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Book Description: Transform your mathematics course into an engaging and mind-opening experience for even your most math-phobic students. The Heart of Mathematics: An invitation to effective thinking --now in its third edition--succeeds at reaching non-math, non-science-oriented majors and encouraging them to discover the mathematics inherent in the world around them. Infused throughout with the authors' humor and enthusiasm, The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking. |
Description: This math class is a pre-algebra math class. We use the book Discovering Algebra which is a book that uses real life situations to teach algebra. In addition to this book we will use hands on activities and activities using the TI-73 calculator. Although our focus will be algebra we will also be covering the 7th grade curriculum which also covers number sense and geometry. Every activity chosen will focus on challenging the students to think creatively.
Class: Math
Description: This general math class covers such topics as number sense, geometry, and algebra. I will use many different learning strategies such as hands on activities, higher order thinking strategies, and using TI-73 activities that correlate with the curriculum.
Class: Science
Description: The 7th grade science curriculum covers a wide variety of topics such as: atmosphere and the weather, biology of the human body, heredity and genetics, and motion and forces.
Students have created a bracket for the NCAA Championships and have found statistics for each team in their region. Between 3/15 & 4/3 the students will answer questions about the statistics they have already found.
Students will be able to take a math concept (statistics) and connect it to the real world. |
Success in math includes mastery of geometry skills and requires children to make connections between the real world and geometry concepts in order to solve problems. Successful problem solvers will be ready for the challenges of mathematics as they advance to more complex topics. The activities in this workbook are designed...
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What exactly is the Golden Ratio? How was it discovered? Where is it found? These questions and more are thoroughly explained in this engaging tour of one of mathematics' most interesting phenomena. Veteran educators and prolific mathematics writers trace the appearance of the Golden Ratio throughout history and demonstrate a variety...
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Many of us trained mainly in the humanities and liberal arts may respect mathematics as an essential scientific discipline, but have done very little mathematics and often feel intimidated by its rigors. If you've ever wondered what mathematicians mean by the aesthetic elegance of their subject, here is your chance to...
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A Tour of the Calculus
Written by David Berlinski
Format: eBook
ISBN: 9780307789730
Our Price: $11.99Success in math requires children to make connections between the real world and math concepts in order to solve problems. Successful problem solvers will be ready for the challenges of mathematics as they advance to more complex topics. The activities in this workbook are designed to help your children catch up...
Read more > |
I don't get the difference between Math level 1 and level 2...
Do the universities prefer one over another?
People seem to take Math level 2 more than level 1.
And if I'm not planning to major anything close to Math or Science, is it okay to take Math level 1?
Math Level I does not allow a calcultor im pretty sure, and tests you on algebra, simple algebraic functions, plane geometry, solid geometry, coordinate geometry, basic trig, elementary stats, and misc. Math level II DOES allow calculators, and has all ^ and includes advanced forms of many of the above, with number theory as well)
The main difference is that Math II covers pre-calc while Math I does not. Math II has a much better curve, so it's better to take that unless you haven't taken precalc yet. In addition, most colleges prefer Math II. |
We will give the appropriated tools for dealing with massive geometric data (algorithms and data structures) and will exploit the geometric properties of the problems posed, to find optimal solutions.
Specific goals
Knowledges
Learn the several kinds of problems in Computational Geometry, as well as their applications.
Learn the capacity of combining geometric tools with the appropriated data structures and algorithmic paradigms.
See in action several algorithmic paradigms and data structures useful in geometric problems.
Apply geometric results to real problems.
Abilities
Ability to solve basic problems that appear in computational geometry.
Ability to implement the solutions proposed in the class, as well as those that can be found in the basic references of the course.
Ability to recognize the geometric problems behind the applications, and to propose adequate algorithmic tools to solve them.
Competences
Ability to create and use real 2D and 3D geometric models.
Ability to analize problems: distinguish the geometric characteristics of the problem, properly describe the data (input) and precisely define the geometric information that we want to obtain (output).
Ability to tackle new problems by consciously using strategies that have proved useful in solving previous problems.
Triangulation of monotone polygons, decomposition of a polygon into monotone polygons.
7.
Proximity
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7,0
2,0
2,0
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3,0
6,0
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20,0
Voronoi Diagram
8.
Triangulations of point sets
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P
L
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5,0
2,0
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10,0
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Delaunay triangulation
9.
Arrangements
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1,0
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Arrengements of lines. Arrengements of segments.
10.
Location
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2,0
0
0
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Students presentations of other problems
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Avaluation additional hours
7,0
Total work hours for student
147,0
Docent Methodolgy
Theory classes will set out the contents of the course, oriented to the resolution of examples and applications.
Exercise classes will be centered in the resolution of problems by the students. Students will be assigned problems and will have enough time to think about them in advance, so that they will be able to propose their solutions during the class, or explain the troubles that they have encountered while trying to solve them. The problems will be mainly algorithmic (not theoretical).
The purpose of the lab classes is to implement the solutions discussed in the theory and exercices classes, the effective solution of problems being one of the goals of the course. The problems to be solved in the lab classes will start being of elementary complexity, and will end with the resolution of a problems, preferably applied and real, to be chosen by each student.
Evaluation Methodgy
COURSE OPTION:
The evaluation will be based on the work done by the student along the course. The four components to be considered will be:
If they ask for it, students who wish can take a final exam, consisting in solving problems in which the knowledge and skills acquired in class will be applied. In this case, the final course grade will be computed combining the grade of the fre lab exercise (L2) and the grade of the final exam (E), in the following way: |
Assessment Rules
CMod description
This
module follows up the foundation course Numerical Skills for Environmental
Sciences I and provides a more complete understanding of the numerical skills
required for studying environmental science. The course will consist of
lectures and workshops which will cover environmental case studies in greater
detail.
Curriculum Design: Outline Syllabus
Lecture 1 (SJL) Algebra:this most essential of numerical skills often causes great difficulty and impedes progress in other areas of numerical work. Here we concentrate on the components of equations (constants, variables, operators) and how to manipulate them into more useful forms.
Curriculum Design: Pre-requisites/Co-requisites/Exclusions
The module follows up the foundation module (ENV 111) in this subject and is designed to give undergraduate students a fuller grounding in the numerical skills required for studying environmental science. The course will consist of lectures and workshops, which will cover environmental case studies in greater detail. Environmental examples from other ENV modules are employed throughout the course |
From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
For departments of computer science offering Sophomore through Junior-level courses in Algorithms or Design and Analysis of Algorithms.
This is an introductory-level algorithm text. It includes worked-out examples and detailed proofs. Presents Algorithms by type rather than application.
Table of contents
1. Preliminaries.
2. Elementary Algorithmicss.
3. Asymptotic Notation.
4. Analysis of Algorithms.
5. Some Data Structures.
6. Greedy Algorithms.
7. Divide-And-Conquer.
8. Dynamic Programming.
9. Exploring Graphs.
10. Probabilistic Algorithms.
11. Parallel Algorithms.
12. Computational Complexity.
13. Heuristic and Approximate Algorithms.
References.
Index.
Features & benefits
structures material by techniques employed, not by the application area, so students can progress from the underlying abstract concepts to the concrete application essentials.
begins with a compact, but complete introduction to some necessary math, and also includes a long introduction to proofs by contradiction and mathematical induction. This serves to fill the gaps that many undergraduates have in their mathematical knowledge.
gives a paced, thorough introduction to the analysis of algorithms, and uses coherent notation and unusually detailed treatment of solving recurrences.
includes a chapter on probabilistic algorithms, and an introduction to parallel algorithms, both of which are becoming increasingly important.
approaches the analysis and design of algorithms by type rather than by application. |
Mathematics Enrichment Center
Two Locations:
Main Campus W117
Fairview Park Plaza Extension Center
(1485 W. King St., Decatur, IL)
Welcome to the Mathematics Enrichment Center! We help students to advance more quickly through developmental mathematics — General Math Skills, Pre Algebra, Basic Algebra, and Intermediate Algebra — in a self paced approach by using technology enhanced individual instruction.
If you need to take:
Then you are eligible to choose:
MATH 087 (General Math Skills)
MATH 090 (Pre-Algebra)
MATH 091 (Basic Algebra)
MATH 098 (Intermediate Algebra)
MATH 096
How
In MATH 096, students begin by taking a mastery test for the level they placed into.
Students are then given personalized assignments covering only the topics needed.
Students have a flexible schedule and choose which four hours to go to class on a walk in basis.
Once students score 70% or higher on the final mastery test, they may start the next course in the same semester saving time and money.
Students can focus on learning the mathematics concepts without the pressure of grades because class is credit/non-credit.
A variety of help is available 24/7 in the form of individual instruction, software called MyMathLab.
Contact Information
Bring your Richland student ID with you to scan in and out of the Mathematics Enrichment Center - if you don't have a Richland student ID, go to the Student Services Office (N127) to get one - you will need a photo ID and your student ID number to get your Richland ID. |
Additional product details
Students get the applied math skills they need for the modern farming industry with MATHEMATICAL APPLICATIONS IN AGRICULTURE, 2nd Edition. Invaluable in any area of agriculture-from livestock and dairy production to horticulture and agronomy -the text focuses on methods for solving problems students will encounter in the real world using math and logic skills. Clearly written and thoughtfully organized, the stand-alone chapters on mathematics involved in crop production, livestock production, horticulture, and financial management allow instructors flexibility in selecting the topics most appropriate to a given region, while line drawings, charts, graphs, case studies, examples, and sample problems help students grasp the concepts and hone their critical thinking skills. |
Pencils (for homework, quizzes, and tests) and a pen (for correcting and notes)
Folder for organization of papers
Ruler, protractor, and compass (for home use - do have some for classroom use)
A good attitude!
Grade Makeup:
Homework 40%
Tests 40%
Quizzes 20%
Algebra I (2nd Semester)Graph paper
A good attitude!
Grade Makeup:
Homework 40%
Quizzes/ Tests 60%
College Alg. & Trig.A good attitude!
Grade Makeup:
Homework 25%
Quizzes/ Tests 75%
Applied Math I
Materials Needed:
A scientific calculator
3-ring binder
Pencils (for homework, quizzes, and tests) and a pen (for correcting and notes)
Standard/metric ruler
A good attitude!
Grade Makeup:
Homework 20%
Quizzes/Tests 50%
Labs 30%
Classroom Rules and Expectations:
In order to make these classes the best they can be I expect students to follow these simple guidelines:
¸ Respect yourself, classmates, and the teacher.
¸ Come to class prepared to learn.
¸ Listen to and follow all verbal and written instructions.
¸ Ask questions!
Grading Procedure:
Progress reports will be sent home every 6 weeks. The semester grade or final grade for my classes is the result of running totals from the 3 6-week grading periods and a final exam (10%), if given; It is not an average. This cumulative grade can be found on the gradesheets sent home and is designated as S1. (G1, G2, and G3 are the individual 6-week grades). |
Featuring hundreds of exercises, this book offers plenty of opportunities for practice on the math found in sixth, seventh, eighth, and ninth grade curriculums. It gives your child the tools to master: integers rational numbers; patterns equations; graphing functions and more. |
On May 15, 2012, Canaa Lee published a collection of math stories entitled, "Algebra for the Urban Student." She is the daughter of Kathleen D. Lee and the late Travis O. Lee. Canaa has been a teacher for 11 years now and loves algebra. She has watched students struggle in math year after year and wanted to help students wrap their minds around abstract math concepts. Lee is an expert algebra teacher. She also orchestrated a math enrichment program, Project (Educating and Diversifying to Grow Exponentially) EDGE in Garland, TX. Not only do students struggle with math but they also struggling with reading.
In addition to Algebra for the Urban Student, Canaa Lee is writing a sequel that is due to be released August 2012. After the release of the sequel to Algebra for the Urban Student, she also plans to write a series of children math books. Please visit
The students Lee teaches inspired "Algebra for the Urban Student." The book started off as just a collection of units for her students so she could ensure that were exposed to and master all the topics for the assessments. Parents commented on homework assignments because they could now help their children with their homework because it was easy to understand and and easy to follow without stepping foot in the classroom! For the first time, many students understood their homework and could complete their assignments. Also, students were improving their reading and comprehension skills in both English and algebra! Canaa Lee has written enough units and assessments to write a book. Textbooks are designed for math teachers and professors; "Algebra for the Urban Student" is intended for the common student. |
Problems in Solutions.
ABSTRACT
Whether it has been a fault in the public school system of America or simply a trait more common to certain minds, the matter of simple proportions is—to the average nurse and oftentimes to physicians as well—most confounding. When the student nurse encounters the elementary arithmetic associated with the making of solutions or the primary problems of chemistry, she is generally, in the language of the street, "up against it." Possibly no one has realized it better than Miss Sullivan, who has had broad teaching experience. How much good such a book will do is problematic. In the first place, one who explains these arithmetical problems should have a reader with patience and a certain amount of intelligence. After this step is successfully passed, Miss Sullivan may get her book across. To one, on the other hand, who has even a fair acquaintance with lower mathematics, the book seems unnecessarily |
Course Detail
Mathematics: Fundamentals of Mathematics
MATH 052 Z1(CRN: 60921)
Emphasizing proofs, fundamental mathematical concepts and techniques are investigated within the context of number theory and other topics. Credit not given for both MATH 052 and MATH 054. Co-requisite: MATH 021. |
Find a Centennial, CO Trigon
...CheersPre-algebra is one of the most important sections in math. It is the basic fundamentals for all high-school and college math. The fundamentals include the following: Operations of real numbers, polynomials, rational expressions, radicals and exponents, equations/functions, area and volume... |
M2= Math Mediator
Algebra 2 High School Lesson Plans
Teachers!
- High School Math
- Lesson Plans
- Algebra 2
- Applications
- Examples
- Activities
- 55 Minute
- Click "Lessons"
Bridging the Gap
Math Mediator instruction lesson plans bridge the gap between classic Algebra 2 and real life. Our
goal is to give high school math teachers a ready-to-use lesson plan as well as help students
with their Algebra 2 homework and understanding of various Algebra 2 concepts.
While primarily a high school math teacher resource for stand alone lesson plans,
parents, students and tutors with a bit of Algebra 2 background will find these lesson plans helpful.
These Algebra 2 high school lesson plans emphasize relevant applications
that motivate students. Currently, we are
only offering Algebra 2 lesson plans in pdf form, with some PowerPoint lessons being added.
Algebra 2 Lesson Plans
LESSONS -- FREE!
We have developed application enriched stand alone Algebra 2 high school
lesson plans that incorporate real life examples. In addition, many of these
Algebra 2 lesson plans raise awareness to pertinent issues on
health, career choices, budgeting and social responsibilities. These lesson
plans have been structured for a 55 minute class duration with detailed
instruction, projects and examples. Please take a look at our list of lesson
plans and an example by clicking on "Lessons" and view the examples. |
College Algebra, ALC + MXL your students dislike carrying a big textbook around campus? We can provide an unbound, three-hole-punched version of the traditional text so that your students can carry just what they need. This unbound version comes with access to MyMaythLab or MyStatLab at a significant discount from the price of the regular text. understand the material. |
Problem Solving Using Dimensional Analysis 10 problem worksheet on dimensional analysis. There is nothing earth shattering here, just good old practice on a concept that many students find difficult. Students who understand the idea of dimensional analysis have far fewer difficulties when they are trying to work stoichiometry problems later in the course. This worksheet can be used as test review, homework, or as a quiz.
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1803.17 |
PracticeMakesPerfect: Trigonometry is a comprehensive guide and workbook that covers all the basics of trigonometry that you need to understand this subject. Each chapter focuses on one major topic, with thorough explanations and many illustrative examples, so you can learn at your own pace and really absorb the information. You get to apply your knowledge and practice what you've learned through a variety of exercises, with an answer key for instant feedback. Offering a winning solution for getting a handle on math right away, PracticeMakesPerfect: Trigonometry is your ultimate resource for building a solid understanding of trigonometry fundamentals.
With more than 1,000,000 copies sold, PracticeMakesPerfect has established itself as a reliable practical workbook series in the language-learning category. Now, with PracticeMakesPerfect: Statistics, students will enjoy the same clear, concise approach and extensive exercises to key fields they've come to expect from the series--but now within mathematics.
PracticeMakesPerfect helps you put your French vocabulary and grammar skills together! You may have all the vocabulary down pat and every grammar point nailed--but without the skill of knowing how to put these elements together, communicating in your second language would be nearly impossible. PracticeMakesPerfect: French English vocabulary and grammar skills together! You may have all the vocabulary down pat and every grammar point nailed--but without the skill of knowing how to put these elements together, communicating in your second language would be nearly impossible. PracticeMakesPerfect: English German vocabulary and grammar skills together! You may have all the vocabulary down pat and every grammar point nailed--but without the skill of knowing how to put these elements together, communicating in your second language would be nearly impossible. PracticeMakesPerfect: German Spanish vocabulary and grammar skills together! You may have all the vocabulary down pat and every grammar point nailed--but without the skill of knowing how to put these elements together, communicating in your second language would be nearly impossible. PracticeMakesPerfect: Spanish Italian vocabulary and grammar skills together! You may have all the vocabulary down pat and every grammar point nailed--but without the skill of knowing how to put these elements together, communicating in your second language would be nearly impossible. PracticeMakesPerfect: Italian has established itself as a reliable practical workbook series in the language-learning category. PracticeMakesPerfect: Algebra, provides students with the same clear, concise approach and extensive exercises to key fields they've come to expect from the series-but now within mathematics. This book presents thorough coverage of skills, such as handling decimals and fractions, functions, and linear and quadratic equations.
PracticeMakesPerfect has established itself as a reliable practical workbook series in the language-learning category. Now, with PracticeMakesPerfect: Calculus, students will enjoy the same clear, concise approach and extensive exercises to key fields they've come to expect from the series--but now within mathematics. PracticeMakesPerfect: Calculus is not focused on any particular test or exam, but complementary to most calculus curricula. Because of this approach, the book can be used by struggling students needing extra help, readers who need to firm up skills for an exam, or those who are returning to the subject years after they first studied it.
Go beyond arrivedirci and add thousands of words to your Italian vocabulary
To communicate comfortably in Italian, you need access to a variety of words that are more than just the basics. In PracticeMakesPerfect: Italian Vocabulary you get the tools you need to expand your lexicon and sharpen your speaking and writing skills. And how do you this? PRACTICE, PRACTICE, PRACTICE!
PracticeMakesPerfect has established itself as a reliable practical workbook series in the language-learning category. Now, with PracticeMakesPerfect: Geometry, students will enjoy the same clear, concise approach and extensive exercises to key fields they've come to expect from the series--but now within mathematics. PracticeMakesPerfect: Geometry is not focused on any particular test or exam, but is complementary to most curricula. Because of this approach, the book can be used by struggling students needing extra help, readers who need to firm up skills for an exam, or those who are returning to the subject years after they first studied it. Its all-encompassing approach will appeal to both U.S. and international students.
The only way to build your skills in a second language is to practice, practice, practice. Following the successful PracticeMakesPerfect approach, this book gives you clear explanations and all the tools you need to learn Italian pronouns and prepositions. A valuable resource for beginning- to intermediate-level Italian learners, PracticeMakesPerfect: Italian Pronouns and Prepositions enables you to: Successfully grasp Italian pronoun and preposition usage Review and compare different types of pronouns and prepositions using easy-to-read tables Build your language skills with more than 100 exercises and an answer key
With PracticeMakesPerfect: Basic Spanish, you'll get just the basics--essential vocabulary and grammar to get you to the next stage of learning Spanish. Inside, three-page units cover each subject, which can be completed in a mere 10 to 15 minutes! Also included are engaging and humorous exercises to keep you focused and interested while you gain confidence in your new language. |
FL Students
Course Name:
Liberal Arts Mathematics
Course Code:
1208300
Honors Course Code:
AP Course Code:
Description:
The total weight of two beluga whales and three orca whales is 36,000 pounds. The weight of each whale could be determined with just one additional fact. The Liberal Arts Math course provides all the math tools needed to answer this weighty question. The setting for this course is an amusement park with animals, rides, and games. The students' job is to apply what they learn to dozens of real-world scenarios. .
Equations, geometric relationships, and statistical probabilities can sometimes be dull, but not in this class! The park guide (teacher) takes each student on a grand tour of problems and puzzles that show how things work and how mathematics provides valuable tools for everyday living.
Students should come ready to reinforce and grow their existing algebra and geometry skills to learn complex algebraic and geometric concepts they will need needed for further study of mathematics.
Note: This course does not meet the academic core requirement for math for entry into the State University System of Florida or eligibility requirements for some Bright Futures Scholarships.
Access the site link below to view |
The EL-W535XB performs over 335 advanced scientific functions and utilizes WriteView Technology, 4-line display and Multi-Line Playback to make scientific equations easier for students to solve. It is ideal for students studying general math, algebra, geometry, trigonometry, statistics, biology, chemistry and general science.
This Sharp calculator, part of the Advanced D.A.L. series, is a scientific model that's ideal for high school students, college students, and professionals. It features Direct Algebraic Logic, an extra large display, 12-digit two-line display, Multi-Line playback, differential and integral calculus functions, 440 mathematical functions and a protec..
Sharp Electronics Scientific Calculator, 272 Functions, LCD Display, BK/WE Scientific calculator performs 272 functions and features Direct Algebraic Logic (DAL) to simplify the entry of equations. DAL allows students to enter the elements of an expression in the exact order they appear in the textbook. The Multi-Line Playback makes scientific equa..
The EL-W516XBSL Sharp scientific calculator is a math scientific calculator that performs over 535 scientific functions in seven different modes. This engineering calculator features Write View technology, which displays formulas as they are written in textbooks eliminating any confusion on how to enter each formula into the calculator. The Sharp s |
Course Communities
Riemann Sum Exploration
This Java applet allows users to see a visual representation of numerical approximations for the area under a sample curve. Users can select from a list of numerical approximation techniques and can manipulate the number of intervals, but they cannot alter the function. |
Practice Makes Perfect Pre-Algebra
Helpful instruction and plenty of practice for your child to understand the basics of pre-algebraUnderstanding pre-algebra is essential for your child to do math problems with confidence.Practice Makes Perfect: Pre-Algebragives your child bite-sized explanations of the subject, with engaging exercises that keep her or him motivated and excited to learn.
They can practice the problems they find challenging, polish skills they've mastered, and stretch themselves to explore skills they have not yet attempted. This book features exercises that increase in difficulty as your child proceeds through it.This book is appropriate for a 6th grade student working above his or her grade level, or as a great review and practice for a struggling 7th or 8th grader.
show more show less
List price:
$16.00
Edition:
2013
Publisher:
McGraw-Hill Companies, The
Binding:
Trade Paper
Pages:
160
Size:
8.70" wide x 10.90" long x 0.40 |
...Major topics studied include: probability, combinatorics, set theory and graph theory. Set theory is the study of sets, both infinite and finite. Some basic operations of set theory include the union and intersection of sets |
The last thing student should have to pay for is some decent software. There are plenty of good free software that offer the same kind of features as the boxed software you can buy at the store or online. Here , I provide you the list of free software for students Let's begin with it.
1)Dreamspark Software FREE For Students Only
You can download software from Microsoft DreamSpark for no charge , only if you're student , type country , name of school , school email and you'll receive activation mail .
1.1 Microsoft Mathematics
Microsoft Mathematics provides a set of mathematical tools that help students get school work done quickly and easily. With Microsoft Mathematics, students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and calculus.
Microsoft Mathematics includes a full-featured graphing calculator that's designed to work just like a handheld calculator. Additional math tools help you evaluate triangles, convert from one system of units to another, and solve systems of equations.2) GNU Octave
Think Matlab is way overpriced? So do we. The good news is that there is a solid free replacement called Octave. Almost everything that can be done in Matlab can be done in Octave. Plus, if you used to have to go to a special computer lab to do your Matlab work, now you can do it on your own computer.
Just like Matlab, Octave is an ultra high-level programming language that is used to solve mathematical problems and plot their solutions. The nice thing about Octave is that it is modular and can easily be expanded to perform additional functions.
3)Eclipse
For those of you who are computer science majors or just taking a few computer programming courses (which is becoming more and more common, especially for science majors) the good news is that you don't have to buy an uber expensive C++ IDE package like Visual Studio. Instead, Eclipse gives you an amazing, fully featured development platform with all the plugins you could possibly want. Plus, Eclipse runs on Windows, Mac OSX, and Linux.
4)FreeMat
A is FreeMat is good free math program with a GUI and M-file editor . It is similar to commercial systems such as MATLAB from Mathworks, and IDL from Research Systems, but is Open Source. FreeMat is available under the GPL license.
5) Netbeans
6)OpenOffice
For those of you that can't afford to pay the premium price in purchasing Microsoft Office, even with a generous student discount, you'll want to look at OpenOffice.org. It is a free piece of software that works on any operating system, and includes access to software to write documents, create spreadsheets, presentations, and more.
8) Art of Illusion
9) Marble
Marble is a geographical atlas and a virtual desktop globe which lets you quickly explore other places on our home planet. You can use Marble to look up places, to easily create maps, measure distances and to retrieve detail information about locations that you have just heard about in the news or on the Internet. The user interface is clean, simple and easy to use.
Let me start by saying wonderful post. Im not positive if it has been talked about, but when utilizing Chrome I can by no means get the complete website to load with out refreshing many times. Could just be my computer. Thanks. |
This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of Cartesian coordinates ... read more
Theories of Figures of Celestial Bodies by Wenceslas S. Jardetzky Suitable for upper-level undergraduates and graduate students, this text explores exact methods used in the theory of figures of equilibrium and examines problems concerning figures of celestial bodies. 1958Relativity: The Special and General Theory by Albert Einstein The great physicist's own explanation of relativity, written for readers unfamiliar with theoretical physics, outlines the special and general theories and presents the ideas in their simplest, most intelligible form.
Geometry and Light: The Science of Invisibility by Ulf Leonhardt, Thomas Philbin Suitable for advanced undergraduate and graduate students of engineering, physics, and mathematics and scientific researchers of all types, this is the first authoritative text on invisibility and the science behind it. More than 100 full-color illustrations, plus exercises with solutions. 2010This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of Cartesian coordinates before undertaking the theory of tensors in curved spaces and its application to general relativity theory. Additional topics include black holes, gravitational waves, and a sound background in applying the principles of general relativity to cosmology. Numerous exercises advance the theoretical developments of the main text, thus enhancing this volume's appeal to students of applied mathematics and physics at both undergraduate and postgraduate levels.
Reprint of the John Wiley & Sons, New York, 1982 edition.
A solutions manual to accompany this text is available for free download. Click here to download PDF version now |
books.google.ca - Finally...
Algebra
Finally with good pedagogy. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises.
User ratings
Review: Algebra
Review: Algebra
User Review - Geoffrey Lee - Goodreads
Hungerford's Algebra is a beautifully written book which covers a wide range of material. Unlike Serge Lang's book on Algebra, which is more like a technical reference guide, Hungerford's book ...Read full review
ALGEBRA–II is a rigorous introduction to all those topics in basic algebra that every ... They also form the syllabus for the Ph.D. Algebra comprehensive exam. The ... ~araghur/ spring2008/ math5623/ syllabus.pdf
620-222 The course will discuss various aspects of modern algebra concentrating on extending the linear algebra that you have already done and introducing some new ... ~s620222/
Less
About the author (1974)
Thomas W. Hungerford received his M.S. and Ph.D. from the University of Chicago. He has taught at the University of Washington and at Cleveland State University, and is now at St. Louis University. His research fields are algebra and mathematics education. He is the author of many notable books for undergraduate and graduate level courses. In addition to ABSTRACT ALGEBRA: AN INTRODUCTION, these include: ALGEBRA (Springer, Graduate Texts in Mathematics, #73. 1974); MATHEMATICS WITH APPLICATIONS, Tenth Edition (Pearson, 2011; with M. Lial and J. Holcomb); and CONTEMPORARY PRECALCULUS, Fifth Edition (Cengage, 2009; with D. Shaw). |
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GENERAL CHEMISTRY: QUANTUM CHEMISTRYCHAPTER NINE1INTRODUCTION.Most of the major theories of atomic and molecular structure that are used inmodern chemistry owe their origin to the application of quantum mechanics.What is now called old qua
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Homework #2, AST 203, Spring 2009Due in class (i.e., by 4:20 pm), Tuesday February 24 To receive full credit, you must give the correct answer and show that you understand it. This requires writing your explanations in full, complete English senten
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Professors Blume, Halpern and Easley 10 December 2004Econ 476/676 CIS 576Final ExamINSTRUCTIONS, SCORING: The points for each question are specified at the beginning of the question. There are 160 points on the exam. Please show your work, so th
333 Project a simulation of reality: building a substantial system in groups of 3 or 4 people "three-tier" system for any application you like 3 major pieces graphical user interface ("presentation layer") processing in the middle ("business |
@inbook {MATHEDUC.05865082,
author = {K\'antor, T\"unde},
title = {J. K\"ursch\'ak a world-famous scholar teacher (1864-1933).},
year = {2010},
booktitle = {Problem solving in mathematics education. Proceedings of the 11th ProMath conference, Budapest, Hungary, September 3--5, 2009},
pages = {76-86},
publisher = {Budapest: Univ. Budapest, Mathematics Teaching and Education Center},
abstract = {},
msc2010 = {A30xx},
identifier = {2011b.00015},
} |
Student must have access to a high-speed (DSL or cable) Internet connection at least 4 days weekly, including a computer with sound and Java applet support, Microsoft Excel or Google Spreadsheet, a media player that can play Windows Media Player (WMA) and RealPlayer (RM) files, and a scanner or digital camera/phone. Access to a Graphing Calculator and experience using it is essential.
Description:
This one semester Pre-Calculus/Advanced Trig course prepares students for eventual work in Calculus. Topics covered include:
· trigonometric and circular functions; their inverses and graphs;
· relations among the parts of a triangle;
· trigonometric identities and equations;
· solutions of right and oblique triangles and their applications to real-world problems.
Graphing calculators are used frequently in each lesson to familiarize students with the basics of graphing calculator use, to demonstrate concepts, to facilitate problem solving, and to verify results of problems solved algebraically. SAT practice topics and problems provide students a review of the prerequisite courses. |
In this course we will revisit some basic notions of
calculus. The emphasis will be on the conceptual understanding of calculus and
on mathematical rigor. One of the goals of the course is helping students to refine their capacities of reading
abstract mathematical texts and develop facilities to write brief mathematical
proofsThe
book by Strichartz provides very good explanations and motivations for many
subjects. (2) (recommended) Introduction to Analysis, by Maxwell
Rosenlicht, Dover
1986 ISBN 0-486-65038-3. This very brief (and cheap) book gives a different
perspective on the subject;
(3) (recommended) Principles of
Mathematical Analysis, 3rd Edition, by W. Rudin, McGraw-Hill., Cambridge. The book by
Rudin (affectionately known as "Baby Rudin") gives a very concise
(about 300 pages) presentation of everything you need to know about rigorous
mathematical analysis. The definitions and statements of the principal results
are very clear, and the problems are excellent. This particular edition of the
book has been in print for about 35 years and you can get it on-line at a very
low price. You should keep in mind, however, that the proofs are exceedingly
brief and the motivations are frequently skipped.
Homework: There will be
weekly homework assignments that will be collected and graded. The Rensselaer
Handbook of Student Rights and Responsibilities define various forms of
Academic Dishonesty and you should make yourself familiar with these. In this
class, all assignments that are turned in for a grade must represent the student's
own work. In cases where help was received, or teamwork was allowed, a notation
on the assignment should indicate your collaboration. Submission of any
assignment that is in violation of this policy may result in a penalty of a
grade of F. If you have any question concerning this policy before submitting
an assignment, please ask for clarification.
Test
questions will be similar to the shorter homework questions from the homework
assignments 1-5. You will be allowed the use of one sheet of hand-written
notes.
To give
you some idea of the possible length and degree of difficulty of the test, here
is a copy of the test I gave when I last taught this course several years ago. Test 1 fall 2006. The scope of your test
will be different, however.
Test 2 will be given on Friday, 9 November
2012, in class. The test will cover the following topics; references are to the
Textbook:
-Definition
of the derivative. Big O and little o. (5.1)
-Local behavior of a function and the sign of its
derivative. The mean value theorem. (5.2)
The test
format will be similar to that of Test 1; questions will be similar to the
shorter homework questions from the homework assignments 6-9. You will be
allowed the use of one sheet of hand-written notes. Test 2 solutions
Final Exam will be given on Wednesday 12/12, 3-6pm in WALKER 5113. The test
will cover the material included in the homework assignments; that is, the last
two topics of the course outline (power series and elementary functions) are
not included.
Here is a
copy of the final exam given in 2006. Your exam will
be at least one problem shorter than this one. You will be allowed the use of
one sheet of hand-written notes. † |
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4434Applied Discrete Structures
Applied Discrete Structures by Al Doerr and Ken Levasseur is a free open content textbook in discrete mathematics. Originally published in 1984 & 1989 by Pearson, the book has been updated to include references to Mathematica and Sage, the open source computer algebra system. Contents:Front Matter: Contents and IntroductionChapter 1: Set Theory I Chapter 2: Combinatorics Chapter 3: Logic Chapter 4: More on Sets Chapter 5: Introduction to Matrix Algebra Chapter 6: Relations and Graphs Chapter 7: Functions Chapter 8: Recursion and Recurrence Relations Chapter 9: Graph Theory Chapter 10: Trees Chapter 11: Algebraic Systems Chapter 12: More Matrix Algebra Chapter 13: Boolean Algebra Chapter 14: Monoids and Automata Chapter 15: Group Theory and Applications Chapter 16: An Introduction to Rings and Fields Solutions to Odd-Numbered Exercises Blast into math
This is a free online textbook offered by BookBoon.'Blast into Math! A fun and rigorous introduction to pure mathematics, is suitable for both students and a general audience interested in learning what pure mathematics is all about. Pure mathematics is presented in a friendly, accessible, and nonetheless rigorous style. Definitions, theorems, and proofs are accompanied by creative analogies and illustrations to convey the meaning and intuition behind the abstract math. The key to reading and understanding this book is doing the exercises. You don't need much background for the first few chapters, but the material builds upon itself, and if you don't do the exercises, eventually you'll have trouble understanding. The book begins by introducing fundamental concepts in logic and continues on to set theory and basic topics in number theory. The sixth chapter shows how we can change our mathematical perspective by writing numbers in bases other than the usual base 10. The last chapter introduces analysis. Readers will be both challenged and encouraged. A parallel is drawn between the process of working through the book and the process of mathematics research. If you read this book and do all the exercises, you will not only learn how to prove theorems, you'll also experience what mathematics research is like: exciting, challenging, and fun!'A First Course in Linear Algebra
'A First Course in Linear Algebra is an introductory textbook designed for university sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form. Along the way, determinants and eigenvalues get fair time.״Calculus in Context
This introductory calculus text covering the topics of: Successive approximations, The derivative, Differential equations, Techniques of differentiation, The integral, Periodicity, Dynamical systems, Functions of several variables, Series and approximations, and Techniques of integration.Ant Colony Optimization - Techniques and Applications
This is a free textbook offered by InTech.'Ant Colony Optimization (ACO) is the best example of how studies aimed at understanding and modeling the behavior of ants and other social insects can provide inspiration for the development of computational algorithms for the solution of difficult mathematical problems. Introduced by Marco Dorigo in his PhD thesis (1992) and initially applied to the travelling salesman problem, the ACO field has experienced a tremendous growth, standing today as an important nature-inspired stochastic metaheuristic for hard optimization problems. This book presents state-of-the-art ACO methods and is divided into two parts: (I) Techniques, which includes parallel implementations, and (II) Applications, where recent contributions of ACO to diverse fields, such as traffic congestion and control, structural optimization, manufacturing, and genomics are presented.'Adaptive Filtering - Theories and Applications
This is a free textbook offered by InTech.'Adaptive filtering can be used to characterize unknown systems in time-variant environments. The main objective of this approach is to meet a difficult comprise: maximum convergence speed with maximum accuracy. Each application requires a certain approach which determines the filter structure, the cost function to minimize the estimation error, the adaptive algorithm, and other parameters; and each selection involves certain cost in computational terms, that in any case should consume less time than the time required by the application working in real-time. Theory and application are not, therefore, isolated entities but an imbricated whole that requires a holistic vision. This book collects some theoretical approaches and practical applications in different areas that support expanding of adaptive systems.'Introductory Maths for Chemists - Chemistry Maths 1
'Chemistry Maths 1 teaches Maths from a "chemical" perspective and is the first are used and structured on a weekly basis to help the students to self-pace themselves. Coloured molecular structures, graphs and diagrams bring the text alive. Navigation between questions and their solutions is by page numbers for use with your PDF reader.'Elementary Algebra Exercise Book I
This is a free textbook from BookBoon.'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.'Algebra is Vital
This is a virtual edition of a developmental algebra textbook. It follows the Suffolk County Community College Mathematics syllabus for our developmental algebra courses, with a few additions.This book has a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License and may be used and shared for any non-profit educational purpose.Intermediate Algebra
This is an online Intermediate Algebra Textbook. In each section there are many videos and exercises. |
Introduction
Let's get into plane curves. In some ways, the following formulas are easier for mathematicians to derive than the ones for figures consisting of lines and angles; in other ways, the formulas for curves are more troublesome. ...
Introduction to System of Units
Units are devices that scientists use to indicate, estimate, and calculate aspects of the world and the universe. Numbers by themselves are abstract. Try to envision the number 5 in your mind. You think of a set or object: five ...
Introduction to Base Units in SI
In all systems of measurement, the base units are those from which all the others can be derived. Base units represent some of the most elementary properties or phenomena we observe in nature.
Introduction to Other Units in Physics
The preceding seven units can be combined in various ways, usually as products and ratios, to generate many other units. Sometimes these derived units are expressed in terms of the base units, ...
Introduction to Physics and Constants
Constants are characteristics of the physical and mathematical world that can be "taken for granted." They don't change, at least not within an ordinary human lifetime, unless certain other factors ...
Introduction to Unit Conversions
With all the different systems of units in use throughout the world, the business of conversion from one system to another has become the subject matter for whole books. Web sites are devoted to this task; at ...
Precedence
Mathematicians, scientists, and engineers have all agreed on a certain order in which operations should be performed when they appear together in an expression. This prevents confusion and ambiguity. When various operations such as ... |
NEWTON: An interactive environment for exploring mathematics
Abstract
In recent years there has been much interest in incorporating computers into the teaching of college-level mathematics classes. In particular, there has been great interest in the use of computer algebra systems in introductory calculus. While educators believe that computer algebra systems have great potential in the calculus sequence, the limitations of such software have often made them difficult to use in the classroom.^ This thesis discusses the design, implementation, and evolution of a computational environment for use in teaching introductory mathematics. This system, called N scEWTON, runs on Macintosh computers and consists of a user-friendly interface to the symbolic mathematics package Maple, supplemented by an extensive library of Maple code. As an interface specifically designed for educational purposes, the driving motivation behind its development was to allow students access to the full power of a computer algebra system, without mastering the idiosyncrasies of the system and struggling to enter information. To accomplish this, we have provided the student with an easy to use interface which adds features specific to its intended educational use.^ A prototype system entered classroom use in 1991. While well received, the system was limited in functionality and proved to be difficult to maintain. The system has been completely redesigned, reimplemented, and extended for use in linear algebra, numerical analysis, and differential equations courses.^ The current N scEWTON interface is based on the concept of an interactive book, and the user can freely mix text, formulas and graphics in collapsible sections on worksheets. Multiple windows allow users and work with several formulas at once. A syntax-directed editor is provided which allows easy two-dimensional input and editing of mathematical formulas. Formulas are easily constructed and modified, appearing as they do in textbooks. Users do not interact with Maple directly and need know nothing of Maple's syntax and command structure. Mathematical operations are selected from menus, and plots and animations are generated from dialog boxes. Other unique aspects of the system include dialogs for techniques of integration and differentiation and the automatic documentation of solutions. ^ |
MAA Review
[Reviewed by Allen Stenger, on 10/06/2008]
This is a beautifully clear exposition of the main points of Lie theory, aimed at undergraduates who have studied calculus and linear algebra. The book is modeled after (and named in homage to) Halmos's Naive Set Theory. The key simplification is that it deals only with matrix groups.
The book is well equipped with examples, and it always ties the matrix groups back to concrete examples, especially the complex numbers and the quaternions. The book has a very strong geometric flavor, both in the use of rotation groups and in the connection between Lie algebras and Lie groups.
The book's most conspicuous weakness is that it treats Lie theory in isolation. It doesn't give any clue where the subject came from or what it is used for today. Each chapter ends with a very informative "Discussion" section, but what is discussed is the portions of Lie theory that we couldn't get to in this book. There's no mention of differential equations, Klein's Erlangen Program, or representation theory.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis. |
Stage 1
Modules include calculus, matrices, numerical methods, an introduction to computer software for mathematics and statistics, and probability and statistical inference
Stage 2
A range of modules gives you further high level mathematical and statistical skills
You have the opportunity to work with school children on mathematics enrichment activities
Work placement year (optional)
An optional placement provides valuable paid professional experience and helps make your CV stand out
Final stage
A placement module offers you real school experience working alongside a mathematics teacher
You will also carry out a mathematics education project in an area of your choice
MODULES
STAGE 1
MATH1501 Calculus This module covers key topics in calculus and will prepare students for the rest of their degree. It has a greater emphasis on proof and rigour than at A-level. It also introduces some more advanced multi-dimensional calculus. The module contains various applications from pure and applied mathematics as well as from statistics, physics and finance.
MATH1502 Reasoning and Analysis This module introduces the basic reasoning skills needed for the development and applications of modern mathematics. The utility of clear logical thinking will be explored in various important topics and current applications, including security on the internet, fractal geometry and the continuous nature of real numbers.
MATH1503 Linear Algebra and Complex Numbers This module explores the concepts and applications of vectors, matrices and complex numbers. The deep connection between algebra and geometry will be explored. The techniques that will be presented in this module are at the foundation of many areas of mathematics, statistics, physics, and several other applications.
MATH1504 Numerical and Computational Methods This module provides an introduction to the Maple and Matlab software, computational mathematics and creating simple computer programs. Students will use Maple/Matlab interactively and also write procedures in the Maple/Matlab computer languages. The elementary numerical methods which underly industrial and scientific applications will be studied.
MATH1505 Probability and Statistical Inference This module provides a mathematical treatment of basic probability and statistical techniques including random variables, estimation, hypothesis testing, as well as regression and correlation. It also covers exploratory data analysis. All methods are implemented using real data and professional computer software such as R.
MATH1506 Geometry, Graphs, and Groups This module introduces three important areas of pure mathematics. First, key theorems and constructions from classical Euclidean geometry will be introduced. Then we study elementary graph theory, and consider graphs on surfaces, and polynomial invariants of graphs. Finally, we define groups, explore a wide variety of examples, and study cosets and quotients.
STAGE 2
MATH2501 Advanced Calculus and Transforms This module extends the differential and integral calculus of severable variables and uses them to solve a wide range of problems. The important techniques of Laplace transforms, Fourier series and transforms are introduced. The applications explored in this module include the solutions of important differential equations and the construction of functions and generalised functions in terms of a basis of orthogonal functions.
MATH2502 Vector Calculus and its Applications This module introduces the student to the methods of vector calculus and the fundamental integration theorems. The applications include a range of important scientific problems primarily from classical mechanics and cosmology. The module also introduces the idea of curvature and applies this to the geometry of bubbles and minimal surfaces.
MATH2503 Ordinary Differential Equations The module aims to provide an introduction to different types of ordinary differential equations and analytical and numerical methods to obtain their solutions. Extensive use will be made of computational mathematics packages. Applications to mechanical and chemical systems are considered as well as the chaotic behaviour seen in climate models.
MATH2504 Operational Research and Monte Carlo Methods This module gives students the opportunity to work on open-ended case studies in operational research (OR) and Monte Carlo methods, both of which are important methods in, for example, industry and finance. It allows students to work on their own and in teams to develop specific skills in OR and programming as well as refining their presentation and communication skills.
MATH2505 Real and Complex Analysis This module deepens the student's understanding of real analysis and introduces complex analysis. The important distinction between real and complex analysis is explored and the utility of the complex framework is demonstrated. The central role of power series and their convergence properties are studied in depth. Applications include the evaluation of improper integrals and the construction of harmonic functions.
MATH2506 Advanced Probability and Statistical Inference This module extends the probability theory covered in the first year. It discusses and demonstrates the links between various distributions, with a focus on some standard continuous distributions. The module also covers topics from the theory of statistical inference and methods of maximum likelihood estimation. This will equip students with the skills required to perform a number of statistical tests, including randomisation tests, using statistical packages where appropriate.
MATH2507 Regression Modelling This module develops your understanding of advanced regression modelling by extending and generalising the linear model. The module will develop your understanding of the underlying mathematical theory by careful use of case studies in a variety of applications using professional software.
Modules studied by current students |
Statistics actually has concepts that are important to learn - without the sophisticated graphing calculators we use now, students would be sort of lost in the forest among all the trees. They can get bogged down in those computations. West Valley College is a two-year community college, from which many students go on to get four-year degrees through the California State University system or the University of California. Our faculty are generally very positive about using technology in courses. Some faculty, however, did resist the change to graphing calculators, because they wanted to focus on teaching the skills required to get the calculations, as opposed to the value of the results. But I've found that when my students aren't making those arithmetical errors anymore, we can get right from the problem to the interpretation of the results.
The courses
I use the graphing calculators in Math 10, "Elementary Statistics and Probability," which has 300 students per semester, and in Math 12, "Calculus for Business Majors," with about 60 students per semester. Actually, the calculators are used in all of our math courses at West Valley. Most of my students are from the social sciences - psychology and sociology - and from business. To graduate from the four-year schools in California, students are required to take one course beyond intermediate algebra, and statistics is the one that most students elect to take. It's perhaps the most useful of the math course options, as opposed to trigonometry, for example, which they may never use. At least statistics opens some doors for them. Both Math 10 and Math 12 are lecture courses, with no sections.
The learning technology
The graphing calculator I require for my classes is the TI 83, from Texas Instruments. The folks at Texas Instruments have programmed in almost all of the statistics formulas that we need. The students have to understand the problem and recognize what formula to use. Then they call up the formula on the screen and the calculator prompts them to enter the right numbers in the proper "order of operations." The calculator does a lot for you, but unless you enter the numbers in the right order, with parentheses and such, you won't get the correct result. The calculator will also display the corresponding graph for the formula; students don't need to laboriously repeat the calculations with different parameters to hand-plot a graph.
One example of how we use the graphing calculator is to construct confidence intervals. If the students want to estimate the confidence interval for the mean of a selected sample, they enter the sample mean, the sample size, the sample standard deviation, and call up the right button; the graphing calculator provides the result. It displays the Bell curve and finds the area under the curve that corresponds to the probability of an event.
To apply that to a real-life problem, say we have to figure the probability that a randomly selected cup in a coffee vending machine will fill up with 5 oz or 6 oz of coffee, when it's supposed to be filled with 8 oz of liquid. When we enter the correct values, the calculator will draw us the bell-shaped curve for that, display it on the calculator screen, and show us exactly what we need. For the real-life application of this, if the vending machine company comes out, takes a sample, and their sample mean varies from what it's supposed to be, the company knows that machine needs to be adjusted. It's actually a quality control issue. Moreover, we don't need a computer and statistical software package to do this kind of work anymore - the calculator does it all
You can find out more about the TI 83 and how it's used at the Texas Instruments web site. |
Pre Algebra 6C
Course Description:
The math curriculum reflects the standards of the National Council of the Teachers of Mathematics and the Illinois Learning Standards. Areas covered include patterns, number sense and algebraic thinking, decimal and fraction operations, data and statistics, integers, geometry, ratios and proportions, percents, and probability, with an emphasis on transferring knowledge into a variety of contexts.
Enduring Understandings:
Mathematics can help us make more informed decisions, work efficiently, solve problems, and appreciate its relevance in the world around us.
Geometric methods can help us to make connections and draw conclusions from the world.
Functions and number operations play fundamental roles in helping us to make sense of the world.
Using prior knowledge, appropriate technology, and logical thinking, we can analyze data and effectively communicate the reasonableness of solutions.
Multiple mathematical approaches and strategies can be used to reach a desired outcome. Algebraic models and graphical representations are tools that can help us make meaningful connections to solve How can you use math tools and technology to solve problems? |
Search Course Communities:
Course Communities
Lesson 41: Conic Sections: Ellipses
Course Topic(s):
Developmental Math | Conic Sections
Beginning with a general introduction to conics and how they are formed, circles are first presented and then ellipses are motivated by looking at the general equation of a circle centered at the origin. After central ellipses, translated ellipses are discussed, followed by a procedure for writing the equation of the ellipse in standard form. The lesson concludes with a procedure for finding the equation of an ellipse given its vertices. |
11th Grade Math: Derivatives Help
Related Subjects
11th Grade Math: Derivatives
In calculus, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position).
A closely related notion is the differential of a function.
The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.
The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.
Many 11th grade math students find derivatives difficult. They feel overwhelmed with derivatives homework, tests and projects. And it is not always easy to find derivatives tutor who is both good and affordable. Now finding derivatives help is easy. For your derivatives homework, derivatives tests, derivatives projects, and derivatives tutoring needs, TuLyn is a one-stop solution. You can master hundreds of math topics by using TuLyn.
Our derivatives videos replace text-based tutorials in 11th grade math books and give you better, step-by-step explanations of derivatives. Watch each video repeatedly until you understand how to approach derivatives problems and how to solve them.
Tons of video tutorials on derivatives make it easy for you to better understand the concept.
Tons of word problems on derivatives give you all the practice you need.
Tons of printable worksheets on derivatives let you practice what you have learned in your 11th grade math class by watching the video tutorials.
How to do better on derivatives: TuLyn makes derivatives easy for 11th grade math students.
Let f(t) be the weight (in grams) of a solid
Let f(t) be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the wieght of the solid at any time t can be be determined from the weight using the formula:
f'(t)=-5(t)(2+f(t))
If there is 3 grams of solid at time t=2...
A one-product firm estimates
(cost and profit) a one-product firm estimates that its daily total cost function (in suitable units) is
C(x)=x3-6x2+13x+15
and its total revenue function is
R(x) = 28... It will help me understand derivatives much better. As a math coach I am always trying to improve upon my own skills before implementing ideas in staff development. I also have an 11th grader who needs my help. finding derivatives of radical functions |
braint... read more
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The Red Book of Mathematical Problems by Kenneth S. Williams, Kenneth Hardy Handy compilation of 100 practice problems, hints, and solutions indispensable for students preparing for the William Lowell Putnam and other mathematical competitions. Preface to the First Edition. Sources. 1988 edition.
Product Description:
brainteasers vary in difficulty, ranging from playful puzzles involving games to tough questions of probability. Complete solutions appear at the end |
Elsevier Science and Technology, July 2009, Pages: 744
Partial Differential Equations and Boundary Value Problems with Maple presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. The Maple commands are so intuitive and easy to learn, students can learn what they need to know about the software in a matter of hours- an investment that provides substantial returns. Maple's animation capabilities allow students and practitioners to see real-time displays of the solutions of partial differential equations. Maple files can be found on the books website.
- Provides a quick overview of the software w/simple commands needed to get started - Includes review material on linear algebra and Ordinary Differential equations, and their contribution in solving partial differential equations - Incorporates an early introduction to Sturm-Liouville boundary problems and generalized eigenfunction expansions - Numerous example problems and end of each chapter exercises
Articolo, George A. Dr. George A. Articolo has 35 years of teaching experience in physics and applied mathematics at Rutgers University, and has been a consultant for several government research laboratories and aerospace corporations. He has a Ph.D. in mathematical physics with degrees from Temple University and Rensselaer Polytechnic Institute. |
The class meets six times a week: four times in lecture, once in conference,
and once in the computer laboratory. You are responsible for any and all
material discussed in lecture, conference, and lab.
Aside from the 6 hours that you spend in class each week, you should devote
at least another 8-10 hours to studying on your
own: reading the book, reading and organizing your notes, solving problems.
Conferences:
In the Friday conference sessions, you will meet with the Peer Learning
Assistant (PLA) for the class. You will be able to ask the PLA questions on the material
covered and homework. The PLA may lso give you in-class assignments and
review course material.
Homework: Written Homework: Problems will be assigned for each section of the
book covered and will be posted on the class web page. It is necessary to do,
at a minimum, the assigned problems so that you can learn and understand
the mathematics. You should do additional problems for further practice.
Working the exercises will help you learn, and give you some perspective
on your progress. You are welcome to
discuss homework problems with one another but you must write up your homework
solutions on your own. Be mindful of your academic integrity.
Your homework will be collected at the beginning of each Monday's class.
Late homework will not be accepted.
If you must miss Monday's class, you should have your work turned in before class time in order for it to be graded.
Do not wait until the weekend to start your homework.
Work on the problems daily.
Your work should be very legible and done neatly. If the work is not
presentable, and is illegible, you will not receive credit for it. Please staple
the sheets of your assignment together. Do not use paper torn out of spiral
bound notebooks. In the upper right hand corner of your assignment you
should write your name, the class section number, and the list
of book sections for the assignment.
Discipline yourself to write clear readable solutions,
they will be of great value as review.
You
need to show both your answer and the work leading to it. Merely having
the right answer gets no credit - we can all look them up in the back of the
book.
Online Homework: There will be homework using the online tool
WebWork. This is the same software that you used for the Math Placement Exam
that you took during the summer. There will occasionally be 7-15 questions
on WebWork that must be done. Go to
Do not use the WebWork system to email for help on problems; such an
email will be sent to all the professors and assistants for all the sections of
MA1022! Instead, see Prof. Weekes or your PLA as soon as possible.
Quizzes:
Each Monday, there will be a 15-20 minute in-class quiz emphasizing the
most recently covered topics. If you miss a quiz for any reason (illness, travel,
etc.), you will receive a score of zero. However, don't worry, the lowest quiz
score will be dropped. Make-up quizzes will, thus, never be given.
Labs:
Each week, students will meet in the computer lab (SH003) with the
Instructor's Assistant (IA) who is Jane Bouchard. We will use the computer
algebra system, Maple V, as a visual and computational aid to help you explore
the mathematical theory and ideas of the calculus. You will not be given
credit for a lab report if you did not attend the lab.
There are no make-up labs.
The first lab will be on Oct. 31st/Nov. 1st.
The final lab will be on Dec. 5th/Dec. 6th.
Final Exam/Basic Skills Test:
On Wednesday 12th December from 7-9 pm, you will have a 2 hour comprehensive final examination. Make arrangements now so that there are no
con
flicts with the final exam.
The Final Assessment will consist of two parts. The first part is the Final
Exam which is used in determining your course average score as detailed before.
The other part is the Basic Skills Exam. You cannot pass the course if you do not pass the
Basic Skills Exam.
Students with failing averages in the course are given grades of NR, whether
or not they passed the Basic Skills exam.
If you pass the Basic Skills component, then your course average will be used
by the professor to determine your grade for the course.
If you fail the Basic Skills Exam, yet have what the instructor determines
to be a course average high enough to pass the course, you will be given a
grade of I (incomplete). You will be given the opportunity to re-take the
Basic Skills exam at a later date; if you pass it, you will receive the grade
that is based on your course average.
Mathematics Tutoring Center:
The Mathematics Tutoring Center is available for any WPI student taking a
course in calculus, differential equations, statistics, and linear algebra.
Stratton Hall 002A
Monday-Thursday 10am-8pm and Friday 10am-4pm.
No appointment needed - drop in any time!
Academic Dishonesty
Please read WPI's
Academic Honesty Policy
and all its pages. Make note of the examples of
academic dishonesty; i.e. acts that interfere with the process of evaluation
by misrepresentation
of the relation between the work being evaluated (or the resulting evaluation)
and the student's actual state of knowledge.
Each student is responsible for familiarizing him/herself with
academic integrity issues and policies at WPI.
All suspected cases of dishonesty will be fully investigated.
Ask Prof. Weekes if you are in any way unsure whether your proposed actions/collaborations will be considered academically honest or not.
Students with Disabilities
Students with disabilities who believe that they may need accommodations in this class are encouraged to contact the Disability Services Office (DSO), as soon as possible to ensure that such accommodations are implemented in a timely fashion. The DSO is located in the Student Development and Counseling Center and the phone number is 508-831-4908,
e-mail is DSO@WPI.
If you are eligible for course adaptations or accommodations because of a disability (whether or not you choose to use these accommodations), or if you have medical information that I should know about please make an appointment with me immediately. |
Find a Birdsboro PrealgebraThe course also introduces students to absolute value, the coordinate plane and different algebraic properties. The students will build on this basic knowledge by learning how to solve multi-step equations and inequalities and the complex algebraic functions that accompany them, such as exponent... |
Algebra 1 Advanced is an in-depth study of algebraic symbolism, systems of equations, graphing, problem-solving, and probability and statistics. The students will build upon their previous knowledge to further understand the characteristics and representations of various functions and relations, including first degree equations and inequalities, polynomials, exponential and radical expressions, quadratic equations complex numbers, and rational algebraic expressions. This course is designed for highly motivated and mathematically talented students.
Homework/Assignments
If you are not receiving the 8th grade weekly parent email which lists the planned homework assignments for the week and any special announcements, please email [email protected] to get your email address added to the mailing list.
Please check upcoming events (see above right or above left) to ensure I do not have a conflict/appointment at the time you are thinking of coming in. I am usually at school by 7:45 am and here until at least 4:30 pm. |
Department of Mathematics
MATH 1RA. Developmental Mathematics I
(3 units)
The first semester in a two semester sequence preparing students for college
level mathematics. See the online Schedule
of Courses for restrictions on enrollment based on the Entry Level Math
test. Properties of ordinary arithmetic, integers, rational numbers and
linear equations. CR/NC grading only; not applicable towards baccalaureate
degree requirements. F
MATH 10A. Structure and Concepts in Mathematics
I (3 units)
Prerequisite: students must meet the ELM requirement. Designed for prospective
elementary school teachers. Development of real numbers including integers,
rational and irrational numbers, computation, prime numbers and factorizations,
and problem-solving strategies. Meets B4 G.E. requirement only for liberal
studies majors. FS
MATH 45. What Is Mathematics? (3 units)
Prerequisite: students must meet the ELM requirement. Covers topics from
the following areas: (I) The Mathematics of Social Choice; (II) Management
Science and Optimization; (III) The Mathematics of Growth and Symmetry;
and (IV) Statistics and Probability. G.E. Foundation B4. FS
MATH 70. Calculus for Life Sciences (4
units)
No credit if taken after MATH 75 or 75A and B. Prerequisite: students must
meet the ELM requirement. Functions and graphs, limits, derivatives, antiderivatives,
differential equations, and partial derivatives with applications in the
Life Sciences. FS
MATH 90. Directed Study (1-3; max total
3 units)
Independently arranged course of study in some limited area of mathematics
either to remove a deficiency or to investigate a topic in more depth. (1-3
hours, to be arranged)
MATH 111. Transition to Advanced Mathematics
(3 units)
Prerequisite: MATH 76. Introduction to the language and problems of mathematics.
Topics include set theory, symbolic logic, types of proofs, and mathematical
induction. Special emphasis is given to improving the student's ability
to construct, explain, and justify mathematical arguments. FS
MATH 133. Number Theory for Liberal Studies
(3 units)
Prerequisite: MATH 10B or permission of instructor. The historical development
of the concept of number and arithmetic algorithms. The magnitude of numbers.
Basic number theory. Special numbers and sequences. Number patterns. Modular
arithmetic. F
MATH 134. Geometry for Liberal Studies
(3 units) Prerequisite: MATH 10B or permission of instructor. The use of computer
technology to study and explore concepts in Euclidean geometry. Topics include,
but are not restricted to, properties of polygons, tilings, and polyhedra.
S
MATH 137. Exploring Statistics (3 units)
Prerequisite: MATH 10B or permission of instructor. Descriptive and inferential
statistics with a focus on applications to mathematics education. Use of
technology and activities for student discovery and understanding of data
organization, collection, analysis, and inference. F
MATH 138. Exploring Algebra (3 units)
Prerequisite: MATH 10B or permission of instructor. Designed for prospective
school teachers who wish to develop a deeper conceptual understanding of
algebraic themes and ideas needed to become competent and effective mathematics
teachers. S
MATH 143. History of Mathematics (4 units)
Prerequisite: MATH 75 or 75A and B. History of the development of mathematical
concepts in algebra, geometry, number theory, analytical geometry, and calculus
from ancient times through modern times. Theorems with historical significance
will be studied as they relate to the development of modern mathematics.
S
MATH 149. Capstone Mathematics for Teachers
(4 units)
Prerequisites: MATH 151, 161, and 171. (MATH 161 and MATH 171 may be taken
concurrently.) Secondary school mathematics from an advanced viewpoint.
Builds on students' work in upper-division mathematics to deepen their understanding
of the mathematics taught in secondary school. Students will actively explore
topics in number theory, algebra, analysis, geometry.
MATH 161. Principles of Geometry (3 units)
Prerequisite: MATH 111. The classical elliptic, parabolic, and hyperbolic
geometries developed on a framework of incidence, order and separation,
congruence; coordinatization. Theory of parallels for parabolic and hyperbolic
geometries. Selected topics of modern Euclidean geometry. S
MATH 165. Differential Geometry (3 units)
Prerequisite: MATH 77 and 111 or permission of instructor. Study of geometry
in Euclidean space by means of calculus, including theory of curves and
surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.
F
MATH 232. Mathematical Models with Technology
(3 units)
Prerequisite: graduate standing in mathematics or permission of instructor.
A technology-assisted study of the mathematics used to model phenomena in
statistics, natural science, and engineering.
MATH 250. Perspectives in Algebra (3 units)
Prerequisite: graduate standing in mathematics or permission of instructor.
Study of advanced topics in algebra, providing a higher perspective to concepts
in the high school curriculum. Topics selected from, but not limited to,
groups, rings, fields, and vector spaces.
MATH 260. Perspectives in Geometry (3
units)
Prerequisite: graduate standing in mathematics or permission of instructor.
Geometry from a transformations point of view. Euclidean and noneuclidean
geometries in two and three dimensions. Problem solving and proofs using
transformations. Topics chosen to be relevant to geometrical concepts in
the high school curriculum.
MATH 270. Perspectives in Analysis (3 units)
Prerequisite: graduate standing in mathematics or permission of instructor.
An overview of the development of mathematical analysis, both real and complex.
Emphasizes interrelation of the various areas of study , the use of technology,
and relevance to the high school mathematics curriculum.
MATH 298. Research Project in Mathematics
(3 units)*
Prerequisite: graduate standing. Independent investigation of advanced character
as the culminating requirement for the master's degree. Approved for RP
grading.
MATH 299. Thesis in Mathematics (3 units)
Prerequisite: See Criteria for Thesis
and Project. Preparation, completion, and submission of an acceptable
thesis for the master's degree. Approved for RP grading. |
From the Publisher: Authors Wayne Winston and Munirpallam Venkataramanan emphasize model-formulation and model-building skills as well as interpretation of computer software output. Focusing on deterministic models, this book is designed for the first half of an operations research sequence. A subset of Winston's best-selling OPERATIONS RESEARCH, INTRODUCTION TO MATHEMATICAL PROGRAMMING offers self-contained chapters that make it flexible enough for one- or two-semester courses ranging from advanced beginning to intermediate in level. The book has a strong computer orientation and emphasizes model-formulation and model-building skills. Every topic includes a corresponding computer-based modeling and solution method and every chapter presents the software tools needed to solve realistic problems. LINDO, LINGO, and Premium Solver for Education software packages are available with the book.
Description:
Introduction to Technical Mathematics, Fifth Edition, has been thoroughly revised
and modernized with up to date applications, an expanded art program, and new pedagogy to help today''s students relate to the mathematics they are learning. The new edition continues ...
Description:
This introduction to aspects of semidefinite programming and its use
in approximation algorithms develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others.
Description:
Stochastic programming the science that provides us with tools to
design and control stochastic systems with the aid of mathematical programming techniques lies at the intersection of statistics and mathematical programming. The book Stochastic Programming is a comprehensive introduction ... |
What Information and Communication Technology resources - both hardware and software - are available for maths teachers? How can they be used to extend and enrich students' learning across the maths curriculum? How can teachers incorporate ICT effectively into their lesson and course planning? Why should maths teachers incorporate ICT into their teaching? What developments are likely in the future? Providing clear and practical answers to questions such as these, covering number, algebra, geometry, trigonometry, statistics, modelling and advanced maths, Teaching Mathematics with ICT the most user-friendly, comprehensive guide to the subject.
"Refreshingly practical and honest… The great boon is the CD provided with the book, which contains the necessary software to allow the reader to replicate what is in the text… Not only does the book fascinate mathematically, it forces reflection on the use of ICT… This is an excellent book and should be a part of the armoury of every teacher of maths." -- Times Education Supplement
Contents: 1. What resources are available? 2. ICT and the school curriculum 3. How to plan for effective ICT use 4. Why integrate ICT into maths teaching? 5. Where is it all going? |
USB port for computer connectivity, unit-to-unit communication with other TI-Nspire family handhelds, and more.
TI-Nspire Applications
CALCULATOR – Enter and view expressions, equations and formulas exactly as they appear in math textbooks. Quickly and easily select the proper syntax, symbols and variables from a template that supports standard mathematical notation. Scroll through previous entries to explore outcomes and patterns.
GRAPHS – Graph and explore functions, animate points on objects or graphs and explain their behavior, and much more.
GEOMETRY – Create and explore geometric shapes.
LISTS & SPREADSHEET – Capture and track the values of a graph and collected data, and observe number patterns. Organize the results of statistical analysis. Capabilities similar to using computer spreadsheets: label columns, insert formulas into cells, select individual cells and change their size, and more.
NOTES – Put the math in writing. Include the word problem with its solutions and explain problem-solving approaches – right in the handheld or computer software. Question-and-answer templates allow educators to prompt students to show solutions.
VERNIER DATAQUEST – Create a hypothesis graphically and replay data collection experiments. Use in conjunction with the TI-Nspire Lab Cradle.
Features and Functionality
Utilize images (jpeg, jpg, bmp, png file formats), including photos
3D graphing
Includes an easy-glide Touchpad that works more like a computer with a mouse.
See multiple representations of a single problem - algebraic, graphical, geometric, numeric and written.
Explore individual representations, one at a time, or as many as four on the same screen.
Grab a graphed function and move it to see the effect on corresponding equations and data lists
"Link" representations: Manipulate the properties of one and observe instant updates to others without switching screens
Solve equations while avoiding arithmetic errors
Receive a TI-84 Plus Keypad by mail. Request it from education.ti.com/84keypad at no added cost.
Save and review work - create, edit and save problem solving in documents and pages similar to the word processing and file storage features of a computer.
Connectivity – easily link with another TI-Nspire family handheld or a PC for easy file transfer
Features a dedicated programming environment as well as programming libraries for global access to user-defined functions & programs
Research
Students learn mathematical concepts more readily with deeper understanding when they learn across different forms of representation
Students using TI-Nspire handhelds have demonstrated deeper understanding and greater abilities in drawing inferences
Appropriate use of TI-Nspire technology can facilitate use of shared resources for collaborative learning, high student engagement, and a novel, integrated format for instructional units. Beliefs that the calculator is an aid to learning mathematics (not just an efficiency device)
Visit for more information about research on TI-Nspire technology
In the package:
Handheld
AAA batteries
Slide case
TI-Nspire Software (student or teacher, trial or perpetual depending on the configuration purchased)
USB computer cable
Unit-to-unit cable
Complementary Technology & Activities
Connect the TI-Nspire ViewScreen Panel to any TI-Nspire handheld for use with an overhead projector.
Use TI-Nspire Teacher Software on a computer connected to an LCD projector to explore the software's built-in SmartView™ Emulator that can be used for classroom demonstration
Upgrade operating systems and transfer files using the TI-Nspire Docking Station |
Equations, Roots & Exponents Mastery DVD A 12-lesson pre-algebra program that teaches selected critical concepts, skills, and problem-solving strategies needed to recognize and work with different types of equations problems.
Statistics and Data Analysis in Sports VHS Using only a calculator, a stat book, and some custom equations, a new generation of baseball statisticians believes it's possible to accurately predict a player's true value to his team. |
Geometry - Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to geometry, including classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory. Many results in this area are basic in either the sense of simple, or useful, or both. History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus.
more>>
Geometry Formulas and Facts - Silvio Levy; The Geometry Center
Rewritten and updated excerpts from the 30th Edition of the CRC Standard Mathematical Tables and Formulas. Covers all of geometry, minus differential geometry. Very complete collection of definitions, formulas, tables and diagrams, divided into two- and three- dimensional geometry, and further into 16 subdivisions such as transformations, polygons, coordinate systems, isometries, polyhedra and spheres.
more>>
1ucasvb's lab
Diagrams and animations from a longtime contributor to Wikipedia's math and physics articles. Posts, which date back to February 2013 and feature graphics coded in POV-Ray and PHP, include "Experimenting with sound: the polygonal trigonometric functions
...more>>
All Ivy Tutoring
Phoenix-based tutoring service providing one-on-one, in-home tutoring, offering help in all math subjects from basic math through college level differential equations and linear algebra.
...more>>
alt.math.undergrad - Math Forum
An unmoderated discussion forum for issues and problems pertaining to college undergraduate mathematics. Read and search archived messages; and register to post to the discussions.
...more>>
Blacks Academy - Blacks Academy
Extensive database of high school level resources for mathematics, suitable for teachers and independent students. Free with registration required. British-based curricula are served, but could be of great interest to American students, including sophomores.
...more>>
Cabri-géomètre - Jean-Marie Laborde
The French site for Cabri Geometry, the geometry software program. The site includes a brief history of the program, a list of new things on the site (see also Cabri-Java), Cabri product information, book and website references, and contact informationComputer Lab Courseware - University of Toronto
Courseware to help in the teaching of various mathematical concepts. Most is made up of Mathematica notebooks and packages, but some consists of C programs written to run under X-Windows. A lab manual in TeX beginning with an introduction to UNIX and
...more>>
DigiArea Group - Helen Golovina
Makers of atlas™, a Maple package for differential geometry calculations that works with manifolds, mappings, embeddings, submersions, p-forms, and tensor fields; and LdeApprox™, a toolbox for Maple or Mathematica that finds analytical polynomial
...more>>
Elsevier Science
"Information Provider to the World." Elsevier's mission is "to advance science, technology and medical science by fulfilling, on a sound commercial basis, the communication needs specific to the international community of scientists, engineers and associated
...more>>
Expert Math Tutoring in New York - Expert Tutor NY
Ivy League chemistry graduate and certified test prep course teacher offers private instruction for math at high school and college levels in metropolitan New York area -- Regents, SAT, SAT II, algebra, geometry, trigonometry, calculus, precalculus.
...more>>
ExploreLearning.com - ExploreLearning
A library of virtual manipulatives called "Gizmos", primarily for grades 3-12, in math and science. Math topics include the major strands of instruction according to national standards, as well as developmental math, college algebra and college-level
...more>>
Foundations of Computational Mathematics
The FoCM's primary aim is to further the understanding of the deep relationships between mathematical analysis, topology, geometry and algebra and the computational process as they are evolving together with the modern computer. The meetings are unified
...more>>
Fundamentals of Geometry - Oleg A. Belyaev
A work in progress to create a free online book on geometry--from exposition of elementary absolute geometry based on Hilbert's axioms to more advanced topics. The site contains a current draft of the book. This text is also intended to provide math background
...more>> |
The present book is intended primarily for an undergraduate audience. The authors believes that a sound grounding in Hilbert space theory is the best way how to approach functional analysis. It consists of sixteen chapters dealing with the following topics: Inner product spaces, Normed spaces, Hilbert and Banach spaces, Orthogonal expansions, Classical Fourier series, Dual spaces, Linear operators, Compact operators, Sturm- Liouville systems, Green's functions, Eigenfunction expansions, Positive operators and contractions, Hardy spaces, Approximation by analytic functions and approximation by meromorphic functions. This last chapter and the one concerning the positive operators may be of interest to electrical engineers, since some recent developments, particularly in control and filter design, require familiarity with this aspect of operator theory. The book presupposes introductory courses in real analysis, linear algebra, topology of metric spaces and elementary complex analysis. The chapter concerning Hardy spaces requires a certain familiarity with Lebesgue measure. [L.Janos] |
Mathematics scares and depresses most of us, but politicians, journalists and everyone in power use numbers all the time to bamboozle us. Most maths is really simple - as easy as 2+2 in fact. Better still it can be understood without any jargon, any formulas - and in fact not even many numbers. Most of it is commonsense, and by using a few really simple... more... The result is a must-have for all those needing to apply the methods in... more...
Assuming only basic algebra and Galois theory, this book develops the method of 'algebraic patching' to realize finite groups and, more generally, to solve finite split embedding problems over fields. The method succeeds over rational function fields of one variable over 'ample fields'. Among others, it leads to the solution of twoThis book concentrates on the mathematics of photonic crystals, which form an important class of physical structures investigated in nanotechnology. Photonic crystals are materials which are composed of two or more different dielectrics or metals, and which exhibit a spatially periodic structure, typically at the length scale of hundred nanometers.... more...
Delves into the world of ideas, explores the spell mathematics casts on our lives, and helps you discover mathematics where you least expect it. Be spellbound by the mathematical designs found in nature. Learn how knots may untie the mysteries of life. Be mesmerized by the computer revolution. Discover how the hidden forces of mathematics -hold architectural... more...
Part of the joy of mathematics is that it is everywhere-in soap bubbles, electricity, da Vinci's masterpieces, even in an ocean wave. Written by the well-known mathematics teacher consultant, this volume's collection of over 200 clearly illustrated mathematical ideas, concepts, puzzles, and games shows where they turn up in the "real" world. You'll... more... fields of physics, engineering and chemistry with an interest in fluid dynamics... more...
This volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems. The first part of the book introduces the mathematical concept required for treating the manifolds |
Dunstable Al provides a review and extension of the concepts taught in Algebra1. Topics include, but are not limited to equations and inequalities, coordinates and graphs, general functions, polynomials and rational functions, exponential and logarithmic functions, trigonometric functions of angles |
Many students worry about starting algebra. Pre-Algebra Essentials For Dummies provides an overview of critical pre-algebra concepts to help new algebra students (and their parents) take the next step without...
$ 6.99
Whether studying chemistry as part of a degree requirement or as part of a core curriculum, students will find Chemistry Essentials For Dummies to be an invaluable quick reference guide to the fundamentals of... |
Intermediate AlgebraExplorations in College Algebra, 5/e and its accompanying ancillaries are designed to make algebra interesting and relevant to the student. The text adopts a problem-solving approach that motivates students to grasp abstract ideas by solving real-world problems.
Sheldon Axler brings a brand new approach to the study of advanced algebra. While many students will bypass the book and go straight for the solutions manual, this text integrates the two in order to engage students in the text itself.
Algebra is fundamental to the working of modern society, yet its origins are as old as the beginnings of civilization. Algebraic equations describe the laws of science, the principles of engineering, and the rules of business.
Algebra is fundamental to the working of modern society, yet its origins are as old as the beginnings of civilization. Algebraic equations describe the laws of science, the principles of engineering, and the rules of businessAlgebra is fundamental to the working of modern society, yet its origins are as old as the beginnings of civilization. Algebraic equations describe the laws of science, the principles of engineering, and the rules of business. |
The Motion of a Pendulum
Pendulum
Project Brief
This investigation provides opportunities for Year 11 Mathematics students to use simple trigonometric functions to model the periodic motion of a pendulum using data that is provided, or using data that the students themselves collect using a calculator based ranger. This process of mathematical modelling mirrors that used by mathematicians to model a wide range of natural and artificial phenomena. These models help us to understand these processes, give a sense about how we might control them and allow us to make predictions about them.
There are countless areas in which mathematicians apply their skills, including the study of the environment, biological and medical processes and in the business world.
Project Files:
INTRODUCTION >> pendulum_introduction.ppt A powerpoint slide show introducing a Directed Investigation on the Motion of a Pendulum.
SAMPLE DATA >> sample_data.xls An Excel file of sample data. This data can be used to do the Directed Investigation when schools do not have a CBR. The sample data can be transferred to lists in a TI or CASIO graphics calculator. The instructions for this data transfer can be obtained via the link DATA TRANSFER INSTRUCTIONS on this home page.
Additional Instructions & Software:
TI CONNECT SOFTWARE >> ticonnect_installation.exe This file can be used to install TIConnect on your computer. It can be downloaded free of charge from the internet at The TIConnect software enables you to exchange data between your GC and your PC. You can connect your GC to your PC using the USB cable that came with your GC. Be sure to install the TIConnect software before you connect your GC to your PC.
CASIO FA-124 SOFTWARE >> install_casio.zip The file can be used to install Casio FA-124 on your computer. It can be downloaded free of charge from the internet at The FA-124 software enables you to exchange data between your GC and your PC. You can connect your GC to your PC using the USB cable that came with your GC.
Solutions to the Project:
DIRECTED INVESTIGATION SOLUTIONS FOR THE SAMPLE DATA >> solutions.doc A word file providing solutions to the Directed Investigation on the Motion of a Pendulum, for the sample data provided.
Additional Resources:
MEET DR. RICHARD CLARKE, APPLIED MATHEMATICIAN Dr. Richard Clarke, ARC Research Associate at the Adelaide University School of Mathematical Sciences talks about the role of an applied mathematician. View the video on Youtube
University Senior College
Address
University Senior College
at Adelaide University Inc
North Terrace
SA 5005 AUSTRALIA |
34385419
ISBN-13:
9780534385415
Publisher:
Brooks/Cole Pub Co
Release Date:
December, 2001
Length:
877 Pages
Weight:
Unavailable
Dimensions:
9.5 X 8.6 X 1.5 inches
Language:
English
Precalculus: Mathematics for Calculus... Read more invests the same attention to detail and clarity as Jim Stewart does in his market-leading Calculus text.
Sorry this edition is not currently available. Click on the author link to see if another edition is available or you can add this edition to your wishlist by clicking the link above.
55
Customer Reviews
great help
Posted by Wendie Cao on 10/08/2008
This book is really a great help for those wanting to exercise their math skills or even challenging themselves in precalculus. It provides answers in the back of the book so you can check your problems and in the beginning of the book it has some review chapters on basic algebra. I use it for my college precalc class it really does help and explains mathematics beautifully.
Precalculus -
09/12/2003
I'd say it's a rather interesting book, especially concerning the set of problems at the end of each chapter. Covers basics of linear algebra and preparation for calculus, with reasonable amount of problems and examples. Clear prose. Warns about mistakes to avoid.
Brief biographies of prominent mathematicians are introduced in the margin sometimes. Occasionally refers to external sources that you may find interesting.
I'm giving it 5 stars because this book is rather non-traditional. The authors undoubtedly wanted to introduce it as an interesting subject and does cover needed preparatory material for calculus.
Very understandable and interesting, a rarity in math texts.
06/15/1999
The book is easy to read and interesting. I am able to teach high school students out of this book, but it has enough depth to still be useful to a first year university student. It has good write-ups on the predominant mathematicians. It's the best pre-calc book I've come across.
One of the top two precalculus texts available
Posted by Charles Ashbacher on 03/14/2005
There is a mind-numbing sameness to precalculus and calculus textbooks, and this book is more of the same. The coverage starts with real numbers, exponents, expressions and solving equations. The basic principles of functions, polynomial, exponential, logarithmic and trigonometric functions, solving systems of equations, sequences, series, counting and probability, analytic geometry and limits follow this. I personally can do without the chapter on limits, when I teach precalculus, I am hard pressed to cover the other material. There is plenty of time to cover limits in calculus and it provides a better context. There are a large number of exercises at the end of each section and solutions to the odd-numbered ones are included in an appendix. As appears to be the case with many books, some of the exercises could have been left out with no decline in quality. At times I suspect there is the mathematical equivalent of an "arms race" to see how many exercises can be included at the end of a chapter. The previous paragraph could be used to describe nearly every precalculus text on the planet, so it fits into the category of obvious, but necessary. Therefore, the key point is what makes this book different from the competition. The answer is not much. The approach is the standard statement of the new material followed by a series of worked examples, which is also the fundamental strategy used in all lower level math books. Short biographical asides of some of the major historical figures in mathematics are interjected on a regular basis. I like that, but wonder how often the students read them. What is different about this book is that the quality of the writing is somewhat better than most. In a field where there is very little to differentiate the texts, that is enough to make me rank this book in the top two precalculus books that are available.
Precalculus: Matheatics for Calculus: 5th Edition (with CD-rom)
Posted by Daniel S. Breit on 02/26/2008
I have never used the CD, but the book is okay. If you are going to take calculus in the future than you should get a cheaper precalculus book and save some money. This is essentially the same as the 4th edition, except the price tag. Only get if you absolutely must for a class. |
Kingwood, TX PrecalculusMany They are mostly used for solving a system of equations in order to determine the answers |
Students performing at the partially proficient level have limited recognition and understanding of and inconsistently apply basic mathematical concepts, skills, and vocabulary to theoretical and real world situations.
These students may understand that a quantity can be represented numerically in various ways. Partially proficient students perform basic computational procedures with inconsistent accuracy.
Partially proficient students have difficulty using informal algebraic concepts and processes.
Partially proficient students inconsistently read, construct, and interpret data and graphs. They inconsistently apply the concepts and methods of discrete mathematics.
These students will occasionally infer, reason and estimate while problem solving. Partially proficient students are frequently ineffectual in selecting a successful process or strategy. These students have difficulty demonstrating a basic understanding of mathematical concepts through written expression and/or symbolic representation.
Proficient
Students performing at the proficient level recognize and understand basic mathematical concepts, skills, and vocabulary and apply them to theoretical and real world situations.
Proficient students understand that a quantity can be represented numerically in various ways. These students perform basic computational procedures.
Proficient students read, construct, and interpret data and graphs. They apply the concepts and methods of discrete mathematics.
These students infer, reason, and estimate while problem solving. Proficient students are flexible in selecting a successful process or strategy. These students demonstrate a basic understanding of mathematical concepts through written expression and/or symbolic representation.
Advanced Proficient
Students performing at the advanced proficient level consistently demonstrate the qualities outlined for proficient performance. In addition, advanced proficient students analyze methods for appropriateness, synthesize processes, and evaluate mathematical relationships. Advanced proficient students demonstrate conceptual understanding by consistently providing clear and complete explanations. These students demonstrate the ability to transfer mathematical concepts to other applications and successfully form conjectures. |
Please do not start Life of Fred: Beginning Algebra without doing the two pre-algebra books first.
The two pre-algebra books are an essential introduction to the material of beginning algebra.
They teach, among many other things,
Graphing
An introduction to word problems. For example, one of the problems in a Your Turn to Play is: Let's suppose on some day Fred sold x Gourmet Gauss Dogs and made a profit of $3 per dog. On that day he also sold x - 2 Double Dogs and made a profit of $2 per dog. And on that day he sold x - 8 Cold Dogs and made a profit of $1 per dog. If he made $168 on that day, how many Gourmet Gauss Dogs did he sell?
We are serious about learning how to do word problems.
Life of Fred: Beginning Algebra is paired with its study guide, Fred's Home Companion: Beginning Algebra. For more information about the guide, click here.
Life of Fred: Beginning Algebra offers more material than is normally taught in a year of high school algebra. It also chronicles five days of Fred's life when he is drafted by accident (at the age of 6) and sent off to an army camp in Texas. |
Show Your Work! 2-step equations: Help for early algebra students battle begins! Students are learning how to solve basic equations in one variable, and the problems are "so easy" they can "do it in their heads" and don't want to show work! Here is a strategy that has helped me get students to show the work I really want. Very adaptable - especially useful as a supplement to typical textbook worksheets that don't provide students any room to show the work we so desperately want to see!
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1983.05Great! I always have two-sided copies of these laying around and offer them to students when I assign a textbook worksheet (since they never give enough room for the work we model and expect of our students!) |
MTS7001 Mathematics Tertiary Preparation (KUMBN)
Synopsis
Using concepts of self-paced instruction the course guides students through a carefully sequenced series of topics, which will provide the foundation for mathematics that will be encountered in tertiary studies, detailed above. The self-paced structure allows students to work at their own pace developing confidence with mathematics and general problem solving. |
In addition to the step-by-step multi-media solutions manual, parents may purchase another set of CDs that contains both lectures—one for each of the 110 lessons in the textbook—and step-by-step multimedia explanations to the 5 practice problems that begin each problem set. These practice problems serve as hints to the very toughest problems in the book.
Friendly Text/More Explanation
Multimedia solutions and multimedia lectures are not the only features which set the Teaching Textbook™ program apart from the com-petition. The textbook itself, since it was designed specifically for independent learners, contains far more explanation than any on the market, and the tone is friendly and conversational. Important portions of the text are also highlighted to enhance reading comprehension.
Review Method
Another advantage of the Geometry Teaching Textbook™ is that it employs the well-known review method. This helps students master difficult concepts and increases long-term retention. Each lesson includes 5 practice problems (examples) and 20-24 assigned problems, for a total of almost 3,500 problems in the book.
SAT and ACT Prep Built In
In addition to covering all the standard school geometry topics, the book puts great emphasis on problems found on the SAT and ACT. In fact, nearly every problem set includes several problems that were modeled after those found on actual SAT and ACT exams. And since all Teaching Textbooks use the review method, students become better and better at solving these important problem types each day.
Funny Examples and Illustrations
The Geometry Teaching Textbook™ is also full of fascinating and entertaining real-world examples that make the math concepts crystal clear. And the book contains many humorous illustrations which put students at ease and keep them totally engaged in the learning process.
Easy-to-Use CD-ROM's
The Geometry Teaching Textbook™ CD-ROM's are also incredibly easy to use. Unlike most software, which often comes with detailed instruction manuals, the Teaching Textbook CDs are designed to be as easy to use as a videotape. You just set the CD in the tray, click on the lesson and lecture you want and that's it!
What Parents and Students are Saying
Parents love the Geometry Teaching Textbook™ program primarily because it relieves them from having to figure out tough math problems on their own, but if you ask students what makes the Teaching Textbook the best, they almost always say the same thing: It's more fun. Students also say that reading the text is like having a friendly tutor or coach gently guiding them through each concept and problem type.
Never Get Stumped Again
The two sets of CD-ROMs offer far more teaching than any other math product on the market. In fact, they even contain more teaching than is available in most traditional classes. With this unprecedented CD-ROM package, the frustration of missing a problem and not being able to figure out what you did wrong is over. Students (and their parents) finally have a powerful yet affordable way to teach themselves math! |
Search Course Communities:
Course Communities
Direction Fields (2D)
Course Topic(s):
Ordinary Differential Equations | Graphic Methods
The user is given a first order differential equation and four plots of direction fields and asked to choose the field that matches the given differential equation. There is also a version of this applet that runs on the University of South Carolina server. |
nonlinear programminglinear programming is discussed in the following articles:
major reference
mathematical programming
...a variety of purposes. If the basic descriptions involved take the form of linear algebraic equations, the technique is described as linear programming. If more complex forms are required, the term
nonlinear programming is applied. Mathematical programming is used in planning production schedules, in transportation, in military logistics, and in calculating economic growth, by inserting assumed |
Head Start Maths
The Mathematics Learning Centre (MLC) in the University of Limerick (UL) deals with large numbers of adult learners of mathematics. These students have different learning issues/requirements compared to traditional age learners who tend to be much more homogeneous as regards age, mathematical background and knowledge etc. 'Front-end' tutorials are organised and taught in the first 2 weeks of the academic year to help adult learners catch up on mathematics fundamentals they will need in their courses. These students can find it tremendously difficult to catch up and keep up with other studies at the same time. An intervention before they start their studies would be more beneficial and a lot less stressful.
The main aim of such a programme is to ease the transition to third level education for Adult learners who are intending on returning to/starting college by:
revising essential mathematics skills they will need,
revising essential study skills by being part of a lecture/tutorial group,
meeting other Adult learners who are 'in the same boat',
giving the learners an induction to life at third level before crowds of students arrive back i.e. lectures, tutorials, finding their way around campus etc.
In 2008, Dr Olivia Gill (University of Limerick) was seconded to work with sigma's Professor Duncan Lawson to create a teaching and learning package consisting of a book of notes/manual that students and teachers of mathematics can take home and keep. This programme, entitled "Head Start Maths", can then be taught to mature students in any third level institution over a period of a week before the students enter college. |
TestSMART® Student Practice Books for Mathematics contain pretests and reproducible practice sheets that help diagnose the standards that students need to master
The multiple worksheets for each skill also provide opportunities for further practiceFamiliarizes students with both the content and format of state-mandated testsAll materials are research-basedAn extensive master skills list represents a synthesis of reading skills from major state test specifications and can easily be correlated from one state to anotherA complete answer key is providedCovers all math objectives thoroughly with the Concepts book and the Operations and Problem Solving book for each grade levelEach book contains comprehensive reproducible practice exercises for word analysis, vocabulary, comprehension and study skillsGrade 3 |
Find a Metuchen PrecalculusAn introduction to variables. The number-line is labeled and the different types of numbers are defined. Students manipulate simple equations, and practice constructing equations based on real world applications |
Program Resources
Founded in 1888 to further mathematical research and scholarship, the American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.
The Mathematical Association of America (MAA) is the world's largest organization devoted to the interests of collegiate mathematics. Members of the MAA receive many valuable benefits for modest dues. These benefits are designed to stimulate interest in mathematics by providing expository books and articles on contemporary mathematics and on recent developments at the frontiers of mathematical research, and by exchanging information about important events in the mathematical world. A major emphasis of the MAA is the teaching of mathematics at the collegiate level, but anyone who is interested in mathematics is welcome to join.
To ensure the strongest interactions between mathematics and other scientific and technological communities, it remains the policy of SIAM to advance the application of mathematics and computational science to engineering, industry, science, and society; promote research that will lead to effective new mathematical and computational methods and techniques for science, engineering, industry, and society; and provide media for the exchange of information and ideas among mathematicians, engineers, and scientists.
The American Statistical Association (ASA) is a scientific and educational society founded in 1839 with the following mission: To promote excellence in the application of statistical science across the wealth of human endeavor.
Michigan Council of Teachers of Mathematics is organized to encourage an active interest in mathematics and its teachings and to work toward the improvement of mathematics education programs in Michigan.
The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students.
Kappa Mu Epsilon an honor society in Mathematics promotes the interest of mathematics among undergraduate students. The chapters' members are selected from students of mathematics and other closely related fields who have maintained standards of scholarship, have professional merit, and have attained academic distinction.
Math Horizons is intended primarily for undergraduates interested in mathematics. Its purpose is to introduce students to the world of mathematics outside the classroom including stories of mathematical people, the history of an idea or circle of ideas, applications, fiction, folklore, traditions, institutions, humor, puzzles, games, book reviews, student math club activities, and career opportunities and advice. Get a copy in the departmental office!
Pi in the Sky is a semi-annual periodical designated for high school students in Alberta and British Columbia, Canada, with the purpose of promoting mathematics, establishing direct contact with teachers and students, increasing the involvement of high school students in mathematical activities, and promoting careers in mathematical sciences.
MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. Like the vibrant and constantly evolving discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries.
The Math Forum is a leading center for mathematics and mathematics education on the Internet. The Math Forum's mission is to provide resources, materials, activities, person-to-person interactions, and educational products and services that enrich and support teaching and learning in an increasingly technological world. |
Category Archives: TI-84 Lessons
The table feature allows you to quickly scroll the an x vs. y chart on your TI-84. This lesson demonstrates how the table can be used from beginning algebra graphs to finding a limit in calculus |
Self-Taught Calculus
Self-Taught Calculus"The Calculus Lifesaver" by Adrian Banner is way better, I used it for Calculus 1 and Calculus 2, and i even refer to it if I need to look something up, it even has online videos.
Why aren't you doing Mathematics SL? I can't imagine it being very hard (I was in HL for a while, then switched to A-Levels), especially for someone who intends on going into physics. Studies SL is fluffy. Too fluffy. Oh, now I get it. You want an easy 7?
Self-Taught Calculus
What about Calculus by Gilbert Strang Spivak's Calculus or Paul's Online Math Notes? Are those good for self study compared to The Calculus Lifesaver and A First Course In Calculus? I really want something that contains proofs and really give me a good understanding and masterI spent 11 years in the Navy and had a 12 year break in math. I had to reteach myself College Algebra and PreCalc. For PreCalc I found out what book they were using at the college I wanted to go to and ordered it from Amazon. If I had questions about it the internet has so many resources to answers those questions. WolframsAlpha is a great tool, also YouTube. Just some food for thought! |
Abstract: Undergraduate mathematics research students
from Summer 2008 will give 15-minute presentations about their
work. We will also give details about summer research
opportunities in mathematics coming up in 2009.
A discrete dynamical system is a
function in which the output from a point is repeatedly reinserted into
the function so as to form a sequence - which is called an orbit.
When a computer is used to generate these sequences, there is
inevitable round off error, so that what is in fact generated is called
a d - pseudo orbit, where d is the amount of possible error. If a
dynamical system has the property that there is an actual orbit close
to every d - pseudo orbit, then the system is said to have the
shadowing property. After giving explanation and examples of the above,
we provide conditions for a decreasing continuous function on the unit
interval to have the shadowing property. Work supported by
NSF-REU #0645887 and an HMMI grant.
How does a bicycle rider keep from
falling? Anyone learning to ride a bicycle realizes that this is
not easy. One must continually steer so that the resulting centrifugal
force (inertia) counteracts the pull of gravity on a slightly tipped
bike and rider. But one must also steer so that the desired direction
of travel is achieved. In this project, students are led through the
process of forming a mathematical model of bicycle tipping, and then
they use the model to discover a way to keep a bicycle upright and
traveling on a straight path. Work supported by an HMMI
grant.
The focus of my research project is
curriculum development for MATH 231 and 232 (sophomore-level math
courses) at Hope College. My responsibilities include writing exercises
for Darin Stephenson's MATH 231 and 232 textbook, creating labs and
projects to supplement traditional lectures, and developing assessments
to evaluate these labs and projects. One lab exercise that will be
discussed in detail involves a model of a population of bacteria, which
is altered each time a generation of genes replicates. Students use
their background of linear algebra and vector calculus to model the
behavior of this population with stochastic matrices. Furthermore, this
example of gene replication can also be utilized at the high school
level due to the relative simplicity of the model. Such an activity
would test the students' understanding of limits, probability, and
systems of equations.
Please join us for refreshments
outside VWF 104 at 3:45 p.m.
The
semester's last colloquium will take a look at Fermat's Last Theorem
Title:
The Proof
Speaker:
Andrew Wiles and others
Time:
Tuesday,
December 2 at 6:30
p.m.
Place:
VWF
104
Abstract: Fermat's Last
Theorem was first conjectured in 1637, but only
recently proved (in 1995) by Princeton mathematician Andrew Wiles.
In
spite of the simplicity of its statement, Fermat's Last Theorem puzzled
even the greatest mathematicians for over 350 years. During this
colloquium, we will watch a NOVA documentary that describes Wiles'
quest to prove Fermat's Last Theorem. The documentary provides
insights into not only the mathematics behind Fermat's Last Theorem,
but also some of the interesting individuals who contributed to the
proof.
Pizza will be served at around 6:15
pm, and a brief discussion will take
place immediately following the video. So that we can order the
right
amount of pizza, please put your name on the signup sheet on Prof.
Hodge's door (VWF 205) if you plan to attend.
(Note that the
time and day for this colloquium are different than usual.)
The end
is near!
With the end of the semester right around
the corner, the ways to receive colloquium credit are also coming to an
end. The last problem of the fortnight appears below. There
will be two more colloquia, one tomorrow and one the week after
Thanksgiving. Details on each of these is given above.
If you have put off completing your colloquium credit, your
opportunities are quickly fading away. Like Greg Meyer shown to
the left, hopefully you will have a strong finish this semester.
The
Problem of the Fortnight
The last
Problem of the Fortnight for this
semester:
A square is divided into three pieces of
equal area by two parallel cuts as shown. The distance between
the parallel lines is 6 inches. What is the area of the square in
square inches?
Write your solution (not just the answer!)
on a sanitized square piece of paper and drop it by Dr.
Pearson's office (VWF 212) by noon
on Wednesday, November 26.
As always, be sure
to write your name, the name(s) of your
professor(s), and your math class(es) on your solution (e.g. Trap
E. Zoid, Prof. Hy Potenuse, Math 225).
Good luck, and have
fun!
Problem
Solvers of the Fortnight
How many
of the positive factors of
36,000,000 are not perfect
squares? |
About This Course
This course, Math 102, covers a broad range of mathematical topics and provides you with a solid overview of concepts that non-mathematics majors need to know. Whether you need to review some key concepts or learn them for the first time, instructors Luke Winspur, Jennifer Scharf and Kathryn Maloney use easy-to-follow examples to take concepts like the Pythagorean theorem and standard deviation and make them accessible.
Though it's designed for students who don't intend to major in mathematics or hard sciences, Math 102 contains foundational knowledge that can prepare you for advanced math courses.
You can take all the lessons in this course in sequence, or just focus on the concepts you need to learn and skip the ones you don't. Course topics covered in Math 102 include: logic, probability and statistics, real number systems, functions and their graphs and more.
All topics are broken down into bite-sized videos and our self-grading online quizzes allow you to see how much you've learned and what you may need to review. |
Math
This course provides an intensive study of the structure of algebra through the field properties of the real number system.
Algebra I (A)
This course emphasizes the structure of the real number system and the techniques of algebra as a reflection of this structure.
Algebra I (B)
This algebra course begins informally, giving students the opportunity to strengthen fundamental skills and concepts.
Geometry (H)
This course develops geometry as a mathematical system. A vigorous approach to deductive reasoning is included.Symbolic logic is studied.
Geometry (A)
This course is a study of points, lines, and planes in space. The course specializes in the use of deduction, induction, and formal proof.
Geometry (B)
This course develops geometry more informally using inductive reasoning.A hands on approach is used.
Algebra II (H)
A more intensive study of the topics described in Algebra II is made. Greater emphasis is placed on theory and more challenging problems.
Algebra II (A)
This course is an extended study of basic concepts, skills, and application of algebra through the development of a mathematical system and the examination of the real and complex numbers.
Algebra II (A3)
This course is designed to study the concepts and applications of second year algebra in less depth than a traditional algebra II course.
Pre-Calculus (H)
The advanced mathematics course is designed for students with above average mathematical ability and motivation. The three main areas of study will be Trigonometry, Analytical Geometry, and an introduction to Calculus.
Trigonometry/Analytic Geometry (A)
This course covers the 6 trigonometric functions, inverse functions, graphing, solving triangles and other topics in trig and analytic geometry.
Trigonometry (A3)
This full-year course in Trigonometry will include a thorough study of the Trig functions and their application to problem solving.
Calculus (H)
This standard one-year course in the calculus of one variable develops the topics of functions, and limits, differentiation, integration, and special functions.
Calculus (AP)
This Advanced Placement Course will cover the first two semesters of a typical college Calculus course.It will prepare students for the AP Calculus AB test.The topics of limits, differentiation, and integration using polynomial, rational, radical, logarithmic, exponential, and trigonometric functions will be studied in depth.
Capital Campaign
Support the Capital Campaign securely online by clicking above or by visiting the Advancement page for additional ways to give.
Donations
Make a gift to the Annual Fund securely online by clicking above or visit the Advancement page for additional ways to give. |
Author: Delbert L. Hall Email: [email protected] Publisher: Spring Knoll Press Trim Size: 8.5" x 11" Pages: 146 ISBN: 0615747795 ISBN-13: 978-0615747798 Price: $19.95 Tentative Date of Release: March 1, 2013 Description: The job of
an entertainment rigger is to safely suspend objects (scenery, lights,
sound equipment, platforms, and even performers) at very specific
locations above the ground. The type, size and location of the
structural members from which these objects must be suspended vary
greatly from venue to venue. Additionally, the size, weight, and
location of each object varies from object to object. To ensure that
each object is safely suspended at the proper location, math is
essential. If you want to be a top-notch rigger, you have to know
math. Math does not have to be hard. It is a lot like baking - you
need a good recipe, and then you just have to follow it - EXACTLY. The
purpose of this book is to provide you with the recipe for solving
rigging problems. Once you learn the recipes, you will be able figure
out many rigging problems. This book is more than a list of formulas -
it will also help users grasp some of the principles behind the physics
of rigging. By understanding these principles and the math behind
them, entertainment riggers should be able to look at many rigging
situations and determine if it is "obviously safe," or "obviously
unsafe," without actually doing any math. However, there are many
cases where the load is just uncertain, or the answer is not obvious, and
the math needs to be done. This book may be of particular interest to
individuals who wish to become a certified rigger. Many of the
mathematical problems and other information presented in this book are
intended to prepare individual for the types of questions they might
encounter on a certification exam – in both theatre and arena rigging.
Table of Contents Part I. Conversions Lesson 1: Converting between Imperial and Metric
Part IV. Truss Lesson 10: Center of Gravity for Two Loads on a Beam Lesson 11: Uniformly Distributed Loads on a Beam Lesson 12: Dead-hang Tension on One End of a Truss Lesson 13: Simple Load on a Beam Lesson 14: Distributed Load on a Beam Lesson 15: Cantilevered Load on a Beam Lesson 16: Chain Hoists, and Truss, and Lights, Oh My! |
This new edition of a classic American Tech textbook presents
basic mathematic concepts typically applied in the industrial,
business, construction and craft trades. By combining
comprehensive text with
illustrated examples of mathematics problems, this book
offers easy-to-understand instructions for solving math-based
problems encountered on the job. Many different trade areas are
represented throughout the book.
Each of the twelve chapters contains an Introduction providing an
overview of the chapter content. Examples of specific mathematic
problems are displayed in illustrated, step-by-step formats.
Following visual as well as written processes provides the reader
with a sequenced opportunity to learn each concept. Learned
knowledge is then applied in Practice Problems, which immediately
follow Examples in the book. Students are encouraged to use the
space provided in the margins to answer these questions.
Tips located throughout the text assist in the development of
mathematics skills.
Calculator tips are also provided in each chapter to offer an
alternative method of solving problems and equations. Points
to Know are included to enhance the learner's understanding of how
mathematics principles are applied to the trade professions.
Additionally, photographs provide visual examples of how these
principles relate to on-the-job skills.
Practical Math is designed to be a basis for a mathematics course
or as a supplement to many other American Tech books and training
products. |
The Mathematics Survival Kit
The Mathematics Survival Kit
Professor Jack Weiner taught at the University of Guelph from 1974 to 1976. He spent the next five years at Parkside High School in Dundas, Ontario. In 1982, he was re-recruited by Guelph and has been happily teaching and writing there ever since. He has won both the University of Guelph Professorial Teaching Award and the prestigious Ontario Conderation of University Faculty Associations Teaching Award. He has been listed as ...
How to get an 'A' in Math!
1) After class, DON'T do your homework! Instead, read over your class notes. When you come to an example done in class...
2) DON'T read the example. Copy out the question, set your notes aside, and do the question yourself. Maybe you will get stuck. Even if you thought you understood the example completely when the teacher went over it in class, you may get stuck.
And this is GOOD NEWS! Now, you know what you don't know. So, consult your notes, look in the text, see your teacher/professor. Do whatever is necessary to figure out the steps in the example that troubled you.
From the Best of Our Knowledge
ALBANY, NY (2007-11-12) THE MATHEMATICS SURVIVAL KIT ,
Pt. 1 of 2 -
If you listen to public radio on the weekends, you have likely heard a university math professor who is also the Math Guy. But if your tastes run more to television, you may have also seen the Friday night CBS show, Numbers, in which a curious young math wiz named Charlie, solves crimes using mathematics. Regardless of your viewing or listening habits, it's apparent more emphasis is being placed on math.
Now, comes The Mathematics Survival Kit. It's written by Professor Jack Weiner from the University of Guelph in Ontario, Canada. Weiner has partnered with education software provider, Maple, to produce an interactive e-book version of his math survival book. The University of Guelph has taken the lead introducing e-books , intelligent assessment systems, and podcasts nto its math curriculum. This next generation of educational technology provides teachers with ore time to motivate students and improve their comprehensive retention. |
Learn . Create . Succeed: Algebra
This site is dedicated to assisting Science, Technology, Engineering and Mathematics (STEM) instructors in incorporating SolidWorks into their course curriculum.en-USFri, 13 Mar 2009 17:04:03 -0400 with My Yahoo!Subscribe with NewsGatorSubscribe with My AOLSubscribe with BloglinesSubscribe with NetvibesSubscribe with GoogleSubscribe with PageflakesMath2Go Accelerated Curriculum
Accelerated Curriculum applies science and mathematics concepts to designing a race car. With 10th scale RC Cars, students maximize performance and drive farther, faster, and all while exploring concepts in physics. The Upside Down Wings exercise takes a student...]]>AlgebraCalculusPhysicsSTEM CourseMatthew WestWed, 23 May 2007 11:37:53 -0400 with SolidWorks
exercise will assist students in visualizing and discovering how to find the formula to figure the sum of the interior angles of an n-gon. For both algebra and geometry students for an exercise in discovery learning. Contributed by Paul...
]]>AlgebraGeometryMarie PlanchardThu, 16 Nov 2006 16:13:57 -0500 and Extuded Cuts
lesson examines the extrusion of a cube then removes the material from each side using different geometrical shapes. Applications in basic math, algebra and drafting. Contributed by Gary Mills. Lesson Plan: solidworksextrusion_hatc2.doc Sample Parts: Cube1.SLDPRT ; Cube2.SLDPRT This lesson examines the extrusion of a cube then removes the material from each side using different geometrical shapes. Applications in basic math, algebra and drafting. Contributed by Gary Mills.
]]>AlgebraMechanical EngineeringMarie PlanchardWed, 19 Jul 2006 13:55:40 -0400 Plan: Parabola from Algebra to Calculus
lesson develops two examples with a parabola. The first example explores graphing a parabola from an equation, finding the area bounded by the x axis, center of mass, and volume of a solid in revolution about a line parallel... This lesson develops two examples with a parabola. The first example explores graphing a parabola from an equation, finding the area bounded by the x axis, center of mass, and volume of a solid in revolution about a line parallel to the x-axis. Understanding a parabola in algebra can lead to success in Calculus and Calculus based science classes such as physics, engineering mechanics and dynamics. The second examples bounds the parabola by both the x and y axis, explores the centroid and volume in revolution. |
This book is a collection of 375 completely solved exercises on differentiable manifolds, Lie groups, fibre bundles, and Riemannian manifolds. The exercises go from elementary computations to rather sophisticated tools. It is the first book consisting of completely solved problems on differentiable manifolds, and therefore will be a complement to the books on theory. A 42-page formulary is included which will be useful as an aide-memoire, especially for teachers and researchers on these topics. The book includes 50 figures. Audience: The book will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics, and some branches of engineering.
DJVUThis book follows the successful Mathematical Activities by the same author and has been written to provide teachers with further ideas to enrich their teaching of mathematics. It contains 127 investigations, puzzles, games and practical activities together with a commentary which provides solutions and additional ideas. These activities are suitable for use with children from age nine upwards to stimulate their interest and encourage mathematical thinking. ...
A complete preparation course for IELTS offers topic development to encourage students to think critically. A Language for Writing syllabus covers grammar and vocabulary. Help Yourself pages with activities to support self-study are provided.
This text gathers together 81 classroom proven strategies designed to build or reinforce the langiage skills of intermediate to advanced students.
This is an overall great "idea" book for the ESL classroom.It is basically a book of activities for each different aspect of language teaching. It devotes entire units to more difficult-to-teach topics such as developing listening skills. Most of the activities are extremely interactional and would enhance student's communicative abilities, either directly or indirectly. This book is a great resource for finding just the right activity for your lesson plan. All activities are also easily adaptable for level and for subject area.
This popular First Certificate course has been updated to prepare students for the new examination syllabus introduced from December 2008. A clear organisation and fresh approach have already made it a popular route to success at FCE in many countries: thirty short units provide thorough training in exam skills, solid language development, and lively class discussion. The course is written by experienced authors who have an in-depth knowledge of the FCE exam, and understand the needs of both students and teachers alike. |
Skowhegan Area High School Mathematics Department
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Objective
of Curriculum
The curriculum for "Contemporary Mathematics in
Context" builds upon the theme of mathematics as sense-making. Investigations
of real-life contexts leads to discovery of important mathematics that make sense
to students and, in turn, enable them to make sense out of new situations and
problems. Each year the four strands of mathematics are studied: algebra and functions,
geometry and trigonometry, statistics and probability, and discrete mathematics.
Problems are integrated across curriculum areas: math, science, social studies,
English, and business. Topics are multicultural as well. All areas of conventional
high school math are covered in this curriculum.
Learning is done collaboratively in small teams with
assistance from teachers. Students are assessed in a number of ways, including
daily work, homework, projects, and tests and quizzes. While the TI-83 graphing
calculator is used extensively, students learn to solve problems numerically,
graphically, and symbolically.
Objective
of this Project
As a final project, Seniors from the Class
of 2003, who have taken Math I,II,III,IV have reviewed each unit that is listed
at the left and have created web pages that summarize the relevant topics. These
web pages may be used by students, teachers, and parents who are interested in
review, preview, and learning.
MEA and Sat Results
Maine Learning Results
The Integrated Math curriculum demonstrates the Guiding
Principles identified by the Department of Education
Each Maine
student must leave school as:
I.
A clear and effective communicator
II. A
self-directed and life-long learner
III. A
creative and practical problem solver
IV. A
responsible and involved citizen
V. A
collaborative and quality worker
VI. An
integrative and informed thinker
Furthermore, the Integrated Math curriculum requires
proficiency in all topics described in the Maine Learning Results (specifics are
identified with each unit): |
This easy-to-use workbook is full of stimulating activities that will jumpstart your students' interest in algebra while reinforcing the major algebra concepts A variety of puzzles, mazes and games will challenge students to think creatively as they sharpen their algebra skills A special assessment section is also included to help prepare students for standardized tests
Product Information
Subject :
Algebra
Grade Level(s) :
Grade 7-9
Usage Ideas :
This easy-to-use workbook is chock full of stimulating activities that will jumpstart your students interest in algebra while reinforcing the major algebra concepts. A variety of puzzles, mazes, and games will challenge students to think creatively as the |
Mathematics AEA
The Mathematics AEA is aimed at the top candidates studying the A-level Mathematics course, regardless of examination board and syllabus. The Mathematics AEA is set by Edexcel, with exam code 9801, although all boards participate in forming the paper and questions. It is accessible to all students but is mainly aimed at those who are predicted an A grade for their mathematics A-level. It is only based on the core of A-level mathematics, i.e. C1, C2, C3, C4; there are no applied mathematics questions nor anything from the further mathematics syllabus (unlike STEP).
Contents
Structure
It is one three-hour long paper that consists of about seven questions. Questions may be multi-step with confidence building parts or unstructured. Some may be of an unusual nature that might include topics from GCSE and logic based items. Questions may be open-ended.
Seven percent of the marks will be assigned for style and clarity of mathematical presentation. The examiners will seek to reward elegance of solution, insight in reaching a solution, rigour in developing a mathematical argument and excellent use of notation.
The use of scientific or graphical calculators will NOT be allowed nor will computer algebra systems.
Candidates will be required to remember the same formulae as for GCE Advanced level Mathematics. They will also be expected to be familiar with the Mathematical Notation agreed for GCE Advanced level Mathematics.
Grading
Assessment materials and mark schemes will lead to awards on a two-point scale: Distinction and Merit, with Distinction being the higher. Candidates who do not reach the minimum standard for Merit will be recorded as ungraded.
Performance level descriptors have been developed to indicate the level of attainment that is characteristic of Distinction and Merit. They give a general indication of the required learning outcomes at each level. The grade awarded will depend in practice upon the extent to which the candidate has met the assessment objective overall. Shortcomings in some aspects of the examination may be balanced by better performances in others.
Candidates who achieve Distinction will demonstrate understanding and command of most of the topics tested.
Candidates who achieve a Merit will demonstrate understanding and command of many of the topics tested.
Future
The AEA in Mathematics has been extended to June 2015, as confirmed by Edexcel here. It is the sole AEA to be available after June 2009 (when the other AEAs were withdrawn), presumably reflecting the fact that the new Mathematics GCE A-level would not be taught until September 2011, and the first new AS exams would be in the 2012 sessions. It will probably be withdrawn after summer 2015. |
Algebra Through Symmetry is a visual method that was prevalent at the dawn of the mathematical sciences. Symmetry is crucial in the formulation of the method and it is symmetry that enables a visual solution of the algebraic equations. The visual method achieves symmetry through a process of axiomatic reformulation. Reformulating an equation to obtain symmetry requires less symbolic logic, fewer arithmetic operations and activates the student's most primitive concepts of one-to-one matching and the symmetry of the human form.
FEEDBACK
perrycream
This is an interesting document. I fully agree that many items, such as completing the square as the only way to solve the quadradic equation, need to be changed in our teaching of math .... this is one of the several alternatives. However, as writen, I don't think visually oriented student thinkers could use it ... I will try it in a class with some modification ... or perhaps as a video or powerpoint.
An excellent approach to helping students develop an intuitive understanding of basic algebra.
August 8, 2010
mcullen
Symmetry is a great addition to the algebra pedagogy. I teach both the axiomatic and visual methods to my HS classes. I allow the students to choose the method that is easiest for them during a test. Most of them choose the visual method because it requires less symbolic logic and it takes less time to obtain a solution.
My teaching style is similar; it has definitely benefited the at-risk students
September 21, 2008
djtotten
I am now retired and formerly taught mathematics in HS and Middle school. I am very impressed with Algebra Through Symmetry and regret that it was not available during my teaching years. Many of the students that I taught had extreme difficulty with the traditional axiomatic method. Based upon outside classroom contact with current students, they learn algebra quickly and easily by using the visual method.
I was raised in the axiomatic method but my experience has taught that the only way to reach the at-risk students is to simplify the procedures. I read The Elements more than 40 years ago, so when I was looking for an easier method, visual algebra surfaced from memory.
September 21, 2008
BruceKWill
If you find that your students just don't "get" algebra by classical methods, this approach may lead them to a fundamental understanding.
Mathematical ability is normally distributed in the K14 student population consequently most students do not learn algebra if the axiomatic method is used exclusively. I teach both methods in parallel, that is, the students are required to solve each equation twice by employing the axiomatic mantra, "whatever you do to one side of an equation you must do to the other", and the visual mantra, "rewrite the equation until both sides look the same." Such a mathematical pedagogy more closely matches the profile of a typical class and enables many more students to learn algebra.
November 23, 2006
OrangeMath
I use this approach in my continuing education program in high school. The standard way failed for these students. The symmetry approach baffles them somewhat, but they also find it fun (relatively) to try this. I consider this paper too short for a regular class, but great for shorter classes that can only cover some standards in depth, such as California.
Most of my students appreciate an intuitive method that enables them to do and understand algebra, many for the first time. I teach both methods in parallel, explicit-axiomatic and implicit-visual. One of my teaching techniques consists in demonstrating that every example problem in the textbook, which uses the axiomatic method exclusively, can be solved more easily through the use of numerical techniques that produce symmetry. The state-mandated criteria require the solution of at least 70% of the algebra problems to pass with a "C" or better. The criteria do not require that the students solve the equations in the most difficult way possible. The visual method is within the scope of any student skilled in arithmetic and may be the only means available to "rescue" those students whose circumstance place them at high risk of failure in abstract mathematics. |
Algebra Through Practice, Book I
9780521272858
ISBN:
0521272858
Publisher: Cambridge University Press
Summary: Problem solving is an art that is central to understanding and ability in mathematics. With this series of books the authors have provided a selection of problems with complete solutions and test papers designed to be used with or instead of standard textbooks on algebra.
This item is printed on demand. Problem solving is an art that is central to understanding and ability in mathematics. With this series of books the authors have provided a s [more]
This item is printed on demand. Problem solving is an art that is central to understanding and ability in mathematics. With this series of books the authors have provided a selection of problems with complete solutions and test papers designed to be used.[less]
pp. 112 Problem solving is an art that is central to understanding and ability in mathematics. With this series of books the authors have provided a selection of problems wit [more]
pp. 112 Problem solving is an art that is central to understanding and ability in mathematics. With this series of books the authors have provided a selection of problems with complete solutions and test papers designed to be used with or instead of standard textbooks on algebra. For the convenience of the reader, a key explaining how the present books may be used in conjunction with some of the major textbooks is included. Each book of problems is divided into chapters that begin with some notes on notation and prerequisites. The majority of the material is aimed at the student of average ability but.[less] |
handyCalc Calculator
handyCalc is a powerful calculator with automatic suggestion and solving which makes it easier to learn and use. With almost all the features you can imagine on a calculator, waiting for you to explore.
Really good just a couple of notes I really like this it is really easy to use and intuitive. Just two things. 1- when performing another calculation it won't recognize a decimal symbol as the start of the new calculation. 2- the arrow keys just look stretchy and bad but that's more personal. |
for a course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course. The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics.
The Includes 295 completely worked out examples to illustrate and clarify all major theorems and definitions. |
There are 10 Standards. Five focus on Content: Number and Operations, Algebra, Geometry, Measurement, and Data Analysis. And five focus on Process: Problem Solving, Reasoning and Proof, Communication, Connections, and Representation.
It is important not only to master the traditional basics, but also the "expanded basics" such as data analysis. Reasoning skills are essential for resourceful problem solving and strategic thinking.
Almost all MathStart books address the educational goals of multiple Standards. However, we thought it would be more useful to highlight the one or two Standards that each book most strongly addresses. |
Precalculus with Trigonometry and Analytical Geometry
Description
This text provides a strong foundation for work with functions that culminates with an introduction to the calculus topics of the derivative and the integral. Beginning with a review of basic trignometry, the study progresses to advanced topics including functions, identities, and trigonometric equations. Development of analytical geometry topics include a logical approach to the study of lines, conics, quadric surfaces, polar coordinates, and parametric equations. Colorful graphs in one, two, and three dimensions illustrate the concepts and provide a frame of reference for discussion. Helpful tips and example problems show step-by-step solutions that aid in understanding and problem solving. Balanced exercises in each chapter provide ample opportunity for students to understand both the algebraic solution and practical application of problem solving |
Understanding Elementary Algebra with Geometry A Course for College Students (with CD-ROM and iirsch and Goodman offer a mathematically sound, rigorous text to those instructors who believe students should be challenged. The text prepares students for future study in higher-level courses by gradually building students' confidence without sacrificing rigor. To help students move beyond the "how" of algebra (computational proficiency) to the "why" (conceptual understanding), the authors introduce topics at an elementary level and return to them at increasing levels of complexity. Their gradual introduction of concepts, rules, and definiti... MOREons through a wealth of illustrative examples -- both numerical and algebraic--helps students compare and contrast related ideas and understand the sometimes-subtle distinctions among a variety of situations. This author team carefully prepares students to succeed in higher-level mathematics. |
Attention
Grades 6–8 Mathematics—Introduction to Connected Mathematics
The educators of the Connected Mathematics Project (CMP) have a collective, overarching goal: to help students understand how important it is to be proficient in the various disciplines of mathematics. Students the world over will one day compete for resources to get ahead. With technological advances and global commerce framing their paths to success, students who have insight and a creative attitude, and who have been disciplined in mathematical reasoning and communication will be better positioned respond to the needs of a global economy.
Specifically, educators of Connected Mathematics seek to guide students to proficiency in these areas of mathematics: reason and communication, vocabulary, forms of representation, materials, tools, techniques, intellectual methods.
Connected Mathematics is a comprehensive middle school mathematics curriculum that provides students with multiple opportunities to develop knowledge and fluency with skills and concepts across mathematical strands. Because the program's mathematical ideas develop and deepen over the course of three years, students must have opportunities to study mathematics in each strand every year.
Support for Connected Mathematics is demonstrated by these results in 10 years of testing:
CMP is an effective middle school curriculum that is accessible to all students.
CMP students do as well as, or better than, non-CMP students on tests of basic skills, and outperform non-CMP students on tests of problem- solving ability, conceptual understanding, and proportional reasoning.
CMP students can use basic skills to solve important mathematical problems and are able to communicate their reasoning and understanding.
By the end of grade 8, CMP students show considerable ability to solve non-routine algebra problems and demonstrate a strong understanding of linear functions and a beginning understanding of exponential and quadratic functions. |
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~ Issues and Implications
The challenge of numbers in the foregoing chapters is clear: How can the nation's
growing need for mathematically skilled workers be met in the face of shrinking
populations from which these workers have been traditionally drawn? General
options include increasing the proportion of such workers from both traditional and
nontraditional sources and increasing the utilization and the effectiveness of available
workers. Specific actions that would lead to improvements are more difficult to
identify, and formulations of these will be left to the final report of the MS 2000
Committee. However, some segments of the general challenge are formulated below.
The five previous chapters describe the people in the
mathematical sciences from the perspective of college and
university programs. These people- students, teachers,
and other workers are scattered throughout the educa-
tional system and the nonacademic workplace. The picture
that emerges is strongly influenced by two general facts:
.
The workplace is changing as jobs require higher-level
skills and greater adaptability. Mathematics-based jobs
are leading the way in increased demand.
If present patterns persist, most socioeconomic and
demographic trends indicate that fewer students will study
mathematics and choose mathematics-based careers.
These trends point to an increased demand for and a
shrinking supply of mathematical scientists and other
mathematically educated workers. The nation must recog-
nize this critical condition, and understand the major
college and university mathematical sciences in particular.
Educating workers for business and industry and teachers
for all levels of education may require fundamental changes
in a system already stressed by the events of the past three
decades.
Several issues that require the nation's attention are
apparent. These are raised by the following questions:
l
~ How can national needsfor mathematically educated
workers be met? How can the expected shortage of mathe-
matically trained workers be averted? How can available
workers be better utilized? What incentives will attract
more interest in mathematics-based occupations, espe-
cially among women, blacks, and Hispanics?
· What changes are necessary to attract more students
to the study of mathematics? How can the mathematical
sciences respond to the change in the traditional pool of
U.S. college students? What and who will stimulate stu
challenge it poses for U.S. education in general and for dents to study the mathematical sciences? How can a more
73
OCR for page 74
A Challenge of Numbers
diverse group of students be attracted to mathematics,
reducing the heavy dependence on white males? What are
the consequences of heavy dependence on non-U.S. stu-
dents in graduate programs? lIow can teaching become
more effective and stimulating?
· What can be done to improve the success rate of stu-
dents dunning the transition from high school mathemat-
ics to college mathematics? How can high school prepa-
ration and college expectations be better reconciled? What
effects are remedial programs and overlaps between the
content of high school courses and college courses having
on student attntion?
· What can colleges and universities do to meet the na-
fional r~eedforschool mathematics teachers? What is the
appropriate education for secondary school mathematics
teachers and for elementary school teachers? How can the
college and university faculty assist in implementing new
standards for school mathematics? What program of con-
tinuing education for teachers will enhance school mathe-
matics instruction?
· What actions will spur renewal and revitalizafion of
the mathematical sciencesfaculty? What steps should be
taken to ensure replacements for the aging collegiate fac-
ulty? What is appropriate preparation for collegiate teach-
ing? What continuing program of scholarship for the non-
research faculty is necessary to maintain the intellectual
vitality of the profession? How can better compensation,
incentives, and working conditions be achieved and main-
tained? What is necessary to maintain and enhance the
research production of the faculty?
74
· Now can better monitoring of the mathematical sci-
ences be implemented? How can both professional or-
ganizations and government agencies cooperate in the col-
lection end reporting of information? How can date tee col-
lected, organized, and disaggregated to provide a compre-
hensive view of the mathematical sciences community?
How can mathematical scientists be identified in the non-
academic workplace?
· How can colleges and universities prepare graduates
who are more valuable and effecizve in the nonacademic
workplace? What changes would make mathematics
graduates more valuable to business and industry? How
can the full potential of the contributions of mathematical
scientists be explored? What new educational programs
could diversify the employment opportunities for mathe-
matical scientists? Are there unrecognized opportunities
for the Ph.D. in the mathematical sciences?
Although these issues center on the mathematical sci-
ences enterprise in U.S. colleges and universities, they
have implications for all of society. Monitoring and
maintaining the health of this administratively decentral-
ized and diverse enterprise transcend the nonnal roles and
responsibilities of academic systems. These concerns and
the importance of a continued healthy flow of mathemati-
cal talent are the reasons that the MS 2000 project and, in
particular, this report were begun. The forthcoming de-
scriptive reports on curriculum and resources and the pre-
scnptive final report of the MS 2000 Committee will pro-
vide the nation with an agenda for revitalization of college
and university mathematical sciences and with recommen-
dations for continued monitoring and assessment. |
Assignment: Your group must choose one of the "Group" problems
in the book and solve it, write your solution up in a formal paper and
present it to the class. In addition each student must write a short
(< 1 page typed) narrative detailing his/her participation in the project.
Your problem choice must be approved and no two groups can work on the
same problem. Problems will be approved on a first come first served basis.
Presentation dates will be assigned. You may chose, alternatively, to look
for an appropriate problem in another source. I must approve the problem
choice in this case as well. For accounting purposes, you must reserve
a problem by reaching me in my office or in class (when I will have the
list with me--a message on my OFFICE answering machine will reserve the
problem IF IT HAS NOT ALREADY BEEN TAKEN)
Purpose: To encourage and promote independent and group thinking and
work. To provide some valuable experience in problem solving in a group
setting, formal mathematical writing and presenting to a peer group.
Format: Presentations can be informal but should be thorough. Your exposition
should be directed at the other students in the class and should be clear
to them. Papers should be written in journal format. For examples look
in The College Mathematics Journal, available in the seminar room or the library.
Length: The papers should be long enough to cover the problem statement,
background and solution and should contain references. For example, if you
choose a problem from the text book, the text book should be referenced
and a footnote should appear on the problem statement. If you use another
text to help you get going, reference it, etc. If in doubt about references,
consult the Holt Handbook or ask me. Most papers should be 3 to 5 pages.
Grading: Weight will be given to the complexity of the problem and the
elegance with which it is solved. Superficial analysis and or relatively
simple problems will receive lower grades than will the more difficult
problems or particularly elegant solutions. Narratives will be graded on
form, style and content. Included in form are grammar, spelling and punctuation
as well as the physical presentation of the paper. Included in content
is proper citation of sources, accuracy, interest and mathematical or scientific
value. Included in style are readability, transitions and all the intangible
things that make a good piece of prose. The presentation will contribute
approximately 20% to your grade the paper 70% and the remaining 10% the
accompanying narrative. These percentages are somewhat arbitrary and may
vary from project to project. Projects will be graded holistically, in
other words I will not give you three separate grades, only one. However,
it is possible (although unlikely) for two people from the same group to
receive different grades.
Deadlines: Initial discussion of the first problem choice should occur
as soon as possible. Problem choices must be approved by October 5
Presentation dates and final due dates will be announced in class. |
Based on the International Commission on Mathematical Instruction conference held in early 1987, this volume consists of a number of key papers presented by international authorities on the role of mathematics in applied subjects, such as engineering, computer science, and mathematical modelling. The importance of certain mathematical ideas, such as geometry and discrete mathematics is stressed, as well as the more classical methods. The book includes a long article by the editor synthesising some of the main themes and trends debated at the conference. |
Communicate mathematical ideas verbally, in writing, and through mathematical representations to various audiences. [1120]
Develop the ability to perceive how people interact with their cultural and natural environments, through their own worldview and through the worldviews of others, in order to analyze how individuals and groups function in local and global contexts. [1986] |
Short Description for Complex Numbers from A to ...Z Helps you to learn how complex numbers may be used to solve algebraic equations, as well as their geometric interpretation. This book covers a selection of Olympiad problems solved by employing the methods presented. Full description
Full description for Complex Numbers from A to ...Z
* Learn how complex numbers may be used to solve algebraic equations, as well as their geometric interpretation * Theoretical aspects are augmented with rich exercises and problems at various levels of difficulty * A special feature is a selection of outstanding Olympiad problems solved by employing the methods presented * May serve as an engaging supplemental text for an introductory undergrad course on complex numbers or number theory |
Course Descriptions
Mathematics Course Listings - Spring 2011
MATH 112 The Language of Mathematics (4 credits) Barhorst, Garry
Prerequisite: Math Placement Level 22 or higher
This is an introduction to mathematics at the beginning college level. MATH 112 will explore topics in contemporary mathematics with a problem-solving approach.
The class meetings will include lectures, problem-solving sessions, and group work. The final grade will be based on quizzes, exams, a project, and/or a comprehensive final. This course is not intended to prepare students for further courses in mathematics. Mathematical-reasoning intensive.
MATH 118 Mathematics for Elementary and Middle School Teachers (4 credits) Monke, Lowell Prerequisite: Math Placement Level 22 or higher
Study of number systems, number theory, patterns, functions, measurement, algebra, logic, probability, and statistics with a special emphasis on the processes of mathematics: problem solving, reasoning and proof, communicating mathematically, and making connections within mathematics and between mathematics and other disciplines. Open only to students intending to major in education. Every year. Mathematical-reasoning intensive.
Prerequisite: MATH 118
Study of basic concepts of plane and solid geometry, including topics from Euclidean, transformational, and projective geometry with a special emphasis on the processes of mathematics: problem solving, reasoning and proof, communicating mathematically, and making connections among mathematical ideas, real-world experiences, and other disciplines. Includes computer lab experiences using Geometer's Sketchpad. Open only to students majoring in education. Every year. Mathematical-reasoning intensive.
Prerequisite: Math Placement Level 24 or higher
This is a standard pre-calculus mathematics course that explores the functions common to the study of calculus. Examination of polynomial, rational, exponential, logarithmic, and trigonometric functions will be done using algebraic, numeric, and graphical techniques. Applications of these functions in formulating and solving real-world problems will also be discussed.
The final grade in the course will be based on homework, quizzes, tests, and a comprehensive final exam. Students are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class and for homework assignments. Mathematical-reasoning intensive.
MATH 127 Introductory Statistics (4 credits) Andrews, Douglas
Prerequisites: Math Placement Level 23 or higher
A study of statistics as the science of using data to glean insight into real-world problems. Includes principles and methods for describing and summarizing data, sampling procedures and experimental design, inferences about the real-world processes that underlie the data, and student projects for collecting and analyzing data. Open to non-majors only.
Note: A student may receive credit for only one of the following statistics courses: MATH 127, MATH 227, PSYC 107, or MGT 210. Mathematical-reasoning intensive.
MATH 131 Essentials of Calculus (4 credits) Shelburne, Brian
Prerequisite: MATH 120 or Math Placement Level 25
This one semester calculus course is an introduction to the techniques and applications of differential and integral calculus. The applications come primarily from the economics and bio-sciences and do not involve any trigonometric models. The final grade in the course will be based on homework, quizzes, tests, and a comprehensive final exam.
Students are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class and for homework assignments. Mathematical-reasoning intensive.
Notes: 1. Students may not receive credit for both MATH 131 and MATH 201
2. MATH 131 does not satisfy the prerequisite for MATH 202.
3. Take MATH 131 only if you are POSITIVE that you will take only one semester of calculus at Wittenberg. Otherwise, you should take MATH 201.
MATH 201 Calculus I (4 credits) Higgins, William
Prerequisite: MATH 120 or Math Placement Level 25
Calculus is the mathematical tool used to analyze changes in physical quantities. This is the first course in the standard calculus sequence. It develops the notion of "derivative", which is used for studying rates of change, and then introduces the concept of "definite integral", which is related to area problems. The overall approach will emphasize the concepts of calculus using graphical, numerical, and symbolic methods.
The two-semester calculus sequence, MATH 201/202, is required for all students majoring or minoring in mathematics, computer science, physics, or chemistry. MATH 201 and MATH 202 can also count as supporting science courses for the BA and BS programs in Biology, Geology, and Biochemistry/Molecular Biology. Students who are sure they will take only one semester of calculus may be better served in the single-semester introduction to calculus, MATH 131: "Essentials of Calculus". Talk with your advisor or with any math professor for advice on which calculus course is most appropriate for you.
Students could be based on homework, quizzes, tests, and a comprehensive final exam. Mathematical-reasoning intensive.
NOTE: Students may not receive credit for both MATH 131 and MATH 201.
MATH 202 Calculus II (4 credits) Stickney, Alan
Prerequisite: MATH 201
This is the second course in Wittenberg's three semester calculus sequence. MATH 202 is primarily concerned with integration and power series representations of functions. Topics covered include indefinite and definite integrals, the Fundamental Theorem of Calculus, integration techniques, approximations of definite integrals, improper integrals, applications of integrals, power series, Taylor's Series, geometric series, and convergence tests for series.
Normally, students will be based on quizzes, tests, and a comprehensive final exam. Mathematical-reasoning intensive.
MATH 210 Fundamentals of Analysis (4 credits) Parker, Adam
Prerequisite: MATH 202
Functions, set theory, sequences, the topology of the real line, and methods of mathematical proof. Particular emphasis is given to careful, accurate definition and proof of mathematical concepts. Grades may be based on several tests, quizzes, homework assignments, and a final examination.
Writing intensive. Mathematical-reasoning intensive.
MATH 212 Multivariable Calculus (4 credits) Stickney, Alan
Prerequisite: MATH 202
This course completes the basic calculus sequence. It covers the calculus of functions of several variables and associated analytic geometry. Students are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class, for homework assignments, and for tests. The final grade in the course is based on quizzes, tests, and a comprehensive final exam. Mathematical-reasoning intensive.
MATH modeling project as part of the course. Laboratory required. This course is cross-listed as COMP 260. Students may enroll in either COMP 260 or MATH 260, but not both. Mathematical-reasoning intensive.
MATH 337 Statistical Design (4 credits) Andrews, Douglas
Prerequisite: MATH 227
Whereas the introductory statistics sequence focuses primarily on exploratory and formal analysis of data that have already been observed, this course focuses primarily on how to design the comparative observational and experimental studies in which data is collected for formal analysis. Students will learn: 1) to choose sound and suitable design structures, 2) to recognize the structure of any balanced design built from crossing and nesting, 3) to assess how well standard analysis assumptions fit the given data and to choose a suitable remedy or alternative when appropriate, 4) to decompose any balanced dataset into components corresponding to the factors of a design, 5) to construct appropriate interval estimates and significance tests from such data, and 6) to interpret patterns and formal inferences in relation to the relevant applied context. Students are required to collaborate on projects in which they design studies, collect and analyze data, and present their findings orally and in writing. Students who have taken a different introductory statistics course may be admitted with permission of instructor. Mathematical-reasoning intensive.
MATH 365 Abstract Algebra (4 credits) Higgins, William
Prerequisite: MATH 205 and MATH 210
This course will focus on abstract algebraic structures such as groups, rings, and fields with particular attention to groups. There will be an emphasis on presenting arguments with a full explanation of the reasoning. Grades will be based on written homework, work done in class, quizzes, and exams.
Writing intensive. Mathematical-reasoning intensive.
MATH 380 Computational Algebraic Geometry (4 credits) Parker, Adam
Prerequisite: MATH 205
Algebraic geometry is the study of systems of polynomial equations in one or more variables. The solution to such a system is a geometric object called a variety and much of algebraic geometry is concerned with how algebraic properties of the system are related to geometric properties of the variety. Results in this field tend to be quite abstract and complexity increases rapidly, making examples hard to compute.
This changed in the mid 1960's with the discovery of a generalized division algorithm called Buchberger's algorithm. With this new computational tool, it became possible to manipulate systems of polynomials efficiently. As a result, algebraic geometry has become accessible to a wider audience of both students and researchers.
This course will concentrate on the algorithmic and computational aspects of algebraic geometry. Topics covered will include, but not be limited to Hilbert Basis Theorem, the Nullstellensatz, resultants, and Buchberger's algorithm. The final course grade will be based on homework, tests, class participation, and a computer project. Mathematical-reasoning intensive. |
Basic College Math, Student Support Edition - 9th edition
Summary: With its complete, interactive, objective-based approach, Basic College Mathematics is the best-seller in this market. The Eighth Edition provides mathematically sound and comprehensive coverage of the topics considered essential in a basic college math course. Furthermore, the Instructor's Annotated Edition features a comprehensive selection of instructor support material. The Aufmann Interactive Method is incorporated throughout the text, ensuring that students int...show moreeract with and master the concepts as they are presented. This approach is especially important in the context of rapidly growing distance-learning and self-paced laboratory situationsVernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Joanne Lockwood is co-author with Dick Aufmann and Vernon Barker on the hardback developmental series, Business Mathematics, Algebra with Trigonometry for College Students, and numerous software ancillaries that accompany Aufmann titles. She is also the co-author of Mathematical Excursions with Aufmann |
algebraexThis post uses MathJax. If you see formulas in unrendered TeX, try refreshing the screen.
A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another. Conceptual blending is a major tool used by our brain to understand the world.
The concept of conceptual blend includes special cases, such as representations,images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned. The Wikipedia article is a good starting place for understanding conceptual blending.
In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus. I omit the connections with programs, which I will discuss in a separate post.
A particular function
Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.
FORMULA:
$h(t)$ is defined by the formula $4-(t-2)^2$.
The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
The formula defines the function, which is a stronger statement than saying it represents the function.
The formula is in standard algebraic notation. (See Note 1)
To use the formula requires one of these:
Understand and use the rules of algebra
Use a calculator
Use an algebraic programming language.
Other formulas could be used, for example $4t-t^2$.
That formula encapsulates a different computation of the value of $h$.
TREE:
$h(t)$ is also defined by this tree (right).
The tree makes explicit the computation needed to evaluate the function.
The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
Both formula and tree require knowledge of conventions.
The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
Other formulas correspond to other trees. In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
OBJECT:
Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$.
Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
The mathematical object $h(t)$ is a particular set of ordered pairs.
It just sits there.
When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
I did not say that math objects are inert and timeless, I said you think of them that way. This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].
DEFINITION
A definition of the concept of function provides a way of thinking about the function.
One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
A concept can have many different definitions.
A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties. But it can be defined as a set with a ternary operation, as well.
A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties. But a partition is an equivalence relation and an equivalence relation is a partition. You have just picked different primitives to spell out the definition.
If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
The definition of group doesn't mention that it has linear representations.
SPECIFICATION
If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.
This tells you everything you need to know to use the function $h$.
It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.
Notes
1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.
2. There are various ways to give a pictorial image of the function. The usual way to do this is presenting the graph as shown above. But you can also show its cograph and its endograph, which are other ways of representing a function pictorially. They are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:
In a recent post, I wrote about defining "category" in a way that (I hope) makes it accessible to undergraduate math majors at an early stage. I have several more things to say about this.
Early intro to categories
The idea is to define a category as a directed graph equipped with an additional structure of composition of paths subject to some axioms. By giving several small finite examples of categories drawn in that way that gives you an understanding of "category" that has several desirable properties:
You get the idea of what a category is in one lecture.
With the right choice of examples you get several fine points cleared up:
The composition is added structure.
A loop doesn't have to be an identity.
Associativity is a genuine requirement – it is not automatic.
You get immediate access to what is by far the most common notation used to work with a category — objects (nodes) and arrows.
You don't have to cope with the difficult chunking required when the first examples given are sets-with-structure and structure-preserving functions. It's quite hard to focus on a couple of dots on the paper each representing a group or a topological space and arrows each representing a whole function (not the value of the function!).
Introduce more examples
Then the teacher can go on with the examples that motivated categories in the first place: the big deal categories such as sets, groups and topological spaces. But they can be introduced using special cases so they don't require much background.
Draw some finite sets and functions between them. (As an exercise, get the students to find some finite sets and functions that make the picture a category with $f=kh$ as the composite and $f\neq g$.)
If the students have had calculus, introduce them to the category whose objects are real finite nonempty intervals with continuous or differentiable mappings between them. (Later you can prove that this category is a groupoid!)
Find all the groups on a two element set and figure out which maps preserve group multiplication. (You don't have to use the word "group" — you can simply show both of them and work out which maps preserve multiplication — and discover isomorphism!.) This introduces the idea of the arrows being structure-preserving mape. You can get more complicated and use semigroups as well. If the students know Mathematica you could even do magmas. Well, maybe not.
All this sounds like a project you could do with high school students.
Large and small
If all this were just a high school (or intro-to-math-for-math-majors) project you wouldn't have to talk about large vs. small. However, I have some ideas about approaching this topic.
In the first place, you can define category, or any other mathematical object that might involve a proper class, using the syntactic approach I described in Just-in-time foundations. You don't say "A category consists of a set of objects and a set of arrows such that …". Instead you say something like "A category $\mathcal{C}$ has objects $A,\,B,\,C\ldots$ such that…".
This can be understood as meaning "For any $A$, the statement $A$ is an object of $\mathcal{C}$ is either true or false", and so on.
This approach is used in the Wikibook on category theory. (Note: this is a permanent link to the November 28 version of the section defining categories, which is mostly my work. As always with Wikimedia things it may be entirely different when you read this.)
If I were dictator of the math world (not the same thing as dictator of MathWorld) I would want definitions written in this syntactic style. The trouble is that mathematicians are now so used to mathematical objects having to be sets-with-structure that wording the definition as I did above may leave them feeling unmoored. Yet the technique avoids having to mention large vs. small until a problem comes up. (In category theory it sometimes comes up when you want to quantify over all objects.)
The ideas outlined in this subsection could be a project for math majors. You would have to introduce Russell's Paradox. But for an early-on intro to categories you could just use the syntactic wording and avoid large vs. small altogether.
The concept of category is typically taught later in undergrad math than the concept of group is. It is supposedly a more advanced concept. Indeed, the typical examples of categories used in applications are more advanced than some of those in group theory (for example, symmetries of geometric shapes and operations on numbers).
Here are some thoughts on how categories could be taught as early as groups, if not earlier.
Nodes and arrows
Small finite categories can be pictured as a graph using nodes and arrows, together with a specification of the identity arrows and a definition of the composition. (I am using the word "graph" the way category people use it: a directed graph with possible multiple edges and loops.)
An example is the category pictured below with three objects and seven arrows. The composition is forced except for $kh$, which I hereby define to be $f$.
This way of picturing a category is easy to grasp. The composite $kh$ visibly has to be either $f$ or $g$. There is only one choice for the composite of any other composable pair. Still, the choice of composite is not deducible directly by looking at the graph.
A first class in category theory using graphs as examples could start with this example, or the example in Note 1 below. This example is nontrivial (never start any subject with trivial examples!) and easy to grasp, in this case using the extraordinary preprocessing your brain does with the input from your eyes. The definition of category is complicated enough that you should probably present the graph and then give the definition while pointing to what each clause says about the graph.
Most abstract structures have several different ways of representing them. In contrast, when you discuss categorial concepts the standard object-and-arrow notation is the overwhelming favorite. It reveals domains and codomains and composable pairs, in fact almost everything except which of several possible arrows the composite actually is. If for example you try to define category using sets and functions as your running example, the student has to do a lot of on-the-go chunking — thinking of a set as a single object, of a set function (which may involve lots of complicated data) as a single chunk with a domain and a codomain, and so on. But an example shown as a graph comes already chunked and in a picture that is guaranteed to be the most common kind of display they will see in discussions of categories.
After you do these examples, you can introduce trivial and simple graph examples in which the composition is entirely induced; for example these three:
(In case you are wondering, one of them is the empty category.) I expect that you should also introduce another graph non-example in which associativity fails.
Multiplication tables
The multiplication table for a group is easy to understand, too, in the sense that it gives you a simple method of calculating the product of any two elements. But it doesn't provide a visual way to see the product as a category-as-graph does. Of course, the graph representation works only for finite categories, just as the multiplication table works only for finite groups.
You can give a multiplication table for a small finite category, too, like the one below for the category above. ("iA" means the identity arrow on A and composition, as usual in category theory, is right to left.) This is certainly more abstract than the graph picture, but it does hit you in the face with the fact that the multiplication is partial.
Notes
1. My suggested example of a category given as a graph shows clearly that you can define two different categorial structures on the graph. One problem is that the two different structures are isomorphic categories. In fact, if you engage the students in a discussion about these examples someone may notice that! So you should probably also use the graph below,where you can define several different category structures that are not all isomorphic.
2. Multiplication tables and categories-as-graphs-with-composition are extensional presentations. This means they are presented with all their parts laid out in front of you. Most groups and categories are given by definitions as accumulations of properties (see concept in the Handbook of Mathematical Discourse). These definitions tend to make some requirements such as associativity obvious.
Students are sometimes bothered by extensional definitions. "What areh and k (in the category above)? What are a, b and c?" (in a group given as a set of letters and a multiplication table).
Note: To manipulate the diagrams in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The Mathematica notebooks used here are listed in the references below.
Pictures, metaphors and etymology
Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept. I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of "secant" that use pictures, metaphors and etymology to describe the concept.
The exposition is interlarded with comments about what I am doing and why. An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!
Secant Line
The word "secant" is used in various related ways in math. To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:
The word "secant" comes from the Latin word for "cut", which came from the Indo-European root "sek", meaning "cut". The IE root also came directly into English via various Germanic sound changes to give us "saw" and "sedge".
The picture
Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept. The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points. You also get a very strong understanding of how the secant line is a function of the two given points. I don't think that is obvious to someone without some experience with such things.
This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects. (Math books are full of such pictures.) So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds. This is the sort of claim that is amenable to field testing.
The metaphor
Most metaphors are based on a physical phenomenon. The mathematical meanings of "secant" use the metaphor of cutting. When the word "secant" was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor. In those days essentially every European scholar read Latin. To them "secant" would transparently mean "cutting". This is not transparent to many of us these days, so the metaphor may be hidden.
If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.
The straight line does not really cut the curve. Indeed, the curve itself is both an abstract object that is not physical, so can't be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it? Cut the screen? The line can't do that.
You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
The metaphor is restricted further by saying that it is determined by two points on the curve. This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines. You could define such a family by using one point on the curve and a slope, for example. This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.
Secant on circle
Another use of the word "secant" is the red line in this picture:
This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve. This means it corresponds to a number, and that number is what we mean by "secant" in trigonometry.
The Definition
The secant of an angle $latex \theta$ is usually defined as $latex \frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.
Different equivalent definitions can give you a very different understanding of the concept.
The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of "secant". You can tell a lot from its behavior right off (it goes to infinity near $latex \pi/2$, for example).
The definition $latex \sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $latex \sec\theta$. It also reduces the definition of $latex \sec\theta$ to a previously known concept.
It used to be common to give only the $latex \frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it. That is a crime. Yes, I know many students don't want to "understand" stuff, they only want to know how to do the problems. Teachers need to talk them out of that attitude. One way to do that in this case is to test them on the geometric definition.
Etymology
This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in "Geometria Rotundi" (1583), where the first known use of the word "secant" occurs. I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.
It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on. A cursory search of the internet gave me the current name in Arabic for secant but nothing else.
Graph of the secant function
The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.
This is my first experiment at posting an active Mathematica CDF document on my blog. To manipulate the graph below, you must have Wolfram CDF Player installed on your computer. It is available free from their website.
This is a new presentation of old work. It is a graph of a certain fifth degree polynomial and its first four derivatives.
The buttons allow you to choose how many derivatives to show and the slider allows you to show the graphs from $latex x=-4$ up to a certain point.
How graphs like this could be used for teaching purposes
You could show this in class, but the best way to learn from it would be to make it part of a discussion in which each student had access to a private copy of the graph. (But you may have other ideas about how to use a graph like this. Share them!)
Some possible discussion questions:
Click button 1. Now you see the function and the derivative. Move the slider all the way to the left and then slowly move it to the right. When the function goes up the derivative is positive. What other things do you notice when you do this?
If you were told only that one of the functions is the derivative of the other, how would you rule out the wrong possibility?
What can you tell about the zeroes of the function by looking at the derivative?
Look at the interval between $latex x=1.5$ and $latex x=1.75$. Does the function have one or two zeroes in that interval? On my screen it looks as if the curve just barely gets above the $latex x$ axis in that interval. What does that say about it having one or two zeroes? How could you verify your answer?
Click button 2. Now you have the function and first and second derivatives. What can you say about maxima, minima and concavity of the function?
Find relationships between the first and second derivatives.
Now click button 4. Evidently the 4th derivative is a straight line with positive slope. Assume that it is. What does that tell you about the graph of the third derivative?
What characteristics of the graph of the function can you tell from knowing that the fourth derivative is a straight line of positive slope?
What can you say about the formula for the function knowing that the fourth derivative is a straight line of positive slope?
Suppose you were given this graph and told that it was a graph of a function and its first four derivatives and nothing else. Specifically, you do not know that the fourth derivative is a straight line. Give a detailed explanation of how to tell which curve is the function and which curve is each specific derivative.
Making this manipulable graph
I posted this graph and a lot of others several years ago on abstractmath.org. (It is the ninth graph down). I fiddled with this polynomial until I got the function and all four derivatives to be separated from each other. All the roots of the function and all its derivatives are real and all are shown. Isn't this gorgeous?
To get it to show up properly on the abmath site I had to thicken the graph line. Otherwise it still showed up on the screen but when I printed it on my inkjet printer the curves disappeared. That seems to be unnecessary now.
Mathematica 8.0 has default colors for graphs, but I kept the old colors because they are easier to distinguish, for me anyway (and I am not color blind).
Inserting CDF documents into html
A Wolfram document explains how to do this. I used the CDF plugin for WordPress. WordPress requires that, to use the plugin, you operate your blog from your own server, not from WordPress.com. That is the main reason for the recent change of site.
Here are some notes and questions on the process. When I find learn more about any of these points I will post the information.
At the moment I don't know how to get rid of the extra space at the top of the graph.
I was surprised that I could not click on the picture and shrink or expand it.
It might be annoying for a student to read the questions above and have to go up and down the screen to see the graph. I had envisioned that the teacher would ask the questions and have the students play with the graph and erupt with questions and opinions. But you could open two copies of the .cdf file (or this blog) and keep one window showing the graph while the other window showed the questions.
Which raises a question: Could it be possible to program the graph with a button that when pushed would make the graph (only) appear in another window?
Other approaches
I have experimented with Khan Academy type videos using CDF files. I made a screen shot and at a certain point I pressed a button and the graph appropriately changed. I expect to produce an example video which I can make appear on this blog (which supposedly can show videos, but I haven't tried that yet.)
It should be possible to have a CDF in which the student saw the graph with instructional text underneath it equipped with next and back buttons. The next button would trigger changes in the picture and replace the text with another sentence or two. This could be instead of spoken stuff or additional to it (which would be a lot of work). Has anyone tried this?
Note
My reaction to Khan Academy was mostly positive. One thing that struck me that no one seems to have commented on is that the lectures are short. They cover one aspect (one definition or one example or what one theorem says) in what felt to me like ten or fifteen minutes. This means that you can watch it and easily go back and forth using the controls on the video display. If it were a 50-minute lecture it would be much harder to find your way around.
I think most students are grasshoppers: When reading text, they jump back and forth, getting the gist of some idea, looking ahead to see where it goes, looking back to read something again, and so on. Short videos allow you to do this with spoken lectures. That seems to me remarkably useful.Introduction
In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of $latex {y=x^2}&fg=000000$ swoops up to the right.
Many of these same metaphors have realizations in actual mathematical representations.
Two images (not mentioned in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.
A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)
The endograph of a function
In principle, the endograph (Note [2]) of a function $latex {f}&fg=000000$ has a dot for each element of the domain and of the codomain, and an arrow from $latex {x}&fg=000000$ to $latex {f(x)}&fg=000000$ for each $latex {x}&fg=000000$ in the domain. For example, this is the endograph of the function $latex {n\mapsto n^2+1 \pmod 11}&fg=000000$ from the set $latex {\{0,1,\ldots,10\}}&fg=000000$ to itself:
"In principle" means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as "the graph of the squaring function" when in fact the infinite tails are cut off.
Real endographs
I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.
Here is the endograph of the function $latex {y=x^2}&fg=000000$ on the reals:
I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.
The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with $latex {x}&fg=000000$.(See note [3]).
I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.
For the one shown, you could ask these questions, probably for class discussion rather that on homework.
Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
Why do the arrows cross over each other? (Tricky question — they wouldn't cross over if you drew the arrows with negative input below the line instead of above.)
What does it say about the function that every arrowhead except two has two curves going into it?
Real Cographs
The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).
Here is the cograph representation of the function $latex {y=x^2}&fg=000000$. Compare it with the endograph representation above.
Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.
More examples
Absolute value function
Arctangent function
Notes
[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can't evaluate or modify the file without Mathematica.
The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.
[2] I call them "endographs" to avoid confusion with the usual graphs of functions — – drawings of (some of) the set of ordered pairs $latex {x,f(x)}&fg=000000$ of the function.
[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.
Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.
[4] The text [1] uses the cograph representation extensively. The word "cograph" is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.
[5] It would also be possible to show the domain codomain in the usual $latex {x-y}&fg=000000$ plane arrangement, with the domain the $latex {x}&fg=000000$ axis and the codomain the $latex {y}&fg=000000$ axis. I have not written the code for this yet.
References
[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003 definition |
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