
and give the domain, range, asymptote, intercepts.
6.6 Compound Interest Formula – define continuous and finite, explain and give an
example of each symbol
6.7 Inverse Functions – write the definition, explain one-to-one correspondence, give an
example to show the test to determine when two functions are inverses, graph the
inverse of a function, find the line of symmetry and the domain and range, explain how
to find inverse analytically and how to draw an inverse on the calculator.
6.8 Logarithm – write the definition and explain the symbols used, define common logs,
characteristic, and mantissa, and list the properties of logarithms.
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6.9 Laws of Logs and Change of Base Formula – list the laws and the change of base
formula and give examples of each.
6.10 Solving Logarithmic Equations – explain rules for solving equations, identify the
domain for an equation, find log28 and log25125, and solve each of these equations for x:
logx 9 = 2, log4 x = 2, log4(x – 3)+log4 x=1).
6.11 Logarithmic Function Base a – write the definition, graph y = logax with a < 1 and
a > 1 and locate three ordered pairs, identify the domain, range, intercepts, and
asymptotes, and find the domain of y = log(x2 + 7x + 10).
6.12 Natural Logarithm Function – write the definition and give the approximate value of e,
graph y = ln x and give the domain, range, and asymptote, and locate three ordered
pairs, solve ln x = 2 for x.
6.13 Exponential Growth and Decay define half-life and solve an example problem, give
and solve an example of population growth using A(t) = Pert.
Activity 1: Fractional Exponents (GLEs: Grade 9: 2, 6, 8; Grade 10: 1; Grade 11/12: 1, 2)
Materials List: paper, pencil, graphing calculator, Math Log Bellringer BLM
In this activity, students will review properties of numbers with integral exponents first
discussed in Unit 3 and extend them to simplify and evaluate expressions with fractional
exponents.
Math Log Bellringer:
Simplify the following and explain in words the law of exponents used.
(1) a2a3
b7
(2)
b3
(3) (c3)4
(4) 2x5 + 3x5
(5) (2x)3
(6) (a + b)2
(7) x0
(8) 2–1
Solutions:
(1) a5, Law: When you multiply 2 variables with the same base, add
exponents.
(2) b4, Law: When you divide two variables with the same base, subtract the
exponents.
(3) c12, Law: When you raise a variable with an exponent to a power, multiply
the exponents.
(4) 5x5, Law: When you add two expressions that have the same variable
raised to the same exponent, add the coefficients.
(5) 8x3, Law: When you raise a product to a power, each of the factors are
raised to that power.
(6) a2 + 2ab + b2, Rule: When you raise a sum to a power, FOIL.
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(7) 1, Law: Any variable or number ≠ 0 raised to the zero power = 1.
(8) ½, Law: A number or variable raised to a negative exponent is the
reciprocal of the number.
Activity:
Overview of the Math Log Bellringers:
As in previous units, each in-class activity in Unit 6 Blackline Masters have not been created for
each of them. Write them on the board before students enter class, paste them into an
enlarged Word document or PowerPoint slide, and project using a TV or digital
projector, or print and display using a document or overhead projector. A sample
enlarged Math Log Bellringer Word document has been included in the Blackline
Masters Choose students to write the Laws of Exponents used in the Bellringers on an overhead
transparency or on the board. Critique the wording as a class, stressing the need for a
common base in #1 and #2, a common base and exponent in #4, and a common exponent
in #5.
Have the students discover the equivalency of the following in their calculators and write
a rule for fractional exponents. This can be done by getting decimal representations, or
the students can use the TEST feature on the TI-83 and TI84 to determine equivalency.
Enter 5 5 ^ 1/ 2 (The "=" sign is found under 2ND , [TEST], (above the MATH
button). If the calculator returns a "1", then the statement is true; if it returns a "0", then
the statement is false.
1
(1) 5 and 5 2
1
3
(2) 6 and 6 3
3
(3) 4
23 and 2 4
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Solutions:
n
All are equivalent. The rule for fractional exponents is if a is a real number,
a
b
b
then a c a b
c c
Have students practice changing radicals to fractional exponents and vice versa using the
laws of exponents by simplifying complex radicals. Have students simplify problems
such as the following without calculators and use the properties in the Bellringers to
simplify similar problems with fractional exponents:
1
1 2
(1)
100
1
3
(2) 8
1
5
(3) 625
(4) 43
1
Solutions: (1) , (2) 2, (3) 5, (4) 8
10
Assign additional problems from the math textbook.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 2: Graphs of Exponential Function (GLEs: Grade 9: 6, 36; Grade 10: 1; Grade
11/12: 2, 4, 6, 7, 8, 19, 25, 27, 28, 29)
Materials List: paper, pencil, graphing calculator, Graphing Exponential Functions Discovery
Worksheet BLM
In this activity, the students will discover the graph of an exponential function and its
domain, range, intercepts, shifts, and effects of differing bases, and will use the graph to
explain irrational exponents.
Math Log Bellringer:
(1) Graph y = x2 and y = 2x on your graphing calculator individually
with a window of x: [10, 10] and y: [10, 10] and describe the
similarities and differences.
(2) Graph them on the same screen with a window of x: [10, 10]
and y: [10, 100] and describe any additional differences.
Solutions:
(1) Both have the same domain, all reals, but the range of y
= x2 is y > 0 and the range of y = 2x is y > 0. There are
different yintercepts, (0, 0) and (0, 1). The end-
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behavior is the same as x approaches , but as x approaches , the
end-behavior of y = x2 approaches and the end- behavior of y = 2x
approaches 0.
(2) y = 2x grows faster than y = x2.
Activity:
Discuss the Bellringer in terms of how fast the functions increase. Show how fast
exponential functions increase by the following demonstration:
Place 1 penny on the first square of a checker board, double it and place two pennies
on the second square, 4 on the next, 8 on the next, and so forth until the piles are
extremely high. Have the students determine how many pennies would be on the last
square, tracing to that number on their calculators. Measure smaller piles to determine
the height of the last pile and compare it to the distance to the sun, which is
93,000,000 miles.
Graphing Exponential Functions Discovery Worksheet BLM:
On this worksheet, the students will use their graphing calculators to graph the
exponential functions f(x) = bx with various changes in the constants to determine
how these changes affect the graph.
The students can graph each equation individually or use the Transformation APP on
the TI 83 and TI 84 as they did in Activity 7 in Unit 5.
To use the Transformation APPS:
Turn on the application by pressing APPS , Transfrm
ENTER ENTER
Enter the equation y1 = Bx
Set the window by pressing WINDOW and cursor to
SETTINGS, set where B will start, in this example
B = 2, and adjust the step for B to Step = 1.
GRAPH and use the cursor to change the values
of B.
When finished, uninstall the transformation APP by
pressing APPS , Transfrm, 1:Uninstall
For more information see the TI 83/TI84
Transformation App Guidebook at
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Distribute Graphing Exponential Functions Discovery Worksheet BLM. Graph the first
equation together having the students locate the yintercept and trace to high and low x
values to determine end-behavior and that there is a horizontal asymptote at y = 0. (This
is not obvious on the graph.) Have them sketch the graph and dot the horizontal
asymptote on the xaxis.
Arrange the students in pairs to complete the graphs and answer the questions.
Circulate to make sure they are graphing correctly.
When the students finish the worksheet, go over the answers to the questions making
sure they have all come to the correct conclusions.
Examine the graph of f(x) = 2x in #1 and discuss its continuity by using the trace function
3
on the calculator to determine f , f 3 , and f 2 . Because it is a continuous
2
function, a number can be raised to any real exponent, rational and irrational, and have a
value. Discuss irrational exponents with the students and have them apply the Laws of
Exponents to simplify the following expressions:
(1) 5 3 56 3
65 2
(2)
6 2
8
(3)
2
2 5 43 5
(4) 1
16 4 82 5
Solutions: (1) 57 3 , (2) 64 2 , (3) 4 , (4) 21 5
Assign additional graphing problems and irrational exponent problems from the math
textbook.
Activity 3: Regression Equation for an Exponential Function (GLEs: Grade 9: 10, 15,
29; Grade 10: 20, 27; Grade 11/12: 2, 4, 6, 7, 8, 10, 19, 22, 27, 28, 29)
Materials List: paper, pencil, graphing calculator, Exponential Regression Equations BLM
In this activity, the students will enter data into their calculators and change all the
parameters for an exponential equation of the form, y = Abx–C + D, to find the best regression
equation. They then will use the equation to interpolate and extrapolate.
Math Log Bellringer:
Use what you know about shifts and translations to graph the following without a
calculator locating asymptotes and yintercepts.
(1) f(x) = 3x
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(2) f(x) = –3x
(3) f(x) = 3–x
(4) Describe the translations in #2 and #3
(5) f(x) = 3x – 4
(6) f(x) = 3x – 4
(7) Describe the shifts in #5 and #6
(8) f(x) = 5(3x)
Solutions:
(1) horizontal asymptote at y = 0,
(2) horizontal asymptote at y = 0,
(3) horizontal asymptote at y = 0,
(4) #2 reflects the parent function across the xaxis and #3 reflects it across
the yaxis
1
(5) 0, , horizontal asymptote at y = 0
81
(6) horizontal asymptote at y = 4
(7) #5 shifted the parent function to the right 4 and #6 shifted it down 4
(8) horizontal asymptote at y = 0
Algebra IIUnit 6Exponential and Logarithmic Functions 129
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Activity:
Use the Bellringer to check for understanding of translations.
Exponential Regression Equations BLM:
In the first section on this Exponential Regression Equations BLM, the students will
enter real-world data into their calculators to create a scatter plot, find an exponential
regression (prediction) equation, and use the model to interpolate and extrapolate
points to answer real-world questions. In the second section, they will be using the
method of finite differences to determine which data is exponential and to find its
regression equation.
Distribute the Exponential Regression Equations BLM and have students work in
pairs.
If necessary, review with students the steps for making a scatter plot. (To enter data
on a TI 84 calculator: STAT, 1:Edit, enter data into L1 and L2 . To set up the plot of the
data: 2nd , [STAT PLOT] (above Y= ), 1:PLOT1, ENTER, On, Type: Scatter Plot, Xlist:
L1, Ylist: L2, Mark (any). To graph the scatter plot: ZOOM , 9: ZoomStat).
When all the students have found an equation in Section 1, Real World Exponential
Data, write all the equations on the board and have the students determine which
equation is the best fit.
Have students use that best fit equation to answer the interpolation and extrapolation
questions in #3.
Discuss how they determined the answer to #4. Since the calculator cannot trace to a
dependent variable, the best method is to graph y = 25 and find the point of
intersection. Review this process with the students. On the TI84, use 2nd [CALC]
(above TRACE ), 5: intersect, enter a lower and upper bound on either side of the point
of intersection and ENTER .
Review the Method of Finite Differences from Unit 2, Activity 8, and have students
apply it to determine which data in Section 2 is exponential then to find a regression
equation for each set of data.
When all students have completed the BLM discuss their answers.
Activity 4: Exponential Data Research (GLEs: Grade 9: 10, 15; Grade 10: 20, 27;
Grade 11/12: 4, 6, 7, 8, 10, 19, 22, 24, 27, 29)
Materials List: paper, pencil, graphing calculator (or computer), Exponential Data Research
Project BLM
Activity:
This is an out-of-class activity in which the students will find data that is best modeled by
an exponential curve.
Exponential Data Research Project:
Distribute the Exponential Data Research Project BLM and discuss the directions
with the students.
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State that this is an individual project and each person must have different data, so
they should be the first to print out the data and claim the topic. Possible topics
include: US Bureau of Statistics, Census, Stocks, Disease, Bacteria Growth,
Investments, Land Value, Animal Population, number of stamps produced each year.
Give the students approximately one week to complete the project.
When the students hand in their projects have each student present his/her findings to
the class.
Activity 5: Solving Exponential Equations with Common Bases (GLEs: 2, 4, 10)
Materials List: paper, pencil, graphing calculator
In this activity, students will use their properties of exponents to solve exponential equations
with similar bases.
Math Log Bellringer:
Graph y = 2x+1 and y = 82x+1 on your graphing calculator. Zoom in and find the point
of intersection. Define point of intersection.
Solution:
A point of intersection is an ordered pair that is a solution for both equations.
Activity:
Define exponential equation as any equation in which a variable appears in the exponent
and have students discuss a method for solving the Bellringer analytically.
Students have a difficult time understanding that a point of intersection is a shared x
and yvalue; therefore, to solve for a point of intersection analytically, the students
should solve the set of equations simultaneously, meaning set y = 2x+1 and y = 82x+1
equal to each other, 2x+1 = 82x+1 and solve for x.
They should develop the property, necessitating getting the same base and setting the
exponents equal to each other.
Solution:
2x+1 = 82x+1
2x+1 = (23)2x+1
2x+1 = 26x+3
2
x + 1 = 6x + 3 x
5
Use the property above to solve the following equations. 3x+2 = 92x
(1) 3–x = 81
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x 1 x
3 27
(2)
2 8
(3) 8x = 4
x
1
(4) 81
27
2 2 4
Solutions: (1) x , (2) x = –4, (3) x = ½, (4) x , (5) x
3 3 3
Assign addtional problems from the math textbook.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 6: Inverse Functions and Logarithmic Functions (GLEs: Grade 9: 4, 35, 36;
Grade 11/12: 2, 3, 4, 8, 25, 27)
Materials List: paper, pencil, graph paper, graphing calculator
In this activity, students will review the concept of inverse functions in order to develop the
logarithmic function which is the inverse of an exponential function.
Math Log Bellringer:
2
(1) Find the domain and range of f ( x)
x 1
2
(2) Find the inverse f–1(x) of f ( x) and state its domain and range.
x 1
(3) Discuss what you remember about inverse functions.
Solutions:
(1) D: x ≠ 1, R: y ≠ 0
2 x
(2) f 1 x D: x ≠ 0, R: y ≠ 1
x
(3) The students should generate these statements:
o Definition: f1(x) is an inverse function of f(x) if and only if
f f 1 x f 1 f x x .
o You find the inverse of a function by swapping the x and y and solving
for y.
o The graphs of a function and its inverse are symmetric over the line
y = x.
o You swap the domains and ranges.
o In all ordered pairs, the abscissa and ordinate are swapped.
o If an inverse relation is going to be an inverse function, then the
original function must have a onetoone correspondence.
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o You can tell if an inverse relation is going to be an inverse function
from the graph if the original function passes both the vertical and
horizontal line test.
Activity:
Review the concepts of an inverse function from Unit 1, Activity 12, and have the
students practice finding an inverse function on the following problem:
(1) Analytically find the inverse of f(x) = x2 + 3 on the restricted domain x > 0
(2) Prove they are inverses using the definition f f 1 x f 1 f x x
1
(3) What is the domain and range of f(x) and f (x)?
(4) Graph both by hand on the same graph labeling x and yintercepts.
(5) Graph the line y = x on the same graph and locate one pair of points that are
symmetric across the line y = x.
(6) Why is the domain of f(x) restricted?
Solution:
(1) f 1 x x 3
2
(2) x 3 3 x2 3 3 x x2 x if x 0
(3) f(x): domain x > 0, range y > 3, f1(x): domain x > 3, range y > 0
(4) y intercept of f(x) is (0, 3), xintercept of f1(x) is (3, 0)
(5) Ordered pairs may vary.
1
f(2) = 7, f (7) = 2
(6) f(x) would not have a onetoone
correspondence and the inverse would not
be a function.
Give the students graph paper and have them discover the
inverse of the exponential function in the following manner:
Graph f(x) = 2x dotting the horizontal asymptote by hand
and label the ordered pairs at x = 2, 1, 0, 1, 2, 3.
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Is this function a onetoone correspondence?
(Solution: yes, therefore an inverse function must exist)
Graph y = x on the same graph and draw the inverse
function by plotting ordered pairs on the inverse and
dotting the vertical asymptote. Discuss the graph of the
inverse – domain, range, increasing and decreasing,
intercepts, and asymptote.
On the calculator graph y1 = 2x and y2 = x. Use the
calculator function, ZOOM, 5:ZSquare. Draw the graph of
the inverse on graphing calculator ( 2nd , [DRAW], (above
PRGM ), 8: DrawInv, VARS , YVARS, 1:Function, 1:Y1).
Have students try to find the inverse of y = 2x analytically by
swapping x and y and attempting to isolate y.
Use this discussion to define logarithm and its relationship
to exponents: logba = c if and only if bc = a
Use the definition to rewrite log28 = 3 as an exponential equation. (Solution: 23 = 8)
Find log525 by thinking exponentially: "5 raised to what power = 25?"
(Solution: 52 = 25 therefore log525 = 2)
Define common logarithm as logarithm with base 10 in which the base is understood:
f(x) = log x. The calculator only finds log base 10. On the calculator, have the
students ZOOM Square and graph y1 = 10x, y2 = log x, y3 = x to see that y1 and y2 are
symmetric across the line y = x.
Have the students find log 100 without a calculator (Solution: log 100 = 2 because
102 = 100) and use the definition of logarithm to evaluate the following logarithmic
expressions. Have students write "because" and the exponential equivalent after each
problem:
(1) log5125
(2) log 0.001
(3) log 1 16
4
(5) log381
(6) log 3 312
Solutions:
(1) log5125 = 3 because 53 =125
(2) log .001 = 3 because 103 = .001
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2
1
(3) log 1 16 2 because 16
4 4
4
(4) log381=4 because 3 = 81
24
1
3
24
(5) log 3 3 24 because
12
32 312
Applying the definition of inverses f f x f f x x to logs implies
1 1
b logb x log b b x x . Use the definition of inverse to simplify the following
expressions:
(1) 3log3 8
log5 2
(2) 5
(3) log 3 317
(4) log15 15 13
Solutions: (1) 8, (2) 2 , (3) 17, (4) 13
Assign additional problems from the math textbook to practice these skills.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 7: Graphing Logarithmic Functions (GLEs: Grade 9: 4, 36: Grade 11/12: 3, 4,
6, 7, 8, 10, 19, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Graphing Logarithmic Functions
Discovery Worksheet BLM
In this activity, students will learn how to graph logarithmic functions, determine the
properties of logarithmic functions, and apply shifts and translations.
Math Log Bellringer:
Evaluate the following: If there is no solution, discuss why.
(1) log 100000 =
(2) log232 =
(3) log 1 243
9
(4) log2 4
Solutions:
(1) 5 , (2) 5, (3) 5
2
(4) no solution, 2 raised to any power will be a positive number.
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Activity:
Use the Bellringer to check for understanding of evaluating logarithms in different bases.
Graphing Logarithmic Functions:
In the Graphing Logarithmic Functions Discovery Worksheet, the students will first
graph f(x) = log x by hand by plotting points and discuss its local and global
characteristics, then use their knowledge of shifts to graph additional log functions by
hand.
Distribute the Graphing Logarithmic Functions Discovery Worksheet BLM. Have
students work in pairs to complete the first section of the worksheet. This is a
noncalculator worksheet so students can get a better understanding of the logarithm
function. Circulate to make sure they are plotting the points correctly. When they
have finished the first section, review the answers to the questions.
Have students complete the worksheet and review answers to the questions.
When they have finished, have students individually graph the following by hand to
check for understanding.
(1) Graph f(x) = log2 x plotting and labeling five ordered pairs.
(2) Graph f(x) = log2 (x 3) + 4
Solutions:
(1) Ordered pairs: (½, 1), (1, 0), (2, 1), (4, 2), (8, 3)
(2)
Activity 8: Laws of Logarithms and Solving Logarithmic Equations (GLEs: Grade 9:
2, 4, 5, 10; Grade 11/12: 2, 3, 10)
Materials List: paper, pencil, graphing calculator
In this activity, the students will express logarithms in expanded form and as a single log in
order to solve logarithmic equations.
Math Log Bellringer:
Solve for x. If there is no solution, discuss why.
(1) log2x = 3
(2) log525 = x
(3) logx16 = 4
(4) log3(log273)=log4x
(5) logx (36) = 2
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Solutions:
(1) x = 8
(2) x = 2
(3) x = 2
(4) x = ¼
(5) no solution. Bases must be positive so a positive number raised to any
power will be positive.
Activity:
Use the Bellringer to discuss how to solve different types of logarithmic equations by
changing them into exponential equations.
Give students additional practice problems from the math textbook.
Have the students discover the Laws of Logarithms using the following modified directed
readingthinking activity (DRTA) (view literacy strategy descriptions). DR-TA is an
instructional approach that invites students to make predictions, and then to check their
predictions during and after the reading. DR-TA provides a frame for self-monitoring
because of the pauses throughout the reading to ask students questions. This is a
modified a DRTA because the students will be calculating not reading.
In DRTA, first activate and build background knowledge for the content to be read.
This often takes the form of a discussion eliciting information the students may
already have, including personal experience, prior to reading. Ask the students to
reiterate the first three Laws of Exponents developed in Activity 1 and write the
words for the Law on the board.
Solutions:
(1) When you multiply two variables with the same base, add exponents.
(2) When you divide two variables with the same base, subtract the exponents.
(3) When you raise a variable with an exponent to a power, multiply the
exponents.
Next in DRTA, students are encouraged to make predictions about the text content.
Ask the students to list what they think will happen with logarithms and list these on
the board.
Then in DRTA, guide students through a section of text, stopping at predetermined
places to ask students to check and revise their predictions. This is a crucial step in
DR-TA instruction. When a stopping point is reached, the teacher asks students to
reread the predictions they wrote and change them, if necessary, in light of new
evidence that has influenced their thinking. Have the students find the following
values in their calculators rounding three places behind the decimal. Once they have
finished, have them reread the predictions to see if they want to change one.
(1) log 4 + log 8 (2) log 32, (3) log ½ + log 100, (4) log 50
Solutions : (1 & 2) 1.505, (3 & 4) 1.699
Continue this cycle with the next set of problems stopping after #8 and #12 to rewrite
predictions.
(5) log 16 log 2 (6) log 8 (7) log 4 – log 8 (8) log 0.5
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Solutions: (5 & 6) 0.903, (7& 8) 0.301
(9) 2log 4 (10) log 16 (11) ½ log 9 (12) log 3
Solutions: (9 & 10) 1.204, (11 & 12) 0 .477
When the students are finished, their revised predictions should be the Laws of
Logarithms. Write the Laws symbolically and verbally. Stress the need for the same
base and relate the Laws of Logs back to the Laws of Exponents.
(1) logb a + logb c = logb ac. Adding two logs with the same base is equivalent to
taking the log of the product the inverse operation of the first Law of
Exponents.
a
(2) logb a logb c logb . Multiplying two logs with the same base is equivalent to
c
taking the log of the quotient the inverse operation of the second Law of
Exponents.
(3) a logb c = logb ca. Multipling a log by a constant is equivalent to taking the log of
the number raised to that exponent the inverse operation of the third Law of
Exponents.
Check for understanding by asking the students to solve the following problems
without a calculator:
(1) log 4 + log 25
(2) log3 24 log38
(3) ½ log2 64
Solutions: (1) 10, (2) 1, (3) 3
Give guided practice problems solving exponential equations by applying the Laws of
Logs. Remind students that the domain of logarithms is x > 0; therefore, all answers
should satisfy this domain.
(1) log x + log (x 3) = 1
(2) log4 x log4 (x 1) = ½
(3) log5 (x 2) + log5 (x 1) = log5 (4x 8)
Solutions:
(1) x = 5 is the solution because x = 2 is not in the domain
(2) x = 2
(3) x = 5 is the solution because x = 2 is not in the domain of log5 (x 2)
Assign additional problems from the math textbook.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
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Activity 9: Solving Exponential Equations with Unlike Bases (GLEs: Grade 9: 4;
Grade 11/12: 2, 3, 10)
Materials List: paper, pencil, graphing calculator
Students will use logarithms to solve exponential equations of unlike bases and will develop
the change of base formula for logarithms.
Math Log Bellringer:
Solve for x: If it cannot be solved by hand, discuss why.
(1) 32x = 27 x+ 1 by hand.
(2) 23x = 64x
Solution:
(1) x = –3
(2) This problem cannot be solved by hand because 2 and 6 cannot be
converted to the same base.
Activity:
Use the Bellringer to review solving exponential equations which have the same base.
Have students find log10 62 on the calculator, then change log10 62 = x to the exponential
equation 10x = 62, noting that this is an exponential equation with different bases, 10 and
6. Develop the process for solving exponential equations with different bases using
logarithms.
(1) When x is in the exponent, take the log of both sides using base 10 because that
base is on the calculator.
(2) Apply the 3rd Law of Logarithms to bring the exponent down to the coefficient.
(3) Isolate x.
Guided Practice:
4(x+3) = 7
log 4(x + 3) = log 7
(x + 3) log 4 = log 7
log 7
x3
log 4
log 7
x 3
log 4
Use the calculator to find the point of intersection of y = 4x+3 and y = 7.
Discuss this alternate process for solving the equation 4x+3 = 7.
Compare the decimal answer to the decimal equivalent of the exact
answer above, and discuss the difference in an exact answer and
log 7
decimal approximation. (Solution: x 3 1.596 )
log 4
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Application:
Have students work in pairs to solve the following application problem. When they
finish the problem, have several groups describe the steps they used to solve the problem
and what properties they used.
A biologist wants to determine the time t in hours needed for a given culture to grow
to 567 bacteria. If the number N of bacteria in the culture is given by the formula
N=7(2)t, find t. Discuss the steps used to solve this problem and the properties you
used. Find both the exact answer and decimal approximation.
Solution: 6.3 hours
log10 8
Have students determine log2 8 by hand and on the calculator, then formulate a
log10 2
log b a
formula for changing the base: log c a . Verify the formula by solving the
log b c
equation log5 6 = x in the following manner:
log5 6 = x
5x = 6
log 5x = log 6
x log 5 = log 6
log10 6
x
log10 5
Assign additional problems from the math textbook solving exponential equations and
changing base of logarithms.
Activity 10: Exponential Growth and Decay (GLEs: Grade 9: 10, 15, 29; Grade 10: 27;
Grade 11/12: 2, 3, 4, 7, 8, 10, 17, 19, 20, 24, 29)
Materials List: paper, pencil, graphing calculator, Skittles (50 per group), Exponential
Growth and Decay Lab BLM, 1 cup per group
Students will model exponential growth and apply logarithms to solve the problems.
Math Log Bellringer:
A millionaire philanthropist walks into class and offers to either pay you one cent on
the first day, two cents on the second day, and double your salary every day thereafter
for thirty days or to pay you one lump sum of exactly one million dollars. Write the
exponential equation that models the daily pay and determine which choice you will
take.
Solution: y = 2x 1 if x starts with 1 and ends with 30, y = 2x if x starts with 0
and ends with 29. If you took the first option, after 30 days you would have
$10,737,418.23.
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Activity:
Have students explain the process they used to generate the pay for each of the thirty days
to find the answer. Discuss the following calculator skills.
Most students will have written down the 30 days of pay and added them up. Show the
different calculator methods for generating and adding a list of numbers.
(1) Iteration Method: On the home screen type 1 ENTER . Then type X
2 ENTER . Continue to press ENTER and count thirty days
recording the numbers and adding them up.
(2) List Method: STAT , EDIT. Put the numbers 1 through 30 in L1. In
L2, move the cursor up to highlight L2 and enter 2^(L1 1) ENTER
and L2 will fill with the daily salary. On the home screen, type 2nd
STAT (LIST), MATH, 5:sum (L2) and it will add all the numbers
in List 2 and give the answer in cents.
(3) Summing a Sequence: On the home screen, type 2ND , [LIST]
(above STAT), MATH, 5:sum(, 2nd [LIST] (above STAT), OPS,
5:seq(, 2^(x1), x, 1, 30)
Exponential Growth and Decay Lab:
In this lab the students will simulate exponential growth and decay using Skittles® (or
M & M's®) to find a regression equation and use that equation to predict the future.
Review, if necessary, how to enter data into a calculator and enter a regression
equation. (steps in the Activity 3 Exponential Regression Equations BLM)
Introduce the correlation coefficient. The correlation coefficient, r2,
is the measure of the fraction of total variation in the values of y.
This concept will be covered in depth in Advanced Math
Statistics, so it is sufficient to refer to r2 simply as the percentage of
points that are clustered in a small band about the regression
equation. Therefore, a higher percentage would be a better fit
regression equation. It is interesting to show the students the
formula that determines r, but the calculator will automatically
calculate this value. The feature must be turned on. 2ND ,
[CATALOG], (above 0. ), DiagnosticOn, ENTER . When the regression equation is
created, it will display the correlation coefficient.
n xy x y
r
n x2 x n y 2 y
2 2
Divide the students in groups of four. Give each group a cup with approximately 50
candies in each cup and the Exponential Growth and Decay Lab BLM.
As the groups finish the Exponential Growth section, circulate and have each group
explain the method they used to solve the related questions.
When the groups have finished both sets of data, combine the statistics and have half
of the groups find a regression equation and correlation coefficient for the whole set
of growth data. The other groups will find the regression equation and correlation
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coefficient for the decay data. Discuss the differences in a sample (the 50 candies
each group has) and a population (the entire bag of candies), then discuss the
accuracy of predictions based on the size of the sample.
Activity 11: Compound Interest and Half-Life Applications (GLEs: Grade 9: 4, 5, 10,
15, 35; Grade 10: 27; Grade 11/12: 2, 3, 10, 19, 24, 29)
Materials List: paper, pencil, graphing calculator
Students will develop the compound interest and half-life formulas then
use them to solve application problems.
Math Log Bellringer:
If you have $2000 dollars and you earn 6% interest in one year, how much money
will you have at the end of a year? Explain the process you used.
Solution: $2120. Students will have different discussions of how they came up
with the answer.
Activity:
Use the Bellringer to review the concept of multiplying by 1.06 to get the final amount in
a one-step process.
Discuss the meaning of compounding interest semiannually and quarterly. Draw an
empty chart similar to the one below on the board or visual presenter. Guide students
through its completion to develop a process to find the value of an account after 2 years.
o $2000 is invested at 6% APR (annual percentage rate) compounded semiannually
(thus 3% each 6 months = 2 times per year). What is the account value after t years?
o While filling in the chart, record on the board the questions the students ask such as:
1. Why do you divide .06 by 2?
2. Why do you have an exponent of 2t?
3. How did you come up with the pattern?
Time Do the Math Developing the Formula Account
years Value
0 $2000 $2000 $2000.00
½ $2000(1.03) $2000(1+.06/2) $2060.00
1 $2060(1.03) $2000(1+.06/2)(1+.06/2) $2121.80
1½ $2121.80(1.03) $2000(1+.06/2)(1+.06/2)(1+.06/2) $2185.454
2 $2185.454(1.03) $2000(1+.06/2)(1+.06/2)(1+.06/2)(1+.06/2) $2251.01762
t $2000(1+.06/2)2t
r
Use the pattern to derive the formula for finding compound interest: A t P(1 )nt .
n
A(t) represents the value of the account in t years,
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P the principal invested,
r the APR or annual percentage rate,
t the time in years,
n the number of times compounded in a year.
.06 2t
Have students test the formula A t 2000(1 ) by finding A(10), then using
2
the iteration feature of the calculator to find the value after 10 years.
Have the students use a modified form of questioning the author (QtA) (view literacy
strategy descriptions) to work additional problems.
The goals of QtA are to construct meaning of text, to help students go beyond the
words on the page, and to relate outside experiences to the texts being read.
Participate in QtA as a facilitator, guide, initiator, and responder. Students need to be
taught that they can, and should, ask questions of authors as they read.
In this modified form of QtA, the student is the author. Assign different rows of
students to do the calculations for investing $2000 with APR of 6% for ten years if
compounded (1) yearly, (2) quarterly, (3) monthly, and (4) daily. Then have the
students swap problems with other students and ask the questions developed earlier.
Once each student is sure that his/her partner has answered the questions and solved
the problem correctly, ask for volunteers to work the problem on the board.
Solutions:
.06 1(10)
(1) yearly: A t 2000(1 ) $3581.70
1
.06 4(10)
(2) quarterly: A t 2000(1 ) $3628.04
4
.06 12(10)
(3) monthly: A t 2000(1 ) $3638.79
12
.06 365(10)
(4) daily: A t 2000(1 ) $3644.06
365
Have students solve the following problem for their situations: How long will it
take to double your money in these situations? Again swap problems and once
again facilitate the QtA process.
Solutions:
.06 1(t )
(1) yearly: 4000 2000(1 ) t =11.896 years
1
.06 4(t )
(2) quarterly: $4000 2000(1 ) t =11.639 years
4
.06 12(t )
(3) monthly: 4000 2000(1 ) t =11.581 years
12
.06 365(t )
(4) daily: 4000 2000(1 ) t =11.553 years
365
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t
Define half-life, develop the exponential decay formula, A A0 1 where k is the
k
2
halflife, and use it to solve the following problem:
A certain substance in the book bag deteriorates from 1000g to 400g in 10 days. Find
its half-life.
Solution:
10
1 k
400 1000
2
t
1k
0.4
2
t
1k
log 0.4 log
2
10 1
log 0.4 log
k 2
log 0.4 10
log 0.5 k
10 log 0.5
k 7.565 days
log 0.4
Assign additional problems on compound interest and halflife from the math textbook.
Activity 12: Natural Logarithms (GLEs: Grade 9: 4, 15, 35, 36; Grade 10: 27; Grade
11/12: 2, 3, 4, 6, 8, 10, 24, 27, 29)
Materials List: paper, pencil, graphing calculator
The students will determine the value of e and define natural logarithm.
Math Log Bellringer:
Use your calculator to determine log 10 and ln e. Draw conclusions.
Solution: log 10 = 1 and ln e = 1. ln must be a logarithm with a base e.
Activity:
Define ln as a natural logarithm base e. Have students do the following activity to
discover the approximation of e. Let students use their calculators to complete the
following table. Have them put the equation in y1 and use the home screen and the
notation y1(1000) to find the values.
n 10 100 1000 10,000 100,000 1,000,000 1,000,000,000
n
1
1 2.05937 2.07048 2.7169 2.7181 2.7182682 2.718280469 2.718281827
n
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Define e as the value that this series approaches as n gets larger and larger. It is
approximately equal to 2.72 and was named after Leonard Euler in 1750. Stress that e is a
transcendental number similar to . Although it looks as if it repeats, the calculator has
limitations. The number is really 2.71828182845904590… and is irrational.
Graph y = ln x and y = ex and discuss inverses and the domain and range of y = ln x. Locate the
xintercept at (1, 0) which establishes the fact that ln e = 1.
n
Compare 1 1 to the compound interest formula, A(t) = Pert, which is derived by
n
increasing the number of times that compounding occurs until interest has been
theoretically compounded an infinite number of times.
Revisit the problem from Activity 11 in which the students invested $2000 at 6%
APR, but this time compound it continuously for one year and discuss the difference.
Solution: $3644.24
Revisit the problem in Activity 11 of how long it will take to double money. When
the students take the log of both sides to solve for t, they should use the natural
logarithm because ln e = 1.
Solution:
$4000 = $2000e.06t
2 = e.06t
ln 2 = ln e.06t
ln 2 = .06t ln e
ln 2 = .06t (1)
ln 2
t
.06
t = 11.552 years
Discuss use of this formula in population growth. Work with the students on the following
two part problem: If the population in Logtown, USA, is 1500 in 2000 and 3000 in 2005,
what would the population be in 2010?
o Most students will answer 4500. Take this opportunity to explain the difference in a
proportion, which is a linear equation having a constant slope, and population growth
which is an exponential equation that follows the A(t) = Pert formula.
o Part I: Find the rate of growth (r)
A(t) = Pert
3000 = 1500(er(5))
2 = e5r
ln 2 = le e5r
ln 2 = (5r) ln e
ln 2 = 5r
ln 2
r . Have students store this decimal representation in a
5
letter in the calculator such as R. Discuss how the error can
be magnified if a rounded number is used in the middle of a
problem.
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o Part II: Use the rate to solve the problem.
A(t) = Pert
A(10) = 1500(eR(10)) using the rate stored in R
A(10) = 6000
o Discuss the difference in what they thought was the answer (4500), which added
1500 every 5 years (linear), and the real answer (6000) which multiplied by 2
every 5 years (exponential).
Assign additional problems from the math textbook.
Critical Thinking Writing Assessment: (See Activity-Specific Assessments at end of unit.)
Activity 13: Comparing Interest Rates (GLEs: 2, 10, 24, 29)
Materials List: paper, pencil, graphing calculator, Money in the Bank Research Project BLM
This is an out-of-class activity. Distribute the Money in the Bank Research Project BLM.
Have students choose a financial institution in town or on the Internet. If possible, have each
student in a class choose a different bank. Have them contact the bank or go online to find
out information about the interest rates available for two different types of accounts and how
they are compounded. Have students fill in the following information and solve the following
problems. When all projects are in, have students report to the class.
Money in the Bank Research Project
Information Sheet: Name of bank, name of person you spoke to, bank address and phone
number or the URL if online, types of accounts, interest rates, and how funds are
compounded.
Problem: Create a hypothetical situation in which you invest $500.
(1) Find the equation to model two different accounts for your bank.
(2) Determine how much you will have at the end of high school and at the end of college for
each account. (Assume you finish high school in one year and college four years later.)
(3) Determine how many years it will take you to double your money for each account.
(4) Determine in which account you will put your money and discuss why.
(5) Display all information on a poster board and report to the class.
Algebra IIUnit 6Exponential and Logarithmic Functions 146
Louisiana Comprehensive Curriculum, Revised 2008 solving exponential equations with same base
(2) graphing y = ex and y = log x with shifts
(3) evaluating logs such as log2 8
Administer two comprehensive assessments:
(1) exponential equations and graphs, evaluating logs, properties of logs and
logarithmic graphs
(2) solving exponential equations with the same base and different bases, and
application problems
Activity-Specific Assessments
Teacher Note: Critical Thinking Writings are used as activity-specific assessments in
many of the activities in every unit. Post the following grading rubric on the wall for
students to refer Critical Thinking Writing
9
2
(1) Simplify .
2
(2) Simplify 9 .
(3) Discuss why the answers to problems 1 and 2 are different.
a , does not apply to
b
b
(4) Discuss why one of the Laws of Exponents, a c a b c c
this problem.
Solutions:
(1) 9
(2) –9
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(3) By order of operations, in problem 1 you have to square the expression
first to get 81 and then take the square root to get 9. In problem 2 you
have to take the square root first to get 3i, then square it to get 9.
(4) This Law of Exponents only applies when a > 0.
Activity 4: Evaluate the Exponential Data Research Project (see activity) using the
following rubric:
Grading Rubric for Data Research Project
10 pts. table of data with proper documentation (source and date of data)
10 pts. scatterplot with model equation from the calculator or spreadsheet (not
by hand)
10 pts. equations, domain, range,
10 pts. real world problem using extrapolation with correct answer
10 pts. discussion of subject and limitations of the prediction
10 pts. poster - neatness, completeness, readability
10 pts. class presentation
Activity 5: Critical Thinking Writing
(1) Solve the two equations: (a) x2 = 9 and (b) 3x = 9
(2) Discuss the family of equations to which they belong.
(3) Discuss how the equations are alike and how they are different.
(4) Discuss the two different processes used to solve for x.
Solutions:
(1) (a) x = ±3, (b) x = 2
(2) x2 belongs to the family of polynomial equations and 3x is an exponential
equation
(3) Both equations have exponent; but in the first the exponent is a number,
and in the 2nd the exponent is a variable
(4) (a) Take the square root of both sides. (b) Find the exponent for which you
can raise 3 to that power to get 9.
Activity 6: Critical Thinking Writing
The value of log316 is not a number you can evaluate easily in your head. Discuss
how you can determine a good approximation.
Solution:
Answers will vary but should discuss the fact that the answer to a log problem
is an exponent and 32 = 9 and 33 = 27 so log316 is between 2 and 3.
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Activity 8: Critical Thinking Writing
The decibel scale measures the relative intensity of a sound. One formula for the
I
decibel level, D, of sound is D 10log , where I is the intensity level in watts
I0
per square meter and I0 is the intensity of barely audible sound.
(1) If the intensity level of a jet is 1014 watts per square meter times the intensity of
barely audible sound (1014I0), what is the decibel level of a jet take-off.
(2) The decibel level of loud music with amplifiers is 120. How many times more
intense is this sound than a barely audible sound?
(3) Compare the decibel levels of jets and loud music.
(4) Are there any ordinances in your town about the acceptable decibel level of
sound?
Solutions: (1) 140 decibels, (2) 1012I0
Activity 12: Critical Thinking Writing
In 1990 statistical data estimated the world population at 5.3 billion with a growth
rate of approximately 1.9% each year.
(1) Let 1990 be time 0 and determine the equation that best models population
growth.
(2) What will the population be in the year 2010?
(3) What was the population in 1980?
(4) In what year will the population be 10 billion?
(5) Discuss the validity of using the data to predict the future.
Solution: (1) A = 5.3e.019t, (2) 7.8 billion, (3) 4.4 billion, (4) 2023
Activity 13: Evaluate the Money in the Bank Research Project (see activity) using the
following rubric:
Grading Rubric for Money in the Bank Research Project
10 pts. Information sheet: Name of bank, name of person you spoke to, bank
address and phone number or the URL if online, types of accounts,
interest rates, and how funds are compounded (source and date of data)
10 pts. Compound interest equation for each situation; account value for both
accounts at the end of high school, college, and when you retire in 50
years (show all your work)
10 pts. Solution showing your work of how long it will take you to double your
money in each account
10 pts. Discussion of where you will put your money and why
10 pts. Poster - neatness, completeness, readability
10 pts. Class presentation
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Algebra II
Unit 7: Advanced Functions
Time Frame: Approximately four weeks
Unit Description
This unit ties together all the functions studied throughout the year. It categorizes them,
graphs them, translates them, and models data with them.
Student Understandings
The students will demonstrate how the rules affecting change of degree, coefficient, and
constants apply to all functions. They will be able to quickly graph the basic functions and
make connections between the graphical representation of a function and the mathematical
description of change. They will be able to translate easily among the equation of a function,
its graph, its verbal representation, and its numerical representation.
Guiding Questions
1. Can students quickly graph lines, power functions, radicals, logarithmic,
exponential, step, rational, and absolute value functions?
2. Can students determine the intervals on which a function is continuous,
increasing, decreasing, or constant?
3. Can students determine the domains, ranges, zeroes, asymptotes, and global
characteristics of these functions?
4. Can students use translations, reflections, and dilations to graph new functions
from parent functions?
5. Can students determine domain and range changes for translated and dilated
abstract functions?
6. Can students graph piecewise defined functions, which are composed of several
types of functions?
7. Can students identify the symmetry of these functions and define even and odd
functions?
8. Can students analyze a set of data and match the data set to the best function
graph?
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Unit 7Algebra
characteristics of the function (A-3-H)
7. Explain, using technology, how the graph of a function is affected by change of
degree, coefficient, and constants in polynomial, rational, radical, exponential,
and logarithmic functions (A-3-H)
8. Categorize non-linear graphs and their equations as quadratic, cubic,
exponential, logarithmic, step function, rational, trigonometric, or absolute
value (A-3-H) (P-5-H)
10. Model and solve problems involving quadratic, polynomial, exponential,
logarithmic, step function, rational, and absolute value equations using
technology (A-4-H)
Geometry
16. Represent translations, reflections, rotations, and dilations of plane figures using
sketches, coordinates, vectors, and matrices (G-3-H)
Data Analysis, Probability, and Discrete Math
19. Correlate/match data sets or graphs and their representations and classify them
as exponential, logarithmic, or polynomial functions (D-2-H)
20. Interpret and explain, with the use of technology, the regression coefficient and
the correlation coefficient for a set of data (D-2-H)
22. Explain the limitations of predictions based on organized sample sets of data
(D-7-H)
Patterns, Relations, and Functions
Grade 9
35. Determine if a relation is a function and use appropriate function notation(P-1-H)
36. Identify the domain and range of functions (P-1Algebra IIUnit 7Advanced Functions 151
Louisiana Comprehensive Curriculum, Revised 2008
GLE # GLE Text and Benchmarks
29. Determine the family or families of functions that can be used to represent a
given set of real-life data, with and without technology (P-5-H)
Sample Activities
Ongoing Activity: Little Black Book of Algebra II Properties
Materials List: black 7Advanced Functions
1
7.1 Basic Graphs Graph and locate f(1): y = x, x2, x3, x , 3 x , x , , x , log x, 2x.
x
7.2 Continuity – provide an informal definition and give examples of continuous and
discontinuous functions.
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7.3 Increasing, Decreasing, and Constant Functions – write definitions and draw example
graphs such as y 9 x 2 , state the intervals on which the graphs are increasing and
decreasing.
7.4 Even and Odd Functions – write definitions and give examples, illustrate properties of
symmetry, and explain how to prove that a function is even or odd (e.g., prove that
y = x4 + x2 + 2 is even and y = x3 + x is odd).
7.5 General Piecewise Function – write the definition and then graph, find the domain and
range, and solve the following example f ( x) R 1
2
Sxx if x 5
for f (4) and f (1).
T 2
if x 5
For properties 7.6 7.9 below, do the following:
Explain in words the effect on the graph.
Give an example of the graph of a given abstract function and then the
function transformed (do not use y = x as your example).
Explain in words the effect on the domain and range of a given function. Use
the domain [–2, 6] and the range [–8, 4] to find the new domain and range of
the transformed function.
7.6 Translations (x + k) and (x k), (x) + k and (x) k
7.7 Reflections (–x) and –(x)
7.8 Dilations (kx), (|k|<1 and |k|>1), k(x) (|k|<1 and |k|>1)
7.9 Reflections (|x|) and |(x)|
Activity 1: Basic Graphs and their Characteristics (GLEs: 6, 8, 25, 27)
Materials List: paper, pencil, graphing calculator, Math Log Bellringer BLM
In this activity, the students will work in groups to review the characteristics of all the basic
graphs they have studied throughout the year. They will also develop a definition for the
continuous, increasing, decreasing, and constant functions.
Math Log Bellringer:
Graph the following by hand, locate zeroes and f(1), and identify the function.
(1) f(x) = x
(2) f(x) = x2
(3) f ( x ) x
(4) f(x) = x3
(5) f(x) = |x|
(6) f(x) = 2x
1
(7) f ( x)
x
(8) f x 3 x
(9) f(x) = log x
(10) f ( x) x
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Solutions:
(1) (6)
f (1) = 1, linear function, f(1) = 2, exponential function
zero (0,0) no zeroes
(2) (7)
f (1) =1, quadratic function f(1) = 1, rational function
also polynomial function, no zeroes
zero (0, 0)
(3) (8)
f(1) = 1, radical function f(1) = 1 , radical function
square root function, zero (0, 0) cube root function, zero (0, 0)
(4) (9)
f(1) = 1, cubic function f(1) = 0, logarithmic function,
also polynomial function, zero (1, 0)
zero (0, 0)
(5) (10)
f(1) = 1, f(1) = 1,
absolute value function, zero (0, 0) greatest integer function,
zeroes: 0 < x < 1
Activity:
Overview of the Math Log Bellringers:
As in previous units, each in-class activity in Unit 7 is started with an activity called
a MathAlgebra IIUnit 7Advanced Functions 154
Louisiana Comprehensive Curriculum, Revised 2008 Function Calisthenics: Use the Bellringer to review the ten basic parent graphs. Then
have the students stand up, call out a parent function, and ask them to form the shape of
the graph with their arms.
Increasing/decreasing/constant functions:
o Ask students to come up with a definition of continuity. (An informal definition of
continuity is sufficient for Algebra II.)
o Then have them develop definitions for increasing, decreasing, and constant
functions.
o Have students look at the abstract graph to the right
and determine if it is continuous and the intervals
in which it is increasing and decreasing. (Stress the
concept that when intervals are asked for, students
should always give intervals of the independent
variable, x in this case, and the intervals should
always be open intervals.)
Solution: Increasing , 1 0,
Decreasing (–1, 0)
o Have each student graph any kind of graph he/she desires on the graphing calculator
and write down the interval on which the graph is increasing and decreasing. Have
students trade calculators with a neighbor and answer the same question for the
neighbor's graph, then compare answers
Flash that Function: Divide students into groups of four and give each student ten blank
5 X 7" cards to create vocabulary cards (view literacy strategy descriptions). When
students create vocabulary cards, they see connections between words, examples of the
word, and the critical attributes associated with the word such as a mathematical formula
or theorem. Have them choose assignments – Grapher, Symbol Maker, Data Driver, and
Verbalizer. Have each member of the group create flash cards of the ten basic graphs in
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the Bellringer activity, but the front of each will be different based on his/her
assignment. (They can use their Little Black Books to review the information.) The front
of Grapher's card will have a graph of the function. The front of the Symbol Maker's
card will have the symbolic equation of the function. The front of the Data Driver's card
will have a table of data that models the function. The front of the Verbalizer's card will
have a verbal description of the function. The back of the card will have all of the
following information: graph, function, the category of parent functions, family, table of
data, domain, range, asymptotes, intercepts, zeroes, end-behavior, and increasing or
decreasing. Once all the cards are complete, have students practice flashing the cards in
the group asking questions about the function, then set up a competition between groups.
Activity 2: Horizontal and Vertical Shifts of Abstract Functions (GLEs: Grade 9: 36;
Grade 11/12: 4, 6, 7, 8, 16, 19, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Translations BLM
In this activity, the students will review horizontal and vertical translations, apply them to
abstract functions, and determine the effects on the domain and range.
Math Log Bellringer:
Graph the following without a calculator: Discuss how the shifts in #25 change the
domain, range, and vertex of the parent function.
(1) f(x) = x2
(2) f(x) = x2 + 4
(3) f(x) = x2 – 5
(4) f(x) = (x + 4)2
(5) f(x) = (x – 5)2
Solutions:
(1) (4)
no change in domain and range,
vertex moves left
(2)
changes the range,
vertex moves up (5)
no change in domain or range,
vertex moves right
(3)
changes the range,
vertex moves down
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Activity:
Have the students check the Bellringer graphs with their calculators and use the
Bellringer to ascertain how much they remember about translations.
Vertical Shifts: f x k
o Have the students refer to Bellringer problems 1 through 3 to develop the rule that
f(x) + k shifts the functions up and f(x) – k shifts the functions down.
o Determine if this shift affects the domain or range. (Solution: range)
o For practice, have students graph the following:
(1) f(x) = x3
(2) f(x) = x3 + 4
(3) f(x) = x3 – 6
Solutions:
(1) (2) (3)
Horizontal Shifts: f x k
o Have the students refer to Bellringer problems 1, 4, and 5 to develop the rule that +k
inside the parentheses shifts the function left and – k shifts the function right,
stressing that it is the opposite of what seems logical when shown in the parentheses.
o Determine if this shift affects the domain or range. (Solution: domain)
o For practice, have students graph the following:
(1) f(x) = x3
(2) f(x) = (x + 4)3
(3) f(x) = (x – 6)3
Solutions:
(1) (2) (3)
Abstract Translations
Divide students into groups of two or three and distribute the Translations BLM.
Have students work the first section shifting an abstract graph vertically and
horizontally. Stop after this section to check their answers.
Have students complete the Translations BLM graphing by hand, applying the shifts
to known parent functions. After they have finished, they should check their answers
with a graphing calculator.
Check for understanding by having students individually graph the following:
(1) f(x) = 4x
(2) g(x) = 4x 2
(3) h(x) = 4x 2
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Solutions:
(1) (2) (3)
Finish the class with Function Calisthenics again, but this time call out the basic
functions with vertical and horizontal shifts.
(e.g. x2, x2 + 2, x3, x3 – 4, x , x 4 , x 5 )
Activity 3: How Coefficients Change Families of Functions (GLEs: Grade 9: 35, 36;
Grade 11/12: 4, 6, 7, 8, 16, 19, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Reflections Discovery Worksheet BLM,
Dilations Discovery Worksheet BLM, Abstract Reflections & Dilations BLM
In this activity, the students will determine the effects of a negative coefficient, coefficients
with different magnitudes on the graphs, and the domains and ranges of functions.
Math Log Bellringer:
Graph the following on your calculator. Discuss what effect the negative sign has.
(1) f x x
(2) f x x
(3) f x x
Solutions:
(1)
(2) reflects graph across the xaxis, affects range
(3) reflects graph across the yaxis, affects domain
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Activity:
Discovering Reflections:
Distribute the Reflections Discovery Worksheet BLM. This BLM is designed to be
teacherguided discovery with the individual students working small sections of the
worksheet at a time, stopping after each section to discuss the concept.
Negating the function: –f(x).
o Have the students sketch their Bellringer problems on the Reflections & Dilations
Discovery Worksheet BLM and refer to Bellringer problems #1 and #2 to develop
the rule, "that a negative sign in front of the function reflects the graph across the
x-axis" (i.e., all positive y-values become negative and all negative y-values
become positive). Have students write the rule in their notebooks.
o Determine if this affects the domain or range. (Solution: range)
o Allow students time to complete the practice on problems #1 6. Check their
answers.
Negating the x within the function: f(–x)
o Have the student refer to Bellringer problems #1 and #3 to develop the rule, "that
the negative sign in front of the x reflects the graph across the y-axis" (i.e., all
positive x-values become negative and all negative x-values become positive).
Have students write the rule in their notebooks.
o Determine if this affects the domain or range. (Solution: domain)
o Allow students time to complete the practice on problems #713. Check their
answers.
Some changes do not seem to make a difference. Have the students examine the
following situations and answer the questions in their notebooks:
(1) Draw the graphs of f(x) = –x2 and h(x) = (–x)2.
(2) Discuss the difference in the graphs. Explain what effect the
parentheses have.
(3) Draw the graphs of f(x) = –x3 and h(x) = (–x)3. Find f(2) and h(2).
(4) Discuss order of operations. Discuss the difference in the graphs.
Explain what effect the parentheses have.
(5) Why do the parentheses affect one set of graphs and not the other?
Discovering Dilations Discovery Worksheet BLM:
Distribute the Dilations Discovery Worksheet BLM. This BLM is designed to be
teacher-guided discovery with the individual students working small sections of the
worksheet at a time, stopping after each section to discuss the concept.
Continue the guided discovery using the problems on the Dilations Discovery
Worksheet BLM, problems #1418.
Coefficients in front of the function: k f(x) (k > 0)
o Have the students refer to problems #14, 15, and 16 to develop the rule for the
graph of k f(x): If k > 1, the graph is stretched vertically compared to the graph of
f(x); and if 0 < k < 1, the graph is compressed vertically compared to the graph of
f(x). Write the rule in #19.
o Ask students to determine if this affects the domain or range. (Solution: range)
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Coefficients in front of the x: f(kx) (k > 0)
o Have the students refer to problems #14, 17, and 18 to develop the rule for the
graph of f(kx): If k > 1, the graph is compressed horizontally compared to the
graph of f(x); and if 0 < k < 1, the graph is stretched horizontally compared to the
graph of f(x). (When the change is inside the parentheses, the graph does the
opposite of what seems logical.) Write the rule in #20.
o Determine if this change affects the domain or range. (Solution: domain) Write
the rule in #21.
o Allow students to complete the practice on this section in problems #2228.
Abstract Reflections and Dilations:
Distribute the Abstract Reflections & Dilations BLM. Divide students into groups of
two or three to complete this BLM, problems #2934.
When the students have completed this BLM, have them swap papers with another
group. If they do not agree, have them justify their transformations.
More Function Calisthenics: Have the students stand up, call out a function, and have
them show the shape of the graph with their arms. This time have one row make the
parent graph and the other rows make graphs with positive and negative coefficients
(i.e., x2, –x2, 2x2, x3, –x3, x , – x , x ).
Activity 4: How Absolute Value Changes Families of Functions (GLEs: Grade 9: 35,
36; Grade 11/12: 4, 6, 7, 8, 16, 19, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Abstract Reflections and Dilations BLM in
Activity 3
In this activity, students will discover how a graph changes when an absolute value sign is
placed around the entire function or placed just around the variable.
Math Log Bellringer:
(1) Graph f(x) = x2 – 4 by hand and locate the zeroes.
(2) Use the graph to solve x2 – 4 > 0.
(3) Use the graph to solve x2 – 4 < 0.
(4) Discuss how the graph can help you solve #2 and #3.
Solutions:
(1) zeroes: {2, 2}
(2) x < –2 or x > 2,
(3) –2 < x < 2
(4) Since y = f(x) = x2 4, the xvalues that make the yvalues positive solve
#2. The xvalues that make the yvalues negative solve #3. Use the
zeroes as the endpoints of the intervals.
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Activity:
x if x 0
Review the definition of absolute value: x and review the rules for
x if x 0
writing an absolute value as a piecewise function: What is inside the absolute value is
both positive and negative. What is inside the absolute value affects the domain.
Absolute Value of a Function: |f(x)|
o Have students use the definition of absolute value to write |f(x)| as a piecewise
f ( x ) if f ( x ) 0
function f ( x)
f ( x ) if f ( x ) 0
o Have the students write |x2 – 4| as a piecewise function and use the Bellringer to
simplify the domains.
x2 4
if x 2 4 0 x 2 4
if x 2 or x 2
Solution: x 4
2
= )
x 4 if x 4 0 x 4 if 2 x 2
2 2 2
o Have the students graph the piecewise function by hand reviewing what –f(x) does to
a graph and find the domain and range.
Solution: D: all reals, R: y > 0
o Have the students check the graph f(x) = |x2 – 4| on the graphing
calculator.
o Have students develop the rule for graphing the absolute value of
a function: Make all y-values positive. More specifically, keep the portions of the
graphs in Quadrants I and II and reflect the graphs in Quadrant III and IV into
Quadrants I and II.
o Ask students to determine if this affects the domain or range. (Solution: range)
o Have students practice on the following graphing by hand first, then checking on the
calculator:
(1) Graph g(x) = |x3| and find the domain and range.
(2) Graph f(x) = |log x| and find the domain and range.
(3) If the function h(x) has a domain [–4, 6] and range [–3, 10], find the domain and
range of |h(x)|.
(4) If the function j(x) has a domain [–4, 6] and range [–13, 10], find the domain and
range of |j(x)|.
Solutions:
(1) D: all reals, R: y >0
(2) D: x > 0, R: y > 0
(3) D: same, R: [0, 10]
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(4) D: same, R: [0, 13]
Absolute Value only on the x: f(|x|)
Have the students write g(x) = (|x| – 4)2 – 9 as a piecewise function.
x 4 2 9 if x 0
Solution: g(x) = x 4 9
2
x 4 9 if x 0
2
o Have the students graph the piecewise function for g(x) by hand reviewing what
the negative only on the x does to a graph.
Solution:
o Have students find the domain and range of g(x). Discuss the fact that negative
xvalues are allowed and negative y-values may result. The range is determined
by the lowest y-value in Quadrant I and IV, in this case the vertex.
Solution: D: all reals, R: y > 9
o Have the students graph y1 = (x – 4)2 – 9 and y2 = (|x| –4)2 – 9 on the graphing
calculator. Turn off y1 and discuss what part of the graph disappeared and why.
o Have students develop the rule for graphing a function with only the x in the
absolute value. Graph the function without the absolute value first. Keep the
portions of the graph in Quadrants I and IV, discard the portion of the graph in
Quadrants II and III, and reflect Quadrants I and IV into II and III. Basically, the
y-output of a positive x-input is the same y-output of a negative x-input.
Have students practice on the following:
(1) Graph y = (|x| + 2)2 and find the domain and range.
(2) Graph y = (|x| – 1)(|x| 5)(|x| – 3) and find the domain and range.
(3) Graph y x 3 and find the domain and range.
(4) If the function h(x) has a domain [–4, 6] and range [–3, 10], find the domain and
range of h(|x|).
(5) If the function j(x) has a domain [–8, 6] and range [–3, 10], find the domain and
range of j(|x|).
Solutions:
(1) D: (∞, ∞), R: y > 4
(2) D: (∞, ∞), R: y > 15, this value cannot be
determined without a calculator until Calculus
because another minimum value may be lower than
the y-intercept
(3) D: x < –3 or x > 3, R: y > 0
(4) D: [–6, 6], R: cannot be determined
(5) D: [–10, 10], R: cannot be determined
Use the practice problems above to determine if f(|x|) affects the domain or range.
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Solution: f(|x|) affects both the domain and possibly the range. To find the new
domain, keep the domain for positive x-values and change the signs to include
the reflected negative x-values. The range cannot be determined unless the
maximum and minimum values of y in Quadrants I and IV can be determined.
Abstract Absolute Value Reflections: Have students draw in their notebook the same
abstract graph from the Abstract Reflections & Dilations BLM from Activity 3, then
sketch |g(x)| and g(|x|) putting solutions on the board.
Solutions:
(4, 8) (4, 8) (4, 8) (4, 8)
4 (–5, 3) 4 4
(1, 2) (1, 2) (1, 2) (1, 2)
(–5, –3) g(x) |g(x)| g(|x|)
Activity 5: Functions - Tying It All Together (GLEs: Grade 9: 35, 36; Grade 11/12: 4,
6, 7, 16, 25, 27, 28)
Materials List: paper, pencil, graphing calculators, Tying It All Together BLM, ½ sheet
poster paper for each group, index cards with one parent graph equation on each card
In this activity, students pull together all the rules of translations, shifts, and dilations.
Math Log Bellringer:
Graph the following by hand labeling h(1). Discuss the change in the graph and
whether the domain or range is affected.
(1) h(x) = 3x (4) h(x) = 3x + 1 (7) h(x) = 3|x|
(2) h(x) = 3x (5) h(x) = 3x + 1 (8) h(x) = 32x
x
(3) h(x) = (3x) (6) h(x) = |3 | (9) h(x) = 2(3x)
Solutions:
(1) (2) reflect across y-axis (3) reflects across x-axis
no change in D or R range changes
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(4) shift left 1 (5) shifts up 1, (6) no change in graph,
no change D or R range changes no change in D or R
(7) discard graph in Q II & III (8) horizontal compression, (9) vertical stretch,
and reflect Q I into Q II, yintercept stayed the same, yintercept changed,
no change in D or R. no change in D or R no change in D or R
Tying It All Together:
Divide students into groups of two or three and distribute the Tying It All Together
BLM.
Have students complete I. GRAPHING and review answers.
Have students complete II. DOMAINS AND RANGES and review answers.
When students have completed the worksheet, enact the professor knowitall
strategy (view literacy strategy descriptions). Explain that each group will draw one
graph and the other groups will come to the front of the class to be a team of Math
Wizards (or any other appropriate name). This team is to come up with the equation
of the graphs.
Distribute ½ sheet of poster paper to each group. Pass out an index card with one
parent graph equation: f(x) = x, f(x) = x2, f ( x ) x , f(x) = x3, f(x) = |x|, f ( x ) 1 ,
x
f(x) = 2 , f x x , f(x) = log x, f ( x) x , to secretly assign each group a
x 3
parent graph. Tell them to draw an x and yaxis and their parent graphs with two
(or three if it is an advanced class) dilations, translations or reflections on one side
of the poster, and write the equation of the graph on the back. They should draw
very accurately and label the x and yintercepts and three other ordered pairs,
and then they should use their graphing calculators to make sure the equation
matches the graph. Circulate to make sure graphs and equations are accurate.
Tape all the posters to the board and give the groups several minutes to confer and
to decide which poster matches which parent graph. Students should not use their
graphing calculators at this time.
Call one group to the front and give it an index card to assign a parent graph. The
group should first model the parent graph using "Function Calisthenics", then find
the poster with that graph, explain why it chose that graph, and discuss what
translations, dilations or reflections have been applied. The group should write the
equation under the graph. Do not evaluate the correctness of the equation until all
groups are finished. Three other groups are allowed to ask the Math Wizards
leading questions about the choice of equations, such as, "Why did you use a
negative? Why do you think your graph belongs to that parent graph?"
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When all groups are finished, ask if there are any changes the groups want to
make in their equations after hearing the other discussions. Calculators should not
be used to check. Turn over the graphs to verify correctness.
Students and the teacher should hold the Math Wizards accountable for their
answers to the questions by assigning points.
Activity 6: More Piecewise Functions (GLEs: Grade 9: 35, 36; Grade 11/12: 4, 6, 7, 8,
10, 16, 19, 24, 25, 27, 28, 29)
Materials List: paper, pencil, Picture the Pieces BLM
In this activity, the students will use piecewise functions to review the translations of all
basic functions.
Math Log Bellringer:
2 x 5 if x 0
(1) Graph f x without a calculator
x if x 0
(2) Find f(3) and f(4)
(3) Find the domain and range
Solutions:
(1)
(2) f(3) = 1, f(4) = 4
(3) D: all reals, R: y < 5
Activity:
Use the bellringer to review the definition of a piecewise function begun in Unit 1 a
g ( x) if x Domain 1
function made of two or more functions and written as f ( x)
h( x) if x Domain 2
where Domain 1 Domain 2 .
Picture the Pieces:
Divide students into groups of two or three and distribute the Picture the Pieces BLM.
Have the students work the section Graphing Piecewise Functions and circulate to
check for accuracy.
Have the students work the section Analyzing Graphs of Piecewise Functions, then
have one student write the equation of g(x) on the board and the other students
analyze it for accuracy.
Discuss the application problem as a group, discussing what the students should look
for when trying to graph: how many functions are involved, what types of functions
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are involved, what translations are involved, and what are the restricted domains for
each piece of the function?
When students have finished, assign the application problem in the ActivitySpecific
Assessments to be completed individually.
Activity 7: Symmetry of Graphs (GLEs: 4, 6, 7, 8, 16, 25, 27, 28)
Materials List: paper, pencil, graphing calculator, Even & Odd Functions Discovery
Worksheet BLM
In this activity, students will discover how to determine if a function is symmetric to the
y-axis, the origin, or other axes of symmetry.
Math Log Bellringer:
Graph without a calculator.
(1) f(x): y = (x)2 , f(–x): y = (–x)2 f(x): y = –x2
(2) f(x): y = log x, f(–x): y = log (–x) –f(x): y = –log x
(3) Discuss the translations made by f(x) and f(x).
Solutions:
(1) , ,
(2) , ,
(3) f(x) reflects the parent graph across the y-axis and f(x) reflects the
parent graph across the x-axis
Activity:
Use the Bellringer to review the reflections f(–x) and –f(x) covered in Activity 3.
Even and Odd Functions:
Distribute the Even & Odd Functions Discovery Worksheet BLM.
This is a guided discovery worksheet. Give the students an opportunity to graph in
their notebooks the functions in the Reflections Revisited section. Circulate to make
sure they have mastered the concept.
Even & Odd Functions Graphically: Ask the students which of the parent functions in
the Bellringer and the worksheet have the property that the graphs of f(–x) and f(x)
match. (Solutions: f(x) = x2 and f(x) = |x|.) Define these as even functions and note
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that this does not necessarily mean that every variable has an even power. Ask what
kind of symmetry they have in common. (Solution: symmetric to the y-axis)
Ask the students which of the parent functions in the Bellringer and the worksheet
have the property that the graphs of f(–x) and –f(x) match. (Solutions: f(x) = x3,
1
f x 3 x , f x , f(x) = x). Define these as odd functions. Ask what kind of
x
symmetry they have in common. (Solution: symmetric to the origin) Discuss what
symmetry to the origin means (i.e. same distance along a line through the origin.)
Have students graph y = x3 + 1 and note that just because it has an odd power does not
mean it is an odd function. Ask the students which of the parent functions do not have
any symmetry and are said to be neither even nor odd.
Solution: f(x) = log x, f(x) = 2x, f x x
Even & Odd Functions Numerically: Have students work this section and ask for
answers and justifications. Discuss whether the seven sets of ordered pairs are
enough to prove that a function is even or odd. For example in h(x), h(–3) = h(3), but
the rest of the sets do not follow this concept.
Even & Odd Functions Analytically: In order to prove whether a function is even or
odd, the student must substitute (–x) for every x and determine if f(–x) = f(x), if
f(–x) = –f(x), or if neither substitution works. Demonstrate the process on the first
problem and allow students to complete the worksheet circulating to make sure the
students are simplifying correctly after substituting x.
Activity 8: History, Data Analysis, and Future Predictions Using Statistics (GLEs: 4, 6,
8, 10, 19, 20, 22, 24, 28, 29)
Materials List: paper, pencil, graphing calculator, Modeling to Predict the Future BLM,
Modeling to Predict the Future Rubric BLM
This activity culminates the study of the ten families of functions. Students will collect
current real world data and decide which function best matches the data, then use that model
to extrapolate to predict the future.
Math Log Bellringer:
Enter the following data into your calculator. Enter 98 for 1998 and 100 for 2000,
etc., making year the independent variable and # of stock in millions, (i.e., use 4.551
million for 4,550,678), the dependent variable. Sketch a scatter plot and find the
linear regression and correlation coefficient. Discuss whether a linear model is good
for this data. Use the model to find the number of stocks that will be traded in 2012.
(i.e., Find f (112).)
year 1998 1999 2000 2001 2002 2003
# of GoMath 4, 550,678 4, 619,700 4,805,230 5, 250, 100 5,923,010 7, 000, 300
stocks traded
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Solution:
The linear model does not follow the data very well and the correlation coefficient is
only 0.932. It should be closer to 1. In 2012, 10,812,124 stocks will be traded.
Activity:
Use the Bellringer to review the processes of entering data, plotting the data, turning on
Diagnostics to see the correlation coefficient, and finding a regression equation. Review
the meaning of the correlation coefficient.
Discuss why use 98 instead of 1998 and 4.551 instead of 4, 550,678 the calculator will
round off, too, using large numbers. Students could also use 8 for 1998 and 10 for 2000.
Have each row of students find a different regression equation to determine which one
best models the data, graph it with ZOOM , Zoom Stat and on a domain of 80 to 120 (i.e.
1980 2020), and use their models to predict how many GoMath stocks will be traded in
2012.
Solutions:
In 2012, 26,960,314 stocks will be traded.
In 2012, 45,164,048 stocks will be traded.
R2 = .99987079. In 2012, 56,229,191 stocks will be traded.
In 2012, 10,513,331 stocks will be traded.
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In 2012, 14,122,248 stocks will be traded.
In 2012, 13,387,785 stocks will be traded.
Discuss which model is the best, based on the correlation coefficient. (Solution: quartic)
Discuss real-world consequences and what model would be the best based on end
behavior. Discuss extrapolation and its reasonableness.
Have students add the following scenario to their data: In 1997, only 1 million shares of
stock were traded the first year they went public.
(1) Have students find quartic regression and the number of stocks traded in 2012 and
discuss the correlation.
Solution:
R2 = .9918924557.. The correlation
coefficient is good, but the leading coefficient
is negative indicating that end-behavior is
down and hopefully the stock will not go
down in the future. In 2012, 597,220,566 stocks will be traded
(2) Have students find the cubic regression and the number of stocks traded in 2012 and
discuss the correlation.
Solution:
The R2 is not as good but the trend seems to
match better because of the endbehavior. In
2012, 181,754,238 stocks will be traded.
(3) Discuss how outliers may throw off a model and should possibly be deleted to get a
more realistic trend.
Modeling to Predict the Future Data Analysis Project:
This is an outofclass endofunit activity. The students may work alone or in
pairs. They will collect data for the past twenty years concerning statistics for their
city, parish, state, or US, trace the history of the statistics discussing reasons for
outliers, evaluate the economic impact, and find a regression equation that best
models the data. They should use either the regression equation on the calculator or
the trendline on an Excel® spreadsheet. They will create a PowerPoint® presentation
of the data including pictures, history, economic impact, spreadsheet or the calculator
graph of regression line and equation, and future predictions.
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Distribute the Modeling to Predict the Future BLM with the directions for the data
analysis project and the Modeling to Predict the Future Rubric BLM. Then discuss
the objectives of the project and the list of possible data topics.
Timeline:
1. Have students bring data to class along with a problem statement (why they are
examining this data) three days after assigned, so it can be approved and they can
begin working on it under teacher direction.
2. The students will utilize one to two weeks of individual time in research and
project compilation, and two to three days of class time for analysis and computer
use if necessary.
Discuss each of the headings on the blackline master:
1. Research: Ask each group to choose a different topic concerning statistical data
for their city, parish, state, or for the US. List the topics on the board and have
each group select one. The independent variable should be years, and there must
be at least twenty years of data with the youngest data no more than five years
ago. The groups should collect the data, analyze the data, research the history of
the data, and take relevant pictures with a digital camera.
2. Calculator/Computer Data Analysis: Students should enter the data into their
graphing calculators, link their graphing calculators to the computer, and
download the data into a spreadsheet, or they should enter their data directly into
the spreadsheet. They should create a scatterplot and regression equation or
trendline of the data points using the correlation coefficient (called Rsquared
value in a spreadsheet) to determine if the function they chose is reliable. They
should be able to explain why they chose this function, based on the correlation
coefficient as well as function characteristics. (e.g., end-behavior, increasing
decreasing, zeroes).
3. Extrapolation: Using critical thinking skills concerning the facts, have the
students make predictions for the next five years and explain the limitations of the
predictions.
4. Presentation: Have students create a PowerPoint® presentation including the
graph, digital pictures, economic analysis, historical synopsis, and future
predictions.
5. Project Analysis: Ask each student to type a journal entry indicating what he/she
learned mathematically, historically, and technologically, and express his/her
opinion of how to improve the project. If students are working in pairs, each
student in the pair must have his/her own journal.
Final Product: Each group must submit:
1. A disk containing the PowerPoint® presentation with the slides listed in BLM.
2. A print out of the slides in the presentation.
3. Release forms signed by all people in the photographs.
4. Project Analysis
5. Rubric
Have students present the information to the class. Either require the students to also
present in another one of their classes or award bonus points for presenting in another
class. As the students present, use the opportunity to review all the characteristics of
the functions studied during the year.
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Sample Assessments
General Assessments
Use Math Log basic graphs
(2) vertical and horizontal shifts
(3) coefficient changes to graphs
(4) absolute value changes to graphs
(5) even and odd functions
Administer one comprehensive assessment about translations, reflections, shifts of
functions, and graphing piecewise functions.
Activity-Specific Assessments
Teacher Note: Critical Thinking Writings are used as activity-specific assessments in many
of the activities in every unit. Post the following grading rubric on the wall for students to
refer
Evaluate the Flash That Function flash cards for accuracy and completeness.
Activity 2: Critical Thinking Writing
Graph the following and discuss the parent function and whether there is a horizontal
shift or vertical shift.
(1) k(x) = x + 5
(2) g x x 2
(3) h x x 2
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Solutions:
(1) The parent function is the line f(x) = x, and the graph of
k(x) is the same whether you shifted it vertically up 5 or
horizontally to the left 5.
(2) and (3)The parent function is greatest integer
f x x , and both graphs are the same even though
g(x) is shifted up 2 and h(x) is shifted to the right 2.
Activity 6: Critical Thinking Writing
Mary is diabetic and takes long-acting insulin shots. Her blood sugar level starts at
100 units at 6:00 a.m. She takes her insulin shot, and the blood sugar increase is
modeled by the exponential function f(t) = Io(1.5t) where Io is the initial amount in the
blood stream and rises for two hours. The insulin reaches its peak effect on the blood
sugar level and remains constant for five hours. Then it begins to decline for five
hours at a constant rate and remains at Io until the next injection the next morning. Let
the function i(t) represent the blood sugar level at time t measured in hours from the
time of injection. Write a piecewise function to represent Mary's blood sugar level.
Graph i(t) and find the blood sugar level at (a) 7:00 a.m. (b) 10:00 a.m. (c) 5:00 p.m.
(d) midnight. (e) Discuss the times in which the function is increasing, decreasing and
constant.
Solution:
100 1.5t if 0 t 2
225 if 2 t 7
i (t )
25(t 7) 225 if 7 t 12
100
if 12 t 24
(a) 150 units, (b) 225 units, (c) 125 units, (d) 100 units,
(e) The function is increasing from6:00 a.m. to 8:00 a.m., constant from 8:00
a.m. to 1:00 p.m., decreasing from 1:00 p.m. to 6:00 p.m. and constant
from 6:00 p.m. to 6:00 a.m.
Activity 7: Critical Thinking Writing
Discuss other symmetry you have learned in previous units, such as the axis of
symmetry in a parabola or an absolute value function and the symmetry of inverse
functions. Give some example equations and graphs and find the lines of symmetry.
Activity 8: Modeling to Predict the Future Data Research Project
Use the Modeling to Predict the Future Rubric BLM to evaluate the research project
discussed in Activity 8.
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Algebra II
Unit 8: Conic Sections
Time Frame: Approximately four weeks
Unit Description
This unit focuses on the analysis and synthesis of graphs and equations of conic sections and
their real-world applications.
Student Understandings
The study of conics helps students relate the cross-curriculum concepts of art and
architecture to math. They define parabolas, circles, ellipses, and hyperbolas in terms of the
distance of points from the foci and describe the relationship of the plane and the double-
napped cone that forms each conic. Students identify various conic sections in real-life
examples and in symbolic equations. Students solve systems of conic and linear equations
with and without technology.
Guiding Questions
1. Can students use the distance formula to define and generate the equation of each
conic?
2. Can students complete the square in a quadratic equation?
3. Can students transform the standard form of the equations of parabolas, circles,
ellipses, and hyperbolas to graphing form?
4. Can students identify the major parts of each of the conics from their graphing
equations and can they graph the conics?
5. Can students formulate the equations of each of these conics from their graphs?
6. Can students find real-life examples of these conics, determine their equations,
and use the equations to solve real-life problems?
7. Can students identify these conics given their stand and graphing equations?
8. Can the students predict how the graphs will be transformed when certain
parameters are changed?
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Unit 86. Simplify and perform basic operations on numerical expressions involving radicals
(e.g., 2 3 + 5 3 = 7 3 ) (N-5-H)
Grade 10
1. Simplify and determine the value of radical expressions (N-2-H) (N-7-H)
Algebra
Grade 9
13. Translate between the characteristics defining a line (i.e., slope, intercepts, points)
and both its equation and graph (A-2-H) (G-3-H)
16. Interpret and solve systems of linear equations using graphing, substitution,
elimination, with and without technology, and matrices using technology (A-4-H)
Grade 10
6. Write the equation of a line parallel or perpendicular to a given line through a
specific point (A-3-H) (G-3-H)
Grade 11/12
4. Translate and show the relationships among non-linear graphs, related tables ofGrade 9
24. Graph a line when the slope and a point or when two points are known (G-3-H)
Grade 10
12. Apply the Pythagorean theorem in both abstract and real-life settings (G-2-H)
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GLE # GLE Text and Benchmarks
13. Solve problems and determine measurements involving chords, radii, arcs, angles,
secants, and tangents of a circle (G-2-H)
Grade 11/12
15. Identify conic sections, including the degenerate conics, and describe the
relationship of the plane and double-napped cone that forms each conic (G-1-H)
16. Represent translations, reflections, rotations, and dilations of plane figures using
sketches, coordinates, vectors, and matrices (G-3-H)
Patterns, Relations, and Functions 8. This is a list of properties in the order in which they will be
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8.1 Circle – write the definition, provide examples of both the standard and
graphing forms of the equation of a circle, show how to graph circles, and
provide a real-life example in which circles are used.
8.2 Parabola – write the definition, give the standard and graphing forms of the
equation of a parabola and show how to graph them in both forms, find the
vertex from the equation and from the graph, give examples of the equations of
both vertical and horizontal parabolas and their graphs, find equations for the
directrix and axis of symmetry, identify the focus, and provide real-life
examples in which parabolas are used
8.3 Ellipse – write the definition, write standard and graphing forms of the equation
of an ellipse and graph both vertical and horizontal, locate and identify foci,
vertices, major and minor axes, explain the relationship of a, b, and c, and
provide a real-life example in which an ellipse is used.
8.4 Hyperbola – write the definition, write the standard and graphing forms of the
equation of a hyperbola and draw graph both vertical and horizontal, identify
vertices, identify transverse and conjugate axes and provide an example of each,
explain the relationships between a, b, and c, find foci and asymptotes, and give
a real-life example in which a hyperbola is used.
8.5 Conic Sections – define each, explain the derivation of the names, and draw
each as a slice from a cone.
8.6 Degenerate Cases of Conics – give examples of equations for each and draw the picture
representations from cones.
Activity 1: Deriving the Equation of a Circle (GLEs: Grade 9: 6; Grade 10: 1, 12;
Grade 11/12: 4, 5, 7, 9, 10, 15, 16, 27, 28)
Materials List: paper, pencil, graphing calculator, Math Log Bellringer BLM
In this activity, students will review the concepts of the Pythagorean theorem and the
distance formula studied in Algebra I in order to derive the equation of a circle from its
definition.
Math Log Bellringer:
(1) Draw a right triangle with sides 6 and 7 and find the length of the hypotenuse.
Algebra IIUnit 8Conic Sections 176
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(2) Find the distance between the points (x, y) and (1, 3).
(3) Define a circle.
Solutions:
(1) 85
6 x
7
(2) d x 1 y 3
2 2
,
(3) Set of all points in a plane equidistant from a fixed point.
Activity:
Overview of the Math Log Bellringers:
As in previous units, each in-class activity in Unit 8 is started with an activity called a
Math Log Bellringer that either reviews past concepts to check for understanding (i.e.
reflective thinking about what was learned in previous classes or previous courses) or
sets the stage for an upcoming concept (i.e. predictive thinking for that day's lesson).
A math log is a form of a learning log (view literacy strategy descriptions) that students
keep combines
writing and reading with content learning. The Math Log Bellringers will include
mathematics preceding the
upcoming lesson during beginningofclass record keeping, and then circulate to
give individual attention to students who are weak in that area.
Compare the Pythagorean theorem used in the Bellringer to the distance formula, and
have students use this to derive the graphing form of the equation of a circle with the
center at the origin.
x 0 y 0
2 2 (x, y)
r
r
r x y 2 2
r 2 x2 y 2
Apply the translations learned in Unit 7 to create the graphing form of equation of a circle
with the center at (h, k) and radius = r: (x – h)2 + (y – k)2 = r2.
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Use the math textbook for practice problems: (1) finding the equation of a circle given
the center and radius, (2) graphing circles given the equation in graphing form.
Have students expand the graphing form of a circle with center (5, 3) and radius = ½
to derive the standard form of an equation of a circle. Ax2 + By2 + Cx + Dy + E = 0
where A = B.
Solution: (x + 5)2 + (y – 3)2 = (½)2
x2 + 10x + 25 + y2 6y + 9 = ¼
4x2 + 40x + 100 + 4y2 24y + 36 = 1
4x2 + 4y2 + 40x 24y + 135 = 0
Review the method of completing the square introduced in Unit 5, Activity 3. Have
students use the method of completing the square to transform the standard form of the
circle above back to graphing form in order to graph the circle.
Solution: 4x2 + 4y2 + 40x 24y + 135 = 0
rearrange grouping variables 4x2 + 40x + 4y2 24y = 135
factor coefficient on squared terms 4(x2 + 10x) + 4(y2 6y) = 135
complete the square 4(x2 + 10x + 25) + 4(y2 6y + 9) = 1
4(x + 5)2 + 4(y 3)2 = 1
divide by coefficient (x + 5)2 + (y 3)2 = ¼
identify the center and radius center (5, 3), radius = ½
Use the math textbook for practice problems finding the graphing form of the equation of
a circle given the standard form.
Discuss degenerate cases of a circle:
1. If the equation is in graphing form and r2 = 0, then the graph is a point, the center
(e.g., (x + 3)2 + (y 7)2 = 0). The graph is the point (3, 7))
2. If the equation is in graphing form and r2 is negative, then the graph is the empty
set (e.g., (x + 3)2 + (y 7)2 = 8. There is no graph.).
Have students graph a circle on their graphing calculators. This should include a
discussion of the following:
1. Functions: The calculator is a function grapher and a circle is not a function.
2. Radicals: In order to graph a circle, isolate y and take
the square root of both sides creating two functions.
Graph both y1 = positive radical and y2 = negative
radical or enter y2 = y1
3. Calculator Settings:
o ZOOM , 5:ZSquare to set the window so the graph looks circular. The circle
may not look like it touches the xaxis because there are only a finite number
of pixels (94 pixels on the TI83 and TI84 calculators) that the graph
evaluates. The x-intercepts may not be one of these.
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o Set the MODE for SIMUL to allow both halves
of the circle to graph simultaneously and HORIZ
to see the graph and equations at the same time.
Have students bring in pictures of something in the real-life world with a circular shape
for an application problem in Activity 2.
Activity 2: Circles - Algebraically and Geometrically (GLEs: Grade 9: 6, 13, 24; Grade
10: 1, 6, 12, 13; Grade 11/12: 9, 10, 16, 24, 28)
Materials List: paper, pencil, graphing calculators, pictures of real-world circles, Circles &
Lines Discovery Worksheet BLM, one copy of Circles in the Real World Math Story Chain
Example BLM for an example
In this activity, students will review geometric properties of a circle and equations of lines to
find equations of circles and apply to real-life situations.
Math Log Bellringer:
(1) Draw a circle and draw a tangent, secant, and chord for the circle and define each.
(2) What is the relationship of a tangent line to a radius?
(3) What is the relationship of a radius perpendicular to a chord?
(4) Find the equation of a line perpendicular to y = 2x and through the
point (6, 10).
Solutions:
(1) tangent line ≡ A line in the same plane as the circle
which intersects the circle at one point.
secant line ≡ A line that intersects the circle at two points.
chord ≡ A segment that connects two points on a circle.
(2) The tangent line is perpendicular to the radius of the circle at the point of
tangency.
(3) A radius which is perpendicular to a chord also bisects the chord.
(4) y = – ½ x + 13
Activity:
Use the Bellringer to review relationships between lines and circles and finding equations
of lines. Give the following problem to practice:
Graph the circle x2 + y2 = 25 and find the equation of the tangent line in point slope
form through the point (3, 4). Graph the circle and the line on the graphing calculator
to check.
3
Solution: y 4 x 3
4
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Graphing Circles & Lines:
Put students in groups of four and distribute the Circles & Lines Discovery
Worksheet BLM. On this worksheet, the students will combine their knowledge of
the distance formula and relationships of circles to tangent lines to find equations of
circles and to graph them.
When the students get to problem #7, they will use the real-world pictures of circles
they brought in to write a math story chain (view literacy strategy descriptions). Story
chains are especially useful in teaching math concepts, while at the same time
promoting writing and reading. The process involves a small group of students
writing a story problem using the math concepts being learned and then solving the
problem. Writing out the problem in a story provides students a reflection of their
understanding. This is reinforced as students attempt to answer the story problem. In
this story chain the first student initiates the story. The next must solve the first
student's problem to add a second problem, the next, a third problem, etc. All group
members should be prepared to revise the story based on the last student's input as to
whether it was clear or not. Model the process for the students before they begin with
the Circles in the Real World Math Story Chain Example BLM.
When the story chains are complete, check for understanding of circle and line
concepts and correctness by swapping stories with other groups.
Activity 3: Developing Equations of Parabolas (GLEs: Grade 10: 1, 12, 27; Grade
11/12: 4, 5, 6, 7, 9, 10, 15, 16, 24, 27, 28)
Materials List: paper, pencil, graphing calculator, graph paper, string, Parabola Discovery
Worksheet BLM
In this activity, students will apply the concept of distance to the definition of a parabola to
derive the equations of parabolas, to graph parabolas, and to apply them to real-life
situations.
Math Log Bellringer:
Graph the following by hand:
(1) y = x2
(2) y = x2 + 6
(3) y = (x + 6)2
(4) y = x2 + 2x – 24
(5) Discuss the translations made and why.
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Solutions:
(1) (2) (3) (4)
(5) #2 is a vertical translation up because the constant is on the y as in f(x)+k.
#3 is a horizontal translation to the left of the form f(x +k).
#3 is translated both horizontally and vertically.
Activity:
Use the Bellringer to review the graphs of parabolas as studied in Unit 5 on quadratic
functions. Review horizontal and vertical translations in Bellringer #2 and #3. Review
b b
finding the vertex in Bellringer #4 using , f and finding the zeroes by
2a 2a
factoring.
Have students complete the square in Bellringer #4 to put the equation of the parabola in
graphing form, y = a(x h)2 + k, and discuss translations from this formula that locate the
vertex at (h, k). (Solution: y = (x 1)2 25).
Have the students practice transforming quadratic equations into graphing form and
locating the vertex using the following equations. Compare vertex answers to values of
b b
2a , f 2a . Graph both problem equation and solution equation to determine if the
graphs are coincident. Examine the graphs to determine the effect of a ± leading
coefficient.
(1) y = 2x2 + 12x + 7
(2) y = 3x2 + 24x 42
Solutions:
(1) y = 2(x + 3)2 11, vertex (3, 11), opens up
(2) y = 3(x 4)2 + 6, vertex (4, 6), opens down
Define a parabola ≡ set of points in a plane equidistant from a point called the focus and a
line called the directrix. Identify these terms on a sketch. Parabolas can be both vertical
and horizontal. Demonstrate this definition using the website,
(x, y)
(8, 4)
(x, 2) y=2
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Discuss real-life parabolas. If a ray of light or a sound wave travels
in a path parallel to the axis of symmetry and strikes a parabolic
dish, it will be reflected to the focus where the receiver is located in
satellite dishes, radio telescopes, and reflecting telescopes.
Discovering Parabolas:
Divide students in pairs and distribute two sheets of graph paper,
a piece of string, and the Parabola Discovery Worksheet BLM. This is a guided
discovery sheet with the students stopping at intervals to make sure they are making
the correct assumptions.
In I. Vertical Parabolas, the students will use the definition of parabola and two equal
lengths on the string to plot points that form a parabola. Demonstrate finding several
of the points to help the students begin. Locate the vertex.
Label one of the points on the parabola (x, y) and the corresponding point on the
directrix (x, 2). Discuss the definition of parabola and how to use the distance formula
to find the equation of the parabola.
Solution:
The distance from the focus to any point on the parabola (x, y) equals the
distance from that point (x, y) to the directrix;
x 8 y 4 x x y 2
2 2 2 2
therefore, .
Have students expand this equation and isolate y to write the equation in standard
form. Use completing the square to write the equation in graphing form and to find
the vertex.
1 1
Solution: y x 2 4 x 19 , y x 8 3 , vertex (8, 3)
2
4 4
In II. Horizontal Parabolas, the students should use the string to sketch the horizontal
parabola and to find the equation without assistance. Check for understanding when
they have completed this section.
Help students come to conclusions about the standard form and graphing form of
vertical and horizontal parabolas and how to find the vertex in each.
o Vertical parabola:
b b
Standard form: y = Ax2 + Bx + C, vertex: , f
2a 2a
Graphing form: y = A(x h) + k, vertex (h, k)
2
o Horizontal parabola:
b b
Standard form: x = Ay2 + By + C, vertex: f , .
2a 2a
This is not a function of x but it is a function of y.
Graphing form: x = A(y k)2 + h, vertex (h, k)
In III. Finding the Focus, have the students answer questions #1 relating the leading
coefficient to the location of the focus and #2 helping students come to the conclusion
that the closer the focus is to the vertex, the narrower the graph. Allow students to
complete the worksheet.
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Check for understanding by giving the students the following application problem. (If
an old satellite dish is available, use the dimensions on it to find the location of its
receiver.)
A satellite is 18 inches wide and 2 inches at its deepest part. What is the equation
of the parabola? (Hint: Locate the vertex at the origin and write the equation in
the form y = ax2.) Where should the receiver be located to have the best
reception? Hand in a graph and its equation showing all work. Be sure to answer
the question in a complete sentence and justify the location.
1
Solution: y x 2 . The receiver should be located 4½ inches above the vertex.
18
Activity 4: Discovering the Graphing Form of the Equation of an Ellipse (GLEs: Grade
10: 12, 27; Grade 11/12: 4, 5, 7, 9, 10, 15, 16, 24, 27, 28)
Materials List: graph paper on cardboard, two tacks and string for each group, Ellipse
Discovery Worksheet BLM, paper, pencil
In this activity, students will apply the definition of an ellipse to sketch the graph of an
ellipse and to discover the relationships between the lengths of the focal radii and axes of
symmetry. They will also find examples of ellipses in the real world.
Math Log Bellringer:
(1) Draw an isosceles triangle with base = 8 and legs = 5. Find the length of the
altitude.
(2) Discuss several properties of isosceles triangles.
Solutions:
5 3 5
(1)
8
(2) An isosceles triangle has congruent sides and congruent base angles. The
altitude to the base of the isosceles triangle bisects the vertex angle and
the base.
minor axis
Activity: focal radii
Define ellipse ≡ set of all points in a plane in which the
sum of the focal radii is constant. Draw an ellipse and focus focus major axis
locate the major axis, minor axis, foci, and focal radii.
Ask for some examples of ellipses in the real world, such as
the orbit of the earth around the sun.
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Discovering Ellipses:
Divide students into groups of three. Give each group a
piece of graph paper glued to a piece of cardboard. On
the cardboard are two points on one of the axes, evenly
spaced from the origin, and a piece of string with tacks
at each end. Each group should have a different set of
points and a length of string. On the back of each
cardboard write the equation of the ellipse that will be sketched. Sample foci, string
sizes and equations below:
x2 y 2
Group 1: foci (±3, 0), string 10 units, equation 1
25 16
x2 y 2
Group 2: foci (0, ±3), string 10 units, equation 1
16 25
x2 y 2
Group 3: foci (±4, 0), string 10 units, equation 1
25 9
x2 y 2
Group 4: foci (0, ±4), string 10 units, equation 1
9 25
x2 y 2
Group 5: foci (±6, 0), string 20 units, equation 1
100 64
x2 y 2
Group 6: foci (0, ±6), string 20 units, equation 1
64 100
x2 y 2
Group 7: foci (±8, 0), string 20 units, equation 1
36 100
x2 y 2
Group 8: foci (0, ±8), string 20 units, equation 1
100 36
Distribute the Ellipse Discovery Worksheet BLM and have groups follow directions
independently to draw an ellipse. After all ellipses are taped to the board, review the
answers to the questions to make sure they have come to the correct conclusions.
Use the graphs on the board to draw conclusions about the location of major and
minor axes and the relationships with the foci and focal radii. Clarify the graphing
form for the equation of an ellipse with center at the origin. (i.e. horizontal ellipse:
x2 y 2 x2 y 2
2 1 , vertical ellipse: 2 2 1 )
a2 b b a
Discuss how the graphing form will change if the center is moved away from the
origin and to a center at (h, k) relating the new equations to the translations studied in
x h y k
2 2
previous units. (i.e. horizontal ellipse: 1 , vertical ellipse:
a2 b2
x h y k
2 2
1)
b2 a2
Algebra IIUnit 8Conic Sections 184
Louisiana Comprehensive Curriculum, Revised 2008
Demonstrate the definition of ellipse by having the students use the website,
, to discover what the distance between foci does to the shape
of the ellipse. (i.e., The closer the foci, the more circular the ellipse.)
Critical Thinking Writing Activity: Assign each group one real-life application to
research, find pictures of, and discuss the importance of the foci (e.g., elliptical orbits,
machine gears, optics, telescopes, sports tracks, lithotripsy, and whisper chambers).
Activity 5: Equations of Ellipses in Standard Form (GLEs: 4, 5, 7, 9, 10, 15, 16, 24, 27,
28)
Materials List: paper, pencil
In this activity, students will determine the standard form of the equation of an ellipse and
will complete the square to transform the equation of an ellipse from standard to graphing
form.
Math Log Bellringer:
x 2 y 3
2 2
(1) Graph 1 by hand.
25 9
(2) Find the foci.
(3) Expand the equation so that there are no fractions and isolate zero.
(4) Discuss the difference in this expanded form and the expanded of a circle.
Solutions:
(1)
(2) (6, 3) and (2, 3)
(3) 9x2 + 25y2 36x + 150y + 36 = 0
(4) The coefficients of x2 and y2 on a circle are equal.
On an ellipse, the coefficients are the same sign
but not equal.
Activity:
Use the Bellringer to check for understanding of graphing ellipses and finding foci.
Use the expanded equation in the Bellringer to have students determine the general
characteristics of the standard form of the equation of an ellipse. Compare the standard
form of an ellipse to the standard forms of equations of lines, parabolas, and circles.
Algebra IIUnit 8Conic Sections 185
Louisiana Comprehensive Curriculum, Revised 2008
Line: Ax + By + C = 0 (x and y are raised only to the first power. Coefficients
may be equal or not or one of them may be zero.)
Parabola: Ax2 + Bx + Cy + D = 0 or Ay2 + By + Cx + D = 0 (only one variable is
squared)
Circle: Ax2 + Ay2 + Bx + Cy + D = 0 (both variables are squared with the same
coefficients)
Ellipse: Ax2 + By2 + Cx + Dy + E = 0 (both variables are squared with different
coefficients which have the same sign)
Have students determine how to transform the standard form into the graphing form of an
ellipse by completing the square. Assign the Bellringer solution #3 to see if they can
transform it into the Bellringer problem.
Discuss degenerate cases of an ellipse:
1. If an equation is in graphing form and equals 0 instead of 1, then the graph is a
point, the center.
x 2 y 3
2 2
(e.g., 0 The graph is the point (2, 3))
25 9
2. If an equation is in graphing form and equals negative 1, then the graph is the
x 2 y 3
2 2
empty set. (e.g., 1 . There is no graph.)
25 9
Have students give their reports on the real-life application assigned in Activity 4.
Assign additional problems in the math textbook.
Activity 6: Determining the Equations and Graphs of Hyperbolas (GLEs: Grade 10:
12; Grade 11/12: 4, 5, 6, 7, 9, 10, 15, 16, 27, 28)
Materials List: paper, pencil, graphing calculator
In this activity, students will apply what they have learned about ellipses to the graphing of
hyperbolas.
Math Log Bellringer:
Determine which of the following equations is a circle, parabola, line, hyperbola or
ellipse. Discuss the differences.
(1) 9x2 + 16y2 + 18x – 64y – 71=0
(2) 9x + 16y – 36 = 0
(3) 9x2 + 16y – 36 = 0
(4) 9x – 16y2 –36 = 0
(5) 9x2 + 9y2 – 36 = 0
(6) 9x2 + 4y2 – 36 = 0
(7) 9x2 – 4y2 – 36 = 0
Algebra IIUnit 8Conic Sections 186
Louisiana Comprehensive Curriculum, Revised 2008
Solutions:
(1) ellipse, different coefficients on x2 and y2 but same sign
(2) line, x and y are raised only to the first power
(3) parabola, only one of the variables is squared
(4) parabola, only one of the variables is squared
(5) circle, equal coefficients on the x2 and y2
(6) ellipse, different coefficients on x2 and y2
(7) hyperbola, opposite signs on the x2 and y2
Activity:
Use the Bellringer to check for understanding in problems 1 through 5.
Students will be unfamiliar with the equation in problem 7. Have the students graph the
two halves on their graphing calculators by
isolating y. Reinforce the concept that the
calculator is a function grapher and because both
variables are squared, this is not a function.
Define hyperbola ≡ set of all points in a plane in which the difference in the focal radii is
constant. Compare the definition of a hyperbola to the definition of an ellipse and ask what
is different about the standard form of the hyperbola. Demonstrate the definition using the
website,
Have students transform the equation in Bellringer problem #6 into the graphing form of
an ellipse and graph it by hand. Then have the students transform the equation in
Bellringer problem #7 in the same way by isolating 1. Have students graph both on the
calculator isolating y and graphing ±y.
Solutions:
x2 y 2
(6) 1
4 9
x2 y 2
(7) 1
4 9
Determine the relationships of the numbers in the equation of the hyperbola to the graph.
(i.e. The square root of the denominator under the x2 is the distance from the center to the
vertex.)
Have students graph 9y2 – 4x2 = 36 on their calculators and determine how the graph is
different from the graph generated by the equation in Bellringer problem 7.
Algebra IIUnit 8Conic Sections 187
Louisiana Comprehensive Curriculum, Revised 2008
Solution:
If x2 has the positive coefficient, the vertices are located on the xaxis. If y2
has the positive coefficient, the vertices are located on the yaxis.
Isolate 1 in the equation above and compare to Bellringer problem #7. Develop the
graphing form of the equation of a hyperbola with the center on the origin:
x2 y 2 y 2 x2
1. horizontal hyperbola: 2 2 1 2. vertical hyperbola: 2 2 1 .
a b a b
Discuss transformations and develop the graphing form of the equations of a hyperbola
with the center at (h, k):
x h y k
2 2
1. horizontal hyperbola: 1
a2 b2
y k x h 1 .
2 2
2. vertical hyperbola:
a2 b2
Locate vertices and foci on the graph. Define and locate: conjugate axis
1. transverse axis ≡ the axis of symmetry
connecting the vertices. focus focus
2. conjugate axis ≡ the axis of symmetry not
connecting the vertices transverse axis
Label ½ the transverse axis as a, ½ the conjugate
axis as b, and the distance from the center of the
hyperbola to the focus as c. Have students draw a
right triangle with a right angle at the center and the
ends of the hypotenuse at the ends of the transverse
c b
and conjugate axes. Demonstrate with string how the
length of the hypotenuse is equal to the length of the a
segment from the center of the hyperbola to the
focus. Let the students determine the relationship
between a, b, and c. c
Solution: a2 + b2 = c2
Algebra IIUnit 8Conic Sections 188
Louisiana Comprehensive Curriculum, Revised 2008
Draw the asymptotes through the corners of the box formed by the conjugate and
transverse axes and explain how these are graphing aids, then find their equations. The
general forms of equations of asymptotes are given below, but it is easier to simply find
the equations of the lines using the center of the hyperbola and the corners of the box.
x2 y 2
1. horizontal hyperbola with center at origin: 2 2 1 ,
a b
b
asymptotes: y x
a
y 2 x2
2. vertical hyperbola with center at origin: 2 2 1 ,
a b
a
asymptotes: y x
b
3. horizontal hyperbola with center at (h, k):
x h y k
2 2
1
a2 b2
b
asymptotes: y k x h
a
y k x h
2 2
4. vertical hyperbola with center at (h, k): 1
a2 b2
a
asymptotes: y k x h
b
Discuss the degenerate form of the equation of a hyperbola: If the equation is in graphing
form and equals 0 instead of 1, then the graph is two lines, the asymptotes.
x 2 y 3
2 2
3
(e.g. 0 . Graphs: y 3 x 2 )
25 9 5
Discuss the applications of a hyperbola: the path of a comet often takes the shape of a
hyperbola, the use of hyperbolic (hyperbola-shaped) lenses in some telescopes, the use of
hyperbolic gears in many machines and in industry, the use of the hyperbolas in
navigation since sound waves travel in hyperbolic paths, etc. Some very interesting
activities using the hyperbola are available at:
Activity 7: Saga of the Roaming Conic (GLEs: 7, 15, 16, 24, 27, 28)
Materials List: paper, pencil, graphing calculator, Saga of the Roaming Conic BLM
This can be an open or closed-book quiz or in-class or at-home creative writing assignment
making students verbalize the characteristics of a particular conic.
Algebra IIUnit 8Conic Sections 189
Louisiana Comprehensive Curriculum, Revised 2008
Math Log Bellringer:
Graph the following pairs of equations on the graphing calculator. ( ZOOM , 2:Zoom
In, 5:ZSquare)
(1) y = x2 and y = 9x2
(2) 2x2 + y2 = 1 and 9x2 + y2 = 1
(3) x2 – y2 = 1 and 9x2 – y2 = 1
(4) Discuss what the size of the coefficients on the x2 does to the shape of the graph
Solutions :
(1)
(2)
(3)
(4) A larger coefficient on the x2 makes a narrower graph because 9x2 is
actually (3x)2 creating a transformation in the form f(kx) which shrinks
the domain.
Activity:
Discuss answers to the Bellringer.
Saga of the Roaming Conic:
Have the students demonstrate their understanding of the transformations of conic
graphs by completing the following RAFT writing (view literacy strategy
descriptions). RAFT writing gives students the freedom to project themselves into
unique roles and look at content from unique perspectives. In this assignment,
students are in the Role of a conic of their choice and the Audience is an Algebra II
student. The Form of the writing is a story of the exploits of the Algebra II student
and the Topic is transformations of the conic graph.
Distribute the Saga of the Roaming Conic BLM giving each student one sheet of
paper with a full size ellipse, hyperbola or parabola drawn on it and the following
directions: You are an ellipse (or parabola or hyperbola). Your owner is an Algebra II
student who moves you and stretches you. Using all you know about yourself,
describe what is happening to you while the Algebra II student is doing his/her
homework. You must include ten facts or properties of an ellipse (or parabola or
hyperbola) in your discussion. Discuss all the changes in your shape and how these
changes affect your equation. Write a small number (e.g. 1, 2, etc.) next to each
Algebra IIUnit 8Conic Sections 190
Louisiana Comprehensive Curriculum, Revised 2008
property in the story to make sure you have covered ten properties. (See sample story
in Unit 1.)
Have students share their stories with the class to review properties. Students should
listen for accuracy and logic in their peers' RAFTs.
Activity 8: Comparison of all Conics and the Double-Napped Cone (GLEs: 5, 7, 9, 10,
15, 16, 27, 28)
Materials List: paper, pencil, graphing calculator, eight cone-shaped pieces of Styrofoam®,
four pieces of cardboard with graph paper pasted to it, four plastic knives
In this activity, students will compare and contrast all conics – their equations, their shapes,
their degenerate forms, their relationships in the plane and double-napped cone that forms
each conic, and their applications.
Math Log Bellringer:
The following are degenerate cases of conics. Complete the square to put each
equation in graphing form, describe the graph, and determine which conic is
involved.
(1) 2x2 + y2 + 6 = 0
(2) x2 + y2 + 4x – 6y + 13 = 0
(3) x2 – 6x – y2 + 9 = 0
(4) 3x2 + x = 0
(5) y2 = 4
Solutions:
x2 y 2
(1) 1 . The sum of two squares cannot be negative, therefore there
3 6
is no graph. This is a degenerate case of an ellipse.
(2) (x + 2)2+ (y 3)2 = 0. The graph is the center point (2, 3), a degenerate
case of a circle.
(3) (x 3)2 y2 = 0. The graph is two intersecting lines, y = ±(x 3), a
degenerate case of a hyperbola.
(4) There is no y variable so the graph is two parallel vertical lines, x = 0
and x 1 , sometimes considered a degenerate case of a parabola.
3
(5) There is no x variable so the graph is two parallel horizontal lines,
y = ±2, sometimes considered a degenerate case of a parabola.
Activity:
Use the Bellringer to check for understanding of recognizing possible conics and their
degenerate cases.
Algebra IIUnit 8Conic Sections 191
Louisiana Comprehensive Curriculum, Revised 2008
Students often think that a parabola and half of a hyperbola are the same. Give them the
y 2 x2
equations 1 and y = .06x2 + 3, which both have a vertex of (0, 3). Have them
9 16
graph both equations on the same screen of their calculators. Then zoom standard, zoom
out, find the points of intersection, and view in the window x: [1, 8] and y: [2.5, 7].
Discuss the differences.
ZOOM Standard ZOOM Out Intersection Set Window
Solution: Between the points of intersection, the parabola is below the
hyperbola and flatter. Outside the points of intersection, the parabola is above
the hyperbola and steeper.
Conics and the Double-Napped Cone Lab:
A plane intersecting a double-napped
cone can be used to determine each
conic and its degenerate case.
Divide students into four groups and
assign each group a different conic
circle, parabola, ellipse, and
hyperbola. Give each group two cone-shaped pieces of Styrofoam®, a piece of
cardboard with graph paper pasted to it, and a plastic knife. Each member of the
group will have a responsibility:
(1) Student A will cut one Styrofoam® cone in the shape of the conic.
(2) Student B will trace the conic formed after cutting the Styrofoam® on the
plane (cardboard with graph paper).
(3) Student C will determine the equation of the graph.
(4) Student D will determine how to cut the second cone of Styrofoam® to
create the degenerate cases of the conic.
(5) Student E will present the findings to the class.
Use the ActivitySpecific Assessment to evaluate the lab.
Activity 9: Solving Systems of Equations Involving Conics (GLEs: Grade 9: 16; Grade
11/12: 5, 6, 7, 9, 10, 15, 16, 28)
Materials List: paper, pencil, graphing calculator
Is this activity, students will review the processes for solving systems of equations begun in
the unit on Systems of Linear Equations in the Algebra I curriculum. They will apply some
of these strategies to solving systems involving conics.
Algebra IIUnit 8Conic Sections 192
Louisiana Comprehensive Curriculum, Revised 2008
Math Log Bellringer:
(1) Graph y = 3x + 6 and 2x – 6y = 9 by hand.
(2) Find the point of intersection by hand.
(3) What actually is a point of intersection?
Solutions:
(1)
45 39
(2) - -
16, 16
(3) A point of intersection is the point at which the two graphs have the same
x- and y-value.
Activity:
Use the Bellringer to determine if the students remember that finding a point of
intersection and solving a system of equations are synonymous. Review solving systems
of equations from Algebra I by substitution and elimination (addition).
Give the students the equations x2 + y2 = 25 and y = x – 1 and
have them work in pairs to solve analytically. Then have them
graph on their calculators ( ZOOM , 5:ZSquare) to find points of
intersection.
Solution: (4, 3) and (4, 3)
Assign the system x2 + y2 = 25 and y = x + 8 that has no solutions. Assign the system
3
x2 + y2 = 25 and y x 6 that has one solution. Solve analytically and graphically.
4
Assign the following systems which require simultaneous solving of two conic equations.
Have students graph the equations first by hand to determine how many points of
intersection exist, and then have the students solve them analytically using the most
appropriate method.
y 2 x2
(1) x2 + y2 = 25 and 1
9 16
x2 y 2
(2) x2 + y2 = 25 and 1
9 16
Algebra IIUnit 8Conic Sections 193
Louisiana Comprehensive Curriculum, Revised 2008
x2 y 2
(3) x2 + y2 = 25 and 1
9 25
Solutions:
(1)
16 369 16 369 16 369 16 369
,
5 , , , , , ,
5 5
5 5
5 5
5
(2) no solutions
(3) (0, 5), (0, 5)
Assign additional problems in the math textbook for practice.
Activity 10: Graphing Art Project (GLEs: 4, 6, 7, 9, 10, 15, 16, 24, 27, 28, 29)
Materials List: paper, pencil, graphing calculator, Graphing Art Bellringer BLM, Graphing
Art Sailboat Graph BLM, Graphing Art Sailboat Equations BLM, Graphing Art Project
Directions BLM, Graphing Art Graph Paper BLM, Graphing Art Project Equations BLM,
Graphing Art Evaluation BLM, Overhead projector-graph transparencies BLM, Optional:
Math Type®, EquationWriter®, Graphmatica® and TI Interactive® computer software
In this Graphing Art Project, students will analyze equations to synthesize graphs and then
analyze graphs to synthesize equations. The students will draw their own pictures composed
of familiar functions, write the equation of each part of the picture finding the points of
intersection, and learn to express their creativity mathematically.
Math Log Bellringer:
Distribute the Graphing Art Bellringer BLM in which the students will individually
graph a set of equations to produce the picture of a heart.
Solution:
Algebra IIUnit 8Conic Sections 194
Louisiana Comprehensive Curriculum, Revised 2008
Activity:
This culminating activity is taken from the February, 1995, issue of Mathematics Teacher in
an article by Fan Disher entitled "Graphing Art" reprinted in Using Activities from the
Mathematics Teacher to Support Principles and Standards, (2004) NCTM. It uses two days
of in-class time and one week of individual time. It follows the unit on conics but involves all
functions learned throughout the year.
Use the Bellringer to review the graphs of lines and absolute value relations, the writing
of restricted domains in various forms, and finding points of intersection. The Bellringer
models the types of answers that will be expected in the next part of the activity. Use the
Bellringer to also review graphing equations on a calculator with restricted domains.
Divide students into five member cooperative groups and distribute the Graphing Art
Sailboat BLM and the Graph and Graphing Art Sailboat Equations BLM. Have group
members determine the equation of each part of the picture and the restrictions on either
the domain or range. This group work will promote some very interesting discussions
concerning the forms of the equations and how to find the restrictions.
The students are now ready to begin the individual portion of their projects.
Distribute Graphing Art Project Directions BLM, Graphing Art Graph Paper BLM
and the Graphing Art Project Equations BLM. In the directions, students are
instructed to use graph paper either vertically or horizontally to draw a picture
containing graphs of any function discussed this year. On the Graphing Art Project
Equations BLM, the students will record a minimum of ten equations, one for each
portion of the picture see Graphing Art Project Directions BLM for equation
requirements. There is no maximum number of equations, which gives individual
students much flexibility. The poorer students can draw the basic picture and
equations and achieve while the creative students can draw more complex pictures.
Distribute the Graphing Art Evaluation BLM and explain how the project will be
graded.
At this point, this is now an out-of-class project in which the students are monitored
halfway through, using a rough draft. Give the students a deadline to hand in the
numbered rough draft and equations. At that time, they should exchange equations
and see if they can graph their partner's picture.
Later, have students turn in final copies of pictures and equations and their
Graphing Art Evaluation BLMs. After all the equations have been checked for
accuracy, appoint an editor from the class to oversee the compilation of the graphs
and equations into a booklet to be distributed to other mathematics teachers for use
in their classes. The students enjoy seeing their names and creations in print and
Algebra IIUnit 8Conic Sections 195
Louisiana Comprehensive Curriculum, Revised 2008
gain a feeling of pride in their creations.
Have students write a journal stating what they learned in the project, what they
liked and disliked about the project, and how they feel the project can be improved. a small quiz after each conic to check for
understanding.
Administer two comprehensive assessments:
(1) circles and parabolas
(2) all conic sections
Activity-Specific Assessments
Activity 6:
Determine which of the following equations is a circle, parabola, line, hyperbola or
ellipse.
(1) 8x2 + 8y2 + 18x – 64y – 71=0
(2) 8x + 7y – 81 = 0
(3) 4x2 + 3y – 6 = 0
(4) 2x + 6y2 –26 = 0
(5) 8x2 8y2 – 6 = 0
(6) 7x2 + y2 – 45 = 0
(7) x2 – y2 – 36 = 0
Solutions :
(1) circle
(2) line
(3) parabola
(4) parabola
(5) hyperbola
(6) ellipse
(7) hyperbola
Algebra IIUnit 8Conic Sections 196
Louisiana Comprehensive Curriculum, Revised 2008
Activity 8:
Evaluate the Double-Napped Cone Lab (see activity) using the following rubric:
10 pts correctly sliced the cone
10 pts. correct graphs and equations
10 pts. slices of degenerate cases
10 pts. presentation
Activity 10:
Evaluate the Graphing Art project using several assessments during the project to
check progress.
(1) The group members should assess each other's rough drafts to catch mistakes
before the project is graded for accuracy.
(2) Evaluate the final picture and equations using the Graphing Art Evaluation BLM.
(3) Evaluate the opinion journal to decide whether to change or modify the unit for
next year.
Algebra IIUnit 8Conic Sections 197 | 677.169 | 1 |
Discrete Mathematics Using a Computer
9781846282416
ISBN:
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Pub Date: 2006 Publisher: Springer
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Details about The Joy of Sets:
This book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. The book is written in a fairly easy-going style, with minimal formalism. In Chapter 1, the basic principles of set theory are developed in a 'naive' manner. Here the notions of 'set', 'union', 'intersection', 'power set', 'rela tion', 'function', etc., are defined and discussed. One assumption in writing Chapter 1 has been that, whereas the reader may have met all of these 1 concepts before and be familiar with their usage, she may not have con sidered the various notions as forming part of the continuous development of a pure subject (namely, set theory). Consequently, the presentation is at the same time rigorous and fast."
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4.
Newnes
An imprint of Elsevier Science
Linacre House, Jordan Hill, Oxford OX2 8DP
200 Wheeler Rd, Burlington, MA 01803
First published 1999
Second edition 2000
Reprinted 2001
Third edition 2002
Copyright 1999, 2000, 2002. John Bird. All rights reserved
The right of John Bird to be identified as the author of this work has been
asserted in accordance with the Copyright, Designs and Patents Act 1988
No part of this publication may be reproduced in
any material form (including photocopying or storing
in any medium by electronic means and whether or not
transiently or incidentally to some other use of
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10.
Preface
Basic Engineering Mathematics, 3rd edition introduces and then consolidates basic mathematical principles and promotes
awareness of mathematical concepts for students needing a broad base for further vocational studies. In this third edition, new
material has been added on the introduction to differential and integral calculus.
The text covers:
(i) the Applied Mathematics content of the GNVQ mandatory unit 'Applied Science and Mathematics for Engineering'
at Intermediate level (i.e. GNVQ 2)
(ii) the mandatory 'Mathematics for Engineering' at Advanced level (i.e. GNVQ 3) in Engineering
(iii) the optional 'Applied Mathematics for Engineering' at Advanced level (i.e. GNVQ 3) in Engineering
(iv) the Mathematics content of 'Applied Science and Mathematics for Technicians' for Edexcel/BTEC First Certificate
(v) the mandatory 'Mathematics for Technicians' for National Certificate and National Diploma in Engineering
(vi) Mathematics 1 for City & Guilds Technician Certificate in Telecommunications and Electronics Engineering
(vii) basic mathematics for a wide range of introductory/access/foundation mathematics courses
(viii) GCSE revision and for similar mathematics courses in English-speaking countries world-wide.
Basic Engineering Mathematics, 3rd edition provides a lead into Engineering Mathematics.
Each topic considered in the text is presented in a way that assumes in the reader little previous knowledge of that topic.
Theory is introduced in each chapter by a brief outline of essential theory, definitions, formulae, laws and procedures. However
these are kept to a minimum, for problem solving is extensively used to establish and exemplify the theory. It is intended that
readers will gain real understanding through seeing problems solved and then solving similar problems themselves.
This textbook contains some 575 worked problems, followed by over 1000 further problems (all with answers – at the
end of the book). The further problems are contained within some 118 Exercises; each Exercise follows on directly from the
relevant section of work. 260 line diagrams enhance the understanding of the theory. Where at all possible the problems
mirror practical situations found in engineering and science.
At regular intervals throughout the text are fifteen Assignments to check understanding. For example, Assignment 1 covers
material contained in Chapters 1 and 2, Assignment 2 covers the material contained in Chapters 3 and 4, and so on. These
Assignments do not have answers given since it is envisaged that lecturers could set the Assignments for students to attempt
as part of their course structure. Lecturers' may obtain a complimentary set of solutions of the Assignments in an Instructor's
Manual available from the publishers via the internet – see below.
At the end of the book a list of relevant formulae contained within the text is included for convenience of reference.
'Learning by Example' is at the heart of Early Engineering Mathematics
John Bird
University of Portsmouth
Instructur's Manual
Full worked solutions and mark scheme for all the Assignments are contained in this Manual which is available to lecturers
only. To obtain a password please e-mail J.Blackford@Elsevier.com with the following details: course title, number of students,
your job title and work postal address.
To download the Instructor's Manual visit and enter the book title in the search box, or use the
following direct URL:
19.
Fractions, decimals and percentages 9
Problem 13. A gear wheel having 80 teeth is in mesh
with a 25 tooth gear. What is the gear ratio?
Gear ratio D 80:25 D
80
25
D
16
5
D 3.2
i.e. gear ratio D 16:5 or 3.2:1
Problem 14. Express 25p as a ratio of £4.25
Working in quantities of the same kind, the required ratio is
25
425
i.e.
1
17
That is, 25p is
1
17
th of £4.25. This may be written either as:
25:425: :1:17 (stated as '25 is to 425 as 1 is to 17') or as
25
425
=
1
17
Problem 15. An alloy is made up of metals A and B
in the ratio 2.5:1 by mass. How much of A has to be
added to 6 kg of B to make the alloy?
Ratio A:B: :2.5:1 i.e.
A
B
D
2.5
1
D 2.5
When B D 6 kg,
A
6
D 2.5 from which, A D 6 ð 2.5 D 15 kg
Problem 16. If 3 people can complete a task in
4 hours, find how long it will take 5 people to complete
the same task, assuming the rate of work remains
constant.
The more the number of people, the more quickly the task is
done, hence inverse proportion exists.
3 people complete the task in 4 hours,
1 person takes three times as long, i.e. 4 ð 3 D 12 hours,
5 people can do it in one fifth of the time that one person
takes, that is
12
5
hours or 2 hours 24 minutes.
Now try the following exercise
Exercise 5 Further problems on ratio and propor-
tion (Answers on page 252)
1. Divide 312 mm in the ratio of 7 to 17.
2. Divide 621 cm in the ratio of 3 to 7 to 13.
3. £4.94 is to be divided between two people in the ratio
of 9 to 17. Determine how much each person will
receive.
4. When mixing a quantity of paints, dyes of four dif-
ferent colours are used in the ratio of 7:3:19:5. If the
mass of the first dye used is 31
2
g, determine the total
mass of the dyes used.
5. Determine how much copper and how much zinc is
needed to make a 99 kg brass ingot if they have to be
in the proportions copper:zinc: :8:3 by mass.
6. It takes 21 hours for 12 men to resurface a stretch of
road. Find how many men it takes to resurface a sim-
ilar stretch of road in 50 hours 24 minutes, assuming
the work rate remains constant.
7. It takes 3 hours 15 minutes to fly from city A to city B
at a constant speed. Find how long the journey takes if
(a) the speed is 11
2
times that of the original speed
and
(b) if the speed is three-quarters of the original speed.
2.3 Decimals
The decimal system of numbers is based on the digits 0 to
9. A number such as 53.17 is called a decimal fraction, a
decimal point separating the integer part, i.e. 53, from the
fractional part, i.e. 0.17
A number which can be expressed exactly as a decimal
fraction is called a terminating decimal and those which
cannot be expressed exactly as a decimal fraction are called
non-terminating decimals. Thus, 3
2
D 1.5 is a terminating
decimal, but 4
3
D 1.33333. . . is a non-terminating decimal.
1.33333. . . can be written as 1.P3, called 'one point-three
recurring'.
The answer to a non-terminating decimal may be expressed
in two ways, depending on the accuracy required:
(i) correct to a number of significant figures, that is, figures
which signify something, and
(ii) correct to a number of decimal places, that is, the
number of figures after the decimal point.
The last digit in the answer is unaltered if the next digit on the
right is in the group of numbers 0, 1, 2, 3 or 4, but is increased
by 1 if the next digit on the right is in the group of numbers
5, 6, 7, 8 or 9. Thus the non-terminating decimal 7.6183. . .
becomes 7.62, correct to 3 significant figures, since the next
digit on the right is 8, which is in the group of numbers 5, 6,
7, 8 or 9. Also 7.6183. . . becomes 7.618, correct to 3 decimal
places, since the next digit on the right is 3, which is in the
group of numbers 0, 1, 2, 3 or 4.
Problem 17. Evaluate 42.7 C 3.04 C 8.7 C 0.06
20.
10 Basic Engineering Mathematics
The numbers are written so that the decimal points are under
each other. Each column is added, starting from the right.
42.7
3.04
8.7
0.06
54.50
Thus 42.7 + 3.04 + 8.7 + 0.06 = 54.50
Problem 18. Take 81.70 from 87.23
The numbers are written with the decimal points under each
other.
87.23
81.70
5.53
Thus 87.23 − 81.70 = 5.53
Problem 19. Find the value of
23.4 17.83 57.6 C 32.68
The sum of the positive decimal fractions is
23.4 C 32.68 D 56.08
The sum of the negative decimal fractions is
17.83 C 57.6 D 75.43
Taking the sum of the negative decimal fractions from the
sum of the positive decimal fractions gives:
56.08 75.43
i.e. 75.43 56.08 D −19.35
Problem 20. Determine the value of 74.3 ð 3.8
When multiplying decimal fractions: (i) the numbers are
multiplied as if they are integers, and (ii) the position of the
decimal point in the answer is such that there are as many
digits to the right of it as the sum of the digits to the right
of the decimal points of the two numbers being multiplied
together. Thus
(i) 743
38
5944
22 290
28 234
(ii) As there are 1C1 D 2 digits to the right of the decimal
points of the two numbers being multiplied together,
(74.3 ð 3.8), then
74.3 U 3.8 = 282.34
Problem 21. Evaluate 37.81 ł 1.7, correct to (i) 4
significant figures and (ii) 4 decimal places.
37.81 ł 1.7 D
37.81
1.7
The denominator is changed into an integer by multiplying
by 10. The numerator is also multiplied by 10 to keep the
fraction the same. Thus
37.81 ł 1.7 D
37.81 ð 10
1.7 ð 10
D
378.1
17
The long division is similar to the long division of integers
and the first four steps are as shown:
22.24117..
17 378.100000
34
38
34
41
34
70
68
20
(i) 37.81 ÷ 1.7 = 22.24, correct to 4 significant figures,
and
(ii) 37.81 ÷ 1.7 = 22.2412, correct to 4 decimal places.
Problem 22. Convert (a) 0.4375 to a proper fraction
and (b) 4.285 to a mixed number.
(a) 0.4375 can be written as
0.4375 ð 10 000
10 000
without chang-
ing its value,
i.e. 0.4375 D
4375
10 000
By cancelling
4375
10 000
D
875
2000
D
175
400
D
35
80
D
7
16
i.e. 0.4375 =
7
16
(b) Similarly, 4.285 D 4
285
1000
D 4
57
200
24.
3
Indices and standard form
3.1 Indices
The lowest factors of 2000 are 2 ð 2 ð 2 ð 2 ð 5 ð 5 ð 5.
These factors are written as 24
ð53
, where 2 and 5 are called
bases and the numbers 4 and 3 are called indices.
When an index is an integer it is called a power. Thus, 24
is called 'two to the power of four', and has a base of 2 and
an index of 4. Similarly, 53
is called 'five to the power of 3'
and has a base of 5 and an index of 3.
Special names may be used when the indices are 2 and 3,
these being called 'squared' and 'cubed', respectively. Thus
72
is called 'seven squared' and 93
is called 'nine cubed'.
When no index is shown, the power is 1, i.e. 2 means 21
.
Reciprocal
The reciprocal of a number is when the index is 1 and its
value is given by 1 divided by the base. Thus the reciprocal
of 2 is 2 1
and its value is 1
2
or 0.5. Similarly, the reciprocal
of 5 is 5 1
which means 1
5
or 0.2
Square root
The square root of a number is when the index is 1
2
, and
the square root of 2 is written as 21/2
or
p
2. The value of a
square root is the value of the base which when multiplied
by itself gives the number. Since 3 ð 3 D 9, then
p
9 D 3.
However, 3 ð 3 D 9, so
p
9 D 3. There are always
two answers when finding the square root of a number and
this is shown by putting both a C and a sign in front of
the answer to a square root problem. Thus
p
9 D š3 and
41/2
D
p
4 D š2, and so on.
Laws of indices
When simplifying calculations involving indices, certain
basic rules or laws can be applied, called the laws of indices.
These are given below.
(i) When multiplying two or more numbers having the
same base, the indices are added. Thus
32
ð 34
D 32C4
D 36
(ii) When a number is divided by a number having the same
base, the indices are subtracted. Thus
35
32
D 35 2
D 33
(iii) When a number which is raised to a power is raised to
a further power, the indices are multiplied. Thus
35 2
D 35ð2
D 310
(iv) When a number has an index of 0, its value is 1. Thus
30
D 1
(v) A number raised to a negative power is the reciprocal of
that number raised to a positive power. Thus 3 4
D
1
34
Similarly,
1
2 3
D 23
(vi) When a number is raised to a fractional power the
denominator of the fraction is the root of the number
and the numerator is the power.
Thus 82/3
D
3
p
82 D 2 2
D 4
and 251/2
D
2
p
251 D
p
251 D š5
(Note that
p
Á 2
p
)
3.2 Worked problems on indices
Problem 1. Evaluate: (a) 52
ð 53
, (b) 32
ð 34
ð 3 and
(c) 2 ð 22
ð 25
30.
4
Calculations and evaluation of formulae
4.1 Errors and approximations
(i) In all problems in which the measurement of distance,
time, mass or other quantities occurs, an exact answer
cannot be given; only an answer which is correct to a
stated degree of accuracy can be given. To take account
of this an error due to measurement is said to exist.
(ii) To take account of measurement errors it is usual to
limit answers so that the result given is not more than
one significant figure greater than the least accurate
number given in the data.
(iii) Rounding-off errors can exist with decimal fractions.
For example, to state that p D 3.142 is not strictly
correct, but 'p D 3.142 correct to 4 significant figures'
is a true statement.
(Actually, p D 3.14159265 . . .)
(iv) It is possible, through an incorrect procedure, to obtain
the wrong answer to a calculation. This type of error is
known as a blunder.
(v) An order of magnitude error is said to exist if incorrect
positioning of the decimal point occurs after a calcula-
tion has been completed.
(vi) Blunders and order of magnitude errors can be
reduced by determining approximate values of
calculations. Answers which do not seem feasible
must be checked and the calculation must be
repeated as necessary. An engineer will often need
to make a quick mental approximation for a
calculation. For example,
49.1 ð 18.4 ð 122.1
61.2 ð 38.1
may
be approximated to
50 ð 20 ð 120
60 ð 40
and then, by
cancelling,
50 ð
1
20 ð 120 2 1
1 60 ð 40 2 1
D 50. An accurate
answer somewhere between 45 and 55 could therefore
be expected. Certainly an answer around 500 or
5 would not be expected. Actually, by calculator
49.1 ð 18.4 ð 122.1
61.2 ð 38.1
D 47.31, correct to 4 significant
figures.
Problem 1. The area A of a triangle is given by
A D 1
2
bh. The base b when measured is found to be
3.26 cm, and the perpendicular height h is 7.5 cm. Deter-
mine the area of the triangle.
Area of triangle D 1
2
bh D 1
2
ð 3.26 ð 7.5 D 12.225 cm2
(by
calculator).
The approximate value is 1
2
ð3ð8 D 12 cm2
, so there are
no obvious blunder or magnitude errors. However, it is not
usual in a measurement type problem to state the answer to
an accuracy greater than 1 significant figure more than the
least accurate number in the data: this is 7.5 cm, so the result
should not have more than 3 significant figures
Thus area of triangle = 12.2 cm2
Problem 2. State which type of error has been made
in the following statements:
(a) 72 ð 31.429 D 2262.9
(b) 16 ð 0.08 ð 7 D 89.6
(c) 11.714 ð 0.0088 D 0.3247, correct to 4 decimal
places.
(d)
29.74 ð 0.0512
11.89
D 0.12, correct to 2 significant
figures.
35.
Calculations and evaluation of formulae 25
2. Below is a list of some metric to imperial conversions.
Length 2.54 cm D 1 inch
1.61 km D 1 mile
Weight 1 kg D 2.2 lb 1 lb D 16 ounces
Capacity 1 litre D 1.76 pints
8 pints D 1 gallon
Use the list to determine (a) the number of millimetres
in 15 inches, (b) a speed of 35 mph in km/h, (c) the
number of kilometres in 235 miles, (d) the number of
pounds and ounces in 24 kg (correct to the nearest
ounce), (e) the number of kilograms in 15 lb, (f) the
number of litres in 12 gallons and (g) the number of
gallons in 25 litres.
3. Deduce the following information from the BR train
timetable shown in Table 4.3:
(a) At what time should a man catch a train at Mossley
Hill to enable him to be in Manchester Piccadilly
by 8.15 a.m.?
Table 4.3 Liverpool, Hunt's Cross and Warrington ! Manchester
Reproduced with permission of British Rail
36.
26 Basic Engineering Mathematics
(b) A girl leaves Hunts Cross at 8.17 a.m. and travels
to Manchester Oxford Road. How long does the
journey take. What is the average speed of the
journey?
(c) A man living at Edge Hill has to be at work at
Trafford Park by 8.45 a.m. It takes him 10 minutes
to walk to his work from Trafford Park station.
What time train should he catch from Edge Hill ?
4.4 Evaluation of formulae
The statement v D u C at is said to be a formula for v in
terms of u, a and t.
v, u, a and t are called symbols.
The single term on the left-hand side of the equation, v, is
called the subject of the formulae.
Provided values are given for all the symbols in a for-
mula except one, the remaining symbol can be made the
subject of the formula and may be evaluated by using a cal-
culator.
Problem 16. In an electrical circuit the voltage V is
given by Ohm's law, i.e. V D IR. Find, correct to 4
significant figures, the voltage when I D 5.36 A and
R D 14.76 .
V D IR D 5.36 14.76
Hence voltage V = 79.11 V, correct to 4 significant
figures.
Problem 17. The surface area A of a hollow cone is
given by A D prl. Determine, correct to 1 decimal
place, the surface area when r D 3.0 cm and l D 8.5 cm.
A D prl D p 3.0 8.5 cm2
Hence surface area A = 80.1 cm2
, correct to 1 decimal
place.
Problem 18. Velocity v is given by v D u C at. If
u D 9.86 m/s, a D 4.25 m/s2
and t D 6.84 s, find v,
correct to 3 significant figures.
v D u C at D 9.86 C 4.25 6.84
D 9.86 C 29.07
D 38.93
Hence velocity v = 38.9 m=s, correct to 3 significant
figures.
Problem 19. The area, A, of a circle is given by
A D pr2
. Determine the area correct to 2 decimal places,
given radius r D 5.23 m.
A D pr2
D p 5.23 2
D p 27.3529
Hence area, A = 85.93 m2
, correct to 2 decimal places.
Problem 20. The power P watts dissipated in an elec-
trical circuit may be expressed by the formula P D
V2
R
.
Evaluate the power, correct to 3 significant figures,
given that V D 17.48 V and R D 36.12 .
P D
V2
R
D
17.48 2
36.12
D
305.5504
36.12
Hence power, P = 8.46 W, correct to 3 significant fig-
ures.
Problem 21. The volume V cm3
of a right circular
cone is given by V D 1
3
pr2
h. Given that r D 4.321 cm
and h D 18.35 cm, find the volume, correct to 4
significant figures.
V D 1
3
pr2
h D 1
3
p 4.321 2
18.35
D 1
3
p 18.671041 18.35
Hence volume, V = 358.8 cm3
, correct to 4 significant
figures.
Problem 22. Force F newtons is given by the formula
F D
Gm1m2
d2
, where m1 and m2 are masses, d their
distance apart and G is a constant. Find the value of the
force given that G D 6.67 ð 10 11
, m1 D 7.36, m2 D
15.5 and d D 22.6. Express the answer in standard form,
correct to 3 significant figures.
F D
Gm1m2
d2
D
6.67 ð 10 11
7.36 15.5
22.6 2
D
6.67 7.36 15.5
1011 510.76
D
1.490
1011
Hence force F = 1.49 U 10−11
newtons, correct to 3 sig-
nificant figures.
37.
Calculations and evaluation of formulae 27
Problem 23. The time of swing t seconds, of a simple
pendulum is given by t D 2p
l
g
. Determine the time,
correct to 3 decimal places, given that l D 12.0 and
g D 9.81
t D 2p
l
g
D 2 p
12.0
9.81
D 2 p
p
1.22324159
D 2 p 1.106002527
Hence time t = 6.950 seconds, correct to 3 decimal places.
Problem 24. Resistance, R , varies with temperature
according to the formula R D R0 1 C at . Evaluate
R, correct to 3 significant figures, given R0 D 14.59,
a D 0.0043 and t D 80.
R D R0 1 C at D 14.59[1 C 0.0043 80 ]
D 14.59 1 C 0.344 D 14.59 1.344
Hence resistance, R = 19.6 Z, correct to 3 significant
figures.
Now try the following exercise
Exercise 15 Further problems on evaluation of for-
mulae (Answers on page 254)
1. The area A of a rectangle is given by the formula
A D lb. Evaluate the area when l D 12.4 cm and
b D 5.37 cm.
2. The circumference C of a circle is given by the
formula C D 2pr. Determine the circumference
given p D 3.14 and r D 8.40 mm.
3. A formula used in connection with gases is
R D PV /T. Evaluate R when P D 1500, V D 5
and T D 200.
4. The velocity of a body is given by v D u C at.
The initial velocity u is measured when time t is
15 seconds and found to be 12 m/s. If the acceleration
a is 9.81 m/s2
calculate the final velocity v.
5. Calculate the current I in an electrical circuit, when
I D V/R amperes when the voltage V is measured
and found to be 7.2 V and the resistance R is 17.7 .
6. Find the distance s, given that s D 1
2
gt2
. Time
t D 0.032 seconds and acceleration due to gravity
g D 9.81 m/s2
.
7. The energy stored in a capacitor is given by
E D 1
2
CV2
joules. Determine the energy when capac-
itance C D 5 ð 10 6
farads and voltage V D 240 V.
8. Find the area A of a triangle, given A D 1
2
bh, when
the base length b is 23.42 m and the height h is
53.7 m.
9. Resistance R2 is given by R2 D R1 1 C at . Find
R2, correct to 4 significant figures, when R1 D 220,
a D 0.00027 and t D 75.6
10. Density D
mass
volume
. Find the density when the mass
is 2.462 kg and the volume is 173 cm3
. Give the
answer in units of kg/m3
.
11. Velocity D frequencyðwavelength. Find the velocity
when the frequency is 1825 Hz and the wavelength
is 0.154 m.
12. Evaluate resistance RT, given
1
RT
D
1
R1
C
1
R2
C
1
R3
when R1 D 5.5 , R2 D 7.42 and R3 D 12.6 .
13. Find the total cost of 37 calculators costing £12.65
each and 19 drawing sets costing £6.38 each.
14. Power D
force ð distance
time
. Find the power when a
force of 3760 N raises an object a distance of 4.73 m
in 35 s.
15. The potential difference, V volts, available at battery
terminals is given by V D E Ir. Evaluate V when
E D 5.62, I D 0.70 and R D 4.30
16. Given force F D 1
2
m v2
u2
, find F when m D 18.3,
v D 12.7 and u D 8.24
17. The current I amperes flowing in a number of cells
is given by I D
nE
R C nr
. Evaluate the current when
n D 36, E D 2.20, R D 2.80 and r D 0.50
18. The time, t seconds, of oscillation for a simple
pendulum is given by t D 2p
l
g
. Determine the
time when p D 3.142, l D 54.32 and g D 9.81
19. Energy, E joules, is given by the formula E D 1
2
LI2
.
Evaluate the energy when L D 5.5 and I D 1.2
20. The current I amperes in an a.c. circuit is given
by I D
V
R2 C X2
. Evaluate the current when
V D 250, R D 11.0 and X D 16.2
21. Distance s metres is given by the formula
s D ut C 1
2
at2
. If u D 9.50, t D 4.60 and a D 2.50,
evaluate the distance. | 677.169 | 1 |
Mathematics
(offered by the Department of Mathematical Sciences) Telephone number 012 429 6202
1
Introduction
All attempts to give a definition of Mathematics have suffered from one shortcoming or another, so we will not try to define the subject here. There is never any doubt, however, as to whether something is mathematical or not. As aspirant Mathematics students, you have already had some contact with Mathematics and so you know something about the subject. The best that we can do here is to say that a mathematician is someone who asks questions of a certain nature and then attempts to find answers to these questions. The aim of the mathematician is to find answers to increasingly difficult questions and to systematise and generalise existing Mathematics so that an ever larger class of questions might be answered in a simple fashion. The questions which a mathematician asks might have their origins in a wide variety of disciplines. They might originate in Mathematics itself or in any of the applied sciences such as physics, mechanics, chemistry, economics, etc. In the teaching of Mathematics, we are guided by the following considerations: (i) the work we discuss must link up with the student's existing knowledge; (ii) the work must be of fundamental importance to both mathematics in its own right and to its fields of application; (iii) the work must be presented in a systematic manner so that as many problems as possible might be solved with the least possible effort. In view of these considerations, undergraduate Mathematics consists mainly of algebra, analysis and discrete mathematics. These topics are of fundamental importance to both the applications of Mathematics and to further developments of the subject as a discipline in its own right.
1.1 Mathematics on first- and second-year level
There are five modules on first level, MAT112, 113 and 103 as well as two introductory modules, MAT110 and 111. As from 2001, the modules MAT112 and 113 replaced the modules MAT101 and MAT102. The four modules MAT111, 112, 113 and 103 are compulsory for a 'full' first-year course in Mathematics and, depending on your results in Mathematics at Matriculation level, it may be required that MAT110 first be passed before registering for MAT112, 113 and 103. The first-level modules APM111 and 112 in Applied Mathematics contain important applications of the calculus discussed in MAT112 and 113. APM113 contains numerical applications of the linear algebra dealt with in MAT103. Hence you are strongly advised to include at least the abovementioned three APM modules in your curriculum, especially if you intend to take Mathematics on second level or as a major subject. On second level there are seven modules, MAT211, 212, 213, 215, 216, 217 and 219, of which any four form a full second-year course. If you wish to take Mathematics at third level, you must ensure that you satisfy the necessary Prerequisite. If you wish to take Mathematics as a major subject, MAT213 and 211 are compulsory. The other module(s) which you take at second level will depend on your choice at third level. The modules MAT211, 215 and 216 are important for applications in physics and other natural sciences.
1.2 Requirements for admission to postgraduate studies
To qualify for admission to studies for the Honours BSc degree in Mathematics a student must hold a Bachelor's degree and, inter alia, have passed and obtained good results in ONE of the following: (a) four third-level module in Mathematics (b) three third-level module in Mathematics and two third-level module in Applied Mathematics (c) Mathematics III Admission can be refused on grounds of an unsatisfactory undergraduate study record. Further particulars will be found in the departmental brochure on postgraduate studies.
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General Information
Credit for the BSc degree and/or the National Certificate in Datametrics is given for five first-level MAT modules. To register for one or more of MAT110 and 111 a student must have satisfied the requirements of Sc1(1)(b)(i)–(iv) in Part 7 of the Calendar. Students who passed (or were exempted from) one or more of MAT111, 101, 102, 103 prior to 1993, may complete the remaining MAT modules without MAT110. If MAT111 has not yet been passed, it must be completed together with the remaining first-level MAT modules. Students who were registered for one or more of MAT111, 101, 102 and 103 prior to 1993 but who did not pass any of them may re-enrol for MAT modules. MAT110 and 111 are NOT available to students who passed all three of MAT101–103 prior to 1990. Such students may now register for second-level MAT modules and, where applicable, second-level APM modules. MAT110 is also NOT available to students who passed all three of MAT101–103 (with or without MAT111) prior to 1993. The use of a pocket calculator is permissible in the examination for MAT217. Credit for a degree is granted for: (i) not more than two of MAT101, 102, 112, 113 (ii) either MAT111 or MAT101, 102 and 103 if all three were passed prior to 1990 (iii) either MAT110 or MAT101, 102, 103 if all three were passed prior to 1993 (iv) not more than one of MAT216, APM201 and 211 (v) not more than one of MAT215, APM202, and 212 (vi) not more than two of MAT215 (or APM212) or MAT216 (or APM211) and MAT214 (or 203)
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(vii) (viii) (ix) (x) (xi) (xii)
not more than two of MAT201, 202 and 211 either MAT212 or COS201 passed prior to 1991 either MAT214 or MAT203 either MAT311 or MAT303 and/or 304 either MAT217 or APM214 either MAT218 or APM215
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Mathematics as a Major Subject
If you intend to obtain a BSc degree with Mathematics as a major subject, please note that: Because some of the most important non-trivial applications of mathematics are found in the field of mechanics and physics, we strongly recommend that you include at least all the first-level modules in Applied Mathematics and/or Physics in your curriculum. If you choose Mathematics as a major subject, you are also strongly advised to include modules in Statistics and Computer Science in your curriculum. A basic knowledge of these Subjects is important for your training as a mathematician. If you plan to take a SECOND major together with Mathematics as a major, you will find that almost any subject offered as a major for the BSc degree can be sensibly combined with Mathematics, eg Applied Mathematics, Physics, Chemistry, Statistics, Astronomy, Computer Science, Geography, etc. Further recommendations are given in this chapter. If you plan to take Mathematics as your ONLY major subject, you should seriously consider including more than the required minimum of four third-level MAT modules in your curriculum. We further recommend that you include all the first- and second-level modules in at least one other subject. Note that your curriculum must contain a total of EIGHT third-level module (see Sc5(1) in Part 7 of the Calendar). NB Should you require any further information or advice regarding the composition of your curriculum, please write to the Registrar or, if possible, discuss the matter in person with the staff of the Department of Student Admissions and Registrations (Tel. 0861 670 411), or one of the Provincial Centres or Regional Offices.
Compulsory modules for a major subject combination:
NB At least two further second-level MAT modules will be required, depending on your choice at third level. It is strongly recommended that you first pass the second-level modules before attempting the corresponding third-level module.
First level: MAT111, 103 and TWO of (MAT101, 102 prior to 2002), 112, 113 Second level: MAT211, 213 Third level: FOUR of the following: (a) MAT301 (b) MAT302 (c) MAT305, 215 (d) MAT306, 216 (e) MAT307, 212 (f) MAT311, 215 (g) APM301, MAT217
REQUIREMENTS FOR THE BSc DEGREE
and A pass in Mathematics (not Mathematical Literacy) with a rating of 5 or higher (NSC) or at least 50% (D-symbol) in Mathematics HIGHER GRADE or 80% (A-symbol) on STANDARD GRADE at Matriculation level prior to 2008, or equivalent. Students who do not meet the requirements may register for MAT111.
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Syllabus
NB All modules in this subject are offered as YEAR MODULES except MAT103N, MAT111N, MAT112P and MAT113Q, which are offered as SEMESTER modules.
ACCESS MODULE
MAT011K Access to Mathematics (year module) Advice: This module may NOT be taken towards degree studies – see Sc1(1)(c) in Part 7 of the Calendar. Purpose: to enable students to demonstrate the understanding of the real number system, ratio, proportion, percentage, integral exponents, scientific notation and estimation, roots, units, algebraic expressions, sequences, linear and quadratic equations and inequalities, systems of equations in two unknowns, exponents, logarithms, functions, straight lines, parabolas, hyperbolas, circles, introduction to elementary statistics, basic geometry (angles, triangles, quadrilaterals) and calculation of areas and volumes.
FIRST-LEVEL MODULES (NQF LEVEL 5)
MAT110M Precalculus mathematics A (year module)* Prerequisite: Mathematics as in Sc1(1)(b)(i)–(iv) in Part 7 of the Calendar. Students who obtained at least 75% in MAT011 or at least 50% in Mathematics HIGHER GRADE, 80% in Mathematics STANDARD GRADE at matriculation level will not be required to take MAT110 on first level. Purpose: to acquire the knowledge and skills that will enable students to draw and interpret graphs of linear, absolute value, quadratic, exponential, logarithmic and trigonometric functions, and to solve related equations and inequalities.
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MAT111N Precalculus mathematics B (S1 and S2)* Prerequisite: Mathematics as in Sc1(1)(b)(i)–(iv) in Part 7 of the Calendar. Purpose: to acquire practical experience of vectors, polar co-ordinates, the complex number system, theory of polynomials, systems of linear equations, sequences, mathematical induction, binomial theorem, conic sections. MAT103N Linear algebra – see Sc1(1)(b)(iv) in Part 7 of the Calendar. (d) MAT110 (e) MAT011 passed with at least 75% Purpose: to obtain knowledge of systems of linear equations, Gaussian elimination and homogeneous systems; matrix algebra, partitioning of matrices, matrix inverses and elementary matrices; determinants, Laplace expansion and cofactor matrices; vector geometry, orthogonality, distance and the dot product, planes and the cross product, and least squares polynomial fitting. MAT112P Calculus A see Sc1(1)(b)(iv) in Part 7 of the Calendar (d) MAT110 (e) MAT011 passed with at least 75% Purpose: to equip students with those basic skills in differential and integral calculus which are essential for the physical, life and economic sciences. Some simple applications are covered. More advanced techniques and further applications are dealt with in module MAT113. MAT113Q Calculus B (S1 and S2)* Prerequisite: MAT112P Purpose: to enable students to continue to obtain basic skills in differentiation and integration, and build on the knowledge provided by module MAT112. More advanced techniques and further basic applications are covered. Together, the modules MAT112 and MAT113 constitute a first course in Calculus which is essential for students taking Mathematics as a major subject.
SECOND-LEVEL MODULES (NQF LEVEL 6)
MAT211R Linear algebra* Prerequisite: MAT103 Purpose: to understand and apply the following linear algebra concepts: vector spaces, rank of a matrix, eigenvalues and eigenvectors, diagonalisation of matrices, orthogonality in Rn, Gram-Schmidt algorithm, orthogonal diagonalisation of symmetric matrices, least squares polynomial fitting, linear transformations, change of basis, invariant subspaces and direct sums, block triangular form. MAT212S Introduction to discrete mathematics* Prerequisite: Any ONE of COS101, MAT101, 102, 103, 112, 113 Advice: For parts of this module the following modules contain useful background: MAT103, COS101. Purpose: to acquaint students with the theory and applications of the following aspects of discrete mathematics: counting principles, relations and digraphs, (including equivalence relations), functions, the pigeonhole principle, order relations and structures (eg partially ordered sets, lattices, Boolean algebras), the principle of induction. MAT213T Real analysis*
NB This module is purely theoretical and is intended mainly for students who wish to take Mathematics as a major subject.
Prerequisite: Any TWO of MAT101, 102, 112, 113 Purpose: to enable students to master and apply the fundamental concepts and techniques of real analysis as they occur in an elementary discussion of the real number system, sequences and series; limits, continuity and differentiability of functions; the Bolzano-Weierstrass property, continuous and uniformly continuous functions, the mean value theorem, Taylor's theorem; the Riemann integral, the fundamental theorem of calculus, improper integrals, and the power series. MAT215V Calculus in higher dimensions Prerequisite: MAT111 (or 103) and any TWO of MAT101, 102, 112, 113 Purpose: to gain clear knowledge and an understanding of vectors in n-space, functions from n-space to m-space, various types of derivatives (grad, div, curl, directional derivatives), higher-order partial derivatives, inverse and implicit functions, double integrals, triple integrals, line integrals and surface integrals, theorems of Green, Gauss and Stokes.
THIRD-LEVEL MODULES (NQF LEVEL 7)
Prerequisite: Any TWO MAT or APM modules on second level MAT301S Linear algebra* Advice: Linear algebra, as dealt with in MAT211 is assumed as known in this module. Purpose: to acquire a basic knowledge concerning inner product spaces, invariant subspaces, cyclic subspaces, operators and their canonical forms. MAT302T Algebra* Advice: Aspects of linear algebra, as dealt with in MAT211, are used in this module.
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Purpose: to enable students to master and practise the applications of the concepts, results and methods necessary to construct mathematical arguments and solve problems independently as they occur in an elementary treatment of algebraic structures, groups, homomorphism theorems, factor groups, permutation groups, the main theorem for Abelian groups, Euclidean rings, divisibility in Euclidean rings, fields, finite fields, and the characteristics of a field. MAT305W Complex analysis* Prerequisite: MAT213 or MAT215 Advice: Real analysis and Calculus in Higher dimensions, as dealt with in MAT213 and MAT215 respectively are assumed as known in this module. Purpose: to introduce students to the following topics in complex analysis: functions of a complex variable, continuity, uniform convergence, complex differentiation, power series and the exponential function, integration, Cauchy's theorem, singularities and residues. MAT306X Ordinary differential equations* Advice: Aspects of linear algebra, as dealt with in MAT211, are used in this module. The content of module MAT216 (or APM211) is assumed as known. Purpose: to enable students to master the fundamental concepts and apply the methods for the solution of homogeneous and non-homogeneous systems of differential equations, as well as Gronwall's inequality, qualitative theory, and the linearisation of nonlinear systems. MAT307Y Discrete mathematics: Combinatorics* Advice: For parts of this module the following modules contain useful background: MAT212, COS201. Purpose: to enable students to understand and apply the following concepts: (a) In graph theory: isomorphism, planar graphs, Euler tours, Hamilton cycles, colouring problems, trees, networks; (b) In enumeration: basic counting principles, distributions, binomial identities, generating functions, recurrence relations, inclusion-exclusion. MAT311U Real analysis Advice: Thorough knowledge of the content of MAT213 and MAT215 is essential for this module. Purpose: to enable students to understand metric spaces, continuity, convergence, completeness, compactness, connectedness, Banach's fixed point theorem and its applications, the Riemann-Stieljes integral, normed linear spaces, and the Stone-Weierstrass theorem.
5
Practical Work and Admission Requirements
Practical work in MAT219 mainly comprises the writing of computer programs. The programs have to be developed on suitable computers using prescribed computer packages. Access to a suitable computer is an admission requirement for all modules with a practical component. Students can gain access as follows: (i) by purchasing a computer for their own use; or (ii) by using a computer belonging to a study group, friend, computer bureau, or employer; or (iii) by reserving time on a computer at one of Unisa's microcomputer laboratories in Pretoria, Polokwane, Cape Town, and Durban. The minimum configuration of a 'suitable' computer is described as follows: An IBM or IBM-compatible personal computer which is year 2000-compliant and has the following minimum configuration: Pentium 75 MHz processor or faster VGA or higher graphics Windows 95 or later version Hard disk 1 Gigabyte or bigger 16 MB RAM (32 MB or higher recommended by some software) A CD-ROM drive A 3.5 inch high-density (1.44 MB) diskette drive A printer that can print both text and graphics (minimum A4 paper size) A compiler for any suitable programming language is an additional requirement for APM213 and APM311. NB Unisa CANNOT supply any of the commercial software packages mentioned. Students are required to either obtain their own copy of the software, or make use of the microcomputer laboratories. Full particulars of the microcomputer laboratories are supplied in a tutorial letter sent to students upon registration.
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"Mathematics Mathematics offered by the Department of Mathematical Sciences" | 677.169 | 1 |
Includes 612 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. In this title, you will have access to 25 detailed videos featuring Math instructors who explain | 677.169 | 1 |
Math past, present and future
Math has never been one of my strongest subjects in school, period. For some people in my classes the numbers just seemed to always jump off the page and they could always find the solution quickly. Me, well, that's a different story. It's always difficult for me to grasp the concept of whatever I am learning as I am taking notes. Heck, sometimes even when I am doing my homework I have no clue what I am doing. However, no adays I've learned to become more disciplined. Whether it means going to weekly math help sessions or meeting with a tutor. I think the only way one can be successful is if you put in the time and effort and don't take short cuts.
Now, I may not have the best grade in this class but for some reason, I feel like more of the concepts are beginning to click. For the first time, I find myself excited to sit down and start to do a problem. I like the challenge of knowing that a problem for evaluating a limit using direct substitution only has one solution. I LOVE DIRECT SUBSTITUION!! There is no gray area, no theorem you have to explain, but just simply find the answer. There's nothing better than getting down to the last step of the problem, plugging in the value for x and getting the correct answer! Plus, I think that it's nice to have a teacher that is a grad student because then I think its less intimidating to ask a question because they were in our shoes not that long ago.
I like taking a new concept that I've learned for evaluating a limit and applying it. Limits have a lot to do with all of chapter 2. So I feel like once you've learned the initial concept you're set because it never goes away, it'll come back later in the chapter. Often times, it's difficult for me to retain information if it just all of a sudden goes away after a test or quiz. So, it's been interesting to see how limits and derivatives have contined to play a role in daily notes. I am excited and interested to learn the more complex concept because I feel as though I already have a grasp of what was originally taught. For example, today in class we started learning Chapter 3 and derivatives came up once again. I am also interested to see how complex and complicated a limit or derivative could get. I feel like once we've learned how to do it one way, that that's the only thing that you could do with a limit or derivative. But in reality, it's just a small piece of the puzzle, so im curious to see how many pieces of the puzzle there are because we learn all the complex concepts of limits and derivatives. | 677.169 | 1 |
Practice solving linear equations with these fifty problems in elementary algebra. The student selects a single variable linear equation, solves for the variable, and checks the answer by viewing the step-by-step solution. Problems start with low difficulty and gradually increase to challenging. Most appropriate for 6th to 8th grade students.
Practice solving linear equations with these fifty basic problems in elementary algebra. The student selects a single variable linear equation, solves for the variable, and checks the answer by viewing the step-by-step solution. Problems start with low difficulty and gradually increase to challenging. Most appropriate for 4th and 5th grade students.This book is exclusively designed for Kids. The book consists of several questions,puzzles,tricky questions exclusively made for Kids. The book both entertains children and also provides knowledge on various topics.The book is real fun for Kids. Knowledge and fun provided in one single book for kids. The book is designed highly keeping in mind about kids and their way of learning.
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Free Algebra Worksheets
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Solutions for Higher Education
Uses for Education
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Using and Understanding Mathematics: A Quantitative Reasoning Approach (4th Edition)
9780321458209
ISBN:
0321458206
Edition: 4 Pub Date: 2007 Publisher: Addison Wesley
Summary: Most students taking this course do so to fulfill a requirement, but the true benefit of the course is learning how to use and understand mathematics in daily life. This quantitative reasoning text is written expressly for those students, providing them with the mathematical reasoning and quantitative literacy skills they'll need to make good decisions throughout their lives. Common-sense applications of mathematics ...engage students while underscoring the practical, essential uses of math.
Jeffrey O. Bennett is the author of Using and Understanding Mathematics: A Quantitative Reasoning Approach (4th Edition), published 2007 under ISBN 9780321458209 and 0321458206. One hundred fifty six Using and Understanding Mathematics: A Quantitative Reasoning Approach (4th Edition) textbooks are available for sale on ValoreBooks.com, thirty four used from the cheapest price of $4.66, or buy new starting at $26GREAT REBIND CANDIDATE. NO ACCESS CODE. Heavy wear and creasing to cover/edges/corners. Cover is separating from binding or completely separated/missing. Cardboard showing on [more]
GREAT REBIND CANDIDATE. NO ACCESS CODE. Heavy wear and creasing to cover/edges/corners. Cover is separating from binding or completely separated/missing. Cardboard showing on corners, edges and spine of book. Title page intact. Binding WE HAVE NUMEROUS COPIES.[less]
HARDCOVER. DOES NOT INCLUDE ACCESS CODE. Mild rippling on one inch of top edge of last ten pages from liquid exposure, but none that detracts from usability. Light to moderate [more]
HARDCOVER. DOES NOT INCLUDE ACCESS CODE. Mild rippling on one inch of top edge of last ten pages from liquid exposure, but none that detracts from usability. Light to moderate wear to cover/edges/corners. School markings and writing on/inside covers. Minimal markings/highlighting on/inside book, none of which detracts from content. Binding and cover solidly connectedEverything about this book was absolutely useful and helpful in working on my assignments and doing homework. I was able to highlight key elements in the book as well to help me prepare for my tests. The diagrams and examples were very self explanatory. I would highly recommend this book to anyone I know that is taking a general college math course. I actually bought this book so I can use it as a reference when working towards my bachelors degree in the near future.
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Synopses & Reviews
Publisher Comments:
This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition of mathematic fundamentals. Rather than a system of theorems with completely developed proofs or examples of applications, readers will encounter a coherent presentation of mathematical ideas that begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of the calculus.
Contents include examinations of the various types of numbers and a criticism of the extension of numbers; arithmetic, geometry, and the rigorous construction of the theory of integers; the rational numbers, the foundation of the arithmetic of natural numbers, and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers.
In issues of mathematical philosophy, the author explores basic theoretical differences that have been a source of debate among the most prominent scholars and on which contemporary mathematicians remain divided. "With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics." — Carl B. Boyer, Brooklyn College. 27 figures. Index.
Book News Annotation:
This survey of mathematical concept formation begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of calculus. No formal training in mathematics is necessary. This is an unabridged republication of the edition published by Harper & Row, New York, 1959. Annotation (c)2003 Book News, Inc., Portland, OR (booknews.com)
Synopsis:Synopsis:
"Synopsis"
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You're in MathematicsElementaryStatistics
ELEMENTARY STATISTICS: A BRIEF VERSION is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step......... more...
This solid text presents ideas and concepts more clearly for students who have little or no background in statistics. The Twelveth Edition retains all the elements and style that educators nationwide have come to expect--clear prose, excellent problems...... more
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DeMYSTiFieD is your solution for tricky subjects like trigonometry
If you think a Cartesian coordinate is something from science fiction or a hyperbolic tangent is an extreme exaggeration, you need Trigonometry DeMYSTiFieD , Second Edition, to unravel this topic's fundamental concepts and theories at your own pace.
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Most math and science study guides are a reflection of the college professors who write them-dry, difficult, and pretentious.
The Humongous Book of Trigonometry Problems is the exception. Author Mike Kelley has taken what appears to be a typical trigonometry workbook, chock full of solved problems-more than 750!-and made notes in the marginsCliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. CliffsQuickReview Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry ? whether you need a supplement to your textbookThis classic text encompasses the most important aspects of plane and spherical trigonometry in a question-and-answer format. Its 913 specially selected questions appear with detailed answers that help readers refresh their trigonometry skills or clear up difficulties in particular areas. Questions and answers in the first part discuss plane trigonometry,... more...
Originally published over a century ago, this work remains among the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics. The subsequent growth of science into a diverse range of specialties has enhanced the value of this classic, whose thorough, basic treatment presents material that is... more...
This single-volume compilation of three books centers on Hyperbolic Functions, an introduction to the relationship between the hyperbolic sine, cosine, and tangent, and the geometric properties of the hyperbola. The development of the hyperbolic functions, in addition to those of the trigonometric (circular) functions, appears in parallel columns... more... | 677.169 | 1 |
A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS, 10th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations.
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THE SOLUTION MANUAL as well as the classes and textbooks are essentially two sides of the same coin. The classes and textbook help you build a solid foundation on which to be examined on.
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These study notes are very easy to understand elementary discrete math and very helpful to built a concept about the foundation of computers.The key points discuss in these notes are:Strings and Languages, Sequences and Strings, Sequence...
These study notes are very easy to understand elementary discrete math and very helpful to built a concept about the foundation of computers.The key points discuss in these notes are:Combinatorics, Elementary Counting, Techniques, Licens...
These study notes are very easy to understand elementary discrete math and very helpful to built a concept about the foundation of computers.The key points discuss in these notes are:Discrete Mathematics, Consisting of Unconnected Parts,... | 677.169 | 1 |
MA150 Precalculus Mathematics
for F1W1) Graphing calculator is required.
2) Laptop computer is recommended.
3) Download and install a free copy of LiveMath Viewer from LiveMath.com. This math application software is used by the instructor to demonstrate to students complex definition and simulation of mathematical concepts.
4) Download and install free copies of Java and Prolog compilers from and respectively. These compilers are used by the instructor to demonstrate to students complex definition and simulation of mathematical concepts.
Course Description: A consideration of those topics in algebra and trigonometry necessary for the calculus. Topics include: mathematical analysis of the line, the conic sections, exponential and logarithmic functions, circular functions, polynomial and rational functions, mathematical induction, and theory of equations. PREREQUISITE: MA 131or equivalent. 3:0:3
Educational Philosophy:
TheLearning Outcomes: Core Learning Outcomes
Demonstrate the basic features of the Cartesian coordinate plane
Analyze and graph the defining features of linear equations
Analyze and graph the defining features of circles, parabolas, ellipses, and hyperbolas
Explain, graph, and apply logarithmic and exponential functions
Demonstrate the fundamental properties of trigonometric definitions, theorems and equations | 677.169 | 1 |
/r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics. Submissions should state and outline problems or questions about a given field or link to an especially insightful article about a mathematical concept.
I am a 16 year old high school student in grade 11. With the high school reform going on in my country, it is now possible for students who are good at Maths to miss out on 2/3 of their math classes and work by themselves instead.
My teacher asked me if I wanted to work on something specific and told me I had to decide on a topic myself.(theoretical or practical mathematics)
So, mathematicians of reddit, have you ever come across an interesting topic in your career that you would recommend to a high schooler?
It would be useful to know what you have covered already, since math education is different from country to country.
That being said, here are some recommendations for classes that I think are useful regardless of whether you pursue mathematics or not:
Calculus and differential equations: I saw that you mentioned derivatives and integrals in one of your other posts, so I assume that you have at least a little knowledge of single variable calculus. The next logical step would be to finish the single variable stuff and move into some multivariate calculus. If that's done, then I'd take a look at differential equations, which is required for most students in a math/physics/engineering program at the college level. You could also explore functions of a complex variable a little bit (since they are really just functions of multiple variables) although some of the theorems may not make a ton of sense without knowledge of set theory/topology.
Linear algebra: I also saw that you have an interest in learning programming so this might be worth studying a little bit more in depth than you might have previously(I know I only had a cursory introduction to it when I was in high school, as did most of my friends). A lot of my computer science friends said that this class helped them a ton when they were going through school, but I only really know about how it gets used in numerical analysis. (That is as far as areas that use a lot of computing are concerned, it's useful in a lot of other areas too.)
Set theory: This one may not be too big if you don't plan on being a mathematician. But, if you want to look into more advanced areas and really understand what is going on, you're going to need to know some set theory. Maybe you won't need a full class on it, but some sort of introduction is useful and most math books begin with a section devoted to the set theory that you will need to understand to make it through the book.
These are all really generic answers though, and you will probably see them in any thread asking this question. However, they are generic because they really are the foundation of mathematics, to the point that they are all prerequisites in many math programs. With a little more insight into your background, it would be easier to give more specific areas to look into, but those are three areas that will benefit you in most STEM fields.
I strongly, strongly recommend studying naive logic/naive set theory, over any other possible choices. They're usually taught at the same time, as either "Introduction to Abstract Mathematics," "Proofwriting," "Introduction to Logic," and so on.
This is the class where most people first learn to write proofs and do rigorous mathematical thinking, and the first class where you can be a lot more creative. There's very little memorization, and a whole lot of hard thinking.
Even for topics like calculus and linear algebra, you will be able to read and understand the best books on these subjects only after having a good grounding in logic/set theory. All of the commonly mentioned books, with clear writing and well presented ideas, are written for an audience that understands naive logic and set theory. Calculus, with the average college textbook, only tells you what to do. Once you have some proofwriting experience, you can pick up a book like Spivak and learn why and how it got that way.
Even if you do not wind up doing math, the skills you are developing by studying are entirely analytic in nature. It forces you to think more carefully and more accurately than any other math class you'll have had up to this point, and that's transferable to any other class.
I recommend this book because it's not difficult to read and starts off quite gently. It's abstract algebra, so it investigates a lot of interesting mathematical structure you've probably seen before but never really thought about. I recommend this book as a second course because of the writing style and how easy it is to get into self-study with it - other subjects don't have an equivalently easy to read book, despite being arguably simpler topics.
I'd recommend you pick any subject you're interested in. Pick something 'mathy' of course ;). Then use wikipedia to find the historical 'roots' of the field. Then start reading there, read the papers of the inventors, read some history and follow the trail all the way up till present day. It's fun and informative. Don't worry about complexity, if you go back far enough you'll be at the same mathematical level as the people you're reading about.
Doing this will really, really help you understand why some things are the way they are. It will help your understanding in a fundamental way which isn't offered in most university courses.
A few suggested fields of study:
Thermodynamics. Lot's of interesting math happening here. Historically interesting because the first inventors in this fields were often combinations of Scholar/general/gentleman/mathematician/scientist/artist/pious e.g. Carnot.
development of differentiation and integration. Very important for many engineering sciences.
special relativity. More interesting for the concepts than the actual maths. High school maths will be enough here. (All you really need to understand is Pythagoras's theorem and some basic physics.) If you want a challenge you can continue into general relativity.
This really depends on your own interests. What areas of math interest you the most? What are your interests outside of mathematics? Do you know of any math topics you might study that already seem intriguing? any that don't sound interesting?
Other than mathematics, I also enjoy physics (extremely fascinated by graphene!), chemistry and psychology. I plan on getting a bit into objective C when I get my laptop for Christmas.
So far, algorithms were suggested to me, which seem interesting. I'm pretty much open to anything as long as it doesn't involve too much graphing.
I thought itd be best to start with applied mathematics and maybe later on cover study a more theoretical topic
Algorithms sounds rather broad. Have you narrowed that down? Statistics might be a good topic for psychology. It sounds like you are more interested in chemistry/physics/chemical engineering. Perhaps someone can suggest what field of mathematics will help you the most with those fields.
So far, I have not. I'm really in the first stage of this program and have not had a chance to narrow anything down so far. My schedule is tight during the week, so I'll look further into some topics over Christmas break.
I'm not a fan of statistics, but, while reading your response, an acquainted physician did come to mind who I could consult!
Unless you want to finish up being a programmer, mathematics is mostly really useless without physics. For me the beauty of mathematics comes through understanding of the physical meaning of each equation. I don't understand mathematicians who like looking at 5-line equations and not even knowing what they are for. (I don't mean to offend anyone, that's just not for my taste.)
I would recommend reading on classical mechanics (Lagrangian and Hamiltonian formalism), and then quantum mechanics. If you choose the right book with an appropriate level of mathematics, you will be fascinated by the beauty and the simplicity of these theories.
On a pure mathematical side: if you are familiar with derivatives and integrals, then Complex Analysis is probably one of the most beautiful mathematical theories. With just a little effort it provides amazing tools for other fields of maths or physics. | 677.169 | 1 |
This book is intended to be the text for a "capstone" course, that is, the final course taken by undergraduate math majors, that should in some way tie together all the preceding courses. The book does this by explicitly drawing connections between branches of mathematics, and most chapters relate linear algebra to one other branch of mathematics. The book gives a competent exposition of a variety of subjects, is well-supplied with examples illustrating the definitions, and has a modest number of (rather easy) exercises.
My big gripe with this book is that, after developing relationships between different branches of mathematics, it does not do anything to use these new structures to solve problems in the original branches. (Chapter 1 is an exception: it shows how geometry can be tied to algebra to determine which angles can be trisected, a very old problem that was intractable until these connections were invented.) I'll pick on Chapter 5 as an example. This chapter, "Matrices and Topology," shows how to define a metric on spaces of matrices and examines the topological properties of these spaces, especially investigating their connected components. Then it stops. Do we know any more about matrices, or topology, when we are done? Not really. There are germs of linear operator theory and Lie groups here, subjects that are useful in studying integral and differential equations and many areas of mathematical physics. Making this connection has historically been a fruitful idea, but none of that is mentioned here.
I think much of the reason the book does not work on such specific problems is that it is focused on classification and categorization. The book's attitude is captured on p. 192 when it says "The ideal goal of all areas of pure mathematics is to produce lists of properties that can be attached to the objects being studied such that these lists characterize when two objects are equivalent." Many mathematicians would disagree that this is the only or main goal of pure mathematics. Certainly it doesn't cover such triumphs as the Prime Number Theorem or the Fundamental Theorem of Algebra (to name two problems that were solved by connecting the original problem to other areas of mathematics).
The prose tends to be verbose (the statement I quoted above about the purpose of pure mathematics is typical). I noticed a half-dozen or so minor errors, mostly involving in leaving out parts of a hypothesis or getting things backward.
A very different text for a capstone course, that also emphasizes the unity of mathematics and drawing connections, is Iosevich's A View from the Top: Analysis, Combinatorics, and Number Theory. Iosevich's book takes the opposite approach from the present book: it starts with specific difficult problems and studies how to bring different branches of mathematics to bear on them, rather than bringing together the branches first. This is a much more realistic problem-solving approach and makes the subject much easier to motivate. | 677.169 | 1 |
04866538lements of Real Analysis (Dover Books on Mathematics)
This classic text in introductory analysis delineates and explores the intermediate steps between the basics of calculus and the ultimate stage of mathematics: abstraction and generalization. Since many abstractions and generalizations originate with the real line, the author has made it the unifying theme of the text, constructing the real number system from the point of view of a Cauchy sequence (a step which Dr. Sprecher feels is essential to learn what the real number system is). The material covered in Elements of Real Analysis should be accessible to those who have completed a course in calculus. To help give students a sound footing, Part One of the text reviews the fundamental concepts of sets and functions and the rational numbers. Part Two explores the real line in terms of the real number system, sequences and series of number and the structure of point sets. Part Three examines the functions of a real variable in terms of continuity, differentiability, spaces of continuous functions, measure and integration, and the Fourier series. An especially valuable feature of the book is the exercises which follow each section. There are over five hundred, ranging from the simple to the highly difficult, each focusing on a concept previously introduced. | 677.169 | 1 |
Deck Purpose : A review of chapter 1, Some Basic Concepts of Arithmetic and Algebra, 1.1, Numerical and Algebraic Expressions. Concepts reviewed include differences between arithmetic and algebra, sets, and order of operations.
Deck Purpose : A review of chapter 1, Some Basic Concepts of Arithmetic and Algebra, 1.2, Prime and Composite Numbers . Concepts reviewed include identifying prime and composite numbers, factoring, and finding the greatest common factor and least common multiple.
Deck Purpose : These flashcards are for my upcoming math test. Even if you're in Algebra or above like me, you still should remember some basic statistics! We just began second semester and this is our first quiz. Remember to title and label your graphs every time! :) ♥
Deck Purpose : This is a set of review flash cards to learn concepts from 4-3 to 4-5. They cover relations, equations as relations, and graphing linear equations and go into detail about Standard Form of a Linear Equation and its four rules. I hope you enjoy! Have fun! | 677.169 | 1 |
Functional equations form a modern branch of mathematics. This book provides an elementary yet comprehensive introduction to the field of functional equations and stabilities. Concentrating on functional equations that are real or complex, the authors present many fundamental techniques for solving these functional equations. Topics covered in the text include Cauchy equations, additive functions, functional equations for distance measures, and Pexider's functional equations. Each chapter points to various developments in abstract domains, such as semigroups, groups, or Banach spaces, and includes exercises for both self-study and classroom use.
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The book includes several interesting and fundamental techniques for solving functional equations in real or complex realms. There exist many useful exercises as well as well-organized concluding remarks in each chapter. ... This book is written in a clear and readable style. It is useful for researchers and students working in functional equations and their stability. -Mohammad Sal Moslehian, Mathematical Reviews, Issue 2012b
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You can earn a 5% commission by selling Introduction to Functional Equations | 677.169 | 1 |
Pocket Polynomial
By Vivussoft Technologies
Description
Allows students to factor and solve complex polynomials and helps students understand the methods involved in factoring and solving quadratics. This is one of the most important chapters in high-school mathematics, and pocket polynomial can be used to help students learn this important section of mathematics! | 677.169 | 1 |
Tagged Questions
For questions related to the teaching and learning of mathematics. Note that Mathematics Educators StackExchange may be a better home for narrowly scoped questions on specific issues in mathematics education.
I am a college student and my current situation is that I have an extremely hard time getting myself to start doing math. I feel like math is 'boring,' but ONLY until I start actually doing it. Once I ...
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I am not sure if this is the right place to post this, but here is the situation.
In about two weeks or so I will be giving a 2-3 hours lecture on some topic in mathematics to freshman and sophomore ...
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I'm looking for a truly basic example of probabilistic method proof which could be presented without a board (i.e. speaking only), that is, even moderately complicated calculations are not allowed.
...
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Do any of you have a fun or interesting Lagrange multiplier problem that would be suitable for undergraduate calculus students? I'm planning on working through a standard Lagrange multiplier problem ...
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A constant function in my case would be:
$f(0) = 1,$
...
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I'm looking ...
I'm studying progressions in math class in Portugal, and I don't know the words/translation in english for certain things so I'll try to explain.
I have this arithmetic progression: 2, 8, 18, 32 ...
...
I have heard and seen several references to "obvious questions", "obvious axioms" and other "obvious" things (I am not referring to obvious results!). Now, in a seminar I am taking, at the end of each ...
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A little background: I'm a high school student enrolling at a local university next fall. I plan to pursue a mathematics degree, have studied this Calculus book. During the next semester, I plan to ...
It is well-known that a sequence has a limit if and only if it is bounded and has a unique limit point. I think this is a better definition of the limit of a sequnece, comparing with the $\epsilon-N$ ...
During the demonstration of the theorem of the convergence of the series of fourier, my teacher wrote :$$ \frac{1}{2}+ \sum_{k=1}^{n} \cos(ky)=\frac{\sin((n+\frac{1}{2})y)}{2\sin(\frac{y}{2})} $$
he ...
I have just finished the Final for my Computer Hardware course, and I'm trying to figure out where my grade currently stands. The way the class is broken up is 50% weight for the homework, 25% for the ... | 677.169 | 1 |
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